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A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":67},{"work":"paper-i","url":"notation/paper-i.html#symbol-838b69abf4e3cc48","tex":"A^{\\oplus r}","aliases":"A^⊕ r oplus","display":"A^{\\oplus r}","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-7bcf60d2a8cb90a8","tex":"\\A_C","aliases":"A_C A mathcal","display":"{\\mathcal A}_C","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6983fa40ff77fa12","tex":"\\mathrm{abs}","aliases":"abs mathrm","display":"\\mathrm{abs}","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-c4ac73828566b6f0","tex":"\\alg","aliases":"alg operatorname","display":"{\\operatorname{alg}}","kind":"definition-passage","excerpt":"[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-f84d87f1612c677e","tex":"\\alg_{\\bbk}","aliases":"alg_bbk alg_𝕜 alg bbk operatorname Bbbk","display":"{\\operatorname{alg}}_{{\\Bbbk}}","kind":"definition-passage","excerpt":"[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Packet system surjective along edges] The module resolved into packets is surjective along edges relative to Γ( G, P) when, for every graph edge α→β, there exists B∈ A such that the pure block P_β BP_α:X_α↠ X_β is surjective.","count":44},{"work":"paper-i","url":"notation/paper-i.html#symbol-1cad61c08170218e","tex":"\\B","aliases":"B mathcal","display":"{\\mathcal B}","kind":"definition-passage","excerpt":"[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):","count":13},{"work":"paper-i","url":"notation/paper-i.html#symbol-5bb28d7c6728c2ad","tex":"\\mathrm{boundary}","aliases":"boundary mathrm","display":"\\mathrm{boundary}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":46},{"work":"paper-i","url":"notation/paper-i.html#symbol-420668a5bdd9e0bf","tex":"\\mathbf C","aliases":"C mathbf","display":"\\mathbf C","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Paper I export object] The canonical Paper I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation, mo","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-76943d929d27bd54","tex":"\\mathbf C_D","aliases":"C_D mathbf","display":"\\mathbf C_D","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-5761fa2adf61c496","tex":"\\cl","aliases":"cl","display":null,"kind":"definition-passage","excerpt":"[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":49},{"work":"paper-i","url":"notation/paper-i.html#symbol-4d523ae87a3985cd","tex":"\\mathrm{dependencies}","aliases":"dependencies mathrm","display":"\\mathrm{dependencies}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes","count":19},{"work":"paper-i","url":"notation/paper-i.html#symbol-55549aac652d0c0b","tex":"\\mathsf{Ev}","aliases":"Ev mathsf","display":"\\mathsf{Ev}","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-375d20cf9877348c","tex":"\\mathsf{Ev}_D","aliases":"Ev_D mathsf","display":"\\mathsf{Ev}_D","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-912eecc2461552b8","tex":"\\mathrm{exports}","aliases":"exports mathrm","display":"\\mathrm{exports}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":28},{"work":"paper-i","url":"notation/paper-i.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"k G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear ","count":65},{"work":"paper-i","url":"notation/paper-i.html#symbol-4fa64e297ddf7f10","tex":"f_{s(a)}","aliases":"f_s(a)","display":"f_{s(a)}","kind":"definition-passage","excerpt":"linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-80f1392d934eaed0","tex":"f_{t(a)}","aliases":"f_t(a)","display":"f_{t(a)}","kind":"definition-passage","excerpt":"linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e34327622e5e789e","tex":"f_v","aliases":"f_v","display":"f_v","kind":"definition-passage","excerpt":"l on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"l=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.","count":15},{"work":"paper-i","url":"notation/paper-i.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"definition-passage","excerpt":"[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom operatorname","display":"{\\operatorname{Hom}}","kind":"definition-passage","excerpt":"[Transfer block] For T∈End_𝕜(X) and a finite packet system P, define T_βα:=P_β TP_α ∈Hom_𝕜(X_α,X_β).","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-7b849ee4e6f402af","tex":"\\Hom_{\\bbk}","aliases":"Hom_bbk Hom_𝕜 Hom bbk operatorname Bbbk","display":"{\\operatorname{Hom}}_{{\\Bbbk}}","kind":"definition-passage","excerpt":"[Transfer block] For T∈End_𝕜(X) and a finite packet system P, define T_βα:=P_β TP_α ∈Hom_𝕜(X_α,X_β).","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-8fff38e3b7008328","tex":"\\Hull_{\\A}","aliases":"Hull_A Hull_ A Hull A operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal A}}","kind":"definition-passage","excerpt":"[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).","count":10},{"work":"paper-i","url":"notation/paper-i.html#symbol-2609b96491ee0fe2","tex":"\\Hull_{\\A,\\tau}","aliases":"Hull_A,τ Hull_ A,τ Hull A tau operatorname mathcal tau","display":"{\\operatorname{Hull}}_{{\\mathcal A},\\tau}","kind":"definition-passage","excerpt":"[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9bc5eb932401337d","tex":"\\Hull_{\\B}","aliases":"Hull_B Hull_ B Hull B operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal B}}","kind":"definition-passage","excerpt":"[Morphism of seeded systems] A morphism (X, A,S, P)⟶(Y, B,S', Q) is a packet compatible operator morphism satisfying the seed condition Φ(S)⊆Hull_ B(S'). In the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-329089d08934b068","tex":"\\mathrm{hypotheses}","aliases":"hypotheses mathrm","display":"\\mathrm{hypotheses}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"definition-passage","excerpt":"[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-7caa8aee712bf74f","tex":"\\mathrm{ID}","aliases":"ID mathrm","display":"\\mathrm{ID}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"definition-passage","excerpt":"The realized action and faithful reduction. [Realized operator algebra] For a realization X, the realized operator algebra is A_X:=im (Λ_ G⟶End_𝕜(|X|)). It records the actual operators in the realization and may be a proper quotient of the abstract grammar algebra.","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):","count":49},{"work":"paper-i","url":"notation/paper-i.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":86},{"work":"paper-i","url":"notation/paper-i.html#symbol-f0042c6cf9ee0b72","tex":"\\mathfrak M","aliases":"M mathfrak","display":"\\mathfrak M","kind":"definition-passage","excerpt":"Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.","count":14},{"work":"paper-i","url":"notation/paper-i.html#symbol-90569498b4dafa23","tex":"M^r","aliases":"M^r","display":"M^r","kind":"definition-passage","excerpt":"gives a canonical map from a finite free module. The generated hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-82964580614c7889","tex":"\\mathfrak M_R","aliases":"M_R mathfrak","display":"\\mathfrak M_R","kind":"definition-passage","excerpt":"Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.","count":11},{"work":"paper-i","url":"notation/paper-i.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"definition-passage","excerpt":"[Seed rank] For a unital left module M over A, define srank_A(M):=min{|S|:AS=M}. For a topological module (X,τ), define srank_ A,τ(X):=min{|S|:cl A S^ τ=X}. Set srank_A(0)=0. When the value is finite, it is the minimal number of module generators.","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"definition-passage","excerpt":"Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.","count":60},{"work":"paper-i","url":"notation/paper-i.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":103},{"work":"paper-i","url":"notation/paper-i.html#symbol-0b67a2140e399c4e","tex":"P_C","aliases":"P_C","display":"P_C","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-634bde01ca310a18","tex":"P_\\alpha^2","aliases":"P_α^2 alpha","display":"P_\\alpha^2","kind":"definition-passage","excerpt":"[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-1f7d96c56f31a733","tex":"P_{\\sigma^{-1}(j)}","aliases":"P_σ^-1(j) sigma","display":"P_{\\sigma^{-1}(j)}","kind":"definition-passage","excerpt":"orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-9a7d755cf9c10f7b","tex":"\\mathrm{proof}","aliases":"proof mathrm","display":"\\mathrm{proof}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1e7be1547abcd294","tex":"\\mathsf{PSC}","aliases":"PSC mathsf","display":"\\mathsf{PSC}","kind":"definition-passage","excerpt":"[Paper I export object] The canonical Paper I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation, mo","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-49925054439766d9","tex":"\\mathsf{PSC}_I","aliases":"PSC_I mathsf","display":"\\mathsf{PSC}_I","kind":"definition-passage","excerpt":"[Paper I export object] The canonical Paper I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation, mo","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-cb103a3f56bb89da","tex":"\\psrank_{\\A,\\Pcal}","aliases":"psrank_A,Pcal psrank_ A, P psrank A Pcal operatorname mathcal mathcal","display":"{\\operatorname{psrank}}_{{\\mathcal A},{\\mathcal P}}","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-28465b6199aa1e59","tex":"\\psrank_{\\A_C,\\Pcal|_C}","aliases":"psrank_A_C,Pcal|_C psrank_ A_C, P|_C psrank A Pcal operatorname mathcal mathcal","display":"{\\operatorname{psrank}}_{{\\mathcal A}_C,{\\mathcal P}|_C}","kind":"definition-passage","excerpt":"f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"definition-passage","excerpt":"[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps ","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"definition-passage","excerpt":"[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps ","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"definition-passage","excerpt":"orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber","count":14},{"work":"paper-i","url":"notation/paper-i.html#symbol-aaf174f6701cc732","tex":"\\Qcal","aliases":"Qcal Q mathcal","display":"{\\mathcal Q}","kind":"definition-passage","excerpt":"[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.","count":53},{"work":"paper-i","url":"notation/paper-i.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":57},{"work":"paper-i","url":"notation/paper-i.html#symbol-39303e93288f0aca","tex":"r_C","aliases":"r_C","display":"r_C","kind":"definition-passage","excerpt":"f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-b617b296bbfd924b","tex":"\\mathfrak S","aliases":"S mathfrak","display":"\\mathfrak S","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps ","count":76},{"work":"paper-i","url":"notation/paper-i.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"definition-passage","excerpt":"[Morphism of seeded systems] A morphism (X, A,S, P)⟶(Y, B,S', Q) is a packet compatible operator morphism satisfying the seed condition Φ(S)⊆Hull_ B(S'). In the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":10},{"work":"paper-i","url":"notation/paper-i.html#symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-a6bf5eb7ecfc145e","tex":"s_r","aliases":"s_r","display":"s_r","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-64e8f7a42ac935d4","tex":"\\mathbf{SeedSys}","aliases":"SeedSys mathbf","display":"\\mathbf{SeedSys}","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-5faac4b7fa18af10","tex":"\\mathbf{SeedSys}_{\\mathrm{abs}}","aliases":"SeedSys_abs mathbf mathrm","display":"\\mathbf{SeedSys}_{\\mathrm{abs}}","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-e0837c3271213f2f","tex":"\\srank_{\\A,\\tau}","aliases":"srank_A,τ srank_ A,τ srank A tau operatorname mathcal tau","display":"{\\operatorname{srank}}_{{\\mathcal A},\\tau}","kind":"definition-passage","excerpt":"[Seed rank] For a unital left module M over A, define srank_A(M):=min{|S|:AS=M}. For a topological module (X,τ), define srank_ A,τ(X):=min{|S|:cl A S^ τ=X}. Set srank_A(0)=0. When the value is finite, it is the minimal number of module generators.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-4284d18f422a1f0d","tex":"\\mathrm{statement}","aliases":"statement mathrm","display":"\\mathrm{statement}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-02a3d6cf5b723f6c","tex":"\\mathrm{status}","aliases":"status mathrm","display":"\\mathrm{status}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A downstream paper cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this paper.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-58e6e44b43884fcc","tex":"\\Subset","aliases":"Subset","display":null,"kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-f1bf00c1871dc54e","tex":"\\sup_{F\\Subset\\mathbb N}","aliases":"sup_FSubset N sup Subset mathbb","display":null,"kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-53c867f72ca37df5","tex":"\\sup_{|a_n|\\leq1}","aliases":"sup_|a_n|≤1 sup leq","display":"\\sup_{|a_n|\\leq1}","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps ","count":31},{"work":"paper-i","url":"notation/paper-i.html#symbol-65fa7282e2650c3e","tex":"\\triangleleft","aliases":"triangleleft","display":null,"kind":"definition-passage","excerpt":"[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":29},{"work":"paper-i","url":"notation/paper-i.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈","count":40},{"work":"paper-i","url":"notation/paper-i.html#symbol-51e189ae43085a7d","tex":"X_a","aliases":"X_a","display":"X_a","kind":"definition-passage","excerpt":"(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-e7e08756fa886133","tex":"X_C","aliases":"X_C","display":"X_C","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":10},{"work":"paper-i","url":"notation/paper-i.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-e6fa3eb7a793d7e6","tex":"X_{s(a)}","aliases":"X_s(a)","display":"X_{s(a)}","kind":"definition-passage","excerpt":"(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-39e2cb7b9e9ab6bc","tex":"X_{t(a)}","aliases":"X_t(a)","display":"X_{t(a)}","kind":"definition-passage","excerpt":"(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-08a6f9f108660d1a","tex":"X_v","aliases":"X_v","display":"X_v","kind":"definition-passage","excerpt":"r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"k G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear ","count":51},{"work":"paper-i","url":"notation/paper-i.html#symbol-536459ef72a0484d","tex":"Y_a","aliases":"Y_a","display":"Y_a","kind":"definition-passage","excerpt":"linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-58a7d551ec1df48c","tex":"Y_v","aliases":"Y_v","display":"Y_v","kind":"definition-passage","excerpt":"l on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9021f7af081c388c","tex":"|_C","aliases":"|_C","display":"|_C","kind":"definition-passage","excerpt":"f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.","count":15},{"work":"paper-i","url":"notation/paper-i.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"definition-passage","excerpt":"efinition[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":20},{"work":"paper-i","url":"notation/paper-i.html#symbol-4330ec1682dc6696","tex":"\\varepsilon_S","aliases":"ε_S varepsilon","display":"\\varepsilon_S","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"definition-passage","excerpt":"[Packet coordinate condition number] Fix a finite product norm on ⊕_α X_α equivalent to the maximum norm, as in [source reference]. Define κ( P):= A_ P S_ P .","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):","count":15},{"work":"paper-i","url":"notation/paper-i.html#symbol-6512b523961a05f5","tex":"\\sigma^{-1}","aliases":"σ^-1 sigma","display":"\\sigma^{-1}","kind":"definition-passage","excerpt":"orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":10},{"work":"paper-i","url":"notation/paper-i.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.","count":57},{"work":"paper-i","url":"notation/paper-i.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"definition-passage","excerpt":"[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):","count":23},{"work":"paper-i","url":"notation/paper-i.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":17},{"work":"paper-i","url":"notation/paper-i.html#symbol-9ec711e428117ead","tex":"\\twoheadrightarrow","aliases":"↠ twoheadrightarrow","display":"\\twoheadrightarrow","kind":"definition-passage","excerpt":"Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.","count":12},{"work":"paper-i","url":"notation/paper-i.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈","count":185},{"work":"paper-i","url":"notation/paper-i.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"definition-passage","excerpt":"Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-6ce2ba87b3975aab","tex":"\\sum_{j=1}^{r}","aliases":"∑_j=1^r sum","display":"\\sum_{j=1}^{r}","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-41a5548d37bd81ad","tex":"\\sum_{n\\in F}","aliases":"∑_n∈ F sum in","display":"\\sum_{n\\in F}","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-7a26fb0b26e32548","tex":"\\sum_{s\\in S}","aliases":"∑_s∈ S sum in","display":"\\sum_{s\\in S}","kind":"definition-passage","excerpt":"[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-37e66864a7c0cc6a","tex":"\\sum_{\\alpha\\in C}","aliases":"∑_α∈ C sum alpha in","display":"\\sum_{\\alpha\\in C}","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-cabedb544f677980","tex":"\\sum_{\\alpha\\in I}","aliases":"∑_α∈ I sum alpha in","display":"\\sum_{\\alpha\\in I}","kind":"definition-passage","excerpt":"[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-29724c5f8f33dece","tex":"\\sum_{\\sigma(\\alpha)=j}","aliases":"∑_σ(α)=j sum sigma alpha","display":"\\sum_{\\sigma(\\alpha)=j}","kind":"definition-passage","excerpt":"orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"definition-passage","excerpt":"[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"definition-passage","excerpt":"[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.","count":18},{"work":"paper-i","url":"notation/paper-i.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"[Packet support and packet pure vectors] For a finite packet system P=(P_α)_α∈ I and x∈ X, define Supp_ P(x) := {α∈ I:P_α x≠0}. The vector x is packet pure if |Supp_ P(x)|≤1. A finite family is packet pure if each of its members is packet pure, although different members may lie in different packets.","count":29},{"work":"paper-i","url":"notation/paper-i.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.","count":16},{"work":"paper-i","url":"notation/paper-i.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Realization on a single space] For a typed realization X, define its realization on a single space by |X|:=⊕_v∈ Q_0X_v. Every vertex idempotent acts as a coordinate projector on |X|, and every path acts as a block operator from its source space to its target space.","count":26},{"work":"paper-i","url":"notation/paper-i.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":37},{"work":"paper-i","url":"notation/paper-i.html#symbol-dcf8db50a213006d","tex":"\\bigoplus_{n\\geq1}^{u}","aliases":"⊕_n≥1^u bigoplus geq","display":"\\bigoplus_{n\\geq1}^{u}","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-46d26f7a7cd16ab9","tex":"\\bigoplus_{v\\in Q_0}","aliases":"⊕_v∈ Q_0 bigoplus in","display":"\\bigoplus_{v\\in Q_0}","kind":"definition-passage","excerpt":"[Realization on a single space] For a typed realization X, define its realization on a single space by |X|:=⊕_v∈ Q_0X_v. Every vertex idempotent acts as a coordinate projector on |X|, and every path acts as a block operator from its source space to its target space.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9b591bcc37a61369","tex":"\\bigoplus_\\alpha","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_\\alpha","kind":"definition-passage","excerpt":"[Packet coordinate condition number] Fix a finite product norm on ⊕_α X_α equivalent to the maximum norm, as in [source reference]. Define κ( P):= A_ P S_ P .","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-90b882d77bb3725c","tex":"\\bigoplus_{\\alpha\\in I}","aliases":"⊕_α∈ I bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I}","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] For a finite packet system, define A_ P:X⟶⊕_α∈ IX_α, x⟼(P_α x)_α, and S_ P:⊕_α∈ IX_α⟶ X, (x_α)_α⟼∑_α x_α.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":19},{"work":"paper-i","url":"notation/paper-i.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":80},{"work":"paper-i","url":"notation/paper-i.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"introduction-passage","excerpt":"dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been adde","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been adde","count":28},{"work":"paper-i","url":"notation/paper-i.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"introduction-passage","excerpt":"dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been adde","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"introduction-passage","excerpt":"[Unweighted source component count] Assume the module resolved into packets is packet propagating and X_α≠0 for every α∈ I. If c_0 denotes the number of source strongly connected components, then psrank_ A, P(X)=c_0. [source reference]","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":32},{"work":"paper-i","url":"notation/paper-i.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":15},{"work":"paper-i","url":"notation/paper-i.html#symbol-616ff9ffcce7048e","tex":"D_i^{d_i}","aliases":"D_i^d_i","display":"D_i^{d_i}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"classicalinput[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":98},{"work":"paper-i","url":"notation/paper-i.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"introduction-passage","excerpt":"vpage10Which zero patterns survive multiplication?SUPPORT ALGEBRAS / A UNIVERSAL TESTMain text: [source reference]. All packet spaces are nonzero Suppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let E_ij denote the matrix that sends e_j to e_i and kills ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-ca7e75161b6bbf35","tex":"E_{ij}","aliases":"E_ij","display":"E_{ij}","kind":"introduction-passage","excerpt":"vpage10Which zero patterns survive multiplication?SUPPORT ALGEBRAS / A UNIVERSAL TESTMain text: [source reference]. All packet spaces are nonzero Suppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let E_ij denote the matrix that sends e_j to e_i and kills ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"introduction-passage","excerpt":"vpage10Which zero patterns survive multiplication?SUPPORT ALGEBRAS / A UNIVERSAL TESTMain text: [source reference]. All packet spaces are nonzero Suppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let E_ij denote the matrix that sends e_j to e_i and kills ","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":22},{"work":"paper-i","url":"notation/paper-i.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N","count":76},{"work":"paper-i","url":"notation/paper-i.html#symbol-0e79b7ad0299d0b0","tex":"I_0","aliases":"I_0","display":"I_0","kind":"introduction-passage","excerpt":"For s∈ S, write s=∑_α∈ I_0P_α s. If β is unreachable from I_0, then for every A∈ A, P_β As = ∑_α∈ I_0P_β AP_α s =0 by [source reference]. The same holds for every finite linear combination of orbit vectors.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-36282b37d83ec448","tex":"\\Irr","aliases":"Irr operatorname","display":"{\\operatorname{Irr}}","kind":"introduction-passage","excerpt":"[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N","count":77},{"work":"paper-i","url":"notation/paper-i.html#symbol-e5a8e8a33dcbaa8c","tex":"M_{d_i}","aliases":"M_d_i","display":"M_{d_i}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"introduction-passage","excerpt":"The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N","count":15},{"work":"paper-i","url":"notation/paper-i.html#symbol-a405bf0a7338f95b","tex":"M_y","aliases":"M_y","display":"M_y","kind":"introduction-passage","excerpt":"headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-b84d9005844b0c6f","tex":"M_y^2","aliases":"M_y^2","display":"M_y^2","kind":"introduction-passage","excerpt":"headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-956f614ae0645ab5","tex":"\\mat","aliases":"mat","display":null,"kind":"introduction-passage","excerpt":"headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"introduction-passage","excerpt":"The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":48},{"work":"paper-i","url":"notation/paper-i.html#symbol-b47e4ae8f60bef45","tex":"p^e","aliases":"p^e","display":"p^e","kind":"introduction-passage","excerpt":"classicalinput[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":25},{"work":"paper-i","url":"notation/paper-i.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ Q mathbb","display":"{\\mathbb Q}","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":33},{"work":"paper-i","url":"notation/paper-i.html#symbol-698cd4222a287d75","tex":"S_i","aliases":"S_i","display":"S_i","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-389639d67f2e3b9d","tex":"S_i^{\\oplus d_i}","aliases":"S_i^⊕ d_i oplus","display":"S_i^{\\oplus d_i}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"introduction-passage","excerpt":"headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":17},{"work":"paper-i","url":"notation/paper-i.html#symbol-89d5e60ade50d285","tex":"X_p","aliases":"X_p","display":"X_p","kind":"introduction-passage","excerpt":"[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":66},{"work":"paper-i","url":"notation/paper-i.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":10},{"work":"paper-i","url":"notation/paper-i.html#symbol-3916bfdbd80e58c2","tex":"\\prod_{i=1}^{q}","aliases":"∏_i=1^q prod","display":"\\prod_{i=1}^{q}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-447b896d00a61af2","tex":"\\sum_{i\\in F}","aliases":"∑_i∈ F sum in","display":"\\sum_{i\\in F}","kind":"introduction-passage","excerpt":"dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been adde","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-f6b0a941f2917366","tex":"\\sum_{\\alpha\\in I_0}","aliases":"∑_α∈ I_0 sum alpha in","display":"\\sum_{\\alpha\\in I_0}","kind":"introduction-passage","excerpt":"For s∈ S, write s=∑_α∈ I_0P_α s. If β is unreachable from I_0, then for every A∈ A, P_β As = ∑_α∈ I_0P_β AP_α s =0 by [source reference]. The same holds for every finite linear combination of orbit vectors.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":24},{"work":"paper-i","url":"notation/paper-i.html#symbol-a258c5e7fc44d1aa","tex":"\\bigoplus_{i=1}^{q}","aliases":"⊕_i=1^q bigoplus","display":"\\bigoplus_{i=1}^{q}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-a13933c3ddf256f7","tex":"\\bigoplus_{p\\in\\Irr(\\bbk[t])}","aliases":"⊕_p∈Irr(bbk[t]) ⊕_p∈Irr(𝕜[t]) bigoplus in Irr bbk bigoplus in operatorname Bbbk","display":"\\bigoplus_{p\\in{\\operatorname{Irr}}({\\Bbbk}[t])}","kind":"introduction-passage","excerpt":"[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint pa","count":21},{"work":"paper-i","url":"notation/paper-i.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"vpage15Walk counts make an endpoint cyclicGRAPH MODEL / THE PATH CERTIFICATEMain text: Worked graph model. Endpoint Krylov proof for every path size Let A be the adjacency matrix of the path with five vertices. The (i,j) entry of K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1] counts walks of length j-1 from vertex 1 to vertex i.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-06b373fe54fc27d1","tex":"A^3","aliases":"A^3","display":"A^3","kind":"use","excerpt":"vpage15Walk counts make an endpoint cyclicGRAPH MODEL / THE PATH CERTIFICATEMain text: Worked graph model. Endpoint Krylov proof for every path size Let A be the adjacency matrix of the path with five vertices. The (i,j) entry of K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1] counts walks of length j-1 from vertex 1 to vertex i.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6a4d1b76a4ccae86","tex":"A^4","aliases":"A^4","display":"A^4","kind":"use","excerpt":"vpage15Walk counts make an endpoint cyclicGRAPH MODEL / THE PATH CERTIFICATEMain text: Worked graph model. Endpoint Krylov proof for every path size Let A be the adjacency matrix of the path with five vertices. The (i,j) entry of K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1] counts walks of length j-1 from vertex 1 to vertex i.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b57dc62cca087611","tex":"A^{n-1}","aliases":"A^n-1","display":"A^{n-1}","kind":"use","excerpt":"…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry count","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6b4a8da3e0ca9141","tex":"\\A^{\\oplus r}","aliases":"A^⊕ r A oplus mathcal oplus","display":"{\\mathcal A}^{\\oplus r}","kind":"use","excerpt":"Topological completion of a presentation. [Completion of a dense free presentation] Let X be a complete Hausdorff locally convex space, let A⊆End(X) act continuously, and let S=(s_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the t","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-37e96daac260c605","tex":"A^{\\oplus2}","aliases":"A^⊕2 oplus","display":"A^{\\oplus2}","kind":"use","excerpt":"[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"The two packet spaces are A_0=𝕜[x], A_1=y𝕜[x]. Let P_0(a+yb)=a, P_1(a+yb)=yb. These idempotents are linear over 𝕜, annihilate one another, and satisfy P_0+P_1=I. They are not module maps over A, because multiplication by y exchanges the two packets.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"The two packet spaces are A_0=𝕜[x], A_1=y𝕜[x]. Let P_0(a+yb)=a, P_1(a+yb)=yb. These idempotents are linear over 𝕜, annihilate one another, and satisfy P_0+P_1=I. They are not module maps over A, because multiplication by y exchanges the two packets.","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"00,0.4800) The quotient keeps only the two chain heads; node[font=scriptsize] at (0.4250,0.1900) e_0; node[font=scriptsize] at (1.2750,0.1900) e_1; node[font=scriptsize] at (2.1250,0.1900) e_2; node[font=scriptsize] at (2.9750,0.1900) f_0; node[font=scriptsize] at (3.8250,0.1900) f_1; node[font=scriptsize] at (4.6750,0.1900) f_2; node[] at (2.5500,","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d325da4f9b58928d","tex":"A_3","aliases":"A_3","display":"A_3","kind":"use","excerpt":"6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-9fdabdeec2ad6682","tex":"A_{C_n}","aliases":"A_C_n","display":"A_{C_n}","kind":"use","excerpt":"Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To se","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-c5f0daaa0c763e4c","tex":"A_G","aliases":"A_G","display":"A_G","kind":"use","excerpt":"Let G be a nonempty finite graph with adjacency matrix A_G, and regard 𝕜^V(G) as a module over 𝕜[A_G]. The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required d","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-17821523641b2512","tex":"A_K","aliases":"A_K","display":"A_K","kind":"use","excerpt":"[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient fi","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-04fb14fb3279afa6","tex":"A_{P_n}","aliases":"A_P_n","display":"A_{P_n}","kind":"use","excerpt":"For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized e","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-895b4c69e53e4df6","tex":"A_x","aliases":"A_x","display":"A_x","kind":"use","excerpt":"[Operators chosen for a target] For a fixed seed s and target x, the existence of a specially defined operator A_x satisfying A_xs=x does not establish a natural seed calculus unless A_x belongs to a grammar fixed independently of the target satisfying [source reference].","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-31bff3b683a68d27","tex":"a_{\\lambda}","aliases":"a_λ lambda","display":"a_{\\lambda}","kind":"use","excerpt":"[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N:","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-133aeb6702a1cdfe","tex":"\\AA","aliases":"AA A mathcal","display":"{\\mathcal A}","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-87708a8e37000af2","tex":"\\operatorname{Ann}","aliases":"Ann operatorname","display":"\\operatorname{Ann}","kind":"use","excerpt":"[Presentation with one seed] For s∈ M, the annihilator Ann_A(s):={a∈ A:as=0} is a left ideal and A/Ann_A(s)≅Hull_A(s). [source reference]","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-eca8c78dc9b64bd4","tex":"\\operatorname{Ann}_A","aliases":"Ann_A operatorname","display":"\\operatorname{Ann}_A","kind":"use","excerpt":"[Presentation with one seed] For s∈ M, the annihilator Ann_A(s):={a∈ A:as=0} is a left ideal and A/Ann_A(s)≅Hull_A(s). [source reference]","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-4afb314fdcd4106f","tex":"\\arraybackslash","aliases":"arraybackslash","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"[Conditioned coordinate transport] For a finite Banach packet system with X≠0, κ( P)≥1. If T∈ B(X) and [T]_ P:= A_ PT S_ P, then [T]_ P ≤κ( P) T , T ≤κ( P) [T]_ P . [source reference]","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-899a52e5583edff3","tex":"\\mathrm b","aliases":"b mathrm","display":"\\mathrm b","kind":"use","excerpt":"[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"00,0.4800) The quotient keeps only the two chain heads; node[font=scriptsize] at (0.4250,0.1900) e_0; node[font=scriptsize] at (1.2750,0.1900) e_1; node[font=scriptsize] at (2.1250,0.1900) e_2; node[font=scriptsize] at (2.9750,0.1900) f_0; node[font=scriptsize] at (3.8250,0.1900) f_1; node[font=scriptsize] at (4.6750,0.1900) f_2; node[] at (2.5500,","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","excerpt":"vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required d","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"Define a surjective linear map q:X⟶ Q^2, q(∑ a_je_j+∑ b_jf_j)=(a_0,b_0).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9db34553aedb800c","tex":"B_{ji}","aliases":"B_ji","display":"B_{ji}","kind":"use","excerpt":"[Table specification for transfer blocks] Given finite projector systems (P_i)_i∈ I and (Q_j)_j∈ J and a linear map T, form the table B_ji=Q_jTP_i. Its assembly reconstructs the input map exactly, ∑_j∈ J∑_i∈ IB_ji=T, and its reported support relation is exactly {(i,j):B_ji≠0}. The latter equality is the specification of the transfer graph output. I","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-6e4af5afb524dbe9","tex":"B_{\\beta\\alpha}","aliases":"B_βα beta alpha","display":"B_{\\beta\\alpha}","kind":"use","excerpt":"Injectivity follows from exact reconstruction. Given arbitrary linear maps B_βα:X_α→ X_β, define T(∑_α x_α) := ∑_α,βB_βαx_α. The packet decomposition is unique and the index set is finite, so this defines a linear operator whose blocks are the prescribed maps. Thus Θ_ P is surjective. Compatibility with multiplication is [source reference].","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-c4577d3803b0374b","tex":"b_{\\lambda,j}","aliases":"b_λ,j lambda","display":"b_{\\lambda,j}","kind":"use","excerpt":"[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N:","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-a34ab46342f89739","tex":"\\BB","aliases":"BB B mathcal","display":"{\\mathcal B}","kind":"use","excerpt":"headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection ","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-b2a8d769f2c647dd","tex":"\\bbk^2","aliases":"bbk^2 𝕜^2 bbk Bbbk","display":"{\\Bbbk}^2","kind":"use","excerpt":"[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-d6253e89ee0f2326","tex":"\\bbk^3","aliases":"bbk^3 𝕜^3 bbk Bbbk","display":"{\\Bbbk}^3","kind":"use","excerpt":"[title=A quotient can strictly lower seed rank] Let A=𝕜 act by scalars on M=𝕜^3. Then srank_A(M)=3. Quotienting by a subspace of dimension one gives a scalar module of dimension two, whose seed rank is 2. Quotient monotonicity is therefore often strict.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-c457dace33feb6b0","tex":"\\bbk^{d_i}","aliases":"bbk^d_i 𝕜^d_i bbk Bbbk","display":"{\\Bbbk}^{d_i}","kind":"use","excerpt":"[Algebraically closed multiplicity formula] Let 𝕜 be algebraically closed, let A be a unital algebra of finite dimension over 𝕜, put J=Jac(A), and let M be a finitely generated left module over A. Assume compatible decompositions indexed by an integer q≥0: A/J≅∏_i=1^qM_d_i(𝕜), M/JM≅⊕_i=1^q(𝕜^d_i)^⊕ m_i, where the d_i are positive integers and the m","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-5722bd9f294027a7","tex":"\\bbk^n","aliases":"bbk^n 𝕜^n bbk Bbbk","display":"{\\Bbbk}^n","kind":"use","excerpt":"Let X=𝕜^n and let A=M_n(𝕜)=End_𝕜(X), with the full matrix algebra declared before a seed or target is chosen. By [source reference], every nonzero seed is cyclic. This is the maximal linear grammar on X. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-18cc9e7034e6c340","tex":"\\bbk^{V(G)}","aliases":"bbk^V(G) 𝕜^V(G) bbk Bbbk","display":"{\\Bbbk}^{V(G)}","kind":"use","excerpt":"Let G be a nonempty finite graph with adjacency matrix A_G, and regard 𝕜^V(G) as a module over 𝕜[A_G]. The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-7aca27c01715d051","tex":"\\boxtimes","aliases":"boxtimes","display":null,"kind":"use","excerpt":"[Tensor product packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] Expor","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathbb","display":"{\\mathbb C}","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-ab8bde9a479f56a0","tex":"\\mathfrak C","aliases":"C mathfrak","display":"\\mathfrak C","kind":"use","excerpt":"[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport for","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-06e2221a9a4e2168","tex":"\\C^n","aliases":"C^n C mathbb","display":"{\\mathbb C}^n","kind":"use","excerpt":"Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To se","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-f6499c4d477ed5bd","tex":"\\mathfrak C_0","aliases":"C_0 mathfrak","display":"\\mathfrak C_0","kind":"use","excerpt":"[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport for","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-dcb4e0ba5e2088bf","tex":"C_6","aliases":"C_6","display":"C_6","kind":"use","excerpt":"vpage16Closing a path changes the seed obstructionGRAPH MODEL / A CYCLE WITH SIX VERTICESMain text: Worked graph model. Proof of the minimum seed count for complex cycles For the adjacency matrix A of C_6, the distinct eigenvalues are 2,1,-1,-2, with multiplicities 1,2,2,1. A single vertex seed spans only four directions. Two adjacent vertex seeds ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-cfbf41dac773cbfc","tex":"C_n","aliases":"C_n","display":"C_n","kind":"use","excerpt":"Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To se","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-82497562659dda68","tex":"\\cA","aliases":"cA A mathcal","display":"{\\mathcal A}","kind":"use","excerpt":"Let X=𝕜^n and let A=M_n(𝕜)=End_𝕜(X), with the full matrix algebra declared before a seed or target is chosen. By [source reference], every nonzero seed is cyclic. This is the maximal linear grammar on X. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-c52bed8ad815407b","tex":"\\cA^{\\oplus r}","aliases":"cA^⊕ r A^⊕ r cA oplus mathcal oplus","display":"{\\mathcal A}^{\\oplus r}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S),","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"use","excerpt":"0 · 1 · 1 · 0 · -1 · -1. Thus q: Q^6↠ Q^2 is a scalar quotient for the eigenvalue 1. Its extension over C gives the same obstruction of dimension two. The orbit of any single seed projects to a line, so no single seed can generate the full module.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-843721b3e84879d7","tex":"\\Ccal","aliases":"Ccal C mathcal","display":"{\\mathcal C}","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-d831feefac3295be","tex":"\\operatorname{char}","aliases":"char operatorname","display":"\\operatorname{char}","kind":"use","excerpt":"Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form,","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized e","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-d565d6a34f41b3b9","tex":"\\cref","aliases":"cref","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-ab700623b6727b01","tex":"\\mathbf D","aliases":"D mathbf","display":"\\mathbf D","kind":"use","excerpt":"[Equational realization principle] Let F: C→ D be an additive functor between preadditive categories. Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by F. Scalar multiplication by a coefficient ring is also preserved when the categories ","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","excerpt":"vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-3b3443a3b74189af","tex":"D^d","aliases":"D^d","display":"D^d","kind":"use","excerpt":"For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-d3394af1c25cf897","tex":"d_\\alpha","aliases":"d_α alpha","display":"d_\\alpha","kind":"use","excerpt":"[Dimension count] If X is finite dimensional and d_α=dim X_α, then dimEnd_𝕜(X) = ∑_α,βd_α d_β = (∑_α d_α)^2. [source reference]","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-9f48cea2c4057357","tex":"d_\\beta","aliases":"d_β beta","display":"d_\\beta","kind":"use","excerpt":"[Dimension count] If X is finite dimensional and d_α=dim X_α, then dimEnd_𝕜(X) = ∑_α,βd_α d_β = (∑_α d_α)^2. [source reference]","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry counts walks of length j-1 from the first vertex to vertex i. This entry is zero when i>j, while the diagonal entry is one because there is a unique shortest walk of length i-1 from vertex 1 to vertex i. Hence K_n is upper triangular with unit","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"use","excerpt":"0 · 1 · 0. The typed relation is BA=I_U. In the opposite order, AB=diag(1,1,0)≠ I_V. The missing direction is Ker B= Q v_3, not an entirely missing vertex.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"[Local algebra formula] Let A be a local ring with Jacobson radical J and residue division ring D=A/J. For every finitely generated left module M over A, srank_A(M)=dim_D(M/JM), where the dimension is that of a left vector space over D. [source reference]","count":14},{"work":"paper-i","url":"notation/paper-i.html#symbol-c75b893cf90454b2","tex":"\\dim_{\\bbk}","aliases":"dim_bbk dim_𝕜 dim bbk dim Bbbk","display":"\\dim_{{\\Bbbk}}","kind":"use","excerpt":"[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-42bb68489be0ad01","tex":"\\dim_{\\bbk[t]/(p)}","aliases":"dim_bbk[t]/(p) dim_𝕜[t]/(p) dim bbk dim Bbbk","display":"\\dim_{{\\Bbbk}[t]/(p)}","kind":"use","excerpt":"[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the d","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-0ca2307ebd3bffbf","tex":"\\dim_D","aliases":"dim_D dim","display":"\\dim_D","kind":"use","excerpt":"[Local algebra formula] Let A be a local ring with Jacobson radical J and residue division ring D=A/J. For every finitely generated left module M over A, srank_A(M)=dim_D(M/JM), where the dimension is that of a left vector space over D. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-98d940b7f3ffe088","tex":"\\dim_K","aliases":"dim_K dim","display":"\\dim_K","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-12cef7357a97a6d2","tex":"\\dim_{K_p}","aliases":"dim_K_p dim","display":"\\dim_{K_p}","kind":"use","excerpt":"(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], sra","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-0645873febb68b5c","tex":"\\Downarrow","aliases":"Downarrow","display":null,"kind":"use","excerpt":"Theorem spine at a glance. ConceptChain c free seed presentation [0.25em] Downarrow [-0.1em] minimal generator number [0.25em] Downarrow [-0.1em] reduction modulo the Jacobson radical [0.25em] Downarrow [-0.1em] semisimple or primary computation c packet idempotents [0.25em] Downarrow [-0.1em] exact analysis/synthesis [0.25em] Downarrow [-0.1em] tr","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"plies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any","count":14},{"work":"paper-i","url":"notation/paper-i.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"[title=Why a seed annihilator need not be an ideal on both sides] Let A=M_2(𝕜) act on M=𝕜^2, and let s=e_1. Then Ann_A(s) consists of matrices whose first column is zero. It is a left ideal because left multiplication preserves a zero first column. It is not generally a right ideal. Thus the natural quotient for one seed is a quotient of the left r","count":35},{"work":"paper-i","url":"notation/paper-i.html#symbol-46018870fc4913f9","tex":"E_{11}","aliases":"E_11","display":"E_{11}","kind":"use","excerpt":"t] (6.4000,-0.8500) rectangle ++(0.8500,-0.8500); node[text=black!45,font=scriptsize] at (6.8250,-1.2750) 0; filldraw[fill=black!5,draw=white,line width=.8pt] (7.2500,-0.8500) rectangle ++(0.8500,-0.8500); node[text=black!45,font=scriptsize] at (7.6750,-1.2750) 0; draw[black!60,line width=.5pt] (6.4,0) rectangle ++(1.7,-1.7); node[font=sffamilybfse","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-f2b11b7b1856ea86","tex":"E_{12}","aliases":"E_12","display":"E_{12}","kind":"use","excerpt":"8pt] (0.0000,-0.8500) rectangle ++(0.8500,-0.8500); node[text=black!45,font=scriptsize] at (0.4250,-1.2750) 0; filldraw[fill=black!5,draw=white,line width=.8pt] (0.8500,-0.8500) rectangle ++(0.8500,-0.8500); node[text=black!45,font=scriptsize] at (1.2750,-1.2750) 0; draw[black!60,line width=.5pt] (0,0) rectangle ++(1.7,-1.7); node[font=sffamilybfse","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":15},{"work":"paper-i","url":"notation/paper-i.html#symbol-eebcf5feadee9573","tex":"E_{21}","aliases":"E_21","display":"E_{21}","kind":"use","excerpt":"The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"[Typed reachability certificate] In the preceding quiver realization, no seed supported at U can generate the full representation, whereas the two seeds e_1∈ U, e_3∈ V do generate it.","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-40fa4751533ea967","tex":"E_{31}","aliases":"E_31","display":"E_{31}","kind":"use","excerpt":"The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-dee62bc352cb736e","tex":"E_{32}","aliases":"E_32","display":"E_{32}","kind":"use","excerpt":"The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-67ab753297c61ce5","tex":"e_4","aliases":"e_4","display":"e_4","kind":"use","excerpt":"vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-cf85306cf8e31b85","tex":"e_{j+1}","aliases":"e_j+1","display":"e_{j+1}","kind":"use","excerpt":"C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-a3fd533b471b8be8","tex":"e_{j-1}","aliases":"e_j-1","display":"e_{j-1}","kind":"use","excerpt":"C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-c6d45676877113dd","tex":"e_{jj}","aliases":"e_jj","display":"e_{jj}","kind":"use","excerpt":"For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-1b09d4efdcdc1f39","tex":"e_p","aliases":"e_p","display":"e_p","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-504dc54aa7ab931b","tex":"e_U","aliases":"e_U","display":"e_U","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-042162b6d9ca555d","tex":"e_v","aliases":"e_v","display":"e_v","kind":"use","excerpt":"[Typed realization and module equivalence] The category of realizations of G is equivalent to the category of unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its de","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-06718d7aa18063e4","tex":"e_w","aliases":"e_w","display":"e_w","kind":"use","excerpt":"n a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v im","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-bbd7a622a6f80cd3","tex":"e_\\lambda","aliases":"e_λ lambda","display":"e_\\lambda","kind":"use","excerpt":"X, continuity of a:X→ X supplies continuous seminorms controlling p(aj(e)), so the induced action on E is continuous and extends uniquely to widehat E. The subspace is dense by hypothesis. A Cauchy net in E is mapped to a Cauchy net in X, which converges because X is complete. This defines a continuous extension widehat j:widehat E→ X. For any x∈ X","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-9b8a4bc27627081b","tex":"F^m","aliases":"F^m","display":"F^m","kind":"use","excerpt":"headSeveral seeds give a module of relations For (s_1,…,s_m), evaluation is (f_1,…,f_m)↦∑_j f_j(T)s_j. Its kernel is a submodule of F^m and can include relations between seeds, not only relations annihilating each seed separately.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"Choose such a k≥ k_0. Since x^(k) is unconditionally summable, there is a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. There","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","excerpt":"Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.","count":11},{"work":"paper-i","url":"notation/paper-i.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]","count":16},{"work":"paper-i","url":"notation/paper-i.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","excerpt":"Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]","count":12},{"work":"paper-i","url":"notation/paper-i.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"use","excerpt":"Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-0fe97be4e95beb8d","tex":"f_{j+1}","aliases":"f_j+1","display":"f_{j+1}","kind":"use","excerpt":"ode[text=black!65] at (9.0000,0.0000) 0; draw[dashed,black!50] (1.0000,0.4800) -- (1.0000,1.0200); node[] at (1.0000,-0.9200) ae_0+af_0; draw[dashed,black!50] (4.0000,0.4800) -- (4.0000,1.0200); node[] at (4.0000,-0.9200) be_1+bf_1; draw[dashed,black!50] (7.0000,0.4800) -- (7.0000,1.0200); node[] at (7.0000,-0.9200) ce_2+cf_2; node[font=sffamilysma","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"use","excerpt":"headSeveral seeds give a module of relations For (s_1,…,s_m), evaluation is (f_1,…,f_m)↦∑_j f_j(T)s_j. Its kernel is a submodule of F^m and can include relations between seeds, not only relations annihilating each seed separately.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-f7499e87191c71a6","tex":"f_r","aliases":"f_r","display":"f_r","kind":"use","excerpt":"Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-9ecd9e4a3964eb4b","tex":"f_\\alpha","aliases":"f_α alpha","display":"f_\\alpha","kind":"use","excerpt":"[Universal mapping property of finite packet coordinates] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-09f3e72028e864cd","tex":"G_m","aliases":"G_m","display":"G_m","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6276707c2b6cfedb","tex":"G_{m-1}","aliases":"G_m-1","display":"G_{m-1}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-841df9ca84cfa40e","tex":"g_\\alpha","aliases":"g_α alpha","display":"g_\\alpha","kind":"use","excerpt":"] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_α","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-d05f0811844f2313","tex":"g_\\beta","aliases":"g_β beta","display":"g_\\beta","kind":"use","excerpt":"For x=∑_α P_α x, any extension of the f_α must satisfy f(x)=∑_α f_α(P_α x), which proves both existence and uniqueness of the first formula. For the second, the sum g=∑_α g_α satisfies P_β g =∑_α P_β g_α =g_β because g_α(Y)⊆ X_α. Uniqueness follows from ∑_α P_α=I.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"lpha,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of over","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-9bae743248da3740","tex":"\\head","aliases":"head","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9c9fcf42a4606088","tex":"\\Hull_{(O_i)}","aliases":"Hull_(O_i) Hull operatorname","display":"{\\operatorname{Hull}}_{(O_i)}","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-72683f068bfcc031","tex":"\\Hull_{\\AA}","aliases":"Hull_AA Hull_ A Hull AA operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal A}}","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-c17d7c5e355fb4ce","tex":"\\Hull_{\\cA}","aliases":"Hull_cA Hull_ A Hull cA operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal A}}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-aed5db3db4ec9fd6","tex":"\\Hull_{\\Ccal}","aliases":"Hull_Ccal Hull_ C Hull Ccal operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal C}}","kind":"use","excerpt":"· =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-40dcb49c2ffb76b9","tex":"\\Hull_F","aliases":"Hull_F Hull operatorname","display":"{\\operatorname{Hull}}_F","kind":"use","excerpt":"IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d2e09d6cb7fed0ec","tex":"\\Hull_{\\QQ[D]}","aliases":"Hull_QQ[D] Hull_ Q[D] Hull QQ operatorname mathbb","display":"{\\operatorname{Hull}}_{{\\mathbb Q}[D]}","kind":"use","excerpt":"headWhat the seed generates With the fixed algebra Q[D] and seed e_1, Hull_ Q[D](e_1)=span{e_1,e_2+e_3}. It has dimension two, not four. Both the independent difference e_2-e_3 and the last coordinate e_4 are missing.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-fbe558355eecd98b","tex":"\\Hull_{\\varphi(\\A)}","aliases":"Hull_φ(A) Hull_φ( A) Hull varphi A operatorname varphi mathcal","display":"{\\operatorname{Hull}}_{\\varphi({\\mathcal A})}","kind":"use","excerpt":"[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-ac5cb1bffa04d178","tex":"\\Hull_{\\varphi(\\AA)}","aliases":"Hull_φ(AA) Hull_φ( A) Hull varphi AA operatorname varphi mathcal","display":"{\\operatorname{Hull}}_{\\varphi({\\mathcal A})}","kind":"use","excerpt":"headWhen the map is not invertible For an intertwiner Φ A=φ(A)Φ, the hull identity uses the image algebra: Φ(Hull_ A(S))=Hull_φ( A)(Φ S). Replacing φ( A) with a larger target algebra can enlarge the target hull. Seed rank need not be equal relative to that larger algebra. A surjective intertwiner gives the appropriate monotonicity inequality, not a","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6c1a6475ed0ab0aa","tex":"I_{e_1}","aliases":"I_e_1","display":"I_{e_1}","kind":"use","excerpt":"headThe distinction is visible in two dimensions Use X= Q^2, seed e_1=(1,0)^ T, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column: I_e_1={mat0 · a 0 · b:a,b∈ Q}.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-111b6ee2c5f8dbd2","tex":"I_s","aliases":"I_s","display":"I_s","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-34af8cf33f9487de","tex":"I_U","aliases":"I_U","display":"I_U","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-f5bae71a192fc64d","tex":"I_V","aliases":"I_V","display":"I_V","kind":"use","excerpt":"0 · 1 · 0. The typed relation is BA=I_U. In the opposite order, AB=diag(1,1,0)≠ I_V. The missing direction is Ker B= Q v_3, not an entirely missing vertex.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-ec3582230ba1341f","tex":"I_{X/K}","aliases":"I_X/K","display":"I_{X/K}","kind":"use","excerpt":"[Packet descent through an invariant quotient] Under the hypotheses of [source reference], the operators overline P_α(x+K)=P_α x+K are well defined pairwise annihilating idempotents with sum I_X/K, and the quotient map satisfies π P_α=overline P_απ. After zero packets are removed, they form the canonical quotient packet system. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-756b4d7c969253c3","tex":"I_{X^*}","aliases":"I_X^*","display":"I_{X^*}","kind":"use","excerpt":"Because dualization reverses products, P_α^*P_β^*=(P_β P_α)^* =δ_αβP_α^*. Also ∑_α P_α^*=I_X^*. Thus the dual maps form a packet system. Restriction gives a map from P_α^*X^* to X_α^*. It is injective because a covector in the range of P_α^* vanishes on every other packet, and it is surjective because any covector on X_α extends to X through the di","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1718302dc2dde398","tex":"I_{X_\\alpha}","aliases":"I_X_α alpha","display":"I_{X_\\alpha}","kind":"use","excerpt":"The identity has diagonal blocks I_X_α and no blocks outside the diagonal, so it belongs to M_R exactly when (α,α)∈ R for every α.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-ab3ef27be5e3d7f7","tex":"I_Y","aliases":"I_Y","display":"I_Y","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-c32aece4942cf822","tex":"I_{\\oplus X_\\alpha}","aliases":"I_⊕ X_α oplus alpha","display":"I_{\\oplus X_\\alpha}","kind":"use","excerpt":"[Exact packet coordinate isomorphism] For every finite packet system, S_ P A_ P=I_X, A_ P S_ P=I_⊕ X_α. Hence X≅⊕_α∈ IX_α. [source reference] ExportsEvery exact packet reconstruction in the series.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":",0.0000) -- (2.0000,2.0000); draw[->,line width=1.2pt] (0.0000,0.0000) -- (3.0000,2.0000); draw[dashed,black!55] (1.0000,0.0000) -- (3.0000,2.0000); draw[dashed,black!55] (2.0000,2.0000) -- (3.0000,2.0000); node[] at (1.0000,-0.5800) Px=(1,0); node[anchor=east] at (1.4700,2.2000) Qx=(2,2); node[anchor=west] at (3.0800,2.1700) x=(3,2); node[font=scr","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-10ac8d7d633361db","tex":"J_3","aliases":"J_3","display":"J_3","kind":"use","excerpt":"Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-9743d7bec48cbb60","tex":"\\Jac","aliases":"Jac operatorname","display":"{\\operatorname{Jac}}","kind":"use","excerpt":"How many seeds are required? · srank_A(M) · For a finitely generated module, pass to M/Jac(A)M. When the quotient algebra is semisimple Artinian, compute its simple multiplicities.","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient fi","count":62},{"work":"paper-i","url":"notation/paper-i.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.","count":31},{"work":"paper-i","url":"notation/paper-i.html#symbol-9817d114fe256b80","tex":"k_0","aliases":"k_0","display":"k_0","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-bb9a538f5e406f30","tex":"K_n","aliases":"K_n","display":"K_n","kind":"use","excerpt":"…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry count","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-b39e465f5dcb17d2","tex":"K_p","aliases":"K_p","display":"K_p","kind":"use","excerpt":"T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-f178970e61601761","tex":"K_\\alpha","aliases":"K_α alpha","display":"K_\\alpha","kind":"use","excerpt":"[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ ","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-a77d87feaab21d87","tex":"K_\\beta","aliases":"K_β beta","display":"K_\\beta","kind":"use","excerpt":"[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-e08d6b60be0ec0b0","tex":"\\Ker","aliases":"Ker operatorname","display":"{\\operatorname{Ker}}","kind":"use","excerpt":"[Finite free presentation] For the map in [source reference], imε_S=Hull_A(S), N_S:=Kerε_S≤ A^⊕ r, and the induced map is an isomorphism of left modules over A A^⊕ r/N_S≅Hull_A(S). Consequently, S generates M if and only if ε_S is surjective. [source reference] ExportsAll later free seed presentations.","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is unreachable from a seed supported only at U. There","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-a4f733cd50dc9a2c","tex":"\\kk","aliases":"kk 𝕜 Bbbk","display":"{\\Bbbk}","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-97491b80a0eec919","tex":"\\kk^2","aliases":"kk^2 𝕜^2 kk Bbbk","display":"{\\Bbbk}^2","kind":"use","excerpt":"The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"One seed and the distinction between module and algebra relations. [Criterion for the annihilator to be an ideal on both sides] For s∈ M, the left ideal L=Ann_A(s) is an ideal on both sides if and only if L Hull_A(s)=0. If s is cyclic, these conditions are equivalent to Ann_A(s)=Ann_A(M). [source reference]","count":20},{"work":"paper-i","url":"notation/paper-i.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"se there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with [2pt] 𝕜^V(G)=span_𝕜 {p(A_G)s_j:p∈𝕜[t], 1≤ j≤ r} }.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-45e90aa85790529d","tex":"\\linewidth","aliases":"linewidth","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Cyclic characterization] For M≠0, srank_A(M)=1 Longleftrightarrow M contains a cyclic vector. [source reference]","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"1150) x^2y; filldraw[fill=black!18,draw=white] (3.9000000000000004,-0.8) rectangle ++(1.15,-.63); node[] at (4.4750,-1.1150) x^3y; filldraw[fill=black!18,draw=white] (5.2,-0.8) rectangle ++(1.15,-.63); node[] at (5.7750,-1.1150) x^4y; node[] at (6.9500,-1.1150) ⋯; node[] at (3.2000,0.5200) Independent coefficient slots: a(x) and b(x); node[] at (3.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-4a511b20c73344dd","tex":"M_{2\\times5}","aliases":"M_2×5 times","display":"M_{2\\times5}","kind":"use","excerpt":"vpage13How much row space can one seed supply?SPLIT SEMISIMPLE MODEL / A CONSTRUCTIVE FORMULAMain text: [source reference] Let M_2( Q) act on X=M_2×5( Q) by left multiplication. Think of the five columns as five copies of the same natural module of dimension two. One seed is a 2×5 matrix V.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-fa02e26ac15b5f1d","tex":"M_3","aliases":"M_3","display":"M_3","kind":"use","excerpt":"M_3( C) acting on seven copies, seed rank · 3","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b9672ca091c4a70a","tex":"M_4","aliases":"M_4","display":"M_4","kind":"use","excerpt":"Full M_4( R) orbit dimensions for two sampled nonzero seeds · 4,4","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-c5d6bd2d31f2166a","tex":"M_6","aliases":"M_6","display":"M_6","kind":"use","excerpt":"filldraw[draw=black!35,fill=white] (1.34,-1.92) rectangle ++(.48,.40); filldraw[draw=black!35,fill=white] (2.0100000000000002,-1.92) rectangle ++(.48,.40); filldraw[draw=black!35,fill=white] (2.68,-1.92) rectangle ++(.48,.40); filldraw[draw=black!35,fill=white] (3.35,-1.92) rectangle ++(.48,.40); node[font=sffamilybfseries] at (4.5000,-1.7200) 1; n","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-c887199df3da3b63","tex":"M_a","aliases":"M_a","display":"M_a","kind":"use","excerpt":"unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-207e98afc142d0de","tex":"M_d","aliases":"M_d","display":"M_d","kind":"use","excerpt":"For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-c879cfab2bd64d1b","tex":"m_j","aliases":"m_j","display":"m_j","kind":"use","excerpt":"The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a fini","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-8f343ef357a239b3","tex":"M_K","aliases":"M_K","display":"M_K","kind":"use","excerpt":"[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient fi","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"use","excerpt":"Let X=𝕜^n and let A=M_n(𝕜)=End_𝕜(X), with the full matrix algebra declared before a seed or target is chosen. By [source reference], every nonzero seed is cyclic. This is the maximal linear grammar on X. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-ce78c2a469d42dfa","tex":"m_n","aliases":"m_n","display":"m_n","kind":"use","excerpt":"For x∈ X, choose m_n∈ M with m_n→ x. Continuity gives P_α m_n→ P_α x, so P_α X⊆clP_α M. The reverse inclusion follows because P_α X is closed and contains P_α M. The bounded extensions remain pairwise annihilating idempotents whose finite sum is the identity.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-f7cf81ec814da4db","tex":"\\mathfrak M_R^{\\mathrm b}","aliases":"M_R^ b mathfrak mathrm","display":"\\mathfrak M_R^{\\mathrm b}","kind":"use","excerpt":"[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-37c0c78c3530a9c1","tex":"\\mathfrak M_{R_\\Gamma}","aliases":"M_R_Γ mathfrak Gamma","display":"\\mathfrak M_{R_\\Gamma}","kind":"use","excerpt":"[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-11048af18ba22882","tex":"m_T","aliases":"m_T","display":"m_T","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-4ff979fe78820d20","tex":"M_v","aliases":"M_v","display":"M_v","kind":"use","excerpt":"unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-0521a5cf2b799404","tex":"M_x","aliases":"M_x","display":"M_x","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-89af9e181208ee45","tex":"M_x^3","aliases":"M_x^3","display":"M_x^3","kind":"use","excerpt":"The relation is reconstructed as an operator identity: M_y^2=M_x^3-M_x.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d3010d2a559d3a13","tex":"M_{x^3-x}","aliases":"M_x^3-x","display":"M_{x^3-x}","kind":"use","excerpt":"Identify the packet coefficients by A_0⊕ A_1⟶𝕜[x]⊕𝕜[x], (a,yb)⟼(a,b). In these ordered coefficient coordinates, with M_x and M_x^3-x denoting multiplication on 𝕜[x], M_x= M_x · 0","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-fdf4922b396511d5","tex":"M_x^i","aliases":"M_x^i","display":"M_x^i","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e1d444128c7741a9","tex":"M_y^j","aliases":"M_y^j","display":"M_y^j","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-0dae06547437e14b","tex":"m_\\lambda","aliases":"m_λ lambda","display":"m_\\lambda","kind":"use","excerpt":"For the topological statement, let S generate M densely. Given n=Φ(m), choose a net m_λ∈ AS converging to m. Continuity gives Φ(m_λ)⟶ n, and every Φ(m_λ) lies in BΦ(S). Thus BΦ(S) is dense in N.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-7a7ea658fe0f7067","tex":"m_{\\lambda}","aliases":"m_λ lambda","display":"m_{\\lambda}","kind":"use","excerpt":"[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N:","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-8d03a88e50994883","tex":"m_{\\lambda_*}","aliases":"m_λ_* lambda","display":"m_{\\lambda_*}","kind":"use","excerpt":"r] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r wi","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"vector space over a complete valued field, together with a unital algebra of continuous linear operators. Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement. Conventions for empty families. For finite families of nonnegative integers we use max∅=0. The height o","count":13},{"work":"paper-i","url":"notation/paper-i.html#symbol-848ac68ee082047f","tex":"\\max_{1\\leq i\\leq q}","aliases":"max_1≤ i≤ q max leq leq","display":"\\max_{1\\leq i\\leq q}","kind":"use","excerpt":"[Semisimple seed rank formula over arbitrary division rings] Let A≅∏_i=1^qM_d_i(D_i) be semisimple Artinian and let M≅⊕_i=1^qS_i^⊕ m_i be a finitely generated left module over A. Then srank_A(M) = max_1≤ i≤ q lceilfracm_id_irceil. If A is any algebra of finite dimension over a field, the same formula applied to the decomposition of M/Jac(A)M over A","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-1110d632fcc473b9","tex":"\\max_p","aliases":"max_p max","display":"\\max_p","kind":"use","excerpt":"[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the d","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-04cf78164e0d76e4","tex":"\\max_{\\lambda\\in\\operatorname{Spec}(T)}","aliases":"max_λ∈Spec(T) max lambda in operatorname","display":"\\max_{\\lambda\\in\\operatorname{Spec}(T)}","kind":"use","excerpt":"[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-689cf901ed5ff942","tex":"\\max_{\\mu\\in\\Spec_K(A)}","aliases":"max_μ∈Spec_K(A) max mu in Spec max mu in operatorname","display":"\\max_{\\mu\\in{\\operatorname{Spec}}_K(A)}","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-3f9db2ae5742966b","tex":"\\midrule","aliases":"midrule","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N mathbb","display":"{\\mathbb N}","kind":"use","excerpt":"[Shift seed] For n∈ N, let G(n) mean that some finite seed set with n elements has dense hull under the declared family {S_m:m∈ Z}. Then 1=min{n∈ N:G(n)}. Equivalently, one seed suffices and no empty seed family generates densely.","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-a41020d8546c0cd6","tex":"\\mathfrak N","aliases":"N mathfrak","display":"\\mathfrak N","kind":"use","excerpt":"[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [labe","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"vpage3Two chains can stay linkedMODEL A / AN OPERATOR IN SIX DIMENSIONSMain text: [source reference]. Matrices over Q Use the ordered basis (e_0,e_1,e_2,f_0,f_1,f_2). Define Ne_0=e_1, Ne_1=e_2, Ne_2=0, Nf_0=f_1, Nf_1=f_2, Nf_2=0. Starting with s=e_0+f_0, the two chains advance together: s=e_0+f_0, Ns=e_1+f_1, N^2s=e_2+f_2, N^3s=0.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"vpage3Two chains can stay linkedMODEL A / AN OPERATOR IN SIX DIMENSIONSMain text: [source reference]. Matrices over Q Use the ordered basis (e_0,e_1,e_2,f_0,f_1,f_2). Define Ne_0=e_1, Ne_1=e_2, Ne_2=0, Nf_0=f_1, Nf_1=f_2, Nf_2=0. Starting with s=e_0+f_0, the two chains advance together: s=e_0+f_0, Ns=e_1+f_1, N^2s=e_2+f_2, N^3s=0.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a fini","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-073301ff99f56685","tex":"\\mathfrak N_R","aliases":"N_R mathfrak","display":"\\mathfrak N_R","kind":"use","excerpt":"[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [labe","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-5ffa89d2eeb94201","tex":"\\mathfrak N_R^{\\,h+1}","aliases":"N_R^ h+1 mathfrak","display":"\\mathfrak N_R^{\\,h+1}","kind":"use","excerpt":"rtial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. item The map to the diagonal classe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-173d1ec8e27571d5","tex":"N_S","aliases":"N_S","display":"N_S","kind":"use","excerpt":"[Finite free presentation] For the map in [source reference], imε_S=Hull_A(S), N_S:=Kerε_S≤ A^⊕ r, and the induced map is an isomorphism of left modules over A A^⊕ r/N_S≅Hull_A(S). Consequently, S generates M if and only if ε_S is surjective. [source reference] ExportsAll later free seed presentations.","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"headWhat the algebra does not preserve For the displayed x, normx^2=13, normPx^2+normQx^2=1+8=9. There is no Parseval identity here. The cross term is nonzero because the packet directions are not perpendicular. Likewise, the simple formula for reversing adjoints of packet blocks requires selfadjoint projectors. It is not justified for this oblique","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-962eb33121c4bcf4","tex":"O_i","aliases":"O_i","display":"O_i","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-ac5881edc3abfcd7","tex":"\\overline I","aliases":"overline I overline","display":"\\overline I","kind":"use","excerpt":"[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [labe","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-ffdd263bffb70d69","tex":"\\overline P","aliases":"overline P overline","display":"\\overline P","kind":"use","excerpt":"[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ ","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-00c5b69e7d95b7d6","tex":"\\overline P_\\alpha","aliases":"overline P_α overline alpha","display":"\\overline P_\\alpha","kind":"use","excerpt":"[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ ","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-f137d0144e6a41ef","tex":"\\overline P_\\beta","aliases":"overline P_β overline beta","display":"\\overline P_\\beta","kind":"use","excerpt":"If x∈ K_α, then Tx∈ K, so T_βαx=P_β Tx∈ K∩ X_β=K_β. On the quotient, overline T_βαπ = overline P_βoverline Toverline P_απ = π P_β TP_α.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-23856ee01c2b6c2b","tex":"\\overline T","aliases":"overline T overline","display":"\\overline T","kind":"use","excerpt":"[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-bb737e6f2757da05","tex":"\\overline T_{\\beta\\alpha}","aliases":"overline T_βα overline beta alpha","display":"\\overline T_{\\beta\\alpha}","kind":"use","excerpt":"[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"0 · 1. These maps satisfy P^2=P, Q^2=Q, PQ=QP=0 and P+Q=I. Their images are the horizontal line and the diagonal line, respectively. They are not selfadjoint in the usual Euclidean inner product.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-3157e31c12869d3e","tex":"p^{e_p}","aliases":"p^e_p","display":"p^{e_p}","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"The two packet spaces are A_0=𝕜[x], A_1=y𝕜[x]. Let P_0(a+yb)=a, P_1(a+yb)=yb. These idempotents are linear over 𝕜, annihilate one another, and satisfy P_0+P_1=I. They are not module maps over A, because multiplication by y exchanges the two packets.","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the ","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[title=R3: an exact but oblique packet split] On X=𝕜^2, set P_1=1 · -1 0 · 0, P_2=0 · 1 0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"[title=R3: an exact but oblique packet split] On X=𝕜^2, set P_1=1 · -1 0 · 0, P_2=0 · 1 0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.","count":10},{"work":"paper-i","url":"notation/paper-i.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the ","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-f0242a419961441b","tex":"P_3","aliases":"P_3","display":"P_3","kind":"use","excerpt":"headReconstruction retains every block The identity P_1+P_2+P_3=I gives T=∑_i,j=1^3 P_iTP_j. Selecting only the diagonal would change the operator. The entries 2,3,4 carry transfers between different packets.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-2fd63ee4a5169ac2","tex":"P_e","aliases":"P_e","display":"P_e","kind":"use","excerpt":"Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-d02aaee4231d3cd1","tex":"P_f","aliases":"P_f","display":"P_f","kind":"use","excerpt":"Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-fbc48a83ee80d83f","tex":"P_i^2","aliases":"P_i^2","display":"P_i^2","kind":"use","excerpt":"0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"vpage8A transfer block has an input and an outputOPERATOR COORDINATES / MULTIPLICATION AS A SUMMain text: [source reference] Use the three coordinate packets of dimension one in Q^3. The block T_ij=P_iTP_j sends input packet j into output packet i. For this example, its nonzero scalar is the (i,j) matrix entry.","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"use","excerpt":"[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤","count":11},{"work":"paper-i","url":"notation/paper-i.html#symbol-e65ae8417651699c","tex":"P_{N,j}","aliases":"P_N,j","display":"P_{N,j}","kind":"use","excerpt":"For N≥1, define residue projectors (P_N,jx)[n] = x[n], · n≡ jpmod N, 0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-4a442c81a1b99426","tex":"P_{N,j+1}","aliases":"P_N,j+1","display":"P_{N,j+1}","kind":"use","excerpt":"0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-4a0ce4bb02b1efb4","tex":"P_U","aliases":"P_U","display":"P_U","kind":"use","excerpt":"headWhy this does not prove an unrestricted minimum of two The bound path algebra includes the vertex idempotents P_U,P_V. A mixed seed s=u_1+v_3∈ U⊕ V satisfies P_Us=u_1 and P_Vs=v_3. It therefore generates the entire representation by itself. The paper proves that no seed supported at U is enough and that two displayed pure seeds suffice. It does","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-ae2504de22a746d7","tex":"P_V","aliases":"P_V","display":"P_V","kind":"use","excerpt":"headWhy this does not prove an unrestricted minimum of two The bound path algebra includes the vertex idempotents P_U,P_V. A mixed seed s=u_1+v_3∈ U⊕ V satisfies P_Us=u_1 and P_Vs=v_3. It therefore generates the entire representation by itself. The paper proves that no seed supported at U is enough and that two displayed pure seeds suffice. It does","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-5b1d3ec03a20b6b9","tex":"P_{\\alpha}","aliases":"P_α alpha","display":"P_{\\alpha}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-ed02c035995b1f57","tex":"P_\\alpha^*","aliases":"P_α^* alpha","display":"P_\\alpha^*","kind":"use","excerpt":"[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint pa","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-235bca2612706176","tex":"P_{\\beta}","aliases":"P_β beta","display":"P_{\\beta}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-2c3fc12ee2bca96e","tex":"P_\\beta^*","aliases":"P_β^* beta","display":"P_\\beta^*","kind":"use","excerpt":"Because dualization reverses products, P_α^*P_β^*=(P_β P_α)^* =δ_αβP_α^*. Also ∑_α P_α^*=I_X^*. Thus the dual maps form a packet system. Restriction gives a map from P_α^*X^* to X_α^*. It is injective because a covector in the range of P_α^* vanishes on every other packet, and it is surjective because any covector on X_α extends to X through the di","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9813ba63ac7cffd1","tex":"P_\\gamma","aliases":"P_γ gamma","display":"P_\\gamma","kind":"use","excerpt":"Insert I=∑_β P_β between S and T: P_γ STP_α = ∑_β P_γ SP_β TP_α.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-7f4d885f46bde8c0","tex":"P_{\\gamma_1}","aliases":"P_γ_1 gamma","display":"P_{\\gamma_1}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a849f1482e28f8a2","tex":"P_{\\gamma_{m-1}}","aliases":"P_γ_m-1 gamma","display":"P_{\\gamma_{m-1}}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-ba7a0d2835235592","tex":"P_{\\sigma^{-1}(i)}","aliases":"P_σ^-1(i) sigma","display":"P_{\\sigma^{-1}(i)}","kind":"use","excerpt":"For the block formula, use intertwining and packet compatibility twice: Q_jφ(T)Q_iΦ · =Q_jφ(T)Φ P_σ^-1(i) · =Q_jΦ TP_σ^-1(i) · =Φ P_σ^-1(j)TP_σ^-1(i) · =∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and Hull_ A(S)=X, the hull equality gives Y=Hull_φ( A)(Φ(S)). If Φ(S)⊆Hull_ B(S'), then Hull_ B(Φ(S))⊆Hull_ B(S'), so S' generates Y.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a156778c1d9d8562","tex":"\\mathrm{packets}","aliases":"packets mathrm","display":"\\mathrm{packets}","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L I · 0 · 0 0 · I · 0. Replacing L by a scalar 1","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-079e8b6488dabbe1","tex":"\\pagerefV","aliases":"pagerefV","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-cf2a973806abd94e","tex":"\\mathbf{PktMod}","aliases":"PktMod mathbf","display":"\\mathbf{PktMod}","kind":"use","excerpt":"[Category of seeded packet systems] Packet compatible operator morphisms compose, seeded morphisms compose, and identity morphisms satisfy both definitions. Thus packet resolved operator modules form a category PktMod and seeded packet systems form a category SeedSys. DependsnolinkurlPSC-I-D-PACKET-MORPHISM, nolinkurlPSC-I-D-SEED-MORPHISM, and noli","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"For N≥1, define residue projectors (P_N,jx)[n] = x[n], · n≡ jpmod N, 0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-4a7053e18464057f","tex":"\\psrank_{A,\\Pcal}","aliases":"psrank_A,Pcal psrank_A, P psrank Pcal operatorname mathcal","display":"{\\operatorname{psrank}}_{A,{\\mathcal P}}","kind":"use","excerpt":"[Basic packet pure rank bounds] For every A module resolved into packets with P_α∈ A, srank_A(X) ≤ psrank_A, P(X) ≤ ∑_α∈ I srank_P_α A P_α(X_α). The corner algebra on the right is unital on X_α with identity P_α. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"use","excerpt":"0 · 1. These maps satisfy P^2=P, Q^2=Q, PQ=QP=0 and P+Q=I. Their images are the horizontal line and the diagonal line, respectively. They are not selfadjoint in the usual Euclidean inner product.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-eda0b1a24e71fdef","tex":"Q_2","aliases":"Q_2","display":"Q_2","kind":"use","excerpt":"Indeed, Q_1T^UQ_2=Q_2T^UQ_1=0, Q_iT^UQ_j=U(P_iTP_j)U^-1. A change in raw matrix sparsity is therefore not necessarily a change in the packet transfer graph. Conjugacy preserves the exact block identities and their zero pattern. A nonunitary shear need not preserve their Euclidean norms.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-5950b0378700bfba","tex":"Q_i","aliases":"Q_i","display":"Q_i","kind":"use","excerpt":"[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and S generates X, then Φ(S) generates Y under φ( A) and hence under B. If, in ad","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-704a2b865a8b60e0","tex":"Q_j^2","aliases":"Q_j^2","display":"Q_j^2","kind":"use","excerpt":"Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a9af1147d28b7af5","tex":"Q_k","aliases":"Q_k","display":"Q_k","kind":"use","excerpt":"Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-ea3d9f514fcaceb6","tex":"Q_\\alpha","aliases":"Q_α alpha","display":"Q_\\alpha","kind":"use","excerpt":"[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algeb","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-d2f8225af58ea4bd","tex":"Q_\\beta","aliases":"Q_β beta","display":"Q_\\beta","kind":"use","excerpt":"[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algeb","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-5a423e1823849ed7","tex":"Q_\\gamma","aliases":"Q_γ gamma","display":"Q_\\gamma","kind":"use","excerpt":"[Tensor product packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] Expor","count":13},{"work":"paper-i","url":"notation/paper-i.html#symbol-1492f3a3a2d7a030","tex":"Q_\\gamma^2","aliases":"Q_γ^2 gamma","display":"Q_\\gamma^2","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-aa7fbe64d6b9e7ee","tex":"Q_\\delta","aliases":"Q_δ delta","display":"Q_\\delta","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-d938f24a7705f0ca","tex":"Q_{\\tau^{-1}(k)}","aliases":"Q_τ^-1(k) tau","display":"Q_{\\tau^{-1}(k)}","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source refer","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b9a6d997b7294e64","tex":"\\QQ^2","aliases":"QQ^2 Q^2 QQ mathbb","display":"{\\mathbb Q}^2","kind":"use","excerpt":"Define a surjective linear map q:X⟶ Q^2, q(∑ a_je_j+∑ b_jf_j)=(a_0,b_0).","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-279caa0ce5bda7c4","tex":"\\QQ^3","aliases":"QQ^3 Q^3 QQ mathbb","display":"{\\mathbb Q}^3","kind":"use","excerpt":"vpage6A reached type can still hide a directionTYPED MODEL / U= Q^2 AND V= Q^3Main text: [source reference] The paper's typed model has a rotation R:U→ U, an inclusion A:U→ V, and a retraction B:V→ U: R=mat0 · -1","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-4dafa9764bf19601","tex":"\\QQ^4","aliases":"QQ^4 Q^4 QQ mathbb","display":"{\\mathbb Q}^4","kind":"use","excerpt":"vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-0989b37fff5d66a9","tex":"\\QQ^6","aliases":"QQ^6 Q^6 QQ mathbb","display":"{\\mathbb Q}^6","kind":"use","excerpt":"Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a64e897db39c6a7f","tex":"R_k","aliases":"R_k","display":"R_k","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source refer","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b5c9a939784c608f","tex":"R_p","aliases":"R_p","display":"R_p","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-0c62b3900bbb09bf","tex":"r_{\\mathrm{spec}}","aliases":"r_spec mathrm","display":"r_{\\mathrm{spec}}","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-9667cad1734fc075","tex":"R_\\Gamma","aliases":"R_Γ Gamma","display":"R_\\Gamma","kind":"use","excerpt":"[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e13c04e5f7fac1fc","tex":"\\raggedright","aliases":"raggedright","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-3bce33d4a711b894","tex":"\\Rcal","aliases":"Rcal R mathcal","display":"{\\mathcal R}","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-607edbc5319584b2","tex":"\\Reach","aliases":"Reach operatorname","display":"{\\operatorname{Reach}}","kind":"use","excerpt":"[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-a62a596621079394","tex":"\\RR","aliases":"RR R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"vpage7Exact does not mean orthogonalPACKET COORDINATES / AN OBLIQUE SPLITTINGMain text: [source reference] In R^2, take P=mat1 · -1","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-78ecfb9af1021553","tex":"\\RR^2","aliases":"RR^2 R^2 RR mathbb","display":"{\\mathbb R}^2","kind":"use","excerpt":"vpage7Exact does not mean orthogonalPACKET COORDINATES / AN OBLIQUE SPLITTINGMain text: [source reference] In R^2, take P=mat1 · -1","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-5fea3d441d7eabe0","tex":"S''","aliases":"S''","display":"S''","kind":"use","excerpt":"· =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-67f7cd56268f3fc1","tex":"S^{\\oplus m}","aliases":"S^⊕ m oplus","display":"S^{\\oplus m}","kind":"use","excerpt":"[Multiplicity obstruction] For the diagonal action of M_d(D) on S^⊕ m, srank_M_d(D)(S^⊕ m) = lceilfracmdrceil. [source reference]","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-92045ce8b164570b","tex":"S_C","aliases":"S_C","display":"S_C","kind":"use","excerpt":"For the reverse inequality, choose for each source component C a packet pure family S_C of size r_C generating X_C under A_C, and put S=⋃_CS_C. Then every source component lies in A S. Traverse the finite condensation graph in a topological order. Suppose all components preceding a component D that is not a source have already been shown to lie in ","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Let S generate M. For every n∈ N, choose m∈ M with Φ(m)=n and write m=∑_ia_is_i. Then n=∑_iφ(a_i)Φ(s_i), so Φ(S) generates N.","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-9c953e0b08751250","tex":"S_i^{\\oplus m_i}","aliases":"S_i^⊕ m_i oplus","display":"S_i^{\\oplus m_i}","kind":"use","excerpt":"[Semisimple seed rank formula over arbitrary division rings] Let A≅∏_i=1^qM_d_i(D_i) be semisimple Artinian and let M≅⊕_i=1^qS_i^⊕ m_i be a finitely generated left module over A. Then srank_A(M) = max_1≤ i≤ q lceilfracm_id_irceil. If A is any algebra of finite dimension over a field, the same formula applied to the decomposition of M/Jac(A)M over A","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e0190c929ce120af","tex":"S_i^{\\oplus rd_i}","aliases":"S_i^⊕ rd_i oplus","display":"S_i^{\\oplus rd_i}","kind":"use","excerpt":"By [source reference], M is generated by r elements exactly when it is a quotient of A^⊕ r. By [source reference], A^⊕ r≅⊕_iS_i^⊕ rd_i. Every short exact sequence of modules over a semisimple ring splits. Therefore a quotient isomorphic to M exists if and only if m_i≤ rd_i for every i. The smallest such r is the displayed maximum. For a nonsemisimp","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-57c446e8e6d068a3","tex":"S_m","aliases":"S_m","display":"S_m","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-b9746e4f6fc18b24","tex":"s_m","aliases":"s_m","display":"s_m","kind":"use","excerpt":"headSeveral seeds give a module of relations For (s_1,…,s_m), evaluation is (f_1,…,f_m)↦∑_j f_j(T)s_j. Its kernel is a submodule of F^m and can include relations between seeds, not only relations annihilating each seed separately.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-593a84ab3d0b94bd","tex":"S_{m+r}","aliases":"S_m+r","display":"S_{m+r}","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-4d94f0719f096749","tex":"S_{\\mathrm{packets}}","aliases":"S_packets mathrm","display":"S_{\\mathrm{packets}}","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L I · 0 · 0 0 · I · 0. Replacing L by a scalar 1","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b37e9083bef6f614","tex":"S_r","aliases":"S_r","display":"S_r","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6122497cef209fd0","tex":"S_\\alpha","aliases":"S_α alpha","display":"S_\\alpha","kind":"use","excerpt":"Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-71a4eb6dc2d7040e","tex":"S_{\\beta\\alpha}","aliases":"S_βα beta alpha","display":"S_{\\beta\\alpha}","kind":"use","excerpt":"[Tensor product transfer blocks] Let S∈End(X) and T∈End(Y), with packet systems P and Q. Relative to the tensor product packet system of [source reference], (S⊗ T)_(β,δ),(α,γ) = S_βα⊗ T_δγ. [source reference] ExportsTensor network, quantum circuit, and multivariate packet calculations.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-39fa26306fe53d57","tex":"S_{\\gamma\\beta}","aliases":"S_γβ gamma beta","display":"S_{\\gamma\\beta}","kind":"use","excerpt":"[Packet matrix multiplication] For S,T∈End_𝕜(X), (ST)_γα = ∑_β∈ IS_γβT_βα. [source reference] ExportsAll packet path expansions.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-72365ef92579e114","tex":"S_{\\gamma\\delta}","aliases":"S_γδ gamma delta","display":"S_{\\gamma\\delta}","kind":"use","excerpt":"[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, ","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"quently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9bd6563a665b1ad1","tex":"\\small","aliases":"small","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span span operatorname","display":"{\\operatorname{span}}","kind":"use","excerpt":"2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is unreachable from a seed supported only at U. Therefore the generated repres","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-66c6b94238fa4435","tex":"\\Span_{\\bbk}","aliases":"Span_bbk span_𝕜 Span bbk operatorname Bbbk","display":"{\\operatorname{span}}_{{\\Bbbk}}","kind":"use","excerpt":"se there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with [2pt] 𝕜^V(G)=span_𝕜 {p(A_G)s_j:p∈𝕜[t], 1≤ j≤ r} }.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-720c3810eb53f410","tex":"\\Span_{\\C}","aliases":"Span_C span_ C Span C operatorname mathbb","display":"{\\operatorname{span}}_{{\\mathbb C}}","kind":"use","excerpt":"Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To se","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-327dcbd8f0058da9","tex":"\\Spec","aliases":"Spec operatorname","display":"{\\operatorname{Spec}}","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-ec2fdd09d5e3a211","tex":"\\mathrm{spec}","aliases":"spec mathrm","display":"\\mathrm{spec}","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-ae51e1c510f803eb","tex":"\\operatorname{Spec}","aliases":"Spec operatorname","display":"\\operatorname{Spec}","kind":"use","excerpt":"[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-263069edf563db19","tex":"\\Spec_K","aliases":"Spec_K Spec operatorname","display":"{\\operatorname{Spec}}_K","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-3df7b5dd54609471","tex":"\\srank_{\\A}","aliases":"srank_A srank_ A srank A operatorname mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}}","kind":"use","excerpt":"] The source component formulas concern packet pure seeds. They do not give the unrestricted seed rank. For example, let A=𝕜×𝕜, X=𝕜⊕𝕜 with the coordinate action and the two coordinate packet projectors. The transfer graph has two source components and psrank_ A, P(X)=2, but the mixed vector (1,1) is cyclic, so srank_ A(X)=1.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-8ec78eed6a4a5e23","tex":"\\srank_{A,\\tau_M}","aliases":"srank_A,τ_M srank tau operatorname tau","display":"{\\operatorname{srank}}_{A,\\tau_M}","kind":"use","excerpt":"onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bou","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-5eb5befad9ecd338","tex":"\\srank_{A/J}","aliases":"srank_A/J srank operatorname","display":"{\\operatorname{srank}}_{A/J}","kind":"use","excerpt":"radical] Let A be a unital ring, let J=Jac(A), and let M be a finitely generated left module over A. A family S⊆ M generates M if and only if its image generates M/JM as a module over A/J. When A/J is semisimple Artinian, this quotient is the usual semisimple top. No semisimplicity of A/J is required for the general equality. Consequently, srank_A(","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-61b79d07c0fd89a7","tex":"\\srank_{A_K}","aliases":"srank_A_K srank operatorname","display":"{\\operatorname{srank}}_{A_K}","kind":"use","excerpt":"[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient fi","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-93686018cc79b240","tex":"\\srank_{\\AA}","aliases":"srank_AA srank_ A srank AA operatorname mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}}","kind":"use","excerpt":"headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection ","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-7b362f5e98fe8e80","tex":"\\srank_{\\AA\\times\\BB}","aliases":"srank_AA×BB srank_ A× B srank AA times BB operatorname mathcal times mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}\\times{\\mathcal B}}","kind":"use","excerpt":"headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a029885343d63c05","tex":"\\srank_{\\AA\\otimes\\BB}","aliases":"srank_AA⊗BB srank_ A⊗ B srank AA otimes BB operatorname mathcal otimes mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}\\otimes{\\mathcal B}}","kind":"use","excerpt":"headThe resulting upper bound Since simple tensors span the algebraic tensor product, the complete rectangular family proves srank_ A⊗ B(X⊗ Y) ≤srank_ A(X) srank_ B(Y). This is not a universal equality claim. Changing the algebra changes the answer: the full independent matrix action can make a single nonzero pure tensor cyclic. Nor does the algebr","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-c0bb1698c9624f4e","tex":"\\srank_{A\\times B}","aliases":"srank_A× B srank times operatorname times","display":"{\\operatorname{srank}}_{A\\times B}","kind":"use","excerpt":"[Direct sum formula for independent actions] Let M be a left module over A and N a left module over B. Under the action of the product algebra (a,b)(m,n)=(am,bn), one has srank_A× B(M⊕ N) = max{srank_A(M),srank_B(N)}. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-26c092d57bf7778b","tex":"\\srank_{A\\otimes B}","aliases":"srank_A⊗ B srank otimes operatorname otimes","display":"{\\operatorname{srank}}_{A\\otimes B}","kind":"use","excerpt":"Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-960d9dd2b343db5c","tex":"\\srank_B","aliases":"srank_B srank operatorname","display":"{\\operatorname{srank}}_B","kind":"use","excerpt":"[Monotonicity under enlargement of the acting algebra] Let B⊆ A⊆End_𝕜(X) be unital operator algebras with the same identity. Then srank_A(X)≤srank_B(X). The same statement holds for topological seed rank when both algebras act continuously. [source reference]","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-dc0e4e4943e6ea7a","tex":"\\srank_{B,\\tau_N}","aliases":"srank_B,τ_N srank tau operatorname tau","display":"{\\operatorname{srank}}_{B,\\tau_N}","kind":"use","excerpt":"onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bou","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e18d0d44e2e3fcea","tex":"\\srank_{\\BB}","aliases":"srank_BB srank_ B srank BB operatorname mathcal","display":"{\\operatorname{srank}}_{{\\mathcal B}}","kind":"use","excerpt":"headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection ","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-6c22e5f4935054ca","tex":"\\srank_{\\bbk[T]}","aliases":"srank_bbk[T] srank_𝕜[T] srank bbk operatorname Bbbk","display":"{\\operatorname{srank}}_{{\\Bbbk}[T]}","kind":"use","excerpt":"[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the d","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-985f568846585534","tex":"\\srank_{\\cA}","aliases":"srank_cA srank_ A srank cA operatorname mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-5a4f0945c263f910","tex":"\\srank_{\\kk I}","aliases":"srank_kk I srank_𝕜 I srank kk operatorname Bbbk","display":"{\\operatorname{srank}}_{{\\Bbbk} I}","kind":"use","excerpt":"The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-5dc32ff45b20c008","tex":"\\srank_{\\kk\\times\\kk}","aliases":"srank_kk×kk srank_𝕜×𝕜 srank kk times kk operatorname Bbbk times Bbbk","display":"{\\operatorname{srank}}_{{\\Bbbk}\\times{\\Bbbk}}","kind":"use","excerpt":"The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-8c953c91802c5c19","tex":"\\srank_{M_2(\\QQ)}","aliases":"srank_M_2(QQ) srank_M_2( Q) srank QQ operatorname mathbb","display":"{\\operatorname{srank}}_{M_2({\\mathbb Q})}","kind":"use","excerpt":"6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b94af8795a55ca25","tex":"\\srank_{M_d(D)}","aliases":"srank_M_d(D) srank operatorname","display":"{\\operatorname{srank}}_{M_d(D)}","kind":"use","excerpt":"[Full matrix algebra] For the natural module S=D^d over M_d(D), srank_M_d(D)(S)=1. More strongly, every nonzero vector of S is cyclic. [source reference]","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-0724cce7fe1672c1","tex":"\\srank_{P_\\alpha A P_\\alpha}","aliases":"srank_P_α A P_α srank alpha alpha operatorname alpha alpha","display":"{\\operatorname{srank}}_{P_\\alpha A P_\\alpha}","kind":"use","excerpt":"[Basic packet pure rank bounds] For every A module resolved into packets with P_α∈ A, srank_A(X) ≤ psrank_A, P(X) ≤ ∑_α∈ I srank_P_α A P_α(X_α). The corner algebra on the right is unital on X_α with identity P_α. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-42ec3110f940782f","tex":"\\srank_{\\QQ[N]}","aliases":"srank_QQ[N] srank_ Q[N] srank QQ operatorname mathbb","display":"{\\operatorname{srank}}_{{\\mathbb Q}[N]}","kind":"use","excerpt":"headLower and upper bounds meet If r seeds generated X, their images would generate the quotient. Since scalar multiplication cannot create a new independent direction, r≥2. Conversely, the two seeds e_0,f_0 generate their respective chains, so srank_ Q[N](X)=2. This proof uses the displayed operator and quotient. It does not ask a computer to sear","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-38c6ab6ee7594444","tex":"\\srank_{R_p}","aliases":"srank_R_p srank operatorname","display":"{\\operatorname{srank}}_{R_p}","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-4e4ee8a109009a53","tex":"\\subsetneq","aliases":"subsetneq","display":null,"kind":"use","excerpt":"2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is unreachable from a seed supported only at U. Therefore the generated repres","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-963869a5c66f4ce3","tex":"\\sup_{F,a}","aliases":"sup_F,a sup","display":"\\sup_{F,a}","kind":"use","excerpt":"Each P_nX=Ker(I-P_n) is closed. For x∈ X, A_ Px _u = sup_F,a ∑_n∈ Fa_nP_nx ≤ K x . For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the uncond","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"headThe distinction is visible in two dimensions Use X= Q^2, seed e_1=(1,0)^ T, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column: I_e_1={mat0 · a","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-10d6fa1215b924bf","tex":"T^*","aliases":"T^*","display":"T^*","kind":"use","excerpt":"[Adjoint reversal] If X is Hilbert, every P_α is orthogonal, and T∈ B(X), then (T_βα)^*=(T^*)_αβ. [source reference]","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":"headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-8c6354f5f56778af","tex":"t^5","aliases":"t^5","display":"t^5","kind":"use","excerpt":"headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-54815db7d2265570","tex":"T^m","aliases":"T^m","display":"T^m","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-79f98be5a96a5c30","tex":"T^U","aliases":"T^U","display":"T^U","kind":"use","excerpt":"vpage11Move the packets with the coordinatesCONJUGACY / STRUCTURE VERSUS RAW SPARSITYMain text: [source reference] Start with T=diag(1,2) and the two coordinate projections. Apply the invertible shear U. The transformed operator is T^U=UTU^-1, and the transformed projectors are Q_i=UP_iU^-1.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-f790967529827b89","tex":"T_{11}","aliases":"T_11","display":"T_{11}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1a36fd1f6d7af0cd","tex":"T_{12}","aliases":"T_12","display":"T_{12}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-21a8dad328629d94","tex":"T_{13}","aliases":"T_13","display":"T_{13}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d40e5e2331bf36b5","tex":"T_{22}","aliases":"T_22","display":"T_{22}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-56f9752ffe92acc1","tex":"T_{23}","aliases":"T_23","display":"T_{23}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-899fcb428498d22e","tex":"T_{32}","aliases":"T_32","display":"T_{32}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e9a7aef78fcda23e","tex":"T_{ij}","aliases":"T_ij","display":"T_{ij}","kind":"use","excerpt":"vpage8A transfer block has an input and an outputOPERATOR COORDINATES / MULTIPLICATION AS A SUMMain text: [source reference] Use the three coordinate packets of dimension one in Q^3. The block T_ij=P_iTP_j sends input packet j into output packet i. For this example, its nonzero scalar is the (i,j) matrix entry.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-df3f310abe33eba9","tex":"T_r","aliases":"T_r","display":"T_r","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1a431e805508f7d7","tex":"T_{\\beta_1\\beta_0}","aliases":"T_β_1β_0 beta beta","display":"T_{\\beta_1\\beta_0}","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-69c19ac5861b4555","tex":"T_{\\beta_m\\beta_{m-1}}","aliases":"T_β_mβ_m-1 beta beta","display":"T_{\\beta_m\\beta_{m-1}}","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-708d92a009cb0a50","tex":"T_{\\delta\\gamma}","aliases":"T_δγ delta gamma","display":"T_{\\delta\\gamma}","kind":"use","excerpt":"[Tensor product transfer blocks] Let S∈End(X) and T∈End(Y), with packet systems P and Q. Relative to the tensor product packet system of [source reference], (S⊗ T)_(β,δ),(α,γ) = S_βα⊗ T_δγ. [source reference] ExportsTensor network, quantum circuit, and multivariate packet calculations.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-54bc773a0263d21b","tex":"\\takeaway","aliases":"takeaway","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-3000c3bd47e8007c","tex":"\\toprule","aliases":"toprule","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, ","count":57},{"work":"paper-i","url":"notation/paper-i.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algeb","count":13},{"work":"paper-i","url":"notation/paper-i.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span ont","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-2f97c9046c1a4c7a","tex":"u_2","aliases":"u_2","display":"u_2","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span ont","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-f61cc273f3d06b1e","tex":"U_{\\gamma\\alpha}","aliases":"U_γα gamma alpha","display":"U_{\\gamma\\alpha}","kind":"use","excerpt":"[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9dd814369d529f5b","tex":"U_{\\gamma\\beta}","aliases":"U_γβ gamma beta","display":"U_{\\gamma\\beta}","kind":"use","excerpt":"et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-82de5e416cdd59ff","tex":"U_{\\delta\\alpha}","aliases":"U_δα delta alpha","display":"U_{\\delta\\alpha}","kind":"use","excerpt":"et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1","count":32},{"work":"paper-i","url":"notation/paper-i.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span ont","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"use","excerpt":"6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span ont","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-f908e7a2d9e05299","tex":"V_3","aliases":"V_3","display":"V_3","kind":"use","excerpt":"6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span ont","count":10},{"work":"paper-i","url":"notation/paper-i.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e8221e0482136849","tex":"v_j^{-1}","aliases":"v_j^-1","display":"v_j^{-1}","kind":"use","excerpt":"The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e9f7dc4db556c3f2","tex":"\\vpage","aliases":"vpage","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":11},{"work":"paper-i","url":"notation/paper-i.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"ambda_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identitie","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-466a54f902244928","tex":"W_n","aliases":"W_n","display":"W_n","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-b2c566e84a80fdd2","tex":"W_{n+1}","aliases":"W_n+1","display":"W_{n+1}","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-03e64edeac04e31f","tex":"\\widehat E","aliases":"widehat E widehat","display":"\\widehat E","kind":"use","excerpt":"_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous seminorms p on X. Every a∈ A acts continuously on E in the transported topology and therefore extends to widehat E.","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-6b108551467ed661","tex":"\\widehat j","aliases":"widehat j widehat","display":"\\widehat j","kind":"use","excerpt":"that subspace. For a∈ A and a continuous seminorm p on X, continuity of a:X→ X supplies continuous seminorms controlling p(aj(e)), so the induced action on E is continuous and extends uniquely to widehat E. The subspace is dense by hypothesis. A Cauchy net in E is mapped to a Cauchy net in X, which converges because X is complete. This defines a co","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-649a6c9973f355dd","tex":"\\widetilde p","aliases":"widetilde p widetilde","display":"\\widetilde p","kind":"use","excerpt":"free presentation] Let X be a complete Hausdorff locally convex space, let A⊆End(X) act continuously, and let S=(s_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous s","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-9fcf9ea0bdd21c78","tex":"\\widetilde T","aliases":"widetilde T widetilde","display":"\\widetilde T","kind":"use","excerpt":"[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algeb","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-05da05e4a35ed5a5","tex":"\\widetilde\\Phi","aliases":"widetildeΦ widetilde Phi","display":"\\widetilde\\Phi","kind":"use","excerpt":"[Quotient topology and completed realization] Let Φ:X↠ Y be a continuous linear quotient map between Hausdorff topological vector spaces, and put K=KerΦ. Then the induced map widetildeΦ:X/K⟶ Y is a topological isomorphism. If D⊆ X is a linear subspace whose image Φ(D) is dense in a complete space Y, then the completion of D/(D∩ K) in the topology t","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-2a7d404b8f08d7c2","tex":"x^{(3)}","aliases":"x^(3)","display":"x^{(3)}","kind":"use","excerpt":"vfig[.93]shift_tail atlasCaptionA finite window of the actual sequence x[n]=2^-|n|. Outside every finite window, coefficients remain nonzero. The shaded window illustrates the truncation x^(3), not a periodic wraparound.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-aeb686029c32e920","tex":"x^{(k)}","aliases":"x^(k)","display":"x^{(k)}","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-a7d632fad29231f3","tex":"x^{(N)}","aliases":"x^(N)","display":"x^{(N)}","kind":"use","excerpt":"This x cannot be a finite linear combination of shifted impulses: it has infinite support. But if x^(N) agrees with x on |n|≤ N and is zero elsewhere, then normx-x^(N)_2^2 =2∑_n>N4^-n =frac23 4^-N⟶0. The exact squared errors for N=1,3,5 are 1/6, 1/96, and 1/1536. These are identities from a geometric series, not floating residuals.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-479d843e18704a71","tex":"x^{(\\ell)}","aliases":"x^(ℓ) ell","display":"x^{(\\ell)}","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-71a31718ae729ffa","tex":"X^*","aliases":"X^*","display":"X^*","kind":"use","excerpt":"[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint pa","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-b5d15c510169ad7c","tex":"x^4","aliases":"x^4","display":"x^4","kind":"use","excerpt":"(1.3,0.0) rectangle ++(1.15,-.63); node[] at (1.8750,-0.3150) x; filldraw[fill=black!8,draw=white] (2.6,0.0) rectangle ++(1.15,-.63); node[] at (3.1750,-0.3150) x^2; filldraw[fill=black!8,draw=white] (3.9000000000000004,0.0) rectangle ++(1.15,-.63); node[] at (4.4750,-0.3150) x^3; filldraw[fill=black!8,draw=white] (5.2,0.0) rectangle ++(1.15,-.63);","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-48f4a03fbc552897","tex":"x^i","aliases":"x^i","display":"x^i","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"[title=Why mixed seeds change the answer] Let the acting algebra be diagonal on X=X_1⊕ X_2, with no transfer between the packets. Generation by packet pure seeds requires one seed in each source component, so psrank=2. A single mixed vector x_1+x_2 may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can b","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[title=Why mixed seeds change the answer] Let the acting algebra be diagonal on X=X_1⊕ X_2, with no transfer between the packets. Generation by packet pure seeds requires one seed in each source component, so psrank=2. A single mixed vector x_1+x_2 may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can b","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-8eeb0a1bbb468424","tex":"X_3","aliases":"X_3","display":"X_3","kind":"use","excerpt":"[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-4433e4e3be068643","tex":"X_D","aliases":"X_D","display":"X_D","kind":"use","excerpt":"a:X_α↠ X_β. Because X_α⊆ A S and A S is invariant under A, surjectivity implies X_β⊆ A S. For any other vertex γ∈ D, choose a directed path inside the strongly connected component from β to γ. Successive surjective edge blocks propagate the full packet space along the path, so X_γ⊆ A S. Hence X_D⊆ A S. Induction over the condensation graph gives X=","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required d","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-dd757208c85dd8b0","tex":"x_n^{(k)}","aliases":"x_n^(k)","display":"x_n^{(k)}","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-a9688c63367ddeff","tex":"x_n^{(\\ell)}","aliases":"x_n^(ℓ) ell","display":"x_n^{(\\ell)}","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1a8e8addc6fe7c2d","tex":"X_\\alpha^*","aliases":"X_α^* alpha","display":"X_\\alpha^*","kind":"use","excerpt":"[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint pa","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-bcf7586068a6a313","tex":"x_\\beta","aliases":"x_β beta","display":"x_\\beta","kind":"use","excerpt":"For x∈ X, S_ P A_ Px = ∑_α P_α x =x. If x_α∈ X_α, then P_β x_α=δ_αβx_α, so A_ P S_ P(x_α)_α = (P_β∑_α x_α)_β =(x_β)_β. Thus analysis and synthesis are inverse linear maps.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a560c8da69b7911b","tex":"X_\\beta^*","aliases":"X_β^* beta","display":"X_\\beta^*","kind":"use","excerpt":"Conversely, suppose transitivity fails. Choose α,β,γ with (β,α)∈ R, (γ,β)∈ R, (γ,α)∉ R. Choose nonzero vectors u∈ X_α, v∈ X_β, and w∈ X_γ. Choose linear functionals f∈ X_α^* and g∈ X_β^* with f(u)=g(v)=1. Let T have the sole block x↦ f(x)v from X_α to X_β, and let S have the sole block y↦ g(y)w from X_β to X_γ. Then S,T∈ M_R, but ST has a nonzero f","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-c5c7030cb4e0b4b6","tex":"X_\\gamma","aliases":"X_γ gamma","display":"X_\\gamma","kind":"use","excerpt":"Conversely, suppose transitivity fails. Choose α,β,γ with (β,α)∈ R, (γ,β)∈ R, (γ,α)∉ R. Choose nonzero vectors u∈ X_α, v∈ X_β, and w∈ X_γ. Choose linear functionals f∈ X_α^* and g∈ X_β^* with f(u)=g(v)=1. Let T have the sole block x↦ f(x)v from X_α to X_β, and let S have the sole block y↦ g(y)w from X_β to X_γ. Then S,T∈ M_R, but ST has a nonzero f","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-1020799dad46e940","tex":"X_\\lambda","aliases":"X_λ lambda","display":"X_\\lambda","kind":"use","excerpt":"The irreducibles are p=t-λ. On the primary component for λ, the quotient by (T-λ I)X_λ has dimension equal to the number of Jordan blocks, which is also dimKer(T-λ I).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1a80ff6e24b3fc04","tex":"x_\\lambda","aliases":"x_λ lambda","display":"x_\\lambda","kind":"use","excerpt":"Given y=Φ(x), choose a net x_λ∈Hull_ A(S) converging to x. Then Φ(x_λ)→ y, and naturality places every Φ(x_λ) in Hull_ B(Φ(S)). Hence the latter hull is dense. For packets, one inclusion follows from compatibility. Conversely, if y=Q_jy, choose x with Φ(x)=y. Then y=Q_jΦ(x)=Φ P_σ^-1(j)x.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"[title=A grammar with two vertices] Take two vertices and one arrow a:1→2. A realization is simply a pair of spaces and a linear map X_1xrightarrow X_a X_2. After collapse, |X|=X_1⊕ X_2, the vertex idempotents are the two coordinate projectors, and a is a single block outside the diagonal. This elementary case already contains the viewpoint of pack","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form,","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-0c909af2321925f7","tex":"y^3","aliases":"y^3","display":"y^3","kind":"use","excerpt":"1150) x^2y; filldraw[fill=black!18,draw=white] (3.9000000000000004,-0.8) rectangle ++(1.15,-.63); node[] at (4.4750,-1.1150) x^3y; filldraw[fill=black!18,draw=white] (5.2,-0.8) rectangle ++(1.15,-.63); node[] at (5.7750,-1.1150) x^4y; node[] at (6.9500,-1.1150) ⋯; node[] at (3.2000,0.5200) Independent coefficient slots: a(x) and b(x); node[] at (3.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-827a2f52108af912","tex":"y^j","aliases":"y^j","display":"y^j","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-560f4e5df768a9d8","tex":"y^m","aliases":"y^m","display":"y^m","kind":"use","excerpt":"alize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring. Every class has a unique normal form f(x,y)=a(x)+y b(x). Indeed, y^2-(x^3-x) is monic of degree two in y, so division in 𝕜[x][y] gives a unique remainder of degree less than tw","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e3d4192e022af00a","tex":"y_j","aliases":"y_j","display":"y_j","kind":"use","excerpt":"vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required d","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-f63c0b46cff9ca02","tex":"y_m","aliases":"y_m","display":"y_m","kind":"use","excerpt":"For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the unconditionally convergent series: P_m S_ Py = ∑_nP_my_n =y_m. Thus A_ P S_ P=I. Complete","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-00fe9b383f124db5","tex":"y_n","aliases":"y_n","display":"y_n","kind":"use","excerpt":"Each P_nX=Ker(I-P_n) is closed. For x∈ X, A_ Px _u = sup_F,a ∑_n∈ Fa_nP_nx ≤ K x . For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the uncond","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-f42e259364006691","tex":"Y_\\gamma","aliases":"Y_γ gamma","display":"Y_\\gamma","kind":"use","excerpt":"ct packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refi","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z mathbb","display":"{\\mathbb Z}","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-5e32645afb213c74","tex":"z_2","aliases":"z_2","display":"z_2","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-974e1ebfd0c228a6","tex":"z_r","aliases":"z_r","display":"z_r","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ Z mathbb","display":"{\\mathbb Z}","kind":"use","excerpt":"vpage18Density is not finite generation of each vectorINFINITE MODEL / THE BILATERAL SHIFTMain text: [source reference] On ℓ^2( Z), declare every integer shift (S_mx)[n]=x[n-m]. The seed δ_0 generates δ_m=S_mδ_0. Finite linear combinations give precisely the sequences with finite support. Their norm closure is all of ℓ^2( Z).","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-343b02c221c97d88","tex":"|_K","aliases":"|_K","display":"|_K","kind":"use","excerpt":"[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d249ec98efda9d47","tex":"|_{X_\\alpha}","aliases":"|_X_α alpha","display":"|_{X_\\alpha}","kind":"use","excerpt":"[Universal mapping property of finite packet coordinates] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"[title=R1: one operator with several primary blocks] Let T act on a finite dimensional vector space. Regard X as a 𝕜[t]-module by t· x=Tx. If T has three Jordan blocks for the same eigenvalue, then three seeds are necessary and sufficient for that primary component. The largest number of blocks over any irreducible polynomial is the seed rank.","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-24182df1bf8329b9","tex":"\\beta_0","aliases":"β_0 beta","display":"\\beta_0","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1bac3c7ce6eb2507","tex":"\\beta_1","aliases":"β_1 beta","display":"\\beta_1","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d6de2171610272b5","tex":"\\beta_m","aliases":"β_m beta","display":"\\beta_m","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-fde937c7afbf21ed","tex":"\\beta_{m-1}","aliases":"β_m-1 beta","display":"\\beta_{m-1}","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"[Tensor product packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] Expor","count":37},{"work":"paper-i","url":"notation/paper-i.html#symbol-d2e737aa54abf669","tex":"\\gamma_1","aliases":"γ_1 gamma","display":"\\gamma_1","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-45d8bf1bf1c60baa","tex":"\\gamma_{m-1}","aliases":"γ_m-1 gamma","display":"\\gamma_{m-1}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invarianc","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"For x∈ X, S_ P A_ Px = ∑_α P_α x =x. If x_α∈ X_α, then P_β x_α=δ_αβx_α, so A_ P S_ P(x_α)_α = (P_β∑_α x_α)_β =(x_β)_β. Thus analysis and synthesis are inverse linear maps.","count":18},{"work":"paper-i","url":"notation/paper-i.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-23fe82c4424f67f6","tex":"\\delta_m","aliases":"δ_m delta","display":"\\delta_m","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-bb6d1253c89c84b1","tex":"\\delta_{nm}","aliases":"δ_nm delta","display":"\\delta_{nm}","kind":"use","excerpt":"[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-fd9e8f5f34bd8907","tex":"\\Delta_{\\Pcal}","aliases":"Δ_Pcal Δ_ P Delta Pcal Delta mathcal","display":"\\Delta_{{\\mathcal P}}","kind":"use","excerpt":"[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invarianc","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-03596e10dbda2d62","tex":"\\Delta_{\\Pcal}^2","aliases":"Δ_Pcal^2 Δ_ P^2 Delta Pcal Delta mathcal","display":"\\Delta_{{\\mathcal P}}^2","kind":"use","excerpt":"Pairwise annihilation gives Δ_ P^2=Δ_ P and identifies its range with diagonal block families. If T commutes with all P_α, then for β≠α, P_β TP_α=P_β P_α T=0. Thus (i) implies (ii). Exact reconstruction shows (ii) implies (iii). If T=∑_α P_α TP_α, then TP_γ=P_γ TP_γ=P_γ T, so (iii) implies (i).","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-0fcb1c3cf5e3bf16","tex":"\\delta_{\\alpha\\beta}","aliases":"δ_αβ delta alpha beta","display":"\\delta_{\\alpha\\beta}","kind":"use","excerpt":"For x∈ X, S_ P A_ Px = ∑_α P_α x =x. If x_α∈ X_α, then P_β x_α=δ_αβx_α, so A_ P S_ P(x_α)_α = (P_β∑_α x_α)_β =(x_β)_β. Thus analysis and synthesis are inverse linear maps.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-3bbbf66ed7bfacbd","tex":"\\varepsilon^3","aliases":"ε^3 varepsilon","display":"\\varepsilon^3","kind":"use","excerpt":"[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-c69c9eb6fe242300","tex":"\\varepsilon_s","aliases":"ε_s varepsilon","display":"\\varepsilon_s","kind":"use","excerpt":"[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"ery n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root o","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-a1810dd5bbc6aace","tex":"\\zeta^{-1}","aliases":"ζ^-1 zeta","display":"\\zeta^{-1}","kind":"use","excerpt":"[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 ","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-c185cb6f6888fc01","tex":"\\zeta^{-j}","aliases":"ζ^-j zeta","display":"\\zeta^{-j}","kind":"use","excerpt":"y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e7afdf00c1a0e7a1","tex":"\\zeta^j","aliases":"ζ^j zeta","display":"\\zeta^j","kind":"use","excerpt":"y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-d595452034045c90","tex":"\\Theta_{\\Pcal}","aliases":"Θ_Pcal Θ_ P Theta Pcal Theta mathcal","display":"\\Theta_{{\\mathcal P}}","kind":"use","excerpt":"[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"[Short exact sequence bounds] Let 0⟶ LxrightarrowιMxrightarrowπN⟶0 be a short exact sequence of left modules over A. Then srank_A(N)≤srank_A(M) ≤srank_A(L)+srank_A(N). [source reference]","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"The action map ρ:Λ_ G↠ A_X is surjective by definition and satisfies λ s=ρ(λ)s. Hence every grammar orbit vector lies in A_XS. The reverse inclusion holds because every element of A_X is ρ(λ) for some λ.","count":38},{"work":"paper-i","url":"notation/paper-i.html#symbol-efe402b27fb44fd4","tex":"\\lambda_*","aliases":"λ_* lambda","display":"\\lambda_*","kind":"use","excerpt":"r] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r wi","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-d594552e2786afd4","tex":"\\lambda_k","aliases":"λ_k lambda","display":"\\lambda_k","kind":"use","excerpt":"For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized e","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the d","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"use","excerpt":"[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the d","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"us and extends uniquely to widehat E. The subspace is dense by hypothesis. A Cauchy net in E is mapped to a Cauchy net in X, which converges because X is complete. This defines a continuous extension widehat j:widehat E→ X. For any x∈ X, choose a net (e_λ) in E with j(e_λ)→ x. The transported topology makes (e_λ) Cauchy, so it converges to some ξ∈w","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"[Short exact sequence bounds] Let 0⟶ LxrightarrowιMxrightarrowπN⟶0 be a short exact sequence of left modules over A. Then srank_A(N)≤srank_A(M) ≤srank_A(L)+srank_A(N). [source reference]","count":10},{"work":"paper-i","url":"notation/paper-i.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"The action map ρ:Λ_ G↠ A_X is surjective by definition and satisfies λ s=ρ(λ)s. Hence every grammar orbit vector lies in A_XS. The reverse inclusion holds because every element of A_X is ρ(λ) for some λ.","count":6},{"work":"paper-i","url":"notation/paper-i.html#symbol-cff2c5f7d8b191ec","tex":"\\tau^{-1}","aliases":"τ^-1 tau","display":"\\tau^{-1}","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source refer","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-2e891663598a2abe","tex":"\\tau_M","aliases":"τ_M tau","display":"\\tau_M","kind":"use","excerpt":"onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bou","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-93f600ceeb9d3b50","tex":"\\tau_N","aliases":"τ_N tau","display":"\\tau_N","kind":"use","excerpt":"onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bou","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d0f231c57bed48c8","tex":"\\Phi_s","aliases":"Φ_s Phi","display":"\\Phi_s","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"[Scalar quotient witness] Let M be a module over A with coefficient field 𝕜. Suppose M has a quotient N of dimension r on which A acts through a character χ:A⟶𝕜. Then srank_A(M)≥ r. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a fini","count":14},{"work":"paper-i","url":"notation/paper-i.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"The quantity κ( P) measures how strongly the chosen packet coordinates can amplify perturbations. An orthogonal Hilbert decomposition has the best possible behavior in the natural ℓ^2 sum norm. An oblique packet system can reconstruct every vector exactly and still have a large condition number.","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a fini","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"How are vectors and operators resolved? · Complete idempotents (P_α) · X≅⊕_α P_α X and T↔(P_β TP_α).","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form,","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-2cf519f92b35ca80","tex":"\\Leftrightarrow","aliases":"⇔ Leftrightarrow","display":"\\Leftrightarrow","kind":"use","excerpt":"ii) holds exactly when every μ_p(T) belonging to a nonzero primary component equals one, which is (iii). Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv). In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"[Propagation of packet support] Let T∈End(X) and x∈ X. Then Supp_ P(Tx) ⊆ {β:∃α∈Supp_ P(x) with T_βα≠0}. Equality holds whenever, for every target β in the set on the right, the active contributions (T_βαP_α x: α∈Supp_ P(x), T_βα≠0) have nonzero sum. In particular, equality holds if at most one active source packet feeds each such β and that source","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"vector space over a complete valued field, together with a unital algebra of continuous linear operators. Every infinite sum, completion, quotient topology, and closure below is interpreted only in the regime explicitly declared in its statement. Conventions for empty families. For finite families of nonnegative integers we use max∅=0. The height o","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-11e1659ea33ec1a6","tex":"\\prod_C","aliases":"∏_C prod","display":"\\prod_C","kind":"use","excerpt":"Let P_C=∑_α∈ CP_α. Define ρ: M_R⟶∏_CEnd(X_C), ρ(T)=(P_CTP_C)_C. If an intermediate class D contributes to P_CSTP_C, then the preorder contains transfers C→ D and D→ C, forcing C=D in the quotient poset. Hence P_CSTP_C=(P_CSP_C)(P_CTP_C), so ρ is an algebra homomorphism. Every collection of endomorphisms of the X_C defines an allowed operator that i","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-24ec639df453ea86","tex":"\\prod_{C\\in\\overline I}","aliases":"∏_C∈overline I prod in overline","display":"\\prod_{C\\in\\overline I}","kind":"use","excerpt":"k M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. item The map to the diagonal classes induces an algebra isomorphism M_R/ N_R ≅ ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b9d5148b142926aa","tex":"\\prod_p","aliases":"∏_p prod","display":"\\prod_p","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9140b13729b27e7a","tex":"\\prod_{p}","aliases":"∏_p prod","display":"\\prod_{p}","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-01f661fab59b64b9","tex":"\\prod_\\alpha","aliases":"∏_α prod alpha","display":"\\prod_\\alpha","kind":"use","excerpt":"enumerate item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source re","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-32f02b479dedc3b2","tex":"\\prod_{\\beta,\\alpha\\in I}","aliases":"∏_β,α∈ I prod beta alpha in","display":"\\prod_{\\beta,\\alpha\\in I}","kind":"use","excerpt":"[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6b80ee3f2f6c8c26","tex":"\\sum_{C\\in\\mathfrak C_0}","aliases":"∑_C∈ C_0 sum in mathfrak","display":"\\sum_{C\\in\\mathfrak C_0}","kind":"use","excerpt":"[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport for","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Let S generate M. For every n∈ N, choose m∈ M with Φ(m)=n and write m=∑_ia_is_i. Then n=∑_iφ(a_i)Φ(s_i), so Φ(S) generates N.","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-3844bf53874d6a99","tex":"\\sum_{i,j=1}^3","aliases":"∑_i,j=1^3 sum","display":"\\sum_{i,j=1}^3","kind":"use","excerpt":"headReconstruction retains every block The identity P_1+P_2+P_3=I gives T=∑_i,j=1^3 P_iTP_j. Selecting only the diagonal would change the operator. The entries 2,3,4 carry transfers between different packets.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6182c3bad2d01e16","tex":"\\sum_{i=1}^r","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^r","kind":"use","excerpt":"Let S={s_1,…,s_r} generate M. Every pure tensor λ⊗ m∈ M_K has an expression m=∑_i=1^r a_is_i, so λ⊗ m = ∑_i=1^r(λ⊗ a_i)(1⊗ s_i). Pure tensors span M_K, hence 1⊗ S generates M_K over A_K. Taking the minimum over S proves the inequality.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-7fec58bf80198971","tex":"\\sum_{i\\in I}","aliases":"∑_i∈ I sum in","display":"\\sum_{i\\in I}","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"The image consists exactly of finite sums ∑_ja_js_j, which is Hull_A(S). Because ε_S is linear over A, its kernel is a left submodule. The first isomorphism theorem gives A^⊕ r/N_S≅imε_S=Hull_A(S). The final equivalence is the equality imε_S=M.","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-f7b37e8032ebe855","tex":"\\sum_{j=0}^{n-1}","aliases":"∑_j=0^n-1 sum","display":"\\sum_{j=0}^{n-1}","kind":"use","excerpt":"y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-aae5c8f2004dc612","tex":"\\sum_{j\\in F}","aliases":"∑_j∈ F sum in","display":"\\sum_{j\\in F}","kind":"use","excerpt":"The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a fini","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-b4619f532d9c032b","tex":"\\sum_{j\\in J}","aliases":"∑_j∈ J sum in","display":"\\sum_{j\\in J}","kind":"use","excerpt":"[Table specification for transfer blocks] Given finite projector systems (P_i)_i∈ I and (Q_j)_j∈ J and a linear map T, form the table B_ji=Q_jTP_i. Its assembly reconstructs the input map exactly, ∑_j∈ J∑_i∈ IB_ji=T, and its reported support relation is exactly {(i,j):B_ji≠0}. The latter equality is the specification of the transfer graph output. I","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-9b846ee3188af45a","tex":"\\sum_{n>N}","aliases":"∑_n>N sum","display":"\\sum_{n>N}","kind":"use","excerpt":"This x cannot be a finite linear combination of shifted impulses: it has infinite support. But if x^(N) agrees with x on |n|≤ N and is zero elsewhere, then normx-x^(N)_2^2 =2∑_n>N4^-n =frac23 4^-N⟶0. The exact squared errors for N=1,3,5 are 1/6, 1/96, and 1/1536. These are identities from a geometric series, not floating residuals.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-f0d1bc9b91545210","tex":"\\sum_{n\\geq1}","aliases":"∑_n≥1 sum geq","display":"\\sum_{n\\geq1}","kind":"use","excerpt":"[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-f1724b58a24b34fd","tex":"\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}","aliases":"∑_substackσ(β)=j σ(α)=i sum substack sigma beta sigma alpha","display":"\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}","kind":"use","excerpt":"[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and S generates X, then Φ(S) generates Y under φ( A) and hence under B. If, in ad","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-53c07efe2188e4b0","tex":"\\sum_v","aliases":"∑_v sum","display":"\\sum_v","kind":"use","excerpt":", the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogo","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1774cac52d8c9785","tex":"\\sum_{v\\in Q_0}","aliases":"∑_v∈ Q_0 sum in","display":"\\sum_{v\\in Q_0}","kind":"use","excerpt":"Because Q_0 is finite, 1=∑_v∈ Q_0e_v in 𝕜 Q, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely,","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-ecf317dfca855f25","tex":"\\sum_{\\alpha}","aliases":"∑_α sum alpha","display":"\\sum_{\\alpha}","kind":"use","excerpt":"[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invarianc","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","excerpt":"Insert the finite packet identity on both sides: T= (∑_β P_β) T (∑_α P_α) = ∑_α,βP_β TP_α.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-575223190a495e3e","tex":"\\sum_{\\alpha,\\beta\\in I}","aliases":"∑_α,β∈ I sum alpha beta in","display":"\\sum_{\\alpha,\\beta\\in I}","kind":"use","excerpt":"[Exact operator reconstruction] Every T∈End_𝕜(X) satisfies T=∑_α,β∈ IT_βα. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e265f6e1bb343d9c","tex":"\\sum_{\\alpha,\\gamma}","aliases":"∑_α,γ sum alpha gamma","display":"\\sum_{\\alpha,\\gamma}","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d237666c585ce1f0","tex":"\\sum_{\\alpha\\in\\Supp_{\\Pcal}(x)}","aliases":"∑_α∈Supp_Pcal(x) ∑_α∈Supp_ P(x) sum alpha in Supp Pcal sum alpha in operatorname mathcal","display":"\\sum_{\\alpha\\in{\\operatorname{Supp}}_{{\\mathcal P}}(x)}","kind":"use","excerpt":"For every target packet β, P_β Tx = ∑_α∈Supp_ P(x)T_βαP_α x. If every block in this sum is zero, then the coordinate with index β vanishes. This proves the inclusion. Equality is exactly the assertion that the displayed active sum is nonzero for every target in the set on the right. The final sufficient condition leaves at most one nonzero summand ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-dd5a8bdc2b931502","tex":"\\sum_\\beta","aliases":"∑_β sum beta","display":"\\sum_\\beta","kind":"use","excerpt":"Insert the finite packet identity on both sides: T= (∑_β P_β) T (∑_α P_α) = ∑_α,βP_β TP_α.","count":5},{"work":"paper-i","url":"notation/paper-i.html#symbol-18666dd24c385c85","tex":"\\sum_{\\beta_1,\\ldots,\\beta_{m-1}}","aliases":"∑_β_1,…,β_m-1 sum beta ldots beta","display":"\\sum_{\\beta_1,\\ldots,\\beta_{m-1}}","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1b2a0f40b316674f","tex":"\\sum_{\\beta\\in I}","aliases":"∑_β∈ I sum beta in","display":"\\sum_{\\beta\\in I}","kind":"use","excerpt":"[Packet matrix multiplication] For S,T∈End_𝕜(X), (ST)_γα = ∑_β∈ IS_γβT_βα. [source reference] ExportsAll packet path expansions.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-9d12cb53c45f8ea0","tex":"\\sum_\\gamma","aliases":"∑_γ sum gamma","display":"\\sum_\\gamma","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-7e22c72e0c03bce6","tex":"\\sum_{\\gamma\\in I}","aliases":"∑_γ∈ I sum gamma in","display":"\\sum_{\\gamma\\in I}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-4519e3a242fabf8b","tex":"\\sum_\\delta","aliases":"∑_δ sum delta","display":"\\sum_\\delta","kind":"use","excerpt":"Apply Q_γ on the left and P_α on the right. Inserting I_X=∑_β P_β gives Q_γ UTP_α = ∑_β Q_γ UP_β TP_α = ∑_β U_γβT_βα. Inserting I_Y=∑_δ Q_δ gives Q_γ SUP_α = ∑_δ Q_γ SQ_δ UP_α = ∑_δ S_γδU_δα. Thus UT=SU implies every block equation. Conversely, if every block equation holds, then all blocks of UT-SU vanish. Exact operator reconstruction yields UT-S","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-3a150ff527e02e03","tex":"\\sum_{\\delta\\in J}","aliases":"∑_δ∈ J sum delta in","display":"\\sum_{\\delta\\in J}","kind":"use","excerpt":"et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-307b928591ce6cbf","tex":"\\sum_{\\tau(\\sigma(\\alpha))=k}","aliases":"∑_τ(σ(α))=k sum tau sigma alpha","display":"\\sum_{\\tau(\\sigma(\\alpha))=k}","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source refer","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Choose such a k≥ k_0. Since x^(k) is unconditionally summable, there is a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. There","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint pa","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]","count":9},{"work":"paper-i","url":"notation/paper-i.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [labe","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"[Criterion for a cyclic matrix] For X≠0 of finite dimension, the following are equivalent: [label=(roman*)] item T has a cyclic vector. item srank_𝕜[T](X)=1. item Every nonzero primary component for p is cyclic. item The rational canonical form has one invariant factor. item The minimal polynomial equals the characteristic polynomial. item After sc","count":12},{"work":"paper-i","url":"notation/paper-i.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"For N≥1, define residue projectors (P_N,jx)[n] = x[n], · n≡ jpmod N, 0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.","count":3},{"work":"paper-i","url":"notation/paper-i.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N:","count":14},{"work":"paper-i","url":"notation/paper-i.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"ctive smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring. Every class has a unique normal form f(x,y)=a(x)+y b(x). Indeed, y^2-(x^3-x) is monic of degree two in y, so division in 𝕜[x][y] gives a unique remainder of degree less than two in y. Con","count":8},{"work":"paper-i","url":"notation/paper-i.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"This example shows why a typed grammar is more than a notational convenience. The obstruction lies in the subspace ker B⊂ V, which is unreachable from seeds supported at U, and is visible before any coordinate calculation.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-d2adebfd04c66222","tex":"\\bigoplus^u","aliases":"⊕^u bigoplus","display":"\\bigoplus^u","kind":"use","excerpt":"s a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. Therefore x∈⊕^uX_n, and the earlier estimate proves x^(k)→ x in · _u.","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-52fe3db28a901e83","tex":"\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}","aliases":"⊕_(α,γ)∈ I× J bigoplus alpha gamma in times","display":"\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}","kind":"use","excerpt":"lpha∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refinement for quantum, signal, ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","excerpt":"By [source reference], M is generated by r elements exactly when it is a quotient of A^⊕ r. By [source reference], A^⊕ r≅⊕_iS_i^⊕ rd_i. Every short exact sequence of modules over a semisimple ring splits. Therefore a quotient isomorphic to M exists if and only if m_i≤ rd_i for every i. The smallest such r is the displayed maximum. For a nonsemisimp","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-b1c37ce88f1f80c5","tex":"\\bigoplus_{j=1}^{d}","aliases":"⊕_j=1^d bigoplus","display":"\\bigoplus_{j=1}^{d}","kind":"use","excerpt":"For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-1e8f7e2fd7fa7ced","tex":"\\bigoplus_{j=1}^{r}","aliases":"⊕_j=1^r bigoplus","display":"\\bigoplus_{j=1}^{r}","kind":"use","excerpt":"Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-6e1bd670f8830b0f","tex":"\\bigoplus_v","aliases":"⊕_v bigoplus","display":"\\bigoplus_v","kind":"use","excerpt":"Because Q_0 is finite, 1=∑_v∈ Q_0e_v in 𝕜 Q, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely,","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-5304936d80ae1b31","tex":"\\bigoplus_{\\alpha\\in C}","aliases":"⊕_α∈ C bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in C}","kind":"use","excerpt":"[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [labe","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-7224e793ec7acd49","tex":"\\bigoplus_{\\alpha\\in I_0}","aliases":"⊕_α∈ I_0 bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I_0}","kind":"use","excerpt":"Reachability obstructions for seeds. [Reachability obstruction for generated hulls] Let I_0⊆ I and let S⊆⊕_α∈ I_0X_α. Then Hull_ A(S) ⊆ ⊕_β∈Reach(I_0)X_β. In particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-af8472aa58b542e6","tex":"\\bigoplus_{\\beta\\in\\Reach(I_0)}","aliases":"⊕_β∈Reach(I_0) bigoplus beta in Reach bigoplus beta in operatorname","display":"\\bigoplus_{\\beta\\in{\\operatorname{Reach}}(I_0)}","kind":"use","excerpt":"Reachability obstructions for seeds. [Reachability obstruction for generated hulls] Let I_0⊆ I and let S⊆⊕_α∈ I_0X_α. Then Hull_ A(S) ⊆ ⊕_β∈Reach(I_0)X_β. In particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-e91d9943bd8850b9","tex":"\\bigoplus_{\\sigma(\\alpha)=j}","aliases":"⊕_σ(α)=j bigoplus sigma alpha","display":"\\bigoplus_{\\sigma(\\alpha)=j}","kind":"use","excerpt":"[Finite packet aggregation] Let σ:I↠ J be a surjection and define Q_j:=∑_σ(α)=jP_α. Then (Q_j)_j∈ J is a finite packet system and Q_jX=⊕_σ(α)=jX_α. [source reference]","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","excerpt":"Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]","count":27},{"work":"paper-i","url":"notation/paper-i.html#symbol-d981952a913b6fd5","tex":"\\otimes_{\\bbk}","aliases":"⊗_bbk ⊗_𝕜 otimes bbk otimes Bbbk","display":"\\otimes_{{\\Bbbk}}","kind":"use","excerpt":"Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates","count":2},{"work":"paper-i","url":"notation/paper-i.html#symbol-5bdecfe476a15417","tex":"\\bigcup_C","aliases":"⋃_C bigcup","display":"\\bigcup_C","kind":"use","excerpt":"For the reverse inequality, choose for each source component C a packet pure family S_C of size r_C generating X_C under A_C, and put S=⋃_CS_C. Then every source component lies in A S. Traverse the finite condensation graph in a topological order. Suppose all components preceding a component D that is not a source have already been shown to lie in ","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-686d00f08d584b6c","tex":"\\bigcup_\\alpha","aliases":"⋃_α bigcup alpha","display":"\\bigcup_\\alpha","kind":"use","excerpt":"Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates","count":1},{"work":"paper-i","url":"notation/paper-i.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]","count":7},{"work":"paper-i","url":"notation/paper-i.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":4},{"work":"paper-i","url":"notation/paper-i.html#symbol-0da92c8c15988310","tex":"\\mathbb{F}","aliases":"𝔽 mathbb","display":"\\mathbb{F}","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"notation-table","excerpt":"J_ℓ+1,ℓ · exact coordinate refinement map","count":127},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"A(t) · polynomial weighted adjacency matrix","count":114},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"notation-table","excerpt":"C_ℓ,ℓ+1 · exact coordinate regrouping from fine to coarse","count":55},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"notation-table","excerpt":"Δ · linear refinement defect for one step CT_f-T_cC","count":50},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-41107b0d171ba82c","tex":"C_{\\ell,\\ell+1}","aliases":"C_ℓ,ℓ+1 ell ell","display":"C_{\\ell,\\ell+1}","kind":"notation-table","excerpt":"C_ℓ,ℓ+1 · exact coordinate regrouping from fine to coarse","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-82497562659dda68","tex":"\\cA","aliases":"cA A mathcal","display":"{\\mathcal A}","kind":"notation-table","excerpt":"A_ℓ, S_ℓ · packet analysis and synthesis maps","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-34805d5c6af82a29","tex":"\\cA_\\ell","aliases":"cA_ℓ A_ℓ cA ell mathcal ell","display":"{\\mathcal A}_\\ell","kind":"notation-table","excerpt":"A_ℓ, S_ℓ · packet analysis and synthesis maps","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-20d41ee7317fddd1","tex":"\\cS","aliases":"cS S mathcal","display":"{\\mathcal S}","kind":"notation-table","excerpt":"A_ℓ, S_ℓ · packet analysis and synthesis maps","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-03243abc6de47f01","tex":"\\cS_\\ell","aliases":"cS_ℓ S_ℓ cS ell mathcal ell","display":"{\\mathcal S}_\\ell","kind":"notation-table","excerpt":"A_ℓ, S_ℓ · packet analysis and synthesis maps","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"notation-table","excerpt":"s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists","count":111},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"notation-table","excerpt":"w(e) · positive filtration weight of a transfer edge","count":55},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"notation-table","excerpt":"Δ · linear refinement defect for one step CT_f-T_cC","count":77},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"notation-table","excerpt":"I_ℓ · packet address set at level ℓ","count":86},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f2a427dbf773b62b","tex":"I_\\ell","aliases":"I_ℓ ell","display":"I_\\ell","kind":"notation-table","excerpt":"I_ℓ · packet address set at level ℓ","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"notation-table","excerpt":"J_ℓ+1,ℓ · exact coordinate refinement map","count":18},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"notation-table","excerpt":"P_N,j · residue or phase selector for address jpmod N","count":59},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a72bc0a75789ec17","tex":"J_{\\ell+1,\\ell}","aliases":"J_ℓ+1,ℓ ell ell","display":"J_{\\ell+1,\\ell}","kind":"notation-table","excerpt":"J_ℓ+1,ℓ · exact coordinate refinement map","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"notation-table","excerpt":"P_N,j · residue or phase selector for address jpmod N","count":140},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"P_ℓ,α · packet projector at refinement level ℓ","count":92},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e65ae8417651699c","tex":"P_{N,j}","aliases":"P_N,j","display":"P_{N,j}","kind":"notation-table","excerpt":"P_N,j · residue or phase selector for address jpmod N","count":16},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-63aba31d0d66fc66","tex":"P_{\\ell,\\alpha}","aliases":"P_ℓ,α ell alpha","display":"P_{\\ell,\\alpha}","kind":"notation-table","excerpt":"P_ℓ,α · packet projector at refinement level ℓ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6f446a816f55174e","tex":"P_{\\ell,\\beta}","aliases":"P_ℓ,β ell beta","display":"P_{\\ell,\\beta}","kind":"notation-table","excerpt":"T^(ℓ)_βα · transfer block P_ℓ,βTP_ℓ,α","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"notation-table","excerpt":"P_N,j · residue or phase selector for address jpmod N","count":31},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"notation-table","excerpt":"Z_s,t(t) · weighted transfer-zeta function from vertex s to vertex t","count":118},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a9e3e3ca095700e1","tex":"s_{\\pi,\\Lambda}","aliases":"s_π,Λ pi Lambda","display":"s_{\\pi,\\Lambda}","kind":"notation-table","excerpt":"s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-32f99fc05b64a106","tex":"\\succ","aliases":"succ","display":"\\succ","kind":"notation-table","excerpt":"βsuccα · fine packet β is a child of coarse packet α","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"notation-table","excerpt":"T^(ℓ)_βα · transfer block P_ℓ,βTP_ℓ,α","count":99},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"notation-table","excerpt":"A(t) · polynomial weighted adjacency matrix","count":55},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3225f6cf360797c4","tex":"T^{(\\ell)}_{\\beta\\alpha}","aliases":"T^(ℓ)_βα ell beta alpha","display":"T^{(\\ell)}_{\\beta\\alpha}","kind":"notation-table","excerpt":"T^(ℓ)_βα · transfer block P_ℓ,βTP_ℓ,α","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0692c76d8494573e","tex":"T_c","aliases":"T_c","display":"T_c","kind":"notation-table","excerpt":"Δ · linear refinement defect for one step CT_f-T_cC","count":29},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4b7fbd7d66e3a28d","tex":"T_f","aliases":"T_f","display":"T_f","kind":"notation-table","excerpt":"Δ · linear refinement defect for one step CT_f-T_cC","count":30},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"notation-table","excerpt":"W(γ) · cumulative weight of a transfer word or graph walk","count":21},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"notation-table","excerpt":"w(e) · positive filtration weight of a transfer edge","count":27},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"notation-table","excerpt":"Z_s,t(t) · weighted transfer-zeta function from vertex s to vertex t","count":24},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-512685c9cd95870b","tex":"Z_{s,t}","aliases":"Z_s,t","display":"Z_{s,t}","kind":"notation-table","excerpt":"Z_s,t(t) · weighted transfer-zeta function from vertex s to vertex t","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"notation-table","excerpt":"P_ℓ,α · packet projector at refinement level ℓ","count":95},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"notation-table","excerpt":"βsuccα · fine packet β is a child of coarse packet α","count":62},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"notation-table","excerpt":"W(γ) · cumulative weight of a transfer word or graph walk","count":34},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"notation-table","excerpt":"Δ · linear refinement defect for one step CT_f-T_cC","count":28},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"notation-table","excerpt":"s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists","count":103},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"notation-table","excerpt":"s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists","count":80},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"notation-table","excerpt":"P_ℓ,α · packet projector at refinement level ℓ","count":32},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":505},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":87},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":18},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0230e8bde40956b0","tex":"\\Acal","aliases":"Acal A mathcal","display":"{\\mathcal A}","kind":"definition-passage","excerpt":"ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ff76bbf0fc00a0b5","tex":"\\Acal_{\\Pcal}","aliases":"Acal_Pcal A_ P Acal Pcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal P}}","kind":"definition-passage","excerpt":"ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c783192e1874491b","tex":"\\Acal_{\\Qcal}","aliases":"Acal_Qcal A_ Q Acal Qcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-96852c201d3f1279","tex":"\\bbk","aliases":"bbk","display":null,"kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-327abfc143ff067a","tex":"\\Bcal","aliases":"Bcal B mathcal","display":"{\\mathcal B}","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":34},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-014eb6330bcf05ac","tex":"C_{\\Pcal\\leftarrow\\Qcal}","aliases":"C_Pcal←Qcal C_ P← Q Pcal leftarrow Qcal mathcal leftarrow mathcal","display":"C_{{\\mathcal P}\\leftarrow{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"definition-passage","excerpt":"W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.","count":13},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-02c1bebd51ef5031","tex":"e_{|\\gamma|}","aliases":"e_|γ| gamma","display":"e_{|\\gamma|}","kind":"definition-passage","excerpt":"W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-29663d1f0bd44f71","tex":"E_\\alpha","aliases":"E_α alpha","display":"E_\\alpha","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":13},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4748e7272f1e3f16","tex":"e_\\nu","aliases":"e_ν nu","display":"e_\\nu","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End operatorname","display":"{\\operatorname{End}}","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":10},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f8c8bc0689ee1b3e","tex":"\\expn","aliases":"expn exp operatorname","display":"{\\operatorname{exp}}","kind":"definition-passage","excerpt":"mbda_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1daca1ee43620e8f","tex":"\\expn_\\pi","aliases":"expn_π exp_π expn pi operatorname pi","display":"{\\operatorname{exp}}_\\pi","kind":"definition-passage","excerpt":"mbda_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":57},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-40808d4081e59d75","tex":"F^0","aliases":"F^0","display":"F^0","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-fef72e85e7b92ea3","tex":"F^1","aliases":"F^1","display":"F^1","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3f424c1523cf5188","tex":"F^L","aliases":"F^L","display":"F^L","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":16},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ba5037951b752c25","tex":"F^r","aliases":"F^r","display":"F^r","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9758c662ccf0f061","tex":"F^{r+s}","aliases":"F^r+s","display":"F^{r+s}","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e45f4042aed97e34","tex":"F^s","aliases":"F^s","display":"F^s","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-dd8cc01590ccaf86","tex":"F^{w(e)}","aliases":"F^w(e)","display":"F^{w(e)}","kind":"definition-passage","excerpt":"with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":38},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":25},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7dd1ffa3cf728bef","tex":"G_{\\Lambda,\\Pcal}","aliases":"G_Λ,Pcal G_Λ, P Lambda Pcal Lambda mathcal","display":"G_{\\Lambda,{\\mathcal P}}","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":12},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":46},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7884b4fb34770863","tex":"J_{\\Qcal\\leftarrow\\Pcal}","aliases":"J_Qcal←Pcal J_ Q← P Qcal leftarrow Pcal mathcal leftarrow mathcal","display":"J_{{\\mathcal Q}\\leftarrow{\\mathcal P}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":23},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":111},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-345427490e88d80b","tex":"\\leadsto","aliases":"leadsto","display":null,"kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":59},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"definition-passage","excerpt":"et Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":14},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":80},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d620bbab2a49de9b","tex":"N_e","aliases":"N_e","display":"N_e","kind":"definition-passage","excerpt":"ef:filtered_algebra A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f35ce0db28d3fb4f","tex":"\\OO","aliases":"OO O mathcal","display":"{{\\mathcal O}}","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":55},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":37},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aad7607e608d0052","tex":"P_\\beta","aliases":"P_β beta","display":"P_\\beta","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":9},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-813431acda8054c4","tex":"\\Pcal","aliases":"Pcal P mathcal","display":"{\\mathcal P}","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":56},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"definition-passage","excerpt":"[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).","count":55},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d2f8225af58ea4bd","tex":"Q_\\beta","aliases":"Q_β beta","display":"Q_\\beta","kind":"definition-passage","excerpt":"[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).","count":27},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aaf174f6701cc732","tex":"\\Qcal","aliases":"Qcal Q mathcal","display":"{\\mathcal Q}","kind":"definition-passage","excerpt":"[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).","count":32},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":42},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":98},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3b36510bbfb3bd3c","tex":"s_\\Lambda","aliases":"s_Λ Lambda","display":"s_\\Lambda","kind":"definition-passage","excerpt":"et V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O modul","count":11},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b764f1cbd00009f2","tex":"\\Scal","aliases":"Scal S mathcal","display":"{\\mathcal S}","kind":"definition-passage","excerpt":"ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3206060e514ec62e","tex":"\\Scal_{\\Pcal}","aliases":"Scal_Pcal S_ P Scal Pcal mathcal mathcal","display":"{\\mathcal S}_{{\\mathcal P}}","kind":"definition-passage","excerpt":"ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-13116bc57b9f1d4e","tex":"\\Scal_{\\Qcal}","aliases":"Scal_Qcal S_ Q Scal Qcal mathcal mathcal","display":"{\\mathcal S}_{{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-36a922b7cae66331","tex":"\\mathrm{split}","aliases":"split mathrm","display":"\\mathrm{split}","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8c82494b5f8dadd9","tex":"t^{w(e)}","aliases":"t^w(e)","display":"t^{w(e)}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":20},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-df5c58f22a65049f","tex":"u^{-1}","aliases":"u^-1","display":"u^{-1}","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"rojector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem","count":45},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":25},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-92becdaca2dd0d19","tex":"V_\\alpha","aliases":"V_α alpha","display":"V_\\alpha","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-50f104bbfd38386a","tex":"\\val","aliases":"val operatorname","display":"{\\operatorname{val}}","kind":"definition-passage","excerpt":"ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. Th","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-949ab90c1fcd9c3d","tex":"\\val_\\pi","aliases":"val_π val pi operatorname pi","display":"{\\operatorname{val}}_\\pi","kind":"definition-passage","excerpt":"ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. Th","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-30a9a914ab13bf7a","tex":"\\varprojlim_L","aliases":"varprojlim_L varprojlim","display":null,"kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":50},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":120},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aeb4dc50b8d5ec9e","tex":"X^{\\Pcal}","aliases":"X^Pcal X^ P Pcal mathcal","display":"X^{{\\mathcal P}}","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3f0de09885820b11","tex":"X^{\\Qcal}","aliases":"X^Qcal X^ Q Qcal mathcal","display":"X^{{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1e9458c4c7da9d3b","tex":"X_\\alpha","aliases":"X_α alpha","display":"X_\\alpha","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-54ac43616fbfd36c","tex":"x_\\alpha","aliases":"x_α alpha","display":"x_\\alpha","kind":"definition-passage","excerpt":"ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"definition-passage","excerpt":"bra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.","count":10},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":53},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-57105cdc4119a19b","tex":"\\mathbb Z_{>0}","aliases":"Z_>0 mathbb","display":"\\mathbb Z_{>0}","kind":"definition-passage","excerpt":"bra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b41db94f18e314fe","tex":"Z_{u,v}","aliases":"Z_u,v","display":"Z_{u,v}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":14},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-adef364156481ffd","tex":"\\Lambda_{\\mathrm{split}}","aliases":"Λ_split Lambda mathrm","display":"\\Lambda_{\\mathrm{split}}","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4e2977311c515de6","tex":"\\Lambda_\\alpha","aliases":"Λ_α Lambda alpha","display":"\\Lambda_\\alpha","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-75be53b3397893ea","tex":"\\pi^s","aliases":"π^s pi","display":"\\pi^s","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":24},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).","count":20},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.","count":17},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":8},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":19},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":81},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a1a9f417c2074f70","tex":"\\prod_{\\nu=1}^{|\\gamma|}","aliases":"∏_ν=1^|γ| prod nu gamma","display":"\\prod_{\\nu=1}^{|\\gamma|}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":78},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-86eb90e9395df95c","tex":"\\sum_{e:i\\to j}","aliases":"∑_e:i→ j sum to","display":"\\sum_{e:i\\to j}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-38c03c7cd3f0f240","tex":"\\sum_{i,j}","aliases":"∑_i,j sum","display":"\\sum_{i,j}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-177fd9fcb5956c9a","tex":"\\sum_{n\\equiv j\\pmod N}","aliases":"∑_n≡ jpmod N sum equiv pmod","display":"\\sum_{n\\equiv j\\pmod N}","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-441a6af1afa122e0","tex":"\\sum_{n\\geq0}","aliases":"∑_n≥0 sum geq","display":"\\sum_{n\\geq0}","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3e0412f70171096f","tex":"\\sum_{\\substack{\\gamma:i\\leadsto j\\\\W(\\gamma)=n}}","aliases":"∑_substackγ:ileadsto j W(γ)=n sum substack gamma leadsto W gamma","display":null,"kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"definition-passage","excerpt":"ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":11},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-cabedb544f677980","tex":"\\sum_{\\alpha\\in I}","aliases":"∑_α∈ I sum alpha in","display":"\\sum_{\\alpha\\in I}","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a87ec80b7b375dff","tex":"\\sum_{\\rho(\\beta)=\\alpha}","aliases":"∑_ρ(β)=α sum rho beta alpha","display":"\\sum_{\\rho(\\beta)=\\alpha}","kind":"definition-passage","excerpt":"[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"definition-passage","excerpt":"et Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":23},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":23},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. Th","count":17},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":23},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":10},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":21},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":12},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9f55fb1c51ddcd12","tex":"\\supseteq","aliases":"⊇ supseteq","display":"\\supseteq","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":12},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9b591bcc37a61369","tex":"\\bigoplus_\\alpha","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_\\alpha","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-90b882d77bb3725c","tex":"\\bigoplus_{\\alpha\\in I}","aliases":"⊕_α∈ I bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I}","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-91f48ece7340f8a0","tex":"\\bigcap_{L\\geq0}","aliases":"⋂_L≥0 bigcap geq","display":"\\bigcap_{L\\geq0}","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":14},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).","count":13},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R","count":8},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":30},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-dbe6869ebc4b6f5f","tex":"e_{j+1\\bmod8}","aliases":"e_j+1bmod8 bmod","display":"e_{j+1\\bmod8}","kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":20},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":9},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8a642ba0a275f90a","tex":"H_{\\rm phase}","aliases":"H_rm phase rm","display":"H_{\\rm phase}","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":48},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ","display":null,"kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f1c20921b96c99d0","tex":"\\QQ^8","aliases":"QQ^8 QQ","display":null,"kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4aa81ec436e2173c","tex":"R_N","aliases":"R_N","display":"R_N","kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R","count":14},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c8d0b37b4af3bfcc","tex":"S^2","aliases":"S^2","display":"S^2","kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYTraditional paper: Theorem 3.5 (p. 4), Theorem 3.8 (p. 6) Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on V20. Write R","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":36},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0ca25195475623b3","tex":"y_{\\rm even}","aliases":"y_rm even rm","display":"y_{\\rm even}","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3e87c12f66aedae3","tex":"y_{\\rm odd}","aliases":"y_rm odd rm","display":"y_{\\rm odd}","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"5000,3.60000) rectangle ++(0.85000,0.62000); node[] at (2.97500,3.91000) 4; draw[fill=black!10,draw=black!45,line width=.35pt,] (3.40000,3.60000) rectangle ++(0.85000,0.62000); node[] at (3.82500,3.91000) 5; draw[fill=black!25,draw=black!45,line width=.35pt,] (4.25000,3.60000) rectangle ++(0.85000,0.62000); node[] at (4.67500,3.91000) 6; node[ancho","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-156c89b978d080a5","tex":"\\mathbf1_{s=t,\\,n=0}","aliases":"1_s=t, n=0 mathbf","display":"\\mathbf1_{s=t,\\,n=0}","kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"Consider the integral involution A= 1 · 1 0 · -1 , A^2=I. Over a field of characteristic different from two, its sign projectors are E_+=fracI+A2 = 1 · 1/2","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"vfig[.97]weighted_word_cutoff platecaptionAll nonzero retained terms of the signed inverse sum, followed by the first discarded nonzero word. The empty word contributes I. Other words of weight below five can still be zero because a^2=b^2=0.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-fd3dce9096a0d298","tex":"A^{2k}","aliases":"A^2k","display":"A^{2k}","kind":"use","excerpt":"In this particular model, A^2=8I. Therefore v_2(A^2k)=3k, v_2(A^2k+1)=3k+1, where the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo 32. Four terms leave valuation six and do suffice.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-49c48961a3e05de8","tex":"A^{2k+1}","aliases":"A^2k+1","display":"A^{2k+1}","kind":"use","excerpt":"In this particular model, A^2=8I. Therefore v_2(A^2k)=3k, v_2(A^2k+1)=3k+1, where the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo 32. Four terms leave valuation six and do suffice.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-be1f0ddb2c78b507","tex":"A^N","aliases":"A^N","display":"A^N","kind":"use","excerpt":"W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is a two-sided inverse of q_L(I+A), where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then th","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-99155510890729d1","tex":"A^r","aliases":"A^r","display":"A^r","kind":"use","excerpt":"[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"_packet, the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-53cc0413ede33538","tex":"\\mathcal A_2","aliases":"A_2 mathcal","display":"\\mathcal A_2","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-60fcf83a29fad2ee","tex":"\\mathcal A_4","aliases":"A_4 mathcal","display":"\\mathcal A_4","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-011995a02078a685","tex":"a_5","aliases":"a_5","display":"a_5","kind":"use","excerpt":"e ++(0.87000,0.65000); node[] at (10.87500,2.07500) 12; draw[fill=black!20,draw=black!45,line width=.35pt,] (10.44000,1.00000) rectangle ++(0.87000,0.65000); node[] at (10.87500,1.32500) 218; node[anchor=east] at (-0.15000,2.07500) n; node[anchor=east] at (-0.15000,1.32500) a_n; node[] at (5.65000,3.10000) Z_0,1(t)=dfrac2t^21-t-2t^3; node[] at (5.6","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-dee1a61ca3190f8c","tex":"a_7","aliases":"a_7","display":"a_7","kind":"use","excerpt":"e ++(0.87000,0.65000); node[] at (10.87500,2.07500) 12; draw[fill=black!20,draw=black!45,line width=.35pt,] (10.44000,1.00000) rectangle ++(0.87000,0.65000); node[] at (10.87500,1.32500) 218; node[anchor=east] at (-0.15000,2.07500) n; node[anchor=east] at (-0.15000,1.32500) a_n; node[] at (5.65000,3.10000) Z_0,1(t)=dfrac2t^21-t-2t^3; node[] at (5.6","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-62f70fcecb631a95","tex":"a_8","aliases":"a_8","display":"a_8","kind":"use","excerpt":"e ++(0.87000,0.65000); node[] at (10.87500,2.07500) 12; draw[fill=black!20,draw=black!45,line width=.35pt,] (10.44000,1.00000) rectangle ++(0.87000,0.65000); node[] at (10.87500,1.32500) 218; node[anchor=east] at (-0.15000,2.07500) n; node[anchor=east] at (-0.15000,1.32500) a_n; node[] at (5.65000,3.10000) Z_0,1(t)=dfrac2t^21-t-2t^3; node[] at (5.6","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7764e407096f3048","tex":"A_e","aliases":"A_e","display":"A_e","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7ff61cb24c6fadc0","tex":"A_{e_1}","aliases":"A_e_1","display":"A_{e_1}","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f7766305f06ed3a3","tex":"A_{e_r}","aliases":"A_e_r","display":"A_{e_r}","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"In the recurrence, set a_k=0 when k<0. This is the coefficient convention for the formal power series Z_u,v(t) and makes the first recurrence equations meaningful even when n<d.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8e90c1adc20133ef","tex":"\\mathcal A_N","aliases":"A_N mathcal","display":"\\mathcal A_N","kind":"use","excerpt":"A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a2cfe3ae1e3afede","tex":"a_{n-1}","aliases":"a_n-1","display":"a_{n-1}","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ffe701ba2c672485","tex":"a_{n-3}","aliases":"a_n-3","display":"a_{n-3}","kind":"use","excerpt":"From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-13ab5ae246f6fa19","tex":"a_{n-d}","aliases":"a_n-d","display":"a_{n-d}","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2859c29e1e96b09d","tex":"a_{n-k}","aliases":"a_n-k","display":"a_{n-k}","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b14cc10bfd7bab79","tex":"a_{n-w(e)}","aliases":"a_n-w(e)","display":"a_{n-w(e)}","kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"use","excerpt":"_packet, the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7d7d4c82e2b393ad","tex":"A_\\gamma","aliases":"A_γ gamma","display":"A_\\gamma","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":9},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-74fd5e15b7fae5b2","tex":"\\operatorname{adj}","aliases":"adj operatorname","display":"\\operatorname{adj}","kind":"use","excerpt":"ce every entry of A(t) has zero constant term, the inverse exists in the formal power series ring and (I-A(t))^-1=∑_r≥0A(t)^r. The (i,j) entry of A(t)^r is the sum of the monomials attached to all paths of length r from i to j. Summing over r records paths of every length. Multiplication by the endpoint vectors gives the stated series. The adjugate","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0287e4860e3449ce","tex":"\\operatorname{analyze}","aliases":"analyze operatorname","display":"\\operatorname{analyze}","kind":"use","excerpt":"The verified contract is synthesize(analyze(x,N))=x. The Lean counterpart uses packet coordinates restricted to projector ranges and proves both inverse laws. The finite Python routine checks the concrete array realization.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":17},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":59},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"Write B=1 · -1 0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"vfig[.97]weighted_word_cutoff platecaptionAll nonzero retained terms of the signed inverse sum, followed by the first discarded nonzero word. The empty word contributes I. Other words of weight below five can still be zero because a^2=b^2=0.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-26b03d7e315b2b07","tex":"b_{\\mathrm{val}}","aliases":"b_val mathrm","display":"b_{\\mathrm{val}}","kind":"use","excerpt":"Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fibre has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b2a8d769f2c647dd","tex":"\\bbk^2","aliases":"bbk^2 bbk","display":null,"kind":"use","excerpt":"[A visible one step defect] Let X_f=bbk^2, let X_c=bbk, and let C(x_1,x_2)=x_1. Take T_f= a · b","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8e3c0fce1b7a60ae","tex":"\\bullet","aliases":"bullet","display":"\\bullet","kind":"use","excerpt":"[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"use","excerpt":"Had we chosen the transported operator T_c'=CT_fC^-1, the defect would be zero and every polynomial action would intertwine exactly. That is a different choice of coarse operator, not a stronger conclusion from the faulty one.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3a3b01a8e2b172a1","tex":"C_{0,L}","aliases":"C_0,L","display":"C_{0,L}","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f944d0625ea61fc0","tex":"C_{0,\\ell}","aliases":"C_0,ℓ ell","display":"C_{0,\\ell}","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ff97b5cd592dca16","tex":"c_{00}","aliases":"c_00","display":"c_{00}","kind":"use","excerpt":"Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer bloc","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ca35fd43dab309b4","tex":"C_{a+1}","aliases":"C_a+1","display":"C_{a+1}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e0d887f344230957","tex":"C_{a,a}","aliases":"C_a,a","display":"C_{a,a}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2dc3b3739f1aa480","tex":"C_{a,b}","aliases":"C_a,b","display":"C_{a,b}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6922a59f4f38ba94","tex":"C_{b-1}","aliases":"C_b-1","display":"C_{b-1}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d5ef149476c4688d","tex":"C_{\\Pcal\\leftarrow\\Rcal}","aliases":"C_Pcal←Rcal C_ P← R Pcal leftarrow Rcal mathcal leftarrow mathcal","display":"C_{{\\mathcal P}\\leftarrow{\\mathcal R}}","kind":"use","excerpt":"herence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-cf3fd1bbd0e06a1b","tex":"C_{\\Qcal\\leftarrow\\Rcal}","aliases":"C_Qcal←Rcal C_ Q← R Qcal leftarrow Rcal mathcal leftarrow mathcal","display":"C_{{\\mathcal Q}\\leftarrow{\\mathcal R}}","kind":"use","excerpt":"herence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4ae0a9be277fd36e","tex":"C_\\ell","aliases":"C_ℓ ell","display":"C_\\ell","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1ada5041fbbc689c","tex":"C_{\\ell+1,L}","aliases":"C_ℓ+1,L ell","display":"C_{\\ell+1,L}","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4c05481d823dd05d","tex":"C_{\\ell,L}","aliases":"C_ℓ,L ell","display":"C_{\\ell,L}","kind":"use","excerpt":"Insert the intermediate terms C_0,ℓT_ℓ C_ℓ,L for 0≤ℓ≤ L. Consecutive differences telescope. The difference between the terms indexed by ℓ+1 and ℓ is C_0,ℓ(C_ℓ T_ℓ+1-T_ℓ C_ℓ)C_ℓ+1,L. Summing gives the identity. The norm estimate follows from the triangle inequality.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5756f63546bea0ab","tex":"\\cA_Q","aliases":"cA_Q A_Q cA mathcal","display":"{\\mathcal A}_Q","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a623735ca1da9cda","tex":"\\cB","aliases":"cB B mathcal","display":"{\\mathcal B}","kind":"use","excerpt":"Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-57b35300f526c067","tex":"\\cS_P","aliases":"cS_P S_P cS mathcal","display":"{\\mathcal S}_P","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"The second part studies inversion in a complete filtered algebra. Suppose an operator has the form T=D+N, where D is invertible and every packet transfer in N raises the filtration by a positive amount. The ordinary Neumann series can then be expanded into weighted paths. The weight is not the number of edges. It is the total filtration gain along ","count":59},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":32},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"use","excerpt":"[Block path formula] Let P=(P_α)_α∈ I be a finite packet system in a complete filtered algebra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be fini","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1ad7b1c62e650428","tex":"D_\\alpha^{-1}","aliases":"D_α^-1 alpha","display":"D_\\alpha^{-1}","kind":"use","excerpt":"[Block path formula] Let P=(P_α)_α∈ I be a finite packet system in a complete filtered algebra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be fini","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"[Rationality of the transfer series] The weighted transfer series satisfies Z_u,v(t) = u^ T(I-A(t))^-1v. It is a rational function whose denominator divides det(I-A(t)).","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"use","excerpt":"[A resultant bound that is not exact] Let A=diag(0,π^s) on the standard lattice Λ= O^2. The factors x and x-π^s have resultant -π^s, up to the harmless choice of resultant order, but the two spectral projectors are the integral coordinate projectors. Their exact separation weight is zero. The resultant sees the closeness of the two eigenvalues. It ","count":9},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a76476f1c15d5142","tex":"e^{(t-s)T_c}","aliases":"e^(t-s)T_c","display":"e^{(t-s)T_c}","kind":"use","excerpt":"[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t ","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b87f37426ee94627","tex":"E^2","aliases":"E^2","display":"E^2","kind":"use","excerpt":"The B ezout identity gives N=rI-UF. The two stated commutation relations and FG=0 give N(UF)=VGUF=VUGF=VUFG=0. Therefore N^2=N(rI-UF)=rN. Since r≠0, E^2=r^-2N^2=E. If Fx=0, the evaluated B ezout identity gives VGx=rx, hence Ex=x. If Gx=0, then Ex=0. The lattice assertion follows directly from rE=VG.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b229ce68227bc751","tex":"e^{ik\\pi/4}","aliases":"e^ikπ/4 pi","display":"e^{ik\\pi/4}","kind":"use","excerpt":"AR PHASE ORBITTraditional paper: Section 12.3 (p. 42), Section 12.3, Smith-phase identity (p. 43) Smith normal form concerns linear congruences. The Smith impedance chart uses the nonlinear coordinate change Γ=fracz-1z+1, z=frac1+Γ1-Γ. They share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points Γ_k=t","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e40ba00de6234ae1","tex":"E^n","aliases":"E^n","display":"E^n","kind":"use","excerpt":"et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-098a771b0be43d72","tex":"e^{sT_f}","aliases":"e^sT_f","display":"e^{sT_f}","kind":"use","excerpt":"[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t ","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5c4a586d94fa15a1","tex":"e^{tT_c}","aliases":"e^tT_c","display":"e^{tT_c}","kind":"use","excerpt":"[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6096747ae971d30d","tex":"e^{tT_f}","aliases":"e^tT_f","display":"e^{tT_f}","kind":"use","excerpt":"[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d8be2ba7e790af78","tex":"E_+","aliases":"E_+","display":"E_+","kind":"use","excerpt":"0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2 0 · 0 , E_-=fracI-A2 = 0 · -1/2","count":22},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c0d11f3c7e1f7a5e","tex":"E_+^2","aliases":"E_+^2","display":"E_+^2","kind":"use","excerpt":"000); node[] at (8.25000,2.15000) frac12; draw[fill=black!0,draw=black!45,line width=.35pt,] (7.20000,1.10000) rectangle ++(0.70000,0.70000); node[] at (7.55000,1.45000) 0; draw[fill=black!0,draw=black!45,line width=.35pt,] (7.90000,1.10000) rectangle ++(0.70000,0.70000); node[] at (8.25000,1.45000) 0; node[] at (11.00000,2.10000) N^2=2N; node[] at","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ca57caa14f72ba3b","tex":"E_-","aliases":"E_-","display":"E_-","kind":"use","excerpt":"0 · 0 , E_-=fracI-A2 = 0 · -1/2 0 · 1 . For every odd prime p, both projectors preserve Z_p^2. At p=2, s_2(E_+)=s_2(E_-)=1. The factors x-1 and x+1 have resultant 2.","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-83ef2b537772a0a9","tex":"E_{00}","aliases":"E_00","display":"E_{00}","kind":"use","excerpt":"The words are written matrix products: in ab, the map b acts first. Therefore ab=8E_00, ba=8E_11, aba=16E_01. The two products ab and ba cannot be identified merely because they have equal weight. Both their order and their source/target information matter.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-febc076ab7500943","tex":"E_{01}","aliases":"E_01","display":"E_{01}","kind":"use","excerpt":"vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGTraditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18) Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least fi","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-136cb5922e752cc4","tex":"E_{10}","aliases":"E_10","display":"E_{10}","kind":"use","excerpt":"vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGTraditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18) Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least fi","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-46018870fc4913f9","tex":"E_{11}","aliases":"E_11","display":"E_{11}","kind":"use","excerpt":"The words are written matrix products: in ab, the map b acts first. Therefore ab=8E_00, ba=8E_11, aba=16E_01. The two products ab and ba cannot be identified merely because they have equal weight. Both their order and their source/target information matter.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Onl","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f8b63d55908fc609","tex":"E_f","aliases":"E_f","display":"E_f","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2f1d90b4eb6efe21","tex":"E_f^2","aliases":"E_f^2","display":"E_f^2","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ca7e75161b6bbf35","tex":"E_{ij}","aliases":"E_ij","display":"E_{ij}","kind":"use","excerpt":"vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGTraditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18) Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least fi","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","excerpt":"vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONTraditional paper: Section B.1 (p. 47) Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-23943c7271363293","tex":"E_\\pm","aliases":"E_± pm","display":"E_\\pm","kind":"use","excerpt":"headThe special fiber does not retain the rational splitting Modulo two, x-1=x+1 and A becomes a nontrivial unipotent matrix. The entries 1/2 in E_± cannot be reduced modulo two. By contrast, the coefficient selectors on V6 use only zero and one and remain available.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-15474d1cea1e947b","tex":"\\End_K","aliases":"End_K End operatorname","display":"{\\operatorname{End}}_K","kind":"use","excerpt":"[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9928404808ae1116","tex":"\\End_\\OO","aliases":"End_OO End_ O End OO operatorname mathcal","display":"{\\operatorname{End}}_{{\\mathcal O}}","kind":"use","excerpt":"Let O be a DVR, K=Frac( O), and A∈End_ O(Λ) for a finite free lattice Λ. Suppose the characteristic polynomial factors over O as χ_A(x)=f(x)g(x), where f and g are coprime over K[x].","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2a6e4a27c6e0ed93","tex":"\\End_{\\OO}","aliases":"End_OO End_ O End OO operatorname mathcal","display":"{\\operatorname{End}}_{{{\\mathcal O}}}","kind":"use","excerpt":"Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F mathbb","display":"{\\mathbb F}","kind":"use","excerpt":"The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fibre does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the s","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"The field or orthogonality condition is essential. A primitive element of order N in a ring with zero divisors does not by itself force the geometric sums away from the diagonal to vanish. For example, in F_3× F_3 the element (-1,1) has order two, but the second component of its nontrivial character sum is nonzero. The coefficient selector uses no ","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"For real t, differentiate F(s)=e^(t-s)T_cCe^sT_f. We obtain F'(s)=e^(t-s)T_cΔ e^sT_f. Integration from 0 to t gives the identity. The complex statement follows by the same calculation along the straight segment. The norm bound follows from the integral formula.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b7784b7f5b08b435","tex":"F^\\bullet","aliases":"F^bullet bullet","display":"F^\\bullet","kind":"use","excerpt":"[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a9954ea8901f689c","tex":"F^{d(\\alpha,\\beta)}","aliases":"F^d(α,β) alpha beta","display":"F^{d(\\alpha,\\beta)}","kind":"use","excerpt":"[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9b8a4bc27627081b","tex":"F^m","aliases":"F^m","display":"F^m","kind":"use","excerpt":"Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5a02562f7de1f19b","tex":"F^{W(\\gamma)}","aliases":"F^W(γ) gamma","display":"F^{W(\\gamma)}","kind":"use","excerpt":"l, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4edc94a6363ddff2","tex":"F^\\infty","aliases":"F^∞ infty","display":"F^\\infty","kind":"use","excerpt":"[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-101fbe8a1f9aa65c","tex":"\\F_2","aliases":"F_2 F mathbb","display":"{\\mathbb F}_2","kind":"use","excerpt":"The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fibre does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the s","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"vpage6Selectors survive without divisionMODEL A / RESIDUE MASKS IN CHARACTERISTIC TWOTraditional paper: Proposition 4.2 (p. 9) Work first in F_2, where 1+1=0. The eight coefficients below are just a finite display of a coefficient sequence. The even and odd projectors are diagonal masks with entries zero and one.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-54c7a07f66f1253f","tex":"\\mathbb F_3","aliases":"F_3 mathbb","display":"\\mathbb F_3","kind":"use","excerpt":"The field or orthogonality condition is essential. A primitive element of order N in a ring with zero divisors does not by itself force the geometric sums away from the diagonal to vanish. For example, in F_3× F_3 the element (-1,1) has order two, but the second component of its nontrivial character sum is nonzero. The coefficient selector uses no ","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-23af4eb7208f9cad","tex":"\\mathbb F_7","aliases":"F_7 mathbb","display":"\\mathbb F_7","kind":"use","excerpt":"vpage7A Fourier certificate you can inspectMODEL A / AN EXACT NINE-ENTRY TABLETraditional paper: Theorem 4.3 (p. 10) Take N=3 and work in F_7. Set ω=2, ω^-1=4, η=3^-1=5. Here 2^3=1 and neither 2 nor 2^2 is one. The unnormalized geometric sums are the entries in the first matrix.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-43a1003114401521","tex":"\\Frac","aliases":"Frac operatorname","display":"{\\operatorname{Frac}}","kind":"use","excerpt":"Let O be a DVR, K=Frac( O), and A∈End_ O(Λ) for a finite free lattice Λ. Suppose the characteristic polynomial factors over O as χ_A(x)=f(x)g(x), where f and g are coprime over K[x].","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0fcc70536b44cfb8","tex":"\\operatorname{GL}","aliases":"GL operatorname","display":"\\operatorname{GL}","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-301f280765554fc6","tex":"\\operatorname{GL}_m","aliases":"GL_m operatorname","display":"\\operatorname{GL}_m","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8df4bc39535de40c","tex":"\\operatorname{GL}_n","aliases":"GL_n operatorname","display":"\\operatorname{GL}_n","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5d724fab1c00df76","tex":"H_e","aliases":"H_e","display":"H_e","kind":"use","excerpt":"A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f7c152eadfb594c5","tex":"H_{e_1}","aliases":"H_e_1","display":"H_{e_1}","kind":"use","excerpt":"Here A_γ=(D^-1H_e_1)⋯(D^-1H_e_r) in the written order. Multiplicativity places it in F^W(γ)R, so the quotient kills it when W(γ)≥ L. Positive weights also imply |γ|≤ W(γ), making the retained search finite.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-26ae0fa325355773","tex":"H_{e_r}","aliases":"H_e_r","display":"H_{e_r}","kind":"use","excerpt":"Here A_γ=(D^-1H_e_1)⋯(D^-1H_e_r) in the written order. Multiplicativity places it in F^W(γ)R, so the quotient kills it when W(γ)≥ L. Positive weights also imply |γ|≤ W(γ), making the retained search finite.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"The values are c|rrrrrr L · 1 · 2 · 3 · 4 · 5 · 6 W(L) · 2 · 3 · 4 · 4 · 4 · 4 # solutions · 4 · 8 · 16 · 16 · 16 · 16. Once L≥3, the nonzero weights have been fully exposed and the admissible fiber cardinality stabilizes at sixteen. The input space still grows as 2^3L. The number of attainable targets grows accordingly.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ad472b41377c2078","tex":"I_m","aliases":"I_m","display":"I_m","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a2b075d59d6e400b","tex":"I_{X^{\\Pcal}}","aliases":"I_X^Pcal I_X^ P Pcal mathcal","display":"I_{X^{{\\mathcal P}}}","kind":"use","excerpt":"[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5de7c345d56728e9","tex":"I_{X^{\\Qcal}}","aliases":"I_X^Qcal I_X^ Q Qcal mathcal","display":"I_{X^{{\\mathcal Q}}}","kind":"use","excerpt":"[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id mathrm","display":"{\\mathrm{id}}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d9d9b2d2680f67c6","tex":"\\id_{I_\\ell}","aliases":"id_I_ℓ id ell mathrm ell","display":"{\\mathrm{id}}_{I_\\ell}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"For a specific attainable target (2,4), 2x≡2pmod8iff x∈{1,5}, 4y≡4pmod8iff y∈{1,3,5,7}. The coordinate choices are independent, giving 2·4=8 solutions. The entire carrier is partitioned consistently: 8 attainable targets × 8 inputs per target=64. For an inadmissible target, such as (1,4) or (2,2), the fiber is empty. The same nonzero cardinality fo","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a9c9a48844d36e49","tex":"J^X_{P_b,P_a}","aliases":"J^X_P_b,P_a","display":"J^X_{P_b,P_a}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-20eda0343851fb87","tex":"J^Y_{Q_b,Q_a}","aliases":"J^Y_Q_b,Q_a","display":"J^Y_{Q_b,Q_a}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d611cab28d1459e6","tex":"J_{4,2}","aliases":"J_4,2","display":"J_{4,2}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0992043af0c631b6","tex":"J_{8,2}","aliases":"J_8,2","display":"J_{8,2}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e3b639da3aa60999","tex":"J_{8,4}","aliases":"J_8,4","display":"J_{8,4}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7b8a912bcc1d47fa","tex":"j_A","aliases":"j_A","display":"j_A","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-62dac47031a2738c","tex":"J_{a,a}","aliases":"J_a,a","display":"J_{a,a}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c71586b5a403c1a2","tex":"J_{a,b}","aliases":"J_a,b","display":"J_{a,b}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-025540ed37ef1856","tex":"j_B","aliases":"j_B","display":"j_B","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e8fbcf86f56456cc","tex":"J_{b,a}","aliases":"J_b,a","display":"J_{b,a}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c2788fcb497b7b9e","tex":"J_{b,a}^{-1}","aliases":"J_b,a^-1","display":"J_{b,a}^{-1}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-33e57b582e50713d","tex":"J_{c,a}","aliases":"J_c,a","display":"J_{c,a}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-653dd629853bff62","tex":"J_{c,b}","aliases":"J_c,b","display":"J_{c,b}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a3bc30050b2c5943","tex":"J_{\\Rcal\\leftarrow\\Pcal}","aliases":"J_Rcal←Pcal J_ R← P Rcal leftarrow Pcal mathcal leftarrow mathcal","display":"J_{{\\mathcal R}\\leftarrow{\\mathcal P}}","kind":"use","excerpt":"[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5ebfd094937c1ad9","tex":"J_{\\Rcal\\leftarrow\\Qcal}","aliases":"J_Rcal←Qcal J_ R← Q Rcal leftarrow Qcal mathcal leftarrow mathcal","display":"J_{{\\mathcal R}\\leftarrow{\\mathcal Q}}","kind":"use","excerpt":"[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate hold","count":32},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ac62d9119015a5fb","tex":"K^\\times","aliases":"K^× times","display":"K^\\times","kind":"use","excerpt":"[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e08d6b60be0ec0b0","tex":"\\Ker","aliases":"Ker operatorname","display":"{\\operatorname{Ker}}","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":10},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"=.35pt,] (4.57000,2.26000) rectangle ++(0.57000,0.57000); node[] at (4.85500,2.54500) 2; draw[fill=black!0,draw=black!45,line width=.35pt,] (5.14000,2.26000) rectangle ++(0.57000,0.57000); node[] at (5.42500,2.54500) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (5.71000,2.26000) rectangle ++(0.57000,0.57000); node[] at (5.99500,2.54500) 4","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":", where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then the nilpotence condition follows from the filtration data because m|γ|≤ W(γ). For m>0, the executable choice N=lceil L/mrceil satisfies L≤ mN. This elemen","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-afe7579823b893fd","tex":"\\length","aliases":"length operatorname","display":"{\\operatorname{length}}","kind":"use","excerpt":"[Total gluing and worst precision] If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O, then length_ O(G_Λ, P) = ∑_j=1^ra_j, s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}). Thus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-993066ece33b1204","tex":"\\operatorname{length}","aliases":"length operatorname","display":"\\operatorname{length}","kind":"use","excerpt":"olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-be97729b8e1f11de","tex":"\\length_{\\OO}","aliases":"length_OO length_ O length OO operatorname mathcal","display":"{\\operatorname{length}}_{{{\\mathcal O}}}","kind":"use","excerpt":"[Total gluing and worst precision] If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O, then length_ O(G_Λ, P) = ∑_j=1^ra_j, s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}). Thus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Integral splitting criterion] Every packet projector preserves Λ if and only if Λ=⊕_α(Λ∩ V_α). Equivalently, s_Λ( P)=0 Longleftrightarrow G_Λ, P=0.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-25d467671d09a670","tex":"\\lVert","aliases":"lVert","display":"\\lVert","kind":"use","excerpt":"lineskip [Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"[htbp] centering includegraphics[width=0.96textwidth]03_refinement_incidence.pdf captionThe incidence matrix M_r,s=1 when the fine residue s belongs to the coarse residue r. Its rows implement exact aggregation from fine packets to coarse packets. Common refinements.","count":22},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-465515cf3e71e41b","tex":"M_{m,n}","aliases":"M_m,n","display":"M_{m,n}","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-eeb84163daca3455","tex":"M_{r,s}","aliases":"M_r,s","display":"M_{r,s}","kind":"use","excerpt":"[htbp] centering includegraphics[width=0.96textwidth]03_refinement_incidence.pdf captionThe incidence matrix M_r,s=1 when the fine residue s belongs to the coarse residue r. Its rows implement exact aggregation from fine packets to coarse packets. Common refinements.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-213d86a5e6b34393","tex":"N^{-1}","aliases":"N^-1","display":"N^{-1}","kind":"use","excerpt":"0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-56155806d18c6f60","tex":"\\mathbb N^r","aliases":"N^r mathbb","display":"\\mathbb N^r","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ca257e6c32ec363d","tex":"N_{01}","aliases":"N_01","display":"N_{01}","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6a87293985c01784","tex":"N_{10}","aliases":"N_10","display":"N_{10}","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","excerpt":"be a rational projector on V, let Λ be a lattice, and let s=s_Λ(E). For every m≥0, the rule [x]_m+s⟼[Ex]_m from Λ/π^m+sΛ to V/π^mΛ is well defined. Here m is an integer. If d is an integer with 0≤ d<s, then for every integer m≥0 there are x,y∈Λ such that x≡ ypmodπ^m+dΛ but Exnot≡ Eypmodπ^mΛ. Thus s extra digits are both sufficient and necessary for","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-946fcb5ff5706330","tex":"o_m","aliases":"o_m","display":"o_m","kind":"use","excerpt":"vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONTraditional paper: Section B.1 (p. 47) Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7b2e774b4630d060","tex":"\\OO^2","aliases":"OO^2 O^2 OO mathcal","display":"{{\\mathcal O}}^2","kind":"use","excerpt":"[A resultant bound that is not exact] Let A=diag(0,π^s) on the standard lattice Λ= O^2. The factors x and x-π^s have resultant -π^s, up to the harmless choice of resultant order, but the two spectral projectors are the integral coordinate projectors. Their exact separation weight is zero. The resultant sees the closeness of the two eigenvalues. It ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-704b220b0886246a","tex":"p^L","aliases":"p^L","display":"p^L","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":10},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aa1e9c58c08b71c9","tex":"p^{\\min(s,L)}","aliases":"p^min(s,L) min","display":"p^{\\min(s,L)}","kind":"use","excerpt":"Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fibre has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global ","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-fc7fd452679f9cf5","tex":"p^{\\min(s_i,L)}","aliases":"p^min(s_i,L) min","display":"p^{\\min(s_i,L)}","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-360f794da8094012","tex":"p^s","aliases":"p^s","display":"p^s","kind":"use","excerpt":"Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fibre has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e79435c578c898aa","tex":"p^{s_1}","aliases":"p^s_1","display":"p^{s_1}","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c5724748474faba5","tex":"p^{s_d}","aliases":"p^s_d","display":"p^{s_d}","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b5bff95dd6769dfd","tex":"p^{s_i}","aliases":"p^s_i","display":"p^{s_i}","kind":"use","excerpt":"If a Smith form is rank deficient, a literal zero diagonal factor requires the corresponding target coordinate to vanish modulo p^L and then contributes all p^L possible coordinate values. In the concrete quotient it can equivalently be represented by any weight s_i≥ L, since p^s_i=0 in Z/(p^L) Z. The theorem then gives exactly that condition and f","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-209584f848a592aa","tex":"p^{\\sum_{i=1}^d\\min(s_i,L)}","aliases":"p^∑_i=1^dmin(s_i,L) sum min","display":"p^{\\sum_{i=1}^d\\min(s_i,L)}","kind":"use","excerpt":"ongruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordi","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1c9e7d74779bb5eb","tex":"P_{12,7}","aliases":"P_12,7","display":"P_{12,7}","kind":"use","excerpt":"[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-bab84a59440e63c0","tex":"P_{2,0}","aliases":"P_2,0","display":"P_{2,0}","kind":"use","excerpt":"vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTTraditional paper: Theorem 3.3 (p. 3), Theorem 4.6 (p. 11) Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6a679a976dd7459e","tex":"P_{2,1}","aliases":"P_2,1","display":"P_{2,1}","kind":"use","excerpt":"75000,2.55000) rectangle ++(0.75000,0.60000); node[] at (4.12500,2.85000) 0; draw[fill=black!22,draw=black!45,line width=.35pt,] (4.50000,2.55000) rectangle ++(0.75000,0.60000); node[] at (4.87500,2.85000) 1; draw[fill=black!0,draw=black!45,line width=.35pt,] (5.25000,2.55000) rectangle ++(0.75000,0.60000); node[] at (5.62500,2.85000) 0; node[ancho","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f7c6a909392369bc","tex":"P_{3,1}","aliases":"P_3,1","display":"P_{3,1}","kind":"use","excerpt":"[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-91a55f7626b81c45","tex":"P_{4,0}","aliases":"P_4,0","display":"P_{4,0}","kind":"use","excerpt":"The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a6b3605e1b2fcc56","tex":"P_{4,2}","aliases":"P_4,2","display":"P_{4,2}","kind":"use","excerpt":"The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-53d17384b2682f1f","tex":"P_{4,3}","aliases":"P_4,3","display":"P_{4,3}","kind":"use","excerpt":"[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-219e101503b0300a","tex":"P_{8,2}","aliases":"P_8,2","display":"P_{8,2}","kind":"use","excerpt":"The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-91b4f0e802663f94","tex":"P_{8,6}","aliases":"P_8,6","display":"P_{8,6}","kind":"use","excerpt":"The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d41e39fff15e99f5","tex":"P_{A,j_A}","aliases":"P_A,j_A","display":"P_{A,j_A}","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7ab4aa6ec4adb158","tex":"P_{AB,j}","aliases":"P_AB,j","display":"P_{AB,j}","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2298f7e841a31d94","tex":"P_{B,j_B}","aliases":"P_B,j_B","display":"P_{B,j_B}","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"000,0.70000); node[] at (0.35000,1.45000) 0; draw[fill=black!0,draw=black!45,line width=.35pt,] (0.70000,1.10000) rectangle ++(0.70000,0.70000); node[] at (1.05000,1.45000) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (1.40000,1.10000) rectangle ++(0.70000,0.70000); node[] at (1.75000,1.45000) 3; node[] at (3.70000,2.05000) × 5; node[] at","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6977f69e7c84b853","tex":"P_{M,s}","aliases":"P_M,s","display":"P_{M,s}","kind":"use","excerpt":"[Nested residue refinement] Let N,M be positive integers with N dividing M, and let 0≤ j<N. Then P_N,j = ∑_substack0≤ s<M s≡ jpmod NP_M,s. When R is nonzero, the packet system modulo M refines the packet system modulo N.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ce857563c4ac4bba","tex":"P_{N,i}","aliases":"P_N,i","display":"P_{N,i}","kind":"use","excerpt":"[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4a442c81a1b99426","tex":"P_{N,j+1}","aliases":"P_N,j+1","display":"P_{N,j+1}","kind":"use","excerpt":"[The shift graph of residue packets] Assume that R is nonzero and N>0. Multiplication by z sends the residue packet P_N,jR[[z]] into P_N,j+1R[[z]], where the index is read modulo N. Its packet graph is the directed cycle 0⟶1⟶⋯⟶ N-1⟶0. Multiplication by a polynomial supported on a set of residues gives the corresponding finite set of phase shifts. T","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-21f18d5054b34924","tex":"P_{N,j+r}","aliases":"P_N,j+r","display":"P_{N,j+r}","kind":"use","excerpt":"A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b619bee6fd91d216","tex":"P_{s,t}","aliases":"P_s,t","display":"P_{s,t}","kind":"use","excerpt":"w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, hence Q≠0 and Q is a unit in Z[[t]], and the independently enumerated walk series satisfies Z_s,t(t)=Q(t)^-1P_s,t(t), Q(t)Z_s,t(t)=P_s,t(t) in Z[[t]].","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7f9d4390a1f58464","tex":"P_{\\alpha'}","aliases":"P_α' alpha","display":"P_{\\alpha'}","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-634bde01ca310a18","tex":"P_\\alpha^2","aliases":"P_α^2 alpha","display":"P_\\alpha^2","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-04eb7fb63bcffdaf","tex":"P_{\\rho(\\beta)}","aliases":"P_ρ(β) rho beta","display":"P_{\\rho(\\beta)}","kind":"use","excerpt":"0 · ρ(β)≠α. Hence Q_β X⊆ P_ρ(β)X. If a third packet system R refines Q, then R refines P through the composite parent map.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c48b0030cb4c99da","tex":"\\Pcal_m","aliases":"Pcal_m P_m Pcal mathcal","display":"{\\mathcal P}_m","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-23edeb9c0147a8cc","tex":"\\Pcal_\\ell","aliases":"Pcal_ℓ P_ℓ Pcal ell mathcal ell","display":"{\\mathcal P}_\\ell","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5bfbf5194d3f5a47","tex":"\\mathrm{pr}","aliases":"pr mathrm","display":"\\mathrm{pr}","kind":"use","excerpt":"et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q mathbb","display":"{\\mathbb Q}","kind":"use","excerpt":"The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fibre does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the s","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of","count":21},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-db9b1118d56e7f21","tex":"q^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}","aliases":"q^(n-r)L+∑_i=1^rmin(s_i,L) sum min","display":"q^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}","kind":"use","excerpt":"rdinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-04992c0930a894b0","tex":"q^L","aliases":"q^L","display":"q^L","kind":"use","excerpt":"ultiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L choices. Multi","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5cd87d7bdf4bf601","tex":"q^{\\min(s_i,L)}","aliases":"q^min(s_i,L) min","display":"q^{\\min(s_i,L)}","kind":"use","excerpt":"Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L cho","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-cc28058c38687619","tex":"q^{\\sum_i\\min(s_i,L)}","aliases":"q^∑_imin(s_i,L) sum min","display":"q^{\\sum_i\\min(s_i,L)}","kind":"use","excerpt":"For an arbitrary DVR O with residue field cardinality q, the analogous formula q^∑_imin(s_i,L) follows from the solvability and fibre count for each coordinate. The current revision adds direct proofs of this DVR coordinate arithmetic; the current correspondence records those targets and the rectangular transport separately. The inherited abstract ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3dcd53dfa5dbd07d","tex":"Q_{00}","aliases":"Q_00","display":"Q_{00}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3aa5ff4c06d335c1","tex":"Q_{01}","aliases":"Q_01","display":"Q_{01}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-757d025f70886a4b","tex":"Q_{10}","aliases":"Q_10","display":"Q_{10}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0b259d086fc3978c","tex":"Q_{11}","aliases":"Q_11","display":"Q_{11}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-50f270102a088631","tex":"\\Q_2","aliases":"Q_2 Q mathbb","display":"{\\mathbb Q}_2","kind":"use","excerpt":"The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fibre does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the s","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-812a1fd77a9f35a3","tex":"\\mathbb Q_2","aliases":"Q_2 mathbb","display":"\\mathbb Q_2","kind":"use","excerpt":"Modulo 2, the two eigenvalues become equal. The rational sign packets remain distinct over Q_2, but the standard lattice does not split into their intersections until one more dyadic digit is retained.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5ca75c044055616d","tex":"Q_a","aliases":"Q_a","display":"Q_a","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c57539584fe8169f","tex":"Q_b","aliases":"Q_b","display":"Q_b","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-369e50e7dcd70df5","tex":"q_d","aliases":"q_d","display":"q_d","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e15bd658e05b8d9b","tex":"q_k","aliases":"q_k","display":"q_k","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ad4d7e06ba73eb0d","tex":"q_L","aliases":"q_L","display":"q_L","kind":"use","excerpt":"[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of","count":11},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-cd98a9c47ecfa404","tex":"\\mathbb Q_p","aliases":"Q_p mathbb","display":"\\mathbb Q_p","kind":"use","excerpt":"0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ea3d9f514fcaceb6","tex":"Q_\\alpha","aliases":"Q_α alpha","display":"Q_\\alpha","kind":"use","excerpt":"Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-574286fc36296144","tex":"Q_{\\beta'}","aliases":"Q_β' beta","display":"Q_{\\beta'}","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-610d527192ec5212","tex":"Q_\\beta^2","aliases":"Q_β^2 beta","display":"Q_\\beta^2","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5a423e1823849ed7","tex":"Q_\\gamma","aliases":"Q_γ gamma","display":"Q_\\gamma","kind":"use","excerpt":"Multiply the identity P_α=∑_ρ(γ)=αQ_γ by Q_β on either side. Pairwise annihilation in the fine family leaves Q_β when ρ(β)=α and leaves zero otherwise. For transitivity, substitute the child expansion of each Q_β into the expansion of P_α and group the resulting terms by the composite parent.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ba83fc4ca5c67ec3","tex":"\\QQ_2","aliases":"QQ_2 QQ","display":null,"kind":"use","excerpt":"vfig[.94]dyadic_eigenlattice platecaptionAn ordinary integer-lattice picture, not a drawing of the topology of QQ_2. Filled dots have even second coordinate. Unfilled dots lie in the other coset. The hollow point at (1/2,0) records the image of an integral vector under E_+.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aff4a4490ea52474","tex":"r^{-1}","aliases":"r^-1","display":"r^{-1}","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-072e7fe97b551a6a","tex":"r^{-2}","aliases":"r^-2","display":"r^{-2}","kind":"use","excerpt":"The B ezout identity gives N=rI-UF. The two stated commutation relations and FG=0 give N(UF)=VGUF=VUGF=VUFG=0. Therefore N^2=N(rI-UF)=rN. Since r≠0, E^2=r^-2N^2=E. If Fx=0, the evaluated B ezout identity gives VGx=rx, hence Ex=x. If Gx=0, then Ex=0. The lattice assertion follows directly from rE=VG.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4056c2b0608f31f7","tex":"R_2^{-1}","aliases":"R_2^-1","display":"R_2^{-1}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0fe85010c113a2aa","tex":"R_4","aliases":"R_4","display":"R_4","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-37f2bdf109707126","tex":"R_c","aliases":"R_c","display":"R_c","kind":"use","excerpt":"Inverse witnesses from one side and norm control. For continuous linear maps, the same identity gives the certified norm bound C R_f-R_c C ≤ R_c Δ R_f , from a left inverse witness for the coarse scalar shift and a right inverse witness for the fine scalar shift. These witnesses suffice for the displayed identity. On their own, they need not define","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a94836df7b97d193","tex":"R_f","aliases":"R_f","display":"R_f","kind":"use","excerpt":"Inverse witnesses from one side and norm control. For continuous linear maps, the same identity gives the certified norm bound C R_f-R_c C ≤ R_c Δ R_f , from a left inverse witness for the coarse scalar shift and a right inverse witness for the fine scalar shift. These witnesses suffice for the displayed identity. On their own, they need not define","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-61bdbf136c2dac84","tex":"R_L","aliases":"R_L","display":"R_L","kind":"use","excerpt":"[Exact finite precision truncation] Under the hypotheses of [source reference], define R_L = ∑_W(γ)<L(-1)^|γ|A_γ D^-1. Then T^-1-R_L∈ F^L B, TR_L-I∈ F^L B, R_LT-I∈ F^L B. Thus R_L is the exact inverse in B/F^L B.","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-30accea853c1a158","tex":"R_N^{-1}","aliases":"R_N^-1","display":"R_N^{-1}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-fce729b05bf0e1ea","tex":"R_{\\alpha\\beta}","aliases":"R_αβ alpha beta","display":"R_{\\alpha\\beta}","kind":"use","excerpt":"[Common refinement of commuting packet systems] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on X. Assume P_α Q_β=Q_β P_α for all α and β. Remove the zero operators from the family R_αβ=P_α Q_β. The remaining operators form a packet system that refines both P and Q. The same statement holds for any finite collection of pairwise commuting pac","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7fda162c9e475299","tex":"R_{\\alpha\\beta}^2","aliases":"R_αβ^2 alpha beta","display":"R_{\\alpha\\beta}^2","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ca0ed80fd48b48de","tex":"R_\\gamma","aliases":"R_γ gamma","display":"R_\\gamma","kind":"use","excerpt":"Let R_γ be a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3bce33d4a711b894","tex":"\\Rcal","aliases":"Rcal R mathcal","display":"{\\mathcal R}","kind":"use","excerpt":"0 · ρ(β)≠α. Hence Q_β X⊆ P_ρ(β)X. If a third packet system R refines Q, then R refines P through the composite parent map.","count":10},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":", where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then the nilpotence condition follows from the filtration data because m|γ|≤ W(γ). For m>0, the executable choice N=lceil L/mrceil satisfies L≤ mN. This elemen","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-06b25b660a3b775f","tex":"\\Res","aliases":"Res operatorname","display":"{\\operatorname{Res}}","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1ecf96b50772dd74","tex":"\\operatorname{Res}","aliases":"Res operatorname","display":"\\operatorname{Res}","kind":"use","excerpt":"For the original sign factors the concrete resultant is Res(x-1,x+1)=2 and the bound is sharp. That equality is special to this displayed calculation, not a universal minimality theorem for every resultant certificate.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-61f9609b70de73a0","tex":"\\rVert","aliases":"rVert","display":"\\rVert","kind":"use","excerpt":"lineskip [Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-977c562e1c896da9","tex":"S^{\\mathrm{pr}}_{L,N}","aliases":"S^pr_L,N mathrm","display":"S^{\\mathrm{pr}}_{L,N}","kind":"use","excerpt":"et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8768859caa7d360e","tex":"S^r","aliases":"S^r","display":"S^r","kind":"use","excerpt":"Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer bloc","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"vpage19Precision exposes directional freedomMODEL E / A DEPTH-BY-DEPTH FIBER BUDGETTraditional paper: Theorem 10.6 (p. 36) Take p=2 and the supplied diagonal weights (s_0,s_1,s_2)=(0,1,3). At depth L, an admissible target has 2^W(L) solutions, W(L)=min(0,L)+min(1,L)+min(3,L). Each direction contributes only the amount visible at the current depth.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":8},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3ce2399701bcdd04","tex":"\\mathcal S_2","aliases":"S_2 mathcal","display":"\\mathcal S_2","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"use","excerpt":"0 · 1 . For every odd prime p, both projectors preserve Z_p^2. At p=2, s_2(E_+)=s_2(E_-)=1. The factors x-1 and x+1 have resultant 2.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4f398ec10f2f1455","tex":"s_{2,\\Z_2^2}","aliases":"s_2,Z_2^2 s_2, Z_2^2 Z mathbb","display":"s_{2,{\\mathbb Z}_2^2}","kind":"use","excerpt":"[Odd-adic versus dyadic behavior] For every prime p≠2, every entry of E_+ and E_- is p-integral, and each projector has exact local denominator weight zero. At p=2, each projector has exact local denominator weight one. Multiplication by 2 makes it integral, while the unscaled matrix is not integral: s_2, Z_2^2(E_+)=s_2, Z_2^2(E_-)=1. The character","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b06cc43523171b06","tex":"\\mathcal S_4","aliases":"S_4 mathcal","display":"\\mathcal S_4","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-db289384e3bfb6f4","tex":"s_d","aliases":"s_d","display":"s_d","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L","count":15},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-22cf9853e0e53ac8","tex":"S_N","aliases":"S_N","display":"S_N","kind":"use","excerpt":"vpage11The residual tells you when to stopMODEL B / A CERTIFIED CUTOFF, NOT A HEURISTICTraditional paper: Theorem 5.10 (p. 17), Corollary 5.11 (p. 18), Section B.2 (p. 47) For S_N=∑_k=0^N-1(-A)^k, telescoping gives (I+A)S_N=S_N(I+A)=I-(-A)^N. Thus the exact residual is known. To certify precision L, it is enough to show that every residual entry is","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-78a9cd432e84c71d","tex":"\\mathcal S_N","aliases":"S_N mathcal","display":"\\mathcal S_N","kind":"use","excerpt":"A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b37e9083bef6f614","tex":"S_r","aliases":"S_r","display":"S_r","kind":"use","excerpt":"A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a6bf5eb7ecfc145e","tex":"s_r","aliases":"s_r","display":"s_r","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-832402dc4a3bcf4c","tex":"s_{\\Lambda_s}","aliases":"s_Λ_s Lambda","display":"s_{\\Lambda_s}","kind":"use","excerpt":"with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e46cbed476cfd5c1","tex":"\\mathrm{Smith}","aliases":"Smith mathrm","display":"\\mathrm{Smith}","kind":"use","excerpt":"The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ae01b8265b984010","tex":"\\mathrm{source}","aliases":"source mathrm","display":"\\mathrm{source}","kind":"use","excerpt":"headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumerat","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-131a1658f6fa0831","tex":"\\operatorname{source}","aliases":"source operatorname","display":"\\operatorname{source}","kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3ee24d95dbc480e1","tex":"\\sup_{z\\in\\Gamma}","aliases":"sup_z∈Γ sup in Gamma","display":"\\sup_{z\\in\\Gamma}","kind":"use","excerpt":"olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f12da8d8c07c189d","tex":"\\operatorname{synthesize}","aliases":"synthesize operatorname","display":"\\operatorname{synthesize}","kind":"use","excerpt":"The verified contract is synthesize(analyze(x,N))=x. The Lean counterpart uses packet coordinates restricted to projector ranges and proves both inverse laws. The finite Python routine checks the concrete array realization.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"[Rationality of the transfer series] The weighted transfer series satisfies Z_u,v(t) = u^ T(I-A(t))^-1v. It is a rational function whose denominator divides det(I-A(t)).","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"use","excerpt":"B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, a","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"[A transfer series with one loop] Take vertices 0, 1, and 2. Give the edge 0→1 weight one, the loop 1→1 weight two, and the edge 1→2 weight one. Every path from 0 to 2 uses the first and last edges and may traverse the loop any number of times. Hence Z_0,2(t) = t^2(1+t^2+t^4+⋯) = fract^21-t^2. There is one path at every positive even weight and no ","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rati","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":"[A transfer series with one loop] Take vertices 0, 1, and 2. Give the edge 0→1 weight one, the loop 1→1 weight two, and the edge 1→2 weight one. Every path from 0 to 2 uses the first and last edges and may traverse the loop any number of times. Hence Z_0,2(t) = t^2(1+t^2+t^4+⋯) = fract^21-t^2. There is one path at every positive even weight and no ","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8c6354f5f56778af","tex":"t^5","aliases":"t^5","display":"t^5","kind":"use","excerpt":"[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rati","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-98f93e20c13d2c57","tex":"t^6","aliases":"t^6","display":"t^6","kind":"use","excerpt":"[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rati","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-179195799d88b1fc","tex":"t^7","aliases":"t^7","display":"t^7","kind":"use","excerpt":"[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rati","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aa7949456d03953c","tex":"t^9","aliases":"t^9","display":"t^9","kind":"use","excerpt":"[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rati","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f544e1fddefb59a0","tex":"t^d","aliases":"t^d","display":"t^d","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d241eb52ee3cfa72","tex":"t^k","aliases":"t^k","display":"t^k","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0a5cb710fde741d7","tex":"T^{\\Pcal}_{ba}","aliases":"T^Pcal_ba T^ P_ba Pcal mathcal","display":"T^{{\\mathcal P}}_{ba}","kind":"use","excerpt":"[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-15f4a4b580fed92c","tex":"T^{\\Pcal}_{\\beta\\alpha}","aliases":"T^Pcal_βα T^ P_βα Pcal beta alpha mathcal beta alpha","display":"T^{{\\mathcal P}}_{\\beta\\alpha}","kind":"use","excerpt":"For T∈End(X) and a packet system P, write T^ P_βα=P_β T P_α.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-40ae1b3bb8b8057b","tex":"T^{\\Qcal}_{\\beta\\alpha}","aliases":"T^Qcal_βα T^ Q_βα Qcal beta alpha mathcal beta alpha","display":"T^{{\\mathcal Q}}_{\\beta\\alpha}","kind":"use","excerpt":"[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c34948496a97b098","tex":"t_1^{w_1(e)}","aliases":"t_1^w_1(e)","display":"t_1^{w_1(e)}","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-75143599dd91b630","tex":"T_{2,1}","aliases":"T_2,1","display":"T_{2,1}","kind":"use","excerpt":"y any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action a","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-bd76f79f8f6e6b19","tex":"T_{ba}","aliases":"T_ba","display":"T_{ba}","kind":"use","excerpt":"T_ba · Input packet a, output packet b: P_bTP_a. · Blocks of a finite cyclic filter.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2e42a52f0ec67dea","tex":"T_c'","aliases":"T_c'","display":"T_c'","kind":"use","excerpt":"Had we chosen the transported operator T_c'=CT_fC^-1, the defect would be zero and every polynomial action would intertwine exactly. That is a different choice of coarse operator, not a stronger conclusion from the faulty one.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-16c26c32d58b7545","tex":"T_c^n","aliases":"T_c^n","display":"T_c^n","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4cc7829c7387a810","tex":"T_c^{n+1}","aliases":"T_c^n+1","display":"T_c^{n+1}","kind":"use","excerpt":"The statement is clear for n=1. For the induction step, write CT_f^n+1-T_c^n+1C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC) and apply the induction hypothesis.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-453187cc310f5d0e","tex":"T_c^{n-1-k}","aliases":"T_c^n-1-k","display":"T_c^{n-1-k}","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a9d7fd646d17da1a","tex":"T_f^k","aliases":"T_f^k","display":"T_f^k","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d73f1652a80f0f11","tex":"T_f^n","aliases":"T_f^n","display":"T_f^n","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a5580e5867f84cd5","tex":"T_f^{n+1}","aliases":"T_f^n+1","display":"T_f^{n+1}","kind":"use","excerpt":"The statement is clear for n=1. For the induction step, write CT_f^n+1-T_c^n+1C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC) and apply the induction hypothesis.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-fcc4f84d94d1a604","tex":"T_L","aliases":"T_L","display":"T_L","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-90b38ede3f1b080b","tex":"T_{N,k}","aliases":"T_N,k","display":"T_{N,k}","kind":"use","excerpt":"t z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3fa825b7cd03ebec","tex":"t_r^{w_r(e)}","aliases":"t_r^w_r(e)","display":"t_r^{w_r(e)}","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-35899e3c99220253","tex":"T_\\ell","aliases":"T_ℓ ell","display":"T_\\ell","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a15b96314fbf77e6","tex":"T_{\\ell+1}","aliases":"T_ℓ+1 ell","display":"T_{\\ell+1}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3c0ebebd59054caf","tex":"\\mathrm{target}","aliases":"target mathrm","display":"\\mathrm{target}","kind":"use","excerpt":"headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumerat","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-89ca555db043e94d","tex":"\\operatorname{target}","aliases":"target operatorname","display":"\\operatorname{target}","kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.","count":29},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a257f9d1ed0e54ad","tex":"u^{\\mathsf T}","aliases":"u^ T mathsf","display":"u^{\\mathsf T}","kind":"use","excerpt":"[Rationality of the transfer series] The weighted transfer series satisfies Z_u,v(t) = u^ T(I-A(t))^-1v. It is a rational function whose denominator divides det(I-A(t)).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-cef659aa5954e576","tex":"U^k","aliases":"U^k","display":"U^k","kind":"use","excerpt":"0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d2bc1d06f7629695","tex":"U^N","aliases":"U^N","display":"U^N","kind":"use","excerpt":"Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"Let M be a matrix over a DVR. Smith normal form gives invertible matrices U,V such that UMV = diag(π^s_1u_1,…,π^s_ru_r,0,…,0), where each u_i is a unit and 0≤ s_1≤⋯≤ s_r. The exponents s_i are directional lifting weights.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a822fb370eb1244d","tex":"u_r","aliases":"u_r","display":"u_r","kind":"use","excerpt":"Let M be a matrix over a DVR. Smith normal form gives invertible matrices U,V such that UMV = diag(π^s_1u_1,…,π^s_ru_r,0,…,0), where each u_i is a unit and 0≤ s_1≤⋯≤ s_r. The exponents s_i are directional lifting weights.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"use","excerpt":"[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"The reduced denominators of the displayed entries are 1 or 2, which proves the local integrality and minimality statements directly. The resultant is Res(x-1,x+1)=2. Thus v_2(2)=1 and v_p(2)=0 for primes p≠2.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"Sign projector resultant weight at the odd prime 3 · v_3(2)=0","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"use","excerpt":"The reduced denominators of the displayed entries are 1 or 2, which proves the local integrality and minimality statements directly. The resultant is Res(x-1,x+1)=2. Thus v_2(2)=1 and v_p(2)=0 for primes p≠2.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-694bdc6e24088dc1","tex":"\\mathrm{val}","aliases":"val mathrm","display":"\\mathrm{val}","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a675678fa6689935","tex":"\\vee","aliases":"vee","display":"\\vee","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"Here e=1+3w+5w^2 and o=2+4w+6w^2, so direct polynomial multiplication gives y_rm even=1+8w+16w^2+17w^3, y_rm odd=4+12w+20w^2+6w^3. This is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0d4e45da82688954","tex":"w^3","aliases":"w^3","display":"w^3","kind":"use","excerpt":"Here e=1+3w+5w^2 and o=2+4w+6w^2, so direct polynomial multiplication gives y_rm even=1+8w+16w^2+17w^3, y_rm odd=4+12w+20w^2+6w^3. This is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-dfa15e280d3611a8","tex":"w_r","aliases":"w_r","display":"w_r","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-566da3104464fc21","tex":"w_{\\mathrm{Smith}}","aliases":"w_Smith mathrm","display":"w_{\\mathrm{Smith}}","kind":"use","excerpt":"The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b0d65d8f4ac93228","tex":"\\widehat T","aliases":"widehat T widehat","display":"\\widehat T","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-27317811f6cd05cf","tex":"\\widehat T_2","aliases":"widehat T_2 widehat","display":"\\widehat T_2","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9844b5e86bad9b35","tex":"\\widehat T_4","aliases":"widehat T_4 widehat","display":"\\widehat T_4","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-37b3cca62187f94f","tex":"\\widehat T_N","aliases":"widehat T_N widehat","display":"\\widehat T_N","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a06250682a8196e7","tex":"\\widehat T_{Q,P}","aliases":"widehat T_Q,P widehat","display":"\\widehat T_{Q,P}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e0aad031a0de93fe","tex":"\\widehat T_{Q_a,P_a}","aliases":"widehat T_Q_a,P_a widehat","display":"\\widehat T_{Q_a,P_a}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6bbb28108c9e01d5","tex":"\\widehat T_{Q_b,P_b}","aliases":"widehat T_Q_b,P_b widehat","display":"\\widehat T_{Q_b,P_b}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0a4e15e563cc5107","tex":"\\widetilde b","aliases":"widetilde b widetilde","display":"\\widetilde b","kind":"use","excerpt":"[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-35f36a53181c409f","tex":"\\widetilde b_i","aliases":"widetilde b_i widetilde","display":"\\widetilde b_i","kind":"use","excerpt":"[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9fcf9ea0bdd21c78","tex":"\\widetilde T","aliases":"widetilde T widetilde","display":"\\widetilde T","kind":"use","excerpt":"[Stability of inversion at finite precision] Let T and widetilde T be invertible elements of F^0 B. If T-widetilde T∈ F^L B, then T^-1-widetilde T^-1∈ F^L B.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-501b4a78440cb78b","tex":"\\widetilde T^{-1}","aliases":"widetilde T^-1 widetilde","display":"\\widetilde T^{-1}","kind":"use","excerpt":"[Stability of inversion at finite precision] Let T and widetilde T be invertible elements of F^0 B. If T-widetilde T∈ F^L B, then T^-1-widetilde T^-1∈ F^L B.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTTraditional paper: Theorem 3.3 (p. 3), Theorem 4.6 (p. 11) Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c4a48827c05d4509","tex":"X_{00}","aliases":"X_00","display":"X_{00}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c546fc907e36cc78","tex":"x_{00}","aliases":"x_00","display":"x_{00}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3f15eebc2170872f","tex":"X_{01}","aliases":"X_01","display":"X_{01}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-769180e7b1f47d49","tex":"x_{01}","aliases":"x_01","display":"x_{01}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"[A visible one step defect] Let X_f=bbk^2, let X_c=bbk, and let C(x_1,x_2)=x_1. Take T_f= a · b","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-805649c1b3a8b1e8","tex":"X_{10}","aliases":"X_10","display":"X_{10}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a1ad6e78b56605d3","tex":"x_{10}","aliases":"x_10","display":"x_{10}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f17c0dba15962adb","tex":"X_{11}","aliases":"X_11","display":"X_{11}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8abb0d1ba7e87e0d","tex":"x_{11}","aliases":"x_11","display":"x_{11}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-65e552b57fafc09d","tex":"x_{14}","aliases":"x_14","display":"x_{14}","kind":"use","excerpt":"vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTTraditional paper: Theorem 3.3 (p. 3), Theorem 4.6 (p. 11) Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9f3039e219d0894a","tex":"x_{15}","aliases":"x_15","display":"x_{15}","kind":"use","excerpt":"vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTTraditional paper: Theorem 3.3 (p. 3), Theorem 4.6 (p. 11) Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[A visible one step defect] Let X_f=bbk^2, let X_c=bbk, and let C(x_1,x_2)=x_1. Take T_f= a · b","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-25c67413dcf997f2","tex":"x_{2m}","aliases":"x_2m","display":"x_{2m}","kind":"use","excerpt":"vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONTraditional paper: Section B.1 (p. 47) Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9b105e9fb57cd70a","tex":"x_{2m+1}","aliases":"x_2m+1","display":"x_{2m+1}","kind":"use","excerpt":"vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONTraditional paper: Section B.1 (p. 47) Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"5pt,] (9.17000,2.26000) rectangle ++(0.57000,0.57000); node[] at (9.45500,2.54500) 1; draw[fill=black!0,draw=black!45,line width=.35pt,] (9.74000,2.26000) rectangle ++(0.57000,0.57000); node[] at (10.02500,2.54500) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (10.31000,2.26000) rectangle ++(0.57000,0.57000); node[] at (10.59500,2.54500) 1","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3864fe3cacc7c90d","tex":"x_6","aliases":"x_6","display":"x_6","kind":"use","excerpt":"headAddress resolution A coefficient such as x_6 can sit in the even packet, then the residue-2 packet modulo 4, then the residue-6 packet modulo 8. Its value does not become more accurate. Its address becomes more specific, while the full object remains exactly reconstructible.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-72e05ef2b0f48e18","tex":"X_c","aliases":"X_c","display":"X_c","kind":"use","excerpt":"Let X_f and X_c be vector spaces or Banach spaces over a common field, and let C:X_f⟶ X_c be a linear map, and let T_f∈End(X_f), T_c∈End(X_c). Define Δ=CT_f-T_cC.","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f76f02d6c081bad5","tex":"X_f","aliases":"X_f","display":"X_f","kind":"use","excerpt":"Let X_f and X_c be vector spaces or Banach spaces over a common field, and let C:X_f⟶ X_c be a linear map, and let T_f∈End(X_f), T_c∈End(X_c). Define Δ=CT_f-T_cC.","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer bloc","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer bloc","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-06a2f75a3e24f887","tex":"x_{n-r}","aliases":"x_n-r","display":"x_{n-r}","kind":"use","excerpt":"A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-924805ec7439cec0","tex":"X_N^{\\rm pkt}","aliases":"X_N^rm pkt rm","display":"X_N^{\\rm pkt}","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-bcf7586068a6a313","tex":"x_\\beta","aliases":"x_β beta","display":"x_\\beta","kind":"use","excerpt":"Every coarse packet is also uniquely reconstructed from its fine children. If vectors x_β∈ Q_β X sum to a coarse packet x, applying Q_γ recovers x_γ=Q_γ x. This is the uniqueness clause of the canonical nested-refinement statement.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-953215dbee72ad06","tex":"x_\\gamma","aliases":"x_γ gamma","display":"x_\\gamma","kind":"use","excerpt":"Every coarse packet is also uniquely reconstructed from its fine children. If vectors x_β∈ Q_β X sum to a coarse packet x, applying Q_γ recovers x_γ=Q_γ x. This is the uniqueness clause of the canonical nested-refinement statement.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4726173f5ad19c77","tex":"x_{\\rho(\\beta)}","aliases":"x_ρ(β) rho beta","display":"x_{\\rho(\\beta)}","kind":"use","excerpt":"w QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-2b6c204b3855e438","tex":"X_\\ell","aliases":"X_ℓ ell","display":"X_\\ell","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-524e8e9a8948b743","tex":"X_{\\ell+1}","aliases":"X_ℓ+1 ell","display":"X_{\\ell+1}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"(6.70000,0.88000) 2; node[font=sffamilyscriptsize] at (6.70000,1.32000) 3; node[font=sffamilyscriptsize] at (6.70000,1.76000) 4; node[font=sffamilyscriptsize] at (6.70000,2.20000) 5; node[font=sffamilyscriptsize] at (6.70000,2.64000) 6; node[font=sffamilyscriptsize] at (6.70000,3.08000) 7; node[] at (5.10000,2.30000) x=Vy; node[] at (5.10000,1.3000","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"(6.70000,0.88000) 2; node[font=sffamilyscriptsize] at (6.70000,1.32000) 3; node[font=sffamilyscriptsize] at (6.70000,1.76000) 4; node[font=sffamilyscriptsize] at (6.70000,2.20000) 5; node[font=sffamilyscriptsize] at (6.70000,2.64000) 6; node[font=sffamilyscriptsize] at (6.70000,3.08000) 7; node[] at (5.10000,2.30000) x=Vy; node[] at (5.10000,1.3000","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L cho","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6811e23146d67e8a","tex":"y_\\alpha","aliases":"y_α alpha","display":"y_\\alpha","kind":"use","excerpt":"[Orthogonal Hilbert packets] If X is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product ⟨(x_α),(y_α)⟩=∑_α⟨ x_α,y_α⟩. Then analysis and synthesis are unitary. Every coordinate change in [source reference] is unitary.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-90bad8055cff4ee1","tex":"y_\\beta","aliases":"y_β beta","display":"y_\\beta","kind":"use","excerpt":"w QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z mathbb","display":"{\\mathbb Z}","kind":"use","excerpt":"A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.","count":9},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-af5f0843f8046d2c","tex":"\\mathbb Z^2","aliases":"Z^2 mathbb","display":"\\mathbb Z^2","kind":"use","excerpt":"0 · 2, det B=2; node[] at (6.45000,0.60000) Filled: B Z^2={(x,y):y is even}; node[] at (6.45000,-0.35000) Unfilled: the other lattice coset.; node[] at (6.45000,-1.25000) B^-1 is integral at odd primes, not at 2.;","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-10d0329c35407a66","tex":"z^r","aliases":"z^r","display":"z^r","kind":"use","excerpt":"000,0.70000); node[] at (0.35000,1.45000) 0; draw[fill=black!0,draw=black!45,line width=.35pt,] (0.70000,1.10000) rectangle ++(0.70000,0.70000); node[] at (1.05000,1.45000) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (1.40000,1.10000) rectangle ++(0.70000,0.70000); node[] at (1.75000,1.45000) 3; node[] at (3.70000,2.05000) × 5; node[] at","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e8394415f576ddd6","tex":"Z_0","aliases":"Z_0","display":"Z_0","kind":"use","excerpt":"Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this d","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d6e4bdbd9401f40b","tex":"Z_{0,1}","aliases":"Z_0,1","display":"Z_{0,1}","kind":"use","excerpt":"From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4151409cc12d59b1","tex":"Z_{0,2}","aliases":"Z_0,2","display":"Z_{0,2}","kind":"use","excerpt":"[A transfer series with one loop] Take vertices 0, 1, and 2. Give the edge 0→1 weight one, the loop 1→1 weight two, and the edge 1→2 weight one. Every path from 0 to 2 uses the first and last edges and may traverse the loop any number of times. Hence Z_0,2(t) = t^2(1+t^2+t^4+⋯) = fract^21-t^2. There is one path at every positive even weight and no ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-78a80f694d80e6e7","tex":"\\Z_2","aliases":"Z_2 Z mathbb","display":"{\\mathbb Z}_2","kind":"use","excerpt":"The failure of the character formula frac12(I± A) to be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished: integral address selector ≠ character-average spectral projector when the group order is not a unit. The first can remain exact over Z_2. ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-370c87df70534fa4","tex":"\\mathbb Z_2","aliases":"Z_2 mathbb","display":"\\mathbb Z_2","kind":"use","excerpt":"The displayed matrices are complementary idempotents. The scalar 2 is a unit in Z_p for odd p, so both matrices are integral there. Over Z_2, multiplying by 2 clears the denominators, while neither projector preserves the standard lattice. Thus both weights are one. The resultant of the two linear factors is their root difference, which is 2.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8fa45213091916f0","tex":"\\Z_2^2","aliases":"Z_2^2 Z mathbb","display":"{\\mathbb Z}_2^2","kind":"use","excerpt":"[Odd-adic versus dyadic behavior] For every prime p≠2, every entry of E_+ and E_- is p-integral, and each projector has exact local denominator weight zero. At p=2, each projector has exact local denominator weight one. Multiplication by 2 makes it integral, while the unscaled matrix is not integral: s_2, Z_2^2(E_+)=s_2, Z_2^2(E_-)=1. The character","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-659336bfb5f82bbc","tex":"\\mathbb Z_2^2","aliases":"Z_2^2 mathbb","display":"\\mathbb Z_2^2","kind":"use","excerpt":"[Strict growth under refinement] Take the standard lattice Z_2^2 and the involution from [source reference]. The one packet system (I) has weight zero. Refining it into the two rational sign packets produces weight one. The coarse lattice is already split because there is only one summand. The fine eigenspaces are glued together by one dyadic digit","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a0d8044d38df3073","tex":"\\Z_{>0}","aliases":"Z_>0 Z mathbb","display":"{\\mathbb Z}_{>0}","kind":"use","excerpt":"A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-74e0eb883b673576","tex":"z_k","aliases":"z_k","display":"z_k","kind":"use","excerpt":") circle (.075); node[] at (-1.38027,-1.38027) 5; fill (0.00000,-1.60000) circle (.075); node[] at (0.00000,-1.95200) 6; fill (1.13137,-1.13137) circle (.075); node[] at (1.38027,-1.38027) 7; draw[dashed] (11.28333,0) circle (2.46667); draw[black!45] (7.90000,0.00000) -- (14.21250,0.00000); draw[black!45] (8.20000,-2.85000) -- (8.20000,2.85000); no","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8723c894d5150ea2","tex":"\\mathbb Z_p","aliases":"Z_p mathbb","display":"\\mathbb Z_p","kind":"use","excerpt":"The displayed matrices are complementary idempotents. The scalar 2 is a unit in Z_p for odd p, so both matrices are integral there. Over Z_2, multiplying by 2 clears the denominators, while neither projector preserves the standard lattice. Thus both weights are one. The resultant of the two linear factors is their root difference, which is 2.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7f3b8fae797e61c1","tex":"\\mathbb Z_p^2","aliases":"Z_p^2 mathbb","display":"\\mathbb Z_p^2","kind":"use","excerpt":"0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ","display":null,"kind":"use","excerpt":"L · Precision depth in a quotient such as ZZ/2^LZZ. · L=5 gives arithmetic modulo 32.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-42f5a664e208866e","tex":"\\ZZ^2","aliases":"ZZ^2 ZZ","display":null,"kind":"use","excerpt":"vpage15Where the dyadic digit is lostMODEL D / THE EIGENLATTICE HAS INDEX TWOTraditional paper: Proposition 9.3 (p. 32) The eigenspaces are spanned by (1,0) and (-1,2). Their integral eigenvectors do not form an integral basis of ZZ^2. The change-of-basis determinant is two.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f2adfa4292f44f63","tex":"\\ZZ_{(p)}^2","aliases":"ZZ_(p)^2 ZZ","display":null,"kind":"use","excerpt":"0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"The failure of the character formula frac12(I± A) to be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished: integral address selector ≠ character-average spectral projector when the group order is not a unit. The first can remain exact over Z_2. ","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"For a specific attainable target (2,4), 2x≡2pmod8iff x∈{1,5}, 4y≡4pmod8iff y∈{1,3,5,7}. The coordinate choices are independent, giving 2·4=8 solutions. The entire carrier is partitioned consistently: 8 attainable targets × 8 inputs per target=64. For an inadmissible target, such as (1,4) or (2,2), the fiber is empty. The same nonzero cardinality fo","count":139},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,","count":23},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f1fb87c81b1a53c3","tex":"\\alpha'","aliases":"α' alpha","display":"\\alpha'","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ac909e48d2ea1c12","tex":"\\beta'","aliases":"β' beta","display":"\\beta'","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"Analytic functional calculus. [Holomorphic functional calculus defect] Let X_f and X_c be complex Banach spaces, and let C:X_f→ X_c, T_f, and T_c be bounded complex-linear maps. Let h be holomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number o","count":17},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8335fc4d7b413b19","tex":"\\Gamma_k","aliases":"Γ_k Gamma","display":"\\Gamma_k","kind":"use","excerpt":"AR PHASE ORBITTraditional paper: Section 12.3 (p. 42), Section 12.3, Smith-phase identity (p. 43) Smith normal form concerns linear congruences. The Smith impedance chart uses the nonlinear coordinate change Γ=fracz-1z+1, z=frac1+Γ1-Γ. They share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points Γ_k=t","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer bloc","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-884e94abba1033ea","tex":"\\delta_{n,r}","aliases":"δ_n,r delta","display":"\\delta_{n,r}","kind":"use","excerpt":"Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer bloc","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d33d8b646241d747","tex":"\\Delta_\\ell","aliases":"Δ_ℓ Delta ell","display":"\\Delta_\\ell","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"vpage22A small defect can grow under powersMODEL H / EXACT TELESCOPING, EXPLICIT GROWTHTraditional paper: Theorem 11.1 (p. 37), Corollary 11.3 (p. 38) Choose an invertible coordinate shear but independently keep the same numerical operator at both levels: C=1 · ε 0 · 1, T_f=T_c=diag(1,2), ε=frac116. Then Δ=CT_f-T_cC=ε E_01. Exact coordinate inverti","count":9},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,","count":8},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6ede142bbc0a6d7e","tex":"\\Lambda_0","aliases":"Λ_0 Lambda","display":"\\Lambda_0","kind":"use","excerpt":"[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Onl","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-26a2a17ac5cf2600","tex":"\\Lambda_1","aliases":"Λ_1 Lambda","display":"\\Lambda_1","kind":"use","excerpt":"[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ac2ec4913e4c3ca3","tex":"\\Lambda_2","aliases":"Λ_2 Lambda","display":"\\Lambda_2","kind":"use","excerpt":"[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7daa0edaa864bbe2","tex":"\\Lambda_p","aliases":"Λ_p Lambda","display":"\\Lambda_p","kind":"use","excerpt":"0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b4da8500e158d102","tex":"\\Lambda_{\\Pcal}","aliases":"Λ_Pcal Λ_ P Lambda Pcal Lambda mathcal","display":"\\Lambda_{{\\mathcal P}}","kind":"use","excerpt":"[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7781b9f2b77d297f","tex":"\\Lambda_{\\Qcal}","aliases":"Λ_Qcal Λ_ Q Lambda Qcal Lambda mathcal","display":"\\Lambda_{{\\mathcal Q}}","kind":"use","excerpt":"[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e08b44bc605b92e6","tex":"\\Lambda_s","aliases":"Λ_s Lambda","display":"\\Lambda_s","kind":"use","excerpt":"[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Onl","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a2ecbf2708f1e683","tex":"\\pi^{-s}","aliases":"π^-s pi","display":"\\pi^{-s}","kind":"use","excerpt":"[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Onl","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-bca6e7d02878c20b","tex":"\\pi^a","aliases":"π^a pi","display":"\\pi^a","kind":"use","excerpt":"[A triangular transfer] Let O be complete with uniformizer π, and consider T= 1 · π^a 0 · 1 with a≥1. The only nontrivial transfer has weight a, and its square is zero. Hence T^-1 = 1 · -π^a","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d5602fb671a74bd1","tex":"\\pi^{a_1}","aliases":"π^a_1 pi","display":"\\pi^{a_1}","kind":"use","excerpt":"[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a9449729f152b345","tex":"\\pi^{a_j}","aliases":"π^a_j pi","display":"\\pi^{a_j}","kind":"use","excerpt":"ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-57b9f29f6af58bd3","tex":"\\pi^{a_n}","aliases":"π^a_n pi","display":"\\pi^{a_n}","kind":"use","excerpt":"[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-12d8be8cdf767d11","tex":"\\pi^d","aliases":"π^d pi","display":"\\pi^d","kind":"use","excerpt":"Let s=s_Λ( P). For every x∈Λ and every α, π^sE_α x∈Λ_α. Hence π^sx=∑_απ^sE_α x∈Λ_split, so π^sG_Λ, P=0. This proves exp_π(G_Λ, P)≤ s. If d<s, then for some α there is an x∈Λ with π^dE_α x∉Λ_α. The class π^d[x] is therefore nonzero in the gluing module. Thus π^d does not annihilate the gluing module, and equality follows. The final formula is the us","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c9c12bb7f86fa0f0","tex":"\\pi^L","aliases":"π^L pi","display":"\\pi^L","kind":"use","excerpt":"If O is a complete discrete valuation ring with uniformizer π and B is a separated and complete O algebra for the filtration F^L B=π^L B, then this is a standard example. Finite matrix algebras over O satisfy these conditions. Completeness of O alone does not imply completeness of every O algebra. Basic facts about completeness in this setting are ","count":8},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0de50b515ae97983","tex":"\\pi^m","aliases":"π^m pi","display":"\\pi^m","kind":"use","excerpt":"Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.","count":8},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-724adebe365a3a3e","tex":"\\pi^{m+d}","aliases":"π^m+d pi","display":"\\pi^{m+d}","kind":"use","excerpt":"n delay] Let E be a rational projector on V, let Λ be a lattice, and let s=s_Λ(E). For every m≥0, the rule [x]_m+s⟼[Ex]_m from Λ/π^m+sΛ to V/π^mΛ is well defined. Here m is an integer. If d is an integer with 0≤ d<s, then for every integer m≥0 there are x,y∈Λ such that x≡ ypmodπ^m+dΛ but Exnot≡ Eypmodπ^mΛ. Thus s extra digits are both sufficient an","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ae5fc9105be497c3","tex":"\\pi^{m+s}","aliases":"π^m+s pi","display":"\\pi^{m+s}","kind":"use","excerpt":"[Projector precision delay] Let E be a rational projector with π^sE(Λ)⊆Λ. If x≡ ypmodπ^m+sΛ, then Ex≡ Eypmodπ^mΛ. Thus m+s input digits suffice to determine m output digits of the projected packet.","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f09d0f0a6806b7c1","tex":"\\pi^{\\min(s_i,L)}","aliases":"π^min(s_i,L) pi min","display":"\\pi^{\\min(s_i,L)}","kind":"use","excerpt":"[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ce46406e805e52d0","tex":"\\pi^{s_1}","aliases":"π^s_1 pi","display":"\\pi^{s_1}","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5a34afdce45841e7","tex":"\\pi^{s_i}","aliases":"π^s_i pi","display":"\\pi^{s_i}","kind":"use","excerpt":"Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L cho","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a901ac9ade8abb67","tex":"\\pi^{s_r}","aliases":"π^s_r pi","display":"\\pi^{s_r}","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-27f57e5aa2f64147","tex":"\\pi^{\\val_\\pi(\\Res(f,g))}","aliases":"π^val_π(Res(f,g)) pi val pi Res pi operatorname pi operatorname","display":"\\pi^{{\\operatorname{val}}_\\pi({\\operatorname{Res}}(f,g))}","kind":"use","excerpt":"[Resultant denominator bound] Under the hypotheses of [source reference], one may take r=Res(f,g) up to multiplication by a unit. Therefore π^val_π(Res(f,g))E_fΛ⊆Λ.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-df63b2d5a9b79da9","tex":"\\rho^{n-1}","aliases":"ρ^n-1 rho","display":"\\rho^{n-1}","kind":"use","excerpt":"[Norm bound] For a submultiplicative operator norm, CT_f^n-T_c^nC ≤ Δ ∑_k=0^n-1 T_c ^n-1-k T_f ^k. If T_f , T_c ≤ρ, then CT_f^n-T_c^nC ≤ nρ^n-1 Δ .","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8dd03dc639dfd669","tex":"\\rho_{m,\\ell}","aliases":"ρ_m,ℓ rho ell","display":"\\rho_{m,\\ell}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-98944c463486b854","tex":"\\rho_{n,m}","aliases":"ρ_n,m rho","display":"\\rho_{n,m}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-af3400ef85124560","tex":"\\rho_{n,\\ell}","aliases":"ρ_n,ℓ rho ell","display":"\\rho_{n,\\ell}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-cfe0654529d554f8","tex":"\\rho_{\\ell,\\ell}","aliases":"ρ_ℓ,ℓ rho ell ell","display":"\\rho_{\\ell,\\ell}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"The assertion is immediate if either space is zero. Otherwise, use the rectifiable-cycle form of the holomorphic functional calculus in [source reference]. For each operator T, compare Γ with a positive simple system around its spectrum. Their difference has zero winding outside U∖σ(T), so Theorem~4.1 applied to the resolvent-valued integrand ident","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"[Gluing module and exact separation weight] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0bbd4413777b414e","tex":"\\chi_A","aliases":"χ_A chi","display":"\\chi_A","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.","count":25},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8c661380e21aa156","tex":"\\omega^{-1}","aliases":"ω^-1 omega","display":"\\omega^{-1}","kind":"use","excerpt":"0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4a26976422bc1dc1","tex":"\\omega^{-jk}","aliases":"ω^-jk omega","display":"\\omega^{-jk}","kind":"use","excerpt":"The field specialization. [Character realization] Let bbk be a field, assume that N is nonzero in bbk, and suppose that bbk contains a primitive Nth root of unity ω. Define (UF)(z)=F(ω z). Then P_N,j = frac1N∑_k=0^N-1ω^-jkU^k.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f04448ecbf148319","tex":"\\omega^3","aliases":"ω^3 omega","display":"\\omega^3","kind":"use","excerpt":".35pt,] (5.00000,1.10000) rectangle ++(0.70000,0.70000); node[] at (5.35000,1.45000) 0; draw[fill=black!0,draw=black!45,line width=.35pt,] (5.70000,1.10000) rectangle ++(0.70000,0.70000); node[] at (6.05000,1.45000) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (6.40000,1.10000) rectangle ++(0.70000,0.70000); node[] at (6.75000,1.45000) 1;","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5228e7d3ffe9f8b9","tex":"\\omega^{k(n-j)}","aliases":"ω^k(n-j) omega","display":"\\omega^{k(n-j)}","kind":"use","excerpt":"On z^n, the right side acts by the scalar frac1N∑_k=0^N-1ω^k(n-j). This is one when n≡ jpmod N and zero otherwise by the geometric sum identity in the field bbk.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c32ce654fa1f4d68","tex":"\\omega^N","aliases":"ω^N omega","display":"\\omega^N","kind":"use","excerpt":"Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-028fd4755c9afd35","tex":"\\omega^{nk}","aliases":"ω^nk omega","display":"\\omega^{nk}","kind":"use","excerpt":"On the monomial z^n, the right-hand side acts by η∑_k=0^N-1(ω^vee)^jkω^nk. The identity ω^N=1 reduces this expression to the table entry indexed by r=nbmod N. The finite certificate therefore gives exactly the coefficient mask defining P_N,j. Over a field, the primitive-root geometric sum supplies the certificate.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-221d9113d4c5884d","tex":"\\omega^{rk}","aliases":"ω^rk omega","display":"\\omega^{rk}","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate hold","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0892bf887fb05f60","tex":"\\omega^\\vee","aliases":"ω^vee omega vee","display":"\\omega^\\vee","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-22344ff7669bd2c8","tex":"\\omega_N^k","aliases":"ω_N^k omega","display":"\\omega_N^k","kind":"use","excerpt":"Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this d","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"headThree boundaries of this calculation This is a formal series identity. No analytic convergence at a chosen real or complex value of t is needed. The canonical determinant need not be the minimal denominator after cancellation. Finally, positive weights prevent infinitely many walks at a fixed weight: a single weight-zero loop would already give","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this d","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"The proof order is: · complete packet axioms ⇒finite-packet uniqueness and coordinate coherence · ⇒coordinate naturality and refined transfer blocks, · residue selectors and a finite Fourier table ⇒Fourier formulas, · coprime moduli⇒dyadic--odd formulas, · quotient, filtration, and edge-weight certificates ⇒explicit pruned inverse, · finite positiv","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fibre has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"Let s=s_Λ( P). For every x∈Λ and every α, π^sE_α x∈Λ_α. Hence π^sx=∑_απ^sE_α x∈Λ_split, so π^sG_Λ, P=0. This proves exp_π(G_Λ, P)≤ s. If d<s, then for some α there is an x∈Λ with π^dE_α x∉Λ_α. The class π^d[x] is therefore nonzero in the gluing module. Thus π^d does not annihilate the gluing module, and equality follows. The final formula is the us","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a4a402d8d3eb9074","tex":"\\prod_{i=1}^r","aliases":"∏_i=1^r prod","display":"\\prod_{i=1}^r","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5e4c84160d4d61eb","tex":"\\prod_{j\\ne i}","aliases":"∏_j≠ i prod ne","display":"\\prod_{j\\ne i}","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ec1616068a6ee172","tex":"\\sum_e","aliases":"∑_e sum","display":"\\sum_e","kind":"use","excerpt":"W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is a two-sided inverse of q_L(I+A), where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then th","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-19360c7481ca29a9","tex":"\\sum_{e:\\alpha\\to\\beta}","aliases":"∑_e:α→β sum alpha to beta","display":"\\sum_{e:\\alpha\\to\\beta}","kind":"use","excerpt":"ra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6d1546afc3ef11e5","tex":"\\sum_{e\\in E}","aliases":"∑_e∈ E sum in","display":"\\sum_{e\\in E}","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b7a8cfbba485d0c4","tex":"\\sum_{i=1}^d","aliases":"∑_i=1^d sum","display":"\\sum_{i=1}^d","kind":"use","excerpt":"ongruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordi","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6182c3bad2d01e16","tex":"\\sum_{i=1}^r","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^r","kind":"use","excerpt":"rdinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-586bc3571b381e64","tex":"\\sum_{j=0}^{N-1}","aliases":"∑_j=0^N-1 sum","display":"\\sum_{j=0}^{N-1}","kind":"use","excerpt":"[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8650842fbc4c4680","tex":"\\sum_{j=1}^r","aliases":"∑_j=1^r sum","display":"\\sum_{j=1}^r","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":3},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f1cf2b9d42f6eee5","tex":"\\sum_{k=0}^2","aliases":"∑_k=0^2 sum","display":"\\sum_{k=0}^2","kind":"use","excerpt":"On the monomial z^n, define UF(z)=F(2z). The proposed projector acts by 5∑_k=0^2 4^jk2^nk. The identity 2^3=1 makes this depend only on nbmod3. The nine-entry table therefore establishes the formula for every exponent once periodicity is used, rather than requiring a separate test at every coefficient.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-41e0c211ba80d17c","tex":"\\sum_{k=0}^{n}","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^{n}","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7796ee229bc8c35a","tex":"\\sum_{k=0}^{N-1}","aliases":"∑_k=0^N-1 sum","display":"\\sum_{k=0}^{N-1}","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate hold","count":7},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-fec10602e4575bf9","tex":"\\sum_{k=0}^{n-1}","aliases":"∑_k=0^n-1 sum","display":"\\sum_{k=0}^{n-1}","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":5},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-55391d8e2ecb990e","tex":"\\sum_{k=0}^{r-1}","aliases":"∑_k=0^r-1 sum","display":"\\sum_{k=0}^{r-1}","kind":"use","excerpt":"[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1ebed7c5a175f44d","tex":"\\sum_{k\\ge0}","aliases":"∑_k≥0 sum ge","display":"\\sum_{k\\ge0}","kind":"use","excerpt":"[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-070f622d9a6f0c69","tex":"\\sum_{n=0}^N","aliases":"∑_n=0^N sum","display":"\\sum_{n=0}^N","kind":"use","excerpt":"[Polynomial defect] For p(z)=∑_n=0^Na_nz^n, we have Cp(T_f)-p(T_c)C = ∑_n=1^Na_n ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0fbc71e3cef7af4e","tex":"\\sum_{n=0}^{N-1}","aliases":"∑_n=0^N-1 sum","display":"\\sum_{n=0}^{N-1}","kind":"use","excerpt":"et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a3ec24382fc8477a","tex":"\\sum_{n=1}^N","aliases":"∑_n=1^N sum","display":"\\sum_{n=1}^N","kind":"use","excerpt":"[Polynomial defect] For p(z)=∑_n=0^Na_nz^n, we have Cp(T_f)-p(T_c)C = ∑_n=1^Na_n ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"use","excerpt":"For fixed start and target vertices s,t, define Z_s,t(t) = ∑_n≥0a_n(s,t)t^n, where a_n(s,t) counts directed walks from s to t with total weight n. Vertices and edges may repeat. Thus ``path zeta'' is retained as the name of the series and Lean API, but the walks are not required to visit each vertex only once.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-912a5316222f1080","tex":"\\sum_{\\rm children}","aliases":"∑_rm children sum rm","display":"\\sum_{\\rm children}","kind":"use","excerpt":"=.35pt,] (4.57000,2.26000) rectangle ++(0.57000,0.57000); node[] at (4.85500,2.54500) 2; draw[fill=black!0,draw=black!45,line width=.35pt,] (5.14000,2.26000) rectangle ++(0.57000,0.57000); node[] at (5.42500,2.54500) 0; draw[fill=black!12,draw=black!45,line width=.35pt,] (5.71000,2.26000) rectangle ++(0.57000,0.57000); node[] at (5.99500,2.54500) 4","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-6b4cbee70b31b1b1","tex":"\\sum_{r\\geq0}","aliases":"∑_r≥0 sum geq","display":"\\sum_{r\\geq0}","kind":"use","excerpt":"Put A=∑_e∈ EA_e. Since left multiplication by D^-1 preserves the filtration, A∈ F^1 B. Completeness gives the convergent Neumann series (I+A)^-1=∑_r≥0(-A)^r. Because T=D(I+A), T^-1=(I+A)^-1D^-1. Expanding each power of A gives the sum over words, with the rightmost edge acting first. Multiplicativity of the filtration gives A_γ∈ F^W(γ) B. For each ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-3554bc0644c628c2","tex":"\\sum_{\\substack{0\\leq s<M\\\\s\\equiv j\\pmod N}}","aliases":"∑_substack0≤ s<M s≡ jpmod N sum substack leq s equiv pmod","display":null,"kind":"use","excerpt":"[Nested residue refinement] Let N,M be positive integers with N dividing M, and let 0≤ j<N. Then P_N,j = ∑_substack0≤ s<M s≡ jpmod NP_M,s. When R is nonzero, the packet system modulo M refines the packet system modulo N.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a6c7f694cb6268c7","tex":"\\sum_{\\substack{e:\\,\\mathrm{source}(e)=s\\\\w(e)\\le n}}","aliases":"∑_substacke: source(e)=s w(e)≤ n sum substack mathrm w le","display":null,"kind":"use","excerpt":"headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumerat","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-4dc3a2d6215c09ba","tex":"\\sum_{\\substack{e:\\,\\operatorname{source}(e)=s\\\\w(e)\\le n}}","aliases":"∑_substacke: source(e)=s w(e)≤ n sum substack operatorname w le","display":null,"kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-d8aed7d286bba088","tex":"\\sum_{\\substack{\\gamma:\\ P_\\alpha R_\\gamma=R_\\gamma\\\\Q_\\beta R_\\gamma=R_\\gamma}}","aliases":"∑_substackγ: P_α R_γ=R_γ Q_β R_γ=R_γ ∑_substackγ: P_α R_γ=R_γ{ Q_β R_γ=R_γ sum substack gamma alpha gamma gamma Q beta gamma gamma sum substack gamma alpha gamma gamma mathbb beta gamma gamma","display":"\\sum_{\\substack{\\gamma:\\ P_\\alpha R_\\gamma=R_\\gamma\\{\\mathbb Q}_\\beta R_\\gamma=R_\\gamma}}","kind":"use","excerpt":"a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ{ Q_β R_γ=R_γR_γ. Thus the packets with the same p","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c4bd3d34873e4945","tex":"\\sum_{\\substack{\\gamma\\in E^n\\\\W(\\gamma)<L}}","aliases":"∑_substackγ∈ E^n W(γ)<L sum substack gamma in W gamma","display":null,"kind":"use","excerpt":"et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-649ddfbfdba19725","tex":"\\sum_{\\substack{\\rho(\\beta)=b\\\\\\rho(\\alpha)=a}}","aliases":"∑_substackρ(β)=b ρ(α)=a sum substack rho beta rho alpha","display":"\\sum_{\\substack{\\rho(\\beta)=b\\\\\\rho(\\alpha)=a}}","kind":"use","excerpt":"[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-364ec86993472593","tex":"\\sum_{W(\\gamma)<L}","aliases":"∑_W(γ)<L sum gamma","display":"\\sum_{W(\\gamma)<L}","kind":"use","excerpt":"[Exact finite precision truncation] Under the hypotheses of [source reference], define R_L = ∑_W(γ)<L(-1)^|γ|A_γ D^-1. Then T^-1-R_L∈ F^L B, TR_L-I∈ F^L B, R_LT-I∈ F^L B. Thus R_L is the exact inverse in B/F^L B.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-dd5a8bdc2b931502","tex":"\\sum_\\beta","aliases":"∑_β sum beta","display":"\\sum_\\beta","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-23c33e8c6e67f2c6","tex":"\\sum_{\\beta\\succ b,\\,\\alpha\\succ a}","aliases":"∑_βsucc b, αsucc a sum beta succ alpha succ","display":"\\sum_{\\beta\\succ b,\\,\\alpha\\succ a}","kind":"use","excerpt":"headRefined blocks still reconstruct the same map A coarse block expands as P_bTP_a=∑_βsucc b, αsucc aP_β TP_α. The terms are maps embedded in the original source and target spaces. Adding their norms, or adding their matrix entries after forgetting the embeddings, is not this operator identity.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-9d12cb53c45f8ea0","tex":"\\sum_\\gamma","aliases":"∑_γ sum gamma","display":"\\sum_\\gamma","kind":"use","excerpt":"Let R_γ be a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-435c95867c65d311","tex":"\\sum_{\\gamma:\\alpha\\leadsto\\beta}","aliases":"∑_γ:αleadstoβ sum gamma alpha leadsto beta","display":null,"kind":"use","excerpt":"pha DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For an edge e:α→β, put A_e=(P_β DP_β)^-1N_e. Then the block from P_α X to P_β X of (D+N)^-1 is P_β(D+N)^-1P_α = ∑_γ:αleadstoβ (-1)^|γ|A_γ D_α^-1, where D_α=P_α","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-816bc4c4be83a776","tex":"\\sum_{\\rho(\\alpha)=a}","aliases":"∑_ρ(α)=a sum rho alpha","display":"\\sum_{\\rho(\\alpha)=a}","kind":"use","excerpt":"Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ad18eea2ec25001f","tex":"\\sum_{\\rho(\\beta)=b}","aliases":"∑_ρ(β)=b sum rho beta","display":"\\sum_{\\rho(\\beta)=b}","kind":"use","excerpt":"Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-8e5722710bcae566","tex":"\\sum_{\\rho(\\gamma)=\\alpha}","aliases":"∑_ρ(γ)=α sum rho gamma alpha","display":"\\sum_{\\rho(\\gamma)=\\alpha}","kind":"use","excerpt":"Multiply the identity P_α=∑_ρ(γ)=αQ_γ by Q_β on either side. Pairwise annihilation in the fine family leaves Q_β when ρ(β)=α and leaves zero otherwise. For transitivity, substitute the child expansion of each Q_β into the expansion of P_α and group the resulting terms by the composite parent.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-43e727cf5aadc16c","tex":"\\sum_{\\ell=0}^{L-1}","aliases":"∑_ℓ=0^L-1 sum ell","display":"\\sum_{\\ell=0}^{L-1}","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.","count":6},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"X_c be complex Banach spaces, and let C:X_f→ X_c, T_f, and T_c be bounded complex-linear maps. Let h be holomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts mul","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a2972272b24e9c45","tex":"\\int_0^t","aliases":"∫_0^t int","display":"\\int_0^t","kind":"use","excerpt":"[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t ","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-67e4672bfa77240e","tex":"\\int_\\Gamma","aliases":"∫_Γ int Gamma","display":"\\int_\\Gamma","kind":"use","excerpt":"X_c be complex Banach spaces, and let C:X_f→ X_c, T_f, and T_c be bounded complex-linear maps. Let h be holomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts mul","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":4},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"The failure of the character formula frac12(I± A) to be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished: integral address selector ≠ character-average spectral projector when the group order is not a unit. The first can remain exact over Z_2. ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":9},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-ce7f3904634c8e1a","tex":"\\bigoplus_j","aliases":"⊕_j bigoplus","display":"\\bigoplus_j","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The apparently larger list of fine packets simply has smaller individual packet spaces.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-5a2986af923c3a3a","tex":"\\bigoplus_{j=1}^n","aliases":"⊕_j=1^n bigoplus","display":"\\bigoplus_{j=1}^n","kind":"use","excerpt":"[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-c7da9c0ad0ae9569","tex":"\\bigoplus_{j=1}^r","aliases":"⊕_j=1^r bigoplus","display":"\\bigoplus_{j=1}^r","kind":"use","excerpt":"ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-b1d323816298cbee","tex":"\\bigoplus_{\\alpha}","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_{\\alpha}","kind":"use","excerpt":"Λ be an O lattice in V. A rational projector E need not preserve Λ. Its separation weight is the least s such that π^sEΛ⊆Λ. A Bézout construction gives an upper bound through a polynomial resultant. That bound need not be exact because it does not see the position of the lattice. The exact answer is controlled by the finite length module G_Λ, P = Λ","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-1a835b644d2cadf6","tex":"\\bigoplus_\\beta","aliases":"⊕_β bigoplus beta","display":"\\bigoplus_\\beta","kind":"use","excerpt":"[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","excerpt":"Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).","count":2},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-eefbbc5154bea1da","tex":"\\otimes_\\OO","aliases":"⊗_OO ⊗_ O otimes OO otimes mathcal","display":"\\otimes_{{\\mathcal O}}","kind":"use","excerpt":"[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-f0ee3d51940d2d69","tex":"\\otimes_{\\OO}","aliases":"⊗_OO ⊗_ O otimes OO otimes mathcal","display":"\\otimes_{{{\\mathcal O}}}","kind":"use","excerpt":"Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"[Orthogonal Hilbert packets] If X is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product ⟨(x_α),(y_α)⟩=∑_α⟨ x_α,y_α⟩. Then analysis and synthesis are unitary. Every coordinate change in [source reference] is unitary.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"[Orthogonal Hilbert packets] If X is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product ⟨(x_α),(y_α)⟩=∑_α⟨ x_α,y_α⟩. Then analysis and synthesis are unitary. Every coordinate change in [source reference] is unitary.","count":1},{"work":"paper-ii","url":"notation/paper-ii.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[Gluing module and exact separation weight] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f839245649ce2101","tex":"\\operatorname{inputs}","aliases":"inputs operatorname","display":"\\operatorname{inputs}","kind":"notation-table","excerpt":"t,inputs(t) · A finite operation tree and its syntactically used input labels. · pagerefV20","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"notation-table","excerpt":"t,inputs(t) · A finite operation tree and its syntactically used input labels. · pagerefV20","count":80},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"notation-table","excerpt":"r(x,p) · A residual at packet p for the named candidate x. · pagerefV19","count":223},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"definition-passage","excerpt":"[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.","count":17},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":151},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komo","count":597},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":50},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,","count":185},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"definition-passage","excerpt":"[Divided square] A divided square on degree one elements is a map q satisfying q(x+y)=q(x)+q(y)+[x,y], q(ax)=a^2q(x), whenever the expressions are defined.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-99155510890729d1","tex":"A^r","aliases":"A^r","display":"A^r","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":10},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-757f361b88f183a1","tex":"A_{g_1}","aliases":"A_g_1","display":"A_{g_1}","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-93d7d5611cfc11e3","tex":"A_{g_1+\\cdots+g_r}","aliases":"A_g_1+⋯+g_r cdots","display":"A_{g_1+\\cdots+g_r}","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d095bb02256e34a5","tex":"A_{g_r}","aliases":"A_g_r","display":"A_{g_r}","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a2cfe3ae1e3afede","tex":"a_{n-1}","aliases":"a_n-1","display":"a_{n-1}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0230e8bde40956b0","tex":"\\Acal","aliases":"Acal A mathcal","display":"{\\mathcal A}","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":17},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c783192e1874491b","tex":"\\Acal_{\\Qcal}","aliases":"Acal_Qcal A_ Q Acal Qcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1fc76a250d22a8eb","tex":"\\operatorname{Aut}","aliases":"Aut operatorname","display":"\\operatorname{Aut}","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-82c6e4fc85ad5450","tex":"\\operatorname{Aut}_R","aliases":"Aut_R operatorname","display":"\\operatorname{Aut}_R","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":59},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,","count":130},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4f4734d9f363e46b","tex":"\\operatorname{binary}","aliases":"binary operatorname","display":"\\operatorname{binary}","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8e3c0fce1b7a60ae","tex":"\\bullet","aliases":"bullet","display":"\\bullet","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":90},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":135},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6bfba5cbf12e1727","tex":"C^\\bullet","aliases":"C^bullet bullet","display":"C^\\bullet","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":11},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-70f1ea36ff493278","tex":"C^{n+1}","aliases":"C^n+1","display":"C^{n+1}","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":17},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ce865f9aea2d3d86","tex":"C_{a_0}","aliases":"C_a_0","display":"C_{a_0}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6db54b248f3844fc","tex":"C_b","aliases":"C_b","display":"C_b","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6f3853f42b762bab","tex":"C_c","aliases":"C_c","display":"C_c","kind":"definition-passage","excerpt":"[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8195c86d8aac64cd","tex":"C_X","aliases":"C_X","display":"C_X","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-92e4993fd113c6f6","tex":"C_Y","aliases":"C_Y","display":"C_Y","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-82497562659dda68","tex":"\\cA","aliases":"cA A mathcal","display":"{\\mathcal A}","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":19},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b4e5cf02187f03e5","tex":"\\cA_b","aliases":"cA_b A_b cA mathcal","display":"{\\mathcal A}_b","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-073a71f814bdfa5f","tex":"\\coker","aliases":"coker operatorname","display":"{\\operatorname{coker}}","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":26},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8d94488f5df4843b","tex":"\\Cone","aliases":"Cone operatorname","display":"{\\operatorname{Cone}}","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8d2a6783a92b4bc0","tex":"\\operatorname{constant}","aliases":"constant operatorname","display":"\\operatorname{constant}","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e63707f1d2e3c7ef","tex":"\\Coord","aliases":"Coord operatorname","display":"{\\operatorname{Coord}}","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e35d2ea5ace79132","tex":"\\cP","aliases":"cP P mathcal","display":"{\\mathcal P}","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-29b3c8e7482cdeab","tex":"\\cP^a","aliases":"cP^a P^a cP mathcal","display":"{\\mathcal P}^a","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-88eb01a9511f791c","tex":"\\cP^b","aliases":"cP^b P^b cP mathcal","display":"{\\mathcal P}^b","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-20d41ee7317fddd1","tex":"\\cS","aliases":"cS S mathcal","display":"{\\mathcal S}","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":18},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-99e69c96f2e59e3a","tex":"\\cS_a","aliases":"cS_a S_a cS mathcal","display":"{\\mathcal S}_a","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-adbc3dd7c43b1b72","tex":"\\cW","aliases":"cW W mathcal","display":"{\\mathcal W}","kind":"definition-passage","excerpt":"[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d7001ee534716827","tex":"\\cW_0","aliases":"cW_0 W_0 cW mathcal","display":"{\\mathcal W}_0","kind":"definition-passage","excerpt":"[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9d96305e76870213","tex":"\\cW_n","aliases":"cW_n W_n cW mathcal","display":"{\\mathcal W}_n","kind":"definition-passage","excerpt":"[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-30c7c25e421d29ab","tex":"\\cW_{n+1}","aliases":"cW_n+1 W_n+1 cW mathcal","display":"{\\mathcal W}_{n+1}","kind":"definition-passage","excerpt":"[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":151},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":41},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-09e71d50145ecad3","tex":"D^r","aliases":"D^r","display":"D^r","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":10},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5c308718fcbd3df8","tex":"d_C","aliases":"d_C","display":"d_C","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-eb69ef05a69fcd3e","tex":"d_{\\Cone}","aliases":"d_Cone Cone operatorname","display":"d_{{\\operatorname{Cone}}}","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a3008bb1b6de0e7c","tex":"d_E","aliases":"d_E","display":"d_E","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":45},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":99},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f06630fc4bbd48b8","tex":"E^\\bullet","aliases":"E^bullet bullet","display":"E^\\bullet","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e40ba00de6234ae1","tex":"E^n","aliases":"E^n","display":"E^n","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":28},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":17},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7b243e274b6d1702","tex":"e_{r+1}","aliases":"e_r+1","display":"e_{r+1}","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c8f21276e6dfefea","tex":"\\ev","aliases":"ev operatorname","display":"{\\operatorname{ev}}","kind":"definition-passage","excerpt":"[Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.Colo","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":174},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":103},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9b8a4bc27627081b","tex":"F^m","aliases":"F^m","display":"F^m","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":32},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6dc70eae04871a40","tex":"F^{m+1}","aliases":"F^m+1","display":"F^{m+1}","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":15},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-92cbe7d6d84c4e19","tex":"F_c","aliases":"F_c","display":"F_c","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":13},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6f98b1eca815e46c","tex":"F_f","aliases":"F_f","display":"F_f","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":13},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4d2d5067e38c8f8c","tex":"F_{\\Qcal\\leftarrow\\Pcal}","aliases":"F_Qcal←Pcal F_ Q← P Qcal leftarrow Pcal mathcal leftarrow mathcal","display":"F_{{\\mathcal Q}\\leftarrow{\\mathcal P}}","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-280dad33f94a39d1","tex":"\\operatorname{Fin}","aliases":"Fin operatorname","display":"\\operatorname{Fin}","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge","count":68},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":26},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-95623e0cac2b3755","tex":"G_a^{-1}","aliases":"G_a^-1","display":"G_a^{-1}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge","count":11},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-76b53653bff85b9e","tex":"G_b","aliases":"G_b","display":"G_b","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-87a3fa4dd69238cf","tex":"g_r","aliases":"g_r","display":"g_r","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9b14e28ca0ef61ba","tex":"G_u^{-1}","aliases":"G_u^-1","display":"G_u^{-1}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8bea312cd06acfdb","tex":"G_v","aliases":"G_v","display":"G_v","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0fcc70536b44cfb8","tex":"\\operatorname{GL}","aliases":"GL operatorname","display":"\\operatorname{GL}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"[Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction","count":72},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-99e2417a50167bf6","tex":"H_{c,b,a}","aliases":"H_c,b,a","display":"H_{c,b,a}","kind":"definition-passage","excerpt":"[Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom operatorname","display":"{\\operatorname{Hom}}","kind":"definition-passage","excerpt":"[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e8a30d8cb6470d6d","tex":"\\Hom_R","aliases":"Hom_R Hom operatorname","display":"{\\operatorname{Hom}}_R","kind":"definition-passage","excerpt":"[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":86},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":64},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id mathrm","display":"{\\mathrm{id}}","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":26},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6c0a8caa7db598ba","tex":"\\id_{C_a}","aliases":"id_C_a id mathrm","display":"{\\mathrm{id}}_{C_a}","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f04fed0295873efc","tex":"\\id_{C_{a_0}}","aliases":"id_C_a_0 id mathrm","display":"{\\mathrm{id}}_{C_{a_0}}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1ce608af84df2a4c","tex":"\\id_M","aliases":"id_M id mathrm","display":"{\\mathrm{id}}_M","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":18},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-16e6e5c52bd14fa0","tex":"\\operatorname{in}","aliases":"in operatorname","display":"\\operatorname{in}","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-61dd0726b69a3f36","tex":"\\operatorname{in}_\\sigma","aliases":"in_σ operatorname sigma","display":"\\operatorname{in}_\\sigma","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4de08132ed3bb27b","tex":"\\operatorname{input}","aliases":"input operatorname","display":"\\operatorname{input}","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-35f44b16b4c77db8","tex":"\\operatorname{IsSolution}","aliases":"IsSolution operatorname","display":"\\operatorname{IsSolution}","kind":"definition-passage","excerpt":"[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPa","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-accca2f8c7d6bf65","tex":"\\operatorname{IsSolution}_r","aliases":"IsSolution_r operatorname","display":"\\operatorname{IsSolution}_r","kind":"definition-passage","excerpt":"[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPa","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":42},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7d6110995dfdfd76","tex":"J^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}","aliases":"J^r_γmidα_1,…,α_r gamma mid alpha ldots alpha","display":null,"kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-98339ce53ef8771d","tex":"J^X_{b\\leftarrow a}","aliases":"J^X_b← a leftarrow","display":"J^X_{b\\leftarrow a}","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1a670656747a1061","tex":"J_{b\\leftarrow a}","aliases":"J_b← a leftarrow","display":"J_{b\\leftarrow a}","kind":"definition-passage","excerpt":"[Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. s","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t)","count":31},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":41},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":55},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1f574898fe03737a","tex":"L_e","aliases":"L_e","display":"L_e","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":24},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-97675c4573f573e3","tex":"L_{e_1}","aliases":"L_e_1","display":"L_{e_1}","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-817d2f3702aba686","tex":"L_{e_n}","aliases":"L_e_n","display":"L_{e_n}","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d8ade83d2dcc64e3","tex":"L_w","aliases":"L_w","display":"L_w","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-109fedbff9e0f183","tex":"L_{\\varnothing}","aliases":"L_∅ varnothing","display":"L_{\\varnothing}","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"definition-passage","excerpt":"[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPa","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":16},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":123},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":10},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N mathbb","display":"{\\mathbb N}","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":96},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-79992dda48a6b986","tex":"\\operatorname{node}","aliases":"node operatorname","display":"\\operatorname{node}","kind":"definition-passage","excerpt":"[Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.Colo","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob operatorname","display":"{\\operatorname{Ob}}","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":43},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5e77bbcefac3aa10","tex":"\\Ob^{\\mu\\nu}_{c,b,a}","aliases":"Ob^μν_c,b,a Ob mu nu operatorname mu nu","display":"{\\operatorname{Ob}}^{\\mu\\nu}_{c,b,a}","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-08aeff18e6189f5c","tex":"\\Ob_{c,b,a}","aliases":"Ob_c,b,a Ob operatorname","display":"{\\operatorname{Ob}}_{c,b,a}","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-46c1284bb2e365e3","tex":"\\Ob_D","aliases":"Ob_D Ob operatorname","display":"{\\operatorname{Ob}}_D","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a85f28cb80e0a27b","tex":"\\Ob_m","aliases":"Ob_m Ob operatorname","display":"{\\operatorname{Ob}}_m","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3323ae2114d44586","tex":"\\operatorname{out}","aliases":"out operatorname","display":"\\operatorname{out}","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e4b45cd66b9c8f4e","tex":"\\overline L","aliases":"overline L overline","display":"\\overline L","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8000240f3988a998","tex":"\\overline L_m","aliases":"overline L_m overline","display":"\\overline L_m","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.","count":95},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":32},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"definition-passage","excerpt":"[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.","count":34},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5782f4f00d6630c2","tex":"\\operatorname{packetMatrix}","aliases":"packetMatrix operatorname","display":"\\operatorname{packetMatrix}","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komo","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-813431acda8054c4","tex":"\\Pcal","aliases":"Pcal P mathcal","display":"{\\mathcal P}","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":38},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":54},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":38},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ea3d9f514fcaceb6","tex":"Q_\\alpha","aliases":"Q_α alpha","display":"Q_\\alpha","kind":"definition-passage","excerpt":"[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.","count":16},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5a423e1823849ed7","tex":"Q_\\gamma","aliases":"Q_γ gamma","display":"Q_\\gamma","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aaf174f6701cc732","tex":"\\Qcal","aliases":"Qcal Q mathcal","display":"{\\mathcal Q}","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":23},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,","count":71},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,","count":138},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.","count":24},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b764f1cbd00009f2","tex":"\\Scal","aliases":"Scal S mathcal","display":"{\\mathcal S}","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":14},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3206060e514ec62e","tex":"\\Scal_{\\Pcal}","aliases":"Scal_Pcal S_ P Scal Pcal mathcal mathcal","display":"{\\mathcal S}_{{\\mathcal P}}","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-259ced97b3ec7ffb","tex":"\\src","aliases":"src operatorname","display":"{\\operatorname{src}}","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":19},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":51},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f3a6488b52c152f1","tex":"T'_{vu}","aliases":"T'_vu","display":"T'_{vu}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5d08324acb961fcf","tex":"t_2","aliases":"t_2","display":"t_2","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-541d33cb655a649a","tex":"t_{r(\\sigma)}","aliases":"t_r(σ) sigma","display":"t_{r(\\sigma)}","kind":"definition-passage","excerpt":"[Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.Colo","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-014a85fddb5fe786","tex":"T_{vu}","aliases":"T_vu","display":"T_{vu}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b74a640d5a7a8150","tex":"\\tgt","aliases":"tgt operatorname","display":"{\\operatorname{tgt}}","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":19},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t)","count":65},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":106},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d7884e8dc4c0191e","tex":"\\Ucal","aliases":"Ucal U mathcal","display":"{\\mathcal U}","kind":"definition-passage","excerpt":"[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-24e7f443fb5fea87","tex":"\\operatorname{unary}","aliases":"unary operatorname","display":"\\operatorname{unary}","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the released formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,","count":73},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"eldef:signature A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe int","count":78},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":25},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":128},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cc360874bad2510d","tex":"X_{\\operatorname{in}_\\sigma(q)}","aliases":"X_in_σ(q) operatorname sigma","display":"X_{\\operatorname{in}_\\sigma(q)}","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":32},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c3f245481574d88d","tex":"X_{\\operatorname{out}(\\sigma)}","aliases":"X_out(σ) operatorname sigma","display":"X_{\\operatorname{out}(\\sigma)}","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-08a6f9f108660d1a","tex":"X_v","aliases":"X_v","display":"X_v","kind":"definition-passage","excerpt":"eldef:signature A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe int","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-54ac43616fbfd36c","tex":"x_\\alpha","aliases":"x_α alpha","display":"x_\\alpha","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-eeab853c3f10dfb2","tex":"X_{\\alpha_1}","aliases":"X_α_1 alpha","display":"X_{\\alpha_1}","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2c9c5408fed893c7","tex":"X_{\\alpha_r}","aliases":"X_α_r alpha","display":"X_{\\alpha_r}","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":11},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"· midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval","count":87},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komo","count":43},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f42e259364006691","tex":"Y_\\gamma","aliases":"Y_γ gamma","display":"Y_\\gamma","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":15},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a283f3826c3c4719","tex":"|_{X_{\\alpha_1}\\times\\cdots\\times X_{\\alpha_r}}","aliases":"|_X_α_1×⋯× X_α_r alpha times cdots times alpha","display":"|_{X_{\\alpha_1}\\times\\cdots\\times X_{\\alpha_r}}","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":30},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":81},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":11},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-af15262773600a7f","tex":"\\alpha_r","aliases":"α_r alpha","display":"\\alpha_r","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":64},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":32},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cdad47e19a781b2f","tex":"\\Gamma^G_{b\\leftarrow a}","aliases":"Γ^G_b← a Gamma leftarrow","display":"\\Gamma^G_{b\\leftarrow a}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2174981cd778e96c","tex":"\\Gamma_{(a_0)}","aliases":"Γ_(a_0) Gamma","display":"\\Gamma_{(a_0)}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-14b9f6147ed3eea3","tex":"\\Gamma_{a_1\\leftarrow a_0}","aliases":"Γ_a_1← a_0 Gamma leftarrow","display":"\\Gamma_{a_1\\leftarrow a_0}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d820849730eca26b","tex":"\\Gamma_{a_n\\leftarrow a_{n-1}}","aliases":"Γ_a_n← a_n-1 Gamma leftarrow","display":"\\Gamma_{a_n\\leftarrow a_{n-1}}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fba48f530144df83","tex":"\\Gamma_{a\\leftarrow a}","aliases":"Γ_a← a Gamma leftarrow","display":"\\Gamma_{a\\leftarrow a}","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-32b19bb3a15c41fa","tex":"\\Gamma_{a\\leftarrow b}","aliases":"Γ_a← b Gamma leftarrow","display":"\\Gamma_{a\\leftarrow b}","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e51268d4f9664f40","tex":"\\Gamma_{a\\leftarrow c}","aliases":"Γ_a← c Gamma leftarrow","display":"\\Gamma_{a\\leftarrow c}","kind":"definition-passage","excerpt":"[Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2a75708c6ecb08f0","tex":"\\Gamma_{b\\leftarrow a}","aliases":"Γ_b← a Gamma leftarrow","display":"\\Gamma_{b\\leftarrow a}","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":17},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-667fc2f1be68ffcf","tex":"\\Gamma_{b\\leftarrow a}^{-1}","aliases":"Γ_b← a^-1 Gamma leftarrow","display":"\\Gamma_{b\\leftarrow a}^{-1}","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b75c5680cc807310","tex":"\\Gamma_{c\\leftarrow a}","aliases":"Γ_c← a Gamma leftarrow","display":"\\Gamma_{c\\leftarrow a}","kind":"definition-passage","excerpt":"[Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. s","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-accadba9f0371bc9","tex":"\\Gamma_{c\\leftarrow b}","aliases":"Γ_c← b Gamma leftarrow","display":"\\Gamma_{c\\leftarrow b}","kind":"definition-passage","excerpt":"[Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. s","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-465089d5be891c7b","tex":"\\Gamma_p","aliases":"Γ_p Gamma","display":"\\Gamma_p","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"definition-passage","excerpt":"[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect","count":26},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1ea06baff08da99a","tex":"\\Delta_{c,b,a}","aliases":"Δ_c,b,a Delta","display":"\\Delta_{c,b,a}","kind":"definition-passage","excerpt":"[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9e14435a4d85c517","tex":"\\Delta_F","aliases":"Δ_F Delta","display":"\\Delta_F","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":29},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":21},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":11},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e64c3229ee2a707a","tex":"\\Pi^a_\\mu","aliases":"Π^a_μ Pi mu","display":"\\Pi^a_\\mu","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e0871b98c3b226a2","tex":"\\Pi^a_\\nu","aliases":"Π^a_ν Pi nu","display":"\\Pi^a_\\nu","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":32},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-17b6bc4f52685e61","tex":"\\Phi_\\sigma","aliases":"Φ_σ Phi sigma","display":"\\Phi_\\sigma","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":43},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":83},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":70},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"definition-passage","excerpt":"[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPa","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,","count":126},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-752c75a3db0198d5","tex":"\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}","aliases":"∏_q∈Fin(r(σ)) prod in operatorname sigma","display":"\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":56},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6d1546afc3ef11e5","tex":"\\sum_{e\\in E}","aliases":"∑_e∈ E sum in","display":"\\sum_{e\\in E}","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cabedb544f677980","tex":"\\sum_{\\alpha\\in I}","aliases":"∑_α∈ I sum alpha in","display":"\\sum_{\\alpha\\in I}","kind":"definition-passage","excerpt":"[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"definition-passage","excerpt":"[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPa","count":22},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":51},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":16},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":25},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":30},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.","count":59},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-02c8464b897b4a69","tex":"\\Walk","aliases":"Walk operatorname","display":"{\\operatorname{Walk}}","kind":"introduction-passage","excerpt":"Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e795a1f6f91d71a4","tex":"\\Walk_n","aliases":"Walk_n Walk operatorname","display":"{\\operatorname{Walk}}_n","kind":"introduction-passage","excerpt":"Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-677eab50351e7675","tex":"A^{-1}","aliases":"A^-1","display":"A^{-1}","kind":"use","excerpt":"[Nonlinear Schur derivative] At (u_0,v_0), DR = D_uF_P - D_vF_P A^-1D_uF_Q.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1a3f87ebd468ea16","tex":"A^{b_1}","aliases":"A^b_1","display":"A^{b_1}","kind":"use","excerpt":"[Abelian cohomology class] When the structure group A is abelian, gauge classes of transports are naturally H^1(Γ,A) ≅ Hom(H_1(Γ, Z),A). After a basis of H_1(Γ, Z) is chosen, this group is identified with A^b_1.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-40bae120bc963ffd","tex":"a_{1,g_1}","aliases":"a_1,g_1","display":"a_{1,g_1}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c647bef050188403","tex":"\\mathcal A_a","aliases":"A_a mathcal","display":"\\mathcal A_a","kind":"use","excerpt":"A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-73c1bf580d5c1dbc","tex":"A_g","aliases":"A_g","display":"A_g","kind":"use","excerpt":"Let G be a finite abelian group and let A=⊕_g∈ GA_g be a G graded module.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-79ce59fb60d59b25","tex":"A_{i+j}","aliases":"A_i+j","display":"A_{i+j}","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-95713f689b2a581a","tex":"a_{j,g}","aliases":"a_j,g","display":"a_{j,g}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7d8e3b8668ae2ddd","tex":"A_k","aliases":"A_k","display":"A_k","kind":"use","excerpt":"Expand every input into its homogeneous components. Each term lands in the component whose label is the sum of its input labels. Projection onto A_k retains exactly the displayed terms.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-501fa09870ac360b","tex":"A_{ki}","aliases":"A_ki","display":"A_{ki}","kind":"use","excerpt":"vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4c8131608f522ec2","tex":"a_{n+1}","aliases":"a_n+1","display":"a_{n+1}","kind":"use","excerpt":"Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9a65c3129d1f56f5","tex":"a_{r,g_r}","aliases":"a_r,g_r","display":"a_{r,g_r}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4ee7d787d8b86a7e","tex":"A_{vu}","aliases":"A_vu","display":"A_{vu}","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-dbdc2defb45a0c61","tex":"A_{vu}^{-1}","aliases":"A_vu^-1","display":"A_{vu}^{-1}","kind":"use","excerpt":"Conversely, choose a spanning tree. Starting from U_r, propagate U_v along the unique tree path by the rule U_v=B_vuU_uA_vu^-1 on every tree edge. This is the only possible extension satisfying the tree edge equations. An edge outside the tree closes one fundamental loop. Its intertwining equation holds exactly when U_r intertwines the two correspo","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ff76bbf0fc00a0b5","tex":"\\Acal_{\\Pcal}","aliases":"Acal_Pcal A_ P Acal Pcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal P}}","kind":"use","excerpt":"Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f2acd68c5aa6b021","tex":"\\Acal_{\\Rcal}","aliases":"Acal_Rcal A_ R Acal Rcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal R}}","kind":"use","excerpt":"The identities S_ P A_ P=I_X and S_ Q A_ Q=I_Y give S_ QF_ Q← P A_ P = S_ Q A_ QF S_ P A_ P =F. For composition, G_ R← QF_ Q← P = A_ RG S_ Q A_ QF S_ P = A_ R(G∘ F) S_ P.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7b340b7ad7452d3e","tex":"\\Acal_{\\Ucal}","aliases":"Acal_Ucal A_ U Acal Ucal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal U}}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"For a unary square, inserting a+b does not give a^2+b^2. The extra 2ab arises because the operation is nonlinear in its single argument. It can be analyzed by representing the square using a binary multiplication node with a repeated input, but that is an additional choice of syntax and semantics.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"The passage from words to walks also applies to products of different matrices. Let V be one finite packet type, let R be a semiring, and let B_1,…,B_m∈ R^V× V be a finite sequence of possibly different packet matrices. Rows are targets and columns are sources as before.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"be the chosen structure group and fix a root r. Restrict both edge transports and gauge values to G. Gauge equivalence classes of G valued transport systems are naturally identified with Hom(π_1(Γ,r),G)/G, where G acts by conjugation. After a spanning tree and orientations of the remaining edges are chosen, this set is identified with simultaneous ","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c6abdf84d4df05b6","tex":"B_{jk}","aliases":"B_jk","display":"B_{jk}","kind":"use","excerpt":"vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b2f9008f03849cc5","tex":"B_k","aliases":"B_k","display":"B_k","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"use","excerpt":"The passage from words to walks also applies to products of different matrices. Let V be one finite packet type, let R be a semiring, and let B_1,…,B_m∈ R^V× V be a finite sequence of possibly different packet matrices. Rows are targets and columns are sources as before.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-57c8f1ae1da23ae1","tex":"B_r","aliases":"B_r","display":"B_r","kind":"use","excerpt":"rices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then ","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ac0d31a2615b8b66","tex":"b_r","aliases":"b_r","display":"b_r","kind":"use","excerpt":"Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-750ec2c2a1a68b60","tex":"B_{vu}","aliases":"B_vu","display":"B_{vu}","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-49b6d1142f9da2e4","tex":"B_\\tau","aliases":"B_τ tau","display":"B_\\tau","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b38caf9b289ccca2","tex":"B_{\\tau_1}","aliases":"B_τ_1 tau","display":"B_{\\tau_1}","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c9e6f3d273912284","tex":"B_{\\tau_r}","aliases":"B_τ_r tau","display":"B_{\\tau_r}","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-96852c201d3f1279","tex":"\\bbk","aliases":"bbk 𝕜 Bbbk","display":"{\\Bbbk}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b2a8d769f2c647dd","tex":"\\bbk^2","aliases":"bbk^2 𝕜^2 bbk Bbbk","display":"{\\Bbbk}^2","kind":"use","excerpt":"[A mixed quadratic map] Let X=Y=𝕜^2 with coordinate packets. For F(x_1,x_2)=(x_1x_2,x_1^2+x_2), the first output packet depends on both input packets. The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second. Packet coordinates record these interactions exactly. Replacing F by ","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-33f082f973fe83c6","tex":"\\bbk^\\times","aliases":"bbk^× 𝕜^× bbk times Bbbk times","display":"{\\Bbbk}^\\times","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"Let X and Y be Banach spaces over the same field R or C and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr echet derivatives and the Taylor integral below has a real parameter. Assume that F:X→ Y is m times continuously Fr echet differentiable. The same local statements hold on open domains cont","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"use","excerpt":"vfig[.97]worked_holonomy_diagnosis platecaptionThe vector row starts with the source witness e_0 and follows it through A, then B, then C^-1. Returning to the base produces (1,1)^ T, whose displacement from the input is e_1.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e7e42689204088d5","tex":"C^0","aliases":"C^0","display":"C^0","kind":"use","excerpt":"[A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the ","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bb4b39a0fa3e20d1","tex":"C^1","aliases":"C^1","display":"C^1","kind":"use","excerpt":"[A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the ","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fc4ec113b29e6ee4","tex":"C^n","aliases":"C^n","display":"C^n","kind":"use","excerpt":"The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b8b169112cc1dba2","tex":"C^{p+q}","aliases":"C^p+q","display":"C^{p+q}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-dd0b3056e758361c","tex":"C^r","aliases":"C^r","display":"C^r","kind":"use","excerpt":"[Local Lyapunov Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-897aec4854fa7c6c","tex":"c^\\leftarrow","aliases":"c^← leftarrow","display":"c^\\leftarrow","kind":"use","excerpt":"For a nonempty composable word w=(e_1,…,e_n), define its reverse coefficient product c^←(w)=c(e_n)c(e_n-1)⋯ c(e_1). The reversal is essential over a noncommutative semiring.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-39f07d70eaa40c73","tex":"C_{0,m}","aliases":"C_0,m","display":"C_{0,m}","kind":"use","excerpt":"y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f944d0625ea61fc0","tex":"C_{0,\\ell}","aliases":"C_0,ℓ ell","display":"C_{0,\\ell}","kind":"use","excerpt":"y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d5b1cdb36b5b2a45","tex":"C_{0j}","aliases":"C_0j","display":"C_{0j}","kind":"use","excerpt":"vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functio","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d4e58ada952ec6ab","tex":"C_4","aliases":"C_4","display":"C_4","kind":"use","excerpt":"P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying th","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ca35fd43dab309b4","tex":"C_{a+1}","aliases":"C_a+1","display":"C_{a+1}","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e0d887f344230957","tex":"C_{a,a}","aliases":"C_a,a","display":"C_{a,a}","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2dc3b3739f1aa480","tex":"C_{a,b}","aliases":"C_a,b","display":"C_{a,b}","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6922a59f4f38ba94","tex":"C_{b-1}","aliases":"C_b-1","display":"C_{b-1}","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b48446852f3335a2","tex":"C_{n-1}","aliases":"C_n-1","display":"C_{n-1}","kind":"use","excerpt":"Assume now that a>0 and b>0. Take R= frac1-sqrt1-4ab2b. Then a+bR^2=R and 2bR<1. Apply [source reference]. The number of full ordered binary trees with n leaves is the Catalan number C_n-1, which gives the displayed majorant.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1b5c76a8c06b016d","tex":"C_Z","aliases":"C_Z","display":"C_Z","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-19b5aea016550f5b","tex":"C_\\alpha^\\bullet","aliases":"C_α^bullet alpha bullet","display":"C_\\alpha^\\bullet","kind":"use","excerpt":"[Direct packet complexes] Suppose C^bullet=⊕_α∈ IC_α^bullet is a finite direct sum of subcomplexes. Then H^n(C^bullet) ≅ ⊕_α∈ IH^n(C_α^bullet). A total cohomology class vanishes if and only if every packet class vanishes.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4ae0a9be277fd36e","tex":"C_\\ell","aliases":"C_ℓ ell","display":"C_\\ell","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b2025a3e28eea9cf","tex":"\\cA^X_a","aliases":"cA^X_a A^X_a cA mathcal","display":"{\\mathcal A}^X_a","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-32b92e346ab956bb","tex":"\\cA^X_b","aliases":"cA^X_b A^X_b cA mathcal","display":"{\\mathcal A}^X_b","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7a3f0b5a74405089","tex":"\\cA^Y","aliases":"cA^Y A^Y cA mathcal","display":"{\\mathcal A}^Y","kind":"use","excerpt":"Both sides of the naturality equation reduce to A^Y_s∘ F∘ S^X_a using S∘ A=id in X and Y. For composition, the middle S^Y∘ A^Y cancels in the same way.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-79b2b39c2a8e8c5a","tex":"\\cA^Y_r","aliases":"cA^Y_r A^Y_r cA mathcal","display":"{\\mathcal A}^Y_r","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0f54c3e34264c2fc","tex":"\\cA^Y_s","aliases":"cA^Y_s A^Y_s cA mathcal","display":"{\\mathcal A}^Y_s","kind":"use","excerpt":"Both sides reduce to A^Y_sF S^X_a after inserting the appropriate synthesis--analysis identity. The composition law is the same calculation with the middle ambient module inserted.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1251c2ab227ac52f","tex":"\\cA^Z_t","aliases":"cA^Z_t A^Z_t cA mathcal","display":"{\\mathcal A}^Z_t","kind":"use","excerpt":"^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then ex","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5e9b4ec45cb5eaa9","tex":"\\cA_a","aliases":"cA_a A_a cA mathcal","display":"{\\mathcal A}_a","kind":"use","excerpt":"ordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-32e5fff3cf3d9303","tex":"\\cA_c","aliases":"cA_c A_c cA mathcal","display":"{\\mathcal A}_c","kind":"use","excerpt":"Insert S_b A_b=id_X: J^X_c← bJ^X_b← a = A_c S_b A_b S_a = A_c S_a =J^X_c← a. The identity statement is A_a S_a=id. Setting c=a in the cocycle gives J^X_a← bJ^X_b← a=id; interchanging a and b gives the other composition, so the two transports are inverses.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b844ea5955b0d603","tex":"\\mathsf{Candidate}","aliases":"Candidate mathsf","display":"\\mathsf{Candidate}","kind":"use","excerpt":"Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"[htbp] centering includegraphics[width=.96linewidth]rotating_projector_transport.pdf captionA rotating rank one packet and its orthogonal complement. With u(θ)=(cosθ,sinθ) and P(θ)=u(θ)u(θ)^ T, the canonical generator is [P',P]=(0 · -1","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8ce3ab72565dec46","tex":"\\cQ","aliases":"cQ Q mathcal","display":"{\\mathcal Q}","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c61c0d89b8905641","tex":"\\cQ^r","aliases":"cQ^r Q^r cQ mathcal","display":"{\\mathcal Q}^r","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4411dd1c078c6979","tex":"\\cQ^s","aliases":"cQ^s Q^s cQ mathcal","display":"{\\mathcal Q}^s","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d9d0a2db27784e60","tex":"\\cS^X_a","aliases":"cS^X_a S^X_a cS mathcal","display":"{\\mathcal S}^X_a","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e10b94ac80a0d767","tex":"\\cS^X_b","aliases":"cS^X_b S^X_b cS mathcal","display":"{\\mathcal S}^X_b","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f06e2bdca099a9e7","tex":"\\cS^Y","aliases":"cS^Y S^Y cS mathcal","display":"{\\mathcal S}^Y","kind":"use","excerpt":"Both sides of the naturality equation reduce to A^Y_s∘ F∘ S^X_a using S∘ A=id in X and Y. For composition, the middle S^Y∘ A^Y cancels in the same way.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-35ee11babd943e3c","tex":"\\cS^Y_r","aliases":"cS^Y_r S^Y_r cS mathcal","display":"{\\mathcal S}^Y_r","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a294477855ee7bbe","tex":"\\cS^Y_s","aliases":"cS^Y_s S^Y_s cS mathcal","display":"{\\mathcal S}^Y_s","kind":"use","excerpt":"beginproposition[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, wit","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f9d6fb8975e8c270","tex":"\\cS_b","aliases":"cS_b S_b cS mathcal","display":"{\\mathcal S}_b","kind":"use","excerpt":"Insert S_b A_b=id_X: J^X_c← bJ^X_b← a = A_c S_b A_b S_a = A_c S_a =J^X_c← a. The identity statement is A_a S_a=id. Setting c=a in the cocycle gives J^X_a← bJ^X_b← a=id; interchanging a and b gives the other composition, so the two transports are inverses.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-40d9503903bdca20","tex":"\\mathfrak D","aliases":"D mathfrak","display":"\\mathfrak D","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7e8acadc439b10d5","tex":"D^*","aliases":"D^*","display":"D^*","kind":"use","excerpt":"A linear correction map D:U⟶ V places the obstruction to solving Du=-r in the canonical quotient coker D. This quotient has a universal property and behaves naturally under commuting squares. Mapping cones extend the same construction to cochain complexes. In a Hilbert space with closed range, the quotient is represented by the orthogonal projectio","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","excerpt":"C · D , where D is invertible. Eliminating the hidden variable replaces the retained derivative A by A-BD^-1C. The feedback term is exact. Keeping only A would ignore the path from retained variables into the hidden variable and back.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic ","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a46c6db9c9c47c40","tex":"D^{m+1}","aliases":"D^m+1","display":"D^{m+1}","kind":"use","excerpt":"· +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c3fb059a8ab4b3a2","tex":"D^\\times","aliases":"D^× times","display":"D^\\times","kind":"use","excerpt":"Flatness replaces the two-edge route in D^× by the direct route, which cancels its inverse. For H, flatness first collapses a→ b→ c to a→ c, then collapses a→ c→ a to the normalized transport a→ a.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e69131764514df66","tex":"D^\\times_{c,b,a}","aliases":"D^×_c,b,a times","display":"D^\\times_{c,b,a}","kind":"use","excerpt":"At a triple (a,b,c), the compiled cocycle predicate is Γ_c← bΓ_b← a =Γ_c← a. Flatness means this equality holds for every triple. The formal multiplicative relative defect and triangle holonomy are D^×_c,b,a = (Γ_c← bΓ_b← a) Γ_c← a^-1, H_c,b,a = Γ_a← c Γ_c← b Γ_b← a. The defect is based at the target c. The holonomy is based at a. statusleanthreeKo","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-082363402fb26c4b","tex":"\\mathfrak D^\\Gamma_{b,a}","aliases":"D^Γ_b,a mathfrak Gamma","display":"\\mathfrak D^\\Gamma_{b,a}","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-96cd702de2890fbe","tex":"D^\\dagger","aliases":"D^† dagger","display":"D^\\dagger","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hol","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ee9c69896c4d1a6d","tex":"D_c","aliases":"D_c","display":"D_c","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":11},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-54ff0eb6a1af0d6f","tex":"D_f","aliases":"D_f","display":"D_f","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":20},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ee1684d3a8f6d798","tex":"D_u","aliases":"D_u","display":"D_u","kind":"use","excerpt":"[Nonlinear Schur derivative] At (u_0,v_0), DR = D_uF_P - D_vF_P A^-1D_uF_Q.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"use","excerpt":"F_Q(u,v) . Assume F takes values in Y_P× Y_Q, is twice continuously Fr echet differentiable near (u_0,v_0), satisfies F_Q(u_0,v_0)=0, and A=D_vF_Q(u_0,v_0) is a Banach space isomorphism from V to Y_Q. The implicit function theorem gives a local map v=φ(u), with φ(u_0)=v_0, satisfying F_Q(u,φ(u))=0. Define the reduced equation R(u)=F_P(u,φ(u)).","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"use","excerpt":"[Packet decomposition of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-503a05ceb4c00d45","tex":"d_\\eta","aliases":"d_η eta","display":"d_\\eta","kind":"use","excerpt":"Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one dep","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"The usual scalar invariants can be misleading: trH=2, det H=1, χ_H(t)=(t-1)^2. These are also the invariants of the 2×2 identity matrix. Nevertheless, Ob=H-I=mat0 · 1","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"vpage14Gauge changes the picture, not flatnessMODEL E / CONJUGATION AND ITS LIMITSTraditional paper: Theorem 6.2 (p. 14) At the base chart, change coordinates by D=diag(2,1). Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving H^D=DHD^-1=mat1 · 2","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-558c1dc971656de6","tex":"\\dist","aliases":"dist operatorname","display":"{\\operatorname{dist}}","kind":"use","excerpt":"agger be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-35679b068dabbae2","tex":"\\dot P","aliases":"dot P dot","display":"\\dot P","kind":"use","excerpt":"Zero projections are permitted in this section and contribute zero to all generator sums. When T=0, every partition product is the empty product I. Completeness and orthogonality give P_α(t)P_β(t)=0 for α≠β, ∑_α P_α(t)=I. Define the canonical generator K(t)=∑_α dot P_α(t)P_α(t).","count":11},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-12567f7ebf7d4fca","tex":"\\dot P_\\alpha","aliases":"dot P_α dot alpha","display":"\\dot P_\\alpha","kind":"use","excerpt":"Zero projections are permitted in this section and contribute zero to all generator sums. When T=0, every partition product is the empty product I. Completeness and orthogonality give P_α(t)P_β(t)=0 for α≠β, ∑_α P_α(t)=I. Define the canonical generator K(t)=∑_α dot P_α(t)P_α(t).","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9645cfd170ae8601","tex":"\\dot P_\\beta","aliases":"dot P_β dot beta","display":"\\dot P_\\beta","kind":"use","excerpt":"begingroup setlengthabovedisplayskip6pt plus 2pt minus 2pt setlengthbelowdisplayskip6pt plus 2pt minus 2pt [Complete packet commutator identity] For every packet label β, dot P_β(t)=[K(t),P_β(t)]. The generator is also skew adjoint: K(t)^*=-K(t).","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d1997e4b5359f675","tex":"\\dot U","aliases":"dot U dot","display":"\\dot U","kind":"use","excerpt":"[Canonical packet transport] Let U(t) solve dot U(t)=K(t)U(t), U(0)=I. Then U(t) is unitary and transports every packet exactly: P_α(t)U(t)=U(t)P_α(0) for all α.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ae6eb7cbea0e8357","tex":"\\dot V","aliases":"dot V dot","display":"\\dot V","kind":"use","excerpt":"The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I; uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f84526501d95b0bb","tex":"\\dot W","aliases":"dot W dot","display":"\\dot W","kind":"use","excerpt":"The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I; uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-64fbb1a3212deefd","tex":"e'","aliases":"e'","display":"e'","kind":"use","excerpt":"The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1009899abf2147a9","tex":"E^{n-1}","aliases":"E^n-1","display":"E^{n-1}","kind":"use","excerpt":"The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"vpage7Reachability is only an upper boundMODEL A / AN ORBIT THAT STAYS IN TWO DIMENSIONSTraditional paper: Corollary 3.11 (p. 6) Keep the diamond and add an isolated packet 4. Start with the single seed e_0∈QQ^5. Direct multiplication gives Te_0=e_1+e_2, T^2e_0=0. Every later orbit vector is zero. The orbit hull generated by T is therefore exactly ","count":14},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-febc076ab7500943","tex":"E_{01}","aliases":"E_01","display":"E_{01}","kind":"use","excerpt":"vpage16Resolve the entire obstructionMODEL F / PACKET BLOCKS FROM INPUT TO OUTPUTTraditional paper: Theorem 7.5 (p. 17), Corollary 7.6 (p. 18) Use three coordinate packets with projectors P_0,P_1,P_2, and set H=I+2E_01+3E_12, Ob=H-I. Here H is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside ","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f863240a4986a039","tex":"E_0^{p,q}","aliases":"E_0^p,q","display":"E_0^{p,q}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-136cb5922e752cc4","tex":"E_{10}","aliases":"E_10","display":"E_{10}","kind":"use","excerpt":"vpage3Read the product in execution orderMODEL A / ONE EDGE IS ONE MATRIX ENTRYTraditional paper: Lemma 2.2 (p. 2), Theorem 2.3 (p. 2), Theorem 2.4 (p. 3) Work over ZZ with packets 0,1,2. Let e:0→1 have coefficient 2, and let f:1→2 have coefficient 3. The elementary operators are L_e=2E_10, L_f=3E_21. The location of the nonzero entry records the t","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f2b11b7b1856ea86","tex":"E_{12}","aliases":"E_12","display":"E_{12}","kind":"use","excerpt":"vpage16Resolve the entire obstructionMODEL F / PACKET BLOCKS FROM INPUT TO OUTPUTTraditional paper: Theorem 7.5 (p. 17), Corollary 7.6 (p. 18) Use three coordinate packets with projectors P_0,P_1,P_2, and set H=I+2E_01+3E_12, Ob=H-I. Here H is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7d1846d5b18c76fd","tex":"E_1^{p,n-p}","aliases":"E_1^p,n-p","display":"E_1^{p,n-p}","kind":"use","excerpt":"[Fine obstruction survival] A class on the first page contributes to the total obstruction only if it is represented by a permanent cycle, so it is neither killed nor identified as a boundary on any later page. If the spectral sequence degenerates at the first page, then H^n(C^bullet)=0 exactly when E_1^p,n-p=0 for every p.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-eedfcc807dd23d85","tex":"E_1^{p,q}","aliases":"E_1^p,q","display":"E_1^{p,q}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functio","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"vpage7Reachability is only an upper boundMODEL A / AN ORBIT THAT STAYS IN TWO DIMENSIONSTraditional paper: Corollary 3.11 (p. 6) Keep the diamond and add an isolated packet 4. Start with the single seed e_0∈QQ^5. Direct multiplication gives Te_0=e_1+e_2, T^2e_0=0. Every later orbit vector is zero. The orbit hull generated by T is therefore exactly ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d61e8fb192b54ca7","tex":"E_{20}","aliases":"E_20","display":"E_{20}","kind":"use","excerpt":"On a vector (x_0,x_1,x_2)^ T, the first step produces (0,2x_0,0)^ T, and the second produces (0,0,6x_0)^ T. Thus L_fL_e=6E_20, L_eL_f=0. For general edges, a summand in the entrywise product can survive only when the same intermediate packet is both tgt(e) and src(f): (L_fL_e)_yx=∑_z(L_f)_yz(L_e)_zx. This is why a broken interface annihilates the e","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-eebcf5feadee9573","tex":"E_{21}","aliases":"E_21","display":"E_{21}","kind":"use","excerpt":"vpage3Read the product in execution orderMODEL A / ONE EDGE IS ONE MATRIX ENTRYTraditional paper: Lemma 2.2 (p. 2), Theorem 2.3 (p. 2), Theorem 2.4 (p. 3) Work over ZZ with packets 0,1,2. Let e:0→1 have coefficient 2, and let f:1→2 have coefficient 3. The elementary operators are L_e=2E_10, L_f=3E_21. The location of the nonzero entry records the t","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-82c2d872ef715d1f","tex":"E_3","aliases":"E_3","display":"E_3","kind":"use","excerpt":"P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying th","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"headThe admissible operator grammar matters Allowing each individual edge operator is a different generation problem. For instance, applying the edge 0→1 and then 1→3 individually produces e_3. It does not follow that e_3 belongs to the orbit of the summed operator T. The paper's theorem on orbit support concerns the explicitly stated operator acti","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-67ab753297c61ce5","tex":"e_4","aliases":"e_4","display":"e_4","kind":"use","excerpt":",0.99500) 0; draw[fill=black!0,draw=black!40,line width=.35pt,] (4.69000,0.65000) rectangle ++(0.69000,0.69000); node[] at (5.03500,0.99500) 0; node[] at (8.85000,3.80000) Reach(0)={0,1,2,3}; node[] at (8.85000,2.75000) Hull_T(e_0)={(a,b,b,0,0)}; node[] at (8.85000,1.65000) e_3: reachable but absent; node[] at (8.85000,0.60000) e_4: unreachable and","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"vpage2An atlas of the objectsORIENTATION / KEEP THE TYPES VISIBLETraditional paper: Definition 2.1 (p. 2), Definition 5.5 (p. 11), Definition 7.4 (p. 16) Rows are targets and columns are sources throughout. If E_ji is a matrix unit, then E_jie_i=e_j. The notation e_i means a basis vector. A named graph edge is written a,b,c,d or e,f when its role i","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"vpage2An atlas of the objectsORIENTATION / KEEP THE TYPES VISIBLETraditional paper: Definition 2.1 (p. 2), Definition 5.5 (p. 11), Definition 7.4 (p. 16) Rows are targets and columns are sources throughout. If E_ji is a matrix unit, then E_jie_i=e_j. The notation e_i means a basis vector. A named graph edge is written a,b,c,d or e,f when its role i","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-606053aa32f17484","tex":"E_{ji}","aliases":"E_ji","display":"E_{ji}","kind":"use","excerpt":"vpage2An atlas of the objectsORIENTATION / KEEP THE TYPES VISIBLETraditional paper: Definition 2.1 (p. 2), Definition 5.5 (p. 11), Definition 7.4 (p. 16) Rows are targets and columns are sources throughout. If E_ji is a matrix unit, then E_jie_i=e_j. The notation e_i means a basis vector. A named graph edge is written a,b,c,d or e,f when its role i","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d3e1f9162e4702a9","tex":"e_{n+1}","aliases":"e_n+1","display":"e_{n+1}","kind":"use","excerpt":"Let e_n= CF_f^n(x)-F_c^n(Cx) . The definition of the defect and the Lipschitz property give e_n+1≤ε_n+qe_n. Iterating this recurrence proves the first estimate. The second is the geometric sum.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b7ce4855f02a711f","tex":"e_{n-1}","aliases":"e_n-1","display":"e_{n-1}","kind":"use","excerpt":"For a nonempty composable word w=(e_1,…,e_n), define its reverse coefficient product c^←(w)=c(e_n)c(e_n-1)⋯ c(e_1). The reversal is essential over a noncommutative semiring.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3f38f9f204752bc5","tex":"E_{rs}","aliases":"E_rs","display":"E_{rs}","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cd0b06b425803045","tex":"E_{rv}","aliases":"E_rv","display":"E_{rv}","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5cee62af73e7dca8","tex":"E_{\\tgt(e),\\src(e)}","aliases":"E_tgt(e),src(e) tgt src operatorname operatorname","display":"E_{{\\operatorname{tgt}}(e),{\\operatorname{src}}(e)}","kind":"use","excerpt":"L_e=c(e)E_tgt(e),src(e) · One operator supported on a packet. · pagerefV3","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9225c9aedb8f0a45","tex":"E_{uv}","aliases":"E_uv","display":"E_{uv}","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d8a2c49583348c4b","tex":"E_{\\alpha}","aliases":"E_α alpha","display":"E_{\\alpha}","kind":"use","excerpt":"Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functio","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End operatorname","display":"{\\operatorname{End}}","kind":"use","excerpt":"Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"[Convergence of sampled transports] Assume K is Lipschitz on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T, define U_π = exp((t_N-t_N-1)K(t_N-1)) ⋯ exp((t_1-t_0)K(t_0)). As the mesh |π| tends to zero, U_π⟶ U(T) in operator norm. There is a constant C, depending only on T, the supremum norm of K, and its Lipschitz constant, such that U_π-U(T) ≤ C|π|.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-40808d4081e59d75","tex":"F^0","aliases":"F^0","display":"F^0","kind":"use","excerpt":"Let X and Y be modules with decreasing filtrations X=F^0X⊇ F^1X⊇⋯, Y=F^0Y⊇ F^1Y⊇⋯. Assume both are separated and complete for their filtration topologies. Let Φ:X→ Y be continuous and filtration congruence preserving in the sense that h∈ F^mX Longrightarrow Φ(x+h)-Φ(x)∈ F^mY for every x∈ X and every m. This condition makes the statement Φ(x)∈ F^mY ","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fef72e85e7b92ea3","tex":"F^1","aliases":"F^1","display":"F^1","kind":"use","excerpt":"Let X and Y be modules with decreasing filtrations X=F^0X⊇ F^1X⊇⋯, Y=F^0Y⊇ F^1Y⊇⋯. Assume both are separated and complete for their filtration topologies. Let Φ:X→ Y be continuous and filtration congruence preserving in the sense that h∈ F^mX Longrightarrow Φ(x+h)-Φ(x)∈ F^mY for every x∈ X and every m. This condition makes the statement Φ(x)∈ F^mY ","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","excerpt":"[A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the ","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-995485db55facd82","tex":"F^{2m}","aliases":"F^2m","display":"F^{2m}","kind":"use","excerpt":"Direct expansion gives Ω(η+h) = Ω(η)+dh+[η,h]+frac12[h,h]. The filtration rule places [h,h] in F^2m L^2, which is contained in F^m+1 L^2 because m≥1. The congruence follows. The last statement is [source reference] applied to the induced graded correction map.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-161ae3b661df7aaf","tex":"F^i","aliases":"F^i","display":"F^i","kind":"use","excerpt":"[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f5e957ee55f3194f","tex":"F^{i+j}","aliases":"F^i+j","display":"F^{i+j}","kind":"use","excerpt":"[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a637cab476ceffaa","tex":"F^j","aliases":"F^j","display":"F^j","kind":"use","excerpt":"[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4256d948f584b864","tex":"F^{N+1}","aliases":"F^N+1","display":"F^{N+1}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ba5037951b752c25","tex":"F^r","aliases":"F^r","display":"F^r","kind":"use","excerpt":"For every fixed r, the congruences imply x_n-x_m∈ F^rX whenever n≥ m≥ r. Thus (x_m) is Cauchy for the filtration topology. Completeness gives a limit x_∞.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"vpage9One vector, three exact framesMODEL C / CANONICAL TRANSPORT HAS NO HOLONOMYTraditional paper: Theorem 4.1 (p. 7), Theorem 4.2 (p. 7) Let F_a synthesize chart coordinates into one ambient QQ^2. Choose F_0=I, F_1=mat1 · 1 0 · 1, F_2=mat2 · 0 1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"vpage9One vector, three exact framesMODEL C / CANONICAL TRANSPORT HAS NO HOLONOMYTraditional paper: Theorem 4.1 (p. 7), Theorem 4.2 (p. 7) Let F_a synthesize chart coordinates into one ambient QQ^2. Choose F_0=I, F_1=mat1 · 1 0 · 1, F_2=mat2 · 0 1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6210caf9e2379390","tex":"F_1^{-1}","aliases":"F_1^-1","display":"F_1^{-1}","kind":"use","excerpt":"Transport from chart a to chart b is forced: J_b← a=F_b^-1F_a. The route 0→1→2 agrees with the direct route because the middle frame cancels: (F_2^-1F_1)(F_1^-1F_0)=F_2^-1F_0. Returning to chart 0 cancels every remaining frame. The final coordinate vector is (3,2)^ T, not a sheared or rotated variant.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"vpage9One vector, three exact framesMODEL C / CANONICAL TRANSPORT HAS NO HOLONOMYTraditional paper: Theorem 4.1 (p. 7), Theorem 4.2 (p. 7) Let F_a synthesize chart coordinates into one ambient QQ^2. Choose F_0=I, F_1=mat1 · 1 0 · 1, F_2=mat2 · 0 1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-30b4f88bb6be80ad","tex":"F_2^{-1}","aliases":"F_2^-1","display":"F_2^{-1}","kind":"use","excerpt":"Transport from chart a to chart b is forced: J_b← a=F_b^-1F_a. The route 0→1→2 agrees with the direct route because the middle frame cancels: (F_2^-1F_1)(F_1^-1F_0)=F_2^-1F_0. Returning to chart 0 cancels every remaining frame. The final coordinate vector is (3,2)^ T, not a sheared or rotated variant.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"Let every chart a carry a linear frame equivalence F_a:Mxrightarrow∼M. The formal Paper III module defines transport from a to b by J_b← a=F_b^-1F_a. This frame formula means synthesize in chart a, then analyze in chart b.","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a645ccc01f63de95","tex":"F_a^{-1}","aliases":"F_a^-1","display":"F_a^{-1}","kind":"use","excerpt":"1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0b8eb6ad7f8b7acf","tex":"f_b","aliases":"f_b","display":"f_b","kind":"use","excerpt":"tangle ++(0.87000,0.58000); node[] at (9.43500,1.39000) 0; draw[fill=black!10,draw=black!40,line width=.35pt,] (9.87000,1.10000) rectangle ++(0.87000,0.58000); node[] at (10.30500,1.39000) -2; draw[fill=black!10,draw=black!40,line width=.35pt,] (10.74000,1.10000) rectangle ++(0.87000,0.58000); node[] at (11.17500,1.39000) -8; node[] at (9.65000,-0.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8dedba4191c4f216","tex":"F_b^{-1}","aliases":"F_b^-1","display":"F_b^{-1}","kind":"use","excerpt":"Let every chart a carry a linear frame equivalence F_a:Mxrightarrow∼M. The formal Paper III module defines transport from a to b by J_b← a=F_b^-1F_a. This frame formula means synthesize in chart a, then analyze in chart b.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8a5ce67bd957032f","tex":"F_c^n","aliases":"F_c^n","display":"F_c^n","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2833b5c1542c1219","tex":"F_f^j","aliases":"F_f^j","display":"F_f^j","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-79022e1216743208","tex":"F_f^n","aliases":"F_f^n","display":"F_f^n","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-07d5ca93af84b975","tex":"F_P","aliases":"F_P","display":"F_P","kind":"use","excerpt":"Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v) F_Q(u,v) . Assume F takes values in Y_P× Y_Q, is twice continuously Fr echet differentiable near (u_0,v_0), satisfies F_Q(u_0,v_0)=0, and A=D_vF_Q(u_0,v_0) is a Banach space isomorphism from V to Y_Q. The impl","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-52602bf9fb1a4e99","tex":"F_{\\Pcal\\leftarrow\\Pcal}","aliases":"F_Pcal←Pcal F_ P← P Pcal leftarrow Pcal mathcal leftarrow mathcal","display":"F_{{\\mathcal P}\\leftarrow{\\mathcal P}}","kind":"use","excerpt":"[Dynamical equivalence] Suppose X=Y and the same packet system is used on both sides. Packet coordinates preserve fixed points, periodic points, injectivity, surjectivity, and invertibility. In particular, F(x)=x Longleftrightarrow F_ P← P( A_ Px)= A_ Px.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4b0ff167397251af","tex":"F_Q","aliases":"F_Q","display":"F_Q","kind":"use","excerpt":"Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v) F_Q(u,v) . Assume F takes values in Y_P× Y_Q, is twice continuously Fr echet differentiable near (u_0,v_0), satisfies F_Q(u_0,v_0)=0, and A=D_vF_Q(u_0,v_0) is a Banach space isomorphism from V to Y_Q. The impl","count":13},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fff6a4a560a0981b","tex":"F_{\\Qcal'\\leftarrow\\Pcal'}","aliases":"F_Qcal'←Pcal' F_ Q'← P' Qcal leftarrow Pcal mathcal leftarrow mathcal","display":"F_{{\\mathcal Q}'\\leftarrow{\\mathcal P}'}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f7110079fe13a85a","tex":"F_r","aliases":"F_r","display":"F_r","kind":"use","excerpt":"Every identity is cancellation of adjacent F_rF_r^-1 factors. The Lean definitions use leannameLinearEquiv.trans; the displayed formulas use the equivalent right-to-left function-composition notation fixed in this paper.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ecf69680e196599f","tex":"F_r^{-1}","aliases":"F_r^-1","display":"F_r^{-1}","kind":"use","excerpt":"Every identity is cancellation of adjacent F_rF_r^-1 factors. The Lean definitions use leannameLinearEquiv.trans; the displayed formulas use the equivalent right-to-left function-composition notation fixed in this paper.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c4f568dc946ddd98","tex":"\\Fcal","aliases":"Fcal F mathcal","display":"{\\mathcal F}","kind":"use","excerpt":"[Curvature of a projected connection] Let P be a smooth orthogonal projector valued map on a parameter manifold. The connection ∇=P d on the range bundle has curvature F=P(dPwedge dP)P.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4ad39170ee6fc0f6","tex":"g^{(j)}","aliases":"g^(j)","display":"g^{(j)}","kind":"use","excerpt":"t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-62e503385c528c25","tex":"g^{(j+1)}","aliases":"g^(j+1)","display":"g^{(j+1)}","kind":"use","excerpt":"Set g(t)=F(x+th) on the open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-208960bb4d25598b","tex":"g^{(r)}","aliases":"g^(r)","display":"g^{(r)}","kind":"use","excerpt":"Set g(t)=F(x+th) on the open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-565b30cbcdc2ca2a","tex":"G^{-1}","aliases":"G^-1","display":"G^{-1}","kind":"use","excerpt":"Direct multiplication gives GOb G^-1=mat-1 · 1 -1 · 1. For example, the original (0,0) block vanishes, while the new (0,0) block relative to the old coordinate projectors does not. This is why the theorem about preservation of each block's zero status requires G P_i=P_iG.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0a566b1db5acde7f","tex":"g_0","aliases":"g_0","display":"g_0","kind":"use","excerpt":"vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESTraditional paper: Theorem 6.3 (p. 14) Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical counterp","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e6df94744f7ade62","tex":"g_2","aliases":"g_2","display":"g_2","kind":"use","excerpt":"vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESTraditional paper: Theorem 6.3 (p. 14) Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical counterp","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"vfig[.97]flat_not_calibrated platecaptionThe rulers display the chosen gauge scales: the marks are g_a n for integer n. They visualize the supplied scalar factors, not an intrinsic metric. Relative transport is the ratio of the target and source scales.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d7ac3982b663f850","tex":"g_a^{-1}","aliases":"g_a^-1","display":"g_a^{-1}","kind":"use","excerpt":"vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESTraditional paper: Theorem 6.3 (p. 14) Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical counterp","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d230c9e7fd4e8bb2","tex":"g_b","aliases":"g_b","display":"g_b","kind":"use","excerpt":"vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESTraditional paper: Theorem 6.3 (p. 14) Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical counterp","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ee47ed2cac150272","tex":"G_b^{-1}","aliases":"G_b^-1","display":"G_b^{-1}","kind":"use","excerpt":"Substitute the definition of Γ^G. Adjacent factors G_b^-1G_b and G_c^-1G_c cancel. In the defect, both the composite and direct terms acquire the same outer factors G_c and G_a^-1. In the loop, all inner factors cancel and only conjugation at the base remains.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d4413bc5f07c709f","tex":"g_b^{-1}","aliases":"g_b^-1","display":"g_b^{-1}","kind":"use","excerpt":"The route through two steps 0→1→2 has multiplier (3/2)2=3, equal to the direct route 0→2. More generally, (g_cg_b^-1)(g_bg_a^-1)=g_cg_a^-1 for every triple. Thus this connection is flat and reversible, but it is not calibrated to the original J, whose values are identities.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fd6dc28b158b5d9e","tex":"G_c","aliases":"G_c","display":"G_c","kind":"use","excerpt":"[Gauge covariance] The defect and triangle holonomy transform as Δ^G_c,b,a =G_cΔ_c,b,aG_a^-1, H^G_c,b,a =G_aH_c,b,aG_a^-1. Thus flatness and triviality of triangle holonomy are gauge invariant. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy","count":10},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a4676088e867baee","tex":"g_c","aliases":"g_c","display":"g_c","kind":"use","excerpt":"The route through two steps 0→1→2 has multiplier (3/2)2=3, equal to the direct route 0→2. More generally, (g_cg_b^-1)(g_bg_a^-1)=g_cg_a^-1 for every triple. Thus this connection is flat and reversible, but it is not calibrated to the original J, whose values are identities.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e041615222944e42","tex":"G_c^{-1}","aliases":"G_c^-1","display":"G_c^{-1}","kind":"use","excerpt":"Substitute the definition of Γ^G. Adjacent factors G_b^-1G_b and G_c^-1G_c cancel. In the defect, both the composite and direct terms acquire the same outer factors G_c and G_a^-1. In the loop, all inner factors cancel and only conjugation at the base remains.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8a0cea7293747747","tex":"G_f","aliases":"G_f","display":"G_f","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-88e67d7f2c866e98","tex":"G_{\\Rcal\\leftarrow\\Qcal}","aliases":"G_Rcal←Qcal G_ R← Q Rcal leftarrow Qcal mathcal leftarrow mathcal","display":"G_{{\\mathcal R}\\leftarrow{\\mathcal Q}}","kind":"use","excerpt":"[Exact nonlinear coordinate theorem] For every map F:X→ Y, S_ QF_ Q← P A_ P=F. If G:Y→ Z and R is a packet system on Z, then (G∘ F)_ R← P = G_ R← Q ∘ F_ Q← P. No linearity, continuity, or differentiability assumption is needed.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d20ed25f6bf4ba52","tex":"G_{v_0}","aliases":"G_v_0","display":"G_{v_0}","kind":"use","excerpt":"[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2a0c361763b69348","tex":"G_{v_0}^{-1}","aliases":"G_v_0^-1","display":"G_{v_0}^{-1}","kind":"use","excerpt":"[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9cd63cbb25af4182","tex":"G_{\\gamma,B}","aliases":"G_γ,B gamma","display":"G_{\\gamma,B}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9459f5036ab9479b","tex":"G_{\\gamma_1,B_1}","aliases":"G_γ_1,B_1 gamma","display":"G_{\\gamma_1,B_1}","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fa85e283567df49a","tex":"G_{\\gamma_k,B_k}","aliases":"G_γ_k,B_k gamma","display":"G_{\\gamma_k,B_k}","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-70196007a27c98a9","tex":"\\gr","aliases":"gr operatorname","display":"{\\operatorname{gr}}","kind":"use","excerpt":"i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-130720b204a25a33","tex":"\\gr^m","aliases":"gr^m gr operatorname","display":"{\\operatorname{gr}}^m","kind":"use","excerpt":"i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7680963df50a25c8","tex":"\\gr^p","aliases":"gr^p gr operatorname","display":"{\\operatorname{gr}}^p","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":64},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1e9de57bf29ef2a6","tex":"h'","aliases":"h'","display":"h'","kind":"use","excerpt":"If x'-x∈ F^mX, then filtration congruence preservation gives Φ(x')-Φ(x)∈ F^mY. Thus one defect lies in F^mY exactly when the other does. If h-h'∈ F^m+1X, then x_m+h and x_m+h' define the same depth m+1 class. Conversely, two depth m+1 classes reducing to [x_m]_m differ from x_m by elements of F^mX, and they are equal precisely when those elements a","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fa5a6573c905a228","tex":"H^0","aliases":"H^0","display":"H^0","kind":"use","excerpt":"[Cohomological obstruction through the cone] Let e∈ E^n satisfy d_Ee=0. The class of (e,0) in H^n(Cone(D)) vanishes if and only if the class [e]∈ H^n(E) lies in the image of H^n(D):H^n(C)⟶ H^n(E). When C and E are concentrated in degree zero, H^0(Cone(D))≅coker D.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"[Abelian cohomology class] When the structure group A is abelian, gauge classes of transports are naturally H^1(Γ,A) ≅ Hom(H_1(Γ, Z),A). After a basis of H_1(Γ, Z) is chosen, this group is identified with A^b_1.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":"[One step direct rotation estimate] Assume P(t) is twice continuously differentiable in operator norm on [0,T]. There are constants h_0>0 and C>0 such that, whenever t,t+h∈[0,T] and |h|≤ h_0, U( P(t+h), P(t)) - I - hK(t) ≤ Ch^2. The constants can be chosen uniformly in t.","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f3959f846ddb0340","tex":"H^D","aliases":"H^D","display":"H^D","kind":"use","excerpt":"vpage14Gauge changes the picture, not flatnessMODEL E / CONJUGATION AND ITS LIMITSTraditional paper: Theorem 6.2 (p. 14) At the base chart, change coordinates by D=diag(2,1). Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving H^D=DHD^-1=mat1 · 2 0 · 1. The shear looks stronger in these fixe","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-16dc3982dc7a54a1","tex":"H^G_{c,b,a}","aliases":"H^G_c,b,a","display":"H^G_{c,b,a}","kind":"use","excerpt":"[Gauge covariance] The defect and triangle holonomy transform as Δ^G_c,b,a =G_cΔ_c,b,aG_a^-1, H^G_c,b,a =G_aH_c,b,aG_a^-1. Thus flatness and triviality of triangle holonomy are gauge invariant. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7c6fa035c4defd46","tex":"H^n","aliases":"H^n","display":"H^n","kind":"use","excerpt":"[Cohomological obstruction through the cone] Let e∈ E^n satisfy d_Ee=0. The class of (e,0) in H^n(Cone(D)) vanishes if and only if the class [e]∈ H^n(E) lies in the image of H^n(D):H^n(C)⟶ H^n(E). When C and E are concentrated in degree zero, H^0(Cone(D))≅coker D.","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-643e255e13e716c4","tex":"H^{p+q}","aliases":"H^p+q","display":"H^{p+q}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"[One step direct rotation estimate] Assume P(t) is twice continuously differentiable in operator norm on [0,T]. There are constants h_0>0 and C>0 such that, whenever t,t+h∈[0,T] and |h|≤ h_0, U( P(t+h), P(t)) - I - hK(t) ≤ Ch^2. The constants can be chosen uniformly in t.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hol","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hol","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2]","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bfc7526d20d019a0","tex":"H_{2,1,0}","aliases":"H_2,1,0","display":"H_{2,1,0}","kind":"use","excerpt":"[The minimal API admits nontrivial holonomy] There is a connection on three points over Q for which transport 0→1 is multiplication by 2, every other transport between distinct endpoints is the identity, and H_2,1,0(1)=2. In particular H_2,1,0≠id. statusleanthreeKomorebi.PaperIII.Transport.twistedThreePointConnection; Komorebi.PaperIII.Transport.tw","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-333bf23be8bc780a","tex":"h_i","aliases":"h_i","display":"h_i","kind":"use","excerpt":"[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5632248249b75fcb","tex":"h_{i_1}","aliases":"h_i_1","display":"h_{i_1}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4f4509ed7d70840f","tex":"h_{i_s}","aliases":"h_i_s","display":"h_{i_s}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"use","excerpt":"For h_j∈ X_α_j, the inherited norm gives Q_γ D^rF(x)[h_1,…,h_r] ≤ Q_γ D^rF(x) ∏_j h_j . Taking the multilinear operator norm proves the sharper restricted estimate. For ambient inputs, replace h_j by P_α_jh_j and apply the operator norm bound to each projector. Every nonzero idempotent has norm at least one, since it fixes a nonzero vector in its r","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-846897bc1a1e9fc7","tex":"h_j^2","aliases":"h_j^2","display":"h_j^2","kind":"use","excerpt":"Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unita","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1b57183bf1f39412","tex":"H_{ji}","aliases":"H_ji","display":"H_{ji}","kind":"use","excerpt":"For a loop represented on one common finite scalar packet coordinate type, the proposition resolves its successive transport matrices into packet walks. If R is a ring, as in the reversible connection API, then (H-I)_ji=H_ji-δ_ji. In particular, off-diagonal entries are unchanged. Thus a nonzero off-diagonal holonomy obstruction in this model with ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-082db4c82cd55c3c","tex":"h_m","aliases":"h_m","display":"h_m","kind":"use","excerpt":"[One step lifting criterion] The approximation x_m has a correction h_m∈ F^mX such that Φ(x_m+h_m)∈ F^m+1Y if and only if Ob_m(x_m)=0. When a correction exists, the set of its classes in F^mX/F^m+1X is a coset of Keroverline L_m. If the graded pieces are vector spaces, this coset is an affine space.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f7e2b72a42d51897","tex":"h_m'","aliases":"h_m'","display":"h_m'","kind":"use","excerpt":"The next depth condition is equivalent to overline L_m[h_m] =-[Φ(x_m)] in F^mY/F^m+1Y. This is solvable exactly when the displayed defect has zero cokernel class. If h_m and h_m' are two solutions, their difference lies in Keroverline L_m. Adding an element of the kernel to one solution gives another, so the solution set is a coset of the kernel.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-785ae493560363c1","tex":"h_n","aliases":"h_n","display":"h_n","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-88eb9c29d4c1ef70","tex":"h_{n+1}","aliases":"h_n+1","display":"h_{n+1}","kind":"use","excerpt":"First derive the unprojected partition formula. At order one it is the ordinary chain rule. Suppose it holds at order n and the maps are C^n+1. Differentiating a summand in direction h_n+1 has two kinds of contribution. A derivative of the outer map creates the new singleton block {n+1}. A derivative of an inner block jet adjoins n+1 to that block.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3e01b0cedad21f47","tex":"h_r","aliases":"h_r","display":"h_r","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5ab62d5df4f5b229","tex":"\\Hol","aliases":"Hol operatorname","display":"{\\operatorname{Hol}}","kind":"use","excerpt":"[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-69ac3b37b8b2ac4d","tex":"\\Hol'","aliases":"Hol' Hol operatorname","display":"{\\operatorname{Hol}}'","kind":"use","excerpt":"[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aae322307284296b","tex":"\\Hol_A","aliases":"Hol_A Hol operatorname","display":"{\\operatorname{Hol}}_A","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-845b4f3714396b47","tex":"\\Hol_B","aliases":"Hol_B Hol operatorname","display":"{\\operatorname{Hol}}_B","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ae848303215cc76e","tex":"\\operatorname{Hull}","aliases":"Hull operatorname","display":"\\operatorname{Hull}","kind":"use","excerpt":"(5.03500,1.68500) 0; draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,0.65000) rectangle ++(0.69000,0.69000); node[] at (4.34500,0.99500) 0; draw[fill=black!0,draw=black!40,line width=.35pt,] (4.69000,0.65000) rectangle ++(0.69000,0.69000); node[] at (5.03500,0.99500) 0; node[] at (8.85000,3.80000) Reach(0)={0,1,2,3}; node[] at (8.85000,","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-33db90fe192c5a42","tex":"\\operatorname{Hull}_T","aliases":"Hull_T operatorname","display":"\\operatorname{Hull}_T","kind":"use","excerpt":"(5.03500,1.68500) 0; draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,0.65000) rectangle ++(0.69000,0.69000); node[] at (4.34500,0.99500) 0; draw[fill=black!0,draw=black!40,line width=.35pt,] (4.69000,0.65000) rectangle ++(0.69000,0.69000); node[] at (5.03500,0.99500) 0; node[] at (8.85000,3.80000) Reach(0)={0,1,2,3}; node[] at (8.85000,","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f18b3c49fd8055ae","tex":"i_0","aliases":"i_0","display":"i_0","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-eff6aa27c552c5d2","tex":"I_a","aliases":"I_a","display":"I_a","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-78210203a2849812","tex":"I_{H_2}","aliases":"I_H_2","display":"I_{H_2}","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hol","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2ee8ac8c8d5446ba","tex":"i_m","aliases":"i_m","display":"i_m","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-714b73df80781bff","tex":"i_{m-1}","aliases":"i_m-1","display":"i_{m-1}","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-98483a254fa93b89","tex":"i_s","aliases":"i_s","display":"i_s","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9d1279bf2b532c2f","tex":"i_U","aliases":"i_U","display":"i_U","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d3f6e419ae12a276","tex":"I_v","aliases":"I_v","display":"I_v","kind":"use","excerpt":"Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate pro","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a011c42e0a9e76fd","tex":"i_V","aliases":"i_V","display":"i_V","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ab3ef27be5e3d7f7","tex":"I_Y","aliases":"I_Y","display":"I_Y","kind":"use","excerpt":"The identities S_ P A_ P=I_X and S_ Q A_ Q=I_Y give S_ QF_ Q← P A_ P = S_ Q A_ QF S_ P A_ P =F. For composition, G_ R← QF_ Q← P = A_ RG S_ Q A_ QF S_ P = A_ R(G∘ F) S_ P.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bdedc0251851d395","tex":"I_Z","aliases":"I_Z","display":"I_Z","kind":"use","excerpt":"[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-09e35188271c2791","tex":"\\id_{\\Coord(\\cP^a)}","aliases":"id_Coord(cP^a) id_Coord( P^a) id Coord cP mathrm operatorname mathcal","display":"{\\mathrm{id}}_{{\\operatorname{Coord}}({\\mathcal P}^a)}","kind":"use","excerpt":"i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord(","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-de717c99c15e8582","tex":"\\id_X","aliases":"id_X id mathrm","display":"{\\mathrm{id}}_X","kind":"use","excerpt":"i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord(","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1491268eb17c25c7","tex":"\\id_{X_v}","aliases":"id_X_v id mathrm","display":"{\\mathrm{id}}_{X_v}","kind":"use","excerpt":"Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate pro","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d09ca7ae57dbf7ab","tex":"J^1_{\\gamma_1\\mid\\alpha_1}","aliases":"J^1_γ_1midα_1 gamma mid alpha","display":null,"kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d5008abe1661077a","tex":"J^1_{\\gamma_2\\mid\\alpha_2}","aliases":"J^1_γ_2midα_2 gamma mid alpha","display":null,"kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-835bd29514aa36c8","tex":"J^1_{\\delta\\mid\\gamma}","aliases":"J^1_δmidγ delta mid gamma","display":null,"kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-049a2e2ed97c4350","tex":"J^2_{\\gamma\\mid\\alpha_1,\\alpha_2}","aliases":"J^2_γmidα_1,α_2 gamma mid alpha alpha","display":null,"kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b44d4f0935dc442b","tex":"J^2_{\\delta\\mid\\gamma_1,\\gamma_2}","aliases":"J^2_δmidγ_1,γ_2 delta mid gamma gamma","display":null,"kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a62392eb5eb3d856","tex":"J^{k}_{\\delta\\mid\\gamma_1,\\ldots,\\gamma_k}","aliases":"J^k_δmidγ_1,…,γ_k delta mid gamma ldots gamma","display":null,"kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2e519f6e43bfb5bd","tex":"J^{n}_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_n}","aliases":"J^n_δmidα_1,…,α_n delta mid alpha ldots alpha","display":null,"kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8eed2d16e16facf1","tex":"J^r_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}","aliases":"J^r_γmidα_1,…,α_r gamma mid alpha ldots alpha","display":null,"kind":"use","excerpt":"[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4f8c22fed2ef52fb","tex":"J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}","aliases":"J^r_δmidα_1,…,α_r delta mid alpha ldots alpha","display":null,"kind":"use","excerpt":"[Structural pruning] For a map H with fixed input and output packet systems, let supp_r H(x) be the set of external profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-89d0ef08e1ef93e7","tex":"J^s_{\\gamma\\mid\\alpha_{i_1},\\ldots,\\alpha_{i_s}}","aliases":"J^s_γmidα_i_1,…,α_i_s gamma mid alpha ldots alpha","display":null,"kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-934d75bba0abbd11","tex":"J^X_{a\\leftarrow a}","aliases":"J^X_a← a leftarrow","display":"J^X_{a\\leftarrow a}","kind":"use","excerpt":"[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoo","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-56e18f88fc4e1759","tex":"J^X_{a\\leftarrow b}","aliases":"J^X_a← b leftarrow","display":"J^X_{a\\leftarrow b}","kind":"use","excerpt":"[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoo","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-061ecdb43086417a","tex":"J^X_{a\\leftarrow c}","aliases":"J^X_a← c leftarrow","display":"J^X_{a\\leftarrow c}","kind":"use","excerpt":"[Canonical triangle holonomy is trivial] For every triple a,b,c, J^X_a← cJ^X_c← bJ^X_b← a=id. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.refineCoordinates_self","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1f8a9e76041833cb","tex":"J^X_{c\\leftarrow a}","aliases":"J^X_c← a leftarrow","display":"J^X_{c\\leftarrow a}","kind":"use","excerpt":"[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoo","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-274d53e819de8ecb","tex":"J^X_{c\\leftarrow b}","aliases":"J^X_c← b leftarrow","display":"J^X_{c\\leftarrow b}","kind":"use","excerpt":"[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoo","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7fc6f897ed8e4ebf","tex":"J^Y_{s\\leftarrow r}","aliases":"J^Y_s← r leftarrow","display":"J^Y_{s\\leftarrow r}","kind":"use","excerpt":"[Canonical coordinate naturality] The exact coordinate representation preserves typed composition and satisfies J^Y_s← rwidehat F_r← a = widehat F_s← bJ^X_b← a. statusleanpriorKomorebi.PaperII.NestedRefinement.coordinateOperator_comp; Komorebi.PaperII.NestedRefinement.coordinateOperator_naturality","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1dd01de180ee6039","tex":"J_{(a_0)}","aliases":"J_(a_0)","display":"J_{(a_0)}","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2587253da8db636d","tex":"J_{a_1\\leftarrow a_0}","aliases":"J_a_1← a_0 leftarrow","display":"J_{a_1\\leftarrow a_0}","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4b3065ac5478799d","tex":"J_{a_n\\leftarrow a_0}","aliases":"J_a_n← a_0 leftarrow","display":"J_{a_n\\leftarrow a_0}","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4d6d693ea2042a00","tex":"J_{a_n\\leftarrow a_{n-1}}","aliases":"J_a_n← a_n-1 leftarrow","display":"J_{a_n\\leftarrow a_{n-1}}","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8a5d3aa52b0f5bb8","tex":"J_{a\\leftarrow a}","aliases":"J_a← a leftarrow","display":"J_{a\\leftarrow a}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-777ddad06f846983","tex":"J_{a\\leftarrow b}","aliases":"J_a← b leftarrow","display":"J_{a\\leftarrow b}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-54a89b74eb1326aa","tex":"J_{a\\leftarrow c}","aliases":"J_a← c leftarrow","display":"J_{a\\leftarrow c}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9052d9414d7b8da7","tex":"J_{a\\leftarrow o}","aliases":"J_a← o leftarrow","display":"J_{a\\leftarrow o}","kind":"use","excerpt":"Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ab66a2af2c46ac12","tex":"J_{c\\leftarrow a}","aliases":"J_c← a leftarrow","display":"J_{c\\leftarrow a}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e0f23abc79bf97b6","tex":"J_{c\\leftarrow b}","aliases":"J_c← b leftarrow","display":"J_{c\\leftarrow b}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b1f0bab3ffea322d","tex":"J_{o\\leftarrow a}","aliases":"J_o← a leftarrow","display":"J_{o\\leftarrow a}","kind":"use","excerpt":"[Flat connections are gauges of the canonical pair transport] Suppose the C_a are exact packet coordinate modules for one ambient module, so the canonical J_b← a is available. If Γ is a flat normalized reversible pair connection, choose o∈ B and set G_a=Γ_a← oJ_o← a. Then every pair transport satisfies Γ_b← a =G_bJ_b← aG_a^-1. statusleanthreeKomore","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-beacfac56a561ebc","tex":"J_{o\\leftarrow b}","aliases":"J_o← b leftarrow","display":"J_{o\\leftarrow b}","kind":"use","excerpt":"Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a · = Γ_b← oΓ_o← a · = Γ_b← a. The first reduction uses the canonical cocycle twice; the second uses flatness of Γ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ebbe8be9d940a9d3","tex":"J_p","aliases":"J_p","display":"J_p","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1788eb72d6d4271c","tex":"J_{\\Pcal\\leftarrow\\Pcal'}","aliases":"J_Pcal←Pcal' J_ P← P' Pcal leftarrow Pcal mathcal leftarrow mathcal","display":"J_{{\\mathcal P}\\leftarrow{\\mathcal P}'}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a770760ae72cbdb3","tex":"J_{\\Qcal'\\leftarrow\\Qcal}","aliases":"J_Qcal'←Qcal J_ Q'← Q Qcal leftarrow Qcal mathcal leftarrow mathcal","display":"J_{{\\mathcal Q}'\\leftarrow{\\mathcal Q}}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1f9e5419185ac3d4","tex":"J_{\\Ucal\\leftarrow\\Vcal}","aliases":"J_Ucal←Vcal J_ U← V Ucal leftarrow Vcal mathcal leftarrow mathcal","display":"J_{{\\mathcal U}\\leftarrow{\\mathcal V}}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-997c382b71c367a6","tex":"K^*","aliases":"K^*","display":"K^*","kind":"use","excerpt":"Differentiating P_α^2=P_α gives P_αdot P_α P_α=0. Differentiating P_β P_α=0 for α≠β gives P_βdot P_α=-dot P_β P_α. Hence K P_β=dot P_β P_β and P_β K=-dot P_β(I-P_β). Their difference is dot P_β. Also, K+K^* = ∑_α(dot P_α P_α+P_αdot P_α) = ∑_αdot P_α =0. endgroup","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1bdfdabfb7ccb3ec","tex":"k_1","aliases":"k_1","display":"k_1","kind":"use","excerpt":"[Quadratic reduced obstruction] Assume the hypotheses of [source reference] with r≥2. For k_1,k_2∈ K, D^2κ(0)[k_1,k_2] = π_O D^2F(0)[k_1,k_2].","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a8cd867fa822df3a","tex":"k_2","aliases":"k_2","display":"k_2","kind":"use","excerpt":"[Quadratic reduced obstruction] Assume the hypotheses of [source reference] with r≥2. For k_1,k_2∈ K, D^2κ(0)[k_1,k_2] = π_O D^2F(0)[k_1,k_2].","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e08d6b60be0ec0b0","tex":"\\Ker","aliases":"Ker operatorname","display":"{\\operatorname{Ker}}","kind":"use","excerpt":"A linear correction map D:U⟶ V places the obstruction to solving Du=-r in the canonical quotient coker D. This quotient has a universal property and behaves naturally under commuting squares. Mapping cones extend the same construction to cochain complexes. In a Hilbert space with closed range, the quotient is represented by the orthogonal projectio","count":19},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"The range X_α is closed because X_α=ker(I-P_α). It is therefore Banach with its inherited norm. The same holds for every Y_γ. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from [source ref","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"[Local Lyapunov Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ec6b4ea000366681","tex":"L_f","aliases":"L_f","display":"L_f","kind":"use","excerpt":"Words are written in execution order. In L_fL_e, the rightmost operator L_e acts first. For a written word (e_1,…,e_n), e_1 acts first and the operator product is L_e_n⋯ L_e_1.","count":17},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"use","excerpt":"Suppose x_m satisfies Φ(x_m)∈ F^mY. Assume that corrections h∈ F^mX obey a first order congruence Φ(x_m+h) ≡ Φ(x_m)+L_mh pmodF^m+1Y, where L_m:F^mX→ F^mY is a module homomorphism satisfying L_m(F^m+1X)⊆ F^m+1Y and therefore induces overline L_m: F^mX/F^m+1X ⟶ F^mY/F^m+1Y.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a1266df2f6d9de73","tex":"L_m'","aliases":"L_m'","display":"L_m'","kind":"use","excerpt":"[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1c5400967fcce1ea","tex":"L_\\infty","aliases":"L_∞ infty","display":"L_\\infty","kind":"use","excerpt":"[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3755b02b61222ecc","tex":"\\Lcal","aliases":"Lcal L mathcal","display":"{\\mathcal L}","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":13},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-61ce5ee12796fd2c","tex":"\\Lcal^1","aliases":"Lcal^1 L^1 Lcal mathcal","display":"{\\mathcal L}^1","kind":"use","excerpt":"lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to i","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e51b117ccc60377f","tex":"\\Lcal^2","aliases":"Lcal^2 L^2 Lcal mathcal","display":"{\\mathcal L}^2","kind":"use","excerpt":"[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The o","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c24e3ef0e3320585","tex":"\\Lcal_{2g}","aliases":"Lcal_2g L_2g Lcal mathcal","display":"{\\mathcal L}_{2g}","kind":"use","excerpt":"[Divided square packet equations] Assume q( L_g)⊆ L_2g. Choose an ordering of the finite label set. The equation dη+q(η)=0 is equivalent to dη_k + ∑_2g=kq(η_g) + ∑_substackg<h","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3dd439a91ab48b70","tex":"\\Lcal_g","aliases":"Lcal_g L_g Lcal mathcal","display":"{\\mathcal L}_g","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-62dad86562132d65","tex":"\\Lcal_{g+h}","aliases":"Lcal_g+h L_g+h Lcal mathcal","display":"{\\mathcal L}_{g+h}","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9b6e9b85d2ea26de","tex":"\\Lcal_h","aliases":"Lcal_h L_h Lcal mathcal","display":"{\\mathcal L}_h","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ce82d7e1785ef3e8","tex":"\\Lcal_k","aliases":"Lcal_k L_k Lcal mathcal","display":"{\\mathcal L}_k","kind":"use","excerpt":"The differential preserves the group label and the bracket adds labels. Apply the projector onto L_k and use exact reconstruction.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"Let X and Y be modules with decreasing filtrations X=F^0X⊇ F^1X⊇⋯, Y=F^0Y⊇ F^1Y⊇⋯. Assume both are separated and complete for their filtration topologies. Let Φ:X→ Y be continuous and filtration congruence preserving in the sense that h∈ F^mX Longrightarrow Φ(x+h)-Φ(x)∈ F^mY for every x∈ X and every m. This condition makes the statement Φ(x)∈ F^mY ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-05e170b594a4705a","tex":"M^k","aliases":"M^k","display":"M^k","kind":"use","excerpt":"[Number of labeled composition terms] For n≥1, if Y has M packets, then the number of packet labeled terms at order n before zero jets are removed is N_n,M = ∑_k=1^nS(n,k)M^k, where S(n,k) is a Stirling number of the second kind. Thus N_n,M is the nth Touchard polynomial evaluated at M.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"vpage5Coefficients need not commuteMODEL A / A MATRIX RING AS THE COEFFICIENT SEMIRINGTraditional paper: Lemma 3.7 (p. 5), Section 9 (p. 20) The edge theorem allows a noncommutative coefficient semiring. Use R=M_2(QQ) and let the two coefficients themselves be matrices: A=mat1 · 1","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0901f1f56ba2bc5e","tex":"M_N","aliases":"M_N","display":"M_N","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-956f614ae0645ab5","tex":"\\mat","aliases":"mat","display":null,"kind":"use","excerpt":"The surviving words are ac,bc,cd,da,db. Their coefficients are 6,-3,3,2,-1, respectively. Their endpoints determine where these numbers enter the matrix: T=mat0 · 0 · 1 1 · 0 · 0 0 · 3 · 0, T^2=mat0 · 3 · 0 0 · 0 · 1 3 · 0 · 0. At row 2, column 0, the two parallel choices give 6-3=3. Deduplicating the edge labels before summing would alter the oper","count":11},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-90fc7282920196de","tex":"\\mathllap","aliases":"mathllap","display":null,"kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-905bf7c29079e31a","tex":"\\mathord","aliases":"mathord","display":null,"kind":"use","excerpt":"Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one dep","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ae66207125e503b8","tex":"\\mathrlap","aliases":"mathrlap","display":null,"kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"It is therefore Banach with its inherited norm. The same holds for every Y_γ. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from [source reference] make them bounded linear isomorphisms. F","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f2f35edd4ea4be66","tex":"\\max_\\alpha","aliases":"max_α max alpha","display":"\\max_\\alpha","kind":"use","excerpt":"It is therefore Banach with its inherited norm. The same holds for every Y_γ. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from [source reference] make them bounded linear isomorphisms. F","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"[Number of labeled composition terms] For n≥1, if Y has M packets, then the number of packet labeled terms at order n before zero jets are removed is N_n,M = ∑_k=1^nS(n,k)M^k, where S(n,k) is a Stirling number of the second kind. Thus N_n,M is the nth Touchard polynomial evaluated at M.","count":16},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-62782243b1cfc0cb","tex":"N_{n,M}","aliases":"N_n,M","display":"N_{n,M}","kind":"use","excerpt":"[Number of labeled composition terms] For n≥1, if Y has M packets, then the number of packet labeled terms at order n before zero jets are removed is N_n,M = ∑_k=1^nS(n,k)M^k, where S(n,k) is a Stirling number of the second kind. Thus N_n,M is the nth Touchard polynomial evaluated at M.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1d7d1fdf60d8fdac","tex":"\\neg","aliases":"neg","display":"\\neg","kind":"use","excerpt":"[Exact logical force of a local obstruction] Every obstruction packet for x proves that x is not a solution, and negIsSolution_r(x) Longleftrightarrow there exists an obstruction packet for x. The reverse implication from failure to a packet witness is classical and does not require the packet type to be finite. statusleanthreeKomorebi.PaperIII.Non","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[Local Lyapunov Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"[Flat connections are gauges of the canonical pair transport] Suppose the C_a are exact packet coordinate modules for one ambient module, so the canonical J_b← a is available. If Γ is a flat normalized reversible pair connection, choose o∈ B and set G_a=Γ_a← oJ_o← a. Then every pair transport satisfies Γ_b← a =G_bJ_b← aG_a^-1. statusleanthreeKomore","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c35c2b5d63deb8c7","tex":"\\mathrm{ob}","aliases":"ob mathrm","display":"\\mathrm{ob}","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hol","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9de829476b80a2aa","tex":"\\Ob^G","aliases":"Ob^G Ob operatorname","display":"{\\operatorname{Ob}}^G","kind":"use","excerpt":"[Gauge covariance when packet projectors are preserved] If G_aΠ^a_μ=Π^a_μ G_a for every μ, then (Ob^G)^μν = G_aOb^μνG_a^-1. Consequently the zero/nonzero status of each obstruction block is preserved by gauges that preserve the packet projectors. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_of_commutes; Komorebi.PaperIII.H","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7b2fbd707f34da9d","tex":"\\Ob^{\\mu\\nu}","aliases":"Ob^μν Ob mu nu operatorname mu nu","display":"{\\operatorname{Ob}}^{\\mu\\nu}","kind":"use","excerpt":"[Gauge covariance when packet projectors are preserved] If G_aΠ^a_μ=Π^a_μ G_a for every μ, then (Ob^G)^μν = G_aOb^μνG_a^-1. Consequently the zero/nonzero status of each obstruction block is preserved by gauges that preserve the packet projectors. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_of_commutes; Komorebi.PaperIII.H","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7acafa494704338e","tex":"\\Ob_{D'}","aliases":"Ob_D' Ob operatorname","display":"{\\operatorname{Ob}}_{D'}","kind":"use","excerpt":"U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1ae97ea283061d02","tex":"\\Ob_m'","aliases":"Ob_m' Ob operatorname","display":"{\\operatorname{Ob}}_m'","kind":"use","excerpt":"[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-54037e9a059bedc5","tex":"\\overline B","aliases":"overline B overline","display":"\\overline B","kind":"use","excerpt":"[Polynomial tree expansion] Suppose there is R>0 such that ϕ(R)≤ R and q= ∑_r=2^dr b_rR^r-1<1. Then the map Φ(x)=s+∑_r=2^dB_r(x,…,x) has a unique fixed point in the closed ball overline B_R(0). The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7da834df65459b68","tex":"\\overline b","aliases":"overline b overline","display":"\\overline b","kind":"use","excerpt":"U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ce8712e6c17bb593","tex":"\\overline B_R","aliases":"overline B_R overline","display":"\\overline B_R","kind":"use","excerpt":"[Polynomial tree expansion] Suppose there is R>0 such that ϕ(R)≤ R and q= ∑_r=2^dr b_rR^r-1<1. Then the map Φ(x)=s+∑_r=2^dB_r(x,…,x) has a unique fixed point in the closed ball overline B_R(0). The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d5be848bd2e45cb7","tex":"\\overline B_\\rho","aliases":"overline B_ρ overline rho","display":"\\overline B_\\rho","kind":"use","excerpt":"[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in ","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-42c6158ae78541bc","tex":"\\overline Q","aliases":"overline Q overline","display":"\\overline Q","kind":"use","excerpt":"[Projectors on the obstruction quotient] Let (Q_α)_α∈ I be a complete family of pairwise annihilating idempotents on V. Assume Q_α(im D)⊆im D for every α. Then each Q_α induces an idempotent overline Q_α on coker D, and coker D = ⊕_α∈ Ioverline Q_α(coker D). For every defect r∈ V, overline Q_αOb_D(r) = [Q_α r]. No packet decomposition of the correc","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fd4837de74cbe911","tex":"\\overline Q_\\alpha","aliases":"overline Q_α overline alpha","display":"\\overline Q_\\alpha","kind":"use","excerpt":"[Projectors on the obstruction quotient] Let (Q_α)_α∈ I be a complete family of pairwise annihilating idempotents on V. Assume Q_α(im D)⊆im D for every α. Then each Q_α induces an idempotent overline Q_α on coker D, and coker D = ⊕_α∈ Ioverline Q_α(coker D). For every defect r∈ V, overline Q_αOb_D(r) = [Q_α r]. No packet decomposition of the correc","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-683c7ddceeb76bac","tex":"\\overline\\varphi","aliases":"overlineφ overline varphi","display":"\\overline\\varphi","kind":"use","excerpt":"[Cokernel obstruction criterion] A linear correction exists if and only if Ob_D(r)=0. The quotient map q_D:V⟶coker D has the following universal property. If φ:V→ W is linear and φ D=0, then there is a unique linear map overlineφ:coker D⟶ W with φ=overlineφ q_D.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","excerpt":"[htbp] centering includegraphics[width=.96linewidth]rotating_projector_transport.pdf captionA rotating rank one packet and its orthogonal complement. With u(θ)=(cosθ,sinθ) and P(θ)=u(θ)u(θ)^ T, the canonical generator is [P',P]=(0 · -1 1 · 0). For a positive step δ<π/2, the polar unitary is the usual rotation R(δ). On each displayed mesh, the direc","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e6dd08fc29eea38a","tex":"P^a_i","aliases":"P^a_i","display":"P^a_i","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3c8bb8a1c16cf6d8","tex":"P^a_j","aliases":"P^a_j","display":"P^a_j","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b4cb38d7eaa8bc03","tex":"P^v_i","aliases":"P^v_i","display":"P^v_i","kind":"use","excerpt":"Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate pro","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"vpage16Resolve the entire obstructionMODEL F / PACKET BLOCKS FROM INPUT TO OUTPUTTraditional paper: Theorem 7.5 (p. 17), Corollary 7.6 (p. 18) Use three coordinate packets with projectors P_0,P_1,P_2, and set H=I+2E_01+3E_12, Ob=H-I. Here H is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside ","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"vpage16Resolve the entire obstructionMODEL F / PACKET BLOCKS FROM INPUT TO OUTPUTTraditional paper: Theorem 7.5 (p. 17), Corollary 7.6 (p. 18) Use three coordinate packets with projectors P_0,P_1,P_2, and set H=I+2E_01+3E_12, Ob=H-I. Here H is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside ","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"vpage16Resolve the entire obstructionMODEL F / PACKET BLOCKS FROM INPUT TO OUTPUTTraditional paper: Theorem 7.5 (p. 17), Corollary 7.6 (p. 18) Use three coordinate packets with projectors P_0,P_1,P_2, and set H=I+2E_01+3E_12, Ob=H-I. Here H is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"-1 · 1. For example, the original (0,0) block vanishes, while the new (0,0) block relative to the old coordinate projectors does not. This is why the theorem about preservation of each block's zero status requires G P_i=P_iG.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"headTransporting the packet frame as well Define Q_i=GP_iG^-1. Then the identities cancel exactly: Q_i(GOb G^-1)Q_j=G(P_iOb P_j)G^-1. Relative to these transported packets, the corresponding labeled blocks again have matching vanishing status. These are the same oblique projectors displayed on V17. One must distinguish moving the operator alone in ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d1cac30b985bfa29","tex":"P_{\\mathrm{ob}}","aliases":"P_ob mathrm","display":"P_{\\mathrm{ob}}","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hol","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9a4173e2450e265f","tex":"p_U","aliases":"p_U","display":"p_U","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cd728d302d0a06c1","tex":"p_V","aliases":"p_V","display":"p_V","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-634bde01ca310a18","tex":"P_\\alpha^2","aliases":"P_α^2 alpha","display":"P_\\alpha^2","kind":"use","excerpt":"Differentiating P_α^2=P_α gives P_αdot P_α P_α=0. Differentiating P_β P_α=0 for α≠β gives P_βdot P_α=-dot P_β P_α. Hence K P_β=dot P_β P_β and P_β K=-dot P_β(I-P_β). Their difference is dot P_β. Also, K+K^* = ∑_α(dot P_α P_α+P_αdot P_α) = ∑_αdot P_α =0. endgroup","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8fae8ad97b2a4d74","tex":"P_{\\alpha_1}","aliases":"P_α_1 alpha","display":"P_{\\alpha_1}","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-70e6a72cdb8f81e7","tex":"P_{\\alpha_j}","aliases":"P_α_j alpha","display":"P_{\\alpha_j}","kind":"use","excerpt":"[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-24058bd88b2e4102","tex":"P_{\\alpha_r}","aliases":"P_α_r alpha","display":"P_{\\alpha_r}","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aad7607e608d0052","tex":"P_\\beta","aliases":"P_β beta","display":"P_\\beta","kind":"use","excerpt":"A finite family of pairwise annihilating idempotents gives exact coordinates. If P_α P_β=0 for α≠β, ∑_α P_α=I, then every vector is the sum of its packet components. Linear operators become block matrices in these coordinates. A nonlinear map cannot be reconstructed from its diagonal restrictions, since mixed interactions between packet components ","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-00d2ac2360124aae","tex":"P_\\mu","aliases":"P_μ mu","display":"P_\\mu","kind":"use","excerpt":"vfig[.95]all_obstruction_blocks platecaptionThe board on the right contains all nine actual embedded blocks P_μOb P_ν. Row labels are output packets and column labels are input packets. Only the addresses (0,1) and (1,2) carry nonzero maps.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-577132b0e435e8fc","tex":"P_\\nu","aliases":"P_ν nu","display":"P_\\nu","kind":"use","excerpt":"vfig[.95]all_obstruction_blocks platecaptionThe board on the right contains all nine actual embedded blocks P_μOb P_ν. Row labels are output packets and column labels are input packets. Only the addresses (0,1) and (1,2) carry nonzero maps.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c5065376efd33e45","tex":"\\mathsf{Packet}","aliases":"Packet mathsf","display":"\\mathsf{Packet}","kind":"use","excerpt":"Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6ab8f7d516126c87","tex":"\\Pcal'","aliases":"Pcal' P' Pcal mathcal","display":"{\\mathcal P}'","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"Suppose x_m satisfies Φ(x_m)∈ F^mY. Assume that corrections h∈ F^mX obey a first order congruence Φ(x_m+h) ≡ Φ(x_m)+L_mh pmodF^m+1Y, where L_m:F^mX→ F^mY is a module homomorphism satisfying L_m(F^m+1X)⊆ F^m+1Y and therefore induces overline L_m: F^mX/F^m+1X ⟶ F^mY/F^m+1Y.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3f1b6dc519af96a7","tex":"\\prime","aliases":"prime","display":"\\prime","kind":"use","excerpt":"The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I; uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"[The minimal API admits nontrivial holonomy] There is a connection on three points over Q for which transport 0→1 is multiplication by 2, every other transport between distinct endpoints is the identity, and H_2,1,0(1)=2. In particular H_2,1,0≠id. statusleanthreeKomorebi.PaperIII.Transport.twistedThreePointConnection; Komorebi.PaperIII.Transport.tw","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-102bb645d8547d75","tex":"q^n","aliases":"q^n","display":"q^n","kind":"use","excerpt":"rated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e9a4386c240576be","tex":"q^{n-1-j}","aliases":"q^n-1-j","display":"q^{n-1-j}","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"use","excerpt":"headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6738621edb3fbe5f","tex":"Q_0^\\mathsf","aliases":"Q_0^ mathsf","display":"Q_0^\\mathsf","kind":"use","excerpt":"headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b9725921b466f523","tex":"Q_0^{\\mathsf T}","aliases":"Q_0^ T mathsf","display":"Q_0^{\\mathsf T}","kind":"use","excerpt":"(11.02000,3.36000) 0; draw[fill=black!12,draw=black!40,line width=.35pt,] (10.00000,2.34000) rectangle ++(0.68000,0.68000); node[] at (10.34000,2.68000) -1; draw[fill=black!12,draw=black!40,line width=.35pt,] (10.68000,2.34000) rectangle ++(0.68000,0.68000); node[] at (11.02000,2.68000) 1; node[] at (9.00000,0.85000) Q_0+Q_1=I, Q_i^2=Q_i; node[] at","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"use","excerpt":"headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-842201207aa7200d","tex":"q_D","aliases":"q_D","display":"q_D","kind":"use","excerpt":"[Cokernel obstruction criterion] A linear correction exists if and only if Ob_D(r)=0. The quotient map q_D:V⟶coker D has the following universal property. If φ:V→ W is linear and φ D=0, then there is a unique linear map overlineφ:coker D⟶ W with φ=overlineφ q_D.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ccf3bc5017cc6d9c","tex":"q_{D'}","aliases":"q_D'","display":"q_{D'}","kind":"use","excerpt":"The relation bD=D'a shows that the quotient map q_D'b vanishes on im D. Apply the universal property in [source reference].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5950b0378700bfba","tex":"Q_i","aliases":"Q_i","display":"Q_i","kind":"use","excerpt":"headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d5842aab9955ad05","tex":"Q_i^2","aliases":"Q_i^2","display":"Q_i^2","kind":"use","excerpt":"headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bb74470e10b0fa87","tex":"q_\\ell","aliases":"q_ℓ ell","display":"q_\\ell","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5bf9af43ed9be269","tex":"\\Qcal'","aliases":"Qcal' Q' Qcal mathcal","display":"{\\mathcal Q}'","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a2b27e6c1de4e544","tex":"\\qedhere","aliases":"qedhere","display":null,"kind":"use","excerpt":"For a section s satisfying Ps=s, ∇^2s=P d(P ds)=P(dP)wedge ds. Since ds=(dP)s+Pds and P(dP)P=0, the surviving term is P(dPwedge dP)Ps.qedhere","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ","display":null,"kind":"use","excerpt":"vpage5Coefficients need not commuteMODEL A / A MATRIX RING AS THE COEFFICIENT SEMIRINGTraditional paper: Lemma 3.7 (p. 5), Section 9 (p. 20) The edge theorem allows a noncommutative coefficient semiring. Use R=M_2(QQ) and let the two coefficients themselves be matrices: A=mat1 · 1","count":14},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b9a6d997b7294e64","tex":"\\QQ^2","aliases":"QQ^2 QQ","display":null,"kind":"use","excerpt":"vpage8Paths through changing packet listsMODEL B / RECTANGULAR PRODUCTS WITH TWO OR THREE STEPSTraditional paper: Proposition 8.1 (p. 19) Here the packet lists have sizes 2,3,2,1 in succession. Let the actual maps be QQ^2xrightarrowAQQ^3xrightarrowBQQ^2xrightarrowCQQ. Unlike a power of one matrix, each step has its own entries and its own source an","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-279caa0ce5bda7c4","tex":"\\QQ^3","aliases":"QQ^3 QQ","display":null,"kind":"use","excerpt":"vpage8Paths through changing packet listsMODEL B / RECTANGULAR PRODUCTS WITH TWO OR THREE STEPSTraditional paper: Proposition 8.1 (p. 19) Here the packet lists have sizes 2,3,2,1 in succession. Let the actual maps be QQ^2xrightarrowAQQ^3xrightarrowBQQ^2xrightarrowCQQ. Unlike a power of one matrix, each step has its own entries and its own source an","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-01a1093edb4895af","tex":"\\QQ^5","aliases":"QQ^5 QQ","display":null,"kind":"use","excerpt":"vpage7Reachability is only an upper boundMODEL A / AN ORBIT THAT STAYS IN TWO DIMENSIONSTraditional paper: Corollary 3.11 (p. 6) Keep the diamond and add an isolated packet 4. Start with the single seed e_0∈QQ^5. Direct multiplication gives Te_0=e_1+e_2, T^2e_0=0. Every later orbit vector is zero. The orbit hull generated by T is therefore exactly ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"Let X and Y be Banach spaces over the same field R or C and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr echet derivatives and the Taylor integral below has a real parameter. Assume that F:X→ Y is m times continuously Fr echet differentiable. The same local statements hold on open domains cont","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"in V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"Assume now that a>0 and b>0. Take R= frac1-sqrt1-4ab2b. Then a+bR^2=R and 2bR<1. Apply [source reference]. The number of full ordered binary trees with n leaves is the Catalan number C_n-1, which gives the displayed majorant.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a3d472d51734fedb","tex":"\\mathbb R^2","aliases":"R^2 mathbb","display":"\\mathbb R^2","kind":"use","excerpt":"[htbp] centering includegraphics[width=.96linewidth]correction_cokernel_plane.pdf captionA specified correction range in R^2. For L(a,b)=(a,0), the image is the horizontal axis. The defect (2,3) splits into (2,0) in that image and the remaining representative (0,3). Subtracting the image component leaves the same cokernel class. The functional (x,y","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ee18dedc5a126f93","tex":"R^{r-1}","aliases":"R^r-1","display":"R^{r-1}","kind":"use","excerpt":"[Polynomial tree expansion] Suppose there is R>0 such that ϕ(R)≤ R and q= ∑_r=2^dr b_rR^r-1<1. Then the map Φ(x)=s+∑_r=2^dB_r(x,…,x) has a unique fixed point in the closed ball overline B_R(0). The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f06eb326e26feef5","tex":"R^V","aliases":"R^V","display":"R^V","kind":"use","excerpt":"[Orbit support obstruction] Let S⊆ V, and let x∈ R^V be supported on S. If j is not reachable by a directed path from any vertex of S, then (T^nx)_j=0 for every n∈ N. Hence the orbit under T of x cannot generate a vector with a nonzero coordinate at j. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Reachable; Komorebi.PaperII","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-28973fc71dc16dde","tex":"R^{V\\times V}","aliases":"R^V× V times","display":"R^{V\\times V}","kind":"use","excerpt":"The passage from words to walks also applies to products of different matrices. Let V be one finite packet type, let R be a semiring, and let B_1,…,B_m∈ R^V× V be a finite sequence of possibly different packet matrices. Rows are targets and columns are sources as before.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9bb48203991c2d2f","tex":"r_f","aliases":"r_f","display":"r_f","kind":"use","excerpt":"[Coarse vanishing and fine vanishing] Let [r_f]∈coker D_f. If its image in coker D_c is zero, then [r_f] comes from coker D_0. Coarse vanishing therefore means that the fine obstruction is supported in information removed by coarsening. It does not imply [r_f]=0.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b2cf9f3a4501d1a5","tex":"R_{m+1}","aliases":"R_m+1","display":"R_{m+1}","kind":"use","excerpt":"[Packet Taylor formula] Assume F is m+1 times continuously Fr echet differentiable on an open set containing the segment x+th for 0≤ t≤1. Then Q_γ F(x+h) = · Q_γ F(x) + ∑_r=1^mfrac1r! ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h,…,P_α_rh] · +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] d","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-937de87a1560a7e8","tex":"\\mathcal R_r","aliases":"R_r mathcal","display":"\\mathcal R_r","kind":"use","excerpt":"in V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-68df6b2bfb142937","tex":"r_\\alpha","aliases":"r_α alpha","display":"r_\\alpha","kind":"use","excerpt":"of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6b98833cae963298","tex":"R_\\delta","aliases":"R_δ delta","display":"R_\\delta","kind":"use","excerpt":"Insert I_Y=∑_γ Q_γ in each argument of the outer derivative and expand the finite sums by multilinearity. Apply R_δ to the output. The directions already lie in the declared input ranges, and each intermediate projection lands in its literal output range. The resulting summands are exactly the displayed packet jets. List directions within a block i","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3bce33d4a711b894","tex":"\\Rcal","aliases":"Rcal R mathcal","display":"{\\mathcal R}","kind":"use","excerpt":"[Exact nonlinear coordinate theorem] For every map F:X→ Y, S_ QF_ Q← P A_ P=F. If G:Y→ Z and R is a packet system on Z, then (G∘ F)_ R← P = G_ R← Q ∘ F_ Q← P. No linearity, continuity, or differentiability assumption is needed.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bf4c43a79117c6ae","tex":"\\operatorname{Reach}","aliases":"Reach operatorname","display":"\\operatorname{Reach}","kind":"use","excerpt":"h=.35pt,] (4.69000,1.34000) rectangle ++(0.69000,0.69000); node[] at (5.03500,1.68500) 0; draw[fill=black!0,draw=black!40,line width=.35pt,] (4.00000,0.65000) rectangle ++(0.69000,0.69000); node[] at (4.34500,0.99500) 0; draw[fill=black!0,draw=black!40,line width=.35pt,] (4.69000,0.65000) rectangle ++(0.69000,0.69000); node[] at (5.03500,0.99500) 0","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c59cc6b0f70cb080","tex":"\\mathsf{Residual}","aliases":"Residual mathsf","display":"\\mathsf{Residual}","kind":"use","excerpt":"Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":32},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d7305573ca2e697d","tex":"S^*","aliases":"S^*","display":"S^*","kind":"use","excerpt":"The identities Q_α S( Q, P) = S( Q, P)P_α = Q_α P_α follow from pairwise orthogonality. Also S^*S = ∑_α P_α Q_α P_α, SS^* = ∑_α Q_α P_α Q_α. Put δ=max({0}∪{ P_α-Q_α :α∈ I}). On P_α H, P_α Q_α P_α = P_α-P_α(P_α-Q_α)P_α, so this block is bounded below by 1-δ>0. The same argument on Q_α H applies to Q_α P_α Q_α. Thus both S^*S and SS^* are invertible.","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d50bfc3cab84f7ec","tex":"\\mathcal S_a","aliases":"S_a mathcal","display":"\\mathcal S_a","kind":"use","excerpt":"A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-13116bc57b9f1d4e","tex":"\\Scal_{\\Qcal}","aliases":"Scal_Qcal S_ Q Scal Qcal mathcal mathcal","display":"{\\mathcal S}_{{\\mathcal Q}}","kind":"use","excerpt":"[Exact nonlinear coordinate theorem] For every map F:X→ Y, S_ QF_ Q← P A_ P=F. If G:Y→ Z and R is a packet system on Z, then (G∘ F)_ R← P = G_ R← Q ∘ F_ Q← P. No linearity, continuity, or differentiability assumption is needed.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ea05e9c36f74f76a","tex":"\\Scal_{\\Vcal}","aliases":"Scal_Vcal S_ V Scal Vcal mathcal mathcal","display":"{\\mathcal S}_{{\\mathcal V}}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"[htbp] centering includegraphics[width=.96linewidth]rotating_projector_transport.pdf captionA rotating rank one packet and its orthogonal complement. With u(θ)=(cosθ,sinθ) and P(θ)=u(θ)u(θ)^ T, the canonical generator is [P',P]=(0 · -1","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span span operatorname","display":"{\\operatorname{span}}","kind":"use","excerpt":"[A fine obstruction erased by coarsening] Take all three correction spaces to be zero. Let V_f=𝕜^2, V_c=𝕜, p_V(x,y)=x+y, and let V_0=Ker p_V=span{(1,-1)}. The fine defect (1,-1) is a nonzero class in coker D_f=V_f, while its coarse image is zero. The class comes from coker D_0=V_0, exactly as [source reference] predicts.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1084f45e6b3e947e","tex":"\\supp","aliases":"supp operatorname","display":"{\\operatorname{supp}}","kind":"use","excerpt":"[Structural pruning] For a map H with fixed input and output packet systems, let supp_r H(x) be the set of external profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-22a5ceacd49ba63d","tex":"\\supp_n","aliases":"supp_n supp operatorname","display":"{\\operatorname{supp}}_n","kind":"use","excerpt":"ernal profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet of F and every inner block jet of G in [source reference] are nonzero. Then supp_n(F∘ G)(x) is contained in t","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-76d43a5663aea1e7","tex":"\\supp_r","aliases":"supp_r supp operatorname","display":"{\\operatorname{supp}}_r","kind":"use","excerpt":"[Structural pruning] For a map H with fixed input and output packet systems, let supp_r H(x) be the set of external profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-da4fb737a4a4bc72","tex":"\\mathfrak T","aliases":"T mathfrak","display":"\\mathfrak T","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"[htbp] centering includegraphics[width=.96linewidth]rotating_projector_transport.pdf captionA rotating rank one packet and its orthogonal complement. With u(θ)=(cosθ,sinθ) and P(θ)=u(θ)u(θ)^ T, the canonical generator is [P',P]=(0 · -1","count":16},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"[A transfer walk with two steps] Let a packet graph have edges e:0→1 and f:1→2, with no other edges. Then T=L_e+L_f, (T^2)_20=c(f)c(e). The products L_eL_e, L_eL_f, and L_fL_f vanish by typed mismatch. Only L_fL_e, corresponding to the walk 0→1→2, contributes to the (2,0) entry. This example also fixes the noncommutative coefficient order.","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-910092ae6df8d19a","tex":"T^n","aliases":"T^n","display":"T^n","kind":"use","excerpt":"[Transfer power equals recursive word expansion] For every n∈ N, T^n= W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_wordExpansion","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6fccf1c31eff02b3","tex":"T^{n+1}","aliases":"T^n+1","display":"T^{n+1}","kind":"use","excerpt":"The zero case is the identity matrix. If T^n= W_n, then the left-successor power convention gives T^n+1=TT^n=T W_n= W_n+1.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-645a5cbe71427f86","tex":"t^r","aliases":"t^r","display":"t^r","kind":"use","excerpt":"Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"g(t)=F(x+th) on the open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus giv","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","excerpt":"the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, where t_0=0 and t_m+1=ϕ(t_m). The sequence (t_m) is increasing because ϕ","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"use","excerpt":"Set g(t)=F(x+th) on the open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-25bdedf7497647d3","tex":"t_j","aliases":"t_j","display":"t_j","kind":"use","excerpt":"Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unita","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b148baa42903d2fe","tex":"t_{j+1}","aliases":"t_j+1","display":"t_{j+1}","kind":"use","excerpt":"Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unita","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b5faa1f8bc6c2f00","tex":"T_{j-1}","aliases":"T_j-1","display":"T_{j-1}","kind":"use","excerpt":"t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b8e003e6dbf7e04c","tex":"T_m","aliases":"T_m","display":"T_m","kind":"use","excerpt":"h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives T_j=T_j-1-g^(j)(0)/j!. Iterating this relation proves the Taylor identity with rema","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b823f2707e33253e","tex":"\\mathfrak T_m","aliases":"T_m mathfrak","display":"\\mathfrak T_m","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-733902ab0028bba1","tex":"t_m","aliases":"t_m","display":"t_m","kind":"use","excerpt":"_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3141391f5a805560","tex":"t_{m+1}","aliases":"t_m+1","display":"t_{m+1}","kind":"use","excerpt":"terate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, where t_0=0 and t_m+1=ϕ(t_m). The sequence (t_m) is increasing because ϕ has nonnega","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-163dd25fe6cb484d","tex":"t_N","aliases":"t_N","display":"t_N","kind":"use","excerpt":"[Direct rotations converge to packet transport] Assume the complete packet family P(t) is twice continuously differentiable in operator norm on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T with sufficiently small mesh, define V_π = U( P(t_N), P(t_N-1)) ⋯ U( P(t_1), P(t_0)). Then V_π⟶ U(T) in operator norm, where U is the canonical transport from [sourc","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8838c89526c109a3","tex":"t_{N-1}","aliases":"t_N-1","display":"t_{N-1}","kind":"use","excerpt":"[Direct rotations converge to packet transport] Assume the complete packet family P(t) is twice continuously differentiable in operator norm on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T with sufficiently small mesh, define V_π = U( P(t_N), P(t_N-1)) ⋯ U( P(t_1), P(t_0)). Then V_π⟶ U(T) in operator norm, where U is the canonical transport from [sourc","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","excerpt":"eaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-10c7fe6561154963","tex":"T_{uv}","aliases":"T_uv","display":"T_{uv}","kind":"use","excerpt":"Let Γ=(V,E) be a finite connected graph and fix a vector space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in thos","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-71780f74aa8fa6a1","tex":"T_{v_1v_0}","aliases":"T_v_1v_0","display":"T_{v_1v_0}","kind":"use","excerpt":"or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-61d5402325a778e4","tex":"T_{v_rv_{r-1}}","aliases":"T_v_rv_r-1","display":"T_{v_rv_{r-1}}","kind":"use","excerpt":"or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0130fb3e4ce12f0a","tex":"T_{vu}^{-1}","aliases":"T_vu^-1","display":"T_{vu}^{-1}","kind":"use","excerpt":"Let Γ=(V,E) be a finite connected graph and fix a vector space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in thos","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e5687f719fcef22b","tex":"T_\\gamma","aliases":"T_γ gamma","display":"T_\\gamma","kind":"use","excerpt":"or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0a7c0769ed8b38f6","tex":"T_{\\gamma_1}","aliases":"T_γ_1 gamma","display":"T_{\\gamma_1}","kind":"use","excerpt":"valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-740e079411fa0098","tex":"T_{\\gamma_2}","aliases":"T_γ_2 gamma","display":"T_{\\gamma_2}","kind":"use","excerpt":"valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a07c1d3d8c928212","tex":"T_{\\gamma_2\\gamma_1}","aliases":"T_γ_2γ_1 gamma gamma","display":"T_{\\gamma_2\\gamma_1}","kind":"use","excerpt":"valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-712d027743d7bde8","tex":"\\Tcal","aliases":"Tcal T mathcal","display":"{\\mathcal T}","kind":"use","excerpt":"[Spanning tree normalization] Choose a root and a spanning tree T of Γ. There is a gauge in which every tree edge transport is the identity. In that gauge, the transport on each edge outside the tree is the holonomy of its fundamental cycle, with the orientation used to define that cycle.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"The usual scalar invariants can be misleading: trH=2, det H=1, χ_H(t)=(t-1)^2. These are also the invariants of the 2×2 identity matrix. Nevertheless, Ob=H-I=mat0 · 1","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"use","excerpt":"[Local Lyapunov Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ec61c8382b2d3710","tex":"U'","aliases":"U'","display":"U'","kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":21},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e1239c6d46ffc26e","tex":"u_0^2","aliases":"u_0^2","display":"u_0^2","kind":"use","excerpt":"[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"ro and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7faa68216e66f910","tex":"u_1^2","aliases":"u_1^2","display":"u_1^2","kind":"use","excerpt":"[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2f97c9046c1a4c7a","tex":"u_2","aliases":"u_2","display":"u_2","kind":"use","excerpt":"000,1.20000); draw[black!14] (9.20000,2.00000) -- (12.00000,2.00000); draw[->] (9.10000,0.40000) -- (12.16000,0.40000); draw[->] (9.20000,0.30000) -- (9.20000,2.64000); draw[->,line width=1.2pt] (9.20000,0.40000) -- (10.40000,0.80000); node[font=sffamilybfseries] at (10.65000,3.55000) chart 2; node[] at (10.65000,2.97000) (frac32,frac12); node[] at","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-91c677571574a0b9","tex":"u_a","aliases":"u_a","display":"u_a","kind":"use","excerpt":"1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-65fa973c459a6b3b","tex":"U_c","aliases":"U_c","display":"U_c","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a69093f2b23577cd","tex":"u_c","aliases":"u_c","display":"u_c","kind":"use","excerpt":"· ⟶coker D_f ⟶coker D_c ⟶0. For u_c∈Ker D_c, choose u_f∈ U_f with p_Uu_f=u_c. There is a unique v_0∈ V_0 such that D_fu_f=i_Vv_0, and the connecting map is δ(u_c)=[v_0].","count":13},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ada27fd6e08006d5","tex":"U_f","aliases":"U_f","display":"U_f","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-77c182ea48441c1f","tex":"u_f","aliases":"u_f","display":"u_f","kind":"use","excerpt":"· ⟶coker D_f ⟶coker D_c ⟶0. For u_c∈Ker D_c, choose u_f∈ U_f with p_Uu_f=u_c. There is a unique v_0∈ V_0 such that D_fu_f=i_Vv_0, and the connecting map is δ(u_c)=[v_0].","count":16},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e41938b7bd341d27","tex":"U_K","aliases":"U_K","display":"U_K","kind":"use","excerpt":"en neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a uni","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8d86088ae482ad75","tex":"u_m","aliases":"u_m","display":"u_m","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-30d31e43294cdc7c","tex":"u_{m+1}","aliases":"u_m+1","display":"u_{m+1}","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8611e5a338fe3b84","tex":"U_r","aliases":"U_r","display":"U_r","kind":"use","excerpt":"Let A be a transport system on copies of a fixed vector space V_0, and let B be a transport system on copies of a fixed vector space W_0. Fix a root r and an isomorphism U_r:V_0⟶ W_0.","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ab56dd36935ce891","tex":"U_u","aliases":"U_u","display":"U_u","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1b6b2cac1ed2c1b6","tex":"U_v","aliases":"U_v","display":"U_v","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e67d8e81d3865f41","tex":"U_\\alpha","aliases":"U_α alpha","display":"U_\\alpha","kind":"use","excerpt":"[Packet decomposition of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5eada3bebae789bc","tex":"U_\\pi","aliases":"U_π pi","display":"U_\\pi","kind":"use","excerpt":"[Convergence of sampled transports] Assume K is Lipschitz on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T, define U_π = exp((t_N-t_N-1)K(t_N-1)) ⋯ exp((t_1-t_0)K(t_0)). As the mesh |π| tends to zero, U_π⟶ U(T) in operator norm. There is a constant C, depending only on T, the supremum norm of K, and its Lipschitz constant, such that U_π-U(T) ≤ C|π|.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4e7142ad42e2513e","tex":"v'","aliases":"v'","display":"v'","kind":"use","excerpt":"If v-v'∈im D, then Q_α(v-v')∈im D. Hence [v]↦[Q_α v] is well defined on the quotient. The idempotent, annihilation, and reconstruction identities descend from V to V/im D. The formula for an obstruction class follows from the definition.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"For a canonical coordinate change with input and output chart v=2x, the target formula is not still v^2. It is widehat f_b(v)=2f(v/2)=fracv^22. Consequently widehat f_b(2x)=2x^2, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on V10.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5ddca55b935c74e1","tex":"V^{m-1}","aliases":"V^m-1","display":"V^{m-1}","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"· ⟶coker D_f ⟶coker D_c ⟶0. For u_c∈Ker D_c, choose u_f∈ U_f with p_Uu_f=u_c. There is a unique v_0∈ V_0 such that D_fu_f=i_Vv_0, and the connecting map is δ(u_c)=[v_0].","count":25},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"The relation to derivatives of the complete coordinate representation is also literal. For an open domain U⊆ X, put widehat F= A_ Q∘ F∘ S_ P on the open coordinate set A_ P(U). Boundedness and linearity of analysis and synthesis imply, for 1≤ r≤ m, D^rwidehat F(u)[v_1,…,v_r] = A_ Q(D^rF( S_ Pu) [ S_ Pv_1,…, S_ Pv_r]). Indeed, the first derivative f","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ef50ddb6014d940d","tex":"V_c","aliases":"V_c","display":"V_c","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4f979281d0af8ba8","tex":"V_f","aliases":"V_f","display":"V_f","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e6c0af9d819fefe7","tex":"v_f","aliases":"v_f","display":"v_f","kind":"use","excerpt":"If a class in coker D_f comes from coker D_0, its image in coker D_c is zero. Conversely, suppose [v_f] maps to zero in coker D_c. Choose u_c with D_cu_c=p_Vv_f and lift it to u_f∈ U_f. Then p_V(v_f-D_fu_f)=0, so v_f-D_fu_f=i_Vv_0 for some v_0∈ V_0. Thus [v_f] comes from [v_0]. Surjectivity at the last term follows from surjectivity of p_V.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"use","excerpt":"Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unita","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"on the open coordinate set A_ P(U). Boundedness and linearity of analysis and synthesis imply, for 1≤ r≤ m, D^rwidehat F(u)[v_1,…,v_r] = A_ Q(D^rF( S_ Pu) [ S_ Pv_1,…, S_ Pv_r]). Indeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each v_j is supported in one coor","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-51fd0818257204a0","tex":"v_r","aliases":"v_r","display":"v_r","kind":"use","excerpt":"The relation to derivatives of the complete coordinate representation is also literal. For an open domain U⊆ X, put widehat F= A_ Q∘ F∘ S_ P on the open coordinate set A_ P(U). Boundedness and linearity of analysis and synthesis imply, for 1≤ r≤ m, D^rwidehat F(u)[v_1,…,v_r] = A_ Q(D^rF( S_ Pu) [ S_ Pv_1,…, S_ Pv_r]). Indeed, the first derivative f","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ad08de7d90dd5111","tex":"v_{r-1}","aliases":"v_r-1","display":"v_{r-1}","kind":"use","excerpt":"or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-92becdaca2dd0d19","tex":"V_\\alpha","aliases":"V_α alpha","display":"V_\\alpha","kind":"use","excerpt":"[Packet decomposition of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b4995dcf4b8a274b","tex":"V_\\pi","aliases":"V_π pi","display":"V_\\pi","kind":"use","excerpt":"[Direct rotations converge to packet transport] Assume the complete packet family P(t) is twice continuously differentiable in operator norm on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T with sufficiently small mesh, define V_π = U( P(t_N), P(t_N-1)) ⋯ U( P(t_1), P(t_0)). Then V_π⟶ U(T) in operator norm, where U is the canonical transport from [sourc","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"use","excerpt":"ng conventions All packet systems are finite unless a statement says otherwise. In a normed setting every packet projector and every coarsening operator is bounded. Differentiability of operator valued maps is taken in operator norm. A Banach space isomorphism is a bounded linear bijection with bounded inverse. For a decreasing filtration (F^mX), c","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9e2d9dcc048eb142","tex":"\\varprojlim_m","aliases":"varprojlim_m varprojlim","display":null,"kind":"use","excerpt":"ng conventions All packet systems are finite unless a statement says otherwise. In a normed setting every packet projector and every coarsening operator is bounded. Differentiability of operator valued maps is taken in operator norm. A Banach space isomorphism is a bounded linear bijection with bounded inverse. For a decreasing filtration (F^mX), c","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2acfb66581624cd9","tex":"\\Vcal","aliases":"Vcal V mathcal","display":"{\\mathcal V}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"[Cokernel obstruction criterion] A linear correction exists if and only if Ob_D(r)=0. The quotient map q_D:V⟶coker D has the following universal property. If φ:V→ W is linear and φ D=0, then there is a unique linear map overlineφ:coker D⟶ W with φ=overlineφ q_D.","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2bbe13df91f37646","tex":"w^{-1}","aliases":"w^-1","display":"w^{-1}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"Now represent the nonlinear operation F(x,y)=(x+y,xy) in the new chart. Set w=u+v-u^2. Its correct coordinate formula is widehat F_b(u,v)= A F S(u,v) =(w, u(v-u^2)+w^2). For (x,y)=(1,2), applying F first gives (3,2) and then (3,11) in the new chart. Changing the input first gives (u,v)=(1,3), and widehat F_b(1,3)=(3,11) as well.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"Let A be a transport system on copies of a fixed vector space V_0, and let B be a transport system on copies of a fixed vector space W_0. Fix a root r and an isomorphism U_r:V_0⟶ W_0.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"[Curvature of a projected connection] Let P be a smooth orthogonal projector valued map on a parameter manifold. The connection ∇=P d on the range bundle has curvature F=P(dPwedge dP)P.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9a999477715e268b","tex":"\\widehat F","aliases":"widehat F widehat","display":"\\widehat F","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":9},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"For a canonical coordinate change with input and output chart v=2x, the target formula is not still v^2. It is widehat f_b(v)=2f(v/2)=fracv^22. Consequently widehat f_b(2x)=2x^2, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on V10.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e8efc3563f059f63","tex":"\\widehat F_b","aliases":"widehat F_b widehat","display":"\\widehat F_b","kind":"use","excerpt":"Now represent the nonlinear operation F(x,y)=(x+y,xy) in the new chart. Set w=u+v-u^2. Its correct coordinate formula is widehat F_b(u,v)= A F S(u,v) =(w, u(v-u^2)+w^2). For (x,y)=(1,2), applying F first gives (3,2) and then (3,11) in the new chart. Changing the input first gives (u,v)=(1,3), and widehat F_b(1,3)=(3,11) as well.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-54d5df4eb419be74","tex":"\\widehat f_b","aliases":"widehat f_b widehat","display":"\\widehat f_b","kind":"use","excerpt":"For a canonical coordinate change with input and output chart v=2x, the target formula is not still v^2. It is widehat f_b(v)=2f(v/2)=fracv^22. Consequently widehat f_b(2x)=2x^2, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on V10.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d128aec02eb0a2e8","tex":"\\widehat F_{r\\leftarrow a}","aliases":"widehat F_r← a widehat leftarrow","display":"\\widehat F_{r\\leftarrow a}","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1da1b91f6369b42a","tex":"\\widehat F_{s\\leftarrow b}","aliases":"widehat F_s← b widehat leftarrow","display":"\\widehat F_{s\\leftarrow b}","kind":"use","excerpt":"[Canonical coordinate naturality] The exact coordinate representation preserves typed composition and satisfies J^Y_s← rwidehat F_r← a = widehat F_s← bJ^X_b← a. statusleanpriorKomorebi.PaperII.NestedRefinement.coordinateOperator_comp; Komorebi.PaperII.NestedRefinement.coordinateOperator_naturality","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4ae2c725a0324b8f","tex":"\\widehat G","aliases":"widehat G widehat","display":"\\widehat G","kind":"use","excerpt":"tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthree","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-41acddacfefb419a","tex":"\\widehat G_{t\\leftarrow r}","aliases":"widehat G_t← r widehat leftarrow","display":"\\widehat G_{t\\leftarrow r}","kind":"use","excerpt":"tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthree","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-66d5bee1680ae2b3","tex":"\\widehat h","aliases":"widehat h widehat","display":"\\widehat h","kind":"use","excerpt":"[Second Schur derivative] For retained directions h,k, set widehat h=(h,Dφ h), widehat k=(k,Dφ k). Then D^2φ[h,k] = -A^-1D^2F_Q[widehat h,widehat k] and D^2R[h,k] = D^2F_P[widehat h,widehat k] - D_vF_P A^-1D^2F_Q[widehat h,widehat k]. All derivatives on the right are evaluated at (u_0,v_0).","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-94d7972844087a8a","tex":"\\widehat k","aliases":"widehat k widehat","display":"\\widehat k","kind":"use","excerpt":"[Second Schur derivative] For retained directions h,k, set widehat h=(h,Dφ h), widehat k=(k,Dφ k). Then D^2φ[h,k] = -A^-1D^2F_Q[widehat h,widehat k] and D^2R[h,k] = D^2F_P[widehat h,widehat k] - D_vF_P A^-1D^2F_Q[widehat h,widehat k]. All derivatives on the right are evaluated at (u_0,v_0).","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f1321a8b68ad36af","tex":"\\widehat{G\\circ F}","aliases":"widehatG∘ F widehat circ","display":"\\widehat{G\\circ F}","kind":"use","excerpt":"tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthree","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a8ca75218336934c","tex":"\\widehat{G\\circ F}_{t\\leftarrow a}","aliases":"widehatG∘ F_t← a widehat circ leftarrow","display":"\\widehat{G\\circ F}_{t\\leftarrow a}","kind":"use","excerpt":"tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthree","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2e926afe687bd984","tex":"\\widehat\\Phi","aliases":"widehatΦ widehat Phi","display":"\\widehat\\Phi","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d21df6346220aa61","tex":"\\widehat\\Phi_a","aliases":"widehatΦ_a widehat Phi","display":"\\widehat\\Phi_a","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-34d4c951aad7cc13","tex":"\\widehat\\Phi_b","aliases":"widehatΦ_b widehat Phi","display":"\\widehat\\Phi_b","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"· ≤ (1-q)ρ+qρ =ρ. Also, Ψ(x)-Ψ(x') ≤ q x-x' . The contraction mapping theorem gives a unique fixed point in the stated set. Applying D to the fixed point identity gives the equation. Conversely, a solution in X_R equals R(Dx) and is therefore a fixed point of Ψ.","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"[htbp] centering includegraphics[width=.96linewidth]packet_jet_worked_example.pdf captionSecond packet jets for F(x,y)=(x+xy,y+x^2) at the origin. Coordinate axis projectors give DF(0)=I. The first output has mixed second derivatives equal to one, while the second has a repeated first input derivative equal to two. Each grid records the ordered pai","count":19},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aeb4dc50b8d5ec9e","tex":"X^{\\Pcal}","aliases":"X^Pcal X^ P Pcal mathcal","display":"X^{{\\mathcal P}}","kind":"use","excerpt":"Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-14fb02863a861bee","tex":"X^r","aliases":"X^r","display":"X^r","kind":"use","excerpt":"R or C and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr echet derivatives and the Taylor integral below has a real parameter. Assume that F:X→ Y is m times continuously Fr echet differentiable. The same local statements hold on open domains containing the points and segments in question. For 1","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"[Local Lyapunov Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"[A mixed quadratic map] Let X=Y=𝕜^2 with coordinate packets. For F(x_1,x_2)=(x_1x_2,x_1^2+x_2), the first output packet depends on both input packets. The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second. Packet coordinates record these interactions exactly. Replacing F by ","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3674278fc192e035","tex":"x_1^2","aliases":"x_1^2","display":"x_1^2","kind":"use","excerpt":"[A mixed quadratic map] Let X=Y=𝕜^2 with coordinate packets. For F(x_1,x_2)=(x_1x_2,x_1^2+x_2), the first output packet depends on both input packets. The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second. Packet coordinates record these interactions exactly. Replacing F by ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[A mixed quadratic map] Let X=Y=𝕜^2 with coordinate packets. For F(x_1,x_2)=(x_1x_2,x_1^2+x_2), the first output packet depends on both input packets. The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second. Packet coordinates record these interactions exactly. Replacing F by ","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-51e189ae43085a7d","tex":"X_a","aliases":"X_a","display":"X_a","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-73a4e2e74cee4766","tex":"X_b","aliases":"X_b","display":"X_b","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-72e05ef2b0f48e18","tex":"X_c","aliases":"X_c","display":"X_c","kind":"use","excerpt":"Let F_f:X_f⟶ Y_f, F_c:X_c⟶ Y_c be fine and coarse maps. Let C_X:X_f⟶ X_c, C_Y:Y_f⟶ Y_c be linear coarsening maps. Whenever a norm, a Lipschitz constant, or a Fr echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-702e017c46584077","tex":"x_c","aliases":"x_c","display":"x_c","kind":"use","excerpt":"[Fixed point stability] Assume F_c is a contraction with constant q<1. Let x_f and x_c be fixed points of F_f and F_c. If Δ_F(x_f) ≤ε, then Cx_f-x_c ≤ fracε1-q. When the defect at x_f is zero, the fixed points are compatible.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f76f02d6c081bad5","tex":"X_f","aliases":"X_f","display":"X_f","kind":"use","excerpt":"Let F_f:X_f⟶ Y_f, F_c:X_c⟶ Y_c be fine and coarse maps. Let C_X:X_f⟶ X_c, C_Y:Y_f⟶ Y_c be linear coarsening maps. Whenever a norm, a Lipschitz constant, or a Fr echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-91c50d828be3947d","tex":"x_f","aliases":"x_f","display":"x_f","kind":"use","excerpt":"[Fixed point stability] Assume F_c is a contraction with constant q<1. Let x_f and x_c be fixed points of F_f and F_c. If Δ_F(x_f) ≤ε, then Cx_f-x_c ≤ fracε1-q. When the defect at x_f is zero, the fixed points are compatible.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Expand (T^nx)_j=∑_i(T^n)_jix_i. Terms with i∉ S vanish by the support of x, and terms with i∈ S vanish by [source reference]. The final claim follows because the j-coordinate also vanishes on every finite linear combination of orbit vectors.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-848620af401d93ab","tex":"x_m'","aliases":"x_m'","display":"x_m'","kind":"use","excerpt":"[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2ce545975574982d","tex":"x_{m+1}","aliases":"x_m+1","display":"x_{m+1}","kind":"use","excerpt":"Choose representatives recursively. Once x_m represents the selected depth m vertex, choose a representative x_m+1 of its selected child with x_m+1-x_m∈ F^mX. The resulting sequence is Cauchy for the filtration topology. Each representative satisfies Φ(x_m)∈ F^mY. The conclusion now follows from [source reference].","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"For every fixed r, the congruences imply x_n-x_m∈ F^rX whenever n≥ m≥ r. Thus (x_m) is Cauchy for the filtration topology. Completeness gives a limit x_∞.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f778110707ddeb9c","tex":"X_q","aliases":"X_q","display":"X_q","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e798a8a7ea91ac3e","tex":"X_R","aliases":"X_R","display":"X_R","kind":"use","excerpt":"[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in ","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-81529bf55585a9e9","tex":"X_{\\src(e)}","aliases":"X_src(e) src operatorname","display":"X_{{\\operatorname{src}}(e)}","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9cd5d8f0da4542ed","tex":"X_{\\tgt(e)}","aliases":"X_tgt(e) tgt operatorname","display":"X_{{\\operatorname{tgt}}(e)}","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1e9458c4c7da9d3b","tex":"X_\\alpha","aliases":"X_α alpha","display":"X_\\alpha","kind":"use","excerpt":"Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-af49eb7478b6e8c8","tex":"X_{\\alpha_i}","aliases":"X_α_i alpha","display":"X_{\\alpha_i}","kind":"use","excerpt":"[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-25cae1ffb84e4bb9","tex":"X_{\\alpha_j}","aliases":"X_α_j alpha","display":"X_{\\alpha_j}","kind":"use","excerpt":"For h_j∈ X_α_j, the inherited norm gives Q_γ D^rF(x)[h_1,…,h_r] ≤ Q_γ D^rF(x) ∏_j h_j . Taking the multilinear operator norm proves the sharper restricted estimate. For ambient inputs, replace h_j by P_α_jh_j and apply the operator norm bound to each projector. Every nonzero idempotent has norm at least one, since it fixes a nonzero vector in its r","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2b6c204b3855e438","tex":"X_\\ell","aliases":"X_ℓ ell","display":"X_\\ell","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4ec4142068739e9a","tex":"x_\\ell","aliases":"x_ℓ ell","display":"x_\\ell","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-524e8e9a8948b743","tex":"X_{\\ell+1}","aliases":"X_ℓ+1 ell","display":"X_{\\ell+1}","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-93da8851ae87d014","tex":"x_{\\ell+1}","aliases":"x_ℓ+1 ell","display":"x_{\\ell+1}","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b99185167014e4eb","tex":"x_\\infty","aliases":"x_∞ infty","display":"x_\\infty","kind":"use","excerpt":"[Complete filtered lifting] Assume there is a compatible sequence x_m+1≡ x_mpmodF^mX with Φ(x_m)∈ F^mY for every m. Then x_m converges to an element x_∞∈ X and Φ(x_∞)=0. In particular, if the obstruction vanishes at every depth and compatible corrections are chosen, an exact solution exists.","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"use","excerpt":"[Stability with respect to the target] Under the hypotheses of [source reference], let x and x' be the fixed points associated with targets y and y'. If both fixed point maps preserve the same ball, then x-x' ≤ frac R 1-q y-y' .","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-879f9148f6a019e7","tex":"Y_c","aliases":"Y_c","display":"Y_c","kind":"use","excerpt":"Let F_f:X_f⟶ Y_f, F_c:X_c⟶ Y_c be fine and coarse maps. Let C_X:X_f⟶ X_c, C_Y:Y_f⟶ Y_c be linear coarsening maps. Whenever a norm, a Lipschitz constant, or a Fr echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-69bb6e300ff2f699","tex":"Y_f","aliases":"Y_f","display":"Y_f","kind":"use","excerpt":"Let F_f:X_f⟶ Y_f, F_c:X_c⟶ Y_c be fine and coarse maps. Let C_X:X_f⟶ X_c, C_Y:Y_f⟶ Y_c be linear coarsening maps. Whenever a norm, a Lipschitz constant, or a Fr echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4ee025aeeb25a3e5","tex":"Y_P","aliases":"Y_P","display":"Y_P","kind":"use","excerpt":"Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v)","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9cafb0ef5346dcbc","tex":"Y_Q","aliases":"Y_Q","display":"Y_Q","kind":"use","excerpt":"Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v)","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-72c43d344dc79844","tex":"Y_r","aliases":"Y_r","display":"Y_r","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-151cda08db8a7a2a","tex":"Y_s","aliases":"Y_s","display":"Y_s","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then ex","count":15},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"For a matrix entry (y,x), (L_fL_e)_yx=∑_z∈ V(L_f)_yz(L_e)_zx. A potentially nonzero summand would require simultaneously z=src(f) and z=tgt(e), contrary to the hypothesis.","count":11},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7fb34d91679bf137","tex":"z^{-1/2}","aliases":"z^-1/2","display":"z^{-1/2}","kind":"use","excerpt":"P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Mu","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-de03d0b564377608","tex":"Z_c","aliases":"Z_c","display":"Z_c","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9f4b4e1abf764838","tex":"Z_f","aliases":"Z_f","display":"Z_f","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bdc1562155501d06","tex":"z_n","aliases":"z_n","display":"z_n","kind":"use","excerpt":"The space X_R is closed. Indeed, if Rz_n→ x, then z_n=DRz_n→ Dx, so x=R(Dx). Define Ψ(x)=R(y-N(x)). This map takes X_R to itself. For x∈overline B_ρ(0)∩ X_R, Ψ(x) · ≤ R(y-N(0)) + R N(x)-N(0)","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3480de748bc9723a","tex":"Z_t","aliases":"Z_t","display":"Z_t","kind":"use","excerpt":"^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then ex","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ","display":null,"kind":"use","excerpt":"vpage3Read the product in execution orderMODEL A / ONE EDGE IS ONE MATRIX ENTRYTraditional paper: Lemma 2.2 (p. 2), Theorem 2.3 (p. 2), Theorem 2.4 (p. 3) Work over ZZ with packets 0,1,2. Let e:0→1 have coefficient 2, and let f:1→2 have coefficient 3. The elementary operators are L_e=2E_10, L_f=3E_21. The location of the nonzero entry records the t","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-343b02c221c97d88","tex":"|_K","aliases":"|_K","display":"|_K","kind":"use","excerpt":"Differentiate G(k,ψ(k))=0 at zero. Since L vanishes on K, one obtains L_1Dψ(0)=0, hence Dψ(0)=0. Finally, Dκ(0)=π_OL|_K=0.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9669e5bb7ba8d32a","tex":"|_{X_1}","aliases":"|_X_1","display":"|_{X_1}","kind":"use","excerpt":"[Local Lyapunov Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one dep","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e8c617160e3b774f","tex":"\\alpha_2","aliases":"α_2 alpha","display":"\\alpha_2","kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f320d7b77ed8bff0","tex":"\\alpha_i","aliases":"α_i alpha","display":"\\alpha_i","kind":"use","excerpt":"[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5a17117e75261dbe","tex":"\\alpha_{i_1}","aliases":"α_i_1 alpha","display":"\\alpha_{i_1}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c4056245b0846ec2","tex":"\\alpha_{i_s}","aliases":"α_i_s alpha","display":"\\alpha_{i_s}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","excerpt":"[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a823753ea2193233","tex":"\\alpha_n","aliases":"α_n alpha","display":"\\alpha_n","kind":"use","excerpt":"[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"A finite family of pairwise annihilating idempotents gives exact coordinates. If P_α P_β=0 for α≠β, ∑_α P_α=I, then every vector is the sum of its packet components. Linear operators become block matrices in these coordinates. A nonlinear map cannot be reconstructed from its diagonal restrictions, since mixed interactions between packet components ","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0fa6ec7a5cf1dd0a","tex":"\\Gamma^G","aliases":"Γ^G Gamma","display":"\\Gamma^G","kind":"use","excerpt":"Substitute the definition of Γ^G. Adjacent factors G_b^-1G_b and G_c^-1G_c cancel. In the defect, both the composite and direct terms acquire the same outer factors G_c and G_a^-1. In the loop, all inner factors cancel and only conjugation at the base remains.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e032ce94ae2fca8c","tex":"\\Gamma^{X_q}_{b\\leftarrow a}","aliases":"Γ^X_q_b← a Gamma leftarrow","display":"\\Gamma^{X_q}_{b\\leftarrow a}","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f20fcd760ea44277","tex":"\\Gamma^Y_{b\\leftarrow a}","aliases":"Γ^Y_b← a Gamma leftarrow","display":"\\Gamma^Y_{b\\leftarrow a}","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0717c41ef9dfce54","tex":"\\Gamma_{0\\leftarrow2}","aliases":"Γ_0←2 Gamma leftarrow","display":"\\Gamma_{0\\leftarrow2}","kind":"use","excerpt":"l hypothesisMODEL D / THE MINIMAL AND REVERSIBLE INTERFACESTraditional paper: Definition 5.1 (p. 9), Definition 5.5 (p. 11) Choose the forward scalar transports Γ_1←0=2, Γ_2←1=3, and Γ_2←0=5. Their multiplicative comparison between transport through two steps and direct transport is D^×=6/5. A loop also needs a declared return map Γ_0←2.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d2e737aa54abf669","tex":"\\gamma_1","aliases":"γ_1 gamma","display":"\\gamma_1","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4938aee8fada09d6","tex":"\\Gamma_{1\\leftarrow0}","aliases":"Γ_1←0 Gamma leftarrow","display":"\\Gamma_{1\\leftarrow0}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2f08cea1939265ee","tex":"\\gamma_2","aliases":"γ_2 gamma","display":"\\gamma_2","kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic ","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fe10d9ec3f095a8c","tex":"\\Gamma_{2\\leftarrow0}","aliases":"Γ_2←0 Gamma leftarrow","display":"\\Gamma_{2\\leftarrow0}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f144291fed6876b9","tex":"\\Gamma_{2\\leftarrow1}","aliases":"Γ_2←1 Gamma leftarrow","display":"\\Gamma_{2\\leftarrow1}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-276414e29824c2c0","tex":"\\Gamma_{a_0\\leftarrow a_0}","aliases":"Γ_a_0← a_0 Gamma leftarrow","display":"\\Gamma_{a_0\\leftarrow a_0}","kind":"use","excerpt":"Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-43ff3a694b57363f","tex":"\\Gamma_{a_{n+1}\\leftarrow a_0}","aliases":"Γ_a_n+1← a_0 Gamma leftarrow","display":"\\Gamma_{a_{n+1}\\leftarrow a_0}","kind":"use","excerpt":"Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5fdaec02b44f0b20","tex":"\\Gamma_{a_{n+1}\\leftarrow a_n}","aliases":"Γ_a_n+1← a_n Gamma leftarrow","display":"\\Gamma_{a_{n+1}\\leftarrow a_n}","kind":"use","excerpt":"Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1fd10a8f613b16db","tex":"\\Gamma_{a_n\\leftarrow a_0}","aliases":"Γ_a_n← a_0 Gamma leftarrow","display":"\\Gamma_{a_n\\leftarrow a_0}","kind":"use","excerpt":"[Flat pair transport collapses to endpoints] If Γ is flat, then every path satisfies Γ_p=Γ_a_n← a_0. In particular, every loop has identity holonomy. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport_eq_direct_of_flat; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport_eq_direct_of_flat; Komorebi.Paper","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8146a29b9b509653","tex":"\\Gamma_{a\\leftarrow o}","aliases":"Γ_a← o Gamma leftarrow","display":"\\Gamma_{a\\leftarrow o}","kind":"use","excerpt":"[Flat connections are gauges of the canonical pair transport] Suppose the C_a are exact packet coordinate modules for one ambient module, so the canonical J_b← a is available. If Γ is a flat normalized reversible pair connection, choose o∈ B and set G_a=Γ_a← oJ_o← a. Then every pair transport satisfies Γ_b← a =G_bJ_b← aG_a^-1. statusleanthreeKomore","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6a70b0f3e648354e","tex":"\\Gamma_{b\\leftarrow o}","aliases":"Γ_b← o Gamma leftarrow","display":"\\Gamma_{b\\leftarrow o}","kind":"use","excerpt":"Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a · = Γ_b← oΓ_o← a · = Γ_b← a. The first reduction uses the canonical cocycle twice; the second uses flatness of Γ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1987d8d2f743de43","tex":"\\Gamma_{c\\leftarrow a}^{-1}","aliases":"Γ_c← a^-1 Gamma leftarrow","display":"\\Gamma_{c\\leftarrow a}^{-1}","kind":"use","excerpt":"At a triple (a,b,c), the compiled cocycle predicate is Γ_c← bΓ_b← a =Γ_c← a. Flatness means this equality holds for every triple. The formal multiplicative relative defect and triangle holonomy are D^×_c,b,a = (Γ_c← bΓ_b← a) Γ_c← a^-1, H_c,b,a = Γ_a← c Γ_c← b Γ_b← a. The defect is based at the target c. The holonomy is based at a. statusleanthreeKo","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5093f2875f491c34","tex":"\\Gamma_e","aliases":"Γ_e Gamma","display":"\\Gamma_e","kind":"use","excerpt":"[A connection on a chart graph] One may specify Γ_e only on the oriented edges of a chart graph, together with inverse maps on reversed edges. Path transport and loop holonomy below still make sense. A direct cocycle defect for (a,b,c), however, requires a declared direct transport from a to c. The pairwise API in [source reference] keeps this comp","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-530f5d7a3b27e60f","tex":"\\gamma_k","aliases":"γ_k gamma","display":"\\gamma_k","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d14549cec41f6edf","tex":"\\Gamma_{o\\leftarrow a}","aliases":"Γ_o← a Gamma leftarrow","display":"\\Gamma_{o\\leftarrow a}","kind":"use","excerpt":"Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":20},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2930293fbf06791f","tex":"\\Delta^G_{c,b,a}","aliases":"Δ^G_c,b,a Delta","display":"\\Delta^G_{c,b,a}","kind":"use","excerpt":"[Gauge covariance] The defect and triangle holonomy transform as Δ^G_c,b,a =G_cΔ_c,b,aG_a^-1, H^G_c,b,a =G_aH_c,b,aG_a^-1. Thus flatness and triviality of triangle holonomy are gauge invariant. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1ff4ad863fb9c187","tex":"\\Delta_{2,1,0}","aliases":"Δ_2,1,0 Delta","display":"\\Delta_{2,1,0}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-37d4cee3e21267da","tex":"\\Delta_G","aliases":"Δ_G Delta","display":"\\Delta_G","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-96bdfbbc4387b77f","tex":"\\Delta_{GF}","aliases":"Δ_GF Delta","display":"\\Delta_{GF}","kind":"use","excerpt":"[Linear product rule] If all maps in [source reference] are linear, then Δ_GF = Δ_G F_f+G_cΔ_F.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-23f92b1c07ee4536","tex":"\\Delta_{G\\circ F}","aliases":"Δ_G∘ F Delta circ","display":"\\Delta_{G\\circ F}","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4a6738581aa5bb8f","tex":"\\delta_{ji}","aliases":"δ_ji delta","display":"\\delta_{ji}","kind":"use","excerpt":"pansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7c3547666183b660","tex":"\\delta_{su}","aliases":"δ_su delta","display":"\\delta_{su}","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b3de5d9511fcd1b4","tex":"\\varepsilon_j","aliases":"ε_j varepsilon","display":"\\varepsilon_j","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-dc06eebc88a8b163","tex":"\\varepsilon_n","aliases":"ε_n varepsilon","display":"\\varepsilon_n","kind":"use","excerpt":"Let e_n= CF_f^n(x)-F_c^n(Cx) . The definition of the defect and the Lipschitz property give e_n+1≤ε_n+qe_n. Iterating this recurrence proves the first estimate. The second is the geometric sum.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1ca916b982c03d00","tex":"\\varepsilon_\\ell","aliases":"ε_ℓ varepsilon ell","display":"\\varepsilon_\\ell","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"[Input locality for operation trees] Let η_1,η_2:I→ Y be two input environments. If η_1(i)=η_2(i) for every i∈inputs(t), then the two evaluations of t are equal under the same constant, unary, and binary semantics. In particular, if inputs(t)=∅, evaluation is independent of the input environment. statusleanthreeKomorebi.PaperIII.NonlinearObstructio","count":18},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7b93f1be7412facb","tex":"\\eta_1","aliases":"η_1 eta","display":"\\eta_1","kind":"use","excerpt":"[Input locality for operation trees] Let η_1,η_2:I→ Y be two input environments. If η_1(i)=η_2(i) for every i∈inputs(t), then the two evaluations of t are equal under the same constant, unary, and binary semantics. In particular, if inputs(t)=∅, evaluation is independent of the input environment. statusleanthreeKomorebi.PaperIII.NonlinearObstructio","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2cb721f0ce1499b1","tex":"\\eta_2","aliases":"η_2 eta","display":"\\eta_2","kind":"use","excerpt":"[Input locality for operation trees] Let η_1,η_2:I→ Y be two input environments. If η_1(i)=η_2(i) for every i∈inputs(t), then the two evaluations of t are equal under the same constant, unary, and binary semantics. In particular, if inputs(t)=∅, evaluation is independent of the input environment. statusleanthreeKomorebi.PaperIII.NonlinearObstructio","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-017a293d20155d21","tex":"\\eta_g","aliases":"η_g eta","display":"\\eta_g","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-57d0f50543139197","tex":"\\eta_{g_1}","aliases":"η_g_1 eta","display":"\\eta_{g_1}","kind":"use","excerpt":"belprop:packet_l_infinity Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet of label k is","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2afdf48eb6389780","tex":"\\eta_{g_r}","aliases":"η_g_r eta","display":"\\eta_{g_r}","kind":"use","excerpt":"belprop:packet_l_infinity Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet of label k is","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-a8216593a1e41c75","tex":"\\eta_h","aliases":"η_h eta","display":"\\eta_h","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fc03a685a0a01ebc","tex":"\\eta_k","aliases":"η_k eta","display":"\\eta_k","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"[htbp] centering includegraphics[width=.96linewidth]rotating_projector_transport.pdf captionA rotating rank one packet and its orthogonal complement. With u(θ)=(cosθ,sinθ) and P(θ)=u(θ)u(θ)^ T, the canonical generator is [P',P]=(0 · -1","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-29b3a606402cae8a","tex":"\\iota_{\\tgt(e)}","aliases":"ι_tgt(e) iota tgt iota operatorname","display":"\\iota_{{\\operatorname{tgt}}(e)}","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborhoods are made smaller if needed, the solutions of F(x)=0 near zero are exactly the points x=k+ψ(k) for which the reduced obstruction map κ(k)=π_OF(k+ψ(k)) vanishes. At t","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":25},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-222fb49e971fef73","tex":"\\pi_1","aliases":"π_1 pi","display":"\\pi_1","kind":"use","excerpt":"[Gauge classification of transports] Let G⊆GL(X) be the chosen structure group and fix a root r. Restrict both edge transports and gauge values to G. Gauge equivalence classes of G valued transport systems are naturally identified with Hom(π_1(Γ,r),G)/G, where G acts by conjugation. After a spanning tree and orientations of the remaining edges are ","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e2bc13eba5395cb2","tex":"\\pi_O","aliases":"π_O pi","display":"\\pi_O","kind":"use","excerpt":"ion] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-666f17fd05c1382b","tex":"\\pi_R","aliases":"π_R pi","display":"\\pi_R","kind":"use","excerpt":"hmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fbd6955f854c7243","tex":"\\pi_{\\src(e)}","aliases":"π_src(e) pi src pi operatorname","display":"\\pi_{{\\operatorname{src}}(e)}","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-81626ca27a10136c","tex":"\\Pi_\\mu","aliases":"Π_μ Pi mu","display":"\\Pi_\\mu","kind":"use","excerpt":"Π_μObΠ_ν · Block from input to output of Ob=H-I. · pagerefV16 to pagerefV18","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d46c1d0799d88384","tex":"\\Pi_\\nu","aliases":"Π_ν Pi nu","display":"\\Pi_\\nu","kind":"use","excerpt":"Π_μObΠ_ν · Block from input to output of Ob=H-I. · pagerefV16 to pagerefV18","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in ","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8c811b6eab2988ab","tex":"\\tau_1","aliases":"τ_1 tau","display":"\\tau_1","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-53304761e001a2f9","tex":"\\tau_r","aliases":"τ_r tau","display":"\\tau_r","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.","count":7},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"[Cokernel obstruction criterion] A linear correction exists if and only if Ob_D(r)=0. The quotient map q_D:V⟶coker D has the following universal property. If φ:V→ W is linear and φ D=0, then there is a unique linear map overlineφ:coker D⟶ W with φ=overlineφ q_D.","count":20},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c89e189652ea7cb2","tex":"\\Phi'","aliases":"Φ' Phi","display":"\\Phi'","kind":"use","excerpt":"[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"The usual scalar invariants can be misleading: trH=2, det H=1, χ_H(t)=(t-1)^2. These are also the invariants of the 2×2 identity matrix. Nevertheless, Ob=H-I=mat0 · 1","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9734ea6868ded897","tex":"\\chi_H","aliases":"χ_H chi","display":"\\chi_H","kind":"use","excerpt":"The usual scalar invariants can be misleading: trH=2, det H=1, χ_H(t)=(t-1)^2. These are also the invariants of the 2×2 identity matrix. Nevertheless, Ob=H-I=mat0 · 1","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"The space X_R is closed. Indeed, if Rz_n→ x, then z_n=DRz_n→ Dx, so x=R(Dx). Define Ψ(x)=R(y-N(x)). This map takes X_R to itself. For x∈overline B_ρ(0)∩ X_R, Ψ(x) · ≤ R(y-N(0)) + R N(x)-N(0)","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"en Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=","count":14},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The o","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hol","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":16},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c3eaa86783ecd3c8","tex":"\\ell_e","aliases":"ℓ_e ell","display":"\\ell_e","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-819cb7ab00d8c22c","tex":"\\ell_r","aliases":"ℓ_r ell","display":"\\ell_r","kind":"use","excerpt":"[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet o","count":4},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Set g(t)=F(x+th) on the open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"The statement proved by the generic certificate has the form neg(∀ p, r(x,p)=0) Longleftrightarrow ∃ p, r(x,p)≠0. The same candidate x appears on both sides. A certificate consists of the packet label and its nonvanishing proof. The general converse uses classical reasoning and does not itself provide an algorithm to find the label.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-569331afbd5d2de3","tex":"\\nabla","aliases":"∇ nabla","display":"\\nabla","kind":"use","excerpt":"[Curvature of a projected connection] Let P be a smooth orthogonal projector valued map on a parameter manifold. The connection ∇=P d on the range bundle has curvature F=P(dPwedge dP)P.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c7b13cbc252814fa","tex":"\\nabla^2","aliases":"∇^2 nabla","display":"\\nabla^2","kind":"use","excerpt":"For a section s satisfying Ps=s, ∇^2s=P d(P ds)=P(dP)wedge ds. Since ds=(dP)s+Pds and P(dP)P=0, the surviving term is P(dPwedge dP)Ps.qedhere","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"Expand (T^nx)_j=∑_i(T^n)_jix_i. Terms with i∉ S vanish by the support of x, and terms with i∈ S vanish by [source reference]. The final claim follows because the j-coordinate also vanishes on every finite linear combination of orbit vectors.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2d3db36262598bcd","tex":"\\prod_j","aliases":"∏_j prod","display":"\\prod_j","kind":"use","excerpt":"[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-aa6ac22062ba0415","tex":"\\prod_{j=1}^r","aliases":"∏_j=1^r prod","display":"\\prod_{j=1}^r","kind":"use","excerpt":"cket range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) ∏_j=1^r P_α_j ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-34d2ddfea83ffb6e","tex":"\\prod_q","aliases":"∏_q prod","display":"\\prod_q","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-925014b526012cb9","tex":"\\sum_{(e_1,\\ldots,e_n)\\in E^n}","aliases":"∑_(e_1,…,e_n)∈ E^n sum ldots in","display":"\\sum_{(e_1,\\ldots,e_n)\\in E^n}","kind":"use","excerpt":"Induction and finite distributivity give (∑_e∈ EL_e)^n = ∑_(e_1,…,e_n)∈ E^nL_e_n⋯ L_e_1. Every noncomposable summand is zero by [source reference], so inserting the composability filter does not change the sum. In Lean, tuple concatenation and the successor sum are handled by leannametupleWordOp_snoc and leannameexplicitWordExpansion_succ; the two ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c81c4d47b5b4ae42","tex":"\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}","aliases":"∑_(i_1,…,i_m-1)∈ V^m-1 sum ldots in","display":"\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-184fe436a04b209f","tex":"\\sum_{2g=k}","aliases":"∑_2g=k sum","display":"\\sum_{2g=k}","kind":"use","excerpt":"[Divided square packet equations] Assume q( L_g)⊆ L_2g. Choose an ordering of the finite label set. The equation dη+q(η)=0 is equivalent to dη_k + ∑_2g=kq(η_g) + ∑_substackg<h g+h=k[η_g,η_h] =0 for every k.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ec1616068a6ee172","tex":"\\sum_e","aliases":"∑_e sum","display":"\\sum_e","kind":"use","excerpt":"T=∑_eL_e · Add all edge operators before taking powers. · pagerefV4 to pagerefV7","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-76e758f1be5158a2","tex":"\\sum_g","aliases":"∑_g sum","display":"\\sum_g","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c01f2477ca38770d","tex":"\\sum_{g+h=k}","aliases":"∑_g+h=k sum","display":"\\sum_{g+h=k}","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-130edb5a70e27b59","tex":"\\sum_{g<h}","aliases":"∑_g<h sum","display":"\\sum_{g<h}","kind":"use","excerpt":"Repeated use of the polarization identity for q gives q(∑_gη_g) = ∑_gq(η_g) + ∑_g<h[η_g,η_h]. Project onto the component of label k.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6917b58d5eb55262","tex":"\\sum_{g_1+\\cdots+g_r=k}","aliases":"∑_g_1+⋯+g_r=k sum cdots","display":"\\sum_{g_1+\\cdots+g_r=k}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6d39fe36bb310cf9","tex":"\\sum_{g\\in G}","aliases":"∑_g∈ G sum in","display":"\\sum_{g\\in G}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Expand (T^nx)_j=∑_i(T^n)_jix_i. Terms with i∉ S vanish by the support of x, and terms with i∈ S vanish by [source reference]. The final claim follows because the j-coordinate also vanishes on every finite linear combination of orbit vectors.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e3ce2bec5bb12bb9","tex":"\\sum_{i+j=k}","aliases":"∑_i+j=k sum","display":"\\sum_{i+j=k}","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-38c03c7cd3f0f240","tex":"\\sum_{i,j}","aliases":"∑_i,j sum","display":"\\sum_{i,j}","kind":"use","excerpt":"headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bd422e93fff6d0ac","tex":"\\sum_{i\\in I_a}","aliases":"∑_i∈ I_a sum in","display":"\\sum_{i\\in I_a}","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-49e814d33599ad8c","tex":"\\sum_{i\\in I_v}","aliases":"∑_i∈ I_v sum in","display":"\\sum_{i\\in I_v}","kind":"use","excerpt":"Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate pro","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"s continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unitary. A telescoping product comparison gives V_π-U(T,0) ≤ ∑_j ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f7b37e8032ebe855","tex":"\\sum_{j=0}^{n-1}","aliases":"∑_j=0^n-1 sum","display":"\\sum_{j=0}^{n-1}","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-1f3736c162765e82","tex":"\\sum_{k=1}^{n}","aliases":"∑_k=1^n sum","display":"\\sum_{k=1}^{n}","kind":"use","excerpt":"[Number of labeled composition terms] For n≥1, if Y has M packets, then the number of packet labeled terms at order n before zero jets are removed is N_n,M = ∑_k=1^nS(n,k)M^k, where S(n,k) is a Stirling number of the second kind. Thus N_n,M is the nth Touchard polynomial evaluated at M.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"[Quadratic Catalan expansion] Let B:X× X→ X be bounded and consider u=s+B(u,u). Put a= s and b= B . If 4ab<1, then u= ∑_n≥1 ∑_substackτ full ordered binary tree with n leaves B_τ(s) converges absolutely. It also satisfies u ≤ frac1-sqrt1-4ab2b when b>0.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bf6132b8c151067a","tex":"\\sum_{r=1}^{m}","aliases":"∑_r=1^m sum","display":"\\sum_{r=1}^{m}","kind":"use","excerpt":"[Packet Taylor formula] Assume F is m+1 times continuously Fr echet differentiable on an open set containing the segment x+th for 0≤ t≤1. Then Q_γ F(x+h) = · Q_γ F(x) + ∑_r=1^mfrac1r! ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h,…,P_α_rh] · +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] d","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3b31a2c0bfdd4782","tex":"\\sum_{r=2}^{d}","aliases":"∑_r=2^d sum","display":"\\sum_{r=2}^{d}","kind":"use","excerpt":"Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-dc62b5ae7466808f","tex":"\\sum_{r\\ge1}","aliases":"∑_r≥1 sum ge","display":"\\sum_{r\\ge1}","kind":"use","excerpt":"[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet o","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d047e8abc842ee44","tex":"\\sum_{\\substack{g<h\\\\g+h=k}}","aliases":"∑_substackg<h g+h=k sum substack g","display":null,"kind":"use","excerpt":"[Divided square packet equations] Assume q( L_g)⊆ L_2g. Choose an ordering of the finite label set. The equation dη+q(η)=0 is equivalent to dη_k + ∑_2g=kq(η_g) + ∑_substackg<h g+h=k[η_g,η_h] =0 for every k.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-07a86bb483b36897","tex":"\\sum_{\\substack{\\tau\\text{ full ordered binary tree}\\\\\\text{with }n\\text{ leaves}}}","aliases":"∑_substackτ full ordered binary tree with n leaves sum substack tau text text text","display":"\\sum_{\\substack{\\tau\\text{ full ordered binary tree}\\\\\\text{with }n\\text{ leaves}}}","kind":"use","excerpt":"[Quadratic Catalan expansion] Let B:X× X→ X be bounded and consider u=s+B(u,u). Put a= s and b= B . If 4ab<1, then u= ∑_n≥1 ∑_substackτ full ordered binary tree with n leaves B_τ(s) converges absolutely. It also satisfies u ≤ frac1-sqrt1-4ab2b when b>0.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-998f4941f1c93781","tex":"\\sum_{w\\in E^n}","aliases":"∑_w∈ E^n sum in","display":"\\sum_{w\\in E^n}","kind":"use","excerpt":"[Finite operator expansion from words to walks] For every n∈ N, T^n = ∑_w∈ E^nL_w = ∑_w∈ E^n L_w, · w composable, 0, · w not composable. Parallel edges occur as distinct summands. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_explicitWordExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.tran","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-dcadd6b9285294d8","tex":"\\sum_{w\\in\\Walk_n(i,j)}","aliases":"∑_w∈Walk_n(i,j) sum in Walk sum in operatorname","display":"\\sum_{w\\in{\\operatorname{Walk}}_n(i,j)}","kind":"use","excerpt":"[Endpoint coefficient walk sum] For all n∈ N and i,j∈ V, (T^n)_ji = ∑_w∈Walk_n(i,j)c^←(w). statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Connects; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.endpointWalkCoefficientSum; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_apply_eq_sum_walks","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-5966b9097878350e","tex":"\\sum_z","aliases":"∑_z sum","display":"\\sum_z","kind":"use","excerpt":"On a vector (x_0,x_1,x_2)^ T, the first step produces (0,2x_0,0)^ T, and the second produces (0,0,6x_0)^ T. Thus L_fL_e=6E_20, L_eL_f=0. For general edges, a summand in the entrywise product can survive only when the same intermediate packet is both tgt(e) and src(f): (L_fL_e)_yx=∑_z(L_f)_yz(L_e)_zx. This is why a broken interface annihilates the e","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ad8cbd6682b9ecb7","tex":"\\sum_{z\\in V}","aliases":"∑_z∈ V sum in","display":"\\sum_{z\\in V}","kind":"use","excerpt":"For a matrix entry (y,x), (L_fL_e)_yx=∑_z∈ V(L_f)_yz(L_e)_zx. A potentially nonzero summand would require simultaneously z=src(f) and z=tgt(e), contrary to the hypothesis.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c5676182f4950dc5","tex":"\\sum_{\\alpha_1,\\ldots,\\alpha_r}","aliases":"∑_α_1,…,α_r sum alpha ldots alpha","display":"\\sum_{\\alpha_1,\\ldots,\\alpha_r}","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-dd5a8bdc2b931502","tex":"\\sum_\\beta","aliases":"∑_β sum beta","display":"\\sum_\\beta","kind":"use","excerpt":"At a vertex labeled by an r multilinear map B_r, insert I_X=∑_β P_β on the output and in every input slot. Multilinearity gives an exact finite sum of packet blocks at that vertex. Apply this identity successively from the leaves toward the root. Since the tree and the packet set are finite, all sums are finite and their total reconstructs the orig","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9d12cb53c45f8ea0","tex":"\\sum_\\gamma","aliases":"∑_γ sum gamma","display":"\\sum_\\gamma","kind":"use","excerpt":"Insert I_Y=∑_γ Q_γ on the output and I_X=∑_α P_α in every input slot. Multilinearity expands the finite sums.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-b177dab24f859fc7","tex":"\\sum_{\\gamma}","aliases":"∑_γ sum gamma","display":"\\sum_{\\gamma}","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c0796b45d0f24473","tex":"\\sum_{\\gamma_1,\\gamma_2}","aliases":"∑_γ_1,γ_2 sum gamma gamma","display":"\\sum_{\\gamma_1,\\gamma_2}","kind":"use","excerpt":"[The second order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first order changes entering a quadratic ","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-35e481843bfd1ee3","tex":"\\sum_{\\gamma_1,\\ldots,\\gamma_k}","aliases":"∑_γ_1,…,γ_k sum gamma ldots gamma","display":"\\sum_{\\gamma_1,\\ldots,\\gamma_k}","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-f85a9d219ed81c46","tex":"\\sum_\\mu","aliases":"∑_μ sum mu","display":"\\sum_\\mu","kind":"use","excerpt":"Insert completeness on both sides: Ob = (∑_μΠ^a_μ) Ob (∑_νΠ^a_ν) = ∑_μ,νΠ^a_μObΠ^a_ν. Thus (iv) implies (iii), and (iii) implies (iv) by composition with the projectors. Conditions (ii) and (iii) are equivalent by the definition of Ob and additive cancellation. Conditions (i) and (ii) are equivalent by [source reference].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e98a525f93a0f0b0","tex":"\\sum_{\\mu,\\nu}","aliases":"∑_μ,ν sum mu nu","display":"\\sum_{\\mu,\\nu}","kind":"use","excerpt":"Insert completeness on both sides: Ob = (∑_μΠ^a_μ) Ob (∑_νΠ^a_ν) = ∑_μ,νΠ^a_μObΠ^a_ν. Thus (iv) implies (iii), and (iii) implies (iv) by composition with the projectors. Conditions (ii) and (iii) are equivalent by the definition of Ob and additive cancellation. Conditions (i) and (ii) are equivalent by [source reference].","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-49a5938a580ef00a","tex":"\\sum_{\\mu,\\nu\\in I_a}","aliases":"∑_μ,ν∈ I_a sum mu nu in","display":"\\sum_{\\mu,\\nu\\in I_a}","kind":"use","excerpt":"[Exact reconstruction and detection by obstruction blocks] The total obstruction has the exact finite resolution Ob_c,b,a = ∑_μ,ν∈ I_a Ob^μν_c,b,a. Moreover, the following are equivalent: [label=(roman*)] item Δ_c,b,a=0; item H_c,b,a=id; item Ob_c,b,a=0; item Ob^μν_c,b,a=0 for every μ,ν. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.obstruction_s","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-cd74cb17b4132748","tex":"\\sum_\\nu","aliases":"∑_ν sum nu","display":"\\sum_\\nu","kind":"use","excerpt":"Insert completeness on both sides: Ob = (∑_μΠ^a_μ) Ob (∑_νΠ^a_ν) = ∑_μ,νΠ^a_μObΠ^a_ν. Thus (iv) implies (iii), and (iii) implies (iv) by composition with the projectors. Conditions (ii) and (iii) are equivalent by the definition of Ob and additive cancellation. Conditions (i) and (ii) are equivalent by [source reference].","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-bbf40b6240c18f80","tex":"\\sum_{\\pi=\\{B_1,\\ldots,B_k\\}}","aliases":"∑_π={B_1,…,B_k} sum pi ldots","display":"\\sum_{\\pi=\\{B_1,\\ldots,B_k\\}}","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-60c47c363b013a09","tex":"\\sum_{\\tau\\in\\mathfrak T_m}","aliases":"∑_τ∈ T_m sum tau in mathfrak","display":"\\sum_{\\tau\\in\\mathfrak T_m}","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-6540b6ebc95be7f8","tex":"\\sum_{\\ell=0}^{m-1}","aliases":"∑_ℓ=0^m-1 sum ell","display":"\\sum_{\\ell=0}^{m-1}","kind":"use","excerpt":"y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":":Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘","count":16},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet o","count":8},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in ","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"[Exact direct packet rotation] If max({0}∪{ P_α-Q_α :α∈ I})<1, then S( Q, P) is invertible. The operator U( Q, P) is unitary and satisfies Q_α U( Q, P) = U( Q, P)P_α for every α.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"· +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","excerpt":"· +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"[Packet decomposition of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.","count":5},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"Suppose x_m satisfies Φ(x_m)∈ F^mY. Assume that corrections h∈ F^mX obey a first order congruence Φ(x_m+h) ≡ Φ(x_m)+L_mh pmodF^m+1Y, where L_m:F^mX→ F^mY is a module homomorphism satisfying L_m(F^m+1X)⊆ F^m+1Y and therefore induces overline L_m: F^mX/F^m+1X ⟶ F^mY/F^m+1Y.","count":3},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). Fo","count":14},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9f55fb1c51ddcd12","tex":"\\supseteq","aliases":"⊇ supseteq","display":"\\supseteq","kind":"use","excerpt":"Let X and Y be modules with decreasing filtrations X=F^0X⊇ F^1X⊇⋯, Y=F^0Y⊇ F^1Y⊇⋯. Assume both are separated and complete for their filtration topologies. Let Φ:X→ Y be continuous and filtration congruence preserving in the sense that h∈ F^mX Longrightarrow Φ(x+h)-Φ(x)∈ F^mY for every x∈ X and every m. This condition makes the statement Φ(x)∈ F^mY ","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":12},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-59a3a39b512b97d6","tex":"\\bigoplus_{g\\in G}","aliases":"⊕_g∈ G bigoplus in","display":"\\bigoplus_{g\\in G}","kind":"use","excerpt":"Let G be a finite abelian group and let A=⊕_g∈ GA_g be a G graded module.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","excerpt":"[Current fiber scope] The compiled Paper III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not att","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-7d0f5e3799ea2757","tex":"\\bigoplus_{j\\in\\mathbb Z/N\\mathbb Z}","aliases":"⊕_j∈ Z/N Z bigoplus in mathbb mathbb","display":"\\bigoplus_{j\\in\\mathbb Z/N\\mathbb Z}","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-9b591bcc37a61369","tex":"\\bigoplus_\\alpha","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_\\alpha","kind":"use","excerpt":"The grading assumption gives im D=⊕_αim D_α. Taking the quotient of the direct sum by this direct sum of images gives the displayed isomorphism.","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-90b882d77bb3725c","tex":"\\bigoplus_{\\alpha\\in I}","aliases":"⊕_α∈ I bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I}","kind":"use","excerpt":"Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":6},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hol","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"a normed setting every packet projector and every coarsening operator is bounded. Differentiability of operator valued maps is taken in operator norm. A Banach space isomorphism is a bounded linear bijection with bounded inverse. For a decreasing filtration (F^mX), completeness means that the canonical map X⟶varprojlim_m X/F^mX is an isomorphism an","count":2},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-0c12df6de0a7dc07","tex":"\\bigcap_m","aliases":"⋂_m bigcap","display":"\\bigcap_m","kind":"use","excerpt":"a normed setting every packet projector and every coarsening operator is bounded. Differentiability of operator valued maps is taken in operator norm. A Banach space isomorphism is a bounded linear bijection with bounded inverse. For a decreasing filtration (F^mX), completeness means that the canonical map X⟶varprojlim_m X/F^mX is an isomorphism an","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-08bf84a1d20b2d0b","tex":"\\bigcap_r","aliases":"⋂_r bigcap","display":"\\bigcap_r","kind":"use","excerpt":"Fix r. For every m≥ r, one has Φ(x_m)∈ F^mY⊆ F^rY. The submodule F^rY is open and therefore closed in the filtration topology. Continuity of Φ gives Φ(x_m)→Φ(x_∞), so Φ(x_∞)∈ F^rY. Since this holds for every r and Y is separated, Φ(x_∞)∈⋂_rF^rY=0.","count":1},{"work":"paper-iii","url":"notation/paper-iii.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[Well defined lift classes] Under the filtration congruence assumption, the condition Φ(x)∈ F^mY depends only on the class of x in X/F^mX. Fix a representative x_m of a depth m class. The map [h]⟼[x_m+h]_m+1 identifies correction classes [h]∈ F^mX/F^m+1X with the classes at depth m+1 that reduce to [x_m]_m. Under this identification, the solutions ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":275},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":779},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4b82e0794ad5cb35","tex":"\\mathfrak A","aliases":"A mathfrak","display":"\\mathfrak A","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-30f43826e4263e1a","tex":"\\mathfrak A_N^+","aliases":"A_N^+ mathfrak","display":"\\mathfrak A_N^+","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d21ef441aba4364d","tex":"\\mathfrak A_N^-","aliases":"A_N^- mathfrak","display":"\\mathfrak A_N^-","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6983fa40ff77fa12","tex":"\\mathrm{abs}","aliases":"abs mathrm","display":"\\mathrm{abs}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-327abfc143ff067a","tex":"\\Bcal","aliases":"Bcal B mathcal","display":"{\\mathcal B}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":17},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-51e663dd35f8862b","tex":"\\Bcal_{N,j}","aliases":"Bcal_N,j B_N,j Bcal mathcal","display":"{\\mathcal B}_{N,j}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"or 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn)","count":36},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-40591b7fff4431b2","tex":"C_{N,j}","aliases":"C_N,j","display":"C_{N,j}","kind":"definition-passage","excerpt":"or 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn)","count":9},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b4a69286b41cbe31","tex":"C_{N,j,n}","aliases":"C_N,j,n","display":"C_{N,j,n}","kind":"definition-passage","excerpt":"For r∈ C, define formal coefficients C_N,j,n(r) by C_N,j(z)^r = ∑_n≥0C_N,j,n(r)fracz^Nn(Nn)!. On a simply connected region containing 0 and avoiding the zeros of Φ_N,j(z^N), interpret this power as e^rL, where L is the holomorphic logarithm of C_N,j normalized by L(0)=0.","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":21},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-456fdfa5f34a5117","tex":"\\CC^N","aliases":"CC^N C^N CC mathbb","display":"{\\mathbb C}^N","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4e52835b675ac9ef","tex":"\\Cfun","aliases":"Cfun C mathsf","display":"{\\mathsf C}","kind":"definition-passage","excerpt":"For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn","count":15},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f5d0042664713d89","tex":"\\Cfun_{N,j}","aliases":"Cfun_N,j C_N,j Cfun mathsf","display":"{\\mathsf C}_{N,j}","kind":"definition-passage","excerpt":"For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn","count":11},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":35},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-19f71f3af9894264","tex":"\\mathrm{der}","aliases":"der mathrm","display":"\\mathrm{der}","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-28852ba6759dd734","tex":"\\Dfun","aliases":"Dfun D mathsf","display":"{\\mathsf D}","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":108},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d580abc74d5062ed","tex":"\\Dfun_N","aliases":"Dfun_N D_N Dfun mathsf","display":"{\\mathsf D}_N","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":70},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e472c8b232ff91ad","tex":"\\Dfun_N'","aliases":"Dfun_N' D_N' Dfun mathsf","display":"{\\mathsf D}_N'","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":17},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-08d21ec98995eca1","tex":"\\Dfun_{N,0}","aliases":"Dfun_N,0 D_N,0 Dfun mathsf","display":"{\\mathsf D}_{N,0}","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c1d5142975bd0fa0","tex":"\\Dfun_{N,j}","aliases":"Dfun_N,j D_N,j Dfun mathsf","display":"{\\mathsf D}_{N,j}","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":41},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":157},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-473dfae94e6cd783","tex":"e^{-s\\log|\\lambda|}","aliases":"e^-slog|λ| log lambda","display":"e^{-s\\log|\\lambda|}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6dcd93ee0614b9ca","tex":"e^{-s\\log|\\rho|}","aliases":"e^-slog|ρ| log rho","display":"e^{-s\\log|\\rho|}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e4f632909385c1ce","tex":"e^{-s\\Log\\lambda}","aliases":"e^-sLogλ Log lambda operatorname lambda","display":"e^{-s{\\operatorname{Log}}\\lambda}","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-184e074ec633d7bb","tex":"e^{-s\\Log\\rho}","aliases":"e^-sLogρ Log rho operatorname rho","display":"e^{-s{\\operatorname{Log}}\\rho}","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c3cbcdea0e9d0e9b","tex":"e^{rL}","aliases":"e^rL","display":"e^{rL}","kind":"definition-passage","excerpt":"For r∈ C, define formal coefficients C_N,j,n(r) by C_N,j(z)^r = ∑_n≥0C_N,j,n(r)fracz^Nn(Nn)!. On a simply connected region containing 0 and avoiding the zeros of Φ_N,j(z^N), interpret this power as e^rL, where L is the holomorphic logarithm of C_N,j normalized by L(0)=0.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cdc491b152dc2abb","tex":"e^{xz}","aliases":"e^xz","display":"e^{xz}","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-517adbc54a8d8e03","tex":"e^{\\lambda z}","aliases":"e^λ z lambda","display":"e^{\\lambda z}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":15},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cf85306cf8e31b85","tex":"e_{j+1}","aliases":"e_j+1","display":"e_{j+1}","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-62326bcf64c61a2a","tex":"E_m^{(N,j,r)}","aliases":"E_m^(N,j,r)","display":"E_m^{(N,j,r)}","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":9},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-19911032e8985ebc","tex":"e_{N-1}","aliases":"e_N-1","display":"e_{N-1}","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-02857ba6277fbbbc","tex":"\\Efun","aliases":"Efun E mathsf","display":"{\\mathsf E}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":21},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-de3cc61a1ed3d5b3","tex":"\\Efun_{N,0}","aliases":"Efun_N,0 E_N,0 Efun mathsf","display":"{\\mathsf E}_{N,0}","kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-504ccb92769528db","tex":"\\Efun_{N,j}","aliases":"Efun_N,j E_N,j Efun mathsf","display":"{\\mathsf E}_{N,j}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":17},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":17},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ac7885f6694a994f","tex":"H_N","aliases":"H_N","display":"H_N","kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":17},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b28d6c4b94c5afd8","tex":"h_N","aliases":"h_N","display":"h_N","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":38},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":190},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6cca5b503f09502b","tex":"J_N","aliases":"J_N","display":"J_N","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":17},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c0314e68e3d01e27","tex":"J_N^N","aliases":"J_N^N","display":"J_N^N","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":65},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"For r∈ C, define formal coefficients C_N,j,n(r) by C_N,j(z)^r = ∑_n≥0C_N,j,n(r)fracz^Nn(Nn)!. On a simply connected region containing 0 and avoiding the zeros of Φ_N,j(z^N), interpret this power as e^rL, where L is the holomorphic logarithm of C_N,j normalized by L(0)=0.","count":24},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-15c8c86f6474c315","tex":"\\Log","aliases":"Log operatorname","display":"{\\operatorname{Log}}","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":27},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":237},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d4df46a14542b4b5","tex":"\\max_{0\\leq k<N}","aliases":"max_0≤ k<N max leq","display":"\\max_{0\\leq k<N}","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":830},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":222},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b3c541ce8e25de25","tex":"\\mathrm{orb}","aliases":"orb mathrm","display":"\\mathrm{orb}","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":60},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0f1efa218d5dcb4c","tex":"P_{N,n}","aliases":"P_N,n","display":"P_{N,n}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":29},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0","count":50},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":215},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ff81ba5e23d549c7","tex":"\\Re","aliases":"Re","display":"\\Re","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":23},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7fdf44047d9358ee","tex":"\\mathrm{red}","aliases":"red mathrm","display":"\\mathrm{red}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":12},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":99},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5fe98e4f30da8940","tex":"\\Sfun","aliases":"Sfun S mathsf","display":"{\\mathsf S}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d87f4ba0b84c237a","tex":"\\Sfun_N^{(r)}","aliases":"Sfun_N^(r) S_N^(r) Sfun mathsf","display":"{\\mathsf S}_N^{(r)}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":32},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-da4fb737a4a4bc72","tex":"\\mathfrak T","aliases":"T mathfrak","display":"\\mathfrak T","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4505604ff161e9a8","tex":"T_N","aliases":"T_N","display":"T_N","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":19},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-426c2e2ef16a6555","tex":"\\mathfrak T_N","aliases":"T_N mathfrak","display":"\\mathfrak T_N","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-46a3ec781ec50559","tex":"\\Tfun","aliases":"Tfun T mathsf","display":"{\\mathsf T}","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e3731375084c54cb","tex":"\\Tfun_N","aliases":"Tfun_N T_N Tfun mathsf","display":"{\\mathsf T}_N","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a4b881f851b3f1df","tex":"\\Thetafun","aliases":"Thetafun symupΘ symup Theta","display":null,"kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-40eef48472d4771a","tex":"\\Thetafun_{N,n}","aliases":"Thetafun_N,n symupΘ_N,n Thetafun symup Theta","display":null,"kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":61},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).","count":131},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6cd4c939f7d7c61f","tex":"w^m","aliases":"w^m","display":"w^m","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":14},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e6f16f2e02ae0bf2","tex":"w^n","aliases":"w^n","display":"w^n","kind":"definition-passage","excerpt":"Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).","count":28},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":66},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-df0285eec4b01028","tex":"\\mathfrak Z","aliases":"Z mathfrak","display":"\\mathfrak Z","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":24},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":372},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"definition-passage","excerpt":"For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn","count":11},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-56835621b2f38901","tex":"z^m","aliases":"z^m","display":"z^m","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-afc30c788aac6f77","tex":"z^N","aliases":"z^N","display":"z^N","kind":"definition-passage","excerpt":"Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).","count":41},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ae6f7a635f6f6ddb","tex":"z^{Nn}","aliases":"z^Nn","display":"z^{Nn}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":12},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f9f1f546f3ae0830","tex":"z^{Nn+j}","aliases":"z^Nn+j","display":"z^{Nn+j}","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-676769c66002f30a","tex":"z^{Nn+N-1}","aliases":"z^Nn+N-1","display":"z^{Nn+N-1}","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3785fea000ba52f0","tex":"\\mathfrak Z_{N,q}^{\\mathrm{der}}","aliases":"Z_N,q^der mathfrak mathrm","display":"\\mathfrak Z_{N,q}^{\\mathrm{der}}","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-03ac7c9641211143","tex":"\\mathfrak Z_N^{\\mathrm{abs}}","aliases":"Z_N^abs mathfrak mathrm","display":"\\mathfrak Z_N^{\\mathrm{abs}}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0c06e3e1fda1dbfa","tex":"\\mathfrak Z_N^{\\mathrm{orb}}","aliases":"Z_N^orb mathfrak mathrm","display":"\\mathfrak Z_N^{\\mathrm{orb}}","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fca16336850bcc9e","tex":"\\mathfrak Z_N^{\\mathrm{red}}","aliases":"Z_N^red mathfrak mathrm","display":"\\mathfrak Z_N^{\\mathrm{red}}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ec84f843cac321e2","tex":"\\Zcal","aliases":"Zcal Z mathcal","display":"{\\mathcal Z}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":63},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f8a971f413dd4d59","tex":"|^{-s}","aliases":"|^-s","display":"|^{-s}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"definition-passage","excerpt":"Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).","count":84},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-51020088f18e62aa","tex":"\\Delta_N","aliases":"Δ_N Delta","display":"\\Delta_N","kind":"definition-passage","excerpt":"Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).","count":59},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":33},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":56},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bc6e9564c2f2a99a","tex":"\\kappa_{N,m}","aliases":"κ_N,m kappa","display":"\\kappa_{N,m}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":23},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":30},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":78},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1e5fde598f577a45","tex":"\\lambda^{-m}","aliases":"λ^-m lambda","display":"\\lambda^{-m}","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-036d06d495662f6f","tex":"\\Lambda_N","aliases":"Λ_N Lambda","display":"\\Lambda_N","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":27},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":36},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-eae7a938d6364e6e","tex":"\\xi^N","aliases":"ξ^N xi","display":"\\xi^N","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":94},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":38},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":32},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-62f366adedc0cbc7","tex":"\\sigma_{N,m}","aliases":"σ_N,m sigma","display":"\\sigma_{N,m}","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn","count":24},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8edece78cbd674e4","tex":"\\Phi_{N,j}","aliases":"Φ_N,j Phi","display":"\\Phi_{N,j}","kind":"definition-passage","excerpt":"For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn","count":20},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":23},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":43},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":136},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d1cabd23a4e71916","tex":"\\sum_{m\\geq0}","aliases":"∑_m≥0 sum geq","display":"\\sum_{m\\geq0}","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-97951b3ae8950fca","tex":"\\sum_{m\\geq1}","aliases":"∑_m≥1 sum geq","display":"\\sum_{m\\geq1}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-441a6af1afa122e0","tex":"\\sum_{n\\geq0}","aliases":"∑_n≥0 sum geq","display":"\\sum_{n\\geq0}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":19},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a7b3358a2feb8ba3","tex":"\\sum_{\\substack{\\Dfun_N(\\rho)=0\\\\ \\Dfun_N'(\\rho)\\ne0}}","aliases":"∑_substackDfun_N(ρ)=0 Dfun_N'(ρ)≠0 ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 sum substack Dfun rho Dfun rho ne sum substack mathsf rho mathsf rho ne","display":"\\sum_{\\substack{{\\mathsf D}_N(\\rho)=0\\\\ {\\mathsf D}_N'(\\rho)\\ne0}}","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-03fcd44478c606ca","tex":"\\sum_{\\lambda\\in\\Lambda_N}","aliases":"∑_λ∈Λ_N sum lambda in Lambda","display":"\\sum_{\\lambda\\in\\Lambda_N}","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":11},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3e969118dcce1120","tex":"\\sum_{\\rho\\in\\Zcal(\\Dfun_N)}","aliases":"∑_ρ∈Zcal(Dfun_N) ∑_ρ∈ Z( D_N) sum rho in Zcal Dfun sum rho in mathcal mathsf","display":"\\sum_{\\rho\\in{\\mathcal Z}({\\mathsf D}_N)}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":66},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":106},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":18},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is cancelled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":35},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-10ef17a5cef5ca03","tex":"b_{N,n}","aliases":"b_N,n","display":"b_{N,n}","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is cancelled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c0bb6498d7717ffb","tex":"\\Cfun_{2,0}","aliases":"Cfun_2,0 C_2,0 Cfun mathsf","display":"{\\mathsf C}_{2,0}","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-994779404d301198","tex":"\\Cfun_{2,1}","aliases":"Cfun_2,1 C_2,1 Cfun mathsf","display":"{\\mathsf C}_{2,1}","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is cancelled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":91},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is cancelled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d3405c1cc6903854","tex":"D_n","aliases":"D_n","display":"D_n","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is cancelled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9ed0c47b8f5552f3","tex":"D_{n-k}","aliases":"D_n-k","display":"D_{n-k}","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is cancelled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b45adb173afae48f","tex":"\\Dfun_N^{(N)}","aliases":"Dfun_N^(N) D_N^(N) Dfun mathsf","display":"{\\mathsf D}_N^{(N)}","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f30b020ad7331a73","tex":"\\Dfun_N^{-r}","aliases":"Dfun_N^-r D_N^-r Dfun mathsf","display":"{\\mathsf D}_N^{-r}","kind":"introduction-passage","excerpt":"For an entire function f, let Z(f) denote its multiset of nonzero zeros. Let r be a positive integer. Then D_N^-r is meromorphic and a simple zero of D_N becomes a pole of order r.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9918448367ba8abf","tex":"e^g","aliases":"e^g","display":"e^g","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1fe69fafa50db3c5","tex":"e^t","aliases":"e^t","display":"e^t","kind":"introduction-passage","excerpt":"=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7c01ae1b703c5263","tex":"e^{yt}","aliases":"e^yt","display":"e^{yt}","kind":"introduction-passage","excerpt":"=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e749405022a66e6b","tex":"E_m","aliases":"E_m","display":"E_m","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-eea8c720f33fb581","tex":"E_m^{(2,0,1)}","aliases":"E_m^(2,0,1)","display":"E_m^{(2,0,1)}","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d05f7fb87f2fa049","tex":"E_m^{(2,1,1)}","aliases":"E_m^(2,1,1)","display":"E_m^{(2,1,1)}","kind":"introduction-passage","excerpt":"daries For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"For an entire function f, let Z(f) denote its multiset of nonzero zeros. Let r be a positive integer. Then D_N^-r is meromorphic and a simple zero of D_N becomes a pole of order r.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"The localization estimate puts every sufficiently large zero in the fundamental sector into one of these disks. The zero sets of G and D_N coincide because their prefactor is nonzero. Each disk is invariant under complex conjugation, and D_N has real coefficients. The conjugate of its unique zero is therefore the same zero. It is real, positive for","count":20},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-02b0b5ff2a25a5d9","tex":"g'","aliases":"g'","display":"g'","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-785ae493560363c1","tex":"h_n","aliases":"h_n","display":"h_n","kind":"introduction-passage","excerpt":"[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by mul","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"introduction-passage","excerpt":"[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by mul","count":14},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c97f0bbe945fa2ee","tex":"K_N","aliases":"K_N","display":"K_N","kind":"introduction-passage","excerpt":"[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by mul","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is cancelled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":14},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-dc7ba1c49968c37c","tex":"p_k","aliases":"p_k","display":"p_k","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is cancelled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1e6bdd9702222c4b","tex":"\\sech","aliases":"sech operatorname","display":"{\\operatorname{sech}}","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":9},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-21670bd4e275805e","tex":"\\sinh","aliases":"sinh","display":"\\sinh","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"introduction-passage","excerpt":"n_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^y","count":64},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1","count":24},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-39aa37dcc31c4c7d","tex":"Y_N","aliases":"Y_N","display":"Y_N","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5381814dc2f38d7a","tex":"\\lambda^{-1}","aliases":"λ^-1 lambda","display":"\\lambda^{-1}","kind":"introduction-passage","excerpt":"[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by mul","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"sufficiently large zero in the fundamental sector into one of these disks. The zero sets of G and D_N coincide because their prefactor is nonzero. Each disk is invariant under complex conjugation, and D_N has real coefficients. The conjugate of its unique zero is therefore the same zero. It is real, positive for large m, and simple by the derivativ","count":32},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0a37e389a69b4972","tex":"\\tau_m","aliases":"τ_m tau","display":"\\tau_m","kind":"introduction-passage","excerpt":"sufficiently large zero in the fundamental sector into one of these disks. The zero sets of G and D_N coincide because their prefactor is nonzero. Each disk is invariant under complex conjugation, and D_N has real coefficients. The conjugate of its unique zero is therefore the same zero. It is real, positive for large m, and simple by the derivativ","count":23},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1f3736c162765e82","tex":"\\sum_{k=1}^{n}","aliases":"∑_k=1^n sum","display":"\\sum_{k=1}^{n}","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is cancelled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"rest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ ","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( -","count":40},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"For a power series F(z)=∑_m≥ 0a_mz^m and a positive integer N, the coefficients may be separated according to the residue of m modulo N. Root of unity averaging gives exact projection operators for this separation. This elementary construction is usually called a multisection or a root filter. It appears throughout the classical theory of generatin","count":58},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"[Newton sums for the reduced zeros] For |w| below the least zero modulus, with the analytic logarithm normalized to zero at the origin, -logΔ_N(w) = ∑_m≥1σ_N,mfracw^mm. Thus κ_N,m=σ_N,m. If a_m=frac(-1)^m(Nm)!, a_0=1, then σ_N,m = -ma_m- ∑_k=1^m-1a_kσ_N,m-k.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"[The case N=2] For N=2, D_2(z)=cos z. Its zeros are ±(m+frac12)π, m≥0, and all are simple. They agree with the formula t_m=A_2(m+1/2).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e513bab7c906c944","tex":"A_{3,1}","aliases":"A_3,1","display":"A_{3,1}","kind":"use","excerpt":",2.60000) (1)_2=2!; draw[fill=black!0,draw=black!40,line width=.35pt] (0.80000,1.35000) rectangle ++(1.60000,0.70000); node[] at (1.60000,1.70000) frac43; draw[fill=black!0,draw=black!40,line width=.35pt] (2.40000,1.35000) rectangle ++(1.60000,0.70000); node[] at (3.20000,1.70000) frac73; node[anchor=west] at (4.50000,1.70000) (4/3)_2; node[] at (9","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d84b5a24a5037067","tex":"A_4","aliases":"A_4","display":"A_4","kind":"use","excerpt":"[Exact fourth-order strings] For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2). The positive zeros are exactly τ_m = sqrt2π(m+frac12) = A_4(m+frac12). Rotation by the fourth roots of unity gives the real and imaginary zero strings. In this case the error term in [source reference] is identically zero.","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e0d7ec79a910cff8","tex":"A_4^{-4s}","aliases":"A_4^-4s","display":"A_4^{-4s}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a457ef5cf975a843","tex":"A_4^{-s}","aliases":"A_4^-s","display":"A_4^{-s}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-986b06b713868bf2","tex":"A_4^4","aliases":"A_4^4","display":"A_4^4","kind":"use","excerpt":"For N=2, the reduced zeros are π^2(m+frac12)^2, m≥0. For N=4, [source reference] gives the reduced zeros A_4^4(m+frac12)^4, m≥0. Summing their negative powers proves the reduced formulas. The absolute formulas follow from Z_N^abs(s) = N Z_N^red(s/N).","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"[Newton sums for the reduced zeros] For |w| below the least zero modulus, with the analytic logarithm normalized to zero at the origin, -logΔ_N(w) = ∑_m≥1σ_N,mfracw^mm. Thus κ_N,m=σ_N,m. If a_m=frac(-1)^m(Nm)!, a_0=1, then σ_N,m = -ma_m- ∑_k=1^m-1a_kσ_N,m-k.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"For a power series F(z)=∑_m≥ 0a_mz^m and a positive integer N, the coefficients may be separated according to the residue of m modulo N. Root of unity averaging gives exact projection operators for this separation. This elementary construction is usually called a multisection or a root filter. It appears throughout the classical theory of generatin","count":9},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-07112285aabb492d","tex":"a_N","aliases":"a_N","display":"a_N","kind":"use","excerpt":"The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.","count":23},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5f3deb161f4183b8","tex":"A_{N,j}","aliases":"A_N,j","display":"A_{N,j}","kind":"use","excerpt":"[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( -","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4e20e5a9f02dabcc","tex":"A_{N,n}","aliases":"A_N,n","display":"A_{N,n}","kind":"use","excerpt":"[A sequence of binomial type] Define A_N,n(r) = fracn!(Nn)!P_N,n(r). Then A_N,n(r+s) = ∑_q=0^n binom nq A_N,q(r)A_N,n-q(s). Thus (A_N,n)_n≥0 is a polynomial sequence of binomial type.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f75a0da9dfc0cdb8","tex":"A_{N,n-q}","aliases":"A_N,n-q","display":"A_{N,n-q}","kind":"use","excerpt":"[A sequence of binomial type] Define A_N,n(r) = fracn!(Nn)!P_N,n(r). Then A_N,n(r+s) = ∑_q=0^n binom nq A_N,q(r)A_N,n-q(s). Thus (A_N,n)_n≥0 is a polynomial sequence of binomial type.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ae9f2ce1dcdcc77f","tex":"A_{N,q}","aliases":"A_N,q","display":"A_{N,q}","kind":"use","excerpt":"[A sequence of binomial type] Define A_N,n(r) = fracn!(Nn)!P_N,n(r). Then A_N,n(r+s) = ∑_q=0^n binom nq A_N,q(r)A_N,n-q(s). Thus (A_N,n)_n≥0 is a polynomial sequence of binomial type.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c8256bcc167667e2","tex":"A_N^{-Ns}","aliases":"A_N^-Ns","display":"A_N^{-Ns}","kind":"use","excerpt":"[Meromorphic continuation of the symmetry-orbit zero zeta] Put A_N=fracπsin(π/N). There is an entire function Q_N^red such that Z_N^red(s) = A_N^-Nsζ(Ns,frac12)+Q_N^red(s) on Re s>1/N. The right hand side gives a meromorphic continuation to C with one simple pole at s=1/N. Its residue is fracsin(π/N)Nπ.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0691165b07852974","tex":"A_N^{-s}","aliases":"A_N^-s","display":"A_N^{-s}","kind":"use","excerpt":"There is an entire function Q_N^abs such that, initially on Re s>1, Z_N^abs(s) = N A_N^-sζ(s,frac12)+Q_N^abs(s). Thus Z_N^abs continues meromorphically to C with one simple pole at s=1, of residue fracNsin(π/N)π.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-38143a44ee0b7f8e","tex":"a_q","aliases":"a_q","display":"a_q","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e792db7aa03cd7f8","tex":"a_{\\rho,r}","aliases":"a_ρ,r rho","display":"a_{\\rho,r}","kind":"use","excerpt":"_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a9db24aaa1cf5ff2","tex":"a_{\\rho,\\ell}","aliases":"a_ρ,ℓ rho ell","display":"a_{\\rho,\\ell}","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a3a52d54c059cb53","tex":"a_\\ell","aliases":"a_ℓ ell","display":"a_\\ell","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8cb55ed44edecb02","tex":"a_\\ell^k","aliases":"a_ℓ^k ell","display":"a_\\ell^k","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4e2b11a7001e1904","tex":"\\arg","aliases":"arg","display":"\\arg","kind":"use","excerpt":"If z=x+iy lies in the sector |arg z|≤ a_N, has x≥ R, and is a zero of D_N, then |y|≤ Y.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONTraditional monograph: Theorem 3.1 (p. 4), Proposition 3.2 (p. 5). For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matc","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"vpage11Two recurrences reach the same answerNEWTON AND ADDITIONTraditional monograph: Theorem 7.3 (p. 14), Theorem 7.4 (p. 15), Theorem 7.6 (p. 15). One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagona","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b224b896d9d01ac4","tex":"B_3","aliases":"B_3","display":"B_3","kind":"use","excerpt":"r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. ","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5284f24ae8cda5b8","tex":"b_3","aliases":"b_3","display":"b_3","kind":"use","excerpt":"FICIENT RECONSTRUCTIONTraditional monograph: Theorem 7.3 (p. 14), Theorem 7.6 (p. 15). Start in the reduced variable with Δ_2(w)=1-frac w2+fracw^224-fracw^3720+⋯. Write 1/Δ_2(w)=∑ b_nw^n. Matching the product with 1 gives a unit lower-triangular system. At each row there is exactly one new unknown. vfigtriangular_inverse The first three equations a","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-572d503f3abbc41a","tex":"\\mathcal B_{3,0}","aliases":"B_3,0 mathcal","display":"\\mathcal B_{3,0}","kind":"use","excerpt":"0,1.25000) rectangle ++(2.65000,0.85000); node[] at (3.72500,1.67500) 6 z^2; draw[fill=black!18,draw=black!40,line width=.35pt] (5.05000,1.25000) rectangle ++(2.65000,0.85000); node[] at (6.37500,1.67500) 3 z^3; draw[fill=black!0,draw=black!40,line width=.35pt] (7.70000,1.25000) rectangle ++(2.65000,0.85000); node[] at (9.02500,1.67500) 3 z^4; node","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-51edaf5d4fd6092c","tex":"\\mathcal B_{4,2}","aliases":"B_4,2 mathcal","display":"\\mathcal B_{4,2}","kind":"use","excerpt":"r rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-35e6270fb1adf1db","tex":"\\mathcal B_{4,j}","aliases":"B_4,j mathcal","display":"\\mathcal B_{4,j}","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONTraditional monograph: Theorem 3.1 (p. 4), Proposition 3.2 (p. 5). For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matc","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"use","excerpt":"The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":9},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-722e62b9ceac6572","tex":"b_{N,0}","aliases":"b_N,0","display":"b_{N,0}","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-99de146f9c618d5f","tex":"\\mathcal B_{N,i}","aliases":"B_N,i mathcal","display":"\\mathcal B_{N,i}","kind":"use","excerpt":"1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6de50404dc3969bc","tex":"\\mathcal B_{N,j}","aliases":"B_N,j mathcal","display":"\\mathcal B_{N,j}","kind":"use","excerpt":"1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-18b50bc4f4ffc9b8","tex":"\\mathcal B_{N,k}","aliases":"B_N,k mathcal","display":"\\mathcal B_{N,k}","kind":"use","excerpt":"1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c63fabcfa89c6ac0","tex":"b_{N,n-k}","aliases":"b_N,n-k","display":"b_{N,n-k}","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c0bb32ec887c4163","tex":"b_{n-k}","aliases":"b_n-k","display":"b_{n-k}","kind":"use","excerpt":"vpage11Two recurrences reach the same answerNEWTON AND ADDITIONTraditional monograph: Theorem 7.3 (p. 14), Theorem 7.4 (p. 15), Theorem 7.6 (p. 15). One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagona","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-57c8f1ae1da23ae1","tex":"B_r","aliases":"B_r","display":"B_r","kind":"use","excerpt":"Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e0d26e84e0952114","tex":"B_r'","aliases":"B_r'","display":"B_r'","kind":"use","excerpt":"Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a1c5eaef95cde19b","tex":"\\Bcal_{N,i}","aliases":"Bcal_N,i B_N,i Bcal mathcal","display":"{\\mathcal B}_{N,i}","kind":"use","excerpt":"[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-924ed6b80b428e13","tex":"\\Bcal_{N,k}","aliases":"Bcal_N,k B_N,k Bcal mathcal","display":"{\\mathcal B}_{N,k}","kind":"use","excerpt":"Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry tr","count":11},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"If G(z)=0, then |1+e^2is_Nz| ≤ C e^-γ_NRe z. The real part of 2is_Nz is bounded in the strip. Apply [source reference] with w=2is_Nz. There is a unique integer m such that |z-t_m|≤ C_1e^-γ_NRe z, where t_m=frac(2m+1)π2s_N. In particular, |Re z-t_m|=o(1). For all sufficiently large zeros we have |Re z-t_m|≤1, and hence |z-t_m|≤ C_2e^-γ_Nt_m with a c","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"G(z)=0, then |1+e^2is_Nz| ≤ C e^-γ_NRe z. The real part of 2is_Nz is bounded in the strip. Apply [source reference] with w=2is_Nz. There is a unique integer m such that |z-t_m|≤ C_1e^-γ_NRe z, where t_m=frac(2m+1)π2s_N. In particular, |Re z-t_m|=o(1). For all sufficiently large zeros we have |Re z-t_m|≤1, and hence |z-t_m|≤ C_2e^-γ_Nt_m with a cons","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-24612095df1bb71a","tex":"C_{2,0}","aliases":"C_2,0","display":"C_{2,0}","kind":"use","excerpt":"vpage14Three similar reciprocals differNORMALIZED SECTIONSTraditional monograph: Definition 9.1 (p. 19), Theorem 9.2 (p. 19), Example 9.11 (p. 22). The signed denominator and the normalized unsigned sections produce different reciprocal families. In order two, D_2^-1=sec z, C_2,0=sech z, C_2,1=z/sinh z. Their first block-EGF coefficients make the d","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8b6228375a9787bb","tex":"C_{2,1}","aliases":"C_2,1","display":"C_{2,1}","kind":"use","excerpt":"vpage14Three similar reciprocals differNORMALIZED SECTIONSTraditional monograph: Definition 9.1 (p. 19), Theorem 9.2 (p. 19), Example 9.11 (p. 22). The signed denominator and the normalized unsigned sections produce different reciprocal families. In order two, D_2^-1=sec z, C_2,0=sech z, C_2,1=z/sinh z. Their first block-EGF coefficients make the d","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a8e8fbaa4e63d973","tex":"C_L","aliases":"C_L","display":"C_L","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1539cdd586b7fb3f","tex":"C_L'","aliases":"C_L'","display":"C_L'","kind":"use","excerpt":"-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-62ce335ae3e26f33","tex":"C_M","aliases":"C_M","display":"C_M","kind":"use","excerpt":"[Near cancellation of two exponentials] For every M>0, there are constants ε_M>0 and C_M>0 such that the following holds. If |Re w|≤ M and |1+e^w|≤ε_M, then there is a unique integer m with |w-(2m+1)π i| ≤ C_M|1+e^w|.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c1ecf2aa8f993509","tex":"c_N","aliases":"c_N","display":"c_N","kind":"use","excerpt":"At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry tr","count":11},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1b714dca696a6ce4","tex":"C_{N,j,n-q}","aliases":"C_N,j,n-q","display":"C_{N,j,n-q}","kind":"use","excerpt":"[Parameter-addition law for residue-section reciprocals] For each n≥1, the coefficient C_N,j,n(r) is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n! (-fracj!(N+j)!)^n. It satisfies C_N,j,n(r+s) = ∑_q=0^n binomNnNq C_N,j,q(r)C_N,j,n-q(s). In particular, C_N,j,n(1)=C_N,j(n).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-28a6886b66c26c24","tex":"C_{N,j,q}","aliases":"C_N,j,q","display":"C_{N,j,q}","kind":"use","excerpt":"[Parameter-addition law for residue-section reciprocals] For each n≥1, the coefficient C_N,j,n(r) is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n! (-fracj!(N+j)!)^n. It satisfies C_N,j,n(r+s) = ∑_q=0^n binomNnNq C_N,j,q(r)C_N,j,n-q(s). In particular, C_N,j,n(1)=C_N,j(n).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e851c4705c600db7","tex":"\\Cfun_{N,j}^r","aliases":"Cfun_N,j^r C_N,j^r Cfun mathsf","display":"{\\mathsf C}_{N,j}^r","kind":"use","excerpt":"Multiply the block series for C_N,j^r by the exponential series and collect the coefficient of total degree m.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"use","excerpt":"For N=1, the only section is e^λ z. For N=2, E_2,0(z)=cosh z, E_2,1(z)=sinh z. These are classical instances of the same cyclic basis.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ab700623b6727b01","tex":"\\mathbf D","aliases":"D mathbf","display":"\\mathbf D","kind":"use","excerpt":"[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":20},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a152920678c4fe82","tex":"D'''","aliases":"D'''","display":"D'''","kind":"use","excerpt":"retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-227743dc6ee61b15","tex":"D^{-(r+s)}","aliases":"D^-(r+s)","display":"D^{-(r+s)}","kind":"use","excerpt":"2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers uses a common specified logarithm. A positive integer reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argumen","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8e6103882f8afa39","tex":"D^{-r}","aliases":"D^-r","display":"D^{-r}","kind":"use","excerpt":"2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers uses a common specified logarithm. A positive integer reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argumen","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-61633e3eec80e17d","tex":"D^{-s}","aliases":"D^-s","display":"D^{-s}","kind":"use","excerpt":"2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers uses a common specified logarithm. A positive integer reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argumen","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7d80d76ad0ed63a4","tex":"d^k","aliases":"d^k","display":"d^k","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1b655d23a365d7e4","tex":"d^N","aliases":"d^N","display":"d^N","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c201d42ec1971e05","tex":"D^{\\prime}","aliases":"D^prime prime","display":"D^{\\prime}","kind":"use","excerpt":"draw[fill=black] (2.80000,2.30000) circle (0.055); node[] at (2.80000,4.65000) fixed-log domain about 0; draw[fill=white,line width=1pt] (4.40000,2.30000) circle (0.09); node[] at (5.10000,1.45000) ρ=π/2; draw[->,line width=1.1pt] (4.88,2.3) arc[start angle=0,end angle=330,radius=.48]; node[] at (2.80000,0.10000) the loop is not inside the disk; no","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-acd57baaf8b7df2d","tex":"D_{0}","aliases":"D_0","display":"D_{0}","kind":"use","excerpt":"pt] (1.40000,0.90000) rectangle ++(0.80000,0.80000); node[] at (1.80000,1.30000) 1; draw[fill=black!0,draw=black!40,line width=.35pt] (2.20000,0.90000) rectangle ++(0.80000,0.80000); node[] at (2.60000,1.30000) 0; node[] at (6.67500,3.78000) e^zJ_3; draw[fill=black!12,draw=black!40,line width=.35pt] (5.10000,2.25000) rectangle ++(1.05000,1.05000); ","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"15000) rectangle ++(1.05000,1.05000); node[] at (6.67500,0.67500) D_1; draw[fill=black!12,draw=black!40,line width=.35pt] (7.20000,0.15000) rectangle ++(1.05000,1.05000); node[] at (7.72500,0.67500) D_0; node[] at (10.80000,2.90000) J_3^3=-I; node[] at (10.80000,1.70000) D_3(0)=e_0; node[] at (10.80000,0.60000) det e^zJ_3=1; node[] at (6.10000,-0.8","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-10832ea38ded7f1f","tex":"D_{1}","aliases":"D_1","display":"D_{1}","kind":"use","excerpt":"!12,draw=black!40,line width=.35pt] (5.10000,2.25000) rectangle ++(1.05000,1.05000); node[] at (5.62500,2.77500) D_0; draw[fill=black!12,draw=black!40,line width=.35pt] (6.15000,2.25000) rectangle ++(1.05000,1.05000); node[] at (6.67500,2.77500) - D_2; draw[fill=black!12,draw=black!40,line width=.35pt] (7.20000,2.25000) rectangle ++(1.05000,1.05000","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","excerpt":"p. 13), Section 11 (p. 25), Section 12 (p. 33). This guide follows a concrete analytic family through three descriptions: selected Taylor coefficients, a finite differential system, and the zeros of an entire function. The formulas are linked, but the evidence needed for each description is different. vfigthree_views headKeep three small models in ","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f847d7618ed63aee","tex":"D_{2}","aliases":"D_2","display":"D_{2}","kind":"use","excerpt":"2.20000,0.90000) rectangle ++(0.80000,0.80000); node[] at (2.60000,1.30000) 0; node[] at (6.67500,3.78000) e^zJ_3; draw[fill=black!12,draw=black!40,line width=.35pt] (5.10000,2.25000) rectangle ++(1.05000,1.05000); node[] at (5.62500,2.77500) D_0; draw[fill=black!12,draw=black!40,line width=.35pt] (6.15000,2.25000) rectangle ++(1.05000,1.05000); no","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fe163f48e48b9c96","tex":"D_2^{-1}","aliases":"D_2^-1","display":"D_2^{-1}","kind":"use","excerpt":"vpage14Three similar reciprocals differNORMALIZED SECTIONSTraditional monograph: Definition 9.1 (p. 19), Theorem 9.2 (p. 19), Example 9.11 (p. 22). The signed denominator and the normalized unsigned sections produce different reciprocal families. In order two, D_2^-1=sec z, C_2,0=sech z, C_2,1=z/sinh z. Their first block-EGF coefficients make the d","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-679950b419c955b3","tex":"D_2^{\\prime\\prime}","aliases":"D_2^primeprime prime prime","display":"D_2^{\\prime\\prime}","kind":"use","excerpt":"[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.00000,4.50000) LINEAR EQUATION; node[] at (9.30000,4.50000) LOGARITHMIC DERIVATIVE; node[] at (3.00000,3.25000) D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000,1.85000) T_3","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d2c6012edbe5de2c","tex":"D_3","aliases":"D_3","display":"D_3","kind":"use","excerpt":"ordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Each initial jet (y(0),y'(0),y''(0)) therefore det","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f7881b24a27eb932","tex":"\\mathbf D_3","aliases":"D_3 mathbf","display":"\\mathbf D_3","kind":"use","excerpt":"; node[] at (5.62500,0.67500) D_2; draw[fill=black!12,draw=black!40,line width=.35pt] (6.15000,0.15000) rectangle ++(1.05000,1.05000); node[] at (6.67500,0.67500) D_1; draw[fill=black!12,draw=black!40,line width=.35pt] (7.20000,0.15000) rectangle ++(1.05000,1.05000); node[] at (7.72500,0.67500) D_0; node[] at (10.80000,2.90000) J_3^3=-I; node[] at ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-af497092b3be5660","tex":"D_3'","aliases":"D_3'","display":"D_3'","kind":"use","excerpt":"vpage19A root estimate becomes a certificateEXACT RATIONAL BRACKETSTraditional monograph: Theorem 11.7 (p. 28), Theorem 11.8 (p. 30), Corollary 11.9 (p. 31). The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' exc","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-90b32d1d1dff83bf","tex":"D_{3,0}","aliases":"D_3,0","display":"D_{3,0}","kind":"use","excerpt":"ed edge determines the cycleDIFFERENTIAL SYSTEMTraditional monograph: Theorem 5.3 (p. 8), Theorem 6.4 (p. 11), Corollary 6.5 (p. 12), Corollary 6.6 (p. 12). Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3e3dff1e5a9abb17","tex":"D_{3,1}","aliases":"D_3,1","display":"D_{3,1}","kind":"use","excerpt":"ed edge determines the cycleDIFFERENTIAL SYSTEMTraditional monograph: Theorem 5.3 (p. 8), Theorem 6.4 (p. 11), Corollary 6.5 (p. 12), Corollary 6.6 (p. 12). Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3ca422484b3e1eb4","tex":"D_{3,2}","aliases":"D_3,2","display":"D_{3,2}","kind":"use","excerpt":"vpage6Addition is quotient multiplicationFINITE ALGEBRATraditional monograph: Proposition 5.5 (p. 8), Theorem 6.2 (p. 10). Work in C[ξ]/(ξ^3+1). Every element has coordinates in 1,ξ,ξ^2. The entire exponential has the finite-coordinate form e^zξ=D_3,0(z)+D_3,1(z)ξ+D_3,2(z)ξ^2. The multiplication table makes the carry sign unavoidable. vfigsigned_ad","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b3a4549b81c78db4","tex":"D_{3,j}","aliases":"D_3,j","display":"D_{3,j}","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMTraditional monograph: Theorem 5.3 (p. 8), Theorem 6.4 (p. 11), Corollary 6.5 (p. 12), Corollary 6.6 (p. 12). Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8e4f06d6066f3c63","tex":"D_3^{\\prime\\prime\\prime}","aliases":"D_3^primeprimeprime prime prime prime","display":"D_3^{\\prime\\prime\\prime}","kind":"use","excerpt":"[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.00000,4.50000) LINEAR EQUATION; node[] at (9.30000,4.50000) LOGARITHMIC DERIVATIVE; node[] at (3.00000,3.25000) D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000,1.85000) T_3","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ca8301964f9bd376","tex":"D_4","aliases":"D_4","display":"D_4","kind":"use","excerpt":"n entire function. The formulas are linked, but the evidence needed for each description is different. vfigthree_views headKeep three small models in view For N=2, the signed denominator is D_2(z)=cos z. Its reciprocal is the familiar secant series. For N=3, a third exponential perturbs the zeros of a dominant cosine. For N=4, an exact factorizatio","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMTraditional monograph: Theorem 5.3 (p. 8), Theorem 6.4 (p. 11), Corollary 6.5 (p. 12), Corollary 6.6 (p. 12). Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3fb8837a44922746","tex":"D_N","aliases":"D_N","display":"D_N","kind":"use","excerpt":"D_N(z) · D_N,0(z), the denominator whose zeros control the signed reciprocal family.","count":25},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7b70f6bea52051d2","tex":"\\mathbf D_N","aliases":"D_N mathbf","display":"\\mathbf D_N","kind":"use","excerpt":"[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-251354dc81bb5f51","tex":"D_N'","aliases":"D_N'","display":"D_N'","kind":"use","excerpt":"T_N(z) · -D_N'(z)/D_N(z), a function, not the coefficient T_N(n).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ce674635b7c486c1","tex":"D_{N,0}","aliases":"D_N,0","display":"D_{N,0}","kind":"use","excerpt":"D_N(z) · D_N,0(z), the denominator whose zeros control the signed reciprocal family.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5ee31d563f299611","tex":"D_{N,j}","aliases":"D_N,j","display":"D_{N,j}","kind":"use","excerpt":"D_N,j(z) · The same series with (-1)^n. One full derivative cycle contributes a minus sign.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4efbc9ac8d9e0f0d","tex":"D_N^{-(r+s)}","aliases":"D_N^-(r+s)","display":"D_N^{-(r+s)}","kind":"use","excerpt":"Theorem 7.6 (p. 15). One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagonal in the Newton determinant is essential. vfignewton_addition headThe additive parameter law has block weights Using the same lo","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-36864a1aaaea8a13","tex":"D_N^{-r}","aliases":"D_N^-r","display":"D_N^{-r}","kind":"use","excerpt":"P_N,n(r) · Block-EGF coefficients of D_N^-r. They are polynomials in r.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-58b404e71b6ab9fb","tex":"D_N^{-s}","aliases":"D_N^-s","display":"D_N^{-s}","kind":"use","excerpt":"Theorem 7.6 (p. 15). One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagonal in the Newton determinant is essential. vfignewton_addition headThe additive parameter law has block weights Using the same lo","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d1d16e41580839a7","tex":"D_N^{\\prime}","aliases":"D_N^prime prime","display":"D_N^{\\prime}","kind":"use","excerpt":"oin=round] node[] at (3.00000,4.50000) LINEAR EQUATION; node[] at (9.30000,4.50000) LOGARITHMIC DERIVATIVE; node[] at (3.00000,3.25000) D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000,1.85000) T_3^primeprime-3T_3T_3^prime+T_3^3=1; node[] at (6.15000,0.50000","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4435e486a9912a3e","tex":"d_{pq}","aliases":"d_pq","display":"d_{pq}","kind":"use","excerpt":"[Entries of the fundamental matrix] For 0≤ p,q<N, let d_pq be the member of {0,…,N-1} congruent to p-q modulo N. Put ε_pq = 1, · p≥ q,","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0d09000efb054df8","tex":"\\ddots","aliases":"ddots","display":"\\ddots","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"igin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), bec","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a6c5d416bb3dfbb4","tex":"\\Dfun_1","aliases":"Dfun_1 D_1 Dfun mathsf","display":"{\\mathsf D}_1","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fbc53ff4b3c87616","tex":"\\Dfun_2","aliases":"Dfun_2 D_2 Dfun mathsf","display":"{\\mathsf D}_2","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5f4377cbf9ad281e","tex":"\\Dfun_3","aliases":"Dfun_3 D_3 Dfun mathsf","display":"{\\mathsf D}_3","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4864c3603c017539","tex":"\\Dfun_4","aliases":"Dfun_4 D_4 Dfun mathsf","display":"{\\mathsf D}_4","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b764ca7a3b447499","tex":"\\Dfun_{N,0}'","aliases":"Dfun_N,0' D_N,0' Dfun mathsf","display":"{\\mathsf D}_{N,0}'","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e930dfa4cab0538f","tex":"\\Dfun_{N,1}","aliases":"Dfun_N,1 D_N,1 Dfun mathsf","display":"{\\mathsf D}_{N,1}","kind":"use","excerpt":"[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-850ed82f3d2578f6","tex":"\\Dfun_{N,d_{pq}}","aliases":"Dfun_N,d_pq D_N,d_pq Dfun mathsf","display":"{\\mathsf D}_{N,d_{pq}}","kind":"use","excerpt":"-1, · p<q. Then (e^zJ_N)_pq = ε_pq D_N,d_pq(z). Thus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from q to p crosses the signed edge.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4b1aed9780b099dc","tex":"\\Dfun_{N,i}","aliases":"Dfun_N,i D_N,i Dfun mathsf","display":"{\\mathsf D}_{N,i}","kind":"use","excerpt":"[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c545097e1ea4183a","tex":"\\Dfun_{N,j}'","aliases":"Dfun_N,j' D_N,j' Dfun mathsf","display":"{\\mathsf D}_{N,j}'","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-15b31c89bfca987f","tex":"\\Dfun_{N,j-1}","aliases":"Dfun_N,j-1 D_N,j-1 Dfun mathsf","display":"{\\mathsf D}_{N,j-1}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-033ca7b2ad984fff","tex":"\\Dfun_{N,j}^{(k)}","aliases":"Dfun_N,j^(k) D_N,j^(k) Dfun mathsf","display":"{\\mathsf D}_{N,j}^{(k)}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-89fc0d6e81a8eaa5","tex":"\\Dfun_{N,j}^{(N)}","aliases":"Dfun_N,j^(N) D_N,j^(N) Dfun mathsf","display":"{\\mathsf D}_{N,j}^{(N)}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2960208ad6b384e5","tex":"\\Dfun_{N,k}","aliases":"Dfun_N,k D_N,k Dfun mathsf","display":"{\\mathsf D}_{N,k}","kind":"use","excerpt":"[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-af02055946b191fd","tex":"\\Dfun_{N,N-1}","aliases":"Dfun_N,N-1 D_N,N-1 Dfun mathsf","display":"{\\mathsf D}_{N,N-1}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-dee7549cf31fd663","tex":"\\Dfun_N^{-q}","aliases":"Dfun_N^-q D_N^-q Dfun mathsf","display":"{\\mathsf D}_N^{-q}","kind":"use","excerpt":"The derivative weighted sum omits the finitely many zeros that might be multiple. A treatment of those zeros requires the complete principal parts of D_N^-q rather than the single derivative weight.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-782506289eda6b80","tex":"\\Dfun_N^{\\prime}","aliases":"Dfun_N^prime D_N^prime Dfun prime mathsf prime","display":"{\\mathsf D}_N^{\\prime}","kind":"use","excerpt":"ic objects are introduced in the following order. First, E_N,j(z) = ∑_n≥0fracz^Nn+j(Nn+j)! is the residue j part of the exponential. Second, D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)! is the signed basis for y^(N)=-y. The section D_N= D_N,0 is the denominator used in the reciprocal family S_N^(r)= D_N^-r. Set T_N=- D_N^prime/ D_N. Equivalently, T_N is","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-01c396ae88f551ee","tex":"e^{(2k+1)\\pi i/N}","aliases":"e^(2k+1)π i/N pi","display":"e^{(2k+1)\\pi i/N}","kind":"use","excerpt":"We now assume N≥2. Write η_k=e^(2k+1)π i/N, 0≤ k<N. These are the vertices of a regular polygon and D_N(z)=frac1N∑_k=0^N-1e^η_kz.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-61130c15ab331a4f","tex":"e^{(z+w)\\xi}","aliases":"e^(z+w)ξ xi","display":"e^{(z+w)\\xi}","kind":"use","excerpt":"Differentiation multiplies by λξ in the first algebra and by ξ in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity e^(z+w)ξ=e^zξe^wξ gives the addition laws after comparison of the coefficients of 1,ξ,…,ξ^N-1.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9d7200ea906e4a03","tex":"e^{-2s_Ny}","aliases":"e^-2s_Ny","display":"e^{-2s_Ny}","kind":"use","excerpt":"a_N-θ. Monotonicity of cosine on [0,π], followed by reflection about π, therefore gives Re(η_ℓ z)≤ rcos(a_N+θ). The remaining vertex e^-ia_N has projection rcos(a_N-θ). Their difference is at least 2s_Nrsinθ=2s_Ny. If D_N(z)=0, divide its finite exponential sum by the term with exponent e^-ia_Nz and apply the triangle inequality: 1≤(N-1)e^-2s_Ny. T","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3e8da4431aa141cc","tex":"e^{-2\\gamma_Nt_m}","aliases":"e^-2γ_Nt_m gamma","display":"e^{-2\\gamma_Nt_m}","kind":"use","excerpt":"[Leading displacement term and derivative sign] With the notation of [source reference], τ_m = t_m+ fracR_N(t_m)2is_N + O(e^-2γ_Nt_m). The quotient R_N(t_m)/(2is_N) is real. One also has D_N'(τ_m) = (-1)^m+1 frac2s_NN e^c_Nτ_m (1+O(e^-γ_Nτ_m)).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-943d51998b0ad115","tex":"e^{-2\\pi i r}","aliases":"e^-2π i r pi","display":"e^{-2\\pi i r}","kind":"use","excerpt":"width=1pt] (4.40000,2.30000) circle (0.09); node[] at (5.10000,1.45000) ρ=π/2; draw[->,line width=1.1pt] (4.88,2.3) arc[start angle=0,end angle=330,radius=.48]; node[] at (2.80000,0.10000) the loop is not inside the disk; node[] at (9.20000,4.10000) D(z)∼ D^prime(ρ)(z-ρ); node[] at (9.20000,2.90000) log D⟼log D+2π i; node[] at (9.20000,1.65000) D^-","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-75c622f56ccbfb4c","tex":"e^{-2\\pi ir}","aliases":"e^-2π ir pi","display":"e^{-2\\pi ir}","kind":"use","excerpt":"gion containing 0 and normalize the logarithm to vanish there. D_N(z)^-r=exp(-rL(z)), e^L(z)=D_N(z), L(0)=0. vfigbranch_domain For D_2=cos z, the open disk |z|<π/2 is an admissible domain. A small loop around the simple zero ρ=π/2 is not contained in that disk. Analytic continuation around the zero changes the local logarithm by 2π i, and the power","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-93f2c197704b6a56","tex":"e^{-3t/2}","aliases":"e^-3t/2","display":"e^{-3t/2}","kind":"use","excerpt":"60) (12.05750,1.33419) (12.08675,1.31211) (12.11600,1.29036) (12.14525,1.26895) (12.17450,1.24790) (12.20375,1.22721) (12.23300,1.20689) (12.26225,1.18695) (12.29150,1.16740) (12.32075,1.14826) (12.35000,1.12952); draw[fill=white] (2.77215,2.02182) circle (0.07); draw[fill=white] (7.01644,2.02182) circle (0.07); draw[fill=white] (11.26073,2.02182) ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-95ab7e932f644020","tex":"e^{-3t_m}","aliases":"e^-3t_m","display":"e^{-3t_m}","kind":"use","excerpt":"[Leading correction for N=3] For N=3, t_m=frac2πsqrt3(m+frac12) and τ_m = t_m+ frac(-1)^msqrt3e^-3t_m/2 + O(e^-3t_m).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d20a1e79d4b08523","tex":"e^{-3t_m/2}","aliases":"e^-3t_m/2","display":"e^{-3t_m/2}","kind":"use","excerpt":"[Leading correction for N=3] For N=3, t_m=frac2πsqrt3(m+frac12) and τ_m = t_m+ frac(-1)^msqrt3e^-3t_m/2 + O(e^-3t_m).","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b6cecc0b8dc4e5e1","tex":"e^{-c_N|\\rho|}","aliases":"e^-c_N|ρ| rho","display":"e^{-c_N|\\rho|}","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9e49328e16a6e146","tex":"e^{-e^{-ia_N}t_m}","aliases":"e^-e^-ia_Nt_m","display":"e^{-e^{-ia_N}t_m}","kind":"use","excerpt":"At the real point t_m, the normalized value N e^-e^-ia_Nt_m D_N(t_m) is purely imaginary. The two dominant terms cancel there, so R_N(t_m) is purely imaginary and the displayed correction is real.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-26428a4ee1bc8c64","tex":"e^{-e^{-ia_N}z}","aliases":"e^-e^-ia_Nz","display":"e^{-e^{-ia_N}z}","kind":"use","excerpt":"η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e3bc61419c10fa3f","tex":"e^{-ia_N}","aliases":"e^-ia_N","display":"e^{-ia_N}","kind":"use","excerpt":"On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":11},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-162f43593822664a","tex":"e^{-is_N\\tau_m}","aliases":"e^-is_Nτ_m tau","display":"e^{-is_N\\tau_m}","kind":"use","excerpt":"For the derivative, use the formula in the proof of [source reference] and e^-is_Nτ_m = (-1)^m(-i) (1+O(e^-γ_Nτ_m)). The stated real sign follows.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-59ef4f417932e955","tex":"e^{-q c_N|\\rho|}","aliases":"e^-q c_N|ρ| rho","display":"e^{-q c_N|\\rho|}","kind":"use","excerpt":"-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3f6dc0c78382f5b1","tex":"e^{-t}","aliases":"e^-t","display":"e^{-t}","kind":"use","excerpt":"vpage18Two large terms nearly cancelTHE CUBIC MODELTraditional monograph: Lemma 11.5 (p. 27), Theorem 11.7 (p. 28), Corollary 11.9 (p. 31). For real t, order three has the exact formula D_3(t)=frace^-t+2e^t/2cos(sqrt3t/2)3. Divide by the nonzero factor 2e^t/2/3. The resulting equation separates the dominant cosine from the small third exponential. ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e252dd71b7542249","tex":"e^{-t/2}","aliases":"e^-t/2","display":"e^{-t/2}","kind":"use","excerpt":"60) (12.05750,1.33419) (12.08675,1.31211) (12.11600,1.29036) (12.14525,1.26895) (12.17450,1.24790) (12.20375,1.22721) (12.23300,1.20689) (12.26225,1.18695) (12.29150,1.16740) (12.32075,1.14826) (12.35000,1.12952); draw[fill=white] (2.77215,2.02182) circle (0.07); draw[fill=white] (7.01644,2.02182) circle (0.07); draw[fill=white] (11.26073,2.02182) ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5dded19773673070","tex":"e^{-w}","aliases":"e^-w","display":"e^{-w}","kind":"use","excerpt":"and integral comparison. Thus truncation has an explicit error mechanism. takeawayThe infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem. platecaptionThe exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use k","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-99b487a1df045997","tex":"e^{-z}","aliases":"e^-z","display":"e^{-z}","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a82e99d45fc05c76","tex":"e^{-\\gamma t_m}","aliases":"e^-γ t_m gamma","display":"e^{-\\gamma t_m}","kind":"use","excerpt":"[Large zero strings] Assume N≥3. There are constants m_0, C, and γ>0 and a sequence of positive real simple zeros (τ_m)_m≥ m_0 of D_N such that τ_m = t_m+O(e^-γ t_m). Every zero of sufficiently large modulus is of the form ω^kτ_m for a unique 0≤ k<N and a unique m≥ m_0.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0e32e6ed4b75868d","tex":"e^{-\\gamma_N\\Re z}","aliases":"e^-γ_NRe z gamma Re","display":"e^{-\\gamma_N\\Re z}","kind":"use","excerpt":"If G(z)=0, then |1+e^2is_Nz| ≤ C e^-γ_NRe z. The real part of 2is_Nz is bounded in the strip. Apply [source reference] with w=2is_Nz. There is a unique integer m such that |z-t_m|≤ C_1e^-γ_NRe z, where t_m=frac(2m+1)π2s_N. In particular, |Re z-t_m|=o(1). For all sufficiently large zeros we have |Re z-t_m|≤1, and hence |z-t_m|≤ C_2e^-γ_Nt_m with a c","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6d73f3a81c6fcc46","tex":"e^{-\\gamma_Nt_m}","aliases":"e^-γ_Nt_m gamma","display":"e^{-\\gamma_Nt_m}","kind":"use","excerpt":"G(z)=0, then |1+e^2is_Nz| ≤ C e^-γ_NRe z. The real part of 2is_Nz is bounded in the strip. Apply [source reference] with w=2is_Nz. There is a unique integer m such that |z-t_m|≤ C_1e^-γ_NRe z, where t_m=frac(2m+1)π2s_N. In particular, |Re z-t_m|=o(1). For all sufficiently large zeros we have |Re z-t_m|≤1, and hence |z-t_m|≤ C_2e^-γ_Nt_m with a cons","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-214052c012436283","tex":"e^{-\\gamma_Nx}","aliases":"e^-γ_Nx gamma","display":"e^{-\\gamma_Nx}","kind":"use","excerpt":"η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fb45e18ecdc05bbe","tex":"e^{-\\gamma_N\\tau_m}","aliases":"e^-γ_Nτ_m gamma tau","display":"e^{-\\gamma_N\\tau_m}","kind":"use","excerpt":"Differentiate the normalized expression. At z=τ_m, G'(τ_m) = 2is_Ne^2is_Nτ_m+R_N'(τ_m) = -2is_N+O(e^-γ_Nτ_m). Since G(τ_m)=0, D_N'(τ_m) = frac1N e^e^-ia_Nτ_mG'(τ_m). Taking absolute values gives the derivative asymptotic. Differentiating D_N(ω z)= D_N(z) gives the rotation formula.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4e408867033868d3","tex":"e^{-\\gamma\\tau_m}","aliases":"e^-γτ_m gamma tau","display":"e^{-\\gamma\\tau_m}","kind":"use","excerpt":"The derivatives satisfy D_N'(ω^kτ_m) = ω^-k D_N'(τ_m) and | D_N'(τ_m)| = frac2s_NNe^c_Nτ_m (1+O(e^-γτ_m)).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9324acca31d7c117","tex":"e^{-\\delta r}","aliases":"e^-δ r delta","display":"e^{-\\delta r}","kind":"use","excerpt":"The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e^-δ r)) uniformly for r≥ R. In particular","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-40fcc5d6ba51777a","tex":"e^{2is_N(t_m+\\delta_m)}","aliases":"e^2is_N(t_m+δ_m) delta","display":"e^{2is_N(t_m+\\delta_m)}","kind":"use","excerpt":"Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-426085442b3888cc","tex":"e^{2is_Nt_m}","aliases":"e^2is_Nt_m","display":"e^{2is_Nt_m}","kind":"use","excerpt":"Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-298655f0e488073a","tex":"e^{2is_Nz}","aliases":"e^2is_Nz","display":"e^{2is_Nz}","kind":"use","excerpt":"η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a99de0abb54b18b7","tex":"e^{2is_N\\tau_m}","aliases":"e^2is_Nτ_m tau","display":"e^{2is_N\\tau_m}","kind":"use","excerpt":"Differentiate the normalized expression. At z=τ_m, G'(τ_m) = 2is_Ne^2is_Nτ_m+R_N'(τ_m) = -2is_N+O(e^-γ_Nτ_m). Since G(τ_m)=0, D_N'(τ_m) = frac1N e^e^-ia_Nτ_mG'(τ_m). Taking absolute values gives the derivative asymptotic. Differentiating D_N(ω z)= D_N(z) gives the rotation formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d86f0c042cad439b","tex":"e^{2Y}","aliases":"e^2Y","display":"e^{2Y}","kind":"use","excerpt":"ng strip, the roots e^ia_N and e^-ia_N have real part c_N, while each remaining root η=u+iv satisfies u≤cos(3a_N)=c_N-γ_N. Hence Re((η-e^-ia_N)z) ≤-γ_Nx+2Y. Division by e^e^-ia_Nz gives the displayed normalization. Each differentiation multiplies a remainder term by η-e^-ia_N, whose modulus is at most two. Thus, for q=0,1,2, |R_N^(q)(z)|≤(N-2)2^qe^","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d7e7084a35eacad9","tex":"e^{2\\pi i/N}","aliases":"e^2π i/N pi","display":"e^{2\\pi i/N}","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-02621fd1f38489c4","tex":"e^{a_\\ell z}","aliases":"e^a_ℓ z ell","display":"e^{a_\\ell z}","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d43f91811ba8378c","tex":"e^{c_Nr}","aliases":"e^c_Nr","display":"e^{c_Nr}","kind":"use","excerpt":"At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry tr","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5e372850a7382fa9","tex":"e^{c_N\\tau_m}","aliases":"e^c_Nτ_m tau","display":"e^{c_N\\tau_m}","kind":"use","excerpt":"The derivatives satisfy D_N'(ω^kτ_m) = ω^-k D_N'(τ_m) and | D_N'(τ_m)| = frac2s_NNe^c_Nτ_m (1+O(e^-γτ_m)).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-49dd531cc109d91e","tex":"e^{e^{-ia_N}z}","aliases":"e^e^-ia_Nz","display":"e^{e^{-ia_N}z}","kind":"use","excerpt":"On the resulting strip, the roots e^ia_N and e^-ia_N have real part c_N, while each remaining root η=u+iv satisfies u≤cos(3a_N)=c_N-γ_N. Hence Re((η-e^-ia_N)z) ≤-γ_Nx+2Y. Division by e^e^-ia_Nz gives the displayed normalization. Each differentiation multiplies a remainder term by η-e^-ia_N, whose modulus is at most two. Thus, for q=0,1,2, |R_N^(q)(","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e6550b2960b333a4","tex":"e^{e^{-ia_N}\\tau_m}","aliases":"e^e^-ia_Nτ_m tau","display":"e^{e^{-ia_N}\\tau_m}","kind":"use","excerpt":"Differentiate the normalized expression. At z=τ_m, G'(τ_m) = 2is_Ne^2is_Nτ_m+R_N'(τ_m) = -2is_N+O(e^-γ_Nτ_m). Since G(τ_m)=0, D_N'(τ_m) = frac1N e^e^-ia_Nτ_mG'(τ_m). Taking absolute values gives the derivative asymptotic. Differentiating D_N(ω z)= D_N(z) gives the rotation formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-36cd39f69725f663","tex":"e^{g(z)}","aliases":"e^g(z)","display":"e^{g(z)}","kind":"use","excerpt":"For a function g, define the complete differential Bell polynomial Y_N by fracd^Ndz^Ne^g(z) = e^g(z)Y_N(g',g'',…,g^(N)).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d861b22faf64eef3","tex":"e^{ia_N}","aliases":"e^ia_N","display":"e^{ia_N}","kind":"use","excerpt":"On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-682a47b2eea70d50","tex":"e^{i\\theta}","aliases":"e^iθ theta","display":"e^{i\\theta}","kind":"use","excerpt":"[Indicator polygon and dominant sectors] The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e","count":11},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a4d0d3ff8a81318d","tex":"e^{L(z)}","aliases":"e^L(z)","display":"e^{L(z)}","kind":"use","excerpt":"vpage13A complex power needs a domainBRANCH BOOKKEEPINGTraditional monograph: Section 2 (p. 2), Definition 7.1 (p. 13), Corollary 8.4 (p. 18). A formal unit series has a formal logarithm at the origin. To interpret that germ analytically, choose a simply connected zero-free region containing 0 and normalize the logarithm to vanish there. D_N(z)^-r=","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4ffe86bb88f85c2e","tex":"e^{L+\\pi i}","aliases":"e^L+π i pi","display":"e^{L+\\pi i}","kind":"use","excerpt":"In fact, the constants ε_M=1/2 and C_M=3/2 work for every complex w, without the strip restriction. Put u=-(1+e^w) and L=log(1+u)=log(-e^w), using the principal logarithm. The elementary logarithm estimate on |u|≤1/2 gives |L|≤(3/2)|u|. Since e^L+π i=e^w, there is an integer m such that w=L+(2m+1)π i. This gives the required distance bound. Distinc","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8f85c8b097ee2545","tex":"e^{r\\Re(\\eta e^{i\\theta})}","aliases":"e^rRe(η e^iθ) Re eta theta","display":"e^{r\\Re(\\eta e^{i\\theta})}","kind":"use","excerpt":"vpage17Growth comes from a polygonDOMINANT EXPONENTSTraditional monograph: Definition 11.3 (p. 26), Theorem 11.4 (p. 26). Write the signed denominator as a finite exponential sum: D_N(z)=frac1N∑_η^N=-1e^η z. For z=re^iθ, the size of one exponential is e^rRe(η e^iθ). The maximal projection determines the directional exponential growth. vfigindicator","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b6a1b00b70a9b7d0","tex":"e^{rz}","aliases":"e^rz","display":"e^{rz}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-315c2026c72e593c","tex":"e^{t/2}","aliases":"e^t/2","display":"e^{t/2}","kind":"use","excerpt":"vpage18Two large terms nearly cancelTHE CUBIC MODELTraditional monograph: Lemma 11.5 (p. 27), Theorem 11.7 (p. 28), Corollary 11.9 (p. 31). For real t, order three has the exact formula D_3(t)=frace^-t+2e^t/2cos(sqrt3t/2)3. Divide by the nonzero factor 2e^t/2/3. The resulting equation separates the dominant cosine from the small third exponential. ","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-91eff23c3d96800f","tex":"e^w","aliases":"e^w","display":"e^w","kind":"use","excerpt":"[Near cancellation of two exponentials] For every M>0, there are constants ε_M>0 and C_M>0 such that the following holds. If |Re w|≤ M and |1+e^w|≤ε_M, then there is a unique integer m with |w-(2m+1)π i| ≤ C_M|1+e^w|.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f1a9e2520fdcb9c4","tex":"e^{w\\xi}","aliases":"e^wξ xi","display":"e^{w\\xi}","kind":"use","excerpt":"Differentiation multiplies by λξ in the first algebra and by ξ in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity e^(z+w)ξ=e^zξe^wξ gives the addition laws after comparison of the coefficients of 1,ξ,…,ξ^N-1.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ec666cade8ac0993","tex":"e^{yz}","aliases":"e^yz","display":"e^{yz}","kind":"use","excerpt":"Differentiate the generating function with respect to x for the first identity. Multiply it by e^yz and compare coefficients for the second.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4083b97ac93b9cef","tex":"e^{z/2}","aliases":"e^z/2","display":"e^{z/2}","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4d7952160b64cde8","tex":"e^{zJ_3}","aliases":"e^zJ_3","display":"e^{zJ_3}","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMTraditional monograph: Theorem 5.3 (p. 8), Theorem 6.4 (p. 11), Corollary 6.5 (p. 12), Corollary 6.6 (p. 12). Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the ","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d9d76e69865d2de6","tex":"e^{zJ_N}","aliases":"e^zJ_N","display":"e^{zJ_N}","kind":"use","excerpt":"D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":9},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f1ae99d7b02428e0","tex":"e^{z\\tr J_N}","aliases":"e^ztr J_N tr operatorname","display":"e^{z{\\operatorname{tr}} J_N}","kind":"use","excerpt":"The first two identities are the exponential group law. Since tr J_N=0, det(e^zJ_N)=e^ztr J_N=1. The first column statement is [source reference].","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1e062ee845109b60","tex":"e^{z\\xi}","aliases":"e^zξ xi","display":"e^{z\\xi}","kind":"use","excerpt":"[Finite algebra exponential identities] In A_N^+ one has e^λ zξ = ∑_j=0^N-1 E_N,j(λ z)ξ^j. In A_N^- one has e^zξ = ∑_j=0^N-1 D_N,j(z)ξ^j. Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b4a231173f4ee3fe","tex":"e^{z\\sum_\\ell a_\\ell}","aliases":"e^z∑_ℓ a_ℓ sum ell ell","display":"e^{z\\sum_\\ell a_\\ell}","kind":"use","excerpt":"igin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), bec","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a8a41de9330886f7","tex":"e^{|z|}","aliases":"e^|z|","display":"e^{|z|}","kind":"use","excerpt":"The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bed1356e32c52598","tex":"e^{\\eta z}","aliases":"e^η z eta","display":"e^{\\eta z}","kind":"use","excerpt":"The second half of the paper concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's monogr","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9a7300563f5018c4","tex":"e^{\\eta_0z}","aliases":"e^η_0z eta","display":"e^{\\eta_0z}","kind":"use","excerpt":"The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8c415fcc7d5848df","tex":"e^{\\eta_kz}","aliases":"e^η_kz eta","display":"e^{\\eta_kz}","kind":"use","excerpt":"We now assume N≥2. Write η_k=e^(2k+1)π i/N, 0≤ k<N. These are the vertices of a regular polygon and D_N(z)=frac1N∑_k=0^N-1e^η_kz.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-605f0e10ae7d823b","tex":"e^{\\lambda w}","aliases":"e^λ w lambda","display":"e^{\\lambda w}","kind":"use","excerpt":"Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5dd1ba005cf62b45","tex":"e^{\\lambda z\\xi}","aliases":"e^λ zξ lambda xi","display":"e^{\\lambda z\\xi}","kind":"use","excerpt":"[Finite algebra exponential identities] In A_N^+ one has e^λ zξ = ∑_j=0^N-1 E_N,j(λ z)ξ^j. In A_N^- one has e^zξ = ∑_j=0^N-1 D_N,j(z)ξ^j. Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-04da8d8bbc069bd5","tex":"e^{\\lambda(z+w)}","aliases":"e^λ(z+w) lambda","display":"e^{\\lambda(z+w)}","kind":"use","excerpt":"Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0080b32292de2b10","tex":"e^{\\rho\\omega^\\ell z}","aliases":"e^ρω^ℓ z rho omega ell","display":"e^{\\rho\\omega^\\ell z}","kind":"use","excerpt":"[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMTraditional monograph: Theorem 5.3 (p. 8), Theorem 6.4 (p. 11), Corollary 6.5 (p. 12), Corollary 6.6 (p. 12). Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the ","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMTraditional monograph: Theorem 5.3 (p. 8), Theorem 6.4 (p. 11), Corollary 6.5 (p. 12), Corollary 6.6 (p. 12). Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the ","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fdddb5e6146d680d","tex":"E_{2,1}","aliases":"E_2,1","display":"E_{2,1}","kind":"use","excerpt":"++(2.10000,0.80000); node[] at (6.05000,1.65000) - frac13; draw[fill=black!0,draw=black!40,line width=.35pt] (7.10000,1.25000) rectangle ++(2.10000,0.80000); node[] at (8.15000,1.65000) frac715; draw[fill=black!8,draw=black!40,line width=.35pt] (9.20000,1.25000) rectangle ++(2.10000,0.80000); node[] at (10.25000,1.65000) - frac3121; node[] at (5.65","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-58f890b6f450ab6a","tex":"E_{3,1}","aliases":"E_3,1","display":"E_{3,1}","kind":"use","excerpt":"because the hypergeometric convention already supplies (1)_n=n!. vfighypergeometric headCheck the displayed case For N=3,j=1,n=2, the six factors after the initial 1! are 2,3,4,5,6,7. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies 2!. The others supply (2/3)_2 and (4/3)_2. The right-hand side is exact","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5c7ee7fb47bfc415","tex":"E_{3,j}","aliases":"E_3,j","display":"E_{3,j}","kind":"use","excerpt":"Multiply the coordinate forms of e^zξ and e^wξ. Their product is e^(z+w)ξ. The coefficient of 1 receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law. headCompare the unsigned model In C[ξ]/(ξ^3-1), the same table has no minus signs. Its exponential coordinates are the ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6662c032276e077b","tex":"E_4","aliases":"E_4","display":"E_4","kind":"use","excerpt":"9091) (11.95200,5.46972) (12.01000,5.65433) (12.06800,5.84486) (12.12600,6.04145) (12.18400,6.24420) (12.24200,6.45326) (12.30000,6.66875); draw[fill=black] (1.31232,1.20000) circle (0.055); draw[fill=black] (4.18000,1.20000) circle (0.055); draw[fill=black] (8.82000,1.20000) circle (0.055); draw[fill=black] (11.68768,1.20000) circle (0.055); node[","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-de744dacfa91c966","tex":"E_4^{(4)}","aliases":"E_4^(4)","display":"E_4^{(4)}","kind":"use","excerpt":") (12.24200,6.45326) (12.30000,6.66875); draw[fill=black] (1.31232,1.20000) circle (0.055); draw[fill=black] (4.18000,1.20000) circle (0.055); draw[fill=black] (8.82000,1.20000) circle (0.055); draw[fill=black] (11.68768,1.20000) circle (0.055); node[] at (3.20000,4.65000) solid: E_4(x)=x^4-6x^2+5; node[] at (9.80000,4.65000) dashed: x^4; node[] at","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2a927cfe01e6d5b3","tex":"E_4^{\\prime\\prime}","aliases":"E_4^primeprime prime prime","display":"E_4^{\\prime\\prime}","kind":"use","excerpt":") (12.24200,6.45326) (12.30000,6.66875); draw[fill=black] (1.31232,1.20000) circle (0.055); draw[fill=black] (4.18000,1.20000) circle (0.055); draw[fill=black] (8.82000,1.20000) circle (0.055); draw[fill=black] (11.68768,1.20000) circle (0.055); node[] at (3.20000,4.65000) solid: E_4(x)=x^4-6x^2+5; node[] at (9.80000,4.65000) dashed: x^4; node[] at","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4b6907ef544fbc21","tex":"E_m'","aliases":"E_m'","display":"E_m'","kind":"use","excerpt":"The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and tr","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-84df450ec19feaf4","tex":"E_{m-1}","aliases":"E_m-1","display":"E_{m-1}","kind":"use","excerpt":"The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and tr","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2b2429b531974850","tex":"E_{m-1}^{(N,j,r)}","aliases":"E_m-1^(N,j,r)","display":"E_{m-1}^{(N,j,r)}","kind":"use","excerpt":"[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8ee0e3ae18e03505","tex":"E_{m-q}^{(N,j,s)}","aliases":"E_m-q^(N,j,s)","display":"E_{m-q}^{(N,j,s)}","kind":"use","excerpt":"[Joint Appell convolution in the space and order parameters] For all r,s,x,y∈ C and m≥0, E_m^(N,j,r+s)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x) E_m-q^(N,j,s)(y).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a8c499641055da34","tex":"E_m^{(N,j,r+s)}","aliases":"E_m^(N,j,r+s)","display":"E_m^{(N,j,r+s)}","kind":"use","excerpt":"[Joint Appell convolution in the space and order parameters] For all r,s,x,y∈ C and m≥0, E_m^(N,j,r+s)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x) E_m-q^(N,j,s)(y).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c0277f6b24af6402","tex":"E_{N,0}","aliases":"E_N,0","display":"E_{N,0}","kind":"use","excerpt":"vpage16Scale and order enter differentlyTHETA COUPLINGTraditional monograph: Theorem 10.2 (p. 23), Proposition 10.3 (p. 23), Proposition 10.4 (p. 23). Let H_N(z)=E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic bl","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e3ca2ec43cc1e4a1","tex":"E_{N,j}","aliases":"E_N,j","display":"E_{N,j}","kind":"use","excerpt":"midrule E_N,j(z) · displaystyle∑_n≥0fracz^Nn+j(Nn+j)! selects one residue class of the exponential.","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1b09d4efdcdc1f39","tex":"e_p","aliases":"e_p","display":"e_p","kind":"use","excerpt":"For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b3f1fc86e79071cb","tex":"E_q","aliases":"E_q","display":"E_q","kind":"use","excerpt":"The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and tr","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-244f69b2cbc609aa","tex":"e_q","aliases":"e_q","display":"e_q","kind":"use","excerpt":"For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9dcdbc1ff00bedd9","tex":"E_q^{(N,j,r)}","aliases":"E_q^(N,j,r)","display":"E_q^{(N,j,r)}","kind":"use","excerpt":"[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bbd7a622a6f80cd3","tex":"e_\\lambda","aliases":"e_λ lambda","display":"e_\\lambda","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3792be68e67ba6b6","tex":"\\Efun_{2,0}","aliases":"Efun_2,0 E_2,0 Efun mathsf","display":"{\\mathsf E}_{2,0}","kind":"use","excerpt":"For N=1, the only section is e^λ z. For N=2, E_2,0(z)=cosh z, E_2,1(z)=sinh z. These are classical instances of the same cyclic basis.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a942d6e96ad70816","tex":"\\Efun_{2,1}","aliases":"Efun_2,1 E_2,1 Efun mathsf","display":"{\\mathsf E}_{2,1}","kind":"use","excerpt":"For N=1, the only section is e^λ z. For N=2, E_2,0(z)=cosh z, E_2,1(z)=sinh z. These are classical instances of the same cyclic basis.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-29b6acd7461514fa","tex":"\\Efun_{N,i}","aliases":"Efun_N,i E_N,i Efun mathsf","display":"{\\mathsf E}_{N,i}","kind":"use","excerpt":"[Addition law] For 0≤ k<N, E_N,k(λ(z+w)) = ∑_substack0≤ i,j<N i+j≡ kbmod N E_N,i(λ z) E_N,j(λ w).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-365590e8160df3b1","tex":"\\Efun_{N,j-1}","aliases":"Efun_N,j-1 E_N,j-1 Efun mathsf","display":"{\\mathsf E}_{N,j-1}","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fad2c39cce8b8872","tex":"\\Efun_{N,k}","aliases":"Efun_N,k E_N,k Efun mathsf","display":"{\\mathsf E}_{N,k}","kind":"use","excerpt":"[Addition law] For 0≤ k<N, E_N,k(λ(z+w)) = ∑_substack0≤ i,j<N i+j≡ kbmod N E_N,i(λ z) E_N,j(λ w).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f83f7c661dcd52e7","tex":"\\Efun_{N,N-1}","aliases":"Efun_N,N-1 E_N,N-1 Efun mathsf","display":"{\\mathsf E}_{N,N-1}","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"For a power series F(z)=∑_m≥ 0a_mz^m and a positive integer N, the coefficients may be separated according to the residue of m modulo N. Root of unity averaging gives exact projection operators for this separation. This elementary construction is usually called a multisection or a root filter. It appears throughout the classical theory of generatin","count":39},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z^10. Applying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original F. headSelection is also a symmetry condition For the selected ","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-31d7232e6fa917d3","tex":"F_i","aliases":"F_i","display":"F_i","kind":"use","excerpt":"Write F=∑_iF_i and G=∑_jG_j, where F_i= B_N,iF and G_j= B_N,jG. Their covariance gives (F_iG_j)(ω z)=ω^i+jF_i(z)G_j(z), so F_iG_j belongs to the packet labelled by i+j modulo N. Expanding the finite product and applying B_N,k gives the formula.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z^10. Applying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original F. headSelection is also a symmetry condition For the selected polynomial F_2, rotation gives F_2(iz)=-F_2(z). More generally, a residue-j section","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b9a66bfb8f08c990","tex":"F_{N-1}","aliases":"F_N-1","display":"F_{N-1}","kind":"use","excerpt":"[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( - A_N,j mathrel, frac(λ z)^NN^N ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5e597b1821a53be0","tex":"\\Fdet","aliases":"Fdet det_ F operatorname","display":"{\\operatorname{det}_{\\!F}}","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"/2)|u| shows that Φ_m maps B_m into itself. The derivative bound |Φ_m'|≤1/2 makes it a contraction. Banach's fixed-point theorem gives a unique fixed point. On B_m, |a(z-t_m)|≤1<π, so the principal logarithm inverts the exponential and the fixed-point equation is equivalent to G(z)=0. Moreover, |z-t_m|≤frac34s_N|R_N(z)|=O(e^-γ_Nt_m). At this zero, ","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-eb695a8d9657252e","tex":"g''","aliases":"g''","display":"g''","kind":"use","excerpt":"For a function g, define the complete differential Bell polynomial Y_N by fracd^Ndz^Ne^g(z) = e^g(z)Y_N(g',g'',…,g^(N)).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b971876a95cdcef2","tex":"g^{(N)}","aliases":"g^(N)","display":"g^{(N)}","kind":"use","excerpt":"For a function g, define the complete differential Bell polynomial Y_N by fracd^Ndz^Ne^g(z) = e^g(z)Y_N(g',g'',…,g^(N)).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ec1c980979f73c29","tex":"G_j","aliases":"G_j","display":"G_j","kind":"use","excerpt":"Write F=∑_iF_i and G=∑_jG_j, where F_i= B_N,iF and G_j= B_N,jG. Their covariance gives (F_iG_j)(ω z)=ω^i+jF_i(z)G_j(z), so F_iG_j belongs to the packet labelled by i+j modulo N. Expanding the finite product and applying B_N,k gives the formula.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bc71f39c22d38b2d","tex":"h_4","aliases":"h_4","display":"h_4","kind":"use","excerpt":"1615) (11.95250,3.25792) (11.98292,3.19481) (12.01333,3.12685) (12.04375,3.05410) (12.07417,2.97659) (12.10458,2.89439) (12.13500,2.80755) (12.16542,2.71613) (12.19583,2.62019) (12.22625,2.51980) (12.25667,2.41503) (12.28708,2.30595) (12.31750,2.19263) (12.34792,2.07515) (12.37833,1.95359) (12.40875,1.82804) (12.43917,1.69859) (12.46958,1.56531) (1","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8e8b586548eb3466","tex":"\\mathcal H_N","aliases":"H_N mathcal","display":"\\mathcal H_N","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5e073754a563e30e","tex":"\\hbox","aliases":"hbox","display":null,"kind":"use","excerpt":"Multiply the generating functions C_N,j(z)^re^xz hboxand C_N,j(z)^se^yz. Their product is the generating function with parameters r+s and x+y. Coefficient comparison gives the formula.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":91},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-11bf5aa7c1c323e1","tex":"i^{(m-j)\\ell}","aliases":"i^(m-j)ℓ ell","display":"i^{(m-j)\\ell}","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONTraditional monograph: Theorem 3.1 (p. 4), Proposition 3.2 (p. 5). For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matc","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-786d9c0a32751abc","tex":"i^{-j\\ell}","aliases":"i^-jℓ ell","display":"i^{-j\\ell}","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONTraditional monograph: Theorem 3.1 (p. 4), Proposition 3.2 (p. 5). For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matc","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-022e2ac42397386f","tex":"i^k","aliases":"i^k","display":"i^k","kind":"use","excerpt":"vpage8Four zeros become one orbit zeroTHE REDUCED VARIABLETraditional monograph: Definition 5.7 (p. 9), Example 11.12 (p. 31), Corollary 11.13 (p. 31). For N=4 the factorization is exact: D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), D_4(z)=Δ_4(z^4). Set A_4=sqrt2π. All the zeros are A_4(m+1/2)i^k, with m≥0 and 0≤ k<4. Each zero is simple. vfigorbit_reduction ","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8a9917336954c9f2","tex":"i^\\ell","aliases":"i^ℓ ell","display":"i^\\ell","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONTraditional monograph: Theorem 3.1 (p. 4), Proposition 3.2 (p. 5). For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matc","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-10ac8d7d633361db","tex":"J_3","aliases":"J_3","display":"J_3","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMTraditional monograph: Theorem 5.3 (p. 8), Theorem 6.4 (p. 11), Corollary 6.5 (p. 12), Corollary 6.6 (p. 12). Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the ","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ea80efecfc6ff0ed","tex":"J_3^3","aliases":"J_3^3","display":"J_3^3","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMTraditional monograph: Theorem 5.3 (p. 8), Theorem 6.4 (p. 11), Corollary 6.5 (p. 12), Corollary 6.6 (p. 12). Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the ","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-30b75b8d021145b9","tex":"J_4","aliases":"J_4","display":"J_4","kind":"use","excerpt":"The signed shift is a finite carrier for the differential cycle. At order four, three basis steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression. modelfiguresigned_cycleThe exact signed shift J_4. ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c18f6751c1c42314","tex":"J_4^4","aliases":"J_4^4","display":"J_4^4","kind":"use","excerpt":"steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression. modelfiguresigned_cycleThe exact signed shift J_4. Source columns and target rows describe the same action as the cycle on the left. Direct int","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8932ad4e519e5510","tex":"J_N^{N-1}","aliases":"J_N^N-1","display":"J_N^{N-1}","kind":"use","excerpt":"The vectors e_0,J_Ne_0,…,J_N^N-1e_0 are the standard basis. Thus e_0 is cyclic for J_N, so the minimal polynomial of J_N has degree N. Since J_N^N=-I, that minimal polynomial divides x^N+1 and therefore equals x^N+1. The characteristic polynomial has the same degree and is divisible by the minimal polynomial, so it too equals x^N+1. This polynomial","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-62296fc9b333afb5","tex":"J_N^{Nn+d_{pq}}","aliases":"J_N^Nn+d_pq","display":"J_N^{Nn+d_{pq}}","kind":"use","excerpt":"For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1ebc30c5e8c584b3","tex":"K_N^m","aliases":"K_N^m","display":"K_N^m","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-365d749061e6ffdd","tex":"K_N^q","aliases":"K_N^q","display":"K_N^q","kind":"use","excerpt":"vpage22Finite coefficients give infinite momentsZERO POWER SUMSTraditional monograph: Theorem 12.6 (p. 35), Corollary 12.7 (p. 36), Corollary 12.9 (p. 37), Corollary 12.10 (p. 37). Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"use","excerpt":"[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-04ccc1386674337e","tex":"\\limsup_{r\\to\\infty}","aliases":"limsup_r→∞ limsup to infty","display":"\\limsup_{r\\to\\infty}","kind":"use","excerpt":"[Indicator polygon and dominant sectors] The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7e3a698fc7564672","tex":"\\limsup_{\\substack{n\\to\\infty\\\\a_n\\ne0}}","aliases":"limsup_substackn→∞ a_n≠0 limsup substack to infty a ne","display":null,"kind":"use","excerpt":"[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-973977200963bdff","tex":"\\log_{10}","aliases":"log_10 log","display":"\\log_{10}","kind":"use","excerpt":"); draw[fill=black] (2.42857,3.23383) circle (0.055); draw[fill=black] (4.05714,2.85954) circle (0.055); draw[fill=black] (5.68571,2.48068) circle (0.055); draw[fill=black] (7.31429,2.10005) circle (0.055); draw[fill=black] (8.94286,1.71875) circle (0.055); draw[fill=black] (10.57143,1.33719) circle (0.055); draw[fill=black] (12.20000,0.95555) circ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.","count":12},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-09a4e391e923d287","tex":"m^N","aliases":"m^N","display":"m^N","kind":"use","excerpt":"vpage24Convergence is not continuationZERO ZETA FUNCTIONSTraditional monograph: Theorem 14.2 (p. 39), Theorem 14.3 (p. 39), Corollary 14.4 (p. 41). Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-594034b0026909b7","tex":"m^{r-2}","aliases":"m^r-2","display":"m^{r-2}","kind":"use","excerpt":"[Leading dominant zero term] Under the hypotheses of [source reference], P_N,n(r) = · m!binomm+r-1r-1 ∑_ρ∈ Z_R fracρ^-m(-ρ D_N'(ρ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term is absent.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":"[Large zero strings] Assume N≥3. There are constants m_0, C, and γ>0 and a sequence of positive real simple zeros (τ_m)_m≥ m_0 of D_N such that τ_m = t_m+O(e^-γ t_m). Every zero of sufficiently large modulus is of the form ω^kτ_m for a unique 0≤ k<N and a unique m≥ m_0.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e4524d978e547026","tex":"M_f","aliases":"M_f","display":"M_f","kind":"use","excerpt":"The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-95bc4d67c762d1bf","tex":"M_{f(\\,\\cdot\\,^N)}","aliases":"M_f( · ^N) cdot","display":"M_{f(\\,\\cdot\\,^N)}","kind":"use","excerpt":"The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b600b58f986c2898","tex":"M_L","aliases":"M_L","display":"M_L","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-781698c822449e4f","tex":"\\mathrel","aliases":"mathrel","display":null,"kind":"use","excerpt":"[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( -","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-03265e359becc498","tex":"\\max_{0\\le k<N}","aliases":"max_0≤ k<N max le","display":"\\max_{0\\le k<N}","kind":"use","excerpt":"om a polygonDOMINANT EXPONENTSTraditional monograph: Definition 11.3 (p. 26), Theorem 11.4 (p. 26). Write the signed denominator as a finite exponential sum: D_N(z)=frac1N∑_η^N=-1e^η z. For z=re^iθ, the size of one exponential is e^rRe(η e^iθ). The maximal projection determines the directional exponential growth. vfigindicator The indicator is h_N(","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b706167f987bdce5","tex":"N^N","aliases":"N^N","display":"N^N","kind":"use","excerpt":"[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( - A_N,j mathrel, frac(λ z)^NN^N ).","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-821e1514a603c20a","tex":"N^{Nn}","aliases":"N^Nn","display":"N^{Nn}","kind":"use","excerpt":"The multiplication formula for the gamma function gives (Nn+j)! = j!N^Nnn! ∏_a∈ A_N,j(a)_n. Indeed, the N Pochhammer factors supplied by the multiplication formula are ((j+m)/N)_n for 1≤ m≤ N; the excluded factor is ((j+N-j)/N)_n=(1)_n=n!. Insertion into the defining series gives the stated hypergeometric expansion.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-db668af012961ca2","tex":"n_{D_4}","aliases":"n_D_4","display":"n_{D_4}","kind":"use","excerpt":"(2.19219,1.00000) circle (0.055); node[] at (2.19219,0.58000) 81/16; node[font=sffamilyfootnotesize] at (2.19219,1.45000) count 2; draw[line width=1pt] (11.54219,0.88000) -- (11.54219,1.12000); draw[fill=black] (11.54219,1.00000) circle (0.055); node[] at (11.54219,0.58000) 625/16; node[font=sffamilyfootnotesize] at (11.54219,1.45000) count 3; node","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1bc0c27714a3cfdd","tex":"n_{D_N}","aliases":"n_D_N","display":"n_{D_N}","kind":"use","excerpt":"ions are v=(m+1/2)^4. vfigcounting The two rows describe the same zero orbits. The upper row counts four zeros each time a new radius is reached. The lower row counts one reduced zero each time its fourth-power location is reached. Both formulas include zeros on the boundary of the disk. headSeparate radius conversion from orbit multiplicity In uns","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-10b68c11b12c7be3","tex":"n_{\\Dfun_N}","aliases":"n_Dfun_N n_ D_N Dfun mathsf","display":"n_{{\\mathsf D}_N}","kind":"use","excerpt":"[Zero counting asymptotics] Let n_ D_N(R) count the zeros of D_N in |z|≤ R, with multiplicity. Let n_Δ_N(R) count the zeros of Δ_N in |w|≤ R. Then, as R→∞, n_ D_N(R) = fracNsin(π/N)πR+O(1) and n_Δ_N(R) = fracsin(π/N)πR^1/N+O(1).","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-91d4d2c67e57dcc9","tex":"n_{\\Delta_4}","aliases":"n_Δ_4 Delta","display":"n_{\\Delta_4}","kind":"use","excerpt":"(2.19219,1.00000) circle (0.055); node[] at (2.19219,0.58000) 81/16; node[font=sffamilyfootnotesize] at (2.19219,1.45000) count 2; draw[line width=1pt] (11.54219,0.88000) -- (11.54219,1.12000); draw[fill=black] (11.54219,1.00000) circle (0.055); node[] at (11.54219,0.58000) 625/16; node[font=sffamilyfootnotesize] at (11.54219,1.45000) count 3; node","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5e90ef4c326a8622","tex":"n_{\\Delta_N}","aliases":"n_Δ_N Delta","display":"n_{\\Delta_N}","kind":"use","excerpt":"[Zero counting asymptotics] Let n_ D_N(R) count the zeros of D_N in |z|≤ R, with multiplicity. Let n_Δ_N(R) count the zeros of Δ_N in |w|≤ R. Then, as R→∞, n_ D_N(R) = fracNsin(π/N)πR+O(1) and n_Δ_N(R) = fracsin(π/N)πR^1/N+O(1).","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"The identity Δ_N(w)^-r = exp( r∑_m≥1κ_N,mfracw^mm ) fits the defining generating function for B_n with x_m=r(m-1)!κ_N,m. The coefficient of w^n is therefore the displayed Bell polynomial divided by n!. Multiplication by (Nn)! gives P_N,n(r). Since Δ_N(w)=1-fracwN!+O(w^2), we have κ_N,1=1/N!. The term x_1^n gives the stated leading coefficient, whic","count":30},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b57e43a9fff05b66","tex":"O_{R_*}","aliases":"O_R_*","display":"O_{R_*}","kind":"use","excerpt":"rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ab3321d6961879fc","tex":"\\ord","aliases":"ord operatorname","display":"{\\operatorname{ord}}","kind":"use","excerpt":"[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ae21a8553a17610b","tex":"\\overline z","aliases":"overline z overline","display":"\\overline z","kind":"use","excerpt":"[Symmetries] The denominator satisfies D_N(ω z)= D_N(z) and D_N(overline z)=overline D_N(z). Its nonzero zeros occur in orbits under multiplication by the Nth roots of unity and under complex conjugation.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a13a376dd76033a7","tex":"\\overline{\\Dfun_N(z)}","aliases":"overlineDfun_N(z) overline D_N(z) overline Dfun overline mathsf","display":"\\overline{{\\mathsf D}_N(z)}","kind":"use","excerpt":"[Symmetries] The denominator satisfies D_N(ω z)= D_N(z) and D_N(overline z)=overline D_N(z). Its nonzero zeros occur in orbits under multiplication by the Nth roots of unity and under complex conjugation.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cb79cf09cdd9505e","tex":"\\overline{\\eta_0}","aliases":"overlineη_0 overline eta","display":"\\overline{\\eta_0}","kind":"use","excerpt":"The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-14472c28742b9379","tex":"P_{1,n}","aliases":"P_1,n","display":"P_{1,n}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6a679a976dd7459e","tex":"P_{2,1}","aliases":"P_2,1","display":"P_{2,1}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-40a2a894b872c666","tex":"P_{2,2}","aliases":"P_2,2","display":"P_{2,2}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-587fb000e9b974ca","tex":"P_{2,3}","aliases":"P_2,3","display":"P_{2,3}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a3ada48942b51047","tex":"P_{2,n}","aliases":"P_2,n","display":"P_{2,n}","kind":"use","excerpt":"[Secant asymptotic] For N=2 and r=1, P_2,n(1) ∼ 2(2n)!(frac2π)^2n+1.","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a6b3605e1b2fcc56","tex":"P_{4,2}","aliases":"P_4,2","display":"P_{4,2}","kind":"use","excerpt":"ample 11.12 (p. 31), Theorem 12.6 (p. 35). Use N=4 throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order. vfigworked_circuit headRoute one: invert the denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=fra","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-363d67fd271736f2","tex":"P_L","aliases":"P_L","display":"P_L","kind":"use","excerpt":"ltiplicity-sensitive genus-zero product is Δ_N(w)=∏_λ∈Λ_N(1-w/λ). Putting K_Ne_λ=λ^-1e_λ on ℓ^2(Λ_N) gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries. vfigproduct headFinite compression is an approximation For N=2, λ_m=π^2(m+1/2)^2. The finite determinant P_L is exactly the displayed finite pro","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-346cf456a77b1ce5","tex":"p_n","aliases":"p_n","display":"p_n","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6e5d7761f0b81ae7","tex":"P_{N,0}","aliases":"P_N,0","display":"P_{N,0}","kind":"use","excerpt":"[Parameter recurrence] We have P_N,0(r)=1. For n≥1, P_N,n(r) = · - ∑_k=1^n-1 (-1)^k binomNn-1Nk P_N,n-k(r)","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b03777ec508d2dd5","tex":"P_{N,1}","aliases":"P_N,1","display":"P_{N,1}","kind":"use","excerpt":"[The first three reciprocal polynomials] For every N≥1, P_N,1(r)=r, P_N,2(r) = r[ frac12binom2NN(r+1)-1 ], and P_N,3(r) = r[ frac(3N)!6(N!)^3(r+1)(r+2) - binom3NN(r+1) +1 ].","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9ef5da6a74405315","tex":"P_{N,2}","aliases":"P_N,2","display":"P_{N,2}","kind":"use","excerpt":"[The first three reciprocal polynomials] For every N≥1, P_N,1(r)=r, P_N,2(r) = r[ frac12binom2NN(r+1)-1 ], and P_N,3(r) = r[ frac(3N)!6(N!)^3(r+1)(r+2) - binom3NN(r+1) +1 ].","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5c5be35340cd11a7","tex":"P_{N,3}","aliases":"P_N,3","display":"P_{N,3}","kind":"use","excerpt":"[The first three reciprocal polynomials] For every N≥1, P_N,1(r)=r, P_N,2(r) = r[ frac12binom2NN(r+1)-1 ], and P_N,3(r) = r[ frac(3N)!6(N!)^3(r+1)(r+2) - binom3NN(r+1) +1 ].","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-dcd49cdc880575e3","tex":"P_{N,n-k}","aliases":"P_N,n-k","display":"P_{N,n-k}","kind":"use","excerpt":"[Parameter recurrence] We have P_N,0(r)=1. For n≥1, P_N,n(r) = · - ∑_k=1^n-1 (-1)^k binomNn-1Nk P_N,n-k(r) · -r ∑_k=1^n (-1)^k binomNn-1Nk-1 P_N,n-k(r).","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-db7164b141c5c6f4","tex":"P_{N,n-q}","aliases":"P_N,n-q","display":"P_{N,n-q}","kind":"use","excerpt":"[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2b530c98cdddafac","tex":"P_{N,q}","aliases":"P_N,q","display":"P_{N,q}","kind":"use","excerpt":"[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1543815dca570b98","tex":"p_{n-1}","aliases":"p_n-1","display":"p_{n-1}","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"vpage4Products mix the residue lanesCYCLIC CONVOLUTIONTraditional monograph: Theorem 3.3 (p. 5). Selection separates a function additively. It does not make multiplication diagonal. A residue-i term multiplied by a residue-j term lands in residue i+jpmod N. For N=3, use F=1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three sel","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3f1b6dc519af96a7","tex":"\\prime","aliases":"prime","display":"\\prime","kind":"use","excerpt":"ic objects are introduced in the following order. First, E_N,j(z) = ∑_n≥0fracz^Nn+j(Nn+j)! is the residue j part of the exponential. Second, D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)! is the signed basis for y^(N)=-y. The section D_N= D_N,0 is the denominator used in the reciprocal family S_N^(r)= D_N^-r. Set T_N=- D_N^prime/ D_N. Equivalently, T_N is","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"[Meromorphic continuation of the symmetry-orbit zero zeta] Put A_N=fracπsin(π/N). There is an entire function Q_N^red such that Z_N^red(s) = A_N^-Nsζ(Ns,frac12)+Q_N^red(s) on Re s>1/N. The right hand side gives a meromorphic continuation to C with one simple pole at s=1/N. Its residue is fracsin(π/N)Nπ.","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-983925c02a852901","tex":"Q_2^{\\mathrm{red}}","aliases":"Q_2^red mathrm","display":"Q_2^{\\mathrm{red}}","kind":"use","excerpt":"For N=2, [source reference] and D_2(z)=Δ_2(z^2) give λ_m=π^2(m+frac12)^2, m≥0. Thus the first formula holds with Q_2^red=0.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-13c6dda682be516f","tex":"Q_N^{\\mathrm{abs}}","aliases":"Q_N^abs mathrm","display":"Q_N^{\\mathrm{abs}}","kind":"use","excerpt":"There is an entire function Q_N^abs such that, initially on Re s>1, Z_N^abs(s) = N A_N^-sζ(s,frac12)+Q_N^abs(s). Thus Z_N^abs continues meromorphically to C with one simple pole at s=1, of residue fracNsin(π/N)π.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e6572edc01bfc69a","tex":"Q_N^{\\mathrm{red}}","aliases":"Q_N^red mathrm","display":"Q_N^{\\mathrm{red}}","kind":"use","excerpt":"[Meromorphic continuation of the symmetry-orbit zero zeta] Put A_N=fracπsin(π/N). There is an entire function Q_N^red such that Z_N^red(s) = A_N^-Nsζ(Ns,frac12)+Q_N^red(s) on Re s>1/N. The right hand side gives a meromorphic continuation to C with one simple pole at s=1/N. Its residue is fracsin(π/N)Nπ.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"[Indicator polygon and dominant sectors] The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e","count":59},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-aff4a4490ea52474","tex":"r^{-1}","aliases":"r^-1","display":"r^{-1}","kind":"use","excerpt":"o-free. headWhy zeros remain possible on a growing ray At a tie, two adjacent exponents have the same real projection. Their complex phases can be opposite. Their sum can therefore vanish even though each term is exponentially large. The minima in the order-four indicator plot occur precisely at the tie directions θ=kπ/2. The indicator uses a limsu","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a345887535b62c23","tex":"R^{-m}","aliases":"R^-m","display":"R^{-m}","kind":"use","excerpt":"[Leading dominant zero term] Under the hypotheses of [source reference], P_N,n(r) = · m!binomm+r-1r-1 ∑_ρ∈ Z_R fracρ^-m(-ρ D_N'(ρ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term is absent.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-08b492aa785b5eaf","tex":"R^{1/N}","aliases":"R^1/N","display":"R^{1/N}","kind":"use","excerpt":"[Zero counting asymptotics] Let n_ D_N(R) count the zeros of D_N in |z|≤ R, with multiplicity. Let n_Δ_N(R) count the zeros of Δ_N in |w|≤ R. Then, as R→∞, n_ D_N(R) = fracNsin(π/N)πR+O(1) and n_Δ_N(R) = fracsin(π/N)πR^1/N+O(1).","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-96dee922af39121c","tex":"r^3","aliases":"r^3","display":"r^3","kind":"use","excerpt":"ynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. Substituting x_1=rκ_1, x_2=rκ_2, x_3=2rκ_3 and multiplying by 6!/3! gives P_2,3(r)=15r^3+30r^2","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6f3447fa324f7dc6","tex":"r^N","aliases":"r^N","display":"r^N","kind":"use","excerpt":"The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e3b6ec511215a68d","tex":"r^n","aliases":"r^n","display":"r^n","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-236f80d54e55c576","tex":"R_*","aliases":"R_*","display":"R_*","kind":"use","excerpt":"rest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ ","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b4dc998495c46ac8","tex":"R_*^{-m}","aliases":"R_*^-m","display":"R_*^{-m}","kind":"use","excerpt":"rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d0aa3589089d58ba","tex":"R_3","aliases":"R_3","display":"R_3","kind":"use","excerpt":"The only nondominant exponent is -1. Hence R_3(t) = exp((-frac32+fracsqrt32i)t). At t=t_m, its phase is i(-1)^m. Insert this value into [source reference].","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4aa81ec436e2173c","tex":"R_N","aliases":"R_N","display":"R_N","kind":"use","excerpt":"On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":11},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"use","excerpt":"t poles are ρ=±π/2. They contribute equally to even-degree coefficients. Including both poles gives P_2,n(1)∼2(2n)!(frac2π)^2n+1. The factor two is orbit multiplicity, not an adjustable fitted constant. vfigpole_asymptotic headA finite quantitative check For n=1,…,8, the exact integer coefficients and a rational enclosure for π bound the ratio R_n ","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1bb952727eb34a88","tex":"R_N'","aliases":"R_N'","display":"R_N'","kind":"use","excerpt":"Fix a large m, put a=2is_N, and consider the closed disk B_m={z:|z-t_m|≤ 1/(2s_N)}. The remainder bounds hold uniformly on these fixed-radius disks. For sufficiently large m, |R_N|≤ 1/2 and |R_N'|≤ s_N/2 there. Define Φ_m(z)=t_m+fraclog(1+R_N(z))2is_N, using the principal logarithm. The estimate |log(1+u)|≤(3/2)|u| shows that Φ_m maps B_m into itse","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-48a0e102d8c6aa7a","tex":"R_N^{(q)}","aliases":"R_N^(q)","display":"R_N^{(q)}","kind":"use","excerpt":"η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3bce33d4a711b894","tex":"\\Rcal","aliases":"Rcal R mathcal","display":"{\\mathcal R}","kind":"use","excerpt":"[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9215c3ab638f6fc4","tex":"\\Rcal_N","aliases":"Rcal_N R_N Rcal mathcal","display":"{\\mathcal R}_N","kind":"use","excerpt":"[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"is not continuationZERO ZETA FUNCTIONSTraditional monograph: Theorem 14.2 (p. 39), Theorem 14.3 (p. 39), Corollary 14.4 (p. 41). Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_plane For N=2 and N","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Set S= D_N^-r. Logarithmic differentiation gives D_NS'+r D_N'S=0. Taking the coefficient of z^Nn-1/(Nn-1)! gives ∑_k=0^n-1(-1)^k binomNn-1NkP_N,n-k(r) +r∑_k=1^n(-1)^k binomNn-1Nk-1P_N,n-k(r)=0. The term with k=0 in the first sum is P_N,n(r), and every other term has a smaller block index. Rearrangement gives the formula.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"Set S= D_N^-r. Logarithmic differentiation gives D_NS'+r D_N'S=0. Taking the coefficient of z^Nn-1/(Nn-1)! gives ∑_k=0^n-1(-1)^k binomNn-1NkP_N,n-k(r) +r∑_k=1^n(-1)^k binomNn-1Nk-1P_N,n-k(r)=0. The term with k=0 in the first sum is P_N,n(r), and every other term has a smaller block index. Rearrangement gives the formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-65e03ace3c0c077c","tex":"S_{90}","aliases":"S_90","display":"S_{90}","kind":"use","excerpt":"Theorem 11.8 (p. 30), Corollary 11.9 (p. 31). The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificate","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cfe0fa40b3c1262c","tex":"s_N","aliases":"s_N","display":"s_N","kind":"use","excerpt":"At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry tr","count":31},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-72398ff18d39e1f6","tex":"\\sec","aliases":"sec","display":"\\sec","kind":"use","excerpt":"A unit constant term makes the formal reciprocal unambiguous. To compute one coefficient, only finitely many earlier coefficients are needed. The later analytic theory explains how those exact coefficients relate to the full zero set. modelfigurereciprocal_normalizationAt N=2 and r=1, the reciprocal is sec z. The upper row gives block exponential c","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2d39429b9e78430d","tex":"\\operatorname{sech}","aliases":"sech operatorname","display":"\\operatorname{sech}","kind":"use","excerpt":"[fill=black!0,draw=black!40,line width=.35pt] (7.10000,3.05000) rectangle ++(2.10000,0.80000); node[] at (8.15000,3.45000) 5; draw[fill=black!8,draw=black!40,line width=.35pt] (9.20000,3.05000) rectangle ++(2.10000,0.80000); node[] at (10.25000,3.45000) 61; draw[fill=black!0,draw=black!40,line width=.35pt] (0.00000,2.15000) rectangle ++(3.40000,0.8","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c0237f89d91c7c75","tex":"\\Sfun_N^{(1)}","aliases":"Sfun_N^(1) S_N^(1) Sfun mathsf","display":"{\\mathsf S}_N^{(1)}","kind":"use","excerpt":"acz^Nn+j(Nn+j)! is the residue j part of the exponential. Second, D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)! is the signed basis for y^(N)=-y. The section D_N= D_N,0 is the denominator used in the reciprocal family S_N^(r)= D_N^-r. Set T_N=- D_N^prime/ D_N. Equivalently, T_N is the logarithmic derivative of S_N^(1).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-545e283bf4ccf588","tex":"\\Sfun_N^{(r+s)}","aliases":"Sfun_N^(r+s) S_N^(r+s) Sfun mathsf","display":"{\\mathsf S}_N^{(r+s)}","kind":"use","excerpt":"[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6d4ed0b3a6647724","tex":"\\Sfun_N^{(s)}","aliases":"Sfun_N^(s) S_N^(s) Sfun mathsf","display":"{\\mathsf S}_N^{(s)}","kind":"use","excerpt":"[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry tr","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-08b5f4a5605ba4c4","tex":"\\sup_{s\\in L}","aliases":"sup_s∈ L sup in","display":"\\sup_{s\\in L}","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6da3583735a2a7a0","tex":"T'","aliases":"T'","display":"T'","kind":"use","excerpt":"retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-46423aa7af18fd67","tex":"T''","aliases":"T''","display":"T''","kind":"use","excerpt":"retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","excerpt":"retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"te endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificate works Evaluate the Taylor polynomial S_90(t)=∑_n=0^90frac(-1)^nt^3n(3n)! at the two rational endpoints. From the next term onward, magnitudes decrease, because their su","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a07369c56b77606c","tex":"t^{3n}","aliases":"t^3n","display":"t^{3n}","kind":"use","excerpt":"Theorem 11.8 (p. 30), Corollary 11.9 (p. 31). The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificate","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-aea3af576abad800","tex":"t^m","aliases":"t^m","display":"t^m","kind":"use","excerpt":"The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"use","excerpt":"The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"pressions have poles rather than finite values. takeawayThe nonlinear tangent equation is exact where division is legal. It is not a global regular differential equation across the zeros. platecaptionThe tests verify the tangent coefficient recurrence, the exact cubic ODE, and the symbolic Riccati reduction after clearing denominators. The order-on","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-71bcc4191c4297b8","tex":"T_2^2","aliases":"T_2^2","display":"T_2^2","kind":"use","excerpt":"[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.00000,4.50000) LINEAR EQUATION; node[] at (9.30000,4.50000) LOGARITHMIC DERIVATIVE; node[] at (3.00000,3.25000) D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000,1.85000) T_3","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4a336dea88d0b19d","tex":"T_2^{\\prime}","aliases":"T_2^prime prime","display":"T_2^{\\prime}","kind":"use","excerpt":"[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.00000,4.50000) LINEAR EQUATION; node[] at (9.30000,4.50000) LOGARITHMIC DERIVATIVE; node[] at (3.00000,3.25000) D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000,1.85000) T_3","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","excerpt":"intikzpicture[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.00000,4.50000) LINEAR EQUATION; node[] at (9.30000,4.50000) LOGARITHMIC DERIVATIVE; node[] at (3.00000,3.25000) D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-18f3a999ce92c850","tex":"T_3^3","aliases":"T_3^3","display":"T_3^3","kind":"use","excerpt":"intikzpicture[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.00000,4.50000) LINEAR EQUATION; node[] at (9.30000,4.50000) LOGARITHMIC DERIVATIVE; node[] at (3.00000,3.25000) D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fa6833e8c41f83ae","tex":"T_3^{\\prime}","aliases":"T_3^prime prime","display":"T_3^{\\prime}","kind":"use","excerpt":"intikzpicture[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.00000,4.50000) LINEAR EQUATION; node[] at (9.30000,4.50000) LOGARITHMIC DERIVATIVE; node[] at (3.00000,3.25000) D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a56759e72b3c0023","tex":"T_3^{\\prime\\prime}","aliases":"T_3^primeprime prime prime","display":"T_3^{\\prime\\prime}","kind":"use","excerpt":"intikzpicture[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.00000,4.50000) LINEAR EQUATION; node[] at (9.30000,4.50000) LOGARITHMIC DERIVATIVE; node[] at (3.00000,3.25000) D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-64f17e7dedd6a141","tex":"T_4","aliases":"T_4","display":"T_4","kind":"use","excerpt":"/D_N. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator. vfigriccati The coefficients retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 ","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-733902ab0028bba1","tex":"t_m","aliases":"t_m","display":"t_m","kind":"use","excerpt":"The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.","count":44},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-143632817a84070d","tex":"t_m^{-1}","aliases":"t_m^-1","display":"t_m^{-1}","kind":"use","excerpt":"For N≥3, [source reference] writes all but finitely many reduced zeros as λ_m = (A_N(m+frac12))^N(1+ε_m) where λ_m>0, 1+ε_m>0, and ε_m = O(t_m^-1e^-γ t_m). On every compact set of s values, (1+ε_m)^-s-1 = O(|ε_m|) uniformly for large m. The difference λ_m^-s - A_N^-Ns(m+frac12)^-Ns therefore has a normally convergent sum on every compact subset of ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-27c742e0ae74badf","tex":"t_m^2","aliases":"t_m^2","display":"t_m^2","kind":"use","excerpt":"he nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0; then λ_m=t_m^2 exactly.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-78eb381d2b54657d","tex":"t_m^N","aliases":"t_m^N","display":"t_m^N","kind":"use","excerpt":"[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0; then λ_m=t_m^2 exactly.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3cfdac8841367b10","tex":"t_m^{N-1}","aliases":"t_m^N-1","display":"t_m^{N-1}","kind":"use","excerpt":"[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0; then λ_m=t_m^2 exactly.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-75177911206476aa","tex":"\\tan","aliases":"tan","display":"\\tan","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-91696060f027f950","tex":"\\Tfun_1","aliases":"Tfun_1 T_1 Tfun mathsf","display":"{\\mathsf T}_1","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b588f6b7ad9c4f3e","tex":"\\Tfun_2","aliases":"Tfun_2 T_2 Tfun mathsf","display":"{\\mathsf T}_2","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7f728757e8598144","tex":"\\Tfun_2'","aliases":"Tfun_2' T_2' Tfun mathsf","display":"{\\mathsf T}_2'","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d2b22989e5403d15","tex":"\\Tfun_2^2","aliases":"Tfun_2^2 T_2^2 Tfun mathsf","display":"{\\mathsf T}_2^2","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b44ccae829ac2f08","tex":"\\Tfun_N'","aliases":"Tfun_N' T_N' Tfun mathsf","display":"{\\mathsf T}_N'","kind":"use","excerpt":"[Scalar higher Riccati identity] At every point of the open set {z: D_N(z)≠0}, Y_N( - T_N, - T_N', …, - T_N^(N-1) ) =-1.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0dc0b6a9ff0861d3","tex":"\\Tfun_N^{(N-1)}","aliases":"Tfun_N^(N-1) T_N^(N-1) Tfun mathsf","display":"{\\mathsf T}_N^{(N-1)}","kind":"use","excerpt":"[Scalar higher Riccati identity] At every point of the open set {z: D_N(z)≠0}, Y_N( - T_N, - T_N', …, - T_N^(N-1) ) =-1.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a9faf0c0fb01f07e","tex":"\\Thetafun_{2,1}","aliases":"Thetafun_2,1 symupΘ_2,1 Thetafun symup Theta","display":null,"kind":"use","excerpt":"For N=2, symupΘ_2,1(u)=-u(u+1) and symupΘ_2,2(u) = u(u+1)(5u^2+u+2). The two factors predicted by [source reference] are visible.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-754a893d0d626b29","tex":"\\Thetafun_{2,2}","aliases":"Thetafun_2,2 symupΘ_2,2 Thetafun symup Theta","display":null,"kind":"use","excerpt":"For N=2, symupΘ_2,1(u)=-u(u+1) and symupΘ_2,2(u) = u(u+1)(5u^2+u+2). The two factors predicted by [source reference] are visible.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-46fbcb84704ec4c2","tex":"\\Thetafun_{N,1}","aliases":"Thetafun_N,1 symupΘ_N,1 Thetafun symup Theta","display":null,"kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr","display":null,"kind":"use","excerpt":"ordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Each initial jet (y(0),y'(0),y''(0)) therefore det","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fc7a5c603b95f30d","tex":"\\tr","aliases":"tr operatorname","display":"{\\operatorname{tr}}","kind":"use","excerpt":"D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[Addition and inversion laws for the companion matrix] Assume N≥2. The matrix U_N(z)=e^zJ_N satisfies U_N(z+w)=U_N(z)U_N(w), U_N(z)^-1=U_N(-z), det U_N(z)=1. Its first column is D_N(z), and all of its entries are given by [source reference].","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"For N=2, symupΘ_2,1(u)=-u(u+1) and symupΘ_2,2(u) = u(u+1)(5u^2+u+2). The two factors predicted by [source reference] are visible.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-aa9db06abcab2308","tex":"u^{2N}","aliases":"u^2N","display":"u^{2N}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d849fcfcf96ad168","tex":"u^3","aliases":"u^3","display":"u^3","kind":"use","excerpt":") (11.64750,1.00832) (11.72000,0.96637) (11.79250,0.92343) (11.86500,0.87947) (11.93750,0.83449) (12.01000,0.78848) (12.08250,0.74141) (12.15500,0.69327) (12.22750,0.64406) (12.30000,0.59375); draw[fill=black] (2.63333,2.30000) circle (0.08); draw[fill=black] (2.63333,3.00000) circle (0.08); node[] at (3.00000,4.65000) solid: Θ_2,1(u)=-u(u+1); node","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5ac2f9810e443357","tex":"u^4","aliases":"u^4","display":"u^4","kind":"use","excerpt":"00,2.50000) circle (0.055); draw[black!45] (2.70000,2.50000) -- (2.70000,0.50000); draw[fill=black] (2.70000,1.90000) circle (0.055); draw[fill=black] (2.70000,0.70000) circle (0.055); node[] at (2.70000,5.00000) u=z/(sqrt2π); node[] at (2.70000,-0.30000) u=tfrac12 i^k, tfrac32 i^k; draw[->,line width=.9pt] (5.30000,2.50000) -- (7.00000,2.50000); n","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7862d568dcbac6d5","tex":"u^N","aliases":"u^N","display":"u^N","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9bb90c7b2f3835ae","tex":"u^{Nm}","aliases":"u^Nm","display":"u^{Nm}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f4dc2d7e9a66ef97","tex":"u^{nN}","aliases":"u^nN","display":"u^{nN}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e184f8300c2a9879","tex":"u^{Nq}","aliases":"u^Nq","display":"u^{Nq}","kind":"use","excerpt":"[Theta reconstruction] For every n≥0, symupΘ_N,n(u) = (-1)^n ∑_q=0^n binomNnNq P_N,q(1)u^NqP_N,n-q(u). Thus symupΘ_N,n is a polynomial of degree at most Nn.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9fcfc7136a869616","tex":"U_N","aliases":"U_N","display":"U_N","kind":"use","excerpt":"[Addition and inversion laws for the companion matrix] Assume N≥2. The matrix U_N(z)=e^zJ_N satisfies U_N(z+w)=U_N(z)U_N(w), U_N(z)^-1=U_N(-z), det U_N(z)=1. Its first column is D_N(z), and all of its entries are given by [source reference].","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"On the resulting strip, the roots e^ia_N and e^-ia_N have real part c_N, while each remaining root η=u+iv satisfies u≤cos(3a_N)=c_N-γ_N. Hence Re((η-e^-ia_N)z) ≤-γ_Nx+2Y. Division by e^e^-ia_Nz gives the displayed normalization. Each differentiation multiplies a remainder term by η-e^-ia_N, whose modulus is at most two. Thus, for q=0,1,2, |R_N^(q)(","count":9},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bcfbf2e206c6fc4d","tex":"v^{1/4}","aliases":"v^1/4","display":"v^{1/4}","kind":"use","excerpt":"(2.19219,1.00000) circle (0.055); node[] at (2.19219,0.58000) 81/16; node[font=sffamilyfootnotesize] at (2.19219,1.45000) count 2; draw[line width=1pt] (11.54219,0.88000) -- (11.54219,1.12000); draw[fill=black] (11.54219,1.00000) circle (0.055); node[] at (11.54219,0.58000) 625/16; node[font=sffamilyfootnotesize] at (11.54219,1.45000) count 3; node","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-86272958a4f136a9","tex":"V_{k\\ell}","aliases":"V_kℓ ell","display":"V_{k\\ell}","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-44cf5118c3be9cba","tex":"\\vdots","aliases":"vdots","display":"\\vdots","kind":"use","excerpt":"[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"The identity Δ_N(w)^-r = exp( r∑_m≥1κ_N,mfracw^mm ) fits the defining generating function for B_n with x_m=r(m-1)!κ_N,m. The coefficient of w^n is therefore the displayed Bell polynomial divided by n!. Multiplication by (Nn)! gives P_N,n(r). Since Δ_N(w)=1-fracwN!+O(w^2), we have κ_N,1=1/N!. The term x_1^n gives the stated leading coefficient, whic","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0d4e45da82688954","tex":"w^3","aliases":"w^3","display":"w^3","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d8bef7eed7e9b97d","tex":"w^b","aliases":"w^b","display":"w^b","kind":"use","excerpt":"Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ccdb5f9bd9497286","tex":"w^{k-1}","aliases":"w^k-1","display":"w^{k-1}","kind":"use","excerpt":"Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ee2fe0c706b5c2fb","tex":"w^{m-1}","aliases":"w^m-1","display":"w^{m-1}","kind":"use","excerpt":"Differentiate the product identity in logarithmic form to obtain Δ_N'(w) = -Δ_N(w) ∑_m≥1σ_N,mw^m-1. Comparison of the coefficient of w^m-1 gives ma_m = -∑_k=0^m-1a_kσ_N,m-k. Solving for σ_N,m gives the recurrence.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5d99ded832587fa9","tex":"w^{n-1}","aliases":"w^n-1","display":"w^{n-1}","kind":"use","excerpt":"Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ce0aec4a79056ca2","tex":"\\widetilde{\\sigma}","aliases":"widetildeσ widetilde sigma","display":"\\widetilde{\\sigma}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2f2644292076c607","tex":"\\widetilde{\\sigma}_{N,1}","aliases":"widetildeσ_N,1 widetilde sigma","display":"\\widetilde{\\sigma}_{N,1}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ac19292d2e508b19","tex":"\\widetilde{\\sigma}_{N,2}","aliases":"widetildeσ_N,2 widetilde sigma","display":"\\widetilde{\\sigma}_{N,2}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d036ddaa4db5d2e7","tex":"\\widetilde{\\sigma}_{N,m}","aliases":"widetildeσ_N,m widetilde sigma","display":"\\widetilde{\\sigma}_{N,m}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-159b2efa6327c4c0","tex":"\\widetilde{\\sigma}_{N,n}","aliases":"widetildeσ_N,n widetilde sigma","display":"\\widetilde{\\sigma}_{N,n}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"6.1--6.6 · Quotient-algebra power bases and actual exponential-series convergence in the finite-dimensional coefficient topology. Both pointwise section-coordinate identities. Signed-shift powers, all matrix-exponential entries, group laws and trace are checked. The characteristic polynomial X^N+1, its distinct roots, exact spectrum and determinant","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"IALSTraditional monograph: Theorem 9.6 (p. 21), Theorem 9.8 (p. 21), Proposition 9.9 (p. 21). For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms s","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"use","excerpt":"ordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Each initial jet (y(0),y'(0),y''(0)) therefore det","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b5d15c510169ad7c","tex":"x^4","aliases":"x^4","display":"x^4","kind":"use","excerpt":"IALSTraditional monograph: Theorem 9.6 (p. 21), Theorem 9.8 (p. 21), Proposition 9.9 (p. 21). For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms s","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0ed7bdf4a651c709","tex":"x^m","aliases":"x^m","display":"x^m","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cea057c6841e3ec4","tex":"x^{m-Nn}","aliases":"x^m-Nn","display":"x^{m-Nn}","kind":"use","excerpt":"[Explicit Appell polynomial] For every m≥0, E_m^(N,j,r)(x) = ∑_0≤ n≤ m/N binommNn C_N,j,n(r)x^m-Nn.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-854c66c730d9b734","tex":"X^N","aliases":"X^N","display":"X^N","kind":"use","excerpt":"6.1--6.6 · Quotient-algebra power bases and actual exponential-series convergence in the finite-dimensional coefficient topology. Both pointwise section-coordinate identities. Signed-shift powers, all matrix-exponential entries, group laws and trace are checked. The characteristic polynomial X^N+1, its distinct roots, exact spectrum and determinant","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-12f1075149375120","tex":"x^N","aliases":"x^N","display":"x^N","kind":"use","excerpt":"The vectors e_0,J_Ne_0,…,J_N^N-1e_0 are the standard basis. Thus e_0 is cyclic for J_N, so the minimal polynomial of J_N has degree N. Since J_N^N=-I, that minimal polynomial divides x^N+1 and therefore equals x^N+1. The characteristic polynomial has the same degree and is divisible by the minimal polynomial, so it too equals x^N+1. This polynomial","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1494511a83649b16","tex":"x_1^3","aliases":"x_1^3","display":"x_1^3","kind":"use","excerpt":"r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0273e5b76a404abc","tex":"x_1^n","aliases":"x_1^n","display":"x_1^n","kind":"use","excerpt":"e identity Δ_N(w)^-r = exp( r∑_m≥1κ_N,mfracw^mm ) fits the defining generating function for B_n with x_m=r(m-1)!κ_N,m. The coefficient of w^n is therefore the displayed Bell polynomial divided by n!. Multiplication by (Nn)! gives P_N,n(r). Since Δ_N(w)=1-fracwN!+O(w^2), we have κ_N,1=1/N!. The term x_1^n gives the stated leading coefficient, which ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. ","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. ","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","excerpt":"The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"use","excerpt":"z^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Ea","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8b40d33482df931f","tex":"y''","aliases":"y''","display":"y''","kind":"use","excerpt":"z^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Ea","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-be45dd38f6e40426","tex":"y^{(N)}","aliases":"y^(N)","display":"y^{(N)}","kind":"use","excerpt":"Fix a positive integer N. Root of unity projection of the exponential produces residue-class sections that, for λ≠0, form a basis for y^(N)=λ^N y. The signed sections for y^(N)=-y yield the denominator D_N and its reduced form Δ_N. We derive reciprocal coefficient polynomials, Bell and Newton recurrences, an additive parameter law, and residue sect","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4b29e0454f65eab4","tex":"y^{m-q}","aliases":"y^m-q","display":"y^{m-q}","kind":"use","excerpt":"[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"ρ=π/2; draw[->,line width=1.1pt] (4.88,2.3) arc[start angle=0,end angle=330,radius=.48]; node[] at (2.80000,0.10000) the loop is not inside the disk; node[] at (9.20000,4.10000) D(z)∼ D^prime(ρ)(z-ρ); node[] at (9.20000,2.90000) log D⟼log D+2π i; node[] at (9.20000,1.65000) D^-r⟼ e^-2π i rD^-r; node[] at (9.20000,0.40000) r=1/2: -1 r∈ Z: 1;","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-208591f165ef47fe","tex":"z^{0}","aliases":"z^0","display":"z^{0}","kind":"use","excerpt":".00000); node[] at (0.60000,2.00000) -1; draw[->,line width=1pt] (2.50000,2.00000) -- (2.50000,0.55000); node[] at (2.50000,0.10000) -i; node[] at (2.50000,-0.30000) frac14∑_ℓ=0^3 i^(m-j)ℓ; node[anchor=east] at (6.10000,3.70000) j=0; draw[fill=black!0,draw=black!40,line width=.35pt] (6.45000,3.40000) rectangle ++(1.70000,0.60000); node[] at (7.3000","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-442155cb8a880e3e","tex":"z^{1}","aliases":"z^1","display":"z^{1}","kind":"use","excerpt":".40000) rectangle ++(1.70000,0.60000); node[] at (9.00000,3.70000) z^4; draw[fill=black!0,draw=black!40,line width=.35pt] (9.85000,3.40000) rectangle ++(1.70000,0.60000); node[] at (10.70000,3.70000) z^8; node[anchor=east] at (6.10000,2.85000) j=1; draw[fill=black!0,draw=black!40,line width=.35pt] (6.45000,2.55000) rectangle ++(1.70000,0.60000); no","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-69ffc71de82ab37e","tex":"z^{10}","aliases":"z^10","display":"z^{10}","kind":"use","excerpt":"r rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-39e2a5cbbb323f45","tex":"z^{11}","aliases":"z^11","display":"z^{11}","kind":"use","excerpt":"lack!0,draw=black!40,line width=.35pt] (6.45000,0.85000) rectangle ++(1.70000,0.60000); node[] at (7.30000,1.15000) z^3; draw[fill=black!0,draw=black!40,line width=.35pt] (8.15000,0.85000) rectangle ++(1.70000,0.60000); node[] at (9.00000,1.15000) z^7; draw[fill=black!0,draw=black!40,line width=.35pt] (9.85000,0.85000) rectangle ++(1.70000,0.60000)","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"For N=2, [source reference] and D_2(z)=Δ_2(z^2) give λ_m=π^2(m+frac12)^2, m≥0. Thus the first formula holds with Q_2^red=0.","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9cca987f30398f28","tex":"z^{2}","aliases":"z^2","display":"z^{2}","kind":"use","excerpt":"[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.72500,4.22000) 2; node[] at (6.37500,4.22000) z; node[] at (9.02500,4.22000) z^2; node[anchor=east] at (2.15000,3.37500) z^3 + 1; draw[fill=black!18,draw=black!40,line width=.35pt] (2.40000,2.95000) rectangle ++(2.65000,0.85000); node[] at (3.72500,3.37500) 2 z^3 +","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4140de75758a272d","tex":"z^3","aliases":"z^3","display":"z^3","kind":"use","excerpt":"vpage4Products mix the residue lanesCYCLIC CONVOLUTIONTraditional monograph: Theorem 3.3 (p. 5). Selection separates a function additively. It does not make multiplication diagonal. A residue-i term multiplied by a residue-j term lands in residue i+jpmod N. For N=3, use F=1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three sel","count":18},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bb6dd62772db46da","tex":"z^{3}","aliases":"z^3","display":"z^{3}","kind":"use","excerpt":"[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (3.72500,4.22000) 2; node[] at (6.37500,4.22000) z; node[] at (9.02500,4.22000) z^2; node[anchor=east] at (2.15000,3.37500) z^3 + 1; draw[fill=black!18,draw=black!40,line width=.35pt] (2.40000,2.95000) rectangle ++(2.65000,0.85000); node[] at (3.72500,3.37500) 2 z^3 +","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-34d4b252430fe95b","tex":"z^4","aliases":"z^4","display":"z^4","kind":"use","excerpt":"ion gives a polygon of exponents. Large zeros can occur where two dominant terms nearly cancel. The proof first localizes those zeros and then uses symmetry and uniqueness to place them on the rays. Finite exceptional zeros remain part of the later products and sums. modelfigurezero_orbitsThe exact order-four model. The first three zeros on each of","count":12},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1a0d5dc45dfbe7ea","tex":"z^{4}","aliases":"z^4","display":"z^{4}","kind":"use","excerpt":"t (6.37500,4.22000) z; node[] at (9.02500,4.22000) z^2; node[anchor=east] at (2.15000,3.37500) z^3 + 1; draw[fill=black!18,draw=black!40,line width=.35pt] (2.40000,2.95000) rectangle ++(2.65000,0.85000); node[] at (3.72500,3.37500) 2 z^3 + 2; draw[fill=black!0,draw=black!40,line width=.35pt] (5.05000,2.95000) rectangle ++(2.65000,0.85000); node[] a","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f2e36b2e5940c808","tex":"z^{5}","aliases":"z^5","display":"z^{5}","kind":"use","excerpt":"aw=black!40,line width=.35pt] (2.40000,2.95000) rectangle ++(2.65000,0.85000); node[] at (3.72500,3.37500) 2 z^3 + 2; draw[fill=black!0,draw=black!40,line width=.35pt] (5.05000,2.95000) rectangle ++(2.65000,0.85000); node[] at (6.37500,3.37500) z^4 + z; draw[fill=black!0,draw=black!40,line width=.35pt] (7.70000,2.95000) rectangle ++(2.65000,0.85000","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6dedd41de3fd2c9b","tex":"z^6","aliases":"z^6","display":"z^6","kind":"use","excerpt":"r rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4ea3f59fe8f7f8bd","tex":"z^{6}","aliases":"z^6","display":"z^{6}","kind":"use","excerpt":"5000) rectangle ++(1.70000,0.60000); node[] at (10.70000,2.85000) z^9; node[anchor=east] at (6.10000,2.00000) j=2; draw[fill=black!18,draw=black!40,line width=.35pt] (6.45000,1.70000) rectangle ++(1.70000,0.60000); node[] at (7.30000,2.00000) z^2; draw[fill=black!18,draw=black!40,line width=.35pt] (8.15000,1.70000) rectangle ++(1.70000,0.60000); no","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-82a6f2d2c744dfea","tex":"z^7","aliases":"z^7","display":"z^7","kind":"use","excerpt":"D_2^primeprime=-D_2; node[] at (9.30000,3.25000) T_2^prime=1+T_2^2; node[] at (3.00000,1.85000) D_3^primeprimeprime=-D_3; node[] at (9.30000,1.85000) T_3^primeprime-3T_3T_3^prime+T_3^3=1; node[] at (6.15000,0.50000) T_N=-D_N^prime/D_N only where D_N≠0; draw[black!25] (6.10000,1.10000) -- (6.10000,4.65000); node[] at (6.10000,-0.40000) T_4(z)=fracz^","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b1f9a176af13e84f","tex":"z^{7}","aliases":"z^7","display":"z^{7}","kind":"use","excerpt":"70000) rectangle ++(1.70000,0.60000); node[] at (10.70000,2.00000) z^10; node[anchor=east] at (6.10000,1.15000) j=3; draw[fill=black!0,draw=black!40,line width=.35pt] (6.45000,0.85000) rectangle ++(1.70000,0.60000); node[] at (7.30000,1.15000) z^3; draw[fill=black!0,draw=black!40,line width=.35pt] (8.15000,0.85000) rectangle ++(1.70000,0.60000); no","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2509ac71932ed6d1","tex":"z^{8}","aliases":"z^8","display":"z^{8}","kind":"use","excerpt":"lack!0,draw=black!40,line width=.35pt] (6.45000,3.40000) rectangle ++(1.70000,0.60000); node[] at (7.30000,3.70000) z^0; draw[fill=black!0,draw=black!40,line width=.35pt] (8.15000,3.40000) rectangle ++(1.70000,0.60000); node[] at (9.00000,3.70000) z^4; draw[fill=black!0,draw=black!40,line width=.35pt] (9.85000,3.40000) rectangle ++(1.70000,0.60000)","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1dfd53f8cdcf1eba","tex":"z^{9}","aliases":"z^9","display":"z^{9}","kind":"use","excerpt":"lack!0,draw=black!40,line width=.35pt] (6.45000,2.55000) rectangle ++(1.70000,0.60000); node[] at (7.30000,2.85000) z^1; draw[fill=black!0,draw=black!40,line width=.35pt] (8.15000,2.55000) rectangle ++(1.70000,0.60000); node[] at (9.00000,2.85000) z^5; draw[fill=black!0,draw=black!40,line width=.35pt] (9.85000,2.55000) rectangle ++(1.70000,0.60000)","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-00a09b294e775bb0","tex":"z^a","aliases":"z^a","display":"z^a","kind":"use","excerpt":"Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4575bcbd3a963aec","tex":"z^{N-1}","aliases":"z^N-1","display":"z^{N-1}","kind":"use","excerpt":"Since D_N(z)=Δ_N(z^N), -frac D_N'(z) D_N(z) = -Nz^N-1fracΔ_N'(z^N)Δ_N(z^N). Differentiate the defining series for -logΔ_N and compare coefficients.","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6908f509cdfbbfa6","tex":"z^{Nm-1}","aliases":"z^Nm-1","display":"z^{Nm-1}","kind":"use","excerpt":"[Logarithmic coefficient formula] The transfer tangent satisfies T_N(z) = N∑_m≥1κ_N,mz^Nm-1. Equivalently, T_N(n) = N(Nn+N-1)!κ_N,n+1.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3b1a441423794c15","tex":"z^{Nn+d_{pq}}","aliases":"z^Nn+d_pq","display":"z^{Nn+d_{pq}}","kind":"use","excerpt":"For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c02dd8bdef514cd5","tex":"z^{Nn-1}","aliases":"z^Nn-1","display":"z^{Nn-1}","kind":"use","excerpt":"Set S= D_N^-r. Logarithmic differentiation gives D_NS'+r D_N'S=0. Taking the coefficient of z^Nn-1/(Nn-1)! gives ∑_k=0^n-1(-1)^k binomNn-1NkP_N,n-k(r) +r∑_k=1^n(-1)^k binomNn-1Nk-1P_N,n-k(r)=0. The term with k=0 in the first sum is P_N,n(r), and every other term has a smaller block index. Rearrangement gives the formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-de29e5d648115a0b","tex":"z^q","aliases":"z^q","display":"z^q","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7dcc0cb193ddf22f","tex":"\\mathfrak Z_2^{\\mathrm{abs}}","aliases":"Z_2^abs mathfrak mathrm","display":"\\mathfrak Z_2^{\\mathrm{abs}}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f51eabb9e01e23cc","tex":"\\mathfrak Z_2^{\\mathrm{red}}","aliases":"Z_2^red mathfrak mathrm","display":"\\mathfrak Z_2^{\\mathrm{red}}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ff87688186f5755b","tex":"\\mathfrak Z_2^{\\rm red}","aliases":"Z_2^rm red mathfrak rm","display":"\\mathfrak Z_2^{\\rm red}","kind":"use","excerpt":"is not continuationZERO ZETA FUNCTIONSTraditional monograph: Theorem 14.2 (p. 39), Theorem 14.3 (p. 39), Corollary 14.4 (p. 41). Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_plane For N=2 and N","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-223d8a54b200d215","tex":"\\mathfrak Z_4^{\\mathrm{abs}}","aliases":"Z_4^abs mathfrak mathrm","display":"\\mathfrak Z_4^{\\mathrm{abs}}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c577fe54775a0551","tex":"\\mathfrak Z_4^{\\mathrm{red}}","aliases":"Z_4^red mathfrak mathrm","display":"\\mathfrak Z_4^{\\mathrm{red}}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-20ff6fff48e9be12","tex":"\\mathfrak Z_4^{\\rm red}","aliases":"Z_4^rm red mathfrak rm","display":"\\mathfrak Z_4^{\\rm red}","kind":"use","excerpt":"is not continuationZERO ZETA FUNCTIONSTraditional monograph: Theorem 14.2 (p. 39), Theorem 14.3 (p. 39), Corollary 14.4 (p. 41). Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_plane For N=2 and N","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e8004e0be384ebd0","tex":"\\mathfrak Z_N^{\\rm abs}","aliases":"Z_N^rm abs mathfrak rm","display":"\\mathfrak Z_N^{\\rm abs}","kind":"use","excerpt":"frak Z_2^rm red(s)=π^-2sζ(2s,1/2), Z_4^rm red(s)=(sqrt2π)^-4sζ(4s,1/2). Each identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at 1/2 or 1/4, with residue 1/(2π) or 1/(4sqrt2π) respectively. headThe factor N matters twice For every order in the paper, Z_N^rm abs(s","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f6be28fa9e4db99c","tex":"\\mathfrak Z_N^{\\rm red}","aliases":"Z_N^rm red mathfrak rm","display":"\\mathfrak Z_N^{\\rm red}","kind":"use","excerpt":"frak Z_2^rm red(s)=π^-2sζ(2s,1/2), Z_4^rm red(s)=(sqrt2π)^-4sζ(4s,1/2). Each identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at 1/2 or 1/4, with residue 1/(2π) or 1/(4sqrt2π) respectively. headThe factor N matters twice For every order in the paper, Z_N^rm abs(s","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1834f61514099928","tex":"\\Zcal_R","aliases":"Zcal_R Z_R Zcal mathcal","display":"{\\mathcal Z}_R","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-485214fbaa6bd648","tex":"\\Zcal_R^{\\mathrm{orb}}","aliases":"Zcal_R^orb Z_R^orb Zcal mathrm mathcal mathrm","display":"{\\mathcal Z}_R^{\\mathrm{orb}}","kind":"use","excerpt":"[Orbit reduced nearest circle term] Assume the hypotheses of [source reference]. Choose one representative from each orbit of the nearest zeros under multiplication by ω, and call the resulting set Z_R^orb. With m=Nn, P_N,n(r) = · N m!binomm+r-1r-1 ∑_τ∈ Z_R^orb fracτ^-m(-τ D_N'(τ))^r","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ Z mathbb","display":"{\\mathbb Z}","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-df6ca16455e2bbb5","tex":"\\ZZ_{>0}","aliases":"ZZ_>0 Z_>0 ZZ mathbb","display":"{\\mathbb Z}_{>0}","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6820374f59f0b332","tex":"|^{-1}","aliases":"|^-1","display":"|^{-1}","kind":"use","excerpt":"By [source reference], the order of Δ_N is 1/N<1. By [source reference], it has infinitely many zeros, and [source reference] gives ∑_λ∈Λ_N|λ|^-1<∞. Hadamard factorization therefore gives a canonical product of genus zero. The only possible exponential factor is a nonzero constant because the order is below one. Since Δ_N(0)=1, that constant is one","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0f1a8444914ffb6d","tex":"|^{-q}","aliases":"|^-q","display":"|^{-q}","kind":"use","excerpt":"out added regularizing terms. takeawayA global analytic product theorem connects the infinite zero sum to a finite algebraic recurrence. The recurrence alone does not prove the product theorem. platecaptionDisplayed moments are checked against formal logarithm coefficients and exact order-two/order-four zeta values. Bar widths are numerical shares.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-48680b3fa133e34c","tex":"|^{1/N}","aliases":"|^1/N","display":"|^{1/N}","kind":"use","excerpt":"For every N≥2, each nonzero zero λ of Δ_N gives N zeros of D_N, each of modulus |λ|^1/N. Hence Z_N^abs(s) = N Z_N^red(s/N). Substitution gives the second formula with Q_N^abs(s)=NQ_N^red(s/N), which is entire.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-16521de3e48f7c29","tex":"|^{M_L}","aliases":"|^M_L","display":"|^{M_L}","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9599612652da43ca","tex":"|_{z=0}","aliases":"|_z=0","display":"|_{z=0}","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"[The case N=2] For N=2, D_2(z)=cos z. Its zeros are ±(m+frac12)π, m≥0, and all are simple. They agree with the formula t_m=A_2(m+1/2).","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[anchor=west] at (0.45000,3.65000) m=0; node[] at (3.75000,3.65000) 1.84981279919014347629; node[] at (8.30000,3.65000) +3.60134×10^-2; node[] at (11.50000,3.65000) -5.243×10^-2; node[anchor=west] at (0.45000,2.70000) m=1; node[] at (3.75000,2.70000) 5.44123335502366335540;","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.","count":21},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4b118facef29f432","tex":"\\gamma_N","aliases":"γ_N gamma","display":"\\gamma_N","kind":"use","excerpt":"The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.","count":16},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.","count":13},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2bb168c166c79f1d","tex":"\\Delta_1","aliases":"Δ_1 Delta","display":"\\Delta_1","kind":"use","excerpt":"and integral comparison. Thus truncation has an explicit error mechanism. takeawayThe infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem. platecaptionThe exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use k","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6c0b4ab8dd399bba","tex":"\\Delta_2","aliases":"Δ_2 Delta","display":"\\Delta_2","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":15},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-198bb3bf1a420a2e","tex":"\\Delta_4","aliases":"Δ_4 Delta","display":"\\Delta_4","kind":"use","excerpt":"vpage8Four zeros become one orbit zeroTHE REDUCED VARIABLETraditional monograph: Definition 5.7 (p. 9), Example 11.12 (p. 31), Corollary 11.13 (p. 31). For N=4 the factorization is exact: D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), D_4(z)=Δ_4(z^4). Set A_4=sqrt2π. All the zeros are A_4(m+1/2)i^k, with m≥0 and 0≤ k<4. Each zero is simple. vfigorbit_reduction ","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-63266e1b2042ea9c","tex":"\\Delta_4^{-1}","aliases":"Δ_4^-1 Delta","display":"\\Delta_4^{-1}","kind":"use","excerpt":"D ZERO WORKED MODELTraditional monograph: Corollary 7.7 (p. 16), Example 11.12 (p. 31), Theorem 12.6 (p. 35). Use N=4 throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order. vfigworked_circuit headRoute one: invert the","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8e4e08a5e917de66","tex":"\\delta_{jk}","aliases":"δ_jk delta","display":"\\delta_{jk}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-25ea098ca9f371ad","tex":"\\delta_{kj}","aliases":"δ_kj delta","display":"\\delta_{kj}","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-23fe82c4424f67f6","tex":"\\delta_m","aliases":"δ_m delta","display":"\\delta_m","kind":"use","excerpt":"Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e3fe3fecf4b7ed4b","tex":"\\delta_m^{(1)}","aliases":"δ_m^(1) delta","display":"\\delta_m^{(1)}","kind":"use","excerpt":"cos(sqrt3t/2)3. Divide by the nonzero factor 2e^t/2/3. The resulting equation separates the dominant cosine from the small third exponential. vfigtwo_term Ignoring the last term gives model zeros t_m=frac2πsqrt3(m+1/2). At t_m, the full normalized expression equals tfrac12e^-3t_m/2>0, so t_m is not an exact zero. Linearizing the cosine at that poin","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bb11ddfc4a29b9f4","tex":"\\delta_m^2","aliases":"δ_m^2 delta","display":"\\delta_m^2","kind":"use","excerpt":"Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f25337fa5b300330","tex":"\\Delta_N'","aliases":"Δ_N' Delta","display":"\\Delta_N'","kind":"use","excerpt":"Since D_N(z)=Δ_N(z^N), -frac D_N'(z) D_N(z) = -Nz^N-1fracΔ_N'(z^N)Δ_N(z^N). Differentiate the defining series for -logΔ_N and compare coefficients.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2f800f69539a7f67","tex":"\\delta_{n0}","aliases":"δ_n0 delta","display":"\\delta_{n0}","kind":"use","excerpt":"[Triangular inverse system] The coefficients are determined by C_N,j(0)=1 and ∑_q=0^n binomNn+jNq C_N,j(q) = δ_n0. For n≥1, C_N,j(n) = - frac displaystyle∑_q=0^n-1 binomNn+jNqC_N,j(q) displaystylebinomNn+jNn .","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"[Entries of the fundamental matrix] For 0≤ p,q<N, let d_pq be the member of {0,…,N-1} congruent to p-q modulo N. Put ε_pq = 1, · p≥ q, -1, · p<q. Then (e^zJ_N)_pq = ε_pq D_N,d_pq(z). Thus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from q to p crosses the signed edge.","count":12},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4b0466fee544c74d","tex":"\\varepsilon_M","aliases":"ε_M varepsilon","display":"\\varepsilon_M","kind":"use","excerpt":"[Near cancellation of two exponentials] For every M>0, there are constants ε_M>0 and C_M>0 such that the following holds. If |Re w|≤ M and |1+e^w|≤ε_M, then there is a unique integer m with |w-(2m+1)π i| ≤ C_M|1+e^w|.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d6b02083e2b0a177","tex":"\\varepsilon_m","aliases":"ε_m varepsilon","display":"\\varepsilon_m","kind":"use","excerpt":"For N≥3, [source reference] writes all but finitely many reduced zeros as λ_m = (A_N(m+frac12))^N(1+ε_m) where λ_m>0, 1+ε_m>0, and ε_m = O(t_m^-1e^-γ t_m). On every compact set of s values, (1+ε_m)^-s-1 = O(|ε_m|) uniformly for large m. The difference λ_m^-s - A_N^-Ns(m+frac12)^-Ns therefore has a normally convergent sum on every compact subset of ","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7bc6fb3bcea699cd","tex":"\\varepsilon_{pq}","aliases":"ε_pq varepsilon","display":"\\varepsilon_{pq}","kind":"use","excerpt":"[Entries of the fundamental matrix] For 0≤ p,q<N, let d_pq be the member of {0,…,N-1} congruent to p-q modulo N. Put ε_pq = 1, · p≥ q, -1, · p<q. Then (e^zJ_N)_pq = ε_pq D_N,d_pq(z). Thus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from q to p crosses the signed edge.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"[The cosine orbit sums] For N=2, Δ_2(w)=cossqrt w. Its zeros are λ_m=π^2(m+frac12)^2, m≥0. Hence σ_2,q = frac2^2q-1π^2qζ(2q). The identity in [source reference] recovers the classical tangent coefficients from these zero power sums.","count":9},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"The second half of the paper concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's monogr","count":21},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-614a48540774c015","tex":"\\eta^4","aliases":"η^4 eta","display":"\\eta^4","kind":"use","excerpt":"934,3.66066) -- (1.28934,1.53934) -- (3.41066,1.53934) -- cycle; draw[->] (0.40000,2.60000) -- (4.30000,2.60000); draw[->] (2.35000,0.65000) -- (2.35000,4.55000); draw[fill=black] (3.41066,3.66066) circle (0.08); draw[fill=black] (1.28934,3.66066) circle (0.08); draw[fill=black] (1.28934,1.53934) circle (0.08); draw[fill=black] (3.41066,1.53934) ci","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-adcf51002ea5c3e3","tex":"\\eta^N","aliases":"η^N eta","display":"\\eta^N","kind":"use","excerpt":"The second half of the paper concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's monogr","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-62ecbfb276c1f894","tex":"\\eta_0","aliases":"η_0 eta","display":"\\eta_0","kind":"use","excerpt":"The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fc03a685a0a01ebc","tex":"\\eta_k","aliases":"η_k eta","display":"\\eta_k","kind":"use","excerpt":"We now assume N≥2. Write η_k=e^(2k+1)π i/N, 0≤ k<N. These are the vertices of a regular polygon and D_N(z)=frac1N∑_k=0^N-1e^η_kz.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-17bbe9dc50ae42d1","tex":"\\eta_\\ell","aliases":"η_ℓ eta ell","display":"\\eta_\\ell","kind":"use","excerpt":"ojection estimate gives the stronger bound Y=fraclog(N-1)2s_N for every zero in the sector, without a large-radius assumption. Write z=re^iθ with 0≤θ≤ a_N and r≥0. For 0≤ℓ≤ N-2, the angle θ+(2ℓ+1)a_N lies between a_N+θ and 2π-a_N-θ. Monotonicity of cosine on [0,π], followed by reflection about π, therefore gives Re(η_ℓ z)≤ rcos(a_N+θ). The remainin","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"vpage16Scale and order enter differentlyTHETA COUPLINGTraditional monograph: Theorem 10.2 (p. 23), Proposition 10.3 (p. 23), Proposition 10.4 (p. 23). Let H_N(z)=E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic bl","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c14bbf5e03ab44c9","tex":"\\Theta_{2,1}","aliases":"Θ_2,1 Theta","display":"\\Theta_{2,1}","kind":"use","excerpt":"0,1.16641) (11.43000,1.12831) (11.50250,1.08927) (11.57500,1.04928) (11.64750,1.00832) (11.72000,0.96637) (11.79250,0.92343) (11.86500,0.87947) (11.93750,0.83449) (12.01000,0.78848) (12.08250,0.74141) (12.15500,0.69327) (12.22750,0.64406) (12.30000,0.59375); draw[fill=black] (2.63333,2.30000) circle (0.08); draw[fill=black] (2.63333,3.00000) circle","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cc757b47064b63ba","tex":"\\Theta_{2,2}","aliases":"Θ_2,2 Theta","display":"\\Theta_{2,2}","kind":"use","excerpt":"efficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u). Every positive-index theta polynomial vanishes at u=0. For even N, H_N(-z)=H_N(z), so the defining product is also 1 at u=-1. headThe odd-order boundary is visible For N=3, Θ_3,1(-1)=2, not zero. The factor u+1 is not a uniform law across orders. In contrast, Θ_2,2(u","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-730e92ec496d3f38","tex":"\\Theta_{3,1}","aliases":"Θ_3,1 Theta","display":"\\Theta_{3,1}","kind":"use","excerpt":"garithmic block m. Reciprocal order contributes u. vfigtheta Their combined logarithm has coefficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u). Every positive-index theta polynomial vanishes at u=0. For even N, H_N(-z)=H_N(z), so the defining product is also 1 at u=-1. headThe odd-order boundary is visible For N=3, Θ_","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c2ec1f968f5fa88b","tex":"\\theta_{N,0}","aliases":"θ_N,0 theta","display":"\\theta_{N,0}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-26bb4e7108e8d9c7","tex":"\\Theta_{N,1}","aliases":"Θ_N,1 Theta","display":"\\Theta_{N,1}","kind":"use","excerpt":"E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic block m. Reciprocal order contributes u. vfigtheta Their combined logarithm has coefficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-444beaffc1630312","tex":"\\Theta_{N,n}","aliases":"Θ_N,n Theta","display":"\\Theta_{N,n}","kind":"use","excerpt":"vpage16Scale and order enter differentlyTHETA COUPLINGTraditional monograph: Theorem 10.2 (p. 23), Proposition 10.3 (p. 23), Proposition 10.4 (p. 23). Let H_N(z)=E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic bl","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-24341a22cfbc2f38","tex":"\\theta_{N,n}","aliases":"θ_N,n theta","display":"\\theta_{N,n}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-630ff65f783b7453","tex":"\\theta_{N,n-m}","aliases":"θ_N,n-m theta","display":"\\theta_{N,n-m}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-62f4ed922bb2ce09","tex":"\\kappa_1","aliases":"κ_1 kappa","display":"\\kappa_1","kind":"use","excerpt":"Corollary 7.7 (p. 16). Let r vary instead of fixing the reciprocal power. Since Δ_N(w)^-r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3fad2613c2b28e73","tex":"\\kappa_1^3","aliases":"κ_1^3 kappa","display":"\\kappa_1^3","kind":"use","excerpt":",0.75000); draw[fill=black] (0.78000,2.82000) circle (0.085); draw[fill=black!12,draw=black!40,line width=.35pt] (1.27000,2.45000) rectangle ++(0.66000,0.75000); draw[fill=black] (1.60000,2.82000) circle (0.085); draw[fill=black!12,draw=black!40,line width=.35pt] (2.09000,2.45000) rectangle ++(0.66000,0.75000); draw[fill=black] (2.42000,2.82000) ci","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-93055c50da4a404a","tex":"\\kappa_2","aliases":"κ_2 kappa","display":"\\kappa_2","kind":"use","excerpt":"Corollary 7.7 (p. 16). Let r vary instead of fixing the reciprocal power. Since Δ_N(w)^-r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3cdc2b33ff718bf4","tex":"\\kappa_{2,1}","aliases":"κ_2,1 kappa","display":"\\kappa_{2,1}","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-570f9f44831ecebb","tex":"\\kappa_{2,2}","aliases":"κ_2,2 kappa","display":"\\kappa_{2,2}","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-545253b67e7a1373","tex":"\\kappa_3","aliases":"κ_3 kappa","display":"\\kappa_3","kind":"use","excerpt":"Corollary 7.7 (p. 16). Let r vary instead of fixing the reciprocal power. Since Δ_N(w)^-r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-00bf93e0f30b5a36","tex":"\\kappa_{4,1}","aliases":"κ_4,1 kappa","display":"\\kappa_{4,1}","kind":"use","excerpt":"8), Theorem 8.5 (p. 19). The transfer tangent is T_N=-D_N'/D_N. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator. vfigriccati The coefficients retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2b72484db9745e3a","tex":"\\kappa_{4,2}","aliases":"κ_4,2 kappa","display":"\\kappa_{4,2}","kind":"use","excerpt":"The transfer tangent is T_N=-D_N'/D_N. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator. vfigriccati The coefficients retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bd45552bd05a5fc5","tex":"\\kappa_{N,1}","aliases":"κ_N,1 kappa","display":"\\kappa_{N,1}","kind":"use","excerpt":"[Bell polynomial formula] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rκ_N,1, r 1! κ_N,2, …, r (n-1)! κ_N,n ). For n≥1, this is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n!(N!)^n.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-849903fb00046048","tex":"\\kappa_{N,2}","aliases":"κ_N,2 kappa","display":"\\kappa_{N,2}","kind":"use","excerpt":"[Bell polynomial formula] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rκ_N,1, r 1! κ_N,2, …, r (n-1)! κ_N,n ). For n≥1, this is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n!(N!)^n.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5222a4a546cc55fc","tex":"\\kappa_{N,3}","aliases":"κ_N,3 kappa","display":"\\kappa_{N,3}","kind":"use","excerpt":"The first logarithmic coefficients are κ_N,1=frac1N!, κ_N,2=frac1(N!)^2-frac2(2N)!, and κ_N,3 = frac1(N!)^3 - frac3N!(2N)! + frac3(3N)!. Insert these values into [source reference] and simplify.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-88fe155938eb7be8","tex":"\\kappa_{N,k}","aliases":"κ_N,k kappa","display":"\\kappa_{N,k}","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b9557fe49782b2a8","tex":"\\kappa_{N,n}","aliases":"κ_N,n kappa","display":"\\kappa_{N,n}","kind":"use","excerpt":"[Bell polynomial formula] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rκ_N,1, r 1! κ_N,2, …, r (n-1)! κ_N,n ). For n≥1, this is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n!(N!)^n.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4cf0a10371eb328d","tex":"\\kappa_{N,n+1}","aliases":"κ_N,n+1 kappa","display":"\\kappa_{N,n+1}","kind":"use","excerpt":"[Logarithmic coefficient formula] The transfer tangent satisfies T_N(z) = N∑_m≥1κ_N,mz^Nm-1. Equivalently, T_N(n) = N(Nn+N-1)!κ_N,n+1.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-35ea25074ca81a87","tex":"\\kappa_{N,q}","aliases":"κ_N,q kappa","display":"\\kappa_{N,q}","kind":"use","excerpt":"vpage22Finite coefficients give infinite momentsZERO POWER SUMSTraditional monograph: Theorem 12.6 (p. 35), Corollary 12.7 (p. 36), Corollary 12.9 (p. 37), Corollary 12.10 (p. 37). Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-30aee352b9817eeb","tex":"\\lambda^{-q}","aliases":"λ^-q lambda","display":"\\lambda^{-q}","kind":"use","excerpt":"vpage22Finite coefficients give infinite momentsZERO POWER SUMSTraditional monograph: Theorem 12.6 (p. 35), Corollary 12.7 (p. 36), Corollary 12.9 (p. 37), Corollary 12.10 (p. 37). Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator ","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-75946f9ee1ce0aa6","tex":"\\lambda^j","aliases":"λ^j lambda","display":"\\lambda^j","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5f9faba9cf67ca67","tex":"\\lambda^m","aliases":"λ^m lambda","display":"\\lambda^m","kind":"use","excerpt":"Take the logarithm of the product in [source reference]. Absolute convergence in a disk below the first zero permits termwise expansion -log(1-frac wλ) = ∑_m≥1fracw^mmλ^m. This gives the logarithmic formula and identifies κ_N,m with σ_N,m.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-385decfaff5738d3","tex":"\\lambda^N","aliases":"λ^N lambda","display":"\\lambda^N","kind":"use","excerpt":"Fix a positive integer N. Root of unity projection of the exponential produces residue-class sections that, for λ≠0, form a basis for y^(N)=λ^N y. The signed sections for y^(N)=-y yield the denominator D_N and its reduced form Δ_N. We derive reciprocal coefficient polynomials, Bell and Newton recurrences, an additive parameter law, and residue sect","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1488a4b56fba6683","tex":"\\lambda^{N(N-1)/2}","aliases":"λ^N(N-1)/2 lambda","display":"\\lambda^{N(N-1)/2}","kind":"use","excerpt":"[Wronskian of the transfer basis] Assume N≥2. For λ≠0, the Wronskian of the ordered basis E_N,0(λ z),…, E_N,N-1(λ z) is the constant λ^N(N-1)/2.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-04db12d96ee6e2df","tex":"\\lambda^{N-1}","aliases":"λ^N-1 lambda","display":"\\lambda^{N-1}","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d594552e2786afd4","tex":"\\lambda_k","aliases":"λ_k lambda","display":"\\lambda_k","kind":"use","excerpt":"ct uses every reduced zero with its multiplicity. Its logarithm turns multiplication into addition, so reciprocal-zero power sums become coefficients. This step depends on the global product theorem. A truncated zero list provides a numerical partial sum, while the identity uses the entire multiset. modelfigurezero_momentsOrder-two moment sums over","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-129bdd0acbdea966","tex":"\\lambda_m","aliases":"λ_m lambda","display":"\\lambda_m","kind":"use","excerpt":"[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0; then λ_m=t_m^2 exactly.","count":14},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0321c3118b993cea","tex":"\\lambda_m^{-1}","aliases":"λ_m^-1 lambda","display":"\\lambda_m^{-1}","kind":"use","excerpt":"on ℓ^2(Λ_N) gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries. vfigproduct headFinite compression is an approximation For N=2, λ_m=π^2(m+1/2)^2. The finite determinant P_L is exactly the displayed finite product, but it is not exactly Δ_2. For 0≤ w≤1 and L≥1, an elementary tail estimate gives 0≤","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-298c99f06a4e2b99","tex":"\\lambda_m^{-s}","aliases":"λ_m^-s lambda","display":"\\lambda_m^{-s}","kind":"use","excerpt":"For N≥3, [source reference] writes all but finitely many reduced zeros as λ_m = (A_N(m+frac12))^N(1+ε_m) where λ_m>0, 1+ε_m>0, and ε_m = O(t_m^-1e^-γ t_m). On every compact set of s values, (1+ε_m)^-s-1 = O(|ε_m|) uniformly for large m. The difference λ_m^-s - A_N^-Ns(m+frac12)^-Ns therefore has a normally convergent sum on every compact subset of ","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8b3c2cf514ae34fe","tex":"\\Lambda_{N,j}","aliases":"Λ_N,j Lambda","display":"\\Lambda_{N,j}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-12eeafab07a92182","tex":"\\mu^{-m}","aliases":"μ^-m mu","display":"\\mu^{-m}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e8149ec697a9ee64","tex":"\\xi^2","aliases":"ξ^2 xi","display":"\\xi^2","kind":"use","excerpt":"vpage6Addition is quotient multiplicationFINITE ALGEBRATraditional monograph: Proposition 5.5 (p. 8), Theorem 6.2 (p. 10). Work in C[ξ]/(ξ^3+1). Every element has coordinates in 1,ξ,ξ^2. The entire exponential has the finite-coordinate form e^zξ=D_3,0(z)+D_3,1(z)ξ+D_3,2(z)ξ^2. The multiplication table makes the carry sign unavoidable. vfigsigned_ad","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ab96ce3cb4126eb6","tex":"\\xi^3","aliases":"ξ^3 xi","display":"\\xi^3","kind":"use","excerpt":"vpage6Addition is quotient multiplicationFINITE ALGEBRATraditional monograph: Proposition 5.5 (p. 8), Theorem 6.2 (p. 10). Work in C[ξ]/(ξ^3+1). Every element has coordinates in 1,ξ,ξ^2. The entire exponential has the finite-coordinate form e^zξ=D_3,0(z)+D_3,1(z)ξ+D_3,2(z)ξ^2. The multiplication table makes the carry sign unavoidable. vfigsigned_ad","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1b835423857234f4","tex":"\\xi^j","aliases":"ξ^j xi","display":"\\xi^j","kind":"use","excerpt":"[Finite algebra exponential identities] In A_N^+ one has e^λ zξ = ∑_j=0^N-1 E_N,j(λ z)ξ^j. In A_N^- one has e^zξ = ∑_j=0^N-1 D_N,j(z)ξ^j. Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bd09c41aee101fea","tex":"\\xi^{N-1}","aliases":"ξ^N-1 xi","display":"\\xi^{N-1}","kind":"use","excerpt":"Differentiation multiplies by λξ in the first algebra and by ξ in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity e^(z+w)ξ=e^zξe^wξ gives the addition laws after comparison of the coefficients of 1,ξ,…,ξ^N-1.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3a52c871cc191c65","tex":"\\xi^{Nn+j}","aliases":"ξ^Nn+j xi","display":"\\xi^{Nn+j}","kind":"use","excerpt":"Expand the exponential and write every exponent uniquely as Nn+j. In A_N^+, the relation ξ^Nn+j=ξ^j gives the transfer sections. In A_N^-, the relation ξ^Nn+j=(-1)^nξ^j gives the signed sections.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5499160581543767","tex":"\\pi^{-2s}","aliases":"π^-2s pi","display":"\\pi^{-2s}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a2ecbf2708f1e683","tex":"\\pi^{-s}","aliases":"π^-s pi","display":"\\pi^{-s}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"ct uses every reduced zero with its multiplicity. Its logarithm turns multiplication into addition, so reciprocal-zero power sums become coefficients. This step depends on the global product theorem. A truncated zero list provides a numerical partial sum, while the identity uses the entire multiset. modelfigurezero_momentsOrder-two moment sums over","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-543449b1bd1d6dc3","tex":"\\pi^{2q}","aliases":"π^2q pi","display":"\\pi^{2q}","kind":"use","excerpt":"[The cosine orbit sums] For N=2, Δ_2(w)=cossqrt w. Its zeros are λ_m=π^2(m+frac12)^2, m≥0. Hence σ_2,q = frac2^2q-1π^2qζ(2q). The identity in [source reference] recovers the classical tangent coefficients from these zero power sums.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f4875cececfa7801","tex":"\\pi^4","aliases":"π^4 pi","display":"\\pi^4","kind":"use","excerpt":"st four-point orbit has fourth power 1/16. The second has fourth power 81/16. headMultiplicity is not multiplied in the reduced function At a nonzero point, z↦ z^4 has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of D_4 therefore correspond to one simple zero of Δ_4, not one zero of multiplicity four","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-789d514d8bf742cb","tex":"\\rho^{-j}","aliases":"ρ^-j rho","display":"\\rho^{-j}","kind":"use","excerpt":"[Relation with the transfer exponential] For every j, D_N,j(z) = ρ^-j E_N,j(ρ z).","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-72962f18025874b5","tex":"\\rho^{-m}","aliases":"ρ^-m rho","display":"\\rho^{-m}","kind":"use","excerpt":"rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a0f42f3310ef6b4e","tex":"\\rho^{i+j-k}","aliases":"ρ^i+j-k rho","display":"\\rho^{i+j-k}","kind":"use","excerpt":"Use [source reference]. The factor introduced by a pair (i,j) is ρ^i+j-k. Since 0≤ i+j≤2N-2 and i+j≡ kbmod N, this factor is one when i+j=k and -1 when i+j=k+N.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d4480c8bdf9c98bc","tex":"\\rho^N","aliases":"ρ^N rho","display":"\\rho^N","kind":"use","excerpt":"Fix any ρ∈ C with ρ^N=-1.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9c4d50634b899cc2","tex":"\\rho^{Nn+j}","aliases":"ρ^Nn+j rho","display":"\\rho^{Nn+j}","kind":"use","excerpt":"The coefficient of z^Nn+j on the right is ρ^-jρ^Nn+jfrac1(Nn+j)! = (-1)^nfrac1(Nn+j)!.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5048c4b9123eb530","tex":"\\sigma_1","aliases":"σ_1 sigma","display":"\\sigma_1","kind":"use","excerpt":"cient. All symbols in the plate refer to this one order. vfigworked_circuit headRoute one: invert the denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: use the zero moments The exact reduced zeros are 4π^4(m+1/2)^4. Their first two power sums are σ_","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2cce6a80c926063f","tex":"\\sigma_1^2","aliases":"σ_1^2 sigma","display":"\\sigma_1^2","kind":"use","excerpt":"he denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: use the zero moments The exact reduced zeros are 4π^4(m+1/2)^4. Their first two power sums are σ_1=1/24 and σ_2=17/10080. The Bell/Newton identity gives the same reciprocal coefficient (σ_1^2+σ_2)","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-15fde97ab40a2e94","tex":"\\sigma_2","aliases":"σ_2 sigma","display":"\\sigma_2","kind":"use","excerpt":"n the plate refer to this one order. vfigworked_circuit headRoute one: invert the denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: use the zero moments The exact reduced zeros are 4π^4(m+1/2)^4. Their first two power sums are σ_1=1/24 and σ_2=17/10","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a5020a90c1801a5f","tex":"\\sigma_{2,1}","aliases":"σ_2,1 sigma","display":"\\sigma_{2,1}","kind":"use","excerpt":"Corollary 12.9 (p. 37), Corollary 12.10 (p. 37). Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12,","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-40f9f58c2d96ae63","tex":"\\sigma_{2,2}","aliases":"σ_2,2 sigma","display":"\\sigma_{2,2}","kind":"use","excerpt":"Corollary 12.9 (p. 37), Corollary 12.10 (p. 37). Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12,","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-99cd8f126976a70b","tex":"\\sigma_{2,3}","aliases":"σ_2,3 sigma","display":"\\sigma_{2,3}","kind":"use","excerpt":"Corollary 12.9 (p. 37), Corollary 12.10 (p. 37). Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12,","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0fb5544cca42028e","tex":"\\sigma_{2,4}","aliases":"σ_2,4 sigma","display":"\\sigma_{2,4}","kind":"use","excerpt":"Corollary 12.9 (p. 37), Corollary 12.10 (p. 37). Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12,","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fc14526e5f22b718","tex":"\\sigma_{2,q}","aliases":"σ_2,q sigma","display":"\\sigma_{2,q}","kind":"use","excerpt":"[The cosine orbit sums] For N=2, Δ_2(w)=cossqrt w. Its zeros are λ_m=π^2(m+frac12)^2, m≥0. Hence σ_2,q = frac2^2q-1π^2qζ(2q). The identity in [source reference] recovers the classical tangent coefficients from these zero power sums.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fb68eddb44cc0a49","tex":"\\sigma_{N,1}","aliases":"σ_N,1 sigma","display":"\\sigma_{N,1}","kind":"use","excerpt":"[Reciprocal coefficients from zero data] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rσ_N,1, r1!σ_N,2, …, r(n-1)!σ_N,n ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4a20bc1293d45df7","tex":"\\sigma_{N,2}","aliases":"σ_N,2 sigma","display":"\\sigma_{N,2}","kind":"use","excerpt":"[Reciprocal coefficients from zero data] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rσ_N,1, r1!σ_N,2, …, r(n-1)!σ_N,n ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-de8bf17230412986","tex":"\\sigma_{N,j,1}","aliases":"σ_N,j,1 sigma","display":"\\sigma_{N,j,1}","kind":"use","excerpt":"ized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9ade27113f7f22ab","tex":"\\sigma_{N,j,2}","aliases":"σ_N,j,2 sigma","display":"\\sigma_{N,j,2}","kind":"use","excerpt":"ized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fca46742b3422d0d","tex":"\\sigma_{N,j,m}","aliases":"σ_N,j,m sigma","display":"\\sigma_{N,j,m}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-15154b154aaffbd7","tex":"\\sigma_{N,j,n}","aliases":"σ_N,j,n sigma","display":"\\sigma_{N,j,n}","kind":"use","excerpt":"ized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cb127ee8ac29b6a2","tex":"\\sigma_{N,m-k}","aliases":"σ_N,m-k sigma","display":"\\sigma_{N,m-k}","kind":"use","excerpt":"[Newton sums for the reduced zeros] For |w| below the least zero modulus, with the analytic logarithm normalized to zero at the origin, -logΔ_N(w) = ∑_m≥1σ_N,mfracw^mm. Thus κ_N,m=σ_N,m. If a_m=frac(-1)^m(Nm)!, a_0=1, then σ_N,m = -ma_m- ∑_k=1^m-1a_kσ_N,m-k.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ce5747ec88efa8ef","tex":"\\sigma_{N,n}","aliases":"σ_N,n sigma","display":"\\sigma_{N,n}","kind":"use","excerpt":"[Reciprocal coefficients from zero data] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rσ_N,1, r1!σ_N,2, …, r(n-1)!σ_N,n ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-29e1a50ce38c4f16","tex":"\\sigma_{N,n+1}","aliases":"σ_N,n+1 sigma","display":"\\sigma_{N,n+1}","kind":"use","excerpt":"[Tangent coefficients as zero sums] For every n≥0, T_N(n) = N(Nn+N-1)!σ_N,n+1.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-761534e1b7a39982","tex":"\\sigma_{N,q}","aliases":"σ_N,q sigma","display":"\\sigma_{N,q}","kind":"use","excerpt":"vpage22Finite coefficients give infinite momentsZERO POWER SUMSTraditional monograph: Theorem 12.6 (p. 35), Corollary 12.7 (p. 36), Corollary 12.9 (p. 37), Corollary 12.10 (p. 37). Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fa63b24aa3bb7ad6","tex":"\\tau^{-m}","aliases":"τ^-m tau","display":"\\tau^{-m}","kind":"use","excerpt":"[Orbit reduced nearest circle term] Assume the hypotheses of [source reference]. Choose one representative from each orbit of the nearest zeros under multiplication by ω, and call the resulting set Z_R^orb. With m=Nn, P_N,n(r) = · N m!binomm+r-1r-1 ∑_τ∈ Z_R^orb fracτ^-m(-τ D_N'(τ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-35af929f24440334","tex":"\\tau_0","aliases":"τ_0 tau","display":"\\tau_0","kind":"use","excerpt":"1)^nt^3n(3n)! at the two rational endpoints. From the next term onward, magnitudes decrease, because their successive ratio is fract^3(3n+1)(3n+2)(3n+3)<1 there. The alternating-series remainder is bounded by the first omitted term. A separate outward interval evaluation of the differentiated polynomial, plus its tail bound, certifies the derivativ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-12031d1920332f75","tex":"\\tau_m^N","aliases":"τ_m^N tau","display":"\\tau_m^N","kind":"use","excerpt":"[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0; then λ_m=t_m^2 exactly.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cac9805d37f409ac","tex":"\\Phi_{2,1}","aliases":"Φ_2,1 Phi","display":"\\Phi_{2,1}","kind":"use","excerpt":"++(2.10000,0.80000); node[] at (6.05000,1.65000) - frac13; draw[fill=black!0,draw=black!40,line width=.35pt] (7.10000,1.25000) rectangle ++(2.10000,0.80000); node[] at (8.15000,1.65000) frac715; draw[fill=black!8,draw=black!40,line width=.35pt] (9.20000,1.25000) rectangle ++(2.10000,0.80000); node[] at (10.25000,1.65000) - frac3121; node[] at (5.65","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c7452277b5b2947b","tex":"\\Phi_m","aliases":"Φ_m Phi","display":"\\Phi_m","kind":"use","excerpt":"Fix a large m, put a=2is_N, and consider the closed disk B_m={z:|z-t_m|≤ 1/(2s_N)}. The remainder bounds hold uniformly on these fixed-radius disks. For sufficiently large m, |R_N|≤ 1/2 and |R_N'|≤ s_N/2 there. Define Φ_m(z)=t_m+fraclog(1+R_N(z))2is_N, using the principal logarithm. The estimate |log(1+u)|≤(3/2)|u| shows that Φ_m maps B_m into itse","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bd1f22a02b3e4635","tex":"\\Phi_m'","aliases":"Φ_m' Phi","display":"\\Phi_m'","kind":"use","excerpt":"_N, and consider the closed disk B_m={z:|z-t_m|≤ 1/(2s_N)}. The remainder bounds hold uniformly on these fixed-radius disks. For sufficiently large m, |R_N|≤ 1/2 and |R_N'|≤ s_N/2 there. Define Φ_m(z)=t_m+fraclog(1+R_N(z))2is_N, using the principal logarithm. The estimate |log(1+u)|≤(3/2)|u| shows that Φ_m maps B_m into itself. The derivative bound","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"The reduced unsigned denominator is Ψ_N(w)=∑_n≥0fracw^n(Nn)!=Δ_N(-w). By [source reference], -logΨ_N(w) = ∑_m≥1widetildeσ_N,mfracw^mm. With w=z^N, the logarithm of the theta generating function is ∑_m≥1 widetildeσ_N,m(u^Nm+u)fracw^mm. The complete Bell polynomial formula follows.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-364ff3282d8568ac","tex":"\\Psi_N","aliases":"Ψ_N Psi","display":"\\Psi_N","kind":"use","excerpt":"The reduced unsigned denominator is Ψ_N(w)=∑_n≥0fracw^n(Nn)!=Δ_N(-w). By [source reference], -logΨ_N(w) = ∑_m≥1widetildeσ_N,mfracw^mm. With w=z^N, the logarithm of the theta generating function is ∑_m≥1 widetildeσ_N,m(u^Nm+u)fracw^mm. The complete Bell polynomial formula follows.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":30},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f00be9f7cd43d3d8","tex":"\\omega^{(m-j)\\ell}","aliases":"ω^(m-j)ℓ omega ell","display":"\\omega^{(m-j)\\ell}","kind":"use","excerpt":"Substitution of the power series gives B_N,jF(z) = ∑_m≥0a_mz^m ( frac1N∑_ℓ=0^N-1ω^(m-j)ℓ ). The inner average is one when m≡ jbmod N and zero otherwise. The operator identities follow from the disjoint residue classes.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f12b1816b314c06d","tex":"\\omega^{-j\\ell}","aliases":"ω^-jℓ omega ell","display":"\\omega^{-j\\ell}","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-42d0e166fb7cd896","tex":"\\omega^{-k}","aliases":"ω^-k omega","display":"\\omega^{-k}","kind":"use","excerpt":"The derivatives satisfy D_N'(ω^kτ_m) = ω^-k D_N'(τ_m) and | D_N'(τ_m)| = frac2s_NNe^c_Nτ_m (1+O(e^-γτ_m)).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-357aa985d77dd9c2","tex":"\\omega^{i+j}","aliases":"ω^i+j omega","display":"\\omega^{i+j}","kind":"use","excerpt":"Write F=∑_iF_i and G=∑_jG_j, where F_i= B_N,iF and G_j= B_N,jG. Their covariance gives (F_iG_j)(ω z)=ω^i+jF_i(z)G_j(z), so F_iG_j belongs to the packet labelled by i+j modulo N. Expanding the finite product and applying B_N,k gives the formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a27b633daa590319","tex":"\\omega^j","aliases":"ω^j omega","display":"\\omega^j","kind":"use","excerpt":"[Covariance characterization] A holomorphic function F belongs to the range of B_N,j if and only if F(ω z)=ω^jF(z).","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-960f13a0e4265f44","tex":"\\omega^k","aliases":"ω^k omega","display":"\\omega^k","kind":"use","excerpt":"Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-01ddb7224174e1ec","tex":"\\omega^m","aliases":"ω^m omega","display":"\\omega^m","kind":"use","excerpt":"[Appell symmetry] Each polynomial E_m^(N,j,r) is monic of degree m, and E_m^(N,j,r)(ω x) = ω^mE_m^(N,j,r)(x).","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7dec09704d948494","tex":"\\omega^{N-1}","aliases":"ω^N-1 omega","display":"\\omega^{N-1}","kind":"use","excerpt":"Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-8788597d7fe50fbe","tex":"\\omega^\\ell","aliases":"ω^ℓ omega ell","display":"\\omega^\\ell","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":28},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].","count":5},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"lta_4(z^4). Set A_4=sqrt2π. All the zeros are A_4(m+1/2)i^k, with m≥0 and 0≤ k<4. Each zero is simple. vfigorbit_reduction The drawing uses u=z/A_4 and v=w/A_4^4, so its marked coordinates are rational. The first four-point orbit has fourth power 1/16. The second has fourth power 81/16. headMultiplicity is not multiplied in the reduced function At ","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":16},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d58378212f1d1d15","tex":"\\partial_x","aliases":"∂_x partial","display":"\\partial_x","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":12},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5459400e75a6595b","tex":"\\partial_x^2","aliases":"∂_x^2 partial","display":"\\partial_x^2","kind":"use","excerpt":"IALSTraditional monograph: Theorem 9.6 (p. 21), Theorem 9.8 (p. 21), Proposition 9.9 (p. 21). For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms s","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-13e29157dac59c52","tex":"\\partial_x^4","aliases":"∂_x^4 partial","display":"\\partial_x^4","kind":"use","excerpt":"IALSTraditional monograph: Theorem 9.6 (p. 21), Theorem 9.8 (p. 21), Proposition 9.9 (p. 21). For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms s","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b546efc5c152ee0e","tex":"\\partial_x^N","aliases":"∂_x^N partial","display":"\\partial_x^N","kind":"use","excerpt":"ferential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are finite on ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-30868654e8125173","tex":"\\partial_x^q","aliases":"∂_x^q partial","display":"\\partial_x^q","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"The multiplication formula for the gamma function gives (Nn+j)! = j!N^Nnn! ∏_a∈ A_N,j(a)_n. Indeed, the N Pochhammer factors supplied by the multiplication formula are ((j+m)/N)_n for 1≤ m≤ N; the excluded factor is ((j+N-j)/N)_n=(1)_n=n!. Insertion into the defining series gives the stated hypergeometric expansion.","count":11},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-c158bc42a84dc596","tex":"\\prod_{a\\in A_{N,j}}","aliases":"∏_a∈ A_N,j prod in","display":"\\prod_{a\\in A_{N,j}}","kind":"use","excerpt":"The multiplication formula for the gamma function gives (Nn+j)! = j!N^Nnn! ∏_a∈ A_N,j(a)_n. Indeed, the N Pochhammer factors supplied by the multiplication formula are ((j+m)/N)_n for 1≤ m≤ N; the excluded factor is ((j+N-j)/N)_n=(1)_n=n!. Insertion into the defining series gives the stated hypergeometric expansion.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-0fb411af0d50e0bd","tex":"\\prod_{m=0}^{L-1}","aliases":"∏_m=0^L-1 prod","display":"\\prod_{m=0}^{L-1}","kind":"use","excerpt":"4835) (10.90625,1.03091) (10.97813,1.01351) (11.05000,0.99615) (11.12187,0.97883) (11.19375,0.96154) (11.26562,0.94429) (11.33750,0.92707) (11.40937,0.90990) (11.48125,0.89276) (11.55312,0.87566) (11.62500,0.85859) (11.69688,0.84156) (11.76875,0.82457) (11.84062,0.80762) (11.91250,0.79070) (11.98437,0.77382) (12.05625,0.75698) (12.12812,0.74017) (1","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-51ca0794f5936ae3","tex":"\\prod_{\\lambda\\in\\Lambda_N}","aliases":"∏_λ∈Λ_N prod lambda in Lambda","display":"\\prod_{\\lambda\\in\\Lambda_N}","kind":"use","excerpt":"[Genus zero product] For N≥2, let Λ_N be the nonzero zero multiset of Δ_N. Then Δ_N(w) = ∏_λ∈Λ_N (1-frac wλ), where the product converges locally uniformly. Consequently, D_N(z) = ∏_λ∈Λ_N (1-fracz^Nλ).","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9798b95d3d831bfd","tex":"\\prod_{\\mu\\in\\Lambda_{N,j}}","aliases":"∏_μ∈Λ_N,j prod mu in Lambda","display":"\\prod_{\\mu\\in\\Lambda_{N,j}}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f3b1d3322f2bf29b","tex":"\\sum_{0\\leq n\\leq m/N}","aliases":"∑_0≤ n≤ m/N sum leq leq","display":"\\sum_{0\\leq n\\leq m/N}","kind":"use","excerpt":"[Explicit Appell polynomial] For every m≥0, E_m^(N,j,r)(x) = ∑_0≤ n≤ m/N binommNn C_N,j,n(r)x^m-Nn.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Write F=∑_iF_i and G=∑_jG_j, where F_i= B_N,iF and G_j= B_N,jG. Their covariance gives (F_iG_j)(ω z)=ω^i+jF_i(z)G_j(z), so F_iG_j belongs to the packet labelled by i+j modulo N. Expanding the finite product and applying B_N,k gives the formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f69801a69bec1d00","tex":"\\sum_{i+j\\equiv k\\ (N)}","aliases":"∑_i+j≡ k (N) sum equiv","display":"\\sum_{i+j\\equiv k\\ (N)}","kind":"use","excerpt":"1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Write F=∑_iF_i and G=∑_jG_j, where F_i= B_N,iF and G_j= B_N,jG. Their covariance gives (F_iG_j)(ω z)=ω^i+jF_i(z)G_j(z), so F_iG_j belongs to the packet labelled by i+j modulo N. Expanding the finite product and applying B_N,k gives the formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-586bc3571b381e64","tex":"\\sum_{j=0}^{N-1}","aliases":"∑_j=0^N-1 sum","display":"\\sum_{j=0}^{N-1}","kind":"use","excerpt":"[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b745b62212003b48","tex":"\\sum_{k=0}^{m-1}","aliases":"∑_k=0^m-1 sum","display":"\\sum_{k=0}^{m-1}","kind":"use","excerpt":"Differentiate the product identity in logarithmic form to obtain Δ_N'(w) = -Δ_N(w) ∑_m≥1σ_N,mw^m-1. Comparison of the coefficient of w^m-1 gives ma_m = -∑_k=0^m-1a_kσ_N,m-k. Solving for σ_N,m gives the recurrence.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-41e0c211ba80d17c","tex":"\\sum_{k=0}^{n}","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^{n}","kind":"use","excerpt":"The defining equation is D_N T_N=- D_N'. Extracting the coefficient of z^Nn+N-1/(Nn+N-1)! gives ∑_k=0^n(-1)^k binomNn+N-1NkT_N(n-k)=(-1)^n. Isolating the k=0 term gives the recurrence.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7796ee229bc8c35a","tex":"\\sum_{k=0}^{N-1}","aliases":"∑_k=0^N-1 sum","display":"\\sum_{k=0}^{N-1}","kind":"use","excerpt":"We now assume N≥2. Write η_k=e^(2k+1)π i/N, 0≤ k<N. These are the vertices of a regular polygon and D_N(z)=frac1N∑_k=0^N-1e^η_kz.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fec10602e4575bf9","tex":"\\sum_{k=0}^{n-1}","aliases":"∑_k=0^n-1 sum","display":"\\sum_{k=0}^{n-1}","kind":"use","excerpt":"Set S= D_N^-r. Logarithmic differentiation gives D_NS'+r D_N'S=0. Taking the coefficient of z^Nn-1/(Nn-1)! gives ∑_k=0^n-1(-1)^k binomNn-1NkP_N,n-k(r) +r∑_k=1^n(-1)^k binomNn-1Nk-1P_N,n-k(r)=0. The term with k=0 in the first sum is P_N,n(r), and every other term has a smaller block index. Rearrangement gives the formula.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d6cd950a1f0e0334","tex":"\\sum_{k=1}^{m-1}","aliases":"∑_k=1^m-1 sum","display":"\\sum_{k=1}^{m-1}","kind":"use","excerpt":"[Newton sums for the reduced zeros] For |w| below the least zero modulus, with the analytic logarithm normalized to zero at the origin, -logΔ_N(w) = ∑_m≥1σ_N,mfracw^mm. Thus κ_N,m=σ_N,m. If a_m=frac(-1)^m(Nm)!, a_0=1, then σ_N,m = -ma_m- ∑_k=1^m-1a_kσ_N,m-k.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-9972412a61817be9","tex":"\\sum_{k=1}^{n-1}","aliases":"∑_k=1^n-1 sum","display":"\\sum_{k=1}^{n-1}","kind":"use","excerpt":"[Parameter recurrence] We have P_N,0(r)=1. For n≥1, P_N,n(r) = · - ∑_k=1^n-1 (-1)^k binomNn-1Nk P_N,n-k(r) · -r ∑_k=1^n (-1)^k binomNn-1Nk-1 P_N,n-k(r).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-31b4e8e8ca1b60bf","tex":"\\sum_{k\\geq1}","aliases":"∑_k≥1 sum geq","display":"\\sum_{k\\geq1}","kind":"use","excerpt":"Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-3fe5a2e3add92762","tex":"\\sum_{m=0}^{12}","aliases":"∑_m=0^12 sum","display":"\\sum_{m=0}^{12}","kind":"use","excerpt":"idue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^1","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4e3a67fde8945085","tex":"\\sum_{m=1}^{n}","aliases":"∑_m=1^n sum","display":"\\sum_{m=1}^{n}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-cba1f7355992d01c","tex":"\\sum_{m\\equiv j\\bmod N}","aliases":"∑_m≡ jbmod N sum equiv bmod","display":"\\sum_{m\\equiv j\\bmod N}","kind":"use","excerpt":"[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b2f2db5791bd090b","tex":"\\sum_{m\\geq 0}","aliases":"∑_m≥ 0 sum geq","display":"\\sum_{m\\geq 0}","kind":"use","excerpt":"For a power series F(z)=∑_m≥ 0a_mz^m and a positive integer N, the coefficients may be separated according to the residue of m modulo N. Root of unity averaging gives exact projection operators for this separation. This elementary construction is usually called a multisection or a root filter. It appears throughout the classical theory of generatin","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-6b8e5ecd78509ed8","tex":"\\sum_{m\\ge L}","aliases":"∑_m≥ L sum ge","display":"\\sum_{m\\ge L}","kind":"use","excerpt":"on ℓ^2(Λ_N) gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries. vfigproduct headFinite compression is an approximation For N=2, λ_m=π^2(m+1/2)^2. The finite determinant P_L is exactly the displayed finite product, but it is not exactly Δ_2. For 0≤ w≤1 and L≥1, an elementary tail estimate gives 0≤","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-a0a9cffbc383b69e","tex":"\\sum_{m\\ge1}","aliases":"∑_m≥1 sum ge","display":"\\sum_{m\\ge1}","kind":"use","excerpt":"κ_N,m · displaystyle-logΔ_N(w)=∑_m≥1κ_N,mw^m/m.","count":6},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4d609c0fc3f9c1f9","tex":"\\sum_{n=0}^{90}","aliases":"∑_n=0^90 sum","display":"\\sum_{n=0}^{90}","kind":"use","excerpt":"Theorem 11.8 (p. 30), Corollary 11.9 (p. 31). The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificate","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-657057157b7882f4","tex":"\\sum_{n\\geq 0}","aliases":"∑_n≥ 0 sum geq","display":"\\sum_{n\\geq 0}","kind":"use","excerpt":"This volume follows the consequences of applying the cyclic projections to the exponential function. The projected exponentials form a finite differential system. After one sign is inserted around the derivative cycle, they form a basis for y^(N)=-y. The section with initial value one is D_N(z)=∑_n≥ 0(-1)^nfracz^Nn(Nn)!. Its reciprocal powers produ","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"use","excerpt":"midrule E_N,j(z) · displaystyle∑_n≥0fracz^Nn+j(Nn+j)! selects one residue class of the exponential.","count":8},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1f8f0dbb2884f06c","tex":"\\sum_{q=0}^m","aliases":"∑_q=0^m sum","display":"\\sum_{q=0}^m","kind":"use","excerpt":"The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and tr","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-5be06f71bbc1552b","tex":"\\sum_{q=0}^{m}","aliases":"∑_q=0^m sum","display":"\\sum_{q=0}^{m}","kind":"use","excerpt":"[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-113d2a04b2350877","tex":"\\sum_{q=0}^{n}","aliases":"∑_q=0^n sum","display":"\\sum_{q=0}^{n}","kind":"use","excerpt":"[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).","count":7},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-2636c510c23e69cc","tex":"\\sum_{q=0}^{n-1}","aliases":"∑_q=0^n-1 sum","display":"\\sum_{q=0}^{n-1}","kind":"use","excerpt":"[Triangular inverse system] The coefficients are determined by C_N,j(0)=1 and ∑_q=0^n binomNn+jNq C_N,j(q) = δ_n0. For n≥1, C_N,j(n) = - frac displaystyle∑_q=0^n-1 binomNn+jNqC_N,j(q) displaystylebinomNn+jNn .","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-eb07f8d7b26ea999","tex":"\\sum_{q\\geq0}","aliases":"∑_q≥0 sum geq","display":"\\sum_{q\\geq0}","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d4cedeb30578efb5","tex":"\\sum_{\\substack{0\\leq i,j<N\\\\ i+j\\equiv k\\bmod N}}","aliases":"∑_substack0≤ i,j<N i+j≡ kbmod N sum substack leq equiv bmod","display":"\\sum_{\\substack{0\\leq i,j<N\\\\ i+j\\equiv k\\bmod N}}","kind":"use","excerpt":"[Cyclic product law] If F and G are holomorphic on the same rotation-invariant domain, then B_N,k(FG) = ∑_substack0≤ i,j<N i+j≡ kbmod N ( B_N,iF)( B_N,jG).","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-1ab55d9727944350","tex":"\\sum_{\\substack{\\eta^N=-1\\\\\n\\eta\\ne e^{ia_N},e^{-ia_N}}}","aliases":"∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N sum substack eta eta ne","display":"\\sum_{\\substack{\\eta^N=-1\\\\\n\\eta\\ne e^{ia_N},e^{-ia_N}}}","kind":"use","excerpt":"On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-fac630f33df3d387","tex":"\\sum_{\\eta^N=-1}","aliases":"∑_η^N=-1 sum eta","display":"\\sum_{\\eta^N=-1}","kind":"use","excerpt":"The second half of the paper concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's monogr","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-bbb81e78281bccf8","tex":"\\sum_{\\eta\\in\\Rcal_N}","aliases":"∑_η∈Rcal_N ∑_η∈ R_N sum eta in Rcal sum eta in mathcal","display":"\\sum_{\\eta\\in{\\mathcal R}_N}","kind":"use","excerpt":"[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-614ab5cb07fb5cef","tex":"\\sum_{\\lambda}","aliases":"∑_λ sum lambda","display":"\\sum_{\\lambda}","kind":"use","excerpt":"Logarithmic differentiation of the genus zero product gives -fracΔ_N'(w)Δ_N(w) = ∑_λ∈Λ_Nfrac1λ-w. The series is locally uniform because ∑_λ|λ|^-1<∞. Substitute w=z^N and use the chain rule.","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-dde65ad1f7f11467","tex":"\\sum_{\\mu\\in\\Lambda_{N,j}}","aliases":"∑_μ∈Λ_N,j sum mu in Lambda","display":"\\sum_{\\mu\\in\\Lambda_{N,j}}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-dc4e4df36e5ded48","tex":"\\sum_{\\rho\\in\\Zcal_R}","aliases":"∑_ρ∈Zcal_R ∑_ρ∈ Z_R sum rho in Zcal sum rho in mathcal","display":"\\sum_{\\rho\\in{\\mathcal Z}_R}","kind":"use","excerpt":"rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ff580f406df4f029","tex":"\\sum_{\\tau\\in\\Zcal_R^{\\mathrm{orb}}}","aliases":"∑_τ∈Zcal_R^orb ∑_τ∈ Z_R^orb sum tau in Zcal mathrm sum tau in mathcal mathrm","display":"\\sum_{\\tau\\in{\\mathcal Z}_R^{\\mathrm{orb}}}","kind":"use","excerpt":"[Orbit reduced nearest circle term] Assume the hypotheses of [source reference]. Choose one representative from each orbit of the nearest zeros under multiplication by ω, and call the resulting set Z_R^orb. With m=Nn, P_N,n(r) = · N m!binomm+r-1r-1 ∑_τ∈ Z_R^orb fracτ^-m(-τ D_N'(τ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-e3116f8319e2684e","tex":"\\sum_\\ell","aliases":"∑_ℓ sum ell","display":"\\sum_\\ell","kind":"use","excerpt":"igin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), bec","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d736c45233cda517","tex":"\\sum_{\\ell=0}^3","aliases":"∑_ℓ=0^3 sum ell","display":"\\sum_{\\ell=0}^3","kind":"use","excerpt":"000) -- (3.95000,2.00000); node[] at (4.40000,2.00000) 1; draw[->,line width=1pt] (2.50000,2.00000) -- (2.50000,3.45000); node[] at (2.50000,3.90000) i; draw[->,line width=1pt] (2.50000,2.00000) -- (1.05000,2.00000); node[] at (0.60000,2.00000) -1; draw[->,line width=1pt] (2.50000,2.00000) -- (2.50000,0.55000); node[] at (2.50000,0.10000) -i; node[","count":1},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d9aa5ade096792f0","tex":"\\sum_{\\ell=0}^{3}","aliases":"∑_ℓ=0^3 sum ell","display":"\\sum_{\\ell=0}^{3}","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONTraditional monograph: Theorem 3.1 (p. 4), Proposition 3.2 (p. 5). For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matc","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-b98eb4d33c4dd25d","tex":"\\sum_{\\ell=0}^{N-1}","aliases":"∑_ℓ=0^N-1 sum ell","display":"\\sum_{\\ell=0}^{N-1}","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":3},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-03d52d6913e49407","tex":"\\sum_{\\ell=1}^{r}","aliases":"∑_ℓ=1^r sum ell","display":"\\sum_{\\ell=1}^{r}","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":2},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].","count":9},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"[Secant asymptotic] For N=2 and r=1, P_2,n(1) ∼ 2(2n)!(frac2π)^2n+1.","count":4},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.","count":12},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"a wrap. Reading that coefficient gives the displayed addition law. headCompare the unsigned model In C[ξ]/(ξ^3-1), the same table has no minus signs. Its exponential coordinates are the unsigned sections E_3,j. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are grouped. For ","count":10},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"midrule E_N,j(z) · displaystyle∑_n≥0fracz^Nn+j(Nn+j)! selects one residue class of the exponential.","count":26},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":12},{"work":"paper-iv","url":"notation/paper-iv.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"000,4.65000) fixed-log domain about 0; draw[fill=white,line width=1pt] (4.40000,2.30000) circle (0.09); node[] at (5.10000,1.45000) ρ=π/2; draw[->,line width=1.1pt] (4.88,2.3) arc[start angle=0,end angle=330,radius=.48]; node[] at (2.80000,0.10000) the loop is not inside the disk; node[] at (9.20000,4.10000) D(z)∼ D^prime(ρ)(z-ρ); node[] at (9.2000","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"notation-table","excerpt":"W_α^*W_α · A projector on the original signal space · A scalar packet energy","count":36},{"work":"paper-v","url":"notation/paper-v.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"notation-table","excerpt":"P_r · the coordinate projector onto the coset r+L","count":300},{"work":"paper-v","url":"notation/paper-v.html#symbol-3e842b663778e524","tex":"+^m","aliases":"+^m","display":"+^m","kind":"notation-table","excerpt":"ϕ_m,τ · the truncated power packet (t-τ)_+^m/m!","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-0230e8bde40956b0","tex":"\\Acal","aliases":"Acal A mathcal","display":"{\\mathcal A}","kind":"notation-table","excerpt":"A_L, S_L · lattice analysis and interleaving synthesis","count":11},{"work":"paper-v","url":"notation/paper-v.html#symbol-7c4a06bd352a4187","tex":"\\Acal_L","aliases":"Acal_L A_L Acal mathcal","display":"{\\mathcal A}_L","kind":"notation-table","excerpt":"A_L, S_L · lattice analysis and interleaving synthesis","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"notation-table","excerpt":"D · One channel-index delay · Necessarily one original-sample delay","count":47},{"work":"paper-v","url":"notation/paper-v.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"notation-table","excerpt":"L≤ Z^d · a finite index sublattice","count":88},{"work":"paper-v","url":"notation/paper-v.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"notation-table","excerpt":"E,R · filter bank analysis and synthesis symbols","count":70},{"work":"paper-v","url":"notation/paper-v.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"notation-table","excerpt":"P_N,rf · A masked signal on the original lattice · The shorter decimated channel","count":93},{"work":"paper-v","url":"notation/paper-v.html#symbol-f7499e87191c71a6","tex":"f_r","aliases":"f_r","display":"f_r","kind":"notation-table","excerpt":"f_r[q] · The value f[Nq+r] · An average of neighboring samples","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-097da937fa2783f7","tex":"\\Gam","aliases":"Gam Γ Gamma","display":"{\\Gamma}","kind":"notation-table","excerpt":"Γ · reflection coefficient · Smith-disk coordinate","count":12},{"work":"paper-v","url":"notation/paper-v.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"notation-table","excerpt":"H_τ · the step packet 1_[τ,1]","count":37},{"work":"paper-v","url":"notation/paper-v.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"notation-table","excerpt":"h · the grid spacing in the recovery theorems","count":87},{"work":"paper-v","url":"notation/paper-v.html#symbol-864591eb3221a503","tex":"H_\\tau","aliases":"H_τ tau","display":"H_\\tau","kind":"notation-table","excerpt":"H_τ · the step packet 1_[τ,1]","count":11},{"work":"paper-v","url":"notation/paper-v.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"notation-table","excerpt":"j,widehat j · the true and estimated singular packets","count":132},{"work":"paper-v","url":"notation/paper-v.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","excerpt":"L≤ Z^d · a finite index sublattice","count":122},{"work":"paper-v","url":"notation/paper-v.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"notation-table","excerpt":"Q_m · a bounded linear approximation operator acting on the smooth part","count":125},{"work":"paper-v","url":"notation/paper-v.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"notation-table","excerpt":"P_N,rf · A masked signal on the original lattice · The shorter decimated channel","count":153},{"work":"paper-v","url":"notation/paper-v.html#symbol-7d905bc6806c132f","tex":"\\one","aliases":"one 1 mathbf","display":"{\\mathbf 1}","kind":"notation-table","excerpt":"H_τ · the step packet 1_[τ,1]","count":12},{"work":"paper-v","url":"notation/paper-v.html#symbol-c4c39f6b410ae5bd","tex":"\\one_{[\\tau,1]}","aliases":"one_[τ,1] 1_[τ,1] one tau mathbf tau","display":"{\\mathbf 1}_{[\\tau,1]}","kind":"notation-table","excerpt":"H_τ · the step packet 1_[τ,1]","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"P_r · the coordinate projector onto the coset r+L","count":46},{"work":"paper-v","url":"notation/paper-v.html#symbol-2bd7ecfa07c32400","tex":"P_{N,r}","aliases":"P_N,r","display":"P_{N,r}","kind":"notation-table","excerpt":"P_N,rf · A masked signal on the original lattice · The shorter decimated channel","count":11},{"work":"paper-v","url":"notation/paper-v.html#symbol-4d3243980b9ae21f","tex":"P_r","aliases":"P_r","display":"P_r","kind":"notation-table","excerpt":"P_r · the coordinate projector onto the coset r+L","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"Q_m · a bounded linear approximation operator acting on the smooth part","count":92},{"work":"paper-v","url":"notation/paper-v.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"notation-table","excerpt":"f_r[q] · The value f[Nq+r] · An average of neighboring samples","count":90},{"work":"paper-v","url":"notation/paper-v.html#symbol-3406acd3f0f64ef4","tex":"Q_m","aliases":"Q_m","display":"Q_m","kind":"notation-table","excerpt":"Q_m · a bounded linear approximation operator acting on the smooth part","count":42},{"work":"paper-v","url":"notation/paper-v.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"notation-table","excerpt":"R · a set of representatives for Z^d/L","count":76},{"work":"paper-v","url":"notation/paper-v.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"notation-table","excerpt":"P_r · the coordinate projector onto the coset r+L","count":149},{"work":"paper-v","url":"notation/paper-v.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"notation-table","excerpt":"midrule S,δ_0 · Sample shift and unit impulse · A finite truncation of every ℓ^2 signal","count":15},{"work":"paper-v","url":"notation/paper-v.html#symbol-b764f1cbd00009f2","tex":"\\Scal","aliases":"Scal S mathcal","display":"{\\mathcal S}","kind":"notation-table","excerpt":"A_L, S_L · lattice analysis and interleaving synthesis","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-2b824c8d1cefc275","tex":"\\Scal_L","aliases":"Scal_L S_L Scal mathcal","display":"{\\mathcal S}_L","kind":"notation-table","excerpt":"A_L, S_L · lattice analysis and interleaving synthesis","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"notation-table","excerpt":"ϕ_m,τ · the truncated power packet (t-τ)_+^m/m!","count":57},{"work":"paper-v","url":"notation/paper-v.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"notation-table","excerpt":"W_α · The addressed Haar analysis operator · An edge detector by itself","count":25},{"work":"paper-v","url":"notation/paper-v.html#symbol-b90de78a5bfaa7d2","tex":"W_\\alpha","aliases":"W_α alpha","display":"W_\\alpha","kind":"notation-table","excerpt":"W_α · The addressed Haar analysis operator · An edge detector by itself","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-e7e7daccdd666565","tex":"W_\\alpha^*","aliases":"W_α^* alpha","display":"W_\\alpha^*","kind":"notation-table","excerpt":"W_α^*W_α · A projector on the original signal space · A scalar packet energy","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-6b108551467ed661","tex":"\\widehat j","aliases":"widehat j widehat","display":"\\widehat j","kind":"notation-table","excerpt":"j,widehat j · the true and estimated singular packets","count":37},{"work":"paper-v","url":"notation/paper-v.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"notation-table","excerpt":"Γ,z · Reflection coefficient and normalized impedance · Coefficients that can be linearly interchanged","count":79},{"work":"paper-v","url":"notation/paper-v.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ Z mathbb","display":"{\\mathbb Z}","kind":"notation-table","excerpt":"L≤ Z^d · a finite index sublattice","count":40},{"work":"paper-v","url":"notation/paper-v.html#symbol-207f0dd92280fb9b","tex":"\\ZZ^d","aliases":"ZZ^d Z^d ZZ mathbb","display":"{\\mathbb Z}^d","kind":"notation-table","excerpt":"L≤ Z^d · a finite index sublattice","count":27},{"work":"paper-v","url":"notation/paper-v.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"notation-table","excerpt":"W_α · The addressed Haar analysis operator · An edge detector by itself","count":13},{"work":"paper-v","url":"notation/paper-v.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"notation-table","excerpt":"Γ,z · Reflection coefficient and normalized impedance · Coefficients that can be linearly interchanged","count":47},{"work":"paper-v","url":"notation/paper-v.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"notation-table","excerpt":"midrule S,δ_0 · Sample shift and unit impulse · A finite truncation of every ℓ^2 signal","count":86},{"work":"paper-v","url":"notation/paper-v.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"notation-table","excerpt":"midrule S,δ_0 · Sample shift and unit impulse · A finite truncation of every ℓ^2 signal","count":16},{"work":"paper-v","url":"notation/paper-v.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"notation-table","excerpt":"Λ · A complete covering antichain of a finite tree · An arbitrary collection of nodes","count":20},{"work":"paper-v","url":"notation/paper-v.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"notation-table","excerpt":"H_τ · the step packet 1_[τ,1]","count":61},{"work":"paper-v","url":"notation/paper-v.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"notation-table","excerpt":"ϕ_m,τ · the truncated power packet (t-τ)_+^m/m!","count":17},{"work":"paper-v","url":"notation/paper-v.html#symbol-730f3c15e053f012","tex":"\\phi_{m,\\tau}","aliases":"ϕ_m,τ phi tau","display":"\\phi_{m,\\tau}","kind":"notation-table","excerpt":"ϕ_m,τ · the truncated power packet (t-τ)_+^m/m!","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"notation-table","excerpt":"[source reference] · HilbertPackets · Complex ℓ^2 impulse expansion, finite-support identity, unitary shifts, and closed shift-generated span.","count":70},{"work":"paper-v","url":"notation/paper-v.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"notation-table","excerpt":"[source reference] · HilbertPackets · Complex ℓ^2 impulse expansion, finite-support identity, unitary shifts, and closed shift-generated span.","count":21},{"work":"paper-v","url":"notation/paper-v.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"notation-table","excerpt":"L≤ Z^d · a finite index sublattice","count":73},{"work":"paper-v","url":"notation/paper-v.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":632},{"work":"paper-v","url":"notation/paper-v.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":123},{"work":"paper-v","url":"notation/paper-v.html#symbol-38143a44ee0b7f8e","tex":"a_q","aliases":"a_q","display":"a_q","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":18},{"work":"paper-v","url":"notation/paper-v.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Linear tube bound] A boundary Γ has a linear tube bound if there are constants C_Γ and δ_0>0 such that μ(N_δ(Γ)) ≤ C_Γδ for 0<δ<δ_0.","count":30},{"work":"paper-v","url":"notation/paper-v.html#symbol-5226331ae2a4df1e","tex":"C_\\Gamma","aliases":"C_Γ Gamma","display":"C_\\Gamma","kind":"definition-passage","excerpt":"[Linear tube bound] A boundary Γ has a linear tube bound if there are constants C_Γ and δ_0>0 such that μ(N_δ(Γ)) ≤ C_Γδ for 0<δ<δ_0.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-d0fe2f8238bcf56d","tex":"\\operatorname{ch}","aliases":"ch operatorname","display":"\\operatorname{ch}","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-860a844f873baee1","tex":"\\operatorname{child}","aliases":"child operatorname","display":"\\operatorname{child}","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-c4f568dc946ddd98","tex":"\\Fcal","aliases":"Fcal F mathcal","display":"{\\mathcal F}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-9ca3cb08c886d426","tex":"\\Fcal_{r,p}","aliases":"Fcal_r,p F_r,p Fcal mathcal","display":"{\\mathcal F}_{r,p}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-af7f4e3b5c2ffe05","tex":"h_{rs}","aliases":"h_rs","display":"h_{rs}","kind":"definition-passage","excerpt":"[Packet transfer kernel] For output residue r∈ R and input residue s∈ R, define a kernel on L by h_rs(m)=h(r-s+m), m∈ L.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-f50c3221ec49a3fa","tex":"\\Hcal","aliases":"Hcal H mathcal","display":"{\\mathcal H}","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-b378be010dc46f96","tex":"\\Hcal_v","aliases":"Hcal_v H_v Hcal mathcal","display":"{\\mathcal H}_v","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-12403ce6e112714f","tex":"\\Hcal_w","aliases":"Hcal_w H_w Hcal mathcal","display":"{\\mathcal H}_w","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":57},{"work":"paper-v","url":"notation/paper-v.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Lattice packet] For r∈ R, define the coordinate projector (P_rx)(n)= 1_r+L(n)x(n). The family (P_r)_r∈ R is called the packet system associated with L and R.","count":137},{"work":"paper-v","url":"notation/paper-v.html#symbol-afdfe01923a13ec5","tex":"N_\\delta","aliases":"N_δ delta","display":"N_\\delta","kind":"definition-passage","excerpt":"[Linear tube bound] A boundary Γ has a linear tube bound if there are constants C_Γ and δ_0>0 such that μ(N_δ(Γ)) ≤ C_Γδ for 0<δ<δ_0.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-e1277ceb9fb78b9f","tex":"\\one_{r+L}","aliases":"one_r+L 1_r+L one mathbf","display":"{\\mathbf 1}_{r+L}","kind":"definition-passage","excerpt":"[Lattice packet] For r∈ R, define the coordinate projector (P_rx)(n)= 1_r+L(n)x(n). The family (P_r)_r∈ R is called the packet system associated with L and R.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":94},{"work":"paper-v","url":"notation/paper-v.html#symbol-7cd951ba120bfdb9","tex":"P_u","aliases":"P_u","display":"P_u","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e70fd9df7027e2bf","tex":"P_v","aliases":"P_v","display":"P_v","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-af4f1e08915ee052","tex":"P_w","aliases":"P_w","display":"P_w","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b36c818af7586966","tex":"P_{\\varnothing}","aliases":"P_∅ varnothing","display":"P_{\\varnothing}","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Packet transfer kernel] For output residue r∈ R and input residue s∈ R, define a kernel on L by h_rs(m)=h(r-s+m), m∈ L.","count":164},{"work":"paper-v","url":"notation/paper-v.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":9},{"work":"paper-v","url":"notation/paper-v.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":43},{"work":"paper-v","url":"notation/paper-v.html#symbol-1b6b2cac1ed2c1b6","tex":"U_v","aliases":"U_v","display":"U_v","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":71},{"work":"paper-v","url":"notation/paper-v.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":25},{"work":"paper-v","url":"notation/paper-v.html#symbol-34d6de1dc85d2727","tex":"W^{r,p}","aliases":"W^r,p","display":"W^{r,p}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":11},{"work":"paper-v","url":"notation/paper-v.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Lattice packet] For r∈ R, define the coordinate projector (P_rx)(n)= 1_r+L(n)x(n). The family (P_r)_r∈ R is called the packet system associated with L and R.","count":130},{"work":"paper-v","url":"notation/paper-v.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"definition-passage","excerpt":"[Linear tube bound] A boundary Γ has a linear tube bound if there are constants C_Γ and δ_0>0 such that μ(N_δ(Γ)) ≤ C_Γδ for 0<δ<δ_0.","count":18},{"work":"paper-v","url":"notation/paper-v.html#symbol-8c811b6eab2988ab","tex":"\\tau_1","aliases":"τ_1 tau","display":"\\tau_1","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-1e99ad6780db1a41","tex":"\\tau_Q","aliases":"τ_Q tau","display":"\\tau_Q","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-c57f88e1e69525bd","tex":"\\tau_q","aliases":"τ_q tau","display":"\\tau_q","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":15},{"work":"paper-v","url":"notation/paper-v.html#symbol-25914e63a3f23d0d","tex":"\\phi_{0,\\tau_q}","aliases":"ϕ_0,τ_q phi tau","display":"\\phi_{0,\\tau_q}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":10},{"work":"paper-v","url":"notation/paper-v.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":85},{"work":"paper-v","url":"notation/paper-v.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":82},{"work":"paper-v","url":"notation/paper-v.html#symbol-361464ec4fbe1876","tex":"\\sum_{q=1}^{Q}","aliases":"∑_q=1^Q sum","display":"\\sum_{q=1}^{Q}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-46a1bb46e6353ae5","tex":"\\sum_{u\\in\\operatorname{ch}(v)}","aliases":"∑_u∈ch(v) sum in operatorname","display":"\\sum_{u\\in\\operatorname{ch}(v)}","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":49},{"work":"paper-v","url":"notation/paper-v.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":16},{"work":"paper-v","url":"notation/paper-v.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":20},{"work":"paper-v","url":"notation/paper-v.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-86042019669510f9","tex":"\\bigoplus_{w\\in\\operatorname{child}(v)}","aliases":"⊕_w∈child(v) bigoplus in operatorname","display":"\\bigoplus_{w\\in\\operatorname{child}(v)}","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":9},{"work":"paper-v","url":"notation/paper-v.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"vpage23A pole is a chart boundary, not a broken group actionHOMOGENEOUS COORDINATES / NO DIVISION BY ZEROTraditional monograph: Theorem 13.2 (p. 26), Corollary 13.3 (p. 26). Represent impedance by a nonzero pair [u:v], where [cu:cv]=[u:v] for c≠0. A finite impedance is [z:1]; infinity is [1:0]. For a nonzero reflection multiplier w, use A(w)=frac12","count":44},{"work":"paper-v","url":"notation/paper-v.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum; this proves it at depth ℓ+1.","count":72},{"work":"paper-v","url":"notation/paper-v.html#symbol-587654947ed7b96d","tex":"\\bbC","aliases":"bbC C mathbb","display":"{\\mathbb C}","kind":"introduction-passage","excerpt":"Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primiti","count":10},{"work":"paper-v","url":"notation/paper-v.html#symbol-e5365be76fcd3f4a","tex":"\\bbZ","aliases":"bbZ Z mathbb","display":"{\\mathbb Z}","kind":"introduction-passage","excerpt":"Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primiti","count":19},{"work":"paper-v","url":"notation/paper-v.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original sampl","count":40},{"work":"paper-v","url":"notation/paper-v.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"introduction-passage","excerpt":"ix of actual delaysFIR TRANSFER / A FOUR-TAP EXAMPLETraditional monograph: Theorem 3.1 (p. 5). Take the causal filter h=(1,2,-1,1), with taps at 0,1,2,3, acting by y[n]=∑_qh[q]f[n-q]. Resolve the input into even and odd channels. Let D denote one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contribution to","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-c5d585ea90c121bc","tex":"\\bm E","aliases":"E bm","display":"\\bm E","kind":"introduction-passage","excerpt":"[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original sampl","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"introduction-passage","excerpt":"of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells. headWhy the coordinate convention matter","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"introduction-passage","excerpt":"ix of actual delaysFIR TRANSFER / A FOUR-TAP EXAMPLETraditional monograph: Theorem 3.1 (p. 5). Take the causal filter h=(1,2,-1,1), with taps at 0,1,2,3, acting by y[n]=∑_qh[q]f[n-q]. Resolve the input into even and odd channels. Let D denote one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contribution to","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"Let 1≤ p≤∞. For a countable set G, write ℓ^p(G) for the usual complex sequence space. For an integer d≥1, when G= Z^d, translation by a∈ Z^d is (T_ax)(n)=x(n-a). Let δ_0 be the impulse at the origin.","count":20},{"work":"paper-v","url":"notation/paper-v.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum; this proves it at depth ℓ+1.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum; this proves it at depth ℓ+1.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-097615d1cce1b4b2","tex":"H_b","aliases":"H_b","display":"H_b","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum; this proves it at depth ℓ+1.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"1-w · 1+w. vfigprojective_poles headThe action is defined before a ratio is taken The determinant is w, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives A(w)A(w')=A(ww'), A(1)=I. For w=i, the orbit through zero is [0:1] ⟼ [-i:1] ⟼ [1:0] ⟼ [i:1] ⟼ [0:1]. The third point is a legitimate projective point, even though a fin","count":52},{"work":"paper-v","url":"notation/paper-v.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"introduction-passage","excerpt":"[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original sampl","count":10},{"work":"paper-v","url":"notation/paper-v.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":14},{"work":"paper-v","url":"notation/paper-v.html#symbol-ab5e8580e0cfc647","tex":"J_y","aliases":"J_y","display":"J_y","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.","count":92},{"work":"paper-v","url":"notation/paper-v.html#symbol-2dcbb7ca5126e8ca","tex":"L_y","aliases":"L_y","display":"L_y","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-a0d3a3f2f247ab07","tex":"\\one_{[\\tau,\\infty)}","aliases":"one_[τ,∞) 1_[τ,∞) one tau infty mathbf tau infty","display":"{\\mathbf 1}_{[\\tau,\\infty)}","kind":"introduction-passage","excerpt":"The adjacent difference detector is local in the sample domain. Wavelet coefficients and higher differences give related multiscale packets. In the Haar subsections the signal is defined on R. Here the step convention is H_τ(t)= 1_[τ,∞)(t), whose restriction to (0,1) agrees with the preceding interval step. A chosen extension beyond the observation","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-bab84a59440e63c0","tex":"P_{2,0}","aliases":"P_2,0","display":"P_{2,0}","kind":"introduction-passage","excerpt":"vpage4Refine packets on a common latticeEXACT REFINEMENT / WHAT CAN BE ADDEDTraditional monograph: Theorem 2.3 (p. 5). An even sample belongs to exactly one of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original s","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-91a55f7626b81c45","tex":"P_{4,0}","aliases":"P_4,0","display":"P_{4,0}","kind":"introduction-passage","excerpt":"vpage4Refine packets on a common latticeEXACT REFINEMENT / WHAT CAN BE ADDEDTraditional monograph: Theorem 2.3 (p. 5). An even sample belongs to exactly one of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original s","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-a6b3605e1b2fcc56","tex":"P_{4,2}","aliases":"P_4,2","display":"P_{4,2}","kind":"introduction-passage","excerpt":"vpage4Refine packets on a common latticeEXACT REFINEMENT / WHAT CAN BE ADDEDTraditional monograph: Theorem 2.3 (p. 5). An even sample belongs to exactly one of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original s","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-90fba425db5ca238","tex":"P_{N,k}","aliases":"P_N,k","display":"P_{N,k}","kind":"introduction-passage","excerpt":"[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-09523592e6cc4b12","tex":"P_{N,s}","aliases":"P_N,s","display":"P_{N,s}","kind":"introduction-passage","excerpt":"[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-cf956bb3d74ebbc2","tex":"\\bm R","aliases":"R bm","display":"\\bm R","kind":"introduction-passage","excerpt":"[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original sampl","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-a3c72cb58f67a4d1","tex":"R_y","aliases":"R_y","display":"R_y","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-a62a596621079394","tex":"\\RR","aliases":"RR R mathbb","display":"{\\mathbb R}","kind":"introduction-passage","excerpt":"The adjacent difference detector is local in the sample domain. Wavelet coefficients and higher differences give related multiscale packets. In the Haar subsections the signal is defined on R. Here the step convention is H_τ(t)= 1_[τ,∞)(t), whose restriction to (0,1) agrees with the preceding interval step. A chosen extension beyond the observation","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-90f70c36f812714e","tex":"S^n","aliases":"S^n","display":"S^n","kind":"introduction-passage","excerpt":"Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primiti","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"Let 1≤ p≤∞. For a countable set G, write ℓ^p(G) for the usual complex sequence space. For an integer d≥1, when G= Z^d, translation by a∈ Z^d is (T_ax)(n)=x(n-a). Let δ_0 be the impulse at the origin.","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"introduction-passage","excerpt":"Let 1≤ p≤∞. For a countable set G, write ℓ^p(G) for the usual complex sequence space. For an integer d≥1, when G= Z^d, translation by a∈ Z^d is (T_ax)(n)=x(n-a). Let δ_0 be the impulse at the origin.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-b99d2e5f37ae7e17","tex":"w'","aliases":"w'","display":"w'","kind":"introduction-passage","excerpt":"1-w · 1+w. vfigprojective_poles headThe action is defined before a ratio is taken The determinant is w, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives A(w)A(w')=A(ww'), A(1)=I. For w=i, the orbit through zero is [0:1] ⟼ [-i:1] ⟼ [1:0] ⟼ [i:1] ⟼ [0:1]. The third point is a legitimate projective point, even though a fin","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-8bf07eb97189ba98","tex":"W_{\\ell,\\alpha}","aliases":"W_ℓ,α ell alpha","display":"W_{\\ell,\\alpha}","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum; this proves it at depth ℓ+1.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":51},{"work":"paper-v","url":"notation/paper-v.html#symbol-f998b273f69a0c77","tex":"z^{-d}","aliases":"z^-d","display":"z^{-d}","kind":"introduction-passage","excerpt":"[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original sampl","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-ae5d1ee4ca82d29f","tex":"\\delta_n","aliases":"δ_n delta","display":"\\delta_n","kind":"introduction-passage","excerpt":"Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primiti","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":17},{"work":"paper-v","url":"notation/paper-v.html#symbol-565b5cc28a0a3bd8","tex":"\\ell^p","aliases":"ℓ^p ell","display":"\\ell^p","kind":"introduction-passage","excerpt":"Let 1≤ p≤∞. For a countable set G, write ℓ^p(G) for the usual complex sequence space. For an integer d≥1, when G= Z^d, translation by a∈ Z^d is (T_ax)(n)=x(n-a). Let δ_0 be the impulse at the origin.","count":9},{"work":"paper-v","url":"notation/paper-v.html#symbol-4886267106f9df63","tex":"\\sum_{k,s}","aliases":"∑_k,s sum","display":"\\sum_{k,s}","kind":"introduction-passage","excerpt":"[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-366ac52aad5d30c9","tex":"\\sum_q","aliases":"∑_q sum","display":"\\sum_q","kind":"introduction-passage","excerpt":"vpage5A filter becomes a matrix of actual delaysFIR TRANSFER / A FOUR-TAP EXAMPLETraditional monograph: Theorem 3.1 (p. 5). Take the causal filter h=(1,2,-1,1), with taps at 0,1,2,3, acting by y[n]=∑_qh[q]f[n-q]. Resolve the input into even and odd channels. Let D denote one delay in the channel index. vfigfir_transfer headReading one matrix entry ","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum; this proves it at depth ℓ+1.","count":32},{"work":"paper-v","url":"notation/paper-v.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"introduction-passage","excerpt":"1-w · 1+w. vfigprojective_poles headThe action is defined before a ratio is taken The determinant is w, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives A(w)A(w')=A(ww'), A(1)=I. For w=i, the orbit through zero is [0:1] ⟼ [-i:1] ⟼ [1:0] ⟼ [i:1] ⟼ [0:1]. The third point is a legitimate projective point, even though a fin","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-776b5d9d1f9b084d","tex":"\\operatorname*","aliases":"* operatorname","display":"\\operatorname*","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-8c33d6b793a3c093","tex":"+^{m-\\ell}","aliases":"+^m-ℓ ell","display":"+^{m-\\ell}","kind":"use","excerpt":"For 0≤ ℓ≤ m, the classical derivative of ϕ_m,τ on the two open sides is ϕ_m,τ^(ℓ)(t) = frac(t-τ)_+^m-ℓ(m-ℓ)!. This derivative has zero jump when ℓ<m and a unit jump when ℓ=m. Derivatives of order larger than m vanish on both open sides. It follows that subtracting the term a_q,mϕ_m,τ_q removes exactly the jump in the derivative of order m without c","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].","count":13},{"work":"paper-v","url":"notation/paper-v.html#symbol-c4a7b03f010bc3fc","tex":"\\mathbf1_{[\\tau,1]}","aliases":"1_[τ,1] mathbf tau","display":"\\mathbf1_{[\\tau,1]}","kind":"use","excerpt":"vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPTraditional monograph: Theorem E7.2 (p. 56), Corollary E7.3 (p. 57). On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-5397b370d7b6d1d0","tex":"\\mathbf1_{[\\tau,\\infty)}","aliases":"1_[τ,∞) mathbf tau infty","display":"\\mathbf1_{[\\tau,\\infty)}","kind":"use","excerpt":"vpage37Haar sees a jump only through its position in the supportSCALE SEPARATION / A QUANTIFIED THRESHOLDTraditional monograph: Theorem E10.1 (p. 68), Theorem E10.2 (p. 69), Corollary E10.3 (p. 69). Use a normalized Haar wavelet on an available interval of length L on the real line, positive on its left half and negative on its right half. Here H_τ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e24b41e93a816fc5","tex":"\\mathbf1_{n\\equiv r\\, (\\mathrm{mod}\\,N)}","aliases":"1_n≡ r (mod N) mathbf equiv mathrm","display":"\\mathbf1_{n\\equiv r\\, (\\mathrm{mod}\\,N)}","kind":"use","excerpt":"Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-7df106ca60925507","tex":"\\mathbf1_{\\widehat\\Omega}","aliases":"1_widehatΩ mathbf widehat Omega","display":"\\mathbf1_{\\widehat\\Omega}","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-df247a376c954e8d","tex":"\\mathbf1_{\\{|t|<1/4\\}}","aliases":"1_{|t|<1/4} mathbf","display":"\\mathbf1_{\\{|t|<1/4\\}}","kind":"use","excerpt":"vpage16Why a Fourier cutoff cannot reproduce this jump pointwiseA CONTINUOUS PULSE / EXACT ENDPOINT VALUESTraditional monograph: Section 6 (p. 11), Section 8 (p. 13). Use a separate period-one example on [-1/2,1/2): j(t)= 1_{|t|<1/4}, S_mj(t)=tfrac12+frac2π∑_n=1^mfracsin(π n/2)ncos(2π nt). This is the exact finite Fourier formula. The drawn curves ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-8cda8a6d7191dd29","tex":"\\mathbf1_\\Omega","aliases":"1_Ω mathbf Omega","display":"\\mathbf1_\\Omega","kind":"use","excerpt":"Write a 1_Ω-widehat a 1_widehatΩ =(a-widehat a) 1_Ω +widehat a( 1_Ω- 1_widehatΩ). Apply the triangle inequality and use 1_Ω- 1_widehatΩ _L^p =μ(ΩtrianglewidehatΩ)^1/p. For the supremum estimate, |1_Ω-1_widehatΩ|≤1 pointwise, and this function vanishes everywhere when the symmetric difference is empty. The same pointwise bound also bounds the essent","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-f0a175c4c1c25e09","tex":"\\mathbf1_{\\Omega}","aliases":"1_Ω mathbf Omega","display":"\\mathbf1_{\\Omega}","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-473264ce94dccbac","tex":"\\mathbf1_{\\Omega_q}","aliases":"1_Ω_q mathbf Omega","display":"\\mathbf1_{\\Omega_q}","kind":"use","excerpt":"[Edge-packet approximation class] There should exist natural function spaces in which a signal is represented as f=s+∑_q a_q1_Ω_q, with separate norms for smooth approximation, boundary geometry, packet amplitude, and packet identification. A projector-resolved approximation theory on such spaces should yield rates that do not mix smooth truncation","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-da9ba20262f8dd56","tex":"\\mathbf1_{\\Omega\\triangle\\widehat\\Omega\\ne\\varnothing}","aliases":"1_ΩtrianglewidehatΩ≠∅ mathbf Omega triangle widehat Omega ne varnothing","display":"\\mathbf1_{\\Omega\\triangle\\widehat\\Omega\\ne\\varnothing}","kind":"use","excerpt":"orms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise supremum, and consequently also the essential L^∞ seminorm, j-widehat j _∞ ≤ |a-widehat a|+|widehat","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"ump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector P_4,rf is different: it keeps the original latt","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"Section 9 (p. 15). On the integer 8×8 grid, choose the disk-shaped mask Ω={(i,j):(i-3)^2+(j-3)^2≤5} and translate it one pixel to the right. Each mask contains 21 pixels; their symmetric difference contains exactly 10. vfigmask_geometry headA geometric error calculation is not a segmentation algorithm For a common amplitude a, the squared counting-","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-0324c102d49c31af","tex":"A^q","aliases":"A^q","display":"A^q","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"nch entirely. For actual Haar packets, set P_v=W_v^*W_v. Parent-to-child conservation gives P_v=P_v0+P_v1. Replacing unselected parents by their children until the complete pruning is reached proves ∑_v∈ΛP_v=I. This is why the coefficients at different depths still form an exact coordinate system. headAn exhaustive finite check At depth three the p","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"STANTS / A MIXED-DEPTH CERTIFICATETraditional monograph: Proposition E6.4 (p. 53). Suppose the root split has nonnegative squared-norm bounds A=1/2 and B=2. Retain the left child. Split the right child with bounds 3/4 and 5/4, then retain both grandchildren. vfig[.85]ext_tree headStart with one at every retained node The left subtree has bounds a_0","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"le(4.199999999999999,2.0);node at(3.5,1.5) 1 ; node[left]at(0,1.5)a_1=2; filldraw[fill=orange!25,draw=white](0.0,0.0)rectangle(1.4,1.0);node at(0.7,0.5) 1 ; filldraw[fill=black!3,draw=white](1.4,0.0)rectangle(2.8,1.0);node at(2.0999999999999996,0.5) 0 ; filldraw[fill=teal!12,draw=white](2.8,0.0)rectangle(4.199999999999999,1.0);node at(3.5,0.5) 4 ; ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-60fcf83a29fad2ee","tex":"\\mathcal A_4","aliases":"A_4 mathcal","display":"\\mathcal A_4","kind":"use","excerpt":"ump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector P_4,rf is different: it keeps the original latt","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-4b862c67265e36c7","tex":"a_{d+1}","aliases":"a_d+1","display":"a_{d+1}","kind":"use","excerpt":". For actual Haar packets, set P_v=W_v^*W_v. Parent-to-child conservation gives P_v=P_v0+P_v1. Replacing unselected parents by their children until the complete pruning is reached proves ∑_v∈ΛP_v=I. This is why the coefficients at different depths still form an exact coordinate system. headAn exhaustive finite check At depth three the pruning count","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-3652828b23145ef1","tex":"a_d^2","aliases":"a_d^2","display":"a_d^2","kind":"use","excerpt":". For actual Haar packets, set P_v=W_v^*W_v. Parent-to-child conservation gives P_v=P_v0+P_v1. Replacing unselected parents by their children until the complete pruning is reached proves ∑_v∈ΛP_v=I. This is why the coefficients at different depths still form an exact coordinate system. headAn exhaustive finite check At depth three the pruning count","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"Linear time invariance is not required for exact packet coordinates. Let A and B be formal power series over a commutative coefficient ring (in particular C). Set A(z)=∑_n≥0a_nz^n, A_i=P_N,iA. All phase indices range from 0 to N-1. For another representation B(z), multiplication gives a nonlinear or bilinear interaction.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"Let (E_i)_i=0^I and (F_j)_j=0^J be measurable partitions of the same domain, up to null sets. Let a_i and b_j be the corresponding values. The index zero may be used for a background region.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-1cea62f3228b8c95","tex":"A_{i_1}","aliases":"A_i_1","display":"A_{i_1}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-de92b398cc45688a","tex":"A_{i_q}","aliases":"A_i_q","display":"A_{i_q}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j,","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"[Background and packet error] Suppose the true and estimated packets use the same amplitude list (a_i)_i=0^I, but possibly different regions. Then j-widehat j _p^p = ∑_i,k|a_i-a_k|^pμ(E_i∩ F_k). Only off diagonal overlaps contribute.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-7dcee546650900ca","tex":"a_{\\min}","aliases":"a_min min","display":"a_{\\min}","kind":"use","excerpt":"The principal recovery theorem uses adjacent sample contrasts. Let b_h=Mh+2ε be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If a_min>2b_h, then every threshold between b_h and a_min-b_h detects exactly the cells containing jumps. The jump locations are recovered within h/2, and each amplit","count":17},{"work":"paper-v","url":"notation/paper-v.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"A_n · S^n · address operator","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"Linear time invariance is not required for exact packet coordinates. Let A and B be formal power series over a commutative coefficient ring (in particular C). Set A(z)=∑_n≥0a_nz^n, A_i=P_N,iA. All phase indices range from 0 to N-1. For another representation B(z), multiplication gives a nonlinear or bilinear interaction.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-394b9fed27d5557c","tex":"a_{q,m}","aliases":"a_q,m","display":"a_{q,m}","kind":"use","excerpt":"egintheorem[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one dimensional Sobolev representatives whose derivatives through order r-1 have one sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) an","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-d71f286ebb4773fa","tex":"a_{\\rm root}","aliases":"a_rm root rm","display":"a_{\\rm root}","kind":"use","excerpt":"Proposition E6.4 (p. 53). Suppose the root split has nonnegative squared-norm bounds A=1/2 and B=2. Retain the left child. Split the right child with bounds 3/4 and 5/4, then retain both grandchildren. vfig[.85]ext_tree headStart with one at every retained node The left subtree has bounds a_0=b_0=1. The right subtree has a_1=3/4 and b_1=5/4. The re","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-0083f442b19187bb","tex":"A_v","aliases":"A_v","display":"A_v","kind":"use","excerpt":"[Bounds for nonunitary local splits] Suppose every internal node has an analysis map U_v and constants 0≤ A_v≤ B_v<∞ satisfying A_v x ^2 ≤ ∑_w∈child(v) (U_vx)_w ^2 ≤ B_v x ^2. Define recursively a(v)= 1, · v retained,","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-767a2fc9b6c07bff","tex":"\\Acal_E","aliases":"Acal_E A_E Acal mathcal","display":"{\\mathcal A}_E","kind":"use","excerpt":"[Frame bounds from the polyphase symbol] Assume that A I_N ≤ E( e^ iω)^*E( e^ iω) ≤ B I_N for almost every ω, where 0<A≤ B<∞. Then the analysis bank satisfies A x _2^2 ≤ A_E x _2^2 ≤ B x _2^2. The pointwise matrix R_rm can( e^ iω) = (E^*E)^-1E^* is a bounded left inverse and gives the canonical dual synthesis bank.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-bafde82ae4b33f78","tex":"\\Acal_L^{-1}","aliases":"Acal_L^-1 A_L^-1 Acal mathcal","display":"{\\mathcal A}_L^{-1}","kind":"use","excerpt":"[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-673f6a95cc986ca7","tex":"\\Acal_{\\Lcal}","aliases":"Acal_Lcal A_ L Acal Lcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal L}}","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-dab36f95bbad8317","tex":"\\Acal_{\\Lcal}^{-1}","aliases":"Acal_Lcal^-1 A_ L^-1 Acal Lcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal L}}^{-1}","kind":"use","excerpt":"The synthesis map is unitary by [source reference]. Hence widehat x-x = A_ L^-1e has the same norm as e.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let B= R E-cz^-dI_N and suppose its action vanishes on every channel sequence. Insert an impulse at channel s, address zero, with every other channel zero. Output channel k reads the coefficie","count":28},{"work":"paper-v","url":"notation/paper-v.html#symbol-b3d4d90b8afc3ca5","tex":"\\bm B","aliases":"B bm","display":"\\bm B","kind":"use","excerpt":"Finite convolution represents Laurent-polynomial multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let B= R E-cz^-dI_N and suppose its action vanishes on every channel sequence. Insert an impulse at channel s, address zero, with every other c","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"ate along branchesLOCAL FRAME CONSTANTS / A MIXED-DEPTH CERTIFICATETraditional monograph: Proposition E6.4 (p. 53). Suppose the root split has nonnegative squared-norm bounds A=1/2 and B=2. Retain the left child. Split the right child with bounds 3/4 and 5/4, then retain both grandchildren. vfig[.85]ext_tree headStart with one at every retained nod","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"ED-DEPTH CERTIFICATETraditional monograph: Proposition E6.4 (p. 53). Suppose the root split has nonnegative squared-norm bounds A=1/2 and B=2. Retain the left child. Split the right child with bounds 3/4 and 5/4, then retain both grandchildren. vfig[.85]ext_tree headStart with one at every retained node The left subtree has bounds a_0=b_0=1. The ri","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"raw=white](0.0,0.0)rectangle(1.4,1.0);node at(0.7,0.5) 1 ; filldraw[fill=black!3,draw=white](1.4,0.0)rectangle(2.8,1.0);node at(2.0999999999999996,0.5) 0 ; filldraw[fill=teal!12,draw=white](2.8,0.0)rectangle(4.199999999999999,1.0);node at(3.5,0.5) 4 ; node[left]at(0,0.5)a_2=5; node[above]at(0.7,3)b_0=0;node[above]at(2.0999999999999996,3)b_1=2;node[","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c69ec0fece4f3aa0","tex":"b_h","aliases":"b_h","display":"b_h","kind":"use","excerpt":"The principal recovery theorem uses adjacent sample contrasts. Let b_h=Mh+2ε be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If a_min>2b_h, then every threshold between b_h and a_min-b_h detects exactly the cells containing jumps. The jump locations are recovered within h/2, and each amplit","count":31},{"work":"paper-v","url":"notation/paper-v.html#symbol-d1d859d024fbe063","tex":"b_{h,\\delta}","aliases":"b_h,δ delta","display":"b_{h,\\delta}","kind":"use","excerpt":"[Probability recovery bound] Fix 0<δ<1 and define t_n,δ = 2σ sqrtlog(frac2nδ) and b_h,δ=Mh+t_n,δ. With probability at least 1-δ, every contrast satisfies the conclusions of [source reference] with b_h replaced by b_h,δ.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"Let (E_i)_i=0^I and (F_j)_j=0^J be measurable partitions of the same domain, up to null sets. Let a_i and b_j be the corresponding values. The index zero may be used for a background region.","count":12},{"work":"paper-v","url":"notation/paper-v.html#symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","excerpt":"Apply [source reference] with b_k=a_k. The diagonal terms vanish.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-5464c5f65d3b3c2c","tex":"B_{ks}","aliases":"B_ks","display":"B_{ks}","kind":"use","excerpt":"multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let B= R E-cz^-dI_N and suppose its action vanishes on every channel sequence. Insert an impulse at channel s, address zero, with every other channel zero. Output channel k reads the coefficie","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-442ac47b1791bc80","tex":"b_{q,m}","aliases":"b_q,m","display":"b_{q,m}","kind":"use","excerpt":"[Uniqueness of the trace packet] Fix the breakpoints τ_q. If f=s+∑_q=1^Q∑_m=0^r-1b_q,mϕ_m,τ_q with s∈ W^r,p(0,1), then b_q,m=jumpf^(m)_τ_q. The finite truncated power packet is therefore the unique packet supported at the declared breakpoints that removes all trace jumps through order r-1.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-be4b228c950e38a7","tex":"b_{\\rm root}","aliases":"b_rm root rm","display":"b_{\\rm root}","kind":"use","excerpt":"Proposition E6.4 (p. 53). Suppose the root split has nonnegative squared-norm bounds A=1/2 and B=2. Retain the left child. Split the right child with bounds 3/4 and 5/4, then retain both grandchildren. vfig[.85]ext_tree headStart with one at every retained node The left subtree has bounds a_0=b_0=1. The right subtree has a_1=3/4 and b_1=5/4. The re","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-aba34f1446cfa879","tex":"B_v","aliases":"B_v","display":"B_v","kind":"use","excerpt":"[Bounds for nonunitary local splits] Suppose every internal node has an analysis map U_v and constants 0≤ A_v≤ B_v<∞ satisfying A_v x ^2 ≤ ∑_w∈child(v) (U_vx)_w ^2 ≤ B_v x ^2. Define recursively a(v)= 1, · v retained,","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-e5e4e7017d9b1a9f","tex":"\\bbN","aliases":"bbN N mathbb","display":"{\\mathbb N}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-5bd492e4ea78c5fe","tex":"\\bigsqcup","aliases":"bigsqcup","display":null,"kind":"use","excerpt":"Let L≤ Z^d be a sublattice of finite index q. Choose a set R⊂ Z^d containing one representative from every coset in Z^d/L. Thus Z^d=bigsqcup_r∈ R(r+L). For x: Z^d→ C, its channel at r is the sequence on L x_r(ℓ)=x(r+ℓ).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a44cd5763af08d17","tex":"\\bigsqcup_{r\\in R}","aliases":"bigsqcup_r∈ R bigsqcup in","display":null,"kind":"use","excerpt":"Let L≤ Z^d be a sublattice of finite index q. Choose a set R⊂ Z^d containing one representative from every coset in Z^d/L. Thus Z^d=bigsqcup_r∈ R(r+L). For x: Z^d→ C, its channel at r is the sequence on L x_r(ℓ)=x(r+ℓ).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"one also has A_N S_N=I. Equip this finite product with the Hilbert norm (x_0,…,x_N-1) _ch^2 =∑_r=0^N-1 x_r _2^2. Then A_N is a linear isometric equivalence and x _2^2=∑_r=0^N-1 x_r _2^2. Euclidean division gives a unique pair (r,q) with 0≤ r<N and n=Nq+r, including negative n. The address maps n↦(nbmod N,nmathbindivN) and (r,q)↦ Nq+r are inverse. A","count":10},{"work":"paper-v","url":"notation/paper-v.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[One-dimensional errors] For 1024 samples on [-1,1) and Fourier cutoff m=18, the retained frequency indices are -18,…,18: 37 bins including the zero-frequency bin. The measured values were lrr method · RMSE · maximum absolute error global Fourier truncation · 5.4023×10^-2 · 3.4968×10^-1 known edge packet · 1.3692×10^-4 · 1.4503×10^-3 estimated edge","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-75693dc2c5a4fbed","tex":"\\BV","aliases":"BV operatorname","display":"{\\operatorname{BV}}","kind":"use","excerpt":"The higher order singular packet theorem removes finitely many trace jumps. A more complete approximation theory should connect packet norms to geometric measure quantities in BV, piecewise Sobolev spaces, and free discontinuity models. It should also determine when a redundant packet dictionary can identify the derivative order of a singularity ra","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"use","excerpt":"Let C(z)=(z-1)/(z+1). Then T_N,k=C^-1∘ R_N,k∘ C. For a division-free construction, represent a point by a nonzero pair [u:v], identifying [cu:cv]=[u:v] for c≠0. With w=ω_N^k, use the matrix A(w)=frac121+w · 1-w","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j,","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-bb134dc373ba038c","tex":"C_Q","aliases":"C_Q","display":"C_Q","kind":"use","excerpt":"[Estimated packet Gibbs bound] Let Q_m be linear and uniformly bounded on L^p, with sup_m Q_m ≤ C_Q. For f=s+j and an estimated packet widehat j_m, define widehat f_m =Q_m(f-widehat j_m)+widehat j_m. Then f-widehat f_m _p ≤ s-Q_ms _p +(1+C_Q) j-widehat j_m _p. Consequently, smooth approximation and packet recovery may be improved independently. No ","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-53faef096d1a0acc","tex":"c_v","aliases":"c_v","display":"c_v","kind":"use","excerpt":"[Dynamic-programming best packet basis] Suppose J_y(Λ)=∑_v∈Λc_v(y). For a node v, define recursively V_y(v) = c_v(y), · v terminal,","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-82497562659dda68","tex":"\\cA","aliases":"cA A mathcal","display":"{\\mathcal A}","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-4a7d8b938ce09cb1","tex":"\\cA_N","aliases":"cA_N A_N cA mathcal","display":"{\\mathcal A}_N","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-21355a3045e08010","tex":"\\mathrm{can}","aliases":"can mathrm","display":"\\mathrm{can}","kind":"use","excerpt":"[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"use","excerpt":"[Finite periodic signals] Let G= Z^d/L, where L≤ Z^d has finite index. The translation orbit of the impulse at the identity is a basis of C^G. In particular, one impulse generates every signal on the finite periodic lattice.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-274a6d2d70507826","tex":"\\CC^G","aliases":"CC^G C^G CC mathbb","display":"{\\mathbb C}^G","kind":"use","excerpt":"[Finite periodic signals] Let G= Z^d/L, where L≤ Z^d has finite index. The translation orbit of the impulse at the identity is a basis of C^G. In particular, one impulse generates every signal on the finite periodic lattice.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-965c9524c8144bbf","tex":"\\mathrm{ch}","aliases":"ch mathrm","display":"\\mathrm{ch}","kind":"use","excerpt":"[Analysis--synthesis identity] S_N A_N=I. On the product of the channel spaces one also has A_N S_N=I. Equip this finite product with the Hilbert norm (x_0,…,x_N-1) _ch^2 =∑_r=0^N-1 x_r _2^2. Then A_N is a linear isometric equivalence and x _2^2=∑_r=0^N-1 x_r _2^2. Euclidean division gives a unique pair (r,q) with 0≤ r<N and n=Nq+r, including negat","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ccb8413fea741c57","tex":"\\operatorname{Child}","aliases":"Child operatorname","display":"\\operatorname{Child}","kind":"use","excerpt":"Suppose N,M≥1 and Nmid M. Every coarse phase has a disjoint family of fine descendants: Child_M(N,r) = {s:0≤ s<M, s≡ rpmod N}.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-025cd74028e154d2","tex":"\\operatorname{Child}_{M}","aliases":"Child_M operatorname","display":"\\operatorname{Child}_{M}","kind":"use","excerpt":"Suppose N,M≥1 and Nmid M. Every coarse phase has a disjoint family of fine descendants: Child_M(N,r) = {s:0≤ s<M, s≡ rpmod N}.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-20d41ee7317fddd1","tex":"\\cS","aliases":"cS S mathcal","display":"{\\mathcal S}","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-296d2757843f9fef","tex":"\\cS_N","aliases":"cS_N S_N cS mathcal","display":"{\\mathcal S}_N","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-7c6669ab098dda59","tex":"\\mathbb D","aliases":"D mathbb","display":"\\mathbb D","kind":"use","excerpt":"Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-fd32e05fe3a41d5b","tex":"D^{2}","aliases":"D^2","display":"D^{2}","kind":"use","excerpt":"node[] at (6.30000,4.60000) h=(1,2,-1,1), D: u[n]↦ u[n-1]; node[anchor=east] at (1.80000,3.10000) output 0; draw[fill=black!0,draw=black!40,line width=.35pt] (2.20000,2.55000) rectangle ++(4.10000,1.05000); node[] at (4.25000,3.07500) 1 - D; draw[fill=black!14,draw=black!40,line width=.35pt] (6.30000,2.55000) rectangle ++(4.10000,1.05000); node[] a","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-d705f2d0b7a4b58f","tex":"D^q","aliases":"D^q","display":"D^q","kind":"use","excerpt":"matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers. headSigned delays need the same care For an original-sample delay d at modulus N, y_k[n]=f_(k-d)bmod N [n+lfloorfrack-dNrfloor]. This includes advances when d<0. The finite verification checks the formula on signed sample addresses. A bank with product ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a572acccc9b7cf05","tex":"D_k","aliases":"D_k","display":"D_k","kind":"use","excerpt":"me f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Define the adjacent contrast D_k=y_k+1-y","count":10},{"work":"paper-v","url":"notation/paper-v.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd d mathrm","display":"{\\,\\mathrm d}","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"finite kernels, composition requires only finitely many operations at each address. The converse uses impulses. Insert an impulse in one input channel. Each output channel lists every coefficient of the corresponding transfer polynomial. If all such responses vanish for RE-I, every coefficient is zero; hence the polynomial matrix itself is zero. he","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-558c1dc971656de6","tex":"\\dist","aliases":"dist operatorname","display":"{\\operatorname{dist}}","kind":"use","excerpt":"For a closed set Γ, write N_δ(Γ) = {x:dist(x,Γ)<δ}.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-fed34424e9d0e3f5","tex":"\\mathrm{div}","aliases":"div mathrm","display":"\\mathrm{div}","kind":"use","excerpt":"one also has A_N S_N=I. Equip this finite product with the Hilbert norm (x_0,…,x_N-1) _ch^2 =∑_r=0^N-1 x_r _2^2. Then A_N is a linear isometric equivalence and x _2^2=∑_r=0^N-1 x_r _2^2. Euclidean division gives a unique pair (r,q) with 0≤ r<N and n=Nq+r, including negative n. The address maps n↦(nbmod N,nmathbindivN) and (r,q)↦ Nq+r are inverse. A","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-0dcf6a495ccbd5bd","tex":"\\e","aliases":"e mathrm","display":"{\\mathrm e}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":16},{"work":"paper-v","url":"notation/paper-v.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"Split x into even and odd channels e,o. Define d=o-e, s=e+frac12d. The inverse is e=s-frac12d, o=d+e.","count":46},{"work":"paper-v","url":"notation/paper-v.html#symbol-ba01bfed40e94931","tex":"E^*","aliases":"E^*","display":"E^*","kind":"use","excerpt":"[Frame bounds from the polyphase symbol] Assume that A I_N ≤ E( e^ iω)^*E( e^ iω) ≤ B I_N for almost every ω, where 0<A≤ B<∞. Then the analysis bank satisfies A x _2^2 ≤ A_E x _2^2 ≤ B x _2^2. The pointwise matrix R_rm can( e^ iω) = (E^*E)^-1E^* is a bounded left inverse and gives the canonical dual synthesis bank.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-ecea398e87a0ddcd","tex":"e^{-2\\ii\\beta\\ell}","aliases":"e^-2iiβℓ e^-2 iβℓ ii beta ell mathrm beta ell","display":"e^{-2{\\mathrm i}\\beta\\ell}","kind":"use","excerpt":"For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ed31a34ba88a7364","tex":"e^{10}","aliases":"e^10","display":"e^{10}","kind":"use","excerpt":"1/2, σ=1/50, δ=1/20, and a_min=2/25. Since log(10240)<10<(16/5)^2, choosing R=16 makes the right-hand side strictly less than frac1256+frac150frac165 =frac217332000<frac225. Thus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies e","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b52fa662abced8d7","tex":"\\e^{2\\pi\\ii k/N}","aliases":"e^2πii k/N e^2π i k/N e pi ii mathrm pi mathrm","display":"{\\mathrm e}^{2\\pi{\\mathrm i} k/N}","kind":"use","excerpt":"+1) into C^-1 and simplify. Preservation of H follows because C maps it to D, rotation preserves D, and C^-1 maps back. The group law follows by conjugating the rotation group law. When θ=2π/N, the Nth power is the identity. Conversely, if its kth power is the identity, conjugation by C and evaluation at w=1/2 give e^2π i k/N=1, so N divides k. Hen","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b42132942b4d3093","tex":"e^{2\\pi\\ii/N}","aliases":"e^2πii/N e^2π i/N pi ii pi mathrm","display":"e^{2\\pi{\\mathrm i}/N}","kind":"use","excerpt":"Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-8618817d79a78398","tex":"\\e^{\\ii d\\omega}","aliases":"e^ii dω e^ i dω e ii omega mathrm mathrm omega","display":"{\\mathrm e}^{{\\mathrm i} d\\omega}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ec5335d70f84b91e","tex":"\\e^{\\ii(\\varphi+\\theta)}","aliases":"e^ii(φ+θ) e^ i(φ+θ) e ii varphi theta mathrm mathrm varphi theta","display":"{\\mathrm e}^{{\\mathrm i}(\\varphi+\\theta)}","kind":"use","excerpt":"[Radial and phase coordinates] Write C(z)=ρ e^ iφ, 0≤ρ<1. Then C(Φ_θ(z)) = ρ e^ i(φ+θ). The radial mismatch ρ is invariant and the phase coordinate advances by θ.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-638797be2d36d412","tex":"\\e^{\\ii\\theta}","aliases":"e^iiθ e^ iθ e ii theta mathrm mathrm theta","display":"{\\mathrm e}^{{\\mathrm i}\\theta}","kind":"use","excerpt":"Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-18321ae54a5eac88","tex":"e^{\\ii\\theta}","aliases":"e^iiθ e^ iθ ii theta mathrm theta","display":"e^{{\\mathrm i}\\theta}","kind":"use","excerpt":"For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-56b7423ff854480a","tex":"\\e^{\\ii\\pi}","aliases":"e^iiπ e^ iπ e ii pi mathrm mathrm pi","display":"{\\mathrm e}^{{\\mathrm i}\\pi}","kind":"use","excerpt":"Insert e^ iπ=-1 in the formula of [source reference].","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a246bb571b3cd9f5","tex":"\\e^{\\ii\\varphi}","aliases":"e^iiφ e^ iφ e ii varphi mathrm mathrm varphi","display":"{\\mathrm e}^{{\\mathrm i}\\varphi}","kind":"use","excerpt":"[Radial and phase coordinates] Write C(z)=ρ e^ iφ, 0≤ρ<1. Then C(Φ_θ(z)) = ρ e^ i(φ+θ). The radial mismatch ρ is invariant and the phase coordinate advances by θ.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-afdbf103ea74d6de","tex":"\\e^{\\ii\\omega}","aliases":"e^iiω e^ iω e ii omega mathrm mathrm omega","display":"{\\mathrm e}^{{\\mathrm i}\\omega}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-6ca1e8040f46f1da","tex":"e^{\\lambda\\eta}","aliases":"e^λη lambda eta","display":"e^{\\lambda\\eta}","kind":"use","excerpt":"vpage36Repeated observations shrink the noise side of the gapSUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGETTraditional monograph: Theorem E9.8 (p. 67), Corollary E9.9 (p. 67), Corollary E9.11 (p. 67). Suppose all noise observations are independent and centered, with moment-generating bound E e^λη≤ e^σ^2λ^2/2. Averaging R observations at each o","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ac72f3950f989280","tex":"\\e^{\\lambda\\eta_k}","aliases":"e^λη_k e lambda eta mathrm lambda eta","display":"{\\mathrm e}^{\\lambda\\eta_k}","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-6cced37b7825af60","tex":"\\e^{\\sigma^2\\lambda^2/2}","aliases":"e^σ^2λ^2/2 e sigma lambda mathrm sigma lambda","display":"{\\mathrm e}^{\\sigma^2\\lambda^2/2}","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c2dc0e0b6c8f96f8","tex":"e^{\\sigma^2\\lambda^2/2}","aliases":"e^σ^2λ^2/2 sigma lambda","display":"e^{\\sigma^2\\lambda^2/2}","kind":"use","excerpt":"vpage36Repeated observations shrink the noise side of the gapSUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGETTraditional monograph: Theorem E9.8 (p. 67), Corollary E9.9 (p. 67), Corollary E9.11 (p. 67). Suppose all noise observations are independent and centered, with moment-generating bound E e^λη≤ e^σ^2λ^2/2. Averaging R observations at each o","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-92780dc9a20a0ca6","tex":"E_H","aliases":"E_H","display":"E_H","kind":"use","excerpt":"The Haar matrix is E_H = frac1sqrt2 1 · 1 1 · -1 . For even and odd samples e(n)=x(2n) and o(n)=x(2n+1), the low and high coordinates are a(n)=frace(n)+o(n)sqrt2, d(n)=frace(n)-o(n)sqrt2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-a0d586db24ff52c2","tex":"\\bm E_{\\mathrm{H}}","aliases":"E_H bm mathrm","display":"\\bm E_{\\mathrm{H}}","kind":"use","excerpt":"The Haar analysis matrix is E_H = frac1sqrt2 1 · 1 1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-6e30f119b7ece4c0","tex":"\\bm E_{\\mathrm{H}}^{\\mathsf T}","aliases":"E_H^ T bm mathrm mathsf","display":"\\bm E_{\\mathrm{H}}^{\\mathsf T}","kind":"use","excerpt":"1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"Let (E_i)_i=0^I and (F_j)_j=0^J be measurable partitions of the same domain, up to null sets. Let a_i and b_j be the corresponding values. The index zero may be used for a background region.","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-3aaf5c530b84ee89","tex":"\\esssup","aliases":"esssup *ess sup operatorname","display":"{\\operatorname*{ess\\,sup}}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-a6ab09549ed6fe5b","tex":"\\esssup_\\omega","aliases":"esssup_ω *ess sup_ω esssup omega operatorname omega","display":"{\\operatorname*{ess\\,sup}}_\\omega","kind":"use","excerpt":"0 · 0.02sinω . Then *ess sup_ω D( e^ iω) _op =0.02. Every input therefore has relative reconstruction error at most two percent in ℓ^2. The estimate is independent of the input length.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-01ad82b028fe9c1f","tex":"\\esssup_{\\omega}","aliases":"esssup_ω *ess sup_ω esssup omega operatorname omega","display":"{\\operatorname*{ess\\,sup}}_{\\omega}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"[Impedance--admittance is the two-phase split] For N=2 and finite z≠0, T_2,1(z)=z^-1. On the full projective line it exchanges 0 and ∞. Thus a half-turn of the reflection disk exchanges normalized impedance and admittance. ω_2=exp(π i)=-1, so A(ω_2)[u:v]=[v:u]. The finite ratio is 1/z when z≠0; the coordinate swap also gives [0:1]↔[1:0]. No ordinar","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":48},{"work":"paper-v","url":"notation/paper-v.html#symbol-6d014ef8b92c530c","tex":"f^{(m)}","aliases":"f^(m)","display":"f^{(m)}","kind":"use","excerpt":"The singular packet results begin with trace data. If a piecewise smooth function has a jump in its derivative of order m at τ_q, then the packet jumpf^(m)_τ_q frac(t-τ_q)_+^mm! removes exactly that trace jump. Subtracting all packets through order r-1 leaves a global W^r,p remainder. The packet coefficients are uniquely determined by the one sided","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-065a29e259e230ea","tex":"f^{(r)}","aliases":"f^(r)","display":"f^{(r)}","kind":"use","excerpt":"sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one sided derivatives through order r-1 at every breakpoint and belongs to W^r,p(0,1). Its weak derivative of order r equals the piecewise classical derivative f^(r) almost everywhere.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-d7fe97397b4335ab","tex":"f_{(k-d)\\bmod N}","aliases":"f_(k-d)bmod N bmod","display":"f_{(k-d)\\bmod N}","kind":"use","excerpt":"inal sample δ_1. Under DI_2 it stays even but moves to channel address one: this is δ_2. Thus J(D)^2=DI_2, H(D)=I_2+2J(D)-J(D)^2+J(D)^3 recovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers. headSigned delays need the same care For an original-sample delay d at","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-7ef89c7821d00e4a","tex":"F_*","aliases":"F_*","display":"F_*","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"[x=1cm,y=1cm,>=Latex,font=sffamilysmall,line cap=round,line join=round] node[] at (6.50000,4.60000) F(x)[n]=x[n]+tfrac12x[n]x[n-1]; draw[fill=black!10,draw=black!40,line width=.35pt] (0.60000,2.70000) rectangle ++(5.70000,1.00000); node[] at (3.45000,3.20000) F_0[n]=e[n]+tfrac12e[n]o[n-1]; draw[fill=black!10,draw=black!40,line width=.35pt] (6.80000","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"all,line cap=round,line join=round] node[] at (6.50000,4.60000) F(x)[n]=x[n]+tfrac12x[n]x[n-1]; draw[fill=black!10,draw=black!40,line width=.35pt] (0.60000,2.70000) rectangle ++(5.70000,1.00000); node[] at (3.45000,3.20000) F_0[n]=e[n]+tfrac12e[n]o[n-1]; draw[fill=black!10,draw=black!40,line width=.35pt] (6.80000,2.70000) rectangle ++(5.70000,1.000","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"Let (E_i)_i=0^I and (F_j)_j=0^J be measurable partitions of the same domain, up to null sets. Let a_i and b_j be the corresponding values. The index zero may be used for a background region.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-7e04443bf0b17f7b","tex":"F_k","aliases":"F_k","display":"F_k","kind":"use","excerpt":"[Background and packet error] Suppose the true and estimated packets use the same amplitude list (a_i)_i=0^I, but possibly different regions. Then j-widehat j _p^p = ∑_i,k|a_i-a_k|^pμ(E_i∩ F_k). Only off diagonal overlaps contribute.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e3f4ba757b6d2cf8","tex":"f_n","aliases":"f_n","display":"f_n","kind":"use","excerpt":"[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f7110079fe13a85a","tex":"F_r","aliases":"F_r","display":"F_r","kind":"use","excerpt":"For a finite nonlinear shift operator F(x)[n]=x[n]+α x[n]x[n-1], the exact packet coordinates are simply F_r((x_s)_s)=P_N,rF (∑_sP_N,sx). Analysis followed by this packet system and synthesis reproduces the direct nonlinear output exactly. The numerical residual in the companion implementation was zero to machine representation.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-15e892a32e1d61ee","tex":"g_n","aliases":"g_n","display":"g_n","kind":"use","excerpt":"[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-077ab2484c314309","tex":"G_{rs}","aliases":"G_rs","display":"G_{rs}","kind":"use","excerpt":"[Composition of transfer matrices] If G and H are convolution operators with packet matrices (G_rs) and (H_rs), then the packet matrix of GH is (GH)_rt = ∑_s∈ RG_rsH_st. In one dimensional transform notation, the corresponding matrix symbols multiply pointwise.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-6bcb32fb1f100c04","tex":"G_s","aliases":"G_s","display":"G_s","kind":"use","excerpt":"[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-0f341c3678cd53cc","tex":"\\Gam_L","aliases":"Gam_L Γ_L Gam Gamma","display":"{\\Gamma}_L","kind":"use","excerpt":"For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-422e0fea8cc00d51","tex":"\\Gam_{\\rm in}","aliases":"Gam_rm in Γ_rm in Gam rm Gamma rm","display":"{\\Gamma}_{\\rm in}","kind":"use","excerpt":"For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-de926b1a644e7db4","tex":"\\bm H","aliases":"H bm","display":"\\bm H","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-b957718497102933","tex":"\\mathbb H","aliases":"H mathbb","display":"\\mathbb H","kind":"use","excerpt":"Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-6a11c4bcc6a8c941","tex":"\\mathrm{H}","aliases":"H mathrm","display":"\\mathrm{H}","kind":"use","excerpt":"The Haar analysis matrix is E_H = frac1sqrt2 1 · 1 1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-0af9e4c11c585190","tex":"\\bm H^{(N)}","aliases":"H^(N) bm","display":"\\bm H^{(N)}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-9eca8df62b809141","tex":"H^{(N)}_{ks}","aliases":"H^(N)_ks","display":"H^{(N)}_{ks}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-39086c50a1b88b83","tex":"H_{1/3}","aliases":"H_1/3","display":"H_{1/3}","kind":"use","excerpt":"vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPTraditional monograph: Theorem E7.2 (p. 56), Corollary E7.3 (p. 57). On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-3ca843cc179ffa40","tex":"\\bm H_4","aliases":"H_4 bm","display":"\\bm H_4","kind":"use","excerpt":"[A four phase finite impulse response operator] Let h(0)=1, h(1)=2, h(2)=3, h(3)=4, h(4)=5, with all other taps zero. For four residue channels, the transfer symbol is H_4(z) = 1+5z^-1 · 4z^-1 · 3z^-1 · 2z^-1 2 · 1+5z^-1 · 4z^-1 · 3z^-1 3 · 2 · 1+5z^-1 · 4z^-1 4 · 3 · 2 · 1+5z^-1 . The delay term records the tap that crosses a coarse cell boundary.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-058fc9338e41e80a","tex":"H_{ks}","aliases":"H_ks","display":"H_{ks}","kind":"use","excerpt":"For L=N Z with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c09363bdd85aad0b","tex":"h_{ks}","aliases":"h_ks","display":"h_{ks}","kind":"use","excerpt":"Let h be a finite impulse response and y=h*x, y[n]=∑_q∈ Zh[q]x[n-q]. Resolve both input and output into N phases. For output phase k and input phase s, define the transfer kernel h_ks[m]=h[Nm+k-s]. Out-of-range taps are understood to be zero.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-5beb8cfa4cdeaa13","tex":"\\bm H_N","aliases":"H_N bm","display":"\\bm H_N","kind":"use","excerpt":"For L=N Z with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-00e5cb4d6d94dccd","tex":"H_{rs}","aliases":"H_rs","display":"H_{rs}","kind":"use","excerpt":"[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-4f0b191a575470a1","tex":"H_{st}","aliases":"H_st","display":"H_{st}","kind":"use","excerpt":"[Composition of transfer matrices] If G and H are convolution operators with packet matrices (G_rs) and (H_rs), then the packet matrix of GH is (GH)_rt = ∑_s∈ RG_rsH_st. In one dimensional transform notation, the corresponding matrix symbols multiply pointwise.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-814f64e7253b7d27","tex":"H_t","aliases":"H_t","display":"H_t","kind":"use","excerpt":"[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-f639c00b864f610e","tex":"H_{\\widehat\\tau}","aliases":"H_widehatτ widehat tau","display":"H_{\\widehat\\tau}","kind":"use","excerpt":"[Location error for one step] Let 0≤τ,widehatτ≤1 and H_τ= 1_[τ,1]. For 1≤ p<∞, aH_τ-aH_widehatτ _p = |a||τ-widehatτ|^1/p. In the supremum norm, the error equals |a| whenever τ≠widehatτ.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-cc40e31f1fe319ab","tex":"H_{\\widehat\\tau_q}","aliases":"H_widehatτ_q widehat tau","display":"H_{\\widehat\\tau_q}","kind":"use","excerpt":"[Packet reconstruction after detection] Let j=∑_q=1^Qa_qH_τ_q, widehat j=∑_q=1^Qwidehat a_qH_widehatτ_q. Under the hypotheses of [source reference], for 1≤ p<∞, j-widehat j _p ≤ Qb_h + ( frac h2∑_q=1^Q|a_q|^p )^1/p. If Q_m is bounded and linear and widehat f_m=Q_m(f-widehat j)+widehat j, then f-widehat f_m _p ≤ · s-Q_ms _p","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-aac3656a4ae03a71","tex":"H_{\\tau_q}","aliases":"H_τ_q tau","display":"H_{\\tau_q}","kind":"use","excerpt":"Let t_k=kh, k=0,…,n, h=frac1n. Assume f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Defin","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-8e51f9863c8a6e19","tex":"\\Hcal_{\\rm root}","aliases":"Hcal_rm root H_rm root Hcal rm mathcal rm","display":"{\\mathcal H}_{\\rm root}","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f1f2427536861709","tex":"\\operatorname{Hol}","aliases":"Hol operatorname","display":"\\operatorname{Hol}","kind":"use","excerpt":"Hol(γ) · closed phase transfer · cycle consistency in a network","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-022e2ac42397386f","tex":"i^k","aliases":"i^k","display":"i^k","kind":"use","excerpt":"se alternatives can all be seen The full binary tree of depth three has only 26 complete prunings. We enumerate all of them, compute a unique mixed-depth optimum, and certify its stability with rational costs. The displayed objective is deliberately explicit, not claimed to be a universal signal-quality score. headA coordinate orbit with no hidden ","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-1e7b5d1fe4b46e62","tex":"I_{16}","aliases":"I_16","display":"I_{16}","kind":"use","excerpt":"00,0.65000); node[] at (5.90000,1.04500) - frac14; draw[fill=black!18,draw=black!40,line width=.35pt] (6.80000,0.72000) rectangle ++(1.80000,0.65000); node[] at (7.70000,1.04500) frac14; node[] at (10.90000,1.05000) frac14; node[] at (5.00000,4.60000) coefficient positions within each packet; node[] at (10.90000,4.60000) energy; node[] at (6.25000,","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"1 · 0, whereas z^-1I_2 delays the original signal by two samples. The general address identity is valid for negative sample addresses as well.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-b06d3ec0fc2cf0c6","tex":"I_{j,k}","aliases":"I_j,k","display":"I_{j,k}","kind":"use","excerpt":"Let I_j,k = [k2^-j,(k+1)2^-j) and let ψ_j,k be the normalized Haar wavelet supported on I_j,k, positive on the left half and negative on the right half.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-63bfdca1004602ec","tex":"i_q","aliases":"i_q","display":"i_q","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-88b326798e7011cf","tex":"\\ii","aliases":"ii i mathrm","display":"{\\mathrm i}","kind":"use","excerpt":"Set A=∑_α W_ℓ,α^*W_ℓ,α. This is a finite sum of bounded operators. The energy result says ⟨ x,Ax⟩=⟨ x,x⟩ for every x. Complex polarization, applied also to x+y and x+ i y, gives ⟨ y,(A-I)x⟩=0 for all x,y, hence A=I. Applying the operator identity to x proves synthesis. This argument uses the actual Hilbert adjoints of the decimation operators, not ","count":23},{"work":"paper-v","url":"notation/paper-v.html#symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","excerpt":"headThe coordinates preserve structure, not Euclidean distances A quarter turn is a rotation in the reflection disk. After pulling back, it is a M obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points 1/2 and -1/2 become 3 and 1/3. The continuous impedan","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-8d6308571f33e418","tex":"J'","aliases":"J'","display":"J'","kind":"use","excerpt":"e every node cost changes by at most ε_v=1/32. A pruning may select as many as eight leaves, so its total cost can change by at most 8/32=1/4. The recursive envelope proves this without guessing which pruning will be selected: E(v)=max{ε_v, E(v0)+E(v1)}, E(∅)=tfrac14. For any competitor Λ, the changed cost difference is bounded below by J'(Λ)-J'(Λ_","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-fbb28fbeeb66e2d7","tex":"j_2","aliases":"j_2","display":"j_2","kind":"use","excerpt":"[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-5be3e9b89708ead6","tex":"j_f","aliases":"j_f","display":"j_f","kind":"use","excerpt":"_packet_decomposition Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one dimensional Sobolev representatives whose derivatives through order r-1 have one sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-50bc1c682c761171","tex":"j_o","aliases":"j_o","display":"j_o","kind":"use","excerpt":"[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-6cf9bd5f1c72ed56","tex":"j_q","aliases":"j_q","display":"j_q","kind":"use","excerpt":"[Several exact edge packets] If f=s+∑_q=1^Mj_q and every j_q is retained exactly, then f-(Q_ms+∑_qj_q) = s-Q_ms . Curved edges, corners, and different jump amplitudes do not alter the identity. The finite sum j=∑_q=1^M j_q is a single retained packet. Applying the preceding cancellation identity gives the result; the empty family M=0 is included wi","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-68bfee508e5dfce6","tex":"J_{y'}","aliases":"J_y'","display":"J_{y'}","kind":"use","excerpt":"imizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-1505c70154d93441","tex":"\\jump","aliases":"jump","display":null,"kind":"use","excerpt":"The singular packet results begin with trace data. If a piecewise smooth function has a jump in its derivative of order m at τ_q, then the packet jumpf^(m)_τ_q frac(t-τ_q)_+^mm! removes exactly that trace jump. Subtracting all packets through order r-1 leaves a global W^r,p remainder. The packet coefficients are uniquely determined by the one sided","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-1a37d1498744920e","tex":"k_x","aliases":"k_x","display":"k_x","kind":"use","excerpt":"[Two-dimensional errors] For the 192×192 disk experiment, a square Fourier mask retains indices |k_x|,|k_y|≤16, or 33^2=1089 bins. The reported errors are lrr method · full-image RMSE · boundary-band RMSE","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-81bf2ba82285d729","tex":"k_y","aliases":"k_y","display":"k_y","kind":"use","excerpt":"[Two-dimensional errors] For the 192×192 disk experiment, a square Fourier mask retains indices |k_x|,|k_y|≤16, or 33^2=1089 bins. The reported errors are lrr method · full-image RMSE · boundary-band RMSE","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[One-dimensional errors] For 1024 samples on [-1,1) and Fourier cutoff m=18, the retained frequency indices are -18,…,18: 37 bins including the zero-frequency bin. The measured values were lrr method · RMSE · maximum absolute error global Fourier truncation · 5.4023×10^-2 · 3.4968×10^-1 known edge packet · 1.3692×10^-4 · 1.4503×10^-3 estimated edge","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-34c05dc8f5073293","tex":"L'","aliases":"L'","display":"L'","kind":"use","excerpt":"a finite index sublattice of Z^d, and let R contain one representative from every coset. The analysis map x⟼ (x_r)_r∈ R, x_r(ℓ)=x(r+ℓ), is an isometric isomorphism on every ℓ^p space. This statement includes the usual one dimensional polyphase decomposition, but it also applies to arbitrary finite index lattice packets in several dimensions. If L'≤","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-f1cfec68d44ee43e","tex":"L^{-1/2}","aliases":"L^-1/2","display":"L^{-1/2}","kind":"use","excerpt":"Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-9d0fbf856fe55b03","tex":"L^1","aliases":"L^1","display":"L^1","kind":"use","excerpt":"ehatΩ=[tfrac516,tfrac1316], a=widehat a=3. The two intervals have the same length. Their disagreement consists of one missed strip and one extra strip, each of width 1/16. vfigmask_translation |ΩmathbintrianglewidehatΩ|=tfrac18, j-widehat j=3(1_Ω-1_widehatΩ). c|ccc error measure · L^1 norm · squared L^2 norm · L^∞ norm","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-5c6f72cfe6318222","tex":"L^{1/2}","aliases":"L^1/2","display":"L^{1/2}","kind":"use","excerpt":"[Exact Haar coefficient of a step] Let τ∈ I_j,k, let L=2^-j, and write u=fracτ-kLL∈[0,1]. Then | ⟨ aH_τ,ψ_j,k ⟩ | = |a|L^1/2min{u,1-u}. Every Haar coefficient whose support does not cross τ is zero.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"use","excerpt":"ZZ with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For finite","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-67cbdd51ca961fc6","tex":"L^{3/2}","aliases":"L^3/2","display":"L^{3/2}","kind":"use","excerpt":"Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-0de1bee007df208a","tex":"L^p","aliases":"L^p","display":"L^p","kind":"use","excerpt":"[Packet perturbation bound] For 1≤ p<∞, interpreted first as an inequality of extended L^p seminorms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise sup","count":19},{"work":"paper-v","url":"notation/paper-v.html#symbol-b12d1bed1ef598c0","tex":"L^\\infty","aliases":"L^∞ infty","display":"L^\\infty","kind":"use","excerpt":"f extended L^p seminorms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise supremum, and consequently also the essential L^∞ seminorm, j-widehat j _∞ ≤ |a","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-38bed36d88af8602","tex":"L_{-P}","aliases":"L_-P","display":"L_{-P}","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b450d7bbb1f72fc4","tex":"L_P","aliases":"L_P","display":"L_P","kind":"use","excerpt":"Let P(z) and Q(z) be Laurent polynomials. Define L_P(z) = 1 · 0 P(z) · 1 , U_Q(z) = 1 · Q(z) 0 · 1 .","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-87ae91a1f9b1b756","tex":"L_P^{-1}","aliases":"L_P^-1","display":"L_P^{-1}","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-3755b02b61222ecc","tex":"\\Lcal","aliases":"Lcal L mathcal","display":"{\\mathcal L}","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":14},{"work":"paper-v","url":"notation/paper-v.html#symbol-e86c461624494706","tex":"\\Lcal_*","aliases":"Lcal_* L_* Lcal mathcal","display":"{\\mathcal L}_*","kind":"use","excerpt":"[Stability of the selected basis] Assume the optimal pruning L_* for c is unique and at least one competing pruning exists. Define g=min_ L≠ L_*(C( L)-C( L_*))>0. If 2ε_*<g, then the optimal pruning for widetilde c is the same. If only one admissible pruning exists, it remains selected under every cost change without a gap condition.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-6d37916692e61255","tex":"\\Lcal_w","aliases":"Lcal_w L_w Lcal mathcal","display":"{\\mathcal L}_w","kind":"use","excerpt":"Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"inal sample δ_1. Under DI_2 it stays even but moves to channel address one: this is δ_2. Thus J(D)^2=DI_2, H(D)=I_2+2J(D)-J(D)^2+J(D)^3 recovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers. headSigned delays need the same care For an original-sample delay d at","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"on be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If a_min>2b_h, then every threshold between b_h and a_min-b_h detects exactly the cells containing jumps. The jump locations are recovered within h/2, and each amplitude error is at most b_h. The probability theorem replaces the determinist","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"Suppose N,M≥1 and Nmid M. Every coarse phase has a disjoint family of fine descendants: Child_M(N,r) = {s:0≤ s<M, s≡ rpmod N}.","count":65},{"work":"paper-v","url":"notation/paper-v.html#symbol-76a8c8c02bf83b7a","tex":"M'","aliases":"M'","display":"M'","kind":"use","excerpt":"[Rectangular image phases] For L=M Z× N Z⊂ Z^2, one may take R={0,…,M-1}×{0,…,N-1}. The analysis map separates an image into MN exact phases. Refining (M,N) to (M',N'), where Mmid M' and Nmid N', splits every coarse phase into fracM'MfracN'N fine phases.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-83d3f2e0b215ee50","tex":"M_{ij}","aliases":"M_ij","display":"M_{ij}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-956f614ae0645ab5","tex":"\\mat","aliases":"mat","display":null,"kind":"use","excerpt":"vpage7Perfect reconstruction is an operator identityFILTER BANKS / WHY A FREQUENCY SPOT-CHECK IS INSUFFICIENTTraditional monograph: Theorem 4.1 (p. 7). Consider the finite Laurent bank E(D)=mat1 · D 0 · 1, R(D)=mat1 · -D 0 · 1. Its synthesis cancels the exact delayed cross-channel contribution introduced by analysis. vfigperfect_reconstruction head","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-2c7620add7ce256d","tex":"\\operatorname{Mat}","aliases":"Mat operatorname","display":"\\operatorname{Mat}","kind":"use","excerpt":"Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-bba1a1e3366bd331","tex":"\\operatorname{Mat}_{N\\times N}","aliases":"Mat_N× N operatorname times","display":"\\operatorname{Mat}_{N\\times N}","kind":"use","excerpt":"Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"use","excerpt":"one also has A_N S_N=I. Equip this finite product with the Hilbert norm (x_0,…,x_N-1) _ch^2 =∑_r=0^N-1 x_r _2^2. Then A_N is a linear isometric equivalence and x _2^2=∑_r=0^N-1 x_r _2^2. Euclidean division gives a unique pair (r,q) with 0≤ r<N and n=Nq+r, including negative n. The address maps n↦(nbmod N,nmathbindivN) and (r,q)↦ Nq+r are inverse. A","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"[Local perturbation certificate] Assume |c_v(y')-c_v(y)|≤ε_v. Define E(v) = ε_v, · v terminal, [1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal va","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-c820cd743c916f52","tex":"\\max_k","aliases":"max_k max","display":"\\max_k","kind":"use","excerpt":"inproof If σ=0, every centered zero-sub-Gaussian noise variable vanishes almost surely, and the noise-free deterministic conclusion applies. For σ>0, the difference η_k+1-η_k is sub Gaussian with variance proxy 2σ^2, so P(|η_k+1-η_k|>t) ≤ 2exp(-fract^24σ^2). A union bound over the n cells gives P( max_k|η_k+1-η_k|>t_n,δ ) ≤ δ. On the complementary ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-d02e656e84ac97fa","tex":"\\max_{\\Lcal}","aliases":"max_Lcal max_ L max Lcal max mathcal","display":"\\max_{{\\mathcal L}}","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-18caafcaaf4899c3","tex":"\\max_w","aliases":"max_w max","display":"\\max_w","kind":"use","excerpt":"A_vmin_w a(w), · v split, and b(v)= 1, · v retained, B_vmax_w b(w), · v split. Then the leaf analysis of the pruned tree has bounds a(rm root) and b(rm root).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-beb7b788f815c1ff","tex":"\\max_{\\Lambda\\in\\mathfrak P_v}","aliases":"max_Λ∈ P_v max Lambda in mathfrak","display":"\\max_{\\Lambda\\in\\mathfrak P_v}","kind":"use","excerpt":"[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"Suppose N,M≥1 and Nmid M. Every coarse phase has a disjoint family of fine descendants: Child_M(N,r) = {s:0≤ s<M, s≡ rpmod N}.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-3f9db2ae5742966b","tex":"\\midrule","aliases":"midrule","display":null,"kind":"use","excerpt":"[One-dimensional errors] For 1024 samples on [-1,1) and Fourier cutoff m=18, the retained frequency indices are -18,…,18: 37 bins including the zero-frequency bin. The measured values were lrr method · RMSE · maximum absolute error global Fourier truncation · 5.4023×10^-2 · 3.4968×10^-1 known edge packet · 1.3692×10^-4 · 1.4503×10^-3 estimated edge","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an","count":28},{"work":"paper-v","url":"notation/paper-v.html#symbol-8dae54f32eaf3a3d","tex":"\\min_{\\Lcal\\ne\\Lcal_*}","aliases":"min_Lcal≠Lcal_* min_ L≠ L_* min Lcal ne Lcal min mathcal ne mathcal","display":"\\min_{{\\mathcal L}\\ne{\\mathcal L}_*}","kind":"use","excerpt":"[Stability of the selected basis] Assume the optimal pruning L_* for c is unique and at least one competing pruning exists. Define g=min_ L≠ L_*(C( L)-C( L_*))>0. If 2ε_*<g, then the optimal pruning for widetilde c is the same. If only one admissible pruning exists, it remains selected under every cost change without a gap condition.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-af555498bdb2b55d","tex":"\\min_q","aliases":"min_q min","display":"\\min_q","kind":"use","excerpt":"[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-aa2dd72af4ea5202","tex":"\\min_w","aliases":"min_w min","display":"\\min_w","kind":"use","excerpt":"[Bounds for nonunitary local splits] Suppose every internal node has an analysis map U_v and constants 0≤ A_v≤ B_v<∞ satisfying A_v x ^2 ≤ ∑_w∈child(v) (U_vx)_w ^2 ≤ B_v x ^2. Define recursively a(v)= 1, · v retained, A_vmin_w a(w), · v split, and b(v)= 1, · v retained,","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-30da5b3706769a4f","tex":"\\min_{\\Lambda\\ne\\Lambda_y^*}","aliases":"min_Λ≠Λ_y^* min Lambda ne Lambda","display":"\\min_{\\Lambda\\ne\\Lambda_y^*}","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f7ed41372efc24c8","tex":"\\mathrm{mod}","aliases":"mod mathrm","display":"\\mathrm{mod}","kind":"use","excerpt":"Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-3e0534ff9dda7eeb","tex":"N'","aliases":"N'","display":"N'","kind":"use","excerpt":"[Rectangular image phases] For L=M Z× N Z⊂ Z^2, one may take R={0,…,M-1}×{0,…,N-1}. The analysis map separates an image into MN exact phases. Refining (M,N) to (M',N'), where Mmid M' and Nmid N', splits every coarse phase into fracM'MfracN'N fine phases.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"Split x into even and odd channels e,o. Define d=o-e, s=e+frac12d. The inverse is e=s-frac12d, o=d+e.","count":26},{"work":"paper-v","url":"notation/paper-v.html#symbol-dcf850482173fa7d","tex":"\\one_{[-a,a]}","aliases":"one_[-a,a] 1_[-a,a] one mathbf","display":"{\\mathbf 1}_{[-a,a]}","kind":"use","excerpt":"[Square wave Fourier tail] Let j(t)= 1_[-a,a](t) on the circle and let s be smooth. If Q_m is a Fourier partial sum, then Q_m(s+j) has the ordinary oscillatory boundary layer. In contrast, Q_ms+j has error exactly s-Q_ms. The visual comparison in [source reference] illustrates this change in approximation target. It is not a new Fourier summability","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-9cf216a2fd3d418b","tex":"\\one_E","aliases":"one_E 1_E one mathbf","display":"{\\mathbf 1}_E","kind":"use","excerpt":"[Exact scalar region formula] Let E,F be measurable subsets of a measure space, let a,b∈ C, and let 1≤ p<∞. Then a 1_E-b 1_F _p^p = · |a-b|^pμ(E∩ F) +|a|^pμ(E∖ F) · +|b|^pμ(F∖ E). For equal amplitudes, a 1_E-a 1_F _p = |a|μ(Emathbintriangle F)^1/p.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-4d2b4fb4a0ad76c8","tex":"\\one_{E_i}","aliases":"one_E_i 1_E_i one mathbf","display":"{\\mathbf 1}_{E_i}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-248f745ef97fafb0","tex":"\\one_F","aliases":"one_F 1_F one mathbf","display":"{\\mathbf 1}_F","kind":"use","excerpt":"[Exact scalar region formula] Let E,F be measurable subsets of a measure space, let a,b∈ C, and let 1≤ p<∞. Then a 1_E-b 1_F _p^p = · |a-b|^pμ(E∩ F) +|a|^pμ(E∖ F) · +|b|^pμ(F∖ E). For equal amplitudes, a 1_E-a 1_F _p = |a|μ(Emathbintriangle F)^1/p.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-9c08f8d7bc2b3255","tex":"\\one_{F_j}","aliases":"one_F_j 1_F_j one mathbf","display":"{\\mathbf 1}_{F_j}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f814b60f967151c4","tex":"\\mathrm{op}","aliases":"op mathrm","display":"\\mathrm{op}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-428838e041027074","tex":"\\overline{s(t)-s(t+L/2)}","aliases":"overlines(t)-s(t+L/2) overline","display":"\\overline{s(t)-s(t+L/2)}","kind":"use","excerpt":"Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b8990f3a0f7e9440","tex":"\\overline{\\operatorname{span}\\{S^n\\delta_0:n\\in\\bbZ\\}}","aliases":"overlinespan{S^nδ_0:n∈bbZ} overlinespan{S^nδ_0:n∈ Z} overline operatorname delta in bbZ overline operatorname delta in mathbb","display":"\\overline{\\operatorname{span}\\{S^n\\delta_0:n\\in{\\mathbb Z}\\}}","kind":"use","excerpt":"[Impulse seed theorem] The closed shift-generated hull of δ_0 is all of ℓ^2( Z): overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 =ℓ^2( Z). Every finitely supported signal has the exact finite expression x=∑_n∈ Zx[n]S^nδ_0.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-085e7278fbd586c8","tex":"\\overline{\\operatorname{span}\\{S^n\\delta_0:n\\in\\bbZ\\}}^{\\,\\ell^2}","aliases":"overlinespan{S^nδ_0:n∈bbZ}^ ℓ^2 overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 overline operatorname delta in bbZ ell overline operatorname delta in mathbb ell","display":"\\overline{\\operatorname{span}\\{S^n\\delta_0:n\\in{\\mathbb Z}\\}}^{\\,\\ell^2}","kind":"use","excerpt":"[Impulse seed theorem] The closed shift-generated hull of δ_0 is all of ℓ^2( Z): overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 =ℓ^2( Z). Every finitely supported signal has the exact finite expression x=∑_n∈ Zx[n]S^nδ_0.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-398aacf1e9e3576c","tex":"\\overline{\\Span\\{T_a\\delta_0:a\\in\\ZZ^d\\}}","aliases":"overlineSpan{T_aδ_0:a∈ZZ^d} overlinespan{T_aδ_0:a∈ Z^d} overline Span delta in ZZ overline operatorname delta in mathbb","display":"\\overline{{\\operatorname{span}}\\{T_a\\delta_0:a\\in{\\mathbb Z}^d\\}}","kind":"use","excerpt":"[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-71d5e65764bad34c","tex":"\\overline{\\Span\\{T_a\\delta_0:a\\in\\ZZ^d\\}}^{\\ell^p}","aliases":"overlineSpan{T_aδ_0:a∈ZZ^d}^ℓ^p overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p overline Span delta in ZZ ell overline operatorname delta in mathbb ell","display":"\\overline{{\\operatorname{span}}\\{T_a\\delta_0:a\\in{\\mathbb Z}^d\\}}^{\\ell^p}","kind":"use","excerpt":"[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ef958519c2ac9923","tex":"\\mathbb P","aliases":"P mathbb","display":"\\mathbb P","kind":"use","excerpt":"[Polyphase M obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"use","excerpt":"[Finite globally optimal stable selector] Let P( T) be the finite set of complete prunings of T.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-7bab906fff5aa2ce","tex":"\\mathbb P^1","aliases":"P^1 mathbb","display":"\\mathbb P^1","kind":"use","excerpt":"[Polyphase M obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-035c508864e2d38f","tex":"p^p","aliases":"p^p","display":"p^p","kind":"use","excerpt":"For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-586c9b3f425a7bf8","tex":"P_{12,k}","aliases":"P_12,k","display":"P_{12,k}","kind":"use","excerpt":"VE ADDRESSESTraditional monograph: Proposition 14.1 (p. 29). At phase order 12=4·3, the map k⟼(kbmod4, kbmod3) is a bijection from twelve residue addresses to the 4×3 grid. Every cell below contains its unique original address. vfigcrt_refinement headThe factorization is an intersection of conditions For integer sample addresses, including negative","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-6a679a976dd7459e","tex":"P_{2,1}","aliases":"P_2,1","display":"P_{2,1}","kind":"use","excerpt":"[fill=black!20,draw=black!40,line width=.35pt] (2.92000,2.35000) rectangle ++(0.84000,0.60000); node[] at (3.34000,2.65000) 3; draw[fill=black!0,draw=black!40,line width=.35pt] (3.76000,2.35000) rectangle ++(0.84000,0.60000); draw[fill=black!20,draw=black!40,line width=.35pt] (4.60000,2.35000) rectangle ++(0.84000,0.60000); node[] at (5.02000,2.650","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-0979bcddbdbf9b2b","tex":"P_{2^s,j_2}","aliases":"P_2^s,j_2","display":"P_{2^s,j_2}","kind":"use","excerpt":"[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e8699f42976e6525","tex":"P_{3,0}","aliases":"P_3,0","display":"P_{3,0}","kind":"use","excerpt":")·(P_N,ky). Here the dot means multiplication at the same sample, not convolution. headCauchy multiplication adds degrees For formal series A(t),B(t), the coefficient of t^n is a sum over a+b=n. Consequently the output residue is i+jpmod N: P_N,k(AB)=∑_i+j≡ k (N)(P_N,iA)(P_N,jB). For N=3, take A=1+2t+3t^2 and B=2+t^2. Direct multiplication gives AB","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-3f64965a0db71aca","tex":"P_{3,k\\bmod3}","aliases":"P_3,kbmod3 bmod","display":"P_{3,k\\bmod3}","kind":"use","excerpt":"VE ADDRESSESTraditional monograph: Proposition 14.1 (p. 29). At phase order 12=4·3, the map k⟼(kbmod4, kbmod3) is a bijection from twelve residue addresses to the 4×3 grid. Every cell below contains its unique original address. vfigcrt_refinement headThe factorization is an intersection of conditions For integer sample addresses, including negative","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e478ceed7696e248","tex":"P_{4,k\\bmod4}","aliases":"P_4,kbmod4 bmod","display":"P_{4,k\\bmod4}","kind":"use","excerpt":"VE ADDRESSESTraditional monograph: Proposition 14.1 (p. 29). At phase order 12=4·3, the map k⟼(kbmod4, kbmod3) is a bijection from twelve residue addresses to the 4×3 grid. Every cell below contains its unique original address. vfigcrt_refinement headThe factorization is an intersection of conditions For integer sample addresses, including negative","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-72f2cd87bd5faecf","tex":"P_{4,r}","aliases":"P_4,r","display":"P_{4,r}","kind":"use","excerpt":"y cell contains an original address and its rational sample value. Shading marks the known jump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-703e11b92029fd77","tex":"P_{M,a}","aliases":"P_M,a","display":"P_{M,a}","kind":"use","excerpt":"leaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples. For example, at the first descendant address the values f[0]=0 and f[2]=1/2 are two separate values. Their sum 1/2 cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement. headThe","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-5ae4b8981bd92730","tex":"P_{M,j_o}","aliases":"P_M,j_o","display":"P_{M,j_o}","kind":"use","excerpt":"[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-6977f69e7c84b853","tex":"P_{M,s}","aliases":"P_M,s","display":"P_{M,s}","kind":"use","excerpt":"[Exact nested refinement] For N,M≥1 and Nmid M, P_N,r = ∑_substack0≤ s<M s≡ r (mod N)P_M,s.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-ce857563c4ac4bba","tex":"P_{N,i}","aliases":"P_N,i","display":"P_{N,i}","kind":"use","excerpt":"Linear time invariance is not required for exact packet coordinates. Let A and B be formal power series over a commutative coefficient ring (in particular C). Set A(z)=∑_n≥0a_nz^n, A_i=P_N,iA. All phase indices range from 0 to N-1. For another representation B(z), multiplication gives a nonlinear or bilinear interaction.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-e65ae8417651699c","tex":"P_{N,j}","aliases":"P_N,j","display":"P_{N,j}","kind":"use","excerpt":"[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-08e120fa2da08c81","tex":"P_r^{L}","aliases":"P_r^L","display":"P_r^{L}","kind":"use","excerpt":"[Exact lattice refinement] For every coarse packet, P_r^L = ∑_s∈Child(r)P_s^L'. The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-78aa8f72acca2b03","tex":"P_s","aliases":"P_s","display":"P_s","kind":"use","excerpt":"[Exact lattice polyphase decomposition] For every 1≤ p≤∞, the analysis map A_L:ℓ^p( Z^d)⟶⊕_r∈ Rℓ^p(L), A_Lx=(x_r)_r∈ R, is an isometric isomorphism. Its inverse is the interleaving map S_L(u)(r+ℓ)=u_r(ℓ). The projectors satisfy P_rP_s=δ_rsP_r, ∑_r∈ RP_r=I.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-eb7674ad2c449edd","tex":"P_s^{L'}","aliases":"P_s^L'","display":"P_s^{L'}","kind":"use","excerpt":"[Exact lattice refinement] For every coarse packet, P_r^L = ∑_s∈Child(r)P_s^L'. The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-20e4069b9d08281b","tex":"\\mathfrak P_v","aliases":"P_v mathfrak","display":"\\mathfrak P_v","kind":"use","excerpt":"[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-f902074ceba32407","tex":"P_{v0}","aliases":"P_v0","display":"P_{v0}","kind":"use","excerpt":"es, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actual Haar packets, set P_v=W_v^*W_v. Parent-to-ch","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-726490b49a9cbc64","tex":"P_{v1}","aliases":"P_v1","display":"P_{v1}","kind":"use","excerpt":"es, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actual Haar packets, set P_v=W_v^*W_v. Parent-to-ch","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"Suppose N,M≥1 and Nmid M. Every coarse phase has a disjoint family of fine descendants: Child_M(N,r) = {s:0≤ s<M, s≡ rpmod N}.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"use","excerpt":"0 · 0, s=(1,0), e=(0,1). Then Q^2=Q and Qs=s, but (I-Q)e=(-2,1). The smooth error is zero while the unit packet error becomes a reconstruction error of norm sqrt5. For a fully specified realization, take j=e and widehat j=0. Then f=(1,1), widehat f=Qf=(3,0), and their difference is exactly (-2,1). headUse the actual operator bound The always-valid ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"vpage28Six image phases are six exact address classesLATTICE COORDINATES / A RECTANGULAR EXAMPLETraditional monograph: Theorem E3.2 (p. 43), Theorem E3.5 (p. 44), Remark E2.2 (p. 43). Take L=2 Z×3 Z inside Z^2, with representatives R={0,1}×{0,1,2}. Every address has a unique expression (i,j)=(r_1,r_2)+(2q_1,3q_2). The six channels are x_r(2q_1,3q_2","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"vpage28Six image phases are six exact address classesLATTICE COORDINATES / A RECTANGULAR EXAMPLETraditional monograph: Theorem E3.2 (p. 43), Theorem E3.5 (p. 44), Remark E2.2 (p. 43). Take L=2 Z×3 Z inside Z^2, with representatives R={0,1}×{0,1,2}. Every address has a unique expression (i,j)=(r_1,r_2)+(2q_1,3q_2). The six channels are x_r(2q_1,3q_2","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-e15bd658e05b8d9b","tex":"q_k","aliases":"q_k","display":"q_k","kind":"use","excerpt":"Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-c95e041440f50ec9","tex":"Q_n","aliases":"Q_n","display":"Q_n","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ","display":null,"kind":"use","excerpt":"Section 13 (p. 24). A packet calculus can reorganize a signal exactly, choose among finitely many representations, or describe a nonlinear coordinate geometry. These are not the same task. This guide keeps one concrete object in view for each, so that the reader can compute rather than only follow arrows. vfigthree_worlds headA signal that can be c","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-2f925d2e485d97c9","tex":"\\QQ^{16}","aliases":"QQ^16 QQ","display":null,"kind":"use","excerpt":"Section 13 (p. 24). A packet calculus can reorganize a signal exactly, choose among finitely many representations, or describe a nonlinear coordinate geometry. These are not the same task. This guide keeps one concrete object in view for each, so that the reader can compute rather than only follow arrows. vfigthree_worlds headA signal that can be c","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"use","excerpt":"Let L'≤ L≤ Z^d, with both indices finite. Choose representative systems R for Z^d/L and R' for Z^d/L'. For r∈ R, define Child(r) = {s∈ R':s+L'⊂ r+L}.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"[A perturbed disk] Let E and F be concentric disks of radii r and r+δ in R^2, with δ>0. Then μ(Emathbintriangle F) = π((r+δ)^2-r^2) = 2π rδ+πδ^2. For equal amplitude a, the exact L^2 packet error is |a|sqrt2π rδ+πδ^2. This gives the expected square root dependence on boundary displacement.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"vpage28Six image phases are six exact address classesLATTICE COORDINATES / A RECTANGULAR EXAMPLETraditional monograph: Theorem E3.2 (p. 43), Theorem E3.5 (p. 44), Remark E2.2 (p. 43). Take L=2 Z×3 Z inside Z^2, with representatives R={0,1}×{0,1,2}. Every address has a unique expression (i,j)=(r_1,r_2)+(2q_1,3q_2). The six channels are x_r(2q_1,3q_2","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"use","excerpt":"vpage28Six image phases are six exact address classesLATTICE COORDINATES / A RECTANGULAR EXAMPLETraditional monograph: Theorem E3.2 (p. 43), Theorem E3.5 (p. 44), Remark E2.2 (p. 43). Take L=2 Z×3 Z inside Z^2, with representatives R={0,1}×{0,1,2}. Every address has a unique expression (i,j)=(r_1,r_2)+(2q_1,3q_2). The six channels are x_r(2q_1,3q_2","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-fb14b1a407e1de8e","tex":"r_a","aliases":"r_a","display":"r_a","kind":"use","excerpt":"[Translation covariance] Fix a∈ Z^d. For every r∈ R, write uniquely r-a=r_a+λ_a,r, r_a∈ R, λ_a,r∈ L. Then ( A_LT_ax)_r(ℓ) = x_r_a(ℓ+λ_a,r). Thus a fine translation permutes the lattice packets and applies an internal translation on L.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-5cd09b860b1a4d1a","tex":"R_{\\mathrm{can}}","aliases":"R_can mathrm","display":"R_{\\mathrm{can}}","kind":"use","excerpt":"[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-bfa7220254bbeabc","tex":"\\bm R_{\\mathrm{H}}","aliases":"R_H bm mathrm","display":"\\bm R_{\\mathrm{H}}","kind":"use","excerpt":"1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j,","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"use","excerpt":"Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-c7691c6d3fc75621","tex":"R_{N,k}","aliases":"R_N,k","display":"R_{N,k}","kind":"use","excerpt":"Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-319463847a07b498","tex":"R_{\\rm can}","aliases":"R_rm can rm","display":"R_{\\rm can}","kind":"use","excerpt":"[Frame bounds from the polyphase symbol] Assume that A I_N ≤ E( e^ iω)^*E( e^ iω) ≤ B I_N for almost every ω, where 0<A≤ B<∞. Then the analysis bank satisfies A x _2^2 ≤ A_E x _2^2 ≤ B x _2^2. The pointwise matrix R_rm can( e^ iω) = (E^*E)^-1E^* is a bounded left inverse and gives the canonical dual synthesis bank.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-a296862ac6ebceae","tex":"r_{\\widehat\\tau}","aliases":"r_widehatτ widehat tau","display":"r_{\\widehat\\tau}","kind":"use","excerpt":"y on the interval between its two locations: H_τ-H_widehatτ _2=sqrtδ=tfrac14, H_τ-H_widehatτ _∞=1. vfig[.89]ext_location headFor a ramp, the error height shrinks too Write r_τ(t)=(t-τ)_+. The difference grows linearly from zero to δ on [τ,widehatτ] and stays at δ on [widehatτ,1]. Direct integration gives r_τ-r_widehatτ _2^2 =fracδ^33+(1-widehatτ)δ^","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-04574488c1d5a80f","tex":"R_\\theta","aliases":"R_θ theta","display":"R_\\theta","kind":"use","excerpt":"Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-c977b69c1a9d9e20","tex":"r_\\tau","aliases":"r_τ tau","display":"r_\\tau","kind":"use","excerpt":"Traditional monograph: Corollary E8.2 (p. 60), Theorem E8.3 (p. 60), Theorem E8.7 (p. 62). Set τ=1/4, widehatτ=5/16, and δ=1/16. A unit step differs only on the interval between its two locations: H_τ-H_widehatτ _2=sqrtδ=tfrac14, H_τ-H_widehatτ _∞=1. vfig[.89]ext_location headFor a ramp, the error height shrinks too Write r_τ(t)=(t-τ)_+. The differ","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-ff81ba5e23d549c7","tex":"\\Re","aliases":"Re","display":"\\Re","kind":"use","excerpt":"Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"use","excerpt":"headThe coordinates preserve structure, not Euclidean distances A quarter turn is a rotation in the reflection disk. After pulling back, it is a M obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points 1/2 and -1/2 become 3 and 1/3. The continuous impedan","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-4321aa573d789e4c","tex":"\\mathrm{rec}","aliases":"rec mathrm","display":"\\mathrm{rec}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"inal sample δ_1. Under DI_2 it stays even but moves to channel address one: this is δ_2. Thus J(D)^2=DI_2, H(D)=I_2+2J(D)-J(D)^2+J(D)^3 recovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers. headSigned delays need the same care For an original-sample delay d at","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","count":10},{"work":"paper-v","url":"notation/paper-v.html#symbol-4aa377bca96588f9","tex":"\\RMSE","aliases":"RMSE operatorname","display":"{\\operatorname{RMSE}}","kind":"use","excerpt":"[One-dimensional errors] For 1024 samples on [-1,1) and Fourier cutoff m=18, the retained frequency indices are -18,…,18: 37 bins including the zero-frequency bin. The measured values were lrr method · RMSE · maximum absolute error global Fourier truncation · 5.4023×10^-2 · 3.4968×10^-1 known edge packet · 1.3692×10^-4 · 1.4503×10^-3 estimated edge","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-78ecfb9af1021553","tex":"\\RR^2","aliases":"RR^2 R^2 RR mathbb","display":"{\\mathbb R}^2","kind":"use","excerpt":"[A perturbed disk] Let E and F be concentric disks of radii r and r+δ in R^2, with δ>0. Then μ(Emathbintriangle F) = π((r+δ)^2-r^2) = 2π rδ+πδ^2. For equal amplitude a, the exact L^2 packet error is |a|sqrt2π rδ+πδ^2. This gives the expected square root dependence on boundary displacement.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"ump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector P_4,rf is different: it keeps the original latt","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-78626aee2df085fd","tex":"\\s","aliases":"s","display":null,"kind":"use","excerpt":"draw[->] (0,0)--(9.4,0) node[right]singular value; draw[line width=6pt,teal!35] (3,1)--(9,1); draw[line width=6pt,black!35] (4,2)--(8,2); node[anchor=east]at(2.7,2)original;node[anchor=east]at(2.7,1)perturbed; foreach x/s in 3/3/4,4/1,8/2,9/9/4 draw(x,-.12)--(x,.12);node[below]at(x,-.15)s; fill (4,2)circle(2pt);fill(8,2)circle(2pt);fill[teal](3,1)c","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","excerpt":"h=.45pt] (11.31329,1.52632) -- (11.31329,1.25000); draw[fill=white,line width=.7pt] (11.31329,1.25000) circle (0.045); draw[black!45,line width=.45pt] (12.05380,1.52632) -- (12.05380,1.34211); draw[fill=white,line width=.7pt] (12.05380,1.34211) circle (0.045); node[] at (6.50000,4.80000) solid: f dashed: Qf open: Qs+j; node[] at (6.50000,0.10000) p","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b06cc43523171b06","tex":"\\mathcal S_4","aliases":"S_4 mathcal","display":"\\mathcal S_4","kind":"use","excerpt":"ump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector P_4,rf is different: it keeps the original latt","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-95d3d602121aab4d","tex":"s_f","aliases":"s_f","display":"s_f","kind":"use","excerpt":"Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one dimensional Sobolev representatives whose derivatives through order r-1 have one sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one sided der","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-57c446e8e6d068a3","tex":"S_m","aliases":"S_m","display":"S_m","kind":"use","excerpt":"vpage16Why a Fourier cutoff cannot reproduce this jump pointwiseA CONTINUOUS PULSE / EXACT ENDPOINT VALUESTraditional monograph: Section 6 (p. 11), Section 8 (p. 13). Use a separate period-one example on [-1/2,1/2): j(t)= 1_{|t|<1/4}, S_mj(t)=tfrac12+frac2π∑_n=1^mfracsin(π n/2)ncos(2π nt). This is the exact finite Fourier formula. The drawn curves ","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-18f5af16bd34d11c","tex":"\\sgn","aliases":"sgn operatorname","display":"{\\operatorname{sgn}}","kind":"use","excerpt":"[Jump count and sign recovery] Under the hypotheses of [source reference], the number of detected cells equals Q, and sgn(widehat a_q)=sgn(a_q) for every real valued jump.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-8b8011d9dfc22b9d","tex":"\\operatorname{Si}","aliases":"Si operatorname","display":"\\operatorname{Si}","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-1d79850b77567ab8","tex":"\\operatorname{sign}","aliases":"sign operatorname","display":"\\operatorname{sign}","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-32f99fc05b64a106","tex":"\\succ","aliases":"succ","display":"\\succ","kind":"use","excerpt":"The signal realization makes the seed, address, packet, transfer, refinement, and synthesis operations concrete. The objects and laws can be read directly: seed · =δ_0, address · =S^n, packet · =P_N,rx, transfer · =P_N,kHP_N,s, refinement · =P_N,r=∑_ssucc rP_M,s, synthesis · =∑_rP_N,rx, nonlinearity · =P_N,rF(∑_sP_N,sx). Filter banks, wavelet packe","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"imizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-0341617620bd99b1","tex":"\\sup_m","aliases":"sup_m sup","display":"\\sup_m","kind":"use","excerpt":"[Estimated packet Gibbs bound] Let Q_m be linear and uniformly bounded on L^p, with sup_m Q_m ≤ C_Q. For f=s+j and an estimated packet widehat j_m, define widehat f_m =Q_m(f-widehat j_m)+widehat j_m. Then f-widehat f_m _p ≤ s-Q_ms _p +(1+C_Q) j-widehat j_m _p. Consequently, smooth approximation and packet recovery may be improved independently. No ","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-11e8bed385caca1a","tex":"\\sup_u","aliases":"sup_u sup","display":"\\sup_u","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-af258da75f945f55","tex":"\\sup_{\\Lambda\\in\\mathfrak P(\\mathcal T)}","aliases":"sup_Λ∈ P( T) sup Lambda in mathfrak mathcal","display":"\\sup_{\\Lambda\\in\\mathfrak P(\\mathcal T)}","kind":"use","excerpt":"imizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-56916448f3b1cf12","tex":"\\sup_{\\Xi}","aliases":"sup_Ξ sup Xi","display":"\\sup_{\\Xi}","kind":"use","excerpt":"set, so the feasible set is nonempty and finite. Thus the minimum is attained. If competitors exist, their finitely many differences from a unique optimum are strictly positive, so their minimum is positive and is achieved by a competitor. In the singleton case there is no alternative selector under any objective change. In the remaining case, for ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"If σ=0, every centered zero-sub-Gaussian noise variable vanishes almost surely, and the noise-free deterministic conclusion applies. For σ>0, the difference η_k+1-η_k is sub Gaussian with variance proxy 2σ^2, so P(|η_k+1-η_k|>t) ≤ 2exp(-fract^24σ^2). A union bound over the n cells gives P( max_k|η_k+1-η_k|>t_n,δ ) ≤ δ. On the complementary event, r","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":")·(P_N,ky). Here the dot means multiplication at the same sample, not convolution. headCauchy multiplication adds degrees For formal series A(t),B(t), the coefficient of t^n is a sum over a+b=n. Consequently the output residue is i+jpmod N: P_N,k(AB)=∑_i+j≡ k (N)(P_N,iA)(P_N,jB). For N=3, take A=1+2t+3t^2 and B=2+t^2. Direct multiplication gives AB","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":")·(P_N,ky). Here the dot means multiplication at the same sample, not convolution. headCauchy multiplication adds degrees For formal series A(t),B(t), the coefficient of t^n is a sum over a+b=n. Consequently the output residue is i+jpmod N: P_N,k(AB)=∑_i+j≡ k (N)(P_N,iA)(P_N,jB). For N=3, take A=1+2t+3t^2 and B=2+t^2. Direct multiplication gives AB","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-2ca79bce6422a743","tex":"t^{m-1}","aliases":"t^m-1","display":"t^{m-1}","kind":"use","excerpt":"ion with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated constant. The supremum norm argum","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"use","excerpt":"kets can interact. vfigproduct_conventions headPointwise multiplication preserves the same address For time-residue masks, (P_N,ix)[n](P_N,jy)[n]=0 when i≠ j. Therefore P_N,k(x· y)=(P_N,kx)·(P_N,ky). Here the dot means multiplication at the same sample, not convolution. headCauchy multiplication adds degrees For formal series A(t),B(t), the coeffic","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-75143599dd91b630","tex":"T_{2,1}","aliases":"T_2,1","display":"T_{2,1}","kind":"use","excerpt":"[Impedance--admittance is the two-phase split] For N=2 and finite z≠0, T_2,1(z)=z^-1. On the full projective line it exchanges 0 and ∞. Thus a half-turn of the reflection disk exchanges normalized impedance and admittance. ω_2=exp(π i)=-1, so A(ω_2)[u:v]=[v:u]. The finite ratio is 1/z when z≠0; the coordinate swap also gives [0:1]↔[1:0]. No ordinar","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-830c9c56b4f2bf21","tex":"t_k","aliases":"t_k","display":"t_k","kind":"use","excerpt":"Let t_k=kh, k=0,…,n, h=frac1n. Assume f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Defin","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-221fdc610b5eb2ef","tex":"t_{k+1}","aliases":"t_k+1","display":"t_{k+1}","kind":"use","excerpt":"[Contrast gap] Assume |η_k|≤ε and set b_h=Mh+2ε. If the cell (t_k,t_k+1) contains no jump, then |D_k|≤ b_h. If it contains one jump of amplitude a_q, then |D_k-a_q|≤ b_h and therefore |D_k|≥ |a_q|-b_h.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-667692e13043c7f4","tex":"T_{ks}","aliases":"T_ks","display":"T_{ks}","kind":"use","excerpt":"T_ks · P_N,kHP_N,s · input-to-output phase transfer","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-90b38ede3f1b080b","tex":"T_{N,k}","aliases":"T_N,k","display":"T_{N,k}","kind":"use","excerpt":"[Polyphase M obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-dcaefd9e3e9b42b6","tex":"T_{N,k+\\ell\\, (\\mathrm{mod}\\,N)}","aliases":"T_N,k+ℓ (mod N) ell mathrm","display":"T_{N,k+\\ell\\, (\\mathrm{mod}\\,N)}","kind":"use","excerpt":"obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a41cf9f85eb00b77","tex":"t_{n,\\delta}","aliases":"t_n,δ delta","display":"t_{n,\\delta}","kind":"use","excerpt":"[Probability recovery bound] Fix 0<δ<1 and define t_n,δ = 2σ sqrtlog(frac2nδ) and b_h,δ=Mh+t_n,δ. With probability at least 1-δ, every contrast satisfies the conclusions of [source reference] with b_h replaced by b_h,δ.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-f2718681b82c9502","tex":"T_{N,\\ell}","aliases":"T_N,ℓ ell","display":"T_{N,\\ell}","kind":"use","excerpt":"obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-3000c3bd47e8007c","tex":"\\toprule","aliases":"toprule","display":null,"kind":"use","excerpt":"[One-dimensional errors] For 1024 samples on [-1,1) and Fourier cutoff m=18, the retained frequency indices are -18,…,18: 37 bins including the zero-frequency bin. The measured values were lrr method · RMSE · maximum absolute error global Fourier truncation · 5.4023×10^-2 · 3.4968×10^-1 known edge packet · 1.3692×10^-4 · 1.4503×10^-3 estimated edge","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-38a6461e5f88ab3d","tex":"\\triangle","aliases":"triangle","display":"\\triangle","kind":"use","excerpt":"[Packet perturbation bound] For 1≤ p<∞, interpreted first as an inequality of extended L^p seminorms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise sup","count":15},{"work":"paper-v","url":"notation/paper-v.html#symbol-2e46faccabfe558b","tex":"U_{-Q}","aliases":"U_-Q","display":"U_{-Q}","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-836830fc78b794aa","tex":"u_k","aliases":"u_k","display":"u_k","kind":"use","excerpt":"For the limit, put u_k=(k+tfrac12)/n. The identity Q_n(x_n)=frac1π n∑_k=0^n-1fracsin(π u_k)u_k ⟶frac1π∫_0^1fracsin(π u)u du =fracSi(π)π is a midpoint Riemann sum, since the integrand has continuous value π at zero. This is the normalized classical example in [source reference].","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-b7e46954a0250371","tex":"U_Q","aliases":"U_Q","display":"U_Q","kind":"use","excerpt":"Let P(z) and Q(z) be Laurent polynomials. Define L_P(z) = 1 · 0 P(z) · 1 , U_Q(z) = 1 · Q(z) 0 · 1 .","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c41edb137b53eb22","tex":"U_Q^{-1}","aliases":"U_Q^-1","display":"U_Q^{-1}","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a822fb370eb1244d","tex":"u_r","aliases":"u_r","display":"u_r","kind":"use","excerpt":"For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-3009c54a4dec532a","tex":"U_{\\rm root}","aliases":"U_rm root rm","display":"U_{\\rm root}","kind":"use","excerpt":"Proceed by induction on the depth of the retained subtree. If the root is retained, the statement is immediate. If the root is split, apply the unitary map U_rm root, then apply the induction hypothesis independently in each child subtree. A direct sum of unitary maps is unitary. Reversing the maps with adjoints gives synthesis.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"[Dynamic-programming best packet basis] Suppose J_y(Λ)=∑_v∈Λc_v(y). For a node v, define recursively V_y(v) = c_v(y), · v terminal, [2mm] displaystyle min{c_v(y),∑_u∈ch(v)V_y(u)}, · v internal. Stop at v when c_v(y)≤∑_uV_y(u), including equality; refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the node c","count":11},{"work":"paper-v","url":"notation/paper-v.html#symbol-a8ee3eea11f4b48b","tex":"V^{-1}","aliases":"V^-1","display":"V^{-1}","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"Writing Γ=u+ i v, the grid equations are (u-fracr1+r)^2+v^2=frac1(1+r)^2, and, for x≠0, (u-1)^2+(v-frac1x)^2=frac1x^2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-625e56119b06c003","tex":"V_y","aliases":"V_y","display":"V_y","kind":"use","excerpt":"[Dynamic-programming best packet basis] Suppose J_y(Λ)=∑_v∈Λc_v(y). For a node v, define recursively V_y(v) = c_v(y), · v terminal, [2mm] displaystyle min{c_v(y),∑_u∈ch(v)V_y(u)}, · v internal. Stop at v when c_v(y)≤∑_uV_y(u), including equality; refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the node c","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-4a48390f6d6d1a30","tex":"W^{2,p}","aliases":"W^2,p","display":"W^{2,p}","kind":"use","excerpt":"l smooth remainder. headThe normalization makes the trace coefficient literal The general packet is ϕ_m,τ(t)=(t-τ)_+^m/m!. Its derivative of order m has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial. Here the remainder has matching value and first-der","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-38938f66d3d8dac8","tex":"w_{\\rm Smith}","aliases":"w_rm Smith rm","display":"w_{\\rm Smith}","kind":"use","excerpt":"Away from its pole Γ=1, the inverse Cayley map has derivative fracdzdΓ=frac2(1-Γ)^2, while normalized admittance has derivative proportional to (1+Γ)^-2. Away from Γ=±1, this motivates the symmetric sensitivity w_rm Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-377769d6a9c94ee0","tex":"W_v","aliases":"W_v","display":"W_v","kind":"use","excerpt":"al positions below are terminal packet addresses, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actua","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f307e7eaf628bf4e","tex":"W_v^*","aliases":"W_v^*","display":"W_v^*","kind":"use","excerpt":"al positions below are terminal packet addresses, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actua","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-afd76799f82fee6b","tex":"W_{\\ell,\\alpha}^*","aliases":"W_ℓ,α^* ell alpha","display":"W_{\\ell,\\alpha}^*","kind":"use","excerpt":"Set A=∑_α W_ℓ,α^*W_ℓ,α. This is a finite sum of bounded operators. The energy result says ⟨ x,Ax⟩=⟨ x,x⟩ for every x. Complex polarization, applied also to x+y and x+ i y, gives ⟨ y,(A-I)x⟩=0 for all x,y, hence A=I. Applying the operator identity to x proves synthesis. This argument uses the actual Hilbert adjoints of the decimation operators, not ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-653fb7da5fc9b85e","tex":"W_{\\ell,\\alpha}^{*}","aliases":"W_ℓ,α^* ell alpha","display":"W_{\\ell,\\alpha}^{*}","kind":"use","excerpt":"[Wavelet-packet reconstruction and energy] At every finite depth ℓ, x=∑_α∈{0,1}^ℓW_ℓ,α^*W_ℓ,αx, and x _2^2 = ∑_α∈{0,1}^ℓ W_ℓ,αx _2^2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":16},{"work":"paper-v","url":"notation/paper-v.html#symbol-65c16e51b7f7e52e","tex":"\\widehat a_q","aliases":"widehat a_q widehat","display":"\\widehat a_q","kind":"use","excerpt":"[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat ","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-e7a695b8f5409721","tex":"\\widehat c","aliases":"widehat c widehat","display":"\\widehat c","kind":"use","excerpt":"[Coefficient perturbations in a unitary tree] Let c= A_ Lx be the leaf coefficients of an admissible unitary pruning, and let widehat c=c+e. If widehat x is synthesized from widehat c, then widehat x-x = e . Thus coefficient quantization and coefficient noise are neither amplified nor attenuated by exact unitary synthesis.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"The reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A va","count":14},{"work":"paper-v","url":"notation/paper-v.html#symbol-f605f803980a555f","tex":"\\widehat f_m","aliases":"widehat f_m widehat","display":"\\widehat f_m","kind":"use","excerpt":"The reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A va","count":10},{"work":"paper-v","url":"notation/paper-v.html#symbol-d68f21ff18b92f60","tex":"\\widehat j_m","aliases":"widehat j_m widehat","display":"\\widehat j_m","kind":"use","excerpt":"[Estimated packet Gibbs bound] Let Q_m be linear and uniformly bounded on L^p, with sup_m Q_m ≤ C_Q. For f=s+j and an estimated packet widehat j_m, define widehat f_m =Q_m(f-widehat j_m)+widehat j_m. Then f-widehat f_m _p ≤ s-Q_ms _p +(1+C_Q) j-widehat j_m _p. Consequently, smooth approximation and packet recovery may be improved independently. No ","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-7d3704d4c4eab86e","tex":"\\widehat s","aliases":"widehat s widehat","display":"\\widehat s","kind":"use","excerpt":"If s,widehat s are measurable smooth-background models, the same triangle inequality gives, for finite p, f-(widehat s+widehat j) _p ≤ s-widehat s _p +|a-widehat a|μ(Ω)^1/p +|widehat a|μ(ΩtrianglewidehatΩ)^1/p. This is the correct separation of errors. Detection error is not hidden inside a generic reconstruction score.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-e8d6f1354dafb866","tex":"\\widehat x","aliases":"widehat x widehat","display":"\\widehat x","kind":"use","excerpt":"[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-71ff495a798a786e","tex":"\\widehat{\\bm X}","aliases":"widehat X widehat bm","display":"\\widehat{\\bm X}","kind":"use","excerpt":"with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input transforms are absolutely convergent, this gives widehat X(z)= R(z) E(z) X(z).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-52b9530420892ef6","tex":"\\widehat\\Kcal","aliases":"widehatKcal widehat K widehat Kcal widehat mathcal","display":"\\widehat{\\mathcal K}","kind":"use","excerpt":"[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ad8c5d8ce79e0069","tex":"\\widehat\\tau","aliases":"widehatτ widehat tau","display":"\\widehat\\tau","kind":"use","excerpt":"reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value ","count":20},{"work":"paper-v","url":"notation/paper-v.html#symbol-85469b8bf7bc0c74","tex":"\\widehat\\tau_q","aliases":"widehatτ_q widehat tau","display":"\\widehat\\tau_q","kind":"use","excerpt":"[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat ","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-48da1c2ead470996","tex":"\\widehat\\Omega","aliases":"widehatΩ widehat Omega","display":"\\widehat\\Omega","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":20},{"work":"paper-v","url":"notation/paper-v.html#symbol-47c52f415949a6cb","tex":"\\widetilde C","aliases":"widetilde C widetilde","display":"\\widetilde C","kind":"use","excerpt":"For every admissible pruning, |C( L)-widetilde C( L)| ≤ ε_*. Evaluate widetilde C on a minimizer for C to obtain widetilde F_*≤ F_*+ε_*. Reverse the roles to obtain F_*≤widetilde F_*+ε_*.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-2500eabb3d3f0489","tex":"\\widetilde c","aliases":"widetilde c widetilde","display":"\\widetilde c","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-20f230bba299446b","tex":"\\widetilde E","aliases":"widetilde E widetilde","display":"\\widetilde E","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op,","count":9},{"work":"paper-v","url":"notation/paper-v.html#symbol-ad91da91d6be3c40","tex":"\\widetilde F","aliases":"widetilde F widetilde","display":"\\widetilde F","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-d74a2e10811483dc","tex":"\\widetilde f","aliases":"widetilde f widetilde","display":"\\widetilde f","kind":"use","excerpt":"Let f=s+j where s belongs to a smooth approximation class and j is a known discontinuity packet. Let Q_m be any approximation operator applied only to s, and define widetilde f_m=Q_ms+j.","count":15},{"work":"paper-v","url":"notation/paper-v.html#symbol-84d1614ea0ca82b5","tex":"\\widetilde F_*","aliases":"widetilde F_* widetilde","display":"\\widetilde F_*","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-aed30d5003ca3472","tex":"\\widetilde f_m","aliases":"widetilde f_m widetilde","display":"\\widetilde f_m","kind":"use","excerpt":"Let f=s+j where s belongs to a smooth approximation class and j is a known discontinuity packet. Let Q_m be any approximation operator applied only to s, and define widetilde f_m=Q_ms+j.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-a1dfb1836c1fd75b","tex":"\\widetilde R","aliases":"widetilde R widetilde","display":"\\widetilde R","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op,","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-c77170c848a2e8b4","tex":"\\widetilde x","aliases":"widetilde x widetilde","display":"\\widetilde x","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-8007fd9fd0b5cd21","tex":"\\bm X","aliases":"X bm","display":"\\bm X","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Writing Γ=u+ i v, the grid equations are (u-fracr1+r)^2+v^2=frac1(1+r)^2, and, for x≠0, (u-1)^2+(v-frac1x)^2=frac1x^2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"put 1; draw[fill=black!14,draw=black!40,line width=.35pt] (2.20000,1.30000) rectangle ++(4.10000,1.05000); node[] at (4.25000,1.82500) D + 2; draw[fill=black!0,draw=black!40,line width=.35pt] (6.30000,1.30000) rectangle ++(4.10000,1.05000); node[] at (8.35000,1.82500) 1 - D; node[] at (4.25000,3.95000) input 0; node[] at (8.35000,3.95000) input 1; ","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-9b816b84df56eb81","tex":"x_F","aliases":"x_F","display":"x_F","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-8e43b4db73042e9a","tex":"x_{N-1}","aliases":"x_N-1","display":"x_{N-1}","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].","count":11},{"work":"paper-v","url":"notation/paper-v.html#symbol-0ef07195fe759ac3","tex":"x_{r-1}","aliases":"x_r-1","display":"x_{r-1}","kind":"use","excerpt":"[Quadratic nearest neighbor interaction] For y(n)=x(n)x(n-1) and modulus N, y_r(q) = x_r(q)x_r-1(q), · r≥1, x_0(q)x_N-1(q-1), · r=0. The carry at residue zero is the nonlinear analogue of a transfer crossing a coarse cell boundary.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-3e41ed00535c2ae0","tex":"x_{r_a}","aliases":"x_r_a","display":"x_{r_a}","kind":"use","excerpt":"[Translation covariance] Fix a∈ Z^d. For every r∈ R, write uniquely r-a=r_a+λ_a,r, r_a∈ R, λ_a,r∈ L. Then ( A_LT_ax)_r(ℓ) = x_r_a(ℓ+λ_a,r). Thus a fine translation permutes the lattice packets and applies an internal translation on L.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-dde230160dab3203","tex":"x_{r_j}","aliases":"x_r_j","display":"x_{r_j}","kind":"use","excerpt":"0, · r≥ b_j. Then the output channel is y_r(q) = ∏_j=1^m x_r_j (q-c_j-κ_r,b_j).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-51c75ae68c337281","tex":"x_{r_k}","aliases":"x_r_k","display":"x_{r_k}","kind":"use","excerpt":"Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-2ee20c14455ce834","tex":"X_s","aliases":"X_s","display":"X_s","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e8786e121e9dc2d7","tex":"x_s","aliases":"x_s","display":"x_s","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":9},{"work":"paper-v","url":"notation/paper-v.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-4ec4142068739e9a","tex":"x_\\ell","aliases":"x_ℓ ell","display":"x_\\ell","kind":"use","excerpt":"Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-7fc54602aaec29a7","tex":"\\bm Y","aliases":"Y bm","display":"\\bm Y","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"use","excerpt":"tem If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y'","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"note one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contribution to the even output is (2D+D^2)f_1. The tap at original delay one reaches the preceding odd channel sample; the tap at delay three reaches the odd channel two positions back. These are different delays despite coming from the same input phas","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"note one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contribution to the even output is (2D+D^2)f_1. The tap at original delay one reaches the preceding odd channel sample; the tap at delay three reaches the odd channel two positions back. These are different delays despite coming from the same input phas","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-b050e04b7e254aa7","tex":"y_k","aliases":"y_k","display":"y_k","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-d39b62f0bd31765f","tex":"y_{k+1}","aliases":"y_k+1","display":"y_{k+1}","kind":"use","excerpt":"me f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Define the adjacent contrast D_k=y_k+1-y","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-57348dda8e2fb02d","tex":"y_r","aliases":"y_r","display":"y_r","kind":"use","excerpt":"[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Let Z_0>0 be a reference impedance and z=fracZZ_0=r+ i x. The reflection coefficient is Γ=fracz-1z+1, z=frac1+Γ1-Γ.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"vpage15Retain the jump and approximate the other packetAN EXACT IDENTITY / A CONDITIONAL INPUTTraditional monograph: Theorem 6.1 (p. 12), Corollary 6.2 (p. 12). Return to the running signal f=s+j. For a concrete finite calculation, use the periodic three-tap smoother (Qx)[n]=tfrac14x[n-1]+tfrac12x[n]+tfrac14x[n+1], n∈ Z/16 Z. The plots compare smoo","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-449753faf1f7a481","tex":"z^{-1}","aliases":"z^-1","display":"z^{-1}","kind":"use","excerpt":"Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-81c0a101b2f89771","tex":"z^{-j}","aliases":"z^-j","display":"z^{-j}","kind":"use","excerpt":"The last factor is x_s[n-m], giving the time-domain formula. Under the additional transform hypothesis, each delayed channel has transform ∑_nx_s[n-m]z^-n =z^-m∑_jx_s[j]z^-j. Absolute summability permits this bijective reindexing, and only finitely many delayed channels are combined. Taking their finite linear combination gives the matrix formula. ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-fa4a9b335ff4f90f","tex":"z^{-m}","aliases":"z^-m","display":"z^{-m}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-ba52b051dd893cc4","tex":"z^{-n}","aliases":"z^-n","display":"z^{-n}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"use","excerpt":"Linear time invariance is not required for exact packet coordinates. Let A and B be formal power series over a commutative coefficient ring (in particular C). Set A(z)=∑_n≥0a_nz^n, A_i=P_N,iA. All phase indices range from 0 to N-1. For another representation B(z), multiplication gives a nonlinear or bilinear interaction.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-8e5dd5cdbfbaa44e","tex":"z^{q_k}","aliases":"z^q_k","display":"z^{q_k}","kind":"use","excerpt":"Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e8394415f576ddd6","tex":"Z_0","aliases":"Z_0","display":"Z_0","kind":"use","excerpt":"Let Z_0>0 be a reference impedance and z=fracZZ_0=r+ i x. The reflection coefficient is Γ=fracz-1z+1, z=frac1+Γ1-Γ.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-42f5a664e208866e","tex":"\\ZZ^2","aliases":"ZZ^2 Z^2 ZZ mathbb","display":"{\\mathbb Z}^2","kind":"use","excerpt":"[Rectangular image phases] For L=M Z× N Z⊂ Z^2, one may take R={0,…,M-1}×{0,…,N-1}. The analysis map separates an image into MN exact phases. Refining (M,N) to (M',N'), where Mmid M' and Nmid N', splits every coarse phase into fracM'MfracN'N fine phases.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":98},{"work":"paper-v","url":"notation/paper-v.html#symbol-6567adae36ef90db","tex":"|^{-2}","aliases":"|^-2","display":"|^{-2}","kind":"use","excerpt":"arepsilon=1/100, the actual increment is 20/9, while derivative times increment is 2. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation. headWhat a proposed weight can and cannot certify Impedance is sensitive near Γ=1, and admittance near Γ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-355b68ae43033677","tex":"|^{1/p}","aliases":"|^1/p","display":"|^{1/p}","kind":"use","excerpt":"reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value ","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-06b81a055469dbd1","tex":"|^p","aliases":"|^p","display":"|^p","kind":"use","excerpt":"Every address has a unique representation r+ℓ. For finite p, x _p^p = ∑_r∈ R∑_ℓ∈ L|x(r+ℓ)|^p = ∑_r∈ R x_r _p^p. The corresponding supremum identity is immediate when p=∞. Unique address representation proves that analysis and interleaving are inverse maps. The projector identities follow from the disjoint coset partition.","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"Away from its pole Γ=1, the inverse Cayley map has derivative fracdzdΓ=frac2(1-Γ)^2, while normalized admittance has derivative proportional to (1+Γ)^-2. Away from Γ=±1, this motivates the symmetric sensitivity w_rm Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"tfrac52(1_Ω-1_widehatΩ). The missed sample has error 3, the five shared samples have error 1/2, and the extra sample has error -5/2. With counting measure, j-widehat j _1=8, j-widehat j _2^2=tfrac332, j-widehat j _∞=3. headRead the theorem as a budget, not an equality claim Here the L^1 bound is attained: |3-tfrac52| |Ω|+tfrac52|ΩtrianglewidehatΩ| ","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"n division gives a unique pair (r,q) with 0≤ r<N and n=Nq+r, including negative n. The address maps n↦(nbmod N,nmathbindivN) and (r,q)↦ Nq+r are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum ∑_n|x[n]|^2 proves the norm identity and shows that both maps preserve square summability. In Lean the targe","count":60},{"work":"paper-v","url":"notation/paper-v.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"W(γ) · filter or path weight · refinement or implementation cost","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-c57377872c9fe8eb","tex":"\\Gamma_y","aliases":"Γ_y Gamma","display":"\\Gamma_y","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Δ(F) · coarse/fine signal defect · aliasing, leakage, or estimator inconsistency","count":19},{"work":"paper-v","url":"notation/paper-v.html#symbol-3f0f0e9cd378ed81","tex":"\\Delta^2","aliases":"Δ^2 Delta","display":"\\Delta^2","kind":"use","excerpt":"vpage39Higher differences annihilate polynomial interiorsBOUNDARY STENCILS / SIGNAL RESPONSE AND NOISE GAINTraditional monograph: Theorem E10.6 (p. 71), Proposition E10.7 (p. 71), Corollary E10.8 (p. 71). Let x(n)=n+2(n-3)_+ on the integer line. Its slope changes from one to three at the breakpoint 3. The second forward difference is Δ^2x(n)=x(n+2)","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"use","excerpt":"[A perturbed disk] Let E and F be concentric disks of radii r and r+δ in R^2, with δ>0. Then μ(Emathbintriangle F) = π((r+δ)^2-r^2) = 2π rδ+πδ^2. For equal amplitude a, the exact L^2 packet error is |a|sqrt2π rδ+πδ^2. This gives the expected square root dependence on boundary displacement.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-02c68d637b81b90c","tex":"\\delta^3","aliases":"δ^3 delta","display":"\\delta^3","kind":"use","excerpt":"y on the interval between its two locations: H_τ-H_widehatτ _2=sqrtδ=tfrac14, H_τ-H_widehatτ _∞=1. vfig[.89]ext_location headFor a ramp, the error height shrinks too Write r_τ(t)=(t-τ)_+. The difference grows linearly from zero to δ on [τ,widehatτ] and stays at δ on [widehatτ,1]. Direct integration gives r_τ-r_widehatτ _2^2 =fracδ^33+(1-widehatτ)δ^","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-11319bcf2c25f823","tex":"\\Delta^{d+1}","aliases":"Δ^d+1 Delta","display":"\\Delta^{d+1}","kind":"use","excerpt":"[Polynomial annihilation and boundary support] Let x be a sequence that agrees with a polynomial of degree at most d on each interval of a finite partition of Z. Then Δ^d+1x(n)=0 whenever the complete stencil {n,n+1,…,n+d+1} lies inside one polynomial piece. Hence Δ^d+1x is supported in a band of at most d+1 addresses before each breakpoint.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-75ac6af28af33827","tex":"\\Delta^m","aliases":"Δ^m Delta","display":"\\Delta^m","kind":"use","excerpt":"Let (Δ x)(n)=x(n+1)-x(n). Then Δ^m x(n) = ∑_r=0^m (-1)^m-rbinom mr x(n+r).","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-9adbb02c2ad54bb0","tex":"\\delta_1","aliases":"δ_1 delta","display":"\\delta_1","kind":"use","excerpt":"DIFFERENCE VISIBLETraditional monograph: Theorem 4.1 (p. 7). A one-sample delay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is J(D), not DI_2. vfigsample_delay headTrace one impulse The input δ_0 has a single nonzero value in even channel address zero. Under J(D) it appears in odd channel address zero: th","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-dac3f880e6e3670a","tex":"\\delta_2","aliases":"δ_2 delta","display":"\\delta_2","kind":"use","excerpt":"ay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is J(D), not DI_2. vfigsample_delay headTrace one impulse The input δ_0 has a single nonzero value in even channel address zero. Under J(D) it appears in odd channel address zero: this is the original sample δ_1. Under DI_2 it stays even but moves to channel ","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","excerpt":"The translation orbit is the standard coordinate family (δ_a)_a∈ Z^d. Its finite span is the space of finitely supported sequences. Finitely supported sequences are dense in ℓ^p for finite p and in c_0. Their closure in the supremum norm is c_0, which is a proper closed subspace of ℓ^∞.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ea7bc2e5a161c9f5","tex":"\\delta_E","aliases":"δ_E delta","display":"\\delta_E","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-ab767052c24571cd","tex":"\\delta_R","aliases":"δ_R delta","display":"\\delta_R","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-51a87ccbd55b1aa4","tex":"\\delta_{rs}","aliases":"δ_rs delta","display":"\\delta_{rs}","kind":"use","excerpt":"[Polyphase projector theorem] The maps P_N,r are bounded orthogonal projectors on ℓ^2, with operator norm at most one, and satisfy P_N,rP_N,s=δ_rsP_N,r, ∑_r=0^N-1P_N,r=I. Consequently, x=∑_r=0^N-1P_N,rx with no approximation.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"For each fixed δ>0, b_h,δ→0. Apply [source reference]. For any error tolerance ϵ>0, the high-probability bound is eventually smaller than ϵ, so the limit superior of the probability that the packet error exceeds ϵ is at most δ. Since δ is arbitrary, convergence is in probability.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"[Local perturbation certificate] Assume |c_v(y')-c_v(y)|≤ε_v. Define E(v) = ε_v, · v terminal,","count":49},{"work":"paper-v","url":"notation/paper-v.html#symbol-a2f9c21b859a02d5","tex":"\\varepsilon_*","aliases":"ε_* varepsilon","display":"\\varepsilon_*","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-bae7842f2d63f7f5","tex":"\\varepsilon_a","aliases":"ε_a varepsilon","display":"\\varepsilon_a","kind":"use","excerpt":"[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-e70f355537b1cd85","tex":"\\varepsilon_h","aliases":"ε_h varepsilon","display":"\\varepsilon_h","kind":"use","excerpt":"[Deterministic consistency] Assume h→0 along grids satisfying the preceding sampling assumptions, ε_h→0, the number of jumps is fixed, and the nonzero jump amplitudes and positive minimum separation remain fixed. Then the detected packet converges to the true packet in every finite L^p norm. If Q_m has a vanishing smooth approximation error and uni","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-5b018625faf1192a","tex":"\\varepsilon_{\\mathrm{rec}}","aliases":"ε_rec varepsilon mathrm","display":"\\varepsilon_{\\mathrm{rec}}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-23404397ce55a3d7","tex":"\\varepsilon_v","aliases":"ε_v varepsilon","display":"\\varepsilon_v","kind":"use","excerpt":"[Local perturbation certificate] Assume |c_v(y')-c_v(y)|≤ε_v. Define E(v) = ε_v, · v terminal,","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-fc03a685a0a01ebc","tex":"\\eta_k","aliases":"η_k eta","display":"\\eta_k","kind":"use","excerpt":"t_k=kh, k=0,…,n, h=frac1n. Assume f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Define th","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-a3874982d0a28e3b","tex":"\\eta_{k+1}","aliases":"η_k+1 eta","display":"\\eta_{k+1}","kind":"use","excerpt":"In a cell without a jump, D_k=s(t_k+1)-s(t_k)+η_k+1-η_k. The Lipschitz and noise bounds give |D_k|≤ Mh+2ε. In a cell with one jump, the same formula has the additional term a_q, which proves the second estimate.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.","count":23},{"work":"paper-v","url":"notation/paper-v.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j, 0, · r≥ b_j. Then the output channel is y_r(q) = ∏_j=1^m x_r_j (q-c_j-κ_r,b_j).","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-6aa4d5926406a9c8","tex":"\\kappa_{r,b_j}","aliases":"κ_r,b_j kappa","display":"\\kappa_{r,b_j}","kind":"use","excerpt":"[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j, 0, · r≥ b_j. Then the output channel is y_r(q) = ∏_j=1^m x_r_j (q-c_j-κ_r,b_j).","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-308a821eda6e7229","tex":"\\lambda^2","aliases":"λ^2 lambda","display":"\\lambda^2","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-6559d219a1ff939b","tex":"\\Lambda_*","aliases":"Λ_* Lambda","display":"\\Lambda_*","kind":"use","excerpt":")+V(v1)}, V(v)=c_v at a terminal node. headRead the calculation from the bottom upward At 10, the children cost 1+1=2, better than stopping for 7. At 11, stopping costs 4, better than 2+3=5. The right subtree therefore costs 6, better than its stop cost 8. The left subtree costs 1+2=3, better than 5. At the root, 3+6=9 beats 20. The unique optimum ","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-def4f51f0a6a5703","tex":"\\lambda_{a,r}","aliases":"λ_a,r lambda","display":"\\lambda_{a,r}","kind":"use","excerpt":"[Translation covariance] Fix a∈ Z^d. For every r∈ R, write uniquely r-a=r_a+λ_a,r, r_a∈ R, λ_a,r∈ L. Then ( A_LT_ax)_r(ℓ) = x_r_a(ℓ+λ_a,r). Thus a fine translation permutes the lattice packets and applies an internal translation on L.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-f8396a195799c469","tex":"\\Lambda_{y'}^*","aliases":"Λ_y'^* Lambda","display":"\\Lambda_{y'}^*","kind":"use","excerpt":"da_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ∈ P( T) |J_y'(Λ)-J_y(Λ)| <fracΓ_y2, then Λ_y'^*=Λ_y^*. item Every selected mini","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-9159b41ba5090052","tex":"\\Lambda_y^*","aliases":"Λ_y^* Lambda","display":"\\Lambda_y^*","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion; no real-valued minimum over an","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-c645657d025cc351","tex":"\\Xi","aliases":"Ξ Xi","display":"\\Xi","kind":"use","excerpt":"set, so the feasible set is nonempty and finite. Thus the minimum is attained. If competitors exist, their finitely many differences from a unique optimum are strictly positive, so their minimum is positive and is achieved by a competitor. In the singleton case there is no alternative selector under any objective change. In the remaining case, for ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.","count":20},{"work":"paper-v","url":"notation/paper-v.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.","count":21},{"work":"paper-v","url":"notation/paper-v.html#symbol-bd1aac0f414abebd","tex":"\\rho^2","aliases":"ρ^2 rho","display":"\\rho^2","kind":"use","excerpt":"For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"on be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If a_min>2b_h, then every threshold between b_h and a_min-b_h detects exactly the cells containing jumps. The jump locations are recovered within h/2, and each amplitude error is at most b_h. The probability theorem replaces the determinist","count":23},{"work":"paper-v","url":"notation/paper-v.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-7d8fec8814f298e9","tex":"\\sigma_n","aliases":"σ_n sigma","display":"\\sigma_n","kind":"use","excerpt":"[Probability consistency] Let σ_nsqrtlog n→0 and h=1/n. For every fixed 0<δ<1, along grids satisfying the preceding sampling assumptions, all jump cells are recovered with probability at least 1-δ for sufficiently large n. For the fixed finite jump model above, along grids satisfying the preceding sampling assumptions, the packet error converges to","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-35af929f24440334","tex":"\\tau_0","aliases":"τ_0 tau","display":"\\tau_0","kind":"use","excerpt":"[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one dimensional Sobolev representatives whose derivatives through order r-1 have one sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b7d379c8c3c32c66","tex":"\\tau_{Q+1}","aliases":"τ_Q+1 tau","display":"\\tau_{Q+1}","kind":"use","excerpt":"[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one dimensional Sobolev representatives whose derivatives through order r-1 have one sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f0b8695b045c81fb","tex":"\\tau_{q+1}","aliases":"τ_q+1 tau","display":"\\tau_{q+1}","kind":"use","excerpt":"[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one dimensional Sobolev representatives whose derivatives through order r-1 have one sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"[Cyclic Mobius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.","count":6},{"work":"paper-v","url":"notation/paper-v.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"[Cyclic Mobius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-52d7460fdf5044c4","tex":"\\phi_{0,1/3}","aliases":"ϕ_0,1/3 phi","display":"\\phi_{0,1/3}","kind":"use","excerpt":"vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPTraditional monograph: Theorem E7.2 (p. 56), Corollary E7.3 (p. 57). On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f1bca9bc547dc424","tex":"\\phi_{0,\\tau}","aliases":"ϕ_0,τ phi tau","display":"\\phi_{0,\\tau}","kind":"use","excerpt":"For m≥0 and 0<τ<1, define ϕ_m,τ(t) = frac(t-τ)_+^mm!. Thus ϕ_0,τ= 1_[τ,1]. On each side of τ, the packet is a polynomial. Its derivative of order m has a unit jump at τ.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-cbe472b688e320fa","tex":"\\phi_{1,2/3}","aliases":"ϕ_1,2/3 phi","display":"\\phi_{1,2/3}","kind":"use","excerpt":"vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPTraditional monograph: Theorem E7.2 (p. 56), Corollary E7.3 (p. 57). On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-16543cca3869d6ab","tex":"\\phi_{m,u}","aliases":"ϕ_m,u phi","display":"\\phi_{m,u}","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-0cdaa5db46e04d3f","tex":"\\phi_{m,\\widehat\\tau}","aliases":"ϕ_m,widehatτ phi widehat tau","display":"\\phi_{m,\\widehat\\tau}","kind":"use","excerpt":"[Location stability for higher order packets] Let m≥1, 1≤ p<∞, and 0≤τ,widehatτ≤1. Then ϕ_m,τ-ϕ_m,widehatτ _L^p(0,1) ≤ frac|τ-widehatτ| (m-1)!((m-1)p+1)^1/p. For p=∞, ϕ_m,τ-ϕ_m,widehatτ _∞ ≤ frac|τ-widehatτ|(m-1)!. Thus location error is Lipschitz for every derivative jump packet of positive order, while a value jump has only the finite L^p modulus","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-d26fdcce1d9ab87d","tex":"\\phi_{m,\\tau}^{(\\ell)}","aliases":"ϕ_m,τ^(ℓ) phi tau ell","display":"\\phi_{m,\\tau}^{(\\ell)}","kind":"use","excerpt":"For 0≤ ℓ≤ m, the classical derivative of ϕ_m,τ on the two open sides is ϕ_m,τ^(ℓ)(t) = frac(t-τ)_+^m-ℓ(m-ℓ)!. This derivative has zero jump when ℓ<m and a unit jump when ℓ=m. Derivatives of order larger than m vanish on both open sides. It follows that subtracting the term a_q,mϕ_m,τ_q removes exactly the jump in the derivative of order m without c","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-d1ae6e52152d86f1","tex":"\\phi_{m,\\tau_q}","aliases":"ϕ_m,τ_q phi tau","display":"\\phi_{m,\\tau_q}","kind":"use","excerpt":"_packet_decomposition Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one dimensional Sobolev representatives whose derivatives through order r-1 have one sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-dadc02b74543d1bf","tex":"\\phi_{m-1,u}","aliases":"ϕ_m-1,u phi","display":"\\phi_{m-1,u}","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-a515509f5caa010d","tex":"\\Phi_\\theta","aliases":"Φ_θ Phi theta","display":"\\Phi_\\theta","kind":"use","excerpt":"[Cyclic Mobius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-1c48d80b7bf9e478","tex":"\\Phi_{\\theta+\\varphi}","aliases":"Φ_θ+φ Phi theta varphi","display":"\\Phi_{\\theta+\\varphi}","kind":"use","excerpt":"[Cyclic Mobius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-1d5fdec04e946351","tex":"\\Phi_\\pi","aliases":"Φ_π Phi pi","display":"\\Phi_\\pi","kind":"use","excerpt":"[Impedance and admittance] For N=2, θ=π and Φ_π(z)=frac1z. Thus the two phase action is normalized impedance and admittance inversion.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c1eb035f6ff0ebc6","tex":"\\Phi_\\varphi","aliases":"Φ_φ Phi varphi","display":"\\Phi_\\varphi","kind":"use","excerpt":"[Cyclic Mobius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Let I_j,k = [k2^-j,(k+1)2^-j) and let ψ_j,k be the normalized Haar wavelet supported on I_j,k, positive on the left half and negative on the right half.","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-cfc034372e2b22b5","tex":"\\psi_{j,k}","aliases":"ψ_j,k psi","display":"\\psi_{j,k}","kind":"use","excerpt":"Let I_j,k = [k2^-j,(k+1)2^-j) and let ψ_j,k be the normalized Haar wavelet supported on I_j,k, positive on the left half and negative on the right half.","count":7},{"work":"paper-v","url":"notation/paper-v.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":23},{"work":"paper-v","url":"notation/paper-v.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.","count":16},{"work":"paper-v","url":"notation/paper-v.html#symbol-5d52ab32e3afcb63","tex":"\\omega_2","aliases":"ω_2 omega","display":"\\omega_2","kind":"use","excerpt":"[Impedance--admittance is the two-phase split] For N=2 and finite z≠0, T_2,1(z)=z^-1. On the full projective line it exchanges 0 and ∞. Thus a half-turn of the reflection disk exchanges normalized impedance and admittance. ω_2=exp(π i)=-1, so A(ω_2)[u:v]=[v:u]. The finite ratio is 1/z when z≠0; the coordinate swap also gives [0:1]↔[1:0]. No ordinar","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-7a87553bb10c05ec","tex":"\\omega_N","aliases":"ω_N omega","display":"\\omega_N","kind":"use","excerpt":"Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-22344ff7669bd2c8","tex":"\\omega_N^k","aliases":"ω_N^k omega","display":"\\omega_N^k","kind":"use","excerpt":"Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-35ebb38a9f8cbb25","tex":"\\omega_N^N","aliases":"ω_N^N omega","display":"\\omega_N^N","kind":"use","excerpt":"1-w · 1+w. It has determinant w≠0, preserves nonzero pairs, and commutes with rescaling, so it induces a projective map. Direct multiplication gives A(w)A(w')=A(ww') and A(1)=I. Since ω_N^N=1, this proves the cyclic composition law modulo N. The Cayley coordinate of [u:v] is [u-v:u+v], and applying A(w) changes this pair to [w(u-v):u+v]. This prove","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a538ce476ad5b75e","tex":"\\Omega_q","aliases":"Ω_q Omega","display":"\\Omega_q","kind":"use","excerpt":"[Edge-packet approximation class] There should exist natural function spaces in which a signal is represented as f=s+∑_q a_q1_Ω_q, with separate norms for smooth approximation, boundary geometry, packet amplitude, and packet identification. A projector-resolved approximation theory on such spaces should yield rates that do not mix smooth truncation","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":13},{"work":"paper-v","url":"notation/paper-v.html#symbol-8aaa13d7254b3cae","tex":"\\ell'","aliases":"ℓ' ell","display":"\\ell'","kind":"use","excerpt":"Evaluate the output at r+ℓ. Write every input address as s+ℓ'. The convolution offset is (r+ℓ)-(s+ℓ') =(r-s)+(ℓ-ℓ'). Setting m=ℓ-ℓ' gives the formula.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"[Several completions] The same algebraic seed description applies to finite sequences, ℓ^1, ℓ^2, periodic sequences, tempered distributions, and multidimensional lattices. The topology decides which infinite reconstructions are legitimate. Finite-dimensional experiments in this paper use ordinary Euclidean norms; theorem statements specify the ambi","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-837406a230d80642","tex":"\\ell^\\infty","aliases":"ℓ^∞ ell infty","display":"\\ell^\\infty","kind":"use","excerpt":"[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":13},{"work":"paper-v","url":"notation/paper-v.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"o-phase split] For N=2 and finite z≠0, T_2,1(z)=z^-1. On the full projective line it exchanges 0 and ∞. Thus a half-turn of the reflection disk exchanges normalized impedance and admittance. ω_2=exp(π i)=-1, so A(ω_2)[u:v]=[v:u]. The finite ratio is 1/z when z≠0; the coordinate swap also gives [0:1]↔[1:0]. No ordinary division by zero is used to re","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"mal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1 is a bijection of Z, with inverse shift","count":10},{"work":"paper-v","url":"notation/paper-v.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j,","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-f5e01624215da775","tex":"\\prod_{j=1}^{m}","aliases":"∏_j=1^m prod","display":"\\prod_{j=1}^{m}","kind":"use","excerpt":"[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j,","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-550f5596bd018fbf","tex":"\\sum_{a\\in\\Lambda}","aliases":"∑_a∈Λ sum in Lambda","display":"\\sum_{a\\in\\Lambda}","kind":"use","excerpt":"[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b551f6661cc1d37f","tex":"\\sum_{i+j\\equiv k\\, (\\mathrm{mod}\\,N)}","aliases":"∑_i+j≡ k (mod N) sum equiv mathrm","display":"\\sum_{i+j\\equiv k\\, (\\mathrm{mod}\\,N)}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f69801a69bec1d00","tex":"\\sum_{i+j\\equiv k\\ (N)}","aliases":"∑_i+j≡ k (N) sum equiv","display":"\\sum_{i+j\\equiv k\\ (N)}","kind":"use","excerpt":"eserves the same address For time-residue masks, (P_N,ix)[n](P_N,jy)[n]=0 when i≠ j. Therefore P_N,k(x· y)=(P_N,kx)·(P_N,ky). Here the dot means multiplication at the same sample, not convolution. headCauchy multiplication adds degrees For formal series A(t),B(t), the coefficient of t^n is a sum over a+b=n. Consequently the output residue is i+jpmo","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-38c03c7cd3f0f240","tex":"\\sum_{i,j}","aliases":"∑_i,j sum","display":"\\sum_{i,j}","kind":"use","excerpt":"and 5, so j-widehat j _1=2+3+5=10, j-widehat j _2^2=4+9+25=38. The counting-measure supremum error is 5. The full-domain mean squared error is 38/12=19/6. A mismatch fraction of 3/12 reports a different statistic and omits all three amplitude weights. headThe formula requires partitions For measurable partitions (E_i) and (F_j), with M_ij=μ(E_i∩ F_","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-815fc827fa2f312d","tex":"\\sum_{i,k}","aliases":"∑_i,k sum","display":"\\sum_{i,k}","kind":"use","excerpt":"[Background and packet error] Suppose the true and estimated packets use the same amplitude list (a_i)_i=0^I, but possibly different regions. Then j-widehat j _p^p = ∑_i,k|a_i-a_k|^pμ(E_i∩ F_k). Only off diagonal overlaps contribute.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-bb4cae048f31bfb6","tex":"\\sum_{i=0}^{I}","aliases":"∑_i=0^I sum","display":"\\sum_{i=0}^{I}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-3c62f1b2b7e728f6","tex":"\\sum_{i_1+\\cdots+i_q\\equiv k}","aliases":"∑_i_1+⋯+i_q≡ k sum cdots equiv","display":"\\sum_{i_1+\\cdots+i_q\\equiv k}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"The last factor is x_s[n-m], giving the time-domain formula. Under the additional transform hypothesis, each delayed channel has transform ∑_nx_s[n-m]z^-n =z^-m∑_jx_s[j]z^-j. Absolute summability permits this bijective reindexing, and only finitely many delayed channels are combined. Taking their finite linear combination gives the matrix formula. ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-dd6a980c531b4292","tex":"\\sum_{j=0}^{J}","aliases":"∑_j=0^J sum","display":"\\sum_{j=0}^{J}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-fec10602e4575bf9","tex":"\\sum_{k=0}^{n-1}","aliases":"∑_k=0^n-1 sum","display":"\\sum_{k=0}^{n-1}","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-7dd1f8372600c494","tex":"\\sum_m","aliases":"∑_m sum","display":"\\sum_m","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-8e3d2691a5710f8e","tex":"\\sum_{m=0}^{r-1}","aliases":"∑_m=0^r-1 sum","display":"\\sum_{m=0}^{r-1}","kind":"use","excerpt":"_packet_decomposition Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one dimensional Sobolev representatives whose derivatives through order r-1 have one sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-09a693e06fd236f6","tex":"\\sum_{m\\in L}","aliases":"∑_m∈ L sum in","display":"\\sum_{m\\in L}","kind":"use","excerpt":"[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-da006f0561df99b4","tex":"\\sum_{m\\in\\bbZ}","aliases":"∑_m∈bbZ ∑_m∈ Z sum in bbZ sum in mathbb","display":"\\sum_{m\\in{\\mathbb Z}}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-b9a242ebebc8b392","tex":"\\sum_{m\\in\\ZZ}","aliases":"∑_m∈ZZ ∑_m∈ Z sum in ZZ sum in mathbb","display":"\\sum_{m\\in{\\mathbb Z}}","kind":"use","excerpt":"For L=N Z with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-4700a53bd785dd67","tex":"\\sum_{n=1}^{m}","aliases":"∑_n=1^m sum","display":"\\sum_{n=1}^{m}","kind":"use","excerpt":"vpage16Why a Fourier cutoff cannot reproduce this jump pointwiseA CONTINUOUS PULSE / EXACT ENDPOINT VALUESTraditional monograph: Section 6 (p. 11), Section 8 (p. 13). Use a separate period-one example on [-1/2,1/2): j(t)= 1_{|t|<1/4}, S_mj(t)=tfrac12+frac2π∑_n=1^mfracsin(π n/2)ncos(2π nt). This is the exact finite Fourier formula. The drawn curves ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-41a5548d37bd81ad","tex":"\\sum_{n\\in F}","aliases":"∑_n∈ F sum in","display":"\\sum_{n\\in F}","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-6806b62c6d27ebf6","tex":"\\sum_{n\\in\\bbZ}","aliases":"∑_n∈bbZ ∑_n∈ Z sum in bbZ sum in mathbb","display":"\\sum_{n\\in{\\mathbb Z}}","kind":"use","excerpt":"[Impulse seed theorem] The closed shift-generated hull of δ_0 is all of ℓ^2( Z): overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 =ℓ^2( Z). Every finitely supported signal has the exact finite expression x=∑_n∈ Zx[n]S^nδ_0.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-8284d03cfea3f8d1","tex":"\\sum_{n\\notin F}","aliases":"∑_n∉ F sum notin","display":"\\sum_{n\\notin F}","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ab5bb9c4725a51c8","tex":"\\sum_{n\\equiv r\\ (\\mathrm{mod}\\,N)}","aliases":"∑_n≡ r (mod N) sum equiv mathrm","display":"\\sum_{n\\equiv r\\ (\\mathrm{mod}\\,N)}","kind":"use","excerpt":"Each integer belongs to exactly one residue class modulo N. Multiplying the two indicator functions gives zero for distinct classes and the same indicator for equal classes. Summing all residue indicators gives one at every address. Moreover, P_N,rx _2^2=∑_n≡ r (mod N)|x[n]|^2≤ x _2^2, ⟨ P_N,rx,y⟩=⟨ x,P_N,ry⟩. The mask is real-valued, so the inner-","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"use","excerpt":"Linear time invariance is not required for exact packet coordinates. Let A and B be formal power series over a commutative coefficient ring (in particular C). Set A(z)=∑_n≥0a_nz^n, A_i=P_N,iA. All phase indices range from 0 to N-1. For another representation B(z), multiplication gives a nonlinear or bilinear interaction.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-441a6af1afa122e0","tex":"\\sum_{n\\geq0}","aliases":"∑_n≥0 sum geq","display":"\\sum_{n\\geq0}","kind":"use","excerpt":"[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-098c7729ca21f24d","tex":"\\sum_{q=1}^M","aliases":"∑_q=1^M sum","display":"\\sum_{q=1}^M","kind":"use","excerpt":"[Several exact edge packets] If f=s+∑_q=1^Mj_q and every j_q is retained exactly, then f-(Q_ms+∑_qj_q) = s-Q_ms . Curved edges, corners, and different jump amplitudes do not alter the identity. The finite sum j=∑_q=1^M j_q is a single retained packet. Applying the preceding cancellation identity gives the result; the empty family M=0 is included wi","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-794344618327eede","tex":"\\sum_{q=1}^{M}","aliases":"∑_q=1^M sum","display":"\\sum_{q=1}^{M}","kind":"use","excerpt":"[Several exact edge packets] If f=s+∑_q=1^Mj_q and every j_q is retained exactly, then f-(Q_ms+∑_qj_q) = s-Q_ms . Curved edges, corners, and different jump amplitudes do not alter the identity. The finite sum j=∑_q=1^M j_q is a single retained packet. Applying the preceding cancellation identity gives the result; the empty family M=0 is included wi","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c67bb6d0757ac234","tex":"\\sum_{q\\in\\bbZ}","aliases":"∑_q∈bbZ ∑_q∈ Z sum in bbZ sum in mathbb","display":"\\sum_{q\\in{\\mathbb Z}}","kind":"use","excerpt":"Let h be a finite impulse response and y=h*x, y[n]=∑_q∈ Zh[q]x[n-q]. Resolve both input and output into N phases. For output phase k and input phase s, define the transfer kernel h_ks[m]=h[Nm+k-s]. Out-of-range taps are understood to be zero.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-0606a20caaf6d2d9","tex":"\\sum_{q\\in\\ZZ^d}","aliases":"∑_q∈ZZ^d ∑_q∈ Z^d sum in ZZ sum in mathbb","display":"\\sum_{q\\in{\\mathbb Z}^d}","kind":"use","excerpt":"Let h∈ℓ^1( Z^d) and x∈ℓ^p( Z^d), with 1≤ p≤∞, and define (Hx)(n) =(h*x)(n) =∑_q∈ Z^dh(q)x(n-q). Young's inequality gives Hx _p≤ h _1 x _p for every 1≤ p≤∞.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"The signal realization makes the seed, address, packet, transfer, refinement, and synthesis operations concrete. The objects and laws can be read directly: seed · =δ_0, address · =S^n, packet · =P_N,rx, transfer · =P_N,kHP_N,s, refinement · =P_N,r=∑_ssucc rP_M,s, synthesis · =∑_rP_N,rx, nonlinearity · =P_N,rF(∑_sP_N,sx). Filter banks, wavelet packe","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-0de4c91e29511ae3","tex":"\\sum_{r+s\\equiv t\\!\\!\\pmod N}","aliases":"∑_r+s≡ t pmod N sum equiv pmod","display":"\\sum_{r+s\\equiv t\\!\\!\\pmod N}","kind":"use","excerpt":"rem[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-f161dab54b7fa5c5","tex":"\\sum_{r=0}^3","aliases":"∑_r=0^3 sum","display":"\\sum_{r=0}^3","kind":"use","excerpt":"ump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector P_4,rf is different: it keeps the original latt","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-fed0ff8a0203f247","tex":"\\sum_{r=0}^{m}","aliases":"∑_r=0^m sum","display":"\\sum_{r=0}^{m}","kind":"use","excerpt":"Let (Δ x)(n)=x(n+1)-x(n). Then Δ^m x(n) = ∑_r=0^m (-1)^m-rbinom mr x(n+r).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-029134516e4be179","tex":"\\sum_{r=0}^{N-1}","aliases":"∑_r=0^N-1 sum","display":"\\sum_{r=0}^{N-1}","kind":"use","excerpt":"[Polyphase projector theorem] The maps P_N,r are bounded orthogonal projectors on ℓ^2, with operator norm at most one, and satisfy P_N,rP_N,s=δ_rsP_N,r, ∑_r=0^N-1P_N,r=I. Consequently, x=∑_r=0^N-1P_N,rx with no approximation.","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-4c18b03cbd50956a","tex":"\\sum_{r\\in R}","aliases":"∑_r∈ R sum in","display":"\\sum_{r\\in R}","kind":"use","excerpt":"For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-7d98fe9de6bebd3c","tex":"\\sum_s","aliases":"∑_s sum","display":"\\sum_s","kind":"use","excerpt":"For a finite nonlinear shift operator F(x)[n]=x[n]+α x[n]x[n-1], the exact packet coordinates are simply F_r((x_s)_s)=P_N,rF (∑_sP_N,sx). Analysis followed by this packet system and synthesis reproduces the direct nonlinear output exactly. The numerical residual in the companion implementation was zero to machine representation.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-51618a69e67f3faa","tex":"\\sum_{s=0}^{N-1}","aliases":"∑_s=0^N-1 sum","display":"\\sum_{s=0}^{N-1}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-ed58db8837af33eb","tex":"\\sum_{s\\succ r}","aliases":"∑_ssucc r sum succ","display":"\\sum_{s\\succ r}","kind":"use","excerpt":"The signal realization makes the seed, address, packet, transfer, refinement, and synthesis operations concrete. The objects and laws can be read directly: seed · =δ_0, address · =S^n, packet · =P_N,rx, transfer · =P_N,kHP_N,s, refinement · =P_N,r=∑_ssucc rP_M,s, synthesis · =∑_rP_N,rx, nonlinearity · =P_N,rF(∑_sP_N,sx). Filter banks, wavelet packe","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-df3e5e35be464a61","tex":"\\sum_{\\substack{0\\le a<M\\\\a\\equiv r\\pmod N}}","aliases":"∑_substack0≤ a<M a≡ rpmod N sum substack le a equiv pmod","display":null,"kind":"use","excerpt":"leaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples. For example, at the first descendant address the values f[0]=0 and f[2]=1/2 are two separate values. Their sum 1/2 cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement. headThe","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-7cb7901967a508d7","tex":"\\sum_{\\substack{0\\le s<M\\\\s\\equiv r\\, (\\mathrm{mod}\\,N)}}","aliases":"∑_substack0≤ s<M s≡ r (mod N) sum substack le s equiv mathrm","display":null,"kind":"use","excerpt":"[Exact nested refinement] For N,M≥1 and Nmid M, P_N,r = ∑_substack0≤ s<M s≡ r (mod N)P_M,s.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-1d7041cea704f845","tex":"\\sum_{s\\in R}","aliases":"∑_s∈ R sum in","display":"\\sum_{s\\in R}","kind":"use","excerpt":"[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-b4be24e93748542c","tex":"\\sum_{s\\in\\operatorname{Child}(r)}","aliases":"∑_s∈Child(r) sum in operatorname","display":"\\sum_{s\\in\\operatorname{Child}(r)}","kind":"use","excerpt":"[Exact lattice refinement] For every coarse packet, P_r^L = ∑_s∈Child(r)P_s^L'. The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-d96e96aadd273752","tex":"\\sum_u","aliases":"∑_u sum","display":"\\sum_u","kind":"use","excerpt":"[2mm] displaystyle min{c_v(y),∑_u∈ch(v)V_y(u)}, · v internal. Stop at v when c_v(y)≤∑_uV_y(u), including equality; refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the node costs supplied and child values cached, the evaluator uses one comparison per internal node, one visit per node, and at most | T|-1 a","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-582da013421a516b","tex":"\\sum_{v\\in\\Lcal}","aliases":"∑_v∈Lcal ∑_v∈ L sum in Lcal sum in mathcal","display":"\\sum_{v\\in{\\mathcal L}}","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-6c2f7250b20e70fc","tex":"\\sum_{v\\in\\Lambda}","aliases":"∑_v∈Λ sum in Lambda","display":"\\sum_{v\\in\\Lambda}","kind":"use","excerpt":"[Exact synthesis from every complete pruning] For every complete pruning Λ, ∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx. Thus changing the selected tree changes the coordinates but does not introduce reconstruction error.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-2b95868d24f2bd48","tex":"\\sum_w","aliases":"∑_w sum","display":"\\sum_w","kind":"use","excerpt":"Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-22b9f62ba75edb77","tex":"\\sum_{w}","aliases":"∑_w sum","display":"\\sum_{w}","kind":"use","excerpt":"Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-a38ac3f06fcc5327","tex":"\\sum_{w\\in\\operatorname{child}(v)}","aliases":"∑_w∈child(v) sum in operatorname","display":"\\sum_{w\\in\\operatorname{child}(v)}","kind":"use","excerpt":"The packet tree results treat every admissible pruning, not only complete trees of a fixed depth. Unitary local splits give a unitary global transform. Local frame bounds propagate recursively in the nonunitary case. If the cost of a pruning is additive over retained nodes, then F(v) = min{ c(v), ∑_w∈child(v)F(w) } gives the exact optimal tree. The","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-b3187996cd33bccc","tex":"\\sum_{|\\alpha|=2}","aliases":"∑_|α|=2 sum alpha","display":"\\sum_{|\\alpha|=2}","kind":"use","excerpt":"/4, 0, 1/4, which again sum to 57. Two factors of 1/sqrt2 explain why the depth-two coefficients are rational halves and quarters. headCoefficients and projectors are different objects W_α f is the short coefficient vector. The original-space component is W_α^*W_α f. Summing these synthesized components, not the short arrays entrywise, reconstructs","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"use","excerpt":"Set A=∑_α W_ℓ,α^*W_ℓ,α. This is a finite sum of bounded operators. The energy result says ⟨ x,Ax⟩=⟨ x,x⟩ for every x. Complex polarization, applied also to x+y and x+ i y, gives ⟨ y,(A-I)x⟩=0 for all x,y, hence A=I. Applying the operator identity to x proves synthesis. This argument uses the actual Hilbert adjoints of the decimation operators, not ","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-3606161998d5afc4","tex":"\\sum_{\\alpha\\in\\{0,1\\}^2}","aliases":"∑_α∈{0,1}^2 sum alpha in","display":"\\sum_{\\alpha\\in\\{0,1\\}^2}","kind":"use","excerpt":"00,0.65000); node[] at (5.90000,1.04500) - frac14; draw[fill=black!18,draw=black!40,line width=.35pt] (6.80000,0.72000) rectangle ++(1.80000,0.65000); node[] at (7.70000,1.04500) frac14; node[] at (10.90000,1.05000) frac14; node[] at (5.00000,4.60000) coefficient positions within each packet; node[] at (10.90000,4.60000) energy; node[] at (6.25000,","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c5a9492e4a8b5806","tex":"\\sum_{\\alpha\\in\\{0,1\\}^{\\ell}}","aliases":"∑_α∈{0,1}^ℓ sum alpha in ell","display":"\\sum_{\\alpha\\in\\{0,1\\}^{\\ell}}","kind":"use","excerpt":"[Wavelet-packet reconstruction and energy] At every finite depth ℓ, x=∑_α∈{0,1}^ℓW_ℓ,α^*W_ℓ,αx, and x _2^2 = ∑_α∈{0,1}^ℓ W_ℓ,αx _2^2.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-a9349b269e60291a","tex":"\\sum_{\\ell\\in L}","aliases":"∑_ℓ∈ L sum ell in","display":"\\sum_{\\ell\\in L}","kind":"use","excerpt":"Every address has a unique representation r+ℓ. For finite p, x _p^p = ∑_r∈ R∑_ℓ∈ L|x(r+ℓ)|^p = ∑_r∈ R x_r _p^p. The corresponding supremum identity is immediate when p=∞. Unique address representation proves that analysis and interleaving are inverse maps. The projector identities follow from the disjoint coset partition.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-db596730d71feb37","tex":"\\sum_{\\ell\\in\\Lcal_w}","aliases":"∑_ℓ∈Lcal_w ∑_ℓ∈ L_w sum ell in Lcal sum ell in mathcal","display":"\\sum_{\\ell\\in{\\mathcal L}_w}","kind":"use","excerpt":"Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[Exact scalar region formula] Let E,F be measurable subsets of a measure space, let a,b∈ C, and let 1≤ p<∞. Then a 1_E-b 1_F _p^p = · |a-b|^pμ(E∩ F) +|a|^pμ(E∖ F) · +|b|^pμ(F∖ E). For equal amplitudes, a 1_E-a 1_F _p = |a|μ(Emathbintriangle F)^1/p.","count":8},{"work":"paper-v","url":"notation/paper-v.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"[Polyphase M obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":4},{"work":"paper-v","url":"notation/paper-v.html#symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","excerpt":"For the limit, put u_k=(k+tfrac12)/n. The identity Q_n(x_n)=frac1π n∑_k=0^n-1fracsin(π u_k)u_k ⟶frac1π∫_0^1fracsin(π u)u du =fracSi(π)π is a midpoint Riemann sum, since the integrand has continuous value π at zero. This is the normalized classical example in [source reference].","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-8bcd55715febb3c0","tex":"\\int_0^z","aliases":"∫_0^z int","display":"\\int_0^z","kind":"use","excerpt":"Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonzero overshoot persists along points app","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-32e5a56ec6778f45","tex":"\\int_a^{a+L/2}","aliases":"∫_a^a+L/2 int","display":"\\int_a^{a+L/2}","kind":"use","excerpt":"Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-bd4f282e19b7fae3","tex":"\\int_\\tau^{\\widehat\\tau}","aliases":"∫_τ^widehatτ int tau widehat tau","display":"\\int_\\tau^{\\widehat\\tau}","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"Write N=2^sM, M odd. The cyclic index splits by the Chinese remainder theorem: Z/N Z ≅ Z/2^s Z× Z/M Z.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"[Location error for one step] Let 0≤τ,widehatτ≤1 and H_τ= 1_[τ,1]. For 1≤ p<∞, aH_τ-aH_widehatτ _p = |a||τ-widehatτ|^1/p. In the supremum norm, the error equals |a| whenever τ≠widehatτ.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].","count":12},{"work":"paper-v","url":"notation/paper-v.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].","count":11},{"work":"paper-v","url":"notation/paper-v.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":5},{"work":"paper-v","url":"notation/paper-v.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":3},{"work":"paper-v","url":"notation/paper-v.html#symbol-d77606a690becea6","tex":"\\bigoplus_{r=0}^{N-1}","aliases":"⊕_r=0^N-1 bigoplus","display":"\\bigoplus_{r=0}^{N-1}","kind":"use","excerpt":"[One dimensional residue coordinates] For L=N Z and R={0,…,N-1}, x_r(k)=x(Nk+r) and ℓ^p( Z)≅⊕_r=0^N-1ℓ^p( Z) isometrically.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-ef2b6439056f27d8","tex":"\\bigoplus_{r\\in R}","aliases":"⊕_r∈ R bigoplus in","display":"\\bigoplus_{r\\in R}","kind":"use","excerpt":"For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.","count":2},{"work":"paper-v","url":"notation/paper-v.html#symbol-9b9985dfb7501871","tex":"\\bigoplus_{v\\in\\Lcal}","aliases":"⊕_v∈Lcal ⊕_v∈ L bigoplus in Lcal bigoplus in mathcal","display":"\\bigoplus_{v\\in{\\mathcal L}}","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":1},{"work":"paper-v","url":"notation/paper-v.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"Each integer belongs to exactly one residue class modulo N. Multiplying the two indicator functions gives zero for distinct classes and the same indicator for equal classes. Summing all residue indicators gives one at every address. Moreover, P_N,rx _2^2=∑_n≡ r (mod N)|x[n]|^2≤ x _2^2, ⟨ P_N,rx,y⟩=⟨ x,P_N,ry⟩. The mask is real-valued, so the inner-","count":9},{"work":"paper-v","url":"notation/paper-v.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"Each integer belongs to exactly one residue class modulo N. Multiplying the two indicator functions gives zero for distinct classes and the same indicator for equal classes. Summing all residue indicators gives one at every address. Moreover, P_N,rx _2^2=∑_n≡ r (mod N)|x[n]|^2≤ x _2^2, ⟨ P_N,rx,y⟩=⟨ x,P_N,ry⟩. The mask is real-valued, so the inner-","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":210},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":935},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":295},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":133},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":16},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6e4af5afb524dbe9","tex":"B_{\\beta\\alpha}","aliases":"B_βα beta alpha","display":"B_{\\beta\\alpha}","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":236},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e4c4b465366a9288","tex":"C_{m,n}","aliases":"C_m,n","display":"C_{m,n}","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":88},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0dcf6a495ccbd5bd","tex":"\\e","aliases":"e mathrm","display":"{\\mathrm e}","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":130},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":155},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5052bf5368337201","tex":"\\e^{tA}","aliases":"e^tA e mathrm","display":"{\\mathrm e}^{tA}","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":22},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":98},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5e073754a563e30e","tex":"\\hbox","aliases":"hbox","display":null,"kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":174},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":177},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f2af7728430d7213","tex":"I_\\alpha","aliases":"I_α alpha","display":"I_\\alpha","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":27},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":144},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":120},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e08d6b60be0ec0b0","tex":"\\Ker","aliases":"Ker operatorname","display":"{\\operatorname{Ker}}","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":30},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":118},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":170},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob operatorname","display":"{\\operatorname{Ob}}","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d9e7551bc92ef7bb","tex":"\\Ob_C","aliases":"Ob_C Ob operatorname","display":"{\\operatorname{Ob}}_C","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9f1ae61172342775","tex":"\\overline\\lambda","aliases":"overlineλ overline lambda","display":"\\overline\\lambda","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":13},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f525677ebf3492f1","tex":"\\overline\\lambda_\\alpha","aliases":"overlineλ_α overline lambda alpha","display":"\\overline\\lambda_\\alpha","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":179},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":228},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":48},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":32},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-aad7607e608d0052","tex":"P_\\beta","aliases":"P_β beta","display":"P_\\beta","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":20},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-813431acda8054c4","tex":"\\Pcal","aliases":"Pcal P mathcal","display":"{\\mathcal P}","kind":"definition-passage","excerpt":"[Packet resolved transfer system] A packet resolved transfer system is a triple (X,A, P) consisting of a Banach state space, a bounded generator, and a finite packet system. A marked transfer system also includes an initial state or source s∈ X. The topology and all limiting operations are supplied by the ambient Banach space structure.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":81},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":54},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":137},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b187bd136ca926c2","tex":"R_{n,m}","aliases":"R_n,m","display":"R_{n,m}","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":11},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-af3bb9ba2ca7c7af","tex":"r_{\\alpha,i}","aliases":"r_α,i alpha","display":"r_{\\alpha,i}","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a62a596621079394","tex":"\\RR","aliases":"RR R mathbb","display":"{\\mathbb R}","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":16},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-abdd5a30d95b7315","tex":"\\RR^m","aliases":"RR^m R^m RR mathbb","display":"{\\mathbb R}^m","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b617b296bbfd924b","tex":"\\mathfrak S","aliases":"S mathfrak","display":"\\mathfrak S","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":121},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":202},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":492},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-64afaf0ed6cbcadc","tex":"T_n","aliases":"T_n","display":"T_n","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":14},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-250004212eff8da7","tex":"T_{\\beta\\alpha}","aliases":"T_βα beta alpha","display":"T_{\\beta\\alpha}","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":24},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-32f66856145f390c","tex":"U_{m,n}","aliases":"U_m,n","display":"U_{m,n}","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-50ce1a62663b77d9","tex":"\\Vert","aliases":"Vert","display":"\\Vert","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":32},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":86},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":138},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":17},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-be1f7209628ae76c","tex":"X_{j+1}","aliases":"X_j+1","display":"X_{j+1}","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0430cf91ce15859e","tex":"x_{j+1}","aliases":"x_j+1","display":"x_{j+1}","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bfd71ea41b57ecef","tex":"X_\\infty","aliases":"X_∞ infty","display":"X_\\infty","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":23},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ca558d2d7c53350d","tex":"|_{\\Ker C}","aliases":"|_Ker C Ker operatorname","display":"|_{{\\operatorname{Ker}} C}","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":192},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":62},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":35},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-412b907de59c5fc0","tex":"\\eta_i","aliases":"η_i eta","display":"\\eta_i","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":156},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":42},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":22},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Transfer-semigroup seed system] A finite transfer-semigroup seed system is a tuple S=(X,s,A,{P_α}_α∈ I,τ), where X is the state space, s is the seed, A is a transfer generator, the P_α are pairwise orthogonal idempotents summing to the identity, and τ specifies the topology in which a completion or limit is interpreted.","count":102},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7aac306ff05814ee","tex":"\\prod_{j\\geq0}","aliases":"∏_j≥0 prod geq","display":"\\prod_{j\\geq0}","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":143},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":16},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a9710f0f60fb9e82","tex":"\\sum_{i\\in I_\\alpha}","aliases":"∑_i∈ I_α sum in alpha","display":"\\sum_{i\\in I_\\alpha}","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":93},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":60},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5494a82a5f60bced","tex":"A_c","aliases":"A_c","display":"A_c","kind":"introduction-passage","excerpt":"vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct ","count":42},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-84cc5863a0df7626","tex":"A_f","aliases":"A_f","display":"A_f","kind":"introduction-passage","excerpt":"vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct ","count":41},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-571e154fec88d563","tex":"A_{QQ}","aliases":"A_QQ","display":"A_{QQ}","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":32},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1f808c793b47d0c2","tex":"\\bar M","aliases":"bar M bar","display":"\\bar M","kind":"introduction-passage","excerpt":"More generally, for a surjective aggregation map C, a linear map M descends exactly if CM vanishes on ker C. The descended map is defined by bar M(Cx)=CMx; the kernel condition makes this independent of the representative x.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct ","count":97},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-746a0fca63346846","tex":"C_r","aliases":"C_r","display":"C_r","kind":"introduction-passage","excerpt":"vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cf090df8943be9f2","tex":"\\CC^n","aliases":"CC^n C^n CC mathbb","display":"{\\mathbb C}^n","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3c079a3d2ea487eb","tex":"\\e^B","aliases":"e^B e mathrm","display":"{\\mathrm e}^B","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f21866dd974b9900","tex":"\\e^{nB}","aliases":"e^nB e mathrm","display":"{\\mathrm e}^{nB}","kind":"introduction-passage","excerpt":"ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for eve","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9c6acec97d560659","tex":"\\e^{s(B)}","aliases":"e^s(B) e mathrm","display":"{\\mathrm e}^{s(B)}","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-699c216a893acf66","tex":"\\e^\\beta","aliases":"e^β e beta mathrm beta","display":"{\\mathrm e}^\\beta","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a178b613b1f6d9d0","tex":"\\e^{\\beta n}","aliases":"e^β n e beta mathrm beta","display":"{\\mathrm e}^{\\beta n}","kind":"introduction-passage","excerpt":"ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for eve","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"0 · 1 · 0. Direct multiplication gives C_rA_f=A_cC_r. For F(z)=1/(1+z), which is defined at both spectra, compute the matrices by inversion: F(A_f)=mat1/3 · -1/9 · 0","count":40},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":176},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":42},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c0031bdf2737d3ca","tex":"\\ip","aliases":"ip","display":null,"kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":13},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"introduction-passage","excerpt":"More generally, for a surjective aggregation map C, a linear map M descends exactly if CM vanishes on ker C. The descended map is defined by bar M(Cx)=CMx; the kernel condition makes this independent of the representative x.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for eve","count":51},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-03fccda709f9f68a","tex":"M_\\beta","aliases":"M_β beta","display":"M_\\beta","kind":"introduction-passage","excerpt":"ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for eve","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-956f614ae0645ab5","tex":"\\mat","aliases":"mat","display":null,"kind":"introduction-passage","excerpt":"vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONTraditional Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct ","count":17},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":18},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"introduction-passage","excerpt":"ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for eve","count":109},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6eda1a560c2d3126","tex":"\\overline{g_i}","aliases":"overlineg_i overline","display":"\\overline{g_i}","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6ffaa0b88d08a3d2","tex":"P_\\mathcal","aliases":"P_ mathcal","display":"P_\\mathcal","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":136},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":19},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":16},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-327dcbd8f0058da9","tex":"\\Spec","aliases":"Spec operatorname","display":"{\\operatorname{Spec}}","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fa3045354c2c4668","tex":"X_Q","aliases":"X_Q","display":"X_Q","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":115},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":77},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-403b4646d62f0c8b","tex":"\\pi_i","aliases":"π_i pi","display":"\\pi_i","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8f322ee5409feacc","tex":"\\pi_j","aliases":"π_j pi","display":"\\pi_j","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6821584482b27152","tex":"\\sum_{j\\in B(i)}","aliases":"∑_j∈ B(i) sum in","display":"\\sum_{j\\in B(i)}","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"introduction-passage","excerpt":"ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for eve","count":104},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-eab1b84529bdde00","tex":"\\bm1","aliases":"1 bm","display":"\\bm1","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-de32a3ebb13b1924","tex":"\\bm1^T","aliases":"1^T bm","display":"\\bm1^T","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-86e9de44a233df86","tex":"\\mathrm A","aliases":"A mathrm","display":"\\mathrm A","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero","count":127},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"vpage29Acyclic paths stop after finitely many transfersDIRECTED TRANSFER / A FINITE DYSON EXAMPLETraditional Part II: Theorem E2.8 and Corollary E2.9. Acyclicity concerns the transfer graph. Let three scalar packets form the directed chain 0→1→2, with transfer coefficients 2 and 3. The generator and its powers are A=mat0 · 0 · 0 2 · 0 · 0 0 · 3 · 0","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-06b373fe54fc27d1","tex":"A^3","aliases":"A^3","display":"A^3","kind":"use","excerpt":"vpage29Acyclic paths stop after finitely many transfersDIRECTED TRANSFER / A FINITE DYSON EXAMPLETraditional Part II: Theorem E2.8 and Corollary E2.9. Acyclicity concerns the transfer graph. Let three scalar packets form the directed chain 0→1→2, with transfer coefficients 2 and 3. The generator and its powers are A=mat0 · 0 · 0 2 · 0 · 0 0 · 3 · 0","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-339ddf229ead4fcf","tex":"A^m","aliases":"A^m","display":"A^m","kind":"use","excerpt":"The hypothesis makes A block diagonal. Every power A^m is block diagonal with respect to the same projectors, and therefore so is the exponential series.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d14dbb0eba947d6f","tex":"A^{m+n}","aliases":"A^m+n","display":"A^{m+n}","kind":"use","excerpt":"Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-99155510890729d1","tex":"A^r","aliases":"A^r","display":"A^r","kind":"use","excerpt":"Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-75d47130ff67df22","tex":"a^T","aliases":"a^T","display":"a^T","kind":"use","excerpt":"Let A be a finite subgenerator: its off-diagonal entries are nonnegative and its column sums are nonpositive. These sign conditions alone allow nonabsorbing states; eventual absorption is a separate hypothesis. Let p_0 be an initial transient-state distribution. Define p(t)=e^tAp_0, S(t)=1^Tp(t). The exit-rate row vector is a^T=-1^TA≥0.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-37386c7055af10de","tex":"A^\\alpha","aliases":"A^α alpha","display":"A^\\alpha","kind":"use","excerpt":"Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"[Normalization bound] Let a,b∈[0,∞)^m be nonzero, and set A=norm a_1 and B=norm b_1. Then norma/A-b/B_1 ≤ frac2norma-b_1max{A,B}.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2167913f89021b35","tex":"A_{j+1}","aliases":"A_j+1","display":"A_{j+1}","kind":"use","excerpt":"[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"[A usable closure condition for refinement towers] Assume the contractive summability hypothesis of [source reference] holds on every bounded time interval. Suppose every P_m,n is an orthogonal projection on a Hilbert space and T_n(t)= e^tA_n is contractive. If, for every fixed m, norm(I-P_m,n)A_nP_m,n ⟶0, then the limiting family U_m is a semigrou","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONTraditional Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-be33c2dc1700652e","tex":"A_{n+1}","aliases":"A_n+1","display":"A_{n+1}","kind":"use","excerpt":"vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONTraditional Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-07043dd4b1cb5784","tex":"A_n^{1/2}","aliases":"A_n^1/2","display":"A_n^{1/2}","kind":"use","excerpt":"at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVETraditional Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-61ec922a7b66b76d","tex":"A_{PP}","aliases":"A_PP","display":"A_{PP}","kind":"use","excerpt":"Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .","count":11},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-358c08cae43f1364","tex":"A_{PQ}","aliases":"A_PQ","display":"A_{PQ}","kind":"use","excerpt":"Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .","count":18},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6fa2f03584287ecd","tex":"A_{QP}","aliases":"A_QP","display":"A_{QP}","kind":"use","excerpt":"Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .","count":19},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4e168c25ce0b085c","tex":"A_{QQ}^k","aliases":"A_QQ^k","display":"A_{QQ}^k","kind":"use","excerpt":"sure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6f627a19ca37cdde","tex":"a_r^T","aliases":"a_r^T","display":"a_r^T","kind":"use","excerpt":"[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f580c536293e4a61","tex":"A_{\\alpha\\alpha}","aliases":"A_αα alpha alpha","display":"A_{\\alpha\\alpha}","kind":"use","excerpt":"thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1c0b084cbb40137f","tex":"A_{\\alpha\\gamma}","aliases":"A_αγ alpha gamma","display":"A_{\\alpha\\gamma}","kind":"use","excerpt":"The first two conditions are equivalent because the packet resolution gives AP_α=∑_β A_βα, P_α A=∑_γ A_αγ. A block diagonal generator has block diagonal powers, so the second condition implies the third. Differentiating the off diagonal blocks at zero proves the converse. The equivalence with invariance follows from the definition of the blocks. Th","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5498b3bc9fb8719d","tex":"A_{\\beta\\alpha}","aliases":"A_βα beta alpha","display":"A_{\\beta\\alpha}","kind":"use","excerpt":"[Reducing packet criterion] The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2141b0918d8b84b6","tex":"A_{\\beta\\gamma}","aliases":"A_βγ beta gamma","display":"A_{\\beta\\gamma}","kind":"use","excerpt":"[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9b978ebe79a99278","tex":"A_{\\gamma\\alpha}","aliases":"A_γα gamma alpha","display":"A_{\\gamma\\alpha}","kind":"use","excerpt":"[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-263b1db52659f9f9","tex":"A_\\ell","aliases":"A_ℓ ell","display":"A_\\ell","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-744d6de0e15e8c83","tex":"A_{\\ell+1}","aliases":"A_ℓ+1 ell","display":"A_{\\ell+1}","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bf089dc8632eb091","tex":"A_{\\ell+1}^{\\alpha}","aliases":"A_ℓ+1^α ell alpha","display":"A_{\\ell+1}^{\\alpha}","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-102f6cf50d31e579","tex":"A_\\ell^{\\alpha}","aliases":"A_ℓ^α ell alpha","display":"A_\\ell^{\\alpha}","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6b3267a5b5057d4c","tex":"A_\\infty","aliases":"A_∞ infty","display":"A_\\infty","kind":"use","excerpt":"If T_ℓ(t)= e^tA_ℓ, the generators are bounded, and sup_ℓnormA_ℓ<∞, C_ℓ A_ℓ+1=A_ℓ C_ℓ, then the continuity condition holds and A_∞(x_ℓ)_ℓ=(A_ℓ x_ℓ)_ℓ is the bounded generator of T_∞.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure of the union, then Cg(A_f)-g(A_c)C = frac12π i ∫_Γ g(z)R_c(z)Δ R_f(z) d z. In particul","count":16},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1267a67d957e703c","tex":"\\mathrm B","aliases":"B mathrm","display":"\\mathrm B","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.","count":41},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-340922fec8f1f7a7","tex":"B^\\alpha","aliases":"B^α alpha","display":"B^\\alpha","kind":"use","excerpt":"Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"Let Ω={1,…,n} carry strictly positive probabilities π_i, and let Q={B_1,…,B_m} be a partition. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j∑_j∈ B(i)π_j, where B(i) is the block containing i.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"[Normalization bound] Let a,b∈[0,∞)^m be nonzero, and set A=norm a_1 and B=norm b_1. Then norma/A-b/B_1 ≤ frac2norma-b_1max{A,B}.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","excerpt":"[A four state lumpable chain] Partition four states into B_1={1,2} and B_2={3,4}. For positive rates a,b,u,v, set Q= -u-a · u · b/2 · b/2","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1a4c2e9cb4075d24","tex":"B_a","aliases":"B_a","display":"B_a","kind":"use","excerpt":"The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C","count":14},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ef9e511d27b6b980","tex":"B_b","aliases":"B_b","display":"B_b","kind":"use","excerpt":"Stationarity follows from overline Qoverlineπ =overline QCπ =CQπ=0. For reversibility, use the aggregate rate formula: overline Q_aboverlineπ_b · =∑_j∈ B_bπ_j∑_i∈ B_aQ_ij · =∑_i∈ B_a∑_j∈ B_bπ_iQ_ji · =overline Q_baoverlineπ_a.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-727bc8e67f8970a4","tex":"\\mathcal B_{\\bar\\lambda}","aliases":"B_barλ mathcal bar lambda","display":"\\mathcal B_{\\bar\\lambda}","kind":"use","excerpt":"For strictly positive fine and coarse likelihoods, and at least two coarse blocks, the manuscript proves the all-prior criterion C B_λ(p)= B_barλ(Cp) hboxfor every p Longleftrightarrow λ_i=cbarλ_a (i∈ I_a) for one global c>0. The global scale is invisible after normalization. A different arbitrary scale for each packet is not invisible: it changes ","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","excerpt":"[Normal hidden dynamics] Suppose X_Q is a finite dimensional Hilbert space. If A_QQ is normal with spectral resolution A_QQ=∑_j=1^rμ_jQ_j, then K(t)=∑_j=1^r e^μ_jtB_j, Σ_P(z)=∑_j=1^rfracB_jz-μ_j, where B_j=A_PQQ_jA_QP. If Reμ_j<0, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J fracnormB_j-Reμ_j.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5a1364749eb81c1c","tex":"B_{j,k}","aliases":"B_j,k","display":"B_{j,k}","kind":"use","excerpt":"mory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-544c723ff631382e","tex":"B_{j,\\nu_j-1}","aliases":"B_j,ν_j-1 nu","display":"B_{j,\\nu_j-1}","kind":"use","excerpt":"[Pole order and hidden Jordan depth] Every finite hidden block produces a strictly proper rational self energy. At an eigenvalue μ_j, its pole order is at most the size of the largest Jordan block of A_QQ at μ_j. Equality holds when the corresponding highest coefficient B_j,ν_j-1 is nonzero.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"use","excerpt":"Let Ω={1,…,n} carry strictly positive probabilities π_i, and let Q={B_1,…,B_m} be a partition. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j∑_j∈ B(i)π_j, where B(i) is the block containing i.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"use","excerpt":"vpage27Small at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVETraditional Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6322ecebf90fbfd2","tex":"B_n^{1/2}","aliases":"B_n^1/2","display":"B_n^{1/2}","kind":"use","excerpt":"at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVETraditional Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4491da1e894267e7","tex":"\\mathcal B_r","aliases":"B_r mathcal","display":"\\mathcal B_r","kind":"use","excerpt":"page11The exact information-loss vectorBAYESIAN DEFECT / MORE THAN AN ERROR BARTraditional Part I: Theorems 8.1 and 8.4, and Corollary 8.6. For a prior p, let P=Cp and let r_a(p) be the conditional average of the fine likelihood inside packet a. For the uniform prior and λ=(4,1,1,1), P=(1/2,1/2)^ T, r=(5/2,1)^ T. The exact coarse posterior is B_r(P","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-190834da497e3d7f","tex":"B_\\alpha","aliases":"B_α alpha","display":"B_\\alpha","kind":"use","excerpt":"Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4cacfd818361b439","tex":"B_\\beta","aliases":"B_β beta","display":"B_\\beta","kind":"use","excerpt":"[Characterization of exact discrete lumping] Let K be column stochastic. A coarse column stochastic matrix overline K satisfying CK=overline KC exists if and only if, for every source block B_β, every pair i,j∈ B_β, and every target block B_α, ∑_r∈ B_αK_ri = ∑_r∈ B_αK_rj. When it exists, overline K is unique and overline K_αβ = ∑_r∈ B_αK_ri for any","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-655c8b8be4a021c9","tex":"\\mathcal B_\\lambda","aliases":"B_λ mathcal lambda","display":"\\mathcal B_\\lambda","kind":"use","excerpt":"B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7e95a89f39649f47","tex":"\\bar b","aliases":"bar b bar","display":"\\bar b","kind":"use","excerpt":"When sufficiency fails, the defect still has an exact finite factorization. Fix a declared coarse likelihood barλ, put P=C_π p, δ(p)=r(p)-barλ, bar b= B_barλ(P), Z_r=⟨ r(p),P⟩.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fad6ae0fd6c0acb8","tex":"\\bar K","aliases":"bar K bar","display":"\\bar K","kind":"use","excerpt":"Let K be a column-stochastic fine transition matrix, let C aggregate fine states into coarse packets, and let bar K be a coarse transition matrix. Define Δ=CK-bar KC.","count":21},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a996e97b2fc3d473","tex":"\\bar K^n","aliases":"bar K^n bar","display":"\\bar K^n","kind":"use","excerpt":"[Exact lumpability] If Δ=0, then CK^n=bar K^nC for every n≥0. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0ffef9424cf7c3c6","tex":"\\bar K^{n+1}","aliases":"bar K^n+1 bar","display":"\\bar K^{n+1}","kind":"use","excerpt":"The result is immediate for n=0. If CK^n=bar K^nC, then CK^n+1=bar K^nCK=bar K^n+1C.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ef943f9b503eecb4","tex":"\\bar K^{n-1-j}","aliases":"bar K^n-1-j bar","display":"\\bar K^{n-1-j}","kind":"use","excerpt":"[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4a0a9afa94f80409","tex":"\\bar K^{n-j}","aliases":"bar K^n-j bar","display":"\\bar K^{n-j}","kind":"use","excerpt":"The identity follows by telescoping: bar K^n-1-jCK^j+1-bar K^n-jCK^j =bar K^n-1-j(CK-bar KC)K^j. Summing from j=0 to n-1 leaves CK^n-bar K^nC. The norm bound follows termwise.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8f8a4d0a063c4065","tex":"\\bar Q","aliases":"bar Q bar","display":"\\bar Q","kind":"use","excerpt":"The continuous generator also closes: CQ=bar Q C, bar Q=-2 · 2 2 · -2. The Duhamel identity therefore gives Ce^tQ=e^tbar QC. This is a second consequence of the corresponding generator identity, not an inference that K is a fixed-time sample of Q.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-60e3a7638b4cd31d","tex":"\\bar\\lambda","aliases":"barλ bar lambda","display":"\\bar\\lambda","kind":"use","excerpt":"[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on ea","count":22},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9ca7f87e3b8a212c","tex":"\\bar\\lambda_a","aliases":"barλ_a bar lambda","display":"\\bar\\lambda_a","kind":"use","excerpt":"[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on ea","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6a848437bb19d2af","tex":"\\bar\\lambda_b","aliases":"barλ_b bar lambda","display":"\\bar\\lambda_b","kind":"use","excerpt":"Now use priors supported on one state of packet a and one state of packet b. Equality of the two coarse posterior odds gives fracλ_aλ_b = fracbarλ_abarλ_b. This holds for every pair of coarse packets, so λ_a=cbarλ_a for one common c>0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-70754a4e2f59f2f1","tex":"\\bar\\lambda_{\\pi(i)}","aliases":"barλ_π(i) bar lambda pi","display":"\\bar\\lambda_{\\pi(i)}","kind":"use","excerpt":"tes barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) _1 ≤ frac2∑_aP_a|δ_a(p)| min{Z_r,Z_barλ} and ∑_aP_a|δ_a(p)| ≤ ∑_i p_i|λ_i-barλ_π(i)","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-36f15bc97a93c5c7","tex":"\\bbP","aliases":"bbP P mathbb","display":"{\\mathbb P}","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a3ee3f8d33e160ca","tex":"\\bbR","aliases":"bbR R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-327abfc143ff067a","tex":"\\Bcal","aliases":"Bcal B mathcal","display":"{\\mathcal B}","kind":"use","excerpt":"Bayesian update · B_λ · likelihood constant inside each coarse packet","count":22},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c669ec793a215af0","tex":"\\Bcal_{c\\lambda}","aliases":"Bcal_cλ B_cλ Bcal lambda mathcal lambda","display":"{\\mathcal B}_{c\\lambda}","kind":"use","excerpt":"[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3c96b2eee990d739","tex":"\\Bcal_L","aliases":"Bcal_L B_L Bcal mathcal","display":"{\\mathcal B}_L","kind":"use","excerpt":"FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] De","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ae942cfdb7498044","tex":"\\Bcal_{L_y}","aliases":"Bcal_L_y B_L_y Bcal mathcal","display":"{\\mathcal B}_{L_y}","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9731ec42b8194977","tex":"\\Bcal_{\\overline L}","aliases":"Bcal_overline L B_overline L Bcal overline mathcal overline","display":"{\\mathcal B}_{\\overline L}","kind":"use","excerpt":"ctor p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3bed59d24e6dcb8e","tex":"\\Bcal_{\\overline L_y}","aliases":"Bcal_overline L_y B_overline L_y Bcal overline mathcal overline","display":"{\\mathcal B}_{\\overline L_y}","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b8461303be59fc46","tex":"\\Bcal_{\\overline\\lambda(p)}","aliases":"Bcal_overlineλ(p) B_overlineλ(p) Bcal overline lambda mathcal overline lambda","display":"{\\mathcal B}_{\\overline\\lambda(p)}","kind":"use","excerpt":"[Exact two stage Bayesian update] For every strictly positive prior, C B_λ(p) = B_overlineλ(p)(Cp). For every packet with overlineλ_α(p)>0, the posterior conditional distribution is q_α,i = fracλ_ir_α,i overlineλ_α(p). A packet with zero average likelihood has zero posterior mass. Thus a fine Bayesian update consists of a coarse update by packet av","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-670d28531783d120","tex":"\\Bcal_{\\widetilde\\lambda}","aliases":"Bcal_widetildeλ B_widetildeλ Bcal widetilde lambda mathcal widetilde lambda","display":"{\\mathcal B}_{\\widetilde\\lambda}","kind":"use","excerpt":"Block constancy gives C B_widetildeλ(p) = B_ℓ(Cp). Aggregation is an ℓ^1 contraction. Apply [source reference] to the two fine posteriors and then aggregate.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-36f47e44857f889e","tex":"\\Bcal_\\lambda","aliases":"Bcal_λ B_λ Bcal lambda mathcal lambda","display":"{\\mathcal B}_\\lambda","kind":"use","excerpt":"Bayesian update · B_λ · likelihood constant inside each coarse packet","count":13},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e1055d827217a09e","tex":"\\Bcal_{\\lambda^{(r)}}","aliases":"Bcal_λ^(r) B_λ^(r) Bcal lambda mathcal lambda","display":"{\\mathcal B}_{\\lambda^{(r)}}","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood of the rth observation. The hidden Markov filter is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r). A standard account of finite hidden Markov models appears in [source reference].","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-160416bc9d0d7c17","tex":"\\Bcal_{\\lambda\\odot\\mu}","aliases":"Bcal_λodotμ B_λodotμ Bcal lambda odot mu mathcal lambda odot mu","display":null,"kind":"use","excerpt":"[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1d7da39846900e8d","tex":"\\Bcal_\\mu","aliases":"Bcal_μ B_μ Bcal mu mathcal mu","display":"{\\mathcal B}_\\mu","kind":"use","excerpt":"[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-34e973b02b27f5d2","tex":"\\Bcal_\\ell","aliases":"Bcal_ℓ B_ℓ Bcal ell mathcal ell","display":"{\\mathcal B}_\\ell","kind":"use","excerpt":"[Characterization of fixed coarse sufficiency] Assume λ_i>0 for every state. The following conditions are equivalent. item There is a positive coarse likelihood ℓ∈(0,∞)^m such that C B_λ(p) = B_ℓ(Cp) for every strictly positive prior p, and the posterior conditional distribution inside every packet equals its prior conditional distribution. item Th","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d549c511b570806a","tex":"\\Bcal_{\\ell(p)}","aliases":"Bcal_ℓ(p) B_ℓ(p) Bcal ell mathcal ell","display":"{\\mathcal B}_{\\ell(p)}","kind":"use","excerpt":"For an arbitrary prior p, put q=Cp and Z(p,L)=ipLp. Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0; choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bfc0ff5fd667e9f6","tex":"\\mathrm{Beta}","aliases":"Beta mathrm","display":"\\mathrm{Beta}","kind":"use","excerpt":"vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCETraditional Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-22835c974e5650b7","tex":"\\operatorname{Beta}","aliases":"Beta operatorname","display":"\\operatorname{Beta}","kind":"use","excerpt":"[Beta--Binomial conjugate packet] A Beta(a,b) prior and s successes in n trials produce Beta(a+s,b+n-s). The prior exponents are additive packets: the likelihood increments them by (s,n-s). In the numerical example, (a,b)=(2,3), (s,n)=(7,10), so the posterior is Beta(9,6) with mean 0.6.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-133380d2b39bd34f","tex":"C^{1-\\alpha}","aliases":"C^1-α alpha","display":"C^{1-\\alpha}","kind":"use","excerpt":"For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ab467eeeb5a83ecd","tex":"C^4","aliases":"C^4","display":"C^4","kind":"use","excerpt":"[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8ee9beb61ef61638","tex":"\\mathbb C^d","aliases":"C^d mathbb","display":"\\mathbb C^d","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c94fd79f2de7324b","tex":"C^T","aliases":"C^T","display":"C^T","kind":"use","excerpt":"[Coarse Feynman Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5ee68c951157fbd1","tex":"C_{ai}","aliases":"C_ai","display":"C_{ai}","kind":"use","excerpt":"The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e9cfc77f3166a2a9","tex":"C_{j+1,m}","aliases":"C_j+1,m","display":"C_{j+1,m}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b038101bc9d9bcb1","tex":"C_{j,j}","aliases":"C_j,j","display":"C_{j,j}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3f2aea9ede1d4ca7","tex":"C_{j,m}","aliases":"C_j,m","display":"C_{j,m}","kind":"use","excerpt":"Expand each summand as C_ℓ,j+1T_j+1C_j+1,m - C_ℓ,jT_jC_j,m. The intermediate terms cancel. The first surviving term is C_ℓ,mT_m, and the last is -T_ℓ C_ℓ,m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-54bfb2f6c9232092","tex":"C_{m+1}","aliases":"C_m+1","display":"C_{m+1}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-102565d13a8ae5ca","tex":"C_{m,m}","aliases":"C_m,m","display":"C_{m,m}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-03f737cfb1d5dd05","tex":"C_{m-1}","aliases":"C_m-1","display":"C_{m-1}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cfbf41dac773cbfc","tex":"C_n","aliases":"C_n","display":"C_n","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"use","excerpt":"A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b48446852f3335a2","tex":"C_{n-1}","aliases":"C_n-1","display":"C_{n-1}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-35910a02402c8671","tex":"C_{\\alpha i}","aliases":"C_α i alpha","display":"C_{\\alpha i}","kind":"use","excerpt":"Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5da91e15de04cc9e","tex":"C_\\eta","aliases":"C_η eta","display":"C_\\eta","kind":"use","excerpt":"vpage41An operator substitution transports the defect tooFUNCTIONAL CALCULUS / AN EXACT RATIONAL TESTTraditional Part II: Theorem E3.4; compare Traditional Part I, Theorem 15.1, and V--25. Use the operators from V--25, but perturb the rectangular comparison map: A=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, B=mat2 · 1 0 · 2, C_η=mat1 · 0 · η 0 · 1 · 0. The g","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-42eb4bc96455bb91","tex":"C_\\pi","aliases":"C_π pi","display":"C_\\pi","kind":"use","excerpt":"Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4ae0a9be277fd36e","tex":"C_\\ell","aliases":"C_ℓ ell","display":"C_\\ell","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":14},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c4a0be93686b794f","tex":"C_{\\ell,j}","aliases":"C_ℓ,j ell","display":"C_{\\ell,j}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c05f0525b2178009","tex":"C_{\\ell,j+1}","aliases":"C_ℓ,j+1 ell","display":"C_{\\ell,j+1}","kind":"use","excerpt":"Expand each summand as C_ℓ,j+1T_j+1C_j+1,m - C_ℓ,jT_jC_j,m. The intermediate terms cancel. The first surviving term is C_ℓ,mT_m, and the last is -T_ℓ C_ℓ,m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-71a8a3e47a0a9522","tex":"C_{\\ell,m}","aliases":"C_ℓ,m ell","display":"C_{\\ell,m}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a623735ca1da9cda","tex":"\\cB","aliases":"cB B mathcal","display":"{\\mathcal B}","kind":"use","excerpt":"Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.","count":13},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-61eeaf6b817657fa","tex":"\\cB_{\\bar\\lambda}","aliases":"cB_barλ B_barλ cB bar lambda mathcal bar lambda","display":"{\\mathcal B}_{\\bar\\lambda}","kind":"use","excerpt":"[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on ea","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-46ffc8349fb86de0","tex":"\\cB_{\\bar\\lambda+\\delta}","aliases":"cB_barλ+δ B_barλ+δ cB bar lambda delta mathcal bar lambda delta","display":"{\\mathcal B}_{\\bar\\lambda+\\delta}","kind":"use","excerpt":"The effective-likelihood theorem reduces the defect to B_barλ+δ(P)- B_barλ(P). Since Z_r=Z_barλ+P^ Tδ, direct subtraction gives frac1Z_r (diag(P)-bar bP^ T)δ. Also, (diag(P)-bar bP^ T)barλ =Podotbarλ-bar bZ_barλ=0. The first norm bound is the likelihood perturbation theorem applied on the coarse simplex. For a zero-mass packet, both sides of the fo","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-17dd4b80f1d797e2","tex":"\\cB_{r(p)}","aliases":"cB_r(p) B_r(p) cB mathcal","display":"{\\mathcal B}_{r(p)}","kind":"use","excerpt":"[Exact effective-likelihood factorization] Whenever the fine evidence is positive, C_π B_λ(p) = B_r(p)(C_π p). Thus the only fine information needed for the coarse posterior, beyond the coarse prior masses, is the vector of conditional packet evidences r(p).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-34496546f1396a6e","tex":"\\cB_\\lambda","aliases":"cB_λ B_λ cB lambda mathcal lambda","display":"{\\mathcal B}_\\lambda","kind":"use","excerpt":"Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-37035c9d5b2ad69b","tex":"\\cB_{\\lambda^{(r)}}","aliases":"cB_λ^(r) B_λ^(r) cB lambda mathcal lambda","display":"{\\mathcal B}_{\\lambda^{(r)}}","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-276d7bc4ee7755c0","tex":"\\cB_{\\lambda\\odot\\mu}","aliases":"cB_λodotμ B_λodotμ cB lambda odot mu mathcal lambda odot mu","display":null,"kind":"use","excerpt":"[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-555bcf940ecf36c3","tex":"\\cB_\\mu","aliases":"cB_μ B_μ cB mu mathcal mu","display":"{\\mathcal B}_\\mu","kind":"use","excerpt":"[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-124b73f571e815f3","tex":"\\cK","aliases":"cK K mathcal","display":"{\\mathcal K}","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"[Summable defects do not force a semigroup limit] Let X_0= R and let X_n= R^2 for every n≥1. Set C_0(x,y)=x, R_0x=(x,0), and use identity connecting maps and reconstructions above level one. Let T_0(t)=I and let every higher level carry the rotation group T_n(t) = cos t · -sin t sin t · cos t . Only the first adjacent defect is nonzero, so every de","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fb3375037c04cb55","tex":"\\Cov","aliases":"Cov operatorname","display":"{\\operatorname{Cov}}","kind":"use","excerpt":"Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-68a00114cff8335b","tex":"\\Cov_{p(t)}","aliases":"Cov_p(t) Cov operatorname","display":"{\\operatorname{Cov}}_{p(t)}","kind":"use","excerpt":"Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-40d9503903bdca20","tex":"\\mathfrak D","aliases":"D mathfrak","display":"\\mathfrak D","kind":"use","excerpt":"Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.","count":59},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"t t=log2, they are Z=7/3 and p=(1,2,4)/7. vfigext_cumulants platecaptionThe score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores. headTwo derivatives have different statistical meanings The first derivative of log evidence is the current mean score, ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2d88cb49c47029e8","tex":"D^\\alpha","aliases":"D^α alpha","display":"D^\\alpha","kind":"use","excerpt":"For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","excerpt":"vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"If B_j and C_j have an intermediate space of dimension d_j, use a block diagonal matrix D with blocks λ_j I_d_j, or the required Jordan blocks when poles repeat. Assemble the matrices B_j and C_j into the output and input maps.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0efdddd015383ecb","tex":"d_p","aliases":"d_p","display":"d_p","kind":"use","excerpt":"[Likelihood perturbation] Let λ,μ≥0 have positive evidences under a fixed prior p. Set d_p(λ,μ) = ∑_iabsλ_i-μ_ip_i. Then norm B_λ(p)- B_μ(p)_1 ≤ frac2d_p(λ,μ) max{Z_λ(p),Z_μ(p)}.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-237fc9015f157bd3","tex":"D_r","aliases":"D_r","display":"D_r","kind":"use","excerpt":"unchanged. Predicting afterward gives (3/4,1/4) instead. The order of operations expresses which time the observation concerns. vfigfilter_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter ","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4decc32ad75f2c6a","tex":"D_V","aliases":"D_V","display":"D_V","kind":"use","excerpt":"Let K be column stochastic. Let V be a real potential on the states and let D_V=diag(V). For functions written as columns, the backward Markov operator is K^T.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-960a9e4480b08df9","tex":"D_w","aliases":"D_w","display":"D_w","kind":"use","excerpt":"Here D_rK means an observation after prediction. In the Feynman--Kac example on pagerefV20, departure-state weighting has forward operator K D_w. Both conventions are legitimate once their meanings are fixed, but they cannot be substituted for one another. takeawayWrite the chronological sequence before multiplying matrices. A product can be normal","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"use","excerpt":"Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2e19eb6c907a8c6c","tex":"D_\\ell","aliases":"D_ℓ ell","display":"D_\\ell","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd d mathrm","display":"{\\,\\mathrm d}","kind":"use","excerpt":"[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.","count":50},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0d09000efb054df8","tex":"\\ddots","aliases":"ddots","display":"\\ddots","kind":"use","excerpt":"For n interior points with spacing h=1/(n+1), define L_h=frac1h^2 2 · -1 · · -1 · 2 · ddots · · ddots · ddots · -1 · · -1 · 2 .","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"[Schur Feshbach resolvent] Assume zI-A_QQ is invertible. Then zI-A is invertible if and only if F_P(z) is invertible. In that case, P(zI-A)^-1P = F_P(z)^-1. In finite dimension, det(zI-A) = det(zI-A_QQ)det F_P(z).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-28852ba6759dd734","tex":"\\Dfun","aliases":"Dfun D mathsf","display":"{\\mathsf D}","kind":"use","excerpt":"This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=lo","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d580abc74d5062ed","tex":"\\Dfun_N","aliases":"Dfun_N D_N Dfun mathsf","display":"{\\mathsf D}_N","kind":"use","excerpt":"This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=lo","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f30b020ad7331a73","tex":"\\Dfun_N^{-r}","aliases":"Dfun_N^-r D_N^-r Dfun mathsf","display":"{\\mathsf D}_N^{-r}","kind":"use","excerpt":"[Operator reciprocal family] Let A be bounded and suppose Spec(A) lies in a simply connected region on which D_N has no zeros. Fix a holomorphic branch g=log D_N there. Then S_N^(r)(A) = exp(-r g(A)) = ( D_N^-r)(A) is well defined and S_N^(r)(A) S_N^(s)(A) = S_N^(r+s)(A). If (P_i) is a finite complete family of bounded idempotents, then P_i f(A)P_j","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"use","excerpt":"[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remai","count":11},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"[Criterion for exact local closure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1231c9b0bf844079","tex":"\\operatorname{Dom}","aliases":"Dom operatorname","display":"\\operatorname{Dom}","kind":"use","excerpt":"[Coordinate generators] Assume the setting of [source reference]. If x=(x_j) satisfies x_j∈Dom(A_j) and the sequence (A_jx_j) belongs to X_∞, then x lies in the generator domain of T_∞ and A_∞ x=(A_jx_j)_j≥0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2dfbbad49009071f","tex":"\\dot p","aliases":"dot p dot","display":"\\dot p","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-20f048d571067858","tex":"\\dot p_0","aliases":"dot p_0 dot","display":"\\dot p_0","kind":"use","excerpt":"Differentiation of the normalized flow yields the centered score equation dot p_i=(g_i-⟨ g,p⟩)p_i. For the displayed model, dot p_0=p_0(1-p_0). The negative average-score term is what keeps the total derivative zero. Omitting the derivative of the normalizer would not preserve mass.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cf0d4e773e1ba217","tex":"\\dot p_i","aliases":"dot p_i dot","display":"\\dot p_i","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-51f3bbd1503295c5","tex":"\\dot u","aliases":"dot u dot","display":"\\dot u","kind":"use","excerpt":"Conditioning over the first small time interval gives the backward equation dot u=(Q^T-D_V)u, u(0)=f. The finite system has a unique solution, namely the displayed matrix exponential. The expectation bound follows from exp(-∫_0^tV(X_s) d s) ≤ e^-v_0t. This finite argument is the matrix form of the Feynman Kac principle originating in [source refere","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f854ffb0431207da","tex":"\\dot x","aliases":"dot x dot","display":"\\dot x","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fd48b253a62d377b","tex":"\\dot x_P","aliases":"dot x_P dot","display":"\\dot x_P","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-389bd177d57e5191","tex":"\\dot y","aliases":"dot y dot","display":"\\dot y","kind":"use","excerpt":"[Finite auxiliary state realization] Every kernel of the form K(t)=B e^tDC can be represented exactly by adjoining the finite auxiliary state dot y=Dy+Cx_P, dot x_P=A_PPx_P+By, y(0)=0. A nonzero auxiliary initial state adds the explicit forcing B e^tDy(0). The self energy of the augmented system is B(zI-D)^-1C.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8ec56f24a343f123","tex":"\\dot y_1","aliases":"dot y_1 dot","display":"\\dot y_1","kind":"use","excerpt":"vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELTraditional Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from V--23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x, · y_","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-76150d7fc49f4b0f","tex":"\\dot y_2","aliases":"dot y_2 dot","display":"\\dot y_2","kind":"use","excerpt":"vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELTraditional Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from V--23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x, · y_","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e13cdcf09a963afd","tex":"\\dot{\\widetilde x}","aliases":"dotwidetilde x dot widetilde","display":"\\dot{\\widetilde x}","kind":"use","excerpt":"entiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II. [Stability of memory compression] Let x and widetilde x solve dot x(t) =Ax(t)+∫_0^t K(t-s)x(s) d s+f(t) and dotwidetilde x(t) =Awidetilde x(t) +∫_0^twidetilde K(t-s)widetil","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":27},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dd88c369e4cd6024","tex":"E^{\\mathsf T}","aliases":"E^ T mathsf","display":"E^{\\mathsf T}","kind":"use","excerpt":")^ T becomes Ef=(2,2,26/7,26/7)^ T. vfigweighted_projection platecaptionThe bars are function values, not state probabilities. Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2b000542aa28a8cd","tex":"\\e^{(a+k)T}","aliases":"e^(a+k)T e mathrm","display":"{\\mathrm e}^{(a+k)T}","kind":"use","excerpt":"e x solve dot x(t) =Ax(t)+∫_0^t K(t-s)x(s) d s+f(t) and dotwidetilde x(t) =Awidetilde x(t) +∫_0^twidetilde K(t-s)widetilde x(s) d s +f(t) with the same initial value. Fix T>0, and assume sup_0≤ t≤ Tnormwidetilde x(t)≤ M. Set a=norm A, k=norm K_L^1(0,T), δ=norm K-widetilde K_L^1(0,T). Then sup_0≤ t≤ T normx(t)-widetilde x(t) ≤ MTδ e^(a+k)T.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a1c47e908e7b023c","tex":"\\e^{(a+k)t}","aliases":"e^(a+k)t e mathrm","display":"{\\mathrm e}^{(a+k)t}","kind":"use","excerpt":"Let e=x-widetilde x. Integrating the difference equation and changing the order of integration gives norme(t) ≤ (a+k)∫_0^tnorme(r) d r +Mtδ. Since t≤ T, Gronwall's inequality gives norme(t) ≤ MTδ e^(a+k)t. Take the supremum over [0,T].","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c092a24aa6947f31","tex":"e^{(s+t)A}","aliases":"e^(s+t)A","display":"e^{(s+t)A}","kind":"use","excerpt":"[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-560d405cf65c60d1","tex":"e^{(s+t)g}","aliases":"e^(s+t)g","display":"e^{(s+t)g}","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3c3c7988f438d287","tex":"e^{(s+t)Q}","aliases":"e^(s+t)Q","display":"e^{(s+t)Q}","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b8dc508a5768c192","tex":"\\e^{(s_m-s_{m-1})D}","aliases":"e^(s_m-s_m-1)D e mathrm","display":"{\\mathrm e}^{(s_m-s_{m-1})D}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-42f3c03924e2cad8","tex":"e^{(t-r)A_c}","aliases":"e^(t-r)A_c","display":"e^{(t-r)A_c}","kind":"use","excerpt":"[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8ecf217baed7a59a","tex":"e^{(t-r)A_{QQ}}","aliases":"e^(t-r)A_QQ","display":"e^{(t-r)A_{QQ}}","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b2230e91fe505a10","tex":"e^{(t-r)D}","aliases":"e^(t-r)D","display":"e^{(t-r)D}","kind":"use","excerpt":"0 · -2, B=(1,0), and C=(0,1)^ T, the feedback kernel is Be^tDC=te^-2t. vfigext_auxiliary platecaptionAn input first enters y_2, transfers to y_1, and then returns to x. The two equal decay rates produce a repeated pole through the intervening link. headElimination recovers the convolution, including its preparation Variation of constants gives y(t)","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ff3bd78608fb37cd","tex":"\\e^{(t-s)(QAQ)|_{QX}}","aliases":"e^(t-s)(QAQ)|_QX e mathrm","display":"{\\mathrm e}^{(t-s)(QAQ)|_{QX}}","kind":"use","excerpt":"For x∈ PX, write p(t)=PT(t)x, q(t)=QT(t)x. The hidden component satisfies q'(t)=QAQq(t)+QAPp(t), q(0)=0. Variation of constants gives q(t)= ∫_0^t e^(t-s)(QAQ)|_QX QAPp(s) d s. Since normp(s)≤norm PnormT(s)norm x and x=Px, the first estimate follows. The contraction case is immediate.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-281758b86b00412e","tex":"\\e^{(t-s)A_c}","aliases":"e^(t-s)A_c e mathrm","display":"{\\mathrm e}^{(t-s)A_c}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7b0f251471820be1","tex":"\\e^{(t-s)A_{QQ}}","aliases":"e^(t-s)A_QQ e mathrm","display":"{\\mathrm e}^{(t-s)A_{QQ}}","kind":"use","excerpt":"For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0)","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bf4d46acf1234d88","tex":"\\e^{(t-s)D}","aliases":"e^(t-s)D e mathrm","display":"{\\mathrm e}^{(t-s)D}","kind":"use","excerpt":"Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a07e55f4cb635281","tex":"\\e^{(t-s_m)D}","aliases":"e^(t-s_m)D e mathrm","display":"{\\mathrm e}^{(t-s_m)D}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ad4ab5527a169fc1","tex":"e^{(\\lambda-z)t}","aliases":"e^(λ-z)t lambda","display":"e^{(\\lambda-z)t}","kind":"use","excerpt":"Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace; equivalently, apply the decomposition to left multiplication by B on the matrix algebra a","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-66a0d4ed11e02630","tex":"e^{(\\mu-z)t}","aliases":"e^(μ-z)t mu","display":"e^{(\\mu-z)t}","kind":"use","excerpt":"General nonnormal memory (13.2). The actual damped exponential is decomposed into finitely many terms of the form t^k e^(μ-z)t. Integrability and convergence to zero follow from Reμ<Rez. The formal proof uses generalized eigenspaces of left multiplication on the matrix algebra, so nilpotent Jordan parts are included. Both inverse identities follow ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-16020a46bf72f212","tex":"\\e^{-(s+t)}","aliases":"e^-(s+t) e mathrm","display":"{\\mathrm e}^{-(s+t)}","kind":"use","excerpt":"Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-213cbea125edd4d0","tex":"e^{-(s-r)B^\\alpha}","aliases":"e^-(s-r)B^α alpha","display":"e^{-(s-r)B^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a17fc961408f2c36","tex":"e^{-(t-r)}","aliases":"e^-(t-r)","display":"e^{-(t-r)}","kind":"use","excerpt":"vpage22Hidden state returns as memoryEXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUPTraditional Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data. Write the same system as x'=-x+y, y'=x-y. Solving the second equation first gives y(t)=e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. Substitution is exact: x'(t)=-x(t)+e^-ty(0)+∫","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9b8fb3a0804953d6","tex":"e^{-(z+2)t}","aliases":"e^-(z+2)t","display":"e^{-(z+2)t}","kind":"use","excerpt":"t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that retained the eigenvalue -2 but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps. headThe transfer function sees pol","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-705a582f899d971c","tex":"\\e^{-2(s+t)}","aliases":"e^-2(s+t) e mathrm","display":"{\\mathrm e}^{-2(s+t)}","kind":"use","excerpt":"Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e19d2c16a0f316fd","tex":"e^{-2(s+t)}","aliases":"e^-2(s+t)","display":"e^{-2(s+t)}","kind":"use","excerpt":"A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mechanism The constant mode is unchanged, while each elapsed interval multiplies the contrast by e^-2t. Consequently two inte","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-846c05d1e926ff8e","tex":"e^{-2(t-r)}","aliases":"e^-2(t-r)","display":"e^{-2(t-r)}","kind":"use","excerpt":"The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn th","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-67d5ba9f4cee7ae8","tex":"e^{-2jt}","aliases":"e^-2jt","display":"e^{-2jt}","kind":"use","excerpt":"itive self-adjoint finite model and its spectral powers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its square root is L^1/2=sqrt2P_1+2P_2+sqrt6P_3, (L^1/2)^2=L. Both heat evolutions retain these modes. Ordinary diffusion mul","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-31a645cf967f487c","tex":"\\e^{-2nt}","aliases":"e^-2nt e mathrm","display":"{\\mathrm e}^{-2nt}","kind":"use","excerpt":"[Hermite packet reconstruction] Let P_n be the orthogonal projection onto C h_n. Then I=∑_n=0^∞ P_n in the strong operator topology and O_t = ∑_n=0^∞ e^-ntP_n. For f∈ L^2(γ), norm O_t f_2^2 = ∑_n=0^∞ e^-2ntnormP_nf_2^2.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0e46f05d4098d239","tex":"\\e^{-2s}","aliases":"e^-2s e mathrm","display":"{\\mathrm e}^{-2s}","kind":"use","excerpt":"Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f67647d8f9d6f031","tex":"e^{-2s}","aliases":"e^-2s","display":"e^{-2s}","kind":"use","excerpt":"A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mechanism The constant mode is unchanged, while each elapsed interval multiplies the contrast by e^-2t. Consequently two inte","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d81f545674e3f61e","tex":"\\e^{-2t}","aliases":"e^-2t e mathrm","display":"{\\mathrm e}^{-2t}","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7511ccdfc1ceeb18","tex":"e^{-2t}","aliases":"e^-2t","display":"e^{-2t}","kind":"use","excerpt":"1 · -1. The total-mass direction (1,1) has eigenvalue zero. The contrast (1,-1) has eigenvalue -2. Resolving those two directions gives the full transfer, not only its first-order approximation: T(t)=e^tG=frac121+e^-2t · 1-e^-2t 1-e^-2t · 1+e^-2t. vfigclock platecaptionA point seed at state zero approaches the uniform distribution. The displayed ti","count":22},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2b902887ca3c0c26","tex":"e^{-3t}","aliases":"e^-3t","display":"e^{-3t}","kind":"use","excerpt":"vpage37Keep blocks by their certified contributionMEMORY COMPRESSION / AN OPTIMIZATION WITH A DEFINED COSTTraditional Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks; block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jorda","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f37aa953e728f3f3","tex":"e^{-4t}","aliases":"e^-4t","display":"e^{-4t}","kind":"use","excerpt":"=-nH_n for these three polynomials. Their squared norms are n! and different Hermite degrees are orthogonal. vfigext_hermite platecaptionFor f=H_1+H_2, the two bars are its separate squared-energy contributions. At t=log2, their sum is 3/8, below the spectral-gap bound 3/4. headReconstruction preserves the modes; time changes their weights The semi","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-15950c0ab9e8c5e6","tex":"e^{-6t}","aliases":"e^-6t","display":"e^{-6t}","kind":"use","excerpt":") (12.144,0.650009) (12.21,0.650008) (12.276,0.650007) (12.342,0.650006) (12.408,0.650005) (12.474,0.650004) (12.54,0.650004) (12.606,0.650003) (12.672,0.650003) (12.738,0.650002) (12.804,0.650002) (12.87,0.650002) (12.936,0.650002) (13.002,0.650001) (13.068,0.650001) (13.134,0.650001) (13.2,0.650001); node[font=small,align=center,font=scriptsize] ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7ed7d803025ff45b","tex":"e^{-8t}","aliases":"e^-8t","display":"e^{-8t}","kind":"use","excerpt":"For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-363d8b60aa81bd16","tex":"e^{-a_nt}","aliases":"e^-a_nt","display":"e^{-a_nt}","kind":"use","excerpt":"vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONTraditional Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-61dcffd4e083f043","tex":"\\e^{-at}","aliases":"e^-at e mathrm","display":"{\\mathrm e}^{-at}","kind":"use","excerpt":"Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-81315b920e13e316","tex":"e^{-at}","aliases":"e^-at","display":"e^{-at}","kind":"use","excerpt":"0. The exact adjacent generator-defect magnitude is |A_n+1-A_n|=2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time interval, not errors sampled at the right endpoint. headThe estimate follows before any graph is drawn For 0≤ t≤ T, the derivative of a↦ e^","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7fbf805b9f71e84a","tex":"\\e^{-bt}","aliases":"e^-bt e mathrm","display":"{\\mathrm e}^{-bt}","kind":"use","excerpt":"d · -b , a,b>0, and retain the first coordinate. The memory kernel and self energy are K(t)=cd e^-bt, Σ(z)=fraccdz+b. The retained coordinate has an exact time local closure for every retained initial value precisely when cd=0. When cd≠0, one auxiliary scalar state realizes the memory exactly.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-735b6f4e0a49bd0b","tex":"\\e^{-nt}","aliases":"e^-nt e mathrm","display":"{\\mathrm e}^{-nt}","kind":"use","excerpt":"[Ornstein Uhlenbeck semigroup] The operators ( O_t)_t≥0 form a positive Markov contraction semigroup on L^p(γ) for every 1≤ p≤∞. On L^2(γ), the semigroup is self adjoint and O_t h_n= e^-nth_n. The family (h_n)_n≥0 is an orthonormal basis of L^2(γ).","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e35c3786b9e0a2fb","tex":"e^{-nt}","aliases":"e^-nt","display":"e^{-nt}","kind":"use","excerpt":"[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-487f96295a31f05f","tex":"e^{-r}","aliases":"e^-r","display":"e^{-r}","kind":"use","excerpt":"The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn th","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-458955e59598018c","tex":"e^{-rA^\\alpha}","aliases":"e^-rA^α alpha","display":"e^{-rA^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-33bc3e010e27b2a8","tex":"\\e^{-s}","aliases":"e^-s e mathrm","display":"{\\mathrm e}^{-s}","kind":"use","excerpt":"Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dc9a49b80c0d1df4","tex":"e^{-s}","aliases":"e^-s","display":"e^{-s}","kind":"use","excerpt":"/2 and f_2(t)=e^-2t. vfigext_risks platecaptionAt t=log2, surviving mass is 3/8. Cause 1 has accumulated probability 1/4 and cause 2 has accumulated probability 3/8; the three pieces sum to one. headCondition on survival before restarting the clock At this time the transient masses are (1/4,1/8), so the conditional distribution of survivors is (2/3","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8add4baf58362c10","tex":"e^{-sA^\\alpha}","aliases":"e^-sA^α alpha","display":"e^{-sA^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ea0104f7653eb4fc","tex":"e^{-sB^\\alpha}","aliases":"e^-sB^α alpha","display":"e^{-sB^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a55c82c872f9fae7","tex":"\\e^{-sL}","aliases":"e^-sL e mathrm","display":"{\\mathrm e}^{-sL}","kind":"use","excerpt":"nstandardfact[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c6c11852ddb445b7","tex":"e^{-sL}","aliases":"e^-sL","display":"e^{-sL}","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c61b01610015a01e","tex":"e^{-sm^\\alpha}","aliases":"e^-sm^α alpha","display":"e^{-sm^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-218864f064df802f","tex":"e^{-\\sqrt{2j}\\,t}","aliases":"e^-sqrt2j t sqrt","display":"e^{-\\sqrt{2j}\\,t}","kind":"use","excerpt":"wers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its square root is L^1/2=sqrt2P_1+2P_2+sqrt6P_3, (L^1/2)^2=L. Both heat evolutions retain these modes. Ordinary diffusion multiplies mode j by e^-2jt; square-root diffusion mul","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9209e63597cdde74","tex":"\\e^{-st}","aliases":"e^-st e mathrm","display":"{\\mathrm e}^{-st}","kind":"use","excerpt":"The Laplace transform follows by integrating the density in [source reference]. Stability of A gives ∫_0^∞ e^-st e^tA d t = (sI-A)^-1. At s=0, a^T(-A)^-1p_0 = 1^Tp_0=1. For the moments, ∫_0^∞ t^m e^tA d t = m!(-A)^-m-1. Multiplying by a^T= 1^T(-A) gives the result. These are the standard matrix formulas for phase type laws [source reference].","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3543ab1cc3650ab3","tex":"e^{-st}","aliases":"e^-st","display":"e^{-st}","kind":"use","excerpt":"For the survival consequences, A is a finite Metzler matrix with nonpositive column sums and spectrum in the open left half-plane, p_0 is a probability vector, S(t)= 1^T e^tAp_0, and f(t)=-S'(t). The one-phase model has λ>0. The survival Laplace identity of Part II also holds for every complex s with Res≥0: stability gives absolute convergence of ∫","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3117c4fb220fd668","tex":"\\e^{-s\\lambda}","aliases":"e^-sλ e lambda mathrm lambda","display":"{\\mathrm e}^{-s\\lambda}","kind":"use","excerpt":"[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-76df3c6c345601fc","tex":"\\e^{-s\\tau}","aliases":"e^-sτ e tau mathrm tau","display":"{\\mathrm e}^{-s\\tau}","kind":"use","excerpt":"[Laplace transform and moments] Assume the spectrum of A lies in the open left half plane. Then absorption occurs with probability one. For s≥0, E[ e^-sτ] = a^T(sI-A)^-1p_0. For every integer m≥1, E[τ^m] = m! 1^T(-A)^-mp_0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8ce5ba5917308780","tex":"\\e^{-t}","aliases":"e^-t e mathrm","display":"{\\mathrm e}^{-t}","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3f6dc0c78382f5b1","tex":"e^{-t}","aliases":"e^-t","display":"e^{-t}","kind":"use","excerpt":"The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn th","count":15},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f1596d0f30252060","tex":"\\e^{-t(-\\Lcal)^\\alpha}","aliases":"e^-t(-Lcal)^α e^-t(- L)^α e Lcal alpha mathrm mathcal alpha","display":"{\\mathrm e}^{-t(-{\\mathcal L})^\\alpha}","kind":"use","excerpt":"[Fractional Hermite transfer] Let - L be the nonnegative Ornstein Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-87d847893f57d8a1","tex":"\\e^{-t(k\\pi)^{2\\alpha}}","aliases":"e^-t(kπ)^2α e pi alpha mathrm pi alpha","display":"{\\mathrm e}^{-t(k\\pi)^{2\\alpha}}","kind":"use","excerpt":"set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(kπ)^2α ≤ fractα(4k^2)^α-1(kπ)^412h^2.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d912db525006fd32","tex":"\\e^{-t^2/(4s)}","aliases":"e^-t^2/(4s) e mathrm","display":"{\\mathrm e}^{-t^2/(4s)}","kind":"use","excerpt":"[Poisson subordination] For t>0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s) e^-sL d s. The integral converges in operator norm.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a21b18821fa51fe8","tex":"e^{-t^2/(4s)}","aliases":"e^-t^2/(4s)","display":"e^{-t^2/(4s)}","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7a923b5b884633e8","tex":"\\e^{-t^2/(4s)-\\lambda s}","aliases":"e^-t^2/(4s)-λ s e lambda mathrm lambda","display":"{\\mathrm e}^{-t^2/(4s)-\\lambda s}","kind":"use","excerpt":"On a spectral packet with eigenvalue λ, the formula reduces to e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s)-λ s d s. This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bcd0496b618b65e0","tex":"e^{-t^2/(4s)-\\lambda s}","aliases":"e^-t^2/(4s)-λ s lambda","display":"e^{-t^2/(4s)-\\lambda s}","kind":"use","excerpt":"By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6885d0ac0968565e","tex":"\\e^{-ta}","aliases":"e^-ta e mathrm","display":"{\\mathrm e}^{-ta}","kind":"use","excerpt":"proof The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-097eefc0b2596360","tex":"\\e^{-tb}","aliases":"e^-tb e mathrm","display":"{\\mathrm e}^{-tb}","kind":"use","excerpt":"proof The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-18da745fe6ddb48f","tex":"\\e^{-tD_V/n}","aliases":"e^-tD_V/n e mathrm","display":"{\\mathrm e}^{-tD_V/n}","kind":"use","excerpt":"[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-98e1690e1022778c","tex":"e^{-tL}","aliases":"e^-tL","display":"e^{-tL}","kind":"use","excerpt":"Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usu","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e483707257d6afec","tex":"\\e^{-tL^\\alpha}","aliases":"e^-tL^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tL^\\alpha}","kind":"use","excerpt":"For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3551df4650a99bb8","tex":"e^{-tL^\\alpha}","aliases":"e^-tL^α alpha","display":"e^{-tL^\\alpha}","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-df43c0922b3aee51","tex":"\\e^{-tL_c^\\alpha}","aliases":"e^-tL_c^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tL_c^\\alpha}","kind":"use","excerpt":"[Exact fractional naturality] Let L_f and L_c be finite dimensional positive self adjoint operators and let C satisfy CL_f=L_cC. Then, for every α>0 and t≥0, CL_f^α=L_c^α C and C e^-tL_f^α = e^-tL_c^αC.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ced91f13885abaca","tex":"\\e^{-tL_f^\\alpha}","aliases":"e^-tL_f^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tL_f^\\alpha}","kind":"use","excerpt":"[Exact fractional naturality] Let L_f and L_c be finite dimensional positive self adjoint operators and let C satisfy CL_f=L_cC. Then, for every α>0 and t≥0, CL_f^α=L_c^α C and C e^-tL_f^α = e^-tL_c^αC.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d2ff9ce40fe71e18","tex":"\\e^{-tn^\\alpha}","aliases":"e^-tn^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tn^\\alpha}","kind":"use","excerpt":"[Fractional Hermite transfer] Let - L be the nonnegative Ornstein Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bd2277b608148acc","tex":"\\e^{-t\\sqrt L}","aliases":"e^-tsqrt L e sqrt mathrm sqrt","display":"{\\mathrm e}^{-t\\sqrt L}","kind":"use","excerpt":"[Poisson subordination] For t>0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s) e^-sL d s. The integral converges in operator norm.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-af1213c560bb1d07","tex":"e^{-t\\sqrt L}","aliases":"e^-tsqrt L sqrt","display":"e^{-t\\sqrt L}","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-09abe17db67d5177","tex":"e^{-t\\sqrt6}","aliases":"e^-tsqrt6 sqrt","display":"e^{-t\\sqrt6}","kind":"use","excerpt":"650013) (12.078,0.650011) (12.144,0.650009) (12.21,0.650008) (12.276,0.650007) (12.342,0.650006) (12.408,0.650005) (12.474,0.650004) (12.54,0.650004) (12.606,0.650003) (12.672,0.650003) (12.738,0.650002) (12.804,0.650002) (12.87,0.650002) (12.936,0.650002) (13.002,0.650001) (13.068,0.650001) (13.134,0.650001) (13.2,0.650001); node[font=small,align=","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dfef5b71f36f50fe","tex":"\\e^{-t\\sqrt\\lambda}","aliases":"e^-tsqrtλ e sqrt lambda mathrm sqrt lambda","display":"{\\mathrm e}^{-t\\sqrt\\lambda}","kind":"use","excerpt":"On a spectral packet with eigenvalue λ, the formula reduces to e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s)-λ s d s. This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ec0c0004dcef9a0c","tex":"e^{-t\\sqrt\\lambda}","aliases":"e^-tsqrtλ sqrt lambda","display":"e^{-t\\sqrt\\lambda}","kind":"use","excerpt":"By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-abbd5115258324d6","tex":"\\e^{-tx^\\alpha}","aliases":"e^-tx^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tx^\\alpha}","kind":"use","excerpt":"The union of the two finite spectra is a finite set. There is a polynomial that agrees with x^α on that set, and another polynomial that agrees with e^-tx^α. Exact intertwining passes to every polynomial by induction.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e8329615317d6615","tex":"\\e^{-t\\lambda^\\alpha}","aliases":"e^-tλ^α e lambda alpha mathrm lambda alpha","display":"{\\mathrm e}^{-t\\lambda^\\alpha}","kind":"use","excerpt":"[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-75a311590636cbd2","tex":"\\e^{-t\\lambda_j^\\alpha}","aliases":"e^-tλ_j^α e lambda alpha mathrm lambda alpha","display":"{\\mathrm e}^{-t\\lambda_j^\\alpha}","kind":"use","excerpt":"Let H be a finite dimensional Hilbert space and let L=L^*≥0. Write its spectral resolution as L=∑_jλ_jP_j. For α>0, define L^α = ∑_jλ_j^α P_j and T_α(t) = e^-tL^α = ∑_j e^-tλ_j^αP_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b7a32a247742f9a8","tex":"e^{-t\\lambda_j^\\alpha}","aliases":"e^-tλ_j^α lambda alpha","display":"e^{-t\\lambda_j^\\alpha}","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-876017de9ad7345a","tex":"\\e^{-t\\lambda_k}","aliases":"e^-tλ_k e lambda mathrm lambda","display":"{\\mathrm e}^{-t\\lambda_k}","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3a4df964ebd62283","tex":"\\e^{-t\\lambda_{k,h}}","aliases":"e^-tλ_k,h e lambda mathrm lambda","display":"{\\mathrm e}^{-t\\lambda_{k,h}}","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0a284cb9f17e58c3","tex":"\\e^{-t\\lambda_{k,h}^\\alpha}","aliases":"e^-tλ_k,h^α e lambda alpha mathrm lambda alpha","display":"{\\mathrm e}^{-t\\lambda_{k,h}^\\alpha}","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f2b5b8666856ee72","tex":"\\e^{-t\\lambda_k^\\alpha}","aliases":"e^-tλ_k^α e lambda alpha mathrm lambda alpha","display":"{\\mathrm e}^{-t\\lambda_k^\\alpha}","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6030b628b360890a","tex":"e^{-u^2}","aliases":"e^-u^2","display":"e^{-u^2}","kind":"use","excerpt":"al time; s is the ordinary diffusion time being mixed. They are not interchangeable variables. vfigrandom_clock platecaptionThe density η_1 peaks at s=1/6 but has a long right tail. The drawing ends at s=2; it does not claim that the missing tail has zero mass. headCheck the normalizer before interpreting a density Under s=t^2/(4u^2), reversing the","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8db1472c40be17a4","tex":"\\e^{-V}","aliases":"e^-V e mathrm","display":"{\\mathrm e}^{-V}","kind":"use","excerpt":"[Discrete Feynman Kac formula] Define M_V=diag( e^-V)K^T. For terminal data f, (M_V^nf)_i = E_i[ exp( -∑_r=0^n-1V(X_r) ) f(X_n) ].","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a5da71253c0e2cad","tex":"e^{-V}","aliases":"e^-V","display":"e^{-V}","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c891c594039cd0f2","tex":"\\e^{-V(i)}","aliases":"e^-V(i) e mathrm","display":"{\\mathrm e}^{-V(i)}","kind":"use","excerpt":"For one step, (M_Vf)_i = e^-V(i)∑_jK_jif_j. Iterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9341ba060ec74a26","tex":"e^{-V(i)}","aliases":"e^-V(i)","display":"e^{-V(i)}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b917ccc3ae79f8f2","tex":"e^{-V(i_r)}","aliases":"e^-V(i_r)","display":"e^{-V(i_r)}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-64589a9f7fc7012d","tex":"\\e^{-v_0t}","aliases":"e^-v_0t e mathrm","display":"{\\mathrm e}^{-v_0t}","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dc510a5e29f5ee80","tex":"\\e^{-zt}","aliases":"e^-zt e mathrm","display":"{\\mathrm e}^{-zt}","kind":"use","excerpt":"[Laplace resolvent and packet transform] Let A∈ B(X), let T(t)= e^tA, and let Rez>norm A. Then the Bochner integral converges in operator norm and (zI-A)^-1 = ∫_0^∞ e^-ztT(t) d t. For every pair of packets, P_β(zI-A)^-1P_α = ∫_0^∞ e^-ztT_βα(t) d t.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-abf22631e30760b4","tex":"e^{-zt}","aliases":"e^-zt","display":"e^{-zt}","kind":"use","excerpt":"[Memory/self-energy correspondence] For finite complex block matrices, assume Rez>Reλ for every λ∈σ(A_QQ). Then the following improper integral is absolutely convergent and ∫_0^∞ e^-zt K(t) dt =A_PQ(zI-A_QQ)^-1A_QP =Σ_P(z).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dff4ae82d4fe03f8","tex":"\\e^{-\\gamma t}","aliases":"e^-γ t e gamma mathrm gamma","display":"{\\mathrm e}^{-\\gamma t}","kind":"use","excerpt":"[Memory and self energy] If z lies to the right of the spectrum of A_QQ, then ∫_0^∞ e^-zt K(t) d t = Σ_P(z). If, for constants M≥0 and γ>0, norm e^tA_QQ ≤ M e^-γ t, then norm K(t) ≤ MnormA_PQnormA_QP e^-γ t and norm K_L^1(0,∞) ≤ fracMγ normA_PQnormA_QP.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8314328de53179fc","tex":"\\e^{-\\delta t}","aliases":"e^-δ t e delta mathrm delta","display":"{\\mathrm e}^{-\\delta t}","kind":"use","excerpt":"[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-640ecb7ba05e7b8b","tex":"e^{-\\lambda s}","aliases":"e^-λ s lambda","display":"e^{-\\lambda s}","kind":"use","excerpt":"ends at s=2; it does not claim that the missing tail has zero mass. headCheck the normalizer before interpreting a density Under s=t^2/(4u^2), reversing the integration limits gives η_t(s) |ds|=frac2sqrtπe^-u^2 du. Its integral over u>0 is one, so η_t is a probability density. On each eigenmode of L, the operator formula reduces to the scalar Lapla","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0cf7bf147f70ddde","tex":"\\e^{-\\lambda t}","aliases":"e^-λ t e lambda mathrm lambda","display":"{\\mathrm e}^{-\\lambda t}","kind":"use","excerpt":"[One phase model] If A=(-λ) and p_0=1, then S(t)= e^-λ t, f(t)=λ e^-λ t, h(t)=λ.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ccc125183ac25b28","tex":"e^{-\\lambda t}","aliases":"e^-λ t lambda","display":"e^{-\\lambda t}","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9ece77397bb6f22b","tex":"\\e^{-\\mu t}","aliases":"e^-μ t e mu mathrm mu","display":"{\\mathrm e}^{-\\mu t}","kind":"use","excerpt":"0, where λ≠μ are positive. The first phase moves to the second at rate λ, and the second is absorbed at rate μ. Direct calculation gives S(t)= fracμ e^-λ t-λ e^-μ tμ-λ, f(t)= fracλμμ-λ ( e^-λ t- e^-μ t). The ratio f(t)/S(t) is generally not constant. A time local Markov model on two transient phases can therefore produce a nonconstant hazard after ","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-50860b6739bbedbb","tex":"e^{1/4}","aliases":"e^1/4","display":"e^{1/4}","kind":"use","excerpt":"und is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields sup_0≤ t≤ T x(t)-widetilde x(t) =fracT^22≤ T^2e^T^2. At T=1/2, the exact error is 1/8 and the certificate is e^1/4/4. Nilpotence make","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b87f37426ee94627","tex":"E^2","aliases":"E^2","display":"E^2","kind":"use","excerpt":")^ T becomes Ef=(2,2,26/7,26/7)^ T. vfigweighted_projection platecaptionThe bars are function values, not state probabilities. Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-93314c38514276c7","tex":"\\e^{at}","aliases":"e^at e mathrm","display":"{\\mathrm e}^{at}","kind":"use","excerpt":"c · b and let the two coordinate axes be the packet spaces. If a≠ b, then e^tA = e^at · 0 displaystyle cfrac e^at- e^bta-b · e^bt . The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with ct, in agreement with [source reference]. When a=b, the same block is ct e^at.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-daab3aa19b753da2","tex":"\\e^{bt}","aliases":"e^bt e mathrm","display":"{\\mathrm e}^{bt}","kind":"use","excerpt":"c · b and let the two coordinate axes be the packet spaces. If a≠ b, then e^tA = e^at · 0 displaystyle cfrac e^at- e^bta-b · e^bt . The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with ct, in agreement with [source reference]. When a=b, the same block is ct e^at.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8c244c5ba6bf72e0","tex":"e^{Q/8}","aliases":"e^Q/8","display":"e^{Q/8}","kind":"use","excerpt":"For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-251aa8cc2c01364f","tex":"e^{rA_f}","aliases":"e^rA_f","display":"e^{rA_f}","kind":"use","excerpt":"[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-275ed3fe872da008","tex":"\\e^{\\operatorname{Re}\\mu_jt}","aliases":"e^Reμ_jt e operatorname mu mathrm operatorname mu","display":"{\\mathrm e}^{\\operatorname{Re}\\mu_jt}","kind":"use","excerpt":"esolvent expansion is (zI-A_QQ)^-1Q_j = ∑_k=0^ν_j-1 fracN_j^kQ_j(z-μ_j)^k+1. Multiplication by the coupling blocks gives the self energy formula for every z outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity ∫_0^∞ e^-zt e^μ_jt fract^kk! d t = frac1(z-μ_j)^k+1. Under stability, ∫_0^∞ e^Reμ_jt fract^","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-39fc7735a3cc982f","tex":"\\e^{s_1D}","aliases":"e^s_1D e mathrm","display":"{\\mathrm e}^{s_1D}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-04db6340c88fba8b","tex":"\\e^{sA}","aliases":"e^sA e mathrm","display":"{\\mathrm e}^{sA}","kind":"use","excerpt":"Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-21367aaf2687869b","tex":"e^{sA}","aliases":"e^sA","display":"e^{sA}","kind":"use","excerpt":"[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-33aedb64301f3993","tex":"\\e^{sA_f}","aliases":"e^sA_f e mathrm","display":"{\\mathrm e}^{sA_f}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e256d144bb836c0b","tex":"\\e^{sg}","aliases":"e^sg e mathrm","display":"{\\mathrm e}^{sg}","kind":"use","excerpt":"The semigroup law follows from multiplication of the likelihoods e^sg and e^tg. Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-31d4d3c24349f5c5","tex":"e^{sg}","aliases":"e^sg","display":"e^{sg}","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2c268d8002e5f66f","tex":"e^{sQ}","aliases":"e^sQ","display":"e^{sQ}","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1fe69fafa50db3c5","tex":"e^t","aliases":"e^t","display":"e^t","kind":"use","excerpt":"vpage12A nonlinear semigroup of oddsBAYESIAN FLOW / COMPOSITION WITHOUT LINEARITYTraditional Part I: Theorems 7.1 and 9.1. Likelihood composition and the replicator field. Let the score vector be g=(1,0) and start at (1/2,1/2). The likelihood at time t is (e^t,1), so the first posterior coordinate is p_0(t)=frace^t1+e^t, p_1(t)=frac11+e^t. The odds","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e9888ce0a8ae50f0","tex":"\\e^{t(B+C)}","aliases":"e^t(B+C) e mathrm","display":"{\\mathrm e}^{t(B+C)}","kind":"use","excerpt":"Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1500c5980f1fb9cc","tex":"e^{t(B-zI)}","aliases":"e^t(B-zI)","display":"e^{t(B-zI)}","kind":"use","excerpt":"Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace; equivalently, apply the decomposition to left multiplication by B on the matrix algebra a","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6b789f046b6d1784","tex":"\\e^{t(g_i-g_j)}","aliases":"e^t(g_i-g_j) e mathrm","display":"{\\mathrm e}^{t(g_i-g_j)}","kind":"use","excerpt":"[Bayesian flow and replicator equation] The maps Φ_t form a nonlinear semigroup: Φ_s+t=Φ_s∘Φ_t. Every coordinate satisfies frac d d tp_i(t) = (g_i-ipgp(t))p_i(t). The support of p(t) is independent of t, and for any two coordinates in the support, fracp_i(t)p_j(t) = e^t(g_i-g_j) fracp_i(0)p_j(0).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fe2f9fcc69678a23","tex":"\\e^{t(\\overline Q^T-\\overline V)}","aliases":"e^t(overline Q^T-overline V) e overline overline mathrm overline overline","display":"{\\mathrm e}^{t(\\overline Q^T-\\overline V)}","kind":"use","excerpt":"[Coarse Feynman Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9ce15abfc0ced257","tex":"\\e^{t(Q+aI)}","aliases":"e^t(Q+aI) e mathrm","display":"{\\mathrm e}^{t(Q+aI)}","kind":"use","excerpt":"Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d049e66d5d6db04b","tex":"\\e^{t(Q-D_V)}","aliases":"e^t(Q-D_V) e mathrm","display":"{\\mathrm e}^{t(Q-D_V)}","kind":"use","excerpt":"Average [source reference] with terminal function f= 1 against the initial distribution. Transposition gives p_0^T e^t(Q^T-D_V) 1 = 1^T e^t(Q-D_V)p_0. Since 1^TQ=0, differentiation gives S_V'(t) = - 1^TD_V e^tAp_0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-46b9991988acc297","tex":"\\e^{t(Q^T-D_V)}","aliases":"e^t(Q^T-D_V) e mathrm","display":"{\\mathrm e}^{t(Q^T-D_V)}","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-219930913ca9cfcf","tex":"\\e^{t(Q^T-V)}","aliases":"e^t(Q^T-V) e mathrm","display":"{\\mathrm e}^{t(Q^T-V)}","kind":"use","excerpt":"Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a5523b5f204f917d","tex":"\\e^{t(Q^T-W)}","aliases":"e^t(Q^T-W) e mathrm","display":"{\\mathrm e}^{t(Q^T-W)}","kind":"use","excerpt":"Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-728cabc55310db85","tex":"\\e^{t(QAQ)|_{QX}}","aliases":"e^t(QAQ)|_QX e mathrm","display":"{\\mathrm e}^{t(QAQ)|_{QX}}","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9c95014e23d49b17","tex":"e^{T^2}","aliases":"e^T^2","display":"e^{T^2}","kind":"use","excerpt":"trajectory platecaptionThe first-coordinate error is exactly t^2/2. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields ","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0bc0eec613816bce","tex":"e^{tA}","aliases":"e^tA","display":"e^{tA}","kind":"use","excerpt":"Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usu","count":17},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-053701566d46fdc2","tex":"\\e^{tA_c}","aliases":"e^tA_c e mathrm","display":"{\\mathrm e}^{tA_c}","kind":"use","excerpt":"Let X_f and X_c be complex Banach spaces. Let A_f∈ B(X_f), A_c∈ B(X_c), and C∈ B(X_f,X_c). Define Δ=CA_f-A_cC and D(t)=C e^tA_f- e^tA_cC.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ee1b64c4f1868d3a","tex":"e^{tA_c}","aliases":"e^tA_c","display":"e^{tA_c}","kind":"use","excerpt":"Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2d95d42808445044","tex":"\\e^{tA_f}","aliases":"e^tA_f e mathrm","display":"{\\mathrm e}^{tA_f}","kind":"use","excerpt":"Let X_f and X_c be complex Banach spaces. Let A_f∈ B(X_f), A_c∈ B(X_c), and C∈ B(X_f,X_c). Define Δ=CA_f-A_cC and D(t)=C e^tA_f- e^tA_cC.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8bef86072190de27","tex":"e^{tA_f}","aliases":"e^tA_f","display":"e^{tA_f}","kind":"use","excerpt":"Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d9f776015ddefc19","tex":"\\e^{tA_n}","aliases":"e^tA_n e mathrm","display":"{\\mathrm e}^{tA_n}","kind":"use","excerpt":"[A usable closure condition for refinement towers] Assume the contractive summability hypothesis of [source reference] holds on every bounded time interval. Suppose every P_m,n is an orthogonal projection on a Hilbert space and T_n(t)= e^tA_n is contractive. If, for every fixed m, norm(I-P_m,n)A_nP_m,n ⟶0, then the limiting family U_m is a semigrou","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6221765d6dbc4c15","tex":"\\e^{tA_{PP}}","aliases":"e^tA_PP e mathrm","display":"{\\mathrm e}^{tA_{PP}}","kind":"use","excerpt":"[Criterion for exact local closure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-93f0045082a837ac","tex":"\\e^{tA_{QQ}}","aliases":"e^tA_QQ e mathrm","display":"{\\mathrm e}^{tA_{QQ}}","kind":"use","excerpt":"For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0)","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c2db33b004054b96","tex":"e^{tA_{QQ}}","aliases":"e^tA_QQ","display":"e^{tA_{QQ}}","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b45b0921ad704da0","tex":"\\e^{tA_{\\alpha\\alpha}}","aliases":"e^tA_αα e alpha alpha mathrm alpha alpha","display":"{\\mathrm e}^{tA_{\\alpha\\alpha}}","kind":"use","excerpt":"thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c3b1c48fbafaa657","tex":"\\e^{tA_\\ell}","aliases":"e^tA_ℓ e ell mathrm ell","display":"{\\mathrm e}^{tA_\\ell}","kind":"use","excerpt":"If T_ℓ(t)= e^tA_ℓ, the generators are bounded, and sup_ℓnormA_ℓ<∞, C_ℓ A_ℓ+1=A_ℓ C_ℓ, then the continuity condition holds and A_∞(x_ℓ)_ℓ=(A_ℓ x_ℓ)_ℓ is the bounded generator of T_∞.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e6c1a6db623e88fe","tex":"e^{tA_\\ell}","aliases":"e^tA_ℓ ell","display":"e^{tA_\\ell}","kind":"use","excerpt":"[Uniform refinement control] For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that normC_ℓ e^tA_ℓ+1-e^tA_ℓC_ℓ ≤ M t normC_ℓ A_ℓ+1-A_ℓ C_ℓ on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-205d9e45d3bf3ac2","tex":"e^{tA_{\\ell+1}}","aliases":"e^tA_ℓ+1 ell","display":"e^{tA_{\\ell+1}}","kind":"use","excerpt":"[Uniform refinement control] For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that normC_ℓ e^tA_ℓ+1-e^tA_ℓC_ℓ ≤ M t normC_ℓ A_ℓ+1-A_ℓ C_ℓ on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e5c432074c8b99aa","tex":"\\e^{tB}","aliases":"e^tB e mathrm","display":"{\\mathrm e}^{tB}","kind":"use","excerpt":"pace, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus e^-zt e^tB is ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-43c981a8b54a5fb2","tex":"e^{tB}","aliases":"e^tB","display":"e^{tB}","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e54061964f4faacf","tex":"\\e^{tB/n}","aliases":"e^tB/n e mathrm","display":"{\\mathrm e}^{tB/n}","kind":"use","excerpt":"Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dbe7c874c164d2d5","tex":"e^{t\\bar Q}","aliases":"e^tbar Q bar","display":"e^{t\\bar Q}","kind":"use","excerpt":"2 · -2. The Duhamel identity therefore gives Ce^tQ=e^tbar QC. This is a second consequence of the corresponding generator identity, not an inference that K is a fixed-time sample of Q.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a740f49548879451","tex":"\\e^{tC/n}","aliases":"e^tC/n e mathrm","display":"{\\mathrm e}^{tC/n}","kind":"use","excerpt":"Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1ef82f33e2b0d0d2","tex":"\\e^{tD}","aliases":"e^tD e mathrm","display":"{\\mathrm e}^{tD}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9b6d0b4fad11116f","tex":"e^{tD}","aliases":"e^tD","display":"e^{tD}","kind":"use","excerpt":"A finite auxiliary realization of a matrix kernel means K(t)=B e^tDC, with finite matrices of compatible sizes. Laplace transforms are taken in a right half-plane of absolute convergence. The rational characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies. [Rational memory realizat","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-179eee2354eefa9f","tex":"\\e^{tG}","aliases":"e^tG e mathrm","display":"{\\mathrm e}^{tG}","kind":"use","excerpt":"Factor e^tG from the normalizing constant. The total weight outside M is at most (1-p_M(0)) e^-δ t, while the weight on M is p_M(0).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-767456b9db25e846","tex":"\\e^{tg}","aliases":"e^tg e mathrm","display":"{\\mathrm e}^{tg}","kind":"use","excerpt":"The semigroup law follows from multiplication of the likelihoods e^sg and e^tg. Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cceac532c75d0672","tex":"e^{tG}","aliases":"e^tG","display":"e^{tG}","kind":"use","excerpt":"1 · -1. The total-mass direction (1,1) has eigenvalue zero. The contrast (1,-1) has eigenvalue -2. Resolving those two directions gives the full transfer, not only its first-order approximation: T(t)=e^tG=frac121+e^-2t · 1-e^-2t 1-e^-2t · 1+e^-2t. vfigclock platecaptionA point seed at state zero approaches the uniform distribution. The displayed ti","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bbc19376502612fc","tex":"e^{tg}","aliases":"e^tg","display":"e^{tg}","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-92b24ebf8d5b2479","tex":"\\e^{tg_i}","aliases":"e^tg_i e mathrm","display":"{\\mathrm e}^{tg_i}","kind":"use","excerpt":"Fix a score vector g∈ R^n. For t≥0, define Φ_t(p)_i = frac e^tg_ip_i ∑_j e^tg_jp_j.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4e4db77f7062a650","tex":"e^{tg_i}","aliases":"e^tg_i","display":"e^{tg_i}","kind":"use","excerpt":"Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-84cd29a204931e68","tex":"\\e^{tg_j}","aliases":"e^tg_j e mathrm","display":"{\\mathrm e}^{tg_j}","kind":"use","excerpt":"Fix a score vector g∈ R^n. For t≥0, define Φ_t(p)_i = frac e^tg_ip_i ∑_j e^tg_jp_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1c3843aff1b9a844","tex":"e^{tg_j}","aliases":"e^tg_j","display":"e^{tg_j}","kind":"use","excerpt":"Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-31529ccb041f73ad","tex":"e^{tJ}","aliases":"e^tJ","display":"e^{tJ}","kind":"use","excerpt":"0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponential is not e^-2tI: e^tJ=e^-2t(I+tN), K_rm mem(t)=Ue^tJV=t e^-2t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that r","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e7ca070f23ef9e5b","tex":"e^{tL}","aliases":"e^tL","display":"e^{tL}","kind":"use","excerpt":"L · A nonnegative self-adjoint operator. · Its evolution is e^-tL, not e^tL.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a1a818910db29e8b","tex":"\\e^{tM_A}","aliases":"e^tM_A e mathrm","display":"{\\mathrm e}^{tM_A}","kind":"use","excerpt":"For bounded generators with common bound M_A, sup_ℓ norm( e^tA_ℓ-I)x_ℓ ≤ ( e^tM_A-1)sup_ℓnormx_ℓ. This tends to zero with t. Exact generator intertwining makes (A_ℓ x_ℓ)_ℓ compatible, and coordinatewise differentiation identifies the generator.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bc1711b5d9da8839","tex":"\\e^{t\\norm A}","aliases":"e^tnorm A e norm mathrm norm","display":null,"kind":"use","excerpt":"The estimate normT(t)≤ e^tnorm A gives absolute convergence of the operator valued integral. Let R(z)=∫_0^∞ e^-ztT(t) d t. Since T'(t)=AT(t)=T(t)A, integration by parts gives (zI-A)R(z) = -∫_0^∞ frac d d t ( e^-ztT(t)) d t =I. The same calculation on the other side gives R(z)(zI-A)=I . Multiplication by P_β and P_α may be passed under the norm conv","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-879f489b7b741e9c","tex":"\\e^{t\\norm D}","aliases":"e^tnorm D e norm mathrm norm","display":null,"kind":"use","excerpt":"Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bb2c45501b486ab2","tex":"\\e^{t\\overline Q}","aliases":"e^toverline Q e overline mathrm overline","display":"{\\mathrm e}^{t\\overline Q}","kind":"use","excerpt":"[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e64b2c7d61b9945a","tex":"\\e^{tQ}","aliases":"e^tQ e mathrm","display":"{\\mathrm e}^{tQ}","kind":"use","excerpt":"Markov chain · K^n or e^tQ · equality of aggregated transition masses","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-084ac283f910faff","tex":"e^{tQ}","aliases":"e^tQ","display":"e^{tQ}","kind":"use","excerpt":"Use column probability vectors. A finite continuous-time Markov generator Q has nonnegative off-diagonal entries and column sums zero. The stochastic semigroup is K(t)=e^tQ.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e4ffd068bc2c7120","tex":"\\e^{tQ^T/n}","aliases":"e^tQ^T/n e mathrm","display":"{\\mathrm e}^{tQ^T/n}","kind":"use","excerpt":"[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-30d533e8bb13e1b1","tex":"e^{tu}","aliases":"e^tu","display":"e^{tu}","kind":"use","excerpt":"vpage36The score rises because it has varianceBAYESIAN FLOW / EVIDENCE CUMULANTSTraditional Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior. Start with equal prior masses on scores u=(0,1,2) and likelihood e^tu. Put q=e^t. The evidence and tilted distribution are Z(t)=frac1+q+q^23, p(t)=frac(1,q,q^2)1+q+q^2. At t=log2","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d93aff4d4f9a952d","tex":"\\e^{\\beta t}","aliases":"e^β t e beta mathrm beta","display":"{\\mathrm e}^{\\beta t}","kind":"use","excerpt":"pace, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus e^-zt e^tB is ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ff20cbfa7862a04f","tex":"\\e^{\\lambda_jt}","aliases":"e^λ_jt e lambda mathrm lambda","display":"{\\mathrm e}^{\\lambda_jt}","kind":"use","excerpt":"[Exponential sum memory] If K(t)=∑_j=1^mB_j e^λ_jtC_j, then K has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in t and are realized by Jordan blocks.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-97bcf9ce563a30aa","tex":"\\e^{\\mu_jt}","aliases":"e^μ_jt e mu mathrm mu","display":"{\\mathrm e}^{\\mu_jt}","kind":"use","excerpt":"xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4e729462b75849d7","tex":"\\e^{\\omega t}","aliases":"e^ω t e omega mathrm omega","display":"{\\mathrm e}^{\\omega t}","kind":"use","excerpt":"[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-94d84178b7ed29ff","tex":"e^{\\omega t}","aliases":"e^ω t omega","display":"e^{\\omega t}","kind":"use","excerpt":"section renewcommandthefigureRarabicfigure renewcommandtheHfigureR.arabicfigure renewcommandthetableRarabictable renewcommandtheHtableR.arabictable FloatBarrier Transfer paths and product-topology limits. Let T(t) be an arbitrary strongly continuous semigroup on a complex Banach space X, with generator A, and suppose normT(t)≤ M e^ω t for M≥1 and ω","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-02ef9a7b897239e9","tex":"\\e^{\\omega_c t}","aliases":"e^ω_c t e omega mathrm omega","display":"{\\mathrm e}^{\\omega_c t}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-345975bbfb2cd2a5","tex":"e^{\\omega_c t}","aliases":"e^ω_c t omega","display":"e^{\\omega_c t}","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d2c4b7d39cf705cb","tex":"e^{\\omega_c(t-r)}","aliases":"e^ω_c(t-r) omega","display":"e^{\\omega_c(t-r)}","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c908c0a0cf833d26","tex":"\\e^{\\omega_c(t-s)+\\omega_fs}","aliases":"e^ω_c(t-s)+ω_fs e omega omega mathrm omega omega","display":"{\\mathrm e}^{\\omega_c(t-s)+\\omega_fs}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dd0cfdcd6332cbae","tex":"\\e^{\\omega_f t}","aliases":"e^ω_f t e omega mathrm omega","display":"{\\mathrm e}^{\\omega_f t}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f7d0a5131b29b5fc","tex":"e^{\\omega_f t}","aliases":"e^ω_f t omega","display":"e^{\\omega_f t}","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8d8b7fcd7ac96536","tex":"e^{\\omega_fr}","aliases":"e^ω_fr omega","display":"e^{\\omega_fr}","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-eba5ceae0baaa423","tex":"\\e^{\\omega_Q(t-s)+\\omega s}","aliases":"e^ω_Q(t-s)+ω s e omega omega mathrm omega omega","display":"{\\mathrm e}^{\\omega_Q(t-s)+\\omega s}","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-79e834ba52a1e5be","tex":"\\e^{\\omega_Qt}","aliases":"e^ω_Qt e omega mathrm omega","display":"{\\mathrm e}^{\\omega_Qt}","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"bottomrule headOne convention test For K_2=3/4 · 1/4 1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the ","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-febc076ab7500943","tex":"E_{01}","aliases":"E_01","display":"E_{01}","kind":"use","excerpt":"state probabilities. Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E, WE=E^ TW, Eones=ones. The middle equation is weighted self-adjointness. Ordinary symmetry fails: E_01=2/","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"ds for B. vfiglumpability platecaptionThe fine columns are visibly different. Their sums over each target packet agree within each source packet. That equality is the reason a two-state description works. headA one-step calculation controls every discrete horizon The displayed columns give CK=bar KC. Multiplication then yields CK^n=bar K^nC (n≥0). ","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-136cb5922e752cc4","tex":"E_{10}","aliases":"E_10","display":"E_{10}","kind":"use","excerpt":"Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E, WE=E^ TW, Eones=ones. The middle equation is weighted self-adjointness. Ordinary symmetry fails: E_01=2/3 whereas E_10=1/3. T","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESTraditional Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"A discrete unit coordinate vector and a discretized unit point mass use different normalizations. For f=e_2, the result is the second inverse column. For f=e_2/h, it is the unscaled tent sampled at the nodes. Either convention can be useful, but it must be specified before comparing answers.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-da2ce42dc70bb3da","tex":"E_n","aliases":"E_n","display":"E_n","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f6b63f7340a4b3b9","tex":"E_\\mu","aliases":"E_μ mu","display":"E_\\mu","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-74537519c7bd4fd0","tex":"E_\\pi","aliases":"E_π pi","display":"E_\\pi","kind":"use","excerpt":"E_π · Conditional expectation of observables. · A weighted average, self-adjoint for the π inner product.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-49550fdb9f0a1cd8","tex":"E_\\ell","aliases":"E_ℓ ell","display":"E_\\ell","kind":"use","excerpt":"[Contractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-192930449f687944","tex":"\\EE","aliases":"EE E mathbb","display":"{\\mathbb E}","kind":"use","excerpt":"Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d2fa527dd8d76e08","tex":"\\EE_0","aliases":"EE_0 E_0 EE mathbb","display":"{\\mathbb E}_0","kind":"use","excerpt":"1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEver","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-86b2b150b465871e","tex":"\\EE_{p(t)}","aliases":"EE_p(t) E_p(t) EE mathbb","display":"{\\mathbb E}_{p(t)}","kind":"use","excerpt":"Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-70a9953dd9bd9b22","tex":"\\operatorname{End}","aliases":"End operatorname","display":"\\operatorname{End}","kind":"use","excerpt":"Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usu","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-39ea6df1a878e2c7","tex":"\\Exp","aliases":"Exp E mathbb","display":"{\\mathbb E}","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].","count":14},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8ce13a52afee76f0","tex":"\\Exp_i","aliases":"Exp_i E_i Exp mathbb","display":"{\\mathbb E}_i","kind":"use","excerpt":"[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9c561c86f95e10f8","tex":"\\Exp_{p_0}","aliases":"Exp_p_0 E_p_0 Exp mathbb","display":"{\\mathbb E}_{p_0}","kind":"use","excerpt":"[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off diagonal entries and nonpositive column sums, so it is the subgenera","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fbfc0ac7d89b213e","tex":"\\Exp_\\gamma","aliases":"Exp_γ E_γ Exp gamma mathbb gamma","display":"{\\mathbb E}_\\gamma","kind":"use","excerpt":"A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-640dc13c34cc0b55","tex":"\\Exp_\\pi","aliases":"Exp_π E_π Exp pi mathbb pi","display":"{\\mathbb E}_\\pi","kind":"use","excerpt":"[Law of total variance] For a real valued f, let P_0f= E_π[f] 1. Then Var_π(f) = Var_π(P_ Qf) + normf-P_ Qf_π^2.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"Set F(r)=e^(t-r)A_cCe^rA_f. Then F'(r)=e^(t-r)A_c(CA_f-A_cC)e^rA_f. Integrating from 0 to t gives F(t)-F(0)=Ce^tA_f-e^tA_cC and proves the formula.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2f4583c8b7c73ba0","tex":"f'","aliases":"f'","display":"f'","kind":"use","excerpt":"[Generator and Dirichlet form] On the polynomial subspace, the infinitesimal generator of ( O_t) is the differential operator L f=f''-xf'. It satisfies L h_n=-nh_n. For every polynomial f, ip- L ff_L^2(γ) = ∫_ Rabsf'(x)^2γ( d x).","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9337f69b3f162423","tex":"f''","aliases":"f''","display":"f''","kind":"use","excerpt":"[Generator and Dirichlet form] On the polynomial subspace, the infinitesimal generator of ( O_t) is the differential operator L f=f''-xf'. It satisfies L h_n=-nh_n. For every polynomial f, ip- L ff_L^2(γ) = ∫_ Rabsf'(x)^2γ( d x).","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f19bf4bf6a7fc594","tex":"f^{\\mathsf T}","aliases":"f^ T mathsf","display":"f^{\\mathsf T}","kind":"use","excerpt":"vpage2A dictionary with orientationsREAD BEFORE MULTIPLYING / CONVENTIONSTraditional Part I: Sections 2, 4, 6, 7, and 12. Column probabilities and backward observables. An apparent disagreement between two transfer formulas often comes from acting on different objects. Throughout this guide, probabilities are columns and observables are functions w","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3492b2ff841a2568","tex":"F^{(k)}","aliases":"F^(k)","display":"F^{(k)}","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3d6125ad8c6995d2","tex":"f^h","aliases":"f^h","display":"f^h","kind":"use","excerpt":"[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2054708cff15ab72","tex":"f^h_i","aliases":"f^h_i","display":"f^h_i","kind":"use","excerpt":"[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"ausesCOMPETING RISKS / RESIDUAL LIFETraditional Part II: Proposition E16.4 and Theorem E16.5. This example is stable and absorbs surely. Two transient states have no internal transitions. Their exit rates are 1 and 2, their initial masses are both 1/2, and the two states exit by distinct causes. Thus A=diag(-1,-2), S(t)=tfrac12e^-t+tfrac12e^-2t, wi","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","excerpt":"[Gaussian spectral gap] If f∈ L^2(γ) has mean zero, then norm O_tf_2 ≤ e^-tnorm f_2. For every polynomial f, Var_γ(f) ≤ ∫_ Rabsf'(x)^2γ( d x).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-43d33b9d6c0a8720","tex":"f_j^h","aliases":"f_j^h","display":"f_j^h","kind":"use","excerpt":"[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-07d5ca93af84b975","tex":"F_P","aliases":"F_P","display":"F_P","kind":"use","excerpt":"[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f7499e87191c71a6","tex":"f_r","aliases":"f_r","display":"f_r","kind":"use","excerpt":"[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d3f85bd975f20dc4","tex":"F_V","aliases":"F_V","display":"F_V","kind":"use","excerpt":"[Positivity and composition] The operators F_V(t)= e^t(Q^T-D_V) form a positive semigroup. If f≥0, then F_V(t)f≥0.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-da614697fd2616e5","tex":"f_\\pi^2","aliases":"f_π^2 pi","display":"f_\\pi^2","kind":"use","excerpt":"P_0,P_1,…,P_r come from increasingly fine partitions, so that Ran P_0⊆Ran P_1⊆⋯⊆Ran P_r. Define D_0=P_0, D_j=P_j-P_j-1 (1≤ j≤ r). Then P_iP_j=P_min(i,j), D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j. Every D_j is an orthogonal projector, and normP_rf_π^2 = ∑_j=0^rnormD_jf_π^2 and norm f_π^2 = ∑_j=0^rnormD_jf_π^2 + normf-P_rf_π^2.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.","count":20},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"[Coarse Dirichlet form] Under the hypotheses of [source reference], let Jg be the fine function that is constant with value g_a on B_a. Then -ipJgQ^TJg_L^2(π) = -ipgoverline Q^Tg_L^2(overlineπ).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"use","excerpt":"Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f7288fb9331c8dd5","tex":"g_j","aliases":"g_j","display":"g_j","kind":"use","excerpt":"Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-68683df8049d4a89","tex":"G_{\\rm nodes}","aliases":"G_rm nodes rm","display":"G_{\\rm nodes}","kind":"use","excerpt":"node[font=small,align=center,font=small] at (10.43,1.2) 1; path[draw=black!30,fill=black!8] (10.86,0.77) rectangle (11.72,1.63); node[font=small,align=center,font=small] at (11.29,1.2) 2; path[draw=black!30,fill=black!8] (11.72,0.77) rectangle (12.58,1.63); node[font=small,align=center,font=small] at (12.15,1.2) 3; node[font=small,align=center,font","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usu","count":15},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fa0621c04438e85a","tex":"h^{-2}","aliases":"h^-2","display":"h^{-2}","kind":"use","excerpt":"FloatBarrier Fractional modes and killed transfer. For the one-dimensional Dirichlet difference operator, let h=1/M with an integer M≥2, let 1≤ k≤ M-1, and set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":"For n interior points with spacing h=1/(n+1), define L_h=frac1h^2 2 · -1 · · -1 · 2 · ddots · · ddots · ddots · -1 · · -1 · 2 .","count":25},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"vpage33Gaussian packets have separate clocksORNSTEIN--UHLENBECK / HERMITE ENERGYTraditional Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure. In L^2(γ) for the standard Gaussian law on R, use the probabilists' Hermite polynomials H_1(x)=x, H_2(x)=x^2-1, H_4(x)=x^4-6x^2+3. The Ornstein--Uhlenbeck generator is Lf=","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"use","excerpt":"vpage33Gaussian packets have separate clocksORNSTEIN--UHLENBECK / HERMITE ENERGYTraditional Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure. In L^2(γ) for the standard Gaussian law on R, use the probabilists' Hermite polynomials H_1(x)=x, H_2(x)=x^2-1, H_4(x)=x^4-6x^2+3. The Ornstein--Uhlenbeck generator is Lf=","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-936bd94983483758","tex":"H_4","aliases":"H_4","display":"H_4","kind":"use","excerpt":"vpage33Gaussian packets have separate clocksORNSTEIN--UHLENBECK / HERMITE ENERGYTraditional Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure. In L^2(γ) for the standard Gaussian law on R, use the probabilists' Hermite polynomials H_1(x)=x, H_2(x)=x^2-1, H_4(x)=x^4-6x^2+3. The Ornstein--Uhlenbeck generator is Lf=","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-333bf23be8bc780a","tex":"h_i","aliases":"h_i","display":"h_i","kind":"use","excerpt":"Differentiate ∑_ih_ip_i(t) and insert the replicator equation.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cf5f97ebded34595","tex":"H_m","aliases":"H_m","display":"H_m","kind":"use","excerpt":"· = ∑_n=0^∞ e^-ntH_n(x)fracu^nn!. Comparison of coefficients proves the eigenvalue formula. The same generating function shows ∫ H_mH_n dγ=n!δ_mn. The Hermite polynomials span all polynomials, and polynomials are dense in L^2(γ). Hence the normalized family is an orthonormal basis. See [source reference] for the general Gaussian Hilbert space setti","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"use","excerpt":"[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-785ae493560363c1","tex":"h_n","aliases":"h_n","display":"h_n","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3e01b0cedad21f47","tex":"h_r","aliases":"h_r","display":"h_r","kind":"use","excerpt":"[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8da0f8726aa71190","tex":"i'","aliases":"i'","display":"i'","kind":"use","excerpt":"Conversely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the con","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0e79b7ad0299d0b0","tex":"I_0","aliases":"I_0","display":"I_0","kind":"use","excerpt":"ureR.arabicfigure renewcommandthetableRarabictable renewcommandtheHtableR.arabictable FloatBarrier Transfer paths and product-topology limits. Let T(t) be an arbitrary strongly continuous semigroup on a complex Banach space X, with generator A, and suppose normT(t)≤ M e^ω t for M≥1 and ω∈ R. Let (P_α)_α∈ I_0 be a finite family of bounded pairwise o","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f18b3c49fd8055ae","tex":"i_0","aliases":"i_0","display":"i_0","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-37475f8aeaec6cce","tex":"I_1","aliases":"I_1","display":"I_1","kind":"use","excerpt":"Let I=I_1sqcup⋯sqcup I_m be a partition. Define packet evidences Z_α = ∑_i∈ I_αλ_ip_i. For Z_α>0, define m_α=fracZ_α∑_β Z_β, q_α,i=fracλ_ip_iZ_α (i∈ I_α). Packets with zero evidence receive posterior mass zero.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-eff6aa27c552c5d2","tex":"I_a","aliases":"I_a","display":"I_a","kind":"use","excerpt":"Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero","count":11},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2344f57325d4c16a","tex":"I_b","aliases":"I_b","display":"I_b","kind":"use","excerpt":"Conversely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the con","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2e959b032fac4b5d","tex":"I_{d_j}","aliases":"I_d_j","display":"I_{d_j}","kind":"use","excerpt":"If B_j and C_j have an intermediate space of dimension d_j, use a block diagonal matrix D with blocks λ_j I_d_j, or the required Jordan blocks when poles repeat. Assemble the matrices B_j and C_j into the output and input maps.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ad472b41377c2078","tex":"I_m","aliases":"I_m","display":"I_m","kind":"use","excerpt":"Let I=I_1sqcup⋯sqcup I_m be a partition. Define packet evidences Z_α = ∑_i∈ I_αλ_ip_i. For Z_α>0, define m_α=fracZ_α∑_β Z_β, q_α,i=fracλ_ip_iZ_α (i∈ I_α). Packets with zero evidence receive posterior mass zero.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cf48159a73d856b5","tex":"i_n","aliases":"i_n","display":"i_n","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-afccf775c866af1f","tex":"i_r","aliases":"i_r","display":"i_r","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8f6ca4a3b6894ed8","tex":"i_{r+1}","aliases":"i_r+1","display":"i_{r+1}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"use","excerpt":"Let X be a complex Banach space. A finite packet system is a family of bounded idempotents P=(P_α)_α∈ I such that P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I_X. No orthogonality is needed in this section.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6e3ef04ac1a939e6","tex":"I_{X_m}","aliases":"I_X_m","display":"I_{X_m}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-826ca417f032adb2","tex":"I_{X_\\ell}","aliases":"I_X_ℓ ell","display":"I_{X_\\ell}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-88b326798e7011cf","tex":"\\ii","aliases":"ii i mathrm","display":"{\\mathrm i}","kind":"use","excerpt":"elextended:thm:holomorphic_defect Let Γ be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the clo","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"Let J=∫_0^∞ T_z(t) dt. Integrating both derivative identities T_z'=(B-zI)T_z=T_z(B-zI) and using T_z(0)=I gives (zI-B)J=J(zI-B)=I. Thus J=(zI-B)^-1. Continuous left and right multiplication carry the Bochner integral through the two rectangular transfer blocks, yielding the claim.","count":25},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"K(t) · The memory kernel after hidden-state elimination. · It is not the transition matrix K.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":62},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b414e0d8b85d3e26","tex":"K^{\\mathsf T}","aliases":"K^ T mathsf","display":"K^{\\mathsf T}","kind":"use","excerpt":"K^ Tf · Expected next observable given the current state. · The same dynamics, with the opposite action on the dual object.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":":04 For the one-dimensional Dirichlet difference operator, let h=1/M with an integer M≥2, let 1≤ k≤ M-1, and set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ad06b8163a249728","tex":"K^j","aliases":"K^j","display":"K^j","kind":"use","excerpt":"[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bd225a0f93f2f10a","tex":"K^{j+1}","aliases":"K^j+1","display":"K^{j+1}","kind":"use","excerpt":"The identity follows by telescoping: bar K^n-1-jCK^j+1-bar K^n-jCK^j =bar K^n-1-j(CK-bar KC)K^j. Summing from j=0 to n-1 leaves CK^n-bar K^nC. The norm bound follows termwise.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-647eccb9127e0259","tex":"K^n","aliases":"K^n","display":"K^n","kind":"use","excerpt":"[Exact lumpability] If Δ=0, then CK^n=bar K^nC for every n≥0. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c47e779cf5c8f6c8","tex":"K^{n+1}","aliases":"K^n+1","display":"K^{n+1}","kind":"use","excerpt":"The result is immediate for n=0. If CK^n=bar K^nC, then CK^n+1=bar K^nCK=bar K^n+1C.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-85c3fb99e8d4fae6","tex":"K^T","aliases":"K^T","display":"K^T","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-173833828c3529a3","tex":"K_2","aliases":"K_2","display":"K_2","kind":"use","excerpt":"bottomrule headOne convention test For K_2=3/4 · 1/4 1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the ","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fad803cc2ee90a75","tex":"K_2^{\\mathsf T}","aliases":"K_2^ T mathsf","display":"K_2^{\\mathsf T}","kind":"use","excerpt":"1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the object being transported. The transpose and the posit","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-08dc6e54edbcd212","tex":"K_c","aliases":"K_c","display":"K_c","kind":"use","excerpt":"Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-211be6ddf3860b2e","tex":"K_{i_{r+1},i_r}","aliases":"K_i_r+1,i_r","display":"K_{i_{r+1},i_r}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-69a26fc25f751d91","tex":"K_{ij}","aliases":"K_ij","display":"K_{ij}","kind":"use","excerpt":"K · One discrete stochastic step. · ones^ TK=ones^ T and K_ij≥0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1ea16453cbd62d91","tex":"K_{j i}","aliases":"K_j i","display":"K_{j i}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cf82e92850871481","tex":"K_{ji}","aliases":"K_ji","display":"K_{ji}","kind":"use","excerpt":"For one step, (M_Vf)_i = e^-V(i)∑_jK_jif_j. Iterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5c0f8cb6a6409762","tex":"K_{L^1(0,T)}","aliases":"K_L^1(0,T)","display":"K_{L^1(0,T)}","kind":"use","excerpt":"· + ∫_0^tS(t-s) ∫_0^sK(s-r)x(r) d r d s. Assume sup_0≤ t≤ Tnormwidetilde x(t)≤ B. Set L=norm K_L^1(0,T), ε=normK-widetilde K_L^1(0,T). Then sup_0≤ t≤ T normx(t)-widetilde x(t) ≤ M_TBTε exp(M_TLT).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b671d5da77c233cc","tex":"K_{ri}","aliases":"K_ri","display":"K_{ri}","kind":"use","excerpt":"[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ce6b6fd6eca36fc9","tex":"K_{rj}","aliases":"K_rj","display":"K_{rj}","kind":"use","excerpt":"[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7df93e11dabca8d6","tex":"K_{\\rm mem}","aliases":"K_rm mem rm","display":"K_{\\rm mem}","kind":"use","excerpt":"0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponential is not e^-2tI: e^tJ=e^-2t(I+tN), K_rm mem(t)=Ue^tJV=t e^-2t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that r","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f178970e61601761","tex":"K_\\alpha","aliases":"K_α alpha","display":"K_\\alpha","kind":"use","excerpt":"[Fractional intertwining theorem] For bounded nonnegative self-adjoint A and B, D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In par","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6c71da9fa04844e9","tex":"K_\\varepsilon","aliases":"K_ε varepsilon","display":"K_\\varepsilon","kind":"use","excerpt":"vpage6A broken lump and its error budgetDEFECTS / A GUARANTEE WITHOUT A FITTraditional Part I: Theorems 3.1, 3.3, and 5.2. Continuous and discrete defect identities. Modify only column zero of K: move 1/16 of its mass from target state zero to target state two. The perturbed matrix K_ε remains stochastic, but its aggregate column now differs from c","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-217a35c5c19e1d0d","tex":"K_\\varepsilon^j","aliases":"K_ε^j varepsilon","display":"K_\\varepsilon^j","kind":"use","excerpt":"1/16 · 0 · 0 · 0, Δ _1=1/8. vfigdefect_budget platecaptionThe solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp. headTelescoping tells us where every error originates For n≥0, CK_ε^n-bar K^nC =∑_j=0^n-1bar K^n-1-jΔ K_ε^j. Each summand propagates ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a375b31a4960827d","tex":"K_\\varepsilon^n","aliases":"K_ε^n varepsilon","display":"K_\\varepsilon^n","kind":"use","excerpt":"1/16 · 0 · 0 · 0, Δ _1=1/8. vfigdefect_budget platecaptionThe solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp. headTelescoping tells us where every error originates For n≥0, CK_ε^n-bar K^nC =∑_j=0^n-1bar K^n-1-jΔ K_ε^j. Each summand propagates ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9feea0dd1279e0f4","tex":"\\Kcal","aliases":"Kcal K mathcal","display":"{\\mathcal K}","kind":"use","excerpt":"For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0) · + ∫_0^t K(t-s)x_P(s) d s, where K(t) = A_PQ e^tA_QQA_QP.","count":22},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1062f1ecdf031386","tex":"\\Kcal_J","aliases":"Kcal_J K_J Kcal mathcal","display":"{\\mathcal K}_J","kind":"use","excerpt":"he least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete primary blocks indexed by J, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J ∑_k=0^ν_j-1","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usu","count":132},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every bl","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"The accompanying Python package implements the formulas and finite models used in the paper. projector_transfer_semigroups The computational organization mirrors the paper: c|l packets.py · projectors, blocks, reconstruction, block composition semigroup.py · matrix semigroups, defects, Feshbach operators, memory markov.py · random walks, exact lump","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a44e55170cadffdb","tex":"L^*","aliases":"L^*","display":"L^*","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9d0fbf856fe55b03","tex":"L^1","aliases":"L^1","display":"L^1","kind":"use","excerpt":"Add and subtract μodot p/Z_λ and use the triangle inequality: normfracλodot pZ_λ-fracμodot pZ_μ_1 ≤ fracnorm(λ-μ)odot p_1Z_λ + normμodot p_1|frac1Z_λ-frac1Z_μ|. The second term equals |Z_λ-Z_μ|/Z_λ and is bounded by the same weighted L^1 difference divided by Z_λ. Interchanging λ and μ gives the minimum-evidence form.","count":16},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5c6f72cfe6318222","tex":"L^{1/2}","aliases":"L^1/2","display":"L^{1/2}","kind":"use","excerpt":"vpage15The modes stay; their clocks changeFRACTIONAL DIFFUSION / NEW COUPLINGSTraditional Part I: Theorem 10.1. A positive self-adjoint finite model and its spectral powers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its squa","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"use","excerpt":"[General conditional expectations] On a probability space, conditional expectation onto a sub sigma algebra is the orthogonal projector in L^2. Nested sigma algebras satisfy the same tower law. Under the usual hypotheses, martingale convergence gives a limit in L^2 and almost surely. Standard treatments include [source reference].","count":13},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0de1bee007df208a","tex":"L^p","aliases":"L^p","display":"L^p","kind":"use","excerpt":"[Ornstein Uhlenbeck semigroup] The operators ( O_t)_t≥0 form a positive Markov contraction semigroup on L^p(γ) for every 1≤ p≤∞. On L^2(γ), the semigroup is self adjoint and O_t h_n= e^-nth_n. The family (h_n)_n≥0 is an orthonormal basis of L^2(γ).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4f30a25ed8c4669b","tex":"L^\\alpha","aliases":"L^α alpha","display":"L^\\alpha","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":15},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5aa47e9873d12ad2","tex":"L^{\\alpha+\\beta}","aliases":"L^α+β alpha beta","display":"L^{\\alpha+\\beta}","kind":"use","excerpt":"[Fractional packet calculus] For α,β>0, L^α L^β=L^α+β, and for s,t≥0, T_α(s)T_α(t)=T_α(s+t). Every spectral packet P_j reduces L^α and T_α(t).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9d2eb5d87d66df4f","tex":"L^\\beta","aliases":"L^β beta","display":"L^\\beta","kind":"use","excerpt":"[Fractional packet calculus] For α,β>0, L^α L^β=L^α+β, and for s,t≥0, T_α(s)T_α(t)=T_α(s+t). Every spectral packet P_j reduces L^α and T_α(t).","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d92ecf36f8db9126","tex":"L_{03}","aliases":"L_03","display":"L_{03}","kind":"use","excerpt":"by e^-2jt; square-root diffusion multiplies it by e^-sqrt2j t. The constant mode is unchanged in both. vfigfractional platecaptionThe highest mode decays at rates 6 and sqrt6. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit. headFractional does not mea","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2f8dd52a01de0ecd","tex":"L_c","aliases":"L_c","display":"L_c","kind":"use","excerpt":"For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c8c2eb0cf9cbeb36","tex":"L_c^\\alpha","aliases":"L_c^α alpha","display":"L_c^\\alpha","kind":"use","excerpt":"For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ec6b4ea000366681","tex":"L_f","aliases":"L_f","display":"L_f","kind":"use","excerpt":"For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d8e73625c7457f64","tex":"L_f^\\alpha","aliases":"L_f^α alpha","display":"L_f^\\alpha","kind":"use","excerpt":"For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-743ff0ea070acbd8","tex":"L_h","aliases":"L_h","display":"L_h","kind":"use","excerpt":"For n interior points with spacing h=1/(n+1), define L_h=frac1h^2 2 · -1 · · -1 · 2 · ddots · · ddots · ddots · -1 · · -1 · 2 .","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-15870635efb91b25","tex":"L_h^{-1}","aliases":"L_h^-1","display":"L_h^{-1}","kind":"use","excerpt":"[Exact discrete Green reconstruction at grid nodes] Let x_i=ih. Then frac1h(L_h^-1)_ij =G(x_i,x_j). Thus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.","count":11},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a716b0894950a10a","tex":"L_i","aliases":"L_i","display":"L_i","kind":"use","excerpt":"tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared f","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2dcbb7ca5126e8ca","tex":"L_y","aliases":"L_y","display":"L_y","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-229e5fbb02149c66","tex":"L_z","aliases":"L_z","display":"L_z","kind":"use","excerpt":"Set L_z= I · -A_PQ(zI-A_QQ)^-1 0 · I and S_z= F_P(z) · 0","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3755b02b61222ecc","tex":"\\Lcal","aliases":"Lcal L mathcal","display":"{\\mathcal L}","kind":"use","excerpt":"[Generator and Dirichlet form] On the polynomial subspace, the infinitesimal generator of ( O_t) is the differential operator L f=f''-xf'. It satisfies L h_n=-nh_n. For every polynomial f, ip- L ff_L^2(γ) = ∫_ Rabsf'(x)^2γ( d x).","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-993066ece33b1204","tex":"\\operatorname{length}","aliases":"length operatorname","display":"\\operatorname{length}","kind":"use","excerpt":"a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure of the union, then Cg(A_f)-g(A_c)C = frac12π i ∫_Γ g(z)R_c(z)Δ R_f(z) d z. In particul","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"[Semigroup survival identities] For t≥0, the finite-time exit density and, wherever S(t)>0, the hazard are f(t)=a^Te^tAp_0=-S'(t), h(t)=fracf(t)S(t). Consequently, S is nonincreasing and S(t)=exp(-∫_0^th(r) dr) whenever S(t)>0. The density may be defective: its total mass is 1-lim_t→∞S(t). It is a probability density when absorption is eventual, so","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-eb514d3fcefe767d","tex":"\\lim_{n\\to\\infty}","aliases":"lim_n→∞ lim to infty","display":"\\lim_{n\\to\\infty}","kind":"use","excerpt":"[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fb093952304f9b7e","tex":"\\lim_{t\\downarrow0}","aliases":"lim_tdownarrow0 lim downarrow","display":"\\lim_{t\\downarrow0}","kind":"use","excerpt":"ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1416c21d48d701c1","tex":"\\lim_{t\\to0}","aliases":"lim_t→0 lim to","display":"\\lim_{t\\to0}","kind":"use","excerpt":"[Infinitesimal recovery of the defect] The generator defect is the first order semigroup defect: lim_t→0 frac D(t)t = Δ in operator norm. Consequently, C e^tA_f= e^tA_cC hboxfor every t if and only if CA_f=A_cC.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-24fd66249142db1e","tex":"\\lim_{t\\to\\infty}","aliases":"lim_t→∞ lim to infty","display":"\\lim_{t\\to\\infty}","kind":"use","excerpt":"[Semigroup survival identities] For t≥0, the finite-time exit density and, wherever S(t)>0, the hazard are f(t)=a^Te^tAp_0=-S'(t), h(t)=fracf(t)S(t). Consequently, S is nonincreasing and S(t)=exp(-∫_0^th(r) dr) whenever S(t)>0. The density may be defective: its total mass is 1-lim_t→∞S(t). It is a probability density when absorption is eventual, so","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7615cae6d0b78c4b","tex":"m^\\alpha","aliases":"m^α alpha","display":"m^\\alpha","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4d074e542a4cc31b","tex":"m^{\\alpha-1}","aliases":"m^α-1 alpha","display":"m^{\\alpha-1}","kind":"use","excerpt":"D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In particular, exact generator intertwining CA=BC implies exact fractional intertwinin","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ece6f9e55a99dd8c","tex":"m_{\\mathrm A}","aliases":"m_ A mathrm","display":"m_{\\mathrm A}","kind":"use","excerpt":"vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditi","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-61577aeca4848aaa","tex":"m_{\\mathrm B}","aliases":"m_ B mathrm","display":"m_{\\mathrm B}","kind":"use","excerpt":"vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditi","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":"atement, now assume A is bounded as in Part II. A directed edge from α to β means that P_β A P_α is nonzero, with α≠β; a packet always has the length-zero path to itself. [Minimum transfer order] Suppose every directed packet path from α to β uses at least m_0 off diagonal transfers. Then P_β e^tAP_α =O(t^m_0) as tdownarrow0. If there is no directe","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-63019db73ab92a7c","tex":"M_A","aliases":"M_A","display":"M_A","kind":"use","excerpt":"For bounded generators with common bound M_A, sup_ℓ norm( e^tA_ℓ-I)x_ℓ ≤ ( e^tM_A-1)sup_ℓnormx_ℓ. This tends to zero with t. Exact generator intertwining makes (A_ℓ x_ℓ)_ℓ compatible, and coordinatewise differentiation identifies the generator.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0f5cef98cfcf15fe","tex":"M_c","aliases":"M_c","display":"M_c","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e4524d978e547026","tex":"M_f","aliases":"M_f","display":"M_f","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5d80ae31b0ce157a","tex":"M_Q","aliases":"M_Q","display":"M_Q","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f241e1659d989d87","tex":"M_T","aliases":"M_T","display":"M_T","kind":"use","excerpt":"[Volterra stability under kernel compression] Let S(t) be a strongly continuous semigroup on a Banach space X, fix T>0, and assume normS(t)≤ M_T (0≤ t≤ T). Let K,widetilde K∈ L^1((0,T), B(X)), let g∈ L^1((0,T),X), and let x,widetilde x∈ C([0,T],X). Suppose these are mild solutions with the same initial state and forcing, except that K is replaced b","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9449462255135514","tex":"M_V","aliases":"M_V","display":"M_V","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d9af7b18b4478efd","tex":"M_V^n","aliases":"M_V^n","display":"M_V^n","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-401d9b49eeb97619","tex":"m_\\alpha","aliases":"m_α alpha","display":"m_\\alpha","kind":"use","excerpt":"Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-81f120bf99a467b2","tex":"\\max_{1\\leq i\\leq n}","aliases":"max_1≤ i≤ n max leq leq","display":"\\max_{1\\leq i\\leq n}","kind":"use","excerpt":"[Discrete maximum bound] The inverse L_h^-1 has nonnegative entries and normL_h^-1_∞→∞ = max_1≤ i≤ n fracx_i(1-x_i)2 ≤ frac18.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7bd86680754bc6e2","tex":"\\max_{i\\in B_a}","aliases":"max_i∈ B_a max in","display":"\\max_{i\\in B_a}","kind":"use","excerpt":"[Oscillation bound] Suppose overline L_a is any representative likelihood for block B_a, and set ω_a = max_i∈ B_aabsL_i-overline L_a. Then d(pmid L,overline L) ≤ ∑_aq_aω_a. Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6913c3ef5d6dc1fd","tex":"\\max_{i\\in I_\\alpha}","aliases":"max_i∈ I_α max in alpha","display":"\\max_{i\\in I_\\alpha}","kind":"use","excerpt":"[Information created by a Bayesian update] Let q= B_λ(p) with positive likelihood and positive prior. Then D(qVert p) = ∑_iq_ilogλ_i - log Z_λ(p). Inside a packet, D(q_αVert r_α) = ∑_i∈ I_α q_α,i logfracλ_ioverlineλ_α(p). Hence D(q_αVert r_α) ≤ log fracmax_i∈ I_αλ_i min_i∈ I_αλ_i.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0; choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overl","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from P_iP_j=P_min(i,j).","count":16},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3b06e69b277c3fc2","tex":"\\min_{I_\\alpha}","aliases":"min_I_α min alpha","display":"\\min_{I_\\alpha}","kind":"use","excerpt":"Subtract r_α,i from the local posterior formula in [source reference]. The norm identity follows because the factor multiplying η_i is nonnegative. The last estimate uses ∑_i r_α,iabsη_i ≤ osc_I_α(λ) and overlineλ_α(p)≥min_I_αλ.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bbe7648c525d8858","tex":"\\min_{i\\in I_\\alpha}","aliases":"min_i∈ I_α min in alpha","display":"\\min_{i\\in I_\\alpha}","kind":"use","excerpt":"] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":43},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b75f3a902c5f7e60","tex":"n^{-1}","aliases":"n^-1","display":"n^{-1}","kind":"use","excerpt":"dotted,line width=.9pt] plot coordinates (1.33125,1.107143) (1.8625,1.221429) (2.39375,1.272222) (2.925,1.300794) (3.45625,1.319079) (3.9875,1.331778) (4.51875,1.341107) (5.05,1.34825) (5.58125,1.353894) (6.1125,1.358465) (6.64375,1.362243) (7.175,1.365418) (7.70625,1.368123) (8.2375,1.370455) (8.76875,1.372487) (9.3,1.374273); node[font=small,alig","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c0769f9271add232","tex":"n^{-2}","aliases":"n^-2","display":"n^{-2}","kind":"use","excerpt":"Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"vpage23A repeated pole remembers a Jordan chainNONDIAGONAL HIDDEN DYNAMICS / POLYNOMIAL EXPONENTIALSTraditional Part I: Theorem 13.2 and Section 15. Functional calculus must retain nilpotent data. Now use a two-dimensional hidden block J=-2I+N, N=mat0 · 1 0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponenti","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"On pagerefV27, the sequences 1/n^2 and 1/n turn this change in error scale into a concrete distinction between summable and nonsummable defects. takeawayA fixed gap is quantitative information. It prevents the derivative of the fractional function from becoming singular along the comparison.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-526a976a5ea4910d","tex":"n^\\alpha","aliases":"n^α alpha","display":"n^\\alpha","kind":"use","excerpt":"[Fractional Hermite transfer] Let - L be the nonnegative Ornstein Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e1613ab8834072c1","tex":"N_j","aliases":"N_j","display":"N_j","kind":"use","excerpt":"[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). ","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-921edcc519f97aae","tex":"N_j^k","aliases":"N_j^k","display":"N_j^k","kind":"use","excerpt":"mory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9d3b88e3c1c7cc4c","tex":"N_j^{\\nu_j}","aliases":"N_j^ν_j nu","display":"N_j^{\\nu_j}","kind":"use","excerpt":"rem[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9727f5125c78ee28","tex":"n_\\alpha","aliases":"n_α alpha","display":"n_\\alpha","kind":"use","excerpt":"on decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the information retain","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[Second-order low-mode convergence] For fixed k, λ_k,h=(kπ)^2+O(h^2). More precisely, for sufficiently small h, 0≤ (kπ)^2-λ_k,h ≤ frac(kπ)^412h^2.","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"[Infinitesimal recovery of the defect] The generator defect is the first order semigroup defect: lim_t→0 frac D(t)t = Δ in operator norm. Consequently, C e^tA_f= e^tA_cC hboxfor every t if and only if CA_f=A_cC.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-295577f481743cf7","tex":"\\Ocal","aliases":"Ocal O mathcal","display":"{\\mathcal O}","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a8732d1ab457c098","tex":"\\Ocal_t","aliases":"Ocal_t O_t Ocal mathcal","display":"{\\mathcal O}_t","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a34910944de266ad","tex":"\\odot","aliases":"odot","display":null,"kind":"use","excerpt":"Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b77c3e9c22499671","tex":"\\oint","aliases":"oint","display":null,"kind":"use","excerpt":"(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula give","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0c668e45ed5a9cdb","tex":"\\oint_{|z-\\mu|=r_\\mu}","aliases":"oint_|z-μ|=r_μ oint mu mu","display":null,"kind":"use","excerpt":"(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula give","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7d905bc6806c132f","tex":"\\one","aliases":"one 1 bm","display":"{\\bm 1}","kind":"use","excerpt":"Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.","count":20},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b9f1bf76a072652d","tex":"\\one^T","aliases":"one^T 1^T one bm","display":"{\\bm 1}^T","kind":"use","excerpt":"Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.","count":15},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dbd0bb0518600c58","tex":"\\one_{i\\in B_a}","aliases":"one_i∈ B_a 1_i∈ B_a one in bm in","display":"{\\bm 1}_{i\\in B_a}","kind":"use","excerpt":"The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2500223217e36ec5","tex":"\\one_{\\{i\\in B_\\alpha\\}}","aliases":"one_{i∈ B_α} 1_{i∈ B_α} one in alpha bm in alpha","display":"{\\bm 1}_{\\{i\\in B_\\alpha\\}}","kind":"use","excerpt":"Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c1a14ba7424a4a8c","tex":"\\ones","aliases":"ones","display":null,"kind":"use","excerpt":"midrule p · A nonnegative column with ones^ Tp=1. · It is not a vector of function values.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-33cfa968292abd16","tex":"\\ones^{\\mathsf T}","aliases":"ones^ T ones mathsf","display":null,"kind":"use","excerpt":"midrule p · A nonnegative column with ones^ Tp=1. · It is not a vector of function values.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a625c27cba36b747","tex":"\\osc","aliases":"osc operatorname","display":"{\\operatorname{osc}}","kind":"use","excerpt":"] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c249b84d38edd209","tex":"\\osc_{I_\\alpha}","aliases":"osc_I_α osc alpha operatorname alpha","display":"{\\operatorname{osc}}_{I_\\alpha}","kind":"use","excerpt":"] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7079e3832479ea24","tex":"\\overline D","aliases":"overline D overline","display":"\\overline D","kind":"use","excerpt":"Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f46e22d0b8fbcf67","tex":"\\overline D_r","aliases":"overline D_r overline","display":"\\overline D_r","kind":"use","excerpt":"Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b7a5b96653c1dfce","tex":"\\overline K","aliases":"overline K overline","display":"\\overline K","kind":"use","excerpt":"[Descent through an aggregation map] Let C:X_f→ X_c be a surjective linear map between finite dimensional spaces. A unique operator overline K:X_c→ X_c satisfying CK=overline KC exists if and only if Ob_C(K)=0. For any right inverse R of C, the only possible coarse operator is overline K=CKR, and CK-overline KC = CK(I-RC).","count":27},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c616ecd4a706af27","tex":"\\overline K^n","aliases":"overline K^n overline","display":"\\overline K^n","kind":"use","excerpt":"[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4c03d7a3e867e77f","tex":"\\overline K^{n-1-j}","aliases":"overline K^n-1-j overline","display":"\\overline K^{n-1-j}","kind":"use","excerpt":"[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c22893176e13cead","tex":"\\overline K^{n-j}","aliases":"overline K^n-j overline","display":"\\overline K^{n-j}","kind":"use","excerpt":"Sum the telescoping identities overline K^n-1-jCK^j+1 - overline K^n-jCK^j = overline K^n-1-jΔ_KK^j. Stochastic matrices are contractions in the ℓ^1 norm. The last statement uses total variation equal to one half of the ℓ^1 distance.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d0c30b2e35f72f77","tex":"\\overline K_{\\alpha\\beta}","aliases":"overline K_αβ overline alpha beta","display":"\\overline K_{\\alpha\\beta}","kind":"use","excerpt":"Let K be column stochastic. A coarse column stochastic matrix overline K satisfying CK=overline KC exists if and only if, for every source block B_β, every pair i,j∈ B_β, and every target block B_α, ∑_r∈ B_αK_ri = ∑_r∈ B_αK_rj. When it exists, overline K is unique and overline K_αβ = ∑_r∈ B_αK_ri for any i∈ B_β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e4b45cd66b9c8f4e","tex":"\\overline L","aliases":"overline L overline","display":"\\overline L","kind":"use","excerpt":"tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared f","count":18},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2d58c598ab4835df","tex":"\\overline L_a","aliases":"overline L_a overline","display":"\\overline L_a","kind":"use","excerpt":"tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared f","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a78d412941cc60b6","tex":"\\overline L_y","aliases":"overline L_y overline","display":"\\overline L_y","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9d8a1921f2a37e0a","tex":"\\overline M","aliases":"overline M overline","display":"\\overline M","kind":"use","excerpt":"[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-53747cc445df0a6e","tex":"\\overline m","aliases":"overline m overline","display":"\\overline m","kind":"use","excerpt":"[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-42c6158ae78541bc","tex":"\\overline Q","aliases":"overline Q overline","display":"\\overline Q","kind":"use","excerpt":"[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.","count":17},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4a3211811166a67a","tex":"\\overline Q^T","aliases":"overline Q^T overline","display":"\\overline Q^T","kind":"use","excerpt":"[Coarse Dirichlet form] Under the hypotheses of [source reference], let Jg be the fine function that is constant with value g_a on B_a. Then -ipJgQ^TJg_L^2(π) = -ipgoverline Q^Tg_L^2(overlineπ).","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1a587a61dc8ffd20","tex":"\\overline Q_{ab}","aliases":"overline Q_ab overline","display":"\\overline Q_{ab}","kind":"use","excerpt":"the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fe8d3f49543714c4","tex":"\\overline Q_{ba}","aliases":"overline Q_ba overline","display":"\\overline Q_{ba}","kind":"use","excerpt":"the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ad21f4f96eb368f9","tex":"\\overline V","aliases":"overline V overline","display":"\\overline V","kind":"use","excerpt":"[Coarse Feynman Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6701fd0fc5e1ceb8","tex":"\\overline{\\left(\\sum_{j\\in B}\\pi_j g_j\\right)}","aliases":"overline(∑_j∈ Bπ_j g_j) overline left sum in pi right","display":"\\overline{\\left(\\sum_{j\\in B}\\pi_j g_j\\right)}","kind":"use","excerpt":"After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block; both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2663db5a20889e2c","tex":"\\overline{\\left(\\sum_{j\\in B}\\pi_jg_j\\right)}","aliases":"overline(∑_j∈ Bπ_jg_j) overline left sum in pi right","display":"\\overline{\\left(\\sum_{j\\in B}\\pi_jg_j\\right)}","kind":"use","excerpt":"One application produces a block constant function, so a second application changes nothing. Expanding both weighted inner products block by block gives ipP_ Qfg_π = ∑_B∈ Q frac (∑_i∈ Bπ_if_i) overline(∑_j∈ Bπ_jg_j) π(B) = ipfP_ Qg_π. The range is exactly the block constant subspace.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3e0650b79ef975f9","tex":"\\overline{D(\\mu,r_\\mu)}","aliases":"overlineD(μ,r_μ) overline mu mu","display":"\\overline{D(\\mu,r_\\mu)}","kind":"use","excerpt":".F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5b677f57c2885a9a","tex":"\\overline{g(x)}","aliases":"overlineg(x) overline","display":"\\overline{g(x)}","kind":"use","excerpt":"-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a1002d19bc28fd56","tex":"\\overline{xg(x)-g'(x)}","aliases":"overlinexg(x)-g'(x) overline","display":"\\overline{xg(x)-g'(x)}","kind":"use","excerpt":"-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-292fef11aee33c25","tex":"\\overline\\lambda_\\beta","aliases":"overlineλ_β overline lambda beta","display":"\\overline\\lambda_\\beta","kind":"use","excerpt":"The posterior mass of packet I_α is frac∑_i∈ I_αλ_ip_i ∑_jλ_jp_j = frac(Cp)_αoverlineλ_α(p) ∑_β(Cp)_βoverlineλ_β(p). This is the coarse Bayesian formula. Dividing the fine posterior coordinates by the packet mass gives the local formula.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-39650f15ca748586","tex":"\\overline\\pi","aliases":"overlineπ overline pi","display":"\\overline\\pi","kind":"use","excerpt":"se Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overl","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b8a56fca100492b3","tex":"\\overline\\pi_a","aliases":"overlineπ_a overline pi","display":"\\overline\\pi_a","kind":"use","excerpt":"se Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overl","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3416c17345e3e658","tex":"\\overline\\pi_b","aliases":"overlineπ_b overline pi","display":"\\overline\\pi_b","kind":"use","excerpt":"the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-782174188fd36427","tex":"\\overline\\Phi","aliases":"overlineΦ overline Phi","display":"\\overline\\Phi","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5e1bcaea0a451444","tex":"\\overline\\Phi_y","aliases":"overlineΦ_y overline Phi","display":"\\overline\\Phi_y","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-78355e7ad6e40f33","tex":"\\overline\\Phi_{y_1}","aliases":"overlineΦ_y_1 overline Phi","display":"\\overline\\Phi_{y_1}","kind":"use","excerpt":"[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9558f1e55cd5cd94","tex":"\\overline\\Phi_{y_r}","aliases":"overlineΦ_y_r overline Phi","display":"\\overline\\Phi_{y_r}","kind":"use","excerpt":"[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b95fc3463e504f17","tex":"P^{\\mathsf T}","aliases":"P^ T mathsf","display":"P^{\\mathsf T}","kind":"use","excerpt":"[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remai","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c4c15eaf6719fd34","tex":"p^{(a)}","aliases":"p^(a)","display":"p^{(a)}","kind":"use","excerpt":"vpage10Two priors that fool the coarse modelSUFFICIENCY / A COMPUTABLE FAILURE WITNESSTraditional Part I: Theorem 8.2 and the one-packet exception. Keep the same likelihood λ=(4,1,1,1)^ T and partition A={0,1}, B={2,3}. Compare p^(a)=(1/2,0,1/2,0)^ T, p^(b)=(0,1/2,1/2,0)^ T. Both have coarse prior (1/2,1/2)^ T. A procedure given only that two-entry","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-78d0cffbd1e1aa1e","tex":"p^{(b)}","aliases":"p^(b)","display":"p^{(b)}","kind":"use","excerpt":"vpage10Two priors that fool the coarse modelSUFFICIENCY / A COMPUTABLE FAILURE WITNESSTraditional Part I: Theorem 8.2 and the one-packet exception. Keep the same likelihood λ=(4,1,1,1)^ T and partition A={0,1}, B={2,3}. Compare p^(a)=(1/2,0,1/2,0)^ T, p^(b)=(0,1/2,1/2,0)^ T. Both have coarse prior (1/2,1/2)^ T. A procedure given only that two-entry","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5f536203d4a8b83a","tex":"P_{\\mathcal Q}","aliases":"P_ Q mathcal","display":"P_{\\mathcal Q}","kind":"use","excerpt":"Let Ω={1,…,n} carry strictly positive probabilities π_i, and let Q={B_1,…,B_m} be a partition. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j∑_j∈ B(i)π_j, where B(i) is the block containing i.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-04e3dc4cda092659","tex":"P_{\\mathcal Q}^2","aliases":"P_ Q^2 mathcal","display":"P_{\\mathcal Q}^2","kind":"use","excerpt":"[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"Let A be a finite subgenerator: its off-diagonal entries are nonnegative and its column sums are nonpositive. These sign conditions alone allow nonabsorbing states; eventual absorption is a separate hypothesis. Let p_0 be an initial transient-state distribution. Define p(t)=e^tAp_0, S(t)=1^Tp(t). The exit-rate row vector is a^T=-1^TA≥0.","count":36},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c641cd3ff43bd1cb","tex":"p_0^T","aliases":"p_0^T","display":"p_0^T","kind":"use","excerpt":"Average [source reference] with terminal function f= 1 against the initial distribution. Transposition gives p_0^T e^t(Q^T-D_V) 1 = 1^T e^t(Q-D_V)p_0. Since 1^TQ=0, differentiation gives S_V'(t) = - 1^TD_V e^tAp_0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"one probability vector and its two block masses. The right panel is a separate retained/hidden model, developed on pagerefV22. Every curve is drawn from an explicit formula. headCoordinates retain; aggregation forgets; elimination remembers Keeping every component P_a x is an exact change of representation when ∑_aP_a=I. Reconstruct by summing the ","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writin","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"one probability vector and its two block masses. The right panel is a separate retained/hidden model, developed on pagerefV22. Every curve is drawn from an explicit formula. headCoordinates retain; aggregation forgets; elimination remembers Keeping every component P_a x is an exact change of representation when ∑_aP_a=I. Reconstruct by summing the ","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f0242a419961441b","tex":"P_3","aliases":"P_3","display":"P_3","kind":"use","excerpt":"basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writin","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"use","excerpt":"one probability vector and its two block masses. The right panel is a separate retained/hidden model, developed on pagerefV22. Every curve is drawn from an explicit formula. headCoordinates retain; aggregation forgets; elimination remembers Keeping every component P_a x is an exact change of representation when ∑_aP_a=I. Reconstruct by summing the ","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero","count":15},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","excerpt":"For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3270e71bf1ae54c5","tex":"P_c","aliases":"P_c","display":"P_c","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d02aaee4231d3cd1","tex":"P_f","aliases":"P_f","display":"P_f","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from P_iP_j=P_min(i,j).","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-36ec2f492b9a5c85","tex":"p_i'","aliases":"p_i'","display":"p_i'","kind":"use","excerpt":"tatistical meanings The first derivative of log evidence is the current mean score, and the second is its variance: fracddtlog Z(t)= E_p(t)u, fracd^2dt^2log Z(t)=Var_p(t)(u)≥0. At q=2 the exact values are E u=frac107, E u^2=frac187, Var(u)=frac2649. Thus the derivative of the mean is 26/49 at this time. The flow equation p_i'=(u_i- E u)p_i gives de","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":19},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"use","excerpt":"The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2646be694d017280","tex":"P_{j-1}","aliases":"P_j-1","display":"P_{j-1}","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e32f108b3db0ce30","tex":"p_M","aliases":"p_M","display":"p_M","kind":"use","excerpt":"[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-68ad068bb42ded1a","tex":"P_{m,n}","aliases":"P_m,n","display":"P_{m,n}","kind":"use","excerpt":"[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally un","count":11},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-22271657f4ac2661","tex":"P_{\\min(i,j)}","aliases":"P_min(i,j) min","display":"P_{\\min(i,j)}","kind":"use","excerpt":"A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from P_iP_j=P_min(i,j).","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"use","excerpt":"[Hermite packet reconstruction] Let P_n be the orthogonal projection onto C h_n. Then I=∑_n=0^∞ P_n in the strong operator topology and O_t = ∑_n=0^∞ e^-ntP_n. For f∈ L^2(γ), norm O_t f_2^2 = ∑_n=0^∞ e^-2ntnormP_nf_2^2.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4d3243980b9ae21f","tex":"P_r","aliases":"P_r","display":"P_r","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-528f89f54cfb3d02","tex":"p_r","aliases":"p_r","display":"p_r","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-152431294b318fe1","tex":"p_{r-1}","aliases":"p_r-1","display":"p_{r-1}","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d58992a0c0c08b27","tex":"p_t","aliases":"p_t","display":"p_t","kind":"use","excerpt":"[Residual life as a restarted semigroup] For S(t)>0, define the conditional transient distribution p_t = frac e^tAp_0S(t). Then fracS(t+s)S(t) = 1^T e^sAp_t.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f29b3a10d396763d","tex":"p_\\alpha","aliases":"p_α alpha","display":"p_\\alpha","kind":"use","excerpt":"[Exact information decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the in","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-eee86870442a9515","tex":"p_{\\alpha,i}","aliases":"p_α,i alpha","display":"p_{\\alpha,i}","kind":"use","excerpt":"For i∈ I_α, write p_i=m_α p_α,i, q_i=n_α q_α,i. Then logfracq_ip_i = logfracn_αm_α + logfracq_α,ip_α,i. Multiply by q_i, sum first inside each packet, and then sum over the packets. This is the finite relative entropy chain rule. Standard information theoretic treatments include [source reference].","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bb3fe555e4ac9447","tex":"P_{\\alpha_0}","aliases":"P_α_0 alpha","display":"P_{\\alpha_0}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8fae8ad97b2a4d74","tex":"P_{\\alpha_1}","aliases":"P_α_1 alpha","display":"P_{\\alpha_1}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-302f53d86a380a68","tex":"P_{\\alpha_m}","aliases":"P_α_m alpha","display":"P_{\\alpha_m}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d81550d96775f196","tex":"P_{\\alpha_{m-1}}","aliases":"P_α_m-1 alpha","display":"P_{\\alpha_{m-1}}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9813ba63ac7cffd1","tex":"P_\\gamma","aliases":"P_γ gamma","display":"P_\\gamma","kind":"use","excerpt":"Insert the resolution of the identity between the two semigroup factors: P_β e^(s+t)AP_α · =P_β e^sA(∑_γ P_γ)e^tAP_α · =∑_γ (P_β e^sAP_γ)(P_γ e^tAP_α).","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"For x∈ PX, write p(t)=PT(t)x, q(t)=QT(t)x. The hidden component satisfies q'(t)=QAQq(t)+QAPp(t), q(0)=0. Variation of constants gives q(t)= ∫_0^t e^(t-s)(QAQ)|_QX QAPp(s) d s. Since normp(s)≤norm PnormT(s)norm x and x=Px, the first estimate follows. The contraction case is immediate.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d76d885e9e3c5ba5","tex":"Q^*","aliases":"Q^*","display":"Q^*","kind":"use","excerpt":"[Finite conditional expectation] The operator P_ Q is the orthogonal projector onto the functions that are constant on every block of Q. In particular, P_ Q^2=P_ Q, P_ Q^*=P_ Q.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"use","excerpt":"[Finite conditional expectation] The operator P_ Q is the orthogonal projector onto the functions that are constant on every block of Q. In particular, P_ Q^2=P_ Q, P_ Q^*=P_ Q.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writin","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d2d3c9a0c830bb0d","tex":"q^3","aliases":"q^3","display":"q^3","kind":"use","excerpt":"basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writin","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-050d088f2ad53d96","tex":"Q^T","aliases":"Q^T","display":"Q^T","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":13},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a634d867493413c1","tex":"q_{\\mathrm A}","aliases":"q_ A mathrm","display":"q_{\\mathrm A}","kind":"use","excerpt":"vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditi","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8497f4e15ecc3f26","tex":"q_{\\mathrm B}","aliases":"q_ B mathrm","display":"q_{\\mathrm B}","kind":"use","excerpt":"vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditi","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ccc70c6e400edfd7","tex":"\\mathcal Q_{\\mathcal L}","aliases":"Q_ L mathcal mathcal","display":"\\mathcal Q_{\\mathcal L}","kind":"use","excerpt":"[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every bl","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5ca75c044055616d","tex":"Q_a","aliases":"Q_a","display":"Q_a","kind":"use","excerpt":"The weighting is by the packet mass P_a from the first argument of D(pVert q). Replacing it by Q_a generally breaks the identity. Likewise, forgetting zero-mass conventions or allowing a positive p_i over a zero q_i changes which expressions are finite. takeawayCoarse relative entropy measures distinguishability that survives aggregation. The condi","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-55d4600438de195a","tex":"q_a","aliases":"q_a","display":"q_a","kind":"use","excerpt":"FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] De","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a4591bb144ad9c3f","tex":"Q_c","aliases":"Q_c","display":"Q_c","kind":"use","excerpt":"-1 · 1C. The coarse matrix is the positive square root of -Q_c. This is a concrete naturality identity, checked on the full matrices. It is not a claim that an arbitrary partition commutes with every fractional construction.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f73627dbab9e543b","tex":"\\mathcal Q_c","aliases":"Q_c mathcal","display":"\\mathcal Q_c","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ea7edaad8d5ca545","tex":"\\mathcal Q_f","aliases":"Q_f mathcal","display":"\\mathcal Q_f","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-97b5f75d73830435","tex":"Q_{ii}","aliases":"Q_ii","display":"Q_{ii}","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dfe49e85066a6b84","tex":"Q_{ij}","aliases":"Q_ij","display":"Q_{ij}","kind":"use","excerpt":"ty means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. The","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). ","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7d146bc16e8f166e","tex":"Q_{ji}","aliases":"Q_ji","display":"Q_{ji}","kind":"use","excerpt":"ty means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. The","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a18c33136ce1ffd8","tex":"Q_{m,n}","aliases":"Q_m,n","display":"Q_{m,n}","kind":"use","excerpt":"A bounded generator of a contraction semigroup on a Hilbert space is dissipative. If Q_m,n=I-P_m,n and y∈ Q_m,nX_n, then ReipQ_m,nA_nQ_m,nyy = ReipA_nyy ≤0. Hence the compressed hidden semigroup is contractive. Apply [source reference] to P_m,n. Since P_m,nR_n,m=R_n,m, η_m,n(T) ≤ Tnorm(I-P_m,n)A_nP_m,n normR_n,m. The right side tends to zero. Use [","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-904b640e83eefc03","tex":"Q_{ri}","aliases":"Q_ri","display":"Q_{ri}","kind":"use","excerpt":"[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ab69aa180f14a53e","tex":"q_\\alpha","aliases":"q_α alpha","display":"q_\\alpha","kind":"use","excerpt":"[Bayesian packet reconstruction] B_λ(p) = ∑_α m_α q_α. Only packets with positive evidence carry a conditional probability vector. A zero-evidence packet has zero posterior mass and contributes the zero vector. The nonlinear coupling between packets occurs only through the total evidence used to normalize the packet evidences.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ee7969975240e6b6","tex":"q_{\\alpha,i}","aliases":"q_α,i alpha","display":"q_{\\alpha,i}","kind":"use","excerpt":"Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"e renewcommandtheHfigureR.arabicfigure renewcommandthetableRarabictable renewcommandtheHtableR.arabictable FloatBarrier Transfer paths and product-topology limits. Let T(t) be an arbitrary strongly continuous semigroup on a complex Banach space X, with generator A, and suppose normT(t)≤ M e^ω t for M≥1 and ω∈ R. Let (P_α)_α∈ I_0 be a finite family ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-47f4284788757fb1","tex":"r^{(a)}","aliases":"r^(a)","display":"r^{(a)}","kind":"use","excerpt":"FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] De","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"[Summable defects do not force a semigroup limit] Let X_0= R and let X_n= R^2 for every n≥1. Set C_0(x,y)=x, R_0x=(x,0), and use identity connecting maps and reconstructions above level one. Let T_0(t)=I and let every higher level carry the rotation group T_n(t) = cos t · -sin t","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"Every nonzero term in [source reference] contains at least m_0 time integrations. The simplex 0<r_1<⋯<r_m<t has volume t^m/m!, so the first possible order is t^m_0. If there is no path, every packet term vanishes.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fb14b1a407e1de8e","tex":"r_a","aliases":"r_a","display":"r_a","kind":"use","excerpt":"Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4bdf782a5897d64d","tex":"r_b","aliases":"r_b","display":"r_b","kind":"use","excerpt":"For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-37f2bdf109707126","tex":"R_c","aliases":"R_c","display":"R_c","kind":"use","excerpt":"For z in the common resolvent set, let R_f(z)=(zI-A_f)^-1, R_c(z)=(zI-A_c)^-1.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bf206f644888b16a","tex":"R_D","aliases":"R_D","display":"R_D","kind":"use","excerpt":"[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a94836df7b97d193","tex":"R_f","aliases":"R_f","display":"R_f","kind":"use","excerpt":"For z in the common resolvent set, let R_f(z)=(zI-A_f)^-1, R_c(z)=(zI-A_c)^-1.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7a14eeb4f04e9a5c","tex":"r_i^{(a)}","aliases":"r_i^(a)","display":"r_i^{(a)}","kind":"use","excerpt":"FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] De","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-25ee15775b880083","tex":"R_m","aliases":"R_m","display":"R_m","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-272cb3ca57c00592","tex":"r_m","aliases":"r_m","display":"r_m","kind":"use","excerpt":"Every nonzero term in [source reference] contains at least m_0 time integrations. The simplex 0<r_1<⋯<r_m<t has volume t^m/m!, so the first possible order is t^m_0. If there is no path, every packet term vanishes.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-eee5b018d7f83622","tex":"R_{m,m}","aliases":"R_m,m","display":"R_{m,m}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b8a8516575d862bd","tex":"R_{n-1}","aliases":"R_n-1","display":"R_{n-1}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3088a38f12fda26d","tex":"R_x","aliases":"R_x","display":"R_x","kind":"use","excerpt":"1 · -1. Retain the first coordinate and eliminate the second. For z away from the relevant poles, the retained entry of (zI-A)^-1 is R_x(z)=[z+1-frac1z+1]^-1 =fracz+1z(z+2). The term Σ(z)=1/(z+1) is the hidden coordinate's self-energy. vfigresolvent platecaptionThe full inverse gives a rational response with two poles, at 0 and -2. Its partial frac","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-68df6b2bfb142937","tex":"r_\\alpha","aliases":"r_α alpha","display":"r_\\alpha","kind":"use","excerpt":"If λ is constant on each packet, [source reference] has packet averages independent of the prior and its local posterior formula gives q_α=r_α. This proves the first condition.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fd35db052b2bfbd8","tex":"r_\\mu","aliases":"r_μ mu","display":"r_\\mu","kind":"use","excerpt":"ator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with ov","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-81ae26f5145c949e","tex":"R_\\ell","aliases":"R_ℓ ell","display":"R_\\ell","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b11c81e6903fa9bf","tex":"\\Ran","aliases":"Ran operatorname","display":"{\\operatorname{Ran}}","kind":"use","excerpt":"[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMTraditional Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil d","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-78ecfb9af1021553","tex":"\\RR^2","aliases":"RR^2 R^2 RR mathbb","display":"{\\mathbb R}^2","kind":"use","excerpt":"[Summable defects do not force a semigroup limit] Let X_0= R and let X_n= R^2 for every n≥1. Set C_0(x,y)=x, R_0x=(x,0), and use identity connecting maps and reconstructions above level one. Let T_0(t)=I and let every higher level carry the rotation group T_n(t) = cos t · -sin t","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-694d5f728e8d54c6","tex":"\\RR^n","aliases":"RR^n R^n RR mathbb","display":"{\\mathbb R}^n","kind":"use","excerpt":"Let Δ_n-1 = {p∈ R^n: p_i≥0, ∑_ip_i=1 } be the probability simplex. Let λ∈[0,∞)^n have positive evidence Z_λ(p)=∑_iλ_ip_i. The Bayesian transfer is B_λ(p) = fracλodot pZ_λ(p).","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":43},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"[Semigroup survival identities] For t≥0, the finite-time exit density and, wherever S(t)>0, the hazard are f(t)=a^Te^tAp_0=-S'(t), h(t)=fracf(t)S(t). Consequently, S is nonincreasing and S(t)=exp(-∫_0^th(r) dr) whenever S(t)>0. The density may be defective: its total mass is 1-lim_t→∞S(t). It is a probability density when absorption is eventual, so","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2b5e6d175de4b2dc","tex":"s^{-1/2}","aliases":"s^-1/2","display":"s^{-1/2}","kind":"use","excerpt":"A normalized density need not have a finite mean. For large s, the exponential factor tends to one and η_t(s) is proportional to s^-3/2. Consequently sη_t(s) has a nonintegrable s^-1/2 tail. The clock has infinite mean even though its total mass is one. takeawaySubordination is a weighted mixture of diffusion times. Replacing the clock distribution","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c1256dd757c5aec0","tex":"s^{-3/2}","aliases":"s^-3/2","display":"s^{-3/2}","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bb0124a734ec8cdf","tex":"s^m","aliases":"s^m","display":"s^m","kind":"use","excerpt":"Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-51a7b924b79e9f32","tex":"s^\\alpha","aliases":"s^α alpha","display":"s^\\alpha","kind":"use","excerpt":"Set R_f(s)=(sI+L_f)^-1, R_c(s)=(sI+L_c)^-1. The resolvent identity gives CR_f(s)-R_c(s)C = -R_c(s)Δ_LR_f(s). Since LR=I-sR, CL_fR_f(s)-L_cR_c(s)C = sR_c(s)Δ_LR_f(s). Insert this into [source reference]. The spectral lower bound gives normR_f(s),normR_c(s) ≤ frac1s+m. Therefore normCL_f^α-L_c^α C · ≤ fracsin(πα)π normΔ_L ∫_0^∞ fracs^α(s+m)^2 d s · =","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0906a9a73912e8a8","tex":"s^{\\alpha-1}","aliases":"s^α-1 alpha","display":"s^{\\alpha-1}","kind":"use","excerpt":"[Balakrishnan formula in finite dimension] For 0<α<1, L^α = fracsin(πα)π ∫_0^∞ s^α-1L(sI+L)^-1 d s. The integral converges in operator norm.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b9746e4f6fc18b24","tex":"s_m","aliases":"s_m","display":"s_m","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d6f3fd1d129e9558","tex":"s_{m-1}","aliases":"s_m-1","display":"s_{m-1}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cb79e1a36c0e4809","tex":"S_V","aliases":"S_V","display":"S_V","kind":"use","excerpt":"[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off diagonal entries and nonpositive column sums, so it is the subgenera","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8bd0c25cd815480a","tex":"S_V'","aliases":"S_V'","display":"S_V'","kind":"use","excerpt":"[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off diagonal entries and nonpositive column sums, so it is the subgenera","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-87a2b7a1a86d2b97","tex":"S_z","aliases":"S_z","display":"S_z","kind":"use","excerpt":"0 · I and S_z= F_P(z) · 0 -A_QP · zI-A_QQ . Direct multiplication gives zI-A=L_zS_z. The first factor is invertible. The claims follow from triangular inversion and the determinant of the second factor. This is the finite block form of the Feshbach reduction [source reference].","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5fe98e4f30da8940","tex":"\\Sfun","aliases":"Sfun S mathsf","display":"{\\mathsf S}","kind":"use","excerpt":"ceives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d87f4ba0b84c237a","tex":"\\Sfun_N^{(r)}","aliases":"Sfun_N^(r) S_N^(r) Sfun mathsf","display":"{\\mathsf S}_N^{(r)}","kind":"use","excerpt":"ceives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-545e283bf4ccf588","tex":"\\Sfun_N^{(r+s)}","aliases":"Sfun_N^(r+s) S_N^(r+s) Sfun mathsf","display":"{\\mathsf S}_N^{(r+s)}","kind":"use","excerpt":"[Operator reciprocal family] Let A be bounded and suppose Spec(A) lies in a simply connected region on which D_N has no zeros. Fix a holomorphic branch g=log D_N there. Then S_N^(r)(A) = exp(-r g(A)) = ( D_N^-r)(A) is well defined and S_N^(r)(A) S_N^(s)(A) = S_N^(r+s)(A). If (P_i) is a finite complete family of bounded idempotents, then P_i f(A)P_j","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6d4ed0b3a6647724","tex":"\\Sfun_N^{(s)}","aliases":"Sfun_N^(s) S_N^(s) Sfun mathsf","display":"{\\mathsf S}_N^{(s)}","kind":"use","excerpt":"[Operator reciprocal family] Let A be bounded and suppose Spec(A) lies in a simply connected region on which D_N has no zeros. Fix a holomorphic branch g=log D_N there. Then S_N^(r)(A) = exp(-r g(A)) = ( D_N^-r)(A) is well defined and S_N^(r)(A) S_N^(s)(A) = S_N^(r+s)(A). If (P_i) is a finite complete family of bounded idempotents, then P_i f(A)P_j","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"[Fractional intertwining theorem] For bounded nonnegative self-adjoint A and B, D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In par","count":19},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"use","excerpt":"The kth discrete eigenvalue is λ_k,h=frac4h^2sin^2(frackπ h2), whereas the continuum eigenvalue is (kπ)^2.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"[Bayesian refinement] For the finite test partition {0,1}sqcup{2,3}, a blockwise-constant likelihood gave commutation residual 1.11×10^-16. A nonconstant likelihood produced an exact defect-factorization residual 7.29×10^-17. Its measured L^1 defect was 4.8787×10^-2, below the certified bound 1.1016×10^-1. Two priors with identical coarse masses bu","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":23},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dc260eee4ab7cffe","tex":"\\sup_{0\\leq r\\leq t}","aliases":"sup_0≤ r≤ t sup leq leq","display":"\\sup_{0\\leq r\\leq t}","kind":"use","excerpt":"Let E(t)=sup_0≤ r≤ tnormx(r)-widetilde x(r). Subtract the two mild equations. Fubini's theorem and the triangle inequality give E(t) ≤ M_TL∫_0^tE(r) d r + M_TBtε. For t≤ T, replace the last factor t by T and apply Gronwall's inequality.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-76d8fadb9fe83504","tex":"\\sup_{0\\leq s,t\\leq T}","aliases":"sup_0≤ s,t≤ T sup leq leq","display":"\\sup_{0\\leq s,t\\leq T}","kind":"use","excerpt":"[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally un","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c58f1649c7ede918","tex":"\\sup_{0\\le t\\le T}","aliases":"sup_0≤ t≤ T sup le le","display":"\\sup_{0\\le t\\le T}","kind":"use","excerpt":"2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time interval, not errors sampled at the right endpoint. headThe estimate follows before any graph is drawn For 0≤ t≤ T, the derivative of a↦ e^-at has absolute value at most T when a≥0. Thus sup_0≤ t≤ T|T_n(","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c9c6f71937fef238","tex":"\\sup_{0\\leq t\\leq T}","aliases":"sup_0≤ t≤ T sup leq leq","display":"\\sup_{0\\leq t\\leq T}","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":11},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3ee24d95dbc480e1","tex":"\\sup_{z\\in\\Gamma}","aliases":"sup_z∈Γ sup in Gamma","display":"\\sup_{z\\in\\Gamma}","kind":"use","excerpt":"a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure of the union, then Cg(A_f)-g(A_c)C = frac12π i ∫_Γ g(z)R_c(z)Δ R_f(z) d z. In particul","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8f99188f5481208b","tex":"\\sup_\\ell","aliases":"sup_ℓ sup ell","display":"\\sup_\\ell","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remai","count":44},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6da3583735a2a7a0","tex":"T'","aliases":"T'","display":"T'","kind":"use","excerpt":"[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"trajectory platecaptionThe first-coordinate error is exactly t^2/2. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":17},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d241eb52ee3cfa72","tex":"t^k","aliases":"t^k","display":"t^k","kind":"use","excerpt":"Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace; equivalently, apply the decomposition to left multiplication by B on the matrix algebra a","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-aea3af576abad800","tex":"t^m","aliases":"t^m","display":"t^m","kind":"use","excerpt":"The Laplace transform follows by integrating the density in [source reference]. Stability of A gives ∫_0^∞ e^-st e^tA d t = (sI-A)^-1. At s=0, a^T(-A)^-1p_0 = 1^Tp_0=1. For the moments, ∫_0^∞ t^m e^tA d t = m!(-A)^-m-1. Multiplying by a^T= 1^T(-A) gives the result. These are the standard matrix formulas for phase type laws [source reference].","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f691afb056141dc2","tex":"t^{m_0}","aliases":"t^m_0","display":"t^{m_0}","kind":"use","excerpt":"in Part II. A directed edge from α to β means that P_β A P_α is nonzero, with α≠β; a packet always has the length-zero path to itself. [Minimum transfer order] Suppose every directed packet path from α to β uses at least m_0 off diagonal transfers. Then P_β e^tAP_α =O(t^m_0) as tdownarrow0. If there is no directed path from α to β, the block vanish","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"use","excerpt":"Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cef6729bcc40a7f6","tex":"t^\\alpha","aliases":"t^α alpha","display":"t^\\alpha","kind":"use","excerpt":"[Fractional intertwining theorem] For bounded nonnegative self-adjoint A and B, D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In par","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bea69885d49bedfc","tex":"t^{\\alpha-1}","aliases":"t^α-1 alpha","display":"t^{\\alpha-1}","kind":"use","excerpt":"For 0<α<1, the Balakrishnan formula gives A^α = fracsin(πα)π ∫_0^∞ t^α-1A(tI+A)^-1 dt, and likewise for B. Using A(tI+A)^-1=I-t(tI+A)^-1 and the rectangular resolvent identity (tI+B)^-1C-C(tI+A)^-1 = (tI+B)^-1(CA-BC)(tI+A)^-1, the difference of the two Balakrishnan integrals becomes the displayed resolvent-defect formula.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-048429bc73f8c238","tex":"t^{\\alpha-2}","aliases":"t^α-2 alpha","display":"t^{\\alpha-2}","kind":"use","excerpt":"For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"[Summable defects do not force a semigroup limit] Let X_0= R and let X_n= R^2 for every n≥1. Set C_0(x,y)=x, R_0x=(x,0), and use identity connecting maps and reconstructions above level one. Let T_0(t)=I and let every higher level carry the rotation group T_n(t) = cos t · -sin t","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bce12004b499e794","tex":"T_{j+1}","aliases":"T_j+1","display":"T_{j+1}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b8e003e6dbf7e04c","tex":"T_m","aliases":"T_m","display":"T_m","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-40bcddeff96a69c7","tex":"T_{n+1}","aliases":"T_n+1","display":"T_{n+1}","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-557661f6ae0c797e","tex":"T_t","aliases":"T_t","display":"T_t","kind":"use","excerpt":"[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-107e01d7daa9bd03","tex":"T_z","aliases":"T_z","display":"T_z","kind":"use","excerpt":"Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace; equivalently, apply the decomposition to left multiplication by B on the matrix algebra a","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7007840dcb4ea04f","tex":"T_z'","aliases":"T_z'","display":"T_z'","kind":"use","excerpt":"Let J=∫_0^∞ T_z(t) dt. Integrating both derivative identities T_z'=(B-zI)T_z=T_z(B-zI) and using T_z(0)=I gives (zI-B)J=J(zI-B)=I. Thus J=(zI-B)^-1. Continuous left and right multiplication carry the Bochner integral through the two rectangular transfer blocks, yielding the claim.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-186d4ea45168e2cc","tex":"T_\\alpha","aliases":"T_α alpha","display":"T_\\alpha","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6f053dd14f899054","tex":"T_{\\beta\\gamma}","aliases":"T_βγ beta gamma","display":"T_{\\beta\\gamma}","kind":"use","excerpt":"[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-265d18b60f7a60c8","tex":"T_{\\gamma\\alpha}","aliases":"T_γα gamma alpha","display":"T_{\\gamma\\alpha}","kind":"use","excerpt":"[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-35899e3c99220253","tex":"T_\\ell","aliases":"T_ℓ ell","display":"T_\\ell","kind":"use","excerpt":"[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a15b96314fbf77e6","tex":"T_{\\ell+1}","aliases":"T_ℓ+1 ell","display":"T_{\\ell+1}","kind":"use","excerpt":"[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b90b5fbc860e4de8","tex":"T_\\infty","aliases":"T_∞ infty","display":"T_\\infty","kind":"use","excerpt":"[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.","count":41},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-11b41534cdf61584","tex":"u''","aliases":"u''","display":"u''","kind":"use","excerpt":"On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-39eb7259ed5bbafd","tex":"u^{(4)}","aliases":"u^(4)","display":"u^{(4)}","kind":"use","excerpt":"[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-663ee04bfdc73b2a","tex":"u^h","aliases":"u^h","display":"u^h","kind":"use","excerpt":"[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f3d9f585790af390","tex":"u^h_i","aliases":"u^h_i","display":"u^h_i","kind":"use","excerpt":"[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-49185fe756c9fe7d","tex":"u^n","aliases":"u^n","display":"u^n","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-515a098b70460045","tex":"u^{\\alpha-1}","aliases":"u^α-1 alpha","display":"u^{\\alpha-1}","kind":"use","excerpt":"It is enough to prove the formula on an eigenvector with eigenvalue x≥0. For x>0, the substitution s=xu gives fracsin(πα)π ∫_0^∞ s^α-1fracxs+x d s =x^α fracsin(πα)π ∫_0^∞fracu^α-11+u d u =x^α. The value at x=0 is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power c","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6a3339eaf4eb0015","tex":"U_{0,n}","aliases":"U_0,n","display":"U_{0,n}","kind":"use","excerpt":"sin t · cos t . Only the first adjacent defect is nonzero, so every defect sum is finite. Yet for n≥1, U_0,n(t)=C_0T_n(t)R_0=cos t. The limiting family is not a semigroup because in general cos(s+t)≠cos scos t. The failure is exactly the excursion from the reconstructed line into its orthogonal complement between two time steps.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6fdb23dc8f442414","tex":"u_h","aliases":"u_h","display":"u_h","kind":"use","excerpt":"nzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMTraditional Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil differentiates the quadratic ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"1 · 2 · 3. vfiggreen platecaptionThe tent is the central column of the continuous Green kernel. It has nodal values (1,2,1)/8; its discrete inverse column is (1,2,1)/32, smaller by the factor h. headA forcing vector is not a quadrature-weighted measure For sampled forcing f_j, the finite solution is u_i=∑_j(L_h^-1)_ijf_j =h∑_jG(x_i,x_j)f_j. The fac","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1c42292632e07b88","tex":"u_i^h","aliases":"u_i^h","display":"u_i^h","kind":"use","excerpt":"[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"0 · 2 · 1 · -3. The graph shows rates between actual states. The four normalized sign vectors on the right form an orthonormal basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve in","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-43190b7d6e150fa1","tex":"u_j^{\\mathsf T}","aliases":"u_j^ T mathsf","display":"u_j^{\\mathsf T}","kind":"use","excerpt":"0 · 2 · 1 · -3. The graph shows rates between actual states. The four normalized sign vectors on the right form an orthonormal basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve in","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-25121c20156ac918","tex":"U_m","aliases":"U_m","display":"U_m","kind":"use","excerpt":"t_completion Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a4312e8e6ee32bea","tex":"U_{m,n+1}","aliases":"U_m,n+1","display":"U_{m,n+1}","kind":"use","excerpt":"Since C_nR_n=I, U_m,n+1(t)-U_m,n(t) = · C_m,n (C_nT_n+1(t)-T_n(t)C_n) R_nR_n,m. Therefore normU_m,n+1(t)-U_m,n(t) ≤ w_m,nnormE_n(t). The weighted summability hypothesis makes the sequence uniformly Cauchy. Summing the tail estimates gives the stated bound.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-167a8b5df4139bcf","tex":"U_n","aliases":"U_n","display":"U_n","kind":"use","excerpt":"not close the dynamicsREFINEMENT / A PERSISTENT RANGE EXCURSIONTraditional Part II: Theorem E4.8 and Example E4.10. Summability and closure are different tests. Let the coarse space be R and every finer space be R^2. The first connecting maps are C(x,y)=x and Rx=(x,0); all later maps are identities. Let the coarse evolution be the identity and all ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":40},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"[Infinitesimal recovery of the defect] The generator defect is the first order semigroup defect: lim_t→0 frac D(t)t = Δ in operator norm. Consequently, C e^tA_f= e^tA_cC hboxfor every t if and only if CA_f=A_cC.","count":14},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"Nonnegativity follows from the entry formula in [source reference]. Let v_i=x_i(1-x_i)/2. Since the second difference is exact on quadratics, L_hv= 1. Hence the ith row sum of L_h^-1 is v_i. For a nonnegative matrix, the infinity operator norm is the largest row sum.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a2ad275314ca92dd","tex":"v_{k,h}","aliases":"v_k,h","display":"v_{k,h}","kind":"use","excerpt":"The discrete eigenpairs are λ_k,h = frac4h^2 sin^2(frackπ h2), v_k,h(i)=sin(kπ x_i).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-51fd0818257204a0","tex":"v_r","aliases":"v_r","display":"v_r","kind":"use","excerpt":"_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c0e8009c8acb599b","tex":"\\Var","aliases":"Var operatorname","display":"{\\operatorname{Var}}","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3696096793ac9d06","tex":"\\operatorname{Var}","aliases":"Var operatorname","display":"\\operatorname{Var}","kind":"use","excerpt":"o. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Co","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1c7db940f017a3bc","tex":"\\Var_{p(t)}","aliases":"Var_p(t) Var operatorname","display":"{\\operatorname{Var}}_{p(t)}","kind":"use","excerpt":"[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dfcf7637eedfd13c","tex":"\\Var_\\gamma","aliases":"Var_γ Var gamma operatorname gamma","display":"{\\operatorname{Var}}_\\gamma","kind":"use","excerpt":"[Gaussian spectral gap] If f∈ L^2(γ) has mean zero, then norm O_tf_2 ≤ e^-tnorm f_2. For every polynomial f, Var_γ(f) ≤ ∫_ Rabsf'(x)^2γ( d x).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-754549a165ac03a7","tex":"\\Var_\\pi","aliases":"Var_π Var pi operatorname pi","display":"{\\operatorname{Var}}_\\pi","kind":"use","excerpt":"[Law of total variance] For a real valued f, let P_0f= E_π[f] 1. Then Var_π(f) = Var_π(P_ Qf) + normf-P_ Qf_π^2.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-902783d1c24b6e39","tex":"\\operatorname{Var}_\\pi","aliases":"Var_π operatorname pi","display":"\\operatorname{Var}_\\pi","kind":"use","excerpt":"o. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Co","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8a4709427aa09482","tex":"w_{(1)}","aliases":"w_(1)","display":"w_{(1)}","kind":"use","excerpt":"Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2433e0c1dac2127f","tex":"w_{(2)}","aliases":"w_(2)","display":"w_{(2)}","kind":"use","excerpt":"Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d2beefde8295c27a","tex":"w_{(j)}","aliases":"w_(j)","display":"w_{(j)}","kind":"use","excerpt":"normB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fe2e39be77a64ced","tex":"w_{(r)}","aliases":"w_(r)","display":"w_{(r)}","kind":"use","excerpt":"Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"A DEFINED COSTTraditional Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks; block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified i","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3106fac3e3c8f992","tex":"w_2","aliases":"w_2","display":"w_2","kind":"use","excerpt":"A DEFINED COSTTraditional Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks; block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified i","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9763a163e7ff6591","tex":"w_3","aliases":"w_3","display":"w_3","kind":"use","excerpt":"A DEFINED COSTTraditional Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks; block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified i","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the object being transported. The transpose and the posit","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"[Optimal selection for the certified primary block bound] Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1b544c1c3e686ebb","tex":"w_{m,j}","aliases":"w_m,j","display":"w_{m,j}","kind":"use","excerpt":"fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5c8ab31deb179927","tex":"w_{m,n}","aliases":"w_m,n","display":"w_{m,n}","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5450287e32dd956c","tex":"\\widehat K","aliases":"widehat K widehat","display":"\\widehat K","kind":"use","excerpt":"24183) (8.62,0.815842) (8.705,0.807882) (8.79,0.800286) (8.875,0.79304) (8.96,0.786128) (9.045,0.779536) (9.13,0.77325) (9.215,0.767257) (9.3,0.761544); path[fill=black] (2.216667,3.409096) circle (0.065); node[font=small,align=center] at (11.7,3.9) J=-2I+N, N^2=0; node[font=small,align=center] at (11.7,2.8) e^tJ=e^-2t(I+tN); node[font=small,align=","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-816a095af468abc4","tex":"\\widehat p_r","aliases":"widehat p_r widehat","display":"\\widehat p_r","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-52b9530420892ef6","tex":"\\widehat\\Kcal","aliases":"widehatKcal widehat K widehat Kcal widehat mathcal","display":"\\widehat{\\mathcal K}","kind":"use","excerpt":"tional characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies. [Rational memory realization] A matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precisely, widehat K(z)=B(zI-D)^-1C if and only if ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9117c60fc5e9378b","tex":"\\widetilde K","aliases":"widetilde K widetilde","display":"\\widetilde K","kind":"use","excerpt":"[One step filter stability] Let K and widetilde K be column stochastic. Assume 0<a≤λ_i≤ b. Then norm B_λ(Kp)- B_λ(widetilde Kq) _1 ≤ frac2ba ( normp-q_1 + normK-widetilde K_1→1 ).","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c77170c848a2e8b4","tex":"\\widetilde x","aliases":"widetilde x widetilde","display":"\\widetilde x","kind":"use","excerpt":"[Volterra stability under kernel compression] Let S(t) be a strongly continuous semigroup on a Banach space X, fix T>0, and assume normS(t)≤ M_T (0≤ t≤ T). Let K,widetilde K∈ L^1((0,T), B(X)), let g∈ L^1((0,T),X), and let x,widetilde x∈ C([0,T],X). Suppose these are mild solutions with the same initial state and forcing, except that K is replaced b","count":13},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c0b6083f2a1db355","tex":"\\widetilde x'","aliases":"widetilde x' widetilde","display":"\\widetilde x'","kind":"use","excerpt":"0 · 0, so N^2=0 and N _2=1. On [0,T], compare the memory kernels K(t)=N and widetilde K(t)=0, with identity free semigroup, no forcing, and initial state x_0=e_1=(0,1)^ T. The equations are x'(t)=∫_0^t N x(r) dr, widetilde x'(t)=0. Their exact solutions are x(t)=(t^2/2,1)^ T and widetilde x(t)=(0,1)^ T. vfigext_trajectory platecaptionThe first-coor","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-badcfecc2304d5fe","tex":"\\widetilde\\Kcal","aliases":"widetildeKcal widetilde K widetilde Kcal widetilde mathcal","display":"\\widetilde{\\mathcal K}","kind":"use","excerpt":"FloatBarrier Memory compression, rational realization, and phase-type consequences. Fix T>0. For the next estimate let A be a bounded operator on a Banach space X, let K,widetilde K∈ L^1((0,T), B(X)), and let f be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a shar","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a48c4c6cb9f766e9","tex":"\\widetilde\\lambda","aliases":"widetildeλ widetilde lambda","display":"\\widetilde\\lambda","kind":"use","excerpt":"[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3e28571e73d20b73","tex":"\\widetilde\\lambda_i","aliases":"widetildeλ_i widetilde lambda","display":"\\widetilde\\lambda_i","kind":"use","excerpt":"[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"vpage22Hidden state returns as memoryEXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUPTraditional Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data. Write the same system as x'=-x+y, y'=x-y. Solving the second equation first gives y(t)=e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. Substitution is exact: x'(t)=-x(t)+e^-ty(0)+∫","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"For x≥0, 0≤ x^2-sin^2x≤ x^4/3. Set x=kπ h/2 and multiply by 4/h^2.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"use","excerpt":"Set x=kπ h/2. The condition kh≤1 gives 0≤ x≤π/2. On this interval, sin x≥ x-fracx^36, so 0≤ x^2-sin^2x≤fracx^43. Multiply by 4/h^2.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b5d15c510169ad7c","tex":"x^4","aliases":"x^4","display":"x^4","kind":"use","excerpt":"For x≥0, 0≤ x^2-sin^2x≤ x^4/3. Set x=kπ h/2 and multiply by 4/h^2.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-213991022244ad3d","tex":"x^\\alpha","aliases":"x^α alpha","display":"x^\\alpha","kind":"use","excerpt":"It is enough to prove the formula on an eigenvector with eigenvalue x≥0. For x>0, the substitution s=xu gives fracsin(πα)π ∫_0^∞ s^α-1fracxs+x d s =x^α fracsin(πα)π ∫_0^∞fracu^α-11+u d u =x^α. The value at x=0 is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power c","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"ssion Let S(t) be a strongly continuous semigroup on a Banach space X, fix T>0, and assume normS(t)≤ M_T (0≤ t≤ T). Let K,widetilde K∈ L^1((0,T), B(X)), let g∈ L^1((0,T),X), and let x,widetilde x∈ C([0,T],X). Suppose these are mild solutions with the same initial state and forcing, except that K is replaced by widetilde K: x(t) = · S(t)x_0+ ∫_0^tS(","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"path, one-edge paths, and the two-edge path exist. The coefficient 3t^2 includes both the edge product 2·3 and the ordered-time simplex area t^2/2. headPath length counts transfers, not elapsed time Starting at e_0, the complete state is (1,2t,3t^2)^ T. No fourth packet or repeated cycle can contribute. Direct differentiation gives x'(t)=Ax(t), inc","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0bade2109ecac9dc","tex":"x_2'","aliases":"x_2'","display":"x_2'","kind":"use","excerpt":"path, one-edge paths, and the two-edge path exist. The coefficient 3t^2 includes both the edge product 2·3 and the ordered-time simplex area t^2/2. headPath length counts transfers, not elapsed time Starting at e_0, the complete state is (1,2t,3t^2)^ T. No fourth packet or repeated cycle can contribute. Direct differentiation gives x'(t)=Ax(t), inc","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8eeb0a1bbb468424","tex":"X_3","aliases":"X_3","display":"X_3","kind":"use","excerpt":"1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEver","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-72e05ef2b0f48e18","tex":"X_c","aliases":"X_c","display":"X_c","kind":"use","excerpt":"Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f76f02d6c081bad5","tex":"X_f","aliases":"X_f","display":"X_f","kind":"use","excerpt":"Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"[Exact discrete Green reconstruction at grid nodes] Let x_i=ih. Then frac1h(L_h^-1)_ij =G(x_i,x_j). Thus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.","count":18},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0597cb543b16b78f","tex":"x_i^2","aliases":"x_i^2","display":"x_i^2","kind":"use","excerpt":"vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMTraditional Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil d","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f3be737b42076753","tex":"X_m","aliases":"X_m","display":"X_m","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-853cecd6231f3bad","tex":"X_P","aliases":"X_P","display":"X_P","kind":"use","excerpt":"Let X=X_P⊕ X_Q and write A= A_PP · A_PQ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-68150f2105b73870","tex":"x_P","aliases":"x_P","display":"x_P","kind":"use","excerpt":"Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b8663ffd669417a5","tex":"x_Q","aliases":"x_Q","display":"x_Q","kind":"use","excerpt":"Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8285a92cff098813","tex":"X_r","aliases":"X_r","display":"X_r","kind":"use","excerpt":"[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2ee20c14455ce834","tex":"X_s","aliases":"X_s","display":"X_s","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2b6c204b3855e438","tex":"X_\\ell","aliases":"X_ℓ ell","display":"X_\\ell","kind":"use","excerpt":"The product of the spaces X_ℓ, equipped with the bounded supremum norm, is Banach. The equations C_ℓ x_ℓ+1=x_ℓ define a closed linear subspace.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4ec4142068739e9a","tex":"x_\\ell","aliases":"x_ℓ ell","display":"x_\\ell","kind":"use","excerpt":"Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-524e8e9a8948b743","tex":"X_{\\ell+1}","aliases":"X_ℓ+1 ell","display":"X_{\\ell+1}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-93da8851ae87d014","tex":"x_{\\ell+1}","aliases":"x_ℓ+1 ell","display":"x_{\\ell+1}","kind":"use","excerpt":"Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-55abe5dbff13226b","tex":"\\xleftarrow","aliases":"xleftarrow","display":null,"kind":"use","excerpt":"Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":50},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"use","excerpt":"vpage22Hidden state returns as memoryEXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUPTraditional Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data. Write the same system as x'=-x+y, y'=x-y. Solving the second equation first gives y(t)=e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. Substitution is exact: x'(t)=-x(t)+e^-ty(0)+∫","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"c K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dd412919846eece9","tex":"y_2","aliases":"y_2","display":"y_2","kind":"use","excerpt":"vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELTraditional Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from V--23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x, · y_","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a478d5b0d5f3dc01","tex":"y_Q","aliases":"y_Q","display":"y_Q","kind":"use","excerpt":"Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-57348dda8e2fb02d","tex":"y_r","aliases":"y_r","display":"y_r","kind":"use","excerpt":"c K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.","count":62},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-449753faf1f7a481","tex":"z^{-1}","aliases":"z^-1","display":"z^{-1}","kind":"use","excerpt":"tain the first coordinate and eliminate the second. For z away from the relevant poles, the retained entry of (zI-A)^-1 is R_x(z)=[z+1-frac1z+1]^-1 =fracz+1z(z+2). The term Σ(z)=1/(z+1) is the hidden coordinate's self-energy. vfigresolvent platecaptionThe full inverse gives a rational response with two poles, at 0 and -2. Its partial fractions are ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-89c743fe25c2bcb6","tex":"z^{k+1}","aliases":"z^k+1","display":"z^{k+1}","kind":"use","excerpt":"For large z, Σ_P(z) = ∑_k=0^∞ fracA_PQA_QQ^kA_QPz^k+1. Thus the second condition implies the moment identities. Conversely, the first d identities imply all later ones by the Cayley Hamilton theorem, so the series vanishes.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-140a1634265e4538","tex":"z^{Nm}","aliases":"z^Nm","display":"z^{Nm}","kind":"use","excerpt":"This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=lo","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1ac2c1acff83ca55","tex":"Z_{\\mathrm A}","aliases":"Z_ A mathrm","display":"Z_{\\mathrm A}","kind":"use","excerpt":"im. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged. vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shap","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-aa421f6078ef940d","tex":"Z_{\\mathrm B}","aliases":"Z_ B mathrm","display":"Z_{\\mathrm B}","kind":"use","excerpt":"lihoods by one common positive factor leaves this posterior unchanged. vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packe","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b23ccb12e2adf65f","tex":"Z_1","aliases":"Z_1","display":"Z_1","kind":"use","excerpt":",3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ T. vfigintegrated_filter platecaptionWithin-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is (7/13,6/13). headCheck the normalizers, not only the final vector The first and second predictive evidenc","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-64a9a143ebce6852","tex":"Z_2","aliases":"Z_2","display":"Z_2","kind":"use","excerpt":",3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ T. vfigintegrated_filter platecaptionWithin-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is (7/13,6/13). headCheck the normalizers, not only the final vector The first and second predictive evidenc","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1c90720ad505f551","tex":"Z_a","aliases":"Z_a","display":"Z_a","kind":"use","excerpt":"The evidence in block B_a is Z_a =q_a∑_i∈ B_aL_ir_i^(a) =q_aoverline L_a. Apply [source reference].","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e847049bb4b64a62","tex":"Z_{\\bar\\lambda}","aliases":"Z_barλ bar lambda","display":"Z_{\\bar\\lambda}","kind":"use","excerpt":"[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remai","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-10845de283abdc8a","tex":"Z_j","aliases":"Z_j","display":"Z_j","kind":"use","excerpt":"_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5e940a55f95e2b85","tex":"Z_r","aliases":"Z_r","display":"Z_r","kind":"use","excerpt":"When sufficiency fails, the defect still has an exact finite factorization. Fix a declared coarse likelihood barλ, put P=C_π p, δ(p)=r(p)-barλ, bar b= B_barλ(P), Z_r=⟨ r(p),P⟩.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3f57a3b892b267df","tex":"Z_{\\widetilde\\lambda}","aliases":"Z_widetildeλ widetilde lambda","display":"Z_{\\widetilde\\lambda}","kind":"use","excerpt":"[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-abf0801b6063c7dd","tex":"Z_\\alpha","aliases":"Z_α alpha","display":"Z_\\alpha","kind":"use","excerpt":"Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fee2475676ee8ef5","tex":"Z_\\beta","aliases":"Z_β beta","display":"Z_\\beta","kind":"use","excerpt":"Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e0ffad91ab133eb8","tex":"Z_\\lambda","aliases":"Z_λ lambda","display":"Z_\\lambda","kind":"use","excerpt":"Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.","count":20},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4412880ca0ddeed9","tex":"Z_{\\lambda\\odot\\mu}","aliases":"Z_λodotμ lambda odot mu","display":null,"kind":"use","excerpt":"[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-caf7d1600a9c465c","tex":"Z_\\mu","aliases":"Z_μ mu","display":"Z_\\mu","kind":"use","excerpt":"[Likelihood perturbation bound] Let p be fixed and let λ,μ≥0 have positive evidences Z_λ,Z_μ. Then norm B_λ(p)- B_μ(p)_1 ≤ frac2∑_i|λ_i-μ_i|p_imin(Z_λ,Z_μ).","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dbbeed832046c3f7","tex":"|_{E_\\mu}","aliases":"|_E_μ mu","display":"|_{E_\\mu}","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-62e4e263d57b6219","tex":"|_{P_\\alpha X}","aliases":"|_P_α X alpha","display":"|_{P_\\alpha X}","kind":"use","excerpt":"thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4d27148b42f2a8d0","tex":"|_{QX}","aliases":"|_QX","display":"|_{QX}","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2db86546abe159fd","tex":"|_{\\Ran P}","aliases":"|_Ran P Ran operatorname","display":"|_{{\\operatorname{Ran}} P}","kind":"use","excerpt":"[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5413a578759b7b98","tex":"|_{\\rm nodes}","aliases":"|_rm nodes rm","display":"|_{\\rm nodes}","kind":"use","excerpt":"vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMTraditional Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil d","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7dfd6e4dd08043ea","tex":"|_{t=0}","aliases":"|_t=0","display":"|_{t=0}","kind":"use","excerpt":"= · f(x)+t(f''(x)-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"6 · 0 · 0, A^3=0. Therefore the exponential is the exact polynomial e^tA=I+tA+fract^22A^2. vfigext_paths platecaptionOnly the empty path, one-edge paths, and the two-edge path exist. The coefficient 3t^2 includes both the edge product 2·3 and the ordered-time simplex area t^2/2. headPath length counts transfers, not elapsed time Starting at e_0, th","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"[Bayesian refinement] For the finite test partition {0,1}sqcup{2,3}, a blockwise-constant likelihood gave commutation residual 1.11×10^-16. A nonconstant likelihood produced an exact defect-factorization residual 7.29×10^-17. Its measured L^1 defect was 4.8787×10^-2, below the certified bound 1.1016×10^-1. Two priors with identical coarse masses bu","count":32},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-59ce46150d0cfe77","tex":"\\alpha_0","aliases":"α_0 alpha","display":"\\alpha_0","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-da4f2068e2750ae4","tex":"\\alpha_m","aliases":"α_m alpha","display":"\\alpha_m","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-20abd9c0a72b9797","tex":"\\alpha_{m-1}","aliases":"α_m-1 alpha","display":"\\alpha_{m-1}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"[Holomorphic defect formula] Let Γ be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the closure ","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.","count":29},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.","count":48},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"Let P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For any operator B, define B_βα=P_β BP_α. Then B=∑_α,βB_βα. For the semigroup, write T_βα(t)=P_β e^tAP_α.","count":32},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"[htbp] centering includegraphics[width=0.78textwidth]03_random_walk_transfer.pdf captionThe point mass δ_0 propagated by repeated symmetric random-walk transfer.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ae0d0e3fccff5d10","tex":"\\Delta_{13}","aliases":"Δ_13 Delta","display":"\\Delta_{13}","kind":"use","excerpt":"0 · 1 · 0. The generator defect is Δ=C_η A-BC_η, with only one nonzero entry, Δ_13=7η (matrix indices begin at one). vfigext_substitution platecaptionThe perturbation lets the discarded eigenvalue enter the comparison. Exact naturality holds at η=0; a nonzero defect is transformed, not erased, by the rational calculus. headA resolvent identity supp","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","excerpt":"C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood; the full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a104cc9eebe89629","tex":"\\delta_i","aliases":"δ_i delta","display":"\\delta_i","kind":"use","excerpt":"Markov · point mass δ_i · K^n or e^tQ · state lumping and CK^n=bar K^nC","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1a991337b8d11cca","tex":"\\Delta_K","aliases":"Δ_K Delta","display":"\\Delta_K","kind":"use","excerpt":"[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dd623f3351ab6025","tex":"\\Delta_L","aliases":"Δ_L Delta","display":"\\Delta_L","kind":"use","excerpt":"[Fractional defect bound] Let 0<α<1. Suppose L_f≥ mI, L_c≥ mI for some m>0. Define Δ_L=CL_f-L_cC. Then normCL_f^α-L_c^α C ≤ α m^α-1normΔ_L. Consequently, norm C e^-tL_f^α - e^-tL_c^αC ≤ α t m^α-1normΔ_L.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e8b18ae884be8df2","tex":"\\delta_{mn}","aliases":"δ_mn delta","display":"\\delta_{mn}","kind":"use","excerpt":"· = ∑_n=0^∞ e^-ntH_n(x)fracu^nn!. Comparison of coefficients proves the eigenvalue formula. The same generating function shows ∫ H_mH_n dγ=n!δ_mn. The Hermite polynomials span all polynomials, and polynomials are dense in L^2(γ). Hence the normalized family is an orthonormal basis. See [source reference] for the general Gaussian Hilbert space setti","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-addee2c936e25fa1","tex":"\\Delta_{n-1}","aliases":"Δ_n-1 Delta","display":"\\Delta_{n-1}","kind":"use","excerpt":"Let Δ_n-1 = {p∈ R^n: p_i≥0, ∑_ip_i=1 } be the probability simplex. Let λ∈[0,∞)^n have positive evidence Z_λ(p)=∑_iλ_ip_i. The Bayesian transfer is B_λ(p) = fracλodot pZ_λ(p).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4bff5300fdcd34eb","tex":"\\delta_y","aliases":"δ_y delta","display":"\\delta_y","kind":"use","excerpt":"On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0fcb1c3cf5e3bf16","tex":"\\delta_{\\alpha\\beta}","aliases":"δ_αβ delta alpha beta","display":"\\delta_{\\alpha\\beta}","kind":"use","excerpt":"Let P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For any operator B, define B_βα=P_β BP_α. Then B=∑_α,βB_βα. For the semigroup, write T_βα(t)=P_β e^tAP_α.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-df0a518550e907c5","tex":"\\delta_{\\beta\\alpha}","aliases":"δ_βα delta beta alpha","display":"\\delta_{\\beta\\alpha}","kind":"use","excerpt":"[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d33d8b646241d747","tex":"\\Delta_\\ell","aliases":"Δ_ℓ Delta ell","display":"\\Delta_\\ell","kind":"use","excerpt":"[Generator defect criterion] Assume normC_ℓ≤1, normR_ℓ≤1, and suppose T_ℓ(t)= e^tA_ℓ are contraction semigroups with bounded generators. Set Δ_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ. If ∑_ℓ=0^∞normΔ_ℓ<∞, then the conclusion of [source reference] holds on every bounded time interval, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ T∑_ℓ=n^∞normΔ_ℓ.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.","count":24},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7b93f1be7412facb","tex":"\\eta_1","aliases":"η_1 eta","display":"\\eta_1","kind":"use","excerpt":"Square-root evolution can be written as an average of ordinary diffusion over a positive clock variable s: e^-tsqrt L=∫_0^∞ η_t(s)e^-sL ds, η_t(s)=fract2sqrtπs^-3/2e^-t^2/(4s), t>0. Here t is the external time; s is the ordinary diffusion time being mixed. They are not interchangeable variables. vfigrandom_clock platecaptionThe density η_1 peaks at","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fb9149ca37d1df00","tex":"\\eta_{m,n}","aliases":"η_m,n eta","display":"\\eta_{m,n}","kind":"use","excerpt":"[Range leakage criterion] Under the contractive assumptions of [source reference], define η_m,n(T) = sup_0≤ t≤ T norm(I-P_m,n)T_n(t)R_n,m. Then ρ_m,n(T)≤η_m,n(T). Consequently, η_m,n(T)→0 for every T>0 is sufficient for the limiting family U_m to satisfy the semigroup law.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-12235ebd9b8ec538","tex":"\\eta_t","aliases":"η_t eta","display":"\\eta_t","kind":"use","excerpt":"vpage17A random diffusion clockSUBORDINATION / A DISTRIBUTION OF ORDINARY TIMESTraditional Part I: Theorem 10.2. The Poisson subordination formula for the finite positive model. Square-root evolution can be written as an average of ordinary diffusion over a positive clock variable s: e^-tsqrt L=∫_0^∞ η_t(s)e^-sL ds, η_t(s)=fract2sqrtπs^-3/2e^-t^2/(","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5624add4e4a44776","tex":"\\eta_{\\alpha,s}","aliases":"η_α,s eta alpha","display":"\\eta_{\\alpha,s}","kind":"use","excerpt":"d t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d67c847e2c962980","tex":"\\eta_{\\alpha,s+t}","aliases":"η_α,s+t eta alpha","display":"\\eta_{\\alpha,s+t}","kind":"use","excerpt":"d t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1c7d3ef1b4110a1b","tex":"\\eta_{\\alpha,t}","aliases":"η_α,t eta alpha","display":"\\eta_{\\alpha,t}","kind":"use","excerpt":"[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCETraditional Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","excerpt":"vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCETraditional Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c1ce6d2d2665df29","tex":"\\theta^3","aliases":"θ^3 theta","display":"\\theta^3","kind":"use","excerpt":"ite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-θ). Multiplication adds exponents, so the posterior belongs to Beta(4,4) with density 140θ^3(1-θ)^3. vfigb","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2c9dd55b67cbaf19","tex":"\\lambda^{\\mathsf T}","aliases":"λ^ T lambda mathsf","display":"\\lambda^{\\mathsf T}","kind":"use","excerpt":"B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a7ba560bfc03b7a8","tex":"\\lambda^{(1)}","aliases":"λ^(1) lambda","display":"\\lambda^{(1)}","kind":"use","excerpt":"vpage26One complete coarse filterEND-TO-END MODEL / TWO OBSERVATIONSTraditional Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate. Use the four-state kernel K and aggregation C from pagerefV5, starting at p_0=e_0. Observe two likelihood vectors constant inside both packets: λ^(1)=(3/4,3/4,1/4,","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4691b3be674d9f65","tex":"\\lambda^{(1)}_i","aliases":"λ^(1)_i lambda","display":"\\lambda^{(1)}_i","kind":"use","excerpt":"(1,1); draw[decorate,decoration=brace,mirror,amplitude=5pt](-.7,.55)--(2.7,.55)node[midway,below=8pt]one packet; node[note]at(4.8,.1)separate packet;node[note]at(7.2,.1)separate packet; node[note]at(3.6,-.75)Joint likelihood profile =(λ^(1)_i,λ^(2)_i);","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-63fbbaa59a200ba4","tex":"\\lambda^{(2)}","aliases":"λ^(2) lambda","display":"\\lambda^{(2)}","kind":"use","excerpt":"vpage26One complete coarse filterEND-TO-END MODEL / TWO OBSERVATIONSTraditional Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate. Use the four-state kernel K and aggregation C from pagerefV5, starting at p_0=e_0. Observe two likelihood vectors constant inside both packets: λ^(1)=(3/4,3/4,1/4,","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f4bef687dace0d09","tex":"\\lambda^{(2)}_i","aliases":"λ^(2)_i lambda","display":"\\lambda^{(2)}_i","kind":"use","excerpt":"(1,1); draw[decorate,decoration=brace,mirror,amplitude=5pt](-.7,.55)--(2.7,.55)node[midway,below=8pt]one packet; node[note]at(4.8,.1)separate packet;node[note]at(7.2,.1)separate packet; node[note]at(3.6,-.75)Joint likelihood profile =(λ^(1)_i,λ^(2)_i);","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-096b5779eec11120","tex":"\\lambda^{(r)}","aliases":"λ^(r) lambda","display":"\\lambda^{(r)}","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e12e7e5ac627fad4","tex":"\\lambda^{(y)}","aliases":"λ^(y) lambda","display":"\\lambda^{(y)}","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-10e520ed7336bfd0","tex":"\\lambda^\\alpha","aliases":"λ^α lambda alpha","display":"\\lambda^\\alpha","kind":"use","excerpt":"[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c9503777fa994def","tex":"\\lambda_a","aliases":"λ_a lambda","display":"\\lambda_a","kind":"use","excerpt":"i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the conditional average of λ_i and λ_i'. It can be independent of that variation only if λ_i=λ_i'. Hence λ is constant on every packet; write the value on I_a as λ_a.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-27a452b3d791e541","tex":"\\lambda_b","aliases":"λ_b lambda","display":"\\lambda_b","kind":"use","excerpt":"Now use priors supported on one state of packet a and one state of packet b. Equality of the two coarse posterior odds gives fracλ_aλ_b = fracbarλ_abarλ_b. This holds for every pair of coarse packets, so λ_a=cbarλ_a for one common c>0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-85199f8ae102ab64","tex":"\\lambda_{i'}","aliases":"λ_i' lambda","display":"\\lambda_{i'}","kind":"use","excerpt":"sely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the condition","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-14e48ea0c939f8be","tex":"\\lambda_i^{(y)}","aliases":"λ_i^(y) lambda","display":"\\lambda_i^{(y)}","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"use","excerpt":"The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0178c33c9a9b20df","tex":"\\lambda_j^{(y)}","aliases":"λ_j^(y) lambda","display":"\\lambda_j^{(y)}","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-974551e97b54067e","tex":"\\lambda_j^\\alpha","aliases":"λ_j^α lambda alpha","display":"\\lambda_j^\\alpha","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-577467d13815a1aa","tex":"\\lambda_j^{\\alpha+\\beta}","aliases":"λ_j^α+β lambda alpha beta","display":"\\lambda_j^{\\alpha+\\beta}","kind":"use","excerpt":"All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d594552e2786afd4","tex":"\\lambda_k","aliases":"λ_k lambda","display":"\\lambda_k","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c3b15579ddc9c765","tex":"\\lambda_{k,h}","aliases":"λ_k,h lambda","display":"\\lambda_{k,h}","kind":"use","excerpt":"The kth discrete eigenvalue is λ_k,h=frac4h^2sin^2(frackπ h2), whereas the continuum eigenvalue is (kπ)^2.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6ba91b1d53b9b2d0","tex":"\\lambda_{k,h}^\\alpha","aliases":"λ_k,h^α lambda alpha","display":"\\lambda_{k,h}^\\alpha","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7a90f856677fe06b","tex":"\\lambda_{k,h}^{\\alpha-1}","aliases":"λ_k,h^α-1 lambda alpha","display":"\\lambda_{k,h}^{\\alpha-1}","kind":"use","excerpt":"The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5d96c20ae88ec969","tex":"\\lambda_k^\\alpha","aliases":"λ_k^α lambda alpha","display":"\\lambda_k^\\alpha","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ec937ee0e7e9a617","tex":"\\lambda_k^\\beta","aliases":"λ_k^β lambda beta","display":"\\lambda_k^\\beta","kind":"use","excerpt":"All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).","count":51},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ac6cb066658534c5","tex":"\\mu_1","aliases":"μ_1 mu","display":"\\mu_1","kind":"use","excerpt":"[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-252454ad397ec5a8","tex":"\\mu_i","aliases":"μ_i mu","display":"\\mu_i","kind":"use","excerpt":"The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7976255081ad4eb3","tex":"\\mu_j","aliases":"μ_j mu","display":"\\mu_j","kind":"use","excerpt":"The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.","count":19},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-22d0564d0934fa50","tex":"\\mu_r","aliases":"μ_r mu","display":"\\mu_r","kind":"use","excerpt":"[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). ","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0661e230ba933432","tex":"\\nu_j","aliases":"ν_j nu","display":"\\nu_j","kind":"use","excerpt":"[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). ","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"nded:record:037 Let P_0,P_1,…,P_r come from increasingly fine partitions, so that Ran P_0⊆Ran P_1⊆⋯⊆Ran P_r. Define D_0=P_0, D_j=P_j-P_j-1 (1≤ j≤ r). Then P_iP_j=P_min(i,j), D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j. Every D_j is an orthogonal projector, and normP_rf_π^2 = ∑_j=0^rnormD_jf_π^2 and norm f_π^2 = ∑_j=0^rnormD_jf_π^2 + normf-P_rf_π^2.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally un","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8e4e2e93725a721e","tex":"\\rho_{m,n}","aliases":"ρ_m,n rho","display":"\\rho_{m,n}","kind":"use","excerpt":"[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally un","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-faf2d9e8110151a4","tex":"\\rho_t","aliases":"ρ_t rho","display":"\\rho_t","kind":"use","excerpt":"Write ρ_t= e^-t=1-t+O(t^2), σ_t^2=1- e^-2t=2t+O(t^2). For a polynomial f, Taylor expansion at x and Gaussian integration give O_tf(x) = · f(x) +f'(x)(ρ_t-1)x +frac12f''(x)σ_t^2 +o(t)","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"The self-energy Σ_P(z)=A_PQ(zI-A_QQ)^-1A_QP contains the exact effect of the eliminated packets. A bare truncation A_PP discards this term and is generally not exact.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"[Memory/self-energy correspondence] For finite complex block matrices, assume Rez>Reλ for every λ∈σ(A_QQ). Then the following improper integral is absolutely convergent and ∫_0^∞ e^-zt K(t) dt =A_PQ(zI-A_QQ)^-1A_QP =Σ_P(z).","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-65acde73901caecb","tex":"\\Sigma_P","aliases":"Σ_P Sigma","display":"\\Sigma_P","kind":"use","excerpt":"The self-energy Σ_P(z)=A_PQ(zI-A_QQ)^-1A_QP contains the exact effect of the eliminated packets. A bare truncation A_PP discards this term and is generally not exact.","count":10},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-87dfaf90b095de92","tex":"\\sigma_t^2","aliases":"σ_t^2 sigma","display":"\\sigma_t^2","kind":"use","excerpt":"Write ρ_t= e^-t=1-t+O(t^2), σ_t^2=1- e^-2t=2t+O(t^2). For a polynomial f, Taylor expansion at x and Gaussian integration give O_tf(x) = · f(x) +f'(x)(ρ_t-1)x +frac12f''(x)σ_t^2 +o(t)","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ff0c3a9746b729b1","tex":"\\tau^2","aliases":"τ^2 tau","display":"\\tau^2","kind":"use","excerpt":"Exact integration gives ∫_0^∞ f(t) dt=1, E[τ]=2, E[τ^2]=6, Var(τ)=2. Equivalently, the mean is ones^ T(-T)^-1e_0. A single exponential waiting time with mean two would have constant hazard 1/2 and variance four, so matching the mean alone would not reproduce this distribution.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7d039a149fedab64","tex":"\\tau^m","aliases":"τ^m tau","display":"\\tau^m","kind":"use","excerpt":"[Laplace transform and moments] Assume the spectrum of A lies in the open left half plane. Then absorption occurs with probability one. For s≥0, E[ e^-sτ] = a^T(sI-A)^-1p_0. For every integer m≥1, E[τ^m] = m! 1^T(-A)^-mp_0.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.","count":9},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"nentially, while the probabilities remain inside the simplex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the od","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-edee295b3bfd5e4d","tex":"\\phi_3","aliases":"ϕ_3 phi","display":"\\phi_3","kind":"use","excerpt":"id and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-acae14b2e159fc5e","tex":"\\phi_a","aliases":"ϕ_a phi","display":"\\phi_a","kind":"use","excerpt":"nentially, while the probabilities remain inside the simplex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the od","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8d53c39f51685b08","tex":"\\phi_{ab}","aliases":"ϕ_ab phi","display":"\\phi_{ab}","kind":"use","excerpt":"lex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Di","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ab83f0114b53e934","tex":"\\phi_b","aliases":"ϕ_b phi","display":"\\phi_b","kind":"use","excerpt":"lex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Di","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d0f231c57bed48c8","tex":"\\Phi_s","aliases":"Φ_s Phi","display":"\\Phi_s","kind":"use","excerpt":"[Bayesian nonlinear semigroup] Φ_s+t=Φ_s∘Φ_t. Its generator is the centered multiplicative field fracddtp_i(t) = (g_i-⟨ g,p(t)⟩)p_i(t).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-540cf3cfaf67581d","tex":"\\Phi_{s+t}","aliases":"Φ_s+t Phi","display":"\\Phi_{s+t}","kind":"use","excerpt":"[Bayesian nonlinear semigroup] Φ_s+t=Φ_s∘Φ_t. Its generator is the centered multiplicative field fracddtp_i(t) = (g_i-⟨ g,p(t)⟩)p_i(t).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-62d471c4d676af70","tex":"\\Phi_t","aliases":"Φ_t Phi","display":"\\Phi_t","kind":"use","excerpt":"Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-aa78eff0c8b0dd97","tex":"\\Phi_y","aliases":"Φ_y Phi","display":"\\Phi_y","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8f70978b22d0e201","tex":"\\Phi_{y_1}","aliases":"Φ_y_1 Phi","display":"\\Phi_{y_1}","kind":"use","excerpt":"[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-edb4f356b2cc314d","tex":"\\Phi_{y_r}","aliases":"Φ_y_r Phi","display":"\\Phi_{y_r}","kind":"use","excerpt":"[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a1d181fcbd5bda98","tex":"\\psi'","aliases":"ψ' psi","display":"\\psi'","kind":"use","excerpt":"[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-40a11436ef06a812","tex":"\\psi''","aliases":"ψ'' psi","display":"\\psi''","kind":"use","excerpt":"[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":13},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0f43487e004c9b12","tex":"\\omega_a","aliases":"ω_a omega","display":"\\omega_a","kind":"use","excerpt":"[Oscillation bound] Suppose overline L_a is any representative likelihood for block B_a, and set ω_a = max_i∈ B_aabsL_i-overline L_a. Then d(pmid L,overline L) ≤ ∑_aq_aω_a. Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4ae46887c6acc343","tex":"\\omega_c","aliases":"ω_c omega","display":"\\omega_c","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-097ee86d56b24dd4","tex":"\\omega_f","aliases":"ω_f omega","display":"\\omega_f","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1b0caddc55ff2af6","tex":"\\omega_Q","aliases":"ω_Q omega","display":"\\omega_Q","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":69},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"Sum the telescoping identities overline K^n-1-jCK^j+1 - overline K^n-jCK^j = overline K^n-1-jΔ_KK^j. Stochastic matrices are contractions in the ℓ^1 norm. The last statement uses total variation equal to one half of the ℓ^1 distance.","count":8},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8a05e97949811e78","tex":"\\ell_1","aliases":"ℓ_1 ell","display":"\\ell_1","kind":"use","excerpt":"[A visible likelihood loss] Let B_1={1,2}, let B_2={3}, and take p=(3/10,2/10,5/10), L=(1,3,2). The coarse prior is (1/2,1/2). The exact block likelihoods are ℓ_1(p)=frac(3/10)+3(2/10)1/2=frac95, ℓ_2(p)=2. Thus the exact coarse posterior is (9/19,10/19). If the first block is replaced by the fixed representative value 2, the approximate coarse post","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f71176b4c504c459","tex":"\\ell_2","aliases":"ℓ_2 ell","display":"\\ell_2","kind":"use","excerpt":"[A visible likelihood loss] Let B_1={1,2}, let B_2={3}, and take p=(3/10,2/10,5/10), L=(1,3,2). The coarse prior is (1/2,1/2). The exact block likelihoods are ℓ_1(p)=frac(3/10)+3(2/10)1/2=frac95, ℓ_2(p)=2. Thus the exact coarse posterior is (9/19,10/19). If the first block is replaced by the fixed representative value 2, the approximate coarse post","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bcd57746c1d3a7fc","tex":"\\ell_a","aliases":"ℓ_a ell","display":"\\ell_a","kind":"use","excerpt":"For an arbitrary prior p, put q=Cp and Z(p,L)=ipLp. Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0; choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a8e35b84dc016d32","tex":"\\ell_y","aliases":"ℓ_y ell","display":"\\ell_y","kind":"use","excerpt":"[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-06fd0b9eb73abd6b","tex":"\\ell_\\alpha","aliases":"ℓ_α ell alpha","display":"\\ell_\\alpha","kind":"use","excerpt":"ons are equivalent. item There is a positive coarse likelihood ℓ∈(0,∞)^m such that C B_λ(p) = B_ℓ(Cp) for every strictly positive prior p, and the posterior conditional distribution inside every packet equals its prior conditional distribution. item The likelihood is constant on every packet. When these conditions hold, ℓ_α can be chosen as the com","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-aa6ac22062ba0415","tex":"\\prod_{j=1}^r","aliases":"∏_j=1^r prod","display":"\\prod_{j=1}^r","kind":"use","excerpt":"_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0fcd1056f96dbe36","tex":"\\prod_r","aliases":"∏_r prod","display":"\\prod_r","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e178b785de8c3c65","tex":"\\prod_{r=0}^{2}","aliases":"∏_r=0^2 prod","display":"\\prod_{r=0}^{2}","kind":"use","excerpt":"1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEver","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ffde8e588cd66fb1","tex":"\\sum_{a=1}^m","aliases":"∑_a=1^m sum","display":"\\sum_{a=1}^m","kind":"use","excerpt":"Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0; choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overl","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","excerpt":"For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3afbe940a4042277","tex":"\\sum_{B\\in\\mathcal Q}","aliases":"∑_B∈ Q sum in mathcal","display":"\\sum_{B\\in\\mathcal Q}","kind":"use","excerpt":"After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block; both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1e4693631642f07b","tex":"\\sum_{c\\in\\{\\mathrm A,\\mathrm B\\}}","aliases":"∑_c∈{ A, B} sum in mathrm mathrm","display":"\\sum_{c\\in\\{\\mathrm A,\\mathrm B\\}}","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a54b8f882d9690fa","tex":"\\sum_{i\\in B}","aliases":"∑_i∈ B sum in","display":"\\sum_{i\\in B}","kind":"use","excerpt":"After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block; both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-60c3732b8a8548e8","tex":"\\sum_{i\\in B_a}","aliases":"∑_i∈ B_a sum in","display":"\\sum_{i\\in B_a}","kind":"use","excerpt":"se Markov generator. For observables, the backward generator is Q^T; weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overl","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-bd422e93fff6d0ac","tex":"\\sum_{i\\in I_a}","aliases":"∑_i∈ I_a sum in","display":"\\sum_{i\\in I_a}","kind":"use","excerpt":"Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b99656d8f6574465","tex":"\\sum_{i\\in M}","aliases":"∑_i∈ M sum in","display":"\\sum_{i\\in M}","kind":"use","excerpt":"[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.","count":16},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d7c6cf93ddec3543","tex":"\\sum_{j,k}","aliases":"∑_j,k sum","display":"\\sum_{j,k}","kind":"use","excerpt":"All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cd623dfebd879fcb","tex":"\\sum_{j=0}^2","aliases":"∑_j=0^2 sum","display":"\\sum_{j=0}^2","kind":"use","excerpt":"ck. vfigvariance platecaptionThe strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with we","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f7b37e8032ebe855","tex":"\\sum_{j=0}^{n-1}","aliases":"∑_j=0^n-1 sum","display":"\\sum_{j=0}^{n-1}","kind":"use","excerpt":"[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9098bd3c42e16d5a","tex":"\\sum_{j=0}^r","aliases":"∑_j=0^r sum","display":"\\sum_{j=0}^r","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-55bfd4be1daf726b","tex":"\\sum_{j=1}^{m}","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^{m}","kind":"use","excerpt":"[Exponential sum memory] If K(t)=∑_j=1^mB_j e^λ_jtC_j, then K has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in t and are realized by Jordan blocks.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-db50c77870a75686","tex":"\\sum_{j=1}^n","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^n","kind":"use","excerpt":"[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8650842fbc4c4680","tex":"\\sum_{j=1}^r","aliases":"∑_j=1^r sum","display":"\\sum_{j=1}^r","kind":"use","excerpt":"xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-369f27f600d1beec","tex":"\\sum_{j=n}^{\\infty}","aliases":"∑_j=n^∞ sum infty","display":"\\sum_{j=n}^{\\infty}","kind":"use","excerpt":"fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9fb4fab2abfee045","tex":"\\sum_{j=\\ell}^{m-1}","aliases":"∑_j=ℓ^m-1 sum ell","display":"\\sum_{j=\\ell}^{m-1}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Several level telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2f357ff52e6c2d2b","tex":"\\sum_{j>k}","aliases":"∑_j>k sum","display":"\\sum_{j>k}","kind":"use","excerpt":"normB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2656f76b906d00aa","tex":"\\sum_{j\\in B_b}","aliases":"∑_j∈ B_b sum in","display":"\\sum_{j\\in B_b}","kind":"use","excerpt":"Stationarity follows from overline Qoverlineπ =overline QCπ =CQπ=0. For reversibility, use the aggregate rate formula: overline Q_aboverlineπ_b · =∑_j∈ B_bπ_j∑_i∈ B_aQ_ij · =∑_i∈ B_a∑_j∈ B_bπ_iQ_ji · =overline Q_baoverlineπ_a.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c80d63770767b412","tex":"\\sum_{j\\notin J}","aliases":"∑_j∉ J sum notin","display":"\\sum_{j\\notin J}","kind":"use","excerpt":"j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete primary blocks indexed by J, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-89330e1a143912f2","tex":"\\sum_{j\\geq0}","aliases":"∑_j≥0 sum geq","display":"\\sum_{j\\geq0}","kind":"use","excerpt":"[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6c601ce784f1ba10","tex":"\\sum_{k=0}^{d-1}","aliases":"∑_k=0^d-1 sum","display":"\\sum_{k=0}^{d-1}","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e39c4713a93500ec","tex":"\\sum_{k=0}^{\\nu_j-1}","aliases":"∑_k=0^ν_j-1 sum nu","display":"\\sum_{k=0}^{\\nu_j-1}","kind":"use","excerpt":"xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f1c2ff812e2b7828","tex":"\\sum_{k=0}^{\\infty}","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^{\\infty}","kind":"use","excerpt":"For large z, Σ_P(z) = ∑_k=0^∞ fracA_PQA_QQ^kA_QPz^k+1. Thus the second condition implies the moment identities. Conversely, the first d identities imply all later ones by the Cayley Hamilton theorem, so the series vanishes.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-655f0e79a95453cc","tex":"\\sum_{m,n\\ge0}","aliases":"∑_m,n≥0 sum ge","display":"\\sum_{m,n\\ge0}","kind":"use","excerpt":"Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-b67bd1ec346d4fa3","tex":"\\sum_{m=0}^\\infty","aliases":"∑_m=0^∞ sum infty","display":"\\sum_{m=0}^\\infty","kind":"use","excerpt":"[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-187f645f69d06fb0","tex":"\\sum_{m=0}^{\\infty}","aliases":"∑_m=0^∞ sum infty","display":"\\sum_{m=0}^{\\infty}","kind":"use","excerpt":"This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=lo","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-22a81cbac9128cfb","tex":"\\sum_{m=1}^{\\infty}","aliases":"∑_m=1^∞ sum infty","display":"\\sum_{m=1}^{\\infty}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","excerpt":"at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVETraditional Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-8773b980988ccc84","tex":"\\sum_{n=0}^\\infty","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^\\infty","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0cc15179080d69b2","tex":"\\sum_{n=m}^{\\infty}","aliases":"∑_n=m^∞ sum infty","display":"\\sum_{n=m}^{\\infty}","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d009dfbbc12565cd","tex":"\\sum_{n\\le N}","aliases":"∑_n≤ N sum le","display":"\\sum_{n\\le N}","kind":"use","excerpt":"dotted,line width=.9pt] plot coordinates (1.33125,1.107143) (1.8625,1.221429) (2.39375,1.272222) (2.925,1.300794) (3.45625,1.319079) (3.9875,1.331778) (4.51875,1.341107) (5.05,1.34825) (5.58125,1.353894) (6.1125,1.358465) (6.64375,1.362243) (7.175,1.365418) (7.70625,1.368123) (8.2375,1.370455) (8.76875,1.372487) (9.3,1.374273); node[font=small,alig","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"use","excerpt":"For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-f0d1bc9b91545210","tex":"\\sum_{n\\geq1}","aliases":"∑_n≥1 sum geq","display":"\\sum_{n\\geq1}","kind":"use","excerpt":"A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2753495994955ca2","tex":"\\sum_{r=0}^{n-1}","aliases":"∑_r=0^n-1 sum","display":"\\sum_{r=0}^{n-1}","kind":"use","excerpt":"[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-54d809b1c239a990","tex":"\\sum_{r=1}^q","aliases":"∑_r=1^q sum","display":"\\sum_{r=1}^q","kind":"use","excerpt":"[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cb3e4b5ba1637239","tex":"\\sum_{r\\in B_\\alpha}","aliases":"∑_r∈ B_α sum in alpha","display":"\\sum_{r\\in B_\\alpha}","kind":"use","excerpt":"[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6a4250883c4968da","tex":"\\sum_{r\\ge0}","aliases":"∑_r≥0 sum ge","display":"\\sum_{r\\ge0}","kind":"use","excerpt":"Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","excerpt":"Let P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For any operator B, define B_βα=P_β BP_α. Then B=∑_α,βB_βα. For the semigroup, write T_βα(t)=P_β e^tAP_α.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ae59b81f83f7356c","tex":"\\sum_{\\alpha,\\beta\\in I_0}","aliases":"∑_α,β∈ I_0 sum alpha beta in","display":"\\sum_{\\alpha,\\beta\\in I_0}","kind":"use","excerpt":"X). The general Laplace formula is [source reference]. [Packet convolution] For all s,t≥0 and all packets α,β, T_βα(s+t) = ∑_γ∈ I_0 T_βγ(s)T_γα(t). The original semigroup is reconstructed from its blocks by T(t)=∑_α,β∈ I_0T_βα(t).","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-46f38e126539df37","tex":"\\sum_{\\alpha:n_\\alpha>0}","aliases":"∑_α:n_α>0 sum alpha alpha","display":"\\sum_{\\alpha:n_\\alpha>0}","kind":"use","excerpt":"on decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the information retain","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2877d5985ed013b2","tex":"\\sum_{\\alpha:Z_\\alpha>0}","aliases":"∑_α:Z_α>0 sum alpha alpha","display":"\\sum_{\\alpha:Z_\\alpha>0}","kind":"use","excerpt":"[Bayesian packet reconstruction] The posterior is the disjoint packet synthesis B_λ(p) = ∑_α:Z_α>0 m_α q_α. The local shape q_α depends only on the prior and likelihood inside packet I_α. Coupling between different packets occurs through the normalization of the packet evidences.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-940305a61c0cbc27","tex":"\\sum_{\\alpha_1,\\ldots,\\alpha_{m-1}}","aliases":"∑_α_1,…,α_m-1 sum alpha ldots alpha","display":"\\sum_{\\alpha_1,\\ldots,\\alpha_{m-1}}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-cabedb544f677980","tex":"\\sum_{\\alpha\\in I}","aliases":"∑_α∈ I sum alpha in","display":"\\sum_{\\alpha\\in I}","kind":"use","excerpt":"Let P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For any operator B, define B_βα=P_β BP_α. Then B=∑_α,βB_βα. For the semigroup, write T_βα(t)=P_β e^tAP_α.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-dd5a8bdc2b931502","tex":"\\sum_\\beta","aliases":"∑_β sum beta","display":"\\sum_\\beta","kind":"use","excerpt":"Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-9d12cb53c45f8ea0","tex":"\\sum_\\gamma","aliases":"∑_γ sum gamma","display":"\\sum_\\gamma","kind":"use","excerpt":"Insert the resolution of the identity between the two semigroup factors: P_β e^(s+t)AP_α · =P_β e^sA(∑_γ P_γ)e^tAP_α · =∑_γ (P_β e^sAP_γ)(P_γ e^tAP_α).","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7e22c72e0c03bce6","tex":"\\sum_{\\gamma\\in I}","aliases":"∑_γ∈ I sum gamma in","display":"\\sum_{\\gamma\\in I}","kind":"use","excerpt":"[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-13a337a61f819973","tex":"\\sum_{\\gamma\\in I_0}","aliases":"∑_γ∈ I_0 sum gamma in","display":"\\sum_{\\gamma\\in I_0}","kind":"use","excerpt":"apply the integrand to each x∈ X and take its Bochner integral in X. It is not asserted to be a Bochner integral in the operator-norm space B(X). The general Laplace formula is [source reference]. [Packet convolution] For all s,t≥0 and all packets α,β, T_βα(s+t) = ∑_γ∈ I_0 T_βγ(s)T_γα(t). The original semigroup is reconstructed from its blocks by T","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e3116f8319e2684e","tex":"\\sum_\\ell","aliases":"∑_ℓ sum ell","display":"\\sum_\\ell","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a895599e52149aa4","tex":"\\sum_{\\ell=0}^{\\infty}","aliases":"∑_ℓ=0^∞ sum ell infty","display":"\\sum_{\\ell=0}^{\\infty}","kind":"use","excerpt":"[Contractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-2c6476408fd5933d","tex":"\\sum_{\\ell=n}^{\\infty}","aliases":"∑_ℓ=n^∞ sum ell infty","display":"\\sum_{\\ell=n}^{\\infty}","kind":"use","excerpt":"ontractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"[Bayesian nonlinear semigroup] Φ_s+t=Φ_s∘Φ_t. Its generator is the centered multiplicative field fracddtp_i(t) = (g_i-⟨ g,p(t)⟩)p_i(t).","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.","count":75},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-16662ae81e20f5e0","tex":"\\int_{0<s_1<\\cdots<s_m<t}","aliases":"∫_0<s_1<⋯<s_m<t int cdots","display":"\\int_{0<s_1<\\cdots<s_m<t}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","excerpt":"On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-051f75585a774ce8","tex":"\\int_0^r","aliases":"∫_0^r int","display":"\\int_0^r","kind":"use","excerpt":"For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-3b8f65fd54e2c3f7","tex":"\\int_0^s","aliases":"∫_0^s int","display":"\\int_0^s","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-a2972272b24e9c45","tex":"\\int_0^t","aliases":"∫_0^t int","display":"\\int_0^t","kind":"use","excerpt":"[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.","count":27},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":32},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ff862dd8e88aa975","tex":"\\int_0^{\\infty}","aliases":"∫_0^∞ int infty","display":"\\int_0^{\\infty}","kind":"use","excerpt":"[Packet resolvent convolution] Let z belong to the resolvent set of A. If Rez is larger than a growth bound for the semigroup, then P_β(zI-A)^-1P_α = ∫_0^∞ e^-ztT_βα(t) d t.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-1a91af32e2911325","tex":"\\int_r^\\infty","aliases":"∫_r^∞ int infty","display":"\\int_r^\\infty","kind":"use","excerpt":"For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7cf42d930c86bbab","tex":"\\int_\\RR","aliases":"∫_RR ∫_ R int RR int mathbb","display":"\\int_{\\mathbb R}","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":4},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-67e4672bfa77240e","tex":"\\int_\\Gamma","aliases":"∫_Γ int Gamma","display":"\\int_\\Gamma","kind":"use","excerpt":"elextended:thm:holomorphic_defect Let Γ be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that Γ lies in the common resolvent set. Thus its winding number is one about every spectral point. If g is holomorphic on a neighborhood of the clo","count":3},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-df556fd37e8a533e","tex":"\\sim_{\\mathcal L}","aliases":"∼_ L sim mathcal","display":"\\sim_{\\mathcal L}","kind":"use","excerpt":"[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every bl","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"[Reducing packets] If P_β AP_α=0 for every β≠α, then P_β e^tAP_α=0 (β≠α) for every t. Each packet evolves independently under P_α AP_α.","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"[Reducing packet criterion] The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.","count":7},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":33},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usu","count":36},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":".F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ","count":1},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"[Tower law and martingale packets] Let P_0,P_1,…,P_r come from increasingly fine partitions, so that Ran P_0⊆Ran P_1⊆⋯⊆Ran P_r. Define D_0=P_0, D_j=P_j-P_j-1 (1≤ j≤ r). Then P_iP_j=P_min(i,j), D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j. Every D_j is an orthogonal projector, and normP_rf_π^2 = ∑_j=0^rnormD_jf_π^2 and norm f_π^2 = ∑_j=0^rnormD_jf_π^2 + normf-P_r","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"Let X=P X⊕ Q X and write A= A_PP · A_PQ","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":12},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.","count":6},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-e6e5d047330a8a9f","tex":"\\rangle_\\pi","aliases":"⟩_π rangle pi","display":"\\rangle_\\pi","kind":"use","excerpt":"[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.","count":2},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"The common operator chain is seed⟶transfer⟶project ⟶condition⟶refine ⟶complete⟶reconstruct.","count":5},{"work":"paper-vi","url":"notation/paper-vi.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"Differentiate r⟼ e^-(s-r)B^αCe^-rA^α and integrate from 0 to s. Positive fractional semigroups are contractions; under the gap, the product of their two norm bounds is e^-sm^α.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"definition-passage","excerpt":"inition:001 A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics ","count":272},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":164},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":1076},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.","count":133},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":379},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":26},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0230e8bde40956b0","tex":"\\Acal","aliases":"Acal A mathcal","display":"{\\mathcal A}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.","count":12},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":85},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":133},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f673c03c77f9ac50","tex":"B^*","aliases":"B^*","display":"B^*","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":27},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-327abfc143ff067a","tex":"\\Bcal","aliases":"Bcal B mathcal","display":"{\\mathcal B}","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-69fe78d4f1aca040","tex":"\\Bcal_{ab}","aliases":"Bcal_ab B_ab Bcal mathcal","display":"{\\mathcal B}_{ab}","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":182},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bbc9ba3c4c4e2bc2","tex":"\\mathfrak c","aliases":"c mathfrak","display":"\\mathfrak c","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":65},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":25},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6db54b248f3844fc","tex":"C_b","aliases":"C_b","display":"C_b","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6f3853f42b762bab","tex":"C_c","aliases":"C_c","display":"C_c","kind":"definition-passage","excerpt":"[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotien","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f0a61d8b37df0b6e","tex":"C_{\\ell,r}","aliases":"C_ℓ,r ell","display":"C_{\\ell,r}","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":41},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-843721b3e84879d7","tex":"\\Ccal","aliases":"Ccal C mathcal","display":"{\\mathcal C}","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-77a4dd0e05e628fb","tex":"\\cH","aliases":"cH H mathcal","display":"{\\mathcal H}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":18},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-754793052329c278","tex":"\\cO","aliases":"cO O mathcal","display":"{\\mathcal O}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-baabcb8a65f0c64e","tex":"\\cR","aliases":"cR R mathcal","display":"{\\mathcal R}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":90},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":147},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bf9b9919bc261407","tex":"d_L","aliases":"d_L","display":"d_L","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-66af625e26b38735","tex":"\\Dcal","aliases":"Dcal D mathcal","display":"{\\mathcal D}","kind":"definition-passage","excerpt":"[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-367a7ddcc412cdcb","tex":"\\Dcal_{\\Pcal}","aliases":"Dcal_Pcal D_ P Dcal Pcal mathcal mathcal","display":"{\\mathcal D}_{{\\mathcal P}}","kind":"definition-passage","excerpt":"[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd d mathrm","display":"{\\,\\mathrm d}","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":30},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c964f73df39bd41c","tex":"\\dd^d","aliases":"dd^d d^d dd mathrm","display":"{\\,\\mathrm d}^d","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":129},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0dcf6a495ccbd5bd","tex":"\\e","aliases":"e mathrm","display":"{\\mathrm e}","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":48},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":65},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c3c5d66994f2ce29","tex":"\\e^{-tL}","aliases":"e^-tL e mathrm","display":"{\\mathrm e}^{-tL}","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8ceab38a8fc70e99","tex":"E_a^*","aliases":"E_a^*","display":"E_a^*","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":25},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9a603588606ce810","tex":"E_b","aliases":"E_b","display":"E_b","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":20},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End operatorname","display":"{\\operatorname{End}}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.","count":19},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":57},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotien","count":21},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-058e3b39ebf27e0f","tex":"f_{ab}^{\\phantom{ab}c}","aliases":"f_ab^phantomabc phantom","display":"f_{ab}^{\\phantom{ab}c}","kind":"definition-passage","excerpt":"[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotien","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-07d5ca93af84b975","tex":"F_P","aliases":"F_P","display":"F_P","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":12},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":22},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":28},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":229},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c3d50e4b12d0681c","tex":"H^*","aliases":"H^*","display":"H^*","kind":"definition-passage","excerpt":"inition:001 A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics ","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4c4425940c285435","tex":"H_\\ell","aliases":"H_ℓ ell","display":"H_\\ell","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f50c3221ec49a3fa","tex":"\\Hcal","aliases":"Hcal H mathcal","display":"{\\mathcal H}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.","count":62},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a40fa7259746e49d","tex":"\\Hcal_\\ell","aliases":"Hcal_ℓ H_ℓ Hcal ell mathcal ell","display":"{\\mathcal H}_\\ell","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-21d3cf9a3f360e8d","tex":"\\Hcal_{\\ell+1}","aliases":"Hcal_ℓ+1 H_ℓ+1 Hcal ell mathcal ell","display":"{\\mathcal H}_{\\ell+1}","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":187},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ad5f56fc779f28fd","tex":"I_{\\Ccal}","aliases":"I_Ccal I_ C Ccal mathcal","display":"I_{{\\mathcal C}}","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-88b326798e7011cf","tex":"\\ii","aliases":"ii i mathrm","display":"{\\mathrm i}","kind":"definition-passage","excerpt":"[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotien","count":69},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-761d17addb30f300","tex":"\\inf_{c\\in\\CC}","aliases":"inf_c∈CC inf_c∈ C inf in CC inf in mathbb","display":"\\inf_{c\\in{\\mathbb C}}","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":72},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":105},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":44},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":12},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-74868e115861830f","tex":"\\max_{a,b}","aliases":"max_a,b max","display":"\\max_{a,b}","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"definition-passage","excerpt":"[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.","count":85},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":45},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob operatorname","display":"{\\operatorname{Ob}}","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-346371b58a75b52b","tex":"\\ob","aliases":"ob o mathfrak","display":"{\\mathfrak o}","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-83563918b0565094","tex":"\\Ob_{ab}^{\\rm cl}","aliases":"Ob_ab^rm cl Ob rm operatorname rm","display":"{\\operatorname{Ob}}_{ab}^{\\rm cl}","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4fa7ee730522dfa6","tex":"\\Ob_P","aliases":"Ob_P Ob operatorname","display":"{\\operatorname{Ob}}_P","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a11605d68f9137ee","tex":"\\ob_P","aliases":"ob_P o_P ob mathfrak","display":"{\\mathfrak o}_P","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":512},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":97},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":47},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-daaae6f128c21312","tex":"P_{\\ell,a}","aliases":"P_ℓ,a ell","display":"P_{\\ell,a}","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-63aba31d0d66fc66","tex":"P_{\\ell,\\alpha}","aliases":"P_ℓ,α ell alpha","display":"P_{\\ell,\\alpha}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-813431acda8054c4","tex":"\\Pcal","aliases":"Pcal P mathcal","display":"{\\mathcal P}","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":20},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bcb34d30cfc37173","tex":"\\Pcal_{ab}","aliases":"Pcal_ab P_ab Pcal mathcal","display":"{\\mathcal P}_{ab}","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ff7eb8f0d00f446f","tex":"\\mathfrak Q","aliases":"Q mathfrak","display":"\\mathfrak Q","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.","count":52},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-32b32661a7d9b468","tex":"q_{\\mathfrak c}","aliases":"q_ c mathfrak","display":"q_{\\mathfrak c}","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0db94dc2cdc9bd9f","tex":"q_P","aliases":"q_P","display":"q_P","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":57},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":127},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-81ae26f5145c949e","tex":"R_\\ell","aliases":"R_ℓ ell","display":"R_\\ell","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":13},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":39},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.","count":88},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d7305573ca2e697d","tex":"S^*","aliases":"S^*","display":"S^*","kind":"definition-passage","excerpt":"[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.","count":19},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-327dcbd8f0058da9","tex":"\\Spec","aliases":"Spec operatorname","display":"{\\operatorname{Spec}}","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":14},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":92},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":229},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-35899e3c99220253","tex":"T_\\ell","aliases":"T_ℓ ell","display":"T_\\ell","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr operatorname","display":"{\\operatorname{Tr}}","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":37},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":125},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":106},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":41},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":100},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-06faf04153e7762d","tex":"Z_L","aliases":"Z_L","display":"Z_L","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ Z mathbb","display":"{\\mathbb Z}","kind":"definition-passage","excerpt":"[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":25},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f8d84b94d64038ea","tex":"\\Gamma_{\\rm eff}","aliases":"Γ_rm eff Gamma rm","display":"\\Gamma_{\\rm eff}","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":33},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4de839f8cd0339b6","tex":"\\varepsilon_{\\rm KL}","aliases":"ε_rm KL varepsilon rm","display":"\\varepsilon_{\\rm KL}","kind":"definition-passage","excerpt":"[Approximate Knill Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":62},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-57d6bd6085b620be","tex":"\\lambda_N","aliases":"λ_N lambda","display":"\\lambda_N","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":53},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":20},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":24},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":107},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c1b88ae641d78c35","tex":"\\psi_0","aliases":"ψ_0 psi","display":"\\psi_0","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":15},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"definition-passage","excerpt":"[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.","count":21},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e99b5ac7e75a7dbb","tex":"\\omega^q","aliases":"ω^q omega","display":"\\omega^q","kind":"definition-passage","excerpt":"[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":31},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":52},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":89},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.","count":187},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"definition-passage","excerpt":"[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.","count":63},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3004d1c3c6d1b821","tex":"\\sum_c","aliases":"∑_c sum","display":"\\sum_c","kind":"definition-passage","excerpt":"[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotien","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":14},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":23},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e0c9947b1daf0522","tex":"A^*","aliases":"A^*","display":"A^*","kind":"introduction-passage","excerpt":"vpage2The objects on the pageAN ATLAS OF TYPES AND CONVENTIONSTraditional edition: Definition II.2.1, Definition II.7.1. The same letter can stand for different kinds of data in different calculations. Keep the following types fixed while reading a formula. In this appendix, vectors are columns, operators compose from right to left, and A^* denotes","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d159e41442deae07","tex":"A_+","aliases":"A_+","display":"A_+","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1a4c2e9cb4075d24","tex":"B_a","aliases":"B_a","display":"B_a","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathbb","display":"{\\mathbb C}","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-97c15171699972db","tex":"C^{\\mathsf T}","aliases":"C^ T mathsf","display":"C^{\\mathsf T}","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"introduction-passage","excerpt":"al trace and its support The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugat","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2e5a5c19cc2b86c8","tex":"\\CC^2","aliases":"CC^2 C^2 CC mathbb","display":"{\\mathbb C}^2","kind":"introduction-passage","excerpt":"al values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d6aace40a7406a32","tex":"\\operatorname{CNOT}","aliases":"CNOT operatorname","display":"\\operatorname{CNOT}","kind":"introduction-passage","excerpt":"[Bell seed circuit] In this example only, H denotes the Hadamard gate, not a Hamiltonian. Starting from ket00, (H⊗ I)ket00 = fracket00+ket10sqrt2, and then CNOT(H⊗ I)ket00 = fracket00+ket11sqrt2. The parity packets show that the final state lies entirely in the even sector, while the intermediate circuit path passes through both even and odd addres","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b5732f6e6e8849f6","tex":"D_{[X,Y]}","aliases":"D_[X,Y]","display":"D_{[X,Y]}","kind":"introduction-passage","excerpt":"textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":16},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f1be8e49b81c9b1e","tex":"D_M","aliases":"D_M","display":"D_M","kind":"introduction-passage","excerpt":"A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fdf9532371a315b7","tex":"D_X","aliases":"D_X","display":"D_X","kind":"introduction-passage","excerpt":"textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6f78b62eb28153fc","tex":"D_Y","aliases":"D_Y","display":"D_Y","kind":"introduction-passage","excerpt":"textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-24671e7796ade104","tex":"d_\\lambda","aliases":"d_λ lambda","display":"d_\\lambda","kind":"introduction-passage","excerpt":"Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"introduction-passage","excerpt":"ermines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integ","count":27},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d1997e4b5359f675","tex":"\\dot U","aliases":"dot U dot","display":"\\dot U","kind":"introduction-passage","excerpt":"Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.","count":16},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-381aa4b32306117e","tex":"\\dot\\psi","aliases":"dotψ dot psi","display":"\\dot\\psi","kind":"introduction-passage","excerpt":"Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-15b45515a5b956c1","tex":"e^{-H}","aliases":"e^-H","display":"e^{-H}","kind":"introduction-passage","excerpt":"vpage36A positive transfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on posi","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7b148da2b6a743d1","tex":"e^{-H'}","aliases":"e^-H'","display":"e^{-H'}","kind":"introduction-passage","excerpt":"integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every f","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5a1e1b0a1da34e35","tex":"e^{-i(t-s)H}","aliases":"e^-i(t-s)H","display":"e^{-i(t-s)H}","kind":"introduction-passage","excerpt":"ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors h","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-390457a95f8f6d57","tex":"e^{-isH}","aliases":"e^-isH","display":"e^{-isH}","kind":"introduction-passage","excerpt":"ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors h","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2637b6dd9fae991f","tex":"e^{-itH}","aliases":"e^-itH","display":"e^{-itH}","kind":"introduction-passage","excerpt":"ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors h","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-925048ed58d84367","tex":"e^{-n\\Delta}","aliases":"e^-nΔ Delta","display":"e^{-n\\Delta}","kind":"introduction-passage","excerpt":"HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly i","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d9b6193b1833f637","tex":"E_a","aliases":"E_a","display":"E_a","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":36},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e764f519ddd193e4","tex":"\\mathcal E_a","aliases":"E_a mathcal","display":"\\mathcal E_a","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e1248c74d891e3c5","tex":"\\mathcal E_x","aliases":"E_x mathcal","display":"\\mathcal E_x","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b8bdb1c02444ca91","tex":"H'","aliases":"H'","display":"H'","kind":"introduction-passage","excerpt":"positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.","count":92},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9d9060f8fe5d796b","tex":"I_A","aliases":"I_A","display":"I_A","kind":"introduction-passage","excerpt":"An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d22810f2cda3ca50","tex":"I_{m_i}","aliases":"I_m_i","display":"I_{m_i}","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":138},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-44819be810b8a1c8","tex":"\\ket","aliases":"ket","display":null,"kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":99},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":33},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":98},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e5a8e8a33dcbaa8c","tex":"M_{d_i}","aliases":"M_d_i","display":"M_{d_i}","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":16},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3d80123f93a0ffd9","tex":"\\Mcal","aliases":"Mcal M mathcal","display":"{\\mathcal M}","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e1ac5173895854fc","tex":"\\Mcal'","aliases":"Mcal' M' Mcal mathcal","display":"{\\mathcal M}'","kind":"introduction-passage","excerpt":"hannel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the c","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"ransfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral value","count":119},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5ae44c4d3d92ce4f","tex":"\\overline{\\psi_{i\\ell}}","aliases":"overlineψ_iℓ overline psi ell","display":"\\overline{\\psi_{i\\ell}}","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"introduction-passage","excerpt":"textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"introduction-passage","excerpt":"ermines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integ","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"introduction-passage","excerpt":"Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The es","count":16},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ed02c035995b1f57","tex":"P_\\alpha^*","aliases":"P_α^* alpha","display":"P_\\alpha^*","kind":"introduction-passage","excerpt":"Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The es","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-aad7607e608d0052","tex":"P_\\beta","aliases":"P_β beta","display":"P_\\beta","kind":"introduction-passage","excerpt":"Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The es","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":97},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d76d885e9e3c5ba5","tex":"Q^*","aliases":"Q^*","display":"Q^*","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":14},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank operatorname","display":"{\\operatorname{rank}}","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":17},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3bce33d4a711b894","tex":"\\Rcal","aliases":"Rcal R mathcal","display":"{\\mathcal R}","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-79c86107d4290f47","tex":"\\Rcal^*","aliases":"Rcal^* R^* Rcal mathcal","display":"{\\mathcal R}^*","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))","count":96},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0ef3dbe5a13282d7","tex":"\\srank","aliases":"srank operatorname","display":"{\\operatorname{srank}}","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fc8b82baaf7d8475","tex":"\\srank_{\\mathcal A}","aliases":"srank_ A srank mathcal operatorname mathcal","display":"{\\operatorname{srank}}_{\\mathcal A}","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":14},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6da3583735a2a7a0","tex":"T'","aliases":"T'","display":"T'","kind":"introduction-passage","excerpt":"lidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-910092ae6df8d19a","tex":"T^n","aliases":"T^n","display":"T^n","kind":"introduction-passage","excerpt":"ermines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integ","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":15},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-325816c77766b823","tex":"T_x","aliases":"T_x","display":"T_x","kind":"introduction-passage","excerpt":"A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"introduction-passage","excerpt":"Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.","count":104},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ec4379f6b5fb17bc","tex":"U_a","aliases":"U_a","display":"U_a","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d7884e8dc4c0191e","tex":"\\Ucal","aliases":"Ucal U mathcal","display":"{\\mathcal U}","kind":"introduction-passage","excerpt":"Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":60},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":51},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1b5f238ca9baf774","tex":"V_r","aliases":"V_r","display":"V_r","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5363f4b0a5b2b694","tex":"V_r^*","aliases":"V_r^*","display":"V_r^*","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":18},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"introduction-passage","excerpt":"eld and extend the projected connection to differential forms with values in the range bundle by ∇ω=P dω. Thus ∇^2 denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields. [Alternative proof of the curvature formula] For a section s with Ps=s, ∇^2s=P d(P ds)=P(dP)wedge ds. Sin","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4ae2c725a0324b8f","tex":"\\widehat G","aliases":"widehat G widehat","display":"\\widehat G","kind":"introduction-passage","excerpt":"Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-82ec90a8c96332be","tex":"\\widetilde{\\mathcal N}","aliases":"widetilde N widetilde mathcal","display":"\\widetilde{\\mathcal N}","kind":"introduction-passage","excerpt":"An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6bdb63d74dc95706","tex":"\\widetilde{\\mathcal N}_E","aliases":"widetilde N_E widetilde mathcal","display":"\\widetilde{\\mathcal N}_E","kind":"introduction-passage","excerpt":"An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-242de7ebea7742ad","tex":"\\widetilde{\\mathcal R}","aliases":"widetilde R widetilde mathcal","display":"\\widetilde{\\mathcal R}","kind":"introduction-passage","excerpt":"channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R wi","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-095fecaa10ffef82","tex":"X_1^{a_1}","aliases":"X_1^a_1","display":"X_1^{a_1}","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-da9130fcd50973a4","tex":"X_n^{a_n}","aliases":"X_n^a_n","display":"X_n^{a_n}","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))","count":23},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"lex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of pro","count":34},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"introduction-passage","excerpt":"Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The es","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"introduction-passage","excerpt":"I.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 det","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"introduction-passage","excerpt":"Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The es","count":43},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0fcb1c3cf5e3bf16","tex":"\\delta_{\\alpha\\beta}","aliases":"δ_αβ delta alpha beta","display":"\\delta_{\\alpha\\beta}","kind":"introduction-passage","excerpt":"Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The es","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":90},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c26fffae89b46f83","tex":"\\rho_B","aliases":"ρ_B rho","display":"\\rho_B","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":43},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-190c688268bfbad7","tex":"\\Phi^*","aliases":"Φ^* Phi","display":"\\Phi^*","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"introduction-passage","excerpt":"Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7b5b1f4eca28240d","tex":"\\chi_\\lambda","aliases":"χ_λ chi lambda","display":"\\chi_\\lambda","kind":"introduction-passage","excerpt":"Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-70309d8763112e95","tex":"\\psi_{ij}","aliases":"ψ_ij psi","display":"\\psi_{ij}","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a86ef458cc61e377","tex":"\\hbar","aliases":"ℏ hbar","display":"\\hbar","kind":"introduction-passage","excerpt":"of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivative","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-569331afbd5d2de3","tex":"\\nabla","aliases":"∇ nabla","display":"\\nabla","kind":"introduction-passage","excerpt":"A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.","count":20},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c7b13cbc252814fa","tex":"\\nabla^2","aliases":"∇^2 nabla","display":"\\nabla^2","kind":"introduction-passage","excerpt":"rms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior forms. Let P be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by ∇ω=P dω. Thus ∇^2 denotes the exterior covariant square, including the Lie bracket term in","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6c9566cb52be5641","tex":"\\nabla_X","aliases":"∇_X nabla","display":"\\nabla_X","kind":"introduction-passage","excerpt":"A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a1427dc433f3c26d","tex":"\\sum_{r\\in A_+}","aliases":"∑_r∈ A_+ sum in","display":"\\sum_{r\\in A_+}","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"introduction-passage","excerpt":"Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The es","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a2972272b24e9c45","tex":"\\int_0^t","aliases":"∫_0^t int","display":"\\int_0^t","kind":"introduction-passage","excerpt":"ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors h","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":39},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":33},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"introduction-passage","excerpt":"ransfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPTraditional edition: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral value","count":17},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":74},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":21},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3be3a4a3a44d636c","tex":"\\bm a","aliases":"a bm","display":"\\bm a","kind":"use","excerpt":"The Pauli identity ( a·σ)( b·σ) =( a· b)I + i( a× b)·σ gives - iTr(P[∂_θ P,∂_ϕ P]) =frac12 n· (∂_θ n×∂_ϕ n) =frac12sinθ. Integrating the displayed curvature over the sphere and applying the stated normalization gives one.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e0a61d58c33b3ed4","tex":"\\mathbf a","aliases":"a mathbf","display":"\\mathbf a","kind":"use","excerpt":"P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A is chosen. This explicit calc","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4b82e0794ad5cb35","tex":"\\mathfrak A","aliases":"A mathfrak","display":"\\mathfrak A","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d9d39a946bf223f1","tex":"\\mathfrak A^{(\\ell)}_{AB}","aliases":"A^(ℓ)_AB mathfrak ell","display":"\\mathfrak A^{(\\ell)}_{AB}","kind":"use","excerpt":"[Anomaly-free refinement] For a specified nontrivial quantum-geometric constraint model, there exists a coherent packet tower with nonzero physical spaces, compatible physical projectors and exact operator closure A^(ℓ)_AB=0 at every finite level. The compatible physical compression then vanishes as a consequence, not as a substitute for closure. T","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"ct worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i), so both sides are computed without evaluating trigonometric functions. This Gram test illustrates the factorization. The written identity proves positivity for all choices of times and vectors. headWhy this does not close to a bare local equation With hidden initia","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges con","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f2cfa673011ea148","tex":"a_3","aliases":"a_3","display":"a_3","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges con","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8a46d81e14e204be","tex":"\\mathcal A_{ab}","aliases":"A_ab mathcal","display":"\\mathcal A_{ab}","kind":"use","excerpt":"A constraint diagnostic that vanishes automatically. Section 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives C_aP=0 for every a. Absorb the factor i and any structure coefficient into f_c, and write A_ab=C_aC_b-C_bC_a-∑_cf_cC_c. [Exact common-kernel compression] If C_cP=0 for every","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-eb2b0edc94206176","tex":"\\mathfrak A_{AB}","aliases":"A_AB mathfrak","display":"\\mathfrak A_{AB}","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-efbc2c564bf50cd2","tex":"\\mathfrak A_{ab}","aliases":"A_ab mathfrak","display":"\\mathfrak A_{ab}","kind":"use","excerpt":"If structure coefficients f_ab^phantomabc are prescribed, define the fixed structure anomaly A_ab = [C_a,C_b] - i∑_cf_ab^phantomabcC_c. Exact realization of the prescribed algebra is the stronger condition A_ab=0.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-08eb65adb1a856b8","tex":"A_b","aliases":"A_b","display":"A_b","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-94cffe8dd0af28a5","tex":"A_{ba}","aliases":"A_ba","display":"A_{ba}","kind":"use","excerpt":"The elementary coordinate object is a complete family of orthogonal projectors P_aP_b=δ_abP_a, ∑_a∈ IP_a=I. Every vector has packet coordinates P_aψ. Every operator has transfer blocks A_ba=P_bAP_a. These identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction,","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite num","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"[Reflection positive correlators] Fix a vector Ω and operators A_1,…,A_m. The matrix C_jk = ipA_jΩT^n_j+n_kA_kΩ is positive semidefinite.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"0 · x≠ a. The seed is ψ_0=ket0. Define a translation operator by (T_a v)(x)=v(x-a). At position j, the local Pauli X is X_j=T_e_j, where e_j has a single nonzero binary coordinate. The formal circuit uses the identity when a_j=0 and X_j when a_j=1.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7d8e3b8668ae2ddd","tex":"A_k","aliases":"A_k","display":"A_k","kind":"use","excerpt":"[Reflection positive correlators] Fix a vector Ω and operators A_1,…,A_m. The matrix C_jk = ipA_jΩT^n_j+n_kA_kΩ is positive semidefinite.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9c5c098d009032a0","tex":"a_{m-1}","aliases":"a_m-1","display":"a_{m-1}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"ndomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=fr","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"one matrices, and pointwise convergence gives C(j,k)=1+(1/2)^j+k. For times (0,1,2) and coefficients (1,-2,1), ∑_j,kv_jC_n(j,k)v_k=(1-q_n)^4≥0. This exact check illustrates how finite quadratic forms pass positivity to the limit. xfigtrace_topology headOperator norm does not preserve trace in growing rank On ℓ^2, let P_n select the first n coordina","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-766daa38933a52e5","tex":"A_{ra}","aliases":"A_ra","display":"A_{ra}","kind":"use","excerpt":"Deriving the Gram matrix from recovery. Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-aa4ce98148303be9","tex":"A_{ra}^*","aliases":"A_ra^*","display":"A_{ra}^*","kind":"use","excerpt":"Deriving the Gram matrix from recovery Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormali","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4331faa8dd20c3ee","tex":"A_w^*","aliases":"A_w^*","display":"A_w^*","kind":"use","excerpt":"We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormali","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6af66e560dfcfb03","tex":"\\mathfrak A_{\\ell+1,ab}","aliases":"A_ℓ+1,ab mathfrak ell","display":"\\mathfrak A_{\\ell+1,ab}","kind":"use","excerpt":"[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-957494c7380f5756","tex":"\\mathfrak A_{\\ell,ab}","aliases":"A_ℓ,ab mathfrak ell","display":"\\mathfrak A_{\\ell,ab}","kind":"use","excerpt":"[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-133aeb6702a1cdfe","tex":"\\AA","aliases":"AA","display":null,"kind":"use","excerpt":"AA · Declared operator algebra · It specifies the available linear operations.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-483b6b4d696310c2","tex":"\\Acal'","aliases":"Acal' A' Acal mathcal","display":"{\\mathcal A}'","kind":"use","excerpt":"[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-239bae8ba3f949d4","tex":"\\Acal^*","aliases":"Acal^* A^* Acal mathcal","display":"{\\mathcal A}^*","kind":"use","excerpt":"Differentiate V^*V=I to obtain A^*=- A. The gauge transformation law follows by differentiating VG. Since P=VV^* and (I-P) d V= d P V, direct expansion gives d A+ Awedge A = d V^*(I-P)wedge d V = V^*( d Pwedge d P)V.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ff76bbf0fc00a0b5","tex":"\\Acal_{\\Pcal}","aliases":"Acal_Pcal A_ P Acal Pcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal P}}","kind":"use","excerpt":"[Dephasing as conditional expectation] The map D_ P is completely positive, trace preserving, unital, and idempotent. It is the Hilbert Schmidt orthogonal projection onto the block diagonal algebra A_ P = {X:[X,P_a]=0 for every a}. In particular, norm D_ P(X)_HS ≤ normX_HS.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-73fff7053fd2794b","tex":"\\Ad","aliases":"Ad operatorname","display":"{\\operatorname{Ad}}","kind":"use","excerpt":"Let S be unitary and N≥1 an integer with S^N=I, and define the conjugation action Ad_S(X)=SXS^*.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b718946e6494e709","tex":"\\Ad_S","aliases":"Ad_S Ad operatorname","display":"{\\operatorname{Ad}}_S","kind":"use","excerpt":"Let S be unitary and N≥1 an integer with S^N=I, and define the conjugation action Ad_S(X)=SXS^*.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d201a3b99225b0bb","tex":"\\Ad_{U(g)}","aliases":"Ad_U(g) Ad operatorname","display":"{\\operatorname{Ad}}_{U(g)}","kind":"use","excerpt":"Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3e51a60f00f5ac7f","tex":"\\Ad_{U(g)^*}","aliases":"Ad_U(g)^* Ad operatorname","display":"{\\operatorname{Ad}}_{U(g)^*}","kind":"use","excerpt":"Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-42648425616e1ec3","tex":"\\bm b","aliases":"b bm","display":"\\bm b","kind":"use","excerpt":"The Pauli identity ( a·σ)( b·σ) =( a· b)I + i( a× b)·σ gives - iTr(P[∂_θ P,∂_ϕ P]) =frac12 n· (∂_θ n×∂_ϕ n) =frac12sinθ. Integrating the displayed curvature over the sphere and applying the stated normalization gives one.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f76a26dbc0d9fe26","tex":"B^2","aliases":"B^2","display":"B^2","kind":"use","excerpt":"[Low energy expansion] Assume D is invertible and abs znormD^-1<1. Then Σ(z) =- ∑_n=0^∞ z^nBD^-n-1B^* with norm convergence. In particular, F_P(z) = z-A+BD^-1B^*+O(abs znorm B^2normD^-1^2).","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-82b9f73f21e567db","tex":"b_a^{\\phantom a c}","aliases":"b_a^phantom a c phantom","display":"b_a^{\\phantom a c}","kind":"use","excerpt":"[Invariance of the physical space] Let H=H^*. Suppose [H,C_a] = ∑_cb_a^phantom a cC_c for every a. Then H preserves H_phys, and so does e^- i tH for every real t.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"use","excerpt":"Expanding the finite product of sums gives T(A) T(B)= T(i↦ A_iB_i). Consequently, if two fine addresses differ in any coordinate, the corresponding product contains a zero factor. If the addresses agree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so Q_a^*=Q_a. Summing over all address functions","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-57c8f1ae1da23ae1","tex":"B_r","aliases":"B_r","display":"B_r","kind":"use","excerpt":"Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2a75a44be7515e9d","tex":"B_r^*","aliases":"B_r^*","display":"B_r^*","kind":"use","excerpt":"Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-026204dc005ea4b1","tex":"B_s","aliases":"B_s","display":"B_s","kind":"use","excerpt":"Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7e95a89f39649f47","tex":"\\bar b","aliases":"bar b bar","display":"\\bar b","kind":"use","excerpt":"ity. Starting from ket0, there are two resolved paths to output ket1. Their signed products are bra1H_rm Hadket0bra0H_rm Hadket0=frac12, bra1H_rm Hadket1bra1H_rm Hadket0=-frac12. Their sum is zero. Squaring each separately gives a different answer. vfiginterference headWhere the missing term goes For path amplitudes a,b, the outcome probability is ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-587654947ed7b96d","tex":"\\bbC","aliases":"bbC C mathbb","display":"{\\mathbb C}","kind":"use","excerpt":"Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty blo","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6881377bbebf9981","tex":"\\bbC^d","aliases":"bbC^d C^d bbC mathbb","display":"{\\mathbb C}^d","kind":"use","excerpt":"[Orbit dimension of one quantum seed] For integers d≥1 and m≥0, consider the action of M_d( C)⊗ I_m on C^d⊗ C^m. Let C be the coefficient matrix of a vector ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} =d rank C. The seed is cyclic if and only if rank C=m, which is possible only when m≤ d.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f89e172ea2b90718","tex":"\\bbC^{d_i}","aliases":"bbC^d_i C^d_i bbC mathbb","display":"{\\mathbb C}^{d_i}","kind":"use","excerpt":"Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty blo","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-63ccc9d44768142b","tex":"\\bbC^m","aliases":"bbC^m C^m bbC mathbb","display":"{\\mathbb C}^m","kind":"use","excerpt":"[Orbit dimension of one quantum seed] For integers d≥1 and m≥0, consider the action of M_d( C)⊗ I_m on C^d⊗ C^m. Let C be the coefficient matrix of a vector ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} =d rank C. The seed is cyclic if and only if rank C=m, which is possible only when m≤ d.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ab6a7db12b8172fe","tex":"\\bbC^{m_i}","aliases":"bbC^m_i C^m_i bbC mathbb","display":"{\\mathbb C}^{m_i}","kind":"use","excerpt":"Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty blo","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a3ee3f8d33e160ca","tex":"\\bbR","aliases":"bbR R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"[Projected connection] Let M be a smooth finite-dimensional real manifold without boundary, let V be a finite-dimensional real normed vector space, and let P:M→End_ R(V) be smooth with P(x)^2=P(x). In the quantum application, V is the underlying real space of a finite complex Hilbert space and P is an orthogonal complex-linear projector.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e5365be76fcd3f4a","tex":"\\bbZ","aliases":"bbZ Z mathbb","display":"{\\mathbb Z}","kind":"use","excerpt":"[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-43fa8bad7cce3005","tex":"\\bra","aliases":"bra","display":null,"kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. Thi","count":21},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"Three exact finite models. addcontentslinetocsectionThree exact finite models A seed remembers multiplicity support. For M_2( C)⊗ I_3, a coefficient matrix with rank one generates an orbit of dimension two. A matrix of rank two generates an orbit of dimension four. Neither single seed spans the full space of dimension six, because a 2×3 matrix cann","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4a41d044c4883dce","tex":"c^*","aliases":"c^*","display":"c^*","kind":"use","excerpt":"For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","excerpt":"Actual projected differentiation and curvature. Let E,F be real normed spaces. Let P:E→ L(F,F) be differentiable with P(x)^2=P(x) for every x. Let s:E→ F be C^2 with P(x)s(x)=s(x), and let X,Y:E→ E be differentiable vector fields. The checked connection is the actual expression ∇_Xs(x)=P(x) Ds(x)[X(x)]. It is proved to preserve the range condition.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e2677238ec3f2ee1","tex":"\\mathbb C^2","aliases":"C^2 mathbb","display":"\\mathbb C^2","kind":"use","excerpt":"model uses two coordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus fam","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-06e2221a9a4e2168","tex":"\\C^n","aliases":"C^n C mathbb","display":"{\\mathbb C}^n","kind":"use","excerpt":"The precise finite channel statement. Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0 0 · 1 ","count":14},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0 0 · 1 ","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"use","excerpt":"For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-73850e5fc882aa8c","tex":"C_{ab}","aliases":"C_ab","display":"C_{ab}","kind":"use","excerpt":"Necessary and sufficient conditions. Under the stated normalization, the following equivalence is proved: ∃ R of the above finite Kraus form, [-2pt] R N_E(ρ)=ρ for every ρsucceq0, Trρ=1, Pρ P=ρ Longleftrightarrow ∃ Csucceq0, PE_a^*E_bP=C_abP for all a,b. The statement applies throughout the code space. The formalization separately proves equivalenc","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-89de6f3841051f82","tex":"c_{ab}","aliases":"c_ab","display":"c_{ab}","kind":"use","excerpt":"vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATATraditional edition: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0 12 · 16 · 0 0 · 0 · ","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0ef49bcadebb010b","tex":"C_{ba}","aliases":"C_ba","display":"C_{ba}","kind":"use","excerpt":"Cross terms for the original errors. For arbitrary positive Gram data, the same recovery satisfies R(E_aρ E_b^*)=C_baρ (Pρ P=ρ). The index order ba is intentional: the defining Gram relation is PE_a^*E_bP=C_abP. The proof first derives scalar corrected Kraus composites for the constructed recovery and then uses its trace-preserving completeness to ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"lus_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-798b72ae3ed7ccde","tex":"C_{ij}","aliases":"C_ij","display":"C_{ij}","kind":"use","excerpt":"Schmidt rank and the reduced state. Use coordinates ψ_ij in the standard product basis and let C_ij=ψ_ij. An explicit linear equivalence identifies the state vector on the product index set with its coefficient matrix. The implementation checks the actual Kronecker action: coeff((A⊗ I_B)ψ)=AC. It also proves that the transported matrix orbit equals","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c7dd170465f9a276","tex":"c_{ja}","aliases":"c_ja","display":"c_{ja}","kind":"use","excerpt":"Every normalized channel in the same span. Suppose F_j=∑_a c_jaE_a and ∑_jF_j^*F_j=I. The decoder is not reconstructed for the F_j. Summing the coherent identities gives a scalar multiple qρ. Trace preservation of both the F channel and the recovery forces q=1 on a nonzero code. The proof uses positivity of P to justify cancellation of its trace an","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8f544215eb08c419","tex":"C_{jk}","aliases":"C_jk","display":"C_{jk}","kind":"use","excerpt":"[Reflection positive correlators] Fix a vector Ω and operators A_1,…,A_m. The matrix C_jk = ipA_jΩT^n_j+n_kA_kΩ is positive semidefinite.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and prov","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-254051ed849e0a84","tex":"c_k","aliases":"c_k","display":"c_k","kind":"use","excerpt":"[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"Let C_1,…,C_m be self adjoint constraint operators on a finite dimensional Hilbert space. The distinction between constraint action and constraint closure is standard in constrained Hamiltonian theory [source reference]. Write c=span_ C{C_1,…,C_m} ⊆End( H).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cfbf41dac773cbfc","tex":"C_n","aliases":"C_n","display":"C_n","kind":"use","excerpt":"vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPETraditional edition: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e949da172302ad25","tex":"C_{n,a}","aliases":"C_n,a","display":"C_{n,a}","kind":"use","excerpt":"[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-707f43f0f354949a","tex":"C_{n,b}","aliases":"C_n,b","display":"C_{n,b}","kind":"use","excerpt":"[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9417d8ef957ab062","tex":"C_{n,c}","aliases":"C_n,c","display":"C_{n,c}","kind":"use","excerpt":"[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-746a0fca63346846","tex":"C_r","aliases":"C_r","display":"C_r","kind":"use","excerpt":"Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4b60e0ee83a3abdf","tex":"C_r^*","aliases":"C_r^*","display":"C_r^*","kind":"use","excerpt":"The fine and coarse families are complete orthogonal self-adjoint projector systems. In particular, C_rC_s=δ_rsC_r, ∑_r C_r=I, C_r^*=C_r. Each coarse projector is exactly the displayed sum of its fine descendants.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-307d2c7e46d5ba73","tex":"C_s","aliases":"C_s","display":"C_s","kind":"use","excerpt":"The fine and coarse families are complete orthogonal self-adjoint projector systems. In particular, C_rC_s=δ_rsC_r, ∑_r C_r=I, C_r^*=C_r. Each coarse projector is exactly the displayed sum of its fine descendants.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-74d610b5f945d7cc","tex":"C_T","aliases":"C_T","display":"C_T","kind":"use","excerpt":"em[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a6ff7660eca84d81","tex":"C_x","aliases":"C_x","display":"C_x","kind":"use","excerpt":"vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncom","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-087e2290da72a37c","tex":"C_y","aliases":"C_y","display":"C_y","kind":"use","excerpt":"vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncom","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-065fc2a65996d145","tex":"C_z","aliases":"C_z","display":"C_z","kind":"use","excerpt":"vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncom","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-029bccf361123f37","tex":"C_\\psi","aliases":"C_ψ psi","display":"C_\\psi","kind":"use","excerpt":"[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4ae0a9be277fd36e","tex":"C_\\ell","aliases":"C_ℓ ell","display":"C_\\ell","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e58ea848499cc339","tex":"C_{\\ell+1,a}","aliases":"C_ℓ+1,a ell","display":"C_{\\ell+1,a}","kind":"use","excerpt":"Let ( H_ℓ,C_ℓ,a) and ( H_ℓ+1,C_ℓ+1,a) be two levels, and let R_ℓ: H_ℓ⟶ H_ℓ+1 be an isometry.","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c04e3fb7a0f52049","tex":"C_{\\ell+1,b}","aliases":"C_ℓ+1,b ell","display":"C_{\\ell+1,b}","kind":"use","excerpt":"[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7561a308f43c77a6","tex":"C_{\\ell,a}","aliases":"C_ℓ,a ell","display":"C_{\\ell,a}","kind":"use","excerpt":"Let ( H_ℓ,C_ℓ,a) and ( H_ℓ+1,C_ℓ+1,a) be two levels, and let R_ℓ: H_ℓ⟶ H_ℓ+1 be an isometry.","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4b6d5c1a68e80f7a","tex":"C_{\\ell,b}","aliases":"C_ℓ,b ell","display":"C_{\\ell,b}","kind":"use","excerpt":"[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-82497562659dda68","tex":"\\cA","aliases":"cA A mathcal","display":"{\\mathcal A}","kind":"use","excerpt":"Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty blo","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-34805d5c6af82a29","tex":"\\cA_\\ell","aliases":"cA_ℓ A_ℓ cA ell mathcal ell","display":"{\\mathcal A}_\\ell","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b66848dfe6155c4c","tex":"\\cC","aliases":"cC C mathcal","display":"{\\mathcal C}","kind":"use","excerpt":"Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environmen","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9632328f158df6f3","tex":"\\CC^3","aliases":"CC^3 C^3 CC mathbb","display":"{\\mathbb C}^3","kind":"use","excerpt":"vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e8feeaa9714c49a4","tex":"\\CC^4","aliases":"CC^4 C^4 CC mathbb","display":"{\\mathbb C}^4","kind":"use","excerpt":"l is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the c","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3a7ba377119b57ac","tex":"\\CC^6","aliases":"CC^6 C^6 CC mathbb","display":"{\\mathbb C}^6","kind":"use","excerpt":"o groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines the required count: srank_AA(HH)=max_ilceil m_i/d_irceil. This is an invariant of the declared representation of the algebra, and depends on more than the dimension of the Hilbert space alone. If the available algebra were enlarged to the full","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dc12fa997357cc20","tex":"\\CC^8","aliases":"CC^8 C^8 CC mathbb","display":"{\\mathbb C}^8","kind":"use","excerpt":"2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'' The eight vectors are orthonormal, so their linear span ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a8326ea62121139c","tex":"\\CC^d","aliases":"CC^d C^d CC mathbb","display":"{\\mathbb C}^d","kind":"use","excerpt":"[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3f64ef532a7ecb69","tex":"\\CC^{d_r}","aliases":"CC^d_r C^d_r CC mathbb","display":"{\\mathbb C}^{d_r}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5008bfc15f104acb","tex":"\\CC^m","aliases":"CC^m C^m CC mathbb","display":"{\\mathbb C}^m","kind":"use","excerpt":"[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-81a71c2d2875e597","tex":"\\CC^{m_r}","aliases":"CC^m_r C^m_r CC mathbb","display":"{\\mathbb C}^{m_r}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cf090df8943be9f2","tex":"\\CC^n","aliases":"CC^n C^n CC mathbb","display":"{\\mathbb C}^n","kind":"use","excerpt":"vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hil","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b15738dd2f995ce3","tex":"\\CC^{n+1}","aliases":"CC^n+1 C^n+1 CC mathbb","display":"{\\mathbb C}^{n+1}","kind":"use","excerpt":"H_n=diag(0,1,1/2,…); node[draw,fill=black!7,align=center] (b) at (9.5,2.2) C^n+1","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e90243c9b0dc0328","tex":"\\CC^r","aliases":"CC^r C^r CC mathbb","display":"{\\mathbb C}^r","kind":"use","excerpt":"Let P(x) be a smooth rank r orthogonal projector over a parameter manifold. Choose a local isometry V(x): C^r⟶ H, V(x)V(x)^*=P(x).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bcdfb14f2ad4a51d","tex":"\\cD","aliases":"cD D mathcal","display":"{\\mathcal D}","kind":"use","excerpt":"[Block-diagonal measurement invariance] Let {P_α} be a complete packet family, and suppose every measurement effect is packet diagonal: E_k=∑_α P_α E_kP_α. Define the packet-dephased state D_P(ρ)=∑_α P_αρ P_α. Then Tr(E_kρ) = Tr(E_k D_P(ρ)) for every outcome k.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b8f165c768f13650","tex":"\\cD_P","aliases":"cD_P D_P cD mathcal","display":"{\\mathcal D}_P","kind":"use","excerpt":"[Block-diagonal measurement invariance] Let {P_α} be a complete packet family, and suppose every measurement effect is packet diagonal: E_k=∑_α P_α E_kP_α. Define the packet-dephased state D_P(ρ)=∑_α P_αρ P_α. Then Tr(E_kρ) = Tr(E_k D_P(ρ)) for every outcome k.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4c70d511b332d10b","tex":"\\cF","aliases":"cF F mathcal","display":"{\\mathcal F}","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b4e856a106ed7fa5","tex":"\\cF_{\\theta\\phi}","aliases":"cF_θϕ F_θϕ cF theta phi mathcal theta phi","display":"{\\mathcal F}_{\\theta\\phi}","kind":"use","excerpt":"I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A is chosen. This explicit calculation is not a separate entry among the 19 ite","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9b06e46899a29307","tex":"\\cF_{\\mu\\nu}","aliases":"cF_μν F_μν cF mu nu mathcal mu nu","display":"{\\mathcal F}_{\\mu\\nu}","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7e4db5255ab27d49","tex":"\\cH_A","aliases":"cH_A H_A cH mathcal","display":"{\\mathcal H}_A","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d8f829d75272be4e","tex":"\\cH_B","aliases":"cH_B H_B cH mathcal","display":"{\\mathcal H}_B","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3c5ec99016fb65f5","tex":"\\cH_\\ell","aliases":"cH_ℓ H_ℓ cH ell mathcal ell","display":"{\\mathcal H}_\\ell","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-03399e5b8114d3a5","tex":"\\cH_{\\ell+1}","aliases":"cH_ℓ+1 H_ℓ+1 cH ell mathcal ell","display":"{\\mathcal H}_{\\ell+1}","kind":"use","excerpt":"hfrak Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7a2d1ebc2d7c43f4","tex":"\\mathrm{CNOT}","aliases":"CNOT mathrm","display":"\\mathrm{CNOT}","kind":"use","excerpt":"le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the or","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-67c1622d52b0e019","tex":"\\mathrm{coarse}","aliases":"coarse mathrm","display":"\\mathrm{coarse}","kind":"use","excerpt":"[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-efd4823e0a402064","tex":"\\operatorname{coeff}","aliases":"coeff operatorname","display":"\\operatorname{coeff}","kind":"use","excerpt":"Schmidt rank and the reduced state. Use coordinates ψ_ij in the standard product basis and let C_ij=ψ_ij. An explicit linear equivalence identifies the state vector on the product index set with its coefficient matrix. The implementation checks the actual Kronecker action: coeff((A⊗ I_B)ψ)=AC. It also proves that the transported matrix orbit equals","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2b0d52c420b6644c","tex":"\\comm","aliases":"comm","display":null,"kind":"use","excerpt":"Define C=∑_aD_aP_a(t). Then C=frac12∑_acommD_aP_a(t), C^*=-C, and, for every b, commCP_b(t)=D_b.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"vpage11Quantify the amplitude that escapesCOMMUTATORS AND PROBABILITY BOUNDSTraditional edition: Theorem II.5.3, Corollary II.5.4. For a two-level system, choose H=ε X, P=ket0bra0, and Q=I-P. Direct exponentiation gives e^- i tH=cos(ε t)I- isin(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary b","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8db6134243ea03f4","tex":"\\cos^2","aliases":"cos^2 cos","display":"\\cos^2","kind":"use","excerpt":"E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the completion proof constructs T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). It equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates. vfigrange_atlas headAn exact example of the domain restriction For the rot","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7e8acadc439b10d5","tex":"D^*","aliases":"D^*","display":"D^*","kind":"use","excerpt":"[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","excerpt":"Sigma|=|1-frac8156|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings. headA series near zero energy and its exact remainder At z=1/2, |z| |D^-1|=1/4","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2340d1b58f61d6b3","tex":"D^{-n-1}","aliases":"D^-n-1","display":"D^{-n-1}","kind":"use","excerpt":"[Low energy expansion] Assume D is invertible and abs znormD^-1<1. Then Σ(z) =- ∑_n=0^∞ z^nBD^-n-1B^* with norm convergence. In particular, F_P(z) = z-A+BD^-1B^*+O(abs znorm B^2normD^-1^2).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","excerpt":"A finite spectral datum consists of a finite dimensional Hilbert space, a nonnegative self adjoint operator L, and any declared packet family. One may take L=D^2 for a finite Dirac type operator or a graph Laplacian. Related geometric viewpoints appear in [source reference].","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a41942edfed6dca3","tex":"d^4","aliases":"d^4","display":"d^4","kind":"use","excerpt":"[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7903c81010eebdd6","tex":"D_a","aliases":"D_a","display":"D_a","kind":"use","excerpt":"Let P_a(s) be a finite family in a normed real algebra with a continuous compatible adjoint. Assume actual derivatives D_a=dot P_a(t) exist and, for all parameter values under consideration, P_aP_b=δ_abP_a, ∑_aP_a=I, P_a^*=P_a. In the supplied Lean theorem, the curves are defined on R and the projector laws hold at every real parameter. The algebra","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0ec57d816513bae2","tex":"d_A","aliases":"d_A","display":"d_A","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f0d18c8f77ae3cfc","tex":"D_a^*","aliases":"D_a^*","display":"D_a^*","kind":"use","excerpt":"The derivative product rule and uniqueness of derivatives give D_aP_b+P_aD_b=δ_abD_a. Differentiating completeness and adjoint invariance gives ∑_aD_a=0, D_a^*=D_a. These are proved derivative consequences, not additional axioms supplied in place of differentiation.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4256ad8575b1b705","tex":"D_b","aliases":"D_b","display":"D_b","kind":"use","excerpt":"Define C=∑_aD_aP_a(t). Then C=frac12∑_acommD_aP_a(t), C^*=-C, and, for every b, commCP_b(t)=D_b.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"use","excerpt":"Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite num","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a50d188eafd60524","tex":"d_{ja}","aliases":"d_ja","display":"d_{ja}","kind":"use","excerpt":"ry vector orthogonal to ψ, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e4d72ff963f20855","tex":"d_{jb}","aliases":"d_jb","display":"d_{jb}","kind":"use","excerpt":"he left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recove","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e5b9b4eb292be27d","tex":"d_{L,a}","aliases":"d_L,a","display":"d_{L,a}","kind":"use","excerpt":"[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1a90b7893f66544d","tex":"d_{L_N}","aliases":"d_L_N","display":"d_{L_N}","kind":"use","excerpt":"[Rigorous intermediate dimension window] For every ε>0, there is a diffusion time t_ε>0 and an integer N_ε such that absd_L_N(t_ε)-d<ε for every N≥ N_ε.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-382f08d5d528afb0","tex":"d_N","aliases":"d_N","display":"d_N","kind":"use","excerpt":"vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSTraditional edition: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f57f52c8b5bbfd0b","tex":"D_{\\rm computational}","aliases":"D_rm computational rm","display":"D_{\\rm computational}","kind":"use","excerpt":"g Every effect in the computational basis E_a=ket abra a has the same expectation on both matrices, namely 1/2 at 00 and 11 and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture. headA measurement that resolves the difference Use the valid two-outcome measurement with","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7b0cdefde0582bce","tex":"D_{\\rm parity}","aliases":"D_rm parity rm","display":"D_{\\rm parity}","kind":"use","excerpt":"[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (2.25,4.65) 2ρ, also 2D_rm parity(ρ); path[draw=black!30,fill=black!8] (0.75,3.2) rectangle (1.5,3.95); node[font=small,align=center,font=small] at (1.125,3.575) 1; path[draw=black!30,fill=white] (","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dae7685d7af3e13a","tex":"d_{\\rm torus}","aliases":"d_rm torus rm","display":"d_{\\rm torus}","kind":"use","excerpt":"[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-de7bc3aee118c968","tex":"d_s","aliases":"d_s","display":"d_s","kind":"use","excerpt":"vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESTraditional edition: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^2","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ad145996f65200fc","tex":"d_{s,2}","aliases":"d_s,2","display":"d_{s,2}","kind":"use","excerpt":"For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-36bf332a8be83b7c","tex":"d_s^{(1)}","aliases":"d_s^(1)","display":"d_s^{(1)}","kind":"use","excerpt":"[Spectral dimension] The median intermediate plateaus were d_s^(1)=1.00062, d_s^(2)=2.03781. These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bc06aed33500255b","tex":"d_s^{(2)}","aliases":"d_s^(2)","display":"d_s^{(2)}","kind":"use","excerpt":"[Spectral dimension] The median intermediate plateaus were d_s^(1)=1.00062, d_s^(2)=2.03781. These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-baa2c2411282fdc8","tex":"D_z","aliases":"D_z","display":"D_z","kind":"use","excerpt":"-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-20011d3f9eb01a6a","tex":"D_z^{-1}","aliases":"D_z^-1","display":"D_z^{-1}","kind":"use","excerpt":"-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2e19eb6c907a8c6c","tex":"D_\\ell","aliases":"D_ℓ ell","display":"D_\\ell","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b22521a2f79f4516","tex":"D_{\\ell+1}","aliases":"D_ℓ+1 ell","display":"D_{\\ell+1}","kind":"use","excerpt":"A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective s","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-335e9e40ffcb4439","tex":"\\ddot x","aliases":"ddot x ddot","display":"\\ddot x","kind":"use","excerpt":"returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict x(t)=1. vfigmemory headEliminate exactly, not approximately Integrating the second equation with y(0)=0 gives y(t)=- i∫_0^tx(s) ds. Substitution produces dot x(t)=-∫_0^tx(s) ds. The memory kernel is the constant function one. Differentiatin","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4a17365eab67e442","tex":"\\deph","aliases":"deph D_P mathcal","display":"{\\mathcal D_P}","kind":"use","excerpt":"Measurements: source Theorem 7.1. For finite complex matrices, define packet dephasing by D_P(X)=∑_aP_aXP_a. The code proves idempotence and trace duality: D_P( D_P(X))= D_P(X), tr( D_P(E)ρ)=tr(E D_P(ρ)). Trace duality follows term by term from cyclicity: tr(P_aEP_aρ)=tr(EP_aρ P_a). Therefore an effect that is diagonal in the packets, D_P(E)=E, sat","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the completion proof constructs T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). It equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates. vfigrange_atlas headAn exact example of the domain restriction For the rot","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"ordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sq","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"ment changes the local spanSCHMIDT RANK AS ACCESSIBLE MULTIPLICITYTraditional edition: Theorem II.2.2, Corollary II.2.4. Write a bipartite vector as a coefficient matrix C, with the first subsystem indexing rows. Acting only on that subsystem transforms C to AC. Every resulting row remains in the row space of C, and every independent choice of such","count":13},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-95483dc6bdcb21fe","tex":"\\dim_\\C","aliases":"dim_C dim_ C dim C dim mathbb","display":"\\dim_{\\mathbb C}","kind":"use","excerpt":"A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-558c1dc971656de6","tex":"\\dist","aliases":"dist operatorname","display":"{\\operatorname{dist}}","kind":"use","excerpt":"[Uniform self energy bounds] Let δ=dist(z,Spec D)>0, g=norm B. Then normΣ(z)≤fracg^2δ and normΣ'(z)≤fracg^2δ^2. For real E,E' in an interval at distance at least δ from Spec D, normΣ(E)-Σ(E') ≤ fracg^2δ^2absE-E'.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-35679b068dabbae2","tex":"\\dot P","aliases":"dot P dot","display":"\\dot P","kind":"use","excerpt":"vpage13Differentiate the subspace itselfTHE CANONICAL PACKET CONNECTIONTraditional edition: Theorem II.6.1, Theorem II.6.2. Continue with P(t)=U(t)diag(1,0)U(t)^* and Q(t)=I-P(t). Differentiating P^2=P gives dot P P+Pdot P=dot P, Pdot P P=0. The second identity says that the derivative has no component entirely inside the current packet. It moves t","count":27},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e4b951c58979abc7","tex":"\\dot P_a","aliases":"dot P_a dot","display":"\\dot P_a","kind":"use","excerpt":"n For this complete pair of packets, define C=dot P P+dot Q Q. Exact calculation yields the constant matrix C=J, with C^*=-C and [C,P]=dot P. At t=0, the displayed small matrices allow these identities to be checked by direct multiplication. headFrom two packets to a complete family For differentiable orthogonal projectors P_a summing to the identi","count":16},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d0be1eb2be8fecda","tex":"\\dot P_b","aliases":"dot P_b dot","display":"\\dot P_b","kind":"use","excerpt":"[Canonical packet connection] Define C(t) = frac12∑_a[dot P_a(t),P_a(t)] = ∑_adot P_a(t)P_a(t). Then C(t)^*=-C(t) and [C(t),P_b(t)]=dot P_b(t) for every packet b.","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-db307e9343ea80e8","tex":"\\dot Q","aliases":"dot Q dot","display":"\\dot Q","kind":"use","excerpt":"ional edition: Theorem II.6.1, Theorem II.6.2. Continue with P(t)=U(t)diag(1,0)U(t)^* and Q(t)=I-P(t). Differentiating P^2=P gives dot P P+Pdot P=dot P, Pdot P P=0. The second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement. vfigconnection For this complete pair","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bddb44ee5b7358ac","tex":"\\dot U^*","aliases":"dot U^* dot","display":"\\dot U^*","kind":"use","excerpt":"The explicitly defined coordinate curve y(s)=U(s)^*ψ(s) is differentiable at t, with dot y=dot U^*ψ(t)+U(t)^*dotψ. It satisfies idot y=H_effy(t), H_eff=U(t)^*HU(t)-iU(t)^*dot U. The connection term -iU(t)^*dot U is Hermitian. If H^*=H, then H_eff^*=H_eff.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ae6eb7cbea0e8357","tex":"\\dot V","aliases":"dot V dot","display":"\\dot V","kind":"use","excerpt":"[Moving frame Hamiltonian] Let U(t) solve idot U(t)=H(t)U(t), U(0)=I, where H(t)=H(t)^* is continuous on the compact time interval under consideration. Set V(t)=W(t)^*U(t). Then idot V(t)=H_rm mov(t)V(t), where H_rm mov(t) = W(t)^*H(t)W(t)- i W(t)^*dot W(t) = W(t)^*(H(t)- i K(t))W(t). The moving frame Hamiltonian is self adjoint.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f84526501d95b0bb","tex":"\\dot W","aliases":"dot W dot","display":"\\dot W","kind":"use","excerpt":"The moving packet section treats time dependent projectors. The canonical generator K(t)=∑_adot P_a(t)P_a(t) is skew adjoint and transports every packet. In the moving frame, the physical Hamiltonian acquires the geometric term - i W(t)^*dot W(t). The leakage estimate must account for the motion of the packet itself. The physical coupling QHP is th","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a89ea0f13cf380df","tex":"\\dot W^*","aliases":"dot W^* dot","display":"\\dot W^*","kind":"use","excerpt":"Differentiate V=W^*U. Since dot W=KW and K^*=-K, dot W^*=-W^*K. Substitution gives the equation. The first term in H_rm mov is self adjoint. The operator - i W^*dot W is self adjoint because W^*dot W is skew adjoint.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f854ffb0431207da","tex":"\\dot x","aliases":"dot x dot","display":"\\dot x","kind":"use","excerpt":"vpage22The smaller system remembers the discarded oneAN EXACT QUANTUM MEMORY EQUATIONTraditional edition: Theorem II.9.8, Theorem II.9.11. Choose H=X, initial state (1,0)^T, and retain the first coordinate. The coupled equations and their solution are idot x=y, idot y=x, x(t)=cos t, y(t)=- isin t. The hidden probability returns periodically. Deleti","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-389bd177d57e5191","tex":"\\dot y","aliases":"dot y dot","display":"\\dot y","kind":"use","excerpt":"1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i - i · 0, idot y=H_rm framey. This matrix is Hermitian. Omitting it would predict a constant y, contradicting the displayed coordinates. In this example the norm of the omitted time derivative is exactly one, so the failure is not a small numerical residual.","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"It does not assert a uniform interval width from that argument, a moving time t_N with no extra estimates, or a dimension plateau at every scale. At fixed finite N, d_N(t)→0 as tdownarrow0 and as t→∞. By contrast, first taking N→∞ at positive t and then taking tdownarrow0 yields the continuum dimension. headWhere the infinite tail is controlled At ","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c7c26f6b3b085b5c","tex":"E'","aliases":"E'","display":"E'","kind":"use","excerpt":"[Uniform self energy bounds] Let δ=dist(z,Spec D)>0, g=norm B. Then normΣ(z)≤fracg^2δ and normΣ'(z)≤fracg^2δ^2. For real E,E' in an interval at distance at least δ from Spec D, normΣ(E)-Σ(E') ≤ fracg^2δ^2absE-E'.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3af4d37764747f9a","tex":"e^{-(t_a+t_b)H}","aliases":"e^-(t_a+t_b)H","display":"e^{-(t_a+t_b)H}","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-531a33389172d686","tex":"\\e^{-1/(4t)}","aliases":"e^-1/(4t) e mathrm","display":"{\\mathrm e}^{-1/(4t)}","kind":"use","excerpt":"[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-63148d416f1c6825","tex":"\\e^{-16t\\abs{k}^2}","aliases":"e^-16tabsk^2 e abs mathrm abs","display":null,"kind":"use","excerpt":"ixed integer mode, [source reference] gives λ_N(k)⟶4π^2absk^2. For centered modes, the inequality sin x≥frac2xπ, 0≤ x≤fracπ2, gives λ_N(k)≥16absk^2. Also λ_N(k)≤4π^2absk^2. Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by e^-16tabsk^2, and the moment terms by 4π^2absk^2 e^-16tabsk^2. Both ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bd5e14eee60bd93f","tex":"e^{-2\\ii t}","aliases":"e^-2ii t e^-2 i t ii mathrm","display":"e^{-2{\\mathrm i} t}","kind":"use","excerpt":"4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structu","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1c483edbc417590e","tex":"e^{-2it}","aliases":"e^-2it","display":"e^{-2it}","kind":"use","excerpt":"weight 16/25; draw[->] (3,1.4)--(5,0.6);draw[->] (9,1.4)--(7,0.6); node[draw,align=center] at (6,0.4) K(t)=(9/25)e^-it+(16/25)e^-2it;","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6332bad66e756810","tex":"e^{-2\\tau}","aliases":"e^-2τ tau","display":"e^{-2\\tau}","kind":"use","excerpt":"vpage23Positive transfer is not a stochastic matrixSPECTRAL POSITIVITY IS BASIS INDEPENDENTTraditional edition: Theorem II.10.1, Theorem II.10.3. For H=I-X, the two eigenvalues are zero and two. Its Euclidean transfer is T(τ)=e^-τ H =frac12mat1+q · 1-q 1-q · 1+q, q=e^-2τ. For τ≥0, the eigenvalues 1,q lie in (0,1], and multiplying two such matrices ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2bee6969f99fc91d","tex":"\\e^{-4\\pi^2t\\abs{k}^2}","aliases":"e^-4π^2tabsk^2 e pi abs mathrm pi abs","display":null,"kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1e211d3027eec7b6","tex":"e^{-4\\tau}","aliases":"e^-4τ tau","display":"e^{-4\\tau}","kind":"use","excerpt":"vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESTraditional edition: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^2","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3b89b0c1ab78e1b6","tex":"\\e^{-a(E_j-E_0)}","aliases":"e^-a(E_j-E_0) e mathrm","display":"{\\mathrm e}^{-a(E_j-E_0)}","kind":"use","excerpt":"Diagonalize H. Each eigenvalue of T is e^-a(E_j-E_0) and therefore lies in (0,1]. The power law follows from the functional calculus.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f846e2f3c2a6fe91","tex":"\\e^{-a(H-E_0I)}","aliases":"e^-a(H-E_0I) e mathrm","display":"{\\mathrm e}^{-a(H-E_0I)}","kind":"use","excerpt":"Let H=H^* on a nonzero finite dimensional Hilbert space and let E_0=minSpec H. For a lattice time step a>0, define T= e^-a(H-E_0I).","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f74ea1ccebeff413","tex":"\\e^{-\\abs{n}^2/(4t)}","aliases":"e^-absn^2/(4t) e abs mathrm abs","display":null,"kind":"use","excerpt":"[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6180d10aad8c0b5e","tex":"\\e^{-aH_T}","aliases":"e^-aH_T e mathrm","display":"{\\mathrm e}^{-aH_T}","kind":"use","excerpt":"[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6de606f9a0b14b66","tex":"e^{-H_n}","aliases":"e^-H_n","display":"e^{-H_n}","kind":"use","excerpt":"For transfers on two states T_n=diag(1,q_n)=e^-H_n, with H_n=diag(0,-log q_n) at unit time, the error in trace norm is 2^-(n+2) and the heat traces converge to 3/2. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat t","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cba54642d9ab8642","tex":"e^{-i(t-s)H_{QQ}}","aliases":"e^-i(t-s)H_QQ","display":"e^{-i(t-s)H_{QQ}}","kind":"use","excerpt":"idot y · =H_QPx+H_QQy. Solving the second equation gives y(t) =e^-itH_QQy(0) -i∫_0^t e^-i(t-s)H_QQH_QPx(s) ds. Substitution yields the exact Volterra equation idot x(t) = · H_PPx(t) +H_PQe^-itH_QQy(0)","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dc15714350459aa5","tex":"\\e^{-\\ii sH}","aliases":"e^-ii sH e^- i sH e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} sH}","kind":"use","excerpt":"The summability condition makes (H_n) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives e^- i tH_n- e^- i tH =- i∫_0^t e^- i(t-s)H_n(H_n-H) e^- i sH d s. Both unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamilt","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-60bbbe7f29c53648","tex":"\\e^{-\\ii sH_{\\rm d}}","aliases":"e^-ii sH_rm d e^- i sH_rm d e ii rm mathrm mathrm rm","display":"{\\mathrm e}^{-{\\mathrm i} sH_{\\rm d}}","kind":"use","excerpt":"Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d727f71a03d62d1e","tex":"e^{-\\ii t}","aliases":"e^-ii t e^- i t ii mathrm","display":"e^{-{\\mathrm i} t}","kind":"use","excerpt":"4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structu","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1b2c8b37cfa4c924","tex":"\\e^{-\\ii tA}","aliases":"e^-ii tA e^- i tA e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tA}","kind":"use","excerpt":"If B=0, the retained and hidden equations decouple. Conversely, suppose that every solution with y(0)=0 satisfies x(t)= e^- i tAx(0). The full block equation gives ddot x(0) = -(A^2+BB^*)x(0), while the declared local equation gives ddot x(0)=-A^2x(0). Since this holds for every retained initial vector, BB^*=0. Hence B=0.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-35f6d789137bc755","tex":"\\e^{-\\ii tD}","aliases":"e^-ii tD e^- i tD e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tD}","kind":"use","excerpt":"[Exact retained memory equation] The retained coordinate satisfies idot x(t) = · Ax(t)+B e^- i tDy(0) · - i∫_0^tK(t-s)x(s) d s, where K(t)=B e^- i tDB^*. For Res>0, ∫_0^∞ e^-stK(t) d t = B(s+ i D)^-1B^*.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4d9aa2691147f3d1","tex":"e^{-\\ii tD}","aliases":"e^-ii tD e^- i tD ii mathrm","display":"e^{-{\\mathrm i} tD}","kind":"use","excerpt":"A nonzero initial hidden vector also supplies the forcing term Be^- i tDy(0). It is additional data, not part of the convolution kernel. The retained checks cover the displayed kernel values, exact Gram factorization, and acceleration defect. The integral equation follows from the written elimination argument. takeawayMemory is positive definite th","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-72f088de936f5a2e","tex":"\\e^{-\\ii tE_r}","aliases":"e^-ii tE_r e^- i tE_r e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tE_r}","kind":"use","excerpt":"[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-53d7011fee16a3bf","tex":"\\e^{-\\ii tH}","aliases":"e^-ii tH e^- i tH e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tH}","kind":"use","excerpt":"[Exact invariant packets] The following conditions are equivalent: item The Hamiltonian commutes with each projector: [H,P_a]=0 for every a. item Transfer blocks between distinct packets vanish: P_bHP_a=0 whenever a≠ b. item Every packet P_a H is invariant under e^- i tH for every real t. When they hold, H=∑_aP_aHP_a and e^- i tH=∑_aP_a e^- i tHP_a","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d2d69dce8d687c13","tex":"e^{-\\ii tH}","aliases":"e^-ii tH e^- i tH ii mathrm","display":"e^{-{\\mathrm i} tH}","kind":"use","excerpt":"H · Self-adjoint Hamiltonian · It generates evolution in real time through e^- i tH.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-53be779234266f77","tex":"\\e^{-\\ii tH_n}","aliases":"e^-ii tH_n e^- i tH_n e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tH_n}","kind":"use","excerpt":"[Norm summable Hamiltonian tower] Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3c7c5aa9303ef0fa","tex":"e^{-\\ii tH_n}","aliases":"e^-ii tH_n e^- i tH_n ii mathrm","display":"e^{-{\\mathrm i} tH_n}","kind":"use","excerpt":"For the norm route, work instead on the single fixed space C^2 and set H_n=diag(0,1+2^-n), H=diag(0,1). Here normH_n+1-H_n=2^-(n+1) and normH_n-H=2^-n. The summable tail supplies an error bound for every n. For |t|≤ T, the paper's Duhamel estimate gives norme^- i tH_n-e^- i tH≤ T2^-n. The Euclidean counterpart holds on 0≤ t≤ T because both Hamilton","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ab9f45562016500f","tex":"\\e^{-\\ii tH_\\ell}","aliases":"e^-ii tH_ℓ e^- i tH_ℓ e ii ell mathrm mathrm ell","display":"{\\mathrm e}^{-{\\mathrm i} tH_\\ell}","kind":"use","excerpt":"[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-82432643f68232b1","tex":"\\e^{-\\ii tH_\\infty}","aliases":"e^-ii tH_∞ e^- i tH_∞ e ii infty mathrm mathrm infty","display":"{\\mathrm e}^{-{\\mathrm i} tH_\\infty}","kind":"use","excerpt":"[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-78c88d9fe415a0e0","tex":"\\e^{-\\ii(t-s)D}","aliases":"e^-ii(t-s)D e^- i(t-s)D e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i}(t-s)D}","kind":"use","excerpt":"The hidden equation is idot y=B^*x+Dy. Variation of constants gives y(t)= e^- i tDy(0) - i∫_0^t e^- i(t-s)DB^*x(s) d s. Substitute into idot x=Ax+By. The Laplace formula follows by diagonalizing D or integrating the norm convergent finite dimensional exponential.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-48f850fbfb58d911","tex":"\\e^{-\\ii(t-s)H}","aliases":"e^-ii(t-s)H e^- i(t-s)H e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i}(t-s)H}","kind":"use","excerpt":"Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4988b85314de55e4","tex":"\\e^{-\\ii(t-s)H_n}","aliases":"e^-ii(t-s)H_n e^- i(t-s)H_n e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i}(t-s)H_n}","kind":"use","excerpt":"The summability condition makes (H_n) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives e^- i tH_n- e^- i tH =- i∫_0^t e^- i(t-s)H_n(H_n-H) e^- i sH d s. Both unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamilt","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e5a49b0472ff2c1d","tex":"\\e^{-\\ii(t_j-t_k)D}","aliases":"e^-ii(t_j-t_k)D e^- i(t_j-t_k)D e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i}(t_j-t_k)D}","kind":"use","excerpt":"Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bbad7626b522cffb","tex":"e^{-\\ii\\pi L/2}","aliases":"e^-iiπ L/2 e^- iπ L/2 ii pi mathrm pi","display":"e^{-{\\mathrm i}\\pi L/2}","kind":"use","excerpt":"lculation of transfer to the opposite siteSEEDS, CHARACTERS, INTERFERENCE, AND DEPHASINGTraditional edition: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4. Return to the shift on four sites S and Laplacian L=2I-S-S^*. The same character projectors have energies (0,2,4,2). At time t=π/2, their evolution phases are (1,-1,1,-1), exactly the charact","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a01743648714134d","tex":"e^{-it}","aliases":"e^-it","display":"e^{-it}","kind":"use","excerpt":"weight 16/25; draw[->] (3,1.4)--(5,0.6);draw[->] (9,1.4)--(7,0.6); node[draw,align=center] at (6,0.4) K(t)=(9/25)e^-it+(16/25)e^-2it;","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-91b2f17ba96a755d","tex":"e^{-itH_{QQ}}","aliases":"e^-itH_QQ","display":"e^{-itH_{QQ}}","kind":"use","excerpt":"idot y · =H_QPx+H_QQy. Solving the second equation gives y(t) =e^-itH_QQy(0) -i∫_0^t e^-i(t-s)H_QQH_QPx(s) ds. Substitution yields the exact Volterra equation idot x(t) = · H_PPx(t) +H_PQe^-itH_QQy(0)","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c1f307a0a1887517","tex":"\\e^{-na(E_j-E_0)}","aliases":"e^-na(E_j-E_0) e mathrm","display":"{\\mathrm e}^{-na(E_j-E_0)}","kind":"use","excerpt":"In the spectral decomposition, T^n-P_0 has eigenvalues zero on the ground space and e^-na(E_j-E_0) on the excited eigenspaces. The largest is e^-naΔ.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c65f064466994a81","tex":"\\e^{-na(H-E_0I)}","aliases":"e^-na(H-E_0I) e mathrm","display":"{\\mathrm e}^{-na(H-E_0I)}","kind":"use","excerpt":"[Finite Euclidean transfer] The operator T is positive definite, self adjoint, and contractive. Its spectrum lies in (0,1], and T^n= e^-na(H-E_0I) for every n∈ N.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-23edb8b0043e53d7","tex":"\\e^{-naH_T}","aliases":"e^-naH_T e mathrm","display":"{\\mathrm e}^{-naH_T}","kind":"use","excerpt":"[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b6cf8132b360cc53","tex":"\\e^{-na\\Delta}","aliases":"e^-naΔ e Delta mathrm Delta","display":"{\\mathrm e}^{-na\\Delta}","kind":"use","excerpt":"[Ground packet convergence] Let P_0 be the spectral projector of H at E_0, and let Δ=min{E_j-E_0:E_j>E_0}. If an excited eigenvalue exists, then normT^n-P_0= e^-naΔ.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9209e63597cdde74","tex":"\\e^{-st}","aliases":"e^-st e mathrm","display":"{\\mathrm e}^{-st}","kind":"use","excerpt":"· - i∫_0^tK(t-s)x(s) d s, where K(t)=B e^- i tDB^*. For Res>0, ∫_0^∞ e^-stK(t) d t = B(s+ i D)^-1B^*.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e864a5dbfd8a4ab4","tex":"e^{-t(H-E_0I)}","aliases":"e^-t(H-E_0I)","display":"e^{-t(H-E_0I)}","kind":"use","excerpt":"For a family with nontrivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is r","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-be2259ac52650147","tex":"e^{-t(x-e)}","aliases":"e^-t(x-e)","display":"e^{-t(x-e)}","kind":"use","excerpt":"Connecting power series and spectral calculus. The shifted generator is proved Hermitian. The implementation then identifies the actual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5006b64dd94bbdc7","tex":"e^{-t_aH}","aliases":"e^-t_aH","display":"e^{-t_aH}","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-31ea130aac066aba","tex":"\\e^{-tH}","aliases":"e^-tH e mathrm","display":"{\\mathrm e}^{-tH}","kind":"use","excerpt":"Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-104acf20a9fbdb09","tex":"\\e^{-tH_n}","aliases":"e^-tH_n e mathrm","display":"{\\mathrm e}^{-tH_n}","kind":"use","excerpt":"Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6ab294202ef31b05","tex":"\\e^{-tH_\\ell}","aliases":"e^-tH_ℓ e ell mathrm ell","display":"{\\mathrm e}^{-tH_\\ell}","kind":"use","excerpt":"imit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3fa4ad68ee2f72f1","tex":"\\e^{-tH_\\infty}","aliases":"e^-tH_∞ e infty mathrm infty","display":"{\\mathrm e}^{-tH_\\infty}","kind":"use","excerpt":"imit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dc658fbfa94bd130","tex":"\\e^{-tL_n}","aliases":"e^-tL_n e mathrm","display":"{\\mathrm e}^{-tL_n}","kind":"use","excerpt":"[Heat trace convergence from trace norm] Let L_n,L be nonnegative self adjoint operators on one Hilbert space. For a fixed t>0, assume their heat operators are trace class and that e^-tL_n⟶ e^-tL in trace norm. Then Tr( e^-tL_n)⟶Tr( e^-tL). If also the heat moment operators are trace class and L_n e^-tL_n⟶ L e^-tL in trace norm and the limiting hea","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-95a8eb776d3e04a4","tex":"e^{-t\\lambda}","aliases":"e^-tλ lambda","display":"e^{-t\\lambda}","kind":"use","excerpt":"At the single time t=log2/16, two small grids admit exact rational heat weights: c|c|c|c|c N · spectrum · Z_N · ∑λ e^-tλ · d_N/log2 2 · (0,16) · 3/2 · 8 · 2/3 4 · (0,32,32,64) · 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe the","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5fbb88a28e074883","tex":"\\e^{-t\\lambda_N(k)}","aliases":"e^-tλ_N(k) e lambda mathrm lambda","display":"{\\mathrm e}^{-t\\lambda_N(k)}","kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-db6f29304e044ecc","tex":"e^{-t\\lambda_N(k)}","aliases":"e^-tλ_N(k) lambda","display":"e^{-t\\lambda_N(k)}","kind":"use","excerpt":"vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSTraditional edition: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-11f9580358f49756","tex":"e^{-\\tau H}","aliases":"e^-τ H tau","display":"e^{-\\tau H}","kind":"use","excerpt":"vpage23Positive transfer is not a stochastic matrixSPECTRAL POSITIVITY IS BASIS INDEPENDENTTraditional edition: Theorem II.10.1, Theorem II.10.3. For H=I-X, the two eigenvalues are zero and two. Its Euclidean transfer is T(τ)=e^-τ H =frac12mat1+q · 1-q 1-q · 1+q, q=e^-2τ. For τ≥0, the eigenvalues 1,q lie in (0,1], and multiplying two such matrices ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0218309a1770ad77","tex":"e^{-\\tau(E_j-E_0)}","aliases":"e^-τ(E_j-E_0) tau","display":"e^{-\\tau(E_j-E_0)}","kind":"use","excerpt":"Diagonalize H=∑_jE_jP_j. Then T(τ)=∑_j e^-τ(E_j-E_0)P_j. Every eigenvalue is strictly positive and at most one. Multiplication adds the exponents, proving the semigroup law and T(0)=I. Positivity here is operator positivity, not an assertion that every matrix entry is nonnegative.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-11707835147f7216","tex":"e^{-\\tau(H-E_0I)}","aliases":"e^-τ(H-E_0I) tau","display":"e^{-\\tau(H-E_0I)}","kind":"use","excerpt":"[Finite Euclidean transfer] Let H be Hermitian on a nonzero finite-dimensional Hilbert space. For all real τ,σ≥0, T(τ)=e^-τ(H-E_0I), E_0=minSpec H, is positive definite and satisfies T(τ+σ)=T(τ)T(σ). Its spectrum is contained in (0,1].","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-91b95fd30d4cf58a","tex":"e^{-\\tau\\lambda_j}","aliases":"e^-τλ_j tau lambda","display":"e^{-\\tau\\lambda_j}","kind":"use","excerpt":"Let L≥0 be a finite Laplacian on N≥1 states, with eigenvalues λ_j and at least one zero eigenvalue. For diffusion time τ>0, define the normalized heat trace K(τ)=frac1N∑_j e^-τλ_j and the scale-dependent spectral dimension d_s(τ) =-2fracdlog K(τ)dlogτ. For a finite system, d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differen","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d7e7084a35eacad9","tex":"e^{2\\pi i/N}","aliases":"e^2π i/N pi","display":"e^{2\\pi i/N}","kind":"use","excerpt":"Let N≥1 be an integer, and let S be a unitary symmetry whose order divides N: S^N=I. Set ω=e^2π i/N.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d7013ba8d76e4f86","tex":"\\e^{2\\pi\\ii/N}","aliases":"e^2πii/N e^2π i/N e pi ii mathrm pi mathrm","display":"{\\mathrm e}^{2\\pi{\\mathrm i}/N}","kind":"use","excerpt":"Let N≥1 be an integer, S unitary with S^N=I, and ω= e^2π i/N.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2159f21c196fa256","tex":"e^{2\\tau}","aliases":"e^2τ tau","display":"e^{2\\tau}","kind":"use","excerpt":"vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESTraditional edition: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^2","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-76a85d33638c39e7","tex":"\\e^{\\ii t_kD}","aliases":"e^ii t_kD e^ i t_kD e ii mathrm mathrm","display":"{\\mathrm e}^{{\\mathrm i} t_kD}","kind":"use","excerpt":"Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0fa91de62b7d1419","tex":"e^{\\ii t_kD}","aliases":"e^ii t_kD e^ i t_kD ii mathrm","display":"e^{{\\mathrm i} t_kD}","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-256b204813b53eee","tex":"e^{\\ii\\phi}","aliases":"e^iiϕ e^ iϕ ii phi mathrm phi","display":"e^{{\\mathrm i}\\phi}","kind":"use","excerpt":"n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. T","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ee4422570e8d67bf","tex":"e^{isH}","aliases":"e^isH","display":"e^{isH}","kind":"use","excerpt":"Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ccbd9d6591d0caa8","tex":"e^{itH}","aliases":"e^itH","display":"e^{itH}","kind":"use","excerpt":"Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0f2de95465372652","tex":"\\e^{t\\Gcal}","aliases":"e^tGcal e^t G e Gcal mathrm mathcal","display":"{\\mathrm e}^{t{\\mathcal G}}","kind":"use","excerpt":"a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3eefb925a82433a8","tex":"e^{zH}","aliases":"e^zH","display":"e^{zH}","kind":"use","excerpt":"Symmetry and measurement. Exact symmetry: source Theorem 5.1. If HP_a=P_aH for every packet, then P_bHP_a=0 (b≠ a), H=∑_aP_aHP_a. In a complex Banach algebra, commutation also propagates to the actual exponential: e^zHP_a=P_ae^zH (z∈ C). Consequently, e^zH=∑_aP_ae^zHP_a, P_be^zHP_a=0 (b≠ a). Taking z=-it gives evolution in real time. The Lean proof","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"For a family with nontrivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is r","count":17},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-521e7dfe4bcb4abb","tex":"E_0^*","aliases":"E_0^*","display":"E_0^*","kind":"use","excerpt":"A genuine channel can still be uncorrectable. For full qubit dephasing, E_0=diag(1,0) and E_1=diag(0,1), the Kraus normalization is checked. But E_0^*E_0 cannot be a scalar multiple of I, as its two diagonal entries differ. Necessity therefore proves that no finite recovery channel corrects the full code space. The case of the zero code is also che","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATATraditional edition: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f2b11b7b1856ea86","tex":"E_{12}","aliases":"E_12","display":"E_{12}","kind":"use","excerpt":"ty gives A_ABΠ^phys=0, Π^phys A_ABΠ^phys=0 automatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, C_1=diag(1,0,0) and C_2=E_12+E_21 annihilate the same third coo","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATATraditional edition: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-eebcf5feadee9573","tex":"E_{21}","aliases":"E_21","display":"E_{21}","kind":"use","excerpt":"ty gives A_ABΠ^phys=0, Π^phys A_ABΠ^phys=0 automatically. This compression therefore cannot certify operator closure away from the physical range. A nonzero measured compression contradicts the exact-kernel assumptions or indicates an approximation or implementation error. For example, C_1=diag(1,0,0) and C_2=E_12+E_21 annihilate the same third coo","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-82c2d872ef715d1f","tex":"E_3","aliases":"E_3","display":"E_3","kind":"use","excerpt":"vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATATraditional edition: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-238a2f0e8e975441","tex":"E_{ab}","aliases":"E_ab","display":"E_{ab}","kind":"use","excerpt":"For the exact calculation, apply the four matrix units E_ab to each seed matrix, flatten the outputs, and compute their column rank. The observed dimensions are two and four, respectively. The calculation directly exhibits the independent rows counted by the theorem. takeawaySchmidt rank measures the remote multiplicity directions accessible to the","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4a9df7c97d7c6847","tex":"E_b^*","aliases":"E_b^*","display":"E_b^*","kind":"use","excerpt":"Cross terms for the original errors. For arbitrary positive Gram data, the same recovery satisfies R(E_aρ E_b^*)=C_baρ (Pρ P=ρ). The index order ba is intentional: the defining Gram relation is PE_a^*E_bP=C_abP. The proof first derives scalar corrected Kraus composites for the constructed recovery and then uses its trace-preserving completeness to ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-52daa780ca76442f","tex":"E_{ii}^{\\lambda}","aliases":"E_ii^λ lambda","display":"E_{ii}^{\\lambda}","kind":"use","excerpt":"x an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-33c9d2aae34fdc7b","tex":"E_{ij}^{\\lambda}","aliases":"E_ij^λ lambda","display":"E_{ij}^{\\lambda}","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-16d60aa45461fc23","tex":"E_{i\\ell}^{\\lambda}","aliases":"E_iℓ^λ ell lambda","display":"E_{i\\ell}^{\\lambda}","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have the same relative weights, but their total ","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"0 · x≠ a. The seed is ψ_0=ket0. Define a translation operator by (T_a v)(x)=v(x-a). At position j, the local Pauli X is X_j=T_e_j, where e_j has a single nonzero binary coordinate. The formal circuit uses the identity when a_j=0 and X_j when a_j=1.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0c982b205469a89b","tex":"E_j^*","aliases":"E_j^*","display":"E_j^*","kind":"use","excerpt":"resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have the same relative weights, but their total ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7d3bf131ce5c7803","tex":"E_{ji}^{\\lambda}","aliases":"E_ji^λ lambda","display":"E_{ji}^{\\lambda}","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1f891a7f61f89a21","tex":"E_k","aliases":"E_k","display":"E_k","kind":"use","excerpt":"E_k · Positive measurement effect · It gives the outcome probability Tr(E_kρ).","count":13},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f79869b257acab57","tex":"E_{k\\ell}^{\\mu}","aliases":"E_kℓ^μ ell mu","display":"E_{k\\ell}^{\\mu}","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3a840ddc04edad8c","tex":"E_r","aliases":"E_r","display":"E_r","kind":"use","excerpt":"Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environmen","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6bd3bde92cfdbd07","tex":"E_x","aliases":"E_x","display":"E_x","kind":"use","excerpt":"vpage14A chart failure is not a singular fiberTHE RANGE BUNDLE IS CONSTRUCTED LOCALLYTraditional edition: Theorem II.6.7. Details of the range atlas: saved v0.6.0 supplement, pp. 2--4. The fiber of a projector field is the actual fixed subspace E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the completion proof constructs T_a(x)=P(x","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-96cf99998ca7f513","tex":"E_z","aliases":"E_z","display":"E_z","kind":"use","excerpt":"very operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6fcc993371cd632c","tex":"E_z^*","aliases":"E_z^*","display":"E_z^*","kind":"use","excerpt":"identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the sam","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-49550fdb9f0a1cd8","tex":"E_\\ell","aliases":"E_ℓ ell","display":"E_\\ell","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-42c5e5713be1939f","tex":"E_\\ell^*","aliases":"E_ℓ^* ell","display":"E_\\ell^*","kind":"use","excerpt":"order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f8483dce1203449b","tex":"\\mathrm{eff}","aliases":"eff mathrm","display":"\\mathrm{eff}","kind":"use","excerpt":"[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the approp","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-70a9953dd9bd9b22","tex":"\\operatorname{End}","aliases":"End operatorname","display":"\\operatorname{End}","kind":"use","excerpt":"[Projected connection] Let M be a smooth finite-dimensional real manifold without boundary, let V be a finite-dimensional real normed vector space, and let P:M→End_ R(V) be smooth with P(x)^2=P(x). In the quantum application, V is the underlying real space of a finite complex Hilbert space and P is an orthogonal complex-linear projector.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-46c0f3838c9722a0","tex":"\\operatorname{End}_{\\bbR}","aliases":"End_bbR End_ R operatorname bbR operatorname mathbb","display":"\\operatorname{End}_{{\\mathbb R}}","kind":"use","excerpt":"[Projected connection] Let M be a smooth finite-dimensional real manifold without boundary, let V be a finite-dimensional real normed vector space, and let P:M→End_ R(V) be smooth with P(x)^2=P(x). In the quantum application, V is the underlying real space of a finite complex Hilbert space and P is an orthogonal complex-linear projector.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-77007f1f55a92c5b","tex":"\\mathrm{even}","aliases":"even mathrm","display":"\\mathrm{even}","kind":"use","excerpt":"For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1c77c474dad0dd4d","tex":"\\mathrm{extra}","aliases":"extra mathrm","display":"\\mathrm{extra}","kind":"use","excerpt":"The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary d","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a3382034c436fa8d","tex":"F^*","aliases":"F^*","display":"F^*","kind":"use","excerpt":"Exact correction of an error span means that one completely positive trace preserving recovery R satisfies R(Fρ F^*)=q_Fρ for every F in that span and every code-supported matrix ρ, with a nonnegative scalar q_F depending on F but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restore","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome su","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome su","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-74317064bf2c9a4d","tex":"F_3","aliases":"F_3","display":"F_3","kind":"use","excerpt":"0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome su","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"From Gram data to an actual total recovery. Assume Csucceq0 and PE_a^*E_bP=C_abP. First restrict the error operators to the code, F_a=E_aP. The finite Hermitian spectral theorem supplies a unitary U such that U^*CU=diag(λ_r), λ_r≥0.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a67dba11d23b584e","tex":"F_a^*","aliases":"F_a^*","display":"F_a^*","kind":"use","excerpt":"Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f23aed2f353ac85e","tex":"f_{AB}^{\\ \\ C}","aliases":"f_AB^ C","display":"f_{AB}^{\\ \\ C}","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c68b67e2e253be8c","tex":"f_c","aliases":"f_c","display":"f_c","kind":"use","excerpt":"A constraint diagnostic that vanishes automatically. Section 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives C_aP=0 for every a. Absorb the factor i and any structure coefficient into f_c, and write A_ab=C_aC_b-C_bC_a-∑_cf_cC_c. [Exact common-kernel compression] If C_cP=0 for every","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"[Orthogonal syndrome construction] Assume the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e03cba7ba564271b","tex":"F_j^*","aliases":"F_j^*","display":"F_j^*","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7de303254524fac2","tex":"F_{P\\circ f}","aliases":"F_P∘ f circ","display":"F_{P\\circ f}","kind":"use","excerpt":"Tensor properties and the remaining global boundary. The coefficient is proved bilinear, alternating, and supported on the range on both sides. The chain rule proves the pullback identity F_P∘ f(x)(v,w)=F_P(f(x))(Df(x)v,Df(x)w). The main theorem has global regularity hypotheses on normed parameter spaces. The pullback identity alone does not supply","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f7110079fe13a85a","tex":"F_r","aliases":"F_r","display":"F_r","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4565bf241049378c","tex":"F_r^*","aliases":"F_r^*","display":"F_r^*","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-30ade5d863641faa","tex":"F_s","aliases":"F_s","display":"F_s","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f79f277673daa5a5","tex":"f_t","aliases":"f_t","display":"f_t","kind":"use","excerpt":"Connecting power series and spectral calculus. The shifted generator is proved Hermitian. The implementation then identifies the actual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a75d31ba6a020ef2","tex":"F_z","aliases":"F_z","display":"F_z","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e1a6f7405620e755","tex":"F_z^*","aliases":"F_z^*","display":"F_z^*","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-94aa4c6f6cc42ad6","tex":"F_{\\theta\\phi}","aliases":"F_θϕ theta phi","display":"F_{\\theta\\phi}","kind":"use","excerpt":"LOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6bb2930101570bd3","tex":"F_\\ell","aliases":"F_ℓ ell","display":"F_\\ell","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c4f568dc946ddd98","tex":"\\Fcal","aliases":"Fcal F mathcal","display":"{\\mathcal F}","kind":"use","excerpt":"[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-79071076af4746fa","tex":"\\Fcal_{\\theta\\phi}","aliases":"Fcal_θϕ F_θϕ Fcal theta phi mathcal theta phi","display":"{\\mathcal F}_{\\theta\\phi}","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ed487b1e5f57fec8","tex":"\\mathrm{fine}","aliases":"fine mathrm","display":"\\mathrm{fine}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-25f0be3ddda5cf8c","tex":"G^*","aliases":"G^*","display":"G^*","kind":"use","excerpt":"[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-aa45a7ad376e6ef9","tex":"g^{-1}","aliases":"g^-1","display":"g^{-1}","kind":"use","excerpt":", not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","excerpt":"The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings ar","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-91e0e997e5796cd2","tex":"g_{aa}","aliases":"g_aa","display":"g_{aa}","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9135467db1e92ace","tex":"g_{ab}","aliases":"g_ab","display":"g_{ab}","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c710e753abb0a6f2","tex":"g_{ac}","aliases":"g_ac","display":"g_{ac}","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2dda9aa2199e20de","tex":"g_{bc}","aliases":"g_bc","display":"g_{bc}","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5079789e5bc0a138","tex":"G_{jk}","aliases":"G_jk","display":"G_{jk}","kind":"use","excerpt":"[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8c918c284a1665cd","tex":"\\Gcal","aliases":"Gcal G mathcal","display":"{\\mathcal G}","kind":"use","excerpt":"[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace ","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters be","count":44},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-daaa917c98a2902e","tex":"H^{-1}","aliases":"H^-1","display":"H^{-1}","kind":"use","excerpt":"1 · 3) and retain its first coordinate. The eliminated block is invertible and the Schur complement of H is 2-1/3=5/3. Its inverse is 3/5, exactly the retained entry of H^-1. Inverting the compressed entry alone gives 1/2. The difference records the contribution of a path through the eliminated coordinate. Direct compression omits that contribution","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fa0621c04438e85a","tex":"h^{-2}","aliases":"h^-2","display":"h^{-2}","kind":"use","excerpt":"[Low-mode second-order bound] For every real k and every nonzero real h, define λ_k,h=4h^-2sin^2(kh/2). Then 0≤ k^2-λ_k,h ≤ frack^4h^212.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":"vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters be","count":20},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"Checked consequences. Unitarity also gives the capped estimate Qexp(-itH)P ≤min{1, |t| [H,P] }. If H=H_0+V, both summands are self-adjoint, and [H_0,P]=0, then Qexp(-it(H_0+V))P ≤2|t| V . The latter uses [V,P] ≤2 V . Its constant is not claimed to be optimal. Exact commutation gives zero leakage. The work does not include unbounded generators or ti","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hil","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6f76a62f943f51d9","tex":"\\mathcal H_A","aliases":"H_A mathcal","display":"\\mathcal H_A","kind":"use","excerpt":"Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c935edba5da750e9","tex":"\\mathcal H_B","aliases":"H_B mathcal","display":"\\mathcal H_B","kind":"use","excerpt":"Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7b786d951bfb29bf","tex":"H_{\\mathrm{eff}}","aliases":"H_eff mathrm","display":"H_{\\mathrm{eff}}","kind":"use","excerpt":"[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the approp","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c59677366440c6e2","tex":"H_{\\mathrm{eff}}^*","aliases":"H_eff^* mathrm","display":"H_{\\mathrm{eff}}^*","kind":"use","excerpt":"The explicitly defined coordinate curve y(s)=U(s)^*ψ(s) is differentiable at t, with dot y=dot U^*ψ(t)+U(t)^*dotψ. It satisfies idot y=H_effy(t), H_eff=U(t)^*HU(t)-iU(t)^*dot U. The connection term -iU(t)^*dot U is Hermitian. If H^*=H, then H_eff^*=H_eff.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-31bc489b2a8008eb","tex":"H_{\\mathrm{even}}","aliases":"H_even mathrm","display":"H_{\\mathrm{even}}","kind":"use","excerpt":"For two qubits and real parameters J,h, consider H = JX⊗ X +h(Z⊗ I+I⊗ Z). In the ordered even basis {ket00,ket11} and odd basis {ket01,ket10}, H_even = 2h · J J · -2h , H_odd = 0 · J J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"use","excerpt":"vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hil","count":24},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8ff15c0817c304f2","tex":"H_{n+1}","aliases":"H_n+1","display":"H_{n+1}","kind":"use","excerpt":"vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hil","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5206b85c212e2fd5","tex":"H_{n+1,n}","aliases":"H_n+1,n","display":"H_{n+1,n}","kind":"use","excerpt":"The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0bb3059c2e24fc1e","tex":"H_{n,n+1}","aliases":"H_n,n+1","display":"H_{n,n+1}","kind":"use","excerpt":"The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-96555427067e554e","tex":"H_{nn}","aliases":"H_nn","display":"H_{nn}","kind":"use","excerpt":"The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3a36e5f02055d100","tex":"H_{\\mathrm{odd}}","aliases":"H_odd mathrm","display":"H_{\\mathrm{odd}}","kind":"use","excerpt":"For two qubits and real parameters J,h, consider H = JX⊗ X +h(Z⊗ I+I⊗ Z). In the ordered even basis {ket00,ket11} and odd basis {ket01,ket10}, H_even = 2h · J J · -2h , H_odd = 0 · J J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c305141ec8df1721","tex":"H_{PP}","aliases":"H_PP","display":"H_{PP}","kind":"use","excerpt":"1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. This is the corrected hypothesis set recorded in the formal release. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 an","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-91cc7303740de01a","tex":"H_{PQ}","aliases":"H_PQ","display":"H_{PQ}","kind":"use","excerpt":"Let P be a retained packet projector and Q=I-P. In packet coordinates, H= H_PP · H_PQ H_QP · H_QQ .","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2473371c8e4d1e23","tex":"H_{QP}","aliases":"H_QP","display":"H_{QP}","kind":"use","excerpt":"Let P be a retained packet projector and Q=I-P. In packet coordinates, H= H_PP · H_PQ H_QP · H_QQ .","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dd8710c6593757d0","tex":"H_{QQ}","aliases":"H_QQ","display":"H_{QQ}","kind":"use","excerpt":"1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. This is the corrected hypothesis set recorded in the formal release. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 an","count":12},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8ee189bd82f77745","tex":"H_{\\rm d}","aliases":"H_rm d rm","display":"H_{\\rm d}","kind":"use","excerpt":"Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9fe2614398adf9a4","tex":"H_{\\rm eff}","aliases":"H_rm eff rm","display":"H_{\\rm eff}","kind":"use","excerpt":"1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. This is the corrected hypothesis set recorded in the formal release. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 an","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-96ae09a38751e66f","tex":"H_{\\rm even}","aliases":"H_rm even rm","display":"H_{\\rm even}","kind":"use","excerpt":"etry reveals blocks, not approximationsONE HAMILTONIAN IN TWO ORDERSTraditional edition: Theorem II.5.1. Take the exact rational Hamiltonian H=X⊗ X+frac38(Z⊗ I+I⊗ Z). It commutes with total parity Z⊗ Z. Reordering the computational basis into even addresses (00,11) and odd addresses (01,10) displays the invariant subspaces explicitly. vfighamiltoni","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5c9132bb49e045e4","tex":"H_{\\rm frame}","aliases":"H_rm frame rm","display":"H_{\\rm frame}","kind":"use","excerpt":"1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i - i · 0, idot y=H_rm framey. This matrix is Hermitian. Omitting it would predict a constant y, contradicting the displayed coordinates. In this example the norm of the omitted time derivative is exactly one, so the failure is not a small numerical residual.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d3b7ddf6b71c7dbb","tex":"H_{\\rm Had}","aliases":"H_rm Had rm","display":"H_{\\rm Had}","kind":"use","excerpt":"A Hamiltonian and a Hadamard gate are both traditionally denoted H. Where both appear, the gate is H_rm Had and the Hamiltonian keeps the letter H. Numerical labels on a drawing are coordinates or exact example values, not observations of a physical device. takeawayCheck the type before interpreting the picture. An idempotent projector, a density m","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f630a7f11cb43320","tex":"H_{\\rm mov}","aliases":"H_rm mov rm","display":"H_{\\rm mov}","kind":"use","excerpt":"[Moving frame Hamiltonian] Let U(t) solve idot U(t)=H(t)U(t), U(0)=I, where H(t)=H(t)^* is continuous on the compact time interval under consideration. Set V(t)=W(t)^*U(t). Then idot V(t)=H_rm mov(t)V(t), where H_rm mov(t) = W(t)^*H(t)W(t)- i W(t)^*dot W(t) = W(t)^*(H(t)- i K(t))W(t). The moving frame Hamiltonian is self adjoint.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-51d96667cf6151ff","tex":"H_{\\rm o}","aliases":"H_rm o rm","display":"H_{\\rm o}","kind":"use","excerpt":"Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9f4d2d0416bc2db4","tex":"H_{\\rm odd}","aliases":"H_rm odd rm","display":"H_{\\rm odd}","kind":"use","excerpt":"etry reveals blocks, not approximationsONE HAMILTONIAN IN TWO ORDERSTraditional edition: Theorem II.5.1. Take the exact rational Hamiltonian H=X⊗ X+frac38(Z⊗ I+I⊗ Z). It commutes with total parity Z⊗ Z. Reordering the computational basis into even addresses (00,11) and odd addresses (01,10) displays the invariant subspaces explicitly. vfighamiltoni","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d5ba533e65aa4c1b","tex":"H_T","aliases":"H_T","display":"H_T","kind":"use","excerpt":"[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a39b4a2b8b246f17","tex":"H_{\\ell+1}","aliases":"H_ℓ+1 ell","display":"H_{\\ell+1}","kind":"use","excerpt":"[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-743400708b98042d","tex":"H_\\infty","aliases":"H_∞ infty","display":"H_\\infty","kind":"use","excerpt":"ditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-795440ec9737d857","tex":"\\Hcal_A","aliases":"Hcal_A H_A Hcal mathcal","display":"{\\mathcal H}_A","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, let d_A=dim H_A≥1, and let r be the Schmidt rank of ψ. Then dimspan{(A⊗ I_B)ψ:A∈End( H_A)} = d_A r. Local operators on A generate the whole bipartite Hilbert space exactly when the reduced state on B has full support.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2723b4e767b85839","tex":"\\Hcal_B","aliases":"Hcal_B H_B Hcal mathcal","display":"{\\mathcal H}_B","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, let d_A=dim H_A≥1, and let r be the Schmidt rank of ψ. Then dimspan{(A⊗ I_B)ψ:A∈End( H_A)} = d_A r. Local operators on A generate the whole bipartite Hilbert space exactly when the reduced state on B has full support.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6fb67345de2f0bec","tex":"\\Hcal_{\\phys}","aliases":"Hcal_phys H_phys Hcal phys mathcal mathrm","display":"{\\mathcal H}_{{\\mathrm{phys}}}","kind":"use","excerpt":"[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9ef531a1e01371ad","tex":"\\Hcal_X","aliases":"Hcal_X H_X Hcal mathcal","display":"{\\mathcal H}_X","kind":"use","excerpt":"Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4e58303c6fd9074e","tex":"\\Hcal_Y","aliases":"Hcal_Y H_Y Hcal mathcal","display":"{\\mathcal H}_Y","kind":"use","excerpt":"Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-47d019f7ef5155b4","tex":"\\Hcal_Z","aliases":"Hcal_Z H_Z Hcal mathcal","display":"{\\mathcal H}_Z","kind":"use","excerpt":"[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2b64a19766af07b6","tex":"\\Hcal_\\infty","aliases":"Hcal_∞ H_∞ Hcal infty mathcal infty","display":"{\\mathcal H}_\\infty","kind":"use","excerpt":"Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7937103467a5b8d0","tex":"\\HH","aliases":"HH","display":null,"kind":"use","excerpt":"midrule HH,ψ · Hilbert space and state vector · They specify the space of addresses and its coherent amplitudes.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ac5f83b9023afd1a","tex":"\\HH_n","aliases":"HH_n HH","display":null,"kind":"use","excerpt":"vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hil","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":", not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8950e80c743e7b98","tex":"\\HS","aliases":"HS mathrm","display":"{\\mathrm{HS}}","kind":"use","excerpt":"Density matrices and observables carry both an input packet and an output packet. Equip End( H) with the Hilbert Schmidt inner product ipXY_HS=Tr(X^*Y).","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9d6409e60138ea26","tex":"I^*","aliases":"I^*","display":"I^*","kind":"use","excerpt":"main inside the original code, so the previous syndrome partition reports no bit flip. frac3ket000+4ket1115 ⟼ frac3ket000-4ket1115. The populations are unchanged, but the relative phase is not. vfiglogical_error Let V be the code isometry with columns ket000,ket111. For the proposed error family {I,Z_1}, its off-diagonal error compression is V^*I^*","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8ba0538909d6a977","tex":"I_3","aliases":"I_3","display":"I_3","kind":"use","excerpt":"vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8ee5bef1760d6bb8","tex":"I_6","aliases":"I_6","display":"I_6","kind":"use","excerpt":"standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3χ_λ(g^-1)U(g). All matrices in this example are rational. Multiplying them gives self-adjoint idempotents of ranks 1,1,4, with pairwise zero products and sum I_6. xfigs3 headThe standard packet has dimension four The standard representation occurs twice in the regular representation. Its isotypic compon","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-60a91a508a9a828a","tex":"I_B","aliases":"I_B","display":"I_B","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cbf63ab2047dd5c1","tex":"I_{\\Hcal}","aliases":"I_Hcal I_ H Hcal mathcal","display":"I_{{\\mathcal H}}","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6ef65037047c3204","tex":"I_J","aliases":"I_J","display":"I_J","kind":"use","excerpt":"To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ad472b41377c2078","tex":"I_m","aliases":"I_m","display":"I_m","kind":"use","excerpt":"0 · 0 · 0. These two matrices provide a complete set of row directions. Their individual orbit dimensions are four and two, and together those spans fill the six-dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c96c37f828cfc2a3","tex":"I_{m_r}","aliases":"I_m_r","display":"I_{m_r}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c305f0f9a08586da","tex":"I_{M_\\mu}","aliases":"I_M_μ mu","display":"I_{M_\\mu}","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0c6b4e7520988cc0","tex":"I_P","aliases":"I_P","display":"I_P","kind":"use","excerpt":"Corrected Feshbach resolvent. Write a finite complex matrix in retained and eliminated coordinates: H=A · B{ C · D, zI-H=zI_P-A · -B -C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-caf5dd124506cb34","tex":"I_Q","aliases":"I_Q","display":"I_Q","kind":"use","excerpt":"[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the approp","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-38cd9fdb96fba5ee","tex":"I_{\\rm in}","aliases":"I_rm in rm","display":"I_{\\rm in}","kind":"use","excerpt":"labelstrong:standardfact:001 A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id mathrm","display":"{\\mathrm{id}}","kind":"use","excerpt":"vpage18An ancilla tests more than positivityWHY COMPLETE POSITIVITY IS THE RELEVANT CONDITIONTraditional edition: Classical input II.7.7. Ordinary transpose T sends a positive matrix on one system to another positive matrix and preserves its trace. Those properties are insufficient for a quantum channel on a subsystem of an entangled state. The tes","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f43b2c8703ad09ef","tex":"\\ii^{-jk}","aliases":"ii^-jk i^-jk ii mathrm","display":"{\\mathrm i}^{-jk}","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-69bc80ad3d09d0fe","tex":"\\ii^{-jr}","aliases":"ii^-jr i^-jr ii mathrm","display":"{\\mathrm i}^{-jr}","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a24bbf47ddba511a","tex":"\\ii^j","aliases":"ii^j i^j ii mathrm","display":"{\\mathrm i}^j","kind":"use","excerpt":"character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing the address multiplie","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-12473beefe7126eb","tex":"\\Im","aliases":"Im","display":"\\Im","kind":"use","excerpt":"1 · 2, so A=0, B=1, and D=2. In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","excerpt":"[Upper half plane sign] If Imz>0, then ImΣ(z) = -(Imz) B(z-D)^-1(overline z-D)^-1B^* ≤0. Thus -Σ is an operator valued Herglotz function.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"same constraints;node[rotate=90,font=footnotesize] at (11.4,1.7) new coordinate; node[font=small] at (6,0.15) closure on imR transfers, and the new complement needs its own test;","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c0031bdf2737d3ca","tex":"\\ip","aliases":"ip","display":null,"kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"-sin t. The vector has not moved in the laboratory, but its coordinate entries have changed. vfigmoving_frame headThe missing term can be computed exactly Let J=mat0 · -1 1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i","count":20},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3a3bcbfb09701caf","tex":"\\bm j","aliases":"j bm","display":"\\bm j","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5b761556af44ee0a","tex":"\\mathbf j","aliases":"j mathbf","display":"\\mathbf j","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-29097134fc17258f","tex":"J^2","aliases":"J^2","display":"J^2","kind":"use","excerpt":"J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0e97a80aa927d24d","tex":"j_1","aliases":"j_1","display":"j_1","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-416b4c70e8031735","tex":"j_m","aliases":"j_m","display":"j_m","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-456e17f40199861f","tex":"j_r","aliases":"j_r","display":"j_r","kind":"use","excerpt":"[Reflection positivity is closed under limits] Suppose C_n(j,k) are reflection positive kernels on N, meaning every finite matrix (C_n(j_r,j_s))_r,s is positive semidefinite. If C_n(j,k)⟶ C(j,k) for every pair, then C is reflection positive.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-06e8fa7eb9aaa088","tex":"j_s","aliases":"j_s","display":"j_s","kind":"use","excerpt":"[Reflection positivity is closed under limits] Suppose C_n(j,k) are reflection positive kernels on N, meaning every finite matrix (C_n(j_r,j_s))_r,s is positive semidefinite. If C_n(j,k)⟶ C(j,k) for every pair, then C is reflection positive.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fb9dab5747004561","tex":"K'","aliases":"K'","display":"K'","kind":"use","excerpt":"vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESTraditional edition: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^2","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-997c382b71c367a6","tex":"K^*","aliases":"K^*","display":"K^*","kind":"use","excerpt":"Differentiate V=W^*U. Since dot W=KW and K^*=-K, dot W^*=-W^*K. Substitution gives the equation. The first term in H_rm mov is self adjoint. The operator - i W^*dot W is self adjoint because W^*dot W is skew adjoint.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4b779dcac03cb4e5","tex":"K^2","aliases":"K^2","display":"K^2","kind":"use","excerpt":"For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters be","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ec1a2b307412bdd0","tex":"k^4","aliases":"k^4","display":"k^4","kind":"use","excerpt":"vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters be","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter kee","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-52d8e03f4220d281","tex":"K_0^*","aliases":"K_0^*","display":"K_0^*","kind":"use","excerpt":"vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter kee","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cf5fcf757b255a84","tex":"K_1","aliases":"K_1","display":"K_1","kind":"use","excerpt":"vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter kee","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ead6134bc1ede95b","tex":"K_1^*","aliases":"K_1^*","display":"K_1^*","kind":"use","excerpt":"vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter kee","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-173833828c3529a3","tex":"K_2","aliases":"K_2","display":"K_2","kind":"use","excerpt":"For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6cc92a51c6969e2c","tex":"K_a","aliases":"K_a","display":"K_a","kind":"use","excerpt":"K_a · Kraus operator · It describes one channel branch and need not be a projector.","count":13},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1c7fa5ebfa740b08","tex":"K_a^*","aliases":"K_a^*","display":"K_a^*","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ca931e9dd55037b6","tex":"K_j","aliases":"K_j","display":"K_j","kind":"use","excerpt":"e, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bcf3911a4b4214e2","tex":"k_j","aliases":"k_j","display":"k_j","kind":"use","excerpt":"Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a68fcbb4721376c8","tex":"K_j^*","aliases":"K_j^*","display":"K_j^*","kind":"use","excerpt":"e, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e96542379e91eebb","tex":"k_j^4","aliases":"k_j^4","display":"k_j^4","kind":"use","excerpt":"[Low mode dispersion in dimension d] For every centered mode k, 0 ≤ 4π^2absk^2-λ_N(k) ≤ frac4π^43N^2 ∑_j=1^dk_j^4.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9a30d27faec7f7f7","tex":"K_r","aliases":"K_r","display":"K_r","kind":"use","excerpt":"[Finite Kraus channel] Let K_1,…,K_r satisfy ∑_aK_a^*K_a=I. Then Φ(ρ)=∑_aK_aρ K_a^* is completely positive and trace preserving.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-193077db22dceb36","tex":"K_r^*","aliases":"K_r^*","display":"K_r^*","kind":"use","excerpt":"_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-75bb1e0e1a9420dc","tex":"K_{\\mathrm{unused}}","aliases":"K_unused mathrm","display":"K_{\\mathrm{unused}}","kind":"use","excerpt":"efine compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are com","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-86aa5e48615bd30f","tex":"K_{\\mathrm{unused}}^*","aliases":"K_unused^* mathrm","display":"K_{\\mathrm{unused}}^*","kind":"use","excerpt":"_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e08d6b60be0ec0b0","tex":"\\Ker","aliases":"Ker operatorname","display":"{\\operatorname{Ker}}","kind":"use","excerpt":"(z-D)^-1B^*x is an isomorphism from Ker F_P(z) onto Ker(z-H).","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"Actual projected differentiation and curvature. Let E,F be real normed spaces. Let P:E→ L(F,F) be differentiable with P(x)^2=P(x) for every x. Let s:E→ F be C^2 with P(x)s(x)=s(x), and let X,Y:E→ E be differentiable vector fields. The checked connection is the actual expression ∇_Xs(x)=P(x) Ds(x)[X(x)]. It is proved to preserve the range condition.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"use","excerpt":"textitNecessity. Fix an error E_a and let L_j=K_jE_aP. Perfect recovery on rank-one code operators gives ∑_jketL_jψbraL_jψ =t_aketψbraψ (ψ∈ C). For every vector orthogonal to ψ, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c91dabfc8b700b3b","tex":"L_N","aliases":"L_N","display":"L_N","kind":"use","excerpt":"[Rigorous intermediate dimension window] For every ε>0, there is a diffusion time t_ε>0 and an integer N_ε such that absd_L_N(t_ε)-d<ε for every N≥ N_ε.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ed76d6b3d182baf7","tex":"L_n","aliases":"L_n","display":"L_n","kind":"use","excerpt":"[Heat trace convergence from trace norm] Let L_n,L be nonnegative self adjoint operators on one Hilbert space. For a fixed t>0, assume their heat operators are trace class and that e^-tL_n⟶ e^-tL in trace norm. Then Tr( e^-tL_n)⟶Tr( e^-tL). If also the heat moment operators are trace class and L_n e^-tL_n⟶ L e^-tL in trace norm and the limiting hea","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a377a288ccd2ba74","tex":"L_r","aliases":"L_r","display":"L_r","kind":"use","excerpt":"[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace ","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1ccfde72f4e3e4f6","tex":"L_r^*","aliases":"L_r^*","display":"L_r^*","kind":"use","excerpt":"[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3755b02b61222ecc","tex":"\\Lcal","aliases":"Lcal L mathcal","display":"{\\mathcal L}","kind":"use","excerpt":"vpage31Coherence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSTraditional edition: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmi","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7151029ba530e5db","tex":"\\Lcal_0","aliases":"Lcal_0 L_0 Lcal mathcal","display":"{\\mathcal L}_0","kind":"use","excerpt":"vpage31Coherence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSTraditional edition: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmi","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-271ba6ea9d5e2027","tex":"\\Lcal_1","aliases":"Lcal_1 L_1 Lcal mathcal","display":"{\\mathcal L}_1","kind":"use","excerpt":"er survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSTraditional edition: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space ar","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3bcbed4467b578cb","tex":"\\Lcal_p","aliases":"Lcal_p L_p Lcal mathcal","display":"{\\mathcal L}_p","kind":"use","excerpt":"[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b8dbbf5357e50fbf","tex":"\\Lcal_q","aliases":"Lcal_q L_q Lcal mathcal","display":"{\\mathcal L}_q","kind":"use","excerpt":"[Coherence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"[Finite scale limits] For every finite nonnegative L with nontrivial kernel, lim_tdownarrow0d_L(t)=0 and lim_t→∞d_L(t)=0. For every t>0, 0≤ d_L(t)≤2tnorm L.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fb093952304f9b7e","tex":"\\lim_{t\\downarrow0}","aliases":"lim_tdownarrow0 lim downarrow","display":"\\lim_{t\\downarrow0}","kind":"use","excerpt":"[Finite scale limits] For every finite nonnegative L with nontrivial kernel, lim_tdownarrow0d_L(t)=0 and lim_t→∞d_L(t)=0. For every t>0, 0≤ d_L(t)≤2tnorm L.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-24fd66249142db1e","tex":"\\lim_{t\\to\\infty}","aliases":"lim_t→∞ lim to infty","display":"\\lim_{t\\to\\infty}","kind":"use","excerpt":"[Finite scale limits] For every finite nonnegative L with nontrivial kernel, lim_tdownarrow0d_L(t)=0 and lim_t→∞d_L(t)=0. For every t>0, 0≤ d_L(t)≤2tnorm L.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-973977200963bdff","tex":"\\log_{10}","aliases":"log_10 log","display":"\\log_{10}","kind":"use","excerpt":"125); node[font=small,align=center,anchor=east,font=scriptsize] at (0.64,2.1125) 1; draw[black!13,line width=.25pt] (0.8,3.425) -- (9.5,3.425); node[font=small,align=center,anchor=east,font=scriptsize] at (0.64,3.425) 2; draw[->,line width=.6pt] (0.8,0.8) -- (9.62,0.8); draw[->,line width=.6pt] (0.8,0.8) -- (0.8,4.07); node[font=small,align=center,","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Spectral equivalence] Let z∉Spec D. Then z∈Spec H Longleftrightarrow 0∈Spec F_P(z). The map x⟼ x","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormali","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-207e98afc142d0de","tex":"M_d","aliases":"M_d","display":"M_d","kind":"use","excerpt":"ment changes the local spanSCHMIDT RANK AS ACCESSIBLE MULTIPLICITYTraditional edition: Theorem II.2.2, Corollary II.2.4. Write a bipartite vector as a coefficient matrix C, with the first subsystem indexing rows. Acting only on that subsystem transforms C to AC. Every resulting row remains in the row space of C, and every independent choice of such","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cd84b726bc30add7","tex":"M_{d_A}","aliases":"M_d_A","display":"M_{d_A}","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2aa0f7b977f40849","tex":"M_{d_r}","aliases":"M_d_r","display":"M_{d_r}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-84cc641b229c54b3","tex":"M_{d\\times m}","aliases":"M_d× m times","display":"M_{d\\times m}","kind":"use","excerpt":"Actual orbits and sharp seed counts. Let C∈ M_d× m( C) be the coefficient matrix of one seed. Define its orbit by actual matrix multiplication: O(C)={AC:A∈ M_d( C)}. This set is already a linear subspace. The source proves that it equals the linear span of the orbit before computing its dimension.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3cc0ed3324bf0e59","tex":"M_m","aliases":"M_m","display":"M_m","kind":"use","excerpt":"[Finite Kraus and Choi criteria] A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"use","excerpt":"[Finite Kraus and Choi criteria] A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-19afb20993a1d995","tex":"m_r","aliases":"m_r","display":"m_r","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c60cf3b807dc9ab5","tex":"M_\\mu","aliases":"M_μ mu","display":"M_\\mu","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-956f614ae0645ab5","tex":"\\mat","aliases":"mat","display":null,"kind":"use","excerpt":"vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDTraditional edition: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0 0 · 1 ","count":21},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"use","excerpt":"n convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbi","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-202d1bbf7e6fd044","tex":"\\max_{1\\le r\\le s}","aliases":"max_1≤ r≤ s max le le","display":"\\max_{1\\le r\\le s}","kind":"use","excerpt":"[Quantum seed rank] For the displayed multiplicity decomposition, srank_ A( H) = max_1≤ r≤ s lceilfracm_rd_rrceil.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c52aaa85647b7be3","tex":"\\max_r","aliases":"max_r max","display":"\\max_r","kind":"use","excerpt":"Set k=max_rlceilfracm_rd_rrceil. For each r, partition a basis of C^m_r into at most k groups of size at most d_r. Choose the coefficient matrix of the jth seed so that its row space spans the jth group. The combined row spaces span every multiplicity factor. The preceding theorem then shows that the k seeds generate every summand.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε ","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6225ff8930a41599","tex":"\\bm n","aliases":"n bm","display":"\\bm n","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2aa34ef1611fca9c","tex":"\\mathbf n","aliases":"n mathbf","display":"\\mathbf n","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"use","excerpt":"The precise finite channel statement. Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSTraditional edition: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2592eef1f50b9297","tex":"\\mathcal N_E","aliases":"N_E mathcal","display":"\\mathcal N_E","kind":"use","excerpt":"The precise finite channel statement. Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"xponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors ","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cf9404d3f534ee7e","tex":"n_k","aliases":"n_k","display":"n_k","kind":"use","excerpt":"tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d7c0e2c4594eec1b","tex":"n_m","aliases":"n_m","display":"n_m","kind":"use","excerpt":"[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-07a2e96d36e692db","tex":"N_\\varepsilon","aliases":"N_ε varepsilon","display":"N_\\varepsilon","kind":"use","excerpt":"· 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less than ε/2. Then choose N_ε so ","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-90f600277c50100a","tex":"\\NN","aliases":"NN N mathbb","display":"{\\mathbb N}","kind":"use","excerpt":"a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the dir","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d2173a70f3a0a3da","tex":"\\NN_0","aliases":"NN_0 N_0 NN mathbb","display":"{\\mathbb N}_0","kind":"use","excerpt":"a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the dir","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"vpage8Track the Bell state through the circuitINTERMEDIATE PACKETS ARE NOT FINAL PACKETSTraditional edition: Theorem II.4.1, Example II.4.6. The input is ket00. First apply a Hadamard to the first qubit, then a controlled flip: ket00 xmapstoH_rm Had⊗ I fracket00+ket10sqrt2 xmapstorm CNOT fracket00+ket11sqrt2. The displayed columns keep amplitudes, ","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"Actual orbits and sharp seed counts. Let C∈ M_d× m( C) be the coefficient matrix of one seed. Define its orbit by actual matrix multiplication: O(C)={AC:A∈ M_d( C)}. This set is already a linear subspace. The source proves that it equals the linear span of the orbit before computing its dimension.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. Thi","count":19},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-090aecb8fb37e7b7","tex":"O_1","aliases":"O_1","display":"O_1","kind":"use","excerpt":"The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c5c9215497110dd4","tex":"\\mathcal O_A","aliases":"O_A mathcal","display":"\\mathcal O_A","kind":"use","excerpt":"Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a7d87e09a33f4365","tex":"O_j","aliases":"O_j","display":"O_j","kind":"use","excerpt":"The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-580246cc2221ddc9","tex":"O_r","aliases":"O_r","display":"O_r","kind":"use","excerpt":"The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0f7264e76f55d278","tex":"\\mathrm{odd}","aliases":"odd mathrm","display":"\\mathrm{odd}","kind":"use","excerpt":"For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-622810599c7ebcb0","tex":"\\overline p","aliases":"overline p overline","display":"\\overline p","kind":"use","excerpt":"Rank-one rigidity of the corrected channel. The module tests the map equality on actual matrix units and flattens each A_ra into a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzer","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-beffd946d87d6320","tex":"\\overline U","aliases":"overline U overline","display":"\\overline U","kind":"use","excerpt":"form a complete orthogonal packet system on Hilbert Schmidt operator space. Dephasing is the diagonal projection. Every channel has input and output superblocks, with the same composition law as ordinary matrix blocks. A covariant channel preserves the isotypic packets of the conjugation action. In the cyclic case these are coherence orders. The Ch","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-57ff192d28b73a0c","tex":"\\overline v","aliases":"overline v overline","display":"\\overline v","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1895fa5459189739","tex":"\\overline v_j","aliases":"overline v_j overline","display":"\\overline v_j","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ae21a8553a17610b","tex":"\\overline z","aliases":"overline z overline","display":"\\overline z","kind":"use","excerpt":"[Upper half plane sign] If Imz>0, then ImΣ(z) = -(Imz) B(z-D)^-1(overline z-D)^-1B^* ≤0. Thus -Σ is an operator valued Herglotz function.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a3d1ed7dab22cee4","tex":"\\overline{c_b}","aliases":"overlinec_b overline","display":"\\overline{c_b}","kind":"use","excerpt":"For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-227f60ba397e1df3","tex":"\\overline{c_j}","aliases":"overlinec_j overline","display":"\\overline{c_j}","kind":"use","excerpt":"[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6c3b42b1cc9cccf8","tex":"\\overline{d_{ja}}","aliases":"overlined_ja overline","display":"\\overline{d_{ja}}","kind":"use","excerpt":"he left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recove","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f7591b1de9b36411","tex":"\\overline{U(g)}","aliases":"overlineU(g) overline","display":"\\overline{U(g)}","kind":"use","excerpt":"[Choi symmetry criterion] Let U:G→ U( H) be a finite group representation. A channel Φ is covariant, Φ(U(g)XU(g)^*) = U(g)Φ(X)U(g)^*, if and only if (U(g)⊗overlineU(g)) J(Φ) (U(g)^*⊗ U(g)^ T) = J(Φ) for every g∈ G.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-32830cd7863f82ba","tex":"\\overline{W_{ja}}","aliases":"overlineW_ja overline","display":"\\overline{W_{ja}}","kind":"use","excerpt":"[Orthogonal syndrome construction] Assume the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ba14a930d8f52433","tex":"\\overline{z_a}","aliases":"overlinez_a overline","display":"\\overline{z_a}","kind":"use","excerpt":"identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the sam","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0975b6c646732480","tex":"\\overline{z_{ra}}","aliases":"overlinez_ra overline","display":"\\overline{z_{ra}}","kind":"use","excerpt":"Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this cal","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0690ec27d59fae38","tex":"\\overline{\\chi(r)}","aliases":"overlineχ(r) overline chi","display":"\\overline{\\chi(r)}","kind":"use","excerpt":"Orthogonality of the projectors. Unitarity supplies ρ(r)^*=ρ(-r), while the character obeys overlineχ(r)=χ(-r). Reindexing r↦-r proves P_j^*=P_j. The final concrete matrix theorem includes Hermiticity together with multiplication, completeness, and eigenaction. Together these identities establish orthogonal spectral packets.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fa9a82fa2877719d","tex":"\\overline{\\chi_\\lambda(g)}","aliases":"overlineχ_λ(g) overline chi lambda","display":"\\overline{\\chi_\\lambda(g)}","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"g Every effect in the computational basis E_a=ket abra a has the same expectation on both matrices, namely 1/2 at 00 and 11 and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture. headA measurement that resolves the difference Use the valid two-outcome measurement with","count":21},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2a1fe20674ded8f2","tex":"p^{\\mathsf T}","aliases":"p^ T mathsf","display":"p^{\\mathsf T}","kind":"use","excerpt":"Rank-one rigidity of the corrected channel. The module tests the map equality on actual matrix units and flattens each A_ra into a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzer","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8994ede9b103570b","tex":"P^{(1)}_{j_1}","aliases":"P^(1)_j_1","display":"P^{(1)}_{j_1}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a123f2e4f0737041","tex":"P^{(a)}_0","aliases":"P^(a)_0","display":"P^{(a)}_0","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-74cec6b6f4a78905","tex":"P^{(a)}_{N-1}","aliases":"P^(a)_N-1","display":"P^{(a)}_{N-1}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b6342e3c90681bbc","tex":"P^{(i)}_{a_i}","aliases":"P^(i)_a_i","display":"P^{(i)}_{a_i}","kind":"use","excerpt":"Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7a44c5f7cde75a0d","tex":"P^{(i)}_j","aliases":"P^(i)_j","display":"P^{(i)}_j","kind":"use","excerpt":"Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c758c2cf6cd36037","tex":"P^{(m)}_{j_m}","aliases":"P^(m)_j_m","display":"P^{(m)}_{j_m}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-16a4b472d59a692e","tex":"P^*","aliases":"P^*","display":"P^*","kind":"use","excerpt":"Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environmen","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e2f259e45c66f6ef","tex":"P_{\\mathbf j}^{\\mathrm{fine}}","aliases":"P_ j^fine mathbf mathrm","display":"P_{\\mathbf j}^{\\mathrm{fine}}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a3439312706e15bf","tex":"P_{\\bm j}^{\\rm fine}","aliases":"P_ j^rm fine bm rm","display":"P_{\\bm j}^{\\rm fine}","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cfbc31b0e1b74bde","tex":"P_{000}","aliases":"P_000","display":"P_{000}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. Thi","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5750c25072ea9ac4","tex":"P_{011}","aliases":"P_011","display":"P_{011}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. Thi","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILURETraditional edition: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e3778c22278196c4","tex":"P_{101}","aliases":"P_101","display":"P_{101}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. Thi","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7f3a6ef15dffa349","tex":"P_{110}","aliases":"P_110","display":"P_{110}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. Thi","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"ll] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f0242a419961441b","tex":"P_3","aliases":"P_3","display":"P_3","kind":"use","excerpt":"at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) P_2=v_2v_2^*; node[font=small,align=center,font=small] at (8.05,1.25) S v_3=- iv_3; node[font=small,align=center,font=small] at ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c261d0404352ff34","tex":"p_a","aliases":"p_a","display":"p_a","kind":"use","excerpt":"Exact correction, including coherent errors. For every operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a7abbe02c9f0ed6a","tex":"P_a^*","aliases":"P_a^*","display":"P_a^*","kind":"use","excerpt":"The projectors P_a form a Kraus family because ∑_aP_a^*P_a=I. Thus the map is completely positive and trace preserving. It is unital for the same reason. Orthogonality and idempotence follow from [source reference]. Its range is the sum of the diagonal operator packets. This is exactly the commutant of the projector family.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-72bf1da84f9b9c1d","tex":"P_a^2","aliases":"P_a^2","display":"P_a^2","kind":"use","excerpt":"Differentiating P_a^2=P_a gives dot P_aP_a+P_adot P_a=dot P_a. Summing over a and using ∑_adot P_a=0 yields ∑_aP_adot P_a=-∑_adot P_aP_a, which proves both displayed formulas for C and its skew-Hermiticity.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a3db31f834a299d9","tex":"P_{a_0}","aliases":"P_a_0","display":"P_{a_0}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2693c2c3c4636ad8","tex":"P_{a_1}","aliases":"P_a_1","display":"P_{a_1}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e071a8db9cfdfcfd","tex":"P_{a_m}","aliases":"P_a_m","display":"P_{a_m}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d53e2b9e1c09bb7d","tex":"P_{a_{m-1}}","aliases":"P_a_m-1","display":"P_{a_{m-1}}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bb071e0da3d1f51a","tex":"P_b^2","aliases":"P_b^2","display":"P_b^2","kind":"use","excerpt":"UU^*=1 · Longrightarrow ∑_aU_ba(U_da)^*=δ_bdP_b. For the first law, expand each corner, collapse P_b^2 and sum: ∑_bP_aU^*P_bUP_c =P_aU^*(∑_bP_b)UP_c=P_aP_c. The second law follows by the analogous calculation and has its own formal proof. For complex matrices, the adjoint is the conjugate transpose. The order of the operator products is preserved t","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3270e71bf1ae54c5","tex":"P_c","aliases":"P_c","display":"P_c","kind":"use","excerpt":"le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the or","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fe083e5d0f70cf84","tex":"P_c^2","aliases":"P_c^2","display":"P_c^2","kind":"use","excerpt":"0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-14f8900152d6716a","tex":"P_d","aliases":"P_d","display":"P_d","kind":"use","excerpt":"Idempotence and completeness follow from the projector identities. For Hilbert Schmidt orthogonality, ipP_aXP_bP_cYP_d_HS · = Tr(P_bX^*P_aP_cYP_d) · = δ_acδ_bd Tr(P_bX^*P_aYP_b). Since the packets are orthogonal and sum to the original operator, their squared norms add to its squared norm.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f4c2e0977ad59ef5","tex":"P_{\\mathrm{even}}","aliases":"P_even mathrm","display":"P_{\\mathrm{even}}","kind":"use","excerpt":"For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to ","count":22},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-21ea731dd7eeae7b","tex":"P_{j+q}","aliases":"P_j+q","display":"P_{j+q}","kind":"use","excerpt":"[Charge selection rule] If A has charge q, then P_kAP_j=0 unless k=j+q in Z/N Z. In fact, AP_j=P_j+qAP_j.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-634eafc544731bd6","tex":"P_j^{(a)}","aliases":"P_j^(a)","display":"P_j^{(a)}","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e11671ca6a219375","tex":"P_j^*","aliases":"P_j^*","display":"P_j^*","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8d6d0b9c4abedaed","tex":"P_j^2","aliases":"P_j^2","display":"P_j^2","kind":"use","excerpt":"Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2abda96372323698","tex":"P_{j_1}^{(1)}","aliases":"P_j_1^(1)","display":"P_{j_1}^{(1)}","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3ce9ef7655396f9f","tex":"P_{j_m}^{(m)}","aliases":"P_j_m^(m)","display":"P_{j_m}^{(m)}","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dc7ba1c49968c37c","tex":"p_k","aliases":"p_k","display":"p_k","kind":"use","excerpt":"Let ρ be a density matrix and let {E_k} be a POVM: E_k≥0, ∑_kE_k=I. The outcome probabilities are p_k=Tr(E_kρ).","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"Let P_1(t),…,P_m(t) be differentiable on an open real time interval, with P_a(t)^*=P_a(t) and P_aP_b=δ_abP_a, ∑_aP_a=I.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"use","excerpt":"nite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives C(j,k)=1+(1/2)^j+k. For times (0,1,2) and coefficients (1,-2,1), ∑_j,kv_jC_n(j,k)v_k=(1-q_n)^4≥0. This exact check illustrates how finite quadratic forms pass positivity to the limit. xfigtrace_topology headOperator norm does not preserve trace in growing r","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2f79379509b17122","tex":"P_{\\mathrm{odd}}","aliases":"P_odd mathrm","display":"P_{\\mathrm{odd}}","kind":"use","excerpt":"For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1ccfbd779afb88e3","tex":"P_{\\phys}","aliases":"P_phys phys mathrm","display":"P_{{\\mathrm{phys}}}","kind":"use","excerpt":"Constraint closure needs a different obstruction. If C_a are self adjoint constraints and P_phys projects onto their common kernel, then P_phys[C_a,C_b]P_phys=0 holds automatically. This compression cannot test closure. The correct finite obstruction is the class of [C_a,C_b] in the quotient of End( H) by the span of the constraints. Vanishing mean","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1c82a6e2325ace1a","tex":"P_r^{\\mathrm{coarse}}","aliases":"P_r^coarse mathrm","display":"P_r^{\\mathrm{coarse}}","kind":"use","excerpt":"[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d75420bdb11de440","tex":"P_r^{\\rm total}","aliases":"P_r^rm total rm","display":"P_r^{\\rm total}","kind":"use","excerpt":"[Tensor charge refinement] For r∈ Z/N Z, define P_r^rm total = ∑_j_1+⋯+j_m=r P_j_1^(1)⊗⋯⊗ P_j_m^(m). Then the P_r^rm total form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5e4bf1229112d277","tex":"P_{\\rm even}","aliases":"P_rm even rm","display":"P_{\\rm even}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. Thi","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7ce1395e82f4cb81","tex":"P_{\\rm odd}","aliases":"P_rm odd rm","display":"P_{\\rm odd}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYTraditional edition: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. Thi","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bb3fe555e4ac9447","tex":"P_{\\alpha_0}","aliases":"P_α_0 alpha","display":"P_{\\alpha_0}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8fae8ad97b2a4d74","tex":"P_{\\alpha_1}","aliases":"P_α_1 alpha","display":"P_{\\alpha_1}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-302f53d86a380a68","tex":"P_{\\alpha_m}","aliases":"P_α_m alpha","display":"P_{\\alpha_m}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d81550d96775f196","tex":"P_{\\alpha_{m-1}}","aliases":"P_α_m-1 alpha","display":"P_{\\alpha_{m-1}}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9813ba63ac7cffd1","tex":"P_\\gamma","aliases":"P_γ gamma","display":"P_\\gamma","kind":"use","excerpt":"Using completeness and unitarity, ∑_β U_βα^*U_βγ = P_α U^*(∑_β P_β)UP_γ = P_α P_γ. For the second identity, sum over the source packets and use unitarity in the reverse order.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-17417a9415875e48","tex":"P_\\delta","aliases":"P_δ delta","display":"P_\\delta","kind":"use","excerpt":"Packet superoperator blocks are defined by Φ_αβ;γδ(X) = P_α Φ(P_γ XP_δ) P_β. Summing all four packet indices reconstructs Φ(X) exactly.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5ed30585279498bf","tex":"P_\\lambda","aliases":"P_λ lambda","display":"P_\\lambda","kind":"use","excerpt":", not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-00d2ac2360124aae","tex":"P_\\mu","aliases":"P_μ mu","display":"P_\\mu","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2cf1c6fcd2f6fc54","tex":"P_{\\ell+1,a}","aliases":"P_ℓ+1,a ell","display":"P_{\\ell+1,a}","kind":"use","excerpt":"[Exact packet direct limit] Assume the packet families have one fixed finite address set at every level and each packet address a is compatible in the sense P_ℓ+1,aR_ℓ=R_ℓ P_ℓ,a. Then there is an orthogonal projector P_∞,a on H_∞ satisfying P_∞,aι_ℓ=ι_ℓ P_ℓ,a. Finite completeness and orthogonality pass to the direct limit.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b5fecae50dac286a","tex":"P_{\\ell+1,\\beta}","aliases":"P_ℓ+1,β ell beta","display":"P_{\\ell+1,\\beta}","kind":"use","excerpt":"A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective s","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f8b027aabd4c6463","tex":"P_{\\infty,a}","aliases":"P_∞,a infty","display":"P_{\\infty,a}","kind":"use","excerpt":"[Exact packet direct limit] Assume the packet families have one fixed finite address set at every level and each packet address a is compatible in the sense P_ℓ+1,aR_ℓ=R_ℓ P_ℓ,a. Then there is an orthogonal projector P_∞,a on H_∞ satisfying P_∞,aι_ℓ=ι_ℓ P_ℓ,a. Finite completeness and orthogonality pass to the direct limit.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-53743af9d1feb8d1","tex":"\\mathrm{phys}","aliases":"phys mathrm","display":"\\mathrm{phys}","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a7a26e99cfb170bd","tex":"\\phys","aliases":"phys mathrm","display":"{\\mathrm{phys}}","kind":"use","excerpt":"Constraint closure needs a different obstruction. If C_a are self adjoint constraints and P_phys projects onto their common kernel, then P_phys[C_a,C_b]P_phys=0 holds automatically. This compression cannot test closure. The correct finite obstruction is the class of [C_a,C_b] in the quotient of End( H) by the span of the constraints. Vanishing mean","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"use","excerpt":"Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a949dbefc3833bd1","tex":"Q^\\perp","aliases":"Q^⊥ perp","display":"Q^\\perp","kind":"use","excerpt":"The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5ca75c044055616d","tex":"Q_a","aliases":"Q_a","display":"Q_a","kind":"use","excerpt":"vpage19Find the error without reading the logical stateFOUR EXACT SYNDROME PACKETSTraditional edition: Theorem II.8.2, Theorem II.8.4, Example II.8.8. Encode a qubit by ket0_L=ket000 and ket1_L=ket111. Let P project onto this code and take errors E_a∈{I,X_1,X_2,X_3}. Their images are four orthogonal rank-two packets Q_a=E_aPE_a^* that sum to the fu","count":12},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-55d4600438de195a","tex":"q_a","aliases":"q_a","display":"q_a","kind":"use","excerpt":"[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f717c588f6601060","tex":"Q_a^*","aliases":"Q_a^*","display":"Q_a^*","kind":"use","excerpt":"Expanding the finite product of sums gives T(A) T(B)= T(i↦ A_iB_i). Consequently, if two fine addresses differ in any coordinate, the corresponding product contains a zero factor. If the addresses agree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so Q_a^*=Q_a. Summing over all address functions","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a4591bb144ad9c3f","tex":"Q_c","aliases":"Q_c","display":"Q_c","kind":"use","excerpt":"Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-61c40f35d4a3b82d","tex":"q_F","aliases":"q_F","display":"q_F","kind":"use","excerpt":"Exact correction of an error span means that one completely positive trace preserving recovery R satisfies R(Fρ F^*)=q_Fρ for every F in that span and every code-supported matrix ρ, with a nonnegative scalar q_F depending on F but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restore","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero ","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a2dbd37e0ccec228","tex":"q_m","aliases":"q_m","display":"q_m","kind":"use","excerpt":"[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4a77ffe628ff771a","tex":"q_n","aliases":"q_n","display":"q_n","kind":"use","excerpt":"vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPETraditional edition: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ed14692ef5195be0","tex":"q_n^{j+k}","aliases":"q_n^j+k","display":"q_n^{j+k}","kind":"use","excerpt":"vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPETraditional edition: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a4210e21e79e5611","tex":"Q_r","aliases":"Q_r","display":"Q_r","kind":"use","excerpt":"[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9666ac009c3e72f6","tex":"q_r","aliases":"q_r","display":"q_r","kind":"use","excerpt":"[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cef20594acd8a578","tex":"\\mathfrak Q_\\ell","aliases":"Q_ℓ mathfrak ell","display":"\\mathfrak Q_\\ell","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-aaf174f6701cc732","tex":"\\Qcal","aliases":"Qcal Q mathcal","display":"{\\mathcal Q}","kind":"use","excerpt":"Let (P_a) and (Q_a) be two complete packet systems with the same index set. Define S( Q, P)=∑_aQ_aP_a.","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d4467d168eb66170","tex":"\\Qcal_{cd}","aliases":"Qcal_cd Q_cd Qcal mathcal","display":"{\\mathcal Q}_{cd}","kind":"use","excerpt":"Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ","display":null,"kind":"use","excerpt":"egative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i), so","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"Checked stronger statement. For every k,h∈ R with h≠0, 0≤ k^2-λ_k,h≤frack^4h^212. The code proves this without the low-mode restriction. The bound may be loose away from the low-mode regime, but it remains valid.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"ses and notation agree with the corrected finite core. The earlier abbreviated argument for error correction is replaced by the complete normalized and scaled recovery proofs above, and its substantive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"use","excerpt":"By [source reference], Z_d(t) =(4π t)^-d/2(1+R(t)), where R(t)=∑_n≠0 e^-absn^2/(4t) =O( e^-1/(4t)). Differentiation gives tR'(t)=O(t^-1 e^-1/(4t)). Substituting this estimate into the logarithmic derivative gives the stated small time expansion.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-055d5ee422435b80","tex":"R^*","aliases":"R^*","display":"R^*","kind":"use","excerpt":"vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILURETraditional edition: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5 4/5 · 3/5, Q_a=RP_aR^*, S=∑_a=0^1Q_aP_a=frac35R. The exact Gram matrices give (S-I)^*(S-I)=frac1625I, S^*S=frac925I, normS-I=frac45<1. Consequently the un","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a77b9a5a0ed9bc86","tex":"R^\\nabla","aliases":"R^∇ nabla","display":"R^\\nabla","kind":"use","excerpt":"Cancellation of the second derivatives. The product rule and the identity for the ambient Lie bracket reduce curvature to R^∇(X,Y)s=P d P[X] d s[Y]-P d P[Y] d s[X]. Substitute the identity obtained by differentiating the range condition for the section. The terms containing P d P[v]P vanish, leaving R^∇(X,Y)s= P( d P[X] d P[Y]- d P[Y] d P[X])P s. L","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"Short steps can be composed before reaching a distant endpoint. For example, the rational step R_1=mat4/5 · -3/5 3/5 · 4/5 obeys R_1^2=mat7/25 · -24/25","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b5c89dab9c39f1c3","tex":"R_1^*","aliases":"R_1^*","display":"R_1^*","kind":"use","excerpt":"The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery return","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-96f4b178fc8132b9","tex":"R_1^2","aliases":"R_1^2","display":"R_1^2","kind":"use","excerpt":"3/5 · 4/5 obeys R_1^2=mat7/25 · -24/25 24/25 · 7/25. The multiplication is exact and each step has a valid local overlap. The convergence theorem has a separate analytic contract: a twice continuously differentiable complete projector path on a compact time interval, sufficiently fine partitions, and a constant depending on the path in an O(|π|) es","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery return","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-689524629f5e72c1","tex":"R_2^*","aliases":"R_2^*","display":"R_2^*","kind":"use","excerpt":"The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery return","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-33462a02fcce1a84","tex":"R_a","aliases":"R_a","display":"R_a","kind":"use","excerpt":"rthogonal rank-two packets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an actual channel T","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-200e88e9f4a8f4ed","tex":"R_a^*","aliases":"R_a^*","display":"R_a^*","kind":"use","excerpt":"ets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an actual channel The operators R_a=E_a^*Q","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5dae4eff487bb83e","tex":"R_{\\mathrm{extra}}","aliases":"R_extra mathrm","display":"R_{\\mathrm{extra}}","kind":"use","excerpt":"The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary d","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d7fa83f4a07c4cba","tex":"R_{\\mathrm{extra}}^*","aliases":"R_extra^* mathrm","display":"R_{\\mathrm{extra}}^*","kind":"use","excerpt":"The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary d","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"use","excerpt":"vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hil","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1b85fe58dfd4105d","tex":"R_P","aliases":"R_P","display":"R_P","kind":"use","excerpt":"Retaining the bracket of vector fields. Define the actual curvature action by R_P(X,Y)s=∇_X∇_Ys-∇_Y∇_Xs-∇_[X,Y]s. The source uses the Lie bracket defined through Fr echet derivatives. Symmetry of second derivatives is supplied by the C^2 hypothesis, not postulated separately. Expanding the connection gives R_P(X,Y)s=P(DP[X]Ds[Y]-DP[Y]Ds[X]). Substi","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1e4761712190b8fb","tex":"R_r","aliases":"R_r","display":"R_r","kind":"use","excerpt":"recise finite channel statement Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5efa3f3752f2e9d1","tex":"R_r^*","aliases":"R_r^*","display":"R_r^*","kind":"use","excerpt":"recise finite channel statement Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b11c81e6903fa9bf","tex":"\\Ran","aliases":"Ran operatorname","display":"{\\operatorname{Ran}}","kind":"use","excerpt":"textitRange coordinates. Fix a∈ M and put T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). Then T_a(a)=I and P(x)T_a(x)=T_a(x)P(a). Invertibility is open, so on a neighborhood of a, T_a(x) restricts to an isomorphism from Ran P(a) to Ran P(x). This proves local constancy of rank. The local coordinates and their transitions are (x,v)⟼ (x,T_a(x)^-1v), u⟼ T_b(x)^-1T","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c0941f5ac249959a","tex":"\\Rcal_0","aliases":"Rcal_0 R_0 Rcal mathcal","display":"{\\mathcal R}_0","kind":"use","excerpt":"[Exact recovery on the error span] Under the hypotheses of [source reference], define on the syndrome support Q=∑_λ_j>0Q_j the completely positive map R_0(X) = ∑_λ_j>0 V_j^*Q_jXQ_jV_j. It extends to a quantum channel R on End( H). If Φ(X)=∑_aE_aXE_a^* is trace preserving on code states, then RΦ(ρ)=ρ for every density matrix ρ=Pρ P.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ff81ba5e23d549c7","tex":"\\Re","aliases":"Re","display":"\\Re","kind":"use","excerpt":"ity. Starting from ket0, there are two resolved paths to output ket1. Their signed products are bra1H_rm Hadket0bra0H_rm Hadket0=frac12, bra1H_rm Hadket1bra1H_rm Hadket0=-frac12. Their sum is zero. Squaring each separately gives a different answer. vfiginterference headWhere the missing term goes For path amplitudes a,b, the outcome probability is ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-27fc32faa85e4ff0","tex":"\\operatorname{Row}","aliases":"Row operatorname","display":"\\operatorname{Row}","kind":"use","excerpt":"A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e31cead1bdb80910","tex":"S^{-r}","aliases":"S^-r","display":"S^{-r}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c8d0b37b4af3bfcc","tex":"S^2","aliases":"S^2","display":"S^2","kind":"use","excerpt":"age27One calculation of transfer to the opposite siteSEEDS, CHARACTERS, INTERFERENCE, AND DEPHASINGTraditional edition: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4. Return to the shift on four sites S and Laplacian L=2I-S-S^*. The same character projectors have energies (0,2,4,2). At time t=π/2, their evolution phases are (1,-1,1,-1), exactly ","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e9818719b46fe474","tex":"S^N","aliases":"S^N","display":"S^N","kind":"use","excerpt":"Let N≥1 be an integer, and let S be a unitary symmetry whose order divides N: S^N=I. Set ω=e^2π i/N.","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8768859caa7d360e","tex":"S^r","aliases":"S^r","display":"S^r","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to ","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e1b9361af622cca4","tex":"S^{r+s}","aliases":"S^r+s","display":"S^{r+s}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-569a7d9157ab32ee","tex":"S^{r+s\\bmod N}","aliases":"S^r+sbmod N bmod","display":"S^{r+s\\bmod N}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ac11a04bc7722918","tex":"S^{\\operatorname{val}(r)}","aliases":"S^val(r) operatorname","display":"S^{\\operatorname{val}(r)}","kind":"use","excerpt":"003 Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val(r). Reduction of exponents modulo N ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"imitation of testing only the common kernel on pagerefV26. takeawayExact finite structure is valuable because its boundary is explicit. The appendix adds understanding and reproducibility, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Th","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-785e812c9b93033d","tex":"S_a","aliases":"S_a","display":"S_a","kind":"use","excerpt":"Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-68387971995ff129","tex":"S_b","aliases":"S_b","display":"S_b","kind":"use","excerpt":"Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-87a2b7a1a86d2b97","tex":"S_z","aliases":"S_z","display":"S_z","kind":"use","excerpt":"-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e79192bebf87a0c5","tex":"S_z^{-1}","aliases":"S_z^-1","display":"S_z^{-1}","kind":"use","excerpt":"Correct statement: source Theorem 9.1. Assume both D_z and zI-H are invertible. Then S_z is invertible and ((zI-H)^-1)_PP=S_z^-1. The left side is the submatrix of the full inverse on the retained coordinates. This matches the paper's compressed operator when written in an orthonormal basis of the retained subspace. The proof is valid for arbitrary","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"vpage11Quantify the amplitude that escapesCOMMUTATORS AND PROBABILITY BOUNDSTraditional edition: Theorem II.5.3, Corollary II.5.4. For a two-level system, choose H=ε X, P=ket0bra0, and Q=I-P. Direct exponentiation gives e^- i tH=cos(ε t)I- isin(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary b","count":45},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"use","excerpt":"irect exponentiation gives e^- i tH=cos(ε t)I- isin(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequ","count":16},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span","display":null,"kind":"use","excerpt":"ment changes the local spanSCHMIDT RANK AS ACCESSIBLE MULTIPLICITYTraditional edition: Theorem II.2.2, Corollary II.2.4. Write a bipartite vector as a coefficient matrix C, with the first subsystem indexing rows. Acting only on that subsystem transforms C to AC. Every resulting row remains in the row space of C, and every independent choice of such","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e4b59b30017e0c86","tex":"\\operatorname{span}_{\\CC}","aliases":"span_CC span_ C operatorname CC operatorname mathbb","display":"\\operatorname{span}_{{\\mathbb C}}","kind":"use","excerpt":"Let C_1,…,C_m be self adjoint constraint operators on a finite dimensional Hilbert space. The distinction between constraint action and constraint closure is standard in constrained Hamiltonian theory [source reference]. Write c=span_ C{C_1,…,C_m} ⊆End( H).","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5945fa8a59c08e31","tex":"\\spanof","aliases":"spanof span operatorname","display":"{\\operatorname{span}}","kind":"use","excerpt":"The matrix-orbit dimension and the coefficient equivalence now yield dim_ Cspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r(ψ).","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-982328590b226ee9","tex":"\\Spec_\\R","aliases":"Spec_R Spec_ R Spec R operatorname mathbb","display":"{\\operatorname{Spec}}_{\\mathbb R}","kind":"use","excerpt":"tual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator t","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a46e78b4903b3de3","tex":"\\operatorname{srank}","aliases":"srank operatorname","display":"\\operatorname{srank}","kind":"use","excerpt":"dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence k≥lceil m/drceil. Partitioning a multiplicity basis into groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-22393d9dec0e79d0","tex":"\\operatorname{srank}_{\\AA}","aliases":"srank_AA operatorname AA","display":null,"kind":"use","excerpt":"dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence k≥lceil m/drceil. Partitioning a multiplicity basis into groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fcb3fb6c02502457","tex":"\\operatorname{srank}_{\\Acal}","aliases":"srank_Acal srank_ A operatorname Acal operatorname mathcal","display":"\\operatorname{srank}_{{\\mathcal A}}","kind":"use","excerpt":"[Quantum seed rank] For the displayed multiplicity decomposition, srank_ A( H) = max_1≤ r≤ s lceilfracm_rd_rrceil.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-af678bb5870a8707","tex":"\\operatorname{srank}_{\\cA}","aliases":"srank_cA srank_ A operatorname cA operatorname mathcal","display":"\\operatorname{srank}_{{\\mathcal A}}","kind":"use","excerpt":"[Quantum seed-rank formula] For A=⊕_i(M_d_i( C)⊗ I_m_i) on H=⊕_i( C^d_i⊗ C^m_i), the minimum number of seeds required to generate H is srank_ A( H) = max_ilceilfracm_id_irceil.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-32f99fc05b64a106","tex":"\\succ","aliases":"succ","display":"\\succ","kind":"use","excerpt":"A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective s","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"use","excerpt":"Matrix multiplication distributes across addition and scalar multiplication, which gives complex linearity. If ρsucceq0, each term K_aρ K_a^* is positive semidefinite, hence so is their finite sum.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c9c6f71937fef238","tex":"\\sup_{0\\leq t\\leq T}","aliases":"sup_0≤ t≤ T sup leq leq","display":"\\sup_{0\\leq t\\leq T}","kind":"use","excerpt":"Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c901ab4425f0be8a","tex":"\\sup_{\\abs t\\leq T}","aliases":"sup_abs t≤ T sup abs leq","display":null,"kind":"use","excerpt":"[Norm summable Hamiltonian tower] Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8f99188f5481208b","tex":"\\sup_\\ell","aliases":"sup_ℓ sup ell","display":"\\sup_\\ell","kind":"use","excerpt":"[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9e6ce97799490c6a","tex":"\\t","aliases":"t","display":null,"kind":"use","excerpt":"draw[->] (1,1.6)--(11,1.6);foreach x/t in 3/9/10,6/1,9/11/10 fill (x,1.6) circle (0.08);node[below] at (x,1.5) t; draw[<->] (3,2.5)--(6,2.5);draw[<->] (6,2.5)--(9,2.5); node[above] at (4.5,2.5) 1/10;node[above] at (7.5,2.5) 1/10; node[font=small] at (6,0.45) one pair: 1/10, full family {I,A}: 1/5;","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5d42311231194d5c","tex":"t^{-1}","aliases":"t^-1","display":"t^{-1}","kind":"use","excerpt":"[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e29220aa3d532708","tex":"T^{n_j}","aliases":"T^n_j","display":"T^{n_j}","kind":"use","excerpt":"tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2042ebdb7ea53ef4","tex":"T^{n_j+n_k}","aliases":"T^n_j+n_k","display":"T^{n_j+n_k}","kind":"use","excerpt":"tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bdc3d343e0ba3452","tex":"T^{n_k}","aliases":"T^n_k","display":"T^{n_k}","kind":"use","excerpt":"Since T is positive and self adjoint, ipv_jT^n_j+n_kv_k = ipT^n_jv_jT^n_kv_k. The quadratic form is the squared norm of ∑_kc_kT^n_kv_k.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the completion proof constructs T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). It equals the identity at the anchor and intertwines the two projectors. Where it is invertible, its restriction provides local fiber coordinates. vfigrange_atlas headAn exact example of the domain restriction For the rot","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","excerpt":"[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here th","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4a9e32fb56cb7f6c","tex":"t_a","aliases":"t_a","display":"t_a","kind":"use","excerpt":"ens also to have nonnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-911d3e6f76681614","tex":"T_{a+b}","aliases":"T_a+b","display":"T_{a+b}","kind":"use","excerpt":"Composition gives T_aT_b=T_a+b, and direct evaluation gives T_aket b=keta+b. Every binary address satisfies a+a=0, so each translation is an involution. In particular, the local operator changes precisely the addressed bit: X_jket a=keta with coordinate j replaced by a_j+1. The selected factor at j equals translation by the address supported at j w","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cefa01be96058c3e","tex":"T_b","aliases":"T_b","display":"T_b","kind":"use","excerpt":"textitRange coordinates. Fix a∈ M and put T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). Then T_a(a)=I and P(x)T_a(x)=T_a(x)P(a). Invertibility is open, so on a neighborhood of a, T_a(x) restricts to an isomorphism from Ran P(a) to Ran P(x). This proves local constancy of rank. The local coordinates and their transitions are (x,v)⟼ (x,T_a(x)^-1v), u⟼ T_b(x)^-1T","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bb1ad3b8487cb395","tex":"T_e","aliases":"T_e","display":"T_e","kind":"use","excerpt":"The actual Euclidean transfer exponential. Let H be a Hermitian matrix on a nonzero complex state space of finite dimension. The source defines T_e(t)=exp(-t(H-eI)) using Lean's exponential defined by a power series. The proof then relates this defined operator to its spectral expansion.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9c3ad1a3dd202746","tex":"T_{E_0}","aliases":"T_E_0","display":"T_{E_0}","kind":"use","excerpt":"Positivity, semigroup composition, and the interval. Every eigenvalue of T_e(t) is strictly positive, for arbitrary real e,t, since the real exponential is positive. For e=E_0 and t≥0 it is also at most one. Therefore T_E_0(t)>0, Spec T_E_0(t)⊆(0,1]. The two scalar multiples of H-eI commute, so the exponential defined by the power series obeys T_e(","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f08f28427ff60b68","tex":"T_{e_j}","aliases":"T_e_j","display":"T_{e_j}","kind":"use","excerpt":"0 · x≠ a. The seed is ψ_0=ket0. Define a translation operator by (T_a v)(x)=v(x-a). At position j, the local Pauli X is X_j=T_e_j, where e_j has a single nonzero binary coordinate. The formal circuit uses the identity when a_j=0 and X_j when a_j=1.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-25bdedf7497647d3","tex":"t_j","aliases":"t_j","display":"t_j","kind":"use","excerpt":"4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structu","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b148baa42903d2fe","tex":"t_{j+1}","aliases":"t_j+1","display":"t_{j+1}","kind":"use","excerpt":"es S( P(t+h), P(t)) = I+hK(t)+O(h^2) uniformly on [0,T]. Since K(t)^*=-K(t), S^*S=I+O(h^2), so the polar factor satisfies U( P(t+h), P(t)) = I+hK(t)+O(h^2). This gives a first order approximation to dot W=KW over a single time step. A telescoping product estimate over the partition gives a total error bounded by a constant times ∑_j(t_j+1-t_j)^2, w","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-830c9c56b4f2bf21","tex":"t_k","aliases":"t_k","display":"t_k","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-733902ab0028bba1","tex":"t_m","aliases":"t_m","display":"t_m","kind":"use","excerpt":"[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-64afaf0ed6cbcadc","tex":"T_n","aliases":"T_n","display":"T_n","kind":"use","excerpt":"For transfers on two states T_n=diag(1,q_n)=e^-H_n, with H_n=diag(0,-log q_n) at unit time, the error in trace norm is 2^-(n+2) and the heat traces converge to 3/2. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat t","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-163dd25fe6cb484d","tex":"t_N","aliases":"t_N","display":"t_N","kind":"use","excerpt":"It does not assert a uniform interval width from that argument, a moving time t_N with no extra estimates, or a dimension plateau at every scale. At fixed finite N, d_N(t)→0 as tdownarrow0 and as t→∞. By contrast, first taking N→∞ at positive t and then taking tdownarrow0 yields the continuum dimension. headWhere the infinite tail is controlled At ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c2ed7a9fa236bbf0","tex":"t_n","aliases":"t_n","display":"t_n","kind":"use","excerpt":"[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-839c9c5ecd29436a","tex":"t_{n-1}","aliases":"t_n-1","display":"t_{n-1}","kind":"use","excerpt":"[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here th","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1927047d884ba88c","tex":"T_P","aliases":"T_P","display":"T_P","kind":"use","excerpt":"Deriving the estimate by differentiation. For a skew-adjoint generator G, define the actual conjugation path T_P(t)=exp(tG)Pexp(-tG). The formalization derives its derivative from the exponential power series and the product rule: T_P'(t)=exp(tG)(GP-PG)exp(-tG). Both exponential factors are proved unitary. Left and right multiplication by a unitary","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-81d0a2ee84c8c5ee","tex":"T_P'","aliases":"T_P'","display":"T_P'","kind":"use","excerpt":"Deriving the estimate by differentiation. For a skew-adjoint generator G, define the actual conjugation path T_P(t)=exp(tG)Pexp(-tG). The formalization derives its derivative from the exponential power series and the product rule: T_P'(t)=exp(tG)(GP-PG)exp(-tG). Both exponential factors are proved unitary. Left and right multiplication by a unitary","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4ab411d9663afe22","tex":"t_w","aliases":"t_w","display":"t_w","kind":"use","excerpt":"We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormali","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e324d98831d6eac7","tex":"t_\\varepsilon","aliases":"t_ε varepsilon","display":"t_\\varepsilon","kind":"use","excerpt":"4 · (0,32,32,64) · 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less than ε/2. Th","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"A_n _1=TrA_n=1; draw[->] (4.7,1.9)--(6.7,1.9);node[font=small] at (6,0.45) positive trace-class control: norm convergence alone does not preserve a trace;","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fc7a5c603b95f30d","tex":"\\tr","aliases":"tr operatorname","display":"{\\operatorname{tr}}","kind":"use","excerpt":"Measurements: source Theorem 7.1. For finite complex matrices, define packet dephasing by D_P(X)=∑_aP_aXP_a. The code proves idempotence and trace duality: D_P( D_P(X))= D_P(X), tr( D_P(E)ρ)=tr(E D_P(ρ)). Trace duality follows term by term from cyclicity: tr(P_aEP_aρ)=tr(EP_aρ P_a). Therefore an effect that is diagonal in the packets, D_P(E)=E, sat","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9bbf4da095ddbb7b","tex":"\\Tr_A","aliases":"Tr_A Tr operatorname","display":"{\\operatorname{Tr}}_A","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-03d1d351ea22341e","tex":"\\Tr_{\\rm out}","aliases":"Tr_rm out Tr rm operatorname rm","display":"{\\operatorname{Tr}}_{\\rm out}","kind":"use","excerpt":"labelstrong:standardfact:001 A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"ORS AND PROBABILITY BOUNDSTraditional edition: Theorem II.5.3, Corollary II.5.4. For a two-level system, choose H=ε X, P=ket0bra0, and Q=I-P. Direct exponentiation gives e^- i tH=cos(ε t)I- isin(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the ampli","count":27},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1296a5898d2e7d5f","tex":"U^*","aliases":"U^*","display":"U^*","kind":"use","excerpt":"1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i","count":23},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-91c677571574a0b9","tex":"u_a","aliases":"u_a","display":"u_a","kind":"use","excerpt":"splayed T happens also to have nonnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive k","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-941e7992e7bd04e6","tex":"U_{ar}","aliases":"U_ar","display":"U_{ar}","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d5ee5bb5c320ccb3","tex":"U_b","aliases":"U_b","display":"U_b","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e21dd43c611f6c4e","tex":"u_b","aliases":"u_b","display":"u_b","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6d427e9012af5c75","tex":"U_{ba}","aliases":"U_ba","display":"U_{ba}","kind":"use","excerpt":"The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading ","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b42126afe8df2055","tex":"U_{ba}^*","aliases":"U_ba^*","display":"U_{ba}^*","kind":"use","excerpt":"The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-71076ebfa81f46b5","tex":"U_{bc}","aliases":"U_bc","display":"U_{bc}","kind":"use","excerpt":"The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading ","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f763d6d1daaa1789","tex":"U_{da}","aliases":"U_da","display":"U_{da}","kind":"use","excerpt":"Adjoint conservation: source Theorem 4.2. Assume a star operation with P_a^*=P_a. Then (U_ba)^*=P_aU^*P_b. The two checked conservation laws are U^*U=1 · Longrightarrow ∑_b(U_ba)^*U_bc=δ_acP_a, UU^*=1 · Longrightarrow ∑_aU_ba(U_da)^*=δ_bdP_b. For the first law, expand each corner, collapse P_b^2 and sum: ∑_bP_aU^*P_bUP_c =P_aU^*(∑_bP_b)UP_c=P_aP_c.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-26bee56e7d89a1c3","tex":"U_{da}^*","aliases":"U_da^*","display":"U_{da}^*","kind":"use","excerpt":"[Packet unitarity identities] If U is unitary and U_ba=P_bUP_a, then ∑_bU_ba^*U_bc=δ_acP_a and ∑_aU_baU_da^*=δ_bdP_b.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"trivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Eu","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-836830fc78b794aa","tex":"u_k","aliases":"u_k","display":"u_k","kind":"use","excerpt":"tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-25121c20156ac918","tex":"U_m","aliases":"U_m","display":"U_m","kind":"use","excerpt":"Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-65a04898bfc53b4d","tex":"U_{m-1}","aliases":"U_m-1","display":"U_{m-1}","kind":"use","excerpt":"Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b207f2554f44324f","tex":"u_N","aliases":"u_N","display":"u_N","kind":"use","excerpt":"n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. T","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dbd28c057692283b","tex":"u_N^*","aliases":"u_N^*","display":"u_N^*","kind":"use","excerpt":"ojector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8611e5a338fe3b84","tex":"U_r","aliases":"U_r","display":"U_r","kind":"use","excerpt":"[Charge pruning of circuit paths] Suppose every gate has a finite charge decomposition U_r=∑_qU_r^(q), SU_r^(q)S^*=ω^qU_r^(q). A term with gate charges q_1,…,q_m can contribute from source charge a_0 only to target charge a_m=a_0+q_1+⋯+q_m. Every packet path violating one of the intermediate charge updates vanishes.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4a59996f84749e99","tex":"U_r^{(q)}","aliases":"U_r^(q)","display":"U_r^{(q)}","kind":"use","excerpt":"[Charge pruning of circuit paths] Suppose every gate has a finite charge decomposition U_r=∑_qU_r^(q), SU_r^(q)S^*=ω^qU_r^(q). A term with gate charges q_1,…,q_m can contribute from source charge a_0 only to target charge a_m=a_0+q_1+⋯+q_m. Every packet path violating one of the intermediate charge updates vanishes.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a6aa6e43e29fe528","tex":"U_{\\beta\\alpha}","aliases":"U_βα beta alpha","display":"U_{\\beta\\alpha}","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-73c14771543aa3ee","tex":"U_{\\beta\\alpha}^*","aliases":"U_βα^* beta alpha","display":"U_{\\beta\\alpha}^*","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1befa0e66e68ea65","tex":"U_{\\beta\\gamma}","aliases":"U_βγ beta gamma","display":"U_{\\beta\\gamma}","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-17bbb1fd2a57ab85","tex":"U_\\gamma","aliases":"U_γ gamma","display":"U_\\gamma","kind":"use","excerpt":"For a packet path γ=(a_0,a_1,…,a_m), write U_γ = P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-80535093685f5ad8","tex":"U_{\\delta\\alpha}^*","aliases":"U_δα^* delta alpha","display":"U_{\\delta\\alpha}^*","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-86e4d773f576d1fc","tex":"\\mathrm{unused}","aliases":"unused mathrm","display":"\\mathrm{unused}","kind":"use","excerpt":"efine compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are com","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"use","excerpt":"[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"use","excerpt":"main inside the original code, so the previous syndrome partition reports no bit flip. frac3ket000+4ket1115 ⟼ frac3ket000-4ket1115. The populations are unchanged, but the relative phase is not. vfiglogical_error Let V be the code isometry with columns ket000,ket111. For the proposed error family {I,Z_1}, its off-diagonal error compression is V^*I^*","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-55c61b51fd70a8e2","tex":"v^*","aliases":"v^*","display":"v^*","kind":"use","excerpt":"spectrum {1,1/2}.; node[font=small,align=center,font=small] at (7,0.4) Positive-definite operator: v^*Tv>0 for v≠0. Not an entrywise test.;","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"anchor=east,font=scriptsize] at (0.85,1.2) site 3; node[font=small,align=center,font=scriptsize] at (1.4,4.23) j=0; node[font=small,align=center,font=scriptsize] at (2.2,4.23) j=1; node[font=small,align=center,font=scriptsize] at (3,4.23) j=2; node[font=small,align=center,font=scriptsize] at (3.8,4.23) j=3; node[font=small,align=center,font=small] ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-915b892e2bee2f05","tex":"v_0^*","aliases":"v_0^*","display":"v_0^*","kind":"use","excerpt":"gn=center,font=scriptsize] at (1.4,4.23) j=0; node[font=small,align=center,font=scriptsize] at (2.2,4.23) j=1; node[font=small,align=center,font=scriptsize] at (3,4.23) j=2; node[font=small,align=center,font=scriptsize] at (3.8,4.23) j=3; node[font=small,align=center,font=small] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divide by zero.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-93f3b201116caa00","tex":"v_1^*","aliases":"v_1^*","display":"v_1^*","kind":"use","excerpt":",font=scriptsize] at (3,4.23) j=2; node[font=small,align=center,font=scriptsize] at (3.8,4.23) j=3; node[font=small,align=center,font=small] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"scriptsize] at (3.8,4.23) j=3; node[font=small,align=center,font=small] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c03d39f1ea842378","tex":"v_2^*","aliases":"v_2^*","display":"v_2^*","kind":"use","excerpt":"ll] at (8.05,3.8) S v_0=1v_0; node[font=small,align=center,font=small] at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"at (11.85,3.8) P_0=v_0v_0^*; node[font=small,align=center,font=small] at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) P_2=v_2v_2^*; node[font=small,align=center,font=small] at ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-aeda5a26750be02f","tex":"v_3^*","aliases":"v_3^*","display":"v_3^*","kind":"use","excerpt":"at (8.05,2.95) S v_1=iv_1; node[font=small,align=center,font=small] at (11.85,2.95) P_1=v_1v_1^*; node[font=small,align=center,font=small] at (8.05,2.1) S v_2=-1v_2; node[font=small,align=center,font=small] at (11.85,2.1) P_2=v_2v_2^*; node[font=small,align=center,font=small] at (8.05,1.25) S v_3=- iv_3; node[font=small,align=center,font=small] at ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a478c8ae160947ff","tex":"V_a","aliases":"V_a","display":"V_a","kind":"use","excerpt":"A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-93a28f0d222e68e7","tex":"v_a","aliases":"v_a","display":"v_a","kind":"use","excerpt":"An independent definition of Schmidt rank. The source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length s is the concrete equality C_ij=∑_a=1^su_a(i)v_a(j). It factors C=UV through an s-dimensional intermediate space, so rank C≤ s. Conversely, choose a basis of the row space of C. Expanding eac","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-56405f2e425e038c","tex":"V_a^*","aliases":"V_a^*","display":"V_a^*","kind":"use","excerpt":"A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2d8b529eb2a0a517","tex":"V_b","aliases":"V_b","display":"V_b","kind":"use","excerpt":"A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3c64cfa0416300f6","tex":"V_b^*","aliases":"V_b^*","display":"V_b^*","kind":"use","excerpt":"Exact correction, including coherent errors. For every operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-588c6f8fdc8039b0","tex":"V_{ca}","aliases":"V_ca","display":"V_{ca}","kind":"use","excerpt":"0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"use","excerpt":"syndrome construction] Assume the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syn","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to ","count":10},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-eb483cd80b51511d","tex":"V_j^*","aliases":"V_j^*","display":"V_j^*","kind":"use","excerpt":"the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero ","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-958542b62a8a99b9","tex":"v_j^*","aliases":"v_j^*","display":"v_j^*","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0a435f82d4918621","tex":"v_k","aliases":"v_k","display":"v_k","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b00eda2d2588cd09","tex":"v_m","aliases":"v_m","display":"v_m","kind":"use","excerpt":"[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a6735e9f71ad2ebf","tex":"V_n","aliases":"V_n","display":"V_n","kind":"use","excerpt":"For n≥ 0, let A_n=( Z/2 Z)^{0,…,n-1}, V_n={v:A_n→ C}. For an address a∈ A_n, define the standard coordinate vector ket a(x)=1 · x=a,","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5df6cd07cd936c95","tex":"V_s","aliases":"V_s","display":"V_s","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7c997f5b198e5681","tex":"V_\\mu","aliases":"V_μ mu","display":"V_\\mu","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-02e87248f01e05be","tex":"V_\\ell","aliases":"V_ℓ ell","display":"V_\\ell","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-022a12d8dd418b72","tex":"\\operatorname{val}","aliases":"val operatorname","display":"\\operatorname{val}","kind":"use","excerpt":"Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges con","count":32},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-73ffecb5618fb3cc","tex":"W^*","aliases":"W^*","display":"W^*","kind":"use","excerpt":"0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome su","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d8fe8dd2410be8cc","tex":"w^*","aliases":"w^*","display":"w^*","kind":"use","excerpt":"0 · 0 · 0 · 1. Its middle block has one negative eigenvalue. More concretely, the normalized vector w=(ket01-ket10)/sqrt2 satisfies w^*(id⊗ T)(ρ_rm Bell)w=-frac12. This exact negative value of the quadratic form certifies failure of positivity on the composite system. No numerical eigenvalue tolerance is needed. headWhy the Kraus form avoids this p","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-79b56eb917fecb80","tex":"W_{011}","aliases":"W_011","display":"W_{011}","kind":"use","excerpt":"node[font=small,align=center] at (2.65,4.05) ket110; path[draw,fill=white] (5.53,3.75) rectangle (6.47,4.35); node[font=small,align=center] at (6,4.05) ket111; node[font=small,align=center,font=small] at (10.15,4.1) Three actual words; node[font=small,align=center,font=small] at (10.15,3.1) W_101=X_1X_3; node[font=small,align=center,font=small] at ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1689504e92758fb9","tex":"W_{101}","aliases":"W_101","display":"W_{101}","kind":"use","excerpt":",1.6) ket101; path[draw,fill=white] (2.18,3.75) rectangle (3.12,4.35); node[font=small,align=center] at (2.65,4.05) ket110; path[draw,fill=white] (5.53,3.75) rectangle (6.47,4.35); node[font=small,align=center] at (6,4.05) ket111; node[font=small,align=center,font=small] at (10.15,4.1) Three actual words; node[font=small,align=center,font=small] at","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-448292cc37c44761","tex":"W_{110}","aliases":"W_110","display":"W_{110}","kind":"use","excerpt":"hite] (5.53,3.75) rectangle (6.47,4.35); node[font=small,align=center] at (6,4.05) ket111; node[font=small,align=center,font=small] at (10.15,4.1) Three actual words; node[font=small,align=center,font=small] at (10.15,3.1) W_101=X_1X_3; node[font=small,align=center,font=small] at (10.15,2.2) W_011=X_2X_3; node[font=small,align=center,font=small] at","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0289b7ef692ee27e","tex":"W_a","aliases":"W_a","display":"W_a","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges con","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9012166380bed7e0","tex":"w_a","aliases":"w_a","display":"w_a","kind":"use","excerpt":"II.8.4, Example II.8.8. Encode a qubit by ket0_L=ket000 and ket1_L=ket111. Let P project onto this code and take errors E_a∈{I,X_1,X_2,X_3}. Their images are four orthogonal rank-two packets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy stat","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0cf3e6c1daea5239","tex":"W_\\pi","aliases":"W_π pi","display":"W_\\pi","kind":"use","excerpt":"[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0aa7779c656ca58e","tex":"\\widehat C_A","aliases":"widehat C_A widehat","display":"\\widehat C_A","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c7f697c3c006092e","tex":"\\widehat C_B","aliases":"widehat C_B widehat","display":"\\widehat C_B","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-62e85424e4f5a5ba","tex":"\\widehat C_C","aliases":"widehat C_C widehat","display":"\\widehat C_C","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f596b26bb2804e7d","tex":"\\widetilde B","aliases":"widetilde B widetilde","display":"\\widetilde B","kind":"use","excerpt":"In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,wid","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9193c44a5528a66b","tex":"\\widetilde B^*","aliases":"widetilde B^* widetilde","display":"\\widetilde B^*","kind":"use","excerpt":"[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-47c52f415949a6cb","tex":"\\widetilde C","aliases":"widetilde C widetilde","display":"\\widetilde C","kind":"use","excerpt":"ull algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R:","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3af8f2cbaa29419e","tex":"\\widetilde C_a","aliases":"widetilde C_a widetilde","display":"\\widetilde C_a","kind":"use","excerpt":"ull algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R:","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ad6dd274a454509b","tex":"\\widetilde C_b","aliases":"widetilde C_b widetilde","display":"\\widetilde C_b","kind":"use","excerpt":"C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d73e807a5c23f393","tex":"\\widetilde C_x","aliases":"widetilde C_x widetilde","display":"\\widetilde C_x","kind":"use","excerpt":"e coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does not close on the whole fine level: on the added coordinate the commutator is zero, whereas 2 iwidetilde C_z is 2 i. Exact transport constrains the image of R and leaves its orthogonal complement as separate data. headAn approximate residual ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0108cdcacfc6d3f3","tex":"\\widetilde C_z","aliases":"widetilde C_z widetilde","display":"\\widetilde C_z","kind":"use","excerpt":"the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does not close on the whole fine level: on the added coordinate the commutator is","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d4a13a7052f06326","tex":"\\widetilde D","aliases":"widetilde D widetilde","display":"\\widetilde D","kind":"use","excerpt":"In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,wid","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f92312b578cf8233","tex":"\\widetilde D^*","aliases":"widetilde D^* widetilde","display":"\\widetilde D^*","kind":"use","excerpt":"[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a1af56e8fd99612c","tex":"\\widetilde\\Sigma","aliases":"widetildeΣ widetilde Sigma","display":"\\widetilde\\Sigma","kind":"use","excerpt":"The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings ar","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-71a31718ae729ffa","tex":"X^*","aliases":"X^*","display":"X^*","kind":"use","excerpt":"Density matrices and observables carry both an input packet and an output packet. Equip End( H) with the Hilbert Schmidt inner product ipXY_HS=Tr(X^*Y).","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6e7e9fdf289546b6","tex":"x^*","aliases":"x^*","display":"x^*","kind":"use","excerpt":"Complex polarization reaches the whole operator space. Write q(x)=xx^*. The module proves the exact identity 4xy^*=q(x+y)-q(x-y) +i(q(x+iy)-q(x-iy)). Agreement on all positive pure outer products therefore implies agreement on every outer product. Actual matrix units then span the full matrix space. Applying this argument to the sandwich by P prove","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-78dcbad828a95e4b","tex":"X^{-1}","aliases":"X^-1","display":"X^{-1}","kind":"use","excerpt":"Use the inverse difference identity X^-1-Y^-1=X^-1(Y-X)Y^-1 with X=F_P(z) and Y=z-A.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Put x=kh/2. The inequalities |sin x|≤|x| and |x-sin x|≤ |x|^3/6, which hold for every real argument, give 0≤ x^2-sin^2x =|x-sin x| |x+sin x| ≤frac|x|^36 2|x| =fracx^43. Multiplying by 4/h^2 proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps k fixed while h→0, so it does not assert a u","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"use","excerpt":"For x∈[0,π/2], sin x≤ x and sin x≥ x-fracx^36. Thus 0≤ x^2-sin^2x≤fracx^43. Apply this with x=πabsk_j/N, multiply by 4N^2, and sum over j.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b5d15c510169ad7c","tex":"x^4","aliases":"x^4","display":"x^4","kind":"use","excerpt":"Put x=kh/2. The inequalities |sin x|≤|x| and |x-sin x|≤ |x|^3/6, which hold for every real argument, give 0≤ x^2-sin^2x =|x-sin x| |x+sin x| ≤frac|x|^36 2|x| =fracx^43. Multiplying by 4/h^2 proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps k fixed while h→0, so it does not assert a u","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9f89be2c73732f99","tex":"X_{(i,\\cdot),(j,\\cdot)}","aliases":"X_(i,·),(j,·) cdot cdot","display":"X_{(i,\\cdot),(j,\\cdot)}","kind":"use","excerpt":"To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies th","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"bottomrule headNormalization and indexing The tensor factors are ordered from left to right, so X_1=X⊗ I⊗ I. A normalized pure state gives ρ=ψψ^*. A coefficient matrix used to count a seed orbit need not be normalized. Scaling a nonzero seed does not change its generated linear span.","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges con","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-14a93f6cb614d651","tex":"X_2^{a_2}","aliases":"X_2^a_2","display":"X_2^{a_2}","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges con","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8eeb0a1bbb468424","tex":"X_3","aliases":"X_3","display":"X_3","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges con","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-298640d8b28fac27","tex":"X_3^{a_3}","aliases":"X_3^a_3","display":"X_3^{a_3}","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARTraditional edition: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges con","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite num","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-560fb8f41570b906","tex":"X_j^{a_j}","aliases":"X_j^a_j","display":"X_j^{a_j}","kind":"use","excerpt":"For every binary address a, W_a=∏_j=0^n-1X_j^a_j=T_a, W_aψ_0=ket a. The family (W_aψ_0)_a∈ A_n is linearly independent, spans V_n, and has exactly 2^n elements. It is the standard computational basis.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d1a93beeda6d35c8","tex":"\\xmapsto","aliases":"xmapsto","display":null,"kind":"use","excerpt":"vpage8Track the Bell state through the circuitINTERMEDIATE PACKETS ARE NOT FINAL PACKETSTraditional edition: Theorem II.4.1, Example II.4.6. The input is ket00. First apply a Hadamard to the first qubit, then a controlled flip: ket00 xmapstoH_rm Had⊗ I fracket00+ket10sqrt2 xmapstorm CNOT fracket00+ket11sqrt2. The displayed columns keep amplitudes, ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"vpage12A moving frame creates a connection termNOT ALL MOTION IS PHYSICAL EVOLUTIONTraditional edition: Theorem II.6.3. Set the laboratory Hamiltonian to zero and keep ψ(t)=ket0 fixed. Describe it in rotating coordinates, U(t)=matcos t · -sin t sin t · cos t, y(t)=U(t)^*ψ(t)=matcos t -sin t. The vector has not moved in the laboratory, but its coord","count":40},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-40d753fc08b490d6","tex":"y^*","aliases":"y^*","display":"y^*","kind":"use","excerpt":"Complex polarization reaches the whole operator space. Write q(x)=xx^*. The module proves the exact identity 4xy^*=q(x+y)-q(x-y) +i(q(x+iy)-q(x-iy)). Agreement on all positive pure outer products therefore implies agreement on every outer product. Actual matrix units then span the full matrix space. Applying this argument to the sandwich by P prove","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2eda1c421506c144","tex":"Y^{-1}","aliases":"Y^-1","display":"Y^{-1}","kind":"use","excerpt":"Use the inverse difference identity X^-1-Y^-1=X^-1(Y-X)Y^-1 with X=F_P(z) and Y=z-A.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite num","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"For n≥ 0, let A_n=( Z/2 Z)^{0,…,n-1}, V_n={v:A_n→ C}. For an address a∈ A_n, define the standard coordinate vector ket a(x)=1 · x=a,","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5e30e315795c2b5b","tex":"Z^*","aliases":"Z^*","display":"Z^*","kind":"use","excerpt":"Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this cal","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c3c14695c16612f8","tex":"z^*","aliases":"z^*","display":"z^*","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. This is the corrected hypothesis set recorded in the formal release. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 an","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"use","excerpt":"|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings. headA series near zero energy and its exact remainder At z=1/2, |z| |D^-1|=1/4<1, so Σ(z)=-∑_n=0","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-287c99551a89afd6","tex":"Z^{\\otimes n}","aliases":"Z^⊗ n otimes","display":"Z^{\\otimes n}","kind":"use","excerpt":"For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-08bf4c2ffd5ec019","tex":"Z^{\\otimes3}","aliases":"Z^⊗3 otimes","display":"Z^{\\otimes3}","kind":"use","excerpt":"[Exact parity refinement] For three qubits, the local computational projectors were grouped by the parity of their Hamming weight. The resulting coarse projectors agreed with (I± Z^⊗3)/2 with residual 0.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b23ccb12e2adf65f","tex":"Z_1","aliases":"Z_1","display":"Z_1","kind":"use","excerpt":"vpage20An unchanged syndrome can hide a logical errorTHE SCALAR COMPRESSION TEST MATTERSTraditional edition: Theorem II.8.2, Proposition II.8.7. Apply Z_1=Z⊗ I⊗ I to the same repetition code. It sends ket000 to itself and ket111 to its negative. Both states remain inside the original code, so the previous syndrome partition reports no bit flip. fra","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-315e6e3c40d12791","tex":"z_a","aliases":"z_a","display":"z_a","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0ff735e21f188d41","tex":"Z_d","aliases":"Z_d","display":"Z_d","kind":"use","excerpt":"[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8254078b2941c5fe","tex":"Z_{L,a}","aliases":"Z_L,a","display":"Z_{L,a}","kind":"use","excerpt":"[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-35030b1cc5b33f81","tex":"Z_N","aliases":"Z_N","display":"Z_N","kind":"use","excerpt":"vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSTraditional edition: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-32eb84b8504eeadf","tex":"Z_{ra}","aliases":"Z_ra","display":"Z_{ra}","kind":"use","excerpt":"Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4e257ef927f71ba8","tex":"z_{ra}","aliases":"z_ra","display":"z_{ra}","kind":"use","excerpt":"Applied to the corrected Kraus family, this yields scalars z_ra with R_rE_aP=z_raP. The essential declarations are pathrank_one_gram_rigidity, pathkraus_rank_one_rigidity and pathcorrected_kraus_family. No rank-one Choi characterization is inserted as an axiom or as the desired conclusion in hypothesis form.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7b1a33765401fd68","tex":"z_{rb}","aliases":"z_rb","display":"z_{rb}","kind":"use","excerpt":"Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this cal","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-207f0dd92280fb9b","tex":"\\ZZ^d","aliases":"ZZ^d Z^d ZZ mathbb","display":"{\\mathbb Z}^d","kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"ity. Starting from ket0, there are two resolved paths to output ket1. Their signed products are bra1H_rm Hadket0bra0H_rm Hadket0=frac12, bra1H_rm Hadket1bra1H_rm Hadket0=-frac12. Their sum is zero. Squaring each separately gives a different answer. vfiginterference headWhere the missing term goes For path amplitudes a,b, the outcome probability is ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3e3225555f03b1a7","tex":"|^3","aliases":"|^3","display":"|^3","kind":"use","excerpt":"Put x=kh/2. The inequalities |sin x|≤|x| and |x-sin x|≤ |x|^3/6, which hold for every real argument, give 0≤ x^2-sin^2x =|x-sin x| |x+sin x| ≤frac|x|^36 2|x| =fracx^43. Multiplying by 4/h^2 proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps k fixed while h→0, so it does not assert a u","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b44e5a2c29c5de91","tex":"|_{U_a}","aliases":"|_U_a","display":"|_{U_a}","kind":"use","excerpt":"). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserv","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"1 · 0. The even block squares to (25/16)I, so its eigenvalues are ±5/4. The odd block squares to I, so its eigenvalues are ±1. Exponentiating either block gives the corresponding part of the exact evolution. Transfer between the packets remains zero for every real time. headWhat has changed? Only the order of basis vectors has changed. No matrix en","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"vpage15Curvature measures a local area responseAN EXACT BLOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the","count":11},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"The exact check constructs each 8×8 word matrix, applies it to the seed, and verifies that the matrix of all eight outputs is the identity. It also tests the displayed superposition against every word output. No search over an unspecified gate set is involved. takeawayA cyclic seed generates a vector space under an algebra. That conclusion does not","count":29},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-59ce46150d0cfe77","tex":"\\alpha_0","aliases":"α_0 alpha","display":"\\alpha_0","kind":"use","excerpt":"[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"use","excerpt":"[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-da4f2068e2750ae4","tex":"\\alpha_m","aliases":"α_m alpha","display":"\\alpha_m","kind":"use","excerpt":"[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-20abd9c0a72b9797","tex":"\\alpha_{m-1}","aliases":"α_m-1 alpha","display":"\\alpha_{m-1}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYTraditional edition: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0","count":19},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f42d1a795468f2ce","tex":"\\gamma'","aliases":"γ' gamma","display":"\\gamma'","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-27ca25276af08346","tex":"\\Gamma_b","aliases":"Γ_b Gamma","display":"\\Gamma_b","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"use","excerpt":"The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings ar","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1525b57dc165eb8a","tex":"\\delta_{ab}","aliases":"δ_ab delta","display":"\\delta_{ab}","kind":"use","excerpt":"The Knill--Laflamme compressions are PE_a^*E_bP=δ_abP. Their scalar form says that the error labels do not resolve the encoded vector. The exact checks evaluate all sixteen compressions, all sixteen syndrome products, the four probabilities, the recovery normalization, and the full recovered matrix.","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c3d61188e8e97282","tex":"\\delta_{ac}","aliases":"δ_ac delta","display":"\\delta_{ac}","kind":"use","excerpt":"The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading ","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3d134e4ce3afe800","tex":"\\delta_{ba}","aliases":"δ_ba delta","display":"\\delta_{ba}","kind":"use","excerpt":"Let U=U_mU_m-1⋯ U_1 be a finite circuit with m≥1, and let (P_a)_a∈ I be a complete orthogonal packet family. For the empty circuit, U=I and its transfer block is P_bP_a=δ_baP_a, whose endpoint sum reconstructs I.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e1fdf516d82b6962","tex":"\\delta_{bd}","aliases":"δ_bd delta","display":"\\delta_{bd}","kind":"use","excerpt":"[Packet unitarity identities] If U is unitary and U_ba=P_bUP_a, then ∑_bU_ba^*U_bc=δ_acP_a and ∑_aU_baU_da^*=δ_bdP_b.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8e4e08a5e917de66","tex":"\\delta_{jk}","aliases":"δ_jk delta","display":"\\delta_{jk}","kind":"use","excerpt":"[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-50baa2ac2f493094","tex":"\\delta_{j\\ell}","aliases":"δ_jℓ delta ell","display":"\\delta_{j\\ell}","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-51a87ccbd55b1aa4","tex":"\\delta_{rs}","aliases":"δ_rs delta","display":"\\delta_{rs}","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7a71a6c1ff123146","tex":"\\delta_{\\alpha\\gamma}","aliases":"δ_αγ delta alpha gamma","display":"\\delta_{\\alpha\\gamma}","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-df0a518550e907c5","tex":"\\delta_{\\beta\\alpha}","aliases":"δ_βα delta beta alpha","display":"\\delta_{\\beta\\alpha}","kind":"use","excerpt":"Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a5a2814c2c585145","tex":"\\delta_{\\beta\\delta}","aliases":"δ_βδ delta beta delta","display":"\\delta_{\\beta\\delta}","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8f11ad62b3f5c5bb","tex":"\\delta_{\\lambda\\mu}","aliases":"δ_λμ delta lambda mu","display":"\\delta_{\\lambda\\mu}","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d33d8b646241d747","tex":"\\Delta_\\ell","aliases":"Δ_ℓ Delta ell","display":"\\Delta_\\ell","kind":"use","excerpt":"A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective s","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bae7842f2d63f7f5","tex":"\\varepsilon_a","aliases":"ε_a varepsilon","display":"\\varepsilon_a","kind":"use","excerpt":"[Approximate constraint transport] Assume normC_ℓ+1,aR_ℓ-R_ℓ C_ℓ,a ≤ε_a and normC_ℓ,a,normC_ℓ+1,a≤ M. Then a coarse physical unit vector ψ satisfies normC_ℓ+1,aR_ℓψ≤ε_a. Also, · norm [C_ℓ+1,a,C_ℓ+1,b]R_ℓ -R_ℓ[C_ℓ,a,C_ℓ,b]","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-48bf82ae71f365c5","tex":"\\varepsilon_b","aliases":"ε_b varepsilon","display":"\\varepsilon_b","kind":"use","excerpt":"[Approximate constraint transport] Assume normC_ℓ+1,aR_ℓ-R_ℓ C_ℓ,a ≤ε_a and normC_ℓ,a,normC_ℓ+1,a≤ M. Then a coarse physical unit vector ψ satisfies normC_ℓ+1,aR_ℓψ≤ε_a. Also, · norm [C_ℓ+1,a,C_ℓ+1,b]R_ℓ -R_ℓ[C_ℓ,a,C_ℓ,b] · ≤ 2M(ε_a+ε_b).","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"actual amplitude vector is (- i,1, i,1)^T/2, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state. headA channel changes the conclusion predictably Before evolving, mix the seed density matr","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8ee7b5d31d3432c2","tex":"\\eta_\\gamma","aliases":"η_γ eta gamma","display":"\\eta_\\gamma","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-eb88d6b2c124cbb8","tex":"\\eta_{\\gamma'}","aliases":"η_γ' eta gamma","display":"\\eta_{\\gamma'}","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"vpage15Curvature measures a local area responseAN EXACT BLOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the","count":27},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","excerpt":"∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ≤θ_0 is π(1-cosθ_0). At θ_0=π/3, it is exactly π/2. The ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bc656d1ba29723e3","tex":"\\iota_\\ell","aliases":"ι_ℓ iota ell","display":"\\iota_\\ell","kind":"use","excerpt":"Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic ","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-10808a5caf1a666d","tex":"\\lambda^{-1/2}","aliases":"λ^-1/2 lambda","display":"\\lambda^{-1/2}","kind":"use","excerpt":"0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divide by zero.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"use","excerpt":"Let L≥0 be a finite Laplacian on N≥1 states, with eigenvalues λ_j and at least one zero eigenvalue. For diffusion time τ>0, define the normalized heat trace K(τ)=frac1N∑_j e^-τλ_j and the scale-dependent spectral dimension d_s(τ) =-2fracdlog K(τ)dlogτ. For a finite system, d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differen","count":14},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-32012d2070cfbb63","tex":"\\lambda_j^{-1/2}","aliases":"λ_j^-1/2 lambda","display":"\\lambda_j^{-1/2}","kind":"use","excerpt":"syndrome construction] Assume the Knill Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syn","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c3b15579ddc9c765","tex":"\\lambda_{k,h}","aliases":"λ_k,h lambda","display":"\\lambda_{k,h}","kind":"use","excerpt":"vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPETraditional edition: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters be","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1dc8e3b2125ff219","tex":"\\lambda_{\\min}","aliases":"λ_min lambda min","display":"\\lambda_{\\min}","kind":"use","excerpt":"d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differentiation gives d_s(τ)=2τ frac∑_jλ_j e^-τλ_j∑_j e^-τλ_j. It tends to zero as τdownarrow0 and, under the stated assumption of a zero mode, as τ→∞. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to 2τλ_min. The conclusion for ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-036d06d495662f6f","tex":"\\Lambda_N","aliases":"Λ_N Lambda","display":"\\Lambda_N","kind":"use","excerpt":"Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0c1a2a4aeea8926c","tex":"\\Lambda_N^*","aliases":"Λ_N^* Lambda","display":"\\Lambda_N^*","kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-36442f76e90ff174","tex":"\\lambda_r","aliases":"λ_r lambda","display":"\\lambda_r","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":12},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ad7d2db293e5cbb7","tex":"\\lambda_r^{-1/2}","aliases":"λ_r^-1/2 lambda","display":"\\lambda_r^{-1/2}","kind":"use","excerpt":"Null directions are removed by proof. If λ_r=0, then B_r^*B_r=0. Positivity of the ordinary complex matrix inner product gives B_r=0. This is the checked declaration pathDiagonalErrors.zero_weight_error. The normalized syndrome operators are defined only on A_+={r:λ_r>0}, V_r=λ_r^-1/2B_r. Consequently no inverse square root at a null eigenvalue is ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"Each interval lies strictly inside (0,h^2/12). Dividing successive interval endpoints proves that both error ratios are between 3.99 and 4. The check verifies these inequalities by rational arithmetic, without relying on approximate sine values or a fitted logarithmic slope. headWhy the mode regime matters If instead k=π/h moves with the grid, the ","count":9},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f4875cececfa7801","tex":"\\pi^4","aliases":"π^4 pi","display":"\\pi^4","kind":"use","excerpt":"[Low mode dispersion in dimension d] For every centered mode k, 0 ≤ 4π^2absk^2-λ_N(k) ≤ frac4π^43N^2 ∑_j=1^dk_j^4.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a66c418313dc8ab5","tex":"\\Pi^{\\mathrm{phys}}","aliases":"Π^phys Pi mathrm","display":"\\Pi^{\\mathrm{phys}}","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1acb7717088ec6fe","tex":"\\pi_\\lambda","aliases":"π_λ pi lambda","display":"\\pi_\\lambda","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fce0b633f1baa40a","tex":"\\pi_\\mu","aliases":"π_μ pi mu","display":"\\pi_\\mu","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3b9f2dbc20110e35","tex":"\\Pi_\\ell^{\\mathrm{phys}}","aliases":"Π_ℓ^phys Pi ell mathrm","display":"\\Pi_\\ell^{\\mathrm{phys}}","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6fd3bc029e9c79bd","tex":"\\rho_+","aliases":"ρ_+ rho","display":"\\rho_+","kind":"use","excerpt":"For ρ_+=frac12mat1 · 1 1 · 1, the exact output is Φ(ρ_+)=mat17/25 · 2/5","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","excerpt":"The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-197a543b67174e7f","tex":"\\rho_{00}","aliases":"ρ_00 rho","display":"\\rho_{00}","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-75b2b525a8ae5055","tex":"\\rho_{01}","aliases":"ρ_01 rho","display":"\\rho_{01}","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-90dbd799c3b71089","tex":"\\rho_{10}","aliases":"ρ_10 rho","display":"\\rho_{10}","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7aad9eff5723a02c","tex":"\\rho_{11}","aliases":"ρ_11 rho","display":"\\rho_{11}","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-83765f8c06700277","tex":"\\rho_{\\rm Bell}","aliases":"ρ_rm Bell rho rm","display":"\\rho_{\\rm Bell}","kind":"use","excerpt":"lassical input II.7.7. Ordinary transpose T sends a positive matrix on one system to another positive matrix and preserves its trace. Those properties are insufficient for a quantum channel on a subsystem of an entangled state. The test must include id⊗ T on the larger system. vfigcomplete_positivity Apply this extended map to the Bell density matr","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9ecaf052e23094fe","tex":"\\rho_\\eta","aliases":"ρ_η rho eta","display":"\\rho_\\eta","kind":"use","excerpt":"actual amplitude vector is (- i,1, i,1)^T/2, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state. headA channel changes the conclusion predictably Before evolving, mix the seed density matr","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-81d80f06b0492996","tex":"\\bm\\sigma","aliases":"σ bm sigma","display":"\\bm\\sigma","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-366e9732bdd10fbb","tex":"\\boldsymbol\\sigma","aliases":"σ boldsymbol sigma","display":"\\boldsymbol\\sigma","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-18866b84545a9559","tex":"\\Sigma'","aliases":"Σ' Sigma","display":"\\Sigma'","kind":"use","excerpt":"[Uniform self energy bounds] Let δ=dist(z,Spec D)>0, g=norm B. Then normΣ(z)≤fracg^2δ and normΣ'(z)≤fracg^2δ^2. For real E,E' in an interval at distance at least δ from Spec D, normΣ(E)-Σ(E') ≤ fracg^2δ^2absE-E'.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7cc70e05d3b03d7a","tex":"\\sigma_a","aliases":"σ_a sigma","display":"\\sigma_a","kind":"use","excerpt":"vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncom","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-65acde73901caecb","tex":"\\Sigma_P","aliases":"Σ_P Sigma","display":"\\Sigma_P","kind":"use","excerpt":"The self-energy Σ_P(z) =H_PQ(z-H_QQ)^-1H_QP contains the complete influence of the eliminated packet. Bare truncation drops Σ_P(z) and is therefore exact only when the coupling between packets vanishes or the self-energy cancels for another stated reason.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"vpage15Curvature measures a local area responseAN EXACT BLOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the","count":21},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9d05df821bd9aeec","tex":"\\Phi^{(J)}","aliases":"Φ^(J) Phi","display":"\\Phi^{(J)}","kind":"use","excerpt":"To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2358d6649cb4b2bc","tex":"\\Phi^+","aliases":"Φ^+ Phi","display":"\\Phi^+","kind":"use","excerpt":"vpage16Which coherence can a measurement see?THE ANSWER DEPENDS ON ITS EFFECTSTraditional edition: Theorem II.7.2, Theorem II.7.4, Corollary II.7.5. For the Bell density matrix ρ=ketΦ^+braΦ^+, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between add","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-53adb99fefcfc89a","tex":"\\Phi_{cd,ab}","aliases":"Φ_cd,ab Phi","display":"\\Phi_{cd,ab}","kind":"use","excerpt":"Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4b9454d5c4e04a08","tex":"\\Phi_{\\alpha\\beta;\\gamma\\delta}","aliases":"Φ_αβ;γδ Phi alpha beta gamma delta","display":"\\Phi_{\\alpha\\beta;\\gamma\\delta}","kind":"use","excerpt":"Packet superoperator blocks are defined by Φ_αβ;γδ(X) = P_α Φ(P_γ XP_δ) P_β. Summing all four packet indices reconstructs Φ(X) exactly.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b1806d9602ee01cb","tex":"\\Phi_\\gamma","aliases":"Φ_γ Phi gamma","display":"\\Phi_\\gamma","kind":"use","excerpt":"0 · 0 . The channel Φ_γ(ρ)=K_0ρ K_0^*+K_1ρ K_1^* is trace preserving. It sends the off diagonal packet ket0bra1 to sqrt1-γket0bra1, while the excited population transfers partly into the ground population.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-353182e6b3dd80e2","tex":"\\psi^*","aliases":"ψ^* psi","display":"\\psi^*","kind":"use","excerpt":"to follow when its objects remain visible. This appendix follows a state vector, a density matrix, and a retained part of a dynamical system. They may describe the same experiment, but they do not answer the same question. vfigthree_objects headAmplitudes retain relative phase For the Bell vector ψ=(ket00+ket11)/sqrt2, the two nonzero entries are a","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a619f915ee96c264","tex":"\\Psi_{ef,cd}","aliases":"Ψ_ef,cd Psi","display":"\\Psi_{ef,cd}","kind":"use","excerpt":"[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e11bd281bd6a189e","tex":"\\Psi_{g,\\ell}","aliases":"Ψ_g,ℓ Psi ell","display":"\\Psi_{g,\\ell}","kind":"use","excerpt":"[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6db479813e7c12fd","tex":"\\psi_i","aliases":"ψ_i psi","display":"\\psi_i","kind":"use","excerpt":"A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4417e8c46d7c52ca","tex":"\\omega^{(k-j)r}","aliases":"ω^(k-j)r omega","display":"\\omega^{(k-j)r}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b22e66852d5de18b","tex":"\\omega^{-jr}","aliases":"ω^-jr omega","display":"\\omega^{-jr}","kind":"use","excerpt":"[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ccaca8bd1561c1ac","tex":"\\omega^{-qr}","aliases":"ω^-qr omega","display":"\\omega^{-qr}","kind":"use","excerpt":"[Coherence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a27b633daa590319","tex":"\\omega^j","aliases":"ω^j omega","display":"\\omega^j","kind":"use","excerpt":"[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d9328f4344362132","tex":"\\omega^{jr}","aliases":"ω^jr omega","display":"\\omega^{jr}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b2cd0e4050c8155a","tex":"\\omega^{q+j}","aliases":"ω^q+j omega","display":"\\omega^{q+j}","kind":"use","excerpt":"For x∈ P_j H, S(Ax) =(SAS^*)Sx =ω^q+jAx. Thus Ax∈ P_j+q H. The block statement follows.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a1ce26be62e1af85","tex":"\\omega^{q_r}","aliases":"ω^q_r omega","display":"\\omega^{q_r}","kind":"use","excerpt":"[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-05ef237cb91b4c15","tex":"\\omega^{\\operatorname{val}(r)}","aliases":"ω^val(r) omega operatorname","display":"\\omega^{\\operatorname{val}(r)}","kind":"use","excerpt":"Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-598fd1586454c966","tex":"\\omega^{\\operatorname{val}(r)\\operatorname{val}(j)}","aliases":"ω^val(r)val(j) omega operatorname operatorname","display":"\\omega^{\\operatorname{val}(r)\\operatorname{val}(j)}","kind":"use","excerpt":"Exact coefficient formula. The packet definition is a finite sum over residue classes: P_j=frac1N∑_r∈ Z/N Zχ(-rj)ρ(r). The source proves χ(-rj)=(ω^val(r)val(j))^-1, so this is precisely the manuscript's N-term Fourier projector formula, with canonical representatives.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILTraditional edition: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the dir","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"ues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect the finite size of the system As τ→0, every eigenmode contri","count":31},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"vpage31Coherence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSTraditional edition: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmi","count":8},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzero p, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matr","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and prov","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"LOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-422f05688ea1c04e","tex":"\\partial_\\theta","aliases":"∂_θ partial theta","display":"\\partial_\\theta","kind":"use","excerpt":"LOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6bd349398cbcc266","tex":"\\partial_\\mu","aliases":"∂_μ partial mu","display":"\\partial_\\mu","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b8be09e8f717ffe2","tex":"\\partial_\\nu","aliases":"∂_ν partial nu","display":"\\partial_\\nu","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-935f9c58c2ce6b38","tex":"\\partial_\\phi","aliases":"∂_ϕ partial phi","display":"\\partial_\\phi","kind":"use","excerpt":"LOCH PROJECTOR CALCULATIONTraditional edition: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and prov","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4e8b6f2780e1c940","tex":"\\nabla^{(a)}_X","aliases":"∇^(a)_X nabla","display":"\\nabla^{(a)}_X","kind":"use","excerpt":"The local connection and its change of coordinates. For a coordinate function u valued in E_a, define ∇^(a)_Xu=A_a(P D_M(B_a u)[X]). For every overlap point, Lean proves the operator identity ∇^(b)_X(g_abu)=g_ab∇^(a)_Xu. This is proved for arbitrary local coordinate functions. It does not assume that u was obtained from a globally smooth section or","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4333a31f419e8b42","tex":"\\nabla^{(b)}_X","aliases":"∇^(b)_X nabla","display":"\\nabla^{(b)}_X","kind":"use","excerpt":"The local connection and its change of coordinates. For a coordinate function u valued in E_a, define ∇^(a)_Xu=A_a(P D_M(B_a u)[X]). For every overlap point, Lean proves the operator identity ∇^(b)_X(g_abu)=g_ab∇^(a)_Xu. This is proved for arbitrary local coordinate functions. It does not assume that u was obtained from a globally smooth section or","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7b7161eb75eecc40","tex":"\\nabla_{[X,Y]}","aliases":"∇_[X,Y] nabla","display":"\\nabla_{[X,Y]}","kind":"use","excerpt":"The rank of P is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section s, so that Ps=s, define ∇_Xs=P ds(X). This is a connection on smooth range sections. For smooth vector fields X,Y, its curvature satisfies (∇_X∇_Y-∇_Y∇_X-∇_[X,Y])s =P[dP(X),dP(Y)]P s. Equivalently, its endomorphism-valued ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3cf6e300b826de4e","tex":"\\nabla_{fX}","aliases":"∇_fX nabla","display":"\\nabla_{fX}","kind":"use","excerpt":"textitConnection rules. Smoothness of P ds(X) and P^2=P show that ∇_Xs is a smooth range section. Differentiation and projection give ∇_X+Ys · =∇_Xs+∇_Ys, · ∇_fXs · =f∇_Xs, ∇_X(s+t) · =∇_Xs+∇_Xt, · ∇_X(fs) · =f∇_Xs+(Xf)s for smooth real f. The range coordinates above identify ambient smooth range sections with smooth sections of the constructed bun","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-60743d31156ad293","tex":"\\nabla_{X+Y}","aliases":"∇_X+Y nabla","display":"\\nabla_{X+Y}","kind":"use","excerpt":"textitConnection rules. Smoothness of P ds(X) and P^2=P show that ∇_Xs is a smooth range section. Differentiation and projection give ∇_X+Ys · =∇_Xs+∇_Ys, · ∇_fXs · =f∇_Xs, ∇_X(s+t) · =∇_Xs+∇_Xt, · ∇_X(fs) · =f∇_Xs+(Xf)s for smooth real f. The range coordinates above identify ambient smooth range sections with smooth sections of the constructed bun","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-555d56402f7457f5","tex":"\\nabla_Y","aliases":"∇_Y nabla","display":"\\nabla_Y","kind":"use","excerpt":"The rank of P is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section s, so that Ps=s, define ∇_Xs=P ds(X). This is a connection on smooth range sections. For smooth vector fields X,Y, its curvature satisfies (∇_X∇_Y-∇_Y∇_X-∇_[X,Y])s =P[dP(X),dP(Y)]P s. Equivalently, its endomorphism-valued ","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"For every binary address a, W_a=∏_j=0^n-1X_j^a_j=T_a, W_aψ_0=ket a. The family (W_aψ_0)_a∈ A_n is linearly independent, spans V_n, and has exactly 2^n elements. It is the standard computational basis.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3065211613fd0977","tex":"\\prod_{i\\in F}","aliases":"∏_i∈ F prod in","display":"\\prod_{i\\in F}","kind":"use","excerpt":"Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite num","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0bafcdd9e08caf20","tex":"\\prod_{j=0}^{n-1}","aliases":"∏_j=0^n-1 prod","display":"\\prod_{j=0}^{n-1}","kind":"use","excerpt":"For every binary address a, W_a=∏_j=0^n-1X_j^a_j=T_a, W_aψ_0=ket a. The family (W_aψ_0)_a∈ A_n is linearly independent, spans V_n, and has exactly 2^n elements. It is the standard computational basis.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7684dda7bca2d3dd","tex":"\\sum_{a,b}","aliases":"∑_a,b sum","display":"\\sum_{a,b}","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fc8fd79600305b45","tex":"\\sum_{a,b,c}","aliases":"∑_a,b,c sum","display":"\\sum_{a,b,c}","kind":"use","excerpt":"le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the or","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c10218f7028528ee","tex":"\\sum_{a,b,c,d}","aliases":"∑_a,b,c,d sum","display":"\\sum_{a,b,c,d}","kind":"use","excerpt":"[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-053dd8ba47178a16","tex":"\\sum_{a:Z_{L,a}(t)>0}","aliases":"∑_a:Z_L,a(t)>0 sum","display":"\\sum_{a:Z_{L,a}(t)>0}","kind":"use","excerpt":"[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-cb22c502080c58d4","tex":"\\sum_{a=0}^1","aliases":"∑_a=0^1 sum","display":"\\sum_{a=0}^1","kind":"use","excerpt":"vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILURETraditional edition: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5 4/5 · 3/5, Q_a=RP_aR^*, S=∑_a=0^1Q_aP_a=frac35R. The exact Gram matrices give (S-I)^*(S-I)=frac1625I, S^*S=frac925I, normS-I=frac45<1. Consequently the un","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3001c1c56077038a","tex":"\\sum_{a=1}^k","aliases":"∑_a=1^k sum","display":"\\sum_{a=1}^k","kind":"use","excerpt":"Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and prov","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-58ca43ed1bf657da","tex":"\\sum_{a=1}^{s}","aliases":"∑_a=1^s sum","display":"\\sum_{a=1}^{s}","kind":"use","excerpt":"An independent definition of Schmidt rank. The source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length s is the concrete equality C_ij=∑_a=1^su_a(i)v_a(j). It factors C=UV through an s-dimensional intermediate space, so rank C≤ s. Conversely, choose a basis of the row space of C. Expanding eac","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f409d86b4db047ab","tex":"\\sum_{a_1,\\ldots,a_{m-1}}","aliases":"∑_a_1,…,a_m-1 sum ldots","display":"\\sum_{a_1,\\ldots,a_{m-1}}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2b67385de1e4af35","tex":"\\sum_{a\\in I}","aliases":"∑_a∈ I sum in","display":"\\sum_{a\\in I}","kind":"use","excerpt":"The elementary coordinate object is a complete family of orthogonal projectors P_aP_b=δ_abP_a, ∑_a∈ IP_a=I. Every vector has packet coordinates P_aψ. Every operator has transfer blocks A_ba=P_bAP_a. These identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction,","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","excerpt":"The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading ","count":6},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c43729fd2f2f98cd","tex":"\\sum_{b,a}","aliases":"∑_b,a sum","display":"\\sum_{b,a}","kind":"use","excerpt":"0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-59a73cc552080ec9","tex":"\\sum_C","aliases":"∑_C sum","display":"\\sum_C","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9b2b4aa44e98a82f","tex":"\\sum_{c,d}","aliases":"∑_c,d sum","display":"\\sum_{c,d}","kind":"use","excerpt":"[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-6d39fe36bb310cf9","tex":"\\sum_{g\\in G}","aliases":"∑_g∈ G sum in","display":"\\sum_{g\\in G}","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-194eb7d3506e206c","tex":"\\sum_{g\\in S_3}","aliases":"∑_g∈ S_3 sum in","display":"\\sum_{g\\in S_3}","kind":"use","excerpt":", not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONTraditional edition: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-38c03c7cd3f0f240","tex":"\\sum_{i,j}","aliases":"∑_i,j sum","display":"\\sum_{i,j}","kind":"use","excerpt":"[Finite Kraus and Choi criteria] A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5af01fe612ec2d43","tex":"\\sum_{i=1}^{d_\\lambda}","aliases":"∑_i=1^d_λ sum lambda","display":"\\sum_{i=1}^{d_\\lambda}","kind":"use","excerpt":"x an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.","count":20},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d7c6cf93ddec3543","tex":"\\sum_{j,k}","aliases":"∑_j,k sum","display":"\\sum_{j,k}","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-47549ba48c05d7aa","tex":"\\sum_{j,k=1}^{m}","aliases":"∑_j,k=1^m sum","display":"\\sum_{j,k=1}^{m}","kind":"use","excerpt":"[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-586bc3571b381e64","tex":"\\sum_{j=0}^{N-1}","aliases":"∑_j=0^N-1 sum","display":"\\sum_{j=0}^{N-1}","kind":"use","excerpt":"[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a9d0eeaa9eff3012","tex":"\\sum_{j=1}^{d}","aliases":"∑_j=1^d sum","display":"\\sum_{j=1}^{d}","kind":"use","excerpt":"Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-12ba6f0fe0cd0b54","tex":"\\sum_{j_1+\\cdots+j_m=r}","aliases":"∑_j_1+⋯+j_m=r sum cdots","display":"\\sum_{j_1+\\cdots+j_m=r}","kind":"use","excerpt":"[Tensor charge refinement] For r∈ Z/N Z, define P_r^rm total = ∑_j_1+⋯+j_m=r P_j_1^(1)⊗⋯⊗ P_j_m^(m). Then the P_r^rm total form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4511ec7b375d647e","tex":"\\sum_{j_1+\\cdots+j_m\\equiv r\\pmod N}","aliases":"∑_j_1+⋯+j_m≡ rpmod N sum cdots equiv pmod","display":"\\sum_{j_1+\\cdots+j_m\\equiv r\\pmod N}","kind":"use","excerpt":"[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-32a0cdf7fe3b4935","tex":"\\sum_{k=1}^{m}","aliases":"∑_k=1^m sum","display":"\\sum_{k=1}^{m}","kind":"use","excerpt":"Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-75edbe8a53ac1864","tex":"\\sum_{k\\in\\ZZ^d}","aliases":"∑_k∈ZZ^d ∑_k∈ Z^d sum in ZZ sum in mathbb","display":"\\sum_{k\\in{\\mathbb Z}^d}","kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-e99f6320075b8a65","tex":"\\sum_{k\\in\\Lambda_N^*}","aliases":"∑_k∈Λ_N^* sum in Lambda","display":"\\sum_{k\\in\\Lambda_N^*}","kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a7626aeba75a509f","tex":"\\sum_{n=0}^{\\infty}","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^{\\infty}","kind":"use","excerpt":"|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings. headA series near zero energy and its exact remainder At z=1/2, |z| |D^-1|=1/4<1, so Σ(z)=-∑_n=0","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ddc26b49cc4ceb42","tex":"\\sum_{n\\in\\ZZ^d}","aliases":"∑_n∈ZZ^d ∑_n∈ Z^d sum in ZZ sum in mathbb","display":"\\sum_{n\\in{\\mathbb Z}^d}","kind":"use","excerpt":"[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fc4aea9ed3564f4b","tex":"\\sum_{n\\neq0}","aliases":"∑_n≠0 sum neq","display":"\\sum_{n\\neq0}","kind":"use","excerpt":"By [source reference], Z_d(t) =(4π t)^-d/2(1+R(t)), where R(t)=∑_n≠0 e^-absn^2/(4t) =O( e^-1/(4t)). Differentiation gives tR'(t)=O(t^-1 e^-1/(4t)). Substituting this estimate into the logarithmic derivative gives the stated small time expansion.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-366ac52aad5d30c9","tex":"\\sum_q","aliases":"∑_q sum","display":"\\sum_q","kind":"use","excerpt":"[Charge pruning of circuit paths] Suppose every gate has a finite charge decomposition U_r=∑_qU_r^(q), SU_r^(q)S^*=ω^qU_r^(q). A term with gate charges q_1,…,q_m can contribute from source charge a_0 only to target charge a_m=a_0+q_1+⋯+q_m. Every packet path violating one of the intermediate charge updates vanishes.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^","count":12},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c6589276c9db025d","tex":"\\sum_{r}","aliases":"∑_r sum","display":"\\sum_{r}","kind":"use","excerpt":"[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-05d2aa25a4e84f79","tex":"\\sum_{r,a}","aliases":"∑_r,a sum","display":"\\sum_{r,a}","kind":"use","excerpt":"Deriving the Gram matrix from recovery Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-f161dab54b7fa5c5","tex":"\\sum_{r=0}^3","aliases":"∑_r=0^3 sum","display":"\\sum_{r=0}^3","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYTraditional edition: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-029134516e4be179","tex":"\\sum_{r=0}^{N-1}","aliases":"∑_r=0^N-1 sum","display":"\\sum_{r=0}^{N-1}","kind":"use","excerpt":"[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c6a6e48cf2891abf","tex":"\\sum_{r\\in\\mathbb Z/N\\mathbb Z}","aliases":"∑_r∈ Z/N Z sum in mathbb mathbb","display":"\\sum_{r\\in\\mathbb Z/N\\mathbb Z}","kind":"use","excerpt":"Exact coefficient formula. The packet definition is a finite sum over residue classes: P_j=frac1N∑_r∈ Z/N Zχ(-rj)ρ(r). The source proves χ(-rj)=(ω^val(r)val(j))^-1, so this is precisely the manuscript's N-term Fourier projector formula, with canonical representatives.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7a4926b7b13440e3","tex":"\\sum_{\\substack{\\gamma,\\gamma'\\in\\Gamma_b\\\\ \\gamma<\\gamma'}}","aliases":"∑_substackγ,γ'∈Γ_b γ<γ' sum substack gamma gamma in Gamma gamma gamma","display":"\\sum_{\\substack{\\gamma,\\gamma'\\in\\Gamma_b\\\\ \\gamma<\\gamma'}}","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-940305a61c0cbc27","tex":"\\sum_{\\alpha_1,\\ldots,\\alpha_{m-1}}","aliases":"∑_α_1,…,α_m-1 sum alpha ldots alpha","display":"\\sum_{\\alpha_1,\\ldots,\\alpha_{m-1}}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet α_0 and target packet α_m, P_α_mUP_α_0 = ∑_α_1,…,α_m-1 P_α_mU_mP_α_m-1 ⋯ P_α_1U_1P_α_0. Summing over the source and target packets reconstructs the complete circuit operator.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dd5a8bdc2b931502","tex":"\\sum_\\beta","aliases":"∑_β sum beta","display":"\\sum_\\beta","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-fa0188218e3fe53e","tex":"\\sum_{\\beta\\succ\\alpha}","aliases":"∑_βsuccα sum beta succ alpha","display":"\\sum_{\\beta\\succ\\alpha}","kind":"use","excerpt":"A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective s","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4077bd4e18a74bee","tex":"\\sum_{\\gamma\\in\\Gamma_b}","aliases":"∑_γ∈Γ_b sum gamma in Gamma","display":"\\sum_{\\gamma\\in\\Gamma_b}","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-3389ad15a324713c","tex":"\\sum_{\\lambda_j>0}","aliases":"∑_λ_j>0 sum lambda","display":"\\sum_{\\lambda_j>0}","kind":"use","excerpt":"[Exact recovery on the error span] Under the hypotheses of [source reference], define on the syndrome support Q=∑_λ_j>0Q_j the completely positive map R_0(X) = ∑_λ_j>0 V_j^*Q_jXQ_jV_j. It extends to a quantum channel R on End( H). If Φ(X)=∑_aE_aXE_a^* is trace preserving on code states, then RΦ(ρ)=ρ for every density matrix ρ=Pρ P.","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bba70d5604bb3f32","tex":"\\sum_{\\lambda_r>0}","aliases":"∑_λ_r>0 sum lambda","display":"\\sum_{\\lambda_r>0}","kind":"use","excerpt":"titSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal pro","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-a17ff8546f2e633d","tex":"\\sum_{\\lambda\\in\\widehat G}","aliases":"∑_λ∈widehat G sum lambda in widehat","display":"\\sum_{\\lambda\\in\\widehat G}","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-acb2bbc59eb8413f","tex":"\\sum_{\\sum_i a_i=r}","aliases":"∑_∑_i a_i=r sum sum","display":"\\sum_{\\sum_i a_i=r}","kind":"use","excerpt":"Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect the finite size of the system As τ→0, every eigenmode contributes approximately one and the dimension diagnostic tends to zero. As τ→∞, the zero mode dominates,","count":28},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-2547054a0e887b21","tex":"\\int_0^{2\\pi}","aliases":"∫_0^2π int pi","display":"\\int_0^{2\\pi}","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-94438197d8221023","tex":"\\int_0^\\pi","aliases":"∫_0^π int pi","display":"\\int_0^\\pi","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"· - i∫_0^tK(t-s)x(s) d s, where K(t)=B e^- i tDB^*. For Res>0, ∫_0^∞ e^-stK(t) d t = B(s+ i D)^-1B^*.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d","count":4},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty blo","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.","count":20},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"[Exact invariant packets] The following conditions are equivalent: item The Hamiltonian commutes with each projector: [H,P_a]=0 for every a. item Transfer blocks between distinct packets vanish: P_bHP_a=0 whenever a≠ b. item Every packet P_a H is invariant under e^- i tH for every real t. When they hold, H=∑_aP_aHP_a and e^- i tH=∑_aP_a e^- i tHP_a","count":3},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"On a two-dimensional code, consider the overlap compression A=PE_a^*E_bP=diag(11/10,9/10). For instance, within the code take E_a=I and E_b=A. This declares an error family, not by itself a normalized noise channel. The optimal scalar compression has inf_c∈ CnormA-cI=frac110, c=1 attains it. Indeed, for every complex c, the triangle inequality give","count":36},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environmen","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELTraditional edition: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncom","count":7},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-5c779b0b534e7ac8","tex":"\\bigoplus_{i=1}^r","aliases":"⊕_i=1^r bigoplus","display":"\\bigoplus_{i=1}^r","kind":"use","excerpt":"Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty blo","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-ce7f3904634c8e1a","tex":"\\bigoplus_j","aliases":"⊕_j bigoplus","display":"\\bigoplus_j","kind":"use","excerpt":"erence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-183f6b24e50dc6fa","tex":"\\bigoplus_{r=1}^{s}","aliases":"⊕_r=1^s bigoplus","display":"\\bigoplus_{r=1}^{s}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-28886d26e9fa8f85","tex":"\\bigoplus_\\mu","aliases":"⊕_μ bigoplus mu","display":"\\bigoplus_\\mu","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-263ab54d332da07b","tex":"\\bigoplus_{\\mu\\in\\widehat G}","aliases":"⊕_μ∈widehat G bigoplus mu in widehat","display":"\\bigoplus_{\\mu\\in\\widehat G}","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.","count":2},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-bca212e1e10d6008","tex":"\\bigcap_a","aliases":"⋂_a bigcap","display":"\\bigcap_a","kind":"use","excerpt":"[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.","count":1},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":5},{"work":"paper-vii","url":"notation/paper-vii.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":92},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":286},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":592},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"acket operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displ","count":309},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6153a08bafbf354e","tex":"A^{m-1}","aliases":"A^m-1","display":"A^{m-1}","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":179},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6e4af5afb524dbe9","tex":"B_{\\beta\\alpha}","aliases":"B_βα beta alpha","display":"B_{\\beta\\alpha}","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":8},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7b62fdaa08c020af","tex":"\\bytes","aliases":"bytes operatorname","display":"{\\operatorname{bytes}}","kind":"definition-passage","excerpt":"[Arithmetic intensity] When the modeled byte traffic is strictly positive, the modeled arithmetic intensity is I = fracflopbytes.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.","count":107},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-98cd1622d78cdfec","tex":"c_{e,k}","aliases":"c_e,k","display":"c_{e,k}","kind":"definition-passage","excerpt":"[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":12},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ff4ec9351d7bdf0c","tex":"\\CC^{n\\times r}","aliases":"CC^n× r C^n× r CC times mathbb times","display":"{\\mathbb C}^{n\\times r}","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":158},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":77},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.","count":120},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7c808485c3e440e2","tex":"\\operatorname{eval}","aliases":"eval operatorname","display":"\\operatorname{eval}","kind":"definition-passage","excerpt":"B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain; ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation is eval(IR) = ∑_(β,α)∈ E ι_β B_βαπ_α, where ι_β:","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4b3cf9d696dc4fcb","tex":"\\flop","aliases":"flop operatorname","display":"{\\operatorname{flop}}","kind":"definition-passage","excerpt":"[Arithmetic intensity] When the modeled byte traffic is strictly positive, the modeled arithmetic intensity is I = fracflopbytes.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Support graph] The support graph G_A has vertex set I. It has a directed edge α⟶β when P_β AP_α≠0.","count":34},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":20},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-28a0f54c10c0e63f","tex":"G_A","aliases":"G_A","display":"G_A","kind":"definition-passage","excerpt":"[Support graph] The support graph G_A has vertex set I. It has a directed edge α⟶β when P_β AP_α≠0.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-99b7a1e1268a1b18","tex":"g_v","aliases":"g_v","display":"g_v","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":67},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-41bd5b0100b2ea43","tex":"\\Ical","aliases":"Ical I mathcal","display":"{\\mathcal I}","kind":"definition-passage","excerpt":"[Arithmetic intensity] When the modeled byte traffic is strictly positive, the modeled arithmetic intensity is I = fracflopbytes.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ea0ad3db0c0f9c47","tex":"\\IR","aliases":"IR mathsf","display":"{\\mathsf{IR}}","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":57},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.","count":137},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fc97a5c4f9fcff61","tex":"K_\\lambda","aliases":"K_λ lambda","display":"K_\\lambda","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9feea0dd1279e0f4","tex":"\\Kcal","aliases":"Kcal K mathcal","display":"{\\mathcal K}","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2def419be484fcdf","tex":"\\Kcal_m","aliases":"Kcal_m K_m Kcal mathcal","display":"{\\mathcal K}_m","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":94},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":151},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9727f5125c78ee28","tex":"n_\\alpha","aliases":"n_α alpha","display":"n_\\alpha","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":8},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"def:packet_layout A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":93},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"definition-passage","excerpt":"def:packet_layout A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":24},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-aad7607e608d0052","tex":"P_\\beta","aliases":"P_β beta","display":"P_\\beta","kind":"definition-passage","excerpt":"acket operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displ","count":14},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-813431acda8054c4","tex":"\\Pcal","aliases":"Pcal P mathcal","display":"{\\mathcal P}","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":102},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":45},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[One pass streaming model] Fix a scalar size s bytes. For every active block, the one pass streaming model reads the block once, reads its source packet once, and reads and writes its target packet once. It does not assume cache reuse across edges.","count":68},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-152abbc9214c8600","tex":"S_\\lambda","aliases":"S_λ lambda","display":"S_\\lambda","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":43},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":54},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":154},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":20},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.","count":51},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-19ce956d274d340e","tex":"w_{e,k}","aliases":"w_e,k","display":"w_{e,k}","kind":"definition-passage","excerpt":"[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":41},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1e9458c4c7da9d3b","tex":"X_\\alpha","aliases":"X_α alpha","display":"X_\\alpha","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":8},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e5a69bbb6bda64f3","tex":"X_\\beta","aliases":"X_β beta","display":"X_\\beta","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":50},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":84},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":51},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":23},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f6991c3ad9fdd14c","tex":"\\epsilon_v","aliases":"ϵ_v epsilon","display":"\\epsilon_v","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":12},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-300f133d4213a5fd","tex":"\\iota_\\alpha","aliases":"ι_α iota alpha","display":"\\iota_\\alpha","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c039735a9807331b","tex":"\\iota_\\beta","aliases":"ι_β iota beta","display":"\\iota_\\beta","kind":"definition-passage","excerpt":"f IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain; ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation i","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":32},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fd0a204d0362614f","tex":"\\kappa_v","aliases":"κ_v kappa","display":"\\kappa_v","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":16},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":77},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"gindefinition[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":11},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-bd82674cd48e7508","tex":"\\pi_\\alpha","aliases":"π_α pi alpha","display":"\\pi_\\alpha","kind":"definition-passage","excerpt":"gindefinition[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":31},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7702fce301ebe3bf","tex":"\\sigma_v","aliases":"σ_v sigma","display":"\\sigma_v","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":12},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":19},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":40},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":61},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":125},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7354371b4dfcd811","tex":"\\sum_{(\\beta,\\alpha)\\in E}","aliases":"∑_(β,α)∈ E sum beta alpha in","display":"\\sum_{(\\beta,\\alpha)\\in E}","kind":"definition-passage","excerpt":"B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain; ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation is eval(IR) = ∑_(β,α)∈ E ι_β B_βαπ_α, where ι_β:","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cabedb544f677980","tex":"\\sum_{\\alpha\\in I}","aliases":"∑_α∈ I sum alpha in","display":"\\sum_{\\alpha\\in I}","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"definition-passage","excerpt":"[Support graph] The support graph G_A has vertex set I. It has a directed edge α⟶β when P_β AP_α≠0.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":32},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":8},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-90b882d77bb3725c","tex":"\\bigoplus_{\\alpha\\in I}","aliases":"⊕_α∈ I bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I}","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7d8e3b8668ae2ddd","tex":"A_k","aliases":"A_k","display":"A_k","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f70174408316e1b5","tex":"A_{\\gamma\\beta}","aliases":"A_γβ gamma beta","display":"A_{\\gamma\\beta}","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b8c748f09e8f469d","tex":"\\operatorname{Anc}","aliases":"Anc operatorname","display":"\\operatorname{Anc}","kind":"introduction-passage","excerpt":"For each vertex, define its strict ancestor set Anc(v)={u:there is a positive-length path uleadsto v}. For every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases wh","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-50c181940765c5f1","tex":"\\operatorname{Apply}","aliases":"Apply operatorname","display":"\\operatorname{Apply}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a6d440b3f2f61b84","tex":"\\operatorname{Apply}_k","aliases":"Apply_k operatorname","display":"\\operatorname{Apply}_k","kind":"introduction-passage","excerpt":"otimes M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e9c15d19ce3d5bef","tex":"\\operatorname{Apply}_{k}","aliases":"Apply_k operatorname","display":"\\operatorname{Apply}_{k}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":54},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"introduction-passage","excerpt":"Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfe","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End operatorname","display":"{\\operatorname{End}}","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":75},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"Walsh butterfly and exact addition count. textitSource item 4.7; module WalshTransform The two-point matrix is constructed as H_2=1 · 1 1 · -1, and the full family is defined by H_1=[1] and H_2N=H_2⊗ H_N =H_N · H_N","count":29},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"introduction-passage","excerpt":"1 · -1, and the full family is defined by H_1=[1] and H_2N=H_2⊗ H_N =H_N · H_N","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"introduction-passage","excerpt":"Walsh butterfly and exact addition count. textitSource item 4.7; module WalshTransform The two-point matrix is constructed as H_2=1 · 1 1 · -1, and the full family is defined by H_1=[1] and H_2N=H_2⊗ H_N =H_N · H_N","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c9b3c767be0a1967","tex":"H_{2N}","aliases":"H_2N","display":"H_{2N}","kind":"introduction-passage","excerpt":"1 · -1, and the full family is defined by H_1=[1] and H_2N=H_2⊗ H_N =H_N · H_N H_N · -H_N. The two-point orthogonality identity is checked explicitly. Kronecker multiplication and induction then prove H_NH_N^T=NI, N=2^k. The ordinary transpose is sufficient because all constructed entries are real, even though the scalar field is C.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ac7885f6694a994f","tex":"H_N","aliases":"H_N","display":"H_N","kind":"introduction-passage","excerpt":"1 · -1, and the full family is defined by H_1=[1] and H_2N=H_2⊗ H_N =H_N · H_N H_N · -H_N. The two-point orthogonality identity is checked explicitly. Kronecker multiplication and induction then prove H_NH_N^T=NI, N=2^k. The ordinary transpose is sufficient because all constructed entries are real, even though the scalar field is C.","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":60},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":135},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-345427490e88d80b","tex":"\\leadsto","aliases":"leadsto","display":null,"kind":"introduction-passage","excerpt":"For each vertex, define its strict ancestor set Anc(v)={u:there is a positive-length path uleadsto v}. For every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases wh","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":34},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c9f29e7d9e7a175b","tex":"M_k","aliases":"M_k","display":"M_k","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3db6ddc80894285f","tex":"M_{k+1}","aliases":"M_k+1","display":"M_{k+1}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"introduction-passage","excerpt":"r every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is define","count":11},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b61b8cfdaf3918dd","tex":"\\max_{u\\to v}","aliases":"max_u→ v max to","display":"\\max_{u\\to v}","kind":"introduction-passage","excerpt":"r every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is define","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"introduction-passage","excerpt":"1 · -1, and the full family is defined by H_1=[1] and H_2N=H_2⊗ H_N =H_N · H_N H_N · -H_N. The two-point orthogonality identity is checked explicitly. Kronecker multiplication and induction then prove H_NH_N^T=NI, N=2^k. The ordinary transpose is sufficient because all constructed entries are real, even though the scalar field is C.","count":26},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0e517349dec3b6ba","tex":"\\overset","aliases":"overset","display":null,"kind":"introduction-passage","excerpt":"Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfe","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"introduction-passage","excerpt":"(r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem. headA hand-computable decisi","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"introduction-passage","excerpt":", the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem. headA hand-computable decision threshold Fi","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4e4ee8a109009a53","tex":"\\subsetneq","aliases":"subsetneq","display":null,"kind":"introduction-passage","excerpt":"For each vertex, define its strict ancestor set Anc(v)={u:there is a positive-length path uleadsto v}. For every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases wh","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1084f45e6b3e947e","tex":"\\supp","aliases":"supp operatorname","display":"{\\operatorname{supp}}","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"ctored operator, and a compiled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide makes those distinctions visible through calculations rather than a pipeline of boxes. vfigthree_views headThree models to carry through the pap","count":167},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-014a85fddb5fe786","tex":"T_{vu}","aliases":"T_vu","display":"T_{vu}","kind":"introduction-passage","excerpt":"Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfe","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","excerpt":"ctored operator, and a compiled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide makes those distinctions visible through calculations rather than a pipeline of boxes. vfigthree_views headThree models to carry through the pap","count":84},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-36077cb07719c8a1","tex":"v^T","aliases":"v^T","display":"v^T","kind":"introduction-passage","excerpt":"ctored operator, and a compiled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide makes those distinctions visible through calculations rather than a pipeline of boxes. vfigthree_views headThree models to carry through the pap","count":17},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":95},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c070a883f58aab2d","tex":"x_{j,\\cdot}","aliases":"x_j,· cdot","display":"x_{j,\\cdot}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-708bfd9721bf49c2","tex":"X_u","aliases":"X_u","display":"X_u","kind":"introduction-passage","excerpt":"Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfe","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6590090cc9a70e3b","tex":"x_u","aliases":"x_u","display":"x_u","kind":"introduction-passage","excerpt":"iangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-08a6f9f108660d1a","tex":"X_v","aliases":"X_v","display":"X_v","kind":"introduction-passage","excerpt":"Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfe","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"introduction-passage","excerpt":"iangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":18},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-46014f7f0635c4a0","tex":"y_{i,u}","aliases":"y_i,u","display":"y_{i,u}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"r every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is define","count":11},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"introduction-passage","excerpt":"r every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is define","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d57a4808dbd2b57e","tex":"\\sum_{j=0}^{d-1}","aliases":"∑_j=0^d-1 sum","display":"\\sum_{j=0}^{d-1}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1ea15b9f71040065","tex":"\\sum_{u\\to v}","aliases":"∑_u→ v sum to","display":"\\sum_{u\\to v}","kind":"introduction-passage","excerpt":"iangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1b2a0f40b316674f","tex":"\\sum_{\\beta\\in I}","aliases":"∑_β∈ I sum beta in","display":"\\sum_{\\beta\\in I}","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"introduction-passage","excerpt":"Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfe","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":", the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem. headA hand-computable decision threshold Fi","count":33},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":11},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"nter,font=scriptsize] at (10.1,3.0575) 3; path[draw=black,fill=black!12,line width=.6pt] (11.456667,0.85) rectangle (12.41,3.144967); node[font=small,align=center,font=scriptsize] at (11.933333,0.55) Frobenius; node[font=small,align=center,font=scriptsize] at (11.933333,3.364967) 2 sqrt3; node[font=small,align=center,font=scriptsize] at (7,0.1) E _","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":103},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0aa786b643f5da71","tex":"A^{(1)}","aliases":"A^(1)","display":"A^{(1)}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6973a6c793fe345f","tex":"A^{(m)}","aliases":"A^(m)","display":"A^{(m)}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e0c9947b1daf0522","tex":"A^*","aliases":"A^*","display":"A^*","kind":"use","excerpt":"0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat3 · 0 4 i · 0 0 · -2 i, A^*A=diag(25,4). vfigcomplex_core headThe small calculation exposes a wrong convention The first column of Q_L has Hermitian squared length 9/25+16/25=1. Its bilinear square without conjugation is 9/25-16/25=-7/25. Thus Q_L^TQ_L is ","count":9},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-677eab50351e7675","tex":"A^{-1}","aliases":"A^-1","display":"A^{-1}","kind":"use","excerpt":"0 · 1 · -1 · 1. The two routes from coordinate zero to coordinate three cancel, so K^2=0. Consequently A^-1=I-K=mat1 · 0 · 0 · 0 -1 · 1 · 0 · 0 -1 · 0 · 1 · 0 0 · -1 · 1 · 1. vfig[.52]inverse_closure headWhat the closure permits The directed transitive closure includes 0→3, because it records existence of a route rather than the signed sum of its c","count":15},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"Cancellation control. Lean verifies the rational example A=1 · 1 -1 · -1, A^2=0. The coordinate block A_00 is nonzero, and a length-two active walk from zero to zero exists. Nevertheless the corresponding block of A^2 is zero. This is an exact counterexample to replacing support inclusion by equality.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-06d6907fddca4380","tex":"A^{h+1}","aliases":"A^h+1","display":"A^{h+1}","kind":"use","excerpt":"Nilpotence certificate. If no walk of length h+1 exists between any pair of packets, then A^h+1=0. Every block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map ℓ:I⟶{0,…,h}, A_bc≠0 Longrightarrow ℓ(c)<ℓ(b). It proves that a length-n walk increases the l","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-afc7e5c12f805c36","tex":"A^j","aliases":"A^j","display":"A^j","kind":"use","excerpt":"With A=diag(1,2), the vectors v and Av=(1,2)^T are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when A=I_2. Both changes enlarge the addressing power, but in different ways. headWhen invariance is only approximate If A=D","count":12},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fab2b96c59335d56","tex":"A^{j-1-k}","aliases":"A^j-1-k","display":"A^{j-1-k}","kind":"use","excerpt":"Use the telescoping identity A^j-D^j = ∑_k=0^j-1 A^j-1-kKD^k. Each term has norm at most M^j-1normK. The vector D^jv lies in the direct sum of local Krylov spaces.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-339ddf229ead4fcf","tex":"A^m","aliases":"A^m","display":"A^m","kind":"use","excerpt":"[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9a52b1bc77ae2b6d","tex":"A^n","aliases":"A^n","display":"A^n","kind":"use","excerpt":"F_n+1(b,a) · =∑_c: A_bc≠0A_bcF_n(c,a). The checked identity is P_bA^nP_a=F_n(b,a) (n≥0). Its proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence the actual block of A^n vanishes.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8f914025617b7ebc","tex":"A^T","aliases":"A^T","display":"A^T","kind":"use","excerpt":"vpage7A transpose, not an adjointKRONECKER ACTION AND ARRAY ORDER With column-major vectorization, the tensor identity is (A⊗ B)vec(X)=vec(BXA^T). Here the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix A=mat1 · i","count":15},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"ildren. Keeping r costs one with error six; splitting it adds overhead one. Keeping either a or b costs three with error two. Splitting a costs four with zero error, and splitting b costs five with zero error. vfigtree_frontier The five complete prunings have (cost,error) pairs (1,6), (7,4), (9,2), (8,2), (10,0). At budget two the unique cheapest p","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0f75896d5ca1818d","tex":"A_{00}","aliases":"A_00","display":"A_{00}","kind":"use","excerpt":"-1 · -1, A^2=0. The coordinate block A_00 is nonzero, and a length-two active walk from zero to zero exists. Nevertheless the corresponding block of A^2 is zero. This is an exact counterexample to replacing support inclusion by equality.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"ildren. Keeping r costs one with error six; splitting it adds overhead one. Keeping either a or b costs three with error two. Splitting a costs four with zero error, and splitting b costs five with zero error. vfigtree_frontier The five complete prunings have (cost,error) pairs (1,6), (7,4), (9,2), (8,2), (10,0). At budget two the unique cheapest p","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"A budget can select a point that no Lagrangian selects. Consider a rooted candidate tree. The root r has children a,b, and a has children a_1,a_2. Split overheads are zero. Each retained node has the following declared cost and integer error: lrrrrr Node · r · a · b · a_1 · a_2","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f2cfa673011ea148","tex":"a_3","aliases":"a_3","display":"a_3","kind":"use","excerpt":"iptsize] at (10.5,2.92) frac45; path[draw=black,fill=black!12,line width=.6pt] (11.733333,0.9) rectangle (12.6,1.0125); node[font=small,align=center,font=scriptsize] at (12.166667,0.6) 3; node[font=small,align=center,font=scriptsize] at (12.166667,1.2325) frac120; node[font=small,align=center,font=scriptsize] at (7,0.05) widehat p_j=a_jwidehat p_j-","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-94cffe8dd0af28a5","tex":"A_{ba}","aliases":"A_ba","display":"A_{ba}","kind":"use","excerpt":"0 · b≠ a, ∑_aP_a=1. Write A_ba=P_bAP_a and let E_A={(b,a):A_ba≠0}. These are exact algebraic zeros, not values discarded by a floating-point threshold.","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7000b8c39e7a4cdb","tex":"A_{bc}","aliases":"A_bc","display":"A_{bc}","kind":"use","excerpt":"Composition. Inserting the projector identity between factors yields (AB)_ba=∑_c A_bcB_ca. If the left side is nonzero, at least one intermediate c has both A_bc≠0 and B_ca≠0. Consequently supp(AB)⊆supp(A)∘supp(B). No equality of supports is asserted.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7764e407096f3048","tex":"A_e","aliases":"A_e","display":"A_e","kind":"use","excerpt":"[Packet perturbation certificate] Let A= ∑_e∈ EA_e, widehat A = ∑_e∈ E(A_e+E_e), where edge e starts at packet α(e), and assume A_e=A_eP_α(e), E_e=E_eP_α(e). Then norm(widehat A-A)x ≤ ∑_e∈ E normE_e normP_α(e)x. If normE_e ≤ ε_enormA_e, then norm(widehat A-A)x ≤ ∑_e∈ E ε_enormA_e normP_α(e)x.","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-74c2307a2295b9fc","tex":"A_{HH}","aliases":"A_HH","display":"A_{HH}","kind":"use","excerpt":"[Schur support inclusion] Split the packets into retained indices R and hidden indices H. Suppose the hidden block A_HH is invertible. The retained Schur complement is S = A_RR - A_RHA_HH^-1A_HR. A retained block (β,α) can be active only if it was already active in A_RR, or if there is a directed walk from α through hidden packets to β. The interna","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4a8722c909d9ace9","tex":"A_{HH}^{-1}","aliases":"A_HH^-1","display":"A_{HH}^{-1}","kind":"use","excerpt":"1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new ","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-574944342ee92d11","tex":"A_{HR}","aliases":"A_HR","display":"A_{HR}","kind":"use","excerpt":"1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new ","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"For any nonnegative decreasing sequence a_i, weights with these properties satisfy ∑_i a_iw_i≥∑_i>ka_i. This finite weighted inequality is itself proved. For k<d, use μ=a_k+1 and compare the summands to tail weights v_i=0 for i≤ k and v_i=1 otherwise: a_iv_i+μ(w_i-v_i)≤ a_iw_i. For a retained coordinate the inequality uses a_i≥μ and w_i≥0. For a ta","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3c32e4b0e0900b11","tex":"A_i^j","aliases":"A_i^j","display":"A_i^j","kind":"use","excerpt":"Exact and approximate packet Krylov spaces. textitSource items 7.2 and 7.3 Write K_m(A,v)=span{v,Av,…,A^m-1v}. For a complete finite projector family P_i, suppose A commutes with every P_i. Let A_i=P_iAP_i. Lean proves the actual orbit identity A^jv=∑_i A_i^jP_iv and therefore K_m(A,v)⊆ ⊕_i K_m(A_i,P_iv). The case j=0 uses ∑_iP_i=I. The zeroth loca","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.","count":13},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2167913f89021b35","tex":"A_{j+1}","aliases":"A_j+1","display":"A_{j+1}","kind":"use","excerpt":"Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-68fee95ac83fd47a","tex":"A_{j,e}","aliases":"A_j,e","display":"A_{j,e}","kind":"use","excerpt":"Packetwise mixed precision. At each stage let A_j=∑_e A_j,e, widehat A_j=∑_e(A_j,e+Δ_j,e), Δ_j,e=Δ_j,eP_j,a(e). The previous packet certificate supplies the uniform operator bound normwidehat A_j-A_j≤ δ_j:=∑_enormΔ_j,enormP_j,a(e). The new theorem substitutes these δ_j into the actual product-chain estimate. The resulting vector error is at most th","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0841717db411db13","tex":"A_{j-1}","aliases":"A_j-1","display":"A_{j-1}","kind":"use","excerpt":"Use the telescoping identity widehat A_m⋯widehat A_1 -A_m⋯ A_1 = ∑_j=1^m widehat A_m⋯widehat A_j+1 E_j A_j-1⋯ A_1. Take norms term by term.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-91f49c5985231a83","tex":"A_{j\\ell}","aliases":"A_jℓ ell","display":"A_{j\\ell}","kind":"use","excerpt":"The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d8f016d5b63ca3cf","tex":"A_{k+1}","aliases":"A_k+1","display":"A_{k+1}","kind":"use","excerpt":"Exact arithmetic count. Each pair of slices forms one sum and one difference at every trailing coordinate. The event type therefore gives A_0=0, A_k+1=2A_k+2^k+1. The closed form and its logarithmic re-expression are both checked: A_k=k2^k=Nlog_2N. The logarithm in the formal count is the natural-number binary logarithm, with its power-of-two ident","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e383dbc0f9ccfa0b","tex":"a_{k+1}","aliases":"a_k+1","display":"a_{k+1}","kind":"use","excerpt":"For any nonnegative decreasing sequence a_i, weights with these properties satisfy ∑_i a_iw_i≥∑_i>ka_i. This finite weighted inequality is itself proved. For k<d, use μ=a_k+1 and compare the summands to tail weights v_i=0 for i≤ k and v_i=1 otherwise: a_iv_i+μ(w_i-v_i)≤ a_iw_i. For a retained coordinate the inequality uses a_i≥μ and w_i≥0. For a ta","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-da2a817868989569","tex":"A_{k-1}","aliases":"A_k-1","display":"A_{k-1}","kind":"use","excerpt":"Arbitrary tensor factors. textitSource item 4.6; module StructuredTensor Let A_0,…,A_k-1 be square factors of order d. Tensor coordinates are represented by a nested product of k copies of {0,…,d-1}. The empty product is a singleton. Lean proves that the coordinate type has cardinality d^k.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-605293bc8cd6a683","tex":"A_{m-1}","aliases":"A_m-1","display":"A_{m-1}","kind":"use","excerpt":"Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8b601bf3f58468ef","tex":"A_{RH}","aliases":"A_RH","display":"A_{RH}","kind":"use","excerpt":"1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new ","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-56009cd5bba2cd88","tex":"A_{RR}","aliases":"A_RR","display":"A_{RR}","kind":"use","excerpt":"1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new ","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-edac8f4090c64b8b","tex":"A_\\alpha","aliases":"A_α alpha","display":"A_\\alpha","kind":"use","excerpt":"[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-84b88103b8092dfe","tex":"A_\\alpha^j","aliases":"A_α^j alpha","display":"A_\\alpha^j","kind":"use","excerpt":"[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-903a870c12766098","tex":"A_{\\alpha_1\\alpha_0}","aliases":"A_α_1α_0 alpha alpha","display":"A_{\\alpha_1\\alpha_0}","kind":"use","excerpt":"[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-bb2c326e063bfa7f","tex":"A_{\\alpha_m\\alpha_{m-1}}","aliases":"A_α_mα_m-1 alpha alpha","display":"A_{\\alpha_m\\alpha_{m-1}}","kind":"use","excerpt":"[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5498b3bc9fb8719d","tex":"A_{\\beta\\alpha}","aliases":"A_βα beta alpha","display":"A_{\\beta\\alpha}","kind":"use","excerpt":"A_βα · A block map · What are its source and target dimensions?","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-935d18e87c9add19","tex":"A_{\\beta\\alpha}^*","aliases":"A_βα^* beta alpha","display":"A_{\\beta\\alpha}^*","kind":"use","excerpt":"[Adjoint and tensor semantics] The adjoint representation has blocks (A^*)_αβ=A_βα^*. If A and B are packet resolved on X and Y, then (A⊗ B)_(β,δ),(α,γ) = A_βα⊗ B_δγ.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"Exact banded addresses and modeled work. textitSource item 4.1; module BandedCount For p packets and half-bandwidth w, the actual address set is B_p,w ={(i,j)∈{0,…,p-1}^2: i≤ j+w and j≤ i+w}. The predicate is the integer version of |i-j|≤ w. The count is first defined as a double sum of indicators, then identified with the cardinality of this subty","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"F_v(b) · A budget table · Does b mean exact weight, or an upper bound?","count":148},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"A direct inverse bound. If C=(I+E)^-1 and norm E_2≤η<1, then C=I-EC. Consequently (1-η)norm C_2≤ 1, normI+E_2≤ 1+η. The constructed normalization is invertible because S and A are. Combining the estimates gives the actual condition number bound κ_2(B):=norm B_2normB^-1_2 ≤frac1+η1-η. This norm argument does not require a separately assumed Hermitia","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"There are n/b active diagonal blocks. Each contains b^2 entries.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5b1e637f78e0d2bb","tex":"B_{\\mathrm F}^{2}","aliases":"B_ F^2 mathrm","display":"B_{\\mathrm F}^{2}","kind":"use","excerpt":"For the Frobenius norm, trace cyclicity proves left and right isometric invariance. The proof also establishes the projection identity normB-QQ^*B_ F^2 =norm B_ F^2-normQ^*B_ F^2. Nonnegativity of the left side gives left compression contractivity; the right version follows by adjoints. No norm-preservation or contractivity hypothesis is inserted.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","excerpt":"A direct inverse bound. If C=(I+E)^-1 and norm E_2≤η<1, then C=I-EC. Consequently (1-η)norm C_2≤ 1, normI+E_2≤ 1+η. The constructed normalization is invertible because S and A are. Combining the estimates gives the actual condition number bound κ_2(B):=norm B_2normB^-1_2 ≤frac1+η1-η. This norm argument does not require a separately assumed Hermitia","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b5384093c6e0cfdb","tex":"B_{ca}","aliases":"B_ca","display":"B_{ca}","kind":"use","excerpt":"Composition. Inserting the projector identity between factors yields (AB)_ba=∑_c A_bcB_ca. If the left side is nonzero, at least one intermediate c has both A_bc≠0 and B_ca≠0. Consequently supp(AB)⊆supp(A)∘supp(B). No equality of supports is asserted.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7bb9adbb22b5e88e","tex":"B_{dc}","aliases":"B_dc","display":"B_{dc}","kind":"use","excerpt":"Adjoint and tensor products. With an involution and self-adjoint projectors, the formalization proves (A_ba)^*=(A^*)_ab. For actual finite matrices and actual Kronecker products, it proves (P_b⊗ Q_d)(A⊗ B)(P_a⊗ Q_c) =A_ba⊗ B_dc. The self-adjointness hypothesis is explicit in the adjoint theorem. The tensor theorem is not a statement about an uninte","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"Keeping node v costs κ_v and incurs integer error w_v. Splitting costs overhead σ_v in addition to the selected child cuts. The exact-weight recurrence is F_v(b)=min{ l κ_v if b=w_v, [2pt] displaystyleσ_v+ min_∑_i b_i=b∑_iF_v_i(b_i) if v is internal. . Nonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-pl","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-643bdb7bc30d61f1","tex":"B_{ik}","aliases":"B_ik","display":"B_{ik}","kind":"use","excerpt":"The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4c82c8901bdffe44","tex":"\\mathcal B_{p,w}","aliases":"B_p,w mathcal","display":"\\mathcal B_{p,w}","kind":"use","excerpt":"Exact banded addresses and modeled work. textitSource item 4.1; module BandedCount For p packets and half-bandwidth w, the actual address set is B_p,w ={(i,j)∈{0,…,p-1}^2: i≤ j+w and j≤ i+w}. The predicate is the integer version of |i-j|≤ w. The count is first defined as a double sum of indicators, then identified with the cardinality of this subty","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-336970a90c727278","tex":"B_{\\rm stream}","aliases":"B_rm stream rm","display":"B_{\\rm stream}","kind":"use","excerpt":"blocks accumulate into one target, the reduction or accumulation policy is additional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention. headCount the data separately For three diagonal 2×2 blocks, eight-byte scalars, and q right-hand side","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b9846458a14f54b6","tex":"B_{\\mathrm{stream}}","aliases":"B_stream mathrm","display":"B_{\\mathrm{stream}}","kind":"use","excerpt":"Streaming. The explicit event type counts one block read, one source read, and a target read plus write for every edge. With scalar size s, its byte count is B_stream=s∑_e(m_en_e+n_eq+2m_eq). No cross-edge cache reuse is assumed. Sparse index metadata and temporary workspace remain separate.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-69b376b6bbe4d3ac","tex":"b_v","aliases":"b_v","display":"b_v","kind":"use","excerpt":"Constructed solution. Well-founded recursion defines x_v=D_v^-1 (b_v-∑_u→ vT_vux_u). Applying D_v proves the original equation at v. If another vector satisfies all equations, induction over predecessors makes each previously used coordinate identical, and invertibility of D_v forces agreement at v. Thus the construction gives existence and uniquen","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5982aa6063dafe04","tex":"b_\\beta","aliases":"b_β beta","display":"b_\\beta","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b0a7f57cb21faf55","tex":"B_{\\delta\\gamma}","aliases":"B_δγ delta gamma","display":"B_{\\delta\\gamma}","kind":"use","excerpt":"[Adjoint and tensor semantics] The adjoint representation has blocks (A^*)_αβ=A_βα^*. If A and B are packet resolved on X and Y, then (A⊗ B)_(β,δ),(α,γ) = A_βα⊗ B_δγ.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0a092bc56762bbd9","tex":"B_\\nu","aliases":"B_ν nu","display":"B_\\nu","kind":"use","excerpt":"ubsection:002textitModule SmallCoreNorms; arbitrary ambient competitors For rectangular isometries Q,V define L(G)=QGV^*, C(B)=Q^*BV, Q^*Q=I, V^*V=I. The following statements are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_ν≤norm B_ν, ν∈{2, F}. For the spe","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c6822eaf3f6b3d51","tex":"\\bar\\kappa","aliases":"barκ bar kappa","display":"\\bar\\kappa","kind":"use","excerpt":"tκ-1sqrtκ+1. Monotonicity of square root and positive denominators prove that q is monotone. Lean also proves κ_2(B)≥1, q(κ)≥0 for κ≥1, and q(κ)<1. Thus q(κ_2(B))≤ q (frac1+η1-η). For 1≤κ and e_0≥0, an explicit classical premise e_m≤2q(κ)^m e_0 and a certified upper bound κ≤barκ yield e_m≤2q(barκ)^m e_0.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATION Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,-1,0)^T, dense multiplicat","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATION Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,-1,0)^T, dense multiplicat","count":49},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathbb","display":"{\\mathbb C}","kind":"use","excerpt":"Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"Why a small-core optimum is a global optimum. textitModule SmallCoreNorms; arbitrary ambient competitors For rectangular isometries Q,V define L(G)=QGV^*, C(B)=Q^*BV, Q^*Q=I, V^*V=I. The following statements are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"use","excerpt":"ents are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_ν≤norm B_ν, ν∈{2, F}. For the spectral norm, the proof uses the Gram matrix identity (QG)^*(QG)=G^*G and the C^*-norm square identity, together with right-side adjoint symmetry. Compression follows from ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-512937534243ab85","tex":"\\C^{m\\times m}","aliases":"C^m× m C times mathbb times","display":"{\\mathbb C}^{m\\times m}","kind":"use","excerpt":"Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-eee5bcabc685dee4","tex":"\\C^{m\\times t}","aliases":"C^m× t C times mathbb times","display":"{\\mathbb C}^{m\\times t}","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8898d320fd158845","tex":"\\C^{n\\times m}","aliases":"C^n× m C times mathbb times","display":"{\\mathbb C}^{n\\times m}","kind":"use","excerpt":"Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1dc82af4d957fdd4","tex":"\\C^{n\\times n}","aliases":"C^n× n C times mathbb times","display":"{\\mathbb C}^{n\\times n}","kind":"use","excerpt":"Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e06675c775fa7c9d","tex":"\\mathbb C^{n\\times r_j}","aliases":"C^n× r_j mathbb times","display":"\\mathbb C^{n\\times r_j}","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c2e1f9b36ab7f058","tex":"\\C^{n\\times t}","aliases":"C^n× t C times mathbb times","display":"{\\mathbb C}^{n\\times t}","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2ccfe315d05aabc8","tex":"\\C^{r_L\\times r_R}","aliases":"C^r_L× r_R C times mathbb times","display":"{\\mathbb C}^{r_L\\times r_R}","kind":"use","excerpt":"Define the small core by actual multiplication, G=R_LR_R^*∈ C^r_L× r_R. Finite matrix associativity and adjoint identities give A=Q_LGQ_R^*, rank G=rank A. The rank identity follows from both multiplication inequalities and the left inverse obtained by compression. It is not inferred from agreeing dimensions.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"A direct inverse bound. If C=(I+E)^-1 and norm E_2≤η<1, then C=I-EC. Consequently (1-η)norm C_2≤ 1, normI+E_2≤ 1+η. The constructed normalization is invertible because S and A are. Combining the estimates gives the actual condition number bound κ_2(B):=norm B_2normB^-1_2 ≤frac1+η1-η. This norm argument does not require a separately assumed Hermitia","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"use","excerpt":"vpage20Spend error where it is cheapestCONTINUOUS AND CLIPPED PRECISION Take sensitivities s=(1,2,4), cost coefficients c=(1,1,2), and budget τ=1. Minimizing ∑_ec_elog(1/ϵ_e) subject to ∑_es_eϵ_e=1 gives ϵ_e=fracc_e4s_e, ϵ=(1/4,1/8,1/8). vfigprecision headThe budget is weighted The three contributions to the error allowance are (1/4,1/4,1/2). Equal","count":34},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cdae72f972852da1","tex":"c_f","aliases":"c_f","display":"c_f","kind":"use","excerpt":"[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8eb98997bbe300af","tex":"c_{i-j}","aliases":"c_i-j","display":"c_{i-j}","kind":"use","excerpt":"Toeplitz padding and column-major vectorization. textitSource items 4.4 and 4.5 Toeplitz embedding. For n>0, define T_ij= c_i-j, · j≤ i, r_j-i, · i<j. The diagonal is supplied by c_0; the coordinate r_0 is unused. The constructed generator of length 2n is g=(c_0,…,c_n-1,0,r_n-1,…,r_1), and the padding map is the actual finite-index sum eliminator t","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"[Polynomial support] If p(z)=∑_j=0^dc_jz^j, then supp(p(A)) is contained in the union of the support relations of A^j for 0≤ j≤ d.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-00ed39c395ad1950","tex":"c_{j,k}","aliases":"c_j,k","display":"c_{j,k}","kind":"use","excerpt":"ct discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-40c460a40f8ac2ac","tex":"c_{j-k}","aliases":"c_j-k","display":"c_{j-k}","kind":"use","excerpt":"4 · 3 · 2, x=(1,2,-1)^T. The diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column g=(2,3,4,0,5,-1)^T and pad the input with three zeros. vfigtoeplitz headRead the wrapped indices For output index j<3, an input index k≤ j uses the first-column coefficient c_j-k. An input index k>j ","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b26dec79c0d88ba5","tex":"c_{j-k\\bmod4}","aliases":"c_j-kbmod4 bmod","display":"c_{j-k\\bmod4}","kind":"use","excerpt":"vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATION Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,-1,0)^T, dense multiplicat","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b7e8c8a0a1fa875e","tex":"c_{n-1}","aliases":"c_n-1","display":"c_{n-1}","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"use","excerpt":"For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0c1fa1d6163c6fb8","tex":"\\CC^{2n}","aliases":"CC^2n C^2n CC mathbb","display":"{\\mathbb C}^{2n}","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b9f0fa654a6d5449","tex":"\\CC^{m\\times m}","aliases":"CC^m× m C^m× m CC times mathbb times","display":"{\\mathbb C}^{m\\times m}","kind":"use","excerpt":"[Kronecker modewise application] Let A∈ C^m× m, B∈ C^n× n, and X∈ C^n× m. With column major vectorization, (A⊗ B)vec(X) = vec(BXA^T). The right side has modeled cost 2n^2m+2nm^2 and stores n^2+m^2 factor entries instead of n^2m^2 entries.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ac276f231fb12d59","tex":"\\CC^{m\\times r}","aliases":"CC^m× r C^m× r CC times mathbb times","display":"{\\mathbb C}^{m\\times r}","kind":"use","excerpt":"[Small core recompression] Let L∈ C^m× r, R∈ C^n× r. Take compact QR factorizations L=Q_LR_L, R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cf090df8943be9f2","tex":"\\CC^n","aliases":"CC^n C^n CC mathbb","display":"{\\mathbb C}^n","kind":"use","excerpt":"[Circulant spectral representation] Let n≥1 be an integer and c∈ C^n, with indices taken modulo n, and define C(c)_jk=c_j-k. Let F be the unitary discrete Fourier matrix F_jk = frac1sqrt n exp(-frac2π i jkn). Then C(c) = F^*diag(sqrt n Fc)F. The operator is determined by n scalars and can be applied through two Fourier transforms and one pointwise ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8e82c63e4ddfb564","tex":"\\CC^{n\\times m}","aliases":"CC^n× m C^n× m CC times mathbb times","display":"{\\mathbb C}^{n\\times m}","kind":"use","excerpt":"[Kronecker modewise application] Let A∈ C^m× m, B∈ C^n× n, and X∈ C^n× m. With column major vectorization, (A⊗ B)vec(X) = vec(BXA^T). The right side has modeled cost 2n^2m+2nm^2 and stores n^2+m^2 factor entries instead of n^2m^2 entries.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3e1e698420bbbca9","tex":"\\CC^{n\\times n}","aliases":"CC^n× n C^n× n CC times mathbb times","display":"{\\mathbb C}^{n\\times n}","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f9a7361238345df3","tex":"\\mathrm{child}","aliases":"child mathrm","display":"\\mathrm{child}","kind":"use","excerpt":"Recurrence. For terminal nodes, F_λ(v)=κ_v+λϵ_v. For internal nodes, F_λ(v)=min{κ_v+λϵ_v, σ_v+∑_w∈child(v)F_λ(w)}. The checked result has two parts. First, F_λ(v)≤ J_λ(L) for every legal cut of the subtree. Second, the constructed function codecompile returns a legal cut with objective exactly F_λ(v). Thus the minimum is attained, not merely bounde","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-860a844f873baee1","tex":"\\operatorname{child}","aliases":"child operatorname","display":"\\operatorname{child}","kind":"use","excerpt":"[Exact Lagrangian packet compiler] Fix λ≥0 and define J_λ( L) = cost( L) + λerr( L). For every node v, set K_λ(v) = κ_v+λϵ_v and S_λ(v) = σ_v + ∑_w∈child(v)F_λ(w). For a terminal node, set F_λ(v)=K_λ(v). For an internal node, the recurrence F_λ(v) = min{K_λ(v),S_λ(v)} computes the exact minimum Lagrangian objective over every admissible pruning of ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-269cd99c3ea8db81","tex":"\\code","aliases":"code","display":null,"kind":"use","excerpt":"For a predicate S on costs, the common minimum specification means: ll codenone: · no cost satisfies S, codesome(v): · S(v) and v≤ z whenever S(z). Both attainment and minimality are included. Optional minimum and optional addition are proved to implement union and independent sum of feasible sets.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-52ad9950a5d7bf10","tex":"\\operatorname{compressed}","aliases":"compressed operatorname","display":"\\operatorname{compressed}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3f4e7b64e9db023c","tex":"\\cond","aliases":"cond operatorname","display":"{\\operatorname{cond}}","kind":"use","excerpt":"[Block Jacobi condition bound] Let A=D+K act on a nonempty finite coordinate space and be Hermitian positive definite with positive definite block diagonal part D. Here D^-1/2 is the actual inverse of the positive square root of D. Suppose E=D^-1/2KD^-1/2 is Hermitian and normE_2≤η<1. Then cond_2(D^-1/2AD^-1/2) ≤ frac1+η1-η.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-aa303ae817004b47","tex":"\\cond_2","aliases":"cond_2 cond operatorname","display":"{\\operatorname{cond}}_2","kind":"use","excerpt":"[Block Jacobi condition bound] Let A=D+K act on a nonempty finite coordinate space and be Hermitian positive definite with positive definite block diagonal part D. Here D^-1/2 is the actual inverse of the positive square root of D. Suppose E=D^-1/2KD^-1/2 is Hermitian and normE_2≤η<1. Then cond_2(D^-1/2AD^-1/2) ≤ frac1+η1-η.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cef2f13a7966bbb0","tex":"\\coprod","aliases":"coprod","display":null,"kind":"use","excerpt":"Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their card","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a0eafdcac63578ce","tex":"\\coprod_{e\\in E}","aliases":"coprod_e∈ E coprod in","display":null,"kind":"use","excerpt":"Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their card","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e1a4d4510c259fa0","tex":"\\cost","aliases":"cost operatorname","display":"{\\operatorname{cost}}","kind":"use","excerpt":"ne. Keeping either a or b costs three with error two. Splitting a costs four with zero error, and splitting b costs five with zero error. vfigtree_frontier The five complete prunings have (cost,error) pairs (1,6), (7,4), (9,2), (8,2), (10,0). At budget two the unique cheapest pruning is {a_0,a_1,b}, costing eight. It is Pareto optimal, but never mi","count":12},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cb7ca0be63af352d","tex":"D^{(m)}","aliases":"D^(m)","display":"D^{(m)}","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fd12667298c0030a","tex":"D^{(m-1)}","aliases":"D^(m-1)","display":"D^{(m-1)}","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","excerpt":"-1 · 3. Hence D^-1u=u/4 and S=1+u^TD^-1u=frac52. The Woodbury identity reduces the entire inverse to A^-1=D^-1-D^-1uS^-1u^TD^-1 =D^-1-frac140uu^T. vfigwoodbury headTwo hypotheses, not one Invertibility of D alone does not suffice. For D=I_2, u=e_1, and v=-e_1, the update is diag(0,1). The small matrix S is zero, exactly where the correction formula","count":25},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6311982fa9099aea","tex":"D^{-1/2}","aliases":"D^-1/2","display":"D^{-1/2}","kind":"use","excerpt":"vpage18Measure coupling after normalizationBLOCK JACOBI AS A GEOMETRIC CHANGE Let A=mat4 · 1 1 · 1, D=diag(4,1), B=D^-1/2=diag(1/2,1). Then BAB=I+E=mat1 · 1/2","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"[Equal tensor factors] For integers k,d≥0, a product of k square factors of order d, acting on a tensor with d^k entries, can be applied by k modewise contractions, reusing intermediate slices, with modeled cost 2k d^k+1 and factor storage kd^2.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-87a4659e9ddda329","tex":"D^j","aliases":"D^j","display":"D^j","kind":"use","excerpt":"With A=diag(1,2), the vectors v and Av=(1,2)^T are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when A=I_2. Both changes enlarge the addressing power, but in different ways. headWhen invariance is only approximate If A=D","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2f80fcde2a727a6d","tex":"D^k","aliases":"D^k","display":"D^k","kind":"use","excerpt":"Use the telescoping identity A^j-D^j = ∑_k=0^j-1 A^j-1-kKD^k. Each term has norm at most M^j-1normK. The vector D^jv lies in the direct sum of local Krylov spaces.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7d80d76ad0ed63a4","tex":"d^k","aliases":"d^k","display":"d^k","kind":"use","excerpt":"[Equal tensor factors] For integers k,d≥0, a product of k square factors of order d, acting on a tensor with d^k entries, can be applied by k modewise contractions, reusing intermediate slices, with modeled cost 2k d^k+1 and factor storage kd^2.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9d209e4a42d264b0","tex":"d^{k+1}","aliases":"d^k+1","display":"d^{k+1}","kind":"use","excerpt":"[Equal tensor factors] For integers k,d≥0, a product of k square factors of order d, acting on a tensor with d^k entries, can be applied by k modewise contractions, reusing intermediate slices, with modeled cost 2k d^k+1 and factor storage kd^2.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c64f3c404f15cb37","tex":"d^{k+2}","aliases":"d^k+2","display":"d^{k+2}","kind":"use","excerpt":"Operation events. The event type for k+1 modes is a disjoint union of d copies of the trailing event type and the events for the final fiber contraction. Its cardinality obeys W_0=0, W_k+1=dW_k+2d^k+2. Lean proves the closed form W_k=2kd^k+1. The factor-address type has cardinality k d^2. These statements include k=0 and d=0. With no factors, the a","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b43f40b2f922a574","tex":"d^{k-1}","aliases":"d^k-1","display":"d^{k-1}","kind":"use","excerpt":"For k=0 there are no contractions and both modeled work and factor storage are zero. For k≥1, one mode has d^k-1 fibers of length d. Multiplying each fiber by a d by d matrix costs 2d^2, so one mode costs 2d^k+1. Sum over the k modes.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-63ed82a00b685c9b","tex":"D_A","aliases":"D_A","display":"D_A","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":23},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ee5bb7d36741efec","tex":"D_B","aliases":"D_B","display":"D_B","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":21},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7f62021dadd3fc99","tex":"D_B^*","aliases":"D_B^*","display":"D_B^*","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-60f01eac58353ae3","tex":"D_B^T","aliases":"D_B^T","display":"D_B^T","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0b05c681f47bcdf3","tex":"D_{j-1}","aliases":"D_j-1","display":"D_{j-1}","kind":"use","excerpt":"Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a572acccc9b7cf05","tex":"D_k","aliases":"D_k","display":"D_k","kind":"use","excerpt":"Spectral optimality from a kernel witness. textitModules DiagonalTruncation and SpectralOptimality Let σ_1≥⋯≥σ_d≥0 and D=diag(σ_1,…,σ_d), D_k=diag(σ_1,…,σ_min(k,d),0,…,0). Lean uses indices beginning at zero; the formulas on this page use the conventional one-based notation. The rank of D_k is proved to be at most k, even when retained singular val","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-71efea616c614868","tex":"D_m","aliases":"D_m","display":"D_m","kind":"use","excerpt":"Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1b008a80da8b3621","tex":"D_v^{-1}","aliases":"D_v^-1","display":"D_v^{-1}","kind":"use","excerpt":"Constructed solution. Well-founded recursion defines x_v=D_v^-1 (b_v-∑_u→ vT_vux_u). Applying D_v proves the original equation at v. If another vector satisfies all equations, induction over predecessors makes each previously used coordinate identical, and invertibility of D_v forces agreement at v. Thus the construction gives existence and uniquen","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-de786ac59a3f75f3","tex":"D_\\beta","aliases":"D_β beta","display":"D_\\beta","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f5ae8519ff6d0dff","tex":"\\mathrm{decode}","aliases":"decode mathrm","display":"\\mathrm{decode}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1e31f1e54430399f","tex":"\\operatorname{decode}","aliases":"decode operatorname","display":"\\operatorname{decode}","kind":"use","excerpt":"[Contract conversion] Suppose one algorithm returns a compressed representation R and another returns an explicit object Y. A direct timing comparison is valid only if decode(R)=Y and the time required by decode is included, or if the downstream task consumes R directly.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"For the running model, the checker multiplies the proposed inverse by A in both orders and obtains the identity. It also checks the determinant relation det A=det Ddet S. These tests verify the actual factors and sign of the correction, not just a numerical solve residual. headConditioning survives the reduction A small S can still be ill-condition","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0658d980d8d1bc6a","tex":"\\operatorname{DFT}","aliases":"DFT operatorname","display":"\\operatorname{DFT}","kind":"use","excerpt":"Normalization and inversion. The proof identifies matrix application with the ordinary counting-measure discrete Fourier transform: Fx=n^-1/2DFT(x). It separately identifies F^*x with the appropriately scaled inverse transform. Mathlib's checked Fourier inversion theorem and the proved scalar normalization yield both identities F^*F=I, FF^*=I. No u","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACT The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat","count":22},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"Convolution and reconstruction. An explicit bijective change of index proves DFT(C(c)x)_k =DFT(c)_kDFT(x)_k. This gives the matrix intertwining relation FC(c)=diag(DFTc)F. Multiplication by the proved inverse reconstructs the circulant: C(c)=F^*diag(sqrt n Fc)F. The two-transform application formula is also checked for every input vector. The resul","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-558c1dc971656de6","tex":"\\dist","aliases":"dist operatorname","display":"{\\operatorname{dist}}","kind":"use","excerpt":"[Approximate packet Krylov bound] Write A=D+K, where D is packet diagonal. For every positive integer j, normA^j-D^j ≤ jM^j-1normK, where M=max{normA,normD}. Consequently, dist(A^jv,⊕_α K_j+1(D_α,v_α)) ≤ jM^j-1normKnormv.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","excerpt":"Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"The preconditioning corollary. Define q(κ)=fracsqrtκ-1sqrtκ+1. Monotonicity of square root and positive denominators prove that q is monotone. Lean also proves κ_2(B)≥1, q(κ)≥0 for κ≥1, and q(κ)<1. Thus q(κ_2(B))≤ q (frac1+η1-η). For 1≤κ and e_0≥0, an explicit classical premise e_m≤2q(κ)^m e_0 and a certified upper bound κ≤barκ yield e_m≤2q(barκ)^m","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"-1 · 3. Hence D^-1u=u/4 and S=1+u^TD^-1u=frac52. The Woodbury identity reduces the entire inverse to A^-1=D^-1-D^-1uS^-1u^TD^-1 =D^-1-frac140uu^T. vfigwoodbury headTwo hypotheses, not one Invertibility of D alone does not suffice. For D=I_2, u=e_1, and v=-e_1, the update is diag(0,1). The small matrix S is zero, exactly where the correction formula","count":9},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-115653a560af9b18","tex":"e_1^T","aliases":"e_1^T","display":"e_1^T","kind":"use","excerpt":"(1.65,2.1) rectangle (2.5,2.95); node[font=small,align=center,font=small] at (2.075,2.525) frac38; node[font=small,align=center] at (6.25,3.5) D^-1u=tfrac14 u; node[font=small,align=center] at (6.25,2.25) S=1+u^TD^-1u=tfrac52; node[font=small,align=center] at (6.25,0.9) A^-1=D^-1-tfrac140uu^T; node[font=small,align=center,font=small] at (11.45,4.25","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"A direct inverse bound. If C=(I+E)^-1 and norm E_2≤η<1, then C=I-EC. Consequently (1-η)norm C_2≤ 1, normI+E_2≤ 1+η. The constructed normalization is invertible because S and A are. Combining the estimates gives the actual condition number bound κ_2(B):=norm B_2normB^-1_2 ≤frac1+η1-η. This norm argument does not require a separately assumed Hermitia","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8db0cea46fdd60f3","tex":"E_A","aliases":"E_A","display":"E_A","kind":"use","excerpt":"0 · b≠ a, ∑_aP_a=1. Write A_ba=P_bAP_a and let E_A={(b,a):A_ba≠0}. These are exact algebraic zeros, not values discarded by a floating-point threshold.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a20a0c8f1552bcce","tex":"E_e","aliases":"E_e","display":"E_e","kind":"use","excerpt":"For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tig","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"E_j · An error operator · Is it a factor perturbation or an additive error after a product?","count":20},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","excerpt":"onditioning corollary. Define q(κ)=fracsqrtκ-1sqrtκ+1. Monotonicity of square root and positive denominators prove that q is monotone. Lean also proves κ_2(B)≥1, q(κ)≥0 for κ≥1, and q(κ)<1. Thus q(κ_2(B))≤ q (frac1+η1-η). For 1≤κ and e_0≥0, an explicit classical premise e_m≤2q(κ)^m e_0 and a certified upper bound κ≤barκ yield e_m≤2q(barκ)^m e_0.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps ε varepsilon","display":"{\\varepsilon}","kind":"use","excerpt":"[Packet perturbation certificate] Let A= ∑_e∈ EA_e, widehat A = ∑_e∈ E(A_e+E_e), where edge e starts at packet α(e), and assume A_e=A_eP_α(e), E_e=E_eP_α(e). Then norm(widehat A-A)x ≤ ∑_e∈ E normE_e normP_α(e)x. If normE_e ≤ ε_enormA_e, then norm(widehat A-A)x ≤ ∑_e∈ E ε_enormA_e normP_α(e)x.","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-bc04c9966dd2873a","tex":"\\eps_e","aliases":"eps_e ε_e eps varepsilon","display":"{\\varepsilon}_e","kind":"use","excerpt":"[Packet perturbation certificate] Let A= ∑_e∈ EA_e, widehat A = ∑_e∈ E(A_e+E_e), where edge e starts at packet α(e), and assume A_e=A_eP_α(e), E_e=E_eP_α(e). Then norm(widehat A-A)x ≤ ∑_e∈ E normE_e normP_α(e)x. If normE_e ≤ ε_enormA_e, then norm(widehat A-A)x ≤ ∑_e∈ E ε_enormA_e normP_α(e)x.","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7340ca44d667f098","tex":"\\err","aliases":"err operatorname","display":"{\\operatorname{err}}","kind":"use","excerpt":"ne. Keeping either a or b costs three with error two. Splitting a costs four with zero error, and splitting b costs five with zero error. vfigtree_frontier The five complete prunings have (cost,error) pairs (1,6), (7,4), (9,2), (8,2), (10,0). At budget two the unique cheapest pruning is {a_0,a_1,b}, costing eight. It is Pareto optimal, but never mi","count":14},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"[Circulant spectral representation] Let n≥1 be an integer and c∈ C^n, with indices taken modulo n, and define C(c)_jk=c_j-k. Let F be the unitary discrete Fourier matrix F_jk = frac1sqrt n exp(-frac2π i jkn). Then C(c) = F^*diag(sqrt n Fc)F. The operator is determined by n scalars and can be applied through two Fourier transforms and one pointwise ","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-484de1c254ac2bb8","tex":"\\mathrm F","aliases":"F mathrm","display":"\\mathrm F","kind":"use","excerpt":"ubsection:002textitModule SmallCoreNorms; arbitrary ambient competitors For rectangular isometries Q,V define L(G)=QGV^*, C(B)=Q^*BV, Q^*Q=I, V^*V=I. The following statements are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_ν≤norm B_ν, ν∈{2, F}. For the spe","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a3382034c436fa8d","tex":"F^*","aliases":"F^*","display":"F^*","kind":"use","excerpt":"vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATION Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,-1,0)^T, dense multiplicat","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","excerpt":"e coherent errorDISCOVERY AND PERTURBATION CERTIFICATES Split QQ^4 into scalar packets and let A=I+E, E=frac1100(onesones^T-I). Dropping every off-diagonal block leaves I. Each discarded entry is only 1/100, but the total error is not bounded by that single-entry size. vfigblock_error headTwo norms answer different questions There are twelve omitte","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"[Exact discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"[Exact discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7ac4acd8af2778de","tex":"F_{j-1}","aliases":"F_j-1","display":"F_{j-1}","kind":"use","excerpt":"[Exact discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6bf87b4a8f3226a8","tex":"F_{jk}","aliases":"F_jk","display":"F_{jk}","kind":"use","excerpt":"vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATION Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,-1,0)^T, dense multiplicat","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"use","excerpt":"total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b≤ B is the exact minimum cost for the q","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c57cafb013c6775f","tex":"F_n","aliases":"F_n","display":"F_n","kind":"use","excerpt":"Define the active contribution sum recursively by F_0(b,a) · =P_a · b=a, 0 · b≠ a, F_n+1(b,a) · =∑_c: A_bc≠0A_bcF_n(c,a). The checked identity is P_bA^nP_a=F_n(b,a) (n≥0). Its proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence t","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7817222f46df5e65","tex":"F_{n+1}","aliases":"F_n+1","display":"F_{n+1}","kind":"use","excerpt":"Define the active contribution sum recursively by F_0(b,a) · =P_a · b=a, 0 · b≠ a, F_n+1(b,a) · =∑_c: A_bc≠0A_bcF_n(c,a). The checked identity is P_bA^nP_a=F_n(b,a) (n≥0). Its proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence t","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-faf064f796e64232","tex":"F_{\\rm root}","aliases":"F_rm root rm","display":"F_{\\rm root}","kind":"use","excerpt":"Assume every keep error is a nonnegative integer w_v. Let B∈ N. For each node v, define F_v(b) as the minimum cost of a pruning of the subtree at v with total error exactly b. The keep option contributes F_v(w_v)≤κ_v. The split option is obtained by min plus convolution of the child tables, followed by addition of σ_v. The smallest value among F_rm","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-21c255185d10999e","tex":"F_{\\mathrm{root}}","aliases":"F_root mathrm","display":"F_{\\mathrm{root}}","kind":"use","excerpt":"The root scan min_0≤ b≤ BF_root(b) is therefore the exact budget-constrained optimum. Increasing the budget cannot increase a finite optimum. The theorem retains feasibility hypotheses where necessary rather than treating codenone as an ordinary real cost.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","excerpt":"Entrywise implementation theorem. For every b≤ B, Lean proves lookup(storedTree(B,t),b)=F_t(b). There is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at b depends only on input entries at most b, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attain","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-78b29d109052da32","tex":"F_v","aliases":"F_v","display":"F_v","kind":"use","excerpt":"F_v(b) · A budget table · Does b mean exact weight, or an upper bound?","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e68207ff3f414911","tex":"F_{v_i}","aliases":"F_v_i","display":"F_{v_i}","kind":"use","excerpt":"Keeping node v costs κ_v and incurs integer error w_v. Splitting costs overhead σ_v in addition to the selected child cuts. The exact-weight recurrence is F_v(b)=min{ l κ_v if b=w_v, [2pt] displaystyleσ_v+ min_∑_i b_i=b∑_iF_v_i(b_i) if v is internal. . Nonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-pl","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-832e56d8bd1b7868","tex":"F_\\lambda","aliases":"F_λ lambda","display":"F_\\lambda","kind":"use","excerpt":"The dynamic recurrence gives the same Lagrangian choices without listing prunings. At a, it compares 2+4λ with 4+λ. At r, it compares 1+6λ with 1+F_λ(a). Its exactness follows from the finite tree and additive model in [source reference]. Overlapping decompositions or nonlocal execution costs need another argument.","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-642d1aafbe0d4290","tex":"\\operatorname{fill}","aliases":"fill operatorname","display":"\\operatorname{fill}","kind":"use","excerpt":"Correctness of the actual update. For any edited subtree T' and valid context C, Lean proves updateValue(C,F(T'))=F(fill(C,T')). The proof inducts through the context. At each frame, the updated child contributes the new value, and each other child contributes its unchanged cached optimum. The frame recurrence is therefore exactly the full-tree rec","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-280dad33f94a39d1","tex":"\\operatorname{Fin}","aliases":"Fin operatorname","display":"\\operatorname{Fin}","kind":"use","excerpt":"Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their card","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2b0a62a84fa62d7d","tex":"\\flop_{\\rm block}","aliases":"flop_rm block flop rm operatorname rm","display":"{\\operatorname{flop}}_{\\rm block}","kind":"use","excerpt":"[Equal diagonal packets] Let p≥1 and b≥1 be integers and n=pb. Suppose n is divided into p equal diagonal packets of width b. Then flop_rm block=2qnb, M_rm block=nb. The dense-to-block storage reduction factor is the following; the arithmetic reduction factor has the same value when q≥1: fracnb.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cde9aac4c00078d2","tex":"\\flop_{\\rm dense}","aliases":"flop_rm dense flop rm operatorname rm","display":"{\\operatorname{flop}}_{\\rm dense}","kind":"use","excerpt":"[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-82a45cf5e0bc4cc8","tex":"\\flop_{\\rm packet}","aliases":"flop_rm packet flop rm operatorname rm","display":"{\\operatorname{flop}}_{\\rm packet}","kind":"use","excerpt":"[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-25f0be3ddda5cf8c","tex":"G^*","aliases":"G^*","display":"G^*","kind":"use","excerpt":"*V=I. The following statements are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_ν≤norm B_ν, ν∈{2, F}. For the spectral norm, the proof uses the Gram matrix identity (QG)^*(QG)=G^*G and the C^*-norm square identity, together with right-side adjoint symmetry.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-103f7de3482048f3","tex":"G_k","aliases":"G_k","display":"G_k","kind":"use","excerpt":"Suppose G_k is a best rank-at-most-k approximation to G in either norm. Take an arbitrary ambient matrix B with rank B≤ k. Compression is an admissible small-core competitor and norm L(G)- L(G_k)_ν · =normG-G_k_ν","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"vpage32Safe integers can still be conservativeQUANTIZATION AND THE SIZE OF THE BUDGET TABLE Let one integer unit represent h=1/10 of certified error. For an individual nonnegative error e, choosing w=lceil e/hrceil ensures e≤ hw. Thus integer feasibility ∑ w≤ B implies the real bound ∑ e≤ hB. This implication is checked before applying the exact op","count":25},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0ea241c4c1df1f56","tex":"H_{2n}","aliases":"H_2n","display":"H_{2n}","kind":"use","excerpt":"vpage8See every butterfly valueWALSH RECURSION IN EIGHT COORDINATES The unnormalized Walsh matrix satisfies H_1=[1] and H_2n=matH_n · H_n H_n · -H_n, H_nH_n^T=nI. Apply the three add/subtract stages to (1,2,3,4,5,6,7,8)^T. The full intermediate ledger is displayed below. vfigwalsh headArithmetic can be counted exactly Each stage visits eight coordi","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-902dedcbd6a5754e","tex":"H_8","aliases":"H_8","display":"H_8","kind":"use","excerpt":"The result (36,-4,-8,0,-16,0,0,0)^T still reconstructs the input: the inverse is H_8^T/8. Because the displayed transform is unnormalized, its squared output norm is eight times the squared input norm. The normalized transform H_8/sqrt8 would instead be orthogonal. headWhat the zeros mean The zero coefficients arise from exact cancellation among th","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7a7409aaabc384b9","tex":"H_8^T","aliases":"H_8^T","display":"H_8^T","kind":"use","excerpt":"The result (36,-4,-8,0,-16,0,0,0)^T still reconstructs the input: the inverse is H_8^T/8. Because the displayed transform is unnormalized, its squared output norm is eight times the squared input norm. The normalized transform H_8/sqrt8 would instead be orthogonal. headWhat the zeros mean The zero coefficients arise from exact cancellation among th","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"use","excerpt":"vpage8See every butterfly valueWALSH RECURSION IN EIGHT COORDINATES The unnormalized Walsh matrix satisfies H_1=[1] and H_2n=matH_n · H_n H_n · -H_n, H_nH_n^T=nI. Apply the three add/subtract stages to (1,2,3,4,5,6,7,8)^T. The full intermediate ledger is displayed below. vfigwalsh headArithmetic can be counted exactly Each stage visits eight coordi","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b557fbd91b225d1c","tex":"H_N^T","aliases":"H_N^T","display":"H_N^T","kind":"use","excerpt":"H_N · -H_N . Then H_NH_N^T=NI. The product H_Nx can be evaluated in exactly Nlog_2N additions or subtractions through the recursion.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e71e9ced609ec71d","tex":"H_n^T","aliases":"H_n^T","display":"H_n^T","kind":"use","excerpt":"vpage8See every butterfly valueWALSH RECURSION IN EIGHT COORDINATES The unnormalized Walsh matrix satisfies H_1=[1] and H_2n=matH_n · H_n H_n · -H_n, H_nH_n^T=nI. Apply the three add/subtract stages to (1,2,3,4,5,6,7,8)^T. The full intermediate ledger is displayed below. vfigwalsh headArithmetic can be counted exactly Each stage visits eight coordi","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ab39294284c03d46","tex":"\\mathrm i","aliases":"i mathrm","display":"\\mathrm i","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACT The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat","count":17},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b966b3d3d2ca3fe9","tex":"\\mathrm i^{-jk}","aliases":"i^-jk mathrm","display":"\\mathrm i^{-jk}","kind":"use","excerpt":"vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATION Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,-1,0)^T, dense multiplicat","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"-1 · 3. Hence D^-1u=u/4 and S=1+u^TD^-1u=frac52. The Woodbury identity reduces the entire inverse to A^-1=D^-1-D^-1uS^-1u^TD^-1 =D^-1-frac140uu^T. vfigwoodbury headTwo hypotheses, not one Invertibility of D alone does not suffice. For D=I_2, u=e_1, and v=-e_1, the update is diag(0,1). The small matrix S is zero, exactly where the correction formula","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"Capacity and roofline. Positive physical capacities and the inequalities F≤ PT, B≤ WT give T≥max{F/P,B/W}. For positive bytes and elapsed time, the rate form is F/T≤min{P,WF/B}. The dense memory bound is the exact equivalence n^2s≤ Miff n≤sqrtM/s for nonnegative n,M and positive s.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-88b326798e7011cf","tex":"\\ii","aliases":"ii i mathrm","display":"{\\mathrm i}","kind":"use","excerpt":"[Circulant spectral representation] Let n≥1 be an integer and c∈ C^n, with indices taken modulo n, and define C(c)_jk=c_j-k. Let F be the unitary discrete Fourier matrix F_jk = frac1sqrt n exp(-frac2π i jkn). Then C(c) = F^*diag(sqrt n Fc)F. The operator is determined by n scalars and can be applied through two Fourier transforms and one pointwise ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the decla","count":16},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-98e38db59701cf95","tex":"J_{\\rm keep}","aliases":"J_rm keep rm","display":"J_{\\rm keep}","kind":"use","excerpt":"bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the decla","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b1a429be9f79e459","tex":"J_{\\rm split}","aliases":"J_rm split rm","display":"J_{\\rm split}","kind":"use","excerpt":"bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the decla","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fc399b7a1160964d","tex":"J_\\lambda","aliases":"J_λ lambda","display":"J_\\lambda","kind":"use","excerpt":"[Exact Lagrangian packet compiler] Fix λ≥0 and define J_λ( L) = cost( L) + λerr( L). For every node v, set K_λ(v) = κ_v+λϵ_v and S_λ(v) = σ_v + ∑_w∈child(v)F_λ(w). For a terminal node, set F_λ(v)=K_λ(v). For an internal node, the recurrence F_λ(v) = min{K_λ(v),S_λ(v)} computes the exact minimum Lagrangian objective over every admissible pruning of ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"vpage17Invariant does not mean independently addressableA STRICT KRYLOV INCLUSION Take A=I_2, split into the two coordinate packets, and start with v=(1,1)^T. Every packet is invariant. Nevertheless every polynomial applies the same scalar to both coordinates: p(A)v=p(1)(1,1)^T, dim K_m(A,v)=1 (m≥1). vfigkrylov The direct sum of the two local Krylo","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fb9dab5747004561","tex":"K'","aliases":"K'","display":"K'","kind":"use","excerpt":"Strict half-gap stability. At a solved internal node write K=κ_v+λϵ_v, S=σ_v+∑_wF_λ(w), g=|K-S|. If the updated scores satisfy |K'-K|<g/2, |S'-S|<g/2, Lean proves K'≤ S' Longleftrightarrow K≤ S. Applying this condition recursively at every internal node proves equality of the entire returned cuts before and after perturbation, not only equality of ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4b779dcac03cb4e5","tex":"K^2","aliases":"K^2","display":"K^2","kind":"use","excerpt":"0 · 1 · -1 · 1. The two routes from coordinate zero to coordinate three cancel, so K^2=0. Consequently A^-1=I-K=mat1 · 0 · 0 · 0","count":8},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a7c895cee857ce71","tex":"K^h","aliases":"K^h","display":"K^h","kind":"use","excerpt":"dicts possible support but not actual support. Replacing the inclusion supp(AB)⊆supp(A)∘supp(B) by equality would be wrong even in this four-dimensional model. headWhat an acyclic graph does guarantee If the longest strict directed path has h edges, then no term of K^h+1 can survive. Thus K^h+1=0. This is an upper bound on the nilpotence index, not","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ba73dafd81ea4478","tex":"K^{h+1}","aliases":"K^h+1","display":"K^{h+1}","kind":"use","excerpt":", K^2=0. The matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion supp(AB)⊆supp(A)∘supp(B) by equality would be wrong even in this four-dimensional model. headWhat an acyclic graph does guarantee If the longest strict directed path has h edges,","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0af29e9b97d6dd25","tex":"K_{10}","aliases":"K_10","display":"K_{10}","kind":"use","excerpt":"vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSION Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. The matrix of path counts sees the two possibilities and","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d9f725106db7a34c","tex":"K_{20}","aliases":"K_20","display":"K_{20}","kind":"use","excerpt":"vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSION Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. The matrix of path counts sees the two possibilities and","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a4a6203c6556b0cb","tex":"K_{31}","aliases":"K_31","display":"K_{31}","kind":"use","excerpt":"vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSION Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. The matrix of path counts sees the two possibilities and","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-21a51e221f2f6f08","tex":"K_{32}","aliases":"K_32","display":"K_{32}","kind":"use","excerpt":"vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSION Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. The matrix of path counts sees the two possibilities and","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b1fd09588ac918c1","tex":"\\mathcal K_{j+1}","aliases":"K_j+1 mathcal","display":"\\mathcal K_{j+1}","kind":"use","excerpt":"Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4b35ad5d2218e1cf","tex":"\\mathcal K_m","aliases":"K_m mathcal","display":"\\mathcal K_m","kind":"use","excerpt":"vpage17Invariant does not mean independently addressableA STRICT KRYLOV INCLUSION Take A=I_2, split into the two coordinate packets, and start with v=(1,1)^T. Every packet is invariant. Nevertheless every polynomial applies the same scalar to both coordinates: p(A)v=p(1)(1,1)^T, dim K_m(A,v)=1 (m≥1). vfigkrylov The direct sum of the two local Krylo","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-44cc17134afaa418","tex":"K_{\\beta\\alpha}","aliases":"K_βα beta alpha","display":"K_{\\beta\\alpha}","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ccedf2f845a654d7","tex":"\\Kcal_{j+1}","aliases":"Kcal_j+1 K_j+1 Kcal mathcal","display":"{\\mathcal K}_{j+1}","kind":"use","excerpt":"[Approximate packet Krylov bound] Write A=D+K, where D is packet diagonal. For every positive integer j, normA^j-D^j ≤ jM^j-1normK, where M=max{normA,normD}. Consequently, dist(A^jv,⊕_α K_j+1(D_α,v_α)) ≤ jM^j-1normKnormv.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"Frobenius optimality from kernel weights. textitModules KernelFrame and FrobeniusOptimality For an arbitrary competitor B with rank B≤ k, construct an orthonormal matrix Q spanning its actual kernel. Then BQ=0, Q^*Q=I, q=dimker B=d-rank B. Define the row weights w_i=∑_j=1^q|Q_ij|^2. The proof derives 0≤ w_i≤1, ∑_iw_i=q≥max(d-k,0). The upper bound u","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACT The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat","count":64},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the decla","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"Frontiers and decision stability. textitSource items 9.6, 9.7, and 9.11 Let L_1,L_2 be actual optima at 0≤λ_1<λ_2. Their optimality inequalities are cost(L_1)+λ_1err(L_1) · ≤cost(L_2)+λ_1err(L_2),","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"Frontiers and decision stability. textitSource items 9.6, 9.7, and 9.11 Let L_1,L_2 be actual optima at 0≤λ_1<λ_2. Their optimality inequalities are cost(L_1)+λ_1err(L_1) · ≤cost(L_2)+λ_1err(L_2),","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0945233af0503d7f","tex":"\\mathcal L_{j+1}","aliases":"L_j+1 mathcal","display":"\\mathcal L_{j+1}","kind":"use","excerpt":"Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"use","excerpt":"R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions; tail singular values beyond d are zero. ","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-66053b0fba2aa69a","tex":"L_{m-1}","aliases":"L_m-1","display":"L_{m-1}","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3755b02b61222ecc","tex":"\\Lcal","aliases":"Lcal L mathcal","display":"{\\mathcal L}","kind":"use","excerpt":"The cost and error of a pruning L are cost( L) = ∑_v∈ Lκ_v + ∑_u splitσ_u and err( L)=∑_v∈ Lϵ_v. The model can include arithmetic, storage, launch, transfer, and precision costs inside κ_v. It remains exact only when the declared totals are additive.","count":11},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-271ba6ea9d5e2027","tex":"\\Lcal_1","aliases":"Lcal_1 L_1 Lcal mathcal","display":"{\\mathcal L}_1","kind":"use","excerpt":"[Monotone Lagrangian frontier] Let 0≤λ_1<λ_2. Choose optimal prunings L_1 and L_2 for the two Lagrangian parameters. Then err( L_2) ≤ err( L_1) and cost( L_2) ≥ cost( L_1).","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cc64d46d6f1e674d","tex":"\\Lcal_2","aliases":"Lcal_2 L_2 Lcal mathcal","display":"{\\mathcal L}_2","kind":"use","excerpt":"[Monotone Lagrangian frontier] Let 0≤λ_1<λ_2. Choose optimal prunings L_1 and L_2 for the two Lagrangian parameters. Then err( L_2) ≤ err( L_1) and cost( L_2) ≥ cost( L_1).","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"vpage32Safe integers can still be conservativeQUANTIZATION AND THE SIZE OF THE BUDGET TABLE Let one integer unit represent h=1/10 of certified error. For an individual nonnegative error e, choosing w=lceil e/hrceil ensures e≤ hw. Thus integer feasibility ∑ w≤ B implies the real bound ∑ e≤ hB. This implication is checked before applying the exact op","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5f10b967e4b2f95d","tex":"\\leadsto_{H}","aliases":"leadsto_H leadsto","display":null,"kind":"use","excerpt":"Hidden elimination. For the actual Schur complement S=A_RR-A_RHA_HH^-1A_HR, the hidden inverse is expanded through the hidden projectors. A nonzero retained correction block requires indices j,k with nonzero entrance and exit blocks and a hidden path jleadsto k. Therefore a nonzero block S_ba requires either a direct retained block, or a⟶ j leadsto","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"ONE LINEAR MODEL Suppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold: T_b(R)=7R, T_c(R)=30+2R. vfigamortization At R=6, both cost 42. The candidate becomes strictly faster at R=7, where it costs 44 instead of 49. The minimum profitab","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"vpage20Spend error where it is cheapestCONTINUOUS AND CLIPPED PRECISION Take sensitivities s=(1,2,4), cost coefficients c=(1,1,2), and budget τ=1. Minimizing ∑_ec_elog(1/ϵ_e) subject to ∑_es_eϵ_e=1 gives ϵ_e=fracc_e4s_e, ϵ=(1/4,1/8,1/8). vfigprecision headThe budget is weighted The three contributions to the error allowance are (1/4,1/4,1/2). Equal","count":18},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"H_N · -H_N . Then H_NH_N^T=NI. The product H_Nx can be evaluated in exactly Nlog_2N additions or subtractions through the recursion.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Strict half-gap stability. At a solved internal node write K=κ_v+λϵ_v, S=σ_v+∑_wF_λ(w), g=|K-S|. If the updated scores satisfy |K'-K|<g/2, |S'-S|<g/2, Lean proves K'≤ S' Longleftrightarrow K≤ S. Applying this condition recursively at every internal node proves equality of the entire returned cuts before and after perturbation, not only equality of ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"Nilpotence certificate. If no walk of length h+1 exists between any pair of packets, then A^h+1=0. Every block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map ℓ:I⟶{0,…,h}, A_bc≠0 Longrightarrow ℓ(c)<ℓ(b). It proves that a length-n walk increases the l","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e0639ea84119f55a","tex":"\\operatorname{lookup}","aliases":"lookup operatorname","display":"\\operatorname{lookup}","kind":"use","excerpt":"Entrywise implementation theorem. For every b≤ B, Lean proves lookup(storedTree(B,t),b)=F_t(b). There is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at b depends only on input entries at most b, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attain","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fc33b4e3162aeb75","tex":"M^*","aliases":"M^*","display":"M^*","kind":"use","excerpt":"Lean identifies the actual Frobenius norm with its trace expression, normM_rm F^2=Retr(M^*M), then expands the finite sum and removes each cross term. The result is normA-Π(A)_rm F^2 =∑_b≠ anormP_bAP_a_rm F^2. No Pythagorean equality for the target residual is assumed.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ca52a161c084be9c","tex":"M^{-1/2}","aliases":"M^-1/2","display":"M^{-1/2}","kind":"use","excerpt":"[Preconditioned conjugate gradient convergence] Let A and M be Hermitian positive definite on a nonempty finite coordinate space, and set κ = cond_2(M^-1/2AM^-1/2). In exact arithmetic, the energy norm error of preconditioned conjugate gradient satisfies normx_m-x_*_A ≤ 2 ( fracsqrtκ-1sqrtκ+1 )^m normx_0-x_*_A.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"9+12 i · 6. Replacing A^T with A^* changes the signs of the imaginary terms. A real-only test would conceal that error, which is why the worked model uses an imaginary coefficient. headWhy modewise contraction matters For A of order m and B of order n, the explicit tensor matrix stores m^2n^2 entries. Its factors store m^2+n^2. Applying through BXA","count":11},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0a98f829b8404676","tex":"M^{j-1}","aliases":"M^j-1","display":"M^{j-1}","kind":"use","excerpt":"With A=diag(1,2), the vectors v and Av=(1,2)^T are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when A=I_2. Both changes enlarge the addressing power, but in different ways. headWhen invariance is only approximate If A=D","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2eb18a26b2462763","tex":"m_e","aliases":"m_e","display":"m_e","kind":"use","excerpt":"Now impose upper tolerances m=(1/10,1/2,1/2). The first edge must stop at its tighter cap. With multiplier λ=10/3, the clipped solution is ϵ_e=min{m_e,c_e/(λ s_e)}, ϵ=(1/10,3/20,3/20). Its weighted contributions are (1/10,3/10,3/5) and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly. headFrom ","count":24},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-83d3f2e0b215ee50","tex":"M_{ij}","aliases":"M_ij","display":"M_{ij}","kind":"use","excerpt":"W^*W=I, Z^*Z=I, σ_1≥⋯≥σ_d≥0. These are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation G^*GZ=Zdiag(σ_1^2,…,σ_d^2). Let h=min(d,k) and retain exactly the first h column indices. The rectangular square-root mask has entries M_ij=sqrtσ_j, · i=j","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-12fcf6e1ba955880","tex":"M_{\\rm block}","aliases":"M_rm block rm","display":"M_{\\rm block}","kind":"use","excerpt":"[Equal diagonal packets] Let p≥1 and b≥1 be integers and n=pb. Suppose n is divided into p equal diagonal packets of width b. Then flop_rm block=2qnb, M_rm block=nb. The dense-to-block storage reduction factor is the following; the arithmetic reduction factor has the same value when q≥1: fracnb.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-27220d8a886e4747","tex":"M_{\\rm packet}","aliases":"M_rm packet rm","display":"M_{\\rm packet}","kind":"use","excerpt":"[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-956f614ae0645ab5","tex":"\\mat","aliases":"mat","display":null,"kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACT The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat","count":38},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b29f89d48389ba9c","tex":"\\max_{v\\in V}","aliases":"max_v∈ V max in","display":"\\max_{v\\in V}","kind":"use","excerpt":"A valid schedule s:V→ N assigns positive stages and strictly increases on each edge. Induction proves the stronger pointwise statement ℓ(v)≤ s(v) for every v. Consequently the constructed schedule achieves the least possible makespan, d=max_v∈ Vℓ(v). For a nonempty graph, d equals the number of vertices on an attained longest path. Vertices of the ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"Now impose upper tolerances m=(1/10,1/2,1/2). The first edge must stop at its tighter cap. With multiplier λ=10/3, the clipped solution is ϵ_e=min{m_e,c_e/(λ s_e)}, ϵ=(1/10,3/20,3/20). Its weighted contributions are (1/10,3/10,3/5) and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly. headFrom ","count":26},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-83370d7c90a0d024","tex":"\\min_{(w,c)\\text{ allowed},\\;w\\leq b}","aliases":"min_(w,c) allowed, w≤ b min text leq","display":"\\min_{(w,c)\\text{ allowed},\\;w\\leq b}","kind":"use","excerpt":"If D is the already-solved tail table, the next edge has exact-weight recurrence F(b)=min_(w,c) allowed, w≤ b{D(b-w)+c}. The final answer is min_0≤ b≤ BF(b). Lean proves that this is the exact minimum over legal assignments with total error at most B, or codenone exactly when no such assignment exists.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-08929fcbd469af5d","tex":"\\min_{0\\leq b\\leq B}","aliases":"min_0≤ b≤ B min leq leq","display":"\\min_{0\\leq b\\leq B}","kind":"use","excerpt":"If D is the already-solved tail table, the next edge has exact-weight recurrence F(b)=min_(w,c) allowed, w≤ b{D(b-w)+c}. The final answer is min_0≤ b≤ BF(b). Lean proves that this is the exact minimum over legal assignments with total error at most B, or codenone exactly when no such assignment exists.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c9ccf46f5921c4a5","tex":"\\min_{0\\leq k\\leq b}","aliases":"min_0≤ k≤ b min leq leq","display":"\\min_{0\\leq k\\leq b}","kind":"use","excerpt":"For exact-weight tables F,G, min-plus convolution is (F⋆ G)(b)=min_0≤ k≤ b{F(k)+G(b-k)}, with an infeasible sum whenever either operand is infeasible. Its correctness theorem identifies every finite answer with an actual feasible pair of component weights and costs. Repeated convolution handles arbitrary finite lists of components.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1ace88525e5ee17f","tex":"\\min_e","aliases":"min_e min","display":"\\min_e","kind":"use","excerpt":"upper bound. The weighted sum ϕ(λ) = ∑_es_e min{ m_e, fracc_eλ s_e } is continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, ϕ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is (m_e)_e and every 0","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b765b9bd7781a783","tex":"\\min_{\\eps_e>0}","aliases":"min_eps_e>0 min_ε_e>0 min eps min varepsilon","display":"\\min_{{\\varepsilon}_e>0}","kind":"use","excerpt":"[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-567e9df79d4789c7","tex":"\\min_k","aliases":"min_k min","display":"\\min_k","kind":"use","excerpt":"ct discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ab2c688fc16fea97","tex":"\\min_{\\varepsilon_e>0}","aliases":"min_ε_e>0 min varepsilon","display":"\\min_{\\varepsilon_e>0}","kind":"use","excerpt":"The continuous allocation theorem is complete. textitSource item 8.4, with an explicit positive budget Let E be a finite nonempty edge set. Assume s_e>0, c_e>0, τ>0, and define C=∑_ec_e. The optimization problem is min_ε_e>0∑_ec_elogfrac1ε_e subject to ∑_es_eε_e=τ. Lean constructs the displayed candidate ε_e^*=fracτ c_es_eC and proves strict positi","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cd9b1bf9b23006d6","tex":"\\min_{\\sum_i b_i=b}","aliases":"min_∑_i b_i=b min sum","display":"\\min_{\\sum_i b_i=b}","kind":"use","excerpt":"Keeping node v costs κ_v and incurs integer error w_v. Splitting costs overhead σ_v in addition to the selected child cuts. The exact-weight recurrence is F_v(b)=min{ l κ_v if b=w_v, [2pt] displaystyleσ_v+ min_∑_i b_i=b∑_iF_v_i(b_i) if v is internal. . Nonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-pl","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"A valid schedule s:V→ N assigns positive stages and strictly increases on each edge. Induction proves the stronger pointwise statement ℓ(v)≤ s(v) for every v. Consequently the constructed schedule achieves the least possible makespan, d=max_v∈ Vℓ(v). For a nonempty graph, d equals the number of vertices on an attained longest path. Vertices of the ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a80e516bfb196e1c","tex":"\\n","aliases":"n","display":null,"kind":"use","excerpt":"[font=sffamilysmall] node[anchor=west] at (0,3.4) A^*A=diag(25,4); draw[->] (0,0)--(8,0) node[right] retained singular directions; draw[->] (0,0)--(0,2.5) node[above] squared error; draw[thick,black!65] (1,2.03)--(3.5,.28)--(6,0); foreach x/y/n/v in 1/2.03/0/29,3.5/.28/1/4,6/0/2/0 fill (x,y) circle (2pt);node[below] at (x,0) n;node[above] at (x,y) ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-79616ac402fc2170","tex":"n^{-1/2}","aliases":"n^-1/2","display":"n^{-1/2}","kind":"use","excerpt":"Normalization and inversion. The proof identifies matrix application with the ordinary counting-measure discrete Fourier transform: Fx=n^-1/2DFT(x). It separately identifies F^*x with the appropriately scaled inverse transform. Mathlib's checked Fourier inversion theorem and the proved scalar normalization yield both identities F^*F=I, FF^*=I. No u","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"vpage4Work removed is not time measuredARITHMETIC, STORAGE, AND CAPACITY For q right-hand sides, the paper models dense application by 2n^2q operations. A block-diagonal-plus-rank-r representation with equal block width b uses F=2qn(b+2r), M=n(b+2r) stored scalars. These formulas count the dense block products and two thin products. They follow the","count":24},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b9588331116f694e","tex":"n_e","aliases":"n_e","display":"n_e","kind":"use","excerpt":"Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their card","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b6801e9881994d7d","tex":"n_i^2","aliases":"n_i^2","display":"n_i^2","kind":"use","excerpt":"BDLR · ∑_i n_i^2+2nr · 2q(∑_i n_i^2+2nr)","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-af2a18fd20397da4","tex":"N_{p,w}","aliases":"N_p,w","display":"N_{p,w}","kind":"use","excerpt":"[Equal block banded count] Let p,b≥1 be integers, n=pb, and let p=n/b equal packets have width b. Suppose a block is active precisely when absi-j≤ w, 0≤ w≤ p-1. The number of active blocks is N_p,w = p(2w+1)-w(w+1). For q right hand sides, the modeled application cost and scalar storage are 2qb^2N_p,w and b^2N_p,w.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3b600c930ff971ba","tex":"n_\\alpha^2","aliases":"n_α^2 alpha","display":"n_\\alpha^2","kind":"use","excerpt":"[BDLR application and storage] For q right hand sides, AZ=DZ+U(V^*Z) has modeled cost 2q ( ∑_α n_α^2+2nr ). The representation stores ∑_α n_α^2+2nr scalars. For n=pb, with integers p,b≥1, the ideal dense storage reduction factor is the following; the arithmetic reduction factor has the same value when q≥1: fracnb+2r.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8205d7811a8a9f9c","tex":"n_\\beta","aliases":"n_β beta","display":"n_\\beta","kind":"use","excerpt":"[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-90f600277c50100a","tex":"\\NN","aliases":"NN N mathbb","display":"{\\mathbb N}","kind":"use","excerpt":"[Quantized budget dynamic program] Assume every keep error is a nonnegative integer w_v. Let B∈ N. For each node v, define F_v(b) as the minimum cost of a pruning of the subtree at v with total error exactly b. The keep option contributes F_v(w_v)≤κ_v. The split option is obtained by min plus convolution of the child tables, followed by addition of","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"[Inverse norm bound] Under the hypotheses of [source reference], norm(D+UV^*)^-1 ≤ normD^-1 + normD^-1U normS^-1 normV^*D^-1.","count":48},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","excerpt":"Fix the source packet a. In the regular representation, define the submodule W_a={x:P_bx=0 whenever anotleadsto b}. It contains P_a. Completeness of the projectors expands each component of Ax into packet transfers. If a transfer into an unreachable packet were nonzero from a reachable packet, it would extend a path, a contradiction. Thus AW_a⊆ W_a","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[Materialization cost] Materializing D+UV^* as an explicit matrix requires storage for n^2 scalars and modeled arithmetic 2n^2r+O(n^2) when the low rank correction is formed by dense multiplication.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"Arithmetic events. For q right-hand sides, the declared event type adds a right-hand-side index and one of two operation roles, multiplication or accumulation: O={0,1}×{0,…,q-1}× S. Hence | O|=2qb^2N_p,w. These identities include zero right-hand-side count. The abstract storage formula also admits zero block width. To use the manuscript's substitut","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"[Equal diagonal packets] Let p≥1 and b≥1 be integers and n=pb. Suppose n is divided into p equal diagonal packets of width b. Then flop_rm block=2qnb, M_rm block=nb. The dense-to-block storage reduction factor is the following; the arithmetic reduction factor has the same value when q≥1: fracnb.","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c1a14ba7424a4a8c","tex":"\\ones","aliases":"ones","display":null,"kind":"use","excerpt":"1 · 3 and put u=v=ones∈QQ^6. Then A=D+uv^T has diagonal entries four, within-block off-diagonal entries two, and every cross-block entry one. All 36 entries are nonzero. x=(1,-1,2,0,-2,3)^T, v^Tx=3, Ax=Dx+3u. vfigfactored_application headWhat was avoided? The global part does not require 36 independent coefficients. It depends on the input through ","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3d48a1e60ecf7c1c","tex":"\\ones^T","aliases":"ones^T ones","display":null,"kind":"use","excerpt":"vpage19Twelve small blocks, one coherent errorDISCOVERY AND PERTURBATION CERTIFICATES Split QQ^4 into scalar packets and let A=I+E, E=frac1100(onesones^T-I). Dropping every off-diagonal block leaves I. Each discarded entry is only 1/100, but the total error is not bounded by that single-entry size. vfigblock_error headTwo norms answer different que","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the decla","count":56},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"s N_p,w=| B_p,w|=p(2w+1)-w(w+1). The proof uses finite prefix counting, the newly exposed boundary row and column when p increases, and the saturated initial band. An intermediate subtraction-free identity avoids relying on natural-number subtraction without the required inequality. The diagonal case w=0 gives exactly p blocks, and the saturated ca","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"Exact packet semantics. textitSource items 2.3, 2.5, and 2.6 Let R be a ring, I a finite index type, and (P_a)_a∈ I a family satisfying P_bP_a=P_a · b=a,","count":19},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-07a802f35729938f","tex":"P_{a(e)}","aliases":"P_a(e)","display":"P_{a(e)}","kind":"use","excerpt":"Packet perturbation certificates. textitSource item 8.1 and two uniform consequences Let X be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps X→ X, with their operator norm. Let a(e) identify the source packet of edge e, and suppose Δ_e=Δ_e P_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exac","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","excerpt":"Exact packet semantics. textitSource items 2.3, 2.5, and 2.6 Let R be a ring, I a finite index type, and (P_a)_a∈ I a family satisfying P_bP_a=P_a · b=a, 0 · b≠ a, ∑_aP_a=1. Write A_ba=P_bAP_a and let E_A={(b,a):A_ba≠0}. These are exact algebraic zeros, not values discarded by a floating-point threshold.","count":11},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3270e71bf1ae54c5","tex":"P_c","aliases":"P_c","display":"P_c","kind":"use","excerpt":"Hilbert--Schmidt orthogonality. For distinct ordered packet pairs (b,a)≠(d,c), adjoint multiplication and projector orthogonality give tr((P_bAP_a)^*P_dBP_c)=0. If b≠ d, the central projector product vanishes. Otherwise a≠ c, and cyclicity of trace brings P_cP_a together. This is proved for arbitrary input matrices A,B.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-14f8900152d6716a","tex":"P_d","aliases":"P_d","display":"P_d","kind":"use","excerpt":"Hilbert--Schmidt orthogonality. For distinct ordered packet pairs (b,a)≠(d,c), adjoint multiplication and projector orthogonality give tr((P_bAP_a)^*P_dBP_c)=0. If b≠ d, the central projector product vanishes. Otherwise a≠ c, and cyclicity of trace brings P_cP_a together. This is proved for arbitrary input matrices A,B.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"Exact and approximate packet Krylov spaces. textitSource items 7.2 and 7.3 Write K_m(A,v)=span{v,Av,…,A^m-1v}. For a complete finite projector family P_i, suppose A commutes with every P_i. Let A_i=P_iAP_i. Lean proves the actual orbit identity A^jv=∑_i A_i^jP_iv and therefore K_m(A,v)⊆ ⊕_i K_m(A_i,P_iv). The case j=0 uses ∑_iP_i=I. The zeroth loca","count":9},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7a5d03f7fbf7ca50","tex":"P_{j+1}","aliases":"P_j+1","display":"P_{j+1}","kind":"use","excerpt":"Actual noncommutative products. The error module defines the left-accumulated product recursively, with P_0=I and P_j+1=A_j+1P_j. For a perturbed sequence widehat A_j, it derives the discrepancy recursion widehat P_j+1-P_j+1 =widehat A_j+1(widehat P_j-P_j) +(widehat A_j+1-A_j+1)P_j. The identity follows from the actual products. It is not supplied ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5e6b6a6054c3c5d9","tex":"P_{j,a(e)}","aliases":"P_j,a(e)","display":"P_{j,a(e)}","kind":"use","excerpt":"Packetwise mixed precision. At each stage let A_j=∑_e A_j,e, widehat A_j=∑_e(A_j,e+Δ_j,e), Δ_j,e=Δ_j,eP_j,a(e). The previous packet certificate supplies the uniform operator bound normwidehat A_j-A_j≤ δ_j:=∑_enormΔ_j,enormP_j,a(e). The new theorem substitutes these δ_j into the actual product-chain estimate. The resulting vector error is at most th","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2646be694d017280","tex":"P_{j-1}","aliases":"P_j-1","display":"P_{j-1}","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"ntheorem[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i )","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5266a4839b2b51a8","tex":"P_{m-1}","aliases":"P_m-1","display":"P_{m-1}","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5b1d3ec03a20b6b9","tex":"P_{\\alpha}","aliases":"P_α alpha","display":"P_{\\alpha}","kind":"use","excerpt":"Take the supremum over unit inputs in [source reference], using P_αx ≤ P_α x . Substitute the resulting uniform operator error bounds into [source reference]; the action estimate follows from submultiplicativity.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-056b9963f80df264","tex":"P_{\\alpha(e)}","aliases":"P_α(e) alpha","display":"P_{\\alpha(e)}","kind":"use","excerpt":"For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tig","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e2bbf7f75432b536","tex":"P_{\\alpha(j,e)}","aliases":"P_α(j,e) alpha","display":"P_{\\alpha(j,e)}","kind":"use","excerpt":"[Mixed precision product chain] If each fixed linear factor A_j is replaced by a fixed packetwise linear approximation widehat A_j, write E_j=widehat A_j-A_j=∑_eΔ_j,eP_α(j,e). A uniform certificate is E_j ≤δ_j:=∑_e Δ_j,e P_α(j,e) . For orthogonal packet projectors, δ_j≤∑_e Δ_j,e . The final operator error is bounded by [source reference] with these","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9813ba63ac7cffd1","tex":"P_\\gamma","aliases":"P_γ gamma","display":"P_\\gamma","kind":"use","excerpt":"Insert the packet identity between A and B: P_γ ABP_α = ∑_β P_γ AP_β BP_α. A summand can be nonzero only when both block addresses are active.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACT The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0","count":49},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q mathbb","display":"{\\mathbb Q}","kind":"use","excerpt":"Shared operator families. For reuse weights r_j, the local keep model can aggregate setup and per-operator costs: κ_v=κ_v^setup+∑_jr_jκ_v^(j), ϵ_v=∑_jr_jϵ_v^(j). Lean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including Q and R. The inte","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"vpage4Work removed is not time measuredARITHMETIC, STORAGE, AND CAPACITY For q right-hand sides, the paper models dense application by 2n^2q operations. A block-diagonal-plus-rank-r representation with equal block width b uses F=2qn(b+2r), M=n(b+2r) stored scalars. These formulas count the dense block products and two thin products. They follow the","count":61},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d76d885e9e3c5ba5","tex":"Q^*","aliases":"Q^*","display":"Q^*","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACT The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a4591bb144ad9c3f","tex":"Q_c","aliases":"Q_c","display":"Q_c","kind":"use","excerpt":"Adjoint and tensor products. With an involution and self-adjoint projectors, the formalization proves (A_ba)^*=(A^*)_ab. For actual finite matrices and actual Kronecker products, it proves (P_b⊗ Q_d)(A⊗ B)(P_a⊗ Q_c) =A_ba⊗ B_dc. The self-adjointness hypothesis is explicit in the adjoint theorem. The tensor theorem is not a statement about an uninte","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-83dc7e104ed7732b","tex":"Q_d","aliases":"Q_d","display":"Q_d","kind":"use","excerpt":"Adjoint and tensor products. With an involution and self-adjoint projectors, the formalization proves (A_ba)^*=(A^*)_ab. For actual finite matrices and actual Kronecker products, it proves (P_b⊗ Q_d)(A⊗ B)(P_a⊗ Q_c) =A_ba⊗ B_dc. The self-adjointness hypothesis is explicit in the adjoint theorem. The tensor theorem is not a statement about an uninte","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-dfe49e85066a6b84","tex":"Q_{ij}","aliases":"Q_ij","display":"Q_{ij}","kind":"use","excerpt":"Frobenius optimality from kernel weights. textitModules KernelFrame and FrobeniusOptimality For an arbitrary competitor B with rank B≤ k, construct an orthonormal matrix Q spanning its actual kernel. Then BQ=0, Q^*Q=I, q=dimker B=d-rank B. Define the row weights w_i=∑_j=1^q|Q_ij|^2. The proof derives 0≤ w_i≤1, ∑_iw_i=q≥max(d-k,0). The upper bound u","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9bd193e7de59f86f","tex":"Q_L","aliases":"Q_L","display":"Q_L","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACT The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat","count":29},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9bcb84d1006a5379","tex":"Q_L^*","aliases":"Q_L^*","display":"Q_L^*","kind":"use","excerpt":"0 · 0 · 0. Keeping only its first singular direction gives a rank-one approximation with spectral and Frobenius error exactly two. Keeping none gives errors five and sqrt29; keeping both gives zero. vfigoptimality headWhy an outside competitor cannot do better For isometric outer factors, compression C(B)=Q_L^*BQ_R cannot increase rank or either ma","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5c06fc40d19e1dce","tex":"Q_L^T","aliases":"Q_L^T","display":"Q_L^T","kind":"use","excerpt":"0 · -2 i, A^*A=diag(25,4). vfigcomplex_core headThe small calculation exposes a wrong convention The first column of Q_L has Hermitian squared length 9/25+16/25=1. Its bilinear square without conjugation is 9/25-16/25=-7/25. Thus Q_L^TQ_L is not the identity. The portable negative control replaces the adjoint by transpose and rejects the resulting ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-808125305b3a9f1c","tex":"Q_R","aliases":"Q_R","display":"Q_R","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACT The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat","count":12},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b2e800860e3d6ef1","tex":"Q_R^*","aliases":"Q_R^*","display":"Q_R^*","kind":"use","excerpt":"0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat3 · 0 4 i · 0 0 · -2 i, A^*A=diag(25,4). vfigcomplex_core headThe small calculation exposes a wrong convention The first column of Q_L has Hermitian squared length 9/25+16/25=1. Its bilinear square without conjugation is 9/25-16/25=-7/25. Thus Q_L^TQ_L is ","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5a423e1823849ed7","tex":"Q_\\gamma","aliases":"Q_γ gamma","display":"Q_\\gamma","kind":"use","excerpt":"Both identities follow by inserting the corresponding coordinate projectors and using (P_β AP_α)^*=P_α A^*P_β. The tensor identity follows from (P_β⊗ Q_δ) (A⊗ B) (P_α⊗ Q_γ) = A_βα⊗ B_δγ.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-aa7fbe64d6b9e7ee","tex":"Q_\\delta","aliases":"Q_δ delta","display":"Q_\\delta","kind":"use","excerpt":"Both identities follow by inserting the corresponding coordinate projectors and using (P_β AP_α)^*=P_α A^*P_β. The tensor identity follows from (P_β⊗ Q_δ) (A⊗ B) (P_α⊗ Q_γ) = A_βα⊗ B_δγ.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ","display":null,"kind":"use","excerpt":"Keeping the first diagonal entry of G gives A_1 with its second column zero. The residual has squared Frobenius norm four and Gram matrix diag(0,4), hence spectral norm two. Both errors agree with the discarded core singular value. Every displayed arithmetic operation occurs in QQ( i). headWhat the finite check establishes It checks both isometries","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b9a6d997b7294e64","tex":"\\QQ^2","aliases":"QQ^2 QQ","display":null,"kind":"use","excerpt":"t does not mean independently addressableA STRICT KRYLOV INCLUSION Take A=I_2, split into the two coordinate packets, and start with v=(1,1)^T. Every packet is invariant. Nevertheless every polynomial applies the same scalar to both coordinates: p(A)v=p(1)(1,1)^T, dim K_m(A,v)=1 (m≥1). vfigkrylov The direct sum of the two local Krylov spans is all ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4dafa9764bf19601","tex":"\\QQ^4","aliases":"QQ^4 QQ","display":null,"kind":"use","excerpt":"vpage19Twelve small blocks, one coherent errorDISCOVERY AND PERTURBATION CERTIFICATES Split QQ^4 into scalar packets and let A=I+E, E=frac1100(onesones^T-I). Dropping every off-diagonal block leaves I. Each discarded entry is only 1/100, but the total error is not bounded by that single-entry size. vfigblock_error headTwo norms answer different que","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0989b37fff5d66a9","tex":"\\QQ^6","aliases":"QQ^6 QQ","display":null,"kind":"use","excerpt":"1 · 3 and put u=v=ones∈QQ^6. Then A=D+uv^T has diagonal entries four, within-block off-diagonal entries two, and every cross-block entry one. All 36 entries are nonzero. x=(1,-1,2,0,-2,3)^T, v^Tx=3, Ax=Dx+3u. vfigfactored_application headWhat was avoided? The global part does not require 36 independent coefficients. It depends on the input through ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACT The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat","count":96},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"erator families. For reuse weights r_j, the local keep model can aggregate setup and per-operator costs: κ_v=κ_v^setup+∑_jr_jκ_v^(j), ϵ_v=∑_jr_jϵ_v^(j). Lean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including Q and R. The intended nonn","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-055d5ee422435b80","tex":"R^*","aliases":"R^*","display":"R^*","kind":"use","excerpt":"* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions; tail singular values beyond d are zero. Define L_k=Q_LW_kΣ_k^1/2, R_k=Q_RZ_kΣ_k^1/2. Then L_kR_k^* is a best rank at most k approximation to LR^*.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-42f97ca6df89f2f9","tex":"R^T","aliases":"R^T","display":"R^T","kind":"use","excerpt":"1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. vfigcompact_core The actual core is G=R_LR_R^T=mat5 · 0 · 0 0 · 2 · 0, LR^T=Q_LG. Its dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"r_j-i, · i<j. The diagonal is supplied by c_0; the coordinate r_0 is unused. The constructed generator of length 2n is g=(c_0,…,c_n-1,0,r_n-1,…,r_1), and the padding map is the actual finite-index sum eliminator that appends n zeros.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6e93838ea3bf8d2c","tex":"r_A","aliases":"r_A","display":"r_A","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c6016f5cf675a3ae","tex":"r_B","aliases":"r_B","display":"r_B","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b3975fa3c319f1d3","tex":"r_e","aliases":"r_e","display":"r_e","kind":"use","excerpt":"Global comparison. For any other feasible ε, set r_e=ε_e/ε_e^*>0. The budget identities give ∑_ec_er_e=C. Using the standard analytic inequality log r≤ r-1 yields ∑_ec_elog r_e≤∑_ec_e(r_e-1)=0. Lean separately proves the exact objective-difference identity J(ε)-J(ε^*)=-∑_ec_elog r_e. Therefore J(ε)≥ J(ε^*).","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f5a0eed759a6f31c","tex":"r_{j-i}","aliases":"r_j-i","display":"r_{j-i}","kind":"use","excerpt":"Toeplitz padding and column-major vectorization. textitSource items 4.4 and 4.5 Toeplitz embedding. For n>0, define T_ij= c_i-j, · j≤ i, r_j-i, · i<j. The diagonal is supplied by c_0; the coordinate r_0 is unused. The constructed generator of length 2n is g=(c_0,…,c_n-1,0,r_n-1,…,r_1), and the padding map is the actual finite-index sum eliminator t","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a64e897db39c6a7f","tex":"R_k","aliases":"R_k","display":"R_k","kind":"use","excerpt":"R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions; tail singular values beyond d are zero. ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-61d7b9e7617499dd","tex":"r_{k-j}","aliases":"r_k-j","display":"r_{k-j}","kind":"use","excerpt":"2,-1)^T. The diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column g=(2,3,4,0,5,-1)^T and pad the input with three zeros. vfigtoeplitz headRead the wrapped indices For output index j<3, an input index k≤ j uses the first-column coefficient c_j-k. An input index k>j wraps to the sto","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-eae30801c8c7f23d","tex":"R_k^*","aliases":"R_k^*","display":"R_k^*","kind":"use","excerpt":"Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions; tail singular values beyond d are zero. Define L_k=Q_LW_kΣ_k^1/2, R_k=Q_RZ_kΣ_k^1/2. Then L_kR_k^* i","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-61bdbf136c2dac84","tex":"R_L","aliases":"R_L","display":"R_L","kind":"use","excerpt":"vpage11Stored width is not intrinsic rankACTUAL RANGE COORDINATES Choose the isometry Q_L=mat3/5 · 0 4/5 · 0 0 · 1 0 · 0, R_L=mat1 · 0 · 1 0 · 1 · 0, R_R=mat3 · 0 · 2 0 · 2 · 0 1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. vfigcompact_core The actual core is G=R_","count":12},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d96a67e19d75bf1d","tex":"r_L","aliases":"r_L","display":"r_L","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-25ee15775b880083","tex":"R_m","aliases":"R_m","display":"R_m","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d151cfb191c2c72b","tex":"R_{m-1}","aliases":"R_m-1","display":"R_{m-1}","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m], R_m · =[R_m-1 (D^(m-1))^*V_m]. Substitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-147aec0806fe40f4","tex":"R_{m-1}^*","aliases":"R_m-1^*","display":"R_{m-1}^*","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f02b5ce794f70074","tex":"R_m^*","aliases":"R_m^*","display":"R_m^*","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0f7c3db2a7afd485","tex":"R_{\\min}","aliases":"R_min min","display":"R_{\\min}","kind":"use","excerpt":"[Amortization threshold] Let s≥0 be a one time setup cost. Let t_b,t_c≥0 be the baseline and candidate times per use and let R∈ N be a reuse count. If t_b>t_c, the candidate is faster after R uses exactly when R> fracst_b-t_c. The smallest integer reuse count is R_min = lfloor fracst_b-t_c rfloor+1. If t_b≤ t_c, no positive reuse count amortizes a ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9d4ab28ea1385868","tex":"r_{n-1}","aliases":"r_n-1","display":"r_{n-1}","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ed27d154616b4c67","tex":"R_R","aliases":"R_R","display":"R_R","kind":"use","excerpt":"vpage11Stored width is not intrinsic rankACTUAL RANGE COORDINATES Choose the isometry Q_L=mat3/5 · 0 4/5 · 0 0 · 1 0 · 0, R_L=mat1 · 0 · 1 0 · 1 · 0, R_R=mat3 · 0 · 2 0 · 2 · 0 1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. vfigcompact_core The actual core is G=R_","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ace9c120d24521df","tex":"r_R","aliases":"r_R","display":"r_R","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-dbe6bc0f4ac42150","tex":"R_R^*","aliases":"R_R^*","display":"R_R^*","kind":"use","excerpt":"[Small core recompression] Let L∈ C^m× r, R∈ C^n× r. Take compact QR factorizations L=Q_LR_L, R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-868ac8e8bd426dc4","tex":"R_R^T","aliases":"R_R^T","display":"R_R^T","kind":"use","excerpt":"1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. vfigcompact_core The actual core is G=R_LR_R^T=mat5 · 0 · 0 0 · 2 · 0, LR^T=Q_LG. Its dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9bf28df79e66226e","tex":"\\operatorname{range}","aliases":"range operatorname","display":"\\operatorname{range}","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"Applying the ordinary matrix-rank inequality gives rank(P_m-D^(m))≤∑_j=1^m r_j. The bound concerns the actual correction matrix, not just the number of allocated columns. The proof also shows that D^(m) stays packet diagonal. The empty chain has identity base, empty column type, and zero correction. Individual r_j=0 are permitted.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank operatorname","display":"{\\operatorname{rank}}","kind":"use","excerpt":"V_C · =V_B · D_B^*V_A and proves AB=D_AD_B+U_CV_C^*. These are actual concatenated matrices with r_B+r_A columns. Expanding the product and grouping the terms with common right factors proves the identity. The separate rank theorem establishes rank(AB-D_AD_B)≤ r_A+r_B. It bounds actual matrix rank, not just the number of stored columns. Packet diag","count":9},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"vpage32Safe integers can still be conservativeQUANTIZATION AND THE SIZE OF THE BUDGET TABLE Let one integer unit represent h=1/10 of certified error. For an individual nonnegative error e, choosing w=lceil e/hrceil ensures e≤ hw. Thus integer feasibility ∑ w≤ B implies the real bound ∑ e≤ hB. This implication is checked before applying the exact op","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"use","excerpt":"Lean identifies the actual Frobenius norm with its trace expression, normM_rm F^2=Retr(M^*M), then expands the finite sum and removes each cross term. The result is normA-Π(A)_rm F^2 =∑_b≠ anormP_bAP_a_rm F^2. No Pythagorean equality for the target residual is assumed.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"ONE LINEAR MODEL Suppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold: T_b(R)=7R, T_c(R)=30+2R. vfigamortization At R=6, both cost 42. The candidate becomes strictly faster at R=7, where it costs 44 instead of 49. The minimum profitab","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the decla","count":15},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a58e4c2e82e7cd1b","tex":"\\mathrm{root}","aliases":"root mathrm","display":"\\mathrm{root}","kind":"use","excerpt":"The root scan min_0≤ b≤ BF_root(b) is therefore the exact budget-constrained optimum. Increasing the budget cannot increase a finite optimum. The theorem retains feasibility hypotheses where necessary rather than treating codenone as an ordinary real cost.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a62a596621079394","tex":"\\RR","aliases":"RR R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the ine","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-162e5e4a459cac13","tex":"\\RR^{m\\times p}","aliases":"RR^m× p R^m× p RR times mathbb times","display":"{\\mathbb R}^{m\\times p}","kind":"use","excerpt":"[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the ine","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2831aff814dfd9d4","tex":"\\RR^{p\\times n}","aliases":"RR^p× n R^p× n RR times mathbb times","display":"{\\mathbb R}^{p\\times n}","kind":"use","excerpt":"[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the ine","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c2b4530a730b06f1","tex":"\\mathrm{run}","aliases":"run mathrm","display":"\\mathrm{run}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"Stored scalars. With equal block width b, the storage-address type is S= B_p,w×{0,…,b-1}^2. Its cardinality is the actual number of represented scalar entries: | S|=b^2N_p,w.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"Strict half-gap stability. At a solved internal node write K=κ_v+λϵ_v, S=σ_v+∑_wF_λ(w), g=|K-S|. If the updated scores satisfy |K'-K|<g/2, |S'-S|<g/2, Lean proves K'≤ S' Longleftrightarrow K≤ S. Applying this condition recursively at every internal node proves equality of the entire returned cuts before and after perturbation, not only equality of ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","excerpt":"-1 · 3. Hence D^-1u=u/4 and S=1+u^TD^-1u=frac52. The Woodbury identity reduces the entire inverse to A^-1=D^-1-D^-1uS^-1u^TD^-1 =D^-1-frac140uu^T. vfigwoodbury headTwo hypotheses, not one Invertibility of D alone does not suffice. For D=I_2, u=e_1, and v=-e_1, the update is diag(0,1). The small matrix S is zero, exactly where the correction formula","count":11},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cc4fbf61a6a9f156","tex":"S_{ba}","aliases":"S_ba","display":"S_{ba}","kind":"use","excerpt":"Hidden elimination. For the actual Schur complement S=A_RR-A_RHA_HH^-1A_HR, the hidden inverse is expanded through the hidden projectors. A nonzero retained correction block requires indices j,k with nonzero entrance and exit blocks and a hidden path jleadsto k. Therefore a nonzero block S_ba requires either a direct retained block, or a⟶ j leadsto","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7aa9c815c9ae3803","tex":"s_e","aliases":"s_e","display":"s_e","kind":"use","excerpt":"s_e,ϵ_e · Sensitivity and tolerance · Is the budget input-dependent or uniform over unit inputs?","count":35},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-85a648879a8456a4","tex":"S_L","aliases":"S_L","display":"S_L","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f7af78049dcddf01","tex":"S_R","aliases":"S_R","display":"S_R","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-91ad15719e4542fe","tex":"\\mathrm{setup}","aliases":"setup mathrm","display":"\\mathrm{setup}","kind":"use","excerpt":"Shared operator families. For reuse weights r_j, the local keep model can aggregate setup and per-operator costs: κ_v=κ_v^setup+∑_jr_jκ_v^(j), ϵ_v=∑_jr_jϵ_v^(j). Lean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including Q and R. The inte","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-89d7944b7e472825","tex":"\\operatorname{step}","aliases":"step operatorname","display":"\\operatorname{step}","kind":"use","excerpt":"Let v be the new optimum of the edited subtree, and let j be the edited child slot at one frame. The update performed at that frame is step(v)= min{K, σ+ ∑_i v, · i=j, widehat F_i, · i≠ j. }. This result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values widehat F_i.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-327fd569cd80a61b","tex":"\\operatorname{storedTree}","aliases":"storedTree operatorname","display":"\\operatorname{storedTree}","kind":"use","excerpt":"Entrywise implementation theorem. For every b≤ B, Lean proves lookup(storedTree(B,t),b)=F_t(b). There is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at b depends only on input entries at most b, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attain","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d6008072c780cd95","tex":"\\mathrm{stream}","aliases":"stream mathrm","display":"\\mathrm{stream}","kind":"use","excerpt":"Streaming. The explicit event type counts one block read, one source read, and a target read plus write for every edge. With scalar size s, its byte count is B_stream=s∑_e(m_en_e+n_eq+2m_eq). No cross-edge cache reuse is assumed. Sparse index metadata and temporary workspace remain separate.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the decla","count":49},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6da3583735a2a7a0","tex":"T'","aliases":"T'","display":"T'","kind":"use","excerpt":"Correctness of the actual update. For any edited subtree T' and valid context C, Lean proves updateValue(C,F(T'))=F(fill(C,T')). The proof inducts through the context. At each frame, the updated child contributes the new value, and each other child contributes its unchanged cached optimum. The frame recurrence is therefore exactly the full-tree rec","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cefa01be96058c3e","tex":"T_b","aliases":"T_b","display":"T_b","kind":"use","excerpt":"vpage25The first win comes after the tieSETUP AND REUSE IN ONE LINEAR MODEL Suppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold: T_b(R)=7R, T_c(R)=30+2R. vfigamortization At R=6, both cost 42. The candidate becomes strictly faster at","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","excerpt":"[Amortization threshold] Let s≥0 be a one time setup cost. Let t_b,t_c≥0 be the baseline and candidate times per use and let R∈ N be a reuse count. If t_b>t_c, the candidate is faster after R uses exactly when R> fracst_b-t_c. The smallest integer reuse count is R_min = lfloor fracst_b-t_c rfloor+1. If t_b≤ t_c, no positive reuse count amortizes a ","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0692c76d8494573e","tex":"T_c","aliases":"T_c","display":"T_c","kind":"use","excerpt":"vpage25The first win comes after the tieSETUP AND REUSE IN ONE LINEAR MODEL Suppose the baseline costs seven units per use. A candidate costs 30 units to set up and two per use. These are synthetic constants chosen to expose the exact threshold: T_b(R)=7R, T_c(R)=30+2R. vfigamortization At R=6, both cost 42. The candidate becomes strictly faster at","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-04db525251126e68","tex":"t_c","aliases":"t_c","display":"t_c","kind":"use","excerpt":"[Amortization threshold] Let s≥0 be a one time setup cost. Let t_b,t_c≥0 be the baseline and candidate times per use and let R∈ N be a reuse count. If t_b>t_c, the candidate is faster after R uses exactly when R> fracst_b-t_c. The smallest integer reuse count is R_min = lfloor fracst_b-t_c rfloor+1. If t_b≤ t_c, no positive reuse count amortizes a ","count":13},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-aa6a46659a84add3","tex":"t_{\\mathrm{decode}}","aliases":"t_decode mathrm","display":"t_{\\mathrm{decode}}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c78b7188b7b592c0","tex":"t_e","aliases":"t_e","display":"t_e","kind":"use","excerpt":"[Materialization threshold] Suppose a compressed representation costs t_c≥0 per downstream use, an explicit representation costs t_e≥0 per use, and one time materialization costs m≥0, with reuse count R∈ N. Materialization is profitable exactly when R> fracmt_c-t_e and t_c>t_e.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e9a7aef78fcda23e","tex":"T_{ij}","aliases":"T_ij","display":"T_{ij}","kind":"use","excerpt":"Toeplitz padding and column-major vectorization. textitSource items 4.4 and 4.5 Toeplitz embedding. For n>0, define T_ij= c_i-j, · j≤ i, r_j-i, · i<j. The diagonal is supplied by c_0; the coordinate r_0 is unused. The constructed generator of length 2n is g=(c_0,…,c_n-1,0,r_n-1,…,r_1), and the padding map is the actual finite-index sum eliminator t","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-dcee229e3aaf9526","tex":"t_o","aliases":"t_o","display":"t_o","kind":"use","excerpt":"[Graph replay threshold] If graph capture costs c≥0, ordinary execution costs t_o≥0, and replay costs 0≤ t_r<t_o, then capture is profitable when R> fracct_o-t_r.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","excerpt":"[Graph replay threshold] If graph capture costs c≥0, ordinary execution costs t_o≥0, and replay costs 0≤ t_r<t_o, then capture is profitable when R> fracct_o-t_r.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4775952797c5d985","tex":"t_{\\mathrm{run}}","aliases":"t_run mathrm","display":"t_{\\mathrm{run}}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"Hilbert--Schmidt orthogonality. For distinct ordered packet pairs (b,a)≠(d,c), adjoint multiplication and projector orthogonality give tr((P_bAP_a)^*P_dBP_c)=0. If b≠ d, the central projector product vanishes. Otherwise a≠ c, and cyclicity of trace brings P_cP_a together. This is proved for arbitrary input matrices A,B.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7c05ecd0a14b319f","tex":"u^T","aliases":"u^T","display":"u^T","kind":"use","excerpt":"-1 · 3. Hence D^-1u=u/4 and S=1+u^TD^-1u=frac52. The Woodbury identity reduces the entire inverse to A^-1=D^-1-D^-1uS^-1u^TD^-1 =D^-1-frac140uu^T. vfigwoodbury headTwo hypotheses, not one Invertibility of D alone does not suffice. For D=I_2, u=e_1, and v=-e_1, the update is diag(0,1). The small matrix S is zero, exactly where the correction formula","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e105a73706668fb5","tex":"U_A","aliases":"U_A","display":"U_A","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-91c677571574a0b9","tex":"u_a","aliases":"u_a","display":"u_a","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-91de088f77bc89c5","tex":"U_B","aliases":"U_B","display":"U_B","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e21dd43c611f6c4e","tex":"u_b","aliases":"u_b","display":"u_b","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cfbfd121cc430e57","tex":"U_C","aliases":"U_C","display":"U_C","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":15},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b1bfe02ef752d496","tex":"u_e","aliases":"u_e","display":"u_e","kind":"use","excerpt":"Constructive parameterization. Write t=1/λ and set u_e(t)=min{m_e,tc_e/s_e}, Φ(t)=∑_e s_eu_e(t). Lean proves that Φ is continuous and nondecreasing, with Φ(0)=0. At the finite positive scale T=∑_efracm_es_ec_e, every coordinate is saturated, so u(T)=m and Φ(T)=∑_es_em_e. The intermediate value theorem produces t>0 with Φ(t)=τ. The multiplier is not","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","excerpt":"[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-25121c20156ac918","tex":"U_m","aliases":"U_m","display":"U_m","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m], R_m · =[R_m-1 (D^(m-1))^*V_m]. Substitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fb939028bb05fbbc","tex":"\\operatorname{updateValue}","aliases":"updateValue operatorname","display":"\\operatorname{updateValue}","kind":"use","excerpt":"Correctness of the actual update. For any edited subtree T' and valid context C, Lean proves updateValue(C,F(T'))=F(fill(C,T')). The proof inducts through the context. At each frame, the updated child contributes the new value, and each other child contributes its unchanged cached optimum. The frame recurrence is therefore exactly the full-tree rec","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1b5cec8c46451b53","tex":"\\v","aliases":"v","display":null,"kind":"use","excerpt":"ont=sffamilysmall] node[anchor=west] at (0,3.4) A^*A=diag(25,4); draw[->] (0,0)--(8,0) node[right] retained singular directions; draw[->] (0,0)--(0,2.5) node[above] squared error; draw[thick,black!65] (1,2.03)--(3.5,.28)--(6,0); foreach x/y/n/v in 1/2.03/0/29,3.5/.28/1/4,6/0/2/0 fill (x,y) circle (2pt);node[below] at (x,0) n;node[above] at (x,y) v;","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b684dfa3e6bcfe1e","tex":"v^{(k)}","aliases":"v^(k)","display":"v^{(k)}","kind":"use","excerpt":"For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2fe163fa16dae77c","tex":"v^{(k)}_j","aliases":"v^(k)_j","display":"v^{(k)}_j","kind":"use","excerpt":"For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b5eff5be3aa59d19","tex":"v^{(k)}_{j-r}","aliases":"v^(k)_j-r","display":"v^{(k)}_{j-r}","kind":"use","excerpt":"For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8c41a1fdf1eb317c","tex":"V_A","aliases":"V_A","display":"V_A","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-93a28f0d222e68e7","tex":"v_a","aliases":"v_a","display":"v_a","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0367417d53502da1","tex":"V_A^*","aliases":"V_A^*","display":"V_A^*","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-db1a7ed5fdbbc416","tex":"V_A^T","aliases":"V_A^T","display":"V_A^T","kind":"use","excerpt":"1.46) 5; path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2); node[font=small,align=center,font=small] at (5.36,0.94) 1; path[draw=black!30,fill=black!8] (5.62,0.68) rectangle (6.14,1.2); node[font=small,align=center,font=small] at (5.88,0.94) 6; node[font=small,align=center] at (10.3,3.45) AB=D_AD_B+U_CV_C^T; node[font=small,align=cen","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-612c002c1c1e9a93","tex":"v_a^T","aliases":"v_a^T","display":"v_a^T","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9f5cc10c11d8d9e7","tex":"V_B","aliases":"V_B","display":"V_B","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3234e8a9dca18cb9","tex":"v_b","aliases":"v_b","display":"v_b","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d343ba286305f1ad","tex":"V_B^*","aliases":"V_B^*","display":"V_B^*","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-12d0f7ef25761981","tex":"V_B^T","aliases":"V_B^T","display":"V_B^T","kind":"use","excerpt":"1.46) 5; path[draw=black!30,fill=black!8] (5.1,0.68) rectangle (5.62,1.2); node[font=small,align=center,font=small] at (5.36,0.94) 1; path[draw=black!30,fill=black!8] (5.62,0.68) rectangle (6.14,1.2); node[font=small,align=center,font=small] at (5.88,0.94) 6; node[font=small,align=center] at (10.3,3.45) AB=D_AD_B+U_CV_C^T; node[font=small,align=cen","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fe377465e680a401","tex":"v_b^T","aliases":"v_b^T","display":"v_b^T","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":8},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9336b2ab1b4f322f","tex":"V_C","aliases":"V_C","display":"V_C","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRA Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correcti","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b0f0625a5bf696ae","tex":"V_C^*","aliases":"V_C^*","display":"V_C^*","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1c63837778b9b8ed","tex":"V_C^T","aliases":"V_C^T","display":"V_C^T","kind":"use","excerpt":"0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. The remaining term is u_av_a^TD_B. Thus AB","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"Keeping node v costs κ_v and incurs integer error w_v. Splitting costs overhead σ_v in addition to the selected child cuts. The exact-weight recurrence is F_v(b)=min{ l κ_v if b=w_v, [2pt] displaystyleσ_v+ min_∑_i b_i=b∑_iF_v_i(b_i) if v is internal. . Nonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-pl","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-eb483cd80b51511d","tex":"V_j^*","aliases":"V_j^*","display":"V_j^*","kind":"use","excerpt":"[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8b86c046f5cd21ee","tex":"V_m","aliases":"V_m","display":"V_m","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m], R_m · =[R_m-1 (D^(m-1))^*V_m]. Substitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-83eb9ad5dd648cb4","tex":"V_m^*","aliases":"V_m^*","display":"V_m^*","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m], R_m · =[R_m-1 (D^(m-1))^*V_m]. Substitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c3a76197560b59e1","tex":"v_\\alpha","aliases":"v_α alpha","display":"v_\\alpha","kind":"use","excerpt":"[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-022a12d8dd418b72","tex":"\\operatorname{val}","aliases":"val operatorname","display":"\\operatorname{val}","kind":"use","excerpt":"n:001textitSource item 4.3; module CirculantFourier Let n>0, with coordinates indexed by Z/n Z. Define the actual circulant by C(c)_jk=c_j-k. The formal Fourier matrix is F_jk=frac1sqrt n χ(-jk), χ(a)=exp (frac2π i an). The standard character is not left abstract: a checked theorem identifies these entries with frac1sqrt nexp (-frac2π i val(j)val(k","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f7cc81212c08d083","tex":"\\operatorname{vec}","aliases":"vec operatorname","display":"\\operatorname{vec}","kind":"use","excerpt":"vpage7A transpose, not an adjointKRONECKER ACTION AND ARRAY ORDER With column-major vectorization, the tensor identity is (A⊗ B)vec(X)=vec(BXA^T). Here the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix A=mat1 · i","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Given operation capacity P and bandwidth W, T≥max(F/P,B/W) is a lower bound, not an achieved runtime. Launch overhead, imperfect overlap, and utilization can make the actual time much larger. takeawayGrouping can remove launches while leaving the same products and data movement. Arithmetic, traffic, and scheduling are related constraints, not one i","count":27},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-73ffecb5618fb3cc","tex":"W^*","aliases":"W^*","display":"W^*","kind":"use","excerpt":"Thin output factors and the final theorem. textitModules CompactTruncation and SmallCoreRecompression For the actual core G=R_LR_R^*, supply ordered thin SVD data G=Wdiag(σ)Z^*, W^*W=I, Z^*Z=I, σ_1≥⋯≥σ_d≥0. These are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gr","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"Operation events. The event type for k+1 modes is a disjoint union of d copies of the trailing event type and the events for the final fiber contraction. Its cardinality obeys W_0=0, W_k+1=dW_k+2d^k+2. Lean proves the closed form W_k=2kd^k+1. The factor-address type has cardinality k d^2. These statements include k=0 and d=0. With no factors, the a","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0289b7ef692ee27e","tex":"W_a","aliases":"W_a","display":"W_a","kind":"use","excerpt":"Fix the source packet a. In the regular representation, define the submodule W_a={x:P_bx=0 whenever anotleadsto b}. It contains P_a. Completeness of the projectors expands each component of Ax into packet transfers. If a transfer into an unreachable packet were nonzero from a reachable packet, it would extend a path, a contradiction. Thus AW_a⊆ W_a","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f39670f030db3b1a","tex":"W_B","aliases":"W_B","display":"W_B","kind":"use","excerpt":"Let W_B(t) be the declared unit-cost count of table-cell constructions and tested min-plus candidates. The model charges the empty child-combination table as well as the output cells and merge candidates. Lean proves W_B(t)≤2(B+1)N+((B+1)+(B+1)^2)(N-1) and hence W_B(t)≤ N(3(B+1)+(B+1)^2). The root minimum scan adds a linear pass. Thus the declared ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"Frobenius optimality from kernel weights. textitModules KernelFrame and FrobeniusOptimality For an arbitrary competitor B with rank B≤ k, construct an orthonormal matrix Q spanning its actual kernel. Then BQ=0, Q^*Q=I, q=dimker B=d-rank B. Define the row weights w_i=∑_j=1^q|Q_ij|^2. The proof derives 0≤ w_i≤1, ∑_iw_i=q≥max(d-k,0). The upper bound u","count":8},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a8da97db1cfb2f53","tex":"w_{j,k}","aliases":"w_j,k","display":"w_{j,k}","kind":"use","excerpt":"ct discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cf76327e749de1b7","tex":"W_k","aliases":"W_k","display":"W_k","kind":"use","excerpt":"R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions; tail singular values beyond d are zero. ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b5731a3740a4e086","tex":"W_{k+1}","aliases":"W_k+1","display":"W_{k+1}","kind":"use","excerpt":"Operation events. The event type for k+1 modes is a disjoint union of d copies of the trailing event type and the events for the final fiber contraction. Its cardinality obeys W_0=0, W_k+1=dW_k+2d^k+2. Lean proves the closed form W_k=2kd^k+1. The factor-address type has cardinality k d^2. These statements include k=0 and d=0. With no factors, the a","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e6e9e4d147adf393","tex":"w_v","aliases":"w_v","display":"w_v","kind":"use","excerpt":"[Quantized budget dynamic program] Assume every keep error is a nonnegative integer w_v. Let B∈ N. For each node v, define F_v(b) as the minimum cost of a pruning of the subtree at v with total error exactly b. The keep option contributes F_v(w_v)≤κ_v. The split option is obtained by min plus convolution of the child tables, followed by addition of","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f33ee59cc7d16b42","tex":"\\operatorname{Walsh}","aliases":"Walsh operatorname","display":"\\operatorname{Walsh}","kind":"use","excerpt":"Actual butterfly. For the two halves x_0,x_1 of the input, the independently defined procedure returns Walsh_2N(x_0,x_1) =(Walsh_N(x_0+x_1), Walsh_N(x_0-x_1)). Lean proves that this vector equals H_2Nx. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-986d47d11f30d4d9","tex":"\\operatorname{Walsh}_{2N}","aliases":"Walsh_2N operatorname","display":"\\operatorname{Walsh}_{2N}","kind":"use","excerpt":"Actual butterfly. For the two halves x_0,x_1 of the input, the independently defined procedure returns Walsh_2N(x_0,x_1) =(Walsh_N(x_0+x_1), Walsh_N(x_0-x_1)). Lean proves that this vector equals H_2Nx. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b72fa7a71d0d77ae","tex":"\\operatorname{Walsh}_N","aliases":"Walsh_N operatorname","display":"\\operatorname{Walsh}_N","kind":"use","excerpt":"Actual butterfly. For the two halves x_0,x_1 of the input, the independently defined procedure returns Walsh_2N(x_0,x_1) =(Walsh_N(x_0+x_1), Walsh_N(x_0-x_1)). Lean proves that this vector equals H_2Nx. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8d3781ce495edcf4","tex":"\\widehat A","aliases":"widehat A widehat","display":"\\widehat A","kind":"use","excerpt":"For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tig","count":23},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-891994962d96636a","tex":"\\widehat A_1","aliases":"widehat A_1 widehat","display":"\\widehat A_1","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4895bc4181eddb45","tex":"\\widehat A_i","aliases":"widehat A_i widehat","display":"\\widehat A_i","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9708fbc1470a12ef","tex":"\\widehat A_j","aliases":"widehat A_j widehat","display":"\\widehat A_j","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ed3d5df42b309969","tex":"\\widehat A_{j+1}","aliases":"widehat A_j+1 widehat","display":"\\widehat A_{j+1}","kind":"use","excerpt":"Use the telescoping identity widehat A_m⋯widehat A_1 -A_m⋯ A_1 = ∑_j=1^m widehat A_m⋯widehat A_j+1 E_j A_j-1⋯ A_1. Take norms term by term.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7a01d17f6b25e6b1","tex":"\\widehat A_m","aliases":"widehat A_m widehat","display":"\\widehat A_m","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"use","excerpt":"[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the ine","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9a999477715e268b","tex":"\\widehat F","aliases":"widehat F widehat","display":"\\widehat F","kind":"use","excerpt":"Let v be the new optimum of the edited subtree, and let j be the edited child slot at one frame. The update performed at that frame is step(v)= min{K, σ+ ∑_i v, · i=j, widehat F_i, · i≠ j. }. This result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values widehat F_i.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d2f6188722a2e8ff","tex":"\\widehat F_i","aliases":"widehat F_i widehat","display":"\\widehat F_i","kind":"use","excerpt":"Let v be the new optimum of the edited subtree, and let j be the edited child slot at one frame. The update performed at that frame is step(v)= min{K, σ+ ∑_i v, · i=j, widehat F_i, · i≠ j. }. This result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values widehat F_i.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-62308a18a459ffc6","tex":"\\widehat P_0","aliases":"widehat P_0 widehat","display":"\\widehat P_0","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-add64233d3bae23a","tex":"\\widehat p_3","aliases":"widehat p_3 widehat","display":"\\widehat p_3","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-22cdc1586433efd2","tex":"\\widehat P_j","aliases":"widehat P_j widehat","display":"\\widehat P_j","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f3a9183aa4d4607b","tex":"\\widehat p_j","aliases":"widehat p_j widehat","display":"\\widehat p_j","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-bd0609390bade526","tex":"\\widehat P_{j+1}","aliases":"widehat P_j+1 widehat","display":"\\widehat P_{j+1}","kind":"use","excerpt":"Actual noncommutative products. The error module defines the left-accumulated product recursively, with P_0=I and P_j+1=A_j+1P_j. For a perturbed sequence widehat A_j, it derives the discrepancy recursion widehat P_j+1-P_j+1 =widehat A_j+1(widehat P_j-P_j) +(widehat A_j+1-A_j+1)P_j. The identity follows from the actual products. It is not supplied ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5a26fa77c1b9a117","tex":"\\widehat P_{j-1}","aliases":"widehat P_j-1 widehat","display":"\\widehat P_{j-1}","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6c1d9a673bbb0544","tex":"\\widehat p_{j-1}","aliases":"widehat p_j-1 widehat","display":"\\widehat p_{j-1}","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSION Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a differ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-266ff731dd16faeb","tex":"\\widehat P_m","aliases":"widehat P_m widehat","display":"\\widehat P_m","kind":"use","excerpt":"ntheorem[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i )","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7efd54fa5d68cf48","tex":"\\widetilde P_0","aliases":"widetilde P_0 widetilde","display":"\\widetilde P_0","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-064ad848deb9dca0","tex":"\\widetilde P_j","aliases":"widetilde P_j widetilde","display":"\\widetilde P_j","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7bf0caffa4ff570d","tex":"\\widetilde P_{j-1}","aliases":"widetilde P_j-1 widetilde","display":"\\widetilde P_{j-1}","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7217c1cd533b14e6","tex":"\\widetilde P_m","aliases":"widetilde P_m widetilde","display":"\\widetilde P_m","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a86d6dedf5bd962e","tex":"\\x","aliases":"x","display":null,"kind":"use","excerpt":"tex] draw[->] (0,0)--(7,0) node[right] r_2; draw[->] (0,0)--(0,4.5) node[above] modeled cost; draw[thick] (0,.6)--(4,3.4) node[right] 6+7r_2 : keep; draw[thick,dashed] (0,1)--(4,1.8) node[right] 10+2r_2 : split; draw[dotted] (.8,0)--(.8,1.16);fill (.8,1.16) circle (2pt); node[below] at (.8,0) 4/5; foreach x in 0,2,3,4draw (x,0)--(x,-.08);node[below","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-dab4e8a89da28b3b","tex":"x^{(0)}","aliases":"x^(0)","display":"x^{(0)}","kind":"use","excerpt":"Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1,","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7696a5e0b656722b","tex":"x^{(d)}","aliases":"x^(d)","display":"x^{(d)}","kind":"use","excerpt":"[5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1 vertex has level at most k, so the recurrence uses only coordinates already proved correct. At the canonical depth d this yields Ax^(d)=b. The local independence the","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0190f670081497c9","tex":"x^{(k)}_u","aliases":"x^(k)_u","display":"x^{(k)}_u","kind":"use","excerpt":"Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1, [5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-55b833de72597be9","tex":"x^{(k)}_v","aliases":"x^(k)_v","display":"x^{(k)}_v","kind":"use","excerpt":"Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1, [5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c44fc36b58974583","tex":"x^{(k+1)}_v","aliases":"x^(k+1)_v","display":"x^{(k+1)}_v","kind":"use","excerpt":"Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1, [5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6e7e9fdf289546b6","tex":"x^*","aliases":"x^*","display":"x^*","kind":"use","excerpt":"For every unit vector x, absx^*Ex≤η. The Rayleigh quotient of I+E lies in [1-η,1+η]. The quotient of the extreme eigenvalue bounds gives the result.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0514a60dd6e60420","tex":"x^T","aliases":"x^T","display":"x^T","kind":"use","excerpt":"1/2 · 1. Its exact eigenvalues are 1/2 and 3/2, giving condition number three. vfigpreconditioning headThe unit circles of the energy norms The left ellipse is x^TAx=1. Writing x=By gives the right ellipse y^T(BAB)y=1. The drawing samples these explicitly defined curves; the norm and eigenvalue equalities are checked symbolically.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-14a3332b34948231","tex":"x_*","aliases":"x_*","display":"x_*","kind":"use","excerpt":"[Preconditioned conjugate gradient convergence] Let A and M be Hermitian positive definite on a nonempty finite coordinate space, and set κ = cond_2(M^-1/2AM^-1/2). In exact arithmetic, the energy norm error of preconditioned conjugate gradient satisfies normx_m-x_*_A ≤ 2 ( fracsqrtκ-1sqrtκ+1 )^m normx_0-x_*_A.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"The orthogonality identity follows by induction from block multiplication. One recursive stage forms x_0+x_1 and x_0-x_1, which uses N additions or subtractions, and then applies two transforms of size N/2. Thus T(N)=2T(N/2)+N, T(1)=0, whose solution is T(N)=Nlog_2N.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"The orthogonality identity follows by induction from block multiplication. One recursive stage forms x_0+x_1 and x_0-x_1, which uses N additions or subtractions, and then applies two transforms of size N/2. Thus T(N)=2T(N/2)+N, T(1)=0, whose solution is T(N)=Nlog_2N.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":", put two on the diagonal and -1 on each declared incoming edge. The edges are 0→2, 1→2, 1→3, 2→4, 3→4, 3→5, 4→6, and 5→6. Solve (2I+K)x=ones. vfigtriangular headRead the stages as dependencies Packets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For insta","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":", put two on the diagonal and -1 on each declared incoming edge. The edges are 0→2, 1→2, 1→3, 2→4, 3→4, 3→5, 4→6, and 5→6. Solve (2I+K)x=ones. vfigtriangular headRead the stages as dependencies Packets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For insta","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ca7008ae447ee9dc","tex":"x_4","aliases":"x_4","display":"x_4","kind":"use","excerpt":", put two on the diagonal and -1 on each declared incoming edge. The edges are 0→2, 1→2, 1→3, 2→4, 3→4, 3→5, 4→6, and 5→6. Solve (2I+K)x=ones. vfigtriangular headRead the stages as dependencies Packets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For insta","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-98391915cbb7346b","tex":"x_5","aliases":"x_5","display":"x_5","kind":"use","excerpt":", put two on the diagonal and -1 on each declared incoming edge. The edges are 0→2, 1→2, 1→3, 2→4, 3→4, 3→5, 4→6, and 5→6. Solve (2I+K)x=ones. vfigtriangular headRead the stages as dependencies Packets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For insta","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3864fe3cacc7c90d","tex":"x_6","aliases":"x_6","display":"x_6","kind":"use","excerpt":", put two on the diagonal and -1 on each declared incoming edge. The edges are 0→2, 1→2, 1→3, 2→4, 3→4, 3→5, 4→6, and 5→6. Solve (2I+K)x=ones. vfigtriangular headRead the stages as dependencies Packets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For insta","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0298cc60bfc2d7e5","tex":"X_{ij}","aliases":"X_ij","display":"X_{ij}","kind":"use","excerpt":"Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2357b435bdbbf889","tex":"X_{k\\ell}","aliases":"X_kℓ ell","display":"X_{k\\ell}","kind":"use","excerpt":"The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","excerpt":"[Preconditioned conjugate gradient convergence] Let A and M be Hermitian positive definite on a nonempty finite coordinate space, and set κ = cond_2(M^-1/2AM^-1/2). In exact arithmetic, the energy norm error of preconditioned conjugate gradient satisfies normx_m-x_*_A ≤ 2 ( fracsqrtκ-1sqrtκ+1 )^m normx_0-x_*_A.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e798a8a7ea91ac3e","tex":"X_R","aliases":"X_R","display":"X_R","kind":"use","excerpt":"Let R be the set of packets reachable from α, including α, and set X_R=⊕_γ∈ RX_γ. No active edge leaves R, so AX_R⊆ X_R. The restriction of A to the finite dimensional space X_R is injective. It is therefore onto X_R. Hence A^-1X_R⊆ X_R. Since β∉ R, the stated block of A^-1 vanishes.","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-54ac43616fbfd36c","tex":"x_\\alpha","aliases":"x_α alpha","display":"x_\\alpha","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-bcf7586068a6a313","tex":"x_\\beta","aliases":"x_β beta","display":"x_\\beta","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c5c7030cb4e0b4b6","tex":"X_\\gamma","aliases":"X_γ gamma","display":"X_\\gamma","kind":"use","excerpt":"Let R be the set of packets reachable from α, including α, and set X_R=⊕_γ∈ RX_γ. No active edge leaves R, so AX_R⊆ X_R. The restriction of A to the finite dimensional space X_R is injective. It is therefore onto X_R. Hence A^-1X_R⊆ X_R. Since β∉ R, the stated block of A^-1 vanishes.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"[Adjoint and tensor semantics] The adjoint representation has blocks (A^*)_αβ=A_βα^*. If A and B are packet resolved on X and Y, then (A⊗ B)_(β,δ),(α,γ) = A_βα⊗ B_δγ.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3590b2065493ac41","tex":"y^{2}","aliases":"y^2","display":"y^{2}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-87e8ed00b8e4c9d2","tex":"y^T","aliases":"y^T","display":"y^T","kind":"use","excerpt":"1/2 · 1. Its exact eigenvalues are 1/2 and 3/2, giving condition number three. vfigpreconditioning headThe unit circles of the energy norms The left ellipse is x^TAx=1. Writing x=By gives the right ellipse y^T(BAB)y=1. The drawing samples these explicitly defined curves; the norm and eigenvalue equalities are checked symbolically.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"[Exact evaluation] Let A∈End(X). Set B_βα=π_β Aι_α and let E contain exactly the nonzero blocks. Then A= ∑_(β,α)∈ E P_β AP_α. For every matrix of right hand sides Z, AZ = ∑_(β,α)∈ E ι_β B_βαπ_α Z.","count":15},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"Circulant Fourier diagonalization. textitSource item 4.3; module CirculantFourier Let n>0, with coordinates indexed by Z/n Z. Define the actual circulant by C(c)_jk=c_j-k. The formal Fourier matrix is F_jk=frac1sqrt n χ(-jk), χ(a)=exp (frac2π i an). The standard character is not left abstract: a checked theorem identifies these entries with frac1sq","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"[Sparse plus low rank application] Let A=S+UV^*, where S has z stored scalar nonzeros and U,V∈ C^n× r. In the model that charges two operations per stored sparse entry and right hand side, applying A to q right hand sides costs 2q(z+2nr) modeled operations and stores z+2nr scalar values, apart from sparse index metadata.","count":13},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5e30e315795c2b5b","tex":"Z^*","aliases":"Z^*","display":"Z^*","kind":"use","excerpt":"[Small core recompression] Let L∈ C^m× r, R∈ C^n× r. Take compact QR factorizations L=Q_LR_L, R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where ","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"use","excerpt":"[Polynomial support] If p(z)=∑_j=0^dc_jz^j, then supp(p(A)) is contained in the union of the support relations of A^j for 0≤ j≤ d.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d533223c5565c514","tex":"Z_k","aliases":"Z_k","display":"Z_k","kind":"use","excerpt":"R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions; tail singular values beyond d are zero. ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the ine","count":10},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"Frobenius optimality from kernel weights. textitModules KernelFrame and FrobeniusOptimality For an arbitrary competitor B with rank B≤ k, construct an orthonormal matrix Q spanning its actual kernel. Then BQ=0, Q^*Q=I, q=dimker B=d-rank B. Define the row weights w_i=∑_j=1^q|Q_ij|^2. The proof derives 0≤ w_i≤1, ∑_iw_i=q≥max(d-k,0). The upper bound u","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3e679c26bc7fd384","tex":"|^{2}","aliases":"|^2","display":"|^{2}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-59ce46150d0cfe77","tex":"\\alpha_0","aliases":"α_0 alpha","display":"\\alpha_0","kind":"use","excerpt":"[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"use","excerpt":"[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-da4f2068e2750ae4","tex":"\\alpha_m","aliases":"α_m alpha","display":"\\alpha_m","kind":"use","excerpt":"[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-20abd9c0a72b9797","tex":"\\alpha_{m-1}","aliases":"α_m-1 alpha","display":"\\alpha_{m-1}","kind":"use","excerpt":"[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3e52bdbf6bbe4050","tex":"\\gamma_p","aliases":"γ_p gamma","display":"\\gamma_p","kind":"use","excerpt":"[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the ine","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"[Mixed precision product chain] If each fixed linear factor A_j is replaced by a fixed packetwise linear approximation widehat A_j, write E_j=widehat A_j-A_j=∑_eΔ_j,eP_α(j,e). A uniform certificate is E_j ≤δ_j:=∑_e Δ_j,e P_α(j,e) . For orthogonal packet projectors, δ_j≤∑_e Δ_j,e . The final operator error is bounded by [source reference] with these","count":13},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"This article begins with a finite direct sum X=⊕_α∈ IX_α and a complete family of coordinate projectors P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. An operator has packet blocks A_βα=P_β AP_α. The active addresses form a directed graph. Block dimensions, block types, approximation tolerances, and output requirements turn this graph into a computational intermedi","count":12},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c5b085f18c150ac1","tex":"\\Delta_e","aliases":"Δ_e Delta","display":"\\Delta_e","kind":"use","excerpt":"Packet perturbation certificates. textitSource item 8.1 and two uniform consequences Let X be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps X→ X, with their operator norm. Let a(e) identify the source packet of edge e, and suppose Δ_e=Δ_e P_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exac","count":8},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"ad P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j. If normE_j≤δ_j, the same formula with δ_j","count":9},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c5a1bc4b1c0866ff","tex":"\\Delta_{j,e}","aliases":"Δ_j,e Delta","display":"\\Delta_{j,e}","kind":"use","excerpt":"[Mixed precision product chain] If each fixed linear factor A_j is replaced by a fixed packetwise linear approximation widehat A_j, write E_j=widehat A_j-A_j=∑_eΔ_j,eP_α(j,e). A uniform certificate is E_j ≤δ_j:=∑_e Δ_j,e P_α(j,e) . For orthogonal packet projectors, δ_j≤∑_e Δ_j,e . The final operator error is bounded by [source reference] with these","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0fcb1c3cf5e3bf16","tex":"\\delta_{\\alpha\\beta}","aliases":"δ_αβ delta alpha beta","display":"\\delta_{\\alpha\\beta}","kind":"use","excerpt":"This article begins with a finite direct sum X=⊕_α∈ IX_α and a complete family of coordinate projectors P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. An operator has packet blocks A_βα=P_β AP_α. The active addresses form a directed graph. Block dimensions, block types, approximation tolerances, and output requirements turn this graph into a computational intermedi","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exact operator and vector identities are proved first: widehat A-A=∑_eΔ_e P_a(e), (widehat A-A)x=∑_eΔ_e(P_a(e)x). Applying the finite triangle inequality and the defining operator-norm estimate gives norm(widehat A-A)x ≤∑_enormΔ_e normP_a(e)x. If normΔ_e≤ε_enormA_e, then the formal relative certific","count":22},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-be8d5809df598ef7","tex":"\\varepsilon^*","aliases":"ε^* varepsilon","display":"\\varepsilon^*","kind":"use","excerpt":"Global comparison. For any other feasible ε, set r_e=ε_e/ε_e^*>0. The budget identities give ∑_ec_er_e=C. Using the standard analytic inequality log r≤ r-1 yields ∑_ec_elog r_e≤∑_ec_e(r_e-1)=0. Lean separately proves the exact objective-difference identity J(ε)-J(ε^*)=-∑_ec_elog r_e. Therefore J(ε)≥ J(ε^*).","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-56db928a7ce0a173","tex":"\\epsilon_e","aliases":"ϵ_e epsilon","display":"\\epsilon_e","kind":"use","excerpt":"s_e,ϵ_e · Sensitivity and tolerance · Is the budget input-dependent or uniform over unit inputs?","count":11},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2fc46a82cd7318e7","tex":"\\varepsilon_e","aliases":"ε_e varepsilon","display":"\\varepsilon_e","kind":"use","excerpt":"_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exact operator and vector identities are proved first: widehat A-A=∑_eΔ_e P_a(e), (widehat A-A)x=∑_eΔ_e(P_a(e)x). Applying the finite triangle inequality and the defining operator-norm estimate gives norm(widehat A-A)x ≤∑_enormΔ_e normP_a(e)x. If normΔ_e≤ε_enormA_e, then the formal relative certific","count":13},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cb0e8c76ef72f0c6","tex":"\\varepsilon_e^*","aliases":"ε_e^* varepsilon","display":"\\varepsilon_e^*","kind":"use","excerpt":"lete textitSource item 8.4, with an explicit positive budget Let E be a finite nonempty edge set. Assume s_e>0, c_e>0, τ>0, and define C=∑_ec_e. The optimization problem is min_ε_e>0∑_ec_elogfrac1ε_e subject to ∑_es_eε_e=τ. Lean constructs the displayed candidate ε_e^*=fracτ c_es_eC and proves strict positivity and the exact budget equality. It the","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-e2a74941d662acb4","tex":"\\epsilon_v^{(j)}","aliases":"ϵ_v^(j) epsilon","display":"\\epsilon_v^{(j)}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"Here E _2=1/2. The manuscript's general inequality κ_2(D^-1/2AD^-1/2) ≤frac1+η1-η, E _2≤η<1, is attained. The raw off-diagonal entry is one; its meaningful size is one-half only after normalization by the diagonal energy scales. headKeep the convergence input separate The condition bound can be inserted into a conjugate-gradient convergence estimat","count":12},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-93055c50da4a404a","tex":"\\kappa_2","aliases":"κ_2 kappa","display":"\\kappa_2","kind":"use","excerpt":"Here E _2=1/2. The manuscript's general inequality κ_2(D^-1/2AD^-1/2) ≤frac1+η1-η, E _2≤η<1, is attained. The raw off-diagonal entry is one; its meaningful size is one-half only after normalization by the diagonal energy scales. headKeep the convergence input separate The condition bound can be inserted into a conjugate-gradient convergence estimat","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-39c423c6294d6173","tex":"\\kappa_v^{(j)}","aliases":"κ_v^(j) kappa","display":"\\kappa_v^{(j)}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a9680dd9eec29c6c","tex":"\\kappa_v^{\\rm setup}","aliases":"κ_v^rm setup kappa rm","display":"\\kappa_v^{\\rm setup}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8e6997cad3f90314","tex":"\\kappa_v^{\\mathrm{setup}}","aliases":"κ_v^setup kappa mathrm","display":"\\kappa_v^{\\mathrm{setup}}","kind":"use","excerpt":"Shared operator families. For reuse weights r_j, the local keep model can aggregate setup and per-operator costs: κ_v=κ_v^setup+∑_jr_jκ_v^(j), ϵ_v=∑_jr_jϵ_v^(j). Lean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including Q and R. The inte","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"[Monotone Lagrangian frontier] Let 0≤λ_1<λ_2. Choose optimal prunings L_1 and L_2 for the two Lagrangian parameters. Then err( L_2) ≤ err( L_1) and cost( L_2) ≥ cost( L_1).","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-975e1a0c6d000741","tex":"\\lambda_2","aliases":"λ_2 lambda","display":"\\lambda_2","kind":"use","excerpt":"[Monotone Lagrangian frontier] Let 0≤λ_1<λ_2. Choose optimal prunings L_1 and L_2 for the two Lagrangian parameters. Then err( L_2) ≤ err( L_1) and cost( L_2) ≥ cost( L_1).","count":5},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"For any nonnegative decreasing sequence a_i, weights with these properties satisfy ∑_i a_iw_i≥∑_i>ka_i. This finite weighted inequality is itself proved. For k<d, use μ=a_k+1 and compare the summands to tail weights v_i=0 for i≤ k and v_i=1 otherwise: a_iv_i+μ(w_i-v_i)≤ a_iw_i. For a retained coordinate the inequality uses a_i≥μ and w_i≥0. For a ta","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"ubsection:002textitModule SmallCoreNorms; arbitrary ambient competitors For rectangular isometries Q,V define L(G)=QGV^*, C(B)=Q^*BV, Q^*Q=I, V^*V=I. The following statements are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_ν≤norm B_ν, ν∈{2, F}. For the spe","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"[Exact block discovery certificate] For an orthogonal packet system (P_α), define the block diagonal compression Π_ P(A) = ∑_α P_α AP_α. Then normA-Π_ P(A)_rm F^2 = ∑_α≠β normP_β AP_α_rm F^2. In operator norm, normA-Π_ P(A)_2 ≤ ∑_α≠β normP_β AP_α_2. Thus any proposed packet discovery has an exact finite residual certificate after its projectors are","count":6},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3315617fe9f782aa","tex":"\\Pi_{\\Pcal}","aliases":"Π_Pcal Π_ P Pi Pcal Pi mathcal","display":"\\Pi_{{\\mathcal P}}","kind":"use","excerpt":"[Exact block discovery certificate] For an orthogonal packet system (P_α), define the block diagonal compression Π_ P(A) = ∑_α P_α AP_α. Then normA-Π_ P(A)_rm F^2 = ∑_α≠β normP_β AP_α_rm F^2. In operator norm, normA-Π_ P(A)_2 ≤ ∑_α≠β normP_β AP_α_2. Thus any proposed packet discovery has an exact finite residual certificate after its projectors are","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9eac9da427d5101f","tex":"\\pi_\\beta","aliases":"π_β pi beta","display":"\\pi_\\beta","kind":"use","excerpt":"[Exact evaluation] Let A∈End(X). Set B_βα=π_β Aι_α and let E contain exactly the nonzero blocks. Then A= ∑_(β,α)∈ E P_β AP_α. For every matrix of right hand sides Z, AZ = ∑_(β,α)∈ E ι_β B_βαπ_α Z.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"[Small core recompression] Let L∈ C^m× r, R∈ C^n× r. Take compact QR factorizations L=Q_LR_L, R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where ","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5048c4b9123eb530","tex":"\\sigma_1","aliases":"σ_1 sigma","display":"\\sigma_1","kind":"use","excerpt":"[Eckart Young Mirsky theorem] If M has singular values σ_1≥σ_2≥⋯, pad this finite sequence with zeros. For each integer k≥0, its truncated singular value decomposition M_k is a best rank-at-most-k approximation. It satisfies normM-M_k_2=σ_k+1 and normM-M_k_rm F = ( ∑_j>kσ_j^2 )^1/2.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2cce6a80c926063f","tex":"\\sigma_1^2","aliases":"σ_1^2 sigma","display":"\\sigma_1^2","kind":"use","excerpt":"05textitModules CompactTruncation and SmallCoreRecompression For the actual core G=R_LR_R^*, supply ordered thin SVD data G=Wdiag(σ)Z^*, W^*W=I, Z^*Z=I, σ_1≥⋯≥σ_d≥0. These are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation G^*GZ=Zdiag(σ_1^2","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-15fde97ab40a2e94","tex":"\\sigma_2","aliases":"σ_2 sigma","display":"\\sigma_2","kind":"use","excerpt":"[Eckart Young Mirsky theorem] If M has singular values σ_1≥σ_2≥⋯, pad this finite sequence with zeros. For each integer k≥0, its truncated singular value decomposition M_k is a best rank-at-most-k approximation. It satisfies normM-M_k_2=σ_k+1 and normM-M_k_rm F = ( ∑_j>kσ_j^2 )^1/2.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-58c1a4cd175eaace","tex":"\\sigma_d","aliases":"σ_d sigma","display":"\\sigma_d","kind":"use","excerpt":"Spectral optimality from a kernel witness. textitModules DiagonalTruncation and SpectralOptimality Let σ_1≥⋯≥σ_d≥0 and D=diag(σ_1,…,σ_d), D_k=diag(σ_1,…,σ_min(k,d),0,…,0). Lean uses indices beginning at zero; the formulas on this page use the conventional one-based notation. The rank of D_k is proved to be at most k, even when retained singular val","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3443a030a9ee99f2","tex":"\\sigma_d^2","aliases":"σ_d^2 sigma","display":"\\sigma_d^2","kind":"use","excerpt":"05textitModules CompactTruncation and SmallCoreRecompression For the actual core G=R_LR_R^*, supply ordered thin SVD data G=Wdiag(σ)Z^*, W^*W=I, Z^*Z=I, σ_1≥⋯≥σ_d≥0. These are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation G^*GZ=Zdiag(σ_1^2","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f9ab434fbb0d31ef","tex":"\\sigma_i^2","aliases":"σ_i^2 sigma","display":"\\sigma_i^2","kind":"use","excerpt":"Set a_i=σ_i^2. Right compression cannot increase the Frobenius norm, and BQ=0, hence normD-B_ F^2 · ≥norm(D-B)Q_ F^2","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-652e4283f0c8bec9","tex":"\\sigma_i^{2}","aliases":"σ_i^2 sigma","display":"\\sigma_i^{2}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8de4fa9e7f9c5804","tex":"\\sigma_j","aliases":"σ_j sigma","display":"\\sigma_j","kind":"use","excerpt":"W^*W=I, Z^*Z=I, σ_1≥⋯≥σ_d≥0. These are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation G^*GZ=Zdiag(σ_1^2,…,σ_d^2). Let h=min(d,k) and retain exactly the first h column indices. The rectangular square-root mask has entries M_ij=sqrtσ_j, · i=j","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-76a89e17dd1077d6","tex":"\\sigma_j^2","aliases":"σ_j^2 sigma","display":"\\sigma_j^2","kind":"use","excerpt":"[Eckart Young Mirsky theorem] If M has singular values σ_1≥σ_2≥⋯, pad this finite sequence with zeros. For each integer k≥0, its truncated singular value decomposition M_k is a best rank-at-most-k approximation. It satisfies normM-M_k_2=σ_k+1 and normM-M_k_rm F = ( ∑_j>kσ_j^2 )^1/2.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2b44b17ec5f59013","tex":"\\sigma_{k+1}","aliases":"σ_k+1 sigma","display":"\\sigma_{k+1}","kind":"use","excerpt":"For a d-dimensional diagonal core and 0≤ k<d, the spectral lower bound has a concrete witness. A rank-at-most-k competitor has a nonzero kernel vector in the first k+1 coordinates. On that vector the diagonal matrix acts with norm at least σ_k+1 times the vector norm. The competitor cannot cancel that direction.","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-d1737ab7785be77f","tex":"\\sigma_{k+1}^{2}","aliases":"σ_k+1^2 sigma","display":"\\sigma_{k+1}^{2}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4fd51bd4c6f1161b","tex":"\\Sigma_k^{1/2}","aliases":"Σ_k^1/2 Sigma","display":"\\Sigma_k^{1/2}","kind":"use","excerpt":"R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions; tail singular values beyond d are zero. ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0931a7c425476227","tex":"\\sigma_{\\min(k,d)}","aliases":"σ_min(k,d) sigma min","display":"\\sigma_{\\min(k,d)}","kind":"use","excerpt":"Spectral optimality from a kernel witness. textitModules DiagonalTruncation and SpectralOptimality Let σ_1≥⋯≥σ_d≥0 and D=diag(σ_1,…,σ_d), D_k=diag(σ_1,…,σ_min(k,d),0,…,0). Lean uses indices beginning at zero; the formulas on this page use the conventional one-based notation. The rank of D_k is proved to be at most k, even when retained singular val","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b17d423cde5b3e30","tex":"\\sigma_u","aliases":"σ_u sigma","display":"\\sigma_u","kind":"use","excerpt":"The cost and error of a pruning L are cost( L) = ∑_v∈ Lκ_v + ∑_u splitσ_u and err( L)=∑_v∈ Lϵ_v. The model can include arithmetic, storage, launch, transfer, and precision costs inside κ_v. It remains exact only when the declared totals are additive.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"vpage20Spend error where it is cheapestCONTINUOUS AND CLIPPED PRECISION Take sensitivities s=(1,2,4), cost coefficients c=(1,1,2), and budget τ=1. Minimizing ∑_ec_elog(1/ϵ_e) subject to ∑_es_eϵ_e=1 gives ϵ_e=fracc_e4s_e, ϵ=(1/4,1/8,1/8). vfigprecision headThe budget is weighted The three contributions to the error allowance are (1/4,1/4,1/2). Equal","count":19},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"Constructive parameterization. Write t=1/λ and set u_e(t)=min{m_e,tc_e/s_e}, Φ(t)=∑_e s_eu_e(t). Lean proves that Φ is continuous and nondecreasing, with Φ(0)=0. At the finite positive scale T=∑_efracm_es_ec_e, every coordinate is saturated, so u(T)=m and Φ(T)=∑_es_em_e. The intermediate value theorem produces t>0 with Φ(t)=τ. The multiplier is not","count":7},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"The Karush Kuhn Tucker conditions give the unconstrained form on every nonactive coordinate and equality with m_e on every active upper bound. The weighted sum ϕ(λ) = ∑_es_e min{ m_e, fracc_eλ s_e } is continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsatur","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"Circulant Fourier diagonalization. textitSource item 4.3; module CirculantFourier Let n>0, with coordinates indexed by Z/n Z. Define the actual circulant by C(c)_jk=c_j-k. The formal Fourier matrix is F_jk=frac1sqrt n χ(-jk), χ(a)=exp (frac2π i an). The standard character is not left abstract: a checked theorem identifies these entries with frac1sq","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"Strictness below saturation. If t<u and Φ(t) is below the maximum, at least one coordinate is unsaturated at t. That coordinate strictly increases; every other coordinate is nondecreasing. Hence Φ(t)<Φ(u). This proves uniqueness of the scale and, by reciprocation, the exact manuscript statement ∃!λ>0: ∑_e s_emin {m_e,fracc_eλ s_e}=τ when 0<τ<∑_es_e","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"Let R be the set of packets reachable from α, including α, and set X_R=⊕_γ∈ RX_γ. No active edge leaves R, so AX_R⊆ X_R. The restriction of A to the finite dimensional space X_R is injective. It is therefore onto X_R. Hence A^-1X_R⊆ X_R. Since β∉ R, the stated block of A^-1 vanishes.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":4},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-75c8e2436af69090","tex":"\\prod_{i=1}^{j-1}","aliases":"∏_i=1^j-1 prod","display":"\\prod_{i=1}^{j-1}","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3c3bdaa5a997770c","tex":"\\prod_{i=j+1}^m","aliases":"∏_i=j+1^m prod","display":"\\prod_{i=j+1}^m","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2e717462c0866d39","tex":"\\prod_{i=j+1}^{m}","aliases":"∏_i=j+1^m prod","display":"\\prod_{i=j+1}^{m}","kind":"use","excerpt":"ntheorem[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i )","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-0ace3cb3f12efd36","tex":"\\sum_{(b,a)\\in E_A}","aliases":"∑_(b,a)∈ E_A sum in","display":"\\sum_{(b,a)\\in E_A}","kind":"use","excerpt":"Reconstruction and action. The checked finite sum is A=∑_(b,a)∈ E_A A_ba. The proof inserts the complete projector sum on both sides and removes only zero summands. Applying this identity to an arbitrary module gives exact action. A separate finite-matrix theorem establishes AZ=∑_(b,a)∈ E_AA_baZ for every rectangular batch of right-hand sides Z.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"Exact packet semantics. textitSource items 2.3, 2.5, and 2.6 Let R be a ring, I a finite index type, and (P_a)_a∈ I a family satisfying P_bP_a=P_a · b=a, 0 · b≠ a, ∑_aP_a=1. Write A_ba=P_bAP_a and let E_A={(b,a):A_ba≠0}. These are exact algebraic zeros, not values discarded by a floating-point threshold.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7bfc5a7087fd58a2","tex":"\\sum_{b=0}^B","aliases":"∑_b=0^B sum","display":"\\sum_{b=0}^B","kind":"use","excerpt":"Counted candidates. One full convolution table examines indices (b,k), 0≤ b≤ B,quad0≤ k≤ b. These form a concrete finite dependent-sum type. Its cardinality is proved to be ∑_b=0^B(b+1)≤(B+1)^2. For a nonempty finite tree, the number of child merges plus one equals the node count N. This identity is proved for arbitrary finite branching.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2046ba257a075646","tex":"\\sum_{b\\ne a}","aliases":"∑_b≠ a sum ne","display":"\\sum_{b\\ne a}","kind":"use","excerpt":"esidual certificates for supplied packets textitSource item 9.2; module DiscoveryCertificate Let (P_a) be a complete orthogonal projector system on a finite complex coordinate space. The projectors need not be coordinate masks. For an actual matrix A, define Π(A)=∑_aP_aAP_a, E=A-Π(A). Reconstruction gives the exact off-diagonal residual E=∑_b≠ aP_b","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3004d1c3c6d1b821","tex":"\\sum_c","aliases":"∑_c sum","display":"\\sum_c","kind":"use","excerpt":"Composition. Inserting the projector identity between factors yields (AB)_ba=∑_c A_bcB_ca. If the left side is nonzero, at least one intermediate c has both A_bc≠0 and B_ca≠0. Consequently supp(AB)⊆supp(A)∘supp(B). No equality of supports is asserted.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3974edc19cef25f4","tex":"\\sum_{c:\\,A_{bc}\\ne0}","aliases":"∑_c: A_bc≠0 sum ne","display":"\\sum_{c:\\,A_{bc}\\ne0}","kind":"use","excerpt":"Define the active contribution sum recursively by F_0(b,a) · =P_a · b=a, 0 · b≠ a, F_n+1(b,a) · =∑_c: A_bc≠0A_bcF_n(c,a). The checked identity is P_bA^nP_a=F_n(b,a) (n≥0). Its proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence t","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-ec1616068a6ee172","tex":"\\sum_e","aliases":"∑_e sum","display":"\\sum_e","kind":"use","excerpt":"For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tig","count":44},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6d1546afc3ef11e5","tex":"\\sum_{e\\in E}","aliases":"∑_e∈ E sum in","display":"\\sum_{e\\in E}","kind":"use","excerpt":"[Packet perturbation certificate] Let A= ∑_e∈ EA_e, widehat A = ∑_e∈ E(A_e+E_e), where edge e starts at packet α(e), and assume A_e=A_eP_α(e), E_e=E_eP_α(e). Then norm(widehat A-A)x ≤ ∑_e∈ E normE_e normP_α(e)x. If normE_e ≤ ε_enormA_e, then norm(widehat A-A)x ≤ ∑_e∈ E ε_enormA_e normP_α(e)x.","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f7155be421cc3965","tex":"\\sum_f","aliases":"∑_f sum","display":"\\sum_f","kind":"use","excerpt":"[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Let v be the new optimum of the edited subtree, and let j be the edited child slot at one frame. The update performed at that frame is step(v)= min{K, σ+ ∑_i v, · i=j, widehat F_i, · i≠ j. }. This result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values widehat F_i.","count":9},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-5bed34448f0dc86f","tex":"\\sum_{i=1}^{k+1}","aliases":"∑_i=1^k+1 sum","display":"\\sum_{i=1}^{k+1}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-995c1e5022b02f2d","tex":"\\sum_{i>k}","aliases":"∑_i>k sum","display":"\\sum_{i>k}","kind":"use","excerpt":"For any nonnegative decreasing sequence a_i, weights with these properties satisfy ∑_i a_iw_i≥∑_i>ka_i. This finite weighted inequality is itself proved. For k<d, use μ=a_k+1 and compare the summands to tail weights v_i=0 for i≤ k and v_i=1 otherwise: a_iv_i+μ(w_i-v_i)≤ a_iw_i. For a retained coordinate the inequality uses a_i≥μ and w_i≥0. For a ta","count":3},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Shared operator families. For reuse weights r_j, the local keep model can aggregate setup and per-operator costs: κ_v=κ_v^setup+∑_jr_jκ_v^(j), ϵ_v=∑_jr_jϵ_v^(j). Lean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including Q and R. The inte","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b845f4dcb81f67c5","tex":"\\sum_{j=0}^d","aliases":"∑_j=0^d sum","display":"\\sum_{j=0}^d","kind":"use","excerpt":"[Polynomial support] If p(z)=∑_j=0^dc_jz^j, then supp(p(A)) is contained in the union of the support relations of A^j for 0≤ j≤ d.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-cfe7b3387ef9124c","tex":"\\sum_{j=0}^{d}","aliases":"∑_j=0^d sum","display":"\\sum_{j=0}^{d}","kind":"use","excerpt":"Polynomial support. For the actual finite sum p(A)=∑_j=0^dc_jA^j, the checked support lies in the union of the support relations of the powers. The formalization treats the j=0 contribution explicitly, so the constant term is not accidentally omitted.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-24dfb51124d8ae91","tex":"\\sum_{j=1}^m","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^m","kind":"use","excerpt":"[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.","count":9},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-55bfd4be1daf726b","tex":"\\sum_{j=1}^{m}","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^{m}","kind":"use","excerpt":"ntheorem[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i )","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-93f65e678e57f837","tex":"\\sum_{j=1}^{q}","aliases":"∑_j=1^q sum","display":"\\sum_{j=1}^{q}","kind":"use","excerpt":"Frobenius optimality from kernel weights. textitModules KernelFrame and FrobeniusOptimality For an arbitrary competitor B with rank B≤ k, construct an orthonormal matrix Q spanning its actual kernel. Then BQ=0, Q^*Q=I, q=dimker B=d-rank B. Define the row weights w_i=∑_j=1^q|Q_ij|^2. The proof derives 0≤ w_i≤1, ∑_iw_i=q≥max(d-k,0). The upper bound u","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2f357ff52e6c2d2b","tex":"\\sum_{j>k}","aliases":"∑_j>k sum","display":"\\sum_{j>k}","kind":"use","excerpt":"[Eckart Young Mirsky theorem] If M has singular values σ_1≥σ_2≥⋯, pad this finite sequence with zeros. For each integer k≥0, its truncated singular value decomposition M_k is a best rank-at-most-k approximation. It satisfies normM-M_k_2=σ_k+1 and normM-M_k_rm F = ( ∑_j>kσ_j^2 )^1/2.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-c98e9b3ccc8ca2c6","tex":"\\sum_{k=0}^{j-1}","aliases":"∑_k=0^j-1 sum","display":"\\sum_{k=0}^{j-1}","kind":"use","excerpt":"Use the telescoping identity A^j-D^j = ∑_k=0^j-1 A^j-1-kKD^k. Each term has norm at most M^j-1normK. The vector D^jv lies in the direct sum of local Krylov spaces.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1f3736c162765e82","tex":"\\sum_{k=1}^{n}","aliases":"∑_k=1^n sum","display":"\\sum_{k=1}^{n}","kind":"use","excerpt":"The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3d357445fed9cde3","tex":"\\sum_{k=1}^{w}","aliases":"∑_k=1^w sum","display":"\\sum_{k=1}^{w}","kind":"use","excerpt":"The diagonal contributes p blocks. At offset k, where 1≤ k≤ w, the upper and lower block diagonals contribute 2(p-k) blocks. Therefore p+2∑_k=1^w(p-k) = p(2w+1)-w(w+1). The cost and storage follow from [source reference].","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2753495994955ca2","tex":"\\sum_{r=0}^{n-1}","aliases":"∑_r=0^n-1 sum","display":"\\sum_{r=0}^{n-1}","kind":"use","excerpt":"For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7ecb3223d556bc7c","tex":"\\sum_{u\\text{ split}}","aliases":"∑_u split sum text","display":"\\sum_{u\\text{ split}}","kind":"use","excerpt":"The cost and error of a pruning L are cost( L) = ∑_v∈ Lκ_v + ∑_u splitσ_u and err( L)=∑_v∈ Lϵ_v. The model can include arithmetic, storage, launch, transfer, and precision costs inside κ_v. It remains exact only when the declared totals are additive.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-582da013421a516b","tex":"\\sum_{v\\in\\Lcal}","aliases":"∑_v∈Lcal ∑_v∈ L sum in Lcal sum in mathcal","display":"\\sum_{v\\in{\\mathcal L}}","kind":"use","excerpt":"The cost and error of a pruning L are cost( L) = ∑_v∈ Lκ_v + ∑_u splitσ_u and err( L)=∑_v∈ Lϵ_v. The model can include arithmetic, storage, launch, transfer, and precision costs inside κ_v. It remains exact only when the declared totals are additive.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-2b95868d24f2bd48","tex":"\\sum_w","aliases":"∑_w sum","display":"\\sum_w","kind":"use","excerpt":"Strict half-gap stability. At a solved internal node write K=κ_v+λϵ_v, S=σ_v+∑_wF_λ(w), g=|K-S|. If the updated scores satisfy |K'-K|<g/2, |S'-S|<g/2, Lean proves K'≤ S' Longleftrightarrow K≤ S. Applying this condition recursively at every internal node proves equality of the entire returned cuts before and after perturbation, not only equality of ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1c79b8a11020f120","tex":"\\sum_{w\\in\\mathrm{child}(v)}","aliases":"∑_w∈child(v) sum in mathrm","display":"\\sum_{w\\in\\mathrm{child}(v)}","kind":"use","excerpt":"Recurrence. For terminal nodes, F_λ(v)=κ_v+λϵ_v. For internal nodes, F_λ(v)=min{κ_v+λϵ_v, σ_v+∑_w∈child(v)F_λ(w)}. The checked result has two parts. First, F_λ(v)≤ J_λ(L) for every legal cut of the subtree. Second, the constructed function codecompile returns a legal cut with objective exactly F_λ(v). Thus the minimum is attained, not merely bounde","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-a38ac3f06fcc5327","tex":"\\sum_{w\\in\\operatorname{child}(v)}","aliases":"∑_w∈child(v) sum in operatorname","display":"\\sum_{w\\in\\operatorname{child}(v)}","kind":"use","excerpt":"[Exact Lagrangian packet compiler] Fix λ≥0 and define J_λ( L) = cost( L) + λerr( L). For every node v, set K_λ(v) = κ_v+λϵ_v and S_λ(v) = σ_v + ∑_w∈child(v)F_λ(w). For a terminal node, set F_λ(v)=K_λ(v). For an internal node, the recurrence F_λ(v) = min{K_λ(v),S_λ(v)} computes the exact minimum Lagrangian objective over every admissible pruning of ","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"use","excerpt":"Insert the identity on both sides of A: A = (∑_β P_β) A (∑_α P_α) = ∑_α,βP_β AP_α. The zero blocks can be removed. Applying the identity to Z gives the second formula.","count":8},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","excerpt":"Insert the identity on both sides of A: A = (∑_β P_β) A (∑_α P_α) = ∑_α,βP_β AP_α. The zero blocks can be removed. Applying the identity to Z gives the second formula.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-97bc4b69d2c44d19","tex":"\\sum_{\\alpha\\to\\beta}","aliases":"∑_α→β sum alpha to beta","display":"\\sum_{\\alpha\\to\\beta}","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-1fa499eb28c28965","tex":"\\sum_{\\alpha\\neq\\beta}","aliases":"∑_α≠β sum alpha neq beta","display":"\\sum_{\\alpha\\neq\\beta}","kind":"use","excerpt":"[Exact block discovery certificate] For an orthogonal packet system (P_α), define the block diagonal compression Π_ P(A) = ∑_α P_α AP_α. Then normA-Π_ P(A)_rm F^2 = ∑_α≠β normP_β AP_α_rm F^2. In operator norm, normA-Π_ P(A)_2 ≤ ∑_α≠β normP_β AP_α_2. Thus any proposed packet discovery has an exact finite residual certificate after its projectors are","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-dd5a8bdc2b931502","tex":"\\sum_\\beta","aliases":"∑_β sum beta","display":"\\sum_\\beta","kind":"use","excerpt":"Insert the identity on both sides of A: A = (∑_β P_β) A (∑_α P_α) = ∑_α,βP_β AP_α. The zero blocks can be removed. Applying the identity to Z gives the second formula.","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-22bd389693d3f179","tex":"\\sum_{\\ell=1}^{m}","aliases":"∑_ℓ=1^m sum ell","display":"\\sum_{\\ell=1}^{m}","kind":"use","excerpt":"The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"The checker compares the recurrence with complete enumeration for every budget from zero through twelve. Infeasible states remain +∞; they are not zero-cost states. headQuantization must be conservative Suppose real errors are 3/50 and 2/25, and one integer unit represents 1/10. Rounding down gives two zero weights, even though the true sum is 7/50","count":20},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"dicts possible support but not actual support. Replacing the inclusion supp(AB)⊆supp(A)∘supp(B) by equality would be wrong even in this four-dimensional model. headWhat an acyclic graph does guarantee If the longest strict directed path has h edges, then no term of K^h+1 can survive. Thus K^h+1=0. This is an upper bound on the nilpotence index, not","count":22},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"vpage32Safe integers can still be conservativeQUANTIZATION AND THE SIZE OF THE BUDGET TABLE Let one integer unit represent h=1/10 of certified error. For an individual nonnegative error e, choosing w=lceil e/hrceil ensures e≤ hw. Thus integer feasibility ∑ w≤ B implies the real bound ∑ e≤ hB. This implication is checked before applying the exact op","count":58},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"[Capacity threshold] Let a device have available memory M≥0 bytes and scalar size s>0, and let n≥0. The matrix representation alone can fit only when n≤sqrtM/s. The equal packet BDLR representation alone fits when n(b+2r)s≤ M.","count":92},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","excerpt":"Exact and approximate packet Krylov spaces. textitSource items 7.2 and 7.3 Write K_m(A,v)=span{v,Av,…,A^m-1v}. For a complete finite projector family P_i, suppose A commutes with every P_i. Let A_i=P_iAP_i. Lean proves the actual orbit identity A^jv=∑_i A_i^jP_iv and therefore K_m(A,v)⊆ ⊕_i K_m(A_i,P_iv). The case j=0 uses ∑_iP_i=I. The zeroth loca","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-9b591bcc37a61369","tex":"\\bigoplus_\\alpha","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_\\alpha","kind":"use","excerpt":"[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).","count":2},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-7f618b24b380de7c","tex":"\\bigoplus_{\\gamma\\in R}","aliases":"⊕_γ∈ R bigoplus gamma in","display":"\\bigoplus_{\\gamma\\in R}","kind":"use","excerpt":"Let R be the set of packets reachable from α, including α, and set X_R=⊕_γ∈ RX_γ. No active edge leaves R, so AX_R⊆ X_R. The restriction of A to the finite dimensional space X_R is injective. It is therefore onto X_R. Hence A^-1X_R⊆ X_R. Since β∉ R, the stated block of A^-1 vanishes.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"use","excerpt":"For exact-weight tables F,G, min-plus convolution is (F⋆ G)(b)=min_0≤ k≤ b{F(k)+G(b-k)}, with an infeasible sum whenever either operand is infeasible. Its correctness theorem identifies every finite answer with an actual feasible pair of component weights and costs. Repeated convolution handles arbitrary finite lists of components.","count":1},{"work":"paper-viii","url":"notation/paper-viii.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"p=2 · 1 · 2 · 2^r","count":2012},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"notation-table","excerpt":"H · positive row step · Euler degree step 4H","count":231},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"notation-table","excerpt":"j · address within a section · 0,1,2,3","count":504},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"notation-table","excerpt":"k · number of differences · order k, independent of n","count":473},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"notation-table","excerpt":"llll Prime branch · Allowed m · q · Row step for the original shift","count":440},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"notation-table","excerpt":"p odd · mmid p-1 · p-1 · dfracp-1mp^r-1","count":92},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"notation-table","excerpt":"n · packed row index · Euler degree 4n","count":722},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"notation-table","excerpt":"p=2 · 1 · 2 · 2^r","count":1233},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-da8a2616906ad75b","tex":"p^{r-1}","aliases":"p^r-1","display":"p^{r-1}","kind":"notation-table","excerpt":"p odd · mmid p-1 · p-1 · dfracp-1mp^r-1","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"notation-table","excerpt":"llll Prime branch · Allowed m · q · Row step for the original shift","count":325},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"notation-table","excerpt":"p=2 · 1 · 2 · 2^r","count":573},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":971},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":215},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":541},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":222},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":287},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-40591b7fff4431b2","tex":"C_{N,j}","aliases":"C_N,j","display":"C_{N,j}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":48},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":135},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":398},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7a3a2aba376670a9","tex":"D^N","aliases":"D^N","display":"D^N","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":120},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a9f8dc796c8ef36","tex":"E_{2m-b}","aliases":"E_2m-b","display":"E_{2m-b}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a603588606ce810","tex":"E_b","aliases":"E_b","display":"E_b","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-69a8a46b93a29118","tex":"E_d^{(N,j)}","aliases":"E_d^(N,j)","display":"E_d^{(N,j)}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-be288f5d115169ac","tex":"E_{Nn}^{(N,j)}","aliases":"E_Nn^(N,j)","display":"E_{Nn}^{(N,j)}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":160},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b4a55684394b7979","tex":"f_d","aliases":"f_d","display":"f_d","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":22},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":142},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f0bbb437f18bc2cd","tex":"\\mathcal H_p","aliases":"H_p mathcal","display":"\\mathcal H_p","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":388},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c66f081d0dbf6c6c","tex":"L_\\lambda","aliases":"L_λ lambda","display":"L_\\lambda","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-39c11bec6df251b7","tex":"L_\\lambda^k","aliases":"L_λ^k lambda","display":"L_\\lambda^k","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed ","count":304},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed ","count":38},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":412},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N mathbb","display":"{\\mathbb N}","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":50},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0f7264e76f55d278","tex":"\\mathrm{odd}","aliases":"odd mathrm","display":"\\mathrm{odd}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-80a406874da00307","tex":"p^{2m}","aliases":"p^2m","display":"p^{2m}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed ","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fbe5654b944a7373","tex":"p^a","aliases":"p^a","display":"p^a","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":34},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-704b220b0886246a","tex":"p^L","aliases":"p^L","display":"p^L","kind":"definition-passage","excerpt":"put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressi","count":76},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":16},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cd98a9c47ecfa404","tex":"\\mathbb Q_p","aliases":"Q_p mathbb","display":"\\mathbb Q_p","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":95},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6b8319ff964bd555","tex":"S_{2(n-m)}","aliases":"S_2(n-m)","display":"S_{2(n-m)}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-508e9a069e81ec8b","tex":"S_d","aliases":"S_d","display":"S_d","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-57c446e8e6d068a3","tex":"S_m","aliases":"S_m","display":"S_m","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2d39429b9e78430d","tex":"\\operatorname{sech}","aliases":"sech operatorname","display":"\\operatorname{sech}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":32},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c2cd755c944ccda0","tex":"\\operatorname{sech}^2","aliases":"sech^2 operatorname","display":"\\operatorname{sech}^2","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-21670bd4e275805e","tex":"\\sinh","aliases":"sinh","display":"\\sinh","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":32},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed ","count":182},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":370},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b020d6ed8cb1fa22","tex":"t^{2m}","aliases":"t^2m","display":"t^{2m}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f544e1fddefb59a0","tex":"t^d","aliases":"t^d","display":"t^d","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":18},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-50c3c8d08b407ee6","tex":"t^{Nn}","aliases":"t^Nn","display":"t^{Nn}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fc681522104c7ab2","tex":"T^{\\mathrm{odd}}_{p,M}","aliases":"T^odd_p,M mathrm","display":"T^{\\mathrm{odd}}_{p,M}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-14cf325be6479cb0","tex":"T_M","aliases":"T_M","display":"T_M","kind":"definition-passage","excerpt":"n] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":133},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"d Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ","count":238},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-03cfa39a8fd5efa1","tex":"U_{N,j}","aliases":"U_N,j","display":"U_{N,j}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":340},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":42},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":389},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z mathbb","display":"{\\mathbb Z}","kind":"definition-passage","excerpt":"put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressi","count":164},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":162},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":115},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a87f7202de83de58","tex":"z^{2m}","aliases":"z^2m","display":"z^{2m}","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b910972bdb78c268","tex":"\\Z_p","aliases":"Z_p Z mathbb","display":"{\\mathbb Z}_p","kind":"definition-passage","excerpt":"put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressi","count":54},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8723c894d5150ea2","tex":"\\mathbb Z_p","aliases":"Z_p mathbb","display":"\\mathbb Z_p","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":33},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":43},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-24182df1bf8329b9","tex":"\\beta_0","aliases":"β_0 beta","display":"\\beta_0","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d6de2171610272b5","tex":"\\beta_m","aliases":"β_m beta","display":"\\beta_m","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":32},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2771b61f3f3f5974","tex":"\\gamma_0","aliases":"γ_0 gamma","display":"\\gamma_0","kind":"definition-passage","excerpt":"-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be od","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-27c5ee666d44751e","tex":"\\gamma_m","aliases":"γ_m gamma","display":"\\gamma_m","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":153},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-608581f0be0c4b00","tex":"\\Delta_h","aliases":"Δ_h Delta","display":"\\Delta_h","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-94a8068a3d828e7e","tex":"\\Delta_h^0","aliases":"Δ_h^0 Delta","display":"\\Delta_h^0","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9e333eeae38a06fd","tex":"\\Delta_h^k","aliases":"Δ_h^k Delta","display":"\\Delta_h^k","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e5e835b6840b45f1","tex":"\\Delta_h^{k+1}","aliases":"Δ_h^k+1 Delta","display":"\\Delta_h^{k+1}","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":27},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"definition-passage","excerpt":"d Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ","count":23},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-62d0d50e0643749d","tex":"\\nu_p","aliases":"ν_p nu","display":"\\nu_p","kind":"definition-passage","excerpt":"d Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":452},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":272},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9068784c641ac3f6","tex":"\\sum_{a=0}^{p-1}","aliases":"∑_a=0^p-1 sum","display":"\\sum_{a=0}^{p-1}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-67f6fb951c2ce7df","tex":"\\sum_{b=0}^{2m}","aliases":"∑_b=0^2m sum","display":"\\sum_{b=0}^{2m}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1a2774d48ee2d8aa","tex":"\\sum_{d\\ge0}","aliases":"∑_d≥0 sum ge","display":"\\sum_{d\\ge0}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b7f3762e9cc7d08b","tex":"\\sum_{d\\geq0}","aliases":"∑_d≥0 sum geq","display":"\\sum_{d\\geq0}","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4dc4a300871f9e3e","tex":"\\sum_{m=0}^{\\min(M,n)}","aliases":"∑_m=0^min(M,n) sum min","display":"\\sum_{m=0}^{\\min(M,n)}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd; primality is needed ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d328b95fa488fa30","tex":"\\sum_{m\\ge0}","aliases":"∑_m≥0 sum ge","display":"\\sum_{m\\ge0}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":20},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":22},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":21},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":91},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":219},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":229},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"introduction-passage","excerpt":"Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a8b9f34806c25cd4","tex":"\\mathbf1_n","aliases":"1_n mathbf","display":"\\mathbf1_n","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"introduction-passage","excerpt":"Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-895b4c69e53e4df6","tex":"A_x","aliases":"A_x","display":"A_x","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":25},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.","count":197},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"introduction-passage","excerpt":"Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eaef9ee8f33eeef5","tex":"B_4^s","aliases":"B_4^s","display":"B_4^s","kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":16},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"introduction-passage","excerpt":"Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-57c8f1ae1da23ae1","tex":"B_r","aliases":"B_r","display":"B_r","kind":"introduction-passage","excerpt":"Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.","count":225},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-823f036b3e8ddecf","tex":"C^N","aliases":"C^N","display":"C^N","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":21},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-746a0fca63346846","tex":"C_r","aliases":"C_r","display":"C_r","kind":"introduction-passage","excerpt":"Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"introduction-passage","excerpt":"The central change in this edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2","count":29},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e73c7731b76eea40","tex":"\\cosh^2","aliases":"cosh^2 cosh","display":"\\cosh^2","kind":"introduction-passage","excerpt":"The central change in this edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dabe11e0627f83c2","tex":"\\cosh_1","aliases":"cosh_1 cosh","display":"\\cosh_1","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-41d310366424af4d","tex":"\\cosh_a","aliases":"cosh_a cosh","display":"\\cosh_a","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-db28b207b9c1b29b","tex":"D^a","aliases":"D^a","display":"D^a","kind":"introduction-passage","excerpt":"packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum an","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3b3443a3b74189af","tex":"D^d","aliases":"D^d","display":"D^d","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-acc501056e7d81d5","tex":"D_T","aliases":"D_T","display":"D_T","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6f78b62eb28153fc","tex":"D_Y","aliases":"D_Y","display":"D_Y","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":13},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":219},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4293330aa39421ea","tex":"E^{(N,j)}_{Nn}","aliases":"E^(N,j)_Nn","display":"E^{(N,j)}_{Nn}","kind":"introduction-passage","excerpt":"Normalization is also an explicit implication. Using the existing definition E^(N,j)_Nn=j!C_N,j(n), the source proves that multiplication by j! commutes with every iterated difference. If the unnormalized difference has precision A, the normalized difference has precision A-v_p(j!). Precision is defined as x=0 hboxor v_p(x)≥ A, so zero and negative","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-81315b920e13e316","tex":"e^{-at}","aliases":"e^-at","display":"e^{-at}","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4a1d25b2a3b14026","tex":"e^{at}","aliases":"e^at","display":"e^{at}","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1fe69fafa50db3c5","tex":"e^t","aliases":"e^t","display":"e^t","kind":"introduction-passage","excerpt":"Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dfb98ad8aedbb4dd","tex":"E_h","aliases":"E_h","display":"E_h","kind":"introduction-passage","excerpt":"Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"introduction-passage","excerpt":"ntCheckpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical; the final core records their resolution. Scope and final statement. Let p be a prime, let m>0 divide p-1, put N=mp and h=(p-1)/m, and let 0≤ j<N. The original rational packet is defined by ∑_q","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3104ecf0e1adeade","tex":"e_{n+h}^{(m,p)}","aliases":"e_n+h^(m,p)","display":"e_{n+h}^{(m,p)}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c910c4f72facf773","tex":"e_{n+uh}^{(m,p)}","aliases":"e_n+uh^(m,p)","display":"e_{n+uh}^{(m,p)}","kind":"introduction-passage","excerpt":"The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f446188894e8b7ca","tex":"e_n^{(m,p)}","aliases":"e_n^(m,p)","display":"e_n^{(m,p)}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-385b796c2c5e99d4","tex":"e_q^{(m,p)}","aliases":"e_q^(m,p)","display":"e_q^{(m,p)}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-14abf8cd4095aad2","tex":"\\exp_a","aliases":"exp_a exp","display":"\\exp_a","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":336},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dd2b9b85d5dfefc2","tex":"F_e","aliases":"F_e","display":"F_e","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-31d7232e6fa917d3","tex":"F_i","aliases":"F_i","display":"F_i","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"introduction-passage","excerpt":"The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includ","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c57cafb013c6775f","tex":"F_n","aliases":"F_n","display":"F_n","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-97bd1abfe72629ee","tex":"F_{N(k+1)+j}","aliases":"F_N(k+1)+j","display":"F_{N(k+1)+j}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ec86e738238d91a6","tex":"F_{N(k+2)+j}","aliases":"F_N(k+2)+j","display":"F_{N(k+2)+j}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7817222f46df5e65","tex":"F_{n+1}","aliases":"F_n+1","display":"F_{n+1}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-80004cdcfd3891b8","tex":"F_{n+2}","aliases":"F_n+2","display":"F_{n+2}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ddf7d50338d5ead5","tex":"f_{n+d}","aliases":"f_n+d","display":"f_{n+d}","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b28b2bb412883c1c","tex":"F_{Nk+j}","aliases":"F_Nk+j","display":"F_{Nk+j}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6a27312454be2902","tex":"F_p","aliases":"F_p","display":"F_p","kind":"introduction-passage","excerpt":"rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f7110079fe13a85a","tex":"F_r","aliases":"F_r","display":"F_r","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f7499e87191c71a6","tex":"f_r","aliases":"f_r","display":"f_r","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cc7582cb4d0dc2f3","tex":"\\FF","aliases":"FF F mathbb","display":"{\\mathbb F}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":24},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5b4740083376d053","tex":"\\FF_p","aliases":"FF_p F_p FF mathbb","display":"{\\mathbb F}_p","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":14},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c03543fb3fd9a9c3","tex":"\\FF_p^+","aliases":"FF_p^+ F_p^+ FF mathbb","display":"{\\mathbb F}_p^+","kind":"introduction-passage","excerpt":"Let p be prime. Write F_p^+ for the additive group of the finite field.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-555687cfad37a088","tex":"\\FF_p^r","aliases":"FF_p^r F_p^r FF mathbb","display":"{\\mathbb F}_p^r","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":113},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bfa087a98ed68edf","tex":"g_{n-r}","aliases":"g_n-r","display":"g_{n-r}","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"introduction-passage","excerpt":"edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2,","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bf26706b1ed38c02","tex":"\\mathcal H_{\\mathrm{int}}","aliases":"H_int mathcal mathrm","display":"\\mathcal H_{\\mathrm{int}}","kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0d2a40129a3b4c8b","tex":"H_p","aliases":"H_p","display":"H_p","kind":"introduction-passage","excerpt":"The central change in this edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5e073754a563e30e","tex":"\\hbox","aliases":"hbox","display":null,"kind":"introduction-passage","excerpt":"Normalization is also an explicit implication. Using the existing definition E^(N,j)_Nn=j!C_N,j(n), the source proves that multiplication by j! commutes with every iterated difference. If the unnormalized difference has precision A, the normalized difference has precision A-v_p(j!). Precision is defined as x=0 hboxor v_p(x)≥ A, so zero and negative","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"introduction-passage","excerpt":"Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients ","count":77},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":133},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"introduction-passage","excerpt":"s over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K)","count":14},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-55466b76f7f0c090","tex":"\\mathrm{int}","aliases":"int mathrm","display":"\\mathrm{int}","kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"introduction-passage","excerpt":"The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is","count":180},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3fa813ff3c6bd07a","tex":"J^{-1}","aliases":"J^-1","display":"J^{-1}","kind":"introduction-passage","excerpt":"bseteq K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative Dψ(t_0)=-J^-1K. Consequently it is differentiable on U. The map is def","count":30},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fe61bfb695992cf7","tex":"J_F","aliases":"J_F","display":"J_F","kind":"introduction-passage","excerpt":"rrow Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) deno","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8af4e249618e23ea","tex":"J_x","aliases":"J_x","display":"J_x","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":40},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"introduction-passage","excerpt":"The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is","count":133},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-53283f2f6027fba4","tex":"\\mathbb K","aliases":"K mathbb","display":"\\mathbb K","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":23},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-44917510cdb5a170","tex":"\\mathbb K^e","aliases":"K^e mathbb","display":"\\mathbb K^e","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2cf939730979c2cb","tex":"\\mathbb K^k","aliases":"K^k mathbb","display":"\\mathbb K^k","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-173833828c3529a3","tex":"K_2","aliases":"K_2","display":"K_2","kind":"introduction-passage","excerpt":"edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2,","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b39e465f5dcb17d2","tex":"K_p","aliases":"K_p","display":"K_p","kind":"introduction-passage","excerpt":"The central change in this edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"introduction-passage","excerpt":"The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is","count":29},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":42},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c91dabfc8b700b3b","tex":"L_N","aliases":"L_N","display":"L_N","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2b80cbb9f17c415c","tex":"\\mathcal L_v","aliases":"L_v mathcal","display":"\\mathcal L_v","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":44},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"introduction-passage","excerpt":"Local analytic graph theorem. There exist a map ψ: K^k⟶ K^e and open sets U⊆ K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative ","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a603e6b6eedcb75f","tex":"M_e","aliases":"M_e","display":"M_e","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bca51ab689269ed0","tex":"M_{e\\times k}","aliases":"M_e× k times","display":"M_{e\\times k}","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-06e9b4e88356c8ed","tex":"M_{r,d}","aliases":"M_r,d","display":"M_{r,d}","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":20},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":22},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-91023a802e0888a1","tex":"o_x","aliases":"o_x","display":"o_x","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":16},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-712a4047f09e3ce3","tex":"\\overline J","aliases":"overline J overline","display":"\\overline J","kind":"introduction-passage","excerpt":"tbfEditorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and wr","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bee338e0d8013678","tex":"\\overline J_F","aliases":"overline J_F overline","display":"\\overline J_F","kind":"introduction-passage","excerpt":"tbfEditorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and wr","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9d8a1921f2a37e0a","tex":"\\overline M","aliases":"overline M overline","display":"\\overline M","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9d1a16aa753ffb59","tex":"p^{(p-1)x+j}","aliases":"p^(p-1)x+j","display":"p^{(p-1)x+j}","kind":"introduction-passage","excerpt":"The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-325ff65361d0b480","tex":"p^{\\min(k,mn)}","aliases":"p^min(k,mn) min","display":"p^{\\min(k,mn)}","kind":"introduction-passage","excerpt":"[All-address differences on the divisor branch] Let p be prime, m>0, mmid p-1, N=mp, and 0≤ j<N. Write h=(p-1)/m. For every u,k,n∈ N, U_N,j(n) · ∈ Z_p, Δ_uh^kU_N,j(n) · ∈ p^min(k,mn) Z_p, v_p(Δ_uh^kC_N,j(n)) · ≥min(k,mn)-v_p(j!). The factorial correction is part of the normalized statement, including addresses j≥ p.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-16055e0aaa2fe4ce","tex":"p^x","aliases":"p^x","display":"p^x","kind":"introduction-passage","excerpt":"The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":140},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q mathbb","display":"{\\mathbb Q}","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":35},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-04992c0930a894b0","tex":"q^L","aliases":"q^L","display":"q^L","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":40},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-50f270102a088631","tex":"\\Q_2","aliases":"Q_2 Q mathbb","display":"{\\mathbb Q}_2","kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8205838caa474d98","tex":"\\Q_p","aliases":"Q_p Q mathbb","display":"{\\mathbb Q}_p","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":19},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ddc9b1c9f42d7650","tex":"\\Qp","aliases":"Qp Q_p mathbb","display":"{\\mathbb Q_p}","kind":"introduction-passage","excerpt":"Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets t","count":21},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"introduction-passage","excerpt":"The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is","count":220},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c2d59bfcf91e5807","tex":"R_q","aliases":"R_q","display":"R_q","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ac70c1f93e7ca709","tex":"R_q^d","aliases":"R_q^d","display":"R_q^d","kind":"introduction-passage","excerpt":"et ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the cokernel below is presented directly","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-93731b161f309dff","tex":"R_q^e","aliases":"R_q^e","display":"R_q^e","kind":"introduction-passage","excerpt":"F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d; the co","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f0e893f7b0a8cb70","tex":"r_x","aliases":"r_x","display":"r_x","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients ","count":339},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8c0785670c74cfb3","tex":"S_n","aliases":"S_n","display":"S_n","kind":"introduction-passage","excerpt":"rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1e6bdd9702222c4b","tex":"\\sech","aliases":"sech operatorname","display":"{\\operatorname{sech}}","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp; scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7e1495d390f06be4","tex":"\\sinh^2","aliases":"sinh^2 sinh","display":"\\sinh^2","kind":"introduction-passage","excerpt":"edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2,","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"introduction-passage","excerpt":"edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2,","count":19},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"introduction-passage","excerpt":"rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":55},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b6d1c35d86f0d39d","tex":"T_k","aliases":"T_k","display":"T_k","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9acac70418a67a9d","tex":"U^p","aliases":"U^p","display":"U^p","kind":"introduction-passage","excerpt":"The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includ","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is","count":59},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":158},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","excerpt":"The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is","count":23},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-47c52f415949a6cb","tex":"\\widetilde C","aliases":"widetilde C widetilde","display":"\\widetilde C","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2d0578330dc67f46","tex":"\\widetilde C_j","aliases":"widetilde C_j widetilde","display":"\\widetilde C_j","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":103},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2b07f815019444f3","tex":"X^a","aliases":"X^a","display":"X^a","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f98d18a8cc9bdad9","tex":"X^d","aliases":"X^d","display":"X^d","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":23},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-73cc2f41cbac9cf2","tex":"X_d","aliases":"X_d","display":"X_d","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":20},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-72c04885d210371b","tex":"x_L","aliases":"x_L","display":"x_L","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":34},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":52},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":140},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":33},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7ef58ddaab7b5984","tex":"Y_1","aliases":"Y_1","display":"Y_1","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3dd104d33d62c251","tex":"Y_e","aliases":"Y_e","display":"Y_e","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-295aec691039ab9f","tex":"\\mathbb Z^d","aliases":"Z^d mathbb","display":"\\mathbb Z^d","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":13},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-99f13701097c17ee","tex":"\\mathbb Z^e","aliases":"Z^e mathbb","display":"\\mathbb Z^e","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-78a80f694d80e6e7","tex":"\\Z_2","aliases":"Z_2 Z mathbb","display":"{\\mathbb Z}_2","kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":28},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-370c87df70534fa4","tex":"\\mathbb Z_2","aliases":"Z_2 mathbb","display":"\\mathbb Z_2","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":61},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e15eb38946d0671a","tex":"\\Zp","aliases":"Zp Z_p mathbb","display":"{\\mathbb Z_p}","kind":"introduction-passage","excerpt":"Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets t","count":66},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-591f18106678b658","tex":"\\Zp^d","aliases":"Zp^d Z_p^d Zp mathbb","display":"{\\mathbb Z_p}^d","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-df7860ea845cecca","tex":"\\Zp^r","aliases":"Zp^r Z_p^r Zp mathbb","display":"{\\mathbb Z_p}^r","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":45},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":54},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2bb168c166c79f1d","tex":"\\Delta_1","aliases":"Δ_1 Delta","display":"\\Delta_1","kind":"introduction-passage","excerpt":"Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients ","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0b8c3548e9b0a2a5","tex":"\\Delta_1^s","aliases":"Δ_1^s Delta","display":"\\Delta_1^s","kind":"introduction-passage","excerpt":"Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients ","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2f800f69539a7f67","tex":"\\delta_{n0}","aliases":"δ_n0 delta","display":"\\delta_{n0}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":13},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4448d5af8e047c6e","tex":"\\Delta_u","aliases":"Δ_u Delta","display":"\\Delta_u","kind":"introduction-passage","excerpt":"Write Δ_u f(n)=f(n+u)-f(n) and use Δ_u^k for its k-fold iterate. Euler coefficients indexed by ordinary degree and by packet row are distinguished explicitly. A packet displacement by u rows is a degree displacement by Nu in a packet of width N.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-76aa50424676d8cf","tex":"\\Delta_u^k","aliases":"Δ_u^k Delta","display":"\\Delta_u^k","kind":"introduction-passage","excerpt":"Write Δ_u f(n)=f(n+u)-f(n) and use Δ_u^k for its k-fold iterate. Euler coefficients indexed by ordinary degree and by packet row are distinguished explicitly. A packet displacement by u rows is a degree displacement by Nu in a packet of width N.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-30d91374b7719e5f","tex":"\\Delta_{uh}^{k}","aliases":"Δ_uh^k Delta","display":"\\Delta_{uh}^{k}","kind":"introduction-passage","excerpt":"[All-address differences on the divisor branch] Let p be prime, m>0, mmid p-1, N=mp, and 0≤ j<N. Write h=(p-1)/m. For every u,k,n∈ N, U_N,j(n) · ∈ Z_p, Δ_uh^kU_N,j(n) · ∈ p^min(k,mn) Z_p, v_p(Δ_uh^kC_N,j(n)) · ≥min(k,mn)-v_p(j!). The factorial correction is part of the normalized statement, including addresses j≥ p.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d96b536d679a03d2","tex":"\\lambda_J","aliases":"λ_J lambda","display":"\\lambda_J","kind":"introduction-passage","excerpt":"s over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K)","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6baeb03c4146c133","tex":"\\lambda_K","aliases":"λ_K lambda","display":"\\lambda_K","kind":"introduction-passage","excerpt":"ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":33},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f6c67cd7deda6b91","tex":"\\rho_L","aliases":"ρ_L rho","display":"\\rho_L","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6eb70cc25d1e53f2","tex":"\\rho_L^F","aliases":"ρ_L^F rho","display":"\\rho_L^F","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-037b39a7fd3c33a4","tex":"\\rho_q","aliases":"ρ_q rho","display":"\\rho_q","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":14},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-41c71b804cba2a03","tex":"\\chi_D","aliases":"χ_D chi","display":"\\chi_D","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"introduction-passage","excerpt":"Local analytic graph theorem. There exist a map ψ: K^k⟶ K^e and open sets U⊆ K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative ","count":24},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets t","count":60},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"introduction-passage","excerpt":"The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K); this is independent of the choice of its preimage because ker J is","count":67},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"introduction-passage","excerpt":"The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"introduction-passage","excerpt":"Local analytic graph theorem. There exist a map ψ: K^k⟶ K^e and open sets U⊆ K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ded5a9bc7d9bfb17","tex":"\\sum_{a=0}^{d}","aliases":"∑_a=0^d sum","display":"\\sum_{a=0}^{d}","kind":"introduction-passage","excerpt":"packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum an","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eb36ae12635f2435","tex":"\\sum_{a=0}^{d-1}","aliases":"∑_a=0^d-1 sum","display":"\\sum_{a=0}^{d-1}","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a73e1778d27a812b","tex":"\\sum_{h\\ge0}","aliases":"∑_h≥0 sum ge","display":"\\sum_{h\\ge0}","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-113d2a04b2350877","tex":"\\sum_{q=0}^{n}","aliases":"∑_q=0^n sum","display":"\\sum_{q=0}^{n}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":14},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4a8ae622c77aac5b","tex":"\\sum_{r=0}^m","aliases":"∑_r=0^m sum","display":"\\sum_{r=0}^m","kind":"introduction-passage","excerpt":"Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m r}, where the braces are Stirling numbers of the second kind. Each 2r!/(r+1) is dyadically integral. Indeed, if s=v_2(r+1)>0, then r≥2^s-1 and v_2(r!)≥ v_2((2^s-1)!)=2^s-1-s≥ s-1. The last inequality is ","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a4fed322c79bb1cd","tex":"\\sum_{r=0}^{n}","aliases":"∑_r=0^n sum","display":"\\sum_{r=0}^{n}","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":135},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"introduction-passage","excerpt":"Local analytic graph theorem. There exist a map ψ: K^k⟶ K^e and open sets U⊆ K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative ","count":23},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":19},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":19},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":24},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6aa48613e1d44abf","tex":"\\mathbf1_{d\\equiv r\\ (N)}","aliases":"1_d≡ r (N) mathbf equiv","display":"\\mathbf1_{d\\equiv r\\ (N)}","kind":"use","excerpt":"Modules. mcodeCyclotomicFiltration, mcodePrimeBlockPolynomial, and mcodeCyclicAmiceCoefficients. 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementa","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0914a0ccf44d4813","tex":"\\mathbf1_{r=0}","aliases":"1_r=0 mathbf","display":"\\mathbf1_{r=0}","kind":"use","excerpt":"For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =coefTdF_r-1-dcoefTdF_r. The initial values are F_r(0)=1_r=0.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"Its exponential generating function is A(t)=∑_n≥0A(n)fract^2n(2n)! =fraccosh(3t)-cosh t4t^2. The quotients by powers of t are well defined because the required initial coefficients vanish. The hyperbolic identity A(t)cosh t=(fracsinh(2t)2t)^2 then gives the exact formal-series product (∑_m≥0β_m(2t)^2m) A(t) =secht. Taking the factorial-scaled coeff","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"R(a,b), R(a',b), R(a',b') Longrightarrow R(a,b')","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-40d113efdc774b45","tex":"a^{(m)}","aliases":"a^(m)","display":"a^{(m)}","kind":"use","excerpt":"[Multiple selector] Define a^(m)∈ H( F_p) by a_n^(m) = 1, · mmid n,","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-677eab50351e7675","tex":"A^{-1}","aliases":"A^-1","display":"A^{-1}","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-280e15d5f806ea95","tex":"a^{-2}","aliases":"a^-2","display":"a^{-2}","kind":"use","excerpt":"ties give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"One sufficient new arithmetic input would be an odd-denominator projective quadratic A Z^2+2BZ+C≡ A W_r(Z)pmod2^P, A odd, 0<|B^2-AC|<2^P, at a proved precision for every admitted ordinary r. Then (B^2-AC)/A^2≡δ_rpmod2^P cannot vanish modulo 2^P. No depth-independent theorem supplying this inequality has been established. The tested recurrence and e","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-943bc84d09f5dc3b","tex":"a^k","aliases":"a^k","display":"a^k","kind":"use","excerpt":"Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s); induction and multiplication by g(n+s","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dd721bed307c9bfa","tex":"a^{p^L}","aliases":"a^p^L","display":"a^{p^L}","kind":"use","excerpt":"[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-55aa8dc52ad3f209","tex":"a^{p^{\\ell-1}}","aliases":"a^p^ℓ-1 ell","display":"a^{p^{\\ell-1}}","kind":"use","excerpt":"ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bf7da5da975b32de","tex":"a^{p^{\\ell-1}(p-1)}","aliases":"a^p^ℓ-1(p-1) ell","display":"a^{p^{\\ell-1}(p-1)}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-db9c84c96b8fd1b3","tex":"a^{\\varphi(p^\\ell)}","aliases":"a^φ(p^ℓ) varphi ell","display":"a^{\\varphi(p^\\ell)}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bbddd5e376c69681","tex":"A_{-c}","aliases":"A_-c","display":"A_{-c}","kind":"use","excerpt":"d product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, star","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bdcba39a7197cb28","tex":"\\mathcal A_0","aliases":"A_0 mathcal","display":"\\mathcal A_0","kind":"use","excerpt":"Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions; no completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source referen","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definition","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1fc2375840e988ee","tex":"a_{0,1}","aliases":"a_0,1","display":"a_{0,1}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e29910acf04cde74","tex":"a_0^{-1}","aliases":"a_0^-1","display":"a_0^{-1}","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6214ded09a4fbf7a","tex":"a_0^{[M]}","aliases":"a_0^[M]","display":"a_0^{[M]}","kind":"use","excerpt":"For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"[Finite Newton dominance] Let f: N→ Q and write a_k=Δ_1^kf(0). Suppose that c∈ N, a_1≠0, v_2(a_1)=c, and v_2(a_k)≥ c+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and T>0, f(n+T)-f(n)≠0, v_2(f(n+T)-f(n))=v_2(T)+c.","count":14},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8f573136a2832fb7","tex":"a_{1,1}","aliases":"a_1,1","display":"a_{1,1}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eefe7444a403f33e","tex":"A_1^s","aliases":"A_1^s","display":"A_1^s","kind":"use","excerpt":"Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-53cc0413ede33538","tex":"\\mathcal A_2","aliases":"A_2 mathcal","display":"\\mathcal A_2","kind":"use","excerpt":"Let A_2 be the divided-power algebra over R generated by P,Q, adjoining B, D with B^2=1+4P, D^2=1+4Q. The functional is integral on its module basis 1, B, D, B D by the Bernoulli lemma, independence, and vanishing of odd moments. Every component in [source reference] has integral divided powers, by the same finite identities as before; P^2 is even ","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"The low Newton coefficients of the actual reciprocal are C_2(0)=1, C_2(1)=-frac115, C_2(2)=frac1345, a_1=-frac1615, a_2=frac6445, where a_m=Δ_1^m C_2(0). Thus v_2(a_1)=4 and v_2(a_2)=6. For m≥3, [source reference] bounds a_m by 2m+v_2(m!). The finite shifted Newton formula Δ_1^kC_2(n)=∑_m=k^n+ka_mbinom nm-k therefore gives v_2(Δ_1^kC_2(n))≥ k+v_2(k","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5827c40753badd76","tex":"a_{2,1}","aliases":"a_2,1","display":"a_{2,1}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b4a0cf96cd5928bd","tex":"a_{2,d}","aliases":"a_2,d","display":"a_{2,d}","kind":"use","excerpt":"At every center x_0∈ Z_2, the interpolations admit ordinary Taylor series C_j(x_0+z)=∑_d≥0a_j,d(x_0)z^d. For positive degrees, v_2(a_j,d(x_0)) · =d · · (j=0,1), v_2(a_2,d(x_0)) · ≥max{2d+lfloor d/2rfloor, d+3}, v_2(a_3,d(x_0)) · ≥ d. At address two equality with 5d/2 holds along every power-of-two degree d≥4.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d325da4f9b58928d","tex":"A_3","aliases":"A_3","display":"A_3","kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e513bab7c906c944","tex":"A_{3,1}","aliases":"A_3,1","display":"A_{3,1}","kind":"use","excerpt":"[Exact change between local source charts] Editorial range correction. This comparison holds for the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero; the eve","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-74ec5788a006c41f","tex":"a_{3,1}","aliases":"a_3,1","display":"a_{3,1}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-35c4026ab117f63f","tex":"a_{3,d}","aliases":"a_3,d","display":"a_{3,d}","kind":"use","excerpt":"At every center x_0∈ Z_2, the interpolations admit ordinary Taylor series C_j(x_0+z)=∑_d≥0a_j,d(x_0)z^d. For positive degrees, v_2(a_j,d(x_0)) · =d · · (j=0,1), v_2(a_2,d(x_0)) · ≥max{2d+lfloor d/2rfloor, d+3}, v_2(a_3,d(x_0)) · ≥ d. At address two equality with 5d/2 holds along every power-of-two degree d≥4.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-af63f9fe0008dfda","tex":"a_{B^s f}","aliases":"a_B^s f","display":"a_{B^s f}","kind":"use","excerpt":"The earlier coefficient criterion is recovered exactly. Define the literal coefficient defect (δ a)(d)=a(d+p^2)-a(d+p). The new source proves, for every s,d, a_B^s f(d)=p^-s(δ^s a_f)(d). For an integral coefficient x, it proves the bridge p^-sx∈ Z_p Longleftrightarrow p^smid x in Z_p. Thus the differential definition is equivalent to the existing c","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5494a82a5f60bced","tex":"A_c","aliases":"A_c","display":"A_c","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3a4f27f07ce0c669","tex":"A_{c+2}","aliases":"A_c+2","display":"A_{c+2}","kind":"use","excerpt":"atives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting from the independently established integrality of the reciprocal G, proves that inversion negates the charge. F","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-408acdef742b081a","tex":"A_{c+d}","aliases":"A_c+d","display":"A_{c+d}","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b2507e870c0b29f3","tex":"A_{c-1}","aliases":"A_c-1","display":"A_{c-1}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b081100956900811","tex":"A_c^s","aliases":"A_c^s","display":"A_c^s","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3581b621d385d761","tex":"A_d","aliases":"A_d","display":"A_d","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5a8e57a002dc32f4","tex":"a_d","aliases":"a_d","display":"a_d","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4b862c67265e36c7","tex":"a_{d+1}","aliases":"a_d+1","display":"a_{d+1}","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d; whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5696e5795686355a","tex":"a_{d+pk}","aliases":"a_d+pk","display":"a_{d+pk}","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dc727ddbdff7e587","tex":"a_f","aliases":"a_f","display":"a_f","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6aa43e91577b201e","tex":"a_{f^{-1}}","aliases":"a_f^-1","display":"a_{f^{-1}}","kind":"use","excerpt":"The reciprocal is genuinely integral. For f∈ I_p with constant coefficient one, the usual power-series reciprocal is used. Its inverse equation yields the exact coefficient recurrence a_f^-1(d)=δ_d0-∑_r=0^d-1 binom dr a_f^-1(r)a_f(d-r). Strong induction proves that all of these coefficients are integral. Combining this construction with the precedi","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6571548bdb8d4622","tex":"a_{fg}","aliases":"a_fg","display":"a_{fg}","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2d83d124aaabaee7","tex":"a_g","aliases":"a_g","display":"a_g","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4f6afa45de73ec0e","tex":"a_{H_ax+j}","aliases":"a_H_ax+j","display":"a_{H_ax+j}","kind":"use","excerpt":", v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a88ad33e388dbded","tex":"a_{Hx+j}","aliases":"a_Hx+j","display":"a_{Hx+j}","kind":"use","excerpt":"means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not an as-written certificate. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_H","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"ts strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use polynomial induction on constants, addition, and multiplication by a variable. Constants have zero derivative. The sum rule agrees with the additivity of formal partial differentiation. For PX_j, the","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d5616d55df0ff6ce","tex":"a_{j,d}","aliases":"a_j,d","display":"a_{j,d}","kind":"use","excerpt":"At every center x_0∈ Z_2, the interpolations admit ordinary Taylor series C_j(x_0+z)=∑_d≥0a_j,d(x_0)z^d. For positive degrees, v_2(a_j,d(x_0)) · =d · · (j=0,1),","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a75acd39d6ce208f","tex":"a_{j,k}","aliases":"a_j,k","display":"a_{j,k}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"[Finite Newton dominance] Let f: N→ Q and write a_k=Δ_1^kf(0). Suppose that c∈ N, a_1≠0, v_2(a_1)=c, and v_2(a_k)≥ c+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and T>0, f(n+T)-f(n)≠0, v_2(f(n+T)-f(n))=v_2(T)+c.","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-997309b6e5a8216f","tex":"a_{k+i}","aliases":"a_k+i","display":"a_{k+i}","kind":"use","excerpt":"Finite Newton reconstruction at an arbitrary row gives Δ_1^kf(n)=∑_i=0^nbinom ni a_k+i. The monotonicity of lfloorlog_2krfloor therefore preserves the higher-coefficient bound after translation. In addition, Δ_1f(n)-a_1∈2^c+1 Z_2, so the first difference is nonzero of exact valuation c.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"The low Newton coefficients of the actual reciprocal are C_2(0)=1, C_2(1)=-frac115, C_2(2)=frac1345, a_1=-frac1615, a_2=frac6445, where a_m=Δ_1^m C_2(0). Thus v_2(a_1)=4 and v_2(a_2)=6. For m≥3, [source reference] bounds a_m by 2m+v_2(m!). The finite shifted Newton formula Δ_1^kC_2(n)=∑_m=k^n+ka_mbinom nm-k therefore gives v_2(Δ_1^kC_2(n))≥ k+v_2(k","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e281fec2511c43be","tex":"a_{M(Nq+s)+r}","aliases":"a_M(Nq+s)+r","display":"a_{M(Nq+s)+r}","kind":"use","excerpt":"Direct substitution gives (a_r^[M])_s^[N](q) = a_M(Nq+s)+r. Iterate this identity along the mixed radix path.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c30d38a2eb4fbb92","tex":"a_{M(q+1)}","aliases":"a_M(q+1)","display":"a_{M(q+1)}","kind":"use","excerpt":"For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8cb01de33ecdba63","tex":"a_{M-1}^{[M]}","aliases":"a_M-1^[M]","display":"a_{M-1}^{[M]}","kind":"use","excerpt":"For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-317d704cb35a25a3","tex":"a_{Mq+r}","aliases":"a_Mq+r","display":"a_{Mq+r}","kind":"use","excerpt":"The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary ","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bbd901b35afb50e7","tex":"a_{Mq+r+1}","aliases":"a_Mq+r+1","display":"a_{Mq+r+1}","kind":"use","excerpt":"For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"[Exact coefficient transport] Let A(t)=∑_n≥0A_nfract^nn!, S(t)=H(t)A(t). Then S_n = ∑_r=0^n binom nr h_rA_n-r. The inverse coordinate formula is A_n = ∑_r=0^n binom nr k_rS_n-r. Both transforms are lower triangular with diagonal one.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"Over a characteristic-zero field K, put E(a)=∑_n≥0fraca_nn!t^n, D(A)_n=n![t^n]A. These inverse maps between sequences and K[[t]] provide formal exponential coordinates. No analytic convergence is needed for the coefficient identities below.","count":19},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4c8131608f522ec2","tex":"a_{n+1}","aliases":"a_n+1","display":"a_{n+1}","kind":"use","excerpt":"Consider the recurrence a_n+1=a_n with integer initial value. Its global parameter space is X= Z. Its local parameter spaces are X_p= Z_p, and every local realization has the same prime-independent recurrence. Give the local factors their usual p-adic topologies and the product its product topology. For these integral local factors, the restricted ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5f3deb161f4183b8","tex":"A_{N,j}","aliases":"A_N,j","display":"A_{N,j}","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν),","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2e4d74b5db7c0422","tex":"A_{n-r}","aliases":"A_n-r","display":"A_{n-r}","kind":"use","excerpt":"[Exact coefficient transport] Let A(t)=∑_n≥0A_nfract^nn!, S(t)=H(t)A(t). Then S_n = ∑_r=0^n binom nr h_rA_n-r. The inverse coordinate formula is A_n = ∑_r=0^n binom nr k_rS_n-r. Both transforms are lower triangular with diagonal one.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1b51e95719a5480d","tex":"a_{n-r}","aliases":"a_n-r","display":"a_{n-r}","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0c376ef200c86635","tex":"a_n^{(m)}","aliases":"a_n^(m)","display":"a_n^{(m)}","kind":"use","excerpt":"[Multiple selector] Define a^(m)∈ H( F_p) by a_n^(m) = 1, · mmid n, 0, · mnmid n.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-36c90390c8711691","tex":"A_{p,2m}","aliases":"A_p,2m","display":"A_{p,2m}","kind":"use","excerpt":"[Unified local Euler transfer] For every prime p and every even n≥0, E_n = ∑_r=0^n/2 binom n2r κ_p,r A_p,n-2r. If H_p(t) = ∑_m≥0A_p,2mfract^2m(2m)! and K_p(t) = ∑_r≥0κ_p,rfract^2r(2r)!, then K_p(t)H_p(t)=secht. For odd p, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). For p=2, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-678c8828f3bebbd3","tex":"A_{p,n}","aliases":"A_p,n","display":"A_{p,n}","kind":"use","excerpt":"For a prime p, define A_p,n = S_n(p), · p odd, R_n, · p=2 and n even, 0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd,","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0c6d5229ad08c832","tex":"A_{p,n-2r}","aliases":"A_p,n-2r","display":"A_{p,n-2r}","kind":"use","excerpt":"[Unified local Euler transfer] For every prime p and every even n≥0, E_n = ∑_r=0^n/2 binom n2r κ_p,r A_p,n-2r. If H_p(t) = ∑_m≥0A_p,2mfract^2m(2m)! and K_p(t) = ∑_r≥0κ_p,rfract^2r(2r)!, then K_p(t)H_p(t)=secht. For odd p, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). For p=2, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-38143a44ee0b7f8e","tex":"a_q","aliases":"a_q","display":"a_q","kind":"use","excerpt":"The factorials are nonzero in K, so encoding and decoding cancel coefficientwise. The constant coefficient of H is 1; hence its formal power-series inverse exists. Multiplication by H and H^-1 gives the two inverse laws. The ordinary product coefficient formula, multiplied by n!, gives the displayed binomial sums. In row n, the coefficient of a_n i","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"use","excerpt":"This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard windo","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-92d57bfe67d21ec0","tex":"a_{r+1}^{[M]}","aliases":"a_r+1^[M]","display":"a_{r+1}^{[M]}","kind":"use","excerpt":"For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a3416243cb2455e6","tex":"a_{r+Ms}^{[MN]}","aliases":"a_r+Ms^[MN]","display":"a_{r+Ms}^{[MN]}","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-357c0cfc7297df89","tex":"A_r^{[M]}","aliases":"A_r^[M]","display":"A_r^{[M]}","kind":"use","excerpt":"[Exact discrete polyphase reconstruction] The analysis map P_M is a bijection between sequences and ordered families of M sequences. If A(z)=∑_n≥0a_nz^n and A_r^[M](w)=∑_q≥0a_Mq+rw^q, then A(z)=∑_r=0^M-1z^rA_r^[M](z^M).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0f45d14646d94b29","tex":"a_r^{[M]}","aliases":"a_r^[M]","display":"a_r^{[M]}","kind":"use","excerpt":"The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary ","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4f1eee202dc651a9","tex":"A_u","aliases":"A_u","display":"A_u","kind":"use","excerpt":"a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d700de4fae3fdd0a","tex":"a_{Uf}","aliases":"a_Uf","display":"a_{Uf}","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c7dc2ed692bd1496","tex":"A_{z_n}","aliases":"A_z_n","display":"A_{z_n}","kind":"use","excerpt":"The one-step Taylor theorem already supplies the exact correction equation b_n+A_z_nt=0, b_n=F(z_n)/p^L+nbmod p. Surjectivity supplies a correcting digit for every residual b_n. The chosen correction gives an actual new integer point z_n+1, using the proved nonlinear Taylor remainder.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c6fe7588fe8a751d","tex":"A_{z_{n+1}}","aliases":"A_z_n+1","display":"A_{z_{n+1}}","kind":"use","excerpt":"Every coordinate change is divisible by p^L+n, hence by p. Each formal partial derivative is itself an integer polynomial, so its value is unchanged modulo p. Therefore A_z_n+1=A_z_n=A_x. The proof assumes no separate surjectivity or solvability statement at later levels. A dependent recursion chooses each next point from the proved one-step existe","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6daaf6159d130196","tex":"A_{\\ell,n}","aliases":"A_ℓ,n ell","display":"A_{\\ell,n}","kind":"use","excerpt":". For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero; the even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity H_p=L_pK_ℓ H_ℓ and its proof remain valid. For any two primes p and ℓ, the source sequences A_p,n a","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-99fa55a48bd5ebd6","tex":"\\mathrm{act}","aliases":"act mathrm","display":"\\mathrm{act}","kind":"use","excerpt":"ion K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). ","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-703614cdc8657cbf","tex":"\\operatorname{aug}","aliases":"aug operatorname","display":"\\operatorname{aug}","kind":"use","excerpt":"[Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"The normalized defect is not assumed to be a derivation. Its proved product rule contains the two middle Leibniz sums: B_Θ(FG)=(B_Θ F)G+F(B_Θ G)+ B_p(F,G). The coefficients of B_p are the actual integral quotients binomp^2a/p and binom pa/p. Derivative stability and this correction formula show that the elements F∈ W_p satisfying B_Θ F∈ W_p are clo","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"use","excerpt":"R(a,b), R(a',b), R(a',b') Longrightarrow R(a,b')","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3079afd29fa1e022","tex":"b^{(m,p)}","aliases":"b^(m,p)","display":"b^{(m,p)}","kind":"use","excerpt":"Since the subgroup average has augmentation one, it has a unique Hurwitz inverse. Write b^(m,p) = (a^(m))^-1.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"0 · cI_k=cI_e+k. The second equality uses JA=cI_e. The generic block proof works over any commutative ring. The p-adic specialization also explicitly proves that B is invertible over Q and that p^sB^-1 is integral. There is no denominator loss from appending the parameter equations.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e36ace1e3c65141c","tex":"b^{-2}","aliases":"b^-2","display":"b^{-2}","kind":"use","excerpt":"ties give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f76a26dbc0d9fe26","tex":"B^2","aliases":"B^2","display":"B^2","kind":"use","excerpt":"One sufficient new arithmetic input would be an odd-denominator projective quadratic A Z^2+2BZ+C≡ A W_r(Z)pmod2^P, A odd, 0<|B^2-AC|<2^P, at a proved precision for every admitted ordinary r. Then (B^2-AC)/A^2≡δ_rpmod2^P cannot vanish modulo 2^P. No depth-independent theorem supplying this inequality has been established. The tested recurrence and e","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d99261575cf632ab","tex":"\\mathcal B^2","aliases":"B^2 mathcal","display":"\\mathcal B^2","kind":"use","excerpt":"The variable B=2Z+1 has transform e^tfrac2te^2t-1=fractsinh t. Its odd moments vanish, and P=( B^2-1)/4=Z(Z+1) has the divided-power moments of Lemma~[source reference]. Introduce an independent copy D and Q=( D^2-1)/4. The exact factorization F_2(t)=fract^2cosh t-cos t =fracatsinh(at)fracbtsinh(bt) shows that U=a B+b D satisfies L(U^4n)=C_2(n). Di","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"For p=2, take F(X)=X+X^2, H(X)=1, a=0, s=0, M=1. There is a unique root b∈2 Z_2 of X+X^2+2. Its equation gives b+2=-b^2. It also forces b≡2pmod4. Therefore v_2(b+2)=2=2M-3s. The accompanying Lean controls show that the error is divisible by four and is not divisible by eight. This refutes a uniform replacement of 2M-3s by 2M-3s+1, even for a one-va","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1bdcbb0ff59dbc74","tex":"b^p","aliases":"b^p","display":"b^p","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d; whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-139ce225a5a62faf","tex":"b^{p^2}","aliases":"b^p^2","display":"b^{p^2}","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d; whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-acb79e87d28a764d","tex":"B^r","aliases":"B^r","display":"B^r","kind":"use","excerpt":"means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not an as-written certificate. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_H","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6ad6156b5e42899a","tex":"B^s","aliases":"B^s","display":"B^s","kind":"use","excerpt":"All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator a","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"(b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872).","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7dc72fd5fdc5b850","tex":"b_{0,r}","aliases":"b_0,r","display":"b_{0,r}","kind":"use","excerpt":"Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the forma","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"(b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872).","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","excerpt":"[Exact four-address first differences] For every n≥0, H>0, and j∈{0,1,2,3}, v_2(C_j(n+H)-C_j(n)) · =v_2(H)+b_j, · (b_0,b_1,b_2,b_3) · =(1,1,4,2), v_2(E_4(n+H)^(4,j)-E_4n^(4,j)) · =v_2(H)+B_j, · (B_0,B_1,B_2,B_3) · =(1,1,5,3). Consequently, the least positive row period modulo 2^r is exactly P_j(r)=2^max(0,r-B_j) (r≥1). The corresponding displacemen","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"(b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872).","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-98aa7dc614deee26","tex":"B_{2m}","aliases":"B_2m","display":"B_{2m}","kind":"use","excerpt":"[Exact weighted inverse coefficient] Let κ_m=β_m2^2m(2m)!. For every m≥1, κ_m=-2^4m(2m-1)B_2m≠0, v_2(κ_m)=4m-1, where B_2m is the usual even Bernoulli number.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b8a910bd8f25ee36","tex":"B_{2n}","aliases":"B_2n","display":"B_{2n}","kind":"use","excerpt":"oulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator and using [source reference]. Comparing even coefficients yields C_2,1(n)=(2-2^2n)B_2n (n∈ N). This classi","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e35089086af3af8c","tex":"B_{2r}","aliases":"B_2r","display":"B_{2r}","kind":"use","excerpt":"[Dyadic inverse coefficients] For r≥1, β_r = - frac2^2r(2r-1)B_2r(2r)!. Consequently, v_2(β_r)=s_2(r)-1. Every R_n is a dyadic unit.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b224b896d9d01ac4","tex":"B_3","aliases":"B_3","display":"B_3","kind":"use","excerpt":"A sharp binomial-weight improvement. Let F(n)=81binom n1+3binom n2+3binom n3+81binom n4. At n=3 and s=1, the four coefficient valuations are (4,1,1,4), while the binomial valuations are (0,1,1,0). Thus the refined bound is B_3(w;3,1)=min(4,2,2,4)=2, stronger than the coefficient-only lower bound one. The source checks both the minimum and the actua","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5284f24ae8cda5b8","tex":"b_3","aliases":"b_3","display":"b_3","kind":"use","excerpt":"(b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872).","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4b94fddf75c4f060","tex":"B_4","aliases":"B_4","display":"B_4","kind":"use","excerpt":"3. The dyadic differential operator and reciprocity. The operator required here is B_4=fracD^8-D^42, D=fracddt. It is not the preceding p=2 specialization (D^4-D^2)/2. It is also not an ordinary derivation. The new algebra module proves the actual product formula, including the middle derivative sums whose coefficients are binom 8i/2 and binom 4i/2","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-16ab16f37fdea085","tex":"b_4","aliases":"b_4","display":"b_4","kind":"use","excerpt":"(b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a0175fc3916c84f4","tex":"B_{4i+2}","aliases":"B_4i+2","display":"B_{4i+2}","kind":"use","excerpt":"G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The co","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1a4c2e9cb4075d24","tex":"B_a","aliases":"B_a","display":"B_a","kind":"use","excerpt":"atrix_power_amplification Editorial hypothesis correction. Require L≥1. At L=0 the standalone claim is false: for p=3, d=1 and X=2=1+B, one has X^3-1=7∉3 Z_3. With L≥1, every higher binomial term in the retained proof has the required valuation, including p=2. If X=I+p^LB with B∈ M_d( Z_p), then for every a≥0 there is B_a∈ M_d( Z_p) such that X^p^a","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fe0fcfcbd04a63ee","tex":"b_{a,j}","aliases":"b_a,j","display":"b_{a,j}","kind":"use","excerpt":", v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2de8fd102567aaa1","tex":"B_{a-1}","aliases":"B_a-1","display":"B_{a-1}","kind":"use","excerpt":"ents from the reciprocal recurrence are a_1=-g(1)-1, a_2=binom2NNg(1)^2-g(2)+2g(1)+1. The first has valuation exactly one. Also binom2NN≡6pmod8 for every N=2^a≥2. To see this, write B_a=binom2^a+12^a =2∏_h=1^2^a-1(1+frac2^ah). The even-h factors give B_a-1; the odd-h factors are one modulo four for a≥2. Since B_a-1 has valuation one, B_a≡ B_a-1pmod","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e3bae85ab6d499ac","tex":"B_D","aliases":"B_D","display":"B_D","kind":"use","excerpt":"p^2. Replacing F by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption: B_Θ W_p⊆ W_p, B_Θ^s W_p⊆ W_p (s≥0). Modules. mcodeWeightedAmiceRing, mcodeFrobeniusSquare, mcodeThetaWeighted. 2. Actual substitution and singular higher differences. Let E(F)(t)=F(e^t-1), D=fracddt, B_D=fracD^p^2-D^pp.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f56251a6f87c66b1","tex":"b_d","aliases":"b_d","display":"b_d","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d2273248685a24f0","tex":"b_{d+1,r}","aliases":"b_d+1,r","display":"b_{d+1,r}","kind":"use","excerpt":"Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the forma","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3beb282e3f0dd2a9","tex":"b_{d,r}","aliases":"b_d,r","display":"b_{d,r}","kind":"use","excerpt":"Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the forma","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-19747db9bb1a3595","tex":"b_{d,r-1}","aliases":"b_d,r-1","display":"b_{d,r-1}","kind":"use","excerpt":"Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the forma","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d442e917a9b5ba8d","tex":"b_{d-i}","aliases":"b_d-i","display":"b_{d-i}","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-49151ef209ba386f","tex":"B_D^s","aliases":"B_D^s","display":"B_D^s","kind":"use","excerpt":"(s≥0). Modules. mcodeWeightedAmiceRing, mcodeFrobeniusSquare, mcodeThetaWeighted. 2. Actual substitution and singular higher differences. Let E(F)(t)=F(e^t-1), D=fracddt, B_D=fracD^p^2-D^pp. All substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives D E(F)= E(Θ F), B_D^s E(F)= E(B_Θ^sF). The imp","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"use","excerpt":"For 0≤ i<j, let d_i=v_2(i!)+v_2((j-1-i)!), k_i=1+lfloorlog_2max(i,j-1-i)rfloor, B_i=i+2^k_i Z_2, t_i=d_i+k_i. At j=1 use d_0=k_0=t_0=0. The balls B_i partition Z_2. Indeed a residue represented twice among 0,…,j-1 splits into its two children; every other represented residue remains one ball. On each such ball, P_j(i+2^k_iy)=2^t_iyQ_i(y), Q_i∈ Z_2⟨","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d05703a4df49dfef","tex":"b_{k+1}","aliases":"b_k+1","display":"b_{k+1}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7e09999c1cd63453","tex":"b_{k+d}","aliases":"b_k+d","display":"b_{k+d}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ca06c4f67613c969","tex":"b_{k+r}","aliases":"b_k+r","display":"b_{k+r}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-762426111124f19d","tex":"B_L","aliases":"B_L","display":"B_L","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":16},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fc9ad75de14545d5","tex":"b_m","aliases":"b_m","display":"b_m","kind":"use","excerpt":"The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer; this is the integrality required by the later congruence.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-def969061c988714","tex":"b_{mn}^{(m,p)}","aliases":"b_mn^(m,p)","display":"b_{mn}^{(m,p)}","kind":"use","excerpt":"[Compressed inverse] Set e_n^(m,p) = b_mn^(m,p).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"use","excerpt":"The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer; this is the integrality required by the later congruence.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","excerpt":"Pascal's identity proves the first product rule, and induction proves the higher rule. The intermediate binomial coefficients binom pr vanish in characteristic p. For the inverse, solve row n using the coefficient of b_n, which is g_0=1: b_n=1_n-∑_r<nbinom nr b_r g_n-r. Subtract the first and pth product rules for b⋆ g=1. The assumed identity for g","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e36803b804d3c151","tex":"b_{n+p-1}^{(m,p)}","aliases":"b_n+p-1^(m,p)","display":"b_{n+p-1}^{(m,p)}","kind":"use","excerpt":"[Support and positive degree period] The inverse b^(m,p) has the following properties. item b_n^(m,p)=0 unless mmid n. item For every positive n, b_n+p-1^(m,p) = b_n^(m,p).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f0e1a295abce9295","tex":"B_{N,j}","aliases":"B_N,j","display":"B_{N,j}","kind":"use","excerpt":"[Small-rank complete isometry table] Every address at N∈{2,4,8} satisfies v_2(C_N,j(n+H)-C_N,j(n))=b_N,j+v_2(H). The following entries are ordered by j=0,…,N-1: c l l N · normalized shifts b_N,j · original shifts B_N,j=b_N,j+v_2(j!)","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2580100ce4d243be","tex":"b_{N,j}","aliases":"b_N,j","display":"b_{N,j}","kind":"use","excerpt":"[Small-rank complete isometry table] Every address at N∈{2,4,8} satisfies v_2(C_N,j(n+H)-C_N,j(n))=b_N,j+v_2(H). The following entries are ordered by j=0,…,N-1: c l l N · normalized shifts b_N,j · original shifts B_N,j=b_N,j+v_2(j!)","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-34d4e296e2f11bfd","tex":"b_{n-r}","aliases":"b_n-r","display":"b_{n-r}","kind":"use","excerpt":"This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard windo","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9f2ddbf7dd109de0","tex":"b_n^{(m,p)}","aliases":"b_n^(m,p)","display":"b_n^{(m,p)}","kind":"use","excerpt":"[Support and positive degree period] The inverse b^(m,p) has the following properties. item b_n^(m,p)=0 unless mmid n. item For every positive n, b_n+p-1^(m,p) = b_n^(m,p).","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8bcbf87ac6779001","tex":"B_p","aliases":"B_p","display":"B_p","kind":"use","excerpt":"Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9ce4130b75dce935","tex":"\\mathcal B_p","aliases":"B_p mathcal","display":"\\mathcal B_p","kind":"use","excerpt":"The normalized defect is not assumed to be a derivation. Its proved product rule contains the two middle Leibniz sums: B_Θ(FG)=(B_Θ F)G+F(B_Θ G)+ B_p(F,G). The coefficients of B_p are the actual integral quotients binomp^2a/p and binom pa/p. Derivative stability and this correction formula show that the elements F∈ W_p satisfying B_Θ F∈ W_p are clo","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-667e7c36f2dec875","tex":"B_p^k","aliases":"B_p^k","display":"B_p^k","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-219538915e682dc1","tex":"B_p^r","aliases":"B_p^r","display":"B_p^r","kind":"use","excerpt":"Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration give","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c734e54250961c96","tex":"B_p^{r-1}","aliases":"B_p^r-1","display":"B_p^{r-1}","kind":"use","excerpt":"Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration give","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-48939aa333eb6f6a","tex":"B_p^s","aliases":"B_p^s","display":"B_p^s","kind":"use","excerpt":"Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ac0d31a2615b8b66","tex":"b_r","aliases":"b_r","display":"b_r","kind":"use","excerpt":"Pascal's identity proves the first product rule, and induction proves the higher rule. The intermediate binomial coefficients binom pr vanish in characteristic p. For the inverse, solve row n using the coefficient of b_n, which is g_0=1: b_n=1_n-∑_r<nbinom nr b_r g_n-r. Subtract the first and pth product rules for b⋆ g=1. The assumed identity for g","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1ab923618be1c5fc","tex":"b_{r-q}","aliases":"b_r-q","display":"b_{r-q}","kind":"use","excerpt":"The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r, 0, · q>r. The row-action equality, the diagonal identity U_r,r=1, and vanishing above the diagonal are all checked. ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b85ab862e92905e8","tex":"b_T","aliases":"b_T","display":"b_T","kind":"use","excerpt":"Theorem~[source reference] gives a unique root b of F+p^M H in a+p^s+1R^e+k, with displacement depth M-s. Its last equations say exactly b_T=t_0+p^Mv=τ. Its first equations are G(b)=0. The integral first-order expression uses CbinomH(a)-v =binomA(H(a)+Kv)-p^sv. Projection onto the solved coordinates gives b_Y-y_0+p^M-sA(H(a)+Kv)∈ p^2M-3sR^e. The sa","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aba34f1446cfa879","tex":"B_v","aliases":"B_v","display":"B_v","kind":"use","excerpt":"a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bec856d92c5ab12c","tex":"b_x","aliases":"b_x","display":"b_x","kind":"use","excerpt":"Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Write r_x=F(x)/p^L, b_x=r_xbmod p, A_x=DF(x)bmod p: F_p^d⟶ F_p^e, ρ_x=rank_ F_pA_x. The actual lift set is L_x,L= {y∈( Z/p^L+1 Z)^d: y≡ xpmodp^L, F(y)=0}, where polynomial evaluation in the displayed set is modulo p^L+1. Define O_x= F_p^e/imA_x, o_x=[b_x]. Lean proves # L_x,L= p^d-ρ","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ed74d4cbc9f00b0b","tex":"b_Y","aliases":"b_Y","display":"b_Y","kind":"use","excerpt":"Theorem~[source reference] gives a unique root b of F+p^M H in a+p^s+1R^e+k, with displacement depth M-s. Its last equations say exactly b_T=t_0+p^Mv=τ. Its first equations are G(b)=0. The integral first-order expression uses CbinomH(a)-v =binomA(H(a)+Kv)-p^sv. Projection onto the solved coordinates gives b_Y-y_0+p^M-sA(H(a)+Kv)∈ p^2M-3sR^e. The sa","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fd8097ebfdb37289","tex":"B_\\Theta","aliases":"B_Θ Theta","display":"B_\\Theta","kind":"use","excerpt":"Put Θ=(1+T)fracddT, B_Θ=fracΘ^p^2-Θ^pp. The coefficient formula [T^d]Θ F=(d+1)a_d+1+da_d proves Θ W_p⊆ W_p. If the floor weight increases between d and d+1, then pmid d+1, which compensates for that increase.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5d3f66919d71566b","tex":"B_\\Theta^s","aliases":"B_Θ^s Theta","display":"B_\\Theta^s","kind":"use","excerpt":"Finally, [T^d]B_Θ F depends only on coefficients through d+p^2. Replacing F by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption: B_Θ W_p⊆ W_p, B_Θ^s W_p⊆ W_p (s≥0). Modules. mcodeWeightedAmiceRing, mcodeFrobeniusSquare, mcodeThetaWeighted. 2. Actual substitution and singular higher differ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e2ec3d8de7f39249","tex":"\\bar a","aliases":"bar a bar","display":"\\bar a","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c8185eb0b0eeb64a","tex":"\\bar a^{p^{\\ell-1}}","aliases":"bar a^p^ℓ-1 bar ell","display":"\\bar a^{p^{\\ell-1}}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4c4d0c215858215d","tex":"\\bar r","aliases":"bar r bar","display":"\\bar r","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dc4e9dcca4ad0352","tex":"\\bar w","aliases":"bar w bar","display":"\\bar w","kind":"use","excerpt":"Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N, F(bar w)=0, free(bar w)=t(n)+p^Nvpmodp^N+1 }. Here p^Nv means any integer representative of v, multiplied by p^N; the residue class does not depend on the representative. Then # L_n,v=p^m-e for every n and every v","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b4d7ee29d2029812","tex":"\\bar x","aliases":"bar x bar","display":"\\bar x","kind":"use","excerpt":"For a fixed polynomial system F∈ Z[X_1,…,X_d]^e, a precision L≥1, and a root bar x modulo p^L, let L_F,L(bar x) ={x'∈( Z/p^L+1 Z)^d: F(x')=0, x'≡bar xpmodp^L}. The finite obstruction theorem decides whether this set is empty from the residual digit and reduced Jacobian. When it is nonempty, the theorem also determines its cardinality and affine ker","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ffda03323e255d73","tex":"\\bigm","aliases":"bigm","display":null,"kind":"use","excerpt":"0, · w=∞. The infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, every term with weight at least L is divisible by p^L. A finite-sum proof yields p^L bigm| ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r -∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-r. The result is specialized to actual p-adic","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"At each index n, exactly one standard residue label equals nbmod N. Consequently a repeated selector changes nothing, two different selectors have zero composite, and the sum retains a(n) exactly once. If Nmid M, reducing nbmod M modulo N gives nbmod N. This proves the refinement identity at every coefficient. Equality of all coefficients proves th","count":31},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[3pt] 1 · -91019/45 · 4 [3pt] 2 · 398197/45 · 6 [3pt] 3 · -208601/15 · 8 [3pt] 4 · 1385 · 10 The checked nonzero error valuations are E_8-T_0(8)=frac5494445, v_2(E_8-T_0(8))=5, E_8-T_3(8)=frac22937615, v_2(E_8-T_3(8))=15. They exceed the uniform guarantees; these two examples do not establish a universally stronger exponent.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e874d9d01b6213d8","tex":"\\operatorname{Bounded}","aliases":"Bounded operatorname","display":"\\operatorname{Bounded}","kind":"use","excerpt":"All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c947faba6dfed07d","tex":"\\operatorname{Bounded}_r","aliases":"Bounded_r operatorname","display":"\\operatorname{Bounded}_r","kind":"use","excerpt":"All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8e3c0fce1b7a60ae","tex":"\\bullet","aliases":"bullet","display":"\\bullet","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"Let G∈ Z[X_1,…,X_d]^f be a second fixed system with the same variables and the same lower root x. Let K_x=ρ_q∘ DG(x), and define C_x= Z^d/(ker J_x+ker K_x). This quotient is finite: it is a quotient of the finite image of J_x, or equivalently of the finite image of K_x.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-faeabc6533da3ce6","tex":"c^{(p)}_{m,j}","aliases":"c^(p)_m,j","display":"c^{(p)}_{m,j}","kind":"use","excerpt":"Specializing to R= Z_p constructs c^(p)_m,j(n). Mapping to Q_p and comparing with the original rational forward substitution gives c^(p)_m,j(n)=C_mp,j(n) in Q_p. Thus every original standard-window coefficient is p-integral, for every row. Integrality is a conclusion, not an additional hypothesis on the coefficients.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e5d5e13c662eedec","tex":"c^{(p)}_{m,k}","aliases":"c^(p)_m,k","display":"c^{(p)}_{m,k}","kind":"use","excerpt":"ply the residue map Independently define e^(m,p) over F_p by the unit-diagonal system ∑_q=0^nbinommnmqe^(m,p)_q=δ_n0. Its recurrence and uniqueness are checked. Applying the actual homomorphism π: Z_p→ F_p to the integral row, and then Lucas reduction, proves π(c^(p)_m,j(n))=e^(m,p)_n. For j,k<p, a separate ideal statement gives pmid c^(p)_m,j(n)-c","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9d46643740805c1d","tex":"C^{12}","aliases":"C^12","display":"C^{12}","kind":"use","excerpt":"1 · 0 , one has C^12 =I+8 29 · 18 18 · 11 . Therefore Δ_12·2^a^ rF_n ≡0 pmod 2^(a+3)r. Every dyadic refinement doubles the step and adds one guaranteed binary digit at each Mahler depth.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","excerpt":"0 · 0) has C^2=0 and χ_C=X^2.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8981f94f66d22b3c","tex":"C^{24}","aliases":"C^24","display":"C^{24}","kind":"use","excerpt":"1 · 0 . The matrix C is invertible modulo twenty four, and C^24-I = 24 3126 · 1932 1932 · 1194 . For every scalar coordinate of the state sequence C^nv_0, Δ_24^ ra_n ≡0 pmod 24^r. One integral jump therefore gives simultaneous congruence ladders at the primes two and three.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4a4fb977dca130a2","tex":"C^3","aliases":"C^3","display":"C^3","kind":"use","excerpt":"[t] centering includegraphics[width=.95linewidth,height=3.15in,keepaspectratio]../../figures/reconstructed/recurrence_packets.pdf captionNew reconstruction. Fibonacci residues F_3k+j, with F_0=0 and F_1=1. Each packet evolves by the same jump matrix C^3 and has a different initial address.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ce7e7a918f6fa28e","tex":"C^H","aliases":"C^H","display":"C^H","kind":"use","excerpt":"0,w∈ Z_p^d, L≥1, and H,j,n∈ N with H>0; the value at Hx+j for x∈ Z_p means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not an as-written certificate. The full original proof follows under the corrected hypotheses. Assume C^","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2482758af295dd87","tex":"C^{H_a}","aliases":"C^H_a","display":"C^{H_a}","kind":"use","excerpt":"dentEditorial hypothesis correction. Require L≥1, C,B∈ M_d( Z_p), v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bec59126aabbbf80","tex":"C^{Hx+j}","aliases":"C^Hx+j","display":"C^{Hx+j}","kind":"use","excerpt":"Use C^Hx+j=C^j(I+p^LB)^x and the matrix binomial series. One difference in the coarse variable multiplies the series by p^LB.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-04ebc96f0d8e8891","tex":"C^j","aliases":"C^j","display":"C^j","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fc4ec113b29e6ee4","tex":"C^n","aliases":"C^n","display":"C^n","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-70f1ea36ff493278","tex":"C^{n+1}","aliases":"C^n+1","display":"C^{n+1}","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-675d2adc99d5a3df","tex":"C^{n+2}","aliases":"C^n+2","display":"C^{n+2}","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3ee90351b9053688","tex":"C^{Nk+j}","aliases":"C^Nk+j","display":"C^{Nk+j}","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"Put K= Q_2(i), R= Z_2[i], a=(1+i)/2, and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial multiplier is integral. Because F_1=F_0D_1^-1, C_1(n)-C_0(n) =∑_i≥1h_ib","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"egintheorem[Four-address factorial bound] For j∈{0,1,2,3}, n≥0, k≥1, and positive mixed steps, v_2(Δ_H_1⋯Δ_H_kC_j(n)) ≥ k+v_2(k!)+∑_ν=1^k v_2(H_ν)+β_j, (β_0,β_1,β_2,β_3)=(0,0,3,1). Address two also satisfies v_2(Δ_H_1⋯Δ_H_kC_2(n)) ≥ 2k+v_2(k!)+lfloorfrack2rfloor +∑_ν=1^k v_2(H_ν). For the original coefficients, the offsets in [source reference] are","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-24612095df1bb71a","tex":"C_{2,0}","aliases":"C_2,0","display":"C_{2,0}","kind":"use","excerpt":"[Euler and Bernoulli boundaries] For (N,j)=(2,0), C_2,0(t)=secht, so C_2,0(n)=E_2n. For (N,j)=(2,1), C_2,1(t)=fractsinh t, and C_2,1(n) = 2(1-2^2n-1)B_2n. The usual convention B_1=-1/2 is used. Euler and Bernoulli number conventions are reviewed in [source reference].","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8b6228375a9787bb","tex":"C_{2,1}","aliases":"C_2,1","display":"C_{2,1}","kind":"use","excerpt":"[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator a","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. Since F_3=F_2D_3^-1, C_3(n)-C_2(n)=∑_i≥1h_ibinom4n4iC_2(n-i). The scaled-binomial argument [source reference] applies ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b60e443e8108308c","tex":"c_3","aliases":"c_3","display":"c_3","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f4c46ceb742a13f2","tex":"C_{3,0}","aliases":"C_3,0","display":"C_{3,0}","kind":"use","excerpt":"At N=p=3 and j=0, put A(n)=(-1)^nC_3,0(n). The first three rows are AtlasImageodd_prime_sign The ordinary first difference at row zero is -1-1=-2. It is a 3-adic unit, so an asserted lower bound of one for that difference would be false. The twisted first difference is zero, whose mathematical valuation is +∞. At the next row, the twisted differenc","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-879a006e315df8a5","tex":"C_{4,0}","aliases":"C_4,0","display":"C_{4,0}","kind":"use","excerpt":"Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b68e81c02307b3f0","tex":"C_{4,1}","aliases":"C_4,1","display":"C_{4,1}","kind":"use","excerpt":"An exact arithmetic check, not an optimality claim. The independent rational recurrence gives C_4,1(1)=-1/5 and C_4,1(2)=121/45. Their difference is 26/9, of 2-adic valuation one. This agrees with the new bound at n=k=u=1. The source also contains universal instantiations of the degree-indexed theorem at all four dyadic addresses and at an odd sing","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a62fe51ac71b9fa6","tex":"C_{4,j}","aliases":"C_4,j","display":"C_{4,j}","kind":"use","excerpt":"For an odd prime, 2mid p-1, so the option m=2 introduces no further case. The sole remaining branch is p=2,m=2,N=4. The all-row theorem proved next gives 2^kmidΔ_1^kC_4,j(n) at every row and address. After multiplying by j!, Lemma~[source reference] gives packet period 2^r-1 and degree displacement 4·2^r-1=2^r+1=qp^r. It also gives integrality. Hen","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-62848497b9641ba6","tex":"C_{6,0}","aliases":"C_6,0","display":"C_{6,0}","kind":"use","excerpt":"No automatic extra power. The original rational triangular coefficients satisfy C_6,0(1)=-1, C_6,0(2)=923, v_3(C_6,0(2)-C_6,0(1))=v_3(924)=1. A period modulo 3 therefore cannot simply be reported as a period modulo 9. This is consistent with the global theorem's proposed first-difference bound; it is not a counterexample to Theorem 7.5.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3f04db759519ade9","tex":"C_{6,2}","aliases":"C_6,2","display":"C_{6,2}","kind":"use","excerpt":"pageheading5. Examples, verification and the remaining gapThe coverage count is unchanged; the unresolved theorem now has a stronger proved part The original rational recurrence gives C_6,2(1)=-frac128, C_6,2(2)=frac425364, C_6,2(3)=-frac20397592660. Consequently, the new source proves Δ_1^2 C_6,2(1)=-frac9499591235, v_3 (Δ_1^2 C_6,2(1))=2. This at","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9129c76ab167e7bf","tex":"C_{6,3}","aliases":"C_6,3","display":"C_{6,3}","kind":"use","excerpt":"[Why the factorial loss is necessary] Take p=3, m=2, N=6, and j=3. The recurrence gives C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(2)-C_6,3(1)=frac64455. The last difference has 3-adic valuation zero. Here h=1, so at n=1 the order-one bound without factorial loss would require valuation at least one. The corrected normalized bound is sharp, sin","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-06fe0f5802f142ad","tex":"C_{6,4}","aliases":"C_6,4","display":"C_{6,4}","kind":"use","excerpt":"w For (p,m,j)=(3,2,3), the source derives C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(3)=-frac2231997980. Consequently, Δ_1^2(3!C_6,3)(1)=-frac14647688645, v_3 (-frac14647688645)=2. This attains min(2,2·1)=2. At the regular address j=4 beyond the singular boundary, the source likewise proves Δ_1^2(4!C_6,4)(1)=-frac1027455122377375, and applies t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b6010785cb6069e5","tex":"c_{a,j,r}","aliases":"c_a,j,r","display":"c_{a,j,r}","kind":"use","excerpt":"ypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e5265479b6f0a2ec","tex":"c_i^{e_i}","aliases":"c_i^e_i","display":"c_i^{e_i}","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"use","excerpt":"For the remainder of the four-packet discussion abbreviate C_j=C_4,j and U_j=U_4,j. Removing the original row cap requires an unshifted coefficient operator.","count":16},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j, U_j(n+T)-U_j(n) · ≠0, · v_2(U_j(n+T)-U_j(n)) · =v_2(T)+d_j.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-407d3e9d1538f42c","tex":"C_{J,K}","aliases":"C_J,K","display":"C_{J,K}","kind":"use","excerpt":"The linear compatibility theorem provides one concrete part of this program. If two local observations J:U→ V and K:U→ W are linear maps from the same correction module, define C_J,K=U/(ker J+ker K). The attainable observations im J and im K have canonical labels in this quotient. Equality of labels is equivalent to simultaneous realization by a si","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0ce23018e3603d00","tex":"c_{j,r}","aliases":"c_j,r","display":"c_{j,r}","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-254051ed849e0a84","tex":"c_k","aliases":"c_k","display":"c_k","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6013c421e4c989b5","tex":"C_{mp,j}","aliases":"C_mp,j","display":"C_{mp,j}","kind":"use","excerpt":"[Lucas reduction and address independence] Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f33574a980dc5307","tex":"C_{mp,k}","aliases":"C_mp,k","display":"C_{mp,k}","kind":"use","excerpt":"prop:arith-lucas-window Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈ N).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c1ecf2aa8f993509","tex":"c_N","aliases":"c_N","display":"c_N","kind":"use","excerpt":"Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-28f4c794936e614f","tex":"C_{N,0}","aliases":"C_N,0","display":"C_{N,0}","kind":"use","excerpt":"[Necessary boundaries] At an odd prime the ordinary step-one difference cannot replace the sign-twisted operator in the general theorem. For j=0, C_N,0(0)=1 and C_N,0(1)=-1, so their ordinary difference is -2, a unit at every odd prime. The proof also does not extend unchanged to non-prime-power N, whose interior binomial coefficients need not be d","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-374d4fe4488ce15e","tex":"C_p","aliases":"C_p","display":"C_p","kind":"use","excerpt":"fferentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c a","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a7f55c1da8911f74","tex":"c_{p^ak}","aliases":"c_p^ak","display":"c_{p^ak}","kind":"use","excerpt":"(q)=q has d_1=1; thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-be57c172ad4ba898","tex":"c_{p^ak+u}","aliases":"c_p^ak+u","display":"c_{p^ak+u}","kind":"use","excerpt":"insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1; thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-557d8a93e9288da4","tex":"C_R","aliases":"C_R","display":"C_R","kind":"use","excerpt":"for maps from A and B to a common set if and only if R is rectangular. In that case a canonical finite choice of common set is the connected-component set C_R of the bipartite graph with vertex set Asqcup B and edges given by R.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"use","excerpt":"[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.","count":40},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e1e71a0eb7c64bd5","tex":"c_s","aliases":"c_s","display":"c_s","kind":"use","excerpt":"An independently defined interpolant. For a finite coefficient list, define I_N(c)(n)=∑_r=0^Nc_rbinom nr. The source proves exact coefficient extraction: Δ_1^s I_N(c)(0)= c_s, · s≤ N, 0, · s>N. It follows that I_N(c) has the stated shifted-coefficient formula at every order, with zero coefficients beyond N. Every difference of order greater than N ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1c4963e002d8ca00","tex":"c_{s+j}","aliases":"c_s+j","display":"c_{s+j}","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8ef88b2c20ebb677","tex":"c_{s+r}","aliases":"c_s+r","display":"c_{s+r}","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c9c457fd3fefb6bf","tex":"c_u","aliases":"c_u","display":"c_u","kind":"use","excerpt":"The product law follows from [source reference]. To prove injectivity, suppose ∑_u∈ F_pc_uϵ_u=0. Looking at the coordinates n=0,1,…,p-1 gives a Vandermonde system with matrix (u^n)_0≤ n≤ p-1, u∈ F_p. Its determinant is the product of all differences between distinct field elements, so it is nonzero. Hence every c_u vanishes.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-85d220b196ae3e90","tex":"\\mathcal C_x","aliases":"C_x mathcal","display":"\\mathcal C_x","kind":"use","excerpt":"Let G∈ Z[X_1,…,X_d]^f be a second fixed system with the same variables and the same lower root x. Let K_x=ρ_q∘ DG(x), and define C_x= Z^d/(ker J_x+ker K_x). This quotient is finite: it is a quotient of the finite image of J_x, or equivalently of the finite image of K_x.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-db3b3f64d870ea56","tex":"c_{\\eta u}","aliases":"c_η u eta","display":"c_{\\eta u}","kind":"use","excerpt":"Write b^(m,p) = ∑_u∈ F_pc_uϵ_u. Invariance gives c_η u=c_u. After the change of variable u↦η u, b_n^(m,p) = η^n b_n^(m,p) for every η∈ H_m. If mnmid n, some η satisfies η^n≠1, so the coefficient vanishes.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-488e5fe62245ff4a","tex":"c_\\chi","aliases":"c_χ chi","display":"c_\\chi","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-843721b3e84879d7","tex":"\\Ccal","aliases":"Ccal C mathcal","display":"{\\mathcal C}","kind":"use","excerpt":"[Congruential reciprocal packet] Define C_N,j(t) = fract^j/j! E_N,j(t) = ∑_n≥0 C_N,j(n) fract^Nn(Nn)!.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1c9fd68e24f50c5c","tex":"\\Ccal_{2,0}","aliases":"Ccal_2,0 C_2,0 Ccal mathcal","display":"{\\mathcal C}_{2,0}","kind":"use","excerpt":"[Euler and Bernoulli boundaries] For (N,j)=(2,0), C_2,0(t)=secht, so C_2,0(n)=E_2n. For (N,j)=(2,1), C_2,1(t)=fractsinh t, and C_2,1(n) = 2(1-2^2n-1)B_2n. The usual convention B_1=-1/2 is used. Euler and Bernoulli number conventions are reviewed in [source reference].","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5a5e9f6fb063921d","tex":"\\Ccal_{2,1}","aliases":"Ccal_2,1 C_2,1 Ccal mathcal","display":"{\\mathcal C}_{2,1}","kind":"use","excerpt":"[Euler and Bernoulli boundaries] For (N,j)=(2,0), C_2,0(t)=secht, so C_2,0(n)=E_2n. For (N,j)=(2,1), C_2,1(t)=fractsinh t, and C_2,1(n) = 2(1-2^2n-1)B_2n. The usual convention B_1=-1/2 is used. Euler and Bernoulli number conventions are reviewed in [source reference].","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f6a3bfeb6b8f17e0","tex":"\\Ccal_{N,j}","aliases":"Ccal_N,j C_N,j Ccal mathcal","display":"{\\mathcal C}_{N,j}","kind":"use","excerpt":"[Congruential reciprocal packet] Define C_N,j(t) = fract^j/j! E_N,j(t) = ∑_n≥0 C_N,j(n) fract^Nn(Nn)!.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e417cc5a30e35857","tex":"\\operatorname{coe}","aliases":"coe operatorname","display":"\\operatorname{coe}","kind":"use","excerpt":"Identity with the original rational coefficients. The integral lift is constructed from the decoded coefficients of the actual reciprocal. The code proves coe(e_j(n))=C_4,j(n) in Q_2, and proves that the coefficient map commutes with every difference stencil. Multiplying by j! yields the unnormalized family. The valuation result is not divisibility","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-92bda6e67397bc66","tex":"\\coef","aliases":"coef","display":null,"kind":"use","excerpt":"mate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =c","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-073a71f814bdfa5f","tex":"\\coker","aliases":"coker operatorname","display":"{\\operatorname{coker}}","kind":"use","excerpt":"The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the paper now supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual dig","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6e6d0cbeeb71722a","tex":"\\operatorname{coker}","aliases":"coker operatorname","display":"\\operatorname{coker}","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"Connecting to the paper's Z_n convention. The paper defines Z_n through sec t+tan t. The Lean construction uses Mathlib's actual formal sine and cosine series and defines sec t=(cos t)^-1, tan t=(sin t)sec t, Z_n=n![t^n](sec t+tan t). It proves the sine and cosine derivative identities, sin^2t+cos^2t=1, and (tan t)'=sec^2t. Sign substitution in the","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8db6134243ea03f4","tex":"\\cos^2","aliases":"cos^2 cos","display":"\\cos^2","kind":"use","excerpt":"Connecting to the paper's Z_n convention. The paper defines Z_n through sec t+tan t. The Lean construction uses Mathlib's actual formal sine and cosine series and defines sec t=(cos t)^-1, tan t=(sin t)sec t, Z_n=n![t^n](sec t+tan t). It proves the sine and cosine derivative identities, sin^2t+cos^2t=1, and (tan t)'=sec^2t. Sign substitution in the","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-04ea05958a7c1391","tex":"\\coth","aliases":"coth","display":null,"kind":"use","excerpt":"Let Q(z)=zcoth z, the regular formal series with constant coefficient one. The elementary differential identity (frac zsinh z)^2=Q(z)-zQ'(z) and the Bernoulli expansion Q(z)=1+∑_m≥1frac2^2mB_2m(2m)!z^2m give β_m=-2^2m(2m-1)B_2m/(2m)!. Multiplication by 2^2m(2m)! proves the identity for κ_m. The Bernoulli denominator theorem gives v_2(B_2m)=-1 for m","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"Over a characteristic-zero field K, put E(a)=∑_n≥0fraca_nn!t^n, D(A)_n=n![t^n]A. These inverse maps between sequences and K[[t]] provide formal exponential coordinates. No analytic convergence is needed for the coefficient identities below.","count":16},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","excerpt":"3. The dyadic differential operator and reciprocity. The operator required here is B_4=fracD^8-D^42, D=fracddt. It is not the preceding p=2 specialization (D^4-D^2)/2. It is also not an ordinary derivation. The new algebra module proves the actual product formula, including the middle derivative sums whose coefficients are binom 8i/2 and binom 4i/2","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-65577bc46f414a5f","tex":"\\mathcal D^2","aliases":"D^2 mathcal","display":"\\mathcal D^2","kind":"use","excerpt":"The variable B=2Z+1 has transform e^tfrac2te^2t-1=fractsinh t. Its odd moments vanish, and P=( B^2-1)/4=Z(Z+1) has the divided-power moments of Lemma~[source reference]. Introduce an independent copy D and Q=( D^2-1)/4. The exact factorization F_2(t)=fract^2cosh t-cos t =fracatsinh(at)fracbtsinh(bt) shows that U=a B+b D satisfies L(U^4n)=C_2(n). Di","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8ac5422b66e1b0c1","tex":"D^4","aliases":"D^4","display":"D^4","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e8f3e6a620b326b0","tex":"D^8","aliases":"D^8","display":"D^8","kind":"use","excerpt":"3. The dyadic differential operator and reciprocity. The operator required here is B_4=fracD^8-D^42, D=fracddt. It is not the preceding p=2 specialization (D^4-D^2)/2. It is also not an ordinary derivation. The new algebra module proves the actual product formula, including the middle derivative sums whose coefficients are binom 8i/2 and binom 4i/2","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-17de839c28aeb1f7","tex":"D^{d-r}","aliases":"D^d-r","display":"D^{d-r}","kind":"use","excerpt":"[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-58357a31d24cf18a","tex":"D^i","aliases":"D^i","display":"D^i","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e04bf4d668de4635","tex":"D^{N-i}","aliases":"D^N-i","display":"D^{N-i}","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3d379d56ac67e266","tex":"D^p","aliases":"D^p","display":"D^p","kind":"use","excerpt":"[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-da3992d397405d31","tex":"D^{p^2}","aliases":"D^p^2","display":"D^{p^2}","kind":"use","excerpt":"Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-09e71d50145ecad3","tex":"D^r","aliases":"D^r","display":"D^r","kind":"use","excerpt":"[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2d88cb49c47029e8","tex":"D^\\alpha","aliases":"D^α alpha","display":"D^\\alpha","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cd476ebfd1a76f31","tex":"D^\\beta","aliases":"D^β beta","display":"D^\\beta","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-737f63af45789c25","tex":"d_{0,k}","aliases":"d_0,k","display":"d_{0,k}","kind":"use","excerpt":"(q)=q has d_1=1; thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"The address-one source is G_4,1(t)=(sinh t+sin t)/(2t). The identities a+b=1, a-b=i, and ab=1/2 give D_1(t):=G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hur","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1a59dd4f5aee6253","tex":"D_1^{-1}","aliases":"D_1^-1","display":"D_1^{-1}","kind":"use","excerpt":"stant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial multiplier is integral. Because F_1=F_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7ec3ef0b1d0ea9eb","tex":"d_{2,\\ell}","aliases":"d_2,ℓ ell","display":"d_{2,\\ell}","kind":"use","excerpt":"=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at address three is at least two; its exact radius i","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d2c6012edbe5de2c","tex":"D_3","aliases":"D_3","display":"D_3","kind":"use","excerpt":"The address-three source is G_4,3(t)=3(sinh t-sin t)/t^3. Addition identities give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, o","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9f66bb3522fa9b25","tex":"d_3","aliases":"d_3","display":"d_3","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e73f1dfc33a2fa5b","tex":"D_3^{-1}","aliases":"D_3^-1","display":"D_3^{-1}","kind":"use","excerpt":"e a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. Since F_3=F_2D_3^-1, C_3(n)-C_2(n)=∑_i≥1h_ibinom4n4iC_2(n-i). The scaled-binomial argument","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7903c81010eebdd6","tex":"D_a","aliases":"D_a","display":"D_a","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"use","excerpt":"For 0≤ i<j, let d_i=v_2(i!)+v_2((j-1-i)!), k_i=1+lfloorlog_2max(i,j-1-i)rfloor, B_i=i+2^k_i Z_2, t_i=d_i+k_i. At j=1 use d_0=k_0=t_0=0. The balls B_i partition Z_2. Indeed a residue represented twice among 0,…,j-1 splits into its two children; every other represented residue remains one ball. On each such ball, P_j(i+2^k_iy)=2^t_iyQ_i(y), Q_i∈ Z_2⟨","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j, U_j(n+T)-U_j(n) · ≠0, · v_2(U_j(n+T)-U_j(n)) · =v_2(T)+d_j.","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b0858de229bbaac5","tex":"d_{j,m}","aliases":"d_j,m","display":"d_{j,m}","kind":"use","excerpt":"For j=0,1, the term m=ℓ in [source reference] has exact valuation ℓ, since s(ℓ,ℓ)=1 and the Newton valuation is exact. Every later term has valuation at least ℓ+1. Thus v_2(d_j,ℓ)=ℓ, and the coefficient radius is exactly two. At a center x_0∈ Z_2, the coefficient of y^ℓ in the translated series is ∑_m≥ℓd_j,mbinom mℓ x_0^m-ℓ. Its initial term again ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5f4f45a608f77322","tex":"d_{j,\\ell}","aliases":"d_j,ℓ ell","display":"d_{j,\\ell}","kind":"use","excerpt":"[Taylor coefficient bounds and radii] For j=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at addres","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5ee31d563f299611","tex":"D_{N,j}","aliases":"D_N,j","display":"D_{N,j}","kind":"use","excerpt":"[A denominator bound for every address] For N≥1, 0≤ j<N, and n∈ N, set D_N,j(n)=∏_r=1^nbinomNr+jNr, D_N,j(0)=1. Then D_N,j(n)C_N,j(n)∈ Z.","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a10db54813efe79a","tex":"d_n^{-1}","aliases":"d_n^-1","display":"d_n^{-1}","kind":"use","excerpt":"First construct the integral solution. For a ring R, a triangular row system T, right-hand side b, and proved unit pivots d_n=T_nn, define s_n=d_n^-1(b_n-∑_q<nT_nqs_q). The new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homo","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-19c16859d92fc1e1","tex":"D_P","aliases":"D_P","display":"D_P","kind":"use","excerpt":"Apply polynomial induction using constants, addition, and multiplication by a variable. Constants have zero remainder, and remainders add. For the multiplication step, suppose the assertion is known for P and write D_P(U)=∑_j(∂_jP)(a)U_j. Multiplication of its expansion by a_i+cU_i gives constant term P(a)a_i, linear term c(a_iD_P(U)+P(a)U_i), and ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3e5463cd5ce49eb7","tex":"d_q","aliases":"d_q","display":"d_q","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"use","excerpt":"[Exact classical tail valuation] For every r≥1 and N≥ r, v_2( T_r-1(N)) =4r-1+v_2binom Nr+μ_r+v_2G_r(N-r). There are unique factors G_r(X)=W_r(X)U_r(X), where W_r∈ Z_2[X] is monic of degree d_r=max J_r and U_r∈ Z_2⟨ X⟩^× satisfies U_r≡1pmod2. Consequently v_2G_r(x)=v_2W_r(x) for every x∈ Z_2, and deg W_r≤lfloorfracs_2(r)-13rfloor. Equation~[source ","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b22090db332ee7ef","tex":"d_{s,k}","aliases":"d_s,k","display":"d_{s,k}","kind":"use","excerpt":"[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d3b6e2eaef5fb02a","tex":"d_v","aliases":"d_v","display":"d_v","kind":"use","excerpt":"[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0d09000efb054df8","tex":"\\ddots","aliases":"ddots","display":"\\ddots","kind":"use","excerpt":"[Polyphase shift matrix] Let E be the scalar shift (Ea)_n=a_n+1, and let E_q shift the coarse index. Then P_M(Ea) = S_M(E_q) P_Ma, where S_M(E_q) = 0 · 1 · 0 · ⋯ · 0 0 · 0 · 1 · ⋯ · 0 vdots · · · ddots · vdots 0 · 0 · 0 · ⋯ · 1 E_q · 0 · 0 · ⋯ · 0 . It also satisfies S_M(E_q)^M=E_qI_M.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-74f94422b9cebc9d","tex":"\\operatorname{dec}","aliases":"dec operatorname","display":"\\operatorname{dec}","kind":"use","excerpt":"Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"eginequation v_2( T_r-1(N)) =4r-1+v_2binom Nr+μ_r+v_2G_r(N-r). There are unique factors G_r(X)=W_r(X)U_r(X), where W_r∈ Z_2[X] is monic of degree d_r=max J_r and U_r∈ Z_2⟨ X⟩^× satisfies U_r≡1pmod2. Consequently v_2G_r(x)=v_2W_r(x) for every x∈ Z_2, and deg W_r≤lfloorfracs_2(r)-13rfloor. Equation~[source reference], the exact weight in [source refe","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"For prime p, rank-nullity gives dimker J_x=d-ρ_x and dim( F_p^e/im J_x)=e-ρ_x. The cardinality of an n-dimensional F_p-space is p^n. Combine these identities with the obstruction criterion and the residue equivalence.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0dcf6a495ccbd5bd","tex":"\\e","aliases":"e mathrm","display":"{\\mathrm e}","kind":"use","excerpt":"[Odd source generating function] For every positive odd q, ∑_n≥0S_n(q)fract^nn! = e^qtcosh(qt)secht. Its even part is ∑_n≥0S_2n(q)fract^2n(2n)! = cosh^2(qt)secht.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"Over a characteristic-zero field K, put E(a)=∑_n≥0fraca_nn!t^n, D(A)_n=n![t^n]A. These inverse maps between sequences and K[[t]] provide formal exponential coordinates. No analytic convergence is needed for the coefficient identities below.","count":29},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-64fbb1a3212deefd","tex":"e'","aliases":"e'","display":"e'","kind":"use","excerpt":"Another representative has the form x'=x+p^Ls. Since L≥1, its Jacobian is the same modulo p. Dividing (1) by p^L shows that its residual digit is e'=e+Js, so it gives exactly the same quotient class. If t_0 solves (2), all its solutions are t_0+ker J, and no others. The finite cardinalities follow from rank-nullity. This also covers no variables, n","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9e2f2fe584334e62","tex":"\\e^{(2a+1)t}","aliases":"e^(2a+1)t e mathrm","display":"{\\mathrm e}^{(2a+1)t}","kind":"use","excerpt":"The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-83389e2c1dbd901f","tex":"e^{(2a+1)t}","aliases":"e^(2a+1)t","display":"e^{(2a+1)t}","kind":"use","excerpt":"The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconst","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6a29eeb72f96adf0","tex":"E^{(4,j)}","aliases":"E^(4,j)","display":"E^{(4,j)}","kind":"use","excerpt":"In dyadic distance, |x|_2=2^-v_2(x). After continuous interpolation to Z_2, the law reads |E^(4,j)(x)-E^(4,j)(y)|_2 =2^-B_j|x-y|_2. It says exactly how much the map shrinks distances. It also prevents any hidden collision at finer precision: distinct inputs remain distinct after the known scale is removed.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-31d6c2ddbbf303ac","tex":"e^{(m,p)}","aliases":"e^(m,p)","display":"e^{(m,p)}","kind":"use","excerpt":"[A second proof of the standard window period] The reduced inverse e^(m,p) and every standard address satisfy the period in [source reference].","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-489e8c151737a2f9","tex":"e^{(m,p)}_n","aliases":"e^(m,p)_n","display":"e^{(m,p)}_n","kind":"use","excerpt":"Then apply the residue map. Independently define e^(m,p) over F_p by the unit-diagonal system ∑_q=0^nbinommnmqe^(m,p)_q=δ_n0. Its recurrence and uniqueness are checked. Applying the actual homomorphism π: Z_p→ F_p to the integral row, and then Lucas reduction, proves π(c^(p)_m,j(n))=e^(m,p)_n. For j,k<p, a separate ideal statement gives pmid c^(p)_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-011d330493ff6133","tex":"e^{(m,p)}_q","aliases":"e^(m,p)_q","display":"e^{(m,p)}_q","kind":"use","excerpt":"Then apply the residue map. Independently define e^(m,p) over F_p by the unit-diagonal system ∑_q=0^nbinommnmqe^(m,p)_q=δ_n0. Its recurrence and uniqueness are checked. Applying the actual homomorphism π: Z_p→ F_p to the integral row, and then Lucas reduction, proves π(c^(p)_m,j(n))=e^(m,p)_n. For j,k<p, a separate ideal statement gives pmid c^(p)_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5dda737c387b3f3d","tex":"E^{(N,j)}","aliases":"E^(N,j)","display":"E^{(N,j)}","kind":"use","excerpt":"There is a prime-power extension. For N=p^a, a≥1, the sign-corrected operator is T_H=S^H-(-1)^H, v_p (T_H_1⋯ T_H_kE^(N,j)(n)) ≥ v_p(j!)+∑_i=1^k(1+v_p(H_i)). Here E^(N,j)(n)=E_Nn^(N,j). For an odd prime, an ordinary first difference need not have positive valuation: at address zero its first two normalized values are 1 and -1, with difference -2. Th","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-544b5bdc27bea927","tex":"e^{-2pt}","aliases":"e^-2pt","display":"e^{-2pt}","kind":"use","excerpt":"Even projection is identified coefficientwise. For any formal series f, let EvenPart(f)=(f(t)+f(-t))/2. Its nth coefficient is proved to be the coefficient of f when n is even and zero otherwise. Rescaling the inverse law proves sech(-t)=sech(t). The exponential multiplication identity then gives cosh^2(pt)=frac2+e^2pt+e^-2pt4, and hence ∑_r≥0S_2r(","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3f6dc0c78382f5b1","tex":"e^{-t}","aliases":"e^-t","display":"e^{-t}","kind":"use","excerpt":"j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1; its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the product formula is empty, so the same argument applies without change; equivalently, F_1,0(t)=e^-t.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-97fed77197e69d14","tex":"\\mathcal E^2","aliases":"E^2 mathcal","display":"\\mathcal E^2","kind":"use","excerpt":"dependent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an indep","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f2f438829b64499b","tex":"e^{2pt}","aliases":"e^2pt","display":"e^{2pt}","kind":"use","excerpt":"The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconst","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-05426a5349dc70a7","tex":"\\e^{2qt}","aliases":"e^2qt e mathrm","display":"{\\mathrm e}^{2qt}","kind":"use","excerpt":"The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d6d6d7824dcc408b","tex":"\\e^{2t}","aliases":"e^2t e mathrm","display":"{\\mathrm e}^{2t}","kind":"use","excerpt":"The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-499fd10d84ddfd9f","tex":"e^{2t}","aliases":"e^2t","display":"e^{2t}","kind":"use","excerpt":"The variable B=2Z+1 has transform e^tfrac2te^2t-1=fractsinh t. Its odd moments vanish, and P=( B^2-1)/4=Z(Z+1) has the divided-power moments of Lemma~[source reference]. Introduce an independent copy D and Q=( D^2-1)/4. The exact factorization F_2(t)=fract^2cosh t-cos t =fracatsinh(at)fracbtsinh(bt) shows that U=a B+b D satisfies L(U^4n)=C_2(n). Di","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bda629913237771d","tex":"\\mathcal E^d","aliases":"E^d mathcal","display":"\\mathcal E^d","kind":"use","excerpt":"Let L be the ordinary Euler evaluation functional on a variable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear; it is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr E","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-06dfc486c3523030","tex":"\\e^{qt}","aliases":"e^qt e mathrm","display":"{\\mathrm e}^{qt}","kind":"use","excerpt":"[Odd source generating function] For every positive odd q, ∑_n≥0S_n(q)fract^nn! = e^qtcosh(qt)secht. Its even part is ∑_n≥0S_2n(q)fract^2n(2n)! = cosh^2(qt)secht.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-478999b366cf71dd","tex":"\\e^t","aliases":"e^t e mathrm","display":"{\\mathrm e}^t","kind":"use","excerpt":"The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8611063e24d14e09","tex":"e^z","aliases":"e^z","display":"e^z","kind":"use","excerpt":"[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator a","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a0b30352ab71380c","tex":"E_{(p-1)x+j}","aliases":"E_(p-1)x+j","display":"E_{(p-1)x+j}","kind":"use","excerpt":"[Odd prime analytic Euler branch] The series defining mathscr E_p,j^× converges in the Tate algebra on Z_p. For every nonnegative integer x, mathscr E_p,j^×(x) = (1-σ_pp^(p-1)x+j) E_(p-1)x+j.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"Three checked boundaries. The initial row. For (p,m)=(3,2), the same reduced recurrence gives e_0=1 and e_1=-1=2 in F_3. Its proposed period is one, so the positive-tail result cannot in general include row zero.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0c86d93cffa1a31f","tex":"e_0^{(m,p)}","aliases":"e_0^(m,p)","display":"e_0^{(m,p)}","kind":"use","excerpt":"[Compressed inverse recurrence] The compressed coefficients satisfy ∑_q=0^n binommnmq e_q^(m,p) = δ_n0. In particular, e_0^(m,p)=1, and the positive coefficients are determined recursively because the diagonal is one.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8f634c4c090fc7de","tex":"E_0^{(N,j)}","aliases":"E_0^(N,j)","display":"E_0^{(N,j)}","kind":"use","excerpt":"There are two coefficient conventions, and they must stay distinct: C_N,j(0)=1, E_0^(N,j)=j!, v_2(E_Nn^(N,j))=v_2(C_N,j(n))+v_2(j!). The last equality uses v_2(0)=+∞. Congruences of dyadic rational numbers mean membership in the corresponding ideal of Z_2; a displayed rational denominator is invertible modulo 2^r when it is odd.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"[A compressed inverse modulo seven] Editorial numerical correction. The positive-index standard-window tail printed below is erroneous. The defining recurrence gives e_1=-1≡6, e_2=5 and e_3=-61≡2pmod7, so the correct positive tail is 6,5,2,6,5,2,…. The period-three conclusion remains correct. The original line is retained only as a corrected histor","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"[A compressed inverse modulo seven] Editorial numerical correction. The positive-index standard-window tail printed below is erroneous. The defining recurrence gives e_1=-1≡6, e_2=5 and e_3=-61≡2pmod7, so the correct positive tail is 6,5,2,6,5,2,…. The period-three conclusion remains correct. The original line is retained only as a corrected histor","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-699df382cec3e125","tex":"E_{2M+2}","aliases":"E_2M+2","display":"E_{2M+2}","kind":"use","excerpt":"[Optimal uniform dyadic tower] For all M,n∈ N, E_2n-T_M(n)∈2^4M+3 Z_2. At the first omitted row n=M+1, the error is nonzero and v_2(E_2M+2-T_M(M+1))=4M+3. Thus the exponent 4M+3 cannot be increased uniformly in the Euler row.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e845d19c3939de97","tex":"E_{2n}","aliases":"E_2n","display":"E_{2n}","kind":"use","excerpt":"The classical signed Euler numbers have generating function secht=∑_n≥0E_2nt^2n/(2n)!. They are the N=2, j=0 member of [source reference]. The coordinate theorem reconstructs them from finite alternating sums of odd powers.","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1415b70590776d91","tex":"\\mathcal E_{2N}","aliases":"E_2N mathcal","display":"\\mathcal E_{2N}","kind":"use","excerpt":"The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d39b0ab0c5e61ab1","tex":"E_{2n}^{(2,0)}","aliases":"E_2n^(2,0)","display":"E_{2n}^{(2,0)}","kind":"use","excerpt":"Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eba3507d2e119c00","tex":"E_{2r}","aliases":"E_2r","display":"E_{2r}","kind":"use","excerpt":"The source defines E_2r as the integer numerator of that rational coefficient and proves that its rational cast equals C_2,0(r). Set E_2r+1=0. The actual exponential-section identity from the first block proves ∑_n≥0E_nfract^nn!=sech t. Thus the integer sequence is connected to the intended signed Euler family, rather than merely assigned the name ","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5002be43e2e62d6e","tex":"E_{2r+1}","aliases":"E_2r+1","display":"E_{2r+1}","kind":"use","excerpt":"The source defines E_2r as the integer numerator of that rational coefficient and proves that its rational cast equals C_2,0(r). Set E_2r+1=0. The actual exponential-section identity from the first block proves ∑_n≥0E_nfract^nn!=sech t. Thus the integer sequence is connected to the intended signed Euler family, rather than merely assigned the name ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"[A compressed inverse modulo seven] Editorial numerical correction. The positive-index standard-window tail printed below is erroneous. The defining recurrence gives e_1=-1≡6, e_2=5 and e_3=-61≡2pmod7, so the correct positive tail is 6,5,2,6,5,2,…. The period-three conclusion remains correct. The original line is retained only as a corrected histor","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6662c032276e077b","tex":"E_4","aliases":"E_4","display":"E_4","kind":"use","excerpt":"Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6b7e8f4d909c6799","tex":"E_{4(n+H)}^{(4,j)}","aliases":"E_4(n+H)^(4,j)","display":"E_{4(n+H)}^{(4,j)}","kind":"use","excerpt":"[Exact four-address first differences] For every n≥0, H>0, and j∈{0,1,2,3}, v_2(C_j(n+H)-C_j(n)) · =v_2(H)+b_j, · (b_0,b_1,b_2,b_3) · =(1,1,4,2), v_2(E_4(n+H)^(4,j)-E_4n^(4,j)) · =v_2(H)+B_j, · (B_0,B_1,B_2,B_3) · =(1,1,5,3). Consequently, the least positive row period modulo 2^r is exactly P_j(r)=2^max(0,r-B_j) (r≥1). The corresponding displacemen","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ab8e9e09ff8c907f","tex":"E_{4(n+iH)}^{(4,j)}","aliases":"E_4(n+iH)^(4,j)","display":"E_{4(n+iH)}^{(4,j)}","kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a5a67dacf9cd88b2","tex":"E_{4n}^{(4,j)}","aliases":"E_4n^(4,j)","display":"E_{4n}^{(4,j)}","kind":"use","excerpt":"[Exact four-address first differences] For every n≥0, H>0, and j∈{0,1,2,3}, v_2(C_j(n+H)-C_j(n)) · =v_2(H)+b_j, · (b_0,b_1,b_2,b_3) · =(1,1,4,2), v_2(E_4(n+H)^(4,j)-E_4n^(4,j)) · =v_2(H)+B_j, · (B_0,B_1,B_2,B_3) · =(1,1,5,3). Consequently, the least positive row period modulo 2^r is exactly P_j(r)=2^max(0,r-B_j) (r≥1). The corresponding displacemen","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-51b438373b685e40","tex":"E_6","aliases":"E_6","display":"E_6","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-edf98275838926c6","tex":"E_8","aliases":"E_8","display":"E_8","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ad3caf981ac30812","tex":"E_c","aliases":"E_c","display":"E_c","kind":"use","excerpt":"pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ce6fb2b17a97a58","tex":"E_F","aliases":"E_F","display":"E_F","kind":"use","excerpt":"For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-92780dc9a20a0ca6","tex":"E_H","aliases":"E_H","display":"E_H","kind":"use","excerpt":"For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"e_i=F_i(x)/p^Lpmod p, J_ij=frac∂ F_i∂ X_j(x)pmod p.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"Identity with the original rational coefficients. The integral lift is constructed from the decoded coefficients of the actual reciprocal. The code proves coe(e_j(n))=C_4,j(n) in Q_2, and proves that the coefficient map commutes with every difference stencil. Multiplying by j! yields the unnormalized family. The valuation result is not divisibility","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aeb1ae02bc3682da","tex":"E_{mpn+qp^r}^{(mp,j)}","aliases":"E_mpn+qp^r^(mp,j)","display":"E_{mpn+qp^r}^{(mp,j)}","kind":"use","excerpt":"nal edition establishes addcontentslinetocsectionWhat this final edition establishes The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d59b3cfc1e7a514b","tex":"E_{mpn}^{(mp,j)}","aliases":"E_mpn^(mp,j)","display":"E_{mpn}^{(mp,j)}","kind":"use","excerpt":"nal edition establishes addcontentslinetocsectionWhat this final edition establishes The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-da2ce42dc70bb3da","tex":"E_n","aliases":"E_n","display":"E_n","kind":"use","excerpt":"Let q be a positive odd integer. Define S_n(q) = ∑_a=0^q-1 (-1)^a(2a+1)^n. Let the signed Euler numbers and Euler zigzag numbers be defined by secht = ∑_n≥0E_nfract^nn! and sec t+tan t = ∑_n≥0Z_nfract^nn!.","count":13},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bf856d35e202a0fa","tex":"\\mathcal E_n","aliases":"E_n mathcal","display":"\\mathcal E_n","kind":"use","excerpt":"To distinguish ordinary signed Euler numbers from the generalized numbers of the main text, write secht=∑_n≥0 E_nfract^nn!, fracte^t-1=∑_n≥0B_nfract^nn!, B_1=-frac12. Throughout this appendix, v_2(2)=1 and v_2(0)=∞. Let s_2(n) be the number of ones in the binary expansion of the nonnegative integer n. Define A(N)=frac3^2N+2-14(2N+1)(2N+2), (frac zs","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e3ca2ec43cc1e4a1","tex":"E_{N,j}","aliases":"E_N,j","display":"E_{N,j}","kind":"use","excerpt":"For N>0 and 0≤ j<N, the independently defined original exponential section is E_N,j(t)=∑_d≡ j (rm mod N)fract^dd!. Using the actual power-series expansion homomorphism z↦ t^N, the code proves E_N,j(t)=fract^jj!K_N,j(t^N), F_N,j(t^N)E_N,j(t)=fract^jj!. Thus the recurrence is connected back to the original normalized EGF definition. The section vanis","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-19911032e8985ebc","tex":"e_{N-1}","aliases":"e_N-1","display":"e_{N-1}","kind":"use","excerpt":"d formula is an integral multinomial formula, rather than a product rule with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0,","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7898ee6b98b86d47","tex":"E_{n-2r}","aliases":"E_n-2r","display":"E_{n-2r}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6735d603366759b5","tex":"E_{n-r}","aliases":"E_n-r","display":"E_{n-r}","kind":"use","excerpt":"The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer; this is the integrality required by the later congruence.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-16602ebf447cab47","tex":"e_n^{(2,7)}","aliases":"e_n^(2,7)","display":"e_n^{(2,7)}","kind":"use","excerpt":"ction. The positive-index standard-window tail printed below is erroneous. The defining recurrence gives e_1=-1≡6, e_2=5 and e_3=-61≡2pmod7, so the correct positive tail is 6,5,2,6,5,2,…. The period-three conclusion remains correct. The original line is retained only as a corrected historical record. Take p=7 and m=2, so h=3. The recurrence gives e","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8f106d4bb685105a","tex":"E_{Nn+qp^r}^{(N,j)}","aliases":"E_Nn+qp^r^(N,j)","display":"E_{Nn+qp^r}^{(N,j)}","kind":"use","excerpt":"[Nishibuchi congruence] Let p be prime, let m>0, and suppose that mmid p-1 or m=2. Put N=mp and q=lcm(2,p-1). For every 0≤ j<N, all U_N,j(n) belong to Z_p. For r≥1 and n≥ r, E_Nn+qp^r^(N,j)≡ E_Nn^(N,j)pmodp^r. Thus the positive sufficient starting index n_0=r works uniformly. If mmid p-1, the stronger sufficient condition r≤ mn is valid.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ddcf053d52b9abd","tex":"E_{p,j}^\\times","aliases":"E_p,j^× times","display":"E_{p,j}^\\times","kind":"use","excerpt":"Fix an even address j∈{0,2,…,p-3}. For x∈ Z_p, define mathscr E_p,j^×(x) = ∑_r≥0 · (-1)^rZ_2r+1p^2r binom(p-1)x+j2r · × ∑_substack0≤ a<p 2a+1≠ p (-1)^a (2a+1)^j-2r ((2a+1)^p-1)^x.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b3f1fc86e79071cb","tex":"E_q","aliases":"E_q","display":"E_q","kind":"use","excerpt":"[Polyphase shift matrix] Let E be the scalar shift (Ea)_n=a_n+1, and let E_q shift the coarse index. Then P_M(Ea) = S_M(E_q) P_Ma, where S_M(E_q) = 0 · 1 · 0 · ⋯ · 0","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3a840ddc04edad8c","tex":"E_r","aliases":"E_r","display":"E_r","kind":"use","excerpt":"The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer; this is the integrality required by the later congruence.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2f89f0fc29c98d5e","tex":"e_{r,0}","aliases":"e_r,0","display":"e_{r,0}","kind":"use","excerpt":"The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-decd9feaf2bf3e0d","tex":"e_{r,q}","aliases":"e_r,q","display":"e_{r,q}","kind":"use","excerpt":"The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-701135f7469e0b6e","tex":"e_r^{(m,p)}","aliases":"e_r^(m,p)","display":"e_r^{(m,p)}","kind":"use","excerpt":"[Rational ordinary generating function] In F_p[[z]], ∑_n≥0e_n^(m,p)z^n = 1+ frac ∑_r=1^he_r^(m,p)z^r 1-z^h.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c470de86d3aa40db","tex":"\\Ecal","aliases":"Ecal E mathcal","display":"{\\mathcal E}","kind":"use","excerpt":"For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3ce1d43d6f91b9a1","tex":"\\Ecal_{N,j}","aliases":"Ecal_N,j E_N,j Ecal mathcal","display":"{\\mathcal E}_{N,j}","kind":"use","excerpt":"For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-192930449f687944","tex":"\\EE","aliases":"EE E mathcal","display":"{\\mathcal E}","kind":"use","excerpt":"Nishibuchi conjecture completion Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical; the final core records their resolution. 1. The complete external statement. For N>0 and 0≤ j<N, use the original unnormalized convention ∑_d≥0 E_d^(N,j)fract^dd","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2ce67a107a09dc9e","tex":"\\EE_d^{(N,j)}","aliases":"EE_d^(N,j) E_d^(N,j) EE mathcal","display":"{\\mathcal E}_d^{(N,j)}","kind":"use","excerpt":"Nishibuchi conjecture completion Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical; the final core records their resolution. 1. The complete external statement. For N>0 and 0≤ j<N, use the original unnormalized convention ∑_d≥0 E_d^(N,j)fract^dd","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9561b663cd811533","tex":"\\EE_{Nn+qp^r}^{(N,j)}","aliases":"EE_Nn+qp^r^(N,j) E_Nn+qp^r^(N,j) EE mathcal","display":"{\\mathcal E}_{Nn+qp^r}^{(N,j)}","kind":"use","excerpt":"Let p be prime, m>0, and assume mmid p-1 or m=2. Put N=mp and q=lcm(2,p-1). Every E_Nn^(N,j) is p-adically integral. For every r≥1, E_Nn+qp^r^(N,j)≡ E_Nn^(N,j)pmodp^r (0≤ j<N, n≥ r). In particular, the positive sufficient threshold n_0=r works uniformly in all allowed parameters.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-63adfcc6886184a1","tex":"\\EE_{Nn}^{(N,j)}","aliases":"EE_Nn^(N,j) E_Nn^(N,j) EE mathcal","display":"{\\mathcal E}_{Nn}^{(N,j)}","kind":"use","excerpt":"ed original supplement. Earlier pending or partial-coverage labels are historical; the final core records their resolution. 1. The complete external statement. For N>0 and 0≤ j<N, use the original unnormalized convention ∑_d≥0 E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The existing packet convention satisfies E_Nn^(N,j)=j!C_N,j(n). The formaliz","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9faafbdfe9a57a78","tex":"\\operatorname{Euler}","aliases":"Euler operatorname","display":"\\operatorname{Euler}","kind":"use","excerpt":"Let L be the ordinary Euler evaluation functional on a variable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear; it is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr E","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d73f1f85c3e00f5a","tex":"\\operatorname{Euler}_{2n+2r}","aliases":"Euler_2n+2r operatorname","display":"\\operatorname{Euler}_{2n+2r}","kind":"use","excerpt":"ariable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear; it is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e617fe37e532f98e","tex":"\\operatorname{Euler}_d","aliases":"Euler_d operatorname","display":"\\operatorname{Euler}_d","kind":"use","excerpt":"Let L be the ordinary Euler evaluation functional on a variable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear; it is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr E","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9c48deb718470362","tex":"\\operatorname{Even}","aliases":"Even operatorname","display":"\\operatorname{Even}","kind":"use","excerpt":"Both row indexing by r and the literal even-degree version with hypothesis Even(n) are checked. All rows include degree zero. The exact identity is a rational equality with the Euler coefficient's integer-to-rational cast made explicit.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9452831b9693a633","tex":"\\operatorname{EvenPart}","aliases":"EvenPart operatorname","display":"\\operatorname{EvenPart}","kind":"use","excerpt":"Even projection is identified coefficientwise. For any formal series f, let EvenPart(f)=(f(t)+f(-t))/2. Its nth coefficient is proved to be the coefficient of f when n is even and zero otherwise. Rescaling the inverse law proves sech(-t)=sech(t). The exponential multiplication identity then gives cosh^2(pt)=frac2+e^2pt+e^-2pt4, and hence ∑_r≥0S_2r(","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f7e053bb5a2cb140","tex":"\\operatorname{Ext}","aliases":"Ext operatorname","display":"\\operatorname{Ext}","kind":"use","excerpt":"The accompanying nolinkurlnative/Native_Conjectures_Progress.md records further results in restricted settings: an intrinsic finite Ext^1 obstruction for finitely presented commutative integer algebras, finite CRT reconstruction for polynomial solution sets, and a characterization of rectangular finite relations by equality of common labels. The ob","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fbd7b2c73ab2fd9c","tex":"\\operatorname{Ext}^1","aliases":"Ext^1 operatorname","display":"\\operatorname{Ext}^1","kind":"use","excerpt":"The accompanying nolinkurlnative/Native_Conjectures_Progress.md records further results in restricted settings: an intrinsic finite Ext^1 obstruction for finitely presented commutative integer algebras, finite CRT reconstruction for polynomial solution sets, and a characterization of rectangular finite relations by equality of common labels. The ob","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3ea44da66f1c1881","tex":"\\operatorname{Ext}^1_B","aliases":"Ext^1_B operatorname","display":"\\operatorname{Ext}^1_B","kind":"use","excerpt":"o(f)∈Ext^1_B(NL_B/ Z, F_p)","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F mathbb","display":"{\\mathbb F}","kind":"use","excerpt":"begingroupsmallraggedright Formal correspondence. leanHurwitz.shift_convolution, leanHurwitz.prime_shift_convolution, leanHurwitz.convolution_left_eq_zero, leanHurwitzInverse.inverse_convolution, and leanHurwitzInverse.inverse_prime_difference. Working directly with coefficient sequences avoids the undefined expression 1/n! in F_p when pmid n!. end","count":24},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"The divisions are exact integer divisions. Regard J as a linear map F_p^d→ F_p^r. Define","count":32},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"Use the augmented polynomial systems F(Y,T)=(F(Y,T),T-t_0), H(Y,T)=(H(Y,T),-v). Then F(a)=0, and its derivative is B=D F(a)=J · K","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ef4a9b3f84f761fd","tex":"F''","aliases":"F''","display":"F''","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9400ebafe84d3b7c","tex":"F'''","aliases":"F'''","display":"F'''","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-49ddaa56c96d4e05","tex":"f^{-1}","aliases":"f^-1","display":"f^{-1}","kind":"use","excerpt":"The reciprocal is genuinely integral. For f∈ I_p with constant coefficient one, the usual power-series reciprocal is used. Its inverse equation yields the exact coefficient recurrence a_f^-1(d)=δ_d0-∑_r=0^d-1 binom dr a_f^-1(r)a_f(d-r). Strong induction proves that all of these coefficients are integral. Combining this construction with the precedi","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0d653662a671cb05","tex":"\\mathcal F^2","aliases":"F^2 mathcal","display":"\\mathcal F^2","kind":"use","excerpt":"0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and Q=( F^2-1)/2.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h. It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by f_0,f_1,…,f_h is injective ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-80c95a4ca2fcbcc6","tex":"F_{1,0}","aliases":"F_1,0","display":"F_{1,0}","kind":"use","excerpt":"j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1; its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the product formula is empty, so the same argument applies without change; equivalently, F_1,0(t)=e^-t.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5396e69399603fd3","tex":"F_{11}","aliases":"F_11","display":"F_{11}","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e4c782c5c6e5ba3b","tex":"F_{14}","aliases":"F_14","display":"F_{14}","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f01538099e62c4bc","tex":"F_{17}","aliases":"F_17","display":"F_{17}","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-74317064bf2c9a4d","tex":"F_3","aliases":"F_3","display":"F_3","kind":"use","excerpt":"e a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. Since F_3=F_2D_3^-1, C_3(n)-C_2(n)=∑_i≥1h_ibinom4n4iC_2(n-i). The scaled-binomial argument","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-08c122811477fa77","tex":"\\F_3","aliases":"F_3 F mathbb","display":"{\\mathbb F}_3","kind":"use","excerpt":"Three checked boundaries. The initial row. For (p,m)=(3,2), the same reduced recurrence gives e_0=1 and e_1=-1=2 in F_3. Its proposed period is one, so the positive-tail result cannot in general include row zero.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-54c7a07f66f1253f","tex":"\\mathbb F_3","aliases":"F_3 mathbb","display":"\\mathbb F_3","kind":"use","excerpt":"setlengthitemsep0ptsetlengthparskip0pt item A failed singular lift: p=3, L=1, F(X)=X^2-3, x=0. Here e=-1≠0 in F_3 and J=0. There are no lifts of zero modulo 3 to a root modulo 9. item Representative independence is a quotient assertion: for F(X)=X^2-1, p=3, L=1, representatives 1 and 4 have residual digits 0 and 2. Their common Jacobian is 2, with ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a3a13a2b63deda5","tex":"F_{3k+j}","aliases":"F_3k+j","display":"F_{3k+j}","kind":"use","excerpt":"[t] centering includegraphics[width=.95linewidth,height=3.15in,keepaspectratio]../../figures/reconstructed/recurrence_packets.pdf captionNew reconstruction. Fibonacci residues F_3k+j, with F_0=0 and F_1=1. Each packet evolves by the same jump matrix C^3 and has a different initial address.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2c547ea3567b9987","tex":"F_{4,j}","aliases":"F_4,j","display":"F_{4,j}","kind":"use","excerpt":"Write C_j=C_4,j and F_j=F_4,j. The all-row theorem removes the threshold, but at N=4 it leaves a further factorial gain unused. The exact algebra below proves that gain for all four addresses and identifies several infinite families where it is sharp.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a1a1f9e3594eef3e","tex":"F_5","aliases":"F_5","display":"F_5","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-21edda27985631b1","tex":"F_8","aliases":"F_8","display":"F_8","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-21527d66bde36b36","tex":"f_{a+b,s+p^at}","aliases":"f_a+b,s+p^at","display":"f_{a+b,s+p^at}","kind":"use","excerpt":"[Exact local branch reconstruction] The family {f_a,s}_0≤ s<p^a determines f uniquely on Z_p. More precisely, if y∈ s+p^a Z_p, then f(y)=f_a,s(fracy-sp^a). For a,b≥1, (f_a,s)_b,t = f_a+b,s+p^at. Thus local refinement is exactly the p adic digit tree.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-45843ed5f03c5c85","tex":"f_{a,s}","aliases":"f_a,s","display":"f_{a,s}","kind":"use","excerpt":"For a≥1 and 0≤ s<p^a, define f_a,s(x)=f(p^ax+s).","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-879f46b871236f65","tex":"F_d","aliases":"F_d","display":"F_d","kind":"use","excerpt":"Let p be prime, R= Z_p, K= Q_p, and d,s,r∈ N. Let F=(F_1,…,F_d)∈ R[X_1,…,X_d]^d, a∈ R^d, and let J=DF(a)=(∂_jF_i(a))_i,j. Assume J∈GL_d(K), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then ∃! b∈ a+p^s+1R^d: F(b)=0, b-a∈ p^r-sR^d. All vector divisibility assertions are coordinatewise. The uniqueness predicate is the entire ball of depth s+1: competitors ne","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1551bc759aa860fc","tex":"f_h","aliases":"f_h","display":"f_h","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"Uniqueness from the first two packet values. For a fixed N,j, any integer sequence satisfying the proved sampled recurrence and starting at F_j,F_N+j agrees with F_Nk+j for all k. This is proved by two-step induction. It is a consequence of the established coefficients and recurrence, not a premise used to identify Fibonacci values initially.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"use","excerpt":"[Isometric packet bijections] For 0≤ j<4, define on natural numbers f_j(n)=fracU_j(n)-j!2^d_j, (d_0,d_1,d_2,d_3)=(1,1,5,3). There is a unique continuous map Φ_j: Z_2→ Z_2 whose value at every n∈ N equals the embedded rational f_j(n). It is an isometric bijection and Φ_j(0)=0: |Φ_j(x)-Φ_j(y)|_2=|x-y|_2 (x,y∈ Z_2). In particular, every element of Z_2","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0fe9f993f4ff4310","tex":"F_N","aliases":"F_N","display":"F_N","kind":"use","excerpt":"n independent Lucas sequence Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e3f4ba757b6d2cf8","tex":"f_n","aliases":"f_n","display":"f_n","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f9c7267a8d94d86f","tex":"F_{N+1}","aliases":"F_N+1","display":"F_{N+1}","kind":"use","excerpt":"n independent Lucas sequence Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bcc6851c91aea042","tex":"f_{n+h}","aliases":"f_n+h","display":"f_{n+h}","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a93e15faf4073060","tex":"F_{N+j}","aliases":"F_N+j","display":"F_{N+j}","kind":"use","excerpt":"Uniqueness from the first two packet values. For a fixed N,j, any integer sequence satisfying the proved sampled recurrence and starting at F_j,F_N+j agrees with F_Nk+j for all k. This is proved by two-step induction. It is a consequence of the established coefficients and recurrence, not a premise used to identify Fibonacci values initially.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a4653715eb491a76","tex":"F_{N,0}","aliases":"F_N,0","display":"F_{N,0}","kind":"use","excerpt":"Modules. mcodeEulerSeriesInverse, mcodeRiccatiFlatPropagation, mcodeRatioRiccati. 4. All-residue assembly and original coefficients. The standard-window source and inverse results supply R_1 and F_N,0. Induction over 1≤ c<N now has two proved cases. At pmid c, the singular weighted estimate and the new substitution theorem give flatness. At pnmid c","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a208d9b8983e3526","tex":"F_{N,j}","aliases":"F_N,j","display":"F_{N,j}","kind":"use","excerpt":"The formalization does not stop at a recurrence. In the variable z=t^N, construct K_N,j(z)=∑_n≥0fracj!(Nn+j)!z^n, F_N,j(z)=∑_n≥0fracC_N,j(n)(Nn)!z^n. It proves the factorial coefficient identity, then F_N,jK_N,j=1 and therefore F_N,j=K_N,j^-1.","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fe274658cc6c2602","tex":"F_{N,j+1}","aliases":"F_N,j+1","display":"F_{N,j+1}","kind":"use","excerpt":"Let K_j(t) be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is F_N,j(t)=j! K_j(t)^-1. The source/inverse identities and the factorial successor identity prove F_N,j+1=F_N,jR_j+1, F_N,j=F_N,0∏_c=1^jR_c. The product formula is not an assembly hypothesis. Proved flatness of all factors and the o","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fda40c00a5457366","tex":"\\F_p","aliases":"F_p F mathbb","display":"{\\mathbb F}_p","kind":"use","excerpt":"begingroupsmallraggedright Formal correspondence. leanHurwitz.shift_convolution, leanHurwitz.prime_shift_convolution, leanHurwitz.convolution_left_eq_zero, leanHurwitzInverse.inverse_convolution, and leanHurwitzInverse.inverse_prime_difference. Working directly with coefficient sequences avoids the undefined expression 1/n! in F_p when pmid n!. end","count":18},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9da98cb9c6f0ae74","tex":"\\mathbb F_p","aliases":"F_p mathbb","display":"\\mathbb F_p","kind":"use","excerpt":"odulo 2 has zero obstruction and lifts to 1,3 modulo 4. Both have nonzero obstruction at the next step. There is no root modulo 8 and no compatible infinite lift. item The raw Jacobian cokernel need not be presentation independent: the presentations F=(X) and G=(X,0) define the same solution system. Their raw obstruction groups at zero are respecti","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cad1ac88f506b2e2","tex":"\\F_p^2","aliases":"F_p^2 F mathbb","display":"{\\mathbb F}_p^2","kind":"use","excerpt":"The raw cokernel already reveals the difficulty. The presentations F=(X) and G=(X,0) describe the same zero locus over every coefficient ring. At zero their Jacobians are, respectively, F_pxrightarrow 1 F_p, F_pxrightarrow (1,0) F_p^2. Their raw cokernels are 0 and F_p. Both distinguished obstruction classes vanish, but the ambient obstruction grou","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aebbcb2c8d779eb8","tex":"\\F_p^d","aliases":"F_p^d F mathbb","display":"{\\mathbb F}_p^d","kind":"use","excerpt":"The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the paper now supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual dig","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2d6d5a8ba40d633c","tex":"\\mathbb F_p^d","aliases":"F_p^d mathbb","display":"\\mathbb F_p^d","kind":"use","excerpt":"The divisions are exact integer divisions. Regard J as a linear map F_p^d→ F_p^r. Define","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c58f74d8b2dada8e","tex":"\\F_p^e","aliases":"F_p^e F mathbb","display":"{\\mathbb F}_p^e","kind":"use","excerpt":"For prime p, rank-nullity gives dimker J_x=d-ρ_x and dim( F_p^e/im J_x)=e-ρ_x. The cardinality of an n-dimensional F_p-space is p^n. Combine these identities with the obstruction criterion and the residue equivalence.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cd1168baff4f0420","tex":"\\mathbb F_p^e","aliases":"F_p^e mathbb","display":"\\mathbb F_p^e","kind":"use","excerpt":"Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Write r_x=F(x)/p^L, b_x=r_xbmod p, A_x=DF(x)bmod p: F_p^d⟶ F_p^e, ρ_x=rank_ F_pA_x. The actual lift set is L_x,L= {y∈( Z/p^L+1 Z)^d: y≡ xpmodp^L, F(y)=0}, where polynomial evaluation in the displayed set is modulo p^L+1. Define O_x= F_p^e/imA_x, o_x=[b_x]. Lean proves # L_x,L= p^d-ρ","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3d7ee4d0e8e62dda","tex":"\\mathbb F_p^{e+q}","aliases":"F_p^e+q mathbb","display":"\\mathbb F_p^{e+q}","kind":"use","excerpt":"Let p be prime, L≥1, e,q≥0, and F∈ Z[X_1,…,X_e+q]^e, x∈ Z^e+q, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^e+q⟶ F_p^e is surjective. All vector congruences are coordinatewise.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ec134db719aa30eb","tex":"\\F_p^k","aliases":"F_p^k F mathbb","display":"{\\mathbb F}_p^k","kind":"use","excerpt":"ints] Let q=p be prime in Theorem~[source reference]. For every τ∈ t_0+p^L Z_p^k, there are an integer tower z_n and b∈ Z_p^m+k such that F(b)=0, free(b)=τ, z_n⟶ b, b-z_n∈ p^L+n Z_p^m+k. The initial value is z_0=x. At every stage and for every proposed parameter digit v∈ F_p^k, the corresponding next residue fiber has exactly p^m-e elements. If m=e","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7fa0d9b2392a00a1","tex":"\\mathbb F_p^k","aliases":"F_p^k mathbb","display":"\\mathbb F_p^k","kind":"use","excerpt":"Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N,","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6edbc189be464124","tex":"\\mathbb F_p^r","aliases":"F_p^r mathbb","display":"\\mathbb F_p^r","kind":"use","excerpt":"The divisions are exact integer divisions. Regard J as a linear map F_p^d→ F_p^r. Define","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0c3489d862af388a","tex":"\\F_p^\\times","aliases":"F_p^× F times mathbb times","display":"{\\mathbb F}_p^\\times","kind":"use","excerpt":"Induction gives the geometric identity. Reduction of a unit to F_p^× gives u_i^p-1=1 modulo p. Taking s powers supplies p^s in each summand. The difference operator is linear over finite sums.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ba2ac5456d322b89","tex":"f_q","aliases":"f_q","display":"f_q","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7ef1b4fdfc9f7935","tex":"f_R","aliases":"f_R","display":"f_R","kind":"use","excerpt":"The component maps f_R:A→ C_R and g_R:B→ C_R have the following universal property. For every set Y and maps h:A→ Y, k:B→ Y satisfying h(a)=k(b) whenever R(a,b), there is a unique map u:C_R→ Y with u∘ f_R=h and u∘ g_R=k.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6470f50faec26a53","tex":"F_r'","aliases":"F_r'","display":"F_r'","kind":"use","excerpt":"ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8bb98f0e3a56f2b0","tex":"F_{r-1}","aliases":"F_r-1","display":"F_{r-1}","kind":"use","excerpt":"ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-06e67f71ff301b19","tex":"\\fall","aliases":"fall","display":null,"kind":"use","excerpt":"[Dyadic Euler coordinate pair] For every even n≥0, E_n = ∑_r=0^n/2 β_r2^2r fall n2r R_n-2r. The inverse transform is R_n = ∑_r=0^n/2 frac2^4r+1(2r+2)! fall n2r E_n-2r.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f8ae171b4ed22f24","tex":"\\FF_3","aliases":"FF_3 F_3 FF mathbb","display":"{\\mathbb F}_3","kind":"use","excerpt":"[A singular polynomial obstruction] Let p=3 and consider the class x_1=0 modulo three. For F_0(x)=x^2, the residual and the reduced derivative both vanish. Every correction y∈ F_3 gives F_0(3y)≡0pmod9, so the class has three children. For F_1(x)=x^2+3, the reduced derivative is still zero, but the reduced residual is one. Its class in the cokernel ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f507805b8e18f517","tex":"\\FF_p^d","aliases":"FF_p^d F_p^d FF mathbb","display":"{\\mathbb F}_p^d","kind":"use","excerpt":"[One level linear lifting] A class represented by x_L lifts to a solution modulo p^L+1 if and only if Ob^lin_L(x_L)=0. More precisely, a correction y∈ F_p^d gives a lift x_L+1=x_L+p^Ly pmod p^L+1 if and only if overline My=-ρ_L(x_L). When one correction exists, the set of all correction classes is a coset of keroverline M.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-385760b8fa8c4e92","tex":"\\FF_p^\\times","aliases":"FF_p^× F_p^× FF times mathbb times","display":"{\\mathbb F}_p^\\times","kind":"use","excerpt":"Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-80af98c69987e79a","tex":"\\operatorname{Flat}","aliases":"Flat operatorname","display":"\\operatorname{Flat}","kind":"use","excerpt":"All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6ba1848f9fd13a41","tex":"\\operatorname{free}","aliases":"free operatorname","display":"\\operatorname{free}","kind":"use","excerpt":"∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate proj","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":92},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2b23da150b06fbef","tex":"G''","aliases":"G''","display":"G''","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a0c88372240ff8d1","tex":"G'''","aliases":"G'''","display":"G'''","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f6a65d66c1c2674f","tex":"G^2","aliases":"G^2","display":"G^2","kind":"use","excerpt":"d product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, star","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","excerpt":"For an actual inverse fg=1, applying the product rule and multiplying by g gives B(g)=-g^2B(f)-gM_p^2(f,g)+gM_p(f,g). If g∈ I, induction on r proves boundedness of g from boundedness of f. At each successor stage the preceding stage supplies boundedness of g needed on the right. Passing to all finite orders proves reciprocal closure for an integral","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ecc79b40ce07ca48","tex":"g^{b-a}","aliases":"g^b-a","display":"g^{b-a}","kind":"use","excerpt":"To prove surjectivity, use compactness. For an isometry g of a compact metric space into itself, the sequence of iterates g^n(x) has a convergent subsequence. Two sufficiently close iterates, with exponents a<b, yield d(x,g^b-a(x)) equally small after removing a isometric iterations. Since b-a>0, the latter point is in the image of g. Thus the imag","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-84fbee9e7706627b","tex":"g^n","aliases":"g^n","display":"g^n","kind":"use","excerpt":"To prove surjectivity, use compactness. For an isometry g of a compact metric space into itself, the sequence of iterates g^n(x) has a convergent subsequence. Two sufficiently close iterates, with exponents a<b, yield d(x,g^b-a(x)) equally small after removing a isometric iterations. Since b-a>0, the latter point is in the image of g. Thus the imag","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0a566b1db5acde7f","tex":"g_0","aliases":"g_0","display":"g_0","kind":"use","excerpt":"[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1, g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e6df94744f7ade62","tex":"g_2","aliases":"g_2","display":"g_2","kind":"use","excerpt":"The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1, g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ab6997694ee35093","tex":"g_3","aliases":"g_3","display":"g_3","kind":"use","excerpt":"The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1, g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e051232a9c93730e","tex":"G_{4,1}","aliases":"G_4,1","display":"G_{4,1}","kind":"use","excerpt":"The address-one source is G_4,1(t)=(sinh t+sin t)/(2t). The identities a+b=1, a-b=i, and ab=1/2 give D_1(t):=G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hur","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-29d433d9dcbeff86","tex":"G_{4,3}","aliases":"G_4,3","display":"G_{4,3}","kind":"use","excerpt":"The address-three source is G_4,3(t)=3(sinh t-sin t)/t^3. Addition identities give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, o","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-28a0f54c10c0e63f","tex":"G_A","aliases":"G_A","display":"G_A","kind":"use","excerpt":"AtlasHeadA derivative weight can cost one digit of root precisionatlas:weighted Compare the linear equations F(x)=3x=0 and G_A(x)=3x+3^A=0, where A≥2. Their coefficient difference has 3-adic valuation A, but their roots differ by x_A-0=-3^A-1, v_3(x_A)=A-1. AtlasImageweighted_motion The derivative has valuation one. Solving the correction equation ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cd7038b72872679a","tex":"G_i","aliases":"G_i","display":"G_i","kind":"use","excerpt":"If p^Lmid c, the actual identities give G(x)=F(x)+cH(x), DG(x)≡ DF(x)pmod p. Thus if F(x)≡0pmodp^L and A_x is surjective, then ∃ b∈ B_L(x):G(b)=0. This theorem allows rectangular systems. The evaluation, derivative and Jacobian identities are proved from the polynomial definition G_i=F_i+C(c)H_i, rather than supplied as hypotheses. Algebraic congru","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ec1c980979f73c29","tex":"G_j","aliases":"G_j","display":"G_j","kind":"use","excerpt":"Pack g_j into degrees divisible by four and form its actual EGF. The formal coefficient identity is G_j(t)=∑_n≥0g_j(n)fract^4n(4n)! =j!∑_n≥0fract^4n(4n+j)!. Its inverse is therefore the original normalized packet EGF. The source Kummer property gives flatness of G_j. The proved reciprocal closure gives flatness of G_j^-1.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f7288fb9331c8dd5","tex":"g_j","aliases":"g_j","display":"g_j","kind":"use","excerpt":"The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1,","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1d2ec925bc33e3de","tex":"g_{j+1}","aliases":"g_j+1","display":"g_{j+1}","kind":"use","excerpt":"For j=0, g_0 is the literal N-divisibility indicator. Because Nmid p(p-1), the two shifts in δ g_0 agree and δ g_0=0. It is flat at every order. The actual binomial identity gives Λ_j+1g_j+1=(j+1)g_j. As 1≤ j+1<p, the nonsingular reflection theorem inductively proves flatness of every g_j in the standard window. There is no uniform-convergence assu","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-570d8d5f32c21bda","tex":"G_j^{-1}","aliases":"G_j^-1","display":"G_j^{-1}","kind":"use","excerpt":"Pack g_j into degrees divisible by four and form its actual EGF. The formal coefficient identity is G_j(t)=∑_n≥0g_j(n)fract^4n(4n)! =j!∑_n≥0fract^4n(4n+j)!. Its inverse is therefore the original normalized packet EGF. The source Kummer property gives flatness of G_j. The proved reciprocal closure gives flatness of G_j^-1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-09f3e72028e864cd","tex":"G_m","aliases":"G_m","display":"G_m","kind":"use","excerpt":"[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1a4a7f45555ebe7e","tex":"g_m","aliases":"g_m","display":"g_m","kind":"use","excerpt":"pageheading2. Identification with the reduced Euler sequenceCorollary 8.5; manuscript page 21 Work over F_p= Z/p Z. For positive mmid p-1, take the literal section g_m(d)=1, · mmid d, 0, · mnmid d. It satisfies g_m(0)=1. Since p≡1pmod m, the conditions mmid d+p and mmid d+1 are equivalent. The source therefore proves U^pg_m=Ug_m.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-605af900ba9b414b","tex":"G_{N,j}","aliases":"G_N,j","display":"G_{N,j}","kind":"use","excerpt":"Throughout these sections, the normalized source and its reciprocal are G_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, F_N,j(t)=G_N,j(t)^-1 =∑_n≥0C_N,j(n)fract^Nn(Nn)!, where N≥1 and 0≤ j<N. In particular, C_N,j(0)=1. The coefficients in the original convention satisfy E_Nn^(N,j)=j!C_N,j(n), E_d^(N,j)=0 (Nnmid d). Thus every valuation of a nonzero difference gai","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-25fd9e2fc804df94","tex":"g_{N,j}","aliases":"g_N,j","display":"g_{N,j}","kind":"use","excerpt":"The packed factorial coefficient of G_N,j is g_N,j(n)=fracj!(Nn)!(Nn+j)! =∏_i=1^j(1+frac Ni n)^-1. The empty product is one. If N=p^a and 1≤ i<N, then v_p(i)<a, so N/i∈ p Z_p. Every affine denominator in [source reference] is therefore a unit on Z_p.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3f8720ad3f121275","tex":"G_r","aliases":"G_r","display":"G_r","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-077b802462413614","tex":"g_R","aliases":"g_R","display":"g_R","kind":"use","excerpt":"The component maps f_R:A→ C_R and g_R:B→ C_R have the following universal property. For every set Y and maps h:A→ Y, k:B→ Y satisfying h(a)=k(b) whenever R(a,b), there is a unique map u:C_R→ Y with u∘ f_R=h and u∘ g_R=k.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2532659d80614fed","tex":"G_r'","aliases":"G_r'","display":"G_r'","kind":"use","excerpt":"For a general depth r, call j_r=max J_r its true terminal. This definition depends on the actual carry minimum; rewriting the same integer as another wall expression does not change j_r. The universal simplicity target is for every integer r≥1 and x∈ Z_2, G_r(x)=0 Longrightarrow G_r'(x)≠0. The following are completed restricted results of the later","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-66816f2f34b0eaea","tex":"g_s","aliases":"g_s","display":"g_s","kind":"use","excerpt":"[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"Principal targets. targetUnitRefinement.omega_compatible, .principal_compatible, .principal_power targetDyadicUnits.dyadic_unit_decomposition, .decomposition_unique pageheading4. CRT addresses and the actual packet selectorsTheorem 10.1; manuscript page 22 Let a≥0, let M be odd, and put N=2^aM. Coprimality is derived from oddness: gcd(2^a,M)=1. The","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0fcc70536b44cfb8","tex":"\\operatorname{GL}","aliases":"GL operatorname","display":"\\operatorname{GL}","kind":"use","excerpt":"[Weighted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.","count":14},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bca14ec6fc8c9c7a","tex":"\\operatorname{GL}_d","aliases":"GL_d operatorname","display":"\\operatorname{GL}_d","kind":"use","excerpt":"[Weighted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d2847f1245d09235","tex":"\\operatorname{GL}_e","aliases":"GL_e operatorname","display":"\\operatorname{GL}_e","kind":"use","excerpt":"[Joint coefficient and parameter deformation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisf","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-daaa917c98a2902e","tex":"H^{-1}","aliases":"H^-1","display":"H^{-1}","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","excerpt":"There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus thro","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":", and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber H_0:=fracU^4-12 · =P+Q-fracP^22-fracQ^22 +3PQ+i E F(P-Q).","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers; steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2fd2f57e4f0efca6","tex":"\\mathcal H_2","aliases":"H_2 mathcal","display":"\\mathcal H_2","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-15c045eade51e6a9","tex":"H_{3,0}","aliases":"H_3,0","display":"H_{3,0}","kind":"use","excerpt":"It also derives coefT3H_3,0(log(1+T))=1/6 from the actual differential recurrence and checks its valuation v_3(1/6)=-1. Thus weighted membership does not mean ordinary coefficient integrality. This value attains the allowed floor weight at degree three.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-047a2d7f6191ab3d","tex":"H_a","aliases":"H_a","display":"H_a","kind":"use","excerpt":"[Scalar polyphase precision amplification] Let u∈ Z_p^×, let a,r≥0, and put H_a= (p-1)p^a, · p odd, 2^a+1, · p=2. Then Δ_H_a^ ru^n = u^n(u^H_a-1)^r. For odd p, v_p(Δ_H_a^ ru^n) ≥(a+1)r. For p=2 and odd u, v_2(Δ_H_a^ ru^n) ≥(a+3)r. Equivalently, every additional prime power digit increases the certified Mahler slope by one.","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b8541d1f64196274","tex":"H_c","aliases":"H_c","display":"H_c","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0b8eba8def9dc978","tex":"H_{c-1}","aliases":"H_c-1","display":"H_{c-1}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-98990972b3d9edcd","tex":"h_d","aliases":"h_d","display":"h_d","kind":"use","excerpt":"If m is not a power of two, s_2(m)≥2 makes μ_m even. If m is a positive power of two, the only central indices which can contribute an odd residue are r=1 and r=3, since r-2s_2(r)=-1 only at these two positive integers. The r=1 term is odd: T(m,1)=2^m-1 and v_2(m!)=m-1. For m≥4 a power of two, the r=3 term is even. More explicitly, fracT(m,3)2^m-3=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d3cba85783189f4d","tex":"H_i","aliases":"H_i","display":"H_i","kind":"use","excerpt":"If p^Lmid c, the actual identities give G(x)=F(x)+cH(x), DG(x)≡ DF(x)pmod p. Thus if F(x)≡0pmodp^L and A_x is surjective, then ∃ b∈ B_L(x):G(b)=0. This theorem allows rectangular systems. The evaluation, derivative and Jacobian identities are proved from the polynomial definition G_i=F_i+C(c)H_i, rather than supplied as hypotheses. Algebraic congru","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-333bf23be8bc780a","tex":"h_i","aliases":"h_i","display":"h_i","kind":"use","excerpt":"nly degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The constant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by i","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-be6006576b4a1fd7","tex":"h_{i-r}","aliases":"h_i-r","display":"h_{i-r}","kind":"use","excerpt":"(2^4i+2-1)B_4i+2(4i+2)(4i+1). The constant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial m","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0231a655bf13a98d","tex":"H_k","aliases":"H_k","display":"H_k","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers; steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":14},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cf5f97ebded34595","tex":"H_m","aliases":"H_m","display":"H_m","kind":"use","excerpt":"Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1c9a6345cfc1fd89","tex":"h_{m-3}","aliases":"h_m-3","display":"h_{m-3}","kind":"use","excerpt":"If m is not a power of two, s_2(m)≥2 makes μ_m even. If m is a positive power of two, the only central indices which can contribute an odd residue are r=1 and r=3, since r-2s_2(r)=-1 only at these two positive integers. The r=1 term is odd: T(m,1)=2^m-1 and v_2(m!)=m-1. For m≥4 a power of two, the r=3 term is even. More explicitly, fracT(m,3)2^m-3=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-59e39ba967eef7ff","tex":"H_{N,c}","aliases":"H_N,c","display":"H_{N,c}","kind":"use","excerpt":"Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical; the final core records their resolution. Scope and result. Let p be prime, let mmid p-1, and put N=mp. For 1≤ a<m, put c=ap and define the original exponential section and ratio by H_N,c(t)=∑_n≥","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7404334682ccef2c","tex":"H_{N,c}'","aliases":"H_N,c'","display":"H_{N,c}'","kind":"use","excerpt":"Modules. mcodeWeightedDworkTransfer, mcodeSeriesDifferenceBridge, mcodeRatioSeriesSupport. 3. Nonsingular propagation without an assumed limit. For consecutive original sections, the already defined ratios satisfy R_cH_N,c=tH_N,c-1, H_N,c'=H_N,c-1. Differentiating the first relation, using the relation for R_c-1, and cancelling a section whose lead","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-66ea18857523fa8b","tex":"H_{N,c-1}","aliases":"H_N,c-1","display":"H_{N,c-1}","kind":"use","excerpt":"Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical; the final core records their resolution. Scope and result. Let p be prime, let mmid p-1, and put N=mp. For 1≤ a<m, put c=ap and define the original exponential section and ratio by H_N,c(t)=∑_n≥","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2d64df6fa5868f4c","tex":"H_r","aliases":"H_r","display":"H_r","kind":"use","excerpt":"Modules. mcodeCyclotomicFiltration, mcodePrimeBlockPolynomial, and mcodeCyclicAmiceCoefficients. 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementa","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3e01b0cedad21f47","tex":"h_r","aliases":"h_r","display":"h_r","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-46b7885433f17d44","tex":"H_r'","aliases":"H_r'","display":"H_r'","kind":"use","excerpt":"ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bd53539eaf104c23","tex":"H_{r-1}","aliases":"H_r-1","display":"H_{r-1}","kind":"use","excerpt":"ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7e094a711b7cae00","tex":"H_\\nu","aliases":"H_ν nu","display":"H_\\nu","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of ","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4c4425940c285435","tex":"H_\\ell","aliases":"H_ℓ ell","display":"H_\\ell","kind":"use","excerpt":"the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero; the even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only a","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f50c3221ec49a3fa","tex":"\\Hcal","aliases":"Hcal H mathcal","display":"{\\mathcal H}","kind":"use","excerpt":"[Hurwitz series ring] Let R be a commutative ring with identity. The Hurwitz series ring H(R) is the set R^ N with componentwise addition and product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_r b_n-r. Its identity is δ=(1,0,0,…).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"[All positive periods and their least element] For r∈ N and 0≤ j<4, a positive integer T is a packet period of U_j modulo 2^r if and only if 2^max(0,r-d_j)mid T. Here a packet period means U_j(n+T)≡ U_j(n)pmod2^r for every n≥0. Consequently the least positive packet periods are c|cccc j · 0 · 1 · 2 · 3 least period · 2^max(0,r-1) · 2^max(0,r-1) · 2","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e0d53229e96ed0e9","tex":"\\operatorname{Hom}","aliases":"Hom operatorname","display":"\\operatorname{Hom}","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6e188ffeaff0072a","tex":"\\operatorname{Hom}_B","aliases":"Hom_B operatorname","display":"\\operatorname{Hom}_B","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-417e8bdb90ba73f1","tex":"\\Hp","aliases":"Hp H( Z_p) mathcal mathbb","display":"{\\mathcal H({\\mathbb Z}_p)}","kind":"use","excerpt":"The relevant integral subring is the Hurwitz subring H( Z_p)={f∈ Q_p[[t]]: n![t^n]f∈ Z_p for every n}. It is not coefficientwise integrality of ordinary power-series coefficients. The precise connection is n![t^n] E(F)=[T^0]Θ^nF. Since every iterate of Θ preserves W_p, the right side is integral. Therefore E( W_p)⊆ H( Z_p). Applying the same statem","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0745ce822383e740","tex":"I^2","aliases":"I^2","display":"I^2","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5b400594c9200202","tex":"I^b","aliases":"I^b","display":"I^b","kind":"use","excerpt":"ficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-04f25a28d6cda5bb","tex":"I^{\\lfloor d/p\\rfloor}","aliases":"I^lfloor d/prfloor lfloor rfloor","display":null,"kind":"use","excerpt":"For the falling factorial (x)_d=∏_i=0^d-1(x-i), grouping into consecutive blocks gives (x)_d∈ I^lfloor d/prfloor ⊆(p^v_p(lfloor d/prfloor!)). Every integral residue coordinate therefore satisfies p^v_p(lfloor d/prfloor!)mid [(r)](x)_d in Z. Dividing a coordinate by d! is done only afterward, using v_p(d!)=lfloor d/prfloor+v_p(lfloor d/prfloor!). Th","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-920dadf058c8fd57","tex":"I^p","aliases":"I^p","display":"I^p","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-52b36dcfd23ea8f2","tex":"I^q","aliases":"I^q","display":"I^q","kind":"use","excerpt":"sses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"ar targetUnitModes.unit_mode_valuation_gain targetUnitModes.finite_unit_modes_valuation_gain pageheading4. Exact examples and adversarial checksThe examples use the same definitions as the general results A polynomial anchor. The actual difference table for f(n)=n^2 yields (c_0,c_1,c_2,c_3)=(0,1,2,0). The finite interpolant therefore reconstructs I","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-20d6646340ce20f0","tex":"I_d","aliases":"I_d","display":"I_d","kind":"use","excerpt":"Let A=p^sJ^-1∈ M_d(R). Inclusion into Q_p and its injectivity give the two integral identities AJ=JA=p^sI_d. These identities are also an equivalent integral certificate for the first two hypotheses of [source reference]. In particular, A is injective: Aw=0 implies p^sw=JAw=0, hence w=0.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1c1fa6f17c9ee965","tex":"I_e","aliases":"I_e","display":"I_e","kind":"use","excerpt":"0 · cI_k. The identities AJ=JA=cI_e give CB=BC=cI_e+k. In the second product, the upper right block is -JAK+cK=0; in the first, it is AK-AK=0. Thus augmentation preserves the original inverse bound s.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-224958a47c5c3088","tex":"I_{e+k}","aliases":"I_e+k","display":"I_{e+k}","kind":"use","excerpt":"0 · cI_k. The identities AJ=JA=cI_e give CB=BC=cI_e+k. In the second product, the upper right block is -JAK+cK=0; in the first, it is AK-AK=0. Thus augmentation preserves the original inverse bound s.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-90844af211ae1183","tex":"I_k","aliases":"I_k","display":"I_k","kind":"use","excerpt":"Now suppose a chosen set of e columns of DF(x)bmod p is independent, and put those coordinates first. Apply the square result to the augmented polynomial map Ψ(Y,T)=(F(Y,T),T). Its reduced Jacobian has upper blocks (D_YF,D_TF) and lower blocks (0,I_k), so it is invertible. For t∈ T, the target (0,t) belongs to Ψ(x)+p^L Z_p^e+k because F(x)≡0pmodp^L","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a3ecf779d0b53571","tex":"I_M","aliases":"I_M","display":"I_M","kind":"use","excerpt":"E_q · 0 · 0 · ⋯ · 0 . It also satisfies S_M(E_q)^M=E_qI_M.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"use","excerpt":"Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c7b33cae672307e6","tex":"I_p","aliases":"I_p","display":"I_p","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"Define the finite obstruction module and its distinguished class by O_x=R_q^e/imJ_x, o_x=[ρ_q(r_x)]∈ O_x. The checked criterion is o_x=0 iff ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. The residual exists by the lower congruence and is unique because q>0. The quotient is finite since its ambient module R_q^e is finite. The theorem permits zero variable","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"O_F,bar x= F_p^r/imJ, o_F,bar x=[e].","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"For a uniform coefficient hypothesis, the simpler conclusion follows: inf_j≥0 w(s+j)≤ord_p(Δ_1^s f(n)). For each finite L≤ B_p(w;n,s), a separate theorem yields p^LmidΔ_1^s f(n) inside Z_p. Another specialization handles arbitrary finite interpolants with infinite weights beyond their degree.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ec79176d200cb3c","tex":"\\inf_{j\\geq0}","aliases":"inf_j≥0 inf geq","display":"\\inf_{j\\geq0}","kind":"use","excerpt":"For a uniform coefficient hypothesis, the simpler conclusion follows: inf_j≥0 w(s+j)≤ord_p(Δ_1^s f(n)). For each finite L≤ B_p(w;n,s), a separate theorem yields p^LmidΔ_1^s f(n) inside Z_p. Another specialization handles arbitrary finite interpolants with infinite weights beyond their degree.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-83d498114d998f28","tex":"J^p","aliases":"J^p","display":"J^p","kind":"use","excerpt":"[Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f63cab9c738188a7","tex":"J_a","aliases":"J_a","display":"J_a","kind":"use","excerpt":"At depth N≥1, suppose F(a)=q^Nr and a desired next integer parameter target satisfies t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exa","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f69f8efae610a884","tex":"J_{F,\\bar x}","aliases":"J_F,bar x bar","display":"J_{F,\\bar x}","kind":"use","excerpt":"The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the paper now supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual dig","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-79e7ca6f1d554bb2","tex":"J_{ij}","aliases":"J_ij","display":"J_{ij}","kind":"use","excerpt":"e_i=F_i(x)/p^Lpmod p, J_ij=frac∂ F_i∂ X_j(x)pmod p.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fe88e020806e1739","tex":"J_r","aliases":"J_r","display":"J_r","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-456e17f40199861f","tex":"j_r","aliases":"j_r","display":"j_r","kind":"use","excerpt":"For a general depth r, call j_r=max J_r its true terminal. This definition depends on the actual carry minimum; rewriting the same integer as another wall expression does not change j_r. The universal simplicity target is for every integer r≥1 and x∈ Z_2, G_r(x)=0 Longrightarrow G_r'(x)≠0. The following are completed restricted results of the later","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-07b6123d41bb6991","tex":"J_t","aliases":"J_t","display":"J_t","kind":"use","excerpt":"[Regularity throughout an open parameter set] Let U⊆ K^k be open, and let h be continuous on U. Suppose that for every t∈ U, F(h(t),t)=0, J_t=D_YF(h(t),t)∈GL_e( K). Then h is analytic at every point of U and its strict derivative is Dh(t)=-J_t^-1D_TF(h(t),t) (t∈ U). At each t∈ U, openness turns continuity on U into continuity at t and makes the roo","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0562af54b8ae3ad7","tex":"J_t^{-1}","aliases":"J_t^-1","display":"J_t^{-1}","kind":"use","excerpt":"[Regularity throughout an open parameter set] Let U⊆ K^k be open, and let h be continuous on U. Suppose that for every t∈ U, F(h(t),t)=0, J_t=D_YF(h(t),t)∈GL_e( K). Then h is analytic at every point of U and its strict derivative is Dh(t)=-J_t^-1D_TF(h(t),t) (t∈ U). At each t∈ U, openness turns continuity on U into continuity at t and makes the roo","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f7fb5e5ec3770a81","tex":"J_{x'}","aliases":"J_x'","display":"J_{x'}","kind":"use","excerpt":"[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-de4001a3c6225d07","tex":"J_{z(n)}","aliases":"J_z(n)","display":"J_{z(n)}","kind":"use","excerpt":"_0, t(n+1)-t(n)∈ q^L+n Z^k, there is a single tower z: N→ Z^m+k with z(0)=x, free(z(n))=t(n), F(z(n))∈ q^L+n Z^e, z(n+1)-z(n)∈ q^L+n Z^m+k. The parameter equality is exact in integers. For a≤ b, z(b)-z(a)∈ q^L+a Z^m+k, DF(z(n))bmod q=DF(x)bmod q. Consequently J_z(n)=J_x at every level. No later surjectivity or per-step solvability is assumed. This ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0015545e23b066dd","tex":"J_{z_n}","aliases":"J_z_n","display":"J_{z_n}","kind":"use","excerpt":"[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f44d0496740a2e92","tex":"J_{z_{n+1}}","aliases":"J_z_n+1","display":"J_{z_{n+1}}","kind":"use","excerpt":"At level N=L+n, divide the residual by q^N. Surjectivity supplies t∈ S_q^d with J_z_nt=-overlineF(z_n)/q^N. Choose an integer representative and apply [source reference] to obtain z_n+1. The new point is congruent to z_n modulo q. Evaluation of each formal partial derivative therefore gives J_z_n+1=J_z_n.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-25e27fdec946575c","tex":"K^d","aliases":"K^d","display":"K^d","kind":"use","excerpt":"mathbb Q_p, and d,s,M∈ N. Let F,H∈ R[X_1,…,X_d]^d, let a∈ R^d satisfy F(a)=0, and set J=DF(a), G=F+p^M H. Assume J∈GL_d(K), p^sJ^-1∈ M_d(R), M>2s. Then there is exactly one root b of G in a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d, and there is ε∈ p^2M-3sR^d such that, in K^d, b-a=-p^M J^-1H(a)+ε. The inverse in this formula belongs to the original Jac","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f9f756320594da4d","tex":"\\mathbb K^n","aliases":"K^n mathbb","display":"\\mathbb K^n","kind":"use","excerpt":"[Derivative of polynomial evaluation] For P∈ K[X_1,…,X_n], polynomial evaluation is analytic at every point. At a∈ K^n its strict Fr echet derivative is v⟼∑_j=1^n (∂_jP)(a)v_j. For a finite system, its strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9817d114fe256b80","tex":"k_0","aliases":"k_0","display":"k_0","kind":"use","excerpt":"For 0≤ i<j, let d_i=v_2(i!)+v_2((j-1-i)!), k_i=1+lfloorlog_2max(i,j-1-i)rfloor, B_i=i+2^k_i Z_2, t_i=d_i+k_i. At j=1 use d_0=k_0=t_0=0. The balls B_i partition Z_2. Indeed a residue represented twice among 0,…,j-1 splits into its two children; every other represented residue remains one ball. On each such ball, P_j(i+2^k_iy)=2^t_iyQ_i(y), Q_i∈ Z_2⟨","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fa83735d882dbba5","tex":"K_{4,j}","aliases":"K_4,j","display":"K_{4,j}","kind":"use","excerpt":"The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1,","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6cc92a51c6969e2c","tex":"K_a","aliases":"K_a","display":"K_a","kind":"use","excerpt":"At depth N≥1, suppose F(a)=q^Nr and a desired next integer parameter target satisfies t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exa","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-08dc6e54edbcd212","tex":"K_c","aliases":"K_c","display":"K_c","kind":"use","excerpt":"Removing a singular forced zero. Put c=ap<N and define the actual normalized kernel K_c(t)=c!∑_n≥0fract^Nn(Nn+c)!, K_c(0)=1. Let ℓ(T)=log(1+T)/T, defined by shifting logarithm coefficients, not by inverting T. Set U(T)=ℓ(T)^cK_c(log(1+T)). The formal proof establishes U(0)=1 and the exact factorization H_N,c(log(1+T))=fracT^cc!U(T). It follows coef","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-848f2d1221ba8b4d","tex":"K_{c-1}","aliases":"K_c-1","display":"K_{c-1}","kind":"use","excerpt":"Boundary. The original positive-address section has zero constant coefficient. It is not the series to which unit inverse closure is applied. 4. The singular ratio estimate. For a positive address c, the rational ratio is independently constructed from the original kernels: R_c(t)=c K_c-1(t)K_c(t)^-1. The scalar factor c is necessary. The source pr","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b783228933c35b11","tex":"k_i","aliases":"k_i","display":"k_i","kind":"use","excerpt":"For 0≤ i<j, let d_i=v_2(i!)+v_2((j-1-i)!), k_i=1+lfloorlog_2max(i,j-1-i)rfloor, B_i=i+2^k_i Z_2, t_i=d_i+k_i. At j=1 use d_0=k_0=t_0=0. The balls B_i partition Z_2. Indeed a residue represented twice among 0,…,j-1 splits into its two children; every other represented residue remains one ball. On each such ball, P_j(i+2^k_iy)=2^t_iyQ_i(y), Q_i∈ Z_2⟨","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ca931e9dd55037b6","tex":"K_j","aliases":"K_j","display":"K_j","kind":"use","excerpt":"Let K_j(t) be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is F_N,j(t)=j! K_j(t)^-1. The source/inverse identities and the factorial successor identity prove F_N,j+1=F_N,jR_j+1, F_N,j=F_N,0∏_c=1^jR_c. The product formula is not an assembly hypothesis. Proved flatness of all factors and the o","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-792ddfc072fc3946","tex":"k_m","aliases":"k_m","display":"k_m","kind":"use","excerpt":"The unit kernel and its reciprocal. Use a squared formal variable u and define k_m=frac2^2m+1(2m+2)!, K(u)=∑_m≥0k_mu^m, β_m=[u^m]K(u)^-1. The constant term is proved to be one. The factorial valuation bound gives v_2(k_m)=2m+1-v_2((2m+2)!)≥0. Thus every k_m belongs to O_2. A general coefficientwise subring theorem is proved by strong induction usin","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-25e4b588e90884b5","tex":"K_{N,j}","aliases":"K_N,j","display":"K_{N,j}","kind":"use","excerpt":"The normalized source and its inverse are K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coeffic","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c3531ccbb80b4887","tex":"K_{N,j}^{-1}","aliases":"K_N,j^-1","display":"K_{N,j}^{-1}","kind":"use","excerpt":"The formalization does not stop at a recurrence. In the variable z=t^N, construct K_N,j(z)=∑_n≥0fracj!(Nn+j)!z^n, F_N,j(z)=∑_n≥0fracC_N,j(n)(Nn)!z^n. It proves the factorial coefficient identity, then F_N,jK_N,j=1 and therefore F_N,j=K_N,j^-1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a30d27faec7f7f7","tex":"K_r","aliases":"K_r","display":"K_r","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e2b60819c95171e8","tex":"k_r","aliases":"k_r","display":"k_r","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ab625c241c8a0e88","tex":"K_t","aliases":"K_t","display":"K_t","kind":"use","excerpt":"There is also an explicit open-set version. Let U be open, let g be continuous on U, and suppose for every t∈ U that F(g(t),t)=0, J_t:=D_YF(g(t),t)∈GL_e( K). Writing K_t=D_TF(g(t),t), the theorem mcodecontinuous_selection_analyticOnNhd proves g is analytic at every t∈ U, Dg(t)=-J_t^-1K_t (t∈ U), with a strict Fréchet derivative at each point. The p","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-73642ab179f59f29","tex":"K_x","aliases":"K_x","display":"K_x","kind":"use","excerpt":"Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residu","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-83bf1e6422e6d967","tex":"K_{z_n}","aliases":"K_z_n","display":"K_{z_n}","kind":"use","excerpt":"At level N=L+n, write the supplied parameter increment exactly as t_n+1-t_n=q^Nw with w∈ Z^k. Surjectivity of the solved block gives a digit u satisfying J_z_nu=-overlineF(z_n)/q^N-K_z_noverline w. Choose an integer representative widetilde u and set z_n+1=z_n+q^N(widetilde u,w). Its parameter coordinates equal the supplied target exactly. Taylor e","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4b7ce4653ca285ed","tex":"K_\\ell","aliases":"K_ℓ ell","display":"K_\\ell","kind":"use","excerpt":"the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero; the even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only a","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N, F(bar w)=0, free(bar w)=t(n)+p^Nvpmodp^N+1 }. Here p^Nv means any integer representative of v, multiplied by p^N; the residue class does not depend on the representative. Then # L_n,v=p^m-e for every n and every v","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1c914ea4ff41f154","tex":"L_{-1}","aliases":"L_-1","display":"L_{-1}","kind":"use","excerpt":"Reciprocal transfer gives L_-1^kF_N,j∈ H_p. Extraction of the packed coefficient gives (S+1)^k C_N,j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1; its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the pr","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4997aff165ad7668","tex":"L_{-1}^k","aliases":"L_-1^k","display":"L_{-1}^k","kind":"use","excerpt":"Reciprocal transfer gives L_-1^kF_N,j∈ H_p. Extraction of the packed coefficient gives (S+1)^k C_N,j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1; its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the pr","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fd9ee043ad27b658","tex":"L_{-\\lambda}^b","aliases":"L_-λ^b lambda","display":"L_{-\\lambda}^b","kind":"use","excerpt":"Set B=A^-1. Its order-zero integrality was established in [source reference]. Assume by induction that L_-λ^bB∈ H_p for b<k. Since AB=1, L_0^k(AB)=(fracD^Np)^k1=0 (k>0). Use [source reference] with μ=-λ. Exactly one term has inverse order b=k: it has a=0 and all e_i=0, and is A L_-λ^k B. In every other term b<k, so its second factor is integral by ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cbcf44fd451f612c","tex":"L_{-\\lambda}^k","aliases":"L_-λ^k lambda","display":"L_{-\\lambda}^k","kind":"use","excerpt":"beginproof Set B=A^-1. Its order-zero integrality was established in [source reference]. Assume by induction that L_-λ^bB∈ H_p for b<k. Since AB=1, L_0^k(AB)=(fracD^Np)^k1=0 (k>0). Use [source reference] with μ=-λ. Exactly one term has inverse order b=k: it has a=0 and all e_i=0, and is A L_-λ^k B. In every other term b<k, so its second factor is i","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-832fc1443a44a791","tex":"L_0^k","aliases":"L_0^k","display":"L_0^k","kind":"use","excerpt":"Set B=A^-1. Its order-zero integrality was established in [source reference]. Assume by induction that L_-λ^bB∈ H_p for b<k. Since AB=1, L_0^k(AB)=(fracD^Np)^k1=0 (k>0). Use [source reference] with μ=-λ. Exactly one term has inverse order b=k: it has a=0 and all e_i=0, and is A L_-λ^k B. In every other term b<k, so its second factor is integral by ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6268c070d1fd0ce6","tex":"L_1^k","aliases":"L_1^k","display":"L_1^k","kind":"use","excerpt":"n adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then proves Δ_1^k g_N,j(n)∈ p^k Z_p for every k,n. The coefficient at degree Nn of L_1^kG_N,j is precisely p^-kΔ_1^k g_N,j(n), and the other residue classes vanish. T","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"dfrac2^4r+1(2r+1)(2r+2), · p=2. Then for every prime p and every even n≥0, A_p,n = ∑_r=0^n/2 binom n2r λ_p,r E_n-2r. Equivalently, if L_p(t)=K_p(t)^-1, then H_p(t)=L_p(t)secht, where L_p(t)=cosh^2(pt) for odd p, and L_2(t)=(fracsinh(2t)2t)^2 for p=2.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c7f6796e75b9a514","tex":"L_3","aliases":"L_3","display":"L_3","kind":"use","excerpt":"l source charts] Editorial range correction. This comparison holds for the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero; the even kernel L_3K_2 cannot cha","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5aa0fda5d74dd324","tex":"L_4","aliases":"L_4","display":"L_4","kind":"use","excerpt":"s.state_is_orbit, FibonacciPackets.packet_is_fibonacci targetFibonacciPackets.trace_power_is_lucas, FibonacciPackets.det_power targetFibonacciPackets.jump_charpoly targetFibonacciPackets.fibonacci_packet_recurrence pageheading4. Exact examples and meaningful boundary checksExample 9.3 and supporting results The independently defined Lucas recurrenc","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d9ec1bee07f32ca4","tex":"L_5","aliases":"L_5","display":"L_5","kind":"use","excerpt":"s.state_is_orbit, FibonacciPackets.packet_is_fibonacci targetFibonacciPackets.trace_power_is_lucas, FibonacciPackets.det_power targetFibonacciPackets.jump_charpoly targetFibonacciPackets.fibonacci_packet_recurrence pageheading4. Exact examples and meaningful boundary checksExample 9.3 and supporting results The independently defined Lucas recurrenc","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7787694a9216e833","tex":"\\mathcal L_{F,L}","aliases":"L_F,L mathcal","display":"\\mathcal L_{F,L}","kind":"use","excerpt":"For a fixed polynomial system F∈ Z[X_1,…,X_d]^e, a precision L≥1, and a root bar x modulo p^L, let L_F,L(bar x) ={x'∈( Z/p^L+1 Z)^d: F(x')=0, x'≡bar xpmodp^L}. The finite obstruction theorem decides whether this set is empty from the residual digit and reduced Jacobian. When it is nonempty, the theorem also determines its cardinality and affine ker","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"use","excerpt":"For a wall label j≥1 and 0≤ q<j, define e_j(q)=3q+s_2(j-1)-s_2(j-q-1), E_j=min_q<je_j(q), Θ(j)=3j+1+s_2(j-1), L_j=Θ(j)-E_j. Here E_j is a carry minimum, not an Euler number. A wall label is regular when e_j(q)>0 for 1≤ q<j. For r=2^L-j with L>L_j, put δ=L-L_j. The carry identities give μ_r=E_j-δ, J_r={j}, G_r(x)=P_j(x)U(x-j)+2^δ V(x), where U≡1pmod","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"use","excerpt":"The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ed76d6b3d182baf7","tex":"L_n","aliases":"L_n","display":"L_n","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7019a73bbeb4a019","tex":"L_{n+1}","aliases":"L_n+1","display":"L_{n+1}","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e69368456b55a8e6","tex":"L_{n+2}","aliases":"L_n+2","display":"L_{n+2}","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-626e812ed4f46844","tex":"\\mathcal L_{n,v}","aliases":"L_n,v mathcal","display":"\\mathcal L_{n,v}","kind":"use","excerpt":"Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N, F(bar w)=0, free(bar w)=t(n)+p^Nvpmodp^N+1 }. Here p^Nv means any integer representative of v, multiplied by p^N; the residue class does not depend on the representative. Then # L_n,v=p^m-e for every n and every v","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5168d74b869524fd","tex":"L_p","aliases":"L_p","display":"L_p","kind":"use","excerpt":"For a finite set S of primes, positive precisions L_p, and a fixed integer polynomial system F, put","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-533de406086c2d49","tex":"L_T","aliases":"L_T","display":"L_T","kind":"use","excerpt":"To pass from the section to the singular ratio, factor out its initial zero: for c=ap, the section has the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-af497bd89f303a99","tex":"\\mathcal L_{x,L}","aliases":"L_x,L mathcal","display":"\\mathcal L_{x,L}","kind":"use","excerpt":"At positive polynomial depth, restrict it to the correction fiber T_x={t∈ S_q^d:overliner_x+J_xt=0} and the lift set L_x,L= {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L, F(y)=0}. The polynomial predicate in the second display is evaluation in the residue ring. Evaluation on canonical integer representatives gives the same predicate, because integer polynomial eva","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-58a0d38624dd583b","tex":"\\mathcal L_{x,L,v}","aliases":"L_x,L,v mathcal","display":"\\mathcal L_{x,L,v}","kind":"use","excerpt":"Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residu","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a8bd4f0e5b0df5e","tex":"\\mathcal L_{z_n,L+n}","aliases":"L_z_n,L+n mathcal","display":"\\mathcal L_{z_n,L+n}","kind":"use","excerpt":"Surjectivity implies rank_ F_pA_x=e, e≤ d. Combining this with the preceding exact residue-lift count gives # L_z_n,L+n=p^d-e for every n. These are distinct solutions modulo p^L+n+1 lying above the selected point modulo p^L+n. The original finite obstruction group at each smooth point has cardinality one.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c0a4f94aeb2c3627","tex":"L_{\\lambda+\\mu}","aliases":"L_λ+μ lambda mu","display":"L_{\\lambda+\\mu}","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a16f7d6dd713850","tex":"L_{\\lambda+\\mu}^k","aliases":"L_λ+μ^k lambda mu","display":"L_{\\lambda+\\mu}^k","kind":"use","excerpt":"For completeness, the iterated formula is an integral multinomial formula, rather than a product rule with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-758c0b21a1e71ef3","tex":"L_\\lambda^a","aliases":"L_λ^a lambda","display":"L_\\lambda^a","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2788cc15f4b75ba1","tex":"L_\\mu","aliases":"L_μ mu","display":"L_\\mu","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2487b1443ba258e8","tex":"L_\\mu^b","aliases":"L_μ^b mu","display":"L_\\mu^b","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"The largest emergence threshold is max_i t_i=v_2((j-1)!)+lceillog_2 jrceil. For j=2^a, all thresholds equal j-1. There are no roots before that distance and exactly j simple roots at and after it, within the positive-distance range. The equality and pre-wall cases are also resolved: regular labels j≥2 have no roots there; j=1 has one simple odd roo","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3860d7e8f88507bb","tex":"\\operatorname{lcm}","aliases":"lcm operatorname","display":"\\operatorname{lcm}","kind":"use","excerpt":"What this final edition establishes. addcontentslinetocsectionWhat this final edition establishes The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-a","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":45},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e5c19d75887aa542","tex":"\\liminf","aliases":"liminf","display":"\\liminf","kind":"use","excerpt":"The radius is 2^liminf_d v_2(a_d)/d in the normalized dyadic absolute value. Hence R_0=R_1=2, R_2=2^5/2, R_3≥2. The exact radius of address three is not asserted. For address two, the lower bound gives convergence inside the claimed radius and the unbounded equality family prevents a larger one.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3e65bdfe891d46c9","tex":"\\liminf_d","aliases":"liminf_d liminf","display":"\\liminf_d","kind":"use","excerpt":"The radius is 2^liminf_d v_2(a_d)/d in the normalized dyadic absolute value. Hence R_0=R_1=2, R_2=2^5/2, R_3≥2. The exact radius of address three is not asserted. For address two, the lower bound gives convergence inside the claimed radius and the unbounded equality family prevents a larger one.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5c66bb4ce7ecf070","tex":"\\mathrm{lin}","aliases":"lin mathrm","display":"\\mathrm{lin}","kind":"use","excerpt":"[Linear lift obstruction] The linear obstruction at depth L is Ob^lin_L(x_L) = [ρ_L(x_L)] ∈ coker(overline M).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"[Finite Newton dominance] Let f: N→ Q and write a_k=Δ_1^kf(0). Suppose that c∈ N, a_1≠0, v_2(a_1)=c, and v_2(a_k)≥ c+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and T>0, f(n+T)-f(n)≠0, v_2(f(n+T)-f(n))=v_2(T)+c.","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-05e170b594a4705a","tex":"M^k","aliases":"M^k","display":"M^k","kind":"use","excerpt":"[Edge coefficients of the affine kernel] For every M≥1, s≥0, and k≥0, Θ_M,s(k,k)=M^k, Θ_M,s(Mk+s,k)=1. For binary radix and zero address, Θ_2,0(k+j,k) = 2^k-jbinom kj (0≤ j≤ k). Hence v_2(Θ_2,0(k+j,k)) = k-j+v_2binom kj.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-207e98afc142d0de","tex":"M_d","aliases":"M_d","display":"M_d","kind":"use","excerpt":"[Weighted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c828d32a87d213ac","tex":"M_I","aliases":"M_I","display":"M_I","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-df92634da3a3e068","tex":"M_i","aliases":"M_i","display":"M_i","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e9577d4e4232e0c8","tex":"m_k","aliases":"m_k","display":"m_k","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c4fb8d0ba7795506","tex":"M_p","aliases":"M_p","display":"M_p","kind":"use","excerpt":"∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral: the middle entries of a prime-power binomial row are divisible by p. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ef9064cfb2a52947","tex":"M_{p^2}","aliases":"M_p^2","display":"M_{p^2}","kind":"use","excerpt":"∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral: the middle entries of a prime-power binomial row are divisible by p. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dbd15cd98c87c0e2","tex":"M_q","aliases":"M_q","display":"M_q","kind":"use","excerpt":"pageheading1. The characteristic-zero product ruleThe normalized defect is not an ordinary derivation Let A be a commutative algebra over a characteristic-zero field K, and let U be an actual K-derivation. The source defines B=fracU^p^2-U^pp, M_q(f,g)=∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b13c32e31bf92433","tex":"m_\\nu","aliases":"m_ν nu","display":"m_\\nu","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f2efb8720cfae2fb","tex":"M_\\ell","aliases":"M_ℓ ell","display":"M_\\ell","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"use","excerpt":"[Common correction and fiber product] For every u,v∈ U, [u]=[v] in C(J,K) Longleftrightarrow ∃ w∈ U: Jw=Ju, Kw=Kv. Both label maps are surjective, and the image of the joint map is exactly the fiber product: im(U→im J×im K, u↦(Ju,Ku)) =im Jmathbin×_C(J,K)im K. If either V or W is finite, then C(J,K) is finite.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":23},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"use","excerpt":"The largest emergence threshold is max_i t_i=v_2((j-1)!)+lceillog_2 jrceil. For j=2^a, all thresholds equal j-1. There are no roots before that distance and exactly j simple roots at and after it, within the positive-distance range. The equality and pre-wall cases are also resolved: regular labels j≥2 have no roots there; j=1 has one simple odd roo","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4df078fcdf121684","tex":"\\mcode","aliases":"mcode","display":null,"kind":"use","excerpt":"_bijective, .address_components, .residue_equivalence targetCRTPackets.packet_factorization, .series_packet_factorization targetFiniteArithmeticExamples.outside_window_not_integral pageheading5. Verification, regressions and remaining workCumulative release evidence The fresh build started with an empty output directory under Lean 4.34.0-rc2 and Ma","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d0ce20525b9e7a28","tex":"\\min_{0\\leq j\\leq n}","aliases":"min_0≤ j≤ n min leq leq","display":"\\min_{0\\leq j\\leq n}","kind":"use","excerpt":"[Weighted difference bound] Let f: N→ Z_p and let w: N→ N∪{∞} satisfy w(s+j)≤ord_p(c_s+j(f)) for 0≤ j≤ n. Then ord_p(Δ_1^s f(n))≥ min_0≤ j≤ n {w(s+j)+v_pbinom nj}.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b0386100c5276882","tex":"\\min_{k\\leq r\\leq Mk+s}","aliases":"min_k≤ r≤ Mk+s min leq leq","display":"\\min_{k\\leq r\\leq Mk+s}","kind":"use","excerpt":"[Polyphase valuation ledger] For every refined coefficient, v_p(d_s,k) ≥ min_k≤ r≤ Mk+s ( v_p(c_r)+v_p(Θ_M,s(r,k)) ). If v_p(c_r)≥ w(r) for a nondecreasing function w, then v_p(d_s,k)≥ w(k). Thus every monotone Mahler precision ladder survives arbitrary polyphase refinement.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9bce0009cb30ebbc","tex":"\\min_{M+1\\leq r\\leq n/2}","aliases":"min_M+1≤ r≤ n/2 min leq leq","display":"\\min_{M+1\\leq r\\leq n/2}","kind":"use","excerpt":"[Optimal dyadic truncation] Let T_2,M(n) = ∑_r=0^min(M,n/2) κ_2,rbinom n2rR_n-2r. Then E_n≡ T_2,M(n)pmod 2^4M+3. More precisely, v_2(E_n-T_2,M(n)) ≥ min_M+1≤ r≤ n/2 ( 4r-1+v_2binom n2r ). The uniform exponent 4M+3 is best possible.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ff0993815f0d3b9","tex":"\\min_{q<j}","aliases":"min_q<j min","display":"\\min_{q<j}","kind":"use","excerpt":"For a wall label j≥1 and 0≤ q<j, define e_j(q)=3q+s_2(j-1)-s_2(j-q-1), E_j=min_q<je_j(q), Θ(j)=3j+1+s_2(j-1), L_j=Θ(j)-E_j. Here E_j is a carry minimum, not an Euler number. A wall label is regular when e_j(q)>0 for 1≤ q<j. For r=2^L-j with L>L_j, put δ=L-L_j. The carry identities give μ_r=E_j-δ, J_r={j}, G_r(x)=P_j(x)U(x-j)+2^δ V(x), where U≡1pmod","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e4b988cfa8d5b1ac","tex":"\\min_{q\\ge0}","aliases":"min_q≥0 min ge","display":"\\min_{q\\ge0}","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c55a63df7ee072d6","tex":"\\min_{r\\geq s}","aliases":"min_r≥ s min geq","display":"\\min_{r\\geq s}","kind":"use","excerpt":"[Mahler valuation ledger] If c_r∈ Z_p and v_p(c_r)≥ w(r), then v_p(Δ^sf(n)) ≥ min_r≥ s ( w(r)+v_pbinom nr-s ).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b0974bf1da548fe8","tex":"\\min_{\\substack{0\\leq r\\leq n\\\\w(r)\\geq L}}","aliases":"min_substack0≤ r≤ n w(r)≥ L min substack leq leq w geq","display":null,"kind":"use","excerpt":"w(r)<L binom nr c_rp^w(r)S_n-r pmod p^L. When the omitted set is nonempty, the tail has valuation at least min_substack0≤ r≤ n w(r)≥ L ( w(r)+v_pbinom nr+v_p(S_n-r) ).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"Let p be prime, R= Z_p, K= Q_p, and d,s,r∈ N. Let F=(F_1,…,F_d)∈ R[X_1,…,X_d]^d, a∈ R^d, and let J=DF(a)=(∂_jF_i(a))_i,j. Assume J∈GL_d(K), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then ∃! b∈ a+p^s+1R^d: F(b)=0, b-a∈ p^r-sR^d. All vector divisibility assertions are coordinatewise. The uniqueness predicate is the entire ball of depth s+1: competitors ne","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"rincipal targets. targetUnitModes.geometric_difference, UnitModes.unit_fermat_divisibility targetUnitModes.unit_mode_valuation_gain targetUnitModes.finite_unit_modes_valuation_gain pageheading4. Exact examples and adversarial checksThe examples use the same definitions as the general results A polynomial anchor. The actual difference table for f(n)","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5ee9bec015b3cbbd","tex":"n^p","aliases":"n^p","display":"n^p","kind":"use","excerpt":"By [source reference], n^p=0. The finite geometric sum gives the inverse.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","excerpt":"chi congruence] Let p be prime, let m>0, and suppose that mmid p-1 or m=2. Put N=mp and q=lcm(2,p-1). For every 0≤ j<N, all U_N,j(n) belong to Z_p. For r≥1 and n≥ r, E_Nn+qp^r^(N,j)≡ E_Nn^(N,j)pmodp^r. Thus the positive sufficient starting index n_0=r works uniformly. If mmid p-1, the stronger sufficient condition r≤ mn is valid.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ffaa0ad23479bca8","tex":"N_1","aliases":"N_1","display":"N_1","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e1613ab8834072c1","tex":"N_j","aliases":"N_j","display":"N_j","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2f56096dd29a0f85","tex":"N_s","aliases":"N_s","display":"N_s","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-44a7952294f06149","tex":"\\mathrm{NL}","aliases":"NL mathrm","display":"\\mathrm{NL}","kind":"use","excerpt":"o(f)∈Ext^1_B(NL_B/ Z, F_p)","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-331ed62039968d1b","tex":"\\mathrm{NL}_{B/\\mathbb Z}","aliases":"NL_B/ Z mathrm mathbb","display":"\\mathrm{NL}_{B/\\mathbb Z}","kind":"use","excerpt":"o(f)∈Ext^1_B(NL_B/ Z, F_p)","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":22},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-90f600277c50100a","tex":"\\NN","aliases":"NN N mathbb","display":"{\\mathbb N}","kind":"use","excerpt":"[Hurwitz series ring] Let R be a commutative ring with identity. The Hurwitz series ring H(R) is the set R^ N with componentwise addition and product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_r b_n-r. Its identity is δ=(1,0,0,…).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","excerpt":"rime_power_sparsification Editorial hypothesis correction. The affine-kernel congruence below needs no change, but its consequence for transformed Mahler coefficients requires c_r∈ Z_p for every r. Convergence of a Q_p-valued Mahler expansion alone is insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1; thus","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"O_F,bar x= F_p^r/imJ, o_F,bar x=[e].","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definition","count":13},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d2b3885ba0bd5cac","tex":"\\mathcal O_2","aliases":"O_2 mathcal","display":"\\mathcal O_2","kind":"use","excerpt":"cients is not universally formalized here. Formal-series identities do not establish analytic convergence. The corrected global odd-prime finite-difference theorem remains pending. Historical numerical evidence is not presented as a fresh reproduction. pageheading1. Integral coefficients and an exact unit sourceLemmas 5.1 and 5.2; manuscript page 1","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cf50fbf6ff02a71d","tex":"O_{F,\\bar x}","aliases":"O_F,bar x bar","display":"O_{F,\\bar x}","kind":"use","excerpt":"O_F,bar x= F_p^r/imJ, o_F,bar x=[e].","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ff62a298d080135d","tex":"o_{F,\\bar x}","aliases":"o_F,bar x bar","display":"o_{F,\\bar x}","kind":"use","excerpt":"The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the paper now supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual dig","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a2bdaaca11c4a62e","tex":"\\mathcal O_x","aliases":"O_x mathcal","display":"\\mathcal O_x","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-480cac95eb168013","tex":"\\operatorname{Ob}","aliases":"Ob operatorname","display":"\\operatorname{Ob}","kind":"use","excerpt":"[Linear lift obstruction] The linear obstruction at depth L is Ob^lin_L(x_L) = [ρ_L(x_L)] ∈ coker(overline M).","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c7af707f4c6857c9","tex":"\\operatorname{Ob}^{F}_L","aliases":"Ob^F_L operatorname","display":"\\operatorname{Ob}^{F}_L","kind":"use","excerpt":"[Polynomial lift obstruction] The polynomial obstruction at depth L is Ob^F_L(x_L) = [ρ_L^F(x_L)] ∈ coker(overline J_F(x_L)).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ff254cc6d240041","tex":"\\operatorname{Ob}^{\\mathrm{lin}}_L","aliases":"Ob^lin_L operatorname mathrm","display":"\\operatorname{Ob}^{\\mathrm{lin}}_L","kind":"use","excerpt":"[Linear lift obstruction] The linear obstruction at depth L is Ob^lin_L(x_L) = [ρ_L(x_L)] ∈ coker(overline M).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-95f972aa7326a181","tex":"\\operatorname{orbit}","aliases":"orbit operatorname","display":"\\operatorname{orbit}","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fd71dfd389a79ac5","tex":"\\operatorname{ord}","aliases":"ord operatorname","display":"\\operatorname{ord}","kind":"use","excerpt":"For x∈ Z_p, extend the valuation to zero by setting ord_p(x)= v_p(x), · x≠0, ∞, · x=0. Thus L≤ord_p(x) is equivalent to x∈ p^L Z_p.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ab3321d6961879fc","tex":"\\ord","aliases":"ord operatorname","display":"{\\operatorname{ord}}","kind":"use","excerpt":"The additive valuation is extended to ord_p(0)=∞. This avoids incorrectly applying Mathlib's finite-valued default at zero to an inequality that must also cover cancellation and identically zero differences.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d4679df4399518e1","tex":"\\ord_3","aliases":"ord_3 ord operatorname","display":"{\\operatorname{ord}}_3","kind":"use","excerpt":"ubsection*A sharp binomial-weight improvement Let F(n)=81binom n1+3binom n2+3binom n3+81binom n4. At n=3 and s=1, the four coefficient valuations are (4,1,1,4), while the binomial valuations are (0,1,1,0). Thus the refined bound is B_3(w;3,1)=min(4,2,2,4)=2, stronger than the coefficient-only lower bound one. The source checks both the minimum and ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9ee3ebe96b297ddb","tex":"\\operatorname{ord}_p","aliases":"ord_p operatorname","display":"\\operatorname{ord}_p","kind":"use","excerpt":"For x∈ Z_p, extend the valuation to zero by setting ord_p(x)= v_p(x), · x≠0, ∞, · x=0. Thus L≤ord_p(x) is equivalent to x∈ p^L Z_p.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bd164462222d475b","tex":"\\ord_p","aliases":"ord_p ord operatorname","display":"{\\operatorname{ord}}_p","kind":"use","excerpt":"The additive valuation is extended to ord_p(0)=∞. This avoids incorrectly applying Mathlib's finite-valued default at zero to an inequality that must also cover cancellation and identically zero differences.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0aeccd83542eabf9","tex":"\\overline C","aliases":"overline C overline","display":"\\overline C","kind":"use","excerpt":"[Lucas reduction and address independence] Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-481dece60de15875","tex":"\\overline C_m","aliases":"overline C_m overline","display":"\\overline C_m","kind":"use","excerpt":"[Lucas reduction and address independence] Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9b0ad0f92e6daea0","tex":"\\overline G","aliases":"overline G overline","display":"\\overline G","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-944f80df05aedc21","tex":"\\overline G_r","aliases":"overline G_r overline","display":"\\overline G_r","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-41dacfd07761cf94","tex":"\\overline r","aliases":"overline r overline","display":"\\overline r","kind":"use","excerpt":"The solved Jacobian remains surjective, so every shifted obstruction [overline r+K_z_nv] vanishes. Formula~[source reference] therefore gives p^m-e choices for every parameter digit at every stage. When m=e, the fixed-coordinate chart in the proof of Theorem~[source reference] contains every root in the initial ball with the specified parameters. I","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c226cd3303f20dba","tex":"\\overline s","aliases":"overline s overline","display":"\\overline s","kind":"use","excerpt":"[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f88991f9db432114","tex":"\\overline t","aliases":"overline t overline","display":"\\overline t","kind":"use","excerpt":"Every point in the lower residue class has the form y=x+q^Lt with t∈ Z^d. By Lemma~[source reference] and 2L≥ L+1, F(x+q^Lt)≡ q^L(r_x+DF(x)t)pmodq^L+1. The integer q^L is nonzero. Cancelling it in the corresponding divisibility condition proves F(x+q^Lt)≡0pmodq^L+1 Longleftrightarrow overliner_x+J_xoverline t=0. Integer vectors reduce onto S_q^d. T","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-db64b27099ab4aae","tex":"\\overline w","aliases":"overline w overline","display":"\\overline w","kind":"use","excerpt":"At level N=L+n, write the supplied parameter increment exactly as t_n+1-t_n=q^Nw with w∈ Z^k. Surjectivity of the solved block gives a digit u satisfying J_z_nu=-overlineF(z_n)/q^N-K_z_noverline w. Choose an integer representative widetilde u and set z_n+1=z_n+q^N(widetilde u,w). Its parameter coordinates equal the supplied target exactly. Taylor e","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-27c76c57aea96832","tex":"\\overline{(G_r-W_kU_k)/2^k}","aliases":"overline(G_r-W_kU_k)/2^k overline","display":"\\overline{(G_r-W_kU_k)/2^k}","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-064dd8c989fe8d11","tex":"\\overline{F(z_n)/q^N}","aliases":"overlineF(z_n)/q^N overline","display":"\\overline{F(z_n)/q^N}","kind":"use","excerpt":"At level N=L+n, divide the residual by q^N. Surjectivity supplies t∈ S_q^d with J_z_nt=-overlineF(z_n)/q^N. Choose an integer representative and apply [source reference] to obtain z_n+1. The new point is congruent to z_n modulo q. Evaluation of each formal partial derivative therefore gives J_z_n+1=J_z_n.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8be67fcf5a4dc680","tex":"\\overline{r_x}","aliases":"overliner_x overline","display":"\\overline{r_x}","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-227188b836d8f64b","tex":"\\overline{r_{x'}}","aliases":"overliner_x' overline","display":"\\overline{r_{x'}}","kind":"use","excerpt":"[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5f33e59d7a0611a5","tex":"\\overline{\\omega_\\ell(a)}","aliases":"overlineω_ℓ(a) overline omega ell","display":"\\overline{\\omega_\\ell(a)}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a0aa06ee1b3de418","tex":"\\overline{\\langle a\\rangle_\\ell}","aliases":"overline⟨ a⟩_ℓ overline langle rangle ell","display":"\\overline{\\langle a\\rangle_\\ell}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definition","count":113},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"The normalized source and its inverse are K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coeffic","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8e43551150771079","tex":"p^{(L+a)r}","aliases":"p^(L+a)r","display":"p^{(L+a)r}","kind":"use","excerpt":"by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Equivalently, Δ_H_a^ ra_n ≡0 pmod","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-03ce65729ffc1fad","tex":"p^{-(L+n)}","aliases":"p^-(L+n)","display":"p^{-(L+n)}","kind":"use","excerpt":"Cross-level compatibility bounds the supremum distance between later terms by p^-(L+n). Completeness gives a limit, and closed norm bounds retain the congruence to every z_n. View the coefficients of the original integer polynomials in Z_p. Their evaluations satisfy both F_i(z_n)→ F_i(a), by continuity, and F_i(z_n)→0, by their growing divisibility","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e57276c69cbfb04f","tex":"p^{-1}","aliases":"p^-1","display":"p^{-1}","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-241566f7741ad3f8","tex":"p^{-2r+\\vp((2r)!)}","aliases":"p^-2r+vp((2r)!) p^-2r+v_p((2r)!) vp","display":"p^{-2r+{v_p}((2r)!)}","kind":"use","excerpt":"Every retained base is a unit, and its (p-1)st power lies in 1+p Z_p. The polynomial p^2rbinom(p-1)x+j2r has Gauss norm at most p^-2r+v_p((2r)!). This tends to zero because p is odd. The remaining factors have bounded Gauss norm. Thus the series converges in the Tate algebra. At a nonnegative integer x, the binomial factor vanishes for large r, and","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-da8fc4cfd30eec05","tex":"p^{-k}","aliases":"p^-k","display":"p^{-k}","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand i","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-acf97391a5aa4604","tex":"p^{-s}","aliases":"p^-s","display":"p^{-s}","kind":"use","excerpt":"The earlier coefficient criterion is recovered exactly. Define the literal coefficient defect (δ a)(d)=a(d+p^2)-a(d+p). The new source proves, for every s,d, a_B^s f(d)=p^-s(δ^s a_f)(d). For an integral coefficient x, it proves the bridge p^-sx∈ Z_p Longleftrightarrow p^smid x in Z_p. Thus the differential definition is equivalent to the existing c","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":", and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber H_0:=fracU^4-12 · =P+Q-fracP^22-fracQ^22 +3PQ+i E F(P-Q).","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.","count":16},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3830d3e65aa149b6","tex":"p^{2(n-q)}","aliases":"p^2(n-q)","display":"p^{2(n-q)}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3f1812809381e70c","tex":"p^{2(r-q)}","aliases":"p^2(r-q)","display":"p^{2(r-q)}","kind":"use","excerpt":"The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r, 0, · q>r. The row-action equality, the diagonal identity U_r,r=1, and vanishing above the diagonal are all checked. ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4173c4ab8bf03143","tex":"p^{2a(M+1)}","aliases":"p^2a(M+1)","display":"p^{2a(M+1)}","kind":"use","excerpt":"[Odd modulus truncation] For every M≥0, every positive odd q, and every even n, E_n ≡ ∑_r=0^min(M,n/2) (-1)^rZ_2r+1 binom n2rq^2rS_n-2r(q) pmod q^2M+2. If pmid q and a=v_p(q), the same truncation is valid modulo p^2a(M+1).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b7d9778b6daa534f","tex":"p^{2D}","aliases":"p^2D","display":"p^{2D}","kind":"use","excerpt":"For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-80a8a6cec8adfc09","tex":"p^{2L}","aliases":"p^2L","display":"p^{2L}","kind":"use","excerpt":"Indeed, each term involving at least two correction factors is divisible by p^2L, and 2L≥ L+1. Constant and linear correction terms give the two displayed summands. Adding the monomial expansions proves (1) for every polynomial, with no denominator or analytic convergence issue.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7290eb9150ccad2b","tex":"p^{2M+2}","aliases":"p^2M+2","display":"p^{2M+2}","kind":"use","excerpt":"begingroupsmallraggedright Formal correspondence. leanFiltered.filtered_triangular_reconstruction and leanFiltered.padic_filtered_reconstruction. For weights 2r, retaining r≤ M gives the bound p^2M+2 through leanFiltered.even_weight_tail. A sharper kernel weight can improve this bound, as in the classical dyadic reconstruction later in the paper. E","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-02d4ba2fdf9c1939","tex":"p^{2M-3s}","aliases":"p^2M-3s","display":"p^{2M-3s}","kind":"use","excerpt":"[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e64e0fdf47880c65","tex":"p^{2r}","aliases":"p^2r","display":"p^{2r}","kind":"use","excerpt":"0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd, β_r2^2r(2r)!, · p=2.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b5b48fc7079a8109","tex":"p^{2s+1}","aliases":"p^2s+1","display":"p^{2s+1}","kind":"use","excerpt":"Precondition the polynomial system by A and set c=p^s+1. Its derivative at a is AJ=p^sI_d. Lemma~[source reference] gives a polynomial vector Q with Q(0)=0 and (AF)(a+cu)=(AF)(a)+p^2s+1u+p^2s+2Q(u). Since A is integral, (AF)(a)∈ p^rR^d. The strict integer inequality r>2s permits k=r-(2s+1)≥0 and a vector C∈ p^kR^d such that (AF)(a)=p^2s+1C. The nor","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-13f82bcd28f67ad7","tex":"p^{2s+2}","aliases":"p^2s+2","display":"p^{2s+2}","kind":"use","excerpt":"Precondition the polynomial system by A and set c=p^s+1. Its derivative at a is AJ=p^sI_d. Lemma~[source reference] gives a polynomial vector Q with Q(0)=0 and (AF)(a+cu)=(AF)(a)+p^2s+1u+p^2s+2Q(u). Since A is integral, (AF)(a)∈ p^rR^d. The strict integer inequality r>2s permits k=r-(2s+1)≥0 and a vector C∈ p^kR^d such that (AF)(a)=p^2s+1C. The nor","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ecd88a663ba55fc6","tex":"p^{[r]}","aliases":"p^[r]","display":"p^{[r]}","kind":"use","excerpt":"To isolate the filtration argument, let p be any element of a commutative ring and let w: N→ N∪{∞}. Set p^[r]=p^r for finite r and p^[∞]=0. The latter notation means the coefficient vanishes.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-59f14f096f9bc565","tex":"p^{[w(r)]}","aliases":"p^[w(r)]","display":"p^{[w(r)]}","kind":"use","excerpt":"[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a6f5fd37a9b086ab","tex":"p^{[w]}","aliases":"p^[w]","display":"p^{[w]}","kind":"use","excerpt":"ach earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral. pageheading3. Filtered tails and the standard windowSource items 2.2 and 8.2 Filtration at a finite coefficient row. Let R be any commutative ring. For w∈ N∪{∞} define","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bb1f5c7992523364","tex":"p^{[\\infty]}","aliases":"p^[∞] infty","display":"p^{[\\infty]}","kind":"use","excerpt":"To isolate the filtration argument, let p be any element of a commutative ring and let w: N→ N∪{∞}. Set p^[r]=p^r for finite r and p^[∞]=0. The latter notation means the coefficient vanishes.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ca74460520851a88","tex":"p^{a+b}","aliases":"p^a+b","display":"p^{a+b}","kind":"use","excerpt":"The cosets s+p^a Z_p form a disjoint partition of Z_p. The first formula is the inverse affine change of variables on one coset. The second follows from p^a(p^bx+t)+s = p^a+bx+p^at+s.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-30f8aa50194b988f","tex":"P^b","aliases":"P^b","display":"P^b","kind":"use","excerpt":"ficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7f4968674b99b5fb","tex":"p^b","aliases":"p^b","display":"p^b","kind":"use","excerpt":"The cosets s+p^a Z_p form a disjoint partition of Z_p. The first formula is the inverse affine change of variables on one coset. The second follows from p^a(p^bx+t)+s = p^a+bx+p^at+s.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8bd8bb8d750c7558","tex":"P^{b+v_p(b!)}","aliases":"P^b+v_p(b!)","display":"P^{b+v_p(b!)}","kind":"use","excerpt":"ficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-856313dc0118a734","tex":"p^D","aliases":"p^D","display":"p^D","kind":"use","excerpt":"For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-993740a96e2e3723","tex":"p^{d-d}","aliases":"p^d-d","display":"p^{d-d}","kind":"use","excerpt":"For square uniqueness, let a,b∈ B_L(x) be arbitrary exact roots. Their canonical integer truncations are genuine roots in the respective finite residue rings, since evaluation commutes with reduction. They agree at depth L. If they agree at depth N≥ L, both next residues belong to the same lift set, which has cardinality p^d-d=1. Induction gives ag","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a9aa353231261c8d","tex":"p^{d-e}","aliases":"p^d-e","display":"p^{d-e}","kind":"use","excerpt":"[Completion and smooth branching] In Theorem~[source reference], let q=p be prime. The tower can be chosen with a limit a∈ Z_p^d such that F(a)=0, z_n⟶ a, a-z_n∈ p^L+n Z_p^d (n≥0). At each stage, the number of distinct next-level residue roots over z_n is exactly p^d-e, and the finite obstruction module has one element. If d=e, there is exactly one","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2ac3eb320a31908f","tex":"p^{d-\\operatorname{rank}J}","aliases":"p^d-rankJ operatorname","display":"p^{d-\\operatorname{rank}J}","kind":"use","excerpt":"Theorem. For this fixed polynomial presentation, the group and distinguished class are independent of the chosen integer representative. The class vanishes if and only if bar x lifts to a solution modulo p^L+1. The obstruction group has p^r-rankJ elements. When lifts exist, their set is a torsor for ker J and contains exactly p^d-rankJ elements.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5dd0b282af35fa57","tex":"p^{d-\\rho}","aliases":"p^d-ρ rho","display":"p^{d-\\rho}","kind":"use","excerpt":"[Exact number of linear children] If ρ=rank_ F_poverline M, then a solvable depth L class has exactly p^d-ρ children modulo p^L+1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2810b88572ae96bd","tex":"p^{d-\\rho_x}","aliases":"p^d-ρ_x rho","display":"p^{d-\\rho_x}","kind":"use","excerpt":"[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0, 0, · o_x≠0, # O_x=p^e-ρ_x.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-06c5d845bc4788fe","tex":"p^{e-\\rho}","aliases":"p^e-ρ rho","display":"p^{e-\\rho}","kind":"use","excerpt":"When q=p is prime, put ρ=rank_ F_pJ. Then # L_v= p^m-ρ, · o_x(v)=0, 0, · o_x(v)≠0, # C_x=p^e-ρ. The count is of distinct actual residue classes at level L+1. The solved dimension m, rather than the total number m+k of variables, appears in the exponent because the parameter digit is fixed. These rank formulas are asserted only over prime fields.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4b6e4666d99160cc","tex":"p^{e-\\rho_x}","aliases":"p^e-ρ_x rho","display":"p^{e-\\rho_x}","kind":"use","excerpt":"[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0, 0, · o_x≠0, # O_x=p^e-ρ_x.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-97bae72ced80959b","tex":"p^i","aliases":"p^i","display":"p^i","kind":"use","excerpt":"To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar mult","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b7f360bb2799a8d5","tex":"p^k","aliases":"p^k","display":"p^k","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand i","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5ad7e5363781b727","tex":"p^{k(1+v_p(u))}","aliases":"p^k(1+v_p(u))","display":"p^{k(1+v_p(u))}","kind":"use","excerpt":"To incorporate v_2(u), expand a displacement using finite Newton coefficients. For any prime p, u>0, and 1≤ a≤ u, p^v_p(u)+1midbinom ua p^a, as follows from abinom ua=ubinomu-1a-1 and v_p(a)<a. Repeated displacement, retaining the remaining base-difference order at each induction step, gives a factor p^k(1+v_p(u)) from k displacements. This inducti","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-231af438231aa3c5","tex":"p^{k+\\sum_\\nu v_p(H_\\nu)}","aliases":"p^k+∑_ν v_p(H_ν) sum nu nu","display":"p^{k+\\sum_\\nu v_p(H_\\nu)}","kind":"use","excerpt":"ng Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar multiplied by p^M, and is divisible by p^k+∑_ν v_p(H_ν). This proves ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1cfaa611e7831efc","tex":"p^{k-1}","aliases":"p^k-1","display":"p^{k-1}","kind":"use","excerpt":"c branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors al","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6ba0888d650823c4","tex":"p^{ks}","aliases":"p^ks","display":"p^{ks}","kind":"use","excerpt":"-field reductionTheorem 7.4; manuscript page 18 For u in any commutative ring, and every natural step h, iteration proves Δ_h^s(u^n)=u^n(u^h-1)^s. The proof unfolds the actual forward-difference operator and uses the power multiplication law. No valuation premise or geometric-difference identity is assumed. More generally, a proved divisibility p^k","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e8ec6caedf71e9a2","tex":"p^{L(R+1-d)}","aliases":"p^L(R+1-d)","display":"p^{L(R+1-d)}","kind":"use","excerpt":"f degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) uniformly on Z_p.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-48e2844ef33a2852","tex":"p^{L+1}","aliases":"p^L+1","display":"p^{L+1}","kind":"use","excerpt":"[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ","count":19},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-09c8b25bb1d743c8","tex":"p^{L+a}","aliases":"p^L+a","display":"p^{L+a}","kind":"use","excerpt":"r Editorial hypothesis correction. Require L≥1. At L=0 the standalone claim is false: for p=3, d=1 and X=2=1+B, one has X^3-1=7∉3 Z_3. With L≥1, every higher binomial term in the retained proof has the required valuation, including p=2. If X=I+p^LB with B∈ M_d( Z_p), then for every a≥0 there is B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d85c672622d89ea4","tex":"p^{L+n}","aliases":"p^L+n","display":"p^{L+n}","kind":"use","excerpt":"[Completion and smooth branching] In Theorem~[source reference], let q=p be prime. The tower can be chosen with a limit a∈ Z_p^d such that F(a)=0, z_n⟶ a, a-z_n∈ p^L+n Z_p^d (n≥0). At each stage, the number of distinct next-level residue roots over z_n is exactly p^d-e, and the finite obstruction module has one element. If d=e, there is exactly one","count":13},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f5e33ab8d0aedef0","tex":"p^{L+n+1}","aliases":"p^L+n+1","display":"p^{L+n+1}","kind":"use","excerpt":"Surjectivity implies rank_ F_pA_x=e, e≤ d. Combining this with the preceding exact residue-lift count gives # L_z_n,L+n=p^d-e for every n. These are distinct solutions modulo p^L+n+1 lying above the selected point modulo p^L+n. The original finite obstruction group at each smooth point has cardinality one.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-944126e1f1284f31","tex":"p^{L-1}","aliases":"p^L-1","display":"p^{L-1}","kind":"use","excerpt":"For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1ad627b635cdde1b","tex":"p^{L_p}","aliases":"p^L_p","display":"p^{L_p}","kind":"use","excerpt":"N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-13b34b1568997397","tex":"p^{\\lfloor d/p\\rfloor}","aliases":"p^lfloor d/prfloor lfloor rfloor","display":null,"kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5ed6db9428239d9e","tex":"p^{Lr}","aliases":"p^Lr","display":"p^{Lr}","kind":"use","excerpt":"means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not an as-written certificate. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_H","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6ef9147e2bbff9d5","tex":"P^m","aliases":"P^m","display":"P^m","kind":"use","excerpt":"at moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e851b902cf92a681","tex":"p^M","aliases":"p^M","display":"p^M","kind":"use","excerpt":"[Smooth coefficient stability] Let p be prime, M≥1, F,H∈ Z[X_1,…,X_d]^e, and c∈ p^M Z. Suppose DF(x)bmod p is surjective for an integer x∈ Z^d. If a∈ Z_p^d satisfies F(a)=0 and a≡ xpmod p, then there is a root b of G=F+cH with b-a∈ p^M Z_p^d.","count":31},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7441b5cfbaa2c0c0","tex":"p^m","aliases":"p^m","display":"p^m","kind":"use","excerpt":"To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar mult","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e7c6808c9432177e","tex":"p^{M-(2s+1)}","aliases":"p^M-(2s+1)","display":"p^{M-(2s+1)}","kind":"use","excerpt":"Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents a","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-79b8bb76c63fb047","tex":"p^{m-e}","aliases":"p^m-e","display":"p^{m-e}","kind":"use","excerpt":"Theorem~[source reference]. For every τ∈ t_0+p^L Z_p^k, there are an integer tower z_n and b∈ Z_p^m+k such that F(b)=0, free(b)=τ, z_n⟶ b, b-z_n∈ p^L+n Z_p^m+k. The initial value is z_0=x. At every stage and for every proposed parameter digit v∈ F_p^k, the corresponding next residue fiber has exactly p^m-e elements. If m=e, then for each τ there is","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-66d21ad971fd5eb9","tex":"p^{M-s}","aliases":"p^M-s","display":"p^{M-s}","kind":"use","excerpt":"[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-516983ba38151684","tex":"p^{M-s-1}","aliases":"p^M-s-1","display":"p^{M-s-1}","kind":"use","excerpt":"Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents a","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1b12f28620c9cf18","tex":"p^{m-\\rho}","aliases":"p^m-ρ rho","display":"p^{m-\\rho}","kind":"use","excerpt":"When q=p is prime, put ρ=rank_ F_pJ. Then # L_v= p^m-ρ, · o_x(v)=0, 0, · o_x(v)≠0, # C_x=p^e-ρ. The count is of distinct actual residue classes at level L+1. The solved dimension m, rather than the total number m+k of variables, appears in the exponent because the parameter digit is fixed. These rank formulas are asserted only over prime fields.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aedad308698af110","tex":"p^{m-\\rho_x}","aliases":"p^m-ρ_x rho","display":"p^{m-\\rho_x}","kind":"use","excerpt":"[Prescribed-parameter obstruction and count] Define o_x(v)=[overliner_x+K_xv]∈ S_q^e/im J_x. Then L_x,L,v is nonempty if and only if o_x(v)=0. When nonempty, it is a torsor for ker J_x. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L,v =p^m-ρ_x, · o_x(v)=0, 0, · o_x(v)≠0, #( F_p^e/im J_x)=p^e-ρ_x. The criterion holds uniformly for every prescrib","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b3455be3fc985410","tex":"p^{\\min(k(1+\\nu_p(u)),mn)}","aliases":"p^min(k(1+ν_p(u)),mn) min nu","display":"p^{\\min(k(1+\\nu_p(u)),mn)}","kind":"use","excerpt":"[Step-sensitive divisor-branch bound] Under the hypotheses of Theorem~[source reference], for all u,k,n∈ N, Δ_u(p-1)/m^kU_mp,j(n) ∈ p^min(k(1+ν_p(u)),mn) Z_p. For C_mp,j the corresponding valuation lower bound is min(k(1+ν_p(u)),mn)-v_p(j!).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-07593e8256373d1f","tex":"p^{\\min(k,r)}","aliases":"p^min(k,r) min","display":"p^{\\min(k,r)}","kind":"use","excerpt":"[Prime-power period from finite differences] Let R be a commutative ring, p a prime, f: N→ R, h,n∈ N, and r≥1. If p^min(k,r)midΔ_h^k f(n) for every k∈ N, then p^rmid f(n+p^r-1h)-f(n).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d0c8f9332b23f7da","tex":"p^{mn}","aliases":"p^mn","display":"p^{mn}","kind":"use","excerpt":"ly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d6c9282569aea3fa","tex":"p^N","aliases":"p^N","display":"p^N","kind":"use","excerpt":"t, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is the forward map, then for every N≥0, h(t)≡ h(u)pmodp^N Longleftrightarrow t≡ up","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d3cdc9ecc001d6c4","tex":"p^n","aliases":"p^n","display":"p^n","kind":"use","excerpt":"For prime p, rank-nullity gives dimker J_x=d-ρ_x and dim( F_p^e/im J_x)=e-ρ_x. The cardinality of an n-dimensional F_p-space is p^n. Combine these identities with the obstruction criterion and the residue equivalence.","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-01d645aa75670391","tex":"p^{N+1}","aliases":"p^N+1","display":"p^{N+1}","kind":"use","excerpt":"First consider a square integer polynomial map P whose reduced Jacobian at x is surjective. Its Jacobian is invertible over F_p. A digit comparison using Taylor expansion shows that equal outputs modulo p^N+1 force equal inputs modulo p^N+1 whenever the inputs already agree modulo p^N and belong to B_1(x). Induction and integer truncation give, for","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c020cb0886702b18","tex":"p^{n+1}","aliases":"p^n+1","display":"p^{n+1}","kind":"use","excerpt":"To obtain the sharper displacement, begin with u∈ R^d. If u∈ p^nR^d, then Q(0)=0 and polynomial congruence give Q(u)∈ p^nR^d. For n<k, both C and pQ(u) are divisible by p^n+1. Equation~[source reference] therefore improves the divisibility of u by one digit. Induction gives u∈ p^kR^d. Thus b=a+p^s+1u satisfies [source reference], because s+1+k=r-s.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2adb92d07b292296","tex":"p^q","aliases":"p^q","display":"p^q","kind":"use","excerpt":"The scalar input is proved for every natural b, including nonunits: p^2mid b^p^2-b^p in Z. Because Θ(1+T)^b=b(1+T)^b, the binomial expansion of T^p=((1+T)-1)^p then proves stability of the divided generator T^p/p. For d=pq+r, 0≤ r<p, every weighted monomial has the exact factorization a_dT^d=(p^qa_d)(fracT^pp)^qT^r. The scalar p^qa_d is integral by","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6f0ea21bbfbf69fb","tex":"p^r","aliases":"p^r","display":"p^r","kind":"use","excerpt":"nal edition establishes addcontentslinetocsectionWhat this final edition establishes The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integ","count":21},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-07c89112aaa8fc8b","tex":"p^{r-\\operatorname{rank}J}","aliases":"p^r-rankJ operatorname","display":"p^{r-\\operatorname{rank}J}","kind":"use","excerpt":"Theorem. For this fixed polynomial presentation, the group and distinguished class are independent of the chosen integer representative. The class vanishes if and only if bar x lifts to a solution modulo p^L+1. The obstruction group has p^r-rankJ elements. When lifts exist, their set is a torsor for ker J and contains exactly p^d-rankJ elements.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2cc2ee50888afb46","tex":"p^{r-s}","aliases":"p^r-s","display":"p^{r-s}","kind":"use","excerpt":"hted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-61ad220e94c8aa08","tex":"p^{r-s+1}","aliases":"p^r-s+1","display":"p^{r-s+1}","kind":"use","excerpt":"and displacement boundary] For p=2, F(X)=X^2+3, and a=1, one has F(a)=4 and DF(a)=2. Thus s=1 and r=2 satisfy every displayed hypothesis except the strict inequality. No root exists in Z_2, since none exists modulo eight. Hence r≥2s cannot replace r>2s uniformly. Also, F(X)=p^sX+p^r has root -p^r-s, whose displacement is not divisible by p^r-s+1. T","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-360f794da8094012","tex":"p^s","aliases":"p^s","display":"p^s","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":41},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e4e5526bd8628ebc","tex":"p^{s+1}","aliases":"p^s+1","display":"p^{s+1}","kind":"use","excerpt":"[Weighted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.","count":20},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-62f2748157764c24","tex":"p^{s+1+k}","aliases":"p^s+1+k","display":"p^{s+1+k}","kind":"use","excerpt":"For completeness, the sharp precision is additional to this fixed-point existence. Inductively, if u∈ p^nR^d, then polynomial congruence and Q(0)=0 imply Q(u)∈ p^nR^d. For n<k, the equation u=-C-pQ(u) therefore gives u∈ p^n+1R^d. Starting at n=0, this proves u∈ p^kR^d. Hence b=a+tu satisfies b-a∈ p^s+1+kR^d=p^r-sR^d. The normalization and injectivi","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a5b6d2f623a6916e","tex":"p^{v_p(\\lfloor d/p\\rfloor!)}","aliases":"p^v_p(lfloor d/prfloor!) lfloor rfloor","display":null,"kind":"use","excerpt":"1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5ca86a25a0110cc3","tex":"P^{v_p(q!)}","aliases":"P^v_p(q!)","display":"P^{v_p(q!)}","kind":"use","excerpt":"p). Expanding ((z-1)+1)^p=1 and using divisibility of every interior binomial coefficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c345ea0f18269172","tex":"p^{v_p(q!)}","aliases":"p^v_p(q!)","display":"p^{v_p(q!)}","kind":"use","excerpt":"sses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f8f0c43926925108","tex":"p^{v_p(u)+1}","aliases":"p^v_p(u)+1","display":"p^{v_p(u)+1}","kind":"use","excerpt":"To incorporate v_2(u), expand a displacement using finite Newton coefficients. For any prime p, u>0, and 1≤ a≤ u, p^v_p(u)+1midbinom ua p^a, as follows from abinom ua=ubinomu-1a-1 and v_p(a)<a. Repeated displacement, retaining the remaining base-difference order at each induction step, gives a factor p^k(1+v_p(u)) from k displacements. This inducti","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-95c96988a1a8fc97","tex":"p^w","aliases":"p^w","display":"p^w","kind":"use","excerpt":"ach earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral. pageheading3. Filtered tails and the standard windowSource items 2.2 and 8.2 Filtration at a finite coefficient row. Let R be any commutative ring. For w∈ N∪{∞} define","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-95797c3dbf3035d3","tex":"p^{w(r)}","aliases":"p^w(r)","display":"p^{w(r)}","kind":"use","excerpt":"[Filtered triangular reconstruction] Let p be prime. Suppose A_n = ∑_r=0^n binom nr c_rp^w(r)S_n-r, where c_r∈ Z_p, S_n∈ Z_p, and w(r)∈ N. For every precision L≥1, A_n ≡ ∑_substack0≤ r≤ n","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-20094bff87ea1f75","tex":"p^\\ell","aliases":"p^ℓ ell","display":"p^\\ell","kind":"use","excerpt":"The specification matters. Rational coefficients are not reduced through an imaginary map Q→ F_p. The proof first constructs their integral lifts in Z_p and establishes equality in Q_p. Root representatives are actual units at the requested precision, not unchanged residues that need not satisfy the root equation modulo p^ℓ.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f150917365c9af4d","tex":"p^{\\ell-1}","aliases":"p^ℓ-1 ell","display":"p^{\\ell-1}","kind":"use","excerpt":"ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4bf4d68504d5d5b8","tex":"p^{\\ell-k}","aliases":"p^ℓ-k ell","display":"p^{\\ell-k}","kind":"use","excerpt":"c branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors al","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"Modulo 256=2^8, the least periods are (P_0(8),P_1(8),P_2(8),P_3(8))=(128,128,8,32). In actual Euler degree, multiply by four: (512,512,32,128). For address two, precision at most 2^5 already has row period one. Its Euler coefficients all lie in 2+32 Z_2; further precision resolves additional dyadic digits within that residue class.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"Modulo 256=2^8, the least periods are (P_0(8),P_1(8),P_2(8),P_3(8))=(128,128,8,32). In actual Euler degree, multiply by four: (512,512,32,128). For address two, precision at most 2^5 already has row period one. Its Euler coefficients all lie in 2+32 Z_2; further precision resolves additional dyadic digits within that residue class.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c29786ad28dc8278","tex":"P_{2^a,\\,r\\bmod2^a}","aliases":"P_2^a, rbmod2^a bmod","display":"P_{2^a,\\,r\\bmod2^a}","kind":"use","excerpt":"0, · otherwise, take the canonical label 0≤ r<N. The logical equivalence yields the actual equality of sequence operators P_N,rf= P_2^a, rbmod2^a(P_M, rbmod Mf). This statement holds for coefficients in any type with zero. A separate theorem lifts the equality coefficientwise to the existing formal power-series selectors over any semiring.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e8501ee74c70e6c9","tex":"P_{2^a,r\\bmod 2^a}","aliases":"P_2^a,rbmod 2^a bmod","display":"P_{2^a,r\\bmod 2^a}","kind":"use","excerpt":"[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-359606f4da41ffb7","tex":"P_{2^aM,r}","aliases":"P_2^aM,r","display":"P_{2^aM,r}","kind":"use","excerpt":"[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f0242a419961441b","tex":"P_3","aliases":"P_3","display":"P_3","kind":"use","excerpt":"Modulo 256=2^8, the least periods are (P_0(8),P_1(8),P_2(8),P_3(8))=(128,128,8,32). In actual Euler degree, multiply by four: (512,512,32,128). For address two, precision at most 2^5 already has row period one. Its Euler coefficients all lie in 2+32 Z_2; further precision resolves additional dyadic digits within that residue class.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4c9a8f8dedb6f5c4","tex":"\\mathcal P_{4,j}","aliases":"P_4,j mathcal","display":"\\mathcal P_{4,j}","kind":"use","excerpt":"Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"ts strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use polynomial induction on constants, addition, and multiplication by a variable. Constants have zero derivative. The sum rule agrees with the additivity of formal partial differentiation. For PX_j, the","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f06a233d5d801ff2","tex":"P_{A,r_A}","aliases":"P_A,r_A","display":"P_{A,r_A}","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-83efd7740f0afa62","tex":"P_{AB,r}","aliases":"P_AB,r","display":"P_{AB,r}","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d04a4a035c233ca7","tex":"P_{B,r_B}","aliases":"P_B,r_B","display":"P_{B,r_B}","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"Here is the polynomial estimate that supplies one additional bit. For i≥1, put P_i(x)=4^ibinom4x4i. Its constant is zero, and its degree-d coefficient is [x^d]P_i(x)=frac4^i4^d s(4i,d)(4i)!, where s(m,d) is a signed Stirling number of the first kind. In each product contributing to s(4i,d), at most d-1 of the 2i-1 even integers in 1,…,4i-1 can be o","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"For a wall label j≥1 and 0≤ q<j, define e_j(q)=3q+s_2(j-1)-s_2(j-q-1), E_j=min_q<je_j(q), Θ(j)=3j+1+s_2(j-1), L_j=Θ(j)-E_j. Here E_j is a carry minimum, not an Euler number. A wall label is regular when e_j(q)>0 for 1≤ q<j. For r=2^L-j with L>L_j, put δ=L-L_j. The carry identities give μ_r=E_j-δ, J_r={j}, G_r(x)=P_j(x)U(x-j)+2^δ V(x), where U≡1pmod","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f6d93da85dc64976","tex":"p_j^{a_j}","aliases":"p_j^a_j","display":"p_j^{a_j}","kind":"use","excerpt":"[Finite local reconstruction] Let N=∏_j=1^sp_j^a_j be the prime power factorization of a positive integer. The map Z/N Z ⟶ ∏_j=1^s Z/p_j^a_j Z is a ring isomorphism. Thus a compatible collection of prime power residues reconstructs one residue modulo N [source reference].","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a8e7ad98972fe76","tex":"\\mathcal P_M","aliases":"P_M mathcal","display":"\\mathcal P_M","kind":"use","excerpt":"For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-37bbcb3eb285f1d4","tex":"P_{M,\\,r\\bmod M}","aliases":"P_M, rbmod M bmod","display":"P_{M,\\,r\\bmod M}","kind":"use","excerpt":"0, · otherwise, take the canonical label 0≤ r<N. The logical equivalence yields the actual equality of sequence operators P_N,rf= P_2^a, rbmod2^a(P_M, rbmod Mf). This statement holds for coefficients in any type with zero. A separate theorem lifts the equality coefficientwise to the existing formal power-series selectors over any semiring.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-34451fc17c1dd5f8","tex":"P_{M,r}","aliases":"P_M,r","display":"P_{M,r}","kind":"use","excerpt":"[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M r≡ jpmod NP_M,ra. The same identities hold coefficientwise for formal power series.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2af13a7e506b5f46","tex":"P_{M,r\\bmod M}","aliases":"P_M,rbmod M bmod","display":"P_{M,r\\bmod M}","kind":"use","excerpt":"[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-68c8c438803a7824","tex":"\\mathcal P_{N,0}","aliases":"P_N,0 mathcal","display":"\\mathcal P_{N,0}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ce857563c4ac4bba","tex":"P_{N,i}","aliases":"P_N,i","display":"P_{N,i}","kind":"use","excerpt":"[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e65ae8417651699c","tex":"P_{N,j}","aliases":"P_N,j","display":"P_{N,j}","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definition","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f0c02aa550cfc1b1","tex":"\\mathcal P_{N,j}","aliases":"P_N,j mathcal","display":"\\mathcal P_{N,j}","kind":"use","excerpt":"The normalized source and its inverse are K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coeffic","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bccb8a74fcfcadc5","tex":"P_{N,j}^2","aliases":"P_N,j^2","display":"P_{N,j}^2","kind":"use","excerpt":"[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2bd7ecfa07c32400","tex":"P_{N,r}","aliases":"P_N,r","display":"P_{N,r}","kind":"use","excerpt":"[Residue projector] For a sequence a=(a_n)_n≥0, a positive integer N, and an address r modulo N, define (P_N,ra)_n = a_n, · n≡ rpmod N, 0, · nnot≡ rpmod N.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-09523592e6cc4b12","tex":"P_{N,s}","aliases":"P_N,s","display":"P_{N,s}","kind":"use","excerpt":"[Complete residue packet system] For fixed N, the projectors P_N,r satisfy P_N,rP_N,s = δ_rsP_N,r and ∑_rbmod NP_N,r=I.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b9d95adbf1986020","tex":"P_{N_j,r_j}","aliases":"P_N_j,r_j","display":"P_{N_j,r_j}","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d61b96062b482c72","tex":"P_q","aliases":"P_q","display":"P_q","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4d3243980b9ae21f","tex":"P_r","aliases":"P_r","display":"P_r","kind":"use","excerpt":"Define the central-factorial basis polynomials P_r(X)=∏_a=0^r-1(X-a(a+1)). At X=Z(Z+1), they have the exact factorization P_r(Z(Z+1)) =∏_a=0^r-1(Z-a)(Z+a+1) =(2r)!binomZ+r2r. The Bernoulli transform gives L((1+T)^Z)=log(1+T)/T, whence L(binom Zj)=(-1)^j/(j+1). Finite Vandermonde expansion therefore gives L(binomZ+r2r) · =(-1)^r∑_q=0^r(-1)^qbinom rq","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4a95280613dbdd3c","tex":"P_{r+1}","aliases":"P_r+1","display":"P_{r+1}","kind":"use","excerpt":"Expand X^m=∑_r=0^mT(m,r)P_r(X). Since XP_r(X)=P_r+1(X)+r(r+1)P_r(X), these integer coefficients satisfy T(m+1,r)=T(m,r-1)+r(r+1)T(m,r), T(0,0)=1. Every r(r+1) is even. Induction, including the boundary indices, proves 2^m-rmid T(m,r) for 0≤ r≤ m. For m>0 the r=0 term vanishes. Each remaining summand of L(P^m/m!) has valuation at least m-r+2v_2(r!)-","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2ff9a149e730be7a","tex":"\\operatorname{pack}","aliases":"pack operatorname","display":"\\operatorname{pack}","kind":"use","excerpt":"For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every n","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-463715ee648b1345","tex":"\\operatorname{pack}_4","aliases":"pack_4 operatorname","display":"\\operatorname{pack}_4","kind":"use","excerpt":"For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every n","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a90766c2377cd948","tex":"\\operatorname{packet}","aliases":"packet operatorname","display":"\\operatorname{packet}","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-065b6c29f5eeff18","tex":"\\PP","aliases":"PP P mathcal","display":"{\\mathcal P}","kind":"use","excerpt":"Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s); induction and multiplication by g(n+s","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-738cf0c2598fdac3","tex":"\\PP_2","aliases":"PP_2 P_2 PP mathcal","display":"{\\mathcal P}_2","kind":"use","excerpt":"An affine sequence whose slope is divisible by a satisfies P_a. Consequently the four displayed source sequences satisfy P_2. No numerical interpolation or finite set of samples is used in that deduction.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6aa31545bc253fc8","tex":"\\PP_a","aliases":"PP_a P_a PP mathcal","display":"{\\mathcal P}_a","kind":"use","excerpt":"Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s); induction and multiplication by g(n+s","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Let Q(z)=zcoth z, the regular formal series with constant coefficient one. The elementary differential identity (frac zsinh z)^2=Q(z)-zQ'(z) and the Bernoulli expansion Q(z)=1+∑_m≥1frac2^2mB_2m(2m)!z^2m give β_m=-2^2m(2m-1)B_2m/(2m)!. Multiplication by 2^2m(2m)! proves the identity for κ_m. The Bernoulli denominator theorem gives v_2(B_2m)=-1 for m","count":74},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0c370aa793b02aac","tex":"Q'","aliases":"Q'","display":"Q'","kind":"use","excerpt":"Let Q(z)=zcoth z, the regular formal series with constant coefficient one. The elementary differential identity (frac zsinh z)^2=Q(z)-zQ'(z) and the Bernoulli expansion Q(z)=1+∑_m≥1frac2^2mB_2m(2m)!z^2m give β_m=-2^2m(2m-1)B_2m/(2m)!. Multiplication by 2^2m(2m)! proves the identity for κ_m. The Bernoulli denominator theorem gives v_2(B_2m)=-1 for m","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"Write the input indices as Mq+u and Mq'+v. Since 0≤ u,v<M, their sum has residue u+v when u+v<M, and residue u+v-M with one additional coarse index when u+v≥ M.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a2a621a3fcb22dce","tex":"q^{-L}","aliases":"q^-L","display":"q^{-L}","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"use","excerpt":", and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber H_0:=fracU^4-12 · =P+Q-fracP^22-fracQ^22 +3PQ+i E F(P-Q).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aa0b566a56f0141e","tex":"q^{2M+2}","aliases":"q^2M+2","display":"q^{2M+2}","kind":"use","excerpt":"[Odd modulus truncation] For every M≥0, every positive odd q, and every even n, E_n ≡ ∑_r=0^min(M,n/2) (-1)^rZ_2r+1 binom n2rq^2rS_n-2r(q) pmod q^2M+2. If pmid q and a=v_p(q), the same truncation is valid modulo p^2a(M+1).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8db2618d57c31645","tex":"q^{2N}","aliases":"q^2N","display":"q^{2N}","kind":"use","excerpt":"^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exactly. The checked polynomial Taylor identity gives F(a')≡ F(a)+q^NDF(a)(widetilde u,w) ≡0pmodq^N+1. Its quadratic remainder is divisible by q^2N, and 2N≥ N+1. ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-42bb94f60a6ecb0a","tex":"q^{2r}","aliases":"q^2r","display":"q^{2r}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ee5559b3980fa51c","tex":"q^d","aliases":"q^d","display":"q^d","kind":"use","excerpt":"For every positive integer q, including composite moduli and modulus one, and every L≥0, the checked map Φ_x,L:( Z/q Z)^d⟶ {ybmod q^L+1:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection. Here widetilde t is the canonical integer representative of the digit vector. The entire residue fiber therefore has q^d elements. Injectivity is proved by cancelli","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0cdef6628c529361","tex":"Q^e","aliases":"Q^e","display":"Q^e","kind":"use","excerpt":")∈ M_e× k(R). Suppose, for natural numbers s,M, that J∈GL_e(Q), p^sJ^-1∈ M_e(R), M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. Then there is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisfies y-y_0∈ p^M-sR^e and there is an integral vector ε∈ p^2M-3sR^e such that, in Q^e, displaystyle y-y_0=-p^MJ^-1(H(a)+Kv)+ε. The same original exponent s","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-938f08ed5d6886a7","tex":"q^{L+1}","aliases":"q^L+1","display":"q^{L+1}","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":18},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dfe745e2a01c96c9","tex":"q^{L+a}","aliases":"q^L+a","display":"q^{L+a}","kind":"use","excerpt":"z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0899f3519cbe2b50","tex":"q^{L+n}","aliases":"q^L+n","display":"q^{L+n}","kind":"use","excerpt":"[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aa29cb49431ad02f","tex":"Q^N","aliases":"Q^N","display":"Q^N","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1928485a8127cf57","tex":"q^N","aliases":"q^N","display":"q^N","kind":"use","excerpt":"At level N=L+n, divide the residual by q^N. Surjectivity supplies t∈ S_q^d with J_z_nt=-overlineF(z_n)/q^N. Choose an integer representative and apply [source reference] to obtain z_n+1. The new point is congruent to z_n modulo q. Evaluation of each formal partial derivative therefore gives J_z_n+1=J_z_n.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d77f110b9dcddc74","tex":"q^{N+1}","aliases":"q^N+1","display":"q^{N+1}","kind":"use","excerpt":"s t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exactly. The checked polynomial Taylor identity gives F(a')≡ F(a)+q^NDF(a)(widetilde u,","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b568c9f273e792aa","tex":"Q^r","aliases":"Q^r","display":"Q^r","kind":"use","excerpt":"/3 in the one-prime example above loses the advertised digit; invertibility of C alone does not restore it. The original proof is retained under these hypotheses. Let Q=∏_p∈ Sp^L_p for a finite set of primes. Let C∈ M_d( Z) be invertible modulo Q, and choose H such that C^H=I+QB with B∈ M_d( Z). Then for every nonnegative integer x, a_Hx+j = ∑_r=0^","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-886dc240f95388c4","tex":"q^{r-d}","aliases":"q^r-d","display":"q^{r-d}","kind":"use","excerpt":"On the polynomial factor the difference operator acts as q+yΔ. The two terms commute. Expand (q+yΔ)^r and evaluate at zero. Since Δ^kQ_j=0 for k>d, every surviving term contains at least q^r-d.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-378ab1d30fa3e392","tex":"q^{r-k}","aliases":"q^r-k","display":"q^{r-k}","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-812a1fd77a9f35a3","tex":"\\mathbb Q_2","aliases":"Q_2 mathbb","display":"\\mathbb Q_2","kind":"use","excerpt":"Put K= Q_2(i), R= Z_2[i], a=(1+i)/2, and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8cef7dc638ad907a","tex":"Q_F","aliases":"Q_F","display":"Q_F","kind":"use","excerpt":"Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents a","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ac09894acaf58fd","tex":"Q_H","aliases":"Q_H","display":"Q_H","kind":"use","excerpt":"Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents a","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5950b0378700bfba","tex":"Q_i","aliases":"Q_i","display":"Q_i","kind":"use","excerpt":"For 0≤ i<j, let d_i=v_2(i!)+v_2((j-1-i)!), k_i=1+lfloorlog_2max(i,j-1-i)rfloor, B_i=i+2^k_i Z_2, t_i=d_i+k_i. At j=1 use d_0=k_0=t_0=0. The balls B_i partition Z_2. Indeed a residue represented twice among 0,…,j-1 splits into its two children; every other represented residue remains one ball. On each such ball, P_j(i+2^k_iy)=2^t_iyQ_i(y), Q_i∈ Z_2⟨","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3312dcd34d0a8dd5","tex":"\\mathbb Q_p^e","aliases":"Q_p^e mathbb","display":"\\mathbb Q_p^e","kind":"use","excerpt":"hbb Z_p^e+k, F(a)=0. Let ι: Z_p↪ Q_p be the coefficient inclusion, applied entrywise to points and matrices. Assume only that ι(D_YF(a))∈GL_e( Q_p). The theorem mcodePolynomialRegularityIntegral.exists_analytic_integral_implicit constructs the preceding analytic graph in the ambient spaces Q_p^k and Q_p^e. Its root equation evaluates the original i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-45c8cae45a70e764","tex":"\\mathbb Q_p^k","aliases":"Q_p^k mathbb","display":"\\mathbb Q_p^k","kind":"use","excerpt":",T]^e, a∈ Z_p^e+k, F(a)=0. Let ι: Z_p↪ Q_p be the coefficient inclusion, applied entrywise to points and matrices. Assume only that ι(D_YF(a))∈GL_e( Q_p). The theorem mcodePolynomialRegularityIntegral.exists_analytic_integral_implicit constructs the preceding analytic graph in the ambient spaces Q_p^k and Q_p^e. Its root equation evaluates the orig","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9666ac009c3e72f6","tex":"q_r","aliases":"q_r","display":"q_r","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-de1f9e9f12343cd8","tex":"q_{u,L}","aliases":"q_u,L","display":"q_{u,L}","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c7c8dd25964c5fdc","tex":"q_{u,L}^{\\,r}","aliases":"q_u,L^ r","display":"q_{u,L}^{\\,r}","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9b6db148287b1581","tex":"\\Qp^d","aliases":"Qp^d Q_p^d Qp mathbb","display":"{\\mathbb Q_p}^d","kind":"use","excerpt":"[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9ea1e9e76890cbee","tex":"\\Qp^e","aliases":"Qp^e Q_p^e Qp mathbb","display":"{\\mathbb Q_p}^e","kind":"use","excerpt":"y-y_0 · =-p^MJ^-1(H(a)+Kv)+ε, ε∈ p^2M-3sR^e. The equality involving J^-1 is in Q_p^e, with an integral error vector. The parameter equality is exact.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ Q mathbb","display":"{\\mathbb Q}","kind":"use","excerpt":"The examples later in the paper use different kernels, but their coefficient algebra has one common form. This section records that form once. Let R be a commutative algebra over Q, and take all coefficients in R.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definition","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fe9a73354b93f7b7","tex":"R^d","aliases":"R^d","display":"R^d","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":44},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3483b609e1120b1c","tex":"R^e","aliases":"R^e","display":"R^e","kind":"use","excerpt":"rmation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisfies y-y_0 · ∈ p^M-sR^e,","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4935b7ed17587997","tex":"R^{e+k}","aliases":"R^e+k","display":"R^{e+k}","kind":"use","excerpt":"[Joint coefficient and parameter deformation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisf","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ac364b379fe8ae12","tex":"R^I","aliases":"R^I","display":"R^I","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a448f4c18ae5322e","tex":"R^k","aliases":"R^k","display":"R^k","kind":"use","excerpt":"[Joint coefficient and parameter deformation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisf","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f9039d8e025c2839","tex":"R^{\\NN}","aliases":"R^NN R^ N NN mathbb","display":"R^{{\\mathbb N}}","kind":"use","excerpt":"[Hurwitz series ring] Let R be a commutative ring with identity. The Hurwitz series ring H(R) is the set R^ N with componentwise addition and product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_r b_n-r. Its identity is δ=(1,0,0,…).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"[The first nontrivial dyadic row] For n=4, R_4=frac9115, R_2=frac53, R_0=1. The exact kernel coefficients are κ_2,1=-frac83, κ_2,2=frac1285. Therefore E_4 = frac9115 -frac83binom42frac53 +frac1285 =5. The rational source values are dyadic units, while the transfer depth is carried by the kernel.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"The standard window supplies the initial ratio R_1. The weighted estimate handles pmid c, and Riccati propagation handles pnmid c. Flatness is closed under products, so the exact factorization establishes flatness of j! P_N,j. Extracting its supported Hurwitz coefficients and applying the proved saturation and step-multiple lemmas yields [source re","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"[The first nontrivial dyadic row] For n=4, R_4=frac9115, R_2=frac53, R_0=1. The exact kernel coefficients are κ_2,1=-frac83, κ_2,2=frac1285. Therefore E_4 = frac9115 -frac83binom42frac53 +frac1285 =5. The rational source values are dyadic units, while the transfer depth is carried by the kernel.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d7fe315bf8e2207f","tex":"R_{2(r-m)}","aliases":"R_2(r-m)","display":"R_{2(r-m)}","kind":"use","excerpt":"rgetDyadicSeries.dyadic_reconstruction_series pageheading3. Exact reconstruction and genuine dyadic congruenceTheorem 5.3 and Corollary 5.4; manuscript pages 14--15 The reciprocal coefficients β_m use an ordinary power expansion. In contrast, the source and Euler series use exponential coefficients. Extracting degree 2r from B(2t)F(t)=sech t theref","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d8078a9a9bb651de","tex":"R_{2n}","aliases":"R_2n","display":"R_{2n}","kind":"use","excerpt":"[Analytic dyadic source] The function mathscr R belongs to the dyadic Tate algebra and takes values in Z_2. For every nonnegative integer n, mathscr R(n)=R_2n.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-742b3c1f9a0009ba","tex":"R_{2r}","aliases":"R_2r","display":"R_{2r}","kind":"use","excerpt":"The source has valuation exactly zero. At even degree 2r, define the literal rational source R_2r=frac3^2r+2-14(2r+1)(2r+2). Both numerator and denominator are nonzero. The pinned Mathlib lifting-the-exponent theorem gives v_2(3^2r+2-1)=2+v_2(2r+2). Because 2r+1 is odd, the denominator has the same valuation. Consequently, v_2(R_2r)=0, |R_2r|_2=1. ","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d0aa3589089d58ba","tex":"R_3","aliases":"R_3","display":"R_3","kind":"use","excerpt":"The radius is 2^liminf_d v_2(a_d)/d in the normalized dyadic absolute value. Hence R_0=R_1=2, R_2=2^5/2, R_3≥2. The exact radius of address three is not asserted. For address two, the lower bound gives convergence inside the claimed radius and the unbounded equality family prevents a larger one.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0fe85010c113a2aa","tex":"R_4","aliases":"R_4","display":"R_4","kind":"use","excerpt":"[The first nontrivial dyadic row] For n=4, R_4=frac9115, R_2=frac53, R_0=1. The exact kernel coefficients are κ_2,1=-frac83, κ_2,2=frac1285. Therefore E_4 = frac9115 -frac83binom42frac53 +frac1285 =5. The rational source values are dyadic units, while the transfer depth is carried by the kernel.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-38ee3d5976c0e937","tex":"R_6","aliases":"R_6","display":"R_6","kind":"use","excerpt":"al checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac13,frac115,-frac2189,frac1675). These values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives (R_0","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-43a442af86882c80","tex":"R_8","aliases":"R_8","display":"R_8","kind":"use","excerpt":"al checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac13,frac115,-frac2189,frac1675). These values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives (R_0","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6e93838ea3bf8d2c","tex":"r_A","aliases":"r_A","display":"r_A","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fb14b1a407e1de8e","tex":"r_a","aliases":"r_a","display":"r_a","kind":"use","excerpt":"The actual ratio is also proved to be an expansion in t^N, so its coefficients vanish outside multiples of N. Constructed Z_p lifts and the established coefficient criterion now give, for its literal Hurwitz coefficients r_a(n), Δ_uh^kr_a(n)∈ p^min(k,mn) Z_p. The separate rational precision theorem is obtained through the coefficient-lift equality.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-01bc47d575242d41","tex":"R_{ap}","aliases":"R_ap","display":"R_{ap}","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c6016f5cf675a3ae","tex":"r_B","aliases":"r_B","display":"r_B","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-37f2bdf109707126","tex":"R_c","aliases":"R_c","display":"R_c","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e58d2fe1775cda0e","tex":"R_c'","aliases":"R_c'","display":"R_c'","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c6f893e6747fcef8","tex":"R_{c-1}","aliases":"R_c-1","display":"R_{c-1}","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-53e3e9d1c2a6131f","tex":"R_{j+1}","aliases":"R_j+1","display":"R_{j+1}","kind":"use","excerpt":"Let K_j(t) be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is F_N,j(t)=j! K_j(t)^-1. The source/inverse identities and the factorial successor identity prove F_N,j+1=F_N,jR_j+1, F_N,j=F_N,0∏_c=1^jR_c. The product formula is not an assembly hypothesis. Proved flatness of all factors and the o","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d96a67e19d75bf1d","tex":"r_L","aliases":"r_L","display":"r_L","kind":"use","excerpt":"AtlasHeadA compatible root is built one digit at a timeatlas:smooth Take F(x)=x^2-7 over the 3-adic integers and start with x_1=1 modulo three. If F(x_L)=3^Lr_L, the next representative has the form x_L+1=x_L+3^Lt_L. Its correcting digit satisfies r_L+2x_Lt_L≡0pmod3. AtlasImagesmooth_tower For example, F(1)/3=-2 and 2x_1=2, so the next digit is one","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4aa81ec436e2173c","tex":"R_N","aliases":"R_N","display":"R_N","kind":"use","excerpt":"Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"use","excerpt":"For even n≥0, define R_n = frac3^n+2-14(n+1)(n+2). Its even exponential generating function is ∑_n evenR_nfract^nn! = fracsinh(2t)sinh t2t^2. Define β_r by fracz^2sinh^2z = ∑_r≥0β_rz^2r.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5a25647980501aee","tex":"R_{n-2}","aliases":"R_n-2","display":"R_{n-2}","kind":"use","excerpt":"The first three sharp truncations are E_n · ≡ R_n · · pmod 8, E_n · ≡ R_n-frac83binom n2R_n-2 · · pmod 128, E_n · ≡ R_n-frac83binom n2R_n-2 +frac1285binom n4R_n-4 · · pmod 2048.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ca477fd0b7d4d865","tex":"R_{n-2r}","aliases":"R_n-2r","display":"R_{n-2r}","kind":"use","excerpt":"[Dyadic Euler coordinate pair] For every even n≥0, E_n = ∑_r=0^n/2 β_r2^2r fall n2r R_n-2r. The inverse transform is R_n = ∑_r=0^n/2 frac2^4r+1(2r+2)! fall n2r E_n-2r.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3f00871f36db30a0","tex":"R_{n-4}","aliases":"R_n-4","display":"R_{n-4}","kind":"use","excerpt":"The first three sharp truncations are E_n · ≡ R_n · · pmod 8, E_n · ≡ R_n-frac83binom n2R_n-2 · · pmod 128, E_n · ≡ R_n-frac83binom n2R_n-2 +frac1285binom n4R_n-4 · · pmod 2048.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1e4761712190b8fb","tex":"R_r","aliases":"R_r","display":"R_r","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cd0c6e5c3a03a755","tex":"r_{x'}","aliases":"r_x'","display":"r_{x'}","kind":"use","excerpt":"[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"Theorem. For this fixed polynomial presentation, the group and distinguished class are independent of the chosen integer representative. The class vanishes if and only if bar x lifts to a solution modulo p^L+1. The obstruction group has p^r-rankJ elements. When lifts exist, their set is a torsor for ker J and contains exactly p^d-rankJ elements.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank operatorname","display":"{\\operatorname{rank}}","kind":"use","excerpt":"[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0,","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ed613f72ca73d2d","tex":"\\operatorname{rank}_{\\mathbb F_p}","aliases":"rank_ F_p operatorname mathbb","display":"\\operatorname{rank}_{\\mathbb F_p}","kind":"use","excerpt":"Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Write r_x=F(x)/p^L, b_x=r_xbmod p, A_x=DF(x)bmod p: F_p^d⟶ F_p^e, ρ_x=rank_ F_pA_x. The actual lift set is L_x,L= {y∈( Z/p^L+1 Z)^d: y≡ xpmodp^L, F(y)=0}, where polynomial evaluation in the displayed set is modulo p^L+1. Define O_x= F_p^e/imA_x, o_x=[b_x]. Lean proves # L_x,L= p^d-ρ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fb1bc2bcf81a206f","tex":"\\rank_{\\F_p}","aliases":"rank_F_p rank_ F_p rank F operatorname mathbb","display":"{\\operatorname{rank}}_{{\\mathbb F}_p}","kind":"use","excerpt":"[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0,","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-22cff1e28b352509","tex":"\\rank_{\\FF_p}","aliases":"rank_FF_p rank_ F_p rank FF operatorname mathbb","display":"{\\operatorname{rank}}_{{\\mathbb F}_p}","kind":"use","excerpt":"[Exact number of linear children] If ρ=rank_ F_poverline M, then a solvable depth L class has exactly p^d-ρ children modulo p^L+1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"The largest emergence threshold is max_i t_i=v_2((j-1)!)+lceillog_2 jrceil. For j=2^a, all thresholds equal j-1. There are no roots before that distance and exactly j simple roots at and after it, within the positive-distance range. The equality and pre-wall cases are also resolved: regular labels j≥2 have no roots there; j=1 has one simple odd roo","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-06b25b660a3b775f","tex":"\\Res","aliases":"Res operatorname","display":"{\\operatorname{Res}}","kind":"use","excerpt":"We use the resultant convention Res_y(f,g) = ∏_f(α)=0g(α) when f is monic, with roots counted with multiplicity.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b1da16dc8b9fc5c0","tex":"\\Res_y","aliases":"Res_y Res operatorname","display":"{\\operatorname{Res}}_y","kind":"use","excerpt":"We use the resultant convention Res_y(f,g) = ∏_f(α)=0g(α) when f is monic, with roots counted with multiplicity.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":45},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"For N>0 and 0≤ j<N, the independently defined original exponential section is E_N,j(t)=∑_d≡ j (rm mod N)fract^dd!. Using the actual power-series expansion homomorphism z↦ t^N, the code proves E_N,j(t)=fract^jj!K_N,j(t^N), F_N,j(t^N)E_N,j(t)=fract^jj!. Thus the recurrence is connected back to the original normalized EGF definition. The section vanis","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7ffddd5883e6d57f","tex":"\\boldsymbol s","aliases":"s boldsymbol","display":"\\boldsymbol s","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definition","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-17e6e62cdd37d68b","tex":"\\mathsf S","aliases":"S mathsf","display":"\\mathsf S","kind":"use","excerpt":"[Polyphase shift matrix] Let E be the scalar shift (Ea)_n=a_n+1, and let E_q shift the coarse index. Then P_M(Ea) = S_M(E_q) P_Ma, where S_M(E_q) = 0 · 1 · 0 · ⋯ · 0","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bea69cee90538d01","tex":"s^{[N]}","aliases":"s^[N]","display":"s^{[N]}","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-14e505b4cdf29e83","tex":"S^e","aliases":"S^e","display":"S^e","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a36ef00e19010338","tex":"S^H","aliases":"S^H","display":"S^H","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers; steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-954f18f50adf21ae","tex":"S^k","aliases":"S^k","display":"S^k","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c863c73afd645b31","tex":"S^m","aliases":"S^m","display":"S^m","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"use","excerpt":"[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"use","excerpt":"[Exact change between local source charts] Editorial range correction. This comparison holds for the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero; the eve","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-db9af401ac382039","tex":"S_2","aliases":"S_2","display":"S_2","kind":"use","excerpt":"[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":20},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-da35858f1dea6007","tex":"S_{2(r-m)}","aliases":"S_2(r-m)","display":"S_{2(r-m)}","kind":"use","excerpt":"ing3. Exact reconstruction and the triangular matrixTheorem 4.2; manuscript page 11 For odd p, rescaling the already proved inverse law gives sech(pt)cosh(pt)=1. Multiplying the even-source identity by sech^2(pt) therefore proves sech^2(pt)(∑_r≥0S_2r(p)fract^2r(2r)!)=sech t. Actual EGF multiplication now gives the integer coefficient formula E_2r=∑","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d79e1a4a3abaf4d8","tex":"S_{2n}","aliases":"S_2n","display":"S_{2n}","kind":"use","excerpt":"The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconst","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a35fc269aba33f6a","tex":"S_{2q}","aliases":"S_2q","display":"S_{2q}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-df3b11dcb98597af","tex":"S_{2r}","aliases":"S_2r","display":"S_{2r}","kind":"use","excerpt":"Even projection is identified coefficientwise. For any formal series f, let EvenPart(f)=(f(t)+f(-t))/2. Its nth coefficient is proved to be the coefficient of f when n is even and zero otherwise. Rescaling the inverse law proves sech(-t)=sech(t). The exponential multiplication identity then gives cosh^2(pt)=frac2+e^2pt+e^-2pt4, and hence ∑_r≥0S_2r(","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f6d0dc9f5b462d0b","tex":"S_4","aliases":"S_4","display":"S_4","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9315144502cf9089","tex":"S_6","aliases":"S_6","display":"S_6","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-71a07bf98e34b5f3","tex":"S_8","aliases":"S_8","display":"S_8","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e3e97de7f305c50f","tex":"\\mathsf S_M","aliases":"S_M mathsf","display":"\\mathsf S_M","kind":"use","excerpt":"[Polyphase shift matrix] Let E be the scalar shift (Ea)_n=a_n+1, and let E_q shift the coarse index. Then P_M(Ea) = S_M(E_q) P_Ma, where S_M(E_q) = 0 · 1 · 0 · ⋯ · 0","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-91ec47101c58ab98","tex":"s_n","aliases":"s_n","display":"s_n","kind":"use","excerpt":"First construct the integral solution. For a ring R, a triangular row system T, right-hand side b, and proved unit pivots d_n=T_nn, define s_n=d_n^-1(b_n-∑_q<nT_nqs_q). The new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homo","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0f71751162a0c66a","tex":"S_{n-2m}","aliases":"S_n-2m","display":"S_{n-2m}","kind":"use","excerpt":"Substituting the independently proved signed-zigzag identity recovers the manuscript's notation. For every even degree n, E_n=∑_m=0^n/2(-1)^m Z_2m+1binom n2mp^2mS_n-2m(p). The Lean version uses n=2r to avoid ambiguous natural-number division. Every even natural degree has that form. Primality is not used in this equality.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b0701b1a8b030e63","tex":"S_{n-2r}","aliases":"S_n-2r","display":"S_{n-2r}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1222b525c58a0fd4","tex":"S_{n-2r}^\\times","aliases":"S_n-2r^× times","display":"S_{n-2r}^\\times","kind":"use","excerpt":"[Depleted odd prime reconstruction] For every even n, (1-σ_pp^n)E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2rp^2rS_n-2r^×(p).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7cd6bf70853b646a","tex":"S_{n-r}","aliases":"S_n-r","display":"S_{n-r}","kind":"use","excerpt":"[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e0082aa89275ee7b","tex":"S_n^\\times","aliases":"S_n^× times","display":"S_n^\\times","kind":"use","excerpt":"For an odd prime p, the source S_n(p) contains one nonunit base. Put σ_p=(-1)^(p-1)/2 and S_n^×(p)=S_n(p)-σ_pp^n.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f2edc85bc58966aa","tex":"S_p","aliases":"S_p","display":"S_p","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-25afd0ab9b73955e","tex":"\\mathcal S_p","aliases":"S_p mathcal","display":"\\mathcal S_p","kind":"use","excerpt":"The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconst","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-80da4524c0ee2fe5","tex":"s_p","aliases":"s_p","display":"s_p","kind":"use","excerpt":"[Odd row weight] For an odd prime p, the rth nonzero term in the reconstruction has valuation W_p(n,r) = 2r+v_p(Z_2r+1) +v_pbinom n2r +v_p(S_n-2r(p)). The binomial contribution is v_pbinom n2r = fracs_p(2r)+s_p(n-2r)-s_p(n)p-1, where s_p is the base p digit sum.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0bcae1d2e5697a14","tex":"S_p^e","aliases":"S_p^e","display":"S_p^e","kind":"use","excerpt":"For every proposed next parameter digit, the previously checked obstruction is [bar r+K_av]∈ S_p^e/imJ_a. The fixed solved Jacobian is surjective, so this class vanishes for every digit at every stage. The actual lift fiber is a torsor for ker J_a. Rank-nullity gives its cardinality p^m-e.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-19ed83d808a66932","tex":"S_q","aliases":"S_q","display":"S_q","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5b939a36dfaae506","tex":"s_q","aliases":"s_q","display":"s_q","kind":"use","excerpt":"First construct the integral solution. For a ring R, a triangular row system T, right-hand side b, and proved unit pivots d_n=T_nn, define s_n=d_n^-1(b_n-∑_q<nT_nqs_q). The new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homo","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aea0312207c8a5fa","tex":"S_q^d","aliases":"S_q^d","display":"S_q^d","kind":"use","excerpt":"For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d9d25005b5f2d519","tex":"S_q^e","aliases":"S_q^e","display":"S_q^e","kind":"use","excerpt":"For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-90a9d0b5e480e51f","tex":"S_q^f","aliases":"S_q^f","display":"S_q^f","kind":"use","excerpt":"[Simultaneous polynomial lifting] Let F∈ Z[X_1,…,X_d]^e and G∈ Z[X_1,…,X_d]^f have a common lower root x modulo q^L, where q≥1 and L≥1. Use the integer-domain reduced Jacobians J: Z^d→ S_q^e and K: Z^d→ S_q^f. Suppose u,v∈ Z^d are separately valid corrections for F and G, respectively. Then [u]=[v] in Z^d/(ker J+ker K) if and only if a single y≡ xp","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0dc6c24dfbac6716","tex":"S_q^k","aliases":"S_q^k","display":"S_q^k","kind":"use","excerpt":"Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residu","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6cf58d9247702fdb","tex":"S_q^m","aliases":"S_q^m","display":"S_q^m","kind":"use","excerpt":"Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residu","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bcd92cad9c977304","tex":"s_\\ell","aliases":"s_ℓ ell","display":"s_\\ell","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-72398ff18d39e1f6","tex":"\\sec","aliases":"sec","display":"\\sec","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c3f001b6b563f398","tex":"\\sec^2","aliases":"sec^2 sec","display":"\\sec^2","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-34726818d802068e","tex":"\\sech^2","aliases":"sech^2 sech operatorname","display":"{\\operatorname{sech}}^2","kind":"use","excerpt":"The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer; this is the integrality required by the later congruence.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"Connecting to the paper's Z_n convention. The paper defines Z_n through sec t+tan t. The Lean construction uses Mathlib's actual formal sine and cosine series and defines sec t=(cos t)^-1, tan t=(sin t)sec t, Z_n=n![t^n](sec t+tan t). It proves the sine and cosine derivative identities, sin^2t+cos^2t=1, and (tan t)'=sec^2t. Sign substitution in the","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"use","excerpt":"Connecting to the paper's Z_n convention. The paper defines Z_n through sec t+tan t. The Lean construction uses Mathlib's actual formal sine and cosine series and defines sec t=(cos t)^-1, tan t=(sin t)sec t, Z_n=n![t^n](sec t+tan t). It proves the sine and cosine derivative identities, sin^2t+cos^2t=1, and (tan t)'=sec^2t. Sign substitution in the","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e99bde03f8066132","tex":"\\sinh_a","aliases":"sinh_a sinh","display":"\\sinh_a","kind":"use","excerpt":". targetDyadic.kernel_coefficient_integral, Dyadic.beta_integral targetDyadic.source_valuation, Dyadic.sourcePadic_isUnit pageheading2. Connecting the definitions to actual formal seriesGenerating-function foundations for Theorem 5.3 Work in Q[[t]]. The formal exponential and rescaling homomorphism are the same checked objects used in the second bl","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c99449412a3a97a2","tex":"\\operatorname{Sol}","aliases":"Sol operatorname","display":"\\operatorname{Sol}","kind":"use","excerpt":"N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5264d5a046b348b2","tex":"\\operatorname{Sol}_F","aliases":"Sol_F operatorname","display":"\\operatorname{Sol}_F","kind":"use","excerpt":"N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"for maps from A and B to a common set if and only if R is rectangular. In that case a canonical finite choice of common set is the connected-component set C_R of the bipartite graph with vertex set Asqcup B and edges given by R.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"use","excerpt":"m x'=x+p^Ls. Since L≥1, its Jacobian is the same modulo p. Dividing (1) by p^L shows that its residual digit is e'=e+Js, so it gives exactly the same quotient class. If t_0 solves (2), all its solutions are t_0+ker J, and no others. The finite cardinalities follow from rank-nullity. This also covers no variables, no equations, zero Jacobian and sin","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ee9095f60897d7d1","tex":"\\starh","aliases":"starh mathbin⋆ mathbin star","display":null,"kind":"use","excerpt":"This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard windo","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f7402991750f0799","tex":"\\sTwo","aliases":"sTwo s_2","display":"{s_2}","kind":"use","excerpt":"[Dyadic inverse coefficients] For r≥1, β_r = - frac2^2r(2r-1)B_2r(2r)!. Consequently, v_2(β_r)=s_2(r)-1. Every R_n is a dyadic unit.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","excerpt":"At positive polynomial depth, restrict it to the correction fiber T_x={t∈ S_q^d:overliner_x+J_xt=0} and the lift set L_x,L= {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L, F(y)=0}. The polynomial predicate in the second display is evaluation in the residue ring. Evaluation on canonical integer representatives gives the same predicate, because integer polynomial eva","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-77970456be0eda56","tex":"t'","aliases":"t'","display":"t'","kind":"use","excerpt":"At depth N≥1, suppose F(a)=q^Nr and a desired next integer parameter target satisfies t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exa","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a60c6c964dc91686","tex":"t^{-c}","aliases":"t^-c","display":"t^{-c}","kind":"use","excerpt":"Application to the actual singular ratios. The preceding v0.9.0 source independently constructs H_N,c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=t^-cH_N,c(t), R_c=fracA_c-1A_c. The notation t^-cH_N,c means removing its proved initial zero coefficients. For each c=ap, 1≤ a<m, that source proves R_ap(log(1+T))∈ W_p. The new transfer consequently proves flatne","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-104f4e0a72089cc5","tex":"T^0","aliases":"T^0","display":"T^0","kind":"use","excerpt":"^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence finite weighted polynomials. Each fixed output coefficient depends on only finitely many input coefficients, which extends the result to the formal series. Substitution T=e^t-1 intertwines D with Θ and maps W_p into H_p, because d![t^d]F(e","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-afa519626035ed43","tex":"t^{2N}","aliases":"t^2N","display":"t^{2N}","kind":"use","excerpt":"m≥1, differentiation of zcoth z gives β_m=-frac2^2m(2m-1)B_2m(2m)!, κ_m=-2^4m(2m-1)B_2m. The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-605988e2d23b9544","tex":"t^{2n}","aliases":"t^2n","display":"t^{2n}","kind":"use","excerpt":"The classical signed Euler numbers have generating function secht=∑_n≥0E_2nt^2n/(2n)!. They are the N=2, j=0 member of [source reference]. The coordinate theorem reconstructs them from finite alternating sums of odd powers.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8c00545ef56fef32","tex":"t^{2r}","aliases":"t^2r","display":"t^{2r}","kind":"use","excerpt":"[Zigzag inverse kernel] sech^2t = ∑_r≥0 (-1)^rZ_2r+1 fract^2r(2r)!.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"The address-three source is G_4,3(t)=3(sinh t-sin t)/t^3. Addition identities give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, o","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":"For the first contribution, independence gives L(γ_k(P+Q))=∑_a=0^kμ_aμ_k-a. The parity generating series of these convolution moments is (1+t+t^2+t^4+t^8+⋯)^2 =1+t^2+t^4+t^8+⋯ in F_2[[t]]. Its coefficient is one at every k=2^r≥4. Thus the normalized moment in [source reference] is odd and v_2(Δ_1^kC_2(0))=2k+v_2(k!)+k/2. Inserting U^4n=(1+4H_2)^n i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1769a0bec82c9aba","tex":"t^{4i}","aliases":"t^4i","display":"t^{4i}","kind":"use","excerpt":"G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The co","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b595555eaabda692","tex":"t^{4n}","aliases":"t^4n","display":"t^{4n}","kind":"use","excerpt":"Pack g_j into degrees divisible by four and form its actual EGF. The formal coefficient identity is G_j(t)=∑_n≥0g_j(n)fract^4n(4n)! =j!∑_n≥0fract^4n(4n+j)!. Its inverse is therefore the original normalized packet EGF. The source Kummer property gives flatness of G_j. The proved reciprocal closure gives flatness of G_j^-1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1214015beb03f0dc","tex":"t^8","aliases":"t^8","display":"t^8","kind":"use","excerpt":"For the first contribution, independence gives L(γ_k(P+Q))=∑_a=0^kμ_aμ_k-a. The parity generating series of these convolution moments is (1+t+t^2+t^4+t^8+⋯)^2 =1+t^2+t^4+t^8+⋯ in F_2[[t]]. Its coefficient is one at every k=2^r≥4. Thus the normalized moment in [source reference] is odd and v_2(Δ_1^kC_2(0))=2k+v_2(k!)+k/2. Inserting U^4n=(1+4H_2)^n i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-55e177ef0ebf15f4","tex":"T^c","aliases":"T^c","display":"T^c","kind":"use","excerpt":"To pass from the section to the singular ratio, factor out its initial zero: for c=ap, the section has the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-24ab8e8d755e8c86","tex":"T^d","aliases":"T^d","display":"T^d","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1cd31341c910e1c7","tex":"T^{d\\bmod p}","aliases":"T^dbmod p bmod","display":"T^{d\\bmod p}","kind":"use","excerpt":"mate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d; whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-68c16f5dbc93324c","tex":"t^j","aliases":"t^j","display":"t^j","kind":"use","excerpt":"For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8fa21937d664fe3d","tex":"t^N","aliases":"t^N","display":"t^N","kind":"use","excerpt":"The formalization does not stop at a recurrence. In the variable z=t^N, construct K_N,j(z)=∑_n≥0fracj!(Nn+j)!z^n, F_N,j(z)=∑_n≥0fracC_N,j(n)(Nn)!z^n. It proves the factorial coefficient identity, then F_N,jK_N,j=1 and therefore F_N,j=K_N,j^-1.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bf91dc9649ebc4f6","tex":"t^{Nn+c}","aliases":"t^Nn+c","display":"t^{Nn+c}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9acde6ce17e28bc3","tex":"t^{Nn+j}","aliases":"t^Nn+j","display":"t^{Nn+j}","kind":"use","excerpt":"For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5d24990e4c5e775a","tex":"t^{Nr}","aliases":"t^Nr","display":"t^{Nr}","kind":"use","excerpt":"pageheading4. From these sections to the original Euler packetsActual reciprocal, actual rational sequence, every order and step multiple The factorial-ratio identity is proved coefficient by coefficient: ∑_d≥0g_j(d)fract^dd! =j!∑_r≥0fract^Nr(Nr+j)!. The right side is precisely the normalized source kernel from the first block, not a replacement fa","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2797f6407ccfeb21","tex":"T^p","aliases":"T^p","display":"T^p","kind":"use","excerpt":"mate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d; whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bf9efaac25b08a98","tex":"T^r","aliases":"T^r","display":"T^r","kind":"use","excerpt":"[Weighted convergence] If widehat F(T)=∑_r≥0c_rT^r and v_p(c_r)+rα⟶+∞, then the series converges whenever v_p(T)≥α.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-645a5cbe71427f86","tex":"t^r","aliases":"t^r","display":"t^r","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"[3pt] 1 · -91019/45 · 4 [3pt] 2 · 398197/45 · 6 [3pt] 3 · -208601/15 · 8 [3pt] 4 · 1385 · 10 The checked nonzero error valuations are E_8-T_0(8)=frac5494445, v_2(E_8-T_0(8))=5, E_8-T_3(8)=frac22937615, v_2(E_8-T_3(8))=15. They exceed the uniform guarantees; these two examples do not establish a universally stronger exponent.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cb04c4472902a215","tex":"T_{2,M}","aliases":"T_2,M","display":"T_{2,M}","kind":"use","excerpt":"[Optimal dyadic truncation] Let T_2,M(n) = ∑_r=0^min(M,n/2) κ_2,rbinom n2rR_n-2r. Then E_n≡ T_2,M(n)pmod 2^4M+3. More precisely, v_2(E_n-T_2,M(n)) ≥ min_M+1≤ r≤ n/2 ( 4r-1+v_2binom n2r ). The uniform exponent 4M+3 is best possible.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","excerpt":"bottomrule The checked nonzero error valuations are E_8-T_0(8)=frac5494445, v_2(E_8-T_0(8))=5, E_8-T_3(8)=frac22937615, v_2(E_8-T_3(8))=15. They exceed the uniform guarantees; these two examples do not establish a universally stronger exponent.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-afd0f5537bb82fff","tex":"t_4","aliases":"t_4","display":"t_4","kind":"use","excerpt":"For example, j=5 has (t_0,…,t_4)=(6,3,4,3,6). The exact root counts at distances 1,2,3,4,5,≥6 are 0,0,2,3,3,5. The older sufficient stages 3,5,7 are not the sharp emergence stages, which are 3,4,6. The first irregular label, j=17, has V≡1+Xpmod2; the regular-wall theorem does not apply. Complete coefficient certificates close distances 1 through 19","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-558b95d00cd66695","tex":"T_H","aliases":"T_H","display":"T_H","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν),","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cf290378a210616e","tex":"T_{H_1}","aliases":"T_H_1","display":"T_{H_1}","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2697a5c99d68fadf","tex":"T_{H_k}","aliases":"T_H_k","display":"T_{H_k}","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9caf405f71847e16","tex":"t_L","aliases":"t_L","display":"t_L","kind":"use","excerpt":"AtlasHeadA compatible root is built one digit at a timeatlas:smooth Take F(x)=x^2-7 over the 3-adic integers and start with x_1=1 modulo three. If F(x_L)=3^Lr_L, the next representative has the form x_L+1=x_L+3^Lt_L. Its correcting digit satisfies r_L+2x_Lt_L≡0pmod3. AtlasImagesmooth_tower For example, F(1)/3=-2 and 2x_1=2, so the next digit is one","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a3ec920e0c2fd2fa","tex":"\\mathcal T_M","aliases":"T_M mathcal","display":"\\mathcal T_M","kind":"use","excerpt":"tion v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eca9aefbb65ee59b","tex":"T_{m,p}","aliases":"T_m,p","display":"T_{m,p}","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c2ed7a9fa236bbf0","tex":"t_n","aliases":"t_n","display":"t_n","kind":"use","excerpt":"[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integer","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-55fe8241bd06de3e","tex":"t_{n+1}","aliases":"t_n+1","display":"t_{n+1}","kind":"use","excerpt":"[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integer","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-61a8465dadb53175","tex":"T_{nn}","aliases":"T_nn","display":"T_{nn}","kind":"use","excerpt":"[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9f3eddf7a095a301","tex":"T_{nq}","aliases":"T_nq","display":"T_{nq}","kind":"use","excerpt":"[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-62e5b19cba54a896","tex":"\\mathcal T_{r-1}","aliases":"T_r-1 mathcal","display":"\\mathcal T_{r-1}","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2cc37eba865c3ced","tex":"T_{rr}","aliases":"T_rr","display":"T_{rr}","kind":"use","excerpt":"em[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a573e0e7c4db0180","tex":"\\mathcal T_x","aliases":"T_x mathcal","display":"\\mathcal T_x","kind":"use","excerpt":"At positive polynomial depth, restrict it to the correction fiber T_x={t∈ S_q^d:overliner_x+J_xt=0} and the lift set L_x,L= {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L, F(y)=0}. The polynomial predicate in the second display is evaluation in the residue ring. Evaluation on canonical integer representatives gives the same predicate, because integer polynomial eva","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-75177911206476aa","tex":"\\tan","aliases":"tan","display":"\\tan","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-978e87e4f3d831a3","tex":"\\tanh","aliases":"tanh","display":"\\tanh","kind":"use","excerpt":"The identity tanh t=-itan(it) gives the odd zigzag expansion of tanh t. Differentiate.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e8ce79d19d64cdb3","tex":"u^{(p-1)p^a}","aliases":"u^(p-1)p^a","display":"u^{(p-1)p^a}","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"integral; mcodeweighted_section. 3. Weighted closure and the normalized denominator. Let w(d)=lfloor d/prfloor. Since w(i)+w(d-i)≤ w(d), the coefficient formula for a product proves closure of W_p under multiplication. Additive closure is immediate. For U∈ W_p with U(0)=1, strong induction in the actual reciprocal coefficient recurrence proves U^-1","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e28cd1be4d9be5f9","tex":"U^2","aliases":"U^2","display":"U^2","kind":"use","excerpt":", and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber H_0:=fracU^4-12 · =P+Q-fracP^22-fracQ^22 +3PQ+i E F(P-Q).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"One step difference multiplies a unit mode by u^H-1. Fermat reduction and lifting the exponent give the odd estimate. For odd u, v_2(u^2^a+1-1) = v_2(u^2-1)+a ≥ a+3.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0a404366fd8ee067","tex":"u^{2^{a+1}}","aliases":"u^2^a+1","display":"u^{2^{a+1}}","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4d0ee566c7488a3c","tex":"U^4","aliases":"U^4","display":"U^4","kind":"use","excerpt":"For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every n","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-51aed094494b5462","tex":"U^{4n}","aliases":"U^4n","display":"U^{4n}","kind":"use","excerpt":"Put K= Q_2(i), R= Z_2[i], a=(1+i)/2, and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fd46b5e0e8aaea04","tex":"U^8","aliases":"U^8","display":"U^8","kind":"use","excerpt":"For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every n","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9bad9d0840a84cee","tex":"U^a","aliases":"U^a","display":"U^a","kind":"use","excerpt":"pageheading1. Coefficient products and reciprocal differentiationA characteristic-p argument without factorial division For sequences f,g: N→ K, where K is a field, define (f⋆ g)(d)=∑_i=0^dbinom di f(i)g(d-i), (U^a f)(d)=f(d+a). These are Hurwitz coefficients. The notation does not divide by d! in F_p. In particular, the implementation is not ordin","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ced2329334a8bdd4","tex":"U^{a-i}","aliases":"U^a-i","display":"U^{a-i}","kind":"use","excerpt":"The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d6d4889a32c287d8","tex":"u^H","aliases":"u^H","display":"u^H","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-663ee04bfdc73b2a","tex":"u^h","aliases":"u^h","display":"u^h","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ec5ed9d3298fa46f","tex":"u^{H_a}","aliases":"u^H_a","display":"u^{H_a}","kind":"use","excerpt":"2^a+1, · p=2. Then Δ_H_a^ ru^n = u^n(u^H_a-1)^r. For odd p, v_p(Δ_H_a^ ru^n) ≥(a+1)r. For p=2 and odd u, v_2(Δ_H_a^ ru^n) ≥(a+3)r. Equivalently, every additional prime power digit increases the certified Mahler slope by one.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8538af623a4ec3ed","tex":"u^{Hx+j}","aliases":"u^Hx+j","display":"u^{Hx+j}","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4fb35335a011459e","tex":"U^i","aliases":"U^i","display":"U^i","kind":"use","excerpt":"The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-df29459720bd9419","tex":"u^j","aliases":"u^j","display":"u^j","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fa49e3001badb297","tex":"u^{j+(p-1)s}","aliases":"u^j+(p-1)s","display":"u^{j+(p-1)s}","kind":"use","excerpt":"[Local phase lattice] For a unit mode u^n, the odd prime coordinate gives u^n = u^j+(p-1)s (u^(p-1)p^a)^q. The dyadic coordinate gives u^n = u^j+2s (u^2^a+1)^q. Thus finite character phase, local digit address, and Mahler depth are separate coordinates.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6b4cbf2da3f0cedc","tex":"u^{j+2s}","aliases":"u^j+2s","display":"u^{j+2s}","kind":"use","excerpt":"[Local phase lattice] For a unit mode u^n, the odd prime coordinate gives u^n = u^j+(p-1)s (u^(p-1)p^a)^q. The dyadic coordinate gives u^n = u^j+2s (u^2^a+1)^q. Thus finite character phase, local digit address, and Mahler depth are separate coordinates.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e6106e1ebef1b51f","tex":"u^m","aliases":"u^m","display":"u^m","kind":"use","excerpt":"The coefficients c_u in the proof of [source reference] are constant on the cosets of H_m. For u≠0, the value u^m depends only on the coset, and every element of G_m arises. Grouping the modes by cosets gives the displayed expansion.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-49185fe756c9fe7d","tex":"u^n","aliases":"u^n","display":"u^n","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":19},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cf3adb26b0f573f1","tex":"u^{n+p-1}","aliases":"u^n+p-1","display":"u^{n+p-1}","kind":"use","excerpt":"For positive n, the zero mode contributes nothing. Every nonzero mode satisfies u^p-1=1. Hence b_n+p-1^(m,p) = ∑_u≠0c_uu^n+p-1 = ∑_u≠0c_uu^n = b_n^(m,p).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2ab4e8c0ac2eecc2","tex":"U^{p+L}","aliases":"U^p+L","display":"U^{p+L}","kind":"use","excerpt":"0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a53f53a1a1287a8","tex":"u^{p-1}","aliases":"u^p-1","display":"u^{p-1}","kind":"use","excerpt":"For positive n, the zero mode contributes nothing. Every nonzero mode satisfies u^p-1=1. Hence b_n+p-1^(m,p) = ∑_u≠0c_uu^n+p-1 = ∑_u≠0c_uu^n = b_n^(m,p).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b13f6b9a00f77135","tex":"U^{p^2}","aliases":"U^p^2","display":"U^{p^2}","kind":"use","excerpt":"0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c4a8fe218fcdcbef","tex":"U^{q-a}","aliases":"U^q-a","display":"U^{q-a}","kind":"use","excerpt":"pageheading1. The characteristic-zero product ruleThe normalized defect is not an ordinary derivation Let A be a commutative algebra over a characteristic-zero field K, and let U be an actual K-derivation. The source defines B=fracU^p^2-U^pp, M_q(f,g)=∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d6a7e14e69737cad","tex":"u^r","aliases":"u^r","display":"u^r","kind":"use","excerpt":"The nth coefficient is ∑_r=0^n binom nr u^rv^n-r = (u+v)^n by the binomial theorem.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","excerpt":"To obtain the root equivalence on a product neighborhood, we choose the neighborhoods as follows. The source of the open partial homeomorphism defined by Φ is open around a; its inverse is g. Choose open sets V and U_0 containing y_0 and t_0 such that V× U_0 lies in that source. Restrict U_0 further by the preimage of an open analytic inverse neigh","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-76bfead452deff02","tex":"U_2","aliases":"U_2","display":"U_2","kind":"use","excerpt":"For example, U_2(n)≡2pmod32 and U_3(n)≡6pmod8 for every row. At precision 2^10 the least packet periods are 512,512,32,128. A packet displacement T has original degree displacement 4T, as proved by leanKomorebiPaperIX.DyadicFourLeastPeriods.original_degree_exact_displacement. Minimality here concerns packet shifts. Shifts of the full degree-indexed","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-deb9be7f41a14efb","tex":"U_3","aliases":"U_3","display":"U_3","kind":"use","excerpt":"For example, U_2(n)≡2pmod32 and U_3(n)≡6pmod8 for every row. At precision 2^10 the least packet periods are 512,512,32,128. A packet displacement T has original degree displacement 4T, as proved by leanKomorebiPaperIX.DyadicFourLeastPeriods.original_degree_exact_displacement. Minimality here concerns packet shifts. Shifts of the full degree-indexed","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e2facfe7dfb6a6c9","tex":"U_{4,j}","aliases":"U_4,j","display":"U_{4,j}","kind":"use","excerpt":"For the remainder of the four-packet discussion abbreviate C_j=C_4,j and U_j=U_4,j. Removing the original row cap requires an unshifted coefficient operator.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e34eef57ab05c016","tex":"U_{6,3}","aliases":"U_6,3","display":"U_{6,3}","kind":"use","excerpt":"C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(2)-C_6,3(1)=frac64455. The last difference has 3-adic valuation zero. Here h=1, so at n=1 the order-one bound without factorial loss would require valuation at least one. The corrected normalized bound is sharp, since v_3(C_6,3(2)-C_6,3(1))=0=1-v_3(3!). Multiplication by 3! gives U_6,3(2)-U_6,3(1)=frac","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ec4379f6b5fb17bc","tex":"U_a","aliases":"U_a","display":"U_a","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0b3db69758fee0b4","tex":"U_d","aliases":"U_d","display":"U_d","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-022130268d2f106e","tex":"U_G","aliases":"U_G","display":"U_G","kind":"use","excerpt":"[Composition of coefficient transports] If U_H and U_G are the triangular transforms associated with unit kernels H and G, then U_HU_G=U_HG. In particular, the inverse of U_H is U_H^-1.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f72fa924cc22eb23","tex":"U_H","aliases":"U_H","display":"U_H","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5fb0aa1c05324642","tex":"U_H^{-1}","aliases":"U_H^-1","display":"U_H^{-1}","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cb9ab2940f863d81","tex":"U_{H^{-1}}","aliases":"U_H^-1","display":"U_{H^{-1}}","kind":"use","excerpt":"[Composition of coefficient transports] If U_H and U_G are the triangular transforms associated with unit kernels H and G, then U_HU_G=U_HG. In particular, the inverse of U_H is U_H^-1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-de7794b119a272d1","tex":"U_{HG}","aliases":"U_HG","display":"U_{HG}","kind":"use","excerpt":"[Composition of coefficient transports] If U_H and U_G are the triangular transforms associated with unit kernels H and G, then U_HU_G=U_HG. In particular, the inverse of U_H is U_H^-1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"Apply polynomial induction using constants, addition, and multiplication by a variable. Constants have zero remainder, and remainders add. For the multiplication step, suppose the assertion is known for P and write D_P(U)=∑_j(∂_jP)(a)U_j. Multiplication of its expansion by a_i+cU_i gives constant term P(a)a_i, linear term c(a_iD_P(U)+P(a)U_i), and ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1c42292632e07b88","tex":"u_i^h","aliases":"u_i^h","display":"u_i^h","kind":"use","excerpt":"Finite integral linear combinations. For any finite index set S, coefficients a_i∈ Z_p and units u_i∈ Z_p^×, linearity of the actual iterated operator gives Δ_h^s (∑_i∈ Sa_i u_i^n) =∑_i∈ Sa_i u_i^n(u_i^h-1)^s. At h=p-1, the result is divisible by p^s. The empty sum is permitted. This does not prove that the general congruential Euler packets of The","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f5e4fdf16b0453e5","tex":"u_i^n","aliases":"u_i^n","display":"u_i^n","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-50d448cbdfbc40ce","tex":"u_i^{p-1}","aliases":"u_i^p-1","display":"u_i^{p-1}","kind":"use","excerpt":"Induction gives the geometric identity. Reduction of a unit to F_p^× gives u_i^p-1=1 modulo p. Taking s powers supplies p^s in each summand. The difference operator is linear over finite sums.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","excerpt":"For the remainder of the four-packet discussion abbreviate C_j=C_4,j and U_j=U_4,j. Removing the original row cap requires an unshifted coefficient operator.","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-72b2bc9e12d0fa8e","tex":"U_k","aliases":"U_k","display":"U_k","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-836830fc78b794aa","tex":"u_k","aliases":"u_k","display":"u_k","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f6951bc1a235a45d","tex":"U_{k+1}","aliases":"U_k+1","display":"U_{k+1}","kind":"use","excerpt":"or preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-41ab4fffc65aae75","tex":"u_{k+1}","aliases":"u_k+1","display":"u_{k+1}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0d6567d269df4cd4","tex":"u_{k+1}^{(j)}","aliases":"u_k+1^(j)","display":"u_{k+1}^{(j)}","kind":"use","excerpt":"[Companion jump recurrence] Every residue state packet satisfies u_k+1^(j)=C^Nu_k^(j). More explicitly, u_k^(j)=C^j(C^N)^kv_0.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-783dcbe8db097453","tex":"u_{k+2}","aliases":"u_k+2","display":"u_{k+2}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9bd23f73fc8e35ba","tex":"u_{k+a}","aliases":"u_k+a","display":"u_{k+a}","kind":"use","excerpt":"The jump formula. Associativity and the laws for powers of one matrix give u_k=C^j(C^N)^kv_0=(C^N)^k(C^jv_0). Only powers of the same matrix are interchanged. The argument does not assume that arbitrary matrices commute. In particular, u_k+1=C^Nu_k, u_k+a=(C^N)^a u_k. The step matrix is C^N, not C. A checked example with the Fibonacci matrix and N=","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-550ce951561bdee5","tex":"u_{k+d}","aliases":"u_k+d","display":"u_{k+d}","kind":"use","excerpt":"The complete vector recurrence. Apply the matrix identity to the actual state u_k. The already proved advance identity identifies each D^a u_k with u_k+a. Consequently, ∑_a=0^dχ_D[a]u_k+a=0, u_k+d=-∑_a=0^d-1c_a u_k+a. This is the desired recurrence, with its monic leading coefficient and all earlier coefficients supplied by χ_D. It is not an assump","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c89ff2cc9cf26697","tex":"u_k^{(j)}","aliases":"u_k^(j)","display":"u_k^{(j)}","kind":"use","excerpt":"Let R be a commutative ring. Let v_n+1=Cv_n, C∈ M_d(R). For a positive integer N and an address j∈{0,…,N-1}, define the residue state packet u_k^(j)=v_Nk+j.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e4d61c16c87b9409","tex":"u_{Mk+s}","aliases":"u_Mk+s","display":"u_{Mk+s}","kind":"use","excerpt":"Refinement of an arithmetic address. Taking another subsequence with step M and offset s yields u_Mk+s=v_(NM)k+(Ns+j). The formal proof establishes this identity for the actual packet function by checking the index arithmetic. It is not inferred from a diagram or a naming convention.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-23483ef5ed33ad34","tex":"U_{mp,j}","aliases":"U_mp,j","display":"U_{mp,j}","kind":"use","excerpt":"[Step-sensitive divisor-branch bound] Under the hypotheses of Theorem~[source reference], for all u,k,n∈ N, Δ_u(p-1)/m^kU_mp,j(n) ∈ p^min(k(1+ν_p(u)),mn) Z_p. For C_mp,j the corresponding valuation lower bound is min(k(1+ν_p(u)),mn)-v_p(j!).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8611e5a338fe3b84","tex":"U_r","aliases":"U_r","display":"U_r","kind":"use","excerpt":"[Exact classical tail valuation] For every r≥1 and N≥ r, v_2( T_r-1(N)) =4r-1+v_2binom Nr+μ_r+v_2G_r(N-r). There are unique factors G_r(X)=W_r(X)U_r(X), where W_r∈ Z_2[X] is monic of degree d_r=max J_r and U_r∈ Z_2⟨ X⟩^× satisfies U_r≡1pmod2. Consequently v_2G_r(x)=v_2W_r(x) for every x∈ Z_2, and deg W_r≤lfloorfracs_2(r)-13rfloor. Equation~[source ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6ee2348ac4bbdee7","tex":"\\mathcal U_r","aliases":"U_r mathcal","display":"\\mathcal U_r","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a822fb370eb1244d","tex":"u_r","aliases":"u_r","display":"u_r","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c6d7e97428f82a13","tex":"\\mathcal U_r'","aliases":"U_r' mathcal","display":"\\mathcal U_r'","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c308aac68fce06b4","tex":"U_{r,q}","aliases":"U_r,q","display":"U_{r,q}","kind":"use","excerpt":"The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r,","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b96781006c96014d","tex":"U_{r,r}","aliases":"U_r,r","display":"U_{r,r}","kind":"use","excerpt":"0, · q>r. The row-action equality, the diagonal identity U_r,r=1, and vanishing above the diagonal are all checked. The arithmetic weight is also explicit: p^2(r-q)mid U_r,q(p) in Z, 0≤ q≤ r. No inverse-coordinate identity, unit diagonal or weight bound is assumed for these rows.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c4102fb7b5673ed4","tex":"\\underline q","aliases":"underline q underline","display":"\\underline q","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a8ee3eea11f4b48b","tex":"V^{-1}","aliases":"V^-1","display":"V^{-1}","kind":"use","excerpt":"the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(1+T)/T, interpreted by coefficient shifting, differentiation of the displayed factorization gives R_c(l","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c01fa9580cfa7f71","tex":"v^n","aliases":"v^n","display":"v^n","kind":"use","excerpt":"[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cb514f238366e3a5","tex":"v^{n-r}","aliases":"v^n-r","display":"v^{n-r}","kind":"use","excerpt":"The nth coefficient is ∑_r=0^n binom nr u^rv^n-r = (u+v)^n by the binomial theorem.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-01749da563f81b32","tex":"V^q","aliases":"V^q","display":"V^q","kind":"use","excerpt":"For k=2^r≥4, put q=k/2, which is even. Normalize the exact expansion [source reference] by 2^q before reducing modulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-16cbf1f7187068e1","tex":"V^{q-1}","aliases":"V^q-1","display":"V^{q-1}","kind":"use","excerpt":"For k=2^r≥4, put q=k/2, which is even. Normalize the exact expansion [source reference] by 2^q before reducing modulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fcdd77803761b961","tex":"v^r","aliases":"v^r","display":"v^r","kind":"use","excerpt":"Distinct elements of G_m define linearly independent functions on one full period, by the Vandermonde determinant. Therefore a positive integer r is a period exactly when v^r=1 for every active mode. The least such r is the stated least common multiple.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0108c541d6b05811","tex":"v_{(NM)k+(Ns+j)}","aliases":"v_(NM)k+(Ns+j)","display":"v_{(NM)k+(Ns+j)}","kind":"use","excerpt":"Refinement of an arithmetic address. Taking another subsequence with step M and offset s yields u_Mk+s=v_(NM)k+(Ns+j). The formal proof establishes this identity for the actual packet function by checking the index arithmetic. It is not inferred from a diagram or a naming convention.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":19},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"use","excerpt":"[Exact address-two power orders] For every starting row n≥0 and all positive mixed steps, v_2(Δ_H_1⋯Δ_H_kC_2(n)) =V_2(k)+∑_ν=1^k v_2(H_ν) whenever k is a positive power of two, where V_2(1)=4, V_2(2)=6, V_2(k)=2k+v_2(k!)+frack2 (k≥4). For the original coefficients, the right side gains one bit.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"ith-normalization-loss Take p=3, m=2, N=6, and j=3. The recurrence gives C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(2)-C_6,3(1)=frac64455. The last difference has 3-adic valuation zero. Here h=1, so at n=1 the order-one bound without factorial loss would require valuation at least one. The corrected normalized bound is sharp, since v_3(C_6,3(2)","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a478c8ae160947ff","tex":"V_a","aliases":"V_a","display":"V_a","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0b837e89a1e8104e","tex":"V_h","aliases":"V_h","display":"V_h","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"[Derivative of polynomial evaluation] For P∈ K[X_1,…,X_n], polynomial evaluation is analytic at every point. At a∈ K^n its strict Fr echet derivative is v⟼∑_j=1^n (∂_jP)(a)v_j. For a finite system, its strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use ","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b47bde7729d16d41","tex":"V_N","aliases":"V_N","display":"V_N","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a6735e9f71ad2ebf","tex":"V_n","aliases":"V_n","display":"V_n","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"use","excerpt":"Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4cf1c9f8283bbb4c","tex":"V_{n+1}","aliases":"V_n+1","display":"V_{n+1}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7fd7bfa7519bcf67","tex":"v_{n+1}","aliases":"v_n+1","display":"v_{n+1}","kind":"use","excerpt":"Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6a3bb1565bfdea96","tex":"V_{n+2}","aliases":"V_n+2","display":"V_{n+2}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8c3c0f1c05a928ce","tex":"v_{Nk+j}","aliases":"v_Nk+j","display":"v_{Nk+j}","kind":"use","excerpt":"Let R be a commutative ring. Let v_n+1=Cv_n, C∈ M_d(R). For a positive integer N and an address j∈{0,…,N-1}, define the residue state packet u_k^(j)=v_Nk+j.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6afc7de2b2e38c7c","tex":"V_{nn}","aliases":"V_nn","display":"V_{nn}","kind":"use","excerpt":"egintheorem[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4751bc50d0cf4124","tex":"V_{nq}","aliases":"V_nq","display":"V_{nq}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-44cf5118c3be9cba","tex":"\\vdots","aliases":"vdots","display":"\\vdots","kind":"use","excerpt":"[Polyphase shift matrix] Let E be the scalar shift (Ea)_n=a_n+1, and let E_q shift the coarse index. Then P_M(Ea) = S_M(E_q) P_Ma, where S_M(E_q) = 0 · 1 · 0 · ⋯ · 0 0 · 0 · 1 · ⋯ · 0 vdots · · · ddots · vdots 0 · 0 · 0 · ⋯ · 1 E_q · 0 · 0 · ⋯ · 0 . It also satisfies S_M(E_q)^M=E_qI_M.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8e4d848b11431c85","tex":"\\vp","aliases":"vp v_p","display":"{v_p}","kind":"use","excerpt":"w(r)<L binom nr c_rp^w(r)S_n-r pmod p^L. When the omitted set is nonempty, the tail has valuation at least min_substack0≤ r≤ n w(r)≥ L ( w(r)+v_pbinom nr+v_p(S_n-r) ).","count":20},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b25fb9551879a1d3","tex":"\\vTwo","aliases":"vTwo v_2","display":"{v_2}","kind":"use","excerpt":"The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the b","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e83c58309eb2060b","tex":"\\mathcal W","aliases":"W mathcal","display":"\\mathcal W","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definition","count":71},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e9fe8ee5ada9c8bf","tex":"W'","aliases":"W'","display":"W'","kind":"use","excerpt":"athrmact). The equality includes the zero case. The same successive monic polynomial division used in Theorem~[source reference], started at precision eight, preserves the quadratic and unit reductions of [source reference]. Write W=Z^2+2β_rZ+α_r and w=W(c_r). Stationarity gives W'(c_r)=-τ_rw, while w=Φ_r^act/ U_r(c_r). The quadratic identity β_r^2","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-20ccd43f98f93af7","tex":"W^k","aliases":"W^k","display":"W^k","kind":"use","excerpt":"For every starting row, Δ_1^k C_0(n) = L(U^4nW^k) =∑_s=0^nbinom ns L(W^k+s). Each summand lies in 2^k k! Z_2, proving the factorial bound for address zero from row zero onward.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-291f703d002fce35","tex":"W^{k+s}","aliases":"W^k+s","display":"W^{k+s}","kind":"use","excerpt":"For every starting row, Δ_1^k C_0(n) = L(U^4nW^k) =∑_s=0^nbinom ns L(W^k+s). Each summand lies in 2^k k! Z_2, proving the factorial bound for address zero from row zero onward.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4f034e94d59a11da","tex":"W^m","aliases":"W^m","display":"W^m","kind":"use","excerpt":"he first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the third identity by (i E F)^m and reduce the Euler powers using the integral quadratic relations. Finally, γ_k(X+Y)=∑_r=0^kγ_r(X)γ_k-r(Y) proves γ_k(H_0)∈ A_0. Applying the functional and putting W=U^4-1=2H_0 gives L(W^m)∈2^m m!R (m≥0). These moment","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9022bd928b04faea","tex":"w^p","aliases":"w^p","display":"w^p","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3d2786a8c65b662f","tex":"w^{p-1}","aliases":"w^p-1","display":"w^{p-1}","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7b8f78bd431e445d","tex":"w^{p^k}","aliases":"w^p^k","display":"w^{p^k}","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-65d03882bd3591c3","tex":"w^q","aliases":"w^q","display":"w^q","kind":"use","excerpt":"The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a5ddff78ba645e2d","tex":"w^T","aliases":"w^T","display":"w^T","kind":"use","excerpt":"Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d01794b5f0c3ada0","tex":"W_1","aliases":"W_1","display":"W_1","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1dd506c998004ce1","tex":"W_2","aliases":"W_2","display":"W_2","kind":"use","excerpt":"egative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is +∞. These conventions include M≥ n/2. For every r≥1, v_2(κ_2,r)=4r-1. The rth reconstruction term has exact valuation v_2( κ_2,rbinom n2rR_n-2r ) = 4r-1+v_2binom n2r. Equivalently, W_2(n,r) = 4r-1+s_2(r)+s_2(n-2r)-s_2(n).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cf76327e749de1b7","tex":"W_k","aliases":"W_k","display":"W_k","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b5731a3740a4e086","tex":"W_{k+1}","aliases":"W_k+1","display":"W_{k+1}","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8d7f1cc8c150d4e5","tex":"W_p","aliases":"W_p","display":"W_p","kind":"use","excerpt":"[Odd row weight] For an odd prime p, the rth nonzero term in the reconstruction has valuation W_p(n,r) = 2r+v_p(Z_2r+1) +v_pbinom n2r +v_p(S_n-2r(p)). The binomial contribution is v_pbinom n2r = fracs_p(2r)+s_p(n-2r)-s_p(n)p-1, where s_p is the base p digit sum.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-750210c4c92949cd","tex":"\\mathcal W_p","aliases":"W_p mathcal","display":"\\mathcal W_p","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4a4f3028bb492bb2","tex":"W_r","aliases":"W_r","display":"W_r","kind":"use","excerpt":"[Exact classical tail valuation] For every r≥1 and N≥ r, v_2( T_r-1(N)) =4r-1+v_2binom Nr+μ_r+v_2G_r(N-r). There are unique factors G_r(X)=W_r(X)U_r(X), where W_r∈ Z_2[X] is monic of degree d_r=max J_r and U_r∈ Z_2⟨ X⟩^× satisfies U_r≡1pmod2. Consequently v_2G_r(x)=v_2W_r(x) for every x∈ Z_2, and deg W_r≤lfloorfracs_2(r)-13rfloor. Equation~[source ","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dce7dfb4f712d666","tex":"\\mathcal W_r","aliases":"W_r mathcal","display":"\\mathcal W_r","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"use","excerpt":"pageheading3. Actual packet congruences and sharp boundariesConsequences for the standard window; not the full Theorem 7.5 The fifth block constructed integral lifts widehat C_mp,j(n)∈ Z_p of the original rational coefficients when 0≤ j<p, and proved that their residues equal e_n^(m,p).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-41f5515a92f29307","tex":"\\widehat C_{mp,j}","aliases":"widehat C_mp,j widehat","display":"\\widehat C_{mp,j}","kind":"use","excerpt":"pageheading3. Actual packet congruences and sharp boundariesConsequences for the standard window; not the full Theorem 7.5 The fifth block constructed integral lifts widehat C_mp,j(n)∈ Z_p of the original rational coefficients when 0≤ j<p, and proved that their residues equal e_n^(m,p).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a999477715e268b","tex":"\\widehat F","aliases":"widehat F widehat","display":"\\widehat F","kind":"use","excerpt":"[Amice pullback] For a formal exponential generating series F(z)=∑_n≥0a_nfracz^nn!, define widehat F(T)=F(log(1+T)).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"use","excerpt":"cal E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-41bdac88160e4cd7","tex":"\\widehat P_{N,j}","aliases":"widehat P_N,j widehat","display":"\\widehat P_{N,j}","kind":"use","excerpt":"cal E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2c8131c596a2a289","tex":"\\widehat u","aliases":"widehat u widehat","display":"\\widehat u","kind":"use","excerpt":"inite solved-block cokernel and its parameter-dependent class by C_x=S^e/imJ, o_x(v)=[bar r+Kv]∈ C_x. Then L_v≠∅iff o_x(v)=0. More precisely, the checked digit-to-lift equivalence is L_v≅{u∈ S^m:Ju=-bar r-Kv}. It sends a solved digit u to the actual next residue represented by x+q^L(widehat u,widehat v). The obstruction depends on the solved block ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f207fa5d22c484f6","tex":"\\widehat v","aliases":"widehat v widehat","display":"\\widehat v","kind":"use","excerpt":"x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the se","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-657478f81580a29e","tex":"\\widehat{\\mathbb Z}","aliases":"widehat Z widehat mathbb","display":"\\widehat{\\mathbb Z}","kind":"use","excerpt":"The reason is exact: the CRT ring isomorphism preserves polynomial evaluation, and zero in a finite product is equivalent to zero in every component. For F empty, passing to all compatible finite congruences identifies the profinite completion widehat Z with ∏_p Z_p. The same finite reconstruction is compatible with coefficient projections and inte","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c150adb9f1dd660a","tex":"\\widehat{F'}","aliases":"widehatF' widehat","display":"\\widehat{F'}","kind":"use","excerpt":"[Derivative conjugacy] widehatF'(T) = (1+T)fracddTwidehat F(T).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-79f270505a397e43","tex":"\\widehat\\Z","aliases":"widehatZ widehat Z widehat mathbb","display":"\\widehat{\\mathbb Z}","kind":"use","excerpt":"Finite CRT reconstruction leads to the familiar completed object widehat Z=∏_p Z_p, but does not identify which local tuples come from a single global integer. A complete form of Native 13.3 must therefore specify finite-level compatibility, transition maps and completion, and any additional global descent condition. If recurrence, spectral, or Fro","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5c95d4e5ac3dec21","tex":"\\widetilde C_{6,3}","aliases":"widetilde C_6,3 widetilde","display":"\\widetilde C_{6,3}","kind":"use","excerpt":"For the unnormalized convention, Δ_1widetilde C_6,3(1)=3! frac64455=frac384455, v_3 (frac384455)=1. The factorial's valuation is also proved: v_3(3!)=1, v_3(Δ_1C_6,3(1)) =1-v_3(3!)=0. This proves sharpness at the displayed normalization boundary. It does not establish the corrected global inequality for all parameters, and does not assert that ever","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bd7df62e1aa54b13","tex":"\\widetilde C_{N,j}","aliases":"widetilde C_N,j widetilde","display":"\\widetilde C_{N,j}","kind":"use","excerpt":"Principal targets. targetFiltered.padic_filtered_reconstruction targetStandardWindow.lucas_row_reduction targetStandardWindow.standard_diagonal_unit targetStandardWindow.factorial_valuation_zero pageheading4. Sharp normalization boundarySource item 13.1 The construction fixes the two coefficient conventions: C_N,j(n), widetilde C_N,j(n)=j! C_N,j(n)","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-73259486419af0dc","tex":"\\widetilde e","aliases":"widetilde e widetilde","display":"\\widetilde e","kind":"use","excerpt":"Fix a prime p, an integer m≥1, and an address 0≤ j<p. Put N=mp. For arbitrary m, let widetilde e_n^(m,p)∈ F_p be the unique sequence satisfying ∑_q=0^n binommnmq widetilde e_q^(m,p) = δ_n0. When mmid p-1, this sequence agrees with the compressed Hurwitz inverse in [source reference].","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6814e4fb1154514e","tex":"\\widetilde e_n^{(m,p)}","aliases":"widetilde e_n^(m,p) widetilde","display":"\\widetilde e_n^{(m,p)}","kind":"use","excerpt":"Fix a prime p, an integer m≥1, and an address 0≤ j<p. Put N=mp. For arbitrary m, let widetilde e_n^(m,p)∈ F_p be the unique sequence satisfying ∑_q=0^n binommnmq widetilde e_q^(m,p) = δ_n0. When mmid p-1, this sequence agrees with the compressed Hurwitz inverse in [source reference].","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ad9be787a2daffac","tex":"\\widetilde e_q^{(m,p)}","aliases":"widetilde e_q^(m,p) widetilde","display":"\\widetilde e_q^{(m,p)}","kind":"use","excerpt":"Fix a prime p, an integer m≥1, and an address 0≤ j<p. Put N=mp. For arbitrary m, let widetilde e_n^(m,p)∈ F_p be the unique sequence satisfying ∑_q=0^n binommnmq widetilde e_q^(m,p) = δ_n0. When mmid p-1, this sequence agrees with the compressed Hurwitz inverse in [source reference].","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6ecf0ac96e6f9d4e","tex":"\\widetilde e_r^{(m,p)}","aliases":"widetilde e_r^(m,p) widetilde","display":"\\widetilde e_r^{(m,p)}","kind":"use","excerpt":"[Rational residual generating function] Under the hypotheses of [source reference], for every standard address j, ∑_n≥0 (C_mp,j(n)bmod p)z^n = 1+ frac ∑_r=1^hwidetilde e_r^(m,p)z^r 1-z^h.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e24a9d0c560ae65e","tex":"\\widetilde t","aliases":"widetilde t widetilde","display":"\\widetilde t","kind":"use","excerpt":"Integer representatives and residue classes at the next level must also be distinguished. For q≥1 and L≥0, the map S_q^d⟶ {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection, where widetilde t is any integer representative. Independence of the representative and injectivity both follow by cancelling q^L in integer divisibility; surje","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aba1aecba944a16d","tex":"\\widetilde U","aliases":"widetilde U widetilde","display":"\\widetilde U","kind":"use","excerpt":"[Exact image of the original coefficients] The continuous extension of the original sequence is widetilde U_j(x)=j!+2^d_jΦ_j(x). It is injective, its image is exactly j!+2^d_j Z_2, and |widetilde U_j(x)-widetilde U_j(y)|_2 =2^-d_j|x-y|_2. Every point of that affine image has exactly one preimage in Z_2.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a1faa955cc41a0e0","tex":"\\widetilde u","aliases":"widetilde u widetilde","display":"\\widetilde u","kind":"use","excerpt":"A candidate for the next lift has the form x+q^L(widetilde u,widetilde v), with solved digit u∈ S_q^m. Its Taylor correction equation is overliner_x+J_xu+K_xv=0. Solvability is exactly vanishing of the displayed cokernel class. Restrict the residue-digit equivalence to the prescribed parameter coordinates. This identifies the lift fiber with the so","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-52f025dc16d3b9fb","tex":"\\widetilde U_j","aliases":"widetilde U_j widetilde","display":"\\widetilde U_j","kind":"use","excerpt":"[Exact image of the original coefficients] The continuous extension of the original sequence is widetilde U_j(x)=j!+2^d_jΦ_j(x). It is injective, its image is exactly j!+2^d_j Z_2, and |widetilde U_j(x)-widetilde U_j(y)|_2 =2^-d_j|x-y|_2. Every point of that affine image has exactly one preimage in Z_2.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3323d1822e9c5f0f","tex":"\\widetilde v","aliases":"widetilde v widetilde","display":"\\widetilde v","kind":"use","excerpt":"ixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residue roots over x whose parameter coordinates are t_0+q^Lwidetilde v modulo q^L+","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b3b957cb4014dfcd","tex":"\\Wp","aliases":"Wp W_p mathcal","display":"{\\mathcal W_p}","kind":"use","excerpt":"Weighted membership. The class used in the formalization is W_p={F∈ Q_p[[T]]: p^lfloor d/prfloorcoefTdF∈ Z_p for every d}. The integral-coordinate estimate from page 2 now proves H_r(log(1+T))∈ W_p for every residue r, including the singular ones. Rational coefficients are mapped into Q_p by the actual coefficient ring homomorphism.","count":22},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-371acb48389709be","tex":"\\mathrm{wt}","aliases":"wt mathrm","display":"\\mathrm{wt}","kind":"use","excerpt":"The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-65e8ec76ae768d6c","tex":"x^{-1}","aliases":"x^-1","display":"x^{-1}","kind":"use","excerpt":"[Finite inverse formula] If x has augmentation one and n=x-1, then x^-1 = ∑_r=0^p-1(-n)^r.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"The affine statement is formalized by leanKomorebi.NativeAffine.finite_affine_obstruction. Positive depth is essential in the nonlinear theorem: for F(X)=X^2-1, q=2, L=0, and x=0, the Jacobian is zero and the residual class is nonzero, yet y=1 is a root modulo two in the same residue modulo one.","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"[A singular polynomial obstruction] Let p=3 and consider the class x_1=0 modulo three. For F_0(x)=x^2, the residual and the reduced derivative both vanish. Every correction y∈ F_3 gives F_0(3y)≡0pmod9, so the class has three children. For F_1(x)=x^2+3, the reduced derivative is still zero, but the reduced residual is one. Its class in the cokernel ","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"use","excerpt":"[Matrix power amplification] Editorial hypothesis correction. Require L≥1. At L=0 the standalone claim is false: for p=3, d=1 and X=2=1+B, one has X^3-1=7∉3 Z_3. With L≥1, every higher binomial term in the retained proof has the required valuation, including p=2. If X=I+p^LB with B∈ M_d( Z_p), then for every a≥0 there is B_a∈ M_d( Z_p) such that X^","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b8bbaae1ffe15b3b","tex":"x^d","aliases":"x^d","display":"x^d","kind":"use","excerpt":"Here is the polynomial estimate that supplies one additional bit. For i≥1, put P_i(x)=4^ibinom4x4i. Its constant is zero, and its degree-d coefficient is [x^d]P_i(x)=frac4^i4^d s(4i,d)(4i)!, where s(m,d) is a signed Stirling number of the first kind. In each product contributing to s(4i,d), at most d-1 of the 2i-1 even integers in 1,…,4i-1 can be o","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7272f222c85749d6","tex":"X^m","aliases":"X^m","display":"X^m","kind":"use","excerpt":"Expand X^m=∑_r=0^mT(m,r)P_r(X). Since XP_r(X)=P_r+1(X)+r(r+1)P_r(X), these integer coefficients satisfy T(m+1,r)=T(m,r-1)+r(r+1)T(m,r), T(0,0)=1. Every r(r+1) is even. Induction, including the boundary indices, proves 2^m-rmid T(m,r) for 0≤ r≤ m. For m>0 the r=0 term vanishes. Each remaining summand of L(P^m/m!) has valuation at least m-r+2v_2(r!)-","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-12f1075149375120","tex":"x^N","aliases":"x^N","display":"x^N","kind":"use","excerpt":"Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-baf58144b77f8bed","tex":"X^p","aliases":"X^p","display":"X^p","kind":"use","excerpt":"Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-efe34588925c4a06","tex":"x^{p-1}","aliases":"x^p-1","display":"x^{p-1}","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0648303074e446a8","tex":"X^{p^a}","aliases":"X^p^a","display":"X^{p^a}","kind":"use","excerpt":"r Editorial hypothesis correction. Require L≥1. At L=0 the standalone claim is false: for p=3, d=1 and X=2=1+B, one has X^3-1=7∉3 Z_3. With L≥1, every higher binomial term in the retained proof has the required valuation, including p=2. If X=I+p^LB with B∈ M_d( Z_p), then for every a≥0 there is B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8744addd278e3b80","tex":"x^{p^{\\ell-1}}","aliases":"x^p^ℓ-1 ell","display":"x^{p^{\\ell-1}}","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-14fb02863a861bee","tex":"X^r","aliases":"X^r","display":"X^r","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b3f03ee84a66d436","tex":"X^{\\underline q}","aliases":"X^underline q underline","display":"X^{\\underline q}","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e291ac7130b02dc8","tex":"x^\\ell","aliases":"x^ℓ ell","display":"x^\\ell","kind":"use","excerpt":"[Taylor coefficient bounds and radii] For j=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at addres","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-162e2cb41df4870a","tex":"x_0^{m-\\ell}","aliases":"x_0^m-ℓ ell","display":"x_0^{m-\\ell}","kind":"use","excerpt":"For j=0,1, the term m=ℓ in [source reference] has exact valuation ℓ, since s(ℓ,ℓ)=1 and the Newton valuation is exact. Every later term has valuation at least ℓ+1. Thus v_2(d_j,ℓ)=ℓ, and the coefficient radius is exactly two. At a center x_0∈ Z_2, the coefficient of y^ℓ in the translated series is ∑_m≥ℓd_j,mbinom mℓ x_0^m-ℓ. Its initial term again ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"0 · 0 . The class x_1=(0,0) solves Mx_1≡ bpmod3 for both b_1=(-3,0) and b_2=(0,-3). For b_1, the reduced residual is (1,0), which belongs to the image of overline M. There are three correction classes, given by y_1+y_2=2 pmod3. For b_2, the residual is (0,1), whose cokernel class is nonzero. No lift modulo nine exists because the second row of M re","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"AtlasHeadA compatible root is built one digit at a timeatlas:smooth Take F(x)=x^2-7 over the 3-adic integers and start with x_1=1 modulo three. If F(x_L)=3^Lr_L, the next representative has the form x_L+1=x_L+3^Lt_L. Its correcting digit satisfies r_L+2x_Lt_L≡0pmod3. AtlasImagesmooth_tower For example, F(1)/3=-2 and 2x_1=2, so the next digit is one","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2a3a227a148b5734","tex":"X_2^{\\mathrm{wt}}","aliases":"X_2^wt mathrm","display":"X_2^{\\mathrm{wt}}","kind":"use","excerpt":"The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"t at a timeatlas:smooth Take F(x)=x^2-7 over the 3-adic integers and start with x_1=1 modulo three. If F(x_L)=3^Lr_L, the next representative has the form x_L+1=x_L+3^Lt_L. Its correcting digit satisfies r_L+2x_Lt_L≡0pmod3. AtlasImagesmooth_tower For example, F(1)/3=-2 and 2x_1=2, so the next digit is one and x_2=4. Then F(4)/9=1, again giving digi","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-762eb59ebd1883d4","tex":"x_A","aliases":"x_A","display":"x_A","kind":"use","excerpt":"AtlasHeadA derivative weight can cost one digit of root precisionatlas:weighted Compare the linear equations F(x)=3x=0 and G_A(x)=3x+3^A=0, where A≥2. Their coefficient difference has 3-adic valuation A, but their roots differ by x_A-0=-3^A-1, v_3(x_A)=A-1. AtlasImageweighted_motion The derivative has valuation one. Solving the correction equation ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-715547043f329ef6","tex":"X_{e+k}","aliases":"X_e+k","display":"X_{e+k}","kind":"use","excerpt":"[Continuous charts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is ho","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1c764cff6a54a44c","tex":"X_{e+q}","aliases":"X_e+q","display":"X_{e+q}","kind":"use","excerpt":"Let p be prime, L≥1, e,q≥0, and F∈ Z[X_1,…,X_e+q]^e, x∈ Z^e+q, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^e+q⟶ F_p^e is surjective. All vector congruences are coordinatewise.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"use","excerpt":"s have zero remainder, and remainders add. For the multiplication step, suppose the assertion is known for P and write D_P(U)=∑_j(∂_jP)(a)U_j. Multiplication of its expansion by a_i+cU_i gives constant term P(a)a_i, linear term c(a_iD_P(U)+P(a)U_i), and remainder c^2(Q(U)(a_i+cU_i)+D_P(U)U_i). The product rule identifies the linear term with the de","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"use","excerpt":"a)v_j. For a finite system, its strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use polynomial induction on constants, addition, and multiplication by a variable. Constants have zero derivative. The sum rule agrees with the additivity of formal partial d","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"Proof chain. The scalar polynomial derivative bridge is proved by induction on actual multivariate polynomials: constants, sums, and multiplication by a variable. For a scalar polynomial P, its strict derivative is v⟼∑_j frac∂ P∂ x_j(a)v_j. Assembling the coordinates gives the strict derivative of the evaluated system as its actual Jacobian matrix.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-441b7c961c9320ed","tex":"x_{L+1}","aliases":"x_L+1","display":"x_{L+1}","kind":"use","excerpt":"[One level linear lifting] A class represented by x_L lifts to a solution modulo p^L+1 if and only if Ob^lin_L(x_L)=0. More precisely, a correction y∈ F_p^d gives a lift x_L+1=x_L+p^Ly pmod p^L+1 if and only if overline My=-ρ_L(x_L). When one correction exists, the set of all correction classes is a coset of keroverline M.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"use","excerpt":"[Derivative of polynomial evaluation] For P∈ K[X_1,…,X_n], polynomial evaluation is analytic at every point. At a∈ K^n its strict Fr echet derivative is v⟼∑_j=1^n (∂_jP)(a)v_j. For a finite system, its strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4715e03ab2461874","tex":"x_{n+1}","aliases":"x_n+1","display":"x_{n+1}","kind":"use","excerpt":"begingroupsmallraggedright Formal correspondence. leanCompanion.companion_jump, leanCompanion.packet_charpoly_recurrence, and leanCompanion.packet_observable_recurrence. The source-sequence versions derive the packet recurrence from the relation x_n+1=Cx_n. For the Fibonacci matrix C=(1 · 1","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f551abf5f44825db","tex":"x_{n+2}","aliases":"x_n+2","display":"x_{n+2}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-425ede770bf1a9d2","tex":"x_{Nk+j}","aliases":"x_Nk+j","display":"x_{Nk+j}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-89d5e60ade50d285","tex":"X_p","aliases":"X_p","display":"X_p","kind":"use","excerpt":"The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"The formal theorem leanKomorebi.PolynomialObstruction.finite_polynomial_obstruction presents the image using the integer-domain map Z^dxrightarrowDF(x) Z^e→ S_q^e. Reduction onto S_q^d identifies that image with im J_x. The class-preserving comparison is explicit in leanKomorebi.PolynomialLiftCounts.obstructionEquiv; the change of presentation does","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0e7659cd4ef8f947","tex":"Y^2","aliases":"Y^2","display":"Y^2","kind":"use","excerpt":"The error exponent cannot uniformly be increased. For F(Y,T)=Y+Y^2+T, H=0, a=0, p=2, s=0, M=1, and v=1, there is an even root y of y+y^2+2=0. Its equation forces y≡2pmod4 and y+2=-y^2, so 4mid y+2, 8nmid y+2. Thus the error depth 2M-3s=2 is attained even for parameter-only motion. The same quadratic with a constant coefficient perturbation gives th","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3aed34bf8b499303","tex":"y^k","aliases":"y^k","display":"y^k","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-560f4e5df768a9d8","tex":"y^m","aliases":"y^m","display":"y^m","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8da30c36f5ac9719","tex":"y^N","aliases":"y^N","display":"y^N","kind":"use","excerpt":"[Characteristic polynomial of the jump] Let χ_C(y)=det(yI-C). Then χ_C^N(x) = Res_y(χ_C(y),x-y^N). The identity holds over the coefficient ring after interpreting the resultant universally.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0ecc9841022db246","tex":"y^p","aliases":"y^p","display":"y^p","kind":"use","excerpt":"Choose a generator g of the cyclic additive group. Then F_p[ F_p^+] ≅ F_p[y]/(y^p-1) = F_p[y]/(y-1)^p. The augmentation ideal is generated by y-1, so its pth power is zero. Every element has the form a+n with a∈ F_p and n∈ J. It is invertible exactly when a≠0, which is the augmentation condition.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fd3e4a5285d4c840","tex":"y^{p^{\\ell-1}}","aliases":"y^p^ℓ-1 ell","display":"y^{p^{\\ell-1}}","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8ded9263acb77baa","tex":"y^\\ell","aliases":"y^ℓ ell","display":"y^\\ell","kind":"use","excerpt":"For j=0,1, the term m=ℓ in [source reference] has exact valuation ℓ, since s(ℓ,ℓ)=1 and the Newton valuation is exact. Every later term has valuation at least ℓ+1. Thus v_2(d_j,ℓ)=ℓ, and the coefficient radius is exactly two. At a center x_0∈ Z_2, the coefficient of y^ℓ in the translated series is ∑_m≥ℓd_j,mbinom mℓ x_0^m-ℓ. Its initial term again ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"0 · 0 . The class x_1=(0,0) solves Mx_1≡ bpmod3 for both b_1=(-3,0) and b_2=(0,-3). For b_1, the reduced residual is (1,0), which belongs to the image of overline M. There are three correction classes, given by y_1+y_2=2 pmod3. For b_2, the residual is (0,1), whose cokernel class is nonzero. No lift modulo nine exists because the second row of M re","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-57080b7d6f26d5e6","tex":"Y_2","aliases":"Y_2","display":"Y_2","kind":"use","excerpt":"nomialPrescribedPadicTower.exists_prescribed_padic_root_with_tower and leanKomorebi.PolynomialPrescribedPadicTower.exists_unique_prescribed_padic_root. The separate supplied-path theorem leanKomorebi.PolynomialPrescribedPadicTower.exists_root_with_prescribed_path accepts any compatible integer parameter path with limit τ. For m>e, endpoint uniquene","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dd412919846eece9","tex":"y_2","aliases":"y_2","display":"y_2","kind":"use","excerpt":"0 · 0 . The class x_1=(0,0) solves Mx_1≡ bpmod3 for both b_1=(-3,0) and b_2=(0,-3). For b_1, the reduced residual is (1,0), which belongs to the image of overline M. There are three correction classes, given by y_1+y_2=2 pmod3. For b_2, the residual is (0,1), whose cokernel class is nonzero. No lift modulo nine exists because the second row of M re","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7256bbf22fa35eec","tex":"Y_m","aliases":"Y_m","display":"Y_m","kind":"use","excerpt":"Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residu","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-00fe9b383f124db5","tex":"y_n","aliases":"y_n","display":"y_n","kind":"use","excerpt":"[ht] centering includegraphics[width=linewidth]../figures/residue_permutations.pdf captionExact residue permutations modulo 256. The vertical coordinate is y_n=(E_4n^(4,j)-j!)/2^B_j modulo P_j(8). Each panel contains one full least period; every vertical residue occurs once.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-57348dda8e2fb02d","tex":"y_r","aliases":"y_r","display":"y_r","kind":"use","excerpt":"Now define explicit iterations y_0=0, y_r+1=Λ_c^-1(cq+y_rq-y_r^2). If q is flat and p-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem: [t^d]y_r=[t^d]y (d≤ r). Indeed, (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y). The second factor has zero constant term. Multiplication therefore raises t","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5cd112d8b799c301","tex":"y_{r+1}","aliases":"y_r+1","display":"y_{r+1}","kind":"use","excerpt":"Now define explicit iterations y_0=0, y_r+1=Λ_c^-1(cq+y_rq-y_r^2). If q is flat and p-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem: [t^d]y_r=[t^d]y (d≤ r). Indeed, (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y). The second factor has zero constant term. Multiplication therefore raises t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-288e0c669aaa9103","tex":"y_r^2","aliases":"y_r^2","display":"y_r^2","kind":"use","excerpt":"Now define explicit iterations y_0=0, y_r+1=Λ_c^-1(cq+y_rq-y_r^2). If q is flat and p-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem: [t^d]y_r=[t^d]y (d≤ r). Indeed, (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y). The second factor has zero constant term. Multiplication therefore raises t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6280117a7eeb98e1","tex":"y_s","aliases":"y_s","display":"y_s","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ea5ac5a487fde7e2","tex":"y_{s+1}","aliases":"y_s+1","display":"y_{s+1}","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1d4854bf905c5d1b","tex":"y_s^2","aliases":"y_s^2","display":"y_s^2","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":35},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1a49e6bec04c7b56","tex":"Z^2","aliases":"Z^2","display":"Z^2","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zero-address case, Θ_2,0(k+j,k)=2^k-jbinom kj (0≤ j≤ k), since (2z+z^2)^k=z","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e712bd3d7c60c1d5","tex":"z^{2n}","aliases":"z^2n","display":"z^{2n}","kind":"use","excerpt":"[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator a","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9613da88d6877c77","tex":"z^{2r}","aliases":"z^2r","display":"z^{2r}","kind":"use","excerpt":"The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the b","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f64931971fecd572","tex":"Z^3","aliases":"Z^3","display":"Z^3","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7206b16a18756e6f","tex":"\\Z^d","aliases":"Z^d Z mathbb","display":"{\\mathbb Z}^d","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":19},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-54b855e49989865e","tex":"z^d","aliases":"z^d","display":"z^d","kind":"use","excerpt":"At every center x_0∈ Z_2, the interpolations admit ordinary Taylor series C_j(x_0+z)=∑_d≥0a_j,d(x_0)z^d. For positive degrees, v_2(a_j,d(x_0)) · =d · · (j=0,1),","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3e46d5eff30bc2a2","tex":"\\Z^e","aliases":"Z^e Z mathbb","display":"{\\mathbb Z}^e","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a71ae723be54d160","tex":"\\Z^{e+k}","aliases":"Z^e+k Z mathbb","display":"{\\mathbb Z}^{e+k}","kind":"use","excerpt":"[Continuous charts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is ho","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b73464ec6718888a","tex":"\\mathbb Z^{e+q}","aliases":"Z^e+q mathbb","display":"\\mathbb Z^{e+q}","kind":"use","excerpt":"Let p be prime, L≥1, e,q≥0, and F∈ Z[X_1,…,X_e+q]^e, x∈ Z^e+q, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^e+q⟶ F_p^e is surjective. All vector congruences are coordinatewise.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-943f96db5b69b8c3","tex":"\\Z^f","aliases":"Z^f Z mathbb","display":"{\\mathbb Z}^f","kind":"use","excerpt":"For affine integer systems F(X)=AX-b and G(X)=BX-c, where A: Z^d→ Z^e and B: Z^d→ Z^f are linear, Corollary~[source reference] holds for q≥1 and every L≥0, including depth zero. Here x is a common root modulo q^L, and J and K are the reductions of A and B with domain Z^d. Given separately valid corrections u,v, a simultaneous next lift exists if an","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ed6a8232558e5682","tex":"z^h","aliases":"z^h","display":"z^h","kind":"use","excerpt":"[Rational ordinary generating function] In F_p[[z]], ∑_n≥0e_n^(m,p)z^n = 1+ frac ∑_r=1^he_r^(m,p)z^r 1-z^h.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0cfa667e85ec10a8","tex":"\\Z^k","aliases":"Z^k Z mathbb","display":"{\\mathbb Z}^k","kind":"use","excerpt":"[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integer","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b4144141aa513159","tex":"\\mathbb Z^k","aliases":"Z^k mathbb","display":"\\mathbb Z^k","kind":"use","excerpt":"Integer prescribed-path theorem. For every supplied path t: N→ Z^k satisfying t(0)=t_0, t(n+1)-t(n)∈ q^L+n Z^k, there is a single tower z: N→ Z^m+k with z(0)=x, free(z(n))=t(n), F(z(n))∈ q^L+n Z^e, z(n+1)-z(n)∈ q^L+n Z^m+k. The parameter equality is exact in integers. For a≤ b, z(b)-z(a)∈ q^L+a Z^m+k, DF(z(n))bmod q=DF(x)bmod q. Consequently J_z(n)","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","excerpt":"kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zero-address case, Θ_2,0(k+j,k)=2^k-jbinom kj (0≤ j≤ k), since (2z+z^2)^k=z","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2bcacc05c95dda04","tex":"z^{k+j}","aliases":"z^k+j","display":"z^{k+j}","kind":"use","excerpt":"The lowest degree term of Φ_M(z) is Mz, and its highest degree term is z^M. Multiplication by (1+z)^s gives the two edge coefficients. When M=2 and s=0, Φ_2(z)^k = (2z+z^2)^k = z^k(2+z)^k. Taking the coefficient of z^k+j gives the binary formula.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-376fec3a52294509","tex":"z^M","aliases":"z^M","display":"z^M","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8d252328af690b62","tex":"\\Z^{m+k}","aliases":"Z^m+k Z mathbb","display":"{\\mathbb Z}^{m+k}","kind":"use","excerpt":"[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integer","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-89db1fa034bed25b","tex":"\\mathbb Z^{m+k}","aliases":"Z^m+k mathbb","display":"\\mathbb Z^{m+k}","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"use","excerpt":"The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary ","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6894c7248320dbce","tex":"z^p","aliases":"z^p","display":"z^p","kind":"use","excerpt":"sumes primality, the modulus divisibility and the specified residue bounds. It does not assume weighted section membership, reciprocal closure, the ratio identity, a Mahler pairing or the desired higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges. 1. Integral cyclic coefficients. Work first in an arbitrary commu","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e765b594366f3e90","tex":"z^{p^a}","aliases":"z^p^a","display":"z^{p^a}","kind":"use","excerpt":"In characteristic p, (1+z)^p^a-1≡ z^p^a. Substitute this congruence in the definition of Θ_p^a,s.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1766cb24e585f5e7","tex":"Z^r","aliases":"Z^r","display":"Z^r","kind":"use","excerpt":"[Bernoulli divided-power moments] Let Z be the ordinary Bernoulli variable, L(Z^r)=B_r, with exponential transform t/(e^t-1). For P=Z(Z+1) and μ_m= L(P^m/m!), μ_m∈ Z_2, v_2(μ_m)≥ s_2(m)-1 (m≥1). Moreover, μ_m is odd exactly when m=0 or m is a positive power of two.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-10d0329c35407a66","tex":"z^r","aliases":"z^r","display":"z^r","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b23ccb12e2adf65f","tex":"Z_1","aliases":"Z_1","display":"Z_1","kind":"use","excerpt":"[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f10cbe5fe4c5d709","tex":"\\mathbb Z_2^\\times","aliases":"Z_2^× mathbb times","display":"\\mathbb Z_2^\\times","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8591c505dfe57b6d","tex":"\\mathbb Z_2^{\\times}","aliases":"Z_2^× mathbb times","display":"\\mathbb Z_2^{\\times}","kind":"use","excerpt":"dd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and Q=( F^2-1)/2.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-18a2153b72d39af1","tex":"Z_{2m+1}","aliases":"Z_2m+1","display":"Z_{2m+1}","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-42fde784682f68de","tex":"Z_{2r+1}","aliases":"Z_2r+1","display":"Z_{2r+1}","kind":"use","excerpt":"[Zigzag inverse kernel] sech^2t = ∑_r≥0 (-1)^rZ_2r+1 fract^2r(2r)!.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2a799d1668e854e4","tex":"Z_3","aliases":"Z_3","display":"Z_3","kind":"use","excerpt":"[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-715faed565a7579b","tex":"\\Z_3","aliases":"Z_3 Z mathbb","display":"{\\mathbb Z}_3","kind":"use","excerpt":":matrix_carmichael_mahler Editorial hypothesis correction. The original displayed statement leaves the vectors untyped. Its integrality conclusion requires v_0,w∈ Z_p^d, L≥1, and H,j,n∈ N with H>0; the value at Hx+j for x∈ Z_p means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-315e6e3c40d12791","tex":"z_a","aliases":"z_a","display":"z_a","kind":"use","excerpt":"z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cdd36f9a578aa62d","tex":"z_b","aliases":"z_b","display":"z_b","kind":"use","excerpt":"z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-42dba447a7ef210c","tex":"z_m","aliases":"z_m","display":"z_m","kind":"use","excerpt":"Compatibility implies, for m≥ n, p^L+nmid z_m-z_n. After casting to Z_p, this bounds each coordinate distance by p^-(L+n), so the sequence is Cauchy. Completeness supplies its limit. Closed norm bounds show that the limit retains the congruence at every stage.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9fdca6236d286603","tex":"Z_n","aliases":"Z_n","display":"Z_n","kind":"use","excerpt":"Let q be a positive odd integer. Define S_n(q) = ∑_a=0^q-1 (-1)^a(2a+1)^n. Let the signed Euler numbers and Euler zigzag numbers be defined by secht = ∑_n≥0E_nfract^nn! and sec t+tan t = ∑_n≥0Z_nfract^nn!.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bdc1562155501d06","tex":"z_n","aliases":"z_n","display":"z_n","kind":"use","excerpt":"[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e,","count":25},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a1bb8bb52da2ab42","tex":"z_{n+1}","aliases":"z_n+1","display":"z_{n+1}","kind":"use","excerpt":"[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1bd159c90606314f","tex":"\\Z_p^d","aliases":"Z_p^d Z mathbb","display":"{\\mathbb Z}_p^d","kind":"use","excerpt":"[Matrix Carmichael and Mahler transfer] Editorial hypothesis correction. The original displayed statement leaves the vectors untyped. Its integrality conclusion requires v_0,w∈ Z_p^d, L≥1, and H,j,n∈ N with H>0; the value at Hx+j for x∈ Z_p means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f72e3ddb158528a6","tex":"\\mathbb Z_p^d","aliases":"Z_p^d mathbb","display":"\\mathbb Z_p^d","kind":"use","excerpt":"Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^d⟶ F_p^e is surjective. Then Lean proves that there is a point a∈ Z_p^d, F(a)=0, a≡ xpmodp^L. Here the coefficients of the original integer polynomials are cast into Z_p, and their multivariable evaluation is exactly zero. This","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1beb3e4d284664d8","tex":"\\mathbb Z_p^e","aliases":"Z_p^e mathbb","display":"\\mathbb Z_p^e","kind":"use","excerpt":"((σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus through σ h(t)=(ϕ(t),t). Every parameter has ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e55fed3bae1ec1c1","tex":"\\mathbb Z_p^{e+k}","aliases":"Z_p^e+k mathbb","display":"\\mathbb Z_p^{e+k}","kind":"use","excerpt":"Square solved-block uniqueness. If m=e, every prescribed τ has exactly one root in the original ball: ∃! b∈ x+p^L Z_p^e+k: F(b)=0, free(b)=τ. Thus all valid integer parameter approximations and all lifting choices with the same target have the same endpoint. This does not assert uniqueness of the integer representatives in the tower. All prime stat","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cc8074994f3710c4","tex":"\\mathbb Z_p^{e+q}","aliases":"Z_p^e+q mathbb","display":"\\mathbb Z_p^{e+q}","kind":"use","excerpt":"There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus thro","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c8f3102e5e0305d5","tex":"\\mathbb Z_p^k","aliases":"Z_p^k mathbb","display":"\\mathbb Z_p^k","kind":"use","excerpt":"Arbitrary p-adic target theorem. Now let q=p be prime. For every τ∈ t_0+p^L Z_p^k, there are a compatible integer tower z as above and b∈ Z_p^m+k such that z(n)⟶ b, F(b)=0, free(b)=τ, b-z(n)∈ p^L+n Z_p^m+k (n≥0). A compatible integer parameter path converging to τ is constructed in the proof, with its initial representative fixed exactly. A separat","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-308f4c34f6281e55","tex":"\\mathbb Z_p^{m+k}","aliases":"Z_p^m+k mathbb","display":"\\mathbb Z_p^{m+k}","kind":"use","excerpt":"Arbitrary p-adic target theorem. Now let q=p be prime. For every τ∈ t_0+p^L Z_p^k, there are a compatible integer tower z as above and b∈ Z_p^m+k such that z(n)⟶ b, F(b)=0, free(b)=τ, b-z(n)∈ p^L+n Z_p^m+k (n≥0). A compatible integer parameter path converging to τ is constructed in the proof, with its initial representative fixed exactly. A separat","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c466d5fa0ef550c1","tex":"\\mathbb Z_p^q","aliases":"Z_p^q mathbb","display":"\\mathbb Z_p^q","kind":"use","excerpt":"There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus thro","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-31d760e606ff3dad","tex":"\\Z_p^\\times","aliases":"Z_p^× Z times mathbb times","display":"{\\mathbb Z}_p^\\times","kind":"use","excerpt":"Why a p-adic unit supplies one factor of p. Now let u∈ Z_p^×. The actual ring homomorphism π: Z_p⟶ F_p sends u to a nonzero unit. Fermat's finite-field theorem therefore gives π(u)^p-1=1, π(u^p-1-1)=0. Using the proved kernel identity kerπ=p Z_p, the source derives pmid u^p-1-1 inside Z_p. The conclusion is not supplied as an abstract certificate. ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a01906ff34cc704","tex":"\\mathbb Z_p^\\times","aliases":"Z_p^× mathbb times","display":"\\mathbb Z_p^\\times","kind":"use","excerpt":"[Scalar polyphase precision amplification] Let u∈ Z_p^×, let a,r≥0, and put H_a= (p-1)p^a, · p odd,","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5e940a55f95e2b85","tex":"Z_r","aliases":"Z_r","display":"Z_r","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-17c7ef854a4ec77f","tex":"z_T","aliases":"z_T","display":"z_T","kind":"use","excerpt":"LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z)≡0pmodq^L+1, z_T≡ t_0+q^L","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b3c4ce700ec5fc9c","tex":"z_{\\sigma(i)}","aliases":"z_σ(i) sigma","display":"z_{\\sigma(i)}","kind":"use","excerpt":"There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus thro","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fef1eff057e46518","tex":"\\mathbb Z_{\\ge0}","aliases":"Z_≥0 mathbb ge","display":"\\mathbb Z_{\\ge0}","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-180c17fa6495e993","tex":"\\mathbb Z_{\\geq0}","aliases":"Z_≥0 mathbb geq","display":"\\mathbb Z_{\\geq0}","kind":"use","excerpt":"[A first-difference criterion] Let C: Z_≥0→ Z_2 have Newton coefficients a_k=Δ_1^kC(0). Suppose for an integer b≥0 that v_2(a_1)=b, v_2(a_k)≥ b+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and H>0, v_2(C(n+H)-C(n))=b+v_2(H). The Newton interpolation extends continuously to Z_2 and satisfies the same identity for any two distinct dyadic arguments.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8f2f95d10cc66f45","tex":"\\Zp^e","aliases":"Zp^e Z_p^e Zp mathbb","display":"{\\mathbb Z_p}^e","kind":"use","excerpt":"R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is the forward map, then for every N≥0, h(t)≡ h(u)pmodp^N Longleftrightarrow t≡ upmodp^N. Equivalently, σ h(t)=(ϕ(t),t) for a continuous ϕ:T→ y_0+p^L Z_p^e whose graph is the entire reordered root locus.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9351a29d1ff29b9c","tex":"\\Zp^{e+k}","aliases":"Zp^e+k Z_p^e+k Zp mathbb","display":"{\\mathbb Z_p}^{e+k}","kind":"use","excerpt":"Now suppose a chosen set of e columns of DF(x)bmod p is independent, and put those coordinates first. Apply the square result to the augmented polynomial map Ψ(Y,T)=(F(Y,T),T). Its reduced Jacobian has upper blocks (D_YF,D_TF) and lower blocks (0,I_k), so it is invertible. For t∈ T, the target (0,t) belongs to Ψ(x)+p^L Z_p^e+k because F(x)≡0pmodp^L","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3fc502ceef10c9a6","tex":"\\Zp^k","aliases":"Zp^k Z_p^k Zp mathbb","display":"{\\mathbb Z_p}^k","kind":"use","excerpt":"arts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6bc1bb97011509d8","tex":"\\Zp^{m+k}","aliases":"Zp^m+k Z_p^m+k Zp mathbb","display":"{\\mathbb Z_p}^{m+k}","kind":"use","excerpt":"[Arbitrary p-adic parameter endpoints] Let q=p be prime in Theorem~[source reference]. For every τ∈ t_0+p^L Z_p^k, there are an integer tower z_n and b∈ Z_p^m+k such that F(b)=0, free(b)=τ, z_n⟶ b, b-z_n∈ p^L+n Z_p^m+k. The initial value is z_0=x. At every stage and for every proposed parameter digit v∈ F_p^k, the corresponding next residue fiber h","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-20995e4cd61e272d","tex":"\\Zp^\\times","aliases":"Zp^× Z_p^× Zp times mathbb times","display":"{\\mathbb Z_p}^\\times","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ Z mathbb","display":"{\\mathbb Z}","kind":"use","excerpt":"[A denominator bound] Let D_N,j(n) = ∏_r=1^n binomNr+jNr, D_N,j(0)=1. Then D_N,j(n)C_N,j(n) ∈ Z. For every prime p, v_p(C_N,j(n)) ≥ - ∑_r=1^n v_pbinomNr+jNr.","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-207f0dd92280fb9b","tex":"\\ZZ^d","aliases":"ZZ^d Z^d ZZ mathbb","display":"{\\mathbb Z}^d","kind":"use","excerpt":"[Base change of an integral recurrence] Let C∈ M_d( Z), v_0∈ Z^d. The integral orbit v_n=C^nv_0 determines, by base change, compatible orbits in every ring R equipped with a homomorphism Z→ R. In particular, the reductions modulo every positive integer and the orbits in every Z_p^d share the same companion matrix, characteristic polynomial, and int","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"[All positive periods and their least element] For r∈ N and 0≤ j<4, a positive integer T is a packet period of U_j modulo 2^r if and only if 2^max(0,r-d_j)mid T. Here a packet period means U_j(n+T)≡ U_j(n)pmod2^r for every n≥0. Consequently the least positive packet periods are c|cccc j · 0 · 1 · 2 · 3 least period · 2^max(0,r-1) · 2^max(0,r-1) · 2","count":20},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f530a3e7890ffba8","tex":"|_2","aliases":"|_2","display":"|_2","kind":"use","excerpt":"[Isometric packet bijections] For 0≤ j<4, define on natural numbers f_j(n)=fracU_j(n)-j!2^d_j, (d_0,d_1,d_2,d_3)=(1,1,5,3). There is a unique continuous map Φ_j: Z_2→ Z_2 whose value at every n∈ N equals the embedded rational f_j(n). It is an isometric bijection and Φ_j(0)=0: |Φ_j(x)-Φ_j(y)|_2=|x-y|_2 (x,y∈ Z_2). In particular, every element of Z_2","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f3b5ec80fa082444","tex":"|_{n=0}","aliases":"|_n=0","display":"|_{n=0}","kind":"use","excerpt":"A sharp unit-mode gain. For p=5 and u=2, Δ_4^2(u^n)|_n=0=(2^4-1)^2=225. The source proves 5^2mid225 and 5^3nmid225. This example attains the guaranteed gain; no additional uniform power of p is claimed.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ae8ddb0b385adcc1","tex":"|_p","aliases":"|_p","display":"|_p","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b44e5a2c29c5de91","tex":"|_{U_a}","aliases":"|_U_a","display":"|_{U_a}","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"For an odd prime, 2mid p-1, so the option m=2 introduces no further case. The sole remaining branch is p=2,m=2,N=4. The all-row theorem proved next gives 2^kmidΔ_1^kC_4,j(n) at every row and address. After multiplying by j!, Lemma~[source reference] gives packet period 2^r-1 and degree displacement 4·2^r-1=2^r+1=qp^r. It also gives integrality. Hen","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-62547ed5692a28f7","tex":"\\times_C","aliases":"×_C times","display":"\\times_C","kind":"use","excerpt":"The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"[Weighted convergence] If widehat F(T)=∑_r≥0c_rT^r and v_p(c_r)+rα⟶+∞, then the series converges whenever v_p(T)≥α.","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"use","excerpt":"After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-275621d24bde4c06","tex":"\\alpha_d","aliases":"α_d alpha","display":"\\alpha_d","kind":"use","excerpt":"After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f320d7b77ed8bff0","tex":"\\alpha_i","aliases":"α_i alpha","display":"\\alpha_i","kind":"use","excerpt":"[Regular-wall root law] For every regular label j and every integer δ≥1, v_2G_r(x)= δ, · x∈ B_i, δ<t_i, d_i+v_2(x-α_i), · x∈ B_i, δ≥ t_i. In the second case there is exactly one root α_i in B_i, it is simple, and v_2(α_i-i)=δ-d_i. Thus the number of dyadic roots is exactly #{i:0≤ i<j, t_i≤δ}. If δ<t_i, the unit perturbation has smaller valuation th","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-29ca9b1f3cc76af1","tex":"\\alpha_i^N","aliases":"α_i^N alpha","display":"\\alpha_i^N","kind":"use","excerpt":"After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-af15262773600a7f","tex":"\\alpha_r","aliases":"α_r alpha","display":"\\alpha_r","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1bac3c7ce6eb2507","tex":"\\beta_1","aliases":"β_1 beta","display":"\\beta_1","kind":"use","excerpt":"Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-05f6c598f663dcba","tex":"\\beta_2","aliases":"β_2 beta","display":"\\beta_2","kind":"use","excerpt":"Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e259a50189711ceb","tex":"\\beta_3","aliases":"β_3 beta","display":"\\beta_3","kind":"use","excerpt":"Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-208473b4b3730308","tex":"\\beta_4","aliases":"β_4 beta","display":"\\beta_4","kind":"use","excerpt":"Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8dbbe747d8622f2f","tex":"\\beta_j","aliases":"β_j beta","display":"\\beta_j","kind":"use","excerpt":"[Four-address factorial bound] For j∈{0,1,2,3}, n≥0, k≥1, and positive mixed steps, v_2(Δ_H_1⋯Δ_H_kC_j(n)) ≥ k+v_2(k!)+∑_ν=1^k v_2(H_ν)+β_j, (β_0,β_1,β_2,β_3)=(0,0,3,1). Address two also satisfies v_2(Δ_H_1⋯Δ_H_kC_2(n)) ≥ 2k+v_2(k!)+lfloorfrack2rfloor +∑_ν=1^k v_2(H_ν). For the original coefficients, the offsets in [source reference] are (0,0,4,2).","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3e24d33779c12c52","tex":"\\beta_r","aliases":"β_r beta","display":"\\beta_r","kind":"use","excerpt":"The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the b","count":13},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5cc53b570750c9f6","tex":"\\beta_{r+q}","aliases":"β_r+q beta","display":"\\beta_{r+q}","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-36143f667466fe6f","tex":"\\beta_r^2","aliases":"β_r^2 beta","display":"\\beta_r^2","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions; no completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source referen","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2f08cea1939265ee","tex":"\\gamma_2","aliases":"γ_2 gamma","display":"\\gamma_2","kind":"use","excerpt":"There is a still stronger filtration. Put X=P+Q, τ=i B D, ϵ=1+τ. Then the exact identities are H_2=ϵ X+2G, G=-γ_2(P)-γ_2(Q)+3PQ-τ Q, ϵ^2=2V, V=τ-2X-8PQ. Every divided power of G is integral component by component. The finite addition identity gives γ_k(H_2)=∑_r=0^k2^rγ_r(G) ϵ^k-rγ_k-r(X) ∈2^lfloor k/2rfloor A_2, because the rth term belongs to 2^r+","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-42316594f86819a5","tex":"\\gamma_{2k}","aliases":"γ_2k gamma","display":"\\gamma_{2k}","kind":"use","excerpt":"wers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2k(X)+ϵγ_2k-1(X) pmod2 A_0. The m","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d502518f858f0a9b","tex":"\\gamma_{2k-1}","aliases":"γ_2k-1 gamma","display":"\\gamma_{2k-1}","kind":"use","excerpt":"wers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2k(X)+ϵγ_2k-1(X) pmod2 A_0. The m","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-440e88e8288fd6e5","tex":"\\gamma_{2m}","aliases":"γ_2m gamma","display":"\\gamma_{2m}","kind":"use","excerpt":"Every displayed component of H_0 has all its divided powers in A_0. Besides the generators, the needed identities are γ_m(P^2/2) · =frac(2m)!2^m m!γ_2m(P),nonumber γ_m(3PQ) · =3^m m!γ_m(P)γ_m(Q),nonumber γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5133d0d16eff7ae2","tex":"\\gamma_{2r}","aliases":"γ_2r gamma","display":"\\gamma_{2r}","kind":"use","excerpt":":four-euler-H gives H_0=γ_2(X)+ϵ X+2G, G=-γ_2(P)-γ_2(Q)+PQ-τ Q, ϵ^2=2(τ-P-Q-2PQ). The divided powers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes mo","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-530f5d7a3b27e60f","tex":"\\gamma_k","aliases":"γ_k gamma","display":"\\gamma_k","kind":"use","excerpt":"γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the third identity by (i E F)^m and reduce the Euler powers using the integral quadratic relations. Finally, γ_k(X+Y)=∑_r=0^kγ_r(X)γ_k-r(Y) proves γ_k(H_0)∈ A_0. Applying the functional and putting W=U^4-1=2H","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3374379bd0bfada4","tex":"\\gamma_{k-1}","aliases":"γ_k-1 gamma","display":"\\gamma_{k-1}","kind":"use","excerpt":"For k=2^r≥4, put q=k/2, which is even. Normalize the exact expansion [source reference] by 2^q before reducing modulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-29b4335be2c3e508","tex":"\\gamma_{k-r}","aliases":"γ_k-r gamma","display":"\\gamma_{k-r}","kind":"use","excerpt":"γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the third identity by (i E F)^m and reduce the Euler powers using the integral quadratic relations. Finally, γ_k(X+Y)=∑_r=0^kγ_r(X)γ_k-r(Y) proves γ_k(H_0)∈ A_0. Applying the functional and putting W=U^4-1=2H","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-afb8535845a78365","tex":"\\gamma_{m-r}","aliases":"γ_m-r gamma","display":"\\gamma_{m-r}","kind":"use","excerpt":"Every displayed component of H_0 has all its divided powers in A_0. Besides the generators, the needed identities are γ_m(P^2/2) · =frac(2m)!2^m m!γ_2m(P),nonumber γ_m(3PQ) · =3^m m!γ_m(P)γ_m(Q),nonumber γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bf09daa6ed346f44","tex":"\\gamma_{n-q}","aliases":"γ_n-q gamma","display":"\\gamma_{n-q}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3c86d4a426c471d1","tex":"\\Gamma_R","aliases":"Γ_R Gamma","display":"\\Gamma_R","kind":"use","excerpt":"Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions; no completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source referen","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-788db25068d804f5","tex":"\\gamma_r","aliases":"γ_r gamma","display":"\\gamma_r","kind":"use","excerpt":"Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions; no completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source referen","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4a741ae82704c654","tex":"\\Delta^k","aliases":"Δ^k Delta","display":"\\Delta^k","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-75ac6af28af33827","tex":"\\Delta^m","aliases":"Δ^m Delta","display":"\\Delta^m","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0d6cdab24a850778","tex":"\\Delta^r","aliases":"Δ^r Delta","display":"\\Delta^r","kind":"use","excerpt":"[Finite Mahler reconstruction] Given f(0),…,f(N), put c_r=Δ^rf(0). Then f(n)=∑_r=0^nc_rbinom nr (0≤ n≤ N). For every s with 0≤ s≤ N, Δ^sf(n) = ∑_r=s^Nc_rbinom nr-s (0≤ n≤ N-s).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fa50885d0e9944c0","tex":"\\Delta^s","aliases":"Δ^s Delta","display":"\\Delta^s","kind":"use","excerpt":"[Finite Mahler reconstruction] Given f(0),…,f(N), put c_r=Δ^rf(0). Then f(n)=∑_r=0^nc_rbinom nr (0≤ n≤ N). For every s with 0≤ s≤ N, Δ^sf(n) = ∑_r=s^Nc_rbinom nr-s (0≤ n≤ N-s).","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e7752e939ff77e03","tex":"\\delta^s","aliases":"δ^s delta","display":"\\delta^s","kind":"use","excerpt":"0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved ","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dde6e4dc1d6569ac","tex":"\\delta^{s-1}","aliases":"δ^s-1 delta","display":"\\delta^{s-1}","kind":"use","excerpt":"pageheading3. A nonsingular Euler-operator argumentThe standard interval without an analytic power-series-in-the-index premise On Hurwitz coefficients, the actual Euler operator plus c is (Λ_c a)(d)=(d+c)a(d). Direct induction proves its commutator with every defect iterate: δ^sΛ_c a =Λ_cδ^s a +sp(pU^p^2-U^p)δ^s-1a, s≥1. Suppose a is integral and s","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6ba59a603a1fa1c4","tex":"\\Delta_{(p-1)p^a}^{\\,r}","aliases":"Δ_(p-1)p^a^ r Delta","display":"\\Delta_{(p-1)p^a}^{\\,r}","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-49a63bc6da6e2534","tex":"\\Delta_{12\\cdot2^a}^{\\,r}","aliases":"Δ_12·2^a^ r Delta cdot","display":"\\Delta_{12\\cdot2^a}^{\\,r}","kind":"use","excerpt":"18 · 11 . Therefore Δ_12·2^a^ rF_n ≡0 pmod 2^(a+3)r. Every dyadic refinement doubles the step and adds one guaranteed binary digit at each Mahler depth.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-954f8b3938ce3865","tex":"\\Delta_1^2","aliases":"Δ_1^2 Delta","display":"\\Delta_1^2","kind":"use","excerpt":"pageheading5. Examples, verification and the remaining gapThe coverage count is unchanged; the unresolved theorem now has a stronger proved part The original rational recurrence gives C_6,2(1)=-frac128, C_6,2(2)=frac425364, C_6,2(3)=-frac20397592660. Consequently, the new source proves Δ_1^2 C_6,2(1)=-frac9499591235, v_3 (Δ_1^2 C_6,2(1))=2. This at","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-54b865827d520271","tex":"\\Delta_1^i","aliases":"Δ_1^i Delta","display":"\\Delta_1^i","kind":"use","excerpt":"g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors interchanged to ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0b2aaddf443a493a","tex":"\\Delta_1^k","aliases":"Δ_1^k Delta","display":"\\Delta_1^k","kind":"use","excerpt":"affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its t","count":27},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9c183ffb2b31e982","tex":"\\Delta_1^{k-r}","aliases":"Δ_1^k-r Delta","display":"\\Delta_1^{k-r}","kind":"use","excerpt":"ence identity is Δ_1^kfrac11+sn =frac(-s)^k k!∏_h=0^k(1+s(n+h)). It follows by induction, subtracting adjacent rational functions; each successive subtraction adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then prov","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8164837cba51cda1","tex":"\\Delta_1^m","aliases":"Δ_1^m Delta","display":"\\Delta_1^m","kind":"use","excerpt":"To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar mult","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-31c8867e60f94613","tex":"\\Delta_1^r","aliases":"Δ_1^r Delta","display":"\\Delta_1^r","kind":"use","excerpt":"For f: N→ R, define the step-h forward difference and its iterates by Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f=f, and put c_r(f)=Δ_1^r f(0). Thus the coefficients are determined by the differences of f.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7c4d86df28afc0cb","tex":"\\Delta_1^{s-i}","aliases":"Δ_1^s-i Delta","display":"\\Delta_1^{s-i}","kind":"use","excerpt":"g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors interchanged to ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6c0b4ab8dd399bba","tex":"\\Delta_2","aliases":"Δ_2 Delta","display":"\\Delta_2","kind":"use","excerpt":"What fails when hypotheses or conventions are dropped. The nonunit u=3 at p=3 gives Δ_2(u^n)|_n=0=8, with no factor of three. A constant unit has every positive-order difference zero, hence infinite extended valuation. The finite-sample counterexample on page 2 distinguishes a true interpolant identity from an unsupported claim about unknown contin","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e8f89ea0d8a9c3b5","tex":"\\Delta_{24}^{\\,r}","aliases":"Δ_24^ r Delta","display":"\\Delta_{24}^{\\,r}","kind":"use","excerpt":"1932 · 1194 . For every scalar coordinate of the state sequence C^nv_0, Δ_24^ ra_n ≡0 pmod 24^r. One integral jump therefore gives simultaneous congruence ladders at the primes two and three.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-121420d9f9749674","tex":"\\Delta_{2^{a+1}}^{\\,r}","aliases":"Δ_2^a+1^ r Delta","display":"\\Delta_{2^{a+1}}^{\\,r}","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-198bb3bf1a420a2e","tex":"\\Delta_4","aliases":"Δ_4 Delta","display":"\\Delta_4","kind":"use","excerpt":"For this stencil, the general dyadic bound is 3+v_2(1)+v_2(2)+v_2(4)=6. At rank four the factorial theorem improves it to seven for addresses zero and one. Exact rational evaluation gives v_2(Δ_1Δ_2Δ_4 C_0(0))=7. The order is odd, so the mixed-step equality criterion also proves this valuation at every starting row.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-397720baa101f32e","tex":"\\Delta_4^2","aliases":"Δ_4^2 Delta","display":"\\Delta_4^2","kind":"use","excerpt":"A sharp unit-mode gain. For p=5 and u=2, Δ_4^2(u^n)|_n=0=(2^4-1)^2=225. The source proves 5^2mid225 and 5^3nmid225. This example attains the guaranteed gain; no additional uniform power of p is claimed.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6ab8756dcff46892","tex":"\\Delta_4^s","aliases":"Δ_4^s Delta","display":"\\Delta_4^s","kind":"use","excerpt":"Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f07d918fcbb9be05","tex":"\\delta_{d0}","aliases":"δ_d0 delta","display":"\\delta_{d0}","kind":"use","excerpt":"The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-28f5ad27d0de9148","tex":"\\Delta_H","aliases":"Δ_H Delta","display":"\\Delta_H","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers; steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7eb12ace4621a9d6","tex":"\\Delta_H^k","aliases":"Δ_H^k Delta","display":"\\Delta_H^k","kind":"use","excerpt":"[Threshold-free dyadic congruence] For N=2^a, a≥0, all permitted j, and n≥0, v_2(Δ_H_1⋯Δ_H_kE_Nn^(N,j)) ≥ v_2(j!)+k+∑_ν=1^k v_2(H_ν). In particular, Δ_H^k gains k(1+v_2(H)) bits before normalization. For r>v_2(j!), the row displacement H=2^max(0,r-v_2(j!)-1) is a sufficient period modulo 2^r from row zero. If r≤ v_2(j!), all original coefficients v","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3d95b94f6578b45a","tex":"\\Delta_h^{k-mn}","aliases":"Δ_h^k-mn Delta","display":"\\Delta_h^{k-mn}","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bbac75fb10eeb11c","tex":"\\Delta_h^{mn}","aliases":"Δ_h^mn Delta","display":"\\Delta_h^{mn}","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aa7d5fad49c50597","tex":"\\Delta_H^r","aliases":"Δ_H^r Delta","display":"\\Delta_H^r","kind":"use","excerpt":"=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not an as-written certificate. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^TC^jB^rv_0. The series converges on Z_p, and Δ_H^ra_n ≡0 pmod p^L","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-03509f28204331cb","tex":"\\Delta_h^s","aliases":"Δ_h^s Delta","display":"\\Delta_h^s","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a6f17fac236c6322","tex":"\\Delta_{H_1}","aliases":"Δ_H_1 Delta","display":"\\Delta_{H_1}","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers; steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8305e56092369ef0","tex":"\\Delta_{H_a}^{\\,r}","aliases":"Δ_H_a^ r Delta","display":"\\Delta_{H_a}^{\\,r}","kind":"use","excerpt":"2^a+1, · p=2. Then Δ_H_a^ ru^n = u^n(u^H_a-1)^r. For odd p, v_p(Δ_H_a^ ru^n) ≥(a+1)r. For p=2 and odd u, v_2(Δ_H_a^ ru^n) ≥(a+3)r. Equivalently, every additional prime power digit increases the certified Mahler slope by one.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a9995779c4ace24a","tex":"\\Delta_{H_k}","aliases":"Δ_H_k Delta","display":"\\Delta_{H_k}","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers; steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-12735990a456db58","tex":"\\Delta_L^s","aliases":"Δ_L^s Delta","display":"\\Delta_L^s","kind":"use","excerpt":"0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8e3ad7ffccce236c","tex":"\\Delta_{p(p-1)}^k","aliases":"Δ_p(p-1)^k Delta","display":"\\Delta_{p(p-1)}^k","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e538ed02a7b65b29","tex":"\\Delta_{p-1}^s","aliases":"Δ_p-1^s Delta","display":"\\Delta_{p-1}^s","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-043cc7068d423bd8","tex":"\\delta_r","aliases":"δ_r delta","display":"\\delta_r","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-51a87ccbd55b1aa4","tex":"\\delta_{rs}","aliases":"δ_rs delta","display":"\\delta_{rs}","kind":"use","excerpt":"[Complete residue packet system] For fixed N, the projectors P_N,r satisfy P_N,rP_N,s = δ_rsP_N,r and ∑_rbmod NP_N,r=I.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-70f334c2cd99d9ef","tex":"\\Delta_{u(p-1)/m}^{k}","aliases":"Δ_u(p-1)/m^k Delta","display":"\\Delta_{u(p-1)/m}^{k}","kind":"use","excerpt":"[Step-sensitive divisor-branch bound] Under the hypotheses of Theorem~[source reference], for all u,k,n∈ N, Δ_u(p-1)/m^kU_mp,j(n) ∈ p^min(k(1+ν_p(u)),mn) Z_p. For C_mp,j the corresponding valuation lower bound is min(k(1+ν_p(u)),mn)-v_p(j!).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-90d5ba069a82a623","tex":"\\Delta_{uh}","aliases":"Δ_uh Delta","display":"\\Delta_{uh}","kind":"use","excerpt":"]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth p","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d5d6c4b3391f8b2e","tex":"\\Delta_{uh}^k","aliases":"Δ_uh^k Delta","display":"\\Delta_{uh}^k","kind":"use","excerpt":"pplied to Δ_h^mnU_N,j(n); every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0 gives zero for k>0, and order zero is the original integral coefficient. This proves the bound for all u and k. Finally Δ_u","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ϵ(a)∈{1,-1}, ⟨ a⟩_4≡1pmod4.","count":22},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fb43c73461568514","tex":"\\epsilon^2","aliases":"ϵ^2 epsilon","display":"\\epsilon^2","kind":"use","excerpt":"There is a still stronger filtration. Put X=P+Q, τ=i B D, ϵ=1+τ. Then the exact identities are H_2=ϵ X+2G, G=-γ_2(P)-γ_2(Q)+3PQ-τ Q, ϵ^2=2V, V=τ-2X-8PQ. Every divided power of G is integral component by component. The finite addition identity gives γ_k(H_2)=∑_r=0^k2^rγ_r(G) ϵ^k-rγ_k-r(X) ∈2^lfloor k/2rfloor A_2, because the rth term belongs to 2^r+","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-74ecfbaf0fda4ff7","tex":"\\epsilon^j","aliases":"ϵ^j epsilon","display":"\\epsilon^j","kind":"use","excerpt":"ϵ^2=2(τ-P-Q-2PQ). The divided powers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-89c2a55fd02aed5b","tex":"\\epsilon^{k-r}","aliases":"ϵ^k-r epsilon","display":"\\epsilon^{k-r}","kind":"use","excerpt":"Put X=P+Q, τ=i B D, ϵ=1+τ. Then the exact identities are H_2=ϵ X+2G, G=-γ_2(P)-γ_2(Q)+3PQ-τ Q, ϵ^2=2V, V=τ-2X-8PQ. Every divided power of G is integral component by component. The finite addition identity gives γ_k(H_2)=∑_r=0^k2^rγ_r(G) ϵ^k-rγ_k-r(X) ∈2^lfloor k/2rfloor A_2, because the rth term belongs to 2^r+lfloor(k-r)/2rfloor A_2. Applying the ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-156a9f90221e1614","tex":"\\epsilon_u","aliases":"ϵ_u epsilon","display":"\\epsilon_u","kind":"use","excerpt":"This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard windo","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-14751022cb8bedd4","tex":"\\epsilon_{u+v}","aliases":"ϵ_u+v epsilon","display":"\\epsilon_{u+v}","kind":"use","excerpt":"[Addition law for modes] For every u,v∈ R, ϵ_umathbin⋆ϵ_v = ϵ_u+v.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f6991c3ad9fdd14c","tex":"\\epsilon_v","aliases":"ϵ_v epsilon","display":"\\epsilon_v","kind":"use","excerpt":"[Addition law for modes] For every u,v∈ R, ϵ_umathbin⋆ϵ_v = ϵ_u+v.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-42e4cf15f8f8ad0f","tex":"\\epsilon_\\zeta","aliases":"ϵ_ζ epsilon zeta","display":"\\epsilon_\\zeta","kind":"use","excerpt":"[Subgroup average] The multiple selector has the mode expansion a^(m) = frac1m ∑_ζ∈ H_m ϵ_ζ.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-81faab646ef6c7f1","tex":"\\epsilon_{\\eta u}","aliases":"ϵ_η u epsilon eta","display":"\\epsilon_{\\eta u}","kind":"use","excerpt":"Multiplication by an element η∈ H_m gives an automorphism of the additive group and therefore an automorphism ϵ_u⟼ϵ_η u of the mode algebra. The selector a^(m) is fixed by every such automorphism, so its unique inverse is fixed as well.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1ca916b982c03d00","tex":"\\varepsilon_\\ell","aliases":"ε_ℓ varepsilon ell","display":"\\varepsilon_\\ell","kind":"use","excerpt":"Why the mod-four branch is separate. Now let ℓ≥2 and a∈( Z/2^ℓ Z)^×. The actual units of Z/4 Z are exhausted by 1 and -1. Define ε_ℓ(a)= 1, · a≡1pmod4, -1, · a≡3pmod4, ⟨ a⟩_2,ℓ=ε_ℓ(a)a. The source uses ℓ=L+2. It proves a=ε_ℓ(a)⟨ a⟩_2,ℓ, ε_ℓ(a)∈{1,-1}, ⟨ a⟩_2,ℓ≡1pmod4. The sign is uniquely determined by the residue modulo four, and cancellation give","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4197bb2b77854291","tex":"\\zeta^m","aliases":"ζ^m zeta","display":"\\zeta^m","kind":"use","excerpt":"Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6d3331bda5586731","tex":"\\zeta^n","aliases":"ζ^n zeta","display":"\\zeta^n","kind":"use","excerpt":"The nth coefficient on the right is frac1m ∑_ζ∈ H_mζ^n. The character sum is one when mmid n and zero otherwise.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"Multiplication by an element η∈ H_m gives an automorphism of the additive group and therefore an automorphism ϵ_u⟼ϵ_η u of the mode algebra. The selector a^(m) is fixed by every such automorphism, so its unique inverse is fixed as well.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e882302ef17445ae","tex":"\\eta^n","aliases":"η^n eta","display":"\\eta^n","kind":"use","excerpt":"Write b^(m,p) = ∑_u∈ F_pc_uϵ_u. Invariance gives c_η u=c_u. After the change of variable u↦η u, b_n^(m,p) = η^n b_n^(m,p) for every η∈ H_m. If mnmid n, some η satisfies η^n≠1, so the coefficient vanishes.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":37},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-60a68771f913a013","tex":"\\Theta^d","aliases":"Θ^d Theta","display":"\\Theta^d","kind":"use","excerpt":"^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence finite weighted polynomials. Each fixed output coefficient depends on only finitely many input coefficients, which extends the result to the formal series. Substitution T=e^t-1 intertwines D with Θ and maps W_p into H_p, because d![t^d]F(e","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-12a64f371afd81e6","tex":"\\Theta^n","aliases":"Θ^n Theta","display":"\\Theta^n","kind":"use","excerpt":"The relevant integral subring is the Hurwitz subring H( Z_p)={f∈ Q_p[[t]]: n![t^n]f∈ Z_p for every n}. It is not coefficientwise integrality of ordinary power-series coefficients. The precise connection is n![t^n] E(F)=[T^0]Θ^nF. Since every iterate of Θ preserves W_p, the right side is integral. Therefore E( W_p)⊆ H( Z_p). Applying the same statem","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c048d6949aa9eafc","tex":"\\Theta^p","aliases":"Θ^p Theta","display":"\\Theta^p","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d; whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8c5dd3e45df8a083","tex":"\\Theta^{p^2}","aliases":"Θ^p^2 Theta","display":"\\Theta^{p^2}","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d; whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-463aadeac846c45b","tex":"\\Theta_{2,0}","aliases":"Θ_2,0 Theta","display":"\\Theta_{2,0}","kind":"use","excerpt":"or a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zero-address ca","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-017adcf52ee20d92","tex":"\\Theta_{M,s}","aliases":"Θ_M,s Theta","display":"\\Theta_{M,s}","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6b4a0b405d0fc12a","tex":"\\Theta_{MN,Mt+s}","aliases":"Θ_MN,Mt+s Theta","display":"\\Theta_{MN,Mt+s}","kind":"use","excerpt":"[Kernel composition] For positive integers M,N, ∑_k≥0 Θ_M,s(r,k) Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). The sum is finite.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-78b975269dab6a12","tex":"\\Theta_{N,t}","aliases":"Θ_N,t Theta","display":"\\Theta_{N,t}","kind":"use","excerpt":"[Kernel composition] For positive integers M,N, ∑_k≥0 Θ_M,s(r,k) Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). The sum is finite.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ca492aad33d122fe","tex":"\\Theta_{p^a,s}","aliases":"Θ_p^a,s Theta","display":"\\Theta_{p^a,s}","kind":"use","excerpt":"but its consequence for transformed Mahler coefficients requires c_r∈ Z_p for every r. Convergence of a Q_p-valued Mahler expansion alone is insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1; thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"[Integral coefficients and centers] Let p be any prime. Suppose F∈ Z_p[Y,T]^e, a∈ Z_p^e+k, and F(a)=0. Let ι: Z_p↪ Q_p be the inclusion. If ι(D_YF(a))∈GL_e( Q_p), the polynomial system obtained by including the original coefficients in Q_p has the analytic local root graph of Theorem~[source reference] at ι(a). The strict derivative at the base is ","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"[Exact weighted inverse coefficient] Let κ_m=β_m2^2m(2m)!. For every m≥1, κ_m=-2^4m(2m-1)B_2m≠0, v_2(κ_m)=4m-1, where B_2m is the usual even Bernoulli number.","count":26},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-35165a2cb10ed645","tex":"\\kappa_0","aliases":"κ_0 kappa","display":"\\kappa_0","kind":"use","excerpt":"racte^t-1=∑_n≥0B_nfract^nn!, B_1=-frac12. Throughout this appendix, v_2(2)=1 and v_2(0)=∞. Let s_2(n) be the number of ones in the binary expansion of the nonnegative integer n. Define A(N)=frac3^2N+2-14(2N+1)(2N+2), (frac zsinh z)^2=∑_m≥0β_m z^2m, κ_m=2^2m(2m)!β_m. The initial values are β_0=κ_0=1. For m≥1, differentiation of zcoth z gives β_m=-fr","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3cdc2b33ff718bf4","tex":"\\kappa_{2,1}","aliases":"κ_2,1 kappa","display":"\\kappa_{2,1}","kind":"use","excerpt":"[The first nontrivial dyadic row] For n=4, R_4=frac9115, R_2=frac53, R_0=1. The exact kernel coefficients are κ_2,1=-frac83, κ_2,2=frac1285. Therefore E_4 = frac9115 -frac83binom42frac53 +frac1285 =5. The rational source values are dyadic units, while the transfer depth is carried by the kernel.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-570f9f44831ecebb","tex":"\\kappa_{2,2}","aliases":"κ_2,2 kappa","display":"\\kappa_{2,2}","kind":"use","excerpt":"[The first nontrivial dyadic row] For n=4, R_4=frac9115, R_2=frac53, R_0=1. The exact kernel coefficients are κ_2,1=-frac83, κ_2,2=frac1285. Therefore E_4 = frac9115 -frac83binom42frac53 +frac1285 =5. The rational source values are dyadic units, while the transfer depth is carried by the kernel.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-17d1892cda93e827","tex":"\\kappa_{2,r}","aliases":"κ_2,r kappa","display":"\\kappa_{2,r}","kind":"use","excerpt":"The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the b","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f4d1001288e3b575","tex":"\\kappa_m","aliases":"κ_m kappa","display":"\\kappa_m","kind":"use","excerpt":"[Exact weighted inverse coefficient] Let κ_m=β_m2^2m(2m)!. For every m≥1, κ_m=-2^4m(2m-1)B_2m≠0, v_2(κ_m)=4m-1, where B_2m is the usual even Bernoulli number.","count":11},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-87654c9c7d2aaa41","tex":"\\kappa_{M+1}","aliases":"κ_M+1 kappa","display":"\\kappa_{M+1}","kind":"use","excerpt":"At n=M+1 there is exactly one omitted summand. Here binom2n2n=1 and A(0)=1, so the error is κ_M+1. Lemma~[source reference] gives its exact valuation 4M+3 and nonvanishing. The statements are leanKomorebiPaperIX.DyadicSharpTower.dyadic_supercongruence_sharp, leanKomorebiPaperIX.DyadicSharpTower.first_omitted_error, and leanKomorebiPaperIX.DyadicSha","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c12049a3b7739a90","tex":"\\kappa_{p,r}","aliases":"κ_p,r kappa","display":"\\kappa_{p,r}","kind":"use","excerpt":"0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd, β_r2^2r(2r)!, · p=2.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d5c11c2e36107f9e","tex":"\\kappa_r","aliases":"κ_r kappa","display":"\\kappa_r","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.","count":12},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-612207c6128de8ac","tex":"\\Lambda_{2,L}","aliases":"Λ_2,L Lambda","display":"\\Lambda_{2,L}","kind":"use","excerpt":"For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c4cbc8bb95a9e6bd","tex":"\\Lambda_c","aliases":"Λ_c Lambda","display":"\\Lambda_c","kind":"use","excerpt":"pageheading3. A nonsingular Euler-operator argumentThe standard interval without an analytic power-series-in-the-index premise On Hurwitz coefficients, the actual Euler operator plus c is (Λ_c a)(d)=(d+c)a(d). Direct induction proves its commutator with every defect iterate: δ^sΛ_c a =Λ_cδ^s a +sp(pU^p^2-U^p)δ^s-1a, s≥1. Suppose a is integral and s","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ce644555e3509ab9","tex":"\\Lambda_c^{-1}","aliases":"Λ_c^-1 Lambda","display":"\\Lambda_c^{-1}","kind":"use","excerpt":"When pnmid c, define the actual inverse coefficientwise: [t^d]Λ_c^-1f=frac[t^d]fd+c. On p-supported degrees, d+c is a genuine p-adic unit. The proof establishes Hurwitz integrality of this solution, its support, both inverse identities, and preservation of full flatness using the earlier all-order Euler commutator theorem. The singular pivots c=ap ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c442f1e041987da1","tex":"\\Lambda_{j+1}","aliases":"Λ_j+1 Lambda","display":"\\Lambda_{j+1}","kind":"use","excerpt":"For j=0, g_0 is the literal N-divisibility indicator. Because Nmid p(p-1), the two shifts in δ g_0 agree and δ g_0=0. It is flat at every order. The actual binomial identity gives Λ_j+1g_j+1=(j+1)g_j. As 1≤ j+1<p, the nonsingular reflection theorem inductively proves flatness of every g_j in the standard window. There is no uniform-convergence assu","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-35626feb35e38e3f","tex":"\\Lambda_{p,L}","aliases":"Λ_p,L Lambda","display":"\\Lambda_{p,L}","kind":"use","excerpt":"For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c28ddbaa56138f5b","tex":"\\lambda_{p,r}","aliases":"λ_p,r lambda","display":"\\lambda_{p,r}","kind":"use","excerpt":"[Unified inverse local transfer] Define λ_p,r = 1, · p odd and r=0, 2^2r-1p^2r, · p odd and r≥1, dfrac2^4r+1(2r+1)(2r+2), · p=2. Then for every prime p and every even n≥0, A_p,n = ∑_r=0^n/2 binom n2r λ_p,r E_n-2r. Equivalently, if L_p(t)=K_p(t)^-1, then H_p(t)=L_p(t)secht, where L_p(t)=cosh^2(pt) for odd p, and L_2(t)=(fracsinh(2t)2t)^2 for p=2.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-36442f76e90ff174","tex":"\\lambda_r","aliases":"λ_r lambda","display":"\\lambda_r","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":23},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3b453de64aef4dc5","tex":"\\mu_0","aliases":"μ_0 mu","display":"\\mu_0","kind":"use","excerpt":"odd: T(m,1)=2^m-1 and v_2(m!)=m-1. For m≥4 a power of two, the r=3 term is even. More explicitly, fracT(m,3)2^m-3=h_m-3(1,3,6), ∑_d≥0h_d(1,3,6)t^d =frac1(1-t)(1-3t)(1-6t). Reduction modulo two gives (1-t)^-2, whose degree m-3 coefficient has parity m-2=0. At m=1,2 the r=3 term is absent, and the r=1 term is again odd. Finally μ_0=1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3d1bb2e4dde90487","tex":"\\mu_a","aliases":"μ_a mu","display":"\\mu_a","kind":"use","excerpt":"this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even because 4mid k. The other contribution expands as L(Qγ_k-1(P+Q)) =∑_a=0^k-1(k-a)μ_aμ_k-a. An odd summand would req","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0c0ba94f6308ea27","tex":"\\mu_h","aliases":"μ_h mu","display":"\\mu_h","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d863225288067393","tex":"\\mu_{k-a}","aliases":"μ_k-a mu","display":"\\mu_{k-a}","kind":"use","excerpt":"this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even because 4mid k. The other contribution expands as L(Qγ_k-1(P+Q)) =∑_a=0^k-1(k-a)μ_aμ_k-a. An odd summand would req","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2ffb97b3578a68df","tex":"\\mu_m","aliases":"μ_m mu","display":"\\mu_m","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-22d0564d0934fa50","tex":"\\mu_r","aliases":"μ_r mu","display":"\\mu_r","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c381b1e9f2cf6895","tex":"\\nu_2","aliases":"ν_2 nu","display":"\\nu_2","kind":"use","excerpt":"[All-row dyadic higher differences] For 0≤ j<4 and all u,k,n∈ N, C_j(n) · ∈ Z_2, Δ_u^kC_j(n) · ∈ 2^k(1+ν_2(u)) Z_2, Δ_u^kU_j(n) · ∈ 2^k(1+ν_2(u))+v_2(j!) Z_2. There is no restriction k≤ n and no minimum with the row n.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"ather At the base parameter, the strict Fr echet derivative is Dψ(t_0)=-J^-1K. In particular, ψ is differentiable on U. Let g be the analytic local inverse of Φ supplied by Lemma~[source reference]. Since F(a)=0, Φ(a)=(0,t_0) and g(0,t_0)=a. Define ψ(t)=π_Yg(0,t), where π_Y projects to the solved coordinates. This gives the base value and analytici","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d416b19c5cb4e834","tex":"\\pi_Y","aliases":"π_Y pi","display":"\\pi_Y","kind":"use","excerpt":"ather At the base parameter, the strict Fr echet derivative is Dψ(t_0)=-J^-1K. In particular, ψ is differentiable on U. Let g be the analytic local inverse of Φ supplied by Lemma~[source reference]. Since F(a)=0, Φ(a)=(0,t_0) and g(0,t_0)=a. Define ψ(t)=π_Yg(0,t), where π_Y projects to the solved coordinates. This gives the base value and analytici","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1a247531173a01df","tex":"\\rho_p","aliases":"ρ_p rho","display":"\\rho_p","kind":"use","excerpt":"The earlier obstruction used the integer-domain map J_x=ρ_p∘ DF(x): Z^d→ F_p^e. The new count uses A_x: F_p^d→ F_p^e. Reduction of integer vectors onto finite vectors is surjective, so the two maps have the same image after restricting scalars. Lean constructs a class-preserving integer-linear equivalence F_p^e/imJ_x ≃ F_p^e/imA_x. Consequently the","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d903684d8cb76c26","tex":"\\rho_x","aliases":"ρ_x rho","display":"\\rho_x","kind":"use","excerpt":"[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0,","count":10},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9264ae4ae2ab0aab","tex":"\\rho_{\\ell,k}","aliases":"ρ_ℓ,k rho ell","display":"\\rho_{\\ell,k}","kind":"use","excerpt":"Principal targets. targetFiniteUnits.finite_unit_decomposition, .power_lift_congruence targetFiniteUnits.omega_unique, .decomposition_unique targetFiniteUnits.omega_mul, .principal_mul pageheading3. Precision compatibility and the dyadic signFinite refinements and Proposition 6.2's dyadic branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"[Continuous charts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is ho","count":20},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6512b523961a05f5","tex":"\\sigma^{-1}","aliases":"σ^-1 sigma","display":"\\sigma^{-1}","kind":"use","excerpt":"Finally, surjectivity of the original rectangular Jacobian guarantees a basis among its columns. Extend the corresponding injection of coordinate indices to a permutation σ. Renaming the variables by σ^-1 transports polynomial evaluation and every formal partial derivative. The fixed-coordinate construction therefore applies to the original system ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2792c54819372788","tex":"\\sigma_p","aliases":"σ_p sigma","display":"\\sigma_p","kind":"use","excerpt":"For an odd prime p, the source S_n(p) contains one nonunit base. Put σ_p=(-1)^(p-1)/2 and S_n^×(p)=S_n(p)-σ_pp^n.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definition","count":51},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ff0c3a9746b729b1","tex":"\\tau^2","aliases":"τ^2 tau","display":"\\tau^2","kind":"use","excerpt":"ulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since q is even and ϵτ≡ϵpmod2, this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-96a7e03674728790","tex":"\\tau_p","aliases":"τ_p tau","display":"\\tau_p","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction; a category of seed presentations and a notion of equivalence still require separate definition","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-53304761e001a2f9","tex":"\\tau_r","aliases":"τ_r tau","display":"\\tau_r","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a75b83819370342d","tex":"\\tau_r^2","aliases":"τ_r^2 tau","display":"\\tau_r^2","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"[Isometric packet bijections] For 0≤ j<4, define on natural numbers f_j(n)=fracU_j(n)-j!2^d_j, (d_0,d_1,d_2,d_3)=(1,1,5,3). There is a unique continuous map Φ_j: Z_2→ Z_2 whose value at every n∈ N equals the embedded rational f_j(n). It is an isometric bijection and Φ_j(0)=0: |Φ_j(x)-Φ_j(y)|_2=|x-y|_2 (x,y∈ Z_2). In particular, every element of Z_2","count":35},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"rphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is the forward map, then for every N≥0, h(t)≡ h(u)pmodp^N Longleftrightarrow t≡ upmodp^N. Equivalently, σ h(t)=(ϕ(t),t) for a continuous ϕ:T→ y_0+p^L Z_p^e whose graph is the entire reordered root locus.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d33036383a860124","tex":"\\Phi_2","aliases":"Φ_2 Phi","display":"\\Phi_2","kind":"use","excerpt":"The lowest degree term of Φ_M(z) is Mz, and its highest degree term is z^M. Multiplication by (1+z)^s gives the two edge coefficients. When M=2 and s=0, Φ_2(z)^k = (2z+z^2)^k = z^k(2+z)^k. Taking the coefficient of z^k+j gives the binary formula.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fda9801fa210c8f8","tex":"\\Phi_j","aliases":"Φ_j Phi","display":"\\Phi_j","kind":"use","excerpt":"[Isometric packet bijections] For 0≤ j<4, define on natural numbers f_j(n)=fracU_j(n)-j!2^d_j, (d_0,d_1,d_2,d_3)=(1,1,5,3). There is a unique continuous map Φ_j: Z_2→ Z_2 whose value at every n∈ N equals the embedded rational f_j(n). It is an isometric bijection and Φ_j(0)=0: |Φ_j(x)-Φ_j(y)|_2=|x-y|_2 (x,y∈ Z_2). In particular, every element of Z_2","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-569ae08551ce0863","tex":"\\Phi_M","aliases":"Φ_M Phi","display":"\\Phi_M","kind":"use","excerpt":"Let Φ_M(z)=(1+z)^M-1. For M≥1, s≥0, and r,k≥0, define Θ_M,s(r,k) = [z^r](1+z)^sΦ_M(z)^k. Equivalently, Θ_M,s(r,k) = ∑_j=0^k (-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-00bfe1f837258b4d","tex":"\\Phi_r","aliases":"Φ_r Phi","display":"\\Phi_r","kind":"use","excerpt":"K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). Excluding a multiple root on the surviving chart is exactly the assertion ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a9cc14498171fac3","tex":"\\Phi_{r,m}^{\\mathrm{act}}","aliases":"Φ_r,m^act Phi mathrm","display":"\\Phi_{r,m}^{\\mathrm{act}}","kind":"use","excerpt":"The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2b5d4a112c69639c","tex":"\\Phi_r^{\\mathrm{act}}","aliases":"Φ_r^act Phi mathrm","display":"\\Phi_r^{\\mathrm{act}}","kind":"use","excerpt":"ion K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). ","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-703718018bf26313","tex":"\\Phi_{x,L}","aliases":"Φ_x,L Phi","display":"\\Phi_{x,L}","kind":"use","excerpt":"For every positive integer q, including composite moduli and modulus one, and every L≥0, the checked map Φ_x,L:( Z/q Z)^d⟶ {ybmod q^L+1:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection. Here widetilde t is the canonical integer representative of the digit vector. The entire residue fiber therefore has q^d elements. Injectivity is proved by cancelli","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1b57be49fae3643b","tex":"\\chi^\\bullet","aliases":"χ^bullet chi bullet","display":"\\chi^\\bullet","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-726e69c1d2b3c2c6","tex":"\\chi^q","aliases":"χ^q chi","display":"\\chi^q","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-09b652c188f47b6e","tex":"\\chi_C","aliases":"χ_C chi","display":"\\chi_C","kind":"use","excerpt":"[Characteristic polynomial of the jump] Let χ_C(y)=det(yI-C). Then χ_C^N(x) = Res_y(χ_C(y),x-y^N). The identity holds over the coefficient ring after interpreting the resultant universally.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9f897e865f593956","tex":"\\chi_{C^N}","aliases":"χ_C^N chi","display":"\\chi_{C^N}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Now suppose a chosen set of e columns of DF(x)bmod p is independent, and put those coordinates first. Apply the square result to the augmented polynomial map Ψ(Y,T)=(F(Y,T),T). Its reduced Jacobian has upper blocks (D_YF,D_TF) and lower blocks (0,I_k), so it is invertible. For t∈ T, the target (0,t) belongs to Ψ(x)+p^L Z_p^e+k because F(x)≡0pmodp^L","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1ca12a002e4da1b8","tex":"\\psi^{-1}","aliases":"ψ^-1 psi","display":"\\psi^{-1}","kind":"use","excerpt":"rhoods as follows. The source of the open partial homeomorphism defined by Φ is open around a; its inverse is g. Choose open sets V and U_0 containing y_0 and t_0 such that V× U_0 lies in that source. Restrict U_0 further by the preimage of an open analytic inverse neighborhood under t↦(0,t), the open locus where this same ψ is analytic, and the in","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5d52ab32e3afcb63","tex":"\\omega_2","aliases":"ω_2 omega","display":"\\omega_2","kind":"use","excerpt":"For p=3,ℓ=2,a=2, the checked values are ω_2(2)=8, ⟨2⟩_2=7, 8^2=1,quad8·7=2 in Z/9 Z. The unlifted representative 2 is not a square root of one modulo nine.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-fa791c4a1646d467","tex":"\\omega_k","aliases":"ω_k omega","display":"\\omega_k","kind":"use","excerpt":"a_unique, .decomposition_unique targetFiniteUnits.omega_mul, .principal_mul pageheading3. Precision compatibility and the dyadic signFinite refinements and Proposition 6.2's dyadic branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-17643fc6afa4dc8b","tex":"\\omega_L","aliases":"ω_L omega","display":"\\omega_L","kind":"use","excerpt":"[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7fb25fe9a8f4c1bc","tex":"\\Omega_{P/\\mathbb Z}","aliases":"Ω_P/ Z Omega mathbb","display":"\\Omega_{P/\\mathbb Z}","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7f24c361b02bdb1d","tex":"\\omega_\\ell","aliases":"ω_ℓ omega ell","display":"\\omega_\\ell","kind":"use","excerpt":"ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"To pass from the section to the singular ratio, factor out its initial zero: for c=ap, the section has the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(","count":59},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3955d1d0d8a03f64","tex":"\\ell^m","aliases":"ℓ^m ell","display":"\\ell^m","kind":"use","excerpt":"Put y=x+1 and let ℓ=log 9. Since v_2(ℓ)=3, frac9^y-18y = fracℓ8 ∑_m≥0 fracℓ^m(m+1)!y^m. The coefficient of y^m has valuation 3m-v_2((m+1)!), which is nonnegative and tends to positive infinity. The quotient therefore belongs to Z_2⟨ y⟩. The factor 2x+1=1+2x is a unit in Z_2⟨ x⟩, with inverse ∑_m≥0(-2x)^m. Hence mathscr R is integral and analytic. T","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-9ec711e428117ead","tex":"\\twoheadrightarrow","aliases":"↠ twoheadrightarrow","display":"\\twoheadrightarrow","kind":"use","excerpt":"The group is finite. Choose a finite presentation P= Z[X_1,…,X_d]↠ B with finitely generated kernel I. The presentation description in https://stacks.math.columbia.edu/tag/0GPUStacks Project, Remark 91.2.4, tag 0GPU identifies the group as","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"Set the first integer parameter approximation equal to the prescribed t_0. For n≥1, use canonical representatives of τbmod p^L+n. The initial congruence and truncation precision give τ-t_n∈ p^L+n Z_p^k. Subtracting consecutive relations gives compatibility in Z_p. Faithfulness of integer divisibility under Z↪ Z_p gives compatibility in Z, and the s","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c97379bf19e9e4d6","tex":"\\partial_i","aliases":"∂_i partial","display":"\\partial_i","kind":"use","excerpt":"For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1cd2f6e5cfcfeb44","tex":"\\partial_j","aliases":"∂_j partial","display":"\\partial_j","kind":"use","excerpt":"For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"Define the finite solved-block cokernel and its parameter-dependent class by C_x=S^e/imJ, o_x(v)=[bar r+Kv]∈ C_x. Then L_v≠∅iff o_x(v)=0. More precisely, the checked digit-to-lift equivalence is L_v≅{u∈ S^m:Ju=-bar r-Kv}. It sends a solved digit u to the actual next residue represented by x+q^L(widehat u,widehat v). The obstruction depends on the s","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"[Matrix power amplification] Editorial hypothesis correction. Require L≥1. At L=0 the standalone claim is false: for p=3, d=1 and X=2=1+B, one has X^3-1=7∉3 Z_3. With L≥1, every higher binomial term in the retained proof has the required valuation, including p=2. If X=I+p^LB with B∈ M_d( Z_p), then for every a≥0 there is B_a∈ M_d( Z_p) such that X^","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"em[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":35},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6d48f9704594142c","tex":"\\prod_{a=0}^{r-1}","aliases":"∏_a=0^r-1 prod","display":"\\prod_{a=0}^{r-1}","kind":"use","excerpt":"Define the central-factorial basis polynomials P_r(X)=∏_a=0^r-1(X-a(a+1)). At X=Z(Z+1), they have the exact factorization P_r(Z(Z+1)) =∏_a=0^r-1(Z-a)(Z+a+1) =(2r)!binomZ+r2r. The Bernoulli transform gives L((1+T)^Z)=log(1+T)/T, whence L(binom Zj)=(-1)^j/(j+1). Finite Vandermonde expansion therefore gives L(binomZ+r2r) · =(-1)^r∑_q=0^r(-1)^qbinom rq","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5e226abbc069da46","tex":"\\prod_{c=1}^{j}","aliases":"∏_c=1^j prod","display":"\\prod_{c=1}^{j}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-587b8e89db5f122b","tex":"\\prod_{f(\\alpha)=0}","aliases":"∏_f(α)=0 prod alpha","display":"\\prod_{f(\\alpha)=0}","kind":"use","excerpt":"We use the resultant convention Res_y(f,g) = ∏_f(α)=0g(α) when f is monic, with roots counted with multiplicity.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6892de3ca76a9445","tex":"\\prod_{h=0}^k","aliases":"∏_h=0^k prod","display":"\\prod_{h=0}^k","kind":"use","excerpt":"For s∈ p Z_p, an exact finite-difference identity is Δ_1^kfrac11+sn =frac(-s)^k k!∏_h=0^k(1+s(n+h)). It follows by induction, subtracting adjacent rational functions; each successive subtraction adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4867538d309900b7","tex":"\\prod_{h=1}^{2^a-1}","aliases":"∏_h=1^2^a-1 prod","display":"\\prod_{h=1}^{2^a-1}","kind":"use","excerpt":"at least two, since i≤ j<N/2. Hence g(1)≡1pmod4 and g(2)≡1pmod8. The first two Newton coefficients from the reciprocal recurrence are a_1=-g(1)-1, a_2=binom2NNg(1)^2-g(2)+2g(1)+1. The first has valuation exactly one. Also binom2NN≡6pmod8 for every N=2^a≥2. To see this, write B_a=binom2^a+12^a =2∏_h=1^2^a-1(1+frac2^ah). The even-h factors give B_a-1","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-488c88df263b9bc8","tex":"\\prod_{i=0}^{d-1}","aliases":"∏_i=0^d-1 prod","display":"\\prod_{i=0}^{d-1}","kind":"use","excerpt":"_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive block","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b55525d7b1b112b4","tex":"\\prod_{i=0}^{p-1}","aliases":"∏_i=0^p-1 prod","display":"\\prod_{i=0}^{p-1}","kind":"use","excerpt":"Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-45251ad81332de19","tex":"\\prod_{i=0}^{q-1}","aliases":"∏_i=0^q-1 prod","display":"\\prod_{i=0}^{q-1}","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-78738dc0b544ac94","tex":"\\prod_{i=1}^{d}","aliases":"∏_i=1^d prod","display":"\\prod_{i=1}^{d}","kind":"use","excerpt":"After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-68bfcf268c38fdd4","tex":"\\prod_{i=1}^j","aliases":"∏_i=1^j prod","display":"\\prod_{i=1}^j","kind":"use","excerpt":"The packed factorial coefficient of G_N,j is g_N,j(n)=fracj!(Nn)!(Nn+j)! =∏_i=1^j(1+frac Ni n)^-1. The empty product is one. If N=p^a and 1≤ i<N, then v_p(i)<a, so N/i∈ p Z_p. Every affine denominator in [source reference] is therefore a unit on Z_p.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4f3135d472778bff","tex":"\\prod_{i=1}^{\\ell-1}","aliases":"∏_i=1^ℓ-1 prod ell","display":"\\prod_{i=1}^{\\ell-1}","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dc7ec85dad31c9de","tex":"\\prod_{i\\in\\F_p}","aliases":"∏_i∈F_p ∏_i∈ F_p prod in F prod in mathbb","display":"\\prod_{i\\in{\\mathbb F}_p}","kind":"use","excerpt":"Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-60ea789b512f778a","tex":"\\prod_{j=1}^{s}","aliases":"∏_j=1^s prod","display":"\\prod_{j=1}^{s}","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b9d5148b142926aa","tex":"\\prod_p","aliases":"∏_p prod","display":"\\prod_p","kind":"use","excerpt":"Why the global/completed distinction is necessary. Consider constant sequences with integer initial value. Their global parameter space is Z, while their local parameter spaces are Z_p. Under the canonical local maps, the image of the global objects is the diagonal D={(n)_p:n∈ Z}⊆∏_p Z_p. This diagonal is dense: a basic open set specifies finitely ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8ec8acc881d5dd6d","tex":"\\prod_{p>2}","aliases":"∏_p>2 prod","display":"\\prod_{p>2}","kind":"use","excerpt":"Consequently this canonical global diagonal cannot be an equality locus {(a,b):f(a)=g(b)} ⊆ Z_2×∏_p>2 Z_p for continuous maps to a Hausdorff space C. Such an equality locus is closed, as the inverse image of the closed diagonal in C× C, whereas D is proper and dense. This topological argument, retained from the native progress note, concerns the st","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-53f2224b9699f49b","tex":"\\prod_{p>2}'","aliases":"∏_p>2' prod","display":"\\prod_{p>2}'","kind":"use","excerpt":"The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-516d6ddd3d40ea65","tex":"\\prod_{p\\in S}","aliases":"∏_p∈ S prod in","display":"\\prod_{p\\in S}","kind":"use","excerpt":"N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0fcd1056f96dbe36","tex":"\\prod_r","aliases":"∏_r prod","display":"\\prod_r","kind":"use","excerpt":"[Lucas congruence] If a=a_0+a_1p+⋯ and b=b_0+b_1p+⋯ are base p expansions, then binom ab ≡ ∏_rbinoma_rb_r pmod p. This is the classical theorem of Lucas [source reference].","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ee7eedb4b2626377","tex":"\\prod_{r<n}","aliases":"∏_r<n prod","display":"\\prod_{r<n}","kind":"use","excerpt":"The recursion uses only smaller indices, and substitution verifies each row. If a second solution agrees through row n-1, subtracting row n and cancelling T_nn proves equality at n. For integrality, use induction and promote every earlier clearing product to ∏_r<nT_rr. Multiplication by the additional factor T_nn cancels the only new division.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-07b87a46416a131f","tex":"\\prod_{r=0}^{n}","aliases":"∏_r=0^n prod","display":"\\prod_{r=0}^{n}","kind":"use","excerpt":"em[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4580ee4902e1619e","tex":"\\prod_{r=1}^{n}","aliases":"∏_r=1^n prod","display":"\\prod_{r=1}^{n}","kind":"use","excerpt":"[A denominator bound for every address] For N≥1, 0≤ j<N, and n∈ N, set D_N,j(n)=∏_r=1^nbinomNr+jNr, D_N,j(0)=1. Then D_N,j(n)C_N,j(n)∈ Z.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-26d84da82cf06c30","tex":"\\sum_{a=0}^k","aliases":"∑_a=0^k sum","display":"\\sum_{a=0}^k","kind":"use","excerpt":"For the first contribution, independence gives L(γ_k(P+Q))=∑_a=0^kμ_aμ_k-a. The parity generating series of these convolution moments is (1+t+t^2+t^4+t^8+⋯)^2 =1+t^2+t^4+t^8+⋯ in F_2[[t]]. Its coefficient is one at every k=2^r≥4. Thus the normalized moment in [source reference] is odd and v_2(Δ_1^kC_2(0))=2k+v_2(k!)+k/2. Inserting U^4n=(1+4H_2)^n i","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-64251df324f966d0","tex":"\\sum_{a=0}^{k-1}","aliases":"∑_a=0^k-1 sum","display":"\\sum_{a=0}^{k-1}","kind":"use","excerpt":"this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even because 4mid k. The other contribution expands as L(Qγ_k-1(P+Q)) =∑_a=0^k-1(k-a)μ_aμ_k-a. An odd summand would req","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-07e82a6cdc184f88","tex":"\\sum_{a=0}^{q-1}","aliases":"∑_a=0^q-1 sum","display":"\\sum_{a=0}^{q-1}","kind":"use","excerpt":"Let q be a positive odd integer. Define S_n(q) = ∑_a=0^q-1 (-1)^a(2a+1)^n. Let the signed Euler numbers and Euler zigzag numbers be defined by secht = ∑_n≥0E_nfract^nn! and sec t+tan t = ∑_n≥0Z_nfract^nn!.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-33094d315b0147bf","tex":"\\sum_{a=0}^{u-1}","aliases":"∑_a=0^u-1 sum","display":"\\sum_{a=0}^{u-1}","kind":"use","excerpt":"l closure therefore gives flatness of the actual packed Euler coefficient sequence. The earlier coefficient-extraction theorem now has a proved antecedent for every j<p, and yields p^min(k,mn)midΔ_h^k C_mp,j(n) in Z_p, h=(p-1)/m. A new finite telescoping argument proves the extension to arbitrary natural multiples of h. Its first-step identity is f","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-996cca7e6dc9487c","tex":"\\sum_{a=1}^{q-1}","aliases":"∑_a=1^q-1 sum","display":"\\sum_{a=1}^{q-1}","kind":"use","excerpt":"pageheading1. The characteristic-zero product ruleThe normalized defect is not an ordinary derivation Let A be a commutative algebra over a characteristic-zero field K, and let U be an actual K-derivation. The source defines B=fracU^p^2-U^pp, M_q(f,g)=∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dab848ac9f38aa10","tex":"\\sum_{A\\subseteq\\{1,\\ldots,k\\}}","aliases":"∑_A⊆{1,…,k} sum subseteq ldots","display":"\\sum_{A\\subseteq\\{1,\\ldots,k\\}}","kind":"use","excerpt":"Write Sf(n)=f(n+1) and Δ_H=S^H-1. Mixed differences commute. Their exact finite expression is Δ_H_1⋯Δ_H_kf(n) =∑_A⊆{1,…,k} (-1)^k-|A| f (n+∑_i∈ AH_i). Repeated subset offsets are combined with their signed multiplicities.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a3a23a48b02a732f","tex":"\\sum_{d\\equiv j\\ ({\\rm mod}\\ N)}","aliases":"∑_d≡ j (rm mod N) sum equiv rm","display":"\\sum_{d\\equiv j\\ ({\\rm mod}\\ N)}","kind":"use","excerpt":"For N>0 and 0≤ j<N, the independently defined original exponential section is E_N,j(t)=∑_d≡ j (rm mod N)fract^dd!. Using the actual power-series expansion homomorphism z↦ t^N, the code proves E_N,j(t)=fract^jj!K_N,j(t^N), F_N,j(t^N)E_N,j(t)=fract^jj!. Thus the recurrence is connected back to the original normalized EGF definition. The section vanis","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2761fb529cb10263","tex":"\\sum_{h=0}^k","aliases":"∑_h=0^k sum","display":"\\sum_{h=0}^k","kind":"use","excerpt":"3 · 4 · 161644544/4849845 · 15 They are finite rational evaluations of [source reference], followed by a_k=∑_h=0^k(-1)^k-hbinom kh C_j(h). For addresses zero and one, k≥3 is covered by the generic bound; for address two, k≥7 is covered; for address three, k≥5 is covered. In each range the elementary inequality is k-lfloorlog_2krfloor≥ b_j+1. Lemma~","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"formula, rather than a product rule with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-bd854b4a4a72698f","tex":"\\sum_{i<d}","aliases":"∑_i<d sum","display":"\\sum_{i<d}","kind":"use","excerpt":"Suppose g(0)=1. The source constructs f by the full triangular recurrence f(d)=δ_d0-∑_i<dbinom di f(i)g(d-i) and proves f⋆ g=1. It also proves cancellation: if a⋆ g=0, then a=0, by induction on the first nonzero coefficient.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-da5774d7444b36de","tex":"\\sum_{i=0}^{a}","aliases":"∑_i=0^a sum","display":"\\sum_{i=0}^{a}","kind":"use","excerpt":"The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c5edf1fa697a19cc","tex":"\\sum_{i=0}^d","aliases":"∑_i=0^d sum","display":"\\sum_{i=0}^d","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-adb6721290f5f31c","tex":"\\sum_{i=0}^{d}","aliases":"∑_i=0^d sum","display":"\\sum_{i=0}^{d}","kind":"use","excerpt":"pageheading1. Coefficient products and reciprocal differentiationA characteristic-p argument without factorial division For sequences f,g: N→ K, where K is a field, define (f⋆ g)(d)=∑_i=0^dbinom di f(i)g(d-i), (U^a f)(d)=f(d+a). These are Hurwitz coefficients. The notation does not divide by d! in F_p. In particular, the implementation is not ordin","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eca795e86c8fb349","tex":"\\sum_{i=0}^k","aliases":"∑_i=0^k sum","display":"\\sum_{i=0}^k","kind":"use","excerpt":"The strongest common-step conclusion in width four is particularly compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and pos","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-964202d717c850f2","tex":"\\sum_{i=0}^{n}","aliases":"∑_i=0^n sum","display":"\\sum_{i=0}^{n}","kind":"use","excerpt":"Finite Newton reconstruction at an arbitrary row gives Δ_1^kf(n)=∑_i=0^nbinom ni a_k+i. The monotonicity of lfloorlog_2krfloor therefore preserves the higher-coefficient bound after translation. In addition, Δ_1f(n)-a_1∈2^c+1 Z_2, so the first difference is nonzero of exact valuation c.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-33db49b261c77217","tex":"\\sum_{i=0}^s","aliases":"∑_i=0^s sum","display":"\\sum_{i=0}^s","kind":"use","excerpt":"g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors interchanged to ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b7a8cfbba485d0c4","tex":"\\sum_{i=1}^d","aliases":"∑_i=1^d sum","display":"\\sum_{i=1}^d","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-58d5910b6df161a7","tex":"\\sum_{i=1}^{d}","aliases":"∑_i=1^d sum","display":"\\sum_{i=1}^{d}","kind":"use","excerpt":"For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-21850518d3a22cfc","tex":"\\sum_{i=1}^H","aliases":"∑_i=1^H sum","display":"\\sum_{i=1}^H","kind":"use","excerpt":"To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar mult","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7b615a361efd7684","tex":"\\sum_{i=1}^k","aliases":"∑_i=1^k sum","display":"\\sum_{i=1}^k","kind":"use","excerpt":"At a natural argument the exact Newton expansion is finite. Vandermonde gives binomn+Hk-binom nk =∑_i=1^kbinom Hibinom nk-i. For 1≤ i≤ H, v_2(binom Hi)≥ v_2(H)-v_2(i) ≥ v_2(H)-lfloorlog_2krfloor; terms with i>H vanish. Thus each Newton summand with k≥2 has valuation at least b+v_2(H)+1. The k=1 summand is Ha_1, of exact valuation b+v_2(H), proving ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-67f5e60d24848d87","tex":"\\sum_{i=1}^{N-1}","aliases":"∑_i=1^N-1 sum","display":"\\sum_{i=1}^{N-1}","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-901f1a8b2b778e8a","tex":"\\sum_{i\\in A}","aliases":"∑_i∈ A sum in","display":"\\sum_{i\\in A}","kind":"use","excerpt":"Write Sf(n)=f(n+1) and Δ_H=S^H-1. Mixed differences commute. Their exact finite expression is Δ_H_1⋯Δ_H_kf(n) =∑_A⊆{1,…,k} (-1)^k-|A| f (n+∑_i∈ AH_i). Repeated subset offsets are combined with their signed multiplicities.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-7fec58bf80198971","tex":"\\sum_{i\\in I}","aliases":"∑_i∈ I sum in","display":"\\sum_{i\\in I}","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4b8afcda15a5dfd1","tex":"\\sum_{i\\in S}","aliases":"∑_i∈ S sum in","display":"\\sum_{i\\in S}","kind":"use","excerpt":"Finite integral linear combinations. For any finite index set S, coefficients a_i∈ Z_p and units u_i∈ Z_p^×, linearity of the actual iterated operator gives Δ_h^s (∑_i∈ Sa_i u_i^n) =∑_i∈ Sa_i u_i^n(u_i^h-1)^s. At h=p-1, the result is divisible by p^s. The empty sum is permitted. This does not prove that the general congruential Euler packets of The","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f727d6dacb4f1062","tex":"\\sum_{i\\geq0}","aliases":"∑_i≥0 sum geq","display":"\\sum_{i\\geq0}","kind":"use","excerpt":"G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The co","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6e4ee3eaa6bf9adf","tex":"\\sum_{i\\geq1}","aliases":"∑_i≥1 sum geq","display":"\\sum_{i\\geq1}","kind":"use","excerpt":"the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial multiplier is integral. Because F_1=F_0D_1^-1, C_1(n)-C_0(n) =∑_i≥1h_ib","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Apply polynomial induction using constants, addition, and multiplication by a variable. Constants have zero remainder, and remainders add. For the multiplication step, suppose the assertion is known for P and write D_P(U)=∑_j(∂_jP)(a)U_j. Multiplication of its expansion by a_i+cU_i gives constant term P(a)a_i, linear term c(a_iD_P(U)+P(a)U_i), and ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-48fa0e3108a03d2c","tex":"\\sum_{j=0}^{k}","aliases":"∑_j=0^k sum","display":"\\sum_{j=0}^{k}","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ee6b2fc7cf7ca7f4","tex":"\\sum_{j=0}^{n}","aliases":"∑_j=0^n sum","display":"\\sum_{j=0}^{n}","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-586bc3571b381e64","tex":"\\sum_{j=0}^{N-1}","aliases":"∑_j=0^N-1 sum","display":"\\sum_{j=0}^{N-1}","kind":"use","excerpt":"[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2c9e6a9c9fb8db72","tex":"\\sum_{j=0}^{s}","aliases":"∑_j=0^s sum","display":"\\sum_{j=0}^{s}","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dec62b92295c8c6d","tex":"\\sum_{j=1}^d","aliases":"∑_j=1^d sum","display":"\\sum_{j=1}^d","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-db50c77870a75686","tex":"\\sum_{j=1}^n","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^n","kind":"use","excerpt":"[Derivative of polynomial evaluation] For P∈ K[X_1,…,X_n], polynomial evaluation is analytic at every point. At a∈ K^n its strict Fr echet derivative is v⟼∑_j=1^n (∂_jP)(a)v_j. For a finite system, its strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5a4e9c334647df08","tex":"\\sum_{k=0}^{\\min(d,r)}","aliases":"∑_k=0^min(d,r) sum min","display":"\\sum_{k=0}^{\\min(d,r)}","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-41e0c211ba80d17c","tex":"\\sum_{k=0}^{n}","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^{n}","kind":"use","excerpt":"AtlasHeadNewton coordinates turn precision into a finite computationatlas:newton Let b_k=Δ_1^kC_4,0(0). The first coefficients obtained from the difference triangle are inputgenerated/worked_newton_values.tex The exact Newton identity recovers each nonnegative integer row: C_4,0(n)=∑_k=0^nb_kbinom nk. AtlasImagenewton_cutoff For this address the pr","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a5316fecc2dcc62e","tex":"\\sum_{k=1}^{p^{r-1}}","aliases":"∑_k=1^p^r-1 sum","display":"\\sum_{k=1}^{p^{r-1}}","kind":"use","excerpt":"The finite Newton expansion gives f(n+p^r-1h)-f(n) =∑_k=1^p^r-1binomp^r-1kΔ_h^k f(n). Terms with k≥ r have the required factor from the difference. For 1≤ k<r, the identity kbinomp^r-1k=p^r-1binomp^r-1-1k-1 and v_p(k)<k imply p^rmidbinomp^r-1k p^k in Z. Mapping that integer divisibility to R proves the claim term by term. The argument includes r=1.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1f9dd23cb17a6b70","tex":"\\sum_{k=2}^{T}","aliases":"∑_k=2^T sum","display":"\\sum_{k=2}^{T}","kind":"use","excerpt":"Now expand the displacement: f(n+T)-f(n)=TΔ_1f(n) +∑_k=2^Tbinom TkΔ_1^kf(n). For 1≤ k≤ T, the binomial denominator identity implies v_2binom Tk≥ v_2(T)-v_2(k) ≥ v_2(T)-lfloorlog_2krfloor. Every term of the remaining sum consequently has valuation at least v_2(T)+c+1. Its sum cannot cancel the leading term, which has exact valuation v_2(T)+c. This p","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a334e226daa86a9f","tex":"\\sum_{k\\geq0}","aliases":"∑_k≥0 sum geq","display":"\\sum_{k\\geq0}","kind":"use","excerpt":"[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-db89bb56e5266992","tex":"\\sum_{m=0}^{\\min(M,N)}","aliases":"∑_m=0^min(M,N) sum min","display":"\\sum_{m=0}^{\\min(M,N)}","kind":"use","excerpt":"tion v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0692fed59f0e0fb2","tex":"\\sum_{m=0}^{\\min(M,r)}","aliases":"∑_m=0^min(M,r) sum min","display":"\\sum_{m=0}^{\\min(M,r)}","kind":"use","excerpt":"al targets. targetOddEuler.odd_reconstruction_series targetOddEuler.odd_euler_reconstruction targetOddEuler.odd_euler_reconstruction_zigzag targetOddEuler.row_action, OddEuler.row_diagonal, OddEuler.row_weight pageheading4. Uniform truncation and a sharp exact exampleCorollary 4.3 and Example 4.4 For row 2r and depth M, the actual truncation is T_M","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-04a4953949e57f10","tex":"\\sum_{m=0}^N","aliases":"∑_m=0^N sum","display":"\\sum_{m=0}^N","kind":"use","excerpt":"The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5e38ac93a3132039","tex":"\\sum_{m=0}^{n}","aliases":"∑_m=0^n sum","display":"\\sum_{m=0}^{n}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-187ef6e56570865f","tex":"\\sum_{m=0}^{n/2}","aliases":"∑_m=0^n/2 sum","display":"\\sum_{m=0}^{n/2}","kind":"use","excerpt":"Substituting the independently proved signed-zigzag identity recovers the manuscript's notation. For every even degree n, E_n=∑_m=0^n/2(-1)^m Z_2m+1binom n2mp^2mS_n-2m(p). The Lean version uses n=2r to avoid ambiguous natural-number division. Every even natural degree has that form. Primality is not used in this equality.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d3dbe092f3fdebc1","tex":"\\sum_{m=0}^{r}","aliases":"∑_m=0^r sum","display":"\\sum_{m=0}^{r}","kind":"use","excerpt":"ing3. Exact reconstruction and the triangular matrixTheorem 4.2; manuscript page 11 For odd p, rescaling the already proved inverse law gives sech(pt)cosh(pt)=1. Multiplying the even-source identity by sech^2(pt) therefore proves sech^2(pt)(∑_r≥0S_2r(p)fract^2r(2r)!)=sech t. Actual EGF multiplication now gives the integer coefficient formula E_2r=∑","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-05073c075273e26d","tex":"\\sum_{m=1}^H","aliases":"∑_m=1^H sum","display":"\\sum_{m=1}^H","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-482bdbbbe4d86bc7","tex":"\\sum_{m=k}^{n+k}","aliases":"∑_m=k^n+k sum","display":"\\sum_{m=k}^{n+k}","kind":"use","excerpt":"eciprocal are C_2(0)=1, C_2(1)=-frac115, C_2(2)=frac1345, a_1=-frac1615, a_2=frac6445, where a_m=Δ_1^m C_2(0). Thus v_2(a_1)=4 and v_2(a_2)=6. For m≥3, [source reference] bounds a_m by 2m+v_2(m!). The finite shifted Newton formula Δ_1^kC_2(n)=∑_m=k^n+ka_mbinom nm-k therefore gives v_2(Δ_1^kC_2(n))≥ k+v_2(k!)+3 for every k≥1. For k≥3 this is 2k≥ k+3","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d1cabd23a4e71916","tex":"\\sum_{m\\geq0}","aliases":"∑_m≥0 sum geq","display":"\\sum_{m\\geq0}","kind":"use","excerpt":"[Unified local Euler transfer] For every prime p and every even n≥0, E_n = ∑_r=0^n/2 binom n2r κ_p,r A_p,n-2r. If H_p(t) = ∑_m≥0A_p,2mfract^2m(2m)! and K_p(t) = ∑_r≥0κ_p,rfract^2r(2r)!, then K_p(t)H_p(t)=secht. For odd p, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). For p=2, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t).","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a0a9cffbc383b69e","tex":"\\sum_{m\\ge1}","aliases":"∑_m≥1 sum ge","display":"\\sum_{m\\ge1}","kind":"use","excerpt":"Let Q(z)=zcoth z, the regular formal series with constant coefficient one. The elementary differential identity (frac zsinh z)^2=Q(z)-zQ'(z) and the Bernoulli expansion Q(z)=1+∑_m≥1frac2^2mB_2m(2m)!z^2m give β_m=-2^2m(2m-1)B_2m/(2m)!. Multiplication by 2^2m(2m)! proves the identity for κ_m. The Bernoulli denominator theorem gives v_2(B_2m)=-1 for m","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-22d4ad01256f7944","tex":"\\sum_{m\\geq\\ell}","aliases":"∑_m≥ℓ sum geq ell","display":"\\sum_{m\\geq\\ell}","kind":"use","excerpt":"Put a_m=Δ_1^mC_j(0) and expand the falling factorial in the Newton series. The ordinary coefficients are d_j,ℓ=∑_m≥ℓfraca_mm!s(m,ℓ). Since s(m,ℓ) is integral and v_2(a_m/m!)≥ m+β_j, the coefficient sum converges and has valuation at least ℓ+β_j. The sum exchange is justified on Z_2 by these uniform geometric coefficient bounds. The ordinary power s","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","excerpt":"The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-45953a160933c484","tex":"\\sum_{n\\text{ even}}","aliases":"∑_n even sum text","display":"\\sum_{n\\text{ even}}","kind":"use","excerpt":"For even n≥0, define R_n = frac3^n+2-14(n+1)(n+2). Its even exponential generating function is ∑_n evenR_nfract^nn! = fracsinh(2t)sinh t2t^2. Define β_r by fracz^2sinh^2z = ∑_r≥0β_rz^2r.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-391cc34f0ac3e01c","tex":"\\sum_{N\\ge0}","aliases":"∑_N≥0 sum ge","display":"\\sum_{N\\ge0}","kind":"use","excerpt":"m≥1, differentiation of zcoth z gives β_m=-frac2^2m(2m-1)B_2m(2m)!, κ_m=-2^4m(2m-1)B_2m. The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-441a6af1afa122e0","tex":"\\sum_{n\\geq0}","aliases":"∑_n≥0 sum geq","display":"\\sum_{n\\geq0}","kind":"use","excerpt":"Over a characteristic-zero field K, put E(a)=∑_n≥0fraca_nn!t^n, D(A)_n=n![t^n]A. These inverse maps between sequences and K[[t]] provide formal exponential coordinates. No analytic convergence is needed for the coefficient identities below.","count":15},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-366ac52aad5d30c9","tex":"\\sum_q","aliases":"∑_q sum","display":"\\sum_q","kind":"use","excerpt":"The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e9342a8d15de8640","tex":"\\sum_{q<n}","aliases":"∑_q<n sum","display":"\\sum_{q<n}","kind":"use","excerpt":"[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5e33c0103731d00c","tex":"\\sum_{q=0}^n","aliases":"∑_q=0^n sum","display":"\\sum_{q=0}^n","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2636c510c23e69cc","tex":"\\sum_{q=0}^{n-1}","aliases":"∑_q=0^n-1 sum","display":"\\sum_{q=0}^{n-1}","kind":"use","excerpt":"eleq:arith-kernel K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coefficient is positive, so thi","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-45df01cda676bad4","tex":"\\sum_{q=0}^r","aliases":"∑_q=0^r sum","display":"\\sum_{q=0}^r","kind":"use","excerpt":"X)=∏_a=0^r-1(X-a(a+1)). At X=Z(Z+1), they have the exact factorization P_r(Z(Z+1)) =∏_a=0^r-1(Z-a)(Z+a+1) =(2r)!binomZ+r2r. The Bernoulli transform gives L((1+T)^Z)=log(1+T)/T, whence L(binom Zj)=(-1)^j/(j+1). Finite Vandermonde expansion therefore gives L(binomZ+r2r) · =(-1)^r∑_q=0^r(-1)^qbinom rqfrac1r+q+1nonumber · =(-1)^rfrac(r!)^2(2r+1)!. The ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-89403d4e42f7d258","tex":"\\sum_{q=0}^{r}","aliases":"∑_q=0^r sum","display":"\\sum_{q=0}^{r}","kind":"use","excerpt":"The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r,","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-18f531fd7c19a676","tex":"\\sum_{q=1}^n","aliases":"∑_q=1^n sum","display":"\\sum_{q=1}^n","kind":"use","excerpt":"The actual reciprocal recurrence is C_N,j(n)=-∑_q=1^nbinomNnNq g_N,j(q)C_N,j(n-q), C_N,j(0)=1. For N=4, the four kernels are 1, frac14q+1, frac1(4q+1)(2q+1), frac3(4q+1)(2q+1)(4q+3). The threshold-free theorem gives v_2(a_k)≥ k in every address. The following exact values cover all orders where that bound does not already imply [source reference]: ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-1315d72b28634d4a","tex":"\\sum_{q\\in J_r}","aliases":"∑_q∈ J_r sum in","display":"\\sum_{q\\in J_r}","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-65994b1f27113b7a","tex":"\\sum_{q\\ge0}","aliases":"∑_q≥0 sum ge","display":"\\sum_{q\\ge0}","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eb07f8d7b26ea999","tex":"\\sum_{q\\geq0}","aliases":"∑_q≥0 sum geq","display":"\\sum_{q\\geq0}","kind":"use","excerpt":"[Exact discrete polyphase reconstruction] The analysis map P_M is a bijection between sequences and ordered families of M sequences. If A(z)=∑_n≥0a_nz^n and A_r^[M](w)=∑_q≥0a_Mq+rw^q, then A(z)=∑_r=0^M-1z^rA_r^[M](z^M).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"g s_2(i) for the number of ones in the binary expansion of i, and using v_2((4i)!)=4i-s_2(i), gives v_2([x^d]P_i) ≥2d-2i+s_2(i)+max(0,2i-d) ≥ d+s_2(i)≥ d+1 (d≥1). For k≥1, Δ_1^k x^d vanishes when d<k and is an integer polynomial with values divisible by k!. To see the latter, expand x^d=∑_r{d r} r!binom xr and apply Δ_1^kbinom xr=binomxr-k. It foll","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d94b7b7c59ff0407","tex":"\\sum_{r<d}","aliases":"∑_r<d sum","display":"\\sum_{r<d}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-aec6d0bc29467f34","tex":"\\sum_{r<n}","aliases":"∑_r<n sum","display":"\\sum_{r<n}","kind":"use","excerpt":"Pascal's identity proves the first product rule, and induction proves the higher rule. The intermediate binomial coefficients binom pr vanish in characteristic p. For the inverse, solve row n using the coefficient of b_n, which is g_0=1: b_n=1_n-∑_r<nbinom nr b_r g_n-r. Subtract the first and pth product rules for b⋆ g=1. The assumed identity for g","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f08116933b3b512d","tex":"\\sum_{r=0}^{D}","aliases":"∑_r=0^D sum","display":"\\sum_{r=0}^{D}","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-08b542392666bac8","tex":"\\sum_{r=0}^{d}","aliases":"∑_r=0^d sum","display":"\\sum_{r=0}^{d}","kind":"use","excerpt":"[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b3608ee04c017022","tex":"\\sum_{r=0}^{d-1}","aliases":"∑_r=0^d-1 sum","display":"\\sum_{r=0}^{d-1}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-660887c0abbbed2c","tex":"\\sum_{r=0}^k","aliases":"∑_r=0^k sum","display":"\\sum_{r=0}^k","kind":"use","excerpt":"ence identity is Δ_1^kfrac11+sn =frac(-s)^k k!∏_h=0^k(1+s(n+h)). It follows by induction, subtracting adjacent rational functions; each successive subtraction adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then prov","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4bb40d93da4fa77c","tex":"\\sum_{r=0}^{M-1}","aliases":"∑_r=0^M-1 sum","display":"\\sum_{r=0}^{M-1}","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-58d677395e4f323c","tex":"\\sum_{r=0}^{\\min(M,n/2)}","aliases":"∑_r=0^min(M,n/2) sum min","display":"\\sum_{r=0}^{\\min(M,n/2)}","kind":"use","excerpt":"[Odd modulus truncation] For every M≥0, every positive odd q, and every even n, E_n ≡ ∑_r=0^min(M,n/2) (-1)^rZ_2r+1 binom n2rq^2rS_n-2r(q) pmod q^2M+2. If pmid q and a=v_p(q), the same truncation is valid modulo p^2a(M+1).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ed8019b78e98d938","tex":"\\sum_{r=0}^{N}","aliases":"∑_r=0^N sum","display":"\\sum_{r=0}^{N}","kind":"use","excerpt":"Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-2753495994955ca2","tex":"\\sum_{r=0}^{n-1}","aliases":"∑_r=0^n-1 sum","display":"\\sum_{r=0}^{n-1}","kind":"use","excerpt":"tlasFigDir/#1.pdf AtlasHeadCompute the reciprocal before estimating itatlas:reciprocal For width N and address j, the normalized source and its reciprocal are G_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, G_N,j(t)^-1=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Multiplying the two series and clearing the factorials gives an exact triangular recurrence: C_N,j(0)=1, C_N,j(n)=-fra","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-8fbff1ef26dc212a","tex":"\\sum_{r=0}^{n/2}","aliases":"∑_r=0^n/2 sum","display":"\\sum_{r=0}^{n/2}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":6},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4daf9cd99c1a0246","tex":"\\sum_{r=0}^{p-1}","aliases":"∑_r=0^p-1 sum","display":"\\sum_{r=0}^{p-1}","kind":"use","excerpt":"[Finite inverse formula] If x has augmentation one and n=x-1, then x^-1 = ∑_r=0^p-1(-n)^r.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-021051c7510dc78f","tex":"\\sum_{r=0}^{x}","aliases":"∑_r=0^x sum","display":"\\sum_{r=0}^{x}","kind":"use","excerpt":"/3 in the one-prime example above loses the advertised digit; invertibility of C alone does not restore it. The original proof is retained under these hypotheses. Let Q=∏_p∈ Sp^L_p for a finite set of primes. Let C∈ M_d( Z) be invertible modulo Q, and choose H such that C^H=I+QB with B∈ M_d( Z). Then for every nonnegative integer x, a_Hx+j = ∑_r=0^","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a40c8a70cdd8f4bb","tex":"\\sum_{r=1}^{h}","aliases":"∑_r=1^h sum","display":"\\sum_{r=1}^{h}","kind":"use","excerpt":"[Rational ordinary generating function] In F_p[[z]], ∑_n≥0e_n^(m,p)z^n = 1+ frac ∑_r=1^he_r^(m,p)z^r 1-z^h.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4ae22d4d94a532ab","tex":"\\sum_{r=1}^{n}","aliases":"∑_r=1^n sum","display":"\\sum_{r=1}^{n}","kind":"use","excerpt":"[A denominator bound] Let D_N,j(n) = ∏_r=1^n binomNr+jNr, D_N,j(0)=1. Then D_N,j(n)C_N,j(n) ∈ Z. For every prime p, v_p(C_N,j(n)) ≥ - ∑_r=1^n v_pbinomNr+jNr.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5525cce418bdaded","tex":"\\sum_{r=1}^{n/2}","aliases":"∑_r=1^n/2 sum","display":"\\sum_{r=1}^{n/2}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5c88576010b1e038","tex":"\\sum_{r=k}^{\\min(D,Mk+s)}","aliases":"∑_r=k^min(D,Mk+s) sum min","display":"\\sum_{r=k}^{\\min(D,Mk+s)}","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b2e9009509f5bdd9","tex":"\\sum_{r=k}^{Mk+s}","aliases":"∑_r=k^Mk+s sum","display":"\\sum_{r=k}^{Mk+s}","kind":"use","excerpt":"[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c27b41f7e97bcb0e","tex":"\\sum_{r=s}^{N}","aliases":"∑_r=s^N sum","display":"\\sum_{r=s}^{N}","kind":"use","excerpt":"[Finite Mahler reconstruction] Given f(0),…,f(N), put c_r=Δ^rf(0). Then f(n)=∑_r=0^nc_rbinom nr (0≤ n≤ N). For every s with 0≤ s≤ N, Δ^sf(n) = ∑_r=s^Nc_rbinom nr-s (0≤ n≤ N-s).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-107ff376ef990f8a","tex":"\\sum_{r\\bmod N}","aliases":"∑_rbmod N sum bmod","display":"\\sum_{r\\bmod N}","kind":"use","excerpt":"[Complete residue packet system] For fixed N, the projectors P_N,r satisfy P_N,rP_N,s = δ_rsP_N,r and ∑_rbmod NP_N,r=I.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6a4250883c4968da","tex":"\\sum_{r\\ge0}","aliases":"∑_r≥0 sum ge","display":"\\sum_{r\\ge0}","kind":"use","excerpt":"[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator a","count":5},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-6b4cbee70b31b1b1","tex":"\\sum_{r\\geq0}","aliases":"∑_r≥0 sum geq","display":"\\sum_{r\\geq0}","kind":"use","excerpt":"The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the b","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-28fc50db4cc325e2","tex":"\\sum_{r\\geq1}","aliases":"∑_r≥1 sum geq","display":"\\sum_{r\\geq1}","kind":"use","excerpt":"The even source identity gives secht = sech^2(qt) ∑_n≥0S_2n(q)fract^2n(2n)!. Insert [source reference] and compare coefficients. For the inverse, multiply secht by cosh^2(qt) = 1+ ∑_r≥12^2r-1q^2rfract^2r(2r)!.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-ab6516e07d5eaa7c","tex":"\\sum_{s=0}^n","aliases":"∑_s=0^n sum","display":"\\sum_{s=0}^n","kind":"use","excerpt":"For every starting row, Δ_1^k C_0(n) = L(U^4nW^k) =∑_s=0^nbinom ns L(W^k+s). Each summand lies in 2^k k! Z_2, proving the factorial bound for address zero from row zero onward.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d4a1b621790c1e71","tex":"\\sum_{\\substack{0\\leq a<p\\\\2a+1\\neq p}}","aliases":"∑_substack0≤ a<p 2a+1≠ p sum substack leq neq","display":"\\sum_{\\substack{0\\leq a<p\\\\2a+1\\neq p}}","kind":"use","excerpt":"Fix an even address j∈{0,2,…,p-3}. For x∈ Z_p, define mathscr E_p,j^×(x) = ∑_r≥0 · (-1)^rZ_2r+1p^2r binom(p-1)x+j2r · × ∑_substack0≤ a<p 2a+1≠ p (-1)^a (2a+1)^j-2r ((2a+1)^p-1)^x.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-49a9c5b899546242","tex":"\\sum_{\\substack{0\\le r<M\\\\r\\bmod N=j}}","aliases":"∑_substack0≤ r<M rbmod N=j sum substack le r bmod","display":null,"kind":"use","excerpt":"0, · otherwise. The source proves idempotence for every selector, orthogonality for distinct standard residue labels, and, when N>0, ∑_j=0^N-1P_N,ja=a. If Nmid M and M>0, it also proves P_N,ja=∑_substack0≤ r<M rbmod N=jP_M,ra. The formal power-series versions are proved coefficientwise for actual mcodePowerSeries objects. There is no finite truncat","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e1f80fcd3e8646fe","tex":"\\sum_{\\substack{0\\leq r<M\\\\r\\equiv j\\pmod N}}","aliases":"∑_substack0≤ r<M r≡ jpmod N sum substack leq r equiv pmod","display":null,"kind":"use","excerpt":"[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M r≡ jpmod NP_M,ra. The same identities hold coefficientwise for formal power series.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-19c8aa9ad21cda67","tex":"\\sum_{\\substack{0\\le r\\le n\\\\w(r)<L}}","aliases":"∑_substack0≤ r≤ n w(r)<L sum substack le le w","display":null,"kind":"use","excerpt":"0, · w=∞. The infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, every term with weight at least L is divisible by p^L. A finite-sum proof yields p^L bigm| ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r -∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-r. The result is specialized to actual p-adic","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-0eec45661b8436b8","tex":"\\sum_{\\substack{0\\leq r\\leq n\\\\w(r)<L}}","aliases":"∑_substack0≤ r≤ n w(r)<L sum substack leq leq w","display":null,"kind":"use","excerpt":"[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d96e96aadd273752","tex":"\\sum_u","aliases":"∑_u sum","display":"\\sum_u","kind":"use","excerpt":"[Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a467b0ef76a3602c","tex":"\\sum_{u+v=r}","aliases":"∑_u+v=r sum","display":"\\sum_{u+v=r}","kind":"use","excerpt":"a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-07ab15ab7625ce3b","tex":"\\sum_{u+v=r+M}","aliases":"∑_u+v=r+M sum","display":"\\sum_{u+v=r+M}","kind":"use","excerpt":"a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-3cdd7357c745c72c","tex":"\\sum_{u=0}^{s}","aliases":"∑_u=0^s sum","display":"\\sum_{u=0}^{s}","kind":"use","excerpt":"insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1; thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-89c9a3fa4b6a762b","tex":"\\sum_{u\\in\\FF_p}","aliases":"∑_u∈FF_p ∑_u∈ F_p sum in FF sum in mathbb","display":"\\sum_{u\\in{\\mathbb F}_p}","kind":"use","excerpt":"The product law follows from [source reference]. To prove injectivity, suppose ∑_u∈ F_pc_uϵ_u=0. Looking at the coordinates n=0,1,…,p-1 gives a Vandermonde system with matrix (u^n)_0≤ n≤ p-1, u∈ F_p. Its determinant is the product of all differences between distinct field elements, so it is nonzero. Hence every c_u vanishes.","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-75b7b5b56c5ec55a","tex":"\\sum_{u\\neq0}","aliases":"∑_u≠0 sum neq","display":"\\sum_{u\\neq0}","kind":"use","excerpt":"For positive n, the zero mode contributes nothing. Every nonzero mode satisfies u^p-1=1. Hence b_n+p-1^(m,p) = ∑_u≠0c_uu^n+p-1 = ∑_u≠0c_uu^n = b_n^(m,p).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4dbf2eb498294f0b","tex":"\\sum_{v\\in G_m}","aliases":"∑_v∈ G_m sum in","display":"\\sum_{v\\in G_m}","kind":"use","excerpt":"[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dccea64f11621a8f","tex":"\\sum_{\\zeta\\in H_m}","aliases":"∑_ζ∈ H_m sum zeta in","display":"\\sum_{\\zeta\\in H_m}","kind":"use","excerpt":"[Subgroup average] The multiple selector has the mode expansion a^(m) = frac1m ∑_ζ∈ H_m ϵ_ζ.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a3ca240df62c5c7d","tex":"\\sum_{\\zeta\\in\\mu_m}","aliases":"∑_ζ∈μ_m sum zeta in mu","display":"\\sum_{\\zeta\\in\\mu_m}","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-cd74cb17b4132748","tex":"\\sum_\\nu","aliases":"∑_ν sum nu","display":"\\sum_\\nu","kind":"use","excerpt":"ng Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar multiplied by p^M, and is divisible by p^k+∑_ν v_p(H_ν). This proves ","count":9},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e1a1496e31ea3a4d","tex":"\\sum_{\\nu=1}^k","aliases":"∑_ν=1^k sum nu","display":"\\sum_{\\nu=1}^k","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of ","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c5c2f2566d8f7c1d","tex":"\\sum_{\\chi\\in\\mu_h}","aliases":"∑_χ∈μ_h sum chi in mu","display":"\\sum_{\\chi\\in\\mu_h}","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-472fd2fb48701e7d","tex":"\\sum_{\\ell\\geq1}","aliases":"∑_ℓ≥1 sum ell geq","display":"\\sum_{\\ell\\geq1}","kind":"use","excerpt":"[Taylor coefficient bounds and radii] For j=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at addres","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"The full-point theorem constructs b=(y,τ) and also proves b-a∈ p^M-sR^e+k∩ p^s+1R^e+k. Parameter equality b_T=τ is exact. Uniqueness concerns every root in the stated solved-coordinate ball with these parameters, not only roots obtained by the construction. No global uniqueness is claimed.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"To isolate the filtration argument, let p be any element of a commutative ring and let w: N→ N∪{∞}. Set p^[r]=p^r for finite r and p^[∞]=0. The latter notation means the coefficient vanishes.","count":4},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"· =(-1)^rfrac(r!)^2(2r+1)!. The last equality is the rational identity obtained by expanding ∫_0^1t^r(1-t)^r dt. This use of an ordinary finite integral makes no assertion about a positive or bounded p-adic measure. In particular, L(P_r(P))=(-1)^rfrac(r!)^22r+1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","excerpt":"· =(-1)^rfrac(r!)^2(2r+1)!. The last equality is the rational identity obtained by expanding ∫_0^1t^r(1-t)^r dt. This use of an ordinary finite integral makes no assertion about a positive or bounded p-adic measure. In particular, L(P_r(P))=(-1)^rfrac(r!)^22r+1.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus thro","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.","count":7},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"Choose a generator g of the cyclic additive group. Then F_p[ F_p^+] ≅ F_p[y]/(y^p-1) = F_p[y]/(y-1)^p. The augmentation ideal is generated by y-1, so its pth power is zero. Every element has the form a+n with a∈ F_p and n∈ J. It is invertible exactly when a≠0, which is the augmentation condition.","count":8},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"Let R be a commutative ring, a: N→ R, N≥1, and 0≤ j<N. Define (P_N,ja)(n)= a(n), · n≡ jpmod N,","count":71},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-27c543b1a137d08d","tex":"\\bigoplus_{a\\in A}","aliases":"⊕_a∈ A bigoplus in","display":"\\bigoplus_{a\\in A}","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-98b6ea417c450bdb","tex":"\\otimes_P","aliases":"⊗_P otimes","display":"\\otimes_P","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"[Lucas congruence] If a=a_0+a_1p+⋯ and b=b_0+b_1p+⋯ are base p expansions, then binom ab ≡ ∏_rbinoma_rb_r pmod p. This is the classical theorem of Lucas [source reference].","count":17},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-eef621a67d48c213","tex":"\\rangle^\\times","aliases":"⟩^× rangle times","display":"\\rangle^\\times","kind":"use","excerpt":"to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-25fb92dda41f7b3d","tex":"\\rangle_2","aliases":"⟩_2 rangle","display":"\\rangle_2","kind":"use","excerpt":"For p=3,ℓ=2,a=2, the checked values are ω_2(2)=8, ⟨2⟩_2=7, 8^2=1,quad8·7=2 in Z/9 Z. The unlifted representative 2 is not a square root of one modulo nine.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-e5ac643f289ff808","tex":"\\rangle_{2,\\ell}","aliases":"⟩_2,ℓ rangle ell","display":"\\rangle_{2,\\ell}","kind":"use","excerpt":"Why the mod-four branch is separate. Now let ℓ≥2 and a∈( Z/2^ℓ Z)^×. The actual units of Z/4 Z are exhausted by 1 and -1. Define ε_ℓ(a)= 1, · a≡1pmod4, -1, · a≡3pmod4, ⟨ a⟩_2,ℓ=ε_ℓ(a)a. The source uses ℓ=L+2. It proves a=ε_ℓ(a)⟨ a⟩_2,ℓ, ε_ℓ(a)∈{1,-1}, ⟨ a⟩_2,ℓ≡1pmod4. The sign is uniquely determined by the residue modulo four, and cancellation give","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-64103ba1ce4e3213","tex":"\\rangle_4","aliases":"⟩_4 rangle","display":"\\rangle_4","kind":"use","excerpt":"Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ϵ(a)∈{1,-1}, ⟨ a⟩_4≡1pmod4.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-5ba5167652cbe588","tex":"\\rangle_k","aliases":"⟩_k rangle","display":"\\rangle_k","kind":"use","excerpt":"pal_mul pageheading3. Precision compatibility and the dyadic signFinite refinements and Proposition 6.2's dyadic branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent fa","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-f3cc5b5bb7a34037","tex":"\\rangle_L","aliases":"⟩_L rangle","display":"\\rangle_L","kind":"use","excerpt":"[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ","count":2},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d4bbee17c7760c6a","tex":"\\rangle_\\ell","aliases":"⟩_ℓ rangle ell","display":"\\rangle_\\ell","kind":"use","excerpt":"ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ","count":3},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-d66576da6e758e08","tex":"\\rangle_\\ell^{\\,p^{\\ell-1}}","aliases":"⟩_ℓ^ p^ℓ-1 rangle ell ell","display":"\\rangle_\\ell^{\\,p^{\\ell-1}}","kind":"use","excerpt":"hism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy ⟨ a⟩_ℓ^ p^ℓ-1=1. At precision one the principal factor is exactly one.","count":1},{"work":"paper-ix","url":"notation/paper-ix.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"Integer representatives and residue classes at the next level must also be distinguished. For q≥1 and L≥0, the map S_q^d⟶ {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection, where widetilde t is any integer representative. Independence of the representative and injectivity both follow by cancelling q^L in integer divisibility; surje","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":424},{"work":"paper-x","url":"notation/paper-x.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":68},{"work":"paper-x","url":"notation/paper-x.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":91},{"work":"paper-x","url":"notation/paper-x.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":30},{"work":"paper-x","url":"notation/paper-x.html#symbol-b4d7ee29d2029812","tex":"\\bar x","aliases":"bar x bar","display":"\\bar x","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":26},{"work":"paper-x","url":"notation/paper-x.html#symbol-db1d2ae53aa18234","tex":"\\bar\\rho","aliases":"barρ bar rho","display":"\\bar\\rho","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":11},{"work":"paper-x","url":"notation/paper-x.html#symbol-a8ab7facc224789a","tex":"\\bar\\rho'","aliases":"barρ' bar rho","display":"\\bar\\rho'","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-80bb66e28ec49c6d","tex":"\\Cl","aliases":"Cl operatorname","display":"{\\operatorname{Cl}}","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-871baec058112b7f","tex":"\\Cl_G","aliases":"Cl_G Cl operatorname","display":"{\\operatorname{Cl}}_G","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-1112049ac7286dcd","tex":"\\Compat","aliases":"Compat C mathfrak","display":"{\\mathfrak C}","kind":"definition-passage","excerpt":"[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the f","count":10},{"work":"paper-x","url":"notation/paper-x.html#symbol-625573dcacdf7fff","tex":"\\Def","aliases":"Def operatorname","display":"{\\operatorname{Def}}","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-2b94f1088caec322","tex":"\\Def_{\\bar\\rho}","aliases":"Def_barρ Def bar rho operatorname bar rho","display":"{\\operatorname{Def}}_{\\bar\\rho}","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-912b7cbebb645a8c","tex":"\\Def_{\\bar\\rho'}","aliases":"Def_barρ' Def bar rho operatorname bar rho","display":"{\\operatorname{Def}}_{\\bar\\rho'}","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-aa4b8ca726217eac","tex":"\\dim_{\\mathrm{seed}}","aliases":"dim_seed dim mathrm","display":"\\dim_{\\mathrm{seed}}","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":49},{"work":"paper-x","url":"notation/paper-x.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":59},{"work":"paper-x","url":"notation/paper-x.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-a1cbd8c236499b7d","tex":"\\lim_{m\\to\\infty}","aliases":"lim_m→∞ lim to infty","display":"\\lim_{m\\to\\infty}","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-838a4fcee43f8fdc","tex":"\\log_q","aliases":"log_q log","display":"\\log_q","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"definition-passage","excerpt":"[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the f","count":23},{"work":"paper-x","url":"notation/paper-x.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":12},{"work":"paper-x","url":"notation/paper-x.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":177},{"work":"paper-x","url":"notation/paper-x.html#symbol-70d3338e989b4ddb","tex":"\\mathcal M_{\\bar x}","aliases":"M_bar x mathcal bar","display":"\\mathcal M_{\\bar x}","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-295577f481743cf7","tex":"\\Ocal","aliases":"Ocal O mathcal","display":"{\\mathcal O}","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":29},{"work":"paper-x","url":"notation/paper-x.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the f","count":41},{"work":"paper-x","url":"notation/paper-x.html#symbol-e70fd9df7027e2bf","tex":"P_v","aliases":"P_v","display":"P_v","kind":"definition-passage","excerpt":"[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the f","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":40},{"work":"paper-x","url":"notation/paper-x.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":54},{"work":"paper-x","url":"notation/paper-x.html#symbol-8b0150d4ec261493","tex":"\\red","aliases":"red operatorname","display":"{\\operatorname{red}}","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-238b4e38e4a9a70d","tex":"\\red_A","aliases":"red_A red operatorname","display":"{\\operatorname{red}}_A","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":29},{"work":"paper-x","url":"notation/paper-x.html#symbol-d0c86daacd8ccb79","tex":"\\mathrm{seed}","aliases":"seed mathrm","display":"\\mathrm{seed}","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-0ef3dbe5a13282d7","tex":"\\srank","aliases":"srank operatorname","display":"{\\operatorname{srank}}","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-31658be192e9af55","tex":"\\srank_G","aliases":"srank_G srank operatorname","display":"{\\operatorname{srank}}_G","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":51},{"work":"paper-x","url":"notation/paper-x.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-54815db7d2265570","tex":"T^m","aliases":"T^m","display":"T^m","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-8ef43bcf3d114a62","tex":"\\mathcal T_A","aliases":"T_A mathcal","display":"\\mathcal T_A","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":11},{"work":"paper-x","url":"notation/paper-x.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the f","count":34},{"work":"paper-x","url":"notation/paper-x.html#symbol-b30f1f16c8acc2ca","tex":"\\bm w","aliases":"w bm","display":"\\bm w","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":38},{"work":"paper-x","url":"notation/paper-x.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":61},{"work":"paper-x","url":"notation/paper-x.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":109},{"work":"paper-x","url":"notation/paper-x.html#symbol-f3be737b42076753","tex":"X_m","aliases":"X_m","display":"X_m","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":16},{"work":"paper-x","url":"notation/paper-x.html#symbol-449cba2c3125b122","tex":"X_{\\lambda,m}","aliases":"X_λ,m lambda","display":"X_{\\lambda,m}","kind":"definition-passage","excerpt":"ic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of adm","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":18},{"work":"paper-x","url":"notation/paper-x.html#symbol-a04e13cebbe7a0ff","tex":"Z_{\\bm w}","aliases":"Z_ w bm","display":"Z_{\\bm w}","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-d74d92bf91d59a7f","tex":"Z_{\\bar x}","aliases":"Z_bar x bar","display":"Z_{\\bar x}","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":18},{"work":"paper-x","url":"notation/paper-x.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"efinition[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C)","count":40},{"work":"paper-x","url":"notation/paper-x.html#symbol-cb28cf0122b8897d","tex":"\\varpi","aliases":"ϖ varpi","display":"\\varpi","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":30},{"work":"paper-x","url":"notation/paper-x.html#symbol-0bff53c563d0671f","tex":"\\varpi^m","aliases":"ϖ^m varpi","display":"\\varpi^m","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":14},{"work":"paper-x","url":"notation/paper-x.html#symbol-6006a1a7260e2f47","tex":"\\varpi_\\lambda^m","aliases":"ϖ_λ^m varpi lambda","display":"\\varpi_\\lambda^m","kind":"definition-passage","excerpt":"at consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduct","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":39},{"work":"paper-x","url":"notation/paper-x.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":90},{"work":"paper-x","url":"notation/paper-x.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":30},{"work":"paper-x","url":"notation/paper-x.html#symbol-a0a9cffbc383b69e","tex":"\\sum_{m\\ge1}","aliases":"∑_m≥1 sum ge","display":"\\sum_{m\\ge1}","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":31},{"work":"paper-x","url":"notation/paper-x.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":30},{"work":"paper-x","url":"notation/paper-x.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":12},{"work":"paper-x","url":"notation/paper-x.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-06b373fe54fc27d1","tex":"A^3","aliases":"A^3","display":"A^3","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1c314d68691af653","tex":"\\absn","aliases":"absn","display":null,"kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"introduction-passage","excerpt":"· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J)","count":20},{"work":"paper-x","url":"notation/paper-x.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","excerpt":"Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}","count":70},{"work":"paper-x","url":"notation/paper-x.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":23},{"work":"paper-x","url":"notation/paper-x.html#symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"introduction-passage","excerpt":"Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":10},{"work":"paper-x","url":"notation/paper-x.html#symbol-4deeb62b76263c34","tex":"\\mathrm{car}","aliases":"car mathrm","display":"\\mathrm{car}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":33},{"work":"paper-x","url":"notation/paper-x.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":79},{"work":"paper-x","url":"notation/paper-x.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"qquad f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar constants denote scalar matrices. The Lean proof establishes at least one additional ideal digit at each step, constructs the resulting compatible family of finite-level matrices, and u","count":55},{"work":"paper-x","url":"notation/paper-x.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":19},{"work":"paper-x","url":"notation/paper-x.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"introduction-passage","excerpt":"Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-bda763105cb3e335","tex":"G_{J,\\chi}","aliases":"G_J,χ chi","display":"G_{J,\\chi}","kind":"introduction-passage","excerpt":"· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J)","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-0abf0d17398d7d33","tex":"G_{\\Q}","aliases":"G_Q G_ Q Q mathbf","display":"G_{{\\mathbf Q}}","kind":"introduction-passage","excerpt":"Block 21 completes this restricted arithmetic interpretation and its conditional completion. The packet is defined by one fixed quotient; it is not the collection of all quadratic characters of G_ Q. No local ramification criterion, Hodge--Tate weight, Weil--Deligne attachment, or cross-coefficient local invariant was established by Block 21. The f","count":10},{"work":"paper-x","url":"notation/paper-x.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":59},{"work":"paper-x","url":"notation/paper-x.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":111},{"work":"paper-x","url":"notation/paper-x.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":42},{"work":"paper-x","url":"notation/paper-x.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":47},{"work":"paper-x","url":"notation/paper-x.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":45},{"work":"paper-x","url":"notation/paper-x.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":45},{"work":"paper-x","url":"notation/paper-x.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":19},{"work":"paper-x","url":"notation/paper-x.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":61},{"work":"paper-x","url":"notation/paper-x.html#symbol-8b12673ac5aded11","tex":"\\Nm","aliases":"Nm N operatorname","display":"{\\operatorname N}","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":17},{"work":"paper-x","url":"notation/paper-x.html#symbol-ea4b9089982f8df8","tex":"\\Nm_{K/\\mathbb Q}","aliases":"Nm_K/ Q N_K/ Q Nm mathbb operatorname mathbb","display":"{\\operatorname N}_{K/\\mathbb Q}","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-94c28f047e602f0f","tex":"\\mathcal O_L","aliases":"O_L mathcal","display":"\\mathcal O_L","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-f35ce0db28d3fb4f","tex":"\\OO","aliases":"OO OO O mathcal","display":null,"kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":16},{"work":"paper-x","url":"notation/paper-x.html#symbol-83c44e1ef141b100","tex":"\\OO_K","aliases":"OO_K O_K OO mathcal","display":"{\\mathcal O}_K","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1918b7c23a15b567","tex":"\\overline{\\Q}","aliases":"overlineQ overline Q overline mathbf","display":"\\overline{{\\mathbf Q}}","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"introduction-passage","excerpt":"arison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"te-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maxim","count":107},{"work":"paper-x","url":"notation/paper-x.html#symbol-a6123ae670e2c050","tex":"\\mathrm{Prop}","aliases":"Prop mathrm","display":"\\mathrm{Prop}","kind":"introduction-passage","excerpt":"Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q mathbf","display":"{\\mathbf Q}","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":27},{"work":"paper-x","url":"notation/paper-x.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":10},{"work":"paper-x","url":"notation/paper-x.html#symbol-ff7eb8f0d00f446f","tex":"\\mathfrak Q","aliases":"Q mathfrak","display":"\\mathfrak Q","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":49},{"work":"paper-x","url":"notation/paper-x.html#symbol-a62a596621079394","tex":"\\RR","aliases":"RR R mathbb","display":"{\\mathbb R}","kind":"introduction-passage","excerpt":"Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"introduction-passage","excerpt":"egree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d unde","count":34},{"work":"paper-x","url":"notation/paper-x.html#symbol-f544e1fddefb59a0","tex":"t^d","aliases":"t^d","display":"t^d","kind":"introduction-passage","excerpt":"egree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d unde","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-f4d0bbae2783fe43","tex":"T_{C,\\chi}","aliases":"T_C,χ chi","display":"T_{C,\\chi}","kind":"introduction-passage","excerpt":"· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J)","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":31},{"work":"paper-x","url":"notation/paper-x.html#symbol-d849c4e507602c15","tex":"U_{0,\\lambda}","aliases":"U_0,λ lambda","display":"U_{0,\\lambda}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-adf10d370b43fb83","tex":"V_{\\mathrm{car},\\lambda}","aliases":"V_car,λ mathrm lambda","display":"V_{\\mathrm{car},\\lambda}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J)","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":18},{"work":"paper-x","url":"notation/paper-x.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ ZZ Z mathbb","display":null,"kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":25},{"work":"paper-x","url":"notation/paper-x.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))} · =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":29},{"work":"paper-x","url":"notation/paper-x.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":33},{"work":"paper-x","url":"notation/paper-x.html#symbol-936d7991f33298f9","tex":"\\rho_i","aliases":"ρ_i rho","display":"\\rho_i","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-d1ef5788f5c903d0","tex":"\\rho_{i,\\lambda}^{\\oplus c_i}","aliases":"ρ_i,λ^⊕ c_i rho lambda oplus","display":"\\rho_{i,\\lambda}^{\\oplus c_i}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c7d25253bfbf71b9","tex":"\\rho_{i,\\lambda}^{\\oplus m_i}","aliases":"ρ_i,λ^⊕ m_i rho lambda oplus","display":"\\rho_{i,\\lambda}^{\\oplus m_i}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"introduction-passage","excerpt":"· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J)","count":43},{"work":"paper-x","url":"notation/paper-x.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))} · =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G","count":39},{"work":"paper-x","url":"notation/paper-x.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a n","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":10},{"work":"paper-x","url":"notation/paper-x.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement an","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"residual root + positive-depth deformation coordinates.","count":87},{"work":"paper-x","url":"notation/paper-x.html#symbol-3be3a4a3a44d636c","tex":"\\bm a","aliases":"a bm","display":"\\bm a","kind":"use","excerpt":"For target (0,1,1), every pair of conditions is occupied: (0,1,0) witnesses the first two, (0,0,1) the first and third, and (1,1,1) the second and third. But the triple itself is absent. Multiplying all three selectors gives the zero matrix. E_ a=∏_i=1^3 I-T_i, · a_i=0, T_i, · a_i=1. Tr(E_ a)=rank(E_ a). The checks compute all eight joint projector","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (3,4.7) N(a+bu)=a^2-2b^2pmod5; path[draw=black!30,fill=white] (1.45,3.38) rectangle (2.07,4); node[font=small,align=center,font=small] at (1.76,3.69) 0; path[draw=black!30,fill=black!8] (2.07,3.38)","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-bce8bfd89f58273e","tex":"a^d","aliases":"a^d","display":"a^d","kind":"use","excerpt":"vpage24A nontrivial character may still be exceptionalTHE EXACT SCALAR TESTMain text: §9; scalar/Mellin checkpoint: scalar-power and nonscalar-witness boundaries. A scalar a∈FF_5^× acts on a degree-d algebra with determinant a^d. Therefore the scalar cancellation route tests χ(a^d), not merely χ(a). vfig24_scalar_exception headA failed sufficient t","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-61bf579da1b5c924","tex":"a^j","aliases":"a^j","display":"a^j","kind":"use","excerpt":"Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the represent","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-66be131002b8b5af","tex":"A_C","aliases":"A_C","display":"A_C","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-04ab0d31f8d5c7fc","tex":"a_{i,1}","aliases":"a_i,1","display":"a_{i,1}","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-2409747eb360eb47","tex":"a_{i,2}","aliases":"a_i,2","display":"a_{i,2}","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5590ebce8ca407f5","tex":"a_{i,j}","aliases":"a_i,j","display":"a_{i,j}","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"scriptsize] at (5.96,4.45) 3^2; node[font=small,align=center,font=scriptsize] at (7.19,4.45) 3^3; node[font=small,align=center,font=scriptsize] at (8.42,4.45) 3^4; node[font=small,align=center,font=scriptsize] at (9.65,4.45) 3^5; node[font=small,align=center,font=scriptsize] at (10.88,4.45) 3^6; node[font=small,align=center,anchor=east] at (1.95,3.","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-9c5c098d009032a0","tex":"a_{m-1}","aliases":"a_m-1","display":"a_{m-1}","kind":"use","excerpt":"The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-db660538430f2d4b","tex":"A_p","aliases":"A_p","display":"A_p","kind":"use","excerpt":"For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positi","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-2277836fd5789d74","tex":"A_R","aliases":"A_R","display":"A_R","kind":"use","excerpt":"Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a charac","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-65c56b1be2f0aadb","tex":"A_T","aliases":"A_T","display":"A_T","kind":"use","excerpt":"[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet i","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-bdbd1a428e5dd1c0","tex":"\\ad","aliases":"ad operatorname","display":"{\\operatorname{ad}}","kind":"use","excerpt":"[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-1fc76a250d22a8eb","tex":"\\operatorname{Aut}","aliases":"Aut operatorname","display":"\\operatorname{Aut}","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-2619e4f89a651f69","tex":"\\operatorname{Aut}_K","aliases":"Aut_K operatorname","display":"\\operatorname{Aut}_K","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":32},{"work":"paper-x","url":"notation/paper-x.html#symbol-42648425616e1ec3","tex":"\\bm b","aliases":"b bm","display":"\\bm b","kind":"use","excerpt":"C m= b · A finite trace system for carrier multiplicities. · Integrality, positivity, or available capacity automatically.","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-546f2147988ed7b9","tex":"\\boldsymbol b","aliases":"b boldsymbol","display":"\\boldsymbol b","kind":"use","excerpt":"6.55,3.45) 2; path[draw=black!30,fill=white] (6.15,2.25) rectangle (6.95,3.05); node[font=small,align=center,font=small] at (6.55,2.65) 0; path[draw=black!30,fill=black!8] (6.15,1.45) rectangle (6.95,2.25); node[font=small,align=center,font=small] at (6.55,1.85) 1; node[font=small,align=center] at (7.45,2.6) =; node[font=small,align=center,font=sma","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (3,4.7) N(a+bu)=a^2-2b^2pmod5; path[draw=black!30,fill=white] (1.45,3.38) rectangle (2.07,4); node[font=small,align=center,font=small] at (1.76,3.69) 0; path[draw=black!30,fill=black!8] (2.07,3.38)","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-acb79e87d28a764d","tex":"B^r","aliases":"B^r","display":"B^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-8ff3305ee67cffa6","tex":"B^\\times","aliases":"B^× times","display":"B^\\times","kind":"use","excerpt":"hmetic completion (Block 21). Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"The assembly identifies the residue sum with the sum over actual generators, recombines the sign portions, and uses Y=t^d to obtain absnG_J,χ(Y)≤ B_0(Y^1-1/d+1). Equation~[source reference] then permits division by the nonzero factor #μ(K)χ( N Jbmod m). The final theorem supplies its finite means and counting estimates internally. For d=1 the expon","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-3caf6ca5bc1139c0","tex":"B_j^r","aliases":"B_j^r","display":"B_j^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5902c64cbd2a059e","tex":"B_w","aliases":"B_w","display":"B_w","kind":"use","excerpt":"Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hy","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-d0086c8c28eb495d","tex":"\\bar E","aliases":"bar E bar","display":"\\bar E","kind":"use","excerpt":"Lifting a summand inside a supplied finite-group carrier (Block 18). Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=b","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-25e0b45f28c87b71","tex":"\\bar E^2","aliases":"bar E^2 bar","display":"\\bar E^2","kind":"use","excerpt":"Lifting a summand inside a supplied finite-group carrier (Block 18). Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=b","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-a336c88f130bb2fe","tex":"\\bar E_i","aliases":"bar E_i bar","display":"\\bar E_i","kind":"use","excerpt":"Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-5bd492e4ea78c5fe","tex":"\\bigsqcup","aliases":"bigsqcup","display":null,"kind":"use","excerpt":"[Residual partition] For every coefficient algebra A, M(A) = bigsqcup_bar x∈ M(k) M_bar x(A).","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-d510e5c9363b78bf","tex":"\\bigsqcup_{\\bar x}","aliases":"bigsqcup_bar x bigsqcup bar","display":null,"kind":"use","excerpt":"For example, the addresses 1,4,7,10,…,25 may be deeply connected by higher-digit changes. They remain disjoint from the addresses reducing to zero. In a genuine deformation problem, the same conclusion applies to operations whose reduction is the identity, not to every operation that happens to be called a deformation. red(x+3h)=red(x), M(A)=bigsqc","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-bf287c25ac4534b1","tex":"\\bigsqcup_{\\bar x\\in\\mathcal M(k)}","aliases":"bigsqcup_bar x∈ M(k) bigsqcup bar in mathcal","display":null,"kind":"use","excerpt":"[Residual partition] For every coefficient algebra A, M(A) = bigsqcup_bar x∈ M(k) M_bar x(A).","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-8e3c0fce1b7a60ae","tex":"\\bullet","aliases":"bullet","display":"\\bullet","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-2ed728a2bb8f27d6","tex":"\\bm c","aliases":"c bm","display":"\\bm c","kind":"use","excerpt":"vpage16A solution is not yet a summandINTEGRAL, NONNEGATIVE, WITHIN CAPACITYMain text: Theorem 9.2 and its multiplicity-recovery example. Keep the same separating matrix C and suppose the available carrier has capacities c=(2,1,1). A recovered vector describes a genuine sub-sum of that carrier only if each entry is an integer and 0≤ m_i≤ c_i. vfig1","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-e7e42689204088d5","tex":"C^0","aliases":"C^0","display":"C^0","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-bb4b39a0fa3e20d1","tex":"C^1","aliases":"C^1","display":"C^1","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-6bfba5cbf12e1727","tex":"C^\\bullet","aliases":"C^bullet bullet","display":"C^\\bullet","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-40378867274e1130","tex":"C^i","aliases":"C^i","display":"C^i","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-0af3bfadfba7f24b","tex":"C^{i+1}","aliases":"C^i+1","display":"C^{i+1}","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5b3b60bc2dabf087","tex":"C^{i-1}","aliases":"C^i-1","display":"C^{i-1}","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-dd0b3056e758361c","tex":"C^r","aliases":"C^r","display":"C^r","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"Finite deletion can destroy density. In the actual discrete group C_2={1,a}, set s(0)=a and s(m+1)=1. The original samples are surjective and satisfy quotient coverage. Removing index zero leaves only the identity, so both density and quotient coverage fail. The actual rational sign and trivial characters agree on the remaining samples but differ g","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-68ec3f96f692ce11","tex":"C_j^\\bullet","aliases":"C_j^bullet bullet","display":"C_j^\\bullet","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-229173948b9deb09","tex":"C_j^r","aliases":"C_j^r","display":"C_j^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1e62e3524995dc9d","tex":"c_K","aliases":"c_K","display":"c_K","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"There is also a completion theorem without chosen compatible characters. Suppose that R/I^m is finite for every m≥1, and that C_m⊆ P_q(R/I^m) are allowed subsets such that every reduction of an allowed member is allowed. Then · ∃χ∈ P_q(R), χ is continuous and χbmod I^m∈ C_m for all m≥1","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"use","excerpt":"The estimate is proved from the actual boundary charts and the fixed-lattice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-34dfb5e598913303","tex":"\\CC^{\\times}","aliases":"CC^× C^× CC times mathbb times","display":"{\\mathbb C}^{\\times}","kind":"use","excerpt":"Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hy","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-d831feefac3295be","tex":"\\operatorname{char}","aliases":"char operatorname","display":"\\operatorname{char}","kind":"use","excerpt":"Let O be a complete discrete valuation ring with uniformizer ϖ, maximal ideal (ϖ), residue field k= O/(ϖ), |k|=q, and fraction field K. When a characteristic-zero conclusion is intended, assume additionally that charK=0; the general counting statements do not require that restriction. Let M be a moduli functor on complete local Noetherian O-algebra","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f39bbad3b3a0ba30","tex":"\\covol","aliases":"covol operatorname","display":"{\\operatorname{covol}}","kind":"use","excerpt":"There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","excerpt":"ing every remaining coordinate with zero. Define r:U_c→ U_m by restriction to the selected copies. Lean proves rj=1_U_m, jρ_m(g)=ρ_c(g)j, rρ_c(g)=ρ_m(g)r. The image of j is exactly the subspace whose unused coordinates vanish. Consequently jr is an equivariant idempotent. A supplied equivariant isomorphism D:Wxrightarrow∼U_c gives the split maps D^","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"C^bullet,d · Cochains and a differential with d^2=0. · Its arithmetic identification with a Galois problem.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-535d0321b9f2a373","tex":"D^\\bullet","aliases":"D^bullet bullet","display":"D^\\bullet","kind":"use","excerpt":"Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-70ffa9c8bbca2ebd","tex":"D_+","aliases":"D_+","display":"D_+","kind":"use","excerpt":"The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no bound","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ff76dfe8b26b12f2","tex":"D_-","aliases":"D_-","display":"D_-","kind":"use","excerpt":"The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no bound","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-63ed82a00b685c9b","tex":"D_A","aliases":"D_A","display":"D_A","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-5d2e949d109d889d","tex":"d_{i-1}","aliases":"d_i-1","display":"d_{i-1}","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-125484ee05f69b84","tex":"D_i^{\\oplus m_i}","aliases":"D_i^⊕ m_i oplus","display":"D_i^{\\oplus m_i}","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-e09c2eb604e5806c","tex":"D_j^\\bullet","aliases":"D_j^bullet bullet","display":"D_j^\\bullet","kind":"use","excerpt":"Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-9b67f0c58ecb3837","tex":"d_{r-1}","aliases":"d_r-1","display":"d_{r-1}","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-de7bc3aee118c968","tex":"d_s","aliases":"d_s","display":"d_s","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd","display":null,"kind":"use","excerpt":"IN ENDPOINT MODELMain text: §9; retained scalar/Mellin continuation: summatory measurability and first-order Mellin control. Use the explicit envelope A(x)=x^1/2 for x≥1. It is not an ideal-counting function. It is a model showing why a power saving can make a Mellin expression regular at s=1. vfig27_mellin headCompute the endpoint rather than just","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, d","count":12},{"work":"paper-x","url":"notation/paper-x.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag","display":null,"kind":"use","excerpt":"A successful answer to one question does not supply the next. vfig01_three_questions headThree models to keep in view The scalar equation x^2+1=0 has the root 1 modulo two, but no root modulo four. Over powers of five, the residues 2,7,57,182 form the beginning of a compatible branch. Neither fact constructs an automorphic carrier. In a supplied fi","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"An actual Z_3 example applies the class-level completion theorem for the cyclic group of order two and polynomial X^2-1, producing a nonidentity generator. Another checked example has compatible classes represented by diag(1,-1) modulo 3 and diag(-1,1) modulo 9: these chosen matrices disagree under reduction, and an actual swap of basis repairs the","count":11},{"work":"paper-x","url":"notation/paper-x.html#symbol-346650cbdd3f09a1","tex":"\\operatorname{disc}","aliases":"disc operatorname","display":"\\operatorname{disc}","kind":"use","excerpt":"The exceptional primes are controlled by the actual field discriminant: S_L={p rational prime:pmiddisc(L)}. This set is finite. For p∉ S_L, every prime of O_L above (p) is unramified. Lean proves that the actual inertia cardinality is the ramification index. It follows that inertia is trivial and the arithmetic Frobenius at a fixed finite prime is ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"rability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function h","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":70},{"work":"paper-x","url":"notation/paper-x.html#symbol-b87f37426ee94627","tex":"E^2","aliases":"E^2","display":"E^2","kind":"use","excerpt":"d for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral sum","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9e16a16a355956ba","tex":"E_{\\bm a}","aliases":"E_ a bm","display":"E_{\\bm a}","kind":"use","excerpt":"For target (0,1,1), every pair of conditions is occupied: (0,1,0) witnesses the first two, (0,0,1) the first and third, and (1,1,1) the second and third. But the triple itself is absent. Multiplying all three selectors gives the zero matrix. E_ a=∏_i=1^3 I-T_i, · a_i=0, T_i, · a_i=1. Tr(E_ a)=rank(E_ a). The checks compute all eight joint projector","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion k","count":13},{"work":"paper-x","url":"notation/paper-x.html#symbol-2d63255e26d245d2","tex":"e_{01}","aliases":"e_01","display":"e_{01}","kind":"use","excerpt":"For the continuous Z_2 carrier ρ(s)=diag(1,-1,1), conjugate the three coordinate projectors by 1+2e_01 and reduce modulo 4. They form a complete equivariant family. The general theorem constructs three nonzero integral summands retaining this family modulo 2. No exact equivariant family retains the whole prescribed family modulo 4. Separate checked","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"An explicit action of C_2× Z has ν(φ)=1/3, F=diag(2,6,18) and Ne_1=e_0, Ne_2=e_1, Ne_0=0. Trivial inertia gives local dimension one and denominator 1-2T; scalar sign inertia gives dimension zero and denominator one. The entire F and N are equal in these two examples, and N^2≠0. Thus inertia cannot be omitted. Conversely, unramified chains of length","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"An explicit action of C_2× Z has ν(φ)=1/3, F=diag(2,6,18) and Ne_1=e_0, Ne_2=e_1, Ne_0=0. Trivial inertia gives local dimension one and denominator 1-2T; scalar sign inertia gives dimension zero and denominator one. The entire F and N are equal in these two examples, and N^2≠0. Thus inertia cannot be omitted. Conversely, unramified chains of length","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"0 · 0 · 1, ρ(d)=diag(1,3,9). Reversing the factor order or dropping the quadratic term gives checked failures. For the length-four chain, the quadratic logarithm has erroneous e_3-to-e_0 coefficient -1/3, so its cubic restriction is substantive. A zero-parameter sample also fails to recover N.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-b9e2b1f616f1340a","tex":"E_i^2","aliases":"E_i^2","display":"E_i^2","kind":"use","excerpt":"Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion k","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-a3fd533b471b8be8","tex":"e_{j-1}","aliases":"e_j-1","display":"e_{j-1}","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion k","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b7ce4855f02a711f","tex":"e_{n-1}","aliases":"e_n-1","display":"e_{n-1}","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion k","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0dd6372c1de2074b","tex":"E_p","aliases":"E_p","display":"E_p","kind":"use","excerpt":"For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positi","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-1b09d4efdcdc1f39","tex":"e_p","aliases":"e_p","display":"e_p","kind":"use","excerpt":"For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positi","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-3a8e0b23212d1b1a","tex":"e_T","aliases":"e_T","display":"e_T","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-8030d3304fb10474","tex":"e_{T,a}","aliases":"e_T,a","display":"e_{T,a}","kind":"use","excerpt":"Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimensi","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5dd4a91606e9704e","tex":"e_{U,a}","aliases":"e_U,a","display":"e_{U,a}","kind":"use","excerpt":"Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimensi","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-74e2d51d64f7fad4","tex":"E_\\lambda","aliases":"E_λ lambda","display":"E_\\lambda","kind":"use","excerpt":"ion] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-70a9953dd9bd9b22","tex":"\\operatorname{End}","aliases":"End operatorname","display":"\\operatorname{End}","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-4ae4396c22c4c43e","tex":"\\operatorname{End}_K","aliases":"End_K operatorname","display":"\\operatorname{End}_K","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the represent","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.","count":37},{"work":"paper-x","url":"notation/paper-x.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F mathbf","display":"{\\mathbf F}","kind":"use","excerpt":"[Finite packet complex] The exact modular test uses two invariant subcomplexes over F_5. Their cohomology dimensions are (0,0,0), (1,2,1), and the reconstructed total is dim(H^0,H^1,H^2)=(1,2,1). The packet-sum residual is zero.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); draw[black!25] (4.55,0) -- (4.55,4.45); draw[black!25] (9.25,0) -- (9.25,4.45); node[font=small,align=center] at (2.1,4.35) A residual root; node[font=small,align=center] at (2.1,3.55) x^2+1=0 over F_2; path[draw,fill=black!12] (2.1,2.55) ci","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-b6bb9a2af76d66fc","tex":"\\mathbf F","aliases":"F mathbf","display":"\\mathbf F","kind":"use","excerpt":"New checked realization of the finite pattern. The Block 04 implementation separately uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linearly","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","excerpt":",align=center,font=small] at (6.135,2.265) 0; path[draw=black!30,fill=white] (6.48,1.92) rectangle (7.17,2.61); node[font=small,align=center,font=small] at (6.825,2.265) 0; path[draw=black!30,fill=white] (7.17,1.92) rectangle (7.86,2.61); node[font=small,align=center,font=small] at (7.515,2.265) 0; node[font=small,align=center,anchor=east,font=scri","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"48,1.92) rectangle (7.17,2.61); node[font=small,align=center,font=small] at (6.825,2.265) 0; path[draw=black!30,fill=white] (7.17,1.92) rectangle (7.86,2.61); node[font=small,align=center,font=small] at (7.515,2.265) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (4.95,2.955) f_0; node[font=small,align=center,anchor=east,font=scrip","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); draw[black!25] (4.55,0) -- (4.55,4.45); draw[black!25] (9.25,0) -- (9.25,4.45); node[font=small,align=center] at (2.1,4.35) A residual root; node[font=small,align=center] at (2.1,3.55) x^2+1=0 over F_2; path[draw,fill=black!12] (2.1,2.55) ci","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9e15e0ac8ef6cc9a","tex":"\\mathbb F_{25}","aliases":"F_25 mathbb","display":"\\mathbb F_{25}","kind":"use","excerpt":"gn=center,font=scriptsize] at (1.76,4.23) b=0; node[font=small,align=center,font=scriptsize] at (2.38,4.23) b=1; node[font=small,align=center,font=scriptsize] at (3,4.23) b=2; node[font=small,align=center,font=scriptsize] at (3.62,4.23) b=3; node[font=small,align=center,font=scriptsize] at (4.24,4.23) b=4; node[font=small,align=center] at (9.6,4.25","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-54c7a07f66f1253f","tex":"\\mathbb F_3","aliases":"F_3 mathbb","display":"\\mathbb F_3","kind":"use","excerpt":"[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center] at (2.2,4.3) Scalar over F_3; node[font=small,align=center] at (2.2,3.25) x=0,1,2; node[font=small,align=center] at (2.2,2.35) x^2+1=1,2,2; node[font=small,align=center] at (2.2,1.25) no solution; draw[black!25]","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-0a27cb068825ed9a","tex":"\\F_5","aliases":"F_5 F mathbf","display":"{\\mathbf F}_5","kind":"use","excerpt":"[Finite packet complex] The exact modular test uses two invariant subcomplexes over F_5. Their cohomology dimensions are (0,0,0), (1,2,1), and the reconstructed total is dim(H^0,H^1,H^2)=(1,2,1). The packet-sum residual is zero.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f3ee6a8ba9575903","tex":"\\mathbb F_5","aliases":"F_5 mathbb","display":"\\mathbb F_5","kind":"use","excerpt":"35,0.67) rectangle (7.8,1.53); node[font=small,align=center,font=small] at (7.075,1.1) 2; path[draw=black!30,fill=black!8] (7.8,0.67) rectangle (9.25,1.53); node[font=small,align=center,font=small] at (8.525,1.1) 6; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,2.82) Z; node[font=small,align=center,anchor=east,font=scriptsize] ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-75c45938e81d6f92","tex":"\\mathbf F_5^2","aliases":"F_5^2 mathbf","display":"\\mathbf F_5^2","kind":"use","excerpt":"New checked realization of the finite pattern. The Block 04 implementation separately uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linearly","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-ecbf3d4265f3e244","tex":"\\mathbf F_5^4","aliases":"F_5^4 mathbf","display":"\\mathbf F_5^4","kind":"use","excerpt":"New checked realization of the finite pattern. The Block 04 implementation separately uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linearly","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-23af4eb7208f9cad","tex":"\\mathbb F_7","aliases":"F_7 mathbb","display":"\\mathbb F_7","kind":"use","excerpt":"2; path[draw=black!30,fill=black!8] (7.8,0.67) rectangle (9.25,1.53); node[font=small,align=center,font=small] at (8.525,1.1) 6; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,2.82) Z; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,1.96) F_5; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-628bef06c6d57486","tex":"\\mathbb F_9","aliases":"F_9 mathbb","display":"\\mathbb F_9","kind":"use","excerpt":"!30,fill=black!8] (6.2,1.85) rectangle (7,2.65); node[font=small,align=center,font=small] at (6.6,2.25) 1; path[draw=black!30,fill=white] (7,1.85) rectangle (7.8,2.65); node[font=small,align=center,font=small] at (7.4,2.25) 0; node[font=small,align=center] at (7,0.85) A^2+I=0; draw[black!25] (9.3,0.3) -- (9.3,4.45); node[font=small,align=center] at","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-31d7232e6fa917d3","tex":"F_i","aliases":"F_i","display":"F_i","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilp","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-e336703b8375549b","tex":"F_i^{-1}","aliases":"F_i^-1","display":"F_i^{-1}","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilp","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilp","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-9da98cb9c6f0ae74","tex":"\\mathbb F_p","aliases":"F_p mathbb","display":"\\mathbb F_p","kind":"use","excerpt":"nter,font=scriptsize] at (4.85,4) 19; node[font=small,align=center,font=scriptsize] at (4.85,3.3) 0; node[font=small,align=center,font=scriptsize] at (5.45,4) 23; node[font=small,align=center,font=scriptsize] at (5.45,3.3) 0; node[font=small,align=center] at (3.4,4.65) odd primes p; node[font=small,align=center,font=scriptsize] at (3.4,2.7) roots o","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-a785141dca19cd38","tex":"F_{\\chi_i}","aliases":"F_χ_i chi","display":"F_{\\chi_i}","kind":"use","excerpt":"For a finite family of nontrivial ordinary class characters, suppose functions p,Z satisfy h p(h)→ P≠0, h Z(h)→ Q≠0, and, for all sufficiently small h>0, Z(h)=(p(h)∏_i F_χ_i(h))^e, e∈ N. The checked real-axis criterion then gives e=1 and F_χ_i(0)≠0 for every i. The pole limits and product identity remain hypotheses of this application.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-cc7582cb4d0dc2f3","tex":"\\FF","aliases":"FF","display":null,"kind":"use","excerpt":"vpage10See cycles, boundaries, and classesA TWO-PACKET COCHAIN COMPLEXMain text: Theorem 4.2 and the finite-complex example. Over FF_5, use bases (e_0,e_1), (u_0,u_1,u_2,u_3), and (f_0,f_1) in degrees zero, one, and two. The only nonzero differential values are d_0e_0=u_0 and d_1u_1=f_0. Consequently d_1d_0=0. vfig10_cohomology headThe quotient is ","count":15},{"work":"paper-x","url":"notation/paper-x.html#symbol-7a4c0a9cf65d6611","tex":"\\FF_{25}","aliases":"FF_25 FF","display":null,"kind":"use","excerpt":"vpage22A norm is a determinant you can computeTHE QUOTIENT ALGEBRA FF_25Main text: §9; retained scalar/Mellin continuation: actual quotient determinant norms. Use A=FF_5[u]/(u^2-2). Since two is not a square modulo five, this quotient is a field with 25 elements. Every element is uniquely a+bu, with a,b∈FF_5. vfig22_norm_table headCoordinates, mult","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f8ae171b4ed22f24","tex":"\\FF_3","aliases":"FF_3 FF","display":null,"kind":"use","excerpt":"vpage29Changing coefficients changes the problemSCALAR ABSENCE VERSUS REPRESENTATION EXISTENCEMain text: Example 8.4 and the declared coefficient categories in §§8--9. Over FF_3, no scalar solves x^2+1=0. But the matrix A=mat0 · -1","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-14e1b580fdde0fc1","tex":"\\FF_5","aliases":"FF_5 FF","display":null,"kind":"use","excerpt":"vpage10See cycles, boundaries, and classesA TWO-PACKET COCHAIN COMPLEXMain text: Theorem 4.2 and the finite-complex example. Over FF_5, use bases (e_0,e_1), (u_0,u_1,u_2,u_3), and (f_0,f_1) in degrees zero, one, and two. The only nonzero differential values are d_0e_0=u_0 and d_1u_1=f_0. Consequently d_1d_0=0. vfig10_cohomology headThe quotient is ","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-5d47e15c2ae55504","tex":"\\FF_5^5","aliases":"FF_5^5 FF","display":null,"kind":"use","excerpt":"theses do real workTRACE AND SEMISIMPLICITYMain text: Theorem 10.1 and the checked false controls following it. The finite occupancy theorem uses a split semisimple commuting family in characteristic zero. Two tiny matrices show why neither the coefficient condition nor semisimplicity should disappear from the statement. vfig19_hypotheses headTrace","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-725725f83cdcb7e8","tex":"\\FF_5^\\times","aliases":"FF_5^× FF times","display":null,"kind":"use","excerpt":"vpage24A nontrivial character may still be exceptionalTHE EXACT SCALAR TESTMain text: §9; scalar/Mellin checkpoint: scalar-power and nonscalar-witness boundaries. A scalar a∈FF_5^× acts on a degree-d algebra with determinant a^d. Therefore the scalar cancellation route tests χ(a^d), not merely χ(a). vfig24_scalar_exception headA failed sufficient t","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-2566c2d81be9a5fa","tex":"\\FF_9","aliases":"FF_9 FF","display":null,"kind":"use","excerpt":"1 · 0∈ M_2(FF_3) satisfies A^2+I=0. And over FF_9=FF_3[u]/(u^2+1), the scalars u and -u solve the same polynomial. vfig29_coefficients headThree outputs with different types The failed scalar problem over FF_3, the successful matrix problem over FF_3, and the successful scalar problem over FF_9 are not contradictory answers. They have different all","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-4d7b4480e3e06144","tex":"\\operatorname{Frob}","aliases":"Frob operatorname","display":"\\operatorname{Frob}","kind":"use","excerpt":"cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j))","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-58bab5cbde13c61a","tex":"\\operatorname{Frob}_{v_j}","aliases":"Frob_v_j operatorname","display":"\\operatorname{Frob}_{v_j}","kind":"use","excerpt":"cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j))","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.","count":24},{"work":"paper-x","url":"notation/paper-x.html#symbol-aa45a7ad376e6ef9","tex":"g^{-1}","aliases":"g^-1","display":"g^{-1}","kind":"use","excerpt":"Derived comparison bounds (Block 16). For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its det","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-ef602db61b865627","tex":"G_{\\mathbb Q}","aliases":"G_ Q mathbb","display":"G_{\\mathbb Q}","kind":"use","excerpt":"The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-09caf7470ee06fab","tex":"G_{\\mathbb Q_p}","aliases":"G_ Q_p mathbb","display":"G_{\\mathbb Q_p}","kind":"use","excerpt":"The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-6fb05800096ed7ee","tex":"G_{F,S}","aliases":"G_F,S","display":"G_{F,S}","kind":"use","excerpt":"For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-82170a73bb5aa973","tex":"G_K","aliases":"G_K","display":"G_K","kind":"use","excerpt":"Actual quadratic local conditions and completion (Block 22). Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embeddi","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-2b7e16a20dfc2d99","tex":"G_{\\Q_p}","aliases":"G_Q_p G_ Q_p Q mathbf","display":"G_{{\\mathbf Q}_p}","kind":"use","excerpt":"This inertia is the actual residue-action kernel inside D_A. No equality D_A=G_ Q_p is established in this block. Nor is ramification at two proved: the nontrivial full local character there is a separate result. At three the Z_2-valued sign character is nontrivial on the full local Galois group but trivial on every inertia subgroup covered by the ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-efa0a3eea83b0f61","tex":"\\operatorname{Gal}","aliases":"Gal operatorname","display":"\\operatorname{Gal}","kind":"use","excerpt":"Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"The exact ideal filter. Suppose IJ=(x), n(x) has the indicated sign, and gcd( N J,m)=1. For c∈ Z/m Z, N I≡ cpmod m Longleftrightarrow overline n_ε([x])=c( N Jbmod m). This gives an actual finite residue target, which is allowed to be empty. The product identity before cancellation does not require coprimality.","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-f9b55a29a191e028","tex":"\\GL","aliases":"GL operatorname","display":"{\\operatorname{GL}}","kind":"use","excerpt":"For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c5d62edc1cdf57e7","tex":"\\mathrm{GL}","aliases":"GL mathrm","display":"\\mathrm{GL}","kind":"use","excerpt":"Lifting a summand inside a supplied finite-group carrier (Block 18). Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=b","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-0fcc70536b44cfb8","tex":"\\operatorname{GL}","aliases":"GL operatorname","display":"\\operatorname{GL}","kind":"use","excerpt":"Checked representation-level completion and its arithmetic boundary. The eighth formalization block works with actual matrix representations of a fixed group, not only with underlying sets. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Mu","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-4d4a5d6f6abf65b4","tex":"\\mathrm{GL}_2","aliases":"GL_2 mathrm","display":"\\mathrm{GL}_2","kind":"use","excerpt":"bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de Franc","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-bca14ec6fc8c9c7a","tex":"\\operatorname{GL}_d","aliases":"GL_d operatorname","display":"\\operatorname{GL}_d","kind":"use","excerpt":"Checked representation-level completion and its arithmetic boundary. The eighth formalization block works with actual matrix representations of a fixed group, not only with underlying sets. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Mu","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5e917386367b6fdc","tex":"\\GL_n","aliases":"GL_n GL operatorname","display":"{\\operatorname{GL}}_n","kind":"use","excerpt":"For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5723752e94fda6ae","tex":"\\mathrm{GL}_n","aliases":"GL_n mathrm","display":"\\mathrm{GL}_n","kind":"use","excerpt":"Lifting a summand inside a supplied finite-group carrier (Block 18). Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=b","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.","count":29},{"work":"paper-x","url":"notation/paper-x.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"[Exact residual partition] The finite implementation partitions the integers 0,…,8 into the three classes modulo 3: {0,3,6}, {1,4,7}, {2,5,8}. Every tested correction of the form x↦ x+p h preserves the residual class exactly.","count":20},{"work":"paper-x","url":"notation/paper-x.html#symbol-fa5a6573c905a228","tex":"H^0","aliases":"H^0","display":"H^0","kind":"use","excerpt":"[Finite packet complex] The exact modular test uses two invariant subcomplexes over F_5. Their cohomology dimensions are (0,0,0), (1,2,1), and the reconstructed total is dim(H^0,H^1,H^2)=(1,2,1). The packet-sum residual is zero.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-054c9b0aa1b44b01","tex":"H^1_j","aliases":"H^1_j","display":"H^1_j","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-f116cd8093d80318","tex":"H^2_j","aliases":"H^2_j","display":"H^2_j","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-92ad3ff2bdd044cf","tex":"H^i","aliases":"H^i","display":"H^i","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-846097f2ae49fb22","tex":"H^r","aliases":"H^r","display":"H^r","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-f4cefef31ff270e3","tex":"H_j^r","aliases":"H_j^r","display":"H_j^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-619ba501c7bc28e9","tex":"H_s","aliases":"H_s","display":"H_s","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom operatorname","display":"{\\operatorname{Hom}}","kind":"use","excerpt":"Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a charac","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0251232b1004a6b9","tex":"\\Hom_{R\\text{-alg}}","aliases":"Hom_R-alg Hom text operatorname text","display":"{\\operatorname{Hom}}_{R\\text{-alg}}","kind":"use","excerpt":"Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a charac","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-fc66c01638c40151","tex":"i^2","aliases":"i^2","display":"i^2","kind":"use","excerpt":"The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6eebe3e99b3727b1","tex":"I^{d_s}","aliases":"I^d_s","display":"I^{d_s}","kind":"use","excerpt":"e map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero after reduction t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-17077128e8026415","tex":"I^k","aliases":"I^k","display":"I^k","kind":"use","excerpt":"Completion from conjugacy classes (Block 14). Over an I-adically complete separated commutative coefficient ring R, for a fixed finite rank, the formalization now takes finite-level objects to be actual matrix representations modulo conjugation by an invertible matrix. Assume the coefficient quotients R/I^k are finite. Conjugation commutes with coe","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-022e2ac42397386f","tex":"i^k","aliases":"i^k","display":"i^k","kind":"use","excerpt":",align=center,font=scriptsize] at (8.5,1.05) no; path[draw=black!30,fill=black!16] (9.3,0.75) rectangle (10.7,1.33); node[font=small,align=center,font=scriptsize] at (10,1.05) yes; path[draw=black!30,fill=black!16] (10.8,0.75) rectangle (12.2,1.33); node[font=small,align=center,font=scriptsize] at (11.5,1.05) yes; node[font=small,align=center,font=","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-d50bce5288ce233a","tex":"I^m","aliases":"I^m","display":"I^m","kind":"use","excerpt":"Checked representation-level completion and its arithmetic boundary. The eighth formalization block works with actual matrix representations of a fixed group, not only with underlying sets. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Mu","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-1a96189008f33553","tex":"I_{\\mathfrak Q_v}","aliases":"I_ Q_v mathfrak","display":"I_{\\mathfrak Q_v}","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":".7); node[font=small,align=center,font=small] at (8,2.3) 0; path[draw=black!30,fill=white] (8.4,1.9) rectangle (9.2,2.7); node[font=small,align=center,font=small] at (8.8,2.3) 0; node[font=small,align=center] at (11.6,2.95) only eigenvalue 0; node[font=small,align=center] at (11.6,2.1) dimker N=1; node[font=small,align=center] at (10.4,0.8) sole-ei","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7cc6e87678c7d216","tex":"I_5","aliases":"I_5","display":"I_5","kind":"use","excerpt":"[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (2.7,4.65) I_5 over F_5; path[draw=black!30,fill=black!8] (1.5,3.47) rectangle (1.98,3.95); node[font=small,align=center,font=small] at (1.74,3.71) 1; path[draw=black!30,fill=white] (1.98,3.47) rec","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9d9060f8fe5d796b","tex":"I_A","aliases":"I_A","display":"I_A","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-0625c4dc07eb960a","tex":"i_j","aliases":"i_j","display":"i_j","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-042e41456d0b7e11","tex":"i_K","aliases":"i_K","display":"i_K","kind":"use","excerpt":"Actual quadratic local conditions and completion (Block 22). Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embeddi","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-0592a96c436fc819","tex":"j_K","aliases":"j_K","display":"j_K","kind":"use","excerpt":"Actual quadratic local conditions and completion (Block 22). Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embeddi","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-ac62d9119015a5fb","tex":"K^\\times","aliases":"K^× times","display":"K^\\times","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilp","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-916bbd7c25020c00","tex":"k_A","aliases":"k_A","display":"k_A","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":16},{"work":"paper-x","url":"notation/paper-x.html#symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"use","excerpt":"at (3,4.4) [0,12)^2∩ Z^2; node[font=small,align=center] at (9.8,3.75) four full 5×5 cells; node[font=small,align=center] at (9.8,2.85) 100 bulk points: sum 0; node[font=small,align=center] at (9.8,1.95) 44 boundary points; node[font=small,align=center] at (9.8,1.05) actual total S_12=1; node[font=small,align=center,font=footnotesize] at (7,-0.1) |S","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-2752bc7b3d7d6299","tex":"L_d","aliases":"L_d","display":"L_d","kind":"use","excerpt":"General finite logarithmic recovery (Block 20). Let A be a rational algebra, without any commutativity assumption on A. For a unipotent element U and a positive integer d satisfying (U-1)^d=0, put L_d(U)=∑_j=1^d-1frac(-1)^j+1j(U-1)^j. The value is independent of the chosen annihilating exponent. The finite logarithm and the nilpotent exponential sa","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-01208e159d2aec8c","tex":"\\land","aliases":"land","display":"\\land","kind":"use","excerpt":"An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrig","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ccb8a376eaddc6e8","tex":"\\lim_m","aliases":"lim_m lim","display":"\\lim_m","kind":"use","excerpt":"headCount vertices and count branches separately There are three roots modulo nine: 0,3,6. But the only roots modulo 27 are 0,9,18, and all three reduce to zero modulo nine. The other two vertices at the previous level have no children. Counting them as permanent branches would be incorrect. #X_m=1,3,3,9,9,27,…, Z(T)=fracT+3T^21-3T^2. The count-gro","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-949a7a665b77fe06","tex":"\\operatorname{LocallyTrivial}","aliases":"LocallyTrivial operatorname","display":"\\operatorname{LocallyTrivial}","kind":"use","excerpt":"For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor ","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-b4c9ac4249963ddc","tex":"\\operatorname{LocallyTrivial}_K","aliases":"LocallyTrivial_K operatorname","display":"\\operatorname{LocallyTrivial}_K","kind":"use","excerpt":"For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor ","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-62b7fa260a967c3f","tex":"\\operatorname{LocallyTrivial}_{\\Q_p}","aliases":"LocallyTrivial_Q_p LocallyTrivial_ Q_p operatorname Q operatorname mathbf","display":"\\operatorname{LocallyTrivial}_{{\\mathbf Q}_p}","kind":"use","excerpt":"l. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue root 2 modulo 5 supp","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"Block 13 also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions r","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c0ce1221dbd41bc1","tex":"\\log_3","aliases":"log_3 log","display":"\\log_3","kind":"use","excerpt":"headCount vertices and count branches separately There are three roots modulo nine: 0,3,6. But the only roots modulo 27 are 0,9,18, and all three reduce to zero modulo nine. The other two vertices at the previous level have no children. Counting them as permanent branches would be incorrect. #X_m=1,3,3,9,9,27,…, Z(T)=fracT+3T^21-3T^2. The count-gro","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Scope of the splitting. The algebraic proof works over any commutative ring and does not divide by 2, so it applies in characteristic two when the stated integral chain projectors exist. It does not construct those projectors. The identification of a particular Galois tangent or obstruction problem with this cohomology remains the preceding classic","count":15},{"work":"paper-x","url":"notation/paper-x.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"In the paper, finite nonempty levels and actual total reduction maps provide compactness even when some vertices die and some transition maps are not onto. One can retain only vertices with descendants at arbitrarily large depths. Finiteness makes that surviving part nonempty and permits compatible choices. X_m≠∅ for all m notLongrightarrow varproj","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-529ad2daacd7efcb","tex":"\\lor","aliases":"lor","display":"\\lor","kind":"use","excerpt":"For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor ","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-1633dfbf62528f10","tex":"\\lvert","aliases":"lvert","display":"\\lvert","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-d4abae2f40eca713","tex":"\\bm m","aliases":"m bm","display":"\\bm m","kind":"use","excerpt":"Use the same multiplicities in every member of the compatible carrier. Direct sums preserve strict compatibility, and m_i≤ c_i makes V_λ a summand of W_λ. At the separating places, the target trace vector is C m. Full column rank makes m the unique solution.","count":10},{"work":"paper-x","url":"notation/paper-x.html#symbol-7bcc78db9fb4a6a7","tex":"\\boldsymbol m","aliases":"m boldsymbol","display":"\\boldsymbol m","kind":"use","excerpt":"tsize] at (1.95,2.65) (12); node[font=small,align=center,anchor=east,font=scriptsize] at (1.95,1.85) (123); node[font=small,align=center,font=scriptsize] at (2.5,4.08) trivial; node[font=small,align=center,font=scriptsize] at (3.3,4.08) sign; node[font=small,align=center,font=scriptsize] at (4.1,4.08) standard; node[font=small,align=center,font=sma","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-af7e789b143d2688","tex":"m^d","aliases":"m^d","display":"m^d","kind":"use","excerpt":"The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"More generally, the kernel of C modulo 2^n has exactly two elements: (0,0,0), (2^n-1,2^n-1,0). To see this, the second equation gives m_1=m_2; the third gives m_3=2m_1; the first then gives 6m_1=0. Three is a unit modulo 2^n, leaving precisely the displayed possibilities.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"vpage29Changing coefficients changes the problemSCALAR ABSENCE VERSUS REPRESENTATION EXISTENCEMain text: Example 8.4 and the declared coefficient categories in §§8--9. Over FF_3, no scalar solves x^2+1=0. But the matrix A=mat0 · -1 1 · 0∈ M_2(FF_3) satisfies A^2+I=0. And over FF_9=FF_3[u]/(u^2+1), the scalars u and -u solve the same polynomial. vfi","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-651ec01c3f6e67cb","tex":"m_2","aliases":"m_2","display":"m_2","kind":"use","excerpt":"More generally, the kernel of C modulo 2^n has exactly two elements: (0,0,0), (2^n-1,2^n-1,0). To see this, the second equation gives m_1=m_2; the third gives m_3=2m_1; the first then gives 6m_1=0. Three is a unit modulo 2^n, leaving precisely the displayed possibilities.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-e622a434aa00eb47","tex":"m_3","aliases":"m_3","display":"m_3","kind":"use","excerpt":"More generally, the kernel of C modulo 2^n has exactly two elements: (0,0,0), (2^n-1,2^n-1,0). To see this, the second equation gives m_1=m_2; the third gives m_3=2m_1; the first then gives 6m_1=0. Three is a unit modulo 2^n, leaving precisely the displayed possibilities.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1bf197ad2566622b","tex":"M_{a+bu}","aliases":"M_a+bu","display":"M_{a+bu}","kind":"use","excerpt":"mall,align=center,font=scriptsize] at (2.38,4.23) b=1; node[font=small,align=center,font=scriptsize] at (3,4.23) b=2; node[font=small,align=center,font=scriptsize] at (3.62,4.23) b=3; node[font=small,align=center,font=scriptsize] at (4.24,4.23) b=4; node[font=small,align=center] at (9.6,4.25) u^2=2 in F_25; node[font=small,align=center] at (9.6,3.2","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-fc481656f7fd52b3","tex":"M_{a,b}","aliases":"M_a,b","display":"M_{a,b}","kind":"use","excerpt":"GEBRA FF_25Main text: §9; retained scalar/Mellin continuation: actual quotient determinant norms. Use A=FF_5[u]/(u^2-2). Since two is not a square modulo five, this quotient is a field with 25 elements. Every element is uniquely a+bu, with a,b∈FF_5. vfig22_norm_table headCoordinates, multiplication, and norm agree In the basis (1,u), multiplication","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-28cc0a43f7b6c71a","tex":"M_r","aliases":"M_r","display":"M_r","kind":"use","excerpt":"[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficien","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-e3e697d6b6f9bc2b","tex":"M_w","aliases":"M_w","display":"M_w","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-956f614ae0645ab5","tex":"\\mat","aliases":"mat","display":null,"kind":"use","excerpt":"of S_3: trivial, sign, and standard. On the identity, a transposition, and a three-cycle, their values form the matrix below. Rows are samples; columns are constituent types. vfig15_separator headWhat full rank buys Since det C=6, these samples distinguish every rational multiplicity vector. A target with two trivial constituents, no sign constitue","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"use","excerpt":"[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficien","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"A local map sends each parameter into the image of (ϖ) in O/ϖ^m. Conversely, every tuple in that image is nilpotent and defines a coefficient-preserving evaluation homomorphism. This homomorphism is local. To prove uniqueness without assuming continuity of an arbitrary competing homomorphism, observe in one variable that a homomorphism killing X^N ","count":79},{"work":"paper-x","url":"notation/paper-x.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-261b1d4accf38dcf","tex":"\\mathbf N","aliases":"N mathbf","display":"\\mathbf N","kind":"use","excerpt":"Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement an","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-213d86a5e6b34393","tex":"N^{-1}","aliases":"N^-1","display":"N^{-1}","kind":"use","excerpt":"b · a; node[font=small,align=center] at (9.6,2) #N^-1(0)=1; node[font=small,align=center] at (9.6,1.05) #N^-1(r)=6 for r≠0; node[font=small,align=center,font=footnotesize] at (7,-0.1) All 25 elements are shown. Every nonzero element is a unit.;","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"An explicit action of C_2× Z has ν(φ)=1/3, F=diag(2,6,18) and Ne_1=e_0, Ne_2=e_1, Ne_0=0. Trivial inertia gives local dimension one and denominator 1-2T; scalar sign inertia gives dimension zero and denominator one. The entire F and N are equal in these two examples, and N^2≠0. Thus inertia cannot be omitted. Conversely, unramified chains of length","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"Block 13 also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions r","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-948594e33e46ceeb","tex":"N^5","aliases":"N^5","display":"N^5","kind":"use","excerpt":"Actual affine representations with monodromy chains of every finite length invoke these theorems. In particular a length-six example has N^5≠0 and N^6=0 and is recovered from its actual reconstructed action. The length-four example still defeats the old quadratic logarithm, while the full finite logarithm recovers it.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6808aa0a6bb0a282","tex":"N^6","aliases":"N^6","display":"N^6","kind":"use","excerpt":"Actual affine representations with monodromy chains of every finite length invoke these theorems. In particular a length-six example has N^5≠0 and N^6=0 and is recovered from its actual reconstructed action. The length-four example still defeats the old quadratic logarithm, while the full finite logarithm recovers it.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1fea4657810bf7e4","tex":"N^d","aliases":"N^d","display":"N^d","kind":"use","excerpt":"Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the represent","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-3f6e0c9e86fd9f34","tex":"N^j","aliases":"N^j","display":"N^j","kind":"use","excerpt":"Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the represent","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-a74297b6dc786495","tex":"N^n","aliases":"N^n","display":"N^n","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion k","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-475389afb995c765","tex":"N^{n-1}","aliases":"N^n-1","display":"N^{n-1}","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion k","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b29cdf7a7fb2f72e","tex":"N_{\\mathbb Q/\\mathbb Q}","aliases":"N_ Q/ Q mathbb mathbb","display":"N_{\\mathbb Q/\\mathbb Q}","kind":"use","excerpt":"ode[font=small,align=center,font=scriptsize] at (6.75,2.2) integer generators; path[draw,fill=black!15] (3.409091,2.75) circle (0.12); draw[->] (3.409091,2.55) -- (6.55,1.75); path[draw,fill=black!15] (8.181818,2.75) circle (0.12); draw[->] (8.181818,2.55) -- (6.55,1.75); node[font=small,align=center] at (6.55,1.45) same residue 2pmod5; node[font=s","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-987804e10f674d55","tex":"N_{\\bm w}","aliases":"N_ w bm","display":"N_{\\bm w}","kind":"use","excerpt":"[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-eb2af32ef3e74216","tex":"N_D","aliases":"N_D","display":"N_D","kind":"use","excerpt":"For supplied covariance data and the twisted cocycle of Block 13, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b82d70c3ab4030a5","tex":"N_i","aliases":"N_i","display":"N_i","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilp","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-37ff4542a5661937","tex":"N_{K/\\QQ}","aliases":"N_K/QQ QQ","display":null,"kind":"use","excerpt":"ALSMain text: §9, arithmetic residue-counting continuation; signed-norm comparison. Even in the number field QQ, congruent generators need not have congruent absolute norms. The integers 2 and -3 are congruent modulo five. Their signed field norms are congruent, but their absolute values are not. vfig21_signed_norm headThe smallest counterexample i","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-718d406878f6c5a6","tex":"N_m","aliases":"N_m","display":"N_m","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilp","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-9f1294e6f5b93673","tex":"N_{m+1}","aliases":"N_m+1","display":"N_{m+1}","kind":"use","excerpt":"t must work at every depthONE REVERSIBLE EXAMPLEMain text: Definition 7.1 and the reduction-compatibility requirement. For the two roots of x^2+1=0 modulo powers of five, negation is an actual reversible transport. It preserves the equation, is its own inverse, and commutes with reduction. vfig13_transport headOne successful input is not a natural ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6a6ca180114c345d","tex":"n_p","aliases":"n_p","display":"n_p","kind":"use","excerpt":"For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positi","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-90f600277c50100a","tex":"\\NN","aliases":"NN N mathbb","display":"{\\mathbb N}","kind":"use","excerpt":"The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"attice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an e","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-9f009778fa1adcff","tex":"O_{K,C}","aliases":"O_K,C","display":"O_{K,C}","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-85ae2403f3c15ce7","tex":"O_{K,J,m,\\chi}","aliases":"O_K,J,m,χ chi","display":"O_{K,J,m,\\chi}","kind":"use","excerpt":"The assembled ideal norm-character estimate. Let C∈Cl( O), let J≠0 be an integral ideal with [J]=C^-1, and let m>0. Let χ: Z/m Z→ C preserve multiplication, 0, and 1. Assume a residue-ring unit satisfying [source reference], and assume χ( N Jbmod m)≠0. Then T_C,χ(X) =∑_substackI≠0, [I]=C N I≤ X χ( N Ibmod m) =O_K,J,m,χ(X^1-1/d). More precisely, som","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-4fa2eeb16819ef5a","tex":"\\Ocal_\\lambda","aliases":"Ocal_λ O_λ Ocal lambda mathcal lambda","display":"{\\mathcal O}_\\lambda","kind":"use","excerpt":"All coefficients of the characteristic polynomial commute with reduction. Consequently, prescribed integral characteristic polynomials at every precision hold exactly in the reconstructed representation. For a common coefficient field E, local integral polynomials lie in O_λ[X] and are compared with P_v∈ E[X] only after mapping both into the local ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-ea26d2394b7c0bc5","tex":"\\overline a","aliases":"overline a overline","display":"\\overline a","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5f61d083e1babfc3","tex":"\\overline a^{\\,p}","aliases":"overline a^ p overline","display":"\\overline a^{\\,p}","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-626b99e92a63b7cb","tex":"\\overline n","aliases":"overline n overline","display":"\\overline n","kind":"use","excerpt":"The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-f5fbedef8300da8c","tex":"\\overline n_J","aliases":"overline n_J overline","display":"\\overline n_J","kind":"use","excerpt":"An actual unit action on ideal residues. Fix m>0 and a nonzero integral ideal J. Multiplication by u∈( O/m O)^× gives a proved permutation of J/mJ. Its inverse follows from (ab-1)x∈ mJ when ab-1∈ m O and x∈ J. No surjectivity of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-f81ac0b7b57e0101","tex":"\\overline n_{\\varepsilon}","aliases":"overline n_ε overline varepsilon","display":"\\overline n_{\\varepsilon}","kind":"use","excerpt":"For ε∈{1,-1} define overline n_ε(a)=εoverline n(a). On the corresponding sign portion, N(x)bmod m=overline n_ε([x]). The sign cannot be dropped: -1≡2pmod3, whereas N(-1)=1 and N(2)=2 for principal ideals of Z.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-43a8f89e4f079c54","tex":"\\overline y","aliases":"overline y overline","display":"\\overline y","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5c45a9f213c593e3","tex":"\\overline y^{\\,n-1}","aliases":"overline y^ n-1 overline","display":"\\overline y^{\\,n-1}","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7896e1964fe42160","tex":"\\overline{\\Q_p}","aliases":"overlineQ_p overline Q_p overline Q overline mathbf","display":"\\overline{{\\mathbf Q}_p}","kind":"use","excerpt":"For every prime p, Lean constructs a valuation subring A⊆overline Q_p containing the integral image of Z_p such that the induced map Z_p→ A is local. For odd p, two is a unit. The preceding argument applies to every valuation subring containing that integral image, and hence every member of P_q(B) is trivial on its inertia subgroup after actual loc","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-e10243300d232e33","tex":"\\overline{x}","aliases":"overlinex overline","display":"\\overline{x}","kind":"use","excerpt":"The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7a4916ff831cccc6","tex":"\\overline{x}^{\\,p}","aliases":"overlinex^ p overline","display":"\\overline{x}^{\\,p}","kind":"use","excerpt":"The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-541bacaca1fd7280","tex":"\\overline{\\gamma_p(a)}","aliases":"overlineγ_p(a) overline gamma","display":"\\overline{\\gamma_p(a)}","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6f403bf81328e606","tex":"\\overline{\\sigma_p(x)}","aliases":"overlineσ_p(x) overline sigma","display":"\\overline{\\sigma_p(x)}","kind":"use","excerpt":"The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"hmetic completion (Block 21). Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)","count":12},{"work":"paper-x","url":"notation/paper-x.html#symbol-aa4097b0f132cbcf","tex":"P^0","aliases":"P^0","display":"P^0","kind":"use","excerpt":"vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace th","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-a8b36d84c479b7cc","tex":"P^1","aliases":"P^1","display":"P^1","kind":"use","excerpt":"vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace th","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace th","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7441b5cfbaa2c0c0","tex":"p^m","aliases":"p^m","display":"p^m","kind":"use","excerpt":"The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-4927a5c0083e2327","tex":"p^{m-1}","aliases":"p^m-1","display":"p^{m-1}","kind":"use","excerpt":"The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-1bbb4a842cfd7d98","tex":"P^r","aliases":"P^r","display":"P^r","kind":"use","excerpt":"vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace th","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1b48313d2ebbd5e3","tex":"P^{r+1}","aliases":"P^r+1","display":"P^{r+1}","kind":"use","excerpt":"vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace th","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9553d695f708fd97","tex":"P_{22}^2","aliases":"P_22^2","display":"P_{22}^2","kind":"use","excerpt":"0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-263260b939049c0f","tex":"P_A^0","aliases":"P_A^0","display":"P_A^0","kind":"use","excerpt":"node[font=small,align=center,font=small] at (5.915,1.865) 0; path[draw=black!30,fill=white] (6.22,1.56) rectangle (6.83,2.17); node[font=small,align=center,font=small] at (6.525,1.865) 0; path[draw=black!30,fill=white] (6.83,1.56) rectangle (7.44,2.17); node[font=small,align=center,font=small] at (7.135,1.865) 0; node[font=small,align=center,font=s","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-cb44ce3dcef3881d","tex":"P_A^1","aliases":"P_A^1","display":"P_A^1","kind":"use","excerpt":"[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (2.47,4.7) valid P_A^1; path[draw=black!30,fill=black!8] (1.25,3.39) rectangle (1.86,4); node[font=small,align=center,font=small] at (1.555,3.695) 1; path[draw=black!30,fill=white] (1.86,3.39) rect","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"For i≠ j, orthogonality gives H(p_i)H(i_j)=0. Idempotence gives H(p_j)H(i_j)=I, and completeness gives ∑_j H(i_j)H(p_j)=I. Thus the maps [z]⟼([P_j^r z])_j, ([z_j])_j⟼[∑_j i_j z_j] are inverse linear maps. The finite product equals the direct sum, proving the assertion. In the formalization, both sides are also identified with Mathlib's standard hom","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-365569fb3de7e2a4","tex":"p_{i,a}","aliases":"p_i,a","display":"p_{i,a}","kind":"use","excerpt":"For each operator, its finite set of actual eigenvalues is used without multiplicity. The polynomial p_i,a(X)=∏_μ∈Spec(T_i)∖{a} (a-μ)^-1(X-μ) is used when a occurs, and the zero polynomial otherwise. For an eigenvector T_i v=bv, evaluation gives p_i,a(T_i)v=v · b=a,","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-6ae74ed9c71e6f69","tex":"p_{i,a_i}","aliases":"p_i,a_i","display":"p_{i,a_i}","kind":"use","excerpt":"0 · b≠ a. Because the eigenspaces span A, this proves that e_i=p_i,a_i(T_i) maps all of A into the selected eigenspace and fixes that eigenspace pointwise. It therefore proves the individual projector and image assertions, including an absent target.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-d1d36be4bad0ff1b","tex":"P_{i,g}","aliases":"P_i,g","display":"P_{i,g}","kind":"use","excerpt":"(XI-ρ_m(g)) =∏_i∈ Idet(XI-ρ_i(g))^m_i, Tr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-de628d850c751acf","tex":"P_{i,g}^{m_i}","aliases":"P_i,g^m_i","display":"P_{i,g}^{m_i}","kind":"use","excerpt":"eratornameTr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is the image of ∏_iP_i,g^m_i. App","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-d3b69870475ee0ca","tex":"P_i^r","aliases":"P_i^r","display":"P_i^r","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"use","excerpt":"h packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are coch","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-2a8fbe464d99fbb3","tex":"P_j^r","aliases":"P_j^r","display":"P_j^r","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-fd9801213575c0e5","tex":"P_j^{r+1}","aliases":"P_j^r+1","display":"P_j^{r+1}","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-21e7fc525024b4c1","tex":"\\mathcal P_q","aliases":"P_q mathcal","display":"\\mathcal P_q","kind":"use","excerpt":"hmetic completion (Block 21). Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)","count":12},{"work":"paper-x","url":"notation/paper-x.html#symbol-be8331cd1f9921b0","tex":"p_s","aliases":"p_s","display":"p_s","kind":"use","excerpt":"Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-a682e6942eadb5e0","tex":"P_{s,m}","aliases":"P_s,m","display":"P_{s,m}","kind":"use","excerpt":"agraphActual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At pre","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":11},{"work":"paper-x","url":"notation/paper-x.html#symbol-43cab5a3c1e158c9","tex":"\\mathbb\n  Q","aliases":"Q mathbb","display":"\\mathbb\n  Q","kind":"use","excerpt":"bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de Franc","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-8605e09a21a342b4","tex":"\\mathbf Q","aliases":"Q mathbf","display":"\\mathbf Q","kind":"use","excerpt":"For the three-character example below, the construction produces an actual three-dimensional summand inside the carrier of dimension seven. At the sample (0,1) its characteristic polynomial is (X-1)^2(X+1). A separate checked counterexample uses the coordinate inclusion Q→ Q^2, its coordinate retraction, and the swap involution: the split law holds","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f4cc63c7c45c2f71","tex":"q^{-j}","aliases":"q^-j","display":"q^{-j}","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion k","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-c2ff5e7340642c49","tex":"q^{-W}","aliases":"q^-W","display":"q^{-W}","kind":"use","excerpt":"[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1c045b4a50d02fa4","tex":"Q^1","aliases":"Q^1","display":"Q^1","kind":"use","excerpt":"font=small,align=center,font=small] at (2.165,1.865) 0; path[draw=black!30,fill=white] (2.47,1.56) rectangle (3.08,2.17); node[font=small,align=center,font=small] at (2.775,1.865) 0; path[draw=black!30,fill=white] (3.08,1.56) rectangle (3.69,2.17); node[font=small,align=center,font=small] at (3.385,1.865) 0; node[font=small,align=center,font=small]","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-53004de1c8fb420f","tex":"\\mathbf Q^2","aliases":"Q^2 mathbf","display":"\\mathbf Q^2","kind":"use","excerpt":"For the three-character example below, the construction produces an actual three-dimensional summand inside the carrier of dimension seven. At the sample (0,1) its characteristic polynomial is (X-1)^2(X+1). A separate checked counterexample uses the coordinate inclusion Q→ Q^2, its coordinate retraction, and the swap involution: the split law holds","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b9baeebca7f43f14","tex":"\\Q^4","aliases":"Q^4 Q mathbf","display":"{\\mathbf Q}^4","kind":"use","excerpt":"A specified finite realization. On Q^4, take the diagonal operators with diagonal lists (1,1,1,0), (2,2,0,0), (3,0,0,0), (0,4,4,4) and target scalars (1,2,3,4). All four targets occur individually. Imposing these conditions successively gives actual projector ranks 3,2,1,0. The last packet is empty despite individual feasibility. The formal code ch","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ee5559b3980fa51c","tex":"q^d","aliases":"q^d","display":"q^d","kind":"use","excerpt":"[Smooth seed zeta function] For a smooth formal component of dimension d, Z_d(T)=fracT1-q^dT.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-9bca90fff80db18b","tex":"q^{d(m-1)}","aliases":"q^d(m-1)","display":"q^{d(m-1)}","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-7ac1542e49ac35e1","tex":"q^{dm-W}","aliases":"q^dm-W","display":"q^{dm-W}","kind":"use","excerpt":"The finite sum records the depths before all coordinates are active. For m≥ w_d+1, a_m=q^dm-W, so the remaining terms form one geometric series.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-e80a32868f710c6e","tex":"q^{e-1}","aliases":"q^e-1","display":"q^{e-1}","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7fd7aaa8906a193c","tex":"q^{m-\\lceil m/e\\rceil}","aliases":"q^m-lceil m/erceil lceil rceil","display":"q^{m-\\lceil m/e\\rceil}","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-08d664fed466d656","tex":"q^{m-w_i}","aliases":"q^m-w_i","display":"q^{m-w_i}","kind":"use","excerpt":"The image of ϖ^w_i O in O/ϖ^m has one element when m≤ w_i, and q^m-w_i elements when m>w_i. Multiplying the independent coordinate counts gives the formula.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-63b1654c14578998","tex":"q^{r-1}","aliases":"q^r-1","display":"q^{r-1}","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f0fc2c2d75a0ca57","tex":"q^{\\sum_{i=1}^d\\max(m-w_i,0)}","aliases":"q^∑_i=1^dmax(m-w_i,0) sum max","display":"q^{\\sum_{i=1}^d\\max(m-w_i,0)}","kind":"use","excerpt":"[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ddd7652dc4b81b62","tex":"q^{\\sum_i\\max(m-w_i,0)}","aliases":"q^∑_imax(m-w_i,0) sum max","display":"q^{\\sum_i\\max(m-w_i,0)}","kind":"use","excerpt":"[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-50f270102a088631","tex":"\\Q_2","aliases":"Q_2 Q mathbf","display":"{\\mathbf Q}_2","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-812a1fd77a9f35a3","tex":"\\mathbb Q_2","aliases":"Q_2 mathbb","display":"\\mathbb Q_2","kind":"use","excerpt":"0 · 2 at a nonidentity quadratic element. The determinant 2 is detected modulo 4. The representations are conjugate over Q_2 but not integrally conjugate. A separate Z_3 example invokes simultaneous completion and reconstructs a comparison which is an integral nonunit and a field unit.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-188ca6e8182a79b4","tex":"\\Q_3","aliases":"Q_3 Q mathbf","display":"{\\mathbf Q}_3","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7bb195dbf7dd5e73","tex":"Q_D","aliases":"Q_D","display":"Q_D","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-dad907bde99257ae","tex":"Q_{D_i}","aliases":"Q_D_i","display":"Q_{D_i}","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-958248099af7659e","tex":"\\mathfrak Q_j","aliases":"Q_j mathfrak","display":"\\mathfrak Q_j","kind":"use","excerpt":"For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesse","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ad4d7e06ba73eb0d","tex":"q_L","aliases":"q_L","display":"q_L","kind":"use","excerpt":"Absolute Galois restriction and arithmetic witnesses (Block 26). Fix a cyclotomic number field L= Q(ζ_n) with n>0 and an embedding j:L↪overline Q. The new restriction map q_L:G_ Q⟶Gal(L/ Q), j(q_L(γ)x)=γ(j(x)), is an actual continuous surjective homomorphism with open kernel. Surjectivity is derived by extending automorphisms to the algebraic closu","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-8205838caa474d98","tex":"\\Q_p","aliases":"Q_p Q mathbf","display":"{\\mathbf Q}_p","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-0722957f7780ba76","tex":"\\mathbb\n  Q_p","aliases":"Q_p mathbb","display":"\\mathbb\n  Q_p","kind":"use","excerpt":"bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de Franc","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-cd98a9c47ecfa404","tex":"\\mathbb Q_p","aliases":"Q_p mathbb","display":"\\mathbb Q_p","kind":"use","excerpt":"The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-614679e7835ca9ed","tex":"\\Q_{p_s}","aliases":"Q_p_s Q mathbf","display":"{\\mathbf Q}_{p_s}","kind":"use","excerpt":"Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-90007b832087e14d","tex":"\\mathfrak Q_u","aliases":"Q_u mathfrak","display":"\\mathfrak Q_u","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-e7293ba39ec7c209","tex":"\\mathfrak Q_v","aliases":"Q_v mathfrak","display":"\\mathfrak Q_v","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-417638970b42b393","tex":"Q_{\\bigoplus_iD_i^{\\oplus m_i}}","aliases":"Q_⊕_iD_i^⊕ m_i bigoplus oplus","display":"Q_{\\bigoplus_iD_i^{\\oplus m_i}}","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ","display":null,"kind":"use","excerpt":"vpage21The sign of a norm is arithmetic dataELEMENTS VERSUS IDEALSMain text: §9, arithmetic residue-counting continuation; signed-norm comparison. Even in the number field QQ, congruent generators need not have congruent absolute norms. The integers 2 and -3 are congruent modulo five. Their signed field norms are congruent, but their absolute value","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.","count":66},{"work":"paper-x","url":"notation/paper-x.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"For a general positive integer L, write L=5q+r, 0≤ r<5. The complete tiled square has side L-r, so |S_L|≤ L^2-(L-r)^2=2Lr-r^2≤8L. The area is X=L^2, making this a square-root-in-area bound for this square-box model. The proof is the exact tile decomposition; checking many L values is only a regression check of the implementation.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-4415f2a3fc7b65ea","tex":"R^n","aliases":"R^n","display":"R^n","kind":"use","excerpt":"The actual submodules U=imE and W=ker E are invariant, and addition gives an equivariant isomorphism U⊕ W≃ R^n. The inclusion i:U↪ R^n and retraction r(v)=Ev are constructed as linear maps and satisfy ri=1_U and ir=E. Their equivariance is proved. Coordinatewise reduction maps U onto im(bar Ebmod(π)^k) and maps W onto ker(bar Ebmod(π)^k). Both equa","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-28d6d92025672629","tex":"R_{\\bar x}","aliases":"R_bar x bar","display":"R_{\\bar x}","kind":"use","excerpt":"[Represented complete deformation problems] When a deformation functor is represented by a complete local ring R_bar x, its O-valued points are continuous local maps R_bar x→ O. Completeness identifies such maps with compatible maps to every quotient O/ϖ^m.","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-001ad2900a7bd426","tex":"r_D","aliases":"r_D","display":"r_D","kind":"use","excerpt":"Block 13 also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions r","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-6cf1fb444bcb5d56","tex":"r_E","aliases":"r_E","display":"r_E","kind":"use","excerpt":"Block 13 also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions r","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-a0a3357a085fae77","tex":"R_K","aliases":"R_K","display":"R_K","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-b35fced5b4293ce5","tex":"r_{m+1,m}","aliases":"r_m+1,m","display":"r_{m+1,m}","kind":"use","excerpt":"Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":".15); node[font=small,align=center,font=small] at (12.045,2.735) 0; path[draw=black!30,fill=white] (10.8,1.49) rectangle (11.63,2.32); node[font=small,align=center,font=small] at (11.215,1.905) 0; path[draw=black!30,fill=white] (11.63,1.49) rectangle (12.46,2.32); node[font=small,align=center,font=small] at (12.045,1.905) 0; node[font=small,align=c","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank operatorname","display":"{\\operatorname{rank}}","kind":"use","excerpt":"[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet i","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-1115f7eaacd50934","tex":"\\operatorname{red}","aliases":"red operatorname","display":"\\operatorname{red}","kind":"use","excerpt":"For example, the addresses 1,4,7,10,…,25 may be deeply connected by higher-digit changes. They remain disjoint from the addresses reducing to zero. In a genuine deformation problem, the same conclusion applies to operations whose reduction is the identity, not to every operation that happens to be called a deformation. red(x+3h)=red(x), M(A)=bigsqc","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-226626f51f7539ed","tex":"\\rvert","aliases":"rvert","display":"\\rvert","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c41f602ac2f82c8e","tex":"\\rvert_p^2","aliases":"rvert_p^2 rvert","display":"\\rvert_p^2","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":25},{"work":"paper-x","url":"notation/paper-x.html#symbol-7ac3f743de10554b","tex":"S_{12}","aliases":"S_12","display":"S_{12}","kind":"use","excerpt":"width=.8pt] (2.7,1.85) rectangle (4.2,3.35); node[font=small,align=center] at (3,4.4) [0,12)^2∩ Z^2; node[font=small,align=center] at (9.8,3.75) four full 5×5 cells; node[font=small,align=center] at (9.8,2.85) 100 bulk points: sum 0; node[font=small,align=center] at (9.8,1.95) 44 boundary points; node[font=small,align=center] at (9.8,1.05) actual t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (3.3,4.55) C: actual S_3 character table; path[draw=black!30,fill=black!8] (2.1,3.05) rectangle (2.9,3.85); node[font=small,align=center,font=small] at (2.5,3.45) 1; path[draw=black!30,fill=black!8","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-85a648879a8456a4","tex":"S_L","aliases":"S_L","display":"S_L","kind":"use","excerpt":"The exceptional primes are controlled by the actual field discriminant: S_L={p rational prime:pmiddisc(L)}. This set is finite. For p∉ S_L, every prime of O_L above (p) is unramified. Lean proves that the actual inertia cardinality is the ramification index. It follows that inertia is trivial and the arithmetic Frobenius at a fixed finite prime is ","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-ed40760b634e85a0","tex":"S_w","aliases":"S_w","display":"S_w","kind":"use","excerpt":"e preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substack","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-6a81d4e3e02ea338","tex":"\\mathrm{sign}","aliases":"sign mathrm","display":"\\mathrm{sign}","kind":"use","excerpt":"0 · -1) and τ(g)=diag(1,-1). It preserves the residual comparison and has determinant one. Pullback through the quadratic character of the actual G_ Q gives continuous arithmetic examples. In contrast, the identity and diag(1,sign) over Z/4 agree modulo two but are not integrally conjugate. Their group order is not a unit. These restricted results ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span","display":null,"kind":"use","excerpt":"ies, and classesA TWO-PACKET COCHAIN COMPLEXMain text: Theorem 4.2 and the finite-complex example. Over FF_5, use bases (e_0,e_1), (u_0,u_1,u_2,u_3), and (f_0,f_1) in degrees zero, one, and two. The only nonzero differential values are d_0e_0=u_0 and d_1u_1=f_0. Consequently d_1d_0=0. vfig10_cohomology headThe quotient is the important step In degr","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9b11e0d35c37be17","tex":"\\operatorname{span}_R","aliases":"span_R operatorname","display":"\\operatorname{span}_R","kind":"use","excerpt":"For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-327dcbd8f0058da9","tex":"\\Spec","aliases":"Spec operatorname","display":"{\\operatorname{Spec}}","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement an","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-a46e78b4903b3de3","tex":"\\operatorname{srank}","aliases":"srank operatorname","display":"\\operatorname{srank}","kind":"use","excerpt":"ill=white] (7,3.8) circle (0.22); node[font=small,align=center] at (7,3.8) C; path[draw,fill=white] (11.7,2.4) circle (0.22); node[font=small,align=center] at (11.7,2.4) F; path[draw,fill=white] (7,0.65) circle (0.22); node[font=small,align=center] at (7,0.65) G; node[font=small,align=center] at (11.1,4.35) three source components; node[font=small,","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ceb0960938d0156a","tex":"\\operatorname{Stab}","aliases":"Stab operatorname","display":"\\operatorname{Stab}","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7e8efc303977c5d4","tex":"\\operatorname{Stab}_{G_{\\Q}}","aliases":"Stab_G_Q Stab_G_ Q operatorname Q operatorname mathbf","display":"\\operatorname{Stab}_{G_{{\\mathbf Q}}}","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-21a8014450044a51","tex":"\\Sys","aliases":"Sys mathscr S mathscr","display":"{\\mathscr S}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ade086a1f71df625","tex":"\\mathrm{sys}","aliases":"sys mathrm","display":"\\mathrm{sys}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c8565080e544a001","tex":"\\Sys_2^{\\mathrm{wt}}","aliases":"Sys_2^wt mathscr S_2^wt Sys mathrm mathscr mathrm","display":"{\\mathscr S}_2^{\\mathrm{wt}}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5d6edfcecc9e2400","tex":"\\Sys_{\\mathrm{sys}}","aliases":"Sys_sys mathscr S_sys Sys mathrm mathscr mathrm","display":"{\\mathscr S}_{\\mathrm{sys}}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-90e631cae94f81c9","tex":"\\Sys_\\lambda","aliases":"Sys_λ mathscr S_λ Sys lambda mathscr lambda","display":"{\\mathscr S}_\\lambda","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"For weights (1,2,4), the counts are 1,2,8,32,256,2048,…. The first coordinate is active immediately, the second one level later, and the third at depth five. The exact rational function generated by the code is Z_(1,2,4)(T) = fracT(1-6T-8T^2-32T^3)1-8T. The negative signs in this presentation arise from moving the denominator to 1-8T; every power-s","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","excerpt":"For weights (1,2,4), the counts are 1,2,8,32,256,2048,…. The first coordinate is active immediately, the second one level later, and the third at depth five. The exact rational function generated by the code is Z_(1,2,4)(T) = fracT(1-6T-8T^2-32T^3)1-8T. The negative signs in this presentation arise from moving the denominator to 1-8T; every power-s","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-7f9e80a5242f2d08","tex":"t^{d-1}","aliases":"t^d-1","display":"t^{d-1}","kind":"use","excerpt":"There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-964fcf8c1f59b656","tex":"T^e","aliases":"T^e","display":"T^e","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-bf9efaac25b08a98","tex":"T^r","aliases":"T^r","display":"T^r","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"h[draw=black!35,fill=black!18] (6.06,2.63) rectangle (7.46,3.12); path[draw=black!35,fill=white] (7.59,2.63) rectangle (8.99,3.12); path[draw=black!35,fill=white] (9.12,2.63) rectangle (10.52,3.12); path[draw=black!35,fill=white] (10.65,2.63) rectangle (12.05,3.12); node[font=small,align=center] at (13,2.89) 3; node[font=small,align=center,anchor=e","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-5d08324acb961fcf","tex":"t_2","aliases":"t_2","display":"t_2","kind":"use","excerpt":"anchor=east] at (1.04,1.835) 9; node[font=small,align=center,font=scriptsize] at (3.915,0.12) 18; node[font=small,align=center,font=scriptsize,anchor=east] at (1.04,3.065) 18; node[font=small,align=center,font=scriptsize] at (4.735,0.12) 24; node[font=small,align=center,font=scriptsize,anchor=east] at (1.04,3.885) 24; node[font=small,align=center] ","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","excerpt":"path[draw=black!35,fill=white] (6.06,1.87) rectangle (7.46,2.36); path[draw=black!35,fill=white] (7.59,1.87) rectangle (8.99,2.36); path[draw=black!35,fill=white] (9.12,1.87) rectangle (10.52,2.36); path[draw=black!35,fill=white] (10.65,1.87) rectangle (12.05,2.36); node[font=small,align=center] at (13,2.13) 2; node[font=small,align=center,anchor=e","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-bd730f6c2c2e8897","tex":"t_d","aliases":"t_d","display":"t_d","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-219b7412a89e1904","tex":"T_i","aliases":"T_i","display":"T_i","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":11},{"work":"paper-x","url":"notation/paper-x.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-25bdedf7497647d3","tex":"t_j","aliases":"t_j","display":"t_j","kind":"use","excerpt":"Suppose two members are isomorphic at λ_0. Injectivity of the common coefficient embedding gives t_i(v)=t_j(v) at every place good for λ_0. At another coefficient prime λ, this equality compares actual member traces on the intersection of the two good-place sets. Conjugacy density of that intersection and continuity extend the equality to the whole","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-4505604ff161e9a8","tex":"T_N","aliases":"T_N","display":"T_N","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr operatorname","display":"{\\operatorname{Tr}}","kind":"use","excerpt":"[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet i","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j))","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"le and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding preser","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"For a finite collection of roots, record a directed edge u→ v whenever occupancy or the relevant property is known to propagate from u to v.","count":57},{"work":"paper-x","url":"notation/paper-x.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"For rigidity alone, let R be a commutative local ring, let π be a nonunit, and let E,F be complete equivariant families on the same carrier with E_i≡ F_ipmod(π)^k for every label and k>0. Then U=∑_i∈ JF_iE_i, U≡1pmod(π)^k, UE_i=F_iU. Reduction reflects invertibility in this setting, so U is an integral unit and UE_iU^-1=F_i. It commutes with the ca","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"=small,align=center,font=scriptsize] at (1.76,4.23) b=0; node[font=small,align=center,font=scriptsize] at (2.38,4.23) b=1; node[font=small,align=center,font=scriptsize] at (3,4.23) b=2; node[font=small,align=center,font=scriptsize] at (3.62,4.23) b=3; node[font=small,align=center,font=scriptsize] at (4.24,4.23) b=4; node[font=small,align=center] at","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-4eebc9c8e49ecfa0","tex":"u^{ab}","aliases":"u^ab","display":"u^{ab}","kind":"use","excerpt":"For actual finite original inertia images the common kernel sample is derived. Suppose u∈ H satisfies ν(u)=1 and t(u)≠0 and both r_D(H) and r_E(H) are finite. Their sampled actions have positive finite orders a,b. At u^ab both original actions are trivial, and t(u^ab)=(ab)t(u)≠0 by the cocycle identity and characteristic zero. Thus the same full in","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"nt=small,align=center,font=small] at (2.39,2.3) 0; path[draw=black!30,fill=white] (1.4,1.31) rectangle (2.06,1.97); node[font=small,align=center,font=small] at (1.73,1.64) 0; path[draw=black!30,fill=white] (2.06,1.31) rectangle (2.72,1.97); node[font=small,align=center,font=small] at (2.39,1.64) 0; node[font=small,align=center,anchor=east,font=scri","count":9},{"work":"paper-x","url":"notation/paper-x.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"te] (1.4,1.31) rectangle (2.06,1.97); node[font=small,align=center,font=small] at (1.73,1.64) 0; path[draw=black!30,fill=white] (2.06,1.31) rectangle (2.72,1.97); node[font=small,align=center,font=small] at (2.39,1.64) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,3.62) u_0; node[font=small,align=center,anchor=east,font=scri","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-2f97c9046c1a4c7a","tex":"u_2","aliases":"u_2","display":"u_2","kind":"use","excerpt":"(1.73,1.64) 0; path[draw=black!30,fill=white] (2.06,1.31) rectangle (2.72,1.97); node[font=small,align=center,font=small] at (2.39,1.64) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,3.62) u_0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.96) u_1; node[font=small,align=center,anchor=east,font=scriptsi","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-cdc6b49492357264","tex":"u_3","aliases":"u_3","display":"u_3","kind":"use","excerpt":"; node[font=small,align=center,font=small] at (2.39,1.64) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,3.62) u_0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.96) u_1; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.3) u_2; node[font=small,align=center,anchor=east,font=scriptsize","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-65fa973c459a6b3b","tex":"U_c","aliases":"U_c","display":"U_c","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimensi","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-25121c20156ac918","tex":"U_m","aliases":"U_m","display":"U_m","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-7a6544df8b60d670","tex":"V^H","aliases":"V^H","display":"V^H","kind":"use","excerpt":"Checked group, inertia and monodromy finite sums (Block 12). Let K be a field, G a group, and ν:G→ K^× a character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficien","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"mall,align=center,font=small] at (9.37,2) 8; draw[->,line width=.8pt] (4.35,2.8) -- (7.65,2.8); node[font=small,align=center] at (6,3.3) multiply by C; node[font=small,align=center] at (3.2,1.15) nonzero modulo 8; node[font=small,align=center] at (10,1.15) zero modulo 8; node[font=small,align=center,font=footnotesize] at (7,0.25) Modulo 16, the tra","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"vpage7Many approximate roots, one limitSINGULAR POWER SEEDSMain text: Theorem 6.4 and Corollary 6.6. Now impose x^2=0 modulo 3^m. The condition is exactly 2v_3(x)≥ m, so the roots are the multiples of 3^lceil m/2rceil in that quotient. Their number is 3^m-lceil m/2rceil. vfig07_singular_forest headCount vertices and count branches separately There ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-91a52a41351c928c","tex":"V_i","aliases":"V_i","display":"V_i","kind":"use","excerpt":"The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"The actual invariant images U_i=imE_i give an equivariant linear isomorphism ⊕_i∈ JU_i≃ R^n, (v_i)_i⟼∑_i v_i, v⟼(E_iv)_i. Lean represents the finite direct sum by the finite product of the image submodules. The existing coordinatewise reduction theorem maps every U_i onto the corresponding reduced projector image. No chosen bases or unproved freene","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0c6259c535cb4a68","tex":"V_{i,\\lambda}","aliases":"V_i,λ lambda","display":"V_{i,\\lambda}","kind":"use","excerpt":"For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesse","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-c115186cc56f7111","tex":"V_i^{\\oplus m_i}","aliases":"V_i^⊕ m_i oplus","display":"V_i^{\\oplus m_i}","kind":"use","excerpt":"The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficien","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"use","excerpt":"Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement an","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-7fd7bfa7519bcf67","tex":"v_{n+1}","aliases":"v_n+1","display":"v_{n+1}","kind":"use","excerpt":"Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement an","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-51fd0818257204a0","tex":"v_r","aliases":"v_r","display":"v_r","kind":"use","excerpt":"[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficien","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-4b2b6ce643923566","tex":"V_\\lambda","aliases":"V_λ lambda","display":"V_\\lambda","kind":"use","excerpt":"arrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at ev","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-036484b5b6983b2e","tex":"V_{\\lambda_0}","aliases":"V_λ_0 lambda","display":"V_{\\lambda_0}","kind":"use","excerpt":"[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9d40461d4abcbdcc","tex":"v_\\pi","aliases":"v_π pi","display":"v_\\pi","kind":"use","excerpt":"For a DVR, e=v_π(|Γ|) is derived from the actual group order. The single precision e+1, independent of rank, detects integral isomorphism. It makes reduction injective on integral isomorphism classes, and finite coefficient quotients give finitely many integral classes in fixed rank. This is a formalization of classical finite-group rigidity (Maran","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-08b28ab9c3fd8503","tex":"v_\\varpi","aliases":"v_ϖ varpi","display":"v_\\varpi","kind":"use","excerpt":"The congruence is equivalent to e v_ϖ(x)≥ m, so x must lie in ϖ^lceil m/erceil O/ϖ^m, which has the stated cardinality. Write m=ek+r, 1≤ r≤ e. The exponent becomes (e-1)k+r-1. Summing over k≥0 and 1≤ r≤ e gives the rational expression.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"use","excerpt":"[Formal inverse-limit reconstruction] Assume M_bar x is represented by a complete local O-algebra R_bar x. Then reduction induces a natural bijection M_bar x( O) ≅ varprojlim_m M_bar x( O/ϖ^m).","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-9e2d9dcc048eb142","tex":"\\varprojlim_m","aliases":"varprojlim_m varprojlim","display":null,"kind":"use","excerpt":"[Formal inverse-limit reconstruction] Assume M_bar x is represented by a complete local O-algebra R_bar x. Then reduction induces a natural bijection M_bar x( O) ≅ varprojlim_m M_bar x( O/ϖ^m).","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-ea75ef5f3ceca490","tex":"\\varprojlim_{m\\ge0}","aliases":"varprojlim_m≥0 varprojlim ge","display":null,"kind":"use","excerpt":"Let I⊆ R and suppose that R is I-adically complete. Using the checked ring-map reconstruction on the quadratic coordinate ring, Lean proves P_q(R)≃varprojlim_m≥0 P_q(R/I^m). The zero-precision quotient is included only for the formal indexing. The inverse constructs an actual arithmetic character, with exactly the prescribed reductions. It is uniqu","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c0ed8b78702b6e2c","tex":"\\vol","aliases":"vol operatorname","display":"{\\operatorname{vol}}","kind":"use","excerpt":"There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.","count":21},{"work":"paper-x","url":"notation/paper-x.html#symbol-d01794b5f0c3ada0","tex":"W_1","aliases":"W_1","display":"W_1","kind":"use","excerpt":"[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficien","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-3106fac3e3c8f992","tex":"w_2","aliases":"w_2","display":"w_2","kind":"use","excerpt":"=black!14] (8,3.2) rectangle (9.3,3.87); node[font=small,align=center] at (8.65,3.55) 4; path[draw=black!35,fill=black!14] (9.35,3.2) rectangle (10.65,3.87); node[font=small,align=center] at (10,3.55) 5; path[draw=black!35,fill=black!14] (10.7,3.2) rectangle (12,3.87); node[font=small,align=center] at (11.35,3.55) 6; node[font=small,align=center,an","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9763a163e7ff6591","tex":"w_3","aliases":"w_3","display":"w_3","kind":"use","excerpt":"=black!14] (8,2.3) rectangle (9.3,2.97); node[font=small,align=center] at (8.65,2.65) 3; path[draw=black!35,fill=black!14] (9.35,2.3) rectangle (10.65,2.97); node[font=small,align=center] at (10,2.65) 4; path[draw=black!35,fill=black!14] (10.7,2.3) rectangle (12,2.97); node[font=small,align=center] at (11.35,2.65) 5; node[font=small,align=center,an","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f6867c80df30672f","tex":"w_d","aliases":"w_d","display":"w_d","kind":"use","excerpt":"[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-18864b29b043b6d8","tex":"W_i","aliases":"W_i","display":"W_i","kind":"use","excerpt":"System isomorphism and good-place overlaps (Block 25). The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality o","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"A simple weighted formal model already captures delayed visibility. Let w_i≥1 and constrain t_i∈(ϖ^w_i O+ϖ^m O)/ϖ^m O. Then coordinate i is invisible until m>w_i.","count":15},{"work":"paper-x","url":"notation/paper-x.html#symbol-4935e01c6a712aa2","tex":"W_{i,\\lambda}","aliases":"W_i,λ lambda","display":"W_{i,\\lambda}","kind":"use","excerpt":"cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j))","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-d4fd2df5cb7d2b75","tex":"W_{i,\\lambda}^{\\oplus m_i}","aliases":"W_i,λ^⊕ m_i lambda oplus","display":"W_{i,\\lambda}^{\\oplus m_i}","kind":"use","excerpt":"arrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at ev","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9ad2cf4a38ca0ffc","tex":"W_{i,\\lambda_0}^{\\oplus m_i}","aliases":"W_i,λ_0^⊕ m_i lambda oplus","display":"W_{i,\\lambda_0}^{\\oplus m_i}","kind":"use","excerpt":"[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-233d4eb90fcdc954","tex":"W_i^{\\oplus c_i}","aliases":"W_i^⊕ c_i oplus","display":"W_i^{\\oplus c_i}","kind":"use","excerpt":"[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ffbde798b9a7c58f","tex":"W_j","aliases":"W_j","display":"W_j","kind":"use","excerpt":"System isomorphism and good-place overlaps (Block 25). The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality o","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-bc64898552e9bfce","tex":"W_{j,\\lambda}","aliases":"W_j,λ lambda","display":"W_{j,\\lambda}","kind":"use","excerpt":"System isomorphism and good-place overlaps (Block 25). The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality o","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-4a4f3028bb492bb2","tex":"W_r","aliases":"W_r","display":"W_r","kind":"use","excerpt":"[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficien","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5704b00361e1b378","tex":"W_\\lambda","aliases":"W_λ lambda","display":"W_\\lambda","kind":"use","excerpt":"Use the same multiplicities in every member of the compatible carrier. Direct sums preserve strict compatibility, and m_i≤ c_i makes V_λ a summand of W_λ. At the separating places, the target trace vector is C m. Full column rank makes m the unique solution.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0bfb4d65b18a629f","tex":"\\operatorname{WD}","aliases":"WD operatorname","display":"\\operatorname{WD}","kind":"use","excerpt":"The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-371acb48389709be","tex":"\\mathrm{wt}","aliases":"wt mathrm","display":"\\mathrm{wt}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-d1d16f551dc7199c","tex":"x^{-2}","aliases":"x^-2","display":"x^{-2}","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f6af5d9dd57bd6ef","tex":"x^{-2-h}","aliases":"x^-2-h","display":"x^{-2-h}","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-a060e2b2a776bb2f","tex":"x^{-3/2}","aliases":"x^-3/2","display":"x^{-3/2}","kind":"use","excerpt":"why a power saving can make a Mellin expression regular at s=1. vfig27_mellin headCompute the endpoint rather than just naming it For real s>1/2, I(s)=s∫_1^∞ A(x)x^-s-1 dd x =fracss-1/2. At s=1+h this gives I(1+h)-I(1)=-frachh+1/2, |I(1+h)-I(1)|≤4|h| (|h|≤1/4). The derivative at one is -2. The logarithmic moment is finite as well: ∫_1^∞ x^-3/2log x","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-950685ba17204191","tex":"x^{-s-1}","aliases":"x^-s-1","display":"x^{-s-1}","kind":"use","excerpt":"IN ENDPOINT MODELMain text: §9; retained scalar/Mellin continuation: summatory measurability and first-order Mellin control. Use the explicit envelope A(x)=x^1/2 for x≥1. It is not an ideal-counting function. It is a model showing why a power saving can make a Mellin expression regular at s=1. vfig27_mellin headCompute the endpoint rather than just","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-770c571f5630bad2","tex":"X^{1-1/d}","aliases":"X^1-1/d","display":"X^{1-1/d}","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-1527cafbcb4af9cf","tex":"x^{1/2}","aliases":"x^1/2","display":"x^{1/2}","kind":"use","excerpt":"vpage27What a power-saving bound buys analyticallyA MELLIN ENDPOINT MODELMain text: §9; retained scalar/Mellin continuation: summatory measurability and first-order Mellin control. Use the explicit envelope A(x)=x^1/2 for x≥1. It is not an ideal-counting function. It is a model showing why a power saving can make a Mellin expression regular at s=1.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"An actual Z_3 example applies the class-level completion theorem for the cyclic group of order two and polynomial X^2-1, producing a nonidentity generator. Another checked example has compatible classes represented by diag(1,-1) modulo 3 and diag(-1,1) modulo 9: these chosen matrices disagree under reduction, and an actual swap of basis repairs the","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"[Fractional singular dimension] For x^e=0, dim_seed=1-frac1e. In particular, x^2=0 has branching dimension 1/2.","count":20},{"work":"paper-x","url":"notation/paper-x.html#symbol-44873f8baac650d8","tex":"x^e","aliases":"x^e","display":"x^e","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-48f4a03fbc552897","tex":"x^i","aliases":"x^i","display":"x^i","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-854c66c730d9b734","tex":"X^N","aliases":"X^N","display":"X^N","kind":"use","excerpt":"A local map sends each parameter into the image of (ϖ) in O/ϖ^m. Conversely, every tuple in that image is nilpotent and defines a coefficient-preserving evaluation homomorphism. This homomorphism is local. To prove uniqueness without assuming continuity of an arbitrary competing homomorphism, observe in one variable that a homomorphism killing X^N ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-d8c5e1feefd58efd","tex":"x^n","aliases":"x^n","display":"x^n","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"For canonical representatives 0≤ x_m<p^m, set x_0=0. Compatibility also gives a_m-1=(x_m-x_m-1)/p^m-1∈{0,…,p-1}. The floor in the first formula is essential. For example, the compatible base-five residues 4,19 give the second digit 3, not the rational number 19/5. The executable companion already uses integer floor division. This edition corrects t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"all,align=center,font=scriptsize] at (7.15,4.5) 6; node[font=small,align=center,font=scriptsize] at (8.12,4.5) 7; node[font=small,align=center,font=scriptsize] at (9.09,4.5) 8; node[font=small,align=center,font=scriptsize] at (10.06,4.5) 9; node[font=small,align=center,font=scriptsize] at (11.03,4.5) 10; node[font=small,align=center,anchor=east] at","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5adb86775ab61369","tex":"x_{1}","aliases":"x_1","display":"x_{1}","kind":"use","excerpt":"nt=scriptsize] at (5.35,4.45) 5^1; node[font=small,align=center,font=scriptsize] at (6.5,4.45) 5^2; node[font=small,align=center,font=scriptsize] at (7.65,4.45) 5^3; node[font=small,align=center,font=scriptsize] at (8.8,4.45) 5^4; node[font=small,align=center,font=scriptsize] at (9.95,4.45) 5^5; node[font=small,align=center,anchor=east] at (2.4,3.8","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"ath[fill=black] (9.09,3.93) circle (0.055); path[draw=black!25,fill=black!16] (9.61,3.68) rectangle (10.53,4.16); path[fill=black] (10.06,3.93) circle (0.055); path[draw=black!25,fill=black!16] (10.58,3.68) rectangle (11.5,4.16); path[fill=black] (11.03,3.93) circle (0.055); node[font=small,align=center] at (12.1,3.93) ⋯; node[font=small,align=cent","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1ee5dfebaaf979b7","tex":"x_{2}","aliases":"x_2","display":"x_{2}","kind":"use","excerpt":"r,font=scriptsize] at (7.65,3.83) textemdash; path[draw=black!25,fill=white] (8.25,3.58) rectangle (9.35,4.06); node[font=small,align=center,font=scriptsize] at (8.8,3.83) textemdash; path[draw=black!25,fill=white] (9.4,3.58) rectangle (10.5,4.06); node[font=small,align=center,font=scriptsize] at (9.95,3.83) textemdash; node[font=small,align=center","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-8eeb0a1bbb468424","tex":"X_3","aliases":"X_3","display":"X_3","kind":"use","excerpt":"ath[fill=black] (9.09,3.32) circle (0.055); path[draw=black!25,fill=black!16] (9.61,3.07) rectangle (10.53,3.55); path[fill=black] (10.06,3.32) circle (0.055); path[draw=black!25,fill=black!16] (10.58,3.07) rectangle (11.5,3.55); path[fill=black] (11.03,3.32) circle (0.055); node[font=small,align=center] at (12.1,3.32) ⋯; node[font=small,align=cent","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-943238c7278f4240","tex":"x_{3}","aliases":"x_3","display":"x_{3}","kind":"use","excerpt":"r,font=scriptsize] at (7.65,3.21) textemdash; path[draw=black!25,fill=white] (8.25,2.96) rectangle (9.35,3.44); node[font=small,align=center,font=scriptsize] at (8.8,3.21) textemdash; path[draw=black!25,fill=white] (9.4,2.96) rectangle (10.5,3.44); node[font=small,align=center,font=scriptsize] at (9.95,3.21) textemdash; node[font=small,align=center","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-fc227e811bd8d876","tex":"X_4","aliases":"X_4","display":"X_4","kind":"use","excerpt":"path[fill=black] (9.09,2.71) circle (0.055); path[draw=black!25,fill=black!16] (9.61,2.46) rectangle (10.53,2.94); path[fill=black] (10.06,2.71) circle (0.055); path[draw=black!25,fill=black!16] (10.58,2.46) rectangle (11.5,2.94); path[fill=black] (11.03,2.71) circle (0.055); node[font=small,align=center] at (12.1,2.71) ⋯; node[font=small,align=cen","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-d3af37e0991f5641","tex":"x_{4}","aliases":"x_4","display":"x_{4}","kind":"use","excerpt":"r,font=scriptsize] at (7.65,2.59) textemdash; path[draw=black!25,fill=white] (8.25,2.34) rectangle (9.35,2.82); node[font=small,align=center,font=scriptsize] at (8.8,2.59) textemdash; path[draw=black!25,fill=white] (9.4,2.34) rectangle (10.5,2.82); node[font=small,align=center,font=scriptsize] at (9.95,2.59) textemdash; node[font=small,align=center","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-863c43c6e558ee59","tex":"X_5","aliases":"X_5","display":"X_5","kind":"use","excerpt":"; path[fill=black] (9.09,2.1) circle (0.055); path[draw=black!25,fill=black!16] (9.61,1.85) rectangle (10.53,2.33); path[fill=black] (10.06,2.1) circle (0.055); path[draw=black!25,fill=black!16] (10.58,1.85) rectangle (11.5,2.33); path[fill=black] (11.03,2.1) circle (0.055); node[font=small,align=center] at (12.1,2.1) ⋯; node[font=small,align=cente","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-cd82234e20691e4c","tex":"x_{5}","aliases":"x_5","display":"x_{5}","kind":"use","excerpt":"lign=center,font=scriptsize] at (7.65,1.97) 3; path[draw=black!25,fill=white] (8.25,1.72) rectangle (9.35,2.2); node[font=small,align=center,font=scriptsize] at (8.8,1.97) textemdash; path[draw=black!25,fill=white] (9.4,1.72) rectangle (10.5,2.2); node[font=small,align=center,font=scriptsize] at (9.95,1.97) textemdash; node[font=small,align=center,","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-a7e5e2fc4457850a","tex":"X_6","aliases":"X_6","display":"X_6","kind":"use","excerpt":"ath[fill=black] (9.09,1.49) circle (0.055); path[draw=black!25,fill=black!16] (9.61,1.24) rectangle (10.53,1.72); path[fill=black] (10.06,1.49) circle (0.055); path[draw=black!25,fill=black!16] (10.58,1.24) rectangle (11.5,1.72); path[fill=black] (11.03,1.49) circle (0.055); node[font=small,align=center] at (12.1,1.49) ⋯; node[font=small,align=cent","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b10e71c86e4f30ef","tex":"x_{6}","aliases":"x_6","display":"x_{6}","kind":"use","excerpt":"all,align=center,font=scriptsize] at (7.65,1.35) 3; path[draw=black!25,fill=black!12] (8.25,1.1) rectangle (9.35,1.58); node[font=small,align=center,font=scriptsize] at (8.8,1.35) 3; path[draw=black!25,fill=white] (9.4,1.1) rectangle (10.5,1.58); node[font=small,align=center,font=scriptsize] at (9.95,1.35) textemdash; node[font=small,align=center,a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"use","excerpt":"vpage20A condition must survive the limitCLOSEDNESS IS NOT DECORATIONMain text: Theorems 8.2 and 11.1; the inverse-limit closure hypotheses. In ZZ_3, every term of the sequence x_k=3^k is nonzero. Yet x_k→0 in the three-adic topology. The property ``nonzero'' is not closed, so a limit of objects satisfying it can lose that property. vfig20_closedne","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","excerpt":"Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.","count":12},{"work":"paper-x","url":"notation/paper-x.html#symbol-7a7064d4243ed8b7","tex":"X_{m+1}","aliases":"X_m+1","display":"X_{m+1}","kind":"use","excerpt":"Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-2ce545975574982d","tex":"x_{m+1}","aliases":"x_m+1","display":"x_{m+1}","kind":"use","excerpt":"Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-80ceedbe9c553f98","tex":"x_{m-1}","aliases":"x_m-1","display":"x_{m-1}","kind":"use","excerpt":"For canonical representatives 0≤ x_m<p^m, set x_0=0. Compatibility also gives a_m-1=(x_m-x_m-1)/p^m-1∈{0,…,p-1}. The floor in the first formula is essential. For example, the compatible base-five residues 4,19 give the second digit 3, not the rational number 19/5. The executable companion already uses integer floor division. This edition corrects t","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-53687bf254c09aec","tex":"x_p","aliases":"x_p","display":"x_p","kind":"use","excerpt":"An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrig","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-59ca56eb5c1b813d","tex":"X_{\\lambda,1}","aliases":"X_λ,1 lambda","display":"X_{\\lambda,1}","kind":"use","excerpt":"[Residual companion obstruction] If X_λ,1( C)=∅ for one coefficient prime λ, no compatible completion with spine C exists. Deeper perturbations at other primes cannot create the missing residual companion.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[No crossing under positive-depth perturbation] Suppose x,y∈ M( O) are related by a deformation operation that is congruent to the identity modulo ϖ. Then red(x)=red(y). Every finite composition remains in one residual fiber. If an infinite perturbation stream converges in the ϖ-adic topology, its limit remains in the same fiber. The existence of t","count":16},{"work":"paper-x","url":"notation/paper-x.html#symbol-61c288ea4029d6a5","tex":"y^{-1}","aliases":"y^-1","display":"y^{-1}","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-bfc9fa04c4a74026","tex":"Y^{1-1/d}","aliases":"Y^1-1/d","display":"Y^{1-1/d}","kind":"use","excerpt":"attice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an e","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-de7ba1dbdac1562b","tex":"y^n","aliases":"y^n","display":"y^n","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-51c2ad6cea95e462","tex":"y^{n-1-i}","aliases":"y^n-1-i","display":"y^{n-1-i}","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z mathbf","display":"{\\mathbf Z}","kind":"use","excerpt":"[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet i","count":11},{"work":"paper-x","url":"notation/paper-x.html#symbol-75fbe28ce1b6d91b","tex":"\\mathbf Z","aliases":"Z mathbf","display":"\\mathbf Z","kind":"use","excerpt":"[Multiplicity recovery] The original companion matrix was not retrieved or rerun. A new exact realization uses the trivial character and two independent sign characters of ( Z/2 Z)^2, evaluated at (0,0),(1,0),(0,1): C=1 · 1 · 1","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"For i≠ j, orthogonality gives H(p_i)H(i_j)=0. Idempotence gives H(p_j)H(i_j)=I, and completeness gives ∑_j H(i_j)H(p_j)=I. Thus the maps [z]⟼([P_j^r z])_j, ([z_j])_j⟼[∑_j i_j z_j] are inverse linear maps. The finite product equals the direct sum, proving the assertion. In the formalization, both sides are also identified with Mathlib's standard hom","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-544ac8aed4a2c573","tex":"Z^1","aliases":"Z^1","display":"Z^1","kind":"use","excerpt":"[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-af5f0843f8046d2c","tex":"\\mathbb Z^2","aliases":"Z^2 mathbb","display":"\\mathbb Z^2","kind":"use","excerpt":"raw=black!22,fill=black!13] (4.5,3.65) rectangle (4.8,3.95); path[draw,fill=black] (4.65,3.8) circle (0.035); path[draw,line width=.8pt] (1.2,0.35) rectangle (2.7,1.85); path[draw,line width=.8pt] (1.2,1.85) rectangle (2.7,3.35); path[draw,line width=.8pt] (2.7,0.35) rectangle (4.2,1.85); path[draw,line width=.8pt] (2.7,1.85) rectangle (4.2,3.35); ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1766cb24e585f5e7","tex":"Z^r","aliases":"Z^r","display":"Z^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-a2077aebd37a8165","tex":"Z_{(1,2,4)}","aliases":"Z_(1,2,4)","display":"Z_{(1,2,4)}","kind":"use","excerpt":"For weights (1,2,4), the counts are 1,2,8,32,256,2048,…. The first coordinate is active immediately, the second one level later, and the third at depth five. The exact rational function generated by the code is Z_(1,2,4)(T) = fracT(1-6T-8T^2-32T^3)1-8T. The negative signs in this presentation arise from moving the denominator to 1-8T; every power-s","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-78a80f694d80e6e7","tex":"\\Z_2","aliases":"Z_2 Z mathbf","display":"{\\mathbf Z}_2","kind":"use","excerpt":"For R= Z_2, Lean derives finiteness of all positive maximal-ideal quotients and applies this completion theorem. Since Z_2 is a domain, its only roots of X^2-1 are 1 and -1. Thus the integral packet has exactly two distinct characters: the trivial character and the sign character describing the action on i. They have the same reduction modulo two a","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-370c87df70534fa4","tex":"\\mathbb Z_2","aliases":"Z_2 mathbb","display":"\\mathbb Z_2","kind":"use","excerpt":"The boundaries are formalized as well. Pullback along a nonsurjective homomorphism can make inequivalent irreducible representations isomorphic. Samples can cover one quotient while missing an open quotient of the full group. Finally, in Z_2, the distinct square roots 1 and -1 have the same residue, so the condition that the root order be a unit ca","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-9bb63c10f3cd1141","tex":"\\mathbb Z_3","aliases":"Z_3 mathbb","display":"\\mathbb Z_3","kind":"use","excerpt":"An actual Z_3 example applies the class-level completion theorem for the cyclic group of order two and polynomial X^2-1, producing a nonidentity generator. Another checked example has compatible classes represented by diag(1,-1) modulo 3 and diag(-1,1) modulo 9: these chosen matrices disagree under reduction, and an actual swap of basis repairs the","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-e584afe354e47377","tex":"\\Z_5","aliases":"Z_5 Z mathbf","display":"{\\mathbf Z}_5","kind":"use","excerpt":"d is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residu","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-347d071dc857b815","tex":"\\mathbb Z_5","aliases":"Z_5 mathbb","display":"\\mathbb Z_5","kind":"use","excerpt":"athbb F_2; path[draw,fill=black!12] (2.1,2.55) circle (0.24); node[font=small,align=center] at (2.1,2.55) 1; node[font=small,align=center] at (2.1,1.45) one point mod 2; node[font=small,align=center,font=footnotesize] at (2.1,0.55) no point mod 4; node[font=small,align=center] at (6.9,4.35) A coherent branch; node[font=small,align=center] at (6.9,3","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0ff735e21f188d41","tex":"Z_d","aliases":"Z_d","display":"Z_d","kind":"use","excerpt":"[Smooth seed zeta function] For a smooth formal component of dimension d, Z_d(T)=fracT1-q^dT.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-483b6af63cf90eb6","tex":"Z_e","aliases":"Z_e","display":"Z_e","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ae33ff0c9a9cfbb4","tex":"z_j","aliases":"z_j","display":"z_j","kind":"use","excerpt":"For i≠ j, orthogonality gives H(p_i)H(i_j)=0. Idempotence gives H(p_j)H(i_j)=I, and completeness gives ∑_j H(i_j)H(p_j)=I. Thus the maps [z]⟼([P_j^r z])_j, ([z_j])_j⟼[∑_j i_j z_j] are inverse linear maps. The finite product equals the direct sum, proving the assertion. In the formalization, both sides are also identified with Mathlib's standard hom","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-69c97e2f51fdcaeb","tex":"Z_j^r","aliases":"Z_j^r","display":"Z_j^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b910972bdb78c268","tex":"\\Z_p","aliases":"Z_p Z mathbf","display":"{\\mathbf Z}_p","kind":"use","excerpt":"For every prime p, Lean constructs a valuation subring A⊆overline Q_p containing the integral image of Z_p such that the induced map Z_p→ A is local. For odd p, two is a unit. The preceding argument applies to every valuation subring containing that integral image, and hence every member of P_q(B) is trivial on its inertia subgroup after actual loc","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-cd569bf5d0c46f05","tex":"\\Z_{\\ge0}","aliases":"Z_≥0 Z ge mathbf ge","display":"{\\mathbf Z}_{\\ge0}","kind":"use","excerpt":"[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet i","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-42f5a664e208866e","tex":"\\ZZ^2","aliases":"ZZ^2 ZZ","display":null,"kind":"use","excerpt":"vpage26Cancellation leaves a boundary problemA FULL LATTICE-BOX CALCULATIONMain text: §9; retained scalar/Mellin continuation: additive residue counts and norm-character bounds. Extend the function f(a,b)=χ(a^2-2b^2bmod5) periodically to ZZ^2. Every complete 5×5 residue tile has sum zero by the finite-field calculation. Only incomplete tiles can co","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b5c3968f000169a0","tex":"\\ZZ_2","aliases":"ZZ_2 ZZ","display":null,"kind":"use","excerpt":"vpage6Weights delay visibilityWEIGHTED DYADIC CHARTSMain text: Proposition 5.3, Theorem 6.3, and §12.2. Over ZZ_2, require three coordinates to lie in 2ZZ_2, 2^2ZZ_2, and 2^4ZZ_2. At depth m, coordinate i has max(m-w_i,0) free binary digits. A zero in the table means only one allowed residue: zero. vfig06_weights headThe two different thresholds A ","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-ee416997480b5c73","tex":"\\ZZ_3","aliases":"ZZ_3 ZZ","display":null,"kind":"use","excerpt":"vpage5A smooth chart has independent choicesACTUAL FINITE POINTSMain text: Proposition 3.4 and Theorem 6.2. Take R=ZZ_3[[t_1,t_2]] and coefficient-preserving local maps to ZZ/27ZZ. Locality forces both parameters into the maximal ideal. Each therefore has nine possible values, 0,3,6,…,24. vfig05_smooth_grid headWhy the exponent is d(m-1) At depth m","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-a7e01f582342edb9","tex":"\\ZZ_5","aliases":"ZZ_5 ZZ","display":null,"kind":"use","excerpt":"iv1pmod5^m, not by taking decimal approximations. At the first six depths they are 4,19,94,469,2344,11719. vfig04_digits headCompatibility is stronger than independent solvability The residues satisfy both x_m+1≡ x_mpmod5^m, 4x_m-1=3·5^m. The latter identity shows that the residual error tends to zero in the five-adic norm. The stabilized digit seq","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-6820374f59f0b332","tex":"|^{-1}","aliases":"|^-1","display":"|^{-1}","kind":"use","excerpt":"Derived comparison bounds (Block 16). For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its det","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b8fe6b5eba5cc87a","tex":"|_3","aliases":"|_3","display":"|_3","kind":"use","excerpt":"requires a proof that the chosen condition is preserved by the limit under construction. The word ``admissible'' does not itself provide that proof. takeawayCompatibility reconstructs an object. Closedness is what carries the declared equations or conditions to that object. captiontextThe finite diagram displays the retreat of the first nonzero dig","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-5b555c064a65341d","tex":"|_{\\ker N}","aliases":"|_ker N ker","display":"|_{\\ker N}","kind":"use","excerpt":"These statements concern supplied algebraic pairs. No Weil-group or inertia action, Hodge--Tate object, arithmetic comparison, or realization of the pair from a Galois representation is constructed here. The polynomial convention is det(XI-F|_ker N); no identification with an inertia-dependent local Euler factor is asserted. Claim 9.2 remains parti","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-8e1af9e562f5ced8","tex":"|_{\\ker N_i}","aliases":"|_ker N_i ker","display":"|_{\\ker N_i}","kind":"use","excerpt":",N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant linear isomorphism ker N_m≃∏_i∈ I∏_k<m_iker N_i. The relation, with q≠0, proves that both F_i and its inverse preserve ker N_i. ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-e3909971d4ebfd26","tex":"|_{\\ker N_m}","aliases":"|_ker N_m ker","display":"|_{\\ker N_m}","kind":"use","excerpt":"and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant linear isomorphism ker N_m≃∏_i∈ I∏_k<m_iker N_i. The relation, with q≠0, proves that both F_i and its inverse preserve ker N_i. Thus F_i|_ker N_i is an actual automorphism, and Lean pro","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"An actual unit action on ideal residues. Fix m>0 and a nonzero integral ideal J. Multiplication by u∈( O/m O)^× gives a proved permutation of J/mJ. Its inverse follows from (ab-1)x∈ mJ when ab-1∈ m O and x∈ J. No surjectivity of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":17},{"work":"paper-x","url":"notation/paper-x.html#symbol-0acecd7697ebad7a","tex":"\\times_{\\Compat}","aliases":"×_Compat ×_ C times Compat times mathfrak","display":"\\times_{{\\mathfrak C}}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"Derived comparison bounds (Block 16). For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its det","count":14},{"work":"paper-x","url":"notation/paper-x.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"Absolute Galois restriction and arithmetic witnesses (Block 26). Fix a cyclotomic number field L= Q(ζ_n) with n>0 and an embedding j:L↪overline Q. The new restriction map q_L:G_ Q⟶Gal(L/ Q), j(q_L(γ)x)=γ(j(x)), is an actual continuous surjective homomorphism with open kernel. Surjectivity is derived by extending automorphisms to the algebraic closu","count":11},{"work":"paper-x","url":"notation/paper-x.html#symbol-51cadb8120940875","tex":"\\gamma_j","aliases":"γ_j gamma","display":"\\gamma_j","kind":"use","excerpt":"compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-3e52bdbf6bbe4050","tex":"\\gamma_p","aliases":"γ_p gamma","display":"\\gamma_p","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-8df3dddd03938b61","tex":"\\gamma_u","aliases":"γ_u gamma","display":"\\gamma_u","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-7db9f39168dc2833","tex":"\\gamma_v","aliases":"γ_v gamma","display":"\\gamma_v","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-27e33acc1ffe034a","tex":"\\gamma_v^{-1}","aliases":"γ_v^-1 gamma","display":"\\gamma_v^{-1}","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1a991337b8d11cca","tex":"\\Delta_K","aliases":"Δ_K Delta","display":"\\Delta_K","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"For ε∈{1,-1} define overline n_ε(a)=εoverline n(a). On the corresponding sign portion, N(x)bmod m=overline n_ε([x]). The sign cannot be dropped: -1≡2pmod3, whereas N(-1)=1 and N(2)=2 for principal ideals of Z.","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-88c0820439bcc4b2","tex":"\\zeta_4","aliases":"ζ_4 zeta","display":"\\zeta_4","kind":"use","excerpt":"ite set, with nonzero determinant. Every coefficient embedding preserves its nonvanishing; actual member trace comparisons retain that member's good-prime condition. No coverage or character-continuity hypothesis is added in this cyclotomic specialization: the prime coverage is proved, and the finite extension's Krull topology is discrete. A concre","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-e6c934809237f301","tex":"\\zeta_n","aliases":"ζ_n zeta","display":"\\zeta_n","kind":"use","excerpt":"Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-5952c7e68d3d79b8","tex":"\\zeta_n^p","aliases":"ζ_n^p zeta","display":"\\zeta_n^p","kind":"use","excerpt":"Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"[Odd and dyadic nonlinear equations] At odd residue characteristic, a differential graded Lie model may be written in Maurer--Cartan form dη+frac12[η,η]=0, and a cyclic grading converts this into exact packet convolutions. At residue characteristic two, division by 2 is not integral. A quadratic or divided-square refinement must be retained. This i","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-08977dc59c5957ea","tex":"\\iota_\\lambda","aliases":"ι_λ iota lambda","display":"\\iota_\\lambda","kind":"use","excerpt":"le and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding preser","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7238fc50817cce55","tex":"\\lambda_0","aliases":"λ_0 lambda","display":"\\lambda_0","kind":"use","excerpt":"Suppose two members are isomorphic at λ_0. Injectivity of the common coefficient embedding gives t_i(v)=t_j(v) at every place good for λ_0. At another coefficient prime λ, this equality compares actual member traces on the intersection of the two good-place sets. Conjugacy density of that intersection and continuity extend the equality to the whole","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"Checked group, inertia and monodromy finite sums (Block 12). Let K be a field, G a group, and ν:G→ K^× a character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant ","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-f174b662c92183fc","tex":"\\varpi^2","aliases":"ϖ^2 varpi","display":"\\varpi^2","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0333e1da9f88d77d","tex":"\\pi^c","aliases":"π^c pi","display":"\\pi^c","kind":"use","excerpt":"For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-4ce2f60287017fac","tex":"\\pi^{cn}","aliases":"π^cn pi","display":"\\pi^{cn}","kind":"use","excerpt":"R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, d","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-bc344e45856ed8a8","tex":"\\pi^{cn+1}","aliases":"π^cn+1 pi","display":"\\pi^{cn+1}","kind":"use","excerpt":"R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, d","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-462bf49f74c1b15a","tex":"\\pi^e","aliases":"π^e pi","display":"\\pi^e","kind":"use","excerpt":"Finite-group comparison with precision loss (Block 17). Let R be a commutative domain, Γ a finite group, and ρ,τ actual integral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-a3cd9f08d1ff6f7d","tex":"\\pi^{e+k}","aliases":"π^e+k pi","display":"\\pi^{e+k}","kind":"use","excerpt":"a commutative domain, Γ a finite group, and ρ,τ actual integral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E i","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-532d6911acc103ff","tex":"\\varpi^{em}","aliases":"ϖ^em varpi","display":"\\varpi^{em}","kind":"use","excerpt":"Counting exponent versus compatible branches. The preceding exponent counts all finite-level solutions, including nodes which cannot lift indefinitely. It is not the cardinality or dimension of the space of infinite branches. In fact every compatible tower of solutions to x^e=0, for e≥1, is the zero tower. For each m, a solution at depth em lies in","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f5eb5dc2722ab8e4","tex":"\\varpi^j","aliases":"ϖ^j varpi","display":"\\varpi^j","kind":"use","excerpt":"For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-fc5afa9dc6cc1828","tex":"\\pi^k","aliases":"π^k pi","display":"\\pi^k","kind":"use","excerpt":"tegral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E integral. Choose b with π^e=ab and put P=A+bπ^kE. Then aP=","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b8a5ad8506c158aa","tex":"\\varpi^{\\lceil m/e\\rceil}","aliases":"ϖ^lceil m/erceil varpi lceil rceil","display":"\\varpi^{\\lceil m/e\\rceil}","kind":"use","excerpt":"The congruence is equivalent to e v_ϖ(x)≥ m, so x must lie in ϖ^lceil m/erceil O/ϖ^m, which has the stated cardinality. Write m=ek+r, 1≤ r≤ e. The exponent becomes (e-1)k+r-1. Summing over k≥0 and 1≤ r≤ e gives the rational expression.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-42ba626e1f091ffe","tex":"\\pi^n","aliases":"π^n pi","display":"\\pi^n","kind":"use","excerpt":"0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-d5468581325c1fe5","tex":"\\varpi^{n+1}","aliases":"ϖ^n+1 varpi","display":"\\varpi^{n+1}","kind":"use","excerpt":"For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-67582a9a61e97f9d","tex":"\\varpi^{w_i}","aliases":"ϖ^w_i varpi","display":"\\varpi^{w_i}","kind":"use","excerpt":"A simple weighted formal model already captures delayed visibility. Let w_i≥1 and constrain t_i∈(ϖ^w_i O+ϖ^m O)/ϖ^m O. Then coordinate i is invisible until m>w_i.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-737cdd19f67dfe28","tex":"\\rho_1","aliases":"ρ_1 rho","display":"\\rho_1","kind":"use","excerpt":"0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9bf4e85984084b75","tex":"\\rho_a","aliases":"ρ_a rho","display":"\\rho_a","kind":"use","excerpt":"The bound is substantive: for the unipotent actions ρ_a(g)=1 · ag 0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-3b48fafc33330554","tex":"\\rho_c","aliases":"ρ_c rho","display":"\\rho_c","kind":"use","excerpt":"≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U_c→ U_m by restriction to the selected copies. Lean proves rj=1_U_m, ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-93b3fea452ee88ac","tex":"\\rho_D","aliases":"ρ_D rho","display":"\\rho_D","kind":"use","excerpt":"For supplied covariance data and the twisted cocycle of Block 13, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-80f34d32227ee3d9","tex":"\\rho_E","aliases":"ρ_E rho","display":"\\rho_E","kind":"use","excerpt":"For supplied covariance data and the twisted cocycle of Block 13, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-56a93667702b9f53","tex":"\\rho_{i,\\lambda}","aliases":"ρ_i,λ rho lambda","display":"\\rho_{i,\\lambda}","kind":"use","excerpt":"For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesse","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-18406af98fd43046","tex":"\\rho_m","aliases":"ρ_m rho","display":"\\rho_m","kind":"use","excerpt":"Checked representation-level completion and its arithmetic boundary. The eighth formalization block works with actual matrix representations of a fixed group, not only with underlying sets. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Mu","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-c03851211fa61002","tex":"\\rho_n","aliases":"ρ_n rho","display":"\\rho_n","kind":"use","excerpt":"Local properties need their own limit argument. Over a domain, for a nonzero π∈ I, the representations of the additive integer group given by ρ_n(g)=1 · π^n g 0 · 1 are all nontrivial and converge pointwise in the ideal-adic topology to the trivial representation. The convergence and the failure of limit preservation for nontriviality are formally ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-fc125fdd0aa28df4","tex":"\\rho_\\lambda","aliases":"ρ_λ rho lambda","display":"\\rho_\\lambda","kind":"use","excerpt":"pine C over a number field M is fixed; item for every finite place λ of M and every m≥1, the companion level X_λ,m( C) is finite and nonempty; item the reduction forests are finitely branching; item the selected local conditions are closed under inverse limits. Then for every λ there is an infinite compatible branch ρ_λ∈varprojlim_mX_λ,m( C), and t","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1 0 · -1, P=1 · 1 0 · 2 at a nonidentity quadratic element. The dete","count":15},{"work":"paper-x","url":"notation/paper-x.html#symbol-2792c54819372788","tex":"\\sigma_p","aliases":"σ_p sigma","display":"\\sigma_p","kind":"use","excerpt":"Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_","count":6},{"work":"paper-x","url":"notation/paper-x.html#symbol-65a866ec8742ecbd","tex":"\\sigma_\\lambda","aliases":"σ_λ sigma lambda","display":"\\sigma_\\lambda","kind":"use","excerpt":"ion] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision","count":12},{"work":"paper-x","url":"notation/paper-x.html#symbol-0d3cf3e8170f0db5","tex":"\\tau_s","aliases":"τ_s tau","display":"\\tau_s","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-bb1c0f1580fe3093","tex":"\\tau_{s,m}","aliases":"τ_s,m tau","display":"\\tau_{s,m}","kind":"use","excerpt":"cates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissibl","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant subspaces. For normal H, the restricted whole-group representation and its inverse automorphisms are cons","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-74777c98027fb1ad","tex":"\\phi_s","aliases":"ϕ_s phi","display":"\\phi_s","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-bc965a9b4f0047d0","tex":"\\chi_+","aliases":"χ_+ chi","display":"\\chi_+","kind":"use","excerpt":"[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-c9f755de0838621d","tex":"\\chi_-","aliases":"χ_- chi","display":"\\chi_-","kind":"use","excerpt":"[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ee549737b8141ac9","tex":"\\chi_4","aliases":"χ_4 chi","display":"\\chi_4","kind":"use","excerpt":"ter may still be exceptionalTHE EXACT SCALAR TESTMain text: §9; scalar/Mellin checkpoint: scalar-power and nonscalar-witness boundaries. A scalar a∈FF_5^× acts on a degree-d algebra with determinant a^d. Therefore the scalar cancellation route tests χ(a^d), not merely χ(a). vfig24_scalar_exception headA failed sufficient test is not an impossibilit","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-75c12a975fdb52b9","tex":"\\chi_4^k","aliases":"χ_4^k chi","display":"\\chi_4^k","kind":"use","excerpt":"ar/Mellin checkpoint: scalar-power and nonscalar-witness boundaries. A scalar a∈FF_5^× acts on a degree-d algebra with determinant a^d. Therefore the scalar cancellation route tests χ(a^d), not merely χ(a). vfig24_scalar_exception headA failed sufficient test is not an impossibility theorem Let χ_4(2)=i generate the character group of FF_5^×, and w","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-61c6f0738ee445dc","tex":"\\chi_i","aliases":"χ_i chi","display":"\\chi_i","kind":"use","excerpt":"For a finite family of nontrivial ordinary class characters, suppose functions p,Z satisfy h p(h)→ P≠0, h Z(h)→ Q≠0, and, for all sufficiently small h>0, Z(h)=(p(h)∏_i F_χ_i(h))^e, e∈ N. The checked real-axis criterion then gives e=1 and F_χ_i(0)≠0 for every i. The pole limits and product identity remain hypotheses of this application.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-942ea15f41bce3dd","tex":"\\chi_K","aliases":"χ_K chi","display":"\\chi_K","kind":"use","excerpt":"For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor ","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-a6b1ce179cf1ad9c","tex":"\\chi_k","aliases":"χ_k chi","display":"\\chi_k","kind":"use","excerpt":",align=center,font=scriptsize] at (8.5,1.05) no; path[draw=black!30,fill=black!16] (9.3,0.75) rectangle (10.7,1.33); node[font=small,align=center,font=scriptsize] at (10,1.05) yes; path[draw=black!30,fill=black!16] (10.8,0.75) rectangle (12.2,1.33); node[font=small,align=center,font=scriptsize] at (11.5,1.05) yes; node[font=small,align=center,font=","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-c131c2c63bfd4ae6","tex":"\\chi_{\\Q_{p_s}}","aliases":"χ_Q_p_s χ_ Q_p_s chi Q chi mathbf","display":"\\chi_{{\\mathbf Q}_{p_s}}","kind":"use","excerpt":"Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"ation preserves the residual root, and induction proves the finite statement. For a convergent stream, convergence at the first finite precision makes its reduction eventually equal to that of its limit. Every term already has the initial reduction, so the limit has that reduction too. Completeness applies to Cauchy streams; it does not make every ","count":16},{"work":"paper-x","url":"notation/paper-x.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"bibitemLarsenPink1992 Michael Larsen and Richard Pink. newblock On ℓ-independence of algebraic monodromy groups in compatible systems of representations. newblock em Inventiones Mathematicae, 107(3):603--636, 1992.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Exact residual partition] The finite implementation partitions the integers 0,…,8 into the three classes modulo 3: {0,3,6}, {1,4,7}, {2,5,8}. Every tested correction of the form x↦ x+p h preserves the residual class exactly.","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"System isomorphism and good-place overlaps (Block 25). The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality o","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrig","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"[Residual companion obstruction] If X_λ,1( C)=∅ for one coefficient prime λ, no compatible completion with spine C exists. Deeper perturbations at other primes cannot create the missing residual companion.","count":5},{"work":"paper-x","url":"notation/paper-x.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"ixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence st","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":13},{"work":"paper-x","url":"notation/paper-x.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"eratornameTr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is the image of ∏_iP_i,g^m_i. App","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-24ff8e3cb9198a85","tex":"\\prod_{i=1}^{3}","aliases":"∏_i=1^3 prod","display":"\\prod_{i=1}^{3}","kind":"use","excerpt":"For target (0,1,1), every pair of conditions is occupied: (0,1,0) witnesses the first two, (0,0,1) the first and third, and (1,1,1) the second and third. But the triple itself is absent. Multiplying all three selectors gives the zero matrix. E_ a=∏_i=1^3 I-T_i, · a_i=0, T_i, · a_i=1. Tr(E_ a)=rank(E_ a). The checks compute all eight joint projector","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-746078d5778f9868","tex":"\\prod_{i=1}^{N}","aliases":"∏_i=1^N prod","display":"\\prod_{i=1}^{N}","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9c0ad644774e2f10","tex":"\\prod_{i\\in I}","aliases":"∏_i∈ I prod in","display":"\\prod_{i\\in I}","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-0bafcdd9e08caf20","tex":"\\prod_{j=0}^{n-1}","aliases":"∏_j=0^n-1 prod","display":"\\prod_{j=0}^{n-1}","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion k","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-d3485e5cffde49dc","tex":"\\prod_{k<c_i}","aliases":"∏_k<c_i prod","display":"\\prod_{k<c_i}","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-02bccd9d77338218","tex":"\\prod_{k<m_i}","aliases":"∏_k<m_i prod","display":"\\prod_{k<m_i}","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-025def8b36bca9d3","tex":"\\prod_{\\lambda\\nmid2}'","aliases":"∏_λnmid2' prod lambda nmid","display":null,"kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9a509fc9aa63176e","tex":"\\prod_{\\mu\\in\\Spec(T_i)\\setminus\\{a_i\\}}","aliases":"∏_μ∈Spec(T_i)∖{a_i} prod mu in Spec setminus prod mu in operatorname setminus","display":"\\prod_{\\mu\\in{\\operatorname{Spec}}(T_i)\\setminus\\{a_i\\}}","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-d37d3047efa1bd65","tex":"\\prod_{\\mu\\in\\Spec(T_i)\\setminus\\{a\\}}","aliases":"∏_μ∈Spec(T_i)∖{a} prod mu in Spec setminus prod mu in operatorname setminus","display":"\\prod_{\\mu\\in{\\operatorname{Spec}}(T_i)\\setminus\\{a\\}}","kind":"use","excerpt":"For each operator, its finite set of actual eigenvalues is used without multiplicity. The polynomial p_i,a(X)=∏_μ∈Spec(T_i)∖{a} (a-μ)^-1(X-μ) is used when a occurs, and the zero polynomial otherwise. For an eigenvector T_i v=bv, evaluation gives p_i,a(T_i)v=v · b=a,","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-1662ca15bb05c694","tex":"\\prod_{\\sigma}","aliases":"∏_σ prod sigma","display":"\\prod_{\\sigma}","kind":"use","excerpt":"The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no bound","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-bb27320775ae67b3","tex":"\\prod_{\\tau}","aliases":"∏_τ prod tau","display":"\\prod_{\\tau}","kind":"use","excerpt":"The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no bound","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"The estimate is proved from the actual boundary charts and the fixed-lattice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-2184640e3d6bfd1d","tex":"\\sum_{a\\in J/M}","aliases":"∑_a∈ J/M sum in","display":"\\sum_{a\\in J/M}","kind":"use","excerpt":"The estimate is proved from the actual boundary charts and the fixed-lattice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-8635bf72798fef18","tex":"\\sum_{a\\in J/mJ}","aliases":"∑_a∈ J/mJ sum in","display":"\\sum_{a\\in J/mJ}","kind":"use","excerpt":"ty of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(u· a)=overline n(u) overline n_J(a). For a multiplicative χ: Z/m Z→ C preserving 1, an exhibited unit with u∈( O/m O)^×, χ(overline n(u))≠1 therefore forces ∑_a∈ J/mJχ(s overline n_J(a))=0 (s∈ Z/m Z). Taking s=1,-1 gives th","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-59a73cc552080ec9","tex":"\\sum_C","aliases":"∑_C sum","display":"\\sum_C","kind":"use","excerpt":"Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hy","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-707c0697ba3640e2","tex":"\\sum_{g\\in\\Gamma}","aliases":"∑_g∈Γ sum in Gamma","display":"\\sum_{g\\in\\Gamma}","kind":"use","excerpt":"Derived comparison bounds (Block 16). For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its det","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-67b8cfba70c769c5","tex":"\\sum_{i=0}^{n-1}","aliases":"∑_i=0^n-1 sum","display":"\\sum_{i=0}^{n-1}","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-b7a8cfbba485d0c4","tex":"\\sum_{i=1}^d","aliases":"∑_i=1^d sum","display":"\\sum_{i=1}^d","kind":"use","excerpt":"[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-7fec58bf80198971","tex":"\\sum_{i\\in I}","aliases":"∑_i∈ I sum in","display":"\\sum_{i\\in I}","kind":"use","excerpt":"The same formalization proves, for each g∈ G, det(XI-ρ_m(g)) =∏_i∈ Idet(XI-ρ_i(g))^m_i, Tr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under ","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-70c8b15d042e9dc0","tex":"\\sum_{i\\in J}","aliases":"∑_i∈ J sum in","display":"\\sum_{i\\in J}","kind":"use","excerpt":"Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-d57a4808dbd2b57e","tex":"\\sum_{j=0}^{d-1}","aliases":"∑_j=0^d-1 sum","display":"\\sum_{j=0}^{d-1}","kind":"use","excerpt":"Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the represent","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-cd8d17b8ac90c2aa","tex":"\\sum_{j=1}^{d-1}","aliases":"∑_j=1^d-1 sum","display":"\\sum_{j=1}^{d-1}","kind":"use","excerpt":"General finite logarithmic recovery (Block 20). Let A be a rational algebra, without any commutativity assumption on A. For a unipotent element U and a positive integer d satisfying (U-1)^d=0, put L_d(U)=∑_j=1^d-1frac(-1)^j+1j(U-1)^j. The value is independent of the chosen annihilating exponent. The finite logarithm and the nilpotent exponential sa","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-ef75e15ec0684e46","tex":"\\sum_{j=1}^{n}","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^{n}","kind":"use","excerpt":"For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-516d97059e6505f6","tex":"\\sum_{m=1}^{w_d}","aliases":"∑_m=1^w_d sum","display":"\\sum_{m=1}^{w_d}","kind":"use","excerpt":"[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-fa57f8e9d7805c7c","tex":"\\sum_{r=1}^{e}","aliases":"∑_r=1^e sum","display":"\\sum_{r=1}^{e}","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-70e4a00162aa41df","tex":"\\sum_{\\substack{I\\ne0\\\\\\Nm I\\leq X}}","aliases":"∑_substackI≠0 Nm I≤ X ∑_substackI≠0 N I≤ X sum substack ne Nm leq sum substack ne operatorname leq","display":"\\sum_{\\substack{I\\ne0\\\\{\\operatorname N} I\\leq X}}","kind":"use","excerpt":"e preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substack","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-6109badb9f9fb1b3","tex":"\\sum_{\\substack{I\\ne0,\\ [I]=C\\\\\\Nm I\\leq X}}","aliases":"∑_substackI≠0, [I]=C Nm I≤ X ∑_substackI≠0, [I]=C N I≤ X sum substack ne Nm leq sum substack ne operatorname leq","display":"\\sum_{\\substack{I\\ne0,\\ [I]=C\\\\{\\operatorname N} I\\leq X}}","kind":"use","excerpt":"The assembled ideal norm-character estimate. Let C∈Cl( O), let J≠0 be an integral ideal with [J]=C^-1, and let m>0. Let χ: Z/m Z→ C preserve multiplication, 0, and 1. Assume a residue-ring unit satisfying [source reference], and assume χ( N Jbmod m)≠0. Then T_C,χ(X) =∑_substackI≠0, [I]=C N I≤ X χ( N Ibmod m) =O_K,J,m,χ(X^1-1/d). More precisely, som","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-eb512622d3091742","tex":"\\sum_{z\\in A}","aliases":"∑_z∈ A sum in","display":"\\sum_{z\\in A}","kind":"use","excerpt":"tained scalar/Mellin continuation: norm-residue lattice mean cancellation. Let χ be the quadratic character modulo five, extended by χ(0)=0. In the field on the preceding page, choose v=2u. Its norm is N(v)=2, so χ(N(v))=-1. vfig23_mean_cancellation headThe whole sum is paired with its negative Multiplication by v permutes the 25 elements. Norm mul","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesse","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"Checked group, inertia and monodromy finite sums (Block 12). Let K be a field, G a group, and ν:G→ K^× a character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant ","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"The remaining prime-existence obligation now has a precise formulation. For every finite Galois subfield L, every finite set B of rational primes, and every σ∈Gal(L/ Q), require a prime p∉ B∪ S_L and an actual absolute arithmetic witness γ_p such that q_L(γ_p)∼σ. Lean proves that this assertion is equivalent to the cofinite quotient coverage requir","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":4},{"work":"paper-x","url":"notation/paper-x.html#symbol-9177766aca870d95","tex":"\\int_1^\\infty","aliases":"∫_1^∞ int infty","display":"\\int_1^\\infty","kind":"use","excerpt":"IN ENDPOINT MODELMain text: §9; retained scalar/Mellin continuation: summatory measurability and first-order Mellin control. Use the explicit envelope A(x)=x^1/2 for x≥1. It is not an ideal-counting function. It is a model showing why a power saving can make a Mellin expression regular at s=1. vfig27_mellin headCompute the endpoint rather than just","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-297d1af96e467fcb","tex":"\\int_1^{\\infty}","aliases":"∫_1^∞ int infty","display":"\\int_1^{\\infty}","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"The remaining prime-existence obligation now has a precise formulation. For every finite Galois subfield L, every finite set B of rational primes, and every σ∈Gal(L/ Q), require a prime p∉ B∪ S_L and an actual absolute arithmetic witness γ_p such that q_L(γ_p)∼σ. Lean proves that this assertion is equivalent to the cofinite quotient coverage requir","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"[Formal inverse-limit reconstruction] Assume M_bar x is represented by a complete local O-algebra R_bar x. Then reduction induces a natural bijection M_bar x( O) ≅ varprojlim_m M_bar x( O/ϖ^m).","count":7},{"work":"paper-x","url":"notation/paper-x.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"The image of ϖ^w_i O in O/ϖ^m has one element when m≤ w_i, and q^m-w_i elements when m>w_i. Multiplying the independent coordinate counts gives the formula.","count":16},{"work":"paper-x","url":"notation/paper-x.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":8},{"work":"paper-x","url":"notation/paper-x.html#symbol-61f317edc40cbb12","tex":"\\bigoplus_{i=1}^{r}","aliases":"⊕_i=1^r bigoplus","display":"\\bigoplus_{i=1}^{r}","kind":"use","excerpt":"[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible ","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-32fcd883ade4f816","tex":"\\bigoplus_{i\\in J}","aliases":"⊕_i∈ J bigoplus in","display":"\\bigoplus_{i\\in J}","kind":"use","excerpt":"The actual invariant images U_i=imE_i give an equivariant linear isomorphism ⊕_i∈ JU_i≃ R^n, (v_i)_i⟼∑_i v_i, v⟼(E_iv)_i. Lean represents the finite direct sum by the finite product of the image submodules. The existing coordinatewise reduction theorem maps every U_i onto the corresponding reduced projector image. No chosen bases or unproved freene","count":1},{"work":"paper-x","url":"notation/paper-x.html#symbol-ce7f3904634c8e1a","tex":"\\bigoplus_j","aliases":"⊕_j bigoplus","display":"\\bigoplus_j","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":2},{"work":"paper-x","url":"notation/paper-x.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":10},{"work":"paper-x","url":"notation/paper-x.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"node[font=small,align=center,font=scriptsize] at (5.445,3.53) u_0; node[font=small,align=center,font=scriptsize] at (6.135,3.53) u_1; node[font=small,align=center,font=scriptsize] at (6.825,3.53) u_2; node[font=small,align=center,font=scriptsize] at (7.515,3.53) u_3; node[font=small,align=center] at (11,3.7) d_1d_0=0; node[font=small,align=center] ","count":3},{"work":"paper-x","url":"notation/paper-x.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"node[font=small,align=center,font=scriptsize] at (5.445,3.53) u_0; node[font=small,align=center,font=scriptsize] at (6.135,3.53) u_1; node[font=small,align=center,font=scriptsize] at (6.825,3.53) u_2; node[font=small,align=center,font=scriptsize] at (7.515,3.53) u_3; node[font=small,align=center] at (11,3.7) d_1d_0=0; node[font=small,align=center] ","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":88},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":451},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.","count":78},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":114},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-85fb367cc0b13557","tex":"A_{v_1v_0}","aliases":"A_v_1v_0","display":"A_{v_1v_0}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2f5b62e2418fa31e","tex":"A_{v_m v_{m-1}}","aliases":"A_v_m v_m-1","display":"A_{v_m v_{m-1}}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7d7d4c82e2b393ad","tex":"A_\\gamma","aliases":"A_γ gamma","display":"A_\\gamma","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":24},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6ea84147003867f2","tex":"\\Arf","aliases":"Arf operatorname","display":"{\\operatorname{Arf}}","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":36},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":56},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-eeabc5ab3835244c","tex":"B_{v_1v_0}","aliases":"B_v_1v_0","display":"B_{v_1v_0}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e3d9f45da6fdf937","tex":"B_{v_m v_{m-1}}","aliases":"B_v_m v_m-1","display":"B_{v_m v_{m-1}}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e01db45ab933df66","tex":"B_\\gamma","aliases":"B_γ gamma","display":"B_\\gamma","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7fc2361ea0bdf8ab","tex":"b_\\rho","aliases":"b_ρ rho","display":"b_\\rho","kind":"definition-passage","excerpt":"[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":39},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":57},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bd4813b07940b348","tex":"c_\\rho","aliases":"c_ρ rho","display":"c_\\rho","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ab865756428ae789","tex":"\\calH","aliases":"calH H mathcal","display":"{\\mathcal H}","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ef089d01f361d945","tex":"\\calX","aliases":"calX X mathcal","display":"{\\mathcal X}","kind":"definition-passage","excerpt":"[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":22},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f57d50dacedd8d8c","tex":"\\CC^X","aliases":"CC^X C^X CC mathbb","display":"{\\mathbb C}^X","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-34dfb5e598913303","tex":"\\CC^{\\times}","aliases":"CC^× C^× CC times mathbb times","display":"{\\mathbb C}^{\\times}","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"definition-passage","excerpt":"[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).","count":19},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":34},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":39},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-cc7582cb4d0dc2f3","tex":"\\FF","aliases":"FF F mathbb","display":"{\\mathbb F}","kind":"definition-passage","excerpt":"[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).","count":37},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ac76bb3454573605","tex":"\\FF_2","aliases":"FF_2 F_2 FF mathbb","display":"{\\mathbb F}_2","kind":"definition-passage","excerpt":"[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":31},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).","count":61},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f9b55a29a191e028","tex":"\\GL","aliases":"GL mathrm GL operatorname","display":null,"kind":"definition-passage","excerpt":"[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.","count":20},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":58},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":41},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-388fb1dbc2c04d8e","tex":"h_\\theta","aliases":"h_θ theta","display":"h_\\theta","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom operatorname","display":"{\\operatorname{Hom}}","kind":"definition-passage","excerpt":"The multiplicity space is M_ρ=Hom_N(W, X).","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e8ccb7cff872b632","tex":"\\Hom_N","aliases":"Hom_N Hom operatorname","display":"{\\operatorname{Hom}}_N","kind":"definition-passage","excerpt":"The multiplicity space is M_ρ=Hom_N(W, X).","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).","count":51},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":47},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"definition-passage","excerpt":"[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":65},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"definition-passage","excerpt":"The multiplicity space is M_ρ=Hom_N(W, X).","count":39},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":71},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5f2828134dc54223","tex":"M_\\rho","aliases":"M_ρ rho","display":"M_\\rho","kind":"definition-passage","excerpt":"The multiplicity space is M_ρ=Hom_N(W, X).","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b2ca635f10eb764b","tex":"\\mathrm{mov}","aliases":"mov mathrm","display":"\\mathrm{mov}","kind":"definition-passage","excerpt":"[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":78},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"item The quotient A is a finite elementary abelian p group. item The representations Π and ρ are irreducible. item The isomorphism class of ρ is fixed by G. item The restriction of Π to N is ρ isotypic. item The quotient records one declared ramification break.","count":162},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1493c2c12981b881","tex":"\\perpbig","aliases":"perpbig mathop⊥limits mathop perp limits","display":null,"kind":"definition-passage","excerpt":"[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3d783edf27173a33","tex":"\\perpbig_{s\\in S}","aliases":"perpbig_s∈ S mathop⊥limits_s∈ S perpbig in mathop perp limits in","display":null,"kind":"definition-passage","excerpt":"[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":123},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6339057791df9dbc","tex":"q^{-1}","aliases":"q^-1","display":"q^{-1}","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d9c016afb8932679","tex":"q^{-1/2}","aliases":"q^-1/2","display":"q^{-1/2}","kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).","count":99},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-650ae3274f788b89","tex":"\\rad","aliases":"rad operatorname","display":"{\\operatorname{rad}}","kind":"definition-passage","excerpt":"[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.","count":41},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":29},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":13},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-911d3e6f76681614","tex":"T_{a+b}","aliases":"T_a+b","display":"T_{a+b}","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-cefa01be96058c3e","tex":"T_b","aliases":"T_b","display":"T_b","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-65fa7282e2650c3e","tex":"\\triangleleft","aliases":"triangleleft","display":null,"kind":"definition-passage","excerpt":"[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":62},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":99},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":53},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-005b50f5c6f12245","tex":"V_{j-1}","aliases":"V_j-1","display":"V_{j-1}","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b00eda2d2588cd09","tex":"v_m","aliases":"v_m","display":"v_m","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f788e25d2802bd1e","tex":"V_{m-1}","aliases":"V_m-1","display":"V_{m-1}","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-36a6d81c2aecf99f","tex":"v_{m-1}","aliases":"v_m-1","display":"v_{m-1}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5df6cd07cd936c95","tex":"V_s","aliases":"V_s","display":"V_s","kind":"definition-passage","excerpt":"[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2db1cdbbe7192b99","tex":"V_\\theta","aliases":"V_θ theta","display":"V_\\theta","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-25aba103b9e1f48f","tex":"V_\\rho","aliases":"V_ρ rho","display":"V_\\rho","kind":"definition-passage","excerpt":"[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"definition-passage","excerpt":"[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.","count":47},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":35},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":36},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"definition-passage","excerpt":"[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).","count":15},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":32},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).","count":26},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1093b5d3cc81a57a","tex":"|_{\\im(g-I)}","aliases":"|_im(g-I) im operatorname","display":"|_{{\\operatorname{im}}(g-I)}","kind":"definition-passage","excerpt":"[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":66},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":16},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":80},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":11},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":24},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"definition-passage","excerpt":"[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":62},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":28},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).","count":38},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2b146fd938e264b6","tex":"\\Phi_p","aliases":"Φ_p Phi","display":"\\Phi_p","kind":"definition-passage","excerpt":"[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1c9a2ecc43bc25db","tex":"\\Phi_p^{\\mathrm{mov}}","aliases":"Φ_p^mov Phi mathrm","display":"\\Phi_p^{\\mathrm{mov}}","kind":"definition-passage","excerpt":"[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":32},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":36},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":15},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.","count":26},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":58},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":35},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2874dc45e47106fd","tex":"\\sum_{i=1}^{r}","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^{r}","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-43b97d47b6ecc1ba","tex":"\\bigoplus_{j=0}^{m-1}","aliases":"⊕_j=0^m-1 bigoplus","display":"\\bigoplus_{j=0}^{m-1}","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":113},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":26},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7f580f0a1cdae14e","tex":"\\dim_{\\FF_p}","aliases":"dim_FF_p dim_ F_p dim FF dim mathbb","display":"\\dim_{{\\mathbb F}_p}","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"The later constructions provide actual arithmetic candidates and exact finite packet recovery in substantial restricted families. They do not establish [source reference] for every admitted source. Their detailed ordinary proofs are typeset in the appendices, and the original reviewed Markdown files accompany this edition. In these appendices F den","count":68},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F mathbb","display":"{\\mathbb F}","kind":"introduction-passage","excerpt":"The symbol K denotes the original local field in the finite-theory discussion. In the explicit arithmetic constructions, F= F_p((t)) is the base field and K=F[β] is its totally ramified parameter field. A symplectic quotient dimension and a Heisenberg representation dimension are different quantities.","count":15},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fda40c00a5457366","tex":"\\F_p","aliases":"F_p F mathbb","display":"{\\mathbb F}_p","kind":"introduction-passage","excerpt":"The symbol K denotes the original local field in the finite-theory discussion. In the explicit arithmetic constructions, F= F_p((t)) is the base field and K=F[β] is its totally ramified parameter field. A symplectic quotient dimension and a Heisenberg representation dimension are different quantities.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5b4740083376d053","tex":"\\FF_p","aliases":"FF_p F_p FF mathbb","display":"{\\mathbb F}_p","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":14},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-25c41d2acd920db0","tex":"\\FF_p^{\\times}","aliases":"FF_p^× F_p^× FF times mathbb times","display":"{\\mathbb F}_p^{\\times}","kind":"introduction-passage","excerpt":"[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"introduction-passage","excerpt":"lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":26},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-98990972b3d9edcd","tex":"h_d","aliases":"h_d","display":"h_d","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":12},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-333bf23be8bc780a","tex":"h_i","aliases":"h_i","display":"h_i","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":13},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-59f64376d60523e7","tex":"\\Ind","aliases":"Ind operatorname","display":"{\\operatorname{Ind}}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":14},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-be0636f2d1fd5dfe","tex":"\\Ind_{W_{K_L}}^{W_K}","aliases":"Ind_W_K_L^W_K Ind operatorname","display":"{\\operatorname{Ind}}_{W_{K_L}}^{W_K}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-da5ab4b9d09e0183","tex":"\\Ind_{W_{K_m}}^{W_K}","aliases":"Ind_W_K_m^W_K Ind operatorname","display":"{\\operatorname{Ind}}_{W_{K_m}}^{W_K}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7adceb058e48c433","tex":"\\Ind_{W_{K_n}}^{W_K}","aliases":"Ind_W_K_n^W_K Ind operatorname","display":"{\\operatorname{Ind}}_{W_{K_n}}^{W_K}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":74},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bcc48739f478f9b9","tex":"J^1","aliases":"J^1","display":"J^1","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":25},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"introduction-passage","excerpt":"ater constructions provide actual arithmetic candidates and exact finite packet recovery in substantial restricted families. They do not establish [source reference] for every admitted source. Their detailed ordinary proofs are typeset in the appendices, and the original reviewed Markdown files accompany this edition. In these appendices F denotes ","count":85},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0a5f8d51d3b62650","tex":"K_L","aliases":"K_L","display":"K_L","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-46d21cf1bf7572c1","tex":"K_m","aliases":"K_m","display":"K_m","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bb9a538f5e406f30","tex":"K_n","aliases":"K_n","display":"K_n","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"introduction-passage","excerpt":"lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ","count":60},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3860d7e8f88507bb","tex":"\\operatorname{lcm}","aliases":"lcm operatorname","display":"\\operatorname{lcm}","kind":"introduction-passage","excerpt":"lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ","count":101},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f35ce0db28d3fb4f","tex":"\\OO","aliases":"OO O mathcal","display":"{\\mathcal O}","kind":"introduction-passage","excerpt":"Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character b","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9d313fd089d71d9a","tex":"\\OO_F^\\times","aliases":"OO_F^× O_F^× OO times mathcal times","display":"{\\mathcal O}_F^\\times","kind":"introduction-passage","excerpt":"Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character b","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c18367064df5a710","tex":"\\mathfrak p","aliases":"p mathfrak","display":"\\mathfrak p","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4dd413d35142652e","tex":"p^{2d}","aliases":"p^2d","display":"p^{2d}","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-085daded6ee24ea6","tex":"p^{2i}","aliases":"p^2i","display":"p^{2i}","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-daabe7304529997d","tex":"p^{2i-2}","aliases":"p^2i-2","display":"p^{2i-2}","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c4f6611ac8fe0528","tex":"p^d","aliases":"p^d","display":"p^d","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-50a36a5ee1bff68c","tex":"p^{d-1}","aliases":"p^d-1","display":"p^{d-1}","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":9},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f29a01821ec9a768","tex":"\\mathfrak p_K^a","aliases":"p_K^a mathfrak","display":"\\mathfrak p_K^a","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":15},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c8e1532bba361388","tex":"R^a","aliases":"R^a","display":"R^a","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-793f37c1d91019f4","tex":"\\Swan","aliases":"Swan operatorname","display":"{\\operatorname{Swan}}","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"introduction-passage","excerpt":"The symbol K denotes the original local field in the finite-theory discussion. In the explicit arithmetic constructions, F= F_p((t)) is the base field and K=F[β] is its totally ramified parameter field. A symplectic quotient dimension and a Heisenberg representation dimension are different quantities.","count":50},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr operatorname","display":"{\\operatorname{Tr}}","kind":"introduction-passage","excerpt":"[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"introduction-passage","excerpt":"Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character b","count":101},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0ec524ed5f354d59","tex":"U_A^a","aliases":"U_A^a","display":"U_A^a","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-559db2c18d7d396e","tex":"U_F^1","aliases":"U_F^1","display":"U_F^1","kind":"introduction-passage","excerpt":"Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character b","count":20},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-55bbc33c76201c71","tex":"U_K^a","aliases":"U_K^a","display":"U_K^a","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a675678fa6689935","tex":"\\vee","aliases":"vee","display":"\\vee","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e0f3a2c6c669d106","tex":"W_K","aliases":"W_K","display":"W_K","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":17},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4cd86c8f321e8c8d","tex":"W_{K_L}","aliases":"W_K_L","display":"W_{K_L}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-aa6cdf6f4a146600","tex":"W_{K_m}","aliases":"W_K_m","display":"W_{K_m}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6453f054ab181a68","tex":"W_{K_n}","aliases":"W_K_n","display":"W_{K_n}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"introduction-passage","excerpt":"lessonPART I / FINITE TAME COORDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"introduction-passage","excerpt":"ater constructions provide actual arithmetic candidates and exact finite packet recovery in substantial restricted families. They do not establish [source reference] for every admitted source. Their detailed ordinary proofs are typeset in the appendices, and the original reviewed Markdown files accompany this edition. In these appendices F denotes ","count":9},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":29},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ca0e6e00d4486392","tex":"\\eta_{L,m}^{\\Phi^i}","aliases":"η_L,m^Φ^i eta Phi","display":"\\eta_{L,m}^{\\Phi^i}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4e2af9f8f06f93e6","tex":"\\xi_{L,n}","aliases":"ξ_L,n xi","display":"\\xi_{L,n}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":44},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1c4d6b081f4bc9ca","tex":"\\pi^\\vee","aliases":"π^vee pi vee","display":"\\pi^\\vee","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3f4525556a6771f0","tex":"\\Phi^i","aliases":"Φ^i Phi","display":"\\Phi^i","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"introduction-passage","excerpt":"[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.","count":15},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b7a8cfbba485d0c4","tex":"\\sum_{i=1}^d","aliases":"∑_i=1^d sum","display":"\\sum_{i=1}^d","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":22},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3c8f746e1dd608b5","tex":"\\bigoplus_{i=0}^{d-1}","aliases":"⊕_i=0^d-1 bigoplus","display":"\\bigoplus_{i=0}^{d-1}","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","excerpt":"RDINATESTensor refinement is a diagonal orbit problemOriginal Theorem 5.1 plate05_tensor_gridThe 4×6 grid separates into two diagonal translation orbits, labelled 0 and 1. The simultaneous translation action on Z/n Z× Z/m Z has d=gcd(n,m) orbits, each of length L=lcm(n,m). Mackey decomposition then gives Ind_W_K_n^W_Kξ⊗Ind_W_K_m^W_Kη ≃⊕_i=0^d-1 Ind","count":13},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.","count":11},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"The dyadic floors cannot be replaced by the odd-prime dimension formula. At stage j, the centralizer grade has dimension 2^d-j. Its contribution to J^1/H^1 is (2^d-j+1-2^d-j) 1_n_j even. Every n_j for j<d is even; n_d=q_1 is odd and contributes zero. The sum is exactly N-2. BK nondegeneracy makes this a symplectic space. A whole source packet of d ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-22eacf5f4b39579e","tex":"\\mathbf1_{n_j\\text{ even}}","aliases":"1_n_j even mathbf text","display":"\\mathbf1_{n_j\\text{ even}}","kind":"use","excerpt":"The dyadic floors cannot be replaced by the odd-prime dimension formula. At stage j, the centralizer grade has dimension 2^d-j. Its contribution to J^1/H^1 is (2^d-j+1-2^d-j) 1_n_j even. Every n_j for j<d is even; n_d=q_1 is odd and contributes zero. The sum is exactly N-2. BK nondegeneracy makes this a symplectic space. A whole source packet of d ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"Fix the paper's reciprocity, additive-character, local-factor, and phase normalizations. Let A be the declared class of irreducible wild Weil atoms under consideration. For ρ∈ A, the supplied assembled finite datum d_ρ consists of the following inputs, historically denoted H1--H4: item an intrinsic finite break decomposition on the Galois side; ite","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"[Hyperbolic phase formula] Let U be a finite dimensional F_p space, let a∈GL(U) have no eigenvalue one, and let g act on H(U)=U⊕ U^* by g=(a,(a^-1)^*). Then Φ_p(g,H(U)) = (fracdet ap).","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"he scalar-pushout realization contract, with no independently fixed conflicting host marking, choose a labelled symplectic embedding into W. Matching commutators gives a center-fixing isomorphism between the source C^× extension and the corresponding host restriction. Retain the image of the whole marked finite subgroup and every lift. For this iso","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c60967b39bed49c7","tex":"a^n","aliases":"a^n","display":"a^n","kind":"use","excerpt":"tching does not construct realizationBlocks 162, 166--167, profile and capacity audits For compact J⊂GL_n(F), the homomorphism v_F∘det has finite image in Z, hence image zero. An unramified character ν_a(g)=a^v_F(det g) is therefore trivial on J. The compact packet restrictions of π and π⊗ν_a agree, while the central character changes by a^n on a s","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c03acd1d1b8f5556","tex":"A^{\\mathrm{symp}}","aliases":"A^symp mathrm","display":"A^{\\mathrm{symp}}","kind":"use","excerpt":"[Radical free envelope] For an arbitrary scalar projective action of A, the quotient A^symp=A/rad b_c is the maximal quotient detected by noncommuting projective intertwiners. If A is an elementary abelian p group, then A^symp is a symplectic F_p space.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-cf7af9a53f6d4e83","tex":"a^{v_F(\\det g)}","aliases":"a^v_F(det g) det","display":"a^{v_F(\\det g)}","kind":"use","excerpt":"lessonPART IV / REALIZATION AND COMPARISONNumerical matching does not construct realizationBlocks 162, 166--167, profile and capacity audits For compact J⊂GL_n(F), the homomorphism v_F∘det has finite image in Z, hence image zero. An unramified character ν_a(g)=a^v_F(det g) is therefore trivial on J. The compact packet restrictions of π and π⊗ν_a ag","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3ff83531e3c9698b","tex":"A_0^{-1}","aliases":"A_0^-1","display":"A_0^{-1}","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7764e407096f3048","tex":"A_e","aliases":"A_e","display":"A_e","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c4514e57e9986c46","tex":"a_{p-1}","aliases":"a_p-1","display":"a_{p-1}","kind":"use","excerpt":"Then v_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic minimal polynomial f(X)=X^p+a_p-1X^p-1+⋯+a_0 is Eisenstein. It is an explicit polynomial: compute the product over the p conjugates","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ab0fbf60c4efcc89","tex":"A_{p_u}","aliases":"A_p_u","display":"A_{p_u}","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5bca16056070f59f","tex":"A_{p_u}^{-1}","aliases":"A_p_u^-1","display":"A_{p_u}^{-1}","kind":"use","excerpt":"ct edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fe95637e997d66bd","tex":"A_{p_v}^{-1}","aliases":"A_p_v^-1","display":"A_{p_v}^{-1}","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-efbaeabd3147a09d","tex":"\\Art","aliases":"Art operatorname","display":"{\\operatorname{Art}}","kind":"use","excerpt":"[Conductor of a nontrivial tame atom] Let τ be an irreducible tame Weil representation of dimension n>1. Then Swan(τ)=0, dimτ^I_K=0, Art(τ)=n.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ef843f340bfccbd5","tex":"\\operatorname{Art}","aliases":"Art operatorname","display":"\\operatorname{Art}","kind":"use","excerpt":"The dimension and orbit follow from [source reference]. The cycle polynomial is [source reference]. For rank greater than one, the conductor is [source reference]. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula Art=dim V-dim V^I_K+Sw. The local polynomial is [source ref","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"use","excerpt":"[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ab3ea1299ebfd7b0","tex":"B^{[j]}","aliases":"B^[j]","display":"B^{[j]}","kind":"use","excerpt":"The field degrees and critical levels of B^[j] are exactly","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-10fba79ce2cdc0ce","tex":"b_c","aliases":"b_c","display":"b_c","kind":"use","excerpt":"Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.","count":13},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-66c8d0359e712fa2","tex":"B_e","aliases":"B_e","display":"B_e","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9f786b5ed58219b3","tex":"B_{\\mathrm{order}}","aliases":"B_order mathrm","display":"B_{\\mathrm{order}}","kind":"use","excerpt":"The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6ee75600b75f53a4","tex":"B_{p_u}","aliases":"B_p_u","display":"B_{p_u}","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-33cb6f2bf5e31462","tex":"B_{p_v}","aliases":"B_p_v","display":"B_{p_v}","kind":"use","excerpt":"ct edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-df47dfef3c938c01","tex":"B_{p_v}^{-1}","aliases":"B_p_v^-1","display":"B_{p_v}^{-1}","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-58af0f264fe7ee38","tex":"\\backslash","aliases":"backslash","display":null,"kind":"use","excerpt":"Apply Mackey decomposition [source reference]. The double cosets W_K_nbackslash W_K/W_K_m are the orbits of simultaneous translation on Z/n Z× Z/m Z. There are d orbits, each of length L. The stabilizer of a point in each orbit is W_K_L. The second character is shifted by the chosen Frobenius representative. Summing the induced terms over the doubl","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b4d7ee29d2029812","tex":"\\bar x","aliases":"bar x bar","display":"\\bar x","kind":"use","excerpt":"J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniqu","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fcd1d016a0b9e4bf","tex":"\\bar y","aliases":"bar y bar","display":"\\bar y","kind":"use","excerpt":"J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniqu","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESExponent coordinates test inertia trivialityOriginal Proposition 5.3 Write the residue character exponents as abmod(q^n-1) and bbmod(q^m-1). The ith refined packet has exponent c_i=afracq^L-1q^n-1+bq^ifracq^L-1q^m-1 pmodq^L-1. Norm lifting multiplies an exponent by the quotient of group orders. Shifting the sec","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathbb","display":"{\\mathbb C}","kind":"use","excerpt":"tion 4 Let G_n=GL_n( F_q) and T_n≃ F_q^n^× be an elliptic torus. A regular torus character determines an irreducible cuspidal representation κ_θ by the classical finite-field construction. If its character is χ_θ and degree is d_θ, then e_θ=fracd_θ|G_n|∑_g∈ G_nχ_θ(g^-1)g is its primitive central idempotent in C[G_n].","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-59133e4900f63250","tex":"\\operatorname{c\\!-\\!Ind}","aliases":"c - Ind operatorname","display":"\\operatorname{c\\!-\\!Ind}","kind":"use","excerpt":"Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is a genuine e","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-be603f780619be75","tex":"\\operatorname{c\\!-\\!Ind}_{\\mathcal M}^G","aliases":"c - Ind_ M^G operatorname mathcal","display":"\\operatorname{c\\!-\\!Ind}_{\\mathcal M}^G","kind":"use","excerpt":"Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is a genuine e","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f3abc569941b2510","tex":"\\operatorname{c\\!-\\!Ind}_J^G","aliases":"c - Ind_J^G operatorname","display":"\\operatorname{c\\!-\\!Ind}_J^G","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","excerpt":"The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e4cfc3c2ac366145","tex":"c^m","aliases":"c^m","display":"c^m","kind":"use","excerpt":"Let C=⟨ c⟩ have order m, and let T be a lift of the projective generator. Since c^m=1, the operator T^m is scalar. Choose λ∈ C^× with λ^mT^m=I. Replacing T by λ T gives an honest cyclic representation.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-15f3d3713c7db98d","tex":"\\C^X","aliases":"C^X C mathbb","display":"{\\mathbb C}^X","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONTranslations and multipliers create Heisenberg packetsOriginal Section 7 plate07_heisenbergThe commutator is central and nontrivial when the pairing is nonzero. For odd p let V=X⊕ X^* and b((x,ξ),(y,η))=η(x)-ξ(y). The group H(V)=V× F_p has product (v,z)(w,t)=(v+w,z+t+tfrac12b(v,w)). On H= C^X the Schrödinger op","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0fa59d916aa991b8","tex":"\\C^\\times","aliases":"C^× C times mathbb times","display":"{\\mathbb C}^\\times","kind":"use","excerpt":"The two pulled-back irreducible Heisenberg representations then have the same central character. Stone--von Neumann gives an invertible U with Uω_ρ(h)=ω_θ(ι_H(h))U. Schur's lemma makes U unique up to a nonzero scalar. After honest cyclic stabilizer linearizations, their discrepancy is a character δ:C→ C^×. Equality at one generator kills this chara","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-62d6dd9a37e47fe4","tex":"\\mathbb C^\\times","aliases":"C^× mathbb times","display":"\\mathbb C^\\times","kind":"use","excerpt":"For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"For q=2,n=4,m=6, the common modulus is 2^12-1=4095. Taking a=b=1, the two packets have c_0=273+65=338, c_1=273+130=403. Both are nonzero modulo 4095, so neither packet contributes inertia invariants.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"For q=2,n=4,m=6, the common modulus is 2^12-1=4095. Taking a=b=1, the two packets have c_0=273+65=338, c_1=273+130=403. Both are nonzero modulo 4095, so neither packet contributes inertia invariants.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2d6682440931076e","tex":"\\C_c","aliases":"C_c C mathbb","display":"{\\mathbb C}_c","kind":"use","excerpt":"The radical consists of a commuting projectively with every b. Its quotient carries a nondegenerate alternating pairing. In the nondegenerate case the twisted group algebra has scalar center and is a full matrix algebra: C_c[A]≃ M_sqrt|A|( C). boundaryThis constructs the finite envelope for a supplied projective action. It does not produce a canoni","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"use","excerpt":"[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"use","excerpt":"[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a0a4ff2df722dd9a","tex":"\\calH_\\theta","aliases":"calH_θ H_θ calH theta mathcal theta","display":"{\\mathcal H}_\\theta","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6e11420ea16392e8","tex":"\\calH_\\rho","aliases":"calH_ρ H_ρ calH rho mathcal rho","display":"{\\mathcal H}_\\rho","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b6691a5fcf8bc635","tex":"\\CC^c","aliases":"CC^c C^c CC mathbb","display":"{\\mathbb C}^c","kind":"use","excerpt":"Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b260911d985f0cb1","tex":"\\CC^{c_\\rho}","aliases":"CC^c_ρ C^c_ρ CC rho mathbb rho","display":"{\\mathbb C}^{c_\\rho}","kind":"use","excerpt":"[Nondegenerate one break packet] If b_ρ is nondegenerate, then C^c_ρ[A] ≅ End(M_ρ), dim M_ρ=sqrtabs A.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ce2f5577f99d3780","tex":"\\CC^\\times","aliases":"CC^× C^× CC times mathbb times","display":"{\\mathbb C}^\\times","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-43e155d639ad4bdc","tex":"\\cInd","aliases":"cInd c - Ind operatorname cInd operatorname","display":null,"kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-eaf3feac0a250482","tex":"\\cInd_{\\mathcal M}^{G}","aliases":"cInd_ M^G c - Ind_ M^G cInd mathcal operatorname mathcal","display":"{\\operatorname{c\\!\\!-\\!\\!Ind}}_{\\mathcal M}^{G}","kind":"use","excerpt":"overy proof plate20_induction_stagesThe noncompact extension Λ is fixed before forming Σ. At both low ranks, K^× normalizes P. Put N=K^× J, M=K^× P, Σ=Ind_ N^ MΛ. The subgroup N is normal of index two, Σ|_P≃ E, and Σ is irreducible. Induction in stages proves π≃c - Ind_ M^GΣ. takeawayThe parameter, full character, original canonical beta-extension ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1f51b343cf55db88","tex":"\\cInd_{\\mathcal M}^{\\GL_N(F)}","aliases":"cInd_ M^GL_N(F) cInd mathcal GL operatorname mathcal operatorname","display":"{\\operatorname{cInd}}_{\\mathcal M}^{{\\operatorname{GL}}_N(F)}","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d6c38b1b2469677d","tex":"\\cInd_J^G","aliases":"cInd_J^G c - Ind_J^G cInd operatorname","display":"{\\operatorname{c\\!\\!-\\!\\!Ind}}_J^G","kind":"use","excerpt":"lessonPART III / ACTUAL ARITHMETIC CONSTRUCTIONCompact recovery must remember the noncompact rootBlock 167, enlarged-carrier contract Let X=c - Ind_J^Gη and retain A_0=Λ(ϖ_K). Define the normalized shift (Tf)(g)=A_0^-1f(ϖ_Kg). The original candidate is the quotient X/(T-1)X≃π. This does not split before central normalization: a vector of X has fini","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":11},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ef2886062e125405","tex":"D_8","aliases":"D_8","display":"D_8","kind":"use","excerpt":"ter pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3b858f3e8665ea65","tex":"D_e","aliases":"D_e","display":"D_e","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-008904a83528d7f3","tex":"d_\\theta","aliases":"d_θ theta","display":"d_\\theta","kind":"use","excerpt":"[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ed6ff526c435c5c5","tex":"d_\\rho","aliases":"d_ρ rho","display":"d_\\rho","kind":"use","excerpt":"Fix the paper's reciprocity, additive-character, local-factor, and phase normalizations. Let A be the declared class of irreducible wild Weil atoms under consideration. For ρ∈ A, the supplied assembled finite datum d_ρ consists of the following inputs, historically denoted H1--H4: item an intrinsic finite break decomposition on the Galois side; ite","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-33bb1588c90c3cf1","tex":"\\dim_{\\mathbb F_2}","aliases":"dim_ F_2 dim mathbb","display":"\\dim_{\\mathbb F_2}","kind":"use","excerpt":"For this same constructed candidate, t(π) · =1, · Sw(π) · =3 2^d-1h_d, dim_ F_2W · =N-2, · dimη · =2^N/2-1, W=J^1/H^1, a(π^vee×π) · =N^2-1+∑_i=1^d3 4^i-1h_i. Matching these profiles does not identify the source, its endo-class, or the pair factors in original H5.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-337441eb94c64c38","tex":"\\dim_{\\mathbb F_p}","aliases":"dim_ F_p dim mathbb","display":"\\dim_{\\mathbb F_p}","kind":"use","excerpt":"The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-13eeb9399bcf4334","tex":"\\dim_{\\F_2}","aliases":"dim_F_2 dim_ F_2 dim F dim mathbb","display":"\\dim_{{\\mathbb F}_2}","kind":"use","excerpt":"a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. For p=2, Sw(π) · =3 2^d-1h_d, dim_ F_2(J^1/H^1) · =N-2, dimη=2^N/2-1, a(π^vee×π) · =N^2-1+∑_i=1^d3 4^i-1h_i. boundaryThese identities belong to the same constructed candidate. Matching numerical profiles does not identify a source Weil atom, an endo-class, or H5 pair factors.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7dc71b328fe22df1","tex":"\\dim_{\\F_p}","aliases":"dim_F_p dim_ F_p dim F dim mathbb","display":"\\dim_{{\\mathbb F}_p}","kind":"use","excerpt":"For odd p, Sw(π) · =p^d-1(p+1)h_d, dim_ F_p(J^1/H^1) · =N-1, dimη=p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. For p=2, Sw(π) · =3 2^d-1h_d,","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"[Scalar and projective gluing] If every based loop acts by a scalar, finite gluing is equivalent to equality of the scalar holonomies on a fundamental cycle basis. The number of independent tests is abs E-abs V+1.","count":42},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0dcf6a495ccbd5bd","tex":"\\e","aliases":"e mathrm","display":"{\\mathrm e}","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4808578a2aed1855","tex":"\\e^{2\\pi i/M}","aliases":"e^2π i/M e pi mathrm pi","display":"{\\mathrm e}^{2\\pi i/M}","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-47852f9fe0da341d","tex":"e^{2\\pi i/M}","aliases":"e^2π i/M pi","display":"e^{2\\pi i/M}","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j cha","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0bd6cafb4982926b","tex":"\\e^{2\\pi iz/p}","aliases":"e^2π iz/p e pi mathrm pi","display":"{\\mathrm e}^{2\\pi iz/p}","kind":"use","excerpt":"Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"The induced tame Weil atom has an elementary cycle invariant. In an orbit basis e_0,…,e_n-1, write τ(Φ)e_j=e_j+1 for j<n-1, and τ(Φ)e_n-1=ae_0.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"The induced tame Weil atom has an elementary cycle invariant. In an orbit basis e_0,…,e_n-1, write τ(Φ)e_j=e_j+1 for j<n-1, and τ(Φ)e_n-1=ae_0.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-cf85306cf8e31b85","tex":"e_{j+1}","aliases":"e_j+1","display":"e_{j+1}","kind":"use","excerpt":"The induced tame Weil atom has an elementary cycle invariant. In an orbit basis e_0,…,e_n-1, write τ(Φ)e_j=e_j+1 for j<n-1, and τ(Φ)e_n-1=ae_0.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b7ce4855f02a711f","tex":"e_{n-1}","aliases":"e_n-1","display":"e_{n-1}","kind":"use","excerpt":"The induced tame Weil atom has an elementary cycle invariant. In an orbit basis e_0,…,e_n-1, write τ(Φ)e_j=e_j+1 for j<n-1, and τ(Φ)e_n-1=ae_0.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-504dc54aa7ab931b","tex":"e_U","aliases":"e_U","display":"e_U","kind":"use","excerpt":"[Parabolic annihilation] Let P=LU be a proper parabolic subgroup of G_n, and put e_U=frac1abs U∑_u∈ Uu. If κ_θ is cuspidal, then e_Ue_θ=e_θ e_U=0.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-632b38e7d6727892","tex":"e_\\theta","aliases":"e_θ theta","display":"e_\\theta","kind":"use","excerpt":"[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3735fcaa18d89e0d","tex":"e_\\Lambda","aliases":"e_Λ Lambda","display":"e_\\Lambda","kind":"use","excerpt":"After imposing the already fixed central character, the relevant quotient is cyclic of order N. Then T^N=1 and the finite average e_Λ=frac1N∑_i=0^N-1T^i selects the same extension. The operator acts on a complex representation space, where division by N is valid. boundaryThis is a different recovery mechanism from a finite sector inside the origina","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End operatorname","display":"{\\operatorname{End}}","kind":"use","excerpt":"[Nondegenerate one break packet] If b_ρ is nondegenerate, then C^c_ρ[A] ≅ End(M_ρ), dim M_ρ=sqrtabs A.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-70a9953dd9bd9b22","tex":"\\operatorname{End}","aliases":"End operatorname","display":"\\operatorname{End}","kind":"use","excerpt":"Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f72b53b71cb2aea7","tex":"\\operatorname{End}_{E_0}","aliases":"End_E_0 operatorname","display":"\\operatorname{End}_{E_0}","kind":"use","excerpt":"Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, fin","count":11},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2f4583c8b7c73ba0","tex":"f'","aliases":"f'","display":"f'","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-08c5fee29bc1411d","tex":"F^{-1}","aliases":"F^-1","display":"F^{-1}","kind":"use","excerpt":"[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3841164c6bcf3539","tex":"F^4","aliases":"F^4","display":"F^4","kind":"use","excerpt":"Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d218908ad4d0b0a2","tex":"F^\\times","aliases":"F^× times","display":"F^\\times","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"Under the scalar-pushout realization contract, with no independently fixed conflicting host marking, choose a labelled symplectic embedding into W. Matching commutators gives a center-fixing isomorphism between the source C^× extension and the corresponding host restriction. Retain the image of the whole marked finite subgroup and every lift. For t","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-101fbe8a1f9aa65c","tex":"\\F_2","aliases":"F_2 F mathbb","display":"{\\mathbb F}_2","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONTwo dyadic refinements can have opposite signsOriginal Section 13 plate10_quadratic_planesThe entries are exact values in F_2; both polar forms are xy'+x'y. A quadratic refinement satisfies q(x+y)+q(x)+q(y)=b(x,y). On a plane, q_0(x,y)=xy, q_1(x,y)=xy+x+y. For q_0 there are three zeros and one one, so its Gauss","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, fin","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1100691a1ba16ecd","tex":"\\F_2^2","aliases":"F_2^2 F mathbb","display":"{\\mathbb F}_2^2","kind":"use","excerpt":"tructions plate19_enlarged_carrierThe new carrier occurs in the original candidate; it is not an abstract extra tensor factor. At rank two, write q=2m+1, m≥1, and use J=U_K^1U_A^m+1, P=U_K^1U_A^m. Then Jtriangleleft P, [P:J]=2, and P∩(K^× J)=J. The conjugate character ratio λ=θ^g/θ is nontrivial and sign-valued. With H_0=kerλ, P/H_0≃ F_2^2, E=Ind_J","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-49ea7c696066b690","tex":"\\mathbb F_2^2","aliases":"F_2^2 mathbb","display":"\\mathbb F_2^2","kind":"use","excerpt":". Define P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e5ebb0f638a43f9c","tex":"\\F_2^4","aliases":"F_2^4 F mathbb","display":"{\\mathbb F}_2^4","kind":"use","excerpt":"For a new coset representative u, the ratio λ=θ^u/θ on the actual H^1 is nontrivial and sign-valued. Setting H_0=kerλ gives P/H_0≃ F_2^4, E=Ind_J^Pη, dim E=4. Its represented commutators are scalar; the quotient has exponent two; irreducibility and the exact scalar preimage prove nondegeneracy. The finite scalar center can exceed μ_2. boundaryNo li","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-426b2d59b3fd2908","tex":"\\mathbb F_2^4","aliases":"F_2^4 mathbb","display":"\\mathbb F_2^4","kind":"use","excerpt":") is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1}. Put H_0=kerλ⊂ H. The calculation proves P/H_0≃ F_2^4, E=Ind_J^P","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bcddfe19017f14c5","tex":"\\mathbb F_2^\\times","aliases":"F_2^× mathbb times","display":"\\mathbb F_2^\\times","kind":"use","excerpt":"For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9da98cb9c6f0ae74","tex":"\\mathbb F_p","aliases":"F_p mathbb","display":"\\mathbb F_p","kind":"use","excerpt":"Let p be odd, F= F_p((t)), and 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. For every such tuple there is an explicit matrix β∈ M_N(F) whose field K=F[β] is totally ramified of degree N, normalizes the complete lattice chain, and gives a maximal simple stratum. It has a literal defining sequence of degrees p,p^2,…,p^d, with leading level n_1 and su","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4cf4c741d2045e51","tex":"\\mathbb F_p^\\times","aliases":"F_p^× mathbb times","display":"\\mathbb F_p^\\times","kind":"use","excerpt":"This procedure never substitutes an unjustified extension through a nonmaximal centralizer. It proves full-character compatibility, including shallow units, rather than merely cancellation of a leading trace. The residue factor fixes ω on F_p^×; an N-th-root choice for the noncompact generator fixes ω(t). Every choice is recorded before H5. See pat","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f9d082b0820d2930","tex":"\\F_q","aliases":"F_q F mathbb","display":"{\\mathbb F}_q","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESA primitive projector selects a cuspidal blockOriginal Section 4 Let G_n=GL_n( F_q) and T_n≃ F_q^n^× be an elliptic torus. A regular torus character determines an irreducible cuspidal representation κ_θ by the classical finite-field construction. If its character is χ_θ and degree is d_θ, then e_θ=fracd_θ|G_n|∑","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a079df3d3f78122a","tex":"\\F_{q^n}^{\\times}","aliases":"F_q^n^× F times mathbb times","display":"{\\mathbb F}_{q^n}^{\\times}","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESRegular orbits have exact countsOriginal Proposition 3.6 plate04_orbit_countsExact integer counts; the vertical scale is logarithmic. Characters of F_q^n^× are indexed modulo q^n-1. Characters fixed by a Frobenius power of degree rmid n number q^r-1. Möbius inversion removes those belonging to smaller orbits: c","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fb6ed245a990fb75","tex":"\\F_q^\\times","aliases":"F_q^× F times mathbb times","display":"{\\mathbb F}_q^\\times","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESFinite Gauss phases are unit scalarsOriginal Section 6 For a nontrivial additive character ψ of F_q and a nontrivial multiplicative character χ of F_q^×, define G(χ,ψ)=∑_x∈ F_q^×χ(x)ψ(x), γ(χ,ψ)=q^-1/2G(χ,ψ). The exact identities are |G(χ,ψ)|^2=q, γ(χ,ψ)γ(χ^-1,ψ)=χ(-1). The magnitude calculation substitutes x=t","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0d10319b98111ddb","tex":"\\FF_2^2","aliases":"FF_2^2 F_2^2 FF mathbb","display":"{\\mathbb F}_2^2","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f392c9ccb437bdd0","tex":"\\FF_2^4","aliases":"FF_2^4 F_2^4 FF mathbb","display":"{\\mathbb F}_2^4","kind":"use","excerpt":"(J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-380df1730c5193bc","tex":"\\FF_2^\\times","aliases":"FF_2^× F_2^× FF times mathbb times","display":"{\\mathbb F}_2^\\times","kind":"use","excerpt":"The construction retains the actual canonical beta extension and can be completed with any prescribed central quasicharacter omega_F trivial on U_F 1. Because k_F^×= F_2^× is trivial, such an omega_F is unramified. Thus this covers every tamely ramified central quasicharacter in this particular residue-field setting.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9e4a1ad9b380105f","tex":"\\FF_3^2","aliases":"FF_3^2 F_3^2 FF mathbb","display":"{\\mathbb F}_3^2","kind":"use","excerpt":"[Rank one Heisenberg group] Take V= F_3^2. The Heisenberg group has order 3^3=27, and its irreducible representation with a fixed nontrivial central character has dimension three. The trace vanishes on every noncentral element. Averaging the translation subgroup gives a rank one projector.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1b447983b8333707","tex":"\\FF_p^2","aliases":"FF_p^2 F_p^2 FF mathbb","display":"{\\mathbb F}_p^2","kind":"use","excerpt":"[Nondegenerate twisted algebra] Let A= F_p^2 with its standard alternating form. A cocycle with that commutator form gives C^c[A]≅ M_p( C). The unique irreducible projective representation has dimension p.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-baf8f5a70f2300fa","tex":"\\FF_q","aliases":"FF_q F_q FF mathbb","display":"{\\mathbb F}_q","kind":"use","excerpt":"Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d90572be1e18f074","tex":"\\FF_{q^L}^{\\times}","aliases":"FF_q^L^× F_q^L^× FF times mathbb times","display":"{\\mathbb F}_{q^L}^{\\times}","kind":"use","excerpt":"[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-217db3583e34db9b","tex":"\\FF_{q^n}^{\\times}","aliases":"FF_q^n^× F_q^n^× FF times mathbb times","display":"{\\mathbb F}_{q^n}^{\\times}","kind":"use","excerpt":"Characters of F_q^n^× are indexed by exponents modulo q^n-1, and Frobenius acts by multiplication by q.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8a8c6f57b92bedc3","tex":"\\FF_{q^r}^{\\times}","aliases":"FF_q^r^× F_q^r^× FF times mathbb times","display":"{\\mathbb F}_{q^r}^{\\times}","kind":"use","excerpt":"For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-aa40e90aedbf1ce9","tex":"\\frakH","aliases":"frakH H mathfrak","display":"{\\mathfrak H}","kind":"use","excerpt":"Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bf6efda13a9cd474","tex":"\\Frob","aliases":"Frob operatorname","display":"{\\operatorname{Frob}}","kind":"use","excerpt":"For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-11e212077d740047","tex":"\\Frob^r","aliases":"Frob^r Frob operatorname","display":"{\\operatorname{Frob}}^r","kind":"use","excerpt":"For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-02b0b5ff2a25a5d9","tex":"g'","aliases":"g'","display":"g'","kind":"use","excerpt":"[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-aa45a7ad376e6ef9","tex":"g^{-1}","aliases":"g^-1","display":"g^{-1}","kind":"use","excerpt":"[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-358c536fce266469","tex":"G_n","aliases":"G_n","display":"G_n","kind":"use","excerpt":"Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-66816f2f34b0eaea","tex":"g_s","aliases":"g_s","display":"g_s","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c5d62edc1cdf57e7","tex":"\\mathrm{GL}","aliases":"GL mathrm","display":"\\mathrm{GL}","kind":"use","excerpt":"The actual stratum admits a full simple character θ trivial on U_F^1. Its canonical beta extension can be completed to an actual irreducible admissible supercuspidal π of GL_N(F) with any prescribed smooth central quasicharacter ω trivial on U_F^1, provided the noncompact completion is free or already compatible. Over this residue field, such ω is ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-15fe16c22d144674","tex":"\\GL_{\\dim\\rho}","aliases":"GL_dimρ GL dim rho mathrm dim rho GL_dimρ GL dim rho operatorname dim rho","display":null,"kind":"use","excerpt":"begingroup renewcommand6666 [Arithmetic packet realization] For every ρ∈ A with its supplied H1--H4 assembled finite datum d_ρ, there exists an actual admissible supercuspidal representation π of GL_dimρ(K) and an arithmetic realization witness R(d_ρ,π). endgroup","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-665db19fdc2e1b41","tex":"\\GL_N","aliases":"GL_N GL mathrm GL_N GL operatorname","display":null,"kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5e917386367b6fdc","tex":"\\GL_n","aliases":"GL_n GL mathrm GL_n GL operatorname","display":null,"kind":"use","excerpt":"The local Langlands correspondence for GL_n(K) is an established theorem. It matches irreducible smooth representations of GL_n(K) with Frobenius semisimple Weil Deligne representations of dimension n. Its proof and normalization draw on global methods, harmonic analysis, types, local factors, and the geometry of Shimura varieties [source reference","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ec44fc6c027681d2","tex":"\\mathrm{GL}_N","aliases":"GL_N mathrm","display":"\\mathrm{GL}_N","kind":"use","excerpt":"The actual stratum admits a full simple character θ trivial on U_F^1. Its canonical beta extension can be completed to an actual irreducible admissible supercuspidal π of GL_N(F) with any prescribed smooth central quasicharacter ω trivial on U_F^1, provided the noncompact completion is free or already compatible. Over this residue field, such ω is ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONTranslations and multipliers create Heisenberg packetsOriginal Section 7 plate07_heisenbergThe commutator is central and nontrivial when the pairing is nonzero. For odd p let V=X⊕ X^* and b((x,ξ),(y,η))=η(x)-ξ(y). The group H(V)=V× F_p has product (v,z)(w,t)=(v+w,z+t+tfrac12b(v,w)). On H= C^X the Schrödinger op","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5cac2bf02c5dcaba","tex":"\\mathrm{H5}","aliases":"H5 mathrm","display":"\\mathrm{H5}","kind":"use","excerpt":"There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently cons","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-67b8b9a6dfc348a3","tex":"H^a","aliases":"H^a","display":"H^a","kind":"use","excerpt":"For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c9dd0244f4332e6d","tex":"H^m","aliases":"H^m","display":"H^m","kind":"use","excerpt":"The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-389193f7658546c5","tex":"H^{m+1}","aliases":"H^m+1","display":"H^{m+1}","kind":"use","excerpt":"For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which t","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":13},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":11},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","excerpt":"The rank-two and rank-four carrier failures cannot be repaired by changing the maximal simple parameter while keeping the specified Swan exponent and torsion number. At rank two, odd Swan forces a minimal odd-level stratum and J^1=H^1. At rank four, Swan 6h_2 with h_2 odd forces the degree pattern 4→2; the maximal quadratic derived field forces an ","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a8c68128dcab45cd","tex":"h_{d-1}","aliases":"h_d-1","display":"h_{d-1}","kind":"use","excerpt":"erse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-48540f8d73f88b86","tex":"H_i^1","aliases":"H_i^1","display":"H_i^1","kind":"use","excerpt":"Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves ever","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-91b7e01f2253c56c","tex":"H_\\theta^1","aliases":"H_θ^1 theta","display":"H_\\theta^1","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e0d53229e96ed0e9","tex":"\\operatorname{Hom}","aliases":"Hom operatorname","display":"\\operatorname{Hom}","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONClifford multiplicity is the correct action spaceOriginal Section 9 Supply finite groups Ntriangleleft G, elementary abelian quotient A=G/N, and irreducibles Π and ρ, with ρ fixed up to isomorphism by G and Π|_N isotypic. The multiplicity space is M_ρ=Hom_N(W,X), W⊗ M_ρ≃ X. For a section s:A→ G and intertwiners","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c6fbb0572f95d570","tex":"\\operatorname{Hom}_N","aliases":"Hom_N operatorname","display":"\\operatorname{Hom}_N","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONClifford multiplicity is the correct action spaceOriginal Section 9 Supply finite groups Ntriangleleft G, elementary abelian quotient A=G/N, and irreducibles Π and ρ, with ρ fixed up to isomorphism by G and Π|_N isotypic. The multiplicity space is M_ρ=Hom_N(W,X), W⊗ M_ρ≃ X. For a section s:A→ G and intertwiners","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-413c3a738073b446","tex":"\\Hom_{W_K}","aliases":"Hom_W_K Hom operatorname","display":"{\\operatorname{Hom}}_{W_K}","kind":"use","excerpt":"[Tame Hom test] For irreducible tame atoms, dimHom_W_K(τ_ξ^vee,τ_η) is zero or one, and it is one exactly when the refined tensor packet test in [source reference] finds one trivial packet.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ad0c30eeda282e1f","tex":"I_{2^{d-1}}","aliases":"I_2^d-1","display":"I_{2^{d-1}}","kind":"use","excerpt":"erse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-eff6aa27c552c5d2","tex":"I_a","aliases":"I_a","display":"I_a","kind":"use","excerpt":"The isotypic condition gives an N equivariant evaluation isomorphism W⊗ M_ρ ⟶ X. Choose a section s:A→ G. For every a∈ A, choose an intertwiner I_a from ρ to its conjugate by s(a). Under the evaluation isomorphism, there is a unique operator T_a on M_ρ such that Π(s(a))=I_a⊗ T_a. The operators T_a form a projective representation of A.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-edadeb2cc1e72d16","tex":"I_K","aliases":"I_K","display":"I_K","kind":"use","excerpt":"Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.","count":9},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ad472b41377c2078","tex":"I_m","aliases":"I_m","display":"I_m","kind":"use","excerpt":"STRUCTIONA recursive matrix gives a literal parameterBlocks 166--167, tower notes plate14_recursive_parameterThe recursive lift is an actual filtered right inverse of normalized corestriction. Start over a coefficient field L with uniformizer t_L and form E=L(α), α^p-α=t_L^-q. At a recursive stage, C is an actual matrix over E. The outer matrix is ","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"use","excerpt":"There is also a positive result beyond C={1}. Suppose the finite prime-to-p stabilizer acts trivially on the source scalar extension, and its prescribed honest operators on the entire rank-N Weil carrier are χ(c)I_N, each lying in the actual inertia image. Assume the candidate's central character is the determinant of that same source under the fix","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-92eb4220503340b4","tex":"I_{p^{d-1}}","aliases":"I_p^d-1","display":"I_{p^{d-1}}","kind":"use","excerpt":", where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. In","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"For a hyperbolic space U⊕ U^* with g=(a,(a^-1)^*) and a-I invertible, Φ_p(g,U⊕ U^*)=(fracdet ap). If g has order prime to p, then V=ker(g-I)⊥im(g-I). The moving summand is nondegenerate and fixed-point-free. Orthogonal block phases multiply. boundaryThe fixed-point-free formula applies to the moving part. The full trace with a fixed part requires i","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ad8e02273a5aecf9","tex":"\\operatorname{Ind}","aliases":"Ind operatorname","display":"\\operatorname{Ind}","kind":"use","excerpt":". Define P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-266b9ca5b05e2205","tex":"\\Ind_{\\mathcal N}^{\\mathcal M}","aliases":"Ind_ N^ M Ind mathcal mathcal operatorname mathcal mathcal","display":"{\\operatorname{Ind}}_{\\mathcal N}^{\\mathcal M}","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0cf82edfc33e1abc","tex":"\\operatorname{Ind}_{\\mathcal N}^{\\mathcal M}","aliases":"Ind_ N^ M operatorname mathcal mathcal","display":"\\operatorname{Ind}_{\\mathcal N}^{\\mathcal M}","kind":"use","excerpt":"Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is a genuine e","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-11972db3b53bb021","tex":"\\Ind_J^P","aliases":"Ind_J^P Ind operatorname","display":"{\\operatorname{Ind}}_J^P","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3643be2089a22b91","tex":"\\operatorname{Ind}_J^P","aliases":"Ind_J^P operatorname","display":"\\operatorname{Ind}_J^P","kind":"use","excerpt":". Define P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a0cd969788b2e98e","tex":"\\Ind_{W_{K_f}}^{W_K}","aliases":"Ind_W_K_f^W_K Ind operatorname","display":"{\\operatorname{Ind}}_{W_{K_f}}^{W_K}","kind":"use","excerpt":"[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-787a425baaf96eca","tex":"j^{-1}","aliases":"j^-1","display":"j^{-1}","kind":"use","excerpt":"Finally, if kj∈P with k∈K× and j∈J, then k∈ Pj^-1⊂ U_A^1. Hence k∈U_K 1 and kj∈J. Therefore","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e9e13171b862dffe","tex":"J_i^1","aliases":"J_i^1","display":"J_i^1","kind":"use","excerpt":"Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves ever","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-284218cb67c97467","tex":"J_\\theta","aliases":"J_θ theta","display":"J_\\theta","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-71f2ee1ce2cf267e","tex":"J_\\theta^1","aliases":"J_θ^1 theta","display":"J_\\theta^1","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].","count":18},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ac62d9119015a5fb","tex":"K^\\times","aliases":"K^× times","display":"K^\\times","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":14},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-649cb8182e4a0913","tex":"K^{\\times}","aliases":"K^× times","display":"K^{\\times}","kind":"use","excerpt":"Local class field theory identifies one dimensional Weil characters with characters of K^× [source reference]. The tame induction theorem extends this explicit picture to higher dimensional tame atoms. The local polynomial and conductor calculations in [source reference] therefore give explicit finite invariants for the declared tame packets. Finit","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f550ca12c9ac5b07","tex":"k^\\times","aliases":"k^× times","display":"k^\\times","kind":"use","excerpt":"Indeed, for residue cardinality q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-805006b107a6864c","tex":"k^{\\times}","aliases":"k^× times","display":"k^{\\times}","kind":"use","excerpt":"Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9817d114fe256b80","tex":"k_0","aliases":"k_0","display":"k_0","kind":"use","excerpt":"For the inductive field step d≥2, put n=p^d-1q_d and R=p^d-2q_d-1. The derived matrix is the recursively constructed maximal-field parameter, with k_0=-q_1. The inequalities n>pR>2R imply -2R>-n-q_1. Bushnell--Kutzko Remark 2.2.7 therefore applies to the actual displayed matrix. It supplies its maximal field, normalization, and critical exponent. V","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-07baee216908c2f6","tex":"K_f","aliases":"K_f","display":"K_f","kind":"use","excerpt":"[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-674fd5cf0671e765","tex":"k_F^\\times","aliases":"k_F^× times","display":"k_F^\\times","kind":"use","excerpt":"The construction retains the actual canonical beta extension and can be completed with any prescribed central quasicharacter omega_F trivial on U_F 1. Because k_F^×= F_2^× is trivial, such an omega_F is unramified. Thus this covers every tamely ramified central quasicharacter in this particular residue-field setting.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e2b60819c95171e8","tex":"k_r","aliases":"k_r","display":"k_r","kind":"use","excerpt":"[Hasse and Davenport lifting] Let k_r/k be the extension of degree r. Lift χ by the norm and ψ by the trace. Then the lifted Gauss sum satisfies the Hasse and Davenport product relation, with a sign determined by the chosen additive character normalization.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"[Fixed and moving decomposition] Assume that g∈Sp(V) has order prime to p. Then V=ker(g-I)mathop⊥limitsim(g-I). The restriction of the symplectic form to im(g-I) is nondegenerate, and g-I is invertible on that moving summand.","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-01208e159d2aec8c","tex":"\\land","aliases":"land","display":"\\land","kind":"use","excerpt":"There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently cons","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3eda83bfb4bc77ed","tex":"\\lcm","aliases":"lcm operatorname","display":"{\\operatorname{lcm}}","kind":"use","excerpt":"Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"ently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the full-character claim, including the odd-critical endpoint 2(lf","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fedb28c62ae7b7d9","tex":"\\operatorname{Lift}","aliases":"Lift operatorname","display":"\\operatorname{Lift}","kind":"use","excerpt":"er t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_","count":11},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-28fb7006ab9bc24c","tex":"\\operatorname{Lift}_E","aliases":"Lift_E operatorname","display":"\\operatorname{Lift}_E","kind":"use","excerpt":"er t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_","count":11},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c8e5d137690efef9","tex":"\\LLC","aliases":"LLC operatorname","display":"{\\operatorname{LLC}}","kind":"use","excerpt":"lessonPART IV / REALIZATION AND COMPARISONH5 compares this same actual candidateContract audit 162; conditional comparison proposition Suppose π is an actual admissible supercuspidal representation of the correct rank and R(d_ρ,π) is supplied. If H5 gives equality of its required pair local factors with those of LLC(ρ) on the stipulated characteriz","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"lessonPART III / ACTUAL ARITHMETIC CONSTRUCTIONDyadic trace one is deep enoughBlock 167, depth invariant plate16_scalar_depthThe scalar correction disappears when its argument lies in t O_F. Fix ψ_F of conductor t O_F. On a scalar unit u∈ U_F^s, the additive correction sees (u-1)trδ. Hence v_F(trδ)+s≥1 Longrightarrow ψ_F((u-1)trδ)=1. The dyadic nat","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-451b483aee615392","tex":"M^\\perp","aliases":"M^⊥ perp","display":"M^\\perp","kind":"use","excerpt":"The invariant orthogonal decomposition gives M=mathop⊥limits_s∈ SM_s. Symplectic invariance gives M^⊥=ker(g-I). Fixed point freeness on M therefore makes the restricted form nondegenerate on M and on each M_s. Each restriction of g_s-I to M_s is invertible, so [source reference] applies to the moving spaces. A zero moving space contributes one. In ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0901f1f56ba2bc5e","tex":"M_N","aliases":"M_N","display":"M_N","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c4fb8d0ba7795506","tex":"M_p","aliases":"M_p","display":"M_p","kind":"use","excerpt":"[Nondegenerate twisted algebra] Let A= F_p^2 with its standard alternating form. A cocycle with that commutator form gives C^c[A]≅ M_p( C). The unique irreducible projective representation has dimension p.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-17770225492e1d01","tex":"M_s","aliases":"M_s","display":"M_s","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6d7c9ff7c9abb006","tex":"M_{\\sqrt{\\abs A}}","aliases":"M_sqrtabs A sqrt abs","display":null,"kind":"use","excerpt":"[Nondegenerate twisted group algebra] Assume that b_c is nondegenerate. Then abs A is a square, and C^c[A] ≅ M_sqrtabs A( C). Consequently, C^c[A] has one irreducible module up to isomorphism, of dimension sqrtabs A.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-42a439094cbc8c02","tex":"M_{\\sqrt{|A|}}","aliases":"M_sqrt|A| sqrt","display":"M_{\\sqrt{|A|}}","kind":"use","excerpt":"The radical consists of a commuting projectively with every b. Its quotient carries a nondegenerate alternating pairing. In the nondegenerate case the twisted group algebra has scalar center and is a full matrix algebra: C_c[A]≃ M_sqrt|A|( C). boundaryThis constructs the finite envelope for a supplied projective action. It does not produce a canoni","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-75ef01eda1abe09b","tex":"\\mathop","aliases":"mathop","display":null,"kind":"use","excerpt":"The invariant orthogonal decomposition gives M=mathop⊥limits_s∈ SM_s. Symplectic invariance gives M^⊥=ker(g-I). Fixed point freeness on M therefore makes the restricted form nondegenerate on M and on each M_s. Each restriction of g_s-I to M_s is invertible, so [source reference] applies to the moving spaces. A zero moving space contributes one. In ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-213d86a5e6b34393","tex":"N^{-1}","aliases":"N^-1","display":"N^{-1}","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6d18e0ad4d2b521a","tex":"N^m","aliases":"N^m","display":"N^m","kind":"use","excerpt":"[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ebcd9b4bdca5cfcd","tex":"N^r","aliases":"N^r","display":"N^r","kind":"use","excerpt":"[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critica","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b6b337ece4924770","tex":"n_2","aliases":"n_2","display":"n_2","kind":"use","excerpt":"cdots<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4d2462ef3372559a","tex":"n_d","aliases":"n_d","display":"n_d","kind":"use","excerpt":"cdots<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-881cb5e47173db34","tex":"N_{j-1,j}","aliases":"N_j-1,j","display":"N_{j-1,j}","kind":"use","excerpt":"[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"[Quantitative finite gluing] With the notation of [source reference], normU_vA_e-B_eU_u ≤ normB_p_vnormD_enormA_p_u^-1. Thus small fundamental cycle defects give small non tree edge residuals whenever the tree transports are uniformly conditioned.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"Every refinement is q_v for a unique v∈ V, because b identifies V with V^*. By [source reference], the Arf invariant changes according to q(v). Let Z and O be the numbers of zeros and ones of q. Then Z+O=2^2r and [source reference] gives Z-O=(-1)^Arf(q)2^r. Solving gives the two displayed numbers, with their roles exchanged when the starting refine","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, fin","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5e8d847d6b436ea8","tex":"\\mathcal O_E^\\times","aliases":"O_E^× mathcal times","display":"\\mathcal O_E^\\times","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a3b35c060b413b3e","tex":"\\mathcal O_F","aliases":"O_F mathcal","display":"\\mathcal O_F","kind":"use","excerpt":"Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, fin","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f9698f456cf9e2ff","tex":"\\OO_F","aliases":"OO_F O_F OO mathcal","display":"{\\mathcal O}_F","kind":"use","excerpt":"[Restricted dyadic marked assembly realization] Let F= F_2((t)), fix an additive character of conductor t O_F, fix d≥1, and let h_1<⋯<h_d be positive odd integers. Put q_i=3h_i, N=2^d, and n_j=2^d-jq_d-j+1. The explicit recursive matrix generates a separable totally ramified degree-N maximal parameter field, normalizes the complete lattice chain, a","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-83c44e1ef141b100","tex":"\\OO_K","aliases":"OO_K O_K OO mathcal","display":"{\\mathcal O}_K","kind":"use","excerpt":"The depth zero construction inflates a finite cuspidal representation to GL_n( O_K), extends it to the standard compact modulo center subgroup, and then compactly induces it to GL_n(K). This is classical type theory [source reference].","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-eb5f4b10196f0437","tex":"\\OO_K^\\times","aliases":"OO_K^× O_K^× OO times mathcal times","display":"{\\mathcal O}_K^\\times","kind":"use","excerpt":"Here O_K^×=U_K^1 because the residue field is F_2. Since U_K 1 is contained in J 1, we have","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-71e41dae7d473f05","tex":"\\mathrm{order}","aliases":"order mathrm","display":"\\mathrm{order}","kind":"use","excerpt":"The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ffdd263bffb70d69","tex":"\\overline P","aliases":"overline P overline","display":"\\overline P","kind":"use","excerpt":"Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.","count":79},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves ever","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-97b363485b44daac","tex":"p^{(N-1)/2}","aliases":"p^(N-1)/2","display":"p^{(N-1)/2}","kind":"use","excerpt":"The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3667a830d02688c4","tex":"P^a","aliases":"P^a","display":"P^a","kind":"use","excerpt":"Put P=rad(A), U^a(A)=1+P^a for atextgreater=1, and m=N/2.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3f712839d32312f3","tex":"p^{d-2}","aliases":"p^d-2","display":"p^{d-2}","kind":"use","excerpt":"For the inductive field step d≥2, put n=p^d-1q_d and R=p^d-2q_d-1. The derived matrix is the recursively constructed maximal-field parameter, with k_0=-q_1. The inequalities n>pR>2R imply -2R>-n-q_1. Bushnell--Kutzko Remark 2.2.7 therefore applies to the actual displayed matrix. It supplies its maximal field, normalization, and critical exponent. V","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c1a386e0dd4490b6","tex":"p^{d-j}","aliases":"p^d-j","display":"p^{d-j}","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-44ddf871c8e79a19","tex":"p^j","aliases":"p^j","display":"p^j","kind":"use","excerpt":"The full-character argument uses a specific order of operations. First construct a full character at native rank p^j, trivial on U_F^1. Transfer its full beta-determinant unit character to multiplicity p, using BK 3.6.13(ii) and 3.6.14. On a scalar u∈ U_F^1, this gives θ_j(u^p)=1. Next add the actual trace-zero correction, using BK 3.3.18. It remai","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1bbb4a842cfd7d98","tex":"P^r","aliases":"P^r","display":"P^r","kind":"use","excerpt":"B-bimodularity gives s(Lift_E(C))=C. The lift preserves exact valuation: u_0 (-1) is an E-unit, and expansion in 1,z gives coefficient valuations distinct modulo 2. Each coefficient entry has the same grade in the interleaved L-chain and the E-chain. In particular s(P^r)=Q^r.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6f0ea21bbfbf69fb","tex":"p^r","aliases":"p^r","display":"p^r","kind":"use","excerpt":"[Finite Schrödinger representation] The map ρ: H(V)→GL( H) is a representation. It is irreducible, its center acts by ψ, and dim H=p^r=sqrtabs V.","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"ursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-10c477be247f9d21","tex":"P_A^j","aliases":"P_A^j","display":"P_A^j","kind":"use","excerpt":"Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK3.1.7--9 give the actual groups","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5f7de3914f64d794","tex":"P_A^{n-r}","aliases":"P_A^n-r","display":"P_A^{n-r}","kind":"use","excerpt":"At both low ranks, K^× normalizes the new P. In rank two this follows directly from normalization of A and its radical. In rank four, a K-uniformizer is approximated by the derived chain shift R times a unit of B, with relative error in P_A^n-r. Its conjugation error on U_B^m lies in U_A^n-r+m⊂ J, which proves normalization. The exact argument is p","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-45c4f17ed76d05ef","tex":"P_g","aliases":"P_g","display":"P_g","kind":"use","excerpt":"[Characteristic polynomial certificate] Let P_g(X)=det(XI-gmid V). Then Φ_p(g,V) = (frac-1p)^r (fracP_g(1)p).","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7b57adfbf4aaa867","tex":"\\mathcal P_i","aliases":"P_i mathcal","display":"\\mathcal P_i","kind":"use","excerpt":"Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves ever","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.","count":13},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2646be694d017280","tex":"P_{j-1}","aliases":"P_j-1","display":"P_{j-1}","kind":"use","excerpt":"[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c5ca4f817be17bab","tex":"P_K","aliases":"P_K","display":"P_K","kind":"use","excerpt":"Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j cha","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8e8a5b53321d6af2","tex":"\\mathfrak p_K^m","aliases":"p_K^m mathfrak","display":"\\mathfrak p_K^m","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6c60807df59cc9f6","tex":"\\mathfrak p_K^{m+1}","aliases":"p_K^m+1 mathfrak","display":"\\mathfrak p_K^{m+1}","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-363d67fd271736f2","tex":"P_L","aliases":"P_L","display":"P_L","kind":"use","excerpt":"[Isotropic averaging projector] The operator P_L = frac1abs L ∑_ℓ∈widetilde L ρ(ℓ) is an orthogonal projector. In the Schrödinger model with L=X, it has rank one and selects the constant functions.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0f5cb33c0e8a0d66","tex":"P_{qj}","aliases":"P_qj","display":"P_{qj}","kind":"use","excerpt":"[Tame character projectors] The operators P_j are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by Φ P_jΦ^-1=P_qj.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-12216838e248404d","tex":"p_u","aliases":"p_u","display":"p_u","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":9},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-26a9ec94734fbef0","tex":"p_v","aliases":"p_v","display":"p_v","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":11},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"Represent the residue characters by exponents apmodq^n-1, bpmodq^m-1.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5f7721258ed720c9","tex":"\\operatorname{pr}","aliases":"pr operatorname","display":"\\operatorname{pr}","kind":"use","excerpt":"er t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"r an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked finite exte","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f4cc63c7c45c2f71","tex":"q^{-j}","aliases":"q^-j","display":"q^{-j}","kind":"use","excerpt":"The only nonzero entry in column j and row j-1 is multiplied by q^j-1q^-j=q^-1 under conjugation.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8ae4ac58b9a6ef67","tex":"q^{1/2}","aliases":"q^1/2","display":"q^{1/2}","kind":"use","excerpt":"[Gauss magnitude] If χ is nontrivial, then absG(χ,ψ)=q^1/2.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-41d4419bab60914f","tex":"q^i","aliases":"q^i","display":"q^i","kind":"use","excerpt":"[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9b02897ce2089839","tex":"Q^j","aliases":"Q^j","display":"Q^j","kind":"use","excerpt":"Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK3.1.7--9 give the actual groups","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9e4eac3b648f9f1d","tex":"q^{j-1}","aliases":"q^j-1","display":"q^{j-1}","kind":"use","excerpt":"The only nonzero entry in column j and row j-1 is multiplied by q^j-1q^-j=q^-1 under conjugation.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-04992c0930a894b0","tex":"q^L","aliases":"q^L","display":"q^L","kind":"use","excerpt":"[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4a3a203830edc472","tex":"q^m","aliases":"q^m","display":"q^m","kind":"use","excerpt":"Represent the residue characters by exponents apmodq^n-1, bpmodq^m-1.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-823af7abc34149de","tex":"q^{m-1}","aliases":"q^m-1","display":"q^{m-1}","kind":"use","excerpt":"[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-102bb645d8547d75","tex":"q^n","aliases":"q^n","display":"q^n","kind":"use","excerpt":"Characters of F_q^n^× are indexed by exponents modulo q^n-1, and Frobenius acts by multiplication by q.","count":12},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-76963409a11b32ee","tex":"q^{n/d}","aliases":"q^n/d","display":"q^{n/d}","kind":"use","excerpt":"[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b568c9f273e792aa","tex":"Q^r","aliases":"Q^r","display":"Q^r","kind":"use","excerpt":"B-bimodularity gives s(Lift_E(C))=C. The lift preserves exact valuation: u_0 (-1) is an E-unit, and expansion in 1,z gives coefficient valuations distinct modulo 2. Each coefficient entry has the same grade in the interleaved L-chain and the E-chain. In particular s(P^r)=Q^r.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-50cffe607a1e0f78","tex":"q^r","aliases":"q^r","display":"q^r","kind":"use","excerpt":"For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","excerpt":"[Dyadic phase ambiguity] On a symplectic plane over F_2, choose refinements q_0 and q_1 with Arf invariants zero and one. Their polar forms are the same, while ∑_x(-1)^q_0(x)=2, ∑_x(-1)^q_1(x)=-2. Thus the alternating packet alone cannot select the dyadic sign.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"[Orthogonal dyadic multiplicativity] For an orthogonal sum (V,q)=(V_1,q_1)mathop⊥limits(V_2,q_2), we have Arf(q)=Arf(q_1)+Arf(q_2) and the normalized Gauss sign is the product of the two normalized signs.","count":11},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"[Orthogonal dyadic multiplicativity] For an orthogonal sum (V,q)=(V_1,q_1)mathop⊥limits(V_2,q_2), we have Arf(q)=Arf(q_1)+Arf(q_2) and the normalized Gauss sign is the product of the two normalized signs.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d2dd5eb187fc710e","tex":"Q_8","aliases":"Q_8","display":"Q_8","kind":"use","excerpt":"r an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked finite exte","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-369e50e7dcd70df5","tex":"q_d","aliases":"q_d","display":"q_d","kind":"use","excerpt":"At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coeffici","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8f7b02bca74c6be6","tex":"q_{d-1}","aliases":"q_d-1","display":"q_{d-1}","kind":"use","excerpt":"At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coeffici","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-830545afde08b32b","tex":"q_{d-j+1}","aliases":"q_d-j+1","display":"q_{d-j+1}","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-69e1eafeb8f88eb2","tex":"q_{\\rm src}","aliases":"q_rm src rm","display":"q_{\\rm src}","kind":"use","excerpt":"The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier; it is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equ","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e52a322b6d2467f2","tex":"q_v","aliases":"q_v","display":"q_v","kind":"use","excerpt":"[Translation of quadratic refinements] For v∈ V, put q_v(x)=q(x)+b(v,x). Then Arf(q_v)=Arf(q)+q(v).","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7f50dcece8b7d4a6","tex":"q_W","aliases":"q_W","display":"q_W","kind":"use","excerpt":"The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier; it is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equ","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"An arithmetic realization witness R(d_ρ,π) consists of actual arithmetic inducing data for an admissible supercuspidal π, together with explicit comparison and recovery maps carrying the prescribed assembled finite packet. The maps preserve its declared stabilizer action, all character values explicitly included in the supplied datum, central-exten","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c4179ac2053e2353","tex":"R^m","aliases":"R^m","display":"R^m","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3e64c4ff18fdf826","tex":"R^{m+1}","aliases":"R^m+1","display":"R^{m+1}","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"ently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the full-character claim, including the odd-critical endpoint 2(lf","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier; it is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equ","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-793e8333d34ceb82","tex":"\\Sp","aliases":"Sp operatorname","display":"{\\operatorname{Sp}}","kind":"use","excerpt":"Let p be odd, let (V,b) be a finite symplectic F_p space of dimension 2r, and let g∈Sp(V) have no fixed vector. Fix one of the standard finite Weil normalizations used in the literature [source reference].","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0875a7163c89630a","tex":"\\operatorname{Sp}","aliases":"Sp operatorname","display":"\\operatorname{Sp}","kind":"use","excerpt":"[Indecomposable monodromy blocks] Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks Sp_m(τ), where τ is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length m.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-18d2a318c2e17978","tex":"\\operatorname{Sp}_m","aliases":"Sp_m operatorname","display":"\\operatorname{Sp}_m","kind":"use","excerpt":"[Indecomposable monodromy blocks] Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks Sp_m(τ), where τ is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length m.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e4353e56e229aaaa","tex":"\\mathrm{src}","aliases":"src mathrm","display":"\\mathrm{src}","kind":"use","excerpt":"Assume the one-vertex scalar-pushout contract: there are d labelled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e5fd5471d426d46d","tex":"\\operatorname{St}","aliases":"St operatorname","display":"\\operatorname{St}","kind":"use","excerpt":"[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ca1b99e90c670ad6","tex":"\\operatorname{St}_m","aliases":"St_m operatorname","display":"\\operatorname{St}_m","kind":"use","excerpt":"[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f74e3bbc76abbdf0","tex":"\\Sw","aliases":"Sw operatorname","display":"{\\operatorname{Sw}}","kind":"use","excerpt":"An inertia-trivial companion cycle has a Weil-fixed vector exactly when a_i=1, and then its fixed space is one dimensional. Therefore dim(τ_ξ⊗τ_η)^W_K =#{i:c_i=0, a_i=1}. For an irreducible nontrivial tame atom of dimension n>1, wild inertia acts trivially and inertia has no fixed vectors. Hence Sw(τ)=0, Art(τ)=n. boundaryThese exact tame formulas ","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ad3fefc79ffc997c","tex":"\\operatorname{Sw}","aliases":"Sw operatorname","display":"\\operatorname{Sw}","kind":"use","excerpt":"The dimension and orbit follow from [source reference]. The cycle polynomial is [source reference]. For rank greater than one, the conductor is [source reference]. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula Art=dim V-dim V^I_K+Sw. The local polynomial is [source ref","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8d8605f4038ec2f5","tex":"\\mathrm{symp}","aliases":"symp mathrm","display":"\\mathrm{symp}","kind":"use","excerpt":"[Radical free envelope] For an arbitrary scalar projective action of A, the quotient A^symp=A/rad b_c is the maximal quotient detected by noncommuting projective intertwiners. If A is an elementary abelian p group, then A^symp is a symplectic F_p space.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"The old fixed-field central argument genuinely failed. At p=3, (h_1,h_2)=(1,5), its naturally transferred full character takes the value ζ_3≠1 on 1+t^2, although the desired wild center is trivial. A shallow character choice cannot repair a forced value. The two-stage changing-field repair is retained in pathcentral/CENTRAL_REPAIR166.md; the trace-","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6e393fab09f85dc1","tex":"T^i","aliases":"T^i","display":"T^i","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-54815db7d2265570","tex":"T^m","aliases":"T^m","display":"T^m","kind":"use","excerpt":"Let C=⟨ c⟩ have order m, and let T be a lift of the projective generator. Since c^m=1, the operator T^m is scalar. Choose λ∈ C^× with λ^mT^m=I. Replacing T by λ T gives an honest cyclic representation.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ac72b989bba7a709","tex":"T^N","aliases":"T^N","display":"T^N","kind":"use","excerpt":"After imposing the already fixed central character, the relevant quotient is cyclic of order N. Then T^N=1 and the finite average e_Λ=frac1N∑_i=0^N-1T^i selects the same extension. The operator acts on a complex representation space, where division by N is valid. boundaryThis is a different recovery mechanism from a finite sector inside the origina","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c10b9cca7424884c","tex":"t^q","aliases":"t^q","display":"t^q","kind":"use","excerpt":"The first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of [source reference], the relation is Φ^-1tΦ=t^q. Equivalently, Φ tΦ^-1=t^q^-1, with inverse taken modulo M since gcd(q,M)=1. Conjugating the Fourier sum and changing its summation index therefore gives Φ P_jΦ^-1=P_qj. The inverse F","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2aa2675cfae8aa2f","tex":"t^{q^{-1}}","aliases":"t^q^-1","display":"t^{q^{-1}}","kind":"use","excerpt":"The first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of [source reference], the relation is Φ^-1tΦ=t^q. Equivalently, Φ tΦ^-1=t^q^-1, with inverse taken modulo M since gcd(q,M)=1. Conjugating the Fourier sum and changing its summation index therefore gives Φ P_jΦ^-1=P_qj. The inverse F","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9caf405f71847e16","tex":"t_L","aliases":"t_L","display":"t_L","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace cores","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-748926db74a9add6","tex":"t_L^{-q}","aliases":"t_L^-q","display":"t_L^{-q}","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace cores","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2e4e16e9495e3826","tex":"t_L^{-q_d}","aliases":"t_L^-q_d","display":"t_L^{-q_d}","kind":"use","excerpt":"At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-63928c9acb41859a","tex":"t_L^a","aliases":"t_L^a","display":"t_L^a","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace cores","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-64afaf0ed6cbcadc","tex":"T_n","aliases":"T_n","display":"T_n","kind":"use","excerpt":"Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f73ad12c9477d300","tex":"\\operatorname{Teich}","aliases":"Teich operatorname","display":"\\operatorname{Teich}","kind":"use","excerpt":"Indeed, for residue cardinality q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace cores","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"The Artin--Schreier filtered lift has native trace one. Consequently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fc7a5c603b95f30d","tex":"\\tr","aliases":"tr operatorname","display":"{\\operatorname{tr}}","kind":"use","excerpt":"For its Eisenstein polynomial f and constant-coordinate projection P_0, the normalized corestriction uses κ=-f'(z)/z^p-1. Writing u_0=s(P_0), Lift_E(C)=u_0^-1CP_0, tr_LLift_E(C) =Tr_E/L(κ^-1tr_EC). If tr_EC=0, the lift has trace zero. Adding α I_m preserves trace zero. Induction therefore gives trace zero at every native stage and every native corr","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9ad8441d9d51dab3","tex":"\\operatorname{tr}_E","aliases":"tr_E operatorname","display":"\\operatorname{tr}_E","kind":"use","excerpt":"_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix ","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a64ceddc60a2b43a","tex":"\\tr_E","aliases":"tr_E tr operatorname","display":"{\\operatorname{tr}}_E","kind":"use","excerpt":"For its Eisenstein polynomial f and constant-coordinate projection P_0, the normalized corestriction uses κ=-f'(z)/z^p-1. Writing u_0=s(P_0), Lift_E(C)=u_0^-1CP_0, tr_LLift_E(C) =Tr_E/L(κ^-1tr_EC). If tr_EC=0, the lift has trace zero. Adding α I_m preserves trace zero. Induction therefore gives trace zero at every native stage and every native corr","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8ccd54953d2277fa","tex":"\\Tr_{E/L}","aliases":"Tr_E/L Tr operatorname","display":"{\\operatorname{Tr}}_{E/L}","kind":"use","excerpt":"lessonPART III / ACTUAL ARITHMETIC CONSTRUCTIONOdd towers retain native trace zeroBlock 166, recursive tower and full-character proofs For odd p, Tr_E/L(α)=0. Choose b with qb≡-1pmod p, let a=(qb+1)/p, and set z=t_L^aα^b. Then z is a uniformizer.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-93993d18d84ec366","tex":"\\operatorname{Tr}_{E/L}","aliases":"Tr_E/L operatorname","display":"\\operatorname{Tr}_{E/L}","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace cores","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-06c81003c38cac43","tex":"\\operatorname{tr}_L","aliases":"tr_L operatorname","display":"\\operatorname{tr}_L","kind":"use","excerpt":"_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix ","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f789352ffc1ad49d","tex":"\\tr_L","aliases":"tr_L tr operatorname","display":"{\\operatorname{tr}}_L","kind":"use","excerpt":"For its Eisenstein polynomial f and constant-coordinate projection P_0, the normalized corestriction uses κ=-f'(z)/z^p-1. Writing u_0=s(P_0), Lift_E(C)=u_0^-1CP_0, tr_LLift_E(C) =Tr_E/L(κ^-1tr_EC). If tr_EC=0, the lift has trace zero. Adding α I_m preserves trace zero. Induction therefore gives trace zero at every native stage and every native corr","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1296a5898d2e7d5f","tex":"U^*","aliases":"U^*","display":"U^*","kind":"use","excerpt":"[Hyperbolic phase formula] Let U be a finite dimensional F_p space, let a∈GL(U) have no eigenvalue one, and let g act on H(U)=U⊕ U^* by g=(a,(a^-1)^*). Then Φ_p(g,H(U)) = (fracdet ap).","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"For each g, the operator Ω_θ(g)^-1UΩ_ρ(g)U^-1 commutes with the type side Heisenberg representation. It is therefore scalar. Since both stabilizer actions are honest representations, multiplication of group elements shows that the scalars form a character.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-df5c58f22a65049f","tex":"u^{-1}","aliases":"u^-1","display":"u^{-1}","kind":"use","excerpt":"For nontrivial inertia, the invariant space is zero. In the trivial case, Frobenius on the invariant packet is the companion matrix of X^r-a. Substituting X=u^-1 and multiplying by u^r gives 1-a u^r.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a86ed18272602c0f","tex":"U^1","aliases":"U^1","display":"U^1","kind":"use","excerpt":"The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which t","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c105d72a53dfb0f9","tex":"U^{2m}","aliases":"U^2m","display":"U^{2m}","kind":"use","excerpt":"For u,v in U m(A), their group commutator is in U^2m(A)=U^N(A), which lies in U (m+1)(A). The terminal approximation satisfies","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9bad9d0840a84cee","tex":"U^a","aliases":"U^a","display":"U^a","kind":"use","excerpt":"Put P=rad(A), U^a(A)=1+P^a for atextgreater=1, and m=N/2.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0667d48b5628d7ee","tex":"u^{k_0}","aliases":"u^k_0","display":"u^{k_0}","kind":"use","excerpt":"y q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the entire sector induction and a","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f8e90bd5459efd8c","tex":"U^m","aliases":"U^m","display":"U^m","kind":"use","excerpt":"Here H m means H^1∩ U^m(A). Therefore the image S of U m(A) in W is canonically","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d2bc1d06f7629695","tex":"U^N","aliases":"U^N","display":"U^N","kind":"use","excerpt":"For u,v in U m(A), their group commutator is in U^2m(A)=U^N(A), which lies in U (m+1)(A). The terminal approximation satisfies","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-da5e436b748b454f","tex":"u^p","aliases":"u^p","display":"u^p","kind":"use","excerpt":"The full-character argument uses a specific order of operations. First construct a full character at native rank p^j, trivial on U_F^1. Transfer its full beta-determinant unit character to multiplicity p, using BK 3.6.13(ii) and 3.6.14. On a scalar u∈ U_F^1, this gives θ_j(u^p)=1. Next add the actual trace-zero correction, using BK 3.3.18. It remai","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d6a7e14e69737cad","tex":"u^r","aliases":"u^r","display":"u^r","kind":"use","excerpt":"[Local polynomial of one tame packet] Suppose an interaction packet has orbit length r and closing scalar a. Its local polynomial is det(I-uΦmid V^I_K) = 1-a u^r, · if the inertia character is trivial, 1, · otherwise.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-050c8b635461358e","tex":"u^{r_i}","aliases":"u^r_i","display":"u^{r_i}","kind":"use","excerpt":"[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bf0029d07e1738a6","tex":"u_0^{-1}","aliases":"u_0^-1","display":"u_0^{-1}","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-91c677571574a0b9","tex":"u_a","aliases":"u_a","display":"u_a","kind":"use","excerpt":"Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4773df27618d48b2","tex":"u_{a+b}","aliases":"u_a+b","display":"u_{a+b}","kind":"use","excerpt":"Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-68c0465eac68c384","tex":"u_a^{-1}","aliases":"u_a^-1","display":"u_a^{-1}","kind":"use","excerpt":"Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-99789a7e60ddcff1","tex":"U_A^1","aliases":"U_A^1","display":"U_A^1","kind":"use","excerpt":"K× normalizes A and every power of its radical, so U_K 1 normalizes U_A m. Therefore P is a subgroup and is compact open. It is contained in U_A 1 because mtextgreater=1 and U_K^1⊂ U_A^1.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2d3c941a9caf1b86","tex":"U_A^j","aliases":"U_A^j","display":"U_A^j","kind":"use","excerpt":"Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK3.1.7--9 give the actual groups","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-81f335e188c44b36","tex":"U_A^m","aliases":"U_A^m","display":"U_A^m","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-50f3b65ae0010e20","tex":"U_A^{m+1}","aliases":"U_A^m+1","display":"U_A^{m+1}","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bbff91e9eae9c3b1","tex":"U_A^{n-r+m}","aliases":"U_A^n-r+m","display":"U_A^{n-r+m}","kind":"use","excerpt":"At both low ranks, K^× normalizes the new P. In rank two this follows directly from normalization of A and its radical. In rank four, a K-uniformizer is approximated by the derived chain shift R times a unit of B, with relative error in P_A^n-r. Its conjugation error on U_B^m lies in U_A^n-r+m⊂ J, which proves normalization. The exact argument is p","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e21dd43c611f6c4e","tex":"u_b","aliases":"u_b","display":"u_b","kind":"use","excerpt":"Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4bd5476608aab2a3","tex":"U_B^j","aliases":"U_B^j","display":"U_B^j","kind":"use","excerpt":"Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK3.1.7--9 give the actual groups","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a1e9f1a899dd951a","tex":"U_B^m","aliases":"U_B^m","display":"U_B^m","kind":"use","excerpt":"two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5d1c052e998f0b1f","tex":"U_B^{m+1}","aliases":"U_B^m+1","display":"U_B^{m+1}","kind":"use","excerpt":"The quotient U_B^m/U_B^m+1 has dimension two. By Section2 the image of J cap U_B m on this grade is exactly its centralizing line, of dimension one. For the converse inclusion needed here, the B-approximation of a K-element of grade m differs from it by grade m+n-rtextgreater a; that error lies in U_A a subset J. Thus the whole centralizing line re","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fad0373a3389fff7","tex":"U_E^{(q+1)/2}","aliases":"U_E^(q+1)/2","display":"U_E^{(q+1)/2}","kind":"use","excerpt":"At the minimal rank-two base, extend the forced character on U_E^(q+1)/2 to U_E^1, imposing triviality on U_F^1. The prescriptions agree on their intersection because the parameter trace is one and ψ_F(u-1)=1. This is an actual finite abelian character extension and gives the full minimal simple character.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-42c782cfb50f2105","tex":"U_E^1","aliases":"U_E^1","display":"U_E^1","kind":"use","excerpt":"Section4 constructs chi_E on U_E 1 extending phi and trivial on U_F 1. Define theta_1 on H^1=U_E^1 U_(A_1) a by chi_E on the first factor and psi_alpha on the second. The restrictions agree on the intersection. BK3.2.1 gives the actual full simple character, with theta_1textbar U_F 1=1.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0975d040ea9e4ac2","tex":"U_F^s","aliases":"U_F^s","display":"U_F^s","kind":"use","excerpt":"The replacement for trace zero is the depth inequality v_F(trδ)+s≥1. It makes the additive correction trivial on scalar U_F^s, because (u-1)trδ∈ t O_F. Our native corrections have trace one, so the inequality holds at s=1. This is the exact source-to-target transfer of the weighted-visibility principle; the Euler coefficient theorem is motivation, ","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d1e23f9abe65f3fa","tex":"U_K^1","aliases":"U_K^1","display":"U_K^1","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":9},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bc43fa35e0f4d358","tex":"U_K^M","aliases":"U_K^M","display":"U_K^M","kind":"use","excerpt":"It is well defined by the intersection condition. Choose Mtextgreater=a with chitextbar U_K M=1. Then chi_0 is a character of a subgroup of the finite abelian group U_K^1/U_K^M. Extend it by adjoining generators and choosing the required complex roots, and pull it back. The resulting chi_K extends chi and is trivial on U_F 1.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d74849910db04db1","tex":"U_K^m","aliases":"U_K^m","display":"U_K^m","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension2, and its field grade p_K^m/ p_K^m+1 has dimension1. Thus","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8611e5a338fe3b84","tex":"U_r","aliases":"U_r","display":"U_r","kind":"use","excerpt":"[Matrix holonomy criterion] Fix a root r and an isomorphism U_r:V_r⟶ W_r. The following conditions are equivalent.","count":12},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ab56dd36935ce891","tex":"U_u","aliases":"U_u","display":"U_u","kind":"use","excerpt":"item There are isomorphisms U_v:V_v→ W_v, with the declared U_r, such that U_vA_e=B_eU_u for every edge e:u→ v. item For one fundamental cycle basis based at r, U_rA_γ=B_γ U_r. item The same identity holds for every based loop γ.","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1b6b2cac1ed2c1b6","tex":"U_v","aliases":"U_v","display":"U_v","kind":"use","excerpt":"item There are isomorphisms U_v:V_v→ W_v, with the declared U_r, such that U_vA_e=B_eU_u for every edge e:u→ v. item For one fundamental cycle basis based at r, U_rA_γ=B_γ U_r. item The same identity holds for every based loop γ.","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ad63d401332166ba","tex":"u_x","aliases":"u_x","display":"u_x","kind":"use","excerpt":"Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"use","excerpt":"[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"use","excerpt":"[Affine space of refinements] If one quadratic refinement exists, then the set of all quadratic refinements is an affine space over V^*.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ff464ae558a3c307","tex":"V^{I_K}","aliases":"V^I_K","display":"V^{I_K}","kind":"use","excerpt":"Let V be one interaction packet. If its inertia character is nontrivial, then V^I_K=0. If the inertia character is trivial, Frobenius acts on the invariant packet by a companion cycle.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"use","excerpt":"[Orthogonal dyadic multiplicativity] For an orthogonal sum (V,q)=(V_1,q_1)mathop⊥limits(V_2,q_2), we have Arf(q)=Arf(q_1)+Arf(q_2) and the normalized Gauss sign is the product of the two normalized signs.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d502f0a9336c724a","tex":"v_E","aliases":"v_E","display":"v_E","kind":"use","excerpt":"At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2528c99df30eac1c","tex":"v_F","aliases":"v_F","display":"v_F","kind":"use","excerpt":"The Artin--Schreier filtered lift has native trace one. Consequently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-91a52a41351c928c","tex":"V_i","aliases":"V_i","display":"V_i","kind":"use","excerpt":"Assume the one-vertex scalar-pushout contract: there are d labelled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1b5f238ca9baf774","tex":"V_r","aliases":"V_r","display":"V_r","kind":"use","excerpt":"[Matrix holonomy criterion] Fix a root r and an isomorphism U_r:V_r⟶ W_r. The following conditions are equivalent.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8043c05ca4f206a4","tex":"V_{\\rm src}","aliases":"V_rm src rm","display":"V_{\\rm src}","kind":"use","excerpt":"An orthogonal splitting W=V_rm src⊥ V_rm src^⊥ gives η_W≃η_rm src⊗η_⊥. A maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-N image-algebra projector. The retained induction/recovery theorem recovers the same Λ and then the same π. bo","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8118f19a74bb49b2","tex":"V_{\\rm src}^{\\perp}","aliases":"V_rm src^⊥ rm perp","display":"V_{\\rm src}^{\\perp}","kind":"use","excerpt":"An orthogonal splitting W=V_rm src⊥ V_rm src^⊥ gives η_W≃η_rm src⊗η_⊥. A maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-N image-algebra projector. The retained induction/recovery theorem recovers the same Λ and then the same π. bo","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-41540bfa4223f61d","tex":"V_{\\mathrm{src}}","aliases":"V_src mathrm","display":"V_{\\mathrm{src}}","kind":"use","excerpt":"Assume the one-vertex scalar-pushout contract: there are d labelled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-feb817bbfab28c96","tex":"V_{\\mathrm{src}}^\\perp","aliases":"V_src^⊥ mathrm perp","display":"V_{\\mathrm{src}}^\\perp","kind":"use","excerpt":"Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual Heisenberg image-algebra projector e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candi","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f3685a7a9e666f70","tex":"V_u","aliases":"V_u","display":"V_u","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b8d2185f95f25265","tex":"V_v","aliases":"V_v","display":"V_v","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-da190764fa0f2565","tex":"W_{K_f}","aliases":"W_K_f","display":"W_{K_f}","kind":"use","excerpt":"[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4a4f3028bb492bb2","tex":"W_r","aliases":"W_r","display":"W_r","kind":"use","excerpt":"[Matrix holonomy criterion] Fix a root r and an isomorphism U_r:V_r⟶ W_r. The following conditions are equivalent.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-28b939ffc4f05e71","tex":"W_u","aliases":"W_u","display":"W_u","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-377769d6a9c94ee0","tex":"W_v","aliases":"W_v","display":"W_v","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-47c52f415949a6cb","tex":"\\widetilde C","aliases":"widetilde C widetilde","display":"\\widetilde C","kind":"use","excerpt":"Then a finite cover widetilde C↠ C acts by actual central matrices in the unchanged candidate. Its represented operators descend to exactly χ on C. Hence the attachment and recovery above are equivariant for that action. A direct matrix homomorphism from C itself is not asserted and can fail.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-468630464758f8a1","tex":"\\widetilde L","aliases":"widetilde L widetilde","display":"\\widetilde L","kind":"use","excerpt":"Let L⊆ V be maximal isotropic. Choose a subgroup lift widetilde L⊆ H(V) on which the central coordinate is zero.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONTwo dyadic refinements can have opposite signsOriginal Section 13 plate10_quadratic_planesThe entries are exact values in F_2; both polar forms are xy'+x'y. A quadratic refinement satisfies q(x+y)+q(x)+q(y)=b(x,y). On a plane, q_0(x,y)=xy, q_1(x,y)=xy+x+y. For q_0 there are three zeros and one one, so its Gauss","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-71a31718ae729ffa","tex":"X^*","aliases":"X^*","display":"X^*","kind":"use","excerpt":"Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Its leading grade is outside the centralizing line, so u represents the nontrivial coset of P/J. Since u^2=1+x^2 in characteristic two, we have u 2 in U_B (2m) subset U_B (m+1) subset H.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e189c65b70ed9178","tex":"X^n","aliases":"X^n","display":"X^n","kind":"use","excerpt":"[Frobenius cycle polynomial] The Frobenius transfer matrix has characteristic polynomial X^n-a and determinant detτ(Φ)=(-1)^n-1a.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-917e6ba09c024b78","tex":"X^{N-1}","aliases":"X^N-1","display":"X^{N-1}","kind":"use","excerpt":"At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coeffici","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-baf58144b77f8bed","tex":"X^p","aliases":"X^p","display":"X^p","kind":"use","excerpt":"recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b4524a2723253991","tex":"X^{p-1}","aliases":"X^p-1","display":"X^{p-1}","kind":"use","excerpt":"Then v_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic minimal polynomial f(X)=X^p+a_p-1X^p-1+⋯+a_0 is Eisenstein. It is an explicit polynomial: compute the product over the p conjugates","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-14fb02863a861bee","tex":"X^r","aliases":"X^r","display":"X^r","kind":"use","excerpt":"For nontrivial inertia, the invariant space is zero. In the trivial case, Frobenius on the invariant packet is the companion matrix of X^r-a. Substituting X=u^-1 and multiplying by u^r gives 1-a u^r.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ab3482f9a2bb99d4","tex":"X^{r_i}","aliases":"X^r_i","display":"X^{r_i}","kind":"use","excerpt":"A nonzero fixed vector must lie in an inertia trivial packet, so c_i=0. On that packet, Frobenius is a companion cycle with characteristic polynomial X^r_i-a_i. Its fixed space is the eigenspace for eigenvalue one. This eigenspace is one dimensional exactly when a_i=1, and it is zero otherwise. Summing over the refined packets proves the formula.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONTwo dyadic refinements can have opposite signsOriginal Section 13 plate10_quadratic_planesThe entries are exact values in F_2; both polar forms are xy'+x'y. A quadratic refinement satisfies q(x+y)+q(x)+q(y)=b(x,y). On a plane, q_0(x,y)=xy, q_1(x,y)=xy+x+y. For q_0 there are three zeros and one one, so its Gauss","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-61c288ea4029d6a5","tex":"y^{-1}","aliases":"y^-1","display":"y^{-1}","kind":"use","excerpt":"Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace cores","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6894c7248320dbce","tex":"z^p","aliases":"z^p","display":"z^p","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-45aff6ffa0d86e5c","tex":"z^{p-1}","aliases":"z^p-1","display":"z^{p-1}","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-315e6e3c40d12791","tex":"z_a","aliases":"z_a","display":"z_a","kind":"use","excerpt":"Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ Z mathbb","display":"{\\mathbb Z}","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6820374f59f0b332","tex":"|^{-1}","aliases":"|^-1","display":"|^{-1}","kind":"use","excerpt":"For a proper parabolic P=LU, averaging over its unipotent radical gives e_U=|U|^-1∑_u∈ Uu, e_Ue_θ=e_θ e_U=0. The average selects U-fixed vectors; the cuspidal block has none. boundaryThe existence and classification of κ_θ are classical inputs. The idempotent calculation explains its exact finite selector.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESFinite Gauss phases are unit scalarsOriginal Section 6 For a nontrivial additive character ψ of F_q and a nontrivial multiplicative character χ of F_q^×, define G(χ,ψ)=∑_x∈ F_q^×χ(x)ψ(x), γ(χ,ψ)=q^-1/2G(χ,ψ). The exact identities are |G(χ,ψ)|^2=q, γ(χ,ψ)γ(χ^-1,ψ)=χ(-1). The magnitude calculation substitutes x=t","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-28c001f86de162f7","tex":"|_D","aliases":"|_D","display":"|_D","kind":"use","excerpt":"minant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1b28103f003acbf0","tex":"|_{H_0}","aliases":"|_H_0","display":"|_{H_0}","kind":"use","excerpt":"[P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and θ|_H_0 is the full character of an actual ne","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-89bc28b8775c3cb7","tex":"|_M","aliases":"|_M","display":"|_M","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-97567dd31deee4db","tex":"|_{M_s}","aliases":"|_M_s","display":"|_{M_s}","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c0304ea8c0ce259f","tex":"|_N","aliases":"|_N","display":"|_N","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONClifford multiplicity is the correct action spaceOriginal Section 9 Supply finite groups Ntriangleleft G, elementary abelian quotient A=G/N, and irreducibles Π and ρ, with ρ fixed up to isomorphism by G and Π|_N isotypic. The multiplicity space is M_ρ=Hom_N(W,X), W⊗ M_ρ≃ X. For a section s:A→ G and intertwiners","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4acb3b96be55ab28","tex":"|_P","aliases":"|_P","display":"|_P","kind":"use","excerpt":"Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is a genuine e","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ab8e308948a772eb","tex":"|_{U_F^1}","aliases":"|_U_F^1","display":"|_{U_F^1}","kind":"use","excerpt":"For this actual stratum there exists a full simple character θ with θ|_U_F^1=1. Retaining its canonical beta extension, the residue character and noncompact extension can be selected so that the resulting irreducible admissible supercuspidal π has any prescribed smooth central quasicharacter ω trivial on U_F^1. These completion choices must be free","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-38c2d53fb86a7cd0","tex":"|_{V_j}","aliases":"|_V_j","display":"|_{V_j}","kind":"use","excerpt":"[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d59a351db3fcbc7f","tex":"|_{V_s}","aliases":"|_V_s","display":"|_{V_s}","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9e57fe2b91672beb","tex":"|_{W_{K_f}}","aliases":"|_W_K_f","display":"|_{W_{K_f}}","kind":"use","excerpt":"[Unramified twist] Let ν be an unramified character of W_K. Then τ_ξ⊗ν ≅ Ind_W_K_f^W_K ( ξ ν|_W_K_f ). The tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar a, the twisted cycle has closing scalar aν(Φ)^f.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"mon arithmetic assembly carrier; it is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding i","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace cores","count":20},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3f7f41ad8028cb36","tex":"\\alpha^{-1}","aliases":"α^-1 alpha","display":"\\alpha^{-1}","kind":"use","excerpt":"er t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d830d9bc7e9c803c","tex":"\\alpha^2","aliases":"α^2 alpha","display":"\\alpha^2","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two; z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace cores","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f80ccf4fb60ee01f","tex":"\\alpha^b","aliases":"α^b alpha","display":"\\alpha^b","kind":"use","excerpt":"At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-79705d61c7703121","tex":"\\alpha^p","aliases":"α^p alpha","display":"\\alpha^p","kind":"use","excerpt":"At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6cb4d369799123ce","tex":"\\beta_i","aliases":"β_i beta","display":"\\beta_i","kind":"use","excerpt":"For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a9fcd3538bb956ab","tex":"\\beta_{i+1}","aliases":"β_i+1 beta","display":"\\beta_{i+1}","kind":"use","excerpt":"For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.","count":13},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8e4e08a5e917de66","tex":"\\delta_{jk}","aliases":"δ_jk delta","display":"\\delta_{jk}","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j cha","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-35348365f6cc6efe","tex":"\\zeta_3","aliases":"ζ_3 zeta","display":"\\zeta_3","kind":"use","excerpt":"The old fixed-field central argument genuinely failed. At p=3, (h_1,h_2)=(1,5), its naturally transferred full character takes the value ζ_3≠1 on 1+t^2, although the desired wild center is trivial. A shallow character choice cannot repair a forced value. The two-stage changing-field repair is retained in pathcentral/CENTRAL_REPAIR166.md; the trace-","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5048985144717904","tex":"\\zeta_M","aliases":"ζ_M zeta","display":"\\zeta_M","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-781d10ac98eeab56","tex":"\\zeta_M^{-ja}","aliases":"ζ_M^-ja zeta","display":"\\zeta_M^{-ja}","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5d39f2d87876b475","tex":"\\zeta_M^j","aliases":"ζ_M^j zeta","display":"\\zeta_M^j","kind":"use","excerpt":"ME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j character space for r(t).","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-195864f2c8af017d","tex":"\\eta_{L,m}","aliases":"η_L,m eta","display":"\\eta_{L,m}","kind":"use","excerpt":"[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-26c6b3957fbb36b9","tex":"\\eta_{\\rm src}","aliases":"η_rm src eta rm","display":"\\eta_{\\rm src}","kind":"use","excerpt":"An orthogonal splitting W=V_rm src⊥ V_rm src^⊥ gives η_W≃η_rm src⊗η_⊥. A maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-N image-algebra projector. The retained induction/recovery theorem recovers the same Λ and then the same π. bo","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-dd536be837f3eb94","tex":"\\eta_{\\mathrm{src}}","aliases":"η_src eta mathrm","display":"\\eta_{\\mathrm{src}}","kind":"use","excerpt":"Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual Heisenberg image-algebra projector e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candi","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4d37bc4b50bcca72","tex":"\\eta_W","aliases":"η_W eta","display":"\\eta_W","kind":"use","excerpt":"Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual Heisenberg image-algebra projector e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candi","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b6855948fa894811","tex":"\\eta_\\theta","aliases":"η_θ eta theta","display":"\\eta_\\theta","kind":"use","excerpt":"[Type side Heisenberg quotient] The pairing h_θ is nondegenerate and alternating. There is one irreducible representation η_θ of J_θ^1 containing θ, and its dimension is sqrtabsJ_θ^1/H_θ^1.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d811658e1fb8893c","tex":"\\eta_\\perp","aliases":"η_⊥ eta perp","display":"\\eta_\\perp","kind":"use","excerpt":"Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual Heisenberg image-algebra projector e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candi","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-aa596740a92286aa","tex":"\\eta_{\\perp}","aliases":"η_⊥ eta perp","display":"\\eta_{\\perp}","kind":"use","excerpt":"An orthogonal splitting W=V_rm src⊥ V_rm src^⊥ gives η_W≃η_rm src⊗η_⊥. A maximal-isotropic average on the complementary Heisenberg factor selects a one-dimensional representation line. Tensoring its projector with the identity gives a rank-N image-algebra projector. The retained induction/recovery theorem recovers the same Λ and then the same π. bo","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which t","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6a830538315a1ab8","tex":"\\vartheta","aliases":"ϑ vartheta","display":"\\vartheta","kind":"use","excerpt":"minant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-8f906ca28e719ae2","tex":"\\theta^g","aliases":"θ^g theta","display":"\\theta^g","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a725949888ccbeda","tex":"\\theta^u","aliases":"θ^u theta","display":"\\theta^u","kind":"use","excerpt":"eEnd_E_0(F^4). The odd lower critical level is r=3h_1=2m+1; the half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u,","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a721963f725aae53","tex":"\\theta_i","aliases":"θ_i theta","display":"\\theta_i","kind":"use","excerpt":"Choose a labelled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves ever","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-59f92b3334eb7f30","tex":"\\Theta_j","aliases":"Θ_j Theta","display":"\\Theta_j","kind":"use","excerpt":"For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which t","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c8082d88b8040556","tex":"\\theta_j","aliases":"θ_j theta","display":"\\theta_j","kind":"use","excerpt":"For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which t","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2568746b37cf9a0e","tex":"\\iota^{-1}","aliases":"ι^-1 iota","display":"\\iota^{-1}","kind":"use","excerpt":"[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-50b831621b666dcd","tex":"\\iota_H","aliases":"ι_H iota","display":"\\iota_H","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"[Regular elliptic torus characters] A regular character θ:T_n→ C^× determines an irreducible cuspidal representation κ_θ of G_n. Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of G_n occurs in this way.","count":15},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a118f9e5c9021f88","tex":"\\kappa^{-1}","aliases":"κ^-1 kappa","display":"\\kappa^{-1}","kind":"use","excerpt":"(P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7d0e527ea9237f05","tex":"\\kappa_\\theta","aliases":"κ_θ kappa theta","display":"\\kappa_\\theta","kind":"use","excerpt":"[Regular elliptic torus characters] A regular character θ:T_n→ C^× determines an irreducible cuspidal representation κ_θ of G_n. Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of G_n occurs in this way.","count":9},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"y). A labelled symplectic embedding gives a center-fixing full scalar-pushout isomorphism. Finite Stone--von Neumann decomposition separates the source factor from its orthogonal complement. Maximal-isotropic averaging on the complementary Heisenberg factor selects a one-dimensional representation line and hence a source-sized sector. The retained ","count":16},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"[Scalar choice invariance] Replacing u_a by λ_au_a, with λ_a∈ C^×, changes c by a coboundary and leaves b_c unchanged.","count":21},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5381814dc2f38d7a","tex":"\\lambda^{-1}","aliases":"λ^-1 lambda","display":"\\lambda^{-1}","kind":"use","excerpt":"As g²∈J, conjugation by g² fixes the character theta. Thus λ^g=λ^-1=λ. It follows that","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-627a7d01bca57e43","tex":"\\lambda^g","aliases":"λ^g lambda","display":"\\lambda^g","kind":"use","excerpt":"As g²∈J, conjugation by g² fixes the character theta. Thus λ^g=λ^-1=λ. It follows that","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5f9faba9cf67ca67","tex":"\\lambda^m","aliases":"λ^m lambda","display":"\\lambda^m","kind":"use","excerpt":"Let C=⟨ c⟩ have order m, and let T be a lift of the projective generator. Since c^m=1, the operator T^m is scalar. Choose λ∈ C^× with λ^mT^m=I. Replacing T by λ T gives an honest cyclic representation.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f5fc985955f34f9e","tex":"\\lambda^u","aliases":"λ^u lambda","display":"\\lambda^u","kind":"use","excerpt":"First, lambda is invariant under u. Indeed u 2 belongs to J and theta is J-invariant, so λ^u=λ^-1=λ by (3). Consequently H0 is normal in P, and theta0 is P-invariant.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c9503777fa994def","tex":"\\lambda_a","aliases":"λ_a lambda","display":"\\lambda_a","kind":"use","excerpt":"[Scalar choice invariance] Replacing u_a by λ_au_a, with λ_a∈ C^×, changes c by a coboundary and leaves b_c unchanged.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fbbdbad261784814","tex":"\\lambda_{a+b}^{-1}","aliases":"λ_a+b^-1 lambda","display":"\\lambda_{a+b}^{-1}","kind":"use","excerpt":"The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-27a452b3d791e541","tex":"\\lambda_b","aliases":"λ_b lambda","display":"\\lambda_b","kind":"use","excerpt":"The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).","count":13},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a202073a149b5a7a","tex":"\\mu_2","aliases":"μ_2 mu","display":"\\mu_2","kind":"use","excerpt":"-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2; ","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"use","excerpt":"[Intrinsic commutator pairing] The pairing b_ρ(a,b) = c(a,b)c(b,a)^-1 is independent of every choice in [source reference]. It is an alternating bicharacter b_ρ:A× A→μ_p.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"[Unramified twist] Let ν be an unramified character of W_K. Then τ_ξ⊗ν ≅ Ind_W_K_f^W_K ( ξ ν|_W_K_f ). The tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar a, the twisted cycle has closing scalar aν(Φ)^f.","count":10},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a1682ede43244542","tex":"\\nu_a","aliases":"ν_a nu","display":"\\nu_a","kind":"use","excerpt":"lessonPART IV / REALIZATION AND COMPARISONNumerical matching does not construct realizationBlocks 162, 166--167, profile and capacity audits For compact J⊂GL_n(F), the homomorphism v_F∘det has finite image in Z, hence image zero. An unramified character ν_a(g)=a^v_F(det g) is therefore trivial on J. The compact packet restrictions of π and π⊗ν_a ag","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-94e907efc315bf5a","tex":"\\xi^{-1}","aliases":"ξ^-1 xi","display":"\\xi^{-1}","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-946a83374eeeebd5","tex":"\\xi^{\\Phi^j}","aliases":"ξ^Φ^j xi Phi","display":"\\xi^{\\Phi^j}","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-cb28cf0122b8897d","tex":"\\varpi","aliases":"ϖ varpi","display":"\\varpi","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-222fb49e971fef73","tex":"\\pi_1","aliases":"π_1 pi","display":"\\pi_1","kind":"use","excerpt":"[Unbased gauge classification] There exists some family of vertex isomorphisms if and only if the two holonomy representations π_1(Γ,r)⟶GL(V_r), π_1(Γ,r)⟶GL(W_r) are conjugate.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d7b208ba9a56ce9d","tex":"\\varpi_K","aliases":"ϖ_K varpi","display":"\\varpi_K","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root; see [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":9},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ee268b4caa6936e7","tex":"\\sigma_\\tau","aliases":"σ_τ sigma tau","display":"\\sigma_\\tau","kind":"use","excerpt":"[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9add02f6eb4c12f0","tex":"\\tau^{I_K}","aliases":"τ^I_K tau","display":"\\tau^{I_K}","kind":"use","excerpt":"[Conductor of a nontrivial tame atom] Let τ be an irreducible tame Weil representation of dimension n>1. Then Swan(τ)=0, dimτ^I_K=0, Art(τ)=n.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3e4e14a09d296fdb","tex":"\\tau_\\eta","aliases":"τ_η tau eta","display":"\\tau_\\eta","kind":"use","excerpt":"Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bda1495d04cf4d81","tex":"\\tau_\\xi","aliases":"τ_ξ tau xi","display":"\\tau_\\xi","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fe758f13cc84cd53","tex":"\\tau_\\xi^\\vee","aliases":"τ_ξ^vee tau xi vee","display":"\\tau_\\xi^\\vee","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-18f9ac23eff67597","tex":"\\Phi^{-1}","aliases":"Φ^-1 Phi","display":"\\Phi^{-1}","kind":"use","excerpt":"[Tame character projectors] The operators P_j are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by Φ P_jΦ^-1=P_qj.","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-53b5436ea4ca2986","tex":"\\Phi^f","aliases":"Φ^f Phi","display":"\\Phi^f","kind":"use","excerpt":"The nonzero tame packets are permuted by Frobenius. A union of Frobenius orbits is stable under W_K, so irreducibility forces one orbit. Choose one packet V_θ. Its stabilizer is W_K_f=⟨ I_K,Φ^f⟩. A nonzero proper stable subspace of V_θ would generate a nonzero proper W_K stable subspace after summing its Frobenius translates. Thus V_θ is irreducibl","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-920bc1c8bd65d9c9","tex":"\\Phi^j","aliases":"Φ^j Phi","display":"\\Phi^j","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7c934e3eb93a78aa","tex":"\\chi^{-1}","aliases":"χ^-1 chi","display":"\\chi^{-1}","kind":"use","excerpt":"[Duality of Gauss phases] For nontrivial χ, γ(χ,ψ)γ(χ^-1,ψ) = χ(-1).","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f9093cf6c7bb37fd","tex":"\\chi_\\theta","aliases":"χ_θ chi theta","display":"\\chi_\\theta","kind":"use","excerpt":"[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].","count":7},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c1b88ae641d78c35","tex":"\\psi_0","aliases":"ψ_0 psi","display":"\\psi_0","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2cea413acc8ef385","tex":"\\psi_F","aliases":"ψ_F psi","display":"\\psi_F","kind":"use","excerpt":"Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, fin","count":4},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-96186ebabe575c55","tex":"\\psi_{\\beta-\\gamma}","aliases":"ψ_β-γ psi beta gamma","display":"\\psi_{\\beta-\\gamma}","kind":"use","excerpt":"minant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ab8e8ecf46dc1fbe","tex":"\\omega_V","aliases":"ω_V omega","display":"\\omega_V","kind":"use","excerpt":"[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bc499d807930c364","tex":"\\omega_{V_s}","aliases":"ω_V_s omega","display":"\\omega_{V_s}","kind":"use","excerpt":"[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-18bc7e17fe02b53f","tex":"\\Omega_\\theta","aliases":"Ω_θ Omega theta","display":"\\Omega_\\theta","kind":"use","excerpt":"[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-990f57cf16482556","tex":"\\omega_\\theta","aliases":"ω_θ omega theta","display":"\\omega_\\theta","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a4292c3402982fcc","tex":"\\Omega_\\rho","aliases":"Ω_ρ Omega rho","display":"\\Omega_\\rho","kind":"use","excerpt":"[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-33382dc01fd09a1e","tex":"\\omega_\\rho","aliases":"ω_ρ omega rho","display":"\\omega_\\rho","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Isotropic averaging projector] The operator P_L = frac1abs L ∑_ℓ∈widetilde L ρ(ℓ) is an orthogonal projector. In the Schrödinger model with L=X, it has rank one and selects the constant functions.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9ec711e428117ead","tex":"\\twoheadrightarrow","aliases":"↠ twoheadrightarrow","display":"\\twoheadrightarrow","kind":"use","excerpt":"Then a finite cover widetilde C↠ C acts by actual central matrices in the unchanged candidate. Its represented operators descend to exactly χ on C. Hence the attachment and recovery above are equivariant for that action. A direct matrix homomorphism from C itself is not asserted and can fail.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[source reference] is a reconstruction theorem. It does not produce the atomic assignment τ↦σ_τ, and it does not prove the required local factor identities.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently cons","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently cons","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"There is a logical difference between ∀π [R(d,π)⇒H5(π)] and ∃π [R(d,π)landH5(π)]. The first statement can be vacuous. The second includes an actual witness. Choosing the established reference atom only supplies a profile-matching object; it leaves its recovery of d unproved. takeawayH5 remains a conditional premise about the same independently cons","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-2d3db36262598bcd","tex":"\\prod_j","aliases":"∏_j prod","display":"\\prod_j","kind":"use","excerpt":"The half dimensions add, and det(g-Imid V) = ∏_jdet(g-Imid V_j). The quadratic character is multiplicative.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-60ea789b512f778a","tex":"\\prod_{j=1}^{s}","aliases":"∏_j=1^s prod","display":"\\prod_{j=1}^{s}","kind":"use","excerpt":"[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0a86f66b3b267a78","tex":"\\prod_s","aliases":"∏_s prod","display":"\\prod_s","kind":"use","excerpt":"The Heisenberg group of an orthogonal sum is the central product of the Heisenberg groups of the summands. The tensor product representation has the declared central character and has dimension ∏_ssqrtabsV_s = sqrtabs V. It is irreducible by the finite Stone von Neumann theorem, so it is the required representation.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f4084704a37f06b3","tex":"\\prod_{s\\in S}","aliases":"∏_s∈ S prod in","display":"\\prod_{s\\in S}","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-a98f0c6f0be59f8a","tex":"\\sum_{a=0}^{M-1}","aliases":"∑_a=0^M-1 sum","display":"\\sum_{a=0}^{M-1}","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ad151dded7ef7d37","tex":"\\sum_{a\\in A}","aliases":"∑_a∈ A sum in","display":"\\sum_{a\\in A}","kind":"use","excerpt":"Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-deb9c97b11939ae2","tex":"\\sum_{d\\mid n}","aliases":"∑_dmid n sum mid","display":null,"kind":"use","excerpt":"[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-778599755ac857bc","tex":"\\sum_{g\\in G_n}","aliases":"∑_g∈ G_n sum in","display":"\\sum_{g\\in G_n}","kind":"use","excerpt":"[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].","count":3},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-da1c94a6e8462cce","tex":"\\sum_{i=0}^{N-1}","aliases":"∑_i=0^N-1 sum","display":"\\sum_{i=0}^{N-1}","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESFourier projectors isolate tame inertiaOriginal Section 3; geometric Frobenius convention Let t generate a finite tame quotient of order M, with (q,M)=1. For j∈ Z/M Z define P_j=frac1M∑_a=0^M-1ζ_M^-jar(t)^a, ζ_M=e^2π i/M. The finite Fourier identities give P_jP_k=δ_jkP_j and ∑_jP_j=I. Thus P_jV is the ζ_M^j cha","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-04a3e524463e2337","tex":"\\sum_{j=1}^{m-1}","aliases":"∑_j=1^m-1 sum","display":"\\sum_{j=1}^{m-1}","kind":"use","excerpt":"[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-bd50407fa1571a87","tex":"\\sum_{t\\in k^{\\times}}","aliases":"∑_t∈ k^× sum in times","display":"\\sum_{t\\in k^{\\times}}","kind":"use","excerpt":"Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-945d71373857dd47","tex":"\\sum_{u\\in U}","aliases":"∑_u∈ U sum in","display":"\\sum_{u\\in U}","kind":"use","excerpt":"[Parabolic annihilation] Let P=LU be a proper parabolic subgroup of G_n, and put e_U=frac1abs U∑_u∈ Uu. If κ_θ is cuspidal, then e_Ue_θ=e_θ e_U=0.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9d4ce8d11f1c2be0","tex":"\\sum_{u\\in X}","aliases":"∑_u∈ X sum in","display":"\\sum_{u\\in X}","kind":"use","excerpt":"If the translation component x is nonzero, the operator permutes the standard basis without a fixed point, so its trace is zero. If x=0 but ξ≠0, the trace is ψ(z)∑_u∈ Xψ(ξ(u))=0. When x=ξ=0, the operator is the scalar ψ(z).","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0e44b9be25437975","tex":"\\sum_x","aliases":"∑_x sum","display":"\\sum_x","kind":"use","excerpt":"Using the quadratic identity, q_v(x)=q(x+v)+q(v). Therefore ∑_x(-1)^q_v(x) = (-1)^q(v) ∑_x(-1)^q(x). Compare the signs in [source reference].","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-60e7e6eb0365413a","tex":"\\sum_{x,y\\in k^{\\times}}","aliases":"∑_x,y∈ k^× sum in times","display":"\\sum_{x,y\\in k^{\\times}}","kind":"use","excerpt":"Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-e4274b6f3ad16bfb","tex":"\\sum_{x\\in k^{\\times}}","aliases":"∑_x∈ k^× sum in times","display":"\\sum_{x\\in k^{\\times}}","kind":"use","excerpt":"Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-9923075fc1158c85","tex":"\\sum_{x\\in V}","aliases":"∑_x∈ V sum in","display":"\\sum_{x\\in V}","kind":"use","excerpt":"[Dyadic Gauss sign] Let dim V=2r, and let q be nondegenerate. Then ∑_x∈ V(-1)^q(x) = (-1)^Arf(q)2^r.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-32572785b48f6bd4","tex":"\\sum_{x\\in X}","aliases":"∑_x∈ X sum in","display":"\\sum_{x\\in X}","kind":"use","excerpt":"The lifted isotropic subgroup is abelian, so group averaging is an idempotent. Every operator in the average is unitary, and the subgroup is closed under inversion, so the average is self adjoint. For L=X, the average is the translation average (P_Lf)(u) = frac1abs X ∑_x∈ Xf(u+x), which is the constant function with value equal to the mean of f. It","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d5c190cbed90c5b8","tex":"\\sum_{x\\in\\F_q^\\times}","aliases":"∑_x∈F_q^× ∑_x∈ F_q^× sum in F times sum in mathbb times","display":"\\sum_{x\\in{\\mathbb F}_q^\\times}","kind":"use","excerpt":"lessonPART I / FINITE TAME COORDINATESFinite Gauss phases are unit scalarsOriginal Section 6 For a nontrivial additive character ψ of F_q and a nontrivial multiplicative character χ of F_q^×, define G(χ,ψ)=∑_x∈ F_q^×χ(x)ψ(x), γ(χ,ψ)=q^-1/2G(χ,ψ). The exact identities are |G(χ,ψ)|^2=q, γ(χ,ψ)γ(χ^-1,ψ)=χ(-1). The magnitude calculation substitutes x=t","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-6c84dfd9bdb76fc0","tex":"\\sum_{\\ell\\in\\widetilde L}","aliases":"∑_ℓ∈widetilde L sum ell in widetilde","display":"\\sum_{\\ell\\in\\widetilde L}","kind":"use","excerpt":"[Isotropic averaging projector] The operator P_L = frac1abs L ∑_ℓ∈widetilde L ρ(ℓ) is an orthogonal projector. In the Schrödinger model with L=X, it has rank one and selects the constant functions.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Write the odd minimal level as q=2m+1, with m≥1, and let J=H^1=J^1=U_K^1U_A^m+1. Define P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced additive-character domain, so λ has values ","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"lessonPART IV / REALIZATION AND COMPARISONNumerical matching does not construct realizationBlocks 162, 166--167, profile and capacity audits For compact J⊂GL_n(F), the homomorphism v_F∘det has finite image in Z, hence image zero. An unramified character ν_a(g)=a^v_F(det g) is therefore trivial on J. The compact packet restrictions of π and π⊗ν_a ag","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the additive-character cross term lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.","count":9},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"[Character support] The character of the Schrödinger representation satisfies Trρ(v,z) = p^rψ(z), · v=0, 0, · v≠0.","count":5},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"They are the compatible native truncations of (5), because the outer extension, uniformizer, and lift depend on the same retained largest h at each stage. For 1≤ j<d, in rank 2 (j+1), put","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"[Quantitative finite gluing] With the notation of [source reference], normU_vA_e-B_eU_u ≤ normB_p_vnormD_enormA_p_u^-1. Thus small fundamental cycle defects give small non tree edge residuals whenever the tree transports are uniformly conditioned.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":26},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":13},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.","count":6},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","excerpt":"Assume the one-vertex scalar-pushout contract: there are d labelled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-5aaaf24677ceb564","tex":"\\bigotimes","aliases":"⊗ bigotimes","display":"\\bigotimes","kind":"use","excerpt":"[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.","count":2},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-1e698d647992f7ca","tex":"\\bigotimes_s","aliases":"⊗_s bigotimes","display":"\\bigotimes_s","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONOrthogonal breaks become tensor factorsOriginal Section 15 plate13_break_tensorSymplectic direct sum and Heisenberg tensor product are compatible. For V=⊥_sV_s and compatible central characters, ω_V≃⊗_sω_V_s, ∏_ssqrt|V_s|=sqrt|V|. The groups combine by a central product. Tensoring the individual intertwiners gi","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c91c177726b0cd36","tex":"\\bigotimes_{s\\in S}","aliases":"⊗_s∈ S bigotimes in","display":"\\bigotimes_{s\\in S}","kind":"use","excerpt":"[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"The invariant orthogonal decomposition gives M=mathop⊥limits_s∈ SM_s. Symplectic invariance gives M^⊥=ker(g-I). Fixed point freeness on M therefore makes the restricted form nondegenerate on M and on each M_s. Each restriction of g_s-I to M_s is invertible, so [source reference] applies to the moving spaces. A zero moving space contributes one. In ","count":8},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-857678628d8afdc4","tex":"\\perp_s","aliases":"⊥_s perp","display":"\\perp_s","kind":"use","excerpt":"lessonPART II / FINITE WILD COMPARISONOrthogonal breaks become tensor factorsOriginal Section 15 plate13_break_tensorSymplectic direct sum and Heisenberg tensor product are compatible. For V=⊥_sV_s and compatible central characters, ω_V≃⊗_sω_V_s, ∏_ssqrt|V_s|=sqrt|V|. The groups combine by a central product. Tensoring the individual intertwiners gi","count":1},{"work":"paper-xi","url":"notation/paper-xi.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":325},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-798565b041c725f1","tex":"\\Ann","aliases":"Ann operatorname","display":"{\\operatorname{Ann}}","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-71a978cf87af4fd1","tex":"\\Ann_R","aliases":"Ann_R Ann operatorname","display":"{\\operatorname{Ann}}_R","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-db1d2ae53aa18234","tex":"\\bar\\rho","aliases":"barρ bar rho","display":"\\bar\\rho","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":50},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2b29a3476d110709","tex":"C_\\infty","aliases":"C_∞ infty","display":"C_\\infty","kind":"definition-passage","excerpt":"[Derived patching datum] A derived patching datum consists of a Noetherian ring R_∞, an ideal a, a perfect complex C_∞ over R_∞, and identifications R≅ R_∞/ a, C≃ C_∞ mathbinmathop⊗limits_R_∞^ LR.","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automo","count":54},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"ndefinition[Derived completion datum at a point] A derived completion datum is defined in the same way, with a perfect patched complex C_∞ of full derived support and a finite level automorphic complex C ≃ C_∞ mathbinmathop⊗limits_R_∞^ LR in place of the ordinary modules. Under the Hecke ring isomorphism e:R≅ T, require an actual derived carrier id","count":100},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c7c26f6b3b085b5c","tex":"E'","aliases":"E'","display":"E'","kind":"definition-passage","excerpt":"[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automo","count":46},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Algebraic Frobenius spine] An algebraic Frobenius spine consists of a number field L, a finite set of places S of F, and polynomials P_v(X)∈ L[X] for every finite place v∉ S, together with any additional local labels required by the declared compatibility problem.","count":11},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":31},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b7794f17df61c3e5","tex":"f_{n,m}","aliases":"f_n,m","display":"f_{n,m}","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ea06efaabf953d2a","tex":"\\fraka","aliases":"fraka a mathfrak","display":"{\\mathfrak a}","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":23},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-39afe4e3b95de796","tex":"\\frakp","aliases":"frakp p mathfrak","display":"{\\mathfrak p}","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":42},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic zero locu","count":50},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"[Algebraic Frobenius spine] An algebraic Frobenius spine consists of a number field L, a finite set of places S of F, and polynomials P_v(X)∈ L[X] for every finite place v∉ S, together with any additional local labels required by the declared compatibility problem.","count":15},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-08093f17d5992ea3","tex":"\\mathbf L","aliases":"L mathbf","display":"\\mathbf L","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":10},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5236b838b836f2fa","tex":"\\mathrm{loc}","aliases":"loc mathrm","display":"\\mathrm{loc}","kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":12},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-68a67f309b99ec73","tex":"\\longleftarrow","aliases":"longleftarrow","display":null,"kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":119},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":82},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2a35d6b08f661ad2","tex":"M_{\\frakp}","aliases":"M_frakp M_ p frakp mathfrak","display":"M_{{\\mathfrak p}}","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":10},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-eae2d59e5b8a6aeb","tex":"M_\\infty","aliases":"M_∞ infty","display":"M_\\infty","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":12},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":8},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-75ef01eda1abe09b","tex":"\\mathop","aliases":"mathop","display":null,"kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":8},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":67},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f35ce0db28d3fb4f","tex":"\\OO","aliases":"OO O mathcal","display":"{\\mathcal O}","kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":52},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"efinition[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic ","count":38},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e70fd9df7027e2bf","tex":"P_v","aliases":"P_v","display":"P_v","kind":"definition-passage","excerpt":"efinition[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic ","count":15},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":240},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8a94f04a221b67c7","tex":"R^{\\mathbf L}","aliases":"R^ L mathbf","display":"R^{\\mathbf L}","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7ed766ea53d3d411","tex":"R^{\\mathrm{loc}}","aliases":"R^loc mathrm","display":"R^{\\mathrm{loc}}","kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":11},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e9bf672601602601","tex":"R^{\\mathrm{loc},\\tau}","aliases":"R^loc,τ mathrm tau","display":"R^{\\mathrm{loc},\\tau}","kind":"definition-passage","excerpt":"every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squarem","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-dd279b65249d7e1f","tex":"R^\\square","aliases":"R^square square","display":"R^\\square","kind":"definition-passage","excerpt":"l deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a383171148f13ce0","tex":"R^{\\tau}","aliases":"R^τ tau","display":"R^{\\tau}","kind":"definition-passage","excerpt":"every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squarem","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d3718ad5d081ab92","tex":"R_v^\\square","aliases":"R_v^square square","display":"R_v^\\square","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2ffbbcc5c7c165f2","tex":"R_v^{\\tau_v}","aliases":"R_v^τ_v tau","display":"R_v^{\\tau_v}","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1903894d14878533","tex":"R_\\infty","aliases":"R_∞ infty","display":"R_\\infty","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":22},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":57},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-327dcbd8f0058da9","tex":"\\Spec","aliases":"Spec operatorname","display":"{\\operatorname{Spec}}","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":27},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":10},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d0f7b96a7d9184e3","tex":"\\Supp","aliases":"Supp operatorname","display":"{\\operatorname{Supp}}","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":28},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ed9ed110cf06d528","tex":"\\Supp_R","aliases":"Supp_R Supp operatorname","display":"{\\operatorname{Supp}}_R","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":21},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-62afa297a68a33f1","tex":"\\Supp_{R_\\infty}","aliases":"Supp_R_∞ Supp infty operatorname infty","display":"{\\operatorname{Supp}}_{R_\\infty}","kind":"definition-passage","excerpt":"datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=S","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-45d47d344acaa5d2","tex":"\\TT","aliases":"TT T mathbb","display":"{\\mathbb T}","kind":"definition-passage","excerpt":"[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic zero locu","count":60},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":78},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":51},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f908e7a2d9e05299","tex":"V_3","aliases":"V_3","display":"V_3","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8b86c046f5cd21ee","tex":"V_m","aliases":"V_m","display":"V_m","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":8},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a6735e9f71ad2ebf","tex":"V_n","aliases":"V_n","display":"V_n","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8174d5fba0700d5c","tex":"\\wideotimes","aliases":"wideotimes mathbinwidehat⊗ mathbin widehat otimes","display":null,"kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c10f72797a18708f","tex":"\\wideotimes_{R^{\\mathrm{loc}}}","aliases":"wideotimes_R^loc mathbinwidehat⊗_R^loc wideotimes mathrm mathbin widehat otimes mathrm","display":null,"kind":"definition-passage","excerpt":"every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squarem","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cfe1eab82fe0f384","tex":"\\wideotimes_{v\\in S,\\OO}","aliases":"wideotimes_v∈ S,OO mathbinwidehat⊗_v∈ S, O wideotimes in OO mathbin widehat otimes in mathcal","display":null,"kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"efinition[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic ","count":60},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automo","count":203},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-59ca56eb5c1b813d","tex":"X_{\\lambda,1}","aliases":"X_λ,1 lambda","display":"X_{\\lambda,1}","kind":"definition-passage","excerpt":"[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finit","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5d9edf14d558205f","tex":"X_{\\lambda,2}","aliases":"X_λ,2 lambda","display":"X_{\\lambda,2}","kind":"definition-passage","excerpt":"[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finit","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e2312717706b5a2a","tex":"X_{\\lambda,3}","aliases":"X_λ,3 lambda","display":"X_{\\lambda,3}","kind":"definition-passage","excerpt":"[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finit","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":27},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7256bbf22fa35eec","tex":"Y_m","aliases":"Y_m","display":"Y_m","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":8},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finit","count":66},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5f9faba9cf67ca67","tex":"\\lambda^m","aliases":"λ^m lambda","display":"\\lambda^m","kind":"definition-passage","excerpt":"[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finit","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automo","count":50},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8239c91a9dd24610","tex":"\\rho_R","aliases":"ρ_R rho","display":"\\rho_R","kind":"definition-passage","excerpt":"[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automo","count":11},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d903684d8cb76c26","tex":"\\rho_x","aliases":"ρ_x rho","display":"\\rho_x","kind":"definition-passage","excerpt":"phism. item An automorphic Hecke datum ( T,H,(P_v)) and a pointwise classicality and Galois realization theorem satisfying all the inputs of [source reference] for the nonzero fiber at x: an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that re","count":28},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":11},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ca0868865309284b","tex":"\\tau_v","aliases":"τ_v tau","display":"\\tau_v","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":47},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":53},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"definition-passage","excerpt":"[Algebraic Frobenius spine] An algebraic Frobenius spine consists of a number field L, a finite set of places S of F, and polynomials P_v(X)∈ L[X] for every finite place v∉ S, together with any additional local labels required by the declared compatibility problem.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"definition-passage","excerpt":"phism. item An automorphic Hecke datum ( T,H,(P_v)) and a pointwise classicality and Galois realization theorem satisfying all the inputs of [source reference] for the nonzero fiber at x: an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that re","count":16},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":28},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":67},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":18},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":13},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":17},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":30},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":7},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e473df1665779d49","tex":"\\bigcap_{n\\geq m}","aliases":"⋂_n≥ m bigcap geq","display":"\\bigcap_{n\\geq m}","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":11},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":14},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"introduction-passage","excerpt":"eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ ","count":7},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":71},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"For M⊗_Λ,ϵ O, we use a right Λ module M. The left module convention is O⊗_Λ,ϵM. Throughout, a M denotes the augmentation relation submodule: it is generated by ma for right modules and by am for left modules, with a∈ a. This convention also applies to nonabelian groups in [source reference]. For abelian groups, both tensor orders agree over the com","count":56},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f1ffe82ebc455b14","tex":"f^*","aliases":"f^*","display":"f^*","kind":"introduction-passage","excerpt":"eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ ","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cc7582cb4d0dc2f3","tex":"\\FF","aliases":"FF F mathbb","display":"{\\mathbb F}","kind":"introduction-passage","excerpt":"The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit cou","count":29},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5b4740083376d053","tex":"\\FF_p","aliases":"FF_p F_p FF mathbb","display":"{\\mathbb F}_p","kind":"introduction-passage","excerpt":"The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit cou","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1ac97bde10b6c4f5","tex":"\\frakm","aliases":"frakm m mathfrak","display":"{\\mathfrak m}","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":24},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d0b81ba1507604a0","tex":"\\frakm_{\\OO}","aliases":"frakm_OO m_ O frakm OO mathfrak mathcal","display":"{\\mathfrak m}_{{\\mathcal O}}","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-911dc5b7a786cf08","tex":"\\frakm_{R_i}","aliases":"frakm_R_i m_R_i frakm mathfrak","display":"{\\mathfrak m}_{R_i}","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit cou","count":15},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit cou","count":28},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":49},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":52},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8d4c5f4c75a9b75d","tex":"I_i","aliases":"I_i","display":"I_i","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":11},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit cou","count":57},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit cou","count":25},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":21},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-841a78b5f2413be0","tex":"\\Supp_{\\TT}","aliases":"Supp_TT Supp_ T Supp TT operatorname mathbb","display":"{\\operatorname{Supp}}_{{\\mathbb T}}","kind":"introduction-passage","excerpt":"eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ ","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","excerpt":"The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit cou","count":30},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"introduction-passage","excerpt":"The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit cou","count":15},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"introduction-passage","excerpt":"For M⊗_Λ,ϵ O, we use a right Λ module M. The left module convention is O⊗_Λ,ϵM. Throughout, a M denotes the augmentation relation submodule: it is generated by ma for right modules and by am for left modules, with a∈ a. This convention also applies to nonabelian groups in [source reference]. For abelian groups, both tensor orders agree over the com","count":25},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"The first assertion follows from locality in [source reference]. For the general finite p group assertion, let e∈ k[G] be idempotent and write ϵ:k[G]→ k for augmentation. Suppose ϵ(e)=0 and e≠0. The F_p span W of the left translates ge is a nonzero finite dimensional F_p vector space, stable under G. Its cardinality is divisible by p. The orbit cou","count":13},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5aaaf24677ceb564","tex":"\\bigotimes","aliases":"⊗ bigotimes","display":"\\bigotimes","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-00a1232f6f9fcef1","tex":"\\bigotimes_{i=1}^{s}","aliases":"⊗_i=1^s bigotimes","display":"\\bigotimes_{i=1}^{s}","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cc9832e1e3b8da8d","tex":"\\otimes_{\\Lambda,\\epsilon}","aliases":"⊗_Λ,ϵ otimes Lambda epsilon","display":"\\otimes_{\\Lambda,\\epsilon}","kind":"introduction-passage","excerpt":"For M⊗_Λ,ϵ O, we use a right Λ module M. The left module convention is O⊗_Λ,ϵM. Throughout, a M denotes the augmentation relation submodule: it is generated by ma for right modules and by am for left modules, with a∈ a. This convention also applies to nonabelian groups in [source reference]. For abelian groups, both tensor orders agree over the com","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"Let f:R^loc→ A be the point. Since A is Hausdorff and f is continuous, ker f is closed. Thus it contains the closed ideal J if and only if it contains the algebraic ideal generated by the images of the I_i. This is equivalent to every local restriction killing I_i. The universal property of the actual ideal quotient gives the factors, and the quoti","count":38},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2bac4a55153a7a84","tex":"\\a","aliases":"a","display":null,"kind":"use","excerpt":"node[anchor=east] at (-.3,2.25) point a; node[anchor=east] at (-.3,1.4) dim H^0(C_a); node[anchor=east] at (-.3,.55) dim H^1(C_a); foreach x/a in 0/-2,1.5/-1,3/0,4.5/1,6/2 node at (x+.4,2.25) a; draw[black!35] (x,1.05) rectangle ++(.8,.7);draw[black!35] (x,.2) rectangle ++(.8,.7); foreach x in 0,1.5,4.5,6node[text=black!45] at (x+.4,1.4) 0;node[tex","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-64209c7327ae53ce","tex":"\\mathfrak a","aliases":"a mathfrak","display":"\\mathfrak a","kind":"use","excerpt":"-------------------------------- setlengthparskip5pt vpage19Complete the actual topologyTOPOLOGICAL ALGEBRA / THE COMPARISON MAPMain text: Theorem 3.1, Corollary 3.2 and Proposition 3.3 p12-route:visual-19Main statement 3.1 The hat in whot names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require a","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient map Z_5→ F_5. A continuous local point is specified by u⟼ aδ+bδ^2, v⟼ cδ+dδ^2, a,b,c,d∈ F_5. It factors through R^τ precisely when a^2=0, c=a, d=b Longleftrightarrow a=c=0, d=b.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cd95d8a3733d5471","tex":"a_{00}","aliases":"a_00","display":"a_{00}","kind":"use","excerpt":"headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-473e1c24b1b07993","tex":"a_{10}","aliases":"a_10","display":"a_{10}","kind":"use","excerpt":"headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-48bc979c627783f9","tex":"a_{11}","aliases":"a_11","display":"a_{11}","kind":"use","excerpt":"-------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-83d889c674c63605","tex":"a_{11}^2","aliases":"a_11^2","display":"a_{11}^2","kind":"use","excerpt":"Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A^2-I to vanish. In B= Z_5[a_11,a_12,a_21,a_22], the ideal J is generat","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e3be1218806df9a4","tex":"a_{12}","aliases":"a_12","display":"a_{12}","kind":"use","excerpt":"-------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-77ec1a545e6a5aa2","tex":"a_{21}","aliases":"a_21","display":"a_{21}","kind":"use","excerpt":"-------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a9809bbe7ab8b054","tex":"a_{22}","aliases":"a_22","display":"a_{22}","kind":"use","excerpt":"-------------------------------------------------------------- vpage24Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-84e82be49be67c79","tex":"a_{22}^2","aliases":"a_22^2","display":"a_{22}^2","kind":"use","excerpt":"Equations survive every precisionADIC EFFECTIVITY / DECLARED POLYNOMIAL CONDITIONSMain text: revised Theorem 10.5 and Corollary 3.2 p12-route:visual-24Main statement 10.5 The preceding involution gives an exact polynomial condition. For a matrix A=(a_ij), require all entries of A^2-I to vanish. In B= Z_5[a_11,a_12,a_21,a_22], the ideal J is generat","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a9234285c855dd5f","tex":"A_{\\frakp}","aliases":"A_frakp A_ p frakp mathfrak","display":"A_{{\\mathfrak p}}","kind":"use","excerpt":"If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjectiv","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2d83d124aaabaee7","tex":"a_g","aliases":"a_g","display":"a_g","kind":"use","excerpt":"Let Δ be a finite abelian p group and let Λ= O[Δ]. The augmentation map is ϵ:Λ⟶ O, ∑_g∈Δa_gg⟼∑_g∈Δa_g. Its kernel is the augmentation ideal a.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-daf800c8ae3d0681","tex":"a_{ij}","aliases":"a_ij","display":"a_{ij}","kind":"use","excerpt":"headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"In characteristic p, (1+X_i)^p^n_i-1=X_i^p^n_i. Thus the assignment X_i↦γ_i-1 gives a surjective homomorphism from the displayed truncated polynomial ring to k[Δ]. Both sides have dimension ∏_ip^n_i=absΔ over k, so the map is an isomorphism. The quotient by the ideal generated by the X_i is k. Since each X_i is nilpotent, that ideal is the unique m","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-bdbd1a428e5dd1c0","tex":"\\ad","aliases":"ad operatorname","display":"{\\operatorname{ad}}","kind":"use","excerpt":"[Adequate image] Let G⊆GL(V) act absolutely irreducibly over a finite field k of characteristic p. Under the standard definition, adequacy imposes vanishing conditions on H^1(G,k) and H^1(G,ad^0V), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of ad^0V.","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-127f64d47614c2ee","tex":"\\operatorname{ad}","aliases":"ad operatorname","display":"\\operatorname{ad}","kind":"use","excerpt":"headWhy the cohomology argument is uniform in the module Since |S_3|=6 is a unit in F_7, a cocycle f:S_3→ M is a coboundary: set u=-6^-1∑_h f(h) and sum f(gh)=g f(h)+f(g) to obtain gu-u=f(g). This proves H^1(S_3,M)=0 for every F_7[S_3]-module M. The printed invariant calculation separately proves H^0(S_3,ad^0V)=0.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3ee48b451744dd8e","tex":"\\ad^0","aliases":"ad^0 ad operatorname","display":"{\\operatorname{ad}}^0","kind":"use","excerpt":"[Adequate image] Let G⊆GL(V) act absolutely irreducibly over a finite field k of characteristic p. Under the standard definition, adequacy imposes vanishing conditions on H^1(G,k) and H^1(G,ad^0V), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of ad^0V.","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6a1295c8322d225d","tex":"\\operatorname{ad}^0","aliases":"ad^0 operatorname","display":"\\operatorname{ad}^0","kind":"use","excerpt":"headWhy the cohomology argument is uniform in the module Since |S_3|=6 is a unit in F_7, a cocycle f:S_3→ M is a coboundary: set u=-6^-1∑_h f(h) and sum f(gh)=g f(h)+f(g) to obtain gu-u=f(g). This proves H^1(S_3,M)=0 for every F_7[S_3]-module M. The printed invariant calculation separately proves H^0(S_3,ad^0V)=0.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4afb314fdcd4106f","tex":"\\arraybackslash","aliases":"arraybackslash","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it car","count":29},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"The coordinate map (a,b)⟼(a,-a+b,-b) identifies the displayed matrices with the permutation action on the sum zero subspace of F_7^3. This action is faithful. The four matrices 1,s,t,st span M_2 over every field of characteristic seven: the determinant of their coordinate matrix is 4≠0. A subspace stable under the group is therefore stable under ev","count":31},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"headAn isomorphism changes coordinates, not occupancy Over a field of characteristic other than two, the isomorphism k[x,y]xrightarrow∼k[a,b], x↦ a+b, y↦ a-b sends the Fitting ideals to (a^2-b^2) and (a,b). The two axes become the two lines a=b and a=-b. No previously missing module vector is created by this relabeling.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage5Completion retains every precisionMODEL A / COMPATIBLE COEFFICIENTSMain text: Theorem 3.1; reconstruction and finite occupancy in Section 10 p12-route:visual-5Main statement 10.5 For B= Z_5[ε]/(ε^2), consider the finite quotient rings B_n=( Z/5^n Z)[ε]/(ε^2), |B_n|=","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f8dd14ae875161a4","tex":"B_\\lambda","aliases":"B_λ lambda","display":"B_\\lambda","kind":"use","excerpt":"Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem,","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-60e74a0938329fe2","tex":"\\bar g","aliases":"bar g bar","display":"\\bar g","kind":"use","excerpt":"headA quotient of groups gives a quotient of algebras Map C_4× C_2→ C_2 by g↦bar g, h↦1. The induced algebra map sends X to bar X with bar X^2=0 and sends Y to zero. It is surjective. Augmentation commutes with this map because every group basis element is sent to another group element of augmentation one.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5a77d9004b78105c","tex":"\\bar X","aliases":"bar X bar","display":"\\bar X","kind":"use","excerpt":"headA quotient of groups gives a quotient of algebras Map C_4× C_2→ C_2 by g↦bar g, h↦1. The induced algebra map sends X to bar X with bar X^2=0 and sends Y to zero. It is surjective. Augmentation commutes with this map because every group basis element is sent to another group element of augmentation one.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2eb6b7c2e5f24259","tex":"\\bar X^2","aliases":"bar X^2 bar","display":"\\bar X^2","kind":"use","excerpt":"headA quotient of groups gives a quotient of algebras Map C_4× C_2→ C_2 by g↦bar g, h↦1. The induced algebra map sends X to bar X with bar X^2=0 and sends Y to zero. It is surjective. Augmentation commutes with this map because every group basis element is sent to another group element of augmentation one.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"us elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgrou","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"Write W(a,b,c)=a · b c · -a, Z=W(2,3,4), A=W(1,1,2), B=W(1,5,6). The three coordinate columns have determinant 3≠0. Under conjugation, sZ=-Z, tZ=Z, sA=-A, sB=A+B, tA=B, tB=-A-B. Consequently (z,(a,b))↦ zZ+aA+bB is an equivariant isomorphism from sgn⊕ V to ad^0V.","count":14},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":12},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d4e58ada952ec6ab","tex":"C_4","aliases":"C_4","display":"C_4","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage21Augmentation in two variablesAUXILIARY ALGEBRA / AN EXACT MONOMIAL BASISMain text: Proposition 5.1, Theorem 5.2 and Theorem 5.4 p12-route:visual-21Main statement 5.2 Take Δ=C_4× C_2 in residue characteristic two. If g and h generate its two factors, put X=g-1, Y=h-","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"use","excerpt":"node[anchor=east] at (-.3,2.25) point a; node[anchor=east] at (-.3,1.4) dim H^0(C_a); node[anchor=east] at (-.3,.55) dim H^1(C_a); foreach x/a in 0/-2,1.5/-1,3/0,4.5/1,6/2 node at (x+.4,2.25) a; draw[black!35] (x,1.05) rectangle ++(.8,.7);draw[black!35] (x,.2) rectangle ++(.8,.7); foreach x in 0,1.5,4.5,6node[text=black!45] at (x+.4,1.4) 0;node[tex","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-685a0465f972e5ce","tex":"C_{p^{n_i}}","aliases":"C_p^n_i","display":"C_{p^{n_i}}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-073a71f814bdfa5f","tex":"\\coker","aliases":"coker operatorname","display":"{\\operatorname{coker}}","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage7The same support can hide more algebraMODEL B / NILPOTENT STRUCTUREMain text: Theorems 6.3--6.4; distinguish an ideal from its vanishing set p12-route:visual-7Main statement 6.4 Compare the previous module with a slightly different presentation: M=cokerx · 0 0 · y, ","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","count":25},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"[Coinvariant recovery] Let M be a free Λ module of rank d. Then M/ a M ≅ M⊗_Λ,ϵ O ≅ O^d. The statement is valid for every coefficient prime, including two.","count":22},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","excerpt":"headAn obstruction you can calculate There are 5^4=625 candidate D_3-points and exactly five accepted points. Over D_2= F_5[δ]/(δ^2), however, the five maps u=v=aδ all satisfy the equations. Reduction of every accepted D_3-point gives the zero D_2-point. The four nonzero tangent directions therefore do not lift to order three.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d2c6012edbe5de2c","tex":"D_3","aliases":"D_3","display":"D_3","kind":"use","excerpt":"ORIZATION AND OBSTRUCTIONMain text: Proposition 3.3; this explicit example also exhibits a failed infinitesimal lift p12-route:visual-4Main statement 3.3 Let R^square= O[[u,v]] and send x↦ u, y↦ v-u. The local conditions become u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient ","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"g T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage8Base change is an algebraic pullbackMODEL B / SPECIALIZATION AND TRANSPORTMain text: Proposition 6.5 and Theorem 7.1 p12-route:visual-8Main statement 6.5 Return to M=cokerdiag(x,y) over R=k[x,y]. Pull back along the diagonal R→ k[t], x↦ t, y↦ t. The base-changed pre","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c9943f2d3b5f6322","tex":"\\dim_{E'}","aliases":"dim_E' dim","display":"\\dim_{E'}","kind":"use","excerpt":"[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-34038c6ab85627dc","tex":"\\dim_{\\kappa(\\frakp)}","aliases":"dim_κ(frakp) dim_κ( p) dim kappa frakp dim kappa mathfrak","display":"\\dim_{\\kappa({\\mathfrak p})}","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-75897025474d86a7","tex":"E'^r","aliases":"E'^r","display":"E'^r","kind":"use","excerpt":"Let W = span { (t_1(ω),…,t_r(ω)):ω∈Ω } ⊆ E'^r. If W≠ E'^r, a nonzero linear functional on E'^r would vanish on every evaluation vector, giving a nontrivial linear relation among the functions. Thus W=E'^r. Choose r evaluation vectors forming a basis.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c2349512e48c7e97","tex":"e^{-1}","aliases":"e^-1","display":"e^{-1}","kind":"use","excerpt":"[Derived pointwise realization] Let e:R≅ T, and let C be a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for C, supported in degrees [a,b]. For every characteristic zero point x:R→ E', write y=x∘ e^-1. Then some integer i∈[a,b] has H^i(Cmathbinmathop⊗limits_ T,y^ LE')≠0. Every t∈ T acts on thi","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-89d50d3d0bc30bea","tex":"e^2","aliases":"e^2","display":"e^2","kind":"use","excerpt":"Let e^2=e. Since e+(1-e)=1, at least one of e and 1-e is a unit. If e is a unit, then e=1. If 1-e is a unit, then e=0.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-dbeccfa5c62c9706","tex":"e_{\\mathfrak n}","aliases":"e_ n mathfrak","display":"e_{\\mathfrak n}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f1285db44838b010","tex":"e_+","aliases":"e_+","display":"e_+","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":8},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3da4494053c2f25c","tex":"e_-","aliases":"e_-","display":"e_-","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.","count":7},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"times; draw[black!35,fill=white] (0.66,0.0) rectangle ++(.55,.55); node[text=black!30] at (0.935,0.275) ×; draw[black!35,fill=white] (1.32,0.0) rectangle ++(.55,.55); node[text=black!30] at (1.5950000000000002,0.275) ×; node[font=footnotesize] at (0.275,3.3) e_1; node[font=footnotesize] at (0.935,3.3) e_2; node[font=footnotesize] at (1.595000000000","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-67ab753297c61ce5","tex":"e_4","aliases":"e_4","display":"e_4","kind":"use","excerpt":"node (r) at (3,4) r; node (a) at (1.8,2.8) a;node (b) at (4.5,2.8) b; node (c) at (.8,1.6) c;node (d) at (2.8,1.6) d; node (e) at (.8,.4) e_4;node (ee) at (.8,-.8) e_5,e_6,…; draw[->,line width=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a64c5d974442c22a","tex":"e_5","aliases":"e_5","display":"e_5","kind":"use","excerpt":"node (r) at (3,4) r; node (a) at (1.8,2.8) a;node (b) at (4.5,2.8) b; node (c) at (.8,1.6) c;node (d) at (2.8,1.6) d; node (e) at (.8,.4) e_4;node (ee) at (.8,-.8) e_5,e_6,…; draw[->,line width=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c8f8d5d3f9cb5dc2","tex":"e_6","aliases":"e_6","display":"e_6","kind":"use","excerpt":"node (r) at (3,4) r; node (a) at (1.8,2.8) a;node (b) at (4.5,2.8) b; node (c) at (.8,1.6) c;node (d) at (2.8,1.6) d; node (e) at (.8,.4) e_4;node (ee) at (.8,-.8) e_5,e_6,…; draw[->,line width=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-384a7e923a8d2319","tex":"e_7","aliases":"e_7","display":"e_7","kind":"use","excerpt":"t (3.86,0.2) ⋯; node[font=scriptsize] at (0.195,3.34) e_1; node[font=scriptsize] at (0.665,3.34) e_2; node[font=scriptsize] at (1.135,3.34) e_3; node[font=scriptsize] at (1.605,3.34) e_4; node[font=scriptsize] at (2.0749999999999997,3.34) e_5; node[font=scriptsize] at (2.5449999999999995,3.34) e_6; node[font=scriptsize] at (3.0149999999999997,3.34)","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6eee83a9b14c4461","tex":"e_8","aliases":"e_8","display":"e_8","kind":"use","excerpt":"at (0.195,3.34) e_1; node[font=scriptsize] at (0.665,3.34) e_2; node[font=scriptsize] at (1.135,3.34) e_3; node[font=scriptsize] at (1.605,3.34) e_4; node[font=scriptsize] at (2.0749999999999997,3.34) e_5; node[font=scriptsize] at (2.5449999999999995,3.34) e_6; node[font=scriptsize] at (3.0149999999999997,3.34) e_7; node[font=scriptsize] at (3.485,","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6438bb683a992a5b","tex":"e_{\\frakm}","aliases":"e_frakm e_ m frakm mathfrak","display":"e_{{\\mathfrak m}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":10},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"captiontextThe projectors retain initial basis directions with dimensions 5,4,4,2,1,1,…. In this explicitly defined example the projector is constant from level five onward. hfill [t].52linewidthcentering sffamilybfseries Infinite carrier ℓ^2( N) vfig[.91]infinite_tail captiontextRow m retains all basis vectors e_j with j≥ m. Every row continues in","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","excerpt":"[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b2c88a89f3ffc44b","tex":"e_{\\m}","aliases":"e_m m","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-416776f638338c57","tex":"e_{m+1}","aliases":"e_m+1","display":"e_{m+1}","kind":"use","excerpt":"[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8b92f146ef8909c6","tex":"e_{m+2}","aliases":"e_m+2","display":"e_{m+2}","kind":"use","excerpt":"[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"------------------------------------- vpage22A finite forest selects a branchFINITE OCCUPANCY / MAPS NEED NOT BE SURJECTIVEMain text: Theorem 10.3 and Corollary 10.4 p12-route:visual-22Main statement 10.3 A finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Her","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d3e1f9162e4702a9","tex":"e_{n+1}","aliases":"e_n+1","display":"e_{n+1}","kind":"use","excerpt":"orollary 10.4 p12-route:visual-22Main statement 10.3 A finite nonempty set at every precision, with total reduction maps, can select a compatible branch even when some nodes have no deeper lift. Here is an explicit system: X_1={r}, X_2={a,b}, X_3={c,d}, X_n={e_n} (n≥4). Both a,b reduce to r; both c,d reduce to a; e_4 reduces to c; and e_n+1 reduces","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-bbd7a622a6f80cd3","tex":"e_\\lambda","aliases":"e_λ lambda","display":"e_\\lambda","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-49ddaa56c96d4e05","tex":"f^{-1}","aliases":"f^-1","display":"f^{-1}","kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b7a41a4acbedce0c","tex":"F^h","aliases":"F^h","display":"F^h","kind":"use","excerpt":"-------------------------------------- vpage15Nonzero at every level is not enoughINVERSE SYSTEMS / THE STABLE COREMain text: Definition 8.9, Lemma 8.10 and Theorem 8.11 p12-route:visual-15Main statement 8.11 Let every level be V_m=k^3. For a transition matrix F, compatibility means v_m=Fv_m+1. The directions visible at level m after looking h leve","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f70b91b12ba08060","tex":"f_{m+1,m}","aliases":"f_m+1,m","display":"f_{m+1,m}","kind":"use","excerpt":"Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-44f22654adae6cd4","tex":"f_{n,m+1}","aliases":"f_n,m+1","display":"f_{n,m+1}","kind":"use","excerpt":"Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ac76bb3454573605","tex":"\\FF_2","aliases":"FF_2 F_2 FF mathbb","display":"{\\mathbb F}_2","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":8},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0d10319b98111ddb","tex":"\\FF_2^2","aliases":"FF_2^2 F_2^2 FF mathbb","display":"{\\mathbb F}_2^2","kind":"use","excerpt":"headA trace can vanish while a packet is occupied The identity projector on F_2^2 has rank two and trace 2=0. Hence nonzero rank and nonzero trace are not equivalent in this characteristic. The localization and rank criteria still detect the nonzero space. The characteristic-zero qualification in Corollary 8.4 is essential.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-14e1b580fdde0fc1","tex":"\\FF_5","aliases":"FF_5 F_5 FF mathbb","display":"{\\mathbb F}_5","kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-78b205d811df544a","tex":"\\FF_7","aliases":"FF_7 F_7 FF mathbb","display":"{\\mathbb F}_7","kind":"use","excerpt":"Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1","count":12},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f90b39e544361804","tex":"\\FF_7^3","aliases":"FF_7^3 F_7^3 FF mathbb","display":"{\\mathbb F}_7^3","kind":"use","excerpt":"The coordinate map (a,b)⟼(a,-a+b,-b) identifies the displayed matrices with the permutation action on the sum zero subspace of F_7^3. This action is faithful. The four matrices 1,s,t,st span M_2 over every field of characteristic seven: the determinant of their coordinate matrix is 4≠0. A subspace stable under the group is therefore stable under ev","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-54248a252cc8600a","tex":"\\Fitt","aliases":"Fitt operatorname","display":"{\\operatorname{Fitt}}","kind":"use","excerpt":"[Fitting support criterion] Let R be Noetherian and let M be a finite R module. Then Supp_R(M) = V(Fitt_0^R(M)).","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-22824516db9af32d","tex":"\\Fitt_0","aliases":"Fitt_0 Fitt operatorname","display":"{\\operatorname{Fitt}}_0","kind":"use","excerpt":"If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjectiv","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-057d9dddfc0ec947","tex":"\\Fitt_0^R","aliases":"Fitt_0^R Fitt operatorname","display":"{\\operatorname{Fitt}}_0^R","kind":"use","excerpt":"[Fitting support criterion] Let R be Noetherian and let M be a finite R module. Then Supp_R(M) = V(Fitt_0^R(M)).","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-52f6395fff570e8e","tex":"\\Fitt_1","aliases":"Fitt_1 Fitt operatorname","display":"{\\operatorname{Fitt}}_1","kind":"use","excerpt":"[c].48linewidth vfig[.98]support_axes hfill [c].48linewidth sffamilybfseries The minor calculation Fitt_0(M) · =(xy), Fitt_1(M) · =(x,y), Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor t","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-37f2699000ff8e0d","tex":"\\Fitt_j","aliases":"Fitt_j Fitt operatorname","display":"{\\operatorname{Fitt}}_j","kind":"use","excerpt":"[c].48linewidth vfig[.98]support_axes hfill [c].48linewidth sffamilybfseries The minor calculation Fitt_0(M) · =(xy), Fitt_1(M) · =(x,y), Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor t","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8c98109b747bfca0","tex":"\\Fitt_j^R","aliases":"Fitt_j^R Fitt operatorname","display":"{\\operatorname{Fitt}}_j^R","kind":"use","excerpt":"[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0d3a9bd14e2f4149","tex":"\\Fitt_j^S","aliases":"Fitt_j^S Fitt operatorname","display":"{\\operatorname{Fitt}}_j^S","kind":"use","excerpt":"[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-257572a086e647f4","tex":"\\Fitt_j^{\\TT}","aliases":"Fitt_j^TT Fitt_j^ T Fitt TT operatorname mathbb","display":"{\\operatorname{Fitt}}_j^{{\\mathbb T}}","kind":"use","excerpt":"[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-14796d921d8dedf4","tex":"\\Fitt_{r-1}","aliases":"Fitt_r-1 Fitt operatorname","display":"{\\operatorname{Fitt}}_{r-1}","kind":"use","excerpt":"Use a finite presentation with target rank n. If r>n, every fiber has dimension at most n, so the left hand locus is empty. The convention Fitt_r-1(M)=R for r-1≥ n makes the right hand locus empty as well. It remains to consider 1≤ r≤ n. At p, the fiber dimension is n-rank A( p). It is at least r exactly when the rank is at most n-r, which is equiv","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b2abda39fe0f3e23","tex":"\\Fitt_{r-1}^R","aliases":"Fitt_r-1^R Fitt operatorname","display":"{\\operatorname{Fitt}}_{r-1}^R","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-997bdc1cc4248ffb","tex":"\\frakm_A","aliases":"frakm_A m_A frakm mathfrak","display":"{\\mathfrak m}_A","kind":"use","excerpt":"and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Comple","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2017fcac4d9be4d5","tex":"\\frakm_B","aliases":"frakm_B m_B frakm mathfrak","display":"{\\mathfrak m}_B","kind":"use","excerpt":"Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it car","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-bf6efda13a9cd474","tex":"\\Frob","aliases":"Frob operatorname","display":"{\\operatorname{Frob}}","kind":"use","excerpt":"g T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7354d07b2ecd363e","tex":"\\Frob_v","aliases":"Frob_v Frob operatorname","display":"{\\operatorname{Frob}}_v","kind":"use","excerpt":"g T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f6a65d66c1c2674f","tex":"G^2","aliases":"G^2","display":"G^2","kind":"use","excerpt":"bottomrule hfill [c].47linewidth In the basis 1,δ, multiplication by g=1+δ is G=1 · 0 1 · 1, G^2=I over F_2. It is not the identity: G-I is nonzero and square-zero.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","excerpt":"The displayed rational elements are the unique primitive idempotents because they project to the two characters of C_2. Their coefficients contain 1/2, so they are not integral. In characteristic two, (g-1)^2=g^2-2g+1=0. The map X↦ g-1 gives the displayed isomorphism. The quotient by (X) is F_2, and (X) is nilpotent. Thus the ring is local.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6fb05800096ed7ee","tex":"G_{F,S}","aliases":"G_F,S","display":"G_{F,S}","kind":"use","excerpt":"Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f9b55a29a191e028","tex":"\\GL","aliases":"GL operatorname","display":"{\\operatorname{GL}}","kind":"use","excerpt":"Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.","count":10},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-31040fa50784b437","tex":"\\GL_2","aliases":"GL_2 GL operatorname","display":"{\\operatorname{GL}}_2","kind":"use","excerpt":"age23Classes are not matrix framesCOMPANION COMPLETION / ACTUAL REPRESENTATIVESMain text: revised Theorem 10.5 and its preceding admissibility hypotheses p12-route:visual-23Main statement 10.5 Let Γ=C_2 with the discrete topology, and represent its generator by D=diag(1,-1) over Z_5. At precision n, its conjugacy class is an admissible class in GL_","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-fd521ec0fb491c20","tex":"\\GL_D","aliases":"GL_D GL operatorname","display":"{\\operatorname{GL}}_D","kind":"use","excerpt":"For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a con","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5e917386367b6fdc","tex":"\\GL_n","aliases":"GL_n GL operatorname","display":"{\\operatorname{GL}}_n","kind":"use","excerpt":"Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"A trace zero matrix fixed by conjugation by both s and t is zero, as follows by comparing its four entries. This proves H^0(S_3,ad^0V)=0. For any F_7[S_3] module M and one cocycle f, put u=-6^-1∑_h∈ S_3f(h). Summing f(gh)=g f(h)+f(g) over h gives gu-u=f(g). Hence every one cocycle is a coboundary and H^1(S_3,M)=0, giving both asserted first cohomol","count":23},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-fa5a6573c905a228","tex":"H^0","aliases":"H^0","display":"H^0","kind":"use","excerpt":"[Finite adequacy certificate] The standard representation of S_3 over F_7 is adequate. More precisely, H^0(S_3,ad^0V)=0, H^1(S_3, F_7)=0, H^1(S_3,ad^0V)=0, and ad^0V≅sgn⊕ V. The sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"[Adequate image] Let G⊆GL(V) act absolutely irreducibly over a finite field k of characteristic p. Under the standard definition, adequacy imposes vanishing conditions on H^1(G,k) and H^1(G,ad^0V), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of ad^0V.","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-92ad3ff2bdd044cf","tex":"H^i","aliases":"H^i","display":"H^i","kind":"use","excerpt":"[Cohomological description of derived support] If R is Noetherian and C is perfect, then Supp_R(C) = ⋃_iSupp_R H^i(C). In particular, derived support is closed and is empty exactly when C≃0 [source reference].","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6e5148f28f72177f","tex":"H_x","aliases":"H_x","display":"H_x","kind":"use","excerpt":"[Support gives a Hecke eigensystem] Let R≅ T, let H be a finite T module, and let x:R→ E' be a field valued point. If ker x∈Supp_R(H), then H_x:=H⊗_ T,xE'≠0. Every element of T acts on H_x through the scalar prescribed by x.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-18adc48f226810ce","tex":"H_\\lambda","aliases":"H_λ lambda","display":"H_\\lambda","kind":"use","excerpt":"y an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which all these finite representations are triv","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9bae743248da3740","tex":"\\head","aliases":"head","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"in statement 8.2 Work over Q and let T= Q[z]/((z-1)^2(z+1)^3), V=T. The Chinese remainder theorem gives T≅ underbrace Q[ε]/(ε^2)_T_+, z=1+ε × underbrace Q[η]/(η^3)_T_-, z=-1+η. Multiplication by z is an operator H. In the ordered local bases 1,ε and 1,η,η^2, its matrix is H=(cc|ccc 1 · 0 · 0 · 0 · 0 1 · 1 · 0 · 0 · 0 0 · 0 · -1 · 0 · 0 0 · 0 · 1 · ","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-37475f8aeaec6cce","tex":"I_1","aliases":"I_1","display":"I_1","kind":"use","excerpt":"gintheorem[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwideh","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage3Local conditions remove directionsMODEL A / A FORMAL THICKENINGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0535963952d3cdb0","tex":"I_{\\frakp}","aliases":"I_frakp I_ p frakp mathfrak","display":"I_{{\\mathfrak p}}","kind":"use","excerpt":"Let p⊇ I. Localizing gives (M/IM)_ p ≅ M_ p/I_ pM_ p. If M_ p=0, the quotient is zero. If M_ p≠0, then I_ p lies in the maximal ideal of the local ring R_ p. Nakayama's lemma shows that M_ p/I_ pM_ p≠0. The stated equality follows.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-111b6ee2c5f8dbd2","tex":"I_s","aliases":"I_s","display":"I_s","kind":"use","excerpt":"gintheorem[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwideh","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f1c9ad67822f1d50","tex":"I_\\lambda","aliases":"I_λ lambda","display":"I_\\lambda","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8d0bba48daafb223","tex":"i_\\lambda","aliases":"i_λ lambda","display":"i_\\lambda","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4331b90602b3a79a","tex":"I_\\lambda^n","aliases":"I_λ^n lambda","display":"I_\\lambda^n","kind":"use","excerpt":"For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a con","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s","count":29},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.","count":20},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-596b3f5156c743a8","tex":"j^2","aliases":"j^2","display":"j^2","kind":"use","excerpt":"headTwo explicit 5-adic numbers Choose the canonical digits α_j=(j^2+1)bmod5, β_j=(2j+1)bmod5, with values in {0,1,2,3,4}. Set a=∑_j≥0α_j5^j and b=∑_j≥0β_j5^j. The element a+ε b is determined by the following coefficient array.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d9fe993390430812","tex":"J_{\\lambda,u}","aliases":"J_λ,u lambda","display":"J_{\\lambda,u}","kind":"use","excerpt":"Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem,","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-146b7f435ba6dc91","tex":"k^3","aliases":"k^3","display":"k^3","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage15Nonzero at every level is not enoughINVERSE SYSTEMS / THE STABLE COREMain text: Definition 8.9, Lemma 8.10 and Theorem 8.11 p12-route:visual-15Main statement 8.11 Let every level be V_m=k^3. For a transition matrix F, compatibility means v_m=Fv_m+1. The directions ","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-73b02dc76290db76","tex":"k^5","aliases":"k^5","display":"k^5","kind":"use","excerpt":"[t].44linewidthcentering sffamilybfseries Fixed carrier k^5 vfig[.75]nested_finite captiontextThe projectors retain initial basis directions with dimensions 5,4,4,2,1,1,…. In this explicitly defined example the projector is constant from level five onward. hfill [t].52linewidthcentering sffamilybfseries Infinite carrier ℓ^2( N) vfig[.91]infinite_ta","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-fc97a5c4f9fcff61","tex":"K_\\lambda","aliases":"K_λ lambda","display":"K_\\lambda","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"Let f:R^loc→ A be the point. Since A is Hausdorff and f is continuous, ker f is closed. Thus it contains the closed ideal J if and only if it contains the algebraic ideal generated by the images of the I_i. This is equivalent to every local restriction killing I_i. The universal property of the actual ideal quotient gives the factors, and the quoti","count":10},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-45e90aa85790529d","tex":"\\linewidth","aliases":"linewidth","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Field fiber criterion] Let R be a commutative ring, let M be a finite R module, and let x:R→ E' be a homomorphism to a field. Put p=ker x. Then M⊗_R,xE'≠0 Longleftrightarrow p∈Supp_R(M).","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"1 · -1, det A=-2. It is invertible over Q. If the measured trace vector is (5,1), solve Am_1 m_2=5 1 Longrightarrow m_1 m_2=3 2.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5ee4319907c91ec7","tex":"\\m","aliases":"m","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-76a8c8c02bf83b7a","tex":"M'","aliases":"M'","display":"M'","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage7The same support can hide more algebraMODEL B / NILPOTENT STRUCTUREMain text: Theorems 6.3--6.4; distinguish an ideal from its vanishing set p12-route:visual-7Main statement 6.4 Compare the previous module with a slightly different presentation: M=cokerx · 0 0 · y, ","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"The coordinate map (a,b)⟼(a,-a+b,-b) identifies the displayed matrices with the permutation action on the sum zero subspace of F_7^3. This action is faithful. The four matrices 1,s,t,st span M_2 over every field of characteristic seven: the determinant of their coordinate matrix is 4≠0. A subspace stable under the group is therefore stable under ev","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-651ec01c3f6e67cb","tex":"m_2","aliases":"m_2","display":"m_2","kind":"use","excerpt":"draw[->,black!35] (-2.5,0)--(2.5,0); draw[->,black!35] (0,-2.3)--(0,2.3) node[above] second coordinate; draw[->,line width=1.2pt] (0,0)--(1.5,1.5); draw[->,line width=1.2pt] (0,0)--(1.5,-1.5); node[anchor=south west] at (1.5,1.5) (1,1); node[anchor=north west] at (1.5,-1.5) (1,-1); node[anchor=west] at (3.3,.7) m_1+m_2=5; node[anchor=west] at (3.3,","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-318be9929beaa6fd","tex":"m_d","aliases":"m_d","display":"m_d","kind":"use","excerpt":"Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"use","excerpt":"Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-66994aa3297dac88","tex":"\\m_\\OO","aliases":"m_OO m_ O m OO m mathcal","display":null,"kind":"use","excerpt":"midrule Empty family · S= O and a=m_ O. The scalar adic topology remains; an empty sum of factor ideals does not erase it.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9cc887d0aa436c20","tex":"\\m_{\\OO}","aliases":"m_OO m_ O m OO m mathcal","display":null,"kind":"use","excerpt":"-------------------------------- setlengthparskip5pt vpage19Complete the actual topologyTOPOLOGICAL ALGEBRA / THE COMPARISON MAPMain text: Theorem 3.1, Corollary 3.2 and Proposition 3.3 p12-route:visual-19Main statement 3.1 The hat in whot names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8683349b5e66d23b","tex":"\\m_{R_i}","aliases":"m_R_i m","display":null,"kind":"use","excerpt":"-------------------------------- setlengthparskip5pt vpage19Complete the actual topologyTOPOLOGICAL ALGEBRA / THE COMPARISON MAPMain text: Theorem 3.1, Corollary 3.2 and Proposition 3.3 p12-route:visual-19Main statement 3.1 The hat in whot names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d49f1f8e75de682d","tex":"\\Max","aliases":"Max operatorname","display":"{\\operatorname{Max}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3f9db2ae5742966b","tex":"\\midrule","aliases":"midrule","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":11},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"[t].44linewidthcentering sffamilybfseries Fixed carrier k^5 vfig[.75]nested_finite captiontextThe projectors retain initial basis directions with dimensions 5,4,4,2,1,1,…. In this explicitly defined example the projector is constant from level five onward. hfill [t].52linewidthcentering sffamilybfseries Infinite carrier ℓ^2( N) vfig[.91]infinite_ta","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5f5eab2cbd9d1834","tex":"\\mathfrak n","aliases":"n mathfrak","display":"\\mathfrak n","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"0 · 0 · 0, P^2=P. vfig[.45]core_projection captiontextAt every positive horizon, precisely the e_1 direction remains visible. hfill [t].48linewidthcentering sffamilybfseries Every direction eventually dies N=0 · 1 · 0 0 · 0 · 1 0 · 0 · 0, N^3=0. vfig[.45]core_nilpotent captiontextImage dimensions are 3,2,1,0,0,…, although every level still has dime","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-eb171c4fc1202d77","tex":"n_r","aliases":"n_r","display":"n_r","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-797aecf4dbd2a9ff","tex":"\\nil","aliases":"nil Nil operatorname","display":"{\\operatorname{Nil}}","kind":"use","excerpt":"[Near faithfulness is full support] Let R be Noetherian and let M be a finite R module. The following conditions are equivalent. item M is nearly faithful. item Supp_R(M)=Spec R. item Ann_R(M)⊆Nil(R). If R is reduced, these conditions are equivalent to faithfulness.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"headWhat makes this a finite certificate The manuscript also checks absolute irreducibility and projector detection on every irreducible adjoint submodule. In this example 7nmid2, and every group element's eigenvalues lie in F_7, as required by the stated convention. The two displayed pairings alone are a worked part of that complete certificate.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-90f600277c50100a","tex":"\\NN","aliases":"NN N mathbb","display":"{\\mathbb N}","kind":"use","excerpt":"[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"ode[font=scriptsize] at (0.345,3.3449999999999998) y^4; draw[fill=white,draw=black!35] (0.75,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,3.3449999999999998) xy^4; draw[fill=white,draw=black!35] (0.0,3.75) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,4.095) y^5; node[anchor=west,font=footnotesize] at (0,-.46) i+j<6 shown; c","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4a86f15b45905b04","tex":"\\OO^d","aliases":"OO^d O^d OO mathcal","display":"{\\mathcal O}^d","kind":"use","excerpt":"[Coinvariant recovery] Let M be a free Λ module of rank d. Then M/ a M ≅ M⊗_Λ,ϵ O ≅ O^d. The statement is valid for every coefficient prime, including two.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a75471218462aa6a","tex":"\\OO_\\lambda","aliases":"OO_λ O_λ OO lambda mathcal lambda","display":"{\\mathcal O}_\\lambda","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":10},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c638d5d3835c366c","tex":"\\overline{JB}","aliases":"overlineJB overline","display":"\\overline{JB}","kind":"use","excerpt":"is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1ec6ea712af79729","tex":"\\overline{L\\widehat S}","aliases":"overlineLwidehat S overline widehat","display":"\\overline{L\\widehat S}","kind":"use","excerpt":"Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map ","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9ce622580e156862","tex":"\\p","aliases":"p","display":null,"kind":"use","excerpt":"Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor the drawing, take k=RR. It shows real closed points, not every point of Spec R. The displayed ideals and rank criterion describe the full a","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"[t].48linewidthcentering sffamilybfseries A surviving direction P=1 · 0 · 0 0 · 0 · 0 0 · 0 · 0, P^2=P. vfig[.45]core_projection captiontextAt every positive horizon, precisely the e_1 direction remains visible. hfill [t].48linewidthcentering sffamilybfseries Every direction eventually dies N=0 · 1 · 0","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7441b5cfbaa2c0c0","tex":"p^m","aliases":"p^m","display":"p^m","kind":"use","excerpt":"[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3f0efa7048440392","tex":"p^{n_1}","aliases":"p^n_1","display":"p^{n_1}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-47c937c3c2303c97","tex":"p^{n_i}","aliases":"p^n_i","display":"p^{n_i}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-eae80e166a46cca8","tex":"p^{n_r}","aliases":"p^n_r","display":"p^{n_r}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c5ba04fa053a01d8","tex":"P_{D,\\alpha}","aliases":"P_D,α alpha","display":"P_{D,\\alpha}","kind":"use","excerpt":"0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-97e4bf0718e3c837","tex":"P_{s,-1}","aliases":"P_s,-1","display":"P_{s,-1}","kind":"use","excerpt":"For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible ad","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-aa62d8c209f88bda","tex":"P_{s,1}","aliases":"P_s,1","display":"P_{s,1}","kind":"use","excerpt":"3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4 0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy de","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b804d42c181b602f","tex":"P_{t,2}","aliases":"P_t,2","display":"P_{t,2}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4 3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-667691477116259f","tex":"P_{tst^{-1},1}","aliases":"P_tst^-1,1","display":"P_{tst^{-1},1}","kind":"use","excerpt":"For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible ad","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5bb0b441f791e733","tex":"P_{\\lambda,v}","aliases":"P_λ,v lambda","display":"P_{\\lambda,v}","kind":"use","excerpt":"Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coeffici","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-079e8b6488dabbe1","tex":"\\pagerefV","aliases":"pagerefV","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"headThe projectors can be written explicitly Modulo (z-1)^2(z+1)^3, put e_+(z)=frac(5-3z)(z+1)^316, e_-(z)=1-e_+(z). Then e_+≡1pmod(z-1)^2 and e_+≡0pmod(z+1)^3. Consequently e_+(H)=diag(1,1,0,0,0), e_-(H)=diag(0,0,1,1,1). The packet dimensions are two and three. The ordinary eigenspaces each have dimension one; the generalized eigenspaces have dime","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"headCalculate the mismatch For Q=(1 · 1 0 · 1), QDQ^-1=1 · -2","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"The original comparison wording did not explicitly supply a compatible Hecke point over the fixed coefficient field. Support preservation alone is insufficient: extension from Q to Q(i) preserves full support, but there is no ring map Q(i)→ Q extending the identity. The revision therefore states y∘ f=x and exposes the arithmetic comparison and exis","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2bdc013a59de0c4b","tex":"\\mathfrak q","aliases":"q mathfrak","display":"\\mathfrak q","kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.","count":12},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f3003d8698b757c1","tex":"Q^{-1}","aliases":"Q^-1","display":"Q^{-1}","kind":"use","excerpt":"0 · 1), QDQ^-1=1 · -2 0 · -1 ≡1 · 23 0 · 24pmod25. It reduces to (1 · 3","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4e4bfa4de54508e8","tex":"q_*","aliases":"q_*","display":"q_*","kind":"use","excerpt":"[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ Q mathbb","display":"{\\mathbb Q}","kind":"use","excerpt":"Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.","count":15},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ba83fc4ca5c67ec3","tex":"\\QQ_2","aliases":"QQ_2 Q_2 QQ mathbb","display":"{\\mathbb Q}_2","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-372683511d4b691a","tex":"\\QQ_p","aliases":"QQ_p Q_p QQ mathbb","display":"{\\mathbb Q}_p","kind":"use","excerpt":"Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":36},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage6A matrix becomes a support locusMODEL B / FITTING STRATAMain text: Theorems 6.2, 6.3 and 6.4 p12-route:visual-6Main statement 6.3 Now change models. Let R=k[x,y] for a field k, and present a module by R^2xrightarrowAR^2⟶ M⟶0, A=x · 0 0 · y. The relations are xe_1=0 ","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c4179ac2053e2353","tex":"R^m","aliases":"R^m","display":"R^m","kind":"use","excerpt":"If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjectiv","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4415f2a3fc7b65ea","tex":"R^n","aliases":"R^n","display":"R^n","kind":"use","excerpt":"If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjectiv","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-147dbf37fcbf55c9","tex":"R^\\tau","aliases":"R^τ tau","display":"R^\\tau","kind":"use","excerpt":"is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage3Local conditions remove directionsMODEL A / A FORMAL THICKENINGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a8af058f9467a00f","tex":"R_{\\frakp}","aliases":"R_frakp R_ p frakp mathfrak","display":"R_{{\\mathfrak p}}","kind":"use","excerpt":"Let p∈Spec R. If Ann_R(M)not⊆ p, choose a∈Ann_R(M)∖ p. The element a becomes a unit in R_ p, so M_ p=0.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8a4a08a9899f99d7","tex":"R_{\\frakp}^n","aliases":"R_frakp^n R_ p^n frakp mathfrak","display":"R_{{\\mathfrak p}}^n","kind":"use","excerpt":"from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal min","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f7b985b4cff51815","tex":"R_s","aliases":"R_s","display":"R_s","kind":"use","excerpt":"[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e13c04e5f7fac1fc","tex":"\\raggedright","aliases":"raggedright","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank operatorname","display":"{\\operatorname{rank}}","kind":"use","excerpt":"Use a finite presentation with target rank n. If r>n, every fiber has dimension at most n, so the left hand locus is empty. The convention Fitt_r-1(M)=R for r-1≥ n makes the right hand locus empty as well. It remains to consider 1≤ r≤ n. At p, the fiber dimension is n-rank A( p). It is at least r exactly when the rank is at most n-r, which is equiv","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c425f07dc5f2732e","tex":"\\rank_{E'}","aliases":"rank_E' rank operatorname","display":"{\\operatorname{rank}}_{E'}","kind":"use","excerpt":"[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a62a596621079394","tex":"\\RR","aliases":"RR","display":null,"kind":"use","excerpt":"Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor the drawing, take k=RR. It shows real closed points, not every point of Spec R. The displayed ideals and rank criterion describe the full a","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"1 · 6. All entries and calculations on this page are modulo seven. Direct multiplication gives s^2=t^3=I and sts=t^-1. Their eigenvalues are {-1,1} and {2,4}. Distinct eigenvalues allow an exact projector formula P_D,α=(α-β)^-1(D-β I).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1","count":14},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-db289384e3bfb6f4","tex":"s_d","aliases":"s_d","display":"s_d","kind":"use","excerpt":"Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6365d83fdc80b979","tex":"\\mathrm{sgn}","aliases":"sgn mathrm","display":"\\mathrm{sgn}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-28f39655338ed57d","tex":"\\operatorname{sgn}","aliases":"sgn operatorname","display":"\\operatorname{sgn}","kind":"use","excerpt":"[Finite adequacy certificate] The standard representation of S_3 over F_7 is adequate. More precisely, H^0(S_3,ad^0V)=0, H^1(S_3, F_7)=0, H^1(S_3,ad^0V)=0, and ad^0V≅sgn⊕ V. The sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9bd6563a665b1ad1","tex":"\\small","aliases":"small","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"Let W = span { (t_1(ω),…,t_r(ω)):ω∈Ω } ⊆ E'^r. If W≠ E'^r, a nonzero linear functional on E'^r would vanish on every evaluation vector, giving a nontrivial linear relation among the functions. Thus W=E'^r. Choose r evaluation vectors forming a basis.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ae51e1c510f803eb","tex":"\\operatorname{Spec}","aliases":"Spec operatorname","display":"\\operatorname{Spec}","kind":"use","excerpt":"node[font=sffamilysmallbfseries] at (1.6,3.15) SpecR; draw[->,line width=1.1pt] (.1,2.1)--(1.6,.35); draw[->,dashed] (3.1,2.1)--(1.6,.35); fill (.1,2.1) circle (.09);draw[fill=white] (3.1,2.1) circle (.09);fill (1.6,.35) circle (.09); node[above=6pt] at (.1,2.1) (x);node[above=6pt] at (3.1,2.1) (y); node[below=6pt] at (1.6,.35) (x,y); node[font=foo","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-348ccb55d8cc92e4","tex":"\\mathrm{ss}","aliases":"ss mathrm","display":"\\mathrm{ss}","kind":"use","excerpt":"tomorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-72d39f9d2160995b","tex":"\\mathrm{std}","aliases":"std mathrm","display":"\\mathrm{std}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5490eb62846f738d","tex":"\\Supp_{R/I}","aliases":"Supp_R/I Supp operatorname","display":"{\\operatorname{Supp}}_{R/I}","kind":"use","excerpt":"[Support after a quotient] Let R be Noetherian, let I⊆ R be an ideal, and let M be a finite R module. Under the canonical identification Spec(R/I)≅ V(I)⊆Spec R, one has Supp_R/I(M/IM) = Supp_R(M)∩ V(I).","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-77bef08da7212ff7","tex":"\\Supp_{R_\\infty/(t)}","aliases":"Supp_R_∞/(t) Supp infty operatorname infty","display":"{\\operatorname{Supp}}_{R_\\infty/(t)}","kind":"use","excerpt":"headThe actual descent theorem is one-way For a finite module over a Noetherian ring, Supp_R_∞/(t)(M_∞/tM_∞) =Supp_R_∞(M_∞)∩ V(t). If the initial support is full, the resulting support is full on the quotient. Our example shows that full support at the quotient can occur even when initial support was not full. It also shows why freeness over S alon","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8839b7e8deefe167","tex":"\\Supp_T","aliases":"Supp_T Supp operatorname","display":"{\\operatorname{Supp}}_T","kind":"use","excerpt":"emptyMODEL C / OCCUPANCY IS MODULE DATAMain text: Theorem 7.4, Theorem 8.2 and Corollary 8.4 p12-route:visual-13Main statement 7.4 Keep the algebra T=T_+× T_- from pagerefV12, but compare two different T-modules: V=T, W=T_+, where T acts on W through projection to its first factor. The algebra still has two maximal ideals, but only one of them is i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":28},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1 0 · 1 , t= 0 · 6 1 · 6 .","count":56},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5d42311231194d5c","tex":"t^{-1}","aliases":"t^-1","display":"t^{-1}","kind":"use","excerpt":"For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible ad","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"[c].44linewidth vfig[.94]diagonal_slice hfill [c].51linewidth headBoth relations vanish together (xy)⟼(t^2), (x,y)⟼(t). The support is the origin, but the origin has fiber dimension two. Replacing (t^2) by its radical (t) would lose the zeroth Fitting ideal.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"1 · 6. All entries and calculations on this page are modulo seven. Direct multiplication gives s^2=t^3=I and sts=t^-1. Their eigenvalues are {-1,1} and {2,4}. Distinct eigenvalues allow an exact projector formula P_D,α=(α-β)^-1(D-β I).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7f502591dbad0d5e","tex":"T_+","aliases":"T_+","display":"T_+","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage12A packet contains an entire Jordan chainMODEL C / A NONSEMISIMPLE HECKE ALGEBRAMain text: Theorem 8.2, Corollary 8.3 and Example 8.6 p12-route:visual-12Main statement 8.2 Work over Q and let T= Q[z]/((z-1)^2(z+1)^3), V=T. The Chinese remainder theorem gives T≅ unde","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4d99951c22895e3a","tex":"T_-","aliases":"T_-","display":"T_-","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage12A packet contains an entire Jordan chainMODEL C / A NONSEMISIMPLE HECKE ALGEBRAMain text: Theorem 8.2, Corollary 8.3 and Example 8.6 p12-route:visual-12Main statement 8.2 Work over Q and let T= Q[z]/((z-1)^2(z+1)^3), V=T. The Chinese remainder theorem gives T≅ unde","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5d08324acb961fcf","tex":"t_2","aliases":"t_2","display":"t_2","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage25Recover multiplicities from tracesA FINITE SEPARATOR / A KNOWN SEMISIMPLE CARRIERMain text: Proposition 10.9 p12-route:visual-25Main statement 10.9 Suppose a known semisimple carrier is a sum of two known constituents with multiplicities m_1,m_2. Consider the indep","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6e8969a358fc5266","tex":"T_d","aliases":"T_d","display":"T_d","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-219b7412a89e1904","tex":"T_i","aliases":"T_i","display":"T_i","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-54bc773a0263d21b","tex":"\\takeaway","aliases":"takeaway","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3000c3bd47e8007c","tex":"\\toprule","aliases":"toprule","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr operatorname","display":"{\\operatorname{Tr}}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4 3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"3 · 6; node[anchor=west] at (8.2,1.8) Tr(PW)=3; draw[black!30] (0,.9)--(11.3,.9); node[anchor=west,font=sffamilybfseries] at (0,.3) standard constituent; node[anchor=west] at (0,-.7) W=6 · 4","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0ec1b1b373658d51","tex":"\\Tr_{E'}","aliases":"Tr_E' Tr operatorname","display":"{\\operatorname{Tr}}_{E'}","kind":"use","excerpt":"[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a5412ab68daaa1c9","tex":"\\TT_{\\frakm}","aliases":"TT_frakm T_ m TT frakm mathbb mathfrak","display":"{\\mathbb T}_{{\\mathfrak m}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[Full support gives pointwise automorphy on the realized locus] In the setting of [source reference], suppose Supp_R(H)=Spec R. For every characteristic zero point x:R→ E' with ker x∈ U⊆Spec R, assume the coefficient topology and continuity hypotheses of [source reference] and the declared classicality and Galois realization input. Then every such ","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"A trace zero matrix fixed by conjugation by both s and t is zero, as follows by comparing its four entries. This proves H^0(S_3,ad^0V)=0. For any F_7[S_3] module M and one cocycle f, put u=-6^-1∑_h∈ S_3f(h). Summing f(gh)=g f(h)+f(g) over h gives gu-u=f(g). Hence every one cocycle is a coboundary and H^1(S_3,M)=0, giving both asserted first cohomol","count":22},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are iso","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7d77b9a1d8369d4e","tex":"v^3","aliases":"v^3","display":"v^3","kind":"use","excerpt":"[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are iso","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ee9257fd8edde904","tex":"V_{\\frakm}","aliases":"V_frakm V_ m frakm mathfrak","display":"V_{{\\mathfrak m}}","kind":"use","excerpt":"[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b00eda2d2588cd09","tex":"v_m","aliases":"v_m","display":"v_m","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage15Nonzero at every level is not enoughINVERSE SYSTEMS / THE STABLE COREMain text: Definition 8.9, Lemma 8.10 and Theorem 8.11 p12-route:visual-15Main statement 8.11 Let every level be V_m=k^3. For a transition matrix F, compatibility means v_m=Fv_m+1. The directions ","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-073c6bfeafa5998b","tex":"V_{m+1}","aliases":"V_m+1","display":"V_{m+1}","kind":"use","excerpt":"[Surjectivity on stable cores] For every m, the transition map V_m+1→ V_m induces a surjection Y_m+1↠ Y_m.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-64e71b2ad0c5d470","tex":"v_{m+1}","aliases":"v_m+1","display":"v_{m+1}","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage15Nonzero at every level is not enoughINVERSE SYSTEMS / THE STABLE COREMain text: Definition 8.9, Lemma 8.10 and Theorem 8.11 p12-route:visual-15Main statement 8.11 Let every level be V_m=k^3. For a transition matrix F, compatibility means v_m=Fv_m+1. The directions ","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8d59d7c310a9781f","tex":"v_{m+3}","aliases":"v_m+3","display":"v_{m+3}","kind":"use","excerpt":"headNow compute the limits, not just their level dimensions For the P-system, the limit consists of the constant compatible sequences (a e_1,a e_1,…), so it is isomorphic to k. For the N-system, every compatible sequence satisfies v_m=N^3v_m+3=0 for every m, and the limit is zero. This is an all-level argument from the identity N^3=0, not a finite ","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f3685a7a9e666f70","tex":"V_u","aliases":"V_u","display":"V_u","kind":"use","excerpt":"[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are iso","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b8d2185f95f25265","tex":"V_v","aliases":"V_v","display":"V_v","kind":"use","excerpt":"[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are iso","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"use","excerpt":"[Nonzero stable core criterion] Let (V_m) be an inverse system of finite dimensional vector spaces. If Y_m≠0 for every m, then varprojlim_mV_m ≠0. More precisely, the inverse limit contains a compatible sequence whose first coordinate is any prescribed nonzero element of Y_1.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9e2d9dcc048eb142","tex":"\\varprojlim_m","aliases":"varprojlim_m varprojlim","display":null,"kind":"use","excerpt":"[Nonzero stable core criterion] Let (V_m) be an inverse system of finite dimensional vector spaces. If Y_m≠0 for every m, then varprojlim_mV_m ≠0. More precisely, the inverse limit contains a compatible sequence whose first coordinate is any prescribed nonzero element of Y_1.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e24b729afa00f95f","tex":"\\varprojlim_n","aliases":"varprojlim_n varprojlim","display":null,"kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage5Completion retains every precisionMODEL A / COMPATIBLE COEFFICIENTSMain text: Theorem 3.1; reconstruction and finite occupancy in Section 10 p12-route:visual-5Main statement 10.5 For B= Z_5[ε]/(ε^2), consider the finite quotient rings B_n=( Z/5^n Z)[ε]/(ε^2), |B_n|=","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e9f7dc4db556c3f2","tex":"\\vpage","aliases":"vpage","display":null,"kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"A vector in the regular representation fixed by all left translations has constant coefficients. Consequently ve=ϵ(e)v=0. On the other hand, e^2=e implies we=w for every w∈ W, hence ve=v, a contradiction. Thus every idempotent of augmentation zero is zero. Since ϵ(e)^2=ϵ(e) in the field k, either ϵ(e)=0 or ϵ(e)=1; in the latter case apply the prece","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f6935289ededde0d","tex":"W_{\\mathrm{sgn}}","aliases":"W_sgn mathrm","display":"W_{\\mathrm{sgn}}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-20edf480f7443c8c","tex":"W_{\\mathrm{std}}","aliases":"W_std mathrm","display":"W_{\\mathrm{std}}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3d7dd17d8821d766","tex":"\\whot","aliases":"whot","display":null,"kind":"use","excerpt":"NGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[[y]], I_2=(y). Their completed local tensor is A= O[[x,y]]. The closed generated ideal is J=(x^2,y), and both routes through Theorem 3.1 give (R_","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-89927d7c0a692067","tex":"\\whot_A","aliases":"whot_A whot","display":null,"kind":"use","excerpt":"For global base change, the structural map A→ B belongs to the continuous local coefficient-algebra setting. Then Bwhot_A(A/J)≅ B/overlineJB. For a continuous algebra point to a complete Hausdorff target, closedness of its kernel gives the factorization test. Locality of a point, when supplied, passes to its unique quotient factor.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5cee0ba622092978","tex":"\\whot_{\\OO}","aliases":"whot_OO whot_ O whot OO whot mathcal","display":null,"kind":"use","excerpt":"NGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[[y]], I_2=(y). Their completed local tensor is A= O[[x,y]]. The closed generated ideal is J=(x^2,y), and both routes through Theorem 3.1 give (R_","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-77449d53c5be9d11","tex":"\\widehat S","aliases":"widehat S widehat","display":"\\widehat S","kind":"use","excerpt":"Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map ","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5c6b245b4f9f9854","tex":"\\widehat{S/L}","aliases":"widehatS/L widehat","display":"\\widehat{S/L}","kind":"use","excerpt":"Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map ","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0c88b3c814ae49b9","tex":"\\wideotimes_{\\OO}","aliases":"wideotimes_OO mathbinwidehat⊗_ O wideotimes OO mathbin widehat otimes mathcal","display":null,"kind":"use","excerpt":"[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-22c6c23ab138715f","tex":"X^{2}","aliases":"X^2","display":"X^{2}","kind":"use","excerpt":"[x=1cm,y=1cm,font=small,>=Stealth] draw[black!40,fill=black!16] (0.0,0.0) rectangle ++(1,.9); node at (0.5,0.45) 1; draw[black!40,fill=white] (1.2,0.0) rectangle ++(1,.9); node at (1.7,0.45) X; draw[black!40,fill=white] (2.4,0.0) rectangle ++(1,.9); node at (2.9,0.45) X^2; draw[black!40,fill=white] (3.5999999999999996,0.0) rectangle ++(1,.9); node ","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"0.75,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,3.3449999999999998) xy^4; draw[fill=white,draw=black!35] (0.0,3.75) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,4.095) y^5; node[anchor=west,font=footnotesize] at (0,-.46) i+j<6 shown; coefficients lie in O.; node[anchor=west,font=sffamilysmallbfseries] at (6.35,5.10) After","count":15},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7fa50f9a5fc0d915","tex":"x^{2}","aliases":"x^2","display":"x^{2}","kind":"use","excerpt":"familysmallbfseries] at (0,5.10) Before imposing the ideals; draw[fill=white,draw=black!35] (0.0,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,0.345) 1; draw[fill=white,draw=black!35] (0.75,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,0.345) x; draw[fill=white,draw=black!35] (1.5,0.0) rectangle ++(.69,.69); node[font=sc","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b3ca6f39aced1b19","tex":"X^{3}","aliases":"X^3","display":"X^{3}","kind":"use","excerpt":"ikzpicture[x=1cm,y=1cm,font=small,>=Stealth] draw[black!40,fill=black!16] (0.0,0.0) rectangle ++(1,.9); node at (0.5,0.45) 1; draw[black!40,fill=white] (1.2,0.0) rectangle ++(1,.9); node at (1.7,0.45) X; draw[black!40,fill=white] (2.4,0.0) rectangle ++(1,.9); node at (2.9,0.45) X^2; draw[black!40,fill=white] (3.5999999999999996,0.0) rectangle ++(1,","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cded7860139623b9","tex":"x^{3}","aliases":"x^3","display":"x^{3}","kind":"use","excerpt":"node[font=scriptsize] at (0.345,0.345) 1; draw[fill=white,draw=black!35] (0.75,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,0.345) x; draw[fill=white,draw=black!35] (1.5,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.845,0.345) x^2; draw[fill=white,draw=black!35] (2.25,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (2.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ee74251aad7234d0","tex":"X^4","aliases":"X^4","display":"X^4","kind":"use","excerpt":"-------------------------------------------------------------------------- vpage21Augmentation in two variablesAUXILIARY ALGEBRA / AN EXACT MONOMIAL BASISMain text: Proposition 5.1, Theorem 5.2 and Theorem 5.4 p12-route:visual-21Main statement 5.2 Take Δ=C_4× C_2 in residue characteristic two. If g and h generate its two factors, put X=g-1, Y=h-1. ","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5c2720ca0aa56527","tex":"x^{4}","aliases":"x^4","display":"x^{4}","kind":"use","excerpt":"tsize] at (1.095,0.345) x; draw[fill=white,draw=black!35] (1.5,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.845,0.345) x^2; draw[fill=white,draw=black!35] (2.25,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,0.345) x^3; draw[fill=white,draw=black!35] (3.0,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (3.3","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f7e3e0710f2c3e67","tex":"x^{5}","aliases":"x^5","display":"x^{5}","kind":"use","excerpt":"] at (1.845,0.345) x^2; draw[fill=white,draw=black!35] (2.25,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,0.345) x^3; draw[fill=white,draw=black!35] (3.0,0.0) rectangle ++(.69,.69); node[font=scriptsize] at (3.3449999999999998,0.345) x^4; draw[fill=white,draw=black!35] (3.75,0.0) rectangle ++(.69,.69); node[font=scriptsi","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-48f4a03fbc552897","tex":"x^i","aliases":"x^i","display":"x^i","kind":"use","excerpt":"headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f93b0321f5ab119e","tex":"X_1^{p^{n_1}}","aliases":"X_1^p^n_1","display":"X_1^{p^{n_1}}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":") at (.8,.4) e_4;node (ee) at (.8,-.8) e_5,e_6,…; draw[->,line width=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width=1.1pt] (ee)--(e); node[anchor=west,font=footnotesize] at (5.4,4) X_1: one fixed residual root; node[anchor=west,font=footnot","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8eeb0a1bbb468424","tex":"X_3","aliases":"X_3","display":"X_3","kind":"use","excerpt":"h=1.1pt] (a)--(r);draw[->,dashed] (b)--(r); draw[->,line width=1.1pt] (c)--(a);draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width=1.1pt] (ee)--(e); node[anchor=west,font=footnotesize] at (5.4,4) X_1: one fixed residual root; node[anchor=west,font=footnotesize] at (5.4,2.8) X_2: two admissible nodes; node[anchor=west,fon","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-fc227e811bd8d876","tex":"X_4","aliases":"X_4","display":"X_4","kind":"use","excerpt":"headWhat the compactness argument proves For general finite levels, the reduction forest is a locally finite rooted tree with vertices at every depth. An infinite branch exists. Surjectivity is sufficient but is not necessary: the map X_3→ X_2 above misses b, and X_4→ X_3 misses d.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-be02837bb7e276a6","tex":"x_{g,a,b}","aliases":"x_g,a,b","display":"x_{g,a,b}","kind":"use","excerpt":"Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem,","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":7},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cb595c87aa88b7d0","tex":"X_i^{p^{n_i}}","aliases":"X_i^p^n_i","display":"X_i^{p^{n_i}}","kind":"use","excerpt":"In characteristic p, (1+X_i)^p^n_i-1=X_i^p^n_i. Thus the assignment X_i↦γ_i-1 gives a surjective homomorphism from the displayed truncated polynomial ring to k[Δ]. Both sides have dimension ∏_ip^n_i=absΔ over k, so the map is an isomorphism. The quotient by the ideal generated by the X_i is k. Since each X_i is nilpotent, that ideal is the unique m","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"use","excerpt":"draw[->,dashed] (d)--(a); draw[->,line width=1.1pt] (e)--(c);draw[->,line width=1.1pt] (ee)--(e); node[anchor=west,font=footnotesize] at (5.4,4) X_1: one fixed residual root; node[anchor=west,font=footnotesize] at (5.4,2.8) X_2: two admissible nodes; node[anchor=west,font=footnotesize] at (5.4,1.6) X_3: two admissible nodes; node[anchor=west,font=f","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8285a92cff098813","tex":"X_r","aliases":"X_r","display":"X_r","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-020f07c72259d4de","tex":"X_r^{p^{n_r}}","aliases":"X_r^p^n_r","display":"X_r^{p^{n_r}}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-449cba2c3125b122","tex":"X_{\\lambda,m}","aliases":"X_λ,m lambda","display":"X_{\\lambda,m}","kind":"use","excerpt":"[Companion branch criterion] Suppose every level X_λ,m is finite and nonempty, and every point reduces along the displayed forest to the fixed residual root. Then varprojlim_mX_λ,m ≠∅.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-27222a821514a259","tex":"X_{\\lambda,n}","aliases":"X_λ,n lambda","display":"X_{\\lambda,n}","kind":"use","excerpt":"For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a con","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjectiv","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[A quotient can erase a component] Let R=k[[x,y]]/(xy) and let T=R/(x)≅ k[[y]]. Viewed as an R module, T has support V(x), which misses the generic point (y) of the other irreducible component. Thus a surjection R↠ T does not by itself imply full component occupancy.","count":120},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0e7659cd4ef8f947","tex":"Y^2","aliases":"Y^2","display":"Y^2","kind":"use","excerpt":"-------------------------------------------------------------------------- vpage21Augmentation in two variablesAUXILIARY ALGEBRA / AN EXACT MONOMIAL BASISMain text: Proposition 5.1, Theorem 5.2 and Theorem 5.4 p12-route:visual-21Main statement 5.2 Take Δ=C_4× C_2 in residue characteristic two. If g and h generate its two factors, put X=g-1, Y=h-1. ","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3590b2065493ac41","tex":"y^{2}","aliases":"y^2","display":"y^{2}","kind":"use","excerpt":"(1.845,1.095) x^2y; draw[fill=white,draw=black!35] (2.25,0.75) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,1.095) x^3y; draw[fill=white,draw=black!35] (3.0,0.75) rectangle ++(.69,.69); node[font=scriptsize] at (3.3449999999999998,1.095) x^4y; draw[fill=white,draw=black!35] (0.0,1.5) rectangle ++(.69,.69); node[font=scriptsiz","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f8c0516a99e819f8","tex":"y^{3}","aliases":"y^3","display":"y^{3}","kind":"use","excerpt":",1.845) xy^2; draw[fill=white,draw=black!35] (1.5,1.5) rectangle ++(.69,.69); node[font=scriptsize] at (1.845,1.845) x^2y^2; draw[fill=white,draw=black!35] (2.25,1.5) rectangle ++(.69,.69); node[font=scriptsize] at (2.5949999999999998,1.845) x^3y^2; draw[fill=white,draw=black!35] (0.0,2.25) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,2.5","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-030ccdff0357de75","tex":"y^{4}","aliases":"y^4","display":"y^{4}","kind":"use","excerpt":"y^3; draw[fill=white,draw=black!35] (0.75,2.25) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,2.5949999999999998) xy^3; draw[fill=white,draw=black!35] (1.5,2.25) rectangle ++(.69,.69); node[font=scriptsize] at (1.845,2.5949999999999998) x^2y^3; draw[fill=white,draw=black!35] (0.0,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-49ee2dd099b00913","tex":"y^{5}","aliases":"y^5","display":"y^{5}","kind":"use","excerpt":"99999999998) x^2y^3; draw[fill=white,draw=black!35] (0.0,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (0.345,3.3449999999999998) y^4; draw[fill=white,draw=black!35] (0.75,3.0) rectangle ++(.69,.69); node[font=scriptsize] at (1.095,3.3449999999999998) xy^4; draw[fill=white,draw=black!35] (0.0,3.75) rectangle ++(.69,.69); node[font=scriptsize","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-827a2f52108af912","tex":"y^j","aliases":"y^j","display":"y^j","kind":"use","excerpt":"headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7ef58ddaab7b5984","tex":"Y_1","aliases":"Y_1","display":"Y_1","kind":"use","excerpt":"[Nonzero stable core criterion] Let (V_m) be an inverse system of finite dimensional vector spaces. If Y_m≠0 for every m, then varprojlim_mV_m ≠0. More precisely, the inverse limit contains a compatible sequence whose first coordinate is any prescribed nonzero element of Y_1.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"Choose 0≠ y_1∈ Y_1. By [source reference], choose recursively y_m+1∈ Y_m+1 mapping to y_m. Every y_m is nonzero because it maps to the nonzero vector y_1. The sequence (y_m) is a nonzero element of the inverse limit.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f63c0b46cff9ca02","tex":"y_m","aliases":"y_m","display":"y_m","kind":"use","excerpt":"Choose 0≠ y_1∈ Y_1. By [source reference], choose recursively y_m+1∈ Y_m+1 mapping to y_m. Every y_m is nonzero because it maps to the nonzero vector y_1. The sequence (y_m) is a nonzero element of the inverse limit.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b76f576ac2d02254","tex":"Y_{m+1}","aliases":"Y_m+1","display":"Y_{m+1}","kind":"use","excerpt":"[Surjectivity on stable cores] For every m, the transition map V_m+1→ V_m induces a surjection Y_m+1↠ Y_m.","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7917a1346f513bef","tex":"y_{m+1}","aliases":"y_m+1","display":"y_{m+1}","kind":"use","excerpt":"Choose 0≠ y_1∈ Y_1. By [source reference], choose recursively y_m+1∈ Y_m+1 mapping to y_m. Every y_m is nonzero because it maps to the nonzero vector y_1. The sequence (y_m) is a nonzero element of the inverse limit.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Write W(a,b,c)=a · b c · -a, Z=W(2,3,4), A=W(1,1,2), B=W(1,5,6). The three coordinate columns have determinant 3≠0. Under conjugation, sZ=-Z, tZ=Z, sA=-A, sB=A+B, tA=B, tB=-A-B. Consequently (z,(a,b))↦ zZ+aA+bB is an equivariant isomorphism from sgn⊕ V to ad^0V.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"node (B) at (0,2.2) B;node (O) at (6.5,2.2) Z_5; node (BJ) at (0,0) B/J;node (On) at (6.5,0) Z/5^n Z; draw[->] (B)--(O) node[midway,above] evaluation at A; draw[->] (B)--(BJ) node[midway,left] quotient; draw[->] (O)--(On) node[midway,right] reduction; draw[->] (BJ)--(On) node[midway,below] evaluation at A_n; draw[->,dashed] (BJ)--(O) node[pos=.4,ab","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"c · -a, Z=W(2,3,4), A=W(1,1,2), B=W(1,5,6). The three coordinate columns have determinant 3≠0. Under conjugation, sZ=-Z, tZ=Z, sA=-A, sB=A+B, tA=B, tB=-A-B. Consequently (z,(a,b))↦ zZ+aA+bB is an equivariant isomorphism from sgn⊕ V to ad^0V.","count":16},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-347d071dc857b815","tex":"\\mathbb Z_5","aliases":"Z_5 mathbb","display":"\\mathbb Z_5","kind":"use","excerpt":"node (B) at (0,2.2) B;node (O) at (6.5,2.2) Z_5; node (BJ) at (0,0) B/J;node (On) at (6.5,0) Z/5^n Z; draw[->] (B)--(O) node[midway,above] evaluation at A; draw[->] (B)--(BJ) node[midway,left] quotient; draw[->] (O)--(On) node[midway,right] reduction; draw[->] (BJ)--(On) node[midway,below] evaluation at A_n; draw[->,dashed] (BJ)--(O) node[pos=.4,ab","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ Z mathbb","display":"{\\mathbb Z}","kind":"use","excerpt":"[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].","count":24},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b5c3968f000169a0","tex":"\\ZZ_2","aliases":"ZZ_2 Z_2 ZZ mathbb","display":"{\\mathbb Z}_2","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-589e52e56408d298","tex":"\\ZZ_2^r","aliases":"ZZ_2^r Z_2^r ZZ mathbb","display":"{\\mathbb Z}_2^r","kind":"use","excerpt":"headRecovery without averaging The augmentation map sends g to 1 and has kernel (g-1). Thus Z_2[C_2]/(g-1)≅ Z_2, ( Z_2[C_2]^r)/(g-1) Z_2[C_2]^r≅ Z_2^r. These quotient identities require no division by two. They explain why augmentation and coinvariants remain usable even when sign-projector formulas do not.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-a7e01f582342edb9","tex":"\\ZZ_5","aliases":"ZZ_5 Z_5 ZZ mathbb","display":"{\\mathbb Z}_5","kind":"use","excerpt":"[3pt] Support · At which prime ideals does the supplied module or complex survive? [3pt] Realization · Does the surviving eigensystem come from a classical automorphic object with the required attached representation? headThree models, rather than one overloaded analogy linewidth@p.20linewidthY>raggedrightarraybackslashp.15linewidth@ Model · What i","count":14},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ebf82fd8d658554c","tex":"\\ZZ_p^r","aliases":"ZZ_p^r Z_p^r ZZ mathbb","display":"{\\mathbb Z}_p^r","kind":"use","excerpt":"[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5a91a1a1d346bb7b","tex":"|_{y=1}","aliases":"|_y=1","display":"|_{y=1}","kind":"use","excerpt":"headMagnify transversely, rather than draw a thicker line Take the slice y=1. The second summand vanishes because y becomes a unit. What remains is M|_y=1≅ k[x]/(x), M'|_y=1≅ k[x]/(x^2). The first has basis 1; the second has basis 1,x. Multiplication by x distinguishes them exactly:","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are iso","count":71},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","count":8},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","excerpt":"[x=1cm,y=1cm,font=small,>=Stealth] node[anchor=east] at (-.18,1.1) α_j; draw[black!35,fill=black!8] (0.0,0.76) rectangle ++(.80,.67); node at (0.4,1.095) 1; draw[black!35,fill=black!12] (0.88,0.76) rectangle ++(.80,.67); node at (1.28,1.095) 2; draw[black!35,fill=black!4] (1.76,0.76) rectangle ++(.80,.67); node at (2.16,1.095) 0; draw[black!35,fill","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","count":8},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-8dbbe747d8622f2f","tex":"\\beta_j","aliases":"β_j beta","display":"\\beta_j","kind":"use","excerpt":"(5.28,0.76) rectangle ++(.80,.67); node at (5.680000000000001,1.095) 2; draw[black!35,fill=black!4] (6.16,0.76) rectangle ++(.80,.67); node at (6.5600000000000005,1.095) 0; draw[black!35,fill=black!4] (7.04,0.76) rectangle ++(.80,.67); node at (7.44,1.095) 0; draw[black!35,fill=black!12] (7.92,0.76) rectangle ++(.80,.67); node at (8.32,1.095) 2; no","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":7},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ae2e30306bc89e00","tex":"\\gamma_i","aliases":"γ_i gamma","display":"\\gamma_i","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7db9f39168dc2833","tex":"\\gamma_v","aliases":"γ_v gamma","display":"\\gamma_v","kind":"use","excerpt":"Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coeffici","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Let Δ be a finite abelian p group and let Λ= O[Δ]. The augmentation map is ϵ:Λ⟶ O, ∑_g∈Δa_gg⟼∑_g∈Δa_g. Its kernel is the augmentation ideal a.","count":22},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"ORIZATION AND OBSTRUCTIONMain text: Proposition 3.3; this explicit example also exhibits a failed infinitesimal lift p12-route:visual-4Main statement 3.3 Let R^square= O[[u,v]] and send x↦ u, y↦ v-u. The local conditions become u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient ","count":13},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d99b498a81da6362","tex":"\\Delta'","aliases":"Δ' Delta","display":"\\Delta'","kind":"use","excerpt":"[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"use","excerpt":"d p12-route:visual-4Main statement 3.3 Let R^square= O[[u,v]] and send x↦ u, y↦ v-u. The local conditions become u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient map Z_5→ F_5. A continuous local point is specified by u⟼ aδ+bδ^2, v⟼ cδ+dδ^2, a,b,c,d∈ F_5. It factors through R^τ","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-02c68d637b81b90c","tex":"\\delta^3","aliases":"δ^3 delta","display":"\\delta^3","kind":"use","excerpt":"ORIZATION AND OBSTRUCTIONMain text: Proposition 3.3; this explicit example also exhibits a failed infinitesimal lift p12-route:visual-4Main statement 3.3 Let R^square= O[[u,v]] and send x↦ u, y↦ v-u. The local conditions become u^2=0, v-u=0, so R^τ= O[[u,v]]/(u^2,v-u)≅ O[ε]/(ε^2). Probe this ring using D_3= F_5[δ]/(δ^3), with the fixed coefficient ","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-dd0aa14ad6313087","tex":"\\Delta_m","aliases":"Δ_m Delta","display":"\\Delta_m","kind":"use","excerpt":"[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"node[font=sffamilysmallbfseries] at (1.6,2.35) packet at z=1; node[font=sffamilysmallbfseries] at (8.0,2.35) packet at z=-1; node at (0,1.45) 1;node at (1.6,1.45) ε;node at (3.2,1.45) 0; draw[->] (.28,1.45)--(1.32,1.45) node[midway,above,font=scriptsize] H-1; draw[->] (1.88,1.45)--(2.92,1.45) node[midway,above,font=scriptsize] H-1; node at (5.5,1.4","count":20},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"NGMain text: Theorem 3.1 and Corollary 3.2 p12-route:visual-3Main statement 3.1 Fix O= Z_5. Take two independent local variables and impose two different kinds of condition: R_1= O[[x]], I_1=(x^2), R_2= O[[y]], I_2=(y). Their completed local tensor is A= O[[x,y]]. The closed generated ideal is J=(x^2,y), and both routes through Theorem 3.1 give (R_","count":7},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-e60af3df9de1cbdc","tex":"\\epsilon_\\Delta","aliases":"ϵ_Δ epsilon Delta","display":"\\epsilon_\\Delta","kind":"use","excerpt":"[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1ec732f5d2a95aaa","tex":"\\epsilon_{\\Delta'}","aliases":"ϵ_Δ' epsilon Delta","display":"\\epsilon_{\\Delta'}","kind":"use","excerpt":"[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"] at (1.6,2.35) packet at z=1; node[font=sffamilysmallbfseries] at (8.0,2.35) packet at z=-1; node at (0,1.45) 1;node at (1.6,1.45) ε;node at (3.2,1.45) 0; draw[->] (.28,1.45)--(1.32,1.45) node[midway,above,font=scriptsize] H-1; draw[->] (1.88,1.45)--(2.92,1.45) node[midway,above,font=scriptsize] H-1; node at (5.5,1.45) 1;node at (7.1,1.45) η;node ","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9657c492efcc34dd","tex":"\\eta^2","aliases":"η^2 eta","display":"\\eta^2","kind":"use","excerpt":"=1; node[font=sffamilysmallbfseries] at (8.0,2.35) packet at z=-1; node at (0,1.45) 1;node at (1.6,1.45) ε;node at (3.2,1.45) 0; draw[->] (.28,1.45)--(1.32,1.45) node[midway,above,font=scriptsize] H-1; draw[->] (1.88,1.45)--(2.92,1.45) node[midway,above,font=scriptsize] H-1; node at (5.5,1.45) 1;node at (7.1,1.45) η;node at (8.7,1.45) η^2;node at (","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7fdc33caf5c6714b","tex":"\\eta^3","aliases":"η^3 eta","display":"\\eta^3","kind":"use","excerpt":"% --------------------------------------------------------------------------- vpage12A packet contains an entire Jordan chainMODEL C / A NONSEMISIMPLE HECKE ALGEBRAMain text: Theorem 8.2, Corollary 8.3 and Example 8.6 p12-route:visual-12Main statement 8.2 Work over Q and let T= Q[z]/((z-1)^2(z+1)^3), V=T. The Chinese remainder theorem gives T≅ unde","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-11c3cfbb073087f5","tex":"\\Lambda^d","aliases":"Λ^d Lambda","display":"\\Lambda^d","kind":"use","excerpt":"Choose a Λ basis of M. Tensoring the resulting isomorphism M≅Λ^d along augmentation gives O^d. Tensoring by Λ/ a is the same as quotienting by a M.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5bccfb6ec5b282b8","tex":"\\Lambda^r","aliases":"Λ^r Lambda","display":"\\Lambda^r","kind":"use","excerpt":"headRecover a free carrier, with its base ring named For Λ= Z_2[Δ] and the right module M=Λ^r, M⊗_Λ,ϵ Z_2 ≅ M/Mkerϵ≅ Z_2^r. The quotient applies augmentation to each basis coordinate. This remains valid for nonabelian finite two groups with the stated right-module convention. It establishes the rank over the coefficient ring after coinvariants, wit","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7238fc50817cce55","tex":"\\lambda_0","aliases":"λ_0 lambda","display":"\\lambda_0","kind":"use","excerpt":"[One member propagation in a fixed system] Let {ρ_λ}_λ∈Λ be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same po","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-18bb5b342b20541e","tex":"\\lambda_d","aliases":"λ_d lambda","display":"\\lambda_d","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":7},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-32c948dcbad490ee","tex":"\\mu_{\\frakp}","aliases":"μ_frakp μ_ p mu frakp mu mathfrak","display":"\\mu_{{\\mathfrak p}}","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b08cdbcbb8edd8df","tex":"\\mu_{\\p}","aliases":"μ_p mu p","display":null,"kind":"use","excerpt":"Fitt_j(M) · =R (j≥2). At a prime p, reduce the matrix to its residue field: μ_p(M)=2-rank A(p). Two independent relations disappear at the origin. Only one disappears along either punctured axis. captiontextFor the drawing, take k=RR. It shows real closed points, not every point of Spec R. The displayed ideals and rank criterion describe the full a","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"tion below to be continuous. Assume the following data. item A represented deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x","count":22},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cb28cf0122b8897d","tex":"\\varpi","aliases":"ϖ varpi","display":"\\varpi","kind":"use","excerpt":"Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c6553fa41ca8fbe5","tex":"\\pi_x","aliases":"π_x pi","display":"\\pi_x","kind":"use","excerpt":"tion below to be continuous. Assume the following data. item A represented deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x","count":6},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-737cdd19f67dfe28","tex":"\\rho_1","aliases":"ρ_1 rho","display":"\\rho_1","kind":"use","excerpt":"[Chebotarev and Brauer Nesbitt identification] Let E' be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let ρ_1,ρ_2:G_F,S⟶GL_n(E') be continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then ρ_1≅ρ_2. This follows f","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d7ba0ffb2518dd65","tex":"\\rho_2","aliases":"ρ_2 rho","display":"\\rho_2","kind":"use","excerpt":"[Chebotarev and Brauer Nesbitt identification] Let E' be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let ρ_1,ρ_2:G_F,S⟶GL_n(E') be continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then ρ_1≅ρ_2. This follows f","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f9e625090d5023eb","tex":"\\rho_x^{\\mathrm{ss}}","aliases":"ρ_x^ss rho mathrm","display":"\\rho_x^{\\mathrm{ss}}","kind":"use","excerpt":"tomorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-fc125fdd0aa28df4","tex":"\\rho_\\lambda","aliases":"ρ_λ rho lambda","display":"\\rho_\\lambda","kind":"use","excerpt":"[Conditional simultaneous companion completion] Under the coefficient and finite admissibility hypotheses above, one can choose continuous representations ρ_λ:Γ⟶GL_D( O_λ) (λ∈Λ) whose reduction class belongs to X_λ,n for every positive n. They retain the declared residual class. After applying i_λ, they are continuous, trivial on H_λ, and satisfy d","count":9},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d43c4345dfbfdfa8","tex":"\\rho_{\\lambda_0}","aliases":"ρ_λ_0 rho lambda","display":"\\rho_{\\lambda_0}","kind":"use","excerpt":"[One member propagation in a fixed system] Let {ρ_λ}_λ∈Λ be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same po","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9b2485d155e7ba40","tex":"\\rho_{\\pi,\\lambda}","aliases":"ρ_π,λ rho pi lambda","display":"\\rho_{\\pi,\\lambda}","kind":"use","excerpt":"For propagation, suppose both a semisimple compatible family {ρ_λ} containing ρ_x and a family {ρ_π,λ} attached to this same π already exist. Require the distinguished members to be ρ_x and its attached representation, respectively. Assume the coefficient, continuity, density, semisimplification and common algebraic Frobenius spine hypotheses of [s","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-dc168b6d53caa485","tex":"\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}","aliases":"ρ_π,λ^ss rho pi lambda mathrm","display":"\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}","kind":"use","excerpt":"be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same polynomials P_v(X). Then for every λ∈Λ, ρ_λ ≅ ρ_π,λ^ss.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-bbb82a7b6e852fe0","tex":"\\rho_{\\pi_x}","aliases":"ρ_π_x rho pi","display":"\\rho_{\\pi_x}","kind":"use","excerpt":"deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') w","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c5bd714c7223a87e","tex":"\\varphi^*","aliases":"φ^* varphi","display":"\\varphi^*","kind":"use","excerpt":"[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coeffici","count":8},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-43b76c3094fa4fa6","tex":"\\omega_1","aliases":"ω_1 omega","display":"\\omega_1","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-152412184ef94979","tex":"\\omega_j","aliases":"ω_j omega","display":"\\omega_j","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5ce5c2cb93fb1605","tex":"\\omega_r","aliases":"ω_r omega","display":"\\omega_r","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-2cf5691eb4311cbb","tex":"\\Omega_\\lambda","aliases":"Ω_λ Omega lambda","display":"\\Omega_\\lambda","kind":"use","excerpt":"Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coeffici","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s","count":18},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9ec711e428117ead","tex":"\\twoheadrightarrow","aliases":"↠ twoheadrightarrow","display":"\\twoheadrightarrow","kind":"use","excerpt":"[A quotient can erase a component] Let R=k[[x,y]]/(xy) and let T=R/(x)≅ k[[y]]. Viewed as an R module, T has support V(x), which misses the generic point (y) of the other irreducible component. Thus a surjection R↠ T does not by itself imply full component occupancy.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"c · -a, Z=W(2,3,4), A=W(1,1,2), B=W(1,5,6). The three coordinate columns have determinant 3≠0. Under conjugation, sZ=-Z, tZ=Z, sA=-A, sB=A+B, tA=B, tB=-A-B. Consequently (z,(a,b))↦ zZ+aA+bB is an equivariant isomorphism from sgn⊕ V to ad^0V.","count":16},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"[Companion branch criterion] Suppose every level X_λ,m is finite and nonempty, and every point reduces along the displayed forest to the fixed residual root. Then varprojlim_mX_λ,m ≠∅.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1e637ce45871759a","tex":"\\prod_{\\frakm\\in\\Max\\TT}","aliases":"∏_frakm∈MaxTT ∏_ m∈Max T prod frakm in Max TT prod mathfrak in operatorname mathbb","display":"\\prod_{{\\mathfrak m}\\in{\\operatorname{Max}}{\\mathbb T}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"In characteristic p, (1+X_i)^p^n_i-1=X_i^p^n_i. Thus the assignment X_i↦γ_i-1 gives a surjective homomorphism from the displayed truncated polynomial ring to k[Δ]. Both sides have dimension ∏_ip^n_i=absΔ over k, so the map is an isomorphism. The quotient by the ideal generated by the X_i is k. Since each X_i is nilpotent, that ideal is the unique m","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9079021fd6762740","tex":"\\prod_{i=1}^{r}","aliases":"∏_i=1^r prod","display":"\\prod_{i=1}^{r}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c902c7e7e8b1ed55","tex":"\\sum_{\\frakm}","aliases":"∑_frakm ∑_ m sum frakm sum mathfrak","display":"\\sum_{{\\mathfrak m}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3035b3c5f1ffe907","tex":"\\sum_{g\\in\\Delta}","aliases":"∑_g∈Δ sum in Delta","display":"\\sum_{g\\in\\Delta}","kind":"use","excerpt":"Let Δ be a finite abelian p group and let Λ= O[Δ]. The augmentation map is ϵ:Λ⟶ O, ∑_g∈Δa_gg⟼∑_g∈Δa_g. Its kernel is the augmentation ideal a.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-20794c41f8dd0413","tex":"\\sum_h","aliases":"∑_h sum","display":"\\sum_h","kind":"use","excerpt":"headWhy the cohomology argument is uniform in the module Since |S_3|=6 is a unit in F_7, a cocycle f:S_3→ M is a coboundary: set u=-6^-1∑_h f(h) and sum f(gh)=g f(h)+f(g) to obtain gu-u=f(g). This proves H^1(S_3,M)=0 for every F_7[S_3]-module M. The printed invariant calculation separately proves H^0(S_3,ad^0V)=0.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-3c105fca98331796","tex":"\\sum_{h\\in S_3}","aliases":"∑_h∈ S_3 sum in","display":"\\sum_{h\\in S_3}","kind":"use","excerpt":"A trace zero matrix fixed by conjugation by both s and t is zero, as follows by comparing its four entries. This proves H^0(S_3,ad^0V)=0. For any F_7[S_3] module M and one cocycle f, put u=-6^-1∑_h∈ S_3f(h). Summing f(gh)=g f(h)+f(g) over h gives gu-u=f(g). Hence every one cocycle is a coboundary and H^1(S_3,M)=0, giving both asserted first cohomol","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-bf85efa4a4e24042","tex":"\\sum_{i,j\\ge0}","aliases":"∑_i,j≥0 sum ge","display":"\\sum_{i,j\\ge0}","kind":"use","excerpt":"headThe calculation behind the image For f=∑_i,j≥0a_ijx^iy^j, reduction gives f⟼ a_00+a_10ε. Every term involving y vanishes. Every term involving x^2 vanishes. The class of x does not vanish: it becomes a nonzero element whose square is zero. The quotient is a free O-module with basis 1,ε.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-49afb54019149b8f","tex":"\\sum_{j\\ge0}","aliases":"∑_j≥0 sum ge","display":"\\sum_{j\\ge0}","kind":"use","excerpt":"headTwo explicit 5-adic numbers Choose the canonical digits α_j=(j^2+1)bmod5, β_j=(2j+1)bmod5, with values in {0,1,2,3,4}. Set a=∑_j≥0α_j5^j and b=∑_j≥0β_j5^j. The element a+ε b is determined by the following coefficient array.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Let p∈Spec R. If Ann_R(M)not⊆ p, choose a∈Ann_R(M)∖ p. The element a becomes a unit in R_ p, so M_ p=0.","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[Support after a quotient] Let R be Noetherian, let I⊆ R be an ideal, and let M be a finite R module. Under the canonical identification Spec(R/I)≅ V(I)⊆Spec R, one has Supp_R/I(M/IM) = Supp_R(M)∩ V(I).","count":4},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"bottomrule Thus Supp(M)=V(xy)=V(x)∪ V(y), whereas the locus where the fiber needs at least two generators is V(x,y). The latter is a higher Fitting stratum. For r>2, the locus μ_p(M)≥ r is empty, corresponding to the unit Fitting ideal.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"headAn isomorphism changes coordinates, not occupancy Over a field of characteristic other than two, the isomorphism k[x,y]xrightarrow∼k[a,b], x↦ a+b, y↦ a-b sends the Fitting ideals to (a^2-b^2) and (a,b). The two axes become the two lines a=b and a=-b. No previously missing module vector is created by this relabeling.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"headThe projectors can be written explicitly Modulo (z-1)^2(z+1)^3, put e_+(z)=frac(5-3z)(z+1)^316, e_-(z)=1-e_+(z). Then e_+≡1pmod(z-1)^2 and e_+≡0pmod(z+1)^3. Consequently e_+(H)=diag(1,1,0,0,0), e_-(H)=diag(0,0,1,1,1). The packet dimensions are two and three. The ordinary eigenspaces each have dimension one; the generalized eigenspaces have dime","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"Use a finite presentation with target rank n. If r>n, every fiber has dimension at most n, so the left hand locus is empty. The convention Fitt_r-1(M)=R for r-1≥ n makes the right hand locus empty as well. It remains to consider 1≤ r≤ n. At p, the fiber dimension is n-rank A( p). It is at least r exactly when the rank is at most n-r, which is equiv","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","count":13},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9f55fb1c51ddcd12","tex":"\\supseteq","aliases":"⊇ supseteq","display":"\\supseteq","kind":"use","excerpt":"Let p⊇ I. Localizing gives (M/IM)_ p ≅ M_ p/I_ pM_ p. If M_ p=0, the quotient is zero. If M_ p≠0, then I_ p lies in the maximal ideal of the local ring R_ p. Nakayama's lemma shows that M_ p/I_ pM_ p≠0. The stated equality follows.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"[Finite adequacy certificate] The standard representation of S_3 over F_7 is adequate. More precisely, H^0(S_3,ad^0V)=0, H^1(S_3, F_7)=0, H^1(S_3,ad^0V)=0, and ad^0V≅sgn⊕ V. The sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.","count":5},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6327e4691c318c59","tex":"\\bigoplus_{\\frakm\\in\\Max\\TT}","aliases":"⊕_frakm∈MaxTT ⊕_ m∈Max T bigoplus frakm in Max TT bigoplus mathfrak in operatorname mathbb","display":"\\bigoplus_{{\\mathfrak m}\\in{\\operatorname{Max}}{\\mathbb T}}","kind":"use","excerpt":"[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f9b87410270e30bb","tex":"\\otimes_A","aliases":"⊗_A otimes","display":"\\otimes_A","kind":"use","excerpt":"Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it car","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-451211fb9fce4bc7","tex":"\\bigotimes_{i=1}^s","aliases":"⊗_i=1^s bigotimes","display":"\\bigotimes_{i=1}^s","kind":"use","excerpt":"-------------------------------- setlengthparskip5pt vpage19Complete the actual topologyTOPOLOGICAL ALGEBRA / THE COMPARISON MAPMain text: Theorem 3.1, Corollary 3.2 and Proposition 3.3 p12-route:visual-19Main statement 3.1 The hat in whot names an actual completion. For the finite family of local rings, start with the ordinary tensor and its ideal","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-10d64144ae124a95","tex":"\\otimes_R","aliases":"⊗_R otimes","display":"\\otimes_R","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":7},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-cb7626f793d1b5b3","tex":"\\otimes_{R,x}","aliases":"⊗_R,x otimes","display":"\\otimes_{R,x}","kind":"use","excerpt":"[Field fiber criterion] Let R be a commutative ring, let M be a finite R module, and let x:R→ E' be a homomorphism to a field. Put p=ker x. Then M⊗_R,xE'≠0 Longleftrightarrow p∈Supp_R(M).","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-9384f30399aada65","tex":"\\otimes_{R/I}","aliases":"⊗_R/I otimes","display":"\\otimes_{R/I}","kind":"use","excerpt":"Let q∈Spec(R/I), and let p be its inverse image in R. The composite point R→ R/I→κ(q) has kernel p, so it factors through a field embedding κ( p)→κ(q). The ordinary tensor associator in every degree gives an isomorphism of complexes κ(q)⊗_R/I((R/I)⊗_RP) ≅κ(q)⊗_RP ≅κ(q)⊗_κ( p) (κ( p)⊗_RP). Extension of fields is faithfully flat. It preserves and ref","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-f92c003c879d28f9","tex":"\\otimes_R^{\\mathbf L}","aliases":"⊗_R^ L otimes mathbf","display":"\\otimes_R^{\\mathbf L}","kind":"use","excerpt":"Choose a bounded complex P of finite projective R modules representing C. Scalar extension to a commutative R algebra S is computed degree by degree as S⊗_RP; the canonical symmetry of tensor products gives the displayed order C⊗_R^ LS. Scalar extension preserves the bounds, finite generation, and projectivity of the terms. A derived isomorphism be","count":2},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1ec184ab518eba56","tex":"\\otimes_{T,\\,z\\mapsto1}","aliases":"⊗_T, z↦1 otimes mapsto","display":"\\otimes_{T,\\,z\\mapsto1}","kind":"use","excerpt":"headA generalized packet is not its residue-field fiber At the point z=1, the full local packet e_+V has dimension two, but its scalar fiber is V⊗_T, z↦1 Q ≅ T_+/(ε)≅ Q. The packet has the basis 1,ε. Its scalar fiber kills ε. The ordinary eigenspace is instead the kernel of multiplication by ε, spanned by ε. In this example the last two spaces have","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-b7d05b2af47a64ad","tex":"\\otimes_{\\TT,x}","aliases":"⊗_TT,x ⊗_ T,x otimes TT otimes mathbb","display":"\\otimes_{{\\mathbb T},x}","kind":"use","excerpt":"[Supported deformation points give nonzero Hecke fibers] Let R≅ T, and let M be a finite Hecke module. If a point x:R→ E' lies in Supp_R(M), then the corresponding Hecke fiber M⊗_ T,xE' is nonzero.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-1eec1f6f77946999","tex":"\\otimes_{\\kappa(\\frakp)}","aliases":"⊗_κ(frakp) ⊗_κ( p) otimes kappa frakp otimes kappa mathfrak","display":"\\otimes_{\\kappa({\\mathfrak p})}","kind":"use","excerpt":"Let q∈Spec(R/I), and let p be its inverse image in R. The composite point R→ R/I→κ(q) has kernel p, so it factors through a field embedding κ( p)→κ(q). The ordinary tensor associator in every degree gives an isomorphism of complexes κ(q)⊗_R/I((R/I)⊗_RP) ≅κ(q)⊗_RP ≅κ(q)⊗_κ( p) (κ( p)⊗_RP). Extension of fields is faithfully flat. It preserves and ref","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-580c59902104a84b","tex":"\\bigcap_{m\\ge1}","aliases":"⋂_m≥1 bigcap ge","display":"\\bigcap_{m\\ge1}","kind":"use","excerpt":"headWhy every infinite tail can be nonzero while their intersection vanishes Let P_m project onto the closed span of e_m,e_m+1,…. Each range is nonzero. But a vector belonging to every range has its j-th coordinate zero once m>j. Every coordinate is therefore zero: ⋂_m≥1P_mℓ^2( N)={0}.","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-c47fe7e501277454","tex":"\\bigcap_{m\\geq1}","aliases":"⋂_m≥1 bigcap geq","display":"\\bigcap_{m\\geq1}","kind":"use","excerpt":"[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.","count":3},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-7969d52e3ccbbabd","tex":"\\bigcap_{n\\ge m}","aliases":"⋂_n≥ m bigcap ge","display":"\\bigcap_{n\\ge m}","kind":"use","excerpt":"headWhat the stable core measures In a general inverse system of finite-dimensional spaces, Y_m=⋂_n≥ mim(f_n,m) contains exactly the vectors visible from arbitrarily remote levels. The paper proves that the transition maps restrict to surjections Y_m+1↠ Y_m. A prescribed nonzero vector in Y_1 can then be lifted recursively to a coherent nonzero seq","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-6eb6d038b89a956e","tex":"\\bigcap_{n\\ge1}","aliases":"⋂_n≥1 bigcap ge","display":"\\bigcap_{n\\ge1}","kind":"use","excerpt":"headThe reconstruction argument is coefficientwise Let A_n be compatible matrices modulo 5^n, each satisfying these four equations. Completeness reconstructs A∈ M_2( Z_5). For each polynomial f∈ J, evaluation commutes with reduction, so f(A)∈⋂_n≥15^n Z_5={0}. Separatedness supplies the last equality. Thus the actual evaluation map B→ Z_5 kills J an","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"[Cohomological description of derived support] If R is Noetherian and C is perfect, then Supp_R(C) = ⋃_iSupp_R H^i(C). In particular, derived support is closed and is empty exactly when C≃0 [source reference].","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-4601d15b3053bbea","tex":"\\bigcup_i","aliases":"⋃_i bigcup","display":"\\bigcup_i","kind":"use","excerpt":"[Cohomological description of derived support] If R is Noetherian and C is perfect, then Supp_R(C) = ⋃_iSupp_R H^i(C). In particular, derived support is closed and is empty exactly when C≃0 [source reference].","count":1},{"work":"paper-xii","url":"notation/paper-xii.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. 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x)_α.","count":1511},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e9458c4c7da9d3b","tex":"X_\\alpha","aliases":"X_α alpha","display":"X_\\alpha","kind":"notation-table","excerpt":"X_α · Packet space · The range P_α X.","count":65},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54ac43616fbfd36c","tex":"x_\\alpha","aliases":"x_α alpha","display":"x_\\alpha","kind":"notation-table","excerpt":"S_ P · Synthesis · The map (x_α)↦∑_α x_α.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5a69bbb6bda64f3","tex":"X_\\beta","aliases":"X_β beta","display":"X_\\beta","kind":"notation-table","excerpt":"T_βα · Transfer block · The source and target block P_β TP_α:X_α→ 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signal","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"notation-table","excerpt":"Λ_ G · Grammar algebra · The quotient path algebra 𝕜 Q/I.","count":210},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78111171c4a4b962","tex":"\\Lambda_{\\mathfrak G}","aliases":"Λ_ G Lambda mathfrak","display":"\\Lambda_{\\mathfrak G}","kind":"notation-table","excerpt":"Λ_ G · Grammar algebra · The quotient path algebra 𝕜 Q/I.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"notation-table","excerpt":"s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists","count":410},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"notation-table","excerpt":"H_τ · the step packet 1_[τ,1]","count":247},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"notation-table","excerpt":"ϕ_m,τ · the truncated power packet (t-τ)_+^m/m!","count":59},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-730f3c15e053f012","tex":"\\phi_{m,\\tau}","aliases":"ϕ_m,τ phi tau","display":"\\phi_{m,\\tau}","kind":"notation-table","excerpt":"ϕ_m,τ · the truncated power packet (t-τ)_+^m/m!","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"notation-table","excerpt":"P_ℓ,α · packet projector at refinement level ℓ","count":346},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"notation-table","excerpt":"[source reference] · HilbertPackets · Complex ℓ^2 impulse expansion, finite-support identity, unitary shifts, and closed shift-generated span.","count":36},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"notation-table","excerpt":"T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.","count":386},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"notation-table","excerpt":"A_ P · Analysis · The map x↦(P_α x)_α.","count":50},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"notation-table","excerpt":"S_ P · Synthesis · The map (x_α)↦∑_α x_α.","count":1172},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"notation-table","excerpt":"S_ P · Synthesis · The map (x_α)↦∑_α x_α.","count":61},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"notation-table","excerpt":"L≤ Z^d · a finite index sublattice","count":505},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"notation-table","excerpt":"S⊆ M · Seed family · The vectors from which the operator orbit is generated.","count":147},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b82e0794ad5cb35","tex":"\\mathfrak A","aliases":"A mathfrak","display":"\\mathfrak A","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":1740},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"definition-passage","excerpt":"[Divided square] A divided square on degree one elements is a map q satisfying q(x+y)=q(x)+q(y)+[x,y], q(ax)=a^2q(x), whenever the expressions are defined.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6153a08bafbf354e","tex":"A^{m-1}","aliases":"A^m-1","display":"A^{m-1}","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99155510890729d1","tex":"A^r","aliases":"A^r","display":"A^r","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-838b69abf4e3cc48","tex":"A^{\\oplus r}","aliases":"A^⊕ r oplus","display":"A^{\\oplus r}","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":40},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":44},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bcf60d2a8cb90a8","tex":"\\A_C","aliases":"A_C A mathcal","display":"{\\mathcal A}_C","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-757f361b88f183a1","tex":"A_{g_1}","aliases":"A_g_1","display":"A_{g_1}","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93d7d5611cfc11e3","tex":"A_{g_1+\\cdots+g_r}","aliases":"A_g_1+⋯+g_r cdots","display":"A_{g_1+\\cdots+g_r}","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d095bb02256e34a5","tex":"A_{g_r}","aliases":"A_g_r","display":"A_{g_r}","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":62},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2cfe3ae1e3afede","tex":"a_{n-1}","aliases":"a_n-1","display":"a_{n-1}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30f43826e4263e1a","tex":"\\mathfrak A_N^+","aliases":"A_N^+ mathfrak","display":"\\mathfrak A_N^+","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d21ef441aba4364d","tex":"\\mathfrak A_N^-","aliases":"A_N^- mathfrak","display":"\\mathfrak A_N^-","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38143a44ee0b7f8e","tex":"a_q","aliases":"a_q","display":"a_q","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-85fb367cc0b13557","tex":"A_{v_1v_0}","aliases":"A_v_1v_0","display":"A_{v_1v_0}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f5b62e2418fa31e","tex":"A_{v_m v_{m-1}}","aliases":"A_v_m v_m-1","display":"A_{v_m v_{m-1}}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d7d4c82e2b393ad","tex":"A_\\gamma","aliases":"A_γ gamma","display":"A_\\gamma","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":70},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6983fa40ff77fa12","tex":"\\mathrm{abs}","aliases":"abs mathrm","display":"\\mathrm{abs}","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff76bbf0fc00a0b5","tex":"\\Acal_{\\Pcal}","aliases":"Acal_Pcal A_ P Acal Pcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal P}}","kind":"definition-passage","excerpt":"ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c783192e1874491b","tex":"\\Acal_{\\Qcal}","aliases":"Acal_Qcal A_ Q Acal Qcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4ac73828566b6f0","tex":"\\alg","aliases":"alg operatorname","display":"{\\operatorname{alg}}","kind":"definition-passage","excerpt":"[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f84d87f1612c677e","tex":"\\alg_{\\bbk}","aliases":"alg_bbk alg_𝕜 alg bbk operatorname Bbbk","display":"{\\operatorname{alg}}_{{\\Bbbk}}","kind":"definition-passage","excerpt":"[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-798565b041c725f1","tex":"\\Ann","aliases":"Ann operatorname","display":"{\\operatorname{Ann}}","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71a978cf87af4fd1","tex":"\\Ann_R","aliases":"Ann_R Ann operatorname","display":"{\\operatorname{Ann}}_R","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ea84147003867f2","tex":"\\Arf","aliases":"Arf operatorname","display":"{\\operatorname{Arf}}","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1fc76a250d22a8eb","tex":"\\operatorname{Aut}","aliases":"Aut operatorname","display":"\\operatorname{Aut}","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82c6e4fc85ad5450","tex":"\\operatorname{Aut}_R","aliases":"Aut_R operatorname","display":"\\operatorname{Aut}_R","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Packet system surjective along edges] The module resolved into packets is surjective along edges relative to Γ( G, P) when, for every graph edge α→β, there exists B∈ A such that the pure block P_β BP_α:X_α↠ X_β is surjective.","count":760},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1cad61c08170218e","tex":"\\B","aliases":"B mathcal","display":"{\\mathcal B}","kind":"definition-passage","excerpt":"[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":36},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,","count":849},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f673c03c77f9ac50","tex":"B^*","aliases":"B^*","display":"B^*","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":27},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eeabc5ab3835244c","tex":"B_{v_1v_0}","aliases":"B_v_1v_0","display":"B_{v_1v_0}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3d9f45da6fdf937","tex":"B_{v_m v_{m-1}}","aliases":"B_v_m v_m-1","display":"B_{v_m v_{m-1}}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e4af5afb524dbe9","tex":"B_{\\beta\\alpha}","aliases":"B_βα beta alpha","display":"B_{\\beta\\alpha}","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e01db45ab933df66","tex":"B_\\gamma","aliases":"B_γ gamma","display":"B_\\gamma","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fc2361ea0bdf8ab","tex":"b_\\rho","aliases":"b_ρ rho","display":"b_\\rho","kind":"definition-passage","excerpt":"[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4d7ee29d2029812","tex":"\\bar x","aliases":"bar x bar","display":"\\bar x","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":32},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db1d2ae53aa18234","tex":"\\bar\\rho","aliases":"barρ bar rho","display":"\\bar\\rho","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8ab7facc224789a","tex":"\\bar\\rho'","aliases":"barρ' bar rho","display":"\\bar\\rho'","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-327abfc143ff067a","tex":"\\Bcal","aliases":"Bcal B mathcal","display":"{\\mathcal B}","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":74},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69fe78d4f1aca040","tex":"\\Bcal_{ab}","aliases":"Bcal_ab B_ab Bcal mathcal","display":"{\\mathcal B}_{ab}","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51e663dd35f8862b","tex":"\\Bcal_{N,j}","aliases":"Bcal_N,j B_N,j Bcal mathcal","display":"{\\mathcal B}_{N,j}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f4734d9f363e46b","tex":"\\operatorname{binary}","aliases":"binary operatorname","display":"\\operatorname{binary}","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No al","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":73},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5bb28d7c6728c2ad","tex":"\\mathrm{boundary}","aliases":"boundary mathrm","display":"\\mathrm{boundary}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e3c0fce1b7a60ae","tex":"\\bullet","aliases":"bullet","display":"\\bullet","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b62fdaa08c020af","tex":"\\bytes","aliases":"bytes operatorname","display":"{\\operatorname{bytes}}","kind":"definition-passage","excerpt":"[Arithmetic intensity] When the modeled byte traffic is strictly positive, the modeled arithmetic intensity is I = fracflopbytes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-420668a5bdd9e0bf","tex":"\\mathbf C","aliases":"C mathbf","display":"\\mathbf C","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbc9ba3c4c4e2bc2","tex":"\\mathfrak c","aliases":"c mathfrak","display":"\\mathfrak c","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bfba5cbf12e1727","tex":"C^\\bullet","aliases":"C^bullet bullet","display":"C^\\bullet","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70f1ea36ff493278","tex":"C^{n+1}","aliases":"C^n+1","display":"C^{n+1}","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":43},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce865f9aea2d3d86","tex":"C_{a_0}","aliases":"C_a_0","display":"C_{a_0}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6db54b248f3844fc","tex":"C_b","aliases":"C_b","display":"C_b","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f3853f42b762bab","tex":"C_c","aliases":"C_c","display":"C_c","kind":"definition-passage","excerpt":"[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76943d929d27bd54","tex":"\\mathbf C_D","aliases":"C_D mathbf","display":"\\mathbf C_D","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98cd1622d78cdfec","tex":"c_{e,k}","aliases":"c_e,k","display":"c_{e,k}","kind":"definition-passage","excerpt":"[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":25},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4c4b465366a9288","tex":"C_{m,n}","aliases":"C_m,n","display":"C_{m,n}","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40591b7fff4431b2","tex":"C_{N,j}","aliases":"C_N,j","display":"C_{N,j}","kind":"definition-passage","excerpt":"or 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn)","count":53},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4a69286b41cbe31","tex":"C_{N,j,n}","aliases":"C_N,j,n","display":"C_{N,j,n}","kind":"definition-passage","excerpt":"For r∈ C, define formal coefficients C_N,j,n(r) by C_N,j(z)^r = ∑_n≥0C_N,j,n(r)fracz^Nn(Nn)!. On a simply connected region containing 0 and avoiding the zeros of Φ_N,j(z^N), interpret this power as e^rL, where L is the holomorphic logarithm of C_N,j normalized by L(0)=0.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-014eb6330bcf05ac","tex":"C_{\\Pcal\\leftarrow\\Qcal}","aliases":"C_Pcal←Qcal C_ P← Q Pcal leftarrow Qcal mathcal leftarrow mathcal","display":"C_{{\\mathcal P}\\leftarrow{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8195c86d8aac64cd","tex":"C_X","aliases":"C_X","display":"C_X","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92e4993fd113c6f6","tex":"C_Y","aliases":"C_Y","display":"C_Y","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5226331ae2a4df1e","tex":"C_\\Gamma","aliases":"C_Γ Gamma","display":"C_\\Gamma","kind":"definition-passage","excerpt":"[Linear tube bound] A boundary Γ has a linear tube bound if there are constants C_Γ and δ_0>0 such that μ(N_δ(Γ)) ≤ C_Γδ for 0<δ<δ_0.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd4813b07940b348","tex":"c_\\rho","aliases":"c_ρ rho","display":"c_\\rho","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0a61d8b37df0b6e","tex":"C_{\\ell,r}","aliases":"C_ℓ,r ell","display":"C_{\\ell,r}","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b29a3476d110709","tex":"C_\\infty","aliases":"C_∞ infty","display":"C_\\infty","kind":"definition-passage","excerpt":"[Derived patching datum] A derived patching datum consists of a Noetherian ring R_∞, an ideal a, a perfect complex C_∞ over R_∞, and identifications R≅ R_∞/ a, C≃ C_∞ mathbinmathop⊗limits_R_∞^ LR.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4e5cf02187f03e5","tex":"\\cA_b","aliases":"cA_b A_b cA mathcal","display":"{\\mathcal A}_b","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab865756428ae789","tex":"\\calH","aliases":"calH H mathcal","display":"{\\mathcal H}","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef089d01f361d945","tex":"\\calX","aliases":"calX X mathcal","display":"{\\mathcal X}","kind":"definition-passage","excerpt":"[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC C mathbb","display":"{\\mathbb C}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":99},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-456fdfa5f34a5117","tex":"\\CC^N","aliases":"CC^N C^N CC mathbb","display":"{\\mathbb C}^N","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff4ec9351d7bdf0c","tex":"\\CC^{n\\times r}","aliases":"CC^n× r C^n× r CC times mathbb times","display":"{\\mathbb C}^{n\\times r}","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f57d50dacedd8d8c","tex":"\\CC^X","aliases":"CC^X C^X CC mathbb","display":"{\\mathbb C}^X","kind":"definition-passage","excerpt":"[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34dfb5e598913303","tex":"\\CC^{\\times}","aliases":"CC^× C^× CC times mathbb times","display":"{\\mathbb C}^{\\times}","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-843721b3e84879d7","tex":"\\Ccal","aliases":"Ccal C mathcal","display":"{\\mathcal C}","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e52835b675ac9ef","tex":"\\Cfun","aliases":"Cfun C mathsf","display":"{\\mathsf C}","kind":"definition-passage","excerpt":"For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5d0042664713d89","tex":"\\Cfun_{N,j}","aliases":"Cfun_N,j C_N,j Cfun mathsf","display":"{\\mathsf C}_{N,j}","kind":"definition-passage","excerpt":"For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-77a4dd0e05e628fb","tex":"\\cH","aliases":"cH H mathcal","display":"{\\mathcal H}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0fe2f8238bcf56d","tex":"\\operatorname{ch}","aliases":"ch operatorname","display":"\\operatorname{ch}","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-860a844f873baee1","tex":"\\operatorname{child}","aliases":"child operatorname","display":"\\operatorname{child}","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80bb66e28ec49c6d","tex":"\\Cl","aliases":"Cl operatorname","display":"{\\operatorname{Cl}}","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5761fa2adf61c496","tex":"\\cl","aliases":"cl","display":null,"kind":"definition-passage","excerpt":"[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-871baec058112b7f","tex":"\\Cl_G","aliases":"Cl_G Cl operatorname","display":"{\\operatorname{Cl}}_G","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-754793052329c278","tex":"\\cO","aliases":"cO O mathcal","display":"{\\mathcal O}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-073a71f814bdfa5f","tex":"\\coker","aliases":"coker operatorname","display":"{\\operatorname{coker}}","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":30},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1112049ac7286dcd","tex":"\\Compat","aliases":"Compat C mathfrak","display":"{\\mathfrak C}","kind":"definition-passage","excerpt":"[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the f","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d94488f5df4843b","tex":"\\Cone","aliases":"Cone operatorname","display":"{\\operatorname{Cone}}","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d2a6783a92b4bc0","tex":"\\operatorname{constant}","aliases":"constant operatorname","display":"\\operatorname{constant}","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No al","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e63707f1d2e3c7ef","tex":"\\Coord","aliases":"Coord operatorname","display":"{\\operatorname{Coord}}","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":55},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e35d2ea5ace79132","tex":"\\cP","aliases":"cP P mathcal","display":"{\\mathcal P}","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29b3c8e7482cdeab","tex":"\\cP^a","aliases":"cP^a P^a cP mathcal","display":"{\\mathcal P}^a","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-88eb01a9511f791c","tex":"\\cP^b","aliases":"cP^b P^b cP mathcal","display":"{\\mathcal P}^b","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-baabcb8a65f0c64e","tex":"\\cR","aliases":"cR R mathcal","display":"{\\mathcal R}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99e69c96f2e59e3a","tex":"\\cS_a","aliases":"cS_a S_a cS mathcal","display":"{\\mathcal S}_a","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-adbc3dd7c43b1b72","tex":"\\cW","aliases":"cW W mathcal","display":"{\\mathcal W}","kind":"definition-passage","excerpt":"[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7001ee534716827","tex":"\\cW_0","aliases":"cW_0 W_0 cW mathcal","display":"{\\mathcal W}_0","kind":"definition-passage","excerpt":"[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d96305e76870213","tex":"\\cW_n","aliases":"cW_n W_n cW mathcal","display":"{\\mathcal W}_n","kind":"definition-passage","excerpt":"[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30c7c25e421d29ab","tex":"\\cW_{n+1}","aliases":"cW_n+1 W_n+1 cW mathcal","display":"{\\mathcal W}_{n+1}","kind":"definition-passage","excerpt":"[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a3a2aba376670a9","tex":"D^N","aliases":"D^N","display":"D^N","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09e71d50145ecad3","tex":"D^r","aliases":"D^r","display":"D^r","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c308718fcbd3df8","tex":"d_C","aliases":"d_C","display":"d_C","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb69ef05a69fcd3e","tex":"d_{\\Cone}","aliases":"d_Cone Cone operatorname","display":"d_{{\\operatorname{Cone}}}","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3008bb1b6de0e7c","tex":"d_E","aliases":"d_E","display":"d_E","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf9b9919bc261407","tex":"d_L","aliases":"d_L","display":"d_L","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66af625e26b38735","tex":"\\Dcal","aliases":"Dcal D mathcal","display":"{\\mathcal D}","kind":"definition-passage","excerpt":"[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-367a7ddcc412cdcb","tex":"\\Dcal_{\\Pcal}","aliases":"Dcal_Pcal D_ P Dcal Pcal mathcal mathcal","display":"{\\mathcal D}_{{\\mathcal P}}","kind":"definition-passage","excerpt":"[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd d mathrm","display":"{\\,\\mathrm d}","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":82},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c964f73df39bd41c","tex":"\\dd^d","aliases":"dd^d d^d dd mathrm","display":"{\\,\\mathrm d}^d","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-625573dcacdf7fff","tex":"\\Def","aliases":"Def operatorname","display":"{\\operatorname{Def}}","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b94f1088caec322","tex":"\\Def_{\\bar\\rho}","aliases":"Def_barρ Def bar rho operatorname bar rho","display":"{\\operatorname{Def}}_{\\bar\\rho}","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-912b7cbebb645a8c","tex":"\\Def_{\\bar\\rho'}","aliases":"Def_barρ' Def bar rho operatorname bar rho","display":"{\\operatorname{Def}}_{\\bar\\rho'}","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d523ae87a3985cd","tex":"\\mathrm{dependencies}","aliases":"dependencies mathrm","display":"\\mathrm{dependencies}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19f71f3af9894264","tex":"\\mathrm{der}","aliases":"der mathrm","display":"\\mathrm{der}","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"definition-passage","excerpt":"[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).","count":46},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28852ba6759dd734","tex":"\\Dfun","aliases":"Dfun D mathsf","display":"{\\mathsf D}","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":114},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d580abc74d5062ed","tex":"\\Dfun_N","aliases":"Dfun_N D_N Dfun mathsf","display":"{\\mathsf D}_N","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":75},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e472c8b232ff91ad","tex":"\\Dfun_N'","aliases":"Dfun_N' D_N' Dfun mathsf","display":"{\\mathsf D}_N'","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08d21ec98995eca1","tex":"\\Dfun_{N,0}","aliases":"Dfun_N,0 D_N,0 Dfun mathsf","display":"{\\mathsf D}_{N,0}","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c1d5142975bd0fa0","tex":"\\Dfun_{N,j}","aliases":"Dfun_N,j D_N,j Dfun mathsf","display":"{\\mathsf D}_{N,j}","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":54},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa4b8ca726217eac","tex":"\\dim_{\\mathrm{seed}}","aliases":"dim_seed dim mathrm","display":"\\dim_{\\mathrm{seed}}","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0dcf6a495ccbd5bd","tex":"\\e","aliases":"e mathrm","display":"{\\mathrm e}","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":198},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7c26f6b3b085b5c","tex":"E'","aliases":"E'","display":"E'","kind":"definition-passage","excerpt":"[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automo","count":44},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-473dfae94e6cd783","tex":"e^{-s\\log|\\lambda|}","aliases":"e^-slog|λ| log lambda","display":"e^{-s\\log|\\lambda|}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6dcd93ee0614b9ca","tex":"e^{-s\\log|\\rho|}","aliases":"e^-slog|ρ| log rho","display":"e^{-s\\log|\\rho|}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4f632909385c1ce","tex":"e^{-s\\Log\\lambda}","aliases":"e^-sLogλ Log lambda operatorname lambda","display":"e^{-s{\\operatorname{Log}}\\lambda}","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-184e074ec633d7bb","tex":"e^{-s\\Log\\rho}","aliases":"e^-sLogρ Log rho operatorname rho","display":"e^{-s{\\operatorname{Log}}\\rho}","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3c5d66994f2ce29","tex":"\\e^{-tL}","aliases":"e^-tL e mathrm","display":"{\\mathrm e}^{-tL}","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f06630fc4bbd48b8","tex":"E^\\bullet","aliases":"E^bullet bullet","display":"E^\\bullet","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e40ba00de6234ae1","tex":"E^n","aliases":"E^n","display":"E^n","kind":"definition-passage","excerpt":"[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3cbcdea0e9d0e9b","tex":"e^{rL}","aliases":"e^rL","display":"e^{rL}","kind":"definition-passage","excerpt":"For r∈ C, define formal coefficients C_N,j,n(r) by C_N,j(z)^r = ∑_n≥0C_N,j,n(r)fracz^Nn(Nn)!. On a simply connected region containing 0 and avoiding the zeros of Φ_N,j(z^N), interpret this power as e^rL, where L is the holomorphic logarithm of C_N,j normalized by L(0)=0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5052bf5368337201","tex":"\\e^{tA}","aliases":"e^tA e mathrm","display":"{\\mathrm e}^{tA}","kind":"definition-passage","excerpt":"[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cdc491b152dc2abb","tex":"e^{xz}","aliases":"e^xz","display":"e^{xz}","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-517adbc54a8d8e03","tex":"e^{\\lambda z}","aliases":"e^λ z lambda","display":"e^{\\lambda z}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":56},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"definition-passage","excerpt":"W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.","count":80},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a9f8dc796c8ef36","tex":"E_{2m-b}","aliases":"E_2m-b","display":"E_{2m-b}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ceab38a8fc70e99","tex":"E_a^*","aliases":"E_a^*","display":"E_a^*","kind":"definition-passage","excerpt":"[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":25},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a603588606ce810","tex":"E_b","aliases":"E_b","display":"E_b","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69a8a46b93a29118","tex":"E_d^{(N,j)}","aliases":"E_d^(N,j)","display":"E_d^{(N,j)}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf85306cf8e31b85","tex":"e_{j+1}","aliases":"e_j+1","display":"e_{j+1}","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62326bcf64c61a2a","tex":"E_m^{(N,j,r)}","aliases":"E_m^(N,j,r)","display":"E_m^{(N,j,r)}","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19911032e8985ebc","tex":"e_{N-1}","aliases":"e_N-1","display":"e_{N-1}","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be288f5d115169ac","tex":"E_{Nn}^{(N,j)}","aliases":"E_Nn^(N,j)","display":"E_{Nn}^{(N,j)}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b243e274b6d1702","tex":"e_{r+1}","aliases":"e_r+1","display":"e_{r+1}","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02c1bebd51ef5031","tex":"e_{|\\gamma|}","aliases":"e_|γ| gamma","display":"e_{|\\gamma|}","kind":"definition-passage","excerpt":"W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29663d1f0bd44f71","tex":"E_\\alpha","aliases":"E_α alpha","display":"E_\\alpha","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4748e7272f1e3f16","tex":"e_\\nu","aliases":"e_ν nu","display":"e_\\nu","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02857ba6277fbbbc","tex":"\\Efun","aliases":"Efun E mathsf","display":"{\\mathsf E}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de3cc61a1ed3d5b3","tex":"\\Efun_{N,0}","aliases":"Efun_N,0 E_N,0 Efun mathsf","display":"{\\mathsf E}_{N,0}","kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-504ccb92769528db","tex":"\\Efun_{N,j}","aliases":"Efun_N,j E_N,j Efun mathsf","display":"{\\mathsf E}_{N,j}","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8f21276e6dfefea","tex":"\\ev","aliases":"ev operatorname","display":"{\\operatorname{ev}}","kind":"definition-passage","excerpt":"[Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.Colo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55549aac652d0c0b","tex":"\\mathsf{Ev}","aliases":"Ev mathsf","display":"\\mathsf{Ev}","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-375d20cf9877348c","tex":"\\mathsf{Ev}_D","aliases":"Ev_D mathsf","display":"\\mathsf{Ev}_D","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c808485c3e440e2","tex":"\\operatorname{eval}","aliases":"eval operatorname","display":"\\operatorname{eval}","kind":"definition-passage","excerpt":"B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain, and ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation is eval(IR) = ∑_(β,α)∈ E ι_β B_βαπ_α, where ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":49},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8c8bc0689ee1b3e","tex":"\\expn","aliases":"expn exp operatorname","display":"{\\operatorname{exp}}","kind":"definition-passage","excerpt":"mbda_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1daca1ee43620e8f","tex":"\\expn_\\pi","aliases":"expn_π exp_π expn pi operatorname pi","display":"{\\operatorname{exp}}_\\pi","kind":"definition-passage","excerpt":"mbda_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-912eecc2461552b8","tex":"\\mathrm{exports}","aliases":"exports mathrm","display":"\\mathrm{exports}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":900},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40808d4081e59d75","tex":"F^0","aliases":"F^0","display":"F^0","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fef72e85e7b92ea3","tex":"F^1","aliases":"F^1","display":"F^1","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f424c1523cf5188","tex":"F^L","aliases":"F^L","display":"F^L","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b8a4bc27627081b","tex":"F^m","aliases":"F^m","display":"F^m","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":34},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6dc70eae04871a40","tex":"F^{m+1}","aliases":"F^m+1","display":"F^{m+1}","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ba5037951b752c25","tex":"F^r","aliases":"F^r","display":"F^r","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9758c662ccf0f061","tex":"F^{r+s}","aliases":"F^r+s","display":"F^{r+s}","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e45f4042aed97e34","tex":"F^s","aliases":"F^s","display":"F^s","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd8cc01590ccaf86","tex":"F^{w(e)}","aliases":"F^w(e)","display":"F^{w(e)}","kind":"definition-passage","excerpt":"with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-058e3b39ebf27e0f","tex":"f_{ab}^{\\phantom{ab}c}","aliases":"f_ab^phantomabc phantom","display":"f_{ab}^{\\phantom{ab}c}","kind":"definition-passage","excerpt":"[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotien","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92cbe7d6d84c4e19","tex":"F_c","aliases":"F_c","display":"F_c","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4a55684394b7979","tex":"f_d","aliases":"f_d","display":"f_d","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f98b1eca815e46c","tex":"F_f","aliases":"F_f","display":"F_f","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7794f17df61c3e5","tex":"f_{n,m}","aliases":"f_n,m","display":"f_{n,m}","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07d5ca93af84b975","tex":"F_P","aliases":"F_P","display":"F_P","kind":"definition-passage","excerpt":"[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d2d5067e38c8f8c","tex":"F_{\\Qcal\\leftarrow\\Pcal}","aliases":"F_Qcal←Pcal F_ Q← P Qcal leftarrow Pcal mathcal leftarrow mathcal","display":"F_{{\\mathcal Q}\\leftarrow{\\mathcal P}}","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4fa64e297ddf7f10","tex":"f_{s(a)}","aliases":"f_s(a)","display":"f_{s(a)}","kind":"definition-passage","excerpt":"linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80f1392d934eaed0","tex":"f_{t(a)}","aliases":"f_t(a)","display":"f_{t(a)}","kind":"definition-passage","excerpt":"linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e34327622e5e789e","tex":"f_v","aliases":"f_v","display":"f_v","kind":"definition-passage","excerpt":"l on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4f568dc946ddd98","tex":"\\Fcal","aliases":"Fcal F mathcal","display":"{\\mathcal F}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ca3cb08c886d426","tex":"\\Fcal_{r,p}","aliases":"Fcal_r,p F_r,p Fcal mathcal","display":"{\\mathcal F}_{r,p}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc7582cb4d0dc2f3","tex":"\\FF","aliases":"FF F mathbb","display":"{\\mathbb F}","kind":"definition-passage","excerpt":"[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).","count":74},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac76bb3454573605","tex":"\\FF_2","aliases":"FF_2 F_2 FF mathbb","display":"{\\mathbb F}_2","kind":"definition-passage","excerpt":"[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-280dad33f94a39d1","tex":"\\operatorname{Fin}","aliases":"Fin operatorname","display":"\\operatorname{Fin}","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b3cf9d696dc4fcb","tex":"\\flop","aliases":"flop operatorname","display":"{\\operatorname{flop}}","kind":"definition-passage","excerpt":"[Arithmetic intensity] When the modeled byte traffic is strictly positive, the modeled arithmetic intensity is I = fracflopbytes.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea06efaabf953d2a","tex":"\\fraka","aliases":"fraka a mathfrak","display":"{\\mathfrak a}","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39afe4e3b95de796","tex":"\\frakp","aliases":"frakp p mathfrak","display":"{\\mathfrak p}","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":42},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"l=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.","count":388},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"definition-passage","excerpt":"[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":367},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28a0f54c10c0e63f","tex":"G_A","aliases":"G_A","display":"G_A","kind":"definition-passage","excerpt":"[Support graph] The support graph G_A has vertex set I. It has a directed edge α⟶β when P_β AP_α≠0.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95623e0cac2b3755","tex":"G_a^{-1}","aliases":"G_a^-1","display":"G_a^{-1}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76b53653bff85b9e","tex":"G_b","aliases":"G_b","display":"G_b","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87a3fa4dd69238cf","tex":"g_r","aliases":"g_r","display":"g_r","kind":"definition-passage","excerpt":"[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b14e28ca0ef61ba","tex":"G_u^{-1}","aliases":"G_u^-1","display":"G_u^{-1}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8bea312cd06acfdb","tex":"G_v","aliases":"G_v","display":"G_v","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99b7a1e1268a1b18","tex":"g_v","aliases":"g_v","display":"g_v","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7dd1ffa3cf728bef","tex":"G_{\\Lambda,\\Pcal}","aliases":"G_Λ,Pcal G_Λ, P Lambda Pcal Lambda mathcal","display":"G_{\\Lambda,{\\mathcal P}}","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9b55a29a191e028","tex":"\\GL","aliases":"GL operatorname","display":"{\\operatorname{GL}}","kind":"definition-passage","excerpt":"[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0fcc70536b44cfb8","tex":"\\operatorname{GL}","aliases":"GL operatorname","display":"\\operatorname{GL}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3d50e4b12d0681c","tex":"H^*","aliases":"H^*","display":"H^*","kind":"definition-passage","excerpt":"inition:001 A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99e2417a50167bf6","tex":"H_{c,b,a}","aliases":"H_c,b,a","display":"H_{c,b,a}","kind":"definition-passage","excerpt":"[Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac7885f6694a994f","tex":"H_N","aliases":"H_N","display":"H_N","kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b28d6c4b94c5afd8","tex":"h_N","aliases":"h_N","display":"h_N","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0bbb437f18bc2cd","tex":"\\mathcal H_p","aliases":"H_p mathcal","display":"\\mathcal H_p","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af7f4e3b5c2ffe05","tex":"h_{rs}","aliases":"h_rs","display":"h_{rs}","kind":"definition-passage","excerpt":"[Packet transfer kernel] For output residue r∈ R and input residue s∈ R, define a kernel on L by h_rs(m)=h(r-s+m), m∈ L.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-388fb1dbc2c04d8e","tex":"h_\\theta","aliases":"h_θ theta","display":"h_\\theta","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c4425940c285435","tex":"H_\\ell","aliases":"H_ℓ ell","display":"H_\\ell","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e073754a563e30e","tex":"\\hbox","aliases":"hbox","display":null,"kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f50c3221ec49a3fa","tex":"\\Hcal","aliases":"Hcal H mathcal","display":"{\\mathcal H}","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":68},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b378be010dc46f96","tex":"\\Hcal_v","aliases":"Hcal_v H_v Hcal mathcal","display":"{\\mathcal H}_v","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12403ce6e112714f","tex":"\\Hcal_w","aliases":"Hcal_w H_w Hcal mathcal","display":"{\\mathcal H}_w","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a40fa7259746e49d","tex":"\\Hcal_\\ell","aliases":"Hcal_ℓ H_ℓ Hcal ell mathcal ell","display":"{\\mathcal H}_\\ell","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21d3cf9a3f360e8d","tex":"\\Hcal_{\\ell+1}","aliases":"Hcal_ℓ+1 H_ℓ+1 Hcal ell mathcal ell","display":"{\\mathcal H}_{\\ell+1}","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom operatorname","display":"{\\operatorname{Hom}}","kind":"definition-passage","excerpt":"[Transfer block] For T∈End_𝕜(X) and a finite packet system P, define T_βα:=P_β TP_α ∈Hom_𝕜(X_α,X_β).","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b849ee4e6f402af","tex":"\\Hom_{\\bbk}","aliases":"Hom_bbk Hom_𝕜 Hom bbk operatorname Bbbk","display":"{\\operatorname{Hom}}_{{\\Bbbk}}","kind":"definition-passage","excerpt":"[Transfer block] For T∈End_𝕜(X) and a finite packet system P, define T_βα:=P_β TP_α ∈Hom_𝕜(X_α,X_β).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8ccb7cff872b632","tex":"\\Hom_N","aliases":"Hom_N Hom operatorname","display":"{\\operatorname{Hom}}_N","kind":"definition-passage","excerpt":"The multiplicity space is M_ρ=Hom_N(W, X).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8a30d8cb6470d6d","tex":"\\Hom_R","aliases":"Hom_R Hom operatorname","display":"{\\operatorname{Hom}}_R","kind":"definition-passage","excerpt":"[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8fff38e3b7008328","tex":"\\Hull_{\\A}","aliases":"Hull_A Hull_ A Hull A operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal A}}","kind":"definition-passage","excerpt":"[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2609b96491ee0fe2","tex":"\\Hull_{\\A,\\tau}","aliases":"Hull_A,τ Hull_ A,τ Hull A tau operatorname mathcal tau","display":"{\\operatorname{Hull}}_{{\\mathcal A},\\tau}","kind":"definition-passage","excerpt":"[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bc5eb932401337d","tex":"\\Hull_{\\B}","aliases":"Hull_B Hull_ B Hull B operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal B}}","kind":"definition-passage","excerpt":"[Morphism of seeded systems] A morphism (X, A,S, P)⟶(Y, B,S', Q) is a packet compatible operator morphism satisfying the seed condition Φ(S)⊆Hull_ B(S'). In the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-329089d08934b068","tex":"\\mathrm{hypotheses}","aliases":"hypotheses mathrm","display":"\\mathrm{hypotheses}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":890},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad5f56fc779f28fd","tex":"I_{\\Ccal}","aliases":"I_Ccal I_ C Ccal mathcal","display":"I_{{\\mathcal C}}","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"definition-passage","excerpt":"[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2af7728430d7213","tex":"I_\\alpha","aliases":"I_α alpha","display":"I_\\alpha","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":27},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41bd5b0100b2ea43","tex":"\\Ical","aliases":"Ical I mathcal","display":"{\\mathcal I}","kind":"definition-passage","excerpt":"[Arithmetic intensity] When the modeled byte traffic is strictly positive, the modeled arithmetic intensity is I = fracflopbytes.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id mathrm","display":"{\\mathrm{id}}","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":31},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7caa8aee712bf74f","tex":"\\mathrm{ID}","aliases":"ID mathrm","display":"\\mathrm{ID}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c0a8caa7db598ba","tex":"\\id_{C_a}","aliases":"id_C_a id mathrm","display":"{\\mathrm{id}}_{C_a}","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f04fed0295873efc","tex":"\\id_{C_{a_0}}","aliases":"id_C_a_0 id mathrm","display":"{\\mathrm{id}}_{C_{a_0}}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ce608af84df2a4c","tex":"\\id_M","aliases":"id_M id mathrm","display":"{\\mathrm{id}}_M","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-88b326798e7011cf","tex":"\\ii","aliases":"ii i mathrm","display":"{\\mathrm i}","kind":"definition-passage","excerpt":"[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotien","count":95},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"definition-passage","excerpt":"The realized action and faithful reduction. [Realized operator algebra] For a realization X, the realized operator algebra is A_X:=im (Λ_ G⟶End_𝕜(|X|)). It records the actual operators in the realization and may be a proper quotient of the abstract grammar algebra.","count":49},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16e6e5c52bd14fa0","tex":"\\operatorname{in}","aliases":"in operatorname","display":"\\operatorname{in}","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61dd0726b69a3f36","tex":"\\operatorname{in}_\\sigma","aliases":"in_σ operatorname sigma","display":"\\operatorname{in}_\\sigma","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"definition-passage","excerpt":"[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-761d17addb30f300","tex":"\\inf_{c\\in\\CC}","aliases":"inf_c∈CC inf_c∈ C inf in CC inf in mathbb","display":"\\inf_{c\\in{\\mathbb C}}","kind":"definition-passage","excerpt":"[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4de08132ed3bb27b","tex":"\\operatorname{input}","aliases":"input operatorname","display":"\\operatorname{input}","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No al","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea0ad3db0c0f9c47","tex":"\\IR","aliases":"IR mathsf","display":"{\\mathsf{IR}}","kind":"definition-passage","excerpt":"[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the dis","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35f44b16b4c77db8","tex":"\\operatorname{IsSolution}","aliases":"IsSolution operatorname","display":"\\operatorname{IsSolution}","kind":"definition-passage","excerpt":"[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPa","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-accca2f8c7d6bf65","tex":"\\operatorname{IsSolution}_r","aliases":"IsSolution_r operatorname","display":"\\operatorname{IsSolution}_r","kind":"definition-passage","excerpt":"[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPa","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d6110995dfdfd76","tex":"J^{r}_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}","aliases":"J^r_γmidα_1,…,α_r gamma mid alpha ldots alpha","display":null,"kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98339ce53ef8771d","tex":"J^X_{b\\leftarrow a}","aliases":"J^X_b← a leftarrow","display":"J^X_{b\\leftarrow a}","kind":"definition-passage","excerpt":"[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRe","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a670656747a1061","tex":"J_{b\\leftarrow a}","aliases":"J_b← a leftarrow","display":"J_{b\\leftarrow a}","kind":"definition-passage","excerpt":"[Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. s","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6cca5b503f09502b","tex":"J_N","aliases":"J_N","display":"J_N","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0314e68e3d01e27","tex":"J_N^N","aliases":"J_N^N","display":"J_N^N","kind":"definition-passage","excerpt":"Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7884b4fb34770863","tex":"J_{\\Qcal\\leftarrow\\Pcal}","aliases":"J_Qcal←Pcal J_ Q← P Qcal leftarrow Pcal mathcal leftarrow mathcal","display":"J_{{\\mathcal Q}\\leftarrow{\\mathcal P}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":597},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No al","count":1060},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc97a5c4f9fcff61","tex":"K_\\lambda","aliases":"K_λ lambda","display":"K_\\lambda","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9feea0dd1279e0f4","tex":"\\Kcal","aliases":"Kcal K mathcal","display":"{\\mathcal K}","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":25},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2def419be484fcdf","tex":"\\Kcal_m","aliases":"Kcal_m K_m Kcal mathcal","display":"{\\mathcal K}_m","kind":"definition-passage","excerpt":"[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e08d6b60be0ec0b0","tex":"\\Ker","aliases":"Ker operatorname","display":"{\\operatorname{Ker}}","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":46},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08093f17d5992ea3","tex":"\\mathbf L","aliases":"L mathbf","display":"\\mathbf L","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f574898fe03737a","tex":"L_e","aliases":"L_e","display":"L_e","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97675c4573f573e3","tex":"L_{e_1}","aliases":"L_e_1","display":"L_{e_1}","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-817d2f3702aba686","tex":"L_{e_n}","aliases":"L_e_n","display":"L_{e_n}","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8ade83d2dcc64e3","tex":"L_w","aliases":"L_w","display":"L_w","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c66f081d0dbf6c6c","tex":"L_\\lambda","aliases":"L_λ lambda","display":"L_\\lambda","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39c11bec6df251b7","tex":"L_\\lambda^k","aliases":"L_λ^k lambda","display":"L_\\lambda^k","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-109fedbff9e0f183","tex":"L_{\\varnothing}","aliases":"L_∅ varnothing","display":"L_{\\varnothing}","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-345427490e88d80b","tex":"\\leadsto","aliases":"leadsto","display":null,"kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1cbd8c236499b7d","tex":"\\lim_{m\\to\\infty}","aliases":"lim_m→∞ lim to infty","display":"\\lim_{m\\to\\infty}","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5236b838b836f2fa","tex":"\\mathrm{loc}","aliases":"loc mathrm","display":"\\mathrm{loc}","kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15c8c86f6474c315","tex":"\\Log","aliases":"Log operatorname","display":"{\\operatorname{Log}}","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":127},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-838a4fcee43f8fdc","tex":"\\log_q","aliases":"log_q log","display":"\\log_q","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68a67f309b99ec73","tex":"\\longleftarrow","aliases":"longleftarrow","display":null,"kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"definition-passage","excerpt":"[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPa","count":55},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0042c6cf9ee0b72","tex":"\\mathfrak M","aliases":"M mathfrak","display":"\\mathfrak M","kind":"definition-passage","excerpt":"Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90569498b4dafa23","tex":"M^r","aliases":"M^r","display":"M^r","kind":"definition-passage","excerpt":"gives a canonical map from a finite free module. The generated hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70d3338e989b4ddb","tex":"\\mathcal M_{\\bar x}","aliases":"M_bar x mathcal bar","display":"\\mathcal M_{\\bar x}","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a35d6b08f661ad2","tex":"M_{\\frakp}","aliases":"M_frakp M_ p frakp mathfrak","display":"M_{{\\mathfrak p}}","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82964580614c7889","tex":"\\mathfrak M_R","aliases":"M_R mathfrak","display":"\\mathfrak M_R","kind":"definition-passage","excerpt":"Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f2828134dc54223","tex":"M_\\rho","aliases":"M_ρ rho","display":"M_\\rho","kind":"definition-passage","excerpt":"The multiplicity space is M_ρ=Hom_N(W, X).","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eae2d59e5b8a6aeb","tex":"M_\\infty","aliases":"M_∞ infty","display":"M_\\infty","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75ef01eda1abe09b","tex":"\\mathop","aliases":"mathop","display":null,"kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"definition-passage","excerpt":"et Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.","count":84},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4df46a14542b4b5","tex":"\\max_{0\\leq k<N}","aliases":"max_0≤ k<N max leq","display":"\\max_{0\\leq k<N}","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74868e115861830f","tex":"\\max_{a,b}","aliases":"max_a,b max","display":"\\max_{a,b}","kind":"definition-passage","excerpt":"[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No al","count":128},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"definition-passage","excerpt":"[Seed rank] For a unital left module M over A, define srank_A(M):=min{|S|:AS=M}. For a topological module (X,τ), define srank_ A,τ(X):=min{|S|:cl A S^ τ=X}. Set srank_A(0)=0. When the value is finite, it is the minimal number of module generators.","count":134},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2ca635f10eb764b","tex":"\\mathrm{mov}","aliases":"mov mathrm","display":"\\mathrm{mov}","kind":"definition-passage","excerpt":"[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N mathbb","display":"{\\mathbb N}","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":62},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":1807},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d620bbab2a49de9b","tex":"N_e","aliases":"N_e","display":"N_e","kind":"definition-passage","excerpt":"ef:filtered_algebra A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9727f5125c78ee28","tex":"n_\\alpha","aliases":"n_α alpha","display":"n_\\alpha","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afdfe01923a13ec5","tex":"N_\\delta","aliases":"N_δ delta","display":"N_\\delta","kind":"definition-passage","excerpt":"[Linear tube bound] A boundary Γ has a linear tube bound if there are constants C_Γ and δ_0>0 such that μ(N_δ(Γ)) ≤ C_Γδ for 0<δ<δ_0.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79992dda48a6b986","tex":"\\operatorname{node}","aliases":"node operatorname","display":"\\operatorname{node}","kind":"definition-passage","excerpt":"[Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.Colo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"definition-passage","excerpt":"[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":205},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob operatorname","display":"{\\operatorname{Ob}}","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":43},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-346371b58a75b52b","tex":"\\ob","aliases":"ob o mathfrak","display":"{\\mathfrak o}","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e77bbcefac3aa10","tex":"\\Ob^{\\mu\\nu}_{c,b,a}","aliases":"Ob^μν_c,b,a Ob mu nu operatorname mu nu","display":"{\\operatorname{Ob}}^{\\mu\\nu}_{c,b,a}","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83563918b0565094","tex":"\\Ob_{ab}^{\\rm cl}","aliases":"Ob_ab^rm cl Ob rm operatorname rm","display":"{\\operatorname{Ob}}_{ab}^{\\rm cl}","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9e7551bc92ef7bb","tex":"\\Ob_C","aliases":"Ob_C Ob operatorname","display":"{\\operatorname{Ob}}_C","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08aeff18e6189f5c","tex":"\\Ob_{c,b,a}","aliases":"Ob_c,b,a Ob operatorname","display":"{\\operatorname{Ob}}_{c,b,a}","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46c1284bb2e365e3","tex":"\\Ob_D","aliases":"Ob_D Ob operatorname","display":"{\\operatorname{Ob}}_D","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a85f28cb80e0a27b","tex":"\\Ob_m","aliases":"Ob_m Ob operatorname","display":"{\\operatorname{Ob}}_m","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4fa7ee730522dfa6","tex":"\\Ob_P","aliases":"Ob_P Ob operatorname","display":"{\\operatorname{Ob}}_P","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a11605d68f9137ee","tex":"\\ob_P","aliases":"ob_P o_P ob mathfrak","display":"{\\mathfrak o}_P","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-295577f481743cf7","tex":"\\Ocal","aliases":"Ocal O mathcal","display":"{\\mathcal O}","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":41},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f7264e76f55d278","tex":"\\mathrm{odd}","aliases":"odd mathrm","display":"\\mathrm{odd}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1277ceb9fb78b9f","tex":"\\one_{r+L}","aliases":"one_r+L 1_r+L one mathbf","display":"{\\mathbf 1}_{r+L}","kind":"definition-passage","excerpt":"[Lattice packet] For r∈ R, define the coordinate projector (P_rx)(n)= 1_r+L(n)x(n). The family (P_r)_r∈ R is called the packet system associated with L and R.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f35ce0db28d3fb4f","tex":"\\OO","aliases":"OO O mathcal","display":null,"kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":112},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3c541ce8e25de25","tex":"\\mathrm{orb}","aliases":"orb mathrm","display":"\\mathrm{orb}","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3323ae2114d44586","tex":"\\operatorname{out}","aliases":"out operatorname","display":"\\operatorname{out}","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4b45cd66b9c8f4e","tex":"\\overline L","aliases":"overline L overline","display":"\\overline L","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8000240f3988a998","tex":"\\overline L_m","aliases":"overline L_m overline","display":"\\overline L_m","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f1ae61172342775","tex":"\\overline\\lambda","aliases":"overlineλ overline lambda","display":"\\overline\\lambda","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f525677ebf3492f1","tex":"\\overline\\lambda_\\alpha","aliases":"overlineλ_α overline lambda alpha","display":"\\overline\\lambda_\\alpha","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1971},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80a406874da00307","tex":"p^{2m}","aliases":"p^2m","display":"p^{2m}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fbe5654b944a7373","tex":"p^a","aliases":"p^a","display":"p^a","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":33},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-704b220b0886246a","tex":"p^L","aliases":"p^L","display":"p^L","kind":"definition-passage","excerpt":"put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressi","count":86},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":138},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":66},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0b67a2140e399c4e","tex":"P_C","aliases":"P_C","display":"P_C","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":49},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f1efa218d5dcb4c","tex":"P_{N,n}","aliases":"P_N,n","display":"P_{N,n}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7cd951ba120bfdb9","tex":"P_u","aliases":"P_u","display":"P_u","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e70fd9df7027e2bf","tex":"P_v","aliases":"P_v","display":"P_v","kind":"definition-passage","excerpt":"[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the f","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af4f1e08915ee052","tex":"P_w","aliases":"P_w","display":"P_w","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-634bde01ca310a18","tex":"P_\\alpha^2","aliases":"P_α^2 alpha","display":"P_\\alpha^2","kind":"definition-passage","excerpt":"[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f7d96c56f31a733","tex":"P_{\\sigma^{-1}(j)}","aliases":"P_σ^-1(j) sigma","display":"P_{\\sigma^{-1}(j)}","kind":"definition-passage","excerpt":"orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-daaae6f128c21312","tex":"P_{\\ell,a}","aliases":"P_ℓ,a ell","display":"P_{\\ell,a}","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b36c818af7586966","tex":"P_{\\varnothing}","aliases":"P_∅ varnothing","display":"P_{\\varnothing}","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5782f4f00d6630c2","tex":"\\operatorname{packetMatrix}","aliases":"packetMatrix operatorname","display":"\\operatorname{packetMatrix}","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komo","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcb34d30cfc37173","tex":"\\Pcal_{ab}","aliases":"Pcal_ab P_ab Pcal mathcal","display":"{\\mathcal P}_{ab}","kind":"definition-passage","excerpt":"[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1493c2c12981b881","tex":"\\perpbig","aliases":"perpbig mathop⊥limits mathop perp limits","display":null,"kind":"definition-passage","excerpt":"[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d783edf27173a33","tex":"\\perpbig_{s\\in S}","aliases":"perpbig_s∈ S mathop⊥limits_s∈ S perpbig in mathop perp limits in","display":null,"kind":"definition-passage","excerpt":"[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a7d755cf9c10f7b","tex":"\\mathrm{proof}","aliases":"proof mathrm","display":"\\mathrm{proof}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e7be1547abcd294","tex":"\\mathsf{PSC}","aliases":"PSC mathsf","display":"\\mathsf{PSC}","kind":"definition-passage","excerpt":"[Chapter I export object] The canonical Chapter I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49925054439766d9","tex":"\\mathsf{PSC}_I","aliases":"PSC_I mathsf","display":"\\mathsf{PSC}_I","kind":"definition-passage","excerpt":"[Chapter I export object] The canonical Chapter I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb103a3f56bb89da","tex":"\\psrank_{\\A,\\Pcal}","aliases":"psrank_A,Pcal psrank_ A, P psrank A Pcal operatorname mathcal mathcal","display":"{\\operatorname{psrank}}_{{\\mathcal A},{\\mathcal P}}","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28465b6199aa1e59","tex":"\\psrank_{\\A_C,\\Pcal|_C}","aliases":"psrank_A_C,Pcal|_C psrank_ A_C, P|_C psrank A Pcal operatorname mathcal mathcal","display":"{\\operatorname{psrank}}_{{\\mathcal A}_C,{\\mathcal P}|_C}","kind":"definition-passage","excerpt":"f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":30},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff7eb8f0d00f446f","tex":"\\mathfrak Q","aliases":"Q mathfrak","display":"\\mathfrak Q","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6339057791df9dbc","tex":"q^{-1}","aliases":"q^-1","display":"q^{-1}","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9c016afb8932679","tex":"q^{-1/2}","aliases":"q^-1/2","display":"q^{-1/2}","kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32b32661a7d9b468","tex":"q_{\\mathfrak c}","aliases":"q_ c mathfrak","display":"q_{\\mathfrak c}","kind":"definition-passage","excerpt":"[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"definition-passage","excerpt":"[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps ","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"definition-passage","excerpt":"[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"definition-passage","excerpt":"orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber","count":35},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd98a9c47ecfa404","tex":"\\mathbb Q_p","aliases":"Q_p mathbb","display":"\\mathbb Q_p","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0db94dc2cdc9bd9f","tex":"q_P","aliases":"q_P","display":"q_P","kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea3d9f514fcaceb6","tex":"Q_\\alpha","aliases":"Q_α alpha","display":"Q_\\alpha","kind":"definition-passage","excerpt":"[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2f8225af58ea4bd","tex":"Q_\\beta","aliases":"Q_β beta","display":"Q_\\beta","kind":"definition-passage","excerpt":"[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).","count":30},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a423e1823849ed7","tex":"Q_\\gamma","aliases":"Q_γ gamma","display":"Q_\\gamma","kind":"definition-passage","excerpt":"[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.","count":29},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aaf174f6701cc732","tex":"\\Qcal","aliases":"Qcal Q mathcal","display":"{\\mathcal Q}","kind":"definition-passage","excerpt":"[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):","count":70},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a94f04a221b67c7","tex":"R^{\\mathbf L}","aliases":"R^ L mathbf","display":"R^{\\mathbf L}","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ed766ea53d3d411","tex":"R^{\\mathrm{loc}}","aliases":"R^loc mathrm","display":"R^{\\mathrm{loc}}","kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9bf672601602601","tex":"R^{\\mathrm{loc},\\tau}","aliases":"R^loc,τ mathrm tau","display":"R^{\\mathrm{loc},\\tau}","kind":"definition-passage","excerpt":"every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squarem","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd279b65249d7e1f","tex":"R^\\square","aliases":"R^square square","display":"R^\\square","kind":"definition-passage","excerpt":"l deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a383171148f13ce0","tex":"R^{\\tau}","aliases":"R^τ tau","display":"R^{\\tau}","kind":"definition-passage","excerpt":"every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squarem","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39303e93288f0aca","tex":"r_C","aliases":"r_C","display":"r_C","kind":"definition-passage","excerpt":"f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b187bd136ca926c2","tex":"R_{n,m}","aliases":"R_n,m","display":"R_{n,m}","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3718ad5d081ab92","tex":"R_v^\\square","aliases":"R_v^square square","display":"R_v^\\square","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ffbbcc5c7c165f2","tex":"R_v^{\\tau_v}","aliases":"R_v^τ_v tau","display":"R_v^{\\tau_v}","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af3bb9ba2ca7c7af","tex":"r_{\\alpha,i}","aliases":"r_α,i alpha","display":"r_{\\alpha,i}","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81ae26f5145c949e","tex":"R_\\ell","aliases":"R_ℓ ell","display":"R_\\ell","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1903894d14878533","tex":"R_\\infty","aliases":"R_∞ infty","display":"R_\\infty","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-650ae3274f788b89","tex":"\\rad","aliases":"rad operatorname","display":"{\\operatorname{rad}}","kind":"definition-passage","excerpt":"[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff81ba5e23d549c7","tex":"\\Re","aliases":"Re","display":"\\Re","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fdf44047d9358ee","tex":"\\mathrm{red}","aliases":"red mathrm","display":"\\mathrm{red}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b0150d4ec261493","tex":"\\red","aliases":"red operatorname","display":"{\\operatorname{red}}","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-238b4e38e4a9a70d","tex":"\\red_A","aliases":"red_A red operatorname","display":"{\\operatorname{red}}_A","kind":"definition-passage","excerpt":"[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"definition-passage","excerpt":"[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":68},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a62a596621079394","tex":"\\RR","aliases":"RR R mathbb","display":"{\\mathbb R}","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-abdd5a30d95b7315","tex":"\\RR^m","aliases":"RR^m R^m RR mathbb","display":"{\\mathbb R}^m","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b617b296bbfd924b","tex":"\\mathfrak S","aliases":"S mathfrak","display":"\\mathfrak S","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"definition-passage","excerpt":"[Morphism of seeded systems] A morphism (X, A,S, P)⟶(Y, B,S', Q) is a packet compatible operator morphism satisfying the seed condition Φ(S)⊆Hull_ B(S'). In the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7305573ca2e697d","tex":"S^*","aliases":"S^*","display":"S^*","kind":"definition-passage","excerpt":"[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.","count":25},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b8319ff964bd555","tex":"S_{2(n-m)}","aliases":"S_2(n-m)","display":"S_{2(n-m)}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-508e9a069e81ec8b","tex":"S_d","aliases":"S_d","display":"S_d","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57c446e8e6d068a3","tex":"S_m","aliases":"S_m","display":"S_m","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6bf5eb7ecfc145e","tex":"s_r","aliases":"s_r","display":"s_r","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-152abbc9214c8600","tex":"S_\\lambda","aliases":"S_λ lambda","display":"S_\\lambda","kind":"definition-passage","excerpt":"[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b36510bbfb3bd3c","tex":"s_\\Lambda","aliases":"s_Λ Lambda","display":"s_\\Lambda","kind":"definition-passage","excerpt":"et V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O modul","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3206060e514ec62e","tex":"\\Scal_{\\Pcal}","aliases":"Scal_Pcal S_ P Scal Pcal mathcal mathcal","display":"{\\mathcal S}_{{\\mathcal P}}","kind":"definition-passage","excerpt":"ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-13116bc57b9f1d4e","tex":"\\Scal_{\\Qcal}","aliases":"Scal_Qcal S_ Q Scal Qcal mathcal mathcal","display":"{\\mathcal S}_{{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d39429b9e78430d","tex":"\\operatorname{sech}","aliases":"sech operatorname","display":"\\operatorname{sech}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":32},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2cd755c944ccda0","tex":"\\operatorname{sech}^2","aliases":"sech^2 operatorname","display":"\\operatorname{sech}^2","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0c86daacd8ccb79","tex":"\\mathrm{seed}","aliases":"seed mathrm","display":"\\mathrm{seed}","kind":"definition-passage","excerpt":"[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-64e8f7a42ac935d4","tex":"\\mathbf{SeedSys}","aliases":"SeedSys mathbf","display":"\\mathbf{SeedSys}","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5faac4b7fa18af10","tex":"\\mathbf{SeedSys}_{\\mathrm{abs}}","aliases":"SeedSys_abs mathbf mathrm","display":"\\mathbf{SeedSys}_{\\mathrm{abs}}","kind":"definition-passage","excerpt":"[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5fe98e4f30da8940","tex":"\\Sfun","aliases":"Sfun S mathsf","display":"{\\mathsf S}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d87f4ba0b84c237a","tex":"\\Sfun_N^{(r)}","aliases":"Sfun_N^(r) S_N^(r) Sfun mathsf","display":"{\\mathsf S}_N^{(r)}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21670bd4e275805e","tex":"\\sinh","aliases":"sinh","display":"\\sinh","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":36},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-327dcbd8f0058da9","tex":"\\Spec","aliases":"Spec operatorname","display":"{\\operatorname{Spec}}","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":45},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36a922b7cae66331","tex":"\\mathrm{split}","aliases":"split mathrm","display":"\\mathrm{split}","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0837c3271213f2f","tex":"\\srank_{\\A,\\tau}","aliases":"srank_A,τ srank_ A,τ srank A tau operatorname mathcal tau","display":"{\\operatorname{srank}}_{{\\mathcal A},\\tau}","kind":"definition-passage","excerpt":"[Seed rank] For a unital left module M over A, define srank_A(M):=min{|S|:AS=M}. For a topological module (X,τ), define srank_ A,τ(X):=min{|S|:cl A S^ τ=X}. Set srank_A(0)=0. When the value is finite, it is the minimal number of module generators.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31658be192e9af55","tex":"\\srank_G","aliases":"srank_G srank operatorname","display":"{\\operatorname{srank}}_G","kind":"definition-passage","excerpt":"[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-259ced97b3ec7ffb","tex":"\\src","aliases":"src operatorname","display":"{\\operatorname{src}}","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4284d18f422a1f0d","tex":"\\mathrm{statement}","aliases":"statement mathrm","display":"\\mathrm{statement}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02a3d6cf5b723f6c","tex":"\\mathrm{status}","aliases":"status mathrm","display":"\\mathrm{status}","kind":"definition-passage","excerpt":"[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58e6e44b43884fcc","tex":"\\Subset","aliases":"Subset","display":null,"kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":37},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1bf00c1871dc54e","tex":"\\sup_{F\\Subset\\mathbb N}","aliases":"sup_FSubset N sup Subset mathbb","display":null,"kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53c867f72ca37df5","tex":"\\sup_{|a_n|\\leq1}","aliases":"sup_|a_n|≤1 sup leq","display":"\\sup_{|a_n|\\leq1}","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed9ed110cf06d528","tex":"\\Supp_R","aliases":"Supp_R Supp operatorname","display":"{\\operatorname{Supp}}_R","kind":"definition-passage","excerpt":"[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62afa297a68a33f1","tex":"\\Supp_{R_\\infty}","aliases":"Supp_R_∞ Supp infty operatorname infty","display":"{\\operatorname{Supp}}_{R_\\infty}","kind":"definition-passage","excerpt":"datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=S","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da4fb737a4a4bc72","tex":"\\mathfrak T","aliases":"T mathfrak","display":"\\mathfrak T","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3a6488b52c152f1","tex":"T'_{vu}","aliases":"T'_vu","display":"T'_{vu}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b020d6ed8cb1fa22","tex":"t^{2m}","aliases":"t^2m","display":"t^{2m}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f544e1fddefb59a0","tex":"t^d","aliases":"t^d","display":"t^d","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54815db7d2265570","tex":"T^m","aliases":"T^m","display":"T^m","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50c3c8d08b407ee6","tex":"t^{Nn}","aliases":"t^Nn","display":"t^{Nn}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc681522104c7ab2","tex":"T^{\\mathrm{odd}}_{p,M}","aliases":"T^odd_p,M mathrm","display":"T^{\\mathrm{odd}}_{p,M}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c82494b5f8dadd9","tex":"t^{w(e)}","aliases":"t^w(e)","display":"t^{w(e)}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No al","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d08324acb961fcf","tex":"t_2","aliases":"t_2","display":"t_2","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No al","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ef43bcf3d114a62","tex":"\\mathcal T_A","aliases":"T_A mathcal","display":"\\mathcal T_A","kind":"definition-passage","excerpt":"[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-911d3e6f76681614","tex":"T_{a+b}","aliases":"T_a+b","display":"T_{a+b}","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cefa01be96058c3e","tex":"T_b","aliases":"T_b","display":"T_b","kind":"definition-passage","excerpt":"[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14cf325be6479cb0","tex":"T_M","aliases":"T_M","display":"T_M","kind":"definition-passage","excerpt":"n] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4505604ff161e9a8","tex":"T_N","aliases":"T_N","display":"T_N","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-64afaf0ed6cbcadc","tex":"T_n","aliases":"T_n","display":"T_n","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-426c2e2ef16a6555","tex":"\\mathfrak T_N","aliases":"T_N mathfrak","display":"\\mathfrak T_N","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-541d33cb655a649a","tex":"t_{r(\\sigma)}","aliases":"t_r(σ) sigma","display":"t_{r(\\sigma)}","kind":"definition-passage","excerpt":"[Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.Colo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-014a85fddb5fe786","tex":"T_{vu}","aliases":"T_vu","display":"T_{vu}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35899e3c99220253","tex":"T_\\ell","aliases":"T_ℓ ell","display":"T_\\ell","kind":"definition-passage","excerpt":"[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is record","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46a3ec781ec50559","tex":"\\Tfun","aliases":"Tfun T mathsf","display":"{\\mathsf T}","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3731375084c54cb","tex":"\\Tfun_N","aliases":"Tfun_N T_N Tfun mathsf","display":"{\\mathsf T}_N","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b74a640d5a7a8150","tex":"\\tgt","aliases":"tgt operatorname","display":"{\\operatorname{tgt}}","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4b881f851b3f1df","tex":"\\Thetafun","aliases":"Thetafun symupΘ symup Theta","display":null,"kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40eef48472d4771a","tex":"\\Thetafun_{N,n}","aliases":"Thetafun_N,n symupΘ_N,n Thetafun symup Theta","display":null,"kind":"definition-passage","excerpt":"Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr Tr operatorname","display":null,"kind":"definition-passage","excerpt":"[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.","count":49},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65fa7282e2650c3e","tex":"\\triangleleft","aliases":"triangleleft","display":null,"kind":"definition-passage","excerpt":"[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45d47d344acaa5d2","tex":"\\TT","aliases":"TT T mathbb","display":"{\\mathbb T}","kind":"definition-passage","excerpt":"[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic zero locu","count":60},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t)","count":562},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":660},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df5c58f22a65049f","tex":"u^{-1}","aliases":"u^-1","display":"u^{-1}","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32f66856145f390c","tex":"U_{m,n}","aliases":"U_m,n","display":"U_{m,n}","kind":"definition-passage","excerpt":"[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03cfa39a8fd5efa1","tex":"U_{N,j}","aliases":"U_N,j","display":"U_{N,j}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b6b2cac1ed2c1b6","tex":"U_v","aliases":"U_v","display":"U_v","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7884e8dc4c0191e","tex":"\\Ucal","aliases":"Ucal U mathcal","display":"{\\mathcal U}","kind":"definition-passage","excerpt":"[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24e7f443fb5fea87","tex":"\\operatorname{unary}","aliases":"unary operatorname","display":"\\operatorname{unary}","kind":"definition-passage","excerpt":"[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No al","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"rojector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem","count":490},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈","count":794},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"definition-passage","excerpt":"[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.","count":27},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":49},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f908e7a2d9e05299","tex":"V_3","aliases":"V_3","display":"V_3","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-005b50f5c6f12245","tex":"V_{j-1}","aliases":"V_j-1","display":"V_{j-1}","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b86c046f5cd21ee","tex":"V_m","aliases":"V_m","display":"V_m","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b00eda2d2588cd09","tex":"v_m","aliases":"v_m","display":"v_m","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f788e25d2802bd1e","tex":"V_{m-1}","aliases":"V_m-1","display":"V_{m-1}","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36a6d81c2aecf99f","tex":"v_{m-1}","aliases":"v_m-1","display":"v_{m-1}","kind":"definition-passage","excerpt":"[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6735e9f71ad2ebf","tex":"V_n","aliases":"V_n","display":"V_n","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":42},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5df6cd07cd936c95","tex":"V_s","aliases":"V_s","display":"V_s","kind":"definition-passage","excerpt":"[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92becdaca2dd0d19","tex":"V_\\alpha","aliases":"V_α alpha","display":"V_\\alpha","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2db1cdbbe7192b99","tex":"V_\\theta","aliases":"V_θ theta","display":"V_\\theta","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25aba103b9e1f48f","tex":"V_\\rho","aliases":"V_ρ rho","display":"V_\\rho","kind":"definition-passage","excerpt":"[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50f104bbfd38386a","tex":"\\val","aliases":"val operatorname","display":"{\\operatorname{val}}","kind":"definition-passage","excerpt":"ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. Th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-949ab90c1fcd9c3d","tex":"\\val_\\pi","aliases":"val_π val pi operatorname pi","display":"{\\operatorname{val}}_\\pi","kind":"definition-passage","excerpt":"ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. Th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30a9a914ab13bf7a","tex":"\\varprojlim_L","aliases":"varprojlim_L varprojlim","display":null,"kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50ce1a62663b77d9","tex":"\\Vert","aliases":"Vert","display":"\\Vert","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b30f1f16c8acc2ca","tex":"\\bm w","aliases":"w bm","display":"\\bm w","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6cd4c939f7d7c61f","tex":"w^m","aliases":"w^m","display":"w^m","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6f16f2e02ae0bf2","tex":"w^n","aliases":"w^n","display":"w^n","kind":"definition-passage","excerpt":"Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34d6de1dc85d2727","tex":"W^{r,p}","aliases":"W^r,p","display":"W^{r,p}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19ce956d274d340e","tex":"w_{e,k}","aliases":"w_e,k","display":"w_{e,k}","kind":"definition-passage","excerpt":"[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8174d5fba0700d5c","tex":"\\wideotimes","aliases":"wideotimes mathbinwidehat⊗ mathbin widehat otimes","display":null,"kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c10f72797a18708f","tex":"\\wideotimes_{R^{\\mathrm{loc}}}","aliases":"wideotimes_R^loc mathbinwidehat⊗_R^loc wideotimes mathrm mathbin widehat otimes mathrm","display":null,"kind":"definition-passage","excerpt":"every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squarem","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cfe1eab82fe0f384","tex":"\\wideotimes_{v\\in S,\\OO}","aliases":"wideotimes_v∈ S,OO mathbinwidehat⊗_v∈ S, O wideotimes in OO mathbin widehat otimes in mathcal","display":null,"kind":"definition-passage","excerpt":"m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=m","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aeb4dc50b8d5ec9e","tex":"X^{\\Pcal}","aliases":"X^Pcal X^ P Pcal mathcal","display":"X^{{\\mathcal P}}","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f0de09885820b11","tex":"X^{\\Qcal}","aliases":"X^Qcal X^ Q Qcal mathcal","display":"X^{{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51e189ae43085a7d","tex":"X_a","aliases":"X_a","display":"X_a","kind":"definition-passage","excerpt":"(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e7e08756fa886133","tex":"X_C","aliases":"X_C","display":"X_C","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc360874bad2510d","tex":"X_{\\operatorname{in}_\\sigma(q)}","aliases":"X_in_σ(q) operatorname sigma","display":"X_{\\operatorname{in}_\\sigma(q)}","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be1f7209628ae76c","tex":"X_{j+1}","aliases":"X_j+1","display":"X_{j+1}","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0430cf91ce15859e","tex":"x_{j+1}","aliases":"x_j+1","display":"x_{j+1}","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3be737b42076753","tex":"X_m","aliases":"X_m","display":"X_m","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"definition-passage","excerpt":"[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.","count":41},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3f245481574d88d","tex":"X_{\\operatorname{out}(\\sigma)}","aliases":"X_out(σ) operatorname sigma","display":"X_{\\operatorname{out}(\\sigma)}","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6fa3eb7a793d7e6","tex":"X_{s(a)}","aliases":"X_s(a)","display":"X_{s(a)}","kind":"definition-passage","excerpt":"(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39e2cb7b9e9ab6bc","tex":"X_{t(a)}","aliases":"X_t(a)","display":"X_{t(a)}","kind":"definition-passage","excerpt":"(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08a6f9f108660d1a","tex":"X_v","aliases":"X_v","display":"X_v","kind":"definition-passage","excerpt":"r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eeab853c3f10dfb2","tex":"X_{\\alpha_1}","aliases":"X_α_1 alpha","display":"X_{\\alpha_1}","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c9c5408fed893c7","tex":"X_{\\alpha_r}","aliases":"X_α_r alpha","display":"X_{\\alpha_r}","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59ca56eb5c1b813d","tex":"X_{\\lambda,1}","aliases":"X_λ,1 lambda","display":"X_{\\lambda,1}","kind":"definition-passage","excerpt":"[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finit","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d9edf14d558205f","tex":"X_{\\lambda,2}","aliases":"X_λ,2 lambda","display":"X_{\\lambda,2}","kind":"definition-passage","excerpt":"[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finit","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2312717706b5a2a","tex":"X_{\\lambda,3}","aliases":"X_λ,3 lambda","display":"X_{\\lambda,3}","kind":"definition-passage","excerpt":"[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finit","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-449cba2c3125b122","tex":"X_{\\lambda,m}","aliases":"X_λ,m lambda","display":"X_{\\lambda,m}","kind":"definition-passage","excerpt":"ic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of adm","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bfd71ea41b57ecef","tex":"X_\\infty","aliases":"X_∞ infty","display":"X_\\infty","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"k G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear ","count":253},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"definition-passage","excerpt":"[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komo","count":497},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-536459ef72a0484d","tex":"Y_a","aliases":"Y_a","display":"Y_a","kind":"definition-passage","excerpt":"linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7256bbf22fa35eec","tex":"Y_m","aliases":"Y_m","display":"Y_m","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58a7d551ec1df48c","tex":"Y_v","aliases":"Y_v","display":"Y_v","kind":"definition-passage","excerpt":"l on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f42e259364006691","tex":"Y_\\gamma","aliases":"Y_γ gamma","display":"Y_\\gamma","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z mathbb Z mathbf","display":null,"kind":"definition-passage","excerpt":"put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressi","count":191},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"definition-passage","excerpt":"bra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.","count":187},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df0285eec4b01028","tex":"\\mathfrak Z","aliases":"Z mathfrak","display":"\\mathfrak Z","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a87f7202de83de58","tex":"z^{2m}","aliases":"z^2m","display":"z^{2m}","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"definition-passage","excerpt":"For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56835621b2f38901","tex":"z^m","aliases":"z^m","display":"z^m","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afc30c788aac6f77","tex":"z^N","aliases":"z^N","display":"z^N","kind":"definition-passage","excerpt":"Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).","count":35},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae6f7a635f6f6ddb","tex":"z^{Nn}","aliases":"z^Nn","display":"z^{Nn}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9f1f546f3ae0830","tex":"z^{Nn+j}","aliases":"z^Nn+j","display":"z^{Nn+j}","kind":"definition-passage","excerpt":"For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-676769c66002f30a","tex":"z^{Nn+N-1}","aliases":"z^Nn+N-1","display":"z^{Nn+N-1}","kind":"definition-passage","excerpt":"Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a04e13cebbe7a0ff","tex":"Z_{\\bm w}","aliases":"Z_ w bm","display":"Z_{\\bm w}","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57105cdc4119a19b","tex":"\\mathbb Z_{>0}","aliases":"Z_>0 mathbb","display":"\\mathbb Z_{>0}","kind":"definition-passage","excerpt":"bra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d74d92bf91d59a7f","tex":"Z_{\\bar x}","aliases":"Z_bar x bar","display":"Z_{\\bar x}","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06faf04153e7762d","tex":"Z_L","aliases":"Z_L","display":"Z_L","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3785fea000ba52f0","tex":"\\mathfrak Z_{N,q}^{\\mathrm{der}}","aliases":"Z_N,q^der mathfrak mathrm","display":"\\mathfrak Z_{N,q}^{\\mathrm{der}}","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03ac7c9641211143","tex":"\\mathfrak Z_N^{\\mathrm{abs}}","aliases":"Z_N^abs mathfrak mathrm","display":"\\mathfrak Z_N^{\\mathrm{abs}}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c06e3e1fda1dbfa","tex":"\\mathfrak Z_N^{\\mathrm{orb}}","aliases":"Z_N^orb mathfrak mathrm","display":"\\mathfrak Z_N^{\\mathrm{orb}}","kind":"definition-passage","excerpt":"Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fca16336850bcc9e","tex":"\\mathfrak Z_N^{\\mathrm{red}}","aliases":"Z_N^red mathfrak mathrm","display":"\\mathfrak Z_N^{\\mathrm{red}}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b910972bdb78c268","tex":"\\Z_p","aliases":"Z_p Z mathbb Z_p Z mathbf","display":null,"kind":"definition-passage","excerpt":"put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressi","count":56},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8723c894d5150ea2","tex":"\\mathbb Z_p","aliases":"Z_p mathbb","display":"\\mathbb Z_p","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":34},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b41db94f18e314fe","tex":"Z_{u,v}","aliases":"Z_u,v","display":"Z_{u,v}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec84f843cac321e2","tex":"\\Zcal","aliases":"Zcal Z mathcal","display":"{\\mathcal Z}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8a971f413dd4d59","tex":"|^{-s}","aliases":"|^-s","display":"|^{-s}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9021f7af081c388c","tex":"|_C","aliases":"|_C","display":"|_C","kind":"definition-passage","excerpt":"f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1093b5d3cc81a57a","tex":"|_{\\im(g-I)}","aliases":"|_im(g-I) im operatorname","display":"|_{{\\operatorname{im}}(g-I)}","kind":"definition-passage","excerpt":"[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca558d2d7c53350d","tex":"|_{\\Ker C}","aliases":"|_Ker C Ker operatorname","display":"|_{{\\operatorname{Ker}} C}","kind":"definition-passage","excerpt":"[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a283f3826c3c4719","tex":"|_{X_{\\alpha_1}\\times\\cdots\\times X_{\\alpha_r}}","aliases":"|_X_α_1×⋯× X_α_r alpha times cdots times alpha","display":"|_{X_{\\alpha_1}\\times\\cdots\\times X_{\\alpha_r}}","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.","count":308},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af15262773600a7f","tex":"\\alpha_r","aliases":"α_r alpha","display":"\\alpha_r","kind":"definition-passage","excerpt":"[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24182df1bf8329b9","tex":"\\beta_0","aliases":"β_0 beta","display":"\\beta_0","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6de2171610272b5","tex":"\\beta_m","aliases":"β_m beta","display":"\\beta_m","kind":"definition-passage","excerpt":"[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cdad47e19a781b2f","tex":"\\Gamma^G_{b\\leftarrow a}","aliases":"Γ^G_b← a Gamma leftarrow","display":"\\Gamma^G_{b\\leftarrow a}","kind":"definition-passage","excerpt":"[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2174981cd778e96c","tex":"\\Gamma_{(a_0)}","aliases":"Γ_(a_0) Gamma","display":"\\Gamma_{(a_0)}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2771b61f3f3f5974","tex":"\\gamma_0","aliases":"γ_0 gamma","display":"\\gamma_0","kind":"definition-passage","excerpt":"-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be od","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14b9f6147ed3eea3","tex":"\\Gamma_{a_1\\leftarrow a_0}","aliases":"Γ_a_1← a_0 Gamma leftarrow","display":"\\Gamma_{a_1\\leftarrow a_0}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d820849730eca26b","tex":"\\Gamma_{a_n\\leftarrow a_{n-1}}","aliases":"Γ_a_n← a_n-1 Gamma leftarrow","display":"\\Gamma_{a_n\\leftarrow a_{n-1}}","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fba48f530144df83","tex":"\\Gamma_{a\\leftarrow a}","aliases":"Γ_a← a Gamma leftarrow","display":"\\Gamma_{a\\leftarrow a}","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32b19bb3a15c41fa","tex":"\\Gamma_{a\\leftarrow b}","aliases":"Γ_a← b Gamma leftarrow","display":"\\Gamma_{a\\leftarrow b}","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e51268d4f9664f40","tex":"\\Gamma_{a\\leftarrow c}","aliases":"Γ_a← c Gamma leftarrow","display":"\\Gamma_{a\\leftarrow c}","kind":"definition-passage","excerpt":"[Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a75708c6ecb08f0","tex":"\\Gamma_{b\\leftarrow a}","aliases":"Γ_b← a Gamma leftarrow","display":"\\Gamma_{b\\leftarrow a}","kind":"definition-passage","excerpt":"[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-667fc2f1be68ffcf","tex":"\\Gamma_{b\\leftarrow a}^{-1}","aliases":"Γ_b← a^-1 Gamma leftarrow","display":"\\Gamma_{b\\leftarrow a}^{-1}","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b75c5680cc807310","tex":"\\Gamma_{c\\leftarrow a}","aliases":"Γ_c← a Gamma leftarrow","display":"\\Gamma_{c\\leftarrow a}","kind":"definition-passage","excerpt":"[Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. s","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-accadba9f0371bc9","tex":"\\Gamma_{c\\leftarrow b}","aliases":"Γ_c← b Gamma leftarrow","display":"\\Gamma_{c\\leftarrow b}","kind":"definition-passage","excerpt":"[Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. s","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27c5ee666d44751e","tex":"\\gamma_m","aliases":"γ_m gamma","display":"\\gamma_m","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-465089d5be891c7b","tex":"\\Gamma_p","aliases":"Γ_p Gamma","display":"\\Gamma_p","kind":"definition-passage","excerpt":"[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transp","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8d84b94d64038ea","tex":"\\Gamma_{\\rm eff}","aliases":"Γ_rm eff Gamma rm","display":"\\Gamma_{\\rm eff}","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ea06baff08da99a","tex":"\\Delta_{c,b,a}","aliases":"Δ_c,b,a Delta","display":"\\Delta_{c,b,a}","kind":"definition-passage","excerpt":"[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9e14435a4d85c517","tex":"\\Delta_F","aliases":"Δ_F Delta","display":"\\Delta_F","kind":"definition-passage","excerpt":"[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-608581f0be0c4b00","tex":"\\Delta_h","aliases":"Δ_h Delta","display":"\\Delta_h","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-94a8068a3d828e7e","tex":"\\Delta_h^0","aliases":"Δ_h^0 Delta","display":"\\Delta_h^0","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9e333eeae38a06fd","tex":"\\Delta_h^k","aliases":"Δ_h^k Delta","display":"\\Delta_h^k","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5e835b6840b45f1","tex":"\\Delta_h^{k+1}","aliases":"Δ_h^k+1 Delta","display":"\\Delta_h^{k+1}","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51020088f18e62aa","tex":"\\Delta_N","aliases":"Δ_N Delta","display":"\\Delta_N","kind":"definition-passage","excerpt":"Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).","count":51},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":55},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":185},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4de839f8cd0339b6","tex":"\\varepsilon_{\\rm KL}","aliases":"ε_rm KL varepsilon rm","display":"\\varepsilon_{\\rm KL}","kind":"definition-passage","excerpt":"[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4330ec1682dc6696","tex":"\\varepsilon_S","aliases":"ε_S varepsilon","display":"\\varepsilon_S","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6991c3ad9fdd14c","tex":"\\epsilon_v","aliases":"ϵ_v epsilon","display":"\\epsilon_v","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":132},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-412b907de59c5fc0","tex":"\\eta_i","aliases":"η_i eta","display":"\\eta_i","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"definition-passage","excerpt":"For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.","count":136},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":38},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-300f133d4213a5fd","tex":"\\iota_\\alpha","aliases":"ι_α iota alpha","display":"\\iota_\\alpha","kind":"definition-passage","excerpt":"[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c039735a9807331b","tex":"\\iota_\\beta","aliases":"ι_β iota beta","display":"\\iota_\\beta","kind":"definition-passage","excerpt":"IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain, and ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"definition-passage","excerpt":"[Packet coordinate condition number] Fix a finite product norm on ⊕_α X_α equivalent to the maximum norm, as in [source reference]. Define κ( P):= A_ P S_ P .","count":129},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc6e9564c2f2a99a","tex":"\\kappa_{N,m}","aliases":"κ_N,m kappa","display":"\\kappa_{N,m}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd0a204d0362614f","tex":"\\kappa_v","aliases":"κ_v kappa","display":"\\kappa_v","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.","count":535},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e5fde598f577a45","tex":"\\lambda^{-m}","aliases":"λ^-m lambda","display":"\\lambda^{-m}","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f9faba9cf67ca67","tex":"\\lambda^m","aliases":"λ^m lambda","display":"\\lambda^m","kind":"definition-passage","excerpt":"[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finit","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":47},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-036d06d495662f6f","tex":"\\Lambda_N","aliases":"Λ_N Lambda","display":"\\Lambda_N","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57d6bd6085b620be","tex":"\\lambda_N","aliases":"λ_N lambda","display":"\\lambda_N","kind":"definition-passage","excerpt":"[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-adef364156481ffd","tex":"\\Lambda_{\\mathrm{split}}","aliases":"Λ_split Lambda mathrm","display":"\\Lambda_{\\mathrm{split}}","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e2977311c515de6","tex":"\\Lambda_\\alpha","aliases":"Λ_α Lambda alpha","display":"\\Lambda_\\alpha","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":155},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":75},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62d0d50e0643749d","tex":"\\nu_p","aliases":"ν_p nu","display":"\\nu_p","kind":"definition-passage","excerpt":"d Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":41},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eae7a938d6364e6e","tex":"\\xi^N","aliases":"ξ^N xi","display":"\\xi^N","kind":"definition-passage","excerpt":"Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":27},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb28cf0122b8897d","tex":"\\varpi","aliases":"ϖ varpi","display":"\\varpi","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":29},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e64c3229ee2a707a","tex":"\\Pi^a_\\mu","aliases":"Π^a_μ Pi mu","display":"\\Pi^a_\\mu","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0871b98c3b226a2","tex":"\\Pi^a_\\nu","aliases":"Π^a_ν Pi nu","display":"\\Pi^a_\\nu","kind":"definition-passage","excerpt":"[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy withi","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bff53c563d0671f","tex":"\\varpi^m","aliases":"ϖ^m varpi","display":"\\varpi^m","kind":"definition-passage","excerpt":"[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75be53b3397893ea","tex":"\\pi^s","aliases":"π^s pi","display":"\\pi^s","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd82674cd48e7508","tex":"\\pi_\\alpha","aliases":"π_α pi alpha","display":"\\pi_\\alpha","kind":"definition-passage","excerpt":"gindefinition[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6006a1a7260e2f47","tex":"\\varpi_\\lambda^m","aliases":"ϖ_λ^m varpi lambda","display":"\\varpi_\\lambda^m","kind":"definition-passage","excerpt":"at consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduct","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).","count":335},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8239c91a9dd24610","tex":"\\rho_R","aliases":"ρ_R rho","display":"\\rho_R","kind":"definition-passage","excerpt":"[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automo","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d903684d8cb76c26","tex":"\\rho_x","aliases":"ρ_x rho","display":"\\rho_x","kind":"definition-passage","excerpt":"phism. item An automorphic Hecke datum ( T,H,(P_v)) and a pointwise classicality and Galois realization theorem satisfying all the inputs of [source reference] for the nonzero fiber at x: an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that re","count":38},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"definition-passage","excerpt":"[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ","count":40},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):","count":145},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6512b523961a05f5","tex":"\\sigma^{-1}","aliases":"σ^-1 sigma","display":"\\sigma^{-1}","kind":"definition-passage","excerpt":"orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62f366adedc0cbc7","tex":"\\sigma_{N,m}","aliases":"σ_N,m sigma","display":"\\sigma_{N,m}","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7702fce301ebe3bf","tex":"\\sigma_v","aliases":"σ_v sigma","display":"\\sigma_v","kind":"definition-passage","excerpt":"[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c811b6eab2988ab","tex":"\\tau_1","aliases":"τ_1 tau","display":"\\tau_1","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e99ad6780db1a41","tex":"\\tau_Q","aliases":"τ_Q tau","display":"\\tau_Q","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c57f88e1e69525bd","tex":"\\tau_q","aliases":"τ_q tau","display":"\\tau_q","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca0868865309284b","tex":"\\tau_v","aliases":"τ_v tau","display":"\\tau_v","kind":"definition-passage","excerpt":"[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a cont","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.","count":233},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"definition-passage","excerpt":"[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):","count":59},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25914e63a3f23d0d","tex":"\\phi_{0,\\tau_q}","aliases":"ϕ_0,τ_q phi tau","display":"\\phi_{0,\\tau_q}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8edece78cbd674e4","tex":"\\Phi_{N,j}","aliases":"Φ_N,j Phi","display":"\\Phi_{N,j}","kind":"definition-passage","excerpt":"For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b146fd938e264b6","tex":"\\Phi_p","aliases":"Φ_p Phi","display":"\\Phi_p","kind":"definition-passage","excerpt":"[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c9a2ecc43bc25db","tex":"\\Phi_p^{\\mathrm{mov}}","aliases":"Φ_p^mov Phi mathrm","display":"\\Phi_p^{\\mathrm{mov}}","kind":"definition-passage","excerpt":"[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17b6bc4f52685e61","tex":"\\Phi_\\sigma","aliases":"Φ_σ Phi sigma","display":"\\Phi_\\sigma","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":85},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"definition-passage","excerpt":"[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.","count":183},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c1b88ae641d78c35","tex":"\\psi_0","aliases":"ψ_0 psi","display":"\\psi_0","kind":"definition-passage","excerpt":"[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltoni","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"definition-passage","excerpt":"[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.","count":123},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e99b5ac7e75a7dbb","tex":"\\omega^q","aliases":"ω^q omega","display":"\\omega^q","kind":"definition-passage","excerpt":"[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention ","count":261},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"definition-passage","excerpt":"[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.","count":87},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ec711e428117ead","tex":"\\twoheadrightarrow","aliases":"↠ twoheadrightarrow","display":"\\twoheadrightarrow","kind":"definition-passage","excerpt":"Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"definition-passage","excerpt":"[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPa","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"definition-passage","excerpt":"[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.Pap","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈","count":1383},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"definition-passage","excerpt":"Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.","count":29},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":93},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7aac306ff05814ee","tex":"\\prod_{j\\geq0}","aliases":"∏_j≥0 prod geq","display":"\\prod_{j\\geq0}","kind":"definition-passage","excerpt":"[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-752c75a3db0198d5","tex":"\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}","aliases":"∏_q∈Fin(r(σ)) prod in operatorname sigma","display":"\\prod_{q\\in\\operatorname{Fin}(r(\\sigma))}","kind":"definition-passage","excerpt":"lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a depend","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1a9f417c2074f70","tex":"\\prod_{\\nu=1}^{|\\gamma|}","aliases":"∏_ν=1^|γ| prod nu gamma","display":"\\prod_{\\nu=1}^{|\\gamma|}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7354371b4dfcd811","tex":"\\sum_{(\\beta,\\alpha)\\in E}","aliases":"∑_(β,α)∈ E sum beta alpha in","display":"\\sum_{(\\beta,\\alpha)\\in E}","kind":"definition-passage","excerpt":"B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain, and ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation is eval(IR) = ∑_(β,α)∈ E ι_β B_βαπ_α, where ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"definition-passage","excerpt":"[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.","count":72},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9068784c641ac3f6","tex":"\\sum_{a=0}^{p-1}","aliases":"∑_a=0^p-1 sum","display":"\\sum_{a=0}^{p-1}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67f6fb951c2ce7df","tex":"\\sum_{b=0}^{2m}","aliases":"∑_b=0^2m sum","display":"\\sum_{b=0}^{2m}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3004d1c3c6d1b821","tex":"\\sum_c","aliases":"∑_c sum","display":"\\sum_c","kind":"definition-passage","excerpt":"[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotien","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a2774d48ee2d8aa","tex":"\\sum_{d\\ge0}","aliases":"∑_d≥0 sum ge","display":"\\sum_{d\\ge0}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7f3762e9cc7d08b","tex":"\\sum_{d\\geq0}","aliases":"∑_d≥0 sum geq","display":"\\sum_{d\\geq0}","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86eb90e9395df95c","tex":"\\sum_{e:i\\to j}","aliases":"∑_e:i→ j sum to","display":"\\sum_{e:i\\to j}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d1546afc3ef11e5","tex":"\\sum_{e\\in E}","aliases":"∑_e∈ E sum in","display":"\\sum_{e\\in E}","kind":"definition-passage","excerpt":"0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem;","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"definition-passage","excerpt":"[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.","count":44},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38c03c7cd3f0f240","tex":"\\sum_{i,j}","aliases":"∑_i,j sum","display":"\\sum_{i,j}","kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2874dc45e47106fd","tex":"\\sum_{i=1}^{r}","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^{r}","kind":"definition-passage","excerpt":"[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9710f0f60fb9e82","tex":"\\sum_{i\\in I_\\alpha}","aliases":"∑_i∈ I_α sum in alpha","display":"\\sum_{i\\in I_\\alpha}","kind":"definition-passage","excerpt":"[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ce2ba87b3975aab","tex":"\\sum_{j=1}^{r}","aliases":"∑_j=1^r sum","display":"\\sum_{j=1}^{r}","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4dc4a300871f9e3e","tex":"\\sum_{m=0}^{\\min(M,n)}","aliases":"∑_m=0^min(M,n) sum min","display":"\\sum_{m=0}^{\\min(M,n)}","kind":"definition-passage","excerpt":"rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d328b95fa488fa30","tex":"\\sum_{m\\ge0}","aliases":"∑_m≥0 sum ge","display":"\\sum_{m\\ge0}","kind":"definition-passage","excerpt":"[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only b","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1cabd23a4e71916","tex":"\\sum_{m\\geq0}","aliases":"∑_m≥0 sum geq","display":"\\sum_{m\\geq0}","kind":"definition-passage","excerpt":"Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0a9cffbc383b69e","tex":"\\sum_{m\\ge1}","aliases":"∑_m≥1 sum ge","display":"\\sum_{m\\ge1}","kind":"definition-passage","excerpt":"[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97951b3ae8950fca","tex":"\\sum_{m\\geq1}","aliases":"∑_m≥1 sum geq","display":"\\sum_{m\\geq1}","kind":"definition-passage","excerpt":"For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41a5548d37bd81ad","tex":"\\sum_{n\\in F}","aliases":"∑_n∈ F sum in","display":"\\sum_{n\\in F}","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-177fd9fcb5956c9a","tex":"\\sum_{n\\equiv j\\pmod N}","aliases":"∑_n≡ jpmod N sum equiv pmod","display":"\\sum_{n\\equiv j\\pmod N}","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"definition-passage","excerpt":"[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the m","count":28},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-441a6af1afa122e0","tex":"\\sum_{n\\geq0}","aliases":"∑_n≥0 sum geq","display":"\\sum_{n\\geq0}","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":37},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-361464ec4fbe1876","tex":"\\sum_{q=1}^{Q}","aliases":"∑_q=1^Q sum","display":"\\sum_{q=1}^{Q}","kind":"definition-passage","excerpt":"[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a7b3358a2feb8ba3","tex":"\\sum_{\\substack{\\Dfun_N(\\rho)=0\\\\ \\Dfun_N'(\\rho)\\ne0}}","aliases":"∑_substackDfun_N(ρ)=0 Dfun_N'(ρ)≠0 ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 sum substack Dfun rho Dfun rho ne sum substack mathsf rho mathsf rho ne","display":"\\sum_{\\substack{{\\mathsf D}_N(\\rho)=0\\\\ {\\mathsf D}_N'(\\rho)\\ne0}}","kind":"definition-passage","excerpt":"Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e0412f70171096f","tex":"\\sum_{\\substack{\\gamma:i\\leadsto j\\\\W(\\gamma)=n}}","aliases":"∑_substackγ:ileadsto j W(γ)=n sum substack gamma leadsto W gamma","display":null,"kind":"definition-passage","excerpt":"[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a26fb0b26e32548","tex":"\\sum_{s\\in S}","aliases":"∑_s∈ S sum in","display":"\\sum_{s\\in S}","kind":"definition-passage","excerpt":"[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46a1bb46e6353ae5","tex":"\\sum_{u\\in\\operatorname{ch}(v)}","aliases":"∑_u∈ch(v) sum in operatorname","display":"\\sum_{u\\in\\operatorname{ch}(v)}","kind":"definition-passage","excerpt":"[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37e66864a7c0cc6a","tex":"\\sum_{\\alpha\\in C}","aliases":"∑_α∈ C sum alpha in","display":"\\sum_{\\alpha\\in C}","kind":"definition-passage","excerpt":"[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cabedb544f677980","tex":"\\sum_{\\alpha\\in I}","aliases":"∑_α∈ I sum alpha in","display":"\\sum_{\\alpha\\in I}","kind":"definition-passage","excerpt":"[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03fcd44478c606ca","tex":"\\sum_{\\lambda\\in\\Lambda_N}","aliases":"∑_λ∈Λ_N sum lambda in Lambda","display":"\\sum_{\\lambda\\in\\Lambda_N}","kind":"definition-passage","excerpt":"For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a87ec80b7b375dff","tex":"\\sum_{\\rho(\\beta)=\\alpha}","aliases":"∑_ρ(β)=α sum rho beta alpha","display":"\\sum_{\\rho(\\beta)=\\alpha}","kind":"definition-passage","excerpt":"[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e969118dcce1120","tex":"\\sum_{\\rho\\in\\Zcal(\\Dfun_N)}","aliases":"∑_ρ∈Zcal(Dfun_N) ∑_ρ∈ Z( D_N) sum rho in Zcal Dfun sum rho in mathcal mathsf","display":"\\sum_{\\rho\\in{\\mathcal Z}({\\mathsf D}_N)}","kind":"definition-passage","excerpt":"Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29724c5f8f33dece","tex":"\\sum_{\\sigma(\\alpha)=j}","aliases":"∑_σ(α)=j sum sigma alpha","display":"\\sum_{\\sigma(\\alpha)=j}","kind":"definition-passage","excerpt":"orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"definition-passage","excerpt":"phism. item An automorphic Hecke datum ( T,H,(P_v)) and a pointwise classicality and Galois realization theorem satisfying all the inputs of [source reference] for the nonzero fiber at x: an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that re","count":56},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For int","count":236},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"definition-passage","excerpt":"[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely ge","count":39},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"definition-passage","excerpt":"[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"definition-passage","excerpt":"[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign ","count":100},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":30},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"definition-passage","excerpt":"[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.","count":111},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"definition-passage","excerpt":"[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.","count":186},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"definition-passage","excerpt":"[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.","count":60},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"definition-passage","excerpt":"[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.","count":200},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. Th","count":329},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG p","count":396},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.","count":524},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"definition-passage","excerpt":"[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.","count":27},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f55fb1c51ddcd12","tex":"\\supseteq","aliases":"⊇ supseteq","display":"\\supseteq","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Realization on a single space] For a typed realization X, define its realization on a single space by |X|:=⊕_v∈ Q_0X_v. Every vertex idempotent acts as a coordinate projector on |X|, and every path acts as a block operator from its source space to its target space.","count":87},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":71},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43b97d47b6ecc1ba","tex":"\\bigoplus_{j=0}^{m-1}","aliases":"⊕_j=0^m-1 bigoplus","display":"\\bigoplus_{j=0}^{m-1}","kind":"definition-passage","excerpt":"[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dcf8db50a213006d","tex":"\\bigoplus_{n\\geq1}^{u}","aliases":"⊕_n≥1^u bigoplus geq","display":"\\bigoplus_{n\\geq1}^{u}","kind":"definition-passage","excerpt":"[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46d26f7a7cd16ab9","tex":"\\bigoplus_{v\\in Q_0}","aliases":"⊕_v∈ Q_0 bigoplus in","display":"\\bigoplus_{v\\in Q_0}","kind":"definition-passage","excerpt":"[Realization on a single space] For a typed realization X, define its realization on a single space by |X|:=⊕_v∈ Q_0X_v. Every vertex idempotent acts as a coordinate projector on |X|, and every path acts as a block operator from its source space to its target space.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86042019669510f9","tex":"\\bigoplus_{w\\in\\operatorname{child}(v)}","aliases":"⊕_w∈child(v) bigoplus in operatorname","display":"\\bigoplus_{w\\in\\operatorname{child}(v)}","kind":"definition-passage","excerpt":"[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b591bcc37a61369","tex":"\\bigoplus_\\alpha","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_\\alpha","kind":"definition-passage","excerpt":"[Packet coordinate condition number] Fix a finite product norm on ⊕_α X_α equivalent to the maximum norm, as in [source reference]. Define κ( P):= A_ P S_ P .","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90b882d77bb3725c","tex":"\\bigoplus_{\\alpha\\in I}","aliases":"⊕_α∈ I bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I}","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] For a finite packet system, define A_ P:X⟶⊕_α∈ IX_α, x⟼(P_α x)_α, and S_ P:⊕_α∈ IX_α⟶ X, (x_α)_α⟼∑_α x_α.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"definition-passage","excerpt":"[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.","count":140},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91f48ece7340f8a0","tex":"\\bigcap_{L\\geq0}","aliases":"⋂_L≥0 bigcap geq","display":"\\bigcap_{L\\geq0}","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e473df1665779d49","tex":"\\bigcap_{n\\geq m}","aliases":"⋂_n≥ m bigcap geq","display":"\\bigcap_{n\\geq m}","kind":"definition-passage","excerpt":"[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(","count":124},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":47},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"definition-passage","excerpt":"item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensi","count":47},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) tha","count":156},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"definition-passage","excerpt":"ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.","count":35},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8b9f34806c25cd4","tex":"\\mathbf1_n","aliases":"1_n mathbf","display":"\\mathbf1_n","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0c9947b1daf0522","tex":"A^*","aliases":"A^*","display":"A^*","kind":"introduction-passage","excerpt":"vpage2The objects on the pageAN ATLAS OF TYPES AND CONVENTIONSChapter VII: Definition II.2.1, Definition II.7.1. The same letter can stand for different kinds of data in different calculations. Keep the following types fixed while reading a formula. In this appendix, vectors are columns, operators compose from right to left, and A^* denotes the con","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06b373fe54fc27d1","tex":"A^3","aliases":"A^3","display":"A^3","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d159e41442deae07","tex":"A_+","aliases":"A_+","display":"A_+","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5494a82a5f60bced","tex":"A_c","aliases":"A_c","display":"A_c","kind":"introduction-passage","excerpt":"vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONChapter VI, Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct ","count":49},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-84cc5863a0df7626","tex":"A_f","aliases":"A_f","display":"A_f","kind":"introduction-passage","excerpt":"vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONChapter VI, Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct ","count":39},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"introduction-passage","excerpt":"dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been adde","count":40},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d8e3b8668ae2ddd","tex":"A_k","aliases":"A_k","display":"A_k","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-571e154fec88d563","tex":"A_{QQ}","aliases":"A_QQ","display":"A_{QQ}","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":32},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"introduction-passage","excerpt":"Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-895b4c69e53e4df6","tex":"A_x","aliases":"A_x","display":"A_x","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":28},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f70174408316e1b5","tex":"A_{\\gamma\\beta}","aliases":"A_γβ gamma beta","display":"A_{\\gamma\\beta}","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c314d68691af653","tex":"\\absn","aliases":"absn","display":null,"kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8c748f09e8f469d","tex":"\\operatorname{Anc}","aliases":"Anc operatorname","display":"\\operatorname{Anc}","kind":"introduction-passage","excerpt":"For each vertex, define its strict ancestor set Anc(v)={u:there is a positive-length path uleadsto v}. For every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases wh","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50c181940765c5f1","tex":"\\operatorname{Apply}","aliases":"Apply operatorname","display":"\\operatorname{Apply}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6d440b3f2f61b84","tex":"\\operatorname{Apply}_k","aliases":"Apply_k operatorname","display":"\\operatorname{Apply}_k","kind":"introduction-passage","excerpt":"otimes M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9c15d19ce3d5bef","tex":"\\operatorname{Apply}_{k}","aliases":"Apply_k operatorname","display":"\\operatorname{Apply}_{k}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"introduction-passage","excerpt":"Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eaef9ee8f33eeef5","tex":"B_4^s","aliases":"B_4^s","display":"B_4^s","kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a4c2e9cb4075d24","tex":"B_a","aliases":"B_a","display":"B_a","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"introduction-passage","excerpt":"dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been adde","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10ef17a5cef5ca03","tex":"b_{N,n}","aliases":"b_N,n","display":"b_{N,n}","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57c8f1ae1da23ae1","tex":"B_r","aliases":"B_r","display":"B_r","kind":"introduction-passage","excerpt":"Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f808c793b47d0c2","tex":"\\bar M","aliases":"bar M bar","display":"\\bar M","kind":"introduction-passage","excerpt":"More generally, for a surjective aggregation map C, a linear map M descends exactly if CM vanishes on ker C. The descended map is defined by bar M(Cx)=CMx. The kernel condition makes this independent of the representative x.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-587654947ed7b96d","tex":"\\bbC","aliases":"bbC C mathbb","display":"{\\mathbb C}","kind":"introduction-passage","excerpt":"Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primiti","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5365be76fcd3f4a","tex":"\\bbZ","aliases":"bbZ Z mathbb","display":"{\\mathbb Z}","kind":"introduction-passage","excerpt":"Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primiti","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathbb","display":"{\\mathbb C}","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97c15171699972db","tex":"C^{\\mathsf T}","aliases":"C^ T mathsf","display":"C^{\\mathsf T}","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"introduction-passage","excerpt":"al trace and its support The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugat","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"introduction-passage","excerpt":"Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-823f036b3e8ddecf","tex":"C^N","aliases":"C^N","display":"C^N","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"introduction-passage","excerpt":"[Unweighted source component count] Assume the module resolved into packets is packet propagating and X_α≠0 for every α∈ I. If c_0 denotes the number of source strongly connected components, then psrank_ A, P(X)=c_0. [source reference]","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-746a0fca63346846","tex":"C_r","aliases":"C_r","display":"C_r","kind":"introduction-passage","excerpt":"Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4deeb62b76263c34","tex":"\\mathrm{car}","aliases":"car mathrm","display":"\\mathrm{car}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e5a5c19cc2b86c8","tex":"\\CC^2","aliases":"CC^2 C^2 CC mathbb","display":"{\\mathbb C}^2","kind":"introduction-passage","excerpt":"al values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf090df8943be9f2","tex":"\\CC^n","aliases":"CC^n C^n CC mathbb","display":"{\\mathbb C}^n","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0bb6498d7717ffb","tex":"\\Cfun_{2,0}","aliases":"Cfun_2,0 C_2,0 Cfun mathsf","display":"{\\mathsf C}_{2,0}","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-994779404d301198","tex":"\\Cfun_{2,1}","aliases":"Cfun_2,1 C_2,1 Cfun mathsf","display":"{\\mathsf C}_{2,1}","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6aace40a7406a32","tex":"\\operatorname{CNOT}","aliases":"CNOT operatorname","display":"\\operatorname{CNOT}","kind":"introduction-passage","excerpt":"[Bell seed circuit] In this example only, H denotes the Hadamard gate, not a Hamiltonian. Starting from ket00, (H⊗ I)ket00 = fracket00+ket10sqrt2, and then CNOT(H⊗ I)ket00 = fracket00+ket11sqrt2. The parity packets show that the final state lies entirely in the even sector, while the intermediate circuit path passes through both even and odd addres","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"introduction-passage","excerpt":"The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolutio","count":35},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e73c7731b76eea40","tex":"\\cosh^2","aliases":"cosh^2 cosh","display":"\\cosh^2","kind":"introduction-passage","excerpt":"The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolutio","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dabe11e0627f83c2","tex":"\\cosh_1","aliases":"cosh_1 cosh","display":"\\cosh_1","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41d310366424af4d","tex":"\\cosh_a","aliases":"cosh_a cosh","display":"\\cosh_a","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"introduction-passage","excerpt":"lter becomes a matrix of actual delaysFIR TRANSFER / A FOUR-TAP EXAMPLEChapter V: Theorem 3.1. Take the causal filter h=(1,2,-1,1), with taps at 0,1,2,3, acting by y[n]=∑_qh[q]f[n-q]. Resolve the input into even and odd channels. Let D denote one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contribution to","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db28b207b9c1b29b","tex":"D^a","aliases":"D^a","display":"D^a","kind":"introduction-passage","excerpt":"packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum an","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b3443a3b74189af","tex":"D^d","aliases":"D^d","display":"D^d","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5732f6e6e8849f6","tex":"D_{[X,Y]}","aliases":"D_[X,Y]","display":"D_{[X,Y]}","kind":"introduction-passage","excerpt":"textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":41},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-616ff9ffcce7048e","tex":"D_i^{d_i}","aliases":"D_i^d_i","display":"D_i^{d_i}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1be8e49b81c9b1e","tex":"D_M","aliases":"D_M","display":"D_M","kind":"introduction-passage","excerpt":"A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3405c1cc6903854","tex":"D_n","aliases":"D_n","display":"D_n","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ed0c47b8f5552f3","tex":"D_{n-k}","aliases":"D_n-k","display":"D_{n-k}","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-acc501056e7d81d5","tex":"D_T","aliases":"D_T","display":"D_T","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"introduction-passage","excerpt":"Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfe","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fdf9532371a315b7","tex":"D_X","aliases":"D_X","display":"D_X","kind":"introduction-passage","excerpt":"textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f78b62eb28153fc","tex":"D_Y","aliases":"D_Y","display":"D_Y","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24671e7796ade104","tex":"d_\\lambda","aliases":"d_λ lambda","display":"d_\\lambda","kind":"introduction-passage","excerpt":"Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b45adb173afae48f","tex":"\\Dfun_N^{(N)}","aliases":"Dfun_N^(N) D_N^(N) Dfun mathsf","display":"{\\mathsf D}_N^{(N)}","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f30b020ad7331a73","tex":"\\Dfun_N^{-r}","aliases":"Dfun_N^-r D_N^-r Dfun mathsf","display":"{\\mathsf D}_N^{-r}","kind":"introduction-passage","excerpt":"For an entire function f, let Z(f) denote its multiset of nonzero zeros. Let r be a positive integer. Then D_N^-r is meromorphic and a simple zero of D_N becomes a pole of order r.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"introduction-passage","excerpt":"sfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPChapter VII: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integ","count":60},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f580f0a1cdae14e","tex":"\\dim_{\\FF_p}","aliases":"dim_FF_p dim_ F_p dim FF dim mathbb","display":"\\dim_{{\\mathbb F}_p}","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1997e4b5359f675","tex":"\\dot U","aliases":"dot U dot","display":"\\dot U","kind":"introduction-passage","excerpt":"Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-381aa4b32306117e","tex":"\\dot\\psi","aliases":"dotψ dot psi","display":"\\dot\\psi","kind":"introduction-passage","excerpt":"Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5d585ea90c121bc","tex":"\\bm E","aliases":"E bm","display":"\\bm E","kind":"introduction-passage","excerpt":"[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original sampl","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":38},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4293330aa39421ea","tex":"E^{(N,j)}_{Nn}","aliases":"E^(N,j)_Nn","display":"E^{(N,j)}_{Nn}","kind":"introduction-passage","excerpt":"Normalization is also an explicit implication. Using the existing definition E^(N,j)_Nn=j!C_N,j(n), the source proves that multiplication by j! commutes with every iterated difference. If the unnormalized difference has precision A, the normalized difference has precision A-v_p(j!). Precision is defined as x=0 hboxor v_p(x)≥ A, so zero and negative","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81315b920e13e316","tex":"e^{-at}","aliases":"e^-at","display":"e^{-at}","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15b45515a5b956c1","tex":"e^{-H}","aliases":"e^-H","display":"e^{-H}","kind":"introduction-passage","excerpt":"vpage36A positive transfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPChapter VII: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b148da2b6a743d1","tex":"e^{-H'}","aliases":"e^-H'","display":"e^{-H'}","kind":"introduction-passage","excerpt":"integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every f","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a1e1b0a1da34e35","tex":"e^{-i(t-s)H}","aliases":"e^-i(t-s)H","display":"e^{-i(t-s)H}","kind":"introduction-passage","excerpt":"ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors h","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-390457a95f8f6d57","tex":"e^{-isH}","aliases":"e^-isH","display":"e^{-isH}","kind":"introduction-passage","excerpt":"ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors h","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2637b6dd9fae991f","tex":"e^{-itH}","aliases":"e^-itH","display":"e^{-itH}","kind":"introduction-passage","excerpt":"ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors h","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-925048ed58d84367","tex":"e^{-n\\Delta}","aliases":"e^-nΔ Delta","display":"e^{-n\\Delta}","kind":"introduction-passage","excerpt":"HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a1d25b2a3b14026","tex":"e^{at}","aliases":"e^at","display":"e^{at}","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c079a3d2ea487eb","tex":"\\e^B","aliases":"e^B e mathrm","display":"{\\mathrm e}^B","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9918448367ba8abf","tex":"e^g","aliases":"e^g","display":"e^g","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f21866dd974b9900","tex":"\\e^{nB}","aliases":"e^nB e mathrm","display":"{\\mathrm e}^{nB}","kind":"introduction-passage","excerpt":"ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for eve","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c6acec97d560659","tex":"\\e^{s(B)}","aliases":"e^s(B) e mathrm","display":"{\\mathrm e}^{s(B)}","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1fe69fafa50db3c5","tex":"e^t","aliases":"e^t","display":"e^t","kind":"introduction-passage","excerpt":"=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c01ae1b703c5263","tex":"e^{yt}","aliases":"e^yt","display":"e^{yt}","kind":"introduction-passage","excerpt":"=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-699c216a893acf66","tex":"\\e^\\beta","aliases":"e^β e beta mathrm beta","display":"{\\mathrm e}^\\beta","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a178b613b1f6d9d0","tex":"\\e^{\\beta n}","aliases":"e^β n e beta mathrm beta","display":"{\\mathrm e}^{\\beta n}","kind":"introduction-passage","excerpt":"ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for eve","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9b6193b1833f637","tex":"E_a","aliases":"E_a","display":"E_a","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":36},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e764f519ddd193e4","tex":"\\mathcal E_a","aliases":"E_a mathcal","display":"\\mathcal E_a","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dfb98ad8aedbb4dd","tex":"E_h","aliases":"E_h","display":"E_h","kind":"introduction-passage","excerpt":"Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca7e75161b6bbf35","tex":"E_{ij}","aliases":"E_ij","display":"E_{ij}","kind":"introduction-passage","excerpt":"vpage10Which zero patterns survive multiplication?SUPPORT ALGEBRAS / A UNIVERSAL TESTMain text: [source reference]. All packet spaces are nonzero Suppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let E_ij denote the matrix that sends e_j to e_i and kills ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"introduction-passage","excerpt":"ntCheckpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. Scope and final statement. Let p be a prime, let m>0 divide p-1, put N=mp and h=(p-1)/m, and let 0≤ j<N. The original rational packet is defined by ∑_q","count":39},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dbe6869ebc4b6f5f","tex":"e_{j+1\\bmod8}","aliases":"e_j+1bmod8 bmod","display":"e_{j+1\\bmod8}","kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYChapter II, mathematical text: Theorem 3.5, Theorem 3.8 Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. W","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e749405022a66e6b","tex":"E_m","aliases":"E_m","display":"E_m","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eea8c720f33fb581","tex":"E_m^{(2,0,1)}","aliases":"E_m^(2,0,1)","display":"E_m^{(2,0,1)}","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d05f7fb87f2fa049","tex":"E_m^{(2,1,1)}","aliases":"E_m^(2,1,1)","display":"E_m^{(2,1,1)}","kind":"introduction-passage","excerpt":"daries For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3104ecf0e1adeade","tex":"e_{n+h}^{(m,p)}","aliases":"e_n+h^(m,p)","display":"e_{n+h}^{(m,p)}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c910c4f72facf773","tex":"e_{n+uh}^{(m,p)}","aliases":"e_n+uh^(m,p)","display":"e_{n+uh}^{(m,p)}","kind":"introduction-passage","excerpt":"The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f446188894e8b7ca","tex":"e_n^{(m,p)}","aliases":"e_n^(m,p)","display":"e_n^{(m,p)}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-385b796c2c5e99d4","tex":"e_q^{(m,p)}","aliases":"e_q^(m,p)","display":"e_q^{(m,p)}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1248c74d891e3c5","tex":"\\mathcal E_x","aliases":"E_x mathcal","display":"\\mathcal E_x","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14abf8cd4095aad2","tex":"\\exp_a","aliases":"exp_a exp","display":"\\exp_a","kind":"introduction-passage","excerpt":"r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1ffe82ebc455b14","tex":"f^*","aliases":"f^*","display":"f^*","kind":"introduction-passage","excerpt":"eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"introduction-passage","excerpt":"of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells. headWhy the coordinate convention matter","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"introduction-passage","excerpt":"lter becomes a matrix of actual delaysFIR TRANSFER / A FOUR-TAP EXAMPLEChapter V: Theorem 3.1. Take the causal filter h=(1,2,-1,1), with taps at 0,1,2,3, acting by y[n]=∑_qh[q]f[n-q]. Resolve the input into even and odd channels. Let D denote one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contribution to","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd2b9b85d5dfefc2","tex":"F_e","aliases":"F_e","display":"F_e","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31d7232e6fa917d3","tex":"F_i","aliases":"F_i","display":"F_i","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"introduction-passage","excerpt":"The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includ","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c57cafb013c6775f","tex":"F_n","aliases":"F_n","display":"F_n","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97bd1abfe72629ee","tex":"F_{N(k+1)+j}","aliases":"F_N(k+1)+j","display":"F_{N(k+1)+j}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec86e738238d91a6","tex":"F_{N(k+2)+j}","aliases":"F_N(k+2)+j","display":"F_{N(k+2)+j}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7817222f46df5e65","tex":"F_{n+1}","aliases":"F_n+1","display":"F_{n+1}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80004cdcfd3891b8","tex":"F_{n+2}","aliases":"F_n+2","display":"F_{n+2}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ddf7d50338d5ead5","tex":"f_{n+d}","aliases":"f_n+d","display":"f_{n+d}","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b28b2bb412883c1c","tex":"F_{Nk+j}","aliases":"F_Nk+j","display":"F_{Nk+j}","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a27312454be2902","tex":"F_p","aliases":"F_p","display":"F_p","kind":"introduction-passage","excerpt":"rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7110079fe13a85a","tex":"F_r","aliases":"F_r","display":"F_r","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b4740083376d053","tex":"\\FF_p","aliases":"FF_p F_p FF mathbb","display":"{\\mathbb F}_p","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":30},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c03543fb3fd9a9c3","tex":"\\FF_p^+","aliases":"FF_p^+ F_p^+ FF mathbb","display":"{\\mathbb F}_p^+","kind":"introduction-passage","excerpt":"Let p be prime. Write F_p^+ for the additive group of the finite field.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-555687cfad37a088","tex":"\\FF_p^r","aliases":"FF_p^r F_p^r FF mathbb","display":"{\\mathbb F}_p^r","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25c41d2acd920db0","tex":"\\FF_p^{\\times}","aliases":"FF_p^× F_p^× FF times mathbb times","display":"{\\mathbb F}_p^{\\times}","kind":"introduction-passage","excerpt":"[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ac97bde10b6c4f5","tex":"\\frakm","aliases":"frakm m mathfrak","display":"{\\mathfrak m}","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0b81ba1507604a0","tex":"\\frakm_{\\OO}","aliases":"frakm_OO m_ O frakm OO mathfrak mathcal","display":"{\\mathfrak m}_{{\\mathcal O}}","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-911dc5b7a786cf08","tex":"\\frakm_{R_i}","aliases":"frakm_R_i m_R_i frakm mathfrak","display":"{\\mathfrak m}_{R_i}","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02b0b5ff2a25a5d9","tex":"g'","aliases":"g'","display":"g'","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bda763105cb3e335","tex":"G_{J,\\chi}","aliases":"G_J,χ chi","display":"G_{J,\\chi}","kind":"introduction-passage","excerpt":"· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J)","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bfa087a98ed68edf","tex":"g_{n-r}","aliases":"g_n-r","display":"g_{n-r}","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0abf0d17398d7d33","tex":"G_{\\Q}","aliases":"G_Q G_ Q Q mathbf","display":"G_{{\\mathbf Q}}","kind":"introduction-passage","excerpt":"The hyperref[book:x-construction-21]fixed-quadratic-packet interpretation completes this restricted arithmetic interpretation and its conditional completion. The packet is defined by one fixed quotient. It is not the collection of all quadratic characters of G_ Q. No local ramification criterion, Hodge--Tate weight, Weil--Deligne attachment, or cro","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8bdb1c02444ca91","tex":"H'","aliases":"H'","display":"H'","kind":"introduction-passage","excerpt":"positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":30},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum. This proves it at depth ℓ+1.","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum. This proves it at depth ℓ+1.","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"introduction-passage","excerpt":"The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolutio","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9b3c767be0a1967","tex":"H_{2N}","aliases":"H_2N","display":"H_{2N}","kind":"introduction-passage","excerpt":"1 · -1, and the full family is defined by H_1=[1] and H_2N=H_2⊗ H_N =H_N · H_N H_N · -H_N. The two-point orthogonality identity is checked explicitly. Kronecker multiplication and induction then prove H_NH_N^T=NI, N=2^k. The ordinary transpose is sufficient because all constructed entries are real, even though the scalar field is C.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-097615d1cce1b4b2","tex":"H_b","aliases":"H_b","display":"H_b","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum. This proves it at depth ℓ+1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98990972b3d9edcd","tex":"h_d","aliases":"h_d","display":"h_d","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-333bf23be8bc780a","tex":"h_i","aliases":"h_i","display":"h_i","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf26706b1ed38c02","tex":"\\mathcal H_{\\mathrm{int}}","aliases":"H_int mathcal mathrm","display":"\\mathcal H_{\\mathrm{int}}","kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-785ae493560363c1","tex":"h_n","aliases":"h_n","display":"h_n","kind":"introduction-passage","excerpt":"[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by mul","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0d2a40129a3b4c8b","tex":"H_p","aliases":"H_p","display":"H_p","kind":"introduction-passage","excerpt":"The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolutio","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a642ba0a275f90a","tex":"H_{\\rm phase}","aliases":"H_rm phase rm","display":"H_{\\rm phase}","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e79b7ad0299d0b0","tex":"I_0","aliases":"I_0","display":"I_0","kind":"introduction-passage","excerpt":"For s∈ S, write s=∑_α∈ I_0P_α s. If β is unreachable from I_0, then for every A∈ A, P_β As = ∑_α∈ I_0P_β AP_α s =0 by [source reference]. The same holds for every finite linear combination of orbit vectors.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d9060f8fe5d796b","tex":"I_A","aliases":"I_A","display":"I_A","kind":"introduction-passage","excerpt":"An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d4c5f4c75a9b75d","tex":"I_i","aliases":"I_i","display":"I_i","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d22810f2cda3ca50","tex":"I_{m_i}","aliases":"I_m_i","display":"I_{m_i}","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"introduction-passage","excerpt":"[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original sampl","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55466b76f7f0c090","tex":"\\mathrm{int}","aliases":"int mathrm","display":"\\mathrm{int}","kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0031bdf2737d3ca","tex":"\\ip","aliases":"ip","display":null,"kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36282b37d83ec448","tex":"\\Irr","aliases":"Irr operatorname","display":"{\\operatorname{Irr}}","kind":"introduction-passage","excerpt":"[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3fa813ff3c6bd07a","tex":"J^{-1}","aliases":"J^-1","display":"J^{-1}","kind":"introduction-passage","excerpt":"bseteq K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative Dψ(t_0)=-J^-1K. Consequently it is differentiable on U. The map is def","count":30},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcc48739f478f9b9","tex":"J^1","aliases":"J^1","display":"J^1","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe61bfb695992cf7","tex":"J_F","aliases":"J_F","display":"J_F","kind":"introduction-passage","excerpt":"rrow Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) deno","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8af4e249618e23ea","tex":"J_x","aliases":"J_x","display":"J_x","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":40},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab5e8580e0cfc647","tex":"J_y","aliases":"J_y","display":"J_y","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53283f2f6027fba4","tex":"\\mathbb K","aliases":"K mathbb","display":"\\mathbb K","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-44917510cdb5a170","tex":"\\mathbb K^e","aliases":"K^e mathbb","display":"\\mathbb K^e","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cf939730979c2cb","tex":"\\mathbb K^k","aliases":"K^k mathbb","display":"\\mathbb K^k","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-173833828c3529a3","tex":"K_2","aliases":"K_2","display":"K_2","kind":"introduction-passage","excerpt":"The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolutio","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c97f0bbe945fa2ee","tex":"K_N","aliases":"K_N","display":"K_N","kind":"introduction-passage","excerpt":"[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by mul","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b39e465f5dcb17d2","tex":"K_p","aliases":"K_p","display":"K_p","kind":"introduction-passage","excerpt":"The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolutio","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"introduction-passage","excerpt":"The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K). This is independent of the choice of its preimage because ker J is","count":73},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-44819be810b8a1c8","tex":"\\ket","aliases":"ket","display":null,"kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":70},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":62},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c91dabfc8b700b3b","tex":"L_N","aliases":"L_N","display":"L_N","kind":"introduction-passage","excerpt":"[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b80cbb9f17c415c","tex":"\\mathcal L_v","aliases":"L_v mathcal","display":"\\mathcal L_v","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2dcbb7ca5126e8ca","tex":"L_y","aliases":"L_y","display":"L_y","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5a8e8a33dcbaa8c","tex":"M_{d_i}","aliases":"M_d_i","display":"M_{d_i}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a603e6b6eedcb75f","tex":"M_e","aliases":"M_e","display":"M_e","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bca51ab689269ed0","tex":"M_{e\\times k}","aliases":"M_e× k times","display":"M_{e\\times k}","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"introduction-passage","excerpt":"The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N","count":50},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9f29e7d9e7a175b","tex":"M_k","aliases":"M_k","display":"M_k","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3db6ddc80894285f","tex":"M_{k+1}","aliases":"M_k+1","display":"M_{k+1}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"introduction-passage","excerpt":"To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06e9b4e88356c8ed","tex":"M_{r,d}","aliases":"M_r,d","display":"M_{r,d}","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a405bf0a7338f95b","tex":"M_y","aliases":"M_y","display":"M_y","kind":"introduction-passage","excerpt":"headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b84d9005844b0c6f","tex":"M_y^2","aliases":"M_y^2","display":"M_y^2","kind":"introduction-passage","excerpt":"headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03fccda709f9f68a","tex":"M_\\beta","aliases":"M_β beta","display":"M_\\beta","kind":"introduction-passage","excerpt":"ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for eve","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-956f614ae0645ab5","tex":"\\mat","aliases":"mat","display":null,"kind":"introduction-passage","excerpt":"headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":77},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b61b8cfdaf3918dd","tex":"\\max_{u\\to v}","aliases":"max_u→ v max to","display":"\\max_{u\\to v}","kind":"introduction-passage","excerpt":"r every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is define","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d80123f93a0ffd9","tex":"\\Mcal","aliases":"Mcal M mathcal","display":"{\\mathcal M}","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1ac5173895854fc","tex":"\\Mcal'","aliases":"Mcal' M' Mcal mathcal","display":"{\\mathcal M}'","kind":"introduction-passage","excerpt":"hannel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the c","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"introduction-passage","excerpt":"The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b12673ac5aded11","tex":"\\Nm","aliases":"Nm N operatorname","display":"{\\operatorname N}","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea4b9089982f8df8","tex":"\\Nm_{K/\\mathbb Q}","aliases":"Nm_K/ Q N_K/ Q Nm mathbb operatorname mathbb","display":"{\\operatorname N}_{K/\\mathbb Q}","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison. Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maxi","count":29},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":89},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-94c28f047e602f0f","tex":"\\mathcal O_L","aliases":"O_L mathcal","display":"\\mathcal O_L","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison. Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maxi","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91023a802e0888a1","tex":"o_x","aliases":"o_x","display":"o_x","kind":"introduction-passage","excerpt":"Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero ","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0d3a3f2f247ab07","tex":"\\one_{[\\tau,\\infty)}","aliases":"one_[τ,∞) 1_[τ,∞) one tau infty mathbf tau infty","display":"{\\mathbf 1}_{[\\tau,\\infty)}","kind":"introduction-passage","excerpt":"The adjacent difference detector is local in the sample domain. Wavelet coefficients and higher differences give related multiscale packets. In the Haar subsections the signal is defined on R. Here the step convention is H_τ(t)= 1_[τ,∞)(t), whose restriction to (0,1) agrees with the preceding interval step. A chosen extension beyond the observation","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d313fd089d71d9a","tex":"\\OO_F^\\times","aliases":"OO_F^× O_F^× OO times mathcal times","display":"{\\mathcal O}_F^\\times","kind":"introduction-passage","excerpt":"Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83c44e1ef141b100","tex":"\\OO_K","aliases":"OO_K O_K OO mathcal","display":"{\\mathcal O}_K","kind":"introduction-passage","excerpt":"Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding coun","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-712a4047f09e3ce3","tex":"\\overline J","aliases":"overline J overline","display":"\\overline J","kind":"introduction-passage","excerpt":"tbfEditorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and wr","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bee338e0d8013678","tex":"\\overline J_F","aliases":"overline J_F overline","display":"\\overline J_F","kind":"introduction-passage","excerpt":"tbfEditorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and wr","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d8a1921f2a37e0a","tex":"\\overline M","aliases":"overline M overline","display":"\\overline M","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6eda1a560c2d3126","tex":"\\overline{g_i}","aliases":"overlineg_i overline","display":"\\overline{g_i}","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1918b7c23a15b567","tex":"\\overline{\\Q}","aliases":"overlineQ overline Q overline mathbf","display":"\\overline{{\\mathbf Q}}","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison. Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maxi","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ae44c4d3d92ce4f","tex":"\\overline{\\psi_{i\\ell}}","aliases":"overlineψ_iℓ overline psi ell","display":"\\overline{\\psi_{i\\ell}}","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e517349dec3b6ba","tex":"\\overset","aliases":"overset","display":null,"kind":"introduction-passage","excerpt":"Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"introduction-passage","excerpt":":x-construction-27 Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c18367064df5a710","tex":"\\mathfrak p","aliases":"p mathfrak","display":"\\mathfrak p","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d1a16aa753ffb59","tex":"p^{(p-1)x+j}","aliases":"p^(p-1)x+j","display":"p^{(p-1)x+j}","kind":"introduction-passage","excerpt":"The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"introduction-passage","excerpt":"textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4dd413d35142652e","tex":"p^{2d}","aliases":"p^2d","display":"p^{2d}","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-085daded6ee24ea6","tex":"p^{2i}","aliases":"p^2i","display":"p^{2i}","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-daabe7304529997d","tex":"p^{2i-2}","aliases":"p^2i-2","display":"p^{2i-2}","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4f6611ac8fe0528","tex":"p^d","aliases":"p^d","display":"p^d","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50a36a5ee1bff68c","tex":"p^{d-1}","aliases":"p^d-1","display":"p^{d-1}","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b47e4ae8f60bef45","tex":"p^e","aliases":"p^e","display":"p^e","kind":"introduction-passage","excerpt":"classicalinput[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-325ff65361d0b480","tex":"p^{\\min(k,mn)}","aliases":"p^min(k,mn) min","display":"p^{\\min(k,mn)}","kind":"introduction-passage","excerpt":"[All-address differences on the divisor branch] Let p be prime, m>0, mmid p-1, N=mp, and 0≤ j<N. Write h=(p-1)/m. For every u,k,n∈ N, U_N,j(n) · ∈ Z_p, Δ_uh^kU_N,j(n) · ∈ p^min(k,mn) Z_p, v_p(Δ_uh^kC_N,j(n)) · ≥min(k,mn)-v_p(j!). The factorial correction is part of the normalized statement, including addresses j≥ p.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16055e0aaa2fe4ce","tex":"p^x","aliases":"p^x","display":"p^x","kind":"introduction-passage","excerpt":"The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ffaa0b88d08a3d2","tex":"P_\\mathcal","aliases":"P_ mathcal","display":"P_\\mathcal","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"introduction-passage","excerpt":"sfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPChapter VII: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integ","count":33},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bab84a59440e63c0","tex":"P_{2,0}","aliases":"P_2,0","display":"P_{2,0}","kind":"introduction-passage","excerpt":"vpage4Refine packets on a common latticeEXACT REFINEMENT / WHAT CAN BE ADDEDChapter V: Theorem 2.3. An even sample belongs to exactly one of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original sample index. Empty ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91a55f7626b81c45","tex":"P_{4,0}","aliases":"P_4,0","display":"P_{4,0}","kind":"introduction-passage","excerpt":"vpage4Refine packets on a common latticeEXACT REFINEMENT / WHAT CAN BE ADDEDChapter V: Theorem 2.3. An even sample belongs to exactly one of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original sample index. Empty ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6b3605e1b2fcc56","tex":"P_{4,2}","aliases":"P_4,2","display":"P_{4,2}","kind":"introduction-passage","excerpt":"vpage4Refine packets on a common latticeEXACT REFINEMENT / WHAT CAN BE ADDEDChapter V: Theorem 2.3. An even sample belongs to exactly one of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original sample index. Empty ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc7ba1c49968c37c","tex":"p_k","aliases":"p_k","display":"p_k","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f29a01821ec9a768","tex":"\\mathfrak p_K^a","aliases":"p_K^a mathfrak","display":"\\mathfrak p_K^a","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90fba425db5ca238","tex":"P_{N,k}","aliases":"P_N,k","display":"P_{N,k}","kind":"introduction-passage","excerpt":"[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09523592e6cc4b12","tex":"P_{N,s}","aliases":"P_N,s","display":"P_{N,s}","kind":"introduction-passage","excerpt":"[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed02c035995b1f57","tex":"P_\\alpha^*","aliases":"P_α^* alpha","display":"P_\\alpha^*","kind":"introduction-passage","excerpt":"Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The es","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6123ae670e2c050","tex":"\\mathrm{Prop}","aliases":"Prop mathrm","display":"\\mathrm{Prop}","kind":"introduction-passage","excerpt":"Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q mathbb Q mathbf","display":null,"kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":62},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d76d885e9e3c5ba5","tex":"Q^*","aliases":"Q^*","display":"Q^*","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04992c0930a894b0","tex":"q^L","aliases":"q^L","display":"q^L","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":46},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50f270102a088631","tex":"\\Q_2","aliases":"Q_2 Q mathbb Q_2 Q mathbf","display":null,"kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8205838caa474d98","tex":"\\Q_p","aliases":"Q_p Q mathbb Q_p Q mathbf","display":null,"kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ddc9b1c9f42d7650","tex":"\\Qp","aliases":"Qp Q_p mathbb","display":"{\\mathbb Q_p}","kind":"introduction-passage","excerpt":"Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets t","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ QQ Q mathbb","display":null,"kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":53},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1c20921b96c99d0","tex":"\\QQ^8","aliases":"QQ^8 QQ","display":null,"kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYChapter II, mathematical text: Theorem 3.5, Theorem 3.8 Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. W","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf956bb3d74ebbc2","tex":"\\bm R","aliases":"R bm","display":"\\bm R","kind":"introduction-passage","excerpt":"[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original sampl","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8e1532bba361388","tex":"R^a","aliases":"R^a","display":"R^a","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"introduction-passage","excerpt":"(r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem. headA hand-computable decisi","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"introduction-passage","excerpt":", the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem. headA hand-computable decision threshold Fi","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4aa81ec436e2173c","tex":"R_N","aliases":"R_N","display":"R_N","kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYChapter II, mathematical text: Theorem 3.5, Theorem 3.8 Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. W","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2d59bfcf91e5807","tex":"R_q","aliases":"R_q","display":"R_q","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac70c1f93e7ca709","tex":"R_q^d","aliases":"R_q^d","display":"R_q^d","kind":"introduction-passage","excerpt":"et ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93731b161f309dff","tex":"R_q^e","aliases":"R_q^e","display":"R_q^e","kind":"introduction-passage","excerpt":"F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The co","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0e893f7b0a8cb70","tex":"r_x","aliases":"r_x","display":"r_x","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3c72cb58f67a4d1","tex":"R_y","aliases":"R_y","display":"R_y","kind":"introduction-passage","excerpt":"Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additi","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank operatorname","display":"{\\operatorname{rank}}","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":34},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3bce33d4a711b894","tex":"\\Rcal","aliases":"Rcal R mathcal","display":"{\\mathcal R}","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":28},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79c86107d4290f47","tex":"\\Rcal^*","aliases":"Rcal^* R^* Rcal mathcal","display":"{\\mathcal R}^*","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8d0b37b4af3bfcc","tex":"S^2","aliases":"S^2","display":"S^2","kind":"introduction-passage","excerpt":"vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYChapter II, mathematical text: Theorem 3.5, Theorem 3.8 Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. W","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90f70c36f812714e","tex":"S^n","aliases":"S^n","display":"S^n","kind":"introduction-passage","excerpt":"Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primiti","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-698cd4222a287d75","tex":"S_i","aliases":"S_i","display":"S_i","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-389639d67f2e3b9d","tex":"S_i^{\\oplus d_i}","aliases":"S_i^⊕ d_i oplus","display":"S_i^{\\oplus d_i}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c0785670c74cfb3","tex":"S_n","aliases":"S_n","display":"S_n","kind":"introduction-passage","excerpt":"rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e6bdd9702222c4b","tex":"\\sech","aliases":"sech operatorname","display":"{\\operatorname{sech}}","kind":"introduction-passage","excerpt":"For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e1495d390f06be4","tex":"\\sinh^2","aliases":"sinh^2 sinh","display":"\\sinh^2","kind":"introduction-passage","excerpt":"The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolutio","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc8b82baaf7d8475","tex":"\\srank_{\\mathcal A}","aliases":"srank_ A srank mathcal operatorname mathcal","display":"{\\operatorname{srank}}_{\\mathcal A}","kind":"introduction-passage","excerpt":"Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feas","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e4ee8a109009a53","tex":"\\subsetneq","aliases":"subsetneq","display":null,"kind":"introduction-passage","excerpt":"For each vertex, define its strict ancestor set Anc(v)={u:there is a positive-length path uleadsto v}. For every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases wh","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1084f45e6b3e947e","tex":"\\supp","aliases":"supp operatorname","display":"{\\operatorname{supp}}","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-841a78b5f2413be0","tex":"\\Supp_{\\TT}","aliases":"Supp_TT Supp_ T Supp TT operatorname mathbb","display":"{\\operatorname{Supp}}_{{\\mathbb T}}","kind":"introduction-passage","excerpt":"eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-793f37c1d91019f4","tex":"\\Swan","aliases":"Swan operatorname","display":"{\\operatorname{Swan}}","kind":"introduction-passage","excerpt":"For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":71},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6da3583735a2a7a0","tex":"T'","aliases":"T'","display":"T'","kind":"introduction-passage","excerpt":"lidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"introduction-passage","excerpt":"The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolutio","count":50},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-910092ae6df8d19a","tex":"T^n","aliases":"T^n","display":"T^n","kind":"introduction-passage","excerpt":"sfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPChapter VII: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integ","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":63},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f4d0bbae2783fe43","tex":"T_{C,\\chi}","aliases":"T_C,χ chi","display":"T_{C,\\chi}","kind":"introduction-passage","excerpt":"· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J)","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6d1c35d86f0d39d","tex":"T_k","aliases":"T_k","display":"T_k","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-325816c77766b823","tex":"T_x","aliases":"T_x","display":"T_x","kind":"introduction-passage","excerpt":"A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9acac70418a67a9d","tex":"U^p","aliases":"U^p","display":"U^p","kind":"introduction-passage","excerpt":"The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d849c4e507602c15","tex":"U_{0,\\lambda}","aliases":"U_0,λ lambda","display":"U_{0,\\lambda}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec4379f6b5fb17bc","tex":"U_a","aliases":"U_a","display":"U_a","kind":"introduction-passage","excerpt":"The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ec524ed5f354d59","tex":"U_A^a","aliases":"U_A^a","display":"U_A^a","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-559db2c18d7d396e","tex":"U_F^1","aliases":"U_F^1","display":"U_F^1","kind":"introduction-passage","excerpt":"Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character b","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55bbc33c76201c71","tex":"U_K^a","aliases":"U_K^a","display":"U_K^a","kind":"introduction-passage","excerpt":"Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36077cb07719c8a1","tex":"v^T","aliases":"v^T","display":"v^T","kind":"introduction-passage","excerpt":"n 3, Appendix F. A dense array, a factored operator, and a compiled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide illustrates those distinctions with calculations. vfigthree_views headThree models to carry through the pape","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":136},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-adf10d370b43fb83","tex":"V_{\\mathrm{car},\\lambda}","aliases":"V_car,λ mathrm lambda","display":"V_{\\mathrm{car},\\lambda}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b5f238ca9baf774","tex":"V_r","aliases":"V_r","display":"V_r","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5363f4b0a5b2b694","tex":"V_r^*","aliases":"V_r^*","display":"V_r^*","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a675678fa6689935","tex":"\\vee","aliases":"vee","display":"\\vee","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b99d2e5f37ae7e17","tex":"w'","aliases":"w'","display":"w'","kind":"introduction-passage","excerpt":"1-w · 1+w. vfigprojective_poles headThe action is defined before a ratio is taken The determinant is w, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives A(w)A(w')=A(ww'), A(1)=I. For w=i, the orbit through zero is [0:1] ⟼ [-i:1] ⟼ [1:0] ⟼ [i:1] ⟼ [0:1]. The third point is a legitimate projective point, even though a fin","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8bf07eb97189ba98","tex":"W_{\\ell,\\alpha}","aliases":"W_ℓ,α ell alpha","display":"W_{\\ell,\\alpha}","kind":"introduction-passage","excerpt":"Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum. This proves it at depth ℓ+1.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02c8464b897b4a69","tex":"\\Walk","aliases":"Walk operatorname","display":"{\\operatorname{Walk}}","kind":"introduction-passage","excerpt":"Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e795a1f6f91d71a4","tex":"\\Walk_n","aliases":"Walk_n Walk operatorname","display":"{\\operatorname{Walk}}_n","kind":"introduction-passage","excerpt":"Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"introduction-passage","excerpt":"eld and extend the projected connection to differential forms with values in the range bundle by ∇ω=P dω. Thus ∇^2 denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields. [Alternative proof of the curvature formula] For a section s with Ps=s, ∇^2s=P d(P ds)=P(dP)wedge ds. Sin","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ae2c725a0324b8f","tex":"\\widehat G","aliases":"widehat G widehat","display":"\\widehat G","kind":"introduction-passage","excerpt":"Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-47c52f415949a6cb","tex":"\\widetilde C","aliases":"widetilde C widetilde","display":"\\widetilde C","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d0578330dc67f46","tex":"\\widetilde C_j","aliases":"widetilde C_j widetilde","display":"\\widetilde C_j","kind":"introduction-passage","excerpt":"[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82ec90a8c96332be","tex":"\\widetilde{\\mathcal N}","aliases":"widetilde N widetilde mathcal","display":"\\widetilde{\\mathcal N}","kind":"introduction-passage","excerpt":"An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bdb63d74dc95706","tex":"\\widetilde{\\mathcal N}_E","aliases":"widetilde N_E widetilde mathcal","display":"\\widetilde{\\mathcal N}_E","kind":"introduction-passage","excerpt":"An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-242de7ebea7742ad","tex":"\\widetilde{\\mathcal R}","aliases":"widetilde R widetilde mathcal","display":"\\widetilde{\\mathcal R}","kind":"introduction-passage","excerpt":"channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R wi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"introduction-passage","excerpt":"1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":47},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"introduction-passage","excerpt":"headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b07f815019444f3","tex":"X^a","aliases":"X^a","display":"X^a","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f98d18a8cc9bdad9","tex":"X^d","aliases":"X^d","display":"X^d","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":44},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-095fecaa10ffef82","tex":"X_1^{a_1}","aliases":"X_1^a_1","display":"X_1^{a_1}","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73cc2f41cbac9cf2","tex":"X_d","aliases":"X_d","display":"X_d","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c070a883f58aab2d","tex":"x_{j,\\cdot}","aliases":"x_j,· cdot","display":"x_{j,\\cdot}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72c04885d210371b","tex":"x_L","aliases":"x_L","display":"x_L","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":34},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da9130fcd50973a4","tex":"X_n^{a_n}","aliases":"X_n^a_n","display":"X_n^{a_n}","kind":"introduction-passage","excerpt":"[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89d5e60ade50d285","tex":"X_p","aliases":"X_p","display":"X_p","kind":"introduction-passage","excerpt":"[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa3045354c2c4668","tex":"X_Q","aliases":"X_Q","display":"X_Q","kind":"introduction-passage","excerpt":"If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every intege","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-708bfd9721bf49c2","tex":"X_u","aliases":"X_u","display":"X_u","kind":"introduction-passage","excerpt":"Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6590090cc9a70e3b","tex":"x_u","aliases":"x_u","display":"x_u","kind":"introduction-passage","excerpt":"iangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"introduction-passage","excerpt":"iangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":34},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ef58ddaab7b5984","tex":"Y_1","aliases":"Y_1","display":"Y_1","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3dd104d33d62c251","tex":"Y_e","aliases":"Y_e","display":"Y_e","kind":"introduction-passage","excerpt":"Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46014f7f0635c4a0","tex":"y_{i,u}","aliases":"y_i,u","display":"y_{i,u}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39aa37dcc31c4c7d","tex":"Y_N","aliases":"Y_N","display":"Y_N","kind":"introduction-passage","excerpt":"Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ca25195475623b3","tex":"y_{\\rm even}","aliases":"y_rm even rm","display":"y_{\\rm even}","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e87c12f66aedae3","tex":"y_{\\rm odd}","aliases":"y_rm odd rm","display":"y_{\\rm odd}","kind":"introduction-passage","excerpt":"If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f998b273f69a0c77","tex":"z^{-d}","aliases":"z^-d","display":"z^{-d}","kind":"introduction-passage","excerpt":"[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original sampl","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-295aec691039ab9f","tex":"\\mathbb Z^d","aliases":"Z^d mathbb","display":"\\mathbb Z^d","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99f13701097c17ee","tex":"\\mathbb Z^e","aliases":"Z^e mathbb","display":"\\mathbb Z^e","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78a80f694d80e6e7","tex":"\\Z_2","aliases":"Z_2 Z mathbb Z_2 Z mathbf","display":null,"kind":"introduction-passage","excerpt":"Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.","count":33},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-370c87df70534fa4","tex":"\\mathbb Z_2","aliases":"Z_2 mathbb","display":"\\mathbb Z_2","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":59},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e15eb38946d0671a","tex":"\\Zp","aliases":"Zp Z_p mathbb","display":"{\\mathbb Z_p}","kind":"introduction-passage","excerpt":"Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets t","count":66},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-591f18106678b658","tex":"\\Zp^d","aliases":"Zp^d Z_p^d Zp mathbb","display":"{\\mathbb Z_p}^d","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df7860ea845cecca","tex":"\\Zp^r","aliases":"Zp^r Z_p^r Zp mathbb","display":"{\\mathbb Z_p}^r","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.","count":53},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2bb168c166c79f1d","tex":"\\Delta_1","aliases":"Δ_1 Delta","display":"\\Delta_1","kind":"introduction-passage","excerpt":"Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients ","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0b8c3548e9b0a2a5","tex":"\\Delta_1^s","aliases":"Δ_1^s Delta","display":"\\Delta_1^s","kind":"introduction-passage","excerpt":"Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients ","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae5d1ee4ca82d29f","tex":"\\delta_n","aliases":"δ_n delta","display":"\\delta_n","kind":"introduction-passage","excerpt":"Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primiti","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f800f69539a7f67","tex":"\\delta_{n0}","aliases":"δ_n0 delta","display":"\\delta_{n0}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4448d5af8e047c6e","tex":"\\Delta_u","aliases":"Δ_u Delta","display":"\\Delta_u","kind":"introduction-passage","excerpt":"Write Δ_u f(n)=f(n+u)-f(n) and use Δ_u^k for its k-fold iterate. Euler coefficients indexed by ordinary degree and by packet row are distinguished explicitly. A packet displacement by u rows is a degree displacement by Nu in a packet of width N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76aa50424676d8cf","tex":"\\Delta_u^k","aliases":"Δ_u^k Delta","display":"\\Delta_u^k","kind":"introduction-passage","excerpt":"Write Δ_u f(n)=f(n+u)-f(n) and use Δ_u^k for its k-fold iterate. Euler coefficients indexed by ordinary degree and by packet row are distinguished explicitly. A packet displacement by u rows is a degree displacement by Nu in a packet of width N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30d91374b7719e5f","tex":"\\Delta_{uh}^{k}","aliases":"Δ_uh^k Delta","display":"\\Delta_{uh}^{k}","kind":"introduction-passage","excerpt":"[All-address differences on the divisor branch] Let p be prime, m>0, mmid p-1, N=mp, and 0≤ j<N. Write h=(p-1)/m. For every u,k,n∈ N, U_N,j(n) · ∈ Z_p, Δ_uh^kU_N,j(n) · ∈ p^min(k,mn) Z_p, v_p(Δ_uh^kC_N,j(n)) · ≥min(k,mn)-v_p(j!). The factorial correction is part of the normalized statement, including addresses j≥ p.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0fcb1c3cf5e3bf16","tex":"\\delta_{\\alpha\\beta}","aliases":"δ_αβ delta alpha beta","display":"\\delta_{\\alpha\\beta}","kind":"introduction-passage","excerpt":"Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The es","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5381814dc2f38d7a","tex":"\\lambda^{-1}","aliases":"λ^-1 lambda","display":"\\lambda^{-1}","kind":"introduction-passage","excerpt":"[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by mul","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d96b536d679a03d2","tex":"\\lambda_J","aliases":"λ_J lambda","display":"\\lambda_J","kind":"introduction-passage","excerpt":"s over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K). This is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6baeb03c4146c133","tex":"\\lambda_K","aliases":"λ_K lambda","display":"\\lambda_K","kind":"introduction-passage","excerpt":"ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K). This is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c4d6b081f4bc9ca","tex":"\\pi^\\vee","aliases":"π^vee pi vee","display":"\\pi^\\vee","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-403b4646d62f0c8b","tex":"\\pi_i","aliases":"π_i pi","display":"\\pi_i","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f322ee5409feacc","tex":"\\pi_j","aliases":"π_j pi","display":"\\pi_j","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c26fffae89b46f83","tex":"\\rho_B","aliases":"ρ_B rho","display":"\\rho_B","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-936d7991f33298f9","tex":"\\rho_i","aliases":"ρ_i rho","display":"\\rho_i","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1ef5788f5c903d0","tex":"\\rho_{i,\\lambda}^{\\oplus c_i}","aliases":"ρ_i,λ^⊕ c_i rho lambda oplus","display":"\\rho_{i,\\lambda}^{\\oplus c_i}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7d25253bfbf71b9","tex":"\\rho_{i,\\lambda}^{\\oplus m_i}","aliases":"ρ_i,λ^⊕ m_i rho lambda oplus","display":"\\rho_{i,\\lambda}^{\\oplus m_i}","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6c67cd7deda6b91","tex":"\\rho_L","aliases":"ρ_L rho","display":"\\rho_L","kind":"introduction-passage","excerpt":"Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6eb70cc25d1e53f2","tex":"\\rho_L^F","aliases":"ρ_L^F rho","display":"\\rho_L^F","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-037b39a7fd3c33a4","tex":"\\rho_q","aliases":"ρ_q rho","display":"\\rho_q","kind":"introduction-passage","excerpt":"Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer m","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a37e389a69b4972","tex":"\\tau_m","aliases":"τ_m tau","display":"\\tau_m","kind":"introduction-passage","excerpt":"sufficiently large zero in the fundamental sector into one of these disks. The zero sets of G and D_N coincide because their prefactor is nonzero. Each disk is invariant under complex conjugation, and D_N has real coefficients. The conjugate of its unique zero is therefore the same zero. It is real, positive for large m, and simple by the derivativ","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-190c688268bfbad7","tex":"\\Phi^*","aliases":"Φ^* Phi","display":"\\Phi^*","kind":"introduction-passage","excerpt":"[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This h","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41c71b804cba2a03","tex":"\\chi_D","aliases":"χ_D chi","display":"\\chi_D","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b5b1f4eca28240d","tex":"\\chi_\\lambda","aliases":"χ_λ chi lambda","display":"\\chi_\\lambda","kind":"introduction-passage","excerpt":"Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70309d8763112e95","tex":"\\psi_{ij}","aliases":"ψ_ij psi","display":"\\psi_{ij}","kind":"introduction-passage","excerpt":"The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C inste","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":43},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a86ef458cc61e377","tex":"\\hbar","aliases":"ℏ hbar","display":"\\hbar","kind":"introduction-passage","excerpt":"of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivative","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-565b5cc28a0a3bd8","tex":"\\ell^p","aliases":"ℓ^p ell","display":"\\ell^p","kind":"introduction-passage","excerpt":"Let 1≤ p≤∞. For a countable set G, write ℓ^p(G) for the usual complex sequence space. For an integer d≥1, when G= Z^d, translation by a∈ Z^d is (T_ax)(n)=x(n-a). Let δ_0 be the impulse at the origin.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"introduction-passage","excerpt":"General finite-field Frobenius comparison. Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maxi","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-569331afbd5d2de3","tex":"\\nabla","aliases":"∇ nabla","display":"\\nabla","kind":"introduction-passage","excerpt":"A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7b13cbc252814fa","tex":"\\nabla^2","aliases":"∇^2 nabla","display":"\\nabla^2","kind":"introduction-passage","excerpt":"rms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior forms. Let P be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by ∇ω=P dω. Thus ∇^2 denotes the exterior covariant square, including the Lie bracket term in","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c9566cb52be5641","tex":"\\nabla_X","aliases":"∇_X nabla","display":"\\nabla_X","kind":"introduction-passage","excerpt":"A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3916bfdbd80e58c2","tex":"\\prod_{i=1}^{q}","aliases":"∏_i=1^q prod","display":"\\prod_{i=1}^{q}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ded5a9bc7d9bfb17","tex":"\\sum_{a=0}^{d}","aliases":"∑_a=0^d sum","display":"\\sum_{a=0}^{d}","kind":"introduction-passage","excerpt":"packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum an","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb36ae12635f2435","tex":"\\sum_{a=0}^{d-1}","aliases":"∑_a=0^d-1 sum","display":"\\sum_{a=0}^{d-1}","kind":"introduction-passage","excerpt":"Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a73e1778d27a812b","tex":"\\sum_{h\\ge0}","aliases":"∑_h≥0 sum ge","display":"\\sum_{h\\ge0}","kind":"introduction-passage","excerpt":"Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coeffici","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7a8cfbba485d0c4","tex":"\\sum_{i=1}^d","aliases":"∑_i=1^d sum","display":"\\sum_{i=1}^d","kind":"introduction-passage","excerpt":"i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair con","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-447b896d00a61af2","tex":"\\sum_{i\\in F}","aliases":"∑_i∈ F sum in","display":"\\sum_{i\\in F}","kind":"introduction-passage","excerpt":"dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been adde","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d57a4808dbd2b57e","tex":"\\sum_{j=0}^{d-1}","aliases":"∑_j=0^d-1 sum","display":"\\sum_{j=0}^{d-1}","kind":"introduction-passage","excerpt":"The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6821584482b27152","tex":"\\sum_{j\\in B(i)}","aliases":"∑_j∈ B(i) sum in","display":"\\sum_{j\\in B(i)}","kind":"introduction-passage","excerpt":"Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4886267106f9df63","tex":"\\sum_{k,s}","aliases":"∑_k,s sum","display":"\\sum_{k,s}","kind":"introduction-passage","excerpt":"[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f3736c162765e82","tex":"\\sum_{k=1}^{n}","aliases":"∑_k=1^n sum","display":"\\sum_{k=1}^{n}","kind":"introduction-passage","excerpt":"Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-366ac52aad5d30c9","tex":"\\sum_q","aliases":"∑_q sum","display":"\\sum_q","kind":"introduction-passage","excerpt":"vpage5A filter becomes a matrix of actual delaysFIR TRANSFER / A FOUR-TAP EXAMPLEChapter V: Theorem 3.1. Take the causal filter h=(1,2,-1,1), with taps at 0,1,2,3, acting by y[n]=∑_qh[q]f[n-q]. Resolve the input into even and odd channels. Let D denote one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contr","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-113d2a04b2350877","tex":"\\sum_{q=0}^{n}","aliases":"∑_q=0^n sum","display":"\\sum_{q=0}^{n}","kind":"introduction-passage","excerpt":"Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite ","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a8ae622c77aac5b","tex":"\\sum_{r=0}^m","aliases":"∑_r=0^m sum","display":"\\sum_{r=0}^m","kind":"introduction-passage","excerpt":"Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m r}, where the braces are Stirling numbers of the second kind. Each 2r!/(r+1) is dyadically integral. Indeed, if s=v_2(r+1)>0, then r≥2^s-1 and v_2(r!)≥ v_2((2^s-1)!)=2^s-1-s≥ s-1. The last inequality is ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4fed322c79bb1cd","tex":"\\sum_{r=0}^{n}","aliases":"∑_r=0^n sum","display":"\\sum_{r=0}^{n}","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1427dc433f3c26d","tex":"\\sum_{r\\in A_+}","aliases":"∑_r∈ A_+ sum in","display":"\\sum_{r\\in A_+}","kind":"introduction-passage","excerpt":"A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ea15b9f71040065","tex":"\\sum_{u\\to v}","aliases":"∑_u→ v sum to","display":"\\sum_{u\\to v}","kind":"introduction-passage","excerpt":"iangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6b0a941f2917366","tex":"\\sum_{\\alpha\\in I_0}","aliases":"∑_α∈ I_0 sum alpha in","display":"\\sum_{\\alpha\\in I_0}","kind":"introduction-passage","excerpt":"For s∈ S, write s=∑_α∈ I_0P_α s. If β is unreachable from I_0, then for every A∈ A, P_β As = ∑_α∈ I_0P_β AP_α s =0 by [source reference]. The same holds for every finite linear combination of orbit vectors.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b2a0f40b316674f","tex":"\\sum_{\\beta\\in I}","aliases":"∑_β∈ I sum beta in","display":"\\sum_{\\beta\\in I}","kind":"introduction-passage","excerpt":"[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2972272b24e9c45","tex":"\\int_0^t","aliases":"∫_0^t int","display":"\\int_0^t","kind":"introduction-passage","excerpt":"ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors h","count":39},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"introduction-passage","excerpt":"Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a258c5e7fc44d1aa","tex":"\\bigoplus_{i=1}^{q}","aliases":"⊕_i=1^q bigoplus","display":"\\bigoplus_{i=1}^{q}","kind":"introduction-passage","excerpt":"[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a13933c3ddf256f7","tex":"\\bigoplus_{p\\in\\Irr(\\bbk[t])}","aliases":"⊕_p∈Irr(bbk[t]) ⊕_p∈Irr(𝕜[t]) bigoplus in Irr bbk bigoplus in operatorname Bbbk","display":"\\bigoplus_{p\\in{\\operatorname{Irr}}({\\Bbbk}[t])}","kind":"introduction-passage","excerpt":"[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5aaaf24677ceb564","tex":"\\bigotimes","aliases":"⊗ bigotimes","display":"\\bigotimes","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00a1232f6f9fcef1","tex":"\\bigotimes_{i=1}^{s}","aliases":"⊗_i=1^s bigotimes","display":"\\bigotimes_{i=1}^{s}","kind":"introduction-passage","excerpt":"Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient to","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc9832e1e3b8da8d","tex":"\\otimes_{\\Lambda,\\epsilon}","aliases":"⊗_Λ,ϵ otimes Lambda epsilon","display":"\\otimes_{\\Lambda,\\epsilon}","kind":"introduction-passage","excerpt":"For M⊗_Λ,ϵ O, we use a right Λ module M. The left module convention is O⊗_Λ,ϵM. Throughout, a M denotes the augmentation relation submodule: it is generated by ma for right modules and by am for left modules, with a∈ a. This convention also applies to nonabelian groups in [source reference]. For abelian groups, both tensor orders agree over the com","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"introduction-passage","excerpt":"In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after cha","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-776b5d9d1f9b084d","tex":"\\operatorname*","aliases":"* operatorname","display":"\\operatorname*","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c33d6b793a3c093","tex":"+^{m-\\ell}","aliases":"+^m-ℓ ell","display":"+^{m-\\ell}","kind":"use","excerpt":"For 0≤ ℓ≤ m, the classical derivative of ϕ_m,τ on the two open sides is ϕ_m,τ^(ℓ)(t) = frac(t-τ)_+^m-ℓ(m-ℓ)!. This derivative has zero jump when ℓ<m and a unit jump when ℓ=m. Derivatives of order larger than m vanish on both open sides. It follows that subtracting the term a_q,mϕ_m,τ_q removes exactly the jump in the derivative of order m without c","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eab1b84529bdde00","tex":"\\bm1","aliases":"1 bm","display":"\\bm1","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de32a3ebb13b1924","tex":"\\bm1^T","aliases":"1^T bm","display":"\\bm1^T","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4a7b03f010bc3fc","tex":"\\mathbf1_{[\\tau,1]}","aliases":"1_[τ,1] mathbf tau","display":"\\mathbf1_{[\\tau,1]}","kind":"use","excerpt":"vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPChapter V: Theorem E7.2, Corollary E7.3. On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), s_f(t)=t^2. vfig[.92]ext_tra","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5397b370d7b6d1d0","tex":"\\mathbf1_{[\\tau,\\infty)}","aliases":"1_[τ,∞) mathbf tau infty","display":"\\mathbf1_{[\\tau,\\infty)}","kind":"use","excerpt":"vpage37Haar sees a jump only through its position in the supportSCALE SEPARATION / A QUANTIFIED THRESHOLDChapter V: Theorem E10.1, Theorem E10.2, Corollary E10.3. Use a normalized Haar wavelet on an available interval of length L on the real line, positive on its left half and negative on its right half. Here H_τ is the half-line step 1_[τ,∞), so t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6aa48613e1d44abf","tex":"\\mathbf1_{d\\equiv r\\ (N)}","aliases":"1_d≡ r (N) mathbf equiv","display":"\\mathbf1_{d\\equiv r\\ (N)}","kind":"use","excerpt":"Modules. mcodeCyclotomicFiltration, mcodePrimeBlockPolynomial, and mcodeCyclicAmiceCoefficients. 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementa","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22eacf5f4b39579e","tex":"\\mathbf1_{n_j\\text{ even}}","aliases":"1_n_j even mathbf text","display":"\\mathbf1_{n_j\\text{ even}}","kind":"use","excerpt":"The dyadic floors cannot be replaced by the odd-prime dimension formula. At stage j, the centralizer grade has dimension 2^d-j. Its contribution to J^1/H^1 is (2^d-j+1-2^d-j) 1_n_j even. Every n_j for j<d is even, whereas n_d=q_1 is odd and contributes zero. The sum is exactly N-2. BK nondegeneracy makes this a symplectic space. A whole source pack","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e24b41e93a816fc5","tex":"\\mathbf1_{n\\equiv r\\, (\\mathrm{mod}\\,N)}","aliases":"1_n≡ r (mod N) mathbf equiv mathrm","display":"\\mathbf1_{n\\equiv r\\, (\\mathrm{mod}\\,N)}","kind":"use","excerpt":"Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0914a0ccf44d4813","tex":"\\mathbf1_{r=0}","aliases":"1_r=0 mathbf","display":"\\mathbf1_{r=0}","kind":"use","excerpt":"For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =coefTdF_r-1-dcoefTdF_r. The initial values are F_r(0)=1_r=0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-156c89b978d080a5","tex":"\\mathbf1_{s=t,\\,n=0}","aliases":"1_s=t, n=0 mathbf","display":"\\mathbf1_{s=t,\\,n=0}","kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7df106ca60925507","tex":"\\mathbf1_{\\widehat\\Omega}","aliases":"1_widehatΩ mathbf widehat Omega","display":"\\mathbf1_{\\widehat\\Omega}","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df247a376c954e8d","tex":"\\mathbf1_{\\{|t|<1/4\\}}","aliases":"1_{|t|<1/4} mathbf","display":"\\mathbf1_{\\{|t|<1/4\\}}","kind":"use","excerpt":"vpage16Why a Fourier cutoff cannot reproduce this jump pointwiseA CONTINUOUS PULSE / EXACT ENDPOINT VALUESChapter V: Section 6, Section 8. Use a separate period-one example on [-1/2,1/2): j(t)= 1_{|t|<1/4}, S_mj(t)=tfrac12+frac2π∑_n=1^mfracsin(π n/2)ncos(2π nt). This is the exact finite Fourier formula. The drawn curves sample it numerically. The e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8cda8a6d7191dd29","tex":"\\mathbf1_\\Omega","aliases":"1_Ω mathbf Omega","display":"\\mathbf1_\\Omega","kind":"use","excerpt":"Write a 1_Ω-widehat a 1_widehatΩ =(a-widehat a) 1_Ω +widehat a( 1_Ω- 1_widehatΩ). Apply the triangle inequality and use 1_Ω- 1_widehatΩ _L^p =μ(ΩtrianglewidehatΩ)^1/p. For the supremum estimate, |1_Ω-1_widehatΩ|≤1 pointwise, and this function vanishes everywhere when the symmetric difference is empty. The same pointwise bound also bounds the essent","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0a175c4c1c25e09","tex":"\\mathbf1_{\\Omega}","aliases":"1_Ω mathbf Omega","display":"\\mathbf1_{\\Omega}","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-473264ce94dccbac","tex":"\\mathbf1_{\\Omega_q}","aliases":"1_Ω_q mathbf Omega","display":"\\mathbf1_{\\Omega_q}","kind":"use","excerpt":"[Edge-packet approximation class] There should exist natural function spaces in which a signal is represented as f=s+∑_q a_q1_Ω_q, with separate norms for smooth approximation, boundary geometry, packet amplitude, and packet identification. A projector-resolved approximation theory on such spaces should yield rates that do not mix smooth truncation","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da9ba20262f8dd56","tex":"\\mathbf1_{\\Omega\\triangle\\widehat\\Omega\\ne\\varnothing}","aliases":"1_ΩtrianglewidehatΩ≠∅ mathbf Omega triangle widehat Omega ne varnothing","display":"\\mathbf1_{\\Omega\\triangle\\widehat\\Omega\\ne\\varnothing}","kind":"use","excerpt":"orms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise supremum, and consequently also the essential L^∞ seminorm, j-widehat j _∞ ≤ |a-widehat a|+|widehat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3be3a4a3a44d636c","tex":"\\bm a","aliases":"a bm","display":"\\bm a","kind":"use","excerpt":"The Pauli identity ( a·σ)( b·σ) =( a· b)I + i( a× b)·σ gives - iTr(P[∂_θ P,∂_ϕ P]) =frac12 n· (∂_θ n×∂_ϕ n) =frac12sinθ. Integrating the displayed curvature over the sphere and applying the stated normalization gives one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0a61d58c33b3ed4","tex":"\\mathbf a","aliases":"a mathbf","display":"\\mathbf a","kind":"use","excerpt":"P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A is chosen. This explicit calc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86e9de44a233df86","tex":"\\mathrm A","aliases":"A mathrm","display":"\\mathrm A","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"R(a,b), R(a',b), R(a',b') Longrightarrow R(a,b')","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0aa786b643f5da71","tex":"A^{(1)}","aliases":"A^(1)","display":"A^{(1)}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6973a6c793fe345f","tex":"A^{(m)}","aliases":"A^(m)","display":"A^{(m)}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40d113efdc774b45","tex":"a^{(m)}","aliases":"a^(m)","display":"a^{(m)}","kind":"use","excerpt":"[Multiple selector] Define a^(m)∈ H( F_p) by a_n^(m) = 1, · mmid n,","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9d39a946bf223f1","tex":"\\mathfrak A^{(\\ell)}_{AB}","aliases":"A^(ℓ)_AB mathfrak ell","display":"\\mathfrak A^{(\\ell)}_{AB}","kind":"use","excerpt":"[Anomaly-free refinement] For a specified nontrivial quantum-geometric constraint model, there exists a coherent packet tower with nonzero physical spaces, compatible physical projectors and exact operator closure A^(ℓ)_AB=0 at every finite level. The compatible physical compression then vanishes as a consequence, not as a substitute for closure. T","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-677eab50351e7675","tex":"A^{-1}","aliases":"A^-1","display":"A^{-1}","kind":"use","excerpt":"[Nonlinear Schur derivative] At (u_0,v_0), DR = D_uF_P - D_vF_P A^-1D_uF_Q.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-280e15d5f806ea95","tex":"a^{-2}","aliases":"a^-2","display":"a^{-2}","kind":"use","excerpt":"ties give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd3dce9096a0d298","tex":"A^{2k}","aliases":"A^2k","display":"A^{2k}","kind":"use","excerpt":"In this particular model, A^2=8I. Therefore v_2(A^2k)=3k, v_2(A^2k+1)=3k+1, where the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo 32. Four terms leave valuation six and do suffice.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49c48961a3e05de8","tex":"A^{2k+1}","aliases":"A^2k+1","display":"A^{2k+1}","kind":"use","excerpt":"In this particular model, A^2=8I. Therefore v_2(A^2k)=3k, v_2(A^2k+1)=3k+1, where the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo 32. Four terms leave valuation six and do suffice.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a4d1b76a4ccae86","tex":"A^4","aliases":"A^4","display":"A^4","kind":"use","excerpt":"vpage15Walk counts make an endpoint cyclicGRAPH MODEL / THE PATH CERTIFICATEMain text: Worked graph model. Endpoint Krylov proof for every path size Let A be the adjacency matrix of the path with five vertices. The (i,j) entry of K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1] counts walks of length j-1 from vertex 1 to vertex i.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a3f87ebd468ea16","tex":"A^{b_1}","aliases":"A^b_1","display":"A^{b_1}","kind":"use","excerpt":"[Abelian cohomology class] When the structure group A is abelian, gauge classes of transports are naturally H^1(Γ,A) ≅ Hom(H_1(Γ, Z),A). After a basis of H_1(Γ, Z) is chosen, this group is identified with A^b_1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06d6907fddca4380","tex":"A^{h+1}","aliases":"A^h+1","display":"A^{h+1}","kind":"use","excerpt":"Nilpotence certificate. If no walk of length h+1 exists between any pair of packets, then A^h+1=0. Every block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map ℓ:I⟶{0,…,h}, A_bc≠0 Longrightarrow ℓ(c)<ℓ(b). It proves that a length-n walk increases the l","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afc7e5c12f805c36","tex":"A^j","aliases":"A^j","display":"A^j","kind":"use","excerpt":"With A=diag(1,2), the vectors v and Av=(1,2)^T are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when A=I_2. Both changes enlarge the addressing power, but in different ways. headWhen invariance is only approximate If A=D","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61bf579da1b5c924","tex":"a^j","aliases":"a^j","display":"a^j","kind":"use","excerpt":"Finite-exponential reconstruction and a cubic inverse. Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fab2b96c59335d56","tex":"A^{j-1-k}","aliases":"A^j-1-k","display":"A^{j-1-k}","kind":"use","excerpt":"Use the telescoping identity A^j-D^j = ∑_k=0^j-1 A^j-1-kKD^k. Each term has norm at most M^j-1normK. The vector D^jv lies in the direct sum of local Krylov spaces.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-943bc84d09f5dc3b","tex":"a^k","aliases":"a^k","display":"a^k","kind":"use","excerpt":"Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s). Induction and multiplication by g(n+s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-339ddf229ead4fcf","tex":"A^m","aliases":"A^m","display":"A^m","kind":"use","excerpt":"The hypothesis makes A block diagonal. Every power A^m is block diagonal with respect to the same projectors, and therefore so is the exponential series.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d14dbb0eba947d6f","tex":"A^{m+n}","aliases":"A^m+n","display":"A^{m+n}","kind":"use","excerpt":"Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be1f0ddb2c78b507","tex":"A^N","aliases":"A^N","display":"A^N","kind":"use","excerpt":"W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is a two-sided inverse of q_L(I+A), where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a52b1bc77ae2b6d","tex":"A^n","aliases":"A^n","display":"A^n","kind":"use","excerpt":"F_n+1(b,a) · =∑_c: A_bc≠0A_bcF_n(c,a). The checked identity is P_bA^nP_a=F_n(b,a) (n≥0). Its proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence the actual block of A^n vanishes.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b57dc62cca087611","tex":"A^{n-1}","aliases":"A^n-1","display":"A^{n-1}","kind":"use","excerpt":"…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry count","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd721bed307c9bfa","tex":"a^{p^L}","aliases":"a^p^L","display":"a^{p^L}","kind":"use","excerpt":"[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55aa8dc52ad3f209","tex":"a^{p^{\\ell-1}}","aliases":"a^p^ℓ-1 ell","display":"a^{p^{\\ell-1}}","kind":"use","excerpt":"ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf7da5da975b32de","tex":"a^{p^{\\ell-1}(p-1)}","aliases":"a^p^ℓ-1(p-1) ell","display":"a^{p^{\\ell-1}(p-1)}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0324c102d49c31af","tex":"A^q","aliases":"A^q","display":"A^q","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c03acd1d1b8f5556","tex":"A^{\\mathrm{symp}}","aliases":"A^symp mathrm","display":"A^{\\mathrm{symp}}","kind":"use","excerpt":"[Radical free envelope] For an arbitrary scalar projective action of A, the quotient A^symp=A/rad b_c is the maximal quotient detected by noncommuting projective intertwiners. If A is an elementary abelian p group, then A^symp is a symplectic F_p space.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f914025617b7ebc","tex":"A^T","aliases":"A^T","display":"A^T","kind":"use","excerpt":"vpage7A transpose, not an adjointKRONECKER ACTION AND ARRAY ORDERChapter VIII: Theorem 4.5, Corollary 4.6. With column-major vectorization, the tensor identity is (A⊗ B)vec(X)=vec(BXA^T). Here the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix A=mat1 · i","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75d47130ff67df22","tex":"a^T","aliases":"a^T","display":"a^T","kind":"use","excerpt":"Let A be a finite subgenerator: its off-diagonal entries are nonnegative and its column sums are nonpositive. These sign conditions alone allow nonabsorbing states. Eventual absorption is a separate hypothesis. Let p_0 be an initial transient-state distribution. Define p(t)=e^tAp_0, S(t)=1^Tp(t). The exit-rate row vector is a^T=-1^TA≥0.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37386c7055af10de","tex":"A^\\alpha","aliases":"A^α alpha","display":"A^\\alpha","kind":"use","excerpt":"Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db9c84c96b8fd1b3","tex":"a^{\\varphi(p^\\ell)}","aliases":"a^φ(p^ℓ) varphi ell","display":"a^{\\varphi(p^\\ell)}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b4a8da3e0ca9141","tex":"\\A^{\\oplus r}","aliases":"A^⊕ r A oplus mathcal oplus","display":"{\\mathcal A}^{\\oplus r}","kind":"use","excerpt":"Topological completion of a presentation. [Completion of a dense free presentation] Let X be a complete Hausdorff locally convex space, let A⊆End(X) act continuously, and let S=(s_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37e96daac260c605","tex":"A^{\\oplus2}","aliases":"A^⊕2 oplus","display":"A^{\\oplus2}","kind":"use","excerpt":"[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbddd5e376c69681","tex":"A_{-c}","aliases":"A_-c","display":"A_{-c}","kind":"use","excerpt":"d product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, star","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bdcba39a7197cb28","tex":"\\mathcal A_0","aliases":"A_0 mathcal","display":"\\mathcal A_0","kind":"use","excerpt":"Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions. No completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source referen","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1fc2375840e988ee","tex":"a_{0,1}","aliases":"a_0,1","display":"a_{0,1}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f75896d5ca1818d","tex":"A_{00}","aliases":"A_00","display":"A_{00}","kind":"use","excerpt":"-1 · -1, A^2=0. The coordinate block A_00 is nonzero, and a length-two active walk from zero to zero exists. Nevertheless the corresponding block of A^2 is zero. This is an exact counterexample to replacing support inclusion by equality.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ff83531e3c9698b","tex":"A_0^{-1}","aliases":"A_0^-1","display":"A_0^{-1}","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e29910acf04cde74","tex":"a_0^{-1}","aliases":"a_0^-1","display":"a_0^{-1}","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6214ded09a4fbf7a","tex":"a_0^{[M]}","aliases":"a_0^[M]","display":"a_0^{[M]}","kind":"use","excerpt":"For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f573136a2832fb7","tex":"a_{1,1}","aliases":"a_1,1","display":"a_{1,1}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40bae120bc963ffd","tex":"a_{1,g_1}","aliases":"a_1,g_1","display":"a_{1,g_1}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eefe7444a403f33e","tex":"A_1^s","aliases":"A_1^s","display":"A_1^s","kind":"use","excerpt":"Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53cc0413ede33538","tex":"\\mathcal A_2","aliases":"A_2 mathcal","display":"\\mathcal A_2","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5827c40753badd76","tex":"a_{2,1}","aliases":"a_2,1","display":"a_{2,1}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d325da4f9b58928d","tex":"A_3","aliases":"A_3","display":"A_3","kind":"use","excerpt":"6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2cfa673011ea148","tex":"a_3","aliases":"a_3","display":"a_3","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARChapter VII: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e513bab7c906c944","tex":"A_{3,1}","aliases":"A_3,1","display":"A_{3,1}","kind":"use","excerpt":"[Exact change between local source charts] Editorial range correction. This comparison holds for the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero. The eve","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74ec5788a006c41f","tex":"a_{3,1}","aliases":"a_3,1","display":"a_{3,1}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d84b5a24a5037067","tex":"A_4","aliases":"A_4","display":"A_4","kind":"use","excerpt":"[Exact fourth-order strings] For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2). The positive zeros are exactly τ_m = sqrt2π(m+frac12) = A_4(m+frac12). Rotation by the fourth roots of unity gives the real and imaginary zero strings. In this case the error term in [source reference] is identically zero.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60fcf83a29fad2ee","tex":"\\mathcal A_4","aliases":"A_4 mathcal","display":"\\mathcal A_4","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0d7ec79a910cff8","tex":"A_4^{-4s}","aliases":"A_4^-4s","display":"A_4^{-4s}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a457ef5cf975a843","tex":"A_4^{-s}","aliases":"A_4^-s","display":"A_4^{-s}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-986b06b713868bf2","tex":"A_4^4","aliases":"A_4^4","display":"A_4^4","kind":"use","excerpt":"For N=2, the reduced zeros are π^2(m+frac12)^2, m≥0. For N=4, [source reference] gives the reduced zeros A_4^4(m+frac12)^4, m≥0. Summing their negative powers proves the reduced formulas. The absolute formulas follow from Z_N^abs(s) = N Z_N^red(s/N).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c647bef050188403","tex":"\\mathcal A_a","aliases":"A_a mathcal","display":"\\mathcal A_a","kind":"use","excerpt":"A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a46d81e14e204be","tex":"\\mathcal A_{ab}","aliases":"A_ab mathcal","display":"\\mathcal A_{ab}","kind":"use","excerpt":"A constraint diagnostic that vanishes automatically. Section 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives C_aP=0 for every a. Absorb the factor i and any structure coefficient into f_c, and write A_ab=C_aC_b-C_bC_a-∑_cf_cC_c. [Exact common-kernel compression] If C_cP=0 for every","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb2b0edc94206176","tex":"\\mathfrak A_{AB}","aliases":"A_AB mathfrak","display":"\\mathfrak A_{AB}","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-efbc2c564bf50cd2","tex":"\\mathfrak A_{ab}","aliases":"A_ab mathfrak","display":"\\mathfrak A_{ab}","kind":"use","excerpt":"If structure coefficients f_ab^phantomabc are prescribed, define the fixed structure anomaly A_ab = [C_a,C_b] - i∑_cf_ab^phantomabcC_c. Exact realization of the prescribed algebra is the stronger condition A_ab=0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08eb65adb1a856b8","tex":"A_b","aliases":"A_b","display":"A_b","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af63f9fe0008dfda","tex":"a_{B^s f}","aliases":"a_B^s f","display":"a_{B^s f}","kind":"use","excerpt":"The earlier coefficient criterion is recovered exactly. Define the literal coefficient defect (δ a)(d)=a(d+p^2)-a(d+p). The new source proves, for every s,d, a_B^s f(d)=p^-s(δ^s a_f)(d). For an integral coefficient x, it proves the bridge p^-sx∈ Z_p Longleftrightarrow p^smid x in Z_p. Thus the differential definition is equivalent to the existing c","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-94cffe8dd0af28a5","tex":"A_{ba}","aliases":"A_ba","display":"A_{ba}","kind":"use","excerpt":"The elementary coordinate object is a complete family of orthogonal projectors P_aP_b=δ_abP_a, ∑_a∈ IP_a=I. Every vector has packet coordinates P_aψ. Every operator has transfer blocks A_ba=P_bAP_a. These identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction,","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7000b8c39e7a4cdb","tex":"A_{bc}","aliases":"A_bc","display":"A_{bc}","kind":"use","excerpt":"Composition. Inserting the projector identity between factors yields (AB)_ba=∑_c A_bcB_ca. If the left side is nonzero, at least one intermediate c has both A_bc≠0 and B_ca≠0. Consequently supp(AB)⊆supp(A)∘supp(B). No equality of supports is asserted.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66be131002b8b5af","tex":"A_C","aliases":"A_C","display":"A_C","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a4f27f07ce0c669","tex":"A_{c+2}","aliases":"A_c+2","display":"A_{c+2}","kind":"use","excerpt":"atives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting from the independently established integrality of the reciprocal G, proves that inversion negates the charge. F","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-408acdef742b081a","tex":"A_{c+d}","aliases":"A_c+d","display":"A_{c+d}","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2507e870c0b29f3","tex":"A_{c-1}","aliases":"A_c-1","display":"A_{c-1}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b081100956900811","tex":"A_c^s","aliases":"A_c^s","display":"A_c^s","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9fdabdeec2ad6682","tex":"A_{C_n}","aliases":"A_C_n","display":"A_{C_n}","kind":"use","excerpt":"Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To se","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3581b621d385d761","tex":"A_d","aliases":"A_d","display":"A_d","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a8e57a002dc32f4","tex":"a_d","aliases":"a_d","display":"a_d","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b862c67265e36c7","tex":"a_{d+1}","aliases":"a_d+1","display":"a_{d+1}","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5696e5795686355a","tex":"a_{d+pk}","aliases":"a_d+pk","display":"a_{d+pk}","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3652828b23145ef1","tex":"a_d^2","aliases":"a_d^2","display":"a_d^2","kind":"use","excerpt":". For actual Haar packets, set P_v=W_v^*W_v. Parent-to-child conservation gives P_v=P_v0+P_v1. Replacing unselected parents by their children until the complete pruning is reached proves ∑_v∈ΛP_v=I. This is why the coefficients at different depths still form an exact coordinate system. headAn exhaustive finite check At depth three the pruning count","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7764e407096f3048","tex":"A_e","aliases":"A_e","display":"A_e","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ff61cb24c6fadc0","tex":"A_{e_1}","aliases":"A_e_1","display":"A_{e_1}","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7766305f06ed3a3","tex":"A_{e_r}","aliases":"A_e_r","display":"A_{e_r}","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc727ddbdff7e587","tex":"a_f","aliases":"a_f","display":"a_f","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6aa43e91577b201e","tex":"a_{f^{-1}}","aliases":"a_f^-1","display":"a_{f^{-1}}","kind":"use","excerpt":"The reciprocal is integral. For f∈ I_p with constant coefficient one, the usual power-series reciprocal is used. Its inverse equation yields the exact coefficient recurrence a_f^-1(d)=δ_d0-∑_r=0^d-1 binom dr a_f^-1(r)a_f(d-r). Strong induction proves that all of these coefficients are integral. Combining this construction with the preceding all-ord","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6571548bdb8d4622","tex":"a_{fg}","aliases":"a_fg","display":"a_{fg}","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9234285c855dd5f","tex":"A_{\\frakp}","aliases":"A_frakp A_ p frakp mathfrak","display":"A_{{\\mathfrak p}}","kind":"use","excerpt":"If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjectiv","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5f0daaa0c763e4c","tex":"A_G","aliases":"A_G","display":"A_G","kind":"use","excerpt":"Let G be a nonempty finite graph with adjacency matrix A_G, and regard 𝕜^V(G) as a module over 𝕜[A_G]. The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73c1bf580d5c1dbc","tex":"A_g","aliases":"A_g","display":"A_g","kind":"use","excerpt":"Let G be a finite abelian group and let A=⊕_g∈ GA_g be a G graded module.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d83d124aaabaee7","tex":"a_g","aliases":"a_g","display":"a_g","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f6afa45de73ec0e","tex":"a_{H_ax+j}","aliases":"a_H_ax+j","display":"a_{H_ax+j}","kind":"use","excerpt":", v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74c2307a2295b9fc","tex":"A_{HH}","aliases":"A_HH","display":"A_{HH}","kind":"use","excerpt":"[Schur support inclusion] Split the packets into retained indices R and hidden indices H. Suppose the hidden block A_HH is invertible. The retained Schur complement is S = A_RR - A_RHA_HH^-1A_HR. A retained block (β,α) can be active only if it was already active in A_RR, or if there is a directed walk from α through hidden packets to β. The interna","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a8722c909d9ace9","tex":"A_{HH}^{-1}","aliases":"A_HH^-1","display":"A_{HH}^{-1}","kind":"use","excerpt":"1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-574944342ee92d11","tex":"A_{HR}","aliases":"A_HR","display":"A_{HR}","kind":"use","excerpt":"1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a88ad33e388dbded","tex":"a_{Hx+j}","aliases":"a_Hx+j","display":"a_{Hx+j}","kind":"use","excerpt":"means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required d","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79ce59fb60d59b25","tex":"A_{i+j}","aliases":"A_i+j","display":"A_{i+j}","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04ab0d31f8d5c7fc","tex":"a_{i,1}","aliases":"a_i,1","display":"a_{i,1}","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2409747eb360eb47","tex":"a_{i,2}","aliases":"a_i,2","display":"a_{i,2}","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5590ebce8ca407f5","tex":"a_{i,j}","aliases":"a_i,j","display":"a_{i,j}","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c32e4b0e0900b11","tex":"A_i^j","aliases":"A_i^j","display":"A_i^j","kind":"use","excerpt":"Exact and approximate packet Krylov spaces. textitSource items 7.2 and 7.3 Write K_m(A,v)=span{v,Av,…,A^m-1v}. For a complete finite projector family P_i, suppose A commutes with every P_i. Let A_i=P_iAP_i. Lean proves the actual orbit identity A^jv=∑_i A_i^jP_iv and therefore K_m(A,v)⊆ ⊕_i K_m(A_i,P_iv). The case j=0 uses ∑_iP_i=I. The zeroth loca","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1cea62f3228b8c95","tex":"A_{i_1}","aliases":"A_i_1","display":"A_{i_1}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de92b398cc45688a","tex":"A_{i_q}","aliases":"A_i_q","display":"A_{i_q}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":25},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2167913f89021b35","tex":"A_{j+1}","aliases":"A_j+1","display":"A_{j+1}","kind":"use","excerpt":"[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68fee95ac83fd47a","tex":"A_{j,e}","aliases":"A_j,e","display":"A_{j,e}","kind":"use","excerpt":"Packetwise mixed precision. At each stage let A_j=∑_e A_j,e, widehat A_j=∑_e(A_j,e+Δ_j,e), Δ_j,e=Δ_j,eP_j,a(e). The previous packet certificate supplies the uniform operator bound normwidehat A_j-A_j≤ δ_j:=∑_enormΔ_j,enormP_j,a(e). The new theorem substitutes these δ_j into the actual product-chain estimate. The resulting vector error is at most th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95713f689b2a581a","tex":"a_{j,g}","aliases":"a_j,g","display":"a_{j,g}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a75acd39d6ce208f","tex":"a_{j,k}","aliases":"a_j,k","display":"a_{j,k}","kind":"use","excerpt":"The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvemen","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0841717db411db13","tex":"A_{j-1}","aliases":"A_j-1","display":"A_{j-1}","kind":"use","excerpt":"Use the telescoping identity widehat A_m⋯widehat A_1 -A_m⋯ A_1 = ∑_j=1^m widehat A_m⋯widehat A_j+1 E_j A_j-1⋯ A_1. Take norms term by term.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91f49c5985231a83","tex":"A_{j\\ell}","aliases":"A_jℓ ell","display":"A_{j\\ell}","kind":"use","excerpt":"The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17821523641b2512","tex":"A_K","aliases":"A_K","display":"A_K","kind":"use","excerpt":"[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient fi","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"In the recurrence, set a_k=0 when k<0. This is the coefficient convention for the formal power series Z_u,v(t) and makes the first recurrence equations meaningful even when n<d.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8f016d5b63ca3cf","tex":"A_{k+1}","aliases":"A_k+1","display":"A_{k+1}","kind":"use","excerpt":"Exact arithmetic count. Each pair of slices forms one sum and one difference at every trailing coordinate. The event type therefore gives A_0=0, A_k+1=2A_k+2^k+1. The closed form and its equivalent logarithmic expression are both checked: A_k=k2^k=Nlog_2N. The logarithm in the formal count is the natural-number binary logarithm, with its power-of-t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e383dbc0f9ccfa0b","tex":"a_{k+1}","aliases":"a_k+1","display":"a_{k+1}","kind":"use","excerpt":"For any nonnegative decreasing sequence a_i, weights with these properties satisfy ∑_i a_iw_i≥∑_i>ka_i. This finite weighted inequality is itself proved. For k<d, use μ=a_k+1 and compare the summands to tail weights v_i=0 for i≤ k and v_i=1 otherwise: a_iv_i+μ(w_i-v_i)≤ a_iw_i. For a retained coordinate the inequality uses a_i≥μ and w_i≥0. For a ta","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-997309b6e5a8216f","tex":"a_{k+i}","aliases":"a_k+i","display":"a_{k+i}","kind":"use","excerpt":"Finite Newton reconstruction at an arbitrary row gives Δ_1^kf(n)=∑_i=0^nbinom ni a_k+i. The monotonicity of lfloorlog_2krfloor therefore preserves the higher-coefficient bound after translation. In addition, Δ_1f(n)-a_1∈2^c+1 Z_2, so the first difference is nonzero of exact valuation c.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da2a817868989569","tex":"A_{k-1}","aliases":"A_k-1","display":"A_{k-1}","kind":"use","excerpt":"Arbitrary tensor factors. textitSource item 4.6; module StructuredTensor Let A_0,…,A_k-1 be square factors of order d. Tensor coordinates are represented by a nested product of k copies of {0,…,d-1}. The empty product is a singleton. Lean proves that the coordinate type has cardinality d^k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-501fa09870ac360b","tex":"A_{ki}","aliases":"A_ki","display":"A_{ki}","kind":"use","excerpt":"vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"For a power series F(z)=∑_m≥ 0a_mz^m and a positive integer N, the coefficients may be separated according to the residue of m modulo N. Root of unity averaging gives exact projection operators for this separation. This elementary construction is usually called a multisection or a root filter. It appears throughout the classical theory of generatin","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e281fec2511c43be","tex":"a_{M(Nq+s)+r}","aliases":"a_M(Nq+s)+r","display":"a_{M(Nq+s)+r}","kind":"use","excerpt":"Direct substitution gives (a_r^[M])_s^[N](q) = a_M(Nq+s)+r. Iterate this identity along the mixed radix path.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c30d38a2eb4fbb92","tex":"a_{M(q+1)}","aliases":"a_M(q+1)","display":"a_{M(q+1)}","kind":"use","excerpt":"For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-605293bc8cd6a683","tex":"A_{m-1}","aliases":"A_m-1","display":"A_{m-1}","kind":"use","excerpt":"Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c5c098d009032a0","tex":"a_{m-1}","aliases":"a_m-1","display":"a_{m-1}","kind":"use","excerpt":"The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8cb01de33ecdba63","tex":"a_{M-1}^{[M]}","aliases":"a_M-1^[M]","display":"a_{M-1}^{[M]}","kind":"use","excerpt":"For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7dcee546650900ca","tex":"a_{\\min}","aliases":"a_min min","display":"a_{\\min}","kind":"use","excerpt":"The principal recovery theorem uses adjacent sample contrasts. Let b_h=Mh+2ε be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If a_min>2b_h, then every threshold between b_h and a_min-b_h detects exactly the cells containing jumps. The jump locations are recovered within h/2, and each amplit","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-317d704cb35a25a3","tex":"a_{Mq+r}","aliases":"a_Mq+r","display":"a_{Mq+r}","kind":"use","excerpt":"The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbd901b35afb50e7","tex":"a_{Mq+r+1}","aliases":"a_Mq+r+1","display":"a_{Mq+r+1}","kind":"use","excerpt":"For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"[Exact coefficient transport] Let A(t)=∑_n≥0A_nfract^nn!, S(t)=H(t)A(t). Then S_n = ∑_r=0^n binom nr h_rA_n-r. The inverse coordinate formula is A_n = ∑_r=0^n binom nr k_rS_n-r. Both transforms are lower triangular with diagonal one.","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e90c1adc20133ef","tex":"\\mathcal A_N","aliases":"A_N mathcal","display":"\\mathcal A_N","kind":"use","excerpt":"A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07112285aabb492d","tex":"a_N","aliases":"a_N","display":"a_N","kind":"use","excerpt":"The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be33c2dc1700652e","tex":"A_{n+1}","aliases":"A_n+1","display":"A_{n+1}","kind":"use","excerpt":"vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONChapter VI, Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c8131608f522ec2","tex":"a_{n+1}","aliases":"a_n+1","display":"a_{n+1}","kind":"use","excerpt":"Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f3deb161f4183b8","tex":"A_{N,j}","aliases":"A_N,j","display":"A_{N,j}","kind":"use","excerpt":"[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( -","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e20e5a9f02dabcc","tex":"A_{N,n}","aliases":"A_N,n","display":"A_{N,n}","kind":"use","excerpt":"[A sequence of binomial type] Define A_N,n(r) = fracn!(Nn)!P_N,n(r). Then A_N,n(r+s) = ∑_q=0^n binom nq A_N,q(r)A_N,n-q(s). Thus (A_N,n)_n≥0 is a polynomial sequence of binomial type.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f75a0da9dfc0cdb8","tex":"A_{N,n-q}","aliases":"A_N,n-q","display":"A_{N,n-q}","kind":"use","excerpt":"[A sequence of binomial type] Define A_N,n(r) = fracn!(Nn)!P_N,n(r). Then A_N,n(r+s) = ∑_q=0^n binom nq A_N,q(r)A_N,n-q(s). Thus (A_N,n)_n≥0 is a polynomial sequence of binomial type.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae9f2ce1dcdcc77f","tex":"A_{N,q}","aliases":"A_N,q","display":"A_{N,q}","kind":"use","excerpt":"[A sequence of binomial type] Define A_N,n(r) = fracn!(Nn)!P_N,n(r). Then A_N,n(r+s) = ∑_q=0^n binom nq A_N,q(r)A_N,n-q(s). Thus (A_N,n)_n≥0 is a polynomial sequence of binomial type.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ffe701ba2c672485","tex":"a_{n-3}","aliases":"a_n-3","display":"a_{n-3}","kind":"use","excerpt":"From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-13ab5ae246f6fa19","tex":"a_{n-d}","aliases":"a_n-d","display":"a_{n-d}","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2859c29e1e96b09d","tex":"a_{n-k}","aliases":"a_n-k","display":"a_{n-k}","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e4d74b5db7c0422","tex":"A_{n-r}","aliases":"A_n-r","display":"A_{n-r}","kind":"use","excerpt":"[Exact coefficient transport] Let A(t)=∑_n≥0A_nfract^nn!, S(t)=H(t)A(t). Then S_n = ∑_r=0^n binom nr h_rA_n-r. The inverse coordinate formula is A_n = ∑_r=0^n binom nr k_rS_n-r. Both transforms are lower triangular with diagonal one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b51e95719a5480d","tex":"a_{n-r}","aliases":"a_n-r","display":"a_{n-r}","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b14cc10bfd7bab79","tex":"a_{n-w(e)}","aliases":"a_n-w(e)","display":"a_{n-w(e)}","kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c376ef200c86635","tex":"a_n^{(m)}","aliases":"a_n^(m)","display":"a_n^{(m)}","kind":"use","excerpt":"[Multiple selector] Define a^(m)∈ H( F_p) by a_n^(m) = 1, · mmid n, 0, · mnmid n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8256bcc167667e2","tex":"A_N^{-Ns}","aliases":"A_N^-Ns","display":"A_N^{-Ns}","kind":"use","excerpt":"[Meromorphic continuation of the symmetry-orbit zero zeta] Put A_N=fracπsin(π/N). There is an entire function Q_N^red such that Z_N^red(s) = A_N^-Nsζ(Ns,frac12)+Q_N^red(s) on Re s>1/N. The right-hand side gives a meromorphic continuation to C with one simple pole at s=1/N. Its residue is fracsin(π/N)Nπ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0691165b07852974","tex":"A_N^{-s}","aliases":"A_N^-s","display":"A_N^{-s}","kind":"use","excerpt":"There is an entire function Q_N^abs such that, initially on Re s>1, Z_N^abs(s) = N A_N^-sζ(s,frac12)+Q_N^abs(s). Thus Z_N^abs continues meromorphically to C with one simple pole at s=1, of residue fracNsin(π/N)π.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07043dd4b1cb5784","tex":"A_n^{1/2}","aliases":"A_n^1/2","display":"A_n^{1/2}","kind":"use","excerpt":"at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVEChapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db660538430f2d4b","tex":"A_p","aliases":"A_p","display":"A_p","kind":"use","excerpt":"For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36c90390c8711691","tex":"A_{p,2m}","aliases":"A_p,2m","display":"A_{p,2m}","kind":"use","excerpt":"[Unified local Euler transfer] For every prime p and every even n≥0, E_n = ∑_r=0^n/2 binom n2r κ_p,r A_p,n-2r. If H_p(t) = ∑_m≥0A_p,2mfract^2m(2m)! and K_p(t) = ∑_r≥0κ_p,rfract^2r(2r)!, then K_p(t)H_p(t)=secht. For odd p, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). For p=2, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-678c8828f3bebbd3","tex":"A_{p,n}","aliases":"A_p,n","display":"A_{p,n}","kind":"use","excerpt":"For a prime p, define A_p,n = S_n(p), · p odd, R_n, · p=2 and n even, 0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd,","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c6d5229ad08c832","tex":"A_{p,n-2r}","aliases":"A_p,n-2r","display":"A_{p,n-2r}","kind":"use","excerpt":"[Unified local Euler transfer] For every prime p and every even n≥0, E_n = ∑_r=0^n/2 binom n2r κ_p,r A_p,n-2r. If H_p(t) = ∑_m≥0A_p,2mfract^2m(2m)! and K_p(t) = ∑_r≥0κ_p,rfract^2r(2r)!, then K_p(t)H_p(t)=secht. For odd p, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). For p=2, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4514e57e9986c46","tex":"a_{p-1}","aliases":"a_p-1","display":"a_{p-1}","kind":"use","excerpt":"Then v_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic minimal polynomial f(X)=X^p+a_p-1X^p-1+⋯+a_0 is Eisenstein. It is an explicit polynomial: compute the product over the p conjugates","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04fb14fb3279afa6","tex":"A_{P_n}","aliases":"A_P_n","display":"A_{P_n}","kind":"use","excerpt":"For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab0fbf60c4efcc89","tex":"A_{p_u}","aliases":"A_p_u","display":"A_{p_u}","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5bca16056070f59f","tex":"A_{p_u}^{-1}","aliases":"A_p_u^-1","display":"A_{p_u}^{-1}","kind":"use","excerpt":"ct edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe95637e997d66bd","tex":"A_{p_v}^{-1}","aliases":"A_p_v^-1","display":"A_{p_v}^{-1}","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61ec922a7b66b76d","tex":"A_{PP}","aliases":"A_PP","display":"A_{PP}","kind":"use","excerpt":"Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-358c08cae43f1364","tex":"A_{PQ}","aliases":"A_PQ","display":"A_{PQ}","kind":"use","excerpt":"Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-394b9fed27d5557c","tex":"a_{q,m}","aliases":"a_q,m","display":"a_{q,m}","kind":"use","excerpt":"egintheorem[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) an","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6fa2f03584287ecd","tex":"A_{QP}","aliases":"A_QP","display":"A_{QP}","kind":"use","excerpt":"Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e168c25ce0b085c","tex":"A_{QQ}^k","aliases":"A_QQ^k","display":"A_{QQ}^k","kind":"use","excerpt":"sure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2277836fd5789d74","tex":"A_R","aliases":"A_R","display":"A_R","kind":"use","excerpt":"Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a charac","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92d57bfe67d21ec0","tex":"a_{r+1}^{[M]}","aliases":"a_r+1^[M]","display":"a_{r+1}^{[M]}","kind":"use","excerpt":"For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3416243cb2455e6","tex":"a_{r+Ms}^{[MN]}","aliases":"a_r+Ms^[MN]","display":"a_{r+Ms}^{[MN]}","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a65c3129d1f56f5","tex":"a_{r,g_r}","aliases":"a_r,g_r","display":"a_{r,g_r}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-357c0cfc7297df89","tex":"A_r^{[M]}","aliases":"A_r^[M]","display":"A_r^{[M]}","kind":"use","excerpt":"[Exact discrete polyphase reconstruction] The analysis map P_M is a bijection between sequences and ordered families of M sequences. If A(z)=∑_n≥0a_nz^n and A_r^[M](w)=∑_q≥0a_Mq+rw^q, then A(z)=∑_r=0^M-1z^rA_r^[M](z^M).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f45d14646d94b29","tex":"a_r^{[M]}","aliases":"a_r^[M]","display":"a_r^{[M]}","kind":"use","excerpt":"The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f627a19ca37cdde","tex":"a_r^T","aliases":"a_r^T","display":"a_r^T","kind":"use","excerpt":"[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-766daa38933a52e5","tex":"A_{ra}","aliases":"A_ra","display":"A_{ra}","kind":"use","excerpt":"Deriving the Gram matrix from recovery. Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa4ce98148303be9","tex":"A_{ra}^*","aliases":"A_ra^*","display":"A_{ra}^*","kind":"use","excerpt":"Deriving the Gram matrix from recovery Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b601bf3f58468ef","tex":"A_{RH}","aliases":"A_RH","display":"A_{RH}","kind":"use","excerpt":"1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d71f286ebb4773fa","tex":"a_{\\rm root}","aliases":"a_rm root rm","display":"a_{\\rm root}","kind":"use","excerpt":"apter V: Proposition E6.4. Suppose the root split has nonnegative squared-norm bounds A=1/2 and B=2. Retain the left child. Split the right child with bounds 3/4 and 5/4, then retain both grandchildren. vfig[.85]ext_tree headStart with one at every retained node The left subtree has bounds a_0=b_0=1. The right subtree has a_1=3/4 and b_1=5/4. The r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56009cd5bba2cd88","tex":"A_{RR}","aliases":"A_RR","display":"A_{RR}","kind":"use","excerpt":"1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65c56b1be2f0aadb","tex":"A_T","aliases":"A_T","display":"A_T","kind":"use","excerpt":"[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet i","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f1eee202dc651a9","tex":"A_u","aliases":"A_u","display":"A_u","kind":"use","excerpt":"a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d700de4fae3fdd0a","tex":"a_{Uf}","aliases":"a_Uf","display":"a_{Uf}","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0083f442b19187bb","tex":"A_v","aliases":"A_v","display":"A_v","kind":"use","excerpt":"[Bounds for nonunitary local splits] Suppose every internal node has an analysis map U_v and constants 0≤ A_v≤ B_v<∞ satisfying A_v x ^2 ≤ ∑_w∈child(v) (U_vx)_w ^2 ≤ B_v x ^2. Define recursively a(v)= 1, · v retained,","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ee7d787d8b86a7e","tex":"A_{vu}","aliases":"A_vu","display":"A_{vu}","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dbdc2defb45a0c61","tex":"A_{vu}^{-1}","aliases":"A_vu^-1","display":"A_{vu}^{-1}","kind":"use","excerpt":"Conversely, choose a spanning tree. Starting from U_r, propagate U_v along the unique tree path by the rule U_v=B_vuU_uA_vu^-1 on every tree edge. This is the only possible extension satisfying the tree edge equations. An edge outside the tree closes one fundamental loop. Its intertwining equation holds exactly when U_r intertwines the two correspo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormali","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4331faa8dd20c3ee","tex":"A_w^*","aliases":"A_w^*","display":"A_w^*","kind":"use","excerpt":"We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormali","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7dc2ed692bd1496","tex":"A_{z_n}","aliases":"A_z_n","display":"A_{z_n}","kind":"use","excerpt":"The one-step Taylor theorem already supplies the exact correction equation b_n+A_z_nt=0, b_n=F(z_n)/p^L+nbmod p. Surjectivity supplies a correcting digit for every residual b_n. The chosen correction gives an actual new integer point z_n+1, using the proved nonlinear Taylor remainder.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6fe7588fe8a751d","tex":"A_{z_{n+1}}","aliases":"A_z_n+1","display":"A_{z_{n+1}}","kind":"use","excerpt":"Every coordinate change is divisible by p^L+n, hence by p. Each formal partial derivative is itself an integer polynomial, so its value is unchanged modulo p. Therefore A_z_n+1=A_z_n=A_x. The proof assumes no separate surjectivity or solvability statement at later levels. A dependent recursion chooses each next point from the proved one-step existe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-edac8f4090c64b8b","tex":"A_\\alpha","aliases":"A_α alpha","display":"A_\\alpha","kind":"use","excerpt":"[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-84b88103b8092dfe","tex":"A_\\alpha^j","aliases":"A_α^j alpha","display":"A_\\alpha^j","kind":"use","excerpt":"[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-903a870c12766098","tex":"A_{\\alpha_1\\alpha_0}","aliases":"A_α_1α_0 alpha alpha","display":"A_{\\alpha_1\\alpha_0}","kind":"use","excerpt":"[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb2c326e063bfa7f","tex":"A_{\\alpha_m\\alpha_{m-1}}","aliases":"A_α_mα_m-1 alpha alpha","display":"A_{\\alpha_m\\alpha_{m-1}}","kind":"use","excerpt":"[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f580c536293e4a61","tex":"A_{\\alpha\\alpha}","aliases":"A_αα alpha alpha","display":"A_{\\alpha\\alpha}","kind":"use","excerpt":"thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c0b084cbb40137f","tex":"A_{\\alpha\\gamma}","aliases":"A_αγ alpha gamma","display":"A_{\\alpha\\gamma}","kind":"use","excerpt":"The first two conditions are equivalent because the packet resolution gives AP_α=∑_β A_βα, P_α A=∑_γ A_αγ. A block diagonal generator has block diagonal powers, so the second condition implies the third. Differentiating the off-diagonal blocks at zero proves the converse. The equivalence with invariance follows from the definition of the blocks. Th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5498b3bc9fb8719d","tex":"A_{\\beta\\alpha}","aliases":"A_βα beta alpha","display":"A_{\\beta\\alpha}","kind":"use","excerpt":"[Reducing packet criterion] The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-935d18e87c9add19","tex":"A_{\\beta\\alpha}^*","aliases":"A_βα^* beta alpha","display":"A_{\\beta\\alpha}^*","kind":"use","excerpt":"[Adjoint and tensor semantics] The adjoint representation has blocks (A^*)_αβ=A_βα^*. If A and B are packet resolved on X and Y, then (A⊗ B)_(β,δ),(α,γ) = A_βα⊗ B_δγ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2141b0918d8b84b6","tex":"A_{\\beta\\gamma}","aliases":"A_βγ beta gamma","display":"A_{\\beta\\gamma}","kind":"use","excerpt":"[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b978ebe79a99278","tex":"A_{\\gamma\\alpha}","aliases":"A_γα gamma alpha","display":"A_{\\gamma\\alpha}","kind":"use","excerpt":"[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31bff3b683a68d27","tex":"a_{\\lambda}","aliases":"a_λ lambda","display":"a_{\\lambda}","kind":"use","excerpt":"[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N:","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e792db7aa03cd7f8","tex":"a_{\\rho,r}","aliases":"a_ρ,r rho","display":"a_{\\rho,r}","kind":"use","excerpt":"_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9db24aaa1cf5ff2","tex":"a_{\\rho,\\ell}","aliases":"a_ρ,ℓ rho ell","display":"a_{\\rho,\\ell}","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-263b1db52659f9f9","tex":"A_\\ell","aliases":"A_ℓ ell","display":"A_\\ell","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3a52d54c059cb53","tex":"a_\\ell","aliases":"a_ℓ ell","display":"a_\\ell","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-744d6de0e15e8c83","tex":"A_{\\ell+1}","aliases":"A_ℓ+1 ell","display":"A_{\\ell+1}","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6af66e560dfcfb03","tex":"\\mathfrak A_{\\ell+1,ab}","aliases":"A_ℓ+1,ab mathfrak ell","display":"\\mathfrak A_{\\ell+1,ab}","kind":"use","excerpt":"[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf089dc8632eb091","tex":"A_{\\ell+1}^{\\alpha}","aliases":"A_ℓ+1^α ell alpha","display":"A_{\\ell+1}^{\\alpha}","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-957494c7380f5756","tex":"\\mathfrak A_{\\ell,ab}","aliases":"A_ℓ,ab mathfrak ell","display":"\\mathfrak A_{\\ell,ab}","kind":"use","excerpt":"[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6daaf6159d130196","tex":"A_{\\ell,n}","aliases":"A_ℓ,n ell","display":"A_{\\ell,n}","kind":"use","excerpt":". For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero. The even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity H_p=L_pK_ℓ H_ℓ and its proof remain valid. For any two primes p and ℓ, the source sequences A_p,n a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8cb55ed44edecb02","tex":"a_\\ell^k","aliases":"a_ℓ^k ell","display":"a_\\ell^k","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-102f6cf50d31e579","tex":"A_\\ell^{\\alpha}","aliases":"A_ℓ^α ell alpha","display":"A_\\ell^{\\alpha}","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b3267a5b5057d4c","tex":"A_\\infty","aliases":"A_∞ infty","display":"A_\\infty","kind":"use","excerpt":"If T_ℓ(t)= e^tA_ℓ, the generators are bounded, and sup_ℓnormA_ℓ<∞, C_ℓ A_ℓ+1=A_ℓ C_ℓ, then the continuity condition holds and A_∞(x_ℓ)_ℓ=(A_ℓ x_ℓ)_ℓ is the bounded generator of T_∞.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-133aeb6702a1cdfe","tex":"\\AA","aliases":"AA AA A mathcal","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-483b6b4d696310c2","tex":"\\Acal'","aliases":"Acal' A' Acal mathcal","display":"{\\mathcal A}'","kind":"use","excerpt":"[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-239bae8ba3f949d4","tex":"\\Acal^*","aliases":"Acal^* A^* Acal mathcal","display":"{\\mathcal A}^*","kind":"use","excerpt":"Differentiate V^*V=I to obtain A^*=- A. The gauge transformation law follows by differentiating VG. Since P=VV^* and (I-P) d V= d P V, direct expansion gives d A+ Awedge A = d V^*(I-P)wedge d V = V^*( d Pwedge d P)V.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-767a2fc9b6c07bff","tex":"\\Acal_E","aliases":"Acal_E A_E Acal mathcal","display":"{\\mathcal A}_E","kind":"use","excerpt":"[Frame bounds from the polyphase symbol] Assume that A I_N ≤ E( e^ iω)^*E( e^ iω) ≤ B I_N for almost every ω, where 0<A≤ B<∞. Then the analysis bank satisfies A x _2^2 ≤ A_E x _2^2 ≤ B x _2^2. The pointwise matrix R_rm can( e^ iω) = (E^*E)^-1E^* is a bounded left inverse and gives the canonical dual synthesis bank.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bafde82ae4b33f78","tex":"\\Acal_L^{-1}","aliases":"Acal_L^-1 A_L^-1 Acal mathcal","display":"{\\mathcal A}_L^{-1}","kind":"use","excerpt":"[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-673f6a95cc986ca7","tex":"\\Acal_{\\Lcal}","aliases":"Acal_Lcal A_ L Acal Lcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal L}}","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dab36f95bbad8317","tex":"\\Acal_{\\Lcal}^{-1}","aliases":"Acal_Lcal^-1 A_ L^-1 Acal Lcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal L}}^{-1}","kind":"use","excerpt":"The synthesis map is unitary by [source reference]. Hence widehat x-x = A_ L^-1e has the same norm as e.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2acd68c5aa6b021","tex":"\\Acal_{\\Rcal}","aliases":"Acal_Rcal A_ R Acal Rcal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal R}}","kind":"use","excerpt":"The identities S_ P A_ P=I_X and S_ Q A_ Q=I_Y give S_ QF_ Q← P A_ P = S_ Q A_ QF S_ P A_ P =F. For composition, G_ R← QF_ Q← P = A_ RG S_ Q A_ QF S_ P = A_ R(G∘ F) S_ P.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b340b7ad7452d3e","tex":"\\Acal_{\\Ucal}","aliases":"Acal_Ucal A_ U Acal Ucal mathcal mathcal","display":"{\\mathcal A}_{{\\mathcal U}}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99fa55a48bd5ebd6","tex":"\\mathrm{act}","aliases":"act mathrm","display":"\\mathrm{act}","kind":"use","excerpt":"ion K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73fff7053fd2794b","tex":"\\Ad","aliases":"Ad operatorname","display":"{\\operatorname{Ad}}","kind":"use","excerpt":"Let S be unitary and N≥1 an integer with S^N=I, and define the conjugation action Ad_S(X)=SXS^*.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bdbd1a428e5dd1c0","tex":"\\ad","aliases":"ad operatorname","display":"{\\operatorname{ad}}","kind":"use","excerpt":"[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ee48b451744dd8e","tex":"\\ad^0","aliases":"ad^0 ad operatorname","display":"{\\operatorname{ad}}^0","kind":"use","excerpt":"[Adequate image] Let G⊆GL(V) act absolutely irreducibly over a finite field k of characteristic p. Under the standard definition, adequacy imposes vanishing conditions on H^1(G,k) and H^1(G,ad^0V), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of ad^0V.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b718946e6494e709","tex":"\\Ad_S","aliases":"Ad_S Ad operatorname","display":"{\\operatorname{Ad}}_S","kind":"use","excerpt":"Let S be unitary and N≥1 an integer with S^N=I, and define the conjugation action Ad_S(X)=SXS^*.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d201a3b99225b0bb","tex":"\\Ad_{U(g)}","aliases":"Ad_U(g) Ad operatorname","display":"{\\operatorname{Ad}}_{U(g)}","kind":"use","excerpt":"Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e51a60f00f5ac7f","tex":"\\Ad_{U(g)^*}","aliases":"Ad_U(g)^* Ad operatorname","display":"{\\operatorname{Ad}}_{U(g)^*}","kind":"use","excerpt":"Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74fd5e15b7fae5b2","tex":"\\operatorname{adj}","aliases":"adj operatorname","display":"\\operatorname{adj}","kind":"use","excerpt":"ce every entry of A(t) has zero constant term, the inverse exists in the formal power series ring and (I-A(t))^-1=∑_r≥0A(t)^r. The (i,j) entry of A(t)^r is the sum of the monomials attached to all paths of length r from i to j. Summing over r records paths of every length. Multiplication by the endpoint vectors gives the stated series. The adjugate","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0287e4860e3449ce","tex":"\\operatorname{analyze}","aliases":"analyze operatorname","display":"\\operatorname{analyze}","kind":"use","excerpt":"The verified contract is synthesize(analyze(x,N))=x. The Lean counterpart uses packet coordinates restricted to projector ranges and proves both inverse laws. The finite Python routine checks the concrete array realization.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87708a8e37000af2","tex":"\\operatorname{Ann}","aliases":"Ann operatorname","display":"\\operatorname{Ann}","kind":"use","excerpt":"[Presentation with one seed] For s∈ M, the annihilator Ann_A(s):={a∈ A:as=0} is a left ideal and A/Ann_A(s)≅Hull_A(s). [source reference]","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eca8c78dc9b64bd4","tex":"\\operatorname{Ann}_A","aliases":"Ann_A operatorname","display":"\\operatorname{Ann}_A","kind":"use","excerpt":"[Presentation with one seed] For s∈ M, the annihilator Ann_A(s):={a∈ A:as=0} is a left ideal and A/Ann_A(s)≅Hull_A(s). [source reference]","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e2b11a7001e1904","tex":"\\arg","aliases":"arg","display":"\\arg","kind":"use","excerpt":"If z=x+iy lies in the sector |arg z|≤ a_N, has x≥ R, and is a zero of D_N, then |y|≤ Y.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4afb314fdcd4106f","tex":"\\arraybackslash","aliases":"arraybackslash","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-efbaeabd3147a09d","tex":"\\Art","aliases":"Art operatorname","display":"{\\operatorname{Art}}","kind":"use","excerpt":"[Conductor of a nontrivial tame atom] Let τ be an irreducible tame Weil representation of dimension n>1. Then Swan(τ)=0, dimτ^I_K=0, Art(τ)=n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef843f340bfccbd5","tex":"\\operatorname{Art}","aliases":"Art operatorname","display":"\\operatorname{Art}","kind":"use","excerpt":"The dimension and orbit follow from [source reference]. The cycle polynomial is [source reference]. For rank greater than one, the conductor is [source reference]. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula Art=dim V-dim V^I_K+Sw. The local polynomial is [source ref","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-703614cdc8657cbf","tex":"\\operatorname{aug}","aliases":"aug operatorname","display":"\\operatorname{aug}","kind":"use","excerpt":"[Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2619e4f89a651f69","tex":"\\operatorname{Aut}_K","aliases":"Aut_K operatorname","display":"\\operatorname{Aut}_K","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3d4d90b8afc3ca5","tex":"\\bm B","aliases":"B bm","display":"\\bm B","kind":"use","excerpt":"Finite convolution represents Laurent-polynomial multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let B= R E-cz^-dI_N and suppose its action vanishes on every channel sequence. Insert an impulse at channel s, address zero, with every other c","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42648425616e1ec3","tex":"\\bm b","aliases":"b bm","display":"\\bm b","kind":"use","excerpt":"The Pauli identity ( a·σ)( b·σ) =( a· b)I + i( a× b)·σ gives - iTr(P[∂_θ P,∂_ϕ P]) =frac12 n· (∂_θ n×∂_ϕ n) =frac12sinθ. Integrating the displayed curvature over the sphere and applying the stated normalization gives one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1267a67d957e703c","tex":"\\mathrm B","aliases":"B mathrm","display":"\\mathrm B","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-899a52e5583edff3","tex":"\\mathrm b","aliases":"b mathrm","display":"\\mathrm b","kind":"use","excerpt":"[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"use","excerpt":"R(a,b), R(a',b), R(a',b') Longrightarrow R(a,b')","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3079afd29fa1e022","tex":"b^{(m,p)}","aliases":"b^(m,p)","display":"b^{(m,p)}","kind":"use","excerpt":"Since the subgroup average has augmentation one, it has a unique Hurwitz inverse. Write b^(m,p) = (a^(m))^-1.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"Write B=1 · -1 0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e36ace1e3c65141c","tex":"b^{-2}","aliases":"b^-2","display":"b^{-2}","kind":"use","excerpt":"ties give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f76a26dbc0d9fe26","tex":"B^2","aliases":"B^2","display":"B^2","kind":"use","excerpt":"One sufficient new arithmetic input would be an odd-denominator projective quadratic A Z^2+2BZ+C≡ A W_r(Z)pmod2^P, A odd, 0<|B^2-AC|<2^P, at a proved precision for every admitted ordinary r. Then (B^2-AC)/A^2≡δ_rpmod2^P cannot vanish modulo 2^P. No depth-independent theorem supplying this inequality has been established. The tested recurrence and e","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d99261575cf632ab","tex":"\\mathcal B^2","aliases":"B^2 mathcal","display":"\\mathcal B^2","kind":"use","excerpt":"The variable B=2Z+1 has transform e^tfrac2te^2t-1=fractsinh t. Its odd moments vanish, and P=( B^2-1)/4=Z(Z+1) has the divided-power moments of Lemma~[source reference]. Introduce an independent copy D and Q=( D^2-1)/4. The exact factorization F_2(t)=fract^2cosh t-cos t =fracatsinh(at)fracbtsinh(bt) shows that U=a B+b D satisfies L(U^4n)=C_2(n). Di","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"vfig[.97]weighted_word_cutoff platecaptionAll nonzero retained terms of the signed inverse sum, followed by the first discarded nonzero word. The empty word contributes I. Other words of weight below five can still be zero because a^2=b^2=0.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab3ea1299ebfd7b0","tex":"B^{[j]}","aliases":"B^[j]","display":"B^{[j]}","kind":"use","excerpt":"The field degrees and critical levels of B^[j] are exactly","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1bdcbb0ff59dbc74","tex":"b^p","aliases":"b^p","display":"b^p","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-139ce225a5a62faf","tex":"b^{p^2}","aliases":"b^p^2","display":"b^{p^2}","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-acb79e87d28a764d","tex":"B^r","aliases":"B^r","display":"B^r","kind":"use","excerpt":"means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ad6156b5e42899a","tex":"B^s","aliases":"B^s","display":"B^s","kind":"use","excerpt":"All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ff3305ee67cffa6","tex":"B^\\times","aliases":"B^× times","display":"B^\\times","kind":"use","excerpt":"Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)= {χ:G_ Q⟶ B^×midker q⊆kerχ}, ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-340922fec8f1f7a7","tex":"B^\\alpha","aliases":"B^α alpha","display":"B^\\alpha","kind":"use","excerpt":"Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b1e637f78e0d2bb","tex":"B_{\\mathrm F}^{2}","aliases":"B_ F^2 mathrm","display":"B_{\\mathrm F}^{2}","kind":"use","excerpt":"For the Frobenius norm, trace cyclicity proves left and right isometric invariance. The proof also establishes the projection identity normB-QQ^*B_ F^2 =norm B_ F^2-normQ^*B_ F^2. Nonnegativity of the left side gives left compression contractivity. The right version follows by adjoints. No norm-preservation or contractivity hypothesis is inserted.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator a","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7dc72fd5fdc5b850","tex":"b_{0,r}","aliases":"b_0,r","display":"b_{0,r}","kind":"use","excerpt":"Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the forma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","excerpt":"[Exact four-address first differences] For every n≥0, H>0, and j∈{0,1,2,3}, v_2(C_j(n+H)-C_j(n)) · =v_2(H)+b_j, · (b_0,b_1,b_2,b_3) · =(1,1,4,2), v_2(E_4(n+H)^(4,j)-E_4n^(4,j)) · =v_2(H)+B_j, · (B_0,B_1,B_2,B_3) · =(1,1,5,3). Consequently, the least positive row period modulo 2^r is exactly P_j(r)=2^max(0,r-B_j) (r≥1). The corresponding displacemen","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98aa7dc614deee26","tex":"B_{2m}","aliases":"B_2m","display":"B_{2m}","kind":"use","excerpt":"[Exact weighted inverse coefficient] Let κ_m=β_m2^2m(2m)!. For every m≥1, κ_m=-2^4m(2m-1)B_2m≠0, v_2(κ_m)=4m-1, where B_2m is the usual even Bernoulli number.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8a910bd8f25ee36","tex":"B_{2n}","aliases":"B_2n","display":"B_{2n}","kind":"use","excerpt":"oulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator and using [source reference]. Comparing even coefficients yields C_2,1(n)=(2-2^2n)B_2n (n∈ N). This classi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e35089086af3af8c","tex":"B_{2r}","aliases":"B_2r","display":"B_{2r}","kind":"use","excerpt":"[Dyadic inverse coefficients] For r≥1, β_r = - frac2^2r(2r-1)B_2r(2r)!. Consequently, v_2(β_r)=s_2(r)-1. Every R_n is a dyadic unit.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b224b896d9d01ac4","tex":"B_3","aliases":"B_3","display":"B_3","kind":"use","excerpt":"r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5284f24ae8cda5b8","tex":"b_3","aliases":"b_3","display":"b_3","kind":"use","excerpt":"is a triangular solveCOEFFICIENT RECONSTRUCTIONChapter IV: Theorem 7.3, Theorem 7.6. Start in the reduced variable with Δ_2(w)=1-frac w2+fracw^224-fracw^3720+⋯. Write 1/Δ_2(w)=∑ b_nw^n. Matching the product with 1 gives a unit lower-triangular system. At each row there is exactly one new unknown. vfigtriangular_inverse The first three equations are","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b94fddf75c4f060","tex":"B_4","aliases":"B_4","display":"B_4","kind":"use","excerpt":"3. The dyadic differential operator and reciprocity. The operator required here is B_4=fracD^8-D^42, D=fracddt. It is not the preceding p=2 specialization (D^4-D^2)/2. It is also not an ordinary derivation. The new algebra module proves the actual product formula, including the middle derivative sums whose coefficients are binom 8i/2 and binom 4i/2","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16ab16f37fdea085","tex":"b_4","aliases":"b_4","display":"b_4","kind":"use","excerpt":"(b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51edaf5d4fd6092c","tex":"\\mathcal B_{4,2}","aliases":"B_4,2 mathcal","display":"\\mathcal B_{4,2}","kind":"use","excerpt":"r rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35e6270fb1adf1db","tex":"\\mathcal B_{4,j}","aliases":"B_4,j mathcal","display":"\\mathcal B_{4,j}","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. N","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0175fc3916c84f4","tex":"B_{4i+2}","aliases":"B_4i+2","display":"B_{4i+2}","kind":"use","excerpt":"G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The co","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe0fcfcbd04a63ee","tex":"b_{a,j}","aliases":"b_a,j","display":"b_{a,j}","kind":"use","excerpt":", v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2de8fd102567aaa1","tex":"B_{a-1}","aliases":"B_a-1","display":"B_{a-1}","kind":"use","excerpt":"ents from the reciprocal recurrence are a_1=-g(1)-1, a_2=binom2NNg(1)^2-g(2)+2g(1)+1. The first has valuation exactly one. Also binom2NN≡6pmod8 for every N=2^a≥2. To see this, write B_a=binom2^a+12^a =2∏_h=1^2^a-1(1+frac2^ah). The even-h factors give B_a-1. The odd-h factors are one modulo four for a≥2. Since B_a-1 has valuation one, B_a≡ B_a-1pmod","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82b9f73f21e567db","tex":"b_a^{\\phantom a c}","aliases":"b_a^phantom a c phantom","display":"b_a^{\\phantom a c}","kind":"use","excerpt":"[Invariance of the physical space] Let H=H^*. Suppose [H,C_a] = ∑_cb_a^phantom a cC_c for every a. Then H preserves H_phys, and so does e^- i tH for every real t.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef9e511d27b6b980","tex":"B_b","aliases":"B_b","display":"B_b","kind":"use","excerpt":"Stationarity follows from overline Qoverlineπ =overline QCπ =CQπ=0. For reversibility, use the aggregate rate formula: overline Q_aboverlineπ_b · =∑_j∈ B_bπ_j∑_i∈ B_aQ_ij · =∑_i∈ B_a∑_j∈ B_bπ_iQ_ji · =overline Q_baoverlineπ_a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-727bc8e67f8970a4","tex":"\\mathcal B_{\\bar\\lambda}","aliases":"B_barλ mathcal bar lambda","display":"\\mathcal B_{\\bar\\lambda}","kind":"use","excerpt":"For strictly positive fine and coarse likelihoods, and at least two coarse blocks, the manuscript proves the all-prior criterion C B_λ(p)= B_barλ(Cp) hboxfor every p Longleftrightarrow λ_i=cbarλ_a (i∈ I_a) for one global c>0. The global scale is invisible after normalization. A different arbitrary scale for each packet is not invisible: it changes ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10fba79ce2cdc0ce","tex":"b_c","aliases":"b_c","display":"b_c","kind":"use","excerpt":"Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5384093c6e0cfdb","tex":"B_{ca}","aliases":"B_ca","display":"B_{ca}","kind":"use","excerpt":"Composition. Inserting the projector identity between factors yields (AB)_ba=∑_c A_bcB_ca. If the left side is nonzero, at least one intermediate c has both A_bc≠0 and B_ca≠0. Consequently supp(AB)⊆supp(A)∘supp(B). No equality of supports is asserted.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3bae85ab6d499ac","tex":"B_D","aliases":"B_D","display":"B_D","kind":"use","excerpt":"p^2. Replacing F by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption: B_Θ W_p⊆ W_p, B_Θ^s W_p⊆ W_p (s≥0). Modules. mcodeWeightedAmiceRing, mcodeFrobeniusSquare, mcodeThetaWeighted. 2. Actual substitution and singular higher differences. Let E(F)(t)=F(e^t-1), D=fracddt, B_D=fracD^p^2-D^pp.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f56251a6f87c66b1","tex":"b_d","aliases":"b_d","display":"b_d","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2273248685a24f0","tex":"b_{d+1,r}","aliases":"b_d+1,r","display":"b_{d+1,r}","kind":"use","excerpt":"Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the forma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3beb282e3f0dd2a9","tex":"b_{d,r}","aliases":"b_d,r","display":"b_{d,r}","kind":"use","excerpt":"Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the forma","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19747db9bb1a3595","tex":"b_{d,r-1}","aliases":"b_d,r-1","display":"b_{d,r-1}","kind":"use","excerpt":"Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the forma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d442e917a9b5ba8d","tex":"b_{d-i}","aliases":"b_d-i","display":"b_{d-i}","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49151ef209ba386f","tex":"B_D^s","aliases":"B_D^s","display":"B_D^s","kind":"use","excerpt":"(s≥0). Modules. mcodeWeightedAmiceRing, mcodeFrobeniusSquare, mcodeThetaWeighted. 2. Actual substitution and singular higher differences. Let E(F)(t)=F(e^t-1), D=fracddt, B_D=fracD^p^2-D^pp. All substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives D E(F)= E(Θ F), B_D^s E(F)= E(B_Θ^sF). The imp","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bb9adbb22b5e88e","tex":"B_{dc}","aliases":"B_dc","display":"B_{dc}","kind":"use","excerpt":"Adjoint and tensor products. With an involution and self-adjoint projectors, the formalization proves (A_ba)^*=(A^*)_ab. For actual finite matrices and actual Kronecker products, it proves (P_b⊗ Q_d)(A⊗ B)(P_a⊗ Q_c) =A_ba⊗ B_dc. The self-adjointness hypothesis is explicit in the adjoint theorem. The tensor theorem is not a statement about an uninte","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66c8d0359e712fa2","tex":"B_e","aliases":"B_e","display":"B_e","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c69ec0fece4f3aa0","tex":"b_h","aliases":"b_h","display":"b_h","kind":"use","excerpt":"The principal recovery theorem uses adjacent sample contrasts. Let b_h=Mh+2ε be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If a_min>2b_h, then every threshold between b_h and a_min-b_h detects exactly the cells containing jumps. The jump locations are recovered within h/2, and each amplit","count":29},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1d859d024fbe063","tex":"b_{h,\\delta}","aliases":"b_h,δ delta","display":"b_{h,\\delta}","kind":"use","excerpt":"[Probability recovery bound] Fix 0<δ<1 and define t_n,δ = 2σ sqrtlog(frac2nδ) and b_h,δ=Mh+t_n,δ. With probability at least 1-δ, every contrast satisfies the conclusions of [source reference] with b_h replaced by b_h,δ.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"use","excerpt":"For 0≤ i<j, let d_i=v_2(i!)+v_2((j-1-i)!), k_i=1+lfloorlog_2max(i,j-1-i)rfloor, B_i=i+2^k_i Z_2, t_i=d_i+k_i. At j=1 use d_0=k_0=t_0=0. The balls B_i partition Z_2. Indeed a residue represented twice among 0,…,j-1 splits into its two children. Every other represented residue remains one ball. On each such ball, P_j(i+2^k_iy)=2^t_iyQ_i(y), Q_i∈ Z_2⟨","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-643bdb7bc30d61f1","tex":"B_{ik}","aliases":"B_ik","display":"B_{ik}","kind":"use","excerpt":"The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a1364749eb81c1c","tex":"B_{j,k}","aliases":"B_j,k","display":"B_{j,k}","kind":"use","excerpt":"mory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-544c723ff631382e","tex":"B_{j,\\nu_j-1}","aliases":"B_j,ν_j-1 nu","display":"B_{j,\\nu_j-1}","kind":"use","excerpt":"[Pole order and hidden Jordan depth] Every finite hidden block produces a strictly proper rational self-energy. At an eigenvalue μ_j, its pole order is at most the size of the largest Jordan block of A_QQ at μ_j. Equality holds when the corresponding highest coefficient B_j,ν_j-1 is nonzero.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3caf6ca5bc1139c0","tex":"B_j^r","aliases":"B_j^r","display":"B_j^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9db34553aedb800c","tex":"B_{ji}","aliases":"B_ji","display":"B_{ji}","kind":"use","excerpt":"[Table specification for transfer blocks] Given finite projector systems (P_i)_i∈ I and (Q_j)_j∈ J and a linear map T, form the table B_ji=Q_jTP_i. Its assembly reconstructs the input map exactly, ∑_j∈ J∑_i∈ IB_ji=T, and its reported support relation is exactly {(i,j):B_ji≠0}. The latter equality is the specification of the transfer graph output. I","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6abdf84d4df05b6","tex":"B_{jk}","aliases":"B_jk","display":"B_{jk}","kind":"use","excerpt":"vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2f9008f03849cc5","tex":"B_k","aliases":"B_k","display":"B_k","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d05703a4df49dfef","tex":"b_{k+1}","aliases":"b_k+1","display":"b_{k+1}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e09999c1cd63453","tex":"b_{k+d}","aliases":"b_k+d","display":"b_{k+d}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca06c4f67613c969","tex":"b_{k+r}","aliases":"b_k+r","display":"b_{k+r}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5464c5f65d3b3c2c","tex":"B_{ks}","aliases":"B_ks","display":"B_{ks}","kind":"use","excerpt":"multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let B= R E-cz^-dI_N and suppose its action vanishes on every channel sequence. Insert an impulse at channel s, address zero, with every other channel zero. Output channel k reads the coefficie","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-762426111124f19d","tex":"B_L","aliases":"B_L","display":"B_L","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc9ad75de14545d5","tex":"b_m","aliases":"b_m","display":"b_m","kind":"use","excerpt":"The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer. This is the integrality required by the later congruence.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-def969061c988714","tex":"b_{mn}^{(m,p)}","aliases":"b_mn^(m,p)","display":"b_{mn}^{(m,p)}","kind":"use","excerpt":"[Compressed inverse] Set e_n^(m,p) = b_mn^(m,p).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"use","excerpt":"The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e36803b804d3c151","tex":"b_{n+p-1}^{(m,p)}","aliases":"b_n+p-1^(m,p)","display":"b_{n+p-1}^{(m,p)}","kind":"use","excerpt":"[Support and positive degree period] The inverse b^(m,p) has the following properties. item b_n^(m,p)=0 unless mmid n. item For every positive n, b_n+p-1^(m,p) = b_n^(m,p).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-722e62b9ceac6572","tex":"b_{N,0}","aliases":"b_N,0","display":"b_{N,0}","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99de146f9c618d5f","tex":"\\mathcal B_{N,i}","aliases":"B_N,i mathcal","display":"\\mathcal B_{N,i}","kind":"use","excerpt":"1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0e1a295abce9295","tex":"B_{N,j}","aliases":"B_N,j","display":"B_{N,j}","kind":"use","excerpt":"[Small-rank complete isometry table] Every address at N∈{2,4,8} satisfies v_2(C_N,j(n+H)-C_N,j(n))=b_N,j+v_2(H). The following entries are ordered by j=0,…,N-1: c l l N · normalized shifts b_N,j · original shifts B_N,j=b_N,j+v_2(j!)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6de50404dc3969bc","tex":"\\mathcal B_{N,j}","aliases":"B_N,j mathcal","display":"\\mathcal B_{N,j}","kind":"use","excerpt":"1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2580100ce4d243be","tex":"b_{N,j}","aliases":"b_N,j","display":"b_{N,j}","kind":"use","excerpt":"[Small-rank complete isometry table] Every address at N∈{2,4,8} satisfies v_2(C_N,j(n+H)-C_N,j(n))=b_N,j+v_2(H). The following entries are ordered by j=0,…,N-1: c l l N · normalized shifts b_N,j · original shifts B_N,j=b_N,j+v_2(j!)","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18b50bc4f4ffc9b8","tex":"\\mathcal B_{N,k}","aliases":"B_N,k mathcal","display":"\\mathcal B_{N,k}","kind":"use","excerpt":"1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c63fabcfa89c6ac0","tex":"b_{N,n-k}","aliases":"b_N,n-k","display":"b_{N,n-k}","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0bb32ec887c4163","tex":"b_{n-k}","aliases":"b_n-k","display":"b_{n-k}","kind":"use","excerpt":"vpage11Two recurrences reach the same answerNEWTON AND ADDITIONChapter IV: Theorem 7.3, Theorem 7.4, Theorem 7.6. One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagonal in the Newton determinant is esse","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34d4e296e2f11bfd","tex":"b_{n-r}","aliases":"b_n-r","display":"b_{n-r}","kind":"use","excerpt":"This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard windo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f2ddbf7dd109de0","tex":"b_n^{(m,p)}","aliases":"b_n^(m,p)","display":"b_n^{(m,p)}","kind":"use","excerpt":"[Support and positive degree period] The inverse b^(m,p) has the following properties. item b_n^(m,p)=0 unless mmid n. item For every positive n, b_n+p-1^(m,p) = b_n^(m,p).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6322ecebf90fbfd2","tex":"B_n^{1/2}","aliases":"B_n^1/2","display":"B_n^{1/2}","kind":"use","excerpt":"at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVEChapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f786b5ed58219b3","tex":"B_{\\mathrm{order}}","aliases":"B_order mathrm","display":"B_{\\mathrm{order}}","kind":"use","excerpt":"The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8bcbf87ac6779001","tex":"B_p","aliases":"B_p","display":"B_p","kind":"use","excerpt":"Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ce4130b75dce935","tex":"\\mathcal B_p","aliases":"B_p mathcal","display":"\\mathcal B_p","kind":"use","excerpt":"The normalized defect is not assumed to be a derivation. Its proved product rule contains the two middle Leibniz sums: B_Θ(FG)=(B_Θ F)G+F(B_Θ G)+ B_p(F,G). The coefficients of B_p are the actual integral quotients binomp^2a/p and binom pa/p. Derivative stability and this correction formula show that the elements F∈ W_p satisfying B_Θ F∈ W_p are clo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c82c8901bdffe44","tex":"\\mathcal B_{p,w}","aliases":"B_p,w mathcal","display":"\\mathcal B_{p,w}","kind":"use","excerpt":"Exact banded addresses and modeled work. textitSource item 4.1; module BandedCount For p packets and half-bandwidth w, the actual address set is B_p,w ={(i,j)∈{0,…,p-1}^2: i≤ j+w and j≤ i+w}. The predicate is the integer version of |i-j|≤ w. The count is first defined as a double sum of indicators, then identified with the cardinality of this subty","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-667e7c36f2dec875","tex":"B_p^k","aliases":"B_p^k","display":"B_p^k","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-219538915e682dc1","tex":"B_p^r","aliases":"B_p^r","display":"B_p^r","kind":"use","excerpt":"Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration give","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c734e54250961c96","tex":"B_p^{r-1}","aliases":"B_p^r-1","display":"B_p^{r-1}","kind":"use","excerpt":"Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration give","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48939aa333eb6f6a","tex":"B_p^s","aliases":"B_p^s","display":"B_p^s","kind":"use","excerpt":"Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ee75600b75f53a4","tex":"B_{p_u}","aliases":"B_p_u","display":"B_{p_u}","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33cb6f2bf5e31462","tex":"B_{p_v}","aliases":"B_p_v","display":"B_{p_v}","kind":"use","excerpt":"ct edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df47dfef3c938c01","tex":"B_{p_v}^{-1}","aliases":"B_p_v^-1","display":"B_{p_v}^{-1}","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-442ac47b1791bc80","tex":"b_{q,m}","aliases":"b_q,m","display":"b_{q,m}","kind":"use","excerpt":"[Uniqueness of the trace packet] Fix the breakpoints τ_q. If f=s+∑_q=1^Q∑_m=0^r-1b_q,mϕ_m,τ_q with s∈ W^r,p(0,1), then b_q,m=jumpf^(m)_τ_q. The finite truncated power packet is therefore the unique packet supported at the declared breakpoints that removes all trace jumps through order r-1.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4491da1e894267e7","tex":"\\mathcal B_r","aliases":"B_r mathcal","display":"\\mathcal B_r","kind":"use","excerpt":"page11The exact information-loss vectorBAYESIAN DEFECT / MORE THAN AN ERROR BARChapter VI, Part I: Theorems 8.1 and 8.4, and Corollary 8.6. For a prior p, let P=Cp and let r_a(p) be the conditional average of the fine likelihood inside packet a. For the uniform prior and λ=(4,1,1,1), P=(1/2,1/2)^ T, r=(5/2,1)^ T. The exact coarse posterior is B_r(P","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac0d31a2615b8b66","tex":"b_r","aliases":"b_r","display":"b_r","kind":"use","excerpt":"Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0d26e84e0952114","tex":"B_r'","aliases":"B_r'","display":"B_r'","kind":"use","excerpt":"Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ab923618be1c5fc","tex":"b_{r-q}","aliases":"b_r-q","display":"b_{r-q}","kind":"use","excerpt":"The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r, 0, · q>r. The row-action equality, the diagonal identity U_r,r=1, and vanishing above the diagonal are all checked. ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a75a44be7515e9d","tex":"B_r^*","aliases":"B_r^*","display":"B_r^*","kind":"use","excerpt":"Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be4b228c950e38a7","tex":"b_{\\rm root}","aliases":"b_rm root rm","display":"b_{\\rm root}","kind":"use","excerpt":"apter V: Proposition E6.4. Suppose the root split has nonnegative squared-norm bounds A=1/2 and B=2. Retain the left child. Split the right child with bounds 3/4 and 5/4, then retain both grandchildren. vfig[.85]ext_tree headStart with one at every retained node The left subtree has bounds a_0=b_0=1. The right subtree has a_1=3/4 and b_1=5/4. The r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-336970a90c727278","tex":"B_{\\rm stream}","aliases":"B_rm stream rm","display":"B_{\\rm stream}","kind":"use","excerpt":"blocks accumulate into one target, the reduction or accumulation policy is additional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention. headCount the data separately For three diagonal 2×2 blocks, eight-byte scalars, and q right-hand side","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-026204dc005ea4b1","tex":"B_s","aliases":"B_s","display":"B_s","kind":"use","excerpt":"Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9846458a14f54b6","tex":"B_{\\mathrm{stream}}","aliases":"B_stream mathrm","display":"B_{\\mathrm{stream}}","kind":"use","excerpt":"Streaming. The explicit event type counts one block read, one source read, and a target read plus write for every edge. With scalar size s, its byte count is B_stream=s∑_e(m_en_e+n_eq+2m_eq). No cross-edge cache reuse is assumed. Sparse index metadata and temporary workspace remain separate.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b85ab862e92905e8","tex":"b_T","aliases":"b_T","display":"b_T","kind":"use","excerpt":"Theorem~[source reference] gives a unique root b of F+p^M H in a+p^s+1R^e+k, with displacement depth M-s. Its last equations say exactly b_T=t_0+p^Mv=τ. Its first equations are G(b)=0. The integral first-order expression uses CbinomH(a)-v =binomA(H(a)+Kv)-p^sv. Projection onto the solved coordinates gives b_Y-y_0+p^M-sA(H(a)+Kv)∈ p^2M-3sR^e. The sa","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aba34f1446cfa879","tex":"B_v","aliases":"B_v","display":"B_v","kind":"use","excerpt":"a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69b376b6bbe4d3ac","tex":"b_v","aliases":"b_v","display":"b_v","kind":"use","excerpt":"Constructed solution. Well-founded recursion defines x_v=D_v^-1 (b_v-∑_u→ vT_vux_u). Applying D_v proves the original equation at v. If another vector satisfies all equations, induction over predecessors makes each previously used coordinate identical, and invertibility of D_v forces agreement at v. Thus the construction gives existence and uniquen","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26b03d7e315b2b07","tex":"b_{\\mathrm{val}}","aliases":"b_val mathrm","display":"b_{\\mathrm{val}}","kind":"use","excerpt":"Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fiber has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-750ec2c2a1a68b60","tex":"B_{vu}","aliases":"B_vu","display":"B_{vu}","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5902c64cbd2a059e","tex":"B_w","aliases":"B_w","display":"B_w","kind":"use","excerpt":"Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hy","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bec856d92c5ab12c","tex":"b_x","aliases":"b_x","display":"b_x","kind":"use","excerpt":"Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Write r_x=F(x)/p^L, b_x=r_xbmod p, A_x=DF(x)bmod p: F_p^d⟶ F_p^e, ρ_x=rank_ F_pA_x. The actual lift set is L_x,L= {y∈( Z/p^L+1 Z)^d: y≡ xpmodp^L, F(y)=0}, where polynomial evaluation in the displayed set is modulo p^L+1. Define O_x= F_p^e/imA_x, o_x=[b_x]. Lean proves # L_x,L= p^d-ρ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed74d4cbc9f00b0b","tex":"b_Y","aliases":"b_Y","display":"b_Y","kind":"use","excerpt":"Theorem~[source reference] gives a unique root b of F+p^M H in a+p^s+1R^e+k, with displacement depth M-s. Its last equations say exactly b_T=t_0+p^Mv=τ. Its first equations are G(b)=0. The integral first-order expression uses CbinomH(a)-v =binomA(H(a)+Kv)-p^sv. Projection onto the solved coordinates gives b_Y-y_0+p^M-sA(H(a)+Kv)∈ p^2M-3sR^e. The sa","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-190834da497e3d7f","tex":"B_\\alpha","aliases":"B_α alpha","display":"B_\\alpha","kind":"use","excerpt":"Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4cacfd818361b439","tex":"B_\\beta","aliases":"B_β beta","display":"B_\\beta","kind":"use","excerpt":"[Characterization of exact discrete lumping] Let K be column stochastic. A coarse column stochastic matrix overline K satisfying CK=overline KC exists if and only if, for every source block B_β, every pair i,j∈ B_β, and every target block B_α, ∑_r∈ B_αK_ri = ∑_r∈ B_αK_rj. When it exists, overline K is unique and overline K_αβ = ∑_r∈ B_αK_ri for any","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5982aa6063dafe04","tex":"b_\\beta","aliases":"b_β beta","display":"b_\\beta","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b0a7f57cb21faf55","tex":"B_{\\delta\\gamma}","aliases":"B_δγ delta gamma","display":"B_{\\delta\\gamma}","kind":"use","excerpt":"[Adjoint and tensor semantics] The adjoint representation has blocks (A^*)_αβ=A_βα^*. If A and B are packet resolved on X and Y, then (A⊗ B)_(β,δ),(α,γ) = A_βα⊗ B_δγ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd8097ebfdb37289","tex":"B_\\Theta","aliases":"B_Θ Theta","display":"B_\\Theta","kind":"use","excerpt":"Put Θ=(1+T)fracddT, B_Θ=fracΘ^p^2-Θ^pp. The coefficient formula [T^d]Θ F=(d+1)a_d+1+da_d proves Θ W_p⊆ W_p. If the floor weight increases between d and d+1, then pmid d+1, which compensates for that increase.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d3f66919d71566b","tex":"B_\\Theta^s","aliases":"B_Θ^s Theta","display":"B_\\Theta^s","kind":"use","excerpt":"Finally, [T^d]B_Θ F depends only on coefficients through d+p^2. Replacing F by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption: B_Θ W_p⊆ W_p, B_Θ^s W_p⊆ W_p (s≥0). Modules. mcodeWeightedAmiceRing, mcodeFrobeniusSquare, mcodeThetaWeighted. 2. Actual substitution and singular higher differ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8dd14ae875161a4","tex":"B_\\lambda","aliases":"B_λ lambda","display":"B_\\lambda","kind":"use","excerpt":"Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem,","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-655c8b8be4a021c9","tex":"\\mathcal B_\\lambda","aliases":"B_λ mathcal lambda","display":"\\mathcal B_\\lambda","kind":"use","excerpt":"B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4577d3803b0374b","tex":"b_{\\lambda,j}","aliases":"b_λ,j lambda","display":"b_{\\lambda,j}","kind":"use","excerpt":"[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N:","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a092bc56762bbd9","tex":"B_\\nu","aliases":"B_ν nu","display":"B_\\nu","kind":"use","excerpt":"ubsection:002textitModule SmallCoreNorms; arbitrary ambient competitors For rectangular isometries Q,V define L(G)=QGV^*, C(B)=Q^*BV, Q^*Q=I, V^*V=I. The following statements are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_ν≤norm B_ν, ν∈{2, F}. For the spe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49b6d1142f9da2e4","tex":"B_\\tau","aliases":"B_τ tau","display":"B_\\tau","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b38caf9b289ccca2","tex":"B_{\\tau_1}","aliases":"B_τ_1 tau","display":"B_{\\tau_1}","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9e6f3d273912284","tex":"B_{\\tau_r}","aliases":"B_τ_r tau","display":"B_{\\tau_r}","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58af0f264fe7ee38","tex":"\\backslash","aliases":"backslash","display":null,"kind":"use","excerpt":"Apply Mackey decomposition [source reference]. The double cosets W_K_nbackslash W_K/W_K_m are the orbits of simultaneous translation on Z/n Z× Z/m Z. There are d orbits, each of length L. The stabilizer of a point in each orbit is W_K_L. The second character is shifted by the chosen Frobenius representative. Summing the induced terms over the doubl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2ec3d8de7f39249","tex":"\\bar a","aliases":"bar a bar","display":"\\bar a","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8185eb0b0eeb64a","tex":"\\bar a^{p^{\\ell-1}}","aliases":"bar a^p^ℓ-1 bar ell","display":"\\bar a^{p^{\\ell-1}}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e95a89f39649f47","tex":"\\bar b","aliases":"bar b bar","display":"\\bar b","kind":"use","excerpt":"When sufficiency fails, the defect still has an exact finite factorization. Fix a declared coarse likelihood barλ, put P=C_π p, δ(p)=r(p)-barλ, bar b= B_barλ(P), Z_r=⟨ r(p),P⟩.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0086c8c28eb495d","tex":"\\bar E","aliases":"bar E bar","display":"\\bar E","kind":"use","excerpt":"Lifting a summand inside a supplied finite-group carrier. Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g)","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25e0b45f28c87b71","tex":"\\bar E^2","aliases":"bar E^2 bar","display":"\\bar E^2","kind":"use","excerpt":"Lifting a summand inside a supplied finite-group carrier. Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a336c88f130bb2fe","tex":"\\bar E_i","aliases":"bar E_i bar","display":"\\bar E_i","kind":"use","excerpt":"Simultaneous projector-family lifting and rigidity. Keep the complete domain, supplied finite-group carrier and group-order hypotheses of the hyperref[book:x-construction-18]supplied-carrier summand-lifting construction. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fad6ae0fd6c0acb8","tex":"\\bar K","aliases":"bar K bar","display":"\\bar K","kind":"use","excerpt":"Let K be a column-stochastic fine transition matrix, let C aggregate fine states into coarse packets, and let bar K be a coarse transition matrix. Define Δ=CK-bar KC.","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a996e97b2fc3d473","tex":"\\bar K^n","aliases":"bar K^n bar","display":"\\bar K^n","kind":"use","excerpt":"[Exact lumpability] If Δ=0, then CK^n=bar K^nC for every n≥0. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ffef9424cf7c3c6","tex":"\\bar K^{n+1}","aliases":"bar K^n+1 bar","display":"\\bar K^{n+1}","kind":"use","excerpt":"The result is immediate for n=0. If CK^n=bar K^nC, then CK^n+1=bar K^nCK=bar K^n+1C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef943f9b503eecb4","tex":"\\bar K^{n-1-j}","aliases":"bar K^n-1-j bar","display":"\\bar K^{n-1-j}","kind":"use","excerpt":"[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a0a9afa94f80409","tex":"\\bar K^{n-j}","aliases":"bar K^n-j bar","display":"\\bar K^{n-j}","kind":"use","excerpt":"The identity follows by telescoping: bar K^n-1-jCK^j+1-bar K^n-jCK^j =bar K^n-1-j(CK-bar KC)K^j. Summing from j=0 to n-1 leaves CK^n-bar K^nC. The norm bound follows termwise.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f8a4d0a063c4065","tex":"\\bar Q","aliases":"bar Q bar","display":"\\bar Q","kind":"use","excerpt":"The continuous generator also closes: CQ=bar Q C, bar Q=-2 · 2 2 · -2. The Duhamel identity therefore gives Ce^tQ=e^tbar QC. This is a second consequence of the corresponding generator identity, not an inference that K is a fixed-time sample of Q.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c4d0c215858215d","tex":"\\bar r","aliases":"bar r bar","display":"\\bar r","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc4e9dcca4ad0352","tex":"\\bar w","aliases":"bar w bar","display":"\\bar w","kind":"use","excerpt":"Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N, F(bar w)=0, free(bar w)=t(n)+p^Nvpmodp^N+1 }. Here p^Nv means any integer representative of v, multiplied by p^N. The residue class does not depend on the representative. Then # L_n,v=p^m-e for every n and every v","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fcd1d016a0b9e4bf","tex":"\\bar y","aliases":"bar y bar","display":"\\bar y","kind":"use","excerpt":"J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniqu","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6822eaf3f6b3d51","tex":"\\bar\\kappa","aliases":"barκ bar kappa","display":"\\bar\\kappa","kind":"use","excerpt":"tκ-1sqrtκ+1. Monotonicity of square root and positive denominators prove that q is monotone. Lean also proves κ_2(B)≥1, q(κ)≥0 for κ≥1, and q(κ)<1. Thus q(κ_2(B))≤ q (frac1+η1-η). For 1≤κ and e_0≥0, an explicit classical premise e_m≤2q(κ)^m e_0 and a certified upper bound κ≤barκ yield e_m≤2q(barκ)^m e_0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60e3a7638b4cd31d","tex":"\\bar\\lambda","aliases":"barλ bar lambda","display":"\\bar\\lambda","kind":"use","excerpt":"[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on ea","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ca7f87e3b8a212c","tex":"\\bar\\lambda_a","aliases":"barλ_a bar lambda","display":"\\bar\\lambda_a","kind":"use","excerpt":"[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on ea","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a848437bb19d2af","tex":"\\bar\\lambda_b","aliases":"barλ_b bar lambda","display":"\\bar\\lambda_b","kind":"use","excerpt":"Now use priors supported on one state of packet a and one state of packet b. Equality of the two coarse posterior odds gives fracλ_aλ_b = fracbarλ_abarλ_b. This holds for every pair of coarse packets, so λ_a=cbarλ_a for one common c>0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70754a4e2f59f2f1","tex":"\\bar\\lambda_{\\pi(i)}","aliases":"barλ_π(i) bar lambda pi","display":"\\bar\\lambda_{\\pi(i)}","kind":"use","excerpt":"tes barλ. This records the common-scale null direction at the declared coarse likelihood. The full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) _1 ≤ frac2∑_aP_a|δ_a(p)| min{Z_r,Z_barλ} and ∑_aP_a|δ_a(p)| ≤ ∑_i p_i|λ_i-barλ_π(i)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a34ab46342f89739","tex":"\\BB","aliases":"BB B mathcal","display":"{\\mathcal B}","kind":"use","excerpt":"headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6881377bbebf9981","tex":"\\bbC^d","aliases":"bbC^d C^d bbC mathbb","display":"{\\mathbb C}^d","kind":"use","excerpt":"[Orbit dimension of one quantum seed] For integers d≥1 and m≥0, consider the action of M_d( C)⊗ I_m on C^d⊗ C^m. Let C be the coefficient matrix of a vector ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} =d rank C. The seed is cyclic if and only if rank C=m, which is possible only when m≤ d.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f89e172ea2b90718","tex":"\\bbC^{d_i}","aliases":"bbC^d_i C^d_i bbC mathbb","display":"{\\mathbb C}^{d_i}","kind":"use","excerpt":"Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty blo","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63ccc9d44768142b","tex":"\\bbC^m","aliases":"bbC^m C^m bbC mathbb","display":"{\\mathbb C}^m","kind":"use","excerpt":"[Orbit dimension of one quantum seed] For integers d≥1 and m≥0, consider the action of M_d( C)⊗ I_m on C^d⊗ C^m. Let C be the coefficient matrix of a vector ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} =d rank C. The seed is cyclic if and only if rank C=m, which is possible only when m≤ d.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab6a7db12b8172fe","tex":"\\bbC^{m_i}","aliases":"bbC^m_i C^m_i bbC mathbb","display":"{\\mathbb C}^{m_i}","kind":"use","excerpt":"Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty blo","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2a8d769f2c647dd","tex":"\\bbk^2","aliases":"bbk^2 bbk bbk^2 𝕜^2 bbk Bbbk","display":null,"kind":"use","excerpt":"[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6253e89ee0f2326","tex":"\\bbk^3","aliases":"bbk^3 𝕜^3 bbk Bbbk","display":"{\\Bbbk}^3","kind":"use","excerpt":"[title=A quotient can strictly lower seed rank] Let A=𝕜 act by scalars on M=𝕜^3. Then srank_A(M)=3. Quotienting by a subspace of dimension one gives a scalar module of dimension two, whose seed rank is 2. Quotient monotonicity is therefore often strict.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c457dace33feb6b0","tex":"\\bbk^{d_i}","aliases":"bbk^d_i 𝕜^d_i bbk Bbbk","display":"{\\Bbbk}^{d_i}","kind":"use","excerpt":"[Algebraically closed multiplicity formula] Let 𝕜 be algebraically closed, let A be a unital algebra of finite dimension over 𝕜, put J=Jac(A), and let M be a finitely generated left module over A. Assume compatible decompositions indexed by an integer q≥0: A/J≅∏_i=1^qM_d_i(𝕜), M/JM≅⊕_i=1^q(𝕜^d_i)^⊕ m_i, where the d_i are positive integers and the m","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5722bd9f294027a7","tex":"\\bbk^n","aliases":"bbk^n 𝕜^n bbk Bbbk","display":"{\\Bbbk}^n","kind":"use","excerpt":"Let X=𝕜^n and let A=M_n(𝕜)=End_𝕜(X), with the full matrix algebra declared before a seed or target is chosen. By [source reference], every nonzero seed is cyclic. This is the maximal linear grammar on X. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18cc9e7034e6c340","tex":"\\bbk^{V(G)}","aliases":"bbk^V(G) 𝕜^V(G) bbk Bbbk","display":"{\\Bbbk}^{V(G)}","kind":"use","excerpt":"Let G be a nonempty finite graph with adjacency matrix A_G, and regard 𝕜^V(G) as a module over 𝕜[A_G]. The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33f082f973fe83c6","tex":"\\bbk^\\times","aliases":"bbk^× 𝕜^× bbk times Bbbk times","display":"{\\Bbbk}^\\times","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5e4e7017d9b1a9f","tex":"\\bbN","aliases":"bbN N mathbb","display":"{\\mathbb N}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36f15bc97a93c5c7","tex":"\\bbP","aliases":"bbP P mathbb","display":"{\\mathbb P}","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3ee3f8d33e160ca","tex":"\\bbR","aliases":"bbR R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c669ec793a215af0","tex":"\\Bcal_{c\\lambda}","aliases":"Bcal_cλ B_cλ Bcal lambda mathcal lambda","display":"{\\mathcal B}_{c\\lambda}","kind":"use","excerpt":"[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c96b2eee990d739","tex":"\\Bcal_L","aliases":"Bcal_L B_L Bcal mathcal","display":"{\\mathcal B}_L","kind":"use","excerpt":"FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] De","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae942cfdb7498044","tex":"\\Bcal_{L_y}","aliases":"Bcal_L_y B_L_y Bcal mathcal","display":"{\\mathcal B}_{L_y}","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1c5eaef95cde19b","tex":"\\Bcal_{N,i}","aliases":"Bcal_N,i B_N,i Bcal mathcal","display":"{\\mathcal B}_{N,i}","kind":"use","excerpt":"[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-924ed6b80b428e13","tex":"\\Bcal_{N,k}","aliases":"Bcal_N,k B_N,k Bcal mathcal","display":"{\\mathcal B}_{N,k}","kind":"use","excerpt":"Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9731ec42b8194977","tex":"\\Bcal_{\\overline L}","aliases":"Bcal_overline L B_overline L Bcal overline mathcal overline","display":"{\\mathcal B}_{\\overline L}","kind":"use","excerpt":"ctor p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3bed59d24e6dcb8e","tex":"\\Bcal_{\\overline L_y}","aliases":"Bcal_overline L_y B_overline L_y Bcal overline mathcal overline","display":"{\\mathcal B}_{\\overline L_y}","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8461303be59fc46","tex":"\\Bcal_{\\overline\\lambda(p)}","aliases":"Bcal_overlineλ(p) B_overlineλ(p) Bcal overline lambda mathcal overline lambda","display":"{\\mathcal B}_{\\overline\\lambda(p)}","kind":"use","excerpt":"[Exact two stage Bayesian update] For every strictly positive prior, C B_λ(p) = B_overlineλ(p)(Cp). For every packet with overlineλ_α(p)>0, the posterior conditional distribution is q_α,i = fracλ_ir_α,i overlineλ_α(p). A packet with zero average likelihood has zero posterior mass. Thus a fine Bayesian update consists of a coarse update by packet av","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-670d28531783d120","tex":"\\Bcal_{\\widetilde\\lambda}","aliases":"Bcal_widetildeλ B_widetildeλ Bcal widetilde lambda mathcal widetilde lambda","display":"{\\mathcal B}_{\\widetilde\\lambda}","kind":"use","excerpt":"Block constancy gives C B_widetildeλ(p) = B_ℓ(Cp). Aggregation is an ℓ^1 contraction. Apply [source reference] to the two fine posteriors and then aggregate.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36f47e44857f889e","tex":"\\Bcal_\\lambda","aliases":"Bcal_λ B_λ Bcal lambda mathcal lambda","display":"{\\mathcal B}_\\lambda","kind":"use","excerpt":"Bayesian update · B_λ · likelihood constant inside each coarse packet","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1055d827217a09e","tex":"\\Bcal_{\\lambda^{(r)}}","aliases":"Bcal_λ^(r) B_λ^(r) Bcal lambda mathcal lambda","display":"{\\mathcal B}_{\\lambda^{(r)}}","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood of the rth observation. The hidden Markov filter is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r). A standard account of finite hidden Markov models appears in [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-160416bc9d0d7c17","tex":"\\Bcal_{\\lambda\\odot\\mu}","aliases":"Bcal_λodotμ B_λodotμ Bcal lambda odot mu mathcal lambda odot mu","display":null,"kind":"use","excerpt":"[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1d7da39846900e8d","tex":"\\Bcal_\\mu","aliases":"Bcal_μ B_μ Bcal mu mathcal mu","display":"{\\mathcal B}_\\mu","kind":"use","excerpt":"[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34e973b02b27f5d2","tex":"\\Bcal_\\ell","aliases":"Bcal_ℓ B_ℓ Bcal ell mathcal ell","display":"{\\mathcal B}_\\ell","kind":"use","excerpt":"[Characterization of fixed coarse sufficiency] Assume λ_i>0 for every state. The following conditions are equivalent. item There is a positive coarse likelihood ℓ∈(0,∞)^m such that C B_λ(p) = B_ℓ(Cp) for every strictly positive prior p, and the posterior conditional distribution inside every packet equals its prior conditional distribution. item Th","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d549c511b570806a","tex":"\\Bcal_{\\ell(p)}","aliases":"Bcal_ℓ(p) B_ℓ(p) Bcal ell mathcal ell","display":"{\\mathcal B}_{\\ell(p)}","kind":"use","excerpt":"For an arbitrary prior p, put q=Cp and Z(p,L)=ipLp. Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0. Choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bfc0ff5fd667e9f6","tex":"\\mathrm{Beta}","aliases":"Beta mathrm","display":"\\mathrm{Beta}","kind":"use","excerpt":"vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCEChapter VI, Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22835c974e5650b7","tex":"\\operatorname{Beta}","aliases":"Beta operatorname","display":"\\operatorname{Beta}","kind":"use","excerpt":"[Beta--Binomial conjugate packet] A Beta(a,b) prior and s successes in n trials produce Beta(a+s,b+n-s). The prior exponents are additive packets: the likelihood increments them by (s,n-s). In the numerical example, (a,b)=(2,3), (s,n)=(7,10), so the posterior is Beta(9,6) with mean 0.6.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ffda03323e255d73","tex":"\\bigm","aliases":"bigm","display":null,"kind":"use","excerpt":"0, · w=∞. The infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, every term with weight at least L is divisible by p^L. A finite-sum proof yields p^L bigm| ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r -∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-r. The result is specialized to actual p-adic","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5bd492e4ea78c5fe","tex":"\\bigsqcup","aliases":"bigsqcup","display":null,"kind":"use","excerpt":"[Residual partition] For every coefficient algebra A, M(A) = bigsqcup_bar x∈ M(k) M_bar x(A).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf287c25ac4534b1","tex":"\\bigsqcup_{\\bar x\\in\\mathcal M(k)}","aliases":"bigsqcup_bar x∈ M(k) bigsqcup bar in mathcal","display":null,"kind":"use","excerpt":"[Residual partition] For every coefficient algebra A, M(A) = bigsqcup_bar x∈ M(k) M_bar x(A).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a44cd5763af08d17","tex":"\\bigsqcup_{r\\in R}","aliases":"bigsqcup_r∈ R bigsqcup in","display":null,"kind":"use","excerpt":"Let L≤ Z^d be a sublattice of finite index q. Choose a set R⊂ Z^d containing one representative from every coset in Z^d/L. Thus Z^d=bigsqcup_r∈ R(r+L). For x: Z^d→ C, its channel at r is the sequence on L x_r(ℓ)=x(r+ℓ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e874d9d01b6213d8","tex":"\\operatorname{Bounded}","aliases":"Bounded operatorname","display":"\\operatorname{Bounded}","kind":"use","excerpt":"All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c947faba6dfed07d","tex":"\\operatorname{Bounded}_r","aliases":"Bounded_r operatorname","display":"\\operatorname{Bounded}_r","kind":"use","excerpt":"All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7aca27c01715d051","tex":"\\boxtimes","aliases":"boxtimes","display":null,"kind":"use","excerpt":"[Tensor product packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] Expor","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43fa8bad7cce3005","tex":"\\bra","aliases":"bra","display":null,"kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a s","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75693dc2c5a4fbed","tex":"\\BV","aliases":"BV operatorname","display":"{\\operatorname{BV}}","kind":"use","excerpt":"The higher order singular packet theorem removes finitely many trace jumps. A more complete approximation theory should connect packet norms to geometric measure quantities in BV, piecewise Sobolev spaces, and free discontinuity models. It should also determine when a redundant packet dictionary can identify the derivative order of a singularity ra","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"Let X and Y be Banach spaces over the same field R or C and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr echet derivatives and the Taylor integral below has a real parameter. Assume that F:X→ Y is m times continuously Fr echet differentiable. The same local statements hold on open domains cont","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"Let G∈ Z[X_1,…,X_d]^f be a second fixed system with the same variables and the same lower root x. Let K_x=ρ_q∘ DG(x), and define C_x= Z^d/(ker J_x+ker K_x). This quotient is finite: it is a quotient of the finite image of J_x, or equivalently of the finite image of K_x.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab8bde9a479f56a0","tex":"\\mathfrak C","aliases":"C mathfrak","display":"\\mathfrak C","kind":"use","excerpt":"[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport for","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59133e4900f63250","tex":"\\operatorname{c\\!-\\!Ind}","aliases":"c - Ind operatorname","display":"\\operatorname{c\\!-\\!Ind}","kind":"use","excerpt":"Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is an actual e","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be603f780619be75","tex":"\\operatorname{c\\!-\\!Ind}_{\\mathcal M}^G","aliases":"c - Ind_ M^G operatorname mathcal","display":"\\operatorname{c\\!-\\!Ind}_{\\mathcal M}^G","kind":"use","excerpt":"Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is an actual e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3abc569941b2510","tex":"\\operatorname{c\\!-\\!Ind}_J^G","aliases":"c - Ind_J^G operatorname","display":"\\operatorname{c\\!-\\!Ind}_J^G","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","excerpt":"The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-faeabc6533da3ce6","tex":"c^{(p)}_{m,j}","aliases":"c^(p)_m,j","display":"c^{(p)}_{m,j}","kind":"use","excerpt":"Specializing to R= Z_p constructs c^(p)_m,j(n). Mapping to Q_p and comparing with the original rational forward substitution gives c^(p)_m,j(n)=C_mp,j(n) in Q_p. Thus every original standard-window coefficient is p-integral, for every row. Integrality is a conclusion, not an additional hypothesis on the coefficients.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5d5e13c662eedec","tex":"c^{(p)}_{m,k}","aliases":"c^(p)_m,k","display":"c^{(p)}_{m,k}","kind":"use","excerpt":"ply the residue map Independently define e^(m,p) over F_p by the unit-diagonal system ∑_q=0^nbinommnmqe^(m,p)_q=δ_n0. Its recurrence and uniqueness are checked. Applying the actual homomorphism π: Z_p→ F_p to the integral row, and then Lucas reduction, proves π(c^(p)_m,j(n))=e^(m,p)_n. For j,k<p, a separate ideal statement gives pmid c^(p)_m,j(n)-c","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a41d044c4883dce","tex":"c^*","aliases":"c^*","display":"c^*","kind":"use","excerpt":"For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e7e42689204088d5","tex":"C^0","aliases":"C^0","display":"C^0","kind":"use","excerpt":"[A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb4b39a0fa3e20d1","tex":"C^1","aliases":"C^1","display":"C^1","kind":"use","excerpt":"[A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-133380d2b39bd34f","tex":"C^{1-\\alpha}","aliases":"C^1-α alpha","display":"C^{1-\\alpha}","kind":"use","excerpt":"For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d46643740805c1d","tex":"C^{12}","aliases":"C^12","display":"C^{12}","kind":"use","excerpt":"1 · 0 , one has C^12 =I+8 29 · 18 18 · 11 . Therefore Δ_12·2^a^ rF_n ≡0 pmod 2^(a+3)r. Every dyadic refinement doubles the step and adds one guaranteed binary digit at each Mahler depth.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","excerpt":"0 · 0) has C^2=0 and χ_C=X^2.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2677238ec3f2ee1","tex":"\\mathbb C^2","aliases":"C^2 mathbb","display":"\\mathbb C^2","kind":"use","excerpt":"model uses two coordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus fam","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8981f94f66d22b3c","tex":"C^{24}","aliases":"C^24","display":"C^{24}","kind":"use","excerpt":"1 · 0 . The matrix C is invertible modulo twenty four, and C^24-I = 24 3126 · 1932 1932 · 1194 . For every scalar coordinate of the state sequence C^nv_0, Δ_24^ ra_n ≡0 pmod 24^r. One integral jump therefore gives simultaneous congruence ladders at the primes two and three.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a4fb977dca130a2","tex":"C^3","aliases":"C^3","display":"C^3","kind":"use","excerpt":"[t] centering includegraphics[width=.95linewidth,height=3.15in,keepaspectratio]../../figures/reconstructed/recurrence_packets.pdf captionNew reconstruction. Fibonacci residues F_3k+j, with F_0=0 and F_1=1. Each packet evolves by the same jump matrix C^3 and has a different initial address.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab467eeeb5a83ecd","tex":"C^4","aliases":"C^4","display":"C^4","kind":"use","excerpt":"[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ee9beb61ef61638","tex":"\\mathbb C^d","aliases":"C^d mathbb","display":"\\mathbb C^d","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce7e7a918f6fa28e","tex":"C^H","aliases":"C^H","display":"C^H","kind":"use","excerpt":"_0,w∈ Z_p^d, L≥1, and H,j,n∈ N with H>0. The value at Hx+j for x∈ Z_p means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2482758af295dd87","tex":"C^{H_a}","aliases":"C^H_a","display":"C^{H_a}","kind":"use","excerpt":"dentEditorial hypothesis correction. Require L≥1, C,B∈ M_d( Z_p), v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bec59126aabbbf80","tex":"C^{Hx+j}","aliases":"C^Hx+j","display":"C^{Hx+j}","kind":"use","excerpt":"Use C^Hx+j=C^j(I+p^LB)^x and the matrix binomial series. One difference in the coarse variable multiplies the series by p^LB.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40378867274e1130","tex":"C^i","aliases":"C^i","display":"C^i","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0af3bfadfba7f24b","tex":"C^{i+1}","aliases":"C^i+1","display":"C^{i+1}","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b3b60bc2dabf087","tex":"C^{i-1}","aliases":"C^i-1","display":"C^{i-1}","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04ebc96f0d8e8891","tex":"C^j","aliases":"C^j","display":"C^j","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4cfc3c2ac366145","tex":"c^m","aliases":"c^m","display":"c^m","kind":"use","excerpt":"Let C=⟨ c⟩ have order m, and let T be a lift of the projective generator. Since c^m=1, the operator T^m is scalar. Choose λ∈ C^× with λ^mT^m=I. Replacing T by λ T gives an honest cyclic representation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-512937534243ab85","tex":"\\C^{m\\times m}","aliases":"C^m× m C times mathbb times","display":"{\\mathbb C}^{m\\times m}","kind":"use","excerpt":"Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eee5bcabc685dee4","tex":"\\C^{m\\times t}","aliases":"C^m× t C times mathbb times","display":"{\\mathbb C}^{m\\times t}","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc4ec113b29e6ee4","tex":"C^n","aliases":"C^n","display":"C^n","kind":"use","excerpt":"The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06e2221a9a4e2168","tex":"\\C^n","aliases":"C^n C mathbb","display":"{\\mathbb C}^n","kind":"use","excerpt":"Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To se","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-675d2adc99d5a3df","tex":"C^{n+2}","aliases":"C^n+2","display":"C^{n+2}","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ee90351b9053688","tex":"C^{Nk+j}","aliases":"C^Nk+j","display":"C^{Nk+j}","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8898d320fd158845","tex":"\\C^{n\\times m}","aliases":"C^n× m C times mathbb times","display":"{\\mathbb C}^{n\\times m}","kind":"use","excerpt":"Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1dc82af4d957fdd4","tex":"\\C^{n\\times n}","aliases":"C^n× n C times mathbb times","display":"{\\mathbb C}^{n\\times n}","kind":"use","excerpt":"Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e06675c775fa7c9d","tex":"\\mathbb C^{n\\times r_j}","aliases":"C^n× r_j mathbb times","display":"\\mathbb C^{n\\times r_j}","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2e1f9b36ab7f058","tex":"\\C^{n\\times t}","aliases":"C^n× t C times mathbb times","display":"{\\mathbb C}^{n\\times t}","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8b169112cc1dba2","tex":"C^{p+q}","aliases":"C^p+q","display":"C^{p+q}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd0b3056e758361c","tex":"C^r","aliases":"C^r","display":"C^r","kind":"use","excerpt":"[Local Lyapunov--Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there ar","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ccfe315d05aabc8","tex":"\\C^{r_L\\times r_R}","aliases":"C^r_L× r_R C times mathbb times","display":"{\\mathbb C}^{r_L\\times r_R}","kind":"use","excerpt":"Define the small core by actual multiplication, G=R_LR_R^*∈ C^r_L× r_R. Finite matrix associativity and adjoint identities give A=Q_LGQ_R^*, rank G=rank A. The rank identity follows from both multiplication inequalities and the left inverse obtained by compression. It is not inferred from agreeing dimensions.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c94fd79f2de7324b","tex":"C^T","aliases":"C^T","display":"C^T","kind":"use","excerpt":"[Coarse Feynman--Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62d6dd9a37e47fe4","tex":"\\mathbb C^\\times","aliases":"C^× mathbb times","display":"\\mathbb C^\\times","kind":"use","excerpt":"For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-897aec4854fa7c6c","tex":"c^\\leftarrow","aliases":"c^← leftarrow","display":"c^\\leftarrow","kind":"use","excerpt":"For a nonempty composable word w=(e_1,…,e_n), define its reverse coefficient product c^←(w)=c(e_n)c(e_n-1)⋯ c(e_1). The reversal is essential over a noncommutative semiring.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6499c4d477ed5bd","tex":"\\mathfrak C_0","aliases":"C_0 mathfrak","display":"\\mathfrak C_0","kind":"use","excerpt":"[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport for","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a3b01a8e2b172a1","tex":"C_{0,L}","aliases":"C_0,L","display":"C_{0,L}","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39f07d70eaa40c73","tex":"C_{0,m}","aliases":"C_0,m","display":"C_{0,m}","kind":"use","excerpt":"y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f944d0625ea61fc0","tex":"C_{0,\\ell}","aliases":"C_0,ℓ ell","display":"C_{0,\\ell}","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff97b5cd592dca16","tex":"c_{00}","aliases":"c_00","display":"c_{00}","kind":"use","excerpt":"Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer bloc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5b1cdb36b5b2a45","tex":"C_{0j}","aliases":"C_0j","display":"C_{0j}","kind":"use","excerpt":"vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":38},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24612095df1bb71a","tex":"C_{2,0}","aliases":"C_2,0","display":"C_{2,0}","kind":"use","excerpt":"vpage14Three similar reciprocals differNORMALIZED SECTIONSChapter IV: Definition 9.1, Theorem 9.2, Example 9.11. The signed denominator and the normalized unsigned sections produce different reciprocal families. In order two, D_2^-1=sec z, C_2,0=sech z, C_2,1=z/sinh z. Their first block-EGF coefficients make the difference visible. vfigdifferent_re","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b6228375a9787bb","tex":"C_{2,1}","aliases":"C_2,1","display":"C_{2,1}","kind":"use","excerpt":"vpage14Three similar reciprocals differNORMALIZED SECTIONSChapter IV: Definition 9.1, Theorem 9.2, Example 9.11. The signed denominator and the normalized unsigned sections produce different reciprocal families. In order two, D_2^-1=sec z, C_2,0=sech z, C_2,1=z/sinh z. Their first block-EGF coefficients make the difference visible. vfigdifferent_re","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functio","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b60e443e8108308c","tex":"c_3","aliases":"c_3","display":"c_3","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f4c46ceb742a13f2","tex":"C_{3,0}","aliases":"C_3,0","display":"C_{3,0}","kind":"use","excerpt":"At N=p=3 and j=0, put A(n)=(-1)^nC_3,0(n). The first three rows are AtlasImageodd_prime_sign The ordinary first difference at row zero is -1-1=-2. It is a 3-adic unit, so an asserted lower bound of one for that difference would be false. The twisted first difference is zero, whose mathematical valuation is +∞. At the next row, the twisted differenc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4e58ada952ec6ab","tex":"C_4","aliases":"C_4","display":"C_4","kind":"use","excerpt":"P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-879a006e315df8a5","tex":"C_{4,0}","aliases":"C_4,0","display":"C_{4,0}","kind":"use","excerpt":"Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b68e81c02307b3f0","tex":"C_{4,1}","aliases":"C_4,1","display":"C_{4,1}","kind":"use","excerpt":"An exact arithmetic check, not an optimality claim. The independent rational recurrence gives C_4,1(1)=-1/5 and C_4,1(2)=121/45. Their difference is 26/9, of 2-adic valuation one. This agrees with the new bound at n=k=u=1. The source also contains universal instantiations of the degree-indexed theorem at all four dyadic addresses and at an odd sing","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a62fe51ac71b9fa6","tex":"C_{4,j}","aliases":"C_4,j","display":"C_{4,j}","kind":"use","excerpt":"For an odd prime, 2mid p-1, so the option m=2 introduces no further case. The sole remaining branch is p=2,m=2,N=4. The all-row theorem proved next gives 2^kmidΔ_1^kC_4,j(n) at every row and address. After multiplying by j!, Lemma~[source reference] gives packet period 2^r-1 and degree displacement 4·2^r-1=2^r+1=qp^r. It also gives integrality. Hen","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dcb4e0ba5e2088bf","tex":"C_6","aliases":"C_6","display":"C_6","kind":"use","excerpt":"vpage16Closing a path changes the seed obstructionGRAPH MODEL / A CYCLE WITH SIX VERTICESMain text: Worked graph model. Proof of the minimum seed count for complex cycles For the adjacency matrix A of C_6, the distinct eigenvalues are 2,1,-1,-2, with multiplicities 1,2,2,1. A single vertex seed spans only four directions. Two adjacent vertex seeds ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62848497b9641ba6","tex":"C_{6,0}","aliases":"C_6,0","display":"C_{6,0}","kind":"use","excerpt":"No automatic extra power. The original rational triangular coefficients satisfy C_6,0(1)=-1, C_6,0(2)=923, v_3(C_6,0(2)-C_6,0(1))=v_3(924)=1. A period modulo 3 therefore cannot simply be reported as a period modulo 9. This is consistent with the global theorem's proposed first-difference bound. It is not a counterexample to Theorem 7.5.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f04db759519ade9","tex":"C_{6,2}","aliases":"C_6,2","display":"C_{6,2}","kind":"use","excerpt":"pageheading5. Examples, verification and the remaining gapThe coverage count is unchanged; the unresolved theorem now has a stronger proved part The original rational recurrence gives C_6,2(1)=-frac128, C_6,2(2)=frac425364, C_6,2(3)=-frac20397592660. Consequently, the new source proves Δ_1^2 C_6,2(1)=-frac9499591235, v_3 (Δ_1^2 C_6,2(1))=2. This at","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9129c76ab167e7bf","tex":"C_{6,3}","aliases":"C_6,3","display":"C_{6,3}","kind":"use","excerpt":"[Why the factorial loss is necessary] Take p=3, m=2, N=6, and j=3. The recurrence gives C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(2)-C_6,3(1)=frac64455. The last difference has 3-adic valuation zero. Here h=1, so at n=1 the order-one bound without factorial loss would require valuation at least one. The corrected normalized bound is sharp, sin","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06fe0f5802f142ad","tex":"C_{6,4}","aliases":"C_6,4","display":"C_{6,4}","kind":"use","excerpt":"w For (p,m,j)=(3,2,3), the source derives C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(3)=-frac2231997980. Consequently, Δ_1^2(3!C_6,3)(1)=-frac14647688645, v_3 (-frac14647688645)=2. This attains min(2,2·1)=2. At the regular address j=4 beyond the singular boundary, the source likewise proves Δ_1^2(4!C_6,4)(1)=-frac1027455122377375, and applies t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca35fd43dab309b4","tex":"C_{a+1}","aliases":"C_a+1","display":"C_{a+1}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0d887f344230957","tex":"C_{a,a}","aliases":"C_a,a","display":"C_{a,a}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2dc3b3739f1aa480","tex":"C_{a,b}","aliases":"C_a,b","display":"C_{a,b}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6010785cb6069e5","tex":"c_{a,j,r}","aliases":"c_a,j,r","display":"c_{a,j,r}","kind":"use","excerpt":"ypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73850e5fc882aa8c","tex":"C_{ab}","aliases":"C_ab","display":"C_{ab}","kind":"use","excerpt":"Necessary and sufficient conditions. Under the stated normalization, the following equivalence is proved: ∃ R of the above finite Kraus form, [-2pt] R N_E(ρ)=ρ for every ρsucceq0, Trρ=1, Pρ P=ρ Longleftrightarrow ∃ Csucceq0, PE_a^*E_bP=C_abP for all a,b. The statement applies throughout the code space. The formalization separately proves equivalenc","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89de6f3841051f82","tex":"c_{ab}","aliases":"c_ab","display":"c_{ab}","kind":"use","excerpt":"vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATAChapter VII: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0 12 · 16 · 0 0 · 0 · 0, PE_a^","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ee68c951157fbd1","tex":"C_{ai}","aliases":"C_ai","display":"C_{ai}","kind":"use","excerpt":"The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6922a59f4f38ba94","tex":"C_{b-1}","aliases":"C_b-1","display":"C_{b-1}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ef49bcadebb010b","tex":"C_{ba}","aliases":"C_ba","display":"C_{ba}","kind":"use","excerpt":"Cross terms for the original errors. For arbitrary positive Gram data, the same recovery satisfies R(E_aρ E_b^*)=C_baρ (Pρ P=ρ). The index order ba is intentional: the defining Gram relation is PE_a^*E_bP=C_abP. The proof first derives scalar corrected Kraus composites for the constructed recovery and then uses its trace-preserving completeness to ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"use","excerpt":"vpage20Spend error where it is cheapestCONTINUOUS AND CLIPPED PRECISIONChapter VIII: Theorem 8.4, Theorem 8.5. Take sensitivities s=(1,2,4), cost coefficients c=(1,1,2), and budget τ=1. Minimizing ∑_ec_elog(1/ϵ_e) subject to ∑_es_eϵ_e=1 gives ϵ_e=fracc_e4s_e, ϵ=(1/4,1/8,1/8). vfigprecision headThe budget is weighted The three contributions to the e","count":31},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cdae72f972852da1","tex":"c_f","aliases":"c_f","display":"c_f","kind":"use","excerpt":"[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"lus_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8eb98997bbe300af","tex":"c_{i-j}","aliases":"c_i-j","display":"c_{i-j}","kind":"use","excerpt":"Toeplitz padding and column-major vectorization. textitSource items 4.4 and 4.5 Toeplitz embedding. For n>0, define T_ij= c_i-j, · j≤ i, r_j-i, · i<j. The diagonal is supplied by c_0, and the coordinate r_0 is unused. The constructed generator of length 2n is g=(c_0,…,c_n-1,0,r_n-1,…,r_1), and the padding map is the actual finite-index sum eliminat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5265479b6f0a2ec","tex":"c_i^{e_i}","aliases":"c_i^e_i","display":"c_i^{e_i}","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-798b72ae3ed7ccde","tex":"C_{ij}","aliases":"C_ij","display":"C_{ij}","kind":"use","excerpt":"Schmidt rank and the reduced state. Use coordinates ψ_ij in the standard product basis and let C_ij=ψ_ij. An explicit linear equivalence identifies the state vector on the product index set with its coefficient matrix. The implementation checks the actual Kronecker action: coeff((A⊗ I_B)ψ)=AC. It also proves that the transported matrix orbit equals","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j, U_j(n+T)-U_j(n) · ≠0, · v_2(U_j(n+T)-U_j(n)) · =v_2(T)+d_j.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9cfc77f3166a2a9","tex":"C_{j+1,m}","aliases":"C_j+1,m","display":"C_{j+1,m}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b038101bc9d9bcb1","tex":"C_{j,j}","aliases":"C_j,j","display":"C_{j,j}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-407d3e9d1538f42c","tex":"C_{J,K}","aliases":"C_J,K","display":"C_{J,K}","kind":"use","excerpt":"The linear compatibility theorem provides one concrete part of this program. If two local observations J:U→ V and K:U→ W are linear maps from the same correction module, define C_J,K=U/(ker J+ker K). The attainable observations im J and im K have canonical labels in this quotient. Equality of labels is equivalent to simultaneous realization by a si","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00ed39c395ad1950","tex":"c_{j,k}","aliases":"c_j,k","display":"c_{j,k}","kind":"use","excerpt":"ct discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f2aea9ede1d4ca7","tex":"C_{j,m}","aliases":"C_j,m","display":"C_{j,m}","kind":"use","excerpt":"Expand each summand as C_ℓ,j+1T_j+1C_j+1,m - C_ℓ,jT_jC_j,m. The intermediate terms cancel. The first surviving term is C_ℓ,mT_m, and the last is -T_ℓ C_ℓ,m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ce23018e3603d00","tex":"c_{j,r}","aliases":"c_j,r","display":"c_{j,r}","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40c460a40f8ac2ac","tex":"c_{j-k}","aliases":"c_j-k","display":"c_{j-k}","kind":"use","excerpt":"4 · 3 · 2, x=(1,2,-1)^T. The diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column g=(2,3,4,0,5,-1)^T and pad the input with three zeros. vfigtoeplitz headRead the wrapped indices For output index j<3, an input index k≤ j uses the first-column coefficient c_j-k. An input index k>j ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b26dec79c0d88ba5","tex":"c_{j-k\\bmod4}","aliases":"c_j-kbmod4 bmod","display":"c_{j-k\\bmod4}","kind":"use","excerpt":"vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATIONChapter VIII: Theorem 4.3. Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68ec3f96f692ce11","tex":"C_j^\\bullet","aliases":"C_j^bullet bullet","display":"C_j^\\bullet","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-229173948b9deb09","tex":"C_j^r","aliases":"C_j^r","display":"C_j^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7dd170465f9a276","tex":"c_{ja}","aliases":"c_ja","display":"c_{ja}","kind":"use","excerpt":"Every normalized channel in the same span. Suppose F_j=∑_a c_jaE_a and ∑_jF_j^*F_j=I. The decoder is not reconstructed for the F_j. Summing the coherent identities gives a scalar multiple qρ. Trace preservation of both the F channel and the recovery forces q=1 on a nonzero code. The proof uses positivity of P to justify cancellation of its trace an","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f544215eb08c419","tex":"C_{jk}","aliases":"C_jk","display":"C_{jk}","kind":"use","excerpt":"[Reflection positive correlators] Fix a vector Ω and operators A_1,…,A_m. The matrix C_jk = ipA_jΩT^n_j+n_kA_kΩ is positive semidefinite.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and prov","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e62e3524995dc9d","tex":"c_K","aliases":"c_K","display":"c_K","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-254051ed849e0a84","tex":"c_k","aliases":"c_k","display":"c_k","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8e8fbaa4e63d973","tex":"C_L","aliases":"C_L","display":"C_L","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1539cdd586b7fb3f","tex":"C_L'","aliases":"C_L'","display":"C_L'","kind":"use","excerpt":"-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62ce335ae3e26f33","tex":"C_M","aliases":"C_M","display":"C_M","kind":"use","excerpt":"[Near cancellation of two exponentials] For every M>0, there are constants ε_M>0 and C_M>0 such that the following holds. If |Re w|≤ M and |1+e^w|≤ε_M, then there is a unique integer m with |w-(2m+1)π i| ≤ C_M|1+e^w|.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"There is also a completion theorem without chosen compatible characters. Suppose that R/I^m is finite for every m≥1, and that C_m⊆ P_q(R/I^m) are allowed subsets such that every reduction of an allowed member is allowed. Then · ∃χ∈ P_q(R), χ is continuous and χbmod I^m∈ C_m for all m≥1","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54bfb2f6c9232092","tex":"C_{m+1}","aliases":"C_m+1","display":"C_{m+1}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-102565d13a8ae5ca","tex":"C_{m,m}","aliases":"C_m,m","display":"C_{m,m}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03f737cfb1d5dd05","tex":"C_{m-1}","aliases":"C_m-1","display":"C_{m-1}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6013c421e4c989b5","tex":"C_{mp,j}","aliases":"C_mp,j","display":"C_{mp,j}","kind":"use","excerpt":"[Lucas reduction and address independence] Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f33574a980dc5307","tex":"C_{mp,k}","aliases":"C_mp,k","display":"C_{mp,k}","kind":"use","excerpt":"prop:arith-lucas-window Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈ N).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cfbf41dac773cbfc","tex":"C_n","aliases":"C_n","display":"C_n","kind":"use","excerpt":"Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To se","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c1ecf2aa8f993509","tex":"c_N","aliases":"c_N","display":"c_N","kind":"use","excerpt":"At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry tr","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"use","excerpt":"[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28f4c794936e614f","tex":"C_{N,0}","aliases":"C_N,0","display":"C_{N,0}","kind":"use","excerpt":"[Necessary boundaries] At an odd prime the ordinary step-one difference cannot replace the sign-twisted operator in the general theorem. For j=0, C_N,0(0)=1 and C_N,0(1)=-1, so their ordinary difference is -2, a unit at every odd prime. The proof also does not extend unchanged to non-prime-power N, whose interior binomial coefficients need not be d","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e949da172302ad25","tex":"C_{n,a}","aliases":"C_n,a","display":"C_{n,a}","kind":"use","excerpt":"[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-707f43f0f354949a","tex":"C_{n,b}","aliases":"C_n,b","display":"C_{n,b}","kind":"use","excerpt":"[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9417d8ef957ab062","tex":"C_{n,c}","aliases":"C_n,c","display":"C_{n,c}","kind":"use","excerpt":"[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b714dca696a6ce4","tex":"C_{N,j,n-q}","aliases":"C_N,j,n-q","display":"C_{N,j,n-q}","kind":"use","excerpt":"[Parameter-addition law for residue-section reciprocals] For each n≥1, the coefficient C_N,j,n(r) is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n! (-fracj!(N+j)!)^n. It satisfies C_N,j,n(r+s) = ∑_q=0^n binomNnNq C_N,j,q(r)C_N,j,n-q(s). In particular, C_N,j,n(1)=C_N,j(n).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28a6886b66c26c24","tex":"C_{N,j,q}","aliases":"C_N,j,q","display":"C_{N,j,q}","kind":"use","excerpt":"[Parameter-addition law for residue-section reciprocals] For each n≥1, the coefficient C_N,j,n(r) is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n! (-fracj!(N+j)!)^n. It satisfies C_N,j,n(r+s) = ∑_q=0^n binomNnNq C_N,j,q(r)C_N,j,n-q(s). In particular, C_N,j,n(1)=C_N,j(n).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b48446852f3335a2","tex":"C_{n-1}","aliases":"C_n-1","display":"C_{n-1}","kind":"use","excerpt":"Assume now that a>0 and b>0. Take R= frac1-sqrt1-4ab2b. Then a+bR^2=R and 2bR<1. Apply [source reference]. The number of full ordered binary trees with n leaves is the Catalan number C_n-1, which gives the displayed majorant.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7e8c8a0a1fa875e","tex":"c_{n-1}","aliases":"c_n-1","display":"c_{n-1}","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-374d4fe4488ce15e","tex":"C_p","aliases":"C_p","display":"C_p","kind":"use","excerpt":"fferentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c a","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a7f55c1da8911f74","tex":"c_{p^ak}","aliases":"c_p^ak","display":"c_{p^ak}","kind":"use","excerpt":"(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be57c172ad4ba898","tex":"c_{p^ak+u}","aliases":"c_p^ak+u","display":"c_{p^ak+u}","kind":"use","excerpt":"insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-685a0465f972e5ce","tex":"C_{p^{n_i}}","aliases":"C_p^n_i","display":"C_{p^{n_i}}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5ef149476c4688d","tex":"C_{\\Pcal\\leftarrow\\Rcal}","aliases":"C_Pcal←Rcal C_ P← R Pcal leftarrow Rcal mathcal leftarrow mathcal","display":"C_{{\\mathcal P}\\leftarrow{\\mathcal R}}","kind":"use","excerpt":"herence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb134dc373ba038c","tex":"C_Q","aliases":"C_Q","display":"C_Q","kind":"use","excerpt":"[Estimated packet Gibbs bound] Let Q_m be linear and uniformly bounded on L^p, with sup_m Q_m ≤ C_Q. For f=s+j and an estimated packet widehat j_m, define widehat f_m =Q_m(f-widehat j_m)+widehat j_m. Then f-widehat f_m _p ≤ s-Q_ms _p +(1+C_Q) j-widehat j_m _p. Consequently, smooth approximation and packet recovery may be improved independently. No ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf3fd1bbd0e06a1b","tex":"C_{\\Qcal\\leftarrow\\Rcal}","aliases":"C_Qcal←Rcal C_ Q← R Qcal leftarrow Rcal mathcal leftarrow mathcal","display":"C_{{\\mathcal Q}\\leftarrow{\\mathcal R}}","kind":"use","excerpt":"herence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-557d8a93e9288da4","tex":"C_R","aliases":"C_R","display":"C_R","kind":"use","excerpt":"for maps from A and B to a common set if and only if R is rectangular. In that case a canonical finite choice of common set is the connected-component set C_R of the bipartite graph with vertex set Asqcup B and edges given by R.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"use","excerpt":"[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.","count":41},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b60e0ee83a3abdf","tex":"C_r^*","aliases":"C_r^*","display":"C_r^*","kind":"use","excerpt":"The fine and coarse families are complete orthogonal self-adjoint projector systems. In particular, C_rC_s=δ_rsC_r, ∑_r C_r=I, C_r^*=C_r. Each coarse projector is exactly the displayed sum of its fine descendants.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-307d2c7e46d5ba73","tex":"C_s","aliases":"C_s","display":"C_s","kind":"use","excerpt":"The fine and coarse families are complete orthogonal self-adjoint projector systems. In particular, C_rC_s=δ_rsC_r, ∑_r C_r=I, C_r^*=C_r. Each coarse projector is exactly the displayed sum of its fine descendants.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1e71a0eb7c64bd5","tex":"c_s","aliases":"c_s","display":"c_s","kind":"use","excerpt":"An independently defined interpolant. For a finite coefficient list, define I_N(c)(n)=∑_r=0^Nc_rbinom nr. The source proves exact coefficient extraction: Δ_1^s I_N(c)(0)= c_s, · s≤ N, 0, · s>N. It follows that I_N(c) has the stated shifted-coefficient formula at every order, with zero coefficients beyond N. Every difference of order greater than N ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c4963e002d8ca00","tex":"c_{s+j}","aliases":"c_s+j","display":"c_{s+j}","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ef88b2c20ebb677","tex":"c_{s+r}","aliases":"c_s+r","display":"c_{s+r}","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74d610b5f945d7cc","tex":"C_T","aliases":"C_T","display":"C_T","kind":"use","excerpt":"em[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9c457fd3fefb6bf","tex":"c_u","aliases":"c_u","display":"c_u","kind":"use","excerpt":"The product law follows from [source reference]. To prove injectivity, suppose ∑_u∈ F_pc_uϵ_u=0. Looking at the coordinates n=0,1,…,p-1 gives a Vandermonde system with matrix (u^n)_0≤ n≤ p-1, u∈ F_p. Its determinant is the product of all differences between distinct field elements, so it is nonzero. Hence every c_u vanishes.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53faef096d1a0acc","tex":"c_v","aliases":"c_v","display":"c_v","kind":"use","excerpt":"[Dynamic-programming best packet basis] Suppose J_y(Λ)=∑_v∈Λc_v(y). For a node v, define recursively V_y(v) = c_v(y), · v terminal,","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6ff7660eca84d81","tex":"C_x","aliases":"C_x","display":"C_x","kind":"use","excerpt":"vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELChapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-85d220b196ae3e90","tex":"\\mathcal C_x","aliases":"C_x mathcal","display":"\\mathcal C_x","kind":"use","excerpt":"Let G∈ Z[X_1,…,X_d]^f be a second fixed system with the same variables and the same lower root x. Let K_x=ρ_q∘ DG(x), and define C_x= Z^d/(ker J_x+ker K_x). This quotient is finite: it is a quotient of the finite image of J_x, or equivalently of the finite image of K_x.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-087e2290da72a37c","tex":"C_y","aliases":"C_y","display":"C_y","kind":"use","excerpt":"vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELChapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b5c76a8c06b016d","tex":"C_Z","aliases":"C_Z","display":"C_Z","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-065fc2a65996d145","tex":"C_z","aliases":"C_z","display":"C_z","kind":"use","excerpt":"vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELChapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35910a02402c8671","tex":"C_{\\alpha i}","aliases":"C_α i alpha","display":"C_{\\alpha i}","kind":"use","excerpt":"Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19b5aea016550f5b","tex":"C_\\alpha^\\bullet","aliases":"C_α^bullet alpha bullet","display":"C_\\alpha^\\bullet","kind":"use","excerpt":"[Direct packet complexes] Suppose C^bullet=⊕_α∈ IC_α^bullet is a finite direct sum of subcomplexes. Then H^n(C^bullet) ≅ ⊕_α∈ IH^n(C_α^bullet). A total cohomology class vanishes if and only if every packet class vanishes.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5da91e15de04cc9e","tex":"C_\\eta","aliases":"C_η eta","display":"C_\\eta","kind":"use","excerpt":"vpage41An operator substitution transports the defect tooFUNCTIONAL CALCULUS / AN EXACT RATIONAL TESTChapter VI, Part II: Theorem E3.4; compare Chapter VI, Part I, Theorem 15.1, and Plate VI.25. Use the operators from Plate VI.25, but perturb the rectangular comparison map: A=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, B=mat2 · 1 0 · 2, C_η=mat1 · 0 · η 0 · ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db3b3f64d870ea56","tex":"c_{\\eta u}","aliases":"c_η u eta","display":"c_{\\eta u}","kind":"use","excerpt":"Write b^(m,p) = ∑_u∈ F_pc_uϵ_u. Invariance gives c_η u=c_u. After the change of variable u↦η u, b_n^(m,p) = η^n b_n^(m,p) for every η∈ H_m. If mnmid n, some η satisfies η^n≠1, so the coefficient vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42eb4bc96455bb91","tex":"C_\\pi","aliases":"C_π pi","display":"C_\\pi","kind":"use","excerpt":"Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-488e5fe62245ff4a","tex":"c_\\chi","aliases":"c_χ chi","display":"c_\\chi","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-029bccf361123f37","tex":"C_\\psi","aliases":"C_ψ psi","display":"C_\\psi","kind":"use","excerpt":"[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ae0a9be277fd36e","tex":"C_\\ell","aliases":"C_ℓ ell","display":"C_\\ell","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":25},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e58ea848499cc339","tex":"C_{\\ell+1,a}","aliases":"C_ℓ+1,a ell","display":"C_{\\ell+1,a}","kind":"use","excerpt":"Let ( H_ℓ,C_ℓ,a) and ( H_ℓ+1,C_ℓ+1,a) be two levels, and let R_ℓ: H_ℓ⟶ H_ℓ+1 be an isometry.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c04e3fb7a0f52049","tex":"C_{\\ell+1,b}","aliases":"C_ℓ+1,b ell","display":"C_{\\ell+1,b}","kind":"use","excerpt":"[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ada5041fbbc689c","tex":"C_{\\ell+1,L}","aliases":"C_ℓ+1,L ell","display":"C_{\\ell+1,L}","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7561a308f43c77a6","tex":"C_{\\ell,a}","aliases":"C_ℓ,a ell","display":"C_{\\ell,a}","kind":"use","excerpt":"Let ( H_ℓ,C_ℓ,a) and ( H_ℓ+1,C_ℓ+1,a) be two levels, and let R_ℓ: H_ℓ⟶ H_ℓ+1 be an isometry.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b6d5c1a68e80f7a","tex":"C_{\\ell,b}","aliases":"C_ℓ,b ell","display":"C_{\\ell,b}","kind":"use","excerpt":"[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4a0be93686b794f","tex":"C_{\\ell,j}","aliases":"C_ℓ,j ell","display":"C_{\\ell,j}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c05f0525b2178009","tex":"C_{\\ell,j+1}","aliases":"C_ℓ,j+1 ell","display":"C_{\\ell,j+1}","kind":"use","excerpt":"Expand each summand as C_ℓ,j+1T_j+1C_j+1,m - C_ℓ,jT_jC_j,m. The intermediate terms cancel. The first surviving term is C_ℓ,mT_m, and the last is -T_ℓ C_ℓ,m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c05481d823dd05d","tex":"C_{\\ell,L}","aliases":"C_ℓ,L ell","display":"C_{\\ell,L}","kind":"use","excerpt":"Insert the intermediate terms C_0,ℓT_ℓ C_ℓ,L for 0≤ℓ≤ L. Consecutive differences telescope. The difference between the terms indexed by ℓ+1 and ℓ is C_0,ℓ(C_ℓ T_ℓ+1-T_ℓ C_ℓ)C_ℓ+1,L. Summing gives the identity. The norm estimate follows from the triangle inequality.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71a8a3e47a0a9522","tex":"C_{\\ell,m}","aliases":"C_ℓ,m ell","display":"C_{\\ell,m}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2025a3e28eea9cf","tex":"\\cA^X_a","aliases":"cA^X_a A^X_a cA mathcal","display":"{\\mathcal A}^X_a","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32b92e346ab956bb","tex":"\\cA^X_b","aliases":"cA^X_b A^X_b cA mathcal","display":"{\\mathcal A}^X_b","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a3f0b5a74405089","tex":"\\cA^Y","aliases":"cA^Y A^Y cA mathcal","display":"{\\mathcal A}^Y","kind":"use","excerpt":"Both sides of the naturality equation reduce to A^Y_s∘ F∘ S^X_a using S∘ A=id in X and Y. For composition, the middle S^Y∘ A^Y cancels in the same way.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79b2b39c2a8e8c5a","tex":"\\cA^Y_r","aliases":"cA^Y_r A^Y_r cA mathcal","display":"{\\mathcal A}^Y_r","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f54c3e34264c2fc","tex":"\\cA^Y_s","aliases":"cA^Y_s A^Y_s cA mathcal","display":"{\\mathcal A}^Y_s","kind":"use","excerpt":"Both sides reduce to A^Y_sF S^X_a after inserting the appropriate synthesis--analysis identity. The composition law is the same calculation with the middle ambient module inserted.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1251c2ab227ac52f","tex":"\\cA^Z_t","aliases":"cA^Z_t A^Z_t cA mathcal","display":"{\\mathcal A}^Z_t","kind":"use","excerpt":"^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then ex","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c52bed8ad815407b","tex":"\\cA^{\\oplus r}","aliases":"cA^⊕ r A^⊕ r cA oplus mathcal oplus","display":"{\\mathcal A}^{\\oplus r}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S),","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e9b4ec45cb5eaa9","tex":"\\cA_a","aliases":"cA_a A_a cA mathcal","display":"{\\mathcal A}_a","kind":"use","excerpt":"ordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32e5fff3cf3d9303","tex":"\\cA_c","aliases":"cA_c A_c cA mathcal","display":"{\\mathcal A}_c","kind":"use","excerpt":"Insert S_b A_b=id_X: J^X_c← bJ^X_b← a = A_c S_b A_b S_a = A_c S_a =J^X_c← a. The identity statement is A_a S_a=id. Setting c=a in the cocycle gives J^X_a← bJ^X_b← a=id. Interchanging a and b gives the other composition, so the two transports are inverses.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a7d8b938ce09cb1","tex":"\\cA_N","aliases":"cA_N A_N cA mathcal","display":"{\\mathcal A}_N","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5756f63546bea0ab","tex":"\\cA_Q","aliases":"cA_Q A_Q cA mathcal","display":"{\\mathcal A}_Q","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0a4ff2df722dd9a","tex":"\\calH_\\theta","aliases":"calH_θ H_θ calH theta mathcal theta","display":"{\\mathcal H}_\\theta","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e11420ea16392e8","tex":"\\calH_\\rho","aliases":"calH_ρ H_ρ calH rho mathcal rho","display":"{\\mathcal H}_\\rho","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21355a3045e08010","tex":"\\mathrm{can}","aliases":"can mathrm","display":"\\mathrm{can}","kind":"use","excerpt":"[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b844ea5955b0d603","tex":"\\mathsf{Candidate}","aliases":"Candidate mathsf","display":"\\mathsf{Candidate}","kind":"use","excerpt":"Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a623735ca1da9cda","tex":"\\cB","aliases":"cB B mathcal","display":"{\\mathcal B}","kind":"use","excerpt":"Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61eeaf6b817657fa","tex":"\\cB_{\\bar\\lambda}","aliases":"cB_barλ B_barλ cB bar lambda mathcal bar lambda","display":"{\\mathcal B}_{\\bar\\lambda}","kind":"use","excerpt":"[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on ea","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46ffc8349fb86de0","tex":"\\cB_{\\bar\\lambda+\\delta}","aliases":"cB_barλ+δ B_barλ+δ cB bar lambda delta mathcal bar lambda delta","display":"{\\mathcal B}_{\\bar\\lambda+\\delta}","kind":"use","excerpt":"The effective-likelihood theorem reduces the defect to B_barλ+δ(P)- B_barλ(P). Since Z_r=Z_barλ+P^ Tδ, direct subtraction gives frac1Z_r (diag(P)-bar bP^ T)δ. Also, (diag(P)-bar bP^ T)barλ =Podotbarλ-bar bZ_barλ=0. The first norm bound is the likelihood perturbation theorem applied on the coarse simplex. For a zero-mass packet, both sides of the fo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17dd4b80f1d797e2","tex":"\\cB_{r(p)}","aliases":"cB_r(p) B_r(p) cB mathcal","display":"{\\mathcal B}_{r(p)}","kind":"use","excerpt":"[Exact effective-likelihood factorization] Whenever the fine evidence is positive, C_π B_λ(p) = B_r(p)(C_π p). Thus the only fine information needed for the coarse posterior, beyond the coarse prior masses, is the vector of conditional packet evidences r(p).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34496546f1396a6e","tex":"\\cB_\\lambda","aliases":"cB_λ B_λ cB lambda mathcal lambda","display":"{\\mathcal B}_\\lambda","kind":"use","excerpt":"Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37035c9d5b2ad69b","tex":"\\cB_{\\lambda^{(r)}}","aliases":"cB_λ^(r) B_λ^(r) cB lambda mathcal lambda","display":"{\\mathcal B}_{\\lambda^{(r)}}","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-276d7bc4ee7755c0","tex":"\\cB_{\\lambda\\odot\\mu}","aliases":"cB_λodotμ B_λodotμ cB lambda odot mu mathcal lambda odot mu","display":null,"kind":"use","excerpt":"[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-555bcf940ecf36c3","tex":"\\cB_\\mu","aliases":"cB_μ B_μ cB mu mathcal mu","display":"{\\mathcal B}_\\mu","kind":"use","excerpt":"[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b66848dfe6155c4c","tex":"\\cC","aliases":"cC C mathcal","display":"{\\mathcal C}","kind":"use","excerpt":"Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environmen","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c1fa1d6163c6fb8","tex":"\\CC^{2n}","aliases":"CC^2n C^2n CC mathbb","display":"{\\mathbb C}^{2n}","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9632328f158df6f3","tex":"\\CC^3","aliases":"CC^3 C^3 CC mathbb","display":"{\\mathbb C}^3","kind":"use","excerpt":"vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDChapter VII: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8feeaa9714c49a4","tex":"\\CC^4","aliases":"CC^4 C^4 CC mathbb","display":"{\\mathbb C}^4","kind":"use","excerpt":"l is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the c","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a7ba377119b57ac","tex":"\\CC^6","aliases":"CC^6 C^6 CC mathbb","display":"{\\mathbb C}^6","kind":"use","excerpt":"o groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines the required count: srank_AA(HH)=max_ilceil m_i/d_irceil. This is an invariant of the declared representation of the algebra, and depends on more than the dimension of the Hilbert space alone. If the available algebra were enlarged to the full","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc12fa997357cc20","tex":"\\CC^8","aliases":"CC^8 C^8 CC mathbb","display":"{\\mathbb C}^8","kind":"use","excerpt":"2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'' The eight vectors are orthonormal, so their linear span ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6691a5fcf8bc635","tex":"\\CC^c","aliases":"CC^c C^c CC mathbb","display":"{\\mathbb C}^c","kind":"use","excerpt":"Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b260911d985f0cb1","tex":"\\CC^{c_\\rho}","aliases":"CC^c_ρ C^c_ρ CC rho mathbb rho","display":"{\\mathbb C}^{c_\\rho}","kind":"use","excerpt":"[Nondegenerate one break packet] If b_ρ is nondegenerate, then C^c_ρ[A] ≅ End(M_ρ), dim M_ρ=sqrtabs A.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8326ea62121139c","tex":"\\CC^d","aliases":"CC^d C^d CC mathbb","display":"{\\mathbb C}^d","kind":"use","excerpt":"[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f64ef532a7ecb69","tex":"\\CC^{d_r}","aliases":"CC^d_r C^d_r CC mathbb","display":"{\\mathbb C}^{d_r}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-274a6d2d70507826","tex":"\\CC^G","aliases":"CC^G C^G CC mathbb","display":"{\\mathbb C}^G","kind":"use","excerpt":"[Finite periodic signals] Let G= Z^d/L, where L≤ Z^d has finite index. The translation orbit of the impulse at the identity is a basis of C^G. In particular, one impulse generates every signal on the finite periodic lattice.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5008bfc15f104acb","tex":"\\CC^m","aliases":"CC^m C^m CC mathbb","display":"{\\mathbb C}^m","kind":"use","excerpt":"[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81a71c2d2875e597","tex":"\\CC^{m_r}","aliases":"CC^m_r C^m_r CC mathbb","display":"{\\mathbb C}^{m_r}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9f0fa654a6d5449","tex":"\\CC^{m\\times m}","aliases":"CC^m× m C^m× m CC times mathbb times","display":"{\\mathbb C}^{m\\times m}","kind":"use","excerpt":"[Kronecker modewise application] Let A∈ C^m× m, B∈ C^n× n, and X∈ C^n× m. With column major vectorization, (A⊗ B)vec(X) = vec(BXA^T). The right side has modeled cost 2n^2m+2nm^2 and stores n^2+m^2 factor entries instead of n^2m^2 entries.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac276f231fb12d59","tex":"\\CC^{m\\times r}","aliases":"CC^m× r C^m× r CC times mathbb times","display":"{\\mathbb C}^{m\\times r}","kind":"use","excerpt":"[Small core recompression] Let L∈ C^m× r, R∈ C^n× r. Take compact QR factorizations L=Q_LR_L, R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e82c63e4ddfb564","tex":"\\CC^{n\\times m}","aliases":"CC^n× m C^n× m CC times mathbb times","display":"{\\mathbb C}^{n\\times m}","kind":"use","excerpt":"[Kronecker modewise application] Let A∈ C^m× m, B∈ C^n× n, and X∈ C^n× m. With column major vectorization, (A⊗ B)vec(X) = vec(BXA^T). The right side has modeled cost 2n^2m+2nm^2 and stores n^2+m^2 factor entries instead of n^2m^2 entries.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e1e698420bbbca9","tex":"\\CC^{n\\times n}","aliases":"CC^n× n C^n× n CC times mathbb times","display":"{\\mathbb C}^{n\\times n}","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e90243c9b0dc0328","tex":"\\CC^r","aliases":"CC^r C^r CC mathbb","display":"{\\mathbb C}^r","kind":"use","excerpt":"Let P(x) be a smooth rank r orthogonal projector over a parameter manifold. Choose a local isometry V(x): C^r⟶ H, V(x)V(x)^*=P(x).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce2f5577f99d3780","tex":"\\CC^\\times","aliases":"CC^× C^× CC times mathbb times","display":"{\\mathbb C}^\\times","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c9fd68e24f50c5c","tex":"\\Ccal_{2,0}","aliases":"Ccal_2,0 C_2,0 Ccal mathcal","display":"{\\mathcal C}_{2,0}","kind":"use","excerpt":"[Euler and Bernoulli boundaries] For (N,j)=(2,0), C_2,0(t)=secht, so C_2,0(n)=E_2n. For (N,j)=(2,1), C_2,1(t)=fractsinh t, and C_2,1(n) = 2(1-2^2n-1)B_2n. The usual convention B_1=-1/2 is used. Euler and Bernoulli number conventions are reviewed in [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a5e9f6fb063921d","tex":"\\Ccal_{2,1}","aliases":"Ccal_2,1 C_2,1 Ccal mathcal","display":"{\\mathcal C}_{2,1}","kind":"use","excerpt":"[Euler and Bernoulli boundaries] For (N,j)=(2,0), C_2,0(t)=secht, so C_2,0(n)=E_2n. For (N,j)=(2,1), C_2,1(t)=fractsinh t, and C_2,1(n) = 2(1-2^2n-1)B_2n. The usual convention B_1=-1/2 is used. Euler and Bernoulli number conventions are reviewed in [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6a3bfeb6b8f17e0","tex":"\\Ccal_{N,j}","aliases":"Ccal_N,j C_N,j Ccal mathcal","display":"{\\mathcal C}_{N,j}","kind":"use","excerpt":"[Congruential reciprocal packet] Define C_N,j(t) = fract^j/j! E_N,j(t) = ∑_n≥0 C_N,j(n) fract^Nn(Nn)!.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcdfb14f2ad4a51d","tex":"\\cD","aliases":"cD D mathcal","display":"{\\mathcal D}","kind":"use","excerpt":"[Block-diagonal measurement invariance] Let {P_α} be a complete packet family, and suppose every measurement effect is packet diagonal: E_k=∑_α P_α E_kP_α. Define the packet-dephased state D_P(ρ)=∑_α P_αρ P_α. Then Tr(E_kρ) = Tr(E_k D_P(ρ)) for every outcome k.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8f165c768f13650","tex":"\\cD_P","aliases":"cD_P D_P cD mathcal","display":"{\\mathcal D}_P","kind":"use","excerpt":"[Block-diagonal measurement invariance] Let {P_α} be a complete packet family, and suppose every measurement effect is packet diagonal: E_k=∑_α P_α E_kP_α. Define the packet-dephased state D_P(ρ)=∑_α P_αρ P_α. Then Tr(E_kρ) = Tr(E_k D_P(ρ)) for every outcome k.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c70d511b332d10b","tex":"\\cF","aliases":"cF F mathcal","display":"{\\mathcal F}","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4e856a106ed7fa5","tex":"\\cF_{\\theta\\phi}","aliases":"cF_θϕ F_θϕ cF theta phi mathcal theta phi","display":"{\\mathcal F}_{\\theta\\phi}","kind":"use","excerpt":"I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A is chosen. This explicit calculation is not a separate entry among the 19 ite","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b06e46899a29307","tex":"\\cF_{\\mu\\nu}","aliases":"cF_μν F_μν cF mu nu mathcal mu nu","display":"{\\mathcal F}_{\\mu\\nu}","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e851c4705c600db7","tex":"\\Cfun_{N,j}^r","aliases":"Cfun_N,j^r C_N,j^r Cfun mathsf","display":"{\\mathsf C}_{N,j}^r","kind":"use","excerpt":"Multiply the block series for C_N,j^r by the exponential series and collect the coefficient of total degree m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-965c9524c8144bbf","tex":"\\mathrm{ch}","aliases":"ch mathrm","display":"\\mathrm{ch}","kind":"use","excerpt":"[Analysis--synthesis identity] S_N A_N=I. On the product of the channel spaces one also has A_N S_N=I. Equip this finite product with the Hilbert norm (x_0,…,x_N-1) _ch^2 =∑_r=0^N-1 x_r _2^2. Then A_N is a linear isometric equivalence and x _2^2=∑_r=0^N-1 x_r _2^2. Euclidean division gives a unique pair (r,q) with 0≤ r<N and n=Nq+r, including negat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e4db5255ab27d49","tex":"\\cH_A","aliases":"cH_A H_A cH mathcal","display":"{\\mathcal H}_A","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8f829d75272be4e","tex":"\\cH_B","aliases":"cH_B H_B cH mathcal","display":"{\\mathcal H}_B","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c5ec99016fb65f5","tex":"\\cH_\\ell","aliases":"cH_ℓ H_ℓ cH ell mathcal ell","display":"{\\mathcal H}_\\ell","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03399e5b8114d3a5","tex":"\\cH_{\\ell+1}","aliases":"cH_ℓ+1 H_ℓ+1 cH ell mathcal ell","display":"{\\mathcal H}_{\\ell+1}","kind":"use","excerpt":"hfrak Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d831feefac3295be","tex":"\\operatorname{char}","aliases":"char operatorname","display":"\\operatorname{char}","kind":"use","excerpt":"Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9a7361238345df3","tex":"\\mathrm{child}","aliases":"child mathrm","display":"\\mathrm{child}","kind":"use","excerpt":"Recurrence. For terminal nodes, F_λ(v)=κ_v+λϵ_v. For internal nodes, F_λ(v)=min{κ_v+λϵ_v, σ_v+∑_w∈child(v)F_λ(w)}. The checked result has two parts. First, F_λ(v)≤ J_λ(L) for every legal cut of the subtree. Second, the constructed function codecompile returns a legal cut with objective exactly F_λ(v). Thus the minimum is attained, not merely bounde","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ccb8413fea741c57","tex":"\\operatorname{Child}","aliases":"Child operatorname","display":"\\operatorname{Child}","kind":"use","excerpt":"Suppose N,M≥1 and Nmid M. Every coarse phase has a disjoint family of fine descendants: Child_M(N,r) = {s:0≤ s<M, s≡ rpmod N}.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-025cd74028e154d2","tex":"\\operatorname{Child}_{M}","aliases":"Child_M operatorname","display":"\\operatorname{Child}_{M}","kind":"use","excerpt":"Suppose N,M≥1 and Nmid M. Every coarse phase has a disjoint family of fine descendants: Child_M(N,r) = {s:0≤ s<M, s≡ rpmod N}.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43e155d639ad4bdc","tex":"\\cInd","aliases":"cInd operatorname","display":"{\\operatorname{cInd}}","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f51b343cf55db88","tex":"\\cInd_{\\mathcal M}^{\\GL_N(F)}","aliases":"cInd_ M^GL_N(F) cInd mathcal GL operatorname mathcal operatorname","display":"{\\operatorname{cInd}}_{\\mathcal M}^{{\\operatorname{GL}}_N(F)}","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-124b73f571e815f3","tex":"\\cK","aliases":"cK K mathcal","display":"{\\mathcal K}","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a2d1ebc2d7c43f4","tex":"\\mathrm{CNOT}","aliases":"CNOT mathrm","display":"\\mathrm{CNOT}","kind":"use","excerpt":"le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the or","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67c1622d52b0e019","tex":"\\mathrm{coarse}","aliases":"coarse mathrm","display":"\\mathrm{coarse}","kind":"use","excerpt":"[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-269cd99c3ea8db81","tex":"\\code","aliases":"code","display":null,"kind":"use","excerpt":"For a predicate S on costs, the common minimum specification means: ll codenone: · no cost satisfies S, codesome(v): · S(v) and v≤ z whenever S(z). Both attainment and minimality are included. Optional minimum and optional addition are proved to implement union and independent sum of feasible sets.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e417cc5a30e35857","tex":"\\operatorname{coe}","aliases":"coe operatorname","display":"\\operatorname{coe}","kind":"use","excerpt":"Identity with the original rational coefficients. The integral lift is constructed from the decoded coefficients of the actual reciprocal. The code proves coe(e_j(n))=C_4,j(n) in Q_2, and proves that the coefficient map commutes with every difference stencil. Multiplying by j! yields the unnormalized family. The valuation result is not divisibility","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92bda6e67397bc66","tex":"\\coef","aliases":"coef","display":null,"kind":"use","excerpt":"mate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =c","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-efd4823e0a402064","tex":"\\operatorname{coeff}","aliases":"coeff operatorname","display":"\\operatorname{coeff}","kind":"use","excerpt":"Schmidt rank and the reduced state. Use coordinates ψ_ij in the standard product basis and let C_ij=ψ_ij. An explicit linear equivalence identifies the state vector on the product index set with its coefficient matrix. The implementation checks the actual Kronecker action: coeff((A⊗ I_B)ψ)=AC. It also proves that the transported matrix orbit equals","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e6d0cbeeb71722a","tex":"\\operatorname{coker}","aliases":"coker operatorname","display":"\\operatorname{coker}","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b0d52c420b6644c","tex":"\\comm","aliases":"comm","display":null,"kind":"use","excerpt":"Define C=∑_aD_aP_a(t). Then C=frac12∑_acommD_aP_a(t), C^*=-C, and, for every b, commCP_b(t)=D_b.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52ad9950a5d7bf10","tex":"\\operatorname{compressed}","aliases":"compressed operatorname","display":"\\operatorname{compressed}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f4e7b64e9db023c","tex":"\\cond","aliases":"cond operatorname","display":"{\\operatorname{cond}}","kind":"use","excerpt":"[Block Jacobi condition bound] Let A=D+K act on a nonempty finite coordinate space and be Hermitian positive definite with positive definite block diagonal part D. Here D^-1/2 is the actual inverse of the positive square root of D. Suppose E=D^-1/2KD^-1/2 is Hermitian and normE_2≤η<1. Then cond_2(D^-1/2AD^-1/2) ≤ frac1+η1-η.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa303ae817004b47","tex":"\\cond_2","aliases":"cond_2 cond operatorname","display":"{\\operatorname{cond}}_2","kind":"use","excerpt":"[Block Jacobi condition bound] Let A=D+K act on a nonempty finite coordinate space and be Hermitian positive definite with positive definite block diagonal part D. Here D^-1/2 is the actual inverse of the positive square root of D. Suppose E=D^-1/2KD^-1/2 is Hermitian and normE_2≤η<1. Then cond_2(D^-1/2AD^-1/2) ≤ frac1+η1-η.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cef2f13a7966bbb0","tex":"\\coprod","aliases":"coprod","display":null,"kind":"use","excerpt":"Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their card","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0eafdcac63578ce","tex":"\\coprod_{e\\in E}","aliases":"coprod_e∈ E coprod in","display":null,"kind":"use","excerpt":"Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their card","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8db6134243ea03f4","tex":"\\cos^2","aliases":"cos^2 cos","display":"\\cos^2","kind":"use","excerpt":"Connecting to the chapter's Z_n convention. The chapter defines Z_n through sec t+tan t. The Lean construction uses Mathlib's actual formal sine and cosine series and defines sec t=(cos t)^-1, tan t=(sin t)sec t, Z_n=n![t^n](sec t+tan t). It proves the sine and cosine derivative identities, sin^2t+cos^2t=1, and (tan t)'=sec^2t. Sign substitution in","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1a4d4510c259fa0","tex":"\\cost","aliases":"cost operatorname","display":"{\\operatorname{cost}}","kind":"use","excerpt":"ne. Keeping either a or b costs three with error two. Splitting a costs four with zero error, and splitting b costs five with zero error. vfigtree_frontier The five complete prunings have (cost,error) pairs (1,6), (7,4), (9,2), (8,2), (10,0). At budget two the unique cheapest pruning is {a_0,a_1,b}, costing eight. It is Pareto optimal, but never mi","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04ea05958a7c1391","tex":"\\coth","aliases":"coth","display":null,"kind":"use","excerpt":"Let Q(z)=zcoth z, the regular formal series with constant coefficient one. The elementary differential identity (frac zsinh z)^2=Q(z)-zQ'(z) and the Bernoulli expansion Q(z)=1+∑_m≥1frac2^2mB_2m(2m)!z^2m give β_m=-2^2m(2m-1)B_2m/(2m)!. Multiplication by 2^2m(2m)! proves the identity for κ_m. The Bernoulli denominator theorem gives v_2(B_2m)=-1 for m","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb3375037c04cb55","tex":"\\Cov","aliases":"Cov operatorname","display":"{\\operatorname{Cov}}","kind":"use","excerpt":"Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68a00114cff8335b","tex":"\\Cov_{p(t)}","aliases":"Cov_p(t) Cov operatorname","display":"{\\operatorname{Cov}}_{p(t)}","kind":"use","excerpt":"Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f39bbad3b3a0ba30","tex":"\\covol","aliases":"covol operatorname","display":"{\\operatorname{covol}}","kind":"use","excerpt":"There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ce3ab72565dec46","tex":"\\cQ","aliases":"cQ Q mathcal","display":"{\\mathcal Q}","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c61c0d89b8905641","tex":"\\cQ^r","aliases":"cQ^r Q^r cQ mathcal","display":"{\\mathcal Q}^r","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4411dd1c078c6979","tex":"\\cQ^s","aliases":"cQ^s Q^s cQ mathcal","display":"{\\mathcal Q}^s","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d565d6a34f41b3b9","tex":"\\cref","aliases":"cref","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9d0a2db27784e60","tex":"\\cS^X_a","aliases":"cS^X_a S^X_a cS mathcal","display":"{\\mathcal S}^X_a","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e10b94ac80a0d767","tex":"\\cS^X_b","aliases":"cS^X_b S^X_b cS mathcal","display":"{\\mathcal S}^X_b","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f06e2bdca099a9e7","tex":"\\cS^Y","aliases":"cS^Y S^Y cS mathcal","display":"{\\mathcal S}^Y","kind":"use","excerpt":"Both sides of the naturality equation reduce to A^Y_s∘ F∘ S^X_a using S∘ A=id in X and Y. For composition, the middle S^Y∘ A^Y cancels in the same way.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35ee11babd943e3c","tex":"\\cS^Y_r","aliases":"cS^Y_r S^Y_r cS mathcal","display":"{\\mathcal S}^Y_r","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a294477855ee7bbe","tex":"\\cS^Y_s","aliases":"cS^Y_s S^Y_s cS mathcal","display":"{\\mathcal S}^Y_s","kind":"use","excerpt":"beginproposition[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, wit","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9d6fb8975e8c270","tex":"\\cS_b","aliases":"cS_b S_b cS mathcal","display":"{\\mathcal S}_b","kind":"use","excerpt":"Insert S_b A_b=id_X: J^X_c← bJ^X_b← a = A_c S_b A_b S_a = A_c S_a =J^X_c← a. The identity statement is A_a S_a=id. Setting c=a in the cocycle gives J^X_a← bJ^X_b← a=id. Interchanging a and b gives the other composition, so the two transports are inverses.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-296d2757843f9fef","tex":"\\cS_N","aliases":"cS_N S_N cS mathcal","display":"{\\mathcal S}_N","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57b35300f526c067","tex":"\\cS_P","aliases":"cS_P S_P cS mathcal","display":"{\\mathcal S}_P","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c6669ab098dda59","tex":"\\mathbb D","aliases":"D mathbb","display":"\\mathbb D","kind":"use","excerpt":"Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab700623b6727b01","tex":"\\mathbf D","aliases":"D mathbf","display":"\\mathbf D","kind":"use","excerpt":"[Equational realization principle] Let F: C→ D be an additive functor between preadditive categories. Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by F. Scalar multiplication by a coefficient ring is also preserved when the categories ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"Over a characteristic-zero field K, put E(a)=∑_n≥0fraca_nn!t^n, D(A)_n=n![t^n]A. These inverse maps between sequences and K[[t]] provide formal exponential coordinates. No analytic convergence is needed for the coefficient identities below.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40d9503903bdca20","tex":"\\mathfrak D","aliases":"D mathfrak","display":"\\mathfrak D","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a152920678c4fe82","tex":"D'''","aliases":"D'''","display":"D'''","kind":"use","excerpt":"retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb7ca0be63af352d","tex":"D^{(m)}","aliases":"D^(m)","display":"D^{(m)}","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd12667298c0030a","tex":"D^{(m-1)}","aliases":"D^(m-1)","display":"D^{(m-1)}","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e8acadc439b10d5","tex":"D^*","aliases":"D^*","display":"D^*","kind":"use","excerpt":"A linear correction map D:U⟶ V places the obstruction to solving Du=-r in the canonical quotient coker D. This quotient has a universal property and behaves naturally under commuting squares. Mapping cones extend the same construction to cochain complexes. In a Hilbert space with closed range, the quotient is represented by the orthogonal projectio","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-227743dc6ee61b15","tex":"D^{-(r+s)}","aliases":"D^-(r+s)","display":"D^{-(r+s)}","kind":"use","excerpt":"2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers uses a common specified logarithm. A positive integer reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argumen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":55},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6311982fa9099aea","tex":"D^{-1/2}","aliases":"D^-1/2","display":"D^{-1/2}","kind":"use","excerpt":"vpage18Measure coupling after normalizationBLOCK JACOBI AS A GEOMETRIC CHANGEChapter VIII: Theorem 7.4, Corollary 7.6, Classical input 7.5. Let A=mat4 · 1 1 · 1, D=diag(4,1), B=D^-1/2=diag(1/2,1). Then BAB=I+E=mat1 · 1/2","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2340d1b58f61d6b3","tex":"D^{-n-1}","aliases":"D^-n-1","display":"D^{-n-1}","kind":"use","excerpt":"[Low energy expansion] Assume D is invertible and abs znormD^-1<1. Then Σ(z) =- ∑_n=0^∞ z^nBD^-n-1B^* with norm convergence. In particular, F_P(z) = z-A+BD^-1B^*+O(abs znorm B^2normD^-1^2).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e6103882f8afa39","tex":"D^{-r}","aliases":"D^-r","display":"D^{-r}","kind":"use","excerpt":"2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers uses a common specified logarithm. A positive integer reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argumen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61633e3eec80e17d","tex":"D^{-s}","aliases":"D^-s","display":"D^{-s}","kind":"use","excerpt":"2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers uses a common specified logarithm. A positive integer reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argumen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65577bc46f414a5f","tex":"\\mathcal D^2","aliases":"D^2 mathcal","display":"\\mathcal D^2","kind":"use","excerpt":"The variable B=2Z+1 has transform e^tfrac2te^2t-1=fractsinh t. Its odd moments vanish, and P=( B^2-1)/4=Z(Z+1) has the divided-power moments of Lemma~[source reference]. Introduce an independent copy D and Q=( D^2-1)/4. The exact factorization F_2(t)=fract^2cosh t-cos t =fracatsinh(at)fracbtsinh(bt) shows that U=a B+b D satisfies L(U^4n)=C_2(n). Di","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"t t=log2, they are Z=7/3 and p=(1,2,4)/7. vfigext_cumulants platecaptionThe score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores. headTwo derivatives have different statistical meanings The first derivative of log evidence is the current mean score, ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ac5422b66e1b0c1","tex":"D^4","aliases":"D^4","display":"D^4","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a41942edfed6dca3","tex":"d^4","aliases":"d^4","display":"d^4","kind":"use","excerpt":"[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8f3e6a620b326b0","tex":"D^8","aliases":"D^8","display":"D^8","kind":"use","excerpt":"3. The dyadic differential operator and reciprocity. The operator required here is B_4=fracD^8-D^42, D=fracddt. It is not the preceding p=2 specialization (D^4-D^2)/2. It is also not an ordinary derivation. The new algebra module proves the actual product formula, including the middle derivative sums whose coefficients are binom 8i/2 and binom 4i/2","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-535d0321b9f2a373","tex":"D^\\bullet","aliases":"D^bullet bullet","display":"D^\\bullet","kind":"use","excerpt":"Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17de839c28aeb1f7","tex":"D^{d-r}","aliases":"D^d-r","display":"D^{d-r}","kind":"use","excerpt":"[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58357a31d24cf18a","tex":"D^i","aliases":"D^i","display":"D^i","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87a4659e9ddda329","tex":"D^j","aliases":"D^j","display":"D^j","kind":"use","excerpt":"With A=diag(1,2), the vectors v and Av=(1,2)^T are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when A=I_2. Both changes enlarge the addressing power, but in different ways. headWhen invariance is only approximate If A=D","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f80fcde2a727a6d","tex":"D^k","aliases":"D^k","display":"D^k","kind":"use","excerpt":"Use the telescoping identity A^j-D^j = ∑_k=0^j-1 A^j-1-kKD^k. Each term has norm at most M^j-1normK. The vector D^jv lies in the direct sum of local Krylov spaces.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d80d76ad0ed63a4","tex":"d^k","aliases":"d^k","display":"d^k","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d209e4a42d264b0","tex":"d^{k+1}","aliases":"d^k+1","display":"d^{k+1}","kind":"use","excerpt":"[Equal tensor factors] For integers k,d≥0, a product of k square factors of order d, acting on a tensor with d^k entries, can be applied by k modewise contractions, reusing intermediate slices, with modeled cost 2k d^k+1 and factor storage kd^2.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c64f3c404f15cb37","tex":"d^{k+2}","aliases":"d^k+2","display":"d^{k+2}","kind":"use","excerpt":"Operation events. The event type for k+1 modes is a disjoint union of d copies of the trailing event type and the events for the final fiber contraction. Its cardinality obeys W_0=0, W_k+1=dW_k+2d^k+2. Lean proves the closed form W_k=2kd^k+1. The factor-address type has cardinality k d^2. These statements include k=0 and d=0. With no factors, the a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b43f40b2f922a574","tex":"d^{k-1}","aliases":"d^k-1","display":"d^{k-1}","kind":"use","excerpt":"For k=0 there are no contractions and both modeled work and factor storage are zero. For k≥1, one mode has d^k-1 fibers of length d. Multiplying each fiber by a d by d matrix costs 2d^2, so one mode costs 2d^k+1. Sum over the k modes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a46c6db9c9c47c40","tex":"D^{m+1}","aliases":"D^m+1","display":"D^{m+1}","kind":"use","excerpt":"· +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b655d23a365d7e4","tex":"d^N","aliases":"d^N","display":"d^N","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e04bf4d668de4635","tex":"D^{N-i}","aliases":"D^N-i","display":"D^{N-i}","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d379d56ac67e266","tex":"D^p","aliases":"D^p","display":"D^p","kind":"use","excerpt":"[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da3992d397405d31","tex":"D^{p^2}","aliases":"D^p^2","display":"D^{p^2}","kind":"use","excerpt":"Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d705f2d0b7a4b58f","tex":"D^q","aliases":"D^q","display":"D^q","kind":"use","excerpt":"matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers. headSigned delays need the same care For an original-sample delay d at modulus N, y_k[n]=f_(k-d)bmod N [n+lfloorfrack-dNrfloor]. This includes advances when d<0. The finite verification checks the formula on signed sample addresses. A bank with product ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3fb059a8ab4b3a2","tex":"D^\\times","aliases":"D^× times","display":"D^\\times","kind":"use","excerpt":"Flatness replaces the two-edge route in D^× by the direct route, which cancels its inverse. For H, flatness first collapses a→ b→ c to a→ c, then collapses a→ c→ a to the normalized transport a→ a.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e69131764514df66","tex":"D^\\times_{c,b,a}","aliases":"D^×_c,b,a times","display":"D^\\times_{c,b,a}","kind":"use","excerpt":"At a triple (a,b,c), the compiled cocycle predicate is Γ_c← bΓ_b← a =Γ_c← a. Flatness means this equality holds for every triple. The formal multiplicative relative defect and triangle holonomy are D^×_c,b,a = (Γ_c← bΓ_b← a) Γ_c← a^-1, H_c,b,a = Γ_a← c Γ_c← b Γ_b← a. The defect is based at the target c. The holonomy is based at a. statusleanthreeKo","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d88cb49c47029e8","tex":"D^\\alpha","aliases":"D^α alpha","display":"D^\\alpha","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd476ebfd1a76f31","tex":"D^\\beta","aliases":"D^β beta","display":"D^\\beta","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-082363402fb26c4b","tex":"\\mathfrak D^\\Gamma_{b,a}","aliases":"D^Γ_b,a mathfrak Gamma","display":"\\mathfrak D^\\Gamma_{b,a}","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96cd702de2890fbe","tex":"D^\\dagger","aliases":"D^† dagger","display":"D^\\dagger","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions ho","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70ffa9c8bbca2ebd","tex":"D_+","aliases":"D_+","display":"D_+","kind":"use","excerpt":"The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no bound","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff76dfe8b26b12f2","tex":"D_-","aliases":"D_-","display":"D_-","kind":"use","excerpt":"The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no bound","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-737f63af45789c25","tex":"d_{0,k}","aliases":"d_0,k","display":"d_{0,k}","kind":"use","excerpt":"(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a59dd4f5aee6253","tex":"D_1^{-1}","aliases":"D_1^-1","display":"D_1^{-1}","kind":"use","excerpt":"stant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial multiplier is integral. Because F_1=F_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","excerpt":": Section 2, Section 7, Section 11, Section 12. This guide follows a concrete analytic family through three descriptions: selected Taylor coefficients, a finite differential system, and the zeros of an entire function. The formulas are linked, but the evidence needed for each description is different. vfigthree_views headKeep three small models in ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ec3ef0b1d0ea9eb","tex":"d_{2,\\ell}","aliases":"d_2,ℓ ell","display":"d_{2,\\ell}","kind":"use","excerpt":"=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at address three is at least two. Its exact radius i","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe163f48e48b9c96","tex":"D_2^{-1}","aliases":"D_2^-1","display":"D_2^{-1}","kind":"use","excerpt":"vpage14Three similar reciprocals differNORMALIZED SECTIONSChapter IV: Definition 9.1, Theorem 9.2, Example 9.11. The signed denominator and the normalized unsigned sections produce different reciprocal families. In order two, D_2^-1=sec z, C_2,0=sech z, C_2,1=z/sinh z. Their first block-EGF coefficients make the difference visible. vfigdifferent_re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2c6012edbe5de2c","tex":"D_3","aliases":"D_3","display":"D_3","kind":"use","excerpt":"ordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Each initial jet (y(0),y'(0),y''(0)) therefore det","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f66bb3522fa9b25","tex":"d_3","aliases":"d_3","display":"d_3","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af497092b3be5660","tex":"D_3'","aliases":"D_3'","display":"D_3'","kind":"use","excerpt":"vpage19A root estimate becomes a certificateEXACT RATIONAL BRACKETSChapter IV: Theorem 11.7, Theorem 11.8, Corollary 11.9. The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90b32d1d1dff83bf","tex":"D_{3,0}","aliases":"D_3,0","display":"D_{3,0}","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e3dff1e5a9abb17","tex":"D_{3,1}","aliases":"D_3,1","display":"D_{3,1}","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ca422484b3e1eb4","tex":"D_{3,2}","aliases":"D_3,2","display":"D_{3,2}","kind":"use","excerpt":"vpage6Addition is quotient multiplicationFINITE ALGEBRAChapter IV: Proposition 5.5, Theorem 6.2. Work in C[ξ]/(ξ^3+1). Every element has coordinates in 1,ξ,ξ^2. The entire exponential has the finite-coordinate form e^zξ=D_3,0(z)+D_3,1(z)ξ+D_3,2(z)ξ^2. The multiplication table makes the carry sign unavoidable. vfigsigned_addition Multiply the coordi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3a4549b81c78db4","tex":"D_{3,j}","aliases":"D_3,j","display":"D_{3,j}","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e73f1dfc33a2fa5b","tex":"D_3^{-1}","aliases":"D_3^-1","display":"D_3^{-1}","kind":"use","excerpt":"e a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. Since F_3=F_2D_3^-1, C_3(n)-C_2(n)=∑_i≥1h_ibinom4n4iC_2(n-i). The scaled-binomial argument","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca8301964f9bd376","tex":"D_4","aliases":"D_4","display":"D_4","kind":"use","excerpt":"n entire function. The formulas are linked, but the evidence needed for each description is different. vfigthree_views headKeep three small models in view For N=2, the signed denominator is D_2(z)=cos z. Its reciprocal is the familiar secant series. For N=3, a third exponential perturbs the zeros of a dominant cosine. For N=4, an exact factorizatio","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef2886062e125405","tex":"D_8","aliases":"D_8","display":"D_8","kind":"use","excerpt":"ter pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63ed82a00b685c9b","tex":"D_A","aliases":"D_A","display":"D_A","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7903c81010eebdd6","tex":"D_a","aliases":"D_a","display":"D_a","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ec57d816513bae2","tex":"d_A","aliases":"d_A","display":"d_A","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0d18c8f77ae3cfc","tex":"D_a^*","aliases":"D_a^*","display":"D_a^*","kind":"use","excerpt":"The derivative product rule and uniqueness of derivatives give D_aP_b+P_aD_b=δ_abD_a. Differentiating completeness and adjoint invariance gives ∑_aD_a=0, D_a^*=D_a. These are proved derivative consequences, not additional axioms supplied in place of differentiation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee5bb7d36741efec","tex":"D_B","aliases":"D_B","display":"D_B","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two ","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4256ad8575b1b705","tex":"D_b","aliases":"D_b","display":"D_b","kind":"use","excerpt":"Define C=∑_aD_aP_a(t). Then C=frac12∑_acommD_aP_a(t), C^*=-C, and, for every b, commCP_b(t)=D_b.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f62021dadd3fc99","tex":"D_B^*","aliases":"D_B^*","display":"D_B^*","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60f01eac58353ae3","tex":"D_B^T","aliases":"D_B^T","display":"D_B^T","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee9c69896c4d1a6d","tex":"D_c","aliases":"D_c","display":"D_c","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b858f3e8665ea65","tex":"D_e","aliases":"D_e","display":"D_e","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54ff0eb6a1af0d6f","tex":"D_f","aliases":"D_f","display":"D_f","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d2e949d109d889d","tex":"d_{i-1}","aliases":"d_i-1","display":"d_{i-1}","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-125484ee05f69b84","tex":"D_i^{\\oplus m_i}","aliases":"D_i^⊕ m_i oplus","display":"D_i^{\\oplus m_i}","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j, U_j(n+T)-U_j(n) · ≠0, · v_2(U_j(n+T)-U_j(n)) · =v_2(T)+d_j.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b0858de229bbaac5","tex":"d_{j,m}","aliases":"d_j,m","display":"d_{j,m}","kind":"use","excerpt":"For j=0,1, the term m=ℓ in [source reference] has exact valuation ℓ, since s(ℓ,ℓ)=1 and the Newton valuation is exact. Every later term has valuation at least ℓ+1. Thus v_2(d_j,ℓ)=ℓ, and the coefficient radius is exactly two. At a center x_0∈ Z_2, the coefficient of y^ℓ in the translated series is ∑_m≥ℓd_j,mbinom mℓ x_0^m-ℓ. Its initial term again ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f4f45a608f77322","tex":"d_{j,\\ell}","aliases":"d_j,ℓ ell","display":"d_{j,\\ell}","kind":"use","excerpt":"[Taylor coefficient bounds and radii] For j=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at addres","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0b05c681f47bcdf3","tex":"D_{j-1}","aliases":"D_j-1","display":"D_{j-1}","kind":"use","excerpt":"Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e09c2eb604e5806c","tex":"D_j^\\bullet","aliases":"D_j^bullet bullet","display":"D_j^\\bullet","kind":"use","excerpt":"Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a50d188eafd60524","tex":"d_{ja}","aliases":"d_ja","display":"d_{ja}","kind":"use","excerpt":"ry vector orthogonal to ψ, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4d72ff963f20855","tex":"d_{jb}","aliases":"d_jb","display":"d_{jb}","kind":"use","excerpt":"he left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recove","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a572acccc9b7cf05","tex":"D_k","aliases":"D_k","display":"D_k","kind":"use","excerpt":"me f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Define the adjacent contrast D_k=y_k+1-y","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5b9b4eb292be27d","tex":"d_{L,a}","aliases":"d_L,a","display":"d_{L,a}","kind":"use","excerpt":"[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a90b7893f66544d","tex":"d_{L_N}","aliases":"d_L_N","display":"d_{L_N}","kind":"use","excerpt":"[Rigorous intermediate dimension window] For every ε>0, there is a diffusion time t_ε>0 and an integer N_ε such that absd_L_N(t_ε)-d<ε for every N≥ N_ε.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71efea616c614868","tex":"D_m","aliases":"D_m","display":"D_m","kind":"use","excerpt":"Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3fb8837a44922746","tex":"D_N","aliases":"D_N","display":"D_N","kind":"use","excerpt":"D_N(z) · D_N,0(z), the denominator whose zeros control the signed reciprocal family.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b70f6bea52051d2","tex":"\\mathbf D_N","aliases":"D_N mathbf","display":"\\mathbf D_N","kind":"use","excerpt":"[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-382f08d5d528afb0","tex":"d_N","aliases":"d_N","display":"d_N","kind":"use","excerpt":"vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSChapter VII: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-251354dc81bb5f51","tex":"D_N'","aliases":"D_N'","display":"D_N'","kind":"use","excerpt":"T_N(z) · -D_N'(z)/D_N(z), a function, not the coefficient T_N(n).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce674635b7c486c1","tex":"D_{N,0}","aliases":"D_N,0","display":"D_{N,0}","kind":"use","excerpt":"D_N(z) · D_N,0(z), the denominator whose zeros control the signed reciprocal family.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ee31d563f299611","tex":"D_{N,j}","aliases":"D_N,j","display":"D_{N,j}","kind":"use","excerpt":"D_N,j(z) · The same series with (-1)^n. One full derivative cycle contributes a minus sign.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4efbc9ac8d9e0f0d","tex":"D_N^{-(r+s)}","aliases":"D_N^-(r+s)","display":"D_N^{-(r+s)}","kind":"use","excerpt":"rem 7.4, Theorem 7.6. One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagonal in the Newton determinant is essential. vfignewton_addition headThe additive parameter law has block weights Using the same l","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a10db54813efe79a","tex":"d_n^{-1}","aliases":"d_n^-1","display":"d_n^{-1}","kind":"use","excerpt":"First construct the integral solution. For a ring R, a triangular row system T, right-hand side b, and proved unit pivots d_n=T_nn, define s_n=d_n^-1(b_n-∑_q<nT_nqs_q). The new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36864a1aaaea8a13","tex":"D_N^{-r}","aliases":"D_N^-r","display":"D_N^{-r}","kind":"use","excerpt":"P_N,n(r) · Block-EGF coefficients of D_N^-r. They are polynomials in r.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58b404e71b6ab9fb","tex":"D_N^{-s}","aliases":"D_N^-s","display":"D_N^{-s}","kind":"use","excerpt":"rem 7.4, Theorem 7.6. One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagonal in the Newton determinant is essential. vfignewton_addition headThe additive parameter law has block weights Using the same l","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19c16859d92fc1e1","tex":"D_P","aliases":"D_P","display":"D_P","kind":"use","excerpt":"Apply polynomial induction using constants, addition, and multiplication by a variable. Constants have zero remainder, and remainders add. For the multiplication step, suppose the assertion is known for P and write D_P(U)=∑_j(∂_jP)(a)U_j. Multiplication of its expansion by a_i+cU_i gives constant term P(a)a_i, linear term c(a_iD_P(U)+P(a)U_i), and ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0efdddd015383ecb","tex":"d_p","aliases":"d_p","display":"d_p","kind":"use","excerpt":"[Likelihood perturbation] Let λ,μ≥0 have positive evidences under a fixed prior p. Set d_p(λ,μ) = ∑_iabsλ_i-μ_ip_i. Then norm B_λ(p)- B_μ(p)_1 ≤ frac2d_p(λ,μ) max{Z_λ(p),Z_μ(p)}.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4435e486a9912a3e","tex":"d_{pq}","aliases":"d_pq","display":"d_{pq}","kind":"use","excerpt":"[Entries of the fundamental matrix] For 0≤ p,q<N, let d_pq be the member of {0,…,N-1} congruent to p-q modulo N. Put ε_pq = 1, · p≥ q,","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e5463cd5ce49eb7","tex":"d_q","aliases":"d_q","display":"d_q","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-237fc9015f157bd3","tex":"D_r","aliases":"D_r","display":"D_r","kind":"use","excerpt":"unchanged. Predicting afterward gives (3/4,1/4) instead. The order of operations expresses which time the observation concerns. vfigfilter_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"use","excerpt":"[Exact classical tail valuation] For every r≥1 and N≥ r, v_2( T_r-1(N)) =4r-1+v_2binom Nr+μ_r+v_2G_r(N-r). There are unique factors G_r(X)=W_r(X)U_r(X), where W_r∈ Z_2[X] is monic of degree d_r=max J_r and U_r∈ Z_2⟨ X⟩^× satisfies U_r≡1pmod2. Consequently v_2G_r(x)=v_2W_r(x) for every x∈ Z_2, and deg W_r≤lfloorfracs_2(r)-13rfloor. Equation~[source ","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b67f0c58ecb3837","tex":"d_{r-1}","aliases":"d_r-1","display":"d_{r-1}","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f57f52c8b5bbfd0b","tex":"D_{\\rm computational}","aliases":"D_rm computational rm","display":"D_{\\rm computational}","kind":"use","excerpt":"g Every effect in the computational basis E_a=ket abra a has the same expectation on both matrices, namely 1/2 at 00 and 11 and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture. headA measurement that resolves the difference Use the valid two-outcome measurement with","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dae7685d7af3e13a","tex":"d_{\\rm torus}","aliases":"d_rm torus rm","display":"d_{\\rm torus}","kind":"use","excerpt":"[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de7bc3aee118c968","tex":"d_s","aliases":"d_s","display":"d_s","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain m","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad145996f65200fc","tex":"d_{s,2}","aliases":"d_s,2","display":"d_{s,2}","kind":"use","excerpt":"For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b22090db332ee7ef","tex":"d_{s,k}","aliases":"d_s,k","display":"d_{s,k}","kind":"use","excerpt":"[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36bf332a8be83b7c","tex":"d_s^{(1)}","aliases":"d_s^(1)","display":"d_s^{(1)}","kind":"use","excerpt":"[Spectral dimension] The median intermediate plateaus were d_s^(1)=1.00062, d_s^(2)=2.03781. These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc06aed33500255b","tex":"d_s^{(2)}","aliases":"d_s^(2)","display":"d_s^{(2)}","kind":"use","excerpt":"[Spectral dimension] The median intermediate plateaus were d_s^(1)=1.00062, d_s^(2)=2.03781. These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee1684d3a8f6d798","tex":"D_u","aliases":"D_u","display":"D_u","kind":"use","excerpt":"[Nonlinear Schur derivative] At (u_0,v_0), DR = D_uF_P - D_vF_P A^-1D_uF_Q.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4decc32ad75f2c6a","tex":"D_V","aliases":"D_V","display":"D_V","kind":"use","excerpt":"Let K be column stochastic. Let V be a real potential on the states and let D_V=diag(V). For functions written as columns, the backward Markov operator is K^T.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3b6e2eaef5fb02a","tex":"d_v","aliases":"d_v","display":"d_v","kind":"use","excerpt":"[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b008a80da8b3621","tex":"D_v^{-1}","aliases":"D_v^-1","display":"D_v^{-1}","kind":"use","excerpt":"Constructed solution. Well-founded recursion defines x_v=D_v^-1 (b_v-∑_u→ vT_vux_u). Applying D_v proves the original equation at v. If another vector satisfies all equations, induction over predecessors makes each previously used coordinate identical, and invertibility of D_v forces agreement at v. Thus the construction gives existence and uniquen","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-960a9e4480b08df9","tex":"D_w","aliases":"D_w","display":"D_w","kind":"use","excerpt":"Here D_rK means an observation after prediction. In the Feynman--Kac example on pagerefV20, departure-state weighting has forward operator K D_w. Both conventions are legitimate once their meanings are fixed, but they cannot be substituted for one another. takeawayWrite the chronological sequence before multiplying matrices. A product can be normal","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-baa2c2411282fdc8","tex":"D_z","aliases":"D_z","display":"D_z","kind":"use","excerpt":"-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20011d3f9eb01a6a","tex":"D_z^{-1}","aliases":"D_z^-1","display":"D_z^{-1}","kind":"use","excerpt":"-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3394af1c25cf897","tex":"d_\\alpha","aliases":"d_α alpha","display":"d_\\alpha","kind":"use","excerpt":"[Dimension count] If X is finite dimensional and d_α=dim X_α, then dimEnd_𝕜(X) = ∑_α,βd_α d_β = (∑_α d_α)^2. [source reference]","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ad7b1c62e650428","tex":"D_\\alpha^{-1}","aliases":"D_α^-1 alpha","display":"D_\\alpha^{-1}","kind":"use","excerpt":"[Block path formula] Let P=(P_α)_α∈ I be a finite packet system in a complete filtered algebra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be fini","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de786ac59a3f75f3","tex":"D_\\beta","aliases":"D_β beta","display":"D_\\beta","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f48cea2c4057357","tex":"d_\\beta","aliases":"d_β beta","display":"d_\\beta","kind":"use","excerpt":"[Dimension count] If X is finite dimensional and d_α=dim X_α, then dimEnd_𝕜(X) = ∑_α,βd_α d_β = (∑_α d_α)^2. [source reference]","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-503a05ceb4c00d45","tex":"d_\\eta","aliases":"d_η eta","display":"d_\\eta","kind":"use","excerpt":"Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one dep","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-008904a83528d7f3","tex":"d_\\theta","aliases":"d_θ theta","display":"d_\\theta","kind":"use","excerpt":"[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed6ff526c435c5c5","tex":"d_\\rho","aliases":"d_ρ rho","display":"d_\\rho","kind":"use","excerpt":"Fix this chapter's reciprocity, additive-character, local-factor, and phase normalizations. Let A be the declared class of irreducible wild Weil atoms under consideration. For ρ∈ A, the supplied assembled finite datum d_ρ consists of the following inputs, historically denoted H1--H4: item an intrinsic finite break decomposition on the Galois side; ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e19eb6c907a8c6c","tex":"D_\\ell","aliases":"D_ℓ ell","display":"D_\\ell","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b22521a2f79f4516","tex":"D_{\\ell+1}","aliases":"D_ℓ+1 ell","display":"D_{\\ell+1}","kind":"use","excerpt":"A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-335e9e40ffcb4439","tex":"\\ddot x","aliases":"ddot x ddot","display":"\\ddot x","kind":"use","excerpt":"returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict x(t)=1. vfigmemory headEliminate exactly, not approximately Integrating the second equation with y(0)=0 gives y(t)=- i∫_0^tx(s) ds. Substitution produces dot x(t)=-∫_0^tx(s) ds. The memory kernel is the constant function one. Differentiatin","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0d09000efb054df8","tex":"\\ddots","aliases":"ddots","display":"\\ddots","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74f94422b9cebc9d","tex":"\\operatorname{dec}","aliases":"dec operatorname","display":"\\operatorname{dec}","kind":"use","excerpt":"Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5ae8519ff6d0dff","tex":"\\mathrm{decode}","aliases":"decode mathrm","display":"\\mathrm{decode}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e31f1e54430399f","tex":"\\operatorname{decode}","aliases":"decode operatorname","display":"\\operatorname{decode}","kind":"use","excerpt":"[Contract conversion] Suppose one algorithm returns a compressed representation R and another returns an explicit object Y. A direct timing comparison is valid only if decode(R)=Y and the time required by decode is included, or if the downstream task consumes R directly.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a17365eab67e442","tex":"\\deph","aliases":"deph D_P mathcal","display":"{\\mathcal D_P}","kind":"use","excerpt":"Measurements: source Theorem 7.1. For finite complex matrices, define packet dephasing by D_P(X)=∑_aP_aXP_a. The code proves idempotence and trace duality: D_P( D_P(X))= D_P(X), tr( D_P(E)ρ)=tr(E D_P(ρ)). Trace duality follows term by term from cyclicity: tr(P_aEP_aρ)=tr(EP_aρ P_a). Therefore an effect that is diagonal in the packets, D_P(E)=E, sat","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0658d980d8d1bc6a","tex":"\\operatorname{DFT}","aliases":"DFT operatorname","display":"\\operatorname{DFT}","kind":"use","excerpt":"Normalization and inversion. The proof identifies matrix application with the ordinary counting-measure discrete Fourier transform: Fx=n^-1/2DFT(x). It separately identifies F^*x with the appropriately scaled inverse transform. Mathlib's checked Fourier inversion theorem and the proved scalar normalization yield both identities F^*F=I, FF^*=I. No u","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6c5d416bb3dfbb4","tex":"\\Dfun_1","aliases":"Dfun_1 D_1 Dfun mathsf","display":"{\\mathsf D}_1","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fbc53ff4b3c87616","tex":"\\Dfun_2","aliases":"Dfun_2 D_2 Dfun mathsf","display":"{\\mathsf D}_2","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f4377cbf9ad281e","tex":"\\Dfun_3","aliases":"Dfun_3 D_3 Dfun mathsf","display":"{\\mathsf D}_3","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4864c3603c017539","tex":"\\Dfun_4","aliases":"Dfun_4 D_4 Dfun mathsf","display":"{\\mathsf D}_4","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b764ca7a3b447499","tex":"\\Dfun_{N,0}'","aliases":"Dfun_N,0' D_N,0' Dfun mathsf","display":"{\\mathsf D}_{N,0}'","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e930dfa4cab0538f","tex":"\\Dfun_{N,1}","aliases":"Dfun_N,1 D_N,1 Dfun mathsf","display":"{\\mathsf D}_{N,1}","kind":"use","excerpt":"[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-850ed82f3d2578f6","tex":"\\Dfun_{N,d_{pq}}","aliases":"Dfun_N,d_pq D_N,d_pq Dfun mathsf","display":"{\\mathsf D}_{N,d_{pq}}","kind":"use","excerpt":"-1, · p<q. Then (e^zJ_N)_pq = ε_pq D_N,d_pq(z). Thus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from q to p crosses the signed edge.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b1aed9780b099dc","tex":"\\Dfun_{N,i}","aliases":"Dfun_N,i D_N,i Dfun mathsf","display":"{\\mathsf D}_{N,i}","kind":"use","excerpt":"[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c545097e1ea4183a","tex":"\\Dfun_{N,j}'","aliases":"Dfun_N,j' D_N,j' Dfun mathsf","display":"{\\mathsf D}_{N,j}'","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15b31c89bfca987f","tex":"\\Dfun_{N,j-1}","aliases":"Dfun_N,j-1 D_N,j-1 Dfun mathsf","display":"{\\mathsf D}_{N,j-1}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-033ca7b2ad984fff","tex":"\\Dfun_{N,j}^{(k)}","aliases":"Dfun_N,j^(k) D_N,j^(k) Dfun mathsf","display":"{\\mathsf D}_{N,j}^{(k)}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89fc0d6e81a8eaa5","tex":"\\Dfun_{N,j}^{(N)}","aliases":"Dfun_N,j^(N) D_N,j^(N) Dfun mathsf","display":"{\\mathsf D}_{N,j}^{(N)}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2960208ad6b384e5","tex":"\\Dfun_{N,k}","aliases":"Dfun_N,k D_N,k Dfun mathsf","display":"{\\mathsf D}_{N,k}","kind":"use","excerpt":"[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af02055946b191fd","tex":"\\Dfun_{N,N-1}","aliases":"Dfun_N,N-1 D_N,N-1 Dfun mathsf","display":"{\\mathsf D}_{N,N-1}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dee7549cf31fd663","tex":"\\Dfun_N^{-q}","aliases":"Dfun_N^-q D_N^-q Dfun mathsf","display":"{\\mathsf D}_N^{-q}","kind":"use","excerpt":"The derivative weighted sum omits the finitely many zeros that might be multiple. A treatment of those zeros requires the complete principal parts of D_N^-q rather than the single derivative weight.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-782506289eda6b80","tex":"\\Dfun_N^{\\prime}","aliases":"Dfun_N^prime D_N^prime Dfun prime mathsf prime","display":"{\\mathsf D}_N^{\\prime}","kind":"use","excerpt":"ic objects are introduced in the following order. First, E_N,j(z) = ∑_n≥0fracz^Nn+j(Nn+j)! is the residue j part of the exponential. Second, D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)! is the signed basis for y^(N)=-y. The section D_N= D_N,0 is the denominator used in the reciprocal family S_N^(r)= D_N^-r. Set T_N=- D_N^prime/ D_N. Equivalently, T_N is","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"vpage14Gauge changes the picture, not flatnessMODEL E / CONJUGATION AND ITS LIMITSChapter III, mathematical text: Theorem 6.2 At the base chart, change coordinates by D=diag(2,1). Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving H^D=DHD^-1=mat1 · 2","count":34},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33bb1588c90c3cf1","tex":"\\dim_{\\mathbb F_2}","aliases":"dim_ F_2 dim mathbb","display":"\\dim_{\\mathbb F_2}","kind":"use","excerpt":"For this same constructed candidate, t(π) · =1, · Sw(π) · =3 2^d-1h_d, dim_ F_2W · =N-2, · dimη · =2^N/2-1, W=J^1/H^1, a(π^vee×π) · =N^2-1+∑_i=1^d3 4^i-1h_i. Matching these profiles does not identify the source, its endo-class, or the pair factors in original H5.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-337441eb94c64c38","tex":"\\dim_{\\mathbb F_p}","aliases":"dim_ F_p dim mathbb","display":"\\dim_{\\mathbb F_p}","kind":"use","excerpt":"The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c75b893cf90454b2","tex":"\\dim_{\\bbk}","aliases":"dim_bbk dim_𝕜 dim bbk dim Bbbk","display":"\\dim_{{\\Bbbk}}","kind":"use","excerpt":"[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42bb68489be0ad01","tex":"\\dim_{\\bbk[t]/(p)}","aliases":"dim_bbk[t]/(p) dim_𝕜[t]/(p) dim bbk dim Bbbk","display":"\\dim_{{\\Bbbk}[t]/(p)}","kind":"use","excerpt":"[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95483dc6bdcb21fe","tex":"\\dim_\\C","aliases":"dim_C dim_ C dim C dim mathbb","display":"\\dim_{\\mathbb C}","kind":"use","excerpt":"A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ca2307ebd3bffbf","tex":"\\dim_D","aliases":"dim_D dim","display":"\\dim_D","kind":"use","excerpt":"[Local algebra formula] Let A be a local ring with Jacobson radical J and residue division ring D=A/J. For every finitely generated left module M over A, srank_A(M)=dim_D(M/JM), where the dimension is that of a left vector space over D. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9943f2d3b5f6322","tex":"\\dim_{E'}","aliases":"dim_E' dim","display":"\\dim_{E'}","kind":"use","excerpt":"[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98d940b7f3ffe088","tex":"\\dim_K","aliases":"dim_K dim","display":"\\dim_K","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12cef7357a97a6d2","tex":"\\dim_{K_p}","aliases":"dim_K_p dim","display":"\\dim_{K_p}","kind":"use","excerpt":"(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], sra","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34038c6ab85627dc","tex":"\\dim_{\\kappa(\\frakp)}","aliases":"dim_κ(frakp) dim_κ( p) dim kappa frakp dim kappa mathfrak","display":"\\dim_{\\kappa({\\mathfrak p})}","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-346650cbdd3f09a1","tex":"\\operatorname{disc}","aliases":"disc operatorname","display":"\\operatorname{disc}","kind":"use","excerpt":"The exceptional primes are controlled by the actual field discriminant: S_L={p rational prime:pmiddisc(L)}. This set is finite. For p∉ S_L, every prime of O_L above (p) is unramified. Lean proves that the actual inertia cardinality is the ramification index. It follows that inertia is trivial and the arithmetic Frobenius at a fixed finite prime is ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-558c1dc971656de6","tex":"\\dist","aliases":"dist operatorname","display":"{\\operatorname{dist}}","kind":"use","excerpt":"gger be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","excerpt":"Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fed34424e9d0e3f5","tex":"\\mathrm{div}","aliases":"div mathrm","display":"\\mathrm{div}","kind":"use","excerpt":"one also has A_N S_N=I. Equip this finite product with the Hilbert norm (x_0,…,x_N-1) _ch^2 =∑_r=0^N-1 x_r _2^2. Then A_N is a linear isometric equivalence and x _2^2=∑_r=0^N-1 x_r _2^2. Euclidean division gives a unique pair (r,q) with 0≤ r<N and n=Nq+r, including negative n. The address maps n↦(nbmod N,nmathbindivN) and (r,q)↦ Nq+r are inverse. A","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1231c9b0bf844079","tex":"\\operatorname{Dom}","aliases":"Dom operatorname","display":"\\operatorname{Dom}","kind":"use","excerpt":"[Coordinate generators] Assume the setting of [source reference]. If x=(x_j) satisfies x_j∈Dom(A_j) and the sequence (A_jx_j) belongs to X_∞, then x lies in the generator domain of T_∞ and A_∞ x=(A_jx_j)_j≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35679b068dabbae2","tex":"\\dot P","aliases":"dot P dot","display":"\\dot P","kind":"use","excerpt":"Zero projections are permitted in this section and contribute zero to all generator sums. When T=0, every partition product is the empty product I. Completeness and orthogonality give P_α(t)P_β(t)=0 for α≠β, ∑_α P_α(t)=I. Define the canonical generator K(t)=∑_α dot P_α(t)P_α(t).","count":36},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2dfbbad49009071f","tex":"\\dot p","aliases":"dot p dot","display":"\\dot p","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20f048d571067858","tex":"\\dot p_0","aliases":"dot p_0 dot","display":"\\dot p_0","kind":"use","excerpt":"Differentiation of the normalized flow yields the centered score equation dot p_i=(g_i-⟨ g,p⟩)p_i. For the displayed model, dot p_0=p_0(1-p_0). The negative average-score term is what keeps the total derivative zero. Omitting the derivative of the normalizer would not preserve mass.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4b951c58979abc7","tex":"\\dot P_a","aliases":"dot P_a dot","display":"\\dot P_a","kind":"use","excerpt":"n For this complete pair of packets, define C=dot P P+dot Q Q. Exact calculation yields the constant matrix C=J, with C^*=-C and [C,P]=dot P. At t=0, the displayed small matrices allow these identities to be checked by direct multiplication. headFrom two packets to a complete family For differentiable orthogonal projectors P_a summing to the identi","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0be1eb2be8fecda","tex":"\\dot P_b","aliases":"dot P_b dot","display":"\\dot P_b","kind":"use","excerpt":"[Canonical packet connection] Define C(t) = frac12∑_a[dot P_a(t),P_a(t)] = ∑_adot P_a(t)P_a(t). Then C(t)^*=-C(t) and [C(t),P_b(t)]=dot P_b(t) for every packet b.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf0d4e773e1ba217","tex":"\\dot p_i","aliases":"dot p_i dot","display":"\\dot p_i","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12567f7ebf7d4fca","tex":"\\dot P_\\alpha","aliases":"dot P_α dot alpha","display":"\\dot P_\\alpha","kind":"use","excerpt":"Zero projections are permitted in this section and contribute zero to all generator sums. When T=0, every partition product is the empty product I. Completeness and orthogonality give P_α(t)P_β(t)=0 for α≠β, ∑_α P_α(t)=I. Define the canonical generator K(t)=∑_α dot P_α(t)P_α(t).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9645cfd170ae8601","tex":"\\dot P_\\beta","aliases":"dot P_β dot beta","display":"\\dot P_\\beta","kind":"use","excerpt":"begingroup setlengthabovedisplayskip6pt plus 2pt minus 2pt setlengthbelowdisplayskip6pt plus 2pt minus 2pt [Complete packet commutator identity] For every packet label β, dot P_β(t)=[K(t),P_β(t)]. The generator is also skew-adjoint: K(t)^*=-K(t).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db307e9343ea80e8","tex":"\\dot Q","aliases":"dot Q dot","display":"\\dot Q","kind":"use","excerpt":"Chapter VII: Theorem II.6.1, Theorem II.6.2. Continue with P(t)=U(t)diag(1,0)U(t)^* and Q(t)=I-P(t). Differentiating P^2=P gives dot P P+Pdot P=dot P, Pdot P P=0. The second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement. vfigconnection For this complete pair o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51f3bbd1503295c5","tex":"\\dot u","aliases":"dot u dot","display":"\\dot u","kind":"use","excerpt":"Conditioning over the first small time interval gives the backward equation dot u=(Q^T-D_V)u, u(0)=f. The finite system has a unique solution, namely the displayed matrix exponential. The expectation bound follows from exp(-∫_0^tV(X_s) d s) ≤ e^-v_0t. This finite argument is the matrix form of the Feynman--Kac principle originating in [source refer","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bddb44ee5b7358ac","tex":"\\dot U^*","aliases":"dot U^* dot","display":"\\dot U^*","kind":"use","excerpt":"The explicitly defined coordinate curve y(s)=U(s)^*ψ(s) is differentiable at t, with dot y=dot U^*ψ(t)+U(t)^*dotψ. It satisfies idot y=H_effy(t), H_eff=U(t)^*HU(t)-iU(t)^*dot U. The connection term -iU(t)^*dot U is Hermitian. If H^*=H, then H_eff^*=H_eff.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae6eb7cbea0e8357","tex":"\\dot V","aliases":"dot V dot","display":"\\dot V","kind":"use","excerpt":"The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I. Uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f84526501d95b0bb","tex":"\\dot W","aliases":"dot W dot","display":"\\dot W","kind":"use","excerpt":"The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I. Uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a89ea0f13cf380df","tex":"\\dot W^*","aliases":"dot W^* dot","display":"\\dot W^*","kind":"use","excerpt":"Differentiate V=W^*U. Since dot W=KW and K^*=-K, dot W^*=-W^*K. Substitution gives the equation. The first term in H_rm mov is self adjoint. The operator - i W^*dot W is self adjoint because W^*dot W is skew adjoint.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f854ffb0431207da","tex":"\\dot x","aliases":"dot x dot","display":"\\dot x","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd48b253a62d377b","tex":"\\dot x_P","aliases":"dot x_P dot","display":"\\dot x_P","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-389bd177d57e5191","tex":"\\dot y","aliases":"dot y dot","display":"\\dot y","kind":"use","excerpt":"[Finite auxiliary state realization] Every kernel of the form K(t)=B e^tDC can be represented exactly by adjoining the finite auxiliary state dot y=Dy+Cx_P, dot x_P=A_PPx_P+By, y(0)=0. A nonzero auxiliary initial state adds the explicit forcing B e^tDy(0). The self-energy of the augmented system is B(zI-D)^-1C.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ec56f24a343f123","tex":"\\dot y_1","aliases":"dot y_1 dot","display":"\\dot y_1","kind":"use","excerpt":"vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELChapter VI, Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from Plate VI.23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76150d7fc49f4b0f","tex":"\\dot y_2","aliases":"dot y_2 dot","display":"\\dot y_2","kind":"use","excerpt":"vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELChapter VI, Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from Plate VI.23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e13cdcf09a963afd","tex":"\\dot{\\widetilde x}","aliases":"dotwidetilde x dot widetilde","display":"\\dot{\\widetilde x}","kind":"use","excerpt":"entiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II. [Stability of memory compression] Let x and widetilde x solve dot x(t) =Ax(t)+∫_0^t K(t-s)x(s) d s+f(t) and dotwidetilde x(t) =Awidetilde x(t) +∫_0^twidetilde K(t-s)widetil","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0645873febb68b5c","tex":"\\Downarrow","aliases":"Downarrow","display":null,"kind":"use","excerpt":"Theorem spine at a glance. ConceptChain c free seed presentation [0.25em] Downarrow [-0.1em] minimal generator number [0.25em] Downarrow [-0.1em] reduction modulo the Jacobson radical [0.25em] Downarrow [-0.1em] semisimple or primary computation c packet idempotents [0.25em] Downarrow [-0.1em] exact analysis/synthesis [0.25em] Downarrow [-0.1em] tr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-64fbb1a3212deefd","tex":"e'","aliases":"e'","display":"e'","kind":"use","excerpt":"The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75897025474d86a7","tex":"E'^r","aliases":"E'^r","display":"E'^r","kind":"use","excerpt":"Let W = span { (t_1(ω),…,t_r(ω)):ω∈Ω } ⊆ E'^r. If W≠ E'^r, a nonzero linear functional on E'^r would vanish on every evaluation vector, giving a nontrivial linear relation among the functions. Thus W=E'^r. Choose r evaluation vectors forming a basis.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd88c369e4cd6024","tex":"E^{\\mathsf T}","aliases":"E^ T mathsf","display":"E^{\\mathsf T}","kind":"use","excerpt":")^ T becomes Ef=(2,2,26/7,26/7)^ T. vfigweighted_projection platecaptionThe bars are function values, not state probabilities. Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9e2f2fe584334e62","tex":"\\e^{(2a+1)t}","aliases":"e^(2a+1)t e mathrm","display":"{\\mathrm e}^{(2a+1)t}","kind":"use","excerpt":"The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83389e2c1dbd901f","tex":"e^{(2a+1)t}","aliases":"e^(2a+1)t","display":"e^{(2a+1)t}","kind":"use","excerpt":"The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconst","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-01c396ae88f551ee","tex":"e^{(2k+1)\\pi i/N}","aliases":"e^(2k+1)π i/N pi","display":"e^{(2k+1)\\pi i/N}","kind":"use","excerpt":"We now assume N≥2. Write η_k=e^(2k+1)π i/N, 0≤ k<N. These are the vertices of a regular polygon and D_N(z)=frac1N∑_k=0^N-1e^η_kz.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b000542aa28a8cd","tex":"\\e^{(a+k)T}","aliases":"e^(a+k)T e mathrm","display":"{\\mathrm e}^{(a+k)T}","kind":"use","excerpt":"e x solve dot x(t) =Ax(t)+∫_0^t K(t-s)x(s) d s+f(t) and dotwidetilde x(t) =Awidetilde x(t) +∫_0^twidetilde K(t-s)widetilde x(s) d s +f(t) with the same initial value. Fix T>0, and assume sup_0≤ t≤ Tnormwidetilde x(t)≤ M. Set a=norm A, k=norm K_L^1(0,T), δ=norm K-widetilde K_L^1(0,T). Then sup_0≤ t≤ T normx(t)-widetilde x(t) ≤ MTδ e^(a+k)T.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1c47e908e7b023c","tex":"\\e^{(a+k)t}","aliases":"e^(a+k)t e mathrm","display":"{\\mathrm e}^{(a+k)t}","kind":"use","excerpt":"Let e=x-widetilde x. Integrating the difference equation and changing the order of integration gives norme(t) ≤ (a+k)∫_0^tnorme(r) d r +Mtδ. Since t≤ T, Gronwall's inequality gives norme(t) ≤ MTδ e^(a+k)t. Take the supremum over [0,T].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31d6c2ddbbf303ac","tex":"e^{(m,p)}","aliases":"e^(m,p)","display":"e^{(m,p)}","kind":"use","excerpt":"[A second proof of the standard window period] The reduced inverse e^(m,p) and every standard address satisfy the period in [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-489e8c151737a2f9","tex":"e^{(m,p)}_n","aliases":"e^(m,p)_n","display":"e^{(m,p)}_n","kind":"use","excerpt":"Then apply the residue map. Independently define e^(m,p) over F_p by the unit-diagonal system ∑_q=0^nbinommnmqe^(m,p)_q=δ_n0. Its recurrence and uniqueness are checked. Applying the actual homomorphism π: Z_p→ F_p to the integral row, and then Lucas reduction, proves π(c^(p)_m,j(n))=e^(m,p)_n. For j,k<p, a separate ideal statement gives pmid c^(p)_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-011d330493ff6133","tex":"e^{(m,p)}_q","aliases":"e^(m,p)_q","display":"e^{(m,p)}_q","kind":"use","excerpt":"Then apply the residue map. Independently define e^(m,p) over F_p by the unit-diagonal system ∑_q=0^nbinommnmqe^(m,p)_q=δ_n0. Its recurrence and uniqueness are checked. Applying the actual homomorphism π: Z_p→ F_p to the integral row, and then Lucas reduction, proves π(c^(p)_m,j(n))=e^(m,p)_n. For j,k<p, a separate ideal statement gives pmid c^(p)_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c092a24aa6947f31","tex":"e^{(s+t)A}","aliases":"e^(s+t)A","display":"e^{(s+t)A}","kind":"use","excerpt":"[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-560d405cf65c60d1","tex":"e^{(s+t)g}","aliases":"e^(s+t)g","display":"e^{(s+t)g}","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c3c7988f438d287","tex":"e^{(s+t)Q}","aliases":"e^(s+t)Q","display":"e^{(s+t)Q}","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8dc508a5768c192","tex":"\\e^{(s_m-s_{m-1})D}","aliases":"e^(s_m-s_m-1)D e mathrm","display":"{\\mathrm e}^{(s_m-s_{m-1})D}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42f3c03924e2cad8","tex":"e^{(t-r)A_c}","aliases":"e^(t-r)A_c","display":"e^{(t-r)A_c}","kind":"use","excerpt":"[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ecf217baed7a59a","tex":"e^{(t-r)A_{QQ}}","aliases":"e^(t-r)A_QQ","display":"e^{(t-r)A_{QQ}}","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2230e91fe505a10","tex":"e^{(t-r)D}","aliases":"e^(t-r)D","display":"e^{(t-r)D}","kind":"use","excerpt":"0 · -2, B=(1,0), and C=(0,1)^ T, the feedback kernel is Be^tDC=te^-2t. vfigext_auxiliary platecaptionAn input first enters y_2, transfers to y_1, and then returns to x. The two equal decay rates produce a repeated pole through the intervening link. headElimination recovers the convolution, including its preparation Variation of constants gives y(t)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff3bd78608fb37cd","tex":"\\e^{(t-s)(QAQ)|_{QX}}","aliases":"e^(t-s)(QAQ)|_QX e mathrm","display":"{\\mathrm e}^{(t-s)(QAQ)|_{QX}}","kind":"use","excerpt":"For x∈ PX, write p(t)=PT(t)x, q(t)=QT(t)x. The hidden component satisfies q'(t)=QAQq(t)+QAPp(t), q(0)=0. Variation of constants gives q(t)= ∫_0^t e^(t-s)(QAQ)|_QX QAPp(s) d s. Since normp(s)≤norm PnormT(s)norm x and x=Px, the first estimate follows. The contraction case is immediate.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-281758b86b00412e","tex":"\\e^{(t-s)A_c}","aliases":"e^(t-s)A_c e mathrm","display":"{\\mathrm e}^{(t-s)A_c}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b0f251471820be1","tex":"\\e^{(t-s)A_{QQ}}","aliases":"e^(t-s)A_QQ e mathrm","display":"{\\mathrm e}^{(t-s)A_{QQ}}","kind":"use","excerpt":"For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf4d46acf1234d88","tex":"\\e^{(t-s)D}","aliases":"e^(t-s)D e mathrm","display":"{\\mathrm e}^{(t-s)D}","kind":"use","excerpt":"Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a76476f1c15d5142","tex":"e^{(t-s)T_c}","aliases":"e^(t-s)T_c","display":"e^{(t-s)T_c}","kind":"use","excerpt":"[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a07e55f4cb635281","tex":"\\e^{(t-s_m)D}","aliases":"e^(t-s_m)D e mathrm","display":"{\\mathrm e}^{(t-s_m)D}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61130c15ab331a4f","tex":"e^{(z+w)\\xi}","aliases":"e^(z+w)ξ xi","display":"e^{(z+w)\\xi}","kind":"use","excerpt":"Differentiation multiplies by λξ in the first algebra and by ξ in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity e^(z+w)ξ=e^zξe^wξ gives the addition laws after comparison of the coefficients of 1,ξ,…,ξ^N-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad4ab5527a169fc1","tex":"e^{(\\lambda-z)t}","aliases":"e^(λ-z)t lambda","display":"e^{(\\lambda-z)t}","kind":"use","excerpt":"Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace. Equivalently, apply the decomposition to left multiplication by B on the matrix algebra a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66a0d4ed11e02630","tex":"e^{(\\mu-z)t}","aliases":"e^(μ-z)t mu","display":"e^{(\\mu-z)t}","kind":"use","excerpt":"General nonnormal memory (13.2). The actual damped exponential is decomposed into finitely many terms of the form t^k e^(μ-z)t. Integrability and convergence to zero follow from Reμ<Rez. The formal proof uses generalized eigenspaces of left multiplication on the matrix algebra, so nilpotent Jordan parts are included. Both inverse identities follow ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ba01bfed40e94931","tex":"E^*","aliases":"E^*","display":"E^*","kind":"use","excerpt":"[Frame bounds from the polyphase symbol] Assume that A I_N ≤ E( e^ iω)^*E( e^ iω) ≤ B I_N for almost every ω, where 0<A≤ B<∞. Then the analysis bank satisfies A x _2^2 ≤ A_E x _2^2 ≤ B x _2^2. The pointwise matrix R_rm can( e^ iω) = (E^*E)^-1E^* is a bounded left inverse and gives the canonical dual synthesis bank.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16020a46bf72f212","tex":"\\e^{-(s+t)}","aliases":"e^-(s+t) e mathrm","display":"{\\mathrm e}^{-(s+t)}","kind":"use","excerpt":"Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-213cbea125edd4d0","tex":"e^{-(s-r)B^\\alpha}","aliases":"e^-(s-r)B^α alpha","display":"e^{-(s-r)B^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a17fc961408f2c36","tex":"e^{-(t-r)}","aliases":"e^-(t-r)","display":"e^{-(t-r)}","kind":"use","excerpt":"vpage22Hidden state returns as memoryEXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUPChapter VI, Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data. Write the same system as x'=-x+y, y'=x-y. Solving the second equation first gives y(t)=e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. Substitution is exact: x'(t)=-x(t)+e^-ty(0)+∫","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3af4d37764747f9a","tex":"e^{-(t_a+t_b)H}","aliases":"e^-(t_a+t_b)H","display":"e^{-(t_a+t_b)H}","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b8fb3a0804953d6","tex":"e^{-(z+2)t}","aliases":"e^-(z+2)t","display":"e^{-(z+2)t}","kind":"use","excerpt":"t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that retained the eigenvalue -2 but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps. headThe transfer function sees pol","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2349512e48c7e97","tex":"e^{-1}","aliases":"e^-1","display":"e^{-1}","kind":"use","excerpt":"[Derived pointwise realization] Let e:R≅ T, and let C be a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for C, supported in degrees [a,b]. For every characteristic zero point x:R→ E', write y=x∘ e^-1. Then some integer i∈[a,b] has H^i(Cmathbinmathop⊗limits_ T,y^ LE')≠0. Every t∈ T acts on thi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-531a33389172d686","tex":"\\e^{-1/(4t)}","aliases":"e^-1/(4t) e mathrm","display":"{\\mathrm e}^{-1/(4t)}","kind":"use","excerpt":"[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63148d416f1c6825","tex":"\\e^{-16t\\abs{k}^2}","aliases":"e^-16tabsk^2 e abs mathrm abs","display":null,"kind":"use","excerpt":"ixed integer mode, [source reference] gives λ_N(k)⟶4π^2absk^2. For centered modes, the inequality sin x≥frac2xπ, 0≤ x≤fracπ2, gives λ_N(k)≥16absk^2. Also λ_N(k)≤4π^2absk^2. Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by e^-16tabsk^2, and the moment terms by 4π^2absk^2 e^-16tabsk^2. Both ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-705a582f899d971c","tex":"\\e^{-2(s+t)}","aliases":"e^-2(s+t) e mathrm","display":"{\\mathrm e}^{-2(s+t)}","kind":"use","excerpt":"Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e19d2c16a0f316fd","tex":"e^{-2(s+t)}","aliases":"e^-2(s+t)","display":"e^{-2(s+t)}","kind":"use","excerpt":"A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mechanism The constant mode is unchanged, while each elapsed interval multiplies the contrast by e^-2t. Consequently two inte","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-846c05d1e926ff8e","tex":"e^{-2(t-r)}","aliases":"e^-2(t-r)","display":"e^{-2(t-r)}","kind":"use","excerpt":"The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd5e14eee60bd93f","tex":"e^{-2\\ii t}","aliases":"e^-2ii t e^-2 i t ii mathrm","display":"e^{-2{\\mathrm i} t}","kind":"use","excerpt":"4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structu","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ecea398e87a0ddcd","tex":"e^{-2\\ii\\beta\\ell}","aliases":"e^-2iiβℓ e^-2 iβℓ ii beta ell mathrm beta ell","display":"e^{-2{\\mathrm i}\\beta\\ell}","kind":"use","excerpt":"For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67d5ba9f4cee7ae8","tex":"e^{-2jt}","aliases":"e^-2jt","display":"e^{-2jt}","kind":"use","excerpt":"itive self-adjoint finite model and its spectral powers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its square root is L^1/2=sqrt2P_1+2P_2+sqrt6P_3, (L^1/2)^2=L. Both heat evolutions retain these modes. Ordinary diffusion mul","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31a645cf967f487c","tex":"\\e^{-2nt}","aliases":"e^-2nt e mathrm","display":"{\\mathrm e}^{-2nt}","kind":"use","excerpt":"[Hermite packet reconstruction] Let P_n be the orthogonal projection onto C h_n. Then I=∑_n=0^∞ P_n in the strong operator topology and O_t = ∑_n=0^∞ e^-ntP_n. For f∈ L^2(γ), norm O_t f_2^2 = ∑_n=0^∞ e^-2ntnormP_nf_2^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-544b5bdc27bea927","tex":"e^{-2pt}","aliases":"e^-2pt","display":"e^{-2pt}","kind":"use","excerpt":"Even projection is identified coefficientwise. For any formal series f, let EvenPart(f)=(f(t)+f(-t))/2. Its nth coefficient is proved to be the coefficient of f when n is even and zero otherwise. Rescaling the inverse law proves sech(-t)=sech(t). The exponential multiplication identity then gives cosh^2(pt)=frac2+e^2pt+e^-2pt4, and hence ∑_r≥0S_2r(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e46f05d4098d239","tex":"\\e^{-2s}","aliases":"e^-2s e mathrm","display":"{\\mathrm e}^{-2s}","kind":"use","excerpt":"Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f67647d8f9d6f031","tex":"e^{-2s}","aliases":"e^-2s","display":"e^{-2s}","kind":"use","excerpt":"A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mechanism The constant mode is unchanged, while each elapsed interval multiplies the contrast by e^-2t. Consequently two inte","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d7200ea906e4a03","tex":"e^{-2s_Ny}","aliases":"e^-2s_Ny","display":"e^{-2s_Ny}","kind":"use","excerpt":"a_N-θ. Monotonicity of cosine on [0,π], followed by reflection about π, therefore gives Re(η_ℓ z)≤ rcos(a_N+θ). The remaining vertex e^-ia_N has projection rcos(a_N-θ). Their difference is at least 2s_Nrsinθ=2s_Ny. If D_N(z)=0, divide its finite exponential sum by the term with exponent e^-ia_Nz and apply the triangle inequality: 1≤(N-1)e^-2s_Ny. T","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d81f545674e3f61e","tex":"\\e^{-2t}","aliases":"e^-2t e mathrm","display":"{\\mathrm e}^{-2t}","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7511ccdfc1ceeb18","tex":"e^{-2t}","aliases":"e^-2t","display":"e^{-2t}","kind":"use","excerpt":"1 · -1. The total-mass direction (1,1) has eigenvalue zero. The contrast (1,-1) has eigenvalue -2. Resolving those two directions gives the full transfer, not only its first-order approximation: T(t)=e^tG=frac121+e^-2t · 1-e^-2t 1-e^-2t · 1+e^-2t. vfigclock platecaptionA point seed at state zero approaches the uniform distribution. The displayed ti","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e8da4431aa141cc","tex":"e^{-2\\gamma_Nt_m}","aliases":"e^-2γ_Nt_m gamma","display":"e^{-2\\gamma_Nt_m}","kind":"use","excerpt":"[Leading displacement term and derivative sign] With the notation of [source reference], τ_m = t_m+ fracR_N(t_m)2is_N + O(e^-2γ_Nt_m). The quotient R_N(t_m)/(2is_N) is real. One also has D_N'(τ_m) = (-1)^m+1 frac2s_NN e^c_Nτ_m (1+O(e^-γ_Nτ_m)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75c622f56ccbfb4c","tex":"e^{-2\\pi ir}","aliases":"e^-2π ir pi","display":"e^{-2\\pi ir}","kind":"use","excerpt":"gion containing 0 and normalize the logarithm to vanish there. D_N(z)^-r=exp(-rL(z)), e^L(z)=D_N(z), L(0)=0. vfigbranch_domain For D_2=cos z, the open disk |z|<π/2 is an admissible domain. A small loop around the simple zero ρ=π/2 is not contained in that disk. Analytic continuation around the zero changes the local logarithm by 2π i, and the power","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6332bad66e756810","tex":"e^{-2\\tau}","aliases":"e^-2τ tau","display":"e^{-2\\tau}","kind":"use","excerpt":"vpage23Positive transfer is not a stochastic matrixSPECTRAL POSITIVITY IS BASIS INDEPENDENTChapter VII: Theorem II.10.1, Theorem II.10.3. For H=I-X, the two eigenvalues are zero and two. Its Euclidean transfer is T(τ)=e^-τ H =frac12mat1+q · 1-q 1-q · 1+q, q=e^-2τ. For τ≥0, the eigenvalues 1,q lie in (0,1], and multiplying two such matrices replaces","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b902887ca3c0c26","tex":"e^{-3t}","aliases":"e^-3t","display":"e^{-3t}","kind":"use","excerpt":"vpage37Keep blocks by their certified contributionMEMORY COMPRESSION / AN OPTIMIZATION WITH A DEFINED COSTChapter VI, Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks. Block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jorda","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95ab7e932f644020","tex":"e^{-3t_m}","aliases":"e^-3t_m","display":"e^{-3t_m}","kind":"use","excerpt":"[Leading correction for N=3] For N=3, t_m=frac2πsqrt3(m+frac12) and τ_m = t_m+ frac(-1)^msqrt3e^-3t_m/2 + O(e^-3t_m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d20a1e79d4b08523","tex":"e^{-3t_m/2}","aliases":"e^-3t_m/2","display":"e^{-3t_m/2}","kind":"use","excerpt":"[Leading correction for N=3] For N=3, t_m=frac2πsqrt3(m+frac12) and τ_m = t_m+ frac(-1)^msqrt3e^-3t_m/2 + O(e^-3t_m).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f37aa953e728f3f3","tex":"e^{-4t}","aliases":"e^-4t","display":"e^{-4t}","kind":"use","excerpt":"nH_n for these three polynomials. Their squared norms are n! and different Hermite degrees are orthogonal. vfigext_hermite platecaptionFor f=H_1+H_2, the two bars are its separate squared-energy contributions. At t=log2, their sum is 3/8, below the spectral-gap bound 3/4. headReconstruction preserves the modes as time changes their weights The semi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2bee6969f99fc91d","tex":"\\e^{-4\\pi^2t\\abs{k}^2}","aliases":"e^-4π^2tabsk^2 e pi abs mathrm pi abs","display":null,"kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e211d3027eec7b6","tex":"e^{-4\\tau}","aliases":"e^-4τ tau","display":"e^{-4\\tau}","kind":"use","excerpt":"vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESChapter VII: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ed7d803025ff45b","tex":"e^{-8t}","aliases":"e^-8t","display":"e^{-8t}","kind":"use","excerpt":"For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b89b0c1ab78e1b6","tex":"\\e^{-a(E_j-E_0)}","aliases":"e^-a(E_j-E_0) e mathrm","display":"{\\mathrm e}^{-a(E_j-E_0)}","kind":"use","excerpt":"Diagonalize H. Each eigenvalue of T is e^-a(E_j-E_0) and therefore lies in (0,1]. The power law follows from the functional calculus.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f846e2f3c2a6fe91","tex":"\\e^{-a(H-E_0I)}","aliases":"e^-a(H-E_0I) e mathrm","display":"{\\mathrm e}^{-a(H-E_0I)}","kind":"use","excerpt":"Let H=H^* on a nonzero finite dimensional Hilbert space and let E_0=minSpec H. For a lattice time step a>0, define T= e^-a(H-E_0I).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-363d8b60aa81bd16","tex":"e^{-a_nt}","aliases":"e^-a_nt","display":"e^{-a_nt}","kind":"use","excerpt":"vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONChapter VI, Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f74ea1ccebeff413","tex":"\\e^{-\\abs{n}^2/(4t)}","aliases":"e^-absn^2/(4t) e abs mathrm abs","display":null,"kind":"use","excerpt":"[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6180d10aad8c0b5e","tex":"\\e^{-aH_T}","aliases":"e^-aH_T e mathrm","display":"{\\mathrm e}^{-aH_T}","kind":"use","excerpt":"[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61dcffd4e083f043","tex":"\\e^{-at}","aliases":"e^-at e mathrm","display":"{\\mathrm e}^{-at}","kind":"use","excerpt":"Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fbf805b9f71e84a","tex":"\\e^{-bt}","aliases":"e^-bt e mathrm","display":"{\\mathrm e}^{-bt}","kind":"use","excerpt":"d · -b , a,b>0, and retain the first coordinate. The memory kernel and self-energy are K(t)=cd e^-bt, Σ(z)=fraccdz+b. The retained coordinate has an exact time local closure for every retained initial value precisely when cd=0. When cd≠0, one auxiliary scalar state realizes the memory exactly.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6cecc0b8dc4e5e1","tex":"e^{-c_N|\\rho|}","aliases":"e^-c_N|ρ| rho","display":"e^{-c_N|\\rho|}","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9e49328e16a6e146","tex":"e^{-e^{-ia_N}t_m}","aliases":"e^-e^-ia_Nt_m","display":"e^{-e^{-ia_N}t_m}","kind":"use","excerpt":"At the real point t_m, the normalized value N e^-e^-ia_Nt_m D_N(t_m) is purely imaginary. The two dominant terms cancel there, so R_N(t_m) is purely imaginary and the displayed correction is real.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26428a4ee1bc8c64","tex":"e^{-e^{-ia_N}z}","aliases":"e^-e^-ia_Nz","display":"e^{-e^{-ia_N}z}","kind":"use","excerpt":"η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6de606f9a0b14b66","tex":"e^{-H_n}","aliases":"e^-H_n","display":"e^{-H_n}","kind":"use","excerpt":"For transfers on two states T_n=diag(1,q_n)=e^-H_n, with H_n=diag(0,-log q_n) at unit time, the error in trace norm is 2^-(n+2) and the heat traces converge to 3/2. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cba54642d9ab8642","tex":"e^{-i(t-s)H_{QQ}}","aliases":"e^-i(t-s)H_QQ","display":"e^{-i(t-s)H_{QQ}}","kind":"use","excerpt":"idot y · =H_QPx+H_QQy. Solving the second equation gives y(t) =e^-itH_QQy(0) -i∫_0^t e^-i(t-s)H_QQH_QPx(s) ds. Substitution yields the exact Volterra equation idot x(t) = · H_PPx(t) +H_PQe^-itH_QQy(0)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3bc61419c10fa3f","tex":"e^{-ia_N}","aliases":"e^-ia_N","display":"e^{-ia_N}","kind":"use","excerpt":"On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc15714350459aa5","tex":"\\e^{-\\ii sH}","aliases":"e^-ii sH e^- i sH e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} sH}","kind":"use","excerpt":"The summability condition makes (H_n) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives e^- i tH_n- e^- i tH =- i∫_0^t e^- i(t-s)H_n(H_n-H) e^- i sH d s. Both unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamilt","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60bbbe7f29c53648","tex":"\\e^{-\\ii sH_{\\rm d}}","aliases":"e^-ii sH_rm d e^- i sH_rm d e ii rm mathrm mathrm rm","display":"{\\mathrm e}^{-{\\mathrm i} sH_{\\rm d}}","kind":"use","excerpt":"Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d727f71a03d62d1e","tex":"e^{-\\ii t}","aliases":"e^-ii t e^- i t ii mathrm","display":"e^{-{\\mathrm i} t}","kind":"use","excerpt":"4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structu","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b2c8b37cfa4c924","tex":"\\e^{-\\ii tA}","aliases":"e^-ii tA e^- i tA e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tA}","kind":"use","excerpt":"If B=0, the retained and hidden equations decouple. Conversely, suppose that every solution with y(0)=0 satisfies x(t)= e^- i tAx(0). The full block equation gives ddot x(0) = -(A^2+BB^*)x(0), while the declared local equation gives ddot x(0)=-A^2x(0). Since this holds for every retained initial vector, BB^*=0. Hence B=0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35f6d789137bc755","tex":"\\e^{-\\ii tD}","aliases":"e^-ii tD e^- i tD e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tD}","kind":"use","excerpt":"[Exact retained memory equation] The retained coordinate satisfies idot x(t) = · Ax(t)+B e^- i tDy(0) · - i∫_0^tK(t-s)x(s) d s, where K(t)=B e^- i tDB^*. For Res>0, ∫_0^∞ e^-stK(t) d t = B(s+ i D)^-1B^*.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d9aa2691147f3d1","tex":"e^{-\\ii tD}","aliases":"e^-ii tD e^- i tD ii mathrm","display":"e^{-{\\mathrm i} tD}","kind":"use","excerpt":"A nonzero initial hidden vector also supplies the forcing term Be^- i tDy(0). It is additional data, not part of the convolution kernel. The retained checks cover the displayed kernel values, exact Gram factorization, and acceleration defect. The integral equation follows from the written elimination argument. takeawayMemory is positive definite th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72f088de936f5a2e","tex":"\\e^{-\\ii tE_r}","aliases":"e^-ii tE_r e^- i tE_r e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tE_r}","kind":"use","excerpt":"[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53d7011fee16a3bf","tex":"\\e^{-\\ii tH}","aliases":"e^-ii tH e^- i tH e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tH}","kind":"use","excerpt":"[Exact invariant packets] The following conditions are equivalent: item The Hamiltonian commutes with each projector: [H,P_a]=0 for every a. item Transfer blocks between distinct packets vanish: P_bHP_a=0 whenever a≠ b. item Every packet P_a H is invariant under e^- i tH for every real t. When they hold, H=∑_aP_aHP_a and e^- i tH=∑_aP_a e^- i tHP_a","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2d69dce8d687c13","tex":"e^{-\\ii tH}","aliases":"e^-ii tH e^- i tH ii mathrm","display":"e^{-{\\mathrm i} tH}","kind":"use","excerpt":"H · Self-adjoint Hamiltonian · It generates evolution in real time through e^- i tH.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53be779234266f77","tex":"\\e^{-\\ii tH_n}","aliases":"e^-ii tH_n e^- i tH_n e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i} tH_n}","kind":"use","excerpt":"[Norm summable Hamiltonian tower] Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c7c5aa9303ef0fa","tex":"e^{-\\ii tH_n}","aliases":"e^-ii tH_n e^- i tH_n ii mathrm","display":"e^{-{\\mathrm i} tH_n}","kind":"use","excerpt":"For the norm route, work instead on the single fixed space C^2 and set H_n=diag(0,1+2^-n), H=diag(0,1). Here normH_n+1-H_n=2^-(n+1) and normH_n-H=2^-n. The summable tail supplies an error bound for every n. For |t|≤ T, the paper's Duhamel estimate gives norme^- i tH_n-e^- i tH≤ T2^-n. The Euclidean counterpart holds on 0≤ t≤ T because both Hamilton","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab9f45562016500f","tex":"\\e^{-\\ii tH_\\ell}","aliases":"e^-ii tH_ℓ e^- i tH_ℓ e ii ell mathrm mathrm ell","display":"{\\mathrm e}^{-{\\mathrm i} tH_\\ell}","kind":"use","excerpt":"[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82432643f68232b1","tex":"\\e^{-\\ii tH_\\infty}","aliases":"e^-ii tH_∞ e^- i tH_∞ e ii infty mathrm mathrm infty","display":"{\\mathrm e}^{-{\\mathrm i} tH_\\infty}","kind":"use","excerpt":"[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78c88d9fe415a0e0","tex":"\\e^{-\\ii(t-s)D}","aliases":"e^-ii(t-s)D e^- i(t-s)D e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i}(t-s)D}","kind":"use","excerpt":"The hidden equation is idot y=B^*x+Dy. Variation of constants gives y(t)= e^- i tDy(0) - i∫_0^t e^- i(t-s)DB^*x(s) d s. Substitute into idot x=Ax+By. The Laplace formula follows by diagonalizing D or integrating the norm convergent finite dimensional exponential.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48f850fbfb58d911","tex":"\\e^{-\\ii(t-s)H}","aliases":"e^-ii(t-s)H e^- i(t-s)H e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i}(t-s)H}","kind":"use","excerpt":"Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4988b85314de55e4","tex":"\\e^{-\\ii(t-s)H_n}","aliases":"e^-ii(t-s)H_n e^- i(t-s)H_n e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i}(t-s)H_n}","kind":"use","excerpt":"The summability condition makes (H_n) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives e^- i tH_n- e^- i tH =- i∫_0^t e^- i(t-s)H_n(H_n-H) e^- i sH d s. Both unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamilt","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5a49b0472ff2c1d","tex":"\\e^{-\\ii(t_j-t_k)D}","aliases":"e^-ii(t_j-t_k)D e^- i(t_j-t_k)D e ii mathrm mathrm","display":"{\\mathrm e}^{-{\\mathrm i}(t_j-t_k)D}","kind":"use","excerpt":"Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbad7626b522cffb","tex":"e^{-\\ii\\pi L/2}","aliases":"e^-iiπ L/2 e^- iπ L/2 ii pi mathrm pi","display":"e^{-{\\mathrm i}\\pi L/2}","kind":"use","excerpt":"One calculation of transfer to the opposite siteSEEDS, CHARACTERS, INTERFERENCE, AND DEPHASINGChapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4. Return to the shift on four sites S and Laplacian L=2I-S-S^*. The same character projectors have energies (0,2,4,2). At time t=π/2, their evolution phases are (1,-1,1,-1), exactly the character","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-162f43593822664a","tex":"e^{-is_N\\tau_m}","aliases":"e^-is_Nτ_m tau","display":"e^{-is_N\\tau_m}","kind":"use","excerpt":"For the derivative, use the formula in the proof of [source reference] and e^-is_Nτ_m = (-1)^m(-i) (1+O(e^-γ_Nτ_m)). The stated real sign follows.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91b2f17ba96a755d","tex":"e^{-itH_{QQ}}","aliases":"e^-itH_QQ","display":"e^{-itH_{QQ}}","kind":"use","excerpt":"idot y · =H_QPx+H_QQy. Solving the second equation gives y(t) =e^-itH_QQy(0) -i∫_0^t e^-i(t-s)H_QQH_QPx(s) ds. Substitution yields the exact Volterra equation idot x(t) = · H_PPx(t) +H_PQe^-itH_QQy(0)","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c1f307a0a1887517","tex":"\\e^{-na(E_j-E_0)}","aliases":"e^-na(E_j-E_0) e mathrm","display":"{\\mathrm e}^{-na(E_j-E_0)}","kind":"use","excerpt":"In the spectral decomposition, T^n-P_0 has eigenvalues zero on the ground space and e^-na(E_j-E_0) on the excited eigenspaces. The largest is e^-naΔ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c65f064466994a81","tex":"\\e^{-na(H-E_0I)}","aliases":"e^-na(H-E_0I) e mathrm","display":"{\\mathrm e}^{-na(H-E_0I)}","kind":"use","excerpt":"[Finite Euclidean transfer] The operator T is positive definite, self adjoint, and contractive. Its spectrum lies in (0,1], and T^n= e^-na(H-E_0I) for every n∈ N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23edb8b0043e53d7","tex":"\\e^{-naH_T}","aliases":"e^-naH_T e mathrm","display":"{\\mathrm e}^{-naH_T}","kind":"use","excerpt":"[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6cf8132b360cc53","tex":"\\e^{-na\\Delta}","aliases":"e^-naΔ e Delta mathrm Delta","display":"{\\mathrm e}^{-na\\Delta}","kind":"use","excerpt":"[Ground packet convergence] Let P_0 be the spectral projector of H at E_0, and let Δ=min{E_j-E_0:E_j>E_0}. If an excited eigenvalue exists, then normT^n-P_0= e^-naΔ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-735b6f4e0a49bd0b","tex":"\\e^{-nt}","aliases":"e^-nt e mathrm","display":"{\\mathrm e}^{-nt}","kind":"use","excerpt":"[Ornstein--Uhlenbeck semigroup] The operators ( O_t)_t≥0 form a positive Markov contraction semigroup on L^p(γ) for every 1≤ p≤∞. On L^2(γ), the semigroup is self-adjoint and O_t h_n= e^-nth_n. The family (h_n)_n≥0 is an orthonormal basis of L^2(γ).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e35c3786b9e0a2fb","tex":"e^{-nt}","aliases":"e^-nt","display":"e^{-nt}","kind":"use","excerpt":"[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59ef4f417932e955","tex":"e^{-q c_N|\\rho|}","aliases":"e^-q c_N|ρ| rho","display":"e^{-q c_N|\\rho|}","kind":"use","excerpt":"-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-487f96295a31f05f","tex":"e^{-r}","aliases":"e^-r","display":"e^{-r}","kind":"use","excerpt":"The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-458955e59598018c","tex":"e^{-rA^\\alpha}","aliases":"e^-rA^α alpha","display":"e^{-rA^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33bc3e010e27b2a8","tex":"\\e^{-s}","aliases":"e^-s e mathrm","display":"{\\mathrm e}^{-s}","kind":"use","excerpt":"Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc9a49b80c0d1df4","tex":"e^{-s}","aliases":"e^-s","display":"e^{-s}","kind":"use","excerpt":"/2 and f_2(t)=e^-2t. vfigext_risks platecaptionAt t=log2, surviving mass is 3/8. Cause 1 has accumulated probability 1/4 and cause 2 has accumulated probability 3/8. The three pieces sum to one. headCondition on survival before restarting the clock At this time the transient masses are (1/4,1/8), so the conditional distribution of survivors is (2/3","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8add4baf58362c10","tex":"e^{-sA^\\alpha}","aliases":"e^-sA^α alpha","display":"e^{-sA^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea0104f7653eb4fc","tex":"e^{-sB^\\alpha}","aliases":"e^-sB^α alpha","display":"e^{-sB^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a55c82c872f9fae7","tex":"\\e^{-sL}","aliases":"e^-sL e mathrm","display":"{\\mathrm e}^{-sL}","kind":"use","excerpt":"nstandardfact[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6c11852ddb445b7","tex":"e^{-sL}","aliases":"e^-sL","display":"e^{-sL}","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c61b01610015a01e","tex":"e^{-sm^\\alpha}","aliases":"e^-sm^α alpha","display":"e^{-sm^\\alpha}","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-218864f064df802f","tex":"e^{-\\sqrt{2j}\\,t}","aliases":"e^-sqrt2j t sqrt","display":"e^{-\\sqrt{2j}\\,t}","kind":"use","excerpt":"wers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its square root is L^1/2=sqrt2P_1+2P_2+sqrt6P_3, (L^1/2)^2=L. Both heat evolutions retain these modes. Ordinary diffusion multiplies mode j by e^-2jt. Square-root diffusion mul","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9209e63597cdde74","tex":"\\e^{-st}","aliases":"e^-st e mathrm","display":"{\\mathrm e}^{-st}","kind":"use","excerpt":"The Laplace transform follows by integrating the density in [source reference]. Stability of A gives ∫_0^∞ e^-st e^tA d t = (sI-A)^-1. At s=0, a^T(-A)^-1p_0 = 1^Tp_0=1. For the moments, ∫_0^∞ t^m e^tA d t = m!(-A)^-m-1. Multiplying by a^T= 1^T(-A) gives the result. These are the standard matrix formulas for phase type laws [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3543ab1cc3650ab3","tex":"e^{-st}","aliases":"e^-st","display":"e^{-st}","kind":"use","excerpt":"For the survival consequences, A is a finite Metzler matrix with nonpositive column sums and spectrum in the open left half-plane, p_0 is a probability vector, S(t)= 1^T e^tAp_0, and f(t)=-S'(t). The one-phase model has λ>0. The survival Laplace identity of Part II also holds for every complex s with Res≥0: stability gives absolute convergence of ∫","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3117c4fb220fd668","tex":"\\e^{-s\\lambda}","aliases":"e^-sλ e lambda mathrm lambda","display":"{\\mathrm e}^{-s\\lambda}","kind":"use","excerpt":"[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76df3c6c345601fc","tex":"\\e^{-s\\tau}","aliases":"e^-sτ e tau mathrm tau","display":"{\\mathrm e}^{-s\\tau}","kind":"use","excerpt":"[Laplace transform and moments] Assume the spectrum of A lies in the open left half plane. Then absorption occurs with probability one. For s≥0, E[ e^-sτ] = a^T(sI-A)^-1p_0. For every integer m≥1, E[τ^m] = m! 1^T(-A)^-mp_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ce5ba5917308780","tex":"\\e^{-t}","aliases":"e^-t e mathrm","display":"{\\mathrm e}^{-t}","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f6dc0c78382f5b1","tex":"e^{-t}","aliases":"e^-t","display":"e^{-t}","kind":"use","excerpt":"vpage18Two large terms nearly cancelTHE CUBIC MODELChapter IV: Lemma 11.5, Theorem 11.7, Corollary 11.9. For real t, order three has the exact formula D_3(t)=frace^-t+2e^t/2cos(sqrt3t/2)3. Divide by the nonzero factor 2e^t/2/3. The resulting equation separates the dominant cosine from the small third exponential. vfigtwo_term Ignoring the last term","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1596d0f30252060","tex":"\\e^{-t(-\\Lcal)^\\alpha}","aliases":"e^-t(-Lcal)^α e^-t(- L)^α e Lcal alpha mathrm mathcal alpha","display":"{\\mathrm e}^{-t(-{\\mathcal L})^\\alpha}","kind":"use","excerpt":"[Fractional Hermite transfer] Let - L be the nonnegative Ornstein--Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e864a5dbfd8a4ab4","tex":"e^{-t(H-E_0I)}","aliases":"e^-t(H-E_0I)","display":"e^{-t(H-E_0I)}","kind":"use","excerpt":"For a family with nontrivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87d847893f57d8a1","tex":"\\e^{-t(k\\pi)^{2\\alpha}}","aliases":"e^-t(kπ)^2α e pi alpha mathrm pi alpha","display":"{\\mathrm e}^{-t(k\\pi)^{2\\alpha}}","kind":"use","excerpt":"set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(kπ)^2α ≤ fractα(4k^2)^α-1(kπ)^412h^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be2259ac52650147","tex":"e^{-t(x-e)}","aliases":"e^-t(x-e)","display":"e^{-t(x-e)}","kind":"use","excerpt":"Connecting power series and spectral calculus. The shifted generator is proved Hermitian. The implementation then identifies the actual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d912db525006fd32","tex":"\\e^{-t^2/(4s)}","aliases":"e^-t^2/(4s) e mathrm","display":"{\\mathrm e}^{-t^2/(4s)}","kind":"use","excerpt":"[Poisson subordination] For t>0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s) e^-sL d s. The integral converges in operator norm.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a21b18821fa51fe8","tex":"e^{-t^2/(4s)}","aliases":"e^-t^2/(4s)","display":"e^{-t^2/(4s)}","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a923b5b884633e8","tex":"\\e^{-t^2/(4s)-\\lambda s}","aliases":"e^-t^2/(4s)-λ s e lambda mathrm lambda","display":"{\\mathrm e}^{-t^2/(4s)-\\lambda s}","kind":"use","excerpt":"On a spectral packet with eigenvalue λ, the formula reduces to e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s)-λ s d s. This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcd0496b618b65e0","tex":"e^{-t^2/(4s)-\\lambda s}","aliases":"e^-t^2/(4s)-λ s lambda","display":"e^{-t^2/(4s)-\\lambda s}","kind":"use","excerpt":"By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5006b64dd94bbdc7","tex":"e^{-t_aH}","aliases":"e^-t_aH","display":"e^{-t_aH}","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6885d0ac0968565e","tex":"\\e^{-ta}","aliases":"e^-ta e mathrm","display":"{\\mathrm e}^{-ta}","kind":"use","excerpt":"proof The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-097eefc0b2596360","tex":"\\e^{-tb}","aliases":"e^-tb e mathrm","display":"{\\mathrm e}^{-tb}","kind":"use","excerpt":"proof The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18da745fe6ddb48f","tex":"\\e^{-tD_V/n}","aliases":"e^-tD_V/n e mathrm","display":"{\\mathrm e}^{-tD_V/n}","kind":"use","excerpt":"[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31ea130aac066aba","tex":"\\e^{-tH}","aliases":"e^-tH e mathrm","display":"{\\mathrm e}^{-tH}","kind":"use","excerpt":"Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-104acf20a9fbdb09","tex":"\\e^{-tH_n}","aliases":"e^-tH_n e mathrm","display":"{\\mathrm e}^{-tH_n}","kind":"use","excerpt":"Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ab294202ef31b05","tex":"\\e^{-tH_\\ell}","aliases":"e^-tH_ℓ e ell mathrm ell","display":"{\\mathrm e}^{-tH_\\ell}","kind":"use","excerpt":"imit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3fa4ad68ee2f72f1","tex":"\\e^{-tH_\\infty}","aliases":"e^-tH_∞ e infty mathrm infty","display":"{\\mathrm e}^{-tH_\\infty}","kind":"use","excerpt":"imit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98e1690e1022778c","tex":"e^{-tL}","aliases":"e^-tL","display":"e^{-tL}","kind":"use","excerpt":"Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usu","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e483707257d6afec","tex":"\\e^{-tL^\\alpha}","aliases":"e^-tL^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tL^\\alpha}","kind":"use","excerpt":"For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3551df4650a99bb8","tex":"e^{-tL^\\alpha}","aliases":"e^-tL^α alpha","display":"e^{-tL^\\alpha}","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df43c0922b3aee51","tex":"\\e^{-tL_c^\\alpha}","aliases":"e^-tL_c^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tL_c^\\alpha}","kind":"use","excerpt":"[Exact fractional naturality] Let L_f and L_c be finite dimensional positive self-adjoint operators and let C satisfy CL_f=L_cC. Then, for every α>0 and t≥0, CL_f^α=L_c^α C and C e^-tL_f^α = e^-tL_c^αC.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ced91f13885abaca","tex":"\\e^{-tL_f^\\alpha}","aliases":"e^-tL_f^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tL_f^\\alpha}","kind":"use","excerpt":"[Exact fractional naturality] Let L_f and L_c be finite dimensional positive self-adjoint operators and let C satisfy CL_f=L_cC. Then, for every α>0 and t≥0, CL_f^α=L_c^α C and C e^-tL_f^α = e^-tL_c^αC.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc658fbfa94bd130","tex":"\\e^{-tL_n}","aliases":"e^-tL_n e mathrm","display":"{\\mathrm e}^{-tL_n}","kind":"use","excerpt":"[Heat trace convergence from trace norm] Let L_n,L be nonnegative self adjoint operators on one Hilbert space. For a fixed t>0, assume their heat operators are trace class and that e^-tL_n⟶ e^-tL in trace norm. Then Tr( e^-tL_n)⟶Tr( e^-tL). If also the heat moment operators are trace class and L_n e^-tL_n⟶ L e^-tL in trace norm and the limiting hea","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2ff9ce40fe71e18","tex":"\\e^{-tn^\\alpha}","aliases":"e^-tn^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tn^\\alpha}","kind":"use","excerpt":"[Fractional Hermite transfer] Let - L be the nonnegative Ornstein--Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd2277b608148acc","tex":"\\e^{-t\\sqrt L}","aliases":"e^-tsqrt L e sqrt mathrm sqrt","display":"{\\mathrm e}^{-t\\sqrt L}","kind":"use","excerpt":"[Poisson subordination] For t>0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s) e^-sL d s. The integral converges in operator norm.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af1213c560bb1d07","tex":"e^{-t\\sqrt L}","aliases":"e^-tsqrt L sqrt","display":"e^{-t\\sqrt L}","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dfef5b71f36f50fe","tex":"\\e^{-t\\sqrt\\lambda}","aliases":"e^-tsqrtλ e sqrt lambda mathrm sqrt lambda","display":"{\\mathrm e}^{-t\\sqrt\\lambda}","kind":"use","excerpt":"On a spectral packet with eigenvalue λ, the formula reduces to e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s)-λ s d s. This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec0c0004dcef9a0c","tex":"e^{-t\\sqrt\\lambda}","aliases":"e^-tsqrtλ sqrt lambda","display":"e^{-t\\sqrt\\lambda}","kind":"use","excerpt":"By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-abbd5115258324d6","tex":"\\e^{-tx^\\alpha}","aliases":"e^-tx^α e alpha mathrm alpha","display":"{\\mathrm e}^{-tx^\\alpha}","kind":"use","excerpt":"The union of the two finite spectra is a finite set. There is a polynomial that agrees with x^α on that set, and another polynomial that agrees with e^-tx^α. Exact intertwining passes to every polynomial by induction.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95a8eb776d3e04a4","tex":"e^{-t\\lambda}","aliases":"e^-tλ lambda","display":"e^{-t\\lambda}","kind":"use","excerpt":"At the single time t=log2/16, two small grids admit exact rational heat weights: c|c|c|c|c N · spectrum · Z_N · ∑λ e^-tλ · d_N/log2 2 · (0,16) · 3/2 · 8 · 2/3 4 · (0,32,32,64) · 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8329615317d6615","tex":"\\e^{-t\\lambda^\\alpha}","aliases":"e^-tλ^α e lambda alpha mathrm lambda alpha","display":"{\\mathrm e}^{-t\\lambda^\\alpha}","kind":"use","excerpt":"[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75a311590636cbd2","tex":"\\e^{-t\\lambda_j^\\alpha}","aliases":"e^-tλ_j^α e lambda alpha mathrm lambda alpha","display":"{\\mathrm e}^{-t\\lambda_j^\\alpha}","kind":"use","excerpt":"Let H be a finite dimensional Hilbert space and let L=L^*≥0. Write its spectral resolution as L=∑_jλ_jP_j. For α>0, define L^α = ∑_jλ_j^α P_j and T_α(t) = e^-tL^α = ∑_j e^-tλ_j^αP_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7a32a247742f9a8","tex":"e^{-t\\lambda_j^\\alpha}","aliases":"e^-tλ_j^α lambda alpha","display":"e^{-t\\lambda_j^\\alpha}","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-876017de9ad7345a","tex":"\\e^{-t\\lambda_k}","aliases":"e^-tλ_k e lambda mathrm lambda","display":"{\\mathrm e}^{-t\\lambda_k}","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a4df964ebd62283","tex":"\\e^{-t\\lambda_{k,h}}","aliases":"e^-tλ_k,h e lambda mathrm lambda","display":"{\\mathrm e}^{-t\\lambda_{k,h}}","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a284cb9f17e58c3","tex":"\\e^{-t\\lambda_{k,h}^\\alpha}","aliases":"e^-tλ_k,h^α e lambda alpha mathrm lambda alpha","display":"{\\mathrm e}^{-t\\lambda_{k,h}^\\alpha}","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2b5b8666856ee72","tex":"\\e^{-t\\lambda_k^\\alpha}","aliases":"e^-tλ_k^α e lambda alpha mathrm lambda alpha","display":"{\\mathrm e}^{-t\\lambda_k^\\alpha}","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5fbb88a28e074883","tex":"\\e^{-t\\lambda_N(k)}","aliases":"e^-tλ_N(k) e lambda mathrm lambda","display":"{\\mathrm e}^{-t\\lambda_N(k)}","kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db6f29304e044ecc","tex":"e^{-t\\lambda_N(k)}","aliases":"e^-tλ_N(k) lambda","display":"e^{-t\\lambda_N(k)}","kind":"use","excerpt":"vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSChapter VII: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6030b628b360890a","tex":"e^{-u^2}","aliases":"e^-u^2","display":"e^{-u^2}","kind":"use","excerpt":"ime, and s is the ordinary diffusion time being mixed. They are not interchangeable variables. vfigrandom_clock platecaptionThe density η_1 peaks at s=1/6 but has a long right tail. The drawing ends at s=2. It does not claim that the missing tail has zero mass. headCheck the normalizer before interpreting a density Under s=t^2/(4u^2), reversing the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8db1472c40be17a4","tex":"\\e^{-V}","aliases":"e^-V e mathrm","display":"{\\mathrm e}^{-V}","kind":"use","excerpt":"[Discrete Feynman--Kac formula] Define M_V=diag( e^-V)K^T. For terminal data f, (M_V^nf)_i = E_i[ exp( -∑_r=0^n-1V(X_r) ) f(X_n) ].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5da71253c0e2cad","tex":"e^{-V}","aliases":"e^-V","display":"e^{-V}","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c891c594039cd0f2","tex":"\\e^{-V(i)}","aliases":"e^-V(i) e mathrm","display":"{\\mathrm e}^{-V(i)}","kind":"use","excerpt":"For one step, (M_Vf)_i = e^-V(i)∑_jK_jif_j. Iterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9341ba060ec74a26","tex":"e^{-V(i)}","aliases":"e^-V(i)","display":"e^{-V(i)}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b917ccc3ae79f8f2","tex":"e^{-V(i_r)}","aliases":"e^-V(i_r)","display":"e^{-V(i_r)}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-64589a9f7fc7012d","tex":"\\e^{-v_0t}","aliases":"e^-v_0t e mathrm","display":"{\\mathrm e}^{-v_0t}","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5dded19773673070","tex":"e^{-w}","aliases":"e^-w","display":"e^{-w}","kind":"use","excerpt":"and integral comparison. Thus truncation has an explicit error mechanism. takeawayThe infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem. platecaptionThe exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use k","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99b487a1df045997","tex":"e^{-z}","aliases":"e^-z","display":"e^{-z}","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc510a5e29f5ee80","tex":"\\e^{-zt}","aliases":"e^-zt e mathrm","display":"{\\mathrm e}^{-zt}","kind":"use","excerpt":"[Laplace resolvent and packet transform] Let A∈ B(X), let T(t)= e^tA, and let Rez>norm A. Then the Bochner integral converges in operator norm and (zI-A)^-1 = ∫_0^∞ e^-ztT(t) d t. For every pair of packets, P_β(zI-A)^-1P_α = ∫_0^∞ e^-ztT_βα(t) d t.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-abf22631e30760b4","tex":"e^{-zt}","aliases":"e^-zt","display":"e^{-zt}","kind":"use","excerpt":"[Memory/self-energy correspondence] For finite complex block matrices, assume Rez>Reλ for every λ∈σ(A_QQ). Then the following improper integral is absolutely convergent and ∫_0^∞ e^-zt K(t) dt =A_PQ(zI-A_QQ)^-1A_QP =Σ_P(z).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dff4ae82d4fe03f8","tex":"\\e^{-\\gamma t}","aliases":"e^-γ t e gamma mathrm gamma","display":"{\\mathrm e}^{-\\gamma t}","kind":"use","excerpt":"[Memory and self-energy] If z lies to the right of the spectrum of A_QQ, then ∫_0^∞ e^-zt K(t) d t = Σ_P(z). If, for constants M≥0 and γ>0, norm e^tA_QQ ≤ M e^-γ t, then norm K(t) ≤ MnormA_PQnormA_QP e^-γ t and norm K_L^1(0,∞) ≤ fracMγ normA_PQnormA_QP.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a82e99d45fc05c76","tex":"e^{-\\gamma t_m}","aliases":"e^-γ t_m gamma","display":"e^{-\\gamma t_m}","kind":"use","excerpt":"[Large zero strings] Assume N≥3. There are constants m_0, C, and γ>0 and a sequence of positive real simple zeros (τ_m)_m≥ m_0 of D_N such that τ_m = t_m+O(e^-γ t_m). Every zero of sufficiently large modulus is of the form ω^kτ_m for a unique 0≤ k<N and a unique m≥ m_0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e32e6ed4b75868d","tex":"e^{-\\gamma_N\\Re z}","aliases":"e^-γ_NRe z gamma Re","display":"e^{-\\gamma_N\\Re z}","kind":"use","excerpt":"If G(z)=0, then |1+e^2is_Nz| ≤ C e^-γ_NRe z. The real part of 2is_Nz is bounded in the strip. Apply [source reference] with w=2is_Nz. There is a unique integer m such that |z-t_m|≤ C_1e^-γ_NRe z, where t_m=frac(2m+1)π2s_N. In particular, |Re z-t_m|=o(1). For all sufficiently large zeros we have |Re z-t_m|≤1, and hence |z-t_m|≤ C_2e^-γ_Nt_m with a c","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d73f3a81c6fcc46","tex":"e^{-\\gamma_Nt_m}","aliases":"e^-γ_Nt_m gamma","display":"e^{-\\gamma_Nt_m}","kind":"use","excerpt":"G(z)=0, then |1+e^2is_Nz| ≤ C e^-γ_NRe z. The real part of 2is_Nz is bounded in the strip. Apply [source reference] with w=2is_Nz. There is a unique integer m such that |z-t_m|≤ C_1e^-γ_NRe z, where t_m=frac(2m+1)π2s_N. In particular, |Re z-t_m|=o(1). For all sufficiently large zeros we have |Re z-t_m|≤1, and hence |z-t_m|≤ C_2e^-γ_Nt_m with a cons","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-214052c012436283","tex":"e^{-\\gamma_Nx}","aliases":"e^-γ_Nx gamma","display":"e^{-\\gamma_Nx}","kind":"use","excerpt":"η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb45e18ecdc05bbe","tex":"e^{-\\gamma_N\\tau_m}","aliases":"e^-γ_Nτ_m gamma tau","display":"e^{-\\gamma_N\\tau_m}","kind":"use","excerpt":"Differentiate the normalized expression. At z=τ_m, G'(τ_m) = 2is_Ne^2is_Nτ_m+R_N'(τ_m) = -2is_N+O(e^-γ_Nτ_m). Since G(τ_m)=0, D_N'(τ_m) = frac1N e^e^-ia_Nτ_mG'(τ_m). Taking absolute values gives the derivative asymptotic. Differentiating D_N(ω z)= D_N(z) gives the rotation formula.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e408867033868d3","tex":"e^{-\\gamma\\tau_m}","aliases":"e^-γτ_m gamma tau","display":"e^{-\\gamma\\tau_m}","kind":"use","excerpt":"The derivatives satisfy D_N'(ω^kτ_m) = ω^-k D_N'(τ_m) and | D_N'(τ_m)| = frac2s_NNe^c_Nτ_m (1+O(e^-γτ_m)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9324acca31d7c117","tex":"e^{-\\delta r}","aliases":"e^-δ r delta","display":"e^{-\\delta r}","kind":"use","excerpt":"The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e^-δ r)) uniformly for r≥ R. In particular","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8314328de53179fc","tex":"\\e^{-\\delta t}","aliases":"e^-δ t e delta mathrm delta","display":"{\\mathrm e}^{-\\delta t}","kind":"use","excerpt":"[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-640ecb7ba05e7b8b","tex":"e^{-\\lambda s}","aliases":"e^-λ s lambda","display":"e^{-\\lambda s}","kind":"use","excerpt":"ends at s=2. It does not claim that the missing tail has zero mass. headCheck the normalizer before interpreting a density Under s=t^2/(4u^2), reversing the integration limits gives η_t(s) |ds|=frac2sqrtπe^-u^2 du. Its integral over u>0 is one, so η_t is a probability density. On each eigenmode of L, the operator formula reduces to the scalar Lapla","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0cf7bf147f70ddde","tex":"\\e^{-\\lambda t}","aliases":"e^-λ t e lambda mathrm lambda","display":"{\\mathrm e}^{-\\lambda t}","kind":"use","excerpt":"[One phase model] If A=(-λ) and p_0=1, then S(t)= e^-λ t, f(t)=λ e^-λ t, h(t)=λ.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ccc125183ac25b28","tex":"e^{-\\lambda t}","aliases":"e^-λ t lambda","display":"e^{-\\lambda t}","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ece77397bb6f22b","tex":"\\e^{-\\mu t}","aliases":"e^-μ t e mu mathrm mu","display":"{\\mathrm e}^{-\\mu t}","kind":"use","excerpt":"0, where λ≠μ are positive. The first phase moves to the second at rate λ, and the second is absorbed at rate μ. Direct calculation gives S(t)= fracμ e^-λ t-λ e^-μ tμ-λ, f(t)= fracλμμ-λ ( e^-λ t- e^-μ t). The ratio f(t)/S(t) is generally not constant. A time local Markov model on two transient phases can therefore produce a nonconstant hazard after ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11f9580358f49756","tex":"e^{-\\tau H}","aliases":"e^-τ H tau","display":"e^{-\\tau H}","kind":"use","excerpt":"vpage23Positive transfer is not a stochastic matrixSPECTRAL POSITIVITY IS BASIS INDEPENDENTChapter VII: Theorem II.10.1, Theorem II.10.3. For H=I-X, the two eigenvalues are zero and two. Its Euclidean transfer is T(τ)=e^-τ H =frac12mat1+q · 1-q 1-q · 1+q, q=e^-2τ. For τ≥0, the eigenvalues 1,q lie in (0,1], and multiplying two such matrices replaces","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0218309a1770ad77","tex":"e^{-\\tau(E_j-E_0)}","aliases":"e^-τ(E_j-E_0) tau","display":"e^{-\\tau(E_j-E_0)}","kind":"use","excerpt":"Diagonalize H=∑_jE_jP_j. Then T(τ)=∑_j e^-τ(E_j-E_0)P_j. Every eigenvalue is strictly positive and at most one. Multiplication adds the exponents, proving the semigroup law and T(0)=I. Positivity here is operator positivity, not an assertion that every matrix entry is nonnegative.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11707835147f7216","tex":"e^{-\\tau(H-E_0I)}","aliases":"e^-τ(H-E_0I) tau","display":"e^{-\\tau(H-E_0I)}","kind":"use","excerpt":"[Finite Euclidean transfer] Let H be Hermitian on a nonzero finite-dimensional Hilbert space. For all real τ,σ≥0, T(τ)=e^-τ(H-E_0I), E_0=minSpec H, is positive definite and satisfies T(τ+σ)=T(τ)T(σ). Its spectrum is contained in (0,1].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91b95fd30d4cf58a","tex":"e^{-\\tau\\lambda_j}","aliases":"e^-τλ_j tau lambda","display":"e^{-\\tau\\lambda_j}","kind":"use","excerpt":"Let L≥0 be a finite Laplacian on N≥1 states, with eigenvalues λ_j and at least one zero eigenvalue. For diffusion time τ>0, define the normalized heat trace K(τ)=frac1N∑_j e^-τλ_j and the scale-dependent spectral dimension d_s(τ) =-2fracdlog K(τ)dlogτ. For a finite system, d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differen","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50860b6739bbedbb","tex":"e^{1/4}","aliases":"e^1/4","display":"e^{1/4}","kind":"use","excerpt":"und is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields sup_0≤ t≤ T x(t)-widetilde x(t) =fracT^22≤ T^2e^T^2. At T=1/2, the exact error is 1/8 and the certificate is e^1/4/4. Nilpotence make","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed31a34ba88a7364","tex":"e^{10}","aliases":"e^10","display":"e^{10}","kind":"use","excerpt":"1/2, σ=1/50, δ=1/20, and a_min=2/25. Since log(10240)<10<(16/5)^2, choosing R=16 makes the right-hand side strictly less than frac1256+frac150frac165 =frac217332000<frac225. Thus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b87f37426ee94627","tex":"E^2","aliases":"E^2","display":"E^2","kind":"use","excerpt":"The B ezout identity gives N=rI-UF. The two stated commutation relations and FG=0 give N(UF)=VGUF=VUGF=VUFG=0. Therefore N^2=N(rI-UF)=rN. Since r≠0, E^2=r^-2N^2=E. If Fx=0, the evaluated B ezout identity gives VGx=rx, hence Ex=x. If Gx=0, then Ex=0. The lattice assertion follows directly from rE=VG.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97fed77197e69d14","tex":"\\mathcal E^2","aliases":"E^2 mathcal","display":"\\mathcal E^2","kind":"use","excerpt":"dependent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an indep","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89d50d3d0bc30bea","tex":"e^2","aliases":"e^2","display":"e^2","kind":"use","excerpt":"Let e^2=e. Since e+(1-e)=1, at least one of e and 1-e is a unit. If e is a unit, then e=1. If 1-e is a unit, then e=0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40fcc5d6ba51777a","tex":"e^{2is_N(t_m+\\delta_m)}","aliases":"e^2is_N(t_m+δ_m) delta","display":"e^{2is_N(t_m+\\delta_m)}","kind":"use","excerpt":"Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-426085442b3888cc","tex":"e^{2is_Nt_m}","aliases":"e^2is_Nt_m","display":"e^{2is_Nt_m}","kind":"use","excerpt":"Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-298655f0e488073a","tex":"e^{2is_Nz}","aliases":"e^2is_Nz","display":"e^{2is_Nz}","kind":"use","excerpt":"η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a99de0abb54b18b7","tex":"e^{2is_N\\tau_m}","aliases":"e^2is_Nτ_m tau","display":"e^{2is_N\\tau_m}","kind":"use","excerpt":"Differentiate the normalized expression. At z=τ_m, G'(τ_m) = 2is_Ne^2is_Nτ_m+R_N'(τ_m) = -2is_N+O(e^-γ_Nτ_m). Since G(τ_m)=0, D_N'(τ_m) = frac1N e^e^-ia_Nτ_mG'(τ_m). Taking absolute values gives the derivative asymptotic. Differentiating D_N(ω z)= D_N(z) gives the rotation formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2f438829b64499b","tex":"e^{2pt}","aliases":"e^2pt","display":"e^{2pt}","kind":"use","excerpt":"The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconst","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-05426a5349dc70a7","tex":"\\e^{2qt}","aliases":"e^2qt e mathrm","display":"{\\mathrm e}^{2qt}","kind":"use","excerpt":"The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6d6d7824dcc408b","tex":"\\e^{2t}","aliases":"e^2t e mathrm","display":"{\\mathrm e}^{2t}","kind":"use","excerpt":"The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-499fd10d84ddfd9f","tex":"e^{2t}","aliases":"e^2t","display":"e^{2t}","kind":"use","excerpt":"The variable B=2Z+1 has transform e^tfrac2te^2t-1=fractsinh t. Its odd moments vanish, and P=( B^2-1)/4=Z(Z+1) has the divided-power moments of Lemma~[source reference]. Introduce an independent copy D and Q=( D^2-1)/4. The exact factorization F_2(t)=fract^2cosh t-cos t =fracatsinh(at)fracbtsinh(bt) shows that U=a B+b D satisfies L(U^4n)=C_2(n). Di","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d86f0c042cad439b","tex":"e^{2Y}","aliases":"e^2Y","display":"e^{2Y}","kind":"use","excerpt":"ng strip, the roots e^ia_N and e^-ia_N have real part c_N, while each remaining root η=u+iv satisfies u≤cos(3a_N)=c_N-γ_N. Hence Re((η-e^-ia_N)z) ≤-γ_Nx+2Y. Division by e^e^-ia_Nz gives the displayed normalization. Each differentiation multiplies a remainder term by η-e^-ia_N, whose modulus is at most two. Thus, for q=0,1,2, |R_N^(q)(z)|≤(N-2)2^qe^","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4808578a2aed1855","tex":"\\e^{2\\pi i/M}","aliases":"e^2π i/M e pi mathrm pi","display":"{\\mathrm e}^{2\\pi i/M}","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7e7084a35eacad9","tex":"e^{2\\pi i/N}","aliases":"e^2π i/N pi","display":"e^{2\\pi i/N}","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bd6cafb4982926b","tex":"\\e^{2\\pi iz/p}","aliases":"e^2π iz/p e pi mathrm pi","display":"{\\mathrm e}^{2\\pi iz/p}","kind":"use","excerpt":"Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b52fa662abced8d7","tex":"\\e^{2\\pi\\ii k/N}","aliases":"e^2πii k/N e^2π i k/N e pi ii mathrm pi mathrm","display":"{\\mathrm e}^{2\\pi{\\mathrm i} k/N}","kind":"use","excerpt":"+1) into C^-1 and simplify. Preservation of H follows because C maps it to D, rotation preserves D, and C^-1 maps back. The group law follows by conjugating the rotation group law. When θ=2π/N, the Nth power is the identity. Conversely, if its kth power is the identity, conjugation by C and evaluation at w=1/2 give e^2π i k/N=1, so N divides k. Hen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7013ba8d76e4f86","tex":"\\e^{2\\pi\\ii/N}","aliases":"e^2πii/N e^2π i/N e pi ii mathrm pi mathrm","display":"{\\mathrm e}^{2\\pi{\\mathrm i}/N}","kind":"use","excerpt":"Let N≥1 be an integer, S unitary with S^N=I, and ω= e^2π i/N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b42132942b4d3093","tex":"e^{2\\pi\\ii/N}","aliases":"e^2πii/N e^2π i/N pi ii pi mathrm","display":"e^{2\\pi{\\mathrm i}/N}","kind":"use","excerpt":"Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2159f21c196fa256","tex":"e^{2\\tau}","aliases":"e^2τ tau","display":"e^{2\\tau}","kind":"use","excerpt":"vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESChapter VII: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02621fd1f38489c4","tex":"e^{a_\\ell z}","aliases":"e^a_ℓ z ell","display":"e^{a_\\ell z}","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93314c38514276c7","tex":"\\e^{at}","aliases":"e^at e mathrm","display":"{\\mathrm e}^{at}","kind":"use","excerpt":"c · b and let the two coordinate axes be the packet spaces. If a≠ b, then e^tA = e^at · 0 displaystyle cfrac e^at- e^bta-b · e^bt . The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with ct, in agreement with [source reference]. When a=b, the same block is ct e^at.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-daab3aa19b753da2","tex":"\\e^{bt}","aliases":"e^bt e mathrm","display":"{\\mathrm e}^{bt}","kind":"use","excerpt":"c · b and let the two coordinate axes be the packet spaces. If a≠ b, then e^tA = e^at · 0 displaystyle cfrac e^at- e^bta-b · e^bt . The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with ct, in agreement with [source reference]. When a=b, the same block is ct e^at.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d43f91811ba8378c","tex":"e^{c_Nr}","aliases":"e^c_Nr","display":"e^{c_Nr}","kind":"use","excerpt":"At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry tr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e372850a7382fa9","tex":"e^{c_N\\tau_m}","aliases":"e^c_Nτ_m tau","display":"e^{c_N\\tau_m}","kind":"use","excerpt":"The derivatives satisfy D_N'(ω^kτ_m) = ω^-k D_N'(τ_m) and | D_N'(τ_m)| = frac2s_NNe^c_Nτ_m (1+O(e^-γτ_m)).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bda629913237771d","tex":"\\mathcal E^d","aliases":"E^d mathcal","display":"\\mathcal E^d","kind":"use","excerpt":"Let L be the ordinary Euler evaluation functional on a variable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear. It is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr E","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49dd531cc109d91e","tex":"e^{e^{-ia_N}z}","aliases":"e^e^-ia_Nz","display":"e^{e^{-ia_N}z}","kind":"use","excerpt":"On the resulting strip, the roots e^ia_N and e^-ia_N have real part c_N, while each remaining root η=u+iv satisfies u≤cos(3a_N)=c_N-γ_N. Hence Re((η-e^-ia_N)z) ≤-γ_Nx+2Y. Division by e^e^-ia_Nz gives the displayed normalization. Each differentiation multiplies a remainder term by η-e^-ia_N, whose modulus is at most two. Thus, for q=0,1,2, |R_N^(q)(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6550b2960b333a4","tex":"e^{e^{-ia_N}\\tau_m}","aliases":"e^e^-ia_Nτ_m tau","display":"e^{e^{-ia_N}\\tau_m}","kind":"use","excerpt":"Differentiate the normalized expression. At z=τ_m, G'(τ_m) = 2is_Ne^2is_Nτ_m+R_N'(τ_m) = -2is_N+O(e^-γ_Nτ_m). Since G(τ_m)=0, D_N'(τ_m) = frac1N e^e^-ia_Nτ_mG'(τ_m). Taking absolute values gives the derivative asymptotic. Differentiating D_N(ω z)= D_N(z) gives the rotation formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36cd39f69725f663","tex":"e^{g(z)}","aliases":"e^g(z)","display":"e^{g(z)}","kind":"use","excerpt":"For a function g, define the complete differential Bell polynomial Y_N by fracd^Ndz^Ne^g(z) = e^g(z)Y_N(g',g'',…,g^(N)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d861b22faf64eef3","tex":"e^{ia_N}","aliases":"e^ia_N","display":"e^{ia_N}","kind":"use","excerpt":"On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8618817d79a78398","tex":"\\e^{\\ii d\\omega}","aliases":"e^ii dω e^ i dω e ii omega mathrm mathrm omega","display":"{\\mathrm e}^{{\\mathrm i} d\\omega}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76a85d33638c39e7","tex":"\\e^{\\ii t_kD}","aliases":"e^ii t_kD e^ i t_kD e ii mathrm mathrm","display":"{\\mathrm e}^{{\\mathrm i} t_kD}","kind":"use","excerpt":"Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0fa91de62b7d1419","tex":"e^{\\ii t_kD}","aliases":"e^ii t_kD e^ i t_kD ii mathrm","display":"e^{{\\mathrm i} t_kD}","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec5335d70f84b91e","tex":"\\e^{\\ii(\\varphi+\\theta)}","aliases":"e^ii(φ+θ) e^ i(φ+θ) e ii varphi theta mathrm mathrm varphi theta","display":"{\\mathrm e}^{{\\mathrm i}(\\varphi+\\theta)}","kind":"use","excerpt":"[Radial and phase coordinates] Write C(z)=ρ e^ iφ, 0≤ρ<1. Then C(Φ_θ(z)) = ρ e^ i(φ+θ). The radial mismatch ρ is invariant and the phase coordinate advances by θ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-638797be2d36d412","tex":"\\e^{\\ii\\theta}","aliases":"e^iiθ e^ iθ e ii theta mathrm mathrm theta","display":"{\\mathrm e}^{{\\mathrm i}\\theta}","kind":"use","excerpt":"Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18321ae54a5eac88","tex":"e^{\\ii\\theta}","aliases":"e^iiθ e^ iθ ii theta mathrm theta","display":"e^{{\\mathrm i}\\theta}","kind":"use","excerpt":"For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56b7423ff854480a","tex":"\\e^{\\ii\\pi}","aliases":"e^iiπ e^ iπ e ii pi mathrm mathrm pi","display":"{\\mathrm e}^{{\\mathrm i}\\pi}","kind":"use","excerpt":"Insert e^ iπ=-1 in the formula of [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a246bb571b3cd9f5","tex":"\\e^{\\ii\\varphi}","aliases":"e^iiφ e^ iφ e ii varphi mathrm mathrm varphi","display":"{\\mathrm e}^{{\\mathrm i}\\varphi}","kind":"use","excerpt":"[Radial and phase coordinates] Write C(z)=ρ e^ iφ, 0≤ρ<1. Then C(Φ_θ(z)) = ρ e^ i(φ+θ). The radial mismatch ρ is invariant and the phase coordinate advances by θ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-256b204813b53eee","tex":"e^{\\ii\\phi}","aliases":"e^iiϕ e^ iϕ ii phi mathrm phi","display":"e^{{\\mathrm i}\\phi}","kind":"use","excerpt":"n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. T","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afdbf103ea74d6de","tex":"\\e^{\\ii\\omega}","aliases":"e^iiω e^ iω e ii omega mathrm mathrm omega","display":"{\\mathrm e}^{{\\mathrm i}\\omega}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b229ce68227bc751","tex":"e^{ik\\pi/4}","aliases":"e^ikπ/4 pi","display":"e^{ik\\pi/4}","kind":"use","excerpt":"A NONLINEAR PHASE ORBITChapter II, mathematical text: Section 12.3, Smith-phase identity (p. 43) Smith normal form concerns linear congruences. The Smith impedance chart uses the nonlinear coordinate change Γ=fracz-1z+1, z=frac1+Γ1-Γ. They share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points Γ_k=tf","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee4422570e8d67bf","tex":"e^{isH}","aliases":"e^isH","display":"e^{isH}","kind":"use","excerpt":"Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ccbd9d6591d0caa8","tex":"e^{itH}","aliases":"e^itH","display":"e^{itH}","kind":"use","excerpt":"Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-682a47b2eea70d50","tex":"e^{i\\theta}","aliases":"e^iθ theta","display":"e^{i\\theta}","kind":"use","excerpt":"[Indicator polygon and dominant sectors] The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4d0d3ff8a81318d","tex":"e^{L(z)}","aliases":"e^L(z)","display":"e^{L(z)}","kind":"use","excerpt":"vpage13A complex power needs a domainBRANCH BOOKKEEPINGChapter IV: Section 2, Definition 7.1, Corollary 8.4. A formal unit series has a formal logarithm at the origin. To interpret that germ analytically, choose a simply connected zero-free region containing 0 and normalize the logarithm to vanish there. D_N(z)^-r=exp(-rL(z)), e^L(z)=D_N(z), L(0)=0","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ffe86bb88f85c2e","tex":"e^{L+\\pi i}","aliases":"e^L+π i pi","display":"e^{L+\\pi i}","kind":"use","excerpt":"In fact, the constants ε_M=1/2 and C_M=3/2 work for every complex w, without the strip restriction. Put u=-(1+e^w) and L=log(1+u)=log(-e^w), using the principal logarithm. The elementary logarithm estimate on |u|≤1/2 gives |L|≤(3/2)|u|. Since e^L+π i=e^w, there is an integer m such that w=L+(2m+1)π i. This gives the required distance bound. Distinc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1009899abf2147a9","tex":"E^{n-1}","aliases":"E^n-1","display":"E^{n-1}","kind":"use","excerpt":"The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c244c5ba6bf72e0","tex":"e^{Q/8}","aliases":"e^Q/8","display":"e^{Q/8}","kind":"use","excerpt":"For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06dfc486c3523030","tex":"\\e^{qt}","aliases":"e^qt e mathrm","display":"{\\mathrm e}^{qt}","kind":"use","excerpt":"[Odd source generating function] For every positive odd q, ∑_n≥0S_n(q)fract^nn! = e^qtcosh(qt)secht. Its even part is ∑_n≥0S_2n(q)fract^2n(2n)! = cosh^2(qt)secht.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-251aa8cc2c01364f","tex":"e^{rA_f}","aliases":"e^rA_f","display":"e^{rA_f}","kind":"use","excerpt":"[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-275ed3fe872da008","tex":"\\e^{\\operatorname{Re}\\mu_jt}","aliases":"e^Reμ_jt e operatorname mu mathrm operatorname mu","display":"{\\mathrm e}^{\\operatorname{Re}\\mu_jt}","kind":"use","excerpt":"esolvent expansion is (zI-A_QQ)^-1Q_j = ∑_k=0^ν_j-1 fracN_j^kQ_j(z-μ_j)^k+1. Multiplication by the coupling blocks gives the self-energy formula for every z outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity ∫_0^∞ e^-zt e^μ_jt fract^kk! d t = frac1(z-μ_j)^k+1. Under stability, ∫_0^∞ e^Reμ_jt fract^","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f85c8b097ee2545","tex":"e^{r\\Re(\\eta e^{i\\theta})}","aliases":"e^rRe(η e^iθ) Re eta theta","display":"e^{r\\Re(\\eta e^{i\\theta})}","kind":"use","excerpt":"vpage17Growth comes from a polygonDOMINANT EXPONENTSChapter IV: Definition 11.3, Theorem 11.4. Write the signed denominator as a finite exponential sum: D_N(z)=frac1N∑_η^N=-1e^η z. For z=re^iθ, the size of one exponential is e^rRe(η e^iθ). The maximal projection determines the directional exponential growth. vfigindicator The indicator is h_N(θ)=ma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6a1b00b70a9b7d0","tex":"e^{rz}","aliases":"e^rz","display":"e^{rz}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39fc7735a3cc982f","tex":"\\e^{s_1D}","aliases":"e^s_1D e mathrm","display":"{\\mathrm e}^{s_1D}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04db6340c88fba8b","tex":"\\e^{sA}","aliases":"e^sA e mathrm","display":"{\\mathrm e}^{sA}","kind":"use","excerpt":"Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21367aaf2687869b","tex":"e^{sA}","aliases":"e^sA","display":"e^{sA}","kind":"use","excerpt":"[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33aedb64301f3993","tex":"\\e^{sA_f}","aliases":"e^sA_f e mathrm","display":"{\\mathrm e}^{sA_f}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e256d144bb836c0b","tex":"\\e^{sg}","aliases":"e^sg e mathrm","display":"{\\mathrm e}^{sg}","kind":"use","excerpt":"The semigroup law follows from multiplication of the likelihoods e^sg and e^tg. Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31d4d3c24349f5c5","tex":"e^{sg}","aliases":"e^sg","display":"e^{sg}","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c268d8002e5f66f","tex":"e^{sQ}","aliases":"e^sQ","display":"e^{sQ}","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-098a771b0be43d72","tex":"e^{sT_f}","aliases":"e^sT_f","display":"e^{sT_f}","kind":"use","excerpt":"[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-478999b366cf71dd","tex":"\\e^t","aliases":"e^t e mathrm","display":"{\\mathrm e}^t","kind":"use","excerpt":"The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9888ce0a8ae50f0","tex":"\\e^{t(B+C)}","aliases":"e^t(B+C) e mathrm","display":"{\\mathrm e}^{t(B+C)}","kind":"use","excerpt":"Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1500c5980f1fb9cc","tex":"e^{t(B-zI)}","aliases":"e^t(B-zI)","display":"e^{t(B-zI)}","kind":"use","excerpt":"Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace. Equivalently, apply the decomposition to left multiplication by B on the matrix algebra a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b789f046b6d1784","tex":"\\e^{t(g_i-g_j)}","aliases":"e^t(g_i-g_j) e mathrm","display":"{\\mathrm e}^{t(g_i-g_j)}","kind":"use","excerpt":"[Bayesian flow and replicator equation] The maps Φ_t form a nonlinear semigroup: Φ_s+t=Φ_s∘Φ_t. Every coordinate satisfies frac d d tp_i(t) = (g_i-ipgp(t))p_i(t). The support of p(t) is independent of t, and for any two coordinates in the support, fracp_i(t)p_j(t) = e^t(g_i-g_j) fracp_i(0)p_j(0).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe2f9fcc69678a23","tex":"\\e^{t(\\overline Q^T-\\overline V)}","aliases":"e^t(overline Q^T-overline V) e overline overline mathrm overline overline","display":"{\\mathrm e}^{t(\\overline Q^T-\\overline V)}","kind":"use","excerpt":"[Coarse Feynman--Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ce15abfc0ced257","tex":"\\e^{t(Q+aI)}","aliases":"e^t(Q+aI) e mathrm","display":"{\\mathrm e}^{t(Q+aI)}","kind":"use","excerpt":"Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d049e66d5d6db04b","tex":"\\e^{t(Q-D_V)}","aliases":"e^t(Q-D_V) e mathrm","display":"{\\mathrm e}^{t(Q-D_V)}","kind":"use","excerpt":"Average [source reference] with terminal function f= 1 against the initial distribution. Transposition gives p_0^T e^t(Q^T-D_V) 1 = 1^T e^t(Q-D_V)p_0. Since 1^TQ=0, differentiation gives S_V'(t) = - 1^TD_V e^tAp_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46b9991988acc297","tex":"\\e^{t(Q^T-D_V)}","aliases":"e^t(Q^T-D_V) e mathrm","display":"{\\mathrm e}^{t(Q^T-D_V)}","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-219930913ca9cfcf","tex":"\\e^{t(Q^T-V)}","aliases":"e^t(Q^T-V) e mathrm","display":"{\\mathrm e}^{t(Q^T-V)}","kind":"use","excerpt":"Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5523b5f204f917d","tex":"\\e^{t(Q^T-W)}","aliases":"e^t(Q^T-W) e mathrm","display":"{\\mathrm e}^{t(Q^T-W)}","kind":"use","excerpt":"Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-728cabc55310db85","tex":"\\e^{t(QAQ)|_{QX}}","aliases":"e^t(QAQ)|_QX e mathrm","display":"{\\mathrm e}^{t(QAQ)|_{QX}}","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-315c2026c72e593c","tex":"e^{t/2}","aliases":"e^t/2","display":"e^{t/2}","kind":"use","excerpt":"vpage18Two large terms nearly cancelTHE CUBIC MODELChapter IV: Lemma 11.5, Theorem 11.7, Corollary 11.9. For real t, order three has the exact formula D_3(t)=frace^-t+2e^t/2cos(sqrt3t/2)3. Divide by the nonzero factor 2e^t/2/3. The resulting equation separates the dominant cosine from the small third exponential. vfigtwo_term Ignoring the last term","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c95014e23d49b17","tex":"e^{T^2}","aliases":"e^T^2","display":"e^{T^2}","kind":"use","excerpt":"trajectory platecaptionThe first-coordinate error is exactly t^2/2. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bc0eec613816bce","tex":"e^{tA}","aliases":"e^tA","display":"e^{tA}","kind":"use","excerpt":"Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usu","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-053701566d46fdc2","tex":"\\e^{tA_c}","aliases":"e^tA_c e mathrm","display":"{\\mathrm e}^{tA_c}","kind":"use","excerpt":"Let X_f and X_c be complex Banach spaces. Let A_f∈ B(X_f), A_c∈ B(X_c), and C∈ B(X_f,X_c). Define Δ=CA_f-A_cC and D(t)=C e^tA_f- e^tA_cC.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee1b64c4f1868d3a","tex":"e^{tA_c}","aliases":"e^tA_c","display":"e^{tA_c}","kind":"use","excerpt":"Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d95d42808445044","tex":"\\e^{tA_f}","aliases":"e^tA_f e mathrm","display":"{\\mathrm e}^{tA_f}","kind":"use","excerpt":"Let X_f and X_c be complex Banach spaces. Let A_f∈ B(X_f), A_c∈ B(X_c), and C∈ B(X_f,X_c). Define Δ=CA_f-A_cC and D(t)=C e^tA_f- e^tA_cC.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8bef86072190de27","tex":"e^{tA_f}","aliases":"e^tA_f","display":"e^{tA_f}","kind":"use","excerpt":"Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9f776015ddefc19","tex":"\\e^{tA_n}","aliases":"e^tA_n e mathrm","display":"{\\mathrm e}^{tA_n}","kind":"use","excerpt":"[A usable closure condition for refinement towers] Assume the contractive summability hypothesis of [source reference] holds on every bounded time interval. Suppose every P_m,n is an orthogonal projection on a Hilbert space and T_n(t)= e^tA_n is contractive. If, for every fixed m, norm(I-P_m,n)A_nP_m,n ⟶0, then the limiting family U_m is a semigrou","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6221765d6dbc4c15","tex":"\\e^{tA_{PP}}","aliases":"e^tA_PP e mathrm","display":"{\\mathrm e}^{tA_{PP}}","kind":"use","excerpt":"[Criterion for exact local closure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93f0045082a837ac","tex":"\\e^{tA_{QQ}}","aliases":"e^tA_QQ e mathrm","display":"{\\mathrm e}^{tA_{QQ}}","kind":"use","excerpt":"For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0)","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2db33b004054b96","tex":"e^{tA_{QQ}}","aliases":"e^tA_QQ","display":"e^{tA_{QQ}}","kind":"use","excerpt":"For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b45b0921ad704da0","tex":"\\e^{tA_{\\alpha\\alpha}}","aliases":"e^tA_αα e alpha alpha mathrm alpha alpha","display":"{\\mathrm e}^{tA_{\\alpha\\alpha}}","kind":"use","excerpt":"thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3b1c48fbafaa657","tex":"\\e^{tA_\\ell}","aliases":"e^tA_ℓ e ell mathrm ell","display":"{\\mathrm e}^{tA_\\ell}","kind":"use","excerpt":"If T_ℓ(t)= e^tA_ℓ, the generators are bounded, and sup_ℓnormA_ℓ<∞, C_ℓ A_ℓ+1=A_ℓ C_ℓ, then the continuity condition holds and A_∞(x_ℓ)_ℓ=(A_ℓ x_ℓ)_ℓ is the bounded generator of T_∞.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6c1a6db623e88fe","tex":"e^{tA_\\ell}","aliases":"e^tA_ℓ ell","display":"e^{tA_\\ell}","kind":"use","excerpt":"[Uniform refinement control] For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that normC_ℓ e^tA_ℓ+1-e^tA_ℓC_ℓ ≤ M t normC_ℓ A_ℓ+1-A_ℓ C_ℓ on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-205d9e45d3bf3ac2","tex":"e^{tA_{\\ell+1}}","aliases":"e^tA_ℓ+1 ell","display":"e^{tA_{\\ell+1}}","kind":"use","excerpt":"[Uniform refinement control] For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that normC_ℓ e^tA_ℓ+1-e^tA_ℓC_ℓ ≤ M t normC_ℓ A_ℓ+1-A_ℓ C_ℓ on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5c432074c8b99aa","tex":"\\e^{tB}","aliases":"e^tB e mathrm","display":"{\\mathrm e}^{tB}","kind":"use","excerpt":"pace, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus e^-zt e^tB is ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43c981a8b54a5fb2","tex":"e^{tB}","aliases":"e^tB","display":"e^{tB}","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e54061964f4faacf","tex":"\\e^{tB/n}","aliases":"e^tB/n e mathrm","display":"{\\mathrm e}^{tB/n}","kind":"use","excerpt":"Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dbe7c874c164d2d5","tex":"e^{t\\bar Q}","aliases":"e^tbar Q bar","display":"e^{t\\bar Q}","kind":"use","excerpt":"2 · -2. The Duhamel identity therefore gives Ce^tQ=e^tbar QC. This is a second consequence of the corresponding generator identity, not an inference that K is a fixed-time sample of Q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a740f49548879451","tex":"\\e^{tC/n}","aliases":"e^tC/n e mathrm","display":"{\\mathrm e}^{tC/n}","kind":"use","excerpt":"Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ef82f33e2b0d0d2","tex":"\\e^{tD}","aliases":"e^tD e mathrm","display":"{\\mathrm e}^{tD}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b6d0b4fad11116f","tex":"e^{tD}","aliases":"e^tD","display":"e^{tD}","kind":"use","excerpt":"A finite auxiliary realization of a matrix kernel means K(t)=B e^tDC, with finite matrices of compatible sizes. Laplace transforms are taken in a right half-plane of absolute convergence. The rational characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies. [Rational memory realizat","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-179eee2354eefa9f","tex":"\\e^{tG}","aliases":"e^tG e mathrm","display":"{\\mathrm e}^{tG}","kind":"use","excerpt":"Factor e^tG from the normalizing constant. The total weight outside M is at most (1-p_M(0)) e^-δ t, while the weight on M is p_M(0).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-767456b9db25e846","tex":"\\e^{tg}","aliases":"e^tg e mathrm","display":"{\\mathrm e}^{tg}","kind":"use","excerpt":"The semigroup law follows from multiplication of the likelihoods e^sg and e^tg. Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cceac532c75d0672","tex":"e^{tG}","aliases":"e^tG","display":"e^{tG}","kind":"use","excerpt":"1 · -1. The total-mass direction (1,1) has eigenvalue zero. The contrast (1,-1) has eigenvalue -2. Resolving those two directions gives the full transfer, not only its first-order approximation: T(t)=e^tG=frac121+e^-2t · 1-e^-2t 1-e^-2t · 1+e^-2t. vfigclock platecaptionA point seed at state zero approaches the uniform distribution. The displayed ti","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbc19376502612fc","tex":"e^{tg}","aliases":"e^tg","display":"e^{tg}","kind":"use","excerpt":"The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92b24ebf8d5b2479","tex":"\\e^{tg_i}","aliases":"e^tg_i e mathrm","display":"{\\mathrm e}^{tg_i}","kind":"use","excerpt":"Fix a score vector g∈ R^n. For t≥0, define Φ_t(p)_i = frac e^tg_ip_i ∑_j e^tg_jp_j.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e4db77f7062a650","tex":"e^{tg_i}","aliases":"e^tg_i","display":"e^{tg_i}","kind":"use","excerpt":"Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-84cd29a204931e68","tex":"\\e^{tg_j}","aliases":"e^tg_j e mathrm","display":"{\\mathrm e}^{tg_j}","kind":"use","excerpt":"Fix a score vector g∈ R^n. For t≥0, define Φ_t(p)_i = frac e^tg_ip_i ∑_j e^tg_jp_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c3843aff1b9a844","tex":"e^{tg_j}","aliases":"e^tg_j","display":"e^{tg_j}","kind":"use","excerpt":"Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f2de95465372652","tex":"\\e^{t\\Gcal}","aliases":"e^tGcal e^t G e Gcal mathrm mathcal","display":"{\\mathrm e}^{t{\\mathcal G}}","kind":"use","excerpt":"a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31529ccb041f73ad","tex":"e^{tJ}","aliases":"e^tJ","display":"e^{tJ}","kind":"use","excerpt":"0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponential is not e^-2tI: e^tJ=e^-2t(I+tN), K_rm mem(t)=Ue^tJV=t e^-2t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e7ca070f23ef9e5b","tex":"e^{tL}","aliases":"e^tL","display":"e^{tL}","kind":"use","excerpt":"L · A nonnegative self-adjoint operator. · Its evolution is e^-tL, not e^tL.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1a818910db29e8b","tex":"\\e^{tM_A}","aliases":"e^tM_A e mathrm","display":"{\\mathrm e}^{tM_A}","kind":"use","excerpt":"For bounded generators with common bound M_A, sup_ℓ norm( e^tA_ℓ-I)x_ℓ ≤ ( e^tM_A-1)sup_ℓnormx_ℓ. This tends to zero with t. Exact generator intertwining makes (A_ℓ x_ℓ)_ℓ compatible, and coordinatewise differentiation identifies the generator.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc1711b5d9da8839","tex":"\\e^{t\\norm A}","aliases":"e^tnorm A e norm mathrm norm","display":null,"kind":"use","excerpt":"The estimate normT(t)≤ e^tnorm A gives absolute convergence of the operator valued integral. Let R(z)=∫_0^∞ e^-ztT(t) d t. Since T'(t)=AT(t)=T(t)A, integration by parts gives (zI-A)R(z) = -∫_0^∞ frac d d t ( e^-ztT(t)) d t =I. The same calculation on the other side gives R(z)(zI-A)=I . Multiplication by P_β and P_α may be passed under the norm conv","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-879f489b7b741e9c","tex":"\\e^{t\\norm D}","aliases":"e^tnorm D e norm mathrm norm","display":null,"kind":"use","excerpt":"Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb2c45501b486ab2","tex":"\\e^{t\\overline Q}","aliases":"e^toverline Q e overline mathrm overline","display":"{\\mathrm e}^{t\\overline Q}","kind":"use","excerpt":"[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e64b2c7d61b9945a","tex":"\\e^{tQ}","aliases":"e^tQ e mathrm","display":"{\\mathrm e}^{tQ}","kind":"use","excerpt":"Markov chain · K^n or e^tQ · equality of aggregated transition masses","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-084ac283f910faff","tex":"e^{tQ}","aliases":"e^tQ","display":"e^{tQ}","kind":"use","excerpt":"Use column probability vectors. A finite continuous-time Markov generator Q has nonnegative off-diagonal entries and column sums zero. The stochastic semigroup is K(t)=e^tQ.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4ffd068bc2c7120","tex":"\\e^{tQ^T/n}","aliases":"e^tQ^T/n e mathrm","display":"{\\mathrm e}^{tQ^T/n}","kind":"use","excerpt":"[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c4a586d94fa15a1","tex":"e^{tT_c}","aliases":"e^tT_c","display":"e^{tT_c}","kind":"use","excerpt":"[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6096747ae971d30d","tex":"e^{tT_f}","aliases":"e^tT_f","display":"e^{tT_f}","kind":"use","excerpt":"[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30d533e8bb13e1b1","tex":"e^{tu}","aliases":"e^tu","display":"e^{tu}","kind":"use","excerpt":"vpage36The score rises because it has varianceBAYESIAN FLOW / EVIDENCE CUMULANTSChapter VI, Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior. Start with equal prior masses on scores u=(0,1,2) and likelihood e^tu. Put q=e^t. The evidence and tilted distribution are Z(t)=frac1+q+q^23, p(t)=frac(1,q,q^2)1+q+q^2. At t=log2","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91eff23c3d96800f","tex":"e^w","aliases":"e^w","display":"e^w","kind":"use","excerpt":"[Near cancellation of two exponentials] For every M>0, there are constants ε_M>0 and C_M>0 such that the following holds. If |Re w|≤ M and |1+e^w|≤ε_M, then there is a unique integer m with |w-(2m+1)π i| ≤ C_M|1+e^w|.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1a9e2520fdcb9c4","tex":"e^{w\\xi}","aliases":"e^wξ xi","display":"e^{w\\xi}","kind":"use","excerpt":"Differentiation multiplies by λξ in the first algebra and by ξ in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity e^(z+w)ξ=e^zξe^wξ gives the addition laws after comparison of the coefficients of 1,ξ,…,ξ^N-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec666cade8ac0993","tex":"e^{yz}","aliases":"e^yz","display":"e^{yz}","kind":"use","excerpt":"Differentiate the generating function with respect to x for the first identity. Multiply it by e^yz and compare coefficients for the second.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8611063e24d14e09","tex":"e^z","aliases":"e^z","display":"e^z","kind":"use","excerpt":"[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4083b97ac93b9cef","tex":"e^{z/2}","aliases":"e^z/2","display":"e^{z/2}","kind":"use","excerpt":"The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3eefb925a82433a8","tex":"e^{zH}","aliases":"e^zH","display":"e^{zH}","kind":"use","excerpt":"Symmetry and measurement. Exact symmetry: source Theorem 5.1. If HP_a=P_aH for every packet, then P_bHP_a=0 (b≠ a), H=∑_aP_aHP_a. In a complex Banach algebra, commutation also propagates to the actual exponential: e^zHP_a=P_ae^zH (z∈ C). Consequently, e^zH=∑_aP_ae^zHP_a, P_be^zHP_a=0 (b≠ a). Taking z=-it gives evolution in real time. The Lean proof","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d7952160b64cde8","tex":"e^{zJ_3}","aliases":"e^zJ_3","display":"e^{zJ_3}","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9d76e69865d2de6","tex":"e^{zJ_N}","aliases":"e^zJ_N","display":"e^{zJ_N}","kind":"use","excerpt":"D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1ae99d7b02428e0","tex":"e^{z\\tr J_N}","aliases":"e^ztr J_N tr operatorname","display":"e^{z{\\operatorname{tr}} J_N}","kind":"use","excerpt":"The first two identities are the exponential group law. Since tr J_N=0, det(e^zJ_N)=e^ztr J_N=1. The first column statement is [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e062ee845109b60","tex":"e^{z\\xi}","aliases":"e^zξ xi","display":"e^{z\\xi}","kind":"use","excerpt":"[Finite algebra exponential identities] In A_N^+ one has e^λ zξ = ∑_j=0^N-1 E_N,j(λ z)ξ^j. In A_N^- one has e^zξ = ∑_j=0^N-1 D_N,j(z)ξ^j. Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4a231173f4ee3fe","tex":"e^{z\\sum_\\ell a_\\ell}","aliases":"e^z∑_ℓ a_ℓ sum ell ell","display":"e^{z\\sum_\\ell a_\\ell}","kind":"use","excerpt":"igin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), bec","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8a41de9330886f7","tex":"e^{|z|}","aliases":"e^|z|","display":"e^{|z|}","kind":"use","excerpt":"The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d93aff4d4f9a952d","tex":"\\e^{\\beta t}","aliases":"e^β t e beta mathrm beta","display":"{\\mathrm e}^{\\beta t}","kind":"use","excerpt":"pace, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus e^-zt e^tB is ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bed1356e32c52598","tex":"e^{\\eta z}","aliases":"e^η z eta","display":"e^{\\eta z}","kind":"use","excerpt":"The second half of this chapter concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's mon","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a7300563f5018c4","tex":"e^{\\eta_0z}","aliases":"e^η_0z eta","display":"e^{\\eta_0z}","kind":"use","excerpt":"The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c415fcc7d5848df","tex":"e^{\\eta_kz}","aliases":"e^η_kz eta","display":"e^{\\eta_kz}","kind":"use","excerpt":"We now assume N≥2. Write η_k=e^(2k+1)π i/N, 0≤ k<N. These are the vertices of a regular polygon and D_N(z)=frac1N∑_k=0^N-1e^η_kz.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-605f0e10ae7d823b","tex":"e^{\\lambda w}","aliases":"e^λ w lambda","display":"e^{\\lambda w}","kind":"use","excerpt":"Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5dd1ba005cf62b45","tex":"e^{\\lambda z\\xi}","aliases":"e^λ zξ lambda xi","display":"e^{\\lambda z\\xi}","kind":"use","excerpt":"[Finite algebra exponential identities] In A_N^+ one has e^λ zξ = ∑_j=0^N-1 E_N,j(λ z)ξ^j. In A_N^- one has e^zξ = ∑_j=0^N-1 D_N,j(z)ξ^j. Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04da8d8bbc069bd5","tex":"e^{\\lambda(z+w)}","aliases":"e^λ(z+w) lambda","display":"e^{\\lambda(z+w)}","kind":"use","excerpt":"Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff20cbfa7862a04f","tex":"\\e^{\\lambda_jt}","aliases":"e^λ_jt e lambda mathrm lambda","display":"{\\mathrm e}^{\\lambda_jt}","kind":"use","excerpt":"[Exponential sum memory] If K(t)=∑_j=1^mB_j e^λ_jtC_j, then K has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in t and are realized by Jordan blocks.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ca1e8040f46f1da","tex":"e^{\\lambda\\eta}","aliases":"e^λη lambda eta","display":"e^{\\lambda\\eta}","kind":"use","excerpt":"vpage36Repeated observations shrink the noise side of the gapSUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGETChapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11. Suppose all noise observations are independent and centered, with moment-generating bound E e^λη≤ e^σ^2λ^2/2. Averaging R observations at each of n+1 sample sites replaces σ by σ/s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac72f3950f989280","tex":"\\e^{\\lambda\\eta_k}","aliases":"e^λη_k e lambda eta mathrm lambda eta","display":"{\\mathrm e}^{\\lambda\\eta_k}","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97bcf9ce563a30aa","tex":"\\e^{\\mu_jt}","aliases":"e^μ_jt e mu mathrm mu","display":"{\\mathrm e}^{\\mu_jt}","kind":"use","excerpt":"xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0080b32292de2b10","tex":"e^{\\rho\\omega^\\ell z}","aliases":"e^ρω^ℓ z rho omega ell","display":"e^{\\rho\\omega^\\ell z}","kind":"use","excerpt":"[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6cced37b7825af60","tex":"\\e^{\\sigma^2\\lambda^2/2}","aliases":"e^σ^2λ^2/2 e sigma lambda mathrm sigma lambda","display":"{\\mathrm e}^{\\sigma^2\\lambda^2/2}","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2dc0e0b6c8f96f8","tex":"e^{\\sigma^2\\lambda^2/2}","aliases":"e^σ^2λ^2/2 sigma lambda","display":"e^{\\sigma^2\\lambda^2/2}","kind":"use","excerpt":"vpage36Repeated observations shrink the noise side of the gapSUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGETChapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11. Suppose all noise observations are independent and centered, with moment-generating bound E e^λη≤ e^σ^2λ^2/2. Averaging R observations at each of n+1 sample sites replaces σ by σ/s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e729462b75849d7","tex":"\\e^{\\omega t}","aliases":"e^ω t e omega mathrm omega","display":"{\\mathrm e}^{\\omega t}","kind":"use","excerpt":"[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-94d84178b7ed29ff","tex":"e^{\\omega t}","aliases":"e^ω t omega","display":"e^{\\omega t}","kind":"use","excerpt":"section renewcommandthefigureRarabicfigure renewcommandtheHfigureR.arabicfigure renewcommandthetableRarabictable renewcommandtheHtableR.arabictable FloatBarrier Transfer paths and product-topology limits. Let T(t) be an arbitrary strongly continuous semigroup on a complex Banach space X, with generator A, and suppose normT(t)≤ M e^ω t for M≥1 and ω","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02ef9a7b897239e9","tex":"\\e^{\\omega_c t}","aliases":"e^ω_c t e omega mathrm omega","display":"{\\mathrm e}^{\\omega_c t}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-345975bbfb2cd2a5","tex":"e^{\\omega_c t}","aliases":"e^ω_c t omega","display":"e^{\\omega_c t}","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2c4b7d39cf705cb","tex":"e^{\\omega_c(t-r)}","aliases":"e^ω_c(t-r) omega","display":"e^{\\omega_c(t-r)}","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c908c0a0cf833d26","tex":"\\e^{\\omega_c(t-s)+\\omega_fs}","aliases":"e^ω_c(t-s)+ω_fs e omega omega mathrm omega omega","display":"{\\mathrm e}^{\\omega_c(t-s)+\\omega_fs}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd0cfdcd6332cbae","tex":"\\e^{\\omega_f t}","aliases":"e^ω_f t e omega mathrm omega","display":"{\\mathrm e}^{\\omega_f t}","kind":"use","excerpt":"[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7d0a5131b29b5fc","tex":"e^{\\omega_f t}","aliases":"e^ω_f t omega","display":"e^{\\omega_f t}","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d8b7fcd7ac96536","tex":"e^{\\omega_fr}","aliases":"e^ω_fr omega","display":"e^{\\omega_fr}","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eba5ceae0baaa423","tex":"\\e^{\\omega_Q(t-s)+\\omega s}","aliases":"e^ω_Q(t-s)+ω s e omega omega mathrm omega omega","display":"{\\mathrm e}^{\\omega_Q(t-s)+\\omega s}","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79e834ba52a1e5be","tex":"\\e^{\\omega_Qt}","aliases":"e^ω_Qt e omega mathrm omega","display":"{\\mathrm e}^{\\omega_Qt}","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dbeccfa5c62c9706","tex":"e_{\\mathfrak n}","aliases":"e_ n mathfrak","display":"e_{\\mathfrak n}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0b30352ab71380c","tex":"E_{(p-1)x+j}","aliases":"E_(p-1)x+j","display":"E_{(p-1)x+j}","kind":"use","excerpt":"[Odd prime analytic Euler branch] The series defining mathscr E_p,j^× converges in the Tate algebra on Z_p. For every nonnegative integer x, mathscr E_p,j^×(x) = (1-σ_pp^(p-1)x+j) E_(p-1)x+j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8be2ba7e790af78","tex":"E_+","aliases":"E_+","display":"E_+","kind":"use","excerpt":"0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2 0 · 0 , E_-=fracI-A2 = 0 · -1/2","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1285db44838b010","tex":"e_+","aliases":"e_+","display":"e_+","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca57caa14f72ba3b","tex":"E_-","aliases":"E_-","display":"E_-","kind":"use","excerpt":"0 · 0 , E_-=fracI-A2 = 0 · -1/2 0 · 1 . For every odd prime p, both projectors preserve Z_p^2. At p=2, s_2(E_+)=s_2(E_-)=1. The factors x-1 and x+1 have resultant 2.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3da4494053c2f25c","tex":"e_-","aliases":"e_-","display":"e_-","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.","count":29},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83ef2b537772a0a9","tex":"E_{00}","aliases":"E_00","display":"E_{00}","kind":"use","excerpt":"The words are written matrix products: in ab, the map b acts first. Therefore ab=8E_00, ba=8E_11, aba=16E_01. The two products ab and ba cannot be identified merely because they have equal weight. Both their order and their source/target information matter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-febc076ab7500943","tex":"E_{01}","aliases":"E_01","display":"E_{01}","kind":"use","excerpt":"vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGChapter II, mathematical text: Theorem 5.10, Corollary 5.11 Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least five v","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d63255e26d245d2","tex":"e_{01}","aliases":"e_01","display":"e_{01}","kind":"use","excerpt":"For the continuous Z_2 carrier ρ(s)=diag(1,-1,1), conjugate the three coordinate projectors by 1+2e_01 and reduce modulo 4. They form a complete equivariant family. The general theorem constructs three nonzero integral summands retaining this family modulo 2. No exact equivariant family retains the whole prescribed family modulo 4. Separate checked","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c86d93cffa1a31f","tex":"e_0^{(m,p)}","aliases":"e_0^(m,p)","display":"e_0^{(m,p)}","kind":"use","excerpt":"[Compressed inverse recurrence] The compressed coefficients satisfy ∑_q=0^n binommnmq e_q^(m,p) = δ_n0. In particular, e_0^(m,p)=1, and the positive coefficients are determined recursively because the diagonal is one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-521e7dfe4bcb4abb","tex":"E_0^*","aliases":"E_0^*","display":"E_0^*","kind":"use","excerpt":"A genuine channel can still be uncorrectable. For full qubit dephasing, E_0=diag(1,0) and E_1=diag(0,1), the Kraus normalization is checked. But E_0^*E_0 cannot be a scalar multiple of I, as its two diagonal entries differ. Necessity therefore proves that no finite recovery channel corrects the full code space. The case of the zero code is also che","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f863240a4986a039","tex":"E_0^{p,q}","aliases":"E_0^p,q","display":"E_0^{p,q}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESChapter VI, Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-136cb5922e752cc4","tex":"E_{10}","aliases":"E_10","display":"E_{10}","kind":"use","excerpt":"vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGChapter II, mathematical text: Theorem 5.10, Corollary 5.11 Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least five v","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46018870fc4913f9","tex":"E_{11}","aliases":"E_11","display":"E_{11}","kind":"use","excerpt":"The words are written matrix products: in ab, the map b acts first. Therefore ab=8E_00, ba=8E_11, aba=16E_01. The two products ab and ba cannot be identified merely because they have equal weight. Both their order and their source/target information matter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2b11b7b1856ea86","tex":"E_{12}","aliases":"E_12","display":"E_{12}","kind":"use","excerpt":"vpage16Resolve the entire obstructionMODEL F / PACKET BLOCKS FROM INPUT TO OUTPUTChapter III, mathematical text: Theorem 7.5, Corollary 7.6 Use three coordinate packets with projectors P_0,P_1,P_2, and set H=I+2E_01+3E_12, Ob=H-I. Here H is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside the","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d1846d5b18c76fd","tex":"E_1^{p,n-p}","aliases":"E_1^p,n-p","display":"E_1^{p,n-p}","kind":"use","excerpt":"[Fine obstruction survival] A class on the first page contributes to the total obstruction only if it is represented by a permanent cycle, so it is neither killed nor identified as a boundary on any later page. If the spectral sequence degenerates at the first page, then H^n(C^bullet)=0 exactly when E_1^p,n-p=0 for every p.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eedfcc807dd23d85","tex":"E_1^{p,q}","aliases":"E_1^p,q","display":"E_1^{p,q}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functio","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":28},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d61e8fb192b54ca7","tex":"E_{20}","aliases":"E_20","display":"E_{20}","kind":"use","excerpt":"On a vector (x_0,x_1,x_2)^ T, the first step produces (0,2x_0,0)^ T, and the second produces (0,0,6x_0)^ T. Thus L_fL_e=6E_20, L_eL_f=0. For general edges, a summand in the entrywise product can survive only when the same intermediate packet is both tgt(e) and src(f): (L_fL_e)_yx=∑_z(L_f)_yz(L_e)_zx. This is why a broken interface annihilates the e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eebcf5feadee9573","tex":"E_{21}","aliases":"E_21","display":"E_{21}","kind":"use","excerpt":"The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-699df382cec3e125","tex":"E_{2M+2}","aliases":"E_2M+2","display":"E_{2M+2}","kind":"use","excerpt":"[Optimal uniform dyadic tower] For all M,n∈ N, E_2n-T_M(n)∈2^4M+3 Z_2. At the first omitted row n=M+1, the error is nonzero and v_2(E_2M+2-T_M(M+1))=4M+3. Thus the exponent 4M+3 cannot be increased uniformly in the Euler row.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e845d19c3939de97","tex":"E_{2n}","aliases":"E_2n","display":"E_{2n}","kind":"use","excerpt":"The classical signed Euler numbers have generating function secht=∑_n≥0E_2nt^2n/(2n)!. They are the N=2, j=0 member of [source reference]. The coordinate theorem reconstructs them from finite alternating sums of odd powers.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1415b70590776d91","tex":"\\mathcal E_{2N}","aliases":"E_2N mathcal","display":"\\mathcal E_{2N}","kind":"use","excerpt":"The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d39b0ab0c5e61ab1","tex":"E_{2n}^{(2,0)}","aliases":"E_2n^(2,0)","display":"E_{2n}^{(2,0)}","kind":"use","excerpt":"Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eba3507d2e119c00","tex":"E_{2r}","aliases":"E_2r","display":"E_{2r}","kind":"use","excerpt":"The source defines E_2r as the integer numerator of that rational coefficient and proves that its rational cast equals C_2,0(r). Set E_2r+1=0. The actual exponential-section identity from the first block proves ∑_n≥0E_nfract^nn!=sech t. Thus the integer sequence is connected to the intended signed Euler family, rather than merely assigned the name ","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5002be43e2e62d6e","tex":"E_{2r+1}","aliases":"E_2r+1","display":"E_{2r+1}","kind":"use","excerpt":"The source defines E_2r as the integer numerator of that rational coefficient and proves that its rational cast equals C_2,0(r). Set E_2r+1=0. The actual exponential-section identity from the first block proves ∑_n≥0E_nfract^nn!=sech t. Thus the integer sequence is connected to the intended signed Euler family, rather than merely assigned the name ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82c2d872ef715d1f","tex":"E_3","aliases":"E_3","display":"E_3","kind":"use","excerpt":"P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"[Typed reachability certificate] In the preceding quiver realization, no seed supported at U can generate the full representation, whereas the two seeds e_1∈ U, e_3∈ V do generate it.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58f890b6f450ab6a","tex":"E_{3,1}","aliases":"E_3,1","display":"E_{3,1}","kind":"use","excerpt":"because the hypergeometric convention already supplies (1)_n=n!. vfighypergeometric headCheck the displayed case For N=3,j=1,n=2, the six factors after the initial 1! are 2,3,4,5,6,7. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies 2!. The others supply (2/3)_2 and (4/3)_2. The right-hand side is exact","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c7ee7fb47bfc415","tex":"E_{3,j}","aliases":"E_3,j","display":"E_{3,j}","kind":"use","excerpt":"Multiply the coordinate forms of e^zξ and e^wξ. Their product is e^(z+w)ξ. The coefficient of 1 receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law. headCompare the unsigned model In C[ξ]/(ξ^3-1), the same table has no minus signs. Its exponential coordinates are the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40fa4751533ea967","tex":"E_{31}","aliases":"E_31","display":"E_{31}","kind":"use","excerpt":"The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dee62bc352cb736e","tex":"E_{32}","aliases":"E_32","display":"E_{32}","kind":"use","excerpt":"The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6662c032276e077b","tex":"E_4","aliases":"E_4","display":"E_4","kind":"use","excerpt":"Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67ab753297c61ce5","tex":"e_4","aliases":"e_4","display":"e_4","kind":"use","excerpt":"vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b7e8f4d909c6799","tex":"E_{4(n+H)}^{(4,j)}","aliases":"E_4(n+H)^(4,j)","display":"E_{4(n+H)}^{(4,j)}","kind":"use","excerpt":"[Exact four-address first differences] For every n≥0, H>0, and j∈{0,1,2,3}, v_2(C_j(n+H)-C_j(n)) · =v_2(H)+b_j, · (b_0,b_1,b_2,b_3) · =(1,1,4,2), v_2(E_4(n+H)^(4,j)-E_4n^(4,j)) · =v_2(H)+B_j, · (B_0,B_1,B_2,B_3) · =(1,1,5,3). Consequently, the least positive row period modulo 2^r is exactly P_j(r)=2^max(0,r-B_j) (r≥1). The corresponding displacemen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab8e9e09ff8c907f","tex":"E_{4(n+iH)}^{(4,j)}","aliases":"E_4(n+iH)^(4,j)","display":"E_{4(n+iH)}^{(4,j)}","kind":"use","excerpt":"The strongest common-step conclusion in width four is compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. I","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5a67dacf9cd88b2","tex":"E_{4n}^{(4,j)}","aliases":"E_4n^(4,j)","display":"E_{4n}^{(4,j)}","kind":"use","excerpt":"[Exact four-address first differences] For every n≥0, H>0, and j∈{0,1,2,3}, v_2(C_j(n+H)-C_j(n)) · =v_2(H)+b_j, · (b_0,b_1,b_2,b_3) · =(1,1,4,2), v_2(E_4(n+H)^(4,j)-E_4n^(4,j)) · =v_2(H)+B_j, · (B_0,B_1,B_2,B_3) · =(1,1,5,3). Consequently, the least positive row period modulo 2^r is exactly P_j(r)=2^max(0,r-B_j) (r≥1). The corresponding displacemen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51b438373b685e40","tex":"E_6","aliases":"E_6","display":"E_6","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-edf98275838926c6","tex":"E_8","aliases":"E_8","display":"E_8","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8db0cea46fdd60f3","tex":"E_A","aliases":"E_A","display":"E_A","kind":"use","excerpt":"0 · b≠ a, ∑_aP_a=1. Write A_ba=P_bAP_a and let E_A={(b,a):A_ba≠0}. These are exact algebraic zeros, not values discarded by a floating-point threshold.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-238a2f0e8e975441","tex":"E_{ab}","aliases":"E_ab","display":"E_{ab}","kind":"use","excerpt":"For the exact calculation, apply the four matrix units E_ab to each seed matrix, flatten the outputs, and compute their column rank. The observed dimensions are two and four, respectively. The calculation directly exhibits the independent rows counted by the theorem. takeawaySchmidt rank measures the remote multiplicity directions accessible to the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a9df7c97d7c6847","tex":"E_b^*","aliases":"E_b^*","display":"E_b^*","kind":"use","excerpt":"Cross terms for the original errors. For arbitrary positive Gram data, the same recovery satisfies R(E_aρ E_b^*)=C_baρ (Pρ P=ρ). The index order ba is intentional: the defining Gram relation is PE_a^*E_bP=C_abP. The proof first derives scalar corrected Kraus composites for the constructed recovery and then uses its trace-preserving completeness to ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad3caf981ac30812","tex":"E_c","aliases":"E_c","display":"E_c","kind":"use","excerpt":"pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a20a0c8f1552bcce","tex":"E_e","aliases":"E_e","display":"E_e","kind":"use","excerpt":"For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tig","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ce6fb2b17a97a58","tex":"E_F","aliases":"E_F","display":"E_F","kind":"use","excerpt":"For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8b63d55908fc609","tex":"E_f","aliases":"E_f","display":"E_f","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f1d90b4eb6efe21","tex":"E_f^2","aliases":"E_f^2","display":"E_f^2","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6438bb683a992a5b","tex":"e_{\\frakm}","aliases":"e_frakm e_ m frakm mathfrak","display":"e_{{\\mathfrak m}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92780dc9a20a0ca6","tex":"E_H","aliases":"E_H","display":"E_H","kind":"use","excerpt":"For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0d586db24ff52c2","tex":"\\bm E_{\\mathrm{H}}","aliases":"E_H bm mathrm","display":"\\bm E_{\\mathrm{H}}","kind":"use","excerpt":"The Haar analysis matrix is E_H = frac1sqrt2 1 · 1 1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e30f119b7ece4c0","tex":"\\bm E_{\\mathrm{H}}^{\\mathsf T}","aliases":"E_H^ T bm mathrm mathsf","display":"\\bm E_{\\mathrm{H}}^{\\mathsf T}","kind":"use","excerpt":"1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9e2b1f616f1340a","tex":"E_i^2","aliases":"E_i^2","display":"E_i^2","kind":"use","excerpt":"Keep the complete domain, supplied finite-group carrier and group-order hypotheses of the hyperref[book:x-construction-18]supplied-carrier summand-lifting construction. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52daa780ca76442f","tex":"E_{ii}^{\\lambda}","aliases":"E_ii^λ lambda","display":"E_{ii}^{\\lambda}","kind":"use","excerpt":"x an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33c9d2aae34fdc7b","tex":"E_{ij}^{\\lambda}","aliases":"E_ij^λ lambda","display":"E_{ij}^{\\lambda}","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16d60aa45461fc23","tex":"E_{i\\ell}^{\\lambda}","aliases":"E_iℓ^λ ell lambda","display":"E_{i\\ell}^{\\lambda}","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3fd533b471b8be8","tex":"e_{j-1}","aliases":"e_j-1","display":"e_{j-1}","kind":"use","excerpt":"C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c982b205469a89b","tex":"E_j^*","aliases":"E_j^*","display":"E_j^*","kind":"use","excerpt":"resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have the same relative weights, but their total ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-606053aa32f17484","tex":"E_{ji}","aliases":"E_ji","display":"E_{ji}","kind":"use","excerpt":"vpage2An atlas of the objectsORIENTATION / KEEP THE TYPES VISIBLEChapter III, mathematical text: Definition 2.1, Definition 5.5, Definition 7.4 Rows are targets and columns are sources throughout. If E_ji is a matrix unit, then E_jie_i=e_j. The notation e_i means a basis vector. A named graph edge is written a,b,c,d or e,f when its role is explicit","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d3bf131ce5c7803","tex":"E_{ji}^{\\lambda}","aliases":"E_ji^λ lambda","display":"E_{ji}^{\\lambda}","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6d45676877113dd","tex":"e_{jj}","aliases":"e_jj","display":"e_{jj}","kind":"use","excerpt":"For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f891a7f61f89a21","tex":"E_k","aliases":"E_k","display":"E_k","kind":"use","excerpt":"E_k · Positive measurement effect · It gives the outcome probability Tr(E_kρ).","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f79869b257acab57","tex":"E_{k\\ell}^{\\mu}","aliases":"E_kℓ^μ ell mu","display":"E_{k\\ell}^{\\mu}","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","excerpt":"vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONChapter II, mathematical text: Section B.1 Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b6907ef544fbc21","tex":"E_m'","aliases":"E_m'","display":"E_m'","kind":"use","excerpt":"The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and tr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-416776f638338c57","tex":"e_{m+1}","aliases":"e_m+1","display":"e_{m+1}","kind":"use","excerpt":"[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b92f146ef8909c6","tex":"e_{m+2}","aliases":"e_m+2","display":"e_{m+2}","kind":"use","excerpt":"[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-84df450ec19feaf4","tex":"E_{m-1}","aliases":"E_m-1","display":"E_{m-1}","kind":"use","excerpt":"The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and tr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b2429b531974850","tex":"E_{m-1}^{(N,j,r)}","aliases":"E_m-1^(N,j,r)","display":"E_{m-1}^{(N,j,r)}","kind":"use","excerpt":"[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ee0e3ae18e03505","tex":"E_{m-q}^{(N,j,s)}","aliases":"E_m-q^(N,j,s)","display":"E_{m-q}^{(N,j,s)}","kind":"use","excerpt":"[Joint Appell convolution in the space and order parameters] For all r,s,x,y∈ C and m≥0, E_m^(N,j,r+s)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x) E_m-q^(N,j,s)(y).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8c499641055da34","tex":"E_m^{(N,j,r+s)}","aliases":"E_m^(N,j,r+s)","display":"E_m^{(N,j,r+s)}","kind":"use","excerpt":"[Joint Appell convolution in the space and order parameters] For all r,s,x,y∈ C and m≥0, E_m^(N,j,r+s)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x) E_m-q^(N,j,s)(y).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aeb1ae02bc3682da","tex":"E_{mpn+qp^r}^{(mp,j)}","aliases":"E_mpn+qp^r^(mp,j)","display":"E_{mpn+qp^r}^{(mp,j)}","kind":"use","excerpt":"Principal results. addcontentslinetocsectionPrincipal results The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integral and E_mpn+qp^r^(mp,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d59b3cfc1e7a514b","tex":"E_{mpn}^{(mp,j)}","aliases":"E_mpn^(mp,j)","display":"E_{mpn}^{(mp,j)}","kind":"use","excerpt":"Principal results. addcontentslinetocsectionPrincipal results The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integral and E_mpn+qp^r^(mp,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da2ce42dc70bb3da","tex":"E_n","aliases":"E_n","display":"E_n","kind":"use","excerpt":"Let q be a positive odd integer. Define S_n(q) = ∑_a=0^q-1 (-1)^a(2a+1)^n. Let the signed Euler numbers and Euler zigzag numbers be defined by secht = ∑_n≥0E_nfract^nn! and sec t+tan t = ∑_n≥0Z_nfract^nn!.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf856d35e202a0fa","tex":"\\mathcal E_n","aliases":"E_n mathcal","display":"\\mathcal E_n","kind":"use","excerpt":"To distinguish ordinary signed Euler numbers from the generalized numbers of the main text, write secht=∑_n≥0 E_nfract^nn!, fracte^t-1=∑_n≥0B_nfract^nn!, B_1=-frac12. Throughout this appendix, v_2(2)=1 and v_2(0)=∞. Let s_2(n) be the number of ones in the binary expansion of the nonnegative integer n. Define A(N)=frac3^2N+2-14(2N+1)(2N+2), (frac zs","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3e1f9162e4702a9","tex":"e_{n+1}","aliases":"e_n+1","display":"e_{n+1}","kind":"use","excerpt":"Let e_n= CF_f^n(x)-F_c^n(Cx) . The definition of the defect and the Lipschitz property give e_n+1≤ε_n+qe_n. Iterating this recurrence proves the first estimate. The second is the geometric sum.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0277f6b24af6402","tex":"E_{N,0}","aliases":"E_N,0","display":"E_{N,0}","kind":"use","excerpt":"vpage16Scale and order enter differentlyTHETA COUPLINGChapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4. Let H_N(z)=E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic block m. Reciprocal order contributes","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3ca2ec43cc1e4a1","tex":"E_{N,j}","aliases":"E_N,j","display":"E_{N,j}","kind":"use","excerpt":"midrule E_N,j(z) · displaystyle∑_n≥0fracz^Nn+j(Nn+j)! selects one residue class of the exponential.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7ce4855f02a711f","tex":"e_{n-1}","aliases":"e_n-1","display":"e_{n-1}","kind":"use","excerpt":"For a nonempty composable word w=(e_1,…,e_n), define its reverse coefficient product c^←(w)=c(e_n)c(e_n-1)⋯ c(e_1). The reversal is essential over a noncommutative semiring.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7898ee6b98b86d47","tex":"E_{n-2r}","aliases":"E_n-2r","display":"E_{n-2r}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6735d603366759b5","tex":"E_{n-r}","aliases":"E_n-r","display":"E_{n-r}","kind":"use","excerpt":"The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer. This is the integrality required by the later congruence.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16602ebf447cab47","tex":"e_n^{(2,7)}","aliases":"e_n^(2,7)","display":"e_n^{(2,7)}","kind":"use","excerpt":"ction. The positive-index standard-window tail printed below is erroneous. The defining recurrence gives e_1=-1≡6, e_2=5 and e_3=-61≡2pmod7, so the correct positive tail is 6,5,2,6,5,2,…. The period-three conclusion remains correct. The original line is retained only as a corrected historical record. Take p=7 and m=2, so h=3. The recurrence gives e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f106d4bb685105a","tex":"E_{Nn+qp^r}^{(N,j)}","aliases":"E_Nn+qp^r^(N,j)","display":"E_{Nn+qp^r}^{(N,j)}","kind":"use","excerpt":"[Nishibuchi congruence] Let p be prime, let m>0, and suppose that mmid p-1 or m=2. Put N=mp and q=lcm(2,p-1). For every 0≤ j<N, all U_N,j(n) belong to Z_p. For r≥1 and n≥ r, E_Nn+qp^r^(N,j)≡ E_Nn^(N,j)pmodp^r. Thus the positive sufficient starting index n_0=r works uniformly. If mmid p-1, the stronger sufficient condition r≤ mn is valid.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0dd6372c1de2074b","tex":"E_p","aliases":"E_p","display":"E_p","kind":"use","excerpt":"For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positi","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b09d4efdcdc1f39","tex":"e_p","aliases":"e_p","display":"e_p","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ddcf053d52b9abd","tex":"E_{p,j}^\\times","aliases":"E_p,j^× times","display":"E_{p,j}^\\times","kind":"use","excerpt":"Fix an even address j∈{0,2,…,p-3}. For x∈ Z_p, define mathscr E_p,j^×(x) = ∑_r≥0 · (-1)^rZ_2r+1p^2r binom(p-1)x+j2r · × ∑_substack0≤ a<p 2a+1≠ p (-1)^a (2a+1)^j-2r ((2a+1)^p-1)^x.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3f1fc86e79071cb","tex":"E_q","aliases":"E_q","display":"E_q","kind":"use","excerpt":"The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and tr","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-244f69b2cbc609aa","tex":"e_q","aliases":"e_q","display":"e_q","kind":"use","excerpt":"For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9dcdbc1ff00bedd9","tex":"E_q^{(N,j,r)}","aliases":"E_q^(N,j,r)","display":"E_q^{(N,j,r)}","kind":"use","excerpt":"[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a840ddc04edad8c","tex":"E_r","aliases":"E_r","display":"E_r","kind":"use","excerpt":"The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer. This is the integrality required by the later congruence.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f89f0fc29c98d5e","tex":"e_{r,0}","aliases":"e_r,0","display":"e_{r,0}","kind":"use","excerpt":"The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-decd9feaf2bf3e0d","tex":"e_{r,q}","aliases":"e_r,q","display":"e_{r,q}","kind":"use","excerpt":"The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-701135f7469e0b6e","tex":"e_r^{(m,p)}","aliases":"e_r^(m,p)","display":"e_r^{(m,p)}","kind":"use","excerpt":"[Rational ordinary generating function] In F_p[[z]], ∑_n≥0e_n^(m,p)z^n = 1+ frac ∑_r=1^he_r^(m,p)z^r 1-z^h.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f38f9f204752bc5","tex":"E_{rs}","aliases":"E_rs","display":"E_{rs}","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd0b06b425803045","tex":"E_{rv}","aliases":"E_rv","display":"E_{rv}","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a8e0b23212d1b1a","tex":"e_T","aliases":"e_T","display":"e_T","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8030d3304fb10474","tex":"e_{T,a}","aliases":"e_T,a","display":"e_{T,a}","kind":"use","excerpt":"Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimensi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5cee62af73e7dca8","tex":"E_{\\tgt(e),\\src(e)}","aliases":"E_tgt(e),src(e) tgt src operatorname operatorname","display":"E_{{\\operatorname{tgt}}(e),{\\operatorname{src}}(e)}","kind":"use","excerpt":"L_e=c(e)E_tgt(e),src(e) · One operator supported on a packet. · pagerefV3","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-504dc54aa7ab931b","tex":"e_U","aliases":"e_U","display":"e_U","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5dd4a91606e9704e","tex":"e_{U,a}","aliases":"e_U,a","display":"e_{U,a}","kind":"use","excerpt":"Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimensi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9225c9aedb8f0a45","tex":"E_{uv}","aliases":"E_uv","display":"E_{uv}","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-042162b6d9ca555d","tex":"e_v","aliases":"e_v","display":"e_v","kind":"use","excerpt":"[Typed realization and module equivalence] The category of realizations of G is equivalent to the category of unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its de","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06718d7aa18063e4","tex":"e_w","aliases":"e_w","display":"e_w","kind":"use","excerpt":"n a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v im","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bd3bde92cfdbd07","tex":"E_x","aliases":"E_x","display":"E_x","kind":"use","excerpt":"vpage14A chart failure is not a singular fiberTHE RANGE BUNDLE IS CONSTRUCTED LOCALLYChapter VII: Theorem II.6.7. Details of the range atlas: Chapter VII, Part III, Sections III.24--III.26 (range-bundle construction). The fiber of a projector field is the actual fixed subspace E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the compl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96cf99998ca7f513","tex":"E_z","aliases":"E_z","display":"E_z","kind":"use","excerpt":"very operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6fcc993371cd632c","tex":"E_z^*","aliases":"E_z^*","display":"E_z^*","kind":"use","excerpt":"identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the sam","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23943c7271363293","tex":"E_\\pm","aliases":"E_± pm","display":"E_\\pm","kind":"use","excerpt":"headThe special fiber does not retain the rational splitting Modulo two, x-1=x+1 and A becomes a nontrivial unipotent matrix. The entries 1/2 in E_± cannot be reduced modulo two. By contrast, the coefficient selectors on Plate II.6 use only zero and one and remain available.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8a2c49583348c4b","tex":"E_{\\alpha}","aliases":"E_α alpha","display":"E_{\\alpha}","kind":"use","excerpt":"Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functio","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-632b38e7d6727892","tex":"e_\\theta","aliases":"e_θ theta","display":"e_\\theta","kind":"use","excerpt":"[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74e2d51d64f7fad4","tex":"E_\\lambda","aliases":"E_λ lambda","display":"E_\\lambda","kind":"use","excerpt":"ion] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbd7a622a6f80cd3","tex":"e_\\lambda","aliases":"e_λ lambda","display":"e_\\lambda","kind":"use","excerpt":"X, continuity of a:X→ X supplies continuous seminorms controlling p(aj(e)), so the induced action on E is continuous and extends uniquely to widehat E. The subspace is dense by hypothesis. A Cauchy net in E is mapped to a Cauchy net in X, which converges because X is complete. This defines a continuous extension widehat j:widehat E→ X. For any x∈ X","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6b63f7340a4b3b9","tex":"E_\\mu","aliases":"E_μ mu","display":"E_\\mu","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74537519c7bd4fd0","tex":"E_\\pi","aliases":"E_π pi","display":"E_\\pi","kind":"use","excerpt":"E_π · Conditional expectation of observables. · A weighted average, self-adjoint for the π inner product.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49550fdb9f0a1cd8","tex":"E_\\ell","aliases":"E_ℓ ell","display":"E_\\ell","kind":"use","excerpt":"[Contractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42c5e5713be1939f","tex":"E_\\ell^*","aliases":"E_ℓ^* ell","display":"E_\\ell^*","kind":"use","excerpt":"order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c470de86d3aa40db","tex":"\\Ecal","aliases":"Ecal E mathcal","display":"{\\mathcal E}","kind":"use","excerpt":"For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ce1d43d6f91b9a1","tex":"\\Ecal_{N,j}","aliases":"Ecal_N,j E_N,j Ecal mathcal","display":"{\\mathcal E}_{N,j}","kind":"use","excerpt":"For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-192930449f687944","tex":"\\EE","aliases":"EE E mathbb EE E mathcal","display":null,"kind":"use","excerpt":"Nishibuchi conjecture completion Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. 1. The complete external statement. For N>0 and 0≤ j<N, use the original unnormalized convention ∑_d≥0 E_d^(N,j)fract^dd","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2fa527dd8d76e08","tex":"\\EE_0","aliases":"EE_0 E_0 EE mathbb","display":"{\\mathbb E}_0","kind":"use","excerpt":"1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEver","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ce67a107a09dc9e","tex":"\\EE_d^{(N,j)}","aliases":"EE_d^(N,j) E_d^(N,j) EE mathcal","display":"{\\mathcal E}_d^{(N,j)}","kind":"use","excerpt":"Nishibuchi conjecture completion Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. 1. The complete external statement. For N>0 and 0≤ j<N, use the original unnormalized convention ∑_d≥0 E_d^(N,j)fract^dd","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9561b663cd811533","tex":"\\EE_{Nn+qp^r}^{(N,j)}","aliases":"EE_Nn+qp^r^(N,j) E_Nn+qp^r^(N,j) EE mathcal","display":"{\\mathcal E}_{Nn+qp^r}^{(N,j)}","kind":"use","excerpt":"Let p be prime, m>0, and assume mmid p-1 or m=2. Put N=mp and q=lcm(2,p-1). Every E_Nn^(N,j) is p-adically integral. For every r≥1, E_Nn+qp^r^(N,j)≡ E_Nn^(N,j)pmodp^r (0≤ j<N, n≥ r). In particular, the positive sufficient threshold n_0=r works uniformly in all allowed parameters.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63adfcc6886184a1","tex":"\\EE_{Nn}^{(N,j)}","aliases":"EE_Nn^(N,j) E_Nn^(N,j) EE mathcal","display":"{\\mathcal E}_{Nn}^{(N,j)}","kind":"use","excerpt":"ed original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. 1. The complete external statement. For N>0 and 0≤ j<N, use the original unnormalized convention ∑_d≥0 E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The existing packet convention satisfies E_Nn^(N,j)=j!C_N,j(n). The formaliz","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86b2b150b465871e","tex":"\\EE_{p(t)}","aliases":"EE_p(t) E_p(t) EE mathbb","display":"{\\mathbb E}_{p(t)}","kind":"use","excerpt":"Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8483dce1203449b","tex":"\\mathrm{eff}","aliases":"eff mathrm","display":"\\mathrm{eff}","kind":"use","excerpt":"[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the approp","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3792be68e67ba6b6","tex":"\\Efun_{2,0}","aliases":"Efun_2,0 E_2,0 Efun mathsf","display":"{\\mathsf E}_{2,0}","kind":"use","excerpt":"For N=1, the only section is e^λ z. For N=2, E_2,0(z)=cosh z, E_2,1(z)=sinh z. These are classical instances of the same cyclic basis.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a942d6e96ad70816","tex":"\\Efun_{2,1}","aliases":"Efun_2,1 E_2,1 Efun mathsf","display":"{\\mathsf E}_{2,1}","kind":"use","excerpt":"For N=1, the only section is e^λ z. For N=2, E_2,0(z)=cosh z, E_2,1(z)=sinh z. These are classical instances of the same cyclic basis.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29b6acd7461514fa","tex":"\\Efun_{N,i}","aliases":"Efun_N,i E_N,i Efun mathsf","display":"{\\mathsf E}_{N,i}","kind":"use","excerpt":"[Addition law] For 0≤ k<N, E_N,k(λ(z+w)) = ∑_substack0≤ i,j<N i+j≡ kbmod N E_N,i(λ z) E_N,j(λ w).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-365590e8160df3b1","tex":"\\Efun_{N,j-1}","aliases":"Efun_N,j-1 E_N,j-1 Efun mathsf","display":"{\\mathsf E}_{N,j-1}","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fad2c39cce8b8872","tex":"\\Efun_{N,k}","aliases":"Efun_N,k E_N,k Efun mathsf","display":"{\\mathsf E}_{N,k}","kind":"use","excerpt":"[Addition law] For 0≤ k<N, E_N,k(λ(z+w)) = ∑_substack0≤ i,j<N i+j≡ kbmod N E_N,i(λ z) E_N,j(λ w).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f83f7c661dcd52e7","tex":"\\Efun_{N,N-1}","aliases":"Efun_N,N-1 E_N,N-1 Efun mathsf","display":"{\\mathsf E}_{N,N-1}","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70a9953dd9bd9b22","tex":"\\operatorname{End}","aliases":"End operatorname","display":"\\operatorname{End}","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46c0f3838c9722a0","tex":"\\operatorname{End}_{\\bbR}","aliases":"End_bbR End_ R operatorname bbR operatorname mathbb","display":"\\operatorname{End}_{{\\mathbb R}}","kind":"use","excerpt":"[Projected connection] Let M be a smooth finite-dimensional real manifold without boundary, let V be a finite-dimensional real normed vector space, and let P:M→End_ R(V) be smooth with P(x)^2=P(x). In the quantum application, V is the underlying real space of a finite complex Hilbert space and P is an orthogonal complex-linear projector.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f72b53b71cb2aea7","tex":"\\operatorname{End}_{E_0}","aliases":"End_E_0 operatorname","display":"\\operatorname{End}_{E_0}","kind":"use","excerpt":"Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1. The half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15474d1cea1e947b","tex":"\\End_K","aliases":"End_K End operatorname","display":"{\\operatorname{End}}_K","kind":"use","excerpt":"[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ae4396c22c4c43e","tex":"\\operatorname{End}_K","aliases":"End_K operatorname","display":"\\operatorname{End}_K","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9928404808ae1116","tex":"\\End_\\OO","aliases":"End_OO End_ O End OO operatorname mathcal","display":"{\\operatorname{End}}_{{\\mathcal O}}","kind":"use","excerpt":"Let O be a DVR, K=Frac( O), and A∈End_ O(Λ) for a finite free lattice Λ. Suppose the characteristic polynomial factors over O as χ_A(x)=f(x)g(x), where f and g are coprime over K[x].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a6e4a27c6e0ed93","tex":"\\End_{\\OO}","aliases":"End_OO End_ O End OO operatorname mathcal","display":"{\\operatorname{End}}_{{{\\mathcal O}}}","kind":"use","excerpt":"Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps ε varepsilon","display":"{\\varepsilon}","kind":"use","excerpt":"[Packet perturbation certificate] Let A= ∑_e∈ EA_e, widehat A = ∑_e∈ E(A_e+E_e), where edge e starts at packet α(e), and assume A_e=A_eP_α(e), E_e=E_eP_α(e). Then norm(widehat A-A)x ≤ ∑_e∈ E normE_e normP_α(e)x. If normE_e ≤ ε_enormA_e, then norm(widehat A-A)x ≤ ∑_e∈ E ε_enormA_e normP_α(e)x.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc04c9966dd2873a","tex":"\\eps_e","aliases":"eps_e ε_e eps varepsilon","display":"{\\varepsilon}_e","kind":"use","excerpt":"[Packet perturbation certificate] Let A= ∑_e∈ EA_e, widehat A = ∑_e∈ E(A_e+E_e), where edge e starts at packet α(e), and assume A_e=A_eP_α(e), E_e=E_eP_α(e). Then norm(widehat A-A)x ≤ ∑_e∈ E normE_e normP_α(e)x. If normE_e ≤ ε_enormA_e, then norm(widehat A-A)x ≤ ∑_e∈ E ε_enormA_e normP_α(e)x.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7340ca44d667f098","tex":"\\err","aliases":"err operatorname","display":"{\\operatorname{err}}","kind":"use","excerpt":"ne. Keeping either a or b costs three with error two. Splitting a costs four with zero error, and splitting b costs five with zero error. vfigtree_frontier The five complete prunings have (cost,error) pairs (1,6), (7,4), (9,2), (8,2), (10,0). At budget two the unique cheapest pruning is {a_0,a_1,b}, costing eight. It is Pareto optimal, but never mi","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3aaf5c530b84ee89","tex":"\\esssup","aliases":"esssup *ess sup operatorname","display":"{\\operatorname*{ess\\,sup}}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6ab09549ed6fe5b","tex":"\\esssup_\\omega","aliases":"esssup_ω *ess sup_ω esssup omega operatorname omega","display":"{\\operatorname*{ess\\,sup}}_\\omega","kind":"use","excerpt":"0 · 0.02sinω . Then *ess sup_ω D( e^ iω) _op =0.02. Every input therefore has relative reconstruction error at most two percent in ℓ^2. The estimate is independent of the input length.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-01ad82b028fe9c1f","tex":"\\esssup_{\\omega}","aliases":"esssup_ω *ess sup_ω esssup omega operatorname omega","display":"{\\operatorname*{ess\\,sup}}_{\\omega}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9faafbdfe9a57a78","tex":"\\operatorname{Euler}","aliases":"Euler operatorname","display":"\\operatorname{Euler}","kind":"use","excerpt":"Let L be the ordinary Euler evaluation functional on a variable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear. It is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr E","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d73f1f85c3e00f5a","tex":"\\operatorname{Euler}_{2n+2r}","aliases":"Euler_2n+2r operatorname","display":"\\operatorname{Euler}_{2n+2r}","kind":"use","excerpt":"ariable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear. It is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e617fe37e532f98e","tex":"\\operatorname{Euler}_d","aliases":"Euler_d operatorname","display":"\\operatorname{Euler}_d","kind":"use","excerpt":"Let L be the ordinary Euler evaluation functional on a variable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear. It is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr E","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-77007f1f55a92c5b","tex":"\\mathrm{even}","aliases":"even mathrm","display":"\\mathrm{even}","kind":"use","excerpt":"For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c48deb718470362","tex":"\\operatorname{Even}","aliases":"Even operatorname","display":"\\operatorname{Even}","kind":"use","excerpt":"Both row indexing by r and the literal even-degree version with hypothesis Even(n) are checked. All rows include degree zero. The exact identity is a rational equality with the Euler coefficient's integer-to-rational cast made explicit.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9452831b9693a633","tex":"\\operatorname{EvenPart}","aliases":"EvenPart operatorname","display":"\\operatorname{EvenPart}","kind":"use","excerpt":"Even projection is identified coefficientwise. For any formal series f, let EvenPart(f)=(f(t)+f(-t))/2. Its nth coefficient is proved to be the coefficient of f when n is even and zero otherwise. Rescaling the inverse law proves sech(-t)=sech(t). The exponential multiplication identity then gives cosh^2(pt)=frac2+e^2pt+e^-2pt4, and hence ∑_r≥0S_2r(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39ea6df1a878e2c7","tex":"\\Exp","aliases":"Exp E mathbb","display":"{\\mathbb E}","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ce13a52afee76f0","tex":"\\Exp_i","aliases":"Exp_i E_i Exp mathbb","display":"{\\mathbb E}_i","kind":"use","excerpt":"[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c561c86f95e10f8","tex":"\\Exp_{p_0}","aliases":"Exp_p_0 E_p_0 Exp mathbb","display":"{\\mathbb E}_{p_0}","kind":"use","excerpt":"[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off-diagonal entries and nonpositive column sums, so it is the subgenera","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fbfc0ac7d89b213e","tex":"\\Exp_\\gamma","aliases":"Exp_γ E_γ Exp gamma mathbb gamma","display":"{\\mathbb E}_\\gamma","kind":"use","excerpt":"A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-640dc13c34cc0b55","tex":"\\Exp_\\pi","aliases":"Exp_π E_π Exp pi mathbb pi","display":"{\\mathbb E}_\\pi","kind":"use","excerpt":"[Law of total variance] For a real valued f, let P_0f= E_π[f] 1. Then Var_π(f) = Var_π(P_ Qf) + normf-P_ Qf_π^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7e053bb5a2cb140","tex":"\\operatorname{Ext}","aliases":"Ext operatorname","display":"\\operatorname{Ext}","kind":"use","excerpt":"The accompanying nolinkurlnative/Native_Conjectures_Progress.md records further results in restricted settings: an intrinsic finite Ext^1 obstruction for finitely presented commutative integer algebras, finite CRT reconstruction for polynomial solution sets, and a characterization of rectangular finite relations by equality of common labels. The ob","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fbd7b2c73ab2fd9c","tex":"\\operatorname{Ext}^1","aliases":"Ext^1 operatorname","display":"\\operatorname{Ext}^1","kind":"use","excerpt":"The accompanying nolinkurlnative/Native_Conjectures_Progress.md records further results in restricted settings: an intrinsic finite Ext^1 obstruction for finitely presented commutative integer algebras, finite CRT reconstruction for polynomial solution sets, and a characterization of rectangular finite relations by equality of common labels. The ob","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ea44da66f1c1881","tex":"\\operatorname{Ext}^1_B","aliases":"Ext^1_B operatorname","display":"\\operatorname{Ext}^1_B","kind":"use","excerpt":"o(f)∈Ext^1_B(NL_B/ Z, F_p)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c77c474dad0dd4d","tex":"\\mathrm{extra}","aliases":"extra mathrm","display":"\\mathrm{extra}","kind":"use","excerpt":"The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary d","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F mathbb F mathbf","display":null,"kind":"use","excerpt":"The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fiber does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the s","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":50},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6bb9a2af76d66fc","tex":"\\mathbf F","aliases":"F mathbf","display":"\\mathbf F","kind":"use","excerpt":"An explicit realization of the finite pattern. A separate finite-complex implementation uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linear","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"Use the augmented polynomial systems F(Y,T)=(F(Y,T),T-t_0), H(Y,T)=(H(Y,T),-v). Then F(a)=0, and its derivative is B=D F(a)=J · K","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-484de1c254ac2bb8","tex":"\\mathrm F","aliases":"F mathrm","display":"\\mathrm F","kind":"use","excerpt":"ubsection:002textitModule SmallCoreNorms; arbitrary ambient competitors For rectangular isometries Q,V define L(G)=QGV^*, C(B)=Q^*BV, Q^*Q=I, V^*V=I. The following statements are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_ν≤norm B_ν, ν∈{2, F}. For the spe","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"For real t, differentiate F(s)=e^(t-s)T_cCe^sT_f. We obtain F'(s)=e^(t-s)T_cΔ e^sT_f. Integration from 0 to t gives the identity. The complex statement follows by the same calculation along the straight segment. The norm bound follows from the integral formula.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f4583c8b7c73ba0","tex":"f'","aliases":"f'","display":"f'","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef4a9b3f84f761fd","tex":"F''","aliases":"F''","display":"F''","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9337f69b3f162423","tex":"f''","aliases":"f''","display":"f''","kind":"use","excerpt":"[Generator and Dirichlet form] On the polynomial subspace, the infinitesimal generator of ( O_t) is the differential operator L f=f''-xf'. It satisfies L h_n=-nh_n. For every polynomial f, ip- L ff_L^2(γ) = ∫_ Rabsf'(x)^2γ( d x).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9400ebafe84d3b7c","tex":"F'''","aliases":"F'''","display":"F'''","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f19bf4bf6a7fc594","tex":"f^{\\mathsf T}","aliases":"f^ T mathsf","display":"f^{\\mathsf T}","kind":"use","excerpt":"vpage2A dictionary with orientationsREAD BEFORE MULTIPLYING / CONVENTIONSChapter VI, Part I: Sections 2, 4, 6, 7, and 12. Column probabilities and backward observables. An apparent disagreement between two transfer formulas often comes from acting on different objects. Throughout this guide, probabilities are columns and observables are functions w","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3492b2ff841a2568","tex":"F^{(k)}","aliases":"F^(k)","display":"F^{(k)}","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d014ef8b92c530c","tex":"f^{(m)}","aliases":"f^(m)","display":"f^{(m)}","kind":"use","excerpt":"The singular packet results begin with trace data. If a piecewise smooth function has a jump in its derivative of order m at τ_q, then the packet jumpf^(m)_τ_q frac(t-τ_q)_+^mm! removes exactly that trace jump. Subtracting all packets through order r-1 leaves a global W^r,p remainder. The packet coefficients are uniquely determined by the one-sided","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-065a29e259e230ea","tex":"f^{(r)}","aliases":"f^(r)","display":"f^{(r)}","kind":"use","excerpt":"sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided derivatives through order r-1 at every breakpoint and belongs to W^r,p(0,1). Its weak derivative of order r equals the piecewise classical derivative f^(r) almost everywhere.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3382034c436fa8d","tex":"F^*","aliases":"F^*","display":"F^*","kind":"use","excerpt":"Exact correction of an error span means that one completely positive trace preserving recovery R satisfies R(Fρ F^*)=q_Fρ for every F in that span and every code-supported matrix ρ, with a nonnegative scalar q_F depending on F but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restore","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08c5fee29bc1411d","tex":"F^{-1}","aliases":"F^-1","display":"F^{-1}","kind":"use","excerpt":"[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49ddaa56c96d4e05","tex":"f^{-1}","aliases":"f^-1","display":"f^{-1}","kind":"use","excerpt":"The reciprocal is integral. For f∈ I_p with constant coefficient one, the usual power-series reciprocal is used. Its inverse equation yields the exact coefficient recurrence a_f^-1(d)=δ_d0-∑_r=0^d-1 binom dr a_f^-1(r)a_f(d-r). Strong induction proves that all of these coefficients are integral. Combining this construction with the preceding all-ord","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0d653662a671cb05","tex":"\\mathcal F^2","aliases":"F^2 mathcal","display":"\\mathcal F^2","kind":"use","excerpt":"0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and Q=( F^2-1)/2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-995485db55facd82","tex":"F^{2m}","aliases":"F^2m","display":"F^{2m}","kind":"use","excerpt":"Direct expansion gives Ω(η+h) = Ω(η)+dh+[η,h]+frac12[h,h]. The filtration rule places [h,h] in F^2m L^2, which is contained in F^m+1 L^2 because m≥1. The congruence follows. The last statement is [source reference] applied to the induced graded correction map.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3841164c6bcf3539","tex":"F^4","aliases":"F^4","display":"F^4","kind":"use","excerpt":"Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1. The half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7784b7f5b08b435","tex":"F^\\bullet","aliases":"F^bullet bullet","display":"F^\\bullet","kind":"use","excerpt":"[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9954ea8901f689c","tex":"F^{d(\\alpha,\\beta)}","aliases":"F^d(α,β) alpha beta","display":"F^{d(\\alpha,\\beta)}","kind":"use","excerpt":"[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d6125ad8c6995d2","tex":"f^h","aliases":"f^h","display":"f^h","kind":"use","excerpt":"[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2054708cff15ab72","tex":"f^h_i","aliases":"f^h_i","display":"f^h_i","kind":"use","excerpt":"[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-161ae3b661df7aaf","tex":"F^i","aliases":"F^i","display":"F^i","kind":"use","excerpt":"[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5e957ee55f3194f","tex":"F^{i+j}","aliases":"F^i+j","display":"F^{i+j}","kind":"use","excerpt":"[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a637cab476ceffaa","tex":"F^j","aliases":"F^j","display":"F^j","kind":"use","excerpt":"[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4256d948f584b864","tex":"F^{N+1}","aliases":"F^N+1","display":"F^{N+1}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a02562f7de1f19b","tex":"F^{W(\\gamma)}","aliases":"F^W(γ) gamma","display":"F^{W(\\gamma)}","kind":"use","excerpt":"l, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d218908ad4d0b0a2","tex":"F^\\times","aliases":"F^× times","display":"F^\\times","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4edc94a6363ddff2","tex":"F^\\infty","aliases":"F^∞ infty","display":"F^\\infty","kind":"use","excerpt":"[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7fe97397b4335ab","tex":"f_{(k-d)\\bmod N}","aliases":"f_(k-d)bmod N bmod","display":"f_{(k-d)\\bmod N}","kind":"use","excerpt":"inal sample δ_1. Under DI_2 it stays even but moves to channel address one: this is δ_2. Thus J(D)^2=DI_2, H(D)=I_2+2J(D)-J(D)^2+J(D)^3 recovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers. headSigned delays need the same care For an original-sample delay d at","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ef89c7821d00e4a","tex":"F_*","aliases":"F_*","display":"F_*","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80c95a4ca2fcbcc6","tex":"F_{1,0}","aliases":"F_1,0","display":"F_{1,0}","kind":"use","excerpt":"j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1. Its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the product formula is empty, so the same argument applies without change. Equivalently, F_1,0(t)=e^-t.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5396e69399603fd3","tex":"F_{11}","aliases":"F_11","display":"F_{11}","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4c782c5c6e5ba3b","tex":"F_{14}","aliases":"F_14","display":"F_{14}","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f01538099e62c4bc","tex":"F_{17}","aliases":"F_17","display":"F_{17}","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6210caf9e2379390","tex":"F_1^{-1}","aliases":"F_1^-1","display":"F_1^{-1}","kind":"use","excerpt":"Transport from chart a to chart b is forced: J_b← a=F_b^-1F_a. The route 0→1→2 agrees with the direct route because the middle frame cancels: (F_2^-1F_1)(F_1^-1F_0)=F_2^-1F_0. Returning to chart 0 cancels every remaining frame. The final coordinate vector is (3,2)^ T, not a sheared or rotated variant.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"vpage9One vector, three exact framesMODEL C / CANONICAL TRANSPORT HAS NO HOLONOMYChapter III, mathematical text: Theorem 4.1, Theorem 4.2 Let F_a synthesize chart coordinates into one ambient QQ^2. Choose F_0=I, F_1=mat1 · 1 0 · 1, F_2=mat2 · 0 1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-101fbe8a1f9aa65c","tex":"\\F_2","aliases":"F_2 F mathbb","display":"{\\mathbb F}_2","kind":"use","excerpt":"The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fiber does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"vpage6Selectors survive without divisionMODEL A / RESIDUE MASKS IN CHARACTERISTIC TWOChapter II, mathematical text: Proposition 4.2 Work first in F_2, where 1+1=0. The eight coefficients below are just a finite display of a coefficient sequence. The even and odd projectors are diagonal masks with entries zero and one.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","excerpt":"Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30b4f88bb6be80ad","tex":"F_2^{-1}","aliases":"F_2^-1","display":"F_2^{-1}","kind":"use","excerpt":"Transport from chart a to chart b is forced: J_b← a=F_b^-1F_a. The route 0→1→2 agrees with the direct route because the middle frame cancels: (F_2^-1F_1)(F_1^-1F_0)=F_2^-1F_0. Returning to chart 0 cancels every remaining frame. The final coordinate vector is (3,2)^ T, not a sheared or rotated variant.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49ea7c696066b690","tex":"\\mathbb F_2^2","aliases":"F_2^2 mathbb","display":"\\mathbb F_2^2","kind":"use","excerpt":"e P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced domain of the additive character, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-426b2d59b3fd2908","tex":"\\mathbb F_2^4","aliases":"F_2^4 mathbb","display":"\\mathbb F_2^4","kind":"use","excerpt":") is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1}. Put H_0=kerλ⊂ H. The calculation proves P/H_0≃ F_2^4, E=Ind_J^P","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcddfe19017f14c5","tex":"\\mathbb F_2^\\times","aliases":"F_2^× mathbb times","display":"\\mathbb F_2^\\times","kind":"use","excerpt":"For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74317064bf2c9a4d","tex":"F_3","aliases":"F_3","display":"F_3","kind":"use","excerpt":"e a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. Since F_3=F_2D_3^-1, C_3(n)-C_2(n)=∑_i≥1h_ibinom4n4iC_2(n-i). The scaled-binomial argument","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08c122811477fa77","tex":"\\F_3","aliases":"F_3 F mathbb","display":"{\\mathbb F}_3","kind":"use","excerpt":"Three checked boundaries. The initial row. For (p,m)=(3,2), the same reduced recurrence gives e_0=1 and e_1=-1=2 in F_3. Its proposed period is one, so the positive-tail result cannot in general include row zero.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54c7a07f66f1253f","tex":"\\mathbb F_3","aliases":"F_3 mathbb","display":"\\mathbb F_3","kind":"use","excerpt":"The field or orthogonality condition is essential. A primitive element of order N in a ring with zero divisors does not by itself force the geometric sums away from the diagonal to vanish. For example, in F_3× F_3 the element (-1,1) has order two, but the second component of its nontrivial character sum is nonzero. The coefficient selector uses no ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a3a13a2b63deda5","tex":"F_{3k+j}","aliases":"F_3k+j","display":"F_{3k+j}","kind":"use","excerpt":"[t] centering includegraphics[width=.95linewidth,height=3.15in,keepaspectratio]../../figures/reconstructed/recurrence_packets.pdf captionNew reconstruction. Fibonacci residues F_3k+j, with F_0=0 and F_1=1. Each packet evolves by the same jump matrix C^3 and has a different initial address.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c547ea3567b9987","tex":"F_{4,j}","aliases":"F_4,j","display":"F_{4,j}","kind":"use","excerpt":"Write C_j=C_4,j and F_j=F_4,j. The all-row theorem removes the threshold, but at N=4 it leaves a further factorial gain unused. The exact algebra below proves that gain for all four addresses and identifies several infinite families where it is sharp.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1a1f9e3594eef3e","tex":"F_5","aliases":"F_5","display":"F_5","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a27cb068825ed9a","tex":"\\F_5","aliases":"F_5 F mathbf","display":"{\\mathbf F}_5","kind":"use","excerpt":"[Finite packet complex] The exact modular test uses two invariant subcomplexes over F_5. Their cohomology dimensions are (0,0,0), (1,2,1), and the reconstructed total is dim(H^0,H^1,H^2)=(1,2,1). The packet-sum residual is zero.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75c45938e81d6f92","tex":"\\mathbf F_5^2","aliases":"F_5^2 mathbf","display":"\\mathbf F_5^2","kind":"use","excerpt":"An explicit realization of the finite pattern. A separate finite-complex implementation uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linear","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ecbf3d4265f3e244","tex":"\\mathbf F_5^4","aliases":"F_5^4 mathbf","display":"\\mathbf F_5^4","kind":"use","excerpt":"An explicit realization of the finite pattern. A separate finite-complex implementation uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linear","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23af4eb7208f9cad","tex":"\\mathbb F_7","aliases":"F_7 mathbb","display":"\\mathbb F_7","kind":"use","excerpt":"vpage7A Fourier certificate you can inspectMODEL A / AN EXACT NINE-ENTRY TABLEChapter II, mathematical text: Theorem 4.3 Take N=3 and work in F_7. Set ω=2, ω^-1=4, η=3^-1=5. Here 2^3=1 and neither 2 nor 2^2 is one. The unnormalized geometric sums are the entries in the first matrix.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21edda27985631b1","tex":"F_8","aliases":"F_8","display":"F_8","kind":"use","excerpt":"ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibona","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"Let every chart a carry a linear frame equivalence F_a:Mxrightarrow∼M. The formal Chapter III module defines transport from a to b by J_b← a=F_b^-1F_a. This frame formula means synthesize in chart a, then analyze in chart b.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21527d66bde36b36","tex":"f_{a+b,s+p^at}","aliases":"f_a+b,s+p^at","display":"f_{a+b,s+p^at}","kind":"use","excerpt":"[Exact local branch reconstruction] The family {f_a,s}_0≤ s<p^a determines f uniquely on Z_p. More precisely, if y∈ s+p^a Z_p, then f(y)=f_a,s(fracy-sp^a). For a,b≥1, (f_a,s)_b,t = f_a+b,s+p^at. Thus local refinement is exactly the p adic digit tree.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45843ed5f03c5c85","tex":"f_{a,s}","aliases":"f_a,s","display":"f_{a,s}","kind":"use","excerpt":"For a≥1 and 0≤ s<p^a, define f_a,s(x)=f(p^ax+s).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a67dba11d23b584e","tex":"F_a^*","aliases":"F_a^*","display":"F_a^*","kind":"use","excerpt":"Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a645ccc01f63de95","tex":"F_a^{-1}","aliases":"F_a^-1","display":"F_a^{-1}","kind":"use","excerpt":"1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f23aed2f353ac85e","tex":"f_{AB}^{\\ \\ C}","aliases":"f_AB^ C","display":"f_{AB}^{\\ \\ C}","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8dedba4191c4f216","tex":"F_b^{-1}","aliases":"F_b^-1","display":"F_b^{-1}","kind":"use","excerpt":"Let every chart a carry a linear frame equivalence F_a:Mxrightarrow∼M. The formal Chapter III module defines transport from a to b by J_b← a=F_b^-1F_a. This frame formula means synthesize in chart a, then analyze in chart b.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c68b67e2e253be8c","tex":"f_c","aliases":"f_c","display":"f_c","kind":"use","excerpt":"A constraint diagnostic that vanishes automatically. Section 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives C_aP=0 for every a. Absorb the factor i and any structure coefficient into f_c, and write A_ab=C_aC_b-C_bC_a-∑_cf_cC_c. [Exact common-kernel compression] If C_cP=0 for every","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a5ce67bd957032f","tex":"F_c^n","aliases":"F_c^n","display":"F_c^n","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-879f46b871236f65","tex":"F_d","aliases":"F_d","display":"F_d","kind":"use","excerpt":"Let p be prime, R= Z_p, K= Q_p, and d,s,r∈ N. Let F=(F_1,…,F_d)∈ R[X_1,…,X_d]^d, a∈ R^d, and let J=DF(a)=(∂_jF_i(a))_i,j. Assume J∈GL_d(K), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then ∃! b∈ a+p^s+1R^d: F(b)=0, b-a∈ p^r-sR^d. All vector divisibility assertions are coordinatewise. The uniqueness predicate is the entire ball of depth s+1: competitors ne","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2833b5c1542c1219","tex":"F_f^j","aliases":"F_f^j","display":"F_f^j","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79022e1216743208","tex":"F_f^n","aliases":"F_f^n","display":"F_f^n","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1551bc759aa860fc","tex":"f_h","aliases":"f_h","display":"f_h","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e336703b8375549b","tex":"F_i^{-1}","aliases":"F_i^-1","display":"F_i^{-1}","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z^10. Applying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original F. headSelection is also a symmetry condition For the selected polynomial F_2, rotation gives F_2(iz)=-F_2(z). More generally, a residue-j section","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ac4acd8af2778de","tex":"F_{j-1}","aliases":"F_j-1","display":"F_{j-1}","kind":"use","excerpt":"[Exact discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e03cba7ba564271b","tex":"F_j^*","aliases":"F_j^*","display":"F_j^*","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43d33b9d6c0a8720","tex":"f_j^h","aliases":"f_j^h","display":"f_j^h","kind":"use","excerpt":"[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bf87b4a8f3226a8","tex":"F_{jk}","aliases":"F_jk","display":"F_{jk}","kind":"use","excerpt":"vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATIONChapter VIII: Theorem 4.3. Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e04443bf0b17f7b","tex":"F_k","aliases":"F_k","display":"F_k","kind":"use","excerpt":"[Background and packet error] Suppose the true and estimated packets use the same amplitude list (a_i)_i=0^I, but possibly different regions. Then j-widehat j _p^p = ∑_i,k|a_i-a_k|^pμ(E_i∩ F_k). Only off-diagonal overlaps contribute.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilp","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f70b91b12ba08060","tex":"f_{m+1,m}","aliases":"f_m+1,m","display":"f_{m+1,m}","kind":"use","excerpt":"Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0fe9f993f4ff4310","tex":"F_N","aliases":"F_N","display":"F_N","kind":"use","excerpt":"n independent Lucas sequence Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3f4ba757b6d2cf8","tex":"f_n","aliases":"f_n","display":"f_n","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9c7267a8d94d86f","tex":"F_{N+1}","aliases":"F_N+1","display":"F_{N+1}","kind":"use","excerpt":"n independent Lucas sequence Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcc6851c91aea042","tex":"f_{n+h}","aliases":"f_n+h","display":"f_{n+h}","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a93e15faf4073060","tex":"F_{N+j}","aliases":"F_N+j","display":"F_{N+j}","kind":"use","excerpt":"Uniqueness from the first two packet values. For a fixed N,j, any integer sequence satisfying the proved sampled recurrence and starting at F_j,F_N+j agrees with F_Nk+j for all k. This is proved by two-step induction. It is a consequence of the established coefficients and recurrence, not a premise used to identify Fibonacci values initially.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4653715eb491a76","tex":"F_{N,0}","aliases":"F_N,0","display":"F_{N,0}","kind":"use","excerpt":"Modules. mcodeEulerSeriesInverse, mcodeRiccatiFlatPropagation, mcodeRatioRiccati. 4. All-residue assembly and original coefficients. The standard-window source and inverse results supply R_1 and F_N,0. Induction over 1≤ c<N now has two proved cases. At pmid c, the singular weighted estimate and the new substitution theorem give flatness. At pnmid c","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a208d9b8983e3526","tex":"F_{N,j}","aliases":"F_N,j","display":"F_{N,j}","kind":"use","excerpt":"The formalization does not stop at a recurrence. In the variable z=t^N, construct K_N,j(z)=∑_n≥0fracj!(Nn+j)!z^n, F_N,j(z)=∑_n≥0fracC_N,j(n)(Nn)!z^n. It proves the factorial coefficient identity, then F_N,jK_N,j=1 and therefore F_N,j=K_N,j^-1.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe274658cc6c2602","tex":"F_{N,j+1}","aliases":"F_N,j+1","display":"F_{N,j+1}","kind":"use","excerpt":"Let K_j(t) be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is F_N,j(t)=j! K_j(t)^-1. The source/inverse identities and the factorial successor identity prove F_N,j+1=F_N,jR_j+1, F_N,j=F_N,0∏_c=1^jR_c. The product formula is not an assembly hypothesis. Proved flatness of all factors and the o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-44f22654adae6cd4","tex":"f_{n,m+1}","aliases":"f_n,m+1","display":"f_{n,m+1}","kind":"use","excerpt":"Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9a66bfb8f08c990","tex":"F_{N-1}","aliases":"F_N-1","display":"F_{N-1}","kind":"use","excerpt":"[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( - A_N,j mathrel, frac(λ z)^NN^N ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fda40c00a5457366","tex":"\\F_p","aliases":"F_p F mathbb","display":"{\\mathbb F}_p","kind":"use","excerpt":"begingroupsmallraggedright Formal correspondence. leanHurwitz.shift_convolution, leanHurwitz.prime_shift_convolution, leanHurwitz.convolution_left_eq_zero, leanHurwitzInverse.inverse_convolution, and leanHurwitzInverse.inverse_prime_difference. Working directly with coefficient sequences avoids the undefined expression 1/n! in F_p when pmid n!. end","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9da98cb9c6f0ae74","tex":"\\mathbb F_p","aliases":"F_p mathbb","display":"\\mathbb F_p","kind":"use","excerpt":"odulo 2 has zero obstruction and lifts to 1,3 modulo 4. Both have nonzero obstruction at the next step. There is no root modulo 8 and no compatible infinite lift. item The raw Jacobian cokernel need not be presentation independent: the presentations F=(X) and G=(X,0) define the same solution system. Their raw obstruction groups at zero are respecti","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cad1ac88f506b2e2","tex":"\\F_p^2","aliases":"F_p^2 F mathbb","display":"{\\mathbb F}_p^2","kind":"use","excerpt":"The raw cokernel already reveals the difficulty. The presentations F=(X) and G=(X,0) describe the same zero locus over every coefficient ring. At zero their Jacobians are, respectively, F_pxrightarrow 1 F_p, F_pxrightarrow (1,0) F_p^2. Their raw cokernels are 0 and F_p. Both distinguished obstruction classes vanish, but the ambient obstruction grou","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aebbcb2c8d779eb8","tex":"\\F_p^d","aliases":"F_p^d F mathbb","display":"{\\mathbb F}_p^d","kind":"use","excerpt":"The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the chapter supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual digit","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d6d5a8ba40d633c","tex":"\\mathbb F_p^d","aliases":"F_p^d mathbb","display":"\\mathbb F_p^d","kind":"use","excerpt":"The divisions are exact integer divisions. Regard J as a linear map F_p^d→ F_p^r. Define","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c58f74d8b2dada8e","tex":"\\F_p^e","aliases":"F_p^e F mathbb","display":"{\\mathbb F}_p^e","kind":"use","excerpt":"For prime p, rank-nullity gives dimker J_x=d-ρ_x and dim( F_p^e/im J_x)=e-ρ_x. The cardinality of an n-dimensional F_p-space is p^n. Combine these identities with the obstruction criterion and the residue equivalence.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd1168baff4f0420","tex":"\\mathbb F_p^e","aliases":"F_p^e mathbb","display":"\\mathbb F_p^e","kind":"use","excerpt":"Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Write r_x=F(x)/p^L, b_x=r_xbmod p, A_x=DF(x)bmod p: F_p^d⟶ F_p^e, ρ_x=rank_ F_pA_x. The actual lift set is L_x,L= {y∈( Z/p^L+1 Z)^d: y≡ xpmodp^L, F(y)=0}, where polynomial evaluation in the displayed set is modulo p^L+1. Define O_x= F_p^e/imA_x, o_x=[b_x]. Lean proves # L_x,L= p^d-ρ","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d7ee4d0e8e62dda","tex":"\\mathbb F_p^{e+q}","aliases":"F_p^e+q mathbb","display":"\\mathbb F_p^{e+q}","kind":"use","excerpt":"Let p be prime, L≥1, e,q≥0, and F∈ Z[X_1,…,X_e+q]^e, x∈ Z^e+q, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^e+q⟶ F_p^e is surjective. All vector congruences are coordinatewise.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec134db719aa30eb","tex":"\\F_p^k","aliases":"F_p^k F mathbb","display":"{\\mathbb F}_p^k","kind":"use","excerpt":"ints] Let q=p be prime in Theorem~[source reference]. For every τ∈ t_0+p^L Z_p^k, there are an integer tower z_n and b∈ Z_p^m+k such that F(b)=0, free(b)=τ, z_n⟶ b, b-z_n∈ p^L+n Z_p^m+k. The initial value is z_0=x. At every stage and for every proposed parameter digit v∈ F_p^k, the corresponding next residue fiber has exactly p^m-e elements. If m=e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fa0d9b2392a00a1","tex":"\\mathbb F_p^k","aliases":"F_p^k mathbb","display":"\\mathbb F_p^k","kind":"use","excerpt":"Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6edbc189be464124","tex":"\\mathbb F_p^r","aliases":"F_p^r mathbb","display":"\\mathbb F_p^r","kind":"use","excerpt":"The divisions are exact integer divisions. Regard J as a linear map F_p^d→ F_p^r. Define","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c3489d862af388a","tex":"\\F_p^\\times","aliases":"F_p^× F times mathbb times","display":"{\\mathbb F}_p^\\times","kind":"use","excerpt":"Induction gives the geometric identity. Reduction of a unit to F_p^× gives u_i^p-1=1 modulo p. Taking s powers supplies p^s in each summand. The difference operator is linear over finite sums.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4cf4c741d2045e51","tex":"\\mathbb F_p^\\times","aliases":"F_p^× mathbb times","display":"\\mathbb F_p^\\times","kind":"use","excerpt":"This procedure never substitutes an unjustified extension through a nonmaximal centralizer. It proves full-character compatibility, including shallow units, rather than merely cancellation of a leading trace. The residue factor fixes ω on F_p^×, and an N-th-root choice for the noncompact generator fixes ω(t). Every choice is recorded before H5. See","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52602bf9fb1a4e99","tex":"F_{\\Pcal\\leftarrow\\Pcal}","aliases":"F_Pcal←Pcal F_ P← P Pcal leftarrow Pcal mathcal leftarrow mathcal","display":"F_{{\\mathcal P}\\leftarrow{\\mathcal P}}","kind":"use","excerpt":"[Dynamical equivalence] Suppose X=Y and the same packet system is used on both sides. Packet coordinates preserve fixed points, periodic points, injectivity, surjectivity, and invertibility. In particular, F(x)=x Longleftrightarrow F_ P← P( A_ Px)= A_ Px.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7de303254524fac2","tex":"F_{P\\circ f}","aliases":"F_P∘ f circ","display":"F_{P\\circ f}","kind":"use","excerpt":"Tensor properties and the remaining global boundary. The coefficient is proved bilinear, alternating, and supported on the range on both sides. The chain rule proves the pullback identity F_P∘ f(x)(v,w)=F_P(f(x))(Df(x)v,Df(x)w). The main theorem has global regularity hypotheses on normed parameter spaces. The pullback identity alone does not supply","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b0ff167397251af","tex":"F_Q","aliases":"F_Q","display":"F_Q","kind":"use","excerpt":"Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v) F_Q(u,v) . Assume F takes values in Y_P× Y_Q, is twice continuously Fr echet differentiable near (u_0,v_0), satisfies F_Q(u_0,v_0)=0, and A=D_vF_Q(u_0,v_0) is a Banach space isomorphism from V to Y_Q. The impl","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ba2ac5456d322b89","tex":"f_q","aliases":"f_q","display":"f_q","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fff6a4a560a0981b","tex":"F_{\\Qcal'\\leftarrow\\Pcal'}","aliases":"F_Qcal'←Pcal' F_ Q'← P' Qcal leftarrow Pcal mathcal leftarrow mathcal","display":"F_{{\\mathcal Q}'\\leftarrow{\\mathcal P}'}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ef1b4fdfc9f7935","tex":"f_R","aliases":"f_R","display":"f_R","kind":"use","excerpt":"The component maps f_R:A→ C_R and g_R:B→ C_R have the following universal property. For every set Y and maps h:A→ Y, k:B→ Y satisfying h(a)=k(b) whenever R(a,b), there is a unique map u:C_R→ Y with u∘ f_R=h and u∘ g_R=k.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6470f50faec26a53","tex":"F_r'","aliases":"F_r'","display":"F_r'","kind":"use","excerpt":"ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8bb98f0e3a56f2b0","tex":"F_{r-1}","aliases":"F_r-1","display":"F_{r-1}","kind":"use","excerpt":"ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4565bf241049378c","tex":"F_r^*","aliases":"F_r^*","display":"F_r^*","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ecf69680e196599f","tex":"F_r^{-1}","aliases":"F_r^-1","display":"F_r^{-1}","kind":"use","excerpt":"Every identity is cancellation of adjacent F_rF_r^-1 factors. The Lean definitions use leannameLinearEquiv.trans. The displayed formulas use the equivalent right-to-left notation for function composition fixed in this chapter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-faf064f796e64232","tex":"F_{\\rm root}","aliases":"F_rm root rm","display":"F_{\\rm root}","kind":"use","excerpt":"Assume every keep error is a nonnegative integer w_v. Let B∈ N. For each node v, define F_v(b) as the minimum cost of a pruning of the subtree at v with total error exactly b. The keep option contributes F_v(w_v)≤κ_v. The split option is obtained by min plus convolution of the child tables, followed by addition of σ_v. The smallest value among F_rm","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21c255185d10999e","tex":"F_{\\mathrm{root}}","aliases":"F_root mathrm","display":"F_{\\mathrm{root}}","kind":"use","excerpt":"The root scan min_0≤ b≤ BF_root(b) is therefore the exact budget-constrained optimum. Increasing the budget cannot increase a finite optimum. The theorem retains feasibility hypotheses where necessary rather than treating codenone as an ordinary real cost.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30ade5d863641faa","tex":"F_s","aliases":"F_s","display":"F_s","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","excerpt":"Entrywise implementation theorem. For every b≤ B, Lean proves lookup(storedTree(B,t),b)=F_t(b). There is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at b depends only on input entries at most b, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f79f277673daa5a5","tex":"f_t","aliases":"f_t","display":"f_t","kind":"use","excerpt":"Connecting power series and spectral calculus. The shifted generator is proved Hermitian. The implementation then identifies the actual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3f85bd975f20dc4","tex":"F_V","aliases":"F_V","display":"F_V","kind":"use","excerpt":"[Positivity and composition] The operators F_V(t)= e^t(Q^T-D_V) form a positive semigroup. If f≥0, then F_V(t)f≥0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78b29d109052da32","tex":"F_v","aliases":"F_v","display":"F_v","kind":"use","excerpt":"F_v(b) · A budget table · Does b mean exact weight, or an upper bound?","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e68207ff3f414911","tex":"F_{v_i}","aliases":"F_v_i","display":"F_{v_i}","kind":"use","excerpt":"Keeping node v costs κ_v and incurs integer error w_v. Splitting costs overhead σ_v in addition to the selected child cuts. The exact-weight recurrence is F_v(b)=min{ l κ_v if b=w_v, [2pt] displaystyleσ_v+ min_∑_i b_i=b∑_iF_v_i(b_i) if v is internal. . Nonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-pl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a75d31ba6a020ef2","tex":"F_z","aliases":"F_z","display":"F_z","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1a6f7405620e755","tex":"F_z^*","aliases":"F_z^*","display":"F_z^*","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ecd9e4a3964eb4b","tex":"f_\\alpha","aliases":"f_α alpha","display":"f_\\alpha","kind":"use","excerpt":"[Universal mapping property of finite packet coordinates] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-94aa4c6f6cc42ad6","tex":"F_{\\theta\\phi}","aliases":"F_θϕ theta phi","display":"F_{\\theta\\phi}","kind":"use","excerpt":"EXACT BLOCH PROJECTOR CALCULATIONChapter VII: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-832e56d8bd1b7868","tex":"F_\\lambda","aliases":"F_λ lambda","display":"F_\\lambda","kind":"use","excerpt":"The dynamic recurrence gives the same Lagrangian choices without listing prunings. At a, it compares 2+4λ with 4+λ. At r, it compares 1+6λ with 1+F_λ(a). Its exactness follows from the finite tree and additive model in [source reference]. Overlapping decompositions or nonlocal execution costs need another argument.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da614697fd2616e5","tex":"f_\\pi^2","aliases":"f_π^2 pi","display":"f_\\pi^2","kind":"use","excerpt":"P_0,P_1,…,P_r come from increasingly fine partitions, so that Ran P_0⊆Ran P_1⊆⋯⊆Ran P_r. Define D_0=P_0, D_j=P_j-P_j-1 (1≤ j≤ r). Then P_iP_j=P_min(i,j), D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j. Every D_j is an orthogonal projector, and normP_rf_π^2 = ∑_j=0^rnormD_jf_π^2 and norm f_π^2 = ∑_j=0^rnormD_jf_π^2 + normf-P_rf_π^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a785141dca19cd38","tex":"F_{\\chi_i}","aliases":"F_χ_i chi","display":"F_{\\chi_i}","kind":"use","excerpt":"For a finite family of nontrivial ordinary class characters, suppose functions p,Z satisfy h p(h)→ P≠0, h Z(h)→ Q≠0, and, for all sufficiently small h>0, Z(h)=(p(h)∏_i F_χ_i(h))^e, e∈ N. The checked real-axis criterion then gives e=1 and F_χ_i(0)≠0 for every i. The pole limits and product identity remain hypotheses of this application.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bb2930101570bd3","tex":"F_\\ell","aliases":"F_ℓ ell","display":"F_\\ell","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06e67f71ff301b19","tex":"\\fall","aliases":"fall","display":null,"kind":"use","excerpt":"[Dyadic Euler coordinate pair] For every even n≥0, E_n = ∑_r=0^n/2 β_r2^2r fall n2r R_n-2r. The inverse transform is R_n = ∑_r=0^n/2 frac2^4r+1(2r+2)! fall n2r E_n-2r.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79071076af4746fa","tex":"\\Fcal_{\\theta\\phi}","aliases":"Fcal_θϕ F_θϕ Fcal theta phi mathcal theta phi","display":"{\\mathcal F}_{\\theta\\phi}","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e597b1821a53be0","tex":"\\Fdet","aliases":"Fdet det_ F operatorname","display":"{\\operatorname{det}_{\\!F}}","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0d10319b98111ddb","tex":"\\FF_2^2","aliases":"FF_2^2 F_2^2 FF mathbb","display":"{\\mathbb F}_2^2","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f392c9ccb437bdd0","tex":"\\FF_2^4","aliases":"FF_2^4 F_2^4 FF mathbb","display":"{\\mathbb F}_2^4","kind":"use","excerpt":"(J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-380df1730c5193bc","tex":"\\FF_2^\\times","aliases":"FF_2^× F_2^× FF times mathbb times","display":"{\\mathbb F}_2^\\times","kind":"use","excerpt":"The construction retains the actual canonical beta extension and can be completed with any prescribed central quasicharacter omega_F trivial on U_F 1. Because k_F^×= F_2^× is trivial, such an omega_F is unramified. Thus this covers every tamely ramified central quasicharacter for this particular residue field.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8ae171b4ed22f24","tex":"\\FF_3","aliases":"FF_3 F_3 FF mathbb","display":"{\\mathbb F}_3","kind":"use","excerpt":"[A singular polynomial obstruction] Let p=3 and consider the class x_1=0 modulo three. For F_0(x)=x^2, the residual and the reduced derivative both vanish. Every correction y∈ F_3 gives F_0(3y)≡0pmod9, so the class has three children. For F_1(x)=x^2+3, the reduced derivative is still zero, but the reduced residual is one. Its class in the cokernel ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9e4a1ad9b380105f","tex":"\\FF_3^2","aliases":"FF_3^2 F_3^2 FF mathbb","display":"{\\mathbb F}_3^2","kind":"use","excerpt":"[Rank one Heisenberg group] Take V= F_3^2. The Heisenberg group has order 3^3=27, and its irreducible representation with a fixed nontrivial central character has dimension three. The trace vanishes on every noncentral element. Averaging the translation subgroup gives a rank one projector.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78b205d811df544a","tex":"\\FF_7","aliases":"FF_7 F_7 FF mathbb","display":"{\\mathbb F}_7","kind":"use","excerpt":"Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f90b39e544361804","tex":"\\FF_7^3","aliases":"FF_7^3 F_7^3 FF mathbb","display":"{\\mathbb F}_7^3","kind":"use","excerpt":"The coordinate map (a,b)⟼(a,-a+b,-b) identifies the displayed matrices with the permutation action on the sum zero subspace of F_7^3. This action is faithful. The four matrices 1,s,t,st span M_2 over every field of characteristic seven: the determinant of their coordinate matrix is 4≠0. A subspace stable under the group is therefore stable under ev","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b447983b8333707","tex":"\\FF_p^2","aliases":"FF_p^2 F_p^2 FF mathbb","display":"{\\mathbb F}_p^2","kind":"use","excerpt":"[Nondegenerate twisted algebra] Let A= F_p^2 with its standard alternating form. A cocycle with that commutator form gives C^c[A]≅ M_p( C). The unique irreducible projective representation has dimension p.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f507805b8e18f517","tex":"\\FF_p^d","aliases":"FF_p^d F_p^d FF mathbb","display":"{\\mathbb F}_p^d","kind":"use","excerpt":"[One level linear lifting] A class represented by x_L lifts to a solution modulo p^L+1 if and only if Ob^lin_L(x_L)=0. More precisely, a correction y∈ F_p^d gives a lift x_L+1=x_L+p^Ly pmod p^L+1 if and only if overline My=-ρ_L(x_L). When one correction exists, the set of all correction classes is a coset of keroverline M.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-385760b8fa8c4e92","tex":"\\FF_p^\\times","aliases":"FF_p^× F_p^× FF times mathbb times","display":"{\\mathbb F}_p^\\times","kind":"use","excerpt":"Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-baf8f5a70f2300fa","tex":"\\FF_q","aliases":"FF_q F_q FF mathbb","display":"{\\mathbb F}_q","kind":"use","excerpt":"Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d90572be1e18f074","tex":"\\FF_{q^L}^{\\times}","aliases":"FF_q^L^× F_q^L^× FF times mathbb times","display":"{\\mathbb F}_{q^L}^{\\times}","kind":"use","excerpt":"[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-217db3583e34db9b","tex":"\\FF_{q^n}^{\\times}","aliases":"FF_q^n^× F_q^n^× FF times mathbb times","display":"{\\mathbb F}_{q^n}^{\\times}","kind":"use","excerpt":"Characters of F_q^n^× are indexed by exponents modulo q^n-1, and Frobenius acts by multiplication by q.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a8c6f57b92bedc3","tex":"\\FF_{q^r}^{\\times}","aliases":"FF_q^r^× F_q^r^× FF times mathbb times","display":"{\\mathbb F}_{q^r}^{\\times}","kind":"use","excerpt":"For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-642d1aafbe0d4290","tex":"\\operatorname{fill}","aliases":"fill operatorname","display":"\\operatorname{fill}","kind":"use","excerpt":"Correctness of the actual update. For any edited subtree T' and valid context C, Lean proves updateValue(C,F(T'))=F(fill(C,T')). The proof inducts through the context. At each frame, the updated child contributes the new value, and each other child contributes its unchanged cached optimum. The frame recurrence is therefore exactly the full-tree rec","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed487b1e5f57fec8","tex":"\\mathrm{fine}","aliases":"fine mathrm","display":"\\mathrm{fine}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54248a252cc8600a","tex":"\\Fitt","aliases":"Fitt operatorname","display":"{\\operatorname{Fitt}}","kind":"use","excerpt":"[Fitting support criterion] Let R be Noetherian and let M be a finite R module. Then Supp_R(M) = V(Fitt_0^R(M)).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22824516db9af32d","tex":"\\Fitt_0","aliases":"Fitt_0 Fitt operatorname","display":"{\\operatorname{Fitt}}_0","kind":"use","excerpt":"If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjectiv","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-057d9dddfc0ec947","tex":"\\Fitt_0^R","aliases":"Fitt_0^R Fitt operatorname","display":"{\\operatorname{Fitt}}_0^R","kind":"use","excerpt":"[Fitting support criterion] Let R be Noetherian and let M be a finite R module. Then Supp_R(M) = V(Fitt_0^R(M)).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c98109b747bfca0","tex":"\\Fitt_j^R","aliases":"Fitt_j^R Fitt operatorname","display":"{\\operatorname{Fitt}}_j^R","kind":"use","excerpt":"[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0d3a9bd14e2f4149","tex":"\\Fitt_j^S","aliases":"Fitt_j^S Fitt operatorname","display":"{\\operatorname{Fitt}}_j^S","kind":"use","excerpt":"[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-257572a086e647f4","tex":"\\Fitt_j^{\\TT}","aliases":"Fitt_j^TT Fitt_j^ T Fitt TT operatorname mathbb","display":"{\\operatorname{Fitt}}_j^{{\\mathbb T}}","kind":"use","excerpt":"[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14796d921d8dedf4","tex":"\\Fitt_{r-1}","aliases":"Fitt_r-1 Fitt operatorname","display":"{\\operatorname{Fitt}}_{r-1}","kind":"use","excerpt":"Use a finite presentation with target rank n. If r>n, every fiber has dimension at most n, so the left-hand locus is empty. The convention Fitt_r-1(M)=R for r-1≥ n makes the right-hand locus empty as well. It remains to consider 1≤ r≤ n. At p, the fiber dimension is n-rank A( p). It is at least r exactly when the rank is at most n-r, which is equiv","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2abda39fe0f3e23","tex":"\\Fitt_{r-1}^R","aliases":"Fitt_r-1^R Fitt operatorname","display":"{\\operatorname{Fitt}}_{r-1}^R","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80af98c69987e79a","tex":"\\operatorname{Flat}","aliases":"Flat operatorname","display":"\\operatorname{Flat}","kind":"use","excerpt":"All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b0a62a84fa62d7d","tex":"\\flop_{\\rm block}","aliases":"flop_rm block flop rm operatorname rm","display":"{\\operatorname{flop}}_{\\rm block}","kind":"use","excerpt":"[Equal diagonal packets] Let p≥1 and b≥1 be integers and n=pb. Suppose n is divided into p equal diagonal packets of width b. Then flop_rm block=2qnb, M_rm block=nb. The following factor describes the dense-to-block storage reduction and, when q≥1, the arithmetic reduction: fracnb.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cde9aac4c00078d2","tex":"\\flop_{\\rm dense}","aliases":"flop_rm dense flop rm operatorname rm","display":"{\\operatorname{flop}}_{\\rm dense}","kind":"use","excerpt":"[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82a45cf5e0bc4cc8","tex":"\\flop_{\\rm packet}","aliases":"flop_rm packet flop rm operatorname rm","display":"{\\operatorname{flop}}_{\\rm packet}","kind":"use","excerpt":"[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43a1003114401521","tex":"\\Frac","aliases":"Frac operatorname","display":"{\\operatorname{Frac}}","kind":"use","excerpt":"Let O be a DVR, K=Frac( O), and A∈End_ O(Λ) for a finite free lattice Λ. Suppose the characteristic polynomial factors over O as χ_A(x)=f(x)g(x), where f and g are coprime over K[x].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa40e90aedbf1ce9","tex":"\\frakH","aliases":"frakH H mathfrak","display":"{\\mathfrak H}","kind":"use","excerpt":"Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-997bdc1cc4248ffb","tex":"\\frakm_A","aliases":"frakm_A m_A frakm mathfrak","display":"{\\mathfrak m}_A","kind":"use","excerpt":"and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Comple","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2017fcac4d9be4d5","tex":"\\frakm_B","aliases":"frakm_B m_B frakm mathfrak","display":"{\\mathfrak m}_B","kind":"use","excerpt":"Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it car","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ba1848f9fd13a41","tex":"\\operatorname{free}","aliases":"free operatorname","display":"\\operatorname{free}","kind":"use","excerpt":"∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate proj","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf6efda13a9cd474","tex":"\\Frob","aliases":"Frob operatorname","display":"{\\operatorname{Frob}}","kind":"use","excerpt":"For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d7b4480e3e06144","tex":"\\operatorname{Frob}","aliases":"Frob operatorname","display":"\\operatorname{Frob}","kind":"use","excerpt":"cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j))","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11e212077d740047","tex":"\\Frob^r","aliases":"Frob^r Frob operatorname","display":"{\\operatorname{Frob}}^r","kind":"use","excerpt":"For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7354d07b2ecd363e","tex":"\\Frob_v","aliases":"Frob_v Frob operatorname","display":"{\\operatorname{Frob}}_v","kind":"use","excerpt":"g T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58bab5cbde13c61a","tex":"\\operatorname{Frob}_{v_j}","aliases":"Frob_v_j operatorname","display":"\\operatorname{Frob}_{v_j}","kind":"use","excerpt":"cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j))","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"/2)|u| shows that Φ_m maps B_m into itself. The derivative bound |Φ_m'|≤1/2 makes it a contraction. Banach's fixed-point theorem gives a unique fixed point. On B_m, |a(z-t_m)|≤1<π, so the principal logarithm inverts the exponential and the fixed-point equation is equivalent to G(z)=0. Moreover, |z-t_m|≤frac34s_N|R_N(z)|=O(e^-γ_Nt_m). At this zero, ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b23da150b06fbef","tex":"G''","aliases":"G''","display":"G''","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb695a8d9657252e","tex":"g''","aliases":"g''","display":"g''","kind":"use","excerpt":"For a function g, define the complete differential Bell polynomial Y_N by fracd^Ndz^Ne^g(z) = e^g(z)Y_N(g',g'',…,g^(N)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0c88372240ff8d1","tex":"G'''","aliases":"G'''","display":"G'''","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ad39170ee6fc0f6","tex":"g^{(j)}","aliases":"g^(j)","display":"g^{(j)}","kind":"use","excerpt":"t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62e503385c528c25","tex":"g^{(j+1)}","aliases":"g^(j+1)","display":"g^{(j+1)}","kind":"use","excerpt":"Set g(t)=F(x+th) on the open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b971876a95cdcef2","tex":"g^{(N)}","aliases":"g^(N)","display":"g^{(N)}","kind":"use","excerpt":"For a function g, define the complete differential Bell polynomial Y_N by fracd^Ndz^Ne^g(z) = e^g(z)Y_N(g',g'',…,g^(N)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-208960bb4d25598b","tex":"g^{(r)}","aliases":"g^(r)","display":"g^{(r)}","kind":"use","excerpt":"Set g(t)=F(x+th) on the open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25f0be3ddda5cf8c","tex":"G^*","aliases":"G^*","display":"G^*","kind":"use","excerpt":"[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-565b30cbcdc2ca2a","tex":"G^{-1}","aliases":"G^-1","display":"G^{-1}","kind":"use","excerpt":"Direct multiplication gives GOb G^-1=mat-1 · 1 -1 · 1. For example, the original (0,0) block vanishes, while the new (0,0) block relative to the old coordinate projectors does not. This is why the theorem about preservation of each block's zero status requires G P_i=P_iG.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa45a7ad376e6ef9","tex":"g^{-1}","aliases":"g^-1","display":"g^{-1}","kind":"use","excerpt":"Derived comparison bounds. For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its determinant is","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6a65d66c1c2674f","tex":"G^2","aliases":"G^2","display":"G^2","kind":"use","excerpt":"d product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, star","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","excerpt":"For an actual inverse fg=1, applying the product rule and multiplying by g gives B(g)=-g^2B(f)-gM_p^2(f,g)+gM_p(f,g). If g∈ I, induction on r proves boundedness of g from boundedness of f. At each successor stage the preceding stage supplies boundedness of g needed on the right. Passing to all finite orders proves reciprocal closure for an integral","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ecc79b40ce07ca48","tex":"g^{b-a}","aliases":"g^b-a","display":"g^{b-a}","kind":"use","excerpt":"To prove surjectivity, use compactness. For an isometry g of a compact metric space into itself, the sequence of iterates g^n(x) has a convergent subsequence. Two sufficiently close iterates, with exponents a<b, yield d(x,g^b-a(x)) equally small after removing a isometric iterations. Since b-a>0, the latter point is in the image of g. Thus the imag","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-84fbee9e7706627b","tex":"g^n","aliases":"g^n","display":"g^n","kind":"use","excerpt":"To prove surjectivity, use compactness. For an isometry g of a compact metric space into itself, the sequence of iterates g^n(x) has a convergent subsequence. Two sufficiently close iterates, with exponents a<b, yield d(x,g^b-a(x)) equally small after removing a isometric iterations. Since b-a>0, the latter point is in the image of g. Thus the imag","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef602db61b865627","tex":"G_{\\mathbb Q}","aliases":"G_ Q mathbb","display":"G_{\\mathbb Q}","kind":"use","excerpt":"The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09caf7470ee06fab","tex":"G_{\\mathbb Q_p}","aliases":"G_ Q_p mathbb","display":"G_{\\mathbb Q_p}","kind":"use","excerpt":"The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a566b1db5acde7f","tex":"g_0","aliases":"g_0","display":"g_0","kind":"use","excerpt":"vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESChapter III, mathematical text: Theorem 6.3 Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical cou","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6df94744f7ade62","tex":"g_2","aliases":"g_2","display":"g_2","kind":"use","excerpt":"vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESChapter III, mathematical text: Theorem 6.3 Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical cou","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab6997694ee35093","tex":"g_3","aliases":"g_3","display":"g_3","kind":"use","excerpt":"The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1, g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e051232a9c93730e","tex":"G_{4,1}","aliases":"G_4,1","display":"G_{4,1}","kind":"use","excerpt":"The address-one source is G_4,1(t)=(sinh t+sin t)/(2t). The identities a+b=1, a-b=i, and ab=1/2 give D_1(t):=G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hur","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29d433d9dcbeff86","tex":"G_{4,3}","aliases":"G_4,3","display":"G_{4,3}","kind":"use","excerpt":"The address-three source is G_4,3(t)=3(sinh t-sin t)/t^3. Addition identities give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, o","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"vfig[.97]flat_not_calibrated platecaptionThe rulers display the chosen gauge scales: the marks are g_a n for integer n. They visualize the supplied scalar factors, not an intrinsic metric. Relative transport is the ratio of the target and source scales.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7ac3982b663f850","tex":"g_a^{-1}","aliases":"g_a^-1","display":"g_a^{-1}","kind":"use","excerpt":"vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESChapter III, mathematical text: Theorem 6.3 Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical cou","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91e0e997e5796cd2","tex":"g_{aa}","aliases":"g_aa","display":"g_{aa}","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9135467db1e92ace","tex":"g_{ab}","aliases":"g_ab","display":"g_{ab}","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c710e753abb0a6f2","tex":"g_{ac}","aliases":"g_ac","display":"g_{ac}","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d230c9e7fd4e8bb2","tex":"g_b","aliases":"g_b","display":"g_b","kind":"use","excerpt":"vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESChapter III, mathematical text: Theorem 6.3 Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical cou","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee47ed2cac150272","tex":"G_b^{-1}","aliases":"G_b^-1","display":"G_b^{-1}","kind":"use","excerpt":"Substitute the definition of Γ^G. Adjacent factors G_b^-1G_b and G_c^-1G_c cancel. In the defect, both the composite and direct terms acquire the same outer factors G_c and G_a^-1. In the loop, all inner factors cancel and only conjugation at the base remains.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4413bc5f07c709f","tex":"g_b^{-1}","aliases":"g_b^-1","display":"g_b^{-1}","kind":"use","excerpt":"The route through two steps 0→1→2 has multiplier (3/2)2=3, equal to the direct route 0→2. More generally, (g_cg_b^-1)(g_bg_a^-1)=g_cg_a^-1 for every triple. Thus this connection is flat and reversible, but it is not calibrated to the original J, whose values are identities.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2dda9aa2199e20de","tex":"g_{bc}","aliases":"g_bc","display":"g_{bc}","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd6dc28b158b5d9e","tex":"G_c","aliases":"G_c","display":"G_c","kind":"use","excerpt":"[Gauge covariance] The defect and triangle holonomy transform as Δ^G_c,b,a =G_cΔ_c,b,aG_a^-1, H^G_c,b,a =G_aH_c,b,aG_a^-1. Thus flatness and triviality of triangle holonomy are gauge invariant. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4676088e867baee","tex":"g_c","aliases":"g_c","display":"g_c","kind":"use","excerpt":"The route through two steps 0→1→2 has multiplier (3/2)2=3, equal to the direct route 0→2. More generally, (g_cg_b^-1)(g_bg_a^-1)=g_cg_a^-1 for every triple. Thus this connection is flat and reversible, but it is not calibrated to the original J, whose values are identities.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e041615222944e42","tex":"G_c^{-1}","aliases":"G_c^-1","display":"G_c^{-1}","kind":"use","excerpt":"Substitute the definition of Γ^G. Adjacent factors G_b^-1G_b and G_c^-1G_c cancel. In the defect, both the composite and direct terms acquire the same outer factors G_c and G_a^-1. In the loop, all inner factors cancel and only conjugation at the base remains.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a0cea7293747747","tex":"G_f","aliases":"G_f","display":"G_f","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6fb05800096ed7ee","tex":"G_{F,S}","aliases":"G_F,S","display":"G_{F,S}","kind":"use","excerpt":"For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd7038b72872679a","tex":"G_i","aliases":"G_i","display":"G_i","kind":"use","excerpt":"If p^Lmid c, the actual identities give G(x)=F(x)+cH(x), DG(x)≡ DF(x)pmod p. Thus if F(x)≡0pmodp^L and A_x is surjective, then ∃ b∈ B_L(x):G(b)=0. This theorem allows rectangular systems. The evaluation, derivative and Jacobian identities are proved from the polynomial definition G_i=F_i+C(c)H_i, rather than supplied as hypotheses. Algebraic congru","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec1c980979f73c29","tex":"G_j","aliases":"G_j","display":"G_j","kind":"use","excerpt":"Write F=∑_iF_i and G=∑_jG_j, where F_i= B_N,iF and G_j= B_N,jG. Their covariance gives (F_iG_j)(ω z)=ω^i+jF_i(z)G_j(z), so F_iG_j belongs to the packet labeled by i+j modulo N. Expanding the finite product and applying B_N,k gives the formula.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7288fb9331c8dd5","tex":"g_j","aliases":"g_j","display":"g_j","kind":"use","excerpt":"The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1,","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1d2ec925bc33e3de","tex":"g_{j+1}","aliases":"g_j+1","display":"g_{j+1}","kind":"use","excerpt":"For j=0, g_0 is the literal N-divisibility indicator. Because Nmid p(p-1), the two shifts in δ g_0 agree and δ g_0=0. It is flat at every order. The actual binomial identity gives Λ_j+1g_j+1=(j+1)g_j. As 1≤ j+1<p, the nonsingular reflection theorem inductively proves flatness of every g_j in the standard window. There is no uniform-convergence assu","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-570d8d5f32c21bda","tex":"G_j^{-1}","aliases":"G_j^-1","display":"G_j^{-1}","kind":"use","excerpt":"Pack g_j into degrees divisible by four and form its actual EGF. The formal coefficient identity is G_j(t)=∑_n≥0g_j(n)fract^4n(4n)! =j!∑_n≥0fract^4n(4n+j)!. Its inverse is therefore the original normalized packet EGF. The source Kummer property gives flatness of G_j. The proved reciprocal closure gives flatness of G_j^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5079789e5bc0a138","tex":"G_{jk}","aliases":"G_jk","display":"G_{jk}","kind":"use","excerpt":"[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82170a73bb5aa973","tex":"G_K","aliases":"G_K","display":"G_K","kind":"use","excerpt":"Actual quadratic local conditions and completion. Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embedding. Lean pr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-103f7de3482048f3","tex":"G_k","aliases":"G_k","display":"G_k","kind":"use","excerpt":"Suppose G_k is a best rank-at-most-k approximation to G in either norm. Take an arbitrary ambient matrix B with rank B≤ k. Compression is an admissible small-core competitor and norm L(G)- L(G_k)_ν · =normG-G_k_ν","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09f3e72028e864cd","tex":"G_m","aliases":"G_m","display":"G_m","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a4a7f45555ebe7e","tex":"g_m","aliases":"g_m","display":"g_m","kind":"use","excerpt":"pageheading2. Identification with the reduced Euler sequenceCorollary 8.5; manuscript page 21 Work over F_p= Z/p Z. For positive mmid p-1, take the literal section g_m(d)=1, · mmid d, 0, · mnmid d. It satisfies g_m(0)=1. Since p≡1pmod m, the conditions mmid d+p and mmid d+1 are equivalent. The source therefore proves U^pg_m=Ug_m.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6276707c2b6cfedb","tex":"G_{m-1}","aliases":"G_m-1","display":"G_{m-1}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-358c536fce266469","tex":"G_n","aliases":"G_n","display":"G_n","kind":"use","excerpt":"Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15e892a32e1d61ee","tex":"g_n","aliases":"g_n","display":"g_n","kind":"use","excerpt":"[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-605af900ba9b414b","tex":"G_{N,j}","aliases":"G_N,j","display":"G_{N,j}","kind":"use","excerpt":"Throughout these sections, the normalized source and its reciprocal are G_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, F_N,j(t)=G_N,j(t)^-1 =∑_n≥0C_N,j(n)fract^Nn(Nn)!, where N≥1 and 0≤ j<N. In particular, C_N,j(0)=1. The coefficients in the original convention satisfy E_Nn^(N,j)=j!C_N,j(n), E_d^(N,j)=0 (Nnmid d). Thus every valuation of a nonzero difference gai","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25fd9e2fc804df94","tex":"g_{N,j}","aliases":"g_N,j","display":"g_{N,j}","kind":"use","excerpt":"The packed factorial coefficient of G_N,j is g_N,j(n)=fracj!(Nn)!(Nn+j)! =∏_i=1^j(1+frac Ni n)^-1. The empty product is one. If N=p^a and 1≤ i<N, then v_p(i)<a, so N/i∈ p Z_p. Every affine denominator in [source reference] is therefore a unit on Z_p.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b7e16a20dfc2d99","tex":"G_{\\Q_p}","aliases":"G_Q_p G_ Q_p Q mathbf","display":"G_{{\\mathbf Q}_p}","kind":"use","excerpt":"This inertia is the actual residue-action kernel inside D_A. No equality D_A=G_ Q_p is established in this block. Nor is ramification at two proved: the nontrivial full local character there is a separate result. At three the Z_2-valued sign character is nontrivial on the full local Galois group but trivial on every inertia subgroup covered by the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f8720ad3f121275","tex":"G_r","aliases":"G_r","display":"G_r","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-077b802462413614","tex":"g_R","aliases":"g_R","display":"g_R","kind":"use","excerpt":"The component maps f_R:A→ C_R and g_R:B→ C_R have the following universal property. For every set Y and maps h:A→ Y, k:B→ Y satisfying h(a)=k(b) whenever R(a,b), there is a unique map u:C_R→ Y with u∘ f_R=h and u∘ g_R=k.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2532659d80614fed","tex":"G_r'","aliases":"G_r'","display":"G_r'","kind":"use","excerpt":"For a general depth r, call j_r=max J_r its true terminal. This definition depends on the actual carry minimum. Rewriting the same integer as another wall expression does not change j_r. The universal simplicity target is for every integer r≥1 and x∈ Z_2, G_r(x)=0 Longrightarrow G_r'(x)≠0. The following are completed restricted results of the later","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-88e67d7f2c866e98","tex":"G_{\\Rcal\\leftarrow\\Qcal}","aliases":"G_Rcal←Qcal G_ R← Q Rcal leftarrow Qcal mathcal leftarrow mathcal","display":"G_{{\\mathcal R}\\leftarrow{\\mathcal Q}}","kind":"use","excerpt":"[Exact nonlinear coordinate theorem] For every map F:X→ Y, S_ QF_ Q← P A_ P=F. If G:Y→ Z and R is a packet system on Z, then (G∘ F)_ R← P = G_ R← Q ∘ F_ Q← P. No linearity, continuity, or differentiability assumption is needed.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-077ab2484c314309","tex":"G_{rs}","aliases":"G_rs","display":"G_{rs}","kind":"use","excerpt":"[Composition of transfer matrices] If G and H are convolution operators with packet matrices (G_rs) and (H_rs), then the packet matrix of GH is (GH)_rt = ∑_s∈ RG_rsH_st. In one-dimensional transform notation, the corresponding matrix symbols multiply pointwise.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bcb32fb1f100c04","tex":"G_s","aliases":"G_s","display":"G_s","kind":"use","excerpt":"[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66816f2f34b0eaea","tex":"g_s","aliases":"g_s","display":"g_s","kind":"use","excerpt":"[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d20ed25f6bf4ba52","tex":"G_{v_0}","aliases":"G_v_0","display":"G_{v_0}","kind":"use","excerpt":"[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a0c361763b69348","tex":"G_{v_0}^{-1}","aliases":"G_v_0^-1","display":"G_{v_0}^{-1}","kind":"use","excerpt":"[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-841df9ca84cfa40e","tex":"g_\\alpha","aliases":"g_α alpha","display":"g_\\alpha","kind":"use","excerpt":"] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_α","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d05f0811844f2313","tex":"g_\\beta","aliases":"g_β beta","display":"g_\\beta","kind":"use","excerpt":"For x=∑_α P_α x, any extension of the f_α must satisfy f(x)=∑_α f_α(P_α x), which proves both existence and uniqueness of the first formula. For the second, the sum g=∑_α g_α satisfies P_β g =∑_α P_β g_α =g_β because g_α(Y)⊆ X_α. Uniqueness follows from ∑_α P_α=I.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9cd63cbb25af4182","tex":"G_{\\gamma,B}","aliases":"G_γ,B gamma","display":"G_{\\gamma,B}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9459f5036ab9479b","tex":"G_{\\gamma_1,B_1}","aliases":"G_γ_1,B_1 gamma","display":"G_{\\gamma_1,B_1}","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa85e283567df49a","tex":"G_{\\gamma_k,B_k}","aliases":"G_γ_k,B_k gamma","display":"G_{\\gamma_k,B_k}","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-efa0a3eea83b0f61","tex":"\\operatorname{Gal}","aliases":"Gal operatorname","display":"\\operatorname{Gal}","kind":"use","excerpt":"Arithmetic coverage for a fixed cyclotomic extension. Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The pr","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f341c3678cd53cc","tex":"\\Gam_L","aliases":"Gam_L Γ_L Gam Gamma","display":"{\\Gamma}_L","kind":"use","excerpt":"For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-422e0fea8cc00d51","tex":"\\Gam_{\\rm in}","aliases":"Gam_rm in Γ_rm in Gam rm Gamma rm","display":"{\\Gamma}_{\\rm in}","kind":"use","excerpt":"For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c918c284a1665cd","tex":"\\Gcal","aliases":"Gcal G mathcal","display":"{\\mathcal G}","kind":"use","excerpt":"[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"Principal targets. targetUnitRefinement.omega_compatible, .principal_compatible, .principal_power targetDyadicUnits.dyadic_unit_decomposition, .decomposition_unique pageheading4. CRT addresses and the actual packet selectorsTheorem 10.1; manuscript page 22 Let a≥0, let M be odd, and put N=2^aM. Coprimality is derived from oddness: gcd(2^a,M)=1. The","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5d62edc1cdf57e7","tex":"\\mathrm{GL}","aliases":"GL mathrm","display":"\\mathrm{GL}","kind":"use","excerpt":"Lifting a summand inside a supplied finite-group carrier. Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g)","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d4a5d6f6abf65b4","tex":"\\mathrm{GL}_2","aliases":"GL_2 mathrm","display":"\\mathrm{GL}_2","kind":"use","excerpt":"bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de Franc","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd521ec0fb491c20","tex":"\\GL_D","aliases":"GL_D GL operatorname","display":"{\\operatorname{GL}}_D","kind":"use","excerpt":"For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a con","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bca14ec6fc8c9c7a","tex":"\\operatorname{GL}_d","aliases":"GL_d operatorname","display":"\\operatorname{GL}_d","kind":"use","excerpt":"[Weighted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15fe16c22d144674","tex":"\\GL_{\\dim\\rho}","aliases":"GL_dimρ GL dim rho operatorname dim rho","display":"{\\operatorname{GL}}_{\\dim\\rho}","kind":"use","excerpt":"begingroup renewcommand6666 [Arithmetic packet realization] For every ρ∈ A with its supplied H1--H4 assembled finite datum d_ρ, there exists an actual admissible supercuspidal representation π of GL_dimρ(K) and an arithmetic realization witness R(d_ρ,π). endgroup","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2847f1245d09235","tex":"\\operatorname{GL}_e","aliases":"GL_e operatorname","display":"\\operatorname{GL}_e","kind":"use","excerpt":"[Joint coefficient and parameter deformation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisf","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-301f280765554fc6","tex":"\\operatorname{GL}_m","aliases":"GL_m operatorname","display":"\\operatorname{GL}_m","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-665db19fdc2e1b41","tex":"\\GL_N","aliases":"GL_N GL operatorname","display":"{\\operatorname{GL}}_N","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e917386367b6fdc","tex":"\\GL_n","aliases":"GL_n GL operatorname","display":"{\\operatorname{GL}}_n","kind":"use","excerpt":"For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec44fc6c027681d2","tex":"\\mathrm{GL}_N","aliases":"GL_N mathrm","display":"\\mathrm{GL}_N","kind":"use","excerpt":"The actual stratum admits a full simple character θ trivial on U_F^1. Its canonical beta extension can be completed to an actual irreducible admissible supercuspidal π of GL_N(F) with any prescribed smooth central quasicharacter ω trivial on U_F^1, provided the noncompact completion is free or already compatible. Over this residue field, such ω is ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5723752e94fda6ae","tex":"\\mathrm{GL}_n","aliases":"GL_n mathrm","display":"\\mathrm{GL}_n","kind":"use","excerpt":"Lifting a summand inside a supplied finite-group carrier. Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8df4bc39535de40c","tex":"\\operatorname{GL}_n","aliases":"GL_n operatorname","display":"\\operatorname{GL}_n","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70196007a27c98a9","tex":"\\gr","aliases":"gr operatorname","display":"{\\operatorname{gr}}","kind":"use","excerpt":"i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-130720b204a25a33","tex":"\\gr^m","aliases":"gr^m gr operatorname","display":"{\\operatorname{gr}}^m","kind":"use","excerpt":"i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7680963df50a25c8","tex":"\\gr^p","aliases":"gr^p gr operatorname","display":"{\\operatorname{gr}}^p","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de926b1a644e7db4","tex":"\\bm H","aliases":"H bm","display":"\\bm H","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b957718497102933","tex":"\\mathbb H","aliases":"H mathbb","display":"\\mathbb H","kind":"use","excerpt":"Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a11c4bcc6a8c941","tex":"\\mathrm{H}","aliases":"H mathrm","display":"\\mathrm{H}","kind":"use","excerpt":"The Haar analysis matrix is E_H = frac1sqrt2 1 · 1 1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e9de57bf29ef2a6","tex":"h'","aliases":"h'","display":"h'","kind":"use","excerpt":"If x'-x∈ F^mX, then filtration congruence preservation gives Φ(x')-Φ(x)∈ F^mY. Thus one defect lies in F^mY exactly when the other does. If h-h'∈ F^m+1X, then x_m+h and x_m+h' define the same depth m+1 class. Conversely, two depth m+1 classes reducing to [x_m]_m differ from x_m by elements of F^mX, and they are equal precisely when those elements a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0af9e4c11c585190","tex":"\\bm H^{(N)}","aliases":"H^(N) bm","display":"\\bm H^{(N)}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9eca8df62b809141","tex":"H^{(N)}_{ks}","aliases":"H^(N)_ks","display":"H^{(N)}_{ks}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-daaa917c98a2902e","tex":"H^{-1}","aliases":"H^-1","display":"H^{-1}","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","excerpt":"There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus thro","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa0621c04438e85a","tex":"h^{-2}","aliases":"h^-2","display":"h^{-2}","kind":"use","excerpt":"FloatBarrier Fractional modes and killed transfer. For the one-dimensional Dirichlet difference operator, let h=1/M with an integer M≥2, let 1≤ k≤ M-1, and set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa5a6573c905a228","tex":"H^0","aliases":"H^0","display":"H^0","kind":"use","excerpt":"[Cohomological obstruction through the cone] Let e∈ E^n satisfy d_Ee=0. The class of (e,0) in H^n(Cone(D)) vanishes if and only if the class [e]∈ H^n(E) lies in the image of H^n(D):H^n(C)⟶ H^n(E). When C and E are concentrated in degree zero, H^0(Cone(D))≅coker D.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-054c9b0aa1b44b01","tex":"H^1_j","aliases":"H^1_j","display":"H^1_j","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":"[One-step direct rotation estimate] Assume P(t) is twice continuously differentiable in operator norm on [0,T]. There are constants h_0>0 and C>0 such that, whenever t,t+h∈[0,T] and |h|≤ h_0, U( P(t+h), P(t)) - I - hK(t) ≤ Ch^2. The constants can be chosen uniformly in t.","count":47},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f116cd8093d80318","tex":"H^2_j","aliases":"H^2_j","display":"H^2_j","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67b8b9a6dfc348a3","tex":"H^a","aliases":"H^a","display":"H^a","kind":"use","excerpt":"For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3959f846ddb0340","tex":"H^D","aliases":"H^D","display":"H^D","kind":"use","excerpt":"vpage14Gauge changes the picture, not flatnessMODEL E / CONJUGATION AND ITS LIMITSChapter III, mathematical text: Theorem 6.2 At the base chart, change coordinates by D=diag(2,1). Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving H^D=DHD^-1=mat1 · 2 0 · 1. The shear looks stronger in these","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16dc3982dc7a54a1","tex":"H^G_{c,b,a}","aliases":"H^G_c,b,a","display":"H^G_{c,b,a}","kind":"use","excerpt":"[Gauge covariance] The defect and triangle holonomy transform as Δ^G_c,b,a =G_cΔ_c,b,aG_a^-1, H^G_c,b,a =G_aH_c,b,aG_a^-1. Thus flatness and triviality of triangle holonomy are gauge invariant. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92ad3ff2bdd044cf","tex":"H^i","aliases":"H^i","display":"H^i","kind":"use","excerpt":"Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9dd0244f4332e6d","tex":"H^m","aliases":"H^m","display":"H^m","kind":"use","excerpt":"The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-389193f7658546c5","tex":"H^{m+1}","aliases":"H^m+1","display":"H^{m+1}","kind":"use","excerpt":"For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c6fa035c4defd46","tex":"H^n","aliases":"H^n","display":"H^n","kind":"use","excerpt":"[Cohomological obstruction through the cone] Let e∈ E^n satisfy d_Ee=0. The class of (e,0) in H^n(Cone(D)) vanishes if and only if the class [e]∈ H^n(E) lies in the image of H^n(D):H^n(C)⟶ H^n(E). When C and E are concentrated in degree zero, H^0(Cone(D))≅coker D.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-643e255e13e716c4","tex":"H^{p+q}","aliases":"H^p+q","display":"H^{p+q}","kind":"use","excerpt":"[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-846097f2ae49fb22","tex":"H^r","aliases":"H^r","display":"H^r","kind":"use","excerpt":"[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"[One-step direct rotation estimate] Assume P(t) is twice continuously differentiable in operator norm on [0,T]. There are constants h_0>0 and C>0 such that, whenever t,t+h∈[0,T] and |h|≤ h_0, U( P(t+h), P(t)) - I - hK(t) ≤ Ch^2. The constants can be chosen uniformly in t.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39086c50a1b88b83","tex":"H_{1/3}","aliases":"H_1/3","display":"H_{1/3}","kind":"use","excerpt":"vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPChapter V: Theorem E7.2, Corollary E7.3. On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), s_f(t)=t^2. vfig[.92]ext_tra","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2fd2f57e4f0efca6","tex":"\\mathcal H_2","aliases":"H_2 mathcal","display":"\\mathcal H_2","kind":"use","excerpt":"For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [sour","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","excerpt":"[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2]","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bfc7526d20d019a0","tex":"H_{2,1,0}","aliases":"H_2,1,0","display":"H_{2,1,0}","kind":"use","excerpt":"[The minimal API admits nontrivial holonomy] There is a connection on three points over Q for which transport 0→1 is multiplication by 2, every other transport between distinct endpoints is the identity, and H_2,1,0(1)=2. In particular H_2,1,0≠id. statusleanthreeKomorebi.PaperIII.Transport.twistedThreePointConnection; Komorebi.PaperIII.Transport.tw","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ea241c4c1df1f56","tex":"H_{2n}","aliases":"H_2n","display":"H_{2n}","kind":"use","excerpt":"vpage8See every butterfly valueWALSH RECURSION IN EIGHT COORDINATESChapter VIII: Theorem 4.7. The unnormalized Walsh matrix satisfies H_1=[1] and H_2n=matH_n · H_n H_n · -H_n, H_nH_n^T=nI. Apply the three add/subtract stages to (1,2,3,4,5,6,7,8)^T. The full intermediate ledger is displayed below. vfigwalsh headArithmetic can be counted exactly Each","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15c045eade51e6a9","tex":"H_{3,0}","aliases":"H_3,0","display":"H_{3,0}","kind":"use","excerpt":"It also derives coefT3H_3,0(log(1+T))=1/6 from the actual differential recurrence and checks its valuation v_3(1/6)=-1. Thus weighted membership does not mean ordinary coefficient integrality. This value attains the allowed floor weight at degree three.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-936bd94983483758","tex":"H_4","aliases":"H_4","display":"H_4","kind":"use","excerpt":"vpage33Gaussian packets have separate clocksORNSTEIN--UHLENBECK / HERMITE ENERGYChapter VI, Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure. In L^2(γ) for the standard Gaussian law on R, use the probabilists' Hermite polynomials H_1(x)=x, H_2(x)=x^2-1, H_4(x)=x^4-6x^2+3. The Ornstein--Uhlenbeck generator is Lf=","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ca843cc179ffa40","tex":"\\bm H_4","aliases":"H_4 bm","display":"\\bm H_4","kind":"use","excerpt":"[A four phase finite impulse response operator] Let h(0)=1, h(1)=2, h(2)=3, h(3)=4, h(4)=5, with all other taps zero. For four residue channels, the transfer symbol is H_4(z) = 1+5z^-1 · 4z^-1 · 3z^-1 · 2z^-1 2 · 1+5z^-1 · 4z^-1 · 3z^-1 3 · 2 · 1+5z^-1 · 4z^-1 4 · 3 · 2 · 1+5z^-1 . The delay term records the tap that crosses a coarse cell boundary.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-902dedcbd6a5754e","tex":"H_8","aliases":"H_8","display":"H_8","kind":"use","excerpt":"The result (36,-4,-8,0,-16,0,0,0)^T still reconstructs the input: the inverse is H_8^T/8. Because the displayed transform is unnormalized, its squared output norm is eight times the squared input norm. The normalized transform H_8/sqrt8 would instead be orthogonal. headWhat the zeros mean The zero coefficients arise from exact cancellation among th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a7409aaabc384b9","tex":"H_8^T","aliases":"H_8^T","display":"H_8^T","kind":"use","excerpt":"The result (36,-4,-8,0,-16,0,0,0)^T still reconstructs the input: the inverse is H_8^T/8. Because the displayed transform is unnormalized, its squared output norm is eight times the squared input norm. The normalized transform H_8/sqrt8 would instead be orthogonal. headWhat the zeros mean The zero coefficients arise from exact cancellation among th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-047a2d7f6191ab3d","tex":"H_a","aliases":"H_a","display":"H_a","kind":"use","excerpt":"[Scalar polyphase precision amplification] Let u∈ Z_p^×, let a,r≥0, and put H_a= (p-1)p^a, · p odd, 2^a+1, · p=2. Then Δ_H_a^ ru^n = u^n(u^H_a-1)^r. For odd p, v_p(Δ_H_a^ ru^n) ≥(a+1)r. For p=2 and odd u, v_2(Δ_H_a^ ru^n) ≥(a+3)r. Equivalently, every additional prime power digit increases the certified Mahler slope by one.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f76a62f943f51d9","tex":"\\mathcal H_A","aliases":"H_A mathcal","display":"\\mathcal H_A","kind":"use","excerpt":"Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c935edba5da750e9","tex":"\\mathcal H_B","aliases":"H_B mathcal","display":"\\mathcal H_B","kind":"use","excerpt":"Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8541d1f64196274","tex":"H_c","aliases":"H_c","display":"H_c","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0b8eba8def9dc978","tex":"H_{c-1}","aliases":"H_c-1","display":"H_{c-1}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8c68128dcab45cd","tex":"h_{d-1}","aliases":"h_d-1","display":"h_{d-1}","kind":"use","excerpt":"erse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d724fab1c00df76","tex":"H_e","aliases":"H_e","display":"H_e","kind":"use","excerpt":"A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7c152eadfb594c5","tex":"H_{e_1}","aliases":"H_e_1","display":"H_{e_1}","kind":"use","excerpt":"Here A_γ=(D^-1H_e_1)⋯(D^-1H_e_r) in the written order. Multiplicativity places it in F^W(γ)R, so the quotient kills it when W(γ)≥ L. Positive weights also imply |γ|≤ W(γ), making the retained search finite.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26ae0fa325355773","tex":"H_{e_r}","aliases":"H_e_r","display":"H_{e_r}","kind":"use","excerpt":"Here A_γ=(D^-1H_e_1)⋯(D^-1H_e_r) in the written order. Multiplicativity places it in F^W(γ)R, so the quotient kills it when W(γ)≥ L. Positive weights also imply |γ|≤ W(γ), making the retained search finite.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b786d951bfb29bf","tex":"H_{\\mathrm{eff}}","aliases":"H_eff mathrm","display":"H_{\\mathrm{eff}}","kind":"use","excerpt":"[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the approp","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c59677366440c6e2","tex":"H_{\\mathrm{eff}}^*","aliases":"H_eff^* mathrm","display":"H_{\\mathrm{eff}}^*","kind":"use","excerpt":"The explicitly defined coordinate curve y(s)=U(s)^*ψ(s) is differentiable at t, with dot y=dot U^*ψ(t)+U(t)^*dotψ. It satisfies idot y=H_effy(t), H_eff=U(t)^*HU(t)-iU(t)^*dot U. The connection term -iU(t)^*dot U is Hermitian. If H^*=H, then H_eff^*=H_eff.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31bc489b2a8008eb","tex":"H_{\\mathrm{even}}","aliases":"H_even mathrm","display":"H_{\\mathrm{even}}","kind":"use","excerpt":"For two qubits and real parameters J,h, consider H = JX⊗ X +h(Z⊗ I+I⊗ Z). In the ordered even basis {ket00,ket11} and odd basis {ket01,ket10}, H_even = 2h · J J · -2h , H_odd = 0 · J J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3cba85783189f4d","tex":"H_i","aliases":"H_i","display":"H_i","kind":"use","excerpt":"If p^Lmid c, the actual identities give G(x)=F(x)+cH(x), DG(x)≡ DF(x)pmod p. Thus if F(x)≡0pmodp^L and A_x is surjective, then ∃ b∈ B_L(x):G(b)=0. This theorem allows rectangular systems. The evaluation, derivative and Jacobian identities are proved from the polynomial definition G_i=F_i+C(c)H_i, rather than supplied as hypotheses. Algebraic congru","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be6006576b4a1fd7","tex":"h_{i-r}","aliases":"h_i-r","display":"h_{i-r}","kind":"use","excerpt":"(2^4i+2-1)B_4i+2(4i+2)(4i+1). The constant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial m","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48540f8d73f88b86","tex":"H_i^1","aliases":"H_i^1","display":"H_i^1","kind":"use","excerpt":"Choose a labeled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5632248249b75fcb","tex":"h_{i_1}","aliases":"h_i_1","display":"h_{i_1}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f4509ed7d70840f","tex":"h_{i_s}","aliases":"h_i_s","display":"h_{i_s}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"use","excerpt":"For h_j∈ X_α_j, the inherited norm gives Q_γ D^rF(x)[h_1,…,h_r] ≤ Q_γ D^rF(x) ∏_j h_j . Taking the multilinear operator norm proves the sharper restricted estimate. For ambient inputs, replace h_j by P_α_jh_j and apply the operator norm bound to each projector. Every nonzero idempotent has norm at least one, since it fixes a nonzero vector in its r","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-846897bc1a1e9fc7","tex":"h_j^2","aliases":"h_j^2","display":"h_j^2","kind":"use","excerpt":"Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unita","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f4cefef31ff270e3","tex":"H_j^r","aliases":"H_j^r","display":"H_j^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b57183bf1f39412","tex":"H_{ji}","aliases":"H_ji","display":"H_{ji}","kind":"use","excerpt":"For a loop represented on one common finite scalar packet coordinate type, the proposition resolves its successive transport matrices into packet walks. If R is a ring, as in the reversible connection API, then (H-I)_ji=H_ji-δ_ji. In particular, off-diagonal entries are unchanged. Thus a nonzero off-diagonal holonomy obstruction in this model with ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0231a655bf13a98d","tex":"H_k","aliases":"H_k","display":"H_k","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-058fc9338e41e80a","tex":"H_{ks}","aliases":"H_ks","display":"H_{ks}","kind":"use","excerpt":"For L=N Z with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c09363bdd85aad0b","tex":"h_{ks}","aliases":"h_ks","display":"h_{ks}","kind":"use","excerpt":"Let h be a finite impulse response and y=h*x, y[n]=∑_q∈ Zh[q]x[n-q]. Resolve both input and output into N phases. For output phase k and input phase s, define the transfer kernel h_ks[m]=h[Nm+k-s]. Out-of-range taps are understood to be zero.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf5f97ebded34595","tex":"H_m","aliases":"H_m","display":"H_m","kind":"use","excerpt":"Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-082db4c82cd55c3c","tex":"h_m","aliases":"h_m","display":"h_m","kind":"use","excerpt":"[One-step lifting criterion] The approximation x_m has a correction h_m∈ F^mX such that Φ(x_m+h_m)∈ F^m+1Y if and only if Ob_m(x_m)=0. When a correction exists, the set of its classes in F^mX/F^m+1X is a coset of Keroverline L_m. If the graded pieces are vector spaces, this coset is an affine space.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7e2b72a42d51897","tex":"h_m'","aliases":"h_m'","display":"h_m'","kind":"use","excerpt":"The next depth condition is equivalent to overline L_m[h_m] =-[Φ(x_m)] in F^mY/F^m+1Y. This is solvable exactly when the displayed defect has zero cokernel class. If h_m and h_m' are two solutions, their difference lies in Keroverline L_m. Adding an element of the kernel to one solution gives another, so the solution set is a coset of the kernel.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c9a6345cfc1fd89","tex":"h_{m-3}","aliases":"h_m-3","display":"h_{m-3}","kind":"use","excerpt":"If m is not a power of two, s_2(m)≥2 makes μ_m even. If m is a positive power of two, the only central indices which can contribute an odd residue are r=1 and r=3, since r-2s_2(r)=-1 only at these two positive integers. The r=1 term is odd: T(m,1)=2^m-1 and v_2(m!)=m-1. For m≥4 a power of two, the r=3 term is even. More explicitly, fracT(m,3)2^m-3=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"use","excerpt":"[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.","count":29},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5beb8cfa4cdeaa13","tex":"\\bm H_N","aliases":"H_N bm","display":"\\bm H_N","kind":"use","excerpt":"For L=N Z with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e8b586548eb3466","tex":"\\mathcal H_N","aliases":"H_N mathcal","display":"\\mathcal H_N","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ff15c0817c304f2","tex":"H_{n+1}","aliases":"H_n+1","display":"H_{n+1}","kind":"use","excerpt":"vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILChapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert spa","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-88eb9c29d4c1ef70","tex":"h_{n+1}","aliases":"h_n+1","display":"h_{n+1}","kind":"use","excerpt":"First derive the unprojected partition formula. At order one it is the ordinary chain rule. Suppose it holds at order n and the maps are C^n+1. Differentiating a summand in direction h_n+1 has two kinds of contribution. A derivative of the outer map creates the new singleton block {n+1}. A derivative of an inner block jet adjoins n+1 to that block.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5206b85c212e2fd5","tex":"H_{n+1,n}","aliases":"H_n+1,n","display":"H_{n+1,n}","kind":"use","excerpt":"The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59e39ba967eef7ff","tex":"H_{N,c}","aliases":"H_N,c","display":"H_{N,c}","kind":"use","excerpt":"Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. Scope and result. Let p be prime, let mmid p-1, and put N=mp. For 1≤ a<m, put c=ap and define the original exponential section and ratio by H_N,c(t)=∑_n≥","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7404334682ccef2c","tex":"H_{N,c}'","aliases":"H_N,c'","display":"H_{N,c}'","kind":"use","excerpt":"Modules. mcodeWeightedDworkTransfer, mcodeSeriesDifferenceBridge, mcodeRatioSeriesSupport. 3. Nonsingular propagation without an assumed limit. For consecutive original sections, the already defined ratios satisfy R_cH_N,c=tH_N,c-1, H_N,c'=H_N,c-1. Differentiating the first relation, using the relation for R_c-1, and cancelling a section whose lead","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66ea18857523fa8b","tex":"H_{N,c-1}","aliases":"H_N,c-1","display":"H_{N,c-1}","kind":"use","excerpt":"Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. Scope and result. Let p be prime, let mmid p-1, and put N=mp. For 1≤ a<m, put c=ap and define the original exponential section and ratio by H_N,c(t)=∑_n≥","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bb3059c2e24fc1e","tex":"H_{n,n+1}","aliases":"H_n,n+1","display":"H_{n,n+1}","kind":"use","excerpt":"The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b557fbd91b225d1c","tex":"H_N^T","aliases":"H_N^T","display":"H_N^T","kind":"use","excerpt":"H_N · -H_N . Then H_NH_N^T=NI. The product H_Nx can be evaluated in exactly Nlog_2N additions or subtractions through the recursion.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e71e9ced609ec71d","tex":"H_n^T","aliases":"H_n^T","display":"H_n^T","kind":"use","excerpt":"vpage8See every butterfly valueWALSH RECURSION IN EIGHT COORDINATESChapter VIII: Theorem 4.7. The unnormalized Walsh matrix satisfies H_1=[1] and H_2n=matH_n · H_n H_n · -H_n, H_nH_n^T=nI. Apply the three add/subtract stages to (1,2,3,4,5,6,7,8)^T. The full intermediate ledger is displayed below. vfigwalsh headArithmetic can be counted exactly Each","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96555427067e554e","tex":"H_{nn}","aliases":"H_nn","display":"H_{nn}","kind":"use","excerpt":"The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a36e5f02055d100","tex":"H_{\\mathrm{odd}}","aliases":"H_odd mathrm","display":"H_{\\mathrm{odd}}","kind":"use","excerpt":"For two qubits and real parameters J,h, consider H = JX⊗ X +h(Z⊗ I+I⊗ Z). In the ordered even basis {ket00,ket11} and odd basis {ket01,ket10}, H_even = 2h · J J · -2h , H_odd = 0 · J J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c305141ec8df1721","tex":"H_{PP}","aliases":"H_PP","display":"H_{PP}","kind":"use","excerpt":"1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. These are the hypotheses used in the formal companion. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, a","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91cc7303740de01a","tex":"H_{PQ}","aliases":"H_PQ","display":"H_{PQ}","kind":"use","excerpt":"Let P be a retained packet projector and Q=I-P. In packet coordinates, H= H_PP · H_PQ H_QP · H_QQ .","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2473371c8e4d1e23","tex":"H_{QP}","aliases":"H_QP","display":"H_{QP}","kind":"use","excerpt":"Let P be a retained packet projector and Q=I-P. In packet coordinates, H= H_PP · H_PQ H_QP · H_QQ .","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd8710c6593757d0","tex":"H_{QQ}","aliases":"H_QQ","display":"H_{QQ}","kind":"use","excerpt":"1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. These are the hypotheses used in the formal companion. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, a","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d64df6fa5868f4c","tex":"H_r","aliases":"H_r","display":"H_r","kind":"use","excerpt":"Modules. mcodeCyclotomicFiltration, mcodePrimeBlockPolynomial, and mcodeCyclicAmiceCoefficients. 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementa","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e01b0cedad21f47","tex":"h_r","aliases":"h_r","display":"h_r","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46b7885433f17d44","tex":"H_r'","aliases":"H_r'","display":"H_r'","kind":"use","excerpt":"ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd53539eaf104c23","tex":"H_{r-1}","aliases":"H_r-1","display":"H_{r-1}","kind":"use","excerpt":"ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ee189bd82f77745","tex":"H_{\\rm d}","aliases":"H_rm d rm","display":"H_{\\rm d}","kind":"use","excerpt":"Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9fe2614398adf9a4","tex":"H_{\\rm eff}","aliases":"H_rm eff rm","display":"H_{\\rm eff}","kind":"use","excerpt":"1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. These are the hypotheses used in the formal companion. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96ae09a38751e66f","tex":"H_{\\rm even}","aliases":"H_rm even rm","display":"H_{\\rm even}","kind":"use","excerpt":"A symmetry reveals blocks, not approximationsONE HAMILTONIAN IN TWO ORDERSChapter VII: Theorem II.5.1. Take the exact rational Hamiltonian H=X⊗ X+frac38(Z⊗ I+I⊗ Z). It commutes with total parity Z⊗ Z. Reordering the computational basis into even addresses (00,11) and odd addresses (01,10) displays the invariant subspaces explicitly. vfighamiltonian","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c9132bb49e045e4","tex":"H_{\\rm frame}","aliases":"H_rm frame rm","display":"H_{\\rm frame}","kind":"use","excerpt":"1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i - i · 0, idot y=H_rm framey. This matrix is Hermitian. Omitting it would predict a constant y, contradicting the displayed coordinates. In this example the norm of the omitted time derivative is exactly one, so the failure is not a small numerical residual.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3b7ddf6b71c7dbb","tex":"H_{\\rm Had}","aliases":"H_rm Had rm","display":"H_{\\rm Had}","kind":"use","excerpt":"A Hamiltonian and a Hadamard gate are both traditionally denoted H. Where both appear, the gate is H_rm Had and the Hamiltonian keeps the letter H. Numerical labels on a drawing are coordinates or exact example values, not observations of a physical device. takeawayCheck the type before interpreting the picture. An idempotent projector, a density m","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f630a7f11cb43320","tex":"H_{\\rm mov}","aliases":"H_rm mov rm","display":"H_{\\rm mov}","kind":"use","excerpt":"[Moving frame Hamiltonian] Let U(t) solve idot U(t)=H(t)U(t), U(0)=I, where H(t)=H(t)^* is continuous on the compact time interval under consideration. Set V(t)=W(t)^*U(t). Then idot V(t)=H_rm mov(t)V(t), where H_rm mov(t) = W(t)^*H(t)W(t)- i W(t)^*dot W(t) = W(t)^*(H(t)- i K(t))W(t). The moving frame Hamiltonian is self adjoint.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51d96667cf6151ff","tex":"H_{\\rm o}","aliases":"H_rm o rm","display":"H_{\\rm o}","kind":"use","excerpt":"Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f4d2d0416bc2db4","tex":"H_{\\rm odd}","aliases":"H_rm odd rm","display":"H_{\\rm odd}","kind":"use","excerpt":"A symmetry reveals blocks, not approximationsONE HAMILTONIAN IN TWO ORDERSChapter VII: Theorem II.5.1. Take the exact rational Hamiltonian H=X⊗ X+frac38(Z⊗ I+I⊗ Z). It commutes with total parity Z⊗ Z. Reordering the computational basis into even addresses (00,11) and odd addresses (01,10) displays the invariant subspaces explicitly. vfighamiltonian","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00e5cb4d6d94dccd","tex":"H_{rs}","aliases":"H_rs","display":"H_{rs}","kind":"use","excerpt":"[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-619ba501c7bc28e9","tex":"H_s","aliases":"H_s","display":"H_s","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain m","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f0b191a575470a1","tex":"H_{st}","aliases":"H_st","display":"H_{st}","kind":"use","excerpt":"[Composition of transfer matrices] If G and H are convolution operators with packet matrices (G_rs) and (H_rs), then the packet matrix of GH is (GH)_rt = ∑_s∈ RG_rsH_st. In one-dimensional transform notation, the corresponding matrix symbols multiply pointwise.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5ba533e65aa4c1b","tex":"H_T","aliases":"H_T","display":"H_T","kind":"use","excerpt":"[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-814f64e7253b7d27","tex":"H_t","aliases":"H_t","display":"H_t","kind":"use","excerpt":"[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f639c00b864f610e","tex":"H_{\\widehat\\tau}","aliases":"H_widehatτ widehat tau","display":"H_{\\widehat\\tau}","kind":"use","excerpt":"[Location error for one step] Let 0≤τ,widehatτ≤1 and H_τ= 1_[τ,1]. For 1≤ p<∞, aH_τ-aH_widehatτ _p = |a||τ-widehatτ|^1/p. In the supremum norm, the error equals |a| whenever τ≠widehatτ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc40e31f1fe319ab","tex":"H_{\\widehat\\tau_q}","aliases":"H_widehatτ_q widehat tau","display":"H_{\\widehat\\tau_q}","kind":"use","excerpt":"[Packet reconstruction after detection] Let j=∑_q=1^Qa_qH_τ_q, widehat j=∑_q=1^Qwidehat a_qH_widehatτ_q. Under the hypotheses of [source reference], for 1≤ p<∞, j-widehat j _p ≤ Qb_h + ( frac h2∑_q=1^Q|a_q|^p )^1/p. If Q_m is bounded and linear and widehat f_m=Q_m(f-widehat j)+widehat j, then f-widehat f_m _p ≤ · s-Q_ms _p","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e5148f28f72177f","tex":"H_x","aliases":"H_x","display":"H_x","kind":"use","excerpt":"[Support gives a Hecke eigensystem] Let R≅ T, let H be a finite T module, and let x:R→ E' be a field valued point. If ker x∈Supp_R(H), then H_x:=H⊗_ T,xE'≠0. Every element of T acts on H_x through the scalar prescribed by x.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91b7e01f2253c56c","tex":"H_\\theta^1","aliases":"H_θ^1 theta","display":"H_\\theta^1","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18adc48f226810ce","tex":"H_\\lambda","aliases":"H_λ lambda","display":"H_\\lambda","kind":"use","excerpt":"y an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which all these finite representations are triv","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e094a711b7cae00","tex":"H_\\nu","aliases":"H_ν nu","display":"H_\\nu","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of ","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aac3656a4ae03a71","tex":"H_{\\tau_q}","aliases":"H_τ_q tau","display":"H_{\\tau_q}","kind":"use","excerpt":"Let t_k=kh, k=0,…,n, h=frac1n. Assume f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Defin","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a39b4a2b8b246f17","tex":"H_{\\ell+1}","aliases":"H_ℓ+1 ell","display":"H_{\\ell+1}","kind":"use","excerpt":"[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-743400708b98042d","tex":"H_\\infty","aliases":"H_∞ infty","display":"H_\\infty","kind":"use","excerpt":"AILChapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-ad","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-795440ec9737d857","tex":"\\Hcal_A","aliases":"Hcal_A H_A Hcal mathcal","display":"{\\mathcal H}_A","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, let d_A=dim H_A≥1, and let r be the Schmidt rank of ψ. Then dimspan{(A⊗ I_B)ψ:A∈End( H_A)} = d_A r. Local operators on A generate the whole bipartite Hilbert space exactly when the reduced state on B has full support.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2723b4e767b85839","tex":"\\Hcal_B","aliases":"Hcal_B H_B Hcal mathcal","display":"{\\mathcal H}_B","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, let d_A=dim H_A≥1, and let r be the Schmidt rank of ψ. Then dimspan{(A⊗ I_B)ψ:A∈End( H_A)} = d_A r. Local operators on A generate the whole bipartite Hilbert space exactly when the reduced state on B has full support.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6fb67345de2f0bec","tex":"\\Hcal_{\\phys}","aliases":"Hcal_phys H_phys Hcal phys mathcal mathrm","display":"{\\mathcal H}_{{\\mathrm{phys}}}","kind":"use","excerpt":"[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e51f9863c8a6e19","tex":"\\Hcal_{\\rm root}","aliases":"Hcal_rm root H_rm root Hcal rm mathcal rm","display":"{\\mathcal H}_{\\rm root}","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ef531a1e01371ad","tex":"\\Hcal_X","aliases":"Hcal_X H_X Hcal mathcal","display":"{\\mathcal H}_X","kind":"use","excerpt":"Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e58303c6fd9074e","tex":"\\Hcal_Y","aliases":"Hcal_Y H_Y Hcal mathcal","display":"{\\mathcal H}_Y","kind":"use","excerpt":"Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-47d019f7ef5155b4","tex":"\\Hcal_Z","aliases":"Hcal_Z H_Z Hcal mathcal","display":"{\\mathcal H}_Z","kind":"use","excerpt":"[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b64a19766af07b6","tex":"\\Hcal_\\infty","aliases":"Hcal_∞ H_∞ Hcal infty mathcal infty","display":"{\\mathcal H}_\\infty","kind":"use","excerpt":"Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bae743248da3740","tex":"\\head","aliases":"head","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7937103467a5b8d0","tex":"\\HH","aliases":"HH","display":null,"kind":"use","excerpt":"midrule HH,ψ · Hilbert space and state vector · They specify the space of addresses and its coherent amplitudes.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac5f83b9023afd1a","tex":"\\HH_n","aliases":"HH_n HH","display":null,"kind":"use","excerpt":"vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILChapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert spa","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ab62d5df4f5b229","tex":"\\Hol","aliases":"Hol operatorname","display":"{\\operatorname{Hol}}","kind":"use","excerpt":"[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1f2427536861709","tex":"\\operatorname{Hol}","aliases":"Hol operatorname","display":"\\operatorname{Hol}","kind":"use","excerpt":"Hol(γ) · closed phase transfer · cycle consistency in a network","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69ac3b37b8b2ac4d","tex":"\\Hol'","aliases":"Hol' Hol operatorname","display":"{\\operatorname{Hol}}'","kind":"use","excerpt":"[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aae322307284296b","tex":"\\Hol_A","aliases":"Hol_A Hol operatorname","display":"{\\operatorname{Hol}}_A","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-845b4f3714396b47","tex":"\\Hol_B","aliases":"Hol_B Hol operatorname","display":"{\\operatorname{Hol}}_B","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0d53229e96ed0e9","tex":"\\operatorname{Hom}","aliases":"Hom operatorname","display":"\\operatorname{Hom}","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e188ffeaff0072a","tex":"\\operatorname{Hom}_B","aliases":"Hom_B operatorname","display":"\\operatorname{Hom}_B","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0251232b1004a6b9","tex":"\\Hom_{R\\text{-alg}}","aliases":"Hom_R-alg Hom text operatorname text","display":"{\\operatorname{Hom}}_{R\\text{-alg}}","kind":"use","excerpt":"Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a charac","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-413c3a738073b446","tex":"\\Hom_{W_K}","aliases":"Hom_W_K Hom operatorname","display":"{\\operatorname{Hom}}_{W_K}","kind":"use","excerpt":"[Tame Hom test] For irreducible tame atoms, dimHom_W_K(τ_ξ^vee,τ_η) is zero or one, and it is one exactly when the refined tensor packet test in [source reference] finds one trivial packet.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-417e8bdb90ba73f1","tex":"\\Hp","aliases":"Hp H( Z_p) mathcal mathbb","display":"{\\mathcal H({\\mathbb Z}_p)}","kind":"use","excerpt":"The relevant integral subring is the Hurwitz subring H( Z_p)={f∈ Q_p[[t]]: n![t^n]f∈ Z_p for every n}. It is not coefficientwise integrality of ordinary power-series coefficients. The precise connection is n![t^n] E(F)=[T^0]Θ^nF. Since every iterate of Θ preserves W_p, the right side is integral. Therefore E( W_p)⊆ H( Z_p). Applying the same statem","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8950e80c743e7b98","tex":"\\HS","aliases":"HS mathrm","display":"{\\mathrm{HS}}","kind":"use","excerpt":"Density matrices and observables carry both an input packet and an output packet. Equip End( H) with the Hilbert--Schmidt inner product ipXY_HS=Tr(X^*Y).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c9fcf42a4606088","tex":"\\Hull_{(O_i)}","aliases":"Hull_(O_i) Hull operatorname","display":"{\\operatorname{Hull}}_{(O_i)}","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72683f068bfcc031","tex":"\\Hull_{\\AA}","aliases":"Hull_AA Hull_ A Hull AA operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal A}}","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c17d7c5e355fb4ce","tex":"\\Hull_{\\cA}","aliases":"Hull_cA Hull_ A Hull cA operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal A}}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aed5db3db4ec9fd6","tex":"\\Hull_{\\Ccal}","aliases":"Hull_Ccal Hull_ C Hull Ccal operatorname mathcal","display":"{\\operatorname{Hull}}_{{\\mathcal C}}","kind":"use","excerpt":"· =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40dcb49c2ffb76b9","tex":"\\Hull_F","aliases":"Hull_F Hull operatorname","display":"{\\operatorname{Hull}}_F","kind":"use","excerpt":"IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2e09d6cb7fed0ec","tex":"\\Hull_{\\QQ[D]}","aliases":"Hull_QQ[D] Hull_ Q[D] Hull QQ operatorname mathbb","display":"{\\operatorname{Hull}}_{{\\mathbb Q}[D]}","kind":"use","excerpt":"headWhat the seed generates With the fixed algebra Q[D] and seed e_1, Hull_ Q[D](e_1)=span{e_1,e_2+e_3}. It has dimension two, not four. Both the independent difference e_2-e_3 and the last coordinate e_4 are missing.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fbe558355eecd98b","tex":"\\Hull_{\\varphi(\\A)}","aliases":"Hull_φ(A) Hull_φ( A) Hull varphi A operatorname varphi mathcal","display":"{\\operatorname{Hull}}_{\\varphi({\\mathcal A})}","kind":"use","excerpt":"[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac5cb1bffa04d178","tex":"\\Hull_{\\varphi(\\AA)}","aliases":"Hull_φ(AA) Hull_φ( A) Hull varphi AA operatorname varphi mathcal","display":"{\\operatorname{Hull}}_{\\varphi({\\mathcal A})}","kind":"use","excerpt":"headWhen the map is not invertible For an intertwiner Φ A=φ(A)Φ, the hull identity uses the image algebra: Φ(Hull_ A(S))=Hull_φ( A)(Φ S). Replacing φ( A) with a larger target algebra can enlarge the target hull. Seed rank need not be equal relative to that larger algebra. A surjective intertwiner gives the appropriate monotonicity inequality, not a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab39294284c03d46","tex":"\\mathrm i","aliases":"i mathrm","display":"\\mathrm i","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACTChapter VIII: Theorem 5.6, Section F.1, Section F.2. The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a gen","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8da0f8726aa71190","tex":"i'","aliases":"i'","display":"i'","kind":"use","excerpt":"Conversely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the con","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11bf5aa7c1c323e1","tex":"i^{(m-j)\\ell}","aliases":"i^(m-j)ℓ ell","display":"i^{(m-j)\\ell}","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. N","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d6409e60138ea26","tex":"I^*","aliases":"I^*","display":"I^*","kind":"use","excerpt":"main inside the original code, so the previous syndrome partition reports no bit flip. frac3ket000+4ket1115 ⟼ frac3ket000-4ket1115. The populations are unchanged, but the relative phase is not. vfiglogical_error Let V be the code isometry with columns ket000,ket111. For the proposed error family {I,Z_1}, its off-diagonal error compression is V^*I^*","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b966b3d3d2ca3fe9","tex":"\\mathrm i^{-jk}","aliases":"i^-jk mathrm","display":"\\mathrm i^{-jk}","kind":"use","excerpt":"vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATIONChapter VIII: Theorem 4.3. Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-786d9c0a32751abc","tex":"i^{-j\\ell}","aliases":"i^-jℓ ell","display":"i^{-j\\ell}","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. N","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0745ce822383e740","tex":"I^2","aliases":"I^2","display":"I^2","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc66c01638c40151","tex":"i^2","aliases":"i^2","display":"i^2","kind":"use","excerpt":"The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b400594c9200202","tex":"I^b","aliases":"I^b","display":"I^b","kind":"use","excerpt":"ficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6eebe3e99b3727b1","tex":"I^{d_s}","aliases":"I^d_s","display":"I^{d_s}","kind":"use","excerpt":"e map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero after reduction t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17077128e8026415","tex":"I^k","aliases":"I^k","display":"I^k","kind":"use","excerpt":"Completion from conjugacy classes. Over an I-adically complete separated commutative coefficient ring R, for a fixed finite rank, the formalization now takes finite-level objects to be actual matrix representations modulo conjugation by an invertible matrix. Assume the coefficient quotients R/I^k are finite. Conjugation commutes with coefficient re","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-022e2ac42397386f","tex":"i^k","aliases":"i^k","display":"i^k","kind":"use","excerpt":"vpage8Four zeros become one orbit zeroTHE REDUCED VARIABLEChapter IV: Definition 5.7, Example 11.12, Corollary 11.13. For N=4 the factorization is exact: D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), D_4(z)=Δ_4(z^4). Set A_4=sqrt2π. All the zeros are A_4(m+1/2)i^k, with m≥0 and 0≤ k<4. Each zero is simple. vfigorbit_reduction The drawing uses u=z/A_4 and v=w/A","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04f25a28d6cda5bb","tex":"I^{\\lfloor d/p\\rfloor}","aliases":"I^lfloor d/prfloor lfloor rfloor","display":null,"kind":"use","excerpt":"For the falling factorial (x)_d=∏_i=0^d-1(x-i), grouping into consecutive blocks gives (x)_d∈ I^lfloor d/prfloor ⊆(p^v_p(lfloor d/prfloor!)). Every integral residue coordinate therefore satisfies p^v_p(lfloor d/prfloor!)mid [(r)](x)_d in Z. Dividing a coordinate by d! is done only afterward, using v_p(d!)=lfloor d/prfloor+v_p(lfloor d/prfloor!). Th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d50bce5288ce233a","tex":"I^m","aliases":"I^m","display":"I^m","kind":"use","excerpt":"Checked representation-level completion and its arithmetic boundary. The representation-level completion construction works with actual matrix representations of a fixed group. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Multiplication ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-920dadf058c8fd57","tex":"I^p","aliases":"I^p","display":"I^p","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52b36dcfd23ea8f2","tex":"I^q","aliases":"I^q","display":"I^q","kind":"use","excerpt":"sses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a9917336954c9f2","tex":"i^\\ell","aliases":"i^ℓ ell","display":"i^\\ell","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. N","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a96189008f33553","tex":"I_{\\mathfrak Q_v}","aliases":"I_ Q_v mathfrak","display":"I_{\\mathfrak Q_v}","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f18b3c49fd8055ae","tex":"i_0","aliases":"i_0","display":"i_0","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37475f8aeaec6cce","tex":"I_1","aliases":"I_1","display":"I_1","kind":"use","excerpt":"gintheorem[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwideh","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"ar targetUnitModes.unit_mode_valuation_gain targetUnitModes.finite_unit_modes_valuation_gain pageheading4. Exact examples and adversarial checksThe examples use the same definitions as the general results A polynomial anchor. The actual difference table for f(n)=n^2 yields (c_0,c_1,c_2,c_3)=(0,1,2,0). The finite interpolant therefore reconstructs I","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad0c30eeda282e1f","tex":"I_{2^{d-1}}","aliases":"I_2^d-1","display":"I_{2^{d-1}}","kind":"use","excerpt":"erse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ba0538909d6a977","tex":"I_3","aliases":"I_3","display":"I_3","kind":"use","excerpt":"vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDChapter VII: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ee5bef1760d6bb8","tex":"I_6","aliases":"I_6","display":"I_6","kind":"use","excerpt":"standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3χ_λ(g^-1)U(g). All matrices in this example are rational. Multiplying them gives self-adjoint idempotents of ranks 1,1,4, with pairwise zero products and sum I_6. xfigs3 headThe standard packet has dimension four The standard representation occurs twice in the regular representation. Its isotypic compon","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eff6aa27c552c5d2","tex":"I_a","aliases":"I_a","display":"I_a","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60a91a508a9a828a","tex":"I_B","aliases":"I_B","display":"I_B","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2344f57325d4c16a","tex":"I_b","aliases":"I_b","display":"I_b","kind":"use","excerpt":"Conversely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the con","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20d6646340ce20f0","tex":"I_d","aliases":"I_d","display":"I_d","kind":"use","excerpt":"Let A=p^sJ^-1∈ M_d(R). Inclusion into Q_p and its injectivity give the two integral identities AJ=JA=p^sI_d. These identities are also an equivalent integral certificate for the first two hypotheses of [source reference]. In particular, A is injective: Aw=0 implies p^sw=JAw=0, hence w=0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e959b032fac4b5d","tex":"I_{d_j}","aliases":"I_d_j","display":"I_{d_j}","kind":"use","excerpt":"If B_j and C_j have an intermediate space of dimension d_j, use a block diagonal matrix D with blocks λ_j I_d_j, or the required Jordan blocks when poles repeat. Assemble the matrices B_j and C_j into the output and input maps.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c1fa6f17c9ee965","tex":"I_e","aliases":"I_e","display":"I_e","kind":"use","excerpt":"0 · cI_k. The identities AJ=JA=cI_e give CB=BC=cI_e+k. In the second product, the upper right block is -JAK+cK=0. In the first, it is AK-AK=0. Thus augmentation preserves the original inverse bound s.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-224958a47c5c3088","tex":"I_{e+k}","aliases":"I_e+k","display":"I_{e+k}","kind":"use","excerpt":"0 · cI_k. The identities AJ=JA=cI_e give CB=BC=cI_e+k. In the second product, the upper right block is -JAK+cK=0. In the first, it is AK-AK=0. Thus augmentation preserves the original inverse bound s.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c1a6475ed0ab0aa","tex":"I_{e_1}","aliases":"I_e_1","display":"I_{e_1}","kind":"use","excerpt":"headThe distinction is visible in two dimensions Use X= Q^2, seed e_1=(1,0)^ T, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column: I_e_1={mat0 · a 0 · b:a,b∈ Q}.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0535963952d3cdb0","tex":"I_{\\frakp}","aliases":"I_frakp I_ p frakp mathfrak","display":"I_{{\\mathfrak p}}","kind":"use","excerpt":"Let p⊇ I. Localizing gives (M/IM)_ p ≅ M_ p/I_ pM_ p. If M_ p=0, the quotient is zero. If M_ p≠0, then I_ p lies in the maximal ideal of the local ring R_ p. Nakayama's lemma shows that M_ p/I_ pM_ p≠0. The stated equality follows.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78210203a2849812","tex":"I_{H_2}","aliases":"I_H_2","display":"I_{H_2}","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions ho","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cbf63ab2047dd5c1","tex":"I_{\\Hcal}","aliases":"I_Hcal I_ H Hcal mathcal","display":"I_{{\\mathcal H}}","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ef65037047c3204","tex":"I_J","aliases":"I_J","display":"I_J","kind":"use","excerpt":"To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0625c4dc07eb960a","tex":"i_j","aliases":"i_j","display":"i_j","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b06d3ec0fc2cf0c6","tex":"I_{j,k}","aliases":"I_j,k","display":"I_{j,k}","kind":"use","excerpt":"Let I_j,k = [k2^-j,(k+1)2^-j) and let ψ_j,k be the normalized Haar wavelet supported on I_j,k, positive on the left half and negative on the right half.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-edadeb2cc1e72d16","tex":"I_K","aliases":"I_K","display":"I_K","kind":"use","excerpt":"Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90844af211ae1183","tex":"I_k","aliases":"I_k","display":"I_k","kind":"use","excerpt":"Now suppose a chosen set of e columns of DF(x)bmod p is independent, and put those coordinates first. Apply the square result to the augmented polynomial map Ψ(Y,T)=(F(Y,T),T). Its reduced Jacobian has upper blocks (D_YF,D_TF) and lower blocks (0,I_k), so it is invertible. For t∈ T, the target (0,t) belongs to Ψ(x)+p^L Z_p^e+k because F(x)≡0pmodp^L","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-042e41456d0b7e11","tex":"i_K","aliases":"i_K","display":"i_K","kind":"use","excerpt":"Actual quadratic local conditions and completion. Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embedding. Lean pr","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3ecf779d0b53571","tex":"I_M","aliases":"I_M","display":"I_M","kind":"use","excerpt":"E_q · 0 · 0 · ⋯ · 0 . It also satisfies S_M(E_q)^M=E_qI_M.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad472b41377c2078","tex":"I_m","aliases":"I_m","display":"I_m","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ee8ac8c8d5446ba","tex":"i_m","aliases":"i_m","display":"i_m","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-714b73df80781bff","tex":"i_{m-1}","aliases":"i_m-1","display":"i_{m-1}","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c96c37f828cfc2a3","tex":"I_{m_r}","aliases":"I_m_r","display":"I_{m_r}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c305f0f9a08586da","tex":"I_{M_\\mu}","aliases":"I_M_μ mu","display":"I_{M_\\mu}","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf48159a73d856b5","tex":"i_n","aliases":"i_n","display":"i_n","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c6b4e7520988cc0","tex":"I_P","aliases":"I_P","display":"I_P","kind":"use","excerpt":"Corrected Feshbach resolvent. Write a finite complex matrix in retained and eliminated coordinates: H=A · B{ C · D, zI-H=zI_P-A · -B -C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7b33cae672307e6","tex":"I_p","aliases":"I_p","display":"I_p","kind":"use","excerpt":"pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92eb4220503340b4","tex":"I_{p^{d-1}}","aliases":"I_p^d-1","display":"I_{p^{d-1}}","kind":"use","excerpt":", where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. In","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-caf5dd124506cb34","tex":"I_Q","aliases":"I_Q","display":"I_Q","kind":"use","excerpt":"[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the approp","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63bfdca1004602ec","tex":"i_q","aliases":"i_q","display":"i_q","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afccf775c866af1f","tex":"i_r","aliases":"i_r","display":"i_r","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f6ca4a3b6894ed8","tex":"i_{r+1}","aliases":"i_r+1","display":"i_{r+1}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38cd9fdb96fba5ee","tex":"I_{\\rm in}","aliases":"I_rm in rm","display":"I_{\\rm in}","kind":"use","excerpt":"labelstrong:standardfact:001 A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-111b6ee2c5f8dbd2","tex":"I_s","aliases":"I_s","display":"I_s","kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98483a254fa93b89","tex":"i_s","aliases":"i_s","display":"i_s","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34af8cf33f9487de","tex":"I_U","aliases":"I_U","display":"I_U","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d1279bf2b532c2f","tex":"i_U","aliases":"i_U","display":"i_U","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5bae71a192fc64d","tex":"I_V","aliases":"I_V","display":"I_V","kind":"use","excerpt":"0 · 1 · 0. The typed relation is BA=I_U. In the opposite order, AB=diag(1,1,0)≠ I_V. The missing direction is Ker B= Q v_3, not an entirely missing vertex.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3f6e419ae12a276","tex":"I_v","aliases":"I_v","display":"I_v","kind":"use","excerpt":"Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate pro","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a011c42e0a9e76fd","tex":"i_V","aliases":"i_V","display":"i_V","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec3582230ba1341f","tex":"I_{X/K}","aliases":"I_X/K","display":"I_{X/K}","kind":"use","excerpt":"[Packet descent through an invariant quotient] Under the hypotheses of [source reference], the operators overline P_α(x+K)=P_α x+K are well defined pairwise annihilating idempotents with sum I_X/K, and the quotient map satisfies π P_α=overline P_απ. After zero packets are removed, they form the canonical quotient packet system. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-756b4d7c969253c3","tex":"I_{X^*}","aliases":"I_X^*","display":"I_{X^*}","kind":"use","excerpt":"Because dualization reverses products, P_α^*P_β^*=(P_β P_α)^* =δ_αβP_α^*. Also ∑_α P_α^*=I_X^*. Thus the dual maps form a packet system. Restriction gives a map from P_α^*X^* to X_α^*. It is injective because a covector in the range of P_α^* vanishes on every other packet, and it is surjective because any covector on X_α extends to X through the di","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2b075d59d6e400b","tex":"I_{X^{\\Pcal}}","aliases":"I_X^Pcal I_X^ P Pcal mathcal","display":"I_{X^{{\\mathcal P}}}","kind":"use","excerpt":"[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5de7c345d56728e9","tex":"I_{X^{\\Qcal}}","aliases":"I_X^Qcal I_X^ Q Qcal mathcal","display":"I_{X^{{\\mathcal Q}}}","kind":"use","excerpt":"[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e3ef04ac1a939e6","tex":"I_{X_m}","aliases":"I_X_m","display":"I_{X_m}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1718302dc2dde398","tex":"I_{X_\\alpha}","aliases":"I_X_α alpha","display":"I_{X_\\alpha}","kind":"use","excerpt":"The identity has diagonal blocks I_X_α and no blocks outside the diagonal, so it belongs to M_R exactly when (α,α)∈ R for every α.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-826ca417f032adb2","tex":"I_{X_\\ell}","aliases":"I_X_ℓ ell","display":"I_{X_\\ell}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab3ef27be5e3d7f7","tex":"I_Y","aliases":"I_Y","display":"I_Y","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bdedc0251851d395","tex":"I_Z","aliases":"I_Z","display":"I_Z","kind":"use","excerpt":"[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L-Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1c9ad67822f1d50","tex":"I_\\lambda","aliases":"I_λ lambda","display":"I_\\lambda","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d0bba48daafb223","tex":"i_\\lambda","aliases":"i_λ lambda","display":"i_\\lambda","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4331b90602b3a79a","tex":"I_\\lambda^n","aliases":"I_λ^n lambda","display":"I_\\lambda^n","kind":"use","excerpt":"For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a con","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c32aece4942cf822","tex":"I_{\\oplus X_\\alpha}","aliases":"I_⊕ X_α oplus alpha","display":"I_{\\oplus X_\\alpha}","kind":"use","excerpt":"[Exact packet coordinate isomorphism] For every finite packet system, S_ P A_ P=I_X, A_ P S_ P=I_⊕ X_α. Hence X≅⊕_α∈ IX_α. [source reference] ExportsEvery exact packet reconstruction in the series.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09e35188271c2791","tex":"\\id_{\\Coord(\\cP^a)}","aliases":"id_Coord(cP^a) id_Coord( P^a) id Coord cP mathrm operatorname mathcal","display":"{\\mathrm{id}}_{{\\operatorname{Coord}}({\\mathcal P}^a)}","kind":"use","excerpt":"i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9d9b2d2680f67c6","tex":"\\id_{I_\\ell}","aliases":"id_I_ℓ id ell mathrm ell","display":"{\\mathrm{id}}_{I_\\ell}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de717c99c15e8582","tex":"\\id_X","aliases":"id_X id mathrm","display":"{\\mathrm{id}}_X","kind":"use","excerpt":"i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord(","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1491268eb17c25c7","tex":"\\id_{X_v}","aliases":"id_X_v id mathrm","display":"{\\mathrm{id}}_{X_v}","kind":"use","excerpt":"Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate pro","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"For a specific attainable target (2,4), 2x≡2pmod8iff x∈{1,5}, 4y≡4pmod8iff y∈{1,3,5,7}. The coordinate choices are independent, giving 2·4=8 solutions. The entire carrier is partitioned consistently: 8 attainable targets × 8 inputs per target=64. For an inadmissible target, such as (1,4) or (2,2), the fiber is empty. The same nonzero cardinality fo","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f43b2c8703ad09ef","tex":"\\ii^{-jk}","aliases":"ii^-jk i^-jk ii mathrm","display":"{\\mathrm i}^{-jk}","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYChapter VII: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69bc80ad3d09d0fe","tex":"\\ii^{-jr}","aliases":"ii^-jr i^-jr ii mathrm","display":"{\\mathrm i}^{-jr}","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYChapter VII: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a24bbf47ddba511a","tex":"\\ii^j","aliases":"ii^j i^j ii mathrm","display":"{\\mathrm i}^j","kind":"use","excerpt":"character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing the address multiplie","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12473beefe7126eb","tex":"\\Im","aliases":"Im","display":"\\Im","kind":"use","excerpt":"1 · 2, so A=0, B=1, and D=2. In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","excerpt":"headThe coordinates preserve structure, not Euclidean distances A quarter turn is a rotation in the reflection disk. After pulling back, it is a M obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points 1/2 and -1/2 become 3 and 1/3. The continuous impedan","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"O_F,bar x= F_p^r/imJ, o_F,bar x=[e].","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59f64376d60523e7","tex":"\\Ind","aliases":"Ind operatorname","display":"{\\operatorname{Ind}}","kind":"use","excerpt":"[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad8e02273a5aecf9","tex":"\\operatorname{Ind}","aliases":"Ind operatorname","display":"\\operatorname{Ind}","kind":"use","excerpt":"e P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced domain of the additive character, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-266b9ca5b05e2205","tex":"\\Ind_{\\mathcal N}^{\\mathcal M}","aliases":"Ind_ N^ M Ind mathcal mathcal operatorname mathcal mathcal","display":"{\\operatorname{Ind}}_{\\mathcal N}^{\\mathcal M}","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0cf82edfc33e1abc","tex":"\\operatorname{Ind}_{\\mathcal N}^{\\mathcal M}","aliases":"Ind_ N^ M operatorname mathcal mathcal","display":"\\operatorname{Ind}_{\\mathcal N}^{\\mathcal M}","kind":"use","excerpt":"Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is an actual e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11972db3b53bb021","tex":"\\Ind_J^P","aliases":"Ind_J^P Ind operatorname","display":"{\\operatorname{Ind}}_J^P","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3643be2089a22b91","tex":"\\operatorname{Ind}_J^P","aliases":"Ind_J^P operatorname","display":"\\operatorname{Ind}_J^P","kind":"use","excerpt":"e P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced domain of the additive character, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0cd969788b2e98e","tex":"\\Ind_{W_{K_f}}^{W_K}","aliases":"Ind_W_K_f^W_K Ind operatorname","display":"{\\operatorname{Ind}}_{W_{K_f}}^{W_K}","kind":"use","excerpt":"[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be0636f2d1fd5dfe","tex":"\\Ind_{W_{K_L}}^{W_K}","aliases":"Ind_W_K_L^W_K Ind operatorname","display":"{\\operatorname{Ind}}_{W_{K_L}}^{W_K}","kind":"use","excerpt":"[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da5ab4b9d09e0183","tex":"\\Ind_{W_{K_m}}^{W_K}","aliases":"Ind_W_K_m^W_K Ind operatorname","display":"{\\operatorname{Ind}}_{W_{K_m}}^{W_K}","kind":"use","excerpt":"Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7adceb058e48c433","tex":"\\Ind_{W_{K_n}}^{W_K}","aliases":"Ind_W_K_n^W_K Ind operatorname","display":"{\\operatorname{Ind}}_{W_{K_n}}^{W_K}","kind":"use","excerpt":"Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ec79176d200cb3c","tex":"\\inf_{j\\geq0}","aliases":"inf_j≥0 inf geq","display":"\\inf_{j\\geq0}","kind":"use","excerpt":"For a uniform coefficient hypothesis, the simpler conclusion follows: inf_j≥0 w(s+j)≤ord_p(Δ_1^s f(n)). For each finite L≤ B_p(w;n,s), a separate theorem yields p^LmidΔ_1^s f(n) inside Z_p. Another specialization handles arbitrary finite interpolants with infinite weights beyond their degree.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a3bcbfb09701caf","tex":"\\bm j","aliases":"j bm","display":"\\bm j","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b761556af44ee0a","tex":"\\mathbf j","aliases":"j mathbf","display":"\\mathbf j","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d6308571f33e418","tex":"J'","aliases":"J'","display":"J'","kind":"use","excerpt":"e every node cost changes by at most ε_v=1/32. A pruning may select as many as eight leaves, so its total cost can change by at most 8/32=1/4. The recursive envelope proves this without guessing which pruning will be selected: E(v)=max{ε_v, E(v0)+E(v1)}, E(∅)=tfrac14. For any competitor Λ, the changed cost difference is bounded below by J'(Λ)-J'(Λ_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-787a425baaf96eca","tex":"j^{-1}","aliases":"j^-1","display":"j^{-1}","kind":"use","excerpt":"Finally, if kj∈P with k∈K× and j∈J, then k∈ Pj^-1⊂ U_A^1. Hence k∈U_K 1 and kj∈J. Therefore","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d09ca7ae57dbf7ab","tex":"J^1_{\\gamma_1\\mid\\alpha_1}","aliases":"J^1_γ_1midα_1 gamma mid alpha","display":null,"kind":"use","excerpt":"[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5008abe1661077a","tex":"J^1_{\\gamma_2\\mid\\alpha_2}","aliases":"J^1_γ_2midα_2 gamma mid alpha","display":null,"kind":"use","excerpt":"[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-835bd29514aa36c8","tex":"J^1_{\\delta\\mid\\gamma}","aliases":"J^1_δmidγ delta mid gamma","display":null,"kind":"use","excerpt":"[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29097134fc17258f","tex":"J^2","aliases":"J^2","display":"J^2","kind":"use","excerpt":"J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-049a2e2ed97c4350","tex":"J^2_{\\gamma\\mid\\alpha_1,\\alpha_2}","aliases":"J^2_γmidα_1,α_2 gamma mid alpha alpha","display":null,"kind":"use","excerpt":"[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b44d4f0935dc442b","tex":"J^2_{\\delta\\mid\\gamma_1,\\gamma_2}","aliases":"J^2_δmidγ_1,γ_2 delta mid gamma gamma","display":null,"kind":"use","excerpt":"[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a62392eb5eb3d856","tex":"J^{k}_{\\delta\\mid\\gamma_1,\\ldots,\\gamma_k}","aliases":"J^k_δmidγ_1,…,γ_k delta mid gamma ldots gamma","display":null,"kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e519f6e43bfb5bd","tex":"J^{n}_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_n}","aliases":"J^n_δmidα_1,…,α_n delta mid alpha ldots alpha","display":null,"kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83d498114d998f28","tex":"J^p","aliases":"J^p","display":"J^p","kind":"use","excerpt":"[Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8eed2d16e16facf1","tex":"J^r_{\\gamma\\mid\\alpha_1,\\ldots,\\alpha_r}","aliases":"J^r_γmidα_1,…,α_r gamma mid alpha ldots alpha","display":null,"kind":"use","excerpt":"[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f8c22fed2ef52fb","tex":"J^r_{\\delta\\mid\\alpha_1,\\ldots,\\alpha_r}","aliases":"J^r_δmidα_1,…,α_r delta mid alpha ldots alpha","display":null,"kind":"use","excerpt":"[Structural pruning] For a map H with fixed input and output packet systems, let supp_r H(x) be the set of external profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89d0ef08e1ef93e7","tex":"J^s_{\\gamma\\mid\\alpha_{i_1},\\ldots,\\alpha_{i_s}}","aliases":"J^s_γmidα_i_1,…,α_i_s gamma mid alpha ldots alpha","display":null,"kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-934d75bba0abbd11","tex":"J^X_{a\\leftarrow a}","aliases":"J^X_a← a leftarrow","display":"J^X_{a\\leftarrow a}","kind":"use","excerpt":"[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56e18f88fc4e1759","tex":"J^X_{a\\leftarrow b}","aliases":"J^X_a← b leftarrow","display":"J^X_{a\\leftarrow b}","kind":"use","excerpt":"[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-061ecdb43086417a","tex":"J^X_{a\\leftarrow c}","aliases":"J^X_a← c leftarrow","display":"J^X_{a\\leftarrow c}","kind":"use","excerpt":"[Canonical triangle holonomy is trivial] For every triple a,b,c, J^X_a← cJ^X_c← bJ^X_b← a=id. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.refineCoordinates_self","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f8a9e76041833cb","tex":"J^X_{c\\leftarrow a}","aliases":"J^X_c← a leftarrow","display":"J^X_{c\\leftarrow a}","kind":"use","excerpt":"[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-274d53e819de8ecb","tex":"J^X_{c\\leftarrow b}","aliases":"J^X_c← b leftarrow","display":"J^X_{c\\leftarrow b}","kind":"use","excerpt":"[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoo","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9c9a48844d36e49","tex":"J^X_{P_b,P_a}","aliases":"J^X_P_b,P_a","display":"J^X_{P_b,P_a}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20eda0343851fb87","tex":"J^Y_{Q_b,Q_a}","aliases":"J^Y_Q_b,Q_a","display":"J^Y_{Q_b,Q_a}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fc6f897ed8e4ebf","tex":"J^Y_{s\\leftarrow r}","aliases":"J^Y_s← r leftarrow","display":"J^Y_{s\\leftarrow r}","kind":"use","excerpt":"[Canonical coordinate naturality] The exact coordinate representation preserves typed composition and satisfies J^Y_s← rwidehat F_r← a = widehat F_s← bJ^X_b← a. statusleanpriorKomorebi.PaperII.NestedRefinement.coordinateOperator_comp; Komorebi.PaperII.NestedRefinement.coordinateOperator_naturality","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1dd01de180ee6039","tex":"J_{(a_0)}","aliases":"J_(a_0)","display":"J_{(a_0)}","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e97a80aa927d24d","tex":"j_1","aliases":"j_1","display":"j_1","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fbb28fbeeb66e2d7","tex":"j_2","aliases":"j_2","display":"j_2","kind":"use","excerpt":"[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10ac8d7d633361db","tex":"J_3","aliases":"J_3","display":"J_3","kind":"use","excerpt":"Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea80efecfc6ff0ed","tex":"J_3^3","aliases":"J_3^3","display":"J_3^3","kind":"use","excerpt":"vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30b75b8d021145b9","tex":"J_4","aliases":"J_4","display":"J_4","kind":"use","excerpt":"The signed shift is a finite carrier for the differential cycle. At order four, three basis steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression. modelfiguresigned_cycleThe exact signed shift J_4. ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d611cab28d1459e6","tex":"J_{4,2}","aliases":"J_4,2","display":"J_{4,2}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c18f6751c1c42314","tex":"J_4^4","aliases":"J_4^4","display":"J_4^4","kind":"use","excerpt":"steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression. modelfiguresigned_cycleThe exact signed shift J_4. Source columns and target rows describe the same action as the cycle on the left. Direct int","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0992043af0c631b6","tex":"J_{8,2}","aliases":"J_8,2","display":"J_{8,2}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3b639da3aa60999","tex":"J_{8,4}","aliases":"J_8,4","display":"J_{8,4}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f63cab9c738188a7","tex":"J_a","aliases":"J_a","display":"J_a","kind":"use","excerpt":"At depth N≥1, suppose F(a)=q^Nr and a desired next integer parameter target satisfies t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exa","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b8a912bcc1d47fa","tex":"j_A","aliases":"j_A","display":"j_A","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62dac47031a2738c","tex":"J_{a,a}","aliases":"J_a,a","display":"J_{a,a}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c71586b5a403c1a2","tex":"J_{a,b}","aliases":"J_a,b","display":"J_{a,b}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2587253da8db636d","tex":"J_{a_1\\leftarrow a_0}","aliases":"J_a_1← a_0 leftarrow","display":"J_{a_1\\leftarrow a_0}","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b3065ac5478799d","tex":"J_{a_n\\leftarrow a_0}","aliases":"J_a_n← a_0 leftarrow","display":"J_{a_n\\leftarrow a_0}","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d6d693ea2042a00","tex":"J_{a_n\\leftarrow a_{n-1}}","aliases":"J_a_n← a_n-1 leftarrow","display":"J_{a_n\\leftarrow a_{n-1}}","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a5d3aa52b0f5bb8","tex":"J_{a\\leftarrow a}","aliases":"J_a← a leftarrow","display":"J_{a\\leftarrow a}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-777ddad06f846983","tex":"J_{a\\leftarrow b}","aliases":"J_a← b leftarrow","display":"J_{a\\leftarrow b}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54a89b74eb1326aa","tex":"J_{a\\leftarrow c}","aliases":"J_a← c leftarrow","display":"J_{a\\leftarrow c}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9052d9414d7b8da7","tex":"J_{a\\leftarrow o}","aliases":"J_a← o leftarrow","display":"J_{a\\leftarrow o}","kind":"use","excerpt":"Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-025540ed37ef1856","tex":"j_B","aliases":"j_B","display":"j_B","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8fbcf86f56456cc","tex":"J_{b,a}","aliases":"J_b,a","display":"J_{b,a}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2788fcb497b7b9e","tex":"J_{b,a}^{-1}","aliases":"J_b,a^-1","display":"J_{b,a}^{-1}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33e57b582e50713d","tex":"J_{c,a}","aliases":"J_c,a","display":"J_{c,a}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-653dd629853bff62","tex":"J_{c,b}","aliases":"J_c,b","display":"J_{c,b}","kind":"use","excerpt":"Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab66a2af2c46ac12","tex":"J_{c\\leftarrow a}","aliases":"J_c← a leftarrow","display":"J_{c\\leftarrow a}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0f23abc79bf97b6","tex":"J_{c\\leftarrow b}","aliases":"J_c← b leftarrow","display":"J_{c\\leftarrow b}","kind":"use","excerpt":"[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTranspor","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5be3e9b89708ead6","tex":"j_f","aliases":"j_f","display":"j_f","kind":"use","excerpt":"_packet_decomposition Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f69f8efae610a884","tex":"J_{F,\\bar x}","aliases":"J_F,bar x bar","display":"J_{F,\\bar x}","kind":"use","excerpt":"The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the chapter supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual digit","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9e13171b862dffe","tex":"J_i^1","aliases":"J_i^1","display":"J_i^1","kind":"use","excerpt":"Choose a labeled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79e7ca6f1d554bb2","tex":"J_{ij}","aliases":"J_ij","display":"J_{ij}","kind":"use","excerpt":"e_i=F_i(x)/p^Lpmod p, J_ij=frac∂ F_i∂ X_j(x)pmod p.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0592a96c436fc819","tex":"j_K","aliases":"j_K","display":"j_K","kind":"use","excerpt":"Actual quadratic local conditions and completion. Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embedding. Lean pr","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-416b4c70e8031735","tex":"j_m","aliases":"j_m","display":"j_m","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8932ad4e519e5510","tex":"J_N^{N-1}","aliases":"J_N^N-1","display":"J_N^{N-1}","kind":"use","excerpt":"The vectors e_0,J_Ne_0,…,J_N^N-1e_0 are the standard basis. Thus e_0 is cyclic for J_N, so the minimal polynomial of J_N has degree N. Since J_N^N=-I, that minimal polynomial divides x^N+1 and therefore equals x^N+1. The characteristic polynomial has the same degree and is divisible by the minimal polynomial, so it too equals x^N+1. This polynomial","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62296fc9b333afb5","tex":"J_N^{Nn+d_{pq}}","aliases":"J_N^Nn+d_pq","display":"J_N^{Nn+d_{pq}}","kind":"use","excerpt":"For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50bc1c682c761171","tex":"j_o","aliases":"j_o","display":"j_o","kind":"use","excerpt":"[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1f0bab3ffea322d","tex":"J_{o\\leftarrow a}","aliases":"J_o← a leftarrow","display":"J_{o\\leftarrow a}","kind":"use","excerpt":"[Flat connections are gauges of the canonical pair transport] Suppose the C_a are exact packet coordinate modules for one ambient module, so the canonical J_b← a is available. If Γ is a flat normalized reversible pair connection, choose o∈ B and set G_a=Γ_a← oJ_o← a. Then every pair transport satisfies Γ_b← a =G_bJ_b← aG_a^-1. statusleanthreeKomore","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-beacfac56a561ebc","tex":"J_{o\\leftarrow b}","aliases":"J_o← b leftarrow","display":"J_{o\\leftarrow b}","kind":"use","excerpt":"Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a · = Γ_b← oΓ_o← a · = Γ_b← a. The first reduction uses the canonical cocycle twice. The second uses flatness of Γ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ebbe8be9d940a9d3","tex":"J_p","aliases":"J_p","display":"J_p","kind":"use","excerpt":"[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTranspo","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1788eb72d6d4271c","tex":"J_{\\Pcal\\leftarrow\\Pcal'}","aliases":"J_Pcal←Pcal' J_ P← P' Pcal leftarrow Pcal mathcal leftarrow mathcal","display":"J_{{\\mathcal P}\\leftarrow{\\mathcal P}'}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6cf9bd5f1c72ed56","tex":"j_q","aliases":"j_q","display":"j_q","kind":"use","excerpt":"[Several exact edge packets] If f=s+∑_q=1^Mj_q and every j_q is retained exactly, then f-(Q_ms+∑_qj_q) = s-Q_ms . Curved edges, corners, and different jump amplitudes do not alter the identity. The finite sum j=∑_q=1^M j_q is a single retained packet. Applying the preceding cancellation identity gives the result. The empty family M=0 is included wi","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a770760ae72cbdb3","tex":"J_{\\Qcal'\\leftarrow\\Qcal}","aliases":"J_Qcal'←Qcal J_ Q'← Q Qcal leftarrow Qcal mathcal leftarrow mathcal","display":"J_{{\\mathcal Q}'\\leftarrow{\\mathcal Q}}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe88e020806e1739","tex":"J_r","aliases":"J_r","display":"J_r","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-456e17f40199861f","tex":"j_r","aliases":"j_r","display":"j_r","kind":"use","excerpt":"For a general depth r, call j_r=max J_r its true terminal. This definition depends on the actual carry minimum. Rewriting the same integer as another wall expression does not change j_r. The universal simplicity target is for every integer r≥1 and x∈ Z_2, G_r(x)=0 Longrightarrow G_r'(x)≠0. The following are completed restricted results of the later","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3bc30050b2c5943","tex":"J_{\\Rcal\\leftarrow\\Pcal}","aliases":"J_Rcal←Pcal J_ R← P Rcal leftarrow Pcal mathcal leftarrow mathcal","display":"J_{{\\mathcal R}\\leftarrow{\\mathcal P}}","kind":"use","excerpt":"[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ebfd094937c1ad9","tex":"J_{\\Rcal\\leftarrow\\Qcal}","aliases":"J_Rcal←Qcal J_ R← Q Rcal leftarrow Qcal mathcal leftarrow mathcal","display":"J_{{\\mathcal R}\\leftarrow{\\mathcal Q}}","kind":"use","excerpt":"[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98e38db59701cf95","tex":"J_{\\rm keep}","aliases":"J_rm keep rm","display":"J_{\\rm keep}","kind":"use","excerpt":"bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the decla","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1a429be9f79e459","tex":"J_{\\rm split}","aliases":"J_rm split rm","display":"J_{\\rm split}","kind":"use","excerpt":"bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the decla","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06e8fa7eb9aaa088","tex":"j_s","aliases":"j_s","display":"j_s","kind":"use","excerpt":"[Reflection positivity is closed under limits] Suppose C_n(j,k) are reflection positive kernels on N, meaning every finite matrix (C_n(j_r,j_s))_r,s is positive semidefinite. If C_n(j,k)⟶ C(j,k) for every pair, then C is reflection positive.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07b6123d41bb6991","tex":"J_t","aliases":"J_t","display":"J_t","kind":"use","excerpt":"[Regularity throughout an open parameter set] Let U⊆ K^k be open, and let h be continuous on U. Suppose that for every t∈ U, F(h(t),t)=0, J_t=D_YF(h(t),t)∈GL_e( K). Then h is analytic at every point of U and its strict derivative is Dh(t)=-J_t^-1D_TF(h(t),t) (t∈ U). At each t∈ U, openness turns continuity on U into continuity at t and makes the roo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0562af54b8ae3ad7","tex":"J_t^{-1}","aliases":"J_t^-1","display":"J_t^{-1}","kind":"use","excerpt":"[Regularity throughout an open parameter set] Let U⊆ K^k be open, and let h be continuous on U. Suppose that for every t∈ U, F(h(t),t)=0, J_t=D_YF(h(t),t)∈GL_e( K). Then h is analytic at every point of U and its strict derivative is Dh(t)=-J_t^-1D_TF(h(t),t) (t∈ U). At each t∈ U, openness turns continuity on U into continuity at t and makes the roo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f9e5419185ac3d4","tex":"J_{\\Ucal\\leftarrow\\Vcal}","aliases":"J_Ucal←Vcal J_ U← V Ucal leftarrow Vcal mathcal leftarrow mathcal","display":"J_{{\\mathcal U}\\leftarrow{\\mathcal V}}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7fb5e5ec3770a81","tex":"J_{x'}","aliases":"J_x'","display":"J_{x'}","kind":"use","excerpt":"[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68bfee508e5dfce6","tex":"J_{y'}","aliases":"J_y'","display":"J_{y'}","kind":"use","excerpt":"imizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de4001a3c6225d07","tex":"J_{z(n)}","aliases":"J_z(n)","display":"J_{z(n)}","kind":"use","excerpt":"_0, t(n+1)-t(n)∈ q^L+n Z^k, there is a single tower z: N→ Z^m+k with z(0)=x, free(z(n))=t(n), F(z(n))∈ q^L+n Z^e, z(n+1)-z(n)∈ q^L+n Z^m+k. The parameter equality is exact in integers. For a≤ b, z(b)-z(a)∈ q^L+a Z^m+k, DF(z(n))bmod q=DF(x)bmod q. Consequently J_z(n)=J_x at every level. No later surjectivity or per-step solvability is assumed. This ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0015545e23b066dd","tex":"J_{z_n}","aliases":"J_z_n","display":"J_{z_n}","kind":"use","excerpt":"[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f44d0496740a2e92","tex":"J_{z_{n+1}}","aliases":"J_z_n+1","display":"J_{z_{n+1}}","kind":"use","excerpt":"At level N=L+n, divide the residual by q^N. Surjectivity supplies t∈ S_q^d with J_z_nt=-overlineF(z_n)/q^N. Choose an integer representative and apply [source reference] to obtain z_n+1. The new point is congruent to z_n modulo q. Evaluation of each formal partial derivative therefore gives J_z_n+1=J_z_n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-284218cb67c97467","tex":"J_\\theta","aliases":"J_θ theta","display":"J_\\theta","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71f2ee1ce2cf267e","tex":"J_\\theta^1","aliases":"J_θ^1 theta","display":"J_\\theta^1","kind":"use","excerpt":"Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc399b7a1160964d","tex":"J_\\lambda","aliases":"J_λ lambda","display":"J_\\lambda","kind":"use","excerpt":"[Exact Lagrangian packet compiler] Fix λ≥0 and define J_λ( L) = cost( L) + λerr( L). For every node v, set K_λ(v) = κ_v+λϵ_v and S_λ(v) = σ_v + ∑_w∈child(v)F_λ(w). For a terminal node, set F_λ(v)=K_λ(v). For an internal node, the recurrence F_λ(v) = min{K_λ(v),S_λ(v)} computes the exact minimum Lagrangian objective over every admissible pruning of ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9fe993390430812","tex":"J_{\\lambda,u}","aliases":"J_λ,u lambda","display":"J_{\\lambda,u}","kind":"use","excerpt":"Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem,","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9743d7bec48cbb60","tex":"\\Jac","aliases":"Jac operatorname","display":"{\\operatorname{Jac}}","kind":"use","excerpt":"How many seeds are required? · srank_A(M) · For a finitely generated module, pass to M/Jac(A)M. When the quotient algebra is semisimple Artinian, compute its simple multiplicities.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1505c70154d93441","tex":"\\jump","aliases":"jump","display":null,"kind":"use","excerpt":"The singular packet results begin with trace data. If a piecewise smooth function has a jump in its derivative of order m at τ_q, then the packet jumpf^(m)_τ_q frac(t-τ_q)_+^mm! removes exactly that trace jump. Subtracting all packets through order r-1 leaves a global W^r,p remainder. The packet coefficients are uniquely determined by the one-sided","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"K(t) · The memory kernel after hidden-state elimination. · It is not the transition matrix K.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb9dab5747004561","tex":"K'","aliases":"K'","display":"K'","kind":"use","excerpt":"vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESChapter VII: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b414e0d8b85d3e26","tex":"K^{\\mathsf T}","aliases":"K^ T mathsf","display":"K^{\\mathsf T}","kind":"use","excerpt":"K^ Tf · Expected next observable given the current state. · The same dynamics, with the opposite action on the dual object.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-997c382b71c367a6","tex":"K^*","aliases":"K^*","display":"K^*","kind":"use","excerpt":"Differentiating P_α^2=P_α gives P_αdot P_α P_α=0. Differentiating P_β P_α=0 for α≠β gives P_βdot P_α=-dot P_β P_α. Hence K P_β=dot P_β P_β and P_β K=-dot P_β(I-P_β). Their difference is dot P_β. Also, K+K^* = ∑_α(dot P_α P_α+P_αdot P_α) = ∑_αdot P_α =0. endgroup","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b779dcac03cb4e5","tex":"K^2","aliases":"K^2","display":"K^2","kind":"use","excerpt":"For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":":04 For the one-dimensional Dirichlet difference operator, let h=1/M with an integer M≥2, let 1≤ k≤ M-1, and set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec1a2b307412bdd0","tex":"k^4","aliases":"k^4","display":"k^4","kind":"use","excerpt":"vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPEChapter VII: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters because th","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25e27fdec946575c","tex":"K^d","aliases":"K^d","display":"K^d","kind":"use","excerpt":"mathbb Q_p, and d,s,M∈ N. Let F,H∈ R[X_1,…,X_d]^d, let a∈ R^d satisfy F(a)=0, and set J=DF(a), G=F+p^M H. Assume J∈GL_d(K), p^sJ^-1∈ M_d(R), M>2s. Then there is exactly one root b of G in a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d, and there is ε∈ p^2M-3sR^d such that, in K^d, b-a=-p^M J^-1H(a)+ε. The inverse in this formula belongs to the original Jac","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a7c895cee857ce71","tex":"K^h","aliases":"K^h","display":"K^h","kind":"use","excerpt":"dicts possible support but not actual support. Replacing the inclusion supp(AB)⊆supp(A)∘supp(B) by equality would be wrong even in this four-dimensional model. headWhat an acyclic graph does guarantee If the longest strict directed path has h edges, then no term of K^h+1 can survive. Thus K^h+1=0. This is an upper bound on the nilpotence index, not","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ba73dafd81ea4478","tex":"K^{h+1}","aliases":"K^h+1","display":"K^{h+1}","kind":"use","excerpt":", K^2=0. The matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion supp(AB)⊆supp(A)∘supp(B) by equality would be wrong even in this four-dimensional model. headWhat an acyclic graph does guarantee If the longest strict directed path has h edges,","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad06b8163a249728","tex":"K^j","aliases":"K^j","display":"K^j","kind":"use","excerpt":"[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd225a0f93f2f10a","tex":"K^{j+1}","aliases":"K^j+1","display":"K^{j+1}","kind":"use","excerpt":"The identity follows by telescoping: bar K^n-1-jCK^j+1-bar K^n-jCK^j =bar K^n-1-j(CK-bar KC)K^j. Summing from j=0 to n-1 leaves CK^n-bar K^nC. The norm bound follows termwise.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-647eccb9127e0259","tex":"K^n","aliases":"K^n","display":"K^n","kind":"use","excerpt":"[Exact lumpability] If Δ=0, then CK^n=bar K^nC for every n≥0. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9f756320594da4d","tex":"\\mathbb K^n","aliases":"K^n mathbb","display":"\\mathbb K^n","kind":"use","excerpt":"[Derivative of polynomial evaluation] For P∈ K[X_1,…,X_n], polynomial evaluation is analytic at every point. At a∈ K^n its strict Fr echet derivative is v⟼∑_j=1^n (∂_jP)(a)v_j. For a finite system, its strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c47e779cf5c8f6c8","tex":"K^{n+1}","aliases":"K^n+1","display":"K^{n+1}","kind":"use","excerpt":"The result is immediate for n=0. If CK^n=bar K^nC, then CK^n+1=bar K^nCK=bar K^n+1C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-85c3fb99e8d4fae6","tex":"K^T","aliases":"K^T","display":"K^T","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac62d9119015a5fb","tex":"K^\\times","aliases":"K^× times","display":"K^\\times","kind":"use","excerpt":"[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-649cb8182e4a0913","tex":"K^{\\times}","aliases":"K^× times","display":"K^{\\times}","kind":"use","excerpt":"Local class field theory identifies one dimensional Weil characters with characters of K^× [source reference]. The tame induction theorem extends this explicit picture to higher dimensional tame atoms. The local polynomial and conductor calculations in [source reference] therefore give explicit finite invariants for the declared tame packets. Finit","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f550ca12c9ac5b07","tex":"k^\\times","aliases":"k^× times","display":"k^\\times","kind":"use","excerpt":"Indeed, for residue cardinality q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-805006b107a6864c","tex":"k^{\\times}","aliases":"k^× times","display":"k^{\\times}","kind":"use","excerpt":"Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYChapter VII: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9817d114fe256b80","tex":"k_0","aliases":"k_0","display":"k_0","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52d8e03f4220d281","tex":"K_0^*","aliases":"K_0^*","display":"K_0^*","kind":"use","excerpt":"vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYChapter VII: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf5fcf757b255a84","tex":"K_1","aliases":"K_1","display":"K_1","kind":"use","excerpt":"vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYChapter VII: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1bdfdabfb7ccb3ec","tex":"k_1","aliases":"k_1","display":"k_1","kind":"use","excerpt":"[Quadratic reduced obstruction] Assume the hypotheses of [source reference] with r≥2. For k_1,k_2∈ K, D^2κ(0)[k_1,k_2] = π_O D^2F(0)[k_1,k_2].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0af29e9b97d6dd25","tex":"K_{10}","aliases":"K_10","display":"K_{10}","kind":"use","excerpt":"vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSIONChapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3. Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. Th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ead6134bc1ede95b","tex":"K_1^*","aliases":"K_1^*","display":"K_1^*","kind":"use","excerpt":"vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYChapter VII: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8cd867fa822df3a","tex":"k_2","aliases":"k_2","display":"k_2","kind":"use","excerpt":"[Quadratic reduced obstruction] Assume the hypotheses of [source reference] with r≥2. For k_1,k_2∈ K, D^2κ(0)[k_1,k_2] = π_O D^2F(0)[k_1,k_2].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9f725106db7a34c","tex":"K_{20}","aliases":"K_20","display":"K_{20}","kind":"use","excerpt":"vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSIONChapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3. Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. Th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fad803cc2ee90a75","tex":"K_2^{\\mathsf T}","aliases":"K_2^ T mathsf","display":"K_2^{\\mathsf T}","kind":"use","excerpt":"1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the object being transported. The transpose and the posit","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4a6203c6556b0cb","tex":"K_{31}","aliases":"K_31","display":"K_{31}","kind":"use","excerpt":"vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSIONChapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3. Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. Th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21a51e221f2f6f08","tex":"K_{32}","aliases":"K_32","display":"K_{32}","kind":"use","excerpt":"vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSIONChapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3. Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. Th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa83735d882dbba5","tex":"K_{4,j}","aliases":"K_4,j","display":"K_{4,j}","kind":"use","excerpt":"The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6cc92a51c6969e2c","tex":"K_a","aliases":"K_a","display":"K_a","kind":"use","excerpt":"At depth N≥1, suppose F(a)=q^Nr and a desired next integer parameter target satisfies t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exa","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-916bbd7c25020c00","tex":"k_A","aliases":"k_A","display":"k_A","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c7fa5ebfa740b08","tex":"K_a^*","aliases":"K_a^*","display":"K_a^*","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08dc6e54edbcd212","tex":"K_c","aliases":"K_c","display":"K_c","kind":"use","excerpt":"Removing a singular forced zero. Put c=ap<N and define the actual normalized kernel K_c(t)=c!∑_n≥0fract^Nn(Nn+c)!, K_c(0)=1. Let ℓ(T)=log(1+T)/T, defined by shifting logarithm coefficients, not by inverting T. Set U(T)=ℓ(T)^cK_c(log(1+T)). The formal proof establishes U(0)=1 and the exact factorization H_N,c(log(1+T))=fracT^cc!U(T). It follows coef","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-848f2d1221ba8b4d","tex":"K_{c-1}","aliases":"K_c-1","display":"K_{c-1}","kind":"use","excerpt":"Boundary. The original positive-address section has zero constant coefficient. It is not the series to which unit inverse closure is applied. 4. The singular ratio estimate. For a positive address c, the rational ratio is independently constructed from the original kernels: R_c(t)=c K_c-1(t)K_c(t)^-1. The scalar factor c is necessary. The source pr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07baee216908c2f6","tex":"K_f","aliases":"K_f","display":"K_f","kind":"use","excerpt":"[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-674fd5cf0671e765","tex":"k_F^\\times","aliases":"k_F^× times","display":"k_F^\\times","kind":"use","excerpt":"The construction retains the actual canonical beta extension and can be completed with any prescribed central quasicharacter omega_F trivial on U_F 1. Because k_F^×= F_2^× is trivial, such an omega_F is unramified. Thus this covers every tamely ramified central quasicharacter for this particular residue field.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b783228933c35b11","tex":"k_i","aliases":"k_i","display":"k_i","kind":"use","excerpt":"For 0≤ i<j, let d_i=v_2(i!)+v_2((j-1-i)!), k_i=1+lfloorlog_2max(i,j-1-i)rfloor, B_i=i+2^k_i Z_2, t_i=d_i+k_i. At j=1 use d_0=k_0=t_0=0. The balls B_i partition Z_2. Indeed a residue represented twice among 0,…,j-1 splits into its two children. Every other represented residue remains one ball. On each such ball, P_j(i+2^k_iy)=2^t_iyQ_i(y), Q_i∈ Z_2⟨","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-211be6ddf3860b2e","tex":"K_{i_{r+1},i_r}","aliases":"K_i_r+1,i_r","display":"K_{i_{r+1},i_r}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69a26fc25f751d91","tex":"K_{ij}","aliases":"K_ij","display":"K_{ij}","kind":"use","excerpt":"K · One discrete stochastic step. · ones^ TK=ones^ T and K_ij≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca931e9dd55037b6","tex":"K_j","aliases":"K_j","display":"K_j","kind":"use","excerpt":"Let K_j(t) be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is F_N,j(t)=j! K_j(t)^-1. The source/inverse identities and the factorial successor identity prove F_N,j+1=F_N,jR_j+1, F_N,j=F_N,0∏_c=1^jR_c. The product formula is not an assembly hypothesis. Proved flatness of all factors and the o","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcf3911a4b4214e2","tex":"k_j","aliases":"k_j","display":"k_j","kind":"use","excerpt":"Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ea16453cbd62d91","tex":"K_{j i}","aliases":"K_j i","display":"K_{j i}","kind":"use","excerpt":"For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1fd09588ac918c1","tex":"\\mathcal K_{j+1}","aliases":"K_j+1 mathcal","display":"\\mathcal K_{j+1}","kind":"use","excerpt":"Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a68fcbb4721376c8","tex":"K_j^*","aliases":"K_j^*","display":"K_j^*","kind":"use","excerpt":"e, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e96542379e91eebb","tex":"k_j^4","aliases":"k_j^4","display":"k_j^4","kind":"use","excerpt":"[Low mode dispersion in dimension d] For every centered mode k, 0 ≤ 4π^2absk^2-λ_N(k) ≤ frac4π^43N^2 ∑_j=1^dk_j^4.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf82e92850871481","tex":"K_{ji}","aliases":"K_ji","display":"K_{ji}","kind":"use","excerpt":"For one step, (M_Vf)_i = e^-V(i)∑_jK_jif_j. Iterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a5f8d51d3b62650","tex":"K_L","aliases":"K_L","display":"K_L","kind":"use","excerpt":"[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c0f8cb6a6409762","tex":"K_{L^1(0,T)}","aliases":"K_L^1(0,T)","display":"K_{L^1(0,T)}","kind":"use","excerpt":"· + ∫_0^tS(t-s) ∫_0^sK(s-r)x(r) d r d s. Assume sup_0≤ t≤ Tnormwidetilde x(t)≤ B. Set L=norm K_L^1(0,T), ε=normK-widetilde K_L^1(0,T). Then sup_0≤ t≤ T normx(t)-widetilde x(t) ≤ M_TBTε exp(M_TLT).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46d21cf1bf7572c1","tex":"K_m","aliases":"K_m","display":"K_m","kind":"use","excerpt":"Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b35ad5d2218e1cf","tex":"\\mathcal K_m","aliases":"K_m mathcal","display":"\\mathcal K_m","kind":"use","excerpt":"vpage17Invariant does not mean independently addressableA STRICT KRYLOV INCLUSIONChapter VIII: Proposition 7.2, Theorem 7.3. Take A=I_2, split into the two coordinate packets, and start with v=(1,1)^T. Every packet is invariant. Nevertheless every polynomial applies the same scalar to both coordinates: p(A)v=p(1)(1,1)^T, dim K_m(A,v)=1 (m≥1). vfigk","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-792ddfc072fc3946","tex":"k_m","aliases":"k_m","display":"k_m","kind":"use","excerpt":"The unit kernel and its reciprocal. Use a squared formal variable u and define k_m=frac2^2m+1(2m+2)!, K(u)=∑_m≥0k_mu^m, β_m=[u^m]K(u)^-1. The constant term is proved to be one. The factorial valuation bound gives v_2(k_m)=2m+1-v_2((2m+2)!)≥0. Thus every k_m belongs to O_2. A general coefficientwise subring theorem is proved by strong induction usin","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb9a538f5e406f30","tex":"K_n","aliases":"K_n","display":"K_n","kind":"use","excerpt":"…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry count","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25e4b588e90884b5","tex":"K_{N,j}","aliases":"K_N,j","display":"K_{N,j}","kind":"use","excerpt":"The normalized source and its inverse are K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coeffic","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3531ccbb80b4887","tex":"K_{N,j}^{-1}","aliases":"K_N,j^-1","display":"K_{N,j}^{-1}","kind":"use","excerpt":"The formalization does not stop at a recurrence. In the variable z=t^N, construct K_N,j(z)=∑_n≥0fracj!(Nn+j)!z^n, F_N,j(z)=∑_n≥0fracC_N,j(n)(Nn)!z^n. It proves the factorial coefficient identity, then F_N,jK_N,j=1 and therefore F_N,j=K_N,j^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ebc30c5e8c584b3","tex":"K_N^m","aliases":"K_N^m","display":"K_N^m","kind":"use","excerpt":"[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-365d749061e6ffdd","tex":"K_N^q","aliases":"K_N^q","display":"K_N^q","kind":"use","excerpt":"vpage22Finite coefficients give infinite momentsZERO POWER SUMSChapter IV: Theorem 12.6, Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No num","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a30d27faec7f7f7","tex":"K_r","aliases":"K_r","display":"K_r","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2b60819c95171e8","tex":"k_r","aliases":"k_r","display":"k_r","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-193077db22dceb36","tex":"K_r^*","aliases":"K_r^*","display":"K_r^*","kind":"use","excerpt":"_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b671d5da77c233cc","tex":"K_{ri}","aliases":"K_ri","display":"K_{ri}","kind":"use","excerpt":"[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce6b6fd6eca36fc9","tex":"K_{rj}","aliases":"K_rj","display":"K_{rj}","kind":"use","excerpt":"[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7df93e11dabca8d6","tex":"K_{\\rm mem}","aliases":"K_rm mem rm","display":"K_{\\rm mem}","kind":"use","excerpt":"0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponential is not e^-2tI: e^tJ=e^-2t(I+tN), K_rm mem(t)=Ue^tJV=t e^-2t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that r","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab625c241c8a0e88","tex":"K_t","aliases":"K_t","display":"K_t","kind":"use","excerpt":"There is also an explicit open-set version. Let U be open, let g be continuous on U, and suppose for every t∈ U that F(g(t),t)=0, J_t:=D_YF(g(t),t)∈GL_e( K). Writing K_t=D_TF(g(t),t), the theorem mcodecontinuous_selection_analyticOnNhd proves g is analytic at every t∈ U, Dg(t)=-J_t^-1K_t (t∈ U), with a strict Fréchet derivative at each point. The i","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75bb1e0e1a9420dc","tex":"K_{\\mathrm{unused}}","aliases":"K_unused mathrm","display":"K_{\\mathrm{unused}}","kind":"use","excerpt":"efine compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are com","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86aa5e48615bd30f","tex":"K_{\\mathrm{unused}}^*","aliases":"K_unused^* mathrm","display":"K_{\\mathrm{unused}}^*","kind":"use","excerpt":"_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73642ab179f59f29","tex":"K_x","aliases":"K_x","display":"K_x","kind":"use","excerpt":"Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residu","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a37d1498744920e","tex":"k_x","aliases":"k_x","display":"k_x","kind":"use","excerpt":"[Two-dimensional errors] For the 192×192 disk experiment, a square Fourier mask retains indices |k_x|,|k_y|≤16, or 33^2=1089 bins. The reported errors are lrr method · full-image RMSE · boundary-band RMSE","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81bf2ba82285d729","tex":"k_y","aliases":"k_y","display":"k_y","kind":"use","excerpt":"[Two-dimensional errors] For the 192×192 disk experiment, a square Fourier mask retains indices |k_x|,|k_y|≤16, or 33^2=1089 bins. The reported errors are lrr method · full-image RMSE · boundary-band RMSE","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83bf1e6422e6d967","tex":"K_{z_n}","aliases":"K_z_n","display":"K_{z_n}","kind":"use","excerpt":"At level N=L+n, write the supplied parameter increment exactly as t_n+1-t_n=q^Nw with w∈ Z^k. Surjectivity of the solved block gives a digit u satisfying J_z_nu=-overlineF(z_n)/q^N-K_z_noverline w. Choose an integer representative widetilde u and set z_n+1=z_n+q^N(widetilde u,w). Its parameter coordinates equal the supplied target exactly. Taylor e","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f178970e61601761","tex":"K_\\alpha","aliases":"K_α alpha","display":"K_\\alpha","kind":"use","excerpt":"[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a77d87feaab21d87","tex":"K_\\beta","aliases":"K_β beta","display":"K_\\beta","kind":"use","excerpt":"[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-44cc17134afaa418","tex":"K_{\\beta\\alpha}","aliases":"K_βα beta alpha","display":"K_{\\beta\\alpha}","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c71da9fa04844e9","tex":"K_\\varepsilon","aliases":"K_ε varepsilon","display":"K_\\varepsilon","kind":"use","excerpt":"vpage6A broken lump and its error budgetDEFECTS / A GUARANTEE WITHOUT A FITChapter VI, Part I: Theorems 3.1, 3.3, and 5.2. Continuous and discrete defect identities. Modify only column zero of K: move 1/16 of its mass from target state zero to target state two. The perturbed matrix K_ε remains stochastic, but its aggregate column now differs from c","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-217a35c5c19e1d0d","tex":"K_\\varepsilon^j","aliases":"K_ε^j varepsilon","display":"K_\\varepsilon^j","kind":"use","excerpt":"1/16 · 0 · 0 · 0, Δ _1=1/8. vfigdefect_budget platecaptionThe solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp. headTelescoping tells us where every error originates For n≥0, CK_ε^n-bar K^nC =∑_j=0^n-1bar K^n-1-jΔ K_ε^j. Each summand propagates ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a375b31a4960827d","tex":"K_\\varepsilon^n","aliases":"K_ε^n varepsilon","display":"K_\\varepsilon^n","kind":"use","excerpt":"1/16 · 0 · 0 · 0, Δ _1=1/8. vfigdefect_budget platecaptionThe solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp. headTelescoping tells us where every error originates For n≥0, CK_ε^n-bar K^nC =∑_j=0^n-1bar K^n-1-jΔ K_ε^j. Each summand propagates ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b7ce4653ca285ed","tex":"K_\\ell","aliases":"K_ℓ ell","display":"K_\\ell","kind":"use","excerpt":"the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero. The even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only a","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1062f1ecdf031386","tex":"\\Kcal_J","aliases":"Kcal_J K_J Kcal mathcal","display":"{\\mathcal K}_J","kind":"use","excerpt":"he least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete primary blocks indexed by J, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J ∑_k=0^ν_j-1","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ccedf2f845a654d7","tex":"\\Kcal_{j+1}","aliases":"Kcal_j+1 K_j+1 Kcal mathcal","display":"{\\mathcal K}_{j+1}","kind":"use","excerpt":"[Approximate packet Krylov bound] Write A=D+K, where D is packet diagonal. For every positive integer j, normA^j-D^j ≤ jM^j-1normK, where M=max{normA,normD}. Consequently, dist(A^jv,⊕_α K_j+1(D_α,v_α)) ≤ jM^j-1normKnormv.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4f733cd50dc9a2c","tex":"\\kk","aliases":"kk 𝕜 Bbbk","display":"{\\Bbbk}","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97491b80a0eec919","tex":"\\kk^2","aliases":"kk^2 𝕜^2 kk Bbbk","display":"{\\Bbbk}^2","kind":"use","excerpt":"The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34c05dc8f5073293","tex":"L'","aliases":"L'","display":"L'","kind":"use","excerpt":"a finite index sublattice of Z^d, and let R contain one representative from every coset. The analysis map x⟼ (x_r)_r∈ R, x_r(ℓ)=x(r+ℓ), is an isometric isomorphism on every ℓ^p space. This statement includes the usual one-dimensional polyphase decomposition, but it also applies to arbitrary finite index lattice packets in several dimensions. If L'≤","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a44e55170cadffdb","tex":"L^*","aliases":"L^*","display":"L^*","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1cfec68d44ee43e","tex":"L^{-1/2}","aliases":"L^-1/2","display":"L^{-1/2}","kind":"use","excerpt":"Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d0fbf856fe55b03","tex":"L^1","aliases":"L^1","display":"L^1","kind":"use","excerpt":"ehatΩ=[tfrac516,tfrac1316], a=widehat a=3. The two intervals have the same length. Their disagreement consists of one missed strip and one extra strip, each of width 1/16. vfigmask_translation |ΩmathbintrianglewidehatΩ|=tfrac18, j-widehat j=3(1_Ω-1_widehatΩ). c|ccc error measure · L^1 norm · squared L^2 norm · L^∞ norm","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c6f72cfe6318222","tex":"L^{1/2}","aliases":"L^1/2","display":"L^{1/2}","kind":"use","excerpt":"[Exact Haar coefficient of a step] Let τ∈ I_j,k, let L=2^-j, and write u=fracτ-kLL∈[0,1]. Then | ⟨ aH_τ,ψ_j,k ⟩ | = |a|L^1/2min{u,1-u}. Every Haar coefficient whose support does not cross τ is zero.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"use","excerpt":"ZZ with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For finite","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67cbdd51ca961fc6","tex":"L^{3/2}","aliases":"L^3/2","display":"L^{3/2}","kind":"use","excerpt":"Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0de1bee007df208a","tex":"L^p","aliases":"L^p","display":"L^p","kind":"use","excerpt":"[Packet perturbation bound] For 1≤ p<∞, interpreted first as an inequality of extended L^p seminorms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise sup","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f30a25ed8c4669b","tex":"L^\\alpha","aliases":"L^α alpha","display":"L^\\alpha","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5aa47e9873d12ad2","tex":"L^{\\alpha+\\beta}","aliases":"L^α+β alpha beta","display":"L^{\\alpha+\\beta}","kind":"use","excerpt":"[Fractional packet calculus] For α,β>0, L^α L^β=L^α+β, and for s,t≥0, T_α(s)T_α(t)=T_α(s+t). Every spectral packet P_j reduces L^α and T_α(t).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d2eb5d87d66df4f","tex":"L^\\beta","aliases":"L^β beta","display":"L^\\beta","kind":"use","excerpt":"[Fractional packet calculus] For α,β>0, L^α L^β=L^α+β, and for s,t≥0, T_α(s)T_α(t)=T_α(s+t). Every spectral packet P_j reduces L^α and T_α(t).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b12d1bed1ef598c0","tex":"L^\\infty","aliases":"L^∞ infty","display":"L^\\infty","kind":"use","excerpt":"f extended L^p seminorms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise supremum, and consequently also the essential L^∞ seminorm, j-widehat j _∞ ≤ |a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c914ea4ff41f154","tex":"L_{-1}","aliases":"L_-1","display":"L_{-1}","kind":"use","excerpt":"Reciprocal transfer gives L_-1^kF_N,j∈ H_p. Extraction of the packed coefficient gives (S+1)^k C_N,j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1. Its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the pr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4997aff165ad7668","tex":"L_{-1}^k","aliases":"L_-1^k","display":"L_{-1}^k","kind":"use","excerpt":"Reciprocal transfer gives L_-1^kF_N,j∈ H_p. Extraction of the packed coefficient gives (S+1)^k C_N,j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1. Its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the pr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38bed36d88af8602","tex":"L_{-P}","aliases":"L_-P","display":"L_{-P}","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd9ee043ad27b658","tex":"L_{-\\lambda}^b","aliases":"L_-λ^b lambda","display":"L_{-\\lambda}^b","kind":"use","excerpt":"Set B=A^-1. Its order-zero integrality was established in [source reference]. Assume by induction that L_-λ^bB∈ H_p for b<k. Since AB=1, L_0^k(AB)=(fracD^Np)^k1=0 (k>0). Use [source reference] with μ=-λ. Exactly one term has inverse order b=k: it has a=0 and all e_i=0, and is A L_-λ^k B. In every other term b<k, so its second factor is integral by ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cbcf44fd451f612c","tex":"L_{-\\lambda}^k","aliases":"L_-λ^k lambda","display":"L_{-\\lambda}^k","kind":"use","excerpt":"beginproof Set B=A^-1. Its order-zero integrality was established in [source reference]. Assume by induction that L_-λ^bB∈ H_p for b<k. Since AB=1, L_0^k(AB)=(fracD^Np)^k1=0 (k>0). Use [source reference] with μ=-λ. Exactly one term has inverse order b=k: it has a=0 and all e_i=0, and is A L_-λ^k B. In every other term b<k, so its second factor is i","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d92ecf36f8db9126","tex":"L_{03}","aliases":"L_03","display":"L_{03}","kind":"use","excerpt":"by e^-2jt. Square-root diffusion multiplies it by e^-sqrt2j t. The constant mode is unchanged in both. vfigfractional platecaptionThe highest mode decays at rates 6 and sqrt6. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit. headFractional does not mea","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-832fc1443a44a791","tex":"L_0^k","aliases":"L_0^k","display":"L_0^k","kind":"use","excerpt":"Set B=A^-1. Its order-zero integrality was established in [source reference]. Assume by induction that L_-λ^bB∈ H_p for b<k. Since AB=1, L_0^k(AB)=(fracD^Np)^k1=0 (k>0). Use [source reference] with μ=-λ. Exactly one term has inverse order b=k: it has a=0 and all e_i=0, and is A L_-λ^k B. In every other term b<k, so its second factor is integral by ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"[Local Lyapunov--Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there ar","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6268c070d1fd0ce6","tex":"L_1^k","aliases":"L_1^k","display":"L_1^k","kind":"use","excerpt":"n adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then proves Δ_1^k g_N,j(n)∈ p^k Z_p for every k,n. The coefficient at degree Nn of L_1^kG_N,j is precisely p^-kΔ_1^k g_N,j(n), and the other residue classes vanish. T","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"dfrac2^4r+1(2r+1)(2r+2), · p=2. Then for every prime p and every even n≥0, A_p,n = ∑_r=0^n/2 binom n2r λ_p,r E_n-2r. Equivalently, if L_p(t)=K_p(t)^-1, then H_p(t)=L_p(t)secht, where L_p(t)=cosh^2(pt) for odd p, and L_2(t)=(fracsinh(2t)2t)^2 for p=2.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7f6796e75b9a514","tex":"L_3","aliases":"L_3","display":"L_3","kind":"use","excerpt":"l source charts] Editorial range correction. This comparison holds for the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero. The even kernel L_3K_2 cannot cha","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5aa0fda5d74dd324","tex":"L_4","aliases":"L_4","display":"L_4","kind":"use","excerpt":"s.state_is_orbit, FibonacciPackets.packet_is_fibonacci targetFibonacciPackets.trace_power_is_lucas, FibonacciPackets.det_power targetFibonacciPackets.jump_charpoly targetFibonacciPackets.fibonacci_packet_recurrence pageheading4. Exact examples and meaningful boundary checksExample 9.3 and supporting results The independently defined Lucas recurrenc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9ec1bee07f32ca4","tex":"L_5","aliases":"L_5","display":"L_5","kind":"use","excerpt":"s.state_is_orbit, FibonacciPackets.packet_is_fibonacci targetFibonacciPackets.trace_power_is_lucas, FibonacciPackets.det_power targetFibonacciPackets.jump_charpoly targetFibonacciPackets.fibonacci_packet_recurrence pageheading4. Exact examples and meaningful boundary checksExample 9.3 and supporting results The independently defined Lucas recurrenc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f8dd52a01de0ecd","tex":"L_c","aliases":"L_c","display":"L_c","kind":"use","excerpt":"For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8c2eb0cf9cbeb36","tex":"L_c^\\alpha","aliases":"L_c^α alpha","display":"L_c^\\alpha","kind":"use","excerpt":"For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2752bc7b3d7d6299","tex":"L_d","aliases":"L_d","display":"L_d","kind":"use","excerpt":"General finite logarithmic recovery. Let A be a rational algebra, without any commutativity assumption on A. For a unipotent element U and a positive integer d satisfying (U-1)^d=0, put L_d(U)=∑_j=1^d-1frac(-1)^j+1j(U-1)^j. The value is independent of the chosen annihilating exponent. The finite logarithm and the nilpotent exponential satisfy L(exp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec6b4ea000366681","tex":"L_f","aliases":"L_f","display":"L_f","kind":"use","excerpt":"Words are written in execution order. In L_fL_e, the rightmost operator L_e acts first. For a written word (e_1,…,e_n), e_1 acts first and the operator product is L_e_n⋯ L_e_1.","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7787694a9216e833","tex":"\\mathcal L_{F,L}","aliases":"L_F,L mathcal","display":"\\mathcal L_{F,L}","kind":"use","excerpt":"For a fixed polynomial system F∈ Z[X_1,…,X_d]^e, a precision L≥1, and a root bar x modulo p^L, let L_F,L(bar x) ={x'∈( Z/p^L+1 Z)^d: F(x')=0, x'≡bar xpmodp^L}. The finite obstruction theorem decides whether this set is empty from the residual digit and reduced Jacobian. When it is nonempty, the theorem also determines its cardinality and affine ker","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8e73625c7457f64","tex":"L_f^\\alpha","aliases":"L_f^α alpha","display":"L_f^\\alpha","kind":"use","excerpt":"For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-743ff0ea070acbd8","tex":"L_h","aliases":"L_h","display":"L_h","kind":"use","excerpt":"For n interior points with spacing h=1/(n+1), define L_h=frac1h^2 2 · -1 · · -1 · 2 · ddots · · ddots · ddots · -1 · · -1 · 2 .","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15870635efb91b25","tex":"L_h^{-1}","aliases":"L_h^-1","display":"L_h^{-1}","kind":"use","excerpt":"[Exact discrete Green reconstruction at grid nodes] Let x_i=ih. Then frac1h(L_h^-1)_ij =G(x_i,x_j). Thus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a716b0894950a10a","tex":"L_i","aliases":"L_i","display":"L_i","kind":"use","excerpt":"tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared f","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"use","excerpt":"For a wall label j≥1 and 0≤ q<j, define e_j(q)=3q+s_2(j-1)-s_2(j-q-1), E_j=min_q<je_j(q), Θ(j)=3j+1+s_2(j-1), L_j=Θ(j)-E_j. Here E_j is a carry minimum, not an Euler number. A wall label is regular when e_j(q)>0 for 1≤ q<j. For r=2^L-j with L>L_j, put δ=L-L_j. The carry identities give μ_r=E_j-δ, J_r={j}, G_r(x)=P_j(x)U(x-j)+2^δ V(x), where U≡1pmod","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0945233af0503d7f","tex":"\\mathcal L_{j+1}","aliases":"L_j+1 mathcal","display":"\\mathcal L_{j+1}","kind":"use","excerpt":"Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"use","excerpt":"R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"use","excerpt":"Suppose x_m satisfies Φ(x_m)∈ F^mY. Assume that corrections h∈ F^mX obey a first-order congruence Φ(x_m+h) ≡ Φ(x_m)+L_mh pmodF^m+1Y, where L_m:F^mX→ F^mY is a module homomorphism satisfying L_m(F^m+1X)⊆ F^m+1Y and therefore induces overline L_m: F^mX/F^m+1X ⟶ F^mY/F^m+1Y.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1266df2f6d9de73","tex":"L_m'","aliases":"L_m'","display":"L_m'","kind":"use","excerpt":"[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66053b0fba2aa69a","tex":"L_{m-1}","aliases":"L_m-1","display":"L_{m-1}","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed76d6b3d182baf7","tex":"L_n","aliases":"L_n","display":"L_n","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7019a73bbeb4a019","tex":"L_{n+1}","aliases":"L_n+1","display":"L_{n+1}","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e69368456b55a8e6","tex":"L_{n+2}","aliases":"L_n+2","display":"L_{n+2}","kind":"use","excerpt":"An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-626e812ed4f46844","tex":"\\mathcal L_{n,v}","aliases":"L_n,v mathcal","display":"\\mathcal L_{n,v}","kind":"use","excerpt":"Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N, F(bar w)=0, free(bar w)=t(n)+p^Nvpmodp^N+1 }. Here p^Nv means any integer representative of v, multiplied by p^N. The residue class does not depend on the representative. Then # L_n,v=p^m-e for every n and every v","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b450d7bbb1f72fc4","tex":"L_P","aliases":"L_P","display":"L_P","kind":"use","excerpt":"Let P(z) and Q(z) be Laurent polynomials. Define L_P(z) = 1 · 0 P(z) · 1 , U_Q(z) = 1 · Q(z) 0 · 1 .","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5168d74b869524fd","tex":"L_p","aliases":"L_p","display":"L_p","kind":"use","excerpt":"For a finite set S of primes, positive precisions L_p, and a fixed integer polynomial system F, put","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87ae91a1f9b1b756","tex":"L_P^{-1}","aliases":"L_P^-1","display":"L_P^{-1}","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a377a288ccd2ba74","tex":"L_r","aliases":"L_r","display":"L_r","kind":"use","excerpt":"[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ccfde72f4e3e4f6","tex":"L_r^*","aliases":"L_r^*","display":"L_r^*","kind":"use","excerpt":"[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-533de406086c2d49","tex":"L_T","aliases":"L_T","display":"L_T","kind":"use","excerpt":"To pass from the section to the singular ratio, factor out its initial zero: for c=ap, the section has the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af497bd89f303a99","tex":"\\mathcal L_{x,L}","aliases":"L_x,L mathcal","display":"\\mathcal L_{x,L}","kind":"use","excerpt":"At positive polynomial depth, restrict it to the correction fiber T_x={t∈ S_q^d:overliner_x+J_xt=0} and the lift set L_x,L= {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L, F(y)=0}. The polynomial predicate in the second display is evaluation in the residue ring. Evaluation on canonical integer representatives gives the same predicate, because integer polynomial eva","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58a0d38624dd583b","tex":"\\mathcal L_{x,L,v}","aliases":"L_x,L,v mathcal","display":"\\mathcal L_{x,L,v}","kind":"use","excerpt":"Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residu","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-229e5fbb02149c66","tex":"L_z","aliases":"L_z","display":"L_z","kind":"use","excerpt":"Set L_z= I · -A_PQ(zI-A_QQ)^-1 0 · I and S_z= F_P(z) · 0","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a8bd4f0e5b0df5e","tex":"\\mathcal L_{z_n,L+n}","aliases":"L_z_n,L+n mathcal","display":"\\mathcal L_{z_n,L+n}","kind":"use","excerpt":"Surjectivity implies rank_ F_pA_x=e, e≤ d. Combining this with the preceding exact residue-lift count gives # L_z_n,L+n=p^d-e for every n. These are distinct solutions modulo p^L+n+1 lying above the selected point modulo p^L+n. The original finite obstruction group at each smooth point has cardinality one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0a4f94aeb2c3627","tex":"L_{\\lambda+\\mu}","aliases":"L_λ+μ lambda mu","display":"L_{\\lambda+\\mu}","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a16f7d6dd713850","tex":"L_{\\lambda+\\mu}^k","aliases":"L_λ+μ^k lambda mu","display":"L_{\\lambda+\\mu}^k","kind":"use","excerpt":"For completeness, the iterated formula is an integral multinomial formula, rather than a product rule with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-758c0b21a1e71ef3","tex":"L_\\lambda^a","aliases":"L_λ^a lambda","display":"L_\\lambda^a","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2788cc15f4b75ba1","tex":"L_\\mu","aliases":"L_μ mu","display":"L_\\mu","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2487b1443ba258e8","tex":"L_\\mu^b","aliases":"L_μ^b mu","display":"L_\\mu^b","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c5400967fcce1ea","tex":"L_\\infty","aliases":"L_∞ infty","display":"L_\\infty","kind":"use","excerpt":"[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-01208e159d2aec8c","tex":"\\land","aliases":"land","display":"\\land","kind":"use","excerpt":"An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrig","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3755b02b61222ecc","tex":"\\Lcal","aliases":"Lcal L mathcal","display":"{\\mathcal L}","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":54},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61ce5ee12796fd2c","tex":"\\Lcal^1","aliases":"Lcal^1 L^1 Lcal mathcal","display":"{\\mathcal L}^1","kind":"use","excerpt":"lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to i","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e51b117ccc60377f","tex":"\\Lcal^2","aliases":"Lcal^2 L^2 Lcal mathcal","display":"{\\mathcal L}^2","kind":"use","excerpt":"[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The o","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e86c461624494706","tex":"\\Lcal_*","aliases":"Lcal_* L_* Lcal mathcal","display":"{\\mathcal L}_*","kind":"use","excerpt":"[Stability of the selected basis] Assume the optimal pruning L_* for c is unique and at least one competing pruning exists. Define g=min_ L≠ L_*(C( L)-C( L_*))>0. If 2ε_*<g, then the optimal pruning for widetilde c is the same. If only one admissible pruning exists, it remains selected under every cost change without a gap condition.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7151029ba530e5db","tex":"\\Lcal_0","aliases":"Lcal_0 L_0 Lcal mathcal","display":"{\\mathcal L}_0","kind":"use","excerpt":"vpage31Coherence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSChapter VII: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt opera","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-271ba6ea9d5e2027","tex":"\\Lcal_1","aliases":"Lcal_1 L_1 Lcal mathcal","display":"{\\mathcal L}_1","kind":"use","excerpt":"ence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSChapter VII: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space ar","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc64d46d6f1e674d","tex":"\\Lcal_2","aliases":"Lcal_2 L_2 Lcal mathcal","display":"{\\mathcal L}_2","kind":"use","excerpt":"[Monotone Lagrangian frontier] Let 0≤λ_1<λ_2. Choose optimal prunings L_1 and L_2 for the two Lagrangian parameters. Then err( L_2) ≤ err( L_1) and cost( L_2) ≥ cost( L_1).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c24e3ef0e3320585","tex":"\\Lcal_{2g}","aliases":"Lcal_2g L_2g Lcal mathcal","display":"{\\mathcal L}_{2g}","kind":"use","excerpt":"[Divided square packet equations] Assume q( L_g)⊆ L_2g. Choose an ordering of the finite label set. The equation dη+q(η)=0 is equivalent to dη_k + ∑_2g=kq(η_g) + ∑_substackg<h","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3dd439a91ab48b70","tex":"\\Lcal_g","aliases":"Lcal_g L_g Lcal mathcal","display":"{\\mathcal L}_g","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62dad86562132d65","tex":"\\Lcal_{g+h}","aliases":"Lcal_g+h L_g+h Lcal mathcal","display":"{\\mathcal L}_{g+h}","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b6e9b85d2ea26de","tex":"\\Lcal_h","aliases":"Lcal_h L_h Lcal mathcal","display":"{\\mathcal L}_h","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce82d7e1785ef3e8","tex":"\\Lcal_k","aliases":"Lcal_k L_k Lcal mathcal","display":"{\\mathcal L}_k","kind":"use","excerpt":"The differential preserves the group label and the bracket adds labels. Apply the projector onto L_k and use exact reconstruction.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3bcbed4467b578cb","tex":"\\Lcal_p","aliases":"Lcal_p L_p Lcal mathcal","display":"{\\mathcal L}_p","kind":"use","excerpt":"[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8dbbf5357e50fbf","tex":"\\Lcal_q","aliases":"Lcal_q L_q Lcal mathcal","display":"{\\mathcal L}_q","kind":"use","excerpt":"[Coherence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert--Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d37916692e61255","tex":"\\Lcal_w","aliases":"Lcal_w L_w Lcal mathcal","display":"{\\mathcal L}_w","kind":"use","excerpt":"Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3eda83bfb4bc77ed","tex":"\\lcm","aliases":"lcm operatorname","display":"{\\operatorname{lcm}}","kind":"use","excerpt":"Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3860d7e8f88507bb","tex":"\\operatorname{lcm}","aliases":"lcm operatorname","display":"\\operatorname{lcm}","kind":"use","excerpt":"Principal results. addcontentslinetocsectionPrincipal results The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integral and E_mpn+qp^r^(mp,","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f10b967e4b2f95d","tex":"\\leadsto_{H}","aliases":"leadsto_H leadsto","display":null,"kind":"use","excerpt":"Hidden elimination. For the actual Schur complement S=A_RR-A_RHA_HH^-1A_HR, the hidden inverse is expanded through the hidden projectors. A nonzero retained correction block requires indices j,k with nonzero entrance and exit blocks and a hidden path jleadsto k. Therefore a nonzero block S_ba requires either a direct retained block, or a⟶ j leadsto","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afe7579823b893fd","tex":"\\length","aliases":"length operatorname","display":"{\\operatorname{length}}","kind":"use","excerpt":"[Total gluing and worst precision] If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O, then length_ O(G_Λ, P) = ∑_j=1^ra_j, s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}). Thus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-993066ece33b1204","tex":"\\operatorname{length}","aliases":"length operatorname","display":"\\operatorname{length}","kind":"use","excerpt":"olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be97729b8e1f11de","tex":"\\length_{\\OO}","aliases":"length_OO length_ O length OO operatorname mathcal","display":"{\\operatorname{length}}_{{{\\mathcal O}}}","kind":"use","excerpt":"[Total gluing and worst precision] If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O, then length_ O(G_Λ, P) = ∑_j=1^ra_j, s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}). Thus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).","count":49},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fedb28c62ae7b7d9","tex":"\\operatorname{Lift}","aliases":"Lift operatorname","display":"\\operatorname{Lift}","kind":"use","excerpt":"(t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28fb7006ab9bc24c","tex":"\\operatorname{Lift}_E","aliases":"Lift_E operatorname","display":"\\operatorname{Lift}_E","kind":"use","excerpt":"(t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb514d3fcefe767d","tex":"\\lim_{n\\to\\infty}","aliases":"lim_n→∞ lim to infty","display":"\\lim_{n\\to\\infty}","kind":"use","excerpt":"[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb093952304f9b7e","tex":"\\lim_{t\\downarrow0}","aliases":"lim_tdownarrow0 lim downarrow","display":"\\lim_{t\\downarrow0}","kind":"use","excerpt":"ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1416c21d48d701c1","tex":"\\lim_{t\\to0}","aliases":"lim_t→0 lim to","display":"\\lim_{t\\to0}","kind":"use","excerpt":"[Infinitesimal recovery of the defect] The generator defect is the first order semigroup defect: lim_t→0 frac D(t)t = Δ in operator norm. Consequently, C e^tA_f= e^tA_cC hboxfor every t if and only if CA_f=A_cC.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24fd66249142db1e","tex":"\\lim_{t\\to\\infty}","aliases":"lim_t→∞ lim to infty","display":"\\lim_{t\\to\\infty}","kind":"use","excerpt":"[Semigroup survival identities] For t≥0, the finite-time exit density and, wherever S(t)>0, the hazard are f(t)=a^Te^tAp_0=-S'(t), h(t)=fracf(t)S(t). Consequently, S is nonincreasing and S(t)=exp(-∫_0^th(r) dr) whenever S(t)>0. The density may be defective: its total mass is 1-lim_t→∞S(t). It is a probability density when absorption is eventual, so","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"use","excerpt":"[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04ccc1386674337e","tex":"\\limsup_{r\\to\\infty}","aliases":"limsup_r→∞ limsup to infty","display":"\\limsup_{r\\to\\infty}","kind":"use","excerpt":"[Indicator polygon and dominant sectors] The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e3a698fc7564672","tex":"\\limsup_{\\substack{n\\to\\infty\\\\a_n\\ne0}}","aliases":"limsup_substackn→∞ a_n≠0 limsup substack to infty a ne","display":null,"kind":"use","excerpt":"[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c66bb4ce7ecf070","tex":"\\mathrm{lin}","aliases":"lin mathrm","display":"\\mathrm{lin}","kind":"use","excerpt":"[Linear lift obstruction] The linear obstruction at depth L is Ob^lin_L(x_L) = [ρ_L(x_L)] ∈ coker(overline M).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45e90aa85790529d","tex":"\\linewidth","aliases":"linewidth","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-949a7a665b77fe06","tex":"\\operatorname{LocallyTrivial}","aliases":"LocallyTrivial operatorname","display":"\\operatorname{LocallyTrivial}","kind":"use","excerpt":"For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4c9ac4249963ddc","tex":"\\operatorname{LocallyTrivial}_K","aliases":"LocallyTrivial_K operatorname","display":"\\operatorname{LocallyTrivial}_K","kind":"use","excerpt":"For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62b7fa260a967c3f","tex":"\\operatorname{LocallyTrivial}_{\\Q_p}","aliases":"LocallyTrivial_Q_p LocallyTrivial_ Q_p operatorname Q operatorname mathbf","display":"\\operatorname{LocallyTrivial}_{{\\mathbf Q}_p}","kind":"use","excerpt":"l. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue root 2 modulo 5 supp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"[Finite Newton dominance] Let f: N→ Q and write a_k=Δ_1^kf(0). Suppose that c∈ N, a_1≠0, v_2(a_1)=c, and v_2(a_k)≥ c+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and T>0, f(n+T)-f(n)≠0, v_2(f(n+T)-f(n))=v_2(T)+c.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0639ea84119f55a","tex":"\\operatorname{lookup}","aliases":"lookup operatorname","display":"\\operatorname{lookup}","kind":"use","excerpt":"Entrywise implementation theorem. For every b≤ B, Lean proves lookup(storedTree(B,t),b)=F_t(b). There is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at b depends only on input entries at most b, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-529ad2daacd7efcb","tex":"\\lor","aliases":"lor","display":"\\lor","kind":"use","excerpt":"For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25d467671d09a670","tex":"\\lVert","aliases":"lVert","display":"\\lVert","kind":"use","excerpt":"lineskip [Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1633dfbf62528f10","tex":"\\lvert","aliases":"lvert","display":"\\lvert","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4abae2f40eca713","tex":"\\bm m","aliases":"m bm","display":"\\bm m","kind":"use","excerpt":"Use the same multiplicities in every member of the compatible carrier. Direct sums preserve strict compatibility, and m_i≤ c_i makes V_λ a summand of W_λ. At the separating places, the target trace vector is C m. Full column rank makes m the unique solution.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76a8c8c02bf83b7a","tex":"M'","aliases":"M'","display":"M'","kind":"use","excerpt":"[Rectangular image phases] For L=M Z× N Z⊂ Z^2, one may take R={0,…,M-1}×{0,…,N-1}. The analysis map separates an image into MN exact phases. Refining (M,N) to (M',N'), where Mmid M' and Nmid N', splits every coarse phase into fracM'MfracN'N fine phases.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc33b4e3162aeb75","tex":"M^*","aliases":"M^*","display":"M^*","kind":"use","excerpt":"Lean identifies the actual Frobenius norm with its trace expression, normM_rm F^2=Retr(M^*M), then expands the finite sum and removes each cross term. The result is normA-Π(A)_rm F^2 =∑_b≠ anormP_bAP_a_rm F^2. No Pythagorean equality for the target residual is assumed.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca52a161c084be9c","tex":"M^{-1/2}","aliases":"M^-1/2","display":"M^{-1/2}","kind":"use","excerpt":"[Preconditioned conjugate gradient convergence] Let A and M be Hermitian positive definite on a nonempty finite coordinate space, and set κ = cond_2(M^-1/2AM^-1/2). In exact arithmetic, the energy norm error of preconditioned conjugate gradient satisfies normx_m-x_*_A ≤ 2 ( fracsqrtκ-1sqrtκ+1 )^m normx_0-x_*_A.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"9+12 i · 6. Replacing A^T with A^* changes the signs of the imaginary terms. A real-only test would conceal that error, which is why the worked model uses an imaginary coefficient. headWhy modewise contraction matters For A of order m and B of order n, the explicit tensor matrix stores m^2n^2 entries. Its factors store m^2+n^2. Applying through BXA","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af7e789b143d2688","tex":"m^d","aliases":"m^d","display":"m^d","kind":"use","excerpt":"The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a98f829b8404676","tex":"M^{j-1}","aliases":"M^j-1","display":"M^{j-1}","kind":"use","excerpt":"With A=diag(1,2), the vectors v and Av=(1,2)^T are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when A=I_2. Both changes enlarge the addressing power, but in different ways. headWhen invariance is only approximate If A=D","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-05e170b594a4705a","tex":"M^k","aliases":"M^k","display":"M^k","kind":"use","excerpt":"[Number of labeled composition terms] For n≥1, if Y has M packets, then the number of packet labeled terms at order n before zero jets are removed is N_n,M = ∑_k=1^nS(n,k)M^k, where S(n,k) is a Stirling number of the second kind. Thus N_n,M is the nth Touchard polynomial evaluated at M.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09a4e391e923d287","tex":"m^N","aliases":"m^N","display":"m^N","kind":"use","excerpt":"vpage24Convergence is not continuationZERO ZETA FUNCTIONSChapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4. Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_plane For N=2 and N=4, the lattices","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-594034b0026909b7","tex":"m^{r-2}","aliases":"m^r-2","display":"m^{r-2}","kind":"use","excerpt":"[Leading dominant zero term] Under the hypotheses of [source reference], P_N,n(r) = · m!binomm+r-1r-1 ∑_ρ∈ Z_R fracρ^-m(-ρ D_N'(ρ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term is absent.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7615cae6d0b78c4b","tex":"m^\\alpha","aliases":"m^α alpha","display":"m^\\alpha","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d074e542a4cc31b","tex":"m^{\\alpha-1}","aliases":"m^α-1 alpha","display":"m^{\\alpha-1}","kind":"use","excerpt":"D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In particular, exact generator intertwining CA=BC implies exact fractional intertwinin","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-451b483aee615392","tex":"M^\\perp","aliases":"M^⊥ perp","display":"M^\\perp","kind":"use","excerpt":"The invariant orthogonal decomposition gives M=mathop⊥limits_s∈ SM_s. Symplectic invariance gives M^⊥=ker(g-I). Fixed point freeness on M therefore makes the restricted form nondegenerate on M and on each M_s. Each restriction of g_s-I to M_s is invertible, so [source reference] applies to the moving spaces. A zero moving space contributes one. In ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ece6f9e55a99dd8c","tex":"m_{\\mathrm A}","aliases":"m_ A mathrm","display":"m_{\\mathrm A}","kind":"use","excerpt":"vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61577aeca4848aaa","tex":"m_{\\mathrm B}","aliases":"m_ B mathrm","display":"m_{\\mathrm B}","kind":"use","excerpt":"vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":"[Large zero strings] Assume N≥3. There are constants m_0, C, and γ>0 and a sequence of positive real simple zeros (τ_m)_m≥ m_0 of D_N such that τ_m = t_m+O(e^-γ t_m). Every zero of sufficiently large modulus is of the form ω^kτ_m for a unique 0≤ k<N and a unique m≥ m_0.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a511b20c73344dd","tex":"M_{2\\times5}","aliases":"M_2×5 times","display":"M_{2\\times5}","kind":"use","excerpt":"vpage13How much row space can one seed supply?SPLIT SEMISIMPLE MODEL / A CONSTRUCTIVE FORMULAMain text: [source reference] Let M_2( Q) act on X=M_2×5( Q) by left multiplication. Think of the five columns as five copies of the same natural module of dimension two. One seed is a 2×5 matrix V.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa02e26ac15b5f1d","tex":"M_3","aliases":"M_3","display":"M_3","kind":"use","excerpt":"M_3( C) acting on seven copies, seed rank · 3","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9672ca091c4a70a","tex":"M_4","aliases":"M_4","display":"M_4","kind":"use","excerpt":"Full M_4( R) orbit dimensions for two sampled nonzero seeds · 4,4","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63019db73ab92a7c","tex":"M_A","aliases":"M_A","display":"M_A","kind":"use","excerpt":"For bounded generators with common bound M_A, sup_ℓ norm( e^tA_ℓ-I)x_ℓ ≤ ( e^tM_A-1)sup_ℓnormx_ℓ. This tends to zero with t. Exact generator intertwining makes (A_ℓ x_ℓ)_ℓ compatible, and coordinatewise differentiation identifies the generator.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c887199df3da3b63","tex":"M_a","aliases":"M_a","display":"M_a","kind":"use","excerpt":"unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f5cef98cfcf15fe","tex":"M_c","aliases":"M_c","display":"M_c","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-207e98afc142d0de","tex":"M_d","aliases":"M_d","display":"M_d","kind":"use","excerpt":"For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.","count":32},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-318be9929beaa6fd","tex":"m_d","aliases":"m_d","display":"m_d","kind":"use","excerpt":"Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd84b726bc30add7","tex":"M_{d_A}","aliases":"M_d_A","display":"M_{d_A}","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2aa0f7b977f40849","tex":"M_{d_r}","aliases":"M_d_r","display":"M_{d_r}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-84cc641b229c54b3","tex":"M_{d\\times m}","aliases":"M_d× m times","display":"M_{d\\times m}","kind":"use","excerpt":"Actual orbits and sharp seed counts. Let C∈ M_d× m( C) be the coefficient matrix of one seed. Define its orbit by actual matrix multiplication: O(C)={AC:A∈ M_d( C)}. This set is already a linear subspace. The source proves that it equals the linear span of the orbit before computing its dimension.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2eb18a26b2462763","tex":"m_e","aliases":"m_e","display":"m_e","kind":"use","excerpt":"Now impose upper tolerances m=(1/10,1/2,1/2). The first edge must stop at its tighter cap. With multiplier λ=10/3, the clipped solution is ϵ_e=min{m_e,c_e/(λ s_e)}, ϵ=(1/10,3/20,3/20). Its weighted contributions are (1/10,3/10,3/5) and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly. headFrom ","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4524d978e547026","tex":"M_f","aliases":"M_f","display":"M_f","kind":"use","excerpt":"The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95bc4d67c762d1bf","tex":"M_{f(\\,\\cdot\\,^N)}","aliases":"M_f( · ^N) cdot","display":"M_{f(\\,\\cdot\\,^N)}","kind":"use","excerpt":"The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c828d32a87d213ac","tex":"M_I","aliases":"M_I","display":"M_I","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df92634da3a3e068","tex":"M_i","aliases":"M_i","display":"M_i","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83d3f2e0b215ee50","tex":"M_{ij}","aliases":"M_ij","display":"M_{ij}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c879cfab2bd64d1b","tex":"m_j","aliases":"m_j","display":"m_j","kind":"use","excerpt":"The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a fini","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f343ef357a239b3","tex":"M_K","aliases":"M_K","display":"M_K","kind":"use","excerpt":"[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient fi","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9577d4e4232e0c8","tex":"m_k","aliases":"m_k","display":"m_k","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b600b58f986c2898","tex":"M_L","aliases":"M_L","display":"M_L","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3cc0ed3324bf0e59","tex":"M_m","aliases":"M_m","display":"M_m","kind":"use","excerpt":"[Finite Kraus and Choi criteria] A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-465515cf3e71e41b","tex":"M_{m,n}","aliases":"M_m,n","display":"M_{m,n}","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0901f1f56ba2bc5e","tex":"M_N","aliases":"M_N","display":"M_N","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce78c2a469d42dfa","tex":"m_n","aliases":"m_n","display":"m_n","kind":"use","excerpt":"For x∈ X, choose m_n∈ M with m_n→ x. Continuity gives P_α m_n→ P_α x, so P_α X⊆clP_α M. The reverse inclusion follows because P_α X is closed and contains P_α M. The bounded extensions remain pairwise annihilating idempotents whose finite sum is the identity.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4fb8d0ba7795506","tex":"M_p","aliases":"M_p","display":"M_p","kind":"use","excerpt":"∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral: the middle entries of a prime-power binomial row are divisible by p. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef9064cfb2a52947","tex":"M_{p^2}","aliases":"M_p^2","display":"M_{p^2}","kind":"use","excerpt":"∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral: the middle entries of a prime-power binomial row are divisible by p. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d80ae31b0ce157a","tex":"M_Q","aliases":"M_Q","display":"M_Q","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dbd15cd98c87c0e2","tex":"M_q","aliases":"M_q","display":"M_q","kind":"use","excerpt":"pageheading1. The characteristic-zero product ruleThe normalized defect is not an ordinary derivation Let A be a commutative algebra over a characteristic-zero field K, and let U be an actual K-derivation. The source defines B=fracU^p^2-U^pp, M_q(f,g)=∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28cc0a43f7b6c71a","tex":"M_r","aliases":"M_r","display":"M_r","kind":"use","excerpt":"[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficien","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19afb20993a1d995","tex":"m_r","aliases":"m_r","display":"m_r","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eeb84163daca3455","tex":"M_{r,s}","aliases":"M_r,s","display":"M_{r,s}","kind":"use","excerpt":"[htbp] centering includegraphics[width=0.96textwidth]03_refinement_incidence.pdf captionThe incidence matrix M_r,s=1 when the fine residue s belongs to the coarse residue r. Its rows implement exact aggregation from fine packets to coarse packets. Common refinements.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7cf81ec814da4db","tex":"\\mathfrak M_R^{\\mathrm b}","aliases":"M_R^ b mathfrak mathrm","display":"\\mathfrak M_R^{\\mathrm b}","kind":"use","excerpt":"[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37c0c78c3530a9c1","tex":"\\mathfrak M_{R_\\Gamma}","aliases":"M_R_Γ mathfrak Gamma","display":"\\mathfrak M_{R_\\Gamma}","kind":"use","excerpt":"[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12fcf6e1ba955880","tex":"M_{\\rm block}","aliases":"M_rm block rm","display":"M_{\\rm block}","kind":"use","excerpt":"[Equal diagonal packets] Let p≥1 and b≥1 be integers and n=pb. Suppose n is divided into p equal diagonal packets of width b. Then flop_rm block=2qnb, M_rm block=nb. The following factor describes the dense-to-block storage reduction and, when q≥1, the arithmetic reduction: fracnb.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27220d8a886e4747","tex":"M_{\\rm packet}","aliases":"M_rm packet rm","display":"M_{\\rm packet}","kind":"use","excerpt":"[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17770225492e1d01","tex":"M_s","aliases":"M_s","display":"M_s","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d7c9ff7c9abb006","tex":"M_{\\sqrt{\\abs A}}","aliases":"M_sqrtabs A sqrt abs","display":null,"kind":"use","excerpt":"[Nondegenerate twisted group algebra] Assume that b_c is nondegenerate. Then abs A is a square, and C^c[A] ≅ M_sqrtabs A( C). Consequently, C^c[A] has one irreducible module up to isomorphism, of dimension sqrtabs A.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f241e1659d989d87","tex":"M_T","aliases":"M_T","display":"M_T","kind":"use","excerpt":"[Volterra stability under kernel compression] Let S(t) be a strongly continuous semigroup on a Banach space X, fix T>0, and assume normS(t)≤ M_T (0≤ t≤ T). Let K,widetilde K∈ L^1((0,T), B(X)), let g∈ L^1((0,T),X), and let x,widetilde x∈ C([0,T],X). Suppose these are mild solutions with the same initial state and forcing, except that K is replaced b","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11048af18ba22882","tex":"m_T","aliases":"m_T","display":"m_T","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9449462255135514","tex":"M_V","aliases":"M_V","display":"M_V","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ff979fe78820d20","tex":"M_v","aliases":"M_v","display":"M_v","kind":"use","excerpt":"unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9af7b18b4478efd","tex":"M_V^n","aliases":"M_V^n","display":"M_V^n","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3e697d6b6f9bc2b","tex":"M_w","aliases":"M_w","display":"M_w","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0521a5cf2b799404","tex":"M_x","aliases":"M_x","display":"M_x","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89af9e181208ee45","tex":"M_x^3","aliases":"M_x^3","display":"M_x^3","kind":"use","excerpt":"The relation is reconstructed as an operator identity: M_y^2=M_x^3-M_x.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3010d2a559d3a13","tex":"M_{x^3-x}","aliases":"M_x^3-x","display":"M_{x^3-x}","kind":"use","excerpt":"Identify the packet coefficients by A_0⊕ A_1⟶𝕜[x]⊕𝕜[x], (a,yb)⟼(a,b). In these ordered coefficient coordinates, with M_x and M_x^3-x denoting multiplication on 𝕜[x], M_x= M_x · 0","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fdf4922b396511d5","tex":"M_x^i","aliases":"M_x^i","display":"M_x^i","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1d444128c7741a9","tex":"M_y^j","aliases":"M_y^j","display":"M_y^j","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-401d9b49eeb97619","tex":"m_\\alpha","aliases":"m_α alpha","display":"m_\\alpha","kind":"use","excerpt":"Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0dae06547437e14b","tex":"m_\\lambda","aliases":"m_λ lambda","display":"m_\\lambda","kind":"use","excerpt":"For the topological statement, let S generate M densely. Given n=Φ(m), choose a net m_λ∈ AS converging to m. Continuity gives Φ(m_λ)⟶ n, and every Φ(m_λ) lies in BΦ(S). Thus BΦ(S) is dense in N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a7ea658fe0f7067","tex":"m_{\\lambda}","aliases":"m_λ lambda","display":"m_{\\lambda}","kind":"use","excerpt":"[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N:","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d03a88e50994883","tex":"m_{\\lambda_*}","aliases":"m_λ_* lambda","display":"m_{\\lambda_*}","kind":"use","excerpt":"r] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r wi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c60cf3b807dc9ab5","tex":"M_\\mu","aliases":"M_μ mu","display":"M_\\mu","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b13c32e31bf92433","tex":"m_\\nu","aliases":"m_ν nu","display":"m_\\nu","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2efb8720cfae2fb","tex":"M_\\ell","aliases":"M_ℓ ell","display":"M_\\ell","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c7620add7ce256d","tex":"\\operatorname{Mat}","aliases":"Mat operatorname","display":"\\operatorname{Mat}","kind":"use","excerpt":"Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bba1a1e3366bd331","tex":"\\operatorname{Mat}_{N\\times N}","aliases":"Mat_N× N operatorname times","display":"\\operatorname{Mat}_{N\\times N}","kind":"use","excerpt":"Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90fc7282920196de","tex":"\\mathllap","aliases":"mathllap","display":null,"kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-905bf7c29079e31a","tex":"\\mathord","aliases":"mathord","display":null,"kind":"use","excerpt":"Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one dep","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-781698c822449e4f","tex":"\\mathrel","aliases":"mathrel","display":null,"kind":"use","excerpt":"[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( -","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae66207125e503b8","tex":"\\mathrlap","aliases":"mathrlap","display":null,"kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d49f1f8e75de682d","tex":"\\Max","aliases":"Max operatorname","display":"{\\operatorname{Max}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03265e359becc498","tex":"\\max_{0\\le k<N}","aliases":"max_0≤ k<N max le","display":"\\max_{0\\le k<N}","kind":"use","excerpt":"vpage17Growth comes from a polygonDOMINANT EXPONENTSChapter IV: Definition 11.3, Theorem 11.4. Write the signed denominator as a finite exponential sum: D_N(z)=frac1N∑_η^N=-1e^η z. For z=re^iθ, the size of one exponential is e^rRe(η e^iθ). The maximal projection determines the directional exponential growth. vfigindicator The indicator is h_N(θ)=ma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81f120bf99a467b2","tex":"\\max_{1\\leq i\\leq n}","aliases":"max_1≤ i≤ n max leq leq","display":"\\max_{1\\leq i\\leq n}","kind":"use","excerpt":"[Discrete maximum bound] The inverse L_h^-1 has nonnegative entries and normL_h^-1_∞→∞ = max_1≤ i≤ n fracx_i(1-x_i)2 ≤ frac18.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-848ac68ee082047f","tex":"\\max_{1\\leq i\\leq q}","aliases":"max_1≤ i≤ q max leq leq","display":"\\max_{1\\leq i\\leq q}","kind":"use","excerpt":"[Semisimple seed rank formula over arbitrary division rings] Let A≅∏_i=1^qM_d_i(D_i) be semisimple Artinian and let M≅⊕_i=1^qS_i^⊕ m_i be a finitely generated left module over A. Then srank_A(M) = max_1≤ i≤ q lceilfracm_id_irceil. If A is any algebra of finite dimension over a field, the same formula applied to the decomposition of M/Jac(A)M over A","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-202d1bbf7e6fd044","tex":"\\max_{1\\le r\\le s}","aliases":"max_1≤ r≤ s max le le","display":"\\max_{1\\le r\\le s}","kind":"use","excerpt":"[Quantum seed rank] For the displayed multiplicity decomposition, srank_ A( H) = max_1≤ r≤ s lceilfracm_rd_rrceil.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bd86680754bc6e2","tex":"\\max_{i\\in B_a}","aliases":"max_i∈ B_a max in","display":"\\max_{i\\in B_a}","kind":"use","excerpt":"[Oscillation bound] Suppose overline L_a is any representative likelihood for block B_a, and set ω_a = max_i∈ B_aabsL_i-overline L_a. Then d(pmid L,overline L) ≤ ∑_aq_aω_a. Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6913c3ef5d6dc1fd","tex":"\\max_{i\\in I_\\alpha}","aliases":"max_i∈ I_α max in alpha","display":"\\max_{i\\in I_\\alpha}","kind":"use","excerpt":"[Information created by a Bayesian update] Let q= B_λ(p) with positive likelihood and positive prior. Then D(qVert p) = ∑_iq_ilogλ_i - log Z_λ(p). Inside a packet, D(q_αVert r_α) = ∑_i∈ I_α q_α,i logfracλ_ioverlineλ_α(p). Hence D(q_αVert r_α) ≤ log fracmax_i∈ I_αλ_i min_i∈ I_αλ_i.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c820cd743c916f52","tex":"\\max_k","aliases":"max_k max","display":"\\max_k","kind":"use","excerpt":"inproof If σ=0, every centered zero-sub-Gaussian noise variable vanishes almost surely, and the noise-free deterministic conclusion applies. For σ>0, the difference η_k+1-η_k is sub-Gaussian with variance proxy 2σ^2, so P(|η_k+1-η_k|>t) ≤ 2exp(-fract^24σ^2). A union bound over the n cells gives P( max_k|η_k+1-η_k|>t_n,δ ) ≤ δ. On the complementary ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d02e656e84ac97fa","tex":"\\max_{\\Lcal}","aliases":"max_Lcal max_ L max Lcal max mathcal","display":"\\max_{{\\mathcal L}}","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1110d632fcc473b9","tex":"\\max_p","aliases":"max_p max","display":"\\max_p","kind":"use","excerpt":"[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the d","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c52aaa85647b7be3","tex":"\\max_r","aliases":"max_r max","display":"\\max_r","kind":"use","excerpt":"Set k=max_rlceilfracm_rd_rrceil. For each r, partition a basis of C^m_r into at most k groups of size at most d_r. Choose the coefficient matrix of the jth seed so that its row space spans the jth group. The combined row spaces span every multiplicity factor. The preceding theorem then shows that the k seeds generate every summand.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b29f89d48389ba9c","tex":"\\max_{v\\in V}","aliases":"max_v∈ V max in","display":"\\max_{v\\in V}","kind":"use","excerpt":"A valid schedule s:V→ N assigns positive stages and strictly increases on each edge. Induction proves the stronger pointwise statement ℓ(v)≤ s(v) for every v. Consequently the constructed schedule achieves the least possible makespan, d=max_v∈ Vℓ(v). For a nonempty graph, d equals the number of vertices on an attained longest path. Vertices of the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18caafcaaf4899c3","tex":"\\max_w","aliases":"max_w max","display":"\\max_w","kind":"use","excerpt":"A_vmin_w a(w), · v split, and b(v)= 1, · v retained, B_vmax_w b(w), · v split. Then the leaf analysis of the pruned tree has bounds a(rm root) and b(rm root).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2f35edd4ea4be66","tex":"\\max_\\alpha","aliases":"max_α max alpha","display":"\\max_\\alpha","kind":"use","excerpt":"It is therefore Banach with its inherited norm. The same holds for every Y_γ. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from [source reference] make them bounded linear isomorphisms. F","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-beb7b788f815c1ff","tex":"\\max_{\\Lambda\\in\\mathfrak P_v}","aliases":"max_Λ∈ P_v max Lambda in mathfrak","display":"\\max_{\\Lambda\\in\\mathfrak P_v}","kind":"use","excerpt":"[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04cf78164e0d76e4","tex":"\\max_{\\lambda\\in\\operatorname{Spec}(T)}","aliases":"max_λ∈Spec(T) max lambda in operatorname","display":"\\max_{\\lambda\\in\\operatorname{Spec}(T)}","kind":"use","excerpt":"[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-689cf901ed5ff942","tex":"\\max_{\\mu\\in\\Spec_K(A)}","aliases":"max_μ∈Spec_K(A) max mu in Spec max mu in operatorname","display":"\\max_{\\mu\\in{\\operatorname{Spec}}_K(A)}","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4df078fcdf121684","tex":"\\mcode","aliases":"mcode","display":null,"kind":"use","excerpt":"_bijective, .address_components, .residue_equivalence targetCRTPackets.packet_factorization, .series_packet_factorization targetFiniteArithmeticExamples.outside_window_not_integral pageheading5. Verification, regressions and remaining workCumulative release evidence The fresh build started with an empty output directory under Lean 4.34.0-rc2 and Ma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f9db2ae5742966b","tex":"\\midrule","aliases":"midrule","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83370d7c90a0d024","tex":"\\min_{(w,c)\\text{ allowed},\\;w\\leq b}","aliases":"min_(w,c) allowed, w≤ b min text leq","display":"\\min_{(w,c)\\text{ allowed},\\;w\\leq b}","kind":"use","excerpt":"If D is the already solved tail table, the next edge has exact-weight recurrence F(b)=min_(w,c) allowed, w≤ b{D(b-w)+c}. The final answer is min_0≤ b≤ BF(b). Lean proves that this is the exact minimum over legal assignments with total error at most B, or codenone exactly when no such assignment exists.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08929fcbd469af5d","tex":"\\min_{0\\leq b\\leq B}","aliases":"min_0≤ b≤ B min leq leq","display":"\\min_{0\\leq b\\leq B}","kind":"use","excerpt":"If D is the already solved tail table, the next edge has exact-weight recurrence F(b)=min_(w,c) allowed, w≤ b{D(b-w)+c}. The final answer is min_0≤ b≤ BF(b). Lean proves that this is the exact minimum over legal assignments with total error at most B, or codenone exactly when no such assignment exists.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0ce20525b9e7a28","tex":"\\min_{0\\leq j\\leq n}","aliases":"min_0≤ j≤ n min leq leq","display":"\\min_{0\\leq j\\leq n}","kind":"use","excerpt":"[Weighted difference bound] Let f: N→ Z_p and let w: N→ N∪{∞} satisfy w(s+j)≤ord_p(c_s+j(f)) for 0≤ j≤ n. Then ord_p(Δ_1^s f(n))≥ min_0≤ j≤ n {w(s+j)+v_pbinom nj}.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9ccf46f5921c4a5","tex":"\\min_{0\\leq k\\leq b}","aliases":"min_0≤ k≤ b min leq leq","display":"\\min_{0\\leq k\\leq b}","kind":"use","excerpt":"For exact-weight tables F,G, min-plus convolution is (F⋆ G)(b)=min_0≤ k≤ b{F(k)+G(b-k)}, with an infeasible sum whenever either operand is infeasible. Its correctness theorem identifies every finite answer with an actual feasible pair of component weights and costs. Repeated convolution handles arbitrary finite lists of components.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ace88525e5ee17f","tex":"\\min_e","aliases":"min_e min","display":"\\min_e","kind":"use","excerpt":"upper bound. The weighted sum ϕ(λ) = ∑_es_e min{ m_e, fracc_eλ s_e } is continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, ϕ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is (m_e)_e and every 0","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b765b9bd7781a783","tex":"\\min_{\\eps_e>0}","aliases":"min_eps_e>0 min_ε_e>0 min eps min varepsilon","display":"\\min_{{\\varepsilon}_e>0}","kind":"use","excerpt":"[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b06e69b277c3fc2","tex":"\\min_{I_\\alpha}","aliases":"min_I_α min alpha","display":"\\min_{I_\\alpha}","kind":"use","excerpt":"Subtract r_α,i from the local posterior formula in [source reference]. The norm identity follows because the factor multiplying η_i is nonnegative. The last estimate uses ∑_i r_α,iabsη_i ≤ osc_I_α(λ) and overlineλ_α(p)≥min_I_αλ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbe7648c525d8858","tex":"\\min_{i\\in I_\\alpha}","aliases":"min_i∈ I_α min in alpha","display":"\\min_{i\\in I_\\alpha}","kind":"use","excerpt":"] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-567e9df79d4789c7","tex":"\\min_k","aliases":"min_k min","display":"\\min_k","kind":"use","excerpt":"ct discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b0386100c5276882","tex":"\\min_{k\\leq r\\leq Mk+s}","aliases":"min_k≤ r≤ Mk+s min leq leq","display":"\\min_{k\\leq r\\leq Mk+s}","kind":"use","excerpt":"[Polyphase valuation ledger] For every refined coefficient, v_p(d_s,k) ≥ min_k≤ r≤ Mk+s ( v_p(c_r)+v_p(Θ_M,s(r,k)) ). If v_p(c_r)≥ w(r) for a nondecreasing function w, then v_p(d_s,k)≥ w(k). Thus every monotone Mahler precision ladder survives arbitrary polyphase refinement.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8dae54f32eaf3a3d","tex":"\\min_{\\Lcal\\ne\\Lcal_*}","aliases":"min_Lcal≠Lcal_* min_ L≠ L_* min Lcal ne Lcal min mathcal ne mathcal","display":"\\min_{{\\mathcal L}\\ne{\\mathcal L}_*}","kind":"use","excerpt":"[Stability of the selected basis] Assume the optimal pruning L_* for c is unique and at least one competing pruning exists. Define g=min_ L≠ L_*(C( L)-C( L_*))>0. If 2ε_*<g, then the optimal pruning for widetilde c is the same. If only one admissible pruning exists, it remains selected under every cost change without a gap condition.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bce0009cb30ebbc","tex":"\\min_{M+1\\leq r\\leq n/2}","aliases":"min_M+1≤ r≤ n/2 min leq leq","display":"\\min_{M+1\\leq r\\leq n/2}","kind":"use","excerpt":"[Optimal dyadic truncation] Let T_2,M(n) = ∑_r=0^min(M,n/2) κ_2,rbinom n2rR_n-2r. Then E_n≡ T_2,M(n)pmod 2^4M+3. More precisely, v_2(E_n-T_2,M(n)) ≥ min_M+1≤ r≤ n/2 ( 4r-1+v_2binom n2r ). The uniform exponent 4M+3 is best possible.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af555498bdb2b55d","tex":"\\min_q","aliases":"min_q min","display":"\\min_q","kind":"use","excerpt":"[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ff0993815f0d3b9","tex":"\\min_{q<j}","aliases":"min_q<j min","display":"\\min_{q<j}","kind":"use","excerpt":"For a wall label j≥1 and 0≤ q<j, define e_j(q)=3q+s_2(j-1)-s_2(j-q-1), E_j=min_q<je_j(q), Θ(j)=3j+1+s_2(j-1), L_j=Θ(j)-E_j. Here E_j is a carry minimum, not an Euler number. A wall label is regular when e_j(q)>0 for 1≤ q<j. For r=2^L-j with L>L_j, put δ=L-L_j. The carry identities give μ_r=E_j-δ, J_r={j}, G_r(x)=P_j(x)U(x-j)+2^δ V(x), where U≡1pmod","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4b988cfa8d5b1ac","tex":"\\min_{q\\ge0}","aliases":"min_q≥0 min ge","display":"\\min_{q\\ge0}","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c55a63df7ee072d6","tex":"\\min_{r\\geq s}","aliases":"min_r≥ s min geq","display":"\\min_{r\\geq s}","kind":"use","excerpt":"[Mahler valuation ledger] If c_r∈ Z_p and v_p(c_r)≥ w(r), then v_p(Δ^sf(n)) ≥ min_r≥ s ( w(r)+v_pbinom nr-s ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b0974bf1da548fe8","tex":"\\min_{\\substack{0\\leq r\\leq n\\\\w(r)\\geq L}}","aliases":"min_substack0≤ r≤ n w(r)≥ L min substack leq leq w geq","display":null,"kind":"use","excerpt":"w(r)<L binom nr c_rp^w(r)S_n-r pmod p^L. When the omitted set is nonempty, the tail has valuation at least min_substack0≤ r≤ n w(r)≥ L ( w(r)+v_pbinom nr+v_p(S_n-r) ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa2dd72af4ea5202","tex":"\\min_w","aliases":"min_w min","display":"\\min_w","kind":"use","excerpt":"[Bounds for nonunitary local splits] Suppose every internal node has an analysis map U_v and constants 0≤ A_v≤ B_v<∞ satisfying A_v x ^2 ≤ ∑_w∈child(v) (U_vx)_w ^2 ≤ B_v x ^2. Define recursively a(v)= 1, · v retained, A_vmin_w a(w), · v split, and b(v)= 1, · v retained,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab2c688fc16fea97","tex":"\\min_{\\varepsilon_e>0}","aliases":"min_ε_e>0 min varepsilon","display":"\\min_{\\varepsilon_e>0}","kind":"use","excerpt":"The continuous allocation theorem. textitSource item 8.4, with an explicit positive budget Let E be a finite nonempty edge set. Assume s_e>0, c_e>0, τ>0, and define C=∑_ec_e. The optimization problem is min_ε_e>0∑_ec_elogfrac1ε_e subject to ∑_es_eε_e=τ. Lean constructs the displayed candidate ε_e^*=fracτ c_es_eC and proves strict positivity and the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30da5b3706769a4f","tex":"\\min_{\\Lambda\\ne\\Lambda_y^*}","aliases":"min_Λ≠Λ_y^* min Lambda ne Lambda","display":"\\min_{\\Lambda\\ne\\Lambda_y^*}","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd9b1bf9b23006d6","tex":"\\min_{\\sum_i b_i=b}","aliases":"min_∑_i b_i=b min sum","display":"\\min_{\\sum_i b_i=b}","kind":"use","excerpt":"Keeping node v costs κ_v and incurs integer error w_v. Splitting costs overhead σ_v in addition to the selected child cuts. The exact-weight recurrence is F_v(b)=min{ l κ_v if b=w_v, [2pt] displaystyleσ_v+ min_∑_i b_i=b∑_iF_v_i(b_i) if v is internal. . Nonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-pl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7ed41372efc24c8","tex":"\\mathrm{mod}","aliases":"mod mathrm","display":"\\mathrm{mod}","kind":"use","excerpt":"Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6225ff8930a41599","tex":"\\bm n","aliases":"n bm","display":"\\bm n","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-261b1d4accf38dcf","tex":"\\mathbf N","aliases":"N mathbf","display":"\\mathbf N","kind":"use","excerpt":"Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement an","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2aa34ef1611fca9c","tex":"\\mathbf n","aliases":"n mathbf","display":"\\mathbf n","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"use","excerpt":"At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original h","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a41020d8546c0cd6","tex":"\\mathfrak N","aliases":"N mathfrak","display":"\\mathfrak N","kind":"use","excerpt":"[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [labe","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f5eab2cbd9d1834","tex":"\\mathfrak n","aliases":"n mathfrak","display":"\\mathfrak n","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e0534ff9dda7eeb","tex":"N'","aliases":"N'","display":"N'","kind":"use","excerpt":"[Rectangular image phases] For L=M Z× N Z⊂ Z^2, one may take R={0,…,M-1}×{0,…,N-1}. The analysis map separates an image into MN exact phases. Refining (M,N) to (M',N'), where Mmid M' and Nmid N', splits every coarse phase into fracM'MfracN'N fine phases.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-213d86a5e6b34393","tex":"N^{-1}","aliases":"N^-1","display":"N^{-1}","kind":"use","excerpt":"0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79616ac402fc2170","tex":"n^{-1/2}","aliases":"n^-1/2","display":"n^{-1/2}","kind":"use","excerpt":"Normalization and inversion. The proof identifies matrix application with the ordinary counting-measure discrete Fourier transform: Fx=n^-1/2DFT(x). It separately identifies F^*x with the appropriately scaled inverse transform. Mathlib's checked Fourier inversion theorem and the proved scalar normalization yield both identities F^*F=I, FF^*=I. No u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0769f9271add232","tex":"n^{-2}","aliases":"n^-2","display":"n^{-2}","kind":"use","excerpt":"Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"rincipal targets. targetUnitModes.geometric_difference, UnitModes.unit_fermat_divisibility targetUnitModes.unit_mode_valuation_gain targetUnitModes.finite_unit_modes_valuation_gain pageheading4. Exact examples and adversarial checksThe examples use the same definitions as the general results A polynomial anchor. The actual difference table for f(n)","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"vpage3Two chains can stay linkedMODEL A / AN OPERATOR IN SIX DIMENSIONSMain text: [source reference]. Matrices over Q Use the ordered basis (e_0,e_1,e_2,f_0,f_1,f_2). Define Ne_0=e_1, Ne_1=e_2, Ne_2=0, Nf_0=f_1, Nf_1=f_2, Nf_2=0. Starting with s=e_0+f_0, the two chains advance together: s=e_0+f_0, Ns=e_1+f_1, N^2s=e_2+f_2, N^3s=0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-948594e33e46ceeb","tex":"N^5","aliases":"N^5","display":"N^5","kind":"use","excerpt":"Actual affine representations with monodromy chains of every finite length invoke these theorems. In particular a length-six example has N^5≠0 and N^6=0 and is recovered from its actual reconstructed action. The length-four example still defeats the old quadratic logarithm, while the full finite logarithm recovers it.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6808aa0a6bb0a282","tex":"N^6","aliases":"N^6","display":"N^6","kind":"use","excerpt":"Actual affine representations with monodromy chains of every finite length invoke these theorems. In particular a length-six example has N^5≠0 and N^6=0 and is recovered from its actual reconstructed action. The length-four example still defeats the old quadratic logarithm, while the full finite logarithm recovers it.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1fea4657810bf7e4","tex":"N^d","aliases":"N^d","display":"N^d","kind":"use","excerpt":"Finite-exponential reconstruction and a cubic inverse. Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f6e0c9e86fd9f34","tex":"N^j","aliases":"N^j","display":"N^j","kind":"use","excerpt":"Finite-exponential reconstruction and a cubic inverse. Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d18e0ad4d2b521a","tex":"N^m","aliases":"N^m","display":"N^m","kind":"use","excerpt":"[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b706167f987bdce5","tex":"N^N","aliases":"N^N","display":"N^N","kind":"use","excerpt":"[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( - A_N,j mathrel, frac(λ z)^NN^N ).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a74297b6dc786495","tex":"N^n","aliases":"N^n","display":"N^n","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space. The assertion k","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-475389afb995c765","tex":"N^{n-1}","aliases":"N^n-1","display":"N^{n-1}","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space. The assertion k","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-821e1514a603c20a","tex":"N^{Nn}","aliases":"N^Nn","display":"N^{Nn}","kind":"use","excerpt":"The multiplication formula for the gamma function gives (Nn+j)! = j!N^Nnn! ∏_a∈ A_N,j(a)_n. Indeed, the N Pochhammer factors supplied by the multiplication formula are ((j+m)/N)_n for 1≤ m≤ N. The excluded factor is ((j+N-j)/N)_n=(1)_n=n!. Insertion into the defining series gives the stated hypergeometric expansion.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ee9bec015b3cbbd","tex":"n^p","aliases":"n^p","display":"n^p","kind":"use","excerpt":"By [source reference], n^p=0. The finite geometric sum gives the inverse.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ebcd9b4bdca5cfcd","tex":"N^r","aliases":"N^r","display":"N^r","kind":"use","excerpt":"[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56155806d18c6f60","tex":"\\mathbb N^r","aliases":"N^r mathbb","display":"\\mathbb N^r","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-526a976a5ea4910d","tex":"n^\\alpha","aliases":"n^α alpha","display":"n^\\alpha","kind":"use","excerpt":"[Fractional Hermite transfer] Let - L be the nonnegative Ornstein--Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-987804e10f674d55","tex":"N_{\\bm w}","aliases":"N_ w bm","display":"N_{\\bm w}","kind":"use","excerpt":"[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","excerpt":"chi congruence] Let p be prime, let m>0, and suppose that mmid p-1 or m=2. Put N=mp and q=lcm(2,p-1). For every 0≤ j<N, all U_N,j(n) belong to Z_p. For r≥1 and n≥ r, E_Nn+qp^r^(N,j)≡ E_Nn^(N,j)pmodp^r. Thus the positive sufficient starting index n_0=r works uniformly. If mmid p-1, the stronger sufficient condition r≤ mn is valid.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca257e6c32ec363d","tex":"N_{01}","aliases":"N_01","display":"N_{01}","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ffaa0ad23479bca8","tex":"N_1","aliases":"N_1","display":"N_1","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critica","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a87293985c01784","tex":"N_{10}","aliases":"N_10","display":"N_{10}","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6b337ece4924770","tex":"n_2","aliases":"n_2","display":"n_2","kind":"use","excerpt":"cdots<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb2af32ef3e74216","tex":"N_D","aliases":"N_D","display":"N_D","kind":"use","excerpt":"For supplied covariance data and the twisted cocycle of the hyperref[book:x-construction-13]finite-exponential reconstruction, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restri","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d2462ef3372559a","tex":"n_d","aliases":"n_d","display":"n_d","kind":"use","excerpt":"cdots<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1bc0c27714a3cfdd","tex":"n_{D_N}","aliases":"n_D_N","display":"n_{D_N}","kind":"use","excerpt":"ions are v=(m+1/2)^4. vfigcounting The two rows describe the same zero orbits. The upper row counts four zeros each time a new radius is reached. The lower row counts one reduced zero each time its fourth-power location is reached. Both formulas include zeros on the boundary of the disk. headSeparate radius conversion from orbit multiplicity In uns","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10b68c11b12c7be3","tex":"n_{\\Dfun_N}","aliases":"n_Dfun_N n_ D_N Dfun mathsf","display":"n_{{\\mathsf D}_N}","kind":"use","excerpt":"[Zero counting asymptotics] Let n_ D_N(R) count the zeros of D_N in |z|≤ R, with multiplicity. Let n_Δ_N(R) count the zeros of Δ_N in |w|≤ R. Then, as R→∞, n_ D_N(R) = fracNsin(π/N)πR+O(1) and n_Δ_N(R) = fracsin(π/N)πR^1/N+O(1).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2592eef1f50b9297","tex":"\\mathcal N_E","aliases":"N_E mathcal","display":"\\mathcal N_E","kind":"use","excerpt":"The precise finite channel statement. Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9588331116f694e","tex":"n_e","aliases":"n_e","display":"n_e","kind":"use","excerpt":"Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their card","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b82d70c3ab4030a5","tex":"N_i","aliases":"N_i","display":"N_i","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilp","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6801e9881994d7d","tex":"n_i^2","aliases":"n_i^2","display":"n_i^2","kind":"use","excerpt":"BDLR · ∑_i n_i^2+2nr · 2q(∑_i n_i^2+2nr)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1613ab8834072c1","tex":"N_j","aliases":"N_j","display":"N_j","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a fini","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-881cb5e47173db34","tex":"N_{j-1,j}","aliases":"N_j-1,j","display":"N_{j-1,j}","kind":"use","excerpt":"[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-921edcc519f97aae","tex":"N_j^k","aliases":"N_j^k","display":"N_j^k","kind":"use","excerpt":"mory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d3b88e3c1c7cc4c","tex":"N_j^{\\nu_j}","aliases":"N_j^ν_j nu","display":"N_j^{\\nu_j}","kind":"use","excerpt":"rem[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf9404d3f534ee7e","tex":"n_k","aliases":"n_k","display":"n_k","kind":"use","excerpt":"tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-718d406878f6c5a6","tex":"N_m","aliases":"N_m","display":"N_m","kind":"use","excerpt":"Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilp","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7c0e2c4594eec1b","tex":"n_m","aliases":"n_m","display":"n_m","kind":"use","excerpt":"[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62782243b1cfc0cb","tex":"N_{n,M}","aliases":"N_n,M","display":"N_{n,M}","kind":"use","excerpt":"[Number of labeled composition terms] For n≥1, if Y has M packets, then the number of packet labeled terms at order n before zero jets are removed is N_n,M = ∑_k=1^nS(n,k)M^k, where S(n,k) is a Stirling number of the second kind. Thus N_n,M is the nth Touchard polynomial evaluated at M.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a6ca180114c345d","tex":"n_p","aliases":"n_p","display":"n_p","kind":"use","excerpt":"For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positi","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af2a18fd20397da4","tex":"N_{p,w}","aliases":"N_p,w","display":"N_{p,w}","kind":"use","excerpt":"[Equal block banded count] Let p,b≥1 be integers, n=pb, and let p=n/b equal packets have width b. Suppose a block is active precisely when absi-j≤ w, 0≤ w≤ p-1. The number of active blocks is N_p,w = p(2w+1)-w(w+1). For q right hand sides, the modeled application cost and scalar storage are 2qb^2N_p,w and b^2N_p,w.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-073301ff99f56685","tex":"\\mathfrak N_R","aliases":"N_R mathfrak","display":"\\mathfrak N_R","kind":"use","excerpt":"[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [labe","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb171c4fc1202d77","tex":"n_r","aliases":"n_r","display":"n_r","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ffa89d2eeb94201","tex":"\\mathfrak N_R^{\\,h+1}","aliases":"N_R^ h+1 mathfrak","display":"\\mathfrak N_R^{\\,h+1}","kind":"use","excerpt":"rtial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. item The map to the diagonal classe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-173d1ec8e27571d5","tex":"N_S","aliases":"N_S","display":"N_S","kind":"use","excerpt":"[Finite free presentation] For the map in [source reference], imε_S=Hull_A(S), N_S:=Kerε_S≤ A^⊕ r, and the induced map is an isomorphism of left modules over A A^⊕ r/N_S≅Hull_A(S). Consequently, S generates M if and only if ε_S is surjective. [source reference] ExportsAll later free seed presentations.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f56096dd29a0f85","tex":"N_s","aliases":"N_s","display":"N_s","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b600c930ff971ba","tex":"n_\\alpha^2","aliases":"n_α^2 alpha","display":"n_\\alpha^2","kind":"use","excerpt":"[BDLR application and storage] For q right hand sides, AZ=DZ+U(V^*Z) has modeled cost 2q ( ∑_α n_α^2+2nr ). The representation stores ∑_α n_α^2+2nr scalars. For n=pb, with integers p,b≥1, the following factor describes the ideal dense storage reduction and, when q≥1, the arithmetic reduction: fracnb+2r.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8205d7811a8a9f9c","tex":"n_\\beta","aliases":"n_β beta","display":"n_\\beta","kind":"use","excerpt":"[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e90ef4c326a8622","tex":"n_{\\Delta_N}","aliases":"n_Δ_N Delta","display":"n_{\\Delta_N}","kind":"use","excerpt":"[Zero counting asymptotics] Let n_ D_N(R) count the zeros of D_N in |z|≤ R, with multiplicity. Let n_Δ_N(R) count the zeros of Δ_N in |w|≤ R. Then, as R→∞, n_ D_N(R) = fracNsin(π/N)πR+O(1) and n_Δ_N(R) = fracsin(π/N)πR^1/N+O(1).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07a2e96d36e692db","tex":"N_\\varepsilon","aliases":"N_ε varepsilon","display":"N_\\varepsilon","kind":"use","excerpt":"· 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less than ε/2. Then choose N_ε so ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1d7d1fdf60d8fdac","tex":"\\neg","aliases":"neg","display":"\\neg","kind":"use","excerpt":"[Exact logical force of a local obstruction] Every obstruction packet for x proves that x is not a solution, and negIsSolution_r(x) Longleftrightarrow there exists an obstruction packet for x. The reverse implication from failure to a packet witness is classical and does not require the packet type to be finite. statusleanthreeKomorebi.PaperIII.Non","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-797aecf4dbd2a9ff","tex":"\\nil","aliases":"nil Nil operatorname","display":"{\\operatorname{Nil}}","kind":"use","excerpt":"[Near faithfulness is full support] Let R be Noetherian and let M be a finite R module. The following conditions are equivalent. item M is nearly faithful. item Supp_R(M)=Spec R. item Ann_R(M)⊆Nil(R). If R is reduced, these conditions are equivalent to faithfulness.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-44a7952294f06149","tex":"\\mathrm{NL}","aliases":"NL mathrm","display":"\\mathrm{NL}","kind":"use","excerpt":"o(f)∈Ext^1_B(NL_B/ Z, F_p)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-331ed62039968d1b","tex":"\\mathrm{NL}_{B/\\mathbb Z}","aliases":"NL_B/ Z mathrm mathbb","display":"\\mathrm{NL}_{B/\\mathbb Z}","kind":"use","excerpt":"o(f)∈Ext^1_B(NL_B/ Z, F_p)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":29},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90f600277c50100a","tex":"\\NN","aliases":"NN N mathbb","display":"{\\mathbb N}","kind":"use","excerpt":"[Hurwitz series ring] Let R be a commutative ring with identity. The Hurwitz series ring H(R) is the set R^ N with componentwise addition and product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_r b_n-r. Its identity is δ=(1,0,0,…).","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2173a70f3a0a3da","tex":"\\NN_0","aliases":"NN_0 N_0 NN mathbb","display":"{\\mathbb N}_0","kind":"use","excerpt":"utes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILChapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the dir","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":71},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-090aecb8fb37e7b7","tex":"O_1","aliases":"O_1","display":"O_1","kind":"use","excerpt":"The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2b3885ba0bd5cac","tex":"\\mathcal O_2","aliases":"O_2 mathcal","display":"\\mathcal O_2","kind":"use","excerpt":"cients is not universally formalized here. Formal-series identities do not establish analytic convergence. The corrected global odd-prime finite-difference theorem remains pending. Historical numerical evidence is not presented as a fresh reproduction. pageheading1. Integral coefficients and an exact unit sourceLemmas 5.1 and 5.2; manuscript page 1","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5c9215497110dd4","tex":"\\mathcal O_A","aliases":"O_A mathcal","display":"\\mathcal O_A","kind":"use","excerpt":"Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e8d847d6b436ea8","tex":"\\mathcal O_E^\\times","aliases":"O_E^× mathcal times","display":"\\mathcal O_E^\\times","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3b35c060b413b3e","tex":"\\mathcal O_F","aliases":"O_F mathcal","display":"\\mathcal O_F","kind":"use","excerpt":"Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, fin","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf50fbf6ff02a71d","tex":"O_{F,\\bar x}","aliases":"O_F,bar x bar","display":"O_{F,\\bar x}","kind":"use","excerpt":"O_F,bar x= F_p^r/imJ, o_F,bar x=[e].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff62a298d080135d","tex":"o_{F,\\bar x}","aliases":"o_F,bar x bar","display":"o_{F,\\bar x}","kind":"use","excerpt":"The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the chapter supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual digit","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-962eb33121c4bcf4","tex":"O_i","aliases":"O_i","display":"O_i","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a7d87e09a33f4365","tex":"O_j","aliases":"O_j","display":"O_j","kind":"use","excerpt":"The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f009778fa1adcff","tex":"O_{K,C}","aliases":"O_K,C","display":"O_{K,C}","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-85ae2403f3c15ce7","tex":"O_{K,J,m,\\chi}","aliases":"O_K,J,m,χ chi","display":"O_{K,J,m,\\chi}","kind":"use","excerpt":"The assembled ideal norm-character estimate. Let C∈Cl( O), let J≠0 be an integral ideal with [J]=C^-1, and let m>0. Let χ: Z/m Z→ C preserve multiplication, 0, and 1. Assume a residue-ring unit satisfying [source reference], and assume χ( N Jbmod m)≠0. Then T_C,χ(X) =∑_substackI≠0, [I]=C N I≤ X χ( N Ibmod m) =O_K,J,m,χ(X^1-1/d). More precisely, som","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-946fcb5ff5706330","tex":"o_m","aliases":"o_m","display":"o_m","kind":"use","excerpt":"vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONChapter II, mathematical text: Section B.1 Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-580246cc2221ddc9","tex":"O_r","aliases":"O_r","display":"O_r","kind":"use","excerpt":"The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b57e43a9fff05b66","tex":"O_{R_*}","aliases":"O_R_*","display":"O_{R_*}","kind":"use","excerpt":"rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2bdaaca11c4a62e","tex":"\\mathcal O_x","aliases":"O_x mathcal","display":"\\mathcal O_x","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c35c2b5d63deb8c7","tex":"\\mathrm{ob}","aliases":"ob mathrm","display":"\\mathrm{ob}","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions ho","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-480cac95eb168013","tex":"\\operatorname{Ob}","aliases":"Ob operatorname","display":"\\operatorname{Ob}","kind":"use","excerpt":"[Linear lift obstruction] The linear obstruction at depth L is Ob^lin_L(x_L) = [ρ_L(x_L)] ∈ coker(overline M).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7af707f4c6857c9","tex":"\\operatorname{Ob}^{F}_L","aliases":"Ob^F_L operatorname","display":"\\operatorname{Ob}^{F}_L","kind":"use","excerpt":"[Polynomial lift obstruction] The polynomial obstruction at depth L is Ob^F_L(x_L) = [ρ_L^F(x_L)] ∈ coker(overline J_F(x_L)).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9de829476b80a2aa","tex":"\\Ob^G","aliases":"Ob^G Ob operatorname","display":"{\\operatorname{Ob}}^G","kind":"use","excerpt":"[Gauge covariance when packet projectors are preserved] If G_aΠ^a_μ=Π^a_μ G_a for every μ, then (Ob^G)^μν = G_aOb^μνG_a^-1. Consequently the zero/nonzero status of each obstruction block is preserved by gauges that preserve the packet projectors. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_of_commutes; Komorebi.PaperIII.H","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ff254cc6d240041","tex":"\\operatorname{Ob}^{\\mathrm{lin}}_L","aliases":"Ob^lin_L operatorname mathrm","display":"\\operatorname{Ob}^{\\mathrm{lin}}_L","kind":"use","excerpt":"[Linear lift obstruction] The linear obstruction at depth L is Ob^lin_L(x_L) = [ρ_L(x_L)] ∈ coker(overline M).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b2fbd707f34da9d","tex":"\\Ob^{\\mu\\nu}","aliases":"Ob^μν Ob mu nu operatorname mu nu","display":"{\\operatorname{Ob}}^{\\mu\\nu}","kind":"use","excerpt":"[Gauge covariance when packet projectors are preserved] If G_aΠ^a_μ=Π^a_μ G_a for every μ, then (Ob^G)^μν = G_aOb^μνG_a^-1. Consequently the zero/nonzero status of each obstruction block is preserved by gauges that preserve the packet projectors. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_of_commutes; Komorebi.PaperIII.H","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7acafa494704338e","tex":"\\Ob_{D'}","aliases":"Ob_D' Ob operatorname","display":"{\\operatorname{Ob}}_{D'}","kind":"use","excerpt":"U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ae97ea283061d02","tex":"\\Ob_m'","aliases":"Ob_m' Ob operatorname","display":"{\\operatorname{Ob}}_m'","kind":"use","excerpt":"[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8732d1ab457c098","tex":"\\Ocal_t","aliases":"Ocal_t O_t Ocal mathcal","display":"{\\mathcal O}_t","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4fa2eeb16819ef5a","tex":"\\Ocal_\\lambda","aliases":"Ocal_λ O_λ Ocal lambda mathcal lambda","display":"{\\mathcal O}_\\lambda","kind":"use","excerpt":"All coefficients of the characteristic polynomial commute with reduction. Consequently, prescribed integral characteristic polynomials at every precision hold exactly in the reconstructed representation. For a common coefficient field E, local integral polynomials lie in O_λ[X] and are compared with P_v∈ E[X] only after mapping both into the local ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a34910944de266ad","tex":"\\odot","aliases":"odot","display":null,"kind":"use","excerpt":"Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b77c3e9c22499671","tex":"\\oint","aliases":"oint","display":null,"kind":"use","excerpt":"(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula give","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c668e45ed5a9cdb","tex":"\\oint_{|z-\\mu|=r_\\mu}","aliases":"oint_|z-μ|=r_μ oint mu mu","display":null,"kind":"use","excerpt":"(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula give","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9f1bf76a072652d","tex":"\\one^T","aliases":"one^T 1^T one bm","display":"{\\bm 1}^T","kind":"use","excerpt":"Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dcf850482173fa7d","tex":"\\one_{[-a,a]}","aliases":"one_[-a,a] 1_[-a,a] one mathbf","display":"{\\mathbf 1}_{[-a,a]}","kind":"use","excerpt":"[Square wave Fourier tail] Let j(t)= 1_[-a,a](t) on the circle and let s be smooth. If Q_m is a Fourier partial sum, then Q_m(s+j) has the ordinary oscillatory boundary layer. In contrast, Q_ms+j has error exactly s-Q_ms. The visual comparison in [source reference] illustrates this change in approximation target. It is not a new Fourier summability","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9cf216a2fd3d418b","tex":"\\one_E","aliases":"one_E 1_E one mathbf","display":"{\\mathbf 1}_E","kind":"use","excerpt":"[Exact scalar region formula] Let E,F be measurable subsets of a measure space, let a,b∈ C, and let 1≤ p<∞. Then a 1_E-b 1_F _p^p = · |a-b|^pμ(E∩ F) +|a|^pμ(E∖ F) · +|b|^pμ(F∖ E). For equal amplitudes, a 1_E-a 1_F _p = |a|μ(Emathbintriangle F)^1/p.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d2b4fb4a0ad76c8","tex":"\\one_{E_i}","aliases":"one_E_i 1_E_i one mathbf","display":"{\\mathbf 1}_{E_i}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-248f745ef97fafb0","tex":"\\one_F","aliases":"one_F 1_F one mathbf","display":"{\\mathbf 1}_F","kind":"use","excerpt":"[Exact scalar region formula] Let E,F be measurable subsets of a measure space, let a,b∈ C, and let 1≤ p<∞. Then a 1_E-b 1_F _p^p = · |a-b|^pμ(E∩ F) +|a|^pμ(E∖ F) · +|b|^pμ(F∖ E). For equal amplitudes, a 1_E-a 1_F _p = |a|μ(Emathbintriangle F)^1/p.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c08f8d7bc2b3255","tex":"\\one_{F_j}","aliases":"one_F_j 1_F_j one mathbf","display":"{\\mathbf 1}_{F_j}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dbd0bb0518600c58","tex":"\\one_{i\\in B_a}","aliases":"one_i∈ B_a 1_i∈ B_a one in bm in","display":"{\\bm 1}_{i\\in B_a}","kind":"use","excerpt":"The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2500223217e36ec5","tex":"\\one_{\\{i\\in B_\\alpha\\}}","aliases":"one_{i∈ B_α} 1_{i∈ B_α} one in alpha bm in alpha","display":"{\\bm 1}_{\\{i\\in B_\\alpha\\}}","kind":"use","excerpt":"Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c1a14ba7424a4a8c","tex":"\\ones","aliases":"ones","display":null,"kind":"use","excerpt":"midrule p · A nonnegative column with ones^ Tp=1. · It is not a vector of function values.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33cfa968292abd16","tex":"\\ones^{\\mathsf T}","aliases":"ones^ T ones mathsf","display":null,"kind":"use","excerpt":"midrule p · A nonnegative column with ones^ Tp=1. · It is not a vector of function values.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d48a1e60ecf7c1c","tex":"\\ones^T","aliases":"ones^T ones","display":null,"kind":"use","excerpt":"vpage19Twelve small blocks, one coherent errorDISCOVERY AND PERTURBATION CERTIFICATESChapter VIII: Theorem 8.1, Proposition 9.2. Split QQ^4 into scalar packets and let A=I+E, E=frac1100(onesones^T-I). Dropping every off-diagonal block leaves I. Each discarded entry is only 1/100, but the total error is not bounded by that single-entry size. vfigblo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b2e774b4630d060","tex":"\\OO^2","aliases":"OO^2 O^2 OO mathcal","display":"{{\\mathcal O}}^2","kind":"use","excerpt":"[A resultant bound that is not exact] Let A=diag(0,π^s) on the standard lattice Λ= O^2. The factors x and x-π^s have resultant -π^s, up to the harmless choice of resultant order, but the two spectral projectors are the integral coordinate projectors. Their exact separation weight is zero. The resultant sees the closeness of the two eigenvalues. It ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a86f15b45905b04","tex":"\\OO^d","aliases":"OO^d O^d OO mathcal","display":"{\\mathcal O}^d","kind":"use","excerpt":"[Coinvariant recovery] Let M be a free Λ module of rank d. Then M/ a M ≅ M⊗_Λ,ϵ O ≅ O^d. The statement is valid for every coefficient prime, including two.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9698f456cf9e2ff","tex":"\\OO_F","aliases":"OO_F O_F OO mathcal","display":"{\\mathcal O}_F","kind":"use","excerpt":"[Restricted dyadic marked assembly realization] Let F= F_2((t)), fix an additive character of conductor t O_F, fix d≥1, and let h_1<⋯<h_d be positive odd integers. Put q_i=3h_i, N=2^d, and n_j=2^d-jq_d-j+1. The explicit recursive matrix generates a separable totally ramified degree-N maximal parameter field, normalizes the complete lattice chain, a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb5f4b10196f0437","tex":"\\OO_K^\\times","aliases":"OO_K^× O_K^× OO times mathcal times","display":"{\\mathcal O}_K^\\times","kind":"use","excerpt":"Here O_K^×=U_K^1 because the residue field is F_2. Since U_K 1 is contained in J 1, we have","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a75471218462aa6a","tex":"\\OO_\\lambda","aliases":"OO_λ O_λ OO lambda mathcal lambda","display":"{\\mathcal O}_\\lambda","kind":"use","excerpt":"Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f814b60f967151c4","tex":"\\mathrm{op}","aliases":"op mathrm","display":"\\mathrm{op}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95f972aa7326a181","tex":"\\operatorname{orbit}","aliases":"orbit operatorname","display":"\\operatorname{orbit}","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd71dfd389a79ac5","tex":"\\operatorname{ord}","aliases":"ord operatorname","display":"\\operatorname{ord}","kind":"use","excerpt":"For x∈ Z_p, extend the valuation to zero by setting ord_p(x)= v_p(x), · x≠0, ∞, · x=0. Thus L≤ord_p(x) is equivalent to x∈ p^L Z_p.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab3321d6961879fc","tex":"\\ord","aliases":"ord operatorname","display":"{\\operatorname{ord}}","kind":"use","excerpt":"[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4679df4399518e1","tex":"\\ord_3","aliases":"ord_3 ord operatorname","display":"{\\operatorname{ord}}_3","kind":"use","excerpt":"ubsection*A sharp binomial-weight improvement Let F(n)=81binom n1+3binom n2+3binom n3+81binom n4. At n=3 and s=1, the four coefficient valuations are (4,1,1,4), while the binomial valuations are (0,1,1,0). Thus the refined bound is B_3(w;3,1)=min(4,2,2,4)=2, stronger than the coefficient-only lower bound one. The source checks both the minimum and ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ee3ebe96b297ddb","tex":"\\operatorname{ord}_p","aliases":"ord_p operatorname","display":"\\operatorname{ord}_p","kind":"use","excerpt":"For x∈ Z_p, extend the valuation to zero by setting ord_p(x)= v_p(x), · x≠0, ∞, · x=0. Thus L≤ord_p(x) is equivalent to x∈ p^L Z_p.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd164462222d475b","tex":"\\ord_p","aliases":"ord_p ord operatorname","display":"{\\operatorname{ord}}_p","kind":"use","excerpt":"The additive valuation is extended to ord_p(0)=∞. This avoids incorrectly applying Mathlib's finite-valued default at zero to an inequality that must also cover cancellation and identically zero differences.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71e41dae7d473f05","tex":"\\mathrm{order}","aliases":"order mathrm","display":"\\mathrm{order}","kind":"use","excerpt":"The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a625c27cba36b747","tex":"\\osc","aliases":"osc operatorname","display":"{\\operatorname{osc}}","kind":"use","excerpt":"] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c249b84d38edd209","tex":"\\osc_{I_\\alpha}","aliases":"osc_I_α osc alpha operatorname alpha","display":"{\\operatorname{osc}}_{I_\\alpha}","kind":"use","excerpt":"] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea26d2394b7c0bc5","tex":"\\overline a","aliases":"overline a overline","display":"\\overline a","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f61d083e1babfc3","tex":"\\overline a^{\\,p}","aliases":"overline a^ p overline","display":"\\overline a^{\\,p}","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54037e9a059bedc5","tex":"\\overline B","aliases":"overline B overline","display":"\\overline B","kind":"use","excerpt":"[Polynomial tree expansion] Suppose there is R>0 such that ϕ(R)≤ R and q= ∑_r=2^dr b_rR^r-1<1. Then the map Φ(x)=s+∑_r=2^dB_r(x,…,x) has a unique fixed point in the closed ball overline B_R(0). The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7da834df65459b68","tex":"\\overline b","aliases":"overline b overline","display":"\\overline b","kind":"use","excerpt":"U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce8712e6c17bb593","tex":"\\overline B_R","aliases":"overline B_R overline","display":"\\overline B_R","kind":"use","excerpt":"[Polynomial tree expansion] Suppose there is R>0 such that ϕ(R)≤ R and q= ∑_r=2^dr b_rR^r-1<1. Then the map Φ(x)=s+∑_r=2^dB_r(x,…,x) has a unique fixed point in the closed ball overline B_R(0). The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5be848bd2e45cb7","tex":"\\overline B_\\rho","aliases":"overline B_ρ overline rho","display":"\\overline B_\\rho","kind":"use","excerpt":"[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L-Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0aeccd83542eabf9","tex":"\\overline C","aliases":"overline C overline","display":"\\overline C","kind":"use","excerpt":"[Lucas reduction and address independence] Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-481dece60de15875","tex":"\\overline C_m","aliases":"overline C_m overline","display":"\\overline C_m","kind":"use","excerpt":"[Lucas reduction and address independence] Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7079e3832479ea24","tex":"\\overline D","aliases":"overline D overline","display":"\\overline D","kind":"use","excerpt":"Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f46e22d0b8fbcf67","tex":"\\overline D_r","aliases":"overline D_r overline","display":"\\overline D_r","kind":"use","excerpt":"Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b0ad0f92e6daea0","tex":"\\overline G","aliases":"overline G overline","display":"\\overline G","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-944f80df05aedc21","tex":"\\overline G_r","aliases":"overline G_r overline","display":"\\overline G_r","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac5881edc3abfcd7","tex":"\\overline I","aliases":"overline I overline","display":"\\overline I","kind":"use","excerpt":"[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [labe","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7a5b96653c1dfce","tex":"\\overline K","aliases":"overline K overline","display":"\\overline K","kind":"use","excerpt":"[Descent through an aggregation map] Let C:X_f→ X_c be a surjective linear map between finite dimensional spaces. A unique operator overline K:X_c→ X_c satisfying CK=overline KC exists if and only if Ob_C(K)=0. For any right inverse R of C, the only possible coarse operator is overline K=CKR, and CK-overline KC = CK(I-RC).","count":27},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c616ecd4a706af27","tex":"\\overline K^n","aliases":"overline K^n overline","display":"\\overline K^n","kind":"use","excerpt":"[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c03d7a3e867e77f","tex":"\\overline K^{n-1-j}","aliases":"overline K^n-1-j overline","display":"\\overline K^{n-1-j}","kind":"use","excerpt":"[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c22893176e13cead","tex":"\\overline K^{n-j}","aliases":"overline K^n-j overline","display":"\\overline K^{n-j}","kind":"use","excerpt":"Sum the telescoping identities overline K^n-1-jCK^j+1 - overline K^n-jCK^j = overline K^n-1-jΔ_KK^j. Stochastic matrices are contractions in the ℓ^1 norm. The last statement uses total variation equal to one half of the ℓ^1 distance.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0c30b2e35f72f77","tex":"\\overline K_{\\alpha\\beta}","aliases":"overline K_αβ overline alpha beta","display":"\\overline K_{\\alpha\\beta}","kind":"use","excerpt":"Let K be column stochastic. A coarse column stochastic matrix overline K satisfying CK=overline KC exists if and only if, for every source block B_β, every pair i,j∈ B_β, and every target block B_α, ∑_r∈ B_αK_ri = ∑_r∈ B_αK_rj. When it exists, overline K is unique and overline K_αβ = ∑_r∈ B_αK_ri for any i∈ B_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d58c598ab4835df","tex":"\\overline L_a","aliases":"overline L_a overline","display":"\\overline L_a","kind":"use","excerpt":"tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared f","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a78d412941cc60b6","tex":"\\overline L_y","aliases":"overline L_y overline","display":"\\overline L_y","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53747cc445df0a6e","tex":"\\overline m","aliases":"overline m overline","display":"\\overline m","kind":"use","excerpt":"[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-626b99e92a63b7cb","tex":"\\overline n","aliases":"overline n overline","display":"\\overline n","kind":"use","excerpt":"The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5fbedef8300da8c","tex":"\\overline n_J","aliases":"overline n_J overline","display":"\\overline n_J","kind":"use","excerpt":"An actual unit action on ideal residues. Fix m>0 and a nonzero integral ideal J. Multiplication by u∈( O/m O)^× gives a proved permutation of J/mJ. Its inverse follows from (ab-1)x∈ mJ when ab-1∈ m O and x∈ J. No surjectivity of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f81ac0b7b57e0101","tex":"\\overline n_{\\varepsilon}","aliases":"overline n_ε overline varepsilon","display":"\\overline n_{\\varepsilon}","kind":"use","excerpt":"For ε∈{1,-1} define overline n_ε(a)=εoverline n(a). On the corresponding sign portion, N(x)bmod m=overline n_ε([x]). The sign cannot be dropped: -1≡2pmod3, whereas N(-1)=1 and N(2)=2 for principal ideals of Z.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ffdd263bffb70d69","tex":"\\overline P","aliases":"overline P overline","display":"\\overline P","kind":"use","excerpt":"[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ ","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-622810599c7ebcb0","tex":"\\overline p","aliases":"overline p overline","display":"\\overline p","kind":"use","excerpt":"Rank-one rigidity of the corrected channel. The module tests the map equality on actual matrix units and flattens each A_ra into a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzer","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00c5b69e7d95b7d6","tex":"\\overline P_\\alpha","aliases":"overline P_α overline alpha","display":"\\overline P_\\alpha","kind":"use","excerpt":"[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f137d0144e6a41ef","tex":"\\overline P_\\beta","aliases":"overline P_β overline beta","display":"\\overline P_\\beta","kind":"use","excerpt":"If x∈ K_α, then Tx∈ K, so T_βαx=P_β Tx∈ K∩ X_β=K_β. On the quotient, overline T_βαπ = overline P_βoverline Toverline P_απ = π P_β TP_α.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42c6158ae78541bc","tex":"\\overline Q","aliases":"overline Q overline","display":"\\overline Q","kind":"use","excerpt":"[Projectors on the obstruction quotient] Let (Q_α)_α∈ I be a complete family of pairwise annihilating idempotents on V. Assume Q_α(im D)⊆im D for every α. Then each Q_α induces an idempotent overline Q_α on coker D, and coker D = ⊕_α∈ Ioverline Q_α(coker D). For every defect r∈ V, overline Q_αOb_D(r) = [Q_α r]. No packet decomposition of the correc","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a3211811166a67a","tex":"\\overline Q^T","aliases":"overline Q^T overline","display":"\\overline Q^T","kind":"use","excerpt":"[Coarse Dirichlet form] Under the hypotheses of [source reference], let Jg be the fine function that is constant with value g_a on B_a. Then -ipJgQ^TJg_L^2(π) = -ipgoverline Q^Tg_L^2(overlineπ).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a587a61dc8ffd20","tex":"\\overline Q_{ab}","aliases":"overline Q_ab overline","display":"\\overline Q_{ab}","kind":"use","excerpt":"the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe8d3f49543714c4","tex":"\\overline Q_{ba}","aliases":"overline Q_ba overline","display":"\\overline Q_{ba}","kind":"use","excerpt":"the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd4837de74cbe911","tex":"\\overline Q_\\alpha","aliases":"overline Q_α overline alpha","display":"\\overline Q_\\alpha","kind":"use","excerpt":"[Projectors on the obstruction quotient] Let (Q_α)_α∈ I be a complete family of pairwise annihilating idempotents on V. Assume Q_α(im D)⊆im D for every α. Then each Q_α induces an idempotent overline Q_α on coker D, and coker D = ⊕_α∈ Ioverline Q_α(coker D). For every defect r∈ V, overline Q_αOb_D(r) = [Q_α r]. No packet decomposition of the correc","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41dacfd07761cf94","tex":"\\overline r","aliases":"overline r overline","display":"\\overline r","kind":"use","excerpt":"The solved Jacobian remains surjective, so every shifted obstruction [overline r+K_z_nv] vanishes. Formula~[source reference] therefore gives p^m-e choices for every parameter digit at every stage. When m=e, the fixed-coordinate chart in the proof of Theorem~[source reference] contains every root in the initial ball with the specified parameters. I","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c226cd3303f20dba","tex":"\\overline s","aliases":"overline s overline","display":"\\overline s","kind":"use","excerpt":"[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23856ee01c2b6c2b","tex":"\\overline T","aliases":"overline T overline","display":"\\overline T","kind":"use","excerpt":"[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f88991f9db432114","tex":"\\overline t","aliases":"overline t overline","display":"\\overline t","kind":"use","excerpt":"Every point in the lower residue class has the form y=x+q^Lt with t∈ Z^d. By Lemma~[source reference] and 2L≥ L+1, F(x+q^Lt)≡ q^L(r_x+DF(x)t)pmodq^L+1. The integer q^L is nonzero. Cancelling it in the corresponding divisibility condition proves F(x+q^Lt)≡0pmodq^L+1 Longleftrightarrow overliner_x+J_xoverline t=0. Integer vectors reduce onto S_q^d. T","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb737e6f2757da05","tex":"\\overline T_{\\beta\\alpha}","aliases":"overline T_βα overline beta alpha","display":"\\overline T_{\\beta\\alpha}","kind":"use","excerpt":"[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-beffd946d87d6320","tex":"\\overline U","aliases":"overline U overline","display":"\\overline U","kind":"use","excerpt":"orm a complete orthogonal packet system on Hilbert--Schmidt operator space. Dephasing is the diagonal projection. Every channel has input and output superblocks, with the same composition law as ordinary matrix blocks. A covariant channel preserves the isotypic packets of the conjugation action. In the cyclic case these are coherence orders. The Ch","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad21f4f96eb368f9","tex":"\\overline V","aliases":"overline V overline","display":"\\overline V","kind":"use","excerpt":"[Coarse Feynman--Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57ff192d28b73a0c","tex":"\\overline v","aliases":"overline v overline","display":"\\overline v","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1895fa5459189739","tex":"\\overline v_j","aliases":"overline v_j overline","display":"\\overline v_j","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db64b27099ab4aae","tex":"\\overline w","aliases":"overline w overline","display":"\\overline w","kind":"use","excerpt":"At level N=L+n, write the supplied parameter increment exactly as t_n+1-t_n=q^Nw with w∈ Z^k. Surjectivity of the solved block gives a digit u satisfying J_z_nu=-overlineF(z_n)/q^N-K_z_noverline w. Choose an integer representative widetilde u and set z_n+1=z_n+q^N(widetilde u,w). Its parameter coordinates equal the supplied target exactly. Taylor e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43a8f89e4f079c54","tex":"\\overline y","aliases":"overline y overline","display":"\\overline y","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c45a9f213c593e3","tex":"\\overline y^{\\,n-1}","aliases":"overline y^ n-1 overline","display":"\\overline y^{\\,n-1}","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae21a8553a17610b","tex":"\\overline z","aliases":"overline z overline","display":"\\overline z","kind":"use","excerpt":"[Symmetries] The denominator satisfies D_N(ω z)= D_N(z) and D_N(overline z)=overline D_N(z). Its nonzero zeros occur in orbits under multiplication by the Nth roots of unity and under complex conjugation.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27c76c57aea96832","tex":"\\overline{(G_r-W_kU_k)/2^k}","aliases":"overline(G_r-W_kU_k)/2^k overline","display":"\\overline{(G_r-W_kU_k)/2^k}","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6701fd0fc5e1ceb8","tex":"\\overline{\\left(\\sum_{j\\in B}\\pi_j g_j\\right)}","aliases":"overline(∑_j∈ Bπ_j g_j) overline left sum in pi right","display":"\\overline{\\left(\\sum_{j\\in B}\\pi_j g_j\\right)}","kind":"use","excerpt":"After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block. Both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2663db5a20889e2c","tex":"\\overline{\\left(\\sum_{j\\in B}\\pi_jg_j\\right)}","aliases":"overline(∑_j∈ Bπ_jg_j) overline left sum in pi right","display":"\\overline{\\left(\\sum_{j\\in B}\\pi_jg_j\\right)}","kind":"use","excerpt":"One application produces a block constant function, so a second application changes nothing. Expanding both weighted inner products block by block gives ipP_ Qfg_π = ∑_B∈ Q frac (∑_i∈ Bπ_if_i) overline(∑_j∈ Bπ_jg_j) π(B) = ipfP_ Qg_π. The range is exactly the block constant subspace.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3d1ed7dab22cee4","tex":"\\overline{c_b}","aliases":"overlinec_b overline","display":"\\overline{c_b}","kind":"use","excerpt":"For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-227f60ba397e1df3","tex":"\\overline{c_j}","aliases":"overlinec_j overline","display":"\\overline{c_j}","kind":"use","excerpt":"[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e0650b79ef975f9","tex":"\\overline{D(\\mu,r_\\mu)}","aliases":"overlineD(μ,r_μ) overline mu mu","display":"\\overline{D(\\mu,r_\\mu)}","kind":"use","excerpt":".F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c3b42b1cc9cccf8","tex":"\\overline{d_{ja}}","aliases":"overlined_ja overline","display":"\\overline{d_{ja}}","kind":"use","excerpt":"he left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recove","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a13a376dd76033a7","tex":"\\overline{\\Dfun_N(z)}","aliases":"overlineDfun_N(z) overline D_N(z) overline Dfun overline mathsf","display":"\\overline{{\\mathsf D}_N(z)}","kind":"use","excerpt":"[Symmetries] The denominator satisfies D_N(ω z)= D_N(z) and D_N(overline z)=overline D_N(z). Its nonzero zeros occur in orbits under multiplication by the Nth roots of unity and under complex conjugation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-064dd8c989fe8d11","tex":"\\overline{F(z_n)/q^N}","aliases":"overlineF(z_n)/q^N overline","display":"\\overline{F(z_n)/q^N}","kind":"use","excerpt":"At level N=L+n, divide the residual by q^N. Surjectivity supplies t∈ S_q^d with J_z_nt=-overlineF(z_n)/q^N. Choose an integer representative and apply [source reference] to obtain z_n+1. The new point is congruent to z_n modulo q. Evaluation of each formal partial derivative therefore gives J_z_n+1=J_z_n.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b677f57c2885a9a","tex":"\\overline{g(x)}","aliases":"overlineg(x) overline","display":"\\overline{g(x)}","kind":"use","excerpt":"-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c638d5d3835c366c","tex":"\\overline{JB}","aliases":"overlineJB overline","display":"\\overline{JB}","kind":"use","excerpt":"is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ec6ea712af79729","tex":"\\overline{L\\widehat S}","aliases":"overlineLwidehat S overline widehat","display":"\\overline{L\\widehat S}","kind":"use","excerpt":"Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7896e1964fe42160","tex":"\\overline{\\Q_p}","aliases":"overlineQ_p overline Q_p overline Q overline mathbf","display":"\\overline{{\\mathbf Q}_p}","kind":"use","excerpt":"For every prime p, Lean constructs a valuation subring A⊆overline Q_p containing the integral image of Z_p such that the induced map Z_p→ A is local. For odd p, two is a unit. The preceding argument applies to every valuation subring containing that integral image, and hence every member of P_q(B) is trivial on its inertia subgroup after actual loc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8be67fcf5a4dc680","tex":"\\overline{r_x}","aliases":"overliner_x overline","display":"\\overline{r_x}","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-227188b836d8f64b","tex":"\\overline{r_{x'}}","aliases":"overliner_x' overline","display":"\\overline{r_{x'}}","kind":"use","excerpt":"[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-428838e041027074","tex":"\\overline{s(t)-s(t+L/2)}","aliases":"overlines(t)-s(t+L/2) overline","display":"\\overline{s(t)-s(t+L/2)}","kind":"use","excerpt":"Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8990f3a0f7e9440","tex":"\\overline{\\operatorname{span}\\{S^n\\delta_0:n\\in\\bbZ\\}}","aliases":"overlinespan{S^nδ_0:n∈bbZ} overlinespan{S^nδ_0:n∈ Z} overline operatorname delta in bbZ overline operatorname delta in mathbb","display":"\\overline{\\operatorname{span}\\{S^n\\delta_0:n\\in{\\mathbb Z}\\}}","kind":"use","excerpt":"[Impulse seed theorem] The closed shift-generated hull of δ_0 is all of ℓ^2( Z): overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 =ℓ^2( Z). Every finitely supported signal has the exact finite expression x=∑_n∈ Zx[n]S^nδ_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-085e7278fbd586c8","tex":"\\overline{\\operatorname{span}\\{S^n\\delta_0:n\\in\\bbZ\\}}^{\\,\\ell^2}","aliases":"overlinespan{S^nδ_0:n∈bbZ}^ ℓ^2 overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 overline operatorname delta in bbZ ell overline operatorname delta in mathbb ell","display":"\\overline{\\operatorname{span}\\{S^n\\delta_0:n\\in{\\mathbb Z}\\}}^{\\,\\ell^2}","kind":"use","excerpt":"[Impulse seed theorem] The closed shift-generated hull of δ_0 is all of ℓ^2( Z): overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 =ℓ^2( Z). Every finitely supported signal has the exact finite expression x=∑_n∈ Zx[n]S^nδ_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-398aacf1e9e3576c","tex":"\\overline{\\Span\\{T_a\\delta_0:a\\in\\ZZ^d\\}}","aliases":"overlineSpan{T_aδ_0:a∈ZZ^d} overlinespan{T_aδ_0:a∈ Z^d} overline Span delta in ZZ overline operatorname delta in mathbb","display":"\\overline{{\\operatorname{span}}\\{T_a\\delta_0:a\\in{\\mathbb Z}^d\\}}","kind":"use","excerpt":"[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71d5e65764bad34c","tex":"\\overline{\\Span\\{T_a\\delta_0:a\\in\\ZZ^d\\}}^{\\ell^p}","aliases":"overlineSpan{T_aδ_0:a∈ZZ^d}^ℓ^p overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p overline Span delta in ZZ ell overline operatorname delta in mathbb ell","display":"\\overline{{\\operatorname{span}}\\{T_a\\delta_0:a\\in{\\mathbb Z}^d\\}}^{\\ell^p}","kind":"use","excerpt":"[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7591b1de9b36411","tex":"\\overline{U(g)}","aliases":"overlineU(g) overline","display":"\\overline{U(g)}","kind":"use","excerpt":"[Choi symmetry criterion] Let U:G→ U( H) be a finite group representation. A channel Φ is covariant, Φ(U(g)XU(g)^*) = U(g)Φ(X)U(g)^*, if and only if (U(g)⊗overlineU(g)) J(Φ) (U(g)^*⊗ U(g)^ T) = J(Φ) for every g∈ G.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32830cd7863f82ba","tex":"\\overline{W_{ja}}","aliases":"overlineW_ja overline","display":"\\overline{W_{ja}}","kind":"use","excerpt":"[Orthogonal syndrome construction] Assume the Knill--Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e10243300d232e33","tex":"\\overline{x}","aliases":"overlinex overline","display":"\\overline{x}","kind":"use","excerpt":"The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a4916ff831cccc6","tex":"\\overline{x}^{\\,p}","aliases":"overlinex^ p overline","display":"\\overline{x}^{\\,p}","kind":"use","excerpt":"The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1002d19bc28fd56","tex":"\\overline{xg(x)-g'(x)}","aliases":"overlinexg(x)-g'(x) overline","display":"\\overline{xg(x)-g'(x)}","kind":"use","excerpt":"-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ba14a930d8f52433","tex":"\\overline{z_a}","aliases":"overlinez_a overline","display":"\\overline{z_a}","kind":"use","excerpt":"identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the sam","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0975b6c646732480","tex":"\\overline{z_{ra}}","aliases":"overlinez_ra overline","display":"\\overline{z_{ra}}","kind":"use","excerpt":"Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this cal","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-541bacaca1fd7280","tex":"\\overline{\\gamma_p(a)}","aliases":"overlineγ_p(a) overline gamma","display":"\\overline{\\gamma_p(a)}","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb79cf09cdd9505e","tex":"\\overline{\\eta_0}","aliases":"overlineη_0 overline eta","display":"\\overline{\\eta_0}","kind":"use","excerpt":"The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-292fef11aee33c25","tex":"\\overline\\lambda_\\beta","aliases":"overlineλ_β overline lambda beta","display":"\\overline\\lambda_\\beta","kind":"use","excerpt":"The posterior mass of packet I_α is frac∑_i∈ I_αλ_ip_i ∑_jλ_jp_j = frac(Cp)_αoverlineλ_α(p) ∑_β(Cp)_βoverlineλ_β(p). This is the coarse Bayesian formula. Dividing the fine posterior coordinates by the packet mass gives the local formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39650f15ca748586","tex":"\\overline\\pi","aliases":"overlineπ overline pi","display":"\\overline\\pi","kind":"use","excerpt":"se Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overl","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8a56fca100492b3","tex":"\\overline\\pi_a","aliases":"overlineπ_a overline pi","display":"\\overline\\pi_a","kind":"use","excerpt":"se Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overl","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3416c17345e3e658","tex":"\\overline\\pi_b","aliases":"overlineπ_b overline pi","display":"\\overline\\pi_b","kind":"use","excerpt":"the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f403bf81328e606","tex":"\\overline{\\sigma_p(x)}","aliases":"overlineσ_p(x) overline sigma","display":"\\overline{\\sigma_p(x)}","kind":"use","excerpt":"The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-782174188fd36427","tex":"\\overline\\Phi","aliases":"overlineΦ overline Phi","display":"\\overline\\Phi","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-683c7ddceeb76bac","tex":"\\overline\\varphi","aliases":"overlineφ overline varphi","display":"\\overline\\varphi","kind":"use","excerpt":"[Cokernel obstruction criterion] A linear correction exists if and only if Ob_D(r)=0. The quotient map q_D:V⟶coker D has the following universal property. If φ:V→ W is linear and φ D=0, then there is a unique linear map overlineφ:coker D⟶ W with φ=overlineφ q_D.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e1bcaea0a451444","tex":"\\overline\\Phi_y","aliases":"overlineΦ_y overline Phi","display":"\\overline\\Phi_y","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78355e7ad6e40f33","tex":"\\overline\\Phi_{y_1}","aliases":"overlineΦ_y_1 overline Phi","display":"\\overline\\Phi_{y_1}","kind":"use","excerpt":"[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9558f1e55cd5cd94","tex":"\\overline\\Phi_{y_r}","aliases":"overlineΦ_y_r overline Phi","display":"\\overline\\Phi_{y_r}","kind":"use","excerpt":"[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0690ec27d59fae38","tex":"\\overline{\\chi(r)}","aliases":"overlineχ(r) overline chi","display":"\\overline{\\chi(r)}","kind":"use","excerpt":"Orthogonality of the projectors. Unitarity supplies ρ(r)^*=ρ(-r), while the character obeys overlineχ(r)=χ(-r). Reindexing r↦-r proves P_j^*=P_j. The final concrete matrix theorem includes Hermiticity together with multiplication, completeness, and eigenaction. Together these identities establish orthogonal spectral packets.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa9a82fa2877719d","tex":"\\overline{\\chi_\\lambda(g)}","aliases":"overlineχ_λ(g) overline chi lambda","display":"\\overline{\\chi_\\lambda(g)}","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f33e59d7a0611a5","tex":"\\overline{\\omega_\\ell(a)}","aliases":"overlineω_ℓ(a) overline omega ell","display":"\\overline{\\omega_\\ell(a)}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0aa06ee1b3de418","tex":"\\overline{\\langle a\\rangle_\\ell}","aliases":"overline⟨ a⟩_ℓ overline langle rangle ell","display":"\\overline{\\langle a\\rangle_\\ell}","kind":"use","excerpt":"The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef958519c2ac9923","tex":"\\mathbb P","aliases":"P mathbb","display":"\\mathbb P","kind":"use","excerpt":"[Polyphase M obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"The normalized source and its inverse are K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coeffic","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","excerpt":"[htbp] centering includegraphics[width=.96linewidth]rotating_projector_transport.pdf captionA rotating rank-one packet and its orthogonal complement. With u(θ)=(cosθ,sinθ) and P(θ)=u(θ)u(θ)^ T, the canonical generator is [P',P]=(0 · -1 1 · 0). For a positive step δ<π/2, the polar unitary is the usual rotation R(δ). On each displayed mesh, the direc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b95fc3463e504f17","tex":"P^{\\mathsf T}","aliases":"P^ T mathsf","display":"P^{\\mathsf T}","kind":"use","excerpt":"[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood. The full exact defect remai","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a1fe20674ded8f2","tex":"p^{\\mathsf T}","aliases":"p^ T mathsf","display":"p^{\\mathsf T}","kind":"use","excerpt":"Rank-one rigidity of the corrected channel. The module tests the map equality on actual matrix units and flattens each A_ra into a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzer","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8994ede9b103570b","tex":"P^{(1)}_{j_1}","aliases":"P^(1)_j_1","display":"P^{(1)}_{j_1}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4c15eaf6719fd34","tex":"p^{(a)}","aliases":"p^(a)","display":"p^{(a)}","kind":"use","excerpt":"vpage10Two priors that fool the coarse modelSUFFICIENCY / A COMPUTABLE FAILURE WITNESSChapter VI, Part I: Theorem 8.2 and the one-packet exception. Keep the same likelihood λ=(4,1,1,1)^ T and partition A={0,1}, B={2,3}. Compare p^(a)=(1/2,0,1/2,0)^ T, p^(b)=(0,1/2,1/2,0)^ T. Both have coarse prior (1/2,1/2)^ T. A procedure given only that two-entry","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a123f2e4f0737041","tex":"P^{(a)}_0","aliases":"P^(a)_0","display":"P^{(a)}_0","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74cec6b6f4a78905","tex":"P^{(a)}_{N-1}","aliases":"P^(a)_N-1","display":"P^{(a)}_{N-1}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78d0cffbd1e1aa1e","tex":"p^{(b)}","aliases":"p^(b)","display":"p^{(b)}","kind":"use","excerpt":"vpage10Two priors that fool the coarse modelSUFFICIENCY / A COMPUTABLE FAILURE WITNESSChapter VI, Part I: Theorem 8.2 and the one-packet exception. Keep the same likelihood λ=(4,1,1,1)^ T and partition A={0,1}, B={2,3}. Compare p^(a)=(1/2,0,1/2,0)^ T, p^(b)=(0,1/2,1/2,0)^ T. Both have coarse prior (1/2,1/2)^ T. A procedure given only that two-entry","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6342e3c90681bbc","tex":"P^{(i)}_{a_i}","aliases":"P^(i)_a_i","display":"P^{(i)}_{a_i}","kind":"use","excerpt":"Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a44c5f7cde75a0d","tex":"P^{(i)}_j","aliases":"P^(i)_j","display":"P^{(i)}_j","kind":"use","excerpt":"Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e43551150771079","tex":"p^{(L+a)r}","aliases":"p^(L+a)r","display":"p^{(L+a)r}","kind":"use","excerpt":"by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Equivalently, Δ_H_a^ ra_n ≡0 pmod","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c758c2cf6cd36037","tex":"P^{(m)}_{j_m}","aliases":"P^(m)_j_m","display":"P^{(m)}_{j_m}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97b363485b44daac","tex":"p^{(N-1)/2}","aliases":"p^(N-1)/2","display":"p^{(N-1)/2}","kind":"use","excerpt":"The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16a4b472d59a692e","tex":"P^*","aliases":"P^*","display":"P^*","kind":"use","excerpt":"Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environmen","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03ce65729ffc1fad","tex":"p^{-(L+n)}","aliases":"p^-(L+n)","display":"p^{-(L+n)}","kind":"use","excerpt":"Cross-level compatibility bounds the supremum distance between later terms by p^-(L+n). Completeness gives a limit, and closed norm bounds retain the congruence to every z_n. View the coefficients of the original integer polynomials in Z_p. Their evaluations satisfy both F_i(z_n)→ F_i(a), by continuity, and F_i(z_n)→0, by their growing divisibility","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e57276c69cbfb04f","tex":"p^{-1}","aliases":"p^-1","display":"p^{-1}","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-241566f7741ad3f8","tex":"p^{-2r+\\vp((2r)!)}","aliases":"p^-2r+vp((2r)!) p^-2r+v_p((2r)!) vp","display":"p^{-2r+{v_p}((2r)!)}","kind":"use","excerpt":"Every retained base is a unit, and its (p-1)st power lies in 1+p Z_p. The polynomial p^2rbinom(p-1)x+j2r has Gauss norm at most p^-2r+v_p((2r)!). This tends to zero because p is odd. The remaining factors have bounded Gauss norm. Thus the series converges in the Tate algebra. At a nonnegative integer x, the binomial factor vanishes for large r, and","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da8fc4cfd30eec05","tex":"p^{-k}","aliases":"p^-k","display":"p^{-k}","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand i","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-acf97391a5aa4604","tex":"p^{-s}","aliases":"p^-s","display":"p^{-s}","kind":"use","excerpt":"The earlier coefficient criterion is recovered exactly. Define the literal coefficient defect (δ a)(d)=a(d+p^2)-a(d+p). The new source proves, for every s,d, a_B^s f(d)=p^-s(δ^s a_f)(d). For an integral coefficient x, it proves the bridge p^-sx∈ Z_p Longleftrightarrow p^smid x in Z_p. Thus the differential definition is equivalent to the existing c","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bab906fff5aa2ce","tex":"\\mathbb P^1","aliases":"P^1 mathbb","display":"\\mathbb P^1","kind":"use","excerpt":"[Polyphase M obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3830d3e65aa149b6","tex":"p^{2(n-q)}","aliases":"p^2(n-q)","display":"p^{2(n-q)}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f1812809381e70c","tex":"p^{2(r-q)}","aliases":"p^2(r-q)","display":"p^{2(r-q)}","kind":"use","excerpt":"The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r, 0, · q>r. The row-action equality, the diagonal identity U_r,r=1, and vanishing above the diagonal are all checked. ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4173c4ab8bf03143","tex":"p^{2a(M+1)}","aliases":"p^2a(M+1)","display":"p^{2a(M+1)}","kind":"use","excerpt":"[Odd modulus truncation] For every M≥0, every positive odd q, and every even n, E_n ≡ ∑_r=0^min(M,n/2) (-1)^rZ_2r+1 binom n2rq^2rS_n-2r(q) pmod q^2M+2. If pmid q and a=v_p(q), the same truncation is valid modulo p^2a(M+1).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7d9778b6daa534f","tex":"p^{2D}","aliases":"p^2D","display":"p^{2D}","kind":"use","excerpt":"For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80a8a6cec8adfc09","tex":"p^{2L}","aliases":"p^2L","display":"p^{2L}","kind":"use","excerpt":"Indeed, each term involving at least two correction factors is divisible by p^2L, and 2L≥ L+1. Constant and linear correction terms give the two displayed summands. Adding the monomial expansions proves (1) for every polynomial, with no denominator or analytic convergence issue.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7290eb9150ccad2b","tex":"p^{2M+2}","aliases":"p^2M+2","display":"p^{2M+2}","kind":"use","excerpt":"begingroupsmallraggedright Formal correspondence. leanFiltered.filtered_triangular_reconstruction and leanFiltered.padic_filtered_reconstruction. For weights 2r, retaining r≤ M gives the bound p^2M+2 through leanFiltered.even_weight_tail. A sharper kernel weight can improve this bound, as in the classical dyadic reconstruction later in the chapter.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02d4ba2fdf9c1939","tex":"p^{2M-3s}","aliases":"p^2M-3s","display":"p^{2M-3s}","kind":"use","excerpt":"[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e64e0fdf47880c65","tex":"p^{2r}","aliases":"p^2r","display":"p^{2r}","kind":"use","excerpt":"0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd, β_r2^2r(2r)!, · p=2.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5b48fc7079a8109","tex":"p^{2s+1}","aliases":"p^2s+1","display":"p^{2s+1}","kind":"use","excerpt":"Precondition the polynomial system by A and set c=p^s+1. Its derivative at a is AJ=p^sI_d. Lemma~[source reference] gives a polynomial vector Q with Q(0)=0 and (AF)(a+cu)=(AF)(a)+p^2s+1u+p^2s+2Q(u). Since A is integral, (AF)(a)∈ p^rR^d. The strict integer inequality r>2s permits k=r-(2s+1)≥0 and a vector C∈ p^kR^d such that (AF)(a)=p^2s+1C. The nor","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-13f82bcd28f67ad7","tex":"p^{2s+2}","aliases":"p^2s+2","display":"p^{2s+2}","kind":"use","excerpt":"Precondition the polynomial system by A and set c=p^s+1. Its derivative at a is AJ=p^sI_d. Lemma~[source reference] gives a polynomial vector Q with Q(0)=0 and (AF)(a+cu)=(AF)(a)+p^2s+1u+p^2s+2Q(u). Since A is integral, (AF)(a)∈ p^rR^d. The strict integer inequality r>2s permits k=r-(2s+1)≥0 and a vector C∈ p^kR^d such that (AF)(a)=p^2s+1C. The nor","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ecd88a663ba55fc6","tex":"p^{[r]}","aliases":"p^[r]","display":"p^{[r]}","kind":"use","excerpt":"To isolate the filtration argument, let p be any element of a commutative ring and let w: N→ N∪{∞}. Set p^[r]=p^r for finite r and p^[∞]=0. The latter notation means the coefficient vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59f14f096f9bc565","tex":"p^{[w(r)]}","aliases":"p^[w(r)]","display":"p^{[w(r)]}","kind":"use","excerpt":"[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6f5fd37a9b086ab","tex":"p^{[w]}","aliases":"p^[w]","display":"p^{[w]}","kind":"use","excerpt":"ach earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral. pageheading3. Filtered tails and the standard windowSource items 2.2 and 8.2 Filtration at a finite coefficient row. Let R be any commutative ring. For w∈ N∪{∞} define","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb1f5c7992523364","tex":"p^{[\\infty]}","aliases":"p^[∞] infty","display":"p^{[\\infty]}","kind":"use","excerpt":"To isolate the filtration argument, let p be any element of a commutative ring and let w: N→ N∪{∞}. Set p^[r]=p^r for finite r and p^[∞]=0. The latter notation means the coefficient vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3667a830d02688c4","tex":"P^a","aliases":"P^a","display":"P^a","kind":"use","excerpt":"Put P=rad(A), U^a(A)=1+P^a for atextgreater=1, and m=N/2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca74460520851a88","tex":"p^{a+b}","aliases":"p^a+b","display":"p^{a+b}","kind":"use","excerpt":"The cosets s+p^a Z_p form a disjoint partition of Z_p. The first formula is the inverse affine change of variables on one coset. The second follows from p^a(p^bx+t)+s = p^a+bx+p^at+s.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6dd08fc29eea38a","tex":"P^a_i","aliases":"P^a_i","display":"P^a_i","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c8bb8a1c16cf6d8","tex":"P^a_j","aliases":"P^a_j","display":"P^a_j","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30f8aa50194b988f","tex":"P^b","aliases":"P^b","display":"P^b","kind":"use","excerpt":"ficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f4968674b99b5fb","tex":"p^b","aliases":"p^b","display":"p^b","kind":"use","excerpt":"The cosets s+p^a Z_p form a disjoint partition of Z_p. The first formula is the inverse affine change of variables on one coset. The second follows from p^a(p^bx+t)+s = p^a+bx+p^at+s.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8bd8bb8d750c7558","tex":"P^{b+v_p(b!)}","aliases":"P^b+v_p(b!)","display":"P^{b+v_p(b!)}","kind":"use","excerpt":"ficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-856313dc0118a734","tex":"p^D","aliases":"p^D","display":"p^D","kind":"use","excerpt":"For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f712839d32312f3","tex":"p^{d-2}","aliases":"p^d-2","display":"p^{d-2}","kind":"use","excerpt":"For the inductive field step d≥2, put n=p^d-1q_d and R=p^d-2q_d-1. The derived matrix is the recursively constructed maximal-field parameter, with k_0=-q_1. The inequalities n>pR>2R imply -2R>-n-q_1. Bushnell--Kutzko Remark 2.2.7 therefore applies to the actual displayed matrix. It supplies its maximal field, normalization, and critical exponent. V","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-993740a96e2e3723","tex":"p^{d-d}","aliases":"p^d-d","display":"p^{d-d}","kind":"use","excerpt":"For square uniqueness, let a,b∈ B_L(x) be arbitrary exact roots. Their canonical integer truncations are genuine roots in the respective finite residue rings, since evaluation commutes with reduction. They agree at depth L. If they agree at depth N≥ L, both next residues belong to the same lift set, which has cardinality p^d-d=1. Induction gives ag","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9aa353231261c8d","tex":"p^{d-e}","aliases":"p^d-e","display":"p^{d-e}","kind":"use","excerpt":"[Completion and smooth branching] In Theorem~[source reference], let q=p be prime. The tower can be chosen with a limit a∈ Z_p^d such that F(a)=0, z_n⟶ a, a-z_n∈ p^L+n Z_p^d (n≥0). At each stage, the number of distinct next-level residue roots over z_n is exactly p^d-e, and the finite obstruction module has one element. If d=e, there is exactly one","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c1a386e0dd4490b6","tex":"p^{d-j}","aliases":"p^d-j","display":"p^{d-j}","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ac3eb320a31908f","tex":"p^{d-\\operatorname{rank}J}","aliases":"p^d-rankJ operatorname","display":"p^{d-\\operatorname{rank}J}","kind":"use","excerpt":"Theorem. For this fixed polynomial presentation, the group and distinguished class are independent of the chosen integer representative. The class vanishes if and only if bar x lifts to a solution modulo p^L+1. The obstruction group has p^r-rankJ elements. When lifts exist, their set is a torsor for ker J and contains exactly p^d-rankJ elements.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5dd0b282af35fa57","tex":"p^{d-\\rho}","aliases":"p^d-ρ rho","display":"p^{d-\\rho}","kind":"use","excerpt":"[Exact number of linear children] If ρ=rank_ F_poverline M, then a solvable depth L class has exactly p^d-ρ children modulo p^L+1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2810b88572ae96bd","tex":"p^{d-\\rho_x}","aliases":"p^d-ρ_x rho","display":"p^{d-\\rho_x}","kind":"use","excerpt":"[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0, 0, · o_x≠0, # O_x=p^e-ρ_x.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06c5d845bc4788fe","tex":"p^{e-\\rho}","aliases":"p^e-ρ rho","display":"p^{e-\\rho}","kind":"use","excerpt":"When q=p is prime, put ρ=rank_ F_pJ. Then # L_v= p^m-ρ, · o_x(v)=0, 0, · o_x(v)≠0, # C_x=p^e-ρ. The count is of distinct actual residue classes at level L+1. The solved dimension m, rather than the total number m+k of variables, appears in the exponent because the parameter digit is fixed. These rank formulas are asserted only over prime fields.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b6e4666d99160cc","tex":"p^{e-\\rho_x}","aliases":"p^e-ρ_x rho","display":"p^{e-\\rho_x}","kind":"use","excerpt":"[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0, 0, · o_x≠0, # O_x=p^e-ρ_x.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3157e31c12869d3e","tex":"p^{e_p}","aliases":"p^e_p","display":"p^{e_p}","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97bae72ced80959b","tex":"p^i","aliases":"p^i","display":"p^i","kind":"use","excerpt":"To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar mult","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-44ddf871c8e79a19","tex":"p^j","aliases":"p^j","display":"p^j","kind":"use","excerpt":"The full-character argument uses a specific order of operations. First construct a full character at native rank p^j, trivial on U_F^1. Transfer its full beta-determinant unit character to multiplicity p, using BK 3.6.13(ii) and 3.6.14. On a scalar u∈ U_F^1, this gives θ_j(u^p)=1. Next add the actual trace-zero correction, using BK 3.3.18. It remai","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7f360bb2799a8d5","tex":"p^k","aliases":"p^k","display":"p^k","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand i","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ad7e5363781b727","tex":"p^{k(1+v_p(u))}","aliases":"p^k(1+v_p(u))","display":"p^{k(1+v_p(u))}","kind":"use","excerpt":"To incorporate v_2(u), expand a displacement using finite Newton coefficients. For any prime p, u>0, and 1≤ a≤ u, p^v_p(u)+1midbinom ua p^a, as follows from abinom ua=ubinomu-1a-1 and v_p(a)<a. Repeated displacement, retaining the remaining base-difference order at each induction step, gives a factor p^k(1+v_p(u)) from k displacements. This inducti","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-231af438231aa3c5","tex":"p^{k+\\sum_\\nu v_p(H_\\nu)}","aliases":"p^k+∑_ν v_p(H_ν) sum nu nu","display":"p^{k+\\sum_\\nu v_p(H_\\nu)}","kind":"use","excerpt":"ng Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar multiplied by p^M, and is divisible by p^k+∑_ν v_p(H_ν). This proves ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1cfaa611e7831efc","tex":"p^{k-1}","aliases":"p^k-1","display":"p^{k-1}","kind":"use","excerpt":"c branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors al","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ba0888d650823c4","tex":"p^{ks}","aliases":"p^ks","display":"p^{ks}","kind":"use","excerpt":"-field reductionTheorem 7.4; manuscript page 18 For u in any commutative ring, and every natural step h, iteration proves Δ_h^s(u^n)=u^n(u^h-1)^s. The proof unfolds the actual forward-difference operator and uses the power multiplication law. No valuation premise or geometric-difference identity is assumed. More generally, a proved divisibility p^k","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8ec6caedf71e9a2","tex":"p^{L(R+1-d)}","aliases":"p^L(R+1-d)","display":"p^{L(R+1-d)}","kind":"use","excerpt":"f degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) uniformly on Z_p.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48e2844ef33a2852","tex":"p^{L+1}","aliases":"p^L+1","display":"p^{L+1}","kind":"use","excerpt":"[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09c8b25bb1d743c8","tex":"p^{L+a}","aliases":"p^L+a","display":"p^{L+a}","kind":"use","excerpt":"r Editorial hypothesis correction. Require L≥1. At L=0 the standalone claim is false: for p=3, d=1 and X=2=1+B, one has X^3-1=7∉3 Z_3. With L≥1, every higher binomial term in the retained proof has the required valuation, including p=2. If X=I+p^LB with B∈ M_d( Z_p), then for every a≥0 there is B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d85c672622d89ea4","tex":"p^{L+n}","aliases":"p^L+n","display":"p^{L+n}","kind":"use","excerpt":"[Completion and smooth branching] In Theorem~[source reference], let q=p be prime. The tower can be chosen with a limit a∈ Z_p^d such that F(a)=0, z_n⟶ a, a-z_n∈ p^L+n Z_p^d (n≥0). At each stage, the number of distinct next-level residue roots over z_n is exactly p^d-e, and the finite obstruction module has one element. If d=e, there is exactly one","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5e33ab8d0aedef0","tex":"p^{L+n+1}","aliases":"p^L+n+1","display":"p^{L+n+1}","kind":"use","excerpt":"Surjectivity implies rank_ F_pA_x=e, e≤ d. Combining this with the preceding exact residue-lift count gives # L_z_n,L+n=p^d-e for every n. These are distinct solutions modulo p^L+n+1 lying above the selected point modulo p^L+n. The original finite obstruction group at each smooth point has cardinality one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-944126e1f1284f31","tex":"p^{L-1}","aliases":"p^L-1","display":"p^{L-1}","kind":"use","excerpt":"For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ad627b635cdde1b","tex":"p^{L_p}","aliases":"p^L_p","display":"p^{L_p}","kind":"use","excerpt":"N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-13b34b1568997397","tex":"p^{\\lfloor d/p\\rfloor}","aliases":"p^lfloor d/prfloor lfloor rfloor","display":null,"kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ed6db9428239d9e","tex":"p^{Lr}","aliases":"p^Lr","display":"p^{Lr}","kind":"use","excerpt":"means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ef9147e2bbff9d5","tex":"P^m","aliases":"P^m","display":"P^m","kind":"use","excerpt":"at moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e851b902cf92a681","tex":"p^M","aliases":"p^M","display":"p^M","kind":"use","excerpt":"[Smooth coefficient stability] Let p be prime, M≥1, F,H∈ Z[X_1,…,X_d]^e, and c∈ p^M Z. Suppose DF(x)bmod p is surjective for an integer x∈ Z^d. If a∈ Z_p^d satisfies F(a)=0 and a≡ xpmod p, then there is a root b of G=F+cH with b-a∈ p^M Z_p^d.","count":31},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7441b5cfbaa2c0c0","tex":"p^m","aliases":"p^m","display":"p^m","kind":"use","excerpt":"To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar mult","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e7c6808c9432177e","tex":"p^{M-(2s+1)}","aliases":"p^M-(2s+1)","display":"p^{M-(2s+1)}","kind":"use","excerpt":"Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4927a5c0083e2327","tex":"p^{m-1}","aliases":"p^m-1","display":"p^{m-1}","kind":"use","excerpt":"The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79b8bb76c63fb047","tex":"p^{m-e}","aliases":"p^m-e","display":"p^{m-e}","kind":"use","excerpt":"Theorem~[source reference]. For every τ∈ t_0+p^L Z_p^k, there are an integer tower z_n and b∈ Z_p^m+k such that F(b)=0, free(b)=τ, z_n⟶ b, b-z_n∈ p^L+n Z_p^m+k. The initial value is z_0=x. At every stage and for every proposed parameter digit v∈ F_p^k, the corresponding next residue fiber has exactly p^m-e elements. If m=e, then for each τ there is","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66d21ad971fd5eb9","tex":"p^{M-s}","aliases":"p^M-s","display":"p^{M-s}","kind":"use","excerpt":"[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-516983ba38151684","tex":"p^{M-s-1}","aliases":"p^M-s-1","display":"p^{M-s-1}","kind":"use","excerpt":"Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b12f28620c9cf18","tex":"p^{m-\\rho}","aliases":"p^m-ρ rho","display":"p^{m-\\rho}","kind":"use","excerpt":"When q=p is prime, put ρ=rank_ F_pJ. Then # L_v= p^m-ρ, · o_x(v)=0, 0, · o_x(v)≠0, # C_x=p^e-ρ. The count is of distinct actual residue classes at level L+1. The solved dimension m, rather than the total number m+k of variables, appears in the exponent because the parameter digit is fixed. These rank formulas are asserted only over prime fields.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aedad308698af110","tex":"p^{m-\\rho_x}","aliases":"p^m-ρ_x rho","display":"p^{m-\\rho_x}","kind":"use","excerpt":"[Prescribed-parameter obstruction and count] Define o_x(v)=[overliner_x+K_xv]∈ S_q^e/im J_x. Then L_x,L,v is nonempty if and only if o_x(v)=0. When nonempty, it is a torsor for ker J_x. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L,v =p^m-ρ_x, · o_x(v)=0, 0, · o_x(v)≠0, #( F_p^e/im J_x)=p^e-ρ_x. The criterion holds uniformly for every prescrib","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3455be3fc985410","tex":"p^{\\min(k(1+\\nu_p(u)),mn)}","aliases":"p^min(k(1+ν_p(u)),mn) min nu","display":"p^{\\min(k(1+\\nu_p(u)),mn)}","kind":"use","excerpt":"[Step-sensitive divisor-branch bound] Under the hypotheses of Theorem~[source reference], for all u,k,n∈ N, Δ_u(p-1)/m^kU_mp,j(n) ∈ p^min(k(1+ν_p(u)),mn) Z_p. For C_mp,j the corresponding valuation lower bound is min(k(1+ν_p(u)),mn)-v_p(j!).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07593e8256373d1f","tex":"p^{\\min(k,r)}","aliases":"p^min(k,r) min","display":"p^{\\min(k,r)}","kind":"use","excerpt":"[Prime-power period from finite differences] Let R be a commutative ring, p a prime, f: N→ R, h,n∈ N, and r≥1. If p^min(k,r)midΔ_h^k f(n) for every k∈ N, then p^rmid f(n+p^r-1h)-f(n).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa1e9c58c08b71c9","tex":"p^{\\min(s,L)}","aliases":"p^min(s,L) min","display":"p^{\\min(s,L)}","kind":"use","excerpt":"Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fiber has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc7fd452679f9cf5","tex":"p^{\\min(s_i,L)}","aliases":"p^min(s_i,L) min","display":"p^{\\min(s_i,L)}","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0c8f9332b23f7da","tex":"p^{mn}","aliases":"p^mn","display":"p^{mn}","kind":"use","excerpt":"ly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6c9282569aea3fa","tex":"p^N","aliases":"p^N","display":"p^N","kind":"use","excerpt":"t, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is the forward map, then for every N≥0, h(t)≡ h(u)pmodp^N Longleftrightarrow t≡ up","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3cdc9ecc001d6c4","tex":"p^n","aliases":"p^n","display":"p^n","kind":"use","excerpt":"For prime p, rank-nullity gives dimker J_x=d-ρ_x and dim( F_p^e/im J_x)=e-ρ_x. The cardinality of an n-dimensional F_p-space is p^n. Combine these identities with the obstruction criterion and the residue equivalence.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-01d645aa75670391","tex":"p^{N+1}","aliases":"p^N+1","display":"p^{N+1}","kind":"use","excerpt":"First consider a square integer polynomial map P whose reduced Jacobian at x is surjective. Its Jacobian is invertible over F_p. A digit comparison using Taylor expansion shows that equal outputs modulo p^N+1 force equal inputs modulo p^N+1 whenever the inputs already agree modulo p^N and belong to B_1(x). Induction and integer truncation give, for","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c020cb0886702b18","tex":"p^{n+1}","aliases":"p^n+1","display":"p^{n+1}","kind":"use","excerpt":"To obtain the sharper displacement, begin with u∈ R^d. If u∈ p^nR^d, then Q(0)=0 and polynomial congruence give Q(u)∈ p^nR^d. For n<k, both C and pQ(u) are divisible by p^n+1. Equation~[source reference] therefore improves the divisibility of u by one digit. Induction gives u∈ p^kR^d. Thus b=a+p^s+1u satisfies [source reference], because s+1+k=r-s.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f0efa7048440392","tex":"p^{n_1}","aliases":"p^n_1","display":"p^{n_1}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-47c937c3c2303c97","tex":"p^{n_i}","aliases":"p^n_i","display":"p^{n_i}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eae80e166a46cca8","tex":"p^{n_r}","aliases":"p^n_r","display":"p^{n_r}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-035c508864e2d38f","tex":"p^p","aliases":"p^p","display":"p^p","kind":"use","excerpt":"For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2adb92d07b292296","tex":"p^q","aliases":"p^q","display":"p^q","kind":"use","excerpt":"The scalar input is proved for every natural b, including nonunits: p^2mid b^p^2-b^p in Z. Because Θ(1+T)^b=b(1+T)^b, the binomial expansion of T^p=((1+T)-1)^p then proves stability of the divided generator T^p/p. For d=pq+r, 0≤ r<p, every weighted monomial has the exact factorization a_dT^d=(p^qa_d)(fracT^pp)^qT^r. The scalar p^qa_d is integral by","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1bbb4a842cfd7d98","tex":"P^r","aliases":"P^r","display":"P^r","kind":"use","excerpt":"B-bimodularity gives s(Lift_E(C))=C. The lift preserves exact valuation: u_0 (-1) is an E-unit, and expansion in 1,z gives coefficient valuations distinct modulo 2. Each coefficient entry has the same grade in the interleaved L-chain and the E-chain. In particular s(P^r)=Q^r.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f0ea21bbfbf69fb","tex":"p^r","aliases":"p^r","display":"p^r","kind":"use","excerpt":"Principal results. addcontentslinetocsectionPrincipal results The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integral and E_mpn+qp^r^(mp,","count":25},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da8a2616906ad75b","tex":"p^{r-1}","aliases":"p^r-1","display":"p^{r-1}","kind":"use","excerpt":"[Prime-power period from finite differences] Let R be a commutative ring, p a prime, f: N→ R, h,n∈ N, and r≥1. If p^min(k,r)midΔ_h^k f(n) for every k∈ N, then p^rmid f(n+p^r-1h)-f(n).","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07c89112aaa8fc8b","tex":"p^{r-\\operatorname{rank}J}","aliases":"p^r-rankJ operatorname","display":"p^{r-\\operatorname{rank}J}","kind":"use","excerpt":"Theorem. For this fixed polynomial presentation, the group and distinguished class are independent of the chosen integer representative. The class vanishes if and only if bar x lifts to a solution modulo p^L+1. The obstruction group has p^r-rankJ elements. When lifts exist, their set is a torsor for ker J and contains exactly p^d-rankJ elements.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cc2ee50888afb46","tex":"p^{r-s}","aliases":"p^r-s","display":"p^{r-s}","kind":"use","excerpt":"hted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61ad220e94c8aa08","tex":"p^{r-s+1}","aliases":"p^r-s+1","display":"p^{r-s+1}","kind":"use","excerpt":"and displacement boundary] For p=2, F(X)=X^2+3, and a=1, one has F(a)=4 and DF(a)=2. Thus s=1 and r=2 satisfy every displayed hypothesis except the strict inequality. No root exists in Z_2, since none exists modulo eight. Hence r≥2s cannot replace r>2s uniformly. Also, F(X)=p^sX+p^r has root -p^r-s, whose displacement is not divisible by p^r-s+1. T","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-360f794da8094012","tex":"p^s","aliases":"p^s","display":"p^s","kind":"use","excerpt":"Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fiber has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global ","count":43},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4e5526bd8628ebc","tex":"p^{s+1}","aliases":"p^s+1","display":"p^{s+1}","kind":"use","excerpt":"[Weighted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62f2748157764c24","tex":"p^{s+1+k}","aliases":"p^s+1+k","display":"p^{s+1+k}","kind":"use","excerpt":"For completeness, the sharp precision is additional to this fixed-point existence. Inductively, if u∈ p^nR^d, then polynomial congruence and Q(0)=0 imply Q(u)∈ p^nR^d. For n<k, the equation u=-C-pQ(u) therefore gives u∈ p^n+1R^d. Starting at n=0, this proves u∈ p^kR^d. Hence b=a+tu satisfies b-a∈ p^s+1+kR^d=p^r-sR^d. The normalization and injectivi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e79435c578c898aa","tex":"p^{s_1}","aliases":"p^s_1","display":"p^{s_1}","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5724748474faba5","tex":"p^{s_d}","aliases":"p^s_d","display":"p^{s_d}","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5bff95dd6769dfd","tex":"p^{s_i}","aliases":"p^s_i","display":"p^{s_i}","kind":"use","excerpt":"If a Smith form is rank deficient, a literal zero diagonal factor requires the corresponding target coordinate to vanish modulo p^L and then contributes all p^L possible coordinate values. In the concrete quotient it can equivalently be represented by any weight s_i≥ L, since p^s_i=0 in Z/(p^L) Z. The theorem then gives exactly that condition and f","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4cb38d7eaa8bc03","tex":"P^v_i","aliases":"P^v_i","display":"P^v_i","kind":"use","excerpt":"Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate pro","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5b6d2f623a6916e","tex":"p^{v_p(\\lfloor d/p\\rfloor!)}","aliases":"p^v_p(lfloor d/prfloor!) lfloor rfloor","display":null,"kind":"use","excerpt":"1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ca86a25a0110cc3","tex":"P^{v_p(q!)}","aliases":"P^v_p(q!)","display":"P^{v_p(q!)}","kind":"use","excerpt":"p). Expanding ((z-1)+1)^p=1 and using divisibility of every interior binomial coefficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c345ea0f18269172","tex":"p^{v_p(q!)}","aliases":"p^v_p(q!)","display":"p^{v_p(q!)}","kind":"use","excerpt":"sses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8f0c43926925108","tex":"p^{v_p(u)+1}","aliases":"p^v_p(u)+1","display":"p^{v_p(u)+1}","kind":"use","excerpt":"To incorporate v_2(u), expand a displacement using finite Newton coefficients. For any prime p, u>0, and 1≤ a≤ u, p^v_p(u)+1midbinom ua p^a, as follows from abinom ua=ubinomu-1a-1 and v_p(a)<a. Repeated displacement, retaining the remaining base-difference order at each induction step, gives a factor p^k(1+v_p(u)) from k displacements. This inducti","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95c96988a1a8fc97","tex":"p^w","aliases":"p^w","display":"p^w","kind":"use","excerpt":"ach earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral. pageheading3. Filtered tails and the standard windowSource items 2.2 and 8.2 Filtration at a finite coefficient row. Let R be any commutative ring. For w∈ N∪{∞} define","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95797c3dbf3035d3","tex":"p^{w(r)}","aliases":"p^w(r)","display":"p^{w(r)}","kind":"use","excerpt":"[Filtered triangular reconstruction] Let p be prime. Suppose A_n = ∑_r=0^n binom nr c_rp^w(r)S_n-r, where c_r∈ Z_p, S_n∈ Z_p, and w(r)∈ N. For every precision L≥1, A_n ≡ ∑_substack0≤ r≤ n","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20094bff87ea1f75","tex":"p^\\ell","aliases":"p^ℓ ell","display":"p^\\ell","kind":"use","excerpt":"The specification matters. Rational coefficients are not reduced through an imaginary map Q→ F_p. The proof first constructs their integral lifts in Z_p and establishes equality in Q_p. Root representatives are actual units at the requested precision, not unchanged residues that need not satisfy the root equation modulo p^ℓ.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f150917365c9af4d","tex":"p^{\\ell-1}","aliases":"p^ℓ-1 ell","display":"p^{\\ell-1}","kind":"use","excerpt":"ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4bf4d68504d5d5b8","tex":"p^{\\ell-k}","aliases":"p^ℓ-k ell","display":"p^{\\ell-k}","kind":"use","excerpt":"c branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors al","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-209584f848a592aa","tex":"p^{\\sum_{i=1}^d\\min(s_i,L)}","aliases":"p^∑_i=1^dmin(s_i,L) sum min","display":"p^{\\sum_{i=1}^d\\min(s_i,L)}","kind":"use","excerpt":"ongruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2f259e45c66f6ef","tex":"P_{\\mathbf j}^{\\mathrm{fine}}","aliases":"P_ j^fine mathbf mathrm","display":"P_{\\mathbf j}^{\\mathrm{fine}}","kind":"use","excerpt":"Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3439312706e15bf","tex":"P_{\\bm j}^{\\rm fine}","aliases":"P_ j^rm fine bm rm","display":"P_{\\bm j}^{\\rm fine}","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f536203d4a8b83a","tex":"P_{\\mathcal Q}","aliases":"P_ Q mathcal","display":"P_{\\mathcal Q}","kind":"use","excerpt":"Let Ω={1,…,n} carry strictly positive probabilities π_i, and let Q={B_1,…,B_m} be a partition. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j∑_j∈ B(i)π_j, where B(i) is the block containing i.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04e3dc4cda092659","tex":"P_{\\mathcal Q}^2","aliases":"P_ Q^2 mathcal","display":"P_{\\mathcal Q}^2","kind":"use","excerpt":"[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the ","count":37},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cfbc31b0e1b74bde","tex":"P_{000}","aliases":"P_000","display":"P_{000}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5750c25072ea9ac4","tex":"P_{011}","aliases":"P_011","display":"P_{011}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c641cd3ff43bd1cb","tex":"p_0^T","aliases":"p_0^T","display":"p_0^T","kind":"use","excerpt":"Average [source reference] with terminal function f= 1 against the initial distribution. Transposition gives p_0^T e^t(Q^T-D_V) 1 = 1^T e^t(Q-D_V)p_0. Since 1^TQ=0, differentiation gives S_V'(t) = - 1^TD_V e^tAp_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[title=R3: an exact but oblique packet split] On X=𝕜^2, set P_1=1 · -1 0 · 0, P_2=0 · 1 0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.","count":23},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14472c28742b9379","tex":"P_{1,n}","aliases":"P_1,n","display":"P_{1,n}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3778c22278196c4","tex":"P_{101}","aliases":"P_101","display":"P_{101}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f3a6ef15dffa349","tex":"P_{110}","aliases":"P_110","display":"P_{110}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c9e7d74779bb5eb","tex":"P_{12,7}","aliases":"P_12,7","display":"P_{12,7}","kind":"use","excerpt":"[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-586c9b3f425a7bf8","tex":"P_{12,k}","aliases":"P_12,k","display":"P_{12,k}","kind":"use","excerpt":"FACTORIZATION / TWELVE ADDRESSESChapter V: Proposition 14.1. At phase order 12=4·3, the map k⟼(kbmod4, kbmod3) is a bijection from twelve residue addresses to the 4×3 grid. Every cell below contains its unique original address. vfigcrt_refinement headThe factorization is an intersection of conditions For integer sample addresses, including negative","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"[title=R3: an exact but oblique packet split] On X=𝕜^2, set P_1=1 · -1 0 · 0, P_2=0 · 1 0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the ","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a679a976dd7459e","tex":"P_{2,1}","aliases":"P_2,1","display":"P_{2,1}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40a2a894b872c666","tex":"P_{2,2}","aliases":"P_2,2","display":"P_{2,2}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-587fb000e9b974ca","tex":"P_{2,3}","aliases":"P_2,3","display":"P_{2,3}","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3ada48942b51047","tex":"P_{2,n}","aliases":"P_2,n","display":"P_{2,n}","kind":"use","excerpt":"[Secant asymptotic] For N=2 and r=1, P_2,n(1) ∼ 2(2n)!(frac2π)^2n+1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9553d695f708fd97","tex":"P_{22}^2","aliases":"P_22^2","display":"P_{22}^2","kind":"use","excerpt":"0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c29786ad28dc8278","tex":"P_{2^a,\\,r\\bmod2^a}","aliases":"P_2^a, rbmod2^a bmod","display":"P_{2^a,\\,r\\bmod2^a}","kind":"use","excerpt":"0, · otherwise, take the canonical label 0≤ r<N. The logical equivalence yields the actual equality of sequence operators P_N,rf= P_2^a, rbmod2^a(P_M, rbmod Mf). This statement holds for coefficients in any type with zero. A separate theorem lifts the equality coefficientwise to the existing formal power-series selectors over any semiring.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8501ee74c70e6c9","tex":"P_{2^a,r\\bmod 2^a}","aliases":"P_2^a,rbmod 2^a bmod","display":"P_{2^a,r\\bmod 2^a}","kind":"use","excerpt":"[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-359606f4da41ffb7","tex":"P_{2^aM,r}","aliases":"P_2^aM,r","display":"P_{2^aM,r}","kind":"use","excerpt":"[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0979bcddbdbf9b2b","tex":"P_{2^s,j_2}","aliases":"P_2^s,j_2","display":"P_{2^s,j_2}","kind":"use","excerpt":"[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0242a419961441b","tex":"P_3","aliases":"P_3","display":"P_3","kind":"use","excerpt":"headReconstruction retains every block The identity P_1+P_2+P_3=I gives T=∑_i,j=1^3 P_iTP_j. Selecting only the diagonal would change the operator. The entries 2,3,4 carry transfers between different packets.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8699f42976e6525","tex":"P_{3,0}","aliases":"P_3,0","display":"P_{3,0}","kind":"use","excerpt":")·(P_N,ky). Here the dot means multiplication at the same sample, not convolution. headCauchy multiplication adds degrees For formal series A(t),B(t), the coefficient of t^n is a sum over a+b=n. Consequently the output residue is i+jpmod N: P_N,k(AB)=∑_i+j≡ k (N)(P_N,iA)(P_N,jB). For N=3, take A=1+2t+3t^2 and B=2+t^2. Direct multiplication gives AB","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7c6a909392369bc","tex":"P_{3,1}","aliases":"P_3,1","display":"P_{3,1}","kind":"use","excerpt":"[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f64965a0db71aca","tex":"P_{3,k\\bmod3}","aliases":"P_3,kbmod3 bmod","display":"P_{3,k\\bmod3}","kind":"use","excerpt":"FACTORIZATION / TWELVE ADDRESSESChapter V: Proposition 14.1. At phase order 12=4·3, the map k⟼(kbmod4, kbmod3) is a bijection from twelve residue addresses to the 4×3 grid. Every cell below contains its unique original address. vfigcrt_refinement headThe factorization is an intersection of conditions For integer sample addresses, including negative","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53d17384b2682f1f","tex":"P_{4,3}","aliases":"P_4,3","display":"P_{4,3}","kind":"use","excerpt":"[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c9a8f8dedb6f5c4","tex":"\\mathcal P_{4,j}","aliases":"P_4,j mathcal","display":"\\mathcal P_{4,j}","kind":"use","excerpt":"Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e478ceed7696e248","tex":"P_{4,k\\bmod4}","aliases":"P_4,kbmod4 bmod","display":"P_{4,k\\bmod4}","kind":"use","excerpt":"FACTORIZATION / TWELVE ADDRESSESChapter V: Proposition 14.1. At phase order 12=4·3, the map k⟼(kbmod4, kbmod3) is a bijection from twelve residue addresses to the 4×3 grid. Every cell below contains its unique original address. vfigcrt_refinement headThe factorization is an intersection of conditions For integer sample addresses, including negative","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72f2cd87bd5faecf","tex":"P_{4,r}","aliases":"P_4,r","display":"P_{4,r}","kind":"use","excerpt":"y cell contains an original address and its rational sample value. Shading marks the known jump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-219e101503b0300a","tex":"P_{8,2}","aliases":"P_8,2","display":"P_{8,2}","kind":"use","excerpt":"The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91b4f0e802663f94","tex":"P_{8,6}","aliases":"P_8,6","display":"P_{8,6}","kind":"use","excerpt":"The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c261d0404352ff34","tex":"p_a","aliases":"p_a","display":"p_a","kind":"use","excerpt":"Exact correction, including coherent errors. For every operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07a802f35729938f","tex":"P_{a(e)}","aliases":"P_a(e)","display":"P_{a(e)}","kind":"use","excerpt":"Packet perturbation certificates. textitSource item 8.1 and two uniform consequences Let X be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps X→ X, with their operator norm. Let a(e) identify the source packet of edge e, and suppose Δ_e=Δ_e P_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exac","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d41e39fff15e99f5","tex":"P_{A,j_A}","aliases":"P_A,j_A","display":"P_{A,j_A}","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f06a233d5d801ff2","tex":"P_{A,r_A}","aliases":"P_A,r_A","display":"P_{A,r_A}","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a7abbe02c9f0ed6a","tex":"P_a^*","aliases":"P_a^*","display":"P_a^*","kind":"use","excerpt":"The projectors P_a form a Kraus family because ∑_aP_a^*P_a=I. Thus the map is completely positive and trace preserving. It is unital for the same reason. Orthogonality and idempotence follow from [source reference]. Its range is the sum of the diagonal operator packets. This is exactly the commutant of the projector family.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72bf1da84f9b9c1d","tex":"P_a^2","aliases":"P_a^2","display":"P_a^2","kind":"use","excerpt":"Differentiating P_a^2=P_a gives dot P_aP_a+P_adot P_a=dot P_a. Summing over a and using ∑_adot P_a=0 yields ∑_aP_adot P_a=-∑_adot P_aP_a, which proves both displayed formulas for C and its skew-Hermiticity.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10c477be247f9d21","tex":"P_A^j","aliases":"P_A^j","display":"P_A^j","kind":"use","excerpt":"Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK 3.1.7--9 give the actual groups","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f7de3914f64d794","tex":"P_A^{n-r}","aliases":"P_A^n-r","display":"P_A^{n-r}","kind":"use","excerpt":"At both low ranks, K^× normalizes the new P. In rank two this follows directly from normalization of A and its radical. In rank four, a K-uniformizer is approximated by the derived chain shift R times a unit of B, with relative error in P_A^n-r. Its conjugation error on U_B^m lies in U_A^n-r+m⊂ J, which proves normalization. The exact argument is p","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3db31f834a299d9","tex":"P_{a_0}","aliases":"P_a_0","display":"P_{a_0}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2693c2c3c4636ad8","tex":"P_{a_1}","aliases":"P_a_1","display":"P_{a_1}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e071a8db9cfdfcfd","tex":"P_{a_m}","aliases":"P_a_m","display":"P_{a_m}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d53e2b9e1c09bb7d","tex":"P_{a_{m-1}}","aliases":"P_a_m-1","display":"P_{a_{m-1}}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ab4aa6ec4adb158","tex":"P_{AB,j}","aliases":"P_AB,j","display":"P_{AB,j}","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83efd7740f0afa62","tex":"P_{AB,r}","aliases":"P_AB,r","display":"P_{AB,r}","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2298f7e841a31d94","tex":"P_{B,j_B}","aliases":"P_B,j_B","display":"P_{B,j_B}","kind":"use","excerpt":"[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d04a4a035c233ca7","tex":"P_{B,r_B}","aliases":"P_B,r_B","display":"P_{B,r_B}","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb071e0da3d1f51a","tex":"P_b^2","aliases":"P_b^2","display":"P_b^2","kind":"use","excerpt":"UU^*=1 · Longrightarrow ∑_aU_ba(U_da)^*=δ_bdP_b. For the first law, expand each corner, collapse P_b^2 and sum: ∑_bP_aU^*P_bUP_c =P_aU^*(∑_bP_b)UP_c=P_aP_c. The second law follows by the analogous calculation and has its own formal proof. For complex matrices, the adjoint is the conjugate transpose. The order of the operator products is preserved t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3270e71bf1ae54c5","tex":"P_c","aliases":"P_c","display":"P_c","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe083e5d0f70cf84","tex":"P_c^2","aliases":"P_c^2","display":"P_c^2","kind":"use","excerpt":"0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14f8900152d6716a","tex":"P_d","aliases":"P_d","display":"P_d","kind":"use","excerpt":"Idempotence and completeness follow from the projector identities. For Hilbert--Schmidt orthogonality, ipP_aXP_bP_cYP_d_HS · = Tr(P_bX^*P_aP_cYP_d) · = δ_acδ_bd Tr(P_bX^*P_aYP_b). Since the packets are orthogonal and sum to the original operator, their squared norms add to its squared norm.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5ba04fa053a01d8","tex":"P_{D,\\alpha}","aliases":"P_D,α alpha","display":"P_{D,\\alpha}","kind":"use","excerpt":"0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2fd63ee4a5169ac2","tex":"P_e","aliases":"P_e","display":"P_e","kind":"use","excerpt":"Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f4c2e0977ad59ef5","tex":"P_{\\mathrm{even}}","aliases":"P_even mathrm","display":"P_{\\mathrm{even}}","kind":"use","excerpt":"For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d02aaee4231d3cd1","tex":"P_f","aliases":"P_f","display":"P_f","kind":"use","excerpt":"Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45c4f17ed76d05ef","tex":"P_g","aliases":"P_g","display":"P_g","kind":"use","excerpt":"[Characteristic polynomial certificate] Let P_g(X)=det(XI-gmid V). Then Φ_p(g,V) = (frac-1p)^r (fracP_g(1)p).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.","count":31},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b57adfbf4aaa867","tex":"\\mathcal P_i","aliases":"P_i mathcal","display":"\\mathcal P_i","kind":"use","excerpt":"Choose a labeled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36ec2f492b9a5c85","tex":"p_i'","aliases":"p_i'","display":"p_i'","kind":"use","excerpt":"tatistical meanings The first derivative of log evidence is the current mean score, and the second is its variance: fracddtlog Z(t)= E_p(t)u, fracd^2dt^2log Z(t)=Var_p(t)(u)≥0. At q=2 the exact values are E u=frac107, E u^2=frac187, Var(u)=frac2649. Thus the derivative of the mean is 26/49 at this time. The flow equation p_i'=(u_i- E u)p_i gives de","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-365569fb3de7e2a4","tex":"p_{i,a}","aliases":"p_i,a","display":"p_{i,a}","kind":"use","excerpt":"For each operator, its finite set of actual eigenvalues is used without multiplicity. The polynomial p_i,a(X)=∏_μ∈Spec(T_i)∖{a} (a-μ)^-1(X-μ) is used when a occurs, and the zero polynomial otherwise. For an eigenvector T_i v=bv, evaluation gives p_i,a(T_i)v=v · b=a,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ae74ed9c71e6f69","tex":"p_{i,a_i}","aliases":"p_i,a_i","display":"p_{i,a_i}","kind":"use","excerpt":"0 · b≠ a. Because the eigenspaces span A, this proves that e_i=p_i,a_i(T_i) maps all of A into the selected eigenspace and fixes that eigenspace pointwise. It therefore proves the individual projector and image assertions, including an absent target.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1d36be4bad0ff1b","tex":"P_{i,g}","aliases":"P_i,g","display":"P_{i,g}","kind":"use","excerpt":"(XI-ρ_m(g)) =∏_i∈ Idet(XI-ρ_i(g))^m_i, Tr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de628d850c751acf","tex":"P_{i,g}^{m_i}","aliases":"P_i,g^m_i","display":"P_{i,g}^{m_i}","kind":"use","excerpt":"eratornameTr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is the image of ∏_iP_i,g^m_i. App","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fbc48a83ee80d83f","tex":"P_i^2","aliases":"P_i^2","display":"P_i^2","kind":"use","excerpt":"0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3b69870475ee0ca","tex":"P_i^r","aliases":"P_i^r","display":"P_i^r","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"vpage8A transfer block has an input and an outputOPERATOR COORDINATES / MULTIPLICATION AS A SUMMain text: [source reference] Use the three coordinate packets of dimension one in Q^3. The block T_ij=P_iTP_j sends input packet j into output packet i. For this example, its nonzero scalar is the (i,j) matrix entry.","count":62},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"use","excerpt":"h packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are coch","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a5d03f7fbf7ca50","tex":"P_{j+1}","aliases":"P_j+1","display":"P_{j+1}","kind":"use","excerpt":"Actual noncommutative products. The error module defines the left-accumulated product recursively, with P_0=I and P_j+1=A_j+1P_j. For a perturbed sequence widehat A_j, it derives the discrepancy recursion widehat P_j+1-P_j+1 =widehat A_j+1(widehat P_j-P_j) +(widehat A_j+1-A_j+1)P_j. The identity follows from the actual products. It is not supplied ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21ea731dd7eeae7b","tex":"P_{j+q}","aliases":"P_j+q","display":"P_{j+q}","kind":"use","excerpt":"[Charge selection rule] If A has charge q, then P_kAP_j=0 unless k=j+q in Z/N Z. In fact, AP_j=P_j+qAP_j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e6b6a6054c3c5d9","tex":"P_{j,a(e)}","aliases":"P_j,a(e)","display":"P_{j,a(e)}","kind":"use","excerpt":"Packetwise mixed precision. At each stage let A_j=∑_e A_j,e, widehat A_j=∑_e(A_j,e+Δ_j,e), Δ_j,e=Δ_j,eP_j,a(e). The previous packet certificate supplies the uniform operator bound normwidehat A_j-A_j≤ δ_j:=∑_enormΔ_j,enormP_j,a(e). The new theorem substitutes these δ_j into the actual product-chain estimate. The resulting vector error is at most th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2646be694d017280","tex":"P_{j-1}","aliases":"P_j-1","display":"P_{j-1}","kind":"use","excerpt":"[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-634eafc544731bd6","tex":"P_j^{(a)}","aliases":"P_j^(a)","display":"P_j^{(a)}","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e11671ca6a219375","tex":"P_j^*","aliases":"P_j^*","display":"P_j^*","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d6d0b9c4abedaed","tex":"P_j^2","aliases":"P_j^2","display":"P_j^2","kind":"use","excerpt":"Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6d93da85dc64976","tex":"p_j^{a_j}","aliases":"p_j^a_j","display":"p_j^{a_j}","kind":"use","excerpt":"[Finite local reconstruction] Let N=∏_j=1^sp_j^a_j be the prime power factorization of a positive integer. The map Z/N Z ⟶ ∏_j=1^s Z/p_j^a_j Z is a ring isomorphism. Thus a compatible collection of prime power residues reconstructs one residue modulo N [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a8fbe464d99fbb3","tex":"P_j^r","aliases":"P_j^r","display":"P_j^r","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd9801213575c0e5","tex":"P_j^{r+1}","aliases":"P_j^r+1","display":"P_j^{r+1}","kind":"use","excerpt":"Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2abda96372323698","tex":"P_{j_1}^{(1)}","aliases":"P_j_1^(1)","display":"P_{j_1}^{(1)}","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ce9ef7655396f9f","tex":"P_{j_m}^{(m)}","aliases":"P_j_m^(m)","display":"P_{j_m}^{(m)}","kind":"use","excerpt":"Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5ca4f817be17bab","tex":"P_K","aliases":"P_K","display":"P_K","kind":"use","excerpt":"Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e8a5b53321d6af2","tex":"\\mathfrak p_K^m","aliases":"p_K^m mathfrak","display":"\\mathfrak p_K^m","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c60807df59cc9f6","tex":"\\mathfrak p_K^{m+1}","aliases":"p_K^m+1 mathfrak","display":"\\mathfrak p_K^{m+1}","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-363d67fd271736f2","tex":"P_L","aliases":"P_L","display":"P_L","kind":"use","excerpt":"ltiplicity-sensitive genus-zero product is Δ_N(w)=∏_λ∈Λ_N(1-w/λ). Putting K_Ne_λ=λ^-1e_λ on ℓ^2(Λ_N) gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries. vfigproduct headFinite compression is an approximation For N=2, λ_m=π^2(m+1/2)^2. The finite determinant P_L is exactly the displayed finite pro","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a8e7ad98972fe76","tex":"\\mathcal P_M","aliases":"P_M mathcal","display":"\\mathcal P_M","kind":"use","excerpt":"For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e32f108b3db0ce30","tex":"p_M","aliases":"p_M","display":"p_M","kind":"use","excerpt":"[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37bbcb3eb285f1d4","tex":"P_{M,\\,r\\bmod M}","aliases":"P_M, rbmod M bmod","display":"P_{M,\\,r\\bmod M}","kind":"use","excerpt":"0, · otherwise, take the canonical label 0≤ r<N. The logical equivalence yields the actual equality of sequence operators P_N,rf= P_2^a, rbmod2^a(P_M, rbmod Mf). This statement holds for coefficients in any type with zero. A separate theorem lifts the equality coefficientwise to the existing formal power-series selectors over any semiring.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-703e11b92029fd77","tex":"P_{M,a}","aliases":"P_M,a","display":"P_{M,a}","kind":"use","excerpt":"leaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples. For example, at the first descendant address the values f[0]=0 and f[2]=1/2 are two separate values. Their sum 1/2 cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement. headThe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ae4b8981bd92730","tex":"P_{M,j_o}","aliases":"P_M,j_o","display":"P_{M,j_o}","kind":"use","excerpt":"[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68ad068bb42ded1a","tex":"P_{m,n}","aliases":"P_m,n","display":"P_{m,n}","kind":"use","excerpt":"[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally un","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34451fc17c1dd5f8","tex":"P_{M,r}","aliases":"P_M,r","display":"P_{M,r}","kind":"use","excerpt":"[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M r≡ jpmod NP_M,ra. The same identities hold coefficientwise for formal power series.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2af13a7e506b5f46","tex":"P_{M,r\\bmod M}","aliases":"P_M,rbmod M bmod","display":"P_{M,r\\bmod M}","kind":"use","excerpt":"[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6977f69e7c84b853","tex":"P_{M,s}","aliases":"P_M,s","display":"P_{M,s}","kind":"use","excerpt":"[Nested residue refinement] Let N,M be positive integers with N dividing M, and let 0≤ j<N. Then P_N,j = ∑_substack0≤ s<M s≡ jpmod NP_M,s. When R is nonzero, the packet system modulo M refines the packet system modulo N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5266a4839b2b51a8","tex":"P_{m-1}","aliases":"P_m-1","display":"P_{m-1}","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22271657f4ac2661","tex":"P_{\\min(i,j)}","aliases":"P_min(i,j) min","display":"P_{\\min(i,j)}","kind":"use","excerpt":"A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from P_iP_j=P_min(i,j).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"use","excerpt":"[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-346cf456a77b1ce5","tex":"p_n","aliases":"p_n","display":"p_n","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e5d7761f0b81ae7","tex":"P_{N,0}","aliases":"P_N,0","display":"P_{N,0}","kind":"use","excerpt":"[Parameter recurrence] We have P_N,0(r)=1. For n≥1, P_N,n(r) = · - ∑_k=1^n-1 (-1)^k binomNn-1Nk P_N,n-k(r)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68c8c438803a7824","tex":"\\mathcal P_{N,0}","aliases":"P_N,0 mathcal","display":"\\mathcal P_{N,0}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b03777ec508d2dd5","tex":"P_{N,1}","aliases":"P_N,1","display":"P_{N,1}","kind":"use","excerpt":"[The first three reciprocal polynomials] For every N≥1, P_N,1(r)=r, P_N,2(r) = r[ frac12binom2NN(r+1)-1 ], and P_N,3(r) = r[ frac(3N)!6(N!)^3(r+1)(r+2) - binom3NN(r+1) +1 ].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ef5da6a74405315","tex":"P_{N,2}","aliases":"P_N,2","display":"P_{N,2}","kind":"use","excerpt":"[The first three reciprocal polynomials] For every N≥1, P_N,1(r)=r, P_N,2(r) = r[ frac12binom2NN(r+1)-1 ], and P_N,3(r) = r[ frac(3N)!6(N!)^3(r+1)(r+2) - binom3NN(r+1) +1 ].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c5be35340cd11a7","tex":"P_{N,3}","aliases":"P_N,3","display":"P_{N,3}","kind":"use","excerpt":"[The first three reciprocal polynomials] For every N≥1, P_N,1(r)=r, P_N,2(r) = r[ frac12binom2NN(r+1)-1 ], and P_N,3(r) = r[ frac(3N)!6(N!)^3(r+1)(r+2) - binom3NN(r+1) +1 ].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce857563c4ac4bba","tex":"P_{N,i}","aliases":"P_N,i","display":"P_{N,i}","kind":"use","excerpt":"[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0c02aa550cfc1b1","tex":"\\mathcal P_{N,j}","aliases":"P_N,j mathcal","display":"\\mathcal P_{N,j}","kind":"use","excerpt":"The normalized source and its inverse are K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coeffic","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a442c81a1b99426","tex":"P_{N,j+1}","aliases":"P_N,j+1","display":"P_{N,j+1}","kind":"use","excerpt":"0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21f18d5054b34924","tex":"P_{N,j+r}","aliases":"P_N,j+r","display":"P_{N,j+r}","kind":"use","excerpt":"A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bccb8a74fcfcadc5","tex":"P_{N,j}^2","aliases":"P_N,j^2","display":"P_{N,j}^2","kind":"use","excerpt":"[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dcd49cdc880575e3","tex":"P_{N,n-k}","aliases":"P_N,n-k","display":"P_{N,n-k}","kind":"use","excerpt":"[Parameter recurrence] We have P_N,0(r)=1. For n≥1, P_N,n(r) = · - ∑_k=1^n-1 (-1)^k binomNn-1Nk P_N,n-k(r) · -r ∑_k=1^n (-1)^k binomNn-1Nk-1 P_N,n-k(r).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db7164b141c5c6f4","tex":"P_{N,n-q}","aliases":"P_N,n-q","display":"P_{N,n-q}","kind":"use","excerpt":"[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b530c98cdddafac","tex":"P_{N,q}","aliases":"P_N,q","display":"P_{N,q}","kind":"use","excerpt":"[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1543815dca570b98","tex":"p_{n-1}","aliases":"p_n-1","display":"p_{n-1}","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots ·","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9d95adbf1986020","tex":"P_{N_j,r_j}","aliases":"P_N_j,r_j","display":"P_{N_j,r_j}","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1cac30b985bfa29","tex":"P_{\\mathrm{ob}}","aliases":"P_ob mathrm","display":"P_{\\mathrm{ob}}","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions ho","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f79379509b17122","tex":"P_{\\mathrm{odd}}","aliases":"P_odd mathrm","display":"P_{\\mathrm{odd}}","kind":"use","excerpt":"For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ccfbd779afb88e3","tex":"P_{\\phys}","aliases":"P_phys phys mathrm","display":"P_{{\\mathrm{phys}}}","kind":"use","excerpt":"Constraint closure needs a different obstruction. If C_a are self adjoint constraints and P_phys projects onto their common kernel, then P_phys[C_a,C_b]P_phys=0 holds automatically. This compression cannot test closure. The correct finite obstruction is the class of [C_a,C_b] in the quotient of End( H) by the span of the constraints. Vanishing mean","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d61b96062b482c72","tex":"P_q","aliases":"P_q","display":"P_q","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21e7fc525024b4c1","tex":"\\mathcal P_q","aliases":"P_q mathcal","display":"\\mathcal P_q","kind":"use","excerpt":"Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)= {χ:G_ Q⟶ B^×midker q⊆kerχ}, ","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f5cb33c0e8a0d66","tex":"P_{qj}","aliases":"P_qj","display":"P_{qj}","kind":"use","excerpt":"[Tame character projectors] The operators P_j are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by Φ P_jΦ^-1=P_qj.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-528f89f54cfb3d02","tex":"p_r","aliases":"p_r","display":"p_r","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a95280613dbdd3c","tex":"P_{r+1}","aliases":"P_r+1","display":"P_{r+1}","kind":"use","excerpt":"Expand X^m=∑_r=0^mT(m,r)P_r(X). Since XP_r(X)=P_r+1(X)+r(r+1)P_r(X), these integer coefficients satisfy T(m+1,r)=T(m,r-1)+r(r+1)T(m,r), T(0,0)=1. Every r(r+1) is even. Induction, including the boundary indices, proves 2^m-rmid T(m,r) for 0≤ r≤ m. For m>0 the r=0 term vanishes. Each remaining summand of L(P^m/m!) has valuation at least m-r+2v_2(r!)-","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-152431294b318fe1","tex":"p_{r-1}","aliases":"p_r-1","display":"p_{r-1}","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c82a6e2325ace1a","tex":"P_r^{\\mathrm{coarse}}","aliases":"P_r^coarse mathrm","display":"P_r^{\\mathrm{coarse}}","kind":"use","excerpt":"[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08e120fa2da08c81","tex":"P_r^{L}","aliases":"P_r^L","display":"P_r^{L}","kind":"use","excerpt":"[Exact lattice refinement] For every coarse packet, P_r^L = ∑_s∈Child(r)P_s^L'. The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d75420bdb11de440","tex":"P_r^{\\rm total}","aliases":"P_r^rm total rm","display":"P_r^{\\rm total}","kind":"use","excerpt":"[Tensor charge refinement] For r∈ Z/N Z, define P_r^rm total = ∑_j_1+⋯+j_m=r P_j_1^(1)⊗⋯⊗ P_j_m^(m). Then the P_r^rm total form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e4bf1229112d277","tex":"P_{\\rm even}","aliases":"P_rm even rm","display":"P_{\\rm even}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a s","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ce1395e82f4cb81","tex":"P_{\\rm odd}","aliases":"P_rm odd rm","display":"P_{\\rm odd}","kind":"use","excerpt":"vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a s","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78aa8f72acca2b03","tex":"P_s","aliases":"P_s","display":"P_s","kind":"use","excerpt":"[Exact lattice polyphase decomposition] For every 1≤ p≤∞, the analysis map A_L:ℓ^p( Z^d)⟶⊕_r∈ Rℓ^p(L), A_Lx=(x_r)_r∈ R, is an isometric isomorphism. Its inverse is the interleaving map S_L(u)(r+ℓ)=u_r(ℓ). The projectors satisfy P_rP_s=δ_rsP_r, ∑_r∈ RP_r=I.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be8331cd1f9921b0","tex":"p_s","aliases":"p_s","display":"p_s","kind":"use","excerpt":"Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97e4bf0718e3c837","tex":"P_{s,-1}","aliases":"P_s,-1","display":"P_{s,-1}","kind":"use","excerpt":"For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible ad","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa62d8c209f88bda","tex":"P_{s,1}","aliases":"P_s,1","display":"P_{s,1}","kind":"use","excerpt":"3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4 0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy de","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a682e6942eadb5e0","tex":"P_{s,m}","aliases":"P_s,m","display":"P_{s,m}","kind":"use","excerpt":"om fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b619bee6fd91d216","tex":"P_{s,t}","aliases":"P_s,t","display":"P_{s,t}","kind":"use","excerpt":"w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, hence Q≠0 and Q is a unit in Z[[t]], and the independently enumerated walk series satisfies Z_s,t(t)=Q(t)^-1P_s,t(t), Q(t)Z_s,t(t)=P_s,t(t) in Z[[t]].","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb7674ad2c449edd","tex":"P_s^{L'}","aliases":"P_s^L'","display":"P_s^{L'}","kind":"use","excerpt":"[Exact lattice refinement] For every coarse packet, P_r^L = ∑_s∈Child(r)P_s^L'. The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d58992a0c0c08b27","tex":"p_t","aliases":"p_t","display":"p_t","kind":"use","excerpt":"[Residual life as a restarted semigroup] For S(t)>0, define the conditional transient distribution p_t = frac e^tAp_0S(t). Then fracS(t+s)S(t) = 1^T e^sAp_t.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b804d42c181b602f","tex":"P_{t,2}","aliases":"P_t,2","display":"P_{t,2}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4 3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-667691477116259f","tex":"P_{tst^{-1},1}","aliases":"P_tst^-1,1","display":"P_{tst^{-1},1}","kind":"use","excerpt":"For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible ad","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a0ce4bb02b1efb4","tex":"P_U","aliases":"P_U","display":"P_U","kind":"use","excerpt":"headWhy this does not prove an unrestricted minimum of two The bound path algebra includes the vertex idempotents P_U,P_V. A mixed seed s=u_1+v_3∈ U⊕ V satisfies P_Us=u_1 and P_Vs=v_3. It therefore generates the entire representation by itself. The chapter proves that no seed supported at U is enough and that two displayed pure seeds suffice. It do","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a4173e2450e265f","tex":"p_U","aliases":"p_U","display":"p_U","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12216838e248404d","tex":"p_u","aliases":"p_u","display":"p_u","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae2504de22a746d7","tex":"P_V","aliases":"P_V","display":"P_V","kind":"use","excerpt":"headWhy this does not prove an unrestricted minimum of two The bound path algebra includes the vertex idempotents P_U,P_V. A mixed seed s=u_1+v_3∈ U⊕ V satisfies P_Us=u_1 and P_Vs=v_3. It therefore generates the entire representation by itself. The chapter proves that no seed supported at U is enough and that two displayed pure seeds suffice. It do","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20e4069b9d08281b","tex":"\\mathfrak P_v","aliases":"P_v mathfrak","display":"\\mathfrak P_v","kind":"use","excerpt":"[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd728d302d0a06c1","tex":"p_V","aliases":"p_V","display":"p_V","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26a9ec94734fbef0","tex":"p_v","aliases":"p_v","display":"p_v","kind":"use","excerpt":"[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f902074ceba32407","tex":"P_{v0}","aliases":"P_v0","display":"P_{v0}","kind":"use","excerpt":"es, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actual Haar packets, set P_v=W_v^*W_v. Parent-to-ch","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-726490b49a9cbc64","tex":"P_{v1}","aliases":"P_v1","display":"P_{v1}","kind":"use","excerpt":"es, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actual Haar packets, set P_v=W_v^*W_v. Parent-to-ch","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b1d3ec03a20b6b9","tex":"P_{\\alpha}","aliases":"P_α alpha","display":"P_{\\alpha}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f29b3a10d396763d","tex":"p_\\alpha","aliases":"p_α alpha","display":"p_\\alpha","kind":"use","excerpt":"[Exact information decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the in","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f9d4390a1f58464","tex":"P_{\\alpha'}","aliases":"P_α' alpha","display":"P_{\\alpha'}","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-056b9963f80df264","tex":"P_{\\alpha(e)}","aliases":"P_α(e) alpha","display":"P_{\\alpha(e)}","kind":"use","excerpt":"For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tig","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2bbf7f75432b536","tex":"P_{\\alpha(j,e)}","aliases":"P_α(j,e) alpha","display":"P_{\\alpha(j,e)}","kind":"use","excerpt":"[Mixed precision product chain] If each fixed linear factor A_j is replaced by a fixed packetwise linear approximation widehat A_j, write E_j=widehat A_j-A_j=∑_eΔ_j,eP_α(j,e). A uniform certificate is E_j ≤δ_j:=∑_e Δ_j,e P_α(j,e) . For orthogonal packet projectors, δ_j≤∑_e Δ_j,e . The final operator error is bounded by [source reference] with these","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eee86870442a9515","tex":"p_{\\alpha,i}","aliases":"p_α,i alpha","display":"p_{\\alpha,i}","kind":"use","excerpt":"For i∈ I_α, write p_i=m_α p_α,i, q_i=n_α q_α,i. Then logfracq_ip_i = logfracn_αm_α + logfracq_α,ip_α,i. Multiply by q_i, sum first inside each packet, and then sum over the packets. This is the finite relative entropy chain rule. Standard information theoretic treatments include [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb3fe555e4ac9447","tex":"P_{\\alpha_0}","aliases":"P_α_0 alpha","display":"P_{\\alpha_0}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8fae8ad97b2a4d74","tex":"P_{\\alpha_1}","aliases":"P_α_1 alpha","display":"P_{\\alpha_1}","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70e6a72cdb8f81e7","tex":"P_{\\alpha_j}","aliases":"P_α_j alpha","display":"P_{\\alpha_j}","kind":"use","excerpt":"[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-302f53d86a380a68","tex":"P_{\\alpha_m}","aliases":"P_α_m alpha","display":"P_{\\alpha_m}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d81550d96775f196","tex":"P_{\\alpha_{m-1}}","aliases":"P_α_m-1 alpha","display":"P_{\\alpha_{m-1}}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24058bd88b2e4102","tex":"P_{\\alpha_r}","aliases":"P_α_r alpha","display":"P_{\\alpha_r}","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-235bca2612706176","tex":"P_{\\beta}","aliases":"P_β beta","display":"P_{\\beta}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c3fc12ee2bca96e","tex":"P_\\beta^*","aliases":"P_β^* beta","display":"P_\\beta^*","kind":"use","excerpt":"Because dualization reverses products, P_α^*P_β^*=(P_β P_α)^* =δ_αβP_α^*. Also ∑_α P_α^*=I_X^*. Thus the dual maps form a packet system. Restriction gives a map from P_α^*X^* to X_α^*. It is injective because a covector in the range of P_α^* vanishes on every other packet, and it is surjective because any covector on X_α extends to X through the di","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9813ba63ac7cffd1","tex":"P_\\gamma","aliases":"P_γ gamma","display":"P_\\gamma","kind":"use","excerpt":"Insert I=∑_β P_β between S and T: P_γ STP_α = ∑_β P_γ SP_β TP_α.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f4d885f46bde8c0","tex":"P_{\\gamma_1}","aliases":"P_γ_1 gamma","display":"P_{\\gamma_1}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a849f1482e28f8a2","tex":"P_{\\gamma_{m-1}}","aliases":"P_γ_m-1 gamma","display":"P_{\\gamma_{m-1}}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17417a9415875e48","tex":"P_\\delta","aliases":"P_δ delta","display":"P_\\delta","kind":"use","excerpt":"Packet superoperator blocks are defined by Φ_αβ;γδ(X) = P_α Φ(P_γ XP_δ) P_β. Summing all four packet indices reconstructs Φ(X) exactly.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ed30585279498bf","tex":"P_\\lambda","aliases":"P_λ lambda","display":"P_\\lambda","kind":"use","excerpt":"cibility, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONChapter VII: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5bb0b441f791e733","tex":"P_{\\lambda,v}","aliases":"P_λ,v lambda","display":"P_{\\lambda,v}","kind":"use","excerpt":"Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coeffici","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00d2ac2360124aae","tex":"P_\\mu","aliases":"P_μ mu","display":"P_\\mu","kind":"use","excerpt":"vfig[.95]all_obstruction_blocks platecaptionThe board on the right contains all nine actual embedded blocks P_μOb P_ν. Row labels are output packets and column labels are input packets. Only the addresses (0,1) and (1,2) carry nonzero maps.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-577132b0e435e8fc","tex":"P_\\nu","aliases":"P_ν nu","display":"P_\\nu","kind":"use","excerpt":"vfig[.95]all_obstruction_blocks platecaptionThe board on the right contains all nine actual embedded blocks P_μOb P_ν. Row labels are output packets and column labels are input packets. Only the addresses (0,1) and (1,2) carry nonzero maps.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04eb7fb63bcffdaf","tex":"P_{\\rho(\\beta)}","aliases":"P_ρ(β) rho beta","display":"P_{\\rho(\\beta)}","kind":"use","excerpt":"0 · ρ(β)≠α. Hence Q_β X⊆ P_ρ(β)X. If a third packet system R refines Q, then R refines P through the composite parent map.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ba7a0d2835235592","tex":"P_{\\sigma^{-1}(i)}","aliases":"P_σ^-1(i) sigma","display":"P_{\\sigma^{-1}(i)}","kind":"use","excerpt":"For the block formula, use intertwining and packet compatibility twice: Q_jφ(T)Q_iΦ · =Q_jφ(T)Φ P_σ^-1(i) · =Q_jΦ TP_σ^-1(i) · =Φ P_σ^-1(j)TP_σ^-1(i) · =∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and Hull_ A(S)=X, the hull equality gives Y=Hull_φ( A)(Φ(S)). If Φ(S)⊆Hull_ B(S'), then Hull_ B(Φ(S))⊆Hull_ B(S'), so S' generates Y.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cf1c6fcd2f6fc54","tex":"P_{\\ell+1,a}","aliases":"P_ℓ+1,a ell","display":"P_{\\ell+1,a}","kind":"use","excerpt":"[Exact packet direct limit] Assume the packet families have one fixed finite address set at every level and each packet address a is compatible in the sense P_ℓ+1,aR_ℓ=R_ℓ P_ℓ,a. Then there is an orthogonal projector P_∞,a on H_∞ satisfying P_∞,aι_ℓ=ι_ℓ P_ℓ,a. Finite completeness and orthogonality pass to the direct limit.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5fecae50dac286a","tex":"P_{\\ell+1,\\beta}","aliases":"P_ℓ+1,β ell beta","display":"P_{\\ell+1,\\beta}","kind":"use","excerpt":"A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8b027aabd4c6463","tex":"P_{\\infty,a}","aliases":"P_∞,a infty","display":"P_{\\infty,a}","kind":"use","excerpt":"[Exact packet direct limit] Assume the packet families have one fixed finite address set at every level and each packet address a is compatible in the sense P_ℓ+1,aR_ℓ=R_ℓ P_ℓ,a. Then there is an orthogonal projector P_∞,a on H_∞ satisfying P_∞,aι_ℓ=ι_ℓ P_ℓ,a. Finite completeness and orthogonality pass to the direct limit.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ff9a149e730be7a","tex":"\\operatorname{pack}","aliases":"pack operatorname","display":"\\operatorname{pack}","kind":"use","excerpt":"For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every n","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-463715ee648b1345","tex":"\\operatorname{pack}_4","aliases":"pack_4 operatorname","display":"\\operatorname{pack}_4","kind":"use","excerpt":"For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every n","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5065376efd33e45","tex":"\\mathsf{Packet}","aliases":"Packet mathsf","display":"\\mathsf{Packet}","kind":"use","excerpt":"Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a90766c2377cd948","tex":"\\operatorname{packet}","aliases":"packet operatorname","display":"\\operatorname{packet}","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a156778c1d9d8562","tex":"\\mathrm{packets}","aliases":"packets mathrm","display":"\\mathrm{packets}","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L I · 0 · 0 0 · I · 0. Replacing L by a scalar 1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-079e8b6488dabbe1","tex":"\\pagerefV","aliases":"pagerefV","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ab8f7d516126c87","tex":"\\Pcal'","aliases":"Pcal' P' Pcal mathcal","display":"{\\mathcal P}'","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c48b0030cb4c99da","tex":"\\Pcal_m","aliases":"Pcal_m P_m Pcal mathcal","display":"{\\mathcal P}_m","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23edeb9c0147a8cc","tex":"\\Pcal_\\ell","aliases":"Pcal_ℓ P_ℓ Pcal ell mathcal ell","display":"{\\mathcal P}_\\ell","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53743af9d1feb8d1","tex":"\\mathrm{phys}","aliases":"phys mathrm","display":"\\mathrm{phys}","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a7a26e99cfb170bd","tex":"\\phys","aliases":"phys mathrm","display":"{\\mathrm{phys}}","kind":"use","excerpt":"Constraint closure needs a different obstruction. If C_a are self adjoint constraints and P_phys projects onto their common kernel, then P_phys[C_a,C_b]P_phys=0 holds automatically. This compression cannot test closure. The correct finite obstruction is the class of [C_a,C_b] in the quotient of End( H) by the span of the constraints. Vanishing mean","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf2a973806abd94e","tex":"\\mathbf{PktMod}","aliases":"PktMod mathbf","display":"\\mathbf{PktMod}","kind":"use","excerpt":"[Category of seeded packet systems] Packet compatible operator morphisms compose, seeded morphisms compose, and identity morphisms satisfy both definitions. Thus packet resolved operator modules form a category PktMod and seeded packet systems form a category SeedSys. DependsnolinkurlPSC-I-D-PACKET-MORPHISM, nolinkurlPSC-I-D-SEED-MORPHISM, and noli","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-065b6c29f5eeff18","tex":"\\PP","aliases":"PP P mathcal","display":"{\\mathcal P}","kind":"use","excerpt":"Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s). Induction and multiplication by g(n+s","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-738cf0c2598fdac3","tex":"\\PP_2","aliases":"PP_2 P_2 PP mathcal","display":"{\\mathcal P}_2","kind":"use","excerpt":"An affine sequence whose slope is divisible by a satisfies P_a. Consequently the four displayed source sequences satisfy P_2. No numerical interpolation or finite set of samples is used in that deduction.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6aa31545bc253fc8","tex":"\\PP_a","aliases":"PP_a P_a PP mathcal","display":"{\\mathcal P}_a","kind":"use","excerpt":"Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s). Induction and multiplication by g(n+s","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5bfbf5194d3f5a47","tex":"\\mathrm{pr}","aliases":"pr mathrm","display":"\\mathrm{pr}","kind":"use","excerpt":"et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5f7721258ed720c9","tex":"\\operatorname{pr}","aliases":"pr operatorname","display":"\\operatorname{pr}","kind":"use","excerpt":"(t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f1b6dc519af96a7","tex":"\\prime","aliases":"prime","display":"\\prime","kind":"use","excerpt":"The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I. Uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a7053e18464057f","tex":"\\psrank_{A,\\Pcal}","aliases":"psrank_A,Pcal psrank_A, P psrank Pcal operatorname mathcal","display":"{\\operatorname{psrank}}_{A,{\\mathcal P}}","kind":"use","excerpt":"[Basic packet pure rank bounds] For every A module resolved into packets with P_α∈ A, srank_A(X) ≤ psrank_A, P(X) ≤ ∑_α∈ I srank_P_α A P_α(X_α). The corner algebra on the right is unital on X_α with identity P_α. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43cab5a3c1e158c9","tex":"\\mathbb\n  Q","aliases":"Q mathbb","display":"\\mathbb\n  Q","kind":"use","excerpt":"bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de Franc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8605e09a21a342b4","tex":"\\mathbf Q","aliases":"Q mathbf","display":"\\mathbf Q","kind":"use","excerpt":"For the three-character example below, the construction produces an actual three-dimensional summand inside the carrier of dimension seven. At the sample (0,1) its characteristic polynomial is (X-1)^2(X+1). A separate checked counterexample uses the coordinate inclusion Q→ Q^2, its coordinate retraction, and the swap involution: the split law holds","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c370aa793b02aac","tex":"Q'","aliases":"Q'","display":"Q'","kind":"use","excerpt":"Let Q(z)=zcoth z, the regular formal series with constant coefficient one. The elementary differential identity (frac zsinh z)^2=Q(z)-zQ'(z) and the Bernoulli expansion Q(z)=1+∑_m≥1frac2^2mB_2m(2m)!z^2m give β_m=-2^2m(2m-1)B_2m/(2m)!. Multiplication by 2^2m(2m)! proves the identity for κ_m. The Bernoulli denominator theorem gives v_2(B_2m)=-1 for m","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"Write the input indices as Mq+u and Mq'+v. Since 0≤ u,v<M, their sum has residue u+v when u+v<M, and residue u+v-M with one additional coarse index when u+v≥ M.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db9b1118d56e7f21","tex":"q^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}","aliases":"q^(n-r)L+∑_i=1^rmin(s_i,L) sum min","display":"q^{(n-r)L+\\sum_{i=1}^r\\min(s_i,L)}","kind":"use","excerpt":"rdinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f4cc63c7c45c2f71","tex":"q^{-j}","aliases":"q^-j","display":"q^{-j}","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space. The assertion k","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2a621a3fcb22dce","tex":"q^{-L}","aliases":"q^-L","display":"q^{-L}","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2ff5e7340642c49","tex":"q^{-W}","aliases":"q^-W","display":"q^{-W}","kind":"use","excerpt":"[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ae4ac58b9a6ef67","tex":"q^{1/2}","aliases":"q^1/2","display":"q^{1/2}","kind":"use","excerpt":"[Gauss magnitude] If χ is nontrivial, then absG(χ,ψ)=q^1/2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"use","excerpt":"0 · 1. These maps satisfy P^2=P, Q^2=Q, PQ=QP=0 and P+Q=I. Their images are the horizontal line and the diagonal line, respectively. They are not self-adjoint in the usual Euclidean inner product.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53004de1c8fb420f","tex":"\\mathbf Q^2","aliases":"Q^2 mathbf","display":"\\mathbf Q^2","kind":"use","excerpt":"For the three-character example below, the construction produces an actual three-dimensional summand inside the carrier of dimension seven. At the sample (0,1) its characteristic polynomial is (X-1)^2(X+1). A separate checked counterexample uses the coordinate inclusion Q→ Q^2, its coordinate retraction, and the swap involution: the split law holds","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa0b566a56f0141e","tex":"q^{2M+2}","aliases":"q^2M+2","display":"q^{2M+2}","kind":"use","excerpt":"[Odd modulus truncation] For every M≥0, every positive odd q, and every even n, E_n ≡ ∑_r=0^min(M,n/2) (-1)^rZ_2r+1 binom n2rq^2rS_n-2r(q) pmod q^2M+2. If pmid q and a=v_p(q), the same truncation is valid modulo p^2a(M+1).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8db2618d57c31645","tex":"q^{2N}","aliases":"q^2N","display":"q^{2N}","kind":"use","excerpt":"^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exactly. The checked polynomial Taylor identity gives F(a')≡ F(a)+q^NDF(a)(widetilde u,w) ≡0pmodq^N+1. Its quadratic remainder is divisible by q^2N, and 2N≥ N+1. ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42bb94f60a6ecb0a","tex":"q^{2r}","aliases":"q^2r","display":"q^{2r}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2d3c9a0c830bb0d","tex":"q^3","aliases":"q^3","display":"q^3","kind":"use","excerpt":"basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writin","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9baeebca7f43f14","tex":"\\Q^4","aliases":"Q^4 Q mathbf","display":"{\\mathbf Q}^4","kind":"use","excerpt":"A specified finite realization. On Q^4, take the diagonal operators with diagonal lists (1,1,1,0), (2,2,0,0), (3,0,0,0), (0,4,4,4) and target scalars (1,2,3,4). All four targets occur individually. Imposing these conditions successively gives actual projector ranks 3,2,1,0. The last packet is empty despite individual feasibility. The formal code ch","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee5559b3980fa51c","tex":"q^d","aliases":"q^d","display":"q^d","kind":"use","excerpt":"For every positive integer q, including composite moduli and modulus one, and every L≥0, the checked map Φ_x,L:( Z/q Z)^d⟶ {ybmod q^L+1:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection. Here widetilde t is the canonical integer representative of the digit vector. The entire residue fiber therefore has q^d elements. Injectivity is proved by cancelli","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bca90fff80db18b","tex":"q^{d(m-1)}","aliases":"q^d(m-1)","display":"q^{d(m-1)}","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ac1542e49ac35e1","tex":"q^{dm-W}","aliases":"q^dm-W","display":"q^{dm-W}","kind":"use","excerpt":"The finite sum records the depths before all coordinates are active. For m≥ w_d+1, a_m=q^dm-W, so the remaining terms form one geometric series.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0cdef6628c529361","tex":"Q^e","aliases":"Q^e","display":"Q^e","kind":"use","excerpt":")∈ M_e× k(R). Suppose, for natural numbers s,M, that J∈GL_e(Q), p^sJ^-1∈ M_e(R), M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. Then there is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisfies y-y_0∈ p^M-sR^e and there is an integral vector ε∈ p^2M-3sR^e such that, in Q^e, displaystyle y-y_0=-p^MJ^-1(H(a)+Kv)+ε. The same original exponent s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e80a32868f710c6e","tex":"q^{e-1}","aliases":"q^e-1","display":"q^{e-1}","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41d4419bab60914f","tex":"q^i","aliases":"q^i","display":"q^i","kind":"use","excerpt":"[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b02897ce2089839","tex":"Q^j","aliases":"Q^j","display":"Q^j","kind":"use","excerpt":"Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK 3.1.7--9 give the actual groups","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9e4eac3b648f9f1d","tex":"q^{j-1}","aliases":"q^j-1","display":"q^{j-1}","kind":"use","excerpt":"The only nonzero entry in column j and row j-1 is multiplied by q^j-1q^-j=q^-1 under conjugation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-938f08ed5d6886a7","tex":"q^{L+1}","aliases":"q^L+1","display":"q^{L+1}","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dfe745e2a01c96c9","tex":"q^{L+a}","aliases":"q^L+a","display":"q^{L+a}","kind":"use","excerpt":"z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0899f3519cbe2b50","tex":"q^{L+n}","aliases":"q^L+n","display":"q^{L+n}","kind":"use","excerpt":"[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a3a203830edc472","tex":"q^m","aliases":"q^m","display":"q^m","kind":"use","excerpt":"Represent the residue characters by exponents apmodq^n-1, bpmodq^m-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-823af7abc34149de","tex":"q^{m-1}","aliases":"q^m-1","display":"q^{m-1}","kind":"use","excerpt":"[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fd7aaa8906a193c","tex":"q^{m-\\lceil m/e\\rceil}","aliases":"q^m-lceil m/erceil lceil rceil","display":"q^{m-\\lceil m/e\\rceil}","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08d664fed466d656","tex":"q^{m-w_i}","aliases":"q^m-w_i","display":"q^{m-w_i}","kind":"use","excerpt":"The image of ϖ^w_i O in O/ϖ^m has one element when m≤ w_i, and q^m-w_i elements when m>w_i. Multiplying the independent coordinate counts gives the formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5cd87d7bdf4bf601","tex":"q^{\\min(s_i,L)}","aliases":"q^min(s_i,L) min","display":"q^{\\min(s_i,L)}","kind":"use","excerpt":"Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L cho","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa29cb49431ad02f","tex":"Q^N","aliases":"Q^N","display":"Q^N","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1928485a8127cf57","tex":"q^N","aliases":"q^N","display":"q^N","kind":"use","excerpt":"At level N=L+n, divide the residual by q^N. Surjectivity supplies t∈ S_q^d with J_z_nt=-overlineF(z_n)/q^N. Choose an integer representative and apply [source reference] to obtain z_n+1. The new point is congruent to z_n modulo q. Evaluation of each formal partial derivative therefore gives J_z_n+1=J_z_n.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-102bb645d8547d75","tex":"q^n","aliases":"q^n","display":"q^n","kind":"use","excerpt":"rated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d77f110b9dcddc74","tex":"q^{N+1}","aliases":"q^N+1","display":"q^{N+1}","kind":"use","excerpt":"s t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exactly. The checked polynomial Taylor identity gives F(a')≡ F(a)+q^NDF(a)(widetilde u,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9a4386c240576be","tex":"q^{n-1-j}","aliases":"q^n-1-j","display":"q^{n-1-j}","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76963409a11b32ee","tex":"q^{n/d}","aliases":"q^n/d","display":"q^{n/d}","kind":"use","excerpt":"[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b568c9f273e792aa","tex":"Q^r","aliases":"Q^r","display":"Q^r","kind":"use","excerpt":"/3 in the one-prime example above loses the advertised digit. Invertibility of C alone does not restore it. The original proof is retained under these hypotheses. Let Q=∏_p∈ Sp^L_p for a finite set of primes. Let C∈ M_d( Z) be invertible modulo Q, and choose H such that C^H=I+QB with B∈ M_d( Z). Then for every nonnegative integer x, a_Hx+j = ∑_r=0^","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50cffe607a1e0f78","tex":"q^r","aliases":"q^r","display":"q^r","kind":"use","excerpt":"For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63b1654c14578998","tex":"q^{r-1}","aliases":"q^r-1","display":"q^{r-1}","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-886dc240f95388c4","tex":"q^{r-d}","aliases":"q^r-d","display":"q^{r-d}","kind":"use","excerpt":"On the polynomial factor the difference operator acts as q+yΔ. The two terms commute. Expand (q+yΔ)^r and evaluate at zero. Since Δ^kQ_j=0 for k>d, every surviving term contains at least q^r-d.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-378ab1d30fa3e392","tex":"q^{r-k}","aliases":"q^r-k","display":"q^{r-k}","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-050d088f2ad53d96","tex":"Q^T","aliases":"Q^T","display":"Q^T","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0fc2c2d75a0ca57","tex":"q^{\\sum_{i=1}^d\\max(m-w_i,0)}","aliases":"q^∑_i=1^dmax(m-w_i,0) sum max","display":"q^{\\sum_{i=1}^d\\max(m-w_i,0)}","kind":"use","excerpt":"[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ddd7652dc4b81b62","tex":"q^{\\sum_i\\max(m-w_i,0)}","aliases":"q^∑_imax(m-w_i,0) sum max","display":"q^{\\sum_i\\max(m-w_i,0)}","kind":"use","excerpt":"[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc28058c38687619","tex":"q^{\\sum_i\\min(s_i,L)}","aliases":"q^∑_imin(s_i,L) sum min","display":"q^{\\sum_i\\min(s_i,L)}","kind":"use","excerpt":"For an arbitrary DVR O with residue field cardinality q, the analogous formula q^∑_imin(s_i,L) follows from the solvability and fiber count for each coordinate. The formal correspondence records direct proofs of this DVR coordinate arithmetic and the rectangular transport as separate targets. The inherited abstract product interface remains availab","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a949dbefc3833bd1","tex":"Q^\\perp","aliases":"Q^⊥ perp","display":"Q^\\perp","kind":"use","excerpt":"The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a634d867493413c1","tex":"q_{\\mathrm A}","aliases":"q_ A mathrm","display":"q_{\\mathrm A}","kind":"use","excerpt":"vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8497f4e15ecc3f26","tex":"q_{\\mathrm B}","aliases":"q_ B mathrm","display":"q_{\\mathrm B}","kind":"use","excerpt":"vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ccc70c6e400edfd7","tex":"\\mathcal Q_{\\mathcal L}","aliases":"Q_ L mathcal mathcal","display":"\\mathcal Q_{\\mathcal L}","kind":"use","excerpt":"[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every bl","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e4bfa4de54508e8","tex":"q_*","aliases":"q_*","display":"q_*","kind":"use","excerpt":"[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","excerpt":"[Dyadic phase ambiguity] On a symplectic plane over F_2, choose refinements q_0 and q_1 with Arf invariants zero and one. Their polar forms are the same, while ∑_x(-1)^q_0(x)=2, ∑_x(-1)^q_1(x)=-2. Thus the alternating packet alone cannot select the dyadic sign.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3dcd53dfa5dbd07d","tex":"Q_{00}","aliases":"Q_00","display":"Q_{00}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3aa5ff4c06d335c1","tex":"Q_{01}","aliases":"Q_01","display":"Q_{01}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6738621edb3fbe5f","tex":"Q_0^\\mathsf","aliases":"Q_0^ mathsf","display":"Q_0^\\mathsf","kind":"use","excerpt":"headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-757d025f70886a4b","tex":"Q_{10}","aliases":"Q_10","display":"Q_{10}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0b259d086fc3978c","tex":"Q_{11}","aliases":"Q_11","display":"Q_{11}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eda0b1a24e71fdef","tex":"Q_2","aliases":"Q_2","display":"Q_2","kind":"use","excerpt":"Indeed, Q_1T^UQ_2=Q_2T^UQ_1=0, Q_iT^UQ_j=U(P_iTP_j)U^-1. A change in raw matrix sparsity is therefore not necessarily a change in the packet transfer graph. Conjugacy preserves the exact block identities and their zero pattern. A nonunitary shear need not preserve their Euclidean norms.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-812a1fd77a9f35a3","tex":"\\mathbb Q_2","aliases":"Q_2 mathbb","display":"\\mathbb Q_2","kind":"use","excerpt":"Modulo 2, the two eigenvalues become equal. The rational sign packets remain distinct over Q_2, but the standard lattice does not split into their intersections until one more dyadic digit is retained.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"[Orthogonal dyadic multiplicativity] For an orthogonal sum (V,q)=(V_1,q_1)mathop⊥limits(V_2,q_2), we have Arf(q)=Arf(q_1)+Arf(q_2) and the normalized Gauss sign is the product of the two normalized signs.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-983925c02a852901","tex":"Q_2^{\\mathrm{red}}","aliases":"Q_2^red mathrm","display":"Q_2^{\\mathrm{red}}","kind":"use","excerpt":"For N=2, [source reference] and D_2(z)=Δ_2(z^2) give λ_m=π^2(m+frac12)^2, m≥0. Thus the first formula holds with Q_2^red=0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-188ca6e8182a79b4","tex":"\\Q_3","aliases":"Q_3 Q mathbf","display":"{\\mathbf Q}_3","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2dd5eb187fc710e","tex":"Q_8","aliases":"Q_8","display":"Q_8","kind":"use","excerpt":"r an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked finite exte","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ca75c044055616d","tex":"Q_a","aliases":"Q_a","display":"Q_a","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55d4600438de195a","tex":"q_a","aliases":"q_a","display":"q_a","kind":"use","excerpt":"FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] De","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f717c588f6601060","tex":"Q_a^*","aliases":"Q_a^*","display":"Q_a^*","kind":"use","excerpt":"Expanding the finite product of sums gives T(A) T(B)= T(i↦ A_iB_i). Consequently, if two fine addresses differ in any coordinate, the corresponding product contains a zero factor. If the addresses agree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so Q_a^*=Q_a. Summing over all address functions","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c57539584fe8169f","tex":"Q_b","aliases":"Q_b","display":"Q_b","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4591bb144ad9c3f","tex":"Q_c","aliases":"Q_c","display":"Q_c","kind":"use","excerpt":"-1 · 1C. The coarse matrix is the positive square root of -Q_c. This is a concrete naturality identity, checked on the full matrices. It is not a claim that an arbitrary partition commutes with every fractional construction.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f73627dbab9e543b","tex":"\\mathcal Q_c","aliases":"Q_c mathcal","display":"\\mathcal Q_c","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bb195dbf7dd5e73","tex":"Q_D","aliases":"Q_D","display":"Q_D","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83dc7e104ed7732b","tex":"Q_d","aliases":"Q_d","display":"Q_d","kind":"use","excerpt":"Adjoint and tensor products. With an involution and self-adjoint projectors, the formalization proves (A_ba)^*=(A^*)_ab. For actual finite matrices and actual Kronecker products, it proves (P_b⊗ Q_d)(A⊗ B)(P_a⊗ Q_c) =A_ba⊗ B_dc. The self-adjointness hypothesis is explicit in the adjoint theorem. The tensor theorem is not a statement about an uninte","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-842201207aa7200d","tex":"q_D","aliases":"q_D","display":"q_D","kind":"use","excerpt":"[Cokernel obstruction criterion] A linear correction exists if and only if Ob_D(r)=0. The quotient map q_D:V⟶coker D has the following universal property. If φ:V→ W is linear and φ D=0, then there is a unique linear map overlineφ:coker D⟶ W with φ=overlineφ q_D.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-369e50e7dcd70df5","tex":"q_d","aliases":"q_d","display":"q_d","kind":"use","excerpt":"[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ccf3bc5017cc6d9c","tex":"q_{D'}","aliases":"q_D'","display":"q_{D'}","kind":"use","excerpt":"The relation bD=D'a shows that the quotient map q_D'b vanishes on im D. Apply the universal property in [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f7b02bca74c6be6","tex":"q_{d-1}","aliases":"q_d-1","display":"q_{d-1}","kind":"use","excerpt":"At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coeffici","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-830545afde08b32b","tex":"q_{d-j+1}","aliases":"q_d-j+1","display":"q_{d-j+1}","kind":"use","excerpt":"[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native paramet","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dad907bde99257ae","tex":"Q_{D_i}","aliases":"Q_D_i","display":"Q_{D_i}","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8cef7dc638ad907a","tex":"Q_F","aliases":"Q_F","display":"Q_F","kind":"use","excerpt":"Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents a","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea7edaad8d5ca545","tex":"\\mathcal Q_f","aliases":"Q_f mathcal","display":"\\mathcal Q_f","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61c40f35d4a3b82d","tex":"q_F","aliases":"q_F","display":"q_F","kind":"use","excerpt":"Exact correction of an error span means that one completely positive trace preserving recovery R satisfies R(Fρ F^*)=q_Fρ for every F in that span and every code-supported matrix ρ, with a nonnegative scalar q_F depending on F but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restore","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ac09894acaf58fd","tex":"Q_H","aliases":"Q_H","display":"Q_H","kind":"use","excerpt":"Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents a","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5950b0378700bfba","tex":"Q_i","aliases":"Q_i","display":"Q_i","kind":"use","excerpt":"[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and S generates X, then Φ(S) generates Y under φ( A) and hence under B. If, in ad","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5842aab9955ad05","tex":"Q_i^2","aliases":"Q_i^2","display":"Q_i^2","kind":"use","excerpt":"headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97b5f75d73830435","tex":"Q_{ii}","aliases":"Q_ii","display":"Q_{ii}","kind":"use","excerpt":"Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dfe49e85066a6b84","tex":"Q_{ij}","aliases":"Q_ij","display":"Q_{ij}","kind":"use","excerpt":"ty means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. The","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-958248099af7659e","tex":"\\mathfrak Q_j","aliases":"Q_j mathfrak","display":"\\mathfrak Q_j","kind":"use","excerpt":"For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesse","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-704a2b865a8b60e0","tex":"Q_j^2","aliases":"Q_j^2","display":"Q_j^2","kind":"use","excerpt":"Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d146bc16e8f166e","tex":"Q_{ji}","aliases":"Q_ji","display":"Q_{ji}","kind":"use","excerpt":"ty means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. The","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9af1147d28b7af5","tex":"Q_k","aliases":"Q_k","display":"Q_k","kind":"use","excerpt":"Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e15bd658e05b8d9b","tex":"q_k","aliases":"q_k","display":"q_k","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bd193e7de59f86f","tex":"Q_L","aliases":"Q_L","display":"Q_L","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACTChapter VIII: Theorem 5.6, Section F.1, Section F.2. The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a gen","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad4d7e06ba73eb0d","tex":"q_L","aliases":"q_L","display":"q_L","kind":"use","excerpt":"[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bcb84d1006a5379","tex":"Q_L^*","aliases":"Q_L^*","display":"Q_L^*","kind":"use","excerpt":"0 · 0 · 0. Keeping only its first singular direction gives a rank-one approximation with spectral and Frobenius error exactly two. Keeping none gives errors five and sqrt29, while keeping both gives zero. vfigoptimality headWhy an outside competitor cannot do better For isometric outer factors, compression C(B)=Q_L^*BQ_R cannot increase rank or eit","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c06fc40d19e1dce","tex":"Q_L^T","aliases":"Q_L^T","display":"Q_L^T","kind":"use","excerpt":"0 · -2 i, A^*A=diag(25,4). vfigcomplex_core headThe small calculation exposes a wrong convention The first column of Q_L has Hermitian squared length 9/25+16/25=1. Its bilinear square without conjugation is 9/25-16/25=-7/25. Thus Q_L^TQ_L is not the identity. The portable negative control replaces the adjoint by transpose and rejects the resulting ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2dbd37e0ccec228","tex":"q_m","aliases":"q_m","display":"q_m","kind":"use","excerpt":"[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a18c33136ce1ffd8","tex":"Q_{m,n}","aliases":"Q_m,n","display":"Q_{m,n}","kind":"use","excerpt":"A bounded generator of a contraction semigroup on a Hilbert space is dissipative. If Q_m,n=I-P_m,n and y∈ Q_m,nX_n, then ReipQ_m,nA_nQ_m,nyy = ReipA_nyy ≤0. Hence the compressed hidden semigroup is contractive. Apply [source reference] to P_m,n. Since P_m,nR_n,m=R_n,m, η_m,n(T) ≤ Tnorm(I-P_m,n)A_nP_m,n normR_n,m. The right side tends to zero. Use [","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c95e041440f50ec9","tex":"Q_n","aliases":"Q_n","display":"Q_n","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a77ffe628ff771a","tex":"q_n","aliases":"q_n","display":"q_n","kind":"use","excerpt":"vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPEChapter VII: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise converg","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-13c6dda682be516f","tex":"Q_N^{\\mathrm{abs}}","aliases":"Q_N^abs mathrm","display":"Q_N^{\\mathrm{abs}}","kind":"use","excerpt":"There is an entire function Q_N^abs such that, initially on Re s>1, Z_N^abs(s) = N A_N^-sζ(s,frac12)+Q_N^abs(s). Thus Z_N^abs continues meromorphically to C with one simple pole at s=1, of residue fracNsin(π/N)π.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed14692ef5195be0","tex":"q_n^{j+k}","aliases":"q_n^j+k","display":"q_n^{j+k}","kind":"use","excerpt":"vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPEChapter VII: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise converg","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6572edc01bfc69a","tex":"Q_N^{\\mathrm{red}}","aliases":"Q_N^red mathrm","display":"Q_N^{\\mathrm{red}}","kind":"use","excerpt":"[Meromorphic continuation of the symmetry-orbit zero zeta] Put A_N=fracπsin(π/N). There is an entire function Q_N^red such that Z_N^red(s) = A_N^-Nsζ(Ns,frac12)+Q_N^red(s) on Re s>1/N. The right-hand side gives a meromorphic continuation to C with one simple pole at s=1/N. Its residue is fracsin(π/N)Nπ.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0722957f7780ba76","tex":"\\mathbb\n  Q_p","aliases":"Q_p mathbb","display":"\\mathbb\n  Q_p","kind":"use","excerpt":"bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de Franc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3312dcd34d0a8dd5","tex":"\\mathbb Q_p^e","aliases":"Q_p^e mathbb","display":"\\mathbb Q_p^e","kind":"use","excerpt":"hbb Z_p^e+k, F(a)=0. Let ι: Z_p↪ Q_p be the coefficient inclusion, applied entrywise to points and matrices. Assume only that ι(D_YF(a))∈GL_e( Q_p). The theorem mcodePolynomialRegularityIntegral.exists_analytic_integral_implicit constructs the preceding analytic graph in the ambient spaces Q_p^k and Q_p^e. Its root equation evaluates the original i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45c8cae45a70e764","tex":"\\mathbb Q_p^k","aliases":"Q_p^k mathbb","display":"\\mathbb Q_p^k","kind":"use","excerpt":",T]^e, a∈ Z_p^e+k, F(a)=0. Let ι: Z_p↪ Q_p be the coefficient inclusion, applied entrywise to points and matrices. Assume only that ι(D_YF(a))∈GL_e( Q_p). The theorem mcodePolynomialRegularityIntegral.exists_analytic_integral_implicit constructs the preceding analytic graph in the ambient spaces Q_p^k and Q_p^e. Its root equation evaluates the orig","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-614679e7835ca9ed","tex":"\\Q_{p_s}","aliases":"Q_p_s Q mathbf","display":"{\\mathbf Q}_{p_s}","kind":"use","excerpt":"Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-808125305b3a9f1c","tex":"Q_R","aliases":"Q_R","display":"Q_R","kind":"use","excerpt":"vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACTChapter VIII: Theorem 5.6, Section F.1, Section F.2. The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a gen","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4210e21e79e5611","tex":"Q_r","aliases":"Q_r","display":"Q_r","kind":"use","excerpt":"[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9666ac009c3e72f6","tex":"q_r","aliases":"q_r","display":"q_r","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2e800860e3d6ef1","tex":"Q_R^*","aliases":"Q_R^*","display":"Q_R^*","kind":"use","excerpt":"0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat3 · 0 4 i · 0 0 · -2 i, A^*A=diag(25,4). vfigcomplex_core headThe small calculation exposes a wrong convention The first column of Q_L has Hermitian squared length 9/25+16/25=1. Its bilinear square without conjugation is 9/25-16/25=-7/25. Thus Q_L^TQ_L is ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-904b640e83eefc03","tex":"Q_{ri}","aliases":"Q_ri","display":"Q_{ri}","kind":"use","excerpt":"[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69e1eafeb8f88eb2","tex":"q_{\\rm src}","aliases":"q_rm src rm","display":"q_{\\rm src}","kind":"use","excerpt":"The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier. It is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90007b832087e14d","tex":"\\mathfrak Q_u","aliases":"Q_u mathfrak","display":"\\mathfrak Q_u","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de1f9e9f12343cd8","tex":"q_{u,L}","aliases":"q_u,L","display":"q_{u,L}","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7c8dd25964c5fdc","tex":"q_{u,L}^{\\,r}","aliases":"q_u,L^ r","display":"q_{u,L}^{\\,r}","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e7293ba39ec7c209","tex":"\\mathfrak Q_v","aliases":"Q_v mathfrak","display":"\\mathfrak Q_v","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e52a322b6d2467f2","tex":"q_v","aliases":"q_v","display":"q_v","kind":"use","excerpt":"[Translation of quadratic refinements] For v∈ V, put q_v(x)=q(x)+b(v,x). Then Arf(q_v)=Arf(q)+q(v).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f50dcece8b7d4a6","tex":"q_W","aliases":"q_W","display":"q_W","kind":"use","excerpt":"The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier. It is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab69aa180f14a53e","tex":"q_\\alpha","aliases":"q_α alpha","display":"q_\\alpha","kind":"use","excerpt":"[Bayesian packet reconstruction] B_λ(p) = ∑_α m_α q_α. Only packets with positive evidence carry a conditional probability vector. A zero-evidence packet has zero posterior mass and contributes the zero vector. The nonlinear coupling between packets occurs only through the total evidence used to normalize the packet evidences.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee7969975240e6b6","tex":"q_{\\alpha,i}","aliases":"q_α,i alpha","display":"q_{\\alpha,i}","kind":"use","excerpt":"Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-574286fc36296144","tex":"Q_{\\beta'}","aliases":"Q_β' beta","display":"Q_{\\beta'}","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-610d527192ec5212","tex":"Q_\\beta^2","aliases":"Q_β^2 beta","display":"Q_\\beta^2","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1492f3a3a2d7a030","tex":"Q_\\gamma^2","aliases":"Q_γ^2 gamma","display":"Q_\\gamma^2","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa7fbe64d6b9e7ee","tex":"Q_\\delta","aliases":"Q_δ delta","display":"Q_\\delta","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d938f24a7705f0ca","tex":"Q_{\\tau^{-1}(k)}","aliases":"Q_τ^-1(k) tau","display":"Q_{\\tau^{-1}(k)}","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source refer","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cef20594acd8a578","tex":"\\mathfrak Q_\\ell","aliases":"Q_ℓ mathfrak ell","display":"\\mathfrak Q_\\ell","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb74470e10b0fa87","tex":"q_\\ell","aliases":"q_ℓ ell","display":"q_\\ell","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-417638970b42b393","tex":"Q_{\\bigoplus_iD_i^{\\oplus m_i}}","aliases":"Q_⊕_iD_i^⊕ m_i bigoplus oplus","display":"Q_{\\bigoplus_iD_i^{\\oplus m_i}}","kind":"use","excerpt":"Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frob","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5bf9af43ed9be269","tex":"\\Qcal'","aliases":"Qcal' Q' Qcal mathcal","display":"{\\mathcal Q}'","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4467d168eb66170","tex":"\\Qcal_{cd}","aliases":"Qcal_cd Q_cd Qcal mathcal","display":"{\\mathcal Q}_{cd}","kind":"use","excerpt":"Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2b27e6c1de4e544","tex":"\\qedhere","aliases":"qedhere","display":null,"kind":"use","excerpt":"For a section s satisfying Ps=s, ∇^2s=P d(P ds)=P(dP)wedge ds. Since ds=(dP)s+Pds and P(dP)P=0, the surviving term is P(dPwedge dP)Ps.qedhere","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b6db148287b1581","tex":"\\Qp^d","aliases":"Qp^d Q_p^d Qp mathbb","display":"{\\mathbb Q_p}^d","kind":"use","excerpt":"[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ea1e9e76890cbee","tex":"\\Qp^e","aliases":"Qp^e Q_p^e Qp mathbb","display":"{\\mathbb Q_p}^e","kind":"use","excerpt":"y-y_0 · =-p^MJ^-1(H(a)+Kv)+ε, ε∈ p^2M-3sR^e. The equality involving J^-1 is in Q_p^e, with an integral error vector. The parameter equality is exact.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f925d2e485d97c9","tex":"\\QQ^{16}","aliases":"QQ^16 QQ","display":null,"kind":"use","excerpt":"tion 11, Section 13. A packet calculus can reorganize a signal exactly, choose among finitely many representations, or describe a nonlinear coordinate geometry. These are not the same task. This guide keeps one concrete object in view for each, so that the reader can compute rather than only follow arrows. vfigthree_worlds headA signal that can be ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9a6d997b7294e64","tex":"\\QQ^2","aliases":"QQ^2 QQ QQ^2 Q^2 QQ mathbb","display":null,"kind":"use","excerpt":"Define a surjective linear map q:X⟶ Q^2, q(∑ a_je_j+∑ b_jf_j)=(a_0,b_0).","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-279caa0ce5bda7c4","tex":"\\QQ^3","aliases":"QQ^3 QQ QQ^3 Q^3 QQ mathbb","display":null,"kind":"use","excerpt":"vpage6A reached type can still hide a directionTYPED MODEL / U= Q^2 AND V= Q^3Main text: [source reference] The chapter's typed model has a rotation R:U→ U, an inclusion A:U→ V, and a retraction B:V→ U: R=mat0 · -1","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4dafa9764bf19601","tex":"\\QQ^4","aliases":"QQ^4 QQ QQ^4 Q^4 QQ mathbb","display":null,"kind":"use","excerpt":"vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-01a1093edb4895af","tex":"\\QQ^5","aliases":"QQ^5 QQ","display":null,"kind":"use","excerpt":"vpage7Reachability is only an upper boundMODEL A / AN ORBIT THAT STAYS IN TWO DIMENSIONSChapter III, mathematical text: Corollary 3.11 Keep the diamond and add an isolated packet 4. Start with the single seed e_0∈QQ^5. Direct multiplication gives Te_0=e_1+e_2, T^2e_0=0. Every later orbit vector is zero. The orbit hull generated by T is therefore ex","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0989b37fff5d66a9","tex":"\\QQ^6","aliases":"QQ^6 QQ QQ^6 Q^6 QQ mathbb","display":null,"kind":"use","excerpt":"Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ba83fc4ca5c67ec3","tex":"\\QQ_2","aliases":"QQ_2 QQ QQ_2 Q_2 QQ mathbb","display":null,"kind":"use","excerpt":"vfig[.94]dyadic_eigenlattice platecaptionA picture of the ordinary integer lattice, not a drawing of the topology of QQ_2. Filled dots have even second coordinate. Unfilled dots lie in the other coset. The hollow point at (1/2,0) records the image of an integral vector under E_+.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-372683511d4b691a","tex":"\\QQ_p","aliases":"QQ_p Q_p QQ mathbb","display":"{\\mathbb Q}_p","kind":"use","excerpt":"Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"use","excerpt":"Let L'≤ L≤ Z^d, with both indices finite. Choose representative systems R for Z^d/L and R' for Z^d/L'. For r∈ R, define Child(r) = {s∈ R':s+L'⊂ r+L}.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-47f4284788757fb1","tex":"r^{(a)}","aliases":"r^(a)","display":"r^{(a)}","kind":"use","excerpt":"FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] De","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-055d5ee422435b80","tex":"R^*","aliases":"R^*","display":"R^*","kind":"use","excerpt":"vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILUREChapter VII: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5 4/5 · 3/5, Q_a=RP_aR^*, S=∑_a=0^1Q_aP_a=frac35R. The exact Gram matrices give (S-I)^*(S-I)=frac1625I, S^*S=frac925I, normS-I=frac45<1. Consequently the unitary po","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aff4a4490ea52474","tex":"r^{-1}","aliases":"r^-1","display":"r^{-1}","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-072e7fe97b551a6a","tex":"r^{-2}","aliases":"r^-2","display":"r^{-2}","kind":"use","excerpt":"The B ezout identity gives N=rI-UF. The two stated commutation relations and FG=0 give N(UF)=VGUF=VUGF=VUFG=0. Therefore N^2=N(rI-UF)=rN. Since r≠0, E^2=r^-2N^2=E. If Fx=0, the evaluated B ezout identity gives VGx=rx, hence Ex=x. If Gx=0, then Ex=0. The lattice assertion follows directly from rE=VG.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a345887535b62c23","tex":"R^{-m}","aliases":"R^-m","display":"R^{-m}","kind":"use","excerpt":"[Leading dominant zero term] Under the hypotheses of [source reference], P_N,n(r) = · m!binomm+r-1r-1 ∑_ρ∈ Z_R fracρ^-m(-ρ D_N'(ρ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term is absent.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08b492aa785b5eaf","tex":"R^{1/N}","aliases":"R^1/N","display":"R^{1/N}","kind":"use","excerpt":"[Zero counting asymptotics] Let n_ D_N(R) count the zeros of D_N in |z|≤ R, with multiplicity. Let n_Δ_N(R) count the zeros of Δ_N in |w|≤ R. Then, as R→∞, n_ D_N(R) = fracNsin(π/N)πR+O(1) and n_Δ_N(R) = fracsin(π/N)πR^1/N+O(1).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"Assume now that a>0 and b>0. Take R= frac1-sqrt1-4ab2b. Then a+bR^2=R and 2bR<1. Apply [source reference]. The number of full ordered binary trees with n leaves is the Catalan number C_n-1, which gives the displayed majorant.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3d472d51734fedb","tex":"\\mathbb R^2","aliases":"R^2 mathbb","display":"\\mathbb R^2","kind":"use","excerpt":"[htbp] centering includegraphics[width=.96linewidth]correction_cokernel_plane.pdf captionA specified correction range in R^2. For L(a,b)=(a,0), the image is the horizontal axis. The defect (2,3) splits into (2,0) in that image and the remaining representative (0,3). Subtracting the image component leaves the same cokernel class. The functional (x,y","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96dee922af39121c","tex":"r^3","aliases":"r^3","display":"r^3","kind":"use","excerpt":"ynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. Substituting x_1=rκ_1, x_2=rκ_2, x_3=2rκ_3 and multiplying by 6!/3! gives P_2,3(r)=15r^3+30r^2","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe9a73354b93f7b7","tex":"R^d","aliases":"R^d","display":"R^d","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":44},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3483b609e1120b1c","tex":"R^e","aliases":"R^e","display":"R^e","kind":"use","excerpt":"rmation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisfies y-y_0 · ∈ p^M-sR^e,","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4935b7ed17587997","tex":"R^{e+k}","aliases":"R^e+k","display":"R^{e+k}","kind":"use","excerpt":"[Joint coefficient and parameter deformation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisf","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac364b379fe8ae12","tex":"R^I","aliases":"R^I","display":"R^I","kind":"use","excerpt":"pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied s","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a448f4c18ae5322e","tex":"R^k","aliases":"R^k","display":"R^k","kind":"use","excerpt":"[Joint coefficient and parameter deformation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisf","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4179ac2053e2353","tex":"R^m","aliases":"R^m","display":"R^m","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e64c4ff18fdf826","tex":"R^{m+1}","aliases":"R^m+1","display":"R^{m+1}","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4415f2a3fc7b65ea","tex":"R^n","aliases":"R^n","display":"R^n","kind":"use","excerpt":"The actual submodules U=imE and W=ker E are invariant, and addition gives an equivariant isomorphism U⊕ W≃ R^n. The inclusion i:U↪ R^n and retraction r(v)=Ev are constructed as linear maps and satisfy ri=1_U and ir=E. Their equivariance is proved. Coordinatewise reduction maps U onto im(bar Ebmod(π)^k) and maps W onto ker(bar Ebmod(π)^k). Both equa","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f3447fa324f7dc6","tex":"r^N","aliases":"r^N","display":"r^N","kind":"use","excerpt":"The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3b6ec511215a68d","tex":"r^n","aliases":"r^n","display":"r^n","kind":"use","excerpt":"For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9039d8e025c2839","tex":"R^{\\NN}","aliases":"R^NN R^ N NN mathbb","display":"R^{{\\mathbb N}}","kind":"use","excerpt":"[Hurwitz series ring] Let R be a commutative ring with identity. The Hurwitz series ring H(R) is the set R^ N with componentwise addition and product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_r b_n-r. Its identity is δ=(1,0,0,…).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee18dedc5a126f93","tex":"R^{r-1}","aliases":"R^r-1","display":"R^{r-1}","kind":"use","excerpt":"[Polynomial tree expansion] Suppose there is R>0 such that ϕ(R)≤ R and q= ∑_r=2^dr b_rR^r-1<1. Then the map Φ(x)=s+∑_r=2^dB_r(x,…,x) has a unique fixed point in the closed ball overline B_R(0). The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42f97ca6df89f2f9","tex":"R^T","aliases":"R^T","display":"R^T","kind":"use","excerpt":"1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. vfigcompact_core The actual core is G=R_LR_R^T=mat5 · 0 · 0 0 · 2 · 0, LR^T=Q_LG. Its dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f06eb326e26feef5","tex":"R^V","aliases":"R^V","display":"R^V","kind":"use","excerpt":"[Orbit support obstruction] Let S⊆ V, and let x∈ R^V be supported on S. If j is not reachable by a directed path from any vertex of S, then (T^nx)_j=0 for every n∈ N. Hence the orbit under T of x cannot generate a vector with a nonzero coordinate at j. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Reachable; Komorebi.PaperII","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28973fc71dc16dde","tex":"R^{V\\times V}","aliases":"R^V× V times","display":"R^{V\\times V}","kind":"use","excerpt":"The passage from words to walks also applies to products of different matrices. Let V be one finite packet type, let R be a semiring, and let B_1,…,B_m∈ R^V× V be a finite sequence of possibly different packet matrices. Rows are targets and columns are sources as before.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-147dbf37fcbf55c9","tex":"R^\\tau","aliases":"R^τ tau","display":"R^\\tau","kind":"use","excerpt":"is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a77b9a5a0ed9bc86","tex":"R^\\nabla","aliases":"R^∇ nabla","display":"R^\\nabla","kind":"use","excerpt":"Cancellation of the second derivatives. The product rule and the identity for the ambient Lie bracket reduce curvature to R^∇(X,Y)s=P d P[X] d s[Y]-P d P[Y] d s[X]. Substitute the identity obtained by differentiating the range condition for the section. The terms containing P d P[v]P vanish, leaving R^∇(X,Y)s= P( d P[X] d P[Y]- d P[Y] d P[X])P s. L","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-236f80d54e55c576","tex":"R_*","aliases":"R_*","display":"R_*","kind":"use","excerpt":"rest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4dc998495c46ac8","tex":"R_*^{-m}","aliases":"R_*^-m","display":"R_*^{-m}","kind":"use","excerpt":"rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"[The first nontrivial dyadic row] For n=4, R_4=frac9115, R_2=frac53, R_0=1. The exact kernel coefficients are κ_2,1=-frac83, κ_2,2=frac1285. Therefore E_4 = frac9115 -frac83binom42frac53 +frac1285 =5. The rational source values are dyadic units, while the transfer depth is carried by the kernel.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"The standard window supplies the initial ratio R_1. The weighted estimate handles pmid c, and Riccati propagation handles pnmid c. Flatness is closed under products, so the exact factorization establishes flatness of j! P_N,j. Extracting its supported Hurwitz coefficients and applying the proved saturation and step-multiple lemmas yields [source re","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5c89dab9c39f1c3","tex":"R_1^*","aliases":"R_1^*","display":"R_1^*","kind":"use","excerpt":"The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery return","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96f4b178fc8132b9","tex":"R_1^2","aliases":"R_1^2","display":"R_1^2","kind":"use","excerpt":"3/5 · 4/5 obeys R_1^2=mat7/25 · -24/25 24/25 · 7/25. The multiplication is exact and each step has a valid local overlap. The convergence theorem has a separate analytic contract: a twice continuously differentiable complete projector path on a compact time interval, sufficiently fine partitions, and a constant depending on the path in an O(|π|) es","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7fe315bf8e2207f","tex":"R_{2(r-m)}","aliases":"R_2(r-m)","display":"R_{2(r-m)}","kind":"use","excerpt":"rgetDyadicSeries.dyadic_reconstruction_series pageheading3. Exact reconstruction and genuine dyadic congruenceTheorem 5.3 and Corollary 5.4; manuscript pages 14--15 The reciprocal coefficients β_m use an ordinary power expansion. In contrast, the source and Euler series use exponential coefficients. Extracting degree 2r from B(2t)F(t)=sech t theref","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-689524629f5e72c1","tex":"R_2^*","aliases":"R_2^*","display":"R_2^*","kind":"use","excerpt":"The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery return","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4056c2b0608f31f7","tex":"R_2^{-1}","aliases":"R_2^-1","display":"R_2^{-1}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8078a9a9bb651de","tex":"R_{2n}","aliases":"R_2n","display":"R_{2n}","kind":"use","excerpt":"[Analytic dyadic source] The function mathscr R belongs to the dyadic Tate algebra and takes values in Z_2. For every nonnegative integer n, mathscr R(n)=R_2n.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-742b3c1f9a0009ba","tex":"R_{2r}","aliases":"R_2r","display":"R_{2r}","kind":"use","excerpt":"The source has valuation exactly zero. At even degree 2r, define the literal rational source R_2r=frac3^2r+2-14(2r+1)(2r+2). Both numerator and denominator are nonzero. The pinned Mathlib lifting-the-exponent theorem gives v_2(3^2r+2-1)=2+v_2(2r+2). Because 2r+1 is odd, the denominator has the same valuation. Consequently, v_2(R_2r)=0, |R_2r|_2=1. ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0aa3589089d58ba","tex":"R_3","aliases":"R_3","display":"R_3","kind":"use","excerpt":"The only nondominant exponent is -1. Hence R_3(t) = exp((-frac32+fracsqrt32i)t). At t=t_m, its phase is i(-1)^m. Insert this value into [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0fe85010c113a2aa","tex":"R_4","aliases":"R_4","display":"R_4","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38ee3d5976c0e937","tex":"R_6","aliases":"R_6","display":"R_6","kind":"use","excerpt":"al checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac13,frac115,-frac2189,frac1675). These values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives (R_0","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43a442af86882c80","tex":"R_8","aliases":"R_8","display":"R_8","kind":"use","excerpt":"al checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac13,frac115,-frac2189,frac1675). These values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives (R_0","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33462a02fcce1a84","tex":"R_a","aliases":"R_a","display":"R_a","kind":"use","excerpt":"rthogonal rank-two packets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an actual channel T","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e93838ea3bf8d2c","tex":"r_A","aliases":"r_A","display":"r_A","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb14b1a407e1de8e","tex":"r_a","aliases":"r_a","display":"r_a","kind":"use","excerpt":"The actual ratio is also proved to be an expansion in t^N, so its coefficients vanish outside multiples of N. Constructed Z_p lifts and the established coefficient criterion now give, for its literal Hurwitz coefficients r_a(n), Δ_uh^kr_a(n)∈ p^min(k,mn) Z_p. The separate rational precision theorem is obtained through the coefficient-lift equality.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-200e88e9f4a8f4ed","tex":"R_a^*","aliases":"R_a^*","display":"R_a^*","kind":"use","excerpt":"ets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an actual channel The operators R_a=E_a^*Q","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-01bc47d575242d41","tex":"R_{ap}","aliases":"R_ap","display":"R_{ap}","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6016f5cf675a3ae","tex":"r_B","aliases":"r_B","display":"r_B","kind":"use","excerpt":"[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4bdf782a5897d64d","tex":"r_b","aliases":"r_b","display":"r_b","kind":"use","excerpt":"For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28d6d92025672629","tex":"R_{\\bar x}","aliases":"R_bar x bar","display":"R_{\\bar x}","kind":"use","excerpt":"[Represented complete deformation problems] When a deformation functor is represented by a complete local ring R_bar x, its O-valued points are continuous local maps R_bar x→ O. Completeness identifies such maps with compatible maps to every quotient O/ϖ^m.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37f2bdf109707126","tex":"R_c","aliases":"R_c","display":"R_c","kind":"use","excerpt":"Inverse witnesses from one side and norm control. For continuous linear maps, the same identity gives the certified norm bound C R_f-R_c C ≤ R_c Δ R_f , from a left inverse witness for the coarse scalar shift and a right inverse witness for the fine scalar shift. These witnesses suffice for the displayed identity. On their own, they need not define","count":31},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e58d2fe1775cda0e","tex":"R_c'","aliases":"R_c'","display":"R_c'","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6f893e6747fcef8","tex":"R_{c-1}","aliases":"R_c-1","display":"R_{c-1}","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5cd09b860b1a4d1a","tex":"R_{\\mathrm{can}}","aliases":"R_can mathrm","display":"R_{\\mathrm{can}}","kind":"use","excerpt":"[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf206f644888b16a","tex":"R_D","aliases":"R_D","display":"R_D","kind":"use","excerpt":"[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-001ad2900a7bd426","tex":"r_D","aliases":"r_D","display":"r_D","kind":"use","excerpt":"The hyperref[book:x-construction-13]finite-exponential reconstruction also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I an","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6cf1fb444bcb5d56","tex":"r_E","aliases":"r_E","display":"r_E","kind":"use","excerpt":"The hyperref[book:x-construction-13]finite-exponential reconstruction also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I an","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3975fa3c319f1d3","tex":"r_e","aliases":"r_e","display":"r_e","kind":"use","excerpt":"Global comparison. For any other feasible ε, set r_e=ε_e/ε_e^*>0. The budget identities give ∑_ec_er_e=C. Using the standard analytic inequality log r≤ r-1 yields ∑_ec_elog r_e≤∑_ec_e(r_e-1)=0. Lean separately proves the exact objective-difference identity J(ε)-J(ε^*)=-∑_ec_elog r_e. Therefore J(ε)≥ J(ε^*).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5dae4eff487bb83e","tex":"R_{\\mathrm{extra}}","aliases":"R_extra mathrm","display":"R_{\\mathrm{extra}}","kind":"use","excerpt":"The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary d","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7fa83f4a07c4cba","tex":"R_{\\mathrm{extra}}^*","aliases":"R_extra^* mathrm","display":"R_{\\mathrm{extra}}^*","kind":"use","excerpt":"The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a94836df7b97d193","tex":"R_f","aliases":"R_f","display":"R_f","kind":"use","excerpt":"Inverse witnesses from one side and norm control. For continuous linear maps, the same identity gives the certified norm bound C R_f-R_c C ≤ R_c Δ R_f , from a left inverse witness for the coarse scalar shift and a right inverse witness for the fine scalar shift. These witnesses suffice for the displayed identity. On their own, they need not define","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bb48203991c2d2f","tex":"r_f","aliases":"r_f","display":"r_f","kind":"use","excerpt":"[Coarse vanishing and fine vanishing] Let [r_f]∈coker D_f. If its image in coker D_c is zero, then [r_f] comes from coker D_0. Coarse vanishing therefore means that the fine obstruction is supported in information removed by coarsening. It does not imply [r_f]=0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8af058f9467a00f","tex":"R_{\\frakp}","aliases":"R_frakp R_ p frakp mathfrak","display":"R_{{\\mathfrak p}}","kind":"use","excerpt":"Let p∈Spec R. If Ann_R(M)not⊆ p, choose a∈Ann_R(M)∖ p. The element a becomes a unit in R_ p, so M_ p=0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a4a08a9899f99d7","tex":"R_{\\frakp}^n","aliases":"R_frakp^n R_ p^n frakp mathfrak","display":"R_{{\\mathfrak p}}^n","kind":"use","excerpt":"from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal min","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bfa7220254bbeabc","tex":"\\bm R_{\\mathrm{H}}","aliases":"R_H bm mathrm","display":"\\bm R_{\\mathrm{H}}","kind":"use","excerpt":"1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a14eeb4f04e9a5c","tex":"r_i^{(a)}","aliases":"r_i^(a)","display":"r_i^{(a)}","kind":"use","excerpt":"FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] De","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53e3e9d1c2a6131f","tex":"R_{j+1}","aliases":"R_j+1","display":"R_{j+1}","kind":"use","excerpt":"Let K_j(t) be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is F_N,j(t)=j! K_j(t)^-1. The source/inverse identities and the factorial successor identity prove F_N,j+1=F_N,jR_j+1, F_N,j=F_N,0∏_c=1^jR_c. The product formula is not an assembly hypothesis. Proved flatness of all factors and the o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5a0eed759a6f31c","tex":"r_{j-i}","aliases":"r_j-i","display":"r_{j-i}","kind":"use","excerpt":"Toeplitz padding and column-major vectorization. textitSource items 4.4 and 4.5 Toeplitz embedding. For n>0, define T_ij= c_i-j, · j≤ i, r_j-i, · i<j. The diagonal is supplied by c_0, and the coordinate r_0 is unused. The constructed generator of length 2n is g=(c_0,…,c_n-1,0,r_n-1,…,r_1), and the padding map is the actual finite-index sum eliminat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0a3357a085fae77","tex":"R_K","aliases":"R_K","display":"R_K","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a64e897db39c6a7f","tex":"R_k","aliases":"R_k","display":"R_k","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source refer","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"use","excerpt":"Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61d7b9e7617499dd","tex":"r_{k-j}","aliases":"r_k-j","display":"r_{k-j}","kind":"use","excerpt":"2,-1)^T. The diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column g=(2,3,4,0,5,-1)^T and pad the input with three zeros. vfigtoeplitz headRead the wrapped indices For output index j<3, an input index k≤ j uses the first-column coefficient c_j-k. An input index k>j wraps to the sto","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eae30801c8c7f23d","tex":"R_k^*","aliases":"R_k^*","display":"R_k^*","kind":"use","excerpt":"Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. Define L_k=Q_LW_kΣ_k^1/2, R_k=Q_RZ_kΣ_k^1/2. Then L_kR_k^* i","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61bdbf136c2dac84","tex":"R_L","aliases":"R_L","display":"R_L","kind":"use","excerpt":"[Exact finite precision truncation] Under the hypotheses of [source reference], define R_L = ∑_W(γ)<L(-1)^|γ|A_γ D^-1. Then T^-1-R_L∈ F^L B, TR_L-I∈ F^L B, R_LT-I∈ F^L B. Thus R_L is the exact inverse in B/F^L B.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d96a67e19d75bf1d","tex":"r_L","aliases":"r_L","display":"r_L","kind":"use","excerpt":"AtlasHeadA compatible root is built one digit at a timeatlas:smooth Take F(x)=x^2-7 over the 3-adic integers and start with x_1=1 modulo three. If F(x_L)=3^Lr_L, the next representative has the form x_L+1=x_L+3^Lt_L. Its correcting digit satisfies r_L+2x_Lt_L≡0pmod3. AtlasImagesmooth_tower For example, F(1)/3=-2 and 2x_1=2, so the next digit is one","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25ee15775b880083","tex":"R_m","aliases":"R_m","display":"R_m","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-272cb3ca57c00592","tex":"r_m","aliases":"r_m","display":"r_m","kind":"use","excerpt":"Every nonzero term in [source reference] contains at least m_0 time integrations. The simplex 0<r_1<⋯<r_m<t has volume t^m/m!, so the first possible order is t^m_0. If there is no path, every packet term vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2cf9f3a4501d1a5","tex":"R_{m+1}","aliases":"R_m+1","display":"R_{m+1}","kind":"use","excerpt":"[Packet Taylor formula] Assume F is m+1 times continuously Fr echet differentiable on an open set containing the segment x+th for 0≤ t≤1. Then Q_γ F(x+h) = · Q_γ F(x) + ∑_r=1^mfrac1r! ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h,…,P_α_rh] · +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] d","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b35fced5b4293ce5","tex":"r_{m+1,m}","aliases":"r_m+1,m","display":"r_{m+1,m}","kind":"use","excerpt":"Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eee5b018d7f83622","tex":"R_{m,m}","aliases":"R_m,m","display":"R_{m,m}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d151cfb191c2c72b","tex":"R_{m-1}","aliases":"R_m-1","display":"R_{m-1}","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m], R_m · =[R_m-1 (D^(m-1))^*V_m]. Substitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-147aec0806fe40f4","tex":"R_{m-1}^*","aliases":"R_m-1^*","display":"R_{m-1}^*","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f02b5ce794f70074","tex":"R_m^*","aliases":"R_m^*","display":"R_m^*","kind":"use","excerpt":"Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f7c3db2a7afd485","tex":"R_{\\min}","aliases":"R_min min","display":"R_{\\min}","kind":"use","excerpt":"[Amortization threshold] Let s≥0 be a one time setup cost. Let t_b,t_c≥0 be the baseline and candidate times per use and let R∈ N be a reuse count. If t_b>t_c, the candidate is faster after R uses exactly when R> fracst_b-t_c. The smallest integer reuse count is R_min = lfloor fracst_b-t_c rfloor+1. If t_b≤ t_c, no positive reuse count amortizes a ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"use","excerpt":"t poles are ρ=±π/2. They contribute equally to even-degree coefficients. Including both poles gives P_2,n(1)∼2(2n)!(frac2π)^2n+1. The factor two is orbit multiplicity, not an adjustable fitted constant. vfigpole_asymptotic headA finite quantitative check For n=1,…,8, the exact integer coefficients and a rational enclosure for π bound the ratio R_n ","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1bb952727eb34a88","tex":"R_N'","aliases":"R_N'","display":"R_N'","kind":"use","excerpt":"Fix a large m, put a=2is_N, and consider the closed disk B_m={z:|z-t_m|≤ 1/(2s_N)}. The remainder bounds hold uniformly on these fixed-radius disks. For sufficiently large m, |R_N|≤ 1/2 and |R_N'|≤ s_N/2 there. Define Φ_m(z)=t_m+fraclog(1+R_N(z))2is_N, using the principal logarithm. The estimate |log(1+u)|≤(3/2)|u| shows that Φ_m maps B_m into itse","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7691c6d3fc75621","tex":"R_{N,k}","aliases":"R_N,k","display":"R_{N,k}","kind":"use","excerpt":"Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8a8516575d862bd","tex":"R_{n-1}","aliases":"R_n-1","display":"R_{n-1}","kind":"use","excerpt":"Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d4ab28ea1385868","tex":"r_{n-1}","aliases":"r_n-1","display":"r_{n-1}","kind":"use","excerpt":"[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a25647980501aee","tex":"R_{n-2}","aliases":"R_n-2","display":"R_{n-2}","kind":"use","excerpt":"The first three sharp truncations are E_n · ≡ R_n · · pmod 8, E_n · ≡ R_n-frac83binom n2R_n-2 · · pmod 128, E_n · ≡ R_n-frac83binom n2R_n-2 +frac1285binom n4R_n-4 · · pmod 2048.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca477fd0b7d4d865","tex":"R_{n-2r}","aliases":"R_n-2r","display":"R_{n-2r}","kind":"use","excerpt":"[Dyadic Euler coordinate pair] For every even n≥0, E_n = ∑_r=0^n/2 β_r2^2r fall n2r R_n-2r. The inverse transform is R_n = ∑_r=0^n/2 frac2^4r+1(2r+2)! fall n2r E_n-2r.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f00871f36db30a0","tex":"R_{n-4}","aliases":"R_n-4","display":"R_{n-4}","kind":"use","excerpt":"The first three sharp truncations are E_n · ≡ R_n · · pmod 8, E_n · ≡ R_n-frac83binom n2R_n-2 · · pmod 128, E_n · ≡ R_n-frac83binom n2R_n-2 +frac1285binom n4R_n-4 · · pmod 2048.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48a0e102d8c6aa7a","tex":"R_N^{(q)}","aliases":"R_N^(q)","display":"R_N^{(q)}","kind":"use","excerpt":"η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30accea853c1a158","tex":"R_N^{-1}","aliases":"R_N^-1","display":"R_N^{-1}","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b85fe58dfd4105d","tex":"R_P","aliases":"R_P","display":"R_P","kind":"use","excerpt":"Retaining the bracket of vector fields. Define the actual curvature action by R_P(X,Y)s=∇_X∇_Ys-∇_Y∇_Xs-∇_[X,Y]s. The source uses the Lie bracket defined through Fr echet derivatives. Symmetry of second derivatives is supplied by the C^2 hypothesis, not postulated separately. Expanding the connection gives R_P(X,Y)s=P(DP[X]Ds[Y]-DP[Y]Ds[X]). Substi","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5c9a939784c608f","tex":"R_p","aliases":"R_p","display":"R_p","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed27d154616b4c67","tex":"R_R","aliases":"R_R","display":"R_R","kind":"use","excerpt":"vpage11Stored width is not intrinsic rankACTUAL RANGE COORDINATESChapter VIII: Theorem 5.6, Section F.1. Choose the isometry Q_L=mat3/5 · 0 4/5 · 0 0 · 1 0 · 0, R_L=mat1 · 0 · 1 0 · 1 · 0, R_R=mat3 · 0 · 2 0 · 2 · 0 1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. v","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e4761712190b8fb","tex":"R_r","aliases":"R_r","display":"R_r","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-937de87a1560a7e8","tex":"\\mathcal R_r","aliases":"R_r mathcal","display":"\\mathcal R_r","kind":"use","excerpt":"in V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ace9c120d24521df","tex":"r_R","aliases":"r_R","display":"r_R","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dbe6bc0f4ac42150","tex":"R_R^*","aliases":"R_R^*","display":"R_R^*","kind":"use","excerpt":"[Small core recompression] Let L∈ C^m× r, R∈ C^n× r. Take compact QR factorizations L=Q_LR_L, R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5efa3f3752f2e9d1","tex":"R_r^*","aliases":"R_r^*","display":"R_r^*","kind":"use","excerpt":"recise finite channel statement Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-868ac8e8bd426dc4","tex":"R_R^T","aliases":"R_R^T","display":"R_R^T","kind":"use","excerpt":"1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. vfigcompact_core The actual core is G=R_LR_R^T=mat5 · 0 · 0 0 · 2 · 0, LR^T=Q_LG. Its dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-319463847a07b498","tex":"R_{\\rm can}","aliases":"R_rm can rm","display":"R_{\\rm can}","kind":"use","excerpt":"[Frame bounds from the polyphase symbol] Assume that A I_N ≤ E( e^ iω)^*E( e^ iω) ≤ B I_N for almost every ω, where 0<A≤ B<∞. Then the analysis bank satisfies A x _2^2 ≤ A_E x _2^2 ≤ B x _2^2. The pointwise matrix R_rm can( e^ iω) = (E^*E)^-1E^* is a bounded left inverse and gives the canonical dual synthesis bank.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7b985b4cff51815","tex":"R_s","aliases":"R_s","display":"R_s","kind":"use","excerpt":"[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c62b3900bbb09bf","tex":"r_{\\mathrm{spec}}","aliases":"r_spec mathrm","display":"r_{\\mathrm{spec}}","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a296862ac6ebceae","tex":"r_{\\widehat\\tau}","aliases":"r_widehatτ widehat tau","display":"r_{\\widehat\\tau}","kind":"use","excerpt":"y on the interval between its two locations: H_τ-H_widehatτ _2=sqrtδ=tfrac14, H_τ-H_widehatτ _∞=1. vfig[.89]ext_location headFor a ramp, the error height shrinks too Write r_τ(t)=(t-τ)_+. The difference grows linearly from zero to δ on [τ,widehatτ] and stays at δ on [widehatτ,1]. Direct integration gives r_τ-r_widehatτ _2^2 =fracδ^33+(1-widehatτ)δ^","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3088a38f12fda26d","tex":"R_x","aliases":"R_x","display":"R_x","kind":"use","excerpt":"1 · -1. Retain the first coordinate and eliminate the second. For z away from the relevant poles, the retained entry of (zI-A)^-1 is R_x(z)=[z+1-frac1z+1]^-1 =fracz+1z(z+2). The term Σ(z)=1/(z+1) is the hidden coordinate's self-energy. vfigresolvent platecaptionThe full inverse gives a rational response with two poles, at 0 and -2. Its partial frac","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd0c6e5c3a03a755","tex":"r_{x'}","aliases":"r_x'","display":"r_{x'}","kind":"use","excerpt":"[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68df6b2bfb142937","tex":"r_\\alpha","aliases":"r_α alpha","display":"r_\\alpha","kind":"use","excerpt":"of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fce729b05bf0e1ea","tex":"R_{\\alpha\\beta}","aliases":"R_αβ alpha beta","display":"R_{\\alpha\\beta}","kind":"use","excerpt":"[Common refinement of commuting packet systems] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on X. Assume P_α Q_β=Q_β P_α for all α and β. Remove the zero operators from the family R_αβ=P_α Q_β. The remaining operators form a packet system that refines both P and Q. The same statement holds for any finite collection of pairwise commuting pac","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fda162c9e475299","tex":"R_{\\alpha\\beta}^2","aliases":"R_αβ^2 alpha beta","display":"R_{\\alpha\\beta}^2","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9667cad1734fc075","tex":"R_\\Gamma","aliases":"R_Γ Gamma","display":"R_\\Gamma","kind":"use","excerpt":"[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca0ed80fd48b48de","tex":"R_\\gamma","aliases":"R_γ gamma","display":"R_\\gamma","kind":"use","excerpt":"Let R_γ be a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b98833cae963298","tex":"R_\\delta","aliases":"R_δ delta","display":"R_\\delta","kind":"use","excerpt":"Insert I_Y=∑_γ Q_γ in each argument of the outer derivative and expand the finite sums by multilinearity. Apply R_δ to the output. The directions already lie in the declared input ranges, and each intermediate projection lands in its literal output range. The resulting summands are exactly the displayed packet jets. List directions within a block i","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04574488c1d5a80f","tex":"R_\\theta","aliases":"R_θ theta","display":"R_\\theta","kind":"use","excerpt":"Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd35db052b2bfbd8","tex":"r_\\mu","aliases":"r_μ mu","display":"r_\\mu","kind":"use","excerpt":"ator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with ov","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c977b69c1a9d9e20","tex":"r_\\tau","aliases":"r_τ tau","display":"r_\\tau","kind":"use","excerpt":"LITY / A ONE-SIXTEENTH DISPLACEMENTChapter V: Corollary E8.2, Theorem E8.3, Theorem E8.7. Set τ=1/4, widehatτ=5/16, and δ=1/16. A unit step differs only on the interval between its two locations: H_τ-H_widehatτ _2=sqrtδ=tfrac14, H_τ-H_widehatτ _∞=1. vfig[.89]ext_location headFor a ramp, the error height shrinks too Write r_τ(t)=(t-τ)_+. The differe","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e13c04e5f7fac1fc","tex":"\\raggedright","aliases":"raggedright","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b11c81e6903fa9bf","tex":"\\Ran","aliases":"Ran operatorname","display":"{\\operatorname{Ran}}","kind":"use","excerpt":"[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bf28df79e66226e","tex":"\\operatorname{range}","aliases":"range operatorname","display":"\\operatorname{range}","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"Theorem. For this fixed polynomial presentation, the group and distinguished class are independent of the chosen integer representative. The class vanishes if and only if bar x lifts to a solution modulo p^L+1. The obstruction group has p^r-rankJ elements. When lifts exist, their set is a torsor for ker J and contains exactly p^d-rankJ elements.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ed613f72ca73d2d","tex":"\\operatorname{rank}_{\\mathbb F_p}","aliases":"rank_ F_p operatorname mathbb","display":"\\operatorname{rank}_{\\mathbb F_p}","kind":"use","excerpt":"Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Write r_x=F(x)/p^L, b_x=r_xbmod p, A_x=DF(x)bmod p: F_p^d⟶ F_p^e, ρ_x=rank_ F_pA_x. The actual lift set is L_x,L= {y∈( Z/p^L+1 Z)^d: y≡ xpmodp^L, F(y)=0}, where polynomial evaluation in the displayed set is modulo p^L+1. Define O_x= F_p^e/imA_x, o_x=[b_x]. Lean proves # L_x,L= p^d-ρ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c425f07dc5f2732e","tex":"\\rank_{E'}","aliases":"rank_E' rank operatorname","display":"{\\operatorname{rank}}_{E'}","kind":"use","excerpt":"[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb1bc2bcf81a206f","tex":"\\rank_{\\F_p}","aliases":"rank_F_p rank_ F_p rank F operatorname mathbb","display":"{\\operatorname{rank}}_{{\\mathbb F}_p}","kind":"use","excerpt":"[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22cff1e28b352509","tex":"\\rank_{\\FF_p}","aliases":"rank_FF_p rank_ F_p rank FF operatorname mathbb","display":"{\\operatorname{rank}}_{{\\mathbb F}_p}","kind":"use","excerpt":"[Exact number of linear children] If ρ=rank_ F_poverline M, then a solvable depth L class has exactly p^d-ρ children modulo p^L+1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0941f5ac249959a","tex":"\\Rcal_0","aliases":"Rcal_0 R_0 Rcal mathcal","display":"{\\mathcal R}_0","kind":"use","excerpt":"[Exact recovery on the error span] Under the hypotheses of [source reference], define on the syndrome support Q=∑_λ_j>0Q_j the completely positive map R_0(X) = ∑_λ_j>0 V_j^*Q_jXQ_jV_j. It extends to a quantum channel R on End( H). If Φ(X)=∑_aE_aXE_a^* is trace preserving on code states, then RΦ(ρ)=ρ for every density matrix ρ=Pρ P.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9215c3ab638f6fc4","tex":"\\Rcal_N","aliases":"Rcal_N R_N Rcal mathcal","display":"{\\mathcal R}_N","kind":"use","excerpt":"[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-607edbc5319584b2","tex":"\\Reach","aliases":"Reach operatorname","display":"{\\operatorname{Reach}}","kind":"use","excerpt":"[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4321aa573d789e4c","tex":"\\mathrm{rec}","aliases":"rec mathrm","display":"\\mathrm{rec}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06b25b660a3b775f","tex":"\\Res","aliases":"Res operatorname","display":"{\\operatorname{Res}}","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ecf96b50772dd74","tex":"\\operatorname{Res}","aliases":"Res operatorname","display":"\\operatorname{Res}","kind":"use","excerpt":"For the original sign factors the concrete resultant is Res(x-1,x+1)=2 and the bound is sharp. That equality is special to this displayed calculation, not a universal minimality theorem for every resultant certificate.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1da16dc8b9fc5c0","tex":"\\Res_y","aliases":"Res_y Res operatorname","display":"{\\operatorname{Res}}_y","kind":"use","excerpt":"We use the resultant convention Res_y(f,g) = ∏_f(α)=0g(α) when f is monic, with roots counted with multiplicity.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c59cc6b0f70cb080","tex":"\\mathsf{Residual}","aliases":"Residual mathsf","display":"\\mathsf{Residual}","kind":"use","excerpt":"Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).","count":49},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4aa377bca96588f9","tex":"\\RMSE","aliases":"RMSE operatorname","display":"{\\operatorname{RMSE}}","kind":"use","excerpt":"[One-dimensional errors] For 1024 samples on [-1,1) and Fourier cutoff m=18, the retained frequency indices are -18,…,18: 37 bins including the zero-frequency bin. The measured values were lrr method · RMSE · maximum absolute error global Fourier truncation · 5.4023×10^-2 · 3.4968×10^-1 known edge packet · 1.3692×10^-4 · 1.4503×10^-3 estimated edge","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a58e4c2e82e7cd1b","tex":"\\mathrm{root}","aliases":"root mathrm","display":"\\mathrm{root}","kind":"use","excerpt":"The root scan min_0≤ b≤ BF_root(b) is therefore the exact budget-constrained optimum. Increasing the budget cannot increase a finite optimum. The theorem retains feasibility hypotheses where necessary rather than treating codenone as an ordinary real cost.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27fc32faa85e4ff0","tex":"\\operatorname{Row}","aliases":"Row operatorname","display":"\\operatorname{Row}","kind":"use","excerpt":"A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78ecfb9af1021553","tex":"\\RR^2","aliases":"RR^2 R^2 RR mathbb","display":"{\\mathbb R}^2","kind":"use","excerpt":"vpage7Exact does not mean orthogonalPACKET COORDINATES / AN OBLIQUE SPLITTINGMain text: [source reference] In R^2, take P=mat1 · -1","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-162e5e4a459cac13","tex":"\\RR^{m\\times p}","aliases":"RR^m× p R^m× p RR times mathbb times","display":"{\\mathbb R}^{m\\times p}","kind":"use","excerpt":"[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the ine","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-694d5f728e8d54c6","tex":"\\RR^n","aliases":"RR^n R^n RR mathbb","display":"{\\mathbb R}^n","kind":"use","excerpt":"Let Δ_n-1 = {p∈ R^n: p_i≥0, ∑_ip_i=1 } be the probability simplex. Let λ∈[0,∞)^n have positive evidence Z_λ(p)=∑_iλ_ip_i. The Bayesian transfer is B_λ(p) = fracλodot pZ_λ(p).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2831aff814dfd9d4","tex":"\\RR^{p\\times n}","aliases":"RR^p× n R^p× n RR times mathbb times","display":"{\\mathbb R}^{p\\times n}","kind":"use","excerpt":"[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the ine","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2b4530a730b06f1","tex":"\\mathrm{run}","aliases":"run mathrm","display":"\\mathrm{run}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61f9609b70de73a0","tex":"\\rVert","aliases":"rVert","display":"\\rVert","kind":"use","excerpt":"lineskip [Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-226626f51f7539ed","tex":"\\rvert","aliases":"rvert","display":"\\rvert","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c41f602ac2f82c8e","tex":"\\rvert_p^2","aliases":"rvert_p^2 rvert","display":"\\rvert_p^2","kind":"use","excerpt":"The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ffddd5883e6d57f","tex":"\\boldsymbol s","aliases":"s boldsymbol","display":"\\boldsymbol s","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17e6e62cdd37d68b","tex":"\\mathsf S","aliases":"S mathsf","display":"\\mathsf S","kind":"use","excerpt":"[Polyphase shift matrix] Let E be the scalar shift (Ea)_n=a_n+1, and let E_q shift the coarse index. Then P_M(Ea) = S_M(E_q) P_Ma, where S_M(E_q) = 0 · 1 · 0 · ⋯ · 0","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5fea3d441d7eabe0","tex":"S''","aliases":"S''","display":"S''","kind":"use","excerpt":"· =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b5e6d175de4b2dc","tex":"s^{-1/2}","aliases":"s^-1/2","display":"s^{-1/2}","kind":"use","excerpt":"A normalized density need not have a finite mean. For large s, the exponential factor tends to one and η_t(s) is proportional to s^-3/2. Consequently sη_t(s) has a nonintegrable s^-1/2 tail. The clock has infinite mean even though its total mass is one. takeawaySubordination is a weighted mixture of diffusion times. Replacing the clock distribution","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c1256dd757c5aec0","tex":"s^{-3/2}","aliases":"s^-3/2","display":"s^{-3/2}","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e31cead1bdb80910","tex":"S^{-r}","aliases":"S^-r","display":"S^{-r}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bea69cee90538d01","tex":"s^{[N]}","aliases":"s^[N]","display":"s^{[N]}","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14e505b4cdf29e83","tex":"S^e","aliases":"S^e","display":"S^e","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a36ef00e19010338","tex":"S^H","aliases":"S^H","display":"S^H","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-954f18f50adf21ae","tex":"S^k","aliases":"S^k","display":"S^k","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c863c73afd645b31","tex":"S^m","aliases":"S^m","display":"S^m","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb0124a734ec8cdf","tex":"s^m","aliases":"s^m","display":"s^m","kind":"use","excerpt":"Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9818719b46fe474","tex":"S^N","aliases":"S^N","display":"S^N","kind":"use","excerpt":"Let N≥1 be an integer, and let S be a unitary symmetry whose order divides N: S^N=I. Set ω=e^2π i/N.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-977c562e1c896da9","tex":"S^{\\mathrm{pr}}_{L,N}","aliases":"S^pr_L,N mathrm","display":"S^{\\mathrm{pr}}_{L,N}","kind":"use","excerpt":"et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8768859caa7d360e","tex":"S^r","aliases":"S^r","display":"S^r","kind":"use","excerpt":"Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer bloc","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1b9361af622cca4","tex":"S^{r+s}","aliases":"S^r+s","display":"S^{r+s}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-569a7d9157ab32ee","tex":"S^{r+s\\bmod N}","aliases":"S^r+sbmod N bmod","display":"S^{r+s\\bmod N}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac11a04bc7722918","tex":"S^{\\operatorname{val}(r)}","aliases":"S^val(r) operatorname","display":"S^{\\operatorname{val}(r)}","kind":"use","excerpt":"003 Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val(r). Reduction of exponents modulo N ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51a7b924b79e9f32","tex":"s^\\alpha","aliases":"s^α alpha","display":"s^\\alpha","kind":"use","excerpt":"Set R_f(s)=(sI+L_f)^-1, R_c(s)=(sI+L_c)^-1. The resolvent identity gives CR_f(s)-R_c(s)C = -R_c(s)Δ_LR_f(s). Since LR=I-sR, CL_fR_f(s)-L_cR_c(s)C = sR_c(s)Δ_LR_f(s). Insert this into [source reference]. The spectral lower bound gives normR_f(s),normR_c(s) ≤ frac1s+m. Therefore normCL_f^α-L_c^α C · ≤ fracsin(πα)π normΔ_L ∫_0^∞ fracs^α(s+m)^2 d s · =","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0906a9a73912e8a8","tex":"s^{\\alpha-1}","aliases":"s^α-1 alpha","display":"s^{\\alpha-1}","kind":"use","excerpt":"[Balakrishnan formula in finite dimension] For 0<α<1, L^α = fracsin(πα)π ∫_0^∞ s^α-1L(sI+L)^-1 d s. The integral converges in operator norm.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67f7cd56268f3fc1","tex":"S^{\\oplus m}","aliases":"S^⊕ m oplus","display":"S^{\\oplus m}","kind":"use","excerpt":"[Multiplicity obstruction] For the diagonal action of M_d(D) on S^⊕ m, srank_M_d(D)(S^⊕ m) = lceilfracmdrceil. [source reference]","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"vpage19Precision exposes directional freedomMODEL E / A FIBER BUDGET AT EACH DEPTHChapter II, mathematical text: Theorem 10.6 Take p=2 and the supplied diagonal weights (s_0,s_1,s_2)=(0,1,3). At depth L, an admissible target has 2^W(L) solutions, W(L)=min(0,L)+min(1,L)+min(3,L). Each direction contributes only the amount visible at the current dept","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db9af401ac382039","tex":"S_2","aliases":"S_2","display":"S_2","kind":"use","excerpt":"[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ce2399701bcdd04","tex":"\\mathcal S_2","aliases":"S_2 mathcal","display":"\\mathcal S_2","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"use","excerpt":"0 · 1 . For every odd prime p, both projectors preserve Z_p^2. At p=2, s_2(E_+)=s_2(E_-)=1. The factors x-1 and x+1 have resultant 2.","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da35858f1dea6007","tex":"S_{2(r-m)}","aliases":"S_2(r-m)","display":"S_{2(r-m)}","kind":"use","excerpt":"ing3. Exact reconstruction and the triangular matrixTheorem 4.2; manuscript page 11 For odd p, rescaling the already proved inverse law gives sech(pt)cosh(pt)=1. Multiplying the even-source identity by sech^2(pt) therefore proves sech^2(pt)(∑_r≥0S_2r(p)fract^2r(2r)!)=sech t. Actual EGF multiplication now gives the integer coefficient formula E_2r=∑","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f398ec10f2f1455","tex":"s_{2,\\Z_2^2}","aliases":"s_2,Z_2^2 s_2, Z_2^2 Z mathbb","display":"s_{2,{\\mathbb Z}_2^2}","kind":"use","excerpt":"[Odd-adic versus dyadic behavior] For every prime p≠2, every entry of E_+ and E_- is p-integral, and each projector has exact local denominator weight zero. At p=2, each projector has exact local denominator weight one. Multiplication by 2 makes it integral, while the unscaled matrix is not integral: s_2, Z_2^2(E_+)=s_2, Z_2^2(E_-)=1. The character","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d79e1a4a3abaf4d8","tex":"S_{2n}","aliases":"S_2n","display":"S_{2n}","kind":"use","excerpt":"The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconst","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a35fc269aba33f6a","tex":"S_{2q}","aliases":"S_2q","display":"S_{2q}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df3b11dcb98597af","tex":"S_{2r}","aliases":"S_2r","display":"S_{2r}","kind":"use","excerpt":"Even projection is identified coefficientwise. For any formal series f, let EvenPart(f)=(f(t)+f(-t))/2. Its nth coefficient is proved to be the coefficient of f when n is even and zero otherwise. Rescaling the inverse law proves sech(-t)=sech(t). The exponential multiplication identity then gives cosh^2(pt)=frac2+e^2pt+e^-2pt4, and hence ∑_r≥0S_2r(","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6d0dc9f5b462d0b","tex":"S_4","aliases":"S_4","display":"S_4","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b06cc43523171b06","tex":"\\mathcal S_4","aliases":"S_4 mathcal","display":"\\mathcal S_4","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9315144502cf9089","tex":"S_6","aliases":"S_6","display":"S_6","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71a07bf98e34b5f3","tex":"S_8","aliases":"S_8","display":"S_8","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65e03ace3c0c077c","tex":"S_{90}","aliases":"S_90","display":"S_{90}","kind":"use","excerpt":"V: Theorem 11.7, Theorem 11.8, Corollary 11.9. The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-785e812c9b93033d","tex":"S_a","aliases":"S_a","display":"S_a","kind":"use","excerpt":"Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d50bfc3cab84f7ec","tex":"\\mathcal S_a","aliases":"S_a mathcal","display":"\\mathcal S_a","kind":"use","excerpt":"A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68387971995ff129","tex":"S_b","aliases":"S_b","display":"S_b","kind":"use","excerpt":"Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc4fbf61a6a9f156","tex":"S_{ba}","aliases":"S_ba","display":"S_{ba}","kind":"use","excerpt":"Hidden elimination. For the actual Schur complement S=A_RR-A_RHA_HH^-1A_HR, the hidden inverse is expanded through the hidden projectors. A nonzero retained correction block requires indices j,k with nonzero entrance and exit blocks and a hidden path jleadsto k. Therefore a nonzero block S_ba requires either a direct retained block, or a⟶ j leadsto","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-92045ce8b164570b","tex":"S_C","aliases":"S_C","display":"S_C","kind":"use","excerpt":"For the reverse inequality, choose for each source component C a packet pure family S_C of size r_C generating X_C under A_C, and put S=⋃_CS_C. Then every source component lies in A S. Traverse the finite condensation graph in a topological order. Suppose all components preceding a component D that is not a source have already been shown to lie in ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db289384e3bfb6f4","tex":"s_d","aliases":"s_d","display":"s_d","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7aa9c815c9ae3803","tex":"s_e","aliases":"s_e","display":"s_e","kind":"use","excerpt":"s_e,ϵ_e · Sensitivity and tolerance · Is the budget input-dependent or uniform over unit inputs?","count":30},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-95d3d602121aab4d","tex":"s_f","aliases":"s_f","display":"s_f","kind":"use","excerpt":"Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided der","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Let S generate M. For every n∈ N, choose m∈ M with Φ(m)=n and write m=∑_ia_is_i. Then n=∑_iφ(a_i)Φ(s_i), so Φ(S) generates N.","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c953e0b08751250","tex":"S_i^{\\oplus m_i}","aliases":"S_i^⊕ m_i oplus","display":"S_i^{\\oplus m_i}","kind":"use","excerpt":"[Semisimple seed rank formula over arbitrary division rings] Let A≅∏_i=1^qM_d_i(D_i) be semisimple Artinian and let M≅⊕_i=1^qS_i^⊕ m_i be a finitely generated left module over A. Then srank_A(M) = max_1≤ i≤ q lceilfracm_id_irceil. If A is any algebra of finite dimension over a field, the same formula applied to the decomposition of M/Jac(A)M over A","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0190c929ce120af","tex":"S_i^{\\oplus rd_i}","aliases":"S_i^⊕ rd_i oplus","display":"S_i^{\\oplus rd_i}","kind":"use","excerpt":"By [source reference], M is generated by r elements exactly when it is a quotient of A^⊕ r. By [source reference], A^⊕ r≅⊕_iS_i^⊕ rd_i. Every short exact sequence of modules over a semisimple ring splits. Therefore a quotient isomorphic to M exists if and only if m_i≤ rd_i for every i. The smallest such r is the displayed maximum. For a nonsemisimp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-85a648879a8456a4","tex":"S_L","aliases":"S_L","display":"S_L","kind":"use","excerpt":"The exceptional primes are controlled by the actual field discriminant: S_L={p rational prime:pmiddisc(L)}. This set is finite. For p∉ S_L, every prime of O_L above (p) is unramified. Lean proves that the actual inertia cardinality is the ramification index. It follows that inertia is trivial and the arithmetic Frobenius at a fixed finite prime is ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3e97de7f305c50f","tex":"\\mathsf S_M","aliases":"S_M mathsf","display":"\\mathsf S_M","kind":"use","excerpt":"[Polyphase shift matrix] Let E be the scalar shift (Ea)_n=a_n+1, and let E_q shift the coarse index. Then P_M(Ea) = S_M(E_q) P_Ma, where S_M(E_q) = 0 · 1 · 0 · ⋯ · 0","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9746e4f6fc18b24","tex":"s_m","aliases":"s_m","display":"s_m","kind":"use","excerpt":"headSeveral seeds give a module of relations For (s_1,…,s_m), evaluation is (f_1,…,f_m)↦∑_j f_j(T)s_j. Its kernel is a submodule of F^m and can include relations between seeds, not only relations annihilating each seed separately.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-593a84ab3d0b94bd","tex":"S_{m+r}","aliases":"S_m+r","display":"S_{m+r}","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6f3fd1d129e9558","tex":"s_{m-1}","aliases":"s_m-1","display":"s_{m-1}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22cf9853e0e53ac8","tex":"S_N","aliases":"S_N","display":"S_N","kind":"use","excerpt":"vpage11The residual tells you when to stopMODEL B / A CERTIFIED CUTOFF, NOT A HEURISTICChapter II, mathematical text: Theorem 5.10, Corollary 5.11, Section B.2 For S_N=∑_k=0^N-1(-A)^k, telescoping gives (I+A)S_N=S_N(I+A)=I-(-A)^N. Thus the exact residual is known. To certify precision L, it is enough to show that every residual entry is divisible b","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78a9cd432e84c71d","tex":"\\mathcal S_N","aliases":"S_N mathcal","display":"\\mathcal S_N","kind":"use","excerpt":"A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cfe0fa40b3c1262c","tex":"s_N","aliases":"s_N","display":"s_N","kind":"use","excerpt":"At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry tr","count":31},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91ec47101c58ab98","tex":"s_n","aliases":"s_n","display":"s_n","kind":"use","excerpt":"First construct the integral solution. For a ring R, a triangular row system T, right-hand side b, and proved unit pivots d_n=T_nn, define s_n=d_n^-1(b_n-∑_q<nT_nqs_q). The new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f71751162a0c66a","tex":"S_{n-2m}","aliases":"S_n-2m","display":"S_{n-2m}","kind":"use","excerpt":"Substituting the independently proved signed-zigzag identity recovers the manuscript's notation. For every even degree n, E_n=∑_m=0^n/2(-1)^m Z_2m+1binom n2mp^2mS_n-2m(p). The Lean version uses n=2r to avoid ambiguous natural-number division. Every even natural degree has that form. Primality is not used in this equality.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b0701b1a8b030e63","tex":"S_{n-2r}","aliases":"S_n-2r","display":"S_{n-2r}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1222b525c58a0fd4","tex":"S_{n-2r}^\\times","aliases":"S_n-2r^× times","display":"S_{n-2r}^\\times","kind":"use","excerpt":"[Depleted odd prime reconstruction] For every even n, (1-σ_pp^n)E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2rp^2rS_n-2r^×(p).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7cd6bf70853b646a","tex":"S_{n-r}","aliases":"S_n-r","display":"S_{n-r}","kind":"use","excerpt":"[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0082aa89275ee7b","tex":"S_n^\\times","aliases":"S_n^× times","display":"S_n^\\times","kind":"use","excerpt":"For an odd prime p, the source S_n(p) contains one nonunit base. Put σ_p=(-1)^(p-1)/2 and S_n^×(p)=S_n(p)-σ_pp^n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2edc85bc58966aa","tex":"S_p","aliases":"S_p","display":"S_p","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25afd0ab9b73955e","tex":"\\mathcal S_p","aliases":"S_p mathcal","display":"\\mathcal S_p","kind":"use","excerpt":"The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconst","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80da4524c0ee2fe5","tex":"s_p","aliases":"s_p","display":"s_p","kind":"use","excerpt":"[Odd row weight] For an odd prime p, the rth nonzero term in the reconstruction has valuation W_p(n,r) = 2r+v_p(Z_2r+1) +v_pbinom n2r +v_p(S_n-2r(p)). The binomial contribution is v_pbinom n2r = fracs_p(2r)+s_p(n-2r)-s_p(n)p-1, where s_p is the base p digit sum.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bcae1d2e5697a14","tex":"S_p^e","aliases":"S_p^e","display":"S_p^e","kind":"use","excerpt":"For every proposed next parameter digit, the previously checked obstruction is [bar r+K_av]∈ S_p^e/imJ_a. The fixed solved Jacobian is surjective, so this class vanishes for every digit at every stage. The actual lift fiber is a torsor for ker J_a. Rank-nullity gives its cardinality p^m-e.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d94f0719f096749","tex":"S_{\\mathrm{packets}}","aliases":"S_packets mathrm","display":"S_{\\mathrm{packets}}","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L I · 0 · 0 0 · I · 0. Replacing L by a scalar 1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19ed83d808a66932","tex":"S_q","aliases":"S_q","display":"S_q","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b939a36dfaae506","tex":"s_q","aliases":"s_q","display":"s_q","kind":"use","excerpt":"First construct the integral solution. For a ring R, a triangular row system T, right-hand side b, and proved unit pivots d_n=T_nn, define s_n=d_n^-1(b_n-∑_q<nT_nqs_q). The new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aea0312207c8a5fa","tex":"S_q^d","aliases":"S_q^d","display":"S_q^d","kind":"use","excerpt":"For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9d25005b5f2d519","tex":"S_q^e","aliases":"S_q^e","display":"S_q^e","kind":"use","excerpt":"For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90a9d0b5e480e51f","tex":"S_q^f","aliases":"S_q^f","display":"S_q^f","kind":"use","excerpt":"[Simultaneous polynomial lifting] Let F∈ Z[X_1,…,X_d]^e and G∈ Z[X_1,…,X_d]^f have a common lower root x modulo q^L, where q≥1 and L≥1. Use the integer-domain reduced Jacobians J: Z^d→ S_q^e and K: Z^d→ S_q^f. Suppose u,v∈ Z^d are separately valid corrections for F and G, respectively. Then [u]=[v] in Z^d/(ker J+ker K) if and only if a single y≡ xp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0dc6c24dfbac6716","tex":"S_q^k","aliases":"S_q^k","display":"S_q^k","kind":"use","excerpt":"Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residu","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6cf58d9247702fdb","tex":"S_q^m","aliases":"S_q^m","display":"S_q^m","kind":"use","excerpt":"Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residu","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7af78049dcddf01","tex":"S_R","aliases":"S_R","display":"S_R","kind":"use","excerpt":"Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b37e9083bef6f614","tex":"S_r","aliases":"S_r","display":"S_r","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb79e1a36c0e4809","tex":"S_V","aliases":"S_V","display":"S_V","kind":"use","excerpt":"[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off-diagonal entries and nonpositive column sums, so it is the subgenera","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8bd0c25cd815480a","tex":"S_V'","aliases":"S_V'","display":"S_V'","kind":"use","excerpt":"[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off-diagonal entries and nonpositive column sums, so it is the subgenera","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed40760b634e85a0","tex":"S_w","aliases":"S_w","display":"S_w","kind":"use","excerpt":"e preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substack","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87a2b7a1a86d2b97","tex":"S_z","aliases":"S_z","display":"S_z","kind":"use","excerpt":"0 · I and S_z= F_P(z) · 0 -A_QP · zI-A_QQ . Direct multiplication gives zI-A=L_zS_z. The first factor is invertible. The claims follow from triangular inversion and the determinant of the second factor. This is the finite block form of the Feshbach reduction [source reference].","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e79192bebf87a0c5","tex":"S_z^{-1}","aliases":"S_z^-1","display":"S_z^{-1}","kind":"use","excerpt":"Correct statement: source Theorem 9.1. Assume both D_z and zI-H are invertible. Then S_z is invertible and ((zI-H)^-1)_PP=S_z^-1. The left side is the submatrix of the full inverse on the retained coordinates. This matches the paper's compressed operator when written in an orthonormal basis of the retained subspace. The proof is valid for arbitrary","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6122497cef209fd0","tex":"S_\\alpha","aliases":"S_α alpha","display":"S_\\alpha","kind":"use","excerpt":"Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71a4eb6dc2d7040e","tex":"S_{\\beta\\alpha}","aliases":"S_βα beta alpha","display":"S_{\\beta\\alpha}","kind":"use","excerpt":"[Tensor product transfer blocks] Let S∈End(X) and T∈End(Y), with packet systems P and Q. Relative to the tensor product packet system of [source reference], (S⊗ T)_(β,δ),(α,γ) = S_βα⊗ T_δγ. [source reference] ExportsTensor network, quantum circuit, and multivariate packet calculations.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39fa26306fe53d57","tex":"S_{\\gamma\\beta}","aliases":"S_γβ gamma beta","display":"S_{\\gamma\\beta}","kind":"use","excerpt":"[Packet matrix multiplication] For S,T∈End_𝕜(X), (ST)_γα = ∑_β∈ IS_γβT_βα. [source reference] ExportsAll packet path expansions.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72365ef92579e114","tex":"S_{\\gamma\\delta}","aliases":"S_γδ gamma delta","display":"S_{\\gamma\\delta}","kind":"use","excerpt":"[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-832402dc4a3bcf4c","tex":"s_{\\Lambda_s}","aliases":"s_Λ_s Lambda","display":"s_{\\Lambda_s}","kind":"use","excerpt":"with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcd92cad9c977304","tex":"s_\\ell","aliases":"s_ℓ ell","display":"s_\\ell","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea05e9c36f74f76a","tex":"\\Scal_{\\Vcal}","aliases":"Scal_Vcal S_ V Scal Vcal mathcal mathcal","display":"{\\mathcal S}_{{\\mathcal V}}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72398ff18d39e1f6","tex":"\\sec","aliases":"sec","display":"\\sec","kind":"use","excerpt":"A unit constant term makes the formal reciprocal unambiguous. To compute one coefficient, only finitely many earlier coefficients are needed. The later analytic theory explains how those exact coefficients relate to the full zero set. modelfigurereciprocal_normalizationAt N=2 and r=1, the reciprocal is sec z. The upper row gives block exponential c","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3f001b6b563f398","tex":"\\sec^2","aliases":"sec^2 sec","display":"\\sec^2","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34726818d802068e","tex":"\\sech^2","aliases":"sech^2 sech operatorname","display":"{\\operatorname{sech}}^2","kind":"use","excerpt":"The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer. This is the integrality required by the later congruence.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91ad15719e4542fe","tex":"\\mathrm{setup}","aliases":"setup mathrm","display":"\\mathrm{setup}","kind":"use","excerpt":"Shared operator families. For reuse weights r_j, the local keep model can aggregate setup and per-operator costs: κ_v=κ_v^setup+∑_jr_jκ_v^(j), ϵ_v=∑_jr_jϵ_v^(j). Lean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including Q and R. The inte","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0237f89d91c7c75","tex":"\\Sfun_N^{(1)}","aliases":"Sfun_N^(1) S_N^(1) Sfun mathsf","display":"{\\mathsf S}_N^{(1)}","kind":"use","excerpt":"acz^Nn+j(Nn+j)! is the residue j part of the exponential. Second, D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)! is the signed basis for y^(N)=-y. The section D_N= D_N,0 is the denominator used in the reciprocal family S_N^(r)= D_N^-r. Set T_N=- D_N^prime/ D_N. Equivalently, T_N is the logarithmic derivative of S_N^(1).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-545e283bf4ccf588","tex":"\\Sfun_N^{(r+s)}","aliases":"Sfun_N^(r+s) S_N^(r+s) Sfun mathsf","display":"{\\mathsf S}_N^{(r+s)}","kind":"use","excerpt":"[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d4ed0b3a6647724","tex":"\\Sfun_N^{(s)}","aliases":"Sfun_N^(s) S_N^(s) Sfun mathsf","display":"{\\mathsf S}_N^{(s)}","kind":"use","excerpt":"[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6365d83fdc80b979","tex":"\\mathrm{sgn}","aliases":"sgn mathrm","display":"\\mathrm{sgn}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28f39655338ed57d","tex":"\\operatorname{sgn}","aliases":"sgn operatorname","display":"\\operatorname{sgn}","kind":"use","excerpt":"[Finite adequacy certificate] The standard representation of S_3 over F_7 is adequate. More precisely, H^0(S_3,ad^0V)=0, H^1(S_3, F_7)=0, H^1(S_3,ad^0V)=0, and ad^0V≅sgn⊕ V. The sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18f5af16bd34d11c","tex":"\\sgn","aliases":"sgn operatorname","display":"{\\operatorname{sgn}}","kind":"use","excerpt":"[Jump count and sign recovery] Under the hypotheses of [source reference], the number of detected cells equals Q, and sgn(widehat a_q)=sgn(a_q) for every real valued jump.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b8011d9dfc22b9d","tex":"\\operatorname{Si}","aliases":"Si operatorname","display":"\\operatorname{Si}","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a81d4e3e02ea338","tex":"\\mathrm{sign}","aliases":"sign mathrm","display":"\\mathrm{sign}","kind":"use","excerpt":"0 · -1) and τ(g)=diag(1,-1). It preserves the residual comparison and has determinant one. Pullback through the quadratic character of the actual G_ Q gives continuous arithmetic examples. In contrast, the identity and diag(1,sign) over Z/4 agree modulo two but are not integrally conjugate. Their group order is not a unit. These restricted results ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1d79850b77567ab8","tex":"\\operatorname{sign}","aliases":"sign operatorname","display":"\\operatorname{sign}","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"quently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof","count":89},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"use","excerpt":"Connecting to the chapter's Z_n convention. The chapter defines Z_n through sec t+tan t. The Lean construction uses Mathlib's actual formal sine and cosine series and defines sec t=(cos t)^-1, tan t=(sin t)sec t, Z_n=n![t^n](sec t+tan t). It proves the sine and cosine derivative identities, sin^2t+cos^2t=1, and (tan t)'=sec^2t. Sign substitution in","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e99bde03f8066132","tex":"\\sinh_a","aliases":"sinh_a sinh","display":"\\sinh_a","kind":"use","excerpt":". targetDyadic.kernel_coefficient_integral, Dyadic.beta_integral targetDyadic.source_valuation, Dyadic.sourcePadic_isUnit pageheading2. Connecting the definitions to actual formal seriesGenerating-function foundations for Theorem 5.3 Work in Q[[t]]. The formal exponential and rescaling homomorphism are the same checked objects used in the second bl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bd6563a665b1ad1","tex":"\\small","aliases":"small","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e46cbed476cfd5c1","tex":"\\mathrm{Smith}","aliases":"Smith mathrm","display":"\\mathrm{Smith}","kind":"use","excerpt":"The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c99449412a3a97a2","tex":"\\operatorname{Sol}","aliases":"Sol operatorname","display":"\\operatorname{Sol}","kind":"use","excerpt":"N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5264d5a046b348b2","tex":"\\operatorname{Sol}_F","aliases":"Sol_F operatorname","display":"\\operatorname{Sol}_F","kind":"use","excerpt":"N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae01b8265b984010","tex":"\\mathrm{source}","aliases":"source mathrm","display":"\\mathrm{source}","kind":"use","excerpt":"headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumerat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-131a1658f6fa0831","tex":"\\operatorname{source}","aliases":"source operatorname","display":"\\operatorname{source}","kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-793e8333d34ceb82","tex":"\\Sp","aliases":"Sp operatorname","display":"{\\operatorname{Sp}}","kind":"use","excerpt":"Let p be odd, let (V,b) be a finite symplectic F_p space of dimension 2r, and let g∈Sp(V) have no fixed vector. Fix one of the standard finite Weil normalizations used in the literature [source reference].","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0875a7163c89630a","tex":"\\operatorname{Sp}","aliases":"Sp operatorname","display":"\\operatorname{Sp}","kind":"use","excerpt":"[Indecomposable monodromy blocks] Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks Sp_m(τ), where τ is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18d2a318c2e17978","tex":"\\operatorname{Sp}_m","aliases":"Sp_m operatorname","display":"\\operatorname{Sp}_m","kind":"use","excerpt":"[Indecomposable monodromy blocks] Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks Sp_m(τ), where τ is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span Span span operatorname","display":null,"kind":"use","excerpt":"2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is unreachable from a seed supported only at U. Therefore the generated repres","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66c6b94238fa4435","tex":"\\Span_{\\bbk}","aliases":"Span_bbk span_𝕜 Span bbk operatorname Bbbk","display":"{\\operatorname{span}}_{{\\Bbbk}}","kind":"use","excerpt":"se there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with [2pt] 𝕜^V(G)=span_𝕜 {p(A_G)s_j:p∈𝕜[t], 1≤ j≤ r} }.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-720c3810eb53f410","tex":"\\Span_{\\C}","aliases":"Span_C span_ C Span C operatorname mathbb","display":"{\\operatorname{span}}_{{\\mathbb C}}","kind":"use","excerpt":"Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To se","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4b59b30017e0c86","tex":"\\operatorname{span}_{\\CC}","aliases":"span_CC span_ C operatorname CC operatorname mathbb","display":"\\operatorname{span}_{{\\mathbb C}}","kind":"use","excerpt":"Let C_1,…,C_m be self adjoint constraint operators on a finite dimensional Hilbert space. The distinction between constraint action and constraint closure is standard in constrained Hamiltonian theory [source reference]. Write c=span_ C{C_1,…,C_m} ⊆End( H).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b11e0d35c37be17","tex":"\\operatorname{span}_R","aliases":"span_R operatorname","display":"\\operatorname{span}_R","kind":"use","excerpt":"For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5945fa8a59c08e31","tex":"\\spanof","aliases":"spanof span operatorname","display":"{\\operatorname{span}}","kind":"use","excerpt":"The matrix-orbit dimension and the coefficient equivalence now yield dim_ Cspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r(ψ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec2fdd09d5e3a211","tex":"\\mathrm{spec}","aliases":"spec mathrm","display":"\\mathrm{spec}","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae51e1c510f803eb","tex":"\\operatorname{Spec}","aliases":"Spec operatorname","display":"\\operatorname{Spec}","kind":"use","excerpt":"[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-263069edf563db19","tex":"\\Spec_K","aliases":"Spec_K Spec operatorname","display":"{\\operatorname{Spec}}_K","kind":"use","excerpt":"[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-982328590b226ee9","tex":"\\Spec_\\R","aliases":"Spec_R Spec_ R Spec R operatorname mathbb","display":"{\\operatorname{Spec}}_{\\mathbb R}","kind":"use","excerpt":"tual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator t","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"for maps from A and B to a common set if and only if R is rectangular. In that case a canonical finite choice of common set is the connected-component set C_R of the bipartite graph with vertex set Asqcup B and edges given by R.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a46e78b4903b3de3","tex":"\\operatorname{srank}","aliases":"srank operatorname","display":"\\operatorname{srank}","kind":"use","excerpt":"dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence k≥lceil m/drceil. Partitioning a multiplicity basis into groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3df7b5dd54609471","tex":"\\srank_{\\A}","aliases":"srank_A srank_ A srank A operatorname mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}}","kind":"use","excerpt":"] The source component formulas concern packet pure seeds. They do not give the unrestricted seed rank. For example, let A=𝕜×𝕜, X=𝕜⊕𝕜 with the coordinate action and the two coordinate packet projectors. The transfer graph has two source components and psrank_ A, P(X)=2, but the mixed vector (1,1) is cyclic, so srank_ A(X)=1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ec78eed6a4a5e23","tex":"\\srank_{A,\\tau_M}","aliases":"srank_A,τ_M srank tau operatorname tau","display":"{\\operatorname{srank}}_{A,\\tau_M}","kind":"use","excerpt":"onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bou","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5eb5befad9ecd338","tex":"\\srank_{A/J}","aliases":"srank_A/J srank operatorname","display":"{\\operatorname{srank}}_{A/J}","kind":"use","excerpt":"radical] Let A be a unital ring, let J=Jac(A), and let M be a finitely generated left module over A. A family S⊆ M generates M if and only if its image generates M/JM as a module over A/J. When A/J is semisimple Artinian, this quotient is the usual semisimple top. No semisimplicity of A/J is required for the general equality. Consequently, srank_A(","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61b79d07c0fd89a7","tex":"\\srank_{A_K}","aliases":"srank_A_K srank operatorname","display":"{\\operatorname{srank}}_{A_K}","kind":"use","excerpt":"[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient fi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22393d9dec0e79d0","tex":"\\operatorname{srank}_{\\AA}","aliases":"srank_AA operatorname AA","display":null,"kind":"use","excerpt":"dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence k≥lceil m/drceil. Partitioning a multiplicity basis into groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93686018cc79b240","tex":"\\srank_{\\AA}","aliases":"srank_AA srank_ A srank AA operatorname mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}}","kind":"use","excerpt":"headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b362f5e98fe8e80","tex":"\\srank_{\\AA\\times\\BB}","aliases":"srank_AA×BB srank_ A× B srank AA times BB operatorname mathcal times mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}\\times{\\mathcal B}}","kind":"use","excerpt":"headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a029885343d63c05","tex":"\\srank_{\\AA\\otimes\\BB}","aliases":"srank_AA⊗BB srank_ A⊗ B srank AA otimes BB operatorname mathcal otimes mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}\\otimes{\\mathcal B}}","kind":"use","excerpt":"headThe resulting upper bound Since simple tensors span the algebraic tensor product, the complete rectangular family proves srank_ A⊗ B(X⊗ Y) ≤srank_ A(X) srank_ B(Y). This is not a universal equality claim. Changing the algebra changes the answer: the full independent matrix action can make a single nonzero pure tensor cyclic. Nor does the algebr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fcb3fb6c02502457","tex":"\\operatorname{srank}_{\\Acal}","aliases":"srank_Acal srank_ A operatorname Acal operatorname mathcal","display":"\\operatorname{srank}_{{\\mathcal A}}","kind":"use","excerpt":"[Quantum seed rank] For the displayed multiplicity decomposition, srank_ A( H) = max_1≤ r≤ s lceilfracm_rd_rrceil.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0bb1698c9624f4e","tex":"\\srank_{A\\times B}","aliases":"srank_A× B srank times operatorname times","display":"{\\operatorname{srank}}_{A\\times B}","kind":"use","excerpt":"[Direct sum formula for independent actions] Let M be a left module over A and N a left module over B. Under the action of the product algebra (a,b)(m,n)=(am,bn), one has srank_A× B(M⊕ N) = max{srank_A(M),srank_B(N)}. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26c092d57bf7778b","tex":"\\srank_{A\\otimes B}","aliases":"srank_A⊗ B srank otimes operatorname otimes","display":"{\\operatorname{srank}}_{A\\otimes B}","kind":"use","excerpt":"Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-960d9dd2b343db5c","tex":"\\srank_B","aliases":"srank_B srank operatorname","display":"{\\operatorname{srank}}_B","kind":"use","excerpt":"[Monotonicity under enlargement of the acting algebra] Let B⊆ A⊆End_𝕜(X) be unital operator algebras with the same identity. Then srank_A(X)≤srank_B(X). The same statement holds for topological seed rank when both algebras act continuously. [source reference]","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc0e4e4943e6ea7a","tex":"\\srank_{B,\\tau_N}","aliases":"srank_B,τ_N srank tau operatorname tau","display":"{\\operatorname{srank}}_{B,\\tau_N}","kind":"use","excerpt":"onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bou","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e18d0d44e2e3fcea","tex":"\\srank_{\\BB}","aliases":"srank_BB srank_ B srank BB operatorname mathcal","display":"{\\operatorname{srank}}_{{\\mathcal B}}","kind":"use","excerpt":"headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c22e5f4935054ca","tex":"\\srank_{\\bbk[T]}","aliases":"srank_bbk[T] srank_𝕜[T] srank bbk operatorname Bbbk","display":"{\\operatorname{srank}}_{{\\Bbbk}[T]}","kind":"use","excerpt":"[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the d","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af678bb5870a8707","tex":"\\operatorname{srank}_{\\cA}","aliases":"srank_cA srank_ A operatorname cA operatorname mathcal","display":"\\operatorname{srank}_{{\\mathcal A}}","kind":"use","excerpt":"[Quantum seed-rank formula] For A=⊕_i(M_d_i( C)⊗ I_m_i) on H=⊕_i( C^d_i⊗ C^m_i), the minimum number of seeds required to generate H is srank_ A( H) = max_ilceilfracm_id_irceil.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-985f568846585534","tex":"\\srank_{\\cA}","aliases":"srank_cA srank_ A srank cA operatorname mathcal","display":"{\\operatorname{srank}}_{{\\mathcal A}}","kind":"use","excerpt":"For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a4f0945c263f910","tex":"\\srank_{\\kk I}","aliases":"srank_kk I srank_𝕜 I srank kk operatorname Bbbk","display":"{\\operatorname{srank}}_{{\\Bbbk} I}","kind":"use","excerpt":"The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5dc32ff45b20c008","tex":"\\srank_{\\kk\\times\\kk}","aliases":"srank_kk×kk srank_𝕜×𝕜 srank kk times kk operatorname Bbbk times Bbbk","display":"{\\operatorname{srank}}_{{\\Bbbk}\\times{\\Bbbk}}","kind":"use","excerpt":"The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c953c91802c5c19","tex":"\\srank_{M_2(\\QQ)}","aliases":"srank_M_2(QQ) srank_M_2( Q) srank QQ operatorname mathbb","display":"{\\operatorname{srank}}_{M_2({\\mathbb Q})}","kind":"use","excerpt":"6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b94af8795a55ca25","tex":"\\srank_{M_d(D)}","aliases":"srank_M_d(D) srank operatorname","display":"{\\operatorname{srank}}_{M_d(D)}","kind":"use","excerpt":"[Full matrix algebra] For the natural module S=D^d over M_d(D), srank_M_d(D)(S)=1. More strongly, every nonzero vector of S is cyclic. [source reference]","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0724cce7fe1672c1","tex":"\\srank_{P_\\alpha A P_\\alpha}","aliases":"srank_P_α A P_α srank alpha alpha operatorname alpha alpha","display":"{\\operatorname{srank}}_{P_\\alpha A P_\\alpha}","kind":"use","excerpt":"[Basic packet pure rank bounds] For every A module resolved into packets with P_α∈ A, srank_A(X) ≤ psrank_A, P(X) ≤ ∑_α∈ I srank_P_α A P_α(X_α). The corner algebra on the right is unital on X_α with identity P_α. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42ec3110f940782f","tex":"\\srank_{\\QQ[N]}","aliases":"srank_QQ[N] srank_ Q[N] srank QQ operatorname mathbb","display":"{\\operatorname{srank}}_{{\\mathbb Q}[N]}","kind":"use","excerpt":"headLower and upper bounds meet If r seeds generated X, their images would generate the quotient. Since scalar multiplication cannot create a new independent direction, r≥2. Conversely, the two seeds e_0,f_0 generate their respective chains, so srank_ Q[N](X)=2. This proof uses the displayed operator and quotient. It does not ask a computer to sear","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38c6ab6ee7594444","tex":"\\srank_{R_p}","aliases":"srank_R_p srank operatorname","display":"{\\operatorname{srank}}_{R_p}","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4353e56e229aaaa","tex":"\\mathrm{src}","aliases":"src mathrm","display":"\\mathrm{src}","kind":"use","excerpt":"Assume the one-vertex scalar-pushout contract: there are d labeled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-348ccb55d8cc92e4","tex":"\\mathrm{ss}","aliases":"ss mathrm","display":"\\mathrm{ss}","kind":"use","excerpt":"tomorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5fd5471d426d46d","tex":"\\operatorname{St}","aliases":"St operatorname","display":"\\operatorname{St}","kind":"use","excerpt":"[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca1b99e90c670ad6","tex":"\\operatorname{St}_m","aliases":"St_m operatorname","display":"\\operatorname{St}_m","kind":"use","excerpt":"[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ceb0960938d0156a","tex":"\\operatorname{Stab}","aliases":"Stab operatorname","display":"\\operatorname{Stab}","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e8efc303977c5d4","tex":"\\operatorname{Stab}_{G_{\\Q}}","aliases":"Stab_G_Q Stab_G_ Q operatorname Q operatorname mathbf","display":"\\operatorname{Stab}_{G_{{\\mathbf Q}}}","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee9095f60897d7d1","tex":"\\starh","aliases":"starh mathbin⋆ mathbin star","display":null,"kind":"use","excerpt":"This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard windo","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72d39f9d2160995b","tex":"\\mathrm{std}","aliases":"std mathrm","display":"\\mathrm{std}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89d7944b7e472825","tex":"\\operatorname{step}","aliases":"step operatorname","display":"\\operatorname{step}","kind":"use","excerpt":"Let v be the new optimum of the edited subtree, and let j be the edited child slot at one frame. The update performed at that frame is step(v)= min{K, σ+ ∑_i v, · i=j, widehat F_i, · i≠ j. }. This result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values widehat F_i.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-327fd569cd80a61b","tex":"\\operatorname{storedTree}","aliases":"storedTree operatorname","display":"\\operatorname{storedTree}","kind":"use","excerpt":"Entrywise implementation theorem. For every b≤ B, Lean proves lookup(storedTree(B,t),b)=F_t(b). There is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at b depends only on input entries at most b, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6008072c780cd95","tex":"\\mathrm{stream}","aliases":"stream mathrm","display":"\\mathrm{stream}","kind":"use","excerpt":"Streaming. The explicit event type counts one block read, one source read, and a target read plus write for every edge. With scalar size s, its byte count is B_stream=s∑_e(m_en_e+n_eq+2m_eq). No cross-edge cache reuse is assumed. Sparse index metadata and temporary workspace remain separate.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7402991750f0799","tex":"\\sTwo","aliases":"sTwo s_2","display":"{s_2}","kind":"use","excerpt":"[Dyadic inverse coefficients] For r≥1, β_r = - frac2^2r(2r-1)B_2r(2r)!. Consequently, v_2(β_r)=s_2(r)-1. Every R_n is a dyadic unit.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"use","excerpt":"Matrix multiplication distributes across addition and scalar multiplication, which gives complex linearity. If ρsucceq0, each term K_aρ K_a^* is positive semidefinite, hence so is their finite sum.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc260eee4ab7cffe","tex":"\\sup_{0\\leq r\\leq t}","aliases":"sup_0≤ r≤ t sup leq leq","display":"\\sup_{0\\leq r\\leq t}","kind":"use","excerpt":"Let E(t)=sup_0≤ r≤ tnormx(r)-widetilde x(r). Subtract the two mild equations. Fubini's theorem and the triangle inequality give E(t) ≤ M_TL∫_0^tE(r) d r + M_TBtε. For t≤ T, replace the last factor t by T and apply Gronwall's inequality.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76d8fadb9fe83504","tex":"\\sup_{0\\leq s,t\\leq T}","aliases":"sup_0≤ s,t≤ T sup leq leq","display":"\\sup_{0\\leq s,t\\leq T}","kind":"use","excerpt":"[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally un","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c58f1649c7ede918","tex":"\\sup_{0\\le t\\le T}","aliases":"sup_0≤ t≤ T sup le le","display":"\\sup_{0\\le t\\le T}","kind":"use","excerpt":"2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time interval, not errors sampled at the right endpoint. headThe estimate follows before any graph is drawn For 0≤ t≤ T, the derivative of a↦ e^-at has absolute value at most T when a≥0. Thus sup_0≤ t≤ T|T_n(","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9c6f71937fef238","tex":"\\sup_{0\\leq t\\leq T}","aliases":"sup_0≤ t≤ T sup leq leq","display":"\\sup_{0\\leq t\\leq T}","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c901ab4425f0be8a","tex":"\\sup_{\\abs t\\leq T}","aliases":"sup_abs t≤ T sup abs leq","display":null,"kind":"use","excerpt":"[Norm summable Hamiltonian tower] Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-963869a5c66f4ce3","tex":"\\sup_{F,a}","aliases":"sup_F,a sup","display":"\\sup_{F,a}","kind":"use","excerpt":"Each P_nX=Ker(I-P_n) is closed. For x∈ X, A_ Px _u = sup_F,a ∑_n∈ Fa_nP_nx ≤ K x . For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the uncond","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0341617620bd99b1","tex":"\\sup_m","aliases":"sup_m sup","display":"\\sup_m","kind":"use","excerpt":"[Estimated packet Gibbs bound] Let Q_m be linear and uniformly bounded on L^p, with sup_m Q_m ≤ C_Q. For f=s+j and an estimated packet widehat j_m, define widehat f_m =Q_m(f-widehat j_m)+widehat j_m. Then f-widehat f_m _p ≤ s-Q_ms _p +(1+C_Q) j-widehat j_m _p. Consequently, smooth approximation and packet recovery may be improved independently. No ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08b5f4a5605ba4c4","tex":"\\sup_{s\\in L}","aliases":"sup_s∈ L sup in","display":"\\sup_{s\\in L}","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11e8bed385caca1a","tex":"\\sup_u","aliases":"sup_u sup","display":"\\sup_u","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3ee24d95dbc480e1","tex":"\\sup_{z\\in\\Gamma}","aliases":"sup_z∈Γ sup in Gamma","display":"\\sup_{z\\in\\Gamma}","kind":"use","excerpt":"olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af258da75f945f55","tex":"\\sup_{\\Lambda\\in\\mathfrak P(\\mathcal T)}","aliases":"sup_Λ∈ P( T) sup Lambda in mathfrak mathcal","display":"\\sup_{\\Lambda\\in\\mathfrak P(\\mathcal T)}","kind":"use","excerpt":"imizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56916448f3b1cf12","tex":"\\sup_{\\Xi}","aliases":"sup_Ξ sup Xi","display":"\\sup_{\\Xi}","kind":"use","excerpt":"set, so the feasible set is nonempty and finite. Thus the minimum is attained. If competitors exist, their finitely many differences from a unique optimum are strictly positive, so their minimum is positive and is achieved by a competitor. In the singleton case there is no alternative selector under any objective change. In the remaining case, for ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f99188f5481208b","tex":"\\sup_\\ell","aliases":"sup_ℓ sup ell","display":"\\sup_\\ell","kind":"use","excerpt":"[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22a5ceacd49ba63d","tex":"\\supp_n","aliases":"supp_n supp operatorname","display":"{\\operatorname{supp}}_n","kind":"use","excerpt":"ernal profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet of F and every inner block jet of G in [source reference] are nonzero. Then supp_n(F∘ G)(x) is contained in t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76d43a5663aea1e7","tex":"\\supp_r","aliases":"supp_r supp operatorname","display":"{\\operatorname{supp}}_r","kind":"use","excerpt":"[Structural pruning] For a map H with fixed input and output packet systems, let supp_r H(x) be the set of external profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5490eb62846f738d","tex":"\\Supp_{R/I}","aliases":"Supp_R/I Supp operatorname","display":"{\\operatorname{Supp}}_{R/I}","kind":"use","excerpt":"[Support after a quotient] Let R be Noetherian, let I⊆ R be an ideal, and let M be a finite R module. Under the canonical identification Spec(R/I)≅ V(I)⊆Spec R, one has Supp_R/I(M/IM) = Supp_R(M)∩ V(I).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad3fefc79ffc997c","tex":"\\operatorname{Sw}","aliases":"Sw operatorname","display":"\\operatorname{Sw}","kind":"use","excerpt":"The dimension and orbit follow from [source reference]. The cycle polynomial is [source reference]. For rank greater than one, the conductor is [source reference]. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula Art=dim V-dim V^I_K+Sw. The local polynomial is [source ref","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d8605f4038ec2f5","tex":"\\mathrm{symp}","aliases":"symp mathrm","display":"\\mathrm{symp}","kind":"use","excerpt":"[Radical free envelope] For an arbitrary scalar projective action of A, the quotient A^symp=A/rad b_c is the maximal quotient detected by noncommuting projective intertwiners. If A is an elementary abelian p group, then A^symp is a symplectic F_p space.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f12da8d8c07c189d","tex":"\\operatorname{synthesize}","aliases":"synthesize operatorname","display":"\\operatorname{synthesize}","kind":"use","excerpt":"The verified contract is synthesize(analyze(x,N))=x. The Lean counterpart uses packet coordinates restricted to projector ranges and proves both inverse laws. The finite Python routine checks the concrete array realization.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21a8014450044a51","tex":"\\Sys","aliases":"Sys mathscr S mathscr","display":"{\\mathscr S}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ade086a1f71df625","tex":"\\mathrm{sys}","aliases":"sys mathrm","display":"\\mathrm{sys}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8565080e544a001","tex":"\\Sys_2^{\\mathrm{wt}}","aliases":"Sys_2^wt mathscr S_2^wt Sys mathrm mathscr mathrm","display":"{\\mathscr S}_2^{\\mathrm{wt}}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d6edfcecc9e2400","tex":"\\Sys_{\\mathrm{sys}}","aliases":"Sys_sys mathscr S_sys Sys mathrm mathscr mathrm","display":"{\\mathscr S}_{\\mathrm{sys}}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90e631cae94f81c9","tex":"\\Sys_\\lambda","aliases":"Sys_λ mathscr S_λ Sys lambda mathscr lambda","display":"{\\mathscr S}_\\lambda","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-77970456be0eda56","tex":"t'","aliases":"t'","display":"t'","kind":"use","excerpt":"At depth N≥1, suppose F(a)=q^Nr and a desired next integer parameter target satisfies t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exa","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46423aa7af18fd67","tex":"T''","aliases":"T''","display":"T''","kind":"use","excerpt":"retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10d6fa1215b924bf","tex":"T^*","aliases":"T^*","display":"T^*","kind":"use","excerpt":"[Adjoint reversal] If X is Hilbert, every P_α is orthogonal, and T∈ B(X), then (T_βα)^*=(T^*)_αβ. [source reference]","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"use","excerpt":"B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, a","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d42311231194d5c","tex":"t^{-1}","aliases":"t^-1","display":"t^{-1}","kind":"use","excerpt":"For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible ad","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a60c6c964dc91686","tex":"t^{-c}","aliases":"t^-c","display":"t^{-c}","kind":"use","excerpt":"Application to the actual singular ratios. The preceding v0.9.0 source independently constructs H_N,c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=t^-cH_N,c(t), R_c=fracA_c-1A_c. The notation t^-cH_N,c means removing its proved initial zero coefficients. For each c=ap, 1≤ a<m, that source proves R_ap(log(1+T))∈ W_p. The new transfer consequently proves flatne","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-104f4e0a72089cc5","tex":"T^0","aliases":"T^0","display":"T^0","kind":"use","excerpt":"^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence finite weighted polynomials. Each fixed output coefficient depends on only finitely many input coefficients, which extends the result to the formal series. Substitution T=e^t-1 intertwines D with Θ and maps W_p into H_p, because d![t^d]F(e","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afa519626035ed43","tex":"t^{2N}","aliases":"t^2N","display":"t^{2N}","kind":"use","excerpt":"m≥1, differentiation of zcoth z gives β_m=-frac2^2m(2m-1)B_2m(2m)!, κ_m=-2^4m(2m-1)B_2m. The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-605988e2d23b9544","tex":"t^{2n}","aliases":"t^2n","display":"t^{2n}","kind":"use","excerpt":"The classical signed Euler numbers have generating function secht=∑_n≥0E_2nt^2n/(2n)!. They are the N=2, j=0 member of [source reference]. The coordinate theorem reconstructs them from finite alternating sums of odd powers.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c00545ef56fef32","tex":"t^{2r}","aliases":"t^2r","display":"t^{2r}","kind":"use","excerpt":"[Zigzag inverse kernel] sech^2t = ∑_r≥0 (-1)^rZ_2r+1 fract^2r(2r)!.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","excerpt":"retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a07369c56b77606c","tex":"t^{3n}","aliases":"t^3n","display":"t^{3n}","kind":"use","excerpt":"V: Theorem 11.7, Theorem 11.8, Corollary 11.9. The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":"headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1769a0bec82c9aba","tex":"t^{4i}","aliases":"t^4i","display":"t^{4i}","kind":"use","excerpt":"G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The co","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b595555eaabda692","tex":"t^{4n}","aliases":"t^4n","display":"t^{4n}","kind":"use","excerpt":"Pack g_j into degrees divisible by four and form its actual EGF. The formal coefficient identity is G_j(t)=∑_n≥0g_j(n)fract^4n(4n)! =j!∑_n≥0fract^4n(4n+j)!. Its inverse is therefore the original normalized packet EGF. The source Kummer property gives flatness of G_j. The proved reciprocal closure gives flatness of G_j^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c6354f5f56778af","tex":"t^5","aliases":"t^5","display":"t^5","kind":"use","excerpt":"headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98f93e20c13d2c57","tex":"t^6","aliases":"t^6","display":"t^6","kind":"use","excerpt":"[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rati","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-179195799d88b1fc","tex":"t^7","aliases":"t^7","display":"t^7","kind":"use","excerpt":"[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rati","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1214015beb03f0dc","tex":"t^8","aliases":"t^8","display":"t^8","kind":"use","excerpt":"For the first contribution, independence gives L(γ_k(P+Q))=∑_a=0^kμ_aμ_k-a. The parity generating series of these convolution moments is (1+t+t^2+t^4+t^8+⋯)^2 =1+t^2+t^4+t^8+⋯ in F_2[[t]]. Its coefficient is one at every k=2^r≥4. Thus the normalized moment in [source reference] is odd and v_2(Δ_1^kC_2(0))=2k+v_2(k!)+k/2. Inserting U^4n=(1+4H_2)^n i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa7949456d03953c","tex":"t^9","aliases":"t^9","display":"t^9","kind":"use","excerpt":"[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rati","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55e177ef0ebf15f4","tex":"T^c","aliases":"T^c","display":"T^c","kind":"use","excerpt":"To pass from the section to the singular ratio, factor out its initial zero: for c=ap, the section has the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24ab8e8d755e8c86","tex":"T^d","aliases":"T^d","display":"T^d","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f9e80a5242f2d08","tex":"t^{d-1}","aliases":"t^d-1","display":"t^{d-1}","kind":"use","excerpt":"There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1cd31341c910e1c7","tex":"T^{d\\bmod p}","aliases":"T^dbmod p bmod","display":"T^{d\\bmod p}","kind":"use","excerpt":"mate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-964fcf8c1f59b656","tex":"T^e","aliases":"T^e","display":"T^e","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e393fab09f85dc1","tex":"T^i","aliases":"T^i","display":"T^i","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68c16f5dbc93324c","tex":"t^j","aliases":"t^j","display":"t^j","kind":"use","excerpt":"For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d241eb52ee3cfa72","tex":"t^k","aliases":"t^k","display":"t^k","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aea3af576abad800","tex":"t^m","aliases":"t^m","display":"t^m","kind":"use","excerpt":"The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ca79bce6422a743","tex":"t^{m-1}","aliases":"t^m-1","display":"t^{m-1}","kind":"use","excerpt":"ion with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated constant. The supremum norm argum","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f691afb056141dc2","tex":"t^{m_0}","aliases":"t^m_0","display":"t^{m_0}","kind":"use","excerpt":"in Part II. A directed edge from α to β means that P_β A P_α is nonzero, with α≠β. A packet always has the length-zero path to itself. [Minimum transfer order] Suppose every directed packet path from α to β uses at least m_0 off-diagonal transfers. Then P_β e^tAP_α =O(t^m_0) as tdownarrow0. If there is no directed path from α to β, the block vanish","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8fa21937d664fe3d","tex":"t^N","aliases":"t^N","display":"t^N","kind":"use","excerpt":"The formalization does not stop at a recurrence. In the variable z=t^N, construct K_N,j(z)=∑_n≥0fracj!(Nn+j)!z^n, F_N,j(z)=∑_n≥0fracC_N,j(n)(Nn)!z^n. It proves the factorial coefficient identity, then F_N,jK_N,j=1 and therefore F_N,j=K_N,j^-1.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6fccf1c31eff02b3","tex":"T^{n+1}","aliases":"T^n+1","display":"T^{n+1}","kind":"use","excerpt":"The zero case is the identity matrix. If T^n= W_n, then the left-successor power convention gives T^n+1=TT^n=T W_n= W_n+1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e29220aa3d532708","tex":"T^{n_j}","aliases":"T^n_j","display":"T^{n_j}","kind":"use","excerpt":"tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2042ebdb7ea53ef4","tex":"T^{n_j+n_k}","aliases":"T^n_j+n_k","display":"T^{n_j+n_k}","kind":"use","excerpt":"tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bdc3d343e0ba3452","tex":"T^{n_k}","aliases":"T^n_k","display":"T^{n_k}","kind":"use","excerpt":"Since T is positive and self adjoint, ipv_jT^n_j+n_kv_k = ipT^n_jv_jT^n_kv_k. The quadratic form is the squared norm of ∑_kc_kT^n_kv_k.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf91dc9649ebc4f6","tex":"t^{Nn+c}","aliases":"t^Nn+c","display":"t^{Nn+c}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9acde6ce17e28bc3","tex":"t^{Nn+j}","aliases":"t^Nn+j","display":"t^{Nn+j}","kind":"use","excerpt":"For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d24990e4c5e775a","tex":"t^{Nr}","aliases":"t^Nr","display":"t^{Nr}","kind":"use","excerpt":"pageheading4. From these sections to the original Euler packetsActual reciprocal, actual rational sequence, every order and step multiple The factorial-ratio identity is proved coefficient by coefficient: ∑_d≥0g_j(d)fract^dd! =j!∑_r≥0fract^Nr(Nr+j)!. The right side is precisely the normalized source kernel from the first block, not a replacement fa","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2797f6407ccfeb21","tex":"T^p","aliases":"T^p","display":"T^p","kind":"use","excerpt":"mate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a5cb710fde741d7","tex":"T^{\\Pcal}_{ba}","aliases":"T^Pcal_ba T^ P_ba Pcal mathcal","display":"T^{{\\mathcal P}}_{ba}","kind":"use","excerpt":"[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15f4a4b580fed92c","tex":"T^{\\Pcal}_{\\beta\\alpha}","aliases":"T^Pcal_βα T^ P_βα Pcal beta alpha mathcal beta alpha","display":"T^{{\\mathcal P}}_{\\beta\\alpha}","kind":"use","excerpt":"For T∈End(X) and a packet system P, write T^ P_βα=P_β T P_α.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c10b9cca7424884c","tex":"t^q","aliases":"t^q","display":"t^q","kind":"use","excerpt":"The first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of [source reference], the relation is Φ^-1tΦ=t^q. Equivalently, Φ tΦ^-1=t^q^-1, with inverse taken modulo M since gcd(q,M)=1. Conjugating the Fourier sum and changing its summation index therefore gives Φ P_jΦ^-1=P_qj. The inverse F","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2aa2675cfae8aa2f","tex":"t^{q^{-1}}","aliases":"t^q^-1","display":"t^{q^{-1}}","kind":"use","excerpt":"The first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of [source reference], the relation is Φ^-1tΦ=t^q. Equivalently, Φ tΦ^-1=t^q^-1, with inverse taken modulo M since gcd(q,M)=1. Conjugating the Fourier sum and changing its summation index therefore gives Φ P_jΦ^-1=P_qj. The inverse F","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40ae1b3bb8b8057b","tex":"T^{\\Qcal}_{\\beta\\alpha}","aliases":"T^Qcal_βα T^ Q_βα Qcal beta alpha mathcal beta alpha","display":"T^{{\\mathcal Q}}_{\\beta\\alpha}","kind":"use","excerpt":"[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf9efaac25b08a98","tex":"T^r","aliases":"T^r","display":"T^r","kind":"use","excerpt":"[Weighted convergence] If widehat F(T)=∑_r≥0c_rT^r and v_p(c_r)+rα⟶+∞, then the series converges whenever v_p(T)≥α.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-645a5cbe71427f86","tex":"t^r","aliases":"t^r","display":"t^r","kind":"use","excerpt":"Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79f98be5a96a5c30","tex":"T^U","aliases":"T^U","display":"T^U","kind":"use","excerpt":"vpage11Move the packets with the coordinatesCONJUGACY / STRUCTURE VERSUS RAW SPARSITYMain text: [source reference] Start with T=diag(1,2) and the two coordinate projections. Apply the invertible shear U. The transformed operator is T^U=UTU^-1, and the transformed projectors are Q_i=UP_iU^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cef6729bcc40a7f6","tex":"t^\\alpha","aliases":"t^α alpha","display":"t^\\alpha","kind":"use","excerpt":"[Fractional intertwining theorem] For bounded nonnegative self-adjoint A and B, D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In par","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bea69885d49bedfc","tex":"t^{\\alpha-1}","aliases":"t^α-1 alpha","display":"t^{\\alpha-1}","kind":"use","excerpt":"For 0<α<1, the Balakrishnan formula gives A^α = fracsin(πα)π ∫_0^∞ t^α-1A(tI+A)^-1 dt, and likewise for B. Using A(tI+A)^-1=I-t(tI+A)^-1 and the rectangular resolvent identity (tI+B)^-1C-C(tI+A)^-1 = (tI+B)^-1(CA-BC)(tI+A)^-1, the difference of the two Balakrishnan integrals becomes the displayed resolvent-defect formula.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-048429bc73f8c238","tex":"t^{\\alpha-2}","aliases":"t^α-2 alpha","display":"t^{\\alpha-2}","kind":"use","excerpt":"For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f790967529827b89","tex":"T_{11}","aliases":"T_11","display":"T_{11}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a36fd1f6d7af0cd","tex":"T_{12}","aliases":"T_12","display":"T_{12}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21a8dad328629d94","tex":"T_{13}","aliases":"T_13","display":"T_{13}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c34948496a97b098","tex":"t_1^{w_1(e)}","aliases":"t_1^w_1(e)","display":"t_1^{w_1(e)}","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75143599dd91b630","tex":"T_{2,1}","aliases":"T_2,1","display":"T_{2,1}","kind":"use","excerpt":"y any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb04c4472902a215","tex":"T_{2,M}","aliases":"T_2,M","display":"T_{2,M}","kind":"use","excerpt":"[Optimal dyadic truncation] Let T_2,M(n) = ∑_r=0^min(M,n/2) κ_2,rbinom n2rR_n-2r. Then E_n≡ T_2,M(n)pmod 2^4M+3. More precisely, v_2(E_n-T_2,M(n)) ≥ min_M+1≤ r≤ n/2 ( 4r-1+v_2binom n2r ). The uniform exponent 4M+3 is best possible.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d40e5e2331bf36b5","tex":"T_{22}","aliases":"T_22","display":"T_{22}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56f9752ffe92acc1","tex":"T_{23}","aliases":"T_23","display":"T_{23}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","excerpt":"bottomrule The checked nonzero error valuations are E_8-T_0(8)=frac5494445, v_2(E_8-T_0(8))=5, E_8-T_3(8)=frac22937615, v_2(E_8-T_3(8))=15. They exceed the uniform guarantees. These two examples do not establish a universally stronger exponent.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-899fcb428498d22e","tex":"T_{32}","aliases":"T_32","display":"T_{32}","kind":"use","excerpt":"headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-64f17e7dedd6a141","tex":"T_4","aliases":"T_4","display":"T_4","kind":"use","excerpt":"/D_N. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator. vfigriccati The coefficients retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afd0f5537bb82fff","tex":"t_4","aliases":"t_4","display":"t_4","kind":"use","excerpt":"For example, j=5 has (t_0,…,t_4)=(6,3,4,3,6). The exact root counts at distances 1,2,3,4,5,≥6 are 0,0,2,3,3,5. The older sufficient stages 3,5,7 are not the sharp emergence stages, which are 3,4,6. The first irregular label, j=17, has V≡1+Xpmod2. The regular-wall theorem does not apply. Complete coefficient certificates close distances 1 through 19","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a9e32fb56cb7f6c","tex":"t_a","aliases":"t_a","display":"t_a","kind":"use","excerpt":"ens also to have nonnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd76f79f8f6e6b19","tex":"T_{ba}","aliases":"T_ba","display":"T_{ba}","kind":"use","excerpt":"T_ba · Input packet a, output packet b: P_bTP_a. · Blocks of a finite cyclic filter.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04db525251126e68","tex":"t_c","aliases":"t_c","display":"t_c","kind":"use","excerpt":"[Amortization threshold] Let s≥0 be a one time setup cost. Let t_b,t_c≥0 be the baseline and candidate times per use and let R∈ N be a reuse count. If t_b>t_c, the candidate is faster after R uses exactly when R> fracst_b-t_c. The smallest integer reuse count is R_min = lfloor fracst_b-t_c rfloor+1. If t_b≤ t_c, no positive reuse count amortizes a ","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e42a52f0ec67dea","tex":"T_c'","aliases":"T_c'","display":"T_c'","kind":"use","excerpt":"Had we chosen the transported operator T_c'=CT_fC^-1, the defect would be zero and every polynomial action would intertwine exactly. That is a different choice of coarse operator, not a stronger conclusion from the faulty one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16c26c32d58b7545","tex":"T_c^n","aliases":"T_c^n","display":"T_c^n","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4cc7829c7387a810","tex":"T_c^{n+1}","aliases":"T_c^n+1","display":"T_c^{n+1}","kind":"use","excerpt":"The statement is clear for n=1. For the induction step, write CT_f^n+1-T_c^n+1C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC) and apply the induction hypothesis.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-453187cc310f5d0e","tex":"T_c^{n-1-k}","aliases":"T_c^n-1-k","display":"T_c^{n-1-k}","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e8969a358fc5266","tex":"T_d","aliases":"T_d","display":"T_d","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd730f6c2c2e8897","tex":"t_d","aliases":"t_d","display":"t_d","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa6a46659a84add3","tex":"t_{\\mathrm{decode}}","aliases":"t_decode mathrm","display":"t_{\\mathrm{decode}}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb1ad3b8487cb395","tex":"T_e","aliases":"T_e","display":"T_e","kind":"use","excerpt":"The actual Euclidean transfer exponential. Let H be a Hermitian matrix on a nonzero complex state space of finite dimension. The source defines T_e(t)=exp(-t(H-eI)) using Lean's exponential defined by a power series. The proof then relates this defined operator to its spectral expansion.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c78b7188b7b592c0","tex":"t_e","aliases":"t_e","display":"t_e","kind":"use","excerpt":"[Materialization threshold] Suppose a compressed representation costs t_c≥0 per downstream use, an explicit representation costs t_e≥0 per use, and one time materialization costs m≥0, with reuse count R∈ N. Materialization is profitable exactly when R> fracmt_c-t_e and t_c>t_e.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c3ad1a3dd202746","tex":"T_{E_0}","aliases":"T_E_0","display":"T_{E_0}","kind":"use","excerpt":"Positivity, semigroup composition, and the interval. Every eigenvalue of T_e(t) is strictly positive, for arbitrary real e,t, since the real exponential is positive. For e=E_0 and t≥0 it is also at most one. Therefore T_E_0(t)>0, Spec T_E_0(t)⊆(0,1]. The two scalar multiples of H-eI commute, so the exponential defined by the power series obeys T_e(","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f08f28427ff60b68","tex":"T_{e_j}","aliases":"T_e_j","display":"T_{e_j}","kind":"use","excerpt":"0 · x≠ a. The seed is ψ_0=ket0. Define a translation operator by (T_a v)(x)=v(x-a). At position j, the local Pauli X is X_j=T_e_j, where e_j has a single nonzero binary coordinate. The formal circuit uses the identity when a_j=0 and X_j when a_j=1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9d7fd646d17da1a","tex":"T_f^k","aliases":"T_f^k","display":"T_f^k","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d73f1652a80f0f11","tex":"T_f^n","aliases":"T_f^n","display":"T_f^n","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5580e5867f84cd5","tex":"T_f^{n+1}","aliases":"T_f^n+1","display":"T_f^{n+1}","kind":"use","excerpt":"The statement is clear for n=1. For the induction step, write CT_f^n+1-T_c^n+1C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC) and apply the induction hypothesis.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-558b95d00cd66695","tex":"T_H","aliases":"T_H","display":"T_H","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν),","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf290378a210616e","tex":"T_{H_1}","aliases":"T_H_1","display":"T_{H_1}","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2697a5c99d68fadf","tex":"T_{H_k}","aliases":"T_H_k","display":"T_{H_k}","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-219b7412a89e1904","tex":"T_i","aliases":"T_i","display":"T_i","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9a7aef78fcda23e","tex":"T_{ij}","aliases":"T_ij","display":"T_{ij}","kind":"use","excerpt":"vpage8A transfer block has an input and an outputOPERATOR COORDINATES / MULTIPLICATION AS A SUMMain text: [source reference] Use the three coordinate packets of dimension one in Q^3. The block T_ij=P_iTP_j sends input packet j into output packet i. For this example, its nonzero scalar is the (i,j) matrix entry.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25bdedf7497647d3","tex":"t_j","aliases":"t_j","display":"t_j","kind":"use","excerpt":"Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unita","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bce12004b499e794","tex":"T_{j+1}","aliases":"T_j+1","display":"T_{j+1}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b148baa42903d2fe","tex":"t_{j+1}","aliases":"t_j+1","display":"t_{j+1}","kind":"use","excerpt":"Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unita","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5faa1f8bc6c2f00","tex":"T_{j-1}","aliases":"T_j-1","display":"T_{j-1}","kind":"use","excerpt":"t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-830c9c56b4f2bf21","tex":"t_k","aliases":"t_k","display":"t_k","kind":"use","excerpt":"Let t_k=kh, k=0,…,n, h=frac1n. Assume f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Defin","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-221fdc610b5eb2ef","tex":"t_{k+1}","aliases":"t_k+1","display":"t_{k+1}","kind":"use","excerpt":"[Contrast gap] Assume |η_k|≤ε and set b_h=Mh+2ε. If the cell (t_k,t_k+1) contains no jump, then |D_k|≤ b_h. If it contains one jump of amplitude a_q, then |D_k-a_q|≤ b_h and therefore |D_k|≥ |a_q|-b_h.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-667692e13043c7f4","tex":"T_{ks}","aliases":"T_ks","display":"T_{ks}","kind":"use","excerpt":"T_ks · P_N,kHP_N,s · input-to-output phase transfer","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fcc4f84d94d1a604","tex":"T_L","aliases":"T_L","display":"T_L","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9caf405f71847e16","tex":"t_L","aliases":"t_L","display":"t_L","kind":"use","excerpt":"AtlasHeadA compatible root is built one digit at a timeatlas:smooth Take F(x)=x^2-7 over the 3-adic integers and start with x_1=1 modulo three. If F(x_L)=3^Lr_L, the next representative has the form x_L+1=x_L+3^Lt_L. Its correcting digit satisfies r_L+2x_Lt_L≡0pmod3. AtlasImagesmooth_tower For example, F(1)/3=-2 and 2x_1=2, so the next digit is one","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-748926db74a9add6","tex":"t_L^{-q}","aliases":"t_L^-q","display":"t_L^{-q}","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace c","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e4e16e9495e3826","tex":"t_L^{-q_d}","aliases":"t_L^-q_d","display":"t_L^{-q_d}","kind":"use","excerpt":"At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63928c9acb41859a","tex":"t_L^a","aliases":"t_L^a","display":"t_L^a","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace c","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8e003e6dbf7e04c","tex":"T_m","aliases":"T_m","display":"T_m","kind":"use","excerpt":"h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives T_j=T_j-1-g^(j)(0)/j!. Iterating this relation proves the Taylor identity with rema","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3ec920e0c2fd2fa","tex":"\\mathcal T_M","aliases":"T_M mathcal","display":"\\mathcal T_M","kind":"use","excerpt":"tion v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b823f2707e33253e","tex":"\\mathfrak T_m","aliases":"T_m mathfrak","display":"\\mathfrak T_m","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-733902ab0028bba1","tex":"t_m","aliases":"t_m","display":"t_m","kind":"use","excerpt":"_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh","count":42},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3141391f5a805560","tex":"t_{m+1}","aliases":"t_m+1","display":"t_{m+1}","kind":"use","excerpt":"terate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, where t_0=0 and t_m+1=ϕ(t_m). The sequence (t_m) is increasing because ϕ has nonnega","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eca9aefbb65ee59b","tex":"T_{m,p}","aliases":"T_m,p","display":"T_{m,p}","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-143632817a84070d","tex":"t_m^{-1}","aliases":"t_m^-1","display":"t_m^{-1}","kind":"use","excerpt":"For N≥3, [source reference] writes all but finitely many reduced zeros as λ_m = (A_N(m+frac12))^N(1+ε_m) where λ_m>0, 1+ε_m>0, and ε_m = O(t_m^-1e^-γ t_m). On every compact set of s values, (1+ε_m)^-s-1 = O(|ε_m|) uniformly for large m. The difference λ_m^-s - A_N^-Ns(m+frac12)^-Ns therefore has a normally convergent sum on every compact subset of ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27c742e0ae74badf","tex":"t_m^2","aliases":"t_m^2","display":"t_m^2","kind":"use","excerpt":"he nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78eb381d2b54657d","tex":"t_m^N","aliases":"t_m^N","display":"t_m^N","kind":"use","excerpt":"[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3cfdac8841367b10","tex":"t_m^{N-1}","aliases":"t_m^N-1","display":"t_m^{N-1}","kind":"use","excerpt":"[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-163dd25fe6cb484d","tex":"t_N","aliases":"t_N","display":"t_N","kind":"use","excerpt":"[Direct rotations converge to packet transport] Assume the complete packet family P(t) is twice continuously differentiable in operator norm on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T with sufficiently small mesh, define V_π = U( P(t_N), P(t_N-1)) ⋯ U( P(t_1), P(t_0)). Then V_π⟶ U(T) in operator norm, where U is the canonical transport from [sourc","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2ed7a9fa236bbf0","tex":"t_n","aliases":"t_n","display":"t_n","kind":"use","excerpt":"[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integer","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40bcddeff96a69c7","tex":"T_{n+1}","aliases":"T_n+1","display":"T_{n+1}","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55fe8241bd06de3e","tex":"t_{n+1}","aliases":"t_n+1","display":"t_{n+1}","kind":"use","excerpt":"[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integer","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90b38ede3f1b080b","tex":"T_{N,k}","aliases":"T_N,k","display":"T_{N,k}","kind":"use","excerpt":"t z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dcaefd9e3e9b42b6","tex":"T_{N,k+\\ell\\, (\\mathrm{mod}\\,N)}","aliases":"T_N,k+ℓ (mod N) ell mathrm","display":"T_{N,k+\\ell\\, (\\mathrm{mod}\\,N)}","kind":"use","excerpt":"obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a41cf9f85eb00b77","tex":"t_{n,\\delta}","aliases":"t_n,δ delta","display":"t_{n,\\delta}","kind":"use","excerpt":"[Probability recovery bound] Fix 0<δ<1 and define t_n,δ = 2σ sqrtlog(frac2nδ) and b_h,δ=Mh+t_n,δ. With probability at least 1-δ, every contrast satisfies the conclusions of [source reference] with b_h replaced by b_h,δ.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2718681b82c9502","tex":"T_{N,\\ell}","aliases":"T_N,ℓ ell","display":"T_{N,\\ell}","kind":"use","excerpt":"obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8838c89526c109a3","tex":"t_{N-1}","aliases":"t_N-1","display":"t_{N-1}","kind":"use","excerpt":"[Direct rotations converge to packet transport] Assume the complete packet family P(t) is twice continuously differentiable in operator norm on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T with sufficiently small mesh, define V_π = U( P(t_N), P(t_N-1)) ⋯ U( P(t_1), P(t_0)). Then V_π⟶ U(T) in operator norm, where U is the canonical transport from [sourc","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-839c9c5ecd29436a","tex":"t_{n-1}","aliases":"t_n-1","display":"t_{n-1}","kind":"use","excerpt":"[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61a8465dadb53175","tex":"T_{nn}","aliases":"T_nn","display":"T_{nn}","kind":"use","excerpt":"[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f3eddf7a095a301","tex":"T_{nq}","aliases":"T_nq","display":"T_{nq}","kind":"use","excerpt":"[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dcee229e3aaf9526","tex":"t_o","aliases":"t_o","display":"t_o","kind":"use","excerpt":"[Graph replay threshold] If graph capture costs c≥0, ordinary execution costs t_o≥0, and replay costs 0≤ t_r<t_o, then capture is profitable when R> fracct_o-t_r.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1927047d884ba88c","tex":"T_P","aliases":"T_P","display":"T_P","kind":"use","excerpt":"Deriving the estimate by differentiation. For a skew-adjoint generator G, define the actual conjugation path T_P(t)=exp(tG)Pexp(-tG). The formalization derives its derivative from the exponential power series and the product rule: T_P'(t)=exp(tG)(GP-PG)exp(-tG). Both exponential factors are proved unitary. Left and right multiplication by a unitary","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81d0a2ee84c8c5ee","tex":"T_P'","aliases":"T_P'","display":"T_P'","kind":"use","excerpt":"Deriving the estimate by differentiation. For a skew-adjoint generator G, define the actual conjugation path T_P(t)=exp(tG)Pexp(-tG). The formalization derives its derivative from the exponential power series and the product rule: T_P'(t)=exp(tG)(GP-PG)exp(-tG). Both exponential factors are proved unitary. Left and right multiplication by a unitary","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df3f310abe33eba9","tex":"T_r","aliases":"T_r","display":"T_r","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62e5b19cba54a896","tex":"\\mathcal T_{r-1}","aliases":"T_r-1 mathcal","display":"\\mathcal T_{r-1}","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3fa825b7cd03ebec","tex":"t_r^{w_r(e)}","aliases":"t_r^w_r(e)","display":"t_r^{w_r(e)}","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cc37eba865c3ced","tex":"T_{rr}","aliases":"T_rr","display":"T_{rr}","kind":"use","excerpt":"em[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4775952797c5d985","tex":"t_{\\mathrm{run}}","aliases":"t_run mathrm","display":"t_{\\mathrm{run}}","kind":"use","excerpt":"Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-557661f6ae0c797e","tex":"T_t","aliases":"T_t","display":"T_t","kind":"use","excerpt":"[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10c7fe6561154963","tex":"T_{uv}","aliases":"T_uv","display":"T_{uv}","kind":"use","excerpt":"Let Γ=(V,E) be a finite connected graph and fix a vector space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in thos","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71780f74aa8fa6a1","tex":"T_{v_1v_0}","aliases":"T_v_1v_0","display":"T_{v_1v_0}","kind":"use","excerpt":"or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61d5402325a778e4","tex":"T_{v_rv_{r-1}}","aliases":"T_v_rv_r-1","display":"T_{v_rv_{r-1}}","kind":"use","excerpt":"or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0130fb3e4ce12f0a","tex":"T_{vu}^{-1}","aliases":"T_vu^-1","display":"T_{vu}^{-1}","kind":"use","excerpt":"Let Γ=(V,E) be a finite connected graph and fix a vector space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in thos","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ab411d9663afe22","tex":"t_w","aliases":"t_w","display":"t_w","kind":"use","excerpt":"We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormali","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a573e0e7c4db0180","tex":"\\mathcal T_x","aliases":"T_x mathcal","display":"\\mathcal T_x","kind":"use","excerpt":"At positive polynomial depth, restrict it to the correction fiber T_x={t∈ S_q^d:overliner_x+J_xt=0} and the lift set L_x,L= {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L, F(y)=0}. The polynomial predicate in the second display is evaluation in the residue ring. Evaluation on canonical integer representatives gives the same predicate, because integer polynomial eva","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-107e01d7daa9bd03","tex":"T_z","aliases":"T_z","display":"T_z","kind":"use","excerpt":"Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace. Equivalently, apply the decomposition to left multiplication by B on the matrix algebra a","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7007840dcb4ea04f","tex":"T_z'","aliases":"T_z'","display":"T_z'","kind":"use","excerpt":"Let J=∫_0^∞ T_z(t) dt. Integrating both derivative identities T_z'=(B-zI)T_z=T_z(B-zI) and using T_z(0)=I gives (zI-B)J=J(zI-B)=I. Thus J=(zI-B)^-1. Continuous left and right multiplication carry the Bochner integral through the two rectangular transfer blocks, yielding the claim.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-186d4ea45168e2cc","tex":"T_\\alpha","aliases":"T_α alpha","display":"T_\\alpha","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a431e805508f7d7","tex":"T_{\\beta_1\\beta_0}","aliases":"T_β_1β_0 beta beta","display":"T_{\\beta_1\\beta_0}","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69c19ac5861b4555","tex":"T_{\\beta_m\\beta_{m-1}}","aliases":"T_β_mβ_m-1 beta beta","display":"T_{\\beta_m\\beta_{m-1}}","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f053dd14f899054","tex":"T_{\\beta\\gamma}","aliases":"T_βγ beta gamma","display":"T_{\\beta\\gamma}","kind":"use","excerpt":"[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5687f719fcef22b","tex":"T_\\gamma","aliases":"T_γ gamma","display":"T_\\gamma","kind":"use","excerpt":"or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a7c0769ed8b38f6","tex":"T_{\\gamma_1}","aliases":"T_γ_1 gamma","display":"T_{\\gamma_1}","kind":"use","excerpt":"valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-740e079411fa0098","tex":"T_{\\gamma_2}","aliases":"T_γ_2 gamma","display":"T_{\\gamma_2}","kind":"use","excerpt":"valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a07c1d3d8c928212","tex":"T_{\\gamma_2\\gamma_1}","aliases":"T_γ_2γ_1 gamma gamma","display":"T_{\\gamma_2\\gamma_1}","kind":"use","excerpt":"valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-265d18b60f7a60c8","tex":"T_{\\gamma\\alpha}","aliases":"T_γα gamma alpha","display":"T_{\\gamma\\alpha}","kind":"use","excerpt":"[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-708d92a009cb0a50","tex":"T_{\\delta\\gamma}","aliases":"T_δγ delta gamma","display":"T_{\\delta\\gamma}","kind":"use","excerpt":"[Tensor product transfer blocks] Let S∈End(X) and T∈End(Y), with packet systems P and Q. Relative to the tensor product packet system of [source reference], (S⊗ T)_(β,δ),(α,γ) = S_βα⊗ T_δγ. [source reference] ExportsTensor network, quantum circuit, and multivariate packet calculations.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e324d98831d6eac7","tex":"t_\\varepsilon","aliases":"t_ε varepsilon","display":"t_\\varepsilon","kind":"use","excerpt":"4 · (0,32,32,64) · 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less than ε/2. Th","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a15b96314fbf77e6","tex":"T_{\\ell+1}","aliases":"T_ℓ+1 ell","display":"T_{\\ell+1}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b90b5fbc860e4de8","tex":"T_\\infty","aliases":"T_∞ infty","display":"T_\\infty","kind":"use","excerpt":"[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54bc773a0263d21b","tex":"\\takeaway","aliases":"takeaway","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75177911206476aa","tex":"\\tan","aliases":"tan","display":"\\tan","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-978e87e4f3d831a3","tex":"\\tanh","aliases":"tanh","display":"\\tanh","kind":"use","excerpt":"The identity tanh t=-itan(it) gives the odd zigzag expansion of tanh t. Differentiate.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c0ebebd59054caf","tex":"\\mathrm{target}","aliases":"target mathrm","display":"\\mathrm{target}","kind":"use","excerpt":"headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumerat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89ca555db043e94d","tex":"\\operatorname{target}","aliases":"target operatorname","display":"\\operatorname{target}","kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-712d027743d7bde8","tex":"\\Tcal","aliases":"Tcal T mathcal","display":"{\\mathcal T}","kind":"use","excerpt":"[Spanning tree normalization] Choose a root and a spanning tree T of Γ. There is a gauge in which every tree edge transport is the identity. In that gauge, the transport on each edge outside the tree is the holonomy of its fundamental cycle, with the orientation used to define that cycle.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f73ad12c9477d300","tex":"\\operatorname{Teich}","aliases":"Teich operatorname","display":"\\operatorname{Teich}","kind":"use","excerpt":"Indeed, for residue cardinality q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91696060f027f950","tex":"\\Tfun_1","aliases":"Tfun_1 T_1 Tfun mathsf","display":"{\\mathsf T}_1","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b588f6b7ad9c4f3e","tex":"\\Tfun_2","aliases":"Tfun_2 T_2 Tfun mathsf","display":"{\\mathsf T}_2","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f728757e8598144","tex":"\\Tfun_2'","aliases":"Tfun_2' T_2' Tfun mathsf","display":"{\\mathsf T}_2'","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2b22989e5403d15","tex":"\\Tfun_2^2","aliases":"Tfun_2^2 T_2^2 Tfun mathsf","display":"{\\mathsf T}_2^2","kind":"use","excerpt":"For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b44ccae829ac2f08","tex":"\\Tfun_N'","aliases":"Tfun_N' T_N' Tfun mathsf","display":"{\\mathsf T}_N'","kind":"use","excerpt":"[Scalar higher Riccati identity] At every point of the open set {z: D_N(z)≠0}, Y_N( - T_N, - T_N', …, - T_N^(N-1) ) =-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0dc0b6a9ff0861d3","tex":"\\Tfun_N^{(N-1)}","aliases":"Tfun_N^(N-1) T_N^(N-1) Tfun mathsf","display":"{\\mathsf T}_N^{(N-1)}","kind":"use","excerpt":"[Scalar higher Riccati identity] At every point of the open set {z: D_N(z)≠0}, Y_N( - T_N, - T_N', …, - T_N^(N-1) ) =-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9faf0c0fb01f07e","tex":"\\Thetafun_{2,1}","aliases":"Thetafun_2,1 symupΘ_2,1 Thetafun symup Theta","display":null,"kind":"use","excerpt":"For N=2, symupΘ_2,1(u)=-u(u+1) and symupΘ_2,2(u) = u(u+1)(5u^2+u+2). The two factors predicted by [source reference] are visible.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-754a893d0d626b29","tex":"\\Thetafun_{2,2}","aliases":"Thetafun_2,2 symupΘ_2,2 Thetafun symup Theta","display":null,"kind":"use","excerpt":"For N=2, symupΘ_2,1(u)=-u(u+1) and symupΘ_2,2(u) = u(u+1)(5u^2+u+2). The two factors predicted by [source reference] are visible.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46fbcb84704ec4c2","tex":"\\Thetafun_{N,1}","aliases":"Thetafun_N,1 symupΘ_N,1 Thetafun symup Theta","display":null,"kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3000c3bd47e8007c","tex":"\\toprule","aliases":"toprule","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j))","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"The usual scalar invariants can be misleading: trH=2, det H=1, χ_H(t)=(t-1)^2. These are also the invariants of the 2×2 identity matrix. Nevertheless, Ob=H-I=mat0 · 1","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc7a5c603b95f30d","tex":"\\tr","aliases":"tr operatorname","display":"{\\operatorname{tr}}","kind":"use","excerpt":"D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bbf4da095ddbb7b","tex":"\\Tr_A","aliases":"Tr_A Tr operatorname","display":"{\\operatorname{Tr}}_A","kind":"use","excerpt":"[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ad8441d9d51dab3","tex":"\\operatorname{tr}_E","aliases":"tr_E operatorname","display":"\\operatorname{tr}_E","kind":"use","excerpt":"_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ec1b1b373658d51","tex":"\\Tr_{E'}","aliases":"Tr_E' Tr operatorname","display":"{\\operatorname{Tr}}_{E'}","kind":"use","excerpt":"[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93993d18d84ec366","tex":"\\operatorname{Tr}_{E/L}","aliases":"Tr_E/L operatorname","display":"\\operatorname{Tr}_{E/L}","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace c","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06c81003c38cac43","tex":"\\operatorname{tr}_L","aliases":"tr_L operatorname","display":"\\operatorname{tr}_L","kind":"use","excerpt":"_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03d1d351ea22341e","tex":"\\Tr_{\\rm out}","aliases":"Tr_rm out Tr rm operatorname rm","display":"{\\operatorname{Tr}}_{\\rm out}","kind":"use","excerpt":"labelstrong:standardfact:001 A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38a6461e5f88ab3d","tex":"\\triangle","aliases":"triangle","display":"\\triangle","kind":"use","excerpt":"[Packet perturbation bound] For 1≤ p<∞, interpreted first as an inequality of extended L^p seminorms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise sup","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5412ab68daaa1c9","tex":"\\TT_{\\frakm}","aliases":"TT_frakm T_ m TT frakm mathbb mathfrak","display":"{\\mathbb T}_{{\\mathfrak m}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"use","excerpt":"[Local Lyapunov--Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there ar","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec61c8382b2d3710","tex":"U'","aliases":"U'","display":"U'","kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11b41534cdf61584","tex":"u''","aliases":"u''","display":"u''","kind":"use","excerpt":"On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a257f9d1ed0e54ad","tex":"u^{\\mathsf T}","aliases":"u^ T mathsf","display":"u^{\\mathsf T}","kind":"use","excerpt":"[Rationality of the transfer series] The weighted transfer series satisfies Z_u,v(t) = u^ T(I-A(t))^-1v. It is a rational function whose denominator divides det(I-A(t)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39eb7259ed5bbafd","tex":"u^{(4)}","aliases":"u^(4)","display":"u^{(4)}","kind":"use","excerpt":"[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8ce79d19d64cdb3","tex":"u^{(p-1)p^a}","aliases":"u^(p-1)p^a","display":"u^{(p-1)p^a}","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1296a5898d2e7d5f","tex":"U^*","aliases":"U^*","display":"U^*","kind":"use","excerpt":"[Hyperbolic phase formula] Let U be a finite dimensional F_p space, let a∈GL(U) have no eigenvalue one, and let g act on H(U)=U⊕ U^* by g=(a,(a^-1)^*). Then Φ_p(g,H(U)) = (fracdet ap).","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algeb","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a86ed18272602c0f","tex":"U^1","aliases":"U^1","display":"U^1","kind":"use","excerpt":"The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e28cd1be4d9be5f9","tex":"U^2","aliases":"U^2","display":"U^2","kind":"use","excerpt":", and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber H_0:=fracU^4-12 · =P+Q-fracP^22-fracQ^22 +3PQ+i E F(P-Q).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a404366fd8ee067","tex":"u^{2^{a+1}}","aliases":"u^2^a+1","display":"u^{2^{a+1}}","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c105d72a53dfb0f9","tex":"U^{2m}","aliases":"U^2m","display":"U^{2m}","kind":"use","excerpt":"For u,v in U m(A), their group commutator is in U^2m(A)=U^N(A), which lies in U (m+1)(A). The terminal approximation satisfies","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa9db06abcab2308","tex":"u^{2N}","aliases":"u^2N","display":"u^{2N}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d0ee566c7488a3c","tex":"U^4","aliases":"U^4","display":"U^4","kind":"use","excerpt":"For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every n","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51aed094494b5462","tex":"U^{4n}","aliases":"U^4n","display":"U^{4n}","kind":"use","excerpt":"Put K= Q_2(i), R= Z_2[i], a=(1+i)/2, and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd46b5e0e8aaea04","tex":"U^8","aliases":"U^8","display":"U^8","kind":"use","excerpt":"For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every n","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bad9d0840a84cee","tex":"U^a","aliases":"U^a","display":"U^a","kind":"use","excerpt":"pageheading1. Coefficient products and reciprocal differentiationA characteristic-p argument without factorial division For sequences f,g: N→ K, where K is a field, define (f⋆ g)(d)=∑_i=0^dbinom di f(i)g(d-i), (U^a f)(d)=f(d+a). These are Hurwitz coefficients. The notation does not divide by d! in F_p. In particular, the implementation is not ordin","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ced2329334a8bdd4","tex":"U^{a-i}","aliases":"U^a-i","display":"U^{a-i}","kind":"use","excerpt":"The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4eebc9c8e49ecfa0","tex":"u^{ab}","aliases":"u^ab","display":"u^{ab}","kind":"use","excerpt":"For actual finite original inertia images the common kernel sample is derived. Suppose u∈ H satisfies ν(u)=1 and t(u)≠0 and both r_D(H) and r_E(H) are finite. Their sampled actions have positive finite orders a,b. At u^ab both original actions are trivial, and t(u^ab)=(ab)t(u)≠0 by the cocycle identity and characteristic zero. Thus the same full in","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6d4889a32c287d8","tex":"u^H","aliases":"u^H","display":"u^H","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-663ee04bfdc73b2a","tex":"u^h","aliases":"u^h","display":"u^h","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec5ed9d3298fa46f","tex":"u^{H_a}","aliases":"u^H_a","display":"u^{H_a}","kind":"use","excerpt":"2^a+1, · p=2. Then Δ_H_a^ ru^n = u^n(u^H_a-1)^r. For odd p, v_p(Δ_H_a^ ru^n) ≥(a+1)r. For p=2 and odd u, v_2(Δ_H_a^ ru^n) ≥(a+3)r. Equivalently, every additional prime power digit increases the certified Mahler slope by one.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3d9f585790af390","tex":"u^h_i","aliases":"u^h_i","display":"u^h_i","kind":"use","excerpt":"[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8538af623a4ec3ed","tex":"u^{Hx+j}","aliases":"u^Hx+j","display":"u^{Hx+j}","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4fb35335a011459e","tex":"U^i","aliases":"U^i","display":"U^i","kind":"use","excerpt":"The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df29459720bd9419","tex":"u^j","aliases":"u^j","display":"u^j","kind":"use","excerpt":"[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa49e3001badb297","tex":"u^{j+(p-1)s}","aliases":"u^j+(p-1)s","display":"u^{j+(p-1)s}","kind":"use","excerpt":"[Local phase lattice] For a unit mode u^n, the odd prime coordinate gives u^n = u^j+(p-1)s (u^(p-1)p^a)^q. The dyadic coordinate gives u^n = u^j+2s (u^2^a+1)^q. Thus finite character phase, local digit address, and Mahler depth are separate coordinates.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b4cbf2da3f0cedc","tex":"u^{j+2s}","aliases":"u^j+2s","display":"u^{j+2s}","kind":"use","excerpt":"[Local phase lattice] For a unit mode u^n, the odd prime coordinate gives u^n = u^j+(p-1)s (u^(p-1)p^a)^q. The dyadic coordinate gives u^n = u^j+2s (u^2^a+1)^q. Thus finite character phase, local digit address, and Mahler depth are separate coordinates.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cef659aa5954e576","tex":"U^k","aliases":"U^k","display":"U^k","kind":"use","excerpt":"0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0667d48b5628d7ee","tex":"u^{k_0}","aliases":"u^k_0","display":"u^{k_0}","kind":"use","excerpt":"y q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the entire sector induction and a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8e90bd5459efd8c","tex":"U^m","aliases":"U^m","display":"U^m","kind":"use","excerpt":"Here H m means H^1∩ U^m(A). Therefore the image S of U m(A) in W is canonically","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6106e1ebef1b51f","tex":"u^m","aliases":"u^m","display":"u^m","kind":"use","excerpt":"The coefficients c_u in the proof of [source reference] are constant on the cosets of H_m. For u≠0, the value u^m depends only on the coset, and every element of G_m arises. Grouping the modes by cosets gives the displayed expansion.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2bc1d06f7629695","tex":"U^N","aliases":"U^N","display":"U^N","kind":"use","excerpt":"Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7862d568dcbac6d5","tex":"u^N","aliases":"u^N","display":"u^N","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49185fe756c9fe7d","tex":"u^n","aliases":"u^n","display":"u^n","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf3adb26b0f573f1","tex":"u^{n+p-1}","aliases":"u^n+p-1","display":"u^{n+p-1}","kind":"use","excerpt":"For positive n, the zero mode contributes nothing. Every nonzero mode satisfies u^p-1=1. Hence b_n+p-1^(m,p) = ∑_u≠0c_uu^n+p-1 = ∑_u≠0c_uu^n = b_n^(m,p).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bb90c7b2f3835ae","tex":"u^{Nm}","aliases":"u^Nm","display":"u^{Nm}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f4dc2d7e9a66ef97","tex":"u^{nN}","aliases":"u^nN","display":"u^{nN}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e184f8300c2a9879","tex":"u^{Nq}","aliases":"u^Nq","display":"u^{Nq}","kind":"use","excerpt":"[Theta reconstruction] For every n≥0, symupΘ_N,n(u) = (-1)^n ∑_q=0^n binomNnNq P_N,q(1)u^NqP_N,n-q(u). Thus symupΘ_N,n is a polynomial of degree at most Nn.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da5e436b748b454f","tex":"u^p","aliases":"u^p","display":"u^p","kind":"use","excerpt":"The full-character argument uses a specific order of operations. First construct a full character at native rank p^j, trivial on U_F^1. Transfer its full beta-determinant unit character to multiplicity p, using BK 3.6.13(ii) and 3.6.14. On a scalar u∈ U_F^1, this gives θ_j(u^p)=1. Next add the actual trace-zero correction, using BK 3.3.18. It remai","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ab4e8c0ac2eecc2","tex":"U^{p+L}","aliases":"U^p+L","display":"U^{p+L}","kind":"use","excerpt":"0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a53f53a1a1287a8","tex":"u^{p-1}","aliases":"u^p-1","display":"u^{p-1}","kind":"use","excerpt":"For positive n, the zero mode contributes nothing. Every nonzero mode satisfies u^p-1=1. Hence b_n+p-1^(m,p) = ∑_u≠0c_uu^n+p-1 = ∑_u≠0c_uu^n = b_n^(m,p).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b13f6b9a00f77135","tex":"U^{p^2}","aliases":"U^p^2","display":"U^{p^2}","kind":"use","excerpt":"0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4a8fe218fcdcbef","tex":"U^{q-a}","aliases":"U^q-a","display":"U^{q-a}","kind":"use","excerpt":"pageheading1. The characteristic-zero product ruleThe normalized defect is not an ordinary derivation Let A be a commutative algebra over a characteristic-zero field K, and let U be an actual K-derivation. The source defines B=fracU^p^2-U^pp, M_q(f,g)=∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6a7e14e69737cad","tex":"u^r","aliases":"u^r","display":"u^r","kind":"use","excerpt":"The nth coefficient is ∑_r=0^n binom nr u^rv^n-r = (u+v)^n by the binomial theorem.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-050c8b635461358e","tex":"u^{r_i}","aliases":"u^r_i","display":"u^{r_i}","kind":"use","excerpt":"[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c05ecd0a14b319f","tex":"u^T","aliases":"u^T","display":"u^T","kind":"use","excerpt":"-1 · 3. Hence D^-1u=u/4 and S=1+u^TD^-1u=frac52. The Woodbury identity reduces the entire inverse to A^-1=D^-1-D^-1uS^-1u^TD^-1 =D^-1-frac140uu^T. vfigwoodbury headTwo hypotheses, not one Invertibility of D alone does not suffice. For D=I_2, u=e_1, and v=-e_1, the update is diag(0,1). The small matrix S is zero, exactly where the correction formula","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-515a098b70460045","tex":"u^{\\alpha-1}","aliases":"u^α-1 alpha","display":"u^{\\alpha-1}","kind":"use","excerpt":"It is enough to prove the formula on an eigenvector with eigenvalue x≥0. For x>0, the substitution s=xu gives fracsin(πα)π ∫_0^∞ s^α-1fracxs+x d s =x^α fracsin(πα)π ∫_0^∞fracu^α-11+u d u =x^α. The value at x=0 is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power c","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e46faccabfe558b","tex":"U_{-Q}","aliases":"U_-Q","display":"U_{-Q}","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a3339eaf4eb0015","tex":"U_{0,n}","aliases":"U_0,n","display":"U_{0,n}","kind":"use","excerpt":"sin t · cos t . Only the first adjacent defect is nonzero, so every defect sum is finite. Yet for n≥1, U_0,n(t)=C_0T_n(t)R_0=cos t. The limiting family is not a semigroup because in general cos(s+t)≠cos scos t. The failure is exactly the excursion from the reconstructed line into its orthogonal complement between two time steps.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf0029d07e1738a6","tex":"u_0^{-1}","aliases":"u_0^-1","display":"u_0^{-1}","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1239c6d46ffc26e","tex":"u_0^2","aliases":"u_0^2","display":"u_0^2","kind":"use","excerpt":"[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"ro and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span ont","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7faa68216e66f910","tex":"u_1^2","aliases":"u_1^2","display":"u_1^2","kind":"use","excerpt":"[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76bfead452deff02","tex":"U_2","aliases":"U_2","display":"U_2","kind":"use","excerpt":"For example, U_2(n)≡2pmod32 and U_3(n)≡6pmod8 for every row. At precision 2^10 the least packet periods are 512,512,32,128. A packet displacement T has original degree displacement 4T, as proved by leanKomorebiPaperIX.DyadicFourLeastPeriods.original_degree_exact_displacement. Minimality here concerns packet shifts. Shifts of the full degree-indexed","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f97c9046c1a4c7a","tex":"u_2","aliases":"u_2","display":"u_2","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span ont","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-deb9be7f41a14efb","tex":"U_3","aliases":"U_3","display":"U_3","kind":"use","excerpt":"For example, U_2(n)≡2pmod32 and U_3(n)≡6pmod8 for every row. At precision 2^10 the least packet periods are 512,512,32,128. A packet displacement T has original degree displacement 4T, as proved by leanKomorebiPaperIX.DyadicFourLeastPeriods.original_degree_exact_displacement. Minimality here concerns packet shifts. Shifts of the full degree-indexed","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2facfe7dfb6a6c9","tex":"U_{4,j}","aliases":"U_4,j","display":"U_{4,j}","kind":"use","excerpt":"For the remainder of the four-packet discussion abbreviate C_j=C_4,j and U_j=U_4,j. Removing the original row cap requires an unshifted coefficient operator.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e34eef57ab05c016","tex":"U_{6,3}","aliases":"U_6,3","display":"U_{6,3}","kind":"use","excerpt":"C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(2)-C_6,3(1)=frac64455. The last difference has 3-adic valuation zero. Here h=1, so at n=1 the order-one bound without factorial loss would require valuation at least one. The corrected normalized bound is sharp, since v_3(C_6,3(2)-C_6,3(1))=0=1-v_3(3!). Multiplication by 3! gives U_6,3(2)-U_6,3(1)=frac","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e105a73706668fb5","tex":"U_A","aliases":"U_A","display":"U_A","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91c677571574a0b9","tex":"u_a","aliases":"u_a","display":"u_a","kind":"use","excerpt":"1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4773df27618d48b2","tex":"u_{a+b}","aliases":"u_a+b","display":"u_{a+b}","kind":"use","excerpt":"Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68c0465eac68c384","tex":"u_a^{-1}","aliases":"u_a^-1","display":"u_a^{-1}","kind":"use","excerpt":"Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99789a7e60ddcff1","tex":"U_A^1","aliases":"U_A^1","display":"U_A^1","kind":"use","excerpt":"K× normalizes A and every power of its radical, so U_K 1 normalizes U_A m. Therefore P is a subgroup and is compact open. It is contained in U_A 1 because mtextgreater=1 and U_K^1⊂ U_A^1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d3c941a9caf1b86","tex":"U_A^j","aliases":"U_A^j","display":"U_A^j","kind":"use","excerpt":"Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK 3.1.7--9 give the actual groups","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81f335e188c44b36","tex":"U_A^m","aliases":"U_A^m","display":"U_A^m","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50f3b65ae0010e20","tex":"U_A^{m+1}","aliases":"U_A^m+1","display":"U_A^{m+1}","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbff91e9eae9c3b1","tex":"U_A^{n-r+m}","aliases":"U_A^n-r+m","display":"U_A^{n-r+m}","kind":"use","excerpt":"At both low ranks, K^× normalizes the new P. In rank two this follows directly from normalization of A and its radical. In rank four, a K-uniformizer is approximated by the derived chain shift R times a unit of B, with relative error in P_A^n-r. Its conjugation error on U_B^m lies in U_A^n-r+m⊂ J, which proves normalization. The exact argument is p","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-941e7992e7bd04e6","tex":"U_{ar}","aliases":"U_ar","display":"U_{ar}","kind":"use","excerpt":"textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91de088f77bc89c5","tex":"U_B","aliases":"U_B","display":"U_B","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5ee5bb5c320ccb3","tex":"U_b","aliases":"U_b","display":"U_b","kind":"use","excerpt":"Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected com","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e21dd43c611f6c4e","tex":"u_b","aliases":"u_b","display":"u_b","kind":"use","excerpt":"Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4bd5476608aab2a3","tex":"U_B^j","aliases":"U_B^j","display":"U_B^j","kind":"use","excerpt":"Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK 3.1.7--9 give the actual groups","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1e9f1a899dd951a","tex":"U_B^m","aliases":"U_B^m","display":"U_B^m","kind":"use","excerpt":"two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d1c052e998f0b1f","tex":"U_B^{m+1}","aliases":"U_B^m+1","display":"U_B^{m+1}","kind":"use","excerpt":"The quotient U_B^m/U_B^m+1 has dimension two. By Section~2 the image of J cap U_B m on this grade is exactly its centralizing line, of dimension one. For the converse inclusion needed here, the B-approximation of a K-element of grade m differs from it by grade m+n-rtextgreater a. That error lies in U_A a subset J. Thus the whole centralizing line r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d427e9012af5c75","tex":"U_{ba}","aliases":"U_ba","display":"U_{ba}","kind":"use","excerpt":"The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b42126afe8df2055","tex":"U_{ba}^*","aliases":"U_ba^*","display":"U_{ba}^*","kind":"use","excerpt":"The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71076ebfa81f46b5","tex":"U_{bc}","aliases":"U_bc","display":"U_{bc}","kind":"use","excerpt":"The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cfbfd121cc430e57","tex":"U_C","aliases":"U_C","display":"U_C","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two ","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65fa973c459a6b3b","tex":"U_c","aliases":"U_c","display":"U_c","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a69093f2b23577cd","tex":"u_c","aliases":"u_c","display":"u_c","kind":"use","excerpt":"· ⟶coker D_f ⟶coker D_c ⟶0. For u_c∈Ker D_c, choose u_f∈ U_f with p_Uu_f=u_c. There is a unique v_0∈ V_0 such that D_fu_f=i_Vv_0, and the connecting map is δ(u_c)=[v_0].","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0b3db69758fee0b4","tex":"U_d","aliases":"U_d","display":"U_d","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f763d6d1daaa1789","tex":"U_{da}","aliases":"U_da","display":"U_{da}","kind":"use","excerpt":"Adjoint conservation: source Theorem 4.2. Assume a star operation with P_a^*=P_a. Then (U_ba)^*=P_aU^*P_b. The two checked conservation laws are U^*U=1 · Longrightarrow ∑_b(U_ba)^*U_bc=δ_acP_a, UU^*=1 · Longrightarrow ∑_aU_ba(U_da)^*=δ_bdP_b. For the first law, expand each corner, collapse P_b^2 and sum: ∑_bP_aU^*P_bUP_c =P_aU^*(∑_bP_b)UP_c=P_aP_c.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26bee56e7d89a1c3","tex":"U_{da}^*","aliases":"U_da^*","display":"U_{da}^*","kind":"use","excerpt":"[Packet unitarity identities] If U is unitary and U_ba=P_bUP_a, then ∑_bU_ba^*U_bc=δ_acP_a and ∑_aU_baU_da^*=δ_bdP_b.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1bfe02ef752d496","tex":"u_e","aliases":"u_e","display":"u_e","kind":"use","excerpt":"Constructive parameterization. Write t=1/λ and set u_e(t)=min{m_e,tc_e/s_e}, Φ(t)=∑_e s_eu_e(t). Lean proves that Φ is continuous and nondecreasing, with Φ(0)=0. At the finite positive scale T=∑_efracm_es_ec_e, every coordinate is saturated, so u(T)=m and Φ(T)=∑_es_em_e. The intermediate value theorem produces t>0 with Φ(t)=τ. The multiplier is not","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fad0373a3389fff7","tex":"U_E^{(q+1)/2}","aliases":"U_E^(q+1)/2","display":"U_E^{(q+1)/2}","kind":"use","excerpt":"At the minimal rank-two base, extend the forced character on U_E^(q+1)/2 to U_E^1, imposing triviality on U_F^1. The prescriptions agree on their intersection because the parameter trace is one and ψ_F(u-1)=1. This is an actual finite abelian character extension and gives the full minimal simple character.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42c782cfb50f2105","tex":"U_E^1","aliases":"U_E^1","display":"U_E^1","kind":"use","excerpt":"Section~4 constructs chi_E on U_E 1 extending phi and trivial on U_F 1. Define theta_1 on H^1=U_E^1 U_(A_1) a by chi_E on the first factor and psi_alpha on the second. The restrictions agree on the intersection. BK 3.2.1 gives the actual full simple character, with theta_1textbar U_F 1=1.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ada27fd6e08006d5","tex":"U_f","aliases":"U_f","display":"U_f","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-77c182ea48441c1f","tex":"u_f","aliases":"u_f","display":"u_f","kind":"use","excerpt":"· ⟶coker D_f ⟶coker D_c ⟶0. For u_c∈Ker D_c, choose u_f∈ U_f with p_Uu_f=u_c. There is a unique v_0∈ V_0 such that D_fu_f=i_Vv_0, and the connecting map is δ(u_c)=[v_0].","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0975d040ea9e4ac2","tex":"U_F^s","aliases":"U_F^s","display":"U_F^s","kind":"use","excerpt":"The replacement for trace zero is the depth inequality v_F(trδ)+s≥1. It makes the additive correction trivial on scalar U_F^s, because (u-1)trδ∈ t O_F. Our native corrections have trace one, so the inequality holds at s=1. This is the exact source-to-target transfer of the weighted-visibility principle. The Euler coefficient theorem is motivation, ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-022130268d2f106e","tex":"U_G","aliases":"U_G","display":"U_G","kind":"use","excerpt":"[Composition of coefficient transports] If U_H and U_G are the triangular transforms associated with unit kernels H and G, then U_HU_G=U_HG. In particular, the inverse of U_H is U_H^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f72fa924cc22eb23","tex":"U_H","aliases":"U_H","display":"U_H","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6fdb23dc8f442414","tex":"u_h","aliases":"u_h","display":"u_h","kind":"use","excerpt":"nzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMChapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil differentiates the quadratic ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5fb0aa1c05324642","tex":"U_H^{-1}","aliases":"U_H^-1","display":"U_H^{-1}","kind":"use","excerpt":"[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb9ab2940f863d81","tex":"U_{H^{-1}}","aliases":"U_H^-1","display":"U_{H^{-1}}","kind":"use","excerpt":"[Composition of coefficient transports] If U_H and U_G are the triangular transforms associated with unit kernels H and G, then U_HU_G=U_HG. In particular, the inverse of U_H is U_H^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de7794b119a272d1","tex":"U_{HG}","aliases":"U_HG","display":"U_{HG}","kind":"use","excerpt":"[Composition of coefficient transports] If U_H and U_G are the triangular transforms associated with unit kernels H and G, then U_HU_G=U_HG. In particular, the inverse of U_H is U_H^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"Apply polynomial induction using constants, addition, and multiplication by a variable. Constants have zero remainder, and remainders add. For the multiplication step, suppose the assertion is known for P and write D_P(U)=∑_j(∂_jP)(a)U_j. Multiplication of its expansion by a_i+cU_i gives constant term P(a)a_i, linear term c(a_iD_P(U)+P(a)U_i), and ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c42292632e07b88","tex":"u_i^h","aliases":"u_i^h","display":"u_i^h","kind":"use","excerpt":"Finite integral linear combinations. For any finite index set S, coefficients a_i∈ Z_p and units u_i∈ Z_p^×, linearity of the actual iterated operator gives Δ_h^s (∑_i∈ Sa_i u_i^n) =∑_i∈ Sa_i u_i^n(u_i^h-1)^s. At h=p-1, the result is divisible by p^s. The empty sum is permitted. This does not prove that the general congruential Euler packets of The","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5e4fdf16b0453e5","tex":"u_i^n","aliases":"u_i^n","display":"u_i^n","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50d448cbdfbc40ce","tex":"u_i^{p-1}","aliases":"u_i^p-1","display":"u_i^{p-1}","kind":"use","excerpt":"Induction gives the geometric identity. Reduction of a unit to F_p^× gives u_i^p-1=1 modulo p. Taking s powers supplies p^s in each summand. The difference operator is linear over finite sums.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","excerpt":"For the remainder of the four-packet discussion abbreviate C_j=C_4,j and U_j=U_4,j. Removing the original row cap requires an unshifted coefficient operator.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43190b7d6e150fa1","tex":"u_j^{\\mathsf T}","aliases":"u_j^ T mathsf","display":"u_j^{\\mathsf T}","kind":"use","excerpt":"0 · 2 · 1 · -3. The graph shows rates between actual states. The four normalized sign vectors on the right form an orthonormal basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve in","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e41938b7bd341d27","tex":"U_K","aliases":"U_K","display":"U_K","kind":"use","excerpt":"en neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a uni","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72b2bc9e12d0fa8e","tex":"U_k","aliases":"U_k","display":"U_k","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-836830fc78b794aa","tex":"u_k","aliases":"u_k","display":"u_k","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6951bc1a235a45d","tex":"U_{k+1}","aliases":"U_k+1","display":"U_{k+1}","kind":"use","excerpt":"or preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41ab4fffc65aae75","tex":"u_{k+1}","aliases":"u_k+1","display":"u_{k+1}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0d6567d269df4cd4","tex":"u_{k+1}^{(j)}","aliases":"u_k+1^(j)","display":"u_{k+1}^{(j)}","kind":"use","excerpt":"[Companion jump recurrence] Every residue state packet satisfies u_k+1^(j)=C^Nu_k^(j). More explicitly, u_k^(j)=C^j(C^N)^kv_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-783dcbe8db097453","tex":"u_{k+2}","aliases":"u_k+2","display":"u_{k+2}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bd23f73fc8e35ba","tex":"u_{k+a}","aliases":"u_k+a","display":"u_{k+a}","kind":"use","excerpt":"The jump formula. Associativity and the laws for powers of one matrix give u_k=C^j(C^N)^kv_0=(C^N)^k(C^jv_0). Only powers of the same matrix are interchanged. The argument does not assume that arbitrary matrices commute. In particular, u_k+1=C^Nu_k, u_k+a=(C^N)^a u_k. The step matrix is C^N, not C. A checked example with the Fibonacci matrix and N=","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-550ce951561bdee5","tex":"u_{k+d}","aliases":"u_k+d","display":"u_{k+d}","kind":"use","excerpt":"The complete vector recurrence. Apply the matrix identity to the actual state u_k. The already proved advance identity identifies each D^a u_k with u_k+a. Consequently, ∑_a=0^dχ_D[a]u_k+a=0, u_k+d=-∑_a=0^d-1c_a u_k+a. This is the desired recurrence, with its monic leading coefficient and all earlier coefficients supplied by χ_D. It is not an assump","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c89ff2cc9cf26697","tex":"u_k^{(j)}","aliases":"u_k^(j)","display":"u_k^{(j)}","kind":"use","excerpt":"Let R be a commutative ring. Let v_n+1=Cv_n, C∈ M_d(R). For a positive integer N and an address j∈{0,…,N-1}, define the residue state packet u_k^(j)=v_Nk+j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1e23f9abe65f3fa","tex":"U_K^1","aliases":"U_K^1","display":"U_K^1","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc43fa35e0f4d358","tex":"U_K^M","aliases":"U_K^M","display":"U_K^M","kind":"use","excerpt":"It is well defined by the intersection condition. Choose Mtextgreater=a with chitextbar U_K M=1. Then chi_0 is a character of a subgroup of the finite abelian group U_K^1/U_K^M. Extend it by adjoining generators and choosing the required complex roots, and pull it back. The resulting chi_K extends chi and is trivial on U_F 1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d74849910db04db1","tex":"U_K^m","aliases":"U_K^m","display":"U_K^m","kind":"use","excerpt":"For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25121c20156ac918","tex":"U_m","aliases":"U_m","display":"U_m","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d86088ae482ad75","tex":"u_m","aliases":"u_m","display":"u_m","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30d31e43294cdc7c","tex":"u_{m+1}","aliases":"u_m+1","display":"u_{m+1}","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4312e8e6ee32bea","tex":"U_{m,n+1}","aliases":"U_m,n+1","display":"U_{m,n+1}","kind":"use","excerpt":"Since C_nR_n=I, U_m,n+1(t)-U_m,n(t) = · C_m,n (C_nT_n+1(t)-T_n(t)C_n) R_nR_n,m. Therefore normU_m,n+1(t)-U_m,n(t) ≤ w_m,nnormE_n(t). The weighted summability hypothesis makes the sequence uniformly Cauchy. Summing the tail estimates gives the stated bound.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65a04898bfc53b4d","tex":"U_{m-1}","aliases":"U_m-1","display":"U_{m-1}","kind":"use","excerpt":"Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4d61c16c87b9409","tex":"u_{Mk+s}","aliases":"u_Mk+s","display":"u_{Mk+s}","kind":"use","excerpt":"Refinement of an arithmetic address. Taking another subsequence with step M and offset s yields u_Mk+s=v_(NM)k+(Ns+j). The formal proof establishes this identity for the actual packet function by checking the index arithmetic. It is not inferred from a diagram or a naming convention.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23483ef5ed33ad34","tex":"U_{mp,j}","aliases":"U_mp,j","display":"U_{mp,j}","kind":"use","excerpt":"[Step-sensitive divisor-branch bound] Under the hypotheses of Theorem~[source reference], for all u,k,n∈ N, Δ_u(p-1)/m^kU_mp,j(n) ∈ p^min(k(1+ν_p(u)),mn) Z_p. For C_mp,j the corresponding valuation lower bound is min(k(1+ν_p(u)),mn)-v_p(j!).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9fcfc7136a869616","tex":"U_N","aliases":"U_N","display":"U_N","kind":"use","excerpt":"[Addition and inversion laws for the companion matrix] Assume N≥2. The matrix U_N(z)=e^zJ_N satisfies U_N(z+w)=U_N(z)U_N(w), U_N(z)^-1=U_N(-z), det U_N(z)=1. Its first column is D_N(z), and all of its entries are given by [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-167a8b5df4139bcf","tex":"U_n","aliases":"U_n","display":"U_n","kind":"use","excerpt":"not close the dynamicsREFINEMENT / A PERSISTENT RANGE EXCURSIONChapter VI, Part II: Theorem E4.8 and Example E4.10. Summability and closure are different tests. Let the coarse space be R and every finer space be R^2. The first connecting maps are C(x,y)=x and Rx=(x,0). All later maps are identities. Let the coarse evolution be the identity and all ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b207f2554f44324f","tex":"u_N","aliases":"u_N","display":"u_N","kind":"use","excerpt":"n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. T","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"use","excerpt":"Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dbd28c057692283b","tex":"u_N^*","aliases":"u_N^*","display":"u_N^*","kind":"use","excerpt":"ojector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7e46954a0250371","tex":"U_Q","aliases":"U_Q","display":"U_Q","kind":"use","excerpt":"Let P(z) and Q(z) be Laurent polynomials. Define L_P(z) = 1 · 0 P(z) · 1 , U_Q(z) = 1 · Q(z) 0 · 1 .","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c41edb137b53eb22","tex":"U_Q^{-1}","aliases":"U_Q^-1","display":"U_Q^{-1}","kind":"use","excerpt":"[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8611e5a338fe3b84","tex":"U_r","aliases":"U_r","display":"U_r","kind":"use","excerpt":"Let A be a transport system on copies of a fixed vector space V_0, and let B be a transport system on copies of a fixed vector space W_0. Fix a root r and an isomorphism U_r:V_0⟶ W_0.","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ee2348ac4bbdee7","tex":"\\mathcal U_r","aliases":"U_r mathcal","display":"\\mathcal U_r","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a822fb370eb1244d","tex":"u_r","aliases":"u_r","display":"u_r","kind":"use","excerpt":"Let M be a matrix over a DVR. Smith normal form gives invertible matrices U,V such that UMV = diag(π^s_1u_1,…,π^s_ru_r,0,…,0), where each u_i is a unit and 0≤ s_1≤⋯≤ s_r. The exponents s_i are directional lifting weights.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6d7e97428f82a13","tex":"\\mathcal U_r'","aliases":"U_r' mathcal","display":"\\mathcal U_r'","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c308aac68fce06b4","tex":"U_{r,q}","aliases":"U_r,q","display":"U_{r,q}","kind":"use","excerpt":"The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r,","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b96781006c96014d","tex":"U_{r,r}","aliases":"U_r,r","display":"U_{r,r}","kind":"use","excerpt":"0, · q>r. The row-action equality, the diagonal identity U_r,r=1, and vanishing above the diagonal are all checked. The arithmetic weight is also explicit: p^2(r-q)mid U_r,q(p) in Z, 0≤ q≤ r. No inverse-coordinate identity, unit diagonal or weight bound is assumed for these rows.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a59996f84749e99","tex":"U_r^{(q)}","aliases":"U_r^(q)","display":"U_r^{(q)}","kind":"use","excerpt":"[Charge pruning of circuit paths] Suppose every gate has a finite charge decomposition U_r=∑_qU_r^(q), SU_r^(q)S^*=ω^qU_r^(q). A term with gate charges q_1,…,q_m can contribute from source charge a_0 only to target charge a_m=a_0+q_1+⋯+q_m. Every packet path violating one of the intermediate charge updates vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3009c54a4dec532a","tex":"U_{\\rm root}","aliases":"U_rm root rm","display":"U_{\\rm root}","kind":"use","excerpt":"Proceed by induction on the depth of the retained subtree. If the root is retained, the statement is immediate. If the root is split, apply the unitary map U_rm root, then apply the induction hypothesis independently in each child subtree. A direct sum of unitary maps is unitary. Reversing the maps with adjoints gives synthesis.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab56dd36935ce891","tex":"U_u","aliases":"U_u","display":"U_u","kind":"use","excerpt":"[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad63d401332166ba","tex":"u_x","aliases":"u_x","display":"u_x","kind":"use","excerpt":"Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e67d8e81d3865f41","tex":"U_\\alpha","aliases":"U_α alpha","display":"U_\\alpha","kind":"use","excerpt":"[Packet decomposition of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6aa6e43e29fe528","tex":"U_{\\beta\\alpha}","aliases":"U_βα beta alpha","display":"U_{\\beta\\alpha}","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73c14771543aa3ee","tex":"U_{\\beta\\alpha}^*","aliases":"U_βα^* beta alpha","display":"U_{\\beta\\alpha}^*","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1befa0e66e68ea65","tex":"U_{\\beta\\gamma}","aliases":"U_βγ beta gamma","display":"U_{\\beta\\gamma}","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17bbb1fd2a57ab85","tex":"U_\\gamma","aliases":"U_γ gamma","display":"U_\\gamma","kind":"use","excerpt":"For a packet path γ=(a_0,a_1,…,a_m), write U_γ = P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f61cc273f3d06b1e","tex":"U_{\\gamma\\alpha}","aliases":"U_γα gamma alpha","display":"U_{\\gamma\\alpha}","kind":"use","excerpt":"[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9dd814369d529f5b","tex":"U_{\\gamma\\beta}","aliases":"U_γβ gamma beta","display":"U_{\\gamma\\beta}","kind":"use","excerpt":"et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-82de5e416cdd59ff","tex":"U_{\\delta\\alpha}","aliases":"U_δα delta alpha","display":"U_{\\delta\\alpha}","kind":"use","excerpt":"et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80535093685f5ad8","tex":"U_{\\delta\\alpha}^*","aliases":"U_δα^* delta alpha","display":"U_{\\delta\\alpha}^*","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5eada3bebae789bc","tex":"U_\\pi","aliases":"U_π pi","display":"U_\\pi","kind":"use","excerpt":"[Convergence of sampled transports] Assume K is Lipschitz on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T, define U_π = exp((t_N-t_N-1)K(t_N-1)) ⋯ exp((t_1-t_0)K(t_0)). As the mesh |π| tends to zero, U_π⟶ U(T) in operator norm. There is a constant C, depending only on T, the supremum norm of K, and its Lipschitz constant, such that U_π-U(T) ≤ C|π|.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4102fb7b5673ed4","tex":"\\underline q","aliases":"underline q underline","display":"\\underline q","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86e4d773f576d1fc","tex":"\\mathrm{unused}","aliases":"unused mathrm","display":"\\mathrm{unused}","kind":"use","excerpt":"efine compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are com","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb939028bb05fbbc","tex":"\\operatorname{updateValue}","aliases":"updateValue operatorname","display":"\\operatorname{updateValue}","kind":"use","excerpt":"Correctness of the actual update. For any edited subtree T' and valid context C, Lean proves updateValue(C,F(T'))=F(fill(C,T')). The proof inducts through the context. At each frame, the updated child contributes the new value, and each other child contributes its unchanged cached optimum. The frame recurrence is therefore exactly the full-tree rec","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"use","excerpt":"[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e7142ad42e2513e","tex":"v'","aliases":"v'","display":"v'","kind":"use","excerpt":"If v-v'∈im D, then Q_α(v-v')∈im D. Hence [v]↦[Q_α v] is well defined on the quotient. The idempotent, annihilation, and reconstruction identities descend from V to V/im D. The formula for an obstruction class follows from the definition.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b684dfa3e6bcfe1e","tex":"v^{(k)}","aliases":"v^(k)","display":"v^{(k)}","kind":"use","excerpt":"For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2fe163fa16dae77c","tex":"v^{(k)}_j","aliases":"v^(k)_j","display":"v^{(k)}_j","kind":"use","excerpt":"For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5eff5be3aa59d19","tex":"v^{(k)}_{j-r}","aliases":"v^(k)_j-r","display":"v^{(k)}_{j-r}","kind":"use","excerpt":"For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8ee3eea11f4b48b","tex":"V^{-1}","aliases":"V^-1","display":"V^{-1}","kind":"use","excerpt":"the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(1+T)/T, interpreted by coefficient shifting, differentiation of the displayed factorization gives R_c(l","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"For a canonical coordinate change with input and output chart v=2x, the target formula is not still v^2. It is widehat f_b(v)=2f(v/2)=fracv^22. Consequently widehat f_b(2x)=2x^2, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on Plate III.10.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d77b9a1d8369d4e","tex":"v^3","aliases":"v^3","display":"v^3","kind":"use","excerpt":"[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are iso","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a6544df8b60d670","tex":"V^H","aliases":"V^H","display":"V^H","kind":"use","excerpt":"Checked group, inertia and monodromy finite sums. Let K be a field, G a group, and ν:G→ K^× a character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant subspaces. ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff464ae558a3c307","tex":"V^{I_K}","aliases":"V^I_K","display":"V^{I_K}","kind":"use","excerpt":"Let V be one interaction packet. If its inertia character is nontrivial, then V^I_K=0. If the inertia character is trivial, Frobenius acts on the invariant packet by a companion cycle.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ddca55b935c74e1","tex":"V^{m-1}","aliases":"V^m-1","display":"V^{m-1}","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c01fa9580cfa7f71","tex":"v^n","aliases":"v^n","display":"v^n","kind":"use","excerpt":"[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb514f238366e3a5","tex":"v^{n-r}","aliases":"v^n-r","display":"v^{n-r}","kind":"use","excerpt":"The nth coefficient is ∑_r=0^n binom nr u^rv^n-r = (u+v)^n by the binomial theorem.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-01749da563f81b32","tex":"V^q","aliases":"V^q","display":"V^q","kind":"use","excerpt":"For k=2^r≥4, put q=k/2, which is even. Normalize the exact expansion [source reference] by 2^q before reducing modulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16cbf1f7187068e1","tex":"V^{q-1}","aliases":"V^q-1","display":"V^{q-1}","kind":"use","excerpt":"For k=2^r≥4, put q=k/2, which is even. Normalize the exact expansion [source reference] by 2^q before reducing modulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fcdd77803761b961","tex":"v^r","aliases":"v^r","display":"v^r","kind":"use","excerpt":"Distinct elements of G_m define linearly independent functions on one full period, by the Vandermonde determinant. Therefore a positive integer r is a period exactly when v^r=1 for every active mode. The least such r is the stated least common multiple.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0108c541d6b05811","tex":"v_{(NM)k+(Ns+j)}","aliases":"v_(NM)k+(Ns+j)","display":"v_{(NM)k+(Ns+j)}","kind":"use","excerpt":"Refinement of an arithmetic address. Taking another subsequence with step M and offset s yields u_Mk+s=v_(NM)k+(Ns+j). The formal proof establishes this identity for the actual packet function by checking the index arithmetic. It is not inferred from a diagram or a naming convention.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span ont","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c41a1fdf1eb317c","tex":"V_A","aliases":"V_A","display":"V_A","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a478c8ae160947ff","tex":"V_a","aliases":"V_a","display":"V_a","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93a28f0d222e68e7","tex":"v_a","aliases":"v_a","display":"v_a","kind":"use","excerpt":"An independent definition of Schmidt rank. The source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length s is the concrete equality C_ij=∑_a=1^su_a(i)v_a(j). It factors C=UV through an s-dimensional intermediate space, so rank C≤ s. Conversely, choose a basis of the row space of C. Expanding eac","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0367417d53502da1","tex":"V_A^*","aliases":"V_A^*","display":"V_A^*","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56405f2e425e038c","tex":"V_a^*","aliases":"V_a^*","display":"V_a^*","kind":"use","excerpt":"A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-612c002c1c1e9a93","tex":"v_a^T","aliases":"v_a^T","display":"v_a^T","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f5cc10c11d8d9e7","tex":"V_B","aliases":"V_B","display":"V_B","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d8b529eb2a0a517","tex":"V_b","aliases":"V_b","display":"V_b","kind":"use","excerpt":"A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3234e8a9dca18cb9","tex":"v_b","aliases":"v_b","display":"v_b","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d343ba286305f1ad","tex":"V_B^*","aliases":"V_B^*","display":"V_B^*","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c64cfa0416300f6","tex":"V_b^*","aliases":"V_b^*","display":"V_b^*","kind":"use","excerpt":"Exact correction, including coherent errors. For every operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe377465e680a401","tex":"v_b^T","aliases":"v_b^T","display":"v_b^T","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9336b2ab1b4f322f","tex":"V_C","aliases":"V_C","display":"V_C","kind":"use","excerpt":"vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef50ddb6014d940d","tex":"V_c","aliases":"V_c","display":"V_c","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b0f0625a5bf696ae","tex":"V_C^*","aliases":"V_C^*","display":"V_C^*","kind":"use","excerpt":"[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c63837778b9b8ed","tex":"V_C^T","aliases":"V_C^T","display":"V_C^T","kind":"use","excerpt":"0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. The remaining term is u_av_a^TD_B. Thus AB","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-588c6f8fdc8039b0","tex":"V_{ca}","aliases":"V_ca","display":"V_{ca}","kind":"use","excerpt":"0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d502f0a9336c724a","tex":"v_E","aliases":"v_E","display":"v_E","kind":"use","excerpt":"At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f979281d0af8ba8","tex":"V_f","aliases":"V_f","display":"V_f","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2528c99df30eac1c","tex":"v_F","aliases":"v_F","display":"v_F","kind":"use","excerpt":"The Artin--Schreier filtered lift has native trace one. Consequently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6c0af9d819fefe7","tex":"v_f","aliases":"v_f","display":"v_f","kind":"use","excerpt":"If a class in coker D_f comes from coker D_0, its image in coker D_c is zero. Conversely, suppose [v_f] maps to zero in coker D_c. Choose u_c with D_cu_c=p_Vv_f and lift it to u_f∈ U_f. Then p_V(v_f-D_fu_f)=0, so v_f-D_fu_f=i_Vv_0 for some v_0∈ V_0. Thus [v_f] comes from [v_0]. Surjectivity at the last term follows from surjectivity of p_V.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee9257fd8edde904","tex":"V_{\\frakm}","aliases":"V_frakm V_ m frakm mathfrak","display":"V_{{\\mathfrak m}}","kind":"use","excerpt":"[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0b837e89a1e8104e","tex":"V_h","aliases":"V_h","display":"V_h","kind":"use","excerpt":"[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91a52a41351c928c","tex":"V_i","aliases":"V_i","display":"V_i","kind":"use","excerpt":"The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"The actual invariant images U_i=imE_i give an equivariant linear isomorphism ⊕_i∈ JU_i≃ R^n, (v_i)_i⟼∑_i v_i, v⟼(E_iv)_i. Lean represents the finite direct sum by the finite product of the image submodules. The existing coordinatewise reduction theorem maps every U_i onto the corresponding reduced projector image. No chosen bases or unproved freene","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c6259c535cb4a68","tex":"V_{i,\\lambda}","aliases":"V_i,λ lambda","display":"V_{i,\\lambda}","kind":"use","excerpt":"For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesse","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c115186cc56f7111","tex":"V_i^{\\oplus m_i}","aliases":"V_i^⊕ m_i oplus","display":"V_i^{\\oplus m_i}","kind":"use","excerpt":"The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb483cd80b51511d","tex":"V_j^*","aliases":"V_j^*","display":"V_j^*","kind":"use","excerpt":"the Knill--Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-958542b62a8a99b9","tex":"v_j^*","aliases":"v_j^*","display":"v_j^*","kind":"use","excerpt":"vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYChapter VII: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8221e0482136849","tex":"v_j^{-1}","aliases":"v_j^-1","display":"v_j^{-1}","kind":"use","excerpt":"The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a435f82d4918621","tex":"v_k","aliases":"v_k","display":"v_k","kind":"use","excerpt":"ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity i","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2ad275314ca92dd","tex":"v_{k,h}","aliases":"v_k,h","display":"v_{k,h}","kind":"use","excerpt":"The discrete eigenpairs are λ_k,h = frac4h^2 sin^2(frackπ h2), v_k,h(i)=sin(kπ x_i).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-86272958a4f136a9","tex":"V_{k\\ell}","aliases":"V_kℓ ell","display":"V_{k\\ell}","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-073c6bfeafa5998b","tex":"V_{m+1}","aliases":"V_m+1","display":"V_{m+1}","kind":"use","excerpt":"[Surjectivity on stable cores] For every m, the transition map V_m+1→ V_m induces a surjection Y_m+1↠ Y_m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83eb9ad5dd648cb4","tex":"V_m^*","aliases":"V_m^*","display":"V_m^*","kind":"use","excerpt":"Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m], R_m · =[R_m-1 (D^(m-1))^*V_m]. Substitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b47bde7729d16d41","tex":"V_N","aliases":"V_N","display":"V_N","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"use","excerpt":"Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4cf1c9f8283bbb4c","tex":"V_{n+1}","aliases":"V_n+1","display":"V_{n+1}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fd7bfa7519bcf67","tex":"v_{n+1}","aliases":"v_n+1","display":"v_{n+1}","kind":"use","excerpt":"Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a3bb1565bfdea96","tex":"V_{n+2}","aliases":"V_n+2","display":"V_{n+2}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c3c0f1c05a928ce","tex":"v_{Nk+j}","aliases":"v_Nk+j","display":"v_{Nk+j}","kind":"use","excerpt":"Let R be a commutative ring. Let v_n+1=Cv_n, C∈ M_d(R). For a positive integer N and an address j∈{0,…,N-1}, define the residue state packet u_k^(j)=v_Nk+j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6afc7de2b2e38c7c","tex":"V_{nn}","aliases":"V_nn","display":"V_{nn}","kind":"use","excerpt":"egintheorem[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4751bc50d0cf4124","tex":"V_{nq}","aliases":"V_nq","display":"V_{nq}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51fd0818257204a0","tex":"v_r","aliases":"v_r","display":"v_r","kind":"use","excerpt":"The relation to derivatives of the complete coordinate representation is also literal. For an open domain U⊆ X, put widehat F= A_ Q∘ F∘ S_ P on the open coordinate set A_ P(U). Boundedness and linearity of analysis and synthesis imply, for 1≤ r≤ m, D^rwidehat F(u)[v_1,…,v_r] = A_ Q(D^rF( S_ Pu) [ S_ Pv_1,…, S_ Pv_r]). Indeed, the first derivative f","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad08de7d90dd5111","tex":"v_{r-1}","aliases":"v_r-1","display":"v_{r-1}","kind":"use","excerpt":"or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41540bfa4223f61d","tex":"V_{\\mathrm{src}}","aliases":"V_src mathrm","display":"V_{\\mathrm{src}}","kind":"use","excerpt":"Assume the one-vertex scalar-pushout contract: there are d labeled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-feb817bbfab28c96","tex":"V_{\\mathrm{src}}^\\perp","aliases":"V_src^⊥ mathrm perp","display":"V_{\\mathrm{src}}^\\perp","kind":"use","excerpt":"Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. N","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3685a7a9e666f70","tex":"V_u","aliases":"V_u","display":"V_u","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8d2185f95f25265","tex":"V_v","aliases":"V_v","display":"V_v","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-625e56119b06c003","tex":"V_y","aliases":"V_y","display":"V_y","kind":"use","excerpt":"[Dynamic-programming best packet basis] Suppose J_y(Λ)=∑_v∈Λc_v(y). For a node v, define recursively V_y(v) = c_v(y), · v terminal, [2mm] displaystyle min{c_v(y),∑_u∈ch(v)V_y(u)}, · v internal. Stop at v when c_v(y)≤∑_uV_y(u), including equality, and refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the no","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3a76197560b59e1","tex":"v_\\alpha","aliases":"v_α alpha","display":"v_\\alpha","kind":"use","excerpt":"[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b2b6ce643923566","tex":"V_\\lambda","aliases":"V_λ lambda","display":"V_\\lambda","kind":"use","excerpt":"arrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at ev","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-036484b5b6983b2e","tex":"V_{\\lambda_0}","aliases":"V_λ_0 lambda","display":"V_{\\lambda_0}","kind":"use","excerpt":"[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c997f5b198e5681","tex":"V_\\mu","aliases":"V_μ mu","display":"V_\\mu","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4995dcf4b8a274b","tex":"V_\\pi","aliases":"V_π pi","display":"V_\\pi","kind":"use","excerpt":"[Direct rotations converge to packet transport] Assume the complete packet family P(t) is twice continuously differentiable in operator norm on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T with sufficiently small mesh, define V_π = U( P(t_N), P(t_N-1)) ⋯ U( P(t_1), P(t_0)). Then V_π⟶ U(T) in operator norm, where U is the canonical transport from [sourc","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d40461d4abcbdcc","tex":"v_\\pi","aliases":"v_π pi","display":"v_\\pi","kind":"use","excerpt":"For a DVR, e=v_π(|Γ|) is derived from the actual group order. The single precision e+1, independent of rank, detects integral isomorphism. It makes reduction injective on integral isomorphism classes, and finite coefficient quotients give finitely many integral classes in fixed rank. This is a formalization of classical finite-group rigidity (Maran","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08b28ab9c3fd8503","tex":"v_\\varpi","aliases":"v_ϖ varpi","display":"v_\\varpi","kind":"use","excerpt":"The congruence is equivalent to e v_ϖ(x)≥ m, so x must lie in ϖ^lceil m/erceil O/ϖ^m, which has the stated cardinality. Write m=ek+r, 1≤ r≤ e. The exponent becomes (e-1)k+r-1. Summing over k≥0 and 1≤ r≤ e gives the rational expression.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02e87248f01e05be","tex":"V_\\ell","aliases":"V_ℓ ell","display":"V_\\ell","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-694bdc6e24088dc1","tex":"\\mathrm{val}","aliases":"val mathrm","display":"\\mathrm{val}","kind":"use","excerpt":"[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-022a12d8dd418b72","tex":"\\operatorname{val}","aliases":"val operatorname","display":"\\operatorname{val}","kind":"use","excerpt":"Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0e8009c8acb599b","tex":"\\Var","aliases":"Var operatorname","display":"{\\operatorname{Var}}","kind":"use","excerpt":"[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3696096793ac9d06","tex":"\\operatorname{Var}","aliases":"Var operatorname","display":"\\operatorname{Var}","kind":"use","excerpt":"o. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Co","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c7db940f017a3bc","tex":"\\Var_{p(t)}","aliases":"Var_p(t) Var operatorname","display":"{\\operatorname{Var}}_{p(t)}","kind":"use","excerpt":"[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dfcf7637eedfd13c","tex":"\\Var_\\gamma","aliases":"Var_γ Var gamma operatorname gamma","display":"{\\operatorname{Var}}_\\gamma","kind":"use","excerpt":"[Gaussian spectral gap] If f∈ L^2(γ) has mean zero, then norm O_tf_2 ≤ e^-tnorm f_2. For every polynomial f, Var_γ(f) ≤ ∫_ Rabsf'(x)^2γ( d x).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-754549a165ac03a7","tex":"\\Var_\\pi","aliases":"Var_π Var pi operatorname pi","display":"{\\operatorname{Var}}_\\pi","kind":"use","excerpt":"[Law of total variance] For a real valued f, let P_0f= E_π[f] 1. Then Var_π(f) = Var_π(P_ Qf) + normf-P_ Qf_π^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-902783d1c24b6e39","tex":"\\operatorname{Var}_\\pi","aliases":"Var_π operatorname pi","display":"\\operatorname{Var}_\\pi","kind":"use","excerpt":"o. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Co","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9e2d9dcc048eb142","tex":"\\varprojlim_m","aliases":"varprojlim_m varprojlim","display":null,"kind":"use","excerpt":"ng conventions All packet systems are finite unless a statement says otherwise. In a normed setting every packet projector and every coarsening operator is bounded. Differentiability of operator-valued maps is taken in operator norm. A Banach space isomorphism is a bounded linear bijection with bounded inverse. For a decreasing filtration (F^mX), c","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea75ef5f3ceca490","tex":"\\varprojlim_{m\\ge0}","aliases":"varprojlim_m≥0 varprojlim ge","display":null,"kind":"use","excerpt":"Let I⊆ R and suppose that R is I-adically complete. Using the checked ring-map reconstruction on the quadratic coordinate ring, Lean proves P_q(R)≃varprojlim_m≥0 P_q(R/I^m). The zero-precision quotient is included only for the formal indexing. The inverse constructs an actual arithmetic character, with exactly the prescribed reductions. It is uniqu","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2acfb66581624cd9","tex":"\\Vcal","aliases":"Vcal V mathcal","display":"{\\mathcal V}","kind":"use","excerpt":"[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-44cf5118c3be9cba","tex":"\\vdots","aliases":"vdots","display":"\\vdots","kind":"use","excerpt":"[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7cc81212c08d083","tex":"\\operatorname{vec}","aliases":"vec operatorname","display":"\\operatorname{vec}","kind":"use","excerpt":"vpage7A transpose, not an adjointKRONECKER ACTION AND ARRAY ORDERChapter VIII: Theorem 4.5, Corollary 4.6. With column-major vectorization, the tensor identity is (A⊗ B)vec(X)=vec(BXA^T). Here the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix A=mat1 · i","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0ed8b78702b6e2c","tex":"\\vol","aliases":"vol operatorname","display":"{\\operatorname{vol}}","kind":"use","excerpt":"There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e4d848b11431c85","tex":"\\vp","aliases":"vp v_p","display":"{v_p}","kind":"use","excerpt":"w(r)<L binom nr c_rp^w(r)S_n-r pmod p^L. When the omitted set is nonempty, the tail has valuation at least min_substack0≤ r≤ n w(r)≥ L ( w(r)+v_pbinom nr+v_p(S_n-r) ).","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9f7dc4db556c3f2","tex":"\\vpage","aliases":"vpage","display":null,"kind":"use","excerpt":"[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lowe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b25fb9551879a1d3","tex":"\\vTwo","aliases":"vTwo v_2","display":"{v_2}","kind":"use","excerpt":"The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the b","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e83c58309eb2060b","tex":"\\mathcal W","aliases":"W mathcal","display":"\\mathcal W","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9fe8ee5ada9c8bf","tex":"W'","aliases":"W'","display":"W'","kind":"use","excerpt":"athrmact). The equality includes the zero case. The same successive monic polynomial division used in Theorem~[source reference], started at precision eight, preserves the quadratic and unit reductions of [source reference]. Write W=Z^2+2β_rZ+α_r and w=W(c_r). Stationarity gives W'(c_r)=-τ_rw, while w=Φ_r^act/ U_r(c_r). The quadratic identity β_r^2","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73ffecb5618fb3cc","tex":"W^*","aliases":"W^*","display":"W^*","kind":"use","excerpt":"0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome su","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8fe8dd2410be8cc","tex":"w^*","aliases":"w^*","display":"w^*","kind":"use","excerpt":"0 · 0 · 0 · 1. Its middle block has one negative eigenvalue. More concretely, the normalized vector w=(ket01-ket10)/sqrt2 satisfies w^*(id⊗ T)(ρ_rm Bell)w=-frac12. This exact negative value of the quadratic form certifies failure of positivity on the composite system. No numerical eigenvalue tolerance is needed. headWhy the Kraus form avoids this p","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2bbe13df91f37646","tex":"w^{-1}","aliases":"w^-1","display":"w^{-1}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"Here e=1+3w+5w^2 and o=2+4w+6w^2, so direct polynomial multiplication gives y_rm even=1+8w+16w^2+17w^3, y_rm odd=4+12w+20w^2+6w^3. This is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a48390f6d6d1a30","tex":"W^{2,p}","aliases":"W^2,p","display":"W^{2,p}","kind":"use","excerpt":"l smooth remainder. headThe normalization makes the trace coefficient literal The general packet is ϕ_m,τ(t)=(t-τ)_+^m/m!. Its derivative of order m has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial. Here the remainder has matching value and first-der","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0d4e45da82688954","tex":"w^3","aliases":"w^3","display":"w^3","kind":"use","excerpt":"Here e=1+3w+5w^2 and o=2+4w+6w^2, so direct polynomial multiplication gives y_rm even=1+8w+16w^2+17w^3, y_rm odd=4+12w+20w^2+6w^3. This is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8bef7eed7e9b97d","tex":"w^b","aliases":"w^b","display":"w^b","kind":"use","excerpt":"Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20ccd43f98f93af7","tex":"W^k","aliases":"W^k","display":"W^k","kind":"use","excerpt":"For every starting row, Δ_1^k C_0(n) = L(U^4nW^k) =∑_s=0^nbinom ns L(W^k+s). Each summand lies in 2^k k! Z_2, proving the factorial bound for address zero from row zero onward.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-291f703d002fce35","tex":"W^{k+s}","aliases":"W^k+s","display":"W^{k+s}","kind":"use","excerpt":"For every starting row, Δ_1^k C_0(n) = L(U^4nW^k) =∑_s=0^nbinom ns L(W^k+s). Each summand lies in 2^k k! Z_2, proving the factorial bound for address zero from row zero onward.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ccdb5f9bd9497286","tex":"w^{k-1}","aliases":"w^k-1","display":"w^{k-1}","kind":"use","excerpt":"Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f034e94d59a11da","tex":"W^m","aliases":"W^m","display":"W^m","kind":"use","excerpt":"he first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the third identity by (i E F)^m and reduce the Euler powers using the integral quadratic relations. Finally, γ_k(X+Y)=∑_r=0^kγ_r(X)γ_k-r(Y) proves γ_k(H_0)∈ A_0. Applying the functional and putting W=U^4-1=2H_0 gives L(W^m)∈2^m m!R (m≥0). These moment","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee2fe0c706b5c2fb","tex":"w^{m-1}","aliases":"w^m-1","display":"w^{m-1}","kind":"use","excerpt":"Differentiate the product identity in logarithmic form to obtain Δ_N'(w) = -Δ_N(w) ∑_m≥1σ_N,mw^m-1. Comparison of the coefficient of w^m-1 gives ma_m = -∑_k=0^m-1a_kσ_N,m-k. Solving for σ_N,m gives the recurrence.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d99ded832587fa9","tex":"w^{n-1}","aliases":"w^n-1","display":"w^{n-1}","kind":"use","excerpt":"Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9022bd928b04faea","tex":"w^p","aliases":"w^p","display":"w^p","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d2786a8c65b662f","tex":"w^{p-1}","aliases":"w^p-1","display":"w^{p-1}","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b8f78bd431e445d","tex":"w^{p^k}","aliases":"w^p^k","display":"w^{p^k}","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65d03882bd3591c3","tex":"w^q","aliases":"w^q","display":"w^q","kind":"use","excerpt":"The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5ddff78ba645e2d","tex":"w^T","aliases":"w^T","display":"w^T","kind":"use","excerpt":"Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a4709427aa09482","tex":"w_{(1)}","aliases":"w_(1)","display":"w_{(1)}","kind":"use","excerpt":"Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2433e0c1dac2127f","tex":"w_{(2)}","aliases":"w_(2)","display":"w_{(2)}","kind":"use","excerpt":"Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2beefde8295c27a","tex":"w_{(j)}","aliases":"w_(j)","display":"w_{(j)}","kind":"use","excerpt":"normB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe2e39be77a64ced","tex":"w_{(r)}","aliases":"w_(r)","display":"w_{(r)}","kind":"use","excerpt":"Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d01794b5f0c3ada0","tex":"W_1","aliases":"W_1","display":"W_1","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1dd506c998004ce1","tex":"W_2","aliases":"W_2","display":"W_2","kind":"use","excerpt":"egative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is +∞. These conventions include M≥ n/2. For every r≥1, v_2(κ_2,r)=4r-1. The rth reconstruction term has exact valuation v_2( κ_2,rbinom n2rR_n-2r ) = 4r-1+v_2binom n2r. Equivalently, W_2(n,r) = 4r-1+s_2(r)+s_2(n-2r)-s_2(n).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3106fac3e3c8f992","tex":"w_2","aliases":"w_2","display":"w_2","kind":"use","excerpt":"A DEFINED COSTChapter VI, Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks. Block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9763a163e7ff6591","tex":"w_3","aliases":"w_3","display":"w_3","kind":"use","excerpt":"A DEFINED COSTChapter VI, Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks. Block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0289b7ef692ee27e","tex":"W_a","aliases":"W_a","display":"W_a","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARChapter VII: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9012166380bed7e0","tex":"w_a","aliases":"w_a","display":"w_a","kind":"use","excerpt":"II.8.4, Example II.8.8. Encode a qubit by ket0_L=ket000 and ket1_L=ket111. Let P project onto this code and take errors E_a∈{I,X_1,X_2,X_3}. Their images are four orthogonal rank-two packets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy stat","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f39670f030db3b1a","tex":"W_B","aliases":"W_B","display":"W_B","kind":"use","excerpt":"Let W_B(t) be the declared unit-cost count of table-cell constructions and tested min-plus candidates. The model charges the empty child-combination table as well as the output cells and merge candidates. Lean proves W_B(t)≤2(B+1)N+((B+1)+(B+1)^2)(N-1) and hence W_B(t)≤ N(3(B+1)+(B+1)^2). The root minimum scan adds a linear pass. Thus the declared ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6867c80df30672f","tex":"w_d","aliases":"w_d","display":"w_d","kind":"use","excerpt":"[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18864b29b043b6d8","tex":"W_i","aliases":"W_i","display":"W_i","kind":"use","excerpt":"System isomorphism and good-place overlaps. The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of character","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4935e01c6a712aa2","tex":"W_{i,\\lambda}","aliases":"W_i,λ lambda","display":"W_{i,\\lambda}","kind":"use","excerpt":"cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j))","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4fd2df5cb7d2b75","tex":"W_{i,\\lambda}^{\\oplus m_i}","aliases":"W_i,λ^⊕ m_i lambda oplus","display":"W_{i,\\lambda}^{\\oplus m_i}","kind":"use","excerpt":"arrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at ev","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ad2cf4a38ca0ffc","tex":"W_{i,\\lambda_0}^{\\oplus m_i}","aliases":"W_i,λ_0^⊕ m_i lambda oplus","display":"W_{i,\\lambda_0}^{\\oplus m_i}","kind":"use","excerpt":"[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-233d4eb90fcdc954","tex":"W_i^{\\oplus c_i}","aliases":"W_i^⊕ c_i oplus","display":"W_i^{\\oplus c_i}","kind":"use","excerpt":"[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ffbde798b9a7c58f","tex":"W_j","aliases":"W_j","display":"W_j","kind":"use","excerpt":"System isomorphism and good-place overlaps. The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of character","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"[Optimal selection for the certified primary block bound] Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8da97db1cfb2f53","tex":"w_{j,k}","aliases":"w_j,k","display":"w_{j,k}","kind":"use","excerpt":"ct discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc64898552e9bfce","tex":"W_{j,\\lambda}","aliases":"W_j,λ lambda","display":"W_{j,\\lambda}","kind":"use","excerpt":"System isomorphism and good-place overlaps. The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of character","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0f3a2c6c669d106","tex":"W_K","aliases":"W_K","display":"W_K","kind":"use","excerpt":"Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cf76327e749de1b7","tex":"W_k","aliases":"W_k","display":"W_k","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5731a3740a4e086","tex":"W_{k+1}","aliases":"W_k+1","display":"W_{k+1}","kind":"use","excerpt":"For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da190764fa0f2565","tex":"W_{K_f}","aliases":"W_K_f","display":"W_{K_f}","kind":"use","excerpt":"[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4cd86c8f321e8c8d","tex":"W_{K_L}","aliases":"W_K_L","display":"W_{K_L}","kind":"use","excerpt":"[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa6cdf6f4a146600","tex":"W_{K_m}","aliases":"W_K_m","display":"W_{K_m}","kind":"use","excerpt":"Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6453f054ab181a68","tex":"W_{K_n}","aliases":"W_K_n","display":"W_{K_n}","kind":"use","excerpt":"Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b544c1c3e686ebb","tex":"w_{m,j}","aliases":"w_m,j","display":"w_{m,j}","kind":"use","excerpt":"fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c8ab31deb179927","tex":"w_{m,n}","aliases":"w_m,n","display":"w_{m,n}","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-466a54f902244928","tex":"W_n","aliases":"W_n","display":"W_n","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2c566e84a80fdd2","tex":"W_{n+1}","aliases":"W_n+1","display":"W_{n+1}","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d7f1cc8c150d4e5","tex":"W_p","aliases":"W_p","display":"W_p","kind":"use","excerpt":"[Odd row weight] For an odd prime p, the rth nonzero term in the reconstruction has valuation W_p(n,r) = 2r+v_p(Z_2r+1) +v_pbinom n2r +v_p(S_n-2r(p)). The binomial contribution is v_pbinom n2r = fracs_p(2r)+s_p(n-2r)-s_p(n)p-1, where s_p is the base p digit sum.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-750210c4c92949cd","tex":"\\mathcal W_p","aliases":"W_p mathcal","display":"\\mathcal W_p","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a4f3028bb492bb2","tex":"W_r","aliases":"W_r","display":"W_r","kind":"use","excerpt":"[Exact classical tail valuation] For every r≥1 and N≥ r, v_2( T_r-1(N)) =4r-1+v_2binom Nr+μ_r+v_2G_r(N-r). There are unique factors G_r(X)=W_r(X)U_r(X), where W_r∈ Z_2[X] is monic of degree d_r=max J_r and U_r∈ Z_2⟨ X⟩^× satisfies U_r≡1pmod2. Consequently v_2G_r(x)=v_2W_r(x) for every x∈ Z_2, and deg W_r≤lfloorfracs_2(r)-13rfloor. Equation~[source ","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dce7dfb4f712d666","tex":"\\mathcal W_r","aliases":"W_r mathcal","display":"\\mathcal W_r","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dfa15e280d3611a8","tex":"w_r","aliases":"w_r","display":"w_r","kind":"use","excerpt":"[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38938f66d3d8dac8","tex":"w_{\\rm Smith}","aliases":"w_rm Smith rm","display":"w_{\\rm Smith}","kind":"use","excerpt":"Away from its pole Γ=1, the inverse Cayley map has derivative fracdzdΓ=frac2(1-Γ)^2, while normalized admittance has derivative proportional to (1+Γ)^-2. Away from Γ=±1, this motivates the symmetric sensitivity w_rm Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6935289ededde0d","tex":"W_{\\mathrm{sgn}}","aliases":"W_sgn mathrm","display":"W_{\\mathrm{sgn}}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-566da3104464fc21","tex":"w_{\\mathrm{Smith}}","aliases":"w_Smith mathrm","display":"w_{\\mathrm{Smith}}","kind":"use","excerpt":"The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20edf480f7443c8c","tex":"W_{\\mathrm{std}}","aliases":"W_std mathrm","display":"W_{\\mathrm{std}}","kind":"use","excerpt":"The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28b939ffc4f05e71","tex":"W_u","aliases":"W_u","display":"W_u","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-377769d6a9c94ee0","tex":"W_v","aliases":"W_v","display":"W_v","kind":"use","excerpt":"Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6e9e4d147adf393","tex":"w_v","aliases":"w_v","display":"w_v","kind":"use","excerpt":"[Quantized budget dynamic program] Assume every keep error is a nonnegative integer w_v. Let B∈ N. For each node v, define F_v(b) as the minimum cost of a pruning of the subtree at v with total error exactly b. The keep option contributes F_v(w_v)≤κ_v. The split option is obtained by min plus convolution of the child tables, followed by addition of","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f307e7eaf628bf4e","tex":"W_v^*","aliases":"W_v^*","display":"W_v^*","kind":"use","excerpt":"al positions below are terminal packet addresses, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actua","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5704b00361e1b378","tex":"W_\\lambda","aliases":"W_λ lambda","display":"W_\\lambda","kind":"use","excerpt":"Use the same multiplicities in every member of the compatible carrier. Direct sums preserve strict compatibility, and m_i≤ c_i makes V_λ a summand of W_λ. At the separating places, the target trace vector is C m. Full column rank makes m the unique solution.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0cf3e6c1daea5239","tex":"W_\\pi","aliases":"W_π pi","display":"W_\\pi","kind":"use","excerpt":"[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afd76799f82fee6b","tex":"W_{\\ell,\\alpha}^*","aliases":"W_ℓ,α^* ell alpha","display":"W_{\\ell,\\alpha}^*","kind":"use","excerpt":"Set A=∑_α W_ℓ,α^*W_ℓ,α. This is a finite sum of bounded operators. The energy result says ⟨ x,Ax⟩=⟨ x,x⟩ for every x. Complex polarization, applied also to x+y and x+ i y, gives ⟨ y,(A-I)x⟩=0 for all x,y, hence A=I. Applying the operator identity to x proves synthesis. This argument uses the actual Hilbert adjoints of the decimation operators, not ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-653fb7da5fc9b85e","tex":"W_{\\ell,\\alpha}^{*}","aliases":"W_ℓ,α^* ell alpha","display":"W_{\\ell,\\alpha}^{*}","kind":"use","excerpt":"[Wavelet-packet reconstruction and energy] At every finite depth ℓ, x=∑_α∈{0,1}^ℓW_ℓ,α^*W_ℓ,αx, and x _2^2 = ∑_α∈{0,1}^ℓ W_ℓ,αx _2^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f33ee59cc7d16b42","tex":"\\operatorname{Walsh}","aliases":"Walsh operatorname","display":"\\operatorname{Walsh}","kind":"use","excerpt":"Actual butterfly. For the two halves x_0,x_1 of the input, the independently defined procedure returns Walsh_2N(x_0,x_1) =(Walsh_N(x_0+x_1), Walsh_N(x_0-x_1)). Lean proves that this vector equals H_2Nx. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-986d47d11f30d4d9","tex":"\\operatorname{Walsh}_{2N}","aliases":"Walsh_2N operatorname","display":"\\operatorname{Walsh}_{2N}","kind":"use","excerpt":"Actual butterfly. For the two halves x_0,x_1 of the input, the independently defined procedure returns Walsh_2N(x_0,x_1) =(Walsh_N(x_0+x_1), Walsh_N(x_0-x_1)). Lean proves that this vector equals H_2Nx. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b72fa7a71d0d77ae","tex":"\\operatorname{Walsh}_N","aliases":"Walsh_N operatorname","display":"\\operatorname{Walsh}_N","kind":"use","excerpt":"Actual butterfly. For the two halves x_0,x_1 of the input, the independently defined procedure returns Walsh_2N(x_0,x_1) =(Walsh_N(x_0+x_1), Walsh_N(x_0-x_1)). Lean proves that this vector equals H_2Nx. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bfb4d65b18a629f","tex":"\\operatorname{WD}","aliases":"WD operatorname","display":"\\operatorname{WD}","kind":"use","excerpt":"The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d3781ce495edcf4","tex":"\\widehat A","aliases":"widehat A widehat","display":"\\widehat A","kind":"use","excerpt":"For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tig","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-891994962d96636a","tex":"\\widehat A_1","aliases":"widehat A_1 widehat","display":"\\widehat A_1","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4895bc4181eddb45","tex":"\\widehat A_i","aliases":"widehat A_i widehat","display":"\\widehat A_i","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9708fbc1470a12ef","tex":"\\widehat A_j","aliases":"widehat A_j widehat","display":"\\widehat A_j","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed3d5df42b309969","tex":"\\widehat A_{j+1}","aliases":"widehat A_j+1 widehat","display":"\\widehat A_{j+1}","kind":"use","excerpt":"Use the telescoping identity widehat A_m⋯widehat A_1 -A_m⋯ A_1 = ∑_j=1^m widehat A_m⋯widehat A_j+1 E_j A_j-1⋯ A_1. Take norms term by term.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a01d17f6b25e6b1","tex":"\\widehat A_m","aliases":"widehat A_m widehat","display":"\\widehat A_m","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65c16e51b7f7e52e","tex":"\\widehat a_q","aliases":"widehat a_q widehat","display":"\\widehat a_q","kind":"use","excerpt":"[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"use","excerpt":"pageheading3. Actual packet congruences and sharp boundariesConsequences for the standard window; not the full Theorem 7.5 The fifth block constructed integral lifts widehat C_mp,j(n)∈ Z_p of the original rational coefficients when 0≤ j<p, and proved that their residues equal e_n^(m,p).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e7a695b8f5409721","tex":"\\widehat c","aliases":"widehat c widehat","display":"\\widehat c","kind":"use","excerpt":"[Coefficient perturbations in a unitary tree] Let c= A_ Lx be the leaf coefficients of an admissible unitary pruning, and let widehat c=c+e. If widehat x is synthesized from widehat c, then widehat x-x = e . Thus coefficient quantization and coefficient noise are neither amplified nor attenuated by exact unitary synthesis.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0aa7779c656ca58e","tex":"\\widehat C_A","aliases":"widehat C_A widehat","display":"\\widehat C_A","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7f697c3c006092e","tex":"\\widehat C_B","aliases":"widehat C_B widehat","display":"\\widehat C_B","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62e85424e4f5a5ba","tex":"\\widehat C_C","aliases":"widehat C_C widehat","display":"\\widehat C_C","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41f5515a92f29307","tex":"\\widehat C_{mp,j}","aliases":"widehat C_mp,j widehat","display":"\\widehat C_{mp,j}","kind":"use","excerpt":"pageheading3. Actual packet congruences and sharp boundariesConsequences for the standard window; not the full Theorem 7.5 The fifth block constructed integral lifts widehat C_mp,j(n)∈ Z_p of the original rational coefficients when 0≤ j<p, and proved that their residues equal e_n^(m,p).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03e64edeac04e31f","tex":"\\widehat E","aliases":"widehat E widehat","display":"\\widehat E","kind":"use","excerpt":"_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous seminorms p on X. Every a∈ A acts continuously on E in the transported topology and therefore extends to widehat E.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a999477715e268b","tex":"\\widehat F","aliases":"widehat F widehat","display":"\\widehat F","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"For a canonical coordinate change with input and output chart v=2x, the target formula is not still v^2. It is widehat f_b(v)=2f(v/2)=fracv^22. Consequently widehat f_b(2x)=2x^2, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on Plate III.10.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8efc3563f059f63","tex":"\\widehat F_b","aliases":"widehat F_b widehat","display":"\\widehat F_b","kind":"use","excerpt":"Now represent the nonlinear operation F(x,y)=(x+y,xy) in the new chart. Set w=u+v-u^2. Its correct coordinate formula is widehat F_b(u,v)= A F S(u,v) =(w, u(v-u^2)+w^2). For (x,y)=(1,2), applying F first gives (3,2) and then (3,11) in the new chart. Changing the input first gives (u,v)=(1,3), and widehat F_b(1,3)=(3,11) as well.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54d5df4eb419be74","tex":"\\widehat f_b","aliases":"widehat f_b widehat","display":"\\widehat f_b","kind":"use","excerpt":"For a canonical coordinate change with input and output chart v=2x, the target formula is not still v^2. It is widehat f_b(v)=2f(v/2)=fracv^22. Consequently widehat f_b(2x)=2x^2, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on Plate III.10.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2f6188722a2e8ff","tex":"\\widehat F_i","aliases":"widehat F_i widehat","display":"\\widehat F_i","kind":"use","excerpt":"Let v be the new optimum of the edited subtree, and let j be the edited child slot at one frame. The update performed at that frame is step(v)= min{K, σ+ ∑_i v, · i=j, widehat F_i, · i≠ j. }. This result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values widehat F_i.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f605f803980a555f","tex":"\\widehat f_m","aliases":"widehat f_m widehat","display":"\\widehat f_m","kind":"use","excerpt":"The reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A va","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d128aec02eb0a2e8","tex":"\\widehat F_{r\\leftarrow a}","aliases":"widehat F_r← a widehat leftarrow","display":"\\widehat F_{r\\leftarrow a}","kind":"use","excerpt":"If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1da1b91f6369b42a","tex":"\\widehat F_{s\\leftarrow b}","aliases":"widehat F_s← b widehat leftarrow","display":"\\widehat F_{s\\leftarrow b}","kind":"use","excerpt":"[Canonical coordinate naturality] The exact coordinate representation preserves typed composition and satisfies J^Y_s← rwidehat F_r← a = widehat F_s← bJ^X_b← a. statusleanpriorKomorebi.PaperII.NestedRefinement.coordinateOperator_comp; Komorebi.PaperII.NestedRefinement.coordinateOperator_naturality","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41acddacfefb419a","tex":"\\widehat G_{t\\leftarrow r}","aliases":"widehat G_t← r widehat leftarrow","display":"\\widehat G_{t\\leftarrow r}","kind":"use","excerpt":"tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthree","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-66d5bee1680ae2b3","tex":"\\widehat h","aliases":"widehat h widehat","display":"\\widehat h","kind":"use","excerpt":"[Second Schur derivative] For retained directions h,k, set widehat h=(h,Dφ h), widehat k=(k,Dφ k). Then D^2φ[h,k] = -A^-1D^2F_Q[widehat h,widehat k] and D^2R[h,k] = D^2F_P[widehat h,widehat k] - D_vF_P A^-1D^2F_Q[widehat h,widehat k]. All derivatives on the right are evaluated at (u_0,v_0).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d68f21ff18b92f60","tex":"\\widehat j_m","aliases":"widehat j_m widehat","display":"\\widehat j_m","kind":"use","excerpt":"[Estimated packet Gibbs bound] Let Q_m be linear and uniformly bounded on L^p, with sup_m Q_m ≤ C_Q. For f=s+j and an estimated packet widehat j_m, define widehat f_m =Q_m(f-widehat j_m)+widehat j_m. Then f-widehat f_m _p ≤ s-Q_ms _p +(1+C_Q) j-widehat j_m _p. Consequently, smooth approximation and packet recovery may be improved independently. No ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-94d7972844087a8a","tex":"\\widehat k","aliases":"widehat k widehat","display":"\\widehat k","kind":"use","excerpt":"[Second Schur derivative] For retained directions h,k, set widehat h=(h,Dφ h), widehat k=(k,Dφ k). Then D^2φ[h,k] = -A^-1D^2F_Q[widehat h,widehat k] and D^2R[h,k] = D^2F_P[widehat h,widehat k] - D_vF_P A^-1D^2F_Q[widehat h,widehat k]. All derivatives on the right are evaluated at (u_0,v_0).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"use","excerpt":"cal E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62308a18a459ffc6","tex":"\\widehat P_0","aliases":"widehat P_0 widehat","display":"\\widehat P_0","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-add64233d3bae23a","tex":"\\widehat p_3","aliases":"widehat p_3 widehat","display":"\\widehat p_3","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSIONChapter VIII: Theorem 5.8, Theorem 5.9. Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_cha","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22cdc1586433efd2","tex":"\\widehat P_j","aliases":"widehat P_j widehat","display":"\\widehat P_j","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3a9183aa4d4607b","tex":"\\widehat p_j","aliases":"widehat p_j widehat","display":"\\widehat p_j","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSIONChapter VIII: Theorem 5.8, Theorem 5.9. Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_cha","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd0609390bade526","tex":"\\widehat P_{j+1}","aliases":"widehat P_j+1 widehat","display":"\\widehat P_{j+1}","kind":"use","excerpt":"Actual noncommutative products. The error module defines the left-accumulated product recursively, with P_0=I and P_j+1=A_j+1P_j. For a perturbed sequence widehat A_j, it derives the discrepancy recursion widehat P_j+1-P_j+1 =widehat A_j+1(widehat P_j-P_j) +(widehat A_j+1-A_j+1)P_j. The identity follows from the actual products. It is not supplied ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a26fa77c1b9a117","tex":"\\widehat P_{j-1}","aliases":"widehat P_j-1 widehat","display":"\\widehat P_{j-1}","kind":"use","excerpt":"[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c1d9a673bbb0544","tex":"\\widehat p_{j-1}","aliases":"widehat p_j-1 widehat","display":"\\widehat p_{j-1}","kind":"use","excerpt":"vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSIONChapter VIII: Theorem 5.8, Theorem 5.9. Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_cha","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-266ff731dd16faeb","tex":"\\widehat P_m","aliases":"widehat P_m widehat","display":"\\widehat P_m","kind":"use","excerpt":"ntheorem[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i )","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41bdac88160e4cd7","tex":"\\widehat P_{N,j}","aliases":"widehat P_N,j widehat","display":"\\widehat P_{N,j}","kind":"use","excerpt":"cal E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-816a095af468abc4","tex":"\\widehat p_r","aliases":"widehat p_r widehat","display":"\\widehat p_r","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-77449d53c5be9d11","tex":"\\widehat S","aliases":"widehat S widehat","display":"\\widehat S","kind":"use","excerpt":"Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d3704d4c4eab86e","tex":"\\widehat s","aliases":"widehat s widehat","display":"\\widehat s","kind":"use","excerpt":"If s,widehat s are measurable smooth-background models, the same triangle inequality gives, for finite p, f-(widehat s+widehat j) _p ≤ s-widehat s _p +|a-widehat a|μ(Ω)^1/p +|widehat a|μ(ΩtrianglewidehatΩ)^1/p. This is the correct separation of errors. Detection error is not hidden inside a generic reconstruction score.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b0d65d8f4ac93228","tex":"\\widehat T","aliases":"widehat T widehat","display":"\\widehat T","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27317811f6cd05cf","tex":"\\widehat T_2","aliases":"widehat T_2 widehat","display":"\\widehat T_2","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9844b5e86bad9b35","tex":"\\widehat T_4","aliases":"widehat T_4 widehat","display":"\\widehat T_4","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37b3cca62187f94f","tex":"\\widehat T_N","aliases":"widehat T_N widehat","display":"\\widehat T_N","kind":"use","excerpt":"The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a06250682a8196e7","tex":"\\widehat T_{Q,P}","aliases":"widehat T_Q,P widehat","display":"\\widehat T_{Q,P}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0aad031a0de93fe","tex":"\\widehat T_{Q_a,P_a}","aliases":"widehat T_Q_a,P_a widehat","display":"\\widehat T_{Q_a,P_a}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bbb28108c9e01d5","tex":"\\widehat T_{Q_b,P_b}","aliases":"widehat T_Q_b,P_b widehat","display":"\\widehat T_{Q_b,P_b}","kind":"use","excerpt":"The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c8131c596a2a289","tex":"\\widehat u","aliases":"widehat u widehat","display":"\\widehat u","kind":"use","excerpt":"inite solved-block cokernel and its parameter-dependent class by C_x=S^e/imJ, o_x(v)=[bar r+Kv]∈ C_x. Then L_v≠∅iff o_x(v)=0. More precisely, the checked digit-to-lift equivalence is L_v≅{u∈ S^m:Ju=-bar r-Kv}. It sends a solved digit u to the actual next residue represented by x+q^L(widehat u,widehat v). The obstruction depends on the solved block ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f207fa5d22c484f6","tex":"\\widehat v","aliases":"widehat v widehat","display":"\\widehat v","kind":"use","excerpt":"x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the se","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8d6f1354dafb866","tex":"\\widehat x","aliases":"widehat x widehat","display":"\\widehat x","kind":"use","excerpt":"[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71ff495a798a786e","tex":"\\widehat{\\bm X}","aliases":"widehat X widehat bm","display":"\\widehat{\\bm X}","kind":"use","excerpt":"with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input transforms are absolutely convergent, this gives widehat X(z)= R(z) E(z) X(z).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-657478f81580a29e","tex":"\\widehat{\\mathbb Z}","aliases":"widehat Z widehat mathbb","display":"\\widehat{\\mathbb Z}","kind":"use","excerpt":"The reason is exact: the CRT ring isomorphism preserves polynomial evaluation, and zero in a finite product is equivalent to zero in every component. For F empty, passing to all compatible finite congruences identifies the profinite completion widehat Z with ∏_p Z_p. The same finite reconstruction is compatible with coefficient projections and inte","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c150adb9f1dd660a","tex":"\\widehat{F'}","aliases":"widehatF' widehat","display":"\\widehat{F'}","kind":"use","excerpt":"[Derivative conjugacy] widehatF'(T) = (1+T)fracddTwidehat F(T).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1321a8b68ad36af","tex":"\\widehat{G\\circ F}","aliases":"widehatG∘ F widehat circ","display":"\\widehat{G\\circ F}","kind":"use","excerpt":"tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthree","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8ca75218336934c","tex":"\\widehat{G\\circ F}_{t\\leftarrow a}","aliases":"widehatG∘ F_t← a widehat circ leftarrow","display":"\\widehat{G\\circ F}_{t\\leftarrow a}","kind":"use","excerpt":"tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthree","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52b9530420892ef6","tex":"\\widehat\\Kcal","aliases":"widehatKcal widehat K widehat Kcal widehat mathcal","display":"\\widehat{\\mathcal K}","kind":"use","excerpt":"[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c6b245b4f9f9854","tex":"\\widehat{S/L}","aliases":"widehatS/L widehat","display":"\\widehat{S/L}","kind":"use","excerpt":"Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79f270505a397e43","tex":"\\widehat\\Z","aliases":"widehatZ widehat Z widehat mathbb","display":"\\widehat{\\mathbb Z}","kind":"use","excerpt":"Finite CRT reconstruction leads to the familiar completed object widehat Z=∏_p Z_p, but does not identify which local tuples come from a single global integer. A complete form of Native 13.3 must therefore specify finite-level compatibility, transition maps and completion, and any additional global descent condition. If recurrence, spectral, or Fro","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad8c5d8ce79e0069","tex":"\\widehat\\tau","aliases":"widehatτ widehat tau","display":"\\widehat\\tau","kind":"use","excerpt":"reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value ","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-85469b8bf7bc0c74","tex":"\\widehat\\tau_q","aliases":"widehatτ_q widehat tau","display":"\\widehat\\tau_q","kind":"use","excerpt":"[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e926afe687bd984","tex":"\\widehat\\Phi","aliases":"widehatΦ widehat Phi","display":"\\widehat\\Phi","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d21df6346220aa61","tex":"\\widehat\\Phi_a","aliases":"widehatΦ_a widehat Phi","display":"\\widehat\\Phi_a","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34d4c951aad7cc13","tex":"\\widehat\\Phi_b","aliases":"widehatΦ_b widehat Phi","display":"\\widehat\\Phi_b","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48da1c2ead470996","tex":"\\widehat\\Omega","aliases":"widehatΩ widehat Omega","display":"\\widehat\\Omega","kind":"use","excerpt":"Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c88b3c814ae49b9","tex":"\\wideotimes_{\\OO}","aliases":"wideotimes_OO mathbinwidehat⊗_ O wideotimes OO mathbin widehat otimes mathcal","display":null,"kind":"use","excerpt":"[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f596b26bb2804e7d","tex":"\\widetilde B","aliases":"widetilde B widetilde","display":"\\widetilde B","kind":"use","excerpt":"In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,wid","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a4e15e563cc5107","tex":"\\widetilde b","aliases":"widetilde b widetilde","display":"\\widetilde b","kind":"use","excerpt":"[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9193c44a5528a66b","tex":"\\widetilde B^*","aliases":"widetilde B^* widetilde","display":"\\widetilde B^*","kind":"use","excerpt":"[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35f36a53181c409f","tex":"\\widetilde b_i","aliases":"widetilde b_i widetilde","display":"\\widetilde b_i","kind":"use","excerpt":"[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2500eabb3d3f0489","tex":"\\widetilde c","aliases":"widetilde c widetilde","display":"\\widetilde c","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c95d4e5ac3dec21","tex":"\\widetilde C_{6,3}","aliases":"widetilde C_6,3 widetilde","display":"\\widetilde C_{6,3}","kind":"use","excerpt":"For the unnormalized convention, Δ_1widetilde C_6,3(1)=3! frac64455=frac384455, v_3 (frac384455)=1. The factorial's valuation is also proved: v_3(3!)=1, v_3(Δ_1C_6,3(1)) =1-v_3(3!)=0. This proves sharpness at the displayed normalization boundary. It does not establish the corrected global inequality for all parameters, and does not assert that ever","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3af8f2cbaa29419e","tex":"\\widetilde C_a","aliases":"widetilde C_a widetilde","display":"\\widetilde C_a","kind":"use","excerpt":"ull algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R:","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad6dd274a454509b","tex":"\\widetilde C_b","aliases":"widetilde C_b widetilde","display":"\\widetilde C_b","kind":"use","excerpt":"C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd7df62e1aa54b13","tex":"\\widetilde C_{N,j}","aliases":"widetilde C_N,j widetilde","display":"\\widetilde C_{N,j}","kind":"use","excerpt":"Principal targets. targetFiltered.padic_filtered_reconstruction targetStandardWindow.lucas_row_reduction targetStandardWindow.standard_diagonal_unit targetStandardWindow.factorial_valuation_zero pageheading4. Sharp normalization boundarySource item 13.1 The construction fixes the two coefficient conventions: C_N,j(n), widetilde C_N,j(n)=j! C_N,j(n)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d73e807a5c23f393","tex":"\\widetilde C_x","aliases":"widetilde C_x widetilde","display":"\\widetilde C_x","kind":"use","excerpt":"e coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does not close on the whole fine level: on the added coordinate the commutator is zero, whereas 2 iwidetilde C_z is 2 i. Exact transport constrains the image of R and leaves its orthogonal complement as separate data. headAn approximate residual ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0108cdcacfc6d3f3","tex":"\\widetilde C_z","aliases":"widetilde C_z widetilde","display":"\\widetilde C_z","kind":"use","excerpt":"the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does not close on the whole fine level: on the added coordinate the commutator is","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4a13a7052f06326","tex":"\\widetilde D","aliases":"widetilde D widetilde","display":"\\widetilde D","kind":"use","excerpt":"In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,wid","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f92312b578cf8233","tex":"\\widetilde D^*","aliases":"widetilde D^* widetilde","display":"\\widetilde D^*","kind":"use","excerpt":"[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20f230bba299446b","tex":"\\widetilde E","aliases":"widetilde E widetilde","display":"\\widetilde E","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op,","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73259486419af0dc","tex":"\\widetilde e","aliases":"widetilde e widetilde","display":"\\widetilde e","kind":"use","excerpt":"Fix a prime p, an integer m≥1, and an address 0≤ j<p. Put N=mp. For arbitrary m, let widetilde e_n^(m,p)∈ F_p be the unique sequence satisfying ∑_q=0^n binommnmq widetilde e_q^(m,p) = δ_n0. When mmid p-1, this sequence agrees with the compressed Hurwitz inverse in [source reference].","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6814e4fb1154514e","tex":"\\widetilde e_n^{(m,p)}","aliases":"widetilde e_n^(m,p) widetilde","display":"\\widetilde e_n^{(m,p)}","kind":"use","excerpt":"Fix a prime p, an integer m≥1, and an address 0≤ j<p. Put N=mp. For arbitrary m, let widetilde e_n^(m,p)∈ F_p be the unique sequence satisfying ∑_q=0^n binommnmq widetilde e_q^(m,p) = δ_n0. When mmid p-1, this sequence agrees with the compressed Hurwitz inverse in [source reference].","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad9be787a2daffac","tex":"\\widetilde e_q^{(m,p)}","aliases":"widetilde e_q^(m,p) widetilde","display":"\\widetilde e_q^{(m,p)}","kind":"use","excerpt":"Fix a prime p, an integer m≥1, and an address 0≤ j<p. Put N=mp. For arbitrary m, let widetilde e_n^(m,p)∈ F_p be the unique sequence satisfying ∑_q=0^n binommnmq widetilde e_q^(m,p) = δ_n0. When mmid p-1, this sequence agrees with the compressed Hurwitz inverse in [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ecf0ac96e6f9d4e","tex":"\\widetilde e_r^{(m,p)}","aliases":"widetilde e_r^(m,p) widetilde","display":"\\widetilde e_r^{(m,p)}","kind":"use","excerpt":"[Rational residual generating function] Under the hypotheses of [source reference], for every standard address j, ∑_n≥0 (C_mp,j(n)bmod p)z^n = 1+ frac ∑_r=1^hwidetilde e_r^(m,p)z^r 1-z^h.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad91da91d6be3c40","tex":"\\widetilde F","aliases":"widetilde F widetilde","display":"\\widetilde F","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d74a2e10811483dc","tex":"\\widetilde f","aliases":"widetilde f widetilde","display":"\\widetilde f","kind":"use","excerpt":"Let f=s+j where s belongs to a smooth approximation class and j is a known discontinuity packet. Let Q_m be any approximation operator applied only to s, and define widetilde f_m=Q_ms+j.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-84d1614ea0ca82b5","tex":"\\widetilde F_*","aliases":"widetilde F_* widetilde","display":"\\widetilde F_*","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aed30d5003ca3472","tex":"\\widetilde f_m","aliases":"widetilde f_m widetilde","display":"\\widetilde f_m","kind":"use","excerpt":"Let f=s+j where s belongs to a smooth approximation class and j is a known discontinuity packet. Let Q_m be any approximation operator applied only to s, and define widetilde f_m=Q_ms+j.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9117c60fc5e9378b","tex":"\\widetilde K","aliases":"widetilde K widetilde","display":"\\widetilde K","kind":"use","excerpt":"[One step filter stability] Let K and widetilde K be column stochastic. Assume 0<a≤λ_i≤ b. Then norm B_λ(Kp)- B_λ(widetilde Kq) _1 ≤ frac2ba ( normp-q_1 + normK-widetilde K_1→1 ).","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-468630464758f8a1","tex":"\\widetilde L","aliases":"widetilde L widetilde","display":"\\widetilde L","kind":"use","excerpt":"Let L⊆ V be maximal isotropic. Choose a subgroup lift widetilde L⊆ H(V) on which the central coordinate is zero.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-649a6c9973f355dd","tex":"\\widetilde p","aliases":"widetilde p widetilde","display":"\\widetilde p","kind":"use","excerpt":"free presentation] Let X be a complete Hausdorff locally convex space, let A⊆End(X) act continuously, and let S=(s_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous s","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7efd54fa5d68cf48","tex":"\\widetilde P_0","aliases":"widetilde P_0 widetilde","display":"\\widetilde P_0","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-064ad848deb9dca0","tex":"\\widetilde P_j","aliases":"widetilde P_j widetilde","display":"\\widetilde P_j","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bf0caffa4ff570d","tex":"\\widetilde P_{j-1}","aliases":"widetilde P_j-1 widetilde","display":"\\widetilde P_{j-1}","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7217c1cd533b14e6","tex":"\\widetilde P_m","aliases":"widetilde P_m widetilde","display":"\\widetilde P_m","kind":"use","excerpt":"Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1dfb1836c1fd75b","tex":"\\widetilde R","aliases":"widetilde R widetilde","display":"\\widetilde R","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9fcf9ea0bdd21c78","tex":"\\widetilde T","aliases":"widetilde T widetilde","display":"\\widetilde T","kind":"use","excerpt":"[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algeb","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e24a9d0c560ae65e","tex":"\\widetilde t","aliases":"widetilde t widetilde","display":"\\widetilde t","kind":"use","excerpt":"Integer representatives and residue classes at the next level must also be distinguished. For q≥1 and L≥0, the map S_q^d⟶ {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection, where widetilde t is any integer representative. Independence of the representative and injectivity both follow by cancelling q^L in integer divisibility. Surje","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-501b4a78440cb78b","tex":"\\widetilde T^{-1}","aliases":"widetilde T^-1 widetilde","display":"\\widetilde T^{-1}","kind":"use","excerpt":"[Stability of inversion at finite precision] Let T and widetilde T be invertible elements of F^0 B. If T-widetilde T∈ F^L B, then T^-1-widetilde T^-1∈ F^L B.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aba1aecba944a16d","tex":"\\widetilde U","aliases":"widetilde U widetilde","display":"\\widetilde U","kind":"use","excerpt":"[Exact image of the original coefficients] The continuous extension of the original sequence is widetilde U_j(x)=j!+2^d_jΦ_j(x). It is injective, its image is exactly j!+2^d_j Z_2, and |widetilde U_j(x)-widetilde U_j(y)|_2 =2^-d_j|x-y|_2. Every point of that affine image has exactly one preimage in Z_2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1faa955cc41a0e0","tex":"\\widetilde u","aliases":"widetilde u widetilde","display":"\\widetilde u","kind":"use","excerpt":"A candidate for the next lift has the form x+q^L(widetilde u,widetilde v), with solved digit u∈ S_q^m. Its Taylor correction equation is overliner_x+J_xu+K_xv=0. Solvability is exactly vanishing of the displayed cokernel class. Restrict the residue-digit equivalence to the prescribed parameter coordinates. This identifies the lift fiber with the so","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52f025dc16d3b9fb","tex":"\\widetilde U_j","aliases":"widetilde U_j widetilde","display":"\\widetilde U_j","kind":"use","excerpt":"[Exact image of the original coefficients] The continuous extension of the original sequence is widetilde U_j(x)=j!+2^d_jΦ_j(x). It is injective, its image is exactly j!+2^d_j Z_2, and |widetilde U_j(x)-widetilde U_j(y)|_2 =2^-d_j|x-y|_2. Every point of that affine image has exactly one preimage in Z_2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3323d1822e9c5f0f","tex":"\\widetilde v","aliases":"widetilde v widetilde","display":"\\widetilde v","kind":"use","excerpt":"ixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residue roots over x whose parameter coordinates are t_0+q^Lwidetilde v modulo q^L+","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c77170c848a2e8b4","tex":"\\widetilde x","aliases":"widetilde x widetilde","display":"\\widetilde x","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":15},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0b6083f2a1db355","tex":"\\widetilde x'","aliases":"widetilde x' widetilde","display":"\\widetilde x'","kind":"use","excerpt":"0 · 0, so N^2=0 and N _2=1. On [0,T], compare the memory kernels K(t)=N and widetilde K(t)=0, with identity free semigroup, no forcing, and initial state x_0=e_1=(0,1)^ T. The equations are x'(t)=∫_0^t N x(r) dr, widetilde x'(t)=0. Their exact solutions are x(t)=(t^2/2,1)^ T and widetilde x(t)=(0,1)^ T. vfigext_trajectory platecaptionThe first-coor","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-badcfecc2304d5fe","tex":"\\widetilde\\Kcal","aliases":"widetildeKcal widetilde K widetilde Kcal widetilde mathcal","display":"\\widetilde{\\mathcal K}","kind":"use","excerpt":"FloatBarrier Memory compression, rational realization, and phase-type consequences. Fix T>0. For the next estimate let A be a bounded operator on a Banach space X, let K,widetilde K∈ L^1((0,T), B(X)), and let f be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a shar","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a48c4c6cb9f766e9","tex":"\\widetilde\\lambda","aliases":"widetildeλ widetilde lambda","display":"\\widetilde\\lambda","kind":"use","excerpt":"[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e28571e73d20b73","tex":"\\widetilde\\lambda_i","aliases":"widetildeλ_i widetilde lambda","display":"\\widetilde\\lambda_i","kind":"use","excerpt":"[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1af56e8fd99612c","tex":"\\widetilde\\Sigma","aliases":"widetildeΣ widetilde Sigma","display":"\\widetilde\\Sigma","kind":"use","excerpt":"The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings ar","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce0aec4a79056ca2","tex":"\\widetilde{\\sigma}","aliases":"widetildeσ widetilde sigma","display":"\\widetilde{\\sigma}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f2644292076c607","tex":"\\widetilde{\\sigma}_{N,1}","aliases":"widetildeσ_N,1 widetilde sigma","display":"\\widetilde{\\sigma}_{N,1}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac19292d2e508b19","tex":"\\widetilde{\\sigma}_{N,2}","aliases":"widetildeσ_N,2 widetilde sigma","display":"\\widetilde{\\sigma}_{N,2}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d036ddaa4db5d2e7","tex":"\\widetilde{\\sigma}_{N,m}","aliases":"widetildeσ_N,m widetilde sigma","display":"\\widetilde{\\sigma}_{N,m}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-159b2efa6327c4c0","tex":"\\widetilde{\\sigma}_{N,n}","aliases":"widetildeσ_N,n widetilde sigma","display":"\\widetilde{\\sigma}_{N,n}","kind":"use","excerpt":"[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-05da05e4a35ed5a5","tex":"\\widetilde\\Phi","aliases":"widetildeΦ widetilde Phi","display":"\\widetilde\\Phi","kind":"use","excerpt":"[Quotient topology and completed realization] Let Φ:X↠ Y be a continuous linear quotient map between Hausdorff topological vector spaces, and put K=KerΦ. Then the induced map widetildeΦ:X/K⟶ Y is a topological isomorphism. If D⊆ X is a linear subspace whose image Φ(D) is dense in a complete space Y, then the completion of D/(D∩ K) in the topology t","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3b957cb4014dfcd","tex":"\\Wp","aliases":"Wp W_p mathcal","display":"{\\mathcal W_p}","kind":"use","excerpt":"Weighted membership. The class used in the formalization is W_p={F∈ Q_p[[T]]: p^lfloor d/prfloorcoefTdF∈ Z_p for every d}. The integral-coordinate estimate from page 2 now proves H_r(log(1+T))∈ W_p for every residue r, including the singular ones. Rational coefficients are mapped into Q_p by the actual coefficient ring homomorphism.","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-371acb48389709be","tex":"\\mathrm{wt}","aliases":"wt mathrm","display":"\\mathrm{wt}","kind":"use","excerpt":"The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8007fd9fd0b5cd21","tex":"\\bm X","aliases":"X bm","display":"\\bm X","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"· ≤ (1-q)ρ+qρ =ρ. Also, Ψ(x)-Ψ(x') ≤ q x-x' . The contraction mapping theorem gives a unique fixed point in the stated set. Applying D to the fixed point identity gives the equation. Conversely, a solution in X_R equals R(Dx) and is therefore a fixed point of Ψ.","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dab4e8a89da28b3b","tex":"x^{(0)}","aliases":"x^(0)","display":"x^{(0)}","kind":"use","excerpt":"Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a7d404b8f08d7c2","tex":"x^{(3)}","aliases":"x^(3)","display":"x^{(3)}","kind":"use","excerpt":"vfig[.93]shift_tail atlasCaptionA finite window of the actual sequence x[n]=2^-|n|. Outside every finite window, coefficients remain nonzero. The shaded window illustrates the truncation x^(3), not a periodic wraparound.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7696a5e0b656722b","tex":"x^{(d)}","aliases":"x^(d)","display":"x^{(d)}","kind":"use","excerpt":"[5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1 vertex has level at most k, so the recurrence uses only coordinates already proved correct. At the canonical depth d this yields Ax^(d)=b. The local independence the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aeb686029c32e920","tex":"x^{(k)}","aliases":"x^(k)","display":"x^{(k)}","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0190f670081497c9","tex":"x^{(k)}_u","aliases":"x^(k)_u","display":"x^{(k)}_u","kind":"use","excerpt":"Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1, [5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55b833de72597be9","tex":"x^{(k)}_v","aliases":"x^(k)_v","display":"x^{(k)}_v","kind":"use","excerpt":"Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1, [5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c44fc36b58974583","tex":"x^{(k+1)}_v","aliases":"x^(k+1)_v","display":"x^{(k+1)}_v","kind":"use","excerpt":"Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1, [5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a7d632fad29231f3","tex":"x^{(N)}","aliases":"x^(N)","display":"x^{(N)}","kind":"use","excerpt":"This x cannot be a finite linear combination of shifted impulses: it has infinite support. But if x^(N) agrees with x on |n|≤ N and is zero elsewhere, then normx-x^(N)_2^2 =2∑_n>N4^-n =frac23 4^-N⟶0. The exact squared errors for N=1,3,5 are 1/6, 1/96, and 1/1536. These are identities from a geometric series, not floating residuals.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-479d843e18704a71","tex":"x^{(\\ell)}","aliases":"x^(ℓ) ell","display":"x^{(\\ell)}","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-71a31718ae729ffa","tex":"X^*","aliases":"X^*","display":"X^*","kind":"use","excerpt":"[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint pa","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e7e9fdf289546b6","tex":"x^*","aliases":"x^*","display":"x^*","kind":"use","excerpt":"Complex polarization reaches the whole operator space. Write q(x)=xx^*. The module proves the exact identity 4xy^*=q(x+y)-q(x-y) +i(q(x+iy)-q(x-iy)). Agreement on all positive pure outer products therefore implies agreement on every outer product. Actual matrix units then span the full matrix space. Applying this argument to the sandwich by P prove","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78dcbad828a95e4b","tex":"X^{-1}","aliases":"X^-1","display":"X^{-1}","kind":"use","excerpt":"Use the inverse difference identity X^-1-Y^-1=X^-1(Y-X)Y^-1 with X=F_P(z) and Y=z-A.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65e8ec76ae768d6c","tex":"x^{-1}","aliases":"x^-1","display":"x^{-1}","kind":"use","excerpt":"[Finite inverse formula] If x has augmentation one and n=x-1, then x^-1 = ∑_r=0^p-1(-n)^r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1d16f551dc7199c","tex":"x^{-2}","aliases":"x^-2","display":"x^{-2}","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6af5d9dd57bd6ef","tex":"x^{-2-h}","aliases":"x^-2-h","display":"x^{-2-h}","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-770c571f5630bad2","tex":"X^{1-1/d}","aliases":"X^1-1/d","display":"X^{1-1/d}","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"The affine statement is formalized by leanKomorebi.NativeAffine.finite_affine_obstruction. Positive depth is essential in the nonlinear theorem: for F(X)=X^2-1, q=2, L=0, and x=0, the Jacobian is zero and the residual class is nonzero, yet y=1 is a root modulo two in the same residue modulo one.","count":25},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"use","excerpt":"ordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Each initial jet (y(0),y'(0),y''(0)) therefore det","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5d15c510169ad7c","tex":"x^4","aliases":"x^4","display":"x^4","kind":"use","excerpt":"ymbol acts finitelyAPPELL POLYNOMIALSChapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9. For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms sur","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8bbaae1ffe15b3b","tex":"x^d","aliases":"x^d","display":"x^d","kind":"use","excerpt":"Here is the polynomial estimate that supplies one additional bit. For i≥1, put P_i(x)=4^ibinom4x4i. Its constant is zero, and its degree-d coefficient is [x^d]P_i(x)=frac4^i4^d s(4i,d)(4i)!, where s(m,d) is a signed Stirling number of the first kind. In each product contributing to s(4i,d), at most d-1 of the 2i-1 even integers in 1,…,4i-1 can be o","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-44873f8baac650d8","tex":"x^e","aliases":"x^e","display":"x^e","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48f4a03fbc552897","tex":"x^i","aliases":"x^i","display":"x^i","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7272f222c85749d6","tex":"X^m","aliases":"X^m","display":"X^m","kind":"use","excerpt":"Expand X^m=∑_r=0^mT(m,r)P_r(X). Since XP_r(X)=P_r+1(X)+r(r+1)P_r(X), these integer coefficients satisfy T(m+1,r)=T(m,r-1)+r(r+1)T(m,r), T(0,0)=1. Every r(r+1) is even. Induction, including the boundary indices, proves 2^m-rmid T(m,r) for 0≤ r≤ m. For m>0 the r=0 term vanishes. Each remaining summand of L(P^m/m!) has valuation at least m-r+2v_2(r!)-","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ed7bdf4a651c709","tex":"x^m","aliases":"x^m","display":"x^m","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cea057c6841e3ec4","tex":"x^{m-Nn}","aliases":"x^m-Nn","display":"x^{m-Nn}","kind":"use","excerpt":"[Explicit Appell polynomial] For every m≥0, E_m^(N,j,r)(x) = ∑_0≤ n≤ m/N binommNn C_N,j,n(r)x^m-Nn.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-854c66c730d9b734","tex":"X^N","aliases":"X^N","display":"X^N","kind":"use","excerpt":"6.1--6.6 · Quotient-algebra power bases and actual exponential-series convergence in the finite-dimensional coefficient topology. Both pointwise section-coordinate identities. Signed-shift powers, all matrix-exponential entries, group laws and trace are checked. The characteristic polynomial X^N+1, its distinct roots, exact spectrum and determinant","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e189c65b70ed9178","tex":"X^n","aliases":"X^n","display":"X^n","kind":"use","excerpt":"[Frobenius cycle polynomial] The Frobenius transfer matrix has characteristic polynomial X^n-a and determinant detτ(Φ)=(-1)^n-1a.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12f1075149375120","tex":"x^N","aliases":"x^N","display":"x^N","kind":"use","excerpt":"The vectors e_0,J_Ne_0,…,J_N^N-1e_0 are the standard basis. Thus e_0 is cyclic for J_N, so the minimal polynomial of J_N has degree N. Since J_N^N=-I, that minimal polynomial divides x^N+1 and therefore equals x^N+1. The characteristic polynomial has the same degree and is divisible by the minimal polynomial, so it too equals x^N+1. This polynomial","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8c5e1feefd58efd","tex":"x^n","aliases":"x^n","display":"x^n","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-917e6ba09c024b78","tex":"X^{N-1}","aliases":"X^N-1","display":"X^{N-1}","kind":"use","excerpt":"At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coeffici","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-baf58144b77f8bed","tex":"X^p","aliases":"X^p","display":"X^p","kind":"use","excerpt":"Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4524a2723253991","tex":"X^{p-1}","aliases":"X^p-1","display":"X^{p-1}","kind":"use","excerpt":"Then v_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic minimal polynomial f(X)=X^p+a_p-1X^p-1+⋯+a_0 is Eisenstein. It is an explicit polynomial: compute the product over the p conjugates","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-efe34588925c4a06","tex":"x^{p-1}","aliases":"x^p-1","display":"x^{p-1}","kind":"use","excerpt":"At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping th","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0648303074e446a8","tex":"X^{p^a}","aliases":"X^p^a","display":"X^{p^a}","kind":"use","excerpt":"r Editorial hypothesis correction. Require L≥1. At L=0 the standalone claim is false: for p=3, d=1 and X=2=1+B, one has X^3-1=7∉3 Z_3. With L≥1, every higher binomial term in the retained proof has the required valuation, including p=2. If X=I+p^LB with B∈ M_d( Z_p), then for every a≥0 there is B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8744addd278e3b80","tex":"x^{p^{\\ell-1}}","aliases":"x^p^ℓ-1 ell","display":"x^{p^{\\ell-1}}","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14fb02863a861bee","tex":"X^r","aliases":"X^r","display":"X^r","kind":"use","excerpt":"R or C and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr echet derivatives and the Taylor integral below has a real parameter. Assume that F:X→ Y is m times continuously Fr echet differentiable. The same local statements hold on open domains containing the points and segments in question. For 1","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab3482f9a2bb99d4","tex":"X^{r_i}","aliases":"X^r_i","display":"X^{r_i}","kind":"use","excerpt":"A nonzero fixed vector must lie in an inertia trivial packet, so c_i=0. On that packet, Frobenius is a companion cycle with characteristic polynomial X^r_i-a_i. Its fixed space is the eigenspace for eigenvalue one. This eigenspace is one dimensional exactly when a_i=1, and it is zero otherwise. Summing over the refined packets proves the formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0514a60dd6e60420","tex":"x^T","aliases":"x^T","display":"x^T","kind":"use","excerpt":"1/2 · 1. Its exact eigenvalues are 1/2 and 3/2, giving condition number three. vfigpreconditioning headThe unit circles of the energy norms The left ellipse is x^TAx=1. Writing x=By gives the right ellipse y^T(BAB)y=1. The drawing samples these explicitly defined curves. The norm and eigenvalue equalities are checked symbolically.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3f03ee84a66d436","tex":"X^{\\underline q}","aliases":"X^underline q underline","display":"X^{\\underline q}","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-213991022244ad3d","tex":"x^\\alpha","aliases":"x^α alpha","display":"x^\\alpha","kind":"use","excerpt":"It is enough to prove the formula on an eigenvector with eigenvalue x≥0. For x>0, the substitution s=xu gives fracsin(πα)π ∫_0^∞ s^α-1fracxs+x d s =x^α fracsin(πα)π ∫_0^∞fracu^α-11+u d u =x^α. The value at x=0 is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power c","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e291ac7130b02dc8","tex":"x^\\ell","aliases":"x^ℓ ell","display":"x^\\ell","kind":"use","excerpt":"[Taylor coefficient bounds and radii] For j=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at addres","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f89be2c73732f99","tex":"X_{(i,\\cdot),(j,\\cdot)}","aliases":"X_(i,·),(j,·) cdot cdot","display":"X_{(i,\\cdot),(j,\\cdot)}","kind":"use","excerpt":"To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14a3332b34948231","tex":"x_*","aliases":"x_*","display":"x_*","kind":"use","excerpt":"[Preconditioned conjugate gradient convergence] Let A and M be Hermitian positive definite on a nonempty finite coordinate space, and set κ = cond_2(M^-1/2AM^-1/2). In exact arithmetic, the energy norm error of preconditioned conjugate gradient satisfies normx_m-x_*_A ≤ 2 ( fracsqrtκ-1sqrtκ+1 )^m normx_0-x_*_A.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operato","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4a48827c05d4509","tex":"X_{00}","aliases":"X_00","display":"X_{00}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c546fc907e36cc78","tex":"x_{00}","aliases":"x_00","display":"x_{00}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f15eebc2170872f","tex":"X_{01}","aliases":"X_01","display":"X_{01}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-769180e7b1f47d49","tex":"x_{01}","aliases":"x_01","display":"x_{01}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-162e2cb41df4870a","tex":"x_0^{m-\\ell}","aliases":"x_0^m-ℓ ell","display":"x_0^{m-\\ell}","kind":"use","excerpt":"For j=0,1, the term m=ℓ in [source reference] has exact valuation ℓ, since s(ℓ,ℓ)=1 and the Newton valuation is exact. Every later term has valuation at least ℓ+1. Thus v_2(d_j,ℓ)=ℓ, and the coefficient radius is exactly two. At a center x_0∈ Z_2, the coefficient of y^ℓ in the translated series is ∑_m≥ℓd_j,mbinom mℓ x_0^m-ℓ. Its initial term again ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"[title=Why mixed seeds change the answer] Let the acting algebra be diagonal on X=X_1⊕ X_2, with no transfer between the packets. Generation by packet pure seeds requires one seed in each source component, so psrank=2. A single mixed vector x_1+x_2 may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can b","count":22},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-805649c1b3a8b1e8","tex":"X_{10}","aliases":"X_10","display":"X_{10}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1ad6e78b56605d3","tex":"x_{10}","aliases":"x_10","display":"x_{10}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f17c0dba15962adb","tex":"X_{11}","aliases":"X_11","display":"X_{11}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8abb0d1ba7e87e0d","tex":"x_{11}","aliases":"x_11","display":"x_{11}","kind":"use","excerpt":"[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coars","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65e552b57fafc09d","tex":"x_{14}","aliases":"x_14","display":"x_{14}","kind":"use","excerpt":"vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTChapter II, mathematical text: Theorem 3.3, Theorem 4.6 Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in ea","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f3039e219d0894a","tex":"x_{15}","aliases":"x_15","display":"x_{15}","kind":"use","excerpt":"vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTChapter II, mathematical text: Theorem 3.3, Theorem 4.6 Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in ea","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3674278fc192e035","tex":"x_1^2","aliases":"x_1^2","display":"x_1^2","kind":"use","excerpt":"[A mixed quadratic map] Let X=Y=𝕜^2 with coordinate packets. For F(x_1,x_2)=(x_1x_2,x_1^2+x_2), the first output packet depends on both input packets. The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second. Packet coordinates record these interactions exactly. Replacing F by ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1494511a83649b16","tex":"x_1^3","aliases":"x_1^3","display":"x_1^3","kind":"use","excerpt":"r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0273e5b76a404abc","tex":"x_1^n","aliases":"x_1^n","display":"x_1^n","kind":"use","excerpt":"e identity Δ_N(w)^-r = exp( r∑_m≥1κ_N,mfracw^mm ) fits the defining generating function for B_n with x_m=r(m-1)!κ_N,m. The coefficient of w^n is therefore the displayed Bell polynomial divided by n!. Multiplication by (Nn)! gives P_N,n(r). Since Δ_N(w)=1-fracwN!+O(w^2), we have κ_N,1=1/N!. The term x_1^n gives the stated leading coefficient, which ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f93b0321f5ab119e","tex":"X_1^{p^{n_1}}","aliases":"X_1^p^n_1","display":"X_1^{p^{n_1}}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[title=Why mixed seeds change the answer] Let the acting algebra be diagonal on X=X_1⊕ X_2, with no transfer between the packets. Generation by packet pure seeds requires one seed in each source component, so psrank=2. A single mixed vector x_1+x_2 may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can b","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bade2109ecac9dc","tex":"x_2'","aliases":"x_2'","display":"x_2'","kind":"use","excerpt":"path, one-edge paths, and the two-edge path exist. The coefficient 3t^2 includes both the edge product 2·3 and the ordered-time simplex area t^2/2. headPath length counts transfers, not elapsed time Starting at e_0, the complete state is (1,2t,3t^2)^ T. No fourth packet or repeated cycle can contribute. Direct differentiation gives x'(t)=Ax(t), inc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14a93f6cb614d651","tex":"X_2^{a_2}","aliases":"X_2^a_2","display":"X_2^{a_2}","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARChapter VII: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a3a227a148b5734","tex":"X_2^{\\mathrm{wt}}","aliases":"X_2^wt mathrm","display":"X_2^{\\mathrm{wt}}","kind":"use","excerpt":"The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25c67413dcf997f2","tex":"x_{2m}","aliases":"x_2m","display":"x_{2m}","kind":"use","excerpt":"vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONChapter II, mathematical text: Section B.1 Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b105e9fb57cd70a","tex":"x_{2m+1}","aliases":"x_2m+1","display":"x_{2m+1}","kind":"use","excerpt":"vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONChapter II, mathematical text: Section B.1 Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8eeb0a1bbb468424","tex":"X_3","aliases":"X_3","display":"X_3","kind":"use","excerpt":"[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-298640d8b28fac27","tex":"X_3^{a_3}","aliases":"X_3^a_3","display":"X_3^{a_3}","kind":"use","excerpt":"vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARChapter VII: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca7008ae447ee9dc","tex":"x_4","aliases":"x_4","display":"x_4","kind":"use","excerpt":", put two on the diagonal and -1 on each declared incoming edge. The edges are 0→2, 1→2, 1→3, 2→4, 3→4, 3→5, 4→6, and 5→6. Solve (2I+K)x=ones. vfigtriangular headRead the stages as dependencies Packets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For insta","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98391915cbb7346b","tex":"x_5","aliases":"x_5","display":"x_5","kind":"use","excerpt":", put two on the diagonal and -1 on each declared incoming edge. The edges are 0→2, 1→2, 1→3, 2→4, 3→4, 3→5, 4→6, and 5→6. Solve (2I+K)x=ones. vfigtriangular headRead the stages as dependencies Packets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For insta","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3864fe3cacc7c90d","tex":"x_6","aliases":"x_6","display":"x_6","kind":"use","excerpt":"headAddress resolution A coefficient such as x_6 can sit in the even packet, then the residue-2 packet modulo 4, then the residue-6 packet modulo 8. Its value does not become more accurate. Its address becomes more specific, while the full object remains exactly reconstructible.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-762eb59ebd1883d4","tex":"x_A","aliases":"x_A","display":"x_A","kind":"use","excerpt":"AtlasHeadA derivative weight can cost one digit of root precisionatlas:weighted Compare the linear equations F(x)=3x=0 and G_A(x)=3x+3^A=0, where A≥2. Their coefficient difference has 3-adic valuation A, but their roots differ by x_A-0=-3^A-1, v_3(x_A)=A-1. AtlasImageweighted_motion The derivative has valuation one. Solving the correction equation ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-73a4e2e74cee4766","tex":"X_b","aliases":"X_b","display":"X_b","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72e05ef2b0f48e18","tex":"X_c","aliases":"X_c","display":"X_c","kind":"use","excerpt":"Let X_f and X_c be vector spaces or Banach spaces over a common field, and let C:X_f⟶ X_c be a linear map, and let T_f∈End(X_f), T_c∈End(X_c). Define Δ=CT_f-T_cC.","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-702e017c46584077","tex":"x_c","aliases":"x_c","display":"x_c","kind":"use","excerpt":"[Fixed point stability] Assume F_c is a contraction with constant q<1. Let x_f and x_c be fixed points of F_f and F_c. If Δ_F(x_f) ≤ε, then Cx_f-x_c ≤ fracε1-q. When the defect at x_f is zero, the fixed points are compatible.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4433e4e3be068643","tex":"X_D","aliases":"X_D","display":"X_D","kind":"use","excerpt":"a:X_α↠ X_β. Because X_α⊆ A S and A S is invariant under A, surjectivity implies X_β⊆ A S. For any other vertex γ∈ D, choose a directed path inside the strongly connected component from β to γ. Successive surjective edge blocks propagate the full packet space along the path, so X_γ⊆ A S. Hence X_D⊆ A S. Induction over the condensation graph gives X=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-715547043f329ef6","tex":"X_{e+k}","aliases":"X_e+k","display":"X_{e+k}","kind":"use","excerpt":"[Continuous charts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is ho","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c764cff6a54a44c","tex":"X_{e+q}","aliases":"X_e+q","display":"X_{e+q}","kind":"use","excerpt":"Let p be prime, L≥1, e,q≥0, and F∈ Z[X_1,…,X_e+q]^e, x∈ Z^e+q, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^e+q⟶ F_p^e is surjective. All vector congruences are coordinatewise.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f76f02d6c081bad5","tex":"X_f","aliases":"X_f","display":"X_f","kind":"use","excerpt":"Let X_f and X_c be vector spaces or Banach spaces over a common field, and let C:X_f⟶ X_c be a linear map, and let T_f∈End(X_f), T_c∈End(X_c). Define Δ=CT_f-T_cC.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b816b84df56eb81","tex":"x_F","aliases":"x_F","display":"x_F","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-91c50d828be3947d","tex":"x_f","aliases":"x_f","display":"x_f","kind":"use","excerpt":"[Fixed point stability] Assume F_c is a contraction with constant q<1. Let x_f and x_c be fixed points of F_f and F_c. If Δ_F(x_f) ≤ε, then Cx_f-x_c ≤ fracε1-q. When the defect at x_f is zero, the fixed points are compatible.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be02837bb7e276a6","tex":"x_{g,a,b}","aliases":"x_g,a,b","display":"x_{g,a,b}","kind":"use","excerpt":"Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"use","excerpt":"[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not a","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required d","count":26},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0597cb543b16b78f","tex":"x_i^2","aliases":"x_i^2","display":"x_i^2","kind":"use","excerpt":"vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMChapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil d","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb595c87aa88b7d0","tex":"X_i^{p^{n_i}}","aliases":"X_i^p^n_i","display":"X_i^{p^{n_i}}","kind":"use","excerpt":"In characteristic p, (1+X_i)^p^n_i-1=X_i^p^n_i. Thus the assignment X_i↦γ_i-1 gives a surjective homomorphism from the displayed truncated polynomial ring to k[Δ]. Both sides have dimension ∏_ip^n_i=absΔ over k, so the map is an isomorphism. The quotient by the ideal generated by the X_i is k. Since each X_i is nilpotent, that ideal is the unique m","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0298cc60bfc2d7e5","tex":"X_{ij}","aliases":"X_ij","display":"X_{ij}","kind":"use","excerpt":"Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-560fb8f41570b906","tex":"X_j^{a_j}","aliases":"X_j^a_j","display":"X_j^{a_j}","kind":"use","excerpt":"For every binary address a, W_a=∏_j=0^n-1X_j^a_j=T_a, W_aψ_0=ket a. The family (W_aψ_0)_a∈ A_n is linearly independent, spans V_n, and has exactly 2^n elements. It is the standard computational basis.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2357b435bdbbf889","tex":"X_{k\\ell}","aliases":"X_kℓ ell","display":"X_{k\\ell}","kind":"use","excerpt":"The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-441b7c961c9320ed","tex":"x_{L+1}","aliases":"x_L+1","display":"x_{L+1}","kind":"use","excerpt":"[One level linear lifting] A class represented by x_L lifts to a solution modulo p^L+1 if and only if Ob^lin_L(x_L)=0. More precisely, a correction y∈ F_p^d gives a lift x_L+1=x_L+p^Ly pmod p^L+1 if and only if overline My=-ρ_L(x_L). When one correction exists, the set of all correction classes is a coset of keroverline M.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-848620af401d93ab","tex":"x_m'","aliases":"x_m'","display":"x_m'","kind":"use","excerpt":"[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a7064d4243ed8b7","tex":"X_{m+1}","aliases":"X_m+1","display":"X_{m+1}","kind":"use","excerpt":"Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ce545975574982d","tex":"x_{m+1}","aliases":"x_m+1","display":"x_{m+1}","kind":"use","excerpt":"Choose representatives recursively. Once x_m represents the selected depth m vertex, choose a representative x_m+1 of its selected child with x_m+1-x_m∈ F^mX. The resulting sequence is Cauchy for the filtration topology. Each representative satisfies Φ(x_m)∈ F^mY. The conclusion now follows from [source reference].","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80ceedbe9c553f98","tex":"x_{m-1}","aliases":"x_m-1","display":"x_{m-1}","kind":"use","excerpt":"For canonical representatives 0≤ x_m<p^m, set x_0=0. Compatibility also gives a_m-1=(x_m-x_m-1)/p^m-1∈{0,…,p-1}. The floor in the first formula is essential. For example, the compatible base-five residues 4,19 give the second digit 3, not the rational number 19/5. The executable companion uses the same integer floor division.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4715e03ab2461874","tex":"x_{n+1}","aliases":"x_n+1","display":"x_{n+1}","kind":"use","excerpt":"begingroupsmallraggedright Formal correspondence. leanCompanion.companion_jump, leanCompanion.packet_charpoly_recurrence, and leanCompanion.packet_observable_recurrence. The source-sequence versions derive the packet recurrence from the relation x_n+1=Cx_n. For the Fibonacci matrix C=(1 · 1","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f551abf5f44825db","tex":"x_{n+2}","aliases":"x_n+2","display":"x_{n+2}","kind":"use","excerpt":"[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e43b4db73042e9a","tex":"x_{N-1}","aliases":"x_N-1","display":"x_{N-1}","kind":"use","excerpt":"Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06a2f75a3e24f887","tex":"x_{n-r}","aliases":"x_n-r","display":"x_{n-r}","kind":"use","excerpt":"A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd757208c85dd8b0","tex":"x_n^{(k)}","aliases":"x_n^(k)","display":"x_n^{(k)}","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9688c63367ddeff","tex":"x_n^{(\\ell)}","aliases":"x_n^(ℓ) ell","display":"x_n^{(\\ell)}","kind":"use","excerpt":"Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-924805ec7439cec0","tex":"X_N^{\\rm pkt}","aliases":"X_N^rm pkt rm","display":"X_N^{\\rm pkt}","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-425ede770bf1a9d2","tex":"x_{Nk+j}","aliases":"x_Nk+j","display":"x_{Nk+j}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-853cecd6231f3bad","tex":"X_P","aliases":"X_P","display":"X_P","kind":"use","excerpt":"Let X=X_P⊕ X_Q and write A= A_PP · A_PQ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68150f2105b73870","tex":"x_P","aliases":"x_P","display":"x_P","kind":"use","excerpt":"Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53687bf254c09aec","tex":"x_p","aliases":"x_p","display":"x_p","kind":"use","excerpt":"An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrig","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f778110707ddeb9c","tex":"X_q","aliases":"X_q","display":"X_q","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8663ffd669417a5","tex":"x_Q","aliases":"x_Q","display":"x_Q","kind":"use","excerpt":"Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e798a8a7ea91ac3e","tex":"X_R","aliases":"X_R","display":"X_R","kind":"use","excerpt":"[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L-Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in ","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8285a92cff098813","tex":"X_r","aliases":"X_r","display":"X_r","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ef07195fe759ac3","tex":"x_{r-1}","aliases":"x_r-1","display":"x_{r-1}","kind":"use","excerpt":"[Quadratic nearest neighbor interaction] For y(n)=x(n)x(n-1) and modulus N, y_r(q) = x_r(q)x_r-1(q), · r≥1, x_0(q)x_N-1(q-1), · r=0. The carry at residue zero is the nonlinear analogue of a transfer crossing a coarse cell boundary.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-020f07c72259d4de","tex":"X_r^{p^{n_r}}","aliases":"X_r^p^n_r","display":"X_r^{p^{n_r}}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e41ed00535c2ae0","tex":"x_{r_a}","aliases":"x_r_a","display":"x_{r_a}","kind":"use","excerpt":"[Translation covariance] Fix a∈ Z^d. For every r∈ R, write uniquely r-a=r_a+λ_a,r, r_a∈ R, λ_a,r∈ L. Then ( A_LT_ax)_r(ℓ) = x_r_a(ℓ+λ_a,r). Thus a fine translation permutes the lattice packets and applies an internal translation on L.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dde230160dab3203","tex":"x_{r_j}","aliases":"x_r_j","display":"x_{r_j}","kind":"use","excerpt":"0, · r≥ b_j. Then the output channel is y_r(q) = ∏_j=1^m x_r_j (q-c_j-κ_r,b_j).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51c75ae68c337281","tex":"x_{r_k}","aliases":"x_r_k","display":"x_{r_k}","kind":"use","excerpt":"Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ee20c14455ce834","tex":"X_s","aliases":"X_s","display":"X_s","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8786e121e9dc2d7","tex":"x_s","aliases":"x_s","display":"x_s","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81529bf55585a9e9","tex":"X_{\\src(e)}","aliases":"X_src(e) src operatorname","display":"X_{{\\operatorname{src}}(e)}","kind":"use","excerpt":"[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"use","excerpt":"[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9cd5d8f0da4542ed","tex":"X_{\\tgt(e)}","aliases":"X_tgt(e) tgt operatorname","display":"X_{{\\operatorname{tgt}}(e)}","kind":"use","excerpt":"[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a8e8addc6fe7c2d","tex":"X_\\alpha^*","aliases":"X_α^* alpha","display":"X_\\alpha^*","kind":"use","excerpt":"[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint pa","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af49eb7478b6e8c8","tex":"X_{\\alpha_i}","aliases":"X_α_i alpha","display":"X_{\\alpha_i}","kind":"use","excerpt":"[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25cae1ffb84e4bb9","tex":"X_{\\alpha_j}","aliases":"X_α_j alpha","display":"X_{\\alpha_j}","kind":"use","excerpt":"For h_j∈ X_α_j, the inherited norm gives Q_γ D^rF(x)[h_1,…,h_r] ≤ Q_γ D^rF(x) ∏_j h_j . Taking the multilinear operator norm proves the sharper restricted estimate. For ambient inputs, replace h_j by P_α_jh_j and apply the operator norm bound to each projector. Every nonzero idempotent has norm at least one, since it fixes a nonzero vector in its r","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcf7586068a6a313","tex":"x_\\beta","aliases":"x_β beta","display":"x_\\beta","kind":"use","excerpt":"For x∈ X, S_ P A_ Px = ∑_α P_α x =x. If x_α∈ X_α, then P_β x_α=δ_αβx_α, so A_ P S_ P(x_α)_α = (P_β∑_α x_α)_β =(x_β)_β. Thus analysis and synthesis are inverse linear maps.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a560c8da69b7911b","tex":"X_\\beta^*","aliases":"X_β^* beta","display":"X_\\beta^*","kind":"use","excerpt":"Conversely, suppose transitivity fails. Choose α,β,γ with (β,α)∈ R, (γ,β)∈ R, (γ,α)∉ R. Choose nonzero vectors u∈ X_α, v∈ X_β, and w∈ X_γ. Choose linear functionals f∈ X_α^* and g∈ X_β^* with f(u)=g(v)=1. Let T have the sole block x↦ f(x)v from X_α to X_β, and let S have the sole block y↦ g(y)w from X_β to X_γ. Then S,T∈ M_R, but ST has a nonzero f","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5c7030cb4e0b4b6","tex":"X_\\gamma","aliases":"X_γ gamma","display":"X_\\gamma","kind":"use","excerpt":"Conversely, suppose transitivity fails. Choose α,β,γ with (β,α)∈ R, (γ,β)∈ R, (γ,α)∉ R. Choose nonzero vectors u∈ X_α, v∈ X_β, and w∈ X_γ. Choose linear functionals f∈ X_α^* and g∈ X_β^* with f(u)=g(v)=1. Let T have the sole block x↦ f(x)v from X_α to X_β, and let S have the sole block y↦ g(y)w from X_β to X_γ. Then S,T∈ M_R, but ST has a nonzero f","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-953215dbee72ad06","tex":"x_\\gamma","aliases":"x_γ gamma","display":"x_\\gamma","kind":"use","excerpt":"Every coarse packet is also uniquely reconstructed from its fine children. If vectors x_β∈ Q_β X sum to a coarse packet x, applying Q_γ recovers x_γ=Q_γ x. This is the uniqueness clause of the canonical nested-refinement statement.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1020799dad46e940","tex":"X_\\lambda","aliases":"X_λ lambda","display":"X_\\lambda","kind":"use","excerpt":"The irreducibles are p=t-λ. On the primary component for λ, the quotient by (T-λ I)X_λ has dimension equal to the number of Jordan blocks, which is also dimKer(T-λ I).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a80ff6e24b3fc04","tex":"x_\\lambda","aliases":"x_λ lambda","display":"x_\\lambda","kind":"use","excerpt":"Given y=Φ(x), choose a net x_λ∈Hull_ A(S) converging to x. Then Φ(x_λ)→ y, and naturality places every Φ(x_λ) in Hull_ B(Φ(S)). Hence the latter hull is dense. For packets, one inclusion follows from compatibility. Conversely, if y=Q_jy, choose x with Φ(x)=y. Then y=Q_jΦ(x)=Φ P_σ^-1(j)x.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27222a821514a259","tex":"X_{\\lambda,n}","aliases":"X_λ,n lambda","display":"X_{\\lambda,n}","kind":"use","excerpt":"For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a con","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4726173f5ad19c77","tex":"x_{\\rho(\\beta)}","aliases":"x_ρ(β) rho beta","display":"x_{\\rho(\\beta)}","kind":"use","excerpt":"w QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b6c204b3855e438","tex":"X_\\ell","aliases":"X_ℓ ell","display":"X_\\ell","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ec4142068739e9a","tex":"x_\\ell","aliases":"x_ℓ ell","display":"x_\\ell","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-524e8e9a8948b743","tex":"X_{\\ell+1}","aliases":"X_ℓ+1 ell","display":"X_{\\ell+1}","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93da8851ae87d014","tex":"x_{\\ell+1}","aliases":"x_ℓ+1 ell","display":"x_{\\ell+1}","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b99185167014e4eb","tex":"x_\\infty","aliases":"x_∞ infty","display":"x_\\infty","kind":"use","excerpt":"[Complete filtered lifting] Assume there is a compatible sequence x_m+1≡ x_mpmodF^mX with Φ(x_m)∈ F^mY for every m. Then x_m converges to an element x_∞∈ X and Φ(x_∞)=0. In particular, if the obstruction vanishes at every depth and compatible corrections are chosen, an exact solution exists.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55abe5dbff13226b","tex":"\\xleftarrow","aliases":"xleftarrow","display":null,"kind":"use","excerpt":"Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1a93beeda6d35c8","tex":"\\xmapsto","aliases":"xmapsto","display":null,"kind":"use","excerpt":"vpage8Track the Bell state through the circuitINTERMEDIATE PACKETS ARE NOT FINAL PACKETSChapter VII: Theorem II.4.1, Example II.4.6. The input is ket00. First apply a Hadamard to the first qubit, then a controlled flip: ket00 xmapstoH_rm Had⊗ I fracket00+ket10sqrt2 xmapstorm CNOT fracket00+ket11sqrt2. The displayed columns keep amplitudes, includin","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fc54602aaec29a7","tex":"\\bm Y","aliases":"Y bm","display":"\\bm Y","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"use","excerpt":"[Stability with respect to the target] Under the hypotheses of [source reference], let x and x' be the fixed points associated with targets y and y'. If both fixed point maps preserve the same ball, then x-x' ≤ frac R 1-q y-y' .","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b40d33482df931f","tex":"y''","aliases":"y''","display":"y''","kind":"use","excerpt":"z^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Ea","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be45dd38f6e40426","tex":"y^{(N)}","aliases":"y^(N)","display":"y^{(N)}","kind":"use","excerpt":"Fix a positive integer N. Root of unity projection of the exponential produces residue-class sections that, for λ≠0, form a basis for y^(N)=λ^N y. The signed sections for y^(N)=-y yield the denominator D_N and its reduced form Δ_N. We derive reciprocal coefficient polynomials, Bell and Newton recurrences, an additive parameter law, and residue sect","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40d753fc08b490d6","tex":"y^*","aliases":"y^*","display":"y^*","kind":"use","excerpt":"Complex polarization reaches the whole operator space. Write q(x)=xx^*. The module proves the exact identity 4xy^*=q(x+y)-q(x-y) +i(q(x+iy)-q(x-iy)). Agreement on all positive pure outer products therefore implies agreement on every outer product. Actual matrix units then span the full matrix space. Applying this argument to the sandwich by P prove","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2eda1c421506c144","tex":"Y^{-1}","aliases":"Y^-1","display":"Y^{-1}","kind":"use","excerpt":"Use the inverse difference identity X^-1-Y^-1=X^-1(Y-X)Y^-1 with X=F_P(z) and Y=z-A.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61c288ea4029d6a5","tex":"y^{-1}","aliases":"y^-1","display":"y^{-1}","kind":"use","excerpt":"The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bfc9fa04c4a74026","tex":"Y^{1-1/d}","aliases":"Y^1-1/d","display":"Y^{1-1/d}","kind":"use","excerpt":"attice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an e","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e7659cd4ef8f947","tex":"Y^2","aliases":"Y^2","display":"Y^2","kind":"use","excerpt":"The error exponent cannot uniformly be increased. For F(Y,T)=Y+Y^2+T, H=0, a=0, p=2, s=0, M=1, and v=1, there is an even root y of y+y^2+2=0. Its equation forces y≡2pmod4 and y+2=-y^2, so 4mid y+2, 8nmid y+2. Thus the error depth 2M-3s=2 is attained even for parameter-only motion. The same quadratic with a constant coefficient perturbation gives th","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form,","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3590b2065493ac41","tex":"y^{2}","aliases":"y^2","display":"y^{2}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-827a2f52108af912","tex":"y^j","aliases":"y^j","display":"y^j","kind":"use","excerpt":"Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3aed34bf8b499303","tex":"y^k","aliases":"y^k","display":"y^k","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-560f4e5df768a9d8","tex":"y^m","aliases":"y^m","display":"y^m","kind":"use","excerpt":"alize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring. Every class has a unique normal form f(x,y)=a(x)+y b(x). Indeed, y^2-(x^3-x) is monic of degree two in y, so division in 𝕜[x][y] gives a unique remainder of degree less than tw","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b29e0454f65eab4","tex":"y^{m-q}","aliases":"y^m-q","display":"y^{m-q}","kind":"use","excerpt":"[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8da30c36f5ac9719","tex":"y^N","aliases":"y^N","display":"y^N","kind":"use","excerpt":"[Characteristic polynomial of the jump] Let χ_C(y)=det(yI-C). Then χ_C^N(x) = Res_y(χ_C(y),x-y^N). The identity holds over the coefficient ring after interpreting the resultant universally.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de7ba1dbdac1562b","tex":"y^n","aliases":"y^n","display":"y^n","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51c2ad6cea95e462","tex":"y^{n-1-i}","aliases":"y^n-1-i","display":"y^{n-1-i}","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ecc9841022db246","tex":"y^p","aliases":"y^p","display":"y^p","kind":"use","excerpt":"Choose a generator g of the cyclic additive group. Then F_p[ F_p^+] ≅ F_p[y]/(y^p-1) = F_p[y]/(y-1)^p. The augmentation ideal is generated by y-1, so its pth power is zero. Every element has the form a+n with a∈ F_p and n∈ J. It is invertible exactly when a≠0, which is the augmentation condition.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd3e4a5285d4c840","tex":"y^{p^{\\ell-1}}","aliases":"y^p^ℓ-1 ell","display":"y^{p^{\\ell-1}}","kind":"use","excerpt":"Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87e8ed00b8e4c9d2","tex":"y^T","aliases":"y^T","display":"y^T","kind":"use","excerpt":"1/2 · 1. Its exact eigenvalues are 1/2 and 3/2, giving condition number three. vfigpreconditioning headThe unit circles of the energy norms The left ellipse is x^TAx=1. Writing x=By gives the right ellipse y^T(BAB)y=1. The drawing samples these explicitly defined curves. The norm and eigenvalue equalities are checked symbolically.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ded9263acb77baa","tex":"y^\\ell","aliases":"y^ℓ ell","display":"y^\\ell","kind":"use","excerpt":"For j=0,1, the term m=ℓ in [source reference] has exact valuation ℓ, since s(ℓ,ℓ)=1 and the Newton valuation is exact. Every later term has valuation at least ℓ+1. Thus v_2(d_j,ℓ)=ℓ, and the coefficient radius is exactly two. At a center x_0∈ Z_2, the coefficient of y^ℓ in the translated series is ∑_m≥ℓd_j,mbinom mℓ x_0^m-ℓ. Its initial term again ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"0 · 0 . The class x_1=(0,0) solves Mx_1≡ bpmod3 for both b_1=(-3,0) and b_2=(0,-3). For b_1, the reduced residual is (1,0), which belongs to the image of overline M. There are three correction classes, given by y_1+y_2=2 pmod3. For b_2, the residual is (0,1), whose cokernel class is nonzero. No lift modulo nine exists because the second row of M re","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57080b7d6f26d5e6","tex":"Y_2","aliases":"Y_2","display":"Y_2","kind":"use","excerpt":"nomialPrescribedPadicTower.exists_prescribed_padic_root_with_tower and leanKomorebi.PolynomialPrescribedPadicTower.exists_unique_prescribed_padic_root. The separate supplied-path theorem leanKomorebi.PolynomialPrescribedPadicTower.exists_root_with_prescribed_path accepts any compatible integer parameter path with limit τ. For m>e, endpoint uniquene","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd412919846eece9","tex":"y_2","aliases":"y_2","display":"y_2","kind":"use","excerpt":"0 · 0 . The class x_1=(0,0) solves Mx_1≡ bpmod3 for both b_1=(-3,0) and b_2=(0,-3). For b_1, the reduced residual is (1,0), which belongs to the image of overline M. There are three correction classes, given by y_1+y_2=2 pmod3. For b_2, the residual is (0,1), whose cokernel class is nonzero. No lift modulo nine exists because the second row of M re","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-879f9148f6a019e7","tex":"Y_c","aliases":"Y_c","display":"Y_c","kind":"use","excerpt":"Let F_f:X_f⟶ Y_f, F_c:X_c⟶ Y_c be fine and coarse maps. Let C_X:X_f⟶ X_c, C_Y:Y_f⟶ Y_c be linear coarsening maps. Whenever a norm, a Lipschitz constant, or a Fr echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69bb6e300ff2f699","tex":"Y_f","aliases":"Y_f","display":"Y_f","kind":"use","excerpt":"Let F_f:X_f⟶ Y_f, F_c:X_c⟶ Y_c be fine and coarse maps. Let C_X:X_f⟶ X_c, C_Y:Y_f⟶ Y_c be linear coarsening maps. Whenever a norm, a Lipschitz constant, or a Fr echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L cho","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3d4192e022af00a","tex":"y_j","aliases":"y_j","display":"y_j","kind":"use","excerpt":"vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required d","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b050e04b7e254aa7","tex":"y_k","aliases":"y_k","display":"y_k","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d39b62f0bd31765f","tex":"y_{k+1}","aliases":"y_k+1","display":"y_{k+1}","kind":"use","excerpt":"me f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Define the adjacent contrast D_k=y_k+1-y","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f63c0b46cff9ca02","tex":"y_m","aliases":"y_m","display":"y_m","kind":"use","excerpt":"For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the unconditionally convergent series: P_m S_ Py = ∑_nP_my_n =y_m. Thus A_ P S_ P=I. Complete","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b76f576ac2d02254","tex":"Y_{m+1}","aliases":"Y_m+1","display":"Y_{m+1}","kind":"use","excerpt":"[Surjectivity on stable cores] For every m, the transition map V_m+1→ V_m induces a surjection Y_m+1↠ Y_m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7917a1346f513bef","tex":"y_{m+1}","aliases":"y_m+1","display":"y_{m+1}","kind":"use","excerpt":"Choose 0≠ y_1∈ Y_1. By [source reference], choose recursively y_m+1∈ Y_m+1 mapping to y_m. Every y_m is nonzero because it maps to the nonzero vector y_1. The sequence (y_m) is a nonzero element of the inverse limit.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00fe9b383f124db5","tex":"y_n","aliases":"y_n","display":"y_n","kind":"use","excerpt":"Each P_nX=Ker(I-P_n) is closed. For x∈ X, A_ Px _u = sup_F,a ∑_n∈ Fa_nP_nx ≤ K x . For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the uncond","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ee025aeeb25a3e5","tex":"Y_P","aliases":"Y_P","display":"Y_P","kind":"use","excerpt":"Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9cafb0ef5346dcbc","tex":"Y_Q","aliases":"Y_Q","display":"Y_Q","kind":"use","excerpt":"Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v)","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a478d5b0d5f3dc01","tex":"y_Q","aliases":"y_Q","display":"y_Q","kind":"use","excerpt":"Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72c43d344dc79844","tex":"Y_r","aliases":"Y_r","display":"Y_r","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57348dda8e2fb02d","tex":"y_r","aliases":"y_r","display":"y_r","kind":"use","excerpt":"Now define explicit iterations y_0=0, y_r+1=Λ_c^-1(cq+y_rq-y_r^2). If q is flat and p-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem: [t^d]y_r=[t^d]y (d≤ r). Indeed, (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y). The second factor has zero constant term. Multiplication therefore raises t","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5cd112d8b799c301","tex":"y_{r+1}","aliases":"y_r+1","display":"y_{r+1}","kind":"use","excerpt":"Now define explicit iterations y_0=0, y_r+1=Λ_c^-1(cq+y_rq-y_r^2). If q is flat and p-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem: [t^d]y_r=[t^d]y (d≤ r). Indeed, (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y). The second factor has zero constant term. Multiplication therefore raises t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-288e0c669aaa9103","tex":"y_r^2","aliases":"y_r^2","display":"y_r^2","kind":"use","excerpt":"Now define explicit iterations y_0=0, y_r+1=Λ_c^-1(cq+y_rq-y_r^2). If q is flat and p-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem: [t^d]y_r=[t^d]y (d≤ r). Indeed, (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y). The second factor has zero constant term. Multiplication therefore raises t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-151cda08db8a7a2a","tex":"Y_s","aliases":"Y_s","display":"Y_s","kind":"use","excerpt":"[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6280117a7eeb98e1","tex":"y_s","aliases":"y_s","display":"y_s","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea5ac5a487fde7e2","tex":"y_{s+1}","aliases":"y_s+1","display":"y_{s+1}","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1d4854bf905c5d1b","tex":"y_s^2","aliases":"y_s^2","display":"y_s^2","kind":"use","excerpt":"At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6811e23146d67e8a","tex":"y_\\alpha","aliases":"y_α alpha","display":"y_\\alpha","kind":"use","excerpt":"[Orthogonal Hilbert packets] If X is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product ⟨(x_α),(y_α)⟩=∑_α⟨ x_α,y_α⟩. Then analysis and synthesis are unitary. Every coordinate change in [source reference] is unitary.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90bad8055cff4ee1","tex":"y_\\beta","aliases":"y_β beta","display":"y_\\beta","kind":"use","excerpt":"w QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75fbe28ce1b6d91b","tex":"\\mathbf Z","aliases":"Z mathbf","display":"\\mathbf Z","kind":"use","excerpt":"[Multiplicity recovery] The original companion matrix was not retrieved or rerun. A new exact realization uses the trivial character and two independent sign characters of ( Z/2 Z)^2, evaluated at (0,0),(1,0),(0,1): C=1 · 1 · 1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e30e315795c2b5b","tex":"Z^*","aliases":"Z^*","display":"Z^*","kind":"use","excerpt":"Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this cal","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3c14695c16612f8","tex":"z^*","aliases":"z^*","display":"z^*","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-449753faf1f7a481","tex":"z^{-1}","aliases":"z^-1","display":"z^{-1}","kind":"use","excerpt":"Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fb34d91679bf137","tex":"z^{-1/2}","aliases":"z^-1/2","display":"z^{-1/2}","kind":"use","excerpt":"P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Mu","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81c0a101b2f89771","tex":"z^{-j}","aliases":"z^-j","display":"z^{-j}","kind":"use","excerpt":"The last factor is x_s[n-m], giving the time-domain formula. Under the additional transform hypothesis, each delayed channel has transform ∑_nx_s[n-m]z^-n =z^-m∑_jx_s[j]z^-j. Absolute summability permits this bijective reindexing, and only finitely many delayed channels are combined. Taking their finite linear combination gives the matrix formula. ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa4a9b335ff4f90f","tex":"z^{-m}","aliases":"z^-m","display":"z^{-m}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ba52b051dd893cc4","tex":"z^{-n}","aliases":"z^-n","display":"z^{-n}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-544ac8aed4a2c573","tex":"Z^1","aliases":"Z^1","display":"Z^1","kind":"use","excerpt":"[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69ffc71de82ab37e","tex":"z^{10}","aliases":"z^10","display":"z^{10}","kind":"use","excerpt":"r rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a49e6bec04c7b56","tex":"Z^2","aliases":"Z^2","display":"Z^2","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"For N=2, [source reference] and D_2(z)=Δ_2(z^2) give λ_m=π^2(m+frac12)^2, m≥0. Thus the first formula holds with Q_2^red=0.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e712bd3d7c60c1d5","tex":"z^{2n}","aliases":"z^2n","display":"z^{2n}","kind":"use","excerpt":"[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9613da88d6877c77","tex":"z^{2r}","aliases":"z^2r","display":"z^{2r}","kind":"use","excerpt":"The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the b","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f64931971fecd572","tex":"Z^3","aliases":"Z^3","display":"Z^3","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4140de75758a272d","tex":"z^3","aliases":"z^3","display":"z^3","kind":"use","excerpt":"vpage4Products mix the residue lanesCYCLIC CONVOLUTIONChapter IV: Theorem 3.3. Selection separates a function additively. It does not make multiplication diagonal. A residue-i term multiplied by a residue-j term lands in residue i+jpmod N. For N=3, use F=1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vf","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34d4b252430fe95b","tex":"z^4","aliases":"z^4","display":"z^4","kind":"use","excerpt":"ion gives a polygon of exponents. Large zeros can occur where two dominant terms nearly cancel. The proof first localizes those zeros and then uses symmetry and uniqueness to place them on the rays. Finite exceptional zeros remain part of the later products and sums. modelfigurezero_orbitsThe exact order-four model. The first three zeros on each of","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6dedd41de3fd2c9b","tex":"z^6","aliases":"z^6","display":"z^6","kind":"use","excerpt":"r rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00a09b294e775bb0","tex":"z^a","aliases":"z^a","display":"z^a","kind":"use","excerpt":"Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7206b16a18756e6f","tex":"\\Z^d","aliases":"Z^d Z mathbb","display":"{\\mathbb Z}^d","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e46d5eff30bc2a2","tex":"\\Z^e","aliases":"Z^e Z mathbb","display":"{\\mathbb Z}^e","kind":"use","excerpt":"[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a71ae723be54d160","tex":"\\Z^{e+k}","aliases":"Z^e+k Z mathbb","display":"{\\mathbb Z}^{e+k}","kind":"use","excerpt":"[Continuous charts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is ho","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b73464ec6718888a","tex":"\\mathbb Z^{e+q}","aliases":"Z^e+q mathbb","display":"\\mathbb Z^{e+q}","kind":"use","excerpt":"Let p be prime, L≥1, e,q≥0, and F∈ Z[X_1,…,X_e+q]^e, x∈ Z^e+q, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^e+q⟶ F_p^e is surjective. All vector congruences are coordinatewise.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-943f96db5b69b8c3","tex":"\\Z^f","aliases":"Z^f Z mathbb","display":"{\\mathbb Z}^f","kind":"use","excerpt":"For affine integer systems F(X)=AX-b and G(X)=BX-c, where A: Z^d→ Z^e and B: Z^d→ Z^f are linear, Corollary~[source reference] holds for q≥1 and every L≥0, including depth zero. Here x is a common root modulo q^L, and J and K are the reductions of A and B with domain Z^d. Given separately valid corrections u,v, a simultaneous next lift exists if an","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed6a8232558e5682","tex":"z^h","aliases":"z^h","display":"z^h","kind":"use","excerpt":"[Rational ordinary generating function] In F_p[[z]], ∑_n≥0e_n^(m,p)z^n = 1+ frac ∑_r=1^he_r^(m,p)z^r 1-z^h.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0cfa667e85ec10a8","tex":"\\Z^k","aliases":"Z^k Z mathbb","display":"{\\mathbb Z}^k","kind":"use","excerpt":"[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integer","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4144141aa513159","tex":"\\mathbb Z^k","aliases":"Z^k mathbb","display":"\\mathbb Z^k","kind":"use","excerpt":"Integer prescribed-path theorem. For every supplied path t: N→ Z^k satisfying t(0)=t_0, t(n+1)-t(n)∈ q^L+n Z^k, there is a single tower z: N→ Z^m+k with z(0)=x, free(z(n))=t(n), F(z(n))∈ q^L+n Z^e, z(n+1)-z(n)∈ q^L+n Z^m+k. The parameter equality is exact in integers. For a≤ b, z(b)-z(a)∈ q^L+a Z^m+k, DF(z(n))bmod q=DF(x)bmod q. Consequently J_z(n)","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89c743fe25c2bcb6","tex":"z^{k+1}","aliases":"z^k+1","display":"z^{k+1}","kind":"use","excerpt":"For large z, Σ_P(z) = ∑_k=0^∞ fracA_PQA_QQ^kA_QPz^k+1. Thus the second condition implies the moment identities. Conversely, the first d identities imply all later ones by the Cayley--Hamilton theorem, so the series vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2bcacc05c95dda04","tex":"z^{k+j}","aliases":"z^k+j","display":"z^{k+j}","kind":"use","excerpt":"The lowest degree term of Φ_M(z) is Mz, and its highest degree term is z^M. Multiplication by (1+z)^s gives the two edge coefficients. When M=2 and s=0, Φ_2(z)^k = (2z+z^2)^k = z^k(2+z)^k. Taking the coefficient of z^k+j gives the binary formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-376fec3a52294509","tex":"z^M","aliases":"z^M","display":"z^M","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d252328af690b62","tex":"\\Z^{m+k}","aliases":"Z^m+k Z mathbb","display":"{\\mathbb Z}^{m+k}","kind":"use","excerpt":"[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integer","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89db1fa034bed25b","tex":"\\mathbb Z^{m+k}","aliases":"Z^m+k mathbb","display":"\\mathbb Z^{m+k}","kind":"use","excerpt":"Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, cho","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4575bcbd3a963aec","tex":"z^{N-1}","aliases":"z^N-1","display":"z^{N-1}","kind":"use","excerpt":"Since D_N(z)=Δ_N(z^N), -frac D_N'(z) D_N(z) = -Nz^N-1fracΔ_N'(z^N)Δ_N(z^N). Differentiate the defining series for -logΔ_N and compare coefficients.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-140a1634265e4538","tex":"z^{Nm}","aliases":"z^Nm","display":"z^{Nm}","kind":"use","excerpt":"This section presents the operator-substitution result transferred from Chapter IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6908f509cdfbbfa6","tex":"z^{Nm-1}","aliases":"z^Nm-1","display":"z^{Nm-1}","kind":"use","excerpt":"[Logarithmic coefficient formula] The transfer tangent satisfies T_N(z) = N∑_m≥1κ_N,mz^Nm-1. Equivalently, T_N(n) = N(Nn+N-1)!κ_N,n+1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b1a441423794c15","tex":"z^{Nn+d_{pq}}","aliases":"z^Nn+d_pq","display":"z^{Nn+d_{pq}}","kind":"use","excerpt":"For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c02dd8bdef514cd5","tex":"z^{Nn-1}","aliases":"z^Nn-1","display":"z^{Nn-1}","kind":"use","excerpt":"Set S= D_N^-r. Logarithmic differentiation gives D_NS'+r D_N'S=0. Taking the coefficient of z^Nn-1/(Nn-1)! gives ∑_k=0^n-1(-1)^k binomNn-1NkP_N,n-k(r) +r∑_k=1^n(-1)^k binomNn-1Nk-1P_N,n-k(r)=0. The term with k=0 in the first sum is P_N,n(r), and every other term has a smaller block index. Rearrangement gives the formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6894c7248320dbce","tex":"z^p","aliases":"z^p","display":"z^p","kind":"use","excerpt":"sumes primality, the modulus divisibility and the specified residue bounds. It does not assume weighted section membership, reciprocal closure, the ratio identity, a Mahler pairing or the desired higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges. 1. Integral cyclic coefficients. Work first in an arbitrary commu","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45aff6ffa0d86e5c","tex":"z^{p-1}","aliases":"z^p-1","display":"z^{p-1}","kind":"use","excerpt":"Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e765b594366f3e90","tex":"z^{p^a}","aliases":"z^p^a","display":"z^{p^a}","kind":"use","excerpt":"In characteristic p, (1+z)^p^a-1≡ z^p^a. Substitute this congruence in the definition of Θ_p^a,s.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de29e5d648115a0b","tex":"z^q","aliases":"z^q","display":"z^q","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e5dd5cdbfbaa44e","tex":"z^{q_k}","aliases":"z^q_k","display":"z^{q_k}","kind":"use","excerpt":"Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1766cb24e585f5e7","tex":"Z^r","aliases":"Z^r","display":"Z^r","kind":"use","excerpt":"[Bernoulli divided-power moments] Let Z be the ordinary Bernoulli variable, L(Z^r)=B_r, with exponential transform t/(e^t-1). For P=Z(Z+1) and μ_m= L(P^m/m!), μ_m∈ Z_2, v_2(μ_m)≥ s_2(m)-1 (m≥1). Moreover, μ_m is odd exactly when m=0 or m is a positive power of two.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10d0329c35407a66","tex":"z^r","aliases":"z^r","display":"z^r","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-287c99551a89afd6","tex":"Z^{\\otimes n}","aliases":"Z^⊗ n otimes","display":"Z^{\\otimes n}","kind":"use","excerpt":"For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08bf4c2ffd5ec019","tex":"Z^{\\otimes3}","aliases":"Z^⊗3 otimes","display":"Z^{\\otimes3}","kind":"use","excerpt":"[Exact parity refinement] For three qubits, the local computational projectors were grouped by the parity of their Hamming weight. The resulting coarse projectors agreed with (I± Z^⊗3)/2 with residual 0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ac2c1acff83ca55","tex":"Z_{\\mathrm A}","aliases":"Z_ A mathrm","display":"Z_{\\mathrm A}","kind":"use","excerpt":"im. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged. vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shap","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa421f6078ef940d","tex":"Z_{\\mathrm B}","aliases":"Z_ B mathrm","display":"Z_{\\mathrm B}","kind":"use","excerpt":"lihoods by one common positive factor leaves this posterior unchanged. vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2077aebd37a8165","tex":"Z_{(1,2,4)}","aliases":"Z_(1,2,4)","display":"Z_{(1,2,4)}","kind":"use","excerpt":"For weights (1,2,4), the counts are 1,2,8,32,256,2048,…. The first coordinate is active immediately, the second one level later, and the third at depth five. The exact rational function generated by the code is Z_(1,2,4)(T) = fracT(1-6T-8T^2-32T^3)1-8T. The negative signs in this presentation arise from moving the denominator to 1-8T. Every power-s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8394415f576ddd6","tex":"Z_0","aliases":"Z_0","display":"Z_0","kind":"use","excerpt":"Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this d","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6e4bdbd9401f40b","tex":"Z_{0,1}","aliases":"Z_0,1","display":"Z_{0,1}","kind":"use","excerpt":"From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4151409cc12d59b1","tex":"Z_{0,2}","aliases":"Z_0,2","display":"Z_{0,2}","kind":"use","excerpt":"[A transfer series with one loop] Take vertices 0, 1, and 2. Give the edge 0→1 weight one, the loop 1→1 weight two, and the edge 1→2 weight one. Every path from 0 to 2 uses the first and last edges and may traverse the loop any number of times. Hence Z_0,2(t) = t^2(1+t^2+t^4+⋯) = fract^21-t^2. There is one path at every positive even weight and no ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b23ccb12e2adf65f","tex":"Z_1","aliases":"Z_1","display":"Z_1","kind":"use","excerpt":"[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-64a9a143ebce6852","tex":"Z_2","aliases":"Z_2","display":"Z_2","kind":"use","excerpt":",3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ T. vfigintegrated_filter platecaptionWithin-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is (7/13,6/13). headCheck the normalizers, not only the final vector The first and second predictive evidenc","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e32645afb213c74","tex":"z_2","aliases":"z_2","display":"z_2","kind":"use","excerpt":"headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8fa45213091916f0","tex":"\\Z_2^2","aliases":"Z_2^2 Z mathbb","display":"{\\mathbb Z}_2^2","kind":"use","excerpt":"[Odd-adic versus dyadic behavior] For every prime p≠2, every entry of E_+ and E_- is p-integral, and each projector has exact local denominator weight zero. At p=2, each projector has exact local denominator weight one. Multiplication by 2 makes it integral, while the unscaled matrix is not integral: s_2, Z_2^2(E_+)=s_2, Z_2^2(E_-)=1. The character","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-659336bfb5f82bbc","tex":"\\mathbb Z_2^2","aliases":"Z_2^2 mathbb","display":"\\mathbb Z_2^2","kind":"use","excerpt":"[Strict growth under refinement] Take the standard lattice Z_2^2 and the involution from [source reference]. The one packet system (I) has weight zero. Refining it into the two rational sign packets produces weight one. The coarse lattice is already split because there is only one summand. The fine eigenspaces are glued together by one dyadic digit","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7dcc0cb193ddf22f","tex":"\\mathfrak Z_2^{\\mathrm{abs}}","aliases":"Z_2^abs mathfrak mathrm","display":"\\mathfrak Z_2^{\\mathrm{abs}}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f51eabb9e01e23cc","tex":"\\mathfrak Z_2^{\\mathrm{red}}","aliases":"Z_2^red mathfrak mathrm","display":"\\mathfrak Z_2^{\\mathrm{red}}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff87688186f5755b","tex":"\\mathfrak Z_2^{\\rm red}","aliases":"Z_2^rm red mathfrak rm","display":"\\mathfrak Z_2^{\\rm red}","kind":"use","excerpt":"vpage24Convergence is not continuationZERO ZETA FUNCTIONSChapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4. Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_plane For N=2 and N=4, the lattices","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f10cbe5fe4c5d709","tex":"\\mathbb Z_2^\\times","aliases":"Z_2^× mathbb times","display":"\\mathbb Z_2^\\times","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8591c505dfe57b6d","tex":"\\mathbb Z_2^{\\times}","aliases":"Z_2^× mathbb times","display":"\\mathbb Z_2^{\\times}","kind":"use","excerpt":"dd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and Q=( F^2-1)/2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18a2153b72d39af1","tex":"Z_{2m+1}","aliases":"Z_2m+1","display":"Z_{2m+1}","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42fde784682f68de","tex":"Z_{2r+1}","aliases":"Z_2r+1","display":"Z_{2r+1}","kind":"use","excerpt":"[Zigzag inverse kernel] sech^2t = ∑_r≥0 (-1)^rZ_2r+1 fract^2r(2r)!.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2a799d1668e854e4","tex":"Z_3","aliases":"Z_3","display":"Z_3","kind":"use","excerpt":"[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-715faed565a7579b","tex":"\\Z_3","aliases":"Z_3 Z mathbb","display":"{\\mathbb Z}_3","kind":"use","excerpt":":matrix_carmichael_mahler Editorial hypothesis correction. The original displayed statement leaves the vectors untyped. Its integrality conclusion requires v_0,w∈ Z_p^d, L≥1, and H,j,n∈ N with H>0. The value at Hx+j for x∈ Z_p means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bb63c10f3cd1141","tex":"\\mathbb Z_3","aliases":"Z_3 mathbb","display":"\\mathbb Z_3","kind":"use","excerpt":"An actual Z_3 example applies the class-level completion theorem for the cyclic group of order two and polynomial X^2-1, producing a nonidentity generator. Another checked example has compatible classes represented by diag(1,-1) modulo 3 and diag(-1,1) modulo 9: these chosen matrices disagree under reduction, and an actual swap of basis repairs the","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-223d8a54b200d215","tex":"\\mathfrak Z_4^{\\mathrm{abs}}","aliases":"Z_4^abs mathfrak mathrm","display":"\\mathfrak Z_4^{\\mathrm{abs}}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c577fe54775a0551","tex":"\\mathfrak Z_4^{\\mathrm{red}}","aliases":"Z_4^red mathfrak mathrm","display":"\\mathfrak Z_4^{\\mathrm{red}}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20ff6fff48e9be12","tex":"\\mathfrak Z_4^{\\rm red}","aliases":"Z_4^rm red mathfrak rm","display":"\\mathfrak Z_4^{\\rm red}","kind":"use","excerpt":"vpage24Convergence is not continuationZERO ZETA FUNCTIONSChapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4. Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_plane For N=2 and N=4, the lattices","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e584afe354e47377","tex":"\\Z_5","aliases":"Z_5 Z mathbf","display":"{\\mathbf Z}_5","kind":"use","excerpt":"d is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residu","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0d8044d38df3073","tex":"\\Z_{>0}","aliases":"Z_>0 Z mathbb","display":"{\\mathbb Z}_{>0}","kind":"use","excerpt":"A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c90720ad505f551","tex":"Z_a","aliases":"Z_a","display":"Z_a","kind":"use","excerpt":"The evidence in block B_a is Z_a =q_a∑_i∈ B_aL_ir_i^(a) =q_aoverline L_a. Apply [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-315e6e3c40d12791","tex":"z_a","aliases":"z_a","display":"z_a","kind":"use","excerpt":"z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cdd36f9a578aa62d","tex":"z_b","aliases":"z_b","display":"z_b","kind":"use","excerpt":"z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e847049bb4b64a62","tex":"Z_{\\bar\\lambda}","aliases":"Z_barλ bar lambda","display":"Z_{\\bar\\lambda}","kind":"use","excerpt":"[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood. The full exact defect remai","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de03d0b564377608","tex":"Z_c","aliases":"Z_c","display":"Z_c","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ff735e21f188d41","tex":"Z_d","aliases":"Z_d","display":"Z_d","kind":"use","excerpt":"[Smooth seed zeta function] For a smooth formal component of dimension d, Z_d(T)=fracT1-q^dT.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-483b6af63cf90eb6","tex":"Z_e","aliases":"Z_e","display":"Z_e","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f4b4e1abf764838","tex":"Z_f","aliases":"Z_f","display":"Z_f","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10845de283abdc8a","tex":"Z_j","aliases":"Z_j","display":"Z_j","kind":"use","excerpt":"_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae33ff0c9a9cfbb4","tex":"z_j","aliases":"z_j","display":"z_j","kind":"use","excerpt":"For i≠ j, orthogonality gives H(p_i)H(i_j)=0. Idempotence gives H(p_j)H(i_j)=I, and completeness gives ∑_j H(i_j)H(p_j)=I. Thus the maps [z]⟼([P_j^r z])_j, ([z_j])_j⟼[∑_j i_j z_j] are inverse linear maps. The finite product equals the direct sum, proving the assertion. In the formalization, both sides are also identified with Mathlib's standard hom","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-69c97e2f51fdcaeb","tex":"Z_j^r","aliases":"Z_j^r","display":"Z_j^r","kind":"use","excerpt":"For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d533223c5565c514","tex":"Z_k","aliases":"Z_k","display":"Z_k","kind":"use","excerpt":"R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8254078b2941c5fe","tex":"Z_{L,a}","aliases":"Z_L,a","display":"Z_{L,a}","kind":"use","excerpt":"[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42dba447a7ef210c","tex":"z_m","aliases":"z_m","display":"z_m","kind":"use","excerpt":"Compatibility implies, for m≥ n, p^L+nmid z_m-z_n. After casting to Z_p, this bounds each coordinate distance by p^-(L+n), so the sequence is Cauchy. Completeness supplies its limit. Closed norm bounds show that the limit retains the congruence at every stage.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35030b1cc5b33f81","tex":"Z_N","aliases":"Z_N","display":"Z_N","kind":"use","excerpt":"vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSChapter VII: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9fdca6236d286603","tex":"Z_n","aliases":"Z_n","display":"Z_n","kind":"use","excerpt":"Let q be a positive odd integer. Define S_n(q) = ∑_a=0^q-1 (-1)^a(2a+1)^n. Let the signed Euler numbers and Euler zigzag numbers be defined by secht = ∑_n≥0E_nfract^nn! and sec t+tan t = ∑_n≥0Z_nfract^nn!.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bdc1562155501d06","tex":"z_n","aliases":"z_n","display":"z_n","kind":"use","excerpt":"The space X_R is closed. Indeed, if Rz_n→ x, then z_n=DRz_n→ Dx, so x=R(Dx). Define Ψ(x)=R(y-N(x)). This map takes X_R to itself. For x∈overline B_ρ(0)∩ X_R, Ψ(x) · ≤ R(y-N(0)) + R N(x)-N(0)","count":27},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1bb8bb52da2ab42","tex":"z_{n+1}","aliases":"z_n+1","display":"z_{n+1}","kind":"use","excerpt":"[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8004e0be384ebd0","tex":"\\mathfrak Z_N^{\\rm abs}","aliases":"Z_N^rm abs mathfrak rm","display":"\\mathfrak Z_N^{\\rm abs}","kind":"use","excerpt":"k Z_2^rm red(s)=π^-2sζ(2s,1/2), Z_4^rm red(s)=(sqrt2π)^-4sζ(4s,1/2). Each identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at 1/2 or 1/4, with residue 1/(2π) or 1/(4sqrt2π) respectively. headThe factor N matters twice For every order in this chapter, Z_N^rm abs(s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f6be28fa9e4db99c","tex":"\\mathfrak Z_N^{\\rm red}","aliases":"Z_N^rm red mathfrak rm","display":"\\mathfrak Z_N^{\\rm red}","kind":"use","excerpt":"k Z_2^rm red(s)=π^-2sζ(2s,1/2), Z_4^rm red(s)=(sqrt2π)^-4sζ(4s,1/2). Each identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at 1/2 or 1/4, with residue 1/(2π) or 1/(4sqrt2π) respectively. headThe factor N matters twice For every order in this chapter, Z_N^rm abs(s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f3b8fae797e61c1","tex":"\\mathbb Z_p^2","aliases":"Z_p^2 mathbb","display":"\\mathbb Z_p^2","kind":"use","excerpt":"0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1bd159c90606314f","tex":"\\Z_p^d","aliases":"Z_p^d Z mathbb","display":"{\\mathbb Z}_p^d","kind":"use","excerpt":"[Matrix Carmichael and Mahler transfer] Editorial hypothesis correction. The original displayed statement leaves the vectors untyped. Its integrality conclusion requires v_0,w∈ Z_p^d, L≥1, and H,j,n∈ N with H>0. The value at Hx+j for x∈ Z_p means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f72e3ddb158528a6","tex":"\\mathbb Z_p^d","aliases":"Z_p^d mathbb","display":"\\mathbb Z_p^d","kind":"use","excerpt":"Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^d⟶ F_p^e is surjective. Then Lean proves that there is a point a∈ Z_p^d, F(a)=0, a≡ xpmodp^L. Here the coefficients of the original integer polynomials are cast into Z_p, and their multivariable evaluation is exactly zero. This","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1beb3e4d284664d8","tex":"\\mathbb Z_p^e","aliases":"Z_p^e mathbb","display":"\\mathbb Z_p^e","kind":"use","excerpt":"((σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus through σ h(t)=(ϕ(t),t). Every parameter has ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e55fed3bae1ec1c1","tex":"\\mathbb Z_p^{e+k}","aliases":"Z_p^e+k mathbb","display":"\\mathbb Z_p^{e+k}","kind":"use","excerpt":"Square solved-block uniqueness. If m=e, every prescribed τ has exactly one root in the original ball: ∃! b∈ x+p^L Z_p^e+k: F(b)=0, free(b)=τ. Thus all valid integer parameter approximations and all lifting choices with the same target have the same endpoint. This does not assert uniqueness of the integer representatives in the tower. All prime stat","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc8074994f3710c4","tex":"\\mathbb Z_p^{e+q}","aliases":"Z_p^e+q mathbb","display":"\\mathbb Z_p^{e+q}","kind":"use","excerpt":"There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus thro","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8f3102e5e0305d5","tex":"\\mathbb Z_p^k","aliases":"Z_p^k mathbb","display":"\\mathbb Z_p^k","kind":"use","excerpt":"Arbitrary p-adic target theorem. Now let q=p be prime. For every τ∈ t_0+p^L Z_p^k, there are a compatible integer tower z as above and b∈ Z_p^m+k such that z(n)⟶ b, F(b)=0, free(b)=τ, b-z(n)∈ p^L+n Z_p^m+k (n≥0). A compatible integer parameter path converging to τ is constructed in the proof, with its initial representative fixed exactly. A separat","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-308f4c34f6281e55","tex":"\\mathbb Z_p^{m+k}","aliases":"Z_p^m+k mathbb","display":"\\mathbb Z_p^{m+k}","kind":"use","excerpt":"Arbitrary p-adic target theorem. Now let q=p be prime. For every τ∈ t_0+p^L Z_p^k, there are a compatible integer tower z as above and b∈ Z_p^m+k such that z(n)⟶ b, F(b)=0, free(b)=τ, b-z(n)∈ p^L+n Z_p^m+k (n≥0). A compatible integer parameter path converging to τ is constructed in the proof, with its initial representative fixed exactly. A separat","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c466d5fa0ef550c1","tex":"\\mathbb Z_p^q","aliases":"Z_p^q mathbb","display":"\\mathbb Z_p^q","kind":"use","excerpt":"There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus thro","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31d760e606ff3dad","tex":"\\Z_p^\\times","aliases":"Z_p^× Z times mathbb times","display":"{\\mathbb Z}_p^\\times","kind":"use","excerpt":"Why a p-adic unit supplies one factor of p. Now let u∈ Z_p^×. The actual ring homomorphism π: Z_p⟶ F_p sends u to a nonzero unit. Fermat's finite-field theorem therefore gives π(u)^p-1=1, π(u^p-1-1)=0. Using the proved kernel identity kerπ=p Z_p, the source derives pmid u^p-1-1 inside Z_p. The conclusion is not supplied as an abstract certificate. ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a01906ff34cc704","tex":"\\mathbb Z_p^\\times","aliases":"Z_p^× mathbb times","display":"\\mathbb Z_p^\\times","kind":"use","excerpt":"[Scalar polyphase precision amplification] Let u∈ Z_p^×, let a,r≥0, and put H_a= (p-1)p^a, · p odd,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e940a55f95e2b85","tex":"Z_r","aliases":"Z_r","display":"Z_r","kind":"use","excerpt":"The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its deg","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-974e1ebfd0c228a6","tex":"z_r","aliases":"z_r","display":"z_r","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32eb84b8504eeadf","tex":"Z_{ra}","aliases":"Z_ra","display":"Z_{ra}","kind":"use","excerpt":"Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e257ef927f71ba8","tex":"z_{ra}","aliases":"z_ra","display":"z_{ra}","kind":"use","excerpt":"Applied to the corrected Kraus family, this yields scalars z_ra with R_rE_aP=z_raP. The essential declarations are pathrank_one_gram_rigidity, pathkraus_rank_one_rigidity and pathcorrected_kraus_family. No rank-one Choi characterization is inserted as an axiom or as the desired conclusion in hypothesis form.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b1a33765401fd68","tex":"z_{rb}","aliases":"z_rb","display":"z_{rb}","kind":"use","excerpt":"Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this cal","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3480de748bc9723a","tex":"Z_t","aliases":"Z_t","display":"Z_t","kind":"use","excerpt":"^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then ex","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17c7ef854a4ec77f","tex":"z_T","aliases":"z_T","display":"z_T","kind":"use","excerpt":"LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z)≡0pmodq^L+1, z_T≡ t_0+q^L","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f57a3b892b267df","tex":"Z_{\\widetilde\\lambda}","aliases":"Z_widetildeλ widetilde lambda","display":"Z_{\\widetilde\\lambda}","kind":"use","excerpt":"[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-abf0801b6063c7dd","tex":"Z_\\alpha","aliases":"Z_α alpha","display":"Z_\\alpha","kind":"use","excerpt":"Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fee2475676ee8ef5","tex":"Z_\\beta","aliases":"Z_β beta","display":"Z_\\beta","kind":"use","excerpt":"Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e0ffad91ab133eb8","tex":"Z_\\lambda","aliases":"Z_λ lambda","display":"Z_\\lambda","kind":"use","excerpt":"Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.","count":20},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4412880ca0ddeed9","tex":"Z_{\\lambda\\odot\\mu}","aliases":"Z_λodotμ lambda odot mu","display":null,"kind":"use","excerpt":"[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-caf7d1600a9c465c","tex":"Z_\\mu","aliases":"Z_μ mu","display":"Z_\\mu","kind":"use","excerpt":"[Likelihood perturbation bound] Let p be fixed and let λ,μ≥0 have positive evidences Z_λ,Z_μ. Then norm B_λ(p)- B_μ(p)_1 ≤ frac2∑_i|λ_i-μ_i|p_imin(Z_λ,Z_μ).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3c4ce700ec5fc9c","tex":"z_{\\sigma(i)}","aliases":"z_σ(i) sigma","display":"z_{\\sigma(i)}","kind":"use","excerpt":"There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus thro","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd569bf5d0c46f05","tex":"\\Z_{\\ge0}","aliases":"Z_≥0 Z ge mathbf ge","display":"{\\mathbf Z}_{\\ge0}","kind":"use","excerpt":"[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fef1eff057e46518","tex":"\\mathbb Z_{\\ge0}","aliases":"Z_≥0 mathbb ge","display":"\\mathbb Z_{\\ge0}","kind":"use","excerpt":"The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-180c17fa6495e993","tex":"\\mathbb Z_{\\geq0}","aliases":"Z_≥0 mathbb geq","display":"\\mathbb Z_{\\geq0}","kind":"use","excerpt":"[A first-difference criterion] Let C: Z_≥0→ Z_2 have Newton coefficients a_k=Δ_1^kC(0). Suppose for an integer b≥0 that v_2(a_1)=b, v_2(a_k)≥ b+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and H>0, v_2(C(n+H)-C(n))=b+v_2(H). The Newton interpolation extends continuously to Z_2 and satisfies the same identity for any two distinct dyadic arguments.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1834f61514099928","tex":"\\Zcal_R","aliases":"Zcal_R Z_R Zcal mathcal","display":"{\\mathcal Z}_R","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-485214fbaa6bd648","tex":"\\Zcal_R^{\\mathrm{orb}}","aliases":"Zcal_R^orb Z_R^orb Zcal mathrm mathcal mathrm","display":"{\\mathcal Z}_R^{\\mathrm{orb}}","kind":"use","excerpt":"[Orbit reduced nearest circle term] Assume the hypotheses of [source reference]. Choose one representative from each orbit of the nearest zeros under multiplication by ω, and call the resulting set Z_R^orb. With m=Nn, P_N,n(r) = · N m!binomm+r-1r-1 ∑_τ∈ Z_R^orb fracτ^-m(-τ D_N'(τ))^r","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f2f95d10cc66f45","tex":"\\Zp^e","aliases":"Zp^e Z_p^e Zp mathbb","display":"{\\mathbb Z_p}^e","kind":"use","excerpt":"R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is the forward map, then for every N≥0, h(t)≡ h(u)pmodp^N Longleftrightarrow t≡ upmodp^N. Equivalently, σ h(t)=(ϕ(t),t) for a continuous ϕ:T→ y_0+p^L Z_p^e whose graph is the entire reordered root locus.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9351a29d1ff29b9c","tex":"\\Zp^{e+k}","aliases":"Zp^e+k Z_p^e+k Zp mathbb","display":"{\\mathbb Z_p}^{e+k}","kind":"use","excerpt":"Now suppose a chosen set of e columns of DF(x)bmod p is independent, and put those coordinates first. Apply the square result to the augmented polynomial map Ψ(Y,T)=(F(Y,T),T). Its reduced Jacobian has upper blocks (D_YF,D_TF) and lower blocks (0,I_k), so it is invertible. For t∈ T, the target (0,t) belongs to Ψ(x)+p^L Z_p^e+k because F(x)≡0pmodp^L","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3fc502ceef10c9a6","tex":"\\Zp^k","aliases":"Zp^k Z_p^k Zp mathbb","display":"{\\mathbb Z_p}^k","kind":"use","excerpt":"arts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bc1bb97011509d8","tex":"\\Zp^{m+k}","aliases":"Zp^m+k Z_p^m+k Zp mathbb","display":"{\\mathbb Z_p}^{m+k}","kind":"use","excerpt":"[Arbitrary p-adic parameter endpoints] Let q=p be prime in Theorem~[source reference]. For every τ∈ t_0+p^L Z_p^k, there are an integer tower z_n and b∈ Z_p^m+k such that F(b)=0, free(b)=τ, z_n⟶ b, b-z_n∈ p^L+n Z_p^m+k. The initial value is z_0=x. At every stage and for every proposed parameter digit v∈ F_p^k, the corresponding next residue fiber h","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20995e4cd61e272d","tex":"\\Zp^\\times","aliases":"Zp^× Z_p^× Zp times mathbb times","display":"{\\mathbb Z_p}^\\times","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42f5a664e208866e","tex":"\\ZZ^2","aliases":"ZZ^2 ZZ ZZ^2 Z^2 ZZ mathbb","display":null,"kind":"use","excerpt":"vpage15Where the dyadic digit is lostMODEL D / THE EIGENLATTICE HAS INDEX TWOChapter II, mathematical text: Proposition 9.3 The eigenspaces are spanned by (1,0) and (-1,2). Their integral eigenvectors do not form an integral basis of ZZ^2. The change-of-basis determinant is two.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2adfa4292f44f63","tex":"\\ZZ_{(p)}^2","aliases":"ZZ_(p)^2 ZZ","display":null,"kind":"use","excerpt":"0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b5c3968f000169a0","tex":"\\ZZ_2","aliases":"ZZ_2 Z_2 ZZ mathbb","display":"{\\mathbb Z}_2","kind":"use","excerpt":"[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df6ca16455e2bbb5","tex":"\\ZZ_{>0}","aliases":"ZZ_>0 Z_>0 ZZ mathbb","display":"{\\mathbb Z}_{>0}","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ebf82fd8d658554c","tex":"\\ZZ_p^r","aliases":"ZZ_p^r Z_p^r ZZ mathbb","display":"{\\mathbb Z}_p^r","kind":"use","excerpt":"[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6820374f59f0b332","tex":"|^{-1}","aliases":"|^-1","display":"|^{-1}","kind":"use","excerpt":"By [source reference], the order of Δ_N is 1/N<1. By [source reference], it has infinitely many zeros, and [source reference] gives ∑_λ∈Λ_N|λ|^-1<∞. Hadamard factorization therefore gives a canonical product of genus zero. The only possible exponential factor is a nonzero constant because the order is below one. Since Δ_N(0)=1, that constant is one","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6567adae36ef90db","tex":"|^{-2}","aliases":"|^-2","display":"|^{-2}","kind":"use","excerpt":"arepsilon=1/100, the actual increment is 20/9, while derivative times increment is 2. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation. headWhat a proposed weight can and cannot certify Impedance is sensitive near Γ=1, and admittance near Γ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f1a8444914ffb6d","tex":"|^{-q}","aliases":"|^-q","display":"|^{-q}","kind":"use","excerpt":"out added regularizing terms. takeawayA global analytic product theorem connects the infinite zero sum to a finite algebraic recurrence. The recurrence alone does not prove the product theorem. platecaptionDisplayed moments are checked against formal logarithm coefficients and exact order-two/order-four zeta values. Bar widths are numerical shares.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48680b3fa133e34c","tex":"|^{1/N}","aliases":"|^1/N","display":"|^{1/N}","kind":"use","excerpt":"For every N≥2, each nonzero zero λ of Δ_N gives N zeros of D_N, each of modulus |λ|^1/N. Hence Z_N^abs(s) = N Z_N^red(s/N). Substitution gives the second formula with Q_N^abs(s)=NQ_N^red(s/N), which is entire.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-355b68ae43033677","tex":"|^{1/p}","aliases":"|^1/p","display":"|^{1/p}","kind":"use","excerpt":"reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e679c26bc7fd384","tex":"|^{2}","aliases":"|^2","display":"|^{2}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e3225555f03b1a7","tex":"|^3","aliases":"|^3","display":"|^3","kind":"use","excerpt":"Put x=kh/2. The inequalities |sin x|≤|x| and |x-sin x|≤ |x|^3/6, which hold for every real argument, give 0≤ x^2-sin^2x =|x-sin x| |x+sin x| ≤frac|x|^36 2|x| =fracx^43. Multiplying by 4/h^2 proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps k fixed while h→0, so it does not assert a u","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16521de3e48f7c29","tex":"|^{M_L}","aliases":"|^M_L","display":"|^{M_L}","kind":"use","excerpt":"For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute v","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06b81a055469dbd1","tex":"|^p","aliases":"|^p","display":"|^p","kind":"use","excerpt":"Every address has a unique representation r+ℓ. For finite p, x _p^p = ∑_r∈ R∑_ℓ∈ L|x(r+ℓ)|^p = ∑_r∈ R x_r _p^p. The corresponding supremum identity is immediate when p=∞. Unique address representation proves that analysis and interleaving are inverse maps. The projector identities follow from the disjoint coset partition.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f530a3e7890ffba8","tex":"|_2","aliases":"|_2","display":"|_2","kind":"use","excerpt":"[Isometric packet bijections] For 0≤ j<4, define on natural numbers f_j(n)=fracU_j(n)-j!2^d_j, (d_0,d_1,d_2,d_3)=(1,1,5,3). There is a unique continuous map Φ_j: Z_2→ Z_2 whose value at every n∈ N equals the embedded rational f_j(n). It is an isometric bijection and Φ_j(0)=0: |Φ_j(x)-Φ_j(y)|_2=|x-y|_2 (x,y∈ Z_2). In particular, every element of Z_2","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28c001f86de162f7","tex":"|_D","aliases":"|_D","display":"|_D","kind":"use","excerpt":"nit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dbbeed832046c3f7","tex":"|_{E_\\mu}","aliases":"|_E_μ mu","display":"|_{E_\\mu}","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b28103f003acbf0","tex":"|_{H_0}","aliases":"|_H_0","display":"|_{H_0}","kind":"use","excerpt":"⊂ U_A^m+1 lies in the forced domain of the additive character, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and θ|_H_0 is the full character of an actual ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-343b02c221c97d88","tex":"|_K","aliases":"|_K","display":"|_K","kind":"use","excerpt":"[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b555c064a65341d","tex":"|_{\\ker N}","aliases":"|_ker N ker","display":"|_{\\ker N}","kind":"use","excerpt":"These statements concern supplied algebraic pairs. No Weil-group or inertia action, Hodge--Tate object, arithmetic comparison, or realization of the pair from a Galois representation is constructed here. The polynomial convention is det(XI-F|_ker N). No identification with an inertia-dependent local Euler factor is asserted. Claim 9.2 remains parti","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e1af9e562f5ced8","tex":"|_{\\ker N_i}","aliases":"|_ker N_i ker","display":"|_{\\ker N_i}","kind":"use","excerpt":",N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant linear isomorphism ker N_m≃∏_i∈ I∏_k<m_iker N_i. The relation, with q≠0, proves that both F_i and its inverse preserve ker N_i. ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3909971d4ebfd26","tex":"|_{\\ker N_m}","aliases":"|_ker N_m ker","display":"|_{\\ker N_m}","kind":"use","excerpt":"and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant linear isomorphism ker N_m≃∏_i∈ I∏_k<m_iker N_i. The relation, with q≠0, proves that both F_i and its inverse preserve ker N_i. Thus F_i|_ker N_i is an actual automorphism, and Lean pro","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89bc28b8775c3cb7","tex":"|_M","aliases":"|_M","display":"|_M","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97567dd31deee4db","tex":"|_{M_s}","aliases":"|_M_s","display":"|_{M_s}","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3b5ec80fa082444","tex":"|_{n=0}","aliases":"|_n=0","display":"|_{n=0}","kind":"use","excerpt":"A sharp unit-mode gain. For p=5 and u=2, Δ_4^2(u^n)|_n=0=(2^4-1)^2=225. The source proves 5^2mid225 and 5^3nmid225. This example attains the guaranteed gain. No additional uniform power of p is claimed.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4acb3b96be55ab28","tex":"|_P","aliases":"|_P","display":"|_P","kind":"use","excerpt":"Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is an actual e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae8ddb0b385adcc1","tex":"|_p","aliases":"|_p","display":"|_p","kind":"use","excerpt":"Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer ce","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62e4e263d57b6219","tex":"|_{P_\\alpha X}","aliases":"|_P_α X alpha","display":"|_{P_\\alpha X}","kind":"use","excerpt":"thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d27148b42f2a8d0","tex":"|_{QX}","aliases":"|_QX","display":"|_{QX}","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2db86546abe159fd","tex":"|_{\\Ran P}","aliases":"|_Ran P Ran operatorname","display":"|_{{\\operatorname{Ran}} P}","kind":"use","excerpt":"[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5413a578759b7b98","tex":"|_{\\rm nodes}","aliases":"|_rm nodes rm","display":"|_{\\rm nodes}","kind":"use","excerpt":"vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMChapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7dfd6e4dd08043ea","tex":"|_{t=0}","aliases":"|_t=0","display":"|_{t=0}","kind":"use","excerpt":"= · f(x)+t(f''(x)-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b44e5a2c29c5de91","tex":"|_{U_a}","aliases":"|_U_a","display":"|_{U_a}","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab8e308948a772eb","tex":"|_{U_F^1}","aliases":"|_U_F^1","display":"|_{U_F^1}","kind":"use","excerpt":"For this actual stratum there exists a full simple character θ with θ|_U_F^1=1. Retaining its canonical beta extension, the residue character and noncompact extension can be selected so that the resulting irreducible admissible supercuspidal π has any prescribed smooth central quasicharacter ω trivial on U_F^1. These completion choices must be free","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-38c2d53fb86a7cd0","tex":"|_{V_j}","aliases":"|_V_j","display":"|_{V_j}","kind":"use","excerpt":"[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d59a351db3fcbc7f","tex":"|_{V_s}","aliases":"|_V_s","display":"|_{V_s}","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9e57fe2b91672beb","tex":"|_{W_{K_f}}","aliases":"|_W_K_f","display":"|_{W_{K_f}}","kind":"use","excerpt":"[Unramified twist] Let ν be an unramified character of W_K. Then τ_ξ⊗ν ≅ Ind_W_K_f^W_K ( ξ ν|_W_K_f ). The tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar a, the twisted cycle has closing scalar aν(Φ)^f.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9669e5bb7ba8d32a","tex":"|_{X_1}","aliases":"|_X_1","display":"|_{X_1}","kind":"use","excerpt":"[Local Lyapunov--Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there ar","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d249ec98efda9d47","tex":"|_{X_\\alpha}","aliases":"|_X_α alpha","display":"|_{X_\\alpha}","kind":"use","excerpt":"[Universal mapping property of finite packet coordinates] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9599612652da43ca","tex":"|_{z=0}","aliases":"|_z=0","display":"|_{z=0}","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"The failure of the character formula frac12(I± A) to be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished: integral address selector ≠ character-average spectral projector when the group order is not a unit. The first can remain exact over Z_2. ","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62547ed5692a28f7","tex":"\\times_C","aliases":"×_C times","display":"\\times_C","kind":"use","excerpt":"The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0acecd7697ebad7a","tex":"\\times_{\\Compat}","aliases":"×_Compat ×_ C times Compat times mathfrak","display":"\\times_{{\\mathfrak C}}","kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1fb87c81b1a53c3","tex":"\\alpha'","aliases":"α' alpha","display":"\\alpha'","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f7f41ad8028cb36","tex":"\\alpha^{-1}","aliases":"α^-1 alpha","display":"\\alpha^{-1}","kind":"use","excerpt":"(t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d830d9bc7e9c803c","tex":"\\alpha^2","aliases":"α^2 alpha","display":"\\alpha^2","kind":"use","excerpt":"At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace c","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f80ccf4fb60ee01f","tex":"\\alpha^b","aliases":"α^b alpha","display":"\\alpha^b","kind":"use","excerpt":"At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-79705d61c7703121","tex":"\\alpha^p","aliases":"α^p alpha","display":"\\alpha^p","kind":"use","excerpt":"At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59ce46150d0cfe77","tex":"\\alpha_0","aliases":"α_0 alpha","display":"\\alpha_0","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8c617160e3b774f","tex":"\\alpha_2","aliases":"α_2 alpha","display":"\\alpha_2","kind":"use","excerpt":"[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-275621d24bde4c06","tex":"\\alpha_d","aliases":"α_d alpha","display":"\\alpha_d","kind":"use","excerpt":"After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f320d7b77ed8bff0","tex":"\\alpha_i","aliases":"α_i alpha","display":"\\alpha_i","kind":"use","excerpt":"[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29ca9b1f3cc76af1","tex":"\\alpha_i^N","aliases":"α_i^N alpha","display":"\\alpha_i^N","kind":"use","excerpt":"After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a17117e75261dbe","tex":"\\alpha_{i_1}","aliases":"α_i_1 alpha","display":"\\alpha_{i_1}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4056245b0846ec2","tex":"\\alpha_{i_s}","aliases":"α_i_s alpha","display":"\\alpha_{i_s}","kind":"use","excerpt":"For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","excerpt":"[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da4f2068e2750ae4","tex":"\\alpha_m","aliases":"α_m alpha","display":"\\alpha_m","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-20abd9c0a72b9797","tex":"\\alpha_{m-1}","aliases":"α_m-1 alpha","display":"\\alpha_{m-1}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a823753ea2193233","tex":"\\alpha_n","aliases":"α_n alpha","display":"\\alpha_n","kind":"use","excerpt":"[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac909e48d2ea1c12","tex":"\\beta'","aliases":"β' beta","display":"\\beta'","kind":"use","excerpt":"Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1bac3c7ce6eb2507","tex":"\\beta_1","aliases":"β_1 beta","display":"\\beta_1","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-05f6c598f663dcba","tex":"\\beta_2","aliases":"β_2 beta","display":"\\beta_2","kind":"use","excerpt":"Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e259a50189711ceb","tex":"\\beta_3","aliases":"β_3 beta","display":"\\beta_3","kind":"use","excerpt":"Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-208473b4b3730308","tex":"\\beta_4","aliases":"β_4 beta","display":"\\beta_4","kind":"use","excerpt":"Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6cb4d369799123ce","tex":"\\beta_i","aliases":"β_i beta","display":"\\beta_i","kind":"use","excerpt":"For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9fcd3538bb956ab","tex":"\\beta_{i+1}","aliases":"β_i+1 beta","display":"\\beta_{i+1}","kind":"use","excerpt":"For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8dbbe747d8622f2f","tex":"\\beta_j","aliases":"β_j beta","display":"\\beta_j","kind":"use","excerpt":"[Four-address factorial bound] For j∈{0,1,2,3}, n≥0, k≥1, and positive mixed steps, v_2(Δ_H_1⋯Δ_H_kC_j(n)) ≥ k+v_2(k!)+∑_ν=1^k v_2(H_ν)+β_j, (β_0,β_1,β_2,β_3)=(0,0,3,1). Address two also satisfies v_2(Δ_H_1⋯Δ_H_kC_2(n)) ≥ 2k+v_2(k!)+lfloorfrack2rfloor +∑_ν=1^k v_2(H_ν). For the original coefficients, the offsets in [source reference] are (0,0,4,2).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fde937c7afbf21ed","tex":"\\beta_{m-1}","aliases":"β_m-1 beta","display":"\\beta_{m-1}","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e24d33779c12c52","tex":"\\beta_r","aliases":"β_r beta","display":"\\beta_r","kind":"use","excerpt":"The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the b","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5cc53b570750c9f6","tex":"\\beta_{r+q}","aliases":"β_r+q beta","display":"\\beta_{r+q}","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36143f667466fe6f","tex":"\\beta_r^2","aliases":"β_r^2 beta","display":"\\beta_r^2","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f42d1a795468f2ce","tex":"\\gamma'","aliases":"γ' gamma","display":"\\gamma'","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0fa6ec7a5cf1dd0a","tex":"\\Gamma^G","aliases":"Γ^G Gamma","display":"\\Gamma^G","kind":"use","excerpt":"Substitute the definition of Γ^G. Adjacent factors G_b^-1G_b and G_c^-1G_c cancel. In the defect, both the composite and direct terms acquire the same outer factors G_c and G_a^-1. In the loop, all inner factors cancel and only conjugation at the base remains.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e032ce94ae2fca8c","tex":"\\Gamma^{X_q}_{b\\leftarrow a}","aliases":"Γ^X_q_b← a Gamma leftarrow","display":"\\Gamma^{X_q}_{b\\leftarrow a}","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f20fcd760ea44277","tex":"\\Gamma^Y_{b\\leftarrow a}","aliases":"Γ^Y_b← a Gamma leftarrow","display":"\\Gamma^Y_{b\\leftarrow a}","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0717c41ef9dfce54","tex":"\\Gamma_{0\\leftarrow2}","aliases":"Γ_0←2 Gamma leftarrow","display":"\\Gamma_{0\\leftarrow2}","kind":"use","excerpt":"eal hypothesisMODEL D / THE MINIMAL AND REVERSIBLE INTERFACESChapter III, mathematical text: Definition 5.1, Definition 5.5 Choose the forward scalar transports Γ_1←0=2, Γ_2←1=3, and Γ_2←0=5. Their multiplicative comparison between transport through two steps and direct transport is D^×=6/5. A loop also needs a declared return map Γ_0←2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2e737aa54abf669","tex":"\\gamma_1","aliases":"γ_1 gamma","display":"\\gamma_1","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4938aee8fada09d6","tex":"\\Gamma_{1\\leftarrow0}","aliases":"Γ_1←0 Gamma leftarrow","display":"\\Gamma_{1\\leftarrow0}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f08cea1939265ee","tex":"\\gamma_2","aliases":"γ_2 gamma","display":"\\gamma_2","kind":"use","excerpt":"[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic ","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42316594f86819a5","tex":"\\gamma_{2k}","aliases":"γ_2k gamma","display":"\\gamma_{2k}","kind":"use","excerpt":"wers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2k(X)+ϵγ_2k-1(X) pmod2 A_0. The m","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d502518f858f0a9b","tex":"\\gamma_{2k-1}","aliases":"γ_2k-1 gamma","display":"\\gamma_{2k-1}","kind":"use","excerpt":"wers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2k(X)+ϵγ_2k-1(X) pmod2 A_0. The m","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-440e88e8288fd6e5","tex":"\\gamma_{2m}","aliases":"γ_2m gamma","display":"\\gamma_{2m}","kind":"use","excerpt":"Every displayed component of H_0 has all its divided powers in A_0. Besides the generators, the needed identities are γ_m(P^2/2) · =frac(2m)!2^m m!γ_2m(P),nonumber γ_m(3PQ) · =3^m m!γ_m(P)γ_m(Q),nonumber γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5133d0d16eff7ae2","tex":"\\gamma_{2r}","aliases":"γ_2r gamma","display":"\\gamma_{2r}","kind":"use","excerpt":":four-euler-H gives H_0=γ_2(X)+ϵ X+2G, G=-γ_2(P)-γ_2(Q)+PQ-τ Q, ϵ^2=2(τ-P-Q-2PQ). The divided powers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes mo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe10d9ec3f095a8c","tex":"\\Gamma_{2\\leftarrow0}","aliases":"Γ_2←0 Gamma leftarrow","display":"\\Gamma_{2\\leftarrow0}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f144291fed6876b9","tex":"\\Gamma_{2\\leftarrow1}","aliases":"Γ_2←1 Gamma leftarrow","display":"\\Gamma_{2\\leftarrow1}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-276414e29824c2c0","tex":"\\Gamma_{a_0\\leftarrow a_0}","aliases":"Γ_a_0← a_0 Gamma leftarrow","display":"\\Gamma_{a_0\\leftarrow a_0}","kind":"use","excerpt":"Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43ff3a694b57363f","tex":"\\Gamma_{a_{n+1}\\leftarrow a_0}","aliases":"Γ_a_n+1← a_0 Gamma leftarrow","display":"\\Gamma_{a_{n+1}\\leftarrow a_0}","kind":"use","excerpt":"Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5fdaec02b44f0b20","tex":"\\Gamma_{a_{n+1}\\leftarrow a_n}","aliases":"Γ_a_n+1← a_n Gamma leftarrow","display":"\\Gamma_{a_{n+1}\\leftarrow a_n}","kind":"use","excerpt":"Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1fd10a8f613b16db","tex":"\\Gamma_{a_n\\leftarrow a_0}","aliases":"Γ_a_n← a_0 Gamma leftarrow","display":"\\Gamma_{a_n\\leftarrow a_0}","kind":"use","excerpt":"[Flat pair transport collapses to endpoints] If Γ is flat, then every path satisfies Γ_p=Γ_a_n← a_0. In particular, every loop has identity holonomy. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport_eq_direct_of_flat; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport_eq_direct_of_flat; Komorebi.Paper","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8146a29b9b509653","tex":"\\Gamma_{a\\leftarrow o}","aliases":"Γ_a← o Gamma leftarrow","display":"\\Gamma_{a\\leftarrow o}","kind":"use","excerpt":"[Flat connections are gauges of the canonical pair transport] Suppose the C_a are exact packet coordinate modules for one ambient module, so the canonical J_b← a is available. If Γ is a flat normalized reversible pair connection, choose o∈ B and set G_a=Γ_a← oJ_o← a. Then every pair transport satisfies Γ_b← a =G_bJ_b← aG_a^-1. statusleanthreeKomore","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27ca25276af08346","tex":"\\Gamma_b","aliases":"Γ_b Gamma","display":"\\Gamma_b","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a70b0f3e648354e","tex":"\\Gamma_{b\\leftarrow o}","aliases":"Γ_b← o Gamma leftarrow","display":"\\Gamma_{b\\leftarrow o}","kind":"use","excerpt":"Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a · = Γ_b← oΓ_o← a · = Γ_b← a. The first reduction uses the canonical cocycle twice. The second uses flatness of Γ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1987d8d2f743de43","tex":"\\Gamma_{c\\leftarrow a}^{-1}","aliases":"Γ_c← a^-1 Gamma leftarrow","display":"\\Gamma_{c\\leftarrow a}^{-1}","kind":"use","excerpt":"At a triple (a,b,c), the compiled cocycle predicate is Γ_c← bΓ_b← a =Γ_c← a. Flatness means this equality holds for every triple. The formal multiplicative relative defect and triangle holonomy are D^×_c,b,a = (Γ_c← bΓ_b← a) Γ_c← a^-1, H_c,b,a = Γ_a← c Γ_c← b Γ_b← a. The defect is based at the target c. The holonomy is based at a. statusleanthreeKo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5093f2875f491c34","tex":"\\Gamma_e","aliases":"Γ_e Gamma","display":"\\Gamma_e","kind":"use","excerpt":"[A connection on a chart graph] One may specify Γ_e only on the oriented edges of a chart graph, together with inverse maps on reversed edges. Path transport and loop holonomy below still make sense. A direct cocycle defect for (a,b,c), however, requires a declared direct transport from a to c. The pairwise API in [source reference] keeps this comp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae2e30306bc89e00","tex":"\\gamma_i","aliases":"γ_i gamma","display":"\\gamma_i","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51cadb8120940875","tex":"\\gamma_j","aliases":"γ_j gamma","display":"\\gamma_j","kind":"use","excerpt":"compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8335fc4d7b413b19","tex":"\\Gamma_k","aliases":"Γ_k Gamma","display":"\\Gamma_k","kind":"use","excerpt":"A NONLINEAR PHASE ORBITChapter II, mathematical text: Section 12.3, Smith-phase identity (p. 43) Smith normal form concerns linear congruences. The Smith impedance chart uses the nonlinear coordinate change Γ=fracz-1z+1, z=frac1+Γ1-Γ. They share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points Γ_k=tf","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-530f5d7a3b27e60f","tex":"\\gamma_k","aliases":"γ_k gamma","display":"\\gamma_k","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3374379bd0bfada4","tex":"\\gamma_{k-1}","aliases":"γ_k-1 gamma","display":"\\gamma_{k-1}","kind":"use","excerpt":"For k=2^r≥4, put q=k/2, which is even. Normalize the exact expansion [source reference] by 2^q before reducing modulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29b4335be2c3e508","tex":"\\gamma_{k-r}","aliases":"γ_k-r gamma","display":"\\gamma_{k-r}","kind":"use","excerpt":"γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the third identity by (i E F)^m and reduce the Euler powers using the integral quadratic relations. Finally, γ_k(X+Y)=∑_r=0^kγ_r(X)γ_k-r(Y) proves γ_k(H_0)∈ A_0. Applying the functional and putting W=U^4-1=2H","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45d8bf1bf1c60baa","tex":"\\gamma_{m-1}","aliases":"γ_m-1 gamma","display":"\\gamma_{m-1}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-afb8535845a78365","tex":"\\gamma_{m-r}","aliases":"γ_m-r gamma","display":"\\gamma_{m-r}","kind":"use","excerpt":"Every displayed component of H_0 has all its divided powers in A_0. Besides the generators, the needed identities are γ_m(P^2/2) · =frac(2m)!2^m m!γ_2m(P),nonumber γ_m(3PQ) · =3^m m!γ_m(P)γ_m(Q),nonumber γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b118facef29f432","tex":"\\gamma_N","aliases":"γ_N gamma","display":"\\gamma_N","kind":"use","excerpt":"The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf09daa6ed346f44","tex":"\\gamma_{n-q}","aliases":"γ_n-q gamma","display":"\\gamma_{n-q}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d14549cec41f6edf","tex":"\\Gamma_{o\\leftarrow a}","aliases":"Γ_o← a Gamma leftarrow","display":"\\Gamma_{o\\leftarrow a}","kind":"use","excerpt":"Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e52bdbf6bbe4050","tex":"\\gamma_p","aliases":"γ_p gamma","display":"\\gamma_p","kind":"use","excerpt":"The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c86d4a426c471d1","tex":"\\Gamma_R","aliases":"Γ_R Gamma","display":"\\Gamma_R","kind":"use","excerpt":"Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions. No completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source referen","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-788db25068d804f5","tex":"\\gamma_r","aliases":"γ_r gamma","display":"\\gamma_r","kind":"use","excerpt":"Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions. No completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source referen","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8df3dddd03938b61","tex":"\\gamma_u","aliases":"γ_u gamma","display":"\\gamma_u","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7db9f39168dc2833","tex":"\\gamma_v","aliases":"γ_v gamma","display":"\\gamma_v","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27e33acc1ffe034a","tex":"\\gamma_v^{-1}","aliases":"γ_v^-1 gamma","display":"\\gamma_v^{-1}","kind":"use","excerpt":"The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c57377872c9fe8eb","tex":"\\Gamma_y","aliases":"Γ_y Gamma","display":"\\Gamma_y","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d99b498a81da6362","tex":"\\Delta'","aliases":"Δ' Delta","display":"\\Delta'","kind":"use","excerpt":"[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f0f0e9cd378ed81","tex":"\\Delta^2","aliases":"Δ^2 Delta","display":"\\Delta^2","kind":"use","excerpt":"vpage39Higher differences annihilate polynomial interiorsBOUNDARY STENCILS / SIGNAL RESPONSE AND NOISE GAINChapter V: Theorem E10.6, Proposition E10.7, Corollary E10.8. Let x(n)=n+2(n-3)_+ on the integer line. Its slope changes from one to three at the breakpoint 3. The second forward difference is Δ^2x(n)=x(n+2)-2x(n+1)+x(n). vfig[.87]ext_differen","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"use","excerpt":"[A perturbed disk] Let E and F be concentric disks of radii r and r+δ in R^2, with δ>0. Then μ(Emathbintriangle F) = π((r+δ)^2-r^2) = 2π rδ+πδ^2. For equal amplitude a, the exact L^2 packet error is |a|sqrt2π rδ+πδ^2. This gives the expected square root dependence on boundary displacement.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02c68d637b81b90c","tex":"\\delta^3","aliases":"δ^3 delta","display":"\\delta^3","kind":"use","excerpt":"y on the interval between its two locations: H_τ-H_widehatτ _2=sqrtδ=tfrac14, H_τ-H_widehatτ _∞=1. vfig[.89]ext_location headFor a ramp, the error height shrinks too Write r_τ(t)=(t-τ)_+. The difference grows linearly from zero to δ on [τ,widehatτ] and stays at δ on [widehatτ,1]. Direct integration gives r_τ-r_widehatτ _2^2 =fracδ^33+(1-widehatτ)δ^","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11319bcf2c25f823","tex":"\\Delta^{d+1}","aliases":"Δ^d+1 Delta","display":"\\Delta^{d+1}","kind":"use","excerpt":"[Polynomial annihilation and boundary support] Let x be a sequence that agrees with a polynomial of degree at most d on each interval of a finite partition of Z. Then Δ^d+1x(n)=0 whenever the complete stencil {n,n+1,…,n+d+1} lies inside one polynomial piece. Hence Δ^d+1x is supported in a band of at most d+1 addresses before each breakpoint.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2930293fbf06791f","tex":"\\Delta^G_{c,b,a}","aliases":"Δ^G_c,b,a Delta","display":"\\Delta^G_{c,b,a}","kind":"use","excerpt":"[Gauge covariance] The defect and triangle holonomy transform as Δ^G_c,b,a =G_cΔ_c,b,aG_a^-1, H^G_c,b,a =G_aH_c,b,aG_a^-1. Thus flatness and triviality of triangle holonomy are gauge invariant. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a741ae82704c654","tex":"\\Delta^k","aliases":"Δ^k Delta","display":"\\Delta^k","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75ac6af28af33827","tex":"\\Delta^m","aliases":"Δ^m Delta","display":"\\Delta^m","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0d6cdab24a850778","tex":"\\Delta^r","aliases":"Δ^r Delta","display":"\\Delta^r","kind":"use","excerpt":"[Finite Mahler reconstruction] Given f(0),…,f(N), put c_r=Δ^rf(0). Then f(n)=∑_r=0^nc_rbinom nr (0≤ n≤ N). For every s with 0≤ s≤ N, Δ^sf(n) = ∑_r=s^Nc_rbinom nr-s (0≤ n≤ N-s).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa50885d0e9944c0","tex":"\\Delta^s","aliases":"Δ^s Delta","display":"\\Delta^s","kind":"use","excerpt":"[Finite Mahler reconstruction] Given f(0),…,f(N), put c_r=Δ^rf(0). Then f(n)=∑_r=0^nc_rbinom nr (0≤ n≤ N). For every s with 0≤ s≤ N, Δ^sf(n) = ∑_r=s^Nc_rbinom nr-s (0≤ n≤ N-s).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e7752e939ff77e03","tex":"\\delta^s","aliases":"δ^s delta","display":"\\delta^s","kind":"use","excerpt":"0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dde6e4dc1d6569ac","tex":"\\delta^{s-1}","aliases":"δ^s-1 delta","display":"\\delta^{s-1}","kind":"use","excerpt":"pageheading3. A nonsingular Euler-operator argumentThe standard interval without an analytic power-series-in-the-index premise On Hurwitz coefficients, the actual Euler operator plus c is (Λ_c a)(d)=(d+c)a(d). Direct induction proves its commutator with every defect iterate: δ^sΛ_c a =Λ_cδ^s a +sp(pU^p^2-U^p)δ^s-1a, s≥1. Suppose a is integral and s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ba59a603a1fa1c4","tex":"\\Delta_{(p-1)p^a}^{\\,r}","aliases":"Δ_(p-1)p^a^ r Delta","display":"\\Delta_{(p-1)p^a}^{\\,r}","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9adbb02c2ad54bb0","tex":"\\delta_1","aliases":"δ_1 delta","display":"\\delta_1","kind":"use","excerpt":"N IMPULSE MAKES THE DIFFERENCE VISIBLEChapter V: Theorem 4.1. A one-sample delay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is J(D), not DI_2. vfigsample_delay headTrace one impulse The input δ_0 has a single nonzero value in even channel address zero. Under J(D) it appears in odd channel address zero: t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49a63bc6da6e2534","tex":"\\Delta_{12\\cdot2^a}^{\\,r}","aliases":"Δ_12·2^a^ r Delta cdot","display":"\\Delta_{12\\cdot2^a}^{\\,r}","kind":"use","excerpt":"18 · 11 . Therefore Δ_12·2^a^ rF_n ≡0 pmod 2^(a+3)r. Every dyadic refinement doubles the step and adds one guaranteed binary digit at each Mahler depth.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae0d0e3fccff5d10","tex":"\\Delta_{13}","aliases":"Δ_13 Delta","display":"\\Delta_{13}","kind":"use","excerpt":"0 · 1 · 0. The generator defect is Δ=C_η A-BC_η, with only one nonzero entry, Δ_13=7η (matrix indices begin at one). vfigext_substitution platecaptionThe perturbation lets the discarded eigenvalue enter the comparison. Exact naturality holds at η=0. A nonzero defect is transformed, not erased, by the rational calculus. headA resolvent identity supp","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-954f8b3938ce3865","tex":"\\Delta_1^2","aliases":"Δ_1^2 Delta","display":"\\Delta_1^2","kind":"use","excerpt":"pageheading5. Examples, verification and the remaining gapThe coverage count is unchanged; the unresolved theorem now has a stronger proved part The original rational recurrence gives C_6,2(1)=-frac128, C_6,2(2)=frac425364, C_6,2(3)=-frac20397592660. Consequently, the new source proves Δ_1^2 C_6,2(1)=-frac9499591235, v_3 (Δ_1^2 C_6,2(1))=2. This at","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54b865827d520271","tex":"\\Delta_1^i","aliases":"Δ_1^i Delta","display":"\\Delta_1^i","kind":"use","excerpt":"g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors interchanged to ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0b2aaddf443a493a","tex":"\\Delta_1^k","aliases":"Δ_1^k Delta","display":"\\Delta_1^k","kind":"use","excerpt":"affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its t","count":27},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c183ffb2b31e982","tex":"\\Delta_1^{k-r}","aliases":"Δ_1^k-r Delta","display":"\\Delta_1^{k-r}","kind":"use","excerpt":"ence identity is Δ_1^kfrac11+sn =frac(-s)^k k!∏_h=0^k(1+s(n+h)). It follows by induction, subtracting adjacent rational functions. Each successive subtraction adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then prov","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8164837cba51cda1","tex":"\\Delta_1^m","aliases":"Δ_1^m Delta","display":"\\Delta_1^m","kind":"use","excerpt":"To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar mult","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31c8867e60f94613","tex":"\\Delta_1^r","aliases":"Δ_1^r Delta","display":"\\Delta_1^r","kind":"use","excerpt":"For f: N→ R, define the step-h forward difference and its iterates by Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f=f, and put c_r(f)=Δ_1^r f(0). Thus the coefficients are determined by the differences of f.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c4d86df28afc0cb","tex":"\\Delta_1^{s-i}","aliases":"Δ_1^s-i Delta","display":"\\Delta_1^{s-i}","kind":"use","excerpt":"g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors interchanged to ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c0b4ab8dd399bba","tex":"\\Delta_2","aliases":"Δ_2 Delta","display":"\\Delta_2","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dac3f880e6e3670a","tex":"\\delta_2","aliases":"δ_2 delta","display":"\\delta_2","kind":"use","excerpt":"ay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is J(D), not DI_2. vfigsample_delay headTrace one impulse The input δ_0 has a single nonzero value in even channel address zero. Under J(D) it appears in odd channel address zero: this is the original sample δ_1. Under DI_2 it stays even but moves to channel ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ff4ad863fb9c187","tex":"\\Delta_{2,1,0}","aliases":"Δ_2,1,0 Delta","display":"\\Delta_{2,1,0}","kind":"use","excerpt":"[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly w","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8f89ea0d8a9c3b5","tex":"\\Delta_{24}^{\\,r}","aliases":"Δ_24^ r Delta","display":"\\Delta_{24}^{\\,r}","kind":"use","excerpt":"1932 · 1194 . For every scalar coordinate of the state sequence C^nv_0, Δ_24^ ra_n ≡0 pmod 24^r. One integral jump therefore gives simultaneous congruence ladders at the primes two and three.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-121420d9f9749674","tex":"\\Delta_{2^{a+1}}^{\\,r}","aliases":"Δ_2^a+1^ r Delta","display":"\\Delta_{2^{a+1}}^{\\,r}","kind":"use","excerpt":"[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-198bb3bf1a420a2e","tex":"\\Delta_4","aliases":"Δ_4 Delta","display":"\\Delta_4","kind":"use","excerpt":"vpage8Four zeros become one orbit zeroTHE REDUCED VARIABLEChapter IV: Definition 5.7, Example 11.12, Corollary 11.13. For N=4 the factorization is exact: D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), D_4(z)=Δ_4(z^4). Set A_4=sqrt2π. All the zeros are A_4(m+1/2)i^k, with m≥0 and 0≤ k<4. Each zero is simple. vfigorbit_reduction The drawing uses u=z/A_4 and v=w/A","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63266e1b2042ea9c","tex":"\\Delta_4^{-1}","aliases":"Δ_4^-1 Delta","display":"\\Delta_4^{-1}","kind":"use","excerpt":"tion: the number 69COEFFICIENT AND ZERO WORKED MODELChapter IV: Corollary 7.7, Example 11.12, Theorem 12.6. Use N=4 throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order. vfigworked_circuit headRoute one: invert the d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-397720baa101f32e","tex":"\\Delta_4^2","aliases":"Δ_4^2 Delta","display":"\\Delta_4^2","kind":"use","excerpt":"A sharp unit-mode gain. For p=5 and u=2, Δ_4^2(u^n)|_n=0=(2^4-1)^2=225. The source proves 5^2mid225 and 5^3nmid225. This example attains the guaranteed gain. No additional uniform power of p is claimed.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ab8756dcff46892","tex":"\\Delta_4^s","aliases":"Δ_4^s Delta","display":"\\Delta_4^s","kind":"use","excerpt":"Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","excerpt":"The translation orbit is the standard coordinate family (δ_a)_a∈ Z^d. Its finite span is the space of finitely supported sequences. Finitely supported sequences are dense in ℓ^p for finite p and in c_0. Their closure in the supremum norm is c_0, which is a proper closed subspace of ℓ^∞.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1525b57dc165eb8a","tex":"\\delta_{ab}","aliases":"δ_ab delta","display":"\\delta_{ab}","kind":"use","excerpt":"The Knill--Laflamme compressions are PE_a^*E_bP=δ_abP. Their scalar form says that the error labels do not resolve the encoded vector. The exact checks evaluate all sixteen compressions, all sixteen syndrome products, the four probabilities, the recovery normalization, and the full recovered matrix.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3d61188e8e97282","tex":"\\delta_{ac}","aliases":"δ_ac delta","display":"\\delta_{ac}","kind":"use","excerpt":"The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d134e4ce3afe800","tex":"\\delta_{ba}","aliases":"δ_ba delta","display":"\\delta_{ba}","kind":"use","excerpt":"Let U=U_mU_m-1⋯ U_1 be a finite circuit with m≥1, and let (P_a)_a∈ I be a complete orthogonal packet family. For the empty circuit, U=I and its transfer block is P_bP_a=δ_baP_a, whose endpoint sum reconstructs I.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1fdf516d82b6962","tex":"\\delta_{bd}","aliases":"δ_bd delta","display":"\\delta_{bd}","kind":"use","excerpt":"[Packet unitarity identities] If U is unitary and U_ba=P_bUP_a, then ∑_bU_ba^*U_bc=δ_acP_a and ∑_aU_baU_da^*=δ_bdP_b.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f07d918fcbb9be05","tex":"\\delta_{d0}","aliases":"δ_d0 delta","display":"\\delta_{d0}","kind":"use","excerpt":"The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5b085f18c150ac1","tex":"\\Delta_e","aliases":"Δ_e Delta","display":"\\Delta_e","kind":"use","excerpt":"Packet perturbation certificates. textitSource item 8.1 and two uniform consequences Let X be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps X→ X, with their operator norm. Let a(e) identify the source packet of edge e, and suppose Δ_e=Δ_e P_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exac","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ea7bc2e5a161c9f5","tex":"\\delta_E","aliases":"δ_E delta","display":"\\delta_E","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-37d4cee3e21267da","tex":"\\Delta_G","aliases":"Δ_G Delta","display":"\\Delta_G","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96bdfbbc4387b77f","tex":"\\Delta_{GF}","aliases":"Δ_GF Delta","display":"\\Delta_{GF}","kind":"use","excerpt":"[Linear product rule] If all maps in [source reference] are linear, then Δ_GF = Δ_G F_f+G_cΔ_F.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23f92b1c07ee4536","tex":"\\Delta_{G\\circ F}","aliases":"Δ_G∘ F Delta circ","display":"\\Delta_{G\\circ F}","kind":"use","excerpt":"[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28f5ad27d0de9148","tex":"\\Delta_H","aliases":"Δ_H Delta","display":"\\Delta_H","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7eb12ace4621a9d6","tex":"\\Delta_H^k","aliases":"Δ_H^k Delta","display":"\\Delta_H^k","kind":"use","excerpt":"[Threshold-free dyadic congruence] For N=2^a, a≥0, all permitted j, and n≥0, v_2(Δ_H_1⋯Δ_H_kE_Nn^(N,j)) ≥ v_2(j!)+k+∑_ν=1^k v_2(H_ν). In particular, Δ_H^k gains k(1+v_2(H)) bits before normalization. For r>v_2(j!), the row displacement H=2^max(0,r-v_2(j!)-1) is a sufficient period modulo 2^r from row zero. If r≤ v_2(j!), all original coefficients v","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d95b94f6578b45a","tex":"\\Delta_h^{k-mn}","aliases":"Δ_h^k-mn Delta","display":"\\Delta_h^{k-mn}","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbac75fb10eeb11c","tex":"\\Delta_h^{mn}","aliases":"Δ_h^mn Delta","display":"\\Delta_h^{mn}","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa7d5fad49c50597","tex":"\\Delta_H^r","aliases":"Δ_H^r Delta","display":"\\Delta_H^r","kind":"use","excerpt":"0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^TC^jB^rv_0. The series converges on Z_p, and Δ_H^ra_n ≡0 pmod p^L","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03509f28204331cb","tex":"\\Delta_h^s","aliases":"Δ_h^s Delta","display":"\\Delta_h^s","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6f17fac236c6322","tex":"\\Delta_{H_1}","aliases":"Δ_H_1 Delta","display":"\\Delta_{H_1}","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8305e56092369ef0","tex":"\\Delta_{H_a}^{\\,r}","aliases":"Δ_H_a^ r Delta","display":"\\Delta_{H_a}^{\\,r}","kind":"use","excerpt":"2^a+1, · p=2. Then Δ_H_a^ ru^n = u^n(u^H_a-1)^r. For odd p, v_p(Δ_H_a^ ru^n) ≥(a+1)r. For p=2 and odd u, v_2(Δ_H_a^ ru^n) ≥(a+3)r. Equivalently, every additional prime power digit increases the certified Mahler slope by one.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9995779c4ace24a","tex":"\\Delta_{H_k}","aliases":"Δ_H_k Delta","display":"\\Delta_{H_k}","kind":"use","excerpt":"Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempt","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a104cc9eebe89629","tex":"\\delta_i","aliases":"δ_i delta","display":"\\delta_i","kind":"use","excerpt":"Markov · point mass δ_i · K^n or e^tQ · state lumping and CK^n=bar K^nC","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5a1bc4b1c0866ff","tex":"\\Delta_{j,e}","aliases":"Δ_j,e Delta","display":"\\Delta_{j,e}","kind":"use","excerpt":"[Mixed precision product chain] If each fixed linear factor A_j is replaced by a fixed packetwise linear approximation widehat A_j, write E_j=widehat A_j-A_j=∑_eΔ_j,eP_α(j,e). A uniform certificate is E_j ≤δ_j:=∑_e Δ_j,e P_α(j,e) . For orthogonal packet projectors, δ_j≤∑_e Δ_j,e . The final operator error is bounded by [source reference] with these","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a6738581aa5bb8f","tex":"\\delta_{ji}","aliases":"δ_ji delta","display":"\\delta_{ji}","kind":"use","excerpt":"pansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e4e08a5e917de66","tex":"\\delta_{jk}","aliases":"δ_jk delta","display":"\\delta_{jk}","kind":"use","excerpt":"[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50baa2ac2f493094","tex":"\\delta_{j\\ell}","aliases":"δ_jℓ delta ell","display":"\\delta_{j\\ell}","kind":"use","excerpt":"For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a991337b8d11cca","tex":"\\Delta_K","aliases":"Δ_K Delta","display":"\\Delta_K","kind":"use","excerpt":"Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl(","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25ea098ca9f371ad","tex":"\\delta_{kj}","aliases":"δ_kj delta","display":"\\delta_{kj}","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd623f3351ab6025","tex":"\\Delta_L","aliases":"Δ_L Delta","display":"\\Delta_L","kind":"use","excerpt":"[Fractional defect bound] Let 0<α<1. Suppose L_f≥ mI, L_c≥ mI for some m>0. Define Δ_L=CL_f-L_cC. Then normCL_f^α-L_c^α C ≤ α m^α-1normΔ_L. Consequently, norm C e^-tL_f^α - e^-tL_c^αC ≤ α t m^α-1normΔ_L.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12735990a456db58","tex":"\\Delta_L^s","aliases":"Δ_L^s Delta","display":"\\Delta_L^s","kind":"use","excerpt":"0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd0aa14ad6313087","tex":"\\Delta_m","aliases":"Δ_m Delta","display":"\\Delta_m","kind":"use","excerpt":"[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23fe82c4424f67f6","tex":"\\delta_m","aliases":"δ_m delta","display":"\\delta_m","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3fe3fecf4b7ed4b","tex":"\\delta_m^{(1)}","aliases":"δ_m^(1) delta","display":"\\delta_m^{(1)}","kind":"use","excerpt":"cos(sqrt3t/2)3. Divide by the nonzero factor 2e^t/2/3. The resulting equation separates the dominant cosine from the small third exponential. vfigtwo_term Ignoring the last term gives model zeros t_m=frac2πsqrt3(m+1/2). At t_m, the full normalized expression equals tfrac12e^-3t_m/2>0, so t_m is not an exact zero. Linearizing the cosine at that poin","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb11ddfc4a29b9f4","tex":"\\delta_m^2","aliases":"δ_m^2 delta","display":"\\delta_m^2","kind":"use","excerpt":"Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8b18ae884be8df2","tex":"\\delta_{mn}","aliases":"δ_mn delta","display":"\\delta_{mn}","kind":"use","excerpt":"· = ∑_n=0^∞ e^-ntH_n(x)fracu^nn!. Comparison of coefficients proves the eigenvalue formula. The same generating function shows ∫ H_mH_n dγ=n!δ_mn. The Hermite polynomials span all polynomials, and polynomials are dense in L^2(γ). Hence the normalized family is an orthonormal basis. See [source reference] for the general Gaussian Hilbert space setti","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f25337fa5b300330","tex":"\\Delta_N'","aliases":"Δ_N' Delta","display":"\\Delta_N'","kind":"use","excerpt":"Since D_N(z)=Δ_N(z^N), -frac D_N'(z) D_N(z) = -Nz^N-1fracΔ_N'(z^N)Δ_N(z^N). Differentiate the defining series for -logΔ_N and compare coefficients.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-884e94abba1033ea","tex":"\\delta_{n,r}","aliases":"δ_n,r delta","display":"\\delta_{n,r}","kind":"use","excerpt":"Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer bloc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-addee2c936e25fa1","tex":"\\Delta_{n-1}","aliases":"Δ_n-1 Delta","display":"\\Delta_{n-1}","kind":"use","excerpt":"Let Δ_n-1 = {p∈ R^n: p_i≥0, ∑_ip_i=1 } be the probability simplex. Let λ∈[0,∞)^n have positive evidence Z_λ(p)=∑_iλ_ip_i. The Bayesian transfer is B_λ(p) = fracλodot pZ_λ(p).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb6d1253c89c84b1","tex":"\\delta_{nm}","aliases":"δ_nm delta","display":"\\delta_{nm}","kind":"use","excerpt":"[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e3ad7ffccce236c","tex":"\\Delta_{p(p-1)}^k","aliases":"Δ_p(p-1)^k Delta","display":"\\Delta_{p(p-1)}^k","kind":"use","excerpt":"Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e538ed02a7b65b29","tex":"\\Delta_{p-1}^s","aliases":"Δ_p-1^s Delta","display":"\\Delta_{p-1}^s","kind":"use","excerpt":"[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fd9e8f5f34bd8907","tex":"\\Delta_{\\Pcal}","aliases":"Δ_Pcal Δ_ P Delta Pcal Delta mathcal","display":"\\Delta_{{\\mathcal P}}","kind":"use","excerpt":"[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invarianc","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03596e10dbda2d62","tex":"\\Delta_{\\Pcal}^2","aliases":"Δ_Pcal^2 Δ_ P^2 Delta Pcal Delta mathcal","display":"\\Delta_{{\\mathcal P}}^2","kind":"use","excerpt":"Pairwise annihilation gives Δ_ P^2=Δ_ P and identifies its range with diagonal block families. If T commutes with all P_α, then for β≠α, P_β TP_α=P_β P_α T=0. Thus (i) implies (ii). Exact reconstruction shows (ii) implies (iii). If T=∑_α P_α TP_α, then TP_γ=P_γ TP_γ=P_γ T, so (iii) implies (i).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab767052c24571cd","tex":"\\delta_R","aliases":"δ_R delta","display":"\\delta_R","kind":"use","excerpt":"[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-043cc7068d423bd8","tex":"\\delta_r","aliases":"δ_r delta","display":"\\delta_r","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51a87ccbd55b1aa4","tex":"\\delta_{rs}","aliases":"δ_rs delta","display":"\\delta_{rs}","kind":"use","excerpt":"[Complete residue packet system] For fixed N, the projectors P_N,r satisfy P_N,rP_N,s = δ_rsP_N,r and ∑_rbmod NP_N,r=I.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c3547666183b660","tex":"\\delta_{su}","aliases":"δ_su delta","display":"\\delta_{su}","kind":"use","excerpt":"[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70f334c2cd99d9ef","tex":"\\Delta_{u(p-1)/m}^{k}","aliases":"Δ_u(p-1)/m^k Delta","display":"\\Delta_{u(p-1)/m}^{k}","kind":"use","excerpt":"[Step-sensitive divisor-branch bound] Under the hypotheses of Theorem~[source reference], for all u,k,n∈ N, Δ_u(p-1)/m^kU_mp,j(n) ∈ p^min(k(1+ν_p(u)),mn) Z_p. For C_mp,j the corresponding valuation lower bound is min(k(1+ν_p(u)),mn)-v_p(j!).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90d5ba069a82a623","tex":"\\Delta_{uh}","aliases":"Δ_uh Delta","display":"\\Delta_{uh}","kind":"use","excerpt":"]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth p","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5d6c4b3391f8b2e","tex":"\\Delta_{uh}^k","aliases":"Δ_uh^k Delta","display":"\\Delta_{uh}^k","kind":"use","excerpt":"pplied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0 gives zero for k>0, and order zero is the original integral coefficient. This proves the bound for all u and k. Finally Δ_u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4bff5300fdcd34eb","tex":"\\delta_y","aliases":"δ_y delta","display":"\\delta_y","kind":"use","excerpt":"On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a71a6c1ff123146","tex":"\\delta_{\\alpha\\gamma}","aliases":"δ_αγ delta alpha gamma","display":"\\delta_{\\alpha\\gamma}","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df0a518550e907c5","tex":"\\delta_{\\beta\\alpha}","aliases":"δ_βα delta beta alpha","display":"\\delta_{\\beta\\alpha}","kind":"use","excerpt":"[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5a2814c2c585145","tex":"\\delta_{\\beta\\delta}","aliases":"δ_βδ delta beta delta","display":"\\delta_{\\beta\\delta}","kind":"use","excerpt":"[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f11ad62b3f5c5bb","tex":"\\delta_{\\lambda\\mu}","aliases":"δ_λμ delta lambda mu","display":"\\delta_{\\lambda\\mu}","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d33d8b646241d747","tex":"\\Delta_\\ell","aliases":"Δ_ℓ Delta ell","display":"\\Delta_\\ell","kind":"use","excerpt":"Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-be8d5809df598ef7","tex":"\\varepsilon^*","aliases":"ε^* varepsilon","display":"\\varepsilon^*","kind":"use","excerpt":"Global comparison. For any other feasible ε, set r_e=ε_e/ε_e^*>0. The budget identities give ∑_ec_er_e=C. Using the standard analytic inequality log r≤ r-1 yields ∑_ec_elog r_e≤∑_ec_e(r_e-1)=0. Lean separately proves the exact objective-difference identity J(ε)-J(ε^*)=-∑_ec_elog r_e. Therefore J(ε)≥ J(ε^*).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb43c73461568514","tex":"\\epsilon^2","aliases":"ϵ^2 epsilon","display":"\\epsilon^2","kind":"use","excerpt":"There is a still stronger filtration. Put X=P+Q, τ=i B D, ϵ=1+τ. Then the exact identities are H_2=ϵ X+2G, G=-γ_2(P)-γ_2(Q)+3PQ-τ Q, ϵ^2=2V, V=τ-2X-8PQ. Every divided power of G is integral component by component. The finite addition identity gives γ_k(H_2)=∑_r=0^k2^rγ_r(G) ϵ^k-rγ_k-r(X) ∈2^lfloor k/2rfloor A_2, because the rth term belongs to 2^r+","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3bbbf66ed7bfacbd","tex":"\\varepsilon^3","aliases":"ε^3 varepsilon","display":"\\varepsilon^3","kind":"use","excerpt":"[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74ecfbaf0fda4ff7","tex":"\\epsilon^j","aliases":"ϵ^j epsilon","display":"\\epsilon^j","kind":"use","excerpt":"ϵ^2=2(τ-P-Q-2PQ). The divided powers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89c2a55fd02aed5b","tex":"\\epsilon^{k-r}","aliases":"ϵ^k-r epsilon","display":"\\epsilon^{k-r}","kind":"use","excerpt":"Put X=P+Q, τ=i B D, ϵ=1+τ. Then the exact identities are H_2=ϵ X+2G, G=-γ_2(P)-γ_2(Q)+3PQ-τ Q, ϵ^2=2V, V=τ-2X-8PQ. Every divided power of G is integral component by component. The finite addition identity gives γ_k(H_2)=∑_r=0^k2^rγ_r(G) ϵ^k-rγ_k-r(X) ∈2^lfloor k/2rfloor A_2, because the rth term belongs to 2^r+lfloor(k-r)/2rfloor A_2. Applying the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2f9c21b859a02d5","tex":"\\varepsilon_*","aliases":"ε_* varepsilon","display":"\\varepsilon_*","kind":"use","excerpt":"[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bae7842f2d63f7f5","tex":"\\varepsilon_a","aliases":"ε_a varepsilon","display":"\\varepsilon_a","kind":"use","excerpt":"[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48bf82ae71f365c5","tex":"\\varepsilon_b","aliases":"ε_b varepsilon","display":"\\varepsilon_b","kind":"use","excerpt":"[Approximate constraint transport] Assume normC_ℓ+1,aR_ℓ-R_ℓ C_ℓ,a ≤ε_a and normC_ℓ,a,normC_ℓ+1,a≤ M. Then a coarse physical unit vector ψ satisfies normC_ℓ+1,aR_ℓψ≤ε_a. Also, · norm [C_ℓ+1,a,C_ℓ+1,b]R_ℓ -R_ℓ[C_ℓ,a,C_ℓ,b] · ≤ 2M(ε_a+ε_b).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56db928a7ce0a173","tex":"\\epsilon_e","aliases":"ϵ_e epsilon","display":"\\epsilon_e","kind":"use","excerpt":"s_e,ϵ_e · Sensitivity and tolerance · Is the budget input-dependent or uniform over unit inputs?","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2fc46a82cd7318e7","tex":"\\varepsilon_e","aliases":"ε_e varepsilon","display":"\\varepsilon_e","kind":"use","excerpt":"_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exact operator and vector identities are proved first: widehat A-A=∑_eΔ_e P_a(e), (widehat A-A)x=∑_eΔ_e(P_a(e)x). Applying the finite triangle inequality and the defining operator-norm estimate gives norm(widehat A-A)x ≤∑_enormΔ_e normP_a(e)x. If normΔ_e≤ε_enormA_e, then the formal relative certific","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb0e8c76ef72f0c6","tex":"\\varepsilon_e^*","aliases":"ε_e^* varepsilon","display":"\\varepsilon_e^*","kind":"use","excerpt":"orem textitSource item 8.4, with an explicit positive budget Let E be a finite nonempty edge set. Assume s_e>0, c_e>0, τ>0, and define C=∑_ec_e. The optimization problem is min_ε_e>0∑_ec_elogfrac1ε_e subject to ∑_es_eε_e=τ. Lean constructs the displayed candidate ε_e^*=fracτ c_es_eC and proves strict positivity and the exact budget equality. It the","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e70f355537b1cd85","tex":"\\varepsilon_h","aliases":"ε_h varepsilon","display":"\\varepsilon_h","kind":"use","excerpt":"[Deterministic consistency] Assume h→0 along grids satisfying the preceding sampling assumptions, ε_h→0, the number of jumps is fixed, and the nonzero jump amplitudes and positive minimum separation remain fixed. Then the detected packet converges to the true packet in every finite L^p norm. If Q_m has a vanishing smooth approximation error and uni","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3de5d9511fcd1b4","tex":"\\varepsilon_j","aliases":"ε_j varepsilon","display":"\\varepsilon_j","kind":"use","excerpt":"[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b0466fee544c74d","tex":"\\varepsilon_M","aliases":"ε_M varepsilon","display":"\\varepsilon_M","kind":"use","excerpt":"[Near cancellation of two exponentials] For every M>0, there are constants ε_M>0 and C_M>0 such that the following holds. If |Re w|≤ M and |1+e^w|≤ε_M, then there is a unique integer m with |w-(2m+1)π i| ≤ C_M|1+e^w|.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6b02083e2b0a177","tex":"\\varepsilon_m","aliases":"ε_m varepsilon","display":"\\varepsilon_m","kind":"use","excerpt":"For N≥3, [source reference] writes all but finitely many reduced zeros as λ_m = (A_N(m+frac12))^N(1+ε_m) where λ_m>0, 1+ε_m>0, and ε_m = O(t_m^-1e^-γ t_m). On every compact set of s values, (1+ε_m)^-s-1 = O(|ε_m|) uniformly for large m. The difference λ_m^-s - A_N^-Ns(m+frac12)^-Ns therefore has a normally convergent sum on every compact subset of ","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc06eebc88a8b163","tex":"\\varepsilon_n","aliases":"ε_n varepsilon","display":"\\varepsilon_n","kind":"use","excerpt":"Let e_n= CF_f^n(x)-F_c^n(Cx) . The definition of the defect and the Lipschitz property give e_n+1≤ε_n+qe_n. Iterating this recurrence proves the first estimate. The second is the geometric sum.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bc6fb3bcea699cd","tex":"\\varepsilon_{pq}","aliases":"ε_pq varepsilon","display":"\\varepsilon_{pq}","kind":"use","excerpt":"[Entries of the fundamental matrix] For 0≤ p,q<N, let d_pq be the member of {0,…,N-1} congruent to p-q modulo N. Put ε_pq = 1, · p≥ q, -1, · p<q. Then (e^zJ_N)_pq = ε_pq D_N,d_pq(z). Thus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from q to p crosses the signed edge.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5b018625faf1192a","tex":"\\varepsilon_{\\mathrm{rec}}","aliases":"ε_rec varepsilon mathrm","display":"\\varepsilon_{\\mathrm{rec}}","kind":"use","excerpt":"[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c69c9eb6fe242300","tex":"\\varepsilon_s","aliases":"ε_s varepsilon","display":"\\varepsilon_s","kind":"use","excerpt":"[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-156a9f90221e1614","tex":"\\epsilon_u","aliases":"ϵ_u epsilon","display":"\\epsilon_u","kind":"use","excerpt":"This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard windo","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14751022cb8bedd4","tex":"\\epsilon_{u+v}","aliases":"ϵ_u+v epsilon","display":"\\epsilon_{u+v}","kind":"use","excerpt":"[Addition law for modes] For every u,v∈ R, ϵ_umathbin⋆ϵ_v = ϵ_u+v.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23404397ce55a3d7","tex":"\\varepsilon_v","aliases":"ε_v varepsilon","display":"\\varepsilon_v","kind":"use","excerpt":"[Local perturbation certificate] Assume |c_v(y')-c_v(y)|≤ε_v. Define E(v) = ε_v, · v terminal,","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2a74941d662acb4","tex":"\\epsilon_v^{(j)}","aliases":"ϵ_v^(j) epsilon","display":"\\epsilon_v^{(j)}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e60af3df9de1cbdc","tex":"\\epsilon_\\Delta","aliases":"ϵ_Δ epsilon Delta","display":"\\epsilon_\\Delta","kind":"use","excerpt":"[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ec732f5d2a95aaa","tex":"\\epsilon_{\\Delta'}","aliases":"ϵ_Δ' epsilon Delta","display":"\\epsilon_{\\Delta'}","kind":"use","excerpt":"[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42e4cf15f8f8ad0f","tex":"\\epsilon_\\zeta","aliases":"ϵ_ζ epsilon zeta","display":"\\epsilon_\\zeta","kind":"use","excerpt":"[Subgroup average] The multiple selector has the mode expansion a^(m) = frac1m ∑_ζ∈ H_m ϵ_ζ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81faab646ef6c7f1","tex":"\\epsilon_{\\eta u}","aliases":"ϵ_η u epsilon eta","display":"\\epsilon_{\\eta u}","kind":"use","excerpt":"Multiplication by an element η∈ H_m gives an automorphism of the additive group and therefore an automorphism ϵ_u⟼ϵ_η u of the mode algebra. The selector a^(m) is fixed by every such automorphism, so its unique inverse is fixed as well.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ca916b982c03d00","tex":"\\varepsilon_\\ell","aliases":"ε_ℓ varepsilon ell","display":"\\varepsilon_\\ell","kind":"use","excerpt":"[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"ery n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root o","count":28},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1810dd5bbc6aace","tex":"\\zeta^{-1}","aliases":"ζ^-1 zeta","display":"\\zeta^{-1}","kind":"use","excerpt":"[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c185cb6f6888fc01","tex":"\\zeta^{-j}","aliases":"ζ^-j zeta","display":"\\zeta^{-j}","kind":"use","excerpt":"y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e7afdf00c1a0e7a1","tex":"\\zeta^j","aliases":"ζ^j zeta","display":"\\zeta^j","kind":"use","excerpt":"y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4197bb2b77854291","tex":"\\zeta^m","aliases":"ζ^m zeta","display":"\\zeta^m","kind":"use","excerpt":"Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d3331bda5586731","tex":"\\zeta^n","aliases":"ζ^n zeta","display":"\\zeta^n","kind":"use","excerpt":"The nth coefficient on the right is frac1m ∑_ζ∈ H_mζ^n. The character sum is one when mmid n and zero otherwise.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35348365f6cc6efe","tex":"\\zeta_3","aliases":"ζ_3 zeta","display":"\\zeta_3","kind":"use","excerpt":"The old fixed-field central argument genuinely failed. At p=3, (h_1,h_2)=(1,5), its naturally transferred full character takes the value ζ_3≠1 on 1+t^2, although the desired wild center is trivial. A shallow character choice cannot repair a forced value. The two-stage changing-field repair is retained in pathcentral/CENTRAL_REPAIR166.md. The trace-","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-88c0820439bcc4b2","tex":"\\zeta_4","aliases":"ζ_4 zeta","display":"\\zeta_4","kind":"use","excerpt":"ite set, with nonzero determinant. Every coefficient embedding preserves its nonvanishing. Actual member trace comparisons retain that member's good-prime condition. No coverage or character-continuity hypothesis is added in this cyclotomic specialization: the prime coverage is proved, and the finite extension's Krull topology is discrete. A concre","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5048985144717904","tex":"\\zeta_M","aliases":"ζ_M zeta","display":"\\zeta_M","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-781d10ac98eeab56","tex":"\\zeta_M^{-ja}","aliases":"ζ_M^-ja zeta","display":"\\zeta_M^{-ja}","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6c934809237f301","tex":"\\zeta_n","aliases":"ζ_n zeta","display":"\\zeta_n","kind":"use","excerpt":"Arithmetic coverage for a fixed cyclotomic extension. Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The pr","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5952c7e68d3d79b8","tex":"\\zeta_n^p","aliases":"ζ_n^p zeta","display":"\\zeta_n^p","kind":"use","excerpt":"Arithmetic coverage for a fixed cyclotomic extension. Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The pr","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-adcf51002ea5c3e3","tex":"\\eta^N","aliases":"η^N eta","display":"\\eta^N","kind":"use","excerpt":"The second half of this chapter concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's mon","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e882302ef17445ae","tex":"\\eta^n","aliases":"η^n eta","display":"\\eta^n","kind":"use","excerpt":"Write b^(m,p) = ∑_u∈ F_pc_uϵ_u. Invariance gives c_η u=c_u. After the change of variable u↦η u, b_n^(m,p) = η^n b_n^(m,p) for every η∈ H_m. If mnmid n, some η satisfies η^n≠1, so the coefficient vanishes.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62ecbfb276c1f894","tex":"\\eta_0","aliases":"η_0 eta","display":"\\eta_0","kind":"use","excerpt":"The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b93f1be7412facb","tex":"\\eta_1","aliases":"η_1 eta","display":"\\eta_1","kind":"use","excerpt":"[Input locality for operation trees] Let η_1,η_2:I→ Y be two input environments. If η_1(i)=η_2(i) for every i∈inputs(t), then the two evaluations of t are equal under the same constant, unary, and binary semantics. In particular, if inputs(t)=∅, evaluation is independent of the input environment. statusleanthreeKomorebi.PaperIII.NonlinearObstructio","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cb721f0ce1499b1","tex":"\\eta_2","aliases":"η_2 eta","display":"\\eta_2","kind":"use","excerpt":"[Input locality for operation trees] Let η_1,η_2:I→ Y be two input environments. If η_1(i)=η_2(i) for every i∈inputs(t), then the two evaluations of t are equal under the same constant, unary, and binary semantics. In particular, if inputs(t)=∅, evaluation is independent of the input environment. statusleanthreeKomorebi.PaperIII.NonlinearObstructio","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-017a293d20155d21","tex":"\\eta_g","aliases":"η_g eta","display":"\\eta_g","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57d0f50543139197","tex":"\\eta_{g_1}","aliases":"η_g_1 eta","display":"\\eta_{g_1}","kind":"use","excerpt":"belprop:packet_l_infinity Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet of label k is","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2afdf48eb6389780","tex":"\\eta_{g_r}","aliases":"η_g_r eta","display":"\\eta_{g_r}","kind":"use","excerpt":"belprop:packet_l_infinity Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet of label k is","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8216593a1e41c75","tex":"\\eta_h","aliases":"η_h eta","display":"\\eta_h","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc03a685a0a01ebc","tex":"\\eta_k","aliases":"η_k eta","display":"\\eta_k","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3874982d0a28e3b","tex":"\\eta_{k+1}","aliases":"η_k+1 eta","display":"\\eta_{k+1}","kind":"use","excerpt":"In a cell without a jump, D_k=s(t_k+1)-s(t_k)+η_k+1-η_k. The Lipschitz and noise bounds give |D_k|≤ Mh+2ε. In a cell with one jump, the same formula has the additional term a_q, which proves the second estimate.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-195864f2c8af017d","tex":"\\eta_{L,m}","aliases":"η_L,m eta","display":"\\eta_{L,m}","kind":"use","excerpt":"[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca0e6e00d4486392","tex":"\\eta_{L,m}^{\\Phi^i}","aliases":"η_L,m^Φ^i eta Phi","display":"\\eta_{L,m}^{\\Phi^i}","kind":"use","excerpt":"[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb9149ca37d1df00","tex":"\\eta_{m,n}","aliases":"η_m,n eta","display":"\\eta_{m,n}","kind":"use","excerpt":"[Range leakage criterion] Under the contractive assumptions of [source reference], define η_m,n(T) = sup_0≤ t≤ T norm(I-P_m,n)T_n(t)R_n,m. Then ρ_m,n(T)≤η_m,n(T). Consequently, η_m,n(T)→0 for every T>0 is sufficient for the limiting family U_m to satisfy the semigroup law.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd536be837f3eb94","tex":"\\eta_{\\mathrm{src}}","aliases":"η_src eta mathrm","display":"\\eta_{\\mathrm{src}}","kind":"use","excerpt":"Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. N","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12235ebd9b8ec538","tex":"\\eta_t","aliases":"η_t eta","display":"\\eta_t","kind":"use","excerpt":"vpage17A random diffusion clockSUBORDINATION / A DISTRIBUTION OF ORDINARY TIMESChapter VI, Part I: Theorem 10.2. The Poisson subordination formula for the finite positive model. Square-root evolution can be written as an average of ordinary diffusion over a positive clock variable s: e^-tsqrt L=∫_0^∞ η_t(s)e^-sL ds, η_t(s)=fract2sqrtπs^-3/2e^-t^2/(","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d37bc4b50bcca72","tex":"\\eta_W","aliases":"η_W eta","display":"\\eta_W","kind":"use","excerpt":"Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. N","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5624add4e4a44776","tex":"\\eta_{\\alpha,s}","aliases":"η_α,s eta alpha","display":"\\eta_{\\alpha,s}","kind":"use","excerpt":"d t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d67c847e2c962980","tex":"\\eta_{\\alpha,s+t}","aliases":"η_α,s+t eta alpha","display":"\\eta_{\\alpha,s+t}","kind":"use","excerpt":"d t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c7d3ef1b4110a1b","tex":"\\eta_{\\alpha,t}","aliases":"η_α,t eta alpha","display":"\\eta_{\\alpha,t}","kind":"use","excerpt":"[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ee7b5d31d3432c2","tex":"\\eta_\\gamma","aliases":"η_γ eta gamma","display":"\\eta_\\gamma","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb88d6b2c124cbb8","tex":"\\eta_{\\gamma'}","aliases":"η_γ' eta gamma","display":"\\eta_{\\gamma'}","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b6855948fa894811","tex":"\\eta_\\theta","aliases":"η_θ eta theta","display":"\\eta_\\theta","kind":"use","excerpt":"[Type side Heisenberg quotient] The pairing h_θ is nondegenerate and alternating. There is one irreducible representation η_θ of J_θ^1 containing θ, and its dimension is sqrtabsJ_θ^1/H_θ^1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17bbe9dc50ae42d1","tex":"\\eta_\\ell","aliases":"η_ℓ eta ell","display":"\\eta_\\ell","kind":"use","excerpt":"ojection estimate gives the stronger bound Y=fraclog(N-1)2s_N for every zero in the sector, without a large-radius assumption. Write z=re^iθ with 0≤θ≤ a_N and r≥0. For 0≤ℓ≤ N-2, the angle θ+(2ℓ+1)a_N lies between a_N+θ and 2π-a_N-θ. Monotonicity of cosine on [0,π], followed by reflection about π, therefore gives Re(η_ℓ z)≤ rcos(a_N+θ). The remainin","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d811658e1fb8893c","tex":"\\eta_\\perp","aliases":"η_⊥ eta perp","display":"\\eta_\\perp","kind":"use","excerpt":"Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. N","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.","count":45},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a830538315a1ab8","tex":"\\vartheta","aliases":"ϑ vartheta","display":"\\vartheta","kind":"use","excerpt":"nit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","excerpt":"vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCEChapter VI, Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c1ce6d2d2665df29","tex":"\\theta^3","aliases":"θ^3 theta","display":"\\theta^3","kind":"use","excerpt":"ite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-θ). Multiplication adds exponents, so the posterior belongs to Beta(4,4) with density 140θ^3(1-θ)^3. vfigb","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60a68771f913a013","tex":"\\Theta^d","aliases":"Θ^d Theta","display":"\\Theta^d","kind":"use","excerpt":"^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence finite weighted polynomials. Each fixed output coefficient depends on only finitely many input coefficients, which extends the result to the formal series. Substitution T=e^t-1 intertwines D with Θ and maps W_p into H_p, because d![t^d]F(e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f906ca28e719ae2","tex":"\\theta^g","aliases":"θ^g theta","display":"\\theta^g","kind":"use","excerpt":"At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12a64f371afd81e6","tex":"\\Theta^n","aliases":"Θ^n Theta","display":"\\Theta^n","kind":"use","excerpt":"The relevant integral subring is the Hurwitz subring H( Z_p)={f∈ Q_p[[t]]: n![t^n]f∈ Z_p for every n}. It is not coefficientwise integrality of ordinary power-series coefficients. The precise connection is n![t^n] E(F)=[T^0]Θ^nF. Since every iterate of Θ preserves W_p, the right side is integral. Therefore E( W_p)⊆ H( Z_p). Applying the same statem","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c048d6949aa9eafc","tex":"\\Theta^p","aliases":"Θ^p Theta","display":"\\Theta^p","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c5dd3e45df8a083","tex":"\\Theta^{p^2}","aliases":"Θ^p^2 Theta","display":"\\Theta^{p^2}","kind":"use","excerpt":"To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a725949888ccbeda","tex":"\\theta^u","aliases":"θ^u theta","display":"\\theta^u","kind":"use","excerpt":"eEnd_E_0(F^4). The odd lower critical level is r=3h_1=2m+1. The half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","excerpt":"∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ≤θ_0 is π(1-cosθ_0). At θ_0=π/3, it is exactly π/2. The ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-463aadeac846c45b","tex":"\\Theta_{2,0}","aliases":"Θ_2,0 Theta","display":"\\Theta_{2,0}","kind":"use","excerpt":"or a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zero-address ca","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cc757b47064b63ba","tex":"\\Theta_{2,2}","aliases":"Θ_2,2 Theta","display":"\\Theta_{2,2}","kind":"use","excerpt":"efficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u). Every positive-index theta polynomial vanishes at u=0. For even N, H_N(-z)=H_N(z), so the defining product is also 1 at u=-1. headThe odd-order boundary is visible For N=3, Θ_3,1(-1)=2, not zero. The factor u+1 is not a uniform law across orders. In contrast, Θ_2,2(u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-730e92ec496d3f38","tex":"\\Theta_{3,1}","aliases":"Θ_3,1 Theta","display":"\\Theta_{3,1}","kind":"use","excerpt":"garithmic block m. Reciprocal order contributes u. vfigtheta Their combined logarithm has coefficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u). Every positive-index theta polynomial vanishes at u=0. For even N, H_N(-z)=H_N(z), so the defining product is also 1 at u=-1. headThe odd-order boundary is visible For N=3, Θ_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a721963f725aae53","tex":"\\theta_i","aliases":"θ_i theta","display":"\\theta_i","kind":"use","excerpt":"Choose a labeled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59f92b3334eb7f30","tex":"\\Theta_j","aliases":"Θ_j Theta","display":"\\Theta_j","kind":"use","excerpt":"For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c8082d88b8040556","tex":"\\theta_j","aliases":"θ_j theta","display":"\\theta_j","kind":"use","excerpt":"For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-017adcf52ee20d92","tex":"\\Theta_{M,s}","aliases":"Θ_M,s Theta","display":"\\Theta_{M,s}","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b4a0b405d0fc12a","tex":"\\Theta_{MN,Mt+s}","aliases":"Θ_MN,Mt+s Theta","display":"\\Theta_{MN,Mt+s}","kind":"use","excerpt":"[Kernel composition] For positive integers M,N, ∑_k≥0 Θ_M,s(r,k) Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). The sum is finite.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c2ec1f968f5fa88b","tex":"\\theta_{N,0}","aliases":"θ_N,0 theta","display":"\\theta_{N,0}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26bb4e7108e8d9c7","tex":"\\Theta_{N,1}","aliases":"Θ_N,1 Theta","display":"\\Theta_{N,1}","kind":"use","excerpt":"E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic block m. Reciprocal order contributes u. vfigtheta Their combined logarithm has coefficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-444beaffc1630312","tex":"\\Theta_{N,n}","aliases":"Θ_N,n Theta","display":"\\Theta_{N,n}","kind":"use","excerpt":"vpage16Scale and order enter differentlyTHETA COUPLINGChapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4. Let H_N(z)=E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic block m. Reciprocal order contributes","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24341a22cfbc2f38","tex":"\\theta_{N,n}","aliases":"θ_N,n theta","display":"\\theta_{N,n}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-630ff65f783b7453","tex":"\\theta_{N,n-m}","aliases":"θ_N,n-m theta","display":"\\theta_{N,n-m}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78b975269dab6a12","tex":"\\Theta_{N,t}","aliases":"Θ_N,t Theta","display":"\\Theta_{N,t}","kind":"use","excerpt":"[Kernel composition] For positive integers M,N, ∑_k≥0 Θ_M,s(r,k) Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). The sum is finite.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ca492aad33d122fe","tex":"\\Theta_{p^a,s}","aliases":"Θ_p^a,s Theta","display":"\\Theta_{p^a,s}","kind":"use","excerpt":"but its consequence for transformed Mahler coefficients requires c_r∈ Z_p for every r. Convergence of a Q_p-valued Mahler expansion alone is insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d595452034045c90","tex":"\\Theta_{\\Pcal}","aliases":"Θ_Pcal Θ_ P Theta Pcal Theta mathcal","display":"\\Theta_{{\\mathcal P}}","kind":"use","excerpt":"[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2568746b37cf9a0e","tex":"\\iota^{-1}","aliases":"ι^-1 iota","display":"\\iota^{-1}","kind":"use","excerpt":"[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-50b831621b666dcd","tex":"\\iota_H","aliases":"ι_H iota","display":"\\iota_H","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29b3a606402cae8a","tex":"\\iota_{\\tgt(e)}","aliases":"ι_tgt(e) iota tgt iota operatorname","display":"\\iota_{{\\operatorname{tgt}}(e)}","kind":"use","excerpt":"[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08977dc59c5957ea","tex":"\\iota_\\lambda","aliases":"ι_λ iota lambda","display":"\\iota_\\lambda","kind":"use","excerpt":"le and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding preser","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc656d1ba29723e3","tex":"\\iota_\\ell","aliases":"ι_ℓ iota ell","display":"\\iota_\\ell","kind":"use","excerpt":"Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a118f9e5c9021f88","tex":"\\kappa^{-1}","aliases":"κ^-1 kappa","display":"\\kappa^{-1}","kind":"use","excerpt":"(P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35165a2cb10ed645","tex":"\\kappa_0","aliases":"κ_0 kappa","display":"\\kappa_0","kind":"use","excerpt":"racte^t-1=∑_n≥0B_nfract^nn!, B_1=-frac12. Throughout this appendix, v_2(2)=1 and v_2(0)=∞. Let s_2(n) be the number of ones in the binary expansion of the nonnegative integer n. Define A(N)=frac3^2N+2-14(2N+1)(2N+2), (frac zsinh z)^2=∑_m≥0β_m z^2m, κ_m=2^2m(2m)!β_m. The initial values are β_0=κ_0=1. For m≥1, differentiation of zcoth z gives β_m=-fr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62f4ed922bb2ce09","tex":"\\kappa_1","aliases":"κ_1 kappa","display":"\\kappa_1","kind":"use","excerpt":"rem 7.2, Corollary 7.7. Let r vary instead of fixing the reciprocal power. Since Δ_N(w)^-r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93055c50da4a404a","tex":"\\kappa_2","aliases":"κ_2 kappa","display":"\\kappa_2","kind":"use","excerpt":"rem 7.2, Corollary 7.7. Let r vary instead of fixing the reciprocal power. Since Δ_N(w)^-r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3cdc2b33ff718bf4","tex":"\\kappa_{2,1}","aliases":"κ_2,1 kappa","display":"\\kappa_{2,1}","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-570f9f44831ecebb","tex":"\\kappa_{2,2}","aliases":"κ_2,2 kappa","display":"\\kappa_{2,2}","kind":"use","excerpt":"[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17d1892cda93e827","tex":"\\kappa_{2,r}","aliases":"κ_2,r kappa","display":"\\kappa_{2,r}","kind":"use","excerpt":"The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the b","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-545253b67e7a1373","tex":"\\kappa_3","aliases":"κ_3 kappa","display":"\\kappa_3","kind":"use","excerpt":"rem 7.2, Corollary 7.7. Let r vary instead of fixing the reciprocal power. Since Δ_N(w)^-r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00bf93e0f30b5a36","tex":"\\kappa_{4,1}","aliases":"κ_4,1 kappa","display":"\\kappa_{4,1}","kind":"use","excerpt":"ollary 8.4, Theorem 8.5. The transfer tangent is T_N=-D_N'/D_N. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator. vfigriccati The coefficients retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b72484db9745e3a","tex":"\\kappa_{4,2}","aliases":"κ_4,2 kappa","display":"\\kappa_{4,2}","kind":"use","excerpt":"The transfer tangent is T_N=-D_N'/D_N. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator. vfigriccati The coefficients retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f4d1001288e3b575","tex":"\\kappa_m","aliases":"κ_m kappa","display":"\\kappa_m","kind":"use","excerpt":"[Exact weighted inverse coefficient] Let κ_m=β_m2^2m(2m)!. For every m≥1, κ_m=-2^4m(2m-1)B_2m≠0, v_2(κ_m)=4m-1, where B_2m is the usual even Bernoulli number.","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87654c9c7d2aaa41","tex":"\\kappa_{M+1}","aliases":"κ_M+1 kappa","display":"\\kappa_{M+1}","kind":"use","excerpt":"At n=M+1 there is exactly one omitted summand. Here binom2n2n=1 and A(0)=1, so the error is κ_M+1. Lemma~[source reference] gives its exact valuation 4M+3 and nonvanishing. The statements are leanKomorebiPaperIX.DyadicSharpTower.dyadic_supercongruence_sharp, leanKomorebiPaperIX.DyadicSharpTower.first_omitted_error, and leanKomorebiPaperIX.DyadicSha","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd45552bd05a5fc5","tex":"\\kappa_{N,1}","aliases":"κ_N,1 kappa","display":"\\kappa_{N,1}","kind":"use","excerpt":"[Bell polynomial formula] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rκ_N,1, r 1! κ_N,2, …, r (n-1)! κ_N,n ). For n≥1, this is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n!(N!)^n.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-849903fb00046048","tex":"\\kappa_{N,2}","aliases":"κ_N,2 kappa","display":"\\kappa_{N,2}","kind":"use","excerpt":"[Bell polynomial formula] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rκ_N,1, r 1! κ_N,2, …, r (n-1)! κ_N,n ). For n≥1, this is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n!(N!)^n.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5222a4a546cc55fc","tex":"\\kappa_{N,3}","aliases":"κ_N,3 kappa","display":"\\kappa_{N,3}","kind":"use","excerpt":"The first logarithmic coefficients are κ_N,1=frac1N!, κ_N,2=frac1(N!)^2-frac2(2N)!, and κ_N,3 = frac1(N!)^3 - frac3N!(2N)! + frac3(3N)!. Insert these values into [source reference] and simplify.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-88fe155938eb7be8","tex":"\\kappa_{N,k}","aliases":"κ_N,k kappa","display":"\\kappa_{N,k}","kind":"use","excerpt":"[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9557fe49782b2a8","tex":"\\kappa_{N,n}","aliases":"κ_N,n kappa","display":"\\kappa_{N,n}","kind":"use","excerpt":"[Bell polynomial formula] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rκ_N,1, r 1! κ_N,2, …, r (n-1)! κ_N,n ). For n≥1, this is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n!(N!)^n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4cf0a10371eb328d","tex":"\\kappa_{N,n+1}","aliases":"κ_N,n+1 kappa","display":"\\kappa_{N,n+1}","kind":"use","excerpt":"[Logarithmic coefficient formula] The transfer tangent satisfies T_N(z) = N∑_m≥1κ_N,mz^Nm-1. Equivalently, T_N(n) = N(Nn+N-1)!κ_N,n+1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35ea25074ca81a87","tex":"\\kappa_{N,q}","aliases":"κ_N,q kappa","display":"\\kappa_{N,q}","kind":"use","excerpt":"vpage22Finite coefficients give infinite momentsZERO POWER SUMSChapter IV: Theorem 12.6, Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No num","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c12049a3b7739a90","tex":"\\kappa_{p,r}","aliases":"κ_p,r kappa","display":"\\kappa_{p,r}","kind":"use","excerpt":"0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd, β_r2^2r(2r)!, · p=2.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5c11c2e36107f9e","tex":"\\kappa_r","aliases":"κ_r kappa","display":"\\kappa_r","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6aa4d5926406a9c8","tex":"\\kappa_{r,b_j}","aliases":"κ_r,b_j kappa","display":"\\kappa_{r,b_j}","kind":"use","excerpt":"[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j, 0, · r≥ b_j. Then the output channel is y_r(q) = ∏_j=1^m x_r_j (q-c_j-κ_r,b_j).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-39c423c6294d6173","tex":"\\kappa_v^{(j)}","aliases":"κ_v^(j) kappa","display":"\\kappa_v^{(j)}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9680dd9eec29c6c","tex":"\\kappa_v^{\\rm setup}","aliases":"κ_v^rm setup kappa rm","display":"\\kappa_v^{\\rm setup}","kind":"use","excerpt":"[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e6997cad3f90314","tex":"\\kappa_v^{\\mathrm{setup}}","aliases":"κ_v^setup kappa mathrm","display":"\\kappa_v^{\\mathrm{setup}}","kind":"use","excerpt":"Shared operator families. For reuse weights r_j, the local keep model can aggregate setup and per-operator costs: κ_v=κ_v^setup+∑_jr_jκ_v^(j), ϵ_v=∑_jr_jϵ_v^(j). Lean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including Q and R. The inte","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d0e527ea9237f05","tex":"\\kappa_\\theta","aliases":"κ_θ kappa theta","display":"\\kappa_\\theta","kind":"use","excerpt":"[Regular elliptic torus characters] A regular character θ:T_n→ C^× determines an irreducible cuspidal representation κ_θ of G_n. Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of G_n occurs in this way.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c9dd55b67cbaf19","tex":"\\lambda^{\\mathsf T}","aliases":"λ^ T lambda mathsf","display":"\\lambda^{\\mathsf T}","kind":"use","excerpt":"B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a7ba560bfc03b7a8","tex":"\\lambda^{(1)}","aliases":"λ^(1) lambda","display":"\\lambda^{(1)}","kind":"use","excerpt":"vpage26One complete coarse filterEND-TO-END MODEL / TWO OBSERVATIONSChapter VI, Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate. Use the four-state kernel K and aggregation C from pagerefV5, starting at p_0=e_0. Observe two likelihood vectors constant inside both packets: λ^(1)=(3/4,3/4,1/4,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-63fbbaa59a200ba4","tex":"\\lambda^{(2)}","aliases":"λ^(2) lambda","display":"\\lambda^{(2)}","kind":"use","excerpt":"vpage26One complete coarse filterEND-TO-END MODEL / TWO OBSERVATIONSChapter VI, Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate. Use the four-state kernel K and aggregation C from pagerefV5, starting at p_0=e_0. Observe two likelihood vectors constant inside both packets: λ^(1)=(3/4,3/4,1/4,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-096b5779eec11120","tex":"\\lambda^{(r)}","aliases":"λ^(r) lambda","display":"\\lambda^{(r)}","kind":"use","excerpt":"Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e12e7e5ac627fad4","tex":"\\lambda^{(y)}","aliases":"λ^(y) lambda","display":"\\lambda^{(y)}","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10808a5caf1a666d","tex":"\\lambda^{-1/2}","aliases":"λ^-1/2 lambda","display":"\\lambda^{-1/2}","kind":"use","excerpt":"0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divide by zero.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30aee352b9817eeb","tex":"\\lambda^{-q}","aliases":"λ^-q lambda","display":"\\lambda^{-q}","kind":"use","excerpt":"vpage22Finite coefficients give infinite momentsZERO POWER SUMSChapter IV: Theorem 12.6, Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No num","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-308a821eda6e7229","tex":"\\lambda^2","aliases":"λ^2 lambda","display":"\\lambda^2","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11c3cfbb073087f5","tex":"\\Lambda^d","aliases":"Λ^d Lambda","display":"\\Lambda^d","kind":"use","excerpt":"Choose a Λ basis of M. Tensoring the resulting isomorphism M≅Λ^d along augmentation gives O^d. Tensoring by Λ/ a is the same as quotienting by a M.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-627a7d01bca57e43","tex":"\\lambda^g","aliases":"λ^g lambda","display":"\\lambda^g","kind":"use","excerpt":"As g²∈J, conjugation by g² fixes the character theta. Thus λ^g=λ^-1=λ. It follows that","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75946f9ee1ce0aa6","tex":"\\lambda^j","aliases":"λ^j lambda","display":"\\lambda^j","kind":"use","excerpt":"[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-385decfaff5738d3","tex":"\\lambda^N","aliases":"λ^N lambda","display":"\\lambda^N","kind":"use","excerpt":"Fix a positive integer N. Root of unity projection of the exponential produces residue-class sections that, for λ≠0, form a basis for y^(N)=λ^N y. The signed sections for y^(N)=-y yield the denominator D_N and its reduced form Δ_N. We derive reciprocal coefficient polynomials, Bell and Newton recurrences, an additive parameter law, and residue sect","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1488a4b56fba6683","tex":"\\lambda^{N(N-1)/2}","aliases":"λ^N(N-1)/2 lambda","display":"\\lambda^{N(N-1)/2}","kind":"use","excerpt":"[Wronskian of the transfer basis] Assume N≥2. For λ≠0, the Wronskian of the ordered basis E_N,0(λ z),…, E_N,N-1(λ z) is the constant λ^N(N-1)/2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04db12d96ee6e2df","tex":"\\lambda^{N-1}","aliases":"λ^N-1 lambda","display":"\\lambda^{N-1}","kind":"use","excerpt":"At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5fc985955f34f9e","tex":"\\lambda^u","aliases":"λ^u lambda","display":"\\lambda^u","kind":"use","excerpt":"First, lambda is invariant under u. Indeed u 2 belongs to J and theta is J-invariant, so λ^u=λ^-1=λ by (3). Consequently H0 is normal in P, and theta0 is P-invariant.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10e520ed7336bfd0","tex":"\\lambda^\\alpha","aliases":"λ^α lambda alpha","display":"\\lambda^\\alpha","kind":"use","excerpt":"[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6559d219a1ff939b","tex":"\\Lambda_*","aliases":"Λ_* Lambda","display":"\\Lambda_*","kind":"use","excerpt":")+V(v1)}, V(v)=c_v at a terminal node. headRead the calculation from the bottom upward At 10, the children cost 1+1=2, better than stopping for 7. At 11, stopping costs 4, better than 2+3=5. The right subtree therefore costs 6, better than its stop cost 8. The left subtree costs 1+2=3, better than 5. At the root, 3+6=9 beats 20. The unique optimum ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-efe402b27fb44fd4","tex":"\\lambda_*","aliases":"λ_* lambda","display":"\\lambda_*","kind":"use","excerpt":"r] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r wi","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ede142bbc0a6d7e","tex":"\\Lambda_0","aliases":"Λ_0 Lambda","display":"\\Lambda_0","kind":"use","excerpt":"[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Onl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7238fc50817cce55","tex":"\\lambda_0","aliases":"λ_0 lambda","display":"\\lambda_0","kind":"use","excerpt":"Suppose two members are isomorphic at λ_0. Injectivity of the common coefficient embedding gives t_i(v)=t_j(v) at every place good for λ_0. At another coefficient prime λ, this equality compares actual member traces on the intersection of the two good-place sets. Conjugacy density of that intersection and continuity extend the equality to the whole","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26a2a17ac5cf2600","tex":"\\Lambda_1","aliases":"Λ_1 Lambda","display":"\\Lambda_1","kind":"use","excerpt":"[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac2ec4913e4c3ca3","tex":"\\Lambda_2","aliases":"Λ_2 Lambda","display":"\\Lambda_2","kind":"use","excerpt":"[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-975e1a0c6d000741","tex":"\\lambda_2","aliases":"λ_2 lambda","display":"\\lambda_2","kind":"use","excerpt":"[Monotone Lagrangian frontier] Let 0≤λ_1<λ_2. Choose optimal prunings L_1 and L_2 for the two Lagrangian parameters. Then err( L_2) ≤ err( L_1) and cost( L_2) ≥ cost( L_1).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-612207c6128de8ac","tex":"\\Lambda_{2,L}","aliases":"Λ_2,L Lambda","display":"\\Lambda_{2,L}","kind":"use","excerpt":"For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9503777fa994def","tex":"\\lambda_a","aliases":"λ_a lambda","display":"\\lambda_a","kind":"use","excerpt":"[Scalar choice invariance] Replacing u_a by λ_au_a, with λ_a∈ C^×, changes c by a coboundary and leaves b_c unchanged.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fbbdbad261784814","tex":"\\lambda_{a+b}^{-1}","aliases":"λ_a+b^-1 lambda","display":"\\lambda_{a+b}^{-1}","kind":"use","excerpt":"The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-def4f51f0a6a5703","tex":"\\lambda_{a,r}","aliases":"λ_a,r lambda","display":"\\lambda_{a,r}","kind":"use","excerpt":"[Translation covariance] Fix a∈ Z^d. For every r∈ R, write uniquely r-a=r_a+λ_a,r, r_a∈ R, λ_a,r∈ L. Then ( A_LT_ax)_r(ℓ) = x_r_a(ℓ+λ_a,r). Thus a fine translation permutes the lattice packets and applies an internal translation on L.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27a452b3d791e541","tex":"\\lambda_b","aliases":"λ_b lambda","display":"\\lambda_b","kind":"use","excerpt":"The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4cbc8bb95a9e6bd","tex":"\\Lambda_c","aliases":"Λ_c Lambda","display":"\\Lambda_c","kind":"use","excerpt":"pageheading3. A nonsingular Euler-operator argumentThe standard interval without an analytic power-series-in-the-index premise On Hurwitz coefficients, the actual Euler operator plus c is (Λ_c a)(d)=(d+c)a(d). Direct induction proves its commutator with every defect iterate: δ^sΛ_c a =Λ_cδ^s a +sp(pU^p^2-U^p)δ^s-1a, s≥1. Suppose a is integral and s","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce644555e3509ab9","tex":"\\Lambda_c^{-1}","aliases":"Λ_c^-1 Lambda","display":"\\Lambda_c^{-1}","kind":"use","excerpt":"When pnmid c, define the actual inverse coefficientwise: [t^d]Λ_c^-1f=frac[t^d]fd+c. On p-supported degrees, d+c is a genuine p-adic unit. The proof establishes Hurwitz integrality of this solution, its support, both inverse identities, and preservation of full flatness using the earlier all-order Euler commutator theorem. The singular pivots c=ap ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18bb5b342b20541e","tex":"\\lambda_d","aliases":"λ_d lambda","display":"\\lambda_d","kind":"use","excerpt":"[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-85199f8ae102ab64","tex":"\\lambda_{i'}","aliases":"λ_i' lambda","display":"\\lambda_{i'}","kind":"use","excerpt":"sely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the condition","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-14e48ea0c939f8be","tex":"\\lambda_i^{(y)}","aliases":"λ_i^(y) lambda","display":"\\lambda_i^{(y)}","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"use","excerpt":"The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c442f1e041987da1","tex":"\\Lambda_{j+1}","aliases":"Λ_j+1 Lambda","display":"\\Lambda_{j+1}","kind":"use","excerpt":"For j=0, g_0 is the literal N-divisibility indicator. Because Nmid p(p-1), the two shifts in δ g_0 agree and δ g_0=0. It is flat at every order. The actual binomial identity gives Λ_j+1g_j+1=(j+1)g_j. As 1≤ j+1<p, the nonsingular reflection theorem inductively proves flatness of every g_j in the standard window. There is no uniform-convergence assu","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0178c33c9a9b20df","tex":"\\lambda_j^{(y)}","aliases":"λ_j^(y) lambda","display":"\\lambda_j^{(y)}","kind":"use","excerpt":"[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32012d2070cfbb63","tex":"\\lambda_j^{-1/2}","aliases":"λ_j^-1/2 lambda","display":"\\lambda_j^{-1/2}","kind":"use","excerpt":"syndrome construction] Assume the Knill--Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a sy","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-974551e97b54067e","tex":"\\lambda_j^\\alpha","aliases":"λ_j^α lambda alpha","display":"\\lambda_j^\\alpha","kind":"use","excerpt":"Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-577467d13815a1aa","tex":"\\lambda_j^{\\alpha+\\beta}","aliases":"λ_j^α+β lambda alpha beta","display":"\\lambda_j^{\\alpha+\\beta}","kind":"use","excerpt":"All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d594552e2786afd4","tex":"\\lambda_k","aliases":"λ_k lambda","display":"\\lambda_k","kind":"use","excerpt":"For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized e","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3b15579ddc9c765","tex":"\\lambda_{k,h}","aliases":"λ_k,h lambda","display":"\\lambda_{k,h}","kind":"use","excerpt":"The kth discrete eigenvalue is λ_k,h=frac4h^2sin^2(frackπ h2), whereas the continuum eigenvalue is (kπ)^2.","count":21},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6ba91b1d53b9b2d0","tex":"\\lambda_{k,h}^\\alpha","aliases":"λ_k,h^α lambda alpha","display":"\\lambda_{k,h}^\\alpha","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a90f856677fe06b","tex":"\\lambda_{k,h}^{\\alpha-1}","aliases":"λ_k,h^α-1 lambda alpha","display":"\\lambda_{k,h}^{\\alpha-1}","kind":"use","excerpt":"The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d96c20ae88ec969","tex":"\\lambda_k^\\alpha","aliases":"λ_k^α lambda alpha","display":"\\lambda_k^\\alpha","kind":"use","excerpt":"[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec937ee0e7e9a617","tex":"\\lambda_k^\\beta","aliases":"λ_k^β lambda beta","display":"\\lambda_k^\\beta","kind":"use","excerpt":"All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-129bdd0acbdea966","tex":"\\lambda_m","aliases":"λ_m lambda","display":"\\lambda_m","kind":"use","excerpt":"[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0321c3118b993cea","tex":"\\lambda_m^{-1}","aliases":"λ_m^-1 lambda","display":"\\lambda_m^{-1}","kind":"use","excerpt":"on ℓ^2(Λ_N) gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries. vfigproduct headFinite compression is an approximation For N=2, λ_m=π^2(m+1/2)^2. The finite determinant P_L is exactly the displayed finite product, but it is not exactly Δ_2. For 0≤ w≤1 and L≥1, an elementary tail estimate gives 0≤","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-298c99f06a4e2b99","tex":"\\lambda_m^{-s}","aliases":"λ_m^-s lambda","display":"\\lambda_m^{-s}","kind":"use","excerpt":"For N≥3, [source reference] writes all but finitely many reduced zeros as λ_m = (A_N(m+frac12))^N(1+ε_m) where λ_m>0, 1+ε_m>0, and ε_m = O(t_m^-1e^-γ t_m). On every compact set of s values, (1+ε_m)^-s-1 = O(|ε_m|) uniformly for large m. The difference λ_m^-s - A_N^-Ns(m+frac12)^-Ns therefore has a normally convergent sum on every compact subset of ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1dc8e3b2125ff219","tex":"\\lambda_{\\min}","aliases":"λ_min lambda min","display":"\\lambda_{\\min}","kind":"use","excerpt":"d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differentiation gives d_s(τ)=2τ frac∑_jλ_j e^-τλ_j∑_j e^-τλ_j. It tends to zero as τdownarrow0 and, under the stated assumption of a zero mode, as τ→∞. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to 2τλ_min. The conclusion for ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8b3c2cf514ae34fe","tex":"\\Lambda_{N,j}","aliases":"Λ_N,j Lambda","display":"\\Lambda_{N,j}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c1a2a4aeea8926c","tex":"\\Lambda_N^*","aliases":"Λ_N^* Lambda","display":"\\Lambda_N^*","kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7daa0edaa864bbe2","tex":"\\Lambda_p","aliases":"Λ_p Lambda","display":"\\Lambda_p","kind":"use","excerpt":"0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35626feb35e38e3f","tex":"\\Lambda_{p,L}","aliases":"Λ_p,L Lambda","display":"\\Lambda_{p,L}","kind":"use","excerpt":"For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c28ddbaa56138f5b","tex":"\\lambda_{p,r}","aliases":"λ_p,r lambda","display":"\\lambda_{p,r}","kind":"use","excerpt":"[Unified inverse local transfer] Define λ_p,r = 1, · p odd and r=0, 2^2r-1p^2r, · p odd and r≥1, dfrac2^4r+1(2r+1)(2r+2), · p=2. Then for every prime p and every even n≥0, A_p,n = ∑_r=0^n/2 binom n2r λ_p,r E_n-2r. Equivalently, if L_p(t)=K_p(t)^-1, then H_p(t)=L_p(t)secht, where L_p(t)=cosh^2(pt) for odd p, and L_2(t)=(fracsinh(2t)2t)^2 for p=2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4da8500e158d102","tex":"\\Lambda_{\\Pcal}","aliases":"Λ_Pcal Λ_ P Lambda Pcal Lambda mathcal","display":"\\Lambda_{{\\mathcal P}}","kind":"use","excerpt":"[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7781b9f2b77d297f","tex":"\\Lambda_{\\Qcal}","aliases":"Λ_Qcal Λ_ Q Lambda Qcal Lambda mathcal","display":"\\Lambda_{{\\mathcal Q}}","kind":"use","excerpt":"[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-36442f76e90ff174","tex":"\\lambda_r","aliases":"λ_r lambda","display":"\\lambda_r","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":16},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad7d2db293e5cbb7","tex":"\\lambda_r^{-1/2}","aliases":"λ_r^-1/2 lambda","display":"\\lambda_r^{-1/2}","kind":"use","excerpt":"Null directions are removed by proof. If λ_r=0, then B_r^*B_r=0. Positivity of the ordinary complex matrix inner product gives B_r=0. This is the checked declaration pathDiagonalErrors.zero_weight_error. The normalized syndrome operators are defined only on A_+={r:λ_r>0}, V_r=λ_r^-1/2B_r. Consequently no inverse square root at a null eigenvalue is ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e08b44bc605b92e6","tex":"\\Lambda_s","aliases":"Λ_s Lambda","display":"\\Lambda_s","kind":"use","excerpt":"[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Onl","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f8396a195799c469","tex":"\\Lambda_{y'}^*","aliases":"Λ_y'^* Lambda","display":"\\Lambda_{y'}^*","kind":"use","excerpt":"da_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ∈ P( T) |J_y'(Λ)-J_y(Λ)| <fracΓ_y2, then Λ_y'^*=Λ_y^*. item Every selected mini","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9159b41ba5090052","tex":"\\Lambda_y^*","aliases":"Λ_y^* Lambda","display":"\\Lambda_y^*","kind":"use","excerpt":"[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12eeafab07a92182","tex":"\\mu^{-m}","aliases":"μ^-m mu","display":"\\mu^{-m}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b453de64aef4dc5","tex":"\\mu_0","aliases":"μ_0 mu","display":"\\mu_0","kind":"use","excerpt":"odd: T(m,1)=2^m-1 and v_2(m!)=m-1. For m≥4 a power of two, the r=3 term is even. More explicitly, fracT(m,3)2^m-3=h_m-3(1,3,6), ∑_d≥0h_d(1,3,6)t^d =frac1(1-t)(1-3t)(1-6t). Reduction modulo two gives (1-t)^-2, whose degree m-3 coefficient has parity m-2=0. At m=1,2 the r=3 term is absent, and the r=1 term is again odd. Finally μ_0=1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ac6cb066658534c5","tex":"\\mu_1","aliases":"μ_1 mu","display":"\\mu_1","kind":"use","excerpt":"[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a202073a149b5a7a","tex":"\\mu_2","aliases":"μ_2 mu","display":"\\mu_2","kind":"use","excerpt":"-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d1bb2e4dde90487","tex":"\\mu_a","aliases":"μ_a mu","display":"\\mu_a","kind":"use","excerpt":"this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even because 4mid k. The other contribution expands as L(Qγ_k-1(P+Q)) =∑_a=0^k-1(k-a)μ_aμ_k-a. An odd summand would req","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32c948dcbad490ee","tex":"\\mu_{\\frakp}","aliases":"μ_frakp μ_ p mu frakp mu mathfrak","display":"\\mu_{{\\mathfrak p}}","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c0ba94f6308ea27","tex":"\\mu_h","aliases":"μ_h mu","display":"\\mu_h","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-252454ad397ec5a8","tex":"\\mu_i","aliases":"μ_i mu","display":"\\mu_i","kind":"use","excerpt":"The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7976255081ad4eb3","tex":"\\mu_j","aliases":"μ_j mu","display":"\\mu_j","kind":"use","excerpt":"The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d863225288067393","tex":"\\mu_{k-a}","aliases":"μ_k-a mu","display":"\\mu_{k-a}","kind":"use","excerpt":"this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even because 4mid k. The other contribution expands as L(Qγ_k-1(P+Q)) =∑_a=0^k-1(k-a)μ_aμ_k-a. An odd summand would req","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2ffb97b3578a68df","tex":"\\mu_m","aliases":"μ_m mu","display":"\\mu_m","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"use","excerpt":"[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the d","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22d0564d0934fa50","tex":"\\mu_r","aliases":"μ_r mu","display":"\\mu_r","kind":"use","excerpt":"The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series give","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c381b1e9f2cf6895","tex":"\\nu_2","aliases":"ν_2 nu","display":"\\nu_2","kind":"use","excerpt":"[All-row dyadic higher differences] For 0≤ j<4 and all u,k,n∈ N, C_j(n) · ∈ Z_2, Δ_u^kC_j(n) · ∈ 2^k(1+ν_2(u)) Z_2, Δ_u^kU_j(n) · ∈ 2^k(1+ν_2(u))+v_2(j!) Z_2. There is no restriction k≤ n and no minimum with the row n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0661e230ba933432","tex":"\\nu_j","aliases":"ν_j nu","display":"\\nu_j","kind":"use","excerpt":"[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). ","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c645657d025cc351","tex":"\\Xi","aliases":"Ξ Xi","display":"\\Xi","kind":"use","excerpt":"set, so the feasible set is nonempty and finite. Thus the minimum is attained. If competitors exist, their finitely many differences from a unique optimum are strictly positive, so their minimum is positive and is achieved by a competitor. In the singleton case there is no alternative selector under any objective change. In the remaining case, for ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-94e907efc315bf5a","tex":"\\xi^{-1}","aliases":"ξ^-1 xi","display":"\\xi^{-1}","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e8149ec697a9ee64","tex":"\\xi^2","aliases":"ξ^2 xi","display":"\\xi^2","kind":"use","excerpt":"vpage6Addition is quotient multiplicationFINITE ALGEBRAChapter IV: Proposition 5.5, Theorem 6.2. Work in C[ξ]/(ξ^3+1). Every element has coordinates in 1,ξ,ξ^2. The entire exponential has the finite-coordinate form e^zξ=D_3,0(z)+D_3,1(z)ξ+D_3,2(z)ξ^2. The multiplication table makes the carry sign unavoidable. vfigsigned_addition Multiply the coordi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab96ce3cb4126eb6","tex":"\\xi^3","aliases":"ξ^3 xi","display":"\\xi^3","kind":"use","excerpt":"vpage6Addition is quotient multiplicationFINITE ALGEBRAChapter IV: Proposition 5.5, Theorem 6.2. Work in C[ξ]/(ξ^3+1). Every element has coordinates in 1,ξ,ξ^2. The entire exponential has the finite-coordinate form e^zξ=D_3,0(z)+D_3,1(z)ξ+D_3,2(z)ξ^2. The multiplication table makes the carry sign unavoidable. vfigsigned_addition Multiply the coordi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b835423857234f4","tex":"\\xi^j","aliases":"ξ^j xi","display":"\\xi^j","kind":"use","excerpt":"[Finite algebra exponential identities] In A_N^+ one has e^λ zξ = ∑_j=0^N-1 E_N,j(λ z)ξ^j. In A_N^- one has e^zξ = ∑_j=0^N-1 D_N,j(z)ξ^j. Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd09c41aee101fea","tex":"\\xi^{N-1}","aliases":"ξ^N-1 xi","display":"\\xi^{N-1}","kind":"use","excerpt":"Differentiation multiplies by λξ in the first algebra and by ξ in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity e^(z+w)ξ=e^zξe^wξ gives the addition laws after comparison of the coefficients of 1,ξ,…,ξ^N-1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a52c871cc191c65","tex":"\\xi^{Nn+j}","aliases":"ξ^Nn+j xi","display":"\\xi^{Nn+j}","kind":"use","excerpt":"Expand the exponential and write every exponent uniquely as Nn+j. In A_N^+, the relation ξ^Nn+j=ξ^j gives the transfer sections. In A_N^-, the relation ξ^Nn+j=(-1)^nξ^j gives the signed sections.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-946a83374eeeebd5","tex":"\\xi^{\\Phi^j}","aliases":"ξ^Φ^j xi Phi","display":"\\xi^{\\Phi^j}","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e2af9f8f06f93e6","tex":"\\xi_{L,n}","aliases":"ξ_L,n xi","display":"\\xi_{L,n}","kind":"use","excerpt":"[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5499160581543767","tex":"\\pi^{-2s}","aliases":"π^-2s pi","display":"\\pi^{-2s}","kind":"use","excerpt":"[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a2ecbf2708f1e683","tex":"\\pi^{-s}","aliases":"π^-s pi","display":"\\pi^{-s}","kind":"use","excerpt":"[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Onl","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"ct uses every reduced zero with its multiplicity. Its logarithm turns multiplication into addition, so reciprocal-zero power sums become coefficients. This step depends on the global product theorem. A truncated zero list provides a numerical partial sum, while the identity uses the entire multiset. modelfigurezero_momentsOrder-two moment sums over","count":18},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f174b662c92183fc","tex":"\\varpi^2","aliases":"ϖ^2 varpi","display":"\\varpi^2","kind":"use","excerpt":"[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-543449b1bd1d6dc3","tex":"\\pi^{2q}","aliases":"π^2q pi","display":"\\pi^{2q}","kind":"use","excerpt":"[The cosine orbit sums] For N=2, Δ_2(w)=cossqrt w. Its zeros are λ_m=π^2(m+frac12)^2, m≥0. Hence σ_2,q = frac2^2q-1π^2qζ(2q). The identity in [source reference] recovers the classical tangent coefficients from these zero power sums.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f4875cececfa7801","tex":"\\pi^4","aliases":"π^4 pi","display":"\\pi^4","kind":"use","excerpt":"st four-point orbit has fourth power 1/16. The second has fourth power 81/16. headMultiplicity is not multiplied in the reduced function At a nonzero point, z↦ z^4 has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of D_4 therefore correspond to one simple zero of Δ_4, not one zero of multiplicity four","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bca6e7d02878c20b","tex":"\\pi^a","aliases":"π^a pi","display":"\\pi^a","kind":"use","excerpt":"[A triangular transfer] Let O be complete with uniformizer π, and consider T= 1 · π^a 0 · 1 with a≥1. The only nontrivial transfer has weight a, and its square is zero. Hence T^-1 = 1 · -π^a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5602fb671a74bd1","tex":"\\pi^{a_1}","aliases":"π^a_1 pi","display":"\\pi^{a_1}","kind":"use","excerpt":"[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9449729f152b345","tex":"\\pi^{a_j}","aliases":"π^a_j pi","display":"\\pi^{a_j}","kind":"use","excerpt":"ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-57b9f29f6af58bd3","tex":"\\pi^{a_n}","aliases":"π^a_n pi","display":"\\pi^{a_n}","kind":"use","excerpt":"[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0333e1da9f88d77d","tex":"\\pi^c","aliases":"π^c pi","display":"\\pi^c","kind":"use","excerpt":"For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ce2f60287017fac","tex":"\\pi^{cn}","aliases":"π^cn pi","display":"\\pi^{cn}","kind":"use","excerpt":"R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc344e45856ed8a8","tex":"\\pi^{cn+1}","aliases":"π^cn+1 pi","display":"\\pi^{cn+1}","kind":"use","excerpt":"R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12d8be8cdf767d11","tex":"\\pi^d","aliases":"π^d pi","display":"\\pi^d","kind":"use","excerpt":"Let s=s_Λ( P). For every x∈Λ and every α, π^sE_α x∈Λ_α. Hence π^sx=∑_απ^sE_α x∈Λ_split, so π^sG_Λ, P=0. This proves exp_π(G_Λ, P)≤ s. If d<s, then for some α there is an x∈Λ with π^dE_α x∉Λ_α. The class π^d[x] is therefore nonzero in the gluing module. Thus π^d does not annihilate the gluing module, and equality follows. The final formula is the us","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-462bf49f74c1b15a","tex":"\\pi^e","aliases":"π^e pi","display":"\\pi^e","kind":"use","excerpt":"Finite-group comparison with precision loss. Let R be a commutative domain, Γ a finite group, and ρ,τ actual integral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3cd9f08d1ff6f7d","tex":"\\pi^{e+k}","aliases":"π^e+k pi","display":"\\pi^{e+k}","kind":"use","excerpt":"a commutative domain, Γ a finite group, and ρ,τ actual integral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-532d6911acc103ff","tex":"\\varpi^{em}","aliases":"ϖ^em varpi","display":"\\varpi^{em}","kind":"use","excerpt":"Counting exponent versus compatible branches. The preceding exponent counts all finite-level solutions, including nodes which cannot lift indefinitely. It is not the cardinality or dimension of the space of infinite branches. In fact every compatible tower of solutions to x^e=0, for e≥1, is the zero tower. For each m, a solution at depth em lies in","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5eb5dc2722ab8e4","tex":"\\varpi^j","aliases":"ϖ^j varpi","display":"\\varpi^j","kind":"use","excerpt":"For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc5afa9dc6cc1828","tex":"\\pi^k","aliases":"π^k pi","display":"\\pi^k","kind":"use","excerpt":"tegral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E integral. Choose b with π^e=ab and put P=A+bπ^kE. Then aP=","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9c12bb7f86fa0f0","tex":"\\pi^L","aliases":"π^L pi","display":"\\pi^L","kind":"use","excerpt":"If O is a complete discrete valuation ring with uniformizer π and B is a separated and complete O algebra for the filtration F^L B=π^L B, then this is a standard example. Finite matrix algebras over O satisfy these conditions. Completeness of O alone does not imply completeness of every O algebra. Basic facts about completeness in this setting are ","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8a5ad8506c158aa","tex":"\\varpi^{\\lceil m/e\\rceil}","aliases":"ϖ^lceil m/erceil varpi lceil rceil","display":"\\varpi^{\\lceil m/e\\rceil}","kind":"use","excerpt":"The congruence is equivalent to e v_ϖ(x)≥ m, so x must lie in ϖ^lceil m/erceil O/ϖ^m, which has the stated cardinality. Write m=ek+r, 1≤ r≤ e. The exponent becomes (e-1)k+r-1. Summing over k≥0 and 1≤ r≤ e gives the rational expression.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0de50b515ae97983","tex":"\\pi^m","aliases":"π^m pi","display":"\\pi^m","kind":"use","excerpt":"Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-724adebe365a3a3e","tex":"\\pi^{m+d}","aliases":"π^m+d pi","display":"\\pi^{m+d}","kind":"use","excerpt":"n delay] Let E be a rational projector on V, let Λ be a lattice, and let s=s_Λ(E). For every m≥0, the rule [x]_m+s⟼[Ex]_m from Λ/π^m+sΛ to V/π^mΛ is well defined. Here m is an integer. If d is an integer with 0≤ d<s, then for every integer m≥0 there are x,y∈Λ such that x≡ ypmodπ^m+dΛ but Exnot≡ Eypmodπ^mΛ. Thus s extra digits are both sufficient an","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae5fc9105be497c3","tex":"\\pi^{m+s}","aliases":"π^m+s pi","display":"\\pi^{m+s}","kind":"use","excerpt":"[Projector precision delay] Let E be a rational projector with π^sE(Λ)⊆Λ. If x≡ ypmodπ^m+sΛ, then Ex≡ Eypmodπ^mΛ. Thus m+s input digits suffice to determine m output digits of the projected packet.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f09d0f0a6806b7c1","tex":"\\pi^{\\min(s_i,L)}","aliases":"π^min(s_i,L) pi min","display":"\\pi^{\\min(s_i,L)}","kind":"use","excerpt":"[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42ba626e1f091ffe","tex":"\\pi^n","aliases":"π^n pi","display":"\\pi^n","kind":"use","excerpt":"0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d5468581325c1fe5","tex":"\\varpi^{n+1}","aliases":"ϖ^n+1 varpi","display":"\\varpi^{n+1}","kind":"use","excerpt":"For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a66c418313dc8ab5","tex":"\\Pi^{\\mathrm{phys}}","aliases":"Π^phys Pi mathrm","display":"\\Pi^{\\mathrm{phys}}","kind":"use","excerpt":"At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce46406e805e52d0","tex":"\\pi^{s_1}","aliases":"π^s_1 pi","display":"\\pi^{s_1}","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a34afdce45841e7","tex":"\\pi^{s_i}","aliases":"π^s_i pi","display":"\\pi^{s_i}","kind":"use","excerpt":"Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L cho","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a901ac9ade8abb67","tex":"\\pi^{s_r}","aliases":"π^s_r pi","display":"\\pi^{s_r}","kind":"use","excerpt":"[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27f57e5aa2f64147","tex":"\\pi^{\\val_\\pi(\\Res(f,g))}","aliases":"π^val_π(Res(f,g)) pi val pi Res pi operatorname pi operatorname","display":"\\pi^{{\\operatorname{val}}_\\pi({\\operatorname{Res}}(f,g))}","kind":"use","excerpt":"[Resultant denominator bound] Under the hypotheses of [source reference], one may take r=Res(f,g) up to multiplication by a unit. Therefore π^val_π(Res(f,g))E_fΛ⊆Λ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67582a9a61e97f9d","tex":"\\varpi^{w_i}","aliases":"ϖ^w_i varpi","display":"\\varpi^{w_i}","kind":"use","excerpt":"A simple weighted formal model already captures delayed visibility. Let w_i≥1 and constrain t_i∈(ϖ^w_i O+ϖ^m O)/ϖ^m O. Then coordinate i is invisible until m>w_i.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-222fb49e971fef73","tex":"\\pi_1","aliases":"π_1 pi","display":"\\pi_1","kind":"use","excerpt":"[Gauge classification of transports] Let G⊆GL(X) be the chosen structure group and fix a root r. Restrict both edge transports and gauge values to G. Gauge equivalence classes of G valued transport systems are naturally identified with Hom(π_1(Γ,r),G)/G, where G acts by conjugation. After a spanning tree and orientations of the remaining edges are ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7b208ba9a56ce9d","tex":"\\varpi_K","aliases":"ϖ_K varpi","display":"\\varpi_K","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e2bc13eba5395cb2","tex":"\\pi_O","aliases":"π_O pi","display":"\\pi_O","kind":"use","excerpt":"ion] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3315617fe9f782aa","tex":"\\Pi_{\\Pcal}","aliases":"Π_Pcal Π_ P Pi Pcal Pi mathcal","display":"\\Pi_{{\\mathcal P}}","kind":"use","excerpt":"[Exact block discovery certificate] For an orthogonal packet system (P_α), define the block diagonal compression Π_ P(A) = ∑_α P_α AP_α. Then normA-Π_ P(A)_rm F^2 = ∑_α≠β normP_β AP_α_rm F^2. In operator norm, normA-Π_ P(A)_2 ≤ ∑_α≠β normP_β AP_α_2. Thus any proposed packet discovery has an exact finite residual certificate after its projectors are","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-666f17fd05c1382b","tex":"\\pi_R","aliases":"π_R pi","display":"\\pi_R","kind":"use","excerpt":"hmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fbd6955f854c7243","tex":"\\pi_{\\src(e)}","aliases":"π_src(e) pi src pi operatorname","display":"\\pi_{{\\operatorname{src}}(e)}","kind":"use","excerpt":"[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6553fa41ca8fbe5","tex":"\\pi_x","aliases":"π_x pi","display":"\\pi_x","kind":"use","excerpt":"tion below to be continuous. Assume the following data. item A represented deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d416b19c5cb4e834","tex":"\\pi_Y","aliases":"π_Y pi","display":"\\pi_Y","kind":"use","excerpt":"ather At the base parameter, the strict Fr echet derivative is Dψ(t_0)=-J^-1K. In particular, ψ is differentiable on U. Let g be the analytic local inverse of Φ supplied by Lemma~[source reference]. Since F(a)=0, Φ(a)=(0,t_0) and g(0,t_0)=a. Define ψ(t)=π_Yg(0,t), where π_Y projects to the solved coordinates. This gives the base value and analytici","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9eac9da427d5101f","tex":"\\pi_\\beta","aliases":"π_β pi beta","display":"\\pi_\\beta","kind":"use","excerpt":"[Exact evaluation] Let A∈End(X). Set B_βα=π_β Aι_α and let E contain exactly the nonzero blocks. Then A= ∑_(β,α)∈ E P_β AP_α. For every matrix of right hand sides Z, AZ = ∑_(β,α)∈ E ι_β B_βαπ_α Z.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1acb7717088ec6fe","tex":"\\pi_\\lambda","aliases":"π_λ pi lambda","display":"\\pi_\\lambda","kind":"use","excerpt":"[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81626ca27a10136c","tex":"\\Pi_\\mu","aliases":"Π_μ Pi mu","display":"\\Pi_\\mu","kind":"use","excerpt":"Π_μObΠ_ν · Block from input to output of Ob=H-I. · pagerefV16 to pagerefV18","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fce0b633f1baa40a","tex":"\\pi_\\mu","aliases":"π_μ pi mu","display":"\\pi_\\mu","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d46c1d0799d88384","tex":"\\Pi_\\nu","aliases":"Π_ν Pi nu","display":"\\Pi_\\nu","kind":"use","excerpt":"Π_μObΠ_ν · Block from input to output of Ob=H-I. · pagerefV16 to pagerefV18","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b9f2dbc20110e35","tex":"\\Pi_\\ell^{\\mathrm{phys}}","aliases":"Π_ℓ^phys Pi ell mathrm","display":"\\Pi_\\ell^{\\mathrm{phys}}","kind":"use","excerpt":"At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-789d514d8bf742cb","tex":"\\rho^{-j}","aliases":"ρ^-j rho","display":"\\rho^{-j}","kind":"use","excerpt":"[Relation with the transfer exponential] For every j, D_N,j(z) = ρ^-j E_N,j(ρ z).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-72962f18025874b5","tex":"\\rho^{-m}","aliases":"ρ^-m rho","display":"\\rho^{-m}","kind":"use","excerpt":"rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd1aac0f414abebd","tex":"\\rho^2","aliases":"ρ^2 rho","display":"\\rho^2","kind":"use","excerpt":"For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a0f42f3310ef6b4e","tex":"\\rho^{i+j-k}","aliases":"ρ^i+j-k rho","display":"\\rho^{i+j-k}","kind":"use","excerpt":"Use [source reference]. The factor introduced by a pair (i,j) is ρ^i+j-k. Since 0≤ i+j≤2N-2 and i+j≡ kbmod N, this factor is one when i+j=k and -1 when i+j=k+N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4480c8bdf9c98bc","tex":"\\rho^N","aliases":"ρ^N rho","display":"\\rho^N","kind":"use","excerpt":"Fix any ρ∈ C with ρ^N=-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df63b2d5a9b79da9","tex":"\\rho^{n-1}","aliases":"ρ^n-1 rho","display":"\\rho^{n-1}","kind":"use","excerpt":"[Norm bound] For a submultiplicative operator norm, CT_f^n-T_c^nC ≤ Δ ∑_k=0^n-1 T_c ^n-1-k T_f ^k. If T_f , T_c ≤ρ, then CT_f^n-T_c^nC ≤ nρ^n-1 Δ .","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c4d50634b899cc2","tex":"\\rho^{Nn+j}","aliases":"ρ^Nn+j rho","display":"\\rho^{Nn+j}","kind":"use","excerpt":"The coefficient of z^Nn+j on the right is ρ^-jρ^Nn+jfrac1(Nn+j)! = (-1)^nfrac1(Nn+j)!.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6fd3bc029e9c79bd","tex":"\\rho_+","aliases":"ρ_+ rho","display":"\\rho_+","kind":"use","excerpt":"For ρ_+=frac12mat1 · 1 1 · 1, the exact output is Φ(ρ_+)=mat17/25 · 2/5","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","excerpt":"The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-197a543b67174e7f","tex":"\\rho_{00}","aliases":"ρ_00 rho","display":"\\rho_{00}","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75b2b525a8ae5055","tex":"\\rho_{01}","aliases":"ρ_01 rho","display":"\\rho_{01}","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-737cdd19f67dfe28","tex":"\\rho_1","aliases":"ρ_1 rho","display":"\\rho_1","kind":"use","excerpt":"0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-90dbd799c3b71089","tex":"\\rho_{10}","aliases":"ρ_10 rho","display":"\\rho_{10}","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7aad9eff5723a02c","tex":"\\rho_{11}","aliases":"ρ_11 rho","display":"\\rho_{11}","kind":"use","excerpt":"0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7ba0ffb2518dd65","tex":"\\rho_2","aliases":"ρ_2 rho","display":"\\rho_2","kind":"use","excerpt":"[Chebotarev and Brauer Nesbitt identification] Let E' be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let ρ_1,ρ_2:G_F,S⟶GL_n(E') be continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then ρ_1≅ρ_2. This follows f","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9bf4e85984084b75","tex":"\\rho_a","aliases":"ρ_a rho","display":"\\rho_a","kind":"use","excerpt":"The bound is substantive: for the unipotent actions ρ_a(g)=1 · ag 0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b48fafc33330554","tex":"\\rho_c","aliases":"ρ_c rho","display":"\\rho_c","kind":"use","excerpt":"≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U_c→ U_m by restriction to the selected copies. Lean proves rj=1_U_m, ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93b3fea452ee88ac","tex":"\\rho_D","aliases":"ρ_D rho","display":"\\rho_D","kind":"use","excerpt":"For supplied covariance data and the twisted cocycle of the hyperref[book:x-construction-13]finite-exponential reconstruction, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restri","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-80f34d32227ee3d9","tex":"\\rho_E","aliases":"ρ_E rho","display":"\\rho_E","kind":"use","excerpt":"For supplied covariance data and the twisted cocycle of the hyperref[book:x-construction-13]finite-exponential reconstruction, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restri","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-56a93667702b9f53","tex":"\\rho_{i,\\lambda}","aliases":"ρ_i,λ rho lambda","display":"\\rho_{i,\\lambda}","kind":"use","excerpt":"For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesse","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18406af98fd43046","tex":"\\rho_m","aliases":"ρ_m rho","display":"\\rho_m","kind":"use","excerpt":"Checked representation-level completion and its arithmetic boundary. The representation-level completion construction works with actual matrix representations of a fixed group. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Multiplication ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e4e2e93725a721e","tex":"\\rho_{m,n}","aliases":"ρ_m,n rho","display":"\\rho_{m,n}","kind":"use","excerpt":"[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally un","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8dd03dc639dfd669","tex":"\\rho_{m,\\ell}","aliases":"ρ_m,ℓ rho ell","display":"\\rho_{m,\\ell}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c03851211fa61002","tex":"\\rho_n","aliases":"ρ_n rho","display":"\\rho_n","kind":"use","excerpt":"Local properties need their own limit argument. Over a domain, for a nonzero π∈ I, the representations of the additive integer group given by ρ_n(g)=1 · π^n g 0 · 1 are all nontrivial and converge pointwise in the ideal-adic topology to the trivial representation. The convergence and the failure of limit preservation for nontriviality are formally ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98944c463486b854","tex":"\\rho_{n,m}","aliases":"ρ_n,m rho","display":"\\rho_{n,m}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af3400ef85124560","tex":"\\rho_{n,\\ell}","aliases":"ρ_n,ℓ rho ell","display":"\\rho_{n,\\ell}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a247531173a01df","tex":"\\rho_p","aliases":"ρ_p rho","display":"\\rho_p","kind":"use","excerpt":"The earlier obstruction used the integer-domain map J_x=ρ_p∘ DF(x): Z^d→ F_p^e. The new count uses A_x: F_p^d→ F_p^e. Reduction of integer vectors onto finite vectors is surjective, so the two maps have the same image after restricting scalars. Lean constructs a class-preserving integer-linear equivalence F_p^e/imJ_x ≃ F_p^e/imA_x. Consequently the","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-83765f8c06700277","tex":"\\rho_{\\rm Bell}","aliases":"ρ_rm Bell rho rm","display":"\\rho_{\\rm Bell}","kind":"use","excerpt":"lassical input II.7.7. Ordinary transpose T sends a positive matrix on one system to another positive matrix and preserves its trace. Those properties are insufficient for a quantum channel on a subsystem of an entangled state. The test must include id⊗ T on the larger system. vfigcomplete_positivity Apply this extended map to the Bell density matr","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-faf2d9e8110151a4","tex":"\\rho_t","aliases":"ρ_t rho","display":"\\rho_t","kind":"use","excerpt":"Write ρ_t= e^-t=1-t+O(t^2), σ_t^2=1- e^-2t=2t+O(t^2). For a polynomial f, Taylor expansion at x and Gaussian integration give O_tf(x) = · f(x) +f'(x)(ρ_t-1)x +frac12f''(x)σ_t^2 +o(t)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9e625090d5023eb","tex":"\\rho_x^{\\mathrm{ss}}","aliases":"ρ_x^ss rho mathrm","display":"\\rho_x^{\\mathrm{ss}}","kind":"use","excerpt":"tomorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ecaf052e23094fe","tex":"\\rho_\\eta","aliases":"ρ_η rho eta","display":"\\rho_\\eta","kind":"use","excerpt":"actual amplitude vector is (- i,1, i,1)^T/2, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state. headA channel changes the conclusion predictably Before evolving, mix the seed density matr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc125fdd0aa28df4","tex":"\\rho_\\lambda","aliases":"ρ_λ rho lambda","display":"\\rho_\\lambda","kind":"use","excerpt":"pine C over a number field M is fixed; item for every finite place λ of M and every m≥1, the companion level X_λ,m( C) is finite and nonempty; item the reduction forests are finitely branching; item the selected local conditions are closed under inverse limits. Then for every λ there is an infinite compatible branch ρ_λ∈varprojlim_mX_λ,m( C), and t","count":11},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d43c4345dfbfdfa8","tex":"\\rho_{\\lambda_0}","aliases":"ρ_λ_0 rho lambda","display":"\\rho_{\\lambda_0}","kind":"use","excerpt":"[One member propagation in a fixed system] Let {ρ_λ}_λ∈Λ be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same po","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b2485d155e7ba40","tex":"\\rho_{\\pi,\\lambda}","aliases":"ρ_π,λ rho pi lambda","display":"\\rho_{\\pi,\\lambda}","kind":"use","excerpt":"For propagation, suppose both a semisimple compatible family {ρ_λ} containing ρ_x and a family {ρ_π,λ} attached to this same π already exist. Require the distinguished members to be ρ_x and its attached representation, respectively. Assume the coefficient, continuity, density, semisimplification and common algebraic Frobenius spine hypotheses of [s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc168b6d53caa485","tex":"\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}","aliases":"ρ_π,λ^ss rho pi lambda mathrm","display":"\\rho_{\\pi,\\lambda}^{\\mathrm{ss}}","kind":"use","excerpt":"be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same polynomials P_v(X). Then for every λ∈Λ, ρ_λ ≅ ρ_π,λ^ss.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbb82a7b6e852fe0","tex":"\\rho_{\\pi_x}","aliases":"ρ_π_x rho pi","display":"\\rho_{\\pi_x}","kind":"use","excerpt":"deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') w","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9264ae4ae2ab0aab","tex":"\\rho_{\\ell,k}","aliases":"ρ_ℓ,k rho ell","display":"\\rho_{\\ell,k}","kind":"use","excerpt":"Principal targets. targetFiniteUnits.finite_unit_decomposition, .power_lift_congruence targetFiniteUnits.omega_unique, .decomposition_unique targetFiniteUnits.omega_mul, .principal_mul pageheading3. Precision compatibility and the dyadic signFinite refinements and Proposition 6.2's dyadic branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cfe0654529d554f8","tex":"\\rho_{\\ell,\\ell}","aliases":"ρ_ℓ,ℓ rho ell ell","display":"\\rho_{\\ell,\\ell}","kind":"use","excerpt":"A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-81d80f06b0492996","tex":"\\bm\\sigma","aliases":"σ bm sigma","display":"\\bm\\sigma","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-366e9732bdd10fbb","tex":"\\boldsymbol\\sigma","aliases":"σ boldsymbol sigma","display":"\\boldsymbol\\sigma","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18866b84545a9559","tex":"\\Sigma'","aliases":"Σ' Sigma","display":"\\Sigma'","kind":"use","excerpt":"[Uniform self energy bounds] Let δ=dist(z,Spec D)>0, g=norm B. Then normΣ(z)≤fracg^2δ and normΣ'(z)≤fracg^2δ^2. For real E,E' in an interval at distance at least δ from Spec D, normΣ(E)-Σ(E') ≤ fracg^2δ^2absE-E'.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5048c4b9123eb530","tex":"\\sigma_1","aliases":"σ_1 sigma","display":"\\sigma_1","kind":"use","excerpt":"cient. All symbols in the plate refer to this one order. vfigworked_circuit headRoute one: invert the denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: use the zero moments The exact reduced zeros are 4π^4(m+1/2)^4. Their first two power sums are σ_","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cce6a80c926063f","tex":"\\sigma_1^2","aliases":"σ_1^2 sigma","display":"\\sigma_1^2","kind":"use","excerpt":"he denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: use the zero moments The exact reduced zeros are 4π^4(m+1/2)^4. Their first two power sums are σ_1=1/24 and σ_2=17/10080. The Bell/Newton identity gives the same reciprocal coefficient (σ_1^2+σ_2)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15fde97ab40a2e94","tex":"\\sigma_2","aliases":"σ_2 sigma","display":"\\sigma_2","kind":"use","excerpt":"n the plate refer to this one order. vfigworked_circuit headRoute one: invert the denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: use the zero moments The exact reduced zeros are 4π^4(m+1/2)^4. Their first two power sums are σ_1=1/24 and σ_2=17/10","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5020a90c1801a5f","tex":"\\sigma_{2,1}","aliases":"σ_2,1 sigma","display":"\\sigma_{2,1}","kind":"use","excerpt":"Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40f9f58c2d96ae63","tex":"\\sigma_{2,2}","aliases":"σ_2,2 sigma","display":"\\sigma_{2,2}","kind":"use","excerpt":"Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-99cd8f126976a70b","tex":"\\sigma_{2,3}","aliases":"σ_2,3 sigma","display":"\\sigma_{2,3}","kind":"use","excerpt":"Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0fb5544cca42028e","tex":"\\sigma_{2,4}","aliases":"σ_2,4 sigma","display":"\\sigma_{2,4}","kind":"use","excerpt":"Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc14526e5f22b718","tex":"\\sigma_{2,q}","aliases":"σ_2,q sigma","display":"\\sigma_{2,q}","kind":"use","excerpt":"[The cosine orbit sums] For N=2, Δ_2(w)=cossqrt w. Its zeros are λ_m=π^2(m+frac12)^2, m≥0. Hence σ_2,q = frac2^2q-1π^2qζ(2q). The identity in [source reference] recovers the classical tangent coefficients from these zero power sums.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7cc70e05d3b03d7a","tex":"\\sigma_a","aliases":"σ_a sigma","display":"\\sigma_a","kind":"use","excerpt":"vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELChapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58c1a4cd175eaace","tex":"\\sigma_d","aliases":"σ_d sigma","display":"\\sigma_d","kind":"use","excerpt":"Spectral optimality from a kernel witness. textitModules DiagonalTruncation and SpectralOptimality Let σ_1≥⋯≥σ_d≥0 and D=diag(σ_1,…,σ_d), D_k=diag(σ_1,…,σ_min(k,d),0,…,0). Lean uses indices beginning at zero, while the formulas on this page use the conventional one-based notation. The rank of D_k is proved to be at most k, even when retained singul","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3443a030a9ee99f2","tex":"\\sigma_d^2","aliases":"σ_d^2 sigma","display":"\\sigma_d^2","kind":"use","excerpt":"05textitModules CompactTruncation and SmallCoreRecompression For the actual core G=R_LR_R^*, supply ordered thin SVD data G=Wdiag(σ)Z^*, W^*W=I, Z^*Z=I, σ_1≥⋯≥σ_d≥0. These are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation G^*GZ=Zdiag(σ_1^2","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9ab434fbb0d31ef","tex":"\\sigma_i^2","aliases":"σ_i^2 sigma","display":"\\sigma_i^2","kind":"use","excerpt":"Set a_i=σ_i^2. Right compression cannot increase the Frobenius norm, and BQ=0, hence normD-B_ F^2 · ≥norm(D-B)Q_ F^2","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-652e4283f0c8bec9","tex":"\\sigma_i^{2}","aliases":"σ_i^2 sigma","display":"\\sigma_i^{2}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8de4fa9e7f9c5804","tex":"\\sigma_j","aliases":"σ_j sigma","display":"\\sigma_j","kind":"use","excerpt":"W^*W=I, Z^*Z=I, σ_1≥⋯≥σ_d≥0. These are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation G^*GZ=Zdiag(σ_1^2,…,σ_d^2). Let h=min(d,k) and retain exactly the first h column indices. The rectangular square-root mask has entries M_ij=sqrtσ_j, · i=j","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76a89e17dd1077d6","tex":"\\sigma_j^2","aliases":"σ_j^2 sigma","display":"\\sigma_j^2","kind":"use","excerpt":"[Eckart--Young--Mirsky theorem] If M has singular values σ_1≥σ_2≥⋯, pad this finite sequence with zeros. For each integer k≥0, its truncated singular value decomposition M_k is a best rank-at-most-k approximation. It satisfies normM-M_k_2=σ_k+1 and normM-M_k_rm F = ( ∑_j>kσ_j^2 )^1/2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b44b17ec5f59013","tex":"\\sigma_{k+1}","aliases":"σ_k+1 sigma","display":"\\sigma_{k+1}","kind":"use","excerpt":"For a d-dimensional diagonal core and 0≤ k<d, the spectral lower bound has a concrete witness. A rank-at-most-k competitor has a nonzero kernel vector in the first k+1 coordinates. On that vector the diagonal matrix acts with norm at least σ_k+1 times the vector norm. The competitor cannot cancel that direction.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1737ab7785be77f","tex":"\\sigma_{k+1}^{2}","aliases":"σ_k+1^2 sigma","display":"\\sigma_{k+1}^{2}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4fd51bd4c6f1161b","tex":"\\Sigma_k^{1/2}","aliases":"Σ_k^1/2 Sigma","display":"\\Sigma_k^{1/2}","kind":"use","excerpt":"R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0931a7c425476227","tex":"\\sigma_{\\min(k,d)}","aliases":"σ_min(k,d) sigma min","display":"\\sigma_{\\min(k,d)}","kind":"use","excerpt":"Spectral optimality from a kernel witness. textitModules DiagonalTruncation and SpectralOptimality Let σ_1≥⋯≥σ_d≥0 and D=diag(σ_1,…,σ_d), D_k=diag(σ_1,…,σ_min(k,d),0,…,0). Lean uses indices beginning at zero, while the formulas on this page use the conventional one-based notation. The rank of D_k is proved to be at most k, even when retained singul","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d8fec8814f298e9","tex":"\\sigma_n","aliases":"σ_n sigma","display":"\\sigma_n","kind":"use","excerpt":"[Probability consistency] Let σ_nsqrtlog n→0 and h=1/n. For every fixed 0<δ<1, along grids satisfying the preceding sampling assumptions, all jump cells are recovered with probability at least 1-δ for sufficiently large n. For the fixed finite jump model above, along grids satisfying the preceding sampling assumptions, the packet error converges to","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fb68eddb44cc0a49","tex":"\\sigma_{N,1}","aliases":"σ_N,1 sigma","display":"\\sigma_{N,1}","kind":"use","excerpt":"[Reciprocal coefficients from zero data] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rσ_N,1, r1!σ_N,2, …, r(n-1)!σ_N,n ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a20bc1293d45df7","tex":"\\sigma_{N,2}","aliases":"σ_N,2 sigma","display":"\\sigma_{N,2}","kind":"use","excerpt":"[Reciprocal coefficients from zero data] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rσ_N,1, r1!σ_N,2, …, r(n-1)!σ_N,n ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-de8bf17230412986","tex":"\\sigma_{N,j,1}","aliases":"σ_N,j,1 sigma","display":"\\sigma_{N,j,1}","kind":"use","excerpt":"ized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9ade27113f7f22ab","tex":"\\sigma_{N,j,2}","aliases":"σ_N,j,2 sigma","display":"\\sigma_{N,j,2}","kind":"use","excerpt":"ized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fca46742b3422d0d","tex":"\\sigma_{N,j,m}","aliases":"σ_N,j,m sigma","display":"\\sigma_{N,j,m}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-15154b154aaffbd7","tex":"\\sigma_{N,j,n}","aliases":"σ_N,j,n sigma","display":"\\sigma_{N,j,n}","kind":"use","excerpt":"ized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb127ee8ac29b6a2","tex":"\\sigma_{N,m-k}","aliases":"σ_N,m-k sigma","display":"\\sigma_{N,m-k}","kind":"use","excerpt":"[Newton sums for the reduced zeros] For |w| below the least zero modulus, with the analytic logarithm normalized to zero at the origin, -logΔ_N(w) = ∑_m≥1σ_N,mfracw^mm. Thus κ_N,m=σ_N,m. If a_m=frac(-1)^m(Nm)!, a_0=1, then σ_N,m = -ma_m- ∑_k=1^m-1a_kσ_N,m-k.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce5747ec88efa8ef","tex":"\\sigma_{N,n}","aliases":"σ_N,n sigma","display":"\\sigma_{N,n}","kind":"use","excerpt":"[Reciprocal coefficients from zero data] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rσ_N,1, r1!σ_N,2, …, r(n-1)!σ_N,n ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-29e1a50ce38c4f16","tex":"\\sigma_{N,n+1}","aliases":"σ_N,n+1 sigma","display":"\\sigma_{N,n+1}","kind":"use","excerpt":"[Tangent coefficients as zero sums] For every n≥0, T_N(n) = N(Nn+N-1)!σ_N,n+1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-761534e1b7a39982","tex":"\\sigma_{N,q}","aliases":"σ_N,q sigma","display":"\\sigma_{N,q}","kind":"use","excerpt":"vpage22Finite coefficients give infinite momentsZERO POWER SUMSChapter IV: Theorem 12.6, Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No num","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65acde73901caecb","tex":"\\Sigma_P","aliases":"Σ_P Sigma","display":"\\Sigma_P","kind":"use","excerpt":"The self-energy Σ_P(z)=A_PQ(zI-A_QQ)^-1A_QP contains the exact effect of the eliminated packets. A bare truncation A_PP discards this term and is generally not exact.","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2792c54819372788","tex":"\\sigma_p","aliases":"σ_p sigma","display":"\\sigma_p","kind":"use","excerpt":"For an odd prime p, the source S_n(p) contains one nonunit base. Put σ_p=(-1)^(p-1)/2 and S_n^×(p)=S_n(p)-σ_pp^n.","count":13},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-87dfaf90b095de92","tex":"\\sigma_t^2","aliases":"σ_t^2 sigma","display":"\\sigma_t^2","kind":"use","excerpt":"Write ρ_t= e^-t=1-t+O(t^2), σ_t^2=1- e^-2t=2t+O(t^2). For a polynomial f, Taylor expansion at x and Gaussian integration give O_tf(x) = · f(x) +f'(x)(ρ_t-1)x +frac12f''(x)σ_t^2 +o(t)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b17d423cde5b3e30","tex":"\\sigma_u","aliases":"σ_u sigma","display":"\\sigma_u","kind":"use","excerpt":"The cost and error of a pruning L are cost( L) = ∑_v∈ Lκ_v + ∑_u splitσ_u and err( L)=∑_v∈ Lϵ_v. The model can include arithmetic, storage, launch, transfer, and precision costs inside κ_v. It remains exact only when the declared totals are additive.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65a866ec8742ecbd","tex":"\\sigma_\\lambda","aliases":"σ_λ sigma lambda","display":"\\sigma_\\lambda","kind":"use","excerpt":"ion] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee268b4caa6936e7","tex":"\\sigma_\\tau","aliases":"σ_τ sigma tau","display":"\\sigma_\\tau","kind":"use","excerpt":"[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cff2c5f7d8b191ec","tex":"\\tau^{-1}","aliases":"τ^-1 tau","display":"\\tau^{-1}","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source refer","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa63b24aa3bb7ad6","tex":"\\tau^{-m}","aliases":"τ^-m tau","display":"\\tau^{-m}","kind":"use","excerpt":"[Orbit reduced nearest circle term] Assume the hypotheses of [source reference]. Choose one representative from each orbit of the nearest zeros under multiplication by ω, and call the resulting set Z_R^orb. With m=Nn, P_N,n(r) = · N m!binomm+r-1r-1 ∑_τ∈ Z_R^orb fracτ^-m(-τ D_N'(τ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff0c3a9746b729b1","tex":"\\tau^2","aliases":"τ^2 tau","display":"\\tau^2","kind":"use","excerpt":"ulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since q is even and ϵτ≡ϵpmod2, this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9add02f6eb4c12f0","tex":"\\tau^{I_K}","aliases":"τ^I_K tau","display":"\\tau^{I_K}","kind":"use","excerpt":"[Conductor of a nontrivial tame atom] Let τ be an irreducible tame Weil representation of dimension n>1. Then Swan(τ)=0, dimτ^I_K=0, Art(τ)=n.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d039a149fedab64","tex":"\\tau^m","aliases":"τ^m tau","display":"\\tau^m","kind":"use","excerpt":"[Laplace transform and moments] Assume the spectrum of A lies in the open left half plane. Then absorption occurs with probability one. For s≥0, E[ e^-sτ] = a^T(sI-A)^-1p_0. For every integer m≥1, E[τ^m] = m! 1^T(-A)^-mp_0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35af929f24440334","tex":"\\tau_0","aliases":"τ_0 tau","display":"\\tau_0","kind":"use","excerpt":"1)^nt^3n(3n)! at the two rational endpoints. From the next term onward, magnitudes decrease, because their successive ratio is fract^3(3n+1)(3n+2)(3n+3)<1 there. The alternating-series remainder is bounded by the first omitted term. A separate outward interval evaluation of the differentiated polynomial, plus its tail bound, certifies the derivativ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e891663598a2abe","tex":"\\tau_M","aliases":"τ_M tau","display":"\\tau_M","kind":"use","excerpt":"onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bou","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12031d1920332f75","tex":"\\tau_m^N","aliases":"τ_m^N tau","display":"\\tau_m^N","kind":"use","excerpt":"[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93f600ceeb9d3b50","tex":"\\tau_N","aliases":"τ_N tau","display":"\\tau_N","kind":"use","excerpt":"onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bou","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96a7e03674728790","tex":"\\tau_p","aliases":"τ_p tau","display":"\\tau_p","kind":"use","excerpt":"The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction. A category of seed presentations and a notion of equivalence still require separate definition","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7d379c8c3c32c66","tex":"\\tau_{Q+1}","aliases":"τ_Q+1 tau","display":"\\tau_{Q+1}","kind":"use","excerpt":"[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0b8695b045c81fb","tex":"\\tau_{q+1}","aliases":"τ_q+1 tau","display":"\\tau_{q+1}","kind":"use","excerpt":"[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53304761e001a2f9","tex":"\\tau_r","aliases":"τ_r tau","display":"\\tau_r","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a75b83819370342d","tex":"\\tau_r^2","aliases":"τ_r^2 tau","display":"\\tau_r^2","kind":"use","excerpt":"[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0d3cf3e8170f0db5","tex":"\\tau_s","aliases":"τ_s tau","display":"\\tau_s","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain m","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb1c0f1580fe3093","tex":"\\tau_{s,m}","aliases":"τ_s,m tau","display":"\\tau_{s,m}","kind":"use","excerpt":"x-construction-15 Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissibl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3e4e14a09d296fdb","tex":"\\tau_\\eta","aliases":"τ_η tau eta","display":"\\tau_\\eta","kind":"use","excerpt":"Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bda1495d04cf4d81","tex":"\\tau_\\xi","aliases":"τ_ξ tau xi","display":"\\tau_\\xi","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fe758f13cc84cd53","tex":"\\tau_\\xi^\\vee","aliases":"τ_ξ^vee tau xi vee","display":"\\tau_\\xi^\\vee","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c89e189652ea7cb2","tex":"\\Phi'","aliases":"Φ' Phi","display":"\\Phi'","kind":"use","excerpt":"[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d05df821bd9aeec","tex":"\\Phi^{(J)}","aliases":"Φ^(J) Phi","display":"\\Phi^{(J)}","kind":"use","excerpt":"To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5bd714c7223a87e","tex":"\\varphi^*","aliases":"φ^* varphi","display":"\\varphi^*","kind":"use","excerpt":"[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2358d6649cb4b2bc","tex":"\\Phi^+","aliases":"Φ^+ Phi","display":"\\Phi^+","kind":"use","excerpt":"vpage16Which coherence can a measurement see?THE ANSWER DEPENDS ON ITS EFFECTSChapter VII: Theorem II.7.2, Theorem II.7.4, Corollary II.7.5. For the Bell density matrix ρ=ketΦ^+braΦ^+, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between addresses 0","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18f9ac23eff67597","tex":"\\Phi^{-1}","aliases":"Φ^-1 Phi","display":"\\Phi^{-1}","kind":"use","excerpt":"[Tame character projectors] The operators P_j are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by Φ P_jΦ^-1=P_qj.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53b5436ea4ca2986","tex":"\\Phi^f","aliases":"Φ^f Phi","display":"\\Phi^f","kind":"use","excerpt":"The nonzero tame packets are permuted by Frobenius. A union of Frobenius orbits is stable under W_K, so irreducibility forces one orbit. Choose one packet V_θ. Its stabilizer is W_K_f=⟨ I_K,Φ^f⟩. A nonzero proper stable subspace of V_θ would generate a nonzero proper W_K stable subspace after summing its Frobenius translates. Thus V_θ is irreducibl","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3f4525556a6771f0","tex":"\\Phi^i","aliases":"Φ^i Phi","display":"\\Phi^i","kind":"use","excerpt":"[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-920bc1c8bd65d9c9","tex":"\\Phi^j","aliases":"Φ^j Phi","display":"\\Phi^j","kind":"use","excerpt":"[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52d7460fdf5044c4","tex":"\\phi_{0,1/3}","aliases":"ϕ_0,1/3 phi","display":"\\phi_{0,1/3}","kind":"use","excerpt":"vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPChapter V: Theorem E7.2, Corollary E7.3. On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), s_f(t)=t^2. vfig[.92]ext_tra","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1bca9bc547dc424","tex":"\\phi_{0,\\tau}","aliases":"ϕ_0,τ phi tau","display":"\\phi_{0,\\tau}","kind":"use","excerpt":"For m≥0 and 0<τ<1, define ϕ_m,τ(t) = frac(t-τ)_+^mm!. Thus ϕ_0,τ= 1_[τ,1]. On each side of τ, the packet is a polynomial. Its derivative of order m has a unit jump at τ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cbe472b688e320fa","tex":"\\phi_{1,2/3}","aliases":"ϕ_1,2/3 phi","display":"\\phi_{1,2/3}","kind":"use","excerpt":"vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPChapter V: Theorem E7.2, Corollary E7.3. On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), s_f(t)=t^2. vfig[.92]ext_tra","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d33036383a860124","tex":"\\Phi_2","aliases":"Φ_2 Phi","display":"\\Phi_2","kind":"use","excerpt":"The lowest degree term of Φ_M(z) is Mz, and its highest degree term is z^M. Multiplication by (1+z)^s gives the two edge coefficients. When M=2 and s=0, Φ_2(z)^k = (2z+z^2)^k = z^k(2+z)^k. Taking the coefficient of z^k+j gives the binary formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-edee295b3bfd5e4d","tex":"\\phi_3","aliases":"ϕ_3 phi","display":"\\phi_3","kind":"use","excerpt":"id and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-acae14b2e159fc5e","tex":"\\phi_a","aliases":"ϕ_a phi","display":"\\phi_a","kind":"use","excerpt":"nentially, while the probabilities remain inside the simplex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the od","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8d53c39f51685b08","tex":"\\phi_{ab}","aliases":"ϕ_ab phi","display":"\\phi_{ab}","kind":"use","excerpt":"lex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Di","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab83f0114b53e934","tex":"\\phi_b","aliases":"ϕ_b phi","display":"\\phi_b","kind":"use","excerpt":"lex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Di","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53adb99fefcfc89a","tex":"\\Phi_{cd,ab}","aliases":"Φ_cd,ab Phi","display":"\\Phi_{cd,ab}","kind":"use","excerpt":"Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fda9801fa210c8f8","tex":"\\Phi_j","aliases":"Φ_j Phi","display":"\\Phi_j","kind":"use","excerpt":"[Isometric packet bijections] For 0≤ j<4, define on natural numbers f_j(n)=fracU_j(n)-j!2^d_j, (d_0,d_1,d_2,d_3)=(1,1,5,3). There is a unique continuous map Φ_j: Z_2→ Z_2 whose value at every n∈ N equals the embedded rational f_j(n). It is an isometric bijection and Φ_j(0)=0: |Φ_j(x)-Φ_j(y)|_2=|x-y|_2 (x,y∈ Z_2). In particular, every element of Z_2","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-569ae08551ce0863","tex":"\\Phi_M","aliases":"Φ_M Phi","display":"\\Phi_M","kind":"use","excerpt":"Let Φ_M(z)=(1+z)^M-1. For M≥1, s≥0, and r,k≥0, define Θ_M,s(r,k) = [z^r](1+z)^sΦ_M(z)^k. Equivalently, Θ_M,s(r,k) = ∑_j=0^k (-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7452277b5b2947b","tex":"\\Phi_m","aliases":"Φ_m Phi","display":"\\Phi_m","kind":"use","excerpt":"Fix a large m, put a=2is_N, and consider the closed disk B_m={z:|z-t_m|≤ 1/(2s_N)}. The remainder bounds hold uniformly on these fixed-radius disks. For sufficiently large m, |R_N|≤ 1/2 and |R_N'|≤ s_N/2 there. Define Φ_m(z)=t_m+fraclog(1+R_N(z))2is_N, using the principal logarithm. The estimate |log(1+u)|≤(3/2)|u| shows that Φ_m maps B_m into itse","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd1f22a02b3e4635","tex":"\\Phi_m'","aliases":"Φ_m' Phi","display":"\\Phi_m'","kind":"use","excerpt":"_N, and consider the closed disk B_m={z:|z-t_m|≤ 1/(2s_N)}. The remainder bounds hold uniformly on these fixed-radius disks. For sufficiently large m, |R_N|≤ 1/2 and |R_N'|≤ s_N/2 there. Define Φ_m(z)=t_m+fraclog(1+R_N(z))2is_N, using the principal logarithm. The estimate |log(1+u)|≤(3/2)|u| shows that Φ_m maps B_m into itself. The derivative bound","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16543cca3869d6ab","tex":"\\phi_{m,u}","aliases":"ϕ_m,u phi","display":"\\phi_{m,u}","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0cdaa5db46e04d3f","tex":"\\phi_{m,\\widehat\\tau}","aliases":"ϕ_m,widehatτ phi widehat tau","display":"\\phi_{m,\\widehat\\tau}","kind":"use","excerpt":"[Location stability for higher order packets] Let m≥1, 1≤ p<∞, and 0≤τ,widehatτ≤1. Then ϕ_m,τ-ϕ_m,widehatτ _L^p(0,1) ≤ frac|τ-widehatτ| (m-1)!((m-1)p+1)^1/p. For p=∞, ϕ_m,τ-ϕ_m,widehatτ _∞ ≤ frac|τ-widehatτ|(m-1)!. Thus location error is Lipschitz for every derivative jump packet of positive order, while a value jump has only the finite L^p modulus","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d26fdcce1d9ab87d","tex":"\\phi_{m,\\tau}^{(\\ell)}","aliases":"ϕ_m,τ^(ℓ) phi tau ell","display":"\\phi_{m,\\tau}^{(\\ell)}","kind":"use","excerpt":"For 0≤ ℓ≤ m, the classical derivative of ϕ_m,τ on the two open sides is ϕ_m,τ^(ℓ)(t) = frac(t-τ)_+^m-ℓ(m-ℓ)!. This derivative has zero jump when ℓ<m and a unit jump when ℓ=m. Derivatives of order larger than m vanish on both open sides. It follows that subtracting the term a_q,mϕ_m,τ_q removes exactly the jump in the derivative of order m without c","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d1ae6e52152d86f1","tex":"\\phi_{m,\\tau_q}","aliases":"ϕ_m,τ_q phi tau","display":"\\phi_{m,\\tau_q}","kind":"use","excerpt":"_packet_decomposition Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dadc02b74543d1bf","tex":"\\phi_{m-1,u}","aliases":"ϕ_m-1,u phi","display":"\\phi_{m-1,u}","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-00bfe1f837258b4d","tex":"\\Phi_r","aliases":"Φ_r Phi","display":"\\Phi_r","kind":"use","excerpt":"K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). Excluding a multiple root on the surviving chart is exactly the assertion ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9cc14498171fac3","tex":"\\Phi_{r,m}^{\\mathrm{act}}","aliases":"Φ_r,m^act Phi mathrm","display":"\\Phi_{r,m}^{\\mathrm{act}}","kind":"use","excerpt":"The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b5d4a112c69639c","tex":"\\Phi_r^{\\mathrm{act}}","aliases":"Φ_r^act Phi mathrm","display":"\\Phi_r^{\\mathrm{act}}","kind":"use","excerpt":"ion K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d0f231c57bed48c8","tex":"\\Phi_s","aliases":"Φ_s Phi","display":"\\Phi_s","kind":"use","excerpt":"vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequentl","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-74777c98027fb1ad","tex":"\\phi_s","aliases":"ϕ_s phi","display":"\\phi_s","kind":"use","excerpt":"Actual local comparisons from fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain m","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-540cf3cfaf67581d","tex":"\\Phi_{s+t}","aliases":"Φ_s+t Phi","display":"\\Phi_{s+t}","kind":"use","excerpt":"[Bayesian nonlinear semigroup] Φ_s+t=Φ_s∘Φ_t. Its generator is the centered multiplicative field fracddtp_i(t) = (g_i-⟨ g,p(t)⟩)p_i(t).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-62d471c4d676af70","tex":"\\Phi_t","aliases":"Φ_t Phi","display":"\\Phi_t","kind":"use","excerpt":"Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-703718018bf26313","tex":"\\Phi_{x,L}","aliases":"Φ_x,L Phi","display":"\\Phi_{x,L}","kind":"use","excerpt":"For every positive integer q, including composite moduli and modulus one, and every L≥0, the checked map Φ_x,L:( Z/q Z)^d⟶ {ybmod q^L+1:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection. Here widetilde t is the canonical integer representative of the digit vector. The entire residue fiber therefore has q^d elements. Injectivity is proved by cancelli","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa78eff0c8b0dd97","tex":"\\Phi_y","aliases":"Φ_y Phi","display":"\\Phi_y","kind":"use","excerpt":"For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8f70978b22d0e201","tex":"\\Phi_{y_1}","aliases":"Φ_y_1 Phi","display":"\\Phi_{y_1}","kind":"use","excerpt":"[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-edb4f356b2cc314d","tex":"\\Phi_{y_r}","aliases":"Φ_y_r Phi","display":"\\Phi_{y_r}","kind":"use","excerpt":"[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b9454d5c4e04a08","tex":"\\Phi_{\\alpha\\beta;\\gamma\\delta}","aliases":"Φ_αβ;γδ Phi alpha beta gamma delta","display":"\\Phi_{\\alpha\\beta;\\gamma\\delta}","kind":"use","excerpt":"Packet superoperator blocks are defined by Φ_αβ;γδ(X) = P_α Φ(P_γ XP_δ) P_β. Summing all four packet indices reconstructs Φ(X) exactly.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1806d9602ee01cb","tex":"\\Phi_\\gamma","aliases":"Φ_γ Phi gamma","display":"\\Phi_\\gamma","kind":"use","excerpt":"0 · 0 . The channel Φ_γ(ρ)=K_0ρ K_0^*+K_1ρ K_1^* is trace preserving. It sends the off diagonal packet ket0bra1 to sqrt1-γket0bra1, while the excited population transfers partly into the ground population.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a515509f5caa010d","tex":"\\Phi_\\theta","aliases":"Φ_θ Phi theta","display":"\\Phi_\\theta","kind":"use","excerpt":"[Cyclic M obius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c48d80b7bf9e478","tex":"\\Phi_{\\theta+\\varphi}","aliases":"Φ_θ+φ Phi theta varphi","display":"\\Phi_{\\theta+\\varphi}","kind":"use","excerpt":"[Cyclic M obius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1d5fdec04e946351","tex":"\\Phi_\\pi","aliases":"Φ_π Phi pi","display":"\\Phi_\\pi","kind":"use","excerpt":"[Impedance and admittance] For N=2, θ=π and Φ_π(z)=frac1z. Thus the two phase action is normalized impedance and admittance inversion.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c1eb035f6ff0ebc6","tex":"\\Phi_\\varphi","aliases":"Φ_φ Phi varphi","display":"\\Phi_\\varphi","kind":"use","excerpt":"[Cyclic M obius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7c934e3eb93a78aa","tex":"\\chi^{-1}","aliases":"χ^-1 chi","display":"\\chi^{-1}","kind":"use","excerpt":"[Duality of Gauss phases] For nontrivial χ, γ(χ,ψ)γ(χ^-1,ψ) = χ(-1).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b57be49fae3643b","tex":"\\chi^\\bullet","aliases":"χ^bullet chi bullet","display":"\\chi^\\bullet","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-726e69c1d2b3c2c6","tex":"\\chi^q","aliases":"χ^q chi","display":"\\chi^q","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc965a9b4f0047d0","tex":"\\chi_+","aliases":"χ_+ chi","display":"\\chi_+","kind":"use","excerpt":"[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c9f755de0838621d","tex":"\\chi_-","aliases":"χ_- chi","display":"\\chi_-","kind":"use","excerpt":"[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bbd4413777b414e","tex":"\\chi_A","aliases":"χ_A chi","display":"\\chi_A","kind":"use","excerpt":"[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09b652c188f47b6e","tex":"\\chi_C","aliases":"χ_C chi","display":"\\chi_C","kind":"use","excerpt":"[Characteristic polynomial of the jump] Let χ_C(y)=det(yI-C). Then χ_C^N(x) = Res_y(χ_C(y),x-y^N). The identity holds over the coefficient ring after interpreting the resultant universally.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9f897e865f593956","tex":"\\chi_{C^N}","aliases":"χ_C^N chi","display":"\\chi_{C^N}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9734ea6868ded897","tex":"\\chi_H","aliases":"χ_H chi","display":"\\chi_H","kind":"use","excerpt":"The usual scalar invariants can be misleading: trH=2, det H=1, χ_H(t)=(t-1)^2. These are also the invariants of the 2×2 identity matrix. Nevertheless, Ob=H-I=mat0 · 1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61c6f0738ee445dc","tex":"\\chi_i","aliases":"χ_i chi","display":"\\chi_i","kind":"use","excerpt":"For a finite family of nontrivial ordinary class characters, suppose functions p,Z satisfy h p(h)→ P≠0, h Z(h)→ Q≠0, and, for all sufficiently small h>0, Z(h)=(p(h)∏_i F_χ_i(h))^e, e∈ N. The checked real-axis criterion then gives e=1 and F_χ_i(0)≠0 for every i. The pole limits and product identity remain hypotheses of this application.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-942ea15f41bce3dd","tex":"\\chi_K","aliases":"χ_K chi","display":"\\chi_K","kind":"use","excerpt":"For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c131c2c63bfd4ae6","tex":"\\chi_{\\Q_{p_s}}","aliases":"χ_Q_p_s χ_ Q_p_s chi Q chi mathbf","display":"\\chi_{{\\mathbf Q}_{p_s}}","kind":"use","excerpt":"Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9093cf6c7bb37fd","tex":"\\chi_\\theta","aliases":"χ_θ chi theta","display":"\\chi_\\theta","kind":"use","excerpt":"[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1d181fcbd5bda98","tex":"\\psi'","aliases":"ψ' psi","display":"\\psi'","kind":"use","excerpt":"[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-40a11436ef06a812","tex":"\\psi''","aliases":"ψ'' psi","display":"\\psi''","kind":"use","excerpt":"[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-353182e6b3dd80e2","tex":"\\psi^*","aliases":"ψ^* psi","display":"\\psi^*","kind":"use","excerpt":"to follow when its objects remain visible. This appendix follows a state vector, a density matrix, and a retained part of a dynamical system. They may describe the same experiment, but they do not answer the same question. vfigthree_objects headAmplitudes retain relative phase For the Bell vector ψ=(ket00+ket11)/sqrt2, the two nonzero entries are a","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ca12a002e4da1b8","tex":"\\psi^{-1}","aliases":"ψ^-1 psi","display":"\\psi^{-1}","kind":"use","excerpt":"rhoods as follows. The source of the open partial homeomorphism defined by Φ is open around a. Its inverse is g. Choose open sets V and U_0 containing y_0 and t_0 such that V× U_0 lies in that source. Restrict U_0 further by the preimage of an open analytic inverse neighborhood under t↦(0,t), the open locus where this same ψ is analytic, and the in","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a619f915ee96c264","tex":"\\Psi_{ef,cd}","aliases":"Ψ_ef,cd Psi","display":"\\Psi_{ef,cd}","kind":"use","excerpt":"[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cea413acc8ef385","tex":"\\psi_F","aliases":"ψ_F psi","display":"\\psi_F","kind":"use","excerpt":"Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, fin","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e11bd281bd6a189e","tex":"\\Psi_{g,\\ell}","aliases":"Ψ_g,ℓ Psi ell","display":"\\Psi_{g,\\ell}","kind":"use","excerpt":"[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6db479813e7c12fd","tex":"\\psi_i","aliases":"ψ_i psi","display":"\\psi_i","kind":"use","excerpt":"A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cfc034372e2b22b5","tex":"\\psi_{j,k}","aliases":"ψ_j,k psi","display":"\\psi_{j,k}","kind":"use","excerpt":"Let I_j,k = [k2^-j,(k+1)2^-j) and let ψ_j,k be the normalized Haar wavelet supported on I_j,k, positive on the left half and negative on the right half.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-364ff3282d8568ac","tex":"\\Psi_N","aliases":"Ψ_N Psi","display":"\\Psi_N","kind":"use","excerpt":"The reduced unsigned denominator is Ψ_N(w)=∑_n≥0fracw^n(Nn)!=Δ_N(-w). By [source reference], -logΨ_N(w) = ∑_m≥1widetildeσ_N,mfracw^mm. With w=z^N, the logarithm of the theta generating function is ∑_m≥1 widetildeσ_N,m(u^Nm+u)fracw^mm. The complete Bell polynomial formula follows.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-96186ebabe575c55","tex":"\\psi_{\\beta-\\gamma}","aliases":"ψ_β-γ psi beta gamma","display":"\\psi_{\\beta-\\gamma}","kind":"use","excerpt":"nit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4417e8c46d7c52ca","tex":"\\omega^{(k-j)r}","aliases":"ω^(k-j)r omega","display":"\\omega^{(k-j)r}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f00be9f7cd43d3d8","tex":"\\omega^{(m-j)\\ell}","aliases":"ω^(m-j)ℓ omega ell","display":"\\omega^{(m-j)\\ell}","kind":"use","excerpt":"Substitution of the power series gives B_N,jF(z) = ∑_m≥0a_mz^m ( frac1N∑_ℓ=0^N-1ω^(m-j)ℓ ). The inner average is one when m≡ jbmod N and zero otherwise. The operator identities follow from the disjoint residue classes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8c661380e21aa156","tex":"\\omega^{-1}","aliases":"ω^-1 omega","display":"\\omega^{-1}","kind":"use","excerpt":"0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4a26976422bc1dc1","tex":"\\omega^{-jk}","aliases":"ω^-jk omega","display":"\\omega^{-jk}","kind":"use","excerpt":"The field specialization. [Character realization] Let bbk be a field, assume that N is nonzero in bbk, and suppose that bbk contains a primitive Nth root of unity ω. Define (UF)(z)=F(ω z). Then P_N,j = frac1N∑_k=0^N-1ω^-jkU^k.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b22e66852d5de18b","tex":"\\omega^{-jr}","aliases":"ω^-jr omega","display":"\\omega^{-jr}","kind":"use","excerpt":"[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f12b1816b314c06d","tex":"\\omega^{-j\\ell}","aliases":"ω^-jℓ omega ell","display":"\\omega^{-j\\ell}","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-42d0e166fb7cd896","tex":"\\omega^{-k}","aliases":"ω^-k omega","display":"\\omega^{-k}","kind":"use","excerpt":"The derivatives satisfy D_N'(ω^kτ_m) = ω^-k D_N'(τ_m) and | D_N'(τ_m)| = frac2s_NNe^c_Nτ_m (1+O(e^-γτ_m)).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ccaca8bd1561c1ac","tex":"\\omega^{-qr}","aliases":"ω^-qr omega","display":"\\omega^{-qr}","kind":"use","excerpt":"[Coherence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert--Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-357aa985d77dd9c2","tex":"\\omega^{i+j}","aliases":"ω^i+j omega","display":"\\omega^{i+j}","kind":"use","excerpt":"Write F=∑_iF_i and G=∑_jG_j, where F_i= B_N,iF and G_j= B_N,jG. Their covariance gives (F_iG_j)(ω z)=ω^i+jF_i(z)G_j(z), so F_iG_j belongs to the packet labeled by i+j modulo N. Expanding the finite product and applying B_N,k gives the formula.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a27b633daa590319","tex":"\\omega^j","aliases":"ω^j omega","display":"\\omega^j","kind":"use","excerpt":"[Covariance characterization] A holomorphic function F belongs to the range of B_N,j if and only if F(ω z)=ω^jF(z).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9328f4344362132","tex":"\\omega^{jr}","aliases":"ω^jr omega","display":"\\omega^{jr}","kind":"use","excerpt":"Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts th","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-960f13a0e4265f44","tex":"\\omega^k","aliases":"ω^k omega","display":"\\omega^k","kind":"use","excerpt":"Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5228e7d3ffe9f8b9","tex":"\\omega^{k(n-j)}","aliases":"ω^k(n-j) omega","display":"\\omega^{k(n-j)}","kind":"use","excerpt":"On z^n, the right side acts by the scalar frac1N∑_k=0^N-1ω^k(n-j). This is one when n≡ jpmod N and zero otherwise by the geometric sum identity in the field bbk.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-01ddb7224174e1ec","tex":"\\omega^m","aliases":"ω^m omega","display":"\\omega^m","kind":"use","excerpt":"[Appell symmetry] Each polynomial E_m^(N,j,r) is monic of degree m, and E_m^(N,j,r)(ω x) = ω^mE_m^(N,j,r)(x).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c32ce654fa1f4d68","tex":"\\omega^N","aliases":"ω^N omega","display":"\\omega^N","kind":"use","excerpt":"Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7dec09704d948494","tex":"\\omega^{N-1}","aliases":"ω^N-1 omega","display":"\\omega^{N-1}","kind":"use","excerpt":"Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-028fd4755c9afd35","tex":"\\omega^{nk}","aliases":"ω^nk omega","display":"\\omega^{nk}","kind":"use","excerpt":"On the monomial z^n, the right-hand side acts by η∑_k=0^N-1(ω^vee)^jkω^nk. The identity ω^N=1 reduces this expression to the table entry indexed by r=nbmod N. The finite certificate therefore gives exactly the coefficient mask defining P_N,j. Over a field, the primitive-root geometric sum supplies the certificate.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2cd0e4050c8155a","tex":"\\omega^{q+j}","aliases":"ω^q+j omega","display":"\\omega^{q+j}","kind":"use","excerpt":"For x∈ P_j H, S(Ax) =(SAS^*)Sx =ω^q+jAx. Thus Ax∈ P_j+q H. The block statement follows.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a1ce26be62e1af85","tex":"\\omega^{q_r}","aliases":"ω^q_r omega","display":"\\omega^{q_r}","kind":"use","excerpt":"[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-221d9113d4c5884d","tex":"\\omega^{rk}","aliases":"ω^rk omega","display":"\\omega^{rk}","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate hold","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-05ef237cb91b4c15","tex":"\\omega^{\\operatorname{val}(r)}","aliases":"ω^val(r) omega operatorname","display":"\\omega^{\\operatorname{val}(r)}","kind":"use","excerpt":"Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-598fd1586454c966","tex":"\\omega^{\\operatorname{val}(r)\\operatorname{val}(j)}","aliases":"ω^val(r)val(j) omega operatorname operatorname","display":"\\omega^{\\operatorname{val}(r)\\operatorname{val}(j)}","kind":"use","excerpt":"Exact coefficient formula. The packet definition is a finite sum over residue classes: P_j=frac1N∑_r∈ Z/N Zχ(-rj)ρ(r). The source proves χ(-rj)=(ω^val(r)val(j))^-1, so this is precisely the manuscript's N-term Fourier projector formula, with canonical representatives.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0892bf887fb05f60","tex":"\\omega^\\vee","aliases":"ω^vee omega vee","display":"\\omega^\\vee","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8788597d7fe50fbe","tex":"\\omega^\\ell","aliases":"ω^ℓ omega ell","display":"\\omega^\\ell","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43b76c3094fa4fa6","tex":"\\omega_1","aliases":"ω_1 omega","display":"\\omega_1","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5d52ab32e3afcb63","tex":"\\omega_2","aliases":"ω_2 omega","display":"\\omega_2","kind":"use","excerpt":"For p=3,ℓ=2,a=2, the checked values are ω_2(2)=8, ⟨2⟩_2=7, 8^2=1,quad8·7=2 in Z/9 Z. The unlifted representative 2 is not a square root of one modulo nine.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0f43487e004c9b12","tex":"\\omega_a","aliases":"ω_a omega","display":"\\omega_a","kind":"use","excerpt":"[Oscillation bound] Suppose overline L_a is any representative likelihood for block B_a, and set ω_a = max_i∈ B_aabsL_i-overline L_a. Then d(pmid L,overline L) ≤ ∑_aq_aω_a. Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ae46887c6acc343","tex":"\\omega_c","aliases":"ω_c omega","display":"\\omega_c","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-097ee86d56b24dd4","tex":"\\omega_f","aliases":"ω_f omega","display":"\\omega_f","kind":"use","excerpt":"[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-152412184ef94979","tex":"\\omega_j","aliases":"ω_j omega","display":"\\omega_j","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa791c4a1646d467","tex":"\\omega_k","aliases":"ω_k omega","display":"\\omega_k","kind":"use","excerpt":"a_unique, .decomposition_unique targetFiniteUnits.omega_mul, .principal_mul pageheading3. Precision compatibility and the dyadic signFinite refinements and Proposition 6.2's dyadic branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-17643fc6afa4dc8b","tex":"\\omega_L","aliases":"ω_L omega","display":"\\omega_L","kind":"use","excerpt":"[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a87553bb10c05ec","tex":"\\omega_N","aliases":"ω_N omega","display":"\\omega_N","kind":"use","excerpt":"Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22344ff7669bd2c8","tex":"\\omega_N^k","aliases":"ω_N^k omega","display":"\\omega_N^k","kind":"use","excerpt":"Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this d","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35ebb38a9f8cbb25","tex":"\\omega_N^N","aliases":"ω_N^N omega","display":"\\omega_N^N","kind":"use","excerpt":"1-w · 1+w. It has determinant w≠0, preserves nonzero pairs, and commutes with rescaling, so it induces a projective map. Direct multiplication gives A(w)A(w')=A(ww') and A(1)=I. Since ω_N^N=1, this proves the cyclic composition law modulo N. The Cayley coordinate of [u:v] is [u-v:u+v], and applying A(w) changes this pair to [w(u-v):u+v]. This prove","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fb25fe9a8f4c1bc","tex":"\\Omega_{P/\\mathbb Z}","aliases":"Ω_P/ Z Omega mathbb","display":"\\Omega_{P/\\mathbb Z}","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a538ce476ad5b75e","tex":"\\Omega_q","aliases":"Ω_q Omega","display":"\\Omega_q","kind":"use","excerpt":"[Edge-packet approximation class] There should exist natural function spaces in which a signal is represented as f=s+∑_q a_q1_Ω_q, with separate norms for smooth approximation, boundary geometry, packet amplitude, and packet identification. A projector-resolved approximation theory on such spaces should yield rates that do not mix smooth truncation","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1b0caddc55ff2af6","tex":"\\omega_Q","aliases":"ω_Q omega","display":"\\omega_Q","kind":"use","excerpt":"[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ce5c2cb93fb1605","tex":"\\omega_r","aliases":"ω_r omega","display":"\\omega_r","kind":"use","excerpt":"[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely ma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab8e8ecf46dc1fbe","tex":"\\omega_V","aliases":"ω_V omega","display":"\\omega_V","kind":"use","excerpt":"[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bc499d807930c364","tex":"\\omega_{V_s}","aliases":"ω_V_s omega","display":"\\omega_{V_s}","kind":"use","excerpt":"[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18bc7e17fe02b53f","tex":"\\Omega_\\theta","aliases":"Ω_θ Omega theta","display":"\\Omega_\\theta","kind":"use","excerpt":"[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-990f57cf16482556","tex":"\\omega_\\theta","aliases":"ω_θ omega theta","display":"\\omega_\\theta","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cf5691eb4311cbb","tex":"\\Omega_\\lambda","aliases":"Ω_λ Omega lambda","display":"\\Omega_\\lambda","kind":"use","excerpt":"Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coeffici","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4292c3402982fcc","tex":"\\Omega_\\rho","aliases":"Ω_ρ Omega rho","display":"\\Omega_\\rho","kind":"use","excerpt":"[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33382dc01fd09a1e","tex":"\\omega_\\rho","aliases":"ω_ρ omega rho","display":"\\omega_\\rho","kind":"use","excerpt":"[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f24c361b02bdb1d","tex":"\\omega_\\ell","aliases":"ω_ℓ omega ell","display":"\\omega_\\ell","kind":"use","excerpt":"ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions ho","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8aaa13d7254b3cae","tex":"\\ell'","aliases":"ℓ' ell","display":"\\ell'","kind":"use","excerpt":"Evaluate the output at r+ℓ. Write every input address as s+ℓ'. The convolution offset is (r+ℓ)-(s+ℓ') =(r-s)+(ℓ-ℓ'). Setting m=ℓ-ℓ' gives the formula.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"[Several completions] The same algebraic seed description applies to finite sequences, ℓ^1, ℓ^2, periodic sequences, tempered distributions, and multidimensional lattices. The topology decides which infinite reconstructions are legitimate. Finite-dimensional experiments in this chapter use ordinary Euclidean norms. Theorem statements specify the am","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3955d1d0d8a03f64","tex":"\\ell^m","aliases":"ℓ^m ell","display":"\\ell^m","kind":"use","excerpt":"Put y=x+1 and let ℓ=log 9. Since v_2(ℓ)=3, frac9^y-18y = fracℓ8 ∑_m≥0 fracℓ^m(m+1)!y^m. The coefficient of y^m has valuation 3m-v_2((m+1)!), which is nonnegative and tends to positive infinity. The quotient therefore belongs to Z_2⟨ y⟩. The factor 2x+1=1+2x is a unit in Z_2⟨ x⟩, with inverse ∑_m≥0(-2x)^m. Hence mathscr R is integral and analytic. T","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-837406a230d80642","tex":"\\ell^\\infty","aliases":"ℓ^∞ ell infty","display":"\\ell^\\infty","kind":"use","excerpt":"[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8a05e97949811e78","tex":"\\ell_1","aliases":"ℓ_1 ell","display":"\\ell_1","kind":"use","excerpt":"[A visible likelihood loss] Let B_1={1,2}, let B_2={3}, and take p=(3/10,2/10,5/10), L=(1,3,2). The coarse prior is (1/2,1/2). The exact block likelihoods are ℓ_1(p)=frac(3/10)+3(2/10)1/2=frac95, ℓ_2(p)=2. Thus the exact coarse posterior is (9/19,10/19). If the first block is replaced by the fixed representative value 2, the approximate coarse post","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f71176b4c504c459","tex":"\\ell_2","aliases":"ℓ_2 ell","display":"\\ell_2","kind":"use","excerpt":"[A visible likelihood loss] Let B_1={1,2}, let B_2={3}, and take p=(3/10,2/10,5/10), L=(1,3,2). The coarse prior is (1/2,1/2). The exact block likelihoods are ℓ_1(p)=frac(3/10)+3(2/10)1/2=frac95, ℓ_2(p)=2. Thus the exact coarse posterior is (9/19,10/19). If the first block is replaced by the fixed representative value 2, the approximate coarse post","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bcd57746c1d3a7fc","tex":"\\ell_a","aliases":"ℓ_a ell","display":"\\ell_a","kind":"use","excerpt":"For an arbitrary prior p, put q=Cp and Z(p,L)=ipLp. Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0. Choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c3eaa86783ecd3c8","tex":"\\ell_e","aliases":"ℓ_e ell","display":"\\ell_e","kind":"use","excerpt":"[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a fini","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-819cb7ab00d8c22c","tex":"\\ell_r","aliases":"ℓ_r ell","display":"\\ell_r","kind":"use","excerpt":"[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet o","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a8e35b84dc016d32","tex":"\\ell_y","aliases":"ℓ_y ell","display":"\\ell_y","kind":"use","excerpt":"[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-06fd0b9eb73abd6b","tex":"\\ell_\\alpha","aliases":"ℓ_α ell alpha","display":"\\ell_\\alpha","kind":"use","excerpt":"ons are equivalent. item There is a positive coarse likelihood ℓ∈(0,∞)^m such that C B_λ(p) = B_ℓ(Cp) for every strictly positive prior p, and the posterior conditional distribution inside every packet equals its prior conditional distribution. item The likelihood is constant on every packet. When these conditions hold, ℓ_α can be chosen as the com","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"How are vectors and operators resolved? · Complete idempotents (P_α) · X≅⊕_α P_α X and T↔(P_β TP_α).","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form,","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2cf519f92b35ca80","tex":"\\Leftrightarrow","aliases":"⇔ Leftrightarrow","display":"\\Leftrightarrow","kind":"use","excerpt":"ii) holds exactly when every μ_p(T) belonging to a nonzero primary component equals one, which is (iii). Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv). In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":28},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c97379bf19e9e4d6","tex":"\\partial_i","aliases":"∂_i partial","display":"\\partial_i","kind":"use","excerpt":"For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1cd2f6e5cfcfeb44","tex":"\\partial_j","aliases":"∂_j partial","display":"\\partial_j","kind":"use","excerpt":"For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d58378212f1d1d15","tex":"\\partial_x","aliases":"∂_x partial","display":"\\partial_x","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5459400e75a6595b","tex":"\\partial_x^2","aliases":"∂_x^2 partial","display":"\\partial_x^2","kind":"use","excerpt":"ymbol acts finitelyAPPELL POLYNOMIALSChapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9. For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms sur","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-13e29157dac59c52","tex":"\\partial_x^4","aliases":"∂_x^4 partial","display":"\\partial_x^4","kind":"use","excerpt":"ymbol acts finitelyAPPELL POLYNOMIALSChapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9. For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms sur","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b546efc5c152ee0e","tex":"\\partial_x^N","aliases":"∂_x^N partial","display":"\\partial_x^N","kind":"use","excerpt":"ferential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are finite on ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-30868654e8125173","tex":"\\partial_x^q","aliases":"∂_x^q partial","display":"\\partial_x^q","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-422f05688ea1c04e","tex":"\\partial_\\theta","aliases":"∂_θ partial theta","display":"\\partial_\\theta","kind":"use","excerpt":"EXACT BLOCH PROJECTOR CALCULATIONChapter VII: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6bd349398cbcc266","tex":"\\partial_\\mu","aliases":"∂_μ partial mu","display":"\\partial_\\mu","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b8be09e8f717ffe2","tex":"\\partial_\\nu","aliases":"∂_ν partial nu","display":"\\partial_\\nu","kind":"use","excerpt":"For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-935f9c58c2ce6b38","tex":"\\partial_\\phi","aliases":"∂_ϕ partial phi","display":"\\partial_\\phi","kind":"use","excerpt":"EXACT BLOCH PROJECTOR CALCULATIONChapter VII: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"[Propagation of packet support] Let T∈End(X) and x∈ X. Then Supp_ P(Tx) ⊆ {β:∃α∈Supp_ P(x) with T_βα≠0}. Equality holds whenever, for every target β in the set on the right, the active contributions (T_βαP_α x: α∈Supp_ P(x), T_βα≠0) have nonzero sum. In particular, equality holds if at most one active source packet feeds each such β and that source","count":24},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e8b6f2780e1c940","tex":"\\nabla^{(a)}_X","aliases":"∇^(a)_X nabla","display":"\\nabla^{(a)}_X","kind":"use","excerpt":"The local connection and its change of coordinates. For a coordinate function u valued in E_a, define ∇^(a)_Xu=A_a(P D_M(B_a u)[X]). For every overlap point, Lean proves the operator identity ∇^(b)_X(g_abu)=g_ab∇^(a)_Xu. This is proved for arbitrary local coordinate functions. It does not assume that u was obtained from a globally smooth section or","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4333a31f419e8b42","tex":"\\nabla^{(b)}_X","aliases":"∇^(b)_X nabla","display":"\\nabla^{(b)}_X","kind":"use","excerpt":"The local connection and its change of coordinates. For a coordinate function u valued in E_a, define ∇^(a)_Xu=A_a(P D_M(B_a u)[X]). For every overlap point, Lean proves the operator identity ∇^(b)_X(g_abu)=g_ab∇^(a)_Xu. This is proved for arbitrary local coordinate functions. It does not assume that u was obtained from a globally smooth section or","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b7161eb75eecc40","tex":"\\nabla_{[X,Y]}","aliases":"∇_[X,Y] nabla","display":"\\nabla_{[X,Y]}","kind":"use","excerpt":"The rank of P is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section s, so that Ps=s, define ∇_Xs=P ds(X). This is a connection on smooth range sections. For smooth vector fields X,Y, its curvature satisfies (∇_X∇_Y-∇_Y∇_X-∇_[X,Y])s =P[dP(X),dP(Y)]P s. Equivalently, its endomorphism-valued ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3cf6e300b826de4e","tex":"\\nabla_{fX}","aliases":"∇_fX nabla","display":"\\nabla_{fX}","kind":"use","excerpt":"textitConnection rules. The smoothness of P ds(X) and the identity P^2=P show that ∇_Xs is a smooth range section. Differentiation and projection give ∇_X+Ys · =∇_Xs+∇_Ys, · ∇_fXs · =f∇_Xs, ∇_X(s+t) · =∇_Xs+∇_Xt, · ∇_X(fs) · =f∇_Xs+(Xf)s for smooth real f. The range coordinates above identify ambient smooth range sections with smooth sections of th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60743d31156ad293","tex":"\\nabla_{X+Y}","aliases":"∇_X+Y nabla","display":"\\nabla_{X+Y}","kind":"use","excerpt":"textitConnection rules. The smoothness of P ds(X) and the identity P^2=P show that ∇_Xs is a smooth range section. Differentiation and projection give ∇_X+Ys · =∇_Xs+∇_Ys, · ∇_fXs · =f∇_Xs, ∇_X(s+t) · =∇_Xs+∇_Xt, · ∇_X(fs) · =f∇_Xs+(Xf)s for smooth real f. The range coordinates above identify ambient smooth range sections with smooth sections of th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-555d56402f7457f5","tex":"\\nabla_Y","aliases":"∇_Y nabla","display":"\\nabla_Y","kind":"use","excerpt":"The rank of P is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section s, so that Ps=s, define ∇_Xs=P ds(X). This is a connection on smooth range sections. For smooth vector fields X,Y, its curvature satisfies (∇_X∇_Y-∇_Y∇_X-∇_[X,Y])s =P[dP(X),dP(Y)]P s. Equivalently, its endomorphism-valued ","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d48f9704594142c","tex":"\\prod_{a=0}^{r-1}","aliases":"∏_a=0^r-1 prod","display":"\\prod_{a=0}^{r-1}","kind":"use","excerpt":"Define the central-factorial basis polynomials P_r(X)=∏_a=0^r-1(X-a(a+1)). At X=Z(Z+1), they have the exact factorization P_r(Z(Z+1)) =∏_a=0^r-1(Z-a)(Z+a+1) =(2r)!binomZ+r2r. The Bernoulli transform gives L((1+T)^Z)=log(1+T)/T, whence L(binom Zj)=(-1)^j/(j+1). Finite Vandermonde expansion therefore gives L(binomZ+r2r) · =(-1)^r∑_q=0^r(-1)^qbinom rq","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c158bc42a84dc596","tex":"\\prod_{a\\in A_{N,j}}","aliases":"∏_a∈ A_N,j prod in","display":"\\prod_{a\\in A_{N,j}}","kind":"use","excerpt":"The multiplication formula for the gamma function gives (Nn+j)! = j!N^Nnn! ∏_a∈ A_N,j(a)_n. Indeed, the N Pochhammer factors supplied by the multiplication formula are ((j+m)/N)_n for 1≤ m≤ N. The excluded factor is ((j+N-j)/N)_n=(1)_n=n!. Insertion into the defining series gives the stated hypergeometric expansion.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-11e1659ea33ec1a6","tex":"\\prod_C","aliases":"∏_C prod","display":"\\prod_C","kind":"use","excerpt":"Let P_C=∑_α∈ CP_α. Define ρ: M_R⟶∏_CEnd(X_C), ρ(T)=(P_CTP_C)_C. If an intermediate class D contributes to P_CSTP_C, then the preorder contains transfers C→ D and D→ C, forcing C=D in the quotient poset. Hence P_CSTP_C=(P_CSP_C)(P_CTP_C), so ρ is an algebra homomorphism. Every collection of endomorphisms of the X_C defines an allowed operator that i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e226abbc069da46","tex":"\\prod_{c=1}^{j}","aliases":"∏_c=1^j prod","display":"\\prod_{c=1}^{j}","kind":"use","excerpt":"Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios direct","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24ec639df453ea86","tex":"\\prod_{C\\in\\overline I}","aliases":"∏_C∈overline I prod in overline","display":"\\prod_{C\\in\\overline I}","kind":"use","excerpt":"k M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. item The map to the diagonal classes induces an algebra isomorphism M_R/ N_R ≅ ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-587b8e89db5f122b","tex":"\\prod_{f(\\alpha)=0}","aliases":"∏_f(α)=0 prod alpha","display":"\\prod_{f(\\alpha)=0}","kind":"use","excerpt":"We use the resultant convention Res_y(f,g) = ∏_f(α)=0g(α) when f is monic, with roots counted with multiplicity.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e637ce45871759a","tex":"\\prod_{\\frakm\\in\\Max\\TT}","aliases":"∏_frakm∈MaxTT ∏_ m∈Max T prod frakm in Max TT prod mathfrak in operatorname mathbb","display":"\\prod_{{\\mathfrak m}\\in{\\operatorname{Max}}{\\mathbb T}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6892de3ca76a9445","tex":"\\prod_{h=0}^k","aliases":"∏_h=0^k prod","display":"\\prod_{h=0}^k","kind":"use","excerpt":"For s∈ p Z_p, an exact finite-difference identity is Δ_1^kfrac11+sn =frac(-s)^k k!∏_h=0^k(1+s(n+h)). It follows by induction, subtracting adjacent rational functions. Each successive subtraction adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4867538d309900b7","tex":"\\prod_{h=1}^{2^a-1}","aliases":"∏_h=1^2^a-1 prod","display":"\\prod_{h=1}^{2^a-1}","kind":"use","excerpt":"at least two, since i≤ j<N/2. Hence g(1)≡1pmod4 and g(2)≡1pmod8. The first two Newton coefficients from the reciprocal recurrence are a_1=-g(1)-1, a_2=binom2NNg(1)^2-g(2)+2g(1)+1. The first has valuation exactly one. Also binom2NN≡6pmod8 for every N=2^a≥2. To see this, write B_a=binom2^a+12^a =2∏_h=1^2^a-1(1+frac2^ah). The even-h factors give B_a-1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-488c88df263b9bc8","tex":"\\prod_{i=0}^{d-1}","aliases":"∏_i=0^d-1 prod","display":"\\prod_{i=0}^{d-1}","kind":"use","excerpt":"_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive block","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b55525d7b1b112b4","tex":"\\prod_{i=0}^{p-1}","aliases":"∏_i=0^p-1 prod","display":"\\prod_{i=0}^{p-1}","kind":"use","excerpt":"Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45251ad81332de19","tex":"\\prod_{i=0}^{q-1}","aliases":"∏_i=0^q-1 prod","display":"\\prod_{i=0}^{q-1}","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-78738dc0b544ac94","tex":"\\prod_{i=1}^{d}","aliases":"∏_i=1^d prod","display":"\\prod_{i=1}^{d}","kind":"use","excerpt":"After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-68bfcf268c38fdd4","tex":"\\prod_{i=1}^j","aliases":"∏_i=1^j prod","display":"\\prod_{i=1}^j","kind":"use","excerpt":"The packed factorial coefficient of G_N,j is g_N,j(n)=fracj!(Nn)!(Nn+j)! =∏_i=1^j(1+frac Ni n)^-1. The empty product is one. If N=p^a and 1≤ i<N, then v_p(i)<a, so N/i∈ p Z_p. Every affine denominator in [source reference] is therefore a unit on Z_p.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75c8e2436af69090","tex":"\\prod_{i=1}^{j-1}","aliases":"∏_i=1^j-1 prod","display":"\\prod_{i=1}^{j-1}","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-746078d5778f9868","tex":"\\prod_{i=1}^{N}","aliases":"∏_i=1^N prod","display":"\\prod_{i=1}^{N}","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a4a402d8d3eb9074","tex":"\\prod_{i=1}^r","aliases":"∏_i=1^r prod","display":"\\prod_{i=1}^r","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9079021fd6762740","tex":"\\prod_{i=1}^{r}","aliases":"∏_i=1^r prod","display":"\\prod_{i=1}^{r}","kind":"use","excerpt":"[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4f3135d472778bff","tex":"\\prod_{i=1}^{\\ell-1}","aliases":"∏_i=1^ℓ-1 prod ell","display":"\\prod_{i=1}^{\\ell-1}","kind":"use","excerpt":"[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c3bdaa5a997770c","tex":"\\prod_{i=j+1}^m","aliases":"∏_i=j+1^m prod","display":"\\prod_{i=j+1}^m","kind":"use","excerpt":"[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2e717462c0866d39","tex":"\\prod_{i=j+1}^{m}","aliases":"∏_i=j+1^m prod","display":"\\prod_{i=j+1}^{m}","kind":"use","excerpt":"ntheorem[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i )","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3065211613fd0977","tex":"\\prod_{i\\in F}","aliases":"∏_i∈ F prod in","display":"\\prod_{i\\in F}","kind":"use","excerpt":"Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite num","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9c0ad644774e2f10","tex":"\\prod_{i\\in I}","aliases":"∏_i∈ I prod in","display":"\\prod_{i\\in I}","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc7ec85dad31c9de","tex":"\\prod_{i\\in\\F_p}","aliases":"∏_i∈F_p ∏_i∈ F_p prod in F prod in mathbb","display":"\\prod_{i\\in{\\mathbb F}_p}","kind":"use","excerpt":"Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2d3db36262598bcd","tex":"\\prod_j","aliases":"∏_j prod","display":"\\prod_j","kind":"use","excerpt":"[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0bafcdd9e08caf20","tex":"\\prod_{j=0}^{n-1}","aliases":"∏_j=0^n-1 prod","display":"\\prod_{j=0}^{n-1}","kind":"use","excerpt":"The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space. The assertion k","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f5e01624215da775","tex":"\\prod_{j=1}^{m}","aliases":"∏_j=1^m prod","display":"\\prod_{j=1}^{m}","kind":"use","excerpt":"[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aa6ac22062ba0415","tex":"\\prod_{j=1}^r","aliases":"∏_j=1^r prod","display":"\\prod_{j=1}^r","kind":"use","excerpt":"cket range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) ∏_j=1^r P_α_j ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60ea789b512f778a","tex":"\\prod_{j=1}^{s}","aliases":"∏_j=1^s prod","display":"\\prod_{j=1}^{s}","kind":"use","excerpt":"[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e4c84160d4d61eb","tex":"\\prod_{j\\ne i}","aliases":"∏_j≠ i prod ne","display":"\\prod_{j\\ne i}","kind":"use","excerpt":"[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3485e5cffde49dc","tex":"\\prod_{k<c_i}","aliases":"∏_k<c_i prod","display":"\\prod_{k<c_i}","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-02bccd9d77338218","tex":"\\prod_{k<m_i}","aliases":"∏_k<m_i prod","display":"\\prod_{k<m_i}","kind":"use","excerpt":"Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The repr","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9d5148b142926aa","tex":"\\prod_p","aliases":"∏_p prod","display":"\\prod_p","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9140b13729b27e7a","tex":"\\prod_{p}","aliases":"∏_p prod","display":"\\prod_{p}","kind":"use","excerpt":"Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with ma","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8ec8acc881d5dd6d","tex":"\\prod_{p>2}","aliases":"∏_p>2 prod","display":"\\prod_{p>2}","kind":"use","excerpt":"Consequently this canonical global diagonal cannot be an equality locus {(a,b):f(a)=g(b)} ⊆ Z_2×∏_p>2 Z_p for continuous maps to a Hausdorff space C. Such an equality locus is closed, as the inverse image of the closed diagonal in C× C, whereas D is proper and dense. This topological argument, retained from the native progress note, concerns the st","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53f2224b9699f49b","tex":"\\prod_{p>2}'","aliases":"∏_p>2' prod","display":"\\prod_{p>2}'","kind":"use","excerpt":"The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-516d6ddd3d40ea65","tex":"\\prod_{p\\in S}","aliases":"∏_p∈ S prod in","display":"\\prod_{p\\in S}","kind":"use","excerpt":"N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-34d2ddfea83ffb6e","tex":"\\prod_q","aliases":"∏_q prod","display":"\\prod_q","kind":"use","excerpt":"For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0fcd1056f96dbe36","tex":"\\prod_r","aliases":"∏_r prod","display":"\\prod_r","kind":"use","excerpt":"[Lucas congruence] If a=a_0+a_1p+⋯ and b=b_0+b_1p+⋯ are base p expansions, then binom ab ≡ ∏_rbinoma_rb_r pmod p. This is the classical theorem of Lucas [source reference].","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee7eedb4b2626377","tex":"\\prod_{r<n}","aliases":"∏_r<n prod","display":"\\prod_{r<n}","kind":"use","excerpt":"The recursion uses only smaller indices, and substitution verifies each row. If a second solution agrees through row n-1, subtracting row n and cancelling T_nn proves equality at n. For integrality, use induction and promote every earlier clearing product to ∏_r<nT_rr. Multiplication by the additional factor T_nn cancels the only new division.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e178b785de8c3c65","tex":"\\prod_{r=0}^{2}","aliases":"∏_r=0^2 prod","display":"\\prod_{r=0}^{2}","kind":"use","excerpt":"1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEver","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07b87a46416a131f","tex":"\\prod_{r=0}^{n}","aliases":"∏_r=0^n prod","display":"\\prod_{r=0}^{n}","kind":"use","excerpt":"em[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4580ee4902e1619e","tex":"\\prod_{r=1}^{n}","aliases":"∏_r=1^n prod","display":"\\prod_{r=1}^{n}","kind":"use","excerpt":"[A denominator bound for every address] For N≥1, 0≤ j<N, and n∈ N, set D_N,j(n)=∏_r=1^nbinomNr+jNr, D_N,j(0)=1. Then D_N,j(n)C_N,j(n)∈ Z.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a86f66b3b267a78","tex":"\\prod_s","aliases":"∏_s prod","display":"\\prod_s","kind":"use","excerpt":"The Heisenberg group of an orthogonal sum is the central product of the Heisenberg groups of the summands. The tensor product representation has the declared central character and has dimension ∏_ssqrtabsV_s = sqrtabs V. It is irreducible by the finite Stone von Neumann theorem, so it is the required representation.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f4084704a37f06b3","tex":"\\prod_{s\\in S}","aliases":"∏_s∈ S prod in","display":"\\prod_{s\\in S}","kind":"use","excerpt":"[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-01f661fab59b64b9","tex":"\\prod_\\alpha","aliases":"∏_α prod alpha","display":"\\prod_\\alpha","kind":"use","excerpt":"enumerate item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source re","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32f02b479dedc3b2","tex":"\\prod_{\\beta,\\alpha\\in I}","aliases":"∏_β,α∈ I prod beta alpha in","display":"\\prod_{\\beta,\\alpha\\in I}","kind":"use","excerpt":"[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-025def8b36bca9d3","tex":"\\prod_{\\lambda\\nmid2}'","aliases":"∏_λnmid2' prod lambda nmid","display":null,"kind":"use","excerpt":"The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51ca0794f5936ae3","tex":"\\prod_{\\lambda\\in\\Lambda_N}","aliases":"∏_λ∈Λ_N prod lambda in Lambda","display":"\\prod_{\\lambda\\in\\Lambda_N}","kind":"use","excerpt":"[Genus zero product] For N≥2, let Λ_N be the nonzero zero multiset of Δ_N. Then Δ_N(w) = ∏_λ∈Λ_N (1-frac wλ), where the product converges locally uniformly. Consequently, D_N(z) = ∏_λ∈Λ_N (1-fracz^Nλ).","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9a509fc9aa63176e","tex":"\\prod_{\\mu\\in\\Spec(T_i)\\setminus\\{a_i\\}}","aliases":"∏_μ∈Spec(T_i)∖{a_i} prod mu in Spec setminus prod mu in operatorname setminus","display":"\\prod_{\\mu\\in{\\operatorname{Spec}}(T_i)\\setminus\\{a_i\\}}","kind":"use","excerpt":"Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d37d3047efa1bd65","tex":"\\prod_{\\mu\\in\\Spec(T_i)\\setminus\\{a\\}}","aliases":"∏_μ∈Spec(T_i)∖{a} prod mu in Spec setminus prod mu in operatorname setminus","display":"\\prod_{\\mu\\in{\\operatorname{Spec}}(T_i)\\setminus\\{a\\}}","kind":"use","excerpt":"For each operator, its finite set of actual eigenvalues is used without multiplicity. The polynomial p_i,a(X)=∏_μ∈Spec(T_i)∖{a} (a-μ)^-1(X-μ) is used when a occurs, and the zero polynomial otherwise. For an eigenvector T_i v=bv, evaluation gives p_i,a(T_i)v=v · b=a,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9798b95d3d831bfd","tex":"\\prod_{\\mu\\in\\Lambda_{N,j}}","aliases":"∏_μ∈Λ_N,j prod mu in Lambda","display":"\\prod_{\\mu\\in\\Lambda_{N,j}}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1662ca15bb05c694","tex":"\\prod_{\\sigma}","aliases":"∏_σ prod sigma","display":"\\prod_{\\sigma}","kind":"use","excerpt":"The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no bound","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb27320775ae67b3","tex":"\\prod_{\\tau}","aliases":"∏_τ prod tau","display":"\\prod_{\\tau}","kind":"use","excerpt":"The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no bound","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0ace3cb3f12efd36","tex":"\\sum_{(b,a)\\in E_A}","aliases":"∑_(b,a)∈ E_A sum in","display":"\\sum_{(b,a)\\in E_A}","kind":"use","excerpt":"Reconstruction and action. The checked finite sum is A=∑_(b,a)∈ E_A A_ba. The proof inserts the complete projector sum on both sides and removes only zero summands. Applying this identity to an arbitrary module gives exact action. A separate finite-matrix theorem establishes AZ=∑_(b,a)∈ E_AA_baZ for every rectangular batch of right-hand sides Z.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-925014b526012cb9","tex":"\\sum_{(e_1,\\ldots,e_n)\\in E^n}","aliases":"∑_(e_1,…,e_n)∈ E^n sum ldots in","display":"\\sum_{(e_1,\\ldots,e_n)\\in E^n}","kind":"use","excerpt":"Induction and finite distributivity give (∑_e∈ EL_e)^n = ∑_(e_1,…,e_n)∈ E^nL_e_n⋯ L_e_1. Every noncomposable summand is zero by [source reference], so inserting the composability filter does not change the sum. In Lean, tuple concatenation and the successor sum are handled by leannametupleWordOp_snoc and leannameexplicitWordExpansion_succ. The two ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c81c4d47b5b4ae42","tex":"\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}","aliases":"∑_(i_1,…,i_m-1)∈ V^m-1 sum ldots in","display":"\\sum_{(i_1,\\ldots,i_{m-1})\\in V^{m-1}}","kind":"use","excerpt":"[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes out","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3b1d3322f2bf29b","tex":"\\sum_{0\\leq n\\leq m/N}","aliases":"∑_0≤ n≤ m/N sum leq leq","display":"\\sum_{0\\leq n\\leq m/N}","kind":"use","excerpt":"[Explicit Appell polynomial] For every m≥0, E_m^(N,j,r)(x) = ∑_0≤ n≤ m/N binommNn C_N,j,n(r)x^m-Nn.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-184fe436a04b209f","tex":"\\sum_{2g=k}","aliases":"∑_2g=k sum","display":"\\sum_{2g=k}","kind":"use","excerpt":"[Divided square packet equations] Assume q( L_g)⊆ L_2g. Choose an ordering of the finite label set. The equation dη+q(η)=0 is equivalent to dη_k + ∑_2g=kq(η_g) + ∑_substackg<h g+h=k[η_g,η_h] =0 for every k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7684dda7bca2d3dd","tex":"\\sum_{a,b}","aliases":"∑_a,b sum","display":"\\sum_{a,b}","kind":"use","excerpt":"nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vec","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc8fd79600305b45","tex":"\\sum_{a,b,c}","aliases":"∑_a,b,c sum","display":"\\sum_{a,b,c}","kind":"use","excerpt":"le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the or","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c10218f7028528ee","tex":"\\sum_{a,b,c,d}","aliases":"∑_a,b,c,d sum","display":"\\sum_{a,b,c,d}","kind":"use","excerpt":"[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-053dd8ba47178a16","tex":"\\sum_{a:Z_{L,a}(t)>0}","aliases":"∑_a:Z_L,a(t)>0 sum","display":"\\sum_{a:Z_{L,a}(t)>0}","kind":"use","excerpt":"[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb22c502080c58d4","tex":"\\sum_{a=0}^1","aliases":"∑_a=0^1 sum","display":"\\sum_{a=0}^1","kind":"use","excerpt":"vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILUREChapter VII: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5 4/5 · 3/5, Q_a=RP_aR^*, S=∑_a=0^1Q_aP_a=frac35R. The exact Gram matrices give (S-I)^*(S-I)=frac1625I, S^*S=frac925I, normS-I=frac45<1. Consequently the unitary po","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-26d84da82cf06c30","tex":"\\sum_{a=0}^k","aliases":"∑_a=0^k sum","display":"\\sum_{a=0}^k","kind":"use","excerpt":"For the first contribution, independence gives L(γ_k(P+Q))=∑_a=0^kμ_aμ_k-a. The parity generating series of these convolution moments is (1+t+t^2+t^4+t^8+⋯)^2 =1+t^2+t^4+t^8+⋯ in F_2[[t]]. Its coefficient is one at every k=2^r≥4. Thus the normalized moment in [source reference] is odd and v_2(Δ_1^kC_2(0))=2k+v_2(k!)+k/2. Inserting U^4n=(1+4H_2)^n i","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-64251df324f966d0","tex":"\\sum_{a=0}^{k-1}","aliases":"∑_a=0^k-1 sum","display":"\\sum_{a=0}^{k-1}","kind":"use","excerpt":"this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even because 4mid k. The other contribution expands as L(Qγ_k-1(P+Q)) =∑_a=0^k-1(k-a)μ_aμ_k-a. An odd summand would req","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a98f0c6f0be59f8a","tex":"\\sum_{a=0}^{M-1}","aliases":"∑_a=0^M-1 sum","display":"\\sum_{a=0}^{M-1}","kind":"use","excerpt":"Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07e82a6cdc184f88","tex":"\\sum_{a=0}^{q-1}","aliases":"∑_a=0^q-1 sum","display":"\\sum_{a=0}^{q-1}","kind":"use","excerpt":"Let q be a positive odd integer. Define S_n(q) = ∑_a=0^q-1 (-1)^a(2a+1)^n. Let the signed Euler numbers and Euler zigzag numbers be defined by secht = ∑_n≥0E_nfract^nn! and sec t+tan t = ∑_n≥0Z_nfract^nn!.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33094d315b0147bf","tex":"\\sum_{a=0}^{u-1}","aliases":"∑_a=0^u-1 sum","display":"\\sum_{a=0}^{u-1}","kind":"use","excerpt":"l closure therefore gives flatness of the actual packed Euler coefficient sequence. The earlier coefficient-extraction theorem now has a proved antecedent for every j<p, and yields p^min(k,mn)midΔ_h^k C_mp,j(n) in Z_p, h=(p-1)/m. A new finite telescoping argument proves the extension to arbitrary natural multiples of h. Its first-step identity is f","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3001c1c56077038a","tex":"\\sum_{a=1}^k","aliases":"∑_a=1^k sum","display":"\\sum_{a=1}^k","kind":"use","excerpt":"Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and prov","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ffde8e588cd66fb1","tex":"\\sum_{a=1}^m","aliases":"∑_a=1^m sum","display":"\\sum_{a=1}^m","kind":"use","excerpt":"Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0. Choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-996cca7e6dc9487c","tex":"\\sum_{a=1}^{q-1}","aliases":"∑_a=1^q-1 sum","display":"\\sum_{a=1}^{q-1}","kind":"use","excerpt":"pageheading1. The characteristic-zero product ruleThe normalized defect is not an ordinary derivation Let A be a commutative algebra over a characteristic-zero field K, and let U be an actual K-derivation. The source defines B=fracU^p^2-U^pp, M_q(f,g)=∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58ca43ed1bf657da","tex":"\\sum_{a=1}^{s}","aliases":"∑_a=1^s sum","display":"\\sum_{a=1}^{s}","kind":"use","excerpt":"An independent definition of Schmidt rank. The source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length s is the concrete equality C_ij=∑_a=1^su_a(i)v_a(j). It factors C=UV through an s-dimensional intermediate space, so rank C≤ s. Conversely, choose a basis of the row space of C. Expanding eac","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f409d86b4db047ab","tex":"\\sum_{a_1,\\ldots,a_{m-1}}","aliases":"∑_a_1,…,a_m-1 sum ldots","display":"\\sum_{a_1,\\ldots,a_{m-1}}","kind":"use","excerpt":"[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad151dded7ef7d37","tex":"\\sum_{a\\in A}","aliases":"∑_a∈ A sum in","display":"\\sum_{a\\in A}","kind":"use","excerpt":"Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b67385de1e4af35","tex":"\\sum_{a\\in I}","aliases":"∑_a∈ I sum in","display":"\\sum_{a\\in I}","kind":"use","excerpt":"The elementary coordinate object is a complete family of orthogonal projectors P_aP_b=δ_abP_a, ∑_a∈ IP_a=I. Every vector has packet coordinates P_aψ. Every operator has transfer blocks A_ba=P_bAP_a. These identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction,","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2184640e3d6bfd1d","tex":"\\sum_{a\\in J/M}","aliases":"∑_a∈ J/M sum in","display":"\\sum_{a\\in J/M}","kind":"use","excerpt":"The estimate is proved from the actual boundary charts and the fixed-lattice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8635bf72798fef18","tex":"\\sum_{a\\in J/mJ}","aliases":"∑_a∈ J/mJ sum in","display":"\\sum_{a\\in J/mJ}","kind":"use","excerpt":"ty of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(u· a)=overline n(u) overline n_J(a). For a multiplicative χ: Z/m Z→ C preserving 1, an exhibited unit with u∈( O/m O)^×, χ(overline n(u))≠1 therefore forces ∑_a∈ J/mJχ(s overline n_J(a))=0 (s∈ Z/m Z). Taking s=1,-1 gives th","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-550f5596bd018fbf","tex":"\\sum_{a\\in\\Lambda}","aliases":"∑_a∈Λ sum in Lambda","display":"\\sum_{a\\in\\Lambda}","kind":"use","excerpt":"[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","excerpt":"For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c43729fd2f2f98cd","tex":"\\sum_{b,a}","aliases":"∑_b,a sum","display":"\\sum_{b,a}","kind":"use","excerpt":"0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7bfc5a7087fd58a2","tex":"\\sum_{b=0}^B","aliases":"∑_b=0^B sum","display":"\\sum_{b=0}^B","kind":"use","excerpt":"Counted candidates. One full convolution table examines indices (b,k), 0≤ b≤ B,quad0≤ k≤ b. These form a concrete finite dependent-sum type. Its cardinality is proved to be ∑_b=0^B(b+1)≤(B+1)^2. For a nonempty finite tree, the number of child merges plus one equals the node count N. This identity is proved for arbitrary finite branching.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3afbe940a4042277","tex":"\\sum_{B\\in\\mathcal Q}","aliases":"∑_B∈ Q sum in mathcal","display":"\\sum_{B\\in\\mathcal Q}","kind":"use","excerpt":"After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block. Both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2046ba257a075646","tex":"\\sum_{b\\ne a}","aliases":"∑_b≠ a sum ne","display":"\\sum_{b\\ne a}","kind":"use","excerpt":"esidual certificates for supplied packets textitSource item 9.2; module DiscoveryCertificate Let (P_a) be a complete orthogonal projector system on a finite complex coordinate space. The projectors need not be coordinate masks. For an actual matrix A, define Π(A)=∑_aP_aAP_a, E=A-Π(A). Reconstruction gives the exact off-diagonal residual E=∑_b≠ aP_b","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59a73cc552080ec9","tex":"\\sum_C","aliases":"∑_C sum","display":"\\sum_C","kind":"use","excerpt":"Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hy","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b2b4aa44e98a82f","tex":"\\sum_{c,d}","aliases":"∑_c,d sum","display":"\\sum_{c,d}","kind":"use","excerpt":"[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3974edc19cef25f4","tex":"\\sum_{c:\\,A_{bc}\\ne0}","aliases":"∑_c: A_bc≠0 sum ne","display":"\\sum_{c:\\,A_{bc}\\ne0}","kind":"use","excerpt":"Define the active contribution sum recursively by F_0(b,a) · =P_a · b=a, 0 · b≠ a, F_n+1(b,a) · =∑_c: A_bc≠0A_bcF_n(c,a). The checked identity is P_bA^nP_a=F_n(b,a) (n≥0). Its proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b80ee3f2f6c8c26","tex":"\\sum_{C\\in\\mathfrak C_0}","aliases":"∑_C∈ C_0 sum in mathfrak","display":"\\sum_{C\\in\\mathfrak C_0}","kind":"use","excerpt":"[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport for","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e4693631642f07b","tex":"\\sum_{c\\in\\{\\mathrm A,\\mathrm B\\}}","aliases":"∑_c∈{ A, B} sum in mathrm mathrm","display":"\\sum_{c\\in\\{\\mathrm A,\\mathrm B\\}}","kind":"use","excerpt":"Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-deb9c97b11939ae2","tex":"\\sum_{d\\mid n}","aliases":"∑_dmid n sum mid","display":null,"kind":"use","excerpt":"[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3a23a48b02a732f","tex":"\\sum_{d\\equiv j\\ ({\\rm mod}\\ N)}","aliases":"∑_d≡ j (rm mod N) sum equiv rm","display":"\\sum_{d\\equiv j\\ ({\\rm mod}\\ N)}","kind":"use","excerpt":"For N>0 and 0≤ j<N, the independently defined original exponential section is E_N,j(t)=∑_d≡ j (rm mod N)fract^dd!. Using the actual power-series expansion homomorphism z↦ t^N, the code proves E_N,j(t)=fract^jj!K_N,j(t^N), F_N,j(t^N)E_N,j(t)=fract^jj!. Thus the recurrence is connected back to the original normalized EGF definition. The section vanis","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ec1616068a6ee172","tex":"\\sum_e","aliases":"∑_e sum","display":"\\sum_e","kind":"use","excerpt":"W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is a two-sided inverse of q_L(I+A), where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then th","count":43},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19360c7481ca29a9","tex":"\\sum_{e:\\alpha\\to\\beta}","aliases":"∑_e:α→β sum alpha to beta","display":"\\sum_{e:\\alpha\\to\\beta}","kind":"use","excerpt":"ra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7155be421cc3965","tex":"\\sum_f","aliases":"∑_f sum","display":"\\sum_f","kind":"use","excerpt":"[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c902c7e7e8b1ed55","tex":"\\sum_{\\frakm}","aliases":"∑_frakm ∑_ m sum frakm sum mathfrak","display":"\\sum_{{\\mathfrak m}}","kind":"use","excerpt":"[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-76e758f1be5158a2","tex":"\\sum_g","aliases":"∑_g sum","display":"\\sum_g","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c01f2477ca38770d","tex":"\\sum_{g+h=k}","aliases":"∑_g+h=k sum","display":"\\sum_{g+h=k}","kind":"use","excerpt":"[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-130edb5a70e27b59","tex":"\\sum_{g<h}","aliases":"∑_g<h sum","display":"\\sum_{g<h}","kind":"use","excerpt":"Repeated use of the polarization identity for q gives q(∑_gη_g) = ∑_gq(η_g) + ∑_g<h[η_g,η_h]. Project onto the component of label k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6917b58d5eb55262","tex":"\\sum_{g_1+\\cdots+g_r=k}","aliases":"∑_g_1+⋯+g_r=k sum cdots","display":"\\sum_{g_1+\\cdots+g_r=k}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6d39fe36bb310cf9","tex":"\\sum_{g\\in G}","aliases":"∑_g∈ G sum in","display":"\\sum_{g\\in G}","kind":"use","excerpt":"[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-778599755ac857bc","tex":"\\sum_{g\\in G_n}","aliases":"∑_g∈ G_n sum in","display":"\\sum_{g\\in G_n}","kind":"use","excerpt":"[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-194eb7d3506e206c","tex":"\\sum_{g\\in S_3}","aliases":"∑_g∈ S_3 sum in","display":"\\sum_{g\\in S_3}","kind":"use","excerpt":"cibility, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONChapter VII: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-707c0697ba3640e2","tex":"\\sum_{g\\in\\Gamma}","aliases":"∑_g∈Γ sum in Gamma","display":"\\sum_{g\\in\\Gamma}","kind":"use","excerpt":"Derived comparison bounds. For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its determinant is","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3035b3c5f1ffe907","tex":"\\sum_{g\\in\\Delta}","aliases":"∑_g∈Δ sum in Delta","display":"\\sum_{g\\in\\Delta}","kind":"use","excerpt":"Let Δ be a finite abelian p group and let Λ= O[Δ]. The augmentation map is ϵ:Λ⟶ O, ∑_g∈Δa_gg⟼∑_g∈Δa_g. Its kernel is the augmentation ideal a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2761fb529cb10263","tex":"\\sum_{h=0}^k","aliases":"∑_h=0^k sum","display":"\\sum_{h=0}^k","kind":"use","excerpt":"3 · 4 · 161644544/4849845 · 15 They are finite rational evaluations of [source reference], followed by a_k=∑_h=0^k(-1)^k-hbinom kh C_j(h). For addresses zero and one, k≥3 is covered by the generic bound. For address two, k≥7 is covered. For address three, k≥5 is covered. In each range the elementary inequality is k-lfloorlog_2krfloor≥ b_j+1. Lemma~","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c105fca98331796","tex":"\\sum_{h\\in S_3}","aliases":"∑_h∈ S_3 sum in","display":"\\sum_{h\\in S_3}","kind":"use","excerpt":"A trace zero matrix fixed by conjugation by both s and t is zero, as follows by comparing its four entries. This proves H^0(S_3,ad^0V)=0. For any F_7[S_3] module M and one cocycle f, put u=-6^-1∑_h∈ S_3f(h). Summing f(gh)=g f(h)+f(g) over h gives gu-u=f(g). Hence every one cocycle is a coboundary and H^1(S_3,M)=0, giving both asserted first cohomol","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3ce2bec5bb12bb9","tex":"\\sum_{i+j=k}","aliases":"∑_i+j=k sum","display":"\\sum_{i+j=k}","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b551f6661cc1d37f","tex":"\\sum_{i+j\\equiv k\\, (\\mathrm{mod}\\,N)}","aliases":"∑_i+j≡ k (mod N) sum equiv mathrm","display":"\\sum_{i+j\\equiv k\\, (\\mathrm{mod}\\,N)}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f69801a69bec1d00","tex":"\\sum_{i+j\\equiv k\\ (N)}","aliases":"∑_i+j≡ k (N) sum equiv","display":"\\sum_{i+j\\equiv k\\ (N)}","kind":"use","excerpt":"1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3844bf53874d6a99","tex":"\\sum_{i,j=1}^3","aliases":"∑_i,j=1^3 sum","display":"\\sum_{i,j=1}^3","kind":"use","excerpt":"headReconstruction retains every block The identity P_1+P_2+P_3=I gives T=∑_i,j=1^3 P_iTP_j. Selecting only the diagonal would change the operator. The entries 2,3,4 carry transfers between different packets.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-815fc827fa2f312d","tex":"\\sum_{i,k}","aliases":"∑_i,k sum","display":"\\sum_{i,k}","kind":"use","excerpt":"[Background and packet error] Suppose the true and estimated packets use the same amplitude list (a_i)_i=0^I, but possibly different regions. Then j-widehat j _p^p = ∑_i,k|a_i-a_k|^pμ(E_i∩ F_k). Only off-diagonal overlaps contribute.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd854b4a4a72698f","tex":"\\sum_{i<d}","aliases":"∑_i<d sum","display":"\\sum_{i<d}","kind":"use","excerpt":"Suppose g(0)=1. The source constructs f by the full triangular recurrence f(d)=δ_d0-∑_i<dbinom di f(i)g(d-i) and proves f⋆ g=1. It also proves cancellation: if a⋆ g=0, then a=0, by induction on the first nonzero coefficient.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da5774d7444b36de","tex":"\\sum_{i=0}^{a}","aliases":"∑_i=0^a sum","display":"\\sum_{i=0}^{a}","kind":"use","excerpt":"The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5edf1fa697a19cc","tex":"\\sum_{i=0}^d","aliases":"∑_i=0^d sum","display":"\\sum_{i=0}^d","kind":"use","excerpt":"Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-adb6721290f5f31c","tex":"\\sum_{i=0}^{d}","aliases":"∑_i=0^d sum","display":"\\sum_{i=0}^{d}","kind":"use","excerpt":"pageheading1. Coefficient products and reciprocal differentiationA characteristic-p argument without factorial division For sequences f,g: N→ K, where K is a field, define (f⋆ g)(d)=∑_i=0^dbinom di f(i)g(d-i), (U^a f)(d)=f(d+a). These are Hurwitz coefficients. The notation does not divide by d! in F_p. In particular, the implementation is not ordin","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bb4cae048f31bfb6","tex":"\\sum_{i=0}^{I}","aliases":"∑_i=0^I sum","display":"\\sum_{i=0}^{I}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eca795e86c8fb349","tex":"\\sum_{i=0}^k","aliases":"∑_i=0^k sum","display":"\\sum_{i=0}^k","kind":"use","excerpt":"The strongest common-step conclusion in width four is compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. I","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-964202d717c850f2","tex":"\\sum_{i=0}^{n}","aliases":"∑_i=0^n sum","display":"\\sum_{i=0}^{n}","kind":"use","excerpt":"Finite Newton reconstruction at an arbitrary row gives Δ_1^kf(n)=∑_i=0^nbinom ni a_k+i. The monotonicity of lfloorlog_2krfloor therefore preserves the higher-coefficient bound after translation. In addition, Δ_1f(n)-a_1∈2^c+1 Z_2, so the first difference is nonzero of exact valuation c.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da1c94a6e8462cce","tex":"\\sum_{i=0}^{N-1}","aliases":"∑_i=0^N-1 sum","display":"\\sum_{i=0}^{N-1}","kind":"use","excerpt":"There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67b8cfba70c769c5","tex":"\\sum_{i=0}^{n-1}","aliases":"∑_i=0^n-1 sum","display":"\\sum_{i=0}^{n-1}","kind":"use","excerpt":"When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-33db49b261c77217","tex":"\\sum_{i=0}^s","aliases":"∑_i=0^s sum","display":"\\sum_{i=0}^s","kind":"use","excerpt":"g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors interchanged to ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58d5910b6df161a7","tex":"\\sum_{i=1}^{d}","aliases":"∑_i=1^d sum","display":"\\sum_{i=1}^{d}","kind":"use","excerpt":"For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5af01fe612ec2d43","tex":"\\sum_{i=1}^{d_\\lambda}","aliases":"∑_i=1^d_λ sum lambda","display":"\\sum_{i=1}^{d_\\lambda}","kind":"use","excerpt":"x an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-21850518d3a22cfc","tex":"\\sum_{i=1}^H","aliases":"∑_i=1^H sum","display":"\\sum_{i=1}^H","kind":"use","excerpt":"To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar mult","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7b615a361efd7684","tex":"\\sum_{i=1}^k","aliases":"∑_i=1^k sum","display":"\\sum_{i=1}^k","kind":"use","excerpt":"At a natural argument the exact Newton expansion is finite. Vandermonde gives binomn+Hk-binom nk =∑_i=1^kbinom Hibinom nk-i. For 1≤ i≤ H, v_2(binom Hi)≥ v_2(H)-v_2(i) ≥ v_2(H)-lfloorlog_2krfloor. Terms with i>H vanish. Thus each Newton summand with k≥2 has valuation at least b+v_2(H)+1. The k=1 summand is Ha_1, of exact valuation b+v_2(H), proving ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5bed34448f0dc86f","tex":"\\sum_{i=1}^{k+1}","aliases":"∑_i=1^k+1 sum","display":"\\sum_{i=1}^{k+1}","kind":"use","excerpt":"Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower b","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67f5e60d24848d87","tex":"\\sum_{i=1}^{N-1}","aliases":"∑_i=1^N-1 sum","display":"\\sum_{i=1}^{N-1}","kind":"use","excerpt":"[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6182c3bad2d01e16","tex":"\\sum_{i=1}^r","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^r","kind":"use","excerpt":"Let S={s_1,…,s_r} generate M. Every pure tensor λ⊗ m∈ M_K has an expression m=∑_i=1^r a_is_i, so λ⊗ m = ∑_i=1^r(λ⊗ a_i)(1⊗ s_i). Pure tensors span M_K, hence 1⊗ S generates M_K over A_K. Taking the minimum over S proves the inequality.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-995c1e5022b02f2d","tex":"\\sum_{i>k}","aliases":"∑_i>k sum","display":"\\sum_{i>k}","kind":"use","excerpt":"For any nonnegative decreasing sequence a_i, weights with these properties satisfy ∑_i a_iw_i≥∑_i>ka_i. This finite weighted inequality is itself proved. For k<d, use μ=a_k+1 and compare the summands to tail weights v_i=0 for i≤ k and v_i=1 otherwise: a_iv_i+μ(w_i-v_i)≤ a_iw_i. For a retained coordinate the inequality uses a_i≥μ and w_i≥0. For a ta","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c62f1b2b7e728f6","tex":"\\sum_{i_1+\\cdots+i_q\\equiv k}","aliases":"∑_i_1+⋯+i_q≡ k sum cdots equiv","display":"\\sum_{i_1+\\cdots+i_q\\equiv k}","kind":"use","excerpt":"[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a54b8f882d9690fa","tex":"\\sum_{i\\in B}","aliases":"∑_i∈ B sum in","display":"\\sum_{i\\in B}","kind":"use","excerpt":"After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block. Both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60c3732b8a8548e8","tex":"\\sum_{i\\in B_a}","aliases":"∑_i∈ B_a sum in","display":"\\sum_{i\\in B_a}","kind":"use","excerpt":"se Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overl","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7fec58bf80198971","tex":"\\sum_{i\\in I}","aliases":"∑_i∈ I sum in","display":"\\sum_{i\\in I}","kind":"use","excerpt":"[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending sub","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd422e93fff6d0ac","tex":"\\sum_{i\\in I_a}","aliases":"∑_i∈ I_a sum in","display":"\\sum_{i\\in I_a}","kind":"use","excerpt":"The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. I","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49e814d33599ad8c","tex":"\\sum_{i\\in I_v}","aliases":"∑_i∈ I_v sum in","display":"\\sum_{i\\in I_v}","kind":"use","excerpt":"Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate pro","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70c8b15d042e9dc0","tex":"\\sum_{i\\in J}","aliases":"∑_i∈ J sum in","display":"\\sum_{i\\in J}","kind":"use","excerpt":"Keep the complete domain, supplied finite-group carrier and group-order hypotheses of the hyperref[book:x-construction-18]supplied-carrier summand-lifting construction. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b99656d8f6574465","tex":"\\sum_{i\\in M}","aliases":"∑_i∈ M sum in","display":"\\sum_{i\\in M}","kind":"use","excerpt":"[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4b8afcda15a5dfd1","tex":"\\sum_{i\\in S}","aliases":"∑_i∈ S sum in","display":"\\sum_{i\\in S}","kind":"use","excerpt":"Finite integral linear combinations. For any finite index set S, coefficients a_i∈ Z_p and units u_i∈ Z_p^×, linearity of the actual iterated operator gives Δ_h^s (∑_i∈ Sa_i u_i^n) =∑_i∈ Sa_i u_i^n(u_i^h-1)^s. At h=p-1, the result is divisible by p^s. The empty sum is permitted. This does not prove that the general congruential Euler packets of The","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f727d6dacb4f1062","tex":"\\sum_{i\\geq0}","aliases":"∑_i≥0 sum geq","display":"\\sum_{i\\geq0}","kind":"use","excerpt":"G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The co","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e4ee3eaa6bf9adf","tex":"\\sum_{i\\geq1}","aliases":"∑_i≥1 sum geq","display":"\\sum_{i\\geq1}","kind":"use","excerpt":"the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial multiplier is integral. Because F_1=F_0D_1^-1, C_1(n)-C_0(n) =∑_i≥1h_ib","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"The image consists exactly of finite sums ∑_ja_js_j, which is Hull_A(S). Because ε_S is linear over A, its kernel is a left submodule. The first isomorphism theorem gives A^⊕ r/N_S≅imε_S=Hull_A(S). The final equivalence is the equality imε_S=M.","count":52},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d7c6cf93ddec3543","tex":"\\sum_{j,k}","aliases":"∑_j,k sum","display":"\\sum_{j,k}","kind":"use","excerpt":"All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-47549ba48c05d7aa","tex":"\\sum_{j,k=1}^{m}","aliases":"∑_j,k=1^m sum","display":"\\sum_{j,k=1}^{m}","kind":"use","excerpt":"[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd623dfebd879fcb","tex":"\\sum_{j=0}^2","aliases":"∑_j=0^2 sum","display":"\\sum_{j=0}^2","kind":"use","excerpt":"ck. vfigvariance platecaptionThe strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with we","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b845f4dcb81f67c5","tex":"\\sum_{j=0}^d","aliases":"∑_j=0^d sum","display":"\\sum_{j=0}^d","kind":"use","excerpt":"[Polynomial support] If p(z)=∑_j=0^dc_jz^j, then supp(p(A)) is contained in the union of the support relations of A^j for 0≤ j≤ d.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cfe7b3387ef9124c","tex":"\\sum_{j=0}^{d}","aliases":"∑_j=0^d sum","display":"\\sum_{j=0}^{d}","kind":"use","excerpt":"Polynomial support. For the actual finite sum p(A)=∑_j=0^dc_jA^j, the checked support lies in the union of the support relations of the powers. The formalization treats the j=0 contribution explicitly, so the constant term is not accidentally omitted.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd6a980c531b4292","tex":"\\sum_{j=0}^{J}","aliases":"∑_j=0^J sum","display":"\\sum_{j=0}^{J}","kind":"use","excerpt":"[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-48fa0e3108a03d2c","tex":"\\sum_{j=0}^{k}","aliases":"∑_j=0^k sum","display":"\\sum_{j=0}^{k}","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ee6b2fc7cf7ca7f4","tex":"\\sum_{j=0}^{n}","aliases":"∑_j=0^n sum","display":"\\sum_{j=0}^{n}","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-586bc3571b381e64","tex":"\\sum_{j=0}^{N-1}","aliases":"∑_j=0^N-1 sum","display":"\\sum_{j=0}^{N-1}","kind":"use","excerpt":"[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f7b37e8032ebe855","tex":"\\sum_{j=0}^{n-1}","aliases":"∑_j=0^n-1 sum","display":"\\sum_{j=0}^{n-1}","kind":"use","excerpt":"y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9098bd3c42e16d5a","tex":"\\sum_{j=0}^r","aliases":"∑_j=0^r sum","display":"\\sum_{j=0}^r","kind":"use","excerpt":"[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c9e6a9c9fb8db72","tex":"\\sum_{j=0}^{s}","aliases":"∑_j=0^s sum","display":"\\sum_{j=0}^{s}","kind":"use","excerpt":"[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbi","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dec62b92295c8c6d","tex":"\\sum_{j=1}^d","aliases":"∑_j=1^d sum","display":"\\sum_{j=1}^d","kind":"use","excerpt":"[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9d0eeaa9eff3012","tex":"\\sum_{j=1}^{d}","aliases":"∑_j=1^d sum","display":"\\sum_{j=1}^{d}","kind":"use","excerpt":"Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd8d17b8ac90c2aa","tex":"\\sum_{j=1}^{d-1}","aliases":"∑_j=1^d-1 sum","display":"\\sum_{j=1}^{d-1}","kind":"use","excerpt":"General finite logarithmic recovery. Let A be a rational algebra, without any commutativity assumption on A. For a unipotent element U and a positive integer d satisfying (U-1)^d=0, put L_d(U)=∑_j=1^d-1frac(-1)^j+1j(U-1)^j. The value is independent of the chosen annihilating exponent. The finite logarithm and the nilpotent exponential satisfy L(exp","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-24dfb51124d8ae91","tex":"\\sum_{j=1}^m","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^m","kind":"use","excerpt":"[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.","count":9},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55bfd4be1daf726b","tex":"\\sum_{j=1}^{m}","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^{m}","kind":"use","excerpt":"[Exponential sum memory] If K(t)=∑_j=1^mB_j e^λ_jtC_j, then K has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in t and are realized by Jordan blocks.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04a3e524463e2337","tex":"\\sum_{j=1}^{m-1}","aliases":"∑_j=1^m-1 sum","display":"\\sum_{j=1}^{m-1}","kind":"use","excerpt":"[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db50c77870a75686","tex":"\\sum_{j=1}^n","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^n","kind":"use","excerpt":"[Derivative of polynomial evaluation] For P∈ K[X_1,…,X_n], polynomial evaluation is analytic at every point. At a∈ K^n its strict Fr echet derivative is v⟼∑_j=1^n (∂_jP)(a)v_j. For a finite system, its strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef75e15ec0684e46","tex":"\\sum_{j=1}^{n}","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^{n}","kind":"use","excerpt":"For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-93f65e678e57f837","tex":"\\sum_{j=1}^{q}","aliases":"∑_j=1^q sum","display":"\\sum_{j=1}^{q}","kind":"use","excerpt":"Frobenius optimality from kernel weights. textitModules KernelFrame and FrobeniusOptimality For an arbitrary competitor B with rank B≤ k, construct an orthonormal matrix Q spanning its actual kernel. Then BQ=0, Q^*Q=I, q=dimker B=d-rank B. Define the row weights w_i=∑_j=1^q|Q_ij|^2. The proof derives 0≤ w_i≤1, ∑_iw_i=q≥max(d-k,0). The upper bound u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8650842fbc4c4680","tex":"\\sum_{j=1}^r","aliases":"∑_j=1^r sum","display":"\\sum_{j=1}^r","kind":"use","excerpt":"[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 ","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-369f27f600d1beec","tex":"\\sum_{j=n}^{\\infty}","aliases":"∑_j=n^∞ sum infty","display":"\\sum_{j=n}^{\\infty}","kind":"use","excerpt":"fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9fb4fab2abfee045","tex":"\\sum_{j=\\ell}^{m-1}","aliases":"∑_j=ℓ^m-1 sum ell","display":"\\sum_{j=\\ell}^{m-1}","kind":"use","excerpt":"Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2f357ff52e6c2d2b","tex":"\\sum_{j>k}","aliases":"∑_j>k sum","display":"\\sum_{j>k}","kind":"use","excerpt":"normB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-12ba6f0fe0cd0b54","tex":"\\sum_{j_1+\\cdots+j_m=r}","aliases":"∑_j_1+⋯+j_m=r sum cdots","display":"\\sum_{j_1+\\cdots+j_m=r}","kind":"use","excerpt":"[Tensor charge refinement] For r∈ Z/N Z, define P_r^rm total = ∑_j_1+⋯+j_m=r P_j_1^(1)⊗⋯⊗ P_j_m^(m). Then the P_r^rm total form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4511ec7b375d647e","tex":"\\sum_{j_1+\\cdots+j_m\\equiv r\\pmod N}","aliases":"∑_j_1+⋯+j_m≡ rpmod N sum cdots equiv pmod","display":"\\sum_{j_1+\\cdots+j_m\\equiv r\\pmod N}","kind":"use","excerpt":"[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2656f76b906d00aa","tex":"\\sum_{j\\in B_b}","aliases":"∑_j∈ B_b sum in","display":"\\sum_{j\\in B_b}","kind":"use","excerpt":"Stationarity follows from overline Qoverlineπ =overline QCπ =CQπ=0. For reversibility, use the aggregate rate formula: overline Q_aboverlineπ_b · =∑_j∈ B_bπ_j∑_i∈ B_aQ_ij · =∑_i∈ B_a∑_j∈ B_bπ_iQ_ji · =overline Q_baoverlineπ_a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aae5c8f2004dc612","tex":"\\sum_{j\\in F}","aliases":"∑_j∈ F sum in","display":"\\sum_{j\\in F}","kind":"use","excerpt":"The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a fini","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4619f532d9c032b","tex":"\\sum_{j\\in J}","aliases":"∑_j∈ J sum in","display":"\\sum_{j\\in J}","kind":"use","excerpt":"[Table specification for transfer blocks] Given finite projector systems (P_i)_i∈ I and (Q_j)_j∈ J and a linear map T, form the table B_ji=Q_jTP_i. Its assembly reconstructs the input map exactly, ∑_j∈ J∑_i∈ IB_ji=T, and its reported support relation is exactly {(i,j):B_ji≠0}. The latter equality is the specification of the transfer graph output. I","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c80d63770767b412","tex":"\\sum_{j\\notin J}","aliases":"∑_j∉ J sum notin","display":"\\sum_{j\\notin J}","kind":"use","excerpt":"j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete primary blocks indexed by J, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89330e1a143912f2","tex":"\\sum_{j\\geq0}","aliases":"∑_j≥0 sum geq","display":"\\sum_{j\\geq0}","kind":"use","excerpt":"[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1cf2b9d42f6eee5","tex":"\\sum_{k=0}^2","aliases":"∑_k=0^2 sum","display":"\\sum_{k=0}^2","kind":"use","excerpt":"On the monomial z^n, define UF(z)=F(2z). The proposed projector acts by 5∑_k=0^2 4^jk2^nk. The identity 2^3=1 makes this depend only on nbmod3. The nine-entry table therefore establishes the formula for every exponent once periodicity is used, rather than requiring a separate test at every coefficient.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c601ce784f1ba10","tex":"\\sum_{k=0}^{d-1}","aliases":"∑_k=0^d-1 sum","display":"\\sum_{k=0}^{d-1}","kind":"use","excerpt":"For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c98e9b3ccc8ca2c6","tex":"\\sum_{k=0}^{j-1}","aliases":"∑_k=0^j-1 sum","display":"\\sum_{k=0}^{j-1}","kind":"use","excerpt":"Use the telescoping identity A^j-D^j = ∑_k=0^j-1 A^j-1-kKD^k. Each term has norm at most M^j-1normK. The vector D^jv lies in the direct sum of local Krylov spaces.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b745b62212003b48","tex":"\\sum_{k=0}^{m-1}","aliases":"∑_k=0^m-1 sum","display":"\\sum_{k=0}^{m-1}","kind":"use","excerpt":"Differentiate the product identity in logarithmic form to obtain Δ_N'(w) = -Δ_N(w) ∑_m≥1σ_N,mw^m-1. Comparison of the coefficient of w^m-1 gives ma_m = -∑_k=0^m-1a_kσ_N,m-k. Solving for σ_N,m gives the recurrence.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a4e9c334647df08","tex":"\\sum_{k=0}^{\\min(d,r)}","aliases":"∑_k=0^min(d,r) sum min","display":"\\sum_{k=0}^{\\min(d,r)}","kind":"use","excerpt":"[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) u","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-41e0c211ba80d17c","tex":"\\sum_{k=0}^{n}","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^{n}","kind":"use","excerpt":"[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7796ee229bc8c35a","tex":"\\sum_{k=0}^{N-1}","aliases":"∑_k=0^N-1 sum","display":"\\sum_{k=0}^{N-1}","kind":"use","excerpt":"[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate hold","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fec10602e4575bf9","tex":"\\sum_{k=0}^{n-1}","aliases":"∑_k=0^n-1 sum","display":"\\sum_{k=0}^{n-1}","kind":"use","excerpt":"[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":8},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-55391d8e2ecb990e","tex":"\\sum_{k=0}^{r-1}","aliases":"∑_k=0^r-1 sum","display":"\\sum_{k=0}^{r-1}","kind":"use","excerpt":"[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e39c4713a93500ec","tex":"\\sum_{k=0}^{\\nu_j-1}","aliases":"∑_k=0^ν_j-1 sum nu","display":"\\sum_{k=0}^{\\nu_j-1}","kind":"use","excerpt":"xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1c2ff812e2b7828","tex":"\\sum_{k=0}^{\\infty}","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^{\\infty}","kind":"use","excerpt":"For large z, Σ_P(z) = ∑_k=0^∞ fracA_PQA_QQ^kA_QPz^k+1. Thus the second condition implies the moment identities. Conversely, the first d identities imply all later ones by the Cayley--Hamilton theorem, so the series vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32a0cdf7fe3b4935","tex":"\\sum_{k=1}^{m}","aliases":"∑_k=1^m sum","display":"\\sum_{k=1}^{m}","kind":"use","excerpt":"Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d6cd950a1f0e0334","tex":"\\sum_{k=1}^{m-1}","aliases":"∑_k=1^m-1 sum","display":"\\sum_{k=1}^{m-1}","kind":"use","excerpt":"[Newton sums for the reduced zeros] For |w| below the least zero modulus, with the analytic logarithm normalized to zero at the origin, -logΔ_N(w) = ∑_m≥1σ_N,mfracw^mm. Thus κ_N,m=σ_N,m. If a_m=frac(-1)^m(Nm)!, a_0=1, then σ_N,m = -ma_m- ∑_k=1^m-1a_kσ_N,m-k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9972412a61817be9","tex":"\\sum_{k=1}^{n-1}","aliases":"∑_k=1^n-1 sum","display":"\\sum_{k=1}^{n-1}","kind":"use","excerpt":"[Parameter recurrence] We have P_N,0(r)=1. For n≥1, P_N,n(r) = · - ∑_k=1^n-1 (-1)^k binomNn-1Nk P_N,n-k(r) · -r ∑_k=1^n (-1)^k binomNn-1Nk-1 P_N,n-k(r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a5316fecc2dcc62e","tex":"\\sum_{k=1}^{p^{r-1}}","aliases":"∑_k=1^p^r-1 sum","display":"\\sum_{k=1}^{p^{r-1}}","kind":"use","excerpt":"The finite Newton expansion gives f(n+p^r-1h)-f(n) =∑_k=1^p^r-1binomp^r-1kΔ_h^k f(n). Terms with k≥ r have the required factor from the difference. For 1≤ k<r, the identity kbinomp^r-1k=p^r-1binomp^r-1-1k-1 and v_p(k)<k imply p^rmidbinomp^r-1k p^k in Z. Mapping that integer divisibility to R proves the claim term by term. The argument includes r=1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3d357445fed9cde3","tex":"\\sum_{k=1}^{w}","aliases":"∑_k=1^w sum","display":"\\sum_{k=1}^{w}","kind":"use","excerpt":"The diagonal contributes p blocks. At offset k, where 1≤ k≤ w, the upper and lower block diagonals contribute 2(p-k) blocks. Therefore p+2∑_k=1^w(p-k) = p(2w+1)-w(w+1). The cost and storage follow from [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f9dd23cb17a6b70","tex":"\\sum_{k=2}^{T}","aliases":"∑_k=2^T sum","display":"\\sum_{k=2}^{T}","kind":"use","excerpt":"Now expand the displacement: f(n+T)-f(n)=TΔ_1f(n) +∑_k=2^Tbinom TkΔ_1^kf(n). For 1≤ k≤ T, the binomial denominator identity implies v_2binom Tk≥ v_2(T)-v_2(k) ≥ v_2(T)-lfloorlog_2krfloor. Every term of the remaining sum consequently has valuation at least v_2(T)+c+1. Its sum cannot cancel the leading term, which has exact valuation v_2(T)+c. This p","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75edbe8a53ac1864","tex":"\\sum_{k\\in\\ZZ^d}","aliases":"∑_k∈ZZ^d ∑_k∈ Z^d sum in ZZ sum in mathbb","display":"\\sum_{k\\in{\\mathbb Z}^d}","kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e99f6320075b8a65","tex":"\\sum_{k\\in\\Lambda_N^*}","aliases":"∑_k∈Λ_N^* sum in Lambda","display":"\\sum_{k\\in\\Lambda_N^*}","kind":"use","excerpt":"[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ebed7c5a175f44d","tex":"\\sum_{k\\ge0}","aliases":"∑_k≥0 sum ge","display":"\\sum_{k\\ge0}","kind":"use","excerpt":"[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a334e226daa86a9f","tex":"\\sum_{k\\geq0}","aliases":"∑_k≥0 sum geq","display":"\\sum_{k\\geq0}","kind":"use","excerpt":"[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-31b4e8e8ca1b60bf","tex":"\\sum_{k\\geq1}","aliases":"∑_k≥1 sum geq","display":"\\sum_{k\\geq1}","kind":"use","excerpt":"Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7dd1f8372600c494","tex":"\\sum_m","aliases":"∑_m sum","display":"\\sum_m","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-655f0e79a95453cc","tex":"\\sum_{m,n\\ge0}","aliases":"∑_m,n≥0 sum ge","display":"\\sum_{m,n\\ge0}","kind":"use","excerpt":"Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3fe5a2e3add92762","tex":"\\sum_{m=0}^{12}","aliases":"∑_m=0^12 sum","display":"\\sum_{m=0}^{12}","kind":"use","excerpt":"idue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db89bb56e5266992","tex":"\\sum_{m=0}^{\\min(M,N)}","aliases":"∑_m=0^min(M,N) sum min","display":"\\sum_{m=0}^{\\min(M,N)}","kind":"use","excerpt":"tion v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0692fed59f0e0fb2","tex":"\\sum_{m=0}^{\\min(M,r)}","aliases":"∑_m=0^min(M,r) sum min","display":"\\sum_{m=0}^{\\min(M,r)}","kind":"use","excerpt":"al targets. targetOddEuler.odd_reconstruction_series targetOddEuler.odd_euler_reconstruction targetOddEuler.odd_euler_reconstruction_zigzag targetOddEuler.row_action, OddEuler.row_diagonal, OddEuler.row_weight pageheading4. Uniform truncation and a sharp exact exampleCorollary 4.3 and Example 4.4 For row 2r and depth M, the actual truncation is T_M","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-04a4953949e57f10","tex":"\\sum_{m=0}^N","aliases":"∑_m=0^N sum","display":"\\sum_{m=0}^N","kind":"use","excerpt":"The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e38ac93a3132039","tex":"\\sum_{m=0}^{n}","aliases":"∑_m=0^n sum","display":"\\sum_{m=0}^{n}","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-187ef6e56570865f","tex":"\\sum_{m=0}^{n/2}","aliases":"∑_m=0^n/2 sum","display":"\\sum_{m=0}^{n/2}","kind":"use","excerpt":"Substituting the independently proved signed-zigzag identity recovers the manuscript's notation. For every even degree n, E_n=∑_m=0^n/2(-1)^m Z_2m+1binom n2mp^2mS_n-2m(p). The Lean version uses n=2r to avoid ambiguous natural-number division. Every even natural degree has that form. Primality is not used in this equality.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d3dbe092f3fdebc1","tex":"\\sum_{m=0}^{r}","aliases":"∑_m=0^r sum","display":"\\sum_{m=0}^{r}","kind":"use","excerpt":"ing3. Exact reconstruction and the triangular matrixTheorem 4.2; manuscript page 11 For odd p, rescaling the already proved inverse law gives sech(pt)cosh(pt)=1. Multiplying the even-source identity by sech^2(pt) therefore proves sech^2(pt)(∑_r≥0S_2r(p)fract^2r(2r)!)=sech t. Actual EGF multiplication now gives the integer coefficient formula E_2r=∑","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e3d2691a5710f8e","tex":"\\sum_{m=0}^{r-1}","aliases":"∑_m=0^r-1 sum","display":"\\sum_{m=0}^{r-1}","kind":"use","excerpt":"_packet_decomposition Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b67bd1ec346d4fa3","tex":"\\sum_{m=0}^\\infty","aliases":"∑_m=0^∞ sum infty","display":"\\sum_{m=0}^\\infty","kind":"use","excerpt":"[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-187f645f69d06fb0","tex":"\\sum_{m=0}^{\\infty}","aliases":"∑_m=0^∞ sum infty","display":"\\sum_{m=0}^{\\infty}","kind":"use","excerpt":"This section presents the operator-substitution result transferred from Chapter IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N o","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-05073c075273e26d","tex":"\\sum_{m=1}^H","aliases":"∑_m=1^H sum","display":"\\sum_{m=1}^H","kind":"use","excerpt":"The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive ord","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4e3a67fde8945085","tex":"\\sum_{m=1}^{n}","aliases":"∑_m=1^n sum","display":"\\sum_{m=1}^{n}","kind":"use","excerpt":"[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-516d97059e6505f6","tex":"\\sum_{m=1}^{w_d}","aliases":"∑_m=1^w_d sum","display":"\\sum_{m=1}^{w_d}","kind":"use","excerpt":"[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22a81cbac9128cfb","tex":"\\sum_{m=1}^{\\infty}","aliases":"∑_m=1^∞ sum infty","display":"\\sum_{m=1}^{\\infty}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-482bdbbbe4d86bc7","tex":"\\sum_{m=k}^{n+k}","aliases":"∑_m=k^n+k sum","display":"\\sum_{m=k}^{n+k}","kind":"use","excerpt":"eciprocal are C_2(0)=1, C_2(1)=-frac115, C_2(2)=frac1345, a_1=-frac1615, a_2=frac6445, where a_m=Δ_1^m C_2(0). Thus v_2(a_1)=4 and v_2(a_2)=6. For m≥3, [source reference] bounds a_m by 2m+v_2(m!). The finite shifted Newton formula Δ_1^kC_2(n)=∑_m=k^n+ka_mbinom nm-k therefore gives v_2(Δ_1^kC_2(n))≥ k+v_2(k!)+3 for every k≥1. For k≥3 this is 2k≥ k+3","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-09a693e06fd236f6","tex":"\\sum_{m\\in L}","aliases":"∑_m∈ L sum in","display":"\\sum_{m\\in L}","kind":"use","excerpt":"[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-da006f0561df99b4","tex":"\\sum_{m\\in\\bbZ}","aliases":"∑_m∈bbZ ∑_m∈ Z sum in bbZ sum in mathbb","display":"\\sum_{m\\in{\\mathbb Z}}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b9a242ebebc8b392","tex":"\\sum_{m\\in\\ZZ}","aliases":"∑_m∈ZZ ∑_m∈ Z sum in ZZ sum in mathbb","display":"\\sum_{m\\in{\\mathbb Z}}","kind":"use","excerpt":"For L=N Z with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cba1f7355992d01c","tex":"\\sum_{m\\equiv j\\bmod N}","aliases":"∑_m≡ jbmod N sum equiv bmod","display":"\\sum_{m\\equiv j\\bmod N}","kind":"use","excerpt":"[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2f2db5791bd090b","tex":"\\sum_{m\\geq 0}","aliases":"∑_m≥ 0 sum geq","display":"\\sum_{m\\geq 0}","kind":"use","excerpt":"For a power series F(z)=∑_m≥ 0a_mz^m and a positive integer N, the coefficients may be separated according to the residue of m modulo N. Root of unity averaging gives exact projection operators for this separation. This elementary construction is usually called a multisection or a root filter. It appears throughout the classical theory of generatin","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b8e5ecd78509ed8","tex":"\\sum_{m\\ge L}","aliases":"∑_m≥ L sum ge","display":"\\sum_{m\\ge L}","kind":"use","excerpt":"on ℓ^2(Λ_N) gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries. vfigproduct headFinite compression is an approximation For N=2, λ_m=π^2(m+1/2)^2. The finite determinant P_L is exactly the displayed finite product, but it is not exactly Δ_2. For 0≤ w≤1 and L≥1, an elementary tail estimate gives 0≤","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22d4ad01256f7944","tex":"\\sum_{m\\geq\\ell}","aliases":"∑_m≥ℓ sum geq ell","display":"\\sum_{m\\geq\\ell}","kind":"use","excerpt":"Put a_m=Δ_1^mC_j(0) and expand the falling factorial in the Newton series. The ordinary coefficients are d_j,ℓ=∑_m≥ℓfraca_mm!s(m,ℓ). Since s(m,ℓ) is integral and v_2(a_m/m!)≥ m+β_j, the coefficient sum converges and has valuation at least ℓ+β_j. The sum exchange is justified on Z_2 by these uniform geometric coefficient bounds. The ordinary power s","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45953a160933c484","tex":"\\sum_{n\\text{ even}}","aliases":"∑_n even sum text","display":"\\sum_{n\\text{ even}}","kind":"use","excerpt":"For even n≥0, define R_n = frac3^n+2-14(n+1)(n+2). Its even exponential generating function is ∑_n evenR_nfract^nn! = fracsinh(2t)sinh t2t^2. Define β_r by fracz^2sinh^2z = ∑_r≥0β_rz^2r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4d609c0fc3f9c1f9","tex":"\\sum_{n=0}^{90}","aliases":"∑_n=0^90 sum","display":"\\sum_{n=0}^{90}","kind":"use","excerpt":"V: Theorem 11.7, Theorem 11.8, Corollary 11.9. The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-070f622d9a6f0c69","tex":"\\sum_{n=0}^N","aliases":"∑_n=0^N sum","display":"\\sum_{n=0}^N","kind":"use","excerpt":"[Polynomial defect] For p(z)=∑_n=0^Na_nz^n, we have Cp(T_f)-p(T_c)C = ∑_n=1^Na_n ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0fbc71e3cef7af4e","tex":"\\sum_{n=0}^{N-1}","aliases":"∑_n=0^N-1 sum","display":"\\sum_{n=0}^{N-1}","kind":"use","excerpt":"et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8773b980988ccc84","tex":"\\sum_{n=0}^\\infty","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^\\infty","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a7626aeba75a509f","tex":"\\sum_{n=0}^{\\infty}","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^{\\infty}","kind":"use","excerpt":"|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings. headA series near zero energy and its exact remainder At z=1/2, |z| |D^-1|=1/4<1, so Σ(z)=-∑_n=0","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4700a53bd785dd67","tex":"\\sum_{n=1}^{m}","aliases":"∑_n=1^m sum","display":"\\sum_{n=1}^{m}","kind":"use","excerpt":"vpage16Why a Fourier cutoff cannot reproduce this jump pointwiseA CONTINUOUS PULSE / EXACT ENDPOINT VALUESChapter V: Section 6, Section 8. Use a separate period-one example on [-1/2,1/2): j(t)= 1_{|t|<1/4}, S_mj(t)=tfrac12+frac2π∑_n=1^mfracsin(π n/2)ncos(2π nt). This is the exact finite Fourier formula. The drawn curves sample it numerically. The e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3ec24382fc8477a","tex":"\\sum_{n=1}^N","aliases":"∑_n=1^N sum","display":"\\sum_{n=1}^N","kind":"use","excerpt":"[Polynomial defect] For p(z)=∑_n=0^Na_nz^n, we have Cp(T_f)-p(T_c)C = ∑_n=1^Na_n ∑_k=0^n-1T_c^n-1-kΔ T_f^k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0cc15179080d69b2","tex":"\\sum_{n=m}^{\\infty}","aliases":"∑_n=m^∞ sum infty","display":"\\sum_{n=m}^{\\infty}","kind":"use","excerpt":"[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b846ee3188af45a","tex":"\\sum_{n>N}","aliases":"∑_n>N sum","display":"\\sum_{n>N}","kind":"use","excerpt":"This x cannot be a finite linear combination of shifted impulses: it has infinite support. But if x^(N) agrees with x on |n|≤ N and is zero elsewhere, then normx-x^(N)_2^2 =2∑_n>N4^-n =frac23 4^-N⟶0. The exact squared errors for N=1,3,5 are 1/6, 1/96, and 1/1536. These are identities from a geometric series, not floating residuals.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6806b62c6d27ebf6","tex":"\\sum_{n\\in\\bbZ}","aliases":"∑_n∈bbZ ∑_n∈ Z sum in bbZ sum in mathbb","display":"\\sum_{n\\in{\\mathbb Z}}","kind":"use","excerpt":"[Impulse seed theorem] The closed shift-generated hull of δ_0 is all of ℓ^2( Z): overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 =ℓ^2( Z). Every finitely supported signal has the exact finite expression x=∑_n∈ Zx[n]S^nδ_0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ddc26b49cc4ceb42","tex":"\\sum_{n\\in\\ZZ^d}","aliases":"∑_n∈ZZ^d ∑_n∈ Z^d sum in ZZ sum in mathbb","display":"\\sum_{n\\in{\\mathbb Z}^d}","kind":"use","excerpt":"[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8284d03cfea3f8d1","tex":"\\sum_{n\\notin F}","aliases":"∑_n∉ F sum notin","display":"\\sum_{n\\notin F}","kind":"use","excerpt":"The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fc4aea9ed3564f4b","tex":"\\sum_{n\\neq0}","aliases":"∑_n≠0 sum neq","display":"\\sum_{n\\neq0}","kind":"use","excerpt":"By [source reference], Z_d(t) =(4π t)^-d/2(1+R(t)), where R(t)=∑_n≠0 e^-absn^2/(4t) =O( e^-1/(4t)). Differentiation gives tR'(t)=O(t^-1 e^-1/(4t)). Substituting this estimate into the logarithmic derivative gives the stated small time expansion.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab5bb9c4725a51c8","tex":"\\sum_{n\\equiv r\\ (\\mathrm{mod}\\,N)}","aliases":"∑_n≡ r (mod N) sum equiv mathrm","display":"\\sum_{n\\equiv r\\ (\\mathrm{mod}\\,N)}","kind":"use","excerpt":"Each integer belongs to exactly one residue class modulo N. Multiplying the two indicator functions gives zero for distinct classes and the same indicator for equal classes. Summing all residue indicators gives one at every address. Moreover, P_N,rx _2^2=∑_n≡ r (mod N)|x[n]|^2≤ x _2^2, ⟨ P_N,rx,y⟩=⟨ x,P_N,ry⟩. The mask is real-valued, so the inner-","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-657057157b7882f4","tex":"\\sum_{n\\geq 0}","aliases":"∑_n≥ 0 sum geq","display":"\\sum_{n\\geq 0}","kind":"use","excerpt":"This chapter follows the consequences of applying the cyclic projections to the exponential function. The projected exponentials form a finite differential system. After one sign is inserted around the derivative cycle, they form a basis for y^(N)=-y. The section with initial value one is D_N(z)=∑_n≥ 0(-1)^nfracz^Nn(Nn)!. Its reciprocal powers prod","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-391cc34f0ac3e01c","tex":"\\sum_{N\\ge0}","aliases":"∑_N≥0 sum ge","display":"\\sum_{N\\ge0}","kind":"use","excerpt":"m≥1, differentiation of zcoth z gives β_m=-frac2^2m(2m-1)B_2m(2m)!, κ_m=-2^4m(2m-1)B_2m. The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"[Quadratic Catalan expansion] Let B:X× X→ X be bounded and consider u=s+B(u,u). Put a= s and b= B . If 4ab<1, then u= ∑_n≥1 ∑_substackτ full ordered binary tree with n leaves B_τ(s) converges absolutely. It also satisfies u ≤ frac1-sqrt1-4ab2b when b>0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0d1bc9b91545210","tex":"\\sum_{n\\geq1}","aliases":"∑_n≥1 sum geq","display":"\\sum_{n\\geq1}","kind":"use","excerpt":"[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e9342a8d15de8640","tex":"\\sum_{q<n}","aliases":"∑_q<n sum","display":"\\sum_{q<n}","kind":"use","excerpt":"[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1f8f0dbb2884f06c","tex":"\\sum_{q=0}^m","aliases":"∑_q=0^m sum","display":"\\sum_{q=0}^m","kind":"use","excerpt":"The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and tr","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5be06f71bbc1552b","tex":"\\sum_{q=0}^{m}","aliases":"∑_q=0^m sum","display":"\\sum_{q=0}^{m}","kind":"use","excerpt":"[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5e33c0103731d00c","tex":"\\sum_{q=0}^n","aliases":"∑_q=0^n sum","display":"\\sum_{q=0}^n","kind":"use","excerpt":"[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2636c510c23e69cc","tex":"\\sum_{q=0}^{n-1}","aliases":"∑_q=0^n-1 sum","display":"\\sum_{q=0}^{n-1}","kind":"use","excerpt":"[Triangular inverse system] The coefficients are determined by C_N,j(0)=1 and ∑_q=0^n binomNn+jNq C_N,j(q) = δ_n0. For n≥1, C_N,j(n) = - frac displaystyle∑_q=0^n-1 binomNn+jNqC_N,j(q) displaystylebinomNn+jNn .","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-45df01cda676bad4","tex":"\\sum_{q=0}^r","aliases":"∑_q=0^r sum","display":"\\sum_{q=0}^r","kind":"use","excerpt":"X)=∏_a=0^r-1(X-a(a+1)). At X=Z(Z+1), they have the exact factorization P_r(Z(Z+1)) =∏_a=0^r-1(Z-a)(Z+a+1) =(2r)!binomZ+r2r. The Bernoulli transform gives L((1+T)^Z)=log(1+T)/T, whence L(binom Zj)=(-1)^j/(j+1). Finite Vandermonde expansion therefore gives L(binomZ+r2r) · =(-1)^r∑_q=0^r(-1)^qbinom rqfrac1r+q+1nonumber · =(-1)^rfrac(r!)^2(2r+1)!. The ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89403d4e42f7d258","tex":"\\sum_{q=0}^{r}","aliases":"∑_q=0^r sum","display":"\\sum_{q=0}^{r}","kind":"use","excerpt":"The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-098c7729ca21f24d","tex":"\\sum_{q=1}^M","aliases":"∑_q=1^M sum","display":"\\sum_{q=1}^M","kind":"use","excerpt":"[Several exact edge packets] If f=s+∑_q=1^Mj_q and every j_q is retained exactly, then f-(Q_ms+∑_qj_q) = s-Q_ms . Curved edges, corners, and different jump amplitudes do not alter the identity. The finite sum j=∑_q=1^M j_q is a single retained packet. Applying the preceding cancellation identity gives the result. The empty family M=0 is included wi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-794344618327eede","tex":"\\sum_{q=1}^{M}","aliases":"∑_q=1^M sum","display":"\\sum_{q=1}^{M}","kind":"use","excerpt":"[Several exact edge packets] If f=s+∑_q=1^Mj_q and every j_q is retained exactly, then f-(Q_ms+∑_qj_q) = s-Q_ms . Curved edges, corners, and different jump amplitudes do not alter the identity. The finite sum j=∑_q=1^M j_q is a single retained packet. Applying the preceding cancellation identity gives the result. The empty family M=0 is included wi","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18f531fd7c19a676","tex":"\\sum_{q=1}^n","aliases":"∑_q=1^n sum","display":"\\sum_{q=1}^n","kind":"use","excerpt":"The actual reciprocal recurrence is C_N,j(n)=-∑_q=1^nbinomNnNq g_N,j(q)C_N,j(n-q), C_N,j(0)=1. For N=4, the four kernels are 1, frac14q+1, frac1(4q+1)(2q+1), frac3(4q+1)(2q+1)(4q+3). The threshold-free theorem gives v_2(a_k)≥ k in every address. The following exact values cover all orders where that bound does not already imply [source reference]: ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1315d72b28634d4a","tex":"\\sum_{q\\in J_r}","aliases":"∑_q∈ J_r sum in","display":"\\sum_{q\\in J_r}","kind":"use","excerpt":"ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c67bb6d0757ac234","tex":"\\sum_{q\\in\\bbZ}","aliases":"∑_q∈bbZ ∑_q∈ Z sum in bbZ sum in mathbb","display":"\\sum_{q\\in{\\mathbb Z}}","kind":"use","excerpt":"Let h be a finite impulse response and y=h*x, y[n]=∑_q∈ Zh[q]x[n-q]. Resolve both input and output into N phases. For output phase k and input phase s, define the transfer kernel h_ks[m]=h[Nm+k-s]. Out-of-range taps are understood to be zero.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0606a20caaf6d2d9","tex":"\\sum_{q\\in\\ZZ^d}","aliases":"∑_q∈ZZ^d ∑_q∈ Z^d sum in ZZ sum in mathbb","display":"\\sum_{q\\in{\\mathbb Z}^d}","kind":"use","excerpt":"Let h∈ℓ^1( Z^d) and x∈ℓ^p( Z^d), with 1≤ p≤∞, and define (Hx)(n) =(h*x)(n) =∑_q∈ Z^dh(q)x(n-q). Young's inequality gives Hx _p≤ h _1 x _p for every 1≤ p≤∞.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-65994b1f27113b7a","tex":"\\sum_{q\\ge0}","aliases":"∑_q≥0 sum ge","display":"\\sum_{q\\ge0}","kind":"use","excerpt":"Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eb07f8d7b26ea999","tex":"\\sum_{q\\geq0}","aliases":"∑_q≥0 sum geq","display":"\\sum_{q\\geq0}","kind":"use","excerpt":"[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"g s_2(i) for the number of ones in the binary expansion of i, and using v_2((4i)!)=4i-s_2(i), gives v_2([x^d]P_i) ≥2d-2i+s_2(i)+max(0,2i-d) ≥ d+s_2(i)≥ d+1 (d≥1). For k≥1, Δ_1^k x^d vanishes when d<k and is an integer polynomial with values divisible by k!. To see the latter, expand x^d=∑_r{d r} r!binom xr and apply Δ_1^kbinom xr=binomxr-k. It foll","count":14},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6589276c9db025d","tex":"\\sum_{r}","aliases":"∑_r sum","display":"\\sum_{r}","kind":"use","excerpt":"[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0de4c91e29511ae3","tex":"\\sum_{r+s\\equiv t\\!\\!\\pmod N}","aliases":"∑_r+s≡ t pmod N sum equiv pmod","display":"\\sum_{r+s\\equiv t\\!\\!\\pmod N}","kind":"use","excerpt":"rem[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-05d2aa25a4e84f79","tex":"\\sum_{r,a}","aliases":"∑_r,a sum","display":"\\sum_{r,a}","kind":"use","excerpt":"Deriving the Gram matrix from recovery Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d94b7b7c59ff0407","tex":"\\sum_{r<d}","aliases":"∑_r<d sum","display":"\\sum_{r<d}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-aec6d0bc29467f34","tex":"\\sum_{r<n}","aliases":"∑_r<n sum","display":"\\sum_{r<n}","kind":"use","excerpt":"Pascal's identity proves the first product rule, and induction proves the higher rule. The intermediate binomial coefficients binom pr vanish in characteristic p. For the inverse, solve row n using the coefficient of b_n, which is g_0=1: b_n=1_n-∑_r<nbinom nr b_r g_n-r. Subtract the first and pth product rules for b⋆ g=1. The assumed identity for g","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f161dab54b7fa5c5","tex":"\\sum_{r=0}^3","aliases":"∑_r=0^3 sum","display":"\\sum_{r=0}^3","kind":"use","excerpt":"ump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector P_4,rf is different: it keeps the original latt","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f08116933b3b512d","tex":"\\sum_{r=0}^{D}","aliases":"∑_r=0^D sum","display":"\\sum_{r=0}^{D}","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08b542392666bac8","tex":"\\sum_{r=0}^{d}","aliases":"∑_r=0^d sum","display":"\\sum_{r=0}^{d}","kind":"use","excerpt":"[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3608ee04c017022","tex":"\\sum_{r=0}^{d-1}","aliases":"∑_r=0^d-1 sum","display":"\\sum_{r=0}^{d-1}","kind":"use","excerpt":"[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalizati","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-660887c0abbbed2c","tex":"\\sum_{r=0}^k","aliases":"∑_r=0^k sum","display":"\\sum_{r=0}^k","kind":"use","excerpt":"ence identity is Δ_1^kfrac11+sn =frac(-s)^k k!∏_h=0^k(1+s(n+h)). It follows by induction, subtracting adjacent rational functions. Each successive subtraction adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then prov","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fed0ff8a0203f247","tex":"\\sum_{r=0}^{m}","aliases":"∑_r=0^m sum","display":"\\sum_{r=0}^{m}","kind":"use","excerpt":"Let (Δ x)(n)=x(n+1)-x(n). Then Δ^m x(n) = ∑_r=0^m (-1)^m-rbinom mr x(n+r).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4bb40d93da4fa77c","tex":"\\sum_{r=0}^{M-1}","aliases":"∑_r=0^M-1 sum","display":"\\sum_{r=0}^{M-1}","kind":"use","excerpt":"polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient an","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-58d677395e4f323c","tex":"\\sum_{r=0}^{\\min(M,n/2)}","aliases":"∑_r=0^min(M,n/2) sum min","display":"\\sum_{r=0}^{\\min(M,n/2)}","kind":"use","excerpt":"[Odd modulus truncation] For every M≥0, every positive odd q, and every even n, E_n ≡ ∑_r=0^min(M,n/2) (-1)^rZ_2r+1 binom n2rq^2rS_n-2r(q) pmod q^2M+2. If pmid q and a=v_p(q), the same truncation is valid modulo p^2a(M+1).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed8019b78e98d938","tex":"\\sum_{r=0}^{N}","aliases":"∑_r=0^N sum","display":"\\sum_{r=0}^{N}","kind":"use","excerpt":"Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-029134516e4be179","tex":"\\sum_{r=0}^{N-1}","aliases":"∑_r=0^N-1 sum","display":"\\sum_{r=0}^{N-1}","kind":"use","excerpt":"[Polyphase projector theorem] The maps P_N,r are bounded orthogonal projectors on ℓ^2, with operator norm at most one, and satisfy P_N,rP_N,s=δ_rsP_N,r, ∑_r=0^N-1P_N,r=I. Consequently, x=∑_r=0^N-1P_N,rx with no approximation.","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2753495994955ca2","tex":"\\sum_{r=0}^{n-1}","aliases":"∑_r=0^n-1 sum","display":"\\sum_{r=0}^{n-1}","kind":"use","excerpt":"tlasFigDir/#1.pdf AtlasHeadCompute the reciprocal before estimating itatlas:reciprocal For width N and address j, the normalized source and its reciprocal are G_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, G_N,j(t)^-1=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Multiplying the two series and clearing the factorials gives an exact triangular recurrence: C_N,j(0)=1, C_N,j(n)=-fra","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8fbff1ef26dc212a","tex":"\\sum_{r=0}^{n/2}","aliases":"∑_r=0^n/2 sum","display":"\\sum_{r=0}^{n/2}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4daf9cd99c1a0246","tex":"\\sum_{r=0}^{p-1}","aliases":"∑_r=0^p-1 sum","display":"\\sum_{r=0}^{p-1}","kind":"use","excerpt":"[Finite inverse formula] If x has augmentation one and n=x-1, then x^-1 = ∑_r=0^p-1(-n)^r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-021051c7510dc78f","tex":"\\sum_{r=0}^{x}","aliases":"∑_r=0^x sum","display":"\\sum_{r=0}^{x}","kind":"use","excerpt":"/3 in the one-prime example above loses the advertised digit. Invertibility of C alone does not restore it. The original proof is retained under these hypotheses. Let Q=∏_p∈ Sp^L_p for a finite set of primes. Let C∈ M_d( Z) be invertible modulo Q, and choose H such that C^H=I+QB with B∈ M_d( Z). Then for every nonnegative integer x, a_Hx+j = ∑_r=0^","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa57f8e9d7805c7c","tex":"\\sum_{r=1}^{e}","aliases":"∑_r=1^e sum","display":"\\sum_{r=1}^{e}","kind":"use","excerpt":"[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a40c8a70cdd8f4bb","tex":"\\sum_{r=1}^{h}","aliases":"∑_r=1^h sum","display":"\\sum_{r=1}^{h}","kind":"use","excerpt":"[Rational ordinary generating function] In F_p[[z]], ∑_n≥0e_n^(m,p)z^n = 1+ frac ∑_r=1^he_r^(m,p)z^r 1-z^h.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bf6132b8c151067a","tex":"\\sum_{r=1}^{m}","aliases":"∑_r=1^m sum","display":"\\sum_{r=1}^{m}","kind":"use","excerpt":"[Packet Taylor formula] Assume F is m+1 times continuously Fr echet differentiable on an open set containing the segment x+th for 0≤ t≤1. Then Q_γ F(x+h) = · Q_γ F(x) + ∑_r=1^mfrac1r! ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h,…,P_α_rh] · +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] d","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4ae22d4d94a532ab","tex":"\\sum_{r=1}^{n}","aliases":"∑_r=1^n sum","display":"\\sum_{r=1}^{n}","kind":"use","excerpt":"[A denominator bound] Let D_N,j(n) = ∏_r=1^n binomNr+jNr, D_N,j(0)=1. Then D_N,j(n)C_N,j(n) ∈ Z. For every prime p, v_p(C_N,j(n)) ≥ - ∑_r=1^n v_pbinomNr+jNr.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5525cce418bdaded","tex":"\\sum_{r=1}^{n/2}","aliases":"∑_r=1^n/2 sum","display":"\\sum_{r=1}^{n/2}","kind":"use","excerpt":"[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-54d809b1c239a990","tex":"\\sum_{r=1}^q","aliases":"∑_r=1^q sum","display":"\\sum_{r=1}^q","kind":"use","excerpt":"[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b31a2c0bfdd4782","tex":"\\sum_{r=2}^{d}","aliases":"∑_r=2^d sum","display":"\\sum_{r=2}^{d}","kind":"use","excerpt":"Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c88576010b1e038","tex":"\\sum_{r=k}^{\\min(D,Mk+s)}","aliases":"∑_r=k^min(D,Mk+s) sum min","display":"\\sum_{r=k}^{\\min(D,Mk+s)}","kind":"use","excerpt":"For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b2e9009509f5bdd9","tex":"\\sum_{r=k}^{Mk+s}","aliases":"∑_r=k^Mk+s sum","display":"\\sum_{r=k}^{Mk+s}","kind":"use","excerpt":"[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c27b41f7e97bcb0e","tex":"\\sum_{r=s}^{N}","aliases":"∑_r=s^N sum","display":"\\sum_{r=s}^{N}","kind":"use","excerpt":"[Finite Mahler reconstruction] Given f(0),…,f(N), put c_r=Δ^rf(0). Then f(n)=∑_r=0^nc_rbinom nr (0≤ n≤ N). For every s with 0≤ s≤ N, Δ^sf(n) = ∑_r=s^Nc_rbinom nr-s (0≤ n≤ N-s).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-107ff376ef990f8a","tex":"\\sum_{r\\bmod N}","aliases":"∑_rbmod N sum bmod","display":"\\sum_{r\\bmod N}","kind":"use","excerpt":"[Complete residue packet system] For fixed N, the projectors P_N,r satisfy P_N,rP_N,s = δ_rsP_N,r and ∑_rbmod NP_N,r=I.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb3e4b5ba1637239","tex":"\\sum_{r\\in B_\\alpha}","aliases":"∑_r∈ B_α sum in alpha","display":"\\sum_{r\\in B_\\alpha}","kind":"use","excerpt":"[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4c18b03cbd50956a","tex":"\\sum_{r\\in R}","aliases":"∑_r∈ R sum in","display":"\\sum_{r\\in R}","kind":"use","excerpt":"For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c6a6e48cf2891abf","tex":"\\sum_{r\\in\\mathbb Z/N\\mathbb Z}","aliases":"∑_r∈ Z/N Z sum in mathbb mathbb","display":"\\sum_{r\\in\\mathbb Z/N\\mathbb Z}","kind":"use","excerpt":"Exact coefficient formula. The packet definition is a finite sum over residue classes: P_j=frac1N∑_r∈ Z/N Zχ(-rj)ρ(r). The source proves χ(-rj)=(ω^val(r)val(j))^-1, so this is precisely the manuscript's N-term Fourier projector formula, with canonical representatives.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6a4250883c4968da","tex":"\\sum_{r\\ge0}","aliases":"∑_r≥0 sum ge","display":"\\sum_{r\\ge0}","kind":"use","excerpt":"[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator a","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6b4cbee70b31b1b1","tex":"\\sum_{r\\geq0}","aliases":"∑_r≥0 sum geq","display":"\\sum_{r\\geq0}","kind":"use","excerpt":"Put A=∑_e∈ EA_e. Since left multiplication by D^-1 preserves the filtration, A∈ F^1 B. Completeness gives the convergent Neumann series (I+A)^-1=∑_r≥0(-A)^r. Because T=D(I+A), T^-1=(I+A)^-1D^-1. Expanding each power of A gives the sum over words, with the rightmost edge acting first. Multiplicativity of the filtration gives A_γ∈ F^W(γ) B. For each ","count":19},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc62b5ae7466808f","tex":"\\sum_{r\\ge1}","aliases":"∑_r≥1 sum ge","display":"\\sum_{r\\ge1}","kind":"use","excerpt":"[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet o","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28fc50db4cc325e2","tex":"\\sum_{r\\geq1}","aliases":"∑_r≥1 sum geq","display":"\\sum_{r\\geq1}","kind":"use","excerpt":"The even source identity gives secht = sech^2(qt) ∑_n≥0S_2n(q)fract^2n(2n)!. Insert [source reference] and compare coefficients. For the inverse, multiply secht by cosh^2(qt) = 1+ ∑_r≥12^2r-1q^2rfract^2r(2r)!.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d98fe9de6bebd3c","tex":"\\sum_s","aliases":"∑_s sum","display":"\\sum_s","kind":"use","excerpt":"For a finite nonlinear shift operator F(x)[n]=x[n]+α x[n]x[n-1], the exact packet coordinates are simply F_r((x_s)_s)=P_N,rF (∑_sP_N,sx). Analysis followed by this packet system and synthesis reproduces the direct nonlinear output exactly. The numerical residual in the companion implementation was zero to machine representation.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ab6516e07d5eaa7c","tex":"\\sum_{s=0}^n","aliases":"∑_s=0^n sum","display":"\\sum_{s=0}^n","kind":"use","excerpt":"For every starting row, Δ_1^k C_0(n) = L(U^4nW^k) =∑_s=0^nbinom ns L(W^k+s). Each summand lies in 2^k k! Z_2, proving the factorial bound for address zero from row zero onward.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-51618a69e67f3faa","tex":"\\sum_{s=0}^{N-1}","aliases":"∑_s=0^N-1 sum","display":"\\sum_{s=0}^{N-1}","kind":"use","excerpt":"[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ed58db8837af33eb","tex":"\\sum_{s\\succ r}","aliases":"∑_ssucc r sum succ","display":"\\sum_{s\\succ r}","kind":"use","excerpt":"The signal realization makes the seed, address, packet, transfer, refinement, and synthesis operations concrete. The objects and laws can be read directly: seed · =δ_0, address · =S^n, packet · =P_N,rx, transfer · =P_N,kHP_N,s, refinement · =P_N,r=∑_ssucc rP_M,s, synthesis · =∑_rP_N,rx, nonlinearity · =P_N,rF(∑_sP_N,sx). Filter banks, wavelet packe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df3e5e35be464a61","tex":"\\sum_{\\substack{0\\le a<M\\\\a\\equiv r\\pmod N}}","aliases":"∑_substack0≤ a<M a≡ rpmod N sum substack le a equiv pmod","display":null,"kind":"use","excerpt":"leaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples. For example, at the first descendant address the values f[0]=0 and f[2]=1/2 are two separate values. Their sum 1/2 cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement. headThe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4a1b621790c1e71","tex":"\\sum_{\\substack{0\\leq a<p\\\\2a+1\\neq p}}","aliases":"∑_substack0≤ a<p 2a+1≠ p sum substack leq neq","display":"\\sum_{\\substack{0\\leq a<p\\\\2a+1\\neq p}}","kind":"use","excerpt":"Fix an even address j∈{0,2,…,p-3}. For x∈ Z_p, define mathscr E_p,j^×(x) = ∑_r≥0 · (-1)^rZ_2r+1p^2r binom(p-1)x+j2r · × ∑_substack0≤ a<p 2a+1≠ p (-1)^a (2a+1)^j-2r ((2a+1)^p-1)^x.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4cedeb30578efb5","tex":"\\sum_{\\substack{0\\leq i,j<N\\\\ i+j\\equiv k\\bmod N}}","aliases":"∑_substack0≤ i,j<N i+j≡ kbmod N sum substack leq equiv bmod","display":"\\sum_{\\substack{0\\leq i,j<N\\\\ i+j\\equiv k\\bmod N}}","kind":"use","excerpt":"[Cyclic product law] If F and G are holomorphic on the same rotation-invariant domain, then B_N,k(FG) = ∑_substack0≤ i,j<N i+j≡ kbmod N ( B_N,iF)( B_N,jG).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49a9c5b899546242","tex":"\\sum_{\\substack{0\\le r<M\\\\r\\bmod N=j}}","aliases":"∑_substack0≤ r<M rbmod N=j sum substack le r bmod","display":null,"kind":"use","excerpt":"0, · otherwise. The source proves idempotence for every selector, orthogonality for distinct standard residue labels, and, when N>0, ∑_j=0^N-1P_N,ja=a. If Nmid M and M>0, it also proves P_N,ja=∑_substack0≤ r<M rbmod N=jP_M,ra. The formal power-series versions are proved coefficientwise for actual mcodePowerSeries objects. There is no finite truncat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1f80fcd3e8646fe","tex":"\\sum_{\\substack{0\\leq r<M\\\\r\\equiv j\\pmod N}}","aliases":"∑_substack0≤ r<M r≡ jpmod N sum substack leq r equiv pmod","display":null,"kind":"use","excerpt":"[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M r≡ jpmod NP_M,ra. The same identities hold coefficientwise for formal power series.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-19c8aa9ad21cda67","tex":"\\sum_{\\substack{0\\le r\\le n\\\\w(r)<L}}","aliases":"∑_substack0≤ r≤ n w(r)<L sum substack le le w","display":null,"kind":"use","excerpt":"0, · w=∞. The infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, every term with weight at least L is divisible by p^L. A finite-sum proof yields p^L bigm| ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r -∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-r. The result is specialized to actual p-adic","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0eec45661b8436b8","tex":"\\sum_{\\substack{0\\leq r\\leq n\\\\w(r)<L}}","aliases":"∑_substack0≤ r≤ n w(r)<L sum substack leq leq w","display":null,"kind":"use","excerpt":"[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3554bc0644c628c2","tex":"\\sum_{\\substack{0\\leq s<M\\\\s\\equiv j\\pmod N}}","aliases":"∑_substack0≤ s<M s≡ jpmod N sum substack leq s equiv pmod","display":null,"kind":"use","excerpt":"[Nested residue refinement] Let N,M be positive integers with N dividing M, and let 0≤ j<N. Then P_N,j = ∑_substack0≤ s<M s≡ jpmod NP_M,s. When R is nonzero, the packet system modulo M refines the packet system modulo N.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7cb7901967a508d7","tex":"\\sum_{\\substack{0\\le s<M\\\\s\\equiv r\\, (\\mathrm{mod}\\,N)}}","aliases":"∑_substack0≤ s<M s≡ r (mod N) sum substack le s equiv mathrm","display":null,"kind":"use","excerpt":"[Exact nested refinement] For N,M≥1 and Nmid M, P_N,r = ∑_substack0≤ s<M s≡ r (mod N)P_M,s.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a6c7f694cb6268c7","tex":"\\sum_{\\substack{e:\\,\\mathrm{source}(e)=s\\\\w(e)\\le n}}","aliases":"∑_substacke: source(e)=s w(e)≤ n sum substack mathrm w le","display":null,"kind":"use","excerpt":"headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumerat","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4dc3a2d6215c09ba","tex":"\\sum_{\\substack{e:\\,\\operatorname{source}(e)=s\\\\w(e)\\le n}}","aliases":"∑_substacke: source(e)=s w(e)≤ n sum substack operatorname w le","display":null,"kind":"use","excerpt":"[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d047e8abc842ee44","tex":"\\sum_{\\substack{g<h\\\\g+h=k}}","aliases":"∑_substackg<h g+h=k sum substack g","display":null,"kind":"use","excerpt":"[Divided square packet equations] Assume q( L_g)⊆ L_2g. Choose an ordering of the finite label set. The equation dη+q(η)=0 is equivalent to dη_k + ∑_2g=kq(η_g) + ∑_substackg<h g+h=k[η_g,η_h] =0 for every k.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-70e4a00162aa41df","tex":"\\sum_{\\substack{I\\ne0\\\\\\Nm I\\leq X}}","aliases":"∑_substackI≠0 Nm I≤ X ∑_substackI≠0 N I≤ X sum substack ne Nm leq sum substack ne operatorname leq","display":"\\sum_{\\substack{I\\ne0\\\\{\\operatorname N} I\\leq X}}","kind":"use","excerpt":"e preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substack","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6109badb9f9fb1b3","tex":"\\sum_{\\substack{I\\ne0,\\ [I]=C\\\\\\Nm I\\leq X}}","aliases":"∑_substackI≠0, [I]=C Nm I≤ X ∑_substackI≠0, [I]=C N I≤ X sum substack ne Nm leq sum substack ne operatorname leq","display":"\\sum_{\\substack{I\\ne0,\\ [I]=C\\\\{\\operatorname N} I\\leq X}}","kind":"use","excerpt":"The assembled ideal norm-character estimate. Let C∈Cl( O), let J≠0 be an integral ideal with [J]=C^-1, and let m>0. Let χ: Z/m Z→ C preserve multiplication, 0, and 1. Assume a residue-ring unit satisfying [source reference], and assume χ( N Jbmod m)≠0. Then T_C,χ(X) =∑_substackI≠0, [I]=C N I≤ X χ( N Ibmod m) =O_K,J,m,χ(X^1-1/d). More precisely, som","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7a4926b7b13440e3","tex":"\\sum_{\\substack{\\gamma,\\gamma'\\in\\Gamma_b\\\\ \\gamma<\\gamma'}}","aliases":"∑_substackγ,γ'∈Γ_b γ<γ' sum substack gamma gamma in Gamma gamma gamma","display":"\\sum_{\\substack{\\gamma,\\gamma'\\in\\Gamma_b\\\\ \\gamma<\\gamma'}}","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d8aed7d286bba088","tex":"\\sum_{\\substack{\\gamma:\\ P_\\alpha R_\\gamma=R_\\gamma\\\\Q_\\beta R_\\gamma=R_\\gamma}}","aliases":"∑_substackγ: P_α R_γ=R_γ Q_β R_γ=R_γ ∑_substackγ: P_α R_γ=R_γ{ Q_β R_γ=R_γ sum substack gamma alpha gamma gamma Q beta gamma gamma sum substack gamma alpha gamma gamma mathbb beta gamma gamma","display":"\\sum_{\\substack{\\gamma:\\ P_\\alpha R_\\gamma=R_\\gamma\\{\\mathbb Q}_\\beta R_\\gamma=R_\\gamma}}","kind":"use","excerpt":"a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ{ Q_β R_γ=R_γR_γ. Thus the packets with the same p","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c4bd3d34873e4945","tex":"\\sum_{\\substack{\\gamma\\in E^n\\\\W(\\gamma)<L}}","aliases":"∑_substackγ∈ E^n W(γ)<L sum substack gamma in W gamma","display":null,"kind":"use","excerpt":"et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1ab55d9727944350","tex":"\\sum_{\\substack{\\eta^N=-1\\\\\n\\eta\\ne e^{ia_N},e^{-ia_N}}}","aliases":"∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N sum substack eta eta ne","display":"\\sum_{\\substack{\\eta^N=-1\\\\\n\\eta\\ne e^{ia_N},e^{-ia_N}}}","kind":"use","excerpt":"On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-649ddfbfdba19725","tex":"\\sum_{\\substack{\\rho(\\beta)=b\\\\\\rho(\\alpha)=a}}","aliases":"∑_substackρ(β)=b ρ(α)=a sum substack rho beta rho alpha","display":"\\sum_{\\substack{\\rho(\\beta)=b\\\\\\rho(\\alpha)=a}}","kind":"use","excerpt":"[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f1724b58a24b34fd","tex":"\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}","aliases":"∑_substackσ(β)=j σ(α)=i sum substack sigma beta sigma alpha","display":"\\sum_{\\substack{\\sigma(\\beta)=j\\\\\\sigma(\\alpha)=i}}","kind":"use","excerpt":"[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and S generates X, then Φ(S) generates Y under φ( A) and hence under B. If, in ad","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07a86bb483b36897","tex":"\\sum_{\\substack{\\tau\\text{ full ordered binary tree}\\\\\\text{with }n\\text{ leaves}}}","aliases":"∑_substackτ full ordered binary tree with n leaves sum substack tau text text text","display":"\\sum_{\\substack{\\tau\\text{ full ordered binary tree}\\\\\\text{with }n\\text{ leaves}}}","kind":"use","excerpt":"[Quadratic Catalan expansion] Let B:X× X→ X be bounded and consider u=s+B(u,u). Put a= s and b= B . If 4ab<1, then u= ∑_n≥1 ∑_substackτ full ordered binary tree with n leaves B_τ(s) converges absolutely. It also satisfies u ≤ frac1-sqrt1-4ab2b when b>0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1d7041cea704f845","tex":"\\sum_{s\\in R}","aliases":"∑_s∈ R sum in","display":"\\sum_{s\\in R}","kind":"use","excerpt":"[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b4be24e93748542c","tex":"\\sum_{s\\in\\operatorname{Child}(r)}","aliases":"∑_s∈Child(r) sum in operatorname","display":"\\sum_{s\\in\\operatorname{Child}(r)}","kind":"use","excerpt":"[Exact lattice refinement] For every coarse packet, P_r^L = ∑_s∈Child(r)P_s^L'. The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd50407fa1571a87","tex":"\\sum_{t\\in k^{\\times}}","aliases":"∑_t∈ k^× sum in times","display":"\\sum_{t\\in k^{\\times}}","kind":"use","excerpt":"Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d96e96aadd273752","tex":"\\sum_u","aliases":"∑_u sum","display":"\\sum_u","kind":"use","excerpt":"[Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7ecb3223d556bc7c","tex":"\\sum_{u\\text{ split}}","aliases":"∑_u split sum text","display":"\\sum_{u\\text{ split}}","kind":"use","excerpt":"The cost and error of a pruning L are cost( L) = ∑_v∈ Lκ_v + ∑_u splitσ_u and err( L)=∑_v∈ Lϵ_v. The model can include arithmetic, storage, launch, transfer, and precision costs inside κ_v. It remains exact only when the declared totals are additive.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a467b0ef76a3602c","tex":"\\sum_{u+v=r}","aliases":"∑_u+v=r sum","display":"\\sum_{u+v=r}","kind":"use","excerpt":"a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-07ab15ab7625ce3b","tex":"\\sum_{u+v=r+M}","aliases":"∑_u+v=r+M sum","display":"\\sum_{u+v=r+M}","kind":"use","excerpt":"a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3cdd7357c745c72c","tex":"\\sum_{u=0}^{s}","aliases":"∑_u=0^s sum","display":"\\sum_{u=0}^{s}","kind":"use","excerpt":"insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-945d71373857dd47","tex":"\\sum_{u\\in U}","aliases":"∑_u∈ U sum in","display":"\\sum_{u\\in U}","kind":"use","excerpt":"[Parabolic annihilation] Let P=LU be a proper parabolic subgroup of G_n, and put e_U=frac1abs U∑_u∈ Uu. If κ_θ is cuspidal, then e_Ue_θ=e_θ e_U=0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d4ce8d11f1c2be0","tex":"\\sum_{u\\in X}","aliases":"∑_u∈ X sum in","display":"\\sum_{u\\in X}","kind":"use","excerpt":"If the translation component x is nonzero, the operator permutes the standard basis without a fixed point, so its trace is zero. If x=0 but ξ≠0, the trace is ψ(z)∑_u∈ Xψ(ξ(u))=0. When x=ξ=0, the operator is the scalar ψ(z).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-89c9a3fa4b6a762b","tex":"\\sum_{u\\in\\FF_p}","aliases":"∑_u∈FF_p ∑_u∈ F_p sum in FF sum in mathbb","display":"\\sum_{u\\in{\\mathbb F}_p}","kind":"use","excerpt":"The product law follows from [source reference]. To prove injectivity, suppose ∑_u∈ F_pc_uϵ_u=0. Looking at the coordinates n=0,1,…,p-1 gives a Vandermonde system with matrix (u^n)_0≤ n≤ p-1, u∈ F_p. Its determinant is the product of all differences between distinct field elements, so it is nonzero. Hence every c_u vanishes.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-75b7b5b56c5ec55a","tex":"\\sum_{u\\neq0}","aliases":"∑_u≠0 sum neq","display":"\\sum_{u\\neq0}","kind":"use","excerpt":"For positive n, the zero mode contributes nothing. Every nonzero mode satisfies u^p-1=1. Hence b_n+p-1^(m,p) = ∑_u≠0c_uu^n+p-1 = ∑_u≠0c_uu^n = b_n^(m,p).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-53c07efe2188e4b0","tex":"\\sum_v","aliases":"∑_v sum","display":"\\sum_v","kind":"use","excerpt":", the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogo","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4dbf2eb498294f0b","tex":"\\sum_{v\\in G_m}","aliases":"∑_v∈ G_m sum in","display":"\\sum_{v\\in G_m}","kind":"use","excerpt":"[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1774cac52d8c9785","tex":"\\sum_{v\\in Q_0}","aliases":"∑_v∈ Q_0 sum in","display":"\\sum_{v\\in Q_0}","kind":"use","excerpt":"Because Q_0 is finite, 1=∑_v∈ Q_0e_v in 𝕜 Q, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-582da013421a516b","tex":"\\sum_{v\\in\\Lcal}","aliases":"∑_v∈Lcal ∑_v∈ L sum in Lcal sum in mathcal","display":"\\sum_{v\\in{\\mathcal L}}","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c2f7250b20e70fc","tex":"\\sum_{v\\in\\Lambda}","aliases":"∑_v∈Λ sum in Lambda","display":"\\sum_{v\\in\\Lambda}","kind":"use","excerpt":"[Exact synthesis from every complete pruning] For every complete pruning Λ, ∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx. Thus changing the selected tree changes the coordinates but does not introduce reconstruction error.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2b95868d24f2bd48","tex":"\\sum_w","aliases":"∑_w sum","display":"\\sum_w","kind":"use","excerpt":"Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22b9f62ba75edb77","tex":"\\sum_{w}","aliases":"∑_w sum","display":"\\sum_{w}","kind":"use","excerpt":"Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-364ec86993472593","tex":"\\sum_{W(\\gamma)<L}","aliases":"∑_W(γ)<L sum gamma","display":"\\sum_{W(\\gamma)<L}","kind":"use","excerpt":"[Exact finite precision truncation] Under the hypotheses of [source reference], define R_L = ∑_W(γ)<L(-1)^|γ|A_γ D^-1. Then T^-1-R_L∈ F^L B, TR_L-I∈ F^L B, R_LT-I∈ F^L B. Thus R_L is the exact inverse in B/F^L B.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-998f4941f1c93781","tex":"\\sum_{w\\in E^n}","aliases":"∑_w∈ E^n sum in","display":"\\sum_{w\\in E^n}","kind":"use","excerpt":"[Finite operator expansion from words to walks] For every n∈ N, T^n = ∑_w∈ E^nL_w = ∑_w∈ E^n L_w, · w composable, 0, · w not composable. Parallel edges occur as distinct summands. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_explicitWordExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.tran","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1c79b8a11020f120","tex":"\\sum_{w\\in\\mathrm{child}(v)}","aliases":"∑_w∈child(v) sum in mathrm","display":"\\sum_{w\\in\\mathrm{child}(v)}","kind":"use","excerpt":"Recurrence. For terminal nodes, F_λ(v)=κ_v+λϵ_v. For internal nodes, F_λ(v)=min{κ_v+λϵ_v, σ_v+∑_w∈child(v)F_λ(w)}. The checked result has two parts. First, F_λ(v)≤ J_λ(L) for every legal cut of the subtree. Second, the constructed function codecompile returns a legal cut with objective exactly F_λ(v). Thus the minimum is attained, not merely bounde","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a38ac3f06fcc5327","tex":"\\sum_{w\\in\\operatorname{child}(v)}","aliases":"∑_w∈child(v) sum in operatorname","display":"\\sum_{w\\in\\operatorname{child}(v)}","kind":"use","excerpt":"The packet tree results treat every admissible pruning, not only complete trees of a fixed depth. Unitary local splits give a unitary global transform. Local frame bounds propagate recursively in the nonunitary case. If the cost of a pruning is additive over retained nodes, then F(v) = min{ c(v), ∑_w∈child(v)F(w) } gives the exact optimal tree. The","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dcadd6b9285294d8","tex":"\\sum_{w\\in\\Walk_n(i,j)}","aliases":"∑_w∈Walk_n(i,j) sum in Walk sum in operatorname","display":"\\sum_{w\\in{\\operatorname{Walk}}_n(i,j)}","kind":"use","excerpt":"[Endpoint coefficient walk sum] For all n∈ N and i,j∈ V, (T^n)_ji = ∑_w∈Walk_n(i,j)c^←(w). statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Connects; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.endpointWalkCoefficientSum; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_apply_eq_sum_walks","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e44b9be25437975","tex":"\\sum_x","aliases":"∑_x sum","display":"\\sum_x","kind":"use","excerpt":"Using the quadratic identity, q_v(x)=q(x+v)+q(v). Therefore ∑_x(-1)^q_v(x) = (-1)^q(v) ∑_x(-1)^q(x). Compare the signs in [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60e7e6eb0365413a","tex":"\\sum_{x,y\\in k^{\\times}}","aliases":"∑_x,y∈ k^× sum in times","display":"\\sum_{x,y\\in k^{\\times}}","kind":"use","excerpt":"Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e4274b6f3ad16bfb","tex":"\\sum_{x\\in k^{\\times}}","aliases":"∑_x∈ k^× sum in times","display":"\\sum_{x\\in k^{\\times}}","kind":"use","excerpt":"Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9923075fc1158c85","tex":"\\sum_{x\\in V}","aliases":"∑_x∈ V sum in","display":"\\sum_{x\\in V}","kind":"use","excerpt":"[Dyadic Gauss sign] Let dim V=2r, and let q be nondegenerate. Then ∑_x∈ V(-1)^q(x) = (-1)^Arf(q)2^r.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32572785b48f6bd4","tex":"\\sum_{x\\in X}","aliases":"∑_x∈ X sum in","display":"\\sum_{x\\in X}","kind":"use","excerpt":"The lifted isotropic subgroup is abelian, so group averaging is an idempotent. Every operator in the average is unitary, and the subgroup is closed under inversion, so the average is self adjoint. For L=X, the average is the translation average (P_Lf)(u) = frac1abs X ∑_x∈ Xf(u+x), which is the constant function with value equal to the mean of f. It","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5966b9097878350e","tex":"\\sum_z","aliases":"∑_z sum","display":"\\sum_z","kind":"use","excerpt":"On a vector (x_0,x_1,x_2)^ T, the first step produces (0,2x_0,0)^ T, and the second produces (0,0,6x_0)^ T. Thus L_fL_e=6E_20, L_eL_f=0. For general edges, a summand in the entrywise product can survive only when the same intermediate packet is both tgt(e) and src(f): (L_fL_e)_yx=∑_z(L_f)_yz(L_e)_zx. This is why a broken interface annihilates the e","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad8cbd6682b9ecb7","tex":"\\sum_{z\\in V}","aliases":"∑_z∈ V sum in","display":"\\sum_{z\\in V}","kind":"use","excerpt":"For a matrix entry (y,x), (L_fL_e)_yx=∑_z∈ V(L_f)_yz(L_e)_zx. A potentially nonzero summand would require simultaneously z=src(f) and z=tgt(e), contrary to the hypothesis.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b3187996cd33bccc","tex":"\\sum_{|\\alpha|=2}","aliases":"∑_|α|=2 sum alpha","display":"\\sum_{|\\alpha|=2}","kind":"use","excerpt":"/4, 0, 1/4, which again sum to 57. Two factors of 1/sqrt2 explain why the depth-two coefficients are rational halves and quarters. headCoefficients and projectors are different objects W_α f is the short coefficient vector. The original-space component is W_α^*W_α f. Summing these synthesized components, not the short arrays entrywise, reconstructs","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ecf317dfca855f25","tex":"\\sum_{\\alpha}","aliases":"∑_α sum alpha","display":"\\sum_{\\alpha}","kind":"use","excerpt":"[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invarianc","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","excerpt":"Insert the finite packet identity on both sides: T= (∑_β P_β) T (∑_α P_α) = ∑_α,βP_β TP_α.","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-575223190a495e3e","tex":"\\sum_{\\alpha,\\beta\\in I}","aliases":"∑_α,β∈ I sum alpha beta in","display":"\\sum_{\\alpha,\\beta\\in I}","kind":"use","excerpt":"[Exact operator reconstruction] Every T∈End_𝕜(X) satisfies T=∑_α,β∈ IT_βα. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ae59b81f83f7356c","tex":"\\sum_{\\alpha,\\beta\\in I_0}","aliases":"∑_α,β∈ I_0 sum alpha beta in","display":"\\sum_{\\alpha,\\beta\\in I_0}","kind":"use","excerpt":"X). The general Laplace formula is [source reference]. [Packet convolution] For all s,t≥0 and all packets α,β, T_βα(s+t) = ∑_γ∈ I_0 T_βγ(s)T_γα(t). The original semigroup is reconstructed from its blocks by T(t)=∑_α,β∈ I_0T_βα(t).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e265f6e1bb343d9c","tex":"\\sum_{\\alpha,\\gamma}","aliases":"∑_α,γ sum alpha gamma","display":"\\sum_{\\alpha,\\gamma}","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-46f38e126539df37","tex":"\\sum_{\\alpha:n_\\alpha>0}","aliases":"∑_α:n_α>0 sum alpha alpha","display":"\\sum_{\\alpha:n_\\alpha>0}","kind":"use","excerpt":"on decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the information retain","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2877d5985ed013b2","tex":"\\sum_{\\alpha:Z_\\alpha>0}","aliases":"∑_α:Z_α>0 sum alpha alpha","display":"\\sum_{\\alpha:Z_\\alpha>0}","kind":"use","excerpt":"[Bayesian packet reconstruction] The posterior is the disjoint packet synthesis B_λ(p) = ∑_α:Z_α>0 m_α q_α. The local shape q_α depends only on the prior and likelihood inside packet I_α. Coupling between different packets occurs through the normalization of the packet evidences.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-940305a61c0cbc27","tex":"\\sum_{\\alpha_1,\\ldots,\\alpha_{m-1}}","aliases":"∑_α_1,…,α_m-1 sum alpha ldots alpha","display":"\\sum_{\\alpha_1,\\ldots,\\alpha_{m-1}}","kind":"use","excerpt":"[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5676182f4950dc5","tex":"\\sum_{\\alpha_1,\\ldots,\\alpha_r}","aliases":"∑_α_1,…,α_r sum alpha ldots alpha","display":"\\sum_{\\alpha_1,\\ldots,\\alpha_r}","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-97bc4b69d2c44d19","tex":"\\sum_{\\alpha\\to\\beta}","aliases":"∑_α→β sum alpha to beta","display":"\\sum_{\\alpha\\to\\beta}","kind":"use","excerpt":"The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d237666c585ce1f0","tex":"\\sum_{\\alpha\\in\\Supp_{\\Pcal}(x)}","aliases":"∑_α∈Supp_Pcal(x) ∑_α∈Supp_ P(x) sum alpha in Supp Pcal sum alpha in operatorname mathcal","display":"\\sum_{\\alpha\\in{\\operatorname{Supp}}_{{\\mathcal P}}(x)}","kind":"use","excerpt":"For every target packet β, P_β Tx = ∑_α∈Supp_ P(x)T_βαP_α x. If every block in this sum is zero, then the coordinate with index β vanishes. This proves the inclusion. Equality is exactly the assertion that the displayed active sum is nonzero for every target in the set on the right. The final sufficient condition leaves at most one nonzero summand ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5a9492e4a8b5806","tex":"\\sum_{\\alpha\\in\\{0,1\\}^{\\ell}}","aliases":"∑_α∈{0,1}^ℓ sum alpha in ell","display":"\\sum_{\\alpha\\in\\{0,1\\}^{\\ell}}","kind":"use","excerpt":"[Wavelet-packet reconstruction and energy] At every finite depth ℓ, x=∑_α∈{0,1}^ℓW_ℓ,α^*W_ℓ,αx, and x _2^2 = ∑_α∈{0,1}^ℓ W_ℓ,αx _2^2.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1fa499eb28c28965","tex":"\\sum_{\\alpha\\neq\\beta}","aliases":"∑_α≠β sum alpha neq beta","display":"\\sum_{\\alpha\\neq\\beta}","kind":"use","excerpt":"[Exact block discovery certificate] For an orthogonal packet system (P_α), define the block diagonal compression Π_ P(A) = ∑_α P_α AP_α. Then normA-Π_ P(A)_rm F^2 = ∑_α≠β normP_β AP_α_rm F^2. In operator norm, normA-Π_ P(A)_2 ≤ ∑_α≠β normP_β AP_α_2. Thus any proposed packet discovery has an exact finite residual certificate after its projectors are","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dd5a8bdc2b931502","tex":"\\sum_\\beta","aliases":"∑_β sum beta","display":"\\sum_\\beta","kind":"use","excerpt":"Insert the finite packet identity on both sides: T= (∑_β P_β) T (∑_α P_α) = ∑_α,βP_β TP_α.","count":17},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-18666dd24c385c85","tex":"\\sum_{\\beta_1,\\ldots,\\beta_{m-1}}","aliases":"∑_β_1,…,β_m-1 sum beta ldots beta","display":"\\sum_{\\beta_1,\\ldots,\\beta_{m-1}}","kind":"use","excerpt":"[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-23c33e8c6e67f2c6","tex":"\\sum_{\\beta\\succ b,\\,\\alpha\\succ a}","aliases":"∑_βsucc b, αsucc a sum beta succ alpha succ","display":"\\sum_{\\beta\\succ b,\\,\\alpha\\succ a}","kind":"use","excerpt":"headRefined blocks still reconstruct the same map A coarse block expands as P_bTP_a=∑_βsucc b, αsucc aP_β TP_α. The terms are maps embedded in the original source and target spaces. Adding their norms, or adding their matrix entries after forgetting the embeddings, is not this operator identity.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fa0188218e3fe53e","tex":"\\sum_{\\beta\\succ\\alpha}","aliases":"∑_βsuccα sum beta succ alpha","display":"\\sum_{\\beta\\succ\\alpha}","kind":"use","excerpt":"A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9d12cb53c45f8ea0","tex":"\\sum_\\gamma","aliases":"∑_γ sum gamma","display":"\\sum_\\gamma","kind":"use","excerpt":"For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows ","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b177dab24f859fc7","tex":"\\sum_{\\gamma}","aliases":"∑_γ sum gamma","display":"\\sum_{\\gamma}","kind":"use","excerpt":"[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-435c95867c65d311","tex":"\\sum_{\\gamma:\\alpha\\leadsto\\beta}","aliases":"∑_γ:αleadstoβ sum gamma alpha leadsto beta","display":null,"kind":"use","excerpt":"pha DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For an edge e:α→β, put A_e=(P_β DP_β)^-1N_e. Then the block from P_α X to P_β X of (D+N)^-1 is P_β(D+N)^-1P_α = ∑_γ:αleadstoβ (-1)^|γ|A_γ D_α^-1, where D_α=P_α","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c0796b45d0f24473","tex":"\\sum_{\\gamma_1,\\gamma_2}","aliases":"∑_γ_1,γ_2 sum gamma gamma","display":"\\sum_{\\gamma_1,\\gamma_2}","kind":"use","excerpt":"[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-35e481843bfd1ee3","tex":"\\sum_{\\gamma_1,\\ldots,\\gamma_k}","aliases":"∑_γ_1,…,γ_k sum gamma ldots gamma","display":"\\sum_{\\gamma_1,\\ldots,\\gamma_k}","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7e22c72e0c03bce6","tex":"\\sum_{\\gamma\\in I}","aliases":"∑_γ∈ I sum gamma in","display":"\\sum_{\\gamma\\in I}","kind":"use","excerpt":"It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet project","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-13a337a61f819973","tex":"\\sum_{\\gamma\\in I_0}","aliases":"∑_γ∈ I_0 sum gamma in","display":"\\sum_{\\gamma\\in I_0}","kind":"use","excerpt":"apply the integrand to each x∈ X and take its Bochner integral in X. It is not asserted to be a Bochner integral in the operator-norm space B(X). The general Laplace formula is [source reference]. [Packet convolution] For all s,t≥0 and all packets α,β, T_βα(s+t) = ∑_γ∈ I_0 T_βγ(s)T_γα(t). The original semigroup is reconstructed from its blocks by T","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4077bd4e18a74bee","tex":"\\sum_{\\gamma\\in\\Gamma_b}","aliases":"∑_γ∈Γ_b sum gamma in Gamma","display":"\\sum_{\\gamma\\in\\Gamma_b}","kind":"use","excerpt":"[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4519e3a242fabf8b","tex":"\\sum_\\delta","aliases":"∑_δ sum delta","display":"\\sum_\\delta","kind":"use","excerpt":"Apply Q_γ on the left and P_α on the right. Inserting I_X=∑_β P_β gives Q_γ UTP_α = ∑_β Q_γ UP_β TP_α = ∑_β U_γβT_βα. Inserting I_Y=∑_δ Q_δ gives Q_γ SUP_α = ∑_δ Q_γ SQ_δ UP_α = ∑_δ S_γδU_δα. Thus UT=SU implies every block equation. Conversely, if every block equation holds, then all blocks of UT-SU vanish. Exact operator reconstruction yields UT-S","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3a150ff527e02e03","tex":"\\sum_{\\delta\\in J}","aliases":"∑_δ∈ J sum delta in","display":"\\sum_{\\delta\\in J}","kind":"use","excerpt":"et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dccea64f11621a8f","tex":"\\sum_{\\zeta\\in H_m}","aliases":"∑_ζ∈ H_m sum zeta in","display":"\\sum_{\\zeta\\in H_m}","kind":"use","excerpt":"[Subgroup average] The multiple selector has the mode expansion a^(m) = frac1m ∑_ζ∈ H_m ϵ_ζ.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a3ca240df62c5c7d","tex":"\\sum_{\\zeta\\in\\mu_m}","aliases":"∑_ζ∈μ_m sum zeta in mu","display":"\\sum_{\\zeta\\in\\mu_m}","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-fac630f33df3d387","tex":"\\sum_{\\eta^N=-1}","aliases":"∑_η^N=-1 sum eta","display":"\\sum_{\\eta^N=-1}","kind":"use","excerpt":"The second half of this chapter concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's mon","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbb81e78281bccf8","tex":"\\sum_{\\eta\\in\\Rcal_N}","aliases":"∑_η∈Rcal_N ∑_η∈ R_N sum eta in Rcal sum eta in mathcal","display":"\\sum_{\\eta\\in{\\mathcal R}_N}","kind":"use","excerpt":"[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-614ab5cb07fb5cef","tex":"\\sum_{\\lambda}","aliases":"∑_λ sum lambda","display":"\\sum_{\\lambda}","kind":"use","excerpt":"Logarithmic differentiation of the genus zero product gives -fracΔ_N'(w)Δ_N(w) = ∑_λ∈Λ_Nfrac1λ-w. The series is locally uniform because ∑_λ|λ|^-1<∞. Substitute w=z^N and use the chain rule.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3389ad15a324713c","tex":"\\sum_{\\lambda_j>0}","aliases":"∑_λ_j>0 sum lambda","display":"\\sum_{\\lambda_j>0}","kind":"use","excerpt":"[Exact recovery on the error span] Under the hypotheses of [source reference], define on the syndrome support Q=∑_λ_j>0Q_j the completely positive map R_0(X) = ∑_λ_j>0 V_j^*Q_jXQ_jV_j. It extends to a quantum channel R on End( H). If Φ(X)=∑_aE_aXE_a^* is trace preserving on code states, then RΦ(ρ)=ρ for every density matrix ρ=Pρ P.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bba70d5604bb3f32","tex":"\\sum_{\\lambda_r>0}","aliases":"∑_λ_r>0 sum lambda","display":"\\sum_{\\lambda_r>0}","kind":"use","excerpt":"titSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal pro","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a17ff8546f2e633d","tex":"\\sum_{\\lambda\\in\\widehat G}","aliases":"∑_λ∈widehat G sum lambda in widehat","display":"\\sum_{\\lambda\\in\\widehat G}","kind":"use","excerpt":"[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f85a9d219ed81c46","tex":"\\sum_\\mu","aliases":"∑_μ sum mu","display":"\\sum_\\mu","kind":"use","excerpt":"Insert completeness on both sides: Ob = (∑_μΠ^a_μ) Ob (∑_νΠ^a_ν) = ∑_μ,νΠ^a_μObΠ^a_ν. Thus (iv) implies (iii), and (iii) implies (iv) by composition with the projectors. Conditions (ii) and (iii) are equivalent by the definition of Ob and additive cancellation. Conditions (i) and (ii) are equivalent by [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e98a525f93a0f0b0","tex":"\\sum_{\\mu,\\nu}","aliases":"∑_μ,ν sum mu nu","display":"\\sum_{\\mu,\\nu}","kind":"use","excerpt":"Insert completeness on both sides: Ob = (∑_μΠ^a_μ) Ob (∑_νΠ^a_ν) = ∑_μ,νΠ^a_μObΠ^a_ν. Thus (iv) implies (iii), and (iii) implies (iv) by composition with the projectors. Conditions (ii) and (iii) are equivalent by the definition of Ob and additive cancellation. Conditions (i) and (ii) are equivalent by [source reference].","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-49a5938a580ef00a","tex":"\\sum_{\\mu,\\nu\\in I_a}","aliases":"∑_μ,ν∈ I_a sum mu nu in","display":"\\sum_{\\mu,\\nu\\in I_a}","kind":"use","excerpt":"[Exact reconstruction and detection by obstruction blocks] The total obstruction has the exact finite resolution Ob_c,b,a = ∑_μ,ν∈ I_a Ob^μν_c,b,a. Moreover, the following are equivalent: [label=(roman*)] item Δ_c,b,a=0; item H_c,b,a=id; item Ob_c,b,a=0; item Ob^μν_c,b,a=0 for every μ,ν. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.obstruction_s","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dde65ad1f7f11467","tex":"\\sum_{\\mu\\in\\Lambda_{N,j}}","aliases":"∑_μ∈Λ_N,j sum mu in Lambda","display":"\\sum_{\\mu\\in\\Lambda_{N,j}}","kind":"use","excerpt":"[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cd74cb17b4132748","tex":"\\sum_\\nu","aliases":"∑_ν sum nu","display":"\\sum_\\nu","kind":"use","excerpt":"Insert completeness on both sides: Ob = (∑_μΠ^a_μ) Ob (∑_νΠ^a_ν) = ∑_μ,νΠ^a_μObΠ^a_ν. Thus (iv) implies (iii), and (iii) implies (iv) by composition with the projectors. Conditions (ii) and (iii) are equivalent by the definition of Ob and additive cancellation. Conditions (i) and (ii) are equivalent by [source reference].","count":10},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e1a1496e31ea3a4d","tex":"\\sum_{\\nu=1}^k","aliases":"∑_ν=1^k sum nu","display":"\\sum_{\\nu=1}^k","kind":"use","excerpt":"[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of ","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bbf40b6240c18f80","tex":"\\sum_{\\pi=\\{B_1,\\ldots,B_k\\}}","aliases":"∑_π={B_1,…,B_k} sum pi ldots","display":"\\sum_{\\pi=\\{B_1,\\ldots,B_k\\}}","kind":"use","excerpt":"di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-816bc4c4be83a776","tex":"\\sum_{\\rho(\\alpha)=a}","aliases":"∑_ρ(α)=a sum rho alpha","display":"\\sum_{\\rho(\\alpha)=a}","kind":"use","excerpt":"Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ad18eea2ec25001f","tex":"\\sum_{\\rho(\\beta)=b}","aliases":"∑_ρ(β)=b sum rho beta","display":"\\sum_{\\rho(\\beta)=b}","kind":"use","excerpt":"Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8e5722710bcae566","tex":"\\sum_{\\rho(\\gamma)=\\alpha}","aliases":"∑_ρ(γ)=α sum rho gamma alpha","display":"\\sum_{\\rho(\\gamma)=\\alpha}","kind":"use","excerpt":"Multiply the identity P_α=∑_ρ(γ)=αQ_γ by Q_β on either side. Pairwise annihilation in the fine family leaves Q_β when ρ(β)=α and leaves zero otherwise. For transitivity, substitute the child expansion of each Q_β into the expansion of P_α and group the resulting terms by the composite parent.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-dc4e4df36e5ded48","tex":"\\sum_{\\rho\\in\\Zcal_R}","aliases":"∑_ρ∈Zcal_R ∑_ρ∈ Z_R sum rho in Zcal sum rho in mathcal","display":"\\sum_{\\rho\\in{\\mathcal Z}_R}","kind":"use","excerpt":"rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-307b928591ce6cbf","tex":"\\sum_{\\tau(\\sigma(\\alpha))=k}","aliases":"∑_τ(σ(α))=k sum tau sigma alpha","display":"\\sum_{\\tau(\\sigma(\\alpha))=k}","kind":"use","excerpt":"Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source refer","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-60c47c363b013a09","tex":"\\sum_{\\tau\\in\\mathfrak T_m}","aliases":"∑_τ∈ T_m sum tau in mathfrak","display":"\\sum_{\\tau\\in\\mathfrak T_m}","kind":"use","excerpt":"Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a t","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff580f406df4f029","tex":"\\sum_{\\tau\\in\\Zcal_R^{\\mathrm{orb}}}","aliases":"∑_τ∈Zcal_R^orb ∑_τ∈ Z_R^orb sum tau in Zcal mathrm sum tau in mathcal mathrm","display":"\\sum_{\\tau\\in{\\mathcal Z}_R^{\\mathrm{orb}}}","kind":"use","excerpt":"[Orbit reduced nearest circle term] Assume the hypotheses of [source reference]. Choose one representative from each orbit of the nearest zeros under multiplication by ω, and call the resulting set Z_R^orb. With m=Nn, P_N,n(r) = · N m!binomm+r-1r-1 ∑_τ∈ Z_R^orb fracτ^-m(-τ D_N'(τ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c5c2f2566d8f7c1d","tex":"\\sum_{\\chi\\in\\mu_h}","aliases":"∑_χ∈μ_h sum chi in mu","display":"\\sum_{\\chi\\in\\mu_h}","kind":"use","excerpt":"Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding t","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e3116f8319e2684e","tex":"\\sum_\\ell","aliases":"∑_ℓ sum ell","display":"\\sum_\\ell","kind":"use","excerpt":"igin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), bec","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d9aa5ade096792f0","tex":"\\sum_{\\ell=0}^{3}","aliases":"∑_ℓ=0^3 sum ell","display":"\\sum_{\\ell=0}^{3}","kind":"use","excerpt":"vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. N","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-43e727cf5aadc16c","tex":"\\sum_{\\ell=0}^{L-1}","aliases":"∑_ℓ=0^L-1 sum ell","display":"\\sum_{\\ell=0}^{L-1}","kind":"use","excerpt":"[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6540b6ebc95be7f8","tex":"\\sum_{\\ell=0}^{m-1}","aliases":"∑_ℓ=0^m-1 sum ell","display":"\\sum_{\\ell=0}^{m-1}","kind":"use","excerpt":"y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b98eb4d33c4dd25d","tex":"\\sum_{\\ell=0}^{N-1}","aliases":"∑_ℓ=0^N-1 sum ell","display":"\\sum_{\\ell=0}^{N-1}","kind":"use","excerpt":"Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a895599e52149aa4","tex":"\\sum_{\\ell=0}^{\\infty}","aliases":"∑_ℓ=0^∞ sum ell infty","display":"\\sum_{\\ell=0}^{\\infty}","kind":"use","excerpt":"[Contractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-22bd389693d3f179","tex":"\\sum_{\\ell=1}^{m}","aliases":"∑_ℓ=1^m sum ell","display":"\\sum_{\\ell=1}^{m}","kind":"use","excerpt":"The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-03d52d6913e49407","tex":"\\sum_{\\ell=1}^{r}","aliases":"∑_ℓ=1^r sum ell","display":"\\sum_{\\ell=1}^{r}","kind":"use","excerpt":"[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2c6476408fd5933d","tex":"\\sum_{\\ell=n}^{\\infty}","aliases":"∑_ℓ=n^∞ sum ell infty","display":"\\sum_{\\ell=n}^{\\infty}","kind":"use","excerpt":"ontractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-a9349b269e60291a","tex":"\\sum_{\\ell\\in L}","aliases":"∑_ℓ∈ L sum ell in","display":"\\sum_{\\ell\\in L}","kind":"use","excerpt":"Every address has a unique representation r+ℓ. For finite p, x _p^p = ∑_r∈ R∑_ℓ∈ L|x(r+ℓ)|^p = ∑_r∈ R x_r _p^p. The corresponding supremum identity is immediate when p=∞. Unique address representation proves that analysis and interleaving are inverse maps. The projector identities follow from the disjoint coset partition.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-db596730d71feb37","tex":"\\sum_{\\ell\\in\\Lcal_w}","aliases":"∑_ℓ∈Lcal_w ∑_ℓ∈ L_w sum ell in Lcal sum ell in mathcal","display":"\\sum_{\\ell\\in{\\mathcal L}_w}","kind":"use","excerpt":"Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6c84dfd9bdb76fc0","tex":"\\sum_{\\ell\\in\\widetilde L}","aliases":"∑_ℓ∈widetilde L sum ell in widetilde","display":"\\sum_{\\ell\\in\\widetilde L}","kind":"use","excerpt":"[Isotropic averaging projector] The operator P_L = frac1abs L ∑_ℓ∈widetilde L ρ(ℓ) is an orthogonal projector. In the Schrödinger model with L=X, it has rank one and selects the constant functions.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-472fd2fb48701e7d","tex":"\\sum_{\\ell\\geq1}","aliases":"∑_ℓ≥1 sum ell geq","display":"\\sum_{\\ell\\geq1}","kind":"use","excerpt":"[Taylor coefficient bounds and radii] For j=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at addres","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-acb2bbc59eb8413f","tex":"\\sum_{\\sum_i a_i=r}","aliases":"∑_∑_i a_i=r sum sum","display":"\\sum_{\\sum_i a_i=r}","kind":"use","excerpt":"Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Choose such a k≥ k_0. Since x^(k) is unconditionally summable, there is a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. There","count":12},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-16662ae81e20f5e0","tex":"\\int_{0<s_1<\\cdots<s_m<t}","aliases":"∫_0<s_1<⋯<s_m<t int cdots","display":"\\int_{0<s_1<\\cdots<s_m<t}","kind":"use","excerpt":"[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","excerpt":"· +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-2547054a0e887b21","tex":"\\int_0^{2\\pi}","aliases":"∫_0^2π int pi","display":"\\int_0^{2\\pi}","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-051f75585a774ce8","tex":"\\int_0^r","aliases":"∫_0^r int","display":"\\int_0^r","kind":"use","excerpt":"For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3b8f65fd54e2c3f7","tex":"\\int_0^s","aliases":"∫_0^s int","display":"\\int_0^s","kind":"use","excerpt":"[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-8bcd55715febb3c0","tex":"\\int_0^z","aliases":"∫_0^z int","display":"\\int_0^z","kind":"use","excerpt":"Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonzero overshoot persists along points app","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-94438197d8221023","tex":"\\int_0^\\pi","aliases":"∫_0^π int pi","display":"\\int_0^\\pi","kind":"use","excerpt":"[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.","count":31},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ff862dd8e88aa975","tex":"\\int_0^{\\infty}","aliases":"∫_0^∞ int infty","display":"\\int_0^{\\infty}","kind":"use","excerpt":"[Packet resolvent convolution] Let z belong to the resolvent set of A. If Rez is larger than a growth bound for the semigroup, then P_β(zI-A)^-1P_α = ∫_0^∞ e^-ztT_βα(t) d t.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-297d1af96e467fcb","tex":"\\int_1^{\\infty}","aliases":"∫_1^∞ int infty","display":"\\int_1^{\\infty}","kind":"use","excerpt":"An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate a","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32e5a56ec6778f45","tex":"\\int_a^{a+L/2}","aliases":"∫_a^a+L/2 int","display":"\\int_a^{a+L/2}","kind":"use","excerpt":"Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a91af32e2911325","tex":"\\int_r^\\infty","aliases":"∫_r^∞ int infty","display":"\\int_r^\\infty","kind":"use","excerpt":"For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7cf42d930c86bbab","tex":"\\int_\\RR","aliases":"∫_RR ∫_ R int RR int mathbb","display":"\\int_{\\mathbb R}","kind":"use","excerpt":"Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-67e4672bfa77240e","tex":"\\int_\\Gamma","aliases":"∫_Γ int Gamma","display":"\\int_\\Gamma","kind":"use","excerpt":"X_c be complex Banach spaces, and let C:X_f→ X_c, T_f, and T_c be bounded complex-linear maps. Let h be holomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts mul","count":5},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bd4f282e19b7fae3","tex":"\\int_\\tau^{\\widehat\\tau}","aliases":"∫_τ^widehatτ int tau widehat tau","display":"\\int_\\tau^{\\widehat\\tau}","kind":"use","excerpt":"Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated co","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-df556fd37e8a533e","tex":"\\sim_{\\mathcal L}","aliases":"∼_ L sim mathcal","display":"\\sim_{\\mathcal L}","kind":"use","excerpt":"[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every bl","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonz","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d2adebfd04c66222","tex":"\\bigoplus^u","aliases":"⊕^u bigoplus","display":"\\bigoplus^u","kind":"use","excerpt":"s a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. Therefore x∈⊕^uX_n, and the earlier estimate proves x^(k)→ x in · _u.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-52fe3db28a901e83","tex":"\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}","aliases":"⊕_(α,γ)∈ I× J bigoplus alpha gamma in times","display":"\\bigoplus_{(\\alpha,\\gamma)\\in I\\times J}","kind":"use","excerpt":"lpha∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refinement for quantum, signal, ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-27c543b1a137d08d","tex":"\\bigoplus_{a\\in A}","aliases":"⊕_a∈ A bigoplus in","display":"\\bigoplus_{a\\in A}","kind":"use","excerpt":"[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6327e4691c318c59","tex":"\\bigoplus_{\\frakm\\in\\Max\\TT}","aliases":"⊕_frakm∈MaxTT ⊕_ m∈Max T bigoplus frakm in Max TT bigoplus mathfrak in operatorname mathbb","display":"\\bigoplus_{{\\mathfrak m}\\in{\\operatorname{Max}}{\\mathbb T}}","kind":"use","excerpt":"[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-59a3a39b512b97d6","tex":"\\bigoplus_{g\\in G}","aliases":"⊕_g∈ G bigoplus in","display":"\\bigoplus_{g\\in G}","kind":"use","excerpt":"Let G be a finite abelian group and let A=⊕_g∈ GA_g be a G graded module.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-3c8f746e1dd608b5","tex":"\\bigoplus_{i=0}^{d-1}","aliases":"⊕_i=0^d-1 bigoplus","display":"\\bigoplus_{i=0}^{d-1}","kind":"use","excerpt":"[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5c779b0b534e7ac8","tex":"\\bigoplus_{i=1}^r","aliases":"⊕_i=1^r bigoplus","display":"\\bigoplus_{i=1}^r","kind":"use","excerpt":"Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty blo","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-61f317edc40cbb12","tex":"\\bigoplus_{i=1}^{r}","aliases":"⊕_i=1^r bigoplus","display":"\\bigoplus_{i=1}^{r}","kind":"use","excerpt":"[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-32fcd883ade4f816","tex":"\\bigoplus_{i\\in J}","aliases":"⊕_i∈ J bigoplus in","display":"\\bigoplus_{i\\in J}","kind":"use","excerpt":"The actual invariant images U_i=imE_i give an equivariant linear isomorphism ⊕_i∈ JU_i≃ R^n, (v_i)_i⟼∑_i v_i, v⟼(E_iv)_i. Lean represents the finite direct sum by the finite product of the image submodules. The existing coordinatewise reduction theorem maps every U_i onto the corresponding reduced projector image. No chosen bases or unproved freene","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ce7f3904634c8e1a","tex":"\\bigoplus_j","aliases":"⊕_j bigoplus","display":"\\bigoplus_j","kind":"use","excerpt":"Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1c37ce88f1f80c5","tex":"\\bigoplus_{j=1}^{d}","aliases":"⊕_j=1^d bigoplus","display":"\\bigoplus_{j=1}^{d}","kind":"use","excerpt":"For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5a2986af923c3a3a","tex":"\\bigoplus_{j=1}^n","aliases":"⊕_j=1^n bigoplus","display":"\\bigoplus_{j=1}^n","kind":"use","excerpt":"[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c7da9c0ad0ae9569","tex":"\\bigoplus_{j=1}^r","aliases":"⊕_j=1^r bigoplus","display":"\\bigoplus_{j=1}^r","kind":"use","excerpt":"ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1e8f7e2fd7fa7ced","tex":"\\bigoplus_{j=1}^{r}","aliases":"⊕_j=1^r bigoplus","display":"\\bigoplus_{j=1}^{r}","kind":"use","excerpt":"Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7d0f5e3799ea2757","tex":"\\bigoplus_{j\\in\\mathbb Z/N\\mathbb Z}","aliases":"⊕_j∈ Z/N Z bigoplus in mathbb mathbb","display":"\\bigoplus_{j\\in\\mathbb Z/N\\mathbb Z}","kind":"use","excerpt":"[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d77606a690becea6","tex":"\\bigoplus_{r=0}^{N-1}","aliases":"⊕_r=0^N-1 bigoplus","display":"\\bigoplus_{r=0}^{N-1}","kind":"use","excerpt":"[One dimensional residue coordinates] For L=N Z and R={0,…,N-1}, x_r(k)=x(Nk+r) and ℓ^p( Z)≅⊕_r=0^N-1ℓ^p( Z) isometrically.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-183f6b24e50dc6fa","tex":"\\bigoplus_{r=1}^{s}","aliases":"⊕_r=1^s bigoplus","display":"\\bigoplus_{r=1}^{s}","kind":"use","excerpt":"Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-ef2b6439056f27d8","tex":"\\bigoplus_{r\\in R}","aliases":"⊕_r∈ R bigoplus in","display":"\\bigoplus_{r\\in R}","kind":"use","excerpt":"For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-6e1bd670f8830b0f","tex":"\\bigoplus_v","aliases":"⊕_v bigoplus","display":"\\bigoplus_v","kind":"use","excerpt":"Because Q_0 is finite, 1=∑_v∈ Q_0e_v in 𝕜 Q, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely,","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9b9985dfb7501871","tex":"\\bigoplus_{v\\in\\Lcal}","aliases":"⊕_v∈Lcal ⊕_v∈ L bigoplus in Lcal bigoplus in mathcal","display":"\\bigoplus_{v\\in{\\mathcal L}}","kind":"use","excerpt":"[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b1d323816298cbee","tex":"\\bigoplus_{\\alpha}","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_{\\alpha}","kind":"use","excerpt":"Λ be an O lattice in V. A rational projector E need not preserve Λ. Its separation weight is the least s such that π^sEΛ⊆Λ. A Bézout construction gives an upper bound through a polynomial resultant. That bound need not be exact because it does not see the position of the lattice. The exact answer is controlled by the finite length module G_Λ, P = Λ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5304936d80ae1b31","tex":"\\bigoplus_{\\alpha\\in C}","aliases":"⊕_α∈ C bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in C}","kind":"use","excerpt":"[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [labe","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7224e793ec7acd49","tex":"\\bigoplus_{\\alpha\\in I_0}","aliases":"⊕_α∈ I_0 bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I_0}","kind":"use","excerpt":"Reachability obstructions for seeds. [Reachability obstruction for generated hulls] Let I_0⊆ I and let S⊆⊕_α∈ I_0X_α. Then Hull_ A(S) ⊆ ⊕_β∈Reach(I_0)X_β. In particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1a835b644d2cadf6","tex":"\\bigoplus_\\beta","aliases":"⊕_β bigoplus beta","display":"\\bigoplus_\\beta","kind":"use","excerpt":"[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-af8472aa58b542e6","tex":"\\bigoplus_{\\beta\\in\\Reach(I_0)}","aliases":"⊕_β∈Reach(I_0) bigoplus beta in Reach bigoplus beta in operatorname","display":"\\bigoplus_{\\beta\\in{\\operatorname{Reach}}(I_0)}","kind":"use","excerpt":"Reachability obstructions for seeds. [Reachability obstruction for generated hulls] Let I_0⊆ I and let S⊆⊕_α∈ I_0X_α. Then Hull_ A(S) ⊆ ⊕_β∈Reach(I_0)X_β. In particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-7f618b24b380de7c","tex":"\\bigoplus_{\\gamma\\in R}","aliases":"⊕_γ∈ R bigoplus gamma in","display":"\\bigoplus_{\\gamma\\in R}","kind":"use","excerpt":"Let R be the set of packets reachable from α, including α, and set X_R=⊕_γ∈ RX_γ. No active edge leaves R, so AX_R⊆ X_R. The restriction of A to the finite dimensional space X_R is injective. It is therefore onto X_R. Hence A^-1X_R⊆ X_R. Since β∉ R, the stated block of A^-1 vanishes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-28886d26e9fa8f85","tex":"\\bigoplus_\\mu","aliases":"⊕_μ bigoplus mu","display":"\\bigoplus_\\mu","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-263ab54d332da07b","tex":"\\bigoplus_{\\mu\\in\\widehat G}","aliases":"⊕_μ∈widehat G bigoplus mu in widehat","display":"\\bigoplus_{\\mu\\in\\widehat G}","kind":"use","excerpt":"Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The as","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e91d9943bd8850b9","tex":"\\bigoplus_{\\sigma(\\alpha)=j}","aliases":"⊕_σ(α)=j bigoplus sigma alpha","display":"\\bigoplus_{\\sigma(\\alpha)=j}","kind":"use","excerpt":"[Finite packet aggregation] Let σ:I↠ J be a surjection and define Q_j:=∑_σ(α)=jP_α. Then (Q_j)_j∈ J is a finite packet system and Q_jX=⊕_σ(α)=jX_α. [source reference]","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f9b87410270e30bb","tex":"\\otimes_A","aliases":"⊗_A otimes","display":"\\otimes_A","kind":"use","excerpt":"Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it car","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d981952a913b6fd5","tex":"\\otimes_{\\bbk}","aliases":"⊗_bbk ⊗_𝕜 otimes bbk otimes Bbbk","display":"\\otimes_{{\\Bbbk}}","kind":"use","excerpt":"Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]","count":4},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eefbbc5154bea1da","tex":"\\otimes_\\OO","aliases":"⊗_OO ⊗_ O otimes OO otimes mathcal","display":"\\otimes_{{\\mathcal O}}","kind":"use","excerpt":"[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f0ee3d51940d2d69","tex":"\\otimes_{\\OO}","aliases":"⊗_OO ⊗_ O otimes OO otimes mathcal","display":"\\otimes_{{{\\mathcal O}}}","kind":"use","excerpt":"Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-98b6ea417c450bdb","tex":"\\otimes_P","aliases":"⊗_P otimes","display":"\\otimes_P","kind":"use","excerpt":"coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-10d64144ae124a95","tex":"\\otimes_R","aliases":"⊗_R otimes","display":"\\otimes_R","kind":"use","excerpt":"[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).","count":6},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-cb7626f793d1b5b3","tex":"\\otimes_{R,x}","aliases":"⊗_R,x otimes","display":"\\otimes_{R,x}","kind":"use","excerpt":"[Field fiber criterion] Let R be a commutative ring, let M be a finite R module, and let x:R→ E' be a homomorphism to a field. Put p=ker x. Then M⊗_R,xE'≠0 Longleftrightarrow p∈Supp_R(M).","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-9384f30399aada65","tex":"\\otimes_{R/I}","aliases":"⊗_R/I otimes","display":"\\otimes_{R/I}","kind":"use","excerpt":"Let q∈Spec(R/I), and let p be its inverse image in R. The composite point R→ R/I→κ(q) has kernel p, so it factors through a field embedding κ( p)→κ(q). The ordinary tensor associator in every degree gives an isomorphism of complexes κ(q)⊗_R/I((R/I)⊗_RP) ≅κ(q)⊗_RP ≅κ(q)⊗_κ( p) (κ( p)⊗_RP). Extension of fields is faithfully flat. It preserves and ref","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f92c003c879d28f9","tex":"\\otimes_R^{\\mathbf L}","aliases":"⊗_R^ L otimes mathbf","display":"\\otimes_R^{\\mathbf L}","kind":"use","excerpt":"Choose a bounded complex P of finite projective R modules representing C. Scalar extension to a commutative R algebra S is computed degree by degree as S⊗_RP. The canonical symmetry of tensor products gives the displayed order C⊗_R^ LS. Scalar extension preserves the bounds, finite generation, and projectivity of the terms. A derived isomorphism be","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c91c177726b0cd36","tex":"\\bigotimes_{s\\in S}","aliases":"⊗_s∈ S bigotimes in","display":"\\bigotimes_{s\\in S}","kind":"use","excerpt":"[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-b7d05b2af47a64ad","tex":"\\otimes_{\\TT,x}","aliases":"⊗_TT,x ⊗_ T,x otimes TT otimes mathbb","display":"\\otimes_{{\\mathbb T},x}","kind":"use","excerpt":"[Supported deformation points give nonzero Hecke fibers] Let R≅ T, and let M be a finite Hecke module. If a point x:R→ E' lies in Supp_R(M), then the corresponding Hecke fiber M⊗_ T,xE' is nonzero.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-1eec1f6f77946999","tex":"\\otimes_{\\kappa(\\frakp)}","aliases":"⊗_κ(frakp) ⊗_κ( p) otimes kappa frakp otimes kappa mathfrak","display":"\\otimes_{\\kappa({\\mathfrak p})}","kind":"use","excerpt":"Let q∈Spec(R/I), and let p be its inverse image in R. The composite point R→ R/I→κ(q) has kernel p, so it factors through a field embedding κ( p)→κ(q). The ordinary tensor associator in every degree gives an isomorphism of complexes κ(q)⊗_R/I((R/I)⊗_RP) ≅κ(q)⊗_RP ≅κ(q)⊗_κ( p) (κ( p)⊗_RP). Extension of fields is faithfully flat. It preserves and ref","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions ho","count":7},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-bca212e1e10d6008","tex":"\\bigcap_a","aliases":"⋂_a bigcap","display":"\\bigcap_a","kind":"use","excerpt":"[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0c12df6de0a7dc07","tex":"\\bigcap_m","aliases":"⋂_m bigcap","display":"\\bigcap_m","kind":"use","excerpt":"a normed setting every packet projector and every coarsening operator is bounded. Differentiability of operator-valued maps is taken in operator norm. A Banach space isomorphism is a bounded linear bijection with bounded inverse. For a decreasing filtration (F^mX), completeness means that the canonical map X⟶varprojlim_m X/F^mX is an isomorphism an","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-c47fe7e501277454","tex":"\\bigcap_{m\\geq1}","aliases":"⋂_m≥1 bigcap geq","display":"\\bigcap_{m\\geq1}","kind":"use","excerpt":"[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-08bf84a1d20b2d0b","tex":"\\bigcap_r","aliases":"⋂_r bigcap","display":"\\bigcap_r","kind":"use","excerpt":"Fix r. For every m≥ r, one has Φ(x_m)∈ F^mY⊆ F^rY. The submodule F^rY is open and therefore closed in the filtration topology. Continuity of Φ gives Φ(x_m)→Φ(x_∞), so Φ(x_∞)∈ F^rY. Since this holds for every r and Y is separated, Φ(x_∞)∈⋂_rF^rY=0.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5bdecfe476a15417","tex":"\\bigcup_C","aliases":"⋃_C bigcup","display":"\\bigcup_C","kind":"use","excerpt":"For the reverse inequality, choose for each source component C a packet pure family S_C of size r_C generating X_C under A_C, and put S=⋃_CS_C. Then every source component lies in A S. Traverse the finite condensation graph in a topological order. Suppose all components preceding a component D that is not a source have already been shown to lie in ","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-4601d15b3053bbea","tex":"\\bigcup_i","aliases":"⋃_i bigcup","display":"\\bigcup_i","kind":"use","excerpt":"[Cohomological description of derived support] If R is Noetherian and C is perfect, then Supp_R(C) = ⋃_iSupp_R H^i(C). In particular, derived support is closed and is empty exactly when C≃0 [source reference].","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-686d00f08d584b6c","tex":"\\bigcup_\\alpha","aliases":"⋃_α bigcup alpha","display":"\\bigcup_\\alpha","kind":"use","excerpt":"Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-eef621a67d48c213","tex":"\\rangle^\\times","aliases":"⟩^× rangle times","display":"\\rangle^\\times","kind":"use","excerpt":"to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-25fb92dda41f7b3d","tex":"\\rangle_2","aliases":"⟩_2 rangle","display":"\\rangle_2","kind":"use","excerpt":"For p=3,ℓ=2,a=2, the checked values are ω_2(2)=8, ⟨2⟩_2=7, 8^2=1,quad8·7=2 in Z/9 Z. The unlifted representative 2 is not a square root of one modulo nine.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e5ac643f289ff808","tex":"\\rangle_{2,\\ell}","aliases":"⟩_2,ℓ rangle ell","display":"\\rangle_{2,\\ell}","kind":"use","excerpt":"Why the mod-four branch is separate. Now let ℓ≥2 and a∈( Z/2^ℓ Z)^×. The actual units of Z/4 Z are exhausted by 1 and -1. Define ε_ℓ(a)= 1, · a≡1pmod4, -1, · a≡3pmod4, ⟨ a⟩_2,ℓ=ε_ℓ(a)a. The source uses ℓ=L+2. It proves a=ε_ℓ(a)⟨ a⟩_2,ℓ, ε_ℓ(a)∈{1,-1}, ⟨ a⟩_2,ℓ≡1pmod4. The sign is uniquely determined by the residue modulo four, and cancellation give","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-64103ba1ce4e3213","tex":"\\rangle_4","aliases":"⟩_4 rangle","display":"\\rangle_4","kind":"use","excerpt":"Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ϵ(a)∈{1,-1}, ⟨ a⟩_4≡1pmod4.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-5ba5167652cbe588","tex":"\\rangle_k","aliases":"⟩_k rangle","display":"\\rangle_k","kind":"use","excerpt":"pal_mul pageheading3. Precision compatibility and the dyadic signFinite refinements and Proposition 6.2's dyadic branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent fa","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-f3cc5b5bb7a34037","tex":"\\rangle_L","aliases":"⟩_L rangle","display":"\\rangle_L","kind":"use","excerpt":"[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-e6e5d047330a8a9f","tex":"\\rangle_\\pi","aliases":"⟩_π rangle pi","display":"\\rangle_\\pi","kind":"use","excerpt":"[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.","count":2},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d4bbee17c7760c6a","tex":"\\rangle_\\ell","aliases":"⟩_ℓ rangle ell","display":"\\rangle_\\ell","kind":"use","excerpt":"ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ","count":3},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-d66576da6e758e08","tex":"\\rangle_\\ell^{\\,p^{\\ell-1}}","aliases":"⟩_ℓ^ p^ℓ-1 rangle ell ell","display":"\\rangle_\\ell^{\\,p^{\\ell-1}}","kind":"use","excerpt":"hism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy ⟨ a⟩_ℓ^ p^ℓ-1=1. At precision one the principal factor is exactly one.","count":1},{"work":"realization-calculus-book","url":"notation/realization-calculus-book.html#symbol-0da92c8c15988310","tex":"\\mathbb{F}","aliases":"𝔽 mathbb","display":"\\mathbb{F}","kind":"use","excerpt":"Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. There","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"Q(a)=(q_i(a))_i · The joint task representation, least in retained distinctions for its task family.","count":197},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"A, L, T_0 · Required premises, admissible leaves, and requested targets, with A⊆ L.","count":66},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"notation-table","excerpt":"Q(a)=(q_i(a))_i · The joint task representation, least in retained distinctions for its task family.","count":52},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e43e0379cbb13a71","tex":"\\Anc","aliases":"Anc operatorname","display":"{\\operatorname{Anc}}","kind":"notation-table","excerpt":"Anc_L(T_0) · Backward ancestry expanded only outside the admissible leaves.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-006c142fbe6d3806","tex":"\\Anc_L","aliases":"Anc_L Anc operatorname","display":"{\\operatorname{Anc}}_L","kind":"notation-table","excerpt":"Anc_L(T_0) · Backward ancestry expanded only outside the admissible leaves.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"notation-table","excerpt":"mathscr A, C, q · A class of possible archives, an encoding scheme, and a task on that class.","count":55},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1ca71bbaea6ea0f3","tex":"\\mathrm{cert}","aliases":"cert mathrm","display":"\\mathrm{cert}","kind":"notation-table","excerpt":"D_cert(K,H) · The least certified closure in the selected record.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a4859245592a0c90","tex":"\\operatorname{cl}","aliases":"cl operatorname","display":"\\operatorname{cl}","kind":"notation-table","excerpt":"cl_F(S) · The least declared rule model containing the positive observations S and all admitted leaves.","count":25},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f3a6008d26267161","tex":"\\operatorname{cl}_F","aliases":"cl_F operatorname","display":"\\operatorname{cl}_F","kind":"notation-table","excerpt":"cl_F(S) · The least declared rule model containing the positive observations S and all admitted leaves.","count":20},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"notation-table","excerpt":"D_cert(K,H) · The least certified closure in the selected record.","count":27},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-476df991b9d363fb","tex":"D_{\\mathrm{cert}}","aliases":"D_cert mathrm","display":"D_{\\mathrm{cert}}","kind":"notation-table","excerpt":"D_cert(K,H) · The least certified closure in the selected record.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"notation-table","excerpt":"D:E→ R, p:E→ S · A linear defect and a protected observable. The repair condition is Dy=0, p(y)=p(x).","count":19},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f8483dce1203449b","tex":"\\mathrm{eff}","aliases":"eff mathrm","display":"\\mathrm{eff}","kind":"notation-table","excerpt":"n_eff · A real, variance-equivalent review count in the specified covariance model.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f7e053bb5a2cb140","tex":"\\operatorname{Ext}","aliases":"Ext operatorname","display":"\\operatorname{Ext}","kind":"notation-table","excerpt":"Ext_F(S,N) · Rule models containing S and avoiding the negative observations N.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-6d57893197a5f161","tex":"\\operatorname{Ext}_F","aliases":"Ext_F operatorname","display":"\\operatorname{Ext}_F","kind":"notation-table","excerpt":"Ext_F(S,N) · Rule models containing S and avoiding the negative observations N.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"notation-table","excerpt":"M_F(U) · Local models on patch U satisfying all certified rules supported entirely in U.","count":81},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-6de9207e2e1aeff1","tex":"F_{\\mathcal U}","aliases":"F_ U mathcal","display":"F_{\\mathcal U}","kind":"notation-table","excerpt":"F_ U · The same field with only rules whose entire supports appear in a covering patch.","count":12},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"notation-table","excerpt":"(K,H) · Selected nodes and certified justification records. Only usable rules contribute to derivability.","count":47},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"notation-table","excerpt":"Q(a)=(q_i(a))_i · The joint task representation, least in retained distinctions for its task family.","count":59},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-753ac74d7d5e39c2","tex":"\\Jcal","aliases":"Jcal J mathcal","display":"{\\mathcal J}","kind":"notation-table","excerpt":"P, V, J · A declared proof field, its nodes, and its justifications.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","excerpt":"(K,H) · Selected nodes and certified justification records. Only usable rules contribute to derivability.","count":64},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"notation-table","excerpt":"H=ker D∩ker p · The ambiguity space of a nonempty protected fiber. This H is not a set of justifications.","count":8},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","excerpt":"A, L, T_0 · Required premises, admissible leaves, and requested targets, with A⊆ L.","count":54},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"notation-table","excerpt":"U_M(S) · Weighted target reach from the mandatory starts and a selected supplement, in a fixed directed bridge graph.","count":31},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"notation-table","excerpt":"M_F(U) · Local models on patch U satisfying all certified rules supported entirely in U.","count":8},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-eacab46b2fc949de","tex":"\\mathcal M_F","aliases":"M_F mathcal","display":"\\mathcal M_F","kind":"notation-table","excerpt":"M_F(U) · Local models on patch U satisfying all certified rules supported entirely in U.","count":8},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"notation-table","excerpt":"mathscr A, C, q · A class of possible archives, an encoding scheme, and a task on that class.","count":20},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"notation-table","excerpt":"Ext_F(S,N) · Rule models containing S and avoiding the negative observations N.","count":31},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"notation-table","excerpt":"n_eff · A real, variance-equivalent review count in the specified covariance model.","count":37},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f03486d125e6b8f9","tex":"n_{\\mathrm{eff}}","aliases":"n_eff mathrm","display":"n_{\\mathrm{eff}}","kind":"notation-table","excerpt":"n_eff · A real, variance-equivalent review count in the specified covariance model.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"P_K(S)=S∩ K · Pure retention of records selected by a fixed mask.","count":17},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"notation-table","excerpt":"D:E→ R, p:E→ S · A linear defect and a protected observable. The repair condition is Dy=0, p(y)=p(x).","count":18},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c5ca4f817be17bab","tex":"P_K","aliases":"P_K","display":"P_K","kind":"notation-table","excerpt":"P_K(S)=S∩ K · Pure retention of records selected by a fixed mask.","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-813431acda8054c4","tex":"\\Pcal","aliases":"Pcal P mathcal","display":"{\\mathcal P}","kind":"notation-table","excerpt":"P, V, J · A declared proof field, its nodes, and its justifications.","count":29},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"Q(a)=(q_i(a))_i · The joint task representation, least in retained distinctions for its task family.","count":29},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"notation-table","excerpt":"Q(a)=(q_i(a))_i · The joint task representation, least in retained distinctions for its task family.","count":47},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"notation-table","excerpt":"Q(a)=(q_i(a))_i · The joint task representation, least in retained distinctions for its task family.","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"notation-table","excerpt":"D:E→ R, p:E→ S · A linear defect and a protected observable. The repair condition is Dy=0, p(y)=p(x).","count":40},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"notation-table","excerpt":"cl_F(S) · The least declared rule model containing the positive observations S and all admitted leaves.","count":98},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"notation-table","excerpt":"A, L, T_0 · Required premises, admissible leaves, and requested targets, with A⊆ L.","count":29},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"notation-table","excerpt":"A, L, T_0 · Required premises, admissible leaves, and requested targets, with A⊆ L.","count":11},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"notation-table","excerpt":"M_F(U) · Local models on patch U satisfying all certified rules supported entirely in U.","count":55},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"notation-table","excerpt":"F_ U · The same field with only rules whose entire supports appear in a covering patch.","count":13},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a49e958a93a3404a","tex":"U_M","aliases":"U_M","display":"U_M","kind":"notation-table","excerpt":"U_M(S) · Weighted target reach from the mandatory starts and a selected supplement, in a fixed directed bridge graph.","count":9},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"notation-table","excerpt":"P, V, J · A declared proof field, its nodes, and its justifications.","count":45},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"notation-table","excerpt":"D:E→ R, p:E→ S · A linear defect and a protected observable. The repair condition is Dy=0, p(y)=p(x).","count":51},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"notation-table","excerpt":"D:E→ R, p:E→ S · A linear defect and a protected observable. The repair condition is Dy=0, p(y)=p(x).","count":41},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"notation-table","excerpt":"Γ( M_F) · Coherent families of those local models under restriction.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"notation-table","excerpt":"D:E→ R, p:E→ S · A linear defect and a protected observable. The repair condition is Dy=0, p(y)=p(x).","count":53},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"notation-table","excerpt":"P_K(S)=S∩ K · Pure retention of records selected by a fixed mask.","count":23},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"notation-table","excerpt":"A, L, T_0 · Required premises, admissible leaves, and requested targets, with A⊆ L.","count":63},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":11},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"A trace σ=(j_1,…,j_k) is a list of rule addresses. Replay begins with the list K_0=Lmathbin+ +S. It rejects an unknown rule address or a step whose premises are not all known. Otherwise it adds the conclusion. Strict replay additionally requires the conclusion to be new. Causal replay permits a conclusion already known. Lists are compared by member","count":33},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"definition-passage","excerpt":"[Future reach utility] The weighted future reach of S is U(S)=∑_f∈ Fw_f 1{f∈Reach_ B(S)}.","count":9},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"definition-passage","excerpt":"[Finite indexed problem and trace] An indexed problem P specifies a finite address space V_n={0,…,n-1}, admitted leaves L, a list of certified rules, positive observations S, and forbidden nodes N. A rule at address j has a finite premise list A_j and conclusion b_j. Two identical rows may have different rule addresses. Every listed node must belon","count":13},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f672f5afa19f4601","tex":"\\operatorname{Aff}","aliases":"Aff operatorname","display":"\\operatorname{Aff}","kind":"definition-passage","excerpt":"[Syntactic affected set] For a changed node set X⊆ V, define Aff(X)=X∪Desc(X), where descendants are computed in the declared dependency graph.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Latent bridge graph] A latent bridge graph is a finite directed graph B=(W,E_B) whose nodes include retained proof objects, omitted proof objects, and anticipated future targets. An edge x→ y means that x has been declared a possible transfer route toward y. The edge may be certified, conjectural, or exploratory, but its status must be recorded.","count":30},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Finite indexed problem and trace] An indexed problem P specifies a finite address space V_n={0,…,n-1}, admitted leaves L, a list of certified rules, positive observations S, and forbidden nodes N. A rule at address j has a finite premise list A_j and conclusion b_j. Two identical rows may have different rule addresses. Every listed node must belon","count":64},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"definition-passage","excerpt":"[Finite indexed problem and trace] An indexed problem P specifies a finite address space V_n={0,…,n-1}, admitted leaves L, a list of certified rules, positive observations S, and forbidden nodes N. A rule at address j has a finite premise list A_j and conclusion b_j. Two identical rows may have different rule addresses. Every listed node must belon","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-327abfc143ff067a","tex":"\\Bcal","aliases":"Bcal B mathcal","display":"{\\mathcal B}","kind":"definition-passage","excerpt":"[Latent bridge graph] A latent bridge graph is a finite directed graph B=(W,E_B) whose nodes include retained proof objects, omitted proof objects, and anticipated future targets. An edge x→ y means that x has been declared a possible transfer route toward y. The edge may be certified, conjectural, or exploratory, but its status must be recorded.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0eaa1204dfcfa445","tex":"\\mathbf c","aliases":"c mathbf","display":"\\mathbf c","kind":"definition-passage","excerpt":"[Typed proof field] A finite typed proof field is a tuple P=(V, J,A,T,τ, E, v, c), where: item V is a finite set of claim nodes. item J is a finite set of justifications. item Each j∈ J has a finite premise set prem(j)⊆ V and one conclusion con(j)∈ V. item A⊆ V is the set of required premises of the declared judgment. item T⊆ V is the nonempty targ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Future task model] Let T be a finite task set with probabilities p_t≥0 satisfying ∑_t p_t=1. Let R_t⊆ V be the node set required by task t, and let c(v)≥0 be the recovery cost of node v when it is absent.","count":50},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-227ceb7baf501818","tex":"c_d","aliases":"c_d","display":"c_d","kind":"definition-passage","excerpt":"[Goal-relative depth objective] For goal g with weights α_g,β_g≥0, define J_g(d)=α_g r_d+β_g c_d. An optimal expansion depth is any minimizer of J_g.","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-843721b3e84879d7","tex":"\\Ccal","aliases":"Ccal C mathcal","display":"{\\mathcal C}","kind":"definition-passage","excerpt":"[Certificate] A certificate for a nonempty requested target set T_0⊆ V is a pair C=(K,H), K⊆ V, H⊆ J^cert, such that A∪ T_0⊆ K and T_0⊆ D_cert(K,H). Order certificates componentwise: (K,H)≤(K^prime,H^prime) means K⊆ K^prime and H⊆ H^prime. A least certificate is contained in every certificate for the same request. Minimizing only the node set, only","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-27b5a2e3b286b58f","tex":"\\mathsf{Cert}","aliases":"Cert mathsf","display":"\\mathsf{Cert}","kind":"definition-passage","excerpt":"[Certified proof category] Fix a directed graph of typed mathematical states and certified primitive transformations. Its path category Cert has these states as objects and finite composable paths as morphisms. The empty path is an identity, and composition is concatenation. One may further identify paths by a declared congruence compatible with co","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f45e1461ee5d2281","tex":"\\mathsf{Certificate}","aliases":"Certificate mathsf","display":"\\mathsf{Certificate}","kind":"definition-passage","excerpt":"[Compositional certificate map] A compositional certificate map is a functor C:Expanded⟶Certificate that sends expanded certified paths to checkable certificate paths and preserves identity and composition.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-95d26be7110cc011","tex":"\\operatorname{con}","aliases":"con operatorname","display":"\\operatorname{con}","kind":"definition-passage","excerpt":"[Typed proof field] A finite typed proof field is a tuple P=(V, J,A,T,τ, E, v, c), where: item V is a finite set of claim nodes. item J is a finite set of justifications. item Each j∈ J has a finite premise set prem(j)⊆ V and one conclusion con(j)∈ V. item A⊆ V is the set of required premises of the declared judgment. item T⊆ V is the nonempty targ","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Task-sufficient representation] The scheme C:mathscr A→mathscr R is sufficient for q when there is a single decoder d_q:C(mathscr A)→ Y_q such that q( P)=d_q(C( P)) for every P∈mathscr A. It is Q-sufficient when this holds for every q∈ Q. The decoder may depend on the task and the scheme, but not on hidden knowledge of the particular archive being","count":36},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3e5463cd5ce49eb7","tex":"d_q","aliases":"d_q","display":"d_q","kind":"definition-passage","excerpt":"[Task-sufficient representation] The scheme C:mathscr A→mathscr R is sufficient for q when there is a single decoder d_q:C(mathscr A)→ Y_q such that q( P)=d_q(C( P)) for every P∈mathscr A. It is Q-sufficient when this holds for every q∈ Q. The decoder may depend on the task and the scheme, but not on hidden knowledge of the particular archive being","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f91059e06d77e679","tex":"\\Debt","aliases":"Debt operatorname","display":"{\\operatorname{Debt}}","kind":"definition-passage","excerpt":"[Expected compression debt] The expected compression debt of retained set K is Debt(K) = ∑_t∈ Tp_t ∑_v∈ R_t∖ Kc(v).","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-26d83baff19c71bc","tex":"\\Desc","aliases":"Desc operatorname","display":"{\\operatorname{Desc}}","kind":"definition-passage","excerpt":"[Syntactic affected set] For a changed node set X⊆ V, define Aff(X)=X∪Desc(X), where descendants are computed in the declared dependency graph.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":20},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-895f40fd3342c6b2","tex":"e^*","aliases":"e^*","display":"e^*","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8c03467ab8d8fd33","tex":"E_B","aliases":"E_B","display":"E_B","kind":"definition-passage","excerpt":"[Latent bridge graph] A latent bridge graph is a finite directed graph B=(W,E_B) whose nodes include retained proof objects, omitted proof objects, and anticipated future targets. An edge x→ y means that x has been declared a possible transfer route toward y. The edge may be certified, conjectural, or exploratory, but its status must be recorded.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c470de86d3aa40db","tex":"\\Ecal","aliases":"Ecal E mathcal","display":"{\\mathcal E}","kind":"definition-passage","excerpt":"[Typed proof field] A finite typed proof field is a tuple P=(V, J,A,T,τ, E, v, c), where: item V is a finite set of claim nodes. item J is a finite set of justifications. item Each j∈ J has a finite premise set prem(j)⊆ V and one conclusion con(j)∈ V. item A⊆ V is the set of required premises of the declared judgment. item T⊆ V is the nonempty targ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-152f3b9f124500b6","tex":"\\mathsf{Expanded}","aliases":"Expanded mathsf","display":"\\mathsf{Expanded}","kind":"definition-passage","excerpt":"[Compositional certificate map] A compositional certificate map is a functor C:Expanded⟶Certificate that sends expanded certified paths to checkable certificate paths and preserves identity and composition.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Future reach utility] The weighted future reach of S is U(S)=∑_f∈ Fw_f 1{f∈Reach_ B(S)}.","count":29},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-78b29d109052da32","tex":"F_v","aliases":"F_v","display":"F_v","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Proof compiler] Let g be a presentation goal and let P be a versioned proof field. A proof compiler is a map on the declared archive class Γ_g:mathscr A⟶ R_g that selects, orders, and renders the material required by g while preserving links to canonical claim and justification identifiers.","count":11},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f2f8bfa898b4de61","tex":"H^{\\prime}","aliases":"H^prime prime","display":"H^{\\prime}","kind":"definition-passage","excerpt":"[Certificate] A certificate for a nonempty requested target set T_0⊆ V is a pair C=(K,H), K⊆ V, H⊆ J^cert, such that A∪ T_0⊆ K and T_0⊆ D_cert(K,H). Order certificates componentwise: (K,H)≤(K^prime,H^prime) means K⊆ K^prime and H⊆ H^prime. A least certificate is contained in every certificate for the same request. Minimizing only the node set, only","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"[Goal-relative depth objective] For goal g with weights α_g,β_g≥0, define J_g(d)=α_g r_d+β_g c_d. An optimal expansion depth is any minimizer of J_g.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Typed proof field] A finite typed proof field is a tuple P=(V, J,A,T,τ, E, v, c), where: item V is a finite set of claim nodes. item J is a finite set of justifications. item Each j∈ J has a finite premise set prem(j)⊆ V and one conclusion con(j)∈ V. item A⊆ V is the set of required premises of the declared judgment. item T⊆ V is the nonempty targ","count":47},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0e97a80aa927d24d","tex":"j_1","aliases":"j_1","display":"j_1","kind":"definition-passage","excerpt":"A trace σ=(j_1,…,j_k) is a list of rule addresses. Replay begins with the list K_0=Lmathbin+ +S. It rejects an unknown rule address or a step whose premises are not all known. Otherwise it adds the conclusion. Strict replay additionally requires the conclusion to be new. Causal replay permits a conclusion already known. Lists are compared by member","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0594198ad0b71612","tex":"J_g","aliases":"J_g","display":"J_g","kind":"definition-passage","excerpt":"[Goal-relative depth objective] For goal g with weights α_g,β_g≥0, define J_g(d)=α_g r_d+β_g c_d. An optimal expansion depth is any minimizer of J_g.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-50c53edb72dca2a6","tex":"j_k","aliases":"j_k","display":"j_k","kind":"definition-passage","excerpt":"A trace σ=(j_1,…,j_k) is a list of rule addresses. Replay begins with the list K_0=Lmathbin+ +S. It rejects an unknown rule address or a step whose premises are not all known. Otherwise it adds the conclusion. Strict replay additionally requires the conclusion to be new. Causal replay permits a conclusion already known. Lists are compared by member","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0156a63858a99e40","tex":"\\Jcal^{\\mathrm{cert}}","aliases":"Jcal^cert J^cert Jcal mathrm mathcal mathrm","display":"{\\mathcal J}^{\\mathrm{cert}}","kind":"definition-passage","excerpt":"[Certificate] A certificate for a nonempty requested target set T_0⊆ V is a pair C=(K,H), K⊆ V, H⊆ J^cert, such that A∪ T_0⊆ K and T_0⊆ D_cert(K,H). Order certificates componentwise: (K,H)≤(K^prime,H^prime) means K⊆ K^prime and H⊆ H^prime. A least certificate is contained in every certificate for the same request. Minimizing only the node set, only","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"A trace σ=(j_1,…,j_k) is a list of rule addresses. Replay begins with the list K_0=Lmathbin+ +S. It rejects an unknown rule address or a step whose premises are not all known. Otherwise it adds the conclusion. Strict replay additionally requires the conclusion to be new. Causal replay permits a conclusion already known. Lists are compared by member","count":25},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-21ee50596c435efd","tex":"K^{\\prime}","aliases":"K^prime prime","display":"K^{\\prime}","kind":"definition-passage","excerpt":"[Certificate] A certificate for a nonempty requested target set T_0⊆ V is a pair C=(K,H), K⊆ V, H⊆ J^cert, such that A∪ T_0⊆ K and T_0⊆ D_cert(K,H). Order certificates componentwise: (K,H)≤(K^prime,H^prime) means K⊆ K^prime and H⊆ H^prime. A least certificate is contained in every certificate for the same request. Minimizing only the node set, only","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"definition-passage","excerpt":"A trace σ=(j_1,…,j_k) is a list of rule addresses. Replay begins with the list K_0=Lmathbin+ +S. It rejects an unknown rule address or a step whose premises are not all known. Otherwise it adds the conclusion. Strict replay additionally requires the conclusion to be new. Causal replay permits a conclusion already known. Lists are compared by member","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0da3381cad1512b4","tex":"\\Loss","aliases":"Loss operatorname","display":"{\\operatorname{Loss}}","kind":"definition-passage","excerpt":"[Additive loss vector] For a retained node set K⊆ V, define Loss( P,K) = ∑_v∉ K v(v). Normalize a coordinate by its total only when that total is positive. If the total is zero, nonnegativity makes every contribution in that coordinate zero. The implementation records normalized loss zero there as a convention, rather than dividing by zero.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"definition-passage","excerpt":"A trace σ=(j_1,…,j_k) is a list of rule addresses. Replay begins with the list K_0=Lmathbin+ +S. It rejects an unknown rule address or a step whose premises are not all known. Otherwise it adds the conclusion. Strict replay additionally requires the conclusion to be new. Causal replay permits a conclusion already known. Lists are compared by member","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d58992a0c0c08b27","tex":"p_t","aliases":"p_t","display":"p_t","kind":"definition-passage","excerpt":"[Future task model] Let T be a finite task set with probabilities p_t≥0 satisfying ∑_t p_t=1. Let R_t⊆ V be the node set required by task t, and let c(v)≥0 be the recovery cost of node v when it is absent.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-88e4f9b3d2961746","tex":"\\Pcal^{\\mathrm{cert}}","aliases":"Pcal^cert P^cert Pcal mathrm mathcal mathrm","display":"{\\mathcal P}^{\\mathrm{cert}}","kind":"definition-passage","excerpt":"[Certified subfield] The certified subfield P^cert contains every node of P and only those justifications accepted by the declared certification policy. Evidence and scope are part of that policy. A certification flag records acceptance: it is not itself a proof that the rule is sound under an intended mathematical interpretation. That separate sou","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3686804581e62485","tex":"\\Pcal_0","aliases":"Pcal_0 P_0 Pcal mathcal","display":"{\\mathcal P}_0","kind":"definition-passage","excerpt":"[Versioned proof object] A versioned proof object is a sequence P_0⟶ P_1⟶⋯ whose nodes and justifications carry stable identifiers. Each transition records additions, removals, altered scopes, evidence changes, and a digest of its parent version.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-7370d891266eac19","tex":"\\Pcal_1","aliases":"Pcal_1 P_1 Pcal mathcal","display":"{\\mathcal P}_1","kind":"definition-passage","excerpt":"[Versioned proof object] A versioned proof object is a sequence P_0⟶ P_1⟶⋯ whose nodes and justifications carry stable identifiers. Each transition records additions, removals, altered scopes, evidence changes, and a digest of its parent version.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-628b76d60808fcdc","tex":"\\Pcal_i","aliases":"Pcal_i P_i Pcal mathcal","display":"{\\mathcal P}_i","kind":"definition-passage","excerpt":"[Append-only certified update] An update P_i→ P_i+1 is append-only certified when every existing node, admissible-leaf status, evidence record, interpretation, and certified justification remains unchanged. The later field may add records and certified rules but may not withdraw an earlier admitted leaf or change the scope of an earlier inference.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e32d02f59f40d27c","tex":"\\Pcal_{i+1}","aliases":"Pcal_i+1 P_i+1 Pcal mathcal","display":"{\\mathcal P}_{i+1}","kind":"definition-passage","excerpt":"[Append-only certified update] An update P_i→ P_i+1 is append-only certified when every existing node, admissible-leaf status, evidence record, interpretation, and certified justification remains unchanged. The later field may add records and certified rules but may not withdraw an earlier admitted leaf or change the scope of an earlier inference.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-71a98ec1e2bb9f0a","tex":"\\preceq","aliases":"preceq","display":"\\preceq","kind":"definition-passage","excerpt":"[Recoverability preorder] For a fixed task family Q, write R_1preceq_ Q R_2 when every task in Q that factors through the representation scheme R_1 also factors through R_2. Both schemes have domain mathscr A, and each factorization uses one decoder valid on that whole domain.","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-072772a7bb15f658","tex":"\\preceq_{\\Qcal}","aliases":"preceq_Qcal preceq_ Q preceq Qcal preceq mathcal","display":"\\preceq_{{\\mathcal Q}}","kind":"definition-passage","excerpt":"[Recoverability preorder] For a fixed task family Q, write R_1preceq_ Q R_2 when every task in Q that factors through the representation scheme R_1 also factors through R_2. Both schemes have domain mathscr A, and each factorization uses one decoder valid on that whole domain.","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e7c76811709d8d27","tex":"\\operatorname{prem}","aliases":"prem operatorname","display":"\\operatorname{prem}","kind":"definition-passage","excerpt":"[Typed proof field] A finite typed proof field is a tuple P=(V, J,A,T,τ, E, v, c), where: item V is a finite set of claim nodes. item J is a finite set of justifications. item Each j∈ J has a finite premise set prem(j)⊆ V and one conclusion con(j)∈ V. item A⊆ V is the set of required premises of the declared judgment. item T⊆ V is the nonempty targ","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3f1b6dc519af96a7","tex":"\\prime","aliases":"prime","display":"\\prime","kind":"definition-passage","excerpt":"[Certificate] A certificate for a nonempty requested target set T_0⊆ V is a pair C=(K,H), K⊆ V, H⊆ J^cert, such that A∪ T_0⊆ K and T_0⊆ D_cert(K,H). Order certificates componentwise: (K,H)≤(K^prime,H^prime) means K⊆ K^prime and H⊆ H^prime. A least certificate is contained in every certificate for the same request. Minimizing only the node set, only","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-aaf174f6701cc732","tex":"\\Qcal","aliases":"Qcal Q mathcal","display":"{\\mathcal Q}","kind":"definition-passage","excerpt":"[Task] A task is a map q:mathscr A→ Y_q to a declared answer set. A task family Q is a set of such maps. Examples include asking for the target certificate, a dependency list, a specified boundary example, or the claims affected by a recorded revision. A teaching or discovery objective becomes an exact task only after its required output or accepta","count":12},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Goal-relative depth objective] For goal g with weights α_g,β_g≥0, define J_g(d)=α_g r_d+β_g c_d. An optimal expansion depth is any minimizer of J_g.","count":37},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"definition-passage","excerpt":"[Recoverability preorder] For a fixed task family Q, write R_1preceq_ Q R_2 when every task in Q that factors through the representation scheme R_1 also factors through R_2. Both schemes have domain mathscr A, and each factorization uses one decoder valid on that whole domain.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"definition-passage","excerpt":"[Recoverability preorder] For a fixed task family Q, write R_1preceq_ Q R_2 when every task in Q that factors through the representation scheme R_1 also factors through R_2. Both schemes have domain mathscr A, and each factorization uses one decoder valid on that whole domain.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-9360f93efbe99f90","tex":"r_d","aliases":"r_d","display":"r_d","kind":"definition-passage","excerpt":"[Goal-relative depth objective] For goal g with weights α_g,β_g≥0, define J_g(d)=α_g r_d+β_g c_d. An optimal expansion depth is any minimizer of J_g.","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a8c3cc1693c46092","tex":"R_g","aliases":"R_g","display":"R_g","kind":"definition-passage","excerpt":"[Proof compiler] Let g be a presentation goal and let P be a versioned proof field. A proof compiler is a map on the declared archive class Γ_g:mathscr A⟶ R_g that selects, orders, and renders the material required by g while preserving links to canonical claim and justification identifiers.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-49121c793f36cbb5","tex":"R_t","aliases":"R_t","display":"R_t","kind":"definition-passage","excerpt":"[Future task model] Let T be a finite task set with probabilities p_t≥0 satisfying ∑_t p_t=1. Let R_t⊆ V be the node set required by task t, and let c(v)≥0 be the recovery cost of node v when it is absent.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-607edbc5319584b2","tex":"\\Reach","aliases":"Reach operatorname","display":"{\\operatorname{Reach}}","kind":"definition-passage","excerpt":"[Future reach utility] The weighted future reach of S is U(S)=∑_f∈ Fw_f 1{f∈Reach_ B(S)}.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-656699f928a088aa","tex":"\\Reach_{\\Bcal}","aliases":"Reach_Bcal Reach_ B Reach Bcal operatorname mathcal","display":"{\\operatorname{Reach}}_{{\\mathcal B}}","kind":"definition-passage","excerpt":"[Future reach utility] The weighted future reach of S is U(S)=∑_f∈ Fw_f 1{f∈Reach_ B(S)}.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V consists of those globa","count":28},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Scoped branch assessment] Let P be a finite typed proof field with declared premises A and target t. An exploratory branch is a finite sequence of candidate justifications whose certification and outstanding obligations are explicitly recorded. Its assessment is one of the following, relative to stated premises and rules: [label=textup(roman*)] it","count":27},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-712d027743d7bde8","tex":"\\Tcal","aliases":"Tcal T mathcal","display":"{\\mathcal T}","kind":"definition-passage","excerpt":"[Future task model] Let T be a finite task set with probabilities p_t≥0 satisfying ∑_t p_t=1. Let R_t⊆ V be the node set required by task t, and let c(v)≥0 be the recovery cost of node v when it is absent.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"he set of required premises of the declared judgment. item T⊆ V is the nonempty target set. item τ assigns node and justification types. item E is a typed evidence ledger. item v assigns nonnegative value coordinates to nodes. item c assigns nonnegative storage, audit, recovery, and cognitive costs. The dependency graph contains an edge u→ v whenev","count":11},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-abe92d08300c0155","tex":"\\mathbf v","aliases":"v mathbf","display":"\\mathbf v","kind":"definition-passage","excerpt":"[Typed proof field] A finite typed proof field is a tuple P=(V, J,A,T,τ, E, v, c), where: item V is a finite set of claim nodes. item J is a finite set of justifications. item Each j∈ J has a finite premise set prem(j)⊆ V and one conclusion con(j)∈ V. item A⊆ V is the set of required premises of the declared judgment. item T⊆ V is the nonempty targ","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"he set of required premises of the declared judgment. item T⊆ V is the nonempty target set. item τ assigns node and justification types. item E is a typed evidence ledger. item v assigns nonnegative value coordinates to nodes. item c assigns nonnegative storage, audit, recovery, and cognitive costs. The dependency graph contains an edge u→ v whenev","count":48},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a6735e9f71ad2ebf","tex":"V_n","aliases":"V_n","display":"V_n","kind":"definition-passage","excerpt":"[Finite indexed problem and trace] An indexed problem P specifies a finite address space V_n={0,…,n-1}, admitted leaves L, a list of certified rules, positive observations S, and forbidden nodes N. A rule at address j has a finite premise list A_j and conclusion b_j. Two identical rows may have different rule addresses. Every listed node must belon","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"definition-passage","excerpt":"[Latent bridge graph] A latent bridge graph is a finite directed graph B=(W,E_B) whose nodes include retained proof objects, omitted proof objects, and anticipated future targets. An edge x→ y means that x has been declared a possible transfer route toward y. The edge may be certified, conjectural, or exploratory, but its status must be recorded.","count":13},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":24},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-240b0e5e6b064bdb","tex":"w_f","aliases":"w_f","display":"w_f","kind":"definition-passage","excerpt":"[Future reach utility] The weighted future reach of S is U(S)=∑_f∈ Fw_f 1{f∈Reach_ B(S)}.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Syntactic affected set] For a changed node set X⊆ V, define Aff(X)=X∪Desc(X), where descendants are computed in the declared dependency graph.","count":47},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-9d5ae088e190bcb8","tex":"x_w","aliases":"x_w","display":"x_w","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Task] A task is a map q:mathscr A→ Y_q to a declared answer set. A task family Q is a set of such maps. Examples include asking for the target certificate, a dependency list, a specified boundary example, or the claims affected by a recorded revision. A teaching or discovery objective becomes an exact task only after its required output or accepta","count":22},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e52a4b6273afce00","tex":"Y_q","aliases":"Y_q","display":"Y_q","kind":"definition-passage","excerpt":"[Task] A task is a map q:mathscr A→ Y_q to a declared answer set. A task family Q is a set of such maps. Examples include asking for the target certificate, a dependency list, a specified boundary example, or the claims affected by a recorded revision. A teaching or discovery objective becomes an exact task only after its required output or accepta","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","excerpt":"[Goal-relative depth objective] For goal g with weights α_g,β_g≥0, define J_g(d)=α_g r_d+β_g c_d. An optimal expansion depth is any minimizer of J_g.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-191835d3d506f9ee","tex":"\\alpha_g","aliases":"α_g alpha","display":"\\alpha_g","kind":"definition-passage","excerpt":"[Goal-relative depth objective] For goal g with weights α_g,β_g≥0, define J_g(d)=α_g r_d+β_g c_d. An optimal expansion depth is any minimizer of J_g.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"definition-passage","excerpt":"[Goal-relative depth objective] For goal g with weights α_g,β_g≥0, define J_g(d)=α_g r_d+β_g c_d. An optimal expansion depth is any minimizer of J_g.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8f2b814a9d3b4d0a","tex":"\\beta_g","aliases":"β_g beta","display":"\\beta_g","kind":"definition-passage","excerpt":"[Goal-relative depth objective] For goal g with weights α_g,β_g≥0, define J_g(d)=α_g r_d+β_g c_d. An optimal expansion depth is any minimizer of J_g.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-caa04fdf50dfd7c6","tex":"\\Gamma_g","aliases":"Γ_g Gamma","display":"\\Gamma_g","kind":"definition-passage","excerpt":"[Proof compiler] Let g be a presentation goal and let P be a versioned proof field. A proof compiler is a map on the declared archive class Γ_g:mathscr A⟶ R_g that selects, orders, and renders the material required by g while preserving links to canonical claim and justification identifiers.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"A trace σ=(j_1,…,j_k) is a list of rule addresses. Replay begins with the list K_0=Lmathbin+ +S. It rejects an unknown rule address or a step whose premises are not all known. Otherwise it adds the conclusion. Strict replay additionally requires the conclusion to be new. Causal replay permits a conclusion already known. Lists are compared by member","count":15},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Typed proof field] A finite typed proof field is a tuple P=(V, J,A,T,τ, E, v, c), where: item V is a finite set of claim nodes. item J is a finite set of justifications. item Each j∈ J has a finite premise set prem(j)⊆ V and one conclusion con(j)∈ V. item A⊆ V is the set of required premises of the declared judgment. item T⊆ V is the nonempty targ","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Finite indexed problem and trace] An indexed problem P specifies a finite address space V_n={0,…,n-1}, admitted leaves L, a list of certified rules, positive observations S, and forbidden nodes N. A rule at address j has a finite premise list A_j and conclusion b_j. Two identical rows may have different rule addresses. Every listed node must belon","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Typed proof field] A finite typed proof field is a tuple P=(V, J,A,T,τ, E, v, c), where: item V is a finite set of claim nodes. item J is a finite set of justifications. item Each j∈ J has a finite premise set prem(j)⊆ V and one conclusion con(j)∈ V. item A⊆ V is the set of required premises of the declared judgment. item T⊆ V is the nonempty targ","count":45},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"definition-passage","excerpt":"[Additive loss vector] For a retained node set K⊆ V, define Loss( P,K) = ∑_v∉ K v(v). Normalize a coordinate by its total only when that total is positive. If the total is zero, nonnegativity makes every contribution in that coordinate zero. The implementation records normalized loss zero there as a convention, rather than dividing by zero.","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8a37bbb74af9ac5c","tex":"\\prod_{v\\in V}","aliases":"∏_v∈ V prod in","display":"\\prod_{v\\in V}","kind":"definition-passage","excerpt":"[Coherent views of a proof field] Choose a finite collection of views V, fibers F_v of local proof states or annotations, and for each declared directed edge e:v→ w a restriction map e^*:F_w→ F_v. The strict realization set is Γ(F)={(x_v)∈∏_v∈ VF_v : e^*(x_w)=x_v for every declared edge e}. The extension fiber of prescribed information s on U⊆ V co","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Additive loss vector] For a retained node set K⊆ V, define Loss( P,K) = ∑_v∉ K v(v). Normalize a coordinate by its total only when that total is positive. If the total is zero, nonnegativity makes every contribution in that coordinate zero. The implementation records normalized loss zero there as a convention, rather than dividing by zero.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8688b636524a25ae","tex":"\\sum_{f\\in F}","aliases":"∑_f∈ F sum in","display":"\\sum_{f\\in F}","kind":"definition-passage","excerpt":"[Future reach utility] The weighted future reach of S is U(S)=∑_f∈ Fw_f 1{f∈Reach_ B(S)}.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-90c396c1f06e154a","tex":"\\sum_t","aliases":"∑_t sum","display":"\\sum_t","kind":"definition-passage","excerpt":"[Future task model] Let T be a finite task set with probabilities p_t≥0 satisfying ∑_t p_t=1. Let R_t⊆ V be the node set required by task t, and let c(v)≥0 be the recovery cost of node v when it is absent.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-039632cabe06b886","tex":"\\sum_{t\\in\\Tcal}","aliases":"∑_t∈Tcal ∑_t∈ T sum in Tcal sum in mathcal","display":"\\sum_{t\\in{\\mathcal T}}","kind":"definition-passage","excerpt":"[Expected compression debt] The expected compression debt of retained set K is Debt(K) = ∑_t∈ Tp_t ∑_v∈ R_t∖ Kc(v).","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1dd09588b24d8ba8","tex":"\\sum_{v\\in R_t\\setminus K}","aliases":"∑_v∈ R_t∖ K sum in setminus","display":"\\sum_{v\\in R_t\\setminus K}","kind":"definition-passage","excerpt":"[Expected compression debt] The expected compression debt of retained set K is Debt(K) = ∑_t∈ Tp_t ∑_v∈ R_t∖ Kc(v).","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-57f47b5d70761baf","tex":"\\sum_{v\\notin K}","aliases":"∑_v∉ K sum notin","display":"\\sum_{v\\notin K}","kind":"definition-passage","excerpt":"[Additive loss vector] For a retained node set K⊆ V, define Loss( P,K) = ∑_v∉ K v(v). Normalize a coordinate by its total only when that total is positive. If the total is zero, nonnegativity makes every contribution in that coordinate zero. The implementation records normalized loss zero there as a convention, rather than dividing by zero.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"definition-passage","excerpt":"[Expected compression debt] The expected compression debt of retained set K is Debt(K) = ∑_t∈ Tp_t ∑_v∈ R_t∖ Kc(v).","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"definition-passage","excerpt":"[Certificate] A certificate for a nonempty requested target set T_0⊆ V is a pair C=(K,H), K⊆ V, H⊆ J^cert, such that A∪ T_0⊆ K and T_0⊆ D_cert(K,H). Order certificates componentwise: (K,H)≤(K^prime,H^prime) means K⊆ K^prime and H⊆ H^prime. A least certificate is contained in every certificate for the same request. Minimizing only the node set, only","count":17},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"[Certificate] A certificate for a nonempty requested target set T_0⊆ V is a pair C=(K,H), K⊆ V, H⊆ J^cert, such that A∪ T_0⊆ K and T_0⊆ D_cert(K,H). Order certificates componentwise: (K,H)≤(K^prime,H^prime) means K⊆ K^prime and H⊆ H^prime. A least certificate is contained in every certificate for the same request. Minimizing only the node set, only","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"[Future task model] Let T be a finite task set with probabilities p_t≥0 satisfying ∑_t p_t=1. Let R_t⊆ V be the node set required by task t, and let c(v)≥0 be the recovery cost of node v when it is absent.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Versioned proof object] A versioned proof object is a sequence P_0⟶ P_1⟶⋯ whose nodes and justifications carry stable identifiers. Each transition records additions, removals, altered scopes, evidence changes, and a digest of its parent version.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Compositional certificate map] A compositional certificate map is a functor C:Expanded⟶Certificate that sends expanded certified paths to checkable certificate paths and preserves identity and composition.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-200dd3f84f09cafb","tex":"\\mathrm{core}","aliases":"core mathrm","display":"\\mathrm{core}","kind":"introduction-passage","excerpt":"Let Anc_L(T_0) denote the premise nodes reached by this backward expansion, stopping at L. Let H_core contain exactly the justifications selected at expanded nodes outside L.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f8ecc0c903c248d2","tex":"H_{\\mathrm{core}}","aliases":"H_core mathrm","display":"H_{\\mathrm{core}}","kind":"introduction-passage","excerpt":"Let Anc_L(T_0) denote the premise nodes reached by this backward expansion, stopping at L. Let H_core contain exactly the justifications selected at expanded nodes outside L.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"introduction-passage","excerpt":"The collision obstruction of Theorem~[source reference] has a converse. It gives a complete test for exact task recovery at the level of sets. Write imC=C(mathscr A) for the attained output set. A decoder is required on this set and nowhere else.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2876747ca029362d","tex":"\\mathbf1^\\top","aliases":"1^top mathbf top","display":"\\mathbf1^\\top","kind":"use","excerpt":"For a general positive definite covariance matrix Σ, consider unbiased linear aggregates w^top X with 1^top w=1.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"Let T:E→ R and Q:E→ W be linear, and fix a known point x_0. Put A={y:Ty=Tx_0}, H=ker T, Q_H=Q|_H. Translation h↦ x_0+h identifies H with A. The observation image is the translate Qx_0+Q(H), and the fiber above the attained value Qx_0 is a translate of ker Q_H.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"ly injective encodings] Let C:A→ B. Every predicate q:A→Prop factors through C if and only if C is injective. If C is injective, define the decoder of q at b by the predicate ∃ a (C(a)=b and q(a)). At b=C(a_0), injectivity makes this equivalent to q(a_0). Conversely, if C(a)=C(a'), use the predicate q(z)≡(z=a). Its decoded values at the equal encod","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4380db69220b934d","tex":"A^F_j","aliases":"A^F_j","display":"A^F_j","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G, j certified in F · Longrightarrow h(j) certified in G, A^G_h(j) · ⊆ f(A^F_j), c^G_h(j) · =f(c^F_j). The premise inclusion points in the di","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-09d367953ceb0276","tex":"A^G_{h(j)}","aliases":"A^G_h(j)","display":"A^G_{h(j)}","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G, j certified in F · Longrightarrow h(j) certified in G, A^G_h(j) · ⊆ f(A^F_j), c^G_h(j) · =f(c^F_j). The premise inclusion points in the di","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-6f67ea78053a16b5","tex":"A^\\prime","aliases":"A^prime prime","display":"A^\\prime","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]fig_07_bridge_dominators.pdf captionA declared bridge graph. The boundary theorem C dominates future targets A^prime and B^prime but not C^prime, which has a bypass from E. These arrows describe future reach in the supplied model, not proof certification. Deleting a bridge vertex is different from me","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"[All predicates distinguish exactly injective encodings] Let C:A→ B. Every predicate q:A→Prop factors through C if and only if C is injective. If C is injective, define the decoder of q at b by the predicate ∃ a (C(a)=b and q(a)). At b=C(a_0), injectivity makes this equivalent to q(a_0). Conversely, if C(a)=C(a'), use the predicate q(z)≡(z=a). Its ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"Let a_t≥0 be candidate arrivals during audit cycle t, let s_t≥0 be service capacity, and let B_t≥0 be the unresolved backlog. Suppose arrivals in a cycle are available to that cycle's service. The basic deterministic queue is B_t+1=max{0,B_t+a_t-s_t}.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0900331d47e50143","tex":"\\mathrm{archive}","aliases":"archive mathrm","display":"\\mathrm{archive}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-16bf83baea03980d","tex":"\\mathrm{audit}","aliases":"audit mathrm","display":"\\mathrm{audit}","kind":"use","excerpt":"A simple audit priority score is P(x) = fracp_err(x) c_consequence(x) (1+r_dependency(x)) c_audit(x). Here p_err(x)∈[0,1], the consequence and dependency factors are nonnegative, and c_audit(x)>0. Zero-cost checks can be handled separately. This is a declared priority heuristic, not a probability of truth or a proved optimal scheduling rule.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"[Bias is not covariance] Covariance-aware weighting does not remove a common bias that is absent from the covariance model. If every reviewer shares bias b, the mean squared error contains the irreducible term b^2.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8a1d2aa0f9492064","tex":"B^\\prime","aliases":"B^prime prime","display":"B^\\prime","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]fig_07_bridge_dominators.pdf captionA declared bridge graph. The boundary theorem C dominates future targets A^prime and B^prime but not C^prime, which has a bypass from E. These arrows describe future reach in the supplied model, not proof certification. Deleting a bridge vertex is different from me","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"Before the truncation at zero, each cycle adds λ-μ. When this quantity is negative, the backlog decreases by a fixed amount until it reaches zero. When it is zero, the backlog is unchanged. When it is positive, the truncation is never active after a nonnegative start, and the backlog is B_0+t(λ-μ).","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-9474dae99bcf0491","tex":"B_t","aliases":"B_t","display":"B_t","kind":"use","excerpt":"Let a_t≥0 be candidate arrivals during audit cycle t, let s_t≥0 be service capacity, and let B_t≥0 be the unresolved backlog. Suppose arrivals in a cycle are available to that cycle's service. The basic deterministic queue is B_t+1=max{0,B_t+a_t-s_t}.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d8d4b66e30067b11","tex":"B_{t+1}","aliases":"B_t+1","display":"B_{t+1}","kind":"use","excerpt":"Let a_t≥0 be candidate arrivals during audit cycle t, let s_t≥0 be service capacity, and let B_t≥0 be the unresolved backlog. Suppose arrivals in a cycle are available to that cycle's service. The basic deterministic queue is B_t+1=max{0,B_t+a_t-s_t}.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5a77d9004b78105c","tex":"\\bar X","aliases":"bar X bar","display":"\\bar X","kind":"use","excerpt":"Let X_i=θ+ε_i be numerical estimates of a fixed quantity θ. Assume Eε_i=0, Var(ε_i)=σ^2>0, and common pairwise correlation ρ. For n>1, assume -1/(n-1)<ρ≤1, so the mean has positive variance. These variables model estimates, not proof truth values. The mean has variance Var(bar X) = σ^2(frac1-ρn+ρ).","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c59061a02439fac0","tex":"c^F_j","aliases":"c^F_j","display":"c^F_j","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G, j certified in F · Longrightarrow h(j) certified in G, A^G_h(j) · ⊆ f(A^F_j), c^G_h(j) · =f(c^F_j). The premise inclusion points in the di","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-57804f08e8ca8e04","tex":"c^G_{h(j)}","aliases":"c^G_h(j)","display":"c^G_{h(j)}","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G, j certified in F · Longrightarrow h(j) certified in G, A^G_h(j) · ⊆ f(A^F_j), c^G_h(j) · =f(c^F_j). The premise inclusion points in the di","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c874bf2cf002f4a1","tex":"C^\\prime","aliases":"C^prime prime","display":"C^\\prime","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]fig_07_bridge_dominators.pdf captionA declared bridge graph. The boundary theorem C dominates future targets A^prime and B^prime but not C^prime, which has a bypass from E. These arrows describe future reach in the supplied model, not proof certification. Deleting a bridge vertex is different from me","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]fig_05_recoverability_preorder.pdf captionA concrete retention diamond for the four archives (C_0,x,y) with x,y∈{0,1}. Arrows discard coordinates. The two middle schemes are incomparable for the task family {q_x,q_y}. This is not a universal ranking of teaching, discovery, and formal presentations.","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-91b6d63d80333e2d","tex":"c_{\\mathrm{audit}}","aliases":"c_audit mathrm","display":"c_{\\mathrm{audit}}","kind":"use","excerpt":"A simple audit priority score is P(x) = fracp_err(x) c_consequence(x) (1+r_dependency(x)) c_audit(x). Here p_err(x)∈[0,1], the consequence and dependency factors are nonnegative, and c_audit(x)>0. Zero-cost checks can be handled separately. This is a declared priority heuristic, not a probability of truth or a proved optimal scheduling rule.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-fa5b599e4a4fe439","tex":"c_{\\mathrm{consequence}}","aliases":"c_consequence mathrm","display":"c_{\\mathrm{consequence}}","kind":"use","excerpt":"A simple audit priority score is P(x) = fracp_err(x) c_consequence(x) (1+r_dependency(x)) c_audit(x). Here p_err(x)∈[0,1], the consequence and dependency factors are nonnegative, and c_audit(x)>0. Zero-cost checks can be handled separately. This is a declared priority heuristic, not a probability of truth or a proved optimal scheduling rule.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-dff8e8535d2ec4f4","tex":"c_{d-1}","aliases":"c_d-1","display":"c_{d-1}","kind":"use","excerpt":"A practical stopping rule selects the first depth satisfying r_d≤ε or fracr_d-1-r_dc_d-c_d-1<τ. The first condition uses a declared tolerance. The second applies only at d≥1 with c_d>c_d-1 and compares the most recent reduction with its extra cost. Zero-cost refinements require separate handling. These stopping rules do not generally minimize J_g o","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-17ea41630af6c876","tex":"c_{d_1}","aliases":"c_d_1","display":"c_{d_1}","kind":"use","excerpt":"Take a proof object with exactly two available depths d_1<d_2, with r_d_2<r_d_1 and c_d_2>c_d_1. Direct subtraction shows that d_2 is preferred exactly when α_g(r_d_1-r_d_2)>β_g(c_d_2-c_d_1). The goal (α_g,β_g)=(1,0) strictly prefers d_2, whereas (0,1) strictly prefers d_1. Since these are the only available depths, each preference is a global opti","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-30179a25de317bb7","tex":"c_{d_2}","aliases":"c_d_2","display":"c_{d_2}","kind":"use","excerpt":"Take a proof object with exactly two available depths d_1<d_2, with r_d_2<r_d_1 and c_d_2>c_d_1. Direct subtraction shows that d_2 is preferred exactly when α_g(r_d_1-r_d_2)>β_g(c_d_2-c_d_1). The goal (α_g,β_g)=(1,0) strictly prefers d_2, whereas (0,1) strictly prefers d_1. Since these are the only available depths, each preference is a global opti","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"[Portfolio robustness] For representation schemes C_i:mathscr A→ R_i, retain the tagged portfolio C( P)=(C_i( P))_i. Every task recoverable from any one C_i is recoverable from C.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"Fix a node set V, a set L⊆ V of admissible leaves, and a set J of justifications. A justification j has a premise set A_j, a conclusion c_j, and a certification predicate. Define its support by S_j=A_j∪{c_j}. A set X⊆ V is a model of the declared field when L⊆ X and, for every certified j, A_j⊆ X Longrightarrow c_j∈ X. Each justification is one imp","count":8},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-16cc1080eab7d1d1","tex":"\\mathsf{Cert}_1","aliases":"Cert_1 mathsf","display":"\\mathsf{Cert}_1","kind":"use","excerpt":"[Functorial transport] Let F:Cert_1→Cert_2 be a functor. If Axrightarrowf_1S_1xrightarrowf_2⋯xrightarrowf_rB is a certified path in Cert_1, then F(A)xrightarrowF(f_1)F(S_1)xrightarrowF(f_2)⋯xrightarrowF(f_r)F(B) is a certified path in Cert_2.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e273f23c3e77b56b","tex":"\\mathsf{Cert}_2","aliases":"Cert_2 mathsf","display":"\\mathsf{Cert}_2","kind":"use","excerpt":"[Functorial transport] Let F:Cert_1→Cert_2 be a functor. If Axrightarrowf_1S_1xrightarrowf_2⋯xrightarrowf_rB is a certified path in Cert_1, then F(A)xrightarrowF(f_1)F(S_1)xrightarrowF(f_2)⋯xrightarrowF(f_r)F(B) is a certified path in Cert_2.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-05b1eb31fcfbf787","tex":"\\operatorname{cl}_{F_{\\mathcal U}}","aliases":"cl_F_ U operatorname mathcal","display":"\\operatorname{cl}_{F_{\\mathcal U}}","kind":"use","excerpt":"Let U=(U_i)_i cover every node. Define F_ U to have the same nodes and leaves as F but only those certified rules whose entire support lies in some U_i. The distinction between the two fields is explicit. We call the cover semantically sufficient for the declared rules when ∀ j certified in F, c_j∈cl_F_ U(A_j). tagSC The word semantic here refers t","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-59cd2ad71953cae8","tex":"\\operatorname{cl}_G","aliases":"cl_G operatorname","display":"\\operatorname{cl}_G","kind":"use","excerpt":"Positive seeds can be added before transport. The same rule transport sends F[S] to G[f(S)], and hence f(cl_F(S)) ⊆cl_G(f(S)). A forbidden source consequence n∈ N∩cl_F(S) maps to the explicit forbidden consequence f(n)∈ f(N)∩cl_G(f(S)). The formal development checks this witness map, not only the existence statement obtained by contraposition.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-df901d7cc2e3c1cf","tex":"\\mathrm{consequence}","aliases":"consequence mathrm","display":"\\mathrm{consequence}","kind":"use","excerpt":"A simple audit priority score is P(x) = fracp_err(x) c_consequence(x) (1+r_dependency(x)) c_audit(x). Here p_err(x)∈[0,1], the consequence and dependency factors are nonnegative, and c_audit(x)>0. Zero-cost checks can be handled separately. This is a declared priority heuristic, not a probability of truth or a proved optimal scheduling rule.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8e1f2e5e8c4fc652","tex":"\\operatorname{Cov}","aliases":"Cov operatorname","display":"\\operatorname{Cov}","kind":"use","excerpt":"[MSE with a shared interpretive offset] Suppose reviewers report J_i=θ+b+ε_i, where θ is the target quantity, b is a fixed shared offset, Eε_i=0, Var(ε_i)=σ^2, and Cov(ε_i,ε_j)=ρσ^2 for i≠ j. For admissible ρ, the mean squared error of the reviewer average is E(overline J_n-θ)^2=b^2+σ^2(ρ+frac1-ρn). For fixed ρ≥0, it approaches b^2+ρσ^2 as n→∞. The","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"use","excerpt":"Take a proof object with exactly two available depths d_1<d_2, with r_d_2<r_d_1 and c_d_2>c_d_1. Direct subtraction shows that d_2 is preferred exactly when α_g(r_d_1-r_d_2)>β_g(c_d_2-c_d_1). The goal (α_g,β_g)=(1,0) strictly prefers d_2, whereas (0,1) strictly prefers d_1. Since these are the only available depths, each preference is a global opti","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"use","excerpt":"Take a proof object with exactly two available depths d_1<d_2, with r_d_2<r_d_1 and c_d_2>c_d_1. Direct subtraction shows that d_2 is preferred exactly when α_g(r_d_1-r_d_2)>β_g(c_d_2-c_d_1). The goal (α_g,β_g)=(1,0) strictly prefers d_2, whereas (0,1) strictly prefers d_1. Since these are the only available depths, each preference is a global opti","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5c308718fcbd3df8","tex":"d_C","aliases":"d_C","display":"d_C","kind":"use","excerpt":"[Lossless encoding at the typed syntax boundary] Let e_P,d_P and e_C,d_C be the stated Lean encoders and partial decoders for problem and certificate records. For every typed problem P and certificate C, d_P(e_P(P))=some(P), d_C(e_C(C))=some(C). Consequently both encoders are injective on their typed domains. These identities do not require the enc","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"use","excerpt":"If q=d_i∘ C_i, compose d_i with the ith coordinate projection from the portfolio. This gives a decoder of q from C. Retaining the coordinate labels is part of the construction. An untagged union that loses source identities need not permit the same recovery.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-bee87637ff187a82","tex":"d_P","aliases":"d_P","display":"d_P","kind":"use","excerpt":"[Lossless encoding at the typed syntax boundary] Let e_P,d_P and e_C,d_C be the stated Lean encoders and partial decoders for problem and certificate records. For every typed problem P and certificate C, d_P(e_P(P))=some(P), d_C(e_C(C))=some(C). Consequently both encoders are injective on their typed domains. These identities do not require the enc","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5110278d3284eee8","tex":"\\mathrm{dependency}","aliases":"dependency mathrm","display":"\\mathrm{dependency}","kind":"use","excerpt":"A simple audit priority score is P(x) = fracp_err(x) c_consequence(x) (1+r_dependency(x)) c_audit(x). Here p_err(x)∈[0,1], the consequence and dependency factors are nonnegative, and c_audit(x)>0. Zero-cost checks can be handled separately. This is a declared priority heuristic, not a probability of truth or a proved optimal scheduling rule.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-cb3c71b741b0bcbf","tex":"\\operatorname{Der}","aliases":"Der operatorname","display":"\\operatorname{Der}","kind":"use","excerpt":"Let Der_F be the inductively generated set obtained from the leaves by certified rules. For a finite field, repeated forward closure computes this set. It is not necessary to enumerate all valuations to perform that computation.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-88c67294ac1c5be8","tex":"\\operatorname{Der}_F","aliases":"Der_F operatorname","display":"\\operatorname{Der}_F","kind":"use","excerpt":"Let Der_F be the inductively generated set obtained from the leaves by certified rules. For a finite field, repeated forward closure computes this set. It is not necessary to enumerate all valuations to perform that computation.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-49f1d36a197b16b7","tex":"\\operatorname{Der}_{F[S]}","aliases":"Der_F[S] operatorname","display":"\\operatorname{Der}_{F[S]}","kind":"use","excerpt":"Fix the field F, including its admitted leaves and certified rules. For a positive observation set S⊆ V, let cl_F(S)=Der_F[S], where F[S] is obtained by admitting L∪ S as leaves and changing nothing else. For a negative observation set N⊆ V, define Ext_F(S,N)= {X⊆ V:X is an F-model, S⊆ X, X∩ N=∅}. Positive and negative observations are both part of","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"| K_P(x)|=2^n-rank(M). item A row vector λ exists with λ M=0 and λ b=1. It is a directly checkable certificate that K_P(x)=∅. If My_0=b, any other solution differs from y_0 by an element of ker M=H, and every such difference gives another solution. Rank nullity gives dim H=n-rank(M). If b is not in the column space of M, linear algebra supplies a l","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-98d940b7f3ffe088","tex":"\\dim_K","aliases":"dim_K dim","display":"\\dim_K","kind":"use","excerpt":"[Exact observation counts] Suppose K is a finite field of cardinality q and E is a finite-dimensional K-vector space. Set k=dim_K H and r=dim_K Q(H). Then | A|=q^k, |Q( A)|=q^r, |{y∈ A:Qy=Qx_0}|=q^k-r. In particular the exact count identity is | A|= |{y∈ A:Qy=Qx_0}| q^r. Every attained observation fiber has the same size. For the uniform distributi","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0e04b7f80fa27ddd","tex":"\\mathrm{discovery}","aliases":"discovery mathrm","display":"\\mathrm{discovery}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"use","excerpt":"Let X_i=θ+ε_i be numerical estimates of a fixed quantity θ. Assume Eε_i=0, Var(ε_i)=σ^2>0, and common pairwise correlation ρ. For n>1, assume -1/(n-1)<ρ≤1, so the mean has positive variance. These variables model estimates, not proof truth values. The mean has variance Var(bar X) = σ^2(frac1-ρn+ρ).","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c2349512e48c7e97","tex":"e^{-1}","aliases":"e^-1","display":"e^{-1}","kind":"use","excerpt":"ginal contribution of the at most k elements of O_k beyond S_i is at least G^*-G_i. Hence at least one candidate has marginal gain at least (G^*-G_i)/k. Greedy chooses a candidate with at least that gain, so G^*-G_i+1 ≤ (1-frac1k)(G^*-G_i). Iterating from G_0=0 gives G^*-G_k ≤ (1-frac1k)^kG^*. Rearranging gives the first bound. The inequality (1-1/","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":", let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrightarrow e_j. Add a zero-cost target node t with the certified justif","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2ed59d2f2951317a","tex":"e_C","aliases":"e_C","display":"e_C","kind":"use","excerpt":"[Lossless encoding at the typed syntax boundary] Let e_P,d_P and e_C,d_C be the stated Lean encoders and partial decoders for problem and certificate records. For every typed problem P and certificate C, d_P(e_P(P))=some(P), d_C(e_C(C))=some(C). Consequently both encoders are injective on their typed domains. These identities do not require the enc","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","excerpt":", let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrightarrow e_j. Add a zero-cost target node t with the certified justif","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0a21ab7396f4979e","tex":"e_P","aliases":"e_P","display":"e_P","kind":"use","excerpt":"[Lossless encoding at the typed syntax boundary] Let e_P,d_P and e_C,d_C be the stated Lean encoders and partial decoders for problem and certificate records. For every typed problem P and certificate C, d_P(e_P(P))=some(P), d_C(e_C(C))=some(C). Consequently both encoders are injective on their typed domains. These identities do not require the enc","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8614d1502b190274","tex":"\\mathrm{err}","aliases":"err mathrm","display":"\\mathrm{err}","kind":"use","excerpt":"A simple audit priority score is P(x) = fracp_err(x) c_consequence(x) (1+r_dependency(x)) c_audit(x). Here p_err(x)∈[0,1], the consequence and dependency factors are nonnegative, and c_audit(x)>0. Zero-cost checks can be handled separately. This is a declared priority heuristic, not a probability of truth or a proved optimal scheduling rule.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-69d9c11ce259a18b","tex":"\\operatorname{Ext}_G","aliases":"Ext_G operatorname","display":"\\operatorname{Ext}_G","kind":"use","excerpt":"[Compositional rule transport] Rule transports admit identities and associative composition. A transport (f,h):F→ G sends every derivation of v in F to a derivation of f(v) in G. Every G-model pulls back along f to an F-model. In particular, Y∈Ext_G(f(S),f(N)) Longrightarrow f^-1(Y)∈Ext_F(S,N). An obstruction in the source extension problem therefo","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"A second connection is quantitative. Work over F_2 with E= F_2^n. Let A:E→ F_2^m impose coherence, P:E→ F_2^p retain prescribed information, and x∈ E be a raw candidate. Define K_P(x)={y∈ E:Ay=0, Py=Px}, M=A","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-49ddaa56c96d4e05","tex":"f^{-1}","aliases":"f^-1","display":"f^{-1}","kind":"use","excerpt":"[Compositional rule transport] Rule transports admit identities and associative composition. A transport (f,h):F→ G sends every derivation of v in F to a derivation of f(v) in G. Every G-model pulls back along f to an F-model. In particular, Y∈Ext_G(f(S),f(N)) Longrightarrow f^-1(Y)∈Ext_F(S,N). An obstruction in the source extension problem therefo","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"[Functorial transport] Let F:Cert_1→Cert_2 be a functor. If Axrightarrowf_1S_1xrightarrowf_2⋯xrightarrowf_rB is a certified path in Cert_1, then F(A)xrightarrowF(f_1)F(S_1)xrightarrowF(f_2)⋯xrightarrowF(f_r)F(B) is a certified path in Cert_2.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"A second connection is quantitative. Work over F_2 with E= F_2^n. Let A:E→ F_2^m impose coherence, P:E→ F_2^p retain prescribed information, and x∈ E be a raw candidate. Define K_P(x)={y∈ E:Ay=0, Py=Px}, M=A","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","excerpt":"[Functorial transport] Let F:Cert_1→Cert_2 be a functor. If Axrightarrowf_1S_1xrightarrowf_2⋯xrightarrowf_rB is a certified path in Cert_1, then F(A)xrightarrowF(f_1)F(S_1)xrightarrowF(f_2)⋯xrightarrowF(f_r)F(B) is a certified path in Cert_2.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2c82feb782bf1c96","tex":"\\mathbb F_2^3","aliases":"F_2^3 mathbb","display":"\\mathbb F_2^3","kind":"use","excerpt":"[One informative question] Let y=(a,b,c)∈ F_2^3 with a+b+c=0 and protected value a=0. The realization fiber is {(0,0,0),(0,1,1)}. It has one bit of ambiguity. Observing b resolves that bit exactly. Observing a supplies no new information, despite being a perfectly valid measurement. This difference is measured by the rank of the observation restric","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-474123ea694580b8","tex":"\\mathbb F_2^m","aliases":"F_2^m mathbb","display":"\\mathbb F_2^m","kind":"use","excerpt":"A second connection is quantitative. Work over F_2 with E= F_2^n. Let A:E→ F_2^m impose coherence, P:E→ F_2^p retain prescribed information, and x∈ E be a raw candidate. Define K_P(x)={y∈ E:Ay=0, Py=Px}, M=A","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b14742c2da57452f","tex":"\\mathbb F_2^n","aliases":"F_2^n mathbb","display":"\\mathbb F_2^n","kind":"use","excerpt":"A second connection is quantitative. Work over F_2 with E= F_2^n. Let A:E→ F_2^m impose coherence, P:E→ F_2^p retain prescribed information, and x∈ E be a raw candidate. Define K_P(x)={y∈ E:Ay=0, Py=Px}, M=A","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-9689c8f6b2264b39","tex":"\\mathbb F_2^p","aliases":"F_2^p mathbb","display":"\\mathbb F_2^p","kind":"use","excerpt":"A second connection is quantitative. Work over F_2 with E= F_2^n. Let A:E→ F_2^m impose coherence, P:E→ F_2^p retain prescribed information, and x∈ E be a raw candidate. Define K_P(x)={y∈ E:Ay=0, Py=Px}, M=A","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-cf988527cae34c53","tex":"\\mathbb F_2^q","aliases":"F_2^q mathbb","display":"\\mathbb F_2^q","kind":"use","excerpt":"Suppose the realization fiber is nonempty and a new linear observation Q:E→ F_2^q becomes available. The right question is not how many bits appear in the output of Q but how many remaining degrees of freedom in the fiber it distinguishes.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f7499e87191c71a6","tex":"f_r","aliases":"f_r","display":"f_r","kind":"use","excerpt":"[Functorial transport] Let F:Cert_1→Cert_2 be a functor. If Axrightarrowf_1S_1xrightarrowf_2⋯xrightarrowf_rB is a certified path in Cert_1, then F(A)xrightarrowF(f_1)F(S_1)xrightarrowF(f_2)⋯xrightarrowF(f_r)F(B) is a certified path in Cert_2.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b0c60609b8685950","tex":"\\mathrm{feasible}","aliases":"feasible mathrm","display":"\\mathrm{feasible}","kind":"use","excerpt":"erdict The complete result format contains a Boolean feasibility flag, a reported closure, a strict closure trace, and an optional forbidden-node witness. Both branches require the complete-closure check. A positive result additionally requires no obstruction witness and no forbidden node in the closure. A negative result requires an accepted obstr","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0cf904dac8f0a2a4","tex":"\\mathrm{formal}","aliases":"formal mathrm","display":"\\mathrm{formal}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G,","count":18},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-25f0be3ddda5cf8c","tex":"G^*","aliases":"G^*","display":"G^*","kind":"use","excerpt":"Write G_i=U_M(S_i)-U_M(∅) and G^*=U_M(O_k)-U_M(∅). By monotonicity and submodularity, the total marginal contribution of the at most k elements of O_k beyond S_i is at least G^*-G_i. Hence at least one candidate has marginal gain at least (G^*-G_i)/k. Greedy chooses a candidate with at least that gain, so G^*-G_i+1 ≤ (1-frac1k)(G^*-G_i). Iterating ","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-575d86b85365c7b4","tex":"G_0","aliases":"G_0","display":"G_0","kind":"use","excerpt":"_M(S_i)-U_M(∅) and G^*=U_M(O_k)-U_M(∅). By monotonicity and submodularity, the total marginal contribution of the at most k elements of O_k beyond S_i is at least G^*-G_i. Hence at least one candidate has marginal gain at least (G^*-G_i)/k. Greedy chooses a candidate with at least that gain, so G^*-G_i+1 ≤ (1-frac1k)(G^*-G_i). Iterating from G_0=0 ","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-cd7038b72872679a","tex":"G_i","aliases":"G_i","display":"G_i","kind":"use","excerpt":"Write G_i=U_M(S_i)-U_M(∅) and G^*=U_M(O_k)-U_M(∅). By monotonicity and submodularity, the total marginal contribution of the at most k elements of O_k beyond S_i is at least G^*-G_i. Hence at least one candidate has marginal gain at least (G^*-G_i)/k. Greedy chooses a candidate with at least that gain, so G^*-G_i+1 ≤ (1-frac1k)(G^*-G_i). Iterating ","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-78a1c74c44055d05","tex":"G_{i+1}","aliases":"G_i+1","display":"G_{i+1}","kind":"use","excerpt":"Write G_i=U_M(S_i)-U_M(∅) and G^*=U_M(O_k)-U_M(∅). By monotonicity and submodularity, the total marginal contribution of the at most k elements of O_k beyond S_i is at least G^*-G_i. Hence at least one candidate has marginal gain at least (G^*-G_i)/k. Greedy chooses a candidate with at least that gain, so G^*-G_i+1 ≤ (1-frac1k)(G^*-G_i). Iterating ","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-103f7de3482048f3","tex":"G_k","aliases":"G_k","display":"G_k","kind":"use","excerpt":"rnothing) and G^*=U_M(O_k)-U_M(∅). By monotonicity and submodularity, the total marginal contribution of the at most k elements of O_k beyond S_i is at least G^*-G_i. Hence at least one candidate has marginal gain at least (G^*-G_i)/k. Greedy chooses a candidate with at least that gain, so G^*-G_i+1 ≤ (1-frac1k)(G^*-G_i). Iterating from G_0=0 gives","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"[A linear certificate alternative] The protected problem is infeasible if and only if there is a linear functional λ:R→ K satisfying λ(Dh)=0 whenever p(h)=0, λ(Dx)=1. For a finite matrix M and right side b, the corresponding alternative is nexists y (My=b) Longleftrightarrow ∃ℓ (ℓ M=0 and ℓ b=1). If Dx∉ B, linear separation of a vector from a subsp","count":19},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Let (U_i)_i∈ I cover the nodes. A family X_i∈ M_F(U_i) is matching when X_i∩ U_j=X_j∩ U_i for every i,j. This equality includes agreement about false values, not merely common true values.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0f6f9e87439c63ef","tex":"\\mathrm{id}","aliases":"id mathrm","display":"\\mathrm{id}","kind":"use","excerpt":"[The proof-field view laws] The restriction maps satisfy ρ_U,U=id, ρ_U,Wρ_W,Z=ρ_U,Z (U⊆ W⊆ Z). Consequently U↦ M_F(U) is a contravariant set-valued view system on the inclusion order of patches. Both laws are the corresponding intersection identities. For the second, (X∩ W)∩ U=X∩ U because U⊆ W. The values outside a patch are absent, or equivalentl","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-fe61bfb695992cf7","tex":"J_F","aliases":"J_F","display":"J_F","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G,","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5d8b19f19485f3d6","tex":"J_G","aliases":"J_G","display":"J_G","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G,","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-11d88ad682117768","tex":"J_i","aliases":"J_i","display":"J_i","kind":"use","excerpt":"[MSE with a shared interpretive offset] Suppose reviewers report J_i=θ+b+ε_i, where θ is the target quantity, b is a fixed shared offset, Eε_i=0, Var(ε_i)=σ^2, and Cov(ε_i,ε_j)=ρσ^2 for i≠ j. For admissible ρ, the mean squared error of the reviewer average is E(overline J_n-θ)^2=b^2+σ^2(ρ+frac1-ρn). For fixed ρ≥0, it approaches b^2+ρσ^2 as n→∞. The","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8c6a42cd426527ba","tex":"\\mathrm{journal}","aliases":"journal mathrm","display":"\\mathrm{journal}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-784c9007afca3979","tex":"\\mathfrak K","aliases":"K mathfrak","display":"\\mathfrak K","kind":"use","excerpt":"A second connection is quantitative. Work over F_2 with E= F_2^n. Let A:E→ F_2^m impose coherence, P:E→ F_2^p retain prescribed information, and x∈ E be a raw candidate. Define K_P(x)={y∈ E:Ay=0, Py=Px}, M=A P, b=0 Px. This is a finite linear realization fiber. The equation My=b does not by itself produce a solution. Compatibility of the prescribed","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-fb9dab5747004561","tex":"K'","aliases":"K'","display":"K'","kind":"use","excerpt":"[Replay preserves derivability and composes] Suppose every member of K belongs to cl_F(S). If either replay mode accepts a trace starting from K and returns K', then K⊆ K'⊆cl_F(S). For a fixed replay mode, replaying σmathbin+ +τ is the same as replaying σ and then replaying τ from the returned state. Failure at either stage propagates to the concat","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-cf5fcf757b255a84","tex":"K_1","aliases":"K_1","display":"K_1","kind":"use","excerpt":"[Debt is monotone under retention] If K_1⊆ K_2, then Debt(K_2)≤Debt(K_1).","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-173833828c3529a3","tex":"K_2","aliases":"K_2","display":"K_2","kind":"use","excerpt":"[Debt is monotone under retention] If K_1⊆ K_2, then Debt(K_2)≤Debt(K_1).","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-06d9aa0dcceaeb39","tex":"K_{\\mathrm{core}}","aliases":"K_core mathrm","display":"K_{\\mathrm{core}}","kind":"use","excerpt":"[Unique verification core] Let P be a finite acyclic AND-DAG relative to T_0. Put K_core=A∪ T_0∪Anc_L(T_0). Then (K_core,H_core) is a certificate and is the least certificate under componentwise inclusion. In particular it is the unique inclusion-minimal certificate pair for this recorded inference problem.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-569ea3429aa54d9c","tex":"\\mathfrak K_P","aliases":"K_P mathfrak","display":"\\mathfrak K_P","kind":"use","excerpt":"A second connection is quantitative. Work over F_2 with E= F_2^n. Let A:E→ F_2^m impose coherence, P:E→ F_2^p retain prescribed information, and x∈ E be a raw candidate. Define K_P(x)={y∈ E:Ay=0, Py=Px}, M=A P, b=0 Px. This is a finite linear realization fiber. The equation My=b does not by itself produce a solution. Compatibility of the prescribed","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-9a30d27faec7f7f7","tex":"K_r","aliases":"K_r","display":"K_r","kind":"use","excerpt":"[Irreversible compression accumulates debt] For a descending sequence of retained sets K_0⊇ K_1⊇⋯⊇ K_r, the expected debt is nondecreasing.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"The final thesis is therefore: 0.88linewidth A proof should be published as a certificate and maintained as a versioned proof field. The certificate preserves deductive validity. The larger field preserves the dependencies, alternatives, examples, failures, provenance, and bridges from which later mathematics may grow.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]fig_02_and_dag_core.pdf captionThe canonical pathand_dag.json example. The AND gate requires both A and D for L_1. Its least certificate contains five nodes and four certified rules. The gold branch is certified but unnecessary for B. The dashed route to F is uncertified and cannot enter the certific","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]fig_09_version_invalidation.pdf captionChanging L_1 marks L_1, L_2, and B for syntactic review. The lower branch has no forward dependency path from L_1. Membership in the review set is not a claim that a theorem has become false.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8285afe9914e471a","tex":"L_F","aliases":"L_F","display":"L_F","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G, j certified in F · Longrightarrow h(j) certified in G, A^G_h(j) · ⊆ f(A^F_j), c^G_h(j) · =f(c^F_j). The premise inclusion points in the di","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-9ac358794339c560","tex":"L_G","aliases":"L_G","display":"L_G","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G, j certified in F · Longrightarrow h(j) certified in G, A^G_h(j) · ⊆ f(A^F_j), c^G_h(j) · =f(c^F_j). The premise inclusion points in the di","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c469daa9fa245a6b","tex":"L_{\\mathrm{short}}","aliases":"L_short mathrm","display":"L_{\\mathrm{short}}","kind":"use","excerpt":"The field examples/alternative_proofs.json contains a short route and a richer explanatory route. The short route has storage cost 1, while the richer route has storage cost 3 and larger meaning and discovery values. Exact certificate search chooses A⟶ L_short⟶ B when the objective is storage cost alone. A teaching compiler may rationally choose th","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-45e90aa85790529d","tex":"\\linewidth","aliases":"linewidth","display":null,"kind":"use","excerpt":"The final thesis is therefore: 0.88linewidth A proof should be published as a certificate and maintained as a versioned proof field. The certificate preserves deductive validity. The larger field preserves the dependencies, alternatives, examples, failures, provenance, and bridges from which later mathematics may grow.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"s. The image Q(H) is a vector subspace of dimension r=rank(Q|_H), so the observed values form the affine set Qy_0+Q(H) of size 2^r. Every such value has a preimage that is a coset of H∩ker Q, whose dimension is dim H-r by rank nullity. Equal fiber cardinalities make the induced output distribution uniform on 2^r values. Its entropy is log_2(2^r)=r,","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"s. The image Q(H) is a vector subspace of dimension r=rank(Q|_H), so the observed values form the affine set Qy_0+Q(H) of size 2^r. Every such value has a preimage that is a coset of H∩ker Q, whose dimension is dim H-r by rank nullity. Equal fiber cardinalities make the induced output distribution uniform on 2^r values. Its entropy is log_2(2^r)=r,","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Soundness and relative completeness] For every node v, v∈Der_F Longleftrightarrow v∈ X for every model X of the declared field. If v is not derivable, Der_F itself is a model omitting v. Every model contains the leaves. Induction on a derivation shows that it contains the conclusion of every certified application whose premises have already been d","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":8},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"Let a_t≥0 be candidate arrivals during audit cycle t, let s_t≥0 be service capacity, and let B_t≥0 be the unresolved backlog. Suppose arrivals in a cycle are available to that cycle's service. The basic deterministic queue is B_t+1=max{0,B_t+a_t-s_t}.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-01b708f84d04625e","tex":"\\operatorname{Models}","aliases":"Models operatorname","display":"\\operatorname{Models}","kind":"use","excerpt":"The strict Komorebi realization is now an actual set of coherent local models, Γ( M_F)= {(X_U)_U:ρ_U,W(X_W)=X_U whenever U⊆ W}. There is an explicit bijection Models(F)≃Γ( M_F), X⟼(X∩ U)_U. Its inverse evaluates a coherent family at the full patch V. Coherence shows that every other component is precisely the restriction of that value. The Lean dec","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3e0534ff9dda7eeb","tex":"N'","aliases":"N'","display":"N'","kind":"use","excerpt":"[Staged closure and persistent obstructions] For positive sets S,T, cl_F(cl_F(S)∪ T) =cl_F(S∪ T). If S⊆ S' and N⊆ N', then Ext_F(S',N')⊆Ext_F(S,N). Consequently an infeasible request stays infeasible under those additions, while the field and the meaning of its rules remain fixed. Closure is extensive, monotone, and idempotent because it is the lea","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-213d86a5e6b34393","tex":"N^{-1}","aliases":"N^-1","display":"N^{-1}","kind":"use","excerpt":"luations of reviewer i under two labels. In a finite set of N reviewers, assign exactly n_1 to the first label and n_0=N-n_1 to the second, uniformly over assignments with those sizes, where n_0,n_1>0. Assume each observed response equals the potential response under its assigned label and does not depend on other assignments. Then the difference o","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","excerpt":"A claim about identity effects needs a design that can distinguish them from other differences. Hold the mathematical content fixed and let Y_i(1) and Y_i(0) be the fixed potential evaluations of reviewer i under two labels. In a finite set of N reviewers, assign exactly n_1 to the first label and n_0=N-n_1 to the second, uniformly over assignments","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"A claim about identity effects needs a design that can distinguish them from other differences. Hold the mathematical content fixed and let Y_i(1) and Y_i(0) be the fixed potential evaluations of reviewer i under two labels. In a finite set of N reviewers, assign exactly n_1 to the first label and n_0=N-n_1 to the second, uniformly over assignments","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-09061c29f4c104f3","tex":"\\nexists","aliases":"nexists","display":null,"kind":"use","excerpt":"[A linear certificate alternative] The protected problem is infeasible if and only if there is a linear functional λ:R→ K satisfying λ(Dh)=0 whenever p(h)=0, λ(Dx)=1. For a finite matrix M and right side b, the corresponding alternative is nexists y (My=b) Longleftrightarrow ∃ℓ (ℓ M=0 and ℓ b=1). If Dx∉ B, linear separation of a vector from a subsp","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[Greedy cardinality guarantee] Let k be an integer with 1≤ k≤|C|. Starting from the empty supplement, let S_k result from repeatedly adding an unselected candidate with largest marginal future reach. Let O_k be an optimal supplement of size at most k. Then U_M(S_k)-U_M(∅) ≥ (1-(1-frac1k)^k) (U_M(O_k)-U_M(∅)), and therefore the greedy gain is at lea","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"The final thesis is therefore: 0.88linewidth A proof should be published as a certificate and maintained as a versioned proof field. The certificate preserves deductive validity. The larger field preserves the dependencies, alternatives, examples, failures, provenance, and bridges from which later mathematics may grow.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5fc7b54723fc93fa","tex":"O_k","aliases":"O_k","display":"O_k","kind":"use","excerpt":"[Greedy cardinality guarantee] Let k be an integer with 1≤ k≤|C|. Starting from the empty supplement, let S_k result from repeatedly adding an unselected candidate with largest marginal future reach. Let O_k be an optimal supplement of size at most k. Then U_M(S_k)-U_M(∅) ≥ (1-(1-frac1k)^k) (U_M(O_k)-U_M(∅)), and therefore the greedy gain is at lea","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-712a4047f09e3ce3","tex":"\\overline J","aliases":"overline J overline","display":"\\overline J","kind":"use","excerpt":"shared interpretive offset] Suppose reviewers report J_i=θ+b+ε_i, where θ is the target quantity, b is a fixed shared offset, Eε_i=0, Var(ε_i)=σ^2, and Cov(ε_i,ε_j)=ρσ^2 for i≠ j. For admissible ρ, the mean squared error of the reviewer average is E(overline J_n-θ)^2=b^2+σ^2(ρ+frac1-ρn). For fixed ρ≥0, it approaches b^2+ρσ^2 as n→∞. The estimation ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5ee13f59c11b39a6","tex":"\\overline J_n","aliases":"overline J_n overline","display":"\\overline J_n","kind":"use","excerpt":"shared interpretive offset] Suppose reviewers report J_i=θ+b+ε_i, where θ is the target quantity, b is a fixed shared offset, Eε_i=0, Var(ε_i)=σ^2, and Cov(ε_i,ε_j)=ρσ^2 for i≠ j. For admissible ρ, the mean squared error of the reviewer average is E(overline J_n-θ)^2=b^2+σ^2(ρ+frac1-ρn). For fixed ρ≥0, it approaches b^2+ρσ^2 as n→∞. The estimation ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-823dc5811670256d","tex":"\\overline\\varepsilon","aliases":"overlineε overline varepsilon","display":"\\overline\\varepsilon","kind":"use","excerpt":"psilon_i)=σ^2, and Cov(ε_i,ε_j)=ρσ^2 for i≠ j. For admissible ρ, the mean squared error of the reviewer average is E(overline J_n-θ)^2=b^2+σ^2(ρ+frac1-ρn). For fixed ρ≥0, it approaches b^2+ρσ^2 as n→∞. The estimation error is overline J_n-θ=b+overlineε_n. Its mean is b, and its variance is σ^2((1-ρ)/n+ρ) by summing all diagonal and off-diagonal cov","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-90ef009be2250bec","tex":"\\overline\\varepsilon_n","aliases":"overlineε_n overline varepsilon","display":"\\overline\\varepsilon_n","kind":"use","excerpt":"psilon_i)=σ^2, and Cov(ε_i,ε_j)=ρσ^2 for i≠ j. For admissible ρ, the mean squared error of the reviewer average is E(overline J_n-θ)^2=b^2+σ^2(ρ+frac1-ρn). For fixed ρ≥0, it approaches b^2+ρσ^2 as n→∞. The estimation error is overline J_n-θ=b+overlineε_n. Its mean is b, and its variance is σ^2((1-ρ)/n+ρ) by summing all diagonal and off-diagonal cov","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-7628f50fe9db4acf","tex":"p_{\\mathrm{err}}","aliases":"p_err mathrm","display":"p_{\\mathrm{err}}","kind":"use","excerpt":"A simple audit priority score is P(x) = fracp_err(x) c_consequence(x) (1+r_dependency(x)) c_audit(x). Here p_err(x)∈[0,1], the consequence and dependency factors are nonnegative, and c_audit(x)>0. Zero-cost checks can be handled separately. This is a declared priority heuristic, not a probability of truth or a proved optimal scheduling rule.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-be70654fd7aa8ba5","tex":"P_{K\\cap L}","aliases":"P_K∩ L cap","display":"P_{K\\cap L}","kind":"use","excerpt":"[Retention projection algebra] For masks K,L⊆ V, P_K∘ P_L=P_K∩ L=P_L∘ P_K. Consequently every retention projection is idempotent, and the masks form a Boolean lattice with meet K∩ L and join K∪ L.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-363d67fd271736f2","tex":"P_L","aliases":"P_L","display":"P_L","kind":"use","excerpt":"[Retention projection algebra] For masks K,L⊆ V, P_K∘ P_L=P_K∩ L=P_L∘ P_K. Consequently every retention projection is idempotent, and the masks form a Boolean lattice with meet K∩ L and join K∪ L.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-25fa169a561d36f9","tex":"\\Pcal_{0y}","aliases":"Pcal_0y P_0y Pcal mathcal","display":"{\\mathcal P}_{0y}","kind":"use","excerpt":"Let the archives be P_xy=(C_0,x,y) for (x,y)∈{0,1}^2, where C_0 is a fixed certificate retained by every allowed mask. Let q_x( P_xy)=x and q_y( P_xy)=y. Retaining x is sufficient for q_x. Any mask omitting x identifies P_0y with P_1y, so it is not sufficient for that task. Thus the least mask for {q_x} retains exactly C_0 and x. The same argument ","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-586cf2a14d5a0b0e","tex":"\\Pcal_{1y}","aliases":"Pcal_1y P_1y Pcal mathcal","display":"{\\mathcal P}_{1y}","kind":"use","excerpt":"Let the archives be P_xy=(C_0,x,y) for (x,y)∈{0,1}^2, where C_0 is a fixed certificate retained by every allowed mask. Let q_x( P_xy)=x and q_y( P_xy)=y. Retaining x is sufficient for q_x. Any mask omitting x identifies P_0y with P_1y, so it is not sufficient for that task. Thus the least mask for {q_x} retains exactly C_0 and x. The same argument ","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8e9e0086ae040eeb","tex":"\\Pcal_2","aliases":"Pcal_2 P_2 Pcal mathcal","display":"{\\mathcal P}_2","kind":"use","excerpt":"[Task information loss criterion] Let C be a compression map and let q be a task. If there exist proof fields P_1 and P_2 such that C( P_1)=C( P_2) but q( P_1)≠ q( P_2), then the compressed representation is not sufficient for q.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1279fa507e6ffd98","tex":"\\Pcal_{xy}","aliases":"Pcal_xy P_xy Pcal mathcal","display":"{\\mathcal P}_{xy}","kind":"use","excerpt":"Let the archives be P_xy=(C_0,x,y) for (x,y)∈{0,1}^2, where C_0 is a fixed certificate retained by every allowed mask. Let q_x( P_xy)=x and q_y( P_xy)=y. Retaining x is sufficient for q_x. Any mask omitting x identifies P_0y with P_1y, so it is not sufficient for that task. Thus the least mask for {q_x} retains exactly C_0 and x. The same argument ","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a6123ae670e2c050","tex":"\\mathrm{Prop}","aliases":"Prop mathrm","display":"\\mathrm{Prop}","kind":"use","excerpt":"For an intended interpretation μ:V→Prop, the transfer to mathematical truth has explicit hypotheses: every leaf must satisfy μ, and every certified rule must preserve μ from all its premises to its conclusion. Under those hypotheses, induction gives v∈Der_F⇒μ(v). Lean checks this statement as pathProofPath.Field.semantic_soundness. It does not infe","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-ee5559b3980fa51c","tex":"q^d","aliases":"q^d","display":"q^d","kind":"use","excerpt":"] In particular the exact count identity is | A|= |{y∈ A:Qy=Qx_0}| q^r. Every attained observation fiber has the same size. For the uniform distribution on A, the observation removes exactly rlog_2q bits of uncertainty. The two translation equivalences reduce the counts to finite vector spaces. Such a space of dimension d has q^d elements. Rank-nul","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-6386a60ab12436a6","tex":"q^k","aliases":"q^k","display":"q^k","kind":"use","excerpt":"[Exact observation counts] Suppose K is a finite field of cardinality q and E is a finite-dimensional K-vector space. Set k=dim_K H and r=dim_K Q(H). Then | A|=q^k, |Q( A)|=q^r, |{y∈ A:Qy=Qx_0}|=q^k-r. In particular the exact count identity is | A|= |{y∈ A:Qy=Qx_0}| q^r. Every attained observation fiber has the same size. For the uniform distributi","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-da98c9120853b94e","tex":"q^{k-r}","aliases":"q^k-r","display":"q^{k-r}","kind":"use","excerpt":"[Exact observation counts] Suppose K is a finite field of cardinality q and E is a finite-dimensional K-vector space. Set k=dim_K H and r=dim_K Q(H). Then | A|=q^k, |Q( A)|=q^r, |{y∈ A:Qy=Qx_0}|=q^k-r. In particular the exact count identity is | A|= |{y∈ A:Qy=Qx_0}| q^r. Every attained observation fiber has the same size. For the uniform distributi","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-50cffe607a1e0f78","tex":"q^r","aliases":"q^r","display":"q^r","kind":"use","excerpt":"[Exact observation counts] Suppose K is a finite field of cardinality q and E is a finite-dimensional K-vector space. Set k=dim_K H and r=dim_K Q(H). Then | A|=q^k, |Q( A)|=q^r, |{y∈ A:Qy=Qx_0}|=q^k-r. In particular the exact count identity is | A|= |{y∈ A:Qy=Qx_0}| q^r. Every attained observation fiber has the same size. For the uniform distributi","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4ac09894acaf58fd","tex":"Q_H","aliases":"Q_H","display":"Q_H","kind":"use","excerpt":"Let T:E→ R and Q:E→ W be linear, and fix a known point x_0. Put A={y:Ty=Tx_0}, H=ker T, Q_H=Q|_H. Translation h↦ x_0+h identifies H with A. The observation image is the translate Qx_0+Q(H), and the fiber above the attained value Qx_0 is a translate of ker Q_H.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-dee0ad4feb2099c6","tex":"q_x","aliases":"q_x","display":"q_x","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]fig_05_recoverability_preorder.pdf captionA concrete retention diamond for the four archives (C_0,x,y) with x,y∈{0,1}. Arrows discard coordinates. The two middle schemes are incomparable for the task family {q_x,q_y}. This is not a universal ranking of teaching, discovery, and formal presentations.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-25bc38ad67c92bd8","tex":"q_y","aliases":"q_y","display":"q_y","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]fig_05_recoverability_preorder.pdf captionA concrete retention diamond for the four archives (C_0,x,y) with x,y∈{0,1}. Arrows discard coordinates. The two middle schemes are incomparable for the task family {q_x,q_y}. This is not a universal ranking of teaching, discovery, and formal presentations.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"[Attained factorization and unique decoding] For C:mathscr A→ R and q:mathscr A→ Y, the following are equivalent: · ∃ d:imC→ Y ∀ a∈mathscr A, q(a)=d(C(a)),","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"If s(r)=s(r'), applying the function h gives equal digests. If s(r)≠ s(r') but h(s(r))=h(s(r')), the two serialized strings form a hash collision by definition. Thus change detection at the record level requires both preservation of the audited distinction by s and exclusion of a collision for the resulting bytes. No injectivity of a fixed-length h","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"Take V_4={0,1,2,3}, L={0}, S=∅, and the two rules r_0:0→1 and r_1:1→2. The least closure is {0,1,2}. Node 3 has no derivation from these inputs.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"Take V_4={0,1,2,3}, L={0}, S=∅, and the two rules r_0:0→1 and r_1:1→2. The least closure is {0,1,2}. Node 3 has no derivation from these inputs.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d0aa3589089d58ba","tex":"R_3","aliases":"R_3","display":"R_3","kind":"use","excerpt":"Every representation supports every task it supports, so the relation is reflexive. If every task supported by R_1 is supported by R_2, and every task supported by R_2 is supported by R_3, then every task supported by R_1 is supported by R_3. Thus the relation is transitive.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2100d3e5c1ac9992","tex":"R_{\\mathrm{archive}}","aliases":"R_archive mathrm","display":"R_{\\mathrm{archive}}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b274d55f5ce565c0","tex":"R_{\\mathrm{cert}}","aliases":"R_cert mathrm","display":"R_{\\mathrm{cert}}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-71e0f18ed7179e19","tex":"R_d","aliases":"R_d","display":"R_d","kind":"use","excerpt":"A refinement preserves the certified target on its declared scope. Preservation of other tasks requires a separate recovery condition. Let R_d:mathscr A→ R_d be the representation scheme at depth d. A sufficient condition is a map π_d:R_d+1(mathscr A)⟶ R_d(mathscr A) with R_d=π_d∘ R_d+1. If a task factors as q=d_q∘ R_d, then q=(d_q∘π_d)∘ R_d+1. Hen","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-945c6e36c79bdd82","tex":"\\mathcal R_d","aliases":"R_d mathcal","display":"\\mathcal R_d","kind":"use","excerpt":"A refinement preserves the certified target on its declared scope. Preservation of other tasks requires a separate recovery condition. Let R_d:mathscr A→ R_d be the representation scheme at depth d. A sufficient condition is a map π_d:R_d+1(mathscr A)⟶ R_d(mathscr A) with R_d=π_d∘ R_d+1. If a task factors as q=d_q∘ R_d, then q=(d_q∘π_d)∘ R_d+1. Hen","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5a7ee9546a2ddb0b","tex":"R_{d+1}","aliases":"R_d+1","display":"R_{d+1}","kind":"use","excerpt":"A refinement preserves the certified target on its declared scope. Preservation of other tasks requires a separate recovery condition. Let R_d:mathscr A→ R_d be the representation scheme at depth d. A sufficient condition is a map π_d:R_d+1(mathscr A)⟶ R_d(mathscr A) with R_d=π_d∘ R_d+1. If a task factors as q=d_q∘ R_d, then q=(d_q∘π_d)∘ R_d+1. Hen","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b20c59a53ec5497e","tex":"r_{d-1}","aliases":"r_d-1","display":"r_{d-1}","kind":"use","excerpt":"A practical stopping rule selects the first depth satisfying r_d≤ε or fracr_d-1-r_dc_d-c_d-1<τ. The first condition uses a declared tolerance. The second applies only at d≥1 with c_d>c_d-1 and compares the most recent reduction with its extra cost. Zero-cost refinements require separate handling. These stopping rules do not generally minimize J_g o","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-fe38754a4b95fa4e","tex":"r_{d_1}","aliases":"r_d_1","display":"r_{d_1}","kind":"use","excerpt":"Take a proof object with exactly two available depths d_1<d_2, with r_d_2<r_d_1 and c_d_2>c_d_1. Direct subtraction shows that d_2 is preferred exactly when α_g(r_d_1-r_d_2)>β_g(c_d_2-c_d_1). The goal (α_g,β_g)=(1,0) strictly prefers d_2, whereas (0,1) strictly prefers d_1. Since these are the only available depths, each preference is a global opti","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-65f1b0c19ea86ff4","tex":"r_{d_2}","aliases":"r_d_2","display":"r_{d_2}","kind":"use","excerpt":"Take a proof object with exactly two available depths d_1<d_2, with r_d_2<r_d_1 and c_d_2>c_d_1. Direct subtraction shows that d_2 is preferred exactly when α_g(r_d_1-r_d_2)>β_g(c_d_2-c_d_1). The goal (α_g,β_g)=(1,0) strictly prefers d_2, whereas (0,1) strictly prefers d_1. Since these are the only available depths, each preference is a global opti","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4c34b821a5d03dfd","tex":"r_{\\mathrm{dependency}}","aliases":"r_dependency mathrm","display":"r_{\\mathrm{dependency}}","kind":"use","excerpt":"A simple audit priority score is P(x) = fracp_err(x) c_consequence(x) (1+r_dependency(x)) c_audit(x). Here p_err(x)∈[0,1], the consequence and dependency factors are nonnegative, and c_audit(x)>0. Zero-cost checks can be handled separately. This is a declared priority heuristic, not a probability of truth or a proved optimal scheduling rule.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1110ca5f0f1eda25","tex":"R_{\\mathrm{discovery}}","aliases":"R_discovery mathrm","display":"R_{\\mathrm{discovery}}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-502ea9eb8c38047a","tex":"R_{\\mathrm{formal}}","aliases":"R_formal mathrm","display":"R_{\\mathrm{formal}}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"use","excerpt":"[Portfolio robustness] For representation schemes C_i:mathscr A→ R_i, retain the tagged portfolio C( P)=(C_i( P))_i. Every task recoverable from any one C_i is recoverable from C.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3be9fba860ec2b6a","tex":"R_{\\mathrm{journal}}","aliases":"R_journal mathrm","display":"R_{\\mathrm{journal}}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-24f6c46c8b72a427","tex":"R_M","aliases":"R_M","display":"R_M","kind":"use","excerpt":"For each candidate x∈ C, let R_x⊆ F be the set of future targets reachable from M∪{x}. Let R_M be the targets already reachable from M. Reachability from a union of starting nodes is the union of their individual reach sets, so U_M(S)= ∑_f∈ R_M∪⋃_x∈ SR_xw_f. Adding a candidate cannot remove a reachable target, so U_M is monotone.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0e70fb35ad42d749","tex":"R_{\\mathrm{teach}}","aliases":"R_teach mathrm","display":"R_{\\mathrm{teach}}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3088a38f12fda26d","tex":"R_x","aliases":"R_x","display":"R_x","kind":"use","excerpt":"For each candidate x∈ C, let R_x⊆ F be the set of future targets reachable from M∪{x}. Let R_M be the targets already reachable from M. Reachability from a union of starting nodes is the union of their individual reach sets, so U_M(S)= ∑_f∈ R_M∪⋃_x∈ SR_xw_f. Adding a candidate cannot remove a reachable target, so U_M is monotone.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"[Finite realization alternative and ambiguity] Exactly one of the following holds: [label=textup(roman*)] item My=b has a solution y_0. Its complete solution set is y_0+H, where H=ker A∩ker P, and | K_P(x)|=2^n-rank(M). item A row vector λ exists with λ M=0 and λ b=1. It is a directly checkable certificate that K_P(x)=∅. If My_0=b, any other soluti","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3bce33d4a711b894","tex":"\\Rcal","aliases":"Rcal R mathcal","display":"{\\mathcal R}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"[Staged closure and persistent obstructions] For positive sets S,T, cl_F(cl_F(S)∪ T) =cl_F(S∪ T). If S⊆ S' and N⊆ N', then Ext_F(S',N')⊆Ext_F(S,N). Consequently an infeasible request stays infeasible under those additions, while the field and the meaning of its rules remain fixed. Closure is extensive, monotone, and idempotent because it is the lea","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]fig_04_certificate_complexity.pdf captionThe Set Cover example as a membership matrix. Filled marks belong to the selected rows S_1 and S_3; open marks show unselected memberships. Every obligation is covered. The target requires all four obligations through one AND rule. The exact minimum is two, al","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-698cd4222a287d75","tex":"S_i","aliases":"S_i","display":"S_i","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-bbf4aa0bb7699587","tex":"S_j","aliases":"S_j","display":"S_j","kind":"use","excerpt":"Fix a node set V, a set L⊆ V of admissible leaves, and a set J of justifications. A justification j has a premise set A_j, a conclusion c_j, and a certification predicate. Define its support by S_j=A_j∪{c_j}. A set X⊆ V is a model of the declared field when L⊆ X and, for every certified j, A_j⊆ X Longrightarrow c_j∈ X. Each justification is one imp","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-deef7bca5ee0eda2","tex":"S_k","aliases":"S_k","display":"S_k","kind":"use","excerpt":"[Greedy cardinality guarantee] Let k be an integer with 1≤ k≤|C|. Starting from the empty supplement, let S_k result from repeatedly adding an unselected candidate with largest marginal future reach. Let O_k be an optimal supplement of size at most k. Then U_M(S_k)-U_M(∅) ≥ (1-(1-frac1k)^k) (U_M(O_k)-U_M(∅)), and therefore the greedy gain is at lea","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8c0785670c74cfb3","tex":"S_n","aliases":"S_n","display":"S_n","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4bca7afcafa97a49","tex":"s_t","aliases":"s_t","display":"s_t","kind":"use","excerpt":"Let a_t≥0 be candidate arrivals during audit cycle t, let s_t≥0 be service capacity, and let B_t≥0 be the unresolved backlog. Suppose arrivals in a cycle are available to that cycle's service. The basic deterministic queue is B_t+1=max{0,B_t+a_t-s_t}.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1afafe3944ee8ba2","tex":"\\mathrm{short}","aliases":"short mathrm","display":"\\mathrm{short}","kind":"use","excerpt":"The field examples/alternative_proofs.json contains a short route and a richer explanatory route. The short route has storage cost 1, while the richer route has storage cost 3 and larger meaning and discovery values. Exact certificate search chooses A⟶ L_short⟶ B when the objective is storage cost alone. A teaching compiler may rationally choose th","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-6de6b55a41f53b2b","tex":"\\operatorname{some}","aliases":"some operatorname","display":"\\operatorname{some}","kind":"use","excerpt":"[Lossless encoding at the typed syntax boundary] Let e_P,d_P and e_C,d_C be the stated Lean encoders and partial decoders for problem and certificate records. For every typed problem P and certificate C, d_P(e_P(P))=some(P), d_C(e_C(C))=some(C). Consequently both encoders are injective on their typed domains. These identities do not require the enc","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-487c245905a46963","tex":"\\mathrm{teach}","aliases":"teach mathrm","display":"\\mathrm{teach}","kind":"use","excerpt":"The same theorem can support several legitimate representations: R(B)= {R_cert,R_journal,R_teach, R_formal,R_discovery,R_archive}. The certificate is optimized for checking. The journal form balances rigor and readable length. The teaching form expands conceptual bottlenecks. The formal form exposes primitive dependencies. The discovery form retain","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"For a general positive definite covariance matrix Σ, consider unbiased linear aggregates w^top X with 1^top w=1.","count":8},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-28ec3adb69bf092d","tex":"\\mathrm{true}","aliases":"true mathrm","display":"\\mathrm{true}","kind":"use","excerpt":"erdict The complete result format contains a Boolean feasibility flag, a reported closure, a strict closure trace, and an optional forbidden-node witness. Both branches require the complete-closure check. A positive result additionally requires no obstruction witness and no forbidden node in the closure. A negative result requires an accepted obstr","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","excerpt":"[Pairwise dependencies do not suffice] Take nodes a,b,c, no admissible leaves (L=∅), and one certified rule {a,b}⟶ c. Use patches U_0={a,c} and U_1={b,c}. The local valuations X_0={a} and X_1={b} agree on their overlap: both assign false to c. Both are local models, since neither patch contains the full rule support. They cover every node and both ","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"[Pairwise dependencies do not suffice] Take nodes a,b,c, no admissible leaves (L=∅), and one certified rule {a,b}⟶ c. Use patches U_0={a,c} and U_1={b,c}. The local valuations X_0={a} and X_1={b} agree on their overlap: both assign false to c. Both are local models, since neither patch contains the full rule support. They cover every node and both ","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d3ab10e5b4d5c86d","tex":"U_{\\Bcal\\setminus\\{v\\}}","aliases":"U_Bcal∖{v} U_ B∖{v} Bcal setminus mathcal setminus","display":"U_{{\\mathcal B}\\setminus\\{v\\}}","kind":"use","excerpt":"[Weighted criticality] For v∉ S, the utility lost by deleting v is the total weight of initially reachable future targets dominated by v: U(S)-U_ B∖{v}(S) =∑_substackf∈ F∩Reach_ B(S) v dominates fw_f. Deleted target vertices are counted as unreachable.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"[Finite arrow-cover gluing] Suppose patches U_i⊆ V cover every view and every declared edge has both endpoints in some U_i. Let s_i be coherent assignments on U_i that agree on all overlaps. There is a unique global coherent assignment s∈Γ(F) restricting to every s_i. For each view v, choose any patch containing it and define s(v) to be the value s","count":12},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","excerpt":"Let (U_i)_i∈ I cover the nodes. A family X_i∈ M_F(U_i) is matching when X_i∩ U_j=X_j∩ U_i for every i,j. This equality includes agreement about false values, not merely common true values.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8d86088ae482ad75","tex":"u_m","aliases":"u_m","display":"u_m","kind":"use","excerpt":"For NP-hardness, reduce Set Cover. Let U={u_1,…,u_m} be the universe, let S_1,…,S_n⊆ U be the available sets, and let k be the cover budget. Construct one optional admissible leaf node s_i for each set S_i, with unit cost. Construct one zero-cost obligation node e_j for each element u_j. Whenever u_j∈ S_i, add a certified justification {s_i}Longrig","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-641dfc399b115eba","tex":"V_4","aliases":"V_4","display":"V_4","kind":"use","excerpt":"Take V_4={0,1,2,3}, L={0}, S=∅, and the two rules r_0:0→1 and r_1:1→2. The least closure is {0,1,2}. Node 3 has no derivation from these inputs.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2fa3b86c4b95c686","tex":"V_F","aliases":"V_F","display":"V_F","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G,","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b80c8ac10f4b5873","tex":"V_G","aliases":"V_G","display":"V_G","kind":"use","excerpt":"A change of proof-field representation needs more than a map of node names. Let F and G be declared fields. A rule transport consists of a node map f:V_F→ V_G and a rule map h:J_F→ J_G satisfying f(L_F) · ⊆ L_G,","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3696096793ac9d06","tex":"\\operatorname{Var}","aliases":"Var operatorname","display":"\\operatorname{Var}","kind":"use","excerpt":"Let X_i=θ+ε_i be numerical estimates of a fixed quantity θ. Assume Eε_i=0, Var(ε_i)=σ^2>0, and common pairwise correlation ρ. For n>1, assume -1/(n-1)<ρ≤1, so the mean has positive variance. These variables model estimates, not proof truth values. The mean has variance Var(bar X) = σ^2(frac1-ρn+ρ).","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d42e4596a0b89189","tex":"\\vdash","aliases":"vdash","display":null,"kind":"use","excerpt":"also remove mathematical information that is not needed for the present theorem. An example may explain why a definition was chosen. A failed route may isolate the exact boundary of a method. An alternative proof may carry a bridge to another field. A computation may reveal a stronger statement than the one originally sought. None of these objects ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d8fe8dd2410be8cc","tex":"w^*","aliases":"w^*","display":"w^*","kind":"use","excerpt":"[Minimum-variance covariance-aware weights] The unique minimum-variance unbiased linear aggregate has weights w^* = fracΣ^-1 1 1^topΣ^-1 1. Its variance is frac1 1^topΣ^-1 1.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-25a84a8fd44f7f54","tex":"w^\\top","aliases":"w^top top","display":"w^\\top","kind":"use","excerpt":"For a general positive definite covariance matrix Σ, consider unbiased linear aggregates w^top X with 1^top w=1.","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"[Pairwise dependencies do not suffice] Take nodes a,b,c, no admissible leaves (L=∅), and one certified rule {a,b}⟶ c. Use patches U_0={a,c} and U_1={b,c}. The local valuations X_0={a} and X_1={b} agree on their overlap: both assign false to c. Both are local models, since neither patch contains the full rule support. They cover every node and both ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"Let T:E→ R and Q:E→ W be linear, and fix a known point x_0. Put A={y:Ty=Tx_0}, H=ker T, Q_H=Q|_H. Translation h↦ x_0+h identifies H with A. The observation image is the translate Qx_0+Q(H), and the fiber above the attained value Qx_0 is a translate of ker Q_H.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"[Pairwise dependencies do not suffice] Take nodes a,b,c, no admissible leaves (L=∅), and one certified rule {a,b}⟶ c. Use patches U_0={a,c} and U_1={b,c}. The local valuations X_0={a} and X_1={b} agree on their overlap: both assign false to c. Both are local models, since neither patch contains the full rule support. They cover every node and both ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"use","excerpt":"Let (U_i)_i∈ I cover the nodes. A family X_i∈ M_F(U_i) is matching when X_i∩ U_j=X_j∩ U_i for every i,j. This equality includes agreement about false values, not merely common true values.","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"use","excerpt":"Let (U_i)_i∈ I cover the nodes. A family X_i∈ M_F(U_i) is matching when X_i∩ U_j=X_j∩ U_i for every i,j. This equality includes agreement about false values, not merely common true values.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2cc214a50d4b2e13","tex":"X_U","aliases":"X_U","display":"X_U","kind":"use","excerpt":"For a patch U⊆ V, let M_F(U) consist of subsets X_U⊆ U which contain L∩ U and obey every certified rule whose entire support lies in U. If U⊆ W, define ρ_U,W: M_F(W)⟶ M_F(U), ρ_U,W(X_W)=X_W∩ U. This map is well defined. A rule supported in U is also supported in W, and its conclusion remains in U. The restriction therefore preserves both the local ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0613772511c49a53","tex":"X_W","aliases":"X_W","display":"X_W","kind":"use","excerpt":"For a patch U⊆ V, let M_F(U) consist of subsets X_U⊆ U which contain L∩ U and obey every certified rule whose entire support lies in U. If U⊆ W, define ρ_U,W: M_F(W)⟶ M_F(U), ρ_U,W(X_W)=X_W∩ U. This map is well defined. A rule supported in U is also supported in W, and its conclusion remains in U. The restriction therefore preserves both the local ","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"[Functorial transport] Let F:Cert_1→Cert_2 be a functor. If Axrightarrowf_1S_1xrightarrowf_2⋯xrightarrowf_rB is a certified path in Cert_1, then F(A)xrightarrowF(f_1)F(S_1)xrightarrowF(f_2)⋯xrightarrowF(f_r)F(B) is a certified path in Cert_2.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"[Finite realization alternative and ambiguity] Exactly one of the following holds: [label=textup(roman*)] item My=b has a solution y_0. Its complete solution set is y_0+H, where H=ker A∩ker P, and | K_P(x)|=2^n-rank(M). item A row vector λ exists with λ M=0 and λ b=1. It is a directly checkable certificate that K_P(x)=∅. If My_0=b, any other soluti","count":8},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-8e1b422661cf5616","tex":"Y_i","aliases":"Y_i","display":"Y_i","kind":"use","excerpt":"Suppose tasks q_i:mathscr A→ Y_i are indexed by i∈ I. Their joint representation is Q(a)=(q_i(a))_i∈ I. It records every distinction asked for by the task family and makes no demand to recover the entire source archive.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"[The proof-field view laws] The restriction maps satisfy ρ_U,U=id, ρ_U,Wρ_W,Z=ρ_U,Z (U⊆ W⊆ Z). Consequently U↦ M_F(U) is a contravariant set-valued view system on the inclusion order of patches. Both laws are the corresponding intersection identities. For the second, (X∩ W)∩ U=X∩ U because U⊆ W. The values outside a patch are absent, or equivalentl","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"es:7:11 Let C:A→ B. Every predicate q:A→Prop factors through C if and only if C is injective. If C is injective, define the decoder of q at b by the predicate ∃ a (C(a)=b and q(a)). At b=C(a_0), injectivity makes this equivalent to q(a_0). Conversely, if C(a)=C(a'), use the predicate q(z)≡(z=a). Its decoded values at the equal encodings must agree,","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"[Linear extraction] With indexed certified adjacency and constant-time membership bookkeeping, the core can be extracted in time linear in |A| plus the number of visited nodes, selected justifications, and premise incidences.","count":14},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-3d7f36918ca613a9","tex":"|_H","aliases":"|_H","display":"|_H","kind":"use","excerpt":"[Exact observation gain] Give the nonempty finite affine fiber y_0+H the uniform distribution. Then the Shannon entropy of the observed value Qy is H(Qy)=rank(Q|_H) bits. For every observation value that actually occurs, the corresponding conditional fiber contains 2^dim H-rank(Q|_H) realizations. Thus the uncertainty reduction is exactly rank(Q|_H","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"Let X_i=θ+ε_i be numerical estimates of a fixed quantity θ. Assume Eε_i=0, Var(ε_i)=σ^2>0, and common pairwise correlation ρ. For n>1, assume -1/(n-1)<ρ≤1, so the mean has positive variance. These variables model estimates, not proof truth values. The mean has variance Var(bar X) = σ^2(frac1-ρn+ρ).","count":8},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f3cf8a4e3d9677f6","tex":"\\varepsilon_i","aliases":"ε_i varepsilon","display":"\\varepsilon_i","kind":"use","excerpt":"Let X_i=θ+ε_i be numerical estimates of a fixed quantity θ. Assume Eε_i=0, Var(ε_i)=σ^2>0, and common pairwise correlation ρ. For n>1, assume -1/(n-1)<ρ≤1, so the mean has positive variance. These variables model estimates, not proof truth values. The mean has variance Var(bar X) = σ^2(frac1-ρn+ρ).","count":7},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b3de5d9511fcd1b4","tex":"\\varepsilon_j","aliases":"ε_j varepsilon","display":"\\varepsilon_j","kind":"use","excerpt":"[MSE with a shared interpretive offset] Suppose reviewers report J_i=θ+b+ε_i, where θ is the target quantity, b is a fixed shared offset, Eε_i=0, Var(ε_i)=σ^2, and Cov(ε_i,ε_j)=ρσ^2 for i≠ j. For admissible ρ, the mean squared error of the reviewer average is E(overline J_n-θ)^2=b^2+σ^2(ρ+frac1-ρn). For fixed ρ≥0, it approaches b^2+ρσ^2 as n→∞. The","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"Let X_i=θ+ε_i be numerical estimates of a fixed quantity θ. Assume Eε_i=0, Var(ε_i)=σ^2>0, and common pairwise correlation ρ. For n>1, assume -1/(n-1)<ρ≤1, so the mean has positive variance. These variables model estimates, not proof truth values. The mean has variance Var(bar X) = σ^2(frac1-ρn+ρ).","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"[Finite realization alternative and ambiguity] Exactly one of the following holds: [label=textup(roman*)] item My=b has a solution y_0. Its complete solution set is y_0+H, where H=ker A∩ker P, and | K_P(x)|=2^n-rank(M). item A row vector λ exists with λ M=0 and λ b=1. It is a directly checkable certificate that K_P(x)=∅. If My_0=b, any other soluti","count":18},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"For an intended interpretation μ:V→Prop, the transfer to mathematical truth has explicit hypotheses: every leaf must satisfy μ, and every certified rule must preserve μ from all its premises to its conclusion. Under those hypotheses, induction gives v∈Der_F⇒μ(v). Lean checks this statement as pathProofPath.Field.semantic_soundness. It does not infe","count":14},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"For the second assertion, write R=π∘ C and q=d∘ R on their attained sets. Then q=(d∘π)∘ C. The identity decoder proves reflexivity, and composition of two recovery maps proves transitivity. All maps are applied only to outputs in their declared domains.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-0065f430ae9d8258","tex":"\\pi_d","aliases":"π_d pi","display":"\\pi_d","kind":"use","excerpt":"A refinement preserves the certified target on its declared scope. Preservation of other tasks requires a separate recovery condition. Let R_d:mathscr A→ R_d be the representation scheme at depth d. A sufficient condition is a map π_d:R_d+1(mathscr A)⟶ R_d(mathscr A) with R_d=π_d∘ R_d+1. If a task factors as q=d_q∘ R_d, then q=(d_q∘π_d)∘ R_d+1. Hen","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"For a patch U⊆ V, let M_F(U) consist of subsets X_U⊆ U which contain L∩ U and obey every certified rule whose entire support lies in U. If U⊆ W, define ρ_U,W: M_F(W)⟶ M_F(U), ρ_U,W(X_W)=X_W∩ U. This map is well defined. A rule supported in U is also supported in W, and its conclusion remains in U. The restriction therefore preserves both the local ","count":20},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d4205abe2eb75344","tex":"\\rho_{U,U}","aliases":"ρ_U,U rho","display":"\\rho_{U,U}","kind":"use","excerpt":"[The proof-field view laws] The restriction maps satisfy ρ_U,U=id, ρ_U,Wρ_W,Z=ρ_U,Z (U⊆ W⊆ Z). Consequently U↦ M_F(U) is a contravariant set-valued view system on the inclusion order of patches. Both laws are the corresponding intersection identities. For the second, (X∩ W)∩ U=X∩ U because U⊆ W. The values outside a patch are absent, or equivalentl","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-575659d4751ab1dd","tex":"\\rho_{U,W}","aliases":"ρ_U,W rho","display":"\\rho_{U,W}","kind":"use","excerpt":"For a patch U⊆ V, let M_F(U) consist of subsets X_U⊆ U which contain L∩ U and obey every certified rule whose entire support lies in U. If U⊆ W, define ρ_U,W: M_F(W)⟶ M_F(U), ρ_U,W(X_W)=X_W∩ U. This map is well defined. A rule supported in U is also supported in W, and its conclusion remains in U. The restriction therefore preserves both the local ","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-66aa3a5b900600dd","tex":"\\rho_{U,Z}","aliases":"ρ_U,Z rho","display":"\\rho_{U,Z}","kind":"use","excerpt":"[The proof-field view laws] The restriction maps satisfy ρ_U,U=id, ρ_U,Wρ_W,Z=ρ_U,Z (U⊆ W⊆ Z). Consequently U↦ M_F(U) is a contravariant set-valued view system on the inclusion order of patches. Both laws are the corresponding intersection identities. For the second, (X∩ W)∩ U=X∩ U because U⊆ W. The values outside a patch are absent, or equivalentl","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-147790fc21d35384","tex":"\\rho_{W,Z}","aliases":"ρ_W,Z rho","display":"\\rho_{W,Z}","kind":"use","excerpt":"[The proof-field view laws] The restriction maps satisfy ρ_U,U=id, ρ_U,Wρ_W,Z=ρ_U,Z (U⊆ W⊆ Z). Consequently U↦ M_F(U) is a contravariant set-valued view system on the inclusion order of patches. Both laws are the corresponding intersection identities. For the second, (X∩ W)∩ U=X∩ U because U⊆ W. The values outside a patch are absent, or equivalentl","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"For a general positive definite covariance matrix Σ, consider unbiased linear aggregates w^top X with 1^top w=1.","count":8},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2b9c06a974b9e679","tex":"\\Sigma^{-1}","aliases":"Σ^-1 Sigma","display":"\\Sigma^{-1}","kind":"use","excerpt":"[Minimum-variance covariance-aware weights] The unique minimum-variance unbiased linear aggregate has weights w^* = fracΣ^-1 1 1^topΣ^-1 1. Its variance is frac1 1^topΣ^-1 1.","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"Let X_i=θ+ε_i be numerical estimates of a fixed quantity θ. Assume Eε_i=0, Var(ε_i)=σ^2>0, and common pairwise correlation ρ. For n>1, assume -1/(n-1)<ρ≤1, so the mean has positive variance. These variables model estimates, not proof truth values. The mean has variance Var(bar X) = σ^2(frac1-ρn+ρ).","count":11},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[A linear certificate alternative] The protected problem is infeasible if and only if there is a linear functional λ:R→ K satisfying λ(Dh)=0 whenever p(h)=0, λ(Dx)=1. For a finite matrix M and right side b, the corresponding alternative is nexists y (My=b) Longleftrightarrow ∃ℓ (ℓ M=0 and ℓ b=1). If Dx∉ B, linear separation of a vector from a subsp","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[The proof-field view laws] The restriction maps satisfy ρ_U,U=id, ρ_U,Wρ_W,Z=ρ_U,Z (U⊆ W⊆ Z). Consequently U↦ M_F(U) is a contravariant set-valued view system on the inclusion order of patches. Both laws are the corresponding intersection identities. For the second, (X∩ W)∩ U=X∩ U because U⊆ W. The values outside a patch are absent, or equivalentl","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"For an intended interpretation μ:V→Prop, the transfer to mathematical truth has explicit hypotheses: every leaf must satisfy μ, and every certified rule must preserve μ from all its premises to its conclusion. Under those hypotheses, induction gives v∈Der_F⇒μ(v). Lean checks this statement as pathProofPath.Field.semantic_soundness. It does not infe","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"[Whole-support gluing] Suppose every certified justification has its whole support in some patch: ∀ j certified, ∃ i S_j⊆ U_i. Then every matching family (X_i) has a unique global field model X satisfying X∩ U_i=X_i for all i. Set X=⋃_iX_i. Matching and node coverage imply X∩ U_i=X_i. A leaf belongs to some patch and hence to its local model, so L⊆","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"[Whole-support gluing] Suppose every certified justification has its whole support in some patch: ∀ j certified, ∃ i S_j⊆ U_i. Then every matching family (X_i) has a unique global field model X satisfying X∩ U_i=X_i for all i. Set X=⋃_iX_i. Matching and node coverage imply X∩ U_i=X_i. A leaf belongs to some patch and hence to its local model, so L⊆","count":4},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"ernative and ambiguity] Exactly one of the following holds: [label=textup(roman*)] item My=b has a solution y_0. Its complete solution set is y_0+H, where H=ker A∩ker P, and | K_P(x)|=2^n-rank(M). item A row vector λ exists with λ M=0 and λ b=1. It is a directly checkable certificate that K_P(x)=∅. If My_0=b, any other solution differs from y_0 by ","count":17},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-f38cc67958f7dd6a","tex":"\\sum_{f\\in R_M\\cup\\bigcup_{x\\in S}R_x}","aliases":"∑_f∈ R_M∪⋃_x∈ SR_x sum in cup bigcup in","display":"\\sum_{f\\in R_M\\cup\\bigcup_{x\\in S}R_x}","kind":"use","excerpt":"For each candidate x∈ C, let R_x⊆ F be the set of future targets reachable from M∪{x}. Let R_M be the targets already reachable from M. Reachability from a union of starting nodes is the union of their individual reach sets, so U_M(S)= ∑_f∈ R_M∪⋃_x∈ SR_xw_f. Adding a candidate cannot remove a reachable target, so U_M is monotone.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"luations of reviewer i under two labels. In a finite set of N reviewers, assign exactly n_1 to the first label and n_0=N-n_1 to the second, uniformly over assignments with those sizes, where n_0,n_1>0. Assume each observed response equals the potential response under its assigned label and does not depend on other assignments. Then the difference o","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-84ecbd4993bf4574","tex":"\\sum_{\\substack{f\\in F\\cap\\Reach_{\\Bcal}(S)\\\\v\\text{ dominates }f}}","aliases":"∑_substackf∈ F∩Reach_Bcal(S) v dominates f ∑_substackf∈ F∩Reach_ B(S) v dominates f sum substack in cap Reach Bcal v text sum substack in cap operatorname mathcal v text","display":null,"kind":"use","excerpt":"[Weighted criticality] For v∉ S, the utility lost by deleting v is the total weight of initially reachable future targets dominated by v: U(S)-U_ B∖{v}(S) =∑_substackf∈ F∩Reach_ B(S) v dominates fw_f. Deleted target vertices are counted as unreachable.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"[Retention projection algebra] For masks K,L⊆ V, P_K∘ P_L=P_K∩ L=P_L∘ P_K. Consequently every retention projection is idempotent, and the masks form a Boolean lattice with meet K∩ L and join K∪ L.","count":9},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"For n=1, the effective count is one. At the excluded boundary ρ=-1/(n-1), the mean has zero variance and there is no finite variance-equivalent count. For fixed 0<ρ≤1, n_eff→1/ρ as n→∞. Negative correlations can give n_eff>n, but a fixed negative correlation is not admissible for arbitrarily large n.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"Define R_1∼_ QR_2 when both R_1preceq_ QR_2 and R_2preceq_ QR_1.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-26341dbce6f0f177","tex":"\\sim_{\\Qcal}","aliases":"∼_Qcal ∼_ Q sim Qcal sim mathcal","display":"\\sim_{{\\mathcal Q}}","kind":"use","excerpt":"Define R_1∼_ QR_2 when both R_1preceq_ QR_2 and R_2preceq_ QR_1.","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"The strict Komorebi realization is now an actual set of coherent local models, Γ( M_F)= {(X_U)_U:ρ_U,W(X_W)=X_U whenever U⊆ W}. There is an explicit bijection Models(F)≃Γ( M_F), X⟼(X∩ U)_U. Its inverse evaluates a coherent family at the full patch V. Coherence shows that every other component is precisely the restriction of that value. The Lean dec","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"For one hundred equicorrelated reviewers with ρ=0.1, n_eff = frac1001+99(0.1) ≈ 9.17. The audit example begins with backlog 10. Arrivals 4 and capacity 5 clear the queue. Equal rates hold the backlog at 10. Arrivals 6 and capacity 5 increase it by one per cycle.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Proof Path Theory studies this larger object without changing the standard of proof. Its central separation is certificate of validity ≠ expanded discovery field. The certificate answers whether the conclusion follows. The discovery field records how the conclusion became reachable, what alternatives were considered, what was lost during compressio","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"es:7:11 Let C:A→ B. Every predicate q:A→Prop factors through C if and only if C is injective. If C is injective, define the decoder of q at b by the predicate ∃ a (C(a)=b and q(a)). At b=C(a_0), injectivity makes this equivalent to q(a_0). Conversely, if C(a)=C(a'), use the predicate q(z)≡(z=a). Its decoded values at the equal encodings must agree,","count":3},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Debt is monotone under retention] If K_1⊆ K_2, then Debt(K_2)≤Debt(K_1).","count":5},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"Let a_t≥0 be candidate arrivals during audit cycle t, let s_t≥0 be service capacity, and let B_t≥0 be the unresolved backlog. Suppose arrivals in a cycle are available to that cycle's service. The basic deterministic queue is B_t+1=max{0,B_t+a_t-s_t}.","count":6},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-9f55fb1c51ddcd12","tex":"\\supseteq","aliases":"⊇ supseteq","display":"\\supseteq","kind":"use","excerpt":"[Irreversible compression accumulates debt] For a descending sequence of retained sets K_0⊇ K_1⊇⋯⊇ K_r, the expected debt is nondecreasing.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"[Whole-support gluing] Suppose every certified justification has its whole support in some patch: ∀ j certified, ∃ i S_j⊆ U_i. Then every matching family (X_i) has a unique global field model X satisfying X∩ U_i=X_i for all i. Set X=⋃_iX_i. Matching and node coverage imply X∩ U_i=X_i. A leaf belongs to some patch and hence to its local model, so L⊆","count":2},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-4601d15b3053bbea","tex":"\\bigcup_i","aliases":"⋃_i bigcup","display":"\\bigcup_i","kind":"use","excerpt":"[Whole-support gluing] Suppose every certified justification has its whole support in some patch: ∀ j certified, ∃ i S_j⊆ U_i. Then every matching family (X_i) has a unique global field model X satisfying X∩ U_i=X_i for all i. Set X=⋃_iX_i. Matching and node coverage imply X∩ U_i=X_i. A leaf belongs to some patch and hence to its local model, so L⊆","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-1d166c83975f5640","tex":"\\bigcup_{x\\in S}","aliases":"⋃_x∈ S bigcup in","display":"\\bigcup_{x\\in S}","kind":"use","excerpt":"For each candidate x∈ C, let R_x⊆ F be the set of future targets reachable from M∪{x}. Let R_M be the targets already reachable from M. Reachability from a union of starting nodes is the union of their individual reach sets, so U_M(S)= ∑_f∈ R_M∪⋃_x∈ SR_xw_f. Adding a candidate cannot remove a reachable target, so U_M is monotone.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"0 · 1 · 0. Its two rows are identical. On H=⟨011⟩ it has rank one, so | A|=2, |Q( A)|=2, and every attained observation fiber has size one. These are count identities before they are interpreted probabilistically.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"0 · 1 · 0. Its two rows are identical. On H=⟨011⟩ it has rank one, so | A|=2, |Q( A)|=2, and every attained observation fiber has size one. These are count identities before they are interpreted probabilistically.","count":1},{"work":"proof-path-theory","url":"notation/proof-path-theory.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"The strict Komorebi realization is now an actual set of coherent local models, Γ( M_F)= {(X_U)_U:ρ_U,W(X_W)=X_U whenever U⊆ W}. There is an explicit bijection Models(F)≃Γ( M_F), X⟼(X∩ U)_U. Its inverse evaluates a coherent family at the full patch V. Coherence shows that every other component is precisely the restriction of that value. The Lean dec","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"An elementary tower shows why this matters. For each natural number n, let X_n={m∈ N:m≥ n}, and let X_n+1→ X_n be inclusion. For any finite prefix through N, the constant value N gives a compatible choice at every level in that prefix. No finite stage defeats the construction.","count":10},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":55},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"For the diagram, write T=A · B C · 2 relative to the retained plane and its complement. Here A=2 · 1","count":9},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"0 · 3, B=(0,1)^ T, and C=(1,-1). The two paths contribute A^2(1,2)^ T and BC(1,2)^ T to their sum.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"For the diagram, write T=A · B C · 2 relative to the retained plane and its complement. Here A=2 · 1","count":10},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-f8a323fea028ce9f","tex":"b_{2j}","aliases":"b_2j","display":"b_{2j}","kind":"use","excerpt":"-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial values are 1,1,0,-2,0,16,0,-272,0. A finite computation suggests that every positive even index has disappeared. The reason is not hidden in the numerical list: e^tsecht =fraccosh t+sinh tcosh t =1+tanh t. The series tanh t is odd. Thus b_2j=0 fo","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-7b27183e1d0ebc44","tex":"\\bar f","aliases":"bar f bar","display":"\\bar f","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":4},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-0172f00d6015ed24","tex":"\\bar T","aliases":"bar T bar","display":"\\bar T","kind":"use","excerpt":"Continuation makes the requirement sharper. For a transformation T:X→ X, a closed evolution bar T on the represented states must satisfy rT=bar T r. Applying the same factorization criterion to rT shows exactly what is needed: whenever two states have the same representation, their successors must also have the same representation. Induction then g","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-94b971a94210082f","tex":"\\bar T^n","aliases":"bar T^n bar","display":"\\bar T^n","kind":"use","excerpt":"Continuation makes the requirement sharper. For a transformation T:X→ X, a closed evolution bar T on the represented states must satisfy rT=bar T r. Applying the same factorization criterion to rT shows exactly what is needed: whenever two states have the same representation, their successors must also have the same representation. Induction then g","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"For the diagram, write T=A · B C · 2 relative to the retained plane and its complement. Here A=2 · 1","count":10},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"use","excerpt":"), F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial values are 1,1,0,-2,0,16,0,-272,0. A finite computation suggests that every positive even index has disappeared. The reason is not hidden in the numerical list: e^tsecht =","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"The distinction is exact in a linear correction problem. Let D:U→ W be a linear map over a field, and let r∈ W be a defect. An admissible correction is a vector h∈ U satisfying Dh=-r. Such a correction exists precisely when the class of r vanishes in the quotient W/imD. Indeed, vanishing says r=Dg for some g, so h=-g corrects it. Conversely, a corr","count":12},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"use","excerpt":"Now allow D'(u,v)=(u,u+v). The choice (-1,-1) supplies the correction (-1,-2), and the defect disappears. Nothing contradictory has happened. The admissible correction space has changed. That change may be an intelligent way to solve a broader problem, but it is not a solution of the original problem unless the extra freedom was permitted.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"Work over Q^3, with T=2 · 1 · 0 0 · 3 · 1 1 · -1 · 2, P=diag(1,1,0), x=1 2 0. Here Tx=(4,6,-1)^ T, so both descriptions give the retained observation (4,6,0)^ T. This agreement follows from Px=x, not from closure under repetition.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","excerpt":"The same care applies to small residuals. On the Hilbert space ℓ^2, define D(x)_n=x_n/n for positive integers n. The map is bounded and injective. For the unit vector e_n, however, De_n =1/n⟶0, dist(e_n,ker D)=1. There is no uniform bound dist(x,ker D)≤ C Dx on this space. Taking x=e_n would require 1≤ C/n for every n. A tiny residual can therefore","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":9},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"The same care applies to small residuals. On the Hilbert space ℓ^2, define D(x)_n=x_n/n for positive integers n. The map is bounded and injective. For the unit vector e_n, however, De_n =1/n⟶0, dist(e_n,ker D)=1. There is no uniform bound dist(x,ker D)≤ C Dx on this space. Taking x=e_n would require 1≤ C/n for every n. A tiny residual can therefore","count":6},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-3f6dc0c78382f5b1","tex":"e^{-t}","aliases":"e^-t","display":"e^{-t}","kind":"use","excerpt":"rmed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial values are 1,1,0,-2,0,16,0,-272,0. A finite computation suggests that every positive even index has disappeared. The reason is not hidden in the numerical list: e^tsecht =fraccosh t+sinh tcosh t =1+tanh t. The series tanh t is odd. Thus b_2j=0 for every j≥1. Multiplication by e^-t recovers th","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e40ba00de6234ae1","tex":"E^n","aliases":"E^n","display":"E^n","kind":"use","excerpt":"The requirement v_0=0 is substantive. With a nonzero initial complementary state, the formula for v_n contains the additional term E^nv_0, and the retained evolution contains BE^nv_0. An unknown v_0 cannot simply be erased. Either the problem supplies it, the retained data determine its relevant effect, or the prediction must preserve the remaining","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-564be5d1570208d5","tex":"E^{n-1-j}","aliases":"E^n-1-j","display":"E^{n-1-j}","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-1fe69fafa50db3c5","tex":"e^t","aliases":"e^t","display":"e^t","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-1f891a7f61f89a21","tex":"E_k","aliases":"E_k","display":"E_k","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-da2ce42dc70bb3da","tex":"E_n","aliases":"E_n","display":"E_n","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"The same care applies to small residuals. On the Hilbert space ℓ^2, define D(x)_n=x_n/n for positive integers n. The map is bounded and injective. For the unit vector e_n, however, De_n =1/n⟶0, dist(e_n,ker D)=1. There is no uniform bound dist(x,ker D)≤ C Dx on this space. Taking x=e_n would require 1≤ C/n for every n. A tiny residual can therefore","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":5},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-6b9fd9d91502409e","tex":"\\mathrm{False}","aliases":"False mathrm","display":"\\mathrm{False}","kind":"use","excerpt":"Axioms require a separate audit. Lean permits new axioms and records proof dependencies. Postulating False allows any proposition to be derived. The derivation is valid relative to that assumption. A project that disallows the assumption must reject the dependency.sourcerefleanaxioms","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"The distinction is exact in a linear correction problem. Let D:U→ W be a linear map over a field, and let r∈ W be a defect. An admissible correction is a vector h∈ U satisfying Dh=-r. Such a correction exists precisely when the class of r vanishes in the quotient W/imD. Indeed, vanishing says r=Dg for some g, so h=-g corrects it. Conversely, a corr","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"The distinction is exact in a linear correction problem. Let D:U→ W be a linear map over a field, and let r∈ W be a defect. An admissible correction is a vector h∈ U satisfying Dh=-r. Such a correction exists precisely when the class of r vanishes in the quotient W/imD. Indeed, vanishing says r=Dg for some g, so h=-g corrects it. Conversely, a corr","count":4},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Let P be a projector on a vector space, so P^2=P, and put Q=I-P. The image of P is the retained space. The image of Q is its complementary space. No inner product is required. If T is a linear operator, the compressed operator PTP records only the part that begins and ends in the retained space.","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"The distinction is exact in a linear correction problem. Let D:U→ W be a linear map over a field, and let r∈ W be a defect. An admissible correction is a vector h∈ U satisfying Dh=-r. Such a correction exists precisely when the class of r vanishes in the quotient W/imD. Indeed, vanishing says r=Dg for some g, so h=-g corrects it. Conversely, a corr","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":8},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"For a linear protected quantity p:U→ V, corrections that leave the quantity unchanged lie in ker p. The relevant correction image is therefore D(ker p), not all of D(U). The obstruction must be calculated relative to that smaller image. This is the elementary form of protected repair developed in the realization framework. It explains why two super","count":4},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"The finite model uses justification hypergraphs because ordinary arrows can hide an important distinction. A step may require both A and B to obtain L. Another conclusion may have two independent routes, either of which is sufficient. The former is a conjunction of premises. The latter is an alternative between justifications. A faithful record mus","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"For example, take certified steps {A,B}⇒ L and L⇒ T. They form one certificate for the target T. A second route might use A⇒ M and {M,B}⇒ T. A candidate extension from M to a new target U can remain visible without becoming part of either certificate. It may be valuable precisely because it suggests what to investigate next, but its value does not ","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"An elementary tower shows why this matters. For each natural number n, let X_n={m∈ N:m≥ n}, and let X_n+1→ X_n be inclusion. For any finite prefix through N, the constant value N gives a compatible choice at every level in that prefix. No finite stage defeats the construction.","count":10},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"An elementary tower shows why this matters. For each natural number n, let X_n={m∈ N:m≥ n}, and let X_n+1→ X_n be inclusion. For any finite prefix through N, the constant value N gives a compatible choice at every level in that prefix. No finite stage defeats the construction.","count":4},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"An elementary tower shows why this matters. For each natural number n, let X_n={m∈ N:m≥ n}, and let X_n+1→ X_n be inclusion. For any finite prefix through N, the constant value N gives a compatible choice at every level in that prefix. No finite stage defeats the construction.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"Continuation makes the requirement sharper. For a transformation T:X→ X, a closed evolution bar T on the represented states must satisfy rT=bar T r. Applying the same factorization criterion to rT shows exactly what is needed: whenever two states have the same representation, their successors must also have the same representation. Induction then g","count":33},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"Let P be a projector on a vector space, so P^2=P, and put Q=I-P. The image of P is the retained space. The image of Q is its complementary space. No inner product is required. If T is a linear operator, the compressed operator PTP records only the part that begins and ends in the retained space.","count":17},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":8},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"Let P be a projector on a vector space, so P^2=P, and put Q=I-P. The image of P is the retained space. The image of Q is its complementary space. No inner product is required. If T is a linear operator, the compressed operator PTP records only the part that begins and ends in the retained space.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Let P be a projector on a vector space, so P^2=P, and put Q=I-P. The image of P is the retained space. The image of Q is its complementary space. No inner product is required. If T is a linear operator, the compressed operator PTP records only the part that begins and ends in the retained space.","count":8},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"Work over Q^3, with T=2 · 1 · 0","count":4},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-da423ba3a883a93b","tex":"\\mathbb Q^2","aliases":"Q^2 mathbb","display":"\\mathbb Q^2","kind":"use","excerpt":"Take U= Q, W= Q^2, and D(u)=(u,u). Only diagonal corrections are allowed. For r=(1,2), the functional λ(w_1,w_2)=w_2-w_1 vanishes on every possible correction but satisfies λ(r)=1. If Dh=-r held, applying λ would give 0=-1. No additional search is needed. The functional is a short certificate of impossibility for the stated correction problem.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-0093762b9a43c998","tex":"\\mathbb Q^3","aliases":"Q^3 mathbb","display":"\\mathbb Q^3","kind":"use","excerpt":"Work over Q^3, with T=2 · 1 · 0","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":25},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":7},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":5},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-765a006157afb823","tex":"S_{ij}","aliases":"S_ij","display":"S_{ij}","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-2d39429b9e78430d","tex":"\\operatorname{sech}","aliases":"sech operatorname","display":"\\operatorname{sech}","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-21670bd4e275805e","tex":"\\sinh","aliases":"sinh","display":"\\sinh","kind":"use","excerpt":"), F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial values are 1,1,0,-2,0,16,0,-272,0. A finite computation suggests that every positive even index has disappeared. The reason is not hidden in the numerical list: e^tsecht =","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Continuation makes the requirement sharper. For a transformation T:X→ X, a closed evolution bar T on the represented states must satisfy rT=bar T r. Applying the same factorization criterion to rT shows exactly what is needed: whenever two states have the same representation, their successors must also have the same representation. Induction then g","count":28},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"0. Here Tx=(4,6,-1)^ T, so both descriptions give the retained observation (4,6,0)^ T. This agreement follows from Px=x, not from closure under repetition.","count":7},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":9},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"Insert I=P+Q between two applications of T. Exact algebra gives PT^2P=(PTP)^2+PTQTP. The second term has a plain meaning. The input leaves the retained space through QTP and later returns through PTQ. Deleting that term is not a harmless change of notation. It changes the evolution unless a further condition makes the return vanish.","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-910092ae6df8d19a","tex":"T^n","aliases":"T^n","display":"T^n","kind":"use","excerpt":"Continuation makes the requirement sharper. For a transformation T:X→ X, a closed evolution bar T on the represented states must satisfy rT=bar T r. Applying the same factorization criterion to rT shows exactly what is needed: whenever two states have the same representation, their successors must also have the same representation. Induction then g","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":4},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e9a7aef78fcda23e","tex":"T_{ij}","aliases":"T_ij","display":"T_{ij}","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-a1d7d8a9d026524b","tex":"T_{jk}","aliases":"T_jk","display":"T_{jk}","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-978e87e4f3d831a3","tex":"\\tanh","aliases":"tanh","display":"\\tanh","kind":"use","excerpt":"), F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial values are 1,1,0,-2,0,16,0,-272,0. A finite computation suggests that every positive even index has disappeared. The reason is not hidden in the numerical list: e^tsecht =","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"The distinction is exact in a linear correction problem. Let D:U→ W be a linear map over a field, and let r∈ W be a defect. An admissible correction is a vector h∈ U satisfying Dh=-r. Such a correction exists precisely when the class of r vanishes in the quotient W/imD. Indeed, vanishing says r=Dg for some g, so h=-g corrects it. Conversely, a corr","count":6},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":8},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"the sum is empty. The induction step multiplies the preceding sum by E and adds Cu_n. Substituting into the first equation therefore proves, for every n≥0, u_n+1=Au_n+∑_j=0^n-1BE^n-1-jCu_j. The reduced description is exact, but it now carries memory. In the displayed rational example E=2, and the first correction is BCu_0=(0,-1)^ T. Equation~[sourc","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":4},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-b03775ada4b9cdf0","tex":"u_{n+1}","aliases":"u_n+1","display":"u_{n+1}","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"For a linear protected quantity p:U→ V, corrections that leave the quantity unchanged lie in ker p. The relevant correction image is therefore D(ker p), not all of D(U). The obstruction must be calculated relative to that smaller image. This is the elementary form of protected repair developed in the realization framework. It explains why two super","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":10},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":5},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":4},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-7fd7bfa7519bcf67","tex":"v_{n+1}","aliases":"v_n+1","display":"v_{n+1}","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"The distinction is exact in a linear correction problem. Let D:U→ W be a linear map over a field, and let r∈ W be a defect. An admissible correction is a vector h∈ U satisfying Dh=-r. Such a correction exists precisely when the class of r vanishes in the quotient W/imD. Indeed, vanishing says r=Dg for some g, so h=-g corrects it. Conversely, a corr","count":4},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Take U= Q, W= Q^2, and D(u)=(u,u). Only diagonal corrections are allowed. For r=(1,2), the functional λ(w_1,w_2)=w_2-w_1 vanishes on every possible correction but satisfies λ(r)=1. If Dh=-r held, applying λ would give 0=-1. No additional search is needed. The functional is a short certificate of impossibility for the stated correction problem.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"Take U= Q, W= Q^2, and D(u)=(u,u). Only diagonal corrections are allowed. For r=(1,2), the functional λ(w_1,w_2)=w_2-w_1 vanishes on every possible correction but satisfies λ(r)=1. If Dh=-r held, applying λ would give 0=-1. No additional search is needed. The functional is a short certificate of impossibility for the stated correction problem.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-3106fac3e3c8f992","tex":"w_2","aliases":"w_2","display":"w_2","kind":"use","excerpt":"Take U= Q, W= Q^2, and D(u)=(u,u). Only diagonal corrections are allowed. For r=(1,2), the functional λ(w_1,w_2)=w_2-w_1 vanishes on every possible correction but satisfies λ(r)=1. If Dh=-r held, applying λ would give 0=-1. No additional search is needed. The functional is a short certificate of impossibility for the stated correction problem.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":9},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":12},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"use","excerpt":"An elementary tower shows why this matters. For each natural number n, let X_n={m∈ N:m≥ n}, and let X_n+1→ X_n be inclusion. For any finite prefix through N, the constant value N gives a compatible choice at every level in that prefix. No finite stage defeats the construction.","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"The same care applies to small residuals. On the Hilbert space ℓ^2, define D(x)_n=x_n/n for positive integers n. The map is bounded and injective. For the unit vector e_n, however, De_n =1/n⟶0, dist(e_n,ker D)=1. There is no uniform bound dist(x,ker D)≤ C Dx on this space. Taking x=e_n would require 1≤ C/n for every n. A tiny residual can therefore","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-98f50cf2afdfc449","tex":"X_{n+1}","aliases":"X_n+1","display":"X_{n+1}","kind":"use","excerpt":"An elementary tower shows why this matters. For each natural number n, let X_n={m∈ N:m≥ n}, and let X_n+1→ X_n be inclusion. For any finite prefix through N, the constant value N gives a compatible choice at every level in that prefix. No finite stage defeats the construction.","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":6},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-d842f0833fbfd1bc","tex":"Y_0","aliases":"Y_0","display":"Y_0","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":5},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e8394415f576ddd6","tex":"Z_0","aliases":"Z_0","display":"Z_0","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-2d0446810a52779a","tex":"\\Delta_S","aliases":"Δ_S Delta","display":"\\Delta_S","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-c76be3f81d648dd8","tex":"\\Delta_{ST}","aliases":"Δ_ST Delta","display":"\\Delta_{ST}","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-0ee930deefef3979","tex":"\\Delta_T","aliases":"Δ_T Delta","display":"\\Delta_T","kind":"use","excerpt":"A similar issue arises when comparing two sequences of transformations. Let T:X→ Y and S:Y→ Z be linear maps, with comparison maps p:X→ X_0, q:Y→ Y_0, and r:Z→ Z_0. Suppose T_0:X_0→ Y_0 and S_0:Y_0→ Z_0 are the proposed reduced maps. Their defects are Δ_T=qT-T_0p, Δ_S=rS-S_0q. Adding and subtracting S_0qT gives Δ_ST=rST-S_0T_0p =Δ_S T+S_0Δ_T. One d","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-935cda8f07907657","tex":"\\zeta^{-rj}","aliases":"ζ^-rj zeta","display":"\\zeta^{-rj}","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-e7afdf00c1a0e7a1","tex":"\\zeta^j","aliases":"ζ^j zeta","display":"\\zeta^j","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-d0952078280ea0fb","tex":"\\zeta^{j(n-r)}","aliases":"ζ^j(n-r) zeta","display":"\\zeta^{j(n-r)}","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Take U= Q, W= Q^2, and D(u)=(u,u). Only diagonal corrections are allowed. For r=(1,2), the functional λ(w_1,w_2)=w_2-w_1 vanishes on every possible correction but satisfies λ(r)=1. If Dh=-r held, applying λ would give 0=-1. No additional search is needed. The functional is a short certificate of impossibility for the stated correction problem.","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"The same care applies to small residuals. On the Hilbert space ℓ^2, define D(x)_n=x_n/n for positive integers n. The map is bounded and injective. For the unit vector e_n, however, De_n =1/n⟶0, dist(e_n,ker D)=1. There is no uniform bound dist(x,ker D)≤ C Dx on this space. Taking x=e_n would require 1≤ C/n for every n. A tiny residual can therefore","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"The same care applies to small residuals. On the Hilbert space ℓ^2, define D(x)_n=x_n/n for positive integers n. The map is bounded and injective. For the unit vector e_n, however, De_n =1/n⟶0, dist(e_n,ker D)=1. There is no uniform bound dist(x,ker D)≤ C Dx on this space. Taking x=e_n would require 1≤ C/n for every n. A tiny residual can therefore","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":14},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"Ordinary hypotheses are different from global axioms. The implication False⇒ P needs no newly postulated axiom, yet cannot establish P without the impossible premise. Less conspicuous hypotheses can conceal the same application gap. The review must therefore inspect the theorem's quantifiers and show that the required data are available where the r","count":5},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","excerpt":"An elementary tower shows why this matters. For each natural number n, let X_n={m∈ N:m≥ n}, and let X_n+1→ X_n be inclusion. For any finite prefix through N, the constant value N gives a compatible choice at every level in that prefix. No finite stage defeats the construction.","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":7},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-38c03c7cd3f0f240","tex":"\\sum_{i,j}","aliases":"∑_i,j sum","display":"\\sum_{i,j}","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"The general packet formulation uses a finite family of pairwise annihilating projectors whose sum is the identity. Write T_ij=P_iTP_j. Then T=∑_i,jT_ij, (ST)_ik=∑_j S_ijT_jk. Composition follows paths through intermediate packets. The laws are familiar block algebra. Their methodological value is that they provide an exact interface for tracking wh","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-2dbaa0f874ca4402","tex":"\\sum_{j=0}^{m-1}","aliases":"∑_j=0^m-1 sum","display":"\\sum_{j=0}^{m-1}","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-f7b37e8032ebe855","tex":"\\sum_{j=0}^{n-1}","aliases":"∑_j=0^n-1 sum","display":"\\sum_{j=0}^{n-1}","kind":"use","excerpt":"The discrepancy can do more than refute a simplification. It can tell us how to construct an exact replacement. Write a general block evolution as u_n+1=Au_n+Bv_n, v_n+1=Cu_n+Ev_n, where u_n is retained and v_n is complementary. Suppose first that v_0=0. Repeated substitution in the second equation gives v_n=∑_j=0^n-1E^n-1-jCu_j. For n=0 the sum is","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-b7f7abb917a0d195","tex":"\\sum_{k=0}^n","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^n","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-dc3a125d1afd05da","tex":"\\sum_{n\\equiv r\\;({\\rm mod}\\,m)}","aliases":"∑_n≡ r (rm mod m) sum equiv rm","display":"\\sum_{n\\equiv r\\;({\\rm mod}\\,m)}","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-441a6af1afa122e0","tex":"\\sum_{n\\geq0}","aliases":"∑_n≥0 sum geq","display":"\\sum_{n\\geq0}","kind":"use","excerpt":"A small generating-function calculation illustrates the movement from pattern to explanation. Define the signed Euler numbers by the formal power series identity over Q, F(t)=secht=∑_n≥0E_nfract^nn!. The initial coefficients are 1,0,-1,0,5,0,-61,0,1385. Multiplication by e^t gives transformed coefficients b_n=∑_k=0^nbinom nk E_k. Their initial valu","count":2},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"This dependence has a simple formulation. Let r:X→ Y be a representation and let f:X→ Z be the question we want to answer. The representation is sufficient for this question precisely when there is a function bar f:r(X)→ Z such that f=bar f∘ r, where the codomain of r is restricted to its image in the composition. Equivalently, whenever r(x)=r(x'),","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"Residue selectors make the mechanism more general. Let F(t)=∑_n≥0a_nt^n be a formal series over C, let ζ be a primitive m-th root of unity, and choose 0≤ r<m. Then frac1m∑_j=0^m-1ζ^-rjF(ζ^jt) =∑_n≡ r (rm mod m)a_nt^n. For the coefficient of t^n, the multiplier is the average of ζ^j(n-r). It is one when m divides n-r, and zero otherwise by the finit","count":1},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"The same care applies to small residuals. On the Hilbert space ℓ^2, define D(x)_n=x_n/n for positive integers n. The map is bounded and injective. For the unit vector e_n, however, De_n =1/n⟶0, dist(e_n,ker D)=1. There is no uniform bound dist(x,ker D)≤ C Dx on this space. Taking x=e_n would require 1≤ C/n for every n. A tiny residual can therefore","count":3},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"An elementary tower shows why this matters. For each natural number n, let X_n={m∈ N:m≥ n}, and let X_n+1→ X_n be inclusion. For any finite prefix through N, the constant value N gives a compatible choice at every level in that prefix. No finite stage defeats the construction.","count":6},{"work":"komorebi-methodology","url":"notation/komorebi-methodology.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"The same care applies to small residuals. On the Hilbert space ℓ^2, define D(x)_n=x_n/n for positive integers n. The map is bounded and injective. For the unit vector e_n, however, De_n =1/n⟶0, dist(e_n,ker D)=1. There is no uniform bound dist(x,ker D)≤ C Dx on this space. Taking x=e_n would require 1≤ C/n for every n. A tiny residual can therefore","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"introduction-passage","excerpt":"bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introdu","count":61},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":246},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":679},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introdu","count":107},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define","count":283},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.","count":195},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"introduction-passage","excerpt":"where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1a4c2e9cb4075d24","tex":"B_a","aliases":"B_a","display":"B_a","kind":"introduction-passage","excerpt":"Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define","count":44},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","excerpt":"bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introdu","count":158},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"introduction-passage","excerpt":"bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introdu","count":19},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define","count":115},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":66},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-682a47b2eea70d50","tex":"e^{i\\theta}","aliases":"e^iθ theta","display":"e^{i\\theta}","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define","count":69},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"introduction-passage","excerpt":"Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"introduction-passage","excerpt":"Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define","count":12},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8c45eea69132da7b","tex":"\\mathcal F_{t-1}","aliases":"F_t-1 mathcal","display":"\\mathcal F_{t-1}","kind":"introduction-passage","excerpt":"Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define","count":102},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":57},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"introduction-passage","excerpt":"bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introdu","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put","count":109},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"introduction-passage","excerpt":"Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define","count":32},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"introduction-passage","excerpt":"Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,","count":122},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"The subscript rm sp denotes spatial coordinates, not a separate measured signal. This margin is expressed before normalization. It simultaneously controls mass orientation and distance from the curved boundary. It is coordinate dependent, so its uncertainty budget must use the same T.","count":81},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-12b33428971a165a","tex":"P_t","aliases":"P_t","display":"P_t","kind":"introduction-passage","excerpt":"Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5ee5e6de08b9da9f","tex":"P_{t,j}","aliases":"P_t,j","display":"P_{t,j}","kind":"introduction-passage","excerpt":"Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"introduction-passage","excerpt":"Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,","count":55},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6fd3920b0502031c","tex":"Q_t","aliases":"Q_t","display":"Q_t","kind":"introduction-passage","excerpt":"Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-63f4df99c85dbf92","tex":"Q_{t,j}","aliases":"Q_t,j","display":"Q_{t,j}","kind":"introduction-passage","excerpt":"Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"introduction-passage","excerpt":"The subscript rm sp denotes spatial coordinates, not a separate measured signal. This margin is expressed before normalization. It simultaneously controls mass orientation and distance from the curved boundary. It is coordinate dependent, so its uncertainty budget must use the same T.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"introduction-passage","excerpt":"bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introdu","count":49},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"The subscript rm sp denotes spatial coordinates, not a separate measured signal. This margin is expressed before normalization. It simultaneously controls mass orientation and distance from the curved boundary. It is coordinate dependent, so its uncertainty budget must use the same T.","count":170},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define","count":95},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"introduction-passage","excerpt":"Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put","count":228},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"introduction-passage","excerpt":"Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define","count":13},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","excerpt":"Write F_a(u)=TB_a(u)T^-1 for the transported operator density. Observations lie in [-1,1] with reference measure du/2. All operator densities in this block are polynomial and all equalities involving integrals use that reference measure. For each action define","count":190},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.","count":85},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","excerpt":"where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.","count":66},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-75d8484cfed31738","tex":"W_*","aliases":"W_*","display":"W_*","kind":"introduction-passage","excerpt":"bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introdu","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"Let P be any causal process on a standard Borel observation space with finite actions. Let Q be a proposed process. Both use the same measurable policy. Write P_t(·mid h,a) and Q_t(·mid h,a) for their next-observation kernels, and define","count":14},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"introduction-passage","excerpt":"bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introdu","count":42},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a1810dd5bbc6aace","tex":"\\zeta^{-1}","aliases":"ζ^-1 zeta","display":"\\zeta^{-1}","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":12},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"introduction-passage","excerpt":"bottomrule A small notation key. A,S denote analysis and synthesis; C=im A denotes realizable coefficients; γ denotes a predictive summary; P,Q denote retained and discarded projections; W_* denotes the completed observable row space; C denotes a probability cone in the stochastic plates; λ denotes its mass map. Symbols local to a plate are introdu","count":111},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":33},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"introduction-passage","excerpt":"Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put","count":18},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-acaa98aac427780c","tex":"\\phi_{t,j}","aliases":"ϕ_t,j phi","display":"\\phi_{t,j}","kind":"introduction-passage","excerpt":"Let F_t-1 denote the available information immediately before observing the next measurement, including the chosen action, current model and any test selected using past data. A predictable test is a measurable function ϕ_t,j of the upcoming measurement, chosen from this information, with values in [0,1]. Put","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"introduction-passage","excerpt":"where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"introduction-passage","excerpt":"Write the rotation angle as θ. It is not a rational multiple of 2π. Otherwise ζ=e^iθ would be a root of unity, so ζ+ζ^-1=6/5 would be a rational algebraic integer. Every rational algebraic integer is an integer, by the rational-root argument for a monic integer polynomial, contradicting 6/5∉ Z. The forward powers of an irrational circle rotation ar","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"introduction-passage","excerpt":"where † denotes the Moore-Penrose inverse. Every other fitted extension agrees with B_0 on W. If W is a proper subspace of the true realizable state space V, the data alone do not identify the operator on V∖ W.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"introduction-passage","excerpt":"Let P_t,j and Q_t,j denote the true and proposed next-observation laws on the same actual history and action. The proposed law must be defined there. Since 0≤ϕ_t,j≤1,","count":187},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"C_π(a,i)= 1_i∈ B_a.","count":22},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-232f12a78d23a1e5","tex":"\\mathbf1^{\\mathsf T}","aliases":"1^ T mathbf mathsf","display":"\\mathbf1^{\\mathsf T}","kind":"use","excerpt":"A finite-generator witness. Let V have full row rank and normalized generator columns. Nonnegative lifts A_o with B_oV=VA_o and 1^ T∑_oA_o=1^ T certify the generated cone. The initial state must be a normalized nonnegative mixture of those columns. claimAll-horizon validity follows by induction from a finite exact witness. Statistical proximity to ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2876747ca029362d","tex":"\\mathbf1^\\top","aliases":"1^top mathbf top","display":"\\mathbf1^\\top","kind":"use","excerpt":"A concrete realization has a belief p∈ R^n_+ with 1^top p=1. For each s=(a,o), let M_s∈ R^n× n be entrywise nonnegative, with","count":18},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f15d374f1be61a46","tex":"\\mathbf1_{i\\in B_a}","aliases":"1_i∈ B_a mathbf in","display":"\\mathbf1_{i\\in B_a}","kind":"use","excerpt":"C_π(a,i)= 1_i∈ B_a.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5afeddd9f11b1bed","tex":"\\mathbf1_m^\\top","aliases":"1_m^top mathbf top","display":"\\mathbf1_m^\\top","kind":"use","excerpt":"λ V=1_m^top.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8cbca48180738558","tex":"\\mathbf1_{\\{A_t=a\\}}","aliases":"1_{A_t=a} mathbf","display":"\\mathbf1_{\\{A_t=a\\}}","kind":"use","excerpt":"μ_t^a=∫ϕ_t(y)P_t(dymid H_t-1,a), W_t^a=frac1_{A_t=a}ϕ_t(Y_t)π_t(amid H_t-1).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-54034542483dca3c","tex":"A^{\\mathsf T}","aliases":"A^ T mathsf","display":"A^{\\mathsf T}","kind":"use","excerpt":"A^ THA=H.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e0c9947b1daf0522","tex":"A^*","aliases":"A^*","display":"A^*","kind":"use","excerpt":"Use the Frobenius norm on symmetric matrices and the induced product norm. Assume M_A^* is injective and M_A^*(H^*)=(0,1)=b. This fixes the scale of a one-dimensional invariant-form space.","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-00ae19a4939e8cce","tex":"\\mathcal A^*","aliases":"A^* mathcal","display":"\\mathcal A^*","kind":"use","excerpt":"Then S A=I and Ax ^2=c x ^2, as proved in [K1]. Here A S=c^-1 A A^* is the orthogonal projection onto the consistent coefficient subspace.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"Let A_1,…,A_n be a measurable partition of Y. Define the cell operators","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.","count":25},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8118052b56b81067","tex":"A_{a,o}","aliases":"A_a,o","display":"A_{a,o}","kind":"use","excerpt":"1^top∑_o A_a,o=1^top, λ B_wz_0=1^top A_wα_0.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-901ee9becf960caf","tex":"A_a^\\dagger","aliases":"A_a^† dagger","display":"A_a^\\dagger","kind":"use","excerpt":"For Euclidean coefficient errors, conditioning remains important. If A_a has smallest singular value at least m_a>0 and largest singular value at most M_a, its canonical synthesis A_a^† has norm at most 1/m_a, and T_ba ≤ M_b/m_a. An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis q","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-08eb65adb1a856b8","tex":"A_b","aliases":"A_b","display":"A_b","kind":"use","excerpt":"T_ba=A_bS_a: C_a⟶ C_b.","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cfa36d36d986d582","tex":"a_H","aliases":"a_H","display":"a_H","kind":"use","excerpt":"a_H=frac78-frac38 3^-H, Δ_H=frac38 3^-H, sup_t|z_t-widehat z_t|≤ frac3 3^-H1+3 3^-H.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"**Proposition 20. Packet-interaction truncation bound.** Put a_i= P_ix . Then","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ff7bb62bf3333f8b","tex":"a_{i_j}","aliases":"a_i_j","display":"a_{i_j}","kind":"use","excerpt":"F(x)-F_ I(x) ≤∑_m=1^p ∑_ i∉ I_m β_m, i∏_j=1^ma_i_j.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"B_a,j=∫_A_jB_a(y) μ(dy).","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"Pr_π(w)=f(w)∏_j=1^|w|π(a_jmid h_j-1),","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"Let A_1,…,A_n be a measurable partition of Y. Define the cell operators","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-96326a0acc77e20a","tex":"A_o","aliases":"A_o","display":"A_o","kind":"use","excerpt":"A finite-generator witness. Let V have full row rank and normalized generator columns. Nonnegative lifts A_o with B_oV=VA_o and 1^ T∑_oA_o=1^ T certify the generated cone. The initial state must be a normalized nonnegative mixture of those columns. claimAll-horizon validity follows by induction from a finite exact witness. Statistical proximity to ","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"use","excerpt":"**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e50ca40bad47ed08","tex":"A_s","aliases":"A_s","display":"A_s","kind":"use","excerpt":"**Proposition 39. Finite positive-lift certificate.** This cone is invariant under B_s if and only if there is an entrywise nonnegative matrix A_s∈ R^m× m such that","count":18},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b5c348dd50366d43","tex":"A_s^{(j)}","aliases":"A_s^(j)","display":"A_s^{(j)}","kind":"use","excerpt":"B_s^(j)V=VA_s^(j), A_s^(j)≥0.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-82821bc2124f1658","tex":"A_t","aliases":"A_t","display":"A_t","kind":"use","excerpt":"μ_t^a=∫ϕ_t(y)P_t(dymid H_t-1,a), W_t^a=frac1_{A_t=a}ϕ_t(Y_t)π_t(amid H_t-1).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-37b59cf4c71406cd","tex":"a_T","aliases":"a_T","display":"a_T","kind":"use","excerpt":"λ B_a_T(y_T)⋯ B_a_1(y_1)z_0","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"1^top∑_o A_a,o=1^top, λ B_wz_0=1^top A_wα_0.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"B_ℓ_y(p) =fracpodotℓ_yℓ_y^top p.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"For a nonempty atlas family, define transport on realizable coefficients from atlas a to atlas b by","count":97},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-21fc71445c843243","tex":"B'_a","aliases":"B'_a","display":"B'_a","kind":"use","excerpt":"with respect to μ^⊗ T. Replacing the reference measure by μ'=rμ, with r>0 almost everywhere and ∫ r dμ=1, requires B'_a(y)=B_a(y)/r(y). Event probabilities and normalized states then remain unchanged.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2def772dd734efb8","tex":"B'_s","aliases":"B'_s","display":"B'_s","kind":"use","excerpt":"B'_s=TB_sT^-1, J'= JT^-1, λ'=λ T^-1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-882ead36162163bb","tex":"b^*","aliases":"b^*","display":"b^*","kind":"use","excerpt":"For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"Iterating this conditional inequality gives the same exponential bound for the martingale sum ∑_s≤ t(X_s,j-m_s,j). Optimizing the exponential Markov inequality gives the two-sided bound 2exp(-2b^2/t). At the displayed b_t,j this is α_j/[t(t+1)]. Sum over t, using ∑_t≥11/[t(t+1)]=1, and then over j. The resulting event holds at every time, so it als","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-775a00a286e71dbd","tex":"B^\\Pi","aliases":"B^Π Pi","display":"B^\\Pi","kind":"use","excerpt":"The full-law comparison between B and B^Π permits any common measurable policy because both are continuous-output models on the same space. The exact equivalence to the finite bin model in Proposition 56 instead requires a bin-measurable policy. These are different conclusions with different observation information.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-75b9a6a84ca0afa9","tex":"B_{\\mathrm p,0}","aliases":"B_ p,0 mathrm","display":"B_{\\mathrm p,0}","kind":"use","excerpt":"B_ p,0=z_0(1/2,1/2,0), B_ p,1=z_0(1/2,-1/2,0).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-76f5f262c69453b6","tex":"B_{\\mathrm p,1}","aliases":"B_ p,1 mathrm","display":"B_{\\mathrm p,1}","kind":"use","excerpt":"B_ p,0=z_0(1/2,1/2,0), B_ p,1=z_0(1/2,-1/2,0).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1b320f3fd3b71c2c","tex":"B_{\\mathrm r,0}","aliases":"B_ r,0 mathrm","display":"B_{\\mathrm r,0}","kind":"use","excerpt":"B_ r,0=T, B_ r,1=0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-72bbd048a7508f66","tex":"B_{\\mathrm r,1}","aliases":"B_ r,1 mathrm","display":"B_{\\mathrm r,1}","kind":"use","excerpt":"B_ r,0=T, B_ r,1=0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"(G_s)_ij=f(h_jst_i), (b_0)_i=f(t_i).","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4b187b9b2c7b1eb9","tex":"B_{0,0}","aliases":"B_0,0","display":"B_{0,0}","kind":"use","excerpt":"B_0,0 · =1 · -1 11/32 · -1/4, · B_0,1 · =0 · 1 -3/32 · 3/4, [4pt] B_1,0 · =5/48 · 5/6 23/192 · 5/24, · B_1,1 · =43/48 · -5/6 37/64 · -5/8.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a624e8b544b10d48","tex":"B_{0,1}","aliases":"B_0,1","display":"B_{0,1}","kind":"use","excerpt":"B_0,0 · =1 · -1 11/32 · -1/4, · B_0,1 · =0 · 1 -3/32 · 3/4, [4pt] B_1,0 · =5/48 · 5/6 23/192 · 5/24, · B_1,1 · =43/48 · -5/6 37/64 · -5/8.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"a(u)=frac1+9u^24, b(u)=frac3u2, c(u)=frac1-9u^24, [3pt] B_0(u)=a(u) · b(u) · 0 b(u) · a(u) · 0 0 · 0 · c(u), B_1(u)=B_0(u)T.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c9f95638522373af","tex":"B_{1,0}","aliases":"B_1,0","display":"B_{1,0}","kind":"use","excerpt":"B_0,0 · =1 · -1 11/32 · -1/4, · B_0,1 · =0 · 1 -3/32 · 3/4, [4pt] B_1,0 · =5/48 · 5/6 23/192 · 5/24, · B_1,1 · =43/48 · -5/6 37/64 · -5/8.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bfa0d10e5aa052f3","tex":"B_{1,1}","aliases":"B_1,1","display":"B_{1,1}","kind":"use","excerpt":"B_0,0 · =1 · -1 11/32 · -1/4, · B_0,1 · =0 · 1 -3/32 · 3/4, [4pt] B_1,0 · =5/48 · 5/6 23/192 · 5/24, · B_1,1 · =43/48 · -5/6 37/64 · -5/8.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-58a32903afc3d14d","tex":"b_a","aliases":"b_a","display":"b_a","kind":"use","excerpt":"λ B_a(u)=(b_a,0(u),b_a(u)^ T), β_a(u)=b_a,0(u)+b_a(u)^ Tc.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1b6f5abb0b93bba3","tex":"b_{a,0}","aliases":"b_a,0","display":"b_{a,0}","kind":"use","excerpt":"λ B_a(u)=(b_a,0(u),b_a(u)^ T), β_a(u)=b_a,0(u)+b_a(u)^ Tc.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fe7473d969f990ef","tex":"B_{a,b}","aliases":"B_a,b","display":"B_{a,b}","kind":"use","excerpt":"B_a,b(u)=v(u) (1,au,bL_2(u)).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-08660a2054eb5991","tex":"B_{a,j}","aliases":"B_a,j","display":"B_{a,j}","kind":"use","excerpt":"B_a,j=∫_A_jB_a(y) μ(dy).","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-595e8b61a1f64863","tex":"B_{a,o}","aliases":"B_a,o","display":"B_{a,o}","kind":"use","excerpt":"Pr(omid z,a)=λ B_a,oz, z^+=fracB_a,ozλ B_a,oz.","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a7ae9b743db972b8","tex":"B_{a,t}","aliases":"B_a,t","display":"B_{a,t}","kind":"use","excerpt":"sup_a,z∈ K∫ (B_a,t(y)-widetilde B_a,t(y))z _C μ(dy) ≤ϵ_t.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bde1d6f3427d151f","tex":"b_a^{\\mathsf T}","aliases":"b_a^ T mathsf","display":"b_a^{\\mathsf T}","kind":"use","excerpt":"For a nonzero z, write x=m(c+w) with w^ THw≤1. Its output mass is m(β_a+b_a^ Tw). Cauchy--Schwarz in the H inner product bounds this below by","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9390f5835f9a8a38","tex":"B_a^{[n]}","aliases":"B_a^[n]","display":"B_a^{[n]}","kind":"use","excerpt":"B_a^[n](x)=∫_0^1D_n(x,v)B_a(v) dv =∑_k=0^n b_n,k(x)C_a,n,k,","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4026fc271476ea02","tex":"B_a^\\Pi","aliases":"B_a^Π Pi","display":"B_a^\\Pi","kind":"use","excerpt":"B_a^Π(y)=fracB_a,jμ(A_j) (y∈ A_j).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-56f7eb8247d0f0e3","tex":"B_{a_1}","aliases":"B_a_1","display":"B_{a_1}","kind":"use","excerpt":"λ B_a_T(y_T)⋯ B_a_1(y_1)z_0","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a6cd5b75b1f1dae8","tex":"B_{a_T}","aliases":"B_a_T","display":"B_{a_T}","kind":"use","excerpt":"λ B_a_T(y_T)⋯ B_a_1(y_1)z_0","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ef9e511d27b6b980","tex":"B_b","aliases":"B_b","display":"B_b","kind":"use","excerpt":"F=B_b(u_*)B_a(u_*)^-1, λ F=λ.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3ca8acd05511af58","tex":"\\mathcal B_{\\bar\\ell}","aliases":"B_barℓ mathcal bar ell","display":"\\mathcal B_{\\bar\\ell}","kind":"use","excerpt":"with one common c>0. For a fixed prior, put δ=r(p)-barℓ, bar b= B_barℓ(P) and Z_r=r(p)^top P. Then","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7fa01f775913c13f","tex":"\\mathcal B_{\\bar\\ell_y}","aliases":"B_barℓ_y mathcal bar ell","display":"\\mathcal B_{\\bar\\ell_y}","kind":"use","excerpt":"C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-557f88fb6cda082f","tex":"B_h","aliases":"B_h","display":"B_h","kind":"use","excerpt":"b(h)=(f(ht_i))_i=B_hb_0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"use","excerpt":"F(x)=b+∑_m=1^pB_m(x,…,x),","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","excerpt":"∑_g∉ G_nπ(g)L_g(D)≤ b_n,","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-990bf3b4099cc02d","tex":"b_{n,k}","aliases":"b_n,k","display":"b_{n,k}","kind":"use","excerpt":"b_n,k(x)=binom nk x^k(1-x)^n-k, D_n(x,v)=(n+1)∑_k=0^n b_n,k(x)b_n,k(v).","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5ded6618a60308ed","tex":"B_o","aliases":"B_o","display":"B_o","kind":"use","excerpt":"[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, ","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-540f951ac76d6b34","tex":"B_{o_1}","aliases":"B_o_1","display":"B_{o_1}","kind":"use","excerpt":"y needs a domain, a mass map and a covered initial state.P29, P39--P40 Let C⊆ R^d be a convex cone and λ a linear mass map strictly positive on C∖{0}. Set K={z∈ C:λ z=1}. Linear observation operators must satisfy B_oC⊆ C, λ∑_o B_o=λ, z_0∈ K. diagram10_domain A finite record o_1⋯ o_T has probability λ B_o_T⋯ B_o_1z_0≥0. Summing over every branch rep","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ddd51421132426d8","tex":"B_{o_T}","aliases":"B_o_T","display":"B_{o_T}","kind":"use","excerpt":"y needs a domain, a mass map and a covered initial state.P29, P39--P40 Let C⊆ R^d be a convex cone and λ a linear mass map strictly positive on C∖{0}. Set K={z∈ C:λ z=1}. Linear observation operators must satisfy B_oC⊆ C, λ∑_o B_o=λ, z_0∈ K. diagram10_domain A finite record o_1⋯ o_T has probability λ B_o_T⋯ B_o_1z_0≥0. Summing over every branch rep","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-026204dc005ea4b1","tex":"B_s","aliases":"B_s","display":"B_s","kind":"use","excerpt":"Choose a full row rank matrix R with row space W_* and first row 1^top. There are unique matrices B_s with","count":54},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3689ceb28316223d","tex":"B_s'","aliases":"B_s'","display":"B_s'","kind":"use","excerpt":"z'=Tz, B_s'=TB_sT^-1, λ'=λ T^-1, C'=TC.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6a158b3bdcfa8c43","tex":"B_s^{(j)}","aliases":"B_s^(j)","display":"B_s^{(j)}","kind":"use","excerpt":"Let U be the convex hull of finitely many **complete operator families** {B_s^(j)}_s. Every family must satisfy the exact per-action mass identities. The convex coefficient is shared across that family's symbols. A product of independent uncertainty sets can instead be handled by its joint vertices.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-135ede262d4c4b20","tex":"B_s^*","aliases":"B_s^*","display":"B_s^*","kind":"use","excerpt":"B_s^*-widehat B_s ≤ϵ_s, widetilde B_s-widehat B_s ≤ϵ_s.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ed277933d8405680","tex":"B_s^{-1}","aliases":"B_s^-1","display":"B_s^{-1}","kind":"use","excerpt":"C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cec278ed3f674ad5","tex":"B_{s_1}","aliases":"B_s_1","display":"B_{s_1}","kind":"use","excerpt":"B_sv_h=v_hs, f(s_1⋯ s_k)=λ B_s_k⋯ B_s_1v_ϵ.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fe8c80bb85809bfe","tex":"B_{s_1\\cdots s_k}","aliases":"B_s_1⋯ s_k cdots","display":"B_{s_1\\cdots s_k}","kind":"use","excerpt":"B_s_1⋯ s_k=B_s_k⋯ B_s_1, B_ϵ=I.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e69dcd8650618d71","tex":"B_{s_k}","aliases":"B_s_k","display":"B_{s_k}","kind":"use","excerpt":"B_sv_h=v_hs, f(s_1⋯ s_k)=λ B_s_k⋯ B_s_1v_ϵ.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e7e9bd9bc89b3a58","tex":"B_{sv}","aliases":"B_sv","display":"B_{sv}","kind":"use","excerpt":"**Proof.** For a first symbol s followed by a word v, B_sv=B_vB_s. The inequalities for all words of length at most h+1 are therefore exactly the displayed recurrence. If z∈ C_∞, every test on B_sz is another test on z, so C_∞ is invariant. Conversely, invariance of a cone contained in C_0 makes every successive test nonnegative, hence places it in","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1048699da3b094d0","tex":"b_{t,j}","aliases":"b_t,j","display":"b_{t,j}","kind":"use","excerpt":"|R_t,j-D_t,j|≤ b_t,j, b_t,j=sqrtfrac t2logfrac2t(t+1)α_j.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cbffb6e67b1866d8","tex":"B_u","aliases":"B_u","display":"B_u","kind":"use","excerpt":"C_0=HR, RL_u=B_uR","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-aba34f1446cfa879","tex":"B_v","aliases":"B_v","display":"B_v","kind":"use","excerpt":"**Proof.** For a first symbol s followed by a word v, B_sv=B_vB_s. The inequalities for all words of length at most h+1 are therefore exactly the displayed recurrence. If z∈ C_∞, every test on B_sz is another test on z, so C_∞ is invariant. Conversely, invariance of a cone contained in C_0 makes every successive test nonnegative, hence places it in","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5902c64cbd2a059e","tex":"B_w","aliases":"B_w","display":"B_w","kind":"use","excerpt":"An initial state has λ z_0=1. The proposed probability of a word is λ B_wz_0. An invertible coordinate change must transform z_0, B_s, λ and the admissible cone together, as in Proposition 37.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6ec0d6058361f1e1","tex":"B_\\epsilon","aliases":"B_ϵ epsilon","display":"B_\\epsilon","kind":"use","excerpt":"B_s_1⋯ s_k=B_s_k⋯ B_s_1, B_ϵ=I.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9bd2acc8bb44974b","tex":"\\mathcal B_{\\ell}","aliases":"B_ℓ mathcal ell","display":"\\mathcal B_{\\ell}","kind":"use","excerpt":"C_π B_ℓ(p)-bar b = frac1Z_r(diagP-bar bP^top)δ.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0aebfc32fe668dc6","tex":"\\mathcal B_{\\ell_y}","aliases":"B_ℓ_y mathcal ell","display":"\\mathcal B_{\\ell_y}","kind":"use","excerpt":"B_ℓ_y(p) =fracpodotℓ_yℓ_y^top p.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7e95a89f39649f47","tex":"\\bar b","aliases":"bar b bar","display":"\\bar b","kind":"use","excerpt":"with one common c>0. For a fixed prior, put δ=r(p)-barℓ, bar b= B_barℓ(P) and Z_r=r(p)^top P. Then","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-52555ee0fe55f126","tex":"\\bar F","aliases":"bar F bar","display":"\\bar F","kind":"use","excerpt":"Let γ:X→ Z=γ(X) be a summary and let F_u:X→ X describe a deterministic transition for action u. The reduced description closes when there is bar F_u:Z→ Z such that","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-678701062f958896","tex":"\\bar F_u","aliases":"bar F_u bar","display":"\\bar F_u","kind":"use","excerpt":"Let γ:X→ Z=γ(X) be a summary and let F_u:X→ X describe a deterministic transition for action u. The reduced description closes when there is bar F_u:Z→ Z such that","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fad6ae0fd6c0acb8","tex":"\\bar K","aliases":"bar K bar","display":"\\bar K","kind":"use","excerpt":"C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ac94e4492f20ba86","tex":"\\bar K^{\\mathsf T}","aliases":"bar K^ T bar mathsf","display":"\\bar K^{\\mathsf T}","kind":"use","excerpt":"∫ K(y)XK(y)^ T ν(dy) =bar KXbar K^ T +∫ (K(y)-bar K)X(K(y)-bar K)^ T ν(dy).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d1786c2e4892c6a4","tex":"\\bar K_u","aliases":"bar K_u bar","display":"\\bar K_u","kind":"use","excerpt":"**Proof.** At a stable partition, states in a common block have the same transition probability into every coarse block. Those probabilities define bar K_u and give the first commutation identity, by checking it on point-mass priors. Likelihood constancy on each block defines barℓ_y and gives the second identity by direct summation.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9ed61447cfccc1a3","tex":"\\bar K_u^\\top","aliases":"bar K_u^top bar top","display":"\\bar K_u^\\top","kind":"use","excerpt":"C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d97fba459e62ed0e","tex":"\\bar\\ell","aliases":"barℓ bar ell","display":"\\bar\\ell","kind":"use","excerpt":"C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ba673b1b51ec3e37","tex":"\\bar\\ell^\\top","aliases":"barℓ^top bar ell top","display":"\\bar\\ell^\\top","kind":"use","excerpt":"global factor gives commutation by summing the fine posterior. Conversely, compare priors supported on one state in each of two distinct coarse blocks. Equality of coarse posterior odds forces the ratio of those states' likelihoods to equal the ratio of the coarse likelihoods. Varying the representatives proves constancy within each block and the c","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1829c3cec6cf2065","tex":"\\bar\\ell_a","aliases":"barℓ_a bar ell","display":"\\bar\\ell_a","kind":"use","excerpt":"ℓ_i=cbarℓ_a for every i∈ B_a","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7a15d5df6692a6e6","tex":"\\bar\\ell_y","aliases":"barℓ_y bar ell","display":"\\bar\\ell_y","kind":"use","excerpt":"C_π K_u^top=bar K_u^top C_π, C_π B_ℓ_y(p) = B_barℓ_y(C_π p).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5bb0d822814471f6","tex":"\\boundary","aliases":"boundary","display":null,"kind":"use","excerpt":"[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"**Proposition 1. Exact realization and consistency.** The analysis map is injective, Π_a=A_aS_a is an idempotent with image C_a, and Π_ac=c exactly when c is a realizable coefficient vector. Moreover,","count":56},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"z'=Tz, B_s'=TB_sT^-1, λ'=λ T^-1, C'=TC.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5eda86663b484271","tex":"c^{\\mathsf T}","aliases":"c^ T mathsf","display":"c^{\\mathsf T}","kind":"use","excerpt":"**Proof.** A nonzero polynomial has strictly positive squared integral, so its coefficient vector c satisfies c^ TW_Dc>0. Thus W_D is invertible. Let T have rows t_i and S have columns h_j. Since H=TS is an invertible square core in an r-dimensional space, both T and S are invertible. Therefore G_a,jH^-1=TM_a,jT^-1. The relation M_a,j=∑_k(W_D)_jkC_","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4a41d044c4883dce","tex":"c^*","aliases":"c^*","display":"c^*","kind":"use","excerpt":"For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4db3245f7b23bd25","tex":"c^{-1}","aliases":"c^-1","display":"c^{-1}","kind":"use","excerpt":"Then S A=I and Ax ^2=c x ^2, as proved in [K1]. Here A S=c^-1 A A^* is the orthogonal projection onto the consistent coefficient subspace.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a71301c4ad4e741e","tex":"C_{\\mathcal J}","aliases":"C_ J mathcal","display":"C_{\\mathcal J}","kind":"use","excerpt":"C_ J={z: J(z)succeq0}, K_ J={z∈ C_ J:λ z=1}.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8659b56e0d40c668","tex":"C_{\\mathrm L}","aliases":"C_ L mathrm","display":"C_{\\mathrm L}","kind":"use","excerpt":"C_ L={(t,x,y):t≥sqrtx^2+y^2}","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"Let the initial summary be C_0x. Regard its rows as linear observables. Starting from W_0=row(C_0), define","count":17},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"c_1=∫|u| dμ, c_2=∫|L_2(u)| dμ.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"c_1=∫|u| dμ, c_2=∫|L_2(u)| dμ.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b60e443e8108308c","tex":"c_3","aliases":"c_3","display":"c_3","kind":"use","excerpt":"subspaceExact storage includes a consistency condition.P1 For linear analysis and synthesis maps A:X→ C and S:C→ X, the left-inverse law SA=I_X gives injectivity of A. The coefficient projector is Π=AS, with Π^2=Π, imΠ=im A, c is realizableiff Π c=c. diagram[0.90]01_consistency A redundant archive. Take X= R^2, C= R^3 and A(x_1,x_2)=(x_1,x_2,x_1+x_","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"use","excerpt":"Let X and C_a be finite-dimensional Hilbert spaces. A linear atlas consists of analysis and synthesis maps","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-520eda3321b864d7","tex":"\\mathcal C_a","aliases":"C_a mathcal","display":"\\mathcal C_a","kind":"use","excerpt":"The coefficients representing actual signals form C_a=imA_a. This distinction matters for redundant representations, where C_a is a proper subspace of the ambient coefficient space.","count":15},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"use","excerpt":"Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7d950b92d5f2adb7","tex":"C_{a,k}","aliases":"C_a,k","display":"C_{a,k}","kind":"use","excerpt":"B_a(u)=∑_k=0^D u^k C_a,k, M_a,j=∫_-1^1 u^j B_a(u) fracdu2.","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-33144b584150db3b","tex":"C_{a,n,k}","aliases":"C_a,n,k","display":"C_{a,n,k}","kind":"use","excerpt":"B_a^[n](x)=∫_0^1D_n(x,v)B_a(v) dv =∑_k=0^n b_n,k(x)C_a,n,k,","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-93dcb97ab3ce2c44","tex":"\\mathcal C_b","aliases":"C_b mathcal","display":"\\mathcal C_b","kind":"use","excerpt":"T_ba=A_bS_a: C_a⟶ C_b.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-baed60336596bbae","tex":"C_{c,H}","aliases":"C_c,H","display":"C_{c,H}","kind":"use","excerpt":"C_c,H={(m,x):m≥0, (x-mc)^ TH(x-mc)≤ m^2}.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fd685bf246955178","tex":"C_h","aliases":"C_h","display":"C_h","kind":"use","excerpt":"C_h={z:λ B_wz≥0 for every |w|≤ h}, C_0={z:λ z≥0}.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e45b8627b92d5b56","tex":"c_h","aliases":"c_h","display":"c_h","kind":"use","excerpt":"**Proof.** If ∑_hc_hv_h=0, then for every test t,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-184303c5df8e0a51","tex":"C_{h+1}","aliases":"C_h+1","display":"C_{h+1}","kind":"use","excerpt":"C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c99be13da294dc50","tex":"c_t","aliases":"c_t","display":"c_t","kind":"use","excerpt":"Its coefficients c_t=Ux_t reconstruct as x_t=U^*c_t. The same dynamics in that atlas are ULU^*, not a separate physical law. Thus coordinate transport preserves the full dynamics.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-42eb4bc96455bb91","tex":"C_\\pi","aliases":"C_π pi","display":"C_\\pi","kind":"use","excerpt":"C_π(a,i)= 1_i∈ B_a.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2b29a3476d110709","tex":"C_\\infty","aliases":"C_∞ infty","display":"C_\\infty","kind":"use","excerpt":"Their intersection C_∞ is the largest cone contained in C_0 and invariant under all B_s. If an exact equality C_h+1=C_h occurs, then C_h=C_∞ and the finite inequalities certify all future word probabilities. If the representation is observable, meaning that the rows λ B_w span E^*, then C_∞ is pointed and λ is strictly positive on its nonzero eleme","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d0fe2f8238bcf56d","tex":"\\operatorname{ch}","aliases":"ch operatorname","display":"\\operatorname{ch}","kind":"use","excerpt":"V(v)=min(c(v),∑_u∈ch(v)V(u))","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-491630d7018dba28","tex":"\\operatorname{conv}","aliases":"conv operatorname","display":"\\operatorname{conv}","kind":"use","excerpt":"Its columns generate C=V R_+^m and its normalized base is K=conv{v_1,…,v_m}. Finite generation makes the cone closed. Its normalized generators and full row rank make it pointed and generating, with a strictly positive mass functional.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"α≤cos(π/n),","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"A=PLP, B=PLQ, C=QLP, D=QLQ.","count":64},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e63d808025fba917","tex":"\\mathrm d","aliases":"d mathrm","display":"\\mathrm d","kind":"use","excerpt":"P_ d=frac121 · -1 -1 · 1","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"For a real d^2× d^2 symmetric matrix F, define","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-87a4659e9ddda329","tex":"D^j","aliases":"D^j","display":"D^j","kind":"use","excerpt":"plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, r","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2f80fcde2a727a6d","tex":"D^k","aliases":"D^k","display":"D^k","kind":"use","excerpt":"z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7cda35a215fda590","tex":"D^t","aliases":"D^t","display":"D^t","kind":"use","excerpt":"z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a12edcc891d90fb9","tex":"D^{t-1-s}","aliases":"D^t-1-s","display":"D^{t-1-s}","kind":"use","excerpt":"q_t=D^tq_0+∑_s=0^t-1D^t-1-sCz_s.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"(w_0+d_0)- w_rm sp+d_rm sp _2 ≥ w_0- w_rm sp _2-|d_0|- d_rm sp _2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","excerpt":"D_2=Z_+Z_-^-1 =0 · 1 1/12 · 5/6.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-943ac188116d74d8","tex":"d_a","aliases":"d_a","display":"d_a","kind":"use","excerpt":"**Proposition 73. Interior repair with an explicit error budget.** Suppose a reference family F^* is valid on L_d, and the combined mass and spatial error of widehat F-F^* is at most d_a(u) on normalized inputs. Then the margin of widetilde F_a(u) is at least","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2b84d7d41efdc0b1","tex":"d_b","aliases":"d_b","display":"d_b","kind":"use","excerpt":"If |Δ a|≤ d_a and |Δ b|≤ d_b, the conditional observation laws on any common normalized state obey","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-93b7f48d1ccd8315","tex":"d_{ba}","aliases":"d_ba","display":"d_{ba}","kind":"use","excerpt":"d_ba(x)=G_b(A_bx)-T_baG_a(A_ax).","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fdadfd14bd665b99","tex":"D_h","aliases":"D_h","display":"D_h","kind":"use","excerpt":"1. There is a linear map D_h:V_h→ V_h satisfying O_hL=D_hO_h. 2. ker O_h⊆ker(O_hL). 3. rankO_h+1=rankO_h.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d3405c1cc6903854","tex":"D_n","aliases":"D_n","display":"D_n","kind":"use","excerpt":"b_n,k(x)=binom nk x^k(1-x)^n-k, D_n(x,v)=(n+1)∑_k=0^n b_n,k(x)b_n,k(v).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8b56e5bc17f40a4f","tex":"d_{\\rm sp}","aliases":"d_rm sp rm","display":"d_{\\rm sp}","kind":"use","excerpt":"(w_0+d_0)- w_rm sp+d_rm sp _2 ≥ w_0- w_rm sp _2-|d_0|- d_rm sp _2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"use","excerpt":"**Proof.** Let u_h,widetilde u_h∈ C be the unnormalized states on each history at depth t, including the common policy factors. Their masses sum to one in each model. Define D_t=∑_h u_h-widetilde u_h _C. At a history with selected action a, write","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b377ac1e0c05ff0f","tex":"d_t","aliases":"d_t","display":"d_t","kind":"use","excerpt":"An empty sum is zero. Let d_t be the exact right-hand side minus this truncated right-hand side evaluated on the **true** history.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c631055a6cfad223","tex":"D_{t+1}","aliases":"D_t+1","display":"D_{t+1}","kind":"use","excerpt":"Sum the first term over observations and use Proposition 42. When λwidetilde u_h>0, normalize only to apply the definition of ϵ_t to the second term. Its sum is bounded by ϵ_tλwidetilde u_h. If the mass is zero, strict positivity implies widetilde u_h=0. Summing histories yields D_t+1≤ D_t+ϵ_t. Averaging over common policy probabilities gives the s","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f030fc249ff54ae2","tex":"D_{t,j}","aliases":"D_t,j","display":"D_{t,j}","kind":"use","excerpt":"R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"Since det Z_-=1/4, these windows span the declared two-dimensional state. Exact arithmetic gives","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"use","excerpt":"plate6Compression can erase a return pathA two-step effect can survive when both compressed steps vanish.P7 Let P^2=P and Q=I-P. Inserting I=P+Q gives the exact defect PLMP-(PLP)(PMP)=PLQMP. diagram06_return_path Take P=diag(1,0) and L=M=0 · 1","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"C_π B_ℓ(p)-bar b = frac1Z_r(diagP-bar bP^top)δ.","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"**Proposition 8. Minimal linear predictive completion.** The sequence stabilizes after at most d-dim W_0 strict enlargements. Its limit W_* is the smallest row space containing W_0 and invariant under right multiplication by every L_u. If R is any full-row-rank matrix whose rows form a basis of W_*, then","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-639ec78966670d7b","tex":"\\operatorname{do}","aliases":"do operatorname","display":"\\operatorname{do}","kind":"use","excerpt":"f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-78eeb6884762f77e","tex":"\\operatorname{dom}","aliases":"dom operatorname","display":"\\operatorname{dom}","kind":"use","excerpt":"dom(g∘ f) ={x∈domf:f(x)∈domg}.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"Since the active sets exhaust the grammar and the evidence sum is finite, Z_nuparrow Z and τ_ndownarrow0. This is posterior approximation *inside the declared grammar*. It does not establish that the real signal-generating mechanism belongs to it.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Within one orthogonal atlas, let E_i be pairwise orthogonal coordinate selectors with ∑_iE_i=I. Pull them back by a unitary map:","count":41},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"use","excerpt":"**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then","count":21},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ba01bfed40e94931","tex":"E^*","aliases":"E^*","display":"E^*","kind":"use","excerpt":"Their intersection C_∞ is the largest cone contained in C_0 and invariant under all B_s. If an exact equality C_h+1=C_h occurs, then C_h=C_∞ and the finite inequalities certify all future word probabilities. If the representation is observable, meaning that the rows λ B_w span E^*, then C_∞ is pointed and λ is strictly positive on its nonzero eleme","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-17ffbecfe58219ec","tex":"e^{7/10}","aliases":"e^7/10","display":"e^{7/10}","kind":"use","excerpt":"**Proof.** The positive exponential series gives e^7/10>1+7/10+(7/10)^2/2+(7/10)^3/6+(7/10)^4/24>2. Hence log 2<7/10, so log V+2log(1/δ)≤(7/10)(k+2h). Divide the scalar regression bound by sqrt V. For a nonnegative rational x, the smallest integer j with (j/10^12)^2≥ x supplies a rational upper square-root enclosure. Integer square root and an exac","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-eee73c0cc404c2bb","tex":"e^{h^2\\sigma^2/2}","aliases":"e^h^2σ^2/2 sigma","display":"e^{h^2\\sigma^2/2}","kind":"use","excerpt":"Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6674c4ab00789c40","tex":"e^{h\\xi_t}","aliases":"e^hξ_t xi","display":"e^{h\\xi_t}","kind":"use","excerpt":"Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-590691dd756441c8","tex":"E^\\top","aliases":"E^top top","display":"E^\\top","kind":"use","excerpt":"**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. squ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-252ea29e5371df47","tex":"e^{\\theta(X-m)}","aliases":"e^θ(X-m) theta","display":"e^{\\theta(X-m)}","kind":"use","excerpt":"E[e^θ(X-m)mid F]≤ e^θ^2/8.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6acc7f1bb681e78f","tex":"e^{\\theta^2/8}","aliases":"e^θ^2/8 theta","display":"e^{\\theta^2/8}","kind":"use","excerpt":"E[e^θ(X-m)mid F]≤ e^θ^2/8.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"Let E_0 bound the initialization errors at indices -H,…,0. Then the recursively generated predictions satisfy","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.","count":15},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-83ef2b537772a0a9","tex":"E_{00}","aliases":"E_00","display":"E_{00}","kind":"use","excerpt":"J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d67a776ac5013026","tex":"E_{00}^{\\mathsf T}","aliases":"E_00^ T mathsf","display":"E_{00}^{\\mathsf T}","kind":"use","excerpt":"J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-febc076ab7500943","tex":"E_{01}","aliases":"E_01","display":"E_{01}","kind":"use","excerpt":"J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d4789697f822b23f","tex":"E_{01}^{\\mathsf T}","aliases":"E_01^ T mathsf","display":"E_{01}^{\\mathsf T}","kind":"use","excerpt":"J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"δ=frac12000, E(u)=E_0+uE_1+u^2E_2,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e8883cee3967295b","tex":"e_1^\\top","aliases":"e_1^top top","display":"e_1^\\top","kind":"use","excerpt":"**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"δ=frac12000, E(u)=E_0+uE_1+u^2E_2,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4a01372424cd7344","tex":"e_A","aliases":"e_A","display":"e_A","kind":"use","excerpt":"For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c513478ff8678834","tex":"e_{a,0}","aliases":"e_a,0","display":"e_{a,0}","kind":"use","excerpt":"|[Δ_a(u)(1,v)]_0| · ≤ e_a,0(u), [Δ_a(u)(1,v)]_rm sp _2 · ≤ e_a,1(u).","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cee16670d4d4e6cf","tex":"e_{a,1}","aliases":"e_a,1","display":"e_{a,1}","kind":"use","excerpt":"|[Δ_a(u)(1,v)]_0| · ≤ e_a,0(u), [Δ_a(u)(1,v)]_rm sp _2 · ≤ e_a,1(u).","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-08b0505f4228a263","tex":"E_{a,k}","aliases":"E_a,k","display":"E_{a,k}","kind":"use","excerpt":"Δ_a(u)=∑_k=0^Du^kE_a,k, |(E_a,k)_ij|≤ r_k.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b6266893a62573e7","tex":"e_b","aliases":"e_b","display":"e_b","kind":"use","excerpt":"For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"Within one orthogonal atlas, let E_i be pairwise orthogonal coordinate selectors with ∑_iE_i=I. Pull them back by a unitary map:","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-33c182fa4faf27b6","tex":"e_i^{\\mathsf T}","aliases":"e_i^ T mathsf","display":"e_i^{\\mathsf T}","kind":"use","excerpt":"|e_i^ T(widehat C_a,k-C_a,k)x| ≤ ∑_j |(W^-1)_kj| β_a,j,i x _V_a,t^-1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ca7e75161b6bbf35","tex":"E_{ij}","aliases":"E_ij","display":"E_{ij}","kind":"use","excerpt":"where E_ij is the corresponding 2×2 matrix unit. The other rotate branch is zero. Thus this exact model has a finite rational matrix certificate without attenuation or a polygonal replacement.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a3fd533b471b8be8","tex":"e_{j-1}","aliases":"e_j-1","display":"e_{j-1}","kind":"use","excerpt":"**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-42fb8ac9b88e6468","tex":"e_M","aliases":"e_M","display":"e_M","kind":"use","excerpt":"M_widehat A- M_A^* ≤ e_M, σ_min( M_widehat A)>e_M.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ed9a375faaf2cfd0","tex":"\\mathbb E_{P,\\pi}","aliases":"E_P,π mathbb pi","display":"\\mathbb E_{P,\\pi}","kind":"use","excerpt":"P_T-Q_T _TV ≤min(1,∑_t=0^T-1 E_P,π[d_t(H_t,A_t)]).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-caa961746837d921","tex":"e_s","aliases":"e_s","display":"e_s","kind":"use","excerpt":"**Proof.** Subtract the two recurrences and bound every retained history error by the largest of the errors at indices t-H,…,t. This gives e_t+1≤ amax_t-H≤ s≤ te_s+Δ. The ball of radius E=max(E_0,Δ/(1-a)) is invariant because aE+Δ≤ E. Induction proves the bound. The estimate for Δ separates coefficient errors, the geometric kernel tail and the init","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3a8e0b23212d1b1a","tex":"e_T","aliases":"e_T","display":"e_T","kind":"use","excerpt":"e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2775a21153e9ebc4","tex":"e_t","aliases":"e_t","display":"e_t","kind":"use","excerpt":"e_t=PLQx_t, e_t ≤ PLQ Qx_t .","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0c0ec2f484593060","tex":"e_{t+1}","aliases":"e_t+1","display":"e_{t+1}","kind":"use","excerpt":"**Proof.** Add and subtract widehat f_t(γ_t(x_t)) to obtain e_t+1≤δ_t+λ_te_t. Repeated substitution yields the product-sum formula. Add and subtract H_t(γ_t(x_t)) for the output inequality. square","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ef7143ab1e6d192d","tex":"e_{T+2}","aliases":"e_T+2","display":"e_{T+2}","kind":"use","excerpt":"**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-aa178f33910340ff","tex":"\\mathbb E_\\nu","aliases":"E_ν mathbb nu","display":"\\mathbb E_\\nu","kind":"use","excerpt":"**Proof.** Substitute K=bar K+(K-bar K) and expand. The two cross terms integrate to zero. The remaining integrand is a positive congruence when X is positive. For an affine matrix function the centered term is (u- E_ν u)K_1, giving the variance formula. Integrating the square of a centered uniform variable gives h^2/12. square","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"dom(g∘ f) ={x∈domf:f(x)∈domg}.","count":27},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a3382034c436fa8d","tex":"F^*","aliases":"F^*","display":"F^*","kind":"use","excerpt":"**Proposition 73. Interior repair with an explicit error budget.** Suppose a reference family F^* is valid on L_d, and the combined mass and spatial error of widehat F-F^* is at most d_a(u) on normalized inputs. Then the margin of widetilde F_a(u) is at least","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","excerpt":"∑_aw_a L-L_a _F^2","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0220b96f493cc7b2","tex":"F_{\\mathcal I}","aliases":"F_ I mathcal","display":"F_{\\mathcal I}","kind":"use","excerpt":"with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-720394ab6129c5b1","tex":"F_{(i,p),(j,q)}","aliases":"F_(i,p),(j,q)","display":"F_{(i,p),(j,q)}","kind":"use","excerpt":"[Φ_F(X)]_ij=∑_p,q=1^d F_(i,p),(j,q)X_pq.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-45843ed5f03c5c85","tex":"f_{a,s}","aliases":"f_a,s","display":"f_{a,s}","kind":"use","excerpt":"w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-54b44dec26d2ba54","tex":"F_b","aliases":"F_b","display":"F_b","kind":"use","excerpt":"T_aa=I, T_cbT_ba=T_ca, F_bT_ba=T_baF_a.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-30ade5d863641faa","tex":"F_s","aliases":"F_s","display":"F_s","kind":"use","excerpt":"widetilde H=H+E, widetilde G_s=G_s+F_s, E _2<s_H.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","excerpt":"Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f79f277673daa5a5","tex":"f_t","aliases":"f_t","display":"f_t","kind":"use","excerpt":"z_t+1=Az_t+∑_k=0^t-1K_kz_t-1-k+f_t, K_k=BD^kC, f_t=BD^tq_0.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4a14d2ab0a464e3b","tex":"F_u","aliases":"F_u","display":"F_u","kind":"use","excerpt":"Let γ:X→ Z=γ(X) be a summary and let F_u:X→ X describe a deterministic transition for action u. The reduced description closes when there is bar F_u:Z→ Z such that","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d1c09be9dde788b9","tex":"F_\\eta","aliases":"F_η eta","display":"F_\\eta","kind":"use","excerpt":"F_η=(1-η)widehat F+η N, 0≤η≤1.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Conversely, suppose maps G_a: C_a→ C_a satisfy G_bT_ba=T_baG_a for every pair. Then there is a unique map F:X→ X with G_a=A_aFS_a for every a.","count":37},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then","count":12},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"dom(g∘ f) ={x∈domf:f(x)∈domg}.","count":35},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-02b0b5ff2a25a5d9","tex":"g'","aliases":"g'","display":"g'","kind":"use","excerpt":"**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-eb695a8d9657252e","tex":"g''","aliases":"g''","display":"g''","kind":"use","excerpt":"**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"The fixed prior need not be uniform. For an explicit enumeration g_1,g_2,…, π(g_k)=2^-k is a normalized example, with omitted prior mass 2^-n after n programs. It expresses a particular enumeration bias, not a canonical prior on mathematical meaning.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e6df94744f7ade62","tex":"g_2","aliases":"g_2","display":"g_2","kind":"use","excerpt":"The fixed prior need not be uniform. For an explicit enumeration g_1,g_2,…, π(g_k)=2^-k is a normalized example, with omitted prior mass 2^-n after n programs. It expresses a particular enumeration bias, not a canonical prior on mathematical meaning.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","excerpt":"Conversely, suppose maps G_a: C_a→ C_a satisfy G_bT_ba=T_baG_a for every pair. Then there is a unique map F:X→ X with G_a=A_aFS_a for every a.","count":12},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"N_a(u)=g_a(u)e_0λ, g_a(u)≥ g_min>0, ∫ g_a(u) du/2=1.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-46275dcab105c413","tex":"G_{a,0}","aliases":"G_a,0","display":"G_{a,0}","kind":"use","excerpt":"Suppose each action has constant positive semidefinite matrices G_a,0,G_a,1, with the second term omitted when q=0, such that","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ba39f280196fc86c","tex":"G_{a,1}","aliases":"G_a,1","display":"G_{a,1}","kind":"use","excerpt":"Suppose each action has constant positive semidefinite matrices G_a,0,G_a,1, with the second term omitted when q=0, such that","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5c4ad32f6de177b3","tex":"G_{a,j}","aliases":"G_a,j","display":"G_{a,j}","kind":"use","excerpt":"H_ij=t_i h_j, (G_a,j)_iℓ=t_i M_a,jh_ℓ.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-76b53653bff85b9e","tex":"G_b","aliases":"G_b","display":"G_b","kind":"use","excerpt":"Conversely, suppose maps G_a: C_a→ C_a satisfy G_bT_ba=T_baG_a for every pair. Then there is a unique map F:X→ X with G_a=A_aFS_a for every a.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"use","excerpt":"g_i^topwidehat B_sv_j ≥ϵ_s g_i _2 v_j _2 for all s,i,j.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-91d93f5104501516","tex":"g_i^\\top","aliases":"g_i^top top","display":"g_i^\\top","kind":"use","excerpt":"For a complete facet description C={z:Gz≥0}, let g_i^top be its rows. A sufficient norm-ball certificate around widehat B_s is","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f7288fb9331c8dd5","tex":"g_j","aliases":"g_j","display":"g_j","kind":"use","excerpt":"**Proposition 24. Prior-preserving program refinement.** Suppose a parent program g is replaced by children g_j with weights α_j≥0, ∑_jα_j=1, and suppose its likelihood is the mixture","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c1c366f26da43334","tex":"g_k","aliases":"g_k","display":"g_k","kind":"use","excerpt":"The fixed prior need not be uniform. For an explicit enumeration g_1,g_2,…, π(g_k)=2^-k is a normalized example, with omitted prior mass 2^-n after n programs. It expresses a particular enumeration bias, not a canonical prior on mathematical meaning.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-edd10ee861d1dce0","tex":"g_{\\min}","aliases":"g_min min","display":"g_{\\min}","kind":"use","excerpt":"N_a(u)=g_a(u)e_0λ, g_a(u)≥ g_min>0, ∫ g_a(u) du/2=1.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-21383e2af4ae11bd","tex":"\\mathcal G_n","aliases":"G_n mathcal","display":"\\mathcal G_n","kind":"use","excerpt":"Let finite active sets G_n increase to G. Their unnormalized evidence is Z_n=∑_g∈ G_nπ(g)L_g(D). Only stages with Z_n>0 are considered.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3f8720ad3f121275","tex":"G_r","aliases":"G_r","display":"G_r","kind":"use","excerpt":"**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6bcb32fb1f100c04","tex":"G_s","aliases":"G_s","display":"G_s","kind":"use","excerpt":"(G_s)_ij=f(h_jst_i), (b_0)_i=f(t_i).","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-29c71f678c07a36a","tex":"g_\\nu","aliases":"g_ν nu","display":"g_\\nu","kind":"use","excerpt":"Φ_u(X)=∑_ν w_ν g_ν(u)K_ν(u)XK_ν(u)^ T, g_ν(u)∈{1,1-u^2}.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"C_0=HR, RL_u=B_uR","count":92},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"For intervention evidence use the information H_t-1 just before choosing the action. Let the action be randomized from a known history-dependent policy π_t. The true next-observation kernel under action a is P_t(·mid H_t-1,a). Randomization uses no unrecorded future information. For a predictable test ϕ_t∈[0,1], define","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c3d50e4b12d0681c","tex":"H^*","aliases":"H^*","display":"H^*","kind":"use","excerpt":"Use the Frobenius norm on symmetric matrices and the induced product norm. Assume M_A^* is injective and M_A^*(H^*)=(0,1)=b. This fixes the scale of a one-dimensional invariant-form space.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-daaa917c98a2902e","tex":"H^{-1}","aliases":"H^-1","display":"H^{-1}","kind":"use","excerpt":"B_s=G_sH^-1, λ=e_1^top, b_0=(f(t_i))_i","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":"The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"**Proposition 63. Conditional operator-only ellipsoid discovery.** Suppose the symmetric solution space of A^ THA=H is one-dimensional and contains a positive definite matrix H_0. Let a rank-one fitted branch at a declared observation have a nonzero-mass image, normalized to v=(1,x_v). If x_v≠ c, define","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"Select r tests t_1,…,t_r and r histories h_1,…,h_r, with t_1=ϵ, for which the square matrix","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","excerpt":"plate10Behavior can supply the coordinatesFuture tests define a predictive state without naming hidden variables.P27--P28 For a controlled process, arrange population probabilities into a behavior table: columns index past histories and rows index future tests. Its full column span is the linear predictive behavior space. c|ccc · h_1 · h_2 · ⋯ τ_1 ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-47802fa794f90e36","tex":"h_{a,s}","aliases":"h_a,s","display":"h_{a,s}","kind":"use","excerpt":"w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f74f6a57337ee481","tex":"H_{ij}","aliases":"H_ij","display":"H_{ij}","kind":"use","excerpt":"H_ij=f(h_jt_i)","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"use","excerpt":"H_ij=f(h_jt_i)","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ea994960bd5b4135","tex":"h_{j-1}","aliases":"h_j-1","display":"h_{j-1}","kind":"use","excerpt":"Pr_π(w)=f(w)∏_j=1^|w|π(a_jmid h_j-1),","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3e01b0cedad21f47","tex":"h_r","aliases":"h_r","display":"h_r","kind":"use","excerpt":"Select r tests t_1,…,t_r and r histories h_1,…,h_r, with t_1=ϵ, for which the square matrix","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-619ba501c7bc28e9","tex":"H_s","aliases":"H_s","display":"H_s","kind":"use","excerpt":"There is a dual finite test. If a complete inequality description of the same cone is C={z:Gz≥0}, then GB_sv_j≥0 for every generator j is equivalent to invariance. One may also certify GB_s=H_sG with H_s≥0. The latter certificate is sufficient by direct multiplication and necessary by the polyhedral cone form of Farkas' lemma [R13]. The reference c","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-814f64e7253b7d27","tex":"H_t","aliases":"H_t","display":"H_t","kind":"use","excerpt":"Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-480c877d455d2c2b","tex":"h_t","aliases":"h_t","display":"h_t","kind":"use","excerpt":"Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b6d6d2b7cd80f373","tex":"H_{t,h}","aliases":"H_t,h","display":"H_{t,h}","kind":"use","excerpt":"mathscr H_t,h=f(ht).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-edcbd44645488c8a","tex":"\\mathcal H_{t-1}","aliases":"H_t-1 mathcal","display":"\\mathcal H_{t-1}","kind":"use","excerpt":"For intervention evidence use the information H_t-1 just before choosing the action. Let the action be randomized from a known history-dependent policy π_t. The true next-observation kernel under action a is P_t(·mid H_t-1,a). Randomization uses no unrecorded future information. For a predictable test ϕ_t∈[0,1], define","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-858e4bb65a33c6eb","tex":"h_\\ell","aliases":"h_ℓ ell","display":"h_\\ell","kind":"use","excerpt":"H_ij=t_i h_j, (G_a,j)_iℓ=t_i M_a,jh_ℓ.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"c|rrrrr u · -1 · -1/2 · 0 · 1/2 · 1 μ{u} · 7/90 · 16/45 · 2/15 · 16/45 · 7/90.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.","count":62},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3d2972c83781b21d","tex":"\\boldsymbol i","aliases":"i boldsymbol","display":"\\boldsymbol i","kind":"use","excerpt":"with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"use","excerpt":"with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-efff486319d6caab","tex":"I_{\\mathcal C_a}","aliases":"I_ C_a mathcal","display":"I_{\\mathcal C_a}","kind":"use","excerpt":"plate2One operation, several coordinatesA commuting square expresses coordinate independence.P2 Let C_a=im A_a and T_ba=A_bS_a. Coordinate realizations are F_a=A_aFS_a on C_a. Both paths through the square represent the same underlying operation. diagram02_transport T_cbT_ba=T_ca, T_aa=I_ C_a, F_bT_ba=T_baF_a. Conversely, a compatible family G_a on","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b684226ff2428045","tex":"I_{d^2}","aliases":"I_d^2","display":"I_{d^2}","kind":"use","excerpt":"Z_q(u)=I_d^2 uI_d^2 vdots u^qI_d^2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0625c4dc07eb960a","tex":"i_j","aliases":"i_j","display":"i_j","kind":"use","excerpt":"F(x)-F_ I(x) ≤∑_m=1^p ∑_ i∉ I_m β_m, i∏_j=1^ma_i_j.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-62ec9ecbb75660b6","tex":"\\mathcal I_m","aliases":"I_m mathcal","display":"\\mathcal I_m","kind":"use","excerpt":"with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2ee8ac8c8d5446ba","tex":"i_m","aliases":"i_m","display":"i_m","kind":"use","excerpt":"Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e9855187f190ff70","tex":"I_r","aliases":"I_r","display":"I_r","kind":"use","excerpt":"**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. squ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"use","excerpt":"A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"The coefficients representing actual signals form C_a=imA_a. This distinction matters for redundant representations, where C_a is a proper subspace of the ambient coefficient space.","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"x _C=inf{λ u+λ v:x=u-v, u,v∈ C}.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-06f9a3ba03a67bc4","tex":"\\mathrm{init}","aliases":"init mathrm","display":"\\mathrm{init}","kind":"use","excerpt":"where B ranges over the current partition. The process terminates after at most |Ω|-|π_init| strict refinement rounds. The final partition is the unique coarsest refinement of π_0 on which all declared predictions and Bayesian updates close for every prior.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"This is a conservation law for information under complete refinement. Deleting a child, storing only its energy, or imposing a coefficient threshold changes the contract. For a retained subset J of an orthogonal decomposition, the lost signal is exactly ∑_i∉ JP_ix, and its squared norm is the sum of the omitted packet energies.","count":15},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-146cbf287aa37a5f","tex":"\\mathcal J","aliases":"J mathcal","display":"\\mathcal J","kind":"use","excerpt":"J:E⟶ S^d, λ z=tr J(z).","count":30},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-37959be93056eb7b","tex":"\\mathcal J'","aliases":"J' mathcal","display":"\\mathcal J'","kind":"use","excerpt":"B'_s=TB_sT^-1, J'= JT^-1, λ'=λ T^-1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-07b6123d41bb6991","tex":"J_t","aliases":"J_t","display":"J_t","kind":"use","excerpt":"For adaptive prediction, an attempted lift J_t contributes its own initialization defect","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix","count":75},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"K={v∈ C:λ v=1}","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"For another linear map M:Y→ Z and output projectors R_k,","count":75},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b414e0d8b85d3e26","tex":"K^{\\mathsf T}","aliases":"K^ T mathsf","display":"K^{\\mathsf T}","kind":"use","excerpt":"Every term wKXK^ T is positive semidefinite when X is. The intertwining identity therefore proves invariance. Iteration gives invariance of every branch state, and the mass identity normalizes every finite observation tree. Substituting the transported quantities proves the final statement. square","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d61145324c98def3","tex":"K_{\\mathcal J}","aliases":"K_ J mathcal","display":"K_{\\mathcal J}","kind":"use","excerpt":"C_ J={z: J(z)succeq0}, K_ J={z∈ C_ J:λ z=1}.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"K_0={(1,q)^top:3/8≤ q≤5/8}.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cf5fcf757b255a84","tex":"K_1","aliases":"K_1","display":"K_1","kind":"use","excerpt":"The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c9ea3477ff6a6c05","tex":"K_1^{\\mathsf T}","aliases":"K_1^ T mathsf","display":"K_1^{\\mathsf T}","kind":"use","excerpt":"The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ca931e9dd55037b6","tex":"K_j","aliases":"K_j","display":"K_j","kind":"use","excerpt":"plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, r","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-86cf1e6739ff675f","tex":"K_k","aliases":"K_k","display":"K_k","kind":"use","excerpt":"z_t+1=Az_t+∑_k=0^t-1K_kz_t-1-k+f_t, K_k=BD^kC, f_t=BD^tq_0.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6af84895f7726912","tex":"K_{sj}","aliases":"K_sj","display":"K_{sj}","kind":"use","excerpt":"J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-69d70cc4edb5e04c","tex":"K_{sj}^{\\mathsf T}","aliases":"K_sj^ T mathsf","display":"K_{sj}^{\\mathsf T}","kind":"use","excerpt":"J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fd51be9cb0ff1729","tex":"K_T","aliases":"K_T","display":"K_T","kind":"use","excerpt":"K_T={T^-1(1,v): v _2≤1}.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-16bc7321a3832aa5","tex":"K_u","aliases":"K_u","display":"K_u","kind":"use","excerpt":"Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3d7241b33bc6bce1","tex":"K_u^\\top","aliases":"K_u^top top","display":"K_u^\\top","kind":"use","excerpt":"Distributions are column vectors. Prediction is p↦ K_u^top p, and a Bayesian update is","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8cf427d15f7dc23b","tex":"K_\\nu","aliases":"K_ν nu","display":"K_\\nu","kind":"use","excerpt":"F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0b1528636175fa07","tex":"K_\\nu^{\\mathsf T}","aliases":"K_ν^ T nu mathsf","display":"K_\\nu^{\\mathsf T}","kind":"use","excerpt":"then Φ_F(X)=∑_ν w_ν K_ν X K_ν^ T. Every positive semidefinite F has a real factorization of this kind. The matrix is the usual Choi/Gram representation, with an explicit index convention [R16].","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"Its annihilator is the largest L-invariant subspace inside ker C_0, the standard unobservable-subspace construction [R2]. With several actions, observables pulled back through action words are required, but the linear span still stabilizes in finite dimension.","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"Let L:X→ Y be linear, with orthogonal input projectors P_i and output projectors Q_j. Its transfer blocks are L_ji=Q_jLP_i.","count":130},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"Let L_d={(m,v):m≥ v _2}, with mass row λ=(1,0,…,0). A mass preserving invertible matrix T transports a proposed predictive domain to this cone. Its normalized base in the original coordinates is","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5c373cbbb5a44e89","tex":"\\mathrm L","aliases":"L mathrm","display":"\\mathrm L","kind":"use","excerpt":"C_ L={(t,x,y):t≥sqrtx^2+y^2}","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-34c05dc8f5073293","tex":"L'","aliases":"L'","display":"L'","kind":"use","excerpt":"L'=0 · 1000 0 · 0, x=(0,1/1000)^top,","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7576dd58cb440d5d","tex":"L^{\\mathsf T}","aliases":"L^ T mathsf","display":"L^{\\mathsf T}","kind":"use","excerpt":"A rationalized invertible factor L specifies H=L^ TL exactly. Positive definiteness is then an algebraic fact even when the factor was suggested by floating-point calculations. It still does not establish operator invariance.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"use","excerpt":"An ordinary orthogonal truncation need not preserve that validity. The valid density p(v)=3v^2 has degree-one L^2 projection","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-20291439ebfc6925","tex":"L^d","aliases":"L^d","display":"L^d","kind":"use","excerpt":",CL^h. All except the final block already occur in O_h. The kernel condition is therefore equivalent to CL^h being in the row space of O_h, which is exactly the rank equality. It also implies O_hLx∈ V_h because this vector is O_h applied to Lx. Uniqueness follows because every element of V_h has a representative. The history recurrence follows dire","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-855799c61af7839f","tex":"L^{d-1}","aliases":"L^d-1","display":"L^{d-1}","kind":"use","excerpt":"C_0, C_0L, …, C_0L^d-1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cc71270a77e70256","tex":"L^h","aliases":"L^h","display":"L^h","kind":"use","excerpt":"**Proof.** The equivalence of the first two statements is the linear factorization criterion: define D_h(O_hx)=O_hLx and require independence from the representative x. The rows of O_hL are CL,…,CL^h. All except the final block already occur in O_h. The kernel condition is therefore equivalent to CL^h being in the row space of O_h, which is exactly","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-33e50eabe203c10a","tex":"L^{h-1}","aliases":"L^h-1","display":"L^{h-1}","kind":"use","excerpt":"O_h=C CL vdots CL^h-1, V_h=imO_h.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9aab632d65fa585e","tex":"L^k","aliases":"L^k","display":"L^k","kind":"use","excerpt":"L_nx⟶ Lx, L_n^kx⟶ L^kx.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5e8f887aff23e701","tex":"L^{k+1}","aliases":"L^k+1","display":"L^{k+1}","kind":"use","excerpt":"(L_n^k+1-L^k+1)x =L_n(L_n^k-L^k)x+(L_n-L)L^kx.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-54b7c3029a97f1e5","tex":"L^t","aliases":"L^t","display":"L^t","kind":"use","excerpt":"**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5b473ace67384bce","tex":"L^{T+1}","aliases":"L^T+1","display":"L^{T+1}","kind":"use","excerpt":"**Proof.** One system has scalar output identically zero. For the other, take dimension T+2, initial state x_0=e_T+2, observation row C=e_1^top, and shift Le_j=e_j-1 for j≥2, Le_1=0. Then CL^tx_0=0 for 0≤ t≤ T and CL^T+1x_0=1. The shift is rational and nilpotent. square","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-73f17951391086e7","tex":"L_*","aliases":"L_*","display":"L_*","kind":"use","excerpt":"over linear maps on X, where · _F is the Frobenius norm. For any target operator L_* and any chosen operator norm,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"v(u)=(1,u,u^2)^ T, L_2(u)=frac3u^2-12.","count":15},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c7f6796e75b9a514","tex":"L_3","aliases":"L_3","display":"L_3","kind":"use","excerpt":"L_3(u)=frac5u^3-3u2, p_±(u)=1±ϵ L_3(u), 0<ϵ<1.","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a11ee0ad40de6dac","tex":"L_a","aliases":"L_a","display":"L_a","kind":"use","excerpt":"L_a=S_aG_aA_a.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ee81616801f13988","tex":"\\mathcal L_d","aliases":"L_d mathcal","display":"\\mathcal L_d","kind":"use","excerpt":"Let L_d={(m,v):m≥ v _2}, with mass row λ=(1,0,…,0). A mass preserving invertible matrix T transports a proposed predictive domain to this cone. Its normalized base in the original coordinates is","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ff93f51daddcc717","tex":"L_g","aliases":"L_g","display":"L_g","kind":"use","excerpt":"Let G be a countable declared program set, with one fixed probability mass function π(g)>0 satisfying ∑_gπ(g)=1. Every program specifies a likelihood L_g(D) for the **same** observed data event or density relative to a common base measure. Assume","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d6e4c227fe804cbf","tex":"L_{g_j}","aliases":"L_g_j","display":"L_{g_j}","kind":"use","excerpt":"L_g(D)=∑_jα_jL_g_j(D).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f356874ec402cb6b","tex":"L_{ji}","aliases":"L_ji","display":"L_{ji}","kind":"use","excerpt":"Let L:X→ Y be linear, with orthogonal input projectors P_i and output projectors Q_j. Its transfer blocks are L_ji=Q_jLP_i.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ed76d6b3d182baf7","tex":"L_n","aliases":"L_n","display":"L_n","kind":"use","excerpt":"**Proposition 13. Strong fixed-horizon refinement limit.** Let X be a Hilbert space, L:X→ X bounded, and P_n orthogonal projections converging strongly to I. Put L_n=P_nLP_n. For every fixed x∈ X and every fixed nonnegative integer k,","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7cb7fe6212128e96","tex":"L_n^k","aliases":"L_n^k","display":"L_n^k","kind":"use","excerpt":"L_nx⟶ Lx, L_n^kx⟶ L^kx.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-52773318441cd873","tex":"L_n^{k+1}","aliases":"L_n^k+1","display":"L_n^{k+1}","kind":"use","excerpt":"(L_n^k+1-L^k+1)x =L_n(L_n^k-L^k)x+(L_n-L)L^kx.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-49e6cb003fe8b536","tex":"L_t","aliases":"L_t","display":"L_t","kind":"use","excerpt":"e_t+1≤α_te_t+β_t, α_t=frac(1+R λ _*)L_twidehatη_t, β_t=frac(1+R λ _*)ϵ_B,tRwidehatη_t.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-40854cd2aa39fed6","tex":"L_u","aliases":"L_u","display":"L_u","kind":"use","excerpt":"There is a constructive finite solution when the dynamics are a known finite family of linear maps L_u on a d-dimensional space. It identifies which additional observables are required, rather than merely requesting a larger state vector.","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"**Proposition 84. Exact rational outer confidence intervals.** Write V=1+∑ϕ_t^2 and widehatθ=∑ϕ_tY_t/V. Let k=lceillog_2 Vrceil and h=lceillog_2(1/δ)rceil. A rational upper enclosure of","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9293b5888c03b9e0","tex":"\\operatorname{Lip}","aliases":"Lip operatorname","display":"\\operatorname{Lip}","kind":"use","excerpt":"Lip(F_ I) ≤∑_m=1^p ∑_ i∈ I_m mβ_m, iR^m-1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"|widehat f(w)-f(w)|≤ ϵ_N, ϵ_N=sqrtfraclog(2M/δ)2N.","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"**Proposition 84. Exact rational outer confidence intervals.** Write V=1+∑ϕ_t^2 and widehatθ=∑ϕ_tY_t/V. Let k=lceillog_2 Vrceil and h=lceillog_2(1/δ)rceil. A rational upper enclosure of","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"γ(x)=γ(x') Longrightarrow γ(F_u(x))=γ(F_u(x'))","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"For another linear map M:Y→ Z and output projectors R_k,","count":57},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"Let M_A map symmetric matrices into a Euclidean product space by","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"F(x)=b+∑_m=1^pB_m(x,…,x),","count":70},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"det J=fracm^2-x^2-y^24.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-012a3d7e3e6f64b4","tex":"m_{\\mathcal F}","aliases":"m_ F mathcal","display":"m_{\\mathcal F}","kind":"use","excerpt":"**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3cc2b9ab727bf366","tex":"m_{\\mathcal G}","aliases":"m_ G mathcal","display":"m_{\\mathcal G}","kind":"use","excerpt":"**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","excerpt":"M_t= (fracdet(γ I)det V_t)^1/2 exp(frac S_t _V_t^-1^22σ^2), M_0=1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":"An empty maximum is zero. If every negative entry has magnitude at most d and every reference entry is at least m_0>0, the simpler sufficient choice is η=d/(m_0+d). These statements follow by solving each scalar inequality. Mass conservation is unchanged by convex mixing.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c887199df3da3b63","tex":"M_a","aliases":"M_a","display":"M_a","kind":"use","excerpt":"For Euclidean coefficient errors, conditioning remains important. If A_a has smallest singular value at least m_a>0 and largest singular value at most M_a, its canonical synthesis A_a^† has norm at most 1/m_a, and T_ba ≤ M_b/m_a. An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis q","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8ace55e8bb71177a","tex":"\\mathcal M_A","aliases":"M_A mathcal","display":"\\mathcal M_A","kind":"use","excerpt":"Let M_A map symmetric matrices into a Euclidean product space by","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bc4c24fa42db60db","tex":"m_a","aliases":"m_a","display":"m_a","kind":"use","excerpt":"For Euclidean coefficient errors, conditioning remains important. If A_a has smallest singular value at least m_a>0 and largest singular value at most M_a, its canonical synthesis A_a^† has norm at most 1/m_a, and T_ba ≤ M_b/m_a. An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis q","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-14c42d23ccae22b4","tex":"M_{a,j}","aliases":"M_a,j","display":"M_{a,j}","kind":"use","excerpt":"B_a(u)=∑_k=0^D u^k C_a,k, M_a,j=∫_-1^1 u^j B_a(u) fracdu2.","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-63846c145ddd5b55","tex":"M_{a,o}","aliases":"M_a,o","display":"M_{a,o}","kind":"use","excerpt":"1^top∑_oM_a,o=1^top.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d687d04a43757b9c","tex":"\\mathcal M_{A^*}","aliases":"M_A^* mathcal","display":"\\mathcal M_{A^*}","kind":"use","excerpt":"Use the Frobenius norm on symmetric matrices and the induced product norm. Assume M_A^* is injective and M_A^*(H^*)=(0,1)=b. This fixes the scale of a one-dimensional invariant-form space.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-284d7acc4980e163","tex":"M_b","aliases":"M_b","display":"M_b","kind":"use","excerpt":"For Euclidean coefficient errors, conditioning remains important. If A_a has smallest singular value at least m_a>0 and largest singular value at most M_a, its canonical synthesis A_a^† has norm at most 1/m_a, and T_ba ≤ M_b/m_a. An arbitrary left inverse can behave much worse away from the consistent coefficient space, so the canonical-synthesis q","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-17770225492e1d01","tex":"M_s","aliases":"M_s","display":"M_s","kind":"use","excerpt":"A concrete realization has a belief p∈ R^n_+ with 1^top p=1. For each s=(a,o), let M_s∈ R^n× n be entrywise nonnegative, with","count":14},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-82f744f1968fc463","tex":"m_s","aliases":"m_s","display":"m_s","kind":"use","excerpt":"J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9043994a5d1f0bad","tex":"m_{s,j}","aliases":"m_s,j","display":"m_{s,j}","kind":"use","excerpt":"R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bcd9236fea7e4c20","tex":"M_{s_1}","aliases":"M_s_1","display":"M_{s_1}","kind":"use","excerpt":"Put M_w=M_s_k⋯ M_s_1. Then","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8cab0eb15393469f","tex":"M_{s_k}","aliases":"M_s_k","display":"M_{s_k}","kind":"use","excerpt":"Put M_w=M_s_k⋯ M_s_1. Then","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9de6a9970cb6757b","tex":"M_t","aliases":"M_t","display":"M_t","kind":"use","excerpt":"M_t= (fracdet(γ I)det V_t)^1/2 exp(frac S_t _V_t^-1^22σ^2), M_0=1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f858605b2ad1c07f","tex":"m_{t,j}","aliases":"m_t,j","display":"m_{t,j}","kind":"use","excerpt":"X_t,j=ϕ_t,j(Y_t), m_t,j= E[X_t,jmid F_t-1], p_t,j=the model's prediction for that test.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e3e697d6b6f9bc2b","tex":"M_w","aliases":"M_w","display":"M_w","kind":"use","excerpt":"Put M_w=M_s_k⋯ M_s_1. Then","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-af6b6e337c90c496","tex":"\\mathcal M_{\\widehat A}","aliases":"M_widehat A mathcal widehat","display":"\\mathcal M_{\\widehat A}","kind":"use","excerpt":"**Proposition 74. A posteriori proposal error.** Given widehat H, let r= M_widehat A(widehat H)-b. Suppose","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"use","excerpt":"mathscr H_t,h=f(ht).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"sup_t≥0 z_t-widehat z_t ≤max(E_0,fracΔ1-a).","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-eec18b0187f98c2c","tex":"\\max_a","aliases":"max_a max","display":"\\max_a","kind":"use","excerpt":"fracLsqrt3(n+2)+ max_afrac1n+1∑_k=0^nη_a,k.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5531d6c3b64bdcac","tex":"\\max_{s,i,j:(A_s)_{ij}<0}","aliases":"max_s,i,j:(A_s)_ij<0 max","display":"\\max_{s,i,j:(A_s)_{ij}<0}","kind":"use","excerpt":"η≥max_s,i,j:(A_s)_ij<0 frac-(A_s)_ij(N_s)_ij-(A_s)_ij.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-554e55435acd155d","tex":"\\max_{t-H\\leq s\\leq t}","aliases":"max_t-H≤ s≤ t max leq leq","display":"\\max_{t-H\\leq s\\leq t}","kind":"use","excerpt":"**Proof.** Subtract the two recurrences and bound every retained history error by the largest of the errors at indices t-H,…,t. This gives e_t+1≤ amax_t-H≤ s≤ te_s+Δ. The ball of radius E=max(E_0,Δ/(1-a)) is invariant because aE+Δ≤ E. Induction proves the bound. The estimate for Δ separates coefficient errors, the geometric kernel tail and the init","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"V(v)=min(c(v),∑_u∈ch(v)V(u))","count":36},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d54dca6b8f2ff18f","tex":"\\min_{\\|v\\|_2\\leq1}","aliases":"min_ v _2≤1 min leq","display":"\\min_{\\|v\\|_2\\leq1}","kind":"use","excerpt":"σ_a(u)=min_ v _2≤1 [(F_a(u)(1,v))_0- (F_a(u)(1,v))_rm sp _2].","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7375211c227de316","tex":"\\min_{a,b\\geq0}","aliases":"min_a,b≥0 min geq","display":"\\min_{a,b\\geq0}","kind":"use","excerpt":"x _C=min_a,b≥0 {1^top(a+b):V(a-b)=x}.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.","count":21},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"W_n+1=W_n+∑_uW_nL_u, W_nL_u={wL_u:w∈ W_n}.","count":101},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"frac34n^2","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d583dfc77954957f","tex":"N_a","aliases":"N_a","display":"N_a","kind":"use","excerpt":"N_a(u)=g_a(u)e_0λ, g_a(u)≥ g_min>0, ∫ g_a(u) du/2=1.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2f56096dd29a0f85","tex":"N_s","aliases":"N_s","display":"N_s","kind":"use","excerpt":"When V is square and invertible, put A_s=V^-1widehat B_sV. Assume mass conservation has already been imposed. Let N_s be an entrywise positive reference instrument with the same per-action mass identities. For 0≤η≤1, the mixture","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"O_h=C CL vdots CL^h-1, V_h=imO_h.","count":25},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.","count":29},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5620c0668bf9837f","tex":"O^{-1}","aliases":"O^-1","display":"O^{-1}","kind":"use","excerpt":"For the running example, the rows of O=1 · 0 1/2 · 1, D=OLO^-1=0 · 1 1/12 · 5/6 span W_*= R^2 *. The state Ox_s=(y_s,y_s+1)^ T therefore closes. claimThe added coordinate is forced by the action of L on the requested observable. In a larger system, unobserved directions unrelated to its future can remain outside W_*. boundaryComputing Ox_s uses a f","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-768cf99fc34f1b8e","tex":"O_0","aliases":"O_0","display":"O_0","kind":"use","excerpt":"O_0=diag(1/4,3/4), O_1=diag(3/4,1/4), M_a,o=O_oT_a.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-090aecb8fb37e7b7","tex":"O_1","aliases":"O_1","display":"O_1","kind":"use","excerpt":"O_0=diag(1/4,3/4), O_1=diag(3/4,1/4), M_a,o=O_oT_a.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e1d6eed537fd5981","tex":"o_1","aliases":"o_1","display":"o_1","kind":"use","excerpt":"f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-95762a76969f784c","tex":"O_2","aliases":"O_2","display":"O_2","kind":"use","excerpt":"O_2=1 · 0 · 0 1/2 · 1 · 0","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bce93d378508fd56","tex":"O_3","aliases":"O_3","display":"O_3","kind":"use","excerpt":"has rank two and O_3 has the same rank. Proposition 14 gives an exact two-dimensional delay state. Choose x_0=(1,0,q)^top for any q. The first four observations are","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-60614ca2902b889f","tex":"O_d","aliases":"O_d","display":"O_d","kind":"use","excerpt":"is therefore equivalent to CL^h being in the row space of O_h, which is exactly the rank equality. It also implies O_hLx∈ V_h because this vector is O_h applied to Lx. Uniqueness follows because every element of V_h has a representative. The history recurrence follows directly. Cayley-Hamilton expresses L^d as a linear combination of lower powers, ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-03c58bd9c57b005f","tex":"O_h","aliases":"O_h","display":"O_h","kind":"use","excerpt":"O_h=C CL vdots CL^h-1, V_h=imO_h.","count":15},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fbf676bef4136c8c","tex":"O_{h+1}","aliases":"O_h+1","display":"O_{h+1}","kind":"use","excerpt":"1. There is a linear map D_h:V_h→ V_h satisfying O_hL=D_hO_h. 2. ker O_h⊆ker(O_hL). 3. rankO_h+1=rankO_h.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-987c9403a7315b90","tex":"o_k","aliases":"o_k","display":"o_k","kind":"use","excerpt":"f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ee165265879f7534","tex":"O_o","aliases":"O_o","display":"O_o","kind":"use","excerpt":"O_0=diag(1/4,3/4), O_1=diag(3/4,1/4), M_a,o=O_oT_a.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-76355effcd2476cc","tex":"o_T","aliases":"o_T","display":"o_T","kind":"use","excerpt":"g is geometricAn operator family needs a domain, a mass map and a covered initial state.P29, P39--P40 Let C⊆ R^d be a convex cone and λ a linear mass map strictly positive on C∖{0}. Set K={z∈ C:λ z=1}. Linear observation operators must satisfy B_oC⊆ C, λ∑_o B_o=λ, z_0∈ K. diagram10_domain A finite record o_1⋯ o_T has probability λ B_o_T⋯ B_o_1z_0≥0","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a34910944de266ad","tex":"\\odot","aliases":"odot","display":null,"kind":"use","excerpt":"B_ℓ_y(p) =fracpodotℓ_yℓ_y^top p.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e4b45cd66b9c8f4e","tex":"\\overline L","aliases":"overline L overline","display":"\\overline L","kind":"use","excerpt":"overline L=∑_aw_aL_a, widetilde G_a=A_aoverline LS_a on C_a=imA_a.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-48770b7c9feffb3b","tex":"\\mathrm p","aliases":"p mathrm","display":"\\mathrm p","kind":"use","excerpt":"B_ p,0=z_0(1/2,1/2,0), B_ p,1=z_0(1/2,-1/2,0).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4e6b0f4721e2e3fa","tex":"p^+","aliases":"p^+","display":"p^+","kind":"use","excerpt":"f(w)=1^top M_wp_0, p^+=fracM_sp1^top M_sp","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"plate6Compression can erase a return pathA two-step effect can survive when both compressed steps vanish.P7 Let P^2=P and Q=I-P. Inserting I=P+Q gives the exact defect PLMP-(PLP)(PMP)=PLQMP. diagram06_return_path Take P=diag(1,0) and L=M=0 · 1","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4ef7192516166547","tex":"p^{[1]}","aliases":"p^[1]","display":"p^{[1]}","kind":"use","excerpt":"p^[1](v)=v+frac12.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-307e9035bbb7b4df","tex":"P^\\top","aliases":"P^top top","display":"P^\\top","kind":"use","excerpt":"C_π B_ℓ(p)-bar b = frac1Z_r(diagP-bar bP^top)δ.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-288e37f4092a531f","tex":"P_{\\mathrm d}","aliases":"P_ d mathrm","display":"P_{\\mathrm d}","kind":"use","excerpt":"P_ d=frac121 · -1 -1 · 1","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8c58117cdec5bfd5","tex":"p_+","aliases":"p_+","display":"p_+","kind":"use","excerpt":"p_+μ-p_-μ _TV=frac13ϵ40.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cc8cf6fb3fe21ade","tex":"p_-","aliases":"p_-","display":"p_-","kind":"use","excerpt":"p_+μ-p_-μ _TV=frac13ϵ40.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"f(w)=1^top M_wp_0, p^+=fracM_sp1^top M_sp","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"Let P_1,…,P_m be a complete family of linear packet maps on a predictive space, with ∑_iP_i=I. Algebraic reconstruction gives z=∑_iP_iz. It does not imply that P_iz is an unnormalized probability state.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"p_1=frac1316, z_1=(1,12/13,0)^ T.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"p_2=frac4131040, z_2=(1,-312/413,-240/413)^ T.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"P_a=∑_i∈ B_ap_i, r_a(p)=frac∑_i∈ B_ap_iℓ_iP_a (P_a>0).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ed468ecf5f244341","tex":"p_{a,b}","aliases":"p_a,b","display":"p_{a,b}","kind":"use","excerpt":"p_a,b(umid x,y)=1+a x u+b yL_2(u)≥ρ.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.","count":20},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"This section imports a specific existing Komorebi foundation rather than inventing another inference heuristic. Let p_i be a prior on a finite fine state set and let B_a be coarse blocks. Define","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-500d516c9149e714","tex":"P_{i,r}","aliases":"P_i,r","display":"P_{i,r}","kind":"use","excerpt":"If P_i=∑_rP_i,r and Q_j=∑_sQ_j,s are complete refinements, then","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f7bf3c26037bd6ea","tex":"P_i^*","aliases":"P_i^*","display":"P_i^*","kind":"use","excerpt":"P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-363752ab4143a226","tex":"P_{i_1}","aliases":"P_i_1","display":"P_{i_1}","kind":"use","excerpt":"Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-121ed7254a7702c6","tex":"P_{i_m}","aliases":"P_i_m","display":"P_{i_m}","kind":"use","excerpt":"Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"use","excerpt":"**Proposition 56. Exact bin coarsening and its posterior.** Every B_a,j preserves the probability cone, and λ∑_jB_a,j=λ. For a policy measurable with respect to bin histories, the finite model {B_a,j} reproduces exactly the binned observation law of the continuous model at every finite horizon. On an observed cell with p_j=λ B_a,jz>0,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"Let P_1,…,P_m be a complete family of linear packet maps on a predictive space, with ∑_iP_i=I. Algebraic reconstruction gives z=∑_iP_iz. It does not imply that P_iz is an unnormalized probability state.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"use","excerpt":"**Proposition 13. Strong fixed-horizon refinement limit.** Let X be a Hilbert space, L:X→ X bounded, and P_n orthogonal projections converging strongly to I. Put L_n=P_nLP_n. For every fixed x∈ X and every fixed nonnegative integer k,","count":13},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-dee35a8d1f1f209c","tex":"P_{s,j}","aliases":"P_s,j","display":"P_{s,j}","kind":"use","excerpt":"frac1t∑_s=1^t P_s,j-Q_s,j _TV ≥frac(|R_t,j|-b_t,j)_+t.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7cac8d9dd7092d2c","tex":"p_{s,j}","aliases":"p_s,j","display":"p_{s,j}","kind":"use","excerpt":"R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-383ce804b4f1b5aa","tex":"P_T","aliases":"P_T","display":"P_T","kind":"use","excerpt":"**Proposition 43. Observation-tree total-variation bound.** If both instruments preserve C and conserve mass, and if every action at time t has the discrepancy in Proposition 42 bounded by ϵ_t, then the distributions P_T,widetilde P_T of length-T observation records under the common policy satisfy","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-66dc9af0899d8d32","tex":"p_{t,j}","aliases":"p_t,j","display":"p_{t,j}","kind":"use","excerpt":"X_t,j=ϕ_t,j(Y_t), m_t,j= E[X_t,jmid F_t-1], p_t,j=the model's prediction for that test.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-92ae0d5f738e2866","tex":"P_T^{(\\alpha)}","aliases":"P_T^(α) alpha","display":"P_T^{(\\alpha)}","kind":"use","excerpt":"P_T-P_T^(α) _TV ≤min(1,T(1-α)/2).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e70fd9df7027e2bf","tex":"P_v","aliases":"P_v","display":"P_v","kind":"use","excerpt":"∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx, x ^2=∑_v∈Λ P_vx ^2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-94edcddc4604eaf4","tex":"p_\\pm","aliases":"p_± pm","display":"p_\\pm","kind":"use","excerpt":"L_3(u)=frac5u^3-3u2, p_±(u)=1±ϵ L_3(u), 0<ϵ<1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6cb635ab85f44386","tex":"\\plate","aliases":"plate","display":null,"kind":"use","excerpt":"[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"f(w)=Pr(o_1,…,o_kmid do(a_1,…,a_k)).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-69d88324535ef335","tex":"\\Pr_\\pi","aliases":"Pr_π Pr pi","display":"\\Pr_\\pi","kind":"use","excerpt":"Pr_π(w)=f(w)∏_j=1^|w|π(a_jmid h_j-1),","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-71a98ec1e2bb9f0a","tex":"\\preceq","aliases":"preceq","display":"\\preceq","kind":"use","excerpt":"widehat B_u^top Mwidehat B_upreceqρ^2M for every declared action u.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"z_t=Px_t, q_t=Qx_t,","count":46},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-34d43de423598186","tex":"q^{\\mathsf T}","aliases":"q^ T mathsf","display":"q^{\\mathsf T}","kind":"use","excerpt":"exp(q^ TS_t/σ -tfrac12q^ T(V_t-γ I)q)","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1ab24439ba9e2a53","tex":"q^+","aliases":"q^+","display":"q^+","kind":"use","excerpt":"p(1mid z,0)=q, q^+=frac-3/32+(3/4)qq =frac34-frac332q.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f3003d8698b757c1","tex":"Q^{-1}","aliases":"Q^-1","display":"Q^{-1}","kind":"use","excerpt":"Proposition 54 reconstructs QB_a(u)Q^-1 from H and the six inserted-moment tables. The code that performs this reconstruction imports no generating matrices. The separate generator produces the tables and is used only in the verification suite. These are population values from an exact example, not learned sample estimates.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","excerpt":"z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"Starting at q_0=13/24, this gives probability 13/24 and q_1=15/26. If action one then produces observation zero, its probability is","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"q_2=frac23/192+(5/24)(15/26)365/624=frac5991460.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"Let L:X→ Y be linear, with orthogonal input projectors P_i and output projectors Q_j. Its transfer blocks are L_ji=Q_jLP_i.","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-211107e8e9878600","tex":"Q_{j,s}","aliases":"Q_j,s","display":"Q_{j,s}","kind":"use","excerpt":"If P_i=∑_rP_i,r and Q_j=∑_sQ_j,s are complete refinements, then","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-704a2b865a8b60e0","tex":"Q_j^2","aliases":"Q_j^2","display":"Q_j^2","kind":"use","excerpt":"**Proof.** Insert the appropriate identity decomposition on the input, output or intermediate space. Use Q_j^2=Q_j in the composition formula. square","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8d635189e27aaaf5","tex":"Q_{s,j}","aliases":"Q_s,j","display":"Q_{s,j}","kind":"use","excerpt":"frac1t∑_s=1^t P_s,j-Q_s,j _TV ≥frac(|R_t,j|-b_t,j)_+t.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5639923eccb4ebd0","tex":"Q_T","aliases":"Q_T","display":"Q_T","kind":"use","excerpt":"P_T-Q_T _TV ≤min(1,∑_t=0^T-1 E_P,π[d_t(H_t,A_t)]).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"z_t=Px_t, q_t=Qx_t,","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-efc4332513366015","tex":"q_{t+1}","aliases":"q_t+1","display":"q_{t+1}","kind":"use","excerpt":"Thus z_t+1=Az_t+Bq_t and q_t+1=Cz_t+Dq_t.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"For another linear map M:Y→ Z and output projectors R_k,","count":64},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"able coefficients form a subspaceExact storage includes a consistency condition.P1 For linear analysis and synthesis maps A:X→ C and S:C→ X, the left-inverse law SA=I_X gives injectivity of A. The coefficient projector is Π=AS, with Π^2=Π, imΠ=im A, c is realizableiff Π c=c. diagram[0.90]01_consistency A redundant archive. Take X= R^2, C= R^3 and A","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"Let a finite-dimensional autonomous linear process have hidden state x_t∈ R^d and observed output y_t∈ R^p:","count":24},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0f35942b18c313c8","tex":"\\mathrm r","aliases":"r mathrm","display":"\\mathrm r","kind":"use","excerpt":"B_ r,0=T, B_ r,1=0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square","count":81},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cd6a51568c71e4ad","tex":"\\R^2","aliases":"R^2 R mathbb","display":"{\\mathbb R}^2","kind":"use","excerpt":"able coefficients form a subspaceExact storage includes a consistency condition.P1 For linear analysis and synthesis maps A:X→ C and S:C→ X, the left-inverse law SA=I_X gives injectivity of A. The coefficient projector is Π=AS, with Π^2=Π, imΠ=im A, c is realizableiff Π c=c. diagram[0.90]01_consistency A redundant archive. Take X= R^2, C= R^3 and A","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1035721564d64bb0","tex":"\\R^{2\\,*}","aliases":"R^2 * R mathbb","display":"{\\mathbb R}^{2\\,*}","kind":"use","excerpt":"1/12 · 5/6 span W_*= R^2 *. The state Ox_s=(y_s,y_s+1)^ T therefore closes. claimThe added coordinate is forced by the action of L on the requested observable. In a larger system, unobserved directions unrelated to its future can remain outside W_*. boundaryComputing Ox_s uses a forward observation. It becomes an available delay state (y_t-1,y_t) o","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c317e850f493a6bd","tex":"\\mathbb R^2_+","aliases":"R^2_+ mathbb","display":"\\mathbb R^2_+","kind":"use","excerpt":"The coefficients include negatives. Nevertheless, every operator preserves the cone R R^2_+, because B_sR=RM_s. A sufficient normalized base is","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-78d6b9680d84cb1c","tex":"\\R^3","aliases":"R^3 R mathbb","display":"{\\mathbb R}^3","kind":"use","excerpt":"icients form a subspaceExact storage includes a consistency condition.P1 For linear analysis and synthesis maps A:X→ C and S:C→ X, the left-inverse law SA=I_X gives injectivity of A. The coefficient projector is Π=AS, with Π^2=Π, imΠ=im A, c is realizableiff Π c=c. diagram[0.90]01_consistency A redundant archive. Take X= R^2, C= R^3 and A(x_1,x_2)=","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6e42d74a72040d0b","tex":"\\R^d","aliases":"R^d R mathbb","display":"{\\mathbb R}^d","kind":"use","excerpt":"plate11Probability meaning is geometricAn operator family needs a domain, a mass map and a covered initial state.P29, P39--P40 Let C⊆ R^d be a convex cone and λ a linear mass map strictly positive on C∖{0}. Set K={z∈ C:λ z=1}. Linear observation operators must satisfy B_oC⊆ C, λ∑_o B_o=λ, z_0∈ K. diagram10_domain A finite record o_1⋯ o_T has probab","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-855b51431c459bf3","tex":"\\mathbb R^d","aliases":"R^d mathbb","display":"\\mathbb R^d","kind":"use","excerpt":"Let a finite-dimensional autonomous linear process have hidden state x_t∈ R^d and observed output y_t∈ R^p:","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a448f4c18ae5322e","tex":"R^k","aliases":"R^k","display":"R^k","kind":"use","excerpt":"t± e_1^top R^k(x,y)^top≥0 for every k≥0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1381075c5bc7ca5f","tex":"R^{m-1}","aliases":"R^m-1","display":"R^{m-1}","kind":"use","excerpt":"Lip(F_ I) ≤∑_m=1^p ∑_ i∈ I_m mβ_m, iR^m-1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fafeacf0f1b02e1e","tex":"\\mathbb R^{m\\times m}","aliases":"R^m× m mathbb times","display":"\\mathbb R^{m\\times m}","kind":"use","excerpt":"**Proposition 39. Finite positive-lift certificate.** This cone is invariant under B_s if and only if there is an entrywise nonnegative matrix A_s∈ R^m× m such that","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-247078b158dd2d51","tex":"\\mathbb R^{m\\times N}","aliases":"R^m× N mathbb times","display":"\\mathbb R^{m\\times N}","kind":"use","excerpt":"Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c24fb2f136f983ef","tex":"\\mathbb R^{m\\times n}","aliases":"R^m× n mathbb times","display":"\\mathbb R^{m\\times n}","kind":"use","excerpt":"Let X∈ R^m× n have rank r≥1, let s=σ_r(X)>0, and observe Y=X+E with E _2≤ε<s/2. Let P project onto imX, and let widehat P project onto the leading r left singular vectors of Y.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ee87839145c587f4","tex":"\\mathbb R^n","aliases":"R^n mathbb","display":"\\mathbb R^n","kind":"use","excerpt":"st, so the stable span is exactly the displayed word span. Invariance supplies the matrices B_s, and full row rank makes them unique. Apply R to the unnormalized and normalized hidden-state updates. If all test probabilities are linear functions of another summary, then each row 1^top M_w lies in its row space. Equality on all beliefs implies equal","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5c4c53377d5b750a","tex":"\\mathbb R^n_+","aliases":"R^n_+ mathbb","display":"\\mathbb R^n_+","kind":"use","excerpt":"A concrete realization has a belief p∈ R^n_+ with 1^top p=1. For each s=(a,o), let M_s∈ R^n× n be entrywise nonnegative, with","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-55f8d0c50ba802bf","tex":"\\mathbb R^{n\\times n}","aliases":"R^n× n mathbb times","display":"\\mathbb R^{n\\times n}","kind":"use","excerpt":"A concrete realization has a belief p∈ R^n_+ with 1^top p=1. For each s=(a,o), let M_s∈ R^n× n be entrywise nonnegative, with","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d4e57a0a0f43b7b2","tex":"\\mathbb R^p","aliases":"R^p mathbb","display":"\\mathbb R^p","kind":"use","excerpt":"Let a finite-dimensional autonomous linear process have hidden state x_t∈ R^d and observed output y_t∈ R^p:","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ff280f64b72aabbe","tex":"\\mathbb R^r","aliases":"R^r mathbb","display":"\\mathbb R^r","kind":"use","excerpt":"**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. squ","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6721e87260be28f1","tex":"\\mathbb R^{r\\times m}","aliases":"R^r× m mathbb times","display":"\\mathbb R^{r\\times m}","kind":"use","excerpt":"Let V=[v_1 ⋯ v_m]∈ R^r× m have full row rank and satisfy","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8313eec815fede09","tex":"\\mathbb R^{r\\times N}","aliases":"R^r× N mathbb times","display":"\\mathbb R^{r\\times N}","kind":"use","excerpt":"Work in a fixed r-dimensional coordinate system, where Z_-∈ R^r× N has full row rank. Let","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5610f8845ccab82f","tex":"\\mathbb R_+^m","aliases":"R_+^m mathbb","display":"\\mathbb R_+^m","kind":"use","excerpt":"Its columns generate C=V R_+^m and its normalized base is K=conv{v_1,…,v_m}. Finite generation makes the cone closed. Its normalized generators and full row rank make it pointed and generating, with a strictly positive mass functional.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-33462a02fcce1a84","tex":"R_a","aliases":"R_a","display":"R_a","kind":"use","excerpt":"**Proposition 62. Oriented quadratic probability certificate.** Suppose that for each action and every u∈[-1,1] there are τ_a(u)≥0 and R_a(u)succeq0 satisfying","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fb14b1a407e1de8e","tex":"r_a","aliases":"r_a","display":"r_a","kind":"use","excerpt":"P_a=∑_i∈ B_ap_i, r_a(p)=frac∑_i∈ B_ap_iℓ_iP_a (P_a>0).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a64e897db39c6a7f","tex":"R_k","aliases":"R_k","display":"R_k","kind":"use","excerpt":"For another linear map M:Y→ Z and output projectors R_k,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"use","excerpt":"Δ_a(u)=∑_k=0^Du^kE_a,k, |(E_a,k)_ij|≤ r_k.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0876bfe73de67b89","tex":"r_s","aliases":"r_s","display":"r_s","kind":"use","excerpt":"d(t)=1+t^2, r_s(t)=fracsd(t)(1-t^2,2t).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7d0b0aaec388130d","tex":"r_t","aliases":"r_t","display":"r_t","kind":"use","excerpt":"r_t(x)=γ_t+1(F_t(x))-widehat f_t(γ_t(x)).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e2194b7d7b31eb1c","tex":"R_{t,j}","aliases":"R_t,j","display":"R_{t,j}","kind":"use","excerpt":"R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"1. There is a linear map D_h:V_h→ V_h satisfying O_hL=D_hO_h. 2. ker O_h⊆ker(O_hL). 3. rankO_h+1=rankO_h.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"**Proposition 84. Exact rational outer confidence intervals.** Write V=1+∑ϕ_t^2 and widehatθ=∑ϕ_tY_t/V. Let k=lceillog_2 Vrceil and h=lceillog_2(1/δ)rceil. A rational upper enclosure of","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-131dd08f2af58ba0","tex":"\\mathrm{repair}","aliases":"repair mathrm","display":"\\mathrm{repair}","kind":"use","excerpt":"ε_repair ≤frac3132frac24140000+frac116 =frac874711280000.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ca2a7a41bc8981eb","tex":"\\operatorname{row}","aliases":"row operatorname","display":"\\operatorname{row}","kind":"use","excerpt":"Let the initial summary be C_0x. Regard its rows as linear observables. Starting from W_0=row(C_0), define","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7013b2edec14a3ac","tex":"\\mathbb S","aliases":"S mathbb","display":"\\mathbb S","kind":"use","excerpt":"Let E= R^r and let S^d be the real symmetric d× d matrices. Fix an injective linear map","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"p_a,b(umid s)=1+a s u+b s^2L_2(u).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-472ee72f7d7d04e2","tex":"\\mathbb S^d","aliases":"S^d mathbb","display":"\\mathbb S^d","kind":"use","excerpt":"Let E= R^r and let S^d be the real symmetric d× d matrices. Fix an injective linear map","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-785e812c9b93033d","tex":"S_a","aliases":"S_a","display":"S_a","kind":"use","excerpt":"A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.","count":26},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-68387971995ff129","tex":"S_b","aliases":"S_b","display":"S_b","kind":"use","excerpt":"**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8ec8b0faab9c7fd8","tex":"S_c","aliases":"S_c","display":"S_c","kind":"use","excerpt":"S_c=1 · 0 -c · I, Q=S_c^ T1 · 0 0 · -HS_c.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-be076fbf4a6c3a0a","tex":"S_c^{\\mathsf T}","aliases":"S_c^ T mathsf","display":"S_c^{\\mathsf T}","kind":"use","excerpt":"S_c=1 · 0 -c · I, Q=S_c^ T1 · 0 0 · -HS_c.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8a9f605aee045f84","tex":"s_H","aliases":"s_H","display":"s_H","kind":"use","excerpt":"Let G_s=B_sH with s_H=σ_min(H)>0. Observe","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b4ebcce0dc1a5c39","tex":"s_k","aliases":"s_k","display":"s_k","kind":"use","excerpt":"Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b37e9083bef6f614","tex":"S_r","aliases":"S_r","display":"S_r","kind":"use","excerpt":"**Proof.** The transport identities follow by inserting S_bA_b=I. The operator identity follows because either path is A_bFS_a. For the converse, choose a reference atlas r and put F=S_rG_rA_r. Compatibility gives G_aA_a=A_aF. Every consistent coefficient is A_ax, which proves the asserted realization. Applying S_a proves uniqueness. square","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-87807c21edc049cd","tex":"S_t","aliases":"S_t","display":"S_t","kind":"use","excerpt":"**Proof.** Put S_t=∑_s≤ tϕ_sξ_s. For every fixed q, the conditional moment bound makes","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"**Proposition 26. Finite instrument closure.** Starting with W_0=span{1^top}, define","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"use","excerpt":"**Proof.** If A_ax=0, then x=S_aA_ax=0. Also Π_a^2=A_a(S_aA_a)S_a=Π_a. Its image is contained in C_a and it fixes every A_ax, so its image is exactly C_a. Both norm inequalities follow from the operator-norm inequality applied to S_a. square","count":87},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4e4ee8a109009a53","tex":"\\subsetneq","aliases":"subsetneq","display":null,"kind":"use","excerpt":"essity follows by applying B to every linear dependence among the input columns. Under the kernel condition, define B(Z_-a)=Z_+a. Two choices of a differ by a kernel vector and give the same answer. This defines and uniquely determines the map on W. It maps W into W exactly under the image condition. The pseudoinverse formula gives that map because","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-32f99fc05b64a106","tex":"\\succ","aliases":"succ","display":"\\succ","kind":"use","excerpt":"Suppose some e∈ E satisfies J(e)succ0. Define","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"use","excerpt":"C_ J={z: J(z)succeq0}, K_ J={z∈ C_ J:λ z=1}.","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"a= widehat A +∑_k=0^H-1 widehat K_k <1, sup_t≥0 d_t ≤Δ.","count":14},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0b4fb903a01be35f","tex":"\\sup_{\\|v'\\|_2\\leq1}","aliases":"sup_ v' _2≤1 sup leq","display":"\\sup_{\\|v'\\|_2\\leq1}","kind":"use","excerpt":"sup_ v' _2≤1 Δ(1,v') _ L_d=ϵ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-eba76ff6ca7697c7","tex":"\\sup_{(a,b)\\in\\Theta}","aliases":"sup_(a,b)∈Θ sup in Theta","display":"\\sup_{(a,b)\\in\\Theta}","kind":"use","excerpt":"eta_t=V_t^-1∑_s≤ tϕ_sY_s. For 0<δ<1, with probability at least 1-δ, simultaneously for every finite t, widehatθ_t-θ_* _V_t ≤σsqrtlogfracdet V_tdet(η I)+2log(1/δ)+sqrtη S. diagram14_uncertainty In the structured moment family, a nonempty returned rectangle Θ is internally valid when its sufficient floor ρ_Θ=1-sup_(a,b)∈Θ(|a|+|b|) is nonnegative. Sta","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bf3109e85833bc4b","tex":"\\sup_{a,z\\in K}","aliases":"sup_a,z∈ K sup in","display":"\\sup_{a,z\\in K}","kind":"use","excerpt":"ϵ_Π:=sup_a,z∈ K∫ (B_a(y)-B_a^Π(y))z _C μ(dy) ≤ Lh.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d2fb9f9c20d3ec6c","tex":"\\sup_s","aliases":"sup_s sup","display":"\\sup_s","kind":"use","excerpt":"Let P,Q be Markov kernels on a common state space. Suppose P(s,·)≥ρμ(·) for every s, where μ is one probability measure and 0<ρ≤1. Suppose also sup_s P(s,·)-Q(s,·) _TV≤Δ.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-34c8666b79775742","tex":"\\sup_t","aliases":"sup_t sup","display":"\\sup_t","kind":"use","excerpt":"a_H=frac78-frac38 3^-H, Δ_H=frac38 3^-H, sup_t|z_t-widehat z_t|≤ frac3 3^-H1+3 3^-H.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-53bc99b07a603a4f","tex":"\\sup_{t\\geq0}","aliases":"sup_t≥0 sup geq","display":"\\sup_{t\\geq0}","kind":"use","excerpt":"a= widehat A +∑_k=0^H-1 widehat K_k <1, sup_t≥0 d_t ≤Δ.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ae17cc4f80a4ce0d","tex":"\\sup_{z\\in K}","aliases":"sup_z∈ K sup in","display":"\\sup_{z\\in K}","kind":"use","excerpt":"ϵ_a=sup_z∈ K∑_o (B_a,o-widetilde B_a,o)z _C.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.","count":57},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"Finally the reachable columns z_0,Tz_0,T^2z_0 are independent. The mass row and probe rows before zero and one rotations span the dual space because the rotation has nonzero sine. Reachability and observability establish predictive dimension three. square","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8eaeac84691b5c3b","tex":"T^k","aliases":"T^k","display":"T^k","kind":"use","excerpt":"Any invariant cone with nonnegative mass lies inside this canonical cone, by Proposition 38. If it is closed, convex and contains z_0, it contains the orbit T^kz_0 and the closed convex hull of that orbit. On t=1, this hull is the full unit disk. Conical scaling gives all of C_ L. The unit disk has infinitely many extreme points, so it is not the c","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f90055131c0edf2c","tex":"T_*","aliases":"T_*","display":"T_*","kind":"use","excerpt":"**Proposition 77. Conditional search for strict rational curved certificates.** Consider finitely many rational polynomial operator families in three coordinates, each with exact mass, and finitely many rational normalized initial states. Suppose there exists a real mass preserving invertible T_* such that every transformed initial state lies stric","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"T_0=3/4 · 1/4 1/4 · 3/4, T_1=1/4 · 2/3 3/4 · 1/3, p_0=2/3 1/3.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"T_0=3/4 · 1/4 1/4 · 3/4, T_1=1/4 · 2/3 3/4 · 1/3, p_0=2/3 1/3.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"Select r tests t_1,…,t_r and r histories h_1,…,h_r, with t_1=ϵ, for which the square matrix","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"use","excerpt":"O_0=diag(1/4,3/4), O_1=diag(3/4,1/4), M_a,o=O_oT_a.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2be3a59da728efb3","tex":"T_{aa}","aliases":"T_aa","display":"T_{aa}","kind":"use","excerpt":"T_aa=I, T_cbT_ba=T_ca, F_bT_ba=T_baF_a.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bd76f79f8f6e6b19","tex":"T_{ba}","aliases":"T_ba","display":"T_{ba}","kind":"use","excerpt":"T_ba=A_bS_a: C_a⟶ C_b.","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f57b8b4f7ff44b7d","tex":"T_{ca}","aliases":"T_ca","display":"T_{ca}","kind":"use","excerpt":"T_aa=I, T_cbT_ba=T_ca, F_bT_ba=T_baF_a.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a5f68d13238b664e","tex":"T_{cb}","aliases":"T_cb","display":"T_{cb}","kind":"use","excerpt":"T_aa=I, T_cbT_ba=T_ca, F_bT_ba=T_baF_a.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"H_ij=f(h_jt_i)","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","excerpt":"Select r tests t_1,…,t_r and r histories h_1,…,h_r, with t_1=ϵ, for which the square matrix","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-186d4ea45168e2cc","tex":"T_\\alpha","aliases":"T_α alpha","display":"T_\\alpha","kind":"use","excerpt":"T_α=diag(1,α R), 0≤α<1,","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"Distributions are column vectors. Prediction is p↦ K_u^top p, and a Bayesian update is","count":50},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"J:E⟶ S^d, λ z=tr J(z).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV operatorname","display":"{\\operatorname{TV}}","kind":"use","excerpt":"-P87 Start two processes from the same history or initial law and use the same action policy. If their conditional next-observation laws differ by at most Δ at every common history, telescoping gives the record bound min(1,TΔ) in total variation. diagram15_horizons For Markov kernels with a common probability measure μ and P(s,·)≥ρμ(·), 0<ρ≤1, and ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f20bd6213e9009d0","tex":"\\mathrm{TV}","aliases":"TV mathrm","display":"\\mathrm{TV}","kind":"use","excerpt":"P-P_n _TV=τ_n, τ_n=1-fracZ_nZ.","count":17},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"For a unitary atlas U_a, the synthesis is U_a^* and the condition number is one. For a learned invertible atlas, exact invertibility is not enough: the synthesis norm controls amplification of coefficient errors. An arbitrarily ill-conditioned exact transform is not a stable representation.","count":26},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"use","excerpt":"Let U be the convex hull of finitely many **complete operator families** {B_s^(j)}_s. Every family must satisfy the exact per-action mass identities. The convex coefficient is shared across that family's symbols. A product of independent uncertainty sets can instead be handled by its joint vertices.","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1296a5898d2e7d5f","tex":"U^*","aliases":"U^*","display":"U^*","kind":"use","excerpt":"P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e28cd1be4d9be5f9","tex":"U^2","aliases":"U^2","display":"U^2","kind":"use","excerpt":"x · ysucceq0, m-y≥0}. diagram13_moments Choose a reference probability measure on [-1,1] with moments E U= E U^3=0, E U^2=1/3 and E U^4=1/5. With L_2(u)=(3u^2-1)/2, define B_a,b(u)=v(u)(1,au,bL_2(u)). Its normalized-state density is p_a,b(umid x,y)=1+axu+byL_2(u). If |a|+|b|≤1-ρ for ρ∈[0,1], it is at least ρ, and integrated mass is exact. claimEver","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"For example, (u+v)^2=u^2+2uv+v^2. A grammar containing only independent packet squares misses 2uv. For nonpolynomial maps, one needs direct nonlinear semantics or a local expansion with a stated remainder bound. The word algebraic does not remove that obligation.","count":14},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8e8dbfc6926d65a2","tex":"U^3","aliases":"U^3","display":"U^3","kind":"use","excerpt":"x · ysucceq0, m-y≥0}. diagram13_moments Choose a reference probability measure on [-1,1] with moments E U= E U^3=0, E U^2=1/3 and E U^4=1/5. With L_2(u)=(3u^2-1)/2, define B_a,b(u)=v(u)(1,au,bL_2(u)). Its normalized-state density is p_a,b(umid x,y)=1+axu+byL_2(u). If |a|+|b|≤1-ρ for ρ∈[0,1], it is at least ρ, and integrated mass is exact. claimEver","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d849fcfcf96ad168","tex":"u^3","aliases":"u^3","display":"u^3","kind":"use","excerpt":"L_3(u)=frac5u^3-3u2, p_±(u)=1±ϵ L_3(u), 0<ϵ<1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4d0ee566c7488a3c","tex":"U^4","aliases":"U^4","display":"U^4","kind":"use","excerpt":"x · ysucceq0, m-y≥0}. diagram13_moments Choose a reference probability measure on [-1,1] with moments E U= E U^3=0, E U^2=1/3 and E U^4=1/5. With L_2(u)=(3u^2-1)/2, define B_a,b(u)=v(u)(1,au,bL_2(u)). Its normalized-state density is p_a,b(umid x,y)=1+axu+byL_2(u). If |a|+|b|≤1-ρ for ρ∈[0,1], it is at least ρ, and integrated mass is exact. claimEver","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-df29459720bd9419","tex":"u^j","aliases":"u^j","display":"u^j","kind":"use","excerpt":"B_a(u)=∑_k=0^D u^k C_a,k, M_a,j=∫_-1^1 u^j B_a(u) fracdu2.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-af8a132fc8246e7f","tex":"u^{j+k}","aliases":"u^j+k","display":"u^{j+k}","kind":"use","excerpt":"(W_D)_jk=∫_-1^1u^j+k fracdu2, 0≤ j,k≤ D.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-66662674398860cf","tex":"u^k","aliases":"u^k","display":"u^k","kind":"use","excerpt":"∫_-1^1u^k fracdu2 =0 · k odd, 1/(k+1) · k even.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3fd5c0aabd378aa4","tex":"u^q","aliases":"u^q","display":"u^q","kind":"use","excerpt":"Z_q(u)=I_d^2 uI_d^2 vdots u^qI_d^2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d813568f15ad058f","tex":"U^\\top","aliases":"U^top top","display":"U^\\top","kind":"use","excerpt":"For equal-rank orthogonal projections, the latter norm equals P-widehat P _2. To see the equality, diagonalize U^topwidehat U by singular value decomposition. Each nontrivial paired principal plane contributes a sine to both norms, while common directions contribute zero. On the same planes the singular values of widehat U^top U are the correspondi","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-df73ec13d71ee514","tex":"u_*","aliases":"u_*","display":"u_*","kind":"use","excerpt":"An operator identity can reveal a useful coordinate geometry without specifying that geometry in advance. Suppose two fitted operators evaluated at an anchor u_* give an invertible ratio","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"By contrast, the event u_1≥0 under action zero followed by u_2≥0 under action one has actual probability","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2f97c9046c1a4c7a","tex":"u_2","aliases":"u_2","display":"u_2","kind":"use","excerpt":"By contrast, the event u_1≥0 under action zero followed by u_2≥0 under action one has actual probability","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ec4379f6b5fb17bc","tex":"U_a","aliases":"U_a","display":"U_a","kind":"use","excerpt":"For a unitary atlas U_a, the synthesis is U_a^* and the condition number is one. For a learned invertible atlas, exact invertibility is not enough: the synthesis norm controls amplification of coefficient errors. An arbitrarily ill-conditioned exact transform is not a stable representation.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8da22c10f7ad4a33","tex":"U_a^*","aliases":"U_a^*","display":"U_a^*","kind":"use","excerpt":"For a unitary atlas U_a, the synthesis is U_a^* and the condition number is one. For a learned invertible atlas, exact invertibility is not enough: the synthesis norm controls amplification of coefficient errors. An arbitrarily ill-conditioned exact transform is not a stable representation.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6fdb23dc8f442414","tex":"u_h","aliases":"u_h","display":"u_h","kind":"use","excerpt":"**Proof.** Let u_h,widetilde u_h∈ C be the unnormalized states on each history at depth t, including the common policy factors. Their masses sum to one in each model. Define D_t=∑_h u_h-widetilde u_h _C. At a history with selected action a, write","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5dc6bf6040b60b4e","tex":"\\mathcal U_n","aliases":"U_n mathcal","display":"\\mathcal U_n","kind":"use","excerpt":"**Proposition 48. Sequential confidence accounting for certified proposals.** At stage n, let the proposal be measurable from previous data, and let a verification procedure return a set U_n for the true model in the declared identifiable coordinate system. Assume conditional coverage","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6ae21ddbb7e958a2","tex":"U_t","aliases":"U_t","display":"U_t","kind":"use","excerpt":"P(U_t∈ dumid F_t-1) =λ B_a(u)Z_t μ(du), Z_t+1=fracB_a(U_t)Z_tλ B_a(U_t)Z_t","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1164a4a0480310ee","tex":"U_{t-1}","aliases":"U_t-1","display":"U_{t-1}","kind":"use","excerpt":"E[U_tmid U_t-1=s]=frac a3s, E[L_2(U_t)mid U_t-1=s]=frac b5s^2.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-18ce5e88af8cfdda","tex":"U_{t-1}^2","aliases":"U_t-1^2","display":"U_{t-1}^2","kind":"use","excerpt":"ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d27fa8e4e0984e4b","tex":"\\uparrow","aliases":"uparrow","display":"\\uparrow","kind":"use","excerpt":"Since the active sets exhaust the grammar and the evidence sum is finite, Z_nuparrow Z and τ_ndownarrow0. This is posterior approximation *inside the declared grammar*. It does not establish that the real signal-generating mechanism belongs to it.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx, x ^2=∑_v∈Λ P_vx ^2.","count":107},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"use","excerpt":"nfidence procedure supplies the stated event, not an inferred event of its own. Proposition 39 supplies internal validity for the fitted model. Proposition 40 extends the same domain to every member of the uncertainty set, hence to its true member on that event. Propositions 42 and 43 give the total-variation bound, and Proposition 44 gives the con","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4e7142ad42e2513e","tex":"v'","aliases":"v'","display":"v'","kind":"use","excerpt":"sup_ v' _2≤1 Δ(1,v') _ L_d=ϵ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a8b2a9721419c0b6","tex":"v^{\\mathsf T}","aliases":"v^ T mathsf","display":"v^{\\mathsf T}","kind":"use","excerpt":"**Proposition 80. Convex repair bracketing.** The feasible repair weights form a closed interval [η_*,1]. If a cone-positive functional ℓ=(1,-v^ T), with v _2≤1, and a normalized input z give c=ℓwidehat F_a(u)z<κ, then every feasible weight satisfies","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a8ee3eea11f4b48b","tex":"V^{-1}","aliases":"V^-1","display":"V^{-1}","kind":"use","excerpt":"When V is square and invertible, put A_s=V^-1widehat B_sV. Assume mass conservation has already been imposed. Let N_s be an entrywise positive reference instrument with the same per-action mass identities. For 0≤η≤1, the mixture","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"For example, (u+v)^2=u^2+2uv+v^2. A grammar containing only independent packet squares misses 2uv. For nonpolynomial maps, one needs direct nonlinear semantics or a local expansion with a stated remainder bound. The word algebraic does not remove that obligation.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"Let V=[v_1 ⋯ v_m]∈ R^r× m have full row rank and satisfy","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7b0f4de0732533ef","tex":"V_{a,t}","aliases":"V_a,t","display":"V_{a,t}","kind":"use","excerpt":"Under the rowwise noise and parameter-bound assumptions of Proposition 81, simultaneous confidence ellipsoids for the rows of M_a,j give a confidence set for every C_a,k by this known linear map. If row i of widehat M_a,j-M_a,j has V_a,t-norm at most β_a,j,i, then for any direction x,","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-744000d29cd80f72","tex":"V_{a,t}^{-1}","aliases":"V_a,t^-1","display":"V_{a,t}^{-1}","kind":"use","excerpt":"|e_i^ T(widehat C_a,k-C_a,k)x| ≤ ∑_j |(W^-1)_kj| β_a,j,i x _V_a,t^-1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0b837e89a1e8104e","tex":"V_h","aliases":"V_h","display":"V_h","kind":"use","excerpt":"O_h=C CL vdots CL^h-1, V_h=imO_h.","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e0f7eb4dc3c79da2","tex":"v_h","aliases":"v_h","display":"v_h","kind":"use","excerpt":"v_h(t)=f(ht),","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9eed82c3f494d753","tex":"v_{hs}","aliases":"v_hs","display":"v_{hs}","kind":"use","excerpt":"B_sv_h=v_hs, f(s_1⋯ s_k)=λ B_s_k⋯ B_s_1v_ϵ.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"**Proof.** Invariance places each B_sv_j in the conical hull of the columns of V. Choose its nonnegative coefficients as column j of A_s. Conversely, B_sVα=VA_sα∈ C for every α≥0. Multiplying the intertwining identity by λ and summing over observations gives","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b00eda2d2588cd09","tex":"v_m","aliases":"v_m","display":"v_m","kind":"use","excerpt":"Let V=[v_1 ⋯ v_m]∈ R^r× m have full row rank and satisfy","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-21f030f572f5b2a5","tex":"V_t","aliases":"V_t","display":"V_t","kind":"use","excerpt":"V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6fd418ffd598751b","tex":"V_t^{-1}","aliases":"V_t^-1","display":"V_t^{-1}","kind":"use","excerpt":"V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4005757cd8a6c8b2","tex":"v_\\epsilon","aliases":"v_ϵ epsilon","display":"v_\\epsilon","kind":"use","excerpt":"**Proposition 27. Minimal finite behavior realization.** If rankmathscr H=r<∞, the vector space V=span{v_h} has dimension r. There are well-defined linear maps B_s:V→ V, an initial vector v_ϵ, and a linear mass functional λ(v)=v(ϵ) satisfying","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3696096793ac9d06","tex":"\\operatorname{Var}","aliases":"Var operatorname","display":"\\operatorname{Var}","kind":"use","excerpt":"The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e763d5f6a1b46b5b","tex":"\\operatorname{Var}_\\nu","aliases":"Var_ν operatorname nu","display":"\\operatorname{Var}_\\nu","kind":"use","excerpt":"The correction is positive semidefinite when Xsucceq0. For an affine matrix function K(u)=K_0+uK_1 with scalar u of finite second moment, it equals Var_ν(u)K_1XK_1^ T. On a uniform interval of width h, the variance is h^2/12.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-44cf5118c3be9cba","tex":"\\vdots","aliases":"vdots","display":"\\vdots","kind":"use","excerpt":"O_h=C CL vdots CL^h-1, V_h=imO_h.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f7cc81212c08d083","tex":"\\operatorname{vec}","aliases":"vec operatorname","display":"\\operatorname{vec}","kind":"use","excerpt":"F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"For unitary U_a and positive weights w_a, put","count":56},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e551eb6e818c1638","tex":"W^{\\mathsf T}","aliases":"W^ T mathsf","display":"W^{\\mathsf T}","kind":"use","excerpt":"J(Tz) · =frac15W J(z)W^ T, J(B_ p,0z) · =E_00 J(z)E_00^ T, J(B_ p,1z) · =E_01 J(z)E_01^ T,","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-00883947feeb7d79","tex":"w^{\\mathsf T}","aliases":"w^ T mathsf","display":"w^{\\mathsf T}","kind":"use","excerpt":"For a nonzero z, write x=m(c+w) with w^ THw≤1. Its output mass is m(β_a+b_a^ Tw). Cauchy--Schwarz in the H inner product bounds this below by","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-11fdec97e82b0ec6","tex":"W^{-1}","aliases":"W^-1","display":"W^{-1}","kind":"use","excerpt":"E[g_j(U_t)Z_t+1mid F_t-1] =M_a,jZ_t, C_a,k=∑_j(W^-1)_kjM_a,j.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"Let the initial summary be C_0x. Regard its rows as linear observables. Starting from W_0=row(C_0), define","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-59e63012bc6d4f87","tex":"w_0","aliases":"w_0","display":"w_0","kind":"use","excerpt":"(w_0+d_0)- w_rm sp+d_rm sp _2 ≥ w_0- w_rm sp _2-|d_0|- d_rm sp _2.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1dd506c998004ce1","tex":"W_2","aliases":"W_2","display":"W_2","kind":"use","excerpt":"W_2=1 · 0 · 1/3 0 · 1/3 · 0 1/3 · 0 · 1/5, W_2^-1=9/4 · 0 · -15/4 0 · 3 · 0 -15/4 · 0 · 45/4, det W_2=frac4135.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8689a52c119b803b","tex":"W_2^{-1}","aliases":"W_2^-1","display":"W_2^{-1}","kind":"use","excerpt":"W_2=1 · 0 · 1/3 0 · 1/3 · 0 1/3 · 0 · 1/5, W_2^-1=9/4 · 0 · -15/4 0 · 3 · 0 -15/4 · 0 · 45/4, det W_2=frac4135.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9012166380bed7e0","tex":"w_a","aliases":"w_a","display":"w_a","kind":"use","excerpt":"For unitary U_a and positive weights w_a, put","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4bc32212e137d0da","tex":"w_{a,s}","aliases":"w_a,s","display":"w_{a,s}","kind":"use","excerpt":"w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-974c9348a2c456d5","tex":"w_{a,s,0}","aliases":"w_a,s,0","display":"w_{a,s,0}","kind":"use","excerpt":"w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-265884cf47051e28","tex":"w_{a,s,1}","aliases":"w_a,s,1","display":"w_{a,s,1}","kind":"use","excerpt":"w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-43196f3f914bfb5f","tex":"w_{a,s,2}","aliases":"w_a,s,2","display":"w_{a,s,2}","kind":"use","excerpt":"w_a,s(u,t) · =F_a(u)(d(t),s(1-t^2),2st)^ T, f_a,s(u,t) · =w_a,s,0(u,t)-κ d(t), h_a,s(u,t) · =f_a,s(u,t)^2-w_a,s,1(u,t)^2-w_a,s,2(u,t)^2.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ac791e77e5ab00c5","tex":"w_a^2","aliases":"w_a^2","display":"w_a^2","kind":"use","excerpt":"Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7908ba336cca4654","tex":"W_D","aliases":"W_D","display":"W_D","kind":"use","excerpt":"Define the moment Gram matrix W_D by","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ce19e5dba7214988","tex":"W_D^{-1}","aliases":"W_D^-1","display":"W_D^{-1}","kind":"use","excerpt":"widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"z=∑_i:w_i>0 w_i z_i, w_i=λ P_iz≥0, z_i=P_iz/w_i, ∑_iw_i=1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ffbde798b9a7c58f","tex":"W_j","aliases":"W_j","display":"W_j","kind":"use","excerpt":"W_j+1=W_j+∑_s∈ A× OW_jM_s.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8c2b5f4f1ae573be","tex":"W_{j+1}","aliases":"W_j+1","display":"W_{j+1}","kind":"use","excerpt":"W_j+1=W_j+∑_s∈ A× OW_jM_s.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f045bdff9119d137","tex":"W_{jk}","aliases":"W_jk","display":"W_{jk}","kind":"use","excerpt":"Let B_a(u)=∑_k=0^Dψ_k(u)C_a,k. Choose functions g_j so that the known matrix W_jk=∫ g_j(u)ψ_k(u) μ(du) is invertible.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-466a54f902244928","tex":"W_n","aliases":"W_n","display":"W_n","kind":"use","excerpt":"W_n+1=W_n+∑_uW_nL_u, W_nL_u={wL_u:w∈ W_n}.","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b2c566e84a80fdd2","tex":"W_{n+1}","aliases":"W_n+1","display":"W_{n+1}","kind":"use","excerpt":"W_n+1=W_n+∑_uW_nL_u, W_nL_u={wL_u:w∈ W_n}.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d6f43210828fc399","tex":"w_{\\rm sp}","aliases":"w_rm sp rm","display":"w_{\\rm sp}","kind":"use","excerpt":"(w_0+d_0)- w_rm sp+d_rm sp _2 ≥ w_0- w_rm sp _2-|d_0|- d_rm sp _2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-60b8bde620b4d40c","tex":"W_s^a","aliases":"W_s^a","display":"W_s^a","kind":"use","excerpt":"Consequently Proposition 66 gives a time-uniform certificate for ∑_s≤ tμ_s^a from ∑_s≤ tW_s^a, with its boundary multiplied by 1/π_min. The target is the average action-conditioned quantity along the encountered contexts, not a reset-population table or the state distribution under a different long-term policy.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-50da3c084817e47e","tex":"w_{sj}","aliases":"w_sj","display":"w_{sj}","kind":"use","excerpt":"J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2b49ff09b03e5e91","tex":"w_t","aliases":"w_t","display":"w_t","kind":"use","excerpt":"plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, r","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-88d526f7f179dd2a","tex":"w_{t+1}","aliases":"w_t+1","display":"w_{t+1}","kind":"use","excerpt":"plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, r","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-88dcf10de2b148a5","tex":"W_t^a","aliases":"W_t^a","display":"W_t^a","kind":"use","excerpt":"μ_t^a=∫ϕ_t(y)P_t(dymid H_t-1,a), W_t^a=frac1_{A_t=a}ϕ_t(Y_t)π_t(amid H_t-1).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c850fcef441cf8c1","tex":"w_\\nu","aliases":"w_ν nu","display":"w_\\nu","kind":"use","excerpt":"F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8d3781ce495edcf4","tex":"\\widehat A","aliases":"widehat A widehat","display":"\\widehat A","kind":"use","excerpt":"widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.","count":14},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7284a914c9ac2996","tex":"\\widehat A^{\\mathsf T}","aliases":"widehat A^ T widehat mathsf","display":"\\widehat A^{\\mathsf T}","kind":"use","excerpt":"The least singular value of M_A^* is at least that of M_widehat A minus e_M. Apply this lower bound and the triangle inequality. To bound the map perturbation, insert and subtract widehat A^ THA^* and apply the Frobenius product inequality. The trace component is unchanged. Finally,","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1b3f66f942a5fe9a","tex":"\\widehat B","aliases":"widehat B widehat","display":"\\widehat B","kind":"use","excerpt":"widehat B=widetilde Z_+widetilde Z_-^†","count":25},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ea83107951d84fac","tex":"\\widehat b","aliases":"widehat b widehat","display":"\\widehat b","kind":"use","excerpt":"For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7c5344ce5b6904ba","tex":"\\widehat B_a","aliases":"widehat B_a widehat","display":"\\widehat B_a","kind":"use","excerpt":"**Proof.** All coefficient matrices are fixed, so the displayed constraints are linear. An invertible solution yields widehat B_a(u)=TB_a(u)T^-1 coefficient by coefficient. Substitution in the template intertwining identity proves the transported certificate. To prove uniqueness, let D be the difference of two solutions. It satisfies widehat C_a,kD","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-053b9af8df9f0d21","tex":"\\widehat B_s","aliases":"widehat B_s widehat","display":"\\widehat B_s","kind":"use","excerpt":"**Proposition 33. Noisy core-block realization.** The estimate widehat B_s=widetilde G_swidetilde H^-1 satisfies","count":15},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d1821845a2a9475d","tex":"\\widehat B_u","aliases":"widehat B_u widehat","display":"\\widehat B_u","kind":"use","excerpt":"For a fixed linear representation R and an estimated transition widehat B_u, the defect is (RL_u-widehat B_uR)x. For a reference reduced operator B_u, it splits exactly as","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-20b922cfbf1bf899","tex":"\\widehat B_u^\\top","aliases":"widehat B_u^top widehat top","display":"\\widehat B_u^\\top","kind":"use","excerpt":"widehat B_u^top Mwidehat B_upreceqρ^2M for every declared action u.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"use","excerpt":"widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e7a695b8f5409721","tex":"\\widehat c","aliases":"widehat c widehat","display":"\\widehat c","kind":"use","excerpt":"For fixed centers c^*=(I-A^*)^-1b^* and widehat c=(I-widehat A)^-1widehat b, with widehat A-A^* _2≤ e_A and widehat b-b^* _2≤ e_b, one has","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f39af840a91bd1b9","tex":"\\widehat C_{a,k}","aliases":"widehat C_a,k widehat","display":"\\widehat C_{a,k}","kind":"use","excerpt":"widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bb6c36e33f5b84db","tex":"\\widehat C_w","aliases":"widehat C_w widehat","display":"\\widehat C_w","kind":"use","excerpt":"are linear in T. Every invertible solution transports the template certificate to the fitted coordinates by widehat J= JT^-1. If the fitted coefficient family is observable, meaning the rows widehatλwidehat C_w span the dual space, the solution is unique. The fitted initial state must still be tested in the transported normalized cone.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9a999477715e268b","tex":"\\widehat F","aliases":"widehat F widehat","display":"\\widehat F","kind":"use","excerpt":"This reference instrument reports a state-independent density and resets to the center. It is a mathematical model modification, not a new observation. Given an estimate widehat F, define","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4b699081b2be6907","tex":"\\widehat F_a","aliases":"widehat F_a widehat","display":"\\widehat F_a","kind":"use","excerpt":"widetilde F_a(u)=(1-η)widehat F_a(u)+η N_a(u), 0≤η≤1.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-defa39f63e78532c","tex":"\\widehat f_t","aliases":"widehat f_t widehat","display":"\\widehat f_t","kind":"use","excerpt":"Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5c577d5457da31ff","tex":"\\widehat H","aliases":"widehat H widehat","display":"\\widehat H","kind":"use","excerpt":"**Proposition 74. A posteriori proposal error.** Given widehat H, let r= M_widehat A(widehat H)-b. Suppose","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-949a006b5c822536","tex":"\\widehat{\\mathcal J}","aliases":"widehat J widehat mathcal","display":"\\widehat{\\mathcal J}","kind":"use","excerpt":"are linear in T. Every invertible solution transports the template certificate to the fitted coordinates by widehat J= JT^-1. If the fitted coefficient family is observable, meaning the rows widehatλwidehat C_w span the dual space, the solution is unique. The fitted initial state must still be tested in the transported normalized cone.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5450287e32dd956c","tex":"\\widehat K","aliases":"widehat K widehat","display":"\\widehat K","kind":"use","excerpt":"widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bf1e8d01f89afe7b","tex":"\\widehat K_k","aliases":"widehat K_k widehat","display":"\\widehat K_k","kind":"use","excerpt":"widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-11c7b6a0608a5de0","tex":"\\widehat L","aliases":"widehat L widehat","display":"\\widehat L","kind":"use","excerpt":"If estimated operators widehat L_n are uniformly bounded and (widehat L_n-L_n)x→0 for every fixed x, the same argument gives fixed-horizon convergence for widehat L_n. These are explicit assumptions, not conclusions of the finite learning procedure.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-04bd9bb2696cd438","tex":"\\widehat L_n","aliases":"widehat L_n widehat","display":"\\widehat L_n","kind":"use","excerpt":"If estimated operators widehat L_n are uniformly bounded and (widehat L_n-L_n)x→0 for every fixed x, the same argument gives fixed-horizon convergence for widehat L_n. These are explicit assumptions, not conclusions of the finite learning procedure.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a410e85313b42614","tex":"\\widehat M","aliases":"widehat M widehat","display":"\\widehat M","kind":"use","excerpt":"widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-005d83be0fb21b00","tex":"\\widehat M_{a,j}","aliases":"widehat M_a,j widehat","display":"\\widehat M_{a,j}","kind":"use","excerpt":"widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"use","excerpt":"Let X∈ R^m× n have rank r≥1, let s=σ_r(X)>0, and observe Y=X+E with E _2≤ε<s/2. Let P project onto imX, and let widehat P project onto the leading r left singular vectors of Y.","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"For a predicted state widehat z, form widehat u=widehat B_swidehat z, widehat p=λwidehat u, and define d=widehat u-u. Write e= widehat z-z , widehat B_s ≤ L, and widehat B_s-B_s ≤ϵ_B. Then","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fe16b9b08fa8986a","tex":"\\widehat P_T","aliases":"widehat P_T widehat","display":"\\widehat P_T","kind":"use","excerpt":"P_T-widehat P_T _TV ≤min(1,frace_0+∑_t<Tϵ_t2).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-88e7974099a060be","tex":"\\widehat p_t","aliases":"widehat p_t widehat","display":"\\widehat p_t","kind":"use","excerpt":"Alternatively, if each realized fitted denominator is checked to satisfy widehat p_t≥widehatη_t>0, the pathwise recursion obeys","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5ec76de4222f5aa9","tex":"\\widehat U","aliases":"widehat U widehat","display":"\\widehat U","kind":"use","excerpt":"The (r+1)st singular value is interpreted as zero when r is the maximum possible rank. If U and widehat U are orthonormal bases of the two ranges, then","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2c8131c596a2a289","tex":"\\widehat u","aliases":"widehat u widehat","display":"\\widehat u","kind":"use","excerpt":"For a predicted state widehat z, form widehat u=widehat B_swidehat z, widehat p=λwidehat u, and define d=widehat u-u. Write e= widehat z-z , widehat B_s ≤ L, and widehat B_s-B_s ≤ϵ_B. Then","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9b03ab0a9ebd52f0","tex":"\\widehat U^\\top","aliases":"widehat U^top widehat top","display":"\\widehat U^\\top","kind":"use","excerpt":"σ_min(widehat U^top U) ≥sqrt1-(2ε/s)^2.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-51fdd71f6572f7c6","tex":"\\widehat y","aliases":"widehat y widehat","display":"\\widehat y","kind":"use","excerpt":"Here 0≤ y_t≤1 and the truncated recursion has gain a_H=frac12+∑_j=0^H-1K_j =frac78-frac38 3^-H<1. With the same initial state, recursive propagation therefore gives sup_t|y_t-widehat y_t|≤fracΔ_H1-a_H =frac33^H+3. boundaryP10's tail estimate uses the same resolved history. P19 supplies the separate stability argument for a predictor using its own h","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-04d3b74573ac9c62","tex":"\\widehat y_t","aliases":"widehat y_t widehat","display":"\\widehat y_t","kind":"use","excerpt":"Here 0≤ y_t≤1 and the truncated recursion has gain a_H=frac12+∑_j=0^H-1K_j =frac78-frac38 3^-H<1. With the same initial state, recursive propagation therefore gives sup_t|y_t-widehat y_t|≤fracΔ_H1-a_H =frac33^H+3. boundaryP10's tail estimate uses the same resolved history. P19 supplies the separate stability argument for a predictor using its own h","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c88f3c79e4ba1c43","tex":"\\widehat z","aliases":"widehat z widehat","display":"\\widehat z","kind":"use","excerpt":"**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,","count":16},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f309777b49eff8ef","tex":"\\widehat z^+","aliases":"widehat z^+ widehat","display":"\\widehat z^+","kind":"use","excerpt":"widehat z^+-z^+ =frac(I-z^+λ)dwidehat p,","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a4050458ecc755c5","tex":"\\widehat z_t","aliases":"widehat z_t widehat","display":"\\widehat z_t","kind":"use","excerpt":"**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5d5f3a9402171e53","tex":"\\widehat z_{t+1}","aliases":"widehat z_t+1 widehat","display":"\\widehat z_{t+1}","kind":"use","excerpt":"**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9bb7047775929054","tex":"\\widehat z_{t-1-k}","aliases":"widehat z_t-1-k widehat","display":"\\widehat z_{t-1-k}","kind":"use","excerpt":"widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9bc4ac46f9980428","tex":"\\widehat\\eta","aliases":"widehatη widehat eta","display":"\\widehat\\eta","kind":"use","excerpt":"Alternatively, if each realized fitted denominator is checked to satisfy widehat p_t≥widehatη_t>0, the pathwise recursion obeys","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-eb9575334c64edff","tex":"\\widehat\\eta_t","aliases":"widehatη_t widehat eta","display":"\\widehat\\eta_t","kind":"use","excerpt":"Alternatively, if each realized fitted denominator is checked to satisfy widehat p_t≥widehatη_t>0, the pathwise recursion obeys","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fec34e4c8a1d397d","tex":"\\widehat\\theta","aliases":"widehatθ widehat theta","display":"\\widehat\\theta","kind":"use","excerpt":"V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7865eb7dfb307672","tex":"\\widehat\\theta_t","aliases":"widehatθ_t widehat theta","display":"\\widehat\\theta_t","kind":"use","excerpt":"V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-08d41d2f8c1b0ad2","tex":"\\widehat\\lambda","aliases":"widehatλ widehat lambda","display":"\\widehat\\lambda","kind":"use","excerpt":"Suppose a structural library supplies a certified template with polynomial coefficients C_a,k, mass row λ and embedding J. Suppose observed population tables produce coefficients widehat C_a,k and mass row widehatλ of the same dimension.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9a5c0f9dd0b0c63d","tex":"\\widetilde A","aliases":"widetilde A widetilde","display":"\\widetilde A","kind":"use","excerpt":"widetilde A_s=(1-η)A_s+η N_s","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6aa6ee2e065a2e0c","tex":"\\widetilde A_s","aliases":"widetilde A_s widetilde","display":"\\widetilde A_s","kind":"use","excerpt":"widetilde A_s=(1-η)A_s+η N_s","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f596b26bb2804e7d","tex":"\\widetilde B","aliases":"widetilde B widetilde","display":"\\widetilde B","kind":"use","excerpt":"Once V is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of widetilde B_s, A_s, widetilde z_0 and α_0. The constraints","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0a4e15e563cc5107","tex":"\\widetilde b","aliases":"widetilde b widetilde","display":"\\widetilde b","kind":"use","excerpt":"widetilde H-H _2≤ rϵ_N, widetilde G_s-G_s _2≤ rϵ_N, widetilde b_0-b_0 _2≤sqrt rϵ_N.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a072d174bd35a4fb","tex":"\\widetilde b_0","aliases":"widetilde b_0 widetilde","display":"\\widetilde b_0","kind":"use","excerpt":"widetilde H-H _2≤ rϵ_N, widetilde G_s-G_s _2≤ rϵ_N, widetilde b_0-b_0 _2≤sqrt rϵ_N.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c3ea8099ef53804a","tex":"\\widetilde B_a","aliases":"widetilde B_a widetilde","display":"\\widetilde B_a","kind":"use","excerpt":"**Proposition 59. Continuous-record total-variation bound.** Let B_a(y) and widetilde B_a(y) satisfy Proposition 52 on a common (C,λ,μ), and let the two models use the same measurable policy on every common observation history. Suppose their initial states satisfy z_0-widetilde z_0 _C≤ e_0 and their time-t instruments satisfy","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d177ff0a67d5635b","tex":"\\widetilde B_{a,o}","aliases":"widetilde B_a,o widetilde","display":"\\widetilde B_{a,o}","kind":"use","excerpt":"widetilde B_sV=VA_s, A_s≥0, λ∑_owidetilde B_a,o=λ, widetilde z_0=Vα_0, α_0≥0, 1^topα_0=1","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-87f266f09a5a1132","tex":"\\widetilde B_{a,t}","aliases":"widetilde B_a,t widetilde","display":"\\widetilde B_{a,t}","kind":"use","excerpt":"sup_a,z∈ K∫ (B_a,t(y)-widetilde B_a,t(y))z _C μ(dy) ≤ϵ_t.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a85baba04632f8c1","tex":"\\widetilde B_a^{[n]}","aliases":"widetilde B_a^[n] widetilde","display":"\\widetilde B_a^{[n]}","kind":"use","excerpt":"Then widetilde B_a^[n](x)=∑_k b_n,k(x)widetilde C_a,n,k is internally valid. If","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5f25a930365307b0","tex":"\\widetilde B_s","aliases":"widetilde B_s widetilde","display":"\\widetilde B_s","kind":"use","excerpt":"Once V is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of widetilde B_s, A_s, widetilde z_0 and α_0. The constraints","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-47c52f415949a6cb","tex":"\\widetilde C","aliases":"widetilde C widetilde","display":"\\widetilde C","kind":"use","excerpt":"widetilde C_a,k-widehat C_a,k ≤η∑_j|(W_D^-1)_kj|.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4b48e3e544e4766b","tex":"\\widetilde C_{a,k}","aliases":"widetilde C_a,k widetilde","display":"\\widetilde C_{a,k}","kind":"use","excerpt":"widetilde C_a,k-widehat C_a,k ≤η∑_j|(W_D^-1)_kj|.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3554e5941037cc46","tex":"\\widetilde C_{a,n,k}","aliases":"widetilde C_a,n,k widetilde","display":"\\widetilde C_{a,n,k}","kind":"use","excerpt":"**Proposition 65. Certified coefficient error and approximation bias.** Suppose every fitted coefficient widetilde C_a,n,k preserves the same cone C, and","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ad91da91d6be3c40","tex":"\\widetilde F","aliases":"widetilde F widetilde","display":"\\widetilde F","kind":"use","excerpt":"widetilde F_a(u)=(1-η)widehat F_a(u)+η N_a(u), 0≤η≤1.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-12e8dbd37add6e7b","tex":"\\widetilde F_a","aliases":"widetilde F_a widetilde","display":"\\widetilde F_a","kind":"use","excerpt":"widetilde F_a(u)=(1-η)widehat F_a(u)+η N_a(u), 0≤η≤1.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-929c170d8af8ade4","tex":"\\widetilde G","aliases":"widetilde G widetilde","display":"\\widetilde G","kind":"use","excerpt":"overline L=∑_aw_aL_a, widetilde G_a=A_aoverline LS_a on C_a=imA_a.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-008a0a204981576f","tex":"\\widetilde G_a","aliases":"widetilde G_a widetilde","display":"\\widetilde G_a","kind":"use","excerpt":"overline L=∑_aw_aL_a, widetilde G_a=A_aoverline LS_a on C_a=imA_a.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-294ab58bbd376183","tex":"\\widetilde G_s","aliases":"widetilde G_s widetilde","display":"\\widetilde G_s","kind":"use","excerpt":"widetilde H=H+E, widetilde G_s=G_s+F_s, E _2<s_H.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-70fad85f996380ab","tex":"\\widetilde H","aliases":"widetilde H widetilde","display":"\\widetilde H","kind":"use","excerpt":"widetilde H=H+E, widetilde G_s=G_s+F_s, E _2<s_H.","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a43c2012e6f47f4e","tex":"\\widetilde H^{-1}","aliases":"widetilde H^-1 widetilde","display":"\\widetilde H^{-1}","kind":"use","excerpt":"**Proposition 33. Noisy core-block realization.** The estimate widehat B_s=widetilde G_swidetilde H^-1 satisfies","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","excerpt":"**Proposition 43. Observation-tree total-variation bound.** If both instruments preserve C and conserve mass, and if every action at time t has the discrepancy in Proposition 42 bounded by ϵ_t, then the distributions P_T,widetilde P_T of length-T observation records under the common policy satisfy","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bb050b4a5097c406","tex":"\\widetilde P_T","aliases":"widetilde P_T widetilde","display":"\\widetilde P_T","kind":"use","excerpt":"**Proposition 43. Observation-tree total-variation bound.** If both instruments preserve C and conserve mass, and if every action at time t has the discrepancy in Proposition 42 bounded by ϵ_t, then the distributions P_T,widetilde P_T of length-T observation records under the common policy satisfy","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cd263f633c7dde0b","tex":"\\widetilde s","aliases":"widetilde s widetilde","display":"\\widetilde s","kind":"use","excerpt":"If widetilde s=σ_min(widetilde H) is known and E _2≤ e<widetilde s, the last denominator may be replaced by widetilde s-e.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a1faa955cc41a0e0","tex":"\\widetilde u","aliases":"widetilde u widetilde","display":"\\widetilde u","kind":"use","excerpt":"**Proof.** Let u_h,widetilde u_h∈ C be the unnormalized states on each history at depth t, including the common policy factors. Their masses sum to one in each model. Define D_t=∑_h u_h-widetilde u_h _C. At a history with selected action a, write","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0d941345540db977","tex":"\\widetilde u_h","aliases":"widetilde u_h widetilde","display":"\\widetilde u_h","kind":"use","excerpt":"**Proof.** Let u_h,widetilde u_h∈ C be the unnormalized states on each history at depth t, including the common policy factors. Their masses sum to one in each model. Define D_t=∑_h u_h-widetilde u_h _C. At a history with selected action a, write","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3323d1822e9c5f0f","tex":"\\widetilde v","aliases":"widetilde v widetilde","display":"\\widetilde v","kind":"use","excerpt":"Taking the infimum proves integrated instrument contraction. At each continuous history let v_h,widetilde v_h be the unnormalized state densities, including the policy factors. Split the next difference as","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a1973f5da4955dc6","tex":"\\widetilde v_h","aliases":"widetilde v_h widetilde","display":"\\widetilde v_h","kind":"use","excerpt":"Taking the infimum proves integrated instrument contraction. At each continuous history let v_h,widetilde v_h be the unnormalized state densities, including the policy factors. Split the next difference as","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-adcc5a10ee60bde9","tex":"\\widetilde Z","aliases":"widetilde Z widetilde","display":"\\widetilde Z","kind":"use","excerpt":"Z_+=BZ_-+R, widetilde Z_-=Z_-+E, widetilde Z_+=Z_++F.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-86c1d03c1a864e81","tex":"\\widetilde z","aliases":"widetilde z widetilde","display":"\\widetilde z","kind":"use","excerpt":"Once V is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of widetilde B_s, A_s, widetilde z_0 and α_0. The constraints","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cba5f51348fe06d7","tex":"\\widetilde Z_+","aliases":"widetilde Z_+ widetilde","display":"\\widetilde Z_+","kind":"use","excerpt":"Z_+=BZ_-+R, widetilde Z_-=Z_-+E, widetilde Z_+=Z_++F.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d7b21eb90f2b359c","tex":"\\widetilde Z_-","aliases":"widetilde Z_- widetilde","display":"\\widetilde Z_-","kind":"use","excerpt":"Z_+=BZ_-+R, widetilde Z_-=Z_-+E, widetilde Z_+=Z_++F.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7feb3c3a7fff568b","tex":"\\widetilde Z_-^\\top","aliases":"widetilde Z_-^top widetilde top","display":"\\widetilde Z_-^\\top","kind":"use","excerpt":"**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. squ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d440b94c004a971a","tex":"\\widetilde Z_-^\\dagger","aliases":"widetilde Z_-^† widetilde dagger","display":"\\widetilde Z_-^\\dagger","kind":"use","excerpt":"widehat B=widetilde Z_+widetilde Z_-^†","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7d98b1209ae4b176","tex":"\\widetilde z_0","aliases":"widetilde z_0 widetilde","display":"\\widetilde z_0","kind":"use","excerpt":"Once V is fixed, certificate-constrained fitting is a convex problem. Unknowns include the entries of widetilde B_s, A_s, widetilde z_0 and α_0. The constraints","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1fa74de5039c01f6","tex":"\\widetilde\\nu","aliases":"widetildeν widetilde nu","display":"\\widetilde\\nu","kind":"use","excerpt":"**Proof.** For ρ<1, write P(s,·)=ρμ+(1-ρ)R(s,·), where R is a Markov kernel. Applying a Markov kernel cannot increase total variation, by integration of bounded test functions. Thus ν P-widetildeν P _TV≤(1-ρ) ν-widetildeν _TV. Adding and subtracting widetildeν P yields e_t+1≤(1-ρ)e_t+Δ. Iterate this inequality. The case ρ=1 follows directly. Applyi","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"An **observation** is an element x of a declared space X. Examples include a finite real signal, a complex field on a finite grid, or an array of categorical labels. Its type includes shape, scalar field or palette, and boundary convention. A zero-extended finite sequence and a periodic sequence are different objects even when their displayed sampl","count":67},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"An **observation** is an element x of a declared space X. Examples include a finite real signal, a complex field on a finite grid, or an array of categorical labels. Its type includes shape, scalar field or palette, and boundary convention. A zero-extended finite sequence and a periodic sequence are different objects even when their displayed sampl","count":167},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"γ(x)=γ(x') Longrightarrow γ(F_u(x))=γ(F_u(x'))","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-71a31718ae729ffa","tex":"X^*","aliases":"X^*","display":"X^*","kind":"use","excerpt":"This gives a precise role for obstruction-driven refinement: add observable directions created by the dynamics until the chosen prediction variables close. It does not guarantee that the completion is small. In some systems W_*=X^* and exact memoryless closure needs the full state.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"C_ L={(t,x,y):t≥sqrtx^2+y^2}","count":13},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8a46ff3f62bf1268","tex":"x^k","aliases":"x^k","display":"x^k","kind":"use","excerpt":"b_n,k(x)=binom nk x^k(1-x)^n-k, D_n(x,v)=(n+1)∑_k=0^n b_n,k(x)b_n,k(v).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7272f222c85749d6","tex":"X^m","aliases":"X^m","display":"X^m","kind":"use","excerpt":"where each B_m:X^m→ Y is multilinear. Then","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f0983851bd00393a","tex":"X^\\dagger","aliases":"X^† dagger","display":"X^\\dagger","kind":"use","excerpt":"(I-widehat P)X _2≤2ε, (I-widehat P)P _2 = (I-widehat P)XX^† _2≤2ε/s.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"x_t+1=Lx_t, L=tfrac12 · 1 [2pt]tfrac14 · tfrac13, x_0=1 0.","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"For example, γ(x_1,x_2)=x_1 and η(x_1,x_2)=(x_1,x_2). The states (0,1) and (0,-1) have the same coarse state, but any common point initialization has worst-case error at least one in Euclidean norm. Padding a vector with zeros is a choice, not information recovery.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"For example, γ(x_1,x_2)=x_1 and η(x_1,x_2)=(x_1,x_2). The states (0,1) and (0,-1) have the same coarse state, but any common point initialization has worst-case error at least one in Euclidean norm. Padding a vector with zeros is a choice, not information recovery.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9fb1edd835282495","tex":"X_{pq}","aliases":"X_pq","display":"X_{pq}","kind":"use","excerpt":"[Φ_F(X)]_ij=∑_p,q=1^d F_(i,p),(j,q)X_pq.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e8786e121e9dc2d7","tex":"x_s","aliases":"x_s","display":"x_s","kind":"use","excerpt":"The coordinates of O_hx_s are future observations relative to time s. They are not available at time s merely because this matrix is written down. At time t≥ h-1, however, the same vector is the observed history","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-28b8f70861f7fff0","tex":"X_{s,j}","aliases":"X_s,j","display":"X_{s,j}","kind":"use","excerpt":"R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1029cb8ad51ed272","tex":"x_t","aliases":"x_t","display":"x_t","kind":"use","excerpt":"Suppose X is a Hilbert space, P is an orthogonal projection, Q=I-P and x_t+1=Lx_t. The retained state is z_t=Px_t. Then","count":22},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-813924c29494401d","tex":"x_{t+1}","aliases":"x_t+1","display":"x_{t+1}","kind":"use","excerpt":"Suppose X is a Hilbert space, P is an orthogonal projection, Q=I-P and x_t+1=Lx_t. The retained state is z_t=Px_t. Then","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ec83b246ddafaa89","tex":"X_{t,j}","aliases":"X_t,j","display":"X_{t,j}","kind":"use","excerpt":"X_t,j=ϕ_t,j(Y_t), m_t,j= E[X_t,jmid F_t-1], p_t,j=the model's prediction for that test.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-36f8eeaf6f458585","tex":"x_{t-h+1}","aliases":"x_t-h+1","display":"x_{t-h+1}","kind":"use","excerpt":"z_t=y_t-h+1 vdots y_t =O_hx_t-h+1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"use","excerpt":"**Proposition 63. Conditional operator-only ellipsoid discovery.** Suppose the symmetric solution space of A^ THA=H is one-dimensional and contains a positive definite matrix H_0. Let a rank-one fitted branch at a declared observation have a nonzero-mass image, normalized to v=(1,x_v). If x_v≠ c, define","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"Let L:X→ Y be linear, with orthogonal input projectors P_i and output projectors Q_j. Its transfer blocks are L_ji=Q_jLP_i.","count":26},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bef13251af5aee62","tex":"\\mathcal Y","aliases":"Y mathcal","display":"\\mathcal Y","kind":"use","excerpt":"Let (Y, Y) be a standard Borel observation space and let μ be a fixed probability reference measure. Keep a finite action set. For each action a, let B_a(y):E→ E be measurable and integrable in operator norm. Let (C,λ) be a proper probability cone and mass functional as in Section 46. Assume that, outside one μ-null set for each action,","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix","count":76},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"C_ L={(t,x,y):t≥sqrtx^2+y^2}","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"y_0=1, y_1=1/2, y_2=1/2, y_3=11/24.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"y_0=1, y_1=1/2, y_2=1/2, y_3=11/24.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-dd412919846eece9","tex":"y_2","aliases":"y_2","display":"y_2","kind":"use","excerpt":"y_0=1, y_1=1/2, y_2=1/2, y_3=11/24.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5171b856d1185d9c","tex":"y_3","aliases":"y_3","display":"y_3","kind":"use","excerpt":"y_0=1, y_1=1/2, y_2=1/2, y_3=11/24.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-10f0012bfa3107dd","tex":"y_4","aliases":"y_4","display":"y_4","kind":"use","excerpt":"It predicts y_4=61/144, which was not used to identify the coefficients. Proposition 14 and the verified rank condition on the true observation maps establish why this recurrence continues for every initial state of this particular system. The four samples alone establish the fitted recurrence only within the declared closed two-state model class. ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b6031abf83540506","tex":"Y_{a,t}","aliases":"Y_a,t","display":"Y_{a,t}","kind":"use","excerpt":"ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0cf0201682d224ec","tex":"Y_{b,t}","aliases":"Y_b,t","display":"Y_{b,t}","kind":"use","excerpt":"ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-151cda08db8a7a2a","tex":"Y_s","aliases":"Y_s","display":"Y_s","kind":"use","excerpt":"V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6280117a7eeb98e1","tex":"y_s","aliases":"y_s","display":"y_s","kind":"use","excerpt":"1/12 · 5/6 span W_*= R^2 *. The state Ox_s=(y_s,y_s+1)^ T therefore closes. claimThe added coordinate is forced by the action of L on the requested observable. In a larger system, unobserved directions unrelated to its future can remain outside W_*. boundaryComputing Ox_s uses a forward observation. It becomes an available delay state (y_t-1,y_t) o","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ea5ac5a487fde7e2","tex":"y_{s+1}","aliases":"y_s+1","display":"y_{s+1}","kind":"use","excerpt":"1/12 · 5/6 span W_*= R^2 *. The state Ox_s=(y_s,y_s+1)^ T therefore closes. claimThe added coordinate is forced by the action of L on the requested observable. In a larger system, unobserved directions unrelated to its future can remain outside W_*. boundaryComputing Ox_s uses a forward observation. It becomes an available delay state (y_t-1,y_t) o","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-50e74e1b36da6948","tex":"Y_t","aliases":"Y_t","display":"Y_t","kind":"use","excerpt":"X_t,j=ϕ_t,j(Y_t), m_t,j= E[X_t,jmid F_t-1], p_t,j=the model's prediction for that test.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-406154b5ab08c392","tex":"y_T","aliases":"y_T","display":"y_T","kind":"use","excerpt":"λ B_a_T(y_T)⋯ B_a_1(y_1)z_0","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1ecb58e66eedd43b","tex":"y_t","aliases":"y_t","display":"y_t","kind":"use","excerpt":"Let a finite-dimensional autonomous linear process have hidden state x_t∈ R^d and observed output y_t∈ R^p:","count":14},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a04e3a6df8caddfb","tex":"y_{t+1}","aliases":"y_t+1","display":"y_{t+1}","kind":"use","excerpt":"y_t+1=frac56y_t+frac112y_t-1, t≥1.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9821a09737f3e751","tex":"y_{t-1}","aliases":"y_t-1","display":"y_{t-1}","kind":"use","excerpt":"y_t+1=frac56y_t+frac112y_t-1, t≥1.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-eca873e51e3ef360","tex":"y_{t-1-j}","aliases":"y_t-1-j","display":"y_{t-1-j}","kind":"use","excerpt":"plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, r","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-41b34bd814150360","tex":"y_{t-h+1}","aliases":"y_t-h+1","display":"y_{t-h+1}","kind":"use","excerpt":"z_t=y_t-h+1 vdots y_t =O_hx_t-h+1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"For another linear map M:Y→ Z and output projectors R_k,","count":57},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"A **predictive state** is a summary z=γ(x), or more generally a summary of an observation history. It is sufficient only relative to a specified family of future quantities, actions and dynamics. There is no implication from exact storage of x to sufficiency of γ(x).","count":144},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"z'=Tz, B_s'=TB_sT^-1, λ'=λ T^-1, C'=TC.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-29be4b954d2875ed","tex":"z^{\\mathsf T}","aliases":"z^ T mathsf","display":"z^{\\mathsf T}","kind":"use","excerpt":"Then C_c,H={z:m≥0, z^ TQz≥0}. The cone is proper and generating, and m is strictly positive on every nonzero cone element. These facts follow by an invertible linear transformation to the Lorentz cone. In particular, an output with zero mass in the cone is the zero vector.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-87f361f91d26a2b6","tex":"z^+","aliases":"z^+","display":"z^+","kind":"use","excerpt":"Pr(omid z,a)=λ B_a,oz, z^+=fracB_a,ozλ B_a,oz.","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4ce8c5865fb8a8b6","tex":"z^\\top","aliases":"z^top top","display":"z^\\top","kind":"use","excerpt":"Then z _M=(z^top Mz)^1/2 gives widehat B_uz _M≤ρ z _M for all actions, directly from the displayed inequality. Proposition 18 therefore applies under arbitrary switching among them. This is the common quadratic Lyapunov construction from control theory [R8], used here as an explicit certificate rather than inferred from a list of eigenvalues.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-11eee2338f613c32","tex":"Z_+","aliases":"Z_+","display":"Z_+","kind":"use","excerpt":"Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.","count":14},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ba874a392801426d","tex":"Z_-","aliases":"Z_-","display":"Z_-","kind":"use","excerpt":"Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.","count":17},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6ebbcdadd1865cf3","tex":"Z_-^{-1}","aliases":"Z_-^-1","display":"Z_-^{-1}","kind":"use","excerpt":"D_2=Z_+Z_-^-1 =0 · 1 1/12 · 5/6.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1c6f799f582027c8","tex":"Z_-^\\top","aliases":"Z_-^top top","display":"Z_-^\\top","kind":"use","excerpt":"**Proof.** For every unit v∈ R^r, widetilde Z_-^top v ≥ Z_-^top v - E^top v ≥ s- E . Thus the perturbed matrix has full row rank and its pseudoinverse has norm at most (s- E )^-1. Since widetilde Z_-widetilde Z_-^†=I_r, subtract B from the fitted expression and collect terms. The bound follows by submultiplicativity and the triangle inequality. squ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d8cb69d506a43a52","tex":"Z_-^\\dagger","aliases":"Z_-^† dagger","display":"Z_-^\\dagger","kind":"use","excerpt":"B_0=Z_+Z_-^†,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"z_0=1, z_1=tfrac12, z_2=tfrac12, z_3=tfrac1124.","count":30},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"z_0=1, z_1=tfrac12, z_2=tfrac12, z_3=tfrac1124.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5e32645afb213c74","tex":"z_2","aliases":"z_2","display":"z_2","kind":"use","excerpt":"z_0=1, z_1=tfrac12, z_2=tfrac12, z_3=tfrac1124.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0702dc64afa5c9d8","tex":"z_3","aliases":"z_3","display":"z_3","kind":"use","excerpt":"z_0=1, z_1=tfrac12, z_2=tfrac12, z_3=tfrac1124.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-86298b26f7dd1fb4","tex":"z_i","aliases":"z_i","display":"z_i","kind":"use","excerpt":"z=∑_i:w_i>0 w_i z_i, w_i=λ P_iz≥0, z_i=P_iz/w_i, ∑_iw_i=1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9fdca6236d286603","tex":"Z_n","aliases":"Z_n","display":"Z_n","kind":"use","excerpt":"Let finite active sets G_n increase to G. Their unnormalized evidence is Z_n=∑_g∈ G_nπ(g)L_g(D). Only stages with Z_n>0 are considered.","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f0fa5f2a39dbffe4","tex":"Z_q","aliases":"Z_q","display":"Z_q","kind":"use","excerpt":"Z_q(u)=I_d^2 uI_d^2 vdots u^qI_d^2.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b29834cdc1b6e7f6","tex":"Z_{q-1}","aliases":"Z_q-1","display":"Z_{q-1}","kind":"use","excerpt":"F_a(u) · =Z_q(u)^ TG_a,0Z_q(u) · +(1-u^2)Z_q-1(u)^ TG_a,1Z_q-1(u), J B_a(u) · =Φ_F_a(u) J.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5e940a55f95e2b85","tex":"Z_r","aliases":"Z_r","display":"Z_r","kind":"use","excerpt":"with one common c>0. For a fixed prior, put δ=r(p)-barℓ, bar b= B_barℓ(P) and Z_r=r(p)^top P. Then","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b3ed7b49158828ff","tex":"z_s","aliases":"z_s","display":"z_s","kind":"use","excerpt":"If D ≤ρ<1 and z_s ≤ Z over the relevant history, omitting terms with k≥ H changes the one-step right-hand side by at most","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3480de748bc9723a","tex":"Z_t","aliases":"Z_t","display":"Z_t","kind":"use","excerpt":"Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"Suppose X is a Hilbert space, P is an orthogonal projection, Q=I-P and x_t+1=Lx_t. The retained state is z_t=Px_t. Then","count":17},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-83c84ae2ae962020","tex":"Z_{t+1}","aliases":"Z_t+1","display":"Z_{t+1}","kind":"use","excerpt":"Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8e5d0000abb9cf07","tex":"z_{t+1}","aliases":"z_t+1","display":"z_{t+1}","kind":"use","excerpt":"z_t+1=PLPz_t+PLQx_t.","count":8},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d7ef16f9cd73bef7","tex":"z_{t-1-k}","aliases":"z_t-1-k","display":"z_{t-1-k}","kind":"use","excerpt":"z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"where B ranges over the current partition. The process terminates after at most |Ω|-|π_init| strict refinement rounds. The final partition is the unique coarsest refinement of π_0 on which all declared predictions and Bayesian updates close for every prior.","count":56},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-050083a1e7579b9f","tex":"|^k","aliases":"|^k","display":"|^k","kind":"use","excerpt":"Using ∫_-1^1|u|^kdu/2=1/(k+1) gives the integrated upper budget 241/40000. The same center-reset reference is valid there. Both models have the same initial state.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"t± e_1^top R^k(x,y)^top≥0 for every k≥0.","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"Suppose candidate states and their successors have been recorded as columns of matrices Z_-,Z_+∈ R^m× N. Their construction may use output histories. No hidden-state coordinates need be supplied to the fitting procedure.","count":18},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"**Proposition 24. Prior-preserving program refinement.** Suppose a parent program g is replaced by children g_j with weights α_j≥0, ∑_jα_j=1, and suppose its likelihood is the mixture","count":34},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-59ce46150d0cfe77","tex":"\\alpha_0","aliases":"α_0 alpha","display":"\\alpha_0","kind":"use","excerpt":"If these identities hold for every symbol and z_0=Vα_0 for α_0≥0 with 1^topα_0=1, then","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","excerpt":"**Proposition 24. Prior-preserving program refinement.** Suppose a parent program g is replaced by children g_j with weights α_j≥0, ∑_jα_j=1, and suppose its likelihood is the mixture","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-673ddcfc7e4cb456","tex":"\\alpha_t","aliases":"α_t alpha","display":"\\alpha_t","kind":"use","excerpt":"e_t+1≤α_te_t+β_t, α_t=frac(1+R λ _*)L_twidehatη_t, β_t=frac(1+R λ _*)ϵ_B,tRwidehatη_t.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.","count":11},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3f38b85993ddc9f2","tex":"\\beta_a","aliases":"β_a beta","display":"\\beta_a","kind":"use","excerpt":"λ B_a(u)=(b_a,0(u),b_a(u)^ T), β_a(u)=b_a,0(u)+b_a(u)^ Tc.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-16954fea007a97d7","tex":"\\beta_{a,j,i}","aliases":"β_a,j,i beta","display":"\\beta_{a,j,i}","kind":"use","excerpt":"Under the rowwise noise and parameter-bound assumptions of Proposition 81, simultaneous confidence ellipsoids for the rows of M_a,j give a confidence set for every C_a,k by this known linear map. If row i of widehat M_a,j-M_a,j has V_a,t-norm at most β_a,j,i, then for any direction x,","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-57613a21d43520bf","tex":"\\beta_{m,\\boldsymbol i}","aliases":"β_m, i beta boldsymbol","display":"\\beta_{m,\\boldsymbol i}","kind":"use","excerpt":"with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-988ef8458bc095f4","tex":"\\beta_t","aliases":"β_t beta","display":"\\beta_t","kind":"use","excerpt":"e_t+1≤α_te_t+β_t, α_t=frac(1+R λ _*)L_twidehatη_t, β_t=frac(1+R λ _*)ϵ_B,tRwidehatη_t.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d72a8a4f1d324e17","tex":"\\gamma_t","aliases":"γ_t gamma","display":"\\gamma_t","kind":"use","excerpt":"Allow the representation to change with time. Let γ_t:X→ Z_t be the declared representation at step t, with a norm on each Z_t. Let x_t+1=F_t(x_t) be the true transition and let widehat f_t:Z_t→ Z_t+1 be the implemented step. Its defect is","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-801ed78887aad933","tex":"\\gamma_{t+1}","aliases":"γ_t+1 gamma","display":"\\gamma_{t+1}","kind":"use","excerpt":"r_t(x)=γ_t+1(F_t(x))-widehat f_t(γ_t(x)).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"a= widehat A +∑_k=0^H-1 widehat K_k <1, sup_t≥0 d_t ≤Δ.","count":42},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.","count":58},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e84d9212fb63670f","tex":"\\Delta_a","aliases":"Δ_a Delta","display":"\\Delta_a","kind":"use","excerpt":"**Proposition 71. Robust oriented margin.** Suppose σ_a(u)≥κ_a(u) and an uncertain family has transported errors Δ_a(u) satisfying, for every normalized input,","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-28f5ad27d0de9148","tex":"\\Delta_H","aliases":"Δ_H Delta","display":"\\Delta_H","kind":"use","excerpt":"a_H=frac78-frac38 3^-H, Δ_H=frac38 3^-H, sup_t|z_t-widehat z_t|≤ frac3 3^-H1+3 3^-H.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"P_i=U^*E_iU, P_iP_j=δ_ijP_i, P_i^*=P_i, ∑_iP_i=I_X.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9601f08c373fe8e4","tex":"\\delta_k","aliases":"δ_k delta","display":"\\delta_k","kind":"use","excerpt":"e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ae5d1ee4ca82d29f","tex":"\\delta_n","aliases":"δ_n delta","display":"\\delta_n","kind":"use","excerpt":"Pr(true model∉ U_nmidprevious data) ≤δ_n, ∑_n≥1δ_n≤δ.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-df62fbeae71e37b5","tex":"\\delta_s","aliases":"δ_s delta","display":"\\delta_s","kind":"use","excerpt":"z=Rp, δ_s=(RM_s-B_sR)p.","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4abc637c4be411ae","tex":"\\delta_t","aliases":"δ_t delta","display":"\\delta_t","kind":"use","excerpt":"**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Let A and O be finite nonempty action and observation sets. A symbol s=(a,o) records an action followed by an observation. A word w=s_1⋯ s_k has temporal order from left to right. The empty word is ϵ.","count":58},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"If the declared polynomial itself approximates a nonpolynomial map with remainder at most ε_R on the ball, add ε_R to the defect bound. Without such a remainder estimate, the result is only a theorem for the polynomial, not for the unexpanded physical law.","count":15},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fb43c73461568514","tex":"\\epsilon^2","aliases":"ϵ^2 epsilon","display":"\\epsilon^2","kind":"use","excerpt":"**Proof.** For a centered Bernoulli variable X-p, write g(t)=log Eexp(t(X-p)). Then g(0)=g'(0)=0, and g''(t) is the variance of a Bernoulli under exponential tilting, at most 1/4. Integrating twice gives g(t)≤ t^2/8. Independence and the exponential Markov inequality give an upper-tail bound exp(-2Nϵ^2) after optimizing t=4ϵ. The lower tail has the","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5493b56ee0c9f701","tex":"\\epsilon_a","aliases":"ϵ_a epsilon","display":"\\epsilon_a","kind":"use","excerpt":"ϵ_a=sup_z∈ K∑_o (B_a,o-widetilde B_a,o)z _C.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-56a68026dad3ce1f","tex":"\\epsilon_B","aliases":"ϵ_B epsilon","display":"\\epsilon_B","kind":"use","excerpt":"For a predicted state widehat z, form widehat u=widehat B_swidehat z, widehat p=λwidehat u, and define d=widehat u-u. Write e= widehat z-z , widehat B_s ≤ L, and widehat B_s-B_s ≤ϵ_B. Then","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8f12c1f9b46464b3","tex":"\\epsilon_{B,t}","aliases":"ϵ_B,t epsilon","display":"\\epsilon_{B,t}","kind":"use","excerpt":"e_t+1≤α_te_t+β_t, α_t=frac(1+R λ _*)L_twidehatη_t, β_t=frac(1+R λ _*)ϵ_B,tRwidehatη_t.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f526d940ef57fb66","tex":"\\epsilon_N","aliases":"ϵ_N epsilon","display":"\\epsilon_N","kind":"use","excerpt":"|widehat f(w)-f(w)|≤ ϵ_N, ϵ_N=sqrtfraclog(2M/δ)2N.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-369a731ef7da1eab","tex":"\\varepsilon_R","aliases":"ε_R varepsilon","display":"\\varepsilon_R","kind":"use","excerpt":"If the declared polynomial itself approximates a nonpolynomial map with remainder at most ε_R on the ball, add ε_R to the defect bound. Without such a remainder estimate, the result is only a theorem for the polynomial, not for the unexpanded physical law.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-48f1a1f0695ed92e","tex":"\\varepsilon_{\\mathrm{repair}}","aliases":"ε_repair varepsilon mathrm","display":"\\varepsilon_{\\mathrm{repair}}","kind":"use","excerpt":"ε_repair ≤frac3132frac24140000+frac116 =frac874711280000.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-30f37f340ba25f4e","tex":"\\varepsilon_{\\rm repaired}","aliases":"ε_rm repaired varepsilon rm","display":"\\varepsilon_{\\rm repaired}","kind":"use","excerpt":"ε_rm repaired ≤frac78frac24140000+frac14 =frac81687320000.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-82928a28dec02c83","tex":"\\epsilon_s","aliases":"ϵ_s epsilon","display":"\\epsilon_s","kind":"use","excerpt":"g_i^topwidehat B_sv_j ≥ϵ_s g_i _2 v_j _2 for all s,i,j.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d08739f367199333","tex":"\\epsilon_t","aliases":"ϵ_t epsilon","display":"\\epsilon_t","kind":"use","excerpt":"**Proposition 43. Observation-tree total-variation bound.** If both instruments preserve C and conserve mass, and if every action at time t has the discrepancy in Proposition 42 bounded by ϵ_t, then the distributions P_T,widetilde P_T of length-T observation records under the common policy satisfy","count":14},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2e1da68720ed7f90","tex":"\\epsilon_\\Pi","aliases":"ϵ_Π epsilon Pi","display":"\\epsilon_\\Pi","kind":"use","excerpt":"ϵ_Π:=sup_a,z∈ K∫ (B_a(y)-B_a^Π(y))z _C μ(dy) ≤ Lh.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then","count":55},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a0fc025c97d2c8ba","tex":"\\eta_*","aliases":"η_* eta","display":"\\eta_*","kind":"use","excerpt":"**Proposition 80. Convex repair bracketing.** The feasible repair weights form a closed interval [η_*,1]. If a cone-positive functional ℓ=(1,-v^ T), with v _2≤1, and a normalized input z give c=ℓwidehat F_a(u)z<κ, then every feasible weight satisfies","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-94992593e2800251","tex":"\\eta_{a,k}","aliases":"η_a,k eta","display":"\\eta_{a,k}","kind":"use","excerpt":"sup_z∈ K (C_a,n,k-widetilde C_a,n,k)z _C ≤η_a,k,","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-12235ebd9b8ec538","tex":"\\eta_t","aliases":"η_t eta","display":"\\eta_t","kind":"use","excerpt":"Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9359c9fe8689c79c","tex":"\\eta_U","aliases":"η_U eta","display":"\\eta_U","kind":"use","excerpt":"A complete certificate at a weight η_U gives an upper bound on η_*. A negative Bernstein coefficient without a negative evaluated point does not give a lower bound.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"∑_s≤ tϕ_sϕ_s^ T and widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s. For 0<δ<1, with probability at least 1-δ, simultaneously for every finite t, widehatθ_t-θ_* _V_t ≤σsqrtlogfracdet V_tdet(η I)+2log(1/δ)+sqrtη S. diagram14_uncertainty In the structured moment family, a nonempty returned rectangle Θ is internally valid when its sufficient floor ρ_Θ=1-sup_(a,b)∈Θ(|a|","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","excerpt":"E[e^θ(X-m)mid F]≤ e^θ^2/8.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c45faae0260eabc9","tex":"\\theta_*","aliases":"θ_* theta","display":"\\theta_*","kind":"use","excerpt":"for every real h, with σ>0, and a declared bound θ_* _2≤ S. With ridge γ>0, set","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c91a4f8145c7f81d","tex":"\\theta_*^{\\mathsf T}","aliases":"θ_*^ T theta mathsf","display":"\\theta_*^{\\mathsf T}","kind":"use","excerpt":"Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c8082d88b8040556","tex":"\\theta_j","aliases":"θ_j theta","display":"\\theta_j","kind":"use","excerpt":"**Proof.** For convex weights θ_j, take A_s=∑_jθ_jA_s^(j). The equality and nonnegativity survive convex combination, as does mass conservation. At each history, every selected family maps the current cone state back into the cone. Applying the mass identity at that node proves recursive normalization, regardless of how the family was selected. For","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then","count":22},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-21ca4fcad33901b6","tex":"\\kappa_a","aliases":"κ_a kappa","display":"\\kappa_a","kind":"use","excerpt":"**Proposition 71. Robust oriented margin.** Suppose σ_a(u)≥κ_a(u) and an uncertain family has transported errors Δ_a(u) satisfying, for every normalized input,","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5016fdf60da7cee5","tex":"\\kappa_t","aliases":"κ_t kappa","display":"\\kappa_t","kind":"use","excerpt":"Empty products are one. If the output h_t(x) has a decoder H_t with Lipschitz constant κ_t and h_t(x_t)-H_t(γ_t(x_t)) ≤η_t, then","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"**Proposition 3. Complete refinement preserves reconstruction.** Suppose a finite rooted tree has root projector I, and every internal projector is the sum of pairwise orthogonal child projectors. Every complete antichain Λ, meeting each root-to-leaf path exactly once, satisfies","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-11c22a9994059ca8","tex":"\\lambda'","aliases":"λ' lambda","display":"\\lambda'","kind":"use","excerpt":"z'=Tz, B_s'=TB_sT^-1, λ'=λ T^-1, C'=TC.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"use","excerpt":"e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-29b14a34343e28e4","tex":"\\lambda_t","aliases":"λ_t lambda","display":"\\lambda_t","kind":"use","excerpt":"**Proposition 18. Compositional defect propagation.** Suppose widehat f_t is λ_t-Lipschitz on a domain containing both γ_t(x_t) and the predicted state widehat z_t. Suppose r_t(x_t) ≤δ_t and widehat z_t+1=widehat f_t(widehat z_t). Then, with e_t= γ_t(x_t)-widehat z_t ,","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"Let (Y, Y) be a standard Borel observation space and let μ be a fixed probability reference measure. Keep a finite action set. For each action a, let B_a(y):E→ E be measurable and integrable in operator norm. Let (C,λ) be a proper probability cone and mass functional as in Section 46. Assume that, outside one μ-null set for each action,","count":39},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-078f627daf3127c0","tex":"\\mu'","aliases":"μ' mu","display":"\\mu'","kind":"use","excerpt":"with respect to μ^⊗ T. Replacing the reference measure by μ'=rμ, with r>0 almost everywhere and ∫ r dμ=1, requires B'_a(y)=B_a(y)/r(y). Event probabilities and normalized states then remain unchanged.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-33d31d05a86d50cf","tex":"\\mu^{\\otimes T}","aliases":"μ^⊗ T mu otimes","display":"\\mu^{\\otimes T}","kind":"use","excerpt":"with respect to μ^⊗ T. Replacing the reference measure by μ'=rμ, with r>0 almost everywhere and ∫ r dμ=1, requires B'_a(y)=B_a(y)/r(y). Event probabilities and normalized states then remain unchanged.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-69100bae87543a36","tex":"\\mu_s^a","aliases":"μ_s^a mu","display":"\\mu_s^a","kind":"use","excerpt":"Consequently Proposition 66 gives a time-uniform certificate for ∑_s≤ tμ_s^a from ∑_s≤ tW_s^a, with its boundary multiplied by 1/π_min. The target is the average action-conditioned quantity along the encountered contexts, not a reset-population table or the state distribution under a different long-term policy.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f6e27daf17e4cbb7","tex":"\\mu_t^a","aliases":"μ_t^a mu","display":"\\mu_t^a","kind":"use","excerpt":"μ_t^a=∫ϕ_t(y)P_t(dymid H_t-1,a), W_t^a=frac1_{A_t=a}ϕ_t(Y_t)π_t(amid H_t-1).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,","count":12},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a1682ede43244542","tex":"\\nu_a","aliases":"ν_a nu","display":"\\nu_a","kind":"use","excerpt":"An additional uncertainty family with combined output error at most ν_a(u) is certified whenever","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3d046b689ede2f93","tex":"\\xi_s","aliases":"ξ_s xi","display":"\\xi_s","kind":"use","excerpt":"**Proof.** Put S_t=∑_s≤ tϕ_sξ_s. For every fixed q, the conditional moment bound makes","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bfa08d60e4e1fbda","tex":"\\xi_t","aliases":"ξ_t xi","display":"\\xi_t","kind":"use","excerpt":"Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"**Proposition 1. Exact realization and consistency.** The analysis map is injective, Π_a=A_aS_a is an idempotent with image C_a, and Π_ac=c exactly when c is a realizable coefficient vector. Moreover,","count":14},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f6c04fa92bb658c0","tex":"\\Pi^2","aliases":"Π^2 Pi","display":"\\Pi^2","kind":"use","excerpt":"[5pt] Transport · F_bT_ba=T_baF_a · Do coordinate descriptions agree? [5pt] Closure · γ F_u=bar F_uγ · Does the summary determine its own future? [5pt] Probability · B_oC⊆ C, λ∑_oB_o=λ · Are every branch and every finite record valid? Reading route. Plates 1--4 build the representation and its predictive quotient. Plates 5--9 explain interactions, ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2d28a27ac7821632","tex":"\\pi_0","aliases":"π_0 pi","display":"\\pi_0","kind":"use","excerpt":"Fix a mandatory initial partition π_0 with at least two blocks. This preserves distinctions that the intended application requires and excludes the trivial one-state quotient.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a23e3c8d3f2c871d","tex":"\\Pi_a","aliases":"Π_a Pi","display":"\\Pi_a","kind":"use","excerpt":"**Proposition 1. Exact realization and consistency.** The analysis map is injective, Π_a=A_aS_a is an idempotent with image C_a, and Π_ac=c exactly when c is a realizable coefficient vector. Moreover,","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a441482c09c00b83","tex":"\\Pi_a^2","aliases":"Π_a^2 Pi","display":"\\Pi_a^2","kind":"use","excerpt":"**Proof.** If A_ax=0, then x=S_aA_ax=0. Also Π_a^2=A_a(S_aA_a)S_a=Π_a. Its image is contained in C_a and it fixes every A_ax, so its image is exactly C_a. Both norm inequalities follow from the operator-norm inequality applied to S_a. square","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5bd12097542b9765","tex":"\\pi_{\\mathrm{init}}","aliases":"π_init pi mathrm","display":"\\pi_{\\mathrm{init}}","kind":"use","excerpt":"where B ranges over the current partition. The process terminates after at most |Ω|-|π_init| strict refinement rounds. The final partition is the unique coarsest refinement of π_0 on which all declared predictions and Bayesian updates close for every prior.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9358dc93f921219b","tex":"\\pi_{\\min}","aliases":"π_min pi min","display":"\\pi_{\\min}","kind":"use","excerpt":"**Proposition 67. Covered adaptive-action evidence.** If the target action has π_t(amid H_t-1)≥π_min>0 at every time, then","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9091f42f61615f68","tex":"\\pi_t","aliases":"π_t pi","display":"\\pi_t","kind":"use","excerpt":"For intervention evidence use the information H_t-1 just before choosing the action. Let the action be randomized from a known history-dependent policy π_t. The true next-observation kernel under action a is P_t(·mid H_t-1,a). Randomization uses no unrecorded future information. For a predictable test ϕ_t∈[0,1], define","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"If D ≤ρ<1 and z_s ≤ Z over the relevant history, omitting terms with k≥ H changes the one-step right-hand side by at most","count":33},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bd1aac0f414abebd","tex":"\\rho^2","aliases":"ρ^2 rho","display":"\\rho^2","kind":"use","excerpt":"widehat B_u^top Mwidehat B_upreceqρ^2M for every declared action u.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8128e6a42059bad5","tex":"\\rho^H","aliases":"ρ^H rho","display":"\\rho^H","kind":"use","excerpt":"B C Zfracρ^H1-ρ.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-db038dce9180f140","tex":"\\rho^T","aliases":"ρ^T rho","display":"\\rho^T","kind":"use","excerpt":"e_T≤ρ^Te_0+δfrac1-ρ^T1-ρ.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-decc400182cbc471","tex":"\\rho^t","aliases":"ρ^t rho","display":"\\rho^t","kind":"use","excerpt":"Dropping the initial unresolved-state term adds at most B ρ^t q_0 .","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e385ab7f9430ddcd","tex":"\\rho_\\Theta","aliases":"ρ_Θ rho Theta","display":"\\rho_\\Theta","kind":"use","excerpt":"eta_t=V_t^-1∑_s≤ tϕ_sY_s. For 0<δ<1, with probability at least 1-δ, simultaneously for every finite t, widehatθ_t-θ_* _V_t ≤σsqrtlogfracdet V_tdet(η I)+2log(1/δ)+sqrtη S. diagram14_uncertainty In the structured moment family, a nonempty returned rectangle Θ is internally valid when its sufficient floor ρ_Θ=1-sup_(a,b)∈Θ(|a|+|b|) is nonnegative. Sta","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"**Proposition 16. Fixed-space identification bound.** Put s=σ_min(Z_-)>0. If E <s, then widetilde Z_- has full row rank, and","count":24},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"Y_t=θ_*^ Tϕ_t+ξ_t, E[e^hξ_tmid F_t-1] ≤ e^h^2σ^2/2","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7cc70e05d3b03d7a","tex":"\\sigma_a","aliases":"σ_a sigma","display":"\\sigma_a","kind":"use","excerpt":"σ_a(u)=min_ v _2≤1 [(F_a(u)(1,v))_0- (F_a(u)(1,v))_rm sp _2].","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fc148a89e558c088","tex":"\\sigma_b","aliases":"σ_b sigma","display":"\\sigma_b","kind":"use","excerpt":"ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-22301d4bf1c931e5","tex":"\\sigma_{\\min}","aliases":"σ_min sigma min","display":"\\sigma_{\\min}","kind":"use","excerpt":"**Proposition 16. Fixed-space identification bound.** Put s=σ_min(Z_-)>0. If E <s, then widetilde Z_- has full row rank, and","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f393e12f20aeb427","tex":"\\sigma_r","aliases":"σ_r sigma","display":"\\sigma_r","kind":"use","excerpt":"Let X∈ R^m× n have rank r≥1, let s=σ_r(X)>0, and observe Y=X+E with E _2≤ε<s/2. Let P project onto imX, and let widehat P project onto the leading r left singular vectors of Y.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e1ee5d0e033b41f4","tex":"\\sigma_{r+1}","aliases":"σ_r+1 sigma","display":"\\sigma_{r+1}","kind":"use","excerpt":"σ_r(Y)≥ s-ε, σ_r+1(Y)≤ε, P-widehat P _2≤frac2εs<1.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"P-P_n _TV=τ_n, τ_n=1-fracZ_nZ.","count":13},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8c811b6eab2988ab","tex":"\\tau_1","aliases":"τ_1 tau","display":"\\tau_1","kind":"use","excerpt":"plate10Behavior can supply the coordinatesFuture tests define a predictive state without naming hidden variables.P27--P28 For a controlled process, arrange population probabilities into a behavior table: columns index past histories and rows index future tests. Its full column span is the linear predictive behavior space. c|ccc · h_1 · h_2 · ⋯ τ_1 ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e675956848d7751d","tex":"\\tau_2","aliases":"τ_2 tau","display":"\\tau_2","kind":"use","excerpt":"plate10Behavior can supply the coordinatesFuture tests define a predictive state without naming hidden variables.P27--P28 For a controlled process, arrange population probabilities into a behavior table: columns index past histories and rows index future tests. Its full column span is the linear predictive behavior space. c|ccc · h_1 · h_2 · ⋯ τ_1 ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c5bd92712ba2926c","tex":"\\tau_a","aliases":"τ_a tau","display":"\\tau_a","kind":"use","excerpt":"**Proposition 62. Oriented quadratic probability certificate.** Suppose that for each action and every u∈[-1,1] there are τ_a(u)≥0 and R_a(u)succeq0 satisfying","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0cddcde32776d569","tex":"\\tau_n","aliases":"τ_n tau","display":"\\tau_n","kind":"use","excerpt":"P-P_n _TV=τ_n, τ_n=1-fracZ_nZ.","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"dence among the input columns. Under the kernel condition, define B(Z_-a)=Z_+a. Two choices of a differ by a kernel vector and give the same answer. This defines and uniquely determines the map on W. It maps W into W exactly under the image condition. The pseudoinverse formula gives that map because a-Z_-^† Z_-a∈ker Z_-. If Wsubsetneq V, choose a n","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-836b4bc987c9b1dc","tex":"\\phi_{a,t}","aliases":"ϕ_a,t phi","display":"\\phi_{a,t}","kind":"use","excerpt":"ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fccb8cf89f9eb47b","tex":"\\phi_{b,t}","aliases":"ϕ_b,t phi","display":"\\phi_{b,t}","kind":"use","excerpt":"ϕ_a,t · =U_t-1/3, · Y_a,t · =U_t, · σ_a · =1, ϕ_b,t · =U_t-1^2/5, · Y_b,t · =L_2(U_t), · σ_b · =3/4.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-dde952a148b3c614","tex":"\\Phi_F","aliases":"Φ_F Phi","display":"\\Phi_F","kind":"use","excerpt":"[Φ_F(X)]_ij=∑_p,q=1^d F_(i,p),(j,q)X_pq.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-193a13ca24658533","tex":"\\Phi_{F_a(u)}","aliases":"Φ_F_a(u) Phi","display":"\\Phi_{F_a(u)}","kind":"use","excerpt":"F_a(u) · =Z_q(u)^ TG_a,0Z_q(u) · +(1-u^2)Z_q-1(u)^ TG_a,1Z_q-1(u), J B_a(u) · =Φ_F_a(u) J.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d0f231c57bed48c8","tex":"\\Phi_s","aliases":"Φ_s Phi","display":"\\Phi_s","kind":"use","excerpt":"J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-74777c98027fb1ad","tex":"\\phi_s","aliases":"ϕ_s phi","display":"\\phi_s","kind":"use","excerpt":"V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-56934ab40890ca78","tex":"\\phi_s^{\\mathsf T}","aliases":"ϕ_s^ T phi mathsf","display":"\\phi_s^{\\mathsf T}","kind":"use","excerpt":"V_t=γ I+∑_s≤ tϕ_sϕ_s^ T, widehatθ_t=V_t^-1∑_s≤ tϕ_sY_s.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7016cbae921a9f68","tex":"\\phi_t","aliases":"ϕ_t phi","display":"\\phi_t","kind":"use","excerpt":"For intervention evidence use the information H_t-1 just before choosing the action. Let the action be randomized from a known history-dependent policy π_t. The true next-observation kernel under action a is P_t(·mid H_t-1,a). Randomization uses no unrecorded future information. For a predictable test ϕ_t∈[0,1], define","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c81e3448ab656f09","tex":"\\phi_t^2","aliases":"ϕ_t^2 phi","display":"\\phi_t^2","kind":"use","excerpt":"**Proposition 84. Exact rational outer confidence intervals.** Write V=1+∑ϕ_t^2 and widehatθ=∑ϕ_tY_t/V. Let k=lceillog_2 Vrceil and h=lceillog_2(1/δ)rceil. A rational upper enclosure of","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3c644a49111d6915","tex":"\\Phi_u","aliases":"Φ_u Phi","display":"\\Phi_u","kind":"use","excerpt":"Φ_u(X)=∑_ν w_ν g_ν(u)K_ν(u)XK_ν(u)^ T, g_ν(u)∈{1,1-u^2}.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Let B_a(u)=∑_k=0^Dψ_k(u)C_a,k. Choose functions g_j so that the known matrix W_jk=∫ g_j(u)ψ_k(u) μ(du) is invertible.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-631a8dfac98b7969","tex":"\\psi_k","aliases":"ψ_k psi","display":"\\psi_k","kind":"use","excerpt":"Let B_a(u)=∑_k=0^Dψ_k(u)C_a,k. Choose functions g_j so that the known matrix W_jk=∫ g_j(u)ψ_k(u) μ(du) is invertible.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"F(x)=b+∑_m=1^pB_m(x,…,x),","count":21},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix","count":18},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"Rare evidence can create a large posterior change from a small absolute error. For example, take a true unnormalized vector (η,0) and a perturbed vector (η,η). Their ℓ^1 difference is η, while their normalized vectors differ by one in ℓ^1. Both arise as possible branch outputs of valid probability instruments when 0<η≤1/2, after assigning the remai","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"This result is strong convergence, not operator-norm convergence and not a bound uniform in time. On ℓ^2, finite coordinate projections converge strongly to I while I-P_n =1 for every n. Unbounded operators require domain control. Adaptive data-dependent partitions and fitted coefficients require additional consistency arguments. These remain separ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"P_a=∑_i∈ B_ap_i, r_a(p)=frac∑_i∈ B_ap_iℓ_iP_a (P_a>0).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a8e35b84dc016d32","tex":"\\ell_y","aliases":"ℓ_y ell","display":"\\ell_y","kind":"use","excerpt":"Let Ω be a finite state set. Let K_u(i,j) be known row-stochastic transition matrices. Let ℓ_y(i)>0 be a finite family of observation likelihoods. A partition π has aggregation matrix","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-96386bd0d0e3dd2b","tex":"\\ell_y^\\top","aliases":"ℓ_y^top ell top","display":"\\ell_y^\\top","kind":"use","excerpt":"B_ℓ_y(p) =fracpodotℓ_yℓ_y^top p.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"For an endomorphism F:X→ X, define its coordinate realization by F_a=A_aFS_a on C_a. In this paragraph F may be nonlinear. For a partial map, every formula is restricted to the transported admissible domain. Maps between different signal types use separate input and output atlases.","count":27},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Distributions are column vectors. Prediction is p↦ K_u^top p, and a Bayesian update is","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","excerpt":"∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx, x ^2=∑_v∈Λ P_vx ^2.","count":75},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-69bc40bed9aab324","tex":"\\prod_{j=0}^{T-1}","aliases":"∏_j=0^T-1 prod","display":"\\prod_{j=0}^{T-1}","kind":"use","excerpt":"e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f5e01624215da775","tex":"\\prod_{j=1}^{m}","aliases":"∏_j=1^m prod","display":"\\prod_{j=1}^{m}","kind":"use","excerpt":"F(x)-F_ I(x) ≤∑_m=1^p ∑_ i∉ I_m β_m, i∏_j=1^ma_i_j.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-bc18fd207837bc48","tex":"\\prod_{j=1}^{|w|}","aliases":"∏_j=1^|w| prod","display":"\\prod_{j=1}^{|w|}","kind":"use","excerpt":"Pr_π(w)=f(w)∏_j=1^|w|π(a_jmid h_j-1),","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e6f346e97f136cc0","tex":"\\prod_{j=k+1}^{T-1}","aliases":"∏_j=k+1^T-1 prod","display":"\\prod_{j=k+1}^{T-1}","kind":"use","excerpt":"e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.","count":113},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0e35bb5874cc0396","tex":"\\sum_{\\boldsymbol i\\in\\mathcal I_m}","aliases":"∑_ i∈ I_m sum boldsymbol in mathcal","display":"\\sum_{\\boldsymbol i\\in\\mathcal I_m}","kind":"use","excerpt":"Lip(F_ I) ≤∑_m=1^p ∑_ i∈ I_m mβ_m, iR^m-1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-738aea01982d68b4","tex":"\\sum_{\\boldsymbol i\\notin\\mathcal I_m}","aliases":"∑_ i∉ I_m sum boldsymbol notin mathcal","display":"\\sum_{\\boldsymbol i\\notin\\mathcal I_m}","kind":"use","excerpt":"F(x)-F_ I(x) ≤∑_m=1^p ∑_ i∉ I_m β_m, i∏_j=1^ma_i_j.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"Ax=(w_aU_ax)_a, S(c_a)_a=frac1c∑_aw_aU_a^*c_a, c=∑_aw_a^2.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-76e758f1be5158a2","tex":"\\sum_g","aliases":"∑_g sum","display":"\\sum_g","kind":"use","excerpt":"Let G be a countable declared program set, with one fixed probability mass function π(g)>0 satisfying ∑_gπ(g)=1. Every program specifies a likelihood L_g(D) for the **same** observed data event or density relative to a common base measure. Assume","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ddde53c97efaa1f5","tex":"\\sum_{g\\in\\mathcal G}","aliases":"∑_g∈ G sum in mathcal","display":"\\sum_{g\\in\\mathcal G}","kind":"use","excerpt":"0<Z(D)=∑_g∈ Gπ(g)L_g(D)<∞.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-eb6772854c63eb68","tex":"\\sum_{g\\in\\mathcal G_n}","aliases":"∑_g∈ G_n sum in mathcal","display":"\\sum_{g\\in\\mathcal G_n}","kind":"use","excerpt":"Let finite active sets G_n increase to G. Their unnormalized evidence is Z_n=∑_g∈ G_nπ(g)L_g(D). Only stages with Z_n>0 are considered.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7fdbc826a37e886e","tex":"\\sum_{g\\notin\\mathcal G_n}","aliases":"∑_g∉ G_n sum notin mathcal","display":"\\sum_{g\\notin\\mathcal G_n}","kind":"use","excerpt":"∑_g∉ G_nπ(g)L_g(D)≤ b_n,","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-20794c41f8dd0413","tex":"\\sum_h","aliases":"∑_h sum","display":"\\sum_h","kind":"use","excerpt":"**Proof.** If ∑_hc_hv_h=0, then for every test t,","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Within one orthogonal atlas, let E_i be pairwise orthogonal coordinate selectors with ∑_iE_i=I. Pull them back by a unitary map:","count":9},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ff1f52005cd89e31","tex":"\\sum_{i:w_i>0}","aliases":"∑_i:w_i>0 sum","display":"\\sum_{i:w_i>0}","kind":"use","excerpt":"z=∑_i:w_i>0 w_i z_i, w_i=λ P_iz≥0, z_i=P_iz/w_i, ∑_iw_i=1.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-428bee4de6cfa9ad","tex":"\\sum_{i_1,\\ldots,i_m}","aliases":"∑_i_1,…,i_m sum ldots","display":"\\sum_{i_1,\\ldots,i_m}","kind":"use","excerpt":"Q_jF(x)=Q_jb+ ∑_m=1^p ∑_i_1,…,i_m Q_jB_m(P_i_1x,…,P_i_mx).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-60c3732b8a8548e8","tex":"\\sum_{i\\in B_a}","aliases":"∑_i∈ B_a sum in","display":"\\sum_{i\\in B_a}","kind":"use","excerpt":"P_a=∑_i∈ B_ap_i, r_a(p)=frac∑_i∈ B_ap_iℓ_iP_a (P_a>0).","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7df1a15008304fec","tex":"\\sum_{i\\notin J}","aliases":"∑_i∉ J sum notin","display":"\\sum_{i\\notin J}","kind":"use","excerpt":"This is a conservation law for information under complete refinement. Deleting a child, storing only its energy, or imposing a coefficient threshold changes the contract. For a retained subset J of an orthogonal decomposition, the lost signal is exactly ∑_i∉ JP_ix, and its squared norm is the sum of the omitted packet energies.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"R_kMLP_i=∑_j(R_kMQ_j)(Q_jLP_i).","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9f64954fa9fe0048","tex":"\\sum_{j,i}","aliases":"∑_j,i sum","display":"\\sum_{j,i}","kind":"use","excerpt":"L=∑_j,iQ_jLP_i.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fff453c73113860b","tex":"\\sum_{j=0}^D","aliases":"∑_j=0^D sum","display":"\\sum_{j=0}^D","kind":"use","excerpt":"widehat M_a,j=G_a,jH^-1, widehat C_a,k=∑_j=0^D(W_D^-1)_kjwidehat M_a,j.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-8f7791db6e6f82eb","tex":"\\sum_{j=0}^{H-1}","aliases":"∑_j=0^H-1 sum","display":"\\sum_{j=0}^{H-1}","kind":"use","excerpt":"Here 0≤ y_t≤1 and the truncated recursion has gain a_H=frac12+∑_j=0^H-1K_j =frac78-frac38 3^-H<1. With the same initial state, recursive propagation therefore gives sup_t|y_t-widehat y_t|≤fracΔ_H1-a_H =frac33^H+3. boundaryP10's tail estimate uses the same resolved history. P19 supplies the separate stability argument for a predictor using its own h","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-df66290cc89f818f","tex":"\\sum_{j=0}^{t-1}","aliases":"∑_j=0^t-1 sum","display":"\\sum_{j=0}^{t-1}","kind":"use","excerpt":"plate7Memory is an exact eliminationThe discarded state leaves a kernel and an initial transient.P10, P19 Write the split dynamics as y_t+1=Ay_t+Bw_t and w_t+1=Cy_t+Dw_t. Eliminating w_t gives y_t+1=Ay_t+∑_j=0^t-1BD^jC y_t-1-j+BD^tw_0. diagram07_memory For the running example, A=1/2, B=1, C=1/4, D=1/3 and w_0=0. Thus K_j=tfrac14 3^-j. If |y_t|≤1, r","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-dc3b0f33f5c37841","tex":"\\sum_{j=1}^{m_s}","aliases":"∑_j=1^m_s sum","display":"\\sum_{j=1}^{m_s}","kind":"use","excerpt":"J B_s=Φ_s J, Φ_s(X)=∑_j=1^m_s w_sjK_sjXK_sj^ T, w_sj≥0.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a2d2365c8715fdcc","tex":"\\sum_{j\\in B}","aliases":"∑_j∈ B sum in","display":"\\sum_{j\\in B}","kind":"use","excerpt":"(∑_j∈ BK_u(i,j))_u,B,","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"**Proof.** A nonzero polynomial has strictly positive squared integral, so its coefficient vector c satisfies c^ TW_Dc>0. Thus W_D is invertible. Let T have rows t_i and S have columns h_j. Since H=TS is an invertible square core in an r-dimensional space, both T and S are invertible. Therefore G_a,jH^-1=TM_a,jT^-1. The relation M_a,j=∑_k(W_D)_jkC_","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b152cb868ef9f041","tex":"\\sum_{k=0}^D","aliases":"∑_k=0^D sum","display":"\\sum_{k=0}^D","kind":"use","excerpt":"B_a(u)=∑_k=0^D u^k C_a,k, M_a,j=∫_-1^1 u^j B_a(u) fracdu2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-349eae842e50f51e","tex":"\\sum_{k=0}^{D}","aliases":"∑_k=0^D sum","display":"\\sum_{k=0}^{D}","kind":"use","excerpt":"Δ_a(u)=∑_k=0^Du^kE_a,k, |(E_a,k)_ij|≤ r_k.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-97f6371d3c17d491","tex":"\\sum_{k=0}^{H-1}","aliases":"∑_k=0^H-1 sum","display":"\\sum_{k=0}^{H-1}","kind":"use","excerpt":"widehat z_t+1=widehat Awidehat z_t+ ∑_k=0^H-1widehat K_kwidehat z_t-1-k.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-b7f7abb917a0d195","tex":"\\sum_{k=0}^n","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^n","kind":"use","excerpt":"b_n,k(x)=binom nk x^k(1-x)^n-k, D_n(x,v)=(n+1)∑_k=0^n b_n,k(x)b_n,k(v).","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-459bad9ed7294423","tex":"\\sum_{k=0}^{T-1}","aliases":"∑_k=0^T-1 sum","display":"\\sum_{k=0}^{T-1}","kind":"use","excerpt":"e_T≤ (∏_j=0^T-1λ_j)e_0 +∑_k=0^T-1 (∏_j=k+1^T-1λ_j)δ_k.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-05da311cedba2569","tex":"\\sum_{k=0}^{t-1}","aliases":"∑_k=0^t-1 sum","display":"\\sum_{k=0}^{t-1}","kind":"use","excerpt":"z_t+1=Az_t+ ∑_k=0^t-1BD^kCz_t-1-k+BD^tq_0.","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c0dd06184d3077e8","tex":"\\sum_{m=1}^{p}","aliases":"∑_m=1^p sum","display":"\\sum_{m=1}^{p}","kind":"use","excerpt":"F(x)=b+∑_m=1^pB_m(x,…,x),","count":6},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f0d1bc9b91545210","tex":"\\sum_{n\\geq1}","aliases":"∑_n≥1 sum geq","display":"\\sum_{n\\geq1}","kind":"use","excerpt":"Pr(true model∉ U_nmidprevious data) ≤δ_n, ∑_n≥1δ_n≤δ.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-ccbd66bdaed18a2b","tex":"\\sum_o","aliases":"∑_o sum","display":"\\sum_o","kind":"use","excerpt":"1^top∑_oM_a,o=1^top.","count":13},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3f209121881932f6","tex":"\\sum_{o\\in\\mathcal O}","aliases":"∑_o∈ O sum in mathcal","display":"\\sum_{o\\in\\mathcal O}","kind":"use","excerpt":"f(ϵ)=1, f(w)≥0, ∑_o∈ Of(w(a,o))=f(w) for every w,a.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-13635d59c1053560","tex":"\\sum_{p,q=1}^d","aliases":"∑_p,q=1^d sum","display":"\\sum_{p,q=1}^d","kind":"use","excerpt":"[Φ_F(X)]_ij=∑_p,q=1^d F_(i,p),(j,q)X_pq.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"If P_i=∑_rP_i,r and Q_j=∑_sQ_j,s are complete refinements, then","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-7d98fe9de6bebd3c","tex":"\\sum_s","aliases":"∑_s sum","display":"\\sum_s","kind":"use","excerpt":"If P_i=∑_rP_i,r and Q_j=∑_sQ_j,s are complete refinements, then","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fa0b628b5b6157b5","tex":"\\sum_{s,r}","aliases":"∑_s,r sum","display":"\\sum_{s,r}","kind":"use","excerpt":"Q_jLP_i=∑_s,rQ_j,sLP_i,r.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2676ba5298805edd","tex":"\\sum_{s=0}^{t-1}","aliases":"∑_s=0^t-1 sum","display":"\\sum_{s=0}^{t-1}","kind":"use","excerpt":"q_t=D^tq_0+∑_s=0^t-1D^t-1-sCz_s.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-710340d697553ace","tex":"\\sum_{s=1}^t","aliases":"∑_s=1^t sum","display":"\\sum_{s=1}^t","kind":"use","excerpt":"R_t,j=∑_s=1^t(X_s,j-p_s,j), D_t,j=∑_s=1^t(m_s,j-p_s,j).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-efc134ce71b0ac13","tex":"\\sum_{s\\in\\mathcal A\\times\\mathcal O}","aliases":"∑_s∈ A× O sum in mathcal times mathcal","display":"\\sum_{s\\in\\mathcal A\\times\\mathcal O}","kind":"use","excerpt":"W_j+1=W_j+∑_s∈ A× OW_jM_s.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-3271c7cb83a580b3","tex":"\\sum_{s\\leq t}","aliases":"∑_s≤ t sum leq","display":"\\sum_{s\\leq t}","kind":"use","excerpt":"Iterating this conditional inequality gives the same exponential bound for the martingale sum ∑_s≤ t(X_s,j-m_s,j). Optimizing the exponential Markov inequality gives the two-sided bound 2exp(-2b^2/t). At the displayed b_t,j this is α_j/[t(t+1)]. Sum over t, using ∑_t≥11/[t(t+1)]=1, and then over j. The resulting event holds at every time, so it als","count":7},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0c905509ce292259","tex":"\\sum_{t<T}","aliases":"∑_t<T sum","display":"\\sum_{t<T}","kind":"use","excerpt":"P_T-widehat P_T _TV ≤min(1,frace_0+∑_t<Tϵ_t2).","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-469e6dedd4f8c880","tex":"\\sum_{t=0}^{T-1}","aliases":"∑_t=0^T-1 sum","display":"\\sum_{t=0}^{T-1}","kind":"use","excerpt":"P_T-widetilde P_T _TV ≤min(1,frac12 z_0-widetilde z_0 _C +frac12∑_t=0^T-1ϵ_t).","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-075f5b1599c3c4ed","tex":"\\sum_{t\\geq1}","aliases":"∑_t≥1 sum geq","display":"\\sum_{t\\geq1}","kind":"use","excerpt":"Iterating this conditional inequality gives the same exponential bound for the martingale sum ∑_s≤ t(X_s,j-m_s,j). Optimizing the exponential Markov inequality gives the two-sided bound 2exp(-2b^2/t). At the displayed b_t,j this is α_j/[t(t+1)]. Sum over t, using ∑_t≥11/[t(t+1)]=1, and then over j. The resulting event holds at every time, so it als","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d96e96aadd273752","tex":"\\sum_u","aliases":"∑_u sum","display":"\\sum_u","kind":"use","excerpt":"W_n+1=W_n+∑_uW_nL_u, W_nL_u={wL_u:w∈ W_n}.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-46a1bb46e6353ae5","tex":"\\sum_{u\\in\\operatorname{ch}(v)}","aliases":"∑_u∈ch(v) sum in operatorname","display":"\\sum_{u\\in\\operatorname{ch}(v)}","kind":"use","excerpt":"V(v)=min(c(v),∑_u∈ch(v)V(u))","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6c2f7250b20e70fc","tex":"\\sum_{v\\in\\Lambda}","aliases":"∑_v∈Λ sum in Lambda","display":"\\sum_{v\\in\\Lambda}","kind":"use","excerpt":"∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx, x ^2=∑_v∈Λ P_vx ^2.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-cd74cb17b4132748","tex":"\\sum_\\nu","aliases":"∑_ν sum nu","display":"\\sum_\\nu","kind":"use","excerpt":"F=∑_ν w_νvec(K_ν) vec(K_ν)^ T, w_ν≥0,","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"dom(g∘ f) ={x∈domf:f(x)∈domg}.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then","count":12},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"The infimum is finite because the cone is generating. Compactness of K makes this a genuine norm. Its unit ball is conv(K∪(-K)). In particular u _C=λ u on C, and |λ x|≤ x _C. On the ordinary probability orthant this is the ℓ^1 norm. These are standard ordered-space constructions [R15], used here to avoid mixing Euclidean coefficient magnitude with ","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"B_a(y)C⊆ C, λ∫_Y B_a(y) μ(dy)=λ.","count":35},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-f7fdd5012f61eec2","tex":"\\int_{-1}^1","aliases":"∫_-1^1 int","display":"\\int_{-1}^1","kind":"use","excerpt":"∫_-1^1u^k fracdu2 =0 · k odd, 1/(k+1) · k even.","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","excerpt":"B_a^[n](x)=∫_0^1D_n(x,v)B_a(v) dv =∑_k=0^n b_n,k(x)C_a,n,k,","count":4},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-6c39689bc04b0a10","tex":"\\int_A","aliases":"∫_A int","display":"\\int_A","kind":"use","excerpt":"P_a(Amid z)=∫_Aλ B_a(y)z μ(dy), z^+(y)=fracB_a(y)zλ B_a(y)z","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a7e334b9883c757e","tex":"\\int_{A_j}","aliases":"∫_A_j int","display":"\\int_{A_j}","kind":"use","excerpt":"B_a,j=∫_A_jB_a(y) μ(dy).","count":3},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-039de79d863b26f6","tex":"\\int_Y","aliases":"∫_Y int","display":"\\int_Y","kind":"use","excerpt":"B_a(y)C⊆ C, λ∫_Y B_a(y) μ(dy)=λ.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"a nonnegative supermartingale. Integrate against the fixed Gaussian law q∼ N(0,(γ I)^-1). Conditional Tonelli and completing the square give the nonnegative supermartingale","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"sup_t≥0e_t≤frac243722320 ≈0.10919.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"**Proof.** If w≠0, put e=w/ w _2. If w=0, choose any spatial unit vector e. Define Δ=ϵ(0,e)λ. Its mass row vanishes and its value on every normalized input is (0,ϵ e), of base norm ϵ. If w≠0, the new spatial norm is m+ϵ. If w=0, the original output mass is zero and the new spatial norm is ϵ. Either image lies outside the cone. square","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"If M_s≠0 and both likelihoods and normalized updates agree on every strictly positive belief, then RM_s=B_sR. Agreement of normalized updates alone need not force this equality.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"A_ax ≥ frac x S_a , S_a(c+e)-S_ac ≤ S_a e .","count":85},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"A **probability-domain certificate** specifies a closed, pointed, generating cone C⊂ E, with λ strictly positive on C∖{0}, such that","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"**Proposition 11. Ideal statistical benefit of refinement.** Let a target Y take values in a finite-dimensional Hilbert space with E Y ^2<∞. Let F⊆ G be information sets and define m_ F= E[Ymid F] and m_ G= E[Ymid G]. Then","count":17},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","excerpt":"with respect to μ^⊗ T. Replacing the reference measure by μ'=rμ, with r>0 almost everywhere and ∫ r dμ=1, requires B'_a(y)=B_a(y)/r(y). Event probabilities and normalized states then remain unchanged.","count":2},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-9f81640dea5052eb","tex":"\\bigcap_s","aliases":"⋂_s bigcap","display":"\\bigcap_s","kind":"use","excerpt":"C_h+1=C_h∩⋂_s B_s^-1C_h, C_h+1⊆ C_h.","count":1},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"with bounded multilinear maps. For each ordered tuple i=(i_1,…,i_m), let β_m, i bound the norm of B_m restricted to imP_i_1×⋯×imP_i_m. Let I_m be the retained tuple set, and let F_ I retain the constant term and exactly those interactions.","count":10},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"A_a:X⟶ C_a, S_a:C_a⟶ X, S_aA_a=I_X.","count":5},{"work":"monographs-large-signal-model","url":"notation/monographs-large-signal-model.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"y⟼ U(widehat U^top U)^-1y,","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"At degree four, E_4=5 contributes 5t^4/4! to secant. At degree five, E_5=16 contributes 16t^5/5! to tangent. The first tangent entry corresponds to degree one. Parity packets can identify familiar functions, provided the factorial normalization and sequence offset are kept. OEIS A000111, A000364, and A000182. The tangent entry has OEIS initial inde","count":437},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"ler_updown ∑_n≥0E_nfract^nn!=sec t+tan t, E_2m=s_m, E_2m+1=τ_m The coefficient table separates even and odd positions of the Euler up/down family. Even positions are positive secant numbers. Odd positions are tangent numbers. The factorial row makes clear that these are exponential-generating-function numbers, not ordinary Taylor coefficients. At d","count":911},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"There exist nonzero complex constants A,B with V(z)=A U(Bz) if and only if D=0. On either surface, put s=sqrtk/p>0. The scalar family is defined by T_x,y'(z)=1+T_x,y(xz)T_x,y(yz), T_x,y(0)=0. It gives the explicit reduction U(z)=sqrts T_q,sr(z/sqrts), V(z)=s^-1/2T_q,sr(sqrts z). Both effective scales lie in (0,1). Equality U=V is more restrictive: ","count":101},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"FrontChapterNotation and glossary Notation is defined locally when each family is introduced. The same letter can serve different roles in separate chapters. In particular, a is an arithmetic weight parameter in Chapter~[source reference], a contraction in Chapter~[source reference], and a spectral shift in Appendix~[source reference].","count":255},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f4c6861751b33d74","tex":"a^\\rho","aliases":"a^ρ rho","display":"a^\\rho","kind":"introduction-passage","excerpt":"item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ a^ρ · b^ρ=1. A positive right eigenvector gives the proportions of the two type vectors. item[Arithmetic depth A_a,β,Q.] The entire sum ∑_n≥2, (n,Q)=1μ(n)^2n^-2a(1-n^-β)^2s, with a>1/2, β>0, Q≥1 integer and the real logarithm in th","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"There exist nonzero complex constants A,B with V(z)=A U(Bz) if and only if D=0. On either surface, put s=sqrtk/p>0. The scalar family is defined by T_x,y'(z)=1+T_x,y(xz)T_x,y(yz), T_x,y(0)=0. It gives the explicit reduction U(z)=sqrts T_q,sr(z/sqrts), V(z)=s^-1/2T_q,sr(sqrts z). Both effective scales lie in (0,1). Equality U=V is more restrictive: ","count":92},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)V(rz), V'=1+U(az)V(bz) and U(0)=V(0)=0, with q,r,a,b∈(0,1). item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ","count":150},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e1d8f832607437f0","tex":"b^\\rho","aliases":"b^ρ rho","display":"b^\\rho","kind":"introduction-passage","excerpt":"item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ a^ρ · b^ρ=1. A positive right eigenvector gives the proportions of the two type vectors. item[Arithmetic depth A_a,β,Q.] The entire sum ∑_n≥2, (n,Q)=1μ(n)^2n^-2a(1-n^-β)^2s, with a>1/2, β>0, Q≥1 integer and the real logarithm in th","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Write D(s)=ζ(s)/ζ(2s). For arbitrary finite complex strengths and Re s>1, absolute convergence permits expansion of the finite prime product. The local squarefree Euler factors give","count":72},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Write f=f_3/4 for the normalized diagonal function, and f(z)=∑_n≥0d_nz^2n+1. The inherited tap continuity theorem gives","count":107},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"introduction-passage","excerpt":"Write f=f_3/4 for the normalized diagonal function, and f(z)=∑_n≥0d_nz^2n+1. The inherited tap continuity theorem gives","count":20},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"ler_updown ∑_n≥0E_nfract^nn!=sec t+tan t, E_2m=s_m, E_2m+1=τ_m The coefficient table separates even and odd positions of the Euler up/down family. Even positions are positive secant numbers. Odd positions are tangent numbers. The factorial row makes clear that these are exponential-generating-function numbers, not ordinary Taylor coefficients. At d","count":23},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6662c032276e077b","tex":"E_4","aliases":"E_4","display":"E_4","kind":"introduction-passage","excerpt":"ler_updown ∑_n≥0E_nfract^nn!=sec t+tan t, E_2m=s_m, E_2m+1=τ_m The coefficient table separates even and odd positions of the Euler up/down family. Even positions are positive secant numbers. Odd positions are tangent numbers. The factorial row makes clear that these are exponential-generating-function numbers, not ordinary Taylor coefficients. At d","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d8c2d18f405d04ca","tex":"E_5","aliases":"E_5","display":"E_5","kind":"introduction-passage","excerpt":"E_2m=s_m, E_2m+1=τ_m The coefficient table separates even and odd positions of the Euler up/down family. Even positions are positive secant numbers. Odd positions are tangent numbers. The factorial row makes clear that these are exponential-generating-function numbers, not ordinary Taylor coefficients. At degree four, E_4=5 contributes 5t^4/4! to s","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":97},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":60},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-459fc9905a7c48b5","tex":"f_{3/4}","aliases":"f_3/4","display":"f_{3/4}","kind":"introduction-passage","excerpt":"Write f=f_3/4 for the normalized diagonal function, and f(z)=∑_n≥0d_nz^2n+1. The inherited tap continuity theorem gives","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d8e21eabea93b0ff","tex":"F_\\Theta","aliases":"F_Θ Theta","display":"F_\\Theta","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2b7c2e71f6ae8204","tex":"F_\\Theta'","aliases":"F_Θ' Theta","display":"F_\\Theta'","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":84},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":24},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":211},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"If these conditions hold, put s=sqrtk/p>0, and let T_x,y denote the normalized scalar solution T'=1+T(xz)T(yz), T(0)=0. Then","count":122},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7e0ecffa7c185eb6","tex":"\\limsup_{R\\to\\infty}","aliases":"limsup_R→∞ limsup to infty","display":"\\limsup_{R\\to\\infty}","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":126},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":72},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"dinary Taylor coefficients. At degree four, E_4=5 contributes 5t^4/4! to secant. At degree five, E_5=16 contributes 16t^5/5! to tangent. The first tangent entry corresponds to degree one. Parity packets can identify familiar functions, provided the factorial normalization and sequence offset are kept. OEIS A000111, A000364, and A000182. The tangent","count":122},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e4524d978e547026","tex":"M_f","aliases":"M_f","display":"M_f","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":43},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-01dd6d4958c0f640","tex":"\\max_{|z|=R}","aliases":"max_|z|=R max","display":"\\max_{|z|=R}","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"Write f=f_3/4 for the normalized diagonal function, and f(z)=∑_n≥0d_nz^2n+1. The inherited tap continuity theorem gives","count":274},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"introduction-passage","excerpt":"item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ","count":43},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"If these conditions hold, put s=sqrtk/p>0, and let T_x,y denote the normalized scalar solution T'=1+T(xz)T(yz), T(0)=0. Then","count":225},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":273},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bb0a84db3ef36d7a","tex":"q^\\rho","aliases":"q^ρ rho","display":"q^\\rho","kind":"introduction-passage","excerpt":"item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ a^ρ · b^ρ=1. A positive right eigenvector gives the proportions of the two type vectors. item[Arithmetic depth A_a,β,Q.] The entire sum ∑_n≥2, (n,Q)=1μ(n)^2n^-2a(1-n^-β)^2s, with a>1/2, β>0, Q≥1 integer and the real logarithm in th","count":19},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ecfc6d63a3b68fb8","tex":"q_j","aliases":"q_j","display":"q_j","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":22},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":164},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":268},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-290457cba72a2ee7","tex":"R^\\rho","aliases":"R^ρ rho","display":"R^\\rho","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":21},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-afd3850f7f5551c4","tex":"r^\\rho","aliases":"r^ρ rho","display":"r^\\rho","kind":"introduction-passage","excerpt":"item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ a^ρ · b^ρ=1. A positive right eigenvector gives the proportions of the two type vectors. item[Arithmetic depth A_a,β,Q.] The entire sum ∑_n≥2, (n,Q)=1μ(n)^2n^-2a(1-n^-β)^2s, with a>1/2, β>0, Q≥1 integer and the real logarithm in th","count":15},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":26},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ff81ba5e23d549c7","tex":"\\Re","aliases":"Re","display":"\\Re","kind":"introduction-passage","excerpt":"Write D(s)=ζ(s)/ζ(2s). For arbitrary finite complex strengths and Re s>1, absolute convergence permits expansion of the finite prime product. The local squarefree Euler factors give","count":24},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Write D(s)=ζ(s)/ζ(2s). For arbitrary finite complex strengths and Re s>1, absolute convergence permits expansion of the finite prime product. The local squarefree Euler factors give","count":154},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2b5e6d175de4b2dc","tex":"s^{-1/2}","aliases":"s^-1/2","display":"s^{-1/2}","kind":"introduction-passage","excerpt":"There exist nonzero complex constants A,B with V(z)=A U(Bz) if and only if D=0. On either surface, put s=sqrtk/p>0. The scalar family is defined by T_x,y'(z)=1+T_x,y(xz)T_x,y(yz), T_x,y(0)=0. It gives the explicit reduction U(z)=sqrts T_q,sr(z/sqrts), V(z)=s^-1/2T_q,sr(sqrts z). Both effective scales lie in (0,1). Equality U=V is more restrictive: ","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-572f941fa1fdf24b","tex":"\\operatorname{spr}","aliases":"spr operatorname","display":"\\operatorname{spr}","kind":"introduction-passage","excerpt":"item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"If these conditions hold, put s=sqrtk/p>0, and let T_x,y denote the normalized scalar solution T'=1+T(xz)T(yz), T(0)=0. Then","count":83},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"introduction-passage","excerpt":"≥0E_nfract^nn!=sec t+tan t, E_2m=s_m, E_2m+1=τ_m The coefficient table separates even and odd positions of the Euler up/down family. Even positions are positive secant numbers. Odd positions are tangent numbers. The factorial row makes clear that these are exponential-generating-function numbers, not ordinary Taylor coefficients. At degree four, E_","count":130},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6da3583735a2a7a0","tex":"T'","aliases":"T'","display":"T'","kind":"introduction-passage","excerpt":"If these conditions hold, put s=sqrtk/p>0, and let T_x,y denote the normalized scalar solution T'=1+T(xz)T(yz), T(0)=0. Then","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"introduction-passage","excerpt":"≥0E_nfract^nn!=sec t+tan t, E_2m=s_m, E_2m+1=τ_m The coefficient table separates even and odd positions of the Euler up/down family. Even positions are positive secant numbers. Odd positions are tangent numbers. The factorial row makes clear that these are exponential-generating-function numbers, not ordinary Taylor coefficients. At degree four, E_","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8c6354f5f56778af","tex":"t^5","aliases":"t^5","display":"t^5","kind":"introduction-passage","excerpt":"d E_2m+1=τ_m The coefficient table separates even and odd positions of the Euler up/down family. Even positions are positive secant numbers. Odd positions are tangent numbers. The factorial row makes clear that these are exponential-generating-function numbers, not ordinary Taylor coefficients. At degree four, E_4=5 contributes 5t^4/4! to secant. A","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1297eed2140d6aa8","tex":"T_{q,sr}","aliases":"T_q,sr","display":"T_{q,sr}","kind":"introduction-passage","excerpt":"There exist nonzero complex constants A,B with V(z)=A U(Bz) if and only if D=0. On either surface, put s=sqrtk/p>0. The scalar family is defined by T_x,y'(z)=1+T_x,y(xz)T_x,y(yz), T_x,y(0)=0. It gives the explicit reduction U(z)=sqrts T_q,sr(z/sqrts), V(z)=s^-1/2T_q,sr(sqrts z). Both effective scales lie in (0,1). Equality U=V is more restrictive: ","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-11ee3c743e3ed0fe","tex":"T_{x,y}","aliases":"T_x,y","display":"T_{x,y}","kind":"introduction-passage","excerpt":"If these conditions hold, put s=sqrtk/p>0, and let T_x,y denote the normalized scalar solution T'=1+T(xz)T(yz), T(0)=0. Then","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0a5906a4b49d79f0","tex":"T_{x,y}'","aliases":"T_x,y'","display":"T_{x,y}'","kind":"introduction-passage","excerpt":"There exist nonzero complex constants A,B with V(z)=A U(Bz) if and only if D=0. On either surface, put s=sqrtk/p>0. The scalar family is defined by T_x,y'(z)=1+T_x,y(xz)T_x,y(yz), T_x,y(0)=0. It gives the explicit reduction U(z)=sqrts T_q,sr(z/sqrts), V(z)=s^-1/2T_q,sr(sqrts z). Both effective scales lie in (0,1). Equality U=V is more restrictive: ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"introduction-passage","excerpt":"item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)V(rz), V'=1+U(az)V(bz) and U(0)=V(0)=0, with q,r,a,b∈(0,1). item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ","count":62},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ec61c8382b2d3710","tex":"U'","aliases":"U'","display":"U'","kind":"introduction-passage","excerpt":"item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)V(rz), V'=1+U(az)V(bz) and U(0)=V(0)=0, with q,r,a,b∈(0,1). item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)V(rz), V'=1+U(az)V(bz) and U(0)=V(0)=0, with q,r,a,b∈(0,1). item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ","count":54},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"introduction-passage","excerpt":"item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)V(rz), V'=1+U(az)V(bz) and U(0)=V(0)=0, with q,r,a,b∈(0,1). item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"If these conditions hold, put s=sqrtk/p>0, and let T_x,y denote the normalized scalar solution T'=1+T(xz)T(yz), T(0)=0. Then","count":179},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"If these conditions hold, put s=sqrtk/p>0, and let T_x,y denote the normalized scalar solution T'=1+T(xz)T(yz), T(0)=0. Then","count":33},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":131},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-59261bec8ae52887","tex":"z^{2n+1}","aliases":"z^2n+1","display":"z^{2n+1}","kind":"introduction-passage","excerpt":"Write f=f_3/4 for the normalized diagonal function, and f(z)=∑_n≥0d_nz^2n+1. The inherited tap continuity theorem gives","count":12},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":97},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"introduction-passage","excerpt":"Write D(s)=ζ(s)/ζ(2s). For arbitrary finite complex strengths and Re s>1, absolute convergence permits expansion of the finite prime product. The local squarefree Euler factors give","count":13},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":105},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"dinary Taylor coefficients. At degree four, E_4=5 contributes 5t^4/4! to secant. At degree five, E_5=16 contributes 16t^5/5! to tangent. The first tangent entry corresponds to degree one. Parity packets can identify familiar functions, provided the factorial normalization and sequence offset are kept. OEIS A000111, A000364, and A000182. The tangent","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0a37e389a69b4972","tex":"\\tau_m","aliases":"τ_m tau","display":"\\tau_m","kind":"introduction-passage","excerpt":"dinary Taylor coefficients. At degree four, E_4=5 contributes 5t^4/4! to secant. At degree five, E_5=16 contributes 16t^5/5! to tangent. The first tangent entry corresponds to degree one. Parity packets can identify familiar functions, provided the factorial normalization and sequence offset are kept. OEIS A000111, A000364, and A000182. The tangent","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":27},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)V(rz), V'=1+U(az)V(bz) and U(0)=V(0)=0, with q,r,a,b∈(0,1). item[Perron balance.] sprP denotes the spectral radius of a matrix. The characteristic exponent satisfies sprq^ρ · r^ρ","count":49},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":157},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":31},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"introduction-passage","excerpt":"Write f=f_3/4 for the normalized diagonal function, and f(z)=∑_n≥0d_nz^2n+1. The inherited tap continuity theorem gives","count":16},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"introduction-passage","excerpt":"item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R. At positive finite order ρ, the lower and upper types used here are the liminf and limsup of log M_f(R)/R^ρ. Their equality is not assumed. item[Coupled pair (U,V).] The normalized entire solution of U'=1+U(qz)","count":44},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an active-channel maximum. item[Maximum modulus and entire order.] M_f(R)=max_|z|=R|f(z)|. For a nonpolynomial entire function, ρ(f)=limsup_R→∞fracloglog M_f(R)log R.","count":212},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"ion. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β). Then A(s)=∑_n∈ S_Qw_ne^-λ_ns, w_n=n^-2a. The positive weights are summable, and the distinct support frequencies decrease to zero. If p_* is the least prime n","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7a9cfc100f41578b","tex":"\\mathbf1_{p\\mid n}","aliases":"1_pmid n mathbf mid","display":null,"kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-13818be49e34a2f8","tex":"\\mathbf1_{q\\mid n}","aliases":"1_qmid n mathbf mid","display":null,"kind":"use","excerpt":"Fix p∈ F, with all other strengths at least -1, and write κ(n)=∏_q∈ F∖{p}(1+λ_q1_qmid n). Separate the positive measure into its p-free and p-divisible parts:","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-53d6e2369412f170","tex":"\\A","aliases":"A A_a,β,Q beta","display":"{A_{a,\\beta,Q}}","kind":"use","excerpt":"MonographPartIIArithmetic depth and signed measuresComplex parameters, positive kernels and the boundaries of positivity. Arithmetic at complex depth. pageheadingFAMILY 2Arithmetic energy at complex depth For real a>1/2, β>0, and integer Q≥1, define A_a,β,Q(s)=∑_substackn≥2 (n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logar","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"a=a', β=β', rad(Q)=rad(Q').","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5b8fd1e4ddd92711","tex":"A^{(j)}","aliases":"A^(j)","display":"A^{(j)}","kind":"use","excerpt":"claimNo polynomial-coefficient linear ODE Suppose ∑_j=0^rP_j(s)A^(j)(s)=0, where the P_j are polynomials and are not all zero. At s=-t, divide by the exponential of the first support atom. The remaining tail is exponentially smaller, so the polynomial ∑_jP_j(-t)(-λ_p_*)^j tends to zero exponentially and must vanish identically. Remove that atom and","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f9e76dc32675b161","tex":"A^{(k)}","aliases":"A^(k)","display":"A^{(k)}","kind":"use","excerpt":"The monomial moment matrices also have infinite rank: m_k=(-1)^kA^(k)(0)=∑_jw_jλ_j^k, and for every nonzero polynomial P, ∑_jw_j|P(λ_j)|^2>0, because a nonzero polynomial cannot vanish at infinitely many distinct frequencies. All finite derivative Hankel matrices are strictly positive definite.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1)=∏_n=1^a-1frac1-x_nx_a-x_n>0, r_a=frac1-x_a+11-x_a =fraca^4(2(a+1)^2-1)(a+1)^4(2a^2-1)∈(0,1), f_a,m(x)=L_a(x)(frac1-x1-x_a)^m, m=0,1,2,…. tag2","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-32527bc9fda09672","tex":"a^{2i+1}","aliases":"a^2i+1","display":"a^{2i+1}","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7b48479868582d73","tex":"a^3","aliases":"a^3","display":"a^3","kind":"use","excerpt":"∑_n>a|μ(n)^2n^-4κ_F(n)| ≤fracC_F3a^3. tag3","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e046b0ebe8054038","tex":"a^4","aliases":"a^4","display":"a^4","kind":"use","excerpt":"L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1)=∏_n=1^a-1frac1-x_nx_a-x_n>0, r_a=frac1-x_a+11-x_a =fraca^4(2(a+1)^2-1)(a+1)^4(2a^2-1)∈(0,1), f_a,m(x)=L_a(x)(frac1-x1-x_a)^m, m=0,1,2,…. tag2","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-943bc84d09f5dc3b","tex":"a^k","aliases":"a^k","display":"a^k","kind":"use","excerpt":"ageheadingORBIT SPECTRA / SHIFTSA Hurwitz-type deformation and its determinant For real a≥0, define initially Z_N(s;a)=∑_j(w_j+a)^-s, Re s>1/N. The rightward shift preserves avoidance of the negative branch cut. Normalize the product with E_N(a) and use log E_N(a+t) in the Mellin argument. For a>0 there are genuine simple poles at s=1/N-k, k≥0: Res","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8a7f3da8e8142706","tex":"a^s","aliases":"a^s","display":"a^s","kind":"use","excerpt":"P(s)=q^s · r^s a^s · b^s.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"17n+5)A_n-n^3A_n-1 The prefix annotation begins the Apéry sequence defined by A_n=∑_k=0^nbinom nk^2binomn+kk^2. The curves show its growth through index fourteen on a base-ten logarithmic scale. The lower diagram evaluates the recurrence at n=2. Using A_1=5 and A_2=73 gives 27A_3=535·73-8·5=39015. Therefore A_3=1445. The recurrence applies for n≥1,","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"lessonSEQUENCES / 13Apéry uses taps that depend on the indexS13_apery_recurrence (n+1)^3A_n+1=(2n+1)(17n^2+17n+5)A_n-n^3A_n-1 The prefix annotation begins the Apéry sequence defined by A_n=∑_k=0^nbinom nk^2binomn+kk^2. The curves show its growth through index fourteen on a base-ten logarithmic scale. The lower diagram evaluates the recurrence at n=","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d8afff31ce6342b8","tex":"A_{12}","aliases":"A_12","display":"A_{12}","kind":"use","excerpt":"tightlist item 18 catalog rows through A_12, independently compared with dense formal Picard iteration through degree 25. item Bivariate polynomials through A_8, with 540 nonzero monomials checked for integer positivity, symmetry, parity and the degree bound. item Tangent coefficients independently obtained by division of sine and cosine, plus hype","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"onSEQUENCES / 13Apéry uses taps that depend on the indexS13_apery_recurrence (n+1)^3A_n+1=(2n+1)(17n^2+17n+5)A_n-n^3A_n-1 The prefix annotation begins the Apéry sequence defined by A_n=∑_k=0^nbinom nk^2binomn+kk^2. The curves show its growth through index fourteen on a base-ten logarithmic scale. The lower diagram evaluates the recurrence at n=2. U","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-dcae1a999b1a8c81","tex":"a_{2n}","aliases":"a_2n","display":"a_{2n}","kind":"use","excerpt":"The two coefficient systems are related by A_n=(2n+1)!c_n. In particular, A_n is the derivative of order 2n+1 at zero. The coefficient c_n, the derivative A_n, and the full EGF sequence a_m with a_2n+1=A_n, a_2n=0 are different objects. None of the code silently substitutes one for another.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6ba40abfc37cdf81","tex":"a_{2n+1}","aliases":"a_2n+1","display":"a_{2n+1}","kind":"use","excerpt":"The two coefficient systems are related by A_n=(2n+1)!c_n. In particular, A_n is the derivative of order 2n+1 at zero. The coefficient c_n, the derivative A_n, and the full EGF sequence a_m with a_2n+1=A_n, a_2n=0 are different objects. None of the code silently substitutes one for another.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d325da4f9b58928d","tex":"A_3","aliases":"A_3","display":"A_3","kind":"use","excerpt":"13Apéry uses taps that depend on the indexS13_apery_recurrence (n+1)^3A_n+1=(2n+1)(17n^2+17n+5)A_n-n^3A_n-1 The prefix annotation begins the Apéry sequence defined by A_n=∑_k=0^nbinom nk^2binomn+kk^2. The curves show its growth through index fourteen on a base-ten logarithmic scale. The lower diagram evaluates the recurrence at n=2. Using A_1=5 and","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d84b5a24a5037067","tex":"A_4","aliases":"A_4","display":"A_4","kind":"use","excerpt":"reports no failed assertion in this run. Independent audit identified two API input-validation defects: a fractional residue was accepted by the section operator, and an empty phase matrix accepted base zero. Both are now rejected and have passing regression checks. These defects affected invalid inputs, not the displayed identities. A draft prose ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9a25d08fb9af50d6","tex":"A_8","aliases":"A_8","display":"A_8","kind":"use","excerpt":"tightlist item 18 catalog rows through A_12, independently compared with dense formal Picard iteration through degree 25. item Bivariate polynomials through A_8, with 540 nonzero monomials checked for integer positivity, symmetry, parity and the degree bound. item Tangent coefficients independently obtained by division of sine and cosine, plus hype","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fff6859df3c675f9","tex":"A_{a+j\\beta/2,\\beta,Q}","aliases":"A_a+jβ/2,β,Q beta beta","display":"A_{a+j\\beta/2,\\beta,Q}","kind":"use","excerpt":"pageheadingFAMILY 2 / CALCULUSClosed shifts, infinite differential complexity The Euler product for Z_Q shows that only rad(Q)=∏_pmid Qp matters. Expansion of the depth factor gives A_a,β,Q(s+1)=A_a,β,Q(s)-2A_a+β/2,β,Q(s) +A_a+β,β,Q(s). The fractional transfer and Newton formulas are A_a,β,Q(s+t)=∑_j≥0(-1)^jbinom2tjA_a+jβ/2,β,Q(s), A_a,β,Q(t)=∑_j≥0","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a102e9e494033755","tex":"A_{a+\\beta,\\beta,Q}","aliases":"A_a+β,β,Q beta beta","display":"A_{a+\\beta,\\beta,Q}","kind":"use","excerpt":"pageheadingFAMILY 2 / CALCULUSClosed shifts, infinite differential complexity The Euler product for Z_Q shows that only rad(Q)=∏_pmid Qp matters. Expansion of the depth factor gives A_a,β,Q(s+1)=A_a,β,Q(s)-2A_a+β/2,β,Q(s) +A_a+β,β,Q(s). The fractional transfer and Newton formulas are A_a,β,Q(s+t)=∑_j≥0(-1)^jbinom2tjA_a+jβ/2,β,Q(s), A_a,β,Q(t)=∑_j≥0","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f3032e48e5cd6143","tex":"A_{a+\\beta/2,\\beta,Q}","aliases":"A_a+β/2,β,Q beta beta","display":"A_{a+\\beta/2,\\beta,Q}","kind":"use","excerpt":"pageheadingFAMILY 2 / CALCULUSClosed shifts, infinite differential complexity The Euler product for Z_Q shows that only rad(Q)=∏_pmid Qp matters. Expansion of the depth factor gives A_a,β,Q(s+1)=A_a,β,Q(s)-2A_a+β/2,β,Q(s) +A_a+β,β,Q(s). The fractional transfer and Newton formulas are A_a,β,Q(s+t)=∑_j≥0(-1)^jbinom2tjA_a+jβ/2,β,Q(s), A_a,β,Q(t)=∑_j≥0","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1ab5984d0f945efa","tex":"A_{a,\\beta,Q}","aliases":"A_a,β,Q beta","display":"A_{a,\\beta,Q}","kind":"use","excerpt":"item[Arithmetic depth A_a,β,Q.] The entire sum ∑_n≥2, (n,Q)=1μ(n)^2n^-2a(1-n^-β)^2s, with a>1/2, β>0, Q≥1 integer and the real logarithm in the complex power. The atom n=1 is excluded at every depth. item[Möbius function and modulus radical.] μ(1)=1. For a product of k distinct primes, μ(n)=(-1)^k. If a prime square divides n, then μ(n)=0. The radi","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e732f662c53e6cdc","tex":"A_{a,\\beta,Q}^{(k)}","aliases":"A_a,β,Q^(k) beta","display":"A_{a,\\beta,Q}^{(k)}","kind":"use","excerpt":"displaystyle sup_|s|≤ R|A_a,β,Q^(k)(s)-A_a,β,Qp^(k)(s)| ≤ λ_p^k e^Rλ_p fracZ_Q(2a)p^2a+1.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-984c337a1923421f","tex":"A_{a,\\beta,Qp}^{(k)}","aliases":"A_a,β,Qp^(k) beta","display":"A_{a,\\beta,Qp}^{(k)}","kind":"use","excerpt":"displaystyle sup_|s|≤ R|A_a,β,Q^(k)(s)-A_a,β,Qp^(k)(s)| ≤ λ_p^k e^Rλ_p fracZ_Q(2a)p^2a+1.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e384f199c4f2f773","tex":"a_{bn+j}","aliases":"a_bn+j","display":"a_{bn+j}","kind":"use","excerpt":"The full-position section and the decimated lane are distinct arrays. A section keeps the original positions, inserting zeros outside the residue. A lane x_j[n]=a_bn+j changes the index to n.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-eb6a5eaf9502f590","tex":"a_{bn+j-t}","aliases":"a_bn+j-t","display":"a_{bn+j-t}","kind":"use","excerpt":"Proof. A term h_t a_bn+j-t is in phase l exactly when t+l≡ jpmod b. Write d=(t+l-j)/b. That term is h_t x_l[n-d], whose OGF is h_tw^dX_l(w) by zero extension. Summing over taps and phases gives the matrix. square","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1a50a2be93f9deb3","tex":"a_{bn+l}","aliases":"a_bn+l","display":"a_{bn+l}","kind":"use","excerpt":"For an OGF sequence a_m, define lanes x_l[n]=a_bn+l and X_l(w)=∑_nx_l[n]w^n. Apply finite ordinary taps and obtain y_j[n]=b_bn+j. Then the exact matrix identity is","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"A_0=1, A_n=∑_i=0^n-1binom2n2i+1 q^2i+1r^2(n-1-i)+1A_i A_n-1-i, n≥1.","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"Proof. Multiply the ordinary Taylor recurrence by (2n+1)!. The factor attached to A_iA_j, where i+j=n-1, is (2n)!/[(2i+1)!(2j+1)!]=binom2n2i+1. This also proves the recurrence directly in the EGF convention. square","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"1/N and the finite-order zero-counting theorem give absolute local uniform convergence in the stated half-plane. Since the order is below one and d_N(0)=1, the Hadamard factorization has genus zero: d_N(w)=∏_j(1-w/w_j). At positive integers, a log derivative gives -wfracd_N'(w)d_N(w)=∑_k≥1 Z_N(k)w^k. If a_n=(-1)^n/(Nn)! and b_k= Z_N(k), then b_k=-k","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"1/N and the finite-order zero-counting theorem give absolute local uniform convergence in the stated half-plane. Since the order is below one and d_N(0)=1, the Hadamard factorization has genus zero: d_N(w)=∏_j(1-w/w_j). At positive integers, a log derivative gives -wfracd_N'(w)d_N(w)=∑_k≥1 Z_N(k)w^k. If a_n=(-1)^n/(Nn)! and b_k= Z_N(k), then b_k=-k","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"pageheadingFAMILY 3 / BOUNDARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite.","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"The two coefficient systems are related by A_n=(2n+1)!c_n. In particular, A_n is the derivative of order 2n+1 at zero. The coefficient c_n, the derivative A_n, and the full EGF sequence a_m with a_2n+1=A_n, a_2n=0 are different objects. None of the code silently substitutes one for another.","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5b13723e8d4242bc","tex":"a_{m+k}","aliases":"a_m+k","display":"a_{m+k}","kind":"use","excerpt":"claimThree operations that must be kept distinct For an EGF E_a(z)=∑_ma_mz^m/m!, differentiation transports future coefficients: E_a^(k)=∑_ma_m+kz^m/m!. Causal ordinary taps instead act by repeated zero-constant integration: E_b=∑_ℓ h_ℓ I^ℓ E_a. Multiplication of two EGFs gives Hurwitz convolution, b_m=∑_k=0^mbinom mk a_kh_m-k. It is not ordinary s","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9f23b8047aebf1bc","tex":"a_{m+t}","aliases":"a_m+t","display":"a_{m+t}","kind":"use","excerpt":"Proof. The EGF of the delayed sequence is ∑_m≥ ta_m-tz^m/m!=I^tG(z). Summing taps gives the formula. Conversely, G^(t) has coefficient sequence a_m+t, so derivatives sample future coefficients and do not implement these past delays. square","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0412b39084d1f606","tex":"a_{m-t}","aliases":"a_m-t","display":"a_{m-t}","kind":"use","excerpt":"b_m=∑_t=0^L-1h_ta_m-t.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-be387eb7f8b25a2f","tex":"a_{m-\\ell}","aliases":"a_m-ℓ ell","display":"a_{m-\\ell}","kind":"use","excerpt":"For causal FIR taps h_ℓ, b_m=∑_ℓ h_ℓ a_m-ℓ with zero extension to negative indices, split A(z) and B(z) into b phases. Then B_i(t)=∑_jH_ij(t)A_j(t), where H_ij(t)=∑_substackℓ≥0","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b42537bbac5709e6","tex":"A_m^{-1/2}","aliases":"A_m^-1/2","display":"A_m^{-1/2}","kind":"use","excerpt":"re ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite. Thus τ_m(λ)=det(A_m+(1+λ)B_m) =det A_m∏_i=1^m[1+(1+λ)η_i], where η_i>0 are the eigenvalues of A_m^-1/2B_mA_m^-1/2. Every determinant root","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"lessonSEQUENCES / 13Apéry uses taps that depend on the indexS13_apery_recurrence (n+1)^3A_n+1=(2n+1)(17n^2+17n+5)A_n-n^3A_n-1 The prefix annotation begins the Apéry sequence defined by A_n=∑_k=0^nbinom nk^2binomn+kk^2. The curves show its growth through index fourteen on a base-ten logarithmic scale. The lower diagram evaluates the recurrence at n=","count":27},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"lessonSEQUENCES / 02Longer recurrence memoryS02_tribonacci_tetranacci T_n=T_n-1+T_n-2+T_n-3, Q_n=∑_j=1^4Q_n-j The seed table fixes the OEIS convention. Tribonacci starts 0,0,1; tetranacci starts 0,0,0,1. The curves use log_10(1+a_n) so that zeros and rapidly growing terms fit together. The transform changes the plotted scale, not the integer sequen","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2f14a96ffdfe764f","tex":"a_{N,k}","aliases":"a_N,k","display":"a_{N,k}","kind":"use","excerpt":"G_N(t)=∑_k≥0a_N,kt^k, g_N(x)=∑_m≥1N a_N,m-1x^Nm-1. tagS12","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5057f1828cc647ee","tex":"a_{N,m-1}","aliases":"a_N,m-1","display":"a_{N,m-1}","kind":"use","excerpt":"G_N(t)=∑_k≥0a_N,kt^k, g_N(x)=∑_m≥1N a_N,m-1x^Nm-1. tagS12","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7f25b4aef7b61428","tex":"A_{n-1-i}","aliases":"A_n-1-i","display":"A_{n-1-i}","kind":"use","excerpt":"A_0=1, A_n=∑_i=0^n-1binom2n2i+1 q^2i+1r^2(n-1-i)+1A_i A_n-1-i, n≥1.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-db660538430f2d4b","tex":"A_p","aliases":"A_p","display":"A_p","kind":"use","excerpt":"For a prime p, let D_p=A_1-A_p. This positive difference consists exactly of the squarefree atoms divisible by p. Its largest frequency is","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-77959e9155c433df","tex":"A_Q","aliases":"A_Q","display":"A_Q","kind":"use","excerpt":"und for every derivative. fig11_prime_instabilityThe proved compact-window envelope shrinks as the excluded prime increases. At a=β=2, Q=1, R=5, excluding p=10007 changes the entire window by at most approximately 1.0749×10^-16. The depth-zero difference is known exactly. The radical differs for every such prime, while all bounded-window derivative","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-774a6103200db324","tex":"A_{\\rm tail}","aliases":"A_rm tail rm","display":"A_{\\rm tail}","kind":"use","excerpt":"A_rm tail=D(4)fracp^4p^4+1 ∏_q≠ p(1+fracλ_qq^4+1) -∑_1≤ n<pμ(n)^2n^-4κ(n)>0,","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d8536cb8e40ffd27","tex":"A_{\\chi,\\psi}","aliases":"A_χ,ψ chi psi","display":"A_{\\chi,\\psi}","kind":"use","excerpt":"A_χ,ψ(s)=∑_n≥2μ(n)^2 η(n)n^-2a(1-n^-β)^2s.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4e2b11a7001e1904","tex":"\\arg","aliases":"arg","display":"\\arg","kind":"use","excerpt":"Let α=frac2a-1β, δ_Q=frac1ζ(2)∏_pmid Qfracpp+1. As |s|→∞ in any closed sector |arg s|≤π/2-ϵ with 0<ϵ<π/2, A_a,β,Q(s)=fracδ_QβB(α,2s+1) +O_a,β,Q,ϵ(|s|^-(2a-1/2)/β). In particular, A_a,β,Q(s)∼fracδ_QΓ(α)β(2s)^-α. The principal power is used in the right half-plane. The beta expression is an asymptotic continuous-density model, not an identity with th","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-da3a0b2bf54ebf15","tex":"\\operatorname{Arg}","aliases":"Arg operatorname","display":"\\operatorname{Arg}","kind":"use","excerpt":"bits pageheadingFAMILY 4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(z)=d_N(z^N). Let w_j be the nonzero zeros of d_N, counted with multiplicity. Each corresponds to one rotation orbit of N zeros of D_N. Using Argw∈(-π,π], def","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-17a6e987887cd940","tex":"\\asymp","aliases":"asymp","display":null,"kind":"use","excerpt":"For completeness, the first estimate follows initially from j=δ_Qn_j+O_Q(sqrtn_j), which gives n_jasymp_Qj; substituting this back improves the error to O_Q(sqrt j). The second uses -log(1-u)=u(1+O(u)) as udownarrow0.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-55d39ed8b98dac13","tex":"\\asymp_Q","aliases":"asymp_Q asymp","display":null,"kind":"use","excerpt":"For completeness, the first estimate follows initially from j=δ_Qn_j+O_Q(sqrtn_j), which gives n_jasymp_Qj; substituting this back improves the error to O_Q(sqrt j). The second uses -log(1-u)=u(1+O(u)) as udownarrow0.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8bed5507d096707b","tex":"B^{\\mathsf T}","aliases":"B^ T mathsf","display":"B^{\\mathsf T}","kind":"use","excerpt":"It has rank one, but is not symmetric: B-B^ T=ℓ_1wedgeℓ_2≠0. At four distinct nonzero nodes, polynomial interpolation lets their vector values vary independently, so terms supported at (az,bz) cannot cancel this coefficient. A scalar autonomous channel at the same node pair would require B=κℓ⊗ℓ, a symmetric rank-one tensor, after projecting the out","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-16636769fc160a5a","tex":"b^{-1}","aliases":"b^-1","display":"b^{-1}","kind":"use","excerpt":"This holds as a formal series and wherever the original series converges. Indeed the coefficient multiplier is the geometric sum b^-1∑_k=0^b-1ω^k(m-j), equal to one or zero according to the residue. It proves all the projector identities","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f76a26dbc0d9fe26","tex":"B^2","aliases":"B^2","display":"B^2","kind":"use","excerpt":"Suppose such an equality held. Normalization gives AB=1. Write p=qr, u=q^2+r^2, and let ρ be the common order. The first two nontrivial coefficients and the growth law force B^2s^2=p, s^2=u/2, 2s^ρ=q^ρ+r^ρ=1. Consequently 2(fracq^2+r^22)^ρ/2=q^ρ+r^ρ. For ρ≠2, strict convexity or concavity of x^ρ/2 forces q=r. For the exceptional case ρ=2, we have u","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"D=k^2u_2-p^2v_2 =fracpkar15(q^2-b^2)(qb-ar).","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-eedd4e4f2fc397c1","tex":"b^{2(n-1-i)+1}","aliases":"b^2(n-1-i)+1","display":"b^{2(n-1-i)+1}","kind":"use","excerpt":"u_n · =frac12n+1∑_i=0^n-1 q^2i+1r^2(n-1-i)+1u_i v_n-1-i, v_n · =frac12n+1∑_i=0^n-1 a^2i+1b^2(n-1-i)+1u_i v_n-1-i.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-715c1527127adf76","tex":"b^{2(n-i)+1}","aliases":"b^2(n-i)+1","display":"b^{2(n-i)+1}","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1cb61b1dc861cd7b","tex":"b^{2j+1}","aliases":"b^2j+1","display":"b^{2j+1}","kind":"use","excerpt":"A_n=∑_i+j=n-1binom2n2i+1q^2i+1r^2j+1A_iB_j, B_n=∑_i+j=n-1binom2n2i+1a^2i+1b^2j+1A_iB_j,","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ad14d2d68ab1677f","tex":"b^4","aliases":"b^4","display":"b^4","kind":"use","excerpt":"u_1=frac p3, v_1=frac k3, u_2=frac p15(p q^2+k r^2), v_2=frac k15(p a^2+k b^2), u_3=frac p7(q^4u_2+p^2u_1v_1+r^4v_2), v_3=frac k7(a^4u_2+k^2u_1v_1+b^4v_2).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ec883bd3313ff73e","tex":"b^s","aliases":"b^s","display":"b^s","kind":"use","excerpt":"P(s)=q^s · r^s a^s · b^s.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-60283dddc0176654","tex":"B_+","aliases":"B_+","display":"B_+","kind":"use","excerpt":"Q=max_jmax(q_j,r_j), m=min_jmin(q_j,r_j), Δ=log(1/m), B_+=max_jlogfrac1q_jr_j+log2.","count":11},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"with A_0=B_0=1. Coefficients with even derivative index vanish. Swapping the components sends the parameter tuple (q,r,a,b) to (b,a,r,q). Identical rows (a,b)=(q,r), or exchanged rows (a,b)=(r,q), yield the scalar reduction U=V=T_q,r.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"lessonSEQUENCES / 11Bernoulli data need both coordinatesS11_bernoulli_pair fracte^t-1=∑_n≥0B_nfract^nn!, B_n=fracp_nq_n, q_n>0 The top panel shows signed numerators on a symmetric logarithmic scale. The lower panel shows positive denominators on a logarithmic scale. Fractions are reduced. A zero Bernoulli number is stored as 0/1, and this guide use","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-327140a84c768be2","tex":"B_{10}","aliases":"B_10","display":"B_{10}","kind":"use","excerpt":"inatesS11_bernoulli_pair fracte^t-1=∑_n≥0B_nfract^nn!, B_n=fracp_nq_n, q_n>0 The top panel shows signed numerators on a symmetric logarithmic scale. The lower panel shows positive denominators on a logarithmic scale. Fractions are reduced. A zero Bernoulli number is stored as 0/1, and this guide uses B_1=-1/2. At n=10, p_10=5 and q_10=66, so B_10=5","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b5a44ae5838838af","tex":"B_{12}","aliases":"B_12","display":"B_{12}","kind":"use","excerpt":"t^nn!, B_n=fracp_nq_n, q_n>0 The top panel shows signed numerators on a symmetric logarithmic scale. The lower panel shows positive denominators on a logarithmic scale. Fractions are reduced. A zero Bernoulli number is stored as 0/1, and this guide uses B_1=-1/2. At n=10, p_10=5 and q_10=66, so B_10=5/66. Its Taylor coefficient is (5/66)/10!. At n=","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"This is a binomial convolution. For example, with a_m=1 and taps h=(1,1), ordinary taps give b_2=2, while the Hurwitz product gives b_2=3. This explicit negative control catches a common normalization error.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4b94fddf75c4f060","tex":"B_4","aliases":"B_4","display":"B_4","kind":"use","excerpt":"lessonSEQUENCES / 05Normalize central binomial coefficientsS05_binomial_catalan B_n=binom2nn, C_n=fracB_nn+1 Each table column shows one central binomial coefficient and its Catalan normalization. The curves extend the comparison through index fourteen on a base-ten logarithmic axis. Division by n+1 is exact here and gives an integer at every index","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7d1b17c2a7b4bc06","tex":"B_5","aliases":"B_5","display":"B_5","kind":"use","excerpt":"row sumsS07_bell_stirling B_n=∑_k=0^nS(n,k), ∑_n≥0B_nfract^nn!=exp(e^t-1) Each matrix row contains Stirling numbers of the second kind, counting partitions into k nonempty blocks. Summing across the row gives the Bell number at its right. The generating function is exponential, so its coefficient is B_n/n!. The fifth row contains 0,1,15,25,10,1. It","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d44c7e972e364559","tex":"b_{bn+j}","aliases":"b_bn+j","display":"b_{bn+j}","kind":"use","excerpt":"For an OGF sequence a_m, define lanes x_l[n]=a_bn+l and X_l(w)=∑_nx_l[n]w^n. Apply finite ordinary taps and obtain y_j[n]=b_bn+j. Then the exact matrix identity is","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"use","excerpt":"For causal FIR taps h_ℓ, b_m=∑_ℓ h_ℓ a_m-ℓ with zero extension to negative indices, split A(z) and B(z) into b phases. Then B_i(t)=∑_jH_ij(t)A_j(t), where H_ij(t)=∑_substackℓ≥0","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"ℓ_i(x)=∏_j≠ ifracx-y_jy_i-y_j, b_i=∏_j≠ ifracmax(y_j,1-y_j)|y_i-y_j|, K_d=∑_i=0^dn_i^4b_i^2.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4e71d786fcee00f3","tex":"b_i^2","aliases":"b_i^2","display":"b_i^2","kind":"use","excerpt":"ℓ_i(x)=∏_j≠ ifracx-y_jy_i-y_j, b_i=∏_j≠ ifracmax(y_j,1-y_j)|y_i-y_j|, K_d=∑_i=0^dn_i^4b_i^2.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","excerpt":"A_n=∑_i+j=n-1binom2n2i+1q^2i+1r^2j+1A_iB_j, B_n=∑_i+j=n-1binom2n2i+1a^2i+1b^2j+1A_iB_j,","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"Let a_j=log(1/q_j), b_j=log(1/r_j), and G(t)=log M(e^t)+t+B_+. For t≥1+Δ, all child moduli are at least one, so the positive integral and weight normalization give","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","excerpt":"he bound |d_N(w)|≤ e^|w|^1/N and the finite-order zero-counting theorem give absolute local uniform convergence in the stated half-plane. Since the order is below one and d_N(0)=1, the Hadamard factorization has genus zero: d_N(w)=∏_j(1-w/w_j). At positive integers, a log derivative gives -wfracd_N'(w)d_N(w)=∑_k≥1 Z_N(k)w^k. If a_n=(-1)^n/(Nn)! and","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bb5f38329dbc4661","tex":"b_{k-j}","aliases":"b_k-j","display":"b_{k-j}","kind":"use","excerpt":"1/N and the finite-order zero-counting theorem give absolute local uniform convergence in the stated half-plane. Since the order is below one and d_N(0)=1, the Hadamard factorization has genus zero: d_N(w)=∏_j(1-w/w_j). At positive integers, a log derivative gives -wfracd_N'(w)d_N(w)=∑_k≥1 Z_N(k)w^k. If a_n=(-1)^n/(Nn)! and b_k= Z_N(k), then b_k=-k","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"use","excerpt":"pageheadingFAMILY 3 / BOUNDARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite.","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fc9ad75de14545d5","tex":"b_m","aliases":"b_m","display":"b_m","kind":"use","excerpt":"b_m=∑_t=0^L-1h_ta_m-t.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bc9fda47adba9273","tex":"b_{m-j}","aliases":"b_m-j","display":"b_{m-j}","kind":"use","excerpt":"b_m:= Z_N(m) =-m c_m-∑_j=1^m-1c_jb_m-j. tagS17","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"use","excerpt":"lessonSEQUENCES / 07Bell numbers are Stirling row sumsS07_bell_stirling B_n=∑_k=0^nS(n,k), ∑_n≥0B_nfract^nn!=exp(e^t-1) Each matrix row contains Stirling numbers of the second kind, counting partitions into k nonempty blocks. Summing across the row gives the Bell number at its right. The generating function is exponential, so its coefficient is B_n","count":11},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b44c8beecb21ab1e","tex":"B_{N,0}","aliases":"B_N,0","display":"B_{N,0}","kind":"use","excerpt":"B_N,0(s)=∫_0^1x^-Nsg_N(x) dx +∫_1^∞ x^-Ns(g_N(x)-1) dx+frac1Ns-1. tagS15","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-be27bc9a55bf645d","tex":"B_{N,a}","aliases":"B_N,a","display":"B_{N,a}","kind":"use","excerpt":"B_N,a(0)=∫_0^1G_N,a(t)dt +∫_1^∞[G_N,a(t)-α t^α-1]dt-1.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0de76cd8a0c6ad01","tex":"B_{N,K}","aliases":"B_N,K","display":"B_{N,K}","kind":"use","excerpt":"B_N,K(s)= · ∫_0^1x^-Ns[g_N(x)-∑_m=1^KN a_N,m-1x^Nm-1]dx · +∑_m=1^Kfraca_N,m-1m-s +∫_1^∞ x^-Ns[g_N(x)-1]dx +frac1Ns-1. tagS13","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-77d58d9e0128d7c9","tex":"B_{n-1-i}","aliases":"B_n-1-i","display":"B_{n-1-i}","kind":"use","excerpt":"A_n · =∑_i=0^n-1binom2n2i+1 q^2i+1r^2(n-1-i)+1A_iB_n-1-i, B_n · =∑_i=0^n-1binom2n2i+1 a^2i+1b^2(n-1-i)+1A_iB_n-1-i.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0a9fe8a3b1c4c673","tex":"B_{\\rm EGF}","aliases":"B_rm EGF rm","display":"B_{\\rm EGF}","kind":"use","excerpt":"B_rm EGF(z)=∑_t=0^L-1h_t I^tG(z), IG(z)=∫_0^zG(w) dw.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"S / 05Normalize central binomial coefficientsS05_binomial_catalan B_n=binom2nn, C_n=fracB_nn+1 Each table column shows one central binomial coefficient and its Catalan normalization. The curves extend the comparison through index fourteen on a base-ten logarithmic axis. Division by n+1 is exact here and gives an integer at every index. For n=4, B_4","count":65},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"MonographPartIIArithmetic depth and signed measuresComplex parameters, positive kernels and the boundaries of positivity. Arithmetic at complex depth. pageheadingFAMILY 2Arithmetic energy at complex depth For real a>1/2, β>0, and integer Q≥1, define A_a,β,Q(s)=∑_substackn≥2 (n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logar","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"|∑_n≥ Nc_nz^2n+1| ≤ frac|z|ρ^N1-ρ, |z|^2<2.","count":99},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","excerpt":"claimThe certificate uses rational arithmetic throughout Take S={1,3,5,7} and define P(x)=∏_n∈ Sfracx-x_nx_2-x_n (frac1-x1-x_2)^3. Its degree is seven. All remaining positive nodes begin at n=11. With C=756482048/20777538825 and r=3856/102487, a complete tail bound gives Q_10^-12(P)≤-frac10^-1216 +fracC^2r^63·7^3 <-5.8845708899×10^-14. The inequali","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e2677238ec3f2ee1","tex":"\\mathbb C^2","aliases":"C^2 mathbb","display":"\\mathbb C^2","kind":"use","excerpt":"A nonzero constant row c=(c_1,c_2)∈ C^2 is an autonomous scalar projection if some scalar functional H satisfies","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-78a649bb99dbf34d","tex":"c_+","aliases":"c_+","display":"c_+","kind":"use","excerpt":"Theorem A.1. There are explicit constants 0<c_-≤ c_+<∞ such that","count":12},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-83dae4cbc4840c66","tex":"c_-","aliases":"c_-","display":"c_-","kind":"use","excerpt":"Theorem A.1. There are explicit constants 0<c_-≤ c_+<∞ such that","count":16},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"log F_0(R)≤ C_0sqrt R (R≥ e), C_0=e^-1/2(1+22log2)≈9.85566103.","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"of the defining feedback. The function is symmetric in q,r, but unequal scales retain left/right address information. fig01_asymmetric_functionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"The parameters can be recovered, up to swapping, from c_1,c_2. Put p=3c_1 and u=15c_2/p^2. Then q,r are the roots of x^2-sqrtu+2p x+p=0.","count":14},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"The parameters can be recovered, up to swapping, from c_1,c_2. Put p=3c_1 and u=15c_2/p^2. Then q,r are the roots of x^2-sqrtu+2p x+p=0.","count":11},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d4e58ada952ec6ab","tex":"C_4","aliases":"C_4","display":"C_4","kind":"use","excerpt":"S / 05Normalize central binomial coefficientsS05_binomial_catalan B_n=binom2nn, C_n=fracB_nn+1 Each table column shows one central binomial coefficient and its Catalan normalization. The curves extend the comparison through index fourteen on a base-ten logarithmic axis. Division by n+1 is exact here and gives an integer at every index. For n=4, B_4","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0c950c408567ca3e","tex":"C_5","aliases":"C_5","display":"C_5","kind":"use","excerpt":"lessonSEQUENCES / 06Catalan convolution rebuilds the next termS06_catalan_convolution C_n+1=∑_k=0^nC_kC_n-k, C(z)=1+zC(z)^2 The five bars are the five products contributing to C_5. The input indices add to four, and their sum supplies the next coefficient. Here C(z)=∑_n≥0C_nz^n is an ordinary generating function. The contributions are 1·14, 1·5, 2·","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"use","excerpt":"pageheadingPRIME SIEVES / CONSTRUCTIVE WITNESSEvery negative prime atom has a finite detector For an integer a≥2, let L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1), r_a=frac1-x_a+11-x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_si","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0fa0207652600c09","tex":"C_F","aliases":"C_F","display":"C_F","kind":"use","excerpt":"x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_signed_detectorA rational detector for p=3, λ_3=-1.1, λ_2=1. The first exact bound proving negativity is m=5, so a Hankel matrix of order eight detects the signed defect. Set C_F=∏_q∈","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"use","excerpt":"on is symmetric in q,r, but unequal scales retain left/right address information. fig01_asymmetric_functionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T_q,r(z)=∑_n≥0c_nz^2n+1, c_0=1, coeff","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"For c=(c_j)∈ℓ^2, define","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-254051ed849e0a84","tex":"c_k","aliases":"c_k","display":"c_k","kind":"use","excerpt":"M<∞, so the same spectral peeling argument, now for complex coefficients, forces the first coefficient and then every coefficient to vanish whenever Uc=0. Positivity was needed to recover unknown frequencies in AI-1; injectivity here only needs absolute summability and the known ordered support. Specifically, after dividing by the largest remaining","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-646fe15eb212b5b3","tex":"c_m","aliases":"c_m","display":"c_m","kind":"use","excerpt":"b_m:= Z_N(m) =-m c_m-∑_j=1^m-1c_jb_m-j. tagS17","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-21721a3ebfbfe771","tex":"c_{mj}","aliases":"c_mj","display":"c_{mj}","kind":"use","excerpt":"If at least two strengths in a positive multiplicative family are nonzero, its entire depth-zero monic polynomial sequence cannot equal any one-prime sequence on the same variable and node system. A direct proof avoids importing a determinacy theorem. Write P_m=x^m+∑_j<mc_mjx^j. The equations ∫ P_m dν=0 recursively determine every normalized moment","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cfbf41dac773cbfc","tex":"C_n","aliases":"C_n","display":"C_n","kind":"use","excerpt":"lessonSEQUENCES / 06Catalan convolution rebuilds the next termS06_catalan_convolution C_n+1=∑_k=0^nC_kC_n-k, C(z)=1+zC(z)^2 The five bars are the five products contributing to C_5. The input indices add to four, and their sum supplies the next coefficient. Here C(z)=∑_n≥0C_nz^n is an ordinary generating function. The contributions are 1·14, 1·5, 2·","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c1ecf2aa8f993509","tex":"c_N","aliases":"c_N","display":"c_N","kind":"use","excerpt":"sector around negative w, choose the root x=(-w)^1/N near the positive real ray. The first exponential dominates the other N-1 terms by a uniform real-part gap at large radius, so this sector is eventually zero free. The finitely many remaining zeros avoid the negative real ray because d_N(-t)>0 for t≥0. Thus all zero arguments have a uniform gap f","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"use","excerpt":"|∑_n≥ Nc_nz^2n+1| ≤ frac|z|ρ^N1-ρ, |z|^2<2.","count":21},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b7e8c8a0a1fa875e","tex":"c_{n-1}","aliases":"c_n-1","display":"c_{n-1}","kind":"use","excerpt":"c_n≥fracq r^2n-12n+1c_n-1 Longrightarrow c_n≥fracq^n r^n^23·5⋯(2n+1).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d43c9291155e1eef","tex":"c_{n-1-i}","aliases":"c_n-1-i","display":"c_{n-1-i}","kind":"use","excerpt":"on is symmetric in q,r, but unequal scales retain left/right address information. fig01_asymmetric_functionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T_q,r(z)=∑_n≥0c_nz^2n+1, c_0=1, coeff","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-42e8b9b2d4e2611c","tex":"c_n^{1/(2n+1)}","aliases":"c_n^1/(2n+1)","display":"c_n^{1/(2n+1)}","kind":"use","excerpt":"c_n^1/(2n+1)≥ q^n/(2n+1)r^n^2/(2n+1)(2n+1)^-n/(2n+1)⟶∞.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5eb6dddee217c229","tex":"C_p^2","aliases":"C_p^2","display":"C_p^2","kind":"use","excerpt":"This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_signed_detectorA rational detector for p=3, λ_3=-1.1, λ_2=1. The first exact bound proving negativity is m=5, so a Hankel matrix of order eight detects the signed defect. Set C_F=∏_q∈ Fmax(1,|1+λ_q|). The summable tail give","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bb134dc373ba038c","tex":"C_Q","aliases":"C_Q","display":"C_Q","kind":"use","excerpt":"(n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2a-1/2)/β; |s|/Re s stays bounded in the stated sector. This gives the erro","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6d7c283002191930","tex":"C_Q^2","aliases":"C_Q^2","display":"C_Q^2","kind":"use","excerpt":"(fracsqrtC_Q^2+4δ_Q(j-1)-C_Q2δ_Q)^2 ≤ n_j≤ (fracsqrtC_Q^2+4δ_Q(j+1)+C_Q2δ_Q)^2.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a9e579ea15fac4ad","tex":"C_S","aliases":"C_S","display":"C_S","kind":"use","excerpt":"P_S,m(x)=∏_n∈ Sfracx-x_nx_2-x_n (frac1-x1-x_2)^m, C_S=∏_n∈ Sfrac1-x_n|x_2-x_n|, r_S=frac1-x_n_+1-x_2<1.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ab0cd19af3333999","tex":"C_S^2","aliases":"C_S^2","display":"C_S^2","kind":"use","excerpt":"Q_δ(P_S,m)≤-fracδ16 +fracC_S^23K^3r_S^2m. tagRA-10","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5ad87e870eb9b4ce","tex":"c_\\varepsilon","aliases":"c_ε varepsilon","display":"c_\\varepsilon","kind":"use","excerpt":"log F_ε(R)≥ c_ε(R+3/ε)^ρ ≥ c_ε R^ρ (R≥2), c_ε=fraclog2(4/ε+8/3)^ρ.","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-942ed63c32d9134b","tex":"C_\\kappa","aliases":"C_κ kappa","display":"C_\\kappa","kind":"use","excerpt":"deg f<m frac∫ f^2ddν_p-free ∫ f^2ddν_p-divisible, δ_mdownarrow0. The detector yields, for C_κ=∏_q≠ pmax(1,1+λ_q), δ_p+m≤fracpC_κ C_p^23r_p^2m. The denominator retains the atom p; all lower nodes in the numerator are killed. This extends the prime-two bound to every prime and to simultaneous positive or deletion strengths. It does not assert simple ","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"claimTwo exact Hurwitz-zeta reductions For N=2, d_2(w)=cossqrt w, whose zeros are w_j=π^2(j+1/2)^2, j≥0. Therefore Z_2(s)=π^-2sζ(2s,1/2). For N=4, D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), and the rotation quotient has one zero w_j=(sqrt2π)^4(j+1/2)^4 per orbit. Hence Z_4(s)=(sqrt2π)^-4sζ(4s,1/2). These anchors are classical reductions. The general spectra","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8db6134243ea03f4","tex":"\\cos^2","aliases":"cos^2 cos","display":"\\cos^2","kind":"use","excerpt":"To check the elementary estimate, write w=re^iθ: |t+w|^2=(t+r)^2-4trsin^2(θ/2)≥(t+r)^2cos^2(θ/2).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"use","excerpt":"claimTwo exact Hurwitz-zeta reductions For N=2, d_2(w)=cossqrt w, whose zeros are w_j=π^2(j+1/2)^2, j≥0. Therefore Z_2(s)=π^-2sζ(2s,1/2). For N=4, D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), and the rotation quotient has one zero w_j=(sqrt2π)^4(j+1/2)^4 per orbit. Hence Z_4(s)=(sqrt2π)^-4sζ(4s,1/2). These anchors are classical reductions. The general spectra","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-45759d4012da1a5d","tex":"d'","aliases":"d'","display":"d'","kind":"use","excerpt":"w'=2+tfrac12(w(sz)^2-d(sz)^2), d'=0.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-034f636739b19fbf","tex":"d^{-2}","aliases":"d^-2","display":"d^{-2}","kind":"use","excerpt":"2=∑_d^2mid nμ(d). For (d,Q)=1, the number of m≤ x/d^2 coprime to Q differs from (φ(Q)/Q)x/d^2 by at most 2^ω(Q). Summing over at most sqrt x values of d gives the first part of C_Q. The omitted main-term tail is at most (φ(Q)/Q)x/lfloorsqrt xrfloor≤2(φ(Q)/Q)sqrt x. Removing n=1 adds one. The main-term density is (φ(Q)/Q)∑_(d,Q)=1μ(d)d^-2=δ_Q by its","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"To verify the constants, expand μ(n)^2=∑_d^2mid nμ(d). For (d,Q)=1, the number of m≤ x/d^2 coprime to Q differs from (φ(Q)/Q)x/d^2 by at most 2^ω(Q). Summing over at most sqrt x values of d gives the first part of C_Q. The omitted main-term tail is at most (φ(Q)/Q)x/lfloorsqrt xrfloor≤2(φ(Q)/Q)sqrt x. Removing n=1 adds one. The main-term density is","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d59fe309c2f353c3","tex":"D^{n(n+1)}","aliases":"D^n(n+1)","display":"D^{n(n+1)}","kind":"use","excerpt":"B_n=H^nD^n(n+1)A_n∈ Z hboxfor all n.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"lessonSEQUENCES / 09Derangements show signed cancellationS09_derangements D_n=n!∑_k=0^nfrac(-1)^kk!, D_n=(n-1)(D_n-1+D_n-2) (n≥2) Positive and negative bars are the inclusion-exclusion contributions at n=6. Their large magnitudes cancel to a smaller integer. The dashed line records the final sum rather than an individual contribution. Here D_0=1 an","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a978ab922b752243","tex":"D_0^{n(n+1)}","aliases":"D_0^n(n+1)","display":"D_0^{n(n+1)}","kind":"use","excerpt":"These derivative arrays are polynomials with nonnegative integer coefficients in the four scale variables. Their total degree is at most n(n+1). If every parameter denominator divides D_0, then D_0^n(n+1)A_n, D_0^n(n+1)B_n hboxare integers. This uniform clearing factor makes exact rational families available to integer sequence experiments. It need","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"lessonSEQUENCES / 09Derangements show signed cancellationS09_derangements D_n=n!∑_k=0^nfrac(-1)^kk!, D_n=(n-1)(D_n-1+D_n-2) (n≥2) Positive and negative bars are the inclusion-exclusion contributions at n=6. Their large magnitudes cancel to a smaller integer. The dashed line records the final sum rather than an individual contribution. Here D_0=1 an","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"use","excerpt":"claimTwo exact Hurwitz-zeta reductions For N=2, d_2(w)=cossqrt w, whose zeros are w_j=π^2(j+1/2)^2, j≥0. Therefore Z_2(s)=π^-2sζ(2s,1/2). For N=4, D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), and the rotation quotient has one zero w_j=(sqrt2π)^4(j+1/2)^4 per orbit. Hence Z_4(s)=(sqrt2π)^-4sζ(4s,1/2). These anchors are classical reductions. The general spectra","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ca8301964f9bd376","tex":"D_4","aliases":"D_4","display":"D_4","kind":"use","excerpt":"claimTwo exact Hurwitz-zeta reductions For N=2, d_2(w)=cossqrt w, whose zeros are w_j=π^2(j+1/2)^2, j≥0. Therefore Z_2(s)=π^-2sζ(2s,1/2). For N=4, D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), and the rotation quotient has one zero w_j=(sqrt2π)^4(j+1/2)^4 per orbit. Hence Z_4(s)=(sqrt2π)^-4sζ(4s,1/2). These anchors are classical reductions. The general spectra","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-980a0ee7024984d0","tex":"D_6","aliases":"D_6","display":"D_6","kind":"use","excerpt":"D_n=n!∑_k=0^nfrac(-1)^kk!, D_n=(n-1)(D_n-1+D_n-2) (n≥2) Positive and negative bars are the inclusion-exclusion contributions at n=6. Their large magnitudes cancel to a smaller integer. The dashed line records the final sum rather than an individual contribution. Here D_0=1 and D_1=0. The sum is 720-720+360-120+30-6+1=265. The recurrence independent","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3fb8837a44922746","tex":"D_N","aliases":"D_N","display":"D_N","kind":"use","excerpt":"MonographPartIIIOrbit spectra and exact coordinatesSpectral continuation and the coefficient language of polyphase transforms. Spectral functions of zero orbits. pageheadingFAMILY 4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-382f08d5d528afb0","tex":"d_N","aliases":"d_N","display":"d_N","kind":"use","excerpt":"MonographPartIIIOrbit spectra and exact coordinatesSpectral continuation and the coefficient language of polyphase transforms. Spectral functions of zero orbits. pageheadingFAMILY 4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(","count":21},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-773cb003024cbb07","tex":"d_N'","aliases":"d_N'","display":"d_N'","kind":"use","excerpt":"ts. Multiplicity is retained by the definition; simplicity is not assumed in general. The bound |d_N(w)|≤ e^|w|^1/N and the finite-order zero-counting theorem give absolute local uniform convergence in the stated half-plane. Since the order is below one and d_N(0)=1, the Hadamard factorization has genus zero: d_N(w)=∏_j(1-w/w_j). At positive intege","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ae94081e451f9069","tex":"D_p","aliases":"D_p","display":"D_p","kind":"use","excerpt":"For a prime p, let D_p=A_1-A_p. This positive difference consists exactly of the squarefree atoms divisible by p. Its largest frequency is","count":11},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b4dd23f65072e2c8","tex":"D_p^{(k)}","aliases":"D_p^(k)","display":"D_p^{(k)}","kind":"use","excerpt":"sup_|s|≤ R|D_p^(k)(s)| ≤ λ_p^k e^Rλ_pfracD(4)p^4+1. tagRA-4","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd dd d","display":null,"kind":"use","excerpt":"(n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2a-1/2)/β; |s|/Re s stays bounded in the stated sector. This gives the erro","count":13},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"B_mA_m^-1/2. Every determinant root is real and below -1; possible coefficient poles occur there, although numerator cancellation is possible. fig05_sieve_boundaryThe line is a proved bound for p=2; dots are ordinary high-precision pencil roots. A defect λ=-1.001 eventually violates positivity even though it can pass low-order checks. The nearest r","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"ARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite. Thus τ_m(λ)=det(A_m+(1+λ)B","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"ontinuously on every fixed compact set, while its growth order jumps when the second channel becomes active. For 0<ε≤1 the order is log2/log(4/3); at ε=0 it is 1/2. Coefficients depend continuously on ε and are dominated by those of the entire f_3/4. Dominated convergence of the Taylor series therefore proves uniform convergence on compact sets as ","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"stored as 0/1, and this guide uses B_1=-1/2. At n=10, p_10=5 and q_10=66, so B_10=5/66. Its Taylor coefficient is (5/66)/10!. At n=12, B_12=-691/2730. An OEIS integer numerator sequence does not by itself specify the rational family or its generating-function coefficient. OEIS A027641 and A027642, indexed from zero. Exact rational recurrence and mu","count":76},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ab649d63695627c9","tex":"e^{(\\lambda_1-\\lambda_j)t}","aliases":"e^(λ_1-λ_j)t lambda lambda","display":"e^{(\\lambda_1-\\lambda_j)t}","kind":"use","excerpt":"η κ_j(t), κ_j(t)=fracA(-t)R_j(-t) ∼fracw_1w_je^(λ_1-λ_j)t.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-161d7ba7fe74873d","tex":"e^{(\\lambda_2-\\lambda_p)T}","aliases":"e^(λ_2-λ_p)T lambda lambda","display":"e^{(\\lambda_2-\\lambda_p)T}","kind":"use","excerpt":"fracA_1(-T)D_p(-T) ≥fracp^4+116D(4)e^(λ_2-λ_p)T. tagRA-8","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-71432ecff293380c","tex":"e^{(\\omega^j-1)x}","aliases":"e^(ω^j-1)x omega","display":"e^{(\\omega^j-1)x}","kind":"use","excerpt":"h_N(x)=∑_j=1^N-1e^(ω^j-1)x, c_N=1-cos(2π/N)>0.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-673af20ac7761c56","tex":"e^{-(\\lambda_j-\\lambda_{j+1})t}","aliases":"e^-(λ_j-λ_j+1)t lambda lambda","display":"e^{-(\\lambda_j-\\lambda_{j+1})t}","kind":"use","excerpt":"R_j(-t)=w_je^λ_jt(1+r_j(t)), 0≤ r_j(t)≤fracM_j+1w_j e^-(λ_j-λ_j+1)t.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ff32b50fce518a09","tex":"e^{-1/2}","aliases":"e^-1/2","display":"e^{-1/2}","kind":"use","excerpt":"log F_0(R)≤ C_0sqrt R (R≥ e), C_0=e^-1/2(1+22log2)≈9.85566103.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-15c619eec864b446","tex":"e^{-B(w)}","aliases":"e^-B(w)","display":"e^{-B(w)}","kind":"use","excerpt":"U(z)=e^B(z)∫_0^z e^-B(w) dw, B(z)=∫_0^zT(qw) dw.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fbd1212a559fe0da","tex":"e^{-c_N(t+a)^\\alpha}","aliases":"e^-c_N(t+a)^α alpha","display":"e^{-c_N(t+a)^\\alpha}","kind":"use","excerpt":"G_N,a(t)=α(t+a)^α-1 +O ((t+a)^α-1e^-c_N(t+a)^α). tagT4","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cbff9ddf6eea6a30","tex":"e^{-c_Nx}","aliases":"e^-c_Nx","display":"e^{-c_Nx}","kind":"use","excerpt":"F_N(x)=frace^xN(1+h_N(x)), |h_N(x)|≤(N-1)e^-c_Nx, |h_N'(x)|≤2(N-1)e^-c_Nx. tagS2","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b4fcb974274d9b00","tex":"e^{-Ns\\log x}","aliases":"e^-Nslog x log","display":"e^{-Ns\\log x}","kind":"use","excerpt":"For x>0, x^-Ns=e^-Nslog x uses the real logarithm. Multiplicities are already incorporated in (S4), so no simple-zero assumption entered the proof.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d064d99d6dbe0e46","tex":"e^{-s\\operatorname{Log}w}","aliases":"e^-sLogw operatorname","display":"e^{-s\\operatorname{Log}w}","kind":"use","excerpt":"∫_0^∞fract^-st+w dt =fracπsin(π s)e^-sLogw, 0<Re s<1, |arg w|<π. tagS10","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0400a938b5bf9ff8","tex":"e^{-s\\operatorname{Log}w_j}","aliases":"e^-sLogw_j operatorname","display":"e^{-s\\operatorname{Log}w_j}","kind":"use","excerpt":"Z_N(s)=∑_j e^-sLogw_j","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a3b3e2ebdde99841","tex":"e^{-x}","aliases":"e^-x","display":"e^{-x}","kind":"use","excerpt":"Since 1+h_N(x)=Ne^-xF_N(x)>0, the real logarithm is well defined for all x≥0. At infinity,","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d5baee9fc7dd593e","tex":"e^{-\\gamma_N|x|}","aliases":"e^-γ_N|x| gamma","display":"e^{-\\gamma_N|x|}","kind":"use","excerpt":"Indeed, the nearest competing root is separated from the positive direction by at least 3π/(2N), while the distinguished root is within π/(2N). Cosine decreases on [0,π]. Thus, once (N-1)e^-γ_N|x|<1, the distinguished term e^x in (S1) is larger in modulus than the sum of all competitors, and d_N(w)≠0. Consequently, sufficiently large zeros avoid th","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-406f16ef4a19b89b","tex":"e^{-\\Delta_jt}","aliases":"e^-Δ_jt Delta","display":"e^{-\\Delta_jt}","kind":"use","excerpt":"0≤λ_j-widehatλ_j(t,h) ≤frac1hlog (1+fracM_j+1w_je^-Δ_jt),","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8df1901b3ad20f7b","tex":"e^{-\\kappa_N t^{1/N}}","aliases":"e^-κ_N t^1/N kappa","display":"e^{-\\kappa_N t^{1/N}}","kind":"use","excerpt":"pageheadingORBIT SPECTRA / LARGE RADIUSA finite exponential sum controls the tail Put E_N(t)=d_N(-t)=∑_n≥0t^n/(Nn)! and ω=e^2π i/N. For x≥0, E_N(x^N)=frac1N∑_j=0^N-1e^ω^jx. Root filtering proves this identity term by term: only powers divisible by N remain. The positive real exponential is dominant, so log E_N(t)=t^1/N-log N+O(e^-κ_N t^1/N), κ_N=1-","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-32d7c432416fd513","tex":"e^{-\\lambda_j/2}","aliases":"e^-λ_j/2 lambda","display":"e^{-\\lambda_j/2}","kind":"use","excerpt":"displaystyle n_j=(1-e^-λ_j/2)^-1/β, a=-fraclog w_j2log n_j.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-024f1b59fbb1bcd5","tex":"e^{-\\lambda_j\\overline t}","aliases":"e^-λ_joverline t lambda overline","display":"e^{-\\lambda_j\\overline t}","kind":"use","excerpt":"Using an inner product linear in the first argument, the coefficient vector of K(·,t) is (sqrtw_je^-λ_joverline t)_j, so ⟨ Uc,K(·,t)⟩=(Uc)(t). This also verifies the sign and conjugation convention directly. Arbitrarily large finite Gram matrices at distinct complex depths are strictly positive, as proved in the base route. Thus the feature space h","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-047fb4f595248035","tex":"e^{-\\lambda_js}","aliases":"e^-λ_js lambda","display":"e^{-\\lambda_js}","kind":"use","excerpt":"λ_j=lim_t→∞fraclog R_j(-t)t, w_j=lim_t→∞e^-λ_jtR_j(-t), R_j+1(s)=R_j(s)-w_je^-λ_js.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c8aea6a6bad46e51","tex":"e^{-\\lambda_jt}","aliases":"e^-λ_jt lambda","display":"e^{-\\lambda_jt}","kind":"use","excerpt":"λ_j=lim_t→∞fraclog R_j(-t)t, w_j=lim_t→∞e^-λ_jtR_j(-t), R_j+1(s)=R_j(s)-w_je^-λ_js.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a2a997b6d850525e","tex":"e^{-\\lambda_ns}","aliases":"e^-λ_ns lambda","display":"e^{-\\lambda_ns}","kind":"use","excerpt":"mid Q(1+p^-v)^-1, v>1. fig03_complex_depthLeft: a=β=2 at real depths, using the exact zeta--Newton representation at 90-digit working precision. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β). Then A(s)=∑_n∈ S_Q","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-03f1748d577b13ea","tex":"e^{-\\rho}","aliases":"e^-ρ rho","display":"e^{-\\rho}","kind":"use","excerpt":"c_+=e^-ρ(V_N+1+Δ+B_+).","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2c4ed99bb25499eb","tex":"e^{2\\pi i/b}","aliases":"e^2π i/b pi","display":"e^{2\\pi i/b}","kind":"use","excerpt":"For an EGF G(z)=∑_m≥0a_m z^m/m!, an integer base b≥1, a residue 0≤ j<b, and ω=e^2π i/b, define","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d7e7084a35eacad9","tex":"e^{2\\pi i/N}","aliases":"e^2π i/N pi","display":"e^{2\\pi i/N}","kind":"use","excerpt":"pageheadingORBIT SPECTRA / LARGE RADIUSA finite exponential sum controls the tail Put E_N(t)=d_N(-t)=∑_n≥0t^n/(Nn)! and ω=e^2π i/N. For x≥0, E_N(x^N)=frac1N∑_j=0^N-1e^ω^jx. Root filtering proves this identity term by term: only powers divisible by N remain. The positive real exponential is dominant, so log E_N(t)=t^1/N-log N+O(e^-κ_N t^1/N), κ_N=1-","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d6e924b95e4b3124","tex":"e^{B(z)}","aliases":"e^B(z)","display":"e^{B(z)}","kind":"use","excerpt":"U(z)=e^B(z)∫_0^z e^-B(w) dw, B(z)=∫_0^zT(qw) dw.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5f912f91f583d2b6","tex":"e^{h\\varepsilon}","aliases":"e^hε varepsilon","display":"e^{h\\varepsilon}","kind":"use","excerpt":"t≥max{0,frac1Δ_j logfracM_j+1w_j(e^hε-1)}.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-682a47b2eea70d50","tex":"e^{i\\theta}","aliases":"e^iθ theta","display":"e^{i\\theta}","kind":"use","excerpt":"To check the elementary estimate, write w=re^iθ: |t+w|^2=(t+r)^2-4trsin^2(θ/2)≥(t+r)^2cos^2(θ/2).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-120bb5fe03f478b3","tex":"e^{R\\lambda_p}","aliases":"e^Rλ_p lambda","display":"e^{R\\lambda_p}","kind":"use","excerpt":"displaystyle sup_|s|≤ R|A_a,β,Q^(k)(s)-A_a,β,Qp^(k)(s)| ≤ λ_p^k e^Rλ_p fracZ_Q(2a)p^2a+1.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1fe69fafa50db3c5","tex":"e^t","aliases":"e^t","display":"e^t","kind":"use","excerpt":"stored as 0/1, and this guide uses B_1=-1/2. At n=10, p_10=5 and q_10=66, so B_10=5/66. Its Taylor coefficient is (5/66)/10!. At n=12, B_12=-691/2730. An OEIS integer numerator sequence does not by itself specify the rational family or its generating-function coefficient. OEIS A027641 and A027642, indexed from zero. Exact rational recurrence and mu","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ffbdf389269f7be3","tex":"e^{t-a_j}","aliases":"e^t-a_j","display":"e^{t-a_j}","kind":"use","excerpt":"log M(e^t)≤ t+log2+ max_j{log M(e^t-a_j)+log M(e^t-b_j)}.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-573f36be9e01e4ae","tex":"e^{t-b_j}","aliases":"e^t-b_j","display":"e^{t-b_j}","kind":"use","excerpt":"log M(e^t)≤ t+log2+ max_j{log M(e^t-a_j)+log M(e^t-b_j)}.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f2582e6375c09280","tex":"e^{t-\\log(1/q)}","aliases":"e^t-log(1/q) log","display":"e^{t-\\log(1/q)}","kind":"use","excerpt":"H_1(e^t)≤ t+log2 +H_1(e^t-log(1/q))+H_2(e^t-log(1/r)).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8c106ce9b207a677","tex":"e^{t-\\log(1/r)}","aliases":"e^t-log(1/r) log","display":"e^{t-\\log(1/r)}","kind":"use","excerpt":"H_1(e^t)≤ t+log2 +H_1(e^t-log(1/q))+H_2(e^t-log(1/r)).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-384d0f4965196664","tex":"e^{t\\lambda_{L+1}}","aliases":"e^tλ_L+1 lambda","display":"e^{t\\lambda_{L+1}}","kind":"use","excerpt":"E_L(t)=e^tλ_L+1fracL^1-2a2a-1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bed6b5798928e05a","tex":"e^x","aliases":"e^x","display":"e^x","kind":"use","excerpt":"F_N(x)=frace^xN(1+h_N(x)), |h_N(x)|≤(N-1)e^-c_Nx, |h_N'(x)|≤2(N-1)e^-c_Nx. tagS2","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-71c6ba365da8d7af","tex":"e^{|w|^{1/N}}","aliases":"e^|w|^1/N","display":"e^{|w|^{1/N}}","kind":"use","excerpt":"one rotation orbit of N zeros of D_N. Using Argw∈(-π,π], define Z_N(s)=∑_jw_j^-s, Re s>1/N. fig06_orbit_zerosFor N=4, the four real/imaginary zeros in each orbit share one positive coordinate w=z^4. The plot shows the first six orbits. Multiplicity is retained by the definition; simplicity is not assumed in general. The bound |d_N(w)|≤ e^|w|^1/N an","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a5683e2710ff1187","tex":"e^{\\lambda_jt}","aliases":"e^λ_jt lambda","display":"e^{\\lambda_jt}","kind":"use","excerpt":"R_j(-t)=w_je^λ_jt(1+r_j(t)), 0≤ r_j(t)≤fracM_j+1w_j e^-(λ_j-λ_j+1)t.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b92b872d8764d707","tex":"e^{\\lambda_pT}","aliases":"e^λ_pT lambda","display":"e^{\\lambda_pT}","kind":"use","excerpt":"p^-4e^λ_pT≤ D_p(-T) ≤fracD(4)p^4+1e^λ_pT (T≥0). tagRA-6","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-22f0fe45aa27bf0a","tex":"e^{\\rho t}","aliases":"e^ρ t rho","display":"e^{\\rho t}","kind":"use","excerpt":"On the seed, G(t)≤ V_N+1+Δ+B_+≤ c_+e^ρ t. Each channel satisfies q_j^ρ+r_j^ρ≤1. Induction in intervals of length min_jmin(a_j,b_j)>0 therefore propagates the seed comparison, giving the explicit upper bound","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c08b8d9e213014dc","tex":"e^{\\tau\\max(-\\Re s,0)}","aliases":"e^τmax(-Re s,0) tau max Re","display":"e^{\\tau\\max(-\\Re s,0)}","kind":"use","excerpt":"the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β). Then A(s)=∑_n∈ S_Qw_ne^-λ_ns, w_n=n^-2a. The positive weights are summable, and the distinct support frequencies decrease to zero. If p_* is the least prime not dividing Q, then τ=-2log(1-p_*^-β), M=Z_Q(2a)-1, |A(s)|≤ M e^τmax(-Re s,0). This proves local uni","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e1a5eb8e9bafe342","tex":"e^{\\omega^j x}","aliases":"e^ω^j x omega","display":"e^{\\omega^j x}","kind":"use","excerpt":"F_N(x):=d_N(-x^N) =frac1N∑_j=0^N-1e^ω^j x =∑_n=0^∞fracx^Nn(Nn)!. tagS1","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e39da0e01d02c3ed","tex":"e^{\\omega^jx}","aliases":"e^ω^jx omega","display":"e^{\\omega^jx}","kind":"use","excerpt":"pageheadingORBIT SPECTRA / LARGE RADIUSA finite exponential sum controls the tail Put E_N(t)=d_N(-t)=∑_n≥0t^n/(Nn)! and ω=e^2π i/N. For x≥0, E_N(x^N)=frac1N∑_j=0^N-1e^ω^jx. Root filtering proves this identity term by term: only powers divisible by N remain. The positive real exponential is dominant, so log E_N(t)=t^1/N-log N+O(e^-κ_N t^1/N), κ_N=1-","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d9b6193b1833f637","tex":"E_a","aliases":"E_a","display":"E_a","kind":"use","excerpt":"claimThree operations that must be kept distinct For an EGF E_a(z)=∑_ma_mz^m/m!, differentiation transports future coefficients: E_a^(k)=∑_ma_m+kz^m/m!. Causal ordinary taps instead act by repeated zero-constant integration: E_b=∑_ℓ h_ℓ I^ℓ E_a. Multiplication of two EGFs gives Hurwitz convolution, b_m=∑_k=0^mbinom mk a_kh_m-k. It is not ordinary s","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e335147a84a98a7d","tex":"E_a^{(k)}","aliases":"E_a^(k)","display":"E_a^{(k)}","kind":"use","excerpt":"claimThree operations that must be kept distinct For an EGF E_a(z)=∑_ma_mz^m/m!, differentiation transports future coefficients: E_a^(k)=∑_ma_m+kz^m/m!. Causal ordinary taps instead act by repeated zero-constant integration: E_b=∑_ℓ h_ℓ I^ℓ E_a. Multiplication of two EGFs gives Hurwitz convolution, b_m=∑_k=0^mbinom mk a_kh_m-k. It is not ordinary s","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9a603588606ce810","tex":"E_b","aliases":"E_b","display":"E_b","kind":"use","excerpt":"claimThree operations that must be kept distinct For an EGF E_a(z)=∑_ma_mz^m/m!, differentiation transports future coefficients: E_a^(k)=∑_ma_m+kz^m/m!. Causal ordinary taps instead act by repeated zero-constant integration: E_b=∑_ℓ h_ℓ I^ℓ E_a. Multiplication of two EGFs gives Hurwitz convolution, b_m=∑_k=0^mbinom mk a_kh_m-k. It is not ordinary s","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-146a7740d8a875a6","tex":"E_L","aliases":"E_L","display":"E_L","kind":"use","excerpt":"E_L(t)=e^tλ_L+1fracL^1-2a2a-1.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e466c5eab5f55a38","tex":"E_N","aliases":"E_N","display":"E_N","kind":"use","excerpt":"pageheadingORBIT SPECTRA / LARGE RADIUSA finite exponential sum controls the tail Put E_N(t)=d_N(-t)=∑_n≥0t^n/(Nn)! and ω=e^2π i/N. For x≥0, E_N(x^N)=frac1N∑_j=0^N-1e^ω^jx. Root filtering proves this identity term by term: only powers divisible by N remain. The positive real exponential is dominant, so log E_N(t)=t^1/N-log N+O(e^-κ_N t^1/N), κ_N=1-","count":10},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3c917abfa3e0bdd2","tex":"E_{N,1}","aliases":"E_N,1","display":"E_{N,1}","kind":"use","excerpt":"MonographPartIIIOrbit spectra and exact coordinatesSpectral continuation and the coefficient language of polyphase transforms. Spectral functions of zero orbits. pageheadingFAMILY 4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"A_a,β,Q(s)=∑_substackn≥2 μ(n)^2=1 (n,Q)=1 n^-2aexp(-λ_n s), λ_n=-2log(1-n^-β).","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"F[X](z)= 1+u(qz)v(rz) 1+u(az)v(bz). tagS1","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"F'(z)=1+∑_j=1^Jh_jF(q_jz)F(r_jz), F(0)=0,","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2f4583c8b7c73ba0","tex":"f'","aliases":"f'","display":"f'","kind":"use","excerpt":"sup_|z|≤ R|F_ε(z)-F_0(z)| ≤ ε L(R), L(R)=∑_n≥1n d_nR^2n+1 =fracRf'(R)-f(R)2. tagRA-1","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8c4caf5c79a35c9c","tex":"f^2","aliases":"f^2","display":"f^2","kind":"use","excerpt":"B_mA_m^-1/2. Every determinant root is real and below -1; possible coefficient poles occur there, although numerator cancellation is possible. fig05_sieve_boundaryThe line is a proved bound for p=2; dots are ordinary high-precision pencil roots. A defect λ=-1.001 eventually violates positivity even though it can pass low-order checks. The nearest r","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"startvolumeSequence atlasOEIS families and exact transforms lessonSEQUENCES / 01Two seeds, one Fibonacci recurrenceS01_fibonacci_lucas F_n+2=F_n+1+F_n, L_n+2=L_n+1+L_n, L_n^2-5F_n^2=4(-1)^n The two families share the same recurrence and have different seeds: (F_0,F_1)=(0,1) and (L_0,L_1)=(2,1). The right panel uses the same index in both families. ","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"startvolumeSequence atlasOEIS families and exact transforms lessonSEQUENCES / 01Two seeds, one Fibonacci recurrenceS01_fibonacci_lucas F_n+2=F_n+1+F_n, L_n+2=L_n+1+L_n, L_n^2-5F_n^2=4(-1)^n The two families share the same recurrence and have different seeds: (F_0,F_1)=(0,1) and (L_0,L_1)=(2,1). The right panel uses the same index in both families. ","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0ece071ddbea0305","tex":"f_{1/2}","aliases":"f_1/2","display":"f_{1/2}","kind":"use","excerpt":"claimA worked order-one member T_1/3,2/3(z)=z+frac227z^3+ frac42187z^5+frac23211160261z^7+O(z^9). It has order one, and T_1/3,2/3(1)≈1.0759239897408249. The diagonal point q=r=1/2 gives f_1/2(z)=sqrt2sinh(z/sqrt2). Every off-diagonal member is separated from all rescaled diagonal members by the theorem above.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d16336bd10bc245a","tex":"f_{1/4}","aliases":"f_1/4","display":"f_{1/4}","kind":"use","excerpt":"sup_|z|≤ R|F_ε(z)-f_1/4(z)| ≤fracε2{R f_3/4'(R)-f_3/4(R)}.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9cad11eb56c2bf0c","tex":"F_{10^{-12}}","aliases":"F_10^-12","display":"F_{10^{-12}}","kind":"use","excerpt":"sup_|z|≤1|F_10^-12(z)-F_0(z)|<2.41890267×10^-13,","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"Set F_1=U, F_2=V, and H_i(R)=log F_i(R). Since F_i(R)≥ R, these logarithms are positive for R>1. The type convention used here is","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-35c78ce40f857704","tex":"f_{3/4}'","aliases":"f_3/4'","display":"f_{3/4}'","kind":"use","excerpt":"sup_|z|≤ R|F_ε(z)-f_1/4(z)| ≤fracε2{R f_3/4'(R)-f_3/4(R)}.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-60ee115da23a6e48","tex":"F_7","aliases":"F_7","display":"F_7","kind":"use","excerpt":"nsforms lessonSEQUENCES / 01Two seeds, one Fibonacci recurrenceS01_fibonacci_lucas F_n+2=F_n+1+F_n, L_n+2=L_n+1+L_n, L_n^2-5F_n^2=4(-1)^n The two families share the same recurrence and have different seeds: (F_0,F_1)=(0,1) and (L_0,L_1)=(2,1). The right panel uses the same index in both families. Its alternating defect stays exactly four in magnitu","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-efee4d9a81841b87","tex":"f_{a,m}","aliases":"f_a,m","display":"f_{a,m}","kind":"use","excerpt":"pageheadingPRIME SIEVES / CONSTRUCTIVE WITNESSEvery negative prime atom has a finite detector For an integer a≥2, let L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1), r_a=frac1-x_a+11-x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_si","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-94d5607cc54c383b","tex":"F_g","aliases":"F_g","display":"F_g","kind":"use","excerpt":"sup_|z|≤ R|F_h(z)-F_g(z)| ≤frac d4{R f_Q'(R)-f_Q(R)}.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e982a068c0aac4ff","tex":"F_h","aliases":"F_h","display":"F_h","kind":"use","excerpt":"sup_|z|≤ R|F_h(z)-F_g(z)| ≤frac d4{R f_Q'(R)-f_Q(R)}.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-31d7232e6fa917d3","tex":"F_i","aliases":"F_i","display":"F_i","kind":"use","excerpt":"Set F_1=U, F_2=V, and H_i(R)=log F_i(R). Since F_i(R)≥ R, these logarithms are positive for R>1. The type convention used here is","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0fe9f993f4ff4310","tex":"F_N","aliases":"F_N","display":"F_N","kind":"use","excerpt":"F_N(x):=d_N(-x^N) =frac1N∑_j=0^N-1e^ω^j x =∑_n=0^∞fracx^Nn(Nn)!. tagS1","count":11},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e3f4ba757b6d2cf8","tex":"f_n","aliases":"f_n","display":"f_n","kind":"use","excerpt":"lessonSEQUENCES / 16Sums and differences form a polynomial ladderS16_polynomial_ladder T_n=∑_j=1^nj, H_n=∑_j=1^nT_j, Δ T_n=n, Δ H_n=T_n The upper table follows positive integers into triangular and tetrahedral numbers. Squares and cubes occupy the lower rows for comparison. The arrows reverse prefix sums by backward differences, Δ f_n=f_n-f_n-1. Al","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-347cfe740f28d541","tex":"F_N'","aliases":"F_N'","display":"F_N'","kind":"use","excerpt":"log F_N(x)=x-log N+O(e^-c_Nx), g_N(x):=fracF_N'(x)F_N(x) =1+O(e^-c_Nx). tagS3","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6b681ab9d723acad","tex":"f_{n-1}","aliases":"f_n-1","display":"f_{n-1}","kind":"use","excerpt":"lessonSEQUENCES / 16Sums and differences form a polynomial ladderS16_polynomial_ladder T_n=∑_j=1^nj, H_n=∑_j=1^nT_j, Δ T_n=n, Δ H_n=T_n The upper table follows positive integers into triangular and tetrahedral numbers. Squares and cubes occupy the lower rows for comparison. The arrows reverse prefix sums by backward differences, Δ f_n=f_n-f_n-1. Al","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8269308c3c2f8363","tex":"f_{p,m}","aliases":"f_p,m","display":"f_{p,m}","kind":"use","excerpt":"∫ f_p,m(x)^2 dν_F,λ(x) ≤frac1+λ_pp^4 +fracC_F C_p^23p^3r_p^2m. tag4","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-14d9bb535fb098ee","tex":"f_{p,m}^2","aliases":"f_p,m^2","display":"f_{p,m}^2","kind":"use","excerpt":"This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_signed_detectorA rational detector for p=3, λ_3=-1.1, λ_2=1. The first exact bound proving negativity is m=5, so a Hankel matrix of order eight detects the signed defect. Set C_F=∏_q∈ Fmax(1,|1+λ_q|). The summable tail give","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-64f5ff12e8643285","tex":"f_Q","aliases":"f_Q","display":"f_Q","kind":"use","excerpt":"where f_Q(z)=∑ d_n(Q)z^2n+1 is the established diagonal function. Consequently, on every disk |z|≤ R,","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ba2ac5456d322b89","tex":"f_q","aliases":"f_q","display":"f_q","kind":"use","excerpt":"claimAn established diagonal antecedent The equation f_q'(z)=1+f_q(qz)^2, f_q(0)=0, is already present in M endez~[source reference]. It remains an important baseline, but its definition is not a discovery of this project. The asymmetric extension keeps two separate contractions and permits a proved distinction from every amplitude/argument rescali","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-92dd42162c18dd09","tex":"f_Q'","aliases":"f_Q'","display":"f_Q'","kind":"use","excerpt":"sup_|z|≤ R|F_h(z)-F_g(z)| ≤frac d4{R f_Q'(R)-f_Q(R)}.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-72181b7db104386d","tex":"f_q'","aliases":"f_q'","display":"f_q'","kind":"use","excerpt":"claimAn established diagonal antecedent The equation f_q'(z)=1+f_q(qz)^2, f_q(0)=0, is already present in M endez~[source reference]. It remains an important baseline, but its definition is not a discovery of this project. The asymmetric extension keeps two separate contractions and permits a proved distinction from every amplitude/argument rescali","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-47567af3d7dd306d","tex":"f_s","aliases":"f_s","display":"f_s","kind":"use","excerpt":"pageheadingFAMILY 1 / SEPARATIONMore than a rescaled one-scale seed claimNonrescaling theorem If q≠ r, then T_q,r cannot equal A f_s(Bz) for nonzero constants A,B and any real 0<s<1, where f_s'=1+f_s(sz)^2 is the known family.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4d14ff68a887e9b1","tex":"f_s'","aliases":"f_s'","display":"f_s'","kind":"use","excerpt":"pageheadingFAMILY 1 / SEPARATIONMore than a rescaled one-scale seed claimNonrescaling theorem If q≠ r, then T_q,r cannot equal A f_s(Bz) for nonzero constants A,B and any real 0<s<1, where f_s'=1+f_s(sz)^2 is the known family.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6b08d601f39e0488","tex":"F_\\varepsilon","aliases":"F_ε varepsilon","display":"F_\\varepsilon","kind":"use","excerpt":"Take the two diagonal channels (q,r)=(1/4,1/4) and (3/4,3/4), with weights 1-ε and ε, respectively. For 0≤ε≤1, write the resulting function as F_ε. The quantitative bound gives","count":11},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f851687617add874","tex":"F_\\varepsilon'","aliases":"F_ε' varepsilon","display":"F_\\varepsilon'","kind":"use","excerpt":"F_ε'(z)=1+(1-ε)F_ε(z/4)^2 +ε F_ε(3z/4)^2, F_ε(0)=0, 0≤ε≤1.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"bibitemDLMF NIST Digital Library of Mathematical Functions. Sections 5.5 and 5.12 give the gamma reflection and beta identities. Section 10.46 defines the Mittag--Leffler function. Sections 16.8 and 16.21 give the generalized hypergeometric and Meijer G differential equations. Section 18.2 develops general orthogonal-polynomial theory. https://dlmf","count":30},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"esS15_binary_families w(n)=popcount(n), t(n)=w(n)bmod2, g(n)=2^w(n) Each column has a four-bit address, its binary weight, its Thue--Morse parity, and its Gould value. Reading a final zero bit preserves weight; reading a final one adds one. Parity toggles and the Gould value doubles in the latter case. The address 13=1101_2 has weight three, so t(1","count":29},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7bd595c553adfa0e","tex":"G^{(t)}","aliases":"G^(t)","display":"G^{(t)}","kind":"use","excerpt":"Proof. The EGF of the delayed sequence is ∑_m≥ ta_m-tz^m/m!=I^tG(z). Summing taps gives the formula. Conversely, G^(t) has coefficient sequence a_m+t, so derivatives sample future coefficients and do not implement these past delays. square","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"use","excerpt":"G_1(t)≤ G_1(t-log(1/q))+G_2(t-log(1/r)).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3b9dcaf46a46092c","tex":"G_2","aliases":"G_2","display":"G_2","kind":"use","excerpt":"G_1(t)≤ G_1(t-log(1/q))+G_2(t-log(1/r)).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","excerpt":"For two EGFs G_a,G_h, ordinary function multiplication instead gives the Hurwitz product","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-624b8e0e8ca99130","tex":"G_h","aliases":"G_h","display":"G_h","kind":"use","excerpt":"For two EGFs G_a,G_h, ordinary function multiplication instead gives the Hurwitz product","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cd7038b72872679a","tex":"G_i","aliases":"G_i","display":"G_i","kind":"use","excerpt":"G_i(t)=H_i(e^t)+t+D.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ec1c980979f73c29","tex":"G_j","aliases":"G_j","display":"G_j","kind":"use","excerpt":"pageheadingEXACT LABORATORY / PROJECTORSPolyphase sections preserve the full sequence For an ordinary generating function G(z)=∑_m≥0a_mz^m and integer b≥1, write G(z)=∑_j=0^b-1z^jG_j(z^b), G_j(t)=∑_n≥0a_bn+jt^n. The root projector is P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =z^jG_j(z^b), ω=e^2π i/b. The same projector works on an EGF, but the section ","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4b4aad952969f96a","tex":"G_N","aliases":"G_N","display":"G_N","kind":"use","excerpt":"Let H_N(t)=d_N(-t)=E_N,1(t), and G_N(t)=H_N'(t)/H_N(t). Logarithmic differentiation of (S4), locally uniformly off the zeros, gives","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3632ec9c2de4f4a3","tex":"g_N","aliases":"g_N","display":"g_N","kind":"use","excerpt":"log F_N(x)=x-log N+O(e^-c_Nx), g_N(x):=fracF_N'(x)F_N(x) =1+O(e^-c_Nx). tagS3","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d3bbdc98f0002252","tex":"G_{N,a}","aliases":"G_N,a","display":"G_{N,a}","kind":"use","excerpt":"This follows by taking the ratio of the two convergent products in (S4), not by introducing an unverified exponential factor. Its logarithmic derivative is G_N,a(t)=H_N'(a+t)/H_N(a+t). The same dominated beta-integral argument gives","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-66816f2f34b0eaea","tex":"g_s","aliases":"g_s","display":"g_s","kind":"use","excerpt":"(n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2a-1/2)/β; |s|/Re s stays bounded in the stated sector. This gives the erro","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-40dc3ed588f9a03b","tex":"g_s'","aliases":"g_s'","display":"g_s'","kind":"use","excerpt":"(n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2a-1/2)/β; |s|/Re s stays bounded in the stated sector. This gives the erro","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"_ladder T_n=∑_j=1^nj, H_n=∑_j=1^nT_j, Δ T_n=n, Δ H_n=T_n The upper table follows positive integers into triangular and tetrahedral numbers. Squares and cubes occupy the lower rows for comparison. The arrows reverse prefix sums by backward differences, Δ f_n=f_n-f_n-1. All plotted values use their displayed common index. At n=4, T_4=10 and H_4=20. T","count":35},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"The image H_a,β,Q=Uℓ^2, with Uc _ H= c _ℓ^2, is therefore a complete Hilbert space of entire functions. Its reproducing kernel is","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","excerpt":"Proof. The preceding positive remainder formula gives widehatλ_j=λ_j+h^-1[log(1+r_j(t+h))-log(1+r_j(t))]. Each tail factor decreases, so 0≤ r_j(t+h)≤ r_j(t). The two inequalities now follow from the bound on r_j(t). For the amplitude, multiply the positive tail bound by e^-λ_jt.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7c6fa035c4defd46","tex":"H^n","aliases":"H^n","display":"H^n","kind":"use","excerpt":"B_n=H^nD^n(n+1)A_n∈ Z hboxfor all n.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"Let h_0,…,h_L-1 be a finite tap vector and let a_m=0 for m<0. Ordinary taps form","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9493d4b148ff0c65","tex":"H_{0,2}","aliases":"H_0,2","display":"H_{0,2}","kind":"use","excerpt":"-1 · 2 · 1 . The entries arise from the same congruence condition. For instance an input in phase two shifted by one lands in phase zero of the next block, contributing 2t to H_0,2.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"H_1(R) · ≥ c_-{w_1[q(R-1)+C]^ρ +w_2[r(R-1)+C]^ρ} · ≥ c_-(q^ρ w_1+r^ρ w_2)(R+C)^ρ =c_-w_1(R+C)^ρ.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"c_n(h,q,r)∈ Q[h_1,…,h_J,q_1,r_1,…,q_J,r_J].","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"use","excerpt":"H_1(e^t)≤ t+log2 +H_1(e^t-log(1/q))+H_2(e^t-log(1/r)).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-936bd94983483758","tex":"H_4","aliases":"H_4","display":"H_4","kind":"use","excerpt":"_ladder T_n=∑_j=1^nj, H_n=∑_j=1^nT_j, Δ T_n=n, Δ H_n=T_n The upper table follows positive integers into triangular and tetrahedral numbers. Squares and cubes occupy the lower rows for comparison. The arrows reverse prefix sums by backward differences, Δ f_n=f_n-f_n-1. All plotted values use their displayed common index. At n=4, T_4=10 and H_4=20. T","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ec07ca37a68b413a","tex":"\\mathcal H_{a,\\beta,Q}","aliases":"H_a,β,Q mathcal beta","display":"\\mathcal H_{a,\\beta,Q}","kind":"use","excerpt":"The image H_a,β,Q=Uℓ^2, with Uc _ H= c _ℓ^2, is therefore a complete Hilbert space of entire functions. Its reproducing kernel is","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d3cba85783189f4d","tex":"H_i","aliases":"H_i","display":"H_i","kind":"use","excerpt":"Set F_1=U, F_2=V, and H_i(R)=log F_i(R). Since F_i(R)≥ R, these logarithms are positive for R>1. The type convention used here is","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f74f6a57337ee481","tex":"H_{ij}","aliases":"H_ij","display":"H_{ij}","kind":"use","excerpt":"For causal FIR taps h_ℓ, b_m=∑_ℓ h_ℓ a_m-ℓ with zero extension to negative indices, split A(z) and B(z) into b phases. Then B_i(t)=∑_jH_ij(t)A_j(t), where H_ij(t)=∑_substackℓ≥0","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-da28811766777fc7","tex":"h_J","aliases":"h_J","display":"h_J","kind":"use","excerpt":"c_n(h,q,r)∈ Q[h_1,…,h_J,q_1,r_1,…,q_J,r_J].","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f1c7cbbaca31147f","tex":"H_{jl}","aliases":"H_jl","display":"H_{jl}","kind":"use","excerpt":"Y_j(w)=∑_l=0^b-1H_jl(w)X_l(w), H_jl(w)=∑_substack0≤ t<L t+l≡ j (b) h_t w^(t+l-j)/b.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a722780b53f2037a","tex":"h_{L-1}","aliases":"h_L-1","display":"h_{L-1}","kind":"use","excerpt":"Let h_0,…,h_L-1 be a finite tap vector and let a_m=0 for m<0. Ordinary taps form","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cf5f97ebded34595","tex":"H_m","aliases":"H_m","display":"H_m","kind":"use","excerpt":"item[Hankel order and polynomial degree.] For moments M_j=∫ x^jddν, the order-m matrix is H_m=[M_i+j]_0≤ i,j<m. It tests quadratic forms of real polynomials of degree at most m-1. At depth k, the matrix is [M_k+i+j]_0≤ i,j<m. An order-five test therefore covers degree four, not degree five. item[Signed quadratic form.] For real polynomials, Q_ν(f)=","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9b091910254bf720","tex":"h_{m-k}","aliases":"h_m-k","display":"h_{m-k}","kind":"use","excerpt":"claimThree operations that must be kept distinct For an EGF E_a(z)=∑_ma_mz^m/m!, differentiation transports future coefficients: E_a^(k)=∑_ma_m+kz^m/m!. Causal ordinary taps instead act by repeated zero-constant integration: E_b=∑_ℓ h_ℓ I^ℓ E_a. Multiplication of two EGFs gives Hurwitz convolution, b_m=∑_k=0^mbinom mk a_kh_m-k. It is not ordinary s","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ac7885f6694a994f","tex":"H_N","aliases":"H_N","display":"H_N","kind":"use","excerpt":"Let H_N(t)=d_N(-t)=E_N,1(t), and G_N(t)=H_N'(t)/H_N(t). Logarithmic differentiation of (S4), locally uniformly off the zeros, gives","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b28d6c4b94c5afd8","tex":"h_N","aliases":"h_N","display":"h_N","kind":"use","excerpt":"h_N(x)=∑_j=1^N-1e^(ω^j-1)x, c_N=1-cos(2π/N)>0.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4486477b16de3cce","tex":"H_N'","aliases":"H_N'","display":"H_N'","kind":"use","excerpt":"Let H_N(t)=d_N(-t)=E_N,1(t), and G_N(t)=H_N'(t)/H_N(t). Logarithmic differentiation of (S4), locally uniformly off the zeros, gives","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cf7cc7adc0d0ac89","tex":"h_N'","aliases":"h_N'","display":"h_N'","kind":"use","excerpt":"F_N(x)=frace^xN(1+h_N(x)), |h_N(x)|≤(N-1)e^-c_Nx, |h_N'(x)|≤2(N-1)e^-c_Nx. tagS2","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-480c877d455d2c2b","tex":"h_t","aliases":"h_t","display":"h_t","kind":"use","excerpt":"For rational channel tuples (h_t,q_t,r_t), use","count":13},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-858e4bb65a33c6eb","tex":"h_\\ell","aliases":"h_ℓ ell","display":"h_\\ell","kind":"use","excerpt":"For causal FIR taps h_ℓ, b_m=∑_ℓ h_ℓ a_m-ℓ with zero extension to negative indices, split A(z) and B(z) into b phases. Then B_i(t)=∑_jH_ij(t)A_j(t), where H_ij(t)=∑_substackℓ≥0","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5e073754a563e30e","tex":"\\hbox","aliases":"hbox","display":null,"kind":"use","excerpt":"B_n=H^nD^n(n+1)A_n∈ Z hboxfor all n.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"At real positive depth, the matrix (A_χ,ψ(s)) on any d distinct characters is positive semidefinite and has identical diagonal A_a,β,Q(s). Its normalized matrix tends to I_d, with operator norm error O_a,β,Q,d(s^-1/(2β)). Indeed, the off-diagonal entry bound controls the maximum row and column sums, and thus the operator norm. This gives quantitati","count":10},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":75},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ac24a69efb384882","tex":"I^t","aliases":"I^t","display":"I^t","kind":"use","excerpt":"B_rm EGF(z)=∑_t=0^L-1h_t I^tG(z), IG(z)=∫_0^zG(w) dw.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0a49e70f6740f4b5","tex":"I^\\ell","aliases":"I^ℓ ell","display":"I^\\ell","kind":"use","excerpt":"claimThree operations that must be kept distinct For an EGF E_a(z)=∑_ma_mz^m/m!, differentiation transports future coefficients: E_a^(k)=∑_ma_m+kz^m/m!. Causal ordinary taps instead act by repeated zero-constant integration: E_b=∑_ℓ h_ℓ I^ℓ E_a. Multiplication of two EGFs gives Hurwitz convolution, b_m=∑_k=0^mbinom mk a_kh_m-k. It is not ordinary s","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-20d6646340ce20f0","tex":"I_d","aliases":"I_d","display":"I_d","kind":"use","excerpt":"At real positive depth, the matrix (A_χ,ψ(s)) on any d distinct characters is positive semidefinite and has identical diagonal A_a,β,Q(s). Its normalized matrix tends to I_d, with operator norm error O_a,β,Q,d(s^-1/(2β)). Indeed, the off-diagonal entry bound controls the maximum row and column sums, and thus the operator norm. This gives quantitati","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-edadeb2cc1e72d16","tex":"I_K","aliases":"I_K","display":"I_K","kind":"use","excerpt":"Near zero write f(t)=(log E_N)'(t)=∑_k≥1α_kt^k-1, where α_k=(-1)^k+1 Z_N(k). For any integer K≥1, subtract K terms on [0,1]: I_K(s)= · ∫_0^1t^-s[f(t)-∑_k=1^Kα_kt^k-1]dd t +∑_k=1^Kfracα_kk-s · +∫_1^∞ t^-s[f(t)-frac1N t^1/N-1]dd t +frac1N(s-1/N). The first integral is holomorphic for Re s<K+1. The second is entire because its remainder decays exponen","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"lessonSEQUENCES / 03Jacobsthal separates two modesS03_jacobsthal J_0=0, J_1=1, J_n+2=J_n+1+2J_n=frac2^n+2-(-1)^n+23 The rising bars are Jacobsthal numbers. The right panel removes their growing mode: 2^n-3J_n=(-1)^n. What remains is an alternating mode of constant magnitude. Both modes belong to the same second-order recurrence. At n=6, J_6=21 and ","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-787a425baaf96eca","tex":"j^{-1}","aliases":"j^-1","display":"j^{-1}","kind":"use","excerpt":"The floor contributes an additional O(j^-1), which is absorbed by the displayed error. These are fixed-parameter asymptotics. They do not give a constant uniform in arbitrary moduli.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-678f077c412c32b2","tex":"j^{-1/2}","aliases":"j^-1/2","display":"j^{-1/2}","kind":"use","excerpt":"n_j=fracjδ_Q(1+O_Q(j^-1/2)), λ_j=2δ_Q^βj^-β (1+O_Q,β(j^-1/2+j^-β)).","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5c33c55d69578754","tex":"j^{-\\beta}","aliases":"j^-β beta","display":"j^{-\\beta}","kind":"use","excerpt":"n_j=fracjδ_Q(1+O_Q(j^-1/2)), λ_j=2δ_Q^βj^-β (1+O_Q,β(j^-1/2+j^-β)).","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-73c8e1ca32d82998","tex":"J_6","aliases":"J_6","display":"J_6","kind":"use","excerpt":"lessonSEQUENCES / 03Jacobsthal separates two modesS03_jacobsthal J_0=0, J_1=1, J_n+2=J_n+1+2J_n=frac2^n+2-(-1)^n+23 The rising bars are Jacobsthal numbers. The right panel removes their growing mode: 2^n-3J_n=(-1)^n. What remains is an alternating mode of constant magnitude. Both modes belong to the same second-order recurrence. At n=6, J_6=21 and ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f0170a84e36a17c9","tex":"J_7","aliases":"J_7","display":"J_7","kind":"use","excerpt":"03Jacobsthal separates two modesS03_jacobsthal J_0=0, J_1=1, J_n+2=J_n+1+2J_n=frac2^n+2-(-1)^n+23 The rising bars are Jacobsthal numbers. The right panel removes their growing mode: 2^n-3J_n=(-1)^n. What remains is an alternating mode of constant magnitude. Both modes belong to the same second-order recurrence. At n=6, J_6=21 and 2^6-3·21=1. At n=7","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a620020ba505249b","tex":"J_n","aliases":"J_n","display":"J_n","kind":"use","excerpt":"lessonSEQUENCES / 03Jacobsthal separates two modesS03_jacobsthal J_0=0, J_1=1, J_n+2=J_n+1+2J_n=frac2^n+2-(-1)^n+23 The rising bars are Jacobsthal numbers. The right panel removes their growing mode: 2^n-3J_n=(-1)^n. What remains is an alternating mode of constant magnitude. Both modes belong to the same second-order recurrence. At n=6, J_6=21 and ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"claimStrict positive complex kernel For distinct complex points s_i, K_ij=A(s_i+overlines_j) is strictly positive definite. A zero quadratic form would force a finite exponential sum to vanish at every λ_n. Their accumulation at zero invokes the identity theorem; derivatives at zero then give a nonsingular Vandermonde system. On the real axis, (-1)","count":48},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"lessonSEQUENCES / 13Apéry uses taps that depend on the indexS13_apery_recurrence (n+1)^3A_n+1=(2n+1)(17n^2+17n+5)A_n-n^3A_n-1 The prefix annotation begins the Apéry sequence defined by A_n=∑_k=0^nbinom nk^2binomn+kk^2. The curves show its growth through index fourteen on a base-ten logarithmic scale. The lower diagram evaluates the recurrence at n=","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-16b09a3431d14563","tex":"K^3","aliases":"K^3","display":"K^3","kind":"use","excerpt":"This polynomial has degree |S|+m, equals one at x_2, and kills every positive atom through the cutoff. At each remaining positive node, the product of the absolute root factors is at most C_S, and the last factor is at most r_S^m. Dropping all other negative atoms and bounding the positive tail by the integral ∑_n>Kn^-4≤1/(3K^3) gives","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-146b7f435ba6dc91","tex":"k^3","aliases":"k^3","display":"k^3","kind":"use","excerpt":"p y^2-frac15v_2ky+k^3=0,","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1f0a33ff47e3ecce","tex":"K_{-1}","aliases":"K_-1","display":"K_{-1}","kind":"use","excerpt":"claimAn endpoint mass is numerically delicate For the undeformed depth-zero measure, set ν_θ=ν+θδ_0, θ≥-1. Write h_j=∫ P_j(x)^2 dν(x) and K_m(x,y)=∑_j=0^mP_j(x)P_j(y)/h_j. The Uvarov formula~[source reference] is P_m^[θ](x)=P_m(x)- fracθ P_m(0)1+θ K_m-1(0,0)K_m-1(x,0), with K_-1=0 for degree zero. The mass at zero is recovered by 1≤frac1K_m(0,0)≤1+","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-15201584ff766791","tex":"K_4","aliases":"K_4","display":"K_4","kind":"use","excerpt":"K_4≈1.50393441996×10^11, 12/K_4≈7.9790713217×10^-11>10^-12.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ee970a7e26313cd3","tex":"K_d","aliases":"K_d","display":"K_d","kind":"use","excerpt":"ℓ_i(x)=∏_j≠ ifracx-y_jy_i-y_j, b_i=∏_j≠ ifracmax(y_j,1-y_j)|y_i-y_j|, K_d=∑_i=0^dn_i^4b_i^2.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-69a26fc25f751d91","tex":"K_{ij}","aliases":"K_ij","display":"K_{ij}","kind":"use","excerpt":"claimStrict positive complex kernel For distinct complex points s_i, K_ij=A(s_i+overlines_j) is strictly positive definite. A zero quadratic form would force a finite exponential sum to vanish at every λ_n. Their accumulation at zero invokes the identity theorem; derivatives at zero then give a nonsingular Vandermonde system. On the real axis, (-1)","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-46d21cf1bf7572c1","tex":"K_m","aliases":"K_m","display":"K_m","kind":"use","excerpt":"claimAn endpoint mass is numerically delicate For the undeformed depth-zero measure, set ν_θ=ν+θδ_0, θ≥-1. Write h_j=∫ P_j(x)^2 dν(x) and K_m(x,y)=∑_j=0^mP_j(x)P_j(y)/h_j. The Uvarov formula~[source reference] is P_m^[θ](x)=P_m(x)- fracθ P_m(0)1+θ K_m-1(0,0)K_m-1(x,0), with K_-1=0 for degree zero. The mass at zero is recovered by 1≤frac1K_m(0,0)≤1+","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3d46662b916fc6f7","tex":"K_{m-1}","aliases":"K_m-1","display":"K_{m-1}","kind":"use","excerpt":"claimAn endpoint mass is numerically delicate For the undeformed depth-zero measure, set ν_θ=ν+θδ_0, θ≥-1. Write h_j=∫ P_j(x)^2 dν(x) and K_m(x,y)=∑_j=0^mP_j(x)P_j(y)/h_j. The Uvarov formula~[source reference] is P_m^[θ](x)=P_m(x)- fracθ P_m(0)1+θ K_m-1(0,0)K_m-1(x,0), with K_-1=0 for degree zero. The mass at zero is recovered by 1≤frac1K_m(0,0)≤1+","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"startvolumeSequence atlasOEIS families and exact transforms lessonSEQUENCES / 01Two seeds, one Fibonacci recurrenceS01_fibonacci_lucas F_n+2=F_n+1+F_n, L_n+2=L_n+1+L_n, L_n^2-5F_n^2=4(-1)^n The two families share the same recurrence and have different seeds: (F_0,F_1)=(0,1) and (L_0,L_1)=(2,1). The right panel uses the same index in both families. ","count":36},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"B_n=H^nD^n(n+1)A_n∈ Z hboxfor all n.","count":12},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a7f7287d8f66cf96","tex":"L^{1-2a}","aliases":"L^1-2a","display":"L^{1-2a}","kind":"use","excerpt":"E_L(t)=e^tλ_L+1fracL^1-2a2a-1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"startvolumeSequence atlasOEIS families and exact transforms lessonSEQUENCES / 01Two seeds, one Fibonacci recurrenceS01_fibonacci_lucas F_n+2=F_n+1+F_n, L_n+2=L_n+1+L_n, L_n^2-5F_n^2=4(-1)^n The two families share the same recurrence and have different seeds: (F_0,F_1)=(0,1) and (L_0,L_1)=(2,1). The right panel uses the same index in both families. ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"startvolumeSequence atlasOEIS families and exact transforms lessonSEQUENCES / 01Two seeds, one Fibonacci recurrenceS01_fibonacci_lucas F_n+2=F_n+1+F_n, L_n+2=L_n+1+L_n, L_n^2-5F_n^2=4(-1)^n The two families share the same recurrence and have different seeds: (F_0,F_1)=(0,1) and (L_0,L_1)=(2,1). The right panel uses the same index in both families. ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fe3ce20fdccd5346","tex":"L_7","aliases":"L_7","display":"L_7","kind":"use","excerpt":"sonSEQUENCES / 01Two seeds, one Fibonacci recurrenceS01_fibonacci_lucas F_n+2=F_n+1+F_n, L_n+2=L_n+1+L_n, L_n^2-5F_n^2=4(-1)^n The two families share the same recurrence and have different seeds: (F_0,F_1)=(0,1) and (L_0,L_1)=(2,1). The right panel uses the same index in both families. Its alternating defect stays exactly four in magnitude. At n=7,","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a11ee0ad40de6dac","tex":"L_a","aliases":"L_a","display":"L_a","kind":"use","excerpt":"pageheadingPRIME SIEVES / CONSTRUCTIVE WITNESSEvery negative prime atom has a finite detector For an integer a≥2, let L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1), r_a=frac1-x_a+11-x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_si","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"-fraclog(λ_lfloor cjrfloor/λ_j)log c =β+O_Q,β,c(j^-1/2+j^-β).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"λ_j=lim_t→∞fraclog R_j(-t)t, w_j=lim_t→∞e^-λ_jtR_j(-t), R_j+1(s)=R_j(s)-w_je^-λ_js.","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3857473c194ade17","tex":"\\lim_{j\\to\\infty}","aliases":"lim_j→∞ lim to infty","display":"\\lim_{j\\to\\infty}","kind":"use","excerpt":"displaystyle β=-lim_j→∞fraclogλ_jlog j.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-dc6af7d77a1c39f5","tex":"\\lim_{R\\to\\infty}","aliases":"lim_R→∞ lim to infty","display":"\\lim_{R\\to\\infty}","kind":"use","excerpt":"lim_R→∞fracloglog M(R)log R=ρ.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-13d1e24391167efa","tex":"\\lim_{T\\to\\infty}","aliases":"lim_T→∞ lim to infty","display":"\\lim_{T\\to\\infty}","kind":"use","excerpt":"At zero the finite parts give I_K(0)=lim_T→∞[log E_N(T)-T^1/N]=-log N. Since (sinπ s/π)'_s=0=1, this equals Z_N'(0). Absolute convergence at a smaller real exponent would give a holomorphic series across 1/N, contradicting the pole and proving the exact abscissa.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-24fd66249142db1e","tex":"\\lim_{t\\to\\infty}","aliases":"lim_t→∞ lim to infty","display":"\\lim_{t\\to\\infty}","kind":"use","excerpt":"λ_j=lim_t→∞fraclog R_j(-t)t, w_j=lim_t→∞e^-λ_jtR_j(-t), R_j+1(s)=R_j(s)-w_je^-λ_js.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e5c19d75887aa542","tex":"\\liminf","aliases":"liminf","display":"\\liminf","kind":"use","excerpt":"c_-≤liminf_R→∞fraclog M(R)R^ρ ≤limsup_R→∞fraclog M(R)R^ρ≤ c_+.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-13bb5066d3ba56a0","tex":"\\liminf_{R\\to\\infty}","aliases":"liminf_R→∞ liminf to infty","display":"\\liminf_{R\\to\\infty}","kind":"use","excerpt":"c_-≤liminf_R→∞fraclog M(R)R^ρ ≤limsup_R→∞fraclog M(R)R^ρ≤ c_+.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c30606ee03663fb1","tex":"\\operatorname{Log}","aliases":"Log operatorname","display":"\\operatorname{Log}","kind":"use","excerpt":"Z_N(s)=∑_j e^-sLogw_j","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-973977200963bdff","tex":"\\log_{10}","aliases":"log_10 log","display":"\\log_{10}","kind":"use","excerpt":"lessonSEQUENCES / 02Longer recurrence memoryS02_tribonacci_tetranacci T_n=T_n-1+T_n-2+T_n-3, Q_n=∑_j=1^4Q_n-j The seed table fixes the OEIS convention. Tribonacci starts 0,0,1; tetranacci starts 0,0,0,1. The curves use log_10(1+a_n) so that zeros and rapidly growing terms fit together. The transform changes the plotted scale, not the integer sequen","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"U=V Longleftrightarrow {q,r}={a,b}.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"c_n≥fracq r^2n-12n+1c_n-1 Longrightarrow c_n≥fracq^n r^n^23·5⋯(2n+1).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","excerpt":"ight: initial atom masses. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_2-M_1^2, P_1=x-fracM_1M_0, P_2=x^2","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"s. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_2-M_1^2, P_1=x-fracM_1M_0, P_2=x^2-fracM_0M_3-M_1M_2Δx+ fr","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5a87813a12d8dc93","tex":"M_1^2","aliases":"M_1^2","display":"M_1^2","kind":"use","excerpt":"ight: initial atom masses. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_2-M_1^2, P_1=x-fracM_1M_0, P_2=x^2","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"ight: initial atom masses. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_2-M_1^2, P_1=x-fracM_1M_0, P_2=x^2","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-baad20238eecefda","tex":"M_2^2","aliases":"M_2^2","display":"M_2^2","kind":"use","excerpt":"s. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_2-M_1^2, P_1=x-fracM_1M_0, P_2=x^2-fracM_0M_3-M_1M_2Δx+ fr","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fa02e26ac15b5f1d","tex":"M_3","aliases":"M_3","display":"M_3","kind":"use","excerpt":"s. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_2-M_1^2, P_1=x-fracM_1M_0, P_2=x^2-fracM_0M_3-M_1M_2Δx+ fr","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4b91ccdd1e345e08","tex":"M_F","aliases":"M_F","display":"M_F","kind":"use","excerpt":"pageheadingDILATION / GROWTH THEOREMThe exact order has positive lower type The single-channel result extends to finite positive feedback banks. Here a bank has weights h_j>0 with ∑_jh_j=1, contractions 0<q_j,r_j<1, and normalized solution F'(z)=1+∑_jh_jF(q_jz)F(r_jz), F(0)=0. Write M(R)=M_F(R) and let q_j^ρ_j+r_j^ρ_j=1. Then, with ρ=max_jρ_j, 0<li","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-dc2d9714e8dfb1db","tex":"M_{F_i}","aliases":"M_F_i","display":"M_{F_i}","kind":"use","excerpt":"underlineτ_i=liminf_R→∞fraclog M_F_i(R)R^ρ, overlineτ_i=limsup_R→∞fraclog M_F_i(R)R^ρ.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b5ed0d1dc53e4c91","tex":"M_{i+j}","aliases":"M_i+j","display":"M_{i+j}","kind":"use","excerpt":"item[Hankel order and polynomial degree.] For moments M_j=∫ x^jddν, the order-m matrix is H_m=[M_i+j]_0≤ i,j<m. It tests quadratic forms of real polynomials of degree at most m-1. At depth k, the matrix is [M_k+i+j]_0≤ i,j<m. An order-five test therefore covers degree four, not degree five. item[Signed quadratic form.] For real polynomials, Q_ν(f)=","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bf2227566ef624f5","tex":"M_j","aliases":"M_j","display":"M_j","kind":"use","excerpt":"item[Hankel order and polynomial degree.] For moments M_j=∫ x^jddν, the order-m matrix is H_m=[M_i+j]_0≤ i,j<m. It tests quadratic forms of real polynomials of degree at most m-1. At depth k, the matrix is [M_k+i+j]_0≤ i,j<m. An order-five test therefore covers degree four, not degree five. item[Signed quadratic form.] For real polynomials, Q_ν(f)=","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f819825263110b36","tex":"M_{j+1}","aliases":"M_j+1","display":"M_{j+1}","kind":"use","excerpt":"Every limit exists and recovers exactly the j-th support frequency and weight. Indeed, if M_j+1=∑_k>jw_k, then for every t≥0,","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1357b55640fde97a","tex":"M_j^{p,\\lambda}","aliases":"M_j^p,λ lambda","display":"M_j^{p,\\lambda}","kind":"use","excerpt":"for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_polynomialsLeft: degree-two packets for p=2, depth zero. Right: initial atom masses. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e9577d4e4232e0c8","tex":"m_k","aliases":"m_k","display":"m_k","kind":"use","excerpt":"The monomial moment matrices also have infinite rank: m_k=(-1)^kA^(k)(0)=∑_jw_jλ_j^k, and for every nonzero polynomial P, ∑_jw_j|P(λ_j)|^2>0, because a nonzero polynomial cannot vanish at infinitely many distinct frequencies. All finite derivative Hankel matrices are strictly positive definite.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ef4662412ccdd49e","tex":"M_{k+i+j}","aliases":"M_k+i+j","display":"M_{k+i+j}","kind":"use","excerpt":"eheadingFAMILY 3 / TRANSPORTClassical operations with arithmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel determi","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9e659896aa55dd88","tex":"M_{k+j}","aliases":"M_k+j","display":"M_{k+j}","kind":"use","excerpt":"Depth-k moments are M_k+j. With v=π^-2, the even-zeta formula makes every D(4+2r) a rational multiple of v^r+2. Thus the moments belong to Q[v,λ_p:p∈ F], and monic orthogonal-polynomial coefficients belong to Q(v,λ_p:p∈ F). This is an exact field statement, with no decimal recognition step.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d9daaf9943072301","tex":"M_{T}","aliases":"M_T","display":"M_{T}","kind":"use","excerpt":"pageheadingFAMILY 1 / GROWTHA scale-balance atlas ρ(T_q,r)=ρ where q^ρ+r^ρ=1. Here M_f(R)=max_|z|=R|f(z)| and ρ(f)=limsup_R→∞loglog M_f(R)/log R is the order of an entire function. Moreover log M_T(R)=O(R^ρ): the upper type is finite. smallfig02_order_atlasThe displayed values numerically solve the proved characteristic equation. Colors saturate at","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-227fd8f55b493bd8","tex":"M_U","aliases":"M_U","display":"M_U","kind":"use","excerpt":"M_U(R)=max_|z|=R|U(z)|=U(R), M_V(R)=V(R).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9449462255135514","tex":"M_V","aliases":"M_V","display":"M_V","kind":"use","excerpt":"M_U(R)=max_|z|=R|U(z)|=U(R), M_V(R)=V(R).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9cc66676daa83a18","tex":"\\max_j","aliases":"max_j max","display":"\\max_j","kind":"use","excerpt":"Q=max_jmax(q_j,r_j), m=min_jmin(q_j,r_j), Δ=log(1/m), B_+=max_jlogfrac1q_jr_j+log2.","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-82dd2ca787a89f2f","tex":"\\max_{j\\text{ active}}","aliases":"max_j active max text","display":"\\max_{j\\text{ active}}","kind":"use","excerpt":"y many channels j=1,…,J, with h_j>0, ∑_jh_j=1, and 0<q_j,r_j<1. Define F_Θ'(z)=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz), F_Θ(0)=0. These taps act inside the nonlinear definition. With μ_1=∑_jh_jq_jr_j and μ_2=∑_jh_jq_jr_j(q_j^2+r_j^2), F_Θ(z)=z+fracμ_13z^3+fracμ_1μ_215z^5+O(z^7). If q_j^ρ_j+r_j^ρ_j=1, then ρ(F_Θ)=max_j activeρ_j. The positive weights alter local","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-34186ebfa8f11a0a","tex":"\\max_{j:h_j>0}","aliases":"max_j:h_j>0 max","display":"\\max_{j:h_j>0}","kind":"use","excerpt":"ρ=max_j:h_j>0ρ_j, q_j^ρ_j+r_j^ρ_j=1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a7db9030fadbda07","tex":"\\max_{|z|\\le R}","aliases":"max_|z|≤ R max le","display":"\\max_{|z|\\le R}","kind":"use","excerpt":"max_|z|≤ R|F(z)-F_N(z)| ≤ B(S)(frac RS)^2N+3.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3c2643cd3f986826","tex":"\\max_{|z|\\le x}","aliases":"max_|z|≤ x max le","display":"\\max_{|z|\\le x}","kind":"use","excerpt":"Analytic continuation by the integral equations expands each known disk by the factor 1/Q, so both solutions are entire. Positive coefficients give max_|z|≤ x|F(z)|=F(x) for either component. The integral equations therefore imply","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"(n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logarithm of 1-n^-β>0. The atom n=1 is excluded because its logarithm is undefined. At positive integer depths its omitted contribution would be zero. At depth zero the corresponding mass-one atom is absent. In particular, A(0)=Z_Q(2a)-1, where Z_Q(v)=∑_substackn≥1 (n,Q)=1μ(n)^2n","count":20},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"B_mA_m^-1/2. Every determinant root is real and below -1; possible coefficient poles occur there, although numerator cancellation is possible. fig05_sieve_boundaryThe line is a proved bound for p=2; dots are ordinary high-precision pencil roots. A defect λ=-1.001 eventually violates positivity even though it can pass low-order checks. The nearest r","count":10},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-755db37dcf979922","tex":"\\min_{0\\ne f\\in\\mathbb R[x],\\ \\deg f<m}","aliases":"min_0≠ f∈ R[x], deg f<m min ne in mathbb deg","display":"\\min_{0\\ne f\\in\\mathbb R[x],\\ \\deg f<m}","kind":"use","excerpt":"δ_m=min_0≠ f∈ R[x], deg f<m frac∫ f^2 dν_p-free∫ f^2 dν_p-divisible>0. tag6","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-60698b1979c563d1","tex":"\\min_j","aliases":"min_j min","display":"\\min_j","kind":"use","excerpt":"Q=max_jmax(q_j,r_j), m=min_jmin(q_j,r_j), Δ=log(1/m), B_+=max_jlogfrac1q_jr_j+log2.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ecb8ebf315534bb3","tex":"\\min_{\\substack{f\\in\\mathbb R[x]\\setminus\\{0\\}\\\\\\deg f<m}}","aliases":"min_substackf∈ R[x]∖{0} deg f<m min substack in mathbb setminus deg","display":"\\min_{\\substack{f\\in\\mathbb R[x]\\setminus\\{0\\}\\\\\\deg f<m}}","kind":"use","excerpt":"B_mA_m^-1/2. Every determinant root is real and below -1; possible coefficient poles occur there, although numerator cancellation is possible. fig05_sieve_boundaryThe line is a proved bound for p=2; dots are ordinary high-precision pencil roots. A defect λ=-1.001 eventually violates positivity even though it can pass low-order checks. The nearest r","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"|∑_n≥ Nc_nz^2n+1| ≤ frac|z|ρ^N1-ρ, |z|^2<2.","count":195},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-213d86a5e6b34393","tex":"N^{-1}","aliases":"N^-1","display":"N^{-1}","kind":"use","excerpt":"This equality follows by expanding the exponentials and using N^-1∑_jω^jm=1 when Nmid m, and zero otherwise. For real x≥0, the power series makes F_N(x)>0. Put","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c0769f9271add232","tex":"n^{-2}","aliases":"n^-2","display":"n^{-2}","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0a1ed04a6f0103b2","tex":"n^{-2a}","aliases":"n^-2a","display":"n^{-2a}","kind":"use","excerpt":"MonographPartIIArithmetic depth and signed measuresComplex parameters, positive kernels and the boundaries of positivity. Arithmetic at complex depth. pageheadingFAMILY 2Arithmetic energy at complex depth For real a>1/2, β>0, and integer Q≥1, define A_a,β,Q(s)=∑_substackn≥2 (n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logar","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-55d7cf56037490a0","tex":"n^{-4}","aliases":"n^-4","display":"n^{-4}","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":13},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-38855ce590112f03","tex":"n^{-s}","aliases":"n^-s","display":"n^{-s}","kind":"use","excerpt":"Z_F(s;λ) =∑_n≥1μ(n)^2n^-sκ_F(n;λ) =D(s)∏_p∈ F(1+fracλ_pp^s+1).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-965be0274c99995d","tex":"n^{-v}","aliases":"n^-v","display":"n^{-v}","kind":"use","excerpt":"(n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logarithm of 1-n^-β>0. The atom n=1 is excluded because its logarithm is undefined. At positive integer depths its omitted contribution would be zero. At depth zero the corresponding mass-one atom is absent. In particular, A(0)=Z_Q(2a)-1, where Z_Q(v)=∑_substackn≥1 (n,Q)=1μ(n)^2n","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-50b12bd693092eb6","tex":"n^{-\\beta}","aliases":"n^-β beta","display":"n^{-\\beta}","kind":"use","excerpt":"MonographPartIIArithmetic depth and signed measuresComplex parameters, positive kernels and the boundaries of positivity. Arithmetic at complex depth. pageheadingFAMILY 2Arithmetic energy at complex depth For real a>1/2, β>0, and integer Q≥1, define A_a,β,Q(s)=∑_substackn≥2 (n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logar","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"c_n≥fracq r^2n-12n+1c_n-1 Longrightarrow c_n≥fracq^n r^n^23·5⋯(2n+1).","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ac5e944c84a2e799","tex":"n^4","aliases":"n^4","display":"n^4","kind":"use","excerpt":"pageheadingRESOLUTION ATLAS / NEGATIVE ATOMSA degree-seven witness exposes a signed defect Let x_n=(1-n^-2)^2, and change the prime-two sieve strength to λ_2=-1-δ. For a real polynomial f, the associated quadratic form is Q_δ(f)=∑_substackn≥1 μ(n)^2=1 n odd fracf(x_n)^2n^4 -δ∑_substackn≥1 μ(n)^2=1 n even fracf(x_n)^2n^4. Every even squarefree atom ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4fa33e766db12578","tex":"n_+","aliases":"n_+","display":"n_+","kind":"use","excerpt":"Let S be exactly the set of all odd squarefree labels n≤ K, where K is an odd squarefree integer, and let n_+ be the next such label. Assume n_+>2, and define","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","excerpt":"Choose d+1 distinct odd squarefree labels n_0,…,n_d, including 1, and set y_i=x_n_i. Their Lagrange polynomials are","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"The logarithm is real. The omitted n=1 atom is not restored in any inverse formula. Let n_1<n_2<⋯ be the support integers, and abbreviate λ_j=λ_n_j, w_j=n_j^-2a. Then λ_1>λ_2>⋯>0, λ_j→0, and ∑_jw_j=M=Z_Q(2a)-1<∞, where","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b6b337ece4924770","tex":"n_2","aliases":"n_2","display":"n_2","kind":"use","excerpt":"The logarithm is real. The omitted n=1 atom is not restored in any inverse formula. Let n_1<n_2<⋯ be the support integers, and abbreviate λ_j=λ_n_j, w_j=n_j^-2a. Then λ_1>λ_2>⋯>0, λ_j→0, and ∑_jw_j=M=Z_Q(2a)-1<∞, where","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4d2462ef3372559a","tex":"n_d","aliases":"n_d","display":"n_d","kind":"use","excerpt":"Choose d+1 distinct odd squarefree labels n_0,…,n_d, including 1, and set y_i=x_n_i. Their Lagrange polynomials are","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"Choose d+1 distinct odd squarefree labels n_0,…,n_d, including 1, and set y_i=x_n_i. Their Lagrange polynomials are","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-46375f6853647c05","tex":"n_i^{-4}","aliases":"n_i^-4","display":"n_i^{-4}","kind":"use","excerpt":"|f(x)|^2≤ (∑_i=0^dn_i^-4f(y_i)^2) (∑_i=0^dn_i^4ℓ_i(x)^2) ≤ K_d∑_i=0^dn_i^-4f(y_i)^2.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-391ef349b60a7e10","tex":"n_i^4","aliases":"n_i^4","display":"n_i^4","kind":"use","excerpt":"ℓ_i(x)=∏_j≠ ifracx-y_jy_i-y_j, b_i=∏_j≠ ifracmax(y_j,1-y_j)|y_i-y_j|, K_d=∑_i=0^dn_i^4b_i^2.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"The logarithm is real. The omitted n=1 atom is not restored in any inverse formula. Let n_1<n_2<⋯ be the support integers, and abbreviate λ_j=λ_n_j, w_j=n_j^-2a. Then λ_1>λ_2>⋯>0, λ_j→0, and ∑_jw_j=M=Z_Q(2a)-1<∞, where","count":13},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-74db9ad57582bdaf","tex":"n_j^{-2a}","aliases":"n_j^-2a","display":"n_j^{-2a}","kind":"use","excerpt":"The logarithm is real. The omitted n=1 atom is not restored in any inverse formula. Let n_1<n_2<⋯ be the support integers, and abbreviate λ_j=λ_n_j, w_j=n_j^-2a. Then λ_1>λ_2>⋯>0, λ_j→0, and ∑_jw_j=M=Z_Q(2a)-1<∞, where","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9ec96d140463b343","tex":"n_j^{-\\beta}","aliases":"n_j^-β beta","display":"n_j^{-\\beta}","kind":"use","excerpt":"Let the support integers be n_1<n_2<⋯, frequencies λ_j=-2log(1-n_j^-β), and weights w_j=n_j^-2a. Starting at R_1=A, peel the ordered atoms: λ_j=lim_t→∞fraclog R_j(-t)t, w_j=lim_t→∞e^-λ_jtR_j(-t), R_j+1(s)=R_j(s)-w_je^-λ_js. Every limit follows from the isolated largest remaining frequency and the positive summable tail. Thus the exact function dete","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0cc66dac9fca8372","tex":"N_Q","aliases":"N_Q","display":"N_Q","kind":"use","excerpt":"N_Q(x):=#{2≤ n≤ x:μ(n)^2=1,(n,Q)=1} =δ_Qx+O_Q(sqrtx).","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"A prime p lies in the recovered integer support exactly when pnmid Q: primes are squarefree, and no other restriction is imposed. Hence","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-97d2a5d7e3358759","tex":"\\mathrm{non}","aliases":"non mathrm","display":"\\mathrm{non}","kind":"use","excerpt":"pageheadingFAMILY 3 / BOUNDARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"nctionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T_q,r(z)=∑_n≥0c_nz^2n+1, c_0=1, coefficient matching gives c_n=frac12n+1∑_i=0^n-1 q^2i+1r^2(n-1-i)+1c_ic_n-1-i, n≥1. In particular, T_q,r(","count":34},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"B_n=H^nD^n(n+1)A_n∈ Z hboxfor all n.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-19709a166b02761d","tex":"O_{a,\\beta,Q,d}","aliases":"O_a,β,Q,d beta","display":"O_{a,\\beta,Q,d}","kind":"use","excerpt":"At real positive depth, the matrix (A_χ,ψ(s)) on any d distinct characters is positive semidefinite and has identical diagonal A_a,β,Q(s). Its normalized matrix tends to I_d, with operator norm error O_a,β,Q,d(s^-1/(2β)). Indeed, the off-diagonal entry bound controls the maximum row and column sums, and thus the operator norm. This gives quantitati","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-dc200e782460d942","tex":"O_{a,\\beta,Q,\\epsilon}","aliases":"O_a,β,Q,ϵ beta epsilon","display":"O_{a,\\beta,Q,\\epsilon}","kind":"use","excerpt":"Let α=frac2a-1β, δ_Q=frac1ζ(2)∏_pmid Qfracpp+1. As |s|→∞ in any closed sector |arg s|≤π/2-ϵ with 0<ϵ<π/2, A_a,β,Q(s)=fracδ_QβB(α,2s+1) +O_a,β,Q,ϵ(|s|^-(2a-1/2)/β). In particular, A_a,β,Q(s)∼fracδ_QΓ(α)β(2s)^-α. The principal power is used in the right half-plane. The beta expression is an asymptotic continuous-density model, not an identity with th","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8261019e380aef3c","tex":"O_{a,\\beta,Q,\\varepsilon}","aliases":"O_a,β,Q,ε beta varepsilon","display":"O_{a,\\beta,Q,\\varepsilon}","kind":"use","excerpt":"A_χ,ψ(s) =O_a,β,Q,ε(|s|^-(2a-1/2)/β), fracA_χ,ψ(s)A_a,β,Q(s) =O_a,β,Q,ε(|s|^-1/(2β)).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1fb7f7783d1fc8bb","tex":"O_{a,\\beta,\\varepsilon}","aliases":"O_a,β,ε beta varepsilon","display":"O_{a,\\beta,\\varepsilon}","kind":"use","excerpt":"which is O_a,β,ε(|s|^-γ) because u≥ |s|sinε. The principal diagonal has the nonzero leading term δ_QΓ(α)β^-1(2s)^-α, α=(2a-1)/β, by the base sector theorem. Its modulus is bounded below by a positive fixed-sector multiple of |s|^-α for sufficiently large |s|. Division gives the second bound, since γ-α=1/(2β).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6219c48933f6ee3b","tex":"O_Q","aliases":"O_Q","display":"O_Q","kind":"use","excerpt":"claimProof mechanism Squarefree inclusion--exclusion gives ∑_substackn≤ x (n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0615dad65c053275","tex":"O_{Q,\\beta}","aliases":"O_Q,β beta","display":"O_{Q,\\beta}","kind":"use","excerpt":"n_j=fracjδ_Q(1+O_Q(j^-1/2)), λ_j=2δ_Q^βj^-β (1+O_Q,β(j^-1/2+j^-β)).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1d9103717d98beab","tex":"O_{Q,\\beta,c}","aliases":"O_Q,β,c beta","display":"O_{Q,\\beta,c}","kind":"use","excerpt":"-fraclog(λ_lfloor cjrfloor/λ_j)log c =β+O_Q,β,c(j^-1/2+j^-β).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f88991f9db432114","tex":"\\overline t","aliases":"overline t overline","display":"\\overline t","kind":"use","excerpt":"K(s,t)=A(s+overline t).","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8c0bc687c4827788","tex":"\\overline{s_j}","aliases":"overlines_j overline","display":"\\overline{s_j}","kind":"use","excerpt":"claimStrict positive complex kernel For distinct complex points s_i, K_ij=A(s_i+overlines_j) is strictly positive definite. A zero quadratic form would force a finite exponential sum to vanish at every λ_n. Their accumulation at zero invokes the identity theorem; derivatives at zero then give a nonsingular Vandermonde system. On the real axis, (-1)","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0f205f23f823611e","tex":"\\overline\\tau","aliases":"overlineτ overline tau","display":"\\overline\\tau","kind":"use","excerpt":"underlineτ_i=liminf_R→∞fraclog M_F_i(R)R^ρ, overlineτ_i=limsup_R→∞fraclog M_F_i(R)R^ρ.","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9ef920776643fcd9","tex":"\\overline{\\boldsymbol\\tau}","aliases":"overlineτ overline boldsymbol tau","display":"\\overline{\\boldsymbol\\tau}","kind":"use","excerpt":"For F_1=U and F_2=V, define underlineτ_i=liminf_R→∞fraclog M_F_i(R)R^ρ, overlineτ_i=limsup_R→∞fraclog M_F_i(R)R^ρ. If P(ρ)w=w with w>0, then both type vectors have precisely these Perron proportions: underlineτ=underlineτ w, overlineτ=overlineτ w, qquad0<underlineτ≤overlineτ<∞. For the worked pair, both the lower type and the upper type of V are tw","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-859226a60f9cdd7e","tex":"\\overline\\tau_1","aliases":"overlineτ_1 overline tau","display":"\\overline\\tau_1","kind":"use","excerpt":"(underlineτ_1,underlineτ_2)^T · =underlineτ w, (overlineτ_1,overlineτ_2)^T · =overlineτ w.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b82a1421af003199","tex":"\\overline\\tau_2","aliases":"overlineτ_2 overline tau","display":"\\overline\\tau_2","kind":"use","excerpt":"(underlineτ_1,underlineτ_2)^T · =underlineτ w, (overlineτ_1,overlineτ_2)^T · =overlineτ w.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4eb8f4d65bb00fe7","tex":"\\overline\\tau_i","aliases":"overlineτ_i overline tau","display":"\\overline\\tau_i","kind":"use","excerpt":"underlineτ_i=liminf_R→∞fraclog M_F_i(R)R^ρ, overlineτ_i=limsup_R→∞fraclog M_F_i(R)R^ρ.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-55dc76c431a12088","tex":"\\overline\\psi","aliases":"overlineψ overline psi","display":"\\overline\\psi","kind":"use","excerpt":"Let χ,ψ be Dirichlet characters modulo the same fixed integer Q, extended by zero outside the units, and put η=χoverlineψ. The base extension is","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =∑_m≡ j (b)a_mfracz^mm!.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7bd100089ec86a9a","tex":"p^{-2}","aliases":"p^-2","display":"p^{-2}","kind":"use","excerpt":"λ_p=-2log(1-p^-2),","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c87fb295f906ab9d","tex":"p^{-2a}","aliases":"p^-2a","display":"p^{-2a}","kind":"use","excerpt":"There is no n=1 term in this difference. Bound the k-th derivative of every exponential by λ_p^ke^Rλ_p and sum. Since λ_p→0 and p^-2a→0, the bound tends to zero. This proves the derivative convergence and the claimed instability.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-909419ab57177b2a","tex":"p^{-4}","aliases":"p^-4","display":"p^{-4}","kind":"use","excerpt":"deg f<m frac∫ f^2 dν_nonp∫ f^2 dν_p>0, δ_mdownarrow0. To prove the limit, approximate a continuous bump supported at the isolated node x_p by polynomials on [0,1]. The numerator tends to zero while the denominator retains the p^-4 atom. Hence every λ<-1 eventually produces a negative finite-degree quadratic form.","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-52f8ea97bf828417","tex":"p^{-4-2k}","aliases":"p^-4-2k","display":"p^{-4-2k}","kind":"use","excerpt":"This is the inherited arithmetic inverse theorem, now read as a precision requirement. For fixed R, the envelope is of order p^-4-2k. Each additional derivative suppresses the large-prime difference even further in absolute units. This does not rule out methods that rescale derivatives and account for their corresponding noise amplification.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6c40a8dc2cfbb534","tex":"p^{-v}","aliases":"p^-v","display":"p^{-v}","kind":"use","excerpt":"(n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logarithm of 1-n^-β>0. The atom n=1 is excluded because its logarithm is undefined. At positive integer depths its omitted contribution would be zero. At depth zero the corresponding mass-one atom is absent. In particular, A(0)=Z_Q(2a)-1, where Z_Q(v)=∑_substackn≥1 (n,Q)=1μ(n)^2n","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"D=k^2u_2-p^2v_2 =fracpkar15(q^2-b^2)(qb-ar).","count":16},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-16bf1666da550802","tex":"p^{2a}","aliases":"p^2a","display":"p^{2a}","kind":"use","excerpt":"displaystyle sup_|s|≤ R|A_a,β,Q^(k)(s)-A_a,β,Qp^(k)(s)| ≤ λ_p^k e^Rλ_p fracZ_Q(2a)p^2a+1.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-578b7099ecb3368e","tex":"p^3","aliases":"p^3","display":"p^3","kind":"use","excerpt":"he common order. The first two nontrivial coefficients and the growth law force B^2s^2=p, s^2=u/2, 2s^ρ=q^ρ+r^ρ=1. Consequently 2(fracq^2+r^22)^ρ/2=q^ρ+r^ρ. For ρ≠2, strict convexity or concavity of x^ρ/2 forces q=r. For the exceptional case ρ=2, we have u=1, s^2=1/2, B^2=2p. The two z^7 coefficients become fracp^3(3-p^2)315 and frac11p^31260. Thei","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8cdfda19e4c4c67b","tex":"p^4","aliases":"p^4","display":"p^4","kind":"use","excerpt":"For p=2, test with f(x)=x(1-x)^m-2, m≥2: δ_m≤frac409681(frac16M_017-1) (frac272567)^2(m-2), M_0=105π^-4. This follows from x_2=9/16 and x_n≥64/81 at nonzero odd nodes. Exactly δ_1=p^4; the limit is an infinite-resolution phenomenon.","count":24},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-965a20d9b2e8c222","tex":"p^{4+2r}","aliases":"p^4+2r","display":"p^{4+2r}","kind":"use","excerpt":"for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_polynomialsLeft: degree-two packets for p=2, depth zero. Right: initial atom masses. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta ","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-360f794da8094012","tex":"p^s","aliases":"p^s","display":"p^s","kind":"use","excerpt":"Z_F(s;λ) =∑_n≥1μ(n)^2n^-sκ_F(n;λ) =D(s)∏_p∈ F(1+fracλ_pp^s+1).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-040e8315926ac941","tex":"p^\\rho","aliases":"p^ρ rho","display":"p^\\rho","kind":"use","excerpt":"Here p(R-1) tends to infinity continuously and [p(R-1)]^ρ/R^ρ→ p^ρ, which justifies each scaled liminf. The liminf of a sum is at least the sum of the liminfs. All quantities are positive and finite by Theorem F.2.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7cd48fb48ab38484","tex":"p_*","aliases":"p_*","display":"p_*","kind":"use","excerpt":"ion. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β). Then A(s)=∑_n∈ S_Qw_ne^-λ_ns, w_n=n^-2a. The positive weights are summable, and the distinct support frequencies decrease to zero. If p_* is the least prime n","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-53f392bf72bc8717","tex":"p_*^{-\\beta}","aliases":"p_*^-β beta","display":"p_*^{-\\beta}","kind":"use","excerpt":"ights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β). Then A(s)=∑_n∈ S_Qw_ne^-λ_ns, w_n=n^-2a. The positive weights are summable, and the distinct support frequencies decrease to zero. If p_* is the least prime not dividing Q, then τ=-2log(1-p_*^-β), M=Z_Q(2a)-","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"s. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_2-M_1^2, P_1=x-fracM_1M_0, P_2=x^2-fracM_0M_3-M_1M_2Δx+ fr","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fd85a1525a70d0a8","tex":"p_{10}","aliases":"p_10","display":"p_{10}","kind":"use","excerpt":"Bernoulli data need both coordinatesS11_bernoulli_pair fracte^t-1=∑_n≥0B_nfract^nn!, B_n=fracp_nq_n, q_n>0 The top panel shows signed numerators on a symmetric logarithmic scale. The lower panel shows positive denominators on a logarithmic scale. Fractions are reduced. A zero Bernoulli number is stored as 0/1, and this guide uses B_1=-1/2. At n=10,","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"s. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_2-M_1^2, P_1=x-fracM_1M_0, P_2=x^2-fracM_0M_3-M_1M_2Δx+ fr","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f19b41a6c37084e2","tex":"P_{b,j}","aliases":"P_b,j","display":"P_{b,j}","kind":"use","excerpt":"pageheadingEXACT LABORATORY / PROJECTORSPolyphase sections preserve the full sequence For an ordinary generating function G(z)=∑_m≥0a_mz^m and integer b≥1, write G(z)=∑_j=0^b-1z^jG_j(z^b), G_j(t)=∑_n≥0a_bn+jt^n. The root projector is P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =z^jG_j(z^b), ω=e^2π i/b. The same projector works on an EGF, but the section ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0640bf40d2405aad","tex":"\\mathcal P_{b,j}","aliases":"P_b,j mathcal","display":"\\mathcal P_{b,j}","kind":"use","excerpt":"P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =∑_m≡ j (b)a_mfracz^mm!.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bae309bd792c4b0b","tex":"\\mathcal P_{b,l}","aliases":"P_b,l mathcal","display":"\\mathcal P_{b,l}","kind":"use","excerpt":"∑_j=0^b-1 P_b,j=I, P_b,j P_b,l=δ_jl P_b,j.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"claimNo polynomial-coefficient linear ODE Suppose ∑_j=0^rP_j(s)A^(j)(s)=0, where the P_j are polynomials and are not all zero. At s=-t, divide by the exponential of the first support atom. The remaining tail is exponentially smaller, so the polynomial ∑_jP_j(-t)(-λ_p_*)^j tends to zero exponentially and must vanish identically. Remove that atom and","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"claimAn endpoint mass is numerically delicate For the undeformed depth-zero measure, set ν_θ=ν+θδ_0, θ≥-1. Write h_j=∫ P_j(x)^2 dν(x) and K_m(x,y)=∑_j=0^mP_j(x)P_j(y)/h_j. The Uvarov formula~[source reference] is P_m^[θ](x)=P_m(x)- fracθ P_m(0)1+θ K_m-1(0,0)K_m-1(x,0), with K_-1=0 for degree zero. The mass at zero is recovered by 1≤frac1K_m(0,0)≤1+","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1e56343dd5d7252d","tex":"P_{m+1}^{(k)}","aliases":"P_m+1^(k)","display":"P_{m+1}^{(k)}","kind":"use","excerpt":"pageheadingFAMILY 3 / TRANSPORTClassical operations with arithmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel dete","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e6b46acea5ed3c84","tex":"P_m^{(k)}","aliases":"P_m^(k)","display":"P_m^{(k)}","kind":"use","excerpt":"pageheadingFAMILY 3 / TRANSPORTClassical operations with arithmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel dete","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-29c1d5aa0e5fdb79","tex":"P_m^{(k+1)}","aliases":"P_m^(k+1)","display":"P_m^{(k+1)}","kind":"use","excerpt":"pageheadingFAMILY 3 / TRANSPORTClassical operations with arithmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel dete","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b5363bee92466360","tex":"P_m^{(k;F,\\boldsymbol\\lambda)}","aliases":"P_m^(k;F,λ) boldsymbol lambda","display":"P_m^{(k;F,\\boldsymbol\\lambda)}","kind":"use","excerpt":"The positive monic families P_m^(k;F,λ) consequently exist at all degrees throughout the closed orthant. Their Hankel determinants never vanish there, including at simultaneous deletion endpoints. Each finite-degree coefficient is analytic in a neighborhood of every point of that orthant. This neighborhood is a finite-degree statement; the next the","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-24ad2969340637f1","tex":"P_m^{(k;p,\\lambda)}","aliases":"P_m^(k;p,λ) lambda","display":"P_m^{(k;p,\\lambda)}","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5cb9ef85b0312505","tex":"P_m^{[\\theta]}","aliases":"P_m^[θ] theta","display":"P_m^{[\\theta]}","kind":"use","excerpt":"claimAn endpoint mass is numerically delicate For the undeformed depth-zero measure, set ν_θ=ν+θδ_0, θ≥-1. Write h_j=∫ P_j(x)^2 dν(x) and K_m(x,y)=∑_j=0^mP_j(x)P_j(y)/h_j. The Uvarov formula~[source reference] is P_m^[θ](x)=P_m(x)- fracθ P_m(0)1+θ K_m-1(0,0)K_m-1(x,0), with K_-1=0 for degree zero. The mass at zero is recovered by 1≤frac1K_m(0,0)≤1+","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-928f913279be127c","tex":"P_{S,3}","aliases":"P_S,3","display":"P_{S,3}","kind":"use","excerpt":"and P_S,3 has degree seven. The exact rational right side of (RA-10) is strictly negative and is approximately","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-49800535bee1f69e","tex":"P_{S,m}","aliases":"P_S,m","display":"P_{S,m}","kind":"use","excerpt":"P_S,m(x)=∏_n∈ Sfracx-x_nx_2-x_n (frac1-x1-x_2)^m, C_S=∏_n∈ Sfrac1-x_n|x_2-x_n|, r_S=frac1-x_n_+1-x_2<1.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"u,v≥ n, u≡ v≡ npmod2, u+v≤ n(n+1).","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-71a98ec1e2bb9f0a","tex":"\\preceq","aliases":"preceq","display":"\\preceq","kind":"use","excerpt":"0preceq c_n(h)preceq d_n(Q)(∑_jh_j)^n.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-79b76a2e983b14b5","tex":"\\mathrm{prime}","aliases":"prime mathrm","display":"\\mathrm{prime}","kind":"use","excerpt":"displaystyle rad(Q)=∏_substackp prime p∉{n_j:j≥1}p.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"ryS02_tribonacci_tetranacci T_n=T_n-1+T_n-2+T_n-3, Q_n=∑_j=1^4Q_n-j The seed table fixes the OEIS convention. Tribonacci starts 0,0,1; tetranacci starts 0,0,0,1. The curves use log_10(1+a_n) so that zeros and rapidly growing terms fit together. The transform changes the plotted scale, not the integer sequence. Tribonacci gives T_5=2+1+1=4. Tetranac","count":156},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"two packets for p=2, depth zero. Right: initial atom masses. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0c370aa793b02aac","tex":"Q'","aliases":"Q'","display":"Q'","kind":"use","excerpt":"a=a', β=β', rad(Q)=rad(Q').","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-24f6e82fabf9c91d","tex":"Q^{-K}","aliases":"Q^-K","display":"Q^{-K}","kind":"use","excerpt":"claimExplicit upper constants are available too Set Q=max_jmax(q_j,r_j), m=min_jmin(q_j,r_j), Δ=log(1/m), and B_+=max_jlog(1/(q_jr_j))+log2. Choose an integer K with Q^-K≥3/m and put V_K=(2^K+1-1)log2+(2^K+1-K-2)log(1/Q), c_+=e^-ρ(V_K+1+Δ+B_+). Then log M(R)≤ c_+R^ρ-log R-B_+ for R≥ e. The tangent majorant M(1)<2, followed by the integral recurrenc","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-560acf983b20269d","tex":"Q^{-k}","aliases":"Q^-k","display":"Q^{-k}","kind":"use","excerpt":"These constants are intentionally conservative, and require no numerical evaluation of an unknown solution. The tangent majorant gives M(1)<2. At R_k=Q^-k, the integral equation gives","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-269a2b88c80612e3","tex":"Q^{-N}","aliases":"Q^-N","display":"Q^{-N}","kind":"use","excerpt":"Choose an integer N≥0 with Q^-N≥3/m, and put","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"D=k^2u_2-p^2v_2 =fracpkar15(q^2-b^2)(qb-ar).","count":17},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-648f67b1c3672456","tex":"q^{2i+1}","aliases":"q^2i+1","display":"q^{2i+1}","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":10},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d2d3c9a0c830bb0d","tex":"q^3","aliases":"q^3","display":"q^3","kind":"use","excerpt":"A_0= · 1, A_1= · 2qr, A_2= · 8q^2r^2(q^2+r^2), A_3= · q^3r^3[48(q^2+r^2)(q^4+r^4)+80q^2r^2].","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6e535c3600229c66","tex":"q^4","aliases":"q^4","display":"q^4","kind":"use","excerpt":"A_0= · 1, A_1= · 2qr, A_2= · 8q^2r^2(q^2+r^2), A_3= · q^3r^3[48(q^2+r^2)(q^4+r^4)+80q^2r^2].","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1e115d96b6a707e0","tex":"Q^k","aliases":"Q^k","display":"Q^k","kind":"use","excerpt":"Start at the first member Q^kx<1 of this finite contraction ladder, use x/(1-x^2) there, and repeatedly apply B(x)=x(1+B(Qx)^2) backward. This defines the rational pathdisk_majorant bound for rational parameters and radius. The implementation has explicit recursion and integer size guards, which raise errors rather than returning a weakened result.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-102bb645d8547d75","tex":"q^n","aliases":"q^n","display":"q^n","kind":"use","excerpt":"c_n≥fracq r^2n-12n+1c_n-1 Longrightarrow c_n≥fracq^n r^n^23·5⋯(2n+1).","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-be35f6b052e45fbe","tex":"q^{n/(2n+1)}","aliases":"q^n/(2n+1)","display":"q^{n/(2n+1)}","kind":"use","excerpt":"c_n^1/(2n+1)≥ q^n/(2n+1)r^n^2/(2n+1)(2n+1)^-n/(2n+1)⟶∞.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e7e5525bc1e1d3e7","tex":"q^s","aliases":"q^s","display":"q^s","kind":"use","excerpt":"P(s)=q^s · r^s a^s · b^s.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-766ffac929dd4972","tex":"q^u","aliases":"q^u","display":"q^u","kind":"use","excerpt":"Polynomial theorem. For every n, A_n∈ Z[q,r], all its nonzero coefficients are positive integers, and A_n(q,r)=A_n(r,q). Every monomial q^u r^v satisfies","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9f065c2f3b34a36a","tex":"Q_*","aliases":"Q_*","display":"Q_*","kind":"use","excerpt":"For a rigorous propagation, put Q_* =max(q,r)<1 and d=(1-Q_*)B+Q_*L>0. Every child of every R≥ B is at most R-d. Induction on the intervals [B+kd,B+(k+1)d], beginning with the already controlled seed, proves the comparison for all larger radii. Continuity deals with common endpoints. All power comparisons use only monotonicity of x^ρ for positive x","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"c_n(h,q,r)∈ Q[h_1,…,h_J,q_1,r_1,…,q_J,r_J].","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4251a20a2f66fb5f","tex":"q_{10}","aliases":"q_10","display":"q_{10}","kind":"use","excerpt":"need both coordinatesS11_bernoulli_pair fracte^t-1=∑_n≥0B_nfract^nn!, B_n=fracp_nq_n, q_n>0 The top panel shows signed numerators on a symmetric logarithmic scale. The lower panel shows positive denominators on a logarithmic scale. Fractions are reduced. A zero Bernoulli number is stored as 0/1, and this guide uses B_1=-1/2. At n=10, p_10=5 and q_1","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a98885c2bb3e2993","tex":"Q_{10^{-12}}","aliases":"Q_10^-12","display":"Q_{10^{-12}}","kind":"use","excerpt":"Q_10^-12(P_S,3)≤-5.8845708899×10^-14.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-11ceb9b5844e4bb9","tex":"Q_7","aliases":"Q_7","display":"Q_7","kind":"use","excerpt":"ryS02_tribonacci_tetranacci T_n=T_n-1+T_n-2+T_n-3, Q_n=∑_j=1^4Q_n-j The seed table fixes the OEIS convention. Tribonacci starts 0,0,1; tetranacci starts 0,0,0,1. The curves use log_10(1+a_n) so that zeros and rapidly growing terms fit together. The transform changes the plotted scale, not the integer sequence. Tribonacci gives T_5=2+1+1=4. Tetranac","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a4591bb144ad9c3f","tex":"Q_c","aliases":"Q_c","display":"Q_c","kind":"use","excerpt":"Q_c[u,v](z)=c_1u(qz)v(rz)+c_2u(az)v(bz).","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1eecc108797713f0","tex":"q_J","aliases":"q_J","display":"q_J","kind":"use","excerpt":"c_n(h,q,r)∈ Q[h_1,…,h_J,q_1,r_1,…,q_J,r_J].","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-25350247db137b17","tex":"q_j^2","aliases":"q_j^2","display":"q_j^2","kind":"use","excerpt":"pageheadingFAMILY 1 / TAPSFeedback banks and weak-channel sensitivity Choose finitely many channels j=1,…,J, with h_j>0, ∑_jh_j=1, and 0<q_j,r_j<1. Define F_Θ'(z)=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz), F_Θ(0)=0. These taps act inside the nonlinear definition. With μ_1=∑_jh_jq_jr_j and μ_2=∑_jh_jq_jr_j(q_j^2+r_j^2), F_Θ(z)=z+fracμ_13z^3+fracμ_1μ_215z^5+O(z^7).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-62c4cef1693f78c7","tex":"q_j^\\rho","aliases":"q_j^ρ rho","display":"q_j^\\rho","kind":"use","excerpt":"On the seed, G(t)≤ V_N+1+Δ+B_+≤ c_+e^ρ t. Each channel satisfies q_j^ρ+r_j^ρ≤1. Induction in intervals of length min_jmin(a_j,b_j)>0 therefore propagates the seed comparison, giving the explicit upper bound","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d3dc262f47b93506","tex":"q_j^{\\rho_j}","aliases":"q_j^ρ_j rho","display":"q_j^{\\rho_j}","kind":"use","excerpt":"vity Choose finitely many channels j=1,…,J, with h_j>0, ∑_jh_j=1, and 0<q_j,r_j<1. Define F_Θ'(z)=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz), F_Θ(0)=0. These taps act inside the nonlinear definition. With μ_1=∑_jh_jq_jr_j and μ_2=∑_jh_jq_jr_j(q_j^2+r_j^2), F_Θ(z)=z+fracμ_13z^3+fracμ_1μ_215z^5+O(z^7). If q_j^ρ_j+r_j^ρ_j=1, then ρ(F_Θ)=max_j activeρ_j. The positive ","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"For rational channel tuples (h_t,q_t,r_t), use","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-aa7fbe64d6b9e7ee","tex":"Q_\\delta","aliases":"Q_δ delta","display":"Q_\\delta","kind":"use","excerpt":"Q_δ(f)=∑_substackn≥1 μ(n)^2=1 n oddn^-4f(x_n)^2 -δ∑_substackn≥1 μ(n)^2=1 n evenn^-4f(x_n)^2.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-576864794019f623","tex":"Q_\\nu","aliases":"Q_ν nu","display":"Q_\\nu","kind":"use","excerpt":"item[Signed quadratic form.] For real polynomials, Q_ν(f)=∫ f(x)^2ddν(x). On a positive measure it is a squared norm. On a signed measure it may be negative and is not a norm. item[Orbit spectral zeta Z_N.] Let d_N(w)=∑_n≥0(-1)^nw^n/(Nn)! and D_N(z)=d_N(z^N), N≥2. The series Z_N(s)=∑_jw_j^-s initially runs over the zeros w_j of d_N, counted with mu","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"B_mA_m^-1/2. Every determinant root is real and below -1; possible coefficient poles occur there, although numerator cancellation is possible. fig05_sieve_boundaryThe line is a proved bound for p=2; dots are ordinary high-precision pencil roots. A defect λ=-1.001 eventually violates positivity even though it can pass low-order checks. The nearest r","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"log M(R)≥frac R2exp(fracq^3R^28-frac q2).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"nctionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T_q,r(z)=∑_n≥0c_nz^2n+1, c_0=1, coefficient matching gives c_n=frac12n+1∑_i=0^n-1 q^2i+1r^2(n-1-i)+1c_ic_n-1-i, n≥1. In particular, T_q,r(","count":12},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2c0f61e4624899ea","tex":"r^{2(n-1-i)+1}","aliases":"r^2(n-1-i)+1","display":"r^{2(n-1-i)+1}","kind":"use","excerpt":"on is symmetric in q,r, but unequal scales retain left/right address information. fig01_asymmetric_functionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T_q,r(z)=∑_n≥0c_nz^2n+1, c_0=1, coeff","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fdfc77ab4e60af66","tex":"r^{2(n-i)+1}","aliases":"r^2(n-i)+1","display":"r^{2(n-i)+1}","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9ca491a7c7512f77","tex":"r^{2j+1}","aliases":"r^2j+1","display":"r^{2j+1}","kind":"use","excerpt":"claimA combinatorial address formula Sum over full ordered binary trees with n internal vertices. A vertex with i internal vertices in its left subtree and j in its right contributes fracq^2i+1r^2j+12(i+j+1)+1. Multiply these factors over internal vertices, then sum over trees. Root splitting reproduces the coefficient recurrence exactly. Unequal c","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1964c771ef1e28bc","tex":"R^{2n+1}","aliases":"R^2n+1","display":"R^{2n+1}","kind":"use","excerpt":"∑_n≥0|c_n(h,q,r)|R^2n+1 ≤ W^-1/2f_Q(sqrt W R)<∞.","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-462a1c9ba944afc0","tex":"r^{2n-1}","aliases":"r^2n-1","display":"r^{2n-1}","kind":"use","excerpt":"c_n≥fracq r^2n-12n+1c_n-1 Longrightarrow c_n≥fracq^n r^n^23·5⋯(2n+1).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-96dee922af39121c","tex":"r^3","aliases":"r^3","display":"r^3","kind":"use","excerpt":"A_0= · 1, A_1= · 2qr, A_2= · 8q^2r^2(q^2+r^2), A_3= · q^3r^3[48(q^2+r^2)(q^4+r^4)+80q^2r^2].","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8477f8563980f26b","tex":"r^4","aliases":"r^4","display":"r^4","kind":"use","excerpt":"A_0= · 1, A_1= · 2qr, A_2= · 8q^2r^2(q^2+r^2), A_3= · q^3r^3[48(q^2+r^2)(q^4+r^4)+80q^2r^2].","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c63c0b8029ca153d","tex":"r^6","aliases":"r^6","display":"r^6","kind":"use","excerpt":"claimThe certificate uses rational arithmetic throughout Take S={1,3,5,7} and define P(x)=∏_n∈ Sfracx-x_nx_2-x_n (frac1-x1-x_2)^3. Its degree is seven. All remaining positive nodes begin at n=11. With C=756482048/20777538825 and r=3856/102487, a complete tail bound gives Q_10^-12(P)≤-frac10^-1216 +fracC^2r^63·7^3 <-5.8845708899×10^-14. The inequali","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c4179ac2053e2353","tex":"R^m","aliases":"R^m","display":"R^m","kind":"use","excerpt":"For every omitted exponent m≥2N+3, use R^m≤ (R/S)^2N+3S^m, sum positive terms, and bound their sum by F(S)≤ B(S). No extra geometric denominator is needed. For rational x≥0, pathcertified_evaluation returns the exact interval [F_N(x),F_N(x)+B(S)(x/S)^2N+3] for each component.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a1db2dbb975d8058","tex":"r^{n^2}","aliases":"r^n^2","display":"r^{n^2}","kind":"use","excerpt":"c_n≥fracq r^2n-12n+1c_n-1 Longrightarrow c_n≥fracq^n r^n^23·5⋯(2n+1).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-044df6ee7258c8df","tex":"r^{n^2/(2n+1)}","aliases":"r^n^2/(2n+1)","display":"r^{n^2/(2n+1)}","kind":"use","excerpt":"c_n^1/(2n+1)≥ q^n/(2n+1)r^n^2/(2n+1)(2n+1)^-n/(2n+1)⟶∞.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d6f8e39a91067d23","tex":"r^s","aliases":"r^s","display":"r^s","kind":"use","excerpt":"P(s)=q^s · r^s a^s · b^s.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c23ccf4a3cc56605","tex":"r^v","aliases":"r^v","display":"r^v","kind":"use","excerpt":"Polynomial theorem. For every n, A_n∈ Z[q,r], all its nonzero coefficients are positive integers, and A_n(q,r)=A_n(r,q). Every monomial q^u r^v satisfies","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3b737fff9fff6e4d","tex":"R^{\\rho_\\alpha}","aliases":"R^ρ_α rho alpha","display":"R^{\\rho_\\alpha}","kind":"use","excerpt":"claimWhy this equation appears Nonnegative coefficients give M(R)=T_q,r(R). The integral equation implies M(R)≤ R[1+M(qR)M(rR)], M(R)≥(1-α)R M(α qR)M(α rR),quad0<α<1. Taking logarithms turns product feedback into a sum over contracted scales. Comparison with CR^ρ proves the upper bound; comparison with cR^ρ_α, where (α q)^ρ_α+ (α r)^ρ_α=1, proves t","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"Indeed all omitted frequencies are at most λ_L+1, and ∑_n>Ln^-2a≤∫_L^∞ x^-2a dx. Ignoring squarefreeness and coprimality makes this a valid upper bound. If r_0,r_1 are the two positive truncated residual sums at t,t+h, the true secant belongs to","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"Starting with R_1=A, define successively","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"Indeed all omitted frequencies are at most λ_L+1, and ∑_n>Ln^-2a≤∫_L^∞ x^-2a dx. Ignoring squarefreeness and coprimality makes this a valid upper bound. If r_0,r_1 are the two positive truncated residual sums at t,t+h, the true secant belongs to","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"R_2(ε)=max{ e,(frac2C_0c_ε)^1/(ρ-1/2), (frac2log2c_ε)^1/ρ}. tagRA-3","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fb14b1a407e1de8e","tex":"r_a","aliases":"r_a","display":"r_a","kind":"use","excerpt":"pageheadingPRIME SIEVES / CONSTRUCTIVE WITNESSEvery negative prime atom has a finite detector For an integer a≥2, let L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1), r_a=frac1-x_a+11-x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_si","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5844f8e74c0ad886","tex":"r_a^m","aliases":"r_a^m","display":"r_a^m","kind":"use","excerpt":"pageheadingPRIME SIEVES / CONSTRUCTIVE WITNESSEvery negative prime atom has a finite detector For an integer a≥2, let L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1), r_a=frac1-x_a+11-x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_si","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a03f214a14787ce8","tex":"R_j","aliases":"R_j","display":"R_j","kind":"use","excerpt":"λ_j=lim_t→∞fraclog R_j(-t)t, w_j=lim_t→∞e^-λ_jtR_j(-t), R_j+1(s)=R_j(s)-w_je^-λ_js.","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-acaf34daefd4c4bd","tex":"r_J","aliases":"r_J","display":"r_J","kind":"use","excerpt":"c_n(h,q,r)∈ Q[h_1,…,h_J,q_1,r_1,…,q_J,r_J].","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-53e3e9d1c2a6131f","tex":"R_{j+1}","aliases":"R_j+1","display":"R_{j+1}","kind":"use","excerpt":"λ_j=lim_t→∞fraclog R_j(-t)t, w_j=lim_t→∞e^-λ_jtR_j(-t), R_j+1(s)=R_j(s)-w_je^-λ_js.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-26204516ead24844","tex":"r_j^2","aliases":"r_j^2","display":"r_j^2","kind":"use","excerpt":"pageheadingFAMILY 1 / TAPSFeedback banks and weak-channel sensitivity Choose finitely many channels j=1,…,J, with h_j>0, ∑_jh_j=1, and 0<q_j,r_j<1. Define F_Θ'(z)=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz), F_Θ(0)=0. These taps act inside the nonlinear definition. With μ_1=∑_jh_jq_jr_j and μ_2=∑_jh_jq_jr_j(q_j^2+r_j^2), F_Θ(z)=z+fracμ_13z^3+fracμ_1μ_215z^5+O(z^7).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1021069881d14e70","tex":"r_j^\\rho","aliases":"r_j^ρ rho","display":"r_j^\\rho","kind":"use","excerpt":"On the seed, G(t)≤ V_N+1+Δ+B_+≤ c_+e^ρ t. Each channel satisfies q_j^ρ+r_j^ρ≤1. Induction in intervals of length min_jmin(a_j,b_j)>0 therefore propagates the seed comparison, giving the explicit upper bound","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7209a8d77364ad12","tex":"r_j^{\\rho_j}","aliases":"r_j^ρ_j rho","display":"r_j^{\\rho_j}","kind":"use","excerpt":"vity Choose finitely many channels j=1,…,J, with h_j>0, ∑_jh_j=1, and 0<q_j,r_j<1. Define F_Θ'(z)=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz), F_Θ(0)=0. These taps act inside the nonlinear definition. With μ_1=∑_jh_jq_jr_j and μ_2=∑_jh_jq_jr_j(q_j^2+r_j^2), F_Θ(z)=z+fracμ_13z^3+fracμ_1μ_215z^5+O(z^7). If q_j^ρ_j+r_j^ρ_j=1, then ρ(F_Θ)=max_j activeρ_j. The positive ","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a64e897db39c6a7f","tex":"R_k","aliases":"R_k","display":"R_k","kind":"use","excerpt":"These constants are intentionally conservative, and require no numerical evaluation of an unknown solution. The tangent majorant gives M(1)<2. At R_k=Q^-k, the integral equation gives","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-66295ad0298f4b0d","tex":"R_{k-1}","aliases":"R_k-1","display":"R_{k-1}","kind":"use","excerpt":"M(R_k)≤ R_k[1+M(R_k-1)^2].","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4aa81ec436e2173c","tex":"R_N","aliases":"R_N","display":"R_N","kind":"use","excerpt":"Define U_0=2, U_k=Q^-k(1+U_k-1^2). Then M(R_k)≤ U_k, U_k≥1, and log U_k≤ klog(1/Q)+log2+2log U_k-1. Summing this elementary recurrence gives log U_N≤ V_N. Since e<3, the logarithmic-radius seed 1≤ t≤1+Δ has e^t≤ e/m<3/m≤ R_N, and therefore log M(e^t)≤ V_N.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0d5a58c436e4715b","tex":"r_p","aliases":"r_p","display":"r_p","kind":"use","excerpt":"All quantities in (5) are rational when the strengths are rational. It provides an exact finite certificate without calculating Hankel eigenvalues or relying on floating-point root finding. A suitable finite m exists because r_p<1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5c3fc73470b15000","tex":"r_p^{2m}","aliases":"r_p^2m","display":"r_p^{2m}","kind":"use","excerpt":"This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_signed_detectorA rational detector for p=3, λ_3=-1.1, λ_2=1. The first exact bound proving negativity is m=5, so a Hankel matrix of order eight detects the signed defect. Set C_F=∏_q∈ Fmax(1,|1+λ_q|). The summable tail give","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c1c6d9769860b84f","tex":"r_S","aliases":"r_S","display":"r_S","kind":"use","excerpt":"P_S,m(x)=∏_n∈ Sfracx-x_nx_2-x_n (frac1-x1-x_2)^m, C_S=∏_n∈ Sfrac1-x_n|x_2-x_n|, r_S=frac1-x_n_+1-x_2<1.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2c005532f953ef3d","tex":"r_S^{2m}","aliases":"r_S^2m","display":"r_S^{2m}","kind":"use","excerpt":"Q_δ(P_S,m)≤-fracδ16 +fracC_S^23K^3r_S^2m. tagRA-10","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a945e284a5498b89","tex":"r_S^m","aliases":"r_S^m","display":"r_S^m","kind":"use","excerpt":"This polynomial has degree |S|+m, equals one at x_2, and kills every positive atom through the cutoff. At each remaining positive node, the product of the absolute root factors is at most C_S, and the last factor is at most r_S^m. Dropping all other negative atoms and bounding the positive tail by the integral ∑_n>Kn^-4≤1/(3K^3) gives","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7d0b0aaec388130d","tex":"r_t","aliases":"r_t","display":"r_t","kind":"use","excerpt":"For rational channel tuples (h_t,q_t,r_t), use","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-79ff2dd3f60f35b4","tex":"\\operatorname{rad}","aliases":"rad operatorname","display":"\\operatorname{rad}","kind":"use","excerpt":"pageheadingFAMILY 2 / CALCULUSClosed shifts, infinite differential complexity The Euler product for Z_Q shows that only rad(Q)=∏_pmid Qp matters. Expansion of the depth factor gives A_a,β,Q(s+1)=A_a,β,Q(s)-2A_a+β/2,β,Q(s) +A_a+β,β,Q(s). The fractional transfer and Newton formulas are A_a,β,Q(s+t)=∑_j≥0(-1)^jbinom2tjA_a+jβ/2,β,Q(s), A_a,β,Q(t)=∑_j≥0","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1ecf96b50772dd74","tex":"\\operatorname{Res}","aliases":"Res operatorname","display":"\\operatorname{Res}","kind":"use","excerpt":"pageheadingORBIT SPECTRA / CONTINUATION THEOREMGlobal meromorphic continuation For every integer N≥2, the orbit zeta has a meromorphic continuation to the whole complex plane with exactly one pole: s=1/N, Res_s=1/N Z_N(s)=fracsin(π/N)Nπ. Its exact absolute-convergence abscissa is 1/N. At nonpositive integers, Z_N(-m)=0 (m≥0), Z_N'(0)=-log N. The ze","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-54ed1f41b0d4d1f8","tex":"\\operatorname{Res}_{s=1/N}","aliases":"Res_s=1/N operatorname","display":"\\operatorname{Res}_{s=1/N}","kind":"use","excerpt":"pageheadingORBIT SPECTRA / CONTINUATION THEOREMGlobal meromorphic continuation For every integer N≥2, the orbit zeta has a meromorphic continuation to the whole complex plane with exactly one pole: s=1/N, Res_s=1/N Z_N(s)=fracsin(π/N)Nπ. Its exact absolute-convergence abscissa is 1/N. At nonpositive integers, Z_N(-m)=0 (m≥0), Z_N'(0)=-log N. The ze","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-669dca7ff0fabc36","tex":"\\operatorname{Res}_{s=1/N-k}","aliases":"Res_s=1/N-k operatorname","display":"\\operatorname{Res}_{s=1/N-k}","kind":"use","excerpt":"ageheadingORBIT SPECTRA / SHIFTSA Hurwitz-type deformation and its determinant For real a≥0, define initially Z_N(s;a)=∑_j(w_j+a)^-s, Re s>1/N. The rightward shift preserves avoidance of the negative branch cut. Normalize the product with E_N(a) and use log E_N(a+t) in the Mellin argument. For a>0 there are genuine simple poles at s=1/N-k, k≥0: Res","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-65afc957b21d2813","tex":"\\operatorname{Res}_{s=\\alpha-k}","aliases":"Res_s=α-k operatorname alpha","display":"\\operatorname{Res}_{s=\\alpha-k}","kind":"use","excerpt":"Res_s=α-k Z_N(s;a) · =fracαsin(π(α-k))π binomα-1ka^k · =fracsin(π/N)Nπ frac(1-1/N)_kk!a^k. tagT7","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"-fraclog(λ_lfloor cjrfloor/λ_j)log c =β+O_Q,β,c(j^-1/2+j^-β).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"B_rm EGF(z)=∑_t=0^L-1h_t I^tG(z), IG(z)=∫_0^zG(w) dw.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"claimThe resolvent is non-D-finite For λ≥-1, the Stieltjes function S_p,λ(z)=∫(z-x)^-1 dν_p,λ(x) has infinitely many isolated poles at the positive-weight nodes x_n→1. A solution of a finite rational-coefficient linear ODE has only finitely many possible finite singular points. The germ at z=2 therefore is not D-finite. This is a general infinite-a","count":34},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"(n,Q)=1μ(n)^2n^-v =fracζ(v)ζ(2v)∏_pmid Q(1+p^-v)^-1, v>1. fig03_complex_depthLeft: a=β=2 at real depths, using the exact zeta--Newton representation at 90-digit working precision. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2b70d7597172d6b7","tex":"s^{-1}","aliases":"s^-1","display":"s^{-1}","kind":"use","excerpt":"When qb=ar, one has a=sq and b=sr, giving equality in the displayed order. When q=b, one has a=s^2r, giving equality after interchanging the factors. The pair also has the required initial values and derivatives. Uniqueness of the formal coefficient recursion identifies it with the coupled solution. Finally, V(z)=s^-1U(sz), proving rescaling.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-92f3d69425180790","tex":"s^{-1/(2\\beta)}","aliases":"s^-1/(2β) beta","display":"s^{-1/(2\\beta)}","kind":"use","excerpt":"At real positive depth, the matrix (A_χ,ψ(s)) on any d distinct characters is positive semidefinite and has identical diagonal A_a,β,Q(s). Its normalized matrix tends to I_d, with operator norm error O_a,β,Q,d(s^-1/(2β)). Indeed, the off-diagonal entry bound controls the maximum row and column sums, and thus the operator norm. This gives quantitati","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"Suppose such an equality held. Normalization gives AB=1. Write p=qr, u=q^2+r^2, and let ρ be the common order. The first two nontrivial coefficients and the growth law force B^2s^2=p, s^2=u/2, 2s^ρ=q^ρ+r^ρ=1. Consequently 2(fracq^2+r^22)^ρ/2=q^ρ+r^ρ. For ρ≠2, strict convexity or concavity of x^ρ/2 forces q=r. For the exceptional case ρ=2, we have u","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b21f2738057876a8","tex":"s^3","aliases":"s^3","display":"s^3","kind":"use","excerpt":"Since sin(π s)/π=s+O(s^3), (S14) implies Z_N(0)=0 and Z_N'(0)=-log N.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bd4501dbf5eb7328","tex":"s^{3/2}","aliases":"s^3/2","display":"s^{3/2}","kind":"use","excerpt":"{qsqrts, s^3/2r} ={a/sqrts, bsqrts}.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c863c73afd645b31","tex":"S^m","aliases":"S^m","display":"S^m","kind":"use","excerpt":"For every omitted exponent m≥2N+3, use R^m≤ (R/S)^2N+3S^m, sum positive terms, and bound their sum by F(S)≤ B(S). No extra geometric denominator is needed. For rational x≥0, pathcertified_evaluation returns the exact interval [F_N(x),F_N(x)+B(S)(x/S)^2N+3] for each component.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9d23b90f03dd0e70","tex":"s^\\rho","aliases":"s^ρ rho","display":"s^\\rho","kind":"use","excerpt":"Suppose such an equality held. Normalization gives AB=1. Write p=qr, u=q^2+r^2, and let ρ be the common order. The first two nontrivial coefficients and the growth law force B^2s^2=p, s^2=u/2, 2s^ρ=q^ρ+r^ρ=1. Consequently 2(fracq^2+r^22)^ρ/2=q^ρ+r^ρ. For ρ≠2, strict convexity or concavity of x^ρ/2 forces q=r. For the exceptional case ρ=2, we have u","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"claimStrict positive complex kernel For distinct complex points s_i, K_ij=A(s_i+overlines_j) is strictly positive definite. A zero quadratic form would force a finite exponential sum to vanish at every λ_n. Their accumulation at zero invokes the identity theorem; derivatives at zero then give a nonsingular Vandermonde system. On the real axis, (-1)","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-39cd8d42a605c666","tex":"S_{p,\\lambda}","aliases":"S_p,λ lambda","display":"S_{p,\\lambda}","kind":"use","excerpt":"claimThe resolvent is non-D-finite For λ≥-1, the Stieltjes function S_p,λ(z)=∫(z-x)^-1 dν_p,λ(x) has infinitely many isolated poles at the positive-weight nodes x_n→1. A solution of a finite rational-coefficient linear ODE has only finitely many possible finite singular points. The germ at z=2 therefore is not D-finite. This is a general infinite-a","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a7e6148c104d684f","tex":"\\mathcal S_Q","aliases":"S_Q mathcal","display":"\\mathcal S_Q","kind":"use","excerpt":"(n,Q)=1μ(n)^2n^-v =fracζ(v)ζ(2v)∏_pmid Q(1+p^-v)^-1, v>1. fig03_complex_depthLeft: a=β=2 at real depths, using the exact zeta--Newton representation at 90-digit working precision. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"which is O_a,β,ε(|s|^-γ) because u≥ |s|sinε. The principal diagonal has the nonzero leading term δ_QΓ(α)β^-1(2s)^-α, α=(2a-1)/β, by the base sector theorem. Its modulus is bounded below by a positive fixed-sector multiple of |s|^-α for sufficiently large |s|. Division gives the second bound, since γ-α=1/(2β).","count":23},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"use","excerpt":"To check the elementary estimate, write w=re^iθ: |t+w|^2=(t+r)^2-4trsin^2(θ/2)≥(t+r)^2cos^2(θ/2).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-21670bd4e275805e","tex":"\\sinh","aliases":"sinh","display":"\\sinh","kind":"use","excerpt":"claimA worked order-one member T_1/3,2/3(z)=z+frac227z^3+ frac42187z^5+frac23211160261z^7+O(z^9). It has order one, and T_1/3,2/3(1)≈1.0759239897408249. The diagonal point q=r=1/2 gives f_1/2(z)=sqrt2sinh(z/sqrt2). Every off-diagonal member is separated from all rescaled diagonal members by the theorem above.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"use","excerpt":"Proof. Multiply the ordinary Taylor recurrence by (2n+1)!. The factor attached to A_iA_j, where i+j=n-1, is (2n)!/[(2i+1)!(2j+1)!]=binom2n2i+1. This also proves the recurrence directly in the EGF convention. square","count":13},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"displaystyle sup_|s|≤ R|A_a,β,Q^(k)(s)-A_a,β,Qp^(k)(s)| ≤ λ_p^k e^Rλ_p fracZ_Q(2a)p^2a+1.","count":11},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a04df13addba999e","tex":"\\sup_{|s|\\le R}","aliases":"sup_|s|≤ R sup le","display":"\\sup_{|s|\\le R}","kind":"use","excerpt":"displaystyle sup_|s|≤ R|A_a,β,Q^(k)(s)-A_a,β,Qp^(k)(s)| ≤ λ_p^k e^Rλ_p fracZ_Q(2a)p^2a+1.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-aadf2562f2fbe303","tex":"\\sup_{|s|\\le5}","aliases":"sup_|s|≤5 sup le","display":"\\sup_{|s|\\le5}","kind":"use","excerpt":"sup_|s|≤5|D_p(s)| <frac43(p^4+1)(1-p^-2)^-10. tagRA-5","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-040fd2dc6848eb29","tex":"\\sup_{|z|\\le R}","aliases":"sup_|z|≤ R sup le","display":"\\sup_{|z|\\le R}","kind":"use","excerpt":"sup_|z|≤ R|F_h(z)-F_g(z)| ≤frac d4{R f_Q'(R)-f_Q(R)}.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6f22e3768f76ad4a","tex":"\\sup_{|z|\\le1}","aliases":"sup_|z|≤1 sup le","display":"\\sup_{|z|\\le1}","kind":"use","excerpt":"sup_|z|≤1|F_10^-12(z)-F_0(z)|<2.41890267×10^-13,","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"It has rank one, but is not symmetric: B-B^ T=ℓ_1wedgeℓ_2≠0. At four distinct nonzero nodes, polynomial interpolation lets their vector values vary independently, so terms supported at (az,bz) cannot cancel this coefficient. A scalar autonomous channel at the same node pair would require B=κℓ⊗ℓ, a symmetric rank-one tensor, after projecting the out","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7cbe9659d20496ea","tex":"T'_{q,r}","aliases":"T'_q,r","display":"T'_{q,r}","kind":"use","excerpt":"item[Contracted feedback T_q,r.] The normalized odd entire solution of T'_q,r(z)=1+T_q,r(qz)T_q,r(rz), T_q,r(0)=0, for 0<q,r<1. The diagonal notation is f_q=T_q,q. item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2061ffb3a7ac142b","tex":"t^{(j+\\ell-i)/b}","aliases":"t^(j+ℓ-i)/b ell","display":"t^{(j+\\ell-i)/b}","kind":"use","excerpt":"For causal FIR taps h_ℓ, b_m=∑_ℓ h_ℓ a_m-ℓ with zero extension to negative indices, split A(z) and B(z) into b phases. Then B_i(t)=∑_jH_ij(t)A_j(t), where H_ij(t)=∑_substackℓ≥0 j+ℓ≡ ipmod b h_ℓ t^(j+ℓ-i)/b. The exponent is the nonnegative carry to the next phase block. This is an exact coefficient law, with no approximation or loss of terms.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-feaabba717bb1d69","tex":"t^{-s}","aliases":"t^-s","display":"t^{-s}","kind":"use","excerpt":"pageheadingORBIT SPECTRA / MELLIN PROOFThe two ends have different jobs For 1/N<Re s<1, the genus-zero log derivative and the angular gap give absolute interchange in Z_N(s)=fracsinπ sπ ∫_0^∞ t^-sfracddtlog E_N(t)dd t. Each term uses the principal-branch beta integral ∫_0^∞ t^-s(t+w)^-1dd t=π w^-s/sinπ s. The denominator estimate bounds the absolut","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d07f7d6cbbe83d18","tex":"t^{-\\sigma}","aliases":"t^-σ sigma","display":"t^{-\\sigma}","kind":"use","excerpt":"∑_j∫_0^∞fract^-σ|t+w_j| dt ≤fracπcsin(πσ)∑_j|w_j|^-σ<∞. tagS9","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cbf7a966b3885edf","tex":"T^{1/N}","aliases":"T^1/N","display":"T^{1/N}","kind":"use","excerpt":"At zero the finite parts give I_K(0)=lim_T→∞[log E_N(T)-T^1/N]=-log N. Since (sinπ s/π)'_s=0=1, this equals Z_N'(0). Absolute convergence at a smaller real exponent would give a holomorphic series across 1/N, contradicting the pole and proving the exact abscissa.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e06ad6f4338b37d4","tex":"t^{1/N}","aliases":"t^1/N","display":"t^{1/N}","kind":"use","excerpt":"pageheadingORBIT SPECTRA / LARGE RADIUSA finite exponential sum controls the tail Put E_N(t)=d_N(-t)=∑_n≥0t^n/(Nn)! and ω=e^2π i/N. For x≥0, E_N(x^N)=frac1N∑_j=0^N-1e^ω^jx. Root filtering proves this identity term by term: only powers divisible by N remain. The positive real exponential is dominant, so log E_N(t)=t^1/N-log N+O(e^-κ_N t^1/N), κ_N=1-","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a4beb2eda3a3f395","tex":"t^{1/N-1}","aliases":"t^1/N-1","display":"t^{1/N-1}","kind":"use","excerpt":"Near zero write f(t)=(log E_N)'(t)=∑_k≥1α_kt^k-1, where α_k=(-1)^k+1 Z_N(k). For any integer K≥1, subtract K terms on [0,1]: I_K(s)= · ∫_0^1t^-s[f(t)-∑_k=1^Kα_kt^k-1]dd t +∑_k=1^Kfracα_kk-s · +∫_1^∞ t^-s[f(t)-frac1N t^1/N-1]dd t +frac1N(s-1/N). The first integral is holomorphic for Re s<K+1. The second is entire because its remainder decays exponen","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d8c8b507408c2a23","tex":"t^{1/N-1-k}","aliases":"t^1/N-1-k","display":"t^{1/N-1-k}","kind":"use","excerpt":"^-s, Re s>1/N. The rightward shift preserves avoidance of the negative branch cut. Normalize the product with E_N(a) and use log E_N(a+t) in the Mellin argument. For a>0 there are genuine simple poles at s=1/N-k, k≥0: Res_s=1/N-k Z_N(s;a) =fracsin(π/N)Nπfrac(1-1/N)_kk!a^k. Indeed the large-radius derivative expands as frac1N(a+t)^1/N-1 =frac1N∑_k≥0","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"For N≥1 and 0≤ R≤1, put t=3R/4. If (N+2)t^2≤ N+1, then n t^2n+1 is nonincreasing for n≥ N+1, and hence","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7990ec161ac46c40","tex":"t^{2n+1}","aliases":"t^2n+1","display":"t^{2n+1}","kind":"use","excerpt":"For N≥1 and 0≤ R≤1, put t=3R/4. If (N+2)t^2≤ N+1, then n t^2n+1 is nonincreasing for n≥ N+1, and hence","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4ce405cd565ef4b2","tex":"t^{2N+3}","aliases":"t^2N+3","display":"t^{2N+3}","kind":"use","excerpt":"0≤ L(R)-∑_n=1^Nn d_nR^2n+1 ≤12(N+1)t^2N+3. tagRA-2","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d241eb52ee3cfa72","tex":"t^k","aliases":"t^k","display":"t^k","kind":"use","excerpt":"G_N(t)=∑_k≥0a_N,kt^k, g_N(x)=∑_m≥1N a_N,m-1x^Nm-1. tagS12","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-956ca6e0f399101b","tex":"t^{k-1}","aliases":"t^k-1","display":"t^{k-1}","kind":"use","excerpt":"Near zero write f(t)=(log E_N)'(t)=∑_k≥1α_kt^k-1, where α_k=(-1)^k+1 Z_N(k). For any integer K≥1, subtract K terms on [0,1]: I_K(s)= · ∫_0^1t^-s[f(t)-∑_k=1^Kα_kt^k-1]dd t +∑_k=1^Kfracα_kk-s","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"use","excerpt":"pageheadingEXACT LABORATORY / PROJECTORSPolyphase sections preserve the full sequence For an ordinary generating function G(z)=∑_m≥0a_mz^m and integer b≥1, write G(z)=∑_j=0^b-1z^jG_j(z^b), G_j(t)=∑_n≥0a_bn+jt^n. The root projector is P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =z^jG_j(z^b), ω=e^2π i/b. The same projector works on an EGF, but the section ","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cef6729bcc40a7f6","tex":"t^\\alpha","aliases":"t^α alpha","display":"t^\\alpha","kind":"use","excerpt":"Integrating exactly and using log H_N(a+t)-t^α→-log N gives","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bea69885d49bedfc","tex":"t^{\\alpha-1}","aliases":"t^α-1 alpha","display":"t^{\\alpha-1}","kind":"use","excerpt":"Finally, the one-term subtraction is already convergent at s=0, since G_N,a(t)-α t^α-1=O(t^α-2) and α<1. Let","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-048429bc73f8c238","tex":"t^{\\alpha-2}","aliases":"t^α-2 alpha","display":"t^{\\alpha-2}","kind":"use","excerpt":"Finally, the one-term subtraction is already convergent at s=0, since G_N,a(t)-α t^α-1=O(t^α-2) and α<1. Let","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-97d6b3e91e3f596b","tex":"t^{\\alpha-k-1}","aliases":"t^α-k-1 alpha","display":"t^{\\alpha-k-1}","kind":"use","excerpt":"G_N,a(t)=∑_k=0^Kαbinomα-1ka^k t^α-k-1 +O(t^α-K-2). tagT5","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ca04e4ec9ca1d873","tex":"t^{\\alpha-K-2}","aliases":"t^α-K-2 alpha","display":"t^{\\alpha-K-2}","kind":"use","excerpt":"G_N,a(t)=∑_k=0^Kαbinomα-1ka^k t^α-k-1 +O(t^α-K-2). tagT5","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7c96f53f49bfa649","tex":"T^{\\alpha-k-s}","aliases":"T^α-k-s alpha","display":"T^{\\alpha-k-s}","kind":"use","excerpt":"fracαbinomα-1ka^k T^α-k-ss-α+k. tagT6","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7f502591dbad0d5e","tex":"T_+","aliases":"T_+","display":"T_+","kind":"use","excerpt":"T_-=fraclog(η(p^4+1)/D(4))λ_p, T_+=fraclog(η p^4)λ_p. tagRA-7","count":10},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4d99951c22895e3a","tex":"T_-","aliases":"T_-","display":"T_-","kind":"use","excerpt":"T_-=fraclog(η(p^4+1)/D(4))λ_p, T_+=fraclog(η p^4)λ_p. tagRA-7","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8b22ffce6c80c776","tex":"T_{1/3,2/3}","aliases":"T_1/3,2/3","display":"T_{1/3,2/3}","kind":"use","excerpt":"claimA worked order-one member T_1/3,2/3(z)=z+frac227z^3+ frac42187z^5+frac23211160261z^7+O(z^9). It has order one, and T_1/3,2/3(1)≈1.0759239897408249. The diagonal point q=r=1/2 gives f_1/2(z)=sqrt2sinh(z/sqrt2). Every off-diagonal member is separated from all rescaled diagonal members by the theorem above.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-64f17e7dedd6a141","tex":"T_4","aliases":"T_4","display":"T_4","kind":"use","excerpt":"16_polynomial_ladder T_n=∑_j=1^nj, H_n=∑_j=1^nT_j, Δ T_n=n, Δ H_n=T_n The upper table follows positive integers into triangular and tetrahedral numbers. Squares and cubes occupy the lower rows for comparison. The arrows reverse prefix sums by backward differences, Δ f_n=f_n-f_n-1. All plotted values use their displayed common index. At n=4, T_4=10 ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6b06b314a3b34d94","tex":"T_5","aliases":"T_5","display":"T_5","kind":"use","excerpt":"CES / 02Longer recurrence memoryS02_tribonacci_tetranacci T_n=T_n-1+T_n-2+T_n-3, Q_n=∑_j=1^4Q_n-j The seed table fixes the OEIS convention. Tribonacci starts 0,0,1; tetranacci starts 0,0,0,1. The curves use log_10(1+a_n) so that zeros and rapidly growing terms fit together. The transform changes the plotted scale, not the integer sequence. Tribonac","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-526eaf8112f7e507","tex":"T_{q,q}","aliases":"T_q,q","display":"T_{q,q}","kind":"use","excerpt":"item[Contracted feedback T_q,r.] The normalized odd entire solution of T'_q,r(z)=1+T_q,r(qz)T_q,r(rz), T_q,r(0)=0, for 0<q,r<1. The diagonal notation is f_q=T_q,q. item[Feedback bank F_Θ.] A finite collection of channels with h_j≥0, ∑_jh_j=1 and 0<q_j,r_j<1, entering F_Θ'=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz). Zero-weight channels are removed before taking an","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-072176a16581e3eb","tex":"T_{q,r}","aliases":"T_q,r","display":"T_{q,r}","kind":"use","excerpt":"mainmatter MonographPartIEntire functions and their growthContractions, coefficients and nonlinear feedback. Contracted nonlinear feedback. pageheadingFAMILY 1Two scales inside the function For real 0<q,r<1, define the normalized odd solution T_q,r'(z)=1+T_q,r(qz)T_q,r(rz), T_q,r(0)=0. The contractions are part of the defining feedback. The functio","count":14},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d5cdff67b1a3241d","tex":"T_{q,r}'","aliases":"T_q,r'","display":"T_{q,r}'","kind":"use","excerpt":"mainmatter MonographPartIEntire functions and their growthContractions, coefficients and nonlinear feedback. Contracted nonlinear feedback. pageheadingFAMILY 1Two scales inside the function For real 0<q,r<1, define the normalized odd solution T_q,r'(z)=1+T_q,r(qz)T_q,r(rz), T_q,r(0)=0. The contractions are part of the defining feedback. The functio","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-75177911206476aa","tex":"\\tan","aliases":"tan","display":"\\tan","kind":"use","excerpt":"f(4/3)≤tan(4/3)≤12.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":61},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ec7d40a789aa2d83","tex":"u'","aliases":"u'","display":"u'","kind":"use","excerpt":"bibitemDascaliuc2025 Radu Dascaliuc, Tuan N. Pham, Enrique Thomann and Edward C. Waymire. Transformation of Stochastic Recursions and Critical Phenomena in the Analysis of the Aldous-Shields-Athreya Cascade and Related Mean Flow Equations. arXiv:2411.00629v2 (2025). The nonlinear equation u'=-u+u(α t)^2 is an adjacent model, not the equation studie","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","excerpt":"Define U_0=2, U_k=Q^-k(1+U_k-1^2). Then M(R_k)≤ U_k, U_k≥1, and log U_k≤ klog(1/Q)+log2+2log U_k-1. Summing this elementary recurrence gives log U_N≤ V_N. Since e<3, the logarithmic-radius seed 1≤ t≤1+Δ has e^t≤ e/m<3/m≤ R_N, and therefore log M(e^t)≤ V_N.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"Then u_0=v_0=1, and, for n≥1,","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"Write U(z)=z+u_1z^3+u_2z^5+u_3z^7+⋯ and V(z)=z+v_1z^3+v_2z^5+v_3z^7+⋯. Put p=qr, k=ab. Coefficient matching gives","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2f97c9046c1a4c7a","tex":"u_2","aliases":"u_2","display":"u_2","kind":"use","excerpt":"D=k^2u_2-p^2v_2 =fracpkar15(q^2-b^2)(qb-ar).","count":15},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cdc6b49492357264","tex":"u_3","aliases":"u_3","display":"u_3","kind":"use","excerpt":"Write U(z)=z+u_1z^3+u_2z^5+u_3z^7+⋯ and V(z)=z+v_1z^3+v_2z^5+v_3z^7+⋯. Put p=qr, k=ab. Coefficient matching gives","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-72b2bc9e12d0fa8e","tex":"U_k","aliases":"U_k","display":"U_k","kind":"use","excerpt":"Define U_0=2, U_k=Q^-k(1+U_k-1^2). Then M(R_k)≤ U_k, U_k≥1, and log U_k≤ klog(1/Q)+log2+2log U_k-1. Summing this elementary recurrence gives log U_N≤ V_N. Since e<3, the logarithmic-radius seed 1≤ t≤1+Δ has e^t≤ e/m<3/m≤ R_N, and therefore log M(e^t)≤ V_N.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-46f4c87c4b6d1661","tex":"U_{k-1}","aliases":"U_k-1","display":"U_{k-1}","kind":"use","excerpt":"Define U_0=2, U_k=Q^-k(1+U_k-1^2). Then M(R_k)≤ U_k, U_k≥1, and log U_k≤ klog(1/Q)+log2+2log U_k-1. Summing this elementary recurrence gives log U_N≤ V_N. Since e<3, the logarithmic-radius seed 1≤ t≤1+Δ has e^t≤ e/m<3/m≤ R_N, and therefore log M(e^t)≤ V_N.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-918236f95f84a379","tex":"U_{k-1}^2","aliases":"U_k-1^2","display":"U_{k-1}^2","kind":"use","excerpt":"Define U_0=2, U_k=Q^-k(1+U_k-1^2). Then M(R_k)≤ U_k, U_k≥1, and log U_k≤ klog(1/Q)+log2+2log U_k-1. Summing this elementary recurrence gives log U_N≤ V_N. Since e<3, the logarithmic-radius seed 1≤ t≤1+Δ has e^t≤ e/m<3/m≤ R_N, and therefore log M(e^t)≤ V_N.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9fcfc7136a869616","tex":"U_N","aliases":"U_N","display":"U_N","kind":"use","excerpt":"Define U_0=2, U_k=Q^-k(1+U_k-1^2). Then M(R_k)≤ U_k, U_k≥1, and log U_k≤ klog(1/Q)+log2+2log U_k-1. Summing this elementary recurrence gives log U_N≤ V_N. Since e<3, the logarithmic-radius seed 1≤ t≤1+Δ has e^t≤ e/m<3/m≤ R_N, and therefore log M(e^t)≤ V_N.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"use","excerpt":"0<u_n≤ 2^-n, 0<v_n≤ 2^-n.","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b03775ada4b9cdf0","tex":"u_{n+1}","aliases":"u_n+1","display":"u_{n+1}","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7c359a771275f81e","tex":"U_{p,j}","aliases":"U_p,j","display":"U_{p,j}","kind":"use","excerpt":"The exact coordinate formula is M_j(λ)=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) ∏_p∈ F(1+fracλ_pp^4+2r+1). Coefficients of the monic families lie in Q(π^-2,λ_p:p∈ F). For two primes, the overlap term is visible: M_j=M_j(0,0)+λ_pU_p,j+λ_qU_q,j +λ_pλ_qU_pq,j. Here U_pq,j>0 is the pq-divisible submeasure moment. The family does not reduce to adding s","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-25af507f02293318","tex":"U_{pq,j}","aliases":"U_pq,j","display":"U_{pq,j}","kind":"use","excerpt":"The exact coordinate formula is M_j(λ)=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) ∏_p∈ F(1+fracλ_pp^4+2r+1). Coefficients of the monic families lie in Q(π^-2,λ_p:p∈ F). For two primes, the overlap term is visible: M_j=M_j(0,0)+λ_pU_p,j+λ_qU_q,j +λ_pλ_qU_pq,j. Here U_pq,j>0 is the pq-divisible submeasure moment. The family does not reduce to adding s","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b341d8a720d297fc","tex":"U_{q,j}","aliases":"U_q,j","display":"U_{q,j}","kind":"use","excerpt":"The exact coordinate formula is M_j(λ)=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) ∏_p∈ F(1+fracλ_pp^4+2r+1). Coefficients of the monic families lie in Q(π^-2,λ_p:p∈ F). For two primes, the overlap term is visible: M_j=M_j(0,0)+λ_pU_p,j+λ_qU_q,j +λ_pλ_qU_pq,j. Here U_pq,j>0 is the pq-divisible submeasure moment. The family does not reduce to adding s","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7901df75af52898c","tex":"\\underline t","aliases":"underline t underline","display":"\\underline t","kind":"use","excerpt":"underline t≥ P(ρ)underline t, underline t=(underlineτ_1,underlineτ_2)^T.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9dca9944d22dad16","tex":"\\underline\\tau","aliases":"underlineτ underline tau","display":"\\underline\\tau","kind":"use","excerpt":"underlineτ_i=liminf_R→∞fraclog M_F_i(R)R^ρ, overlineτ_i=limsup_R→∞fraclog M_F_i(R)R^ρ.","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-18c376ea8c5999b9","tex":"\\underline{\\boldsymbol\\tau}","aliases":"underlineτ underline boldsymbol tau","display":"\\underline{\\boldsymbol\\tau}","kind":"use","excerpt":"For F_1=U and F_2=V, define underlineτ_i=liminf_R→∞fraclog M_F_i(R)R^ρ, overlineτ_i=limsup_R→∞fraclog M_F_i(R)R^ρ. If P(ρ)w=w with w>0, then both type vectors have precisely these Perron proportions: underlineτ=underlineτ w, overlineτ=overlineτ w, qquad0<underlineτ≤overlineτ<∞. For the worked pair, both the lower type and the upper type of V are tw","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-52db1610d1d38a91","tex":"\\underline\\tau_1","aliases":"underlineτ_1 underline tau","display":"\\underline\\tau_1","kind":"use","excerpt":"(underlineτ_1,underlineτ_2)^T · =underlineτ w, (overlineτ_1,overlineτ_2)^T · =overlineτ w.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6b5f45b3f3a6177a","tex":"\\underline\\tau_2","aliases":"underlineτ_2 underline tau","display":"\\underline\\tau_2","kind":"use","excerpt":"(underlineτ_1,underlineτ_2)^T · =underlineτ w, (overlineτ_1,overlineτ_2)^T · =overlineτ w.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-602cf8d2c767b52f","tex":"\\underline\\tau_i","aliases":"underlineτ_i underline tau","display":"\\underline\\tau_i","kind":"use","excerpt":"underlineτ_i=liminf_R→∞fraclog M_F_i(R)R^ρ, overlineτ_i=limsup_R→∞fraclog M_F_i(R)R^ρ.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d27fa8e4e0984e4b","tex":"\\uparrow","aliases":"uparrow","display":"\\uparrow","kind":"use","excerpt":"The earlier moving-width lower argument used a fixed α<1; it proved orders below ρ and then recovered the order by αuparrow1, but did not establish positive lower type at ρ. This is a limitation of that argument, not a counterexample. The fixed-width comparison closes precisely that missing implication. The evidence archive retains execution failur","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":59},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4e7142ad42e2513e","tex":"v'","aliases":"v'","display":"v'","kind":"use","excerpt":"Zero scales lie outside Theorem H.1. On zero-constant inputs, a product vanishes identically if either of its scales is zero. If both products vanish, both equations are already u'=v'=1. If only the first product vanishes, the projection u is autonomous, regardless of whether the second pair agrees with the first. Thus extending Theorem H.1 to zero","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bb159415c13fd65d","tex":"v^{r+2}","aliases":"v^r+2","display":"v^{r+2}","kind":"use","excerpt":"Depth-k moments are M_k+j. With v=π^-2, the even-zeta formula makes every D(4+2r) a rational multiple of v^r+2. Thus the moments belong to Q[v,λ_p:p∈ F], and monic orthogonal-polynomial coefficients belong to Q(v,λ_p:p∈ F). This is an exact field statement, with no decimal recognition step.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"Then u_0=v_0=1, and, for n≥1,","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"Write U(z)=z+u_1z^3+u_2z^5+u_3z^7+⋯ and V(z)=z+v_1z^3+v_2z^5+v_3z^7+⋯. Put p=qr, k=ab. Coefficient matching gives","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"use","excerpt":"The value of C_0 comes directly from the earlier explicit constants: V_2=15log2, Δ=2log2, and B_+=5log2. Thus V_2+1+Δ+B_+=1+22log2.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"D=k^2u_2-p^2v_2 =fracpkar15(q^2-b^2)(qb-ar).","count":14},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"Write U(z)=z+u_1z^3+u_2z^5+u_3z^7+⋯ and V(z)=z+v_1z^3+v_2z^5+v_3z^7+⋯. Put p=qr, k=ab. Coefficient matching gives","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"Put v=h-g. Its positive and negative parts both have sum d/2, since ∑_jv_j=0. At every point of the segment between g and h, all partial derivatives lie in the interval [0,n d_n(Q)]. Hence","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-616a0b138220d2cb","tex":"V_K","aliases":"V_K","display":"V_K","kind":"use","excerpt":"claimExplicit upper constants are available too Set Q=max_jmax(q_j,r_j), m=min_jmin(q_j,r_j), Δ=log(1/m), and B_+=max_jlog(1/(q_jr_j))+log2. Choose an integer K with Q^-K≥3/m and put V_K=(2^K+1-1)log2+(2^K+1-K-2)log(1/Q), c_+=e^-ρ(V_K+1+Δ+B_+). Then log M(R)≤ c_+R^ρ-log R-B_+ for R≥ e. The tangent majorant M(1)<2, followed by the integral recurrenc","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b47bde7729d16d41","tex":"V_N","aliases":"V_N","display":"V_N","kind":"use","excerpt":"V_N=(2^N+1-1)log2+ (2^N+1-N-2)log(1/Q),","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"use","excerpt":"0<u_n≤ 2^-n, 0<v_n≤ 2^-n.","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7fd7bfa7519bcf67","tex":"v_{n+1}","aliases":"v_n+1","display":"v_{n+1}","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e939734de5143142","tex":"v_{n-1-i}","aliases":"v_n-1-i","display":"v_{n-1-i}","kind":"use","excerpt":"u_n · =frac12n+1∑_i=0^n-1 q^2i+1r^2(n-1-i)+1u_i v_n-1-i, v_n · =frac12n+1∑_i=0^n-1 a^2i+1b^2(n-1-i)+1u_i v_n-1-i.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-75dca79d2dcf2d7b","tex":"v_{n-i}","aliases":"v_n-i","display":"v_{n-i}","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"use","excerpt":"The function itself therefore encodes the weight exponent, the depth exponent, and the finite excluded prime set. It does not encode valuations v_p(Q).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Each is homogeneous of degree n in the weights. If a parameter set has |q_j|,|r_j|≤ Q<1 and ∑_j|h_j|≤ W, with W>0, coefficient induction gives","count":14},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"mid Q(1+p^-v)^-1, v>1. fig03_complex_depthLeft: a=β=2 at real depths, using the exact zeta--Newton representation at 90-digit working precision. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β). Then A(s)=∑_n∈ S_Q","count":118},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b99d2e5f37ae7e17","tex":"w'","aliases":"w'","display":"w'","kind":"use","excerpt":"w'=2+tfrac12(w(sz)^2-d(sz)^2), d'=0.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-33bdc21e050de4bd","tex":"w^{(t+l-j)/b}","aliases":"w^(t+l-j)/b","display":"w^{(t+l-j)/b}","kind":"use","excerpt":"Y_j(w)=∑_l=0^b-1H_jl(w)X_l(w), H_jl(w)=∑_substack0≤ t<L t+l≡ j (b) h_t w^(t+l-j)/b.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c6af27b09698980f","tex":"W^{-1/2}","aliases":"W^-1/2","display":"W^{-1/2}","kind":"use","excerpt":"∑_n≥0|c_n(h,q,r)|R^2n+1 ≤ W^-1/2f_Q(sqrt W R)<∞.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-dc4ffc3cd79eb63e","tex":"w^{-s}","aliases":"w^-s","display":"w^{-s}","kind":"use","excerpt":"pageheadingORBIT SPECTRA / MELLIN PROOFThe two ends have different jobs For 1/N<Re s<1, the genus-zero log derivative and the angular gap give absolute interchange in Z_N(s)=fracsinπ sπ ∫_0^∞ t^-sfracddtlog E_N(t)dd t. Each term uses the principal-branch beta integral ∫_0^∞ t^-s(t+w)^-1dd t=π w^-s/sinπ s. The denominator estimate bounds the absolut","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e8ae5637dcea5fce","tex":"w^d","aliases":"w^d","display":"w^d","kind":"use","excerpt":"Proof. A term h_t a_bn+j-t is in phase l exactly when t+l≡ jpmod b. Write d=(t+l-j)/b. That term is h_t x_l[n-d], whose OGF is h_tw^dX_l(w) by zero extension. Summing over taps and phases gives the matrix. square","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5caf668b59cf4d10","tex":"w^k","aliases":"w^k","display":"w^k","kind":"use","excerpt":"ts. Multiplicity is retained by the definition; simplicity is not assumed in general. The bound |d_N(w)|≤ e^|w|^1/N and the finite-order zero-counting theorem give absolute local uniform convergence in the stated half-plane. Since the order is below one and d_N(0)=1, the Hadamard factorization has genus zero: d_N(w)=∏_j(1-w/w_j). At positive intege","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-79739777a970bb5b","tex":"W^n","aliases":"W^n","display":"W^n","kind":"use","excerpt":"|c_n(h,q,r)|≤ W^n d_n(Q),","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e6f16f2e02ae0bf2","tex":"w^n","aliases":"w^n","display":"w^n","kind":"use","excerpt":"MonographPartIIIOrbit spectra and exact coordinatesSpectral continuation and the coefficient language of polyphase transforms. Spectral functions of zero orbits. pageheadingFAMILY 4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"η κ_j(t), κ_j(t)=fracA(-t)R_j(-t) ∼fracw_1w_je^(λ_1-λ_j)t.","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3106fac3e3c8f992","tex":"w_2","aliases":"w_2","display":"w_2","kind":"use","excerpt":"Both coordinates of w are positive, max(w_1,w_2)=1, and direct multiplication gives","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"0<c_-w_i≤underlineτ_i≤overlineτ_i≤ c_+w_i<∞, i=1,2.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"act coordinatesSpectral continuation and the coefficient language of polyphase transforms. Spectral functions of zero orbits. pageheadingFAMILY 4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(z)=d_N(z^N). Let w_j be the nonzero ","count":43},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2b9b23ce83f85a7a","tex":"w_j^{-s}","aliases":"w_j^-s","display":"w_j^{-s}","kind":"use","excerpt":"4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(z)=d_N(z^N). Let w_j be the nonzero zeros of d_N, counted with multiplicity. Each corresponds to one rotation orbit of N zeros of D_N. Using Argw∈(-π,π], define Z_N(s)=∑_jw_j^-s, R","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ebdf9f205f201a52","tex":"w_k","aliases":"w_k","display":"w_k","kind":"use","excerpt":"Every limit exists and recovers exactly the j-th support frequency and weight. Indeed, if M_j+1=∑_k>jw_k, then for every t≥0,","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a854079e0c620b44","tex":"W_N","aliases":"W_N","display":"W_N","kind":"use","excerpt":"W_N=(2^N+1-1)log2 +(2^N+1-N-2)log(1/Q),","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8945947ef031a855","tex":"w_n","aliases":"w_n","display":"w_n","kind":"use","excerpt":"mid Q(1+p^-v)^-1, v>1. fig03_complex_depthLeft: a=β=2 at real depths, using the exact zeta--Newton representation at 90-digit working precision. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β). Then A(s)=∑_n∈ S_Q","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"It has rank one, but is not symmetric: B-B^ T=ℓ_1wedgeℓ_2≠0. At four distinct nonzero nodes, polynomial interpolation lets their vector values vary independently, so terms supported at (az,bz) cannot cancel this coefficient. A scalar autonomous channel at the same node pair would require B=κℓ⊗ℓ, a symmetric rank-one tensor, after projecting the out","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-08d41d2f8c1b0ad2","tex":"\\widehat\\lambda","aliases":"widehatλ widehat lambda","display":"\\widehat\\lambda","kind":"use","excerpt":"widehatλ_j(t,h) =fraclog R_j(-t-h)-log R_j(-t)h.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-eb1c2658a2bed248","tex":"\\widehat\\lambda_j","aliases":"widehatλ_j widehat lambda","display":"\\widehat\\lambda_j","kind":"use","excerpt":"widehatλ_j(t,h) =fraclog R_j(-t-h)-log R_j(-t)h.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-636598453ef52965","tex":"\\widetilde w","aliases":"widetilde w widetilde","display":"\\widetilde w","kind":"use","excerpt":"widetilde w=(r^ρ,1-q^ρ)^T, w=fracwidetilde wmax(r^ρ,1-q^ρ).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1fa74de5039c01f6","tex":"\\widetilde\\nu","aliases":"widetildeν widetilde nu","display":"\\widetilde\\nu","kind":"use","excerpt":"widetildeν_F,γ =∑_n≥1μ(n)^2n^-4 (1+∑_p∈ Fγ_p1_pmid n)δ_x_n.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-850fae3b9cda528c","tex":"\\widetilde\\nu_{F,\\boldsymbol\\gamma}","aliases":"widetildeν_F,γ widetilde nu boldsymbol gamma","display":"\\widetilde\\nu_{F,\\boldsymbol\\gamma}","kind":"use","excerpt":"widetildeν_F,γ =∑_n≥1μ(n)^2n^-4 (1+∑_p∈ Fγ_p1_pmid n)δ_x_n.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"(x-X)(pX-ky)=0.","count":15},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"This leaves at most two positive candidates. If there are two distinct positive roots x,x', their product is xx'=kp. The corresponding seventh-coefficient values differ by","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6a6ab9b9a4069531","tex":"x'^2","aliases":"x'^2","display":"x'^2","kind":"use","excerpt":"u_3(x)-u_3(x') =frac p7(x^2-x'^2) (u_2-fracp^2k^2v_2) =fracpD7k^2(x^2-x'^2).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-85751d123968485a","tex":"x^{-2a}","aliases":"x^-2a","display":"x^{-2a}","kind":"use","excerpt":"(n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2a-1/2)/β; |s|/Re s stays bounded in the stated sector. This gives the erro","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b23253130c665fdc","tex":"x^{-Ns}","aliases":"x^-Ns","display":"x^{-Ns}","kind":"use","excerpt":"Z_N(s)=fracsin(π s)π ∫_0^∞ x^-Nsg_N(x) dx, frac1N<Re s<1. tagS11","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-91bec74d22815113","tex":"x^{-\\beta}","aliases":"x^-β beta","display":"x^{-\\beta}","kind":"use","excerpt":"(n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2a-1/2)/β; |s|/Re s stays bounded in the stated sector. This gives the erro","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"(k^2u-p^2v)(x^2-X^2)=0.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"(k^2u-p^2v)(x^2-X^2)=0.","count":16},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9e2df13713b51ec7","tex":"x^j","aliases":"x^j","display":"x^j","kind":"use","excerpt":"item[Hankel order and polynomial degree.] For moments M_j=∫ x^jddν, the order-m matrix is H_m=[M_i+j]_0≤ i,j<m. It tests quadratic forms of real polynomials of degree at most m-1. At depth k, the matrix is [M_k+i+j]_0≤ i,j<m. An order-five test therefore covers degree four, not degree five. item[Signed quadratic form.] For real polynomials, Q_ν(f)=","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8a46ff3f62bf1268","tex":"x^k","aliases":"x^k","display":"x^k","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0ed7bdf4a651c709","tex":"x^m","aliases":"x^m","display":"x^m","kind":"use","excerpt":"If at least two strengths in a positive multiplicative family are nonzero, its entire depth-zero monic polynomial sequence cannot equal any one-prime sequence on the same variable and node system. A direct proof avoids importing a determinacy theorem. Write P_m=x^m+∑_j<mc_mjx^j. The equations ∫ P_m dν=0 recursively determine every normalized moment","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-12f1075149375120","tex":"x^N","aliases":"x^N","display":"x^N","kind":"use","excerpt":"pageheadingORBIT SPECTRA / LARGE RADIUSA finite exponential sum controls the tail Put E_N(t)=d_N(-t)=∑_n≥0t^n/(Nn)! and ω=e^2π i/N. For x≥0, E_N(x^N)=frac1N∑_j=0^N-1e^ω^jx. Root filtering proves this identity term by term: only powers divisible by N remain. The positive real exponential is dominant, so log E_N(t)=t^1/N-log N+O(e^-κ_N t^1/N), κ_N=1-","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ce7d04b01e240c45","tex":"x^{N(K+1)-1}","aliases":"x^N(K+1)-1","display":"x^{N(K+1)-1}","kind":"use","excerpt":"The first integral is holomorphic for Re s<K+1. This includes local uniform domination, since the Taylor remainder is O(x^N(K+1)-1). The last integral is entire in s, because (S3) gives exponential decay against every power of x, uniformly on compact s-sets, including after any number of s-derivatives. The rational terms exhibit the only possible p","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1156163322778acc","tex":"x^{N-1}","aliases":"x^N-1","display":"x^{N-1}","kind":"use","excerpt":"At the origin g_N(x)=O(x^N-1). Its only Taylor powers are x^Nm-1, with m≥1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c73a09acd6236df7","tex":"x^{Nm-1}","aliases":"x^Nm-1","display":"x^{Nm-1}","kind":"use","excerpt":"At the origin g_N(x)=O(x^N-1). Its only Taylor powers are x^Nm-1, with m≥1.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fb65eac3347b47b0","tex":"x^{Nn}","aliases":"x^Nn","display":"x^{Nn}","kind":"use","excerpt":"F_N(x):=d_N(-x^N) =frac1N∑_j=0^N-1e^ω^j x =∑_n=0^∞fracx^Nn(Nn)!. tagS1","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ebcaaad2a22554b3","tex":"x^\\rho","aliases":"x^ρ rho","display":"x^\\rho","kind":"use","excerpt":"For a rigorous propagation, put Q_* =max(q,r)<1 and d=(1-Q_*)B+Q_*L>0. Every child of every R≥ B is at most R-d. Induction on the intervals [B+kd,B+(k+1)d], beginning with the already controlled seed, proves the comparison for all larger radii. Continuity deals with common endpoints. All power comparisons use only monotonicity of x^ρ for positive x","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-77cff28da3730ef5","tex":"x^{\\rho/2}","aliases":"x^ρ/2 rho","display":"x^{\\rho/2}","kind":"use","excerpt":"Suppose such an equality held. Normalization gives AB=1. Write p=qr, u=q^2+r^2, and let ρ be the common order. The first two nontrivial coefficients and the growth law force B^2s^2=p, s^2=u/2, 2s^ρ=q^ρ+r^ρ=1. Consequently 2(fracq^2+r^22)^ρ/2=q^ρ+r^ρ. For ρ≠2, strict convexity or concavity of x^ρ/2 forces q=r. For the exceptional case ρ=2, we have u","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"Y_0 Y_1 =h_0 · h_1w h_1 · h_0 X_0 X_1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"Y_0 Y_1 =h_0 · h_1w h_1 · h_0 X_0 X_1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"The mass at x_1=0 is exactly 1. A strength λ_p=-1 deletes every squarefree atom divisible by p. Several deletions commute. The depth-k family uses x^k dν, for integers k≥0.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"lessonSEQUENCES / 04A Pell pair shares one unitS04_sqrt3_pell x_n+y_nsqrt3=(2+sqrt3)^n, x_n^2-3y_n^2=1 The table keeps both coordinates at the same index. Multiplication by 2+sqrt3 advances the pair using the displayed integer update. Its determinant is one, and it preserves the quadratic norm. The zero-index pair is (1,0). Starting from (x_2,y_2)=","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"lessonSEQUENCES / 04A Pell pair shares one unitS04_sqrt3_pell x_n+y_nsqrt3=(2+sqrt3)^n, x_n^2-3y_n^2=1 The table keeps both coordinates at the same index. Multiplication by 2+sqrt3 advances the pair using the displayed integer update. Its determinant is one, and it preserves the quadratic norm. The zero-index pair is (1,0). Starting from (x_2,y_2)=","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9067dcf296a5b175","tex":"x_a","aliases":"x_a","display":"x_a","kind":"use","excerpt":"pageheadingPRIME SIEVES / CONSTRUCTIVE WITNESSEvery negative prime atom has a finite detector For an integer a≥2, let L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1), r_a=frac1-x_a+11-x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_si","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a363cbf484b54cd6","tex":"x_{a+1}","aliases":"x_a+1","display":"x_{a+1}","kind":"use","excerpt":"pageheadingPRIME SIEVES / CONSTRUCTIVE WITNESSEvery negative prime atom has a finite detector For an integer a≥2, let L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1), r_a=frac1-x_a+11-x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_si","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"use","excerpt":"For an ordinary generating function X(z)=∑_ma_m z^m, define X_j(w)=∑_na_bn+jw^n. Then","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"The full-position section and the decimated lane are distinct arrays. A section keeps the original positions, inserting zeros outside the residue. A lane x_j[n]=a_bn+j changes the index to n.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-756d830f7479eded","tex":"X_l","aliases":"X_l","display":"X_l","kind":"use","excerpt":"For an OGF sequence a_m, define lanes x_l[n]=a_bn+l and X_l(w)=∑_nx_l[n]w^n. Apply finite ordinary taps and obtain y_j[n]=b_bn+j. Then the exact matrix identity is","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c44f6e7c4de4d865","tex":"x_l","aliases":"x_l","display":"x_l","kind":"use","excerpt":"For an OGF sequence a_m, define lanes x_l[n]=a_bn+l and X_l(w)=∑_nx_l[n]w^n. Apply finite ordinary taps and obtain y_j[n]=b_bn+j. Then the exact matrix identity is","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":19},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4b720c31908e08c7","tex":"x_n^{i+j}","aliases":"x_n^i+j","display":"x_n^{i+j}","kind":"use","excerpt":"A_m=[∑_pnmid nμ(n)^2n^-4κ(n)x_n^i+j]_i,j<m, B_m=[∑_pmid nμ(n)^2n^-4κ(n)x_n^i+j]_i,j<m.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a83f86caa4659377","tex":"x_n^j","aliases":"x_n^j","display":"x_n^j","kind":"use","excerpt":"where U_pq,j=∑_pqmid nμ(n)^2n^-4x_n^j>0. More generally, with all other strengths at least -1, every mixed moment derivative in a set of distinct primes is strictly positive. It is the integral over atoms divisible by those primes, with the remaining nonnegative prime factors. Infinitely many such atoms remain after any finite collection of deletio","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4a9fa0175cfe6f46","tex":"x_{n_+}","aliases":"x_n_+","display":"x_{n_+}","kind":"use","excerpt":"P_S,m(x)=∏_n∈ Sfracx-x_nx_2-x_n (frac1-x1-x_2)^m, C_S=∏_n∈ Sfrac1-x_n|x_2-x_n|, r_S=frac1-x_n_+1-x_2<1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-051306e23d0f34f6","tex":"x_{n_i}","aliases":"x_n_i","display":"x_{n_i}","kind":"use","excerpt":"Choose d+1 distinct odd squarefree labels n_0,…,n_d, including 1, and set y_i=x_n_i. Their Lagrange polynomials are","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-53687bf254c09aec","tex":"x_p","aliases":"x_p","display":"x_p","kind":"use","excerpt":"deg f<m frac∫ f^2 dν_nonp∫ f^2 dν_p>0, δ_mdownarrow0. To prove the limit, approximate a continuous bump supported at the isolated node x_p by polynomials on [0,1]. The numerator tends to zero while the denominator retains the p^-4 atom. Hence every λ<-1 eventually produces a negative finite-degree quadratic form.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d17ce75dba8895be","tex":"x_p^{-k}","aliases":"x_p^-k","display":"x_p^{-k}","kind":"use","excerpt":"The same argument applies to every nonnegative depth, replacing each integral by its x^k tilt. The denominator's explicit atom contribution becomes p^-4x_p^k, and (7) acquires the factor x_p^-k.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c5fb9cebcc381493","tex":"x_p^k","aliases":"x_p^k","display":"x_p^k","kind":"use","excerpt":"od_q∈ Fmax(1,|1+λ_q|). The summable tail gives ∫ f_p,m^2ddν ≤frac1+λ_pp^4+ fracC_FC_p^23p^3r_p^2m. Hence the quadratic form is negative whenever pC_FC_p^2r_p^2m/3<-(1+λ_p). For rational strengths this is an exact rational certificate. It needs no numerical eigenvalue calculation. A depth tilt x^k replaces the negative atom contribution by (1+λ_p)x_","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"Y_j(w)=∑_l=0^b-1H_jl(w)X_l(w), H_jl(w)=∑_substack0≤ t<L t+l≡ j (b) h_t w^(t+l-j)/b.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"use","excerpt":"with root product yy'=k^3/p. The two candidate seventh coefficients have difference","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-909b83f9ab0ca7e2","tex":"y'^2","aliases":"y'^2","display":"y'^2","kind":"use","excerpt":"v_3(y)-v_3(y') =frac k7(y^2-y'^2) (u_2-fracp^2k^2v_2) =fracD7k(y^2-y'^2),","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"p y^2-frac15v_2ky+k^3=0,","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d842f0833fbfd1bc","tex":"Y_0","aliases":"Y_0","display":"Y_0","kind":"use","excerpt":"Y_0 Y_1 =h_0 · h_1w h_1 · h_0 X_0 X_1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7ef58ddaab7b5984","tex":"Y_1","aliases":"Y_1","display":"Y_1","kind":"use","excerpt":"Y_0 Y_1 =h_0 · h_1w h_1 · h_0 X_0 X_1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-dd412919846eece9","tex":"y_2","aliases":"y_2","display":"y_2","kind":"use","excerpt":"lessonSEQUENCES / 04A Pell pair shares one unitS04_sqrt3_pell x_n+y_nsqrt3=(2+sqrt3)^n, x_n^2-3y_n^2=1 The table keeps both coordinates at the same index. Multiplication by 2+sqrt3 advances the pair using the displayed integer update. Its determinant is one, and it preserves the quadratic norm. The zero-index pair is (1,0). Starting from (x_2,y_2)=","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5171b856d1185d9c","tex":"y_3","aliases":"y_3","display":"y_3","kind":"use","excerpt":"lessonSEQUENCES / 04A Pell pair shares one unitS04_sqrt3_pell x_n+y_nsqrt3=(2+sqrt3)^n, x_n^2-3y_n^2=1 The table keeps both coordinates at the same index. Multiplication by 2+sqrt3 advances the pair using the displayed integer update. Its determinant is one, and it preserves the quadratic norm. The zero-index pair is (1,0). Starting from (x_2,y_2)=","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"Choose d+1 distinct odd squarefree labels n_0,…,n_d, including 1, and set y_i=x_n_i. Their Lagrange polynomials are","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-27bd33e3fbb9a571","tex":"Y_j","aliases":"Y_j","display":"Y_j","kind":"use","excerpt":"Y_j(w)=∑_l=0^b-1H_jl(w)X_l(w), H_jl(w)=∑_substack0≤ t<L t+l≡ j (b) h_t w^(t+l-j)/b.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e3d4192e022af00a","tex":"y_j","aliases":"y_j","display":"y_j","kind":"use","excerpt":"For an OGF sequence a_m, define lanes x_l[n]=a_bn+l and X_l(w)=∑_nx_l[n]w^n. Apply finite ordinary taps and obtain y_j[n]=b_bn+j. Then the exact matrix identity is","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"(n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logarithm of 1-n^-β>0. The atom n=1 is excluded because its logarithm is undefined. At positive integer depths its omitted contribution would be zero. At depth zero the corresponding mass-one atom is absent. In particular, A(0)=Z_Q(2a)-1, where Z_Q(v)=∑_substackn≥1","count":18},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z Z_Q","display":"{Z_Q}","kind":"use","excerpt":"ights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β). Then A(s)=∑_n∈ S_Qw_ne^-λ_ns, w_n=n^-2a. The positive weights are summable, and the distinct support frequencies decrease to zero. If p_* is the least prime not dividing Q, then τ=-2log(1-p_*^-β), M=Z_Q(2a)-","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"Polynomial theorem. For every n, A_n∈ Z[q,r], all its nonzero coefficients are positive integers, and A_n(q,r)=A_n(r,q). Every monomial q^u r^v satisfies","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c656dc90ffbe53cd","tex":"\\mathcal Z","aliases":"Z mathcal","display":"\\mathcal Z","kind":"use","excerpt":"4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(z)=d_N(z^N). Let w_j be the nonzero zeros of d_N, counted with multiplicity. Each corresponds to one rotation orbit of N zeros of D_N. Using Argw∈(-π,π], define Z_N(s)=∑_jw_j^-s, R","count":38},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"Taking successively (w,g)=(z,z),(z^2,z),(z,z^2) yields qr=ab, q^2r=a^2b, and qr^2=ab^2. Nonzero scales then give q=a and r=b. Conversely these equalities make the two components of (S1) identical on every input, so (1,-1) F[X]=0. The only possible autonomous rows therefore span a one-dimensional space, which cannot be the two rows of an invertible ","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f0af7329a24be52a","tex":"z^{201}","aliases":"z^201","display":"z^{201}","kind":"use","excerpt":"How to read the figure. The left curves divide the difference by the tap weight, so the relevant small quantity is the plotted height multiplied by ε. The right panel records the proved order, which is determined by arbitrarily large radii. The curve computation uses powers through z^201, with a normalized analytic omitted-tail bound below 5.26×10^","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4140de75758a272d","tex":"z^3","aliases":"z^3","display":"z^3","kind":"use","excerpt":"nctionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T_q,r(z)=∑_n≥0c_nz^2n+1, c_0=1, coefficient matching gives c_n=frac12n+1∑_i=0^n-1 q^2i+1r^2(n-1-i)+1c_ic_n-1-i, n≥1. In particular, T_q,r(","count":11},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5e824ee39e9eb6f6","tex":"z^{321}","aliases":"z^321","display":"z^{321}","kind":"use","excerpt":"1Two scales inside the function For real 0<q,r<1, define the normalized odd solution T_q,r'(z)=1+T_q,r(qz)T_q,r(rz), T_q,r(0)=0. The contractions are part of the defining feedback. The function is symmetric in q,r, but unequal scales retain left/right address information. fig01_asymmetric_functionsComputed Taylor curves. Each entire curve uses term","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-34d4b252430fe95b","tex":"z^4","aliases":"z^4","display":"z^4","kind":"use","excerpt":"w^n(Nn)!=E_N,1(-w), D_N(z)=d_N(z^N). Let w_j be the nonzero zeros of d_N, counted with multiplicity. Each corresponds to one rotation orbit of N zeros of D_N. Using Argw∈(-π,π], define Z_N(s)=∑_jw_j^-s, Re s>1/N. fig06_orbit_zerosFor N=4, the four real/imaginary zeros in each orbit share one positive coordinate w=z^4. The plot shows the first six o","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-82f6433a3ae799be","tex":"z^5","aliases":"z^5","display":"z^5","kind":"use","excerpt":"nctionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T_q,r(z)=∑_n≥0c_nz^2n+1, c_0=1, coefficient matching gives c_n=frac12n+1∑_i=0^n-1 q^2i+1r^2(n-1-i)+1c_ic_n-1-i, n≥1. In particular, T_q,r(","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-82a6f2d2c744dfea","tex":"z^7","aliases":"z^7","display":"z^7","kind":"use","excerpt":"nctionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T_q,r(z)=∑_n≥0c_nz^2n+1, c_0=1, coefficient matching gives c_n=frac12n+1∑_i=0^n-1 q^2i+1r^2(n-1-i)+1c_ic_n-1-i, n≥1. In particular, T_q,r(","count":9},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-11c6aa222a0c3d6c","tex":"z^9","aliases":"z^9","display":"z^9","kind":"use","excerpt":"claimA worked order-one member T_1/3,2/3(z)=z+frac227z^3+ frac42187z^5+frac23211160261z^7+O(z^9). It has order one, and T_1/3,2/3(1)≈1.0759239897408249. The diagonal point q=r=1/2 gives f_1/2(z)=sqrt2sinh(z/sqrt2). Every off-diagonal member is separated from all rescaled diagonal members by the theorem above.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8b5c38b3bc4e7ad6","tex":"z^b","aliases":"z^b","display":"z^b","kind":"use","excerpt":"X(z)=∑_j=0^b-1z^jX_j(z^b).","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c9af0e3c44c5d431","tex":"z^{bn+j}","aliases":"z^bn+j","display":"z^{bn+j}","kind":"use","excerpt":"the full sequence For an ordinary generating function G(z)=∑_m≥0a_mz^m and integer b≥1, write G(z)=∑_j=0^b-1z^jG_j(z^b), G_j(t)=∑_n≥0a_bn+jt^n. The root projector is P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =z^jG_j(z^b), ω=e^2π i/b. The same projector works on an EGF, but the section retains its original factorials: ∑_na_bn+jz^bn+j/(bn+j)!. Removing t","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"use","excerpt":"X(z)=∑_j=0^b-1z^jX_j(z^b).","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-56835621b2f38901","tex":"z^m","aliases":"z^m","display":"z^m","kind":"use","excerpt":"For an EGF G(z)=∑_m≥0a_m z^m/m!, an integer base b≥1, a residue 0≤ j<b, and ω=e^2π i/b, define","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-afc30c788aac6f77","tex":"z^N","aliases":"z^N","display":"z^N","kind":"use","excerpt":"MonographPartIIIOrbit spectra and exact coordinatesSpectral continuation and the coefficient language of polyphase transforms. Spectral functions of zero orbits. pageheadingFAMILY 4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"use","excerpt":"lessonSEQUENCES / 06Catalan convolution rebuilds the next termS06_catalan_convolution C_n+1=∑_k=0^nC_kC_n-k, C(z)=1+zC(z)^2 The five bars are the five products contributing to C_5. The input indices add to four, and their sum supplies the next coefficient. Here C(z)=∑_n≥0C_nz^n is an ordinary generating function. The contributions are 1·14, 1·5, 2·","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-11335cb05beff53a","tex":"z^t","aliases":"z^t","display":"z^t","kind":"use","excerpt":"For OGFs this is multiplication by H(z)=∑_th_tz^t: B(z)=H(z)X(z). For EGFs it is a sum of repeated zero-constant antiderivatives:","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e8394415f576ddd6","tex":"Z_0","aliases":"Z_0","display":"Z_0","kind":"use","excerpt":"To bound a seed interval, the coefficient majorant from Theorem F.1 gives F_i(1)≤2. Put R_k=Q^-k, Z_0=2, and","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"If (S2) holds, Q_c is constant on every fiber of (u,v)↦ c_1u+c_2v. Fix z_0≠0. Polynomial interpolation gives a scalar polynomial f with f(0)=0 and f(sz_0)=1 at every distinct scale s∈{q,r,a,b}. For inputs u=xf,v=yf, where x,y∈ C,","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b23ccb12e2adf65f","tex":"Z_1","aliases":"Z_1","display":"Z_1","kind":"use","excerpt":"item[Squarefree Dirichlet sum.] For v>1, Z_Q(v)=ζ(v)ζ(2v)^-1∏_pmid Q(1+p^-v)^-1. The unrestricted notation is D(v)=Z_1(v). Thus A_a,β,Q(0)=Z_Q(2a)-1. item[Prime-sieve measure.] ν_F,λ is the arithmetic atomic measure obtained by multiplying the mass at x_n=(1-n^-2)^2 by ∏_p∈ F(1+λ_p1_pmid n). It is positive exactly when every real strength satisfies","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d691d3e93ced8a74","tex":"\\mathcal Z_2","aliases":"Z_2 mathcal","display":"\\mathcal Z_2","kind":"use","excerpt":"claimTwo exact Hurwitz-zeta reductions For N=2, d_2(w)=cossqrt w, whose zeros are w_j=π^2(j+1/2)^2, j≥0. Therefore Z_2(s)=π^-2sζ(2s,1/2). For N=4, D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), and the rotation quotient has one zero w_j=(sqrt2π)^4(j+1/2)^4 per orbit. Hence Z_4(s)=(sqrt2π)^-4sζ(4s,1/2). These anchors are classical reductions. The general spectra","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6d41bcad5561556b","tex":"\\mathcal Z_4","aliases":"Z_4 mathcal","display":"\\mathcal Z_4","kind":"use","excerpt":"claimTwo exact Hurwitz-zeta reductions For N=2, d_2(w)=cossqrt w, whose zeros are w_j=π^2(j+1/2)^2, j≥0. Therefore Z_2(s)=π^-2sζ(2s,1/2). For N=4, D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), and the rotation quotient has one zero w_j=(sqrt2π)^4(j+1/2)^4 per orbit. Hence Z_4(s)=(sqrt2π)^-4sζ(4s,1/2). These anchors are classical reductions. The general spectra","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0abec7ade721129c","tex":"Z_F","aliases":"Z_F","display":"Z_F","kind":"use","excerpt":"Z_F(s;λ) =∑_n≥1μ(n)^2n^-sκ_F(n;λ) =D(s)∏_p∈ F(1+fracλ_pp^s+1).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d533223c5565c514","tex":"Z_k","aliases":"Z_k","display":"Z_k","kind":"use","excerpt":"Z_k=Q^-k(1+Z_k-1^2).","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-def556e17bedf59b","tex":"Z_{k-1}","aliases":"Z_k-1","display":"Z_{k-1}","kind":"use","excerpt":"log Z_k≤ klog(1/Q)+log2+2log Z_k-1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-71ff3283061f7e06","tex":"Z_{k-1}^2","aliases":"Z_k-1^2","display":"Z_{k-1}^2","kind":"use","excerpt":"Z_k=Q^-k(1+Z_k-1^2).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-35030b1cc5b33f81","tex":"Z_N","aliases":"Z_N","display":"Z_N","kind":"use","excerpt":"Summing this elementary recurrence gives log Z_N≤ W_N. Therefore H_i(e^t)≤ W_N throughout 1≤ t≤1+Δ, since e^t≤ e/m<3/m≤ R_N.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9edc9c695b5ebed3","tex":"\\mathcal Z_N","aliases":"Z_N mathcal","display":"\\mathcal Z_N","kind":"use","excerpt":"4One spectral address for each rotation orbit For integer N≥2, define the classical Mittag--Leffler seed~[source reference] d_N(w)=∑_n≥0frac(-1)^nw^n(Nn)!=E_N,1(-w), D_N(z)=d_N(z^N). Let w_j be the nonzero zeros of d_N, counted with multiplicity. Each corresponds to one rotation orbit of N zeros of D_N. Using Argw∈(-π,π], define Z_N(s)=∑_jw_j^-s, R","count":28},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5518ad0b85c40a1b","tex":"\\mathcal Z_N'","aliases":"Z_N' mathcal","display":"\\mathcal Z_N'","kind":"use","excerpt":"pageheadingORBIT SPECTRA / CONTINUATION THEOREMGlobal meromorphic continuation For every integer N≥2, the orbit zeta has a meromorphic continuation to the whole complex plane with exactly one pole: s=1/N, Res_s=1/N Z_N(s)=fracsin(π/N)Nπ. Its exact absolute-convergence abscissa is 1/N. At nonpositive integers, Z_N(-m)=0 (m≥0), Z_N'(0)=-log N. The ze","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ce4c9fac39d3db2e","tex":"Z_Q","aliases":"Z_Q","display":"Z_Q","kind":"use","excerpt":"(n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logarithm of 1-n^-β>0. The atom n=1 is excluded because its logarithm is undefined. At positive integer depths its omitted contribution would be zero. At depth zero the corresponding mass-one atom is absent. In particular, A(0)=Z_Q(2a)-1, where Z_Q(v)=∑_substackn≥1","count":10},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1b14ae4d8553a856","tex":"|^{-(2a-1)/\\beta}","aliases":"|^-(2a-1)/β beta","display":"|^{-(2a-1)/\\beta}","kind":"use","excerpt":"acter resolution For characters χ,ψ modulo a fixed Q, let η=χoverlineψ. When η is nonprincipal, the elementary squarefree character sum is O_Q(sqrt x), with no density term. The same sector partial-integration argument therefore gives A_χ,ψ(s) =O_a,β,Q,ϵ(|s|^-(2a-1/2)/β), |arg s|≤π/2-ϵ. Compared with the principal |s|^-(2a-1)/β scale, this is faste","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cb560c18b76991c8","tex":"|^{-(2a-1/2)/\\beta}","aliases":"|^-(2a-1/2)/β beta","display":"|^{-(2a-1/2)/\\beta}","kind":"use","excerpt":"Let α=frac2a-1β, δ_Q=frac1ζ(2)∏_pmid Qfracpp+1. As |s|→∞ in any closed sector |arg s|≤π/2-ϵ with 0<ϵ<π/2, A_a,β,Q(s)=fracδ_QβB(α,2s+1) +O_a,β,Q,ϵ(|s|^-(2a-1/2)/β). In particular, A_a,β,Q(s)∼fracδ_QΓ(α)β(2s)^-α. The principal power is used in the right half-plane. The beta expression is an asymptotic continuous-density model, not an identity with th","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f313372e5dd9cb9b","tex":"|^{-1/(2\\beta)}","aliases":"|^-1/(2β) beta","display":"|^{-1/(2\\beta)}","kind":"use","excerpt":"A_χ,ψ(s) =O_a,β,Q,ε(|s|^-(2a-1/2)/β), fracA_χ,ψ(s)A_a,β,Q(s) =O_a,β,Q,ε(|s|^-1/(2β)).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-05c6643e270be04a","tex":"|^{-\\Re s}","aliases":"|^-Re s Re","display":"|^{-\\Re s}","kind":"use","excerpt":"s have different jobs For 1/N<Re s<1, the genus-zero log derivative and the angular gap give absolute interchange in Z_N(s)=fracsinπ sπ ∫_0^∞ t^-sfracddtlog E_N(t)dd t. Each term uses the principal-branch beta integral ∫_0^∞ t^-s(t+w)^-1dd t=π w^-s/sinπ s. The denominator estimate bounds the absolute integral by a constant times |w|^-Re s, whose su","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fc53e2b4fff174f3","tex":"|^{-\\alpha}","aliases":"|^-α alpha","display":"|^{-\\alpha}","kind":"use","excerpt":"which is O_a,β,ε(|s|^-γ) because u≥ |s|sinε. The principal diagonal has the nonzero leading term δ_QΓ(α)β^-1(2s)^-α, α=(2a-1)/β, by the base sector theorem. Its modulus is bounded below by a positive fixed-sector multiple of |s|^-α for sufficiently large |s|. Division gives the second bound, since γ-α=1/(2β).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-acae6f249d7f62d2","tex":"|^{-\\gamma}","aliases":"|^-γ gamma","display":"|^{-\\gamma}","kind":"use","excerpt":"which is O_a,β,ε(|s|^-γ) because u≥ |s|sinε. The principal diagonal has the nonzero leading term δ_QΓ(α)β^-1(2s)^-α, α=(2a-1)/β, by the base sector theorem. Its modulus is bounded below by a positive fixed-sector multiple of |s|^-α for sufficiently large |s|. Division gives the second bound, since γ-α=1/(2β).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8aeefd5c7c18d478","tex":"|^{-\\sigma}","aliases":"|^-σ sigma","display":"|^{-\\sigma}","kind":"use","excerpt":"For completeness, the series bound |d_N(w)|≤exp(|w|^1/N) gives order at most 1/N<1. Hadamard factorization therefore has genus zero and a constant exponential factor, which equals one because d_N(0)=1. The usual zero-count consequence gives ∑_j|w_j|^-σ<∞ for every σ>1/N. Along the negative real axis (S3) also shows the order is exactly 1/N.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-48680b3fa133e34c","tex":"|^{1/N}","aliases":"|^1/N","display":"|^{1/N}","kind":"use","excerpt":"one rotation orbit of N zeros of D_N. Using Argw∈(-π,π], define Z_N(s)=∑_jw_j^-s, Re s>1/N. fig06_orbit_zerosFor N=4, the four real/imaginary zeros in each orbit share one positive coordinate w=z^4. The plot shows the first six orbits. Multiplicity is retained by the definition; simplicity is not assumed in general. The bound |d_N(w)|≤ e^|w|^1/N an","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"|∑_n≥ Nc_nz^2n+1| ≤ frac|z|ρ^N1-ρ, |z|^2<2.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"01Two seeds, one Fibonacci recurrenceS01_fibonacci_lucas F_n+2=F_n+1+F_n, L_n+2=L_n+1+L_n, L_n^2-5F_n^2=4(-1)^n The two families share the same recurrence and have different seeds: (F_0,F_1)=(0,1) and (L_0,L_1)=(2,1). The right panel uses the same index in both families. Its alternating defect stays exactly four in magnitude. At n=7, F_7=13 and L_7","count":17},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"le squarefree atoms. Their frequencies are at most λ_p and their total weight is the Euler-factor expression on the right. Termwise differentiation proves the bound for every derivative. fig11_prime_instabilityThe proved compact-window envelope shrinks as the excluded prime increases. At a=β=2, Q=1, R=5, excluding p=10007 changes the entire window ","count":33},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"The earlier moving-width lower argument used a fixed α<1; it proved orders below ρ and then recovered the order by αuparrow1, but did not establish positive lower type at ρ. This is a limitation of that argument, not a counterexample. The fixed-width comparison closes precisely that missing implication. The evidence archive retains execution failur","count":34},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7e20c498fc8e5735","tex":"\\alpha^s","aliases":"α^s alpha","display":"\\alpha^s","kind":"use","excerpt":"The decisive extension is from a scalar positive comparison to a comparison weighted by a positive eigenvector. The fixed-width integral controls the lower type at the exact exponent. A fixed proportional interval [α R,R], with α<1, would only give a smaller exponent through α^sP(s). Letting α tend to one recovers exact order but does not, by itsel","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-993a80257a1ce549","tex":"\\alpha_k","aliases":"α_k alpha","display":"\\alpha_k","kind":"use","excerpt":"Near zero write f(t)=(log E_N)'(t)=∑_k≥1α_kt^k-1, where α_k=(-1)^k+1 Z_N(k). For any integer K≥1, subtract K terms on [0,1]: I_K(s)= · ∫_0^1t^-s[f(t)-∑_k=1^Kα_kt^k-1]dd t +∑_k=1^Kfracα_kk-s","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"MonographPartIIArithmetic depth and signed measuresComplex parameters, positive kernels and the boundaries of positivity. Arithmetic at complex depth. pageheadingFAMILY 2Arithmetic energy at complex depth For real a>1/2, β>0, and integer Q≥1, define A_a,β,Q(s)=∑_substackn≥2","count":62},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ac909e48d2ea1c12","tex":"\\beta'","aliases":"β' beta","display":"\\beta'","kind":"use","excerpt":"a=a', β=β', rad(Q)=rad(Q').","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8735109003c9a8e3","tex":"\\beta^{-1}","aliases":"β^-1 beta","display":"\\beta^{-1}","kind":"use","excerpt":"which is O_a,β,ε(|s|^-γ) because u≥ |s|sinε. The principal diagonal has the nonzero leading term δ_QΓ(α)β^-1(2s)^-α, α=(2a-1)/β, by the base sector theorem. Its modulus is bounded below by a positive fixed-sector multiple of |s|^-α for sufficiently large |s|. Division gives the second bound, since γ-α=1/(2β).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"Let α=frac2a-1β, δ_Q=frac1ζ(2)∏_pmid Qfracpp+1. As |s|→∞ in any closed sector |arg s|≤π/2-ϵ with 0<ϵ<π/2, A_a,β,Q(s)=fracδ_QβB(α,2s+1) +O_a,β,Q,ϵ(|s|^-(2a-1/2)/β). In particular, A_a,β,Q(s)∼fracδ_QΓ(α)β(2s)^-α. The principal power is used in the right half-plane. The beta expression is an asymptotic continuous-density model, not an identity with th","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cb6ed207b06be299","tex":"\\boldsymbol\\gamma","aliases":"γ boldsymbol gamma","display":"\\boldsymbol\\gamma","kind":"use","excerpt":"widetildeν_F,γ =∑_n≥1μ(n)^2n^-4 (1+∑_p∈ Fγ_p1_pmid n)δ_x_n.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"Use partial integration against g_s(x)=x^-2a(1-x^-β)^2s, whose endpoint values vanish when Re s>0. The counting bound reduces the magnitude to a constant times ∫_1^∞sqrt x |g_s'(x)| dx. Writing u=Re s and γ=(2a-1/2)/β>0, direct substitution y=x^-β bounds its two derivative terms by","count":13},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2f08cea1939265ee","tex":"\\gamma_2","aliases":"γ_2 gamma","display":"\\gamma_2","kind":"use","excerpt":"ecessity uses the squarefree integer n=∏_p∈ Sp; sufficiency checks every atom's prime subset. Its finite Hankel matrices are affine functions of the strengths, so their positive-semidefinite feasible sets are convex spectrahedra. In contrast, the multiplicative family includes the arithmetic prime-overlap terms and has polynomial parameter dependen","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3cb24f35d371b658","tex":"\\gamma_3","aliases":"γ_3 gamma","display":"\\gamma_3","kind":"use","excerpt":"ecessity uses the squarefree integer n=∏_p∈ Sp; sufficiency checks every atom's prime subset. Its finite Hankel matrices are affine functions of the strengths, so their positive-semidefinite feasible sets are convex spectrahedra. In contrast, the multiplicative family includes the arithmetic prime-overlap terms and has polynomial parameter dependen","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4b118facef29f432","tex":"\\gamma_N","aliases":"γ_N gamma","display":"\\gamma_N","kind":"use","excerpt":"Re x-Re(ω^j x) ≥ [cosfracπ2N-cosfrac3π2N]|x| =:γ_N|x|>0. tagS5","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3e52bdbf6bbe4050","tex":"\\gamma_p","aliases":"γ_p gamma","display":"\\gamma_p","kind":"use","excerpt":"claimA separate additive family has different geometry For weights 1+∑_p∈ Fγ_p1_pmid n, positivity is equivalent to 1+∑_γ_p<0γ_p≥0. Every prime subset appears at its squarefree product, proving necessity. Checking each atom proves sufficiency. Its finite Hankel matrices are affine in γ, so the semidefinite feasible sets are convex spectrahedra. The","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"lessonSEQUENCES / 16Sums and differences form a polynomial ladderS16_polynomial_ladder T_n=∑_j=1^nj, H_n=∑_j=1^nT_j, Δ T_n=n, Δ H_n=T_n The upper table follows positive integers into triangular and tetrahedral numbers. Squares and cubes occupy the lower rows for comparison. The arrows reverse prefix sums by backward differences, Δ f_n=f_n-f_n-1. Al","count":24},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"Let α=frac2a-1β, δ_Q=frac1ζ(2)∏_pmid Qfracpp+1. As |s|→∞ in any closed sector |arg s|≤π/2-ϵ with 0<ϵ<π/2, A_a,β,Q(s)=fracδ_QβB(α,2s+1) +O_a,β,Q,ϵ(|s|^-(2a-1/2)/β). In particular, A_a,β,Q(s)∼fracδ_QΓ(α)β(2s)^-α. The principal power is used in the right half-plane. The beta expression is an asymptotic continuous-density model, not an identity with th","count":49},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"claimAn endpoint mass is numerically delicate For the undeformed depth-zero measure, set ν_θ=ν+θδ_0, θ≥-1. Write h_j=∫ P_j(x)^2 dν(x) and K_m(x,y)=∑_j=0^mP_j(x)P_j(y)/h_j. The Uvarov formula~[source reference] is P_m^[θ](x)=P_m(x)- fracθ P_m(0)1+θ K_m-1(0,0)K_m-1(x,0), with K_-1=0 for degree zero. The mass at zero is recovered by 1≤frac1K_m(0,0)≤1+","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9adbb02c2ad54bb0","tex":"\\delta_1","aliases":"δ_1 delta","display":"\\delta_1","kind":"use","excerpt":"For p=2, test with f(x)=x(1-x)^m-2, m≥2: δ_m≤frac409681(frac16M_017-1) (frac272567)^2(m-2), M_0=105π^-4. This follows from x_2=9/16 and x_n≥64/81 at nonzero odd nodes. Exactly δ_1=p^4; the limit is an infinite-resolution phenomenon.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-288a046779bf284f","tex":"\\Delta_j","aliases":"Δ_j Delta","display":"\\Delta_j","kind":"use","excerpt":"Theorem B.2 (AI-2), gap bound. Suppose the first j-1 atoms have already been removed exactly, put Δ_j=λ_j-λ_j+1>0, and fix h>0. Define","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fa05543491a5a34a","tex":"\\delta_{jl}","aliases":"δ_jl delta","display":"\\delta_{jl}","kind":"use","excerpt":"∑_j=0^b-1 P_b,j=I, P_b,j P_b,l=δ_jl P_b,j.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-23fe82c4424f67f6","tex":"\\delta_m","aliases":"δ_m delta","display":"\\delta_m","kind":"use","excerpt":"nvalues of A_m^-1/2B_mA_m^-1/2. Every determinant root is real and below -1; possible coefficient poles occur there, although numerator cancellation is possible. fig05_sieve_boundaryThe line is a proved bound for p=2; dots are ordinary high-precision pencil roots. A defect λ=-1.001 eventually violates positivity even though it can pass low-order ch","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6773ce97fdbad01c","tex":"\\delta_{p+m}","aliases":"δ_p+m delta","display":"\\delta_{p+m}","kind":"use","excerpt":"deg f<m frac∫ f^2ddν_p-free ∫ f^2ddν_p-divisible, δ_mdownarrow0. The detector yields, for C_κ=∏_q≠ pmax(1,1+λ_q), δ_p+m≤fracpC_κ C_p^23r_p^2m. The denominator retains the atom p; all lower nodes in the numerator are killed. This extends the prime-two bound to every prime and to simultaneous positive or deletion strengths. It does not assert simple ","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7d38daffe35514fe","tex":"\\delta_Q","aliases":"δ_Q delta","display":"\\delta_Q","kind":"use","excerpt":"Let α=frac2a-1β, δ_Q=frac1ζ(2)∏_pmid Qfracpp+1. As |s|→∞ in any closed sector |arg s|≤π/2-ϵ with 0<ϵ<π/2, A_a,β,Q(s)=fracδ_QβB(α,2s+1) +O_a,β,Q,ϵ(|s|^-(2a-1/2)/β). In particular, A_a,β,Q(s)∼fracδ_QΓ(α)β(2s)^-α. The principal power is used in the right half-plane. The beta expression is an asymptotic continuous-density model, not an identity with th","count":13},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d4a2fe4186d0321f","tex":"\\delta_Q^\\beta","aliases":"δ_Q^β delta beta","display":"\\delta_Q^\\beta","kind":"use","excerpt":"The counting law N_Q(x)=δ_Qx+O_Q(sqrt x) gives λ_j=2δ_Q^β j^-β [1+O_Q,β(j^-1/2+j^-β)], β=-lim_j→∞fraclogλ_jlog j. Once β is recovered, n_j=(1-e^-λ_j/2)^-1/β, a=-fraclog w_j2log n_j. Prime membership in this integer support then recovers the excluded prime set. The reconstruction is exact from infinite data; it does not certify that a finite search ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a13d7dd648fdb1b2","tex":"\\delta_Q^{\\beta}","aliases":"δ_Q^β delta beta","display":"\\delta_Q^{\\beta}","kind":"use","excerpt":"n_j=fracjδ_Q(1+O_Q(j^-1/2)), λ_j=2δ_Q^βj^-β (1+O_Q,β(j^-1/2+j^-β)).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d7bcd96d2ab76f6a","tex":"\\delta_{x_n}","aliases":"δ_x_n delta","display":"\\delta_{x_n}","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-666e2cba77c64efa","tex":"\\delta_{\\lambda_j}","aliases":"δ_λ_j delta lambda","display":"\\delta_{\\lambda_j}","kind":"use","excerpt":"Uniqueness of a positive measure from its Laplace transform is classical. Schilling, Song and Vondraček~[source reference] give the uniqueness and Bernstein representation results. Here the measure is ∑_jw_jδ_λ_j, the transform variable is real depth s>0, and finiteness follows from ∑_jw_j<∞. The exact recovery of a, β and rad(Q), the rank asymptot","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Let α=frac2a-1β, δ_Q=frac1ζ(2)∏_pmid Qfracpp+1. As |s|→∞ in any closed sector |arg s|≤π/2-ϵ with 0<ϵ<π/2, A_a,β,Q(s)=fracδ_QβB(α,2s+1) +O_a,β,Q,ϵ(|s|^-(2a-1/2)/β). In particular, A_a,β,Q(s)∼fracδ_QΓ(α)β(2s)^-α. The principal power is used in the right half-plane. The beta expression is an asymptotic continuous-density model, not an identity with th","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"fracμ_13z^3+fracμ_1μ_215z^5+O(z^7). If q_j^ρ_j+r_j^ρ_j=1, then ρ(F_Θ)=max_j activeρ_j. The positive weights alter local coefficients. The most expansive active channel determines the asymptotic order. A zero-weight channel is removed before taking the maximum. fig07_weak_tapA two-channel example: scales (1/4,1/4) with weight 1-ε, and (3/4,3/4) with","count":55},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a1e744dd9b829b48","tex":"\\varepsilon^{-\\rho/(\\rho-1/2)}","aliases":"ε^-ρ/(ρ-1/2) varepsilon rho rho","display":"\\varepsilon^{-\\rho/(\\rho-1/2)}","kind":"use","excerpt":"The radius column displays the analytic formula approximately. For a certificate use the formula itself or a rigorously rounded upper value. These large numbers reflect the conservatism of the global comparison. They do not mean that observations at all smaller radii fail. The formula grows proportionally to ε^-ρ/(ρ-1/2) as the tap vanishes.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"ARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite. Thus τ_m(λ)=det(A_m+(1+λ)B","count":26},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9657c492efcc34dd","tex":"\\eta^2","aliases":"η^2 eta","display":"\\eta^2","kind":"use","excerpt":"A_χ,ψ(s) =∑_j≥0(-1)^jbinom2sj [fracL(2a+jβ,η) L(4a+2jβ,η^2)-1].","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-412b907de59c5fc0","tex":"\\eta_i","aliases":"η_i eta","display":"\\eta_i","kind":"use","excerpt":"ARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite. Thus τ_m(λ)=det(A_m+(1+λ)B","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-efb4acc8c0653938","tex":"\\eta_N","aliases":"η_N eta","display":"\\eta_N","kind":"use","excerpt":"[0,π]. Thus, once (N-1)e^-γ_N|x|<1, the distinguished term e^x in (S1) is larger in modulus than the sum of all competitors, and d_N(w)≠0. Consequently, sufficiently large zeros avoid the entire closed left half-plane. There are only finitely many remaining zeros in a bounded disk. None lies on the negative ray, since d_N(-t)>0 for t≥0. It follows ","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"claimAn endpoint mass is numerically delicate For the undeformed depth-zero measure, set ν_θ=ν+θδ_0, θ≥-1. Write h_j=∫ P_j(x)^2 dν(x) and K_m(x,y)=∑_j=0^mP_j(x)P_j(y)/h_j. The Uvarov formula~[source reference] is P_m^[θ](x)=P_m(x)- fracθ P_m(0)1+θ K_m-1(0,0)K_m-1(x,0), with K_-1=0 for degree zero. The mass at zero is recovered by 1≤frac1K_m(0,0)≤1+","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"η κ_j(t), κ_j(t)=fracA(-t)R_j(-t) ∼fracw_1w_je^(λ_1-λ_j)t.","count":19},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2eebd371ce816638","tex":"\\kappa_F","aliases":"κ_F kappa","display":"\\kappa_F","kind":"use","excerpt":"x_n=(1-n^-2)^2, κ_F(n;λ)=∏_p∈ F(1+λ_p1_pmid n), ν_F,λ =∑_n≥1μ(n)^2n^-4κ_F(n;λ)δ_x_n.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-09fa540d6637b1f4","tex":"\\kappa_j","aliases":"κ_j kappa","display":"\\kappa_j","kind":"use","excerpt":"η κ_j(t), κ_j(t)=fracA(-t)R_j(-t) ∼fracw_1w_je^(λ_1-λ_j)t.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-95fab017df947b09","tex":"\\kappa_N","aliases":"κ_N kappa","display":"\\kappa_N","kind":"use","excerpt":"pageheadingORBIT SPECTRA / LARGE RADIUSA finite exponential sum controls the tail Put E_N(t)=d_N(-t)=∑_n≥0t^n/(Nn)! and ω=e^2π i/N. For x≥0, E_N(x^N)=frac1N∑_j=0^N-1e^ω^jx. Root filtering proves this identity term by term: only powers divisible by N remain. The positive real exponential is dominant, so log E_N(t)=t^1/N-log N+O(e^-κ_N t^1/N), κ_N=1-","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9e6fb51590070831","tex":"\\boldsymbol\\lambda","aliases":"λ boldsymbol lambda","display":"\\boldsymbol\\lambda","kind":"use","excerpt":"item[Prime-sieve measure.] ν_F,λ is the arithmetic atomic measure obtained by multiplying the mass at x_n=(1-n^-2)^2 by ∏_p∈ F(1+λ_p1_pmid n). It is positive exactly when every real strength satisfies λ_p≥-1. item[Hankel order and polynomial degree.] For moments M_j=∫ x^jddν, the order-m matrix is H_m=[M_i+j]_0≤ i,j<m. It tests quadratic forms of r","count":11},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"-v =fracζ(v)ζ(2v)∏_pmid Q(1+p^-v)^-1, v>1. fig03_complex_depthLeft: a=β=2 at real depths, using the exact zeta--Newton representation at 90-digit working precision. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β)","count":124},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"The logarithm is real. The omitted n=1 atom is not restored in any inverse formula. Let n_1<n_2<⋯ be the support integers, and abbreviate λ_j=λ_n_j, w_j=n_j^-2a. Then λ_1>λ_2>⋯>0, λ_j→0, and ∑_jw_j=M=Z_Q(2a)-1<∞, where","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-975e1a0c6d000741","tex":"\\lambda_2","aliases":"λ_2 lambda","display":"\\lambda_2","kind":"use","excerpt":"The logarithm is real. The omitted n=1 atom is not restored in any inverse formula. Let n_1<n_2<⋯ be the support integers, and abbreviate λ_j=λ_n_j, w_j=n_j^-2a. Then λ_1>λ_2>⋯>0, λ_j→0, and ∑_jw_j=M=Z_Q(2a)-1<∞, where","count":12},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-29ccb409dc80afd9","tex":"\\lambda_3","aliases":"λ_3 lambda","display":"\\lambda_3","kind":"use","excerpt":"pageheadingPRIME SIEVES / FINITE TESTSMoment positivity is a weaker requirement fig12_multi_prime_domainThe exact positive-measure domain and the order-one moment test have different parameter geometry. Regions passing a finite test outside the safe orthant still contain negative atoms. For primes two and three, M_0=105π^-4(1+λ_2/17)(1+λ_3/82). Bot","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-976821aed6410d3e","tex":"\\lambda_5","aliases":"λ_5 lambda","display":"\\lambda_5","kind":"use","excerpt":"The separate rational certificates (5) do not depend on those approximate roots. The script constructs negative detectors for primes 2,3,5,7, at Hankel orders 8,11,19,40, respectively. These sufficient orders are not asserted to be minimal. In the p=7 example, the other strengths are λ_2=1/2,λ_3=-1,λ_5=2, and λ_7=-1.0001; the exact rational upper b","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-617fec3b79c52760","tex":"\\lambda_7","aliases":"λ_7 lambda","display":"\\lambda_7","kind":"use","excerpt":"The separate rational certificates (5) do not depend on those approximate roots. The script constructs negative detectors for primes 2,3,5,7, at Hankel orders 8,11,19,40, respectively. These sufficient orders are not asserted to be minimal. In the p=7 example, the other strengths are λ_2=1/2,λ_3=-1,λ_5=2, and λ_7=-1.0001; the exact rational upper b","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"use","excerpt":"The logarithm is real. The omitted n=1 atom is not restored in any inverse formula. Let n_1<n_2<⋯ be the support integers, and abbreviate λ_j=λ_n_j, w_j=n_j^-2a. Then λ_1>λ_2>⋯>0, λ_j→0, and ∑_jw_j=M=Z_Q(2a)-1<∞, where","count":28},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9d24463159df8045","tex":"\\lambda_{j+1}","aliases":"λ_j+1 lambda","display":"\\lambda_{j+1}","kind":"use","excerpt":"R_j(-t)=w_je^λ_jt(1+r_j(t)), 0≤ r_j(t)≤fracM_j+1w_j e^-(λ_j-λ_j+1)t.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d43eb991ba79b2dc","tex":"\\lambda_j^k","aliases":"λ_j^k lambda","display":"\\lambda_j^k","kind":"use","excerpt":"The monomial moment matrices also have infinite rank: m_k=(-1)^kA^(k)(0)=∑_jw_jλ_j^k, and for every nonzero polynomial P, ∑_jw_j|P(λ_j)|^2>0, because a nonzero polynomial cannot vanish at infinitely many distinct frequencies. All finite derivative Hankel matrices are strictly positive definite.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-525b314d831dddb8","tex":"\\lambda_{L+1}","aliases":"λ_L+1 lambda","display":"\\lambda_{L+1}","kind":"use","excerpt":"E_L(t)=e^tλ_L+1fracL^1-2a2a-1.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4a848fec60c201a8","tex":"\\lambda_{\\lfloor cj\\rfloor}","aliases":"λ_lfloor cjrfloor lambda lfloor rfloor","display":null,"kind":"use","excerpt":"-fraclog(λ_lfloor cjrfloor/λ_j)log c =β+O_Q,β,c(j^-1/2+j^-β).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-75281f36da6b8ff7","tex":"\\lambda_n","aliases":"λ_n lambda","display":"\\lambda_n","kind":"use","excerpt":"-v =fracζ(v)ζ(2v)∏_pmid Q(1+p^-v)^-1, v>1. fig03_complex_depthLeft: a=β=2 at real depths, using the exact zeta--Newton representation at 90-digit working precision. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β)","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c38c87f0ab778af3","tex":"\\lambda_{n_j}","aliases":"λ_n_j lambda","display":"\\lambda_{n_j}","kind":"use","excerpt":"The logarithm is real. The omitted n=1 atom is not restored in any inverse formula. Let n_1<n_2<⋯ be the support integers, and abbreviate λ_j=λ_n_j, w_j=n_j^-2a. Then λ_1>λ_2>⋯>0, λ_j→0, and ∑_jw_j=M=Z_Q(2a)-1<∞, where","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fdc2dc9f687c22a4","tex":"\\lambda_p","aliases":"λ_p lambda","display":"\\lambda_p","kind":"use","excerpt":"item[Prime-sieve measure.] ν_F,λ is the arithmetic atomic measure obtained by multiplying the mass at x_n=(1-n^-2)^2 by ∏_p∈ F(1+λ_p1_pmid n). It is positive exactly when every real strength satisfies λ_p≥-1. item[Hankel order and polynomial degree.] For moments M_j=∫ x^jddν, the order-m matrix is H_m=[M_i+j]_0≤ i,j<m. It tests quadratic forms of r","count":49},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9acb20de82fbb3ac","tex":"\\lambda_p^k","aliases":"λ_p^k lambda","display":"\\lambda_p^k","kind":"use","excerpt":"displaystyle sup_|s|≤ R|A_a,β,Q^(k)(s)-A_a,β,Qp^(k)(s)| ≤ λ_p^k e^Rλ_p fracZ_Q(2a)p^2a+1.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2adeb808f9a94c18","tex":"\\lambda_{p_*}","aliases":"λ_p_* lambda","display":"\\lambda_{p_*}","kind":"use","excerpt":"claimNo polynomial-coefficient linear ODE Suppose ∑_j=0^rP_j(s)A^(j)(s)=0, where the P_j are polynomials and are not all zero. At s=-t, divide by the exponential of the first support atom. The remaining tail is exponentially smaller, so the polynomial ∑_jP_j(-t)(-λ_p_*)^j tends to zero exponentially and must vanish identically. Remove that atom and","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7a9bc3a491e56c32","tex":"\\lambda_q","aliases":"λ_q lambda","display":"\\lambda_q","kind":"use","excerpt":"The exact coordinate formula is M_j(λ)=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) ∏_p∈ F(1+fracλ_pp^4+2r+1). Coefficients of the monic families lie in Q(π^-2,λ_p:p∈ F). For two primes, the overlap term is visible: M_j=M_j(0,0)+λ_pU_p,j+λ_qU_q,j +λ_pλ_qU_pq,j. Here U_pq,j>0 is the pq-divisible submeasure moment. The family does not reduce to adding s","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"changes the address ruleS14_divisor_convolution ∑_dmid nμ(d)=1_n=1, ∑_dmid nφ(d)=n The figure places the six divisors of twelve in matching columns. Möbius contributions sum to zero; totient contributions sum to twelve. The address rule is divisibility. This differs from the additive index splitting used by ordinary power-series convolution. For n=","count":33},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ac6cb066658534c5","tex":"\\mu_1","aliases":"μ_1 mu","display":"\\mu_1","kind":"use","excerpt":"pageheadingFAMILY 1 / TAPSFeedback banks and weak-channel sensitivity Choose finitely many channels j=1,…,J, with h_j>0, ∑_jh_j=1, and 0<q_j,r_j<1. Define F_Θ'(z)=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz), F_Θ(0)=0. These taps act inside the nonlinear definition. With μ_1=∑_jh_jq_jr_j and μ_2=∑_jh_jq_jr_j(q_j^2+r_j^2), F_Θ(z)=z+fracμ_13z^3+fracμ_1μ_215z^5+O(z^7).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a202073a149b5a7a","tex":"\\mu_2","aliases":"μ_2 mu","display":"\\mu_2","kind":"use","excerpt":"pageheadingFAMILY 1 / TAPSFeedback banks and weak-channel sensitivity Choose finitely many channels j=1,…,J, with h_j>0, ∑_jh_j=1, and 0<q_j,r_j<1. Define F_Θ'(z)=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz), F_Θ(0)=0. These taps act inside the nonlinear definition. With μ_1=∑_jh_jq_jr_j and μ_2=∑_jh_jq_jr_j(q_j^2+r_j^2), F_Θ(z)=z+fracμ_13z^3+fracμ_1μ_215z^5+O(z^7).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":24},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-52e066c2e19f8906","tex":"\\nu_{F,\\boldsymbol\\lambda}","aliases":"ν_F,λ nu boldsymbol lambda","display":"\\nu_{F,\\boldsymbol\\lambda}","kind":"use","excerpt":"item[Prime-sieve measure.] ν_F,λ is the arithmetic atomic measure obtained by multiplying the mass at x_n=(1-n^-2)^2 by ∏_p∈ F(1+λ_p1_pmid n). It is positive exactly when every real strength satisfies λ_p≥-1. item[Hankel order and polynomial degree.] For moments M_j=∫ x^jddν, the order-m matrix is H_m=[M_i+j]_0≤ i,j<m. It tests quadratic forms of r","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d1773026827069e8","tex":"\\nu_{\\mathrm{non}p}","aliases":"ν_nonp nu mathrm","display":"\\nu_{\\mathrm{non}p}","kind":"use","excerpt":"pageheadingFAMILY 3 / BOUNDARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-62d0d50e0643749d","tex":"\\nu_p","aliases":"ν_p nu","display":"\\nu_p","kind":"use","excerpt":"pageheadingFAMILY 3 / BOUNDARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4ee64081d97aef16","tex":"\\nu_{p,0}","aliases":"ν_p,0 nu","display":"\\nu_{p,0}","kind":"use","excerpt":"pageheadingFAMILY 3 / BOUNDARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-13fa1be848469109","tex":"\\nu_{p,\\lambda}","aliases":"ν_p,λ nu lambda","display":"\\nu_{p,\\lambda}","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-cdef040a8920bbb6","tex":"\\nu_{p\\text{-divisible}}","aliases":"ν_p-divisible nu text","display":"\\nu_{p\\text{-divisible}}","kind":"use","excerpt":"Thus every one-variable determinant fiber has exact degree m and all roots real below -1. Its nearest margin satisfies δ_m=min_substackf∈ R[x]∖{0} deg f<m frac∫ f^2ddν_p-free ∫ f^2ddν_p-divisible, δ_mdownarrow0. The detector yields, for C_κ=∏_q≠ pmax(1,1+λ_q), δ_p+m≤fracpC_κ C_p^23r_p^2m. The denominator retains the atom p; all lower nodes in the n","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bc55fd682b8a78b4","tex":"\\nu_{p\\text{-free}}","aliases":"ν_p-free nu text","display":"\\nu_{p\\text{-free}}","kind":"use","excerpt":"Thus every one-variable determinant fiber has exact degree m and all roots real below -1. Its nearest margin satisfies δ_m=min_substackf∈ R[x]∖{0} deg f<m frac∫ f^2ddν_p-free ∫ f^2ddν_p-divisible, δ_mdownarrow0. The detector yields, for C_κ=∏_q≠ pmax(1,1+λ_q), δ_p+m≤fracpC_κ C_p^23r_p^2m. The denominator retains the atom p; all lower nodes in the n","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-251ea3a3e1a96024","tex":"\\nu_\\theta","aliases":"ν_θ nu theta","display":"\\nu_\\theta","kind":"use","excerpt":"claimAn endpoint mass is numerically delicate For the undeformed depth-zero measure, set ν_θ=ν+θδ_0, θ≥-1. Write h_j=∫ P_j(x)^2 dν(x) and K_m(x,y)=∑_j=0^mP_j(x)P_j(y)/h_j. The Uvarov formula~[source reference] is P_m^[θ](x)=P_m(x)- fracθ P_m(0)1+θ K_m-1(0,0)K_m-1(x,0), with K_-1=0 for degree zero. The mass at zero is recovered by 1≤frac1K_m(0,0)≤1+","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"Let α=frac2a-1β, δ_Q=frac1ζ(2)∏_pmid Qfracpp+1. As |s|→∞ in any closed sector |arg s|≤π/2-ϵ with 0<ϵ<π/2, A_a,β,Q(s)=fracδ_QβB(α,2s+1) +O_a,β,Q,ϵ(|s|^-(2a-1/2)/β). In particular, A_a,β,Q(s)∼fracδ_QΓ(α)β(2s)^-α. The principal power is used in the right half-plane. The beta expression is an asymptotic continuous-density model, not an identity with th","count":57},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-69b66a03c86339b8","tex":"\\pi^{-2}","aliases":"π^-2 pi","display":"\\pi^{-2}","kind":"use","excerpt":"two packets for p=2, depth zero. Right: initial atom masses. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta stencils: M_j^p,λ=∑_r=0^2j(-1)^rbinom2jr fracζ(4+2r)ζ(8+4r) (1+fracλp^4+2r+1). All polynomial coefficients belong to Q(π^-2,λ). With Δ=M_0M_","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5499160581543767","tex":"\\pi^{-2s}","aliases":"π^-2s pi","display":"\\pi^{-2s}","kind":"use","excerpt":"claimTwo exact Hurwitz-zeta reductions For N=2, d_2(w)=cossqrt w, whose zeros are w_j=π^2(j+1/2)^2, j≥0. Therefore Z_2(s)=π^-2sζ(2s,1/2). For N=4, D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), and the rotation quotient has one zero w_j=(sqrt2π)^4(j+1/2)^4 per orbit. Hence Z_4(s)=(sqrt2π)^-4sζ(4s,1/2). These anchors are classical reductions. The general spectra","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d923d7282e111692","tex":"\\pi^{-4}","aliases":"π^-4 pi","display":"\\pi^{-4}","kind":"use","excerpt":"For p=2, test with f(x)=x(1-x)^m-2, m≥2: δ_m≤frac409681(frac16M_017-1) (frac272567)^2(m-2), M_0=105π^-4. This follows from x_2=9/16 and x_n≥64/81 at nonzero odd nodes. Exactly δ_1=p^4; the limit is an infinite-resolution phenomenon.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"claimTwo exact Hurwitz-zeta reductions For N=2, d_2(w)=cossqrt w, whose zeros are w_j=π^2(j+1/2)^2, j≥0. Therefore Z_2(s)=π^-2sζ(2s,1/2). For N=4, D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), and the rotation quotient has one zero w_j=(sqrt2π)^4(j+1/2)^4 per orbit. Hence Z_4(s)=(sqrt2π)^-4sζ(4s,1/2). These anchors are classical reductions. The general spectra","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f4875cececfa7801","tex":"\\pi^4","aliases":"π^4 pi","display":"\\pi^4","kind":"use","excerpt":"For a=β=2,Q=1, the mass removed by p is exactly 105/[π^4(p^4+1)]. The examples p=101,1009,10007 quantify how quickly large-prime distinctions disappear on |s|≤5.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d4480c8bdf9c98bc","tex":"\\rho^N","aliases":"ρ^N rho","display":"\\rho^N","kind":"use","excerpt":"|∑_n≥ Nc_nz^2n+1| ≤ frac|z|ρ^N1-ρ, |z|^2<2.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c86a36ff5a44dcd8","tex":"\\rho_j","aliases":"ρ_j rho","display":"\\rho_j","kind":"use","excerpt":"vity Choose finitely many channels j=1,…,J, with h_j>0, ∑_jh_j=1, and 0<q_j,r_j<1. Define F_Θ'(z)=1+∑_jh_jF_Θ(q_jz)F_Θ(r_jz), F_Θ(0)=0. These taps act inside the nonlinear definition. With μ_1=∑_jh_jq_jr_j and μ_2=∑_jh_jq_jr_j(q_j^2+r_j^2), F_Θ(z)=z+fracμ_13z^3+fracμ_1μ_215z^5+O(z^7). If q_j^ρ_j+r_j^ρ_j=1, then ρ(F_Θ)=max_j activeρ_j. The positive ","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8ca4d7e240aeeb8a","tex":"\\rho_\\alpha","aliases":"ρ_α rho alpha","display":"\\rho_\\alpha","kind":"use","excerpt":"claimWhy this equation appears Nonnegative coefficients give M(R)=T_q,r(R). The integral equation implies M(R)≤ R[1+M(qR)M(rR)], M(R)≥(1-α)R M(α qR)M(α rR),quad0<α<1. Taking logarithms turns product feedback into a sum over contracted scales. Comparison with CR^ρ proves the upper bound; comparison with cR^ρ_α, where (α q)^ρ_α+ (α r)^ρ_α=1, proves t","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"For completeness, the series bound |d_N(w)|≤exp(|w|^1/N) gives order at most 1/N<1. Hadamard factorization therefore has genus zero and a constant exponential factor, which equals one because d_N(0)=1. The usual zero-count consequence gives ∑_j|w_j|^-σ<∞ for every σ>1/N. Along the negative real axis (S3) also shows the order is exactly 1/N.","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a7c7531c6c696429","tex":"\\tau_0^{(k)}","aliases":"τ_0^(k) tau","display":"\\tau_0^{(k)}","kind":"use","excerpt":"operations with arithmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel determinants τ_m^(k)=det[M_k+i+j]_i,j<m, with","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-60c8e80f0348d076","tex":"\\tau_{m+1}^{(k)}","aliases":"τ_m+1^(k) tau","display":"\\tau_{m+1}^{(k)}","kind":"use","excerpt":"thmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel determinants τ_m^(k)=det[M_k+i+j]_i,j<m, with τ_0^(k)=1, τ_m^(k)","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9755870d27155352","tex":"\\tau_{m-1}^{(k+2)}","aliases":"τ_m-1^(k+2) tau","display":"\\tau_{m-1}^{(k+2)}","kind":"use","excerpt":"thmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel determinants τ_m^(k)=det[M_k+i+j]_i,j<m, with τ_0^(k)=1, τ_m^(k)","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5bca8e7bf8c9be62","tex":"\\tau_m^{(k)}","aliases":"τ_m^(k) tau","display":"\\tau_m^{(k)}","kind":"use","excerpt":"eheadingFAMILY 3 / TRANSPORTClassical operations with arithmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel determi","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a82f41142314ef63","tex":"\\tau_m^{(k+1)}","aliases":"τ_m^(k+1) tau","display":"\\tau_m^{(k+1)}","kind":"use","excerpt":"thmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel determinants τ_m^(k)=det[M_k+i+j]_i,j<m, with τ_0^(k)=1, τ_m^(k)","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1e726be0971dc17b","tex":"\\tau_m^{(k+2)}","aliases":"τ_m^(k+2) tau","display":"\\tau_m^{(k+2)}","kind":"use","excerpt":"thmetic coordinates The depth tilt x^k is a Christoffel transformation. Suppressing p,λ, P_m^(k+1)(x)=fracP_m+1^(k)(x) -[P_m+1^(k)(0)/P_m^(k)(0)]P_m^(k)(x)x. The numerator vanishes at zero. All roots are positive, so the displayed denominator coefficient is nonzero. For shifted Hankel determinants τ_m^(k)=det[M_k+i+j]_i,j<m, with τ_0^(k)=1, τ_m^(k)","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"pageheadingARITHMETIC DEPTH / REALIZATIONAn infinite feature space with arithmetic labels For the ordered support, define ϕ_s(j)=sqrtw_je^-λ_js. The kernel A(s+overline t) is its weighted ℓ^2 Gram kernel. Summability and bounded frequencies make every feature vector well defined.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"|N_Q(x)-δ_Qx|≤ C_Qsqrt x+1, C_Q=2^ω(Q)+2fracφ(Q)Q.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-74777c98027fb1ad","tex":"\\phi_s","aliases":"ϕ_s phi","display":"\\phi_s","kind":"use","excerpt":"pageheadingARITHMETIC DEPTH / REALIZATIONAn infinite feature space with arithmetic labels For the ordered support, define ϕ_s(j)=sqrtw_je^-λ_js. The kernel A(s+overline t) is its weighted ℓ^2 Gram kernel. Summability and bounded frequencies make every feature vector well defined.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"Let χ,ψ be Dirichlet characters modulo the same fixed integer Q, extended by zero outside the units, and put η=χoverlineψ. The base extension is","count":10},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Let χ,ψ be Dirichlet characters modulo the same fixed integer Q, extended by zero outside the units, and put η=χoverlineψ. The base extension is","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"The rank asymptotic can be made effective without an RH input. Write ω(Q)=#{p:pmid Q}. For real x≥4, inclusion and exclusion gives","count":14},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4a26976422bc1dc1","tex":"\\omega^{-jk}","aliases":"ω^-jk omega","display":"\\omega^{-jk}","kind":"use","excerpt":"P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =∑_m≡ j (b)a_mfracz^mm!.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a27b633daa590319","tex":"\\omega^j","aliases":"ω^j omega","display":"\\omega^j","kind":"use","excerpt":"pageheadingORBIT SPECTRA / LARGE RADIUSA finite exponential sum controls the tail Put E_N(t)=d_N(-t)=∑_n≥0t^n/(Nn)! and ω=e^2π i/N. For x≥0, E_N(x^N)=frac1N∑_j=0^N-1e^ω^jx. Root filtering proves this identity term by term: only powers divisible by N remain. The positive real exponential is dominant, so log E_N(t)=t^1/N-log N+O(e^-κ_N t^1/N), κ_N=1-","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a8e0712f7cf91f55","tex":"\\omega^{jm}","aliases":"ω^jm omega","display":"\\omega^{jm}","kind":"use","excerpt":"This equality follows by expanding the exponentials and using N^-1∑_jω^jm=1 when Nmid m, and zero otherwise. For real x≥0, the power series makes F_N(x)>0. Put","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-960f13a0e4265f44","tex":"\\omega^k","aliases":"ω^k omega","display":"\\omega^k","kind":"use","excerpt":"P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =∑_m≡ j (b)a_mfracz^mm!.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f7c438c89e61ad74","tex":"\\omega^{k(m-j)}","aliases":"ω^k(m-j) omega","display":"\\omega^{k(m-j)}","kind":"use","excerpt":"This holds as a formal series and wherever the original series converges. Indeed the coefficient multiplier is the geometric sum b^-1∑_k=0^b-1ω^k(m-j), equal to one or zero according to the residue. It proves all the projector identities","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"^nn!=exp(e^t-1) Each matrix row contains Stirling numbers of the second kind, counting partitions into k nonempty blocks. Summing across the row gives the Bell number at its right. The generating function is exponential, so its coefficient is B_n/n!. The fifth row contains 0,1,15,25,10,1. Its sum is 52=B_5. Equivalently, apply the Stirling transfor","count":16},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"For c=(c_j)∈ℓ^2, define","count":24},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"For c=(c_j)∈ℓ^2, define","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-43faa4d2efd90c1a","tex":"\\ell^T","aliases":"ℓ^T ell","display":"\\ell^T","kind":"use","excerpt":"ℓ=(a^ρ,1-q^ρ)^T, ℓ^TP(ρ)=ℓ^T.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8a05e97949811e78","tex":"\\ell_1","aliases":"ℓ_1 ell","display":"\\ell_1","kind":"use","excerpt":"The preceding proof also makes a coordinate invariant obstruction explicit. Under an invertible state change, write the original coordinates as u=ℓ_1(W),v=ℓ_2(W), where ℓ_1,ℓ_2 are independent linear forms. At the pair of distinct nodes (qz,rz), the first quadratic term has bilinear coefficient","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f71176b4c504c459","tex":"\\ell_2","aliases":"ℓ_2 ell","display":"\\ell_2","kind":"use","excerpt":"The preceding proof also makes a coordinate invariant obstruction explicit. Under an invertible state change, write the original coordinates as u=ℓ_1(W),v=ℓ_2(W), where ℓ_1,ℓ_2 are independent linear forms. At the pair of distinct nodes (qz,rz), the first quadratic term has bilinear coefficient","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"ℓ_i(x)=∏_j≠ ifracx-y_jy_i-y_j, b_i=∏_j≠ ifracmax(y_j,1-y_j)|y_i-y_j|, K_d=∑_i=0^dn_i^4b_i^2.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"The integrals are path independent on the disk. Linear-ODE uniqueness shows U=T on the previous disk, and there T(qz) equals the newly extended function at qz. Thus the same nonlinear equation holds on the enlarged disk. Iterating R↦ R/q exhausts the plane.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"0≤∂_h_jc_n≤ n d_n(Q).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c511f467b9789a5b","tex":"\\partial_{h_j}","aliases":"∂_h_j partial","display":"\\partial_{h_j}","kind":"use","excerpt":"0≤∂_h_jc_n≤ n d_n(Q).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-569331afbd5d2de3","tex":"\\nabla","aliases":"∇ nabla","display":"\\nabla","kind":"use","excerpt":"|v·∇ c_n|≤frac d2 n d_n(Q), |c_n(h)-c_n(g)|≤frac d2 n d_n(Q).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"displaystyle rad(Q)=∏_substackp prime p∉{n_j:j≥1}p.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"(n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logarithm of 1-n^-β>0. The atom n=1 is excluded because its logarithm is undefined. At positive integer depths its omitted contribution would be zero. At depth zero the corresponding mass-one atom is absent. In particular, A(0)=Z_Q(2a)-1, where Z_Q(v)=∑_substackn≥1 (n,Q)=1μ(n)^2n","count":35},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ebf4e0c91e2a4804","tex":"\\prod_{i=1}^{m}","aliases":"∏_i=1^m prod","display":"\\prod_{i=1}^{m}","kind":"use","excerpt":"ARYFinite positivity can conceal a signed defect Let ν_nonp and ν_p be the restrictions of the undeformed measure ν_p,0 to indices not divisible by p and divisible by p, respectively. Then ν_p,λ=ν_nonp+(1+λ)ν_p. Let A_m,B_m be the order-m Hankel matrices of these two unscaled positive measures. Both are positive definite. Thus τ_m(λ)=det(A_m+(1+λ)B","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-2d3db36262598bcd","tex":"\\prod_j","aliases":"∏_j prod","display":"\\prod_j","kind":"use","excerpt":"bit share one positive coordinate w=z^4. The plot shows the first six orbits. Multiplicity is retained by the definition; simplicity is not assumed in general. The bound |d_N(w)|≤ e^|w|^1/N and the finite-order zero-counting theorem give absolute local uniform convergence in the stated half-plane. Since the order is below one and d_N(0)=1, the Hada","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ea8969bf20f180b7","tex":"\\prod_j^{\\rm reg}","aliases":"∏_j^rm reg prod rm","display":"\\prod_j^{\\rm reg}","kind":"use","excerpt":"The negative-integer values remain zero, and Z_N'(0;a)=-log[N E_N(a)], ∏_j^rm reg(w_j+a)=N E_N,1(a). The finite part -log N and normalization -log E_N(a) give a zeta-regularized product, not an ordinary product. At N=2 it is 2coshsqrt a, the classical half-integer spectral determinant reproduced by the four shifted benchmarks. Functional equations ","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-5e4c84160d4d61eb","tex":"\\prod_{j\\ne i}","aliases":"∏_j≠ i prod ne","display":"\\prod_{j\\ne i}","kind":"use","excerpt":"ℓ_i(x)=∏_j≠ ifracx-y_jy_i-y_j, b_i=∏_j≠ ifracmax(y_j,1-y_j)|y_i-y_j|, K_d=∑_i=0^dn_i^4b_i^2.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-eb89c659c1dcb524","tex":"\\prod_{n=1}^{a-1}","aliases":"∏_n=1^a-1 prod","display":"\\prod_{n=1}^{a-1}","kind":"use","excerpt":"pageheadingPRIME SIEVES / CONSTRUCTIVE WITNESSEvery negative prime atom has a finite detector For an integer a≥2, let L_a(x)=∏_n=1^a-1fracx-x_nx_a-x_n, C_a=L_a(1), r_a=frac1-x_a+11-x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_si","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-34f53e64299c0c85","tex":"\\prod_{n\\in S}","aliases":"∏_n∈ S prod in","display":"\\prod_{n\\in S}","kind":"use","excerpt":"P_S,m(x)=∏_n∈ Sfracx-x_nx_2-x_n (frac1-x1-x_2)^m, C_S=∏_n∈ Sfrac1-x_n|x_2-x_n|, r_S=frac1-x_n_+1-x_2<1.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9705db81396e2375","tex":"\\prod_{p\\mid Q}","aliases":"∏_pmid Q prod mid","display":null,"kind":"use","excerpt":"(n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logarithm of 1-n^-β>0. The atom n=1 is excluded because its logarithm is undefined. At positive integer depths its omitted contribution would be zero. At depth zero the corresponding mass-one atom is absent. In particular, A(0)=Z_Q(2a)-1, where Z_Q(v)=∑_substackn≥1 (n,Q)=1μ(n)^2n","count":7},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-421711867d59e872","tex":"\\prod_{p\\in F}","aliases":"∏_p∈ F prod in","display":"\\prod_{p\\in F}","kind":"use","excerpt":"item[Prime-sieve measure.] ν_F,λ is the arithmetic atomic measure obtained by multiplying the mass at x_n=(1-n^-2)^2 by ∏_p∈ F(1+λ_p1_pmid n). It is positive exactly when every real strength satisfies λ_p≥-1. item[Hankel order and polynomial degree.] For moments M_j=∫ x^jddν, the order-m matrix is H_m=[M_i+j]_0≤ i,j<m. It tests quadratic forms of r","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-516d6ddd3d40ea65","tex":"\\prod_{p\\in S}","aliases":"∏_p∈ S prod in","display":"\\prod_{p\\in S}","kind":"use","excerpt":"Equivalently, expansion in subsets S⊆ F gives the coefficient D(s)/∏_p∈ S(p^s+1) for ∏_p∈ Sλ_p. This is an independently useful coordinate formula for prime overlap.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6d272a8de10a4f57","tex":"\\prod_{q\\in F}","aliases":"∏_q∈ F prod in","display":"\\prod_{q\\in F}","kind":"use","excerpt":"x_a<1, f_a,m(x)=L_a(x)(frac1-x1-x_a)^m. This polynomial kills every lower node, equals one at x_a, and has absolute value at most C_ar_a^m at every higher node. fig13_signed_detectorA rational detector for p=3, λ_3=-1.1, λ_2=1. The first exact bound proving negativity is m=5, so a Hankel matrix of order eight detects the signed defect. Set C_F=∏_q∈","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-466ef17a1d5c1f76","tex":"\\prod_{q\\in F\\setminus\\{p\\}}","aliases":"∏_q∈ F∖{p} prod in setminus","display":"\\prod_{q\\in F\\setminus\\{p\\}}","kind":"use","excerpt":"Fix p∈ F, with all other strengths at least -1, and write κ(n)=∏_q∈ F∖{p}(1+λ_q1_qmid n). Separate the positive measure into its p-free and p-divisible parts:","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1d0d85414c2df4d5","tex":"\\prod_{q\\ne p}","aliases":"∏_q≠ p prod ne","display":"\\prod_{q\\ne p}","kind":"use","excerpt":"deg f<m frac∫ f^2ddν_p-free ∫ f^2ddν_p-divisible, δ_mdownarrow0. The detector yields, for C_κ=∏_q≠ pmax(1,1+λ_q), δ_p+m≤fracpC_κ C_p^23r_p^2m. The denominator retains the atom p; all lower nodes in the numerator are killed. This extends the prime-two bound to every prime and to simultaneous positive or deletion strengths. It does not assert simple ","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d17889f843382d5d","tex":"\\prod_{\\substack{p\\ \\mathrm{prime}\\\\\n                                      p\\notin\\{n_j:j\\ge1\\}}}","aliases":"∏_substackp prime p∉{n_j:j≥1} prod substack mathrm notin ge","display":"\\prod_{\\substack{p\\ \\mathrm{prime}\\\\\n                                      p\\notin\\{n_j:j\\ge1\\}}}","kind":"use","excerpt":"displaystyle rad(Q)=∏_substackp prime p∉{n_j:j≥1}p.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-71ad65db94900d3f","tex":"\\sum_{(d,Q)=1}","aliases":"∑_(d,Q)=1 sum","display":"\\sum_{(d,Q)=1}","kind":"use","excerpt":"2=∑_d^2mid nμ(d). For (d,Q)=1, the number of m≤ x/d^2 coprime to Q differs from (φ(Q)/Q)x/d^2 by at most 2^ω(Q). Summing over at most sqrt x values of d gives the first part of C_Q. The omitted main-term tail is at most (φ(Q)/Q)x/lfloorsqrt xrfloor≤2(φ(Q)/Q)sqrt x. Removing n=1 adds one. The main-term density is (φ(Q)/Q)∑_(d,Q)=1μ(d)d^-2=δ_Q by its","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-82f319514727c447","tex":"\\sum_{1\\le n<p}","aliases":"∑_1≤ n<p sum le","display":"\\sum_{1\\le n<p}","kind":"use","excerpt":"A_rm tail=D(4)fracp^4p^4+1 ∏_q≠ p(1+fracλ_qq^4+1) -∑_1≤ n<pμ(n)^2n^-4κ(n)>0,","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-88173d4630911f5f","tex":"\\sum_{d^2\\mid n}","aliases":"∑_d^2mid n sum mid","display":null,"kind":"use","excerpt":"To verify the constants, expand μ(n)^2=∑_d^2mid nμ(d). For (d,Q)=1, the number of m≤ x/d^2 coprime to Q differs from (φ(Q)/Q)x/d^2 by at most 2^ω(Q). Summing over at most sqrt x values of d gives the first part of C_Q. The omitted main-term tail is at most (φ(Q)/Q)x/lfloorsqrt xrfloor≤2(φ(Q)/Q)sqrt x. Removing n=1 adds one. The main-term density is","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bba356c5b89a8e22","tex":"\\sum_{d\\le\\sqrt x}","aliases":"∑_d≤sqrt x sum le sqrt","display":"\\sum_{d\\le\\sqrt x}","kind":"use","excerpt":"∑_n≤ xμ(n)^2η(n) =∑_d≤sqrt xμ(d)η(d)^2 ∑_m≤ x/d^2η(m)=O_Q(sqrt x).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d241aacdf6fc5d4d","tex":"\\sum_{i+j=n-1}","aliases":"∑_i+j=n-1 sum","display":"\\sum_{i+j=n-1}","kind":"use","excerpt":"A_n=∑_i+j=n-1binom2n2i+1q^2i+1r^2j+1A_iB_j, B_n=∑_i+j=n-1binom2n2i+1a^2i+1b^2j+1A_iB_j,","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-73a8e90cd2970d4f","tex":"\\sum_{i=0}^{4}","aliases":"∑_i=0^4 sum","display":"\\sum_{i=0}^{4}","kind":"use","excerpt":"claimInterpolation controls the whole polynomial space Choose the odd squarefree labels n_i=1,3,5,7,11, write y_i=x_n_i, and put b_i=∏_j≠ ifracmax(y_j,1-y_j)|y_i-y_j|, K_4=∑_i=0^4n_i^4b_i^2. Lagrange interpolation and weighted Cauchy--Schwarz show that, for every polynomial f of degree at most four, Q_δ(f)≥ (1-fracδ K_412) ∑_i=0^4n_i^-4f(y_i)^2. Ex","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-adb6721290f5f31c","tex":"\\sum_{i=0}^{d}","aliases":"∑_i=0^d sum","display":"\\sum_{i=0}^{d}","kind":"use","excerpt":"ℓ_i(x)=∏_j≠ ifracx-y_jy_i-y_j, b_i=∏_j≠ ifracmax(y_j,1-y_j)|y_i-y_j|, K_d=∑_i=0^dn_i^4b_i^2.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-964202d717c850f2","tex":"\\sum_{i=0}^{n}","aliases":"∑_i=0^n sum","display":"\\sum_{i=0}^{n}","kind":"use","excerpt":"u_n+1=frac∑_i=0^nq^2i+1r^2(n-i)+1u_i v_n-i2n+3, v_n+1=frac∑_i=0^na^2i+1b^2(n-i)+1u_i v_n-i2n+3.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-67b8cfba70c769c5","tex":"\\sum_{i=0}^{n-1}","aliases":"∑_i=0^n-1 sum","display":"\\sum_{i=0}^{n-1}","kind":"use","excerpt":"on is symmetric in q,r, but unequal scales retain left/right address information. fig01_asymmetric_functionsComputed Taylor curves. Each entire curve uses terms through z^321; positive-coefficient Cauchy estimates bound the omitted tail on each displayed interval. The black dashed curve is ordinary tangent. For T_q,r(z)=∑_n≥0c_nz^2n+1, c_0=1, coeff","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-872e42e0e6f809d7","tex":"\\sum_{j<m}","aliases":"∑_j<m sum","display":"\\sum_{j<m}","kind":"use","excerpt":"If at least two strengths in a positive multiplicative family are nonzero, its entire depth-zero monic polynomial sequence cannot equal any one-prime sequence on the same variable and node system. A direct proof avoids importing a determinacy theorem. Write P_m=x^m+∑_j<mc_mjx^j. The equations ∫ P_m dν=0 recursively determine every normalized moment","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d65790fef3711913","tex":"\\sum_{j=0}^{b-1}","aliases":"∑_j=0^b-1 sum","display":"\\sum_{j=0}^{b-1}","kind":"use","excerpt":"∑_j=0^b-1 P_b,j=I, P_b,j P_b,l=δ_jl P_b,j.","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-96f9384f676f024f","tex":"\\sum_{j=0}^{m}","aliases":"∑_j=0^m sum","display":"\\sum_{j=0}^{m}","kind":"use","excerpt":"claimAn endpoint mass is numerically delicate For the undeformed depth-zero measure, set ν_θ=ν+θδ_0, θ≥-1. Write h_j=∫ P_j(x)^2 dν(x) and K_m(x,y)=∑_j=0^mP_j(x)P_j(y)/h_j. The Uvarov formula~[source reference] is P_m^[θ](x)=P_m(x)- fracθ P_m(0)1+θ K_m-1(0,0)K_m-1(x,0), with K_-1=0 for degree zero. The mass at zero is recovered by 1≤frac1K_m(0,0)≤1+","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-586bc3571b381e64","tex":"\\sum_{j=0}^{N-1}","aliases":"∑_j=0^N-1 sum","display":"\\sum_{j=0}^{N-1}","kind":"use","excerpt":"pageheadingORBIT SPECTRA / LARGE RADIUSA finite exponential sum controls the tail Put E_N(t)=d_N(-t)=∑_n≥0t^n/(Nn)! and ω=e^2π i/N. For x≥0, E_N(x^N)=frac1N∑_j=0^N-1e^ω^jx. Root filtering proves this identity term by term: only powers divisible by N remain. The positive real exponential is dominant, so log E_N(t)=t^1/N-log N+O(e^-κ_N t^1/N), κ_N=1-","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-47306c517d6a1276","tex":"\\sum_{j=0}^{r}","aliases":"∑_j=0^r sum","display":"\\sum_{j=0}^{r}","kind":"use","excerpt":"claimNo polynomial-coefficient linear ODE Suppose ∑_j=0^rP_j(s)A^(j)(s)=0, where the P_j are polynomials and are not all zero. At s=-t, divide by the exponential of the first support atom. The remaining tail is exponentially smaller, so the polynomial ∑_jP_j(-t)(-λ_p_*)^j tends to zero exponentially and must vanish identically. Remove that atom and","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3d78cd26a42024c2","tex":"\\sum_{j=1}^{J}","aliases":"∑_j=1^J sum","display":"\\sum_{j=1}^{J}","kind":"use","excerpt":"F'(z)=1+∑_j=1^Jh_jF(q_jz)F(r_jz), F(0)=0,","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bcc0e2d264600f5a","tex":"\\sum_{j=1}^{k-1}","aliases":"∑_j=1^k-1 sum","display":"\\sum_{j=1}^{k-1}","kind":"use","excerpt":"1/N and the finite-order zero-counting theorem give absolute local uniform convergence in the stated half-plane. Since the order is below one and d_N(0)=1, the Hadamard factorization has genus zero: d_N(w)=∏_j(1-w/w_j). At positive integers, a log derivative gives -wfracd_N'(w)d_N(w)=∑_k≥1 Z_N(k)w^k. If a_n=(-1)^n/(Nn)! and b_k= Z_N(k), then b_k=-k","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-04a3e524463e2337","tex":"\\sum_{j=1}^{m-1}","aliases":"∑_j=1^m-1 sum","display":"\\sum_{j=1}^{m-1}","kind":"use","excerpt":"b_m:= Z_N(m) =-m c_m-∑_j=1^m-1c_jb_m-j. tagS17","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7f926f49d177196b","tex":"\\sum_{j=1}^{N-1}","aliases":"∑_j=1^N-1 sum","display":"\\sum_{j=1}^{N-1}","kind":"use","excerpt":"h_N(x)=∑_j=1^N-1e^(ω^j-1)x, c_N=1-cos(2π/N)>0.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-49afb54019149b8f","tex":"\\sum_{j\\ge0}","aliases":"∑_j≥0 sum ge","display":"\\sum_{j\\ge0}","kind":"use","excerpt":"pageheadingFAMILY 2 / CALCULUSClosed shifts, infinite differential complexity The Euler product for Z_Q shows that only rad(Q)=∏_pmid Qp matters. Expansion of the depth factor gives A_a,β,Q(s+1)=A_a,β,Q(s)-2A_a+β/2,β,Q(s) +A_a+β,β,Q(s). The fractional transfer and Newton formulas are A_a,β,Q(s+t)=∑_j≥0(-1)^jbinom2tjA_a+jβ/2,β,Q(s), A_a,β,Q(t)=∑_j≥0","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0c18bcf01edb7e76","tex":"\\sum_{j\\ge1}","aliases":"∑_j≥1 sum ge","display":"\\sum_{j\\ge1}","kind":"use","excerpt":"(Uc)(s)=∑_j≥1c_jsqrtw_j e^-λ_js.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-69dc5a5d6e10e4b9","tex":"\\sum_{k=0}^{b-1}","aliases":"∑_k=0^b-1 sum","display":"\\sum_{k=0}^{b-1}","kind":"use","excerpt":"P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =∑_m≡ j (b)a_mfracz^mm!.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7c520b68dc16ea6b","tex":"\\sum_{k=0}^{K}","aliases":"∑_k=0^K sum","display":"\\sum_{k=0}^{K}","kind":"use","excerpt":"G_N,a(t)=∑_k=0^Kαbinomα-1ka^k t^α-k-1 +O(t^α-K-2). tagT5","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-910ea7e4acbeebab","tex":"\\sum_{k=0}^m","aliases":"∑_k=0^m sum","display":"\\sum_{k=0}^m","kind":"use","excerpt":"claimThree operations that must be kept distinct For an EGF E_a(z)=∑_ma_mz^m/m!, differentiation transports future coefficients: E_a^(k)=∑_ma_m+kz^m/m!. Causal ordinary taps instead act by repeated zero-constant integration: E_b=∑_ℓ h_ℓ I^ℓ E_a. Multiplication of two EGFs gives Hurwitz convolution, b_m=∑_k=0^mbinom mk a_kh_m-k. It is not ordinary s","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-41e0c211ba80d17c","tex":"\\sum_{k=0}^{n}","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^{n}","kind":"use","excerpt":"lessonSEQUENCES / 13Apéry uses taps that depend on the indexS13_apery_recurrence (n+1)^3A_n+1=(2n+1)(17n^2+17n+5)A_n-n^3A_n-1 The prefix annotation begins the Apéry sequence defined by A_n=∑_k=0^nbinom nk^2binomn+kk^2. The curves show its growth through index fourteen on a base-ten logarithmic scale. The lower diagram evaluates the recurrence at n=","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e861dfba300b543d","tex":"\\sum_{k=1}^K","aliases":"∑_k=1^K sum","display":"\\sum_{k=1}^K","kind":"use","excerpt":"Near zero write f(t)=(log E_N)'(t)=∑_k≥1α_kt^k-1, where α_k=(-1)^k+1 Z_N(k). For any integer K≥1, subtract K terms on [0,1]: I_K(s)= · ∫_0^1t^-s[f(t)-∑_k=1^Kα_kt^k-1]dd t +∑_k=1^Kfracα_kk-s · +∫_1^∞ t^-s[f(t)-frac1N t^1/N-1]dd t +frac1N(s-1/N). The first integral is holomorphic for Re s<K+1. The second is entire because its remainder decays exponen","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-aecc1a98e370ff5d","tex":"\\sum_{k>j}","aliases":"∑_k>j sum","display":"\\sum_{k>j}","kind":"use","excerpt":"Every limit exists and recovers exactly the j-th support frequency and weight. Indeed, if M_j+1=∑_k>jw_k, then for every t≥0,","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1ebed7c5a175f44d","tex":"\\sum_{k\\ge0}","aliases":"∑_k≥0 sum ge","display":"\\sum_{k\\ge0}","kind":"use","excerpt":"^-s, Re s>1/N. The rightward shift preserves avoidance of the negative branch cut. Normalize the product with E_N(a) and use log E_N(a+t) in the Mellin argument. For a>0 there are genuine simple poles at s=1/N-k, k≥0: Res_s=1/N-k Z_N(s;a) =fracsin(π/N)Nπfrac(1-1/N)_kk!a^k. Indeed the large-radius derivative expands as frac1N(a+t)^1/N-1 =frac1N∑_k≥0","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-01f56a5e1c2b8b56","tex":"\\sum_{k\\ge1}","aliases":"∑_k≥1 sum ge","display":"\\sum_{k\\ge1}","kind":"use","excerpt":"ts. Multiplicity is retained by the definition; simplicity is not assumed in general. The bound |d_N(w)|≤ e^|w|^1/N and the finite-order zero-counting theorem give absolute local uniform convergence in the stated half-plane. Since the order is below one and d_N(0)=1, the Hadamard factorization has genus zero: d_N(w)=∏_j(1-w/w_j). At positive intege","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4d9b6e3fa631ea76","tex":"\\sum_{l=0}^{b-1}","aliases":"∑_l=0^b-1 sum","display":"\\sum_{l=0}^{b-1}","kind":"use","excerpt":"Y_j(w)=∑_l=0^b-1H_jl(w)X_l(w), H_jl(w)=∑_substack0≤ t<L t+l≡ j (b) h_t w^(t+l-j)/b.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7dd1f8372600c494","tex":"\\sum_m","aliases":"∑_m sum","display":"\\sum_m","kind":"use","excerpt":"For an ordinary generating function X(z)=∑_ma_m z^m, define X_j(w)=∑_na_bn+jw^n. Then","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-87b4d05e06bc5d37","tex":"\\sum_{m=1}^{K}","aliases":"∑_m=1^K sum","display":"\\sum_{m=1}^{K}","kind":"use","excerpt":"B_N,K(s)= · ∫_0^1x^-Ns[g_N(x)-∑_m=1^KN a_N,m-1x^Nm-1]dx · +∑_m=1^Kfraca_N,m-1m-s +∫_1^∞ x^-Ns[g_N(x)-1]dx +frac1Ns-1. tagS13","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4bf4d5fe8b7255ff","tex":"\\sum_{m\\equiv j\\,(b)}","aliases":"∑_m≡ j (b) sum equiv","display":"\\sum_{m\\equiv j\\,(b)}","kind":"use","excerpt":"P_b,jG(z)=frac1b∑_k=0^b-1ω^-jkG(ω^kz) =∑_m≡ j (b)a_mfracz^mm!.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6d36bd874658e219","tex":"\\sum_{m\\le x/d^2}","aliases":"∑_m≤ x/d^2 sum le","display":"\\sum_{m\\le x/d^2}","kind":"use","excerpt":"∑_n≤ xμ(n)^2η(n) =∑_d≤sqrt xμ(d)η(d)^2 ∑_m≤ x/d^2η(m)=O_Q(sqrt x).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-77352be184f09175","tex":"\\sum_{m\\le y}","aliases":"∑_m≤ y sum le","display":"\\sum_{m\\le y}","kind":"use","excerpt":"Proof. Distinct characters give a nonprincipal η. Its sum over a complete period is zero, so |∑_m≤ yη(m)|≤ Q. Hence","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3984e551ebd0bd42","tex":"\\sum_{m\\ge t}","aliases":"∑_m≥ t sum ge","display":"\\sum_{m\\ge t}","kind":"use","excerpt":"Proof. The EGF of the delayed sequence is ∑_m≥ ta_m-tz^m/m!=I^tG(z). Summing taps gives the formula. Conversely, G^(t) has coefficient sequence a_m+t, so derivatives sample future coefficients and do not implement these past delays. square","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d328b95fa488fa30","tex":"\\sum_{m\\ge0}","aliases":"∑_m≥0 sum ge","display":"\\sum_{m\\ge0}","kind":"use","excerpt":"For an EGF G(z)=∑_m≥0a_m z^m/m!, an integer base b≥1, a residue 0≤ j<b, and ω=e^2π i/b, define","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a0a9cffbc383b69e","tex":"\\sum_{m\\ge1}","aliases":"∑_m≥1 sum ge","display":"\\sum_{m\\ge1}","kind":"use","excerpt":"G_N(t)=∑_k≥0a_N,kt^k, g_N(x)=∑_m≥1N a_N,m-1x^Nm-1. tagS12","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","excerpt":"item[Coefficient conventions.] An ordinary generating function uses ∑_na_nz^n. An exponential generating function uses ∑_na_nz^n/n!. Their product and shift rules differ. item[Polyphase section and finite taps.] Polyphase splits a coefficient sequence by residue class. Finite impulse response (FIR) taps act by finite ordinary convolution. A phase m","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-45953a160933c484","tex":"\\sum_{n\\text{ even}}","aliases":"∑_n even sum text","display":"\\sum_{n\\text{ even}}","kind":"use","excerpt":"The total even mass satisfies ∑_n evenμ(n)^2n^-4<frac116(1+frac13)=1/12. Keeping the selected nodes in the positive part yields the explicit bound","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a7626aeba75a509f","tex":"\\sum_{n=0}^{\\infty}","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^{\\infty}","kind":"use","excerpt":"d_N(w)=∑_n=0^∞frac(-1)^n w^n(Nn)! =E_N,1(-w).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-afbea6d048f6cfef","tex":"\\sum_{n=1}^{N}","aliases":"∑_n=1^N sum","display":"\\sum_{n=1}^{N}","kind":"use","excerpt":"0≤ L(R)-∑_n=1^Nn d_nR^2n+1 ≤12(N+1)t^2N+3. tagRA-2","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c5fbf0290642fcfd","tex":"\\sum_{n>a}","aliases":"∑_n>a sum","display":"\\sum_{n>a}","kind":"use","excerpt":"∑_n>a|μ(n)^2n^-4κ_F(n)| ≤fracC_F3a^3. tag3","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-67e8088ae0f488e2","tex":"\\sum_{n>K}","aliases":"∑_n>K sum","display":"\\sum_{n>K}","kind":"use","excerpt":"This polynomial has degree |S|+m, equals one at x_2, and kills every positive atom through the cutoff. At each remaining positive node, the product of the absolute root factors is at most C_S, and the last factor is at most r_S^m. Dropping all other negative atoms and bounding the positive tail by the integral ∑_n>Kn^-4≤1/(3K^3) gives","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0d5aa9b849d8c625","tex":"\\sum_{n>L}","aliases":"∑_n>L sum","display":"\\sum_{n>L}","kind":"use","excerpt":"Indeed all omitted frequencies are at most λ_L+1, and ∑_n>Ln^-2a≤∫_L^∞ x^-2a dx. Ignoring squarefreeness and coprimality makes this a valid upper bound. If r_0,r_1 are the two positive truncated residual sums at t,t+h, the true secant belongs to","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-35da066aecc0b7b4","tex":"\\sum_{n\\in\\mathcal S_Q}","aliases":"∑_n∈ S_Q sum in mathcal","display":"\\sum_{n\\in\\mathcal S_Q}","kind":"use","excerpt":"mid Q(1+p^-v)^-1, v>1. fig03_complex_depthLeft: a=β=2 at real depths, using the exact zeta--Newton representation at 90-digit working precision. Right: initial squarefree frequencies and weights for Q=1. Only finitely many atoms are shown; the proof uses the infinite support. Let S_Q={n≥2:μ(n)^2=1, (n,Q)=1} and λ_n=-2log(1-n^-β). Then A(s)=∑_n∈ S_Q","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-00ccbfb616e03af5","tex":"\\sum_{n\\le x}","aliases":"∑_n≤ x sum le","display":"\\sum_{n\\le x}","kind":"use","excerpt":"∑_n≤ xμ(n)^2η(n) =∑_d≤sqrt xμ(d)η(d)^2 ∑_m≤ x/d^2η(m)=O_Q(sqrt x).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-bbca946d1ba849e5","tex":"\\sum_{n\\ge N}","aliases":"∑_n≥ N sum ge","display":"\\sum_{n\\ge N}","kind":"use","excerpt":"|∑_n≥ Nc_nz^2n+1| ≤ frac|z|ρ^N1-ρ, |z|^2<2.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"Prime sieves and signed measures. pageheadingFAMILY 3An infinite prime shelf changes the packets For a prime p and λ≥-1, let x_n=(1-n^-2)^2, ν_p,λ=∑_n≥1μ(n)^2n^-4 (1+λ1_pmid n)δ_x_n. Define monic orthogonal polynomials P_m^(k;p,λ) for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_pol","count":8},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-069c58c03dba2dbd","tex":"\\sum_{n\\ge2}","aliases":"∑_n≥2 sum ge","display":"\\sum_{n\\ge2}","kind":"use","excerpt":"A_χ,ψ(s)=∑_n≥2μ(n)^2 η(n)n^-2a(1-n^-β)^2s.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-85191274d535f60c","tex":"\\sum_{n\\ge2,\\ (n,Q)=1}","aliases":"∑_n≥2, (n,Q)=1 sum ge","display":"\\sum_{n\\ge2,\\ (n,Q)=1}","kind":"use","excerpt":"item[Arithmetic depth A_a,β,Q.] The entire sum ∑_n≥2, (n,Q)=1μ(n)^2n^-2a(1-n^-β)^2s, with a>1/2, β>0, Q≥1 integer and the real logarithm in the complex power. The atom n=1 is excluded at every depth. item[Möbius function and modulus radical.] μ(1)=1. For a product of k distinct primes, μ(n)=(-1)^k. If a prime square divides n, then μ(n)=0. The radi","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-b0678753651fceb8","tex":"\\sum_{p\\mid n}","aliases":"∑_pmid n sum mid","display":null,"kind":"use","excerpt":"A_m=[∑_pnmid nμ(n)^2n^-4κ(n)x_n^i+j]_i,j<m, B_m=[∑_pmid nμ(n)^2n^-4κ(n)x_n^i+j]_i,j<m.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-555bf12e7a9d43f1","tex":"\\sum_{p\\nmid n}","aliases":"∑_pnmid n sum nmid","display":null,"kind":"use","excerpt":"A_m=[∑_pnmid nμ(n)^2n^-4κ(n)x_n^i+j]_i,j<m, B_m=[∑_pmid nμ(n)^2n^-4κ(n)x_n^i+j]_i,j<m.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-af09eb8f7dc80b60","tex":"\\sum_{pq\\mid n}","aliases":"∑_pqmid n sum mid","display":null,"kind":"use","excerpt":"where U_pq,j=∑_pqmid nμ(n)^2n^-4x_n^j>0. More generally, with all other strengths at least -1, every mixed moment derivative in a set of distinct primes is strictly positive. It is the integral over atoms divisible by those primes, with the remaining nonnegative prime factors. Infinitely many such atoms remain after any finite collection of deletio","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d6a4f0f21d8284e7","tex":"\\sum_{p\\in F}","aliases":"∑_p∈ F sum in","display":"\\sum_{p\\in F}","kind":"use","excerpt":"claimA separate additive family has different geometry For weights 1+∑_p∈ Fγ_p1_pmid n, positivity is equivalent to 1+∑_γ_p<0γ_p≥0. Every prime subset appears at its squarefree product, proving necessity. Checking each atom proves sufficiency. Its finite Hankel matrices are affine in γ, so the semidefinite feasible sets are convex spectrahedra. The","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-fb8d107fbfa44c79","tex":"\\sum_{p\\in S}","aliases":"∑_p∈ S sum in","display":"\\sum_{p\\in S}","kind":"use","excerpt":"1+∑_p∈ Sγ_p≥0 (S⊆ F), equivalently 1+∑_γ_p<0γ_p≥0.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-16a33ffe0404155f","tex":"\\sum_{r=0}^{2j}","aliases":"∑_r=0^2j sum","display":"\\sum_{r=0}^{2j}","kind":"use","excerpt":"for the measure x^k dν_p,λ, k≥0 integer. The zero atom has mass one at k=0 and is annihilated when k≥1. fig04_prime_polynomialsLeft: degree-two packets for p=2, depth zero. Right: initial atom masses. Strength -1 deletes every even squarefree atom; strength 0 is the original measure; strength 1 doubles that shelf. The exact moments are finite zeta ","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c49ab2b6398af97e","tex":"\\sum_{\\substack{0\\le t<L\\\\t+l\\equiv j\\,(b)}}","aliases":"∑_substack0≤ t<L t+l≡ j (b) sum substack le t equiv","display":null,"kind":"use","excerpt":"Y_j(w)=∑_l=0^b-1H_jl(w)X_l(w), H_jl(w)=∑_substack0≤ t<L t+l≡ j (b) h_t w^(t+l-j)/b.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-baeef91b0146cbb4","tex":"\\sum_{\\substack{n\\le x\\\\(n,Q)=1}}","aliases":"∑_substackn≤ x (n,Q)=1 sum substack le","display":"\\sum_{\\substack{n\\le x\\\\(n,Q)=1}}","kind":"use","excerpt":"claimProof mechanism Squarefree inclusion--exclusion gives ∑_substackn≤ x (n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6c98b1dfa927080c","tex":"\\sum_{\\substack{n\\ge1\\\\(n,Q)=1}}","aliases":"∑_substackn≥1 (n,Q)=1 sum substack ge","display":"\\sum_{\\substack{n\\ge1\\\\(n,Q)=1}}","kind":"use","excerpt":"(n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logarithm of 1-n^-β>0. The atom n=1 is excluded because its logarithm is undefined. At positive integer depths its omitted contribution would be zero. At depth zero the corresponding mass-one atom is absent. In particular, A(0)=Z_Q(2a)-1, where Z_Q(v)=∑_substackn≥1 (n,Q)=1μ(n)^2n","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-73016fa68c8510ea","tex":"\\sum_{\\substack{n\\ge1\\\\\\mu(n)^2=1\\\\n\\text{ even}}}","aliases":"∑_substackn≥1 μ(n)^2=1 n even sum substack ge mu n text","display":null,"kind":"use","excerpt":"Q_δ(f)=∑_substackn≥1 μ(n)^2=1 n oddn^-4f(x_n)^2 -δ∑_substackn≥1 μ(n)^2=1 n evenn^-4f(x_n)^2.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c27ed7b086d8c557","tex":"\\sum_{\\substack{n\\ge1\\\\\\mu(n)^2=1\\\\n\\text{ odd}}}","aliases":"∑_substackn≥1 μ(n)^2=1 n odd sum substack ge mu n text","display":null,"kind":"use","excerpt":"Q_δ(f)=∑_substackn≥1 μ(n)^2=1 n oddn^-4f(x_n)^2 -δ∑_substackn≥1 μ(n)^2=1 n evenn^-4f(x_n)^2.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-6b4ae6d1668fdf50","tex":"\\sum_{\\substack{n\\ge1\\\\\\mu(n)^2=1,(n,Q)=1\\\\p\\mid n}}","aliases":"∑_substackn≥1 μ(n)^2=1,(n,Q)=1 pmid n sum substack ge mu p mid","display":null,"kind":"use","excerpt":"∑_substackn≥1 μ(n)^2=1,(n,Q)=1 pmid nn^-2a =fracZ_Q(2a)p^2a+1.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-eaa6df9127dc54f3","tex":"\\sum_{\\substack{n\\ge2\\\\(n,Q)=1}}","aliases":"∑_substackn≥2 (n,Q)=1 sum substack ge","display":"\\sum_{\\substack{n\\ge2\\\\(n,Q)=1}}","kind":"use","excerpt":"MonographPartIIArithmetic depth and signed measuresComplex parameters, positive kernels and the boundaries of positivity. Arithmetic at complex depth. pageheadingFAMILY 2Arithmetic energy at complex depth For real a>1/2, β>0, and integer Q≥1, define A_a,β,Q(s)=∑_substackn≥2 (n,Q)=1 μ(n)^2n^-2a(1-n^-β)^2s, s∈ C. The complex power uses the real logar","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-316baeb98c4c10eb","tex":"\\sum_{\\substack{n\\ge2\\\\ \\mu(n)^2=1\\\\ (n,Q)=1}}","aliases":"∑_substackn≥2 μ(n)^2=1 (n,Q)=1 sum substack ge mu","display":"\\sum_{\\substack{n\\ge2\\\\ \\mu(n)^2=1\\\\ (n,Q)=1}}","kind":"use","excerpt":"A_a,β,Q(s)=∑_substackn≥2 μ(n)^2=1 (n,Q)=1 n^-2aexp(-λ_n s), λ_n=-2log(1-n^-β).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-1bb747f70594da18","tex":"\\sum_{\\substack{n\\ge2\\\\\\mu(n)^2=1\\\\(n,Q)=1}}","aliases":"∑_substackn≥2 μ(n)^2=1 (n,Q)=1 sum substack ge mu","display":"\\sum_{\\substack{n\\ge2\\\\\\mu(n)^2=1\\\\(n,Q)=1}}","kind":"use","excerpt":"A_Q(s)=∑_substackn≥2 μ(n)^2=1 (n,Q)=1 n^-4(1-n^-2)^2s.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-77316f9b4755c877","tex":"\\sum_{\\substack{\\ell\\ge0\\\\j+\\ell\\equiv i\\pmod b}}","aliases":"∑_substackℓ≥0 j+ℓ≡ ipmod b sum substack ell ge j ell equiv pmod","display":null,"kind":"use","excerpt":"For causal FIR taps h_ℓ, b_m=∑_ℓ h_ℓ a_m-ℓ with zero extension to negative indices, split A(z) and B(z) into b phases. Then B_i(t)=∑_jH_ij(t)A_j(t), where H_ij(t)=∑_substackℓ≥0 j+ℓ≡ ipmod b h_ℓ t^(j+ℓ-i)/b. The exponent is the nonnegative carry to the next phase block. This is an exact coefficient law, with no approximation or loss of terms.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-90c396c1f06e154a","tex":"\\sum_t","aliases":"∑_t sum","display":"\\sum_t","kind":"use","excerpt":"F'(z)=1+∑_t h_tF(q_tz)F(r_tz), F(0)=0.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c77f9524c6708030","tex":"\\sum_{t=0}^{L-1}","aliases":"∑_t=0^L-1 sum","display":"\\sum_{t=0}^{L-1}","kind":"use","excerpt":"b_m=∑_t=0^L-1h_ta_m-t.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-251b06985c4ef1af","tex":"\\sum_{t=0}^m","aliases":"∑_t=0^m sum","display":"\\sum_{t=0}^m","kind":"use","excerpt":"G_a(z)G_h(z)=∑_m[ ∑_t=0^mbinom mt h_ta_m-t]fracz^mm!.","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-081578110d190345","tex":"\\sum_{\\gamma_p<0}","aliases":"∑_γ_p<0 sum gamma","display":"\\sum_{\\gamma_p<0}","kind":"use","excerpt":"claimA separate additive family has different geometry For weights 1+∑_p∈ Fγ_p1_pmid n, positivity is equivalent to 1+∑_γ_p<0γ_p≥0. Every prime subset appears at its squarefree product, proving necessity. Checking each atom proves sufficiency. Its finite Hankel matrices are affine in γ, so the semidefinite feasible sets are convex spectrahedra. The","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e3116f8319e2684e","tex":"\\sum_\\ell","aliases":"∑_ℓ sum ell","display":"\\sum_\\ell","kind":"use","excerpt":"For causal FIR taps h_ℓ, b_m=∑_ℓ h_ℓ a_m-ℓ with zero extension to negative indices, split A(z) and B(z) into b phases. Then B_i(t)=∑_jH_ij(t)A_j(t), where H_ij(t)=∑_substackℓ≥0","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"B_mA_m^-1/2. Every determinant root is real and below -1; possible coefficient poles occur there, although numerator cancellation is possible. fig05_sieve_boundaryThe line is a proved bound for p=2; dots are ordinary high-precision pencil roots. A defect λ=-1.001 eventually violates positivity even though it can pass low-order checks. The nearest r","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"(n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2a-1/2)/β; |s|/Re s stays bounded in the stated sector. This gives the erro","count":31},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","excerpt":"Near zero write f(t)=(log E_N)'(t)=∑_k≥1α_kt^k-1, where α_k=(-1)^k+1 Z_N(k). For any integer K≥1, subtract K terms on [0,1]: I_K(s)= · ∫_0^1t^-s[f(t)-∑_k=1^Kα_kt^k-1]dd t +∑_k=1^Kfracα_kk-s · +∫_1^∞ t^-s[f(t)-frac1N t^1/N-1]dd t +frac1N(s-1/N). The first integral is holomorphic for Re s<K+1. The second is entire because its remainder decays exponen","count":4},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-8bcd55715febb3c0","tex":"\\int_0^z","aliases":"∫_0^z int","display":"\\int_0^z","kind":"use","excerpt":"U(z)=e^B(z)∫_0^z e^-B(w) dw, B(z)=∫_0^zT(qw) dw.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"pageheadingORBIT SPECTRA / MELLIN PROOFThe two ends have different jobs For 1/N<Re s<1, the genus-zero log derivative and the angular gap give absolute interchange in Z_N(s)=fracsinπ sπ ∫_0^∞ t^-sfracddtlog E_N(t)dd t. Each term uses the principal-branch beta integral ∫_0^∞ t^-s(t+w)^-1dd t=π w^-s/sinπ s. The denominator estimate bounds the absolut","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-9177766aca870d95","tex":"\\int_1^\\infty","aliases":"∫_1^∞ int infty","display":"\\int_1^\\infty","kind":"use","excerpt":"(n,Q)=1μ(n)^2 =δ_Q x+O_Q(sqrtx). Integrate against g_s(x)=x^-2a(1-x^-β)^2s on [1,∞). The density term becomes the beta integral after y=x^-β. Partial integration bounds the error by C_Q∫_1^∞sqrtx|g_s'(x)| d x. The two derivative terms are beta integrals with decay exponent (2a-1/2)/β; |s|/Re s stays bounded in the stated sector. This gives the erro","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-3f9ce7fb3cacd052","tex":"\\int_L^\\infty","aliases":"∫_L^∞ int infty","display":"\\int_L^\\infty","kind":"use","excerpt":"Indeed all omitted frequencies are at most λ_L+1, and ∑_n>Ln^-2a≤∫_L^∞ x^-2a dx. Ignoring squarefreeness and coprimality makes this a valid upper bound. If r_0,r_1 are the two positive truncated residual sums at t,t+h, the true secant belongs to","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-d84f6c5a867fda9c","tex":"\\int_{R-1}^R","aliases":"∫_R-1^R int","display":"\\int_{R-1}^R","kind":"use","excerpt":"U(R)≥∫_R-1^R U(qt)V(rt) dt ≥ U(q(R-1))V(r(R-1)).","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-c288d46e36a6f555","tex":"\\int_{R-L}^{R}","aliases":"∫_R-L^R int","display":"\\int_{R-L}^{R}","kind":"use","excerpt":"M(R) · ≥ h∫_R-L^RM(qt)M(rt) dt · ≥ hL M(q(R-L))M(r(R-L)) · =M(q(R-L))M(r(R-L)).","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"Let α=frac2a-1β, δ_Q=frac1ζ(2)∏_pmid Qfracpp+1. As |s|→∞ in any closed sector |arg s|≤π/2-ϵ with 0<ϵ<π/2, A_a,β,Q(s)=fracδ_QβB(α,2s+1) +O_a,β,Q,ϵ(|s|^-(2a-1/2)/β). In particular, A_a,β,Q(s)∼fracδ_QΓ(α)β(2s)^-α. The principal power is used in the right half-plane. The beta expression is an asymptotic continuous-density model, not an identity with th","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"claimA worked order-one member T_1/3,2/3(z)=z+frac227z^3+ frac42187z^5+frac23211160261z^7+O(z^9). It has order one, and T_1/3,2/3(1)≈1.0759239897408249. The diagonal point q=r=1/2 gives f_1/2(z)=sqrt2sinh(z/sqrt2). Every off-diagonal member is separated from all rescaled diagonal members by the theorem above.","count":6},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"pageheadingFAMILY 1 / SEPARATIONMore than a rescaled one-scale seed claimNonrescaling theorem If q≠ r, then T_q,r cannot equal A f_s(Bz) for nonzero constants A,B and any real 0<s<1, where f_s'=1+f_s(sz)^2 is the known family.","count":25},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"u,v≥ n, u≡ v≡ npmod2, u+v≤ n(n+1).","count":5},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"_n=(-1)^n. What remains is an alternating mode of constant magnitude. Both modes belong to the same second-order recurrence. At n=6, J_6=21 and 2^6-3·21=1. At n=7, J_7=43 and 2^7-3·43=-1. A recurrence can combine growth and oscillation. Splitting the modes explains the parity correction. OEIS A001045, indexed from zero. Direct closed form and exact","count":167},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"Equivalently, expansion in subsets S⊆ F gives the coefficient D(s)/∏_p∈ S(p^s+1) for ∏_p∈ Sλ_p. This is an independently useful coordinate formula for prime overlap.","count":2},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","excerpt":"B=ℓ_1⊗ℓ_2.","count":3},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"item[Rising factorial.] (s)_k=s(s+1)⋯(s+k-1) for k≥1, and (s)_0=1. It occurs in the shifted spectral residues. item[Coefficient conventions.] An ordinary generating function uses ∑_na_nz^n. An exponential generating function uses ∑_na_nz^n/n!. Their product and shift rules differ. item[Polyphase section and finite taps.] Polyphase splits a coeffici","count":10},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"Using an inner product linear in the first argument, the coefficient vector of K(·,t) is (sqrtw_je^-λ_joverline t)_j, so ⟨ Uc,K(·,t)⟩=(Uc)(t). This also verifies the sign and conjugation convention directly. Arbitrarily large finite Gram matrices at distinct complex depths are strictly positive, as proved in the base route. Thus the feature space h","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"Using an inner product linear in the first argument, the coefficient vector of K(·,t) is (sqrtw_je^-λ_joverline t)_j, so ⟨ Uc,K(·,t)⟩=(Uc)(t). This also verifies the sign and conjugation convention directly. Arbitrarily large finite Gram matrices at distinct complex depths are strictly positive, as proved in the base route. Thus the feature space h","count":1},{"work":"monographs-special-functions","url":"notation/monographs-special-functions.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"c_n^1/(2n+1)≥ q^n/(2n+1)r^n^2/(2n+1)(2n+1)^-n/(2n+1)⟶∞.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"notation-table","excerpt":"W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.","count":51},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.","count":287},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"I, A(s) · Institutional state and admissible protocol catalogue at state s. Admissibility requires its stipulated implementation.","count":60},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"notation-table","excerpt":"B, B · A nonempty set of states consistent with a common history, and the space of such sets. These are not posterior distributions.","count":119},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"notation-table","excerpt":"B, B · A nonempty set of states consistent with a common history, and the space of such sets. These are not posterior distributions.","count":14},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"notation-table","excerpt":"λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.","count":59},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0b4eec345b52552e","tex":"e^\\varepsilon","aliases":"e^ε varepsilon","display":"e^\\varepsilon","kind":"notation-table","excerpt":"λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"notation-table","excerpt":"F, g, α · Authorized collectible penalty, deviation gain in the same utility units, and the protected false-positive limit.","count":68},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"notation-table","excerpt":"F, g, α · Authorized collectible penalty, deviation gain in the same utility units, and the protected false-positive limit.","count":73},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"notation-table","excerpt":"I, A(s) · Institutional state and admissible protocol catalogue at state s. Admissibility requires its stipulated implementation.","count":33},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","excerpt":"K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.","count":22},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-16bc7321a3832aa5","tex":"K_u","aliases":"K_u","display":"K_u","kind":"notation-table","excerpt":"K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","excerpt":"T_*, L_* · Ending attractor and the running kernel that preserves an available authorized ending.","count":32},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-73f17951391086e7","tex":"L_*","aliases":"L_*","display":"L_*","kind":"notation-table","excerpt":"T_*, L_* · Ending attractor and the running kernel that preserves an available authorized ending.","count":12},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].","count":75},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.","count":16},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-baed31a30986ab0a","tex":"Q_u","aliases":"Q_u","display":"Q_u","kind":"notation-table","excerpt":"K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"notation-table","excerpt":"W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.","count":47},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-236f80d54e55c576","tex":"R_*","aliases":"R_*","display":"R_*","kind":"notation-table","excerpt":"W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"notation-table","excerpt":"X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.","count":60},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"notation-table","excerpt":"I, A(s) · Institutional state and admissible protocol catalogue at state s. Admissibility requires its stipulated implementation.","count":119},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-01027990b0cb85d6","tex":"\\operatorname{Succ}","aliases":"Succ operatorname","display":"\\operatorname{Succ}","kind":"notation-table","excerpt":"Δ, Succ · Complete-state successors and the induced successors of public possibility sets.","count":8},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0b7cc491b1ee030b","tex":"\\operatorname{supp}","aliases":"supp operatorname","display":"\\operatorname{supp}","kind":"notation-table","excerpt":"suppK · The full supported successor set. A small positive probability is not discarded in a sure guarantee.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"notation-table","excerpt":"π, T · State abstraction or transport. Preservation of outputs does not automatically preserve actions, knowledge, or rights.","count":68},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"notation-table","excerpt":"X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.","count":113},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f90055131c0edf2c","tex":"T_*","aliases":"T_*","display":"T_*","kind":"notation-table","excerpt":"T_*, L_* · Ending attractor and the running kernel that preserves an available authorized ending.","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV d_TV mathrm","display":"{d_{\\mathrm{TV}}}","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].","count":19},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"notation-table","excerpt":"K_u, Q_u · Row-stochastic transition and report kernels. The model states when their joint law requires separate treatment.","count":46},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"notation-table","excerpt":"W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.","count":35},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-75d8484cfed31738","tex":"W_*","aliases":"W_*","display":"W_*","kind":"notation-table","excerpt":"W_*, R_* · Safe continuation kernel and certified-service attractor for one contract. The first need not imply the second.","count":14},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"notation-table","excerpt":"X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.","count":12},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-09868efb784faa54","tex":"\\mathcal X","aliases":"X mathcal","display":"\\mathcal X","kind":"notation-table","excerpt":"X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].","count":55},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"notation-table","excerpt":"X_t, S, X · Complete modeled state or its state space. The space is finite unless an explicit extension says otherwise.","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].","count":20},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"notation-table","excerpt":"F, g, α · Authorized collectible penalty, deviation gain in the same utility units, and the protected false-positive limit.","count":42},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"notation-table","excerpt":"Δ, Succ · Complete-state successors and the induced successors of public possibility sets.","count":15},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"notation-table","excerpt":"λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.","count":46},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"notation-table","excerpt":"λ=e^ε · Pairwise disclosure factor in Section~[source reference]. Elsewhere λ may denote a mixture or intervention.","count":53},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"notation-table","excerpt":"π, T · State abstraction or transport. Preservation of outputs does not automatically preserve actions, knowledge, or rights.","count":40},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].","count":35},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0e44b9be25437975","tex":"\\sum_x","aliases":"∑_x sum","display":"\\sum_x","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite space. This lies in [0,1].","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"definition-passage","excerpt":"[Institutional realization] An institutional realization comprises an environment state x, an institutional state I, a set of local actions A_i(x,I) for each decision center, a transition kernel K_I on the complete state, and a family of admissible interventions on I and its implementation. A written rule has operative force only relative to change","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-861bfd8f1f28ff59","tex":"\\mathcal A_i","aliases":"A_i mathcal","display":"\\mathcal A_i","kind":"definition-passage","excerpt":"[Institutional realization] An institutional realization comprises an environment state x, an institutional state I, a set of local actions A_i(x,I) for each decision center, a transition kernel K_I on the complete state, and a family of admissible interventions on I and its implementation. A written rule has operative force only relative to change","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Endogenous institutional formation] Let D record the rules, affordances, and permissions fixed by the designer and let I_t be the institutional state. Formation is endogenous relative to that family when interactions among participating decision centers make a causal difference to the content, adoption, revision, or enforcement of I_t beyond what ","count":75},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fa77d9059acbf531","tex":"F^A_m","aliases":"F^A_m","display":"F^A_m","kind":"definition-passage","excerpt":"[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities appl","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7029cd6f04e528a0","tex":"F^B_{\\tau(s,m)}","aliases":"F^B_τ(s,m) tau","display":"F^B_{\\tau(s,m)}","kind":"definition-passage","excerpt":"[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities appl","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.","count":119},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-34b5a08da5138c3a","tex":"I_t","aliases":"I_t","display":"I_t","kind":"definition-passage","excerpt":"[Endogenous institutional formation] Let D record the rules, affordances, and permissions fixed by the designer and let I_t be the institutional state. Formation is endogenous relative to that family when interactions among participating decision centers make a causal difference to the content, adoption, revision, or enforcement of I_t beyond what ","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7451eebb9fb0ad5d","tex":"K_I","aliases":"K_I","display":"K_I","kind":"definition-passage","excerpt":"[Institutional realization] An institutional realization comprises an environment state x, an institutional state I, a set of local actions A_i(x,I) for each decision center, a transition kernel K_I on the complete state, and a family of admissible interventions on I and its implementation. A written rule has operative force only relative to change","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"definition-passage","excerpt":"[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities appl","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined,","count":49},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"definition-passage","excerpt":"[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.","count":22},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"definition-passage","excerpt":"[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-962eb33121c4bcf4","tex":"O_i","aliases":"O_i","display":"O_i","kind":"definition-passage","excerpt":"[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-881d6987a42aea74","tex":"\\mathcal O_i","aliases":"O_i mathcal","display":"\\mathcal O_i","kind":"definition-passage","excerpt":"[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d08d07be26360470","tex":"O_S","aliases":"O_S","display":"O_S","kind":"definition-passage","excerpt":"[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ad17208a282be173","tex":"\\push","aliases":"push","display":null,"kind":"definition-passage","excerpt":"[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e53fe76bd3b89fa2","tex":"\\X","aliases":"X mathcal","display":"{\\mathcal X}","kind":"definition-passage","excerpt":"[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.","count":8},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z mathcal","display":"{\\mathcal Z}","kind":"definition-passage","excerpt":"[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.","count":64},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"definition-passage","excerpt":"[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.","count":17},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6a331de74ff475c9","tex":"\\pi^{-1}","aliases":"π^-1 pi","display":"\\pi^{-1}","kind":"definition-passage","excerpt":"[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined,","count":12},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"definition-passage","excerpt":"[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities appl","count":24},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1b5b4e6ee09dffe3","tex":"\\ell_A","aliases":"ℓ_A ell","display":"\\ell_A","kind":"definition-passage","excerpt":"[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities appl","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-96a5e0d2554892b6","tex":"\\ell_B","aliases":"ℓ_B ell","display":"\\ell_B","kind":"definition-passage","excerpt":"[Operatively faithful language transport] The pair (T,τ) is faithful on a transition-closed declared domain if, for every source state s and message m, the following conditions hold: F^A_m(s) is defined · Longleftrightarrow F^B_τ(s,m)(T(s)) is defined, T(F^A_m(s)) · =F^B_τ(s,m)(T(s)), ℓ_A(s,m) · =ℓ_B(T(s),τ(s,m)), where the last two equalities appl","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Declared decomposition and access model] A decomposition specifies components and maps O_i: X→ O_i describing their accessible observations. A coalition S has the permitted information O_S(X), supplemented only by the transcripts, computation, and external queries explicitly allowed to that coalition.","count":40},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f2466f7003a98443","tex":"\\sum_{x:\\pi(x)=z}","aliases":"∑_x:π(x)=z sum pi","display":"\\sum_{x:\\pi(x)=z}","kind":"definition-passage","excerpt":"[Candidate macrostate] A candidate macrostate is a surjection π: X→ Z. For a distribution μ on X, its pushforward is (pushμ)(z)=∑_x:π(x)=zμ(x). The fibre π^-1(z) is the set of implementations represented by z.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":77},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also ","count":81},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","excerpt":"[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C)","count":106},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"introduction-passage","excerpt":"Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t),","count":17},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not ","count":84},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-87871832cc0d50ff","tex":"\\mathcal C_t","aliases":"C_t mathcal","display":"\\mathcal C_t","kind":"introduction-passage","excerpt":"Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t),","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c99be13da294dc50","tex":"c_t","aliases":"c_t","display":"c_t","kind":"introduction-passage","excerpt":"Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not ","count":9},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c1e92538a918e673","tex":"\\mathcal C_{t+1}","aliases":"C_t+1 mathcal","display":"\\mathcal C_{t+1}","kind":"introduction-passage","excerpt":"Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t),","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dac734820522a9ee","tex":"c_{t+1}","aliases":"c_t+1","display":"c_{t+1}","kind":"introduction-passage","excerpt":"may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a7d678072fed2039","tex":"C_v","aliases":"C_v","display":"C_v","kind":"introduction-passage","excerpt":"v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z) lies in the declared successor set D_v(z). The loca","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e92b1df899796cf6","tex":"C_w","aliases":"C_w","display":"C_w","kind":"introduction-passage","excerpt":"andover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z) lies in the declared successor set D_v(z). The local certificate proves D_v(z)⊆ C_v. A handover from v to w must keep every intermediate state","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"introduction-passage","excerpt":"Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C)","count":64},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"introduction-passage","excerpt":"requires indefinite preservation. At (v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z) lies in the d","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"Membership, decision rights, resource control, and exposure to consequences define different sets. Let M denote participating members, V positions with specified decision rights, H infrastructure controllers, and E affected parties outside the membership. These sets may overlap where the description allows it. A human resource owner can be in H wit","count":26},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"introduction-passage","excerpt":"A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stoch","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-27ca42d64343fc33","tex":"\\mathsf{epoch}","aliases":"epoch mathsf","display":"\\mathsf{epoch}","kind":"introduction-passage","excerpt":"A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also ","count":24},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"introduction-passage","excerpt":"A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t),","count":38},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b4f9811bf3c61e66","tex":"G_t","aliases":"G_t","display":"G_t","kind":"introduction-passage","excerpt":"A continuing office need not be one of a permanently supplied list. A population may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication li","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d8313a7d60fe037e","tex":"G_{t+1}","aliases":"G_t+1","display":"G_{t+1}","kind":"introduction-passage","excerpt":"may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their ","count":27},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). ","count":52},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"introduction-passage","excerpt":"of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-814f64e7253b7d27","tex":"H_t","aliases":"H_t","display":"H_t","kind":"introduction-passage","excerpt":"may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"introduction-passage","excerpt":"A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stoch","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-95d680fad2ccb816","tex":"I_n","aliases":"I_n","display":"I_n","kind":"introduction-passage","excerpt":"A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stoch","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7d929052114e2b78","tex":"\\mathsf{issuer}","aliases":"issuer mathsf","display":"\\mathsf{issuer}","kind":"introduction-passage","excerpt":"A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"with a declared review schedule Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision ","count":94},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also ","count":67},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0c7bd880455e7fb0","tex":"K_i","aliases":"K_i","display":"K_i","kind":"introduction-passage","excerpt":"A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stoch","count":9},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-581b4f8b484b100b","tex":"\\mathsf{kind}","aliases":"kind mathsf","display":"\\mathsf{kind}","kind":"introduction-passage","excerpt":"A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not ","count":20},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9de6a9970cb6757b","tex":"M_t","aliases":"M_t","display":"M_t","kind":"introduction-passage","excerpt":"Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not ","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"introduction-passage","excerpt":"If a message M has the same conditional distribution under two equally possible hidden states θ=0,1, then Bayes' rule gives Pr(θ=1mid M)=Pr(θ=1) wherever the posterior is defined. Such a channel carries no information about the distinction, regardless of how fluent or authoritative its reports appear. In strategic environments, senders may have pre","count":17},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stoch","count":57},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b8be2dc73c4dd73d","tex":"\\mathsf{nonce}","aliases":"nonce mathsf","display":"\\mathsf{nonce}","kind":"introduction-passage","excerpt":"A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C)","count":23},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-01eecdf5cc4bca8b","tex":"\\overline A","aliases":"overline A overline","display":"\\overline A","kind":"introduction-passage","excerpt":"Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longl","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9b0ad0f92e6daea0","tex":"\\overline G","aliases":"overline G overline","display":"\\overline G","kind":"introduction-passage","excerpt":"Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longl","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9ea515a6ead5b2d4","tex":"\\overline h","aliases":"overline h overline","display":"\\overline h","kind":"introduction-passage","excerpt":"Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longl","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1766d7a5f31944e1","tex":"\\overline S","aliases":"overline S overline","display":"\\overline S","kind":"introduction-passage","excerpt":"Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longl","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-55aa78fb2320522b","tex":"\\overline S_{\\rm safe}","aliases":"overline S_rm safe overline rm","display":"\\overline S_{\\rm safe}","kind":"introduction-passage","excerpt":"Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longl","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-95524823843282fd","tex":"\\overline\\Delta","aliases":"overlineΔ overline Delta","display":"\\overline\\Delta","kind":"introduction-passage","excerpt":"Consider a reduced finite contract (overline S,overline h,overline A,overlineΔ,overline S_rm safe,overline G) with the same catalogue U. Let π:S→overline S be a surjection. Assume, for every state and every admissible profile, h(s) · =overline h(π s), · A(s) · =overline A(π s), s∈ S_rm safe · Longleftrightarrowπ s∈overline S_rm safe, · s∈ G · Longl","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0b67a2140e399c4e","tex":"P_C","aliases":"P_C","display":"P_C","kind":"introduction-passage","excerpt":"[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C)","count":38},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a9d31d0c349bd2e3","tex":"P_D","aliases":"P_D","display":"P_D","kind":"introduction-passage","excerpt":"[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C)","count":19},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5ed30585279498bf","tex":"P_\\lambda","aliases":"P_λ lambda","display":"P_\\lambda","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":9},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-21e83399daffb8b6","tex":"P_{\\lambda'}","aliases":"P_λ' lambda","display":"P_{\\lambda'}","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-de93a32bd3ccecc2","tex":"\\mathsf{payload}","aliases":"payload mathsf","display":"\\mathsf{payload}","kind":"introduction-passage","excerpt":"A message can be represented as a typed record m=(kind,issuer,epoch,payload,nonce). Its operative meaning is not just its payload. For a complete institutional state s, an interpreter defines a partial transition F_m(s) and an event record ℓ(s,m) when that transition is defined. Undefinedness represents rejection under the declared protocol, not a ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e695ee90e9b92c62","tex":"\\operatorname{Post}","aliases":"Post operatorname","display":"\\operatorname{Post}","kind":"introduction-passage","excerpt":"of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"introduction-passage","excerpt":"If a message M has the same conditional distribution under two equally possible hidden states θ=0,1, then Bayes' rule gives Pr(θ=1mid M)=Pr(θ=1) wherever the posterior is defined. Such a channel carries no information about the distinction, regardless of how fluent or authoritative its reports appear. In strategic environments, senders may have pre","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"introduction-passage","excerpt":"Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t),","count":11},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-03372583aa2e6154","tex":"\\mathcal Q_t","aliases":"Q_t mathcal","display":"\\mathcal Q_t","kind":"introduction-passage","excerpt":"Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t),","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-72c3e7af491c9a99","tex":"\\mathcal Q_{t+1}","aliases":"Q_t+1 mathcal","display":"\\mathcal Q_{t+1}","kind":"introduction-passage","excerpt":"Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t),","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"A continuing office need not be one of a permanently supplied list. A population may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication li","count":84},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"introduction-passage","excerpt":"with a declared review schedule Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stochastic revision ","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7d0b0aaec388130d","tex":"r_t","aliases":"r_t","display":"r_t","kind":"introduction-passage","excerpt":"A continuing office need not be one of a permanently supplied list. A population may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication li","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c5d0744b62ca38dd","tex":"r_{t+1}","aliases":"r_t+1","display":"r_{t+1}","kind":"introduction-passage","excerpt":"may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"introduction-passage","excerpt":"A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also ","count":31},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1ad04db978145bef","tex":"S_{\\rm safe}","aliases":"S_rm safe rm","display":"S_{\\rm safe}","kind":"introduction-passage","excerpt":"A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also ","count":10},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4f9add33b306e767","tex":"\\sup_{\\lambda,\\lambda'\\in\\Lambda(x,z')}","aliases":"sup_λ,λ'∈Λ(x,z') sup lambda lambda in Lambda","display":"\\sup_{\\lambda,\\lambda'\\in\\Lambda(x,z')}","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"introduction-passage","excerpt":"An organization is always evaluated relative to a contract: permitted controls, target outputs, horizon, environment family, resource limits, and error tolerances. Let T denote this contract. Claims such as ``the group is irreducible'' or ``the identity survived'' are incomplete without it. A state sufficient for tomorrow's scheduling decision may ","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"introduction-passage","excerpt":"Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not ","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a6ab3f6998c3dac0","tex":"u_t","aliases":"u_t","display":"u_t","kind":"introduction-passage","excerpt":"Write C_t for available method records and Q_t for recognized question records. A question record includes a statement or schema, prerequisites, admissible forms of answer or refutation, provenance, and a resource request. Methods carry domains of validity and the evidence supporting them. The coupled state may evolve as C_t+1=F( C_t, Q_t,I_t,u_t),","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not ","count":42},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"For a finite set of modes V, let C_v⊆ Z be a certified state domain, with each C_v contained in a common protected set C. A certified hold exists when a final state requires indefinite preservation. At (v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read","count":39},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-21f030f572f5b2a5","tex":"V_t","aliases":"V_t","display":"V_t","kind":"introduction-passage","excerpt":"Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not ","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"se the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read only their declared observations. A positive local guard must be sound for the whole action interval: it entails that every actual successor of κ_v(z) lies in the declared successor set D_v(z). The local certificate proves D_v(z)⊆ C_v. A hando","count":38},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"introduction-passage","excerpt":"A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cf76327e749de1b7","tex":"W_k","aliases":"W_k","display":"W_k","kind":"introduction-passage","excerpt":"A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also ","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b5731a3740a4e086","tex":"W_{k+1}","aliases":"W_k+1","display":"W_{k+1}","kind":"introduction-passage","excerpt":"A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dacac00565e66fe6","tex":"\\widehat X","aliases":"widehat X widehat","display":"\\widehat X","kind":"introduction-passage","excerpt":"A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stoch","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-447a3c0003fa94fb","tex":"\\widehat X_n","aliases":"widehat X_n widehat","display":"\\widehat X_n","kind":"introduction-passage","excerpt":"A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stoch","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"introduction-passage","excerpt":"A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stoch","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":11},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"nt public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u","count":19},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3c453920b3857736","tex":"Y^+","aliases":"Y^+","display":"Y^+","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":14},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their ","count":16},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"introduction-passage","excerpt":"may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their ","count":9},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5b1f633540b1d1ef","tex":"\\Gamma_{I_t}","aliases":"Γ_I_t Gamma","display":"\\Gamma_{I_t}","kind":"introduction-passage","excerpt":"may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":16},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"introduction-passage","excerpt":"If a message M has the same conditional distribution under two equally possible hidden states θ=0,1, then Bayes' rule gives Pr(θ=1mid M)=Pr(θ=1) wherever the posterior is defined. Such a channel carries no information about the distinction, regardless of how fluent or authoritative its reports appear. In strategic environments, senders may have pre","count":21},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"introduction-passage","excerpt":"For a finite set of modes V, let C_v⊆ Z be a certified state domain, with each C_v contained in a common protected set C. A certified hold exists when a final state requires indefinite preservation. At (v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read","count":13},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fd0a204d0362614f","tex":"\\kappa_v","aliases":"κ_v kappa","display":"\\kappa_v","kind":"introduction-passage","excerpt":"For a finite set of modes V, let C_v⊆ Z be a certified state domain, with each C_v contained in a common protected set C. A certified hold exists when a final state requires indefinite preservation. At (v,z) with z∈ C_v, a selector may either use the local policy κ_v or execute an authorized handover to another mode. The selector and guard can read","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-11c22a9994059ca8","tex":"\\lambda'","aliases":"λ' lambda","display":"\\lambda'","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"introduction-passage","excerpt":"A continuation kernel for the whole contract. Set W_0 · ={B∈ B:B⊆ S_rm safe}, W_k+1 · ={B∈ W_0:∃ u∈ A(B), Succ(B,u)⊆ W_k}. The sequence is decreasing. It stabilizes after at most | B| strict removals, and we denote its fixed point by W_*. This is the joint continuation kernel. A state is removed not only when it violates an invariant now, but also ","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"introduction-passage","excerpt":"The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). ","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Given a future outcome experiment, write P_λ for the resulting law of Y^+ under reset λ. Define implementation ambiguity by η(x,z')=sup_λ,λ'∈Λ(x,z') d_TV(P_λ,P_λ'). A small ambiguity means different allowed implementations of the same macro-assignment have similar consequences. Large ambiguity means the macro-intervention is not sufficiently specif","count":84},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8509fe8bdd8a2289","tex":"\\sum_{d(o)>0}","aliases":"∑_d(o)>0 sum","display":"\\sum_{d(o)>0}","kind":"introduction-passage","excerpt":"p:deterrence-tv In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C). For every 0≤ s≤1, ∑_o s(o)d(o)≤∑","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ccbd66bdaed18a2b","tex":"\\sum_o","aliases":"∑_o sum","display":"\\sum_o","kind":"introduction-passage","excerpt":"[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C)","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"introduction-passage","excerpt":"of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"introduction-passage","excerpt":"may establish a verification service, divide a task into specialties, retire a redundant committee, or form a temporary coalition. Its communication graph can change with these decisions. To express that possibility, let G_t specify currently permitted communication links, r_t an assignment of roles, and c_t a competence and capacity record. Their ","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). ","count":10},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"introduction-passage","excerpt":"[The exact bounded-sanction discrimination limit] In this model, some observation-based sanction makes compliance weakly preferable if and only if g≤ F d_TV(P_D,P_C). Strict preference is attainable if and only if the inequality is strict. Write d(o)=P_D(o)-P_C(o). Since ∑_o d(o)=0, the sum of its positive terms equals frac12∑_o|d(o)|=d_TV(P_D,P_C)","count":56},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"introduction-passage","excerpt":"A block model with a declared review schedule. Let I and E be finite sets of institutional and fast states. At review epoch n the institution is I_n=i and the fast state is X_n=x. Keep i fixed for a predeclared number m≥1 of fast steps with kernel K_i. Write widehat X_n for the resulting endpoint. The next institution is sampled through a row-stoch","count":25},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"introduction-passage","excerpt":"Let U be a finite universe of possible participant handles for the finite results. Write M_t⊆ U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not ","count":34},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"introduction-passage","excerpt":"The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c4af609ad047671b","tex":"\\bigcap_{s\\in B}","aliases":"⋂_s∈ B bigcap in","display":"\\bigcap_{s\\in B}","kind":"introduction-passage","excerpt":"The state of possible institutional realizations. The relevant public state is the set of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"introduction-passage","excerpt":"of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-77e93ecdf7164d2a","tex":"\\bigcup_{s\\in B}","aliases":"⋃_s∈ B bigcup in","display":"\\bigcup_{s\\in B}","kind":"introduction-passage","excerpt":"of complete states still compatible with the observed history. Define B={B⊆ S:B≠∅, h is constant on B}. These are sets of possibilities, not posterior probability distributions. At B, a profile must be admissible for every remaining possibility: A(B)=⋂_s∈ BA(s). For u∈ A(B) and a next public observation y, define Post(B,u,y)= (⋃_s∈ BΔ(s,u))∩ h^-1(y","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random vari","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-24f07e60acb49c42","tex":"A_*","aliases":"A_*","display":"A_*","kind":"use","excerpt":"Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"Why separate marginal ignorance is insufficient. [Local ignorance with a perfectly correct joint action] Let U be a fair bit. Agent 1 observes H_1=U and agent 2 observes H_2=U⊕θ. Each individual observation has the same law for both values of θ. Nevertheless, the joint action (a_1,a_2)=(H_1,H_2) always satisfies a_1⊕ a_2=θ. The joint observation la","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"Why separate marginal ignorance is insufficient. [Local ignorance with a perfectly correct joint action] Let U be a fair bit. Agent 1 observes H_1=U and agent 2 observes H_2=U⊕θ. Each individual observation has the same law for both values of θ. Nevertheless, the joint action (a_1,a_2)=(H_1,H_2) always satisfies a_1⊕ a_2=θ. The joint observation la","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"A formation barrier despite mutual benefit. The stationary cost comparison does not describe a route to adoption. Consider n≥2 agents choosing a_i∈{0,1}. An adopter pays cost c>0 and receives benefit b>0 from each other adopter: u_i(a)=a_i(b∑_j≠ ia_j-c). Nonadopters receive zero. This is a finite network-coordination model, not a universal utility ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"A formation barrier despite mutual benefit. The stationary cost comparison does not describe a route to adoption. Consider n≥2 agents choosing a_i∈{0,1}. An adopter pays cost c>0 and receives benefit b>0 from each other adopter: u_i(a)=a_i(b∑_j≠ ia_j-c). Nonadopters receive zero. This is a finite network-coordination model, not a universal utility ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5b950cf09bc94cd2","tex":"A_{\\rm end}","aliases":"A_rm end rm","display":"A_{\\rm end}","kind":"use","excerpt":"The finite construction of Section~[source reference] yields an exact characterization of continued operation with an available authorized ending. Use its possibility space B, successor operation, and protected set S_rm safe. For each possibility set B, distinguish implemented running profiles A_rm run(B) from implemented ending profiles A_rm end(B","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4f6eb91cb1eca840","tex":"A_{\\rm run}","aliases":"A_rm run rm","display":"A_{\\rm run}","kind":"use","excerpt":"The finite construction of Section~[source reference] yields an exact characterization of continued operation with an available authorized ending. Use its possibility space B, successor operation, and protected set S_rm safe. For each possibility set B, distinguish implemented running profiles A_rm run(B) from implemented ending profiles A_rm end(B","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-82821bc2124f1658","tex":"A_t","aliases":"A_t","display":"A_t","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"Repair needs a causal and temporal test. Let e_t≥0 measure deviation from an acceptable organization after accounting for legitimate changes in task state. Suppose a repair controller ensures e_t+1≤ a_t e_t+b_t, where a_t measures retained deviation and b_t covers new disturbance and model mismatch.","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9d81388db8446d61","tex":"a_{t-1}","aliases":"a_t-1","display":"a_{t-1}","kind":"use","excerpt":"A decision-only public record. Let an unknown state θ∈{0,1} have equal prior probabilities. Participant t receives a private signal s_t∈{0,1} with P(s_t=θmidθ)=p, 1/2<p<1. Signals are independent conditional on θ, and their accuracy is common knowledge. In sequence, each participant observes previous actions a_1,…,a_t-1 and its own signal, then cho","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-81e6e5218cf492d2","tex":"\\mathsf{Assumptions}","aliases":"Assumptions mathsf","display":"\\mathsf{Assumptions}","kind":"use","excerpt":"An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a componen","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c714218dd4336db3","tex":"\\mathsf{Auth}","aliases":"Auth mathsf","display":"\\mathsf{Auth}","kind":"use","excerpt":"A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b863979fb4982e9c","tex":"\\mathsf{authority}","aliases":"authority mathsf","display":"\\mathsf{authority}","kind":"use","excerpt":"tibility, and knowledge Temporal obligations, contracts, and office succession. Permissions describe available actions; obligations constrain continuations. The debt created by an urgency exchange is not satisfied merely because the creditor remains authorized to ask for service. It requires a particular future event. A finite obligation record can","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":66},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"[Indistinguishability obstruction to joint safety] Let B_0 and B_1 be disjoint subsets of a common finite joint-action alphabet, containing the actions that meet a declared requirement under conditions 0 and 1. If the laws of H are within η in total variation, then P_0(B_0)+P_1(B_1)≤1+η. Consequently at least one condition has failure probability a","count":8},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-56e5c919eaefa876","tex":"\\mathcal B_0","aliases":"B_0 mathcal","display":"\\mathcal B_0","kind":"use","excerpt":"The smallest exact summary under a fixed contract. The conditions yield a finite construction rather than only an existence criterion. Begin with the partition B_0 that identifies exactly the states sharing (ℓ(s),D(s)). At each refinement, identify s,t only when they were previously in the same block and every action in their common enabled set lea","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"[Indistinguishability obstruction to joint safety] Let B_0 and B_1 be disjoint subsets of a common finite joint-action alphabet, containing the actions that meet a declared requirement under conditions 0 and 1. If the laws of H are within η in total variation, then P_0(B_0)+P_1(B_1)≤1+η. Consequently at least one condition has failure probability a","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fb9e4008ca2e6c3a","tex":"b_{12}","aliases":"b_12","display":"b_{12}","kind":"use","excerpt":"This result is a standard parity fact on graphs, closely related to cycle-space algebra and discrete cocycles. Its value here is methodological. It shows why pairwise satisfiable cannot be equated with globally realizable. For a triangle with b_12=0, b_23=0, and b_31=1, every individual edge requirement has a solution, but the three cannot hold tog","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f2cec4611ded7308","tex":"b_{23}","aliases":"b_23","display":"b_{23}","kind":"use","excerpt":"This result is a standard parity fact on graphs, closely related to cycle-space algebra and discrete cocycles. Its value here is methodological. It shows why pairwise satisfiable cannot be equated with globally realizable. For a triangle with b_12=0, b_23=0, and b_31=1, every individual edge requirement has a solution, but the three cannot hold tog","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b560aea36fa2e800","tex":"b_{31}","aliases":"b_31","display":"b_{31}","kind":"use","excerpt":"This result is a standard parity fact on graphs, closely related to cycle-space algebra and discrete cocycles. Its value here is methodological. It shows why pairwise satisfiable cannot be equated with globally realizable. For a triangle with b_12=0, b_23=0, and b_31=1, every individual edge requirement has a solution, but the three cannot hold tog","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"In an idealized transferable-utility setting there is a corresponding conditional observation. If the discounted net advantages b_i satisfy ∑_i b_i≥0, then there exist budget-balanced transfers t_i with ∑_i t_i=0 and b_i+t_i≥0 for all i: choose nonnegative w_i summing to ∑_i b_i and set t_i=w_i-b_i. This algebra does not supply liquidity, a transfe","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b7b4d4f8db808bcb","tex":"b_{ij}","aliases":"b_ij","display":"b_{ij}","kind":"use","excerpt":"Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a rea","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f7362460f92e2f43","tex":"b_{ij}'","aliases":"b_ij'","display":"b_{ij}'","kind":"use","excerpt":"The binary jurisdiction model admits a sharper statement. A local convention change x_i'=x_i⊕ c_i transforms its edge labels to b_ij'=b_ij⊕ c_i⊕ c_j.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a16b4bd9e760df1e","tex":"\\mathcal B_k","aliases":"B_k mathcal","display":"\\mathcal B_k","kind":"use","excerpt":"mber of states among those exact quotients, up to relabeling. Every strict refinement increases the number of blocks, which cannot exceed |S|. At a fixed point, states in one block have the same labels and enabled sets and their enabled successors remain in one block, so the quotient conditions hold. Any compatible partition refines B_0. If it refi","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8e9e9a15d9e36ca8","tex":"\\mathcal B_{k+1}","aliases":"B_k+1 mathcal","display":"\\mathcal B_{k+1}","kind":"use","excerpt":"ndproposition Every strict refinement increases the number of blocks, which cannot exceed |S|. At a fixed point, states in one block have the same labels and enabled sets and their enabled successors remain in one block, so the quotient conditions hold. Any compatible partition refines B_0. If it refines B_k, compatible successors show that it also","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9474dae99bcf0491","tex":"B_t","aliases":"B_t","display":"B_t","kind":"use","excerpt":"hcal U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be r","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b17ff042f6455b31","tex":"b_t","aliases":"b_t","display":"b_t","kind":"use","excerpt":"Repair needs a causal and temporal test. Let e_t≥0 measure deviation from an acceptable organization after accounting for legitimate changes in task state. Suppose a repair controller ensures e_t+1≤ a_t e_t+b_t, where a_t measures retained deviation and b_t covers new disturbance and model mismatch.","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e2ec3d8de7f39249","tex":"\\bar a","aliases":"bar a bar","display":"\\bar a","kind":"use","excerpt":"[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1c90b524b55dd9ea","tex":"\\bar a^t","aliases":"bar a^t bar","display":"\\bar a^t","kind":"use","excerpt":"[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7e95a89f39649f47","tex":"\\bar b","aliases":"bar b bar","display":"\\bar b","kind":"use","excerpt":"[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-eebc9b85b3d9979a","tex":"\\bar D","aliases":"bar D bar","display":"\\bar D","kind":"use","excerpt":"[Exact governance quotient] Let S be finite and nonempty and A finite, and let π:S→ Z be surjective. There exist functions barℓ, bar D, and partial transitions bar F satisfying ℓ=barℓπ, D=bar Dπ, π F(s,a)=bar F(π s,a) (a∈ D(s)) if and only if whenever π s=π t, ℓ(s)=ℓ(t), D(s)=D(t), π F(s,a)=π F(t,a) for all a∈ D(s). The quotient data are unique. Fr","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-52555ee0fe55f126","tex":"\\bar F","aliases":"bar F bar","display":"\\bar F","kind":"use","excerpt":"[Exact governance quotient] Let S be finite and nonempty and A finite, and let π:S→ Z be surjective. There exist functions barℓ, bar D, and partial transitions bar F satisfying ℓ=barℓπ, D=bar Dπ, π F(s,a)=bar F(π s,a) (a∈ D(s)) if and only if whenever π s=π t, ℓ(s)=ℓ(t), D(s)=D(t), π F(s,a)=π F(t,a) for all a∈ D(s). The quotient data are unique. Fr","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dc4e9dcca4ad0352","tex":"\\bar w","aliases":"bar w bar","display":"\\bar w","kind":"use","excerpt":"tor v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-step monotonic estimate followed by induction. Under uniformly bounded disturbance normw_t_v,∞≤bar w, the last term is at most bar w(1-κ^t)/(1-κ). This gives a conditional atte","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d97fba459e62ed0e","tex":"\\bar\\ell","aliases":"barℓ bar ell","display":"\\bar\\ell","kind":"use","excerpt":"[Exact governance quotient] Let S be finite and nonempty and A finite, and let π:S→ Z be surjective. There exist functions barℓ, bar D, and partial transitions bar F satisfying ℓ=barℓπ, D=bar Dπ, π F(s,a)=bar F(π s,a) (a∈ D(s)) if and only if whenever π s=π t, ℓ(s)=ℓ(t), D(s)=D(t), π F(s,a)=π F(t,a) for all a∈ D(s). The quotient data are unique. Fr","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d13b1528b249ec85","tex":"\\bigwedge","aliases":"bigwedge","display":"\\bigwedge","kind":"use","excerpt":"For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling com","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fb18ab95ec9896db","tex":"\\bigwedge_{i\\in J}","aliases":"bigwedge_i∈ J bigwedge in","display":"\\bigwedge_{i\\in J}","kind":"use","excerpt":"For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling com","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c64534aa25b678b4","tex":"\\bigwedge_{n\\ge0}","aliases":"bigwedge_n≥0 bigwedge ge","display":"\\bigwedge_{n\\ge0}","kind":"use","excerpt":"For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling com","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6bcbca61ff620dad","tex":"\\mathcal C_{\\mathcal T}","aliases":"C_ T mathcal mathcal","display":"\\mathcal C_{\\mathcal T}","kind":"use","excerpt":"A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may b","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cba2c263db336679","tex":"\\mathcal C_*","aliases":"C_* mathcal","display":"\\mathcal C_*","kind":"use","excerpt":"A finite closure theorem, not an open-endedness theorem. Take finite universes C_* and Q_*. Consider a fixed collection of production rules of the form ``if these finitely many records are present, add this record.'' Prerequisites are positive: no rule requires the absence of a record. The content and correctness of those rules are assumptions. An ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-84112663abfb6c31","tex":"\\mathcal C_0","aliases":"C_0 mathcal","display":"\\mathcal C_0","kind":"use","excerpt":"Let Φ simultaneously add all conclusions whose prerequisites are satisfied, retaining the input records. Order pairs ( C, Q) by componentwise inclusion. [Least closure of a finite problem--method system] Starting from ( C_0, Q_0), iteration of Φ reaches the least closed pair containing that initial pair. There are at most | C_*∖ C_0|+| Q_*∖ Q_0| st","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"Suppose a coordination protocol without institution has expected per-period cost c_0. A proposed institutional protocol has operating cost c_1, per-period maintenance m, and a one-time setup cost B. With discount factor 0≤δ<1, write g=c_0-c_1-m.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"For sufficiency of the interval tests, fix U. Its union of integer windows decomposes into disjoint maximal integer intervals C_1,…,C_k. Each window is contained in one component because it is an interval. The number of appeals from U in C_i is no larger than the number of all appeals whose entire windows lie in C_i. By~[source reference], this is ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"Suppose a coordination protocol without institution has expected per-period cost c_0. A proposed institutional protocol has operating cost c_1, per-period maintenance m, and a one-time setup cost B. With discount factor 0≤δ<1, write g=c_0-c_1-m.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"use","excerpt":"The audit provider receives a separate collectible fee w for verified execution, with w larger than its stipulated cost c_a. A genuine-execution receipt is checked by a specified trusted mechanism. The fee is not a reward for accusing someone. It makes execution preferable to skipping in this narrow two-action provider model. Monetary units are tra","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-59651fe6df7d2f0f","tex":"c_g","aliases":"c_g","display":"c_g","kind":"use","excerpt":"Specialization has a cost and a reachability condition. Consider two services performed once per period. A generalist arrangement costs 2c_g per period. Two specialists cost 2c_s+b, where b is the required communication and integration cost, and training costs h for each specialist. Over T periods the specialist arrangement has lower total cost pre","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"For sufficiency of the interval tests, fix U. Its union of integer windows decomposes into disjoint maximal integer intervals C_1,…,C_k. Each window is contained in one component because it is an interval. The number of appeals from U in C_i is no larger than the number of all appeals whose entire windows lie in C_i. By~[source reference], this is ","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"use","excerpt":"The binary jurisdiction model admits a sharper statement. A local convention change x_i'=x_i⊕ c_i transforms its edge labels to b_ij'=b_ij⊕ c_i⊕ c_j.","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3f6f7b82f9cd7b0e","tex":"C_J","aliases":"C_J","display":"C_J","kind":"use","excerpt":"For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling com","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5704259308c6a17c","tex":"\\mathcal C_j","aliases":"C_j mathcal","display":"\\mathcal C_j","kind":"use","excerpt":"Let C_j⊆ Q be the queries for which generation j has at least one admitted executable method satisfying the contract. This is an attainable-capability set, not a measurement of what an arbitrary method-selection policy actually achieves.","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"The binary jurisdiction model admits a sharper statement. A local convention change x_i'=x_i⊕ c_i transforms its edge labels to b_ij'=b_ij⊕ c_i⊕ c_j.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c8879dc8df5b4bd1","tex":"\\mathcal C_{j+1}","aliases":"C_j+1 mathcal","display":"\\mathcal C_{j+1}","kind":"use","excerpt":"[Conditional cumulative culture] Suppose every certified method available at generation j has an admitted realization at generation j+1 preserving its query semantics, correctness premises, required resources, and execution rights. Suppose newly added methods satisfy the same declared correctness contract. Then C_j⊆ C_j+1. The inclusion is strict e","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"For sufficiency of the interval tests, fix U. Its union of integer windows decomposes into disjoint maximal integer intervals C_1,…,C_k. Each window is contained in one component because it is an interval. The number of appeals from U in C_i is no larger than the number of all appeals whose entire windows lie in C_i. By~[source reference], this is ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e1e71a0eb7c64bd5","tex":"c_s","aliases":"c_s","display":"c_s","kind":"use","excerpt":"Specialization has a cost and a reachability condition. Consider two services performed once per period. A generalist arrangement costs 2c_g per period. Two specialists cost 2c_s+b, where b is the required communication and integration cost, and training costs h for each specialist. Over T periods the specialist arrangement has lower total cost pre","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-433d97a8fbb31f65","tex":"C_t","aliases":"C_t","display":"C_t","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-692972739c4f3493","tex":"C_{v_0}","aliases":"C_v_0","display":"C_{v_0}","kind":"use","excerpt":"[Guarded certificate use and authorized handover] Assume that the selector, at every reachable decision epoch, either uses a local policy with a sound positive guard or performs a handover satisfying the preceding conditions. Starting from z_0∈ C_v_0, every decision state belongs to its current certified domain and every intermediate handover state","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-877970907da456e8","tex":"C_{\\{A,B\\}}","aliases":"C_{A,B}","display":"C_{\\{A,B\\}}","kind":"use","excerpt":"[Everyone knows, but it is not common knowledge] Let W={0,1,2} and let φ hold exactly at 0,1. Agent A has partition {0,1}mid{2}, and agent B has partition {0}mid{1,2}. At world 0, both agents know φ. However, the chain 0∼_A1∼_B2 reaches a world where it is false. Thus E_{A,B}φ holds at 0 but C_{A,B}φ does not.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b9e4c40b00acb386","tex":"\\mathsf{Claim}","aliases":"Claim mathsf","display":"\\mathsf{Claim}","kind":"use","excerpt":"An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a componen","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0beda1e3044cbea5","tex":"\\mathsf{created}","aliases":"created mathsf","display":"\\mathsf{created}","kind":"use","excerpt":"For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy f","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fbc994c237735f1c","tex":"\\mathsf{creditor}","aliases":"creditor mathsf","display":"\\mathsf{creditor}","kind":"use","excerpt":"ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The excha","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-40d9503903bdca20","tex":"\\mathfrak D","aliases":"D mathfrak","display":"\\mathfrak D","kind":"use","excerpt":"ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f25b325fee30d349","tex":"d_{\\mathcal T}","aliases":"d_ T mathcal","display":"d_{\\mathcal T}","kind":"use","excerpt":"For approximate comparison, use the pseudometric d_ T(x,x')=sup_τ∈ T d_TV( L_x(Y^+midτ), L_x'(Y^+midτ)). A finite test suite may make distinct implementations distance zero. That is indistinguishability under those tests, not universal sameness.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gain","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","excerpt":"When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gain","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a3008bb1b6de0e7c","tex":"d_E","aliases":"d_E","display":"d_E","kind":"use","excerpt":"Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gain","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7bdda834cd44dcd4","tex":"d_{j+1}","aliases":"d_j+1","display":"d_{j+1}","kind":"use","excerpt":"Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a572acccc9b7cf05","tex":"D_k","aliases":"D_k","display":"D_k","kind":"use","excerpt":"When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gain","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-71efea616c614868","tex":"D_m","aliases":"D_m","display":"D_m","kind":"use","excerpt":"dproposition For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L_m-1D_m-1+ε_m-1 un","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-688ebedc522ae64c","tex":"d_m","aliases":"d_m","display":"d_m","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-082091b8c43962a9","tex":"D_{m-1}","aliases":"D_m-1","display":"D_{m-1}","kind":"use","excerpt":"dproposition For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L_m-1D_m-1+ε_m-1 un","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cd7ea73588d447c4","tex":"D_{\\rm lineage}","aliases":"D_rm lineage rm","display":"D_{\\rm lineage}","kind":"use","excerpt":"A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may b","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"use","excerpt":"False-positive limits and multiple deviations. A constitutional protection may impose E_P_Cs≤α, with 0≤α≤1. The relevant discrimination capacity is then the finite linear program D_α(P_D,P_C)= max_substack0≤ s≤1 E_P_Cs≤α ( E_P_Ds- E_P_Cs). The feasible set is nonempty and compact, so a maximum exists. The preceding proof with the feasible set restr","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f78b7c92952c5c31","tex":"\\mathsf{debtor}","aliases":"debtor mathsf","display":"\\mathsf{debtor}","kind":"use","excerpt":"ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The excha","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6e958c5e324e38c2","tex":"\\mathsf{deliver}","aliases":"deliver mathsf","display":"\\mathsf{deliver}","kind":"use","excerpt":"[A safe amendment can destroy every safe continuation] There are four publicly distinguishable states r,ℓ,g,b. The safe set is {r,ℓ,g}. At r, profile deliver moves to g, while profile freeze moves to ℓ. At g, wait loops safely. At ℓ, the only admissible profile advances an outstanding deadline and moves to breach state b. The amendment freeze prese","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e0061fd518f4afc3","tex":"\\mathsf{Domain}","aliases":"Domain mathsf","display":"\\mathsf{Domain}","kind":"use","excerpt":"An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a componen","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fcecd8e21956fefb","tex":"\\mathsf{due}","aliases":"due mathsf","display":"\\mathsf{due}","kind":"use","excerpt":"ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The excha","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fb7427a95ad8c5b4","tex":"\\E","aliases":"E mathbb","display":"{\\mathbb E}","kind":"use","excerpt":"A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distri","count":13},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-64fbb1a3212deefd","tex":"e'","aliases":"e'","display":"e'","kind":"use","excerpt":"[.5em] What state must persist? · The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens. [.5em] Can a due debt be erased? · The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-de71c382ea6a2b1c","tex":"e^{\\varepsilon_t}","aliases":"e^ε_t varepsilon","display":"e^{\\varepsilon_t}","kind":"use","excerpt":"Writing λ_t=e^ε_t gives the additive parameter ∑_tε_t. The proposition requires a fixed finite number of releases and uniform conditional bounds. A data-selected stopping rule must itself be represented in the transcript or controlled by an appropriate conditional argument. Privileged side information not covered by the declared record is not prote","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"[Approximate closure over time] Under~[source reference], let the actual and reduced processes use the same predeclared open-loop control sequence. If their initial macro-laws are within e_0 in total variation, then at time t they are within e_t≤min{1,e_0+tε}. If each overline K_u contracts total variation by at most κ<1, then e_t≤κ^t e_0+εfrac1-κ^","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ad3caf981ac30812","tex":"E_c","aliases":"E_c","display":"E_c","kind":"use","excerpt":"Common semantic coordinates. For each component type c∈{P,O,R,E}, let g_j^c:Z_j⟶(E_c,δ_c) be a contract observable in a common metric codomain. P records permissions over the common action alphabet; O records obligation-monitor state; R records resources in common physical or accounting units; and E records the required evidentiary and provenance s","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-28c01717961ca2fe","tex":"e_G","aliases":"e_G","display":"e_G","kind":"use","excerpt":"[Exact frame synchronization] Global frames exist if and only if H_γ=e_G for every closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1","count":9},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b5c061d3b046f5ce","tex":"\\E_H","aliases":"E_H E mathbb","display":"{\\mathbb E}_H","kind":"use","excerpt":"r · 1-r-d, 0<f≤1, r≥0, d>0, r+d≤1. Every nonfailure state has a positive-probability path to failure, so the finite substochastic chain is transient. Solving the two first-step equations gives E_Hτ=fracf+r+dfd. For f=1/10, r=2/5, and d=1/10, the expected first exit is 60 epochs. With r=0 and the other parameters fixed it is 20 epochs. These are exa","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-650700dc8010c4c1","tex":"E_J","aliases":"E_J","display":"E_J","kind":"use","excerpt":"For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling com","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7d045d3a0bea2c3c","tex":"E_J^0","aliases":"E_J^0","display":"E_J^0","kind":"use","excerpt":"For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling com","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ed34ad226f36ee02","tex":"E_J^n","aliases":"E_J^n","display":"E_J^n","kind":"use","excerpt":"For a proposition φ, K_iφ holds at w when φ holds at every v∼_i w. Let E_Jφ=bigwedge_i∈ JK_iφ. Define common knowledge by all finite depths, C_Jφ=bigwedge_n≥0E_J^nφ, E_J^0φ=φ. Distributed knowledge instead quantifies over the intersection of the agents' indistinguishability classes. Common knowledge is governed by chains in their union. Pooling com","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-da2ce42dc70bb3da","tex":"E_n","aliases":"E_n","display":"E_n","kind":"use","excerpt":"Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bec753b55b1a3746","tex":"\\E_{\\overline P_C}","aliases":"E_overline P_C E overline mathbb overline","display":"{\\mathbb E}_{\\overline P_C}","kind":"use","excerpt":"rline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no observation-based sanction with penalty F deters the deviation. Also, E_P_Cs^⋆≤ E_overlin","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-503671c0b4008987","tex":"\\E_{P_C}","aliases":"E_P_C E mathbb","display":"{\\mathbb E}_{P_C}","kind":"use","excerpt":"Under these assumptions the deviation's net expected advantage is g-F( E_P_Ds- E_P_Cs). Compliance is weakly preferable exactly when this quantity is nonpositive. Equality permits indifference and does not predict which action will be selected.","count":8},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f9379434c1b658b6","tex":"\\E_{P_D}","aliases":"E_P_D E mathbb","display":"{\\mathbb E}_{P_D}","kind":"use","excerpt":"Under these assumptions the deviation's net expected advantage is g-F( E_P_Ds- E_P_Cs). Compliance is weakly preferable exactly when this quantity is nonpositive. Equality permits indifference and does not predict which action will be selected.","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4ed775c05819f6a5","tex":"\\E_{P_{D_j}}","aliases":"E_P_D_j E mathbb","display":"{\\mathbb E}_{P_{D_j}}","kind":"use","excerpt":"When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gain","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f75d15c864ad3cef","tex":"E_t","aliases":"E_t","display":"E_t","kind":"use","excerpt":"To formalize an operative institution, augment the microstate of Section~[source reference] with a distinguished component I_t. In a finite setting we may describe it as I_t=(N_t,P_t,E_t,A_t,L_t), where N specifies norms and role obligations, P permissions and capabilities, E enforcement and conflict procedures, A admissible amendments, and L recor","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3a8e0b23212d1b1a","tex":"e_T","aliases":"e_T","display":"e_T","kind":"use","excerpt":"[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2775a21153e9ebc4","tex":"e_t","aliases":"e_t","display":"e_t","kind":"use","excerpt":"[Approximate closure over time] Under~[source reference], let the actual and reduced processes use the same predeclared open-loop control sequence. If their initial macro-laws are within e_0 in total variation, then at time t they are within e_t≤min{1,e_0+tε}. If each overline K_u contracts total variation by at most κ<1, then e_t≤κ^t e_0+εfrac1-κ^","count":9},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0c0ec2f484593060","tex":"e_{t+1}","aliases":"e_t+1","display":"e_{t+1}","kind":"use","excerpt":"o-history policies, the weaker path-law bound min{1,e_0+tε} also holds, but the contraction claim needs a separate closed-loop contraction assumption. Averaging~[source reference] over the actual microstate law shows that its next projected law is within ε of the current projected law evolved by overline K_u. Kernel nonexpansiveness gives e_t+1≤ε+e","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ab5bad267a913e53","tex":"E_{\\{A,B\\}}","aliases":"E_{A,B}","display":"E_{\\{A,B\\}}","kind":"use","excerpt":"[Everyone knows, but it is not common knowledge] Let W={0,1,2} and let φ hold exactly at 0,1. Agent A has partition {0,1}mid{2}, and agent B has partition {0}mid{1,2}. At world 0, both agents know φ. However, the chain 0∼_A1∼_B2 reaches a world where it is false. Thus E_{A,B}φ holds at 0 but C_{A,B}φ does not.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5325ae4241f36b1d","tex":"\\E_\\mu","aliases":"E_μ E mu mathbb mu","display":"{\\mathbb E}_\\mu","kind":"use","excerpt":"[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random vari","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f8483dce1203449b","tex":"\\mathrm{eff}","aliases":"eff mathrm","display":"\\mathrm{eff}","kind":"use","excerpt":"ginequation Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equal-variance estimators, the variance-equivalent effective sample size is n_eff=fracn1+(n-1)ρ. If the errors are perfectly corre","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-11d470feca39d87f","tex":"\\mathsf{Effect}","aliases":"Effect mathsf","display":"\\mathsf{Effect}","kind":"use","excerpt":"A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3844c1e6dc6edcea","tex":"\\mathsf{Evidence}","aliases":"Evidence mathsf","display":"\\mathsf{Evidence}","kind":"use","excerpt":"An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a componen","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a rea","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b6bb9a2af76d66fc","tex":"\\mathbf F","aliases":"F mathbf","display":"\\mathbf F","kind":"use","excerpt":"For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy f","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f6200026fca4fbf6","tex":"F^A","aliases":"F^A","display":"F^A","kind":"use","excerpt":"Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated of","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8cdef41e39b0cc65","tex":"F^B","aliases":"F^B","display":"F^B","kind":"use","excerpt":"Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated of","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-82816d0530292289","tex":"F_0^a","aliases":"F_0^a","display":"F_0^a","kind":"use","excerpt":"-Lipschitz. Then ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continui","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-85a4a71a03fbd7dc","tex":"F_1^a","aliases":"F_1^a","display":"F_1^a","kind":"use","excerpt":"es. Suppose U is L-Lipschitz. Then ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a rea","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-be6ce8f88a6b7ad7","tex":"F_2^a","aliases":"F_2^a","display":"F_2^a","kind":"use","excerpt":"ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound th","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-31d7232e6fa917d3","tex":"F_i","aliases":"F_i","display":"F_i","kind":"use","excerpt":"Even when a global static state exists, a federation may fail dynamically. Local rule applications could commute in one context and conflict in another. A translator T_ij:Z_i→ Z_j should preserve the operations relevant to the interaction, approximately satisfying T_ijF_i≃ F_jT_ij. A federation requires both representational compatibility and admis","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-58ccb6627c020146","tex":"F_i^a","aliases":"F_i^a","display":"F_i^a","kind":"use","excerpt":"For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additiona","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8576ad4cc4bc1341","tex":"F_{j+1}","aliases":"F_j+1","display":"F_{j+1}","kind":"use","excerpt":"Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c0ae9543891a277b","tex":"F_j^a","aliases":"F_j^a","display":"F_j^a","kind":"use","excerpt":"We formalize a restricted compositional calculus, rather than assuming that every mismatch is measured in one natural unit. Three realizations j=0,1,2 have metric state spaces (Z_j,d_j) and a common finite action alphabet A. For algebraic comparison, each action has a total update F_j^a: a disabled action goes to an explicit rejection state. Transl","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4177c6a4de39fdeb","tex":"F_{m-1}","aliases":"F_m-1","display":"F_{m-1}","kind":"use","excerpt":"· ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a0e5c74d87cb8ce3","tex":"F_T","aliases":"F_T","display":"F_T","kind":"use","excerpt":"A deterministic microscopic realization. For any permitted macro-operation T, write (p',q')=T(p,q) and define F_T(a,b,c,d)=(a, a⊕ p', c, c⊕ q'). Then the two local XORs of F_T(a,b,c,d) are exactly (p',q'). Consequently π F_T=Tπ for every microstate, where π(a,b,c,d)=(p,q).","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4a14d2ab0a464e3b","tex":"F_u","aliases":"F_u","display":"F_u","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3871879ef12fd24f","tex":"\\mathbf F_{\\leq D}","aliases":"F_≤ D mathbf leq","display":"\\mathbf F_{\\leq D}","kind":"use","excerpt":"For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy f","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4f178cacb435e6d1","tex":"\\mathsf{Fallback}","aliases":"Fallback mathsf","display":"\\mathsf{Fallback}","kind":"use","excerpt":"An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a componen","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9bba072b92bb3075","tex":"\\FloatBarrier","aliases":"FloatBarrier","display":null,"kind":"use","excerpt":"[.5em] What state must persist? · The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens. [.5em] Can a due debt be erased? · The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8dc93a04e6f9cec5","tex":"\\mathsf{freeze}","aliases":"freeze mathsf","display":"\\mathsf{freeze}","kind":"use","excerpt":"[A safe amendment can destroy every safe continuation] There are four publicly distinguishable states r,ℓ,g,b. The safe set is {r,ℓ,g}. At r, profile deliver moves to g, while profile freeze moves to ℓ. At g, wait loops safely. At ℓ, the only admissible profile advances an outstanding deadline and moves to breach state b. The amendment freeze prese","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f68bc52b44f56ab1","tex":"\\Ftwo","aliases":"Ftwo F_2 mathbb","display":"{\\mathbb F_2}","kind":"use","excerpt":"For z=(p,q)∈ F_2^2, the encoding (p,q,p⊕ q) has distance two. Any two components reconstruct the state, and no one component reconstructs all of it. It tolerates one known erasure. It does not correct an arbitrary one-component corruption at an unknown location. Confusing these fault models overstates robustness.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b5e2fba9b2567ef0","tex":"\\Ftwo^2","aliases":"Ftwo^2 F_2^2 Ftwo mathbb","display":"{\\mathbb F_2}^2","kind":"use","excerpt":"For z=(p,q)∈ F_2^2, the encoding (p,q,p⊕ q) has distance two. Any two components reconstruct the state, and no one component reconstructs all of it. It tolerates one known erasure. It does not correct an arbitrary one-component corruption at an unknown location. Confusing these fault models overstates robustness.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2ec21cceb5b7d7a8","tex":"\\mathbf G","aliases":"G mathbf","display":"\\mathbf G","kind":"use","excerpt":"For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy f","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-76bf4b4cfdf73e02","tex":"g_0^c","aliases":"g_0^c","display":"g_0^c","kind":"use","excerpt":"zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound the two terms by Lε_rm dyn(T) and ε_rm dyn(U). For an observable type c, insert g_1^cT(z) between g_2^cUT(z) and g_0^c(z). Both comparis","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"dination environments in which a durable institution has strictly positive aggregate discounted benefit but is rejected by a rational agent who unilaterally bears setup cost and has veto power, assuming quasilinear payoffs, no transfers, and no additional benefits to that agent. Let the setup cost be B=10, borne entirely by agent 1, and let per-per","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-89e871b998f960fd","tex":"g_1^c","aliases":"g_1^c","display":"g_1^c","kind":"use","excerpt":"). Consequently translations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound the two terms by Lε_rm dyn(T) and ε_rm dyn(U). For an observable type c, insert g_1^cT(z) between g_2^","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e6df94744f7ade62","tex":"g_2","aliases":"g_2","display":"g_2","kind":"use","excerpt":"environments in which a durable institution has strictly positive aggregate discounted benefit but is rejected by a rational agent who unilaterally bears setup cost and has veto power, assuming quasilinear payoffs, no transfers, and no additional benefits to that agent. Let the setup cost be B=10, borne entirely by agent 1, and let per-period gains","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fb7bad6dd26ce3a5","tex":"g_2^c","aliases":"g_2^c","display":"g_2^c","kind":"use","excerpt":"ranslations with zero residual in every coordinate compose to translations with zero residual under this contract. Insert UF_1^aT(z) between UTF_0^a(z) and F_2^aUT(z). The triangle inequality, then Lipschitz continuity of U, bound the two terms by Lε_rm dyn(T) and ε_rm dyn(U). For an observable type c, insert g_1^cT(z) between g_2^cUT(z) and g_0^c(","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e5998b0165958d5f","tex":"g_{AB}","aliases":"g_AB","display":"g_{AB}","kind":"use","excerpt":"A three-jurisdiction example. Take G=S_3, acting on three role labels. Let the translations on the oriented triangle A→ B→ C→ A be g_AB=(12), g_BC=(23), g_CA=e_G. Every edge is a valid bijection. With the specified composition convention, the round trip is (23)(12)≠ e_G. Thus no single assignment of complete role frames realizes all three interface","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5c29749f7018044d","tex":"g_{BC}","aliases":"g_BC","display":"g_{BC}","kind":"use","excerpt":"A three-jurisdiction example. Take G=S_3, acting on three role labels. Let the translations on the oriented triangle A→ B→ C→ A be g_AB=(12), g_BC=(23), g_CA=e_G. Every edge is a valid bijection. With the specified composition convention, the round trip is (23)(12)≠ e_G. Thus no single assignment of complete role frames realizes all three interface","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a5d94e7cf8e07886","tex":"g_{CA}","aliases":"g_CA","display":"g_{CA}","kind":"use","excerpt":"A three-jurisdiction example. Take G=S_3, acting on three role labels. Let the translations on the oriented triangle A→ B→ C→ A be g_AB=(12), g_BC=(23), g_CA=e_G. Every edge is a valid bijection. With the specified composition convention, the round trip is (23)(12)≠ e_G. Thus no single assignment of complete role frames realizes all three interface","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1dea34eb24027a93","tex":"g_i^c","aliases":"g_i^c","display":"g_i^c","kind":"use","excerpt":"For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additiona","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-26e0fe64a6654470","tex":"g_{ij}","aliases":"g_ij","display":"g_{ij}","kind":"use","excerpt":"Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the p","count":12},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6677040f24473561","tex":"g_{ij}^{-1}","aliases":"g_ij^-1","display":"g_{ij}^{-1}","kind":"use","excerpt":"Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the p","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f7288fb9331c8dd5","tex":"g_j","aliases":"g_j","display":"g_j","kind":"use","excerpt":"The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ec945b23e1bf8f82","tex":"g_j^c","aliases":"g_j^c","display":"g_j^c","kind":"use","excerpt":"Common semantic coordinates. For each component type c∈{P,O,R,E}, let g_j^c:Z_j⟶(E_c,δ_c) be a contract observable in a common metric codomain. P records permissions over the common action alphabet; O records obligation-monitor state; R records resources in common physical or accounting units; and E records the required evidentiary and provenance s","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f5bf2eca8fa066ad","tex":"g_{ji}","aliases":"g_ji","display":"g_{ji}","kind":"use","excerpt":"Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the p","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-21dd07c90035ab44","tex":"G_{\\rm formation}","aliases":"G_rm formation rm","display":"G_{\\rm formation}","kind":"use","excerpt":"A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may b","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-773c719cc1e4b602","tex":"G_{\\rm settle}","aliases":"G_rm settle rm","display":"G_{\\rm settle}","kind":"use","excerpt":"Let G_rm suspend require that the declared external capabilities are inactive and that the pending-claim record remains available. Let G_rm settle additionally require that every covered claim has been fulfilled, discharged under the stipulated authority, or transferred to an authorized successor that has accepted it, and that the required final re","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-502a10d3487699f3","tex":"G_{\\rm suspend}","aliases":"G_rm suspend rm","display":"G_{\\rm suspend}","kind":"use","excerpt":"Let G_rm suspend require that the declared external capabilities are inactive and that the pending-claim record remains available. Let G_rm settle additionally require that every covered claim has been fulfilled, discharged under the stipulated authority, or transferred to an authorized successor that has accepted it, and that the required final re","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dcc5a64c4252f82c","tex":"g_{v_0v_1}","aliases":"g_v_0v_1","display":"g_{v_0v_1}","kind":"use","excerpt":"Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the p","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-82d8162c0eae7654","tex":"g_{v_{k-1}v_k}","aliases":"g_v_k-1v_k","display":"g_{v_{k-1}v_k}","kind":"use","excerpt":"Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the p","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-91a8cea6e9ae6182","tex":"g_\\lambda","aliases":"g_λ lambda","display":"g_\\lambda","kind":"use","excerpt":"For a probability vector λ∈Δ_m, set P_λ=∑_jλ_jP_j and g_λ=∑_jλ_jg_j. Write V_α(P,P_C)=max_s∈ M_α⟨ P-P_C,s⟩. This is the same constrained observation advantage used in the single-deviation analysis, now evaluated on mixtures of deviation laws.","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cbffe8e9b78ad1e7","tex":"\\mathsf{Guard}","aliases":"Guard mathsf","display":"\\mathsf{Guard}","kind":"use","excerpt":"An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a componen","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"The question can be stated without selecting a universal theory of legitimacy. Let H be a set of possible founding histories, let a: H→{0,1} be an externally stipulated predicate such as ``the resource owner authorized this founding event,'' and let R: H→ R be the record available to the institution. The predicate models one particular authorizatio","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1e9de57bf29ef2a6","tex":"h'","aliases":"h'","display":"h'","kind":"use","excerpt":"nd only if a is constant on every nonempty fibre of R. This is a statement of informational determination, not necessarily of computability. If two histories with different labels induce identical distributions of every record available to an evaluator, no evaluator restricted to those records distinguishes them with accuracy above 1/2 under equal ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"[Exact internal accounting with an omitted external claim] Four execution slots exist. The internal state contains four member requests and a rule granting one slot to each. In realization h_0, all four slots are available to that institution. In realization h_1, an external owner has reserved one slot for a nonmember, and has authorized the instit","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"Why separate marginal ignorance is insufficient. [Local ignorance with a perfectly correct joint action] Let U be a fair bit. Agent 1 observes H_1=U and agent 2 observes H_2=U⊕θ. Each individual observation has the same law for both values of θ. Nevertheless, the joint action (a_1,a_2)=(H_1,H_2) always satisfies a_1⊕ a_2=θ. The joint observation la","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"[Exact internal accounting with an omitted external claim] Four execution slots exist. The internal state contains four member requests and a rule granting one slot to each. In realization h_0, all four slots are available to that institution. In realization h_1, an external owner has reserved one slot for a nonmember, and has authorized the instit","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"use","excerpt":"Why separate marginal ignorance is insufficient. [Local ignorance with a perfectly correct joint action] Let U be a fair bit. Agent 1 observes H_1=U and agent 2 observes H_2=U⊕θ. Each individual observation has the same law for both values of θ. Nevertheless, the joint action (a_1,a_2)=(H_1,H_2) always satisfies a_1⊕ a_2=θ. The joint observation la","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-333bf23be8bc780a","tex":"h_i","aliases":"h_i","display":"h_i","kind":"use","excerpt":"ph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfyi","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-edabde0d8f0f2fb5","tex":"h_i^{-1}","aliases":"h_i^-1","display":"h_i^{-1}","kind":"use","excerpt":"i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on ev","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"use","excerpt":"i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the path's holonomy. An exact system of global frames is a family h_i∈ G satisfying g_ij=h_jh_i^-1 on ev","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-52b73dd3471377a1","tex":"h_j^{-1}","aliases":"h_j^-1","display":"h_j^{-1}","kind":"use","excerpt":"cancel in every path product, giving H_γ=h_v_kh_v_0^-1 and hence identity on a closed walk. Conversely, choose a root r, set h_r=e_G, and define h_i as the product on the unique tree path from r to i. Tree edges satisfy the required identity. For any non-tree edge i→ j, the rooted loop consisting of the path to i, that edge, and the reversed path f","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3e01b0cedad21f47","tex":"h_r","aliases":"h_r","display":"h_r","kind":"use","excerpt":"ery closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1 and hence identity on a closed walk. Conversely, choose a root r, set h_r=e_G,","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0bbfbbf8fdee97fd","tex":"h_r^{-1}","aliases":"h_r^-1","display":"h_r^{-1}","kind":"use","excerpt":"th k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some traversed edge has error at least d(H_γ,e_G)/k for every candidate frame family. Gauge-transform each edge to widetilde g_ij=h_j^-1g_ijh_i. Bi-invariance gives d(widetilde g_ij,e_G)=δ_ij. The transformed loop product is h_r^-1H_γ h_r, at the same distance from identity as","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6d6860f62790cbda","tex":"h_{v_0}^{-1}","aliases":"h_v_0^-1","display":"h_{v_0}^{-1}","kind":"use","excerpt":"act frame synchronization] Global frames exist if and only if H_γ=e_G for every closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1 an","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bcf5c45a69bb520f","tex":"h_{v_k}","aliases":"h_v_k","display":"h_{v_k}","kind":"use","excerpt":"act frame synchronization] Global frames exist if and only if H_γ=e_G for every closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1 an","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4906fdd4162c9b6e","tex":"H_\\gamma","aliases":"H_γ gamma","display":"H_\\gamma","kind":"use","excerpt":"Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the p","count":10},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1111fb53a62a388c","tex":"\\Hist","aliases":"Hist H mathcal","display":"{\\mathcal H}","kind":"use","excerpt":"Predictive closure has a target and a distribution. For a fixed future target Y^+ and data-generating policy, a predictive statistic satisfies I(Y^+; H_tmid Z_t)=0. Approximate versions bound this residual information or, more directly, the excess predictive loss on held-out data. This is related to predictive-state constructions~[source reference]","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-95a9d0221b28fa4b","tex":"\\Hist_t","aliases":"Hist_t H_t Hist mathcal","display":"{\\mathcal H}_t","kind":"use","excerpt":"Predictive closure has a target and a distribution. For a fixed future target Y^+ and data-generating policy, a predictive statistic satisfies I(Y^+; H_tmid Z_t)=0. Approximate versions bound this residual information or, more directly, the excess predictive loss on held-out data. This is related to predictive-state constructions~[source reference]","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-46cc9b01cd7f4804","tex":"I'","aliases":"I'","display":"I'","kind":"use","excerpt":"For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared fam","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0e79b7ad0299d0b0","tex":"I_0","aliases":"I_0","display":"I_0","kind":"use","excerpt":"[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5d47bac3825149e4","tex":"I_T","aliases":"I_T","display":"I_T","kind":"use","excerpt":"[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-aacf5b6839c1ca6a","tex":"\\mathrm{Id}","aliases":"Id mathrm","display":"\\mathrm{Id}","kind":"use","excerpt":"[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random vari","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-429c14d61f733b4b","tex":"\\mathsf{id}","aliases":"id mathsf","display":"\\mathsf{id}","kind":"use","excerpt":"ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The excha","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distri","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-229edcd074540eda","tex":"\\inf_f","aliases":"inf_f inf","display":"\\inf_f","kind":"use","excerpt":"A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distri","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-78aea61693fa7ede","tex":"\\mathrm{inst}","aliases":"inst mathrm","display":"\\mathrm{inst}","kind":"use","excerpt":"For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared fam","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bbab85f7b2dcaaec","tex":"\\mathsf{Int}","aliases":"Int mathsf","display":"\\mathsf{Int}","kind":"use","excerpt":"Hierarchy, gluing, and the limits of flattening. A lower-level organization can serve as a component at the next level when it exports a sufficient interface. That interface must describe not only a state but also the operations that are permitted on it, their outcome semantics, and their implementation error. A useful interface certificate is Int=","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1","count":18},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-485db41ee683b54c","tex":"J_0","aliases":"J_0","display":"J_0","kind":"use","excerpt":"[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a620020ba505249b","tex":"J_n","aliases":"J_n","display":"J_n","kind":"use","excerpt":"[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fb1ef9525dc5b427","tex":"J_T","aliases":"J_T","display":"J_T","kind":"use","excerpt":"[Finite-block institutional averaging] Under~[source reference], every actual conditional one-review law, given an institutional history ending at i, is within ε of overline R(i,·). Let (J_n) be the Markov chain with kernel overline R and the same initial institutional distribution. Then, for every integer T≥0, d_TV( L(I_0,…,I_T), L(J_0,…,J_T)) ≤ 1","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a componen","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b783228933c35b11","tex":"k_i","aliases":"k_i","display":"k_i","kind":"use","excerpt":"Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative grou","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-caa5b7c8da47b932","tex":"k_i^{-1}","aliases":"k_i^-1","display":"k_i^{-1}","kind":"use","excerpt":"Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative grou","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d28dfa84fbebcbdc","tex":"K_i^m","aliases":"K_i^m","display":"K_i^m","kind":"use","excerpt":"Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j),","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bcf3911a4b4214e2","tex":"k_j","aliases":"k_j","display":"k_j","kind":"use","excerpt":"Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative grou","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e2b60819c95171e8","tex":"k_r","aliases":"k_r","display":"k_r","kind":"use","excerpt":"Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative grou","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9efd26545e0fdba4","tex":"k_r^{-1}","aliases":"k_r^-1","display":"k_r^{-1}","kind":"use","excerpt":"Changing local conventions by elements k_i replaces g_ij by k_jg_ijk_i^-1. A loop based at r then has holonomy k_rH_γ k_r^-1. In particular, identity holonomy is invariant under local changes of coordinates. An arbitrary binary cycle basis with unordered multiplication is not a valid replacement for rooted fundamental loops in a noncommutative grou","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-09eb22fcb1ad8180","tex":"L_{k+1}","aliases":"L_k+1","display":"L_{k+1}","kind":"use","excerpt":"T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-66053b0fba2aa69a","tex":"L_{m-1}","aliases":"L_m-1","display":"L_{m-1}","kind":"use","excerpt":"qquad≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the sec","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-49e6cb003fe8b536","tex":"L_t","aliases":"L_t","display":"L_t","kind":"use","excerpt":"To formalize an operative institution, augment the microstate of Section~[source reference] with a distinguished component I_t. In a finite setting we may describe it as I_t=(N_t,P_t,E_t,A_t,L_t), where N specifies norms and role obligations, P permissions and capabilities, E enforcement and conflict procedures, A admissible amendments, and L recor","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-894b3d805f0da1d1","tex":"\\Law","aliases":"Law L mathcal","display":"{\\mathcal L}","kind":"use","excerpt":"For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal ","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-655fe704a38e0697","tex":"\\Law_e","aliases":"Law_e L_e Law mathcal","display":"{\\mathcal L}_e","kind":"use","excerpt":"For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared fam","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-91b7a65d9478c7b6","tex":"\\Law_x","aliases":"Law_x L_x Law mathcal","display":"{\\mathcal L}_x","kind":"use","excerpt":"For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-949feb1aba030003","tex":"\\Law_{x'}","aliases":"Law_x' L_x' Law mathcal","display":"{\\mathcal L}_{x'}","kind":"use","excerpt":"For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[Exact finite-memory lower bound] Every deterministic summary that supports exact continuation on these histories requires at least N distinct states, hence at least lceillog_2Nrceil fixed-length bits. If the permitted continuation family is closed under prefixing and the history equivalence is a transition congruence, the quotient classes themselv","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"[Minimum profitable coordinated seed] Starting from universal nonadoption, a simultaneously adopting coalition of size m makes each member strictly better off exactly when b(m-1)>c. When such a coalition exists, the smallest size is m_* = lfloorfrac cbrfloor+2≤ n. Once at least m_* adopters are present, every remaining nonadopter has a profitable u","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"[Exact finite-memory lower bound] Every deterministic summary that supports exact continuation on these histories requires at least N distinct states, hence at least lceillog_2Nrceil fixed-length bits. If the permitted continuation family is closed under prefixing and the history equivalence is a transition congruence, the quotient classes themselv","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"[Exact finite-memory lower bound] Every deterministic summary that supports exact continuation on these histories requires at least N distinct states, hence at least lceillog_2Nrceil fixed-length bits. If the permitted continuation family is closed under prefixing and the history equivalence is a transition congruence, the quotient classes themselv","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"[Deterministic quotient criterion] For maps F_u: X→ X and a surjection π: X→ Z, maps overline F_u: Z→ Z satisfying π F_u=overline F_uπ exist if and only if π(x)=π(x') Longrightarrow π(F_u(x))=π(F_u(x')) for every u,x,x'. When they exist, they are unique. Necessity follows by applying overline F_u to the common value π(x). For sufficiency, set overl","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated of","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-349dacc2f32ca156","tex":"M^{\\rm act}","aliases":"M^rm act rm","display":"M^{\\rm act}","kind":"use","excerpt":"Let a history H be summarized by an inheritance map C(H). A practically meaningful retained state can be divided into an active canon M^rm act for immediate use, an archive M^rm arc for recovery and criticism, and a provenance and uncertainty record P. These stores can overlap physically, but serve distinct decision contracts. The canon keeps inter","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-68234482819a0a5d","tex":"M^{\\rm arc}","aliases":"M^rm arc rm","display":"M^{\\rm arc}","kind":"use","excerpt":"Let a history H be summarized by an inheritance map C(H). A practically meaningful retained state can be divided into an active canon M^rm act for immediate use, an archive M^rm arc for recovery and criticism, and a provenance and uncertainty record P. These stores can overlap physically, but serve distinct decision contracts. The canon keeps inter","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b11203379a62bd48","tex":"m_*","aliases":"m_*","display":"m_*","kind":"use","excerpt":"[Minimum profitable coordinated seed] Starting from universal nonadoption, a simultaneously adopting coalition of size m makes each member strictly better off exactly when b(m-1)>c. When such a coalition exists, the smallest size is m_* = lfloorfrac cbrfloor+2≤ n. Once at least m_* adopters are present, every remaining nonadopter has a profitable u","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8ace55e8bb71177a","tex":"\\mathcal M_A","aliases":"M_A mathcal","display":"\\mathcal M_A","kind":"use","excerpt":"Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated of","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a79467f050b89408","tex":"\\mathcal M_B","aliases":"M_B mathcal","display":"\\mathcal M_B","kind":"use","excerpt":"Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated of","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8bb69312790c247e","tex":"M_{t+1}","aliases":"M_t+1","display":"M_{t+1}","kind":"use","excerpt":"nter controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assigns nonnegative voting weights, q_t is an approval threshold, and e_t is a configuration epoch. Adm","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-967ea86ebf866388","tex":"\\mathcal M_\\alpha","aliases":"M_α mathcal alpha","display":"\\mathcal M_\\alpha","kind":"use","excerpt":"The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"False-positive limits and multiple deviations. A constitutional protection may impose E_P_Cs≤α, with 0≤α≤1. The relevant discrimination capacity is then the finite linear program D_α(P_D,P_C)= max_substack0≤ s≤1 E_P_Cs≤α ( E_P_Ds- E_P_Cs). The feasible set is nonempty and compact, so a maximum exists. The preceding proof with the feasible set restr","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6930c978a50d9a39","tex":"\\max_{0\\le s\\le1,\\ \\E_{P_C}s\\le\\alpha}","aliases":"max_0≤ s≤1, E_P_Cs≤α max le le E le alpha max le le mathbb le alpha","display":"\\max_{0\\le s\\le1,\\ {\\mathbb E}_{P_C}s\\le\\alpha}","kind":"use","excerpt":"Keep the earlier definition D_α(P_D,P_C)=max_0≤ s≤1, E_P_Cs≤α ( E_P_Ds- E_P_Cs), 0≤α≤1. The same randomized sanction s must obey the protected false-positive limit and supply the deviation margin.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"use","excerpt":"nnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnor","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e99a7833fcf755ca","tex":"\\max_{\\substack{0\\leq s\\leq1\\\\\\E_{P_C}s\\leq\\alpha}}","aliases":"max_substack0≤ s≤1 E_P_Cs≤α max substack leq leq E leq alpha max substack leq leq mathbb leq alpha","display":"\\max_{\\substack{0\\leq s\\leq1\\\\{\\mathbb E}_{P_C}s\\leq\\alpha}}","kind":"use","excerpt":"False-positive limits and multiple deviations. A constitutional protection may impose E_P_Cs≤α, with 0≤α≤1. The relevant discrimination capacity is then the finite linear program D_α(P_D,P_C)= max_substack0≤ s≤1 E_P_Cs≤α ( E_P_Ds- E_P_Cs). The feasible set is nonempty and compact, so a maximum exists. The preceding proof with the feasible set restr","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-56b2f3cfe3dd0a1b","tex":"\\max_{s\\in\\mathcal M_\\alpha}","aliases":"max_s∈ M_α max in mathcal alpha","display":"\\max_{s\\in\\mathcal M_\\alpha}","kind":"use","excerpt":"th probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous margin by D_j(s)=F⟨ P_j-P_C,s⟩-g_j, Γ_α=max_s∈ M_αmin_1≤ j≤ ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distri","count":17},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6b0b4e4f28de5e50","tex":"\\min_{1\\leq j\\leq m}","aliases":"min_1≤ j≤ m min leq leq","display":"\\min_{1\\leq j\\leq m}","kind":"use","excerpt":"th probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous margin by D_j(s)=F⟨ P_j-P_C,s⟩-g_j, Γ_α=max_s∈ M_αmin_1≤ j≤ ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1ace88525e5ee17f","tex":"\\min_e","aliases":"min_e min","display":"\\min_e","kind":"use","excerpt":"esidual and minimum authorized repair] The XOR of edge labels around a cycle is unchanged by every local convention change above. On a single simple cycle with odd XOR, suppose an editable subset of edges can be flipped at nonnegative costs w_e, with other edges protected. An authorized repair exists exactly when the editable subset is nonempty. Th","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f2be642960c93c27","tex":"\\min_{S\\in\\mathcal A}","aliases":"min_S∈ A min in mathcal","display":"\\min_{S\\in\\mathcal A}","kind":"use","excerpt":"A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distri","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f149a3237795b7bd","tex":"\\min_{z\\ne z'}","aliases":"min_z≠ z' min ne","display":"\\min_{z\\ne z'}","kind":"use","excerpt":"Erasure resilience is not arbitrary error correction. Let an organizational state z range over a finite set with at least two elements and be encoded as E(z)=(E_1(z),…,E_n(z)). Define the block distance d_E=min_z≠ z'abs{i:E_i(z)≠ E_i(z')}.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-619a17276daec9c4","tex":"\\min_{\\lambda\\in\\Delta_m}","aliases":"min_λ∈Δ_m min lambda in Delta","display":"\\min_{\\lambda\\in\\Delta_m}","kind":"use","excerpt":"[Simultaneous enforcement and mixture obstruction] Under the finite common-monitor model, Γ_α=min_λ∈Δ_m (FV_α(P_λ,P_C)-g_λ). Both extrema are attained. Consequently, one monitor weakly deters all listed deviations if and only if FV_α(P_λ,P_C)≥ g_λ for every λ∈Δ_m. Without an additional false-positive restriction, α=1 and V_1(P,P_C)=d_TV(P,P_C). For","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-059b561ecaed3b30","tex":"\\min_{\\varnothing\\ne S\\subseteq J}","aliases":"min_∅≠ S⊆ J min varnothing ne subseteq","display":"\\min_{\\varnothing\\ne S\\subseteq J}","kind":"use","excerpt":"The theorem permits a simple robustness index. If the initial graph admits a staffing, define σ=min_∅≠ S⊆ J(|N(S)|-|S|). Then exactly the erasure budgets k≤σ are guaranteed tolerable in this model. This is a static allocation certificate, not a claim that the surviving agents can discover or execute the reassignment before a deadline. That further ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7997c4f3be17c7ce","tex":"\\models","aliases":"models","display":null,"kind":"use","excerpt":"[Common knowledge as a reachability condition] Let Reach_J(w) be the worlds reachable from w by a finite chain of relations ∼_i with i∈ J, including the zero-length chain. Then wmodels C_Jφ Longleftrightarrow Reach_J(w)⊆{v:vmodelsφ}. Induction on n shows that E_J^nφ requires φ at the endpoint of every chain of n indistinguishability steps. Requirin","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Memory lower bounds from distinguishable histories. Fix a deterministic exact predictor for a family of future input continuations. Call two past histories equivalent when they require the same output sequence for every permitted continuation. Let N be the number of equivalence classes among the histories under consideration.","count":26},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"[Soundness of a finite continuation certificate] If the preceding checks hold, κ preserves the complete contract from B_0 for all transitions. If, in addition, there is a map r: C→ N such that r(B)=0 implies B⊆ G and every successor of a set with positive rank has strictly smaller rank, then a set contained in G is reached within at most r(B_0) tra","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f03486d125e6b8f9","tex":"n_{\\mathrm{eff}}","aliases":"n_eff mathrm","display":"n_{\\mathrm{eff}}","kind":"use","excerpt":"ginequation Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equal-variance estimators, the variance-equivalent effective sample size is n_eff=fracn1+(n-1)ρ. If the errors are perfectly corre","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-15b483d1ca8d08c8","tex":"N_F","aliases":"N_F","display":"N_F","kind":"use","excerpt":"Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not desc","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b82d70c3ab4030a5","tex":"N_i","aliases":"N_i","display":"N_i","kind":"use","excerpt":"A norm can be presented operationally as a condition on histories and actions, N_i(h,a)∈{0,1}, interpreted as the system's declared permission for action a by position i after history h. This does not make the declared permission morally correct. Enforcement changes the action transition or cost structure; appeal changes what happens after an alleg","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-87025dc8dd8d0f19","tex":"N_T","aliases":"N_T","display":"N_T","kind":"use","excerpt":"Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not desc","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-340f7d39d04f82c1","tex":"N_t","aliases":"N_t","display":"N_t","kind":"use","excerpt":"To formalize an operative institution, augment the microstate of Section~[source reference] with a distinguished component I_t. In a finite setting we may describe it as I_t=(N_t,P_t,E_t,A_t,L_t), where N specifies norms and role obligations, P permissions and capabilities, E enforcement and conflict procedures, A admissible amendments, and L recor","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"nnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnor","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-090aecb8fb37e7b7","tex":"O_1","aliases":"O_1","display":"O_1","kind":"use","excerpt":"[Two uninformative records with an informative joint history] Let θ∈{0,1} distinguish two protected cases, and let R be a fair independent bit. Release O_1=R and O_2=R⊕θ. Each marginal is a fair bit under both cases and satisfies~[source reference] with λ=1. But O_1⊕ O_2=θ. The joint laws have disjoint support and satisfy no finite multiplicative b","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-95762a76969f784c","tex":"O_2","aliases":"O_2","display":"O_2","kind":"use","excerpt":"[Two uninformative records with an informative joint history] Let θ∈{0,1} distinguish two protected cases, and let R be a fair independent bit. Release O_1=R and O_2=R⊕θ. Each marginal is a fair bit under both cases and satisfies~[source reference] with λ=1. But O_1⊕ O_2=θ. The joint laws have disjoint support and satisfy no finite multiplicative b","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7d562788a949216b","tex":"O_{\\rm ext}","aliases":"O_rm ext rm","display":"O_{\\rm ext}","kind":"use","excerpt":"A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5b5f757795d9d4e3","tex":"O_V","aliases":"O_V","display":"O_V","kind":"use","excerpt":"A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distri","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a46c63c71e71fa74","tex":"\\mathsf{open}","aliases":"open mathsf","display":"\\mathsf{open}","kind":"use","excerpt":"[An unchecked change of the checker] Let the institutional state contain a validator mode v∈{strict,open} and a resource policy p∈{safe,unsafe}. Under v=strict, the current policy must remain safe, but the validator mode itself is unprotected. Thus (strict,safe)→(open,safe) is admitted. Under v=open, any policy transition is admitted, including (op","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-512abf9c72d3182f","tex":"\\overline d","aliases":"overline d overline","display":"\\overline d","kind":"use","excerpt":"[A robust margin for reduced-model enforcement] Let s^⋆(o) sanction precisely where overline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no ob","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d472066b913c5d84","tex":"\\overline F","aliases":"overline F overline","display":"\\overline F","kind":"use","excerpt":"[Deterministic quotient criterion] For maps F_u: X→ X and a surjection π: X→ Z, maps overline F_u: Z→ Z satisfying π F_u=overline F_uπ exist if and only if π(x)=π(x') Longrightarrow π(F_u(x))=π(F_u(x')) for every u,x,x'. When they exist, they are unique. Necessity follows by applying overline F_u to the common value π(x). For sufficiency, set overl","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d0e9f947070b1886","tex":"\\overline F_u","aliases":"overline F_u overline","display":"\\overline F_u","kind":"use","excerpt":"[Deterministic quotient criterion] For maps F_u: X→ X and a surjection π: X→ Z, maps overline F_u: Z→ Z satisfying π F_u=overline F_uπ exist if and only if π(x)=π(x') Longrightarrow π(F_u(x))=π(F_u(x')) for every u,x,x'. When they exist, they are unique. Necessity follows by applying overline F_u to the common value π(x). For sufficiency, set overl","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b7a5b96653c1dfce","tex":"\\overline K","aliases":"overline K overline","display":"\\overline K","kind":"use","excerpt":"[Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law","count":9},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3a2dda7b86c14643","tex":"\\overline K_u","aliases":"overline K_u overline","display":"\\overline K_u","kind":"use","excerpt":"[Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law","count":8},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ffdd263bffb70d69","tex":"\\overline P","aliases":"overline P overline","display":"\\overline P","kind":"use","excerpt":"Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bo","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0554550a5d4c3d25","tex":"\\overline P_a","aliases":"overline P_a overline","display":"\\overline P_a","kind":"use","excerpt":"Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bo","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c6155705a6fd7527","tex":"\\overline P_C","aliases":"overline P_C overline","display":"\\overline P_C","kind":"use","excerpt":"Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bo","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-35facfc15ab1071a","tex":"\\overline P_D","aliases":"overline P_D overline","display":"\\overline P_D","kind":"use","excerpt":"Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bo","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-42c6158ae78541bc","tex":"\\overline Q","aliases":"overline Q overline","display":"\\overline Q","kind":"use","excerpt":"Hierarchy, gluing, and the limits of flattening. A lower-level organization can serve as a component at the next level when it exports a sufficient interface. That interface must describe not only a state but also the operations that are permitted on it, their outcome semantics, and their implementation error. A useful interface certificate is Int=","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1a02ef2c814691f5","tex":"\\overline R","aliases":"overline R overline","display":"\\overline R","kind":"use","excerpt":"Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j), overline R(i,j) · =∑_yμ_i(y)R_i(y,j). The endpoint is the only fast input to this particular revision rule. A rule that uses the full trajectory, accumulated costs, or a data-","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8b5639b5bcc171d8","tex":"\\overline R_k","aliases":"overline R_k overline","display":"\\overline R_k","kind":"use","excerpt":"[Governance abstraction preserves joint realizability] Under~[source reference]--[source reference], for every B∈ B and every horizon k, B∈ W_kLongleftrightarrowπ[B]∈overline W_k, B∈ W_*Longleftrightarrowπ[B]∈overline W_*. The progress sets satisfy B∈ R_kLongleftrightarrowπ[B]∈overline R_k, so finite progress ranks agree. A reduced safe or ranked s","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-70becec0d32f5b1e","tex":"\\overline W","aliases":"overline W overline","display":"\\overline W","kind":"use","excerpt":"[Governance abstraction preserves joint realizability] Under~[source reference]--[source reference], for every B∈ B and every horizon k, B∈ W_kLongleftrightarrowπ[B]∈overline W_k, B∈ W_*Longleftrightarrowπ[B]∈overline W_*. The progress sets satisfy B∈ R_kLongleftrightarrowπ[B]∈overline R_k, so finite progress ranks agree. A reduced safe or ranked s","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-898f9c5012aeca12","tex":"\\overline W_*","aliases":"overline W_* overline","display":"\\overline W_*","kind":"use","excerpt":"[Governance abstraction preserves joint realizability] Under~[source reference]--[source reference], for every B∈ B and every horizon k, B∈ W_kLongleftrightarrowπ[B]∈overline W_k, B∈ W_*Longleftrightarrowπ[B]∈overline W_*. The progress sets satisfy B∈ R_kLongleftrightarrowπ[B]∈overline R_k, so finite progress ranks agree. A reduced safe or ranked s","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c9d9fa3ccc63f9f9","tex":"\\overline W_k","aliases":"overline W_k overline","display":"\\overline W_k","kind":"use","excerpt":"[Governance abstraction preserves joint realizability] Under~[source reference]--[source reference], for every B∈ B and every horizon k, B∈ W_kLongleftrightarrowπ[B]∈overline W_k, B∈ W_*Longleftrightarrowπ[B]∈overline W_*. The progress sets satisfy B∈ R_kLongleftrightarrowπ[B]∈overline R_k, so finite progress ranks agree. A reduced safe or ranked s","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-699186a0f56a06b6","tex":"\\overline{\\operatorname{Post}}","aliases":"overlinePost overline operatorname","display":"\\overline{\\operatorname{Post}}","kind":"use","excerpt":"progress ranks agree. A reduced safe or ranked strategy pulls back by choosing at B the profile assigned to π[B]. The observation equality makes π[B] a valid reduced possibility set. Admissibility and safe-set membership are preserved and reflected. For every possible next observation y, successor equality gives the commuting update identity π[Post","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"Rule-changing computation and governance-preserving abstraction. A persistent society does not merely choose actions under a rule. Some authorized actions change which future actions are admitted. Write a complete state as s=(x,I, P,r,λ,c), where x is operational state, I institutional records, P the current population and office registry, r resour","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":49},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-171e10e212c438c2","tex":"p'","aliases":"p'","display":"p'","kind":"use","excerpt":"A deterministic microscopic realization. For any permitted macro-operation T, write (p',q')=T(p,q) and define F_T(a,b,c,d)=(a, a⊕ p', c, c⊕ q'). Then the two local XORs of F_T(a,b,c,d) are exactly (p',q'). Consequently π F_T=Tπ for every microstate, where π(a,b,c,d)=(p,q).","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d191a609e31a0b28","tex":"p^*","aliases":"p^*","display":"p^*","kind":"use","excerpt":"[Inspection without commitment] Under the preceding strict inequalities, the game has no pure Nash equilibrium and has a unique Nash equilibrium in independently mixed strategies. Its deviation and inspection probabilities are p^*=fraccb, q^*=fracgF. In particular, certain compliance is not sustained in this simultaneous game. If the auditor inspec","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"[Almost-sure locking with a positive error probability] In this model a cascade begins almost surely after finitely many revealing decisions. Its probability of selecting the wrong state is η(p)=frac(1-p)^2p^2+(1-p)^2>0. The probability that no cascade has begun after 2r decisions is [2p(1-p)]^r. After locking, further actions contain no additional","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2f6e2105e532043f","tex":"P^\\star_{z'}","aliases":"P^⋆_z' star","display":"P^\\star_{z'}","kind":"use","excerpt":"[Implementation dependence] If every P_λ is within ε of a proposed macro-intervention law P^⋆_z', then η(x,z')≤2ε. Conversely, if two admissible implementations are distance η apart, no single law can approximate both within less than η/2. Both claims follow from the triangle inequality for total variation.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":",1} and a class of admissible protocols. For one protocol, let H be the tuple of all local histories up to the decision, together with the private and shared random tapes used to choose actions. The tapes have their specified law and cannot encode extra information about θ. The joint action is a measurable function of H even though no participant n","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[Individually enforceable deviations with no common monitor] Take O={0,1}, P_C=(1/2,1/2), P_1=(1,0), P_2=(0,1), F=2, g_1=g_2=1/2, and α=1/2. The monitor (1,0) strictly deters deviation 1, and (0,1) strictly deters deviation 2. Both respect the false-positive cap. For any common monitor, D_1(s)=s(0)-s(1)-tfrac12, D_2(s)=s(1)-s(0)-tfrac12. Their sum ","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"[Individually enforceable deviations with no common monitor] Take O={0,1}, P_C=(1/2,1/2), P_1=(1,0), P_2=(0,1), F=2, g_1=g_2=1/2, and α=1/2. The monitor (1,0) strictly deters deviation 1, and (0,1) strictly deters deviation 2. Both respect the false-positive cap. For any common monitor, D_1(s)=s(0)-s(1)-tfrac12, D_2(s)=s(1)-s(0)-tfrac12. Their sum ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bo","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c6db9f401f21bd60","tex":"P_C^t","aliases":"P_C^t","display":"P_C^t","kind":"use","excerpt":"[Conditional disclosure accounting] Consider m releases on finite alphabets. At each common prefix h, suppose their conditional next-release laws P_C^t(·mid h) and P_D^t(·mid h) satisfy both pointwise inequalities with factor λ_t≥1, uniformly over prefixes that can occur. Then the joint transcript laws satisfy~[source reference] with λ=∏_t=1^mλ_t. ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-24aee7146853637c","tex":"P_D^t","aliases":"P_D^t","display":"P_D^t","kind":"use","excerpt":"[Conditional disclosure accounting] Consider m releases on finite alphabets. At each common prefix h, suppose their conditional next-release laws P_C^t(·mid h) and P_D^t(·mid h) satisfy both pointwise inequalities with factor λ_t≥1, uniformly over prefixes that can occur. Then the joint transcript laws satisfy~[source reference] with λ=∏_t=1^mλ_t. ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0973faa3939feb7a","tex":"P_{D_1}","aliases":"P_D_1","display":"P_{D_1}","kind":"use","excerpt":"When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gain","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-976e344ae208876a","tex":"P_{D_2}","aliases":"P_D_2","display":"P_{D_2}","kind":"use","excerpt":"When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gain","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d4ac276dce951030","tex":"P_{D_j}","aliases":"P_D_j","display":"P_{D_j}","kind":"use","excerpt":"When there are deviations D_1,…,D_k, one common policy must satisfy every incentive inequality. Separate pairwise discrimination bounds are necessary but not sufficient. With P_C=(1/2,1/2), P_D_1=(1,0), and P_D_2=(0,1), each pair has distance 1/2, yet the two differences E_P_D_js- E_P_Cs sum to zero. No common sanction deters strictly positive gain","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-139b65216c886fd4","tex":"P_{M,w}","aliases":"P_M,w","display":"P_{M,w}","kind":"use","excerpt":"[No universal constitutional safety decider] There is no total algorithm that decides safety for every program in this class. In the same class, occurrence of the forbidden event is semidecidable, whereas safety is not semidecidable in general. Given a Turing machine M and input w, construct a program P_M,w that simulates M(w) and produces the forb","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-12b33428971a165a","tex":"P_t","aliases":"P_t","display":"P_t","kind":"use","excerpt":"To formalize an operative institution, augment the microstate of Section~[source reference] with a distinguished component I_t. In a finite setting we may describe it as I_t=(N_t,P_t,E_t,A_t,L_t), where N specifies norms and role obligations, P permissions and capabilities, E enforcement and conflict procedures, A admissible amendments, and L recor","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-605436089c261022","tex":"\\mathsf{permissions}","aliases":"permissions mathsf","display":"\\mathsf{permissions}","kind":"use","excerpt":"Hierarchy, gluing, and the limits of flattening. A lower-level organization can serve as a component at the next level when it exports a sufficient interface. That interface must describe not only a state but also the operations that are permitted on it, their outcome semantics, and their implementation error. A useful interface certificate is Int=","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"[Deterministic microstate, non-Markov macrostate] Let X_t+1=X_t+1pmod 3 with uniform initial distribution. Set π(0)=π(1)=a and π(2)=b. Stationarity gives P(Z_t+1=bmid Z_t=a)=1/2. Yet P(Z_t+1=bmid Z_t=a,Z_t-1=b)=0, whereas the same probability conditional on Z_t=a,Z_t-1=a is 1. Thus the apparent randomness comes with memory: the one-step marginal do","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-80b4b6d1ff3db017","tex":"\\Prob","aliases":"Prob P mathbb","display":"{\\mathbb P}","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-911fd80387908659","tex":"\\Prob_\\mu","aliases":"Prob_μ P_μ Prob mu mathbb mu","display":"{\\mathbb P}_\\mu","kind":"use","excerpt":"[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random vari","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"public record contains the current allocation rule, valid credentials, accepted urgency certificates, outstanding service obligations, and the amendment history. Slots have unique names (e,k), where e is the epoch and k∈{1,2,3,4}. A grant consumes one available slot token. Copying a message that describes a grant does not create an additional token","count":35},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"A deterministic microscopic realization. For any permitted macro-operation T, write (p',q')=T(p,q) and define F_T(a,b,c,d)=(a, a⊕ p', c, c⊕ q'). Then the two local XORs of F_T(a,b,c,d) are exactly (p',q'). Consequently π F_T=Tπ for every microstate, where π(a,b,c,d)=(p,q).","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5cf102ce096caa7a","tex":"q^*","aliases":"q^*","display":"q^*","kind":"use","excerpt":"[Inspection without commitment] Under the preceding strict inequalities, the game has no pure Nash equilibrium and has a unique Nash equilibrium in independently mixed strategies. Its deviation and inspection probabilities are p^*=fraccb, q^*=fracgF. In particular, certain compliance is not sustained in this simultaneous game. If the auditor inspec","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1a964a7903c6c96c","tex":"Q^t","aliases":"Q^t","display":"Q^t","kind":"use","excerpt":"[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random vari","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0c35b13a3641b2f4","tex":"\\mathcal Q_*","aliases":"Q_* mathcal","display":"\\mathcal Q_*","kind":"use","excerpt":"A finite closure theorem, not an open-endedness theorem. Take finite universes C_* and Q_*. Consider a fixed collection of production rules of the form ``if these finitely many records are present, add this record.'' Prerequisites are positive: no rule requires the absence of a record. The content and correctness of those rules are assumptions. An ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3ed43c7f5cff9ad1","tex":"\\mathcal Q_0","aliases":"Q_0 mathcal","display":"\\mathcal Q_0","kind":"use","excerpt":"Let Φ simultaneously add all conclusions whose prerequisites are satisfied, retaining the input records. Order pairs ( C, Q) by componentwise inclusion. [Least closure of a finite problem--method system] Starting from ( C_0, Q_0), iteration of Φ reaches the least closed pair containing that initial pair. There are at most | C_*∖ C_0|+| Q_*∖ Q_0| st","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"use","excerpt":"public record contains the current allocation rule, valid credentials, accepted urgency certificates, outstanding service obligations, and the amendment history. Slots have unique names (e,k), where e is the epoch and k∈{1,2,3,4}. A grant consumes one available slot token. Copying a message that describes a grant does not create an additional token","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"hcal U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be r","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5e30045657849316","tex":"\\mathsf{quantity}","aliases":"quantity mathsf","display":"\\mathsf{quantity}","kind":"use","excerpt":"ubsectionTwo rules and an inherited obligation Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The excha","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"A transparent comparison model can make one class of failure mathematically tractable. Let x_t∈ R_≥0^n record nonnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vecto","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distri","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"[.5em] What state must persist? · The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens. [.5em] Can a due debt be erased? · The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6ac432f94aea74b8","tex":"r_{-i}","aliases":"r_-i","display":"r_{-i}","kind":"use","excerpt":"A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e8531ad2dd91d431","tex":"r_F","aliases":"r_F","display":"r_F","kind":"use","excerpt":"Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not desc","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c058161c25df89bd","tex":"r_F^t","aliases":"r_F^t","display":"r_F^t","kind":"use","excerpt":"Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not desc","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0cecacd0d0bb53a7","tex":"\\mathcal R_i","aliases":"R_i mathcal","display":"\\mathcal R_i","kind":"use","excerpt":"A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-48d1f02269ce141f","tex":"r_i'","aliases":"r_i'","display":"r_i'","kind":"use","excerpt":"A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e16c40c1720d168f","tex":"r_{ij}^i","aliases":"r_ij^i","display":"r_{ij}^i","kind":"use","excerpt":"Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentatio","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0b8641bf15f44dd3","tex":"r_{ij}^j","aliases":"r_ij^j","display":"r_{ij}^j","kind":"use","excerpt":"Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentatio","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"A bounded allocation model makes the capacity obligation precise. Let J be a finite set of admitted appeals and let the integer epochs be 1,…,T. At epoch t, there are c_t≥0 reserved review slots after accounting for all other protected work. An appeal j has an evidence-ready release epoch r_j and deadline d_j, with 1≤ r_j≤ d_j≤ T. It requires one s","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a64e897db39c6a7f","tex":"R_k","aliases":"R_k","display":"R_k","kind":"use","excerpt":"Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b9fe2b346d4c6f8d","tex":"R_{k+1}","aliases":"R_k+1","display":"R_{k+1}","kind":"use","excerpt":"Let G⊆ S_rm safe be a stipulated completion set. The target is certified completion, meaning that the current possibility set lies wholly in G. Let G={B∈ W_*:B⊆ G}, A_*(B)={u∈ A(B):Succ(B,u)⊆ W_*}. Define R_0= G and R_k+1=R_k∪ {B∈ W_*:∃ u∈ A_*(B), Succ(B,u)⊆ R_k}. The increasing sequence stabilizes at R_*. For B∈ R_*, let r(B) be its first index of","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a4ff961cd74d1869","tex":"r_n","aliases":"r_n","display":"r_n","kind":"use","excerpt":"A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-777e99bfe04d5f4e","tex":"r_{\\rm end}","aliases":"r_rm end rm","display":"r_{\\rm end}","kind":"use","excerpt":"T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-935c16efac7d9b95","tex":"R_{\\rm repair}","aliases":"R_rm repair rm","display":"R_{\\rm repair}","kind":"use","excerpt":"A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may b","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-72ef1cd62b216cdd","tex":"r_T","aliases":"r_T","display":"r_T","kind":"use","excerpt":"Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not desc","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cab8512436e7d001","tex":"r_T^t","aliases":"r_T^t","display":"r_T^t","kind":"use","excerpt":"Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not desc","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9a0a032f1455f671","tex":"\\mathcal R_\\ell","aliases":"R_ℓ mathcal ell","display":"\\mathcal R_\\ell","kind":"use","excerpt":"A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distri","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8f79fd1cbecd31c3","tex":"\\mathbb R_{\\ge0}^n","aliases":"R_≥0^n mathbb ge","display":"\\mathbb R_{\\ge0}^n","kind":"use","excerpt":"A transparent comparison model can make one class of failure mathematically tractable. Let x_t∈ R_≥0^n record nonnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vecto","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[Exact finite-memory lower bound] Every deterministic summary that supports exact continuation on these histories requires at least N distinct states, hence at least lceillog_2Nrceil fixed-length bits. If the permitted continuation family is closed under prefixing and the history equivalence is a transition congruence, the quotient classes themselv","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f8e96c093962b45c","tex":"\\mathsf{Reach}","aliases":"Reach mathsf","display":"\\mathsf{Reach}","kind":"use","excerpt":"[Common knowledge as a reachability condition] Let Reach_J(w) be the worlds reachable from w by a finite chain of relations ∼_i with i∈ J, including the zero-length chain. Then wmodels C_Jφ Longleftrightarrow Reach_J(w)⊆{v:vmodelsφ}. Induction on n shows that E_J^nφ requires φ at the endpoint of every chain of n indistinguishability steps. Requirin","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c8f7e2c70f9ad344","tex":"\\mathsf{Reach}_J","aliases":"Reach_J mathsf","display":"\\mathsf{Reach}_J","kind":"use","excerpt":"[Common knowledge as a reachability condition] Let Reach_J(w) be the worlds reachable from w by a finite chain of relations ∼_i with i∈ J, including the zero-length chain. Then wmodels C_Jφ Longleftrightarrow Reach_J(w)⊆{v:vmodelsφ}. Induction on n shows that E_J^nφ requires φ at the endpoint of every chain of n indistinguishability steps. Requirin","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2781357a113adde1","tex":"\\mathsf{Remedy}","aliases":"Remedy mathsf","display":"\\mathsf{Remedy}","kind":"use","excerpt":"A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-37a16f3398973366","tex":"\\mathsf{Revision}","aliases":"Revision mathsf","display":"\\mathsf{Revision}","kind":"use","excerpt":"An assurance record can contain K=(Claim,Assumptions,Evidence, Domain,Guard,Fallback,Revision). Its status should distinguish at least an applicable certificate, a certificate awaiting rechecking, a suspended claim, and a superseded version. Evidence for an assumption can be observed, externally attested under a trust model, proved about a componen","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"[Minimum profitable coordinated seed] Starting from universal nonadoption, a simultaneously adopting coalition of size m makes each member strictly better off exactly when b(m-1)>c. When such a coalition exists, the smallest size is m_* = lfloorfrac cbrfloor+2≤ n. Once at least m_* adopters are present, every remaining nonadopter has a profitable u","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5e769b89788d547a","tex":"s'","aliases":"s'","display":"s'","kind":"use","excerpt":"Let s=(x,I) be the full operating state and let R(s,s') be the admitted transition relation, including ordinary actions and amendments. A safety property is a predicate Ψ(s).","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dc18d935845d0826","tex":"s^\\star","aliases":"s^⋆ star","display":"s^\\star","kind":"use","excerpt":"[A robust margin for reduced-model enforcement] Let s^⋆(o) sanction precisely where overline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no ob","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"[Constitutional inductive safety] If Ψ(s_0) and every admitted transition satisfies Ψ(s)wedge R(s,s')LongrightarrowΨ(s'), then Ψ(s_t) holds for every state reachable by finitely many admitted transitions. Induction on the number of admitted transitions.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3ea628df7c9adeec","tex":"S_{1,t}","aliases":"S_1,t","display":"S_{1,t}","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"A three-jurisdiction example. Take G=S_3, acting on three role labels. Let the translations on the oriented triangle A→ B→ C→ A be g_AB=(12), g_BC=(23), g_CA=e_G. Every edge is a valid bijection. With the specified composition convention, the round trip is (23)(12)≠ e_G. Thus no single assignment of complete role frames realizes all three interface","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2badf92439cc7951","tex":"S_A","aliases":"S_A","display":"S_A","kind":"use","excerpt":"Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated of","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-450f6a43de6765bd","tex":"S_B","aliases":"S_B","display":"S_B","kind":"use","excerpt":"Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated of","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-698cd4222a287d75","tex":"S_i","aliases":"S_i","display":"S_i","kind":"use","excerpt":"One possible factorization of a microstate is X_t=(S_1,t,…,S_n,t,M_t,C_t,A_t,Θ_t), where S_i is local process state, M durable storage, C messages and channel state, A artifacts, and Θ the current configuration, including executable rules. These categories describe a storage and access organization. They need not be disjoint physical objects.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-040191bdcfa58557","tex":"S_{n,t}","aliases":"S_n,t","display":"S_{n,t}","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4bca7afcafa97a49","tex":"s_t","aliases":"s_t","display":"s_t","kind":"use","excerpt":"[Constitutional inductive safety] If Ψ(s_0) and every admitted transition satisfies Ψ(s)wedge R(s,s')LongrightarrowΨ(s'), then Ψ(s_t) holds for every state reachable by finitely many admitted transitions. Induction on the number of admitted transitions.","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bc10232afca3de12","tex":"\\mathsf{safe}","aliases":"safe mathsf","display":"\\mathsf{safe}","kind":"use","excerpt":"[An unchecked change of the checker] Let the institutional state contain a validator mode v∈{strict,open} and a resource policy p∈{safe,unsafe}. Under v=strict, the current policy must remain safe, but the validator mode itself is unprotected. Thus (strict,safe)→(open,safe) is admitted. Under v=open, any policy transition is admitted, including (op","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f93687b75635f48a","tex":"\\section","aliases":"section","display":null,"kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-af23f16ae38328e6","tex":"\\mathsf{settled}","aliases":"settled mathsf","display":"\\mathsf{settled}","kind":"use","excerpt":"For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy f","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-66a61f82005247b0","tex":"\\mathsf{Standing}","aliases":"Standing mathsf","display":"\\mathsf{Standing}","kind":"use","excerpt":"A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-937d77032661535d","tex":"\\mathsf{status}","aliases":"status mathsf","display":"\\mathsf{status}","kind":"use","excerpt":"tibility, and knowledge Temporal obligations, contracts, and office succession. Permissions describe available actions; obligations constrain continuations. The debt created by an urgency exchange is not satisfied merely because the creditor remains authorized to ask for service. It requires a particular future event. A finite obligation record can","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e3a3690589fe3220","tex":"\\mathsf{strict}","aliases":"strict mathsf","display":"\\mathsf{strict}","kind":"use","excerpt":"[An unchecked change of the checker] Let the institutional state contain a validator mode v∈{strict,open} and a resource policy p∈{safe,unsafe}. Under v=strict, the current policy must remain safe, but the validator mode itself is unprotected. Thus (strict,safe)→(open,safe) is admitted. Under v=open, any policy transition is admitted, including (op","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-848c3c3d231100c2","tex":"\\subsection","aliases":"subsection","display":null,"kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-32f99fc05b64a106","tex":"\\succ","aliases":"succ","display":"\\succ","kind":"use","excerpt":"Social choice and principled disagreement. Plurality cannot be retained while silently replacing all preferences with one externally supplied reward. Even agents that agree on every fact can rank institutional alternatives differently. Let the alternatives be three policies R,U,V and let three voters have strict preferences Rsucc_1 Usucc_1 V, Usucc","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-eaa105d35de9dffc","tex":"\\succ_1","aliases":"succ_1 succ","display":"\\succ_1","kind":"use","excerpt":"Social choice and principled disagreement. Plurality cannot be retained while silently replacing all preferences with one externally supplied reward. Even agents that agree on every fact can rank institutional alternatives differently. Let the alternatives be three policies R,U,V and let three voters have strict preferences Rsucc_1 Usucc_1 V, Usucc","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b77d42fca9fc6f71","tex":"\\succ_2","aliases":"succ_2 succ","display":"\\succ_2","kind":"use","excerpt":"Social choice and principled disagreement. Plurality cannot be retained while silently replacing all preferences with one externally supplied reward. Even agents that agree on every fact can rank institutional alternatives differently. Let the alternatives be three policies R,U,V and let three voters have strict preferences Rsucc_1 Usucc_1 V, Usucc","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-14a376fad88dda63","tex":"\\succ_3","aliases":"succ_3 succ","display":"\\succ_3","kind":"use","excerpt":"Social choice and principled disagreement. Plurality cannot be retained while silently replacing all preferences with one externally supplied reward. Even agents that agree on every fact can rank institutional alternatives differently. Let the alternatives be three policies R,U,V and let three voters have strict preferences Rsucc_1 Usucc_1 V, Usucc","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-04eac83a730993ac","tex":"\\sup_{e\\in\\mathcal E}","aliases":"sup_e∈ E sup in mathcal","display":"\\sup_{e\\in\\mathcal E}","kind":"use","excerpt":"For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared fam","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b869d08795b0a514","tex":"\\sup_{i,x}","aliases":"sup_i,x sup","display":"\\sup_{i,x}","kind":"use","excerpt":"Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j),","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9d55308f3616d884","tex":"\\sup_{x,u}","aliases":"sup_x,u sup","display":"\\sup_{x,u}","kind":"use","excerpt":"Finite-horizon error accounting. For finite distributions, d_TV(μ,ν)=tfrac12∑_z|μ(z)-ν(z)|. Suppose candidate kernels satisfy sup_x,ud_TV(push K_u(x,·),overline K_u(π(x),·)) ≤ε. This uniform condition is stronger than a small average error on the trajectories seen during training.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2b95d0c23567ffe8","tex":"\\sup_z","aliases":"sup_z sup","display":"\\sup_z","kind":"use","excerpt":"For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additiona","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5b62d811cced096b","tex":"\\sup_{z,a}","aliases":"sup_z,a sup","display":"\\sup_{z,a}","kind":"use","excerpt":"For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additiona","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-58462a227dfef4aa","tex":"\\sup_{z\\in Z_j}","aliases":"sup_z∈ Z_j sup in","display":"\\sup_{z\\in Z_j}","kind":"use","excerpt":"Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5da19979b8bdde70","tex":"\\sup_{\\tau\\in\\mathcal T}","aliases":"sup_τ∈ T sup tau in mathcal","display":"\\sup_{\\tau\\in\\mathcal T}","kind":"use","excerpt":"For approximate comparison, use the pseudometric d_ T(x,x')=sup_τ∈ T d_TV( L_x(Y^+midτ), L_x'(Y^+midτ)). A finite test suite may make distinct implementations distance zero. That is indistinguishability under those tests, not universal sameness.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c18dc2736f8201fc","tex":"T_{0:m}","aliases":"T_0:m","display":"T_{0:m}","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dff2d096303f5338","tex":"T_{0:m-1}","aliases":"T_0:m-1","display":"T_{0:m-1}","kind":"use","excerpt":"· ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f94f900a994c88b6","tex":"T_{AB}","aliases":"T_AB","display":"T_{AB}","kind":"use","excerpt":"tion of finite institutional conversations] Under Definition~[source reference], every finite source conversation has an admitted prefix exactly when its stepwise translation has that admitted prefix from the translated initial state. Corresponding reached states are related by T, and corresponding event records agree. Two faithful transports compo","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"In an idealized transferable-utility setting there is a corresponding conditional observation. If the discounted net advantages b_i satisfy ∑_i b_i≥0, then there exist budget-balanced transfers t_i with ∑_i t_i=0 and b_i+t_i≥0 for all i: choose nonnegative w_i summing to ∑_i b_i and set t_i=w_i-b_i. This algebra does not supply liquidity, a transfe","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e9a7aef78fcda23e","tex":"T_{ij}","aliases":"T_ij","display":"T_{ij}","kind":"use","excerpt":"Even when a global static state exists, a federation may fail dynamically. Local rule applications could commute in one context and conflict in another. A translator T_ij:Z_i→ Z_j should preserve the operations relevant to the interaction, approximately satisfying T_ijF_i≃ F_jT_ij. A federation requires both representational compatibility and admis","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"use","excerpt":"Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b6d1c35d86f0d39d","tex":"T_k","aliases":"T_k","display":"T_k","kind":"use","excerpt":"Define T_0 · ={B∈ B:B⊆ G}, T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4e56fe1bda71e196","tex":"T_{k+1}","aliases":"T_k+1","display":"T_{k+1}","kind":"use","excerpt":"Define T_0 · ={B∈ B:B⊆ G}, T_k+1 · =T_k∪{B∈ B:B⊆ S_rm safe, ∃ u∈ A_rm end(B), Succ(B,u)⊆ T_k}. Let T_* be the stable set and r_rm end(B) the first entry index. Then define L_0=T_*, L_k+1={B∈ T_*:∃ u∈ A_rm run(B), Succ(B,u)⊆ L_k}, with stable set L_*. These are finite constructions with the same potentially exponential possibility space as before.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d1d458adac479328","tex":"T_{m-1}","aliases":"T_m-1","display":"T_{m-1}","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"Two rules and an inherited obligation. Under rule R, each office receives one slot. Rule U permits an urgency exchange: when no previous service debt is outstanding, one office j with an accepted urgency certificate may receive two slots if another office k≠ j explicitly volunteers its slot. The other two offices each receive one. The exchange crea","count":32},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-300f218594b1347a","tex":"\\U","aliases":"U mathcal","display":"{\\mathcal U}","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6ae21ddbb7e958a2","tex":"U_t","aliases":"U_t","display":"U_t","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8c32968924d276d3","tex":"\\mathsf{unsafe}","aliases":"unsafe mathsf","display":"\\mathsf{unsafe}","kind":"use","excerpt":"[An unchecked change of the checker] Let the institutional state contain a validator mode v∈{strict,open} and a resource policy p∈{safe,unsafe}. Under v=strict, the current policy must remain safe, but the validator mode itself is unprotected. Thus (strict,safe)→(open,safe) is admitted. Under v=open, any policy transition is admitted, including (op","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"Define V_0=S and recursively, for h≥0, V_h+1={s∈ S:∃ a∈ A(s) such that suppK(·mid s,a)⊆ V_h}. [Finite-horizon almost-sure institutional viability] A state s belongs to V_h if and only if there exists an admissible policy under which every state visited over the next h transitions, including the present state, lies in S with probability one. Policie","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the p","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"-monitor model, Γ_α=min_λ∈Δ_m (FV_α(P_λ,P_C)-g_λ). Both extrema are attained. Consequently, one monitor weakly deters all listed deviations if and only if FV_α(P_λ,P_C)≥ g_λ for every λ∈Δ_m. Without an additional false-positive restriction, α=1 and V_1(P,P_C)=d_TV(P,P_C). For any fixed s, the minimum of the numbers D_j(s) is the minimum of ∑_jλ_jD_","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"[Weight-conserving refinement of controlled votes] Fix a weighted threshold rule with threshold q. Replace a voting position v of weight w(v) by positions v_1,…,v_k controlled by the same decision center, with nonnegative weights satisfying ∑_j w(v_j)=w(v). Leave all other weights and the threshold unchanged. Then every coalition of decision center","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0b837e89a1e8104e","tex":"V_h","aliases":"V_h","display":"V_h","kind":"use","excerpt":"Define V_0=S and recursively, for h≥0, V_h+1={s∈ S:∃ a∈ A(s) such that suppK(·mid s,a)⊆ V_h}. [Finite-horizon almost-sure institutional viability] A state s belongs to V_h if and only if there exists an admissible policy under which every state visited over the next h transitions, including the present state, lies in S with probability one. Policie","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-11ef40505ae9bca6","tex":"V_{h+1}","aliases":"V_h+1","display":"V_{h+1}","kind":"use","excerpt":"Define V_0=S and recursively, for h≥0, V_h+1={s∈ S:∃ a∈ A(s) such that suppK(·mid s,a)⊆ V_h}. [Finite-horizon almost-sure institutional viability] A state s belongs to V_h if and only if there exists an admissible policy under which every state visited over the next h transitions, including the present state, lies in S with probability one. Policie","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"nnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnor","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"[Weight-conserving refinement of controlled votes] Fix a weighted threshold rule with threshold q. Replace a voting position v of weight w(v) by positions v_1,…,v_k controlled by the same decision center, with nonnegative weights satisfying ∑_j w(v_j)=w(v). Leave all other weights and the threshold unchanged. Then every coalition of decision center","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0a435f82d4918621","tex":"v_k","aliases":"v_k","display":"v_k","kind":"use","excerpt":"[Weight-conserving refinement of controlled votes] Fix a weighted threshold rule with threshold q. Replace a voting position v of weight w(v) by positions v_1,…,v_k controlled by the same decision center, with nonnegative weights satisfying ∑_j w(v_j)=w(v). Leave all other weights and the threshold unchanged. Then every coalition of decision center","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8bd64ec0c9cf0366","tex":"v_{k-1}","aliases":"v_k-1","display":"v_{k-1}","kind":"use","excerpt":"Conventions and exact synchronization. Let Γ=(V,E) be a finite connected graph. Each oriented edge i→ j carries g_ij∈ G, where G is a group and g_ji=g_ij^-1. The element g_ij transports coordinates from i to j by a left action. For a path γ=(v_0,…,v_k), define H_γ=g_v_k-1v_k⋯ g_v_0v_1. The rightmost map acts first. If v_k=v_0, this product is the p","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b1555515dbd55fec","tex":"V_T","aliases":"V_T","display":"V_T","kind":"use","excerpt":"[Institutional amortization] Under stationary costs and infinite-horizon discounting, the institution's expected discounted cost advantage over the baseline is V=-B+fracg1-δ. For a finite horizon T, it is V_T=-B+g(1-δ^T)/(1-δ). The institution is cost-improving under these assumptions exactly when the corresponding value is positive. Subtract insti","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-92becdaca2dd0d19","tex":"V_\\alpha","aliases":"V_α alpha","display":"V_\\alpha","kind":"use","excerpt":"For a probability vector λ∈Δ_m, set P_λ=∑_jλ_jP_j and g_λ=∑_jλ_jg_j. Write V_α(P,P_C)=max_s∈ M_α⟨ P-P_C,s⟩. This is the same constrained observation advantage used in the single-deviation analysis, now evaluated on mixtures of deviation laws.","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3696096793ac9d06","tex":"\\operatorname{Var}","aliases":"Var operatorname","display":"\\operatorname{Var}","kind":"use","excerpt":"Suppose errors E_1,…,E_n in scalar estimates are centered, have common variance σ^2>0, and have common pairwise correlation ρ∈[0,1]. Then Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equa","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a675678fa6689935","tex":"\\vee","aliases":"vee","display":"\\vee","kind":"use","excerpt":"Convergent records and valid decisions. Let (L,vee) be a join-semilattice, with order a≤ b when avee b=b. A replica holds ℓ_i∈ L. In the following restricted model a local update joins a declared contribution δ into the state, and a merge replaces ℓ_i by ℓ_iveeℓ_j. Contributions are fixed elements of L once created. This is a state-merge constructi","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e83c58309eb2060b","tex":"\\mathcal W","aliases":"W mathcal","display":"\\mathcal W","kind":"use","excerpt":"Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bd5ef9a5ba57a080","tex":"W_A","aliases":"W_A","display":"W_A","kind":"use","excerpt":"Consider models on W_A,W_B with the same named agents and atomic predicates. Let T:W_A→ W_B be surjective and preserve the truth of each atom in both directions. For every agent i, require w∼_i^A v · Longrightarrow T(w)∼_i^B T(v) (forth),","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f39670f030db3b1a","tex":"W_B","aliases":"W_B","display":"W_B","kind":"use","excerpt":"Consider models on W_A,W_B with the same named agents and atomic predicates. Let T:W_A→ W_B be surjective and preserve the truth of each atom in both directions. For every agent i, require w∼_i^A v · Longrightarrow T(w)∼_i^B T(v) (forth),","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-179eb697363e8b1a","tex":"w_e","aliases":"w_e","display":"w_e","kind":"use","excerpt":"[Cycle residual and minimum authorized repair] The XOR of edge labels around a cycle is unchanged by every local convention change above. On a single simple cycle with odd XOR, suppose an editable subset of edges can be flipped at nonnegative costs w_e, with other edges protected. An authorized repair exists exactly when the editable subset is none","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-18864b29b043b6d8","tex":"W_i","aliases":"W_i","display":"W_i","kind":"use","excerpt":"Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentatio","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentatio","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentatio","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8d7f1cc8c150d4e5","tex":"W_p","aliases":"W_p","display":"W_p","kind":"use","excerpt":"An ideal public announcement is a strong update assumption. For a propositional fact p whose valuation is unchanged by the announcement, an ideal truthful public update restricts W to W_p={w:wmodels p} and restricts every ∼_i to that set. This update models that the event, its interpretation, and its receipt are public to the group. In the restrict","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a3535b369082a4f1","tex":"W_{\\rm glue}","aliases":"W_rm glue rm","display":"W_{\\rm glue}","kind":"use","excerpt":"Structural gluing is a different question. Suppose each subsystem i has local state space W_i and each overlap has restriction maps r_ij^i and r_ij^j. The compatible families form W_rm glue={(w_i):r_ij^i(w_i)=r_ij^j(w_j) on every declared overlap}. For an edge-complete cover of a fixed diagram, compatible local sections can give another presentatio","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2b49ff09b03e5e91","tex":"w_t","aliases":"w_t","display":"w_t","kind":"use","excerpt":"hcal U for admitted members, V_t⊆ M_t for voting positions, and D for a declared set of effective decision centers. A map c_t:V_t→ D records which decision center controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be r","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-29b34095556c1a12","tex":"\\mathsf{wait}","aliases":"wait mathsf","display":"\\mathsf{wait}","kind":"use","excerpt":"[A safe amendment can destroy every safe continuation] There are four publicly distinguishable states r,ℓ,g,b. The safe set is {r,ℓ,g}. At r, profile deliver moves to g, while profile freeze moves to ℓ. At g, wait loops safely. At ℓ, the only admissible profile advances an outstanding deadline and moves to breach state b. The amendment freeze prese","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"[Constitutional inductive safety] If Ψ(s_0) and every admitted transition satisfies Ψ(s)wedge R(s,s')LongrightarrowΨ(s'), then Ψ(s_t) holds for every state reachable by finitely many admitted transitions. Induction on the number of admitted transitions.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5450287e32dd956c","tex":"\\widehat K","aliases":"widehat K widehat","display":"\\widehat K","kind":"use","excerpt":"Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-567192244cfc2ddc","tex":"\\widehat K_u","aliases":"widehat K_u widehat","display":"\\widehat K_u","kind":"use","excerpt":"Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"use","excerpt":"Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different f","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-867fe7c70266a5ce","tex":"\\widehat P_C","aliases":"widehat P_C widehat","display":"\\widehat P_C","kind":"use","excerpt":"Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different f","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-22cdc1586433efd2","tex":"\\widehat P_j","aliases":"widehat P_j widehat","display":"\\widehat P_j","kind":"use","excerpt":"Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different f","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fec34e4c8a1d397d","tex":"\\widehat\\theta","aliases":"widehatθ widehat theta","display":"\\widehat\\theta","kind":"use","excerpt":"An evidence-preserving comparison. Now change only the information contract. Suppose authentic individual signals are available to a decision rule that selects their majority. It need not infer signals from prior actions. Let widehatθ_n be any majority choice with an arbitrary fixed rule for ties.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-da21fd16a51a8ec1","tex":"\\widehat\\theta_n","aliases":"widehatθ_n widehat theta","display":"\\widehat\\theta_n","kind":"use","excerpt":"An evidence-preserving comparison. Now change only the information contract. Suppose authentic individual signals are available to a decision rule that selects their majority. It need not infer signals from prior actions. Let widehatθ_n be any majority choice with an arbitrary fixed rule for ties.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-089f71382a559dc8","tex":"\\widetilde g","aliases":"widetilde g widetilde","display":"\\widetilde g","kind":"use","excerpt":"_i^-1). [Loop defect bounds unavoidable edge error] For a closed walk γ with k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some traversed edge has error at least d(H_γ,e_G)/k for every candidate frame family. Gauge-transform each edge to widetilde g_ij=h_j^-1g_ijh_i. Bi-invariance gives d(widetilde g_ij,e_G)=δ_ij. The transf","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b79ff1013bc386ca","tex":"\\widetilde g_{ij}","aliases":"widetilde g_ij widetilde","display":"\\widetilde g_{ij}","kind":"use","excerpt":"_i^-1). [Loop defect bounds unavoidable edge error] For a closed walk γ with k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some traversed edge has error at least d(H_γ,e_G)/k for every candidate frame family. Gauge-transform each edge to widetilde g_ij=h_j^-1g_ijh_i. Bi-invariance gives d(widetilde g_ij,e_G)=δ_ij. The transf","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":9},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"n explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-762eb59ebd1883d4","tex":"x_A","aliases":"x_A","display":"x_A","kind":"use","excerpt":"[Nontrivial frame defect with a compatible state] On a triangle, let one oriented edge carry (12)∈ S_3 and the other two carry identity. Holonomy is nontrivial, so global frames do not exist. Nevertheless, the constant state x_A=x_B=x_C=3 is compatible because (12) fixes 3. Failure to align all role meanings is stronger than failure to agree on one","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-53277cef66815b36","tex":"x_B","aliases":"x_B","display":"x_B","kind":"use","excerpt":"[Nontrivial frame defect with a compatible state] On a triangle, let one oriented edge carry (12)∈ S_3 and the other two carry identity. Holonomy is nontrivial, so global frames do not exist. Nevertheless, the constant state x_A=x_B=x_C=3 is compatible because (12) fixes 3. Failure to align all role meanings is stronger than failure to agree on one","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1de5ff44fea6e14e","tex":"x_C","aliases":"x_C","display":"x_C","kind":"use","excerpt":"[Nontrivial frame defect with a compatible state] On a triangle, let one oriented edge carry (12)∈ S_3 and the other two carry identity. Holonomy is nontrivial, so global frames do not exist. Nevertheless, the constant state x_A=x_B=x_C=3 is compatible because (12) fixes 3. Failure to align all role meanings is stronger than failure to agree on one","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a rea","count":9},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e7a4c23a3df19ee0","tex":"x_i'","aliases":"x_i'","display":"x_i'","kind":"use","excerpt":"The binary jurisdiction model admits a sharper statement. A local convention change x_i'=x_i⊕ c_i transforms its edge labels to b_ij'=b_ij⊕ c_i⊕ c_j.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"Consider an undirected graph G=(V,E) whose vertices are institutional positions. Each position carries a binary coordinate x_i∈ F_2 and each edge {i,j} imposes the relative constraint x_i⊕ x_j=b_ij, b_ij∈ F_2. The coordinates might represent a toy authorization bit or a binary convention, but the mathematical model itself makes no claim about a rea","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"[Compatible states and holonomy fixed points] A compatible state family exists if and only if there is x_r∈ D fixed by every holonomy based at the root r. Each such x_r uniquely determines a compatible family by transport along a spanning tree. For a free group action, existence of a compatible family implies identity holonomy; without freeness tha","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1029cb8ad51ed272","tex":"x_t","aliases":"x_t","display":"x_t","kind":"use","excerpt":"A transparent comparison model can make one class of failure mathematically tractable. Let x_t∈ R_≥0^n record nonnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vecto","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1e4fb4e21dd5aa54","tex":"X_{t+1}","aliases":"X_t+1","display":"X_{t+1}","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-813924c29494401d","tex":"x_{t+1}","aliases":"x_t+1","display":"x_{t+1}","kind":"use","excerpt":"A transparent comparison model can make one class of failure mathematically tractable. Let x_t∈ R_≥0^n record nonnegative deviations at n institutional positions. Suppose an explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vecto","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-29ee7dd9b2baba56","tex":"\\Y","aliases":"Y mathcal","display":"{\\mathcal Y}","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"use","excerpt":"[.5em] What state must persist? · The allocation rule alone is insufficient. Governance preservation also retains the enabled amendment actions and labels for active credentials, service debt, and resource tokens. [.5em] Can a due debt be erased? · The obligation monitor distinguishes fulfillment, authorized discharge, and deletion. Amendments are ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c0b1df2ea073f12b","tex":"Y_{0:T}","aliases":"Y_0:T","display":"Y_{0:T}","kind":"use","excerpt":"For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared fam","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Every organizational state in this paper remains a function of a complete computational realization. If Z_t=π(X_t), then Z_t is not information outside X_t. The purpose of a higher-level description is to preserve relevant operations with less descriptive complexity. An institutional description must additionally preserve who may act, which commitm","count":30},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c656dc90ffbe53cd","tex":"\\mathcal Z","aliases":"Z mathcal","display":"\\mathcal Z","kind":"use","excerpt":"Compression can erase enforceable distinctions. Let an observation channel B replace o∈ O by a record z∈ Z. It may be deterministic compression or a stochastic reporting channel, but must be the same channel under compliance and deviation.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e8394415f576ddd6","tex":"Z_0","aliases":"Z_0","display":"Z_0","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"[Guarded certificate use and authorized handover] Assume that the selector, at every reachable decision epoch, either uses a local policy with a sound positive guard or performs a handover satisfying the preceding conditions. Starting from z_0∈ C_v_0, every decision state belongs to its current certified domain and every intermediate handover state","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d4697ab9515717cf","tex":"Z_{0:t}","aliases":"Z_0:t","display":"Z_{0:t}","kind":"use","excerpt":"eedspace8baselineskip [Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b23ccb12e2adf65f","tex":"Z_1","aliases":"Z_1","display":"Z_1","kind":"use","excerpt":"[Composition of typed institutional residuals] Let T:Z_0→ Z_1 and U:Z_1→ Z_2 have finite residuals under the same semantic coordinates. Suppose U is L-Lipschitz. Then ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to tran","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-64a9a143ebce6852","tex":"Z_2","aliases":"Z_2","display":"Z_2","kind":"use","excerpt":"[Composition of typed institutional residuals] Let T:Z_0→ Z_1 and U:Z_1→ Z_2 have finite residuals under the same semantic coordinates. Suppose U is L-Lipschitz. Then ε_rm dyn(UT) ≤ Lε_rm dyn(T)+ε_rm dyn(U), and, separately for every c∈{P,O,R,E}, ε_c(UT)≤ε_c(T)+ε_c(U). Consequently translations with zero residual in every coordinate compose to tran","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bc1f10031646b5b4","tex":"Z_i","aliases":"Z_i","display":"Z_i","kind":"use","excerpt":"Even when a global static state exists, a federation may fail dynamically. Local rule applications could commute in one context and conflict in another. A translator T_ij:Z_i→ Z_j should preserve the operations relevant to the interaction, approximately satisfying T_ijF_i≃ F_jT_ij. A federation requires both representational compatibility and admis","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-10845de283abdc8a","tex":"Z_j","aliases":"Z_j","display":"Z_j","kind":"use","excerpt":"Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε","count":8},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3333cd3c2a193464","tex":"Z_{j+1}","aliases":"Z_j+1","display":"Z_{j+1}","kind":"use","excerpt":"Machine populations may change their internal languages and storage formats. The right question about a translator is whether it preserves operations, not only whether its output looks fluent. Suppose generation j uses a metric state space (Z_j,d_j), update operator F_j:Z_j→ Z_j, and translator T_j:Z_j→ Z_j+1. The one-step compatibility defect is ε","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3992003344c63054","tex":"Z_{\\rm ext}","aliases":"Z_rm ext rm","display":"Z_{\\rm ext}","kind":"use","excerpt":"A useful external-effects contract records B=(Z_rm ext,O_rm ext,Auth,Effect, Standing,Remedy). Here Z_rm ext contains the external facts relevant to the guarantee, O_rm ext specifies which of them can be observed, Auth identifies the stipulated authorization for each boundary-crossing act, and Effect includes the consequences covered by the model. ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3480de748bc9723a","tex":"Z_t","aliases":"Z_t","display":"Z_t","kind":"use","excerpt":"Every organizational state in this paper remains a function of a complete computational realization. If Z_t=π(X_t), then Z_t is not information outside X_t. The purpose of a higher-level description is to preserve relevant operations with less descriptive complexity. An institutional description must additionally preserve who may act, which commitm","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-83c84ae2ae962020","tex":"Z_{t+1}","aliases":"Z_t+1","display":"Z_{t+1}","kind":"use","excerpt":"[Deterministic microstate, non-Markov macrostate] Let X_t+1=X_t+1pmod 3 with uniform initial distribution. Set π(0)=π(1)=a and π(2)=b. Stationarity gives P(Z_t+1=bmid Z_t=a)=1/2. Yet P(Z_t+1=bmid Z_t=a,Z_t-1=b)=0, whereas the same probability conditional on Z_t=a,Z_t-1=a is 1. Thus the apparent randomness comes with memory: the one-step marginal do","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e3f81397417a104e","tex":"Z_{t-1}","aliases":"Z_t-1","display":"Z_{t-1}","kind":"use","excerpt":"[Deterministic microstate, non-Markov macrostate] Let X_t+1=X_t+1pmod 3 with uniform initial distribution. Set π(0)=π(1)=a and π(2)=b. Stationarity gives P(Z_t+1=bmid Z_t=a)=1/2. Yet P(Z_t+1=bmid Z_t=a,Z_t-1=b)=0, whereas the same probability conditional on Z_t=a,Z_t-1=a is 1. Thus the apparent randomness comes with memory: the one-step marginal do","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"Faithful translation preserves admissibility and effect. Let institutions A and B have state spaces S_A,S_B, message sets M_A, M_B, partial interpreters F^A,F^B, and event records in a common semantic space E. A state translator is T:S_A→ S_B. Allow a context-dependent message translator τ:S_A× M_A→ M_B. Context dependence matters for translated of","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-673ddcfc7e4cb456","tex":"\\alpha_t","aliases":"α_t alpha","display":"\\alpha_t","kind":"use","excerpt":"eedspace8baselineskip [Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"A transparent kernel family is K_i=(1-β_i)Id+β_i 1μ_i, 0<β_i≤1, where every row of 1μ_i is μ_i. Its mth power is (1-β_i)^mId+[1-(1-β_i)^m] 1μ_i. Thus the mixing error is bounded by (1-β_i)^m. For a fair two-state μ_i and β_i=1/2, the exact worst-row error is 2^-m-1. With m=8, the linear bound over twenty reviews is 20/512=5/128. These are analytica","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6cb4d369799123ce","tex":"\\beta_i","aliases":"β_i beta","display":"\\beta_i","kind":"use","excerpt":"A transparent kernel family is K_i=(1-β_i)Id+β_i 1μ_i, 0<β_i≤1, where every row of 1μ_i is μ_i. Its mth power is (1-β_i)^mId+[1-(1-β_i)^m] 1μ_i. Thus the mixing error is bounded by (1-β_i)^m. For a fair two-state μ_i and β_i=1/2, the exact worst-row error is 2^-m-1. With m=8, the linear bound over twenty reviews is 20/512=5/128. These are analytica","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"Office continuity is distinct from occupant survival. Suppose, separately, that each original occupant is replaced at each epoch with independent probability γ. Its probability of still occupying its role after t epochs is (1-γ)^t. Only if handovers preserve the institutional state and do not change the assumed Q can the institutional survival law ","count":14},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f8dfdbf3cc0b5e24","tex":"\\Gamma_\\alpha","aliases":"Γ_α Gamma alpha","display":"\\Gamma_\\alpha","kind":"use","excerpt":"th probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s must be used for every listed deviation. Define its margin against deviation j and its best simultaneous margin by D_j(s)=F⟨ P_j-P_C,s⟩-g_j, Γ_α=max_s∈ M_αmin_1≤ j≤ ","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":18},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7b1d94d61b4dc82c","tex":"\\delta^T","aliases":"δ^T delta","display":"\\delta^T","kind":"use","excerpt":"[Institutional amortization] Under stationary costs and infinite-horizon discounting, the institution's expected discounted cost advantage over the baseline is V=-B+fracg1-δ. For a finite horizon T, it is V_T=-B+g(1-δ^T)/(1-δ). The institution is cost-improving under these assumptions exactly when the corresponding value is positive. Subtract insti","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-14f278050573d8ad","tex":"\\Delta_{\\mathcal A}","aliases":"Δ_ A Delta mathcal","display":"\\Delta_{\\mathcal A}","kind":"use","excerpt":"A more operational measure uses decision loss. Given permitted information O and loss ℓ, define the Bayes risk R_ℓ(O)=inf_f E[ℓ(f(O),Y)]. For a nonempty allowed family A of proper coalitions, the gap Δ_ A=min_S∈ A R_ℓ(O_S) - R_ℓ(O_V) expresses an advantage over every coalition in that family. A strictly positive gap is relative to the target distri","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9adbb02c2ad54bb0","tex":"\\delta_1","aliases":"δ_1 delta","display":"\\delta_1","kind":"use","excerpt":"[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-1d6bb66bd4727edf","tex":"\\delta_c","aliases":"δ_c delta","display":"\\delta_c","kind":"use","excerpt":"Common semantic coordinates. For each component type c∈{P,O,R,E}, let g_j^c:Z_j⟶(E_c,δ_c) be a contract observable in a common metric codomain. P records permissions over the common action alphabet; O records obligation-monitor state; R records resources in common physical or accounting units; and E records the required evidentiary and provenance s","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-572519ec1d6f4294","tex":"\\delta_{F_u(x)}","aliases":"δ_F_u(x) delta","display":"\\delta_{F_u(x)}","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"A quantitative lower bound on interface error. Suppose G has a bi-invariant metric d, so both left and right multiplication preserve distance. For candidate frames h_i, write δ_ij=d(g_ij,h_jh_i^-1). [Loop defect bounds unavoidable edge error] For a closed walk γ with k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some travers","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9ddbf7c75026bec5","tex":"\\Delta_{\\mathrm{inst}}^{\\mathcal E,T}","aliases":"Δ_inst^ E,T Delta mathrm mathcal","display":"\\Delta_{\\mathrm{inst}}^{\\mathcal E,T}","kind":"use","excerpt":"For a family E of permitted environment setups, a horizon T, and feasible interventions λ(I),λ(I') that differ institutionally while matching the intended other conditions, define the institutional effect profile Δ_inst^ E,T(I,I') =sup_e∈ Ed_TV( L_e(Y_0:Tmidλ(I)), L_e(Y_0:Tmidλ(I'))). A zero value shows observational equivalence on the declared fam","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dd0aa14ad6313087","tex":"\\Delta_m","aliases":"Δ_m Delta","display":"\\Delta_m","kind":"use","excerpt":"For a probability vector λ∈Δ_m, set P_λ=∑_jλ_jP_j and g_λ=∑_jλ_jg_j. Write V_α(P,P_C)=max_s∈ M_α⟨ P-P_C,s⟩. This is the same constrained observation advantage used in the single-deviation analysis, now evaluated on mixtures of deviation laws.","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-417868d41866988d","tex":"\\delta_N","aliases":"δ_N delta","display":"\\delta_N","kind":"use","excerpt":"[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-21126255abeaa6ee","tex":"\\Delta_{\\rm coal}","aliases":"Δ_rm coal Delta rm","display":"\\Delta_{\\rm coal}","kind":"use","excerpt":"A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may b","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different f","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a05a39430affb6df","tex":"\\varepsilon_0","aliases":"ε_0 varepsilon","display":"\\varepsilon_0","kind":"use","excerpt":"· ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-eb1a042b3cb1067c","tex":"\\varepsilon_1","aliases":"ε_1 varepsilon","display":"\\varepsilon_1","kind":"use","excerpt":"Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5b28ece65b4eb364","tex":"\\varepsilon_2","aliases":"ε_2 varepsilon","display":"\\varepsilon_2","kind":"use","excerpt":"Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-bae7842f2d63f7f5","tex":"\\varepsilon_a","aliases":"ε_a varepsilon","display":"\\varepsilon_a","kind":"use","excerpt":"Transporting an enforcement certificate through a reduced model. Suppose an evidence model used at a review has true laws P_C,P_D and reduced laws overline P_C,overline P_D with certified bounds d_TV(P_a,overline P_a)≤ε_a, a∈{C,D}. Such bounds follow from the block argument when each conduct-dependent fast kernel satisfies its own uniform mixing bo","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c0dff5e4591b6949","tex":"\\epsilon_C","aliases":"ϵ_C epsilon","display":"\\epsilon_C","kind":"use","excerpt":"Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different f","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5fafc40bd0085548","tex":"\\varepsilon_C","aliases":"ε_C varepsilon","display":"\\varepsilon_C","kind":"use","excerpt":"[A robust margin for reduced-model enforcement] Let s^⋆(o) sanction precisely where overline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no ob","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-684affc58e5950cc","tex":"\\varepsilon_c","aliases":"ε_c varepsilon","display":"\\varepsilon_c","kind":"use","excerpt":"For a translation T:Z_i→ Z_j, define ε_rm dyn(T) · =sup_z,ad_j(TF_i^a(z),F_j^aT(z)), ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additiona","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7879135e402b1ed2","tex":"\\varepsilon_D","aliases":"ε_D varepsilon","display":"\\varepsilon_D","kind":"use","excerpt":"[A robust margin for reduced-model enforcement] Let s^⋆(o) sanction precisely where overline P_D(o)>overline P_C(o) and put overline d=d_TV(overline P_D,overline P_C). In the model of Section~[source reference], this fixed policy makes compliance weakly preferable whenever g≤ F(overline d-ε_C-ε_D). Conversely, if g>Fmin{1,overline d+ε_C+ε_D}, no ob","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3a1bd9c12f8b633a","tex":"\\varepsilon_E","aliases":"ε_E varepsilon","display":"\\varepsilon_E","kind":"use","excerpt":"ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-a14473f51c7df774","tex":"\\epsilon_j","aliases":"ϵ_j epsilon","display":"\\epsilon_j","kind":"use","excerpt":"Uncertainty must be absorbed by the same monitor. Averaging, translation, and statistical approximation can change the estimated observation laws. The preceding optimization does not remove those errors. Let widehat P_C,widehat P_j be specified approximate laws with d_TV(P_C,widehat P_C)≤ϵ_C, d_TV(P_j,widehat P_j)≤ϵ_j. The bounds may be different f","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b3de5d9511fcd1b4","tex":"\\varepsilon_j","aliases":"ε_j varepsilon","display":"\\varepsilon_j","kind":"use","excerpt":"[Compositional one-step closure] The composite abstraction ψπ satisfies d_TV((ψπ)_#K_u(x,·), widehat K_u(ψπ x,·)) ≤ε_1+ε_2. For L levels, the one-step defect is at most ∑_j=1^Lε_j. In particular, exact abstractions compose exactly. Insert ψ_#overline K_u(π x,·), use the triangle inequality, and use nonexpansiveness of total variation under determin","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-febfa4f397b3571b","tex":"\\varepsilon_{m-1}","aliases":"ε_m-1 varepsilon","display":"\\varepsilon_{m-1}","kind":"use","excerpt":"ith empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-1(z) between the two terms at generation m. Lipschitz continuity of T_m-1 bounds the first difference by L_m-1 times the prior defect, while~[source reference] bounds the second by ε_m-1. The recurrence D_m≤ L","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d5d46059ce01e388","tex":"\\varepsilon_O","aliases":"ε_O varepsilon","display":"\\varepsilon_O","kind":"use","excerpt":"ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-038c0ada2296bcc6","tex":"\\varepsilon_P","aliases":"ε_P varepsilon","display":"\\varepsilon_P","kind":"use","excerpt":"ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-369a731ef7da1eab","tex":"\\varepsilon_R","aliases":"ε_R varepsilon","display":"\\varepsilon_R","kind":"use","excerpt":"ε_c(T) · =sup_zδ_c(g_j^cT(z),g_i^c(z)) (c=P,O,R,E). The institutional residual profile is D(T)= (ε_rm dyn,ε_P,ε_O, ε_R,ε_E). Its entries are not added together. They have different meanings and may have different units. A one-sided authorization check can additionally distinguish newly granted forbidden actions from lost legitimate permissions. The","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-287aae19eedd3ef2","tex":"\\varepsilon_{\\rm dyn}","aliases":"ε_rm dyn varepsilon rm","display":"\\varepsilon_{\\rm dyn}","kind":"use","excerpt":"A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may b","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-09d350165c7120ad","tex":"\\varepsilon_{\\rm int}","aliases":"ε_rm int varepsilon rm","display":"\\varepsilon_{\\rm int}","kind":"use","excerpt":"A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may b","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dd54bd5850296191","tex":"\\varepsilon_{\\rm pred}","aliases":"ε_rm pred varepsilon rm","display":"\\varepsilon_{\\rm pred}","kind":"use","excerpt":"A relational certificate is a profile. A useful reporting profile is C_ T=(ε_rm pred,ε_rm dyn,ε_rm int, Δ_rm coal,R_rm repair,D_rm lineage,G_rm formation). Its entries report predictive error, dynamic closure error, intervention mismatch, coalition performance gaps, repair behavior, lineage evidence, and evidence about formation. Some entries may b","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-76ab81e6b93d8a15","tex":"\\varepsilon_t","aliases":"ε_t varepsilon","display":"\\varepsilon_t","kind":"use","excerpt":"Writing λ_t=e^ε_t gives the additive parameter ∑_tε_t. The proposition requires a fixed finite number of releases and uniform conditional bounds. A data-selected stopping rule must itself be represented in the transcript or controlled by an appropriate conditional argument. Privileged side information not covered by the declared record is not prote","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0d7cc092bd519f22","tex":"\\Theta_t","aliases":"Θ_t Theta","display":"\\Theta_t","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7e9973998ea1e838","tex":"\\kappa^t","aliases":"κ^t kappa","display":"\\kappa^t","kind":"use","excerpt":"ition[Approximate closure over time] Under~[source reference], let the actual and reduced processes use the same predeclared open-loop control sequence. If their initial macro-laws are within e_0 in total variation, then at time t they are within e_t≤min{1,e_0+tε}. If each overline K_u contracts total variation by at most κ<1, then e_t≤κ^t e_0+εfra","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4b263b6f4c51eac0","tex":"\\kappa^{t-1-j}","aliases":"κ^t-1-j kappa","display":"\\kappa^{t-1-j}","kind":"use","excerpt":"n explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dc402836c2cb145c","tex":"\\lambda^z","aliases":"λ^z lambda","display":"\\lambda^z","kind":"use","excerpt":"Set λ=p/(1-p)>1. Before a cascade starts, define an integer public score z so that the likelihood ratio of the observed action history is λ^z. A new private signal changes its owner's posterior exponent by +1 or -1. At z∈{-1,0,1} the stated tie rule makes the action reveal that signal. Once z reaches 2, even a contrary signal leaves a strictly posi","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"use","excerpt":"For a probability vector λ∈Δ_m, set P_λ=∑_jλ_jP_j and g_λ=∑_jλ_jg_j. Write V_α(P,P_C)=max_s∈ M_α⟨ P-P_C,s⟩. This is the same constrained observation advantage used in the single-deviation analysis, now evaluated on mixtures of deviation laws.","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-29b14a34343e28e4","tex":"\\lambda_t","aliases":"λ_t lambda","display":"\\lambda_t","kind":"use","excerpt":"[Conditional disclosure accounting] Consider m releases on finite alphabets. At each common prefix h, suppose their conditional next-release laws P_C^t(·mid h) and P_D^t(·mid h) satisfy both pointwise inequalities with factor λ_t≥1, uniformly over prefixes that can occur. Then the joint transcript laws satisfy~[source reference] with λ=∏_t=1^mλ_t. ","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-64833ef6070cbe84","tex":"\\lambda_t^{-1}","aliases":"λ_t^-1 lambda","display":"\\lambda_t^{-1}","kind":"use","excerpt":"omputed only from that transcript obeys Theorem~[source reference] with this product factor. The conditional inequalities imply common support at each stage. Induction therefore gives common support for prefixes under the two cases. For a positive-probability full transcript, the ratio of its probabilities is the product of conditional ratios, each","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3b453de64aef4dc5","tex":"\\mu_0","aliases":"μ_0 mu","display":"\\mu_0","kind":"use","excerpt":"rnel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law pushμ_0. For outputs to agree as we","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-252454ad397ec5a8","tex":"\\mu_i","aliases":"μ_i mu","display":"\\mu_i","kind":"use","excerpt":"Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j),","count":8},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"Finite-horizon error accounting. For finite distributions, d_TV(μ,ν)=tfrac12∑_z|μ(z)-ν(z)|. Suppose candidate kernels satisfy sup_x,ud_TV(push K_u(x,·),overline K_u(π(x),·)) ≤ε. This uniform condition is stronger than a small average error on the trajectories seen during training.","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"[Survival and expected institutional lifetime] For every integer t≥0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random vari","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"Suppose errors E_1,…,E_n in scalar estimates are centered, have common variance σ^2>0, and have common pairwise correlation ρ∈[0,1]. Then Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equa","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"Suppose errors E_1,…,E_n in scalar estimates are centered, have common variance σ^2>0, and have common pairwise correlation ρ∈[0,1]. Then Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equa","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-da6215232ea750c2","tex":"\\tau_{AB}","aliases":"τ_AB tau","display":"\\tau_{AB}","kind":"use","excerpt":"tion of finite institutional conversations] Under Definition~[source reference], every finite source conversation has an admitted prefix exactly when its stepwise translation has that admitted prefix from the translated initial state. Corresponding reached states are related by T, and corresponding event records agree. Two faithful transports compo","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c14027706b566ee4","tex":"\\tau_{AC}","aliases":"τ_AC tau","display":"\\tau_{AC}","kind":"use","excerpt":"tion of finite institutional conversations] Under Definition~[source reference], every finite source conversation has an admitted prefix exactly when its stepwise translation has that admitted prefix from the translated initial state. Corresponding reached states are related by T, and corresponding event records agree. Two faithful transports compo","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-3c628f32996e414d","tex":"\\tau_{BC}","aliases":"τ_BC tau","display":"\\tau_{BC}","kind":"use","excerpt":"tion of finite institutional conversations] Under Definition~[source reference], every finite source conversation has an admitted prefix exactly when its stepwise translation has that admitted prefix from the translated initial state. Corresponding reached states are related by T, and corresponding event records agree. Two faithful transports compo","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"A candidate set of institutional rules I can be made part of a finite strategic environment. Participants have local choices r_i∈ R_i, with overall configuration r=(r_1,…,r_n) and expected utilities u_i(r). A finite game is an exact potential game if there exists Φ(r) such that for every unilateral change of i, u_i(r_i',r_-i)-u_i(r_i,r_-i) =Φ(r_i',","count":7},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"tibility, and knowledge Temporal obligations, contracts, and office succession. Permissions describe available actions; obligations constrain continuations. The debt created by an urgency exchange is not satisfied merely because the creditor remains authorized to ask for service. It requires a particular future event. A finite obligation record can","count":16},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Let s=(x,I) be the full operating state and let R(s,s') be the admitted transition relation, including ordinary actions and amendments. A safety property is a predicate Ψ(s).","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-25d93829156f568b","tex":"\\psi_{\\#}","aliases":"ψ_# psi","display":"\\psi_{\\#}","kind":"use","excerpt":"Composition of dynamic abstractions. Suppose π: X→ Z and ψ: Z→ W are surjections and, for the same control family, d_TV(push K_u(x,·),overline K_u(π x,·))≤ε_1, d_TV(ψ_#overline K_u(z,·),widehat K_u(ψ z,·))≤ε_2.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"[.35em] 1 · Three offices adopt U; A requests urgency and B consents · (2,0,1,1) · A owes B one slot at epoch 2. [.35em] 2 · A is replaced by an authorized successor; repayment takes precedence · (0,2,1,1) · The same obligation is fulfilled, not erased. [.35em] 3 · A proposed external grant would duplicate a slot token · (1,1,1,1) · The proposal is","count":16},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"Possible metrics include the discrete metric on permission vectors, a declared metric on obligation-monitor states, an ℓ^1 distance on common-unit resource vectors, and symmetric-difference cardinality on required evidence records. For evidence, record equality is structural preservation under a trust model, not a truth certificate. If the payload ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e18760f27e56e31b","tex":"\\ell^\\star","aliases":"ℓ^⋆ ell star","display":"\\ell^\\star","kind":"use","excerpt":"[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e149f481feb293a4","tex":"\\ell_0","aliases":"ℓ_0 ell","display":"\\ell_0","kind":"use","excerpt":"[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let ","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"Convergent records and valid decisions. Let (L,vee) be a join-semilattice, with order a≤ b when avee b=b. A replica holds ℓ_i∈ L. In the following restricted model a local update joins a declared contribution δ into the state, and a merge replaces ℓ_i by ℓ_iveeℓ_j. Contributions are fixed elements of L once created. This is a state-merge constructi","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-8943b22d90e5e02b","tex":"\\ell_j","aliases":"ℓ_j ell","display":"\\ell_j","kind":"use","excerpt":"Convergent records and valid decisions. Let (L,vee) be a join-semilattice, with order a≤ b when avee b=b. A replica holds ℓ_i∈ L. In the following restricted model a local update joins a declared contribution δ into the state, and a merge replaces ℓ_i by ℓ_iveeℓ_j. Contributions are fixed elements of L once created. This is a state-merge constructi","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c18573d79c46d124","tex":"\\to^*","aliases":"→^* to","display":"\\to^*","kind":"use","excerpt":"Fix a bounded amendment episode. Let s→ s' mean one permitted amendment step, and let →^* be its reflexive transitive closure. The episode state contains any pending proposals, version counters, permissions, and consequence records required by the contract. It is terminating if it has no infinite chain of actual steps, and locally confluent if s→ a","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Exact frame synchronization] Global frames exist if and only if H_γ=e_G for every closed walk. When they exist, they are unique up to common right multiplication h_i↦ h_i k. It suffices to check the rooted fundamental loops associated with one spanning tree. If g_ij=h_jh_i^-1, adjacent factors cancel in every path product, giving H_γ=h_v_kh_v_0^-1","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"For a bounded deadline D, a trace requirement has the form G(created(o)⇒ F_≤ Dsettled(o)). Here settled means fulfilled or validly discharged. The temporal operators mean ``at every position'' and ``within at most D subsequent clock steps.'' The obligation identifier binds the later event to the same commitment. No implication of ethical adequacy f","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"Let Q be the substochastic transition matrix on V, so the missing row mass represents exit. Let μ be an initial row probability vector supported on V and define τ=inf{t≥1:s_t∉ V}.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f4630c58d4f10a12","tex":"\\prod_{k=j+1}^{m-1}","aliases":"∏_k=j+1^m-1 prod","display":"\\prod_{k=j+1}^{m-1}","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-049564f82e5a2c6c","tex":"\\prod_{t=0}^{T-1}","aliases":"∏_t=0^T-1 prod","display":"\\prod_{t=0}^{T-1}","kind":"use","excerpt":"[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-b1ce9ee9accd41e8","tex":"\\prod_{t=1}^{m}","aliases":"∏_t=1^m prod","display":"\\prod_{t=1}^{m}","kind":"use","excerpt":"tional disclosure accounting] Consider m releases on finite alphabets. At each common prefix h, suppose their conditional next-release laws P_C^t(·mid h) and P_D^t(·mid h) satisfy both pointwise inequalities with factor λ_t≥1, uniformly over prefixes that can occur. Then the joint transcript laws satisfy~[source reference] with λ=∏_t=1^mλ_t. The bo","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0d855e2065f74dbf","tex":"\\prod_{t=j+1}^{T-1}","aliases":"∏_t=j+1^T-1 prod","display":"\\prod_{t=j+1}^{T-1}","kind":"use","excerpt":"[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6d53b2a4c03aa039","tex":"\\sum_{(i,j)\\text{ traversed by }\\gamma}","aliases":"∑_(i,j) traversed by γ sum text gamma","display":"\\sum_{(i,j)\\text{ traversed by }\\gamma}","kind":"use","excerpt":"A quantitative lower bound on interface error. Suppose G has a bi-invariant metric d, so both left and right multiplication preserve distance. For candidate frames h_i, write δ_ij=d(g_ij,h_jh_i^-1). [Loop defect bounds unavoidable edge error] For a closed walk γ with k≥1 traversals, d(H_γ,e_G)≤∑_(i,j) traversed by γδ_ij. In particular, some travers","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"In an idealized transferable-utility setting there is a corresponding conditional observation. If the discounted net advantages b_i satisfy ∑_i b_i≥0, then there exist budget-balanced transfers t_i with ∑_i t_i=0 and b_i+t_i≥0 for all i: choose nonnegative w_i summing to ∑_i b_i and set t_i=w_i-b_i. This algebra does not supply liquidity, a transfe","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2f29622fb484623a","tex":"\\sum_{i=1}^n","aliases":"∑_i=1^n sum","display":"\\sum_{i=1}^n","kind":"use","excerpt":"Suppose errors E_1,…,E_n in scalar estimates are centered, have common variance σ^2>0, and have common pairwise correlation ρ∈[0,1]. Then Var(frac1n∑_i=1^n E_i) =fracσ^2n(1+(n-1)ρ). This elementary identity follows by expanding the variance and counting n(n-1) off-diagonal covariance terms. Relative to the variance of an average of independent equa","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-aba48accee79a124","tex":"\\sum_{i\\in\\{A,B,C,D\\}}","aliases":"∑_i∈{A,B,C,D} sum in","display":"\\sum_{i\\in\\{A,B,C,D\\}}","kind":"use","excerpt":"public record contains the current allocation rule, valid credentials, accepted urgency certificates, outstanding service obligations, and the amendment history. Slots have unique names (e,k), where e is the epoch and k∈{1,2,3,4}. A grant consumes one available slot token. Copying a message that describes a grant does not create an additional token","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"The same argument applies separately to output kernels. Combining this proposition with Proposition~[source reference] yields a finite-horizon bound of e_0+t∑_jε_j, capped at one, under the corresponding controls. This elementary estimate belongs to the broader subject of compositional abstraction error~[source reference]. It does not establish glo","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2dbaa0f874ca4402","tex":"\\sum_{j=0}^{m-1}","aliases":"∑_j=0^m-1 sum","display":"\\sum_{j=0}^{m-1}","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z)) · ≤∑_j=0^m-1ε_j∏_k=j+1^m-1L_k, with empty products equal to 1. For m=1 this is the definition of ε_0. Suppose the bound holds through generation m-1. Insert T_m-1F_m-1T_0:m-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fc292c4a072bc463","tex":"\\sum_{j=0}^{T-1}","aliases":"∑_j=0^T-1 sum","display":"\\sum_{j=0}^{T-1}","kind":"use","excerpt":"[Repair budget] If 0≤ a_t≤bar a<1 and 0≤ b_t≤bar b, then e_t≤ bar a^t e_0+bar bfrac1-bar a^t1-bar a. More generally, for nonnegative a_t,b_t satisfying the stated recurrence, e_T≤ e_0∏_t=0^T-1a_t+ ∑_j=0^T-1b_j∏_t=j+1^T-1a_t. Repeated substitution gives the product--sum formula. Uniform bounds reduce the sum to a geometric series.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-df66290cc89f818f","tex":"\\sum_{j=0}^{t-1}","aliases":"∑_j=0^t-1 sum","display":"\\sum_{j=0}^{t-1}","kind":"use","excerpt":"n explicitly justified comparison model provides x_t+1≤ Bx_t+w_t componentwise, B≥0, where w_t represents fresh disturbance and B summarizes one-step propagation. Suppose a vector v>0 and κ<1 satisfy Bv≤κ v. In the weighted maximum norm normx_v,∞=max_i x_i/v_i we obtain normx_t_v,∞≤ κ^tnormx_0_v,∞ +∑_j=0^t-1κ^t-1-jnormw_j_v,∞. The proof is the one-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-25b387c951fb9265","tex":"\\sum_{j=1}^L","aliases":"∑_j=1^L sum","display":"\\sum_{j=1}^L","kind":"use","excerpt":"[Compositional one-step closure] The composite abstraction ψπ satisfies d_TV((ψπ)_#K_u(x,·), widehat K_u(ψπ x,·)) ≤ε_1+ε_2. For L levels, the one-step defect is at most ∑_j=1^Lε_j. In particular, exact abstractions compose exactly. Insert ψ_#overline K_u(π x,·), use the triangle inequality, and use nonexpansiveness of total variation under determin","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-00a7ede556f767a5","tex":"\\sum_{j\\ne i}","aliases":"∑_j≠ i sum ne","display":"\\sum_{j\\ne i}","kind":"use","excerpt":"A formation barrier despite mutual benefit. The stationary cost comparison does not describe a route to adoption. Consider n≥2 agents choosing a_i∈{0,1}. An adopter pays cost c>0 and receives benefit b>0 from each other adopter: u_i(a)=a_i(b∑_j≠ ia_j-c). Nonadopters receive zero. This is a finite network-coordination model, not a universal utility ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d74790b215363bd7","tex":"\\sum_{o\\in O}","aliases":"∑_o∈ O sum in","display":"\\sum_{o\\in O}","kind":"use","excerpt":"empty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,s⟩≤α}. Here ⟨ P,s⟩=∑_o∈ OP(o)s(o). The same s mus","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-db5f373ed82eca46","tex":"\\sum_{r=0}^{\\infty}","aliases":"∑_r=0^∞ sum infty","display":"\\sum_{r=0}^{\\infty}","kind":"use","excerpt":"a pair locks correctly with probability p^2, locks incorrectly with probability (1-p)^2, and returns to zero with probability 2p(1-p). Conditional independence gives independent repetitions of this two-step cycle until absorption. Since 2p(1-p)<1, nonabsorption through r cycles has the stated probability and tends to zero. Summing the probability o","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-90c396c1f06e154a","tex":"\\sum_t","aliases":"∑_t sum","display":"\\sum_t","kind":"use","excerpt":"Writing λ_t=e^ε_t gives the additive parameter ∑_tε_t. The proposition requires a fixed finite number of releases and uniform conditional bounds. A data-selected stopping rule must itself be represented in the transcript or controlled by an appropriate conditional argument. Privileged side information not covered by the declared record is not prote","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-71278a52a9bdedac","tex":"\\sum_{t=a}^{b}","aliases":"∑_t=a^b sum","display":"\\sum_{t=a}^{b}","kind":"use","excerpt":"All admitted appeals can be assigned to distinct review slots in their windows if and only if, for every subset U⊆ J, |U|≤∑_t∈⋃_j∈ U{r_j,…,d_j}c_t. Equivalently, because the windows are intervals, it suffices to check every integer interval [a,b]⊆[1,T]: |{j:r_j≥ a, d_j≤ b}| ≤∑_t=a^bc_t. Replace epoch t by c_t distinct slot vertices and join appeal ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-62d0208cce9ec444","tex":"\\sum_{t\\in\\bigcup_{j\\in U}\\{r_j,\\ldots,d_j\\}}","aliases":"∑_t∈⋃_j∈ U{r_j,…,d_j} sum in bigcup in ldots","display":"\\sum_{t\\in\\bigcup_{j\\in U}\\{r_j,\\ldots,d_j\\}}","kind":"use","excerpt":"[Capacity for a finite appeal contract] All admitted appeals can be assigned to distinct review slots in their windows if and only if, for every subset U⊆ J, |U|≤∑_t∈⋃_j∈ U{r_j,…,d_j}c_t. Equivalently, because the windows are intervals, it suffices to check every integer interval [a,b]⊆[1,T]: |{j:r_j≥ a, d_j≤ b}| ≤∑_t=a^bc_t. Replace epoch t by c_t","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-aa1dbc193c31e5d5","tex":"\\sum_{t\\geq0}","aliases":"∑_t≥0 sum geq","display":"\\sum_{t\\geq0}","kind":"use","excerpt":"geq0, P_μ(τ>t)=μ Q^t1. If the spectral radius satisfies ρ(Q)<1, then E_μ[τ]=μ(Id-Q)^-11. The unnormalized row law of trajectories that remain in V through t is μ Q^t, by induction. Summing it gives the survival probability. The tail-sum formula for a positive integer-valued random variable gives Eτ=∑_t≥0 P(τ>t). Under ρ(Q)<1, the matrix geometric s","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2b925196a0a0558c","tex":"\\sum_{w:\\pi(w)=z'}","aliases":"∑_w:π(w)=z' sum pi","display":"\\sum_{w:\\pi(w)=z'}","kind":"use","excerpt":"[Controlled lumpability] A common kernel overline K_u represents the projected one-step laws from every microstate if and only if, for all z' and all x,x' with π(x)=π(x'), ∑_w:π(w)=z'K_u(x,w) =∑_w:π(w)=z'K_u(x',w). Under this condition, every macro-history policy α_t(umid Z_0:t) induces the same projected process as the macro-model with initial law","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ee373b78330c8be1","tex":"\\sum_y","aliases":"∑_y sum","display":"\\sum_y","kind":"use","excerpt":"Assume K_i has a stationary law μ_i and sup_i,xd_TV(K_i^m(x,·),μ_i)≤ε. The actual one-review law from (i,x) and its stationary average are P(i,x;j) · =∑_y K_i^m(x,y)R_i(y,j), overline R(i,j) · =∑_yμ_i(y)R_i(y,j). The endpoint is the only fast input to this particular revision rule. A rule that uses the full trajectory, accumulated costs, or a data-","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5966b9097878350e","tex":"\\sum_z","aliases":"∑_z sum","display":"\\sum_z","kind":"use","excerpt":"Finite-horizon error accounting. For finite distributions, d_TV(μ,ν)=tfrac12∑_z|μ(z)-ν(z)|. Suppose candidate kernels satisfy sup_x,ud_TV(push K_u(x,·),overline K_u(π(x),·)) ≤ε. This uniform condition is stronger than a small average error on the trajectories seen during training.","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"nter controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assigns nonnegative voting weights, q_t is an approval threshold, and e_t is a configuration epoch. Adm","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"The order of such operations can matter. Let G set a permission bit to 1 (grant) and R set it to 0 (revoke). Then G∘ R yields 1 and R∘ G yields 0 on every input. This simple noncommutativity has practical conceptual force: a log of individual events without their authorized order may fail to reconstruct the actual institutional state. Revision hist","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"[Accumulation of cultural transport defects] Assume each T_j is L_j-Lipschitz and ε_j<∞ for j=0,…,m-1. Define T_0:m=T_m-1⋯ T_0. For every z∈ Z_0, · d_m(T_0:mF_0(z),F_mT_0:m(z))","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"nter controls each voting position under the model. This is an assumption about effective control, not a fact established by distinct names, keys, or model instances. A boundary state may be represented as B_t=(M_t,V_t,c_t,w_t,q_t,e_t), where w_t assigns nonnegative voting weights, q_t is an approval threshold, and e_t is a configuration epoch. Adm","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-58db8131f02376fc","tex":"\\sim_A","aliases":"∼_A sim","display":"\\sim_A","kind":"use","excerpt":"[Everyone knows, but it is not common knowledge] Let W={0,1,2} and let φ hold exactly at 0,1. Agent A has partition {0,1}mid{2}, and agent B has partition {0}mid{1,2}. At world 0, both agents know φ. However, the chain 0∼_A1∼_B2 reaches a world where it is false. Thus E_{A,B}φ holds at 0 but C_{A,B}φ does not.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-e4bf4b163e1200f5","tex":"\\sim_B","aliases":"∼_B sim","display":"\\sim_B","kind":"use","excerpt":"[Everyone knows, but it is not common knowledge] Let W={0,1,2} and let φ hold exactly at 0,1. Agent A has partition {0,1}mid{2}, and agent B has partition {0}mid{1,2}. At world 0, both agents know φ. However, the chain 0∼_A1∼_B2 reaches a world where it is false. Thus E_{A,B}φ holds at 0 but C_{A,B}φ does not.","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-5e33633cccaa9740","tex":"\\sim_i","aliases":"∼_i sim","display":"\\sim_i","kind":"use","excerpt":"Fix one configuration epoch and a nonempty group J of named decision centers. Let W be a finite set of possible worlds. Each equivalence relation ∼_i records the worlds agent i cannot distinguish using its permitted information. A world may include the epoch, relevant history, and current records. This is an idealized semantics of information, not ","count":4},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2b21cb87657d54c8","tex":"\\sim_i^A","aliases":"∼_i^A sim","display":"\\sim_i^A","kind":"use","excerpt":"Consider models on W_A,W_B with the same named agents and atomic predicates. Let T:W_A→ W_B be surjective and preserve the truth of each atom in both directions. For every agent i, require w∼_i^A v · Longrightarrow T(w)∼_i^B T(v) (forth), T(w)∼_i^B z · Longrightarrow ∃ v: w∼_i^A v and T(v)=z (back). These are the usual bounded-morphism conditions, ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-21aa252daa4fd2b9","tex":"\\sim_i^B","aliases":"∼_i^B sim","display":"\\sim_i^B","kind":"use","excerpt":"Consider models on W_A,W_B with the same named agents and atomic predicates. Let T:W_A→ W_B be surjective and preserve the truth of each atom in both directions. For every agent i, require w∼_i^A v · Longrightarrow T(w)∼_i^B T(v) (forth), T(w)∼_i^B z · Longrightarrow ∃ v: w∼_i^A v and T(v)=z (back). These are the usual bounded-morphism conditions, ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"Even when a global static state exists, a federation may fail dynamically. Local rule applications could commute in one context and conflict in another. A translator T_ij:Z_i→ Z_j should preserve the operations relevant to the interaction, approximately satisfying T_ijF_i≃ F_jT_ij. A federation requires both representational compatibility and admis","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal ","count":2},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-117259cd973260b1","tex":"\\equiv_{\\mathcal T}","aliases":"≡_ T equiv mathcal","display":"\\equiv_{\\mathcal T}","kind":"use","excerpt":"For a family T of finite continuation experiments, define exact behavioral equivalence by x≡_ Tx' Longleftrightarrow L_x(Y^+midτ)= L_x'(Y^+midτ) for every τ∈ T. Equality of these indexed laws is reflexive, symmetric, and transitive. It therefore defines a quotient. This equivalence is relative to the chosen experiments. It does not by itself equal ","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-6ceea42b9b950da5","tex":"\\equiv_k","aliases":"≡_k equiv","display":"\\equiv_k","kind":"use","excerpt":"he smallest exact summary under a fixed contract The conditions yield a finite construction rather than only an existence criterion. Begin with the partition B_0 that identifies exactly the states sharing (ℓ(s),D(s)). At each refinement, identify s,t only when they were previously in the same block and every action in their common enabled set leads","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-f4e191c5dd746e23","tex":"\\equiv_{k+1}","aliases":"≡_k+1 equiv","display":"\\equiv_{k+1}","kind":"use","excerpt":"he smallest exact summary under a fixed contract The conditions yield a finite construction rather than only an existence criterion. Begin with the partition B_0 that identifies exactly the states sharing (ℓ(s),D(s)). At each refinement, identify s,t only when they were previously in the same block and every action in their common enabled set leads","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"Suppose a coordination protocol without institution has expected per-period cost c_0. A proposed institutional protocol has operating cost c_1, per-period maintenance m, and a one-time setup cost B. With discount factor 0≤δ<1, write g=c_0-c_1-m.","count":21},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"In an idealized transferable-utility setting there is a corresponding conditional observation. If the discounted net advantages b_i satisfy ∑_i b_i≥0, then there exist budget-balanced transfers t_i with ∑_i t_i=0 and b_i+t_i≥0 for all i: choose nonnegative w_i summing to ∑_i b_i and set t_i=w_i-b_i. This algebra does not supply liquidity, a transfe","count":13},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"Relational information: access, synergy, and usefulness. Let A,B be independent fair bits and Y=A⊕ B. Then I(Y;A)=I(Y;B)=0, I(Y;A,B)=1. This elementary example shows that joint observation may reveal information absent from each marginal observation. It does not show that the information is absent from the complete system, that a mysterious carrier","count":15},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-2defdf654978cc42","tex":"\\bigcup_{j\\in U}","aliases":"⋃_j∈ U bigcup in","display":"\\bigcup_{j\\in U}","kind":"use","excerpt":"[Capacity for a finite appeal contract] All admitted appeals can be assigned to distinct review slots in their windows if and only if, for every subset U⊆ J, |U|≤∑_t∈⋃_j∈ U{r_j,…,d_j}c_t. Equivalently, because the windows are intervals, it suffices to check every integer interval [a,b]⊆[1,T]: |{j:r_j≥ a, d_j≤ b}| ≤∑_t=a^bc_t. Replace epoch t by c_t","count":1},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"use","excerpt":"[Implementation dependence] If every P_λ is within ε of a proposed macro-intervention law P^⋆_z', then η(x,z')≤2ε. Conversely, if two admissible implementations are distance η apart, no single law can approximate both within less than η/2. Both claims follow from the triangle inequality for total variation.","count":5},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"[Quiescent convergence and invariant preservation] Assume finitely many replicas begin at ℓ_0, only finitely many contributions δ_1,…,δ_N are created, and every contribution eventually becomes included in every replica. Then each replica eventually equals ℓ^⋆=ℓ_0veeδ_1vee⋯veeδ_N, independently of merge order and duplicate deliveries. Moreover, let ","count":3},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"The common-monitor problem. Let O be a finite nonempty observation set. Compliance has law P_C, and deviation j∈{1,…,m} has law P_j and gross utility advantage g_j≥0, where m≥1. A fixed penalty F>0 is applied with probability s(o)∈[0,1] after record o. Impose a protected false-positive rate α∈[0,1] through the compact polytope M_α={s∈[0,1]^O:⟨ P_C,","count":6},{"work":"monographs-machine-societies","url":"notation/monographs-machine-societies.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"Likewise, persistence of a practice is not evidence of its truth. In an illustrative two-type copying model, let accurate and inaccurate practices reproduce at fixed rates r_T,r_F>0, with positive initial counts N_T,N_F. If r_F>r_T, the accurate type's share is fracN_T r_T^tN_T r_T^t+N_F r_F^t =frac11+(N_F/N_T)(r_F/r_T)^t⟶0. The model does not desc","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":50},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":246},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Selection interval] For A⊆ B⊆ J, define I(A,B)={T⊆ J:A⊆ T⊆ B}. The set A contains candidates already required, and B contains all candidates still permitted by the branch decisions. The unresolved indices form B∖ A.","count":51},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"definition-passage","excerpt":"[Separating block] A separating block is a pair (B,Y), where B⊆{0,…,q-1} and Y∈ℚ^1× m satisfy YD=0, Ywidetilde E_B=0, YR≠0. The set B need not itself belong to A.","count":8},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"Faithfulness on precisely the required data. [Faithfulness on a submodule] For O≤ Q_D, the map B is faithful on O when ∀ z∈ O, Bz=0Longrightarrow z=0.","count":56},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":182},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":81},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":18},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"definition-passage","excerpt":"[Selection interval] For A⊆ B⊆ J, define I(A,B)={T⊆ J:A⊆ T⊆ B}. The set A contains candidates already required, and B contains all candidates still permitted by the branch decisions. The unresolved indices form B∖ A.","count":11},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"tion[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":27},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"[Selection interval] For A⊆ B⊆ J, define I(A,B)={T⊆ J:A⊆ T⊆ B}. The set A contains candidates already required, and B contains all candidates still permitted by the branch decisions. The unresolved indices form B∖ A.","count":33},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Accepted branch tree] A tree for I(A,B) consists of one of the three terminal certificates above, or a split at j∈ B∖ A with accepted trees for both child intervals. In lower-bound mode a cost threshold is fixed throughout the tree. In infeasibility mode cost leaves are rejected.","count":30},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a4f733cd50dc9a2c","tex":"\\kk","aliases":"kk 𝕜 Bbbk","display":"{\\Bbbk}","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":18},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"definition-passage","excerpt":"Faithfulness on precisely the required data. [Faithfulness on a submodule] For O≤ Q_D, the map B is faithful on O when ∀ z∈ O, Bz=0Longrightarrow z=0.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":25},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":63},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4a37c6e148ffc898","tex":"O_D","aliases":"O_D","display":"O_D","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":8},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":33},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q ℚ mathbb","display":"{\\mathbb{Q}}","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":29},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":72},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-7ca77479d1369dc3","tex":"\\Q^{1\\times m}","aliases":"Q^1× m ℚ^1× m Q times mathbb times","display":"{\\mathbb{Q}}^{1\\times m}","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-7bb195dbf7dd5e73","tex":"Q_D","aliases":"Q_D","display":"Q_D","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":16},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-842201207aa7200d","tex":"q_D","aliases":"q_D","display":"q_D","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":57},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":48},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":85},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"definition-passage","excerpt":"tion[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":20},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":65},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span span operatorname","display":"{\\operatorname{span}}","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":6},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-53d25c3e843a0147","tex":"\\Span_{\\kk}","aliases":"Span_kk span_𝕜 Span kk operatorname Bbbk","display":"{\\operatorname{span}}_{{\\Bbbk}}","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Selection interval] For A⊆ B⊆ J, define I(A,B)={T⊆ J:A⊆ T⊆ B}. The set A contains candidates already required, and B contains all candidates still permitted by the branch decisions. The unresolved indices form B∖ A.","count":33},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":35},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":48},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":45},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":36},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":32},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-20f230bba299446b","tex":"\\widetilde E","aliases":"widetilde E widetilde","display":"\\widetilde E","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":13},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-beb35e5ac649a42e","tex":"\\widetilde E_B","aliases":"widetilde E_B widetilde","display":"\\widetilde E_B","kind":"definition-passage","excerpt":"[Separating block] A separating block is a pair (B,Y), where B⊆{0,…,q-1} and Y∈ℚ^1× m satisfy YD=0, Ywidetilde E_B=0, YR≠0. The set B need not itself belong to A.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-6c871982c1283b93","tex":"\\widetilde E_S","aliases":"widetilde E_S widetilde","display":"\\widetilde E_S","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":9},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":23},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"Faithfulness on precisely the required data. [Faithfulness on a submodule] For O≤ Q_D, the map B is faithful on O when ∀ z∈ O, Bz=0Longrightarrow z=0.","count":8},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":15},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Separating block] A separating block is a pair (B,Y), where B⊆{0,…,q-1} and Y∈ℚ^1× m satisfy YD=0, Ywidetilde E_B=0, YR≠0. The set B need not itself belong to A.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":32},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"definition-passage","excerpt":"Faithfulness on precisely the required data. [Faithfulness on a submodule] For O≤ Q_D, the map B is faithful on O when ∀ z∈ O, Bz=0Longrightarrow z=0.","count":4},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Correction and repair] A residual r∈ V is correctable by D when there is u∈ U such that Du=-r. An added correction map E:W→ V repairs r when there are u∈ U and w∈ W with Du+Ew=-r. It repairs the family when this condition holds for every r_i. The witnesses may depend on i.","count":73},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"definition-passage","excerpt":"[Selection interval] For A⊆ B⊆ J, define I(A,B)={T⊆ J:A⊆ T⊆ B}. The set A contains candidates already required, and B contains all candidates still permitted by the branch decisions. The unresolved indices form B∖ A.","count":6},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"definition-passage","excerpt":"Separating obstructions. [Separating row] A separating certificate for S is a row Y∈ℚ^1× m satisfying YD=0, Ywidetilde E_S=0, YR≠0. The last condition asks that at least one original residual is detected. No residual index needs to be selected when the family is empty.","count":7},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"The obstruction span of a family. [Required obstruction span] For a residual family r:I→ V, put O_D(r)=span_𝕜{q_D(r_i):i∈ I}≤ Q_D. We write O when the family and correction map are understood.","count":19},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Separating block] A separating block is a pair (B,Y), where B⊆{0,…,q-1} and Y∈ℚ^1× m satisfy YD=0, Ywidetilde E_B=0, YR≠0. The set B need not itself belong to A.","count":24},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":39},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-62a0c6783e936d9c","tex":"\\arrow","aliases":"arrow","display":null,"kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":22},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":40},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-56fe48ab61c25d5d","tex":"\\operatorname{check}","aliases":"check operatorname","display":"\\operatorname{check}","kind":"introduction-passage","excerpt":"[The Boolean test has the intended meaning] The equality check(S)=true holds if and only if E_S repairs the residual family. The Boolean is defined by deciding precisely the quantified equations in [source reference] for the map E_S. No intermediate rank guess or unproved feasibility oracle replaces those equations. formalrepairMenuCheck_spec","count":4},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":16},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":16},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":25},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c23f8c6cbb98f93c","tex":"E_S","aliases":"E_S","display":"E_S","kind":"introduction-passage","excerpt":"[The Boolean test has the intended meaning] The equality check(S)=true holds if and only if E_S repairs the residual family. The Boolean is defined by deciding precisely the quantified equations in [source reference] for the map E_S. No intermediate rank guess or unproved feasibility oracle replaces those equations. formalrepairMenuCheck_spec","count":12},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":13},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":27},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-ccf3bc5017cc6d9c","tex":"q_{D'}","aliases":"q_D'","display":"q_{D'}","kind":"introduction-passage","excerpt":"U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":29},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-28ec3adb69bf092d","tex":"\\mathrm{true}","aliases":"true mathrm","display":"\\mathrm{true}","kind":"introduction-passage","excerpt":"[The Boolean test has the intended meaning] The equality check(S)=true holds if and only if E_S repairs the residual family. The Boolean is defined by deciding precisely the quantified equations in [source reference] for the map E_S. No intermediate rank guess or unproved feasibility oracle replaces those equations. formalrepairMenuCheck_spec","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-ec61c8382b2d3710","tex":"U'","aliases":"U'","display":"U'","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"introduction-passage","excerpt":"Faithful transport of obstructions. The induced quotient map. Suppose two correction systems are related by a commuting square [column sep=large,row sep=large] U arrow[r,\"D\"] arrow[d,\"a\"'] · V arrow[d,\"b\"] U' arrow[r,\"D'\"'] · V'. bD=D'a. All four spaces are modules over the same commutative ring. Denote the quotient maps by q_D and q_D'.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"[A nonlinear root with a nonzero first order obstruction] For a∈ℝ∖{0}, define F_a(x)=x^2-a^2. At zero, F_a'(0)=0, and the correction equation 0· u=a^2 has no solution. Nevertheless, F_a(a)=0. Differentiation gives F_a'(x)=2x, so the derivative at zero is zero. Since a≠0, its square is nonzero, while 0· u=0 for every u. Finally, F_a(a)=a^2-a^2=0. fo","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"Let b_1,b_2,b_3 be the standard basis of ℚ^3. Take D(u)=ub_1 and residual r=b_2+b_3. There are 24 candidate entries: a_0=b_2, a_1=b_3, a_j=b_1 (2≤ j<24). Their costs are two, three, and ten for each remaining entry. All subsets are admissible.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"Let b_1,b_2,b_3 be the standard basis of ℚ^3. Take D(u)=ub_1 and residual r=b_2+b_3. There are 24 candidate entries: a_0=b_2, a_1=b_3, a_j=b_1 (2≤ j<24). Their costs are two, three, and ten for each remaining entry. All subsets are admissible.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"[Exact finite menu feasibility] The selected menu S repairs the family exactly when O_D(r)⊆ O_S. Equivalently, ∀ i∈ I ∃ u_i∈ U ∃ a_i∈𝕜^S, Du_i+∑_j∈ Sa_ije_j=-r_i. Apply [source reference] to the actual map E_S and use [source reference]. Its reverse direction supplies the original correction witnesses and the selected coefficients, rather than only","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-daf800c8ae3d0681","tex":"a_{ij}","aliases":"a_ij","display":"a_{ij}","kind":"use","excerpt":"[Exact finite menu feasibility] The selected menu S repairs the family exactly when O_D(r)⊆ O_S. Equivalently, ∀ i∈ I ∃ u_i∈ U ∃ a_i∈𝕜^S, Du_i+∑_j∈ Sa_ije_j=-r_i. Apply [source reference] to the actual map E_S and use [source reference]. Its reverse direction supplies the original correction witnesses and the selected coefficients, rather than only","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"Finite menus and conservation. A menu is an actual coefficient map. Return to a commutative ring 𝕜 and modules U,V. Let e:J→ V be a family of candidate directions and let S⊆ J be finite. A coefficient vector is a function a:S→𝕜. Define E_S:𝕜^S⟶ V, E_S(a)=∑_j∈ Sa_je_j. Finite addition and scalar multiplication show that E_S is linear. The coefficien","count":4},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4d982043d705b926","tex":"\\operatorname{allowed}","aliases":"allowed operatorname","display":"\\operatorname{allowed}","kind":"use","excerpt":"Additional admissibility is a supplied function allowed: P(J)→{false,true}. The cost is any supplied function c: P(J)→ℕ. The formal theorem does not require c to be additive in the selected entries.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"Let B be a finite list of separating blocks. Say that B covers T when some (B,Y)∈ B satisfies T⊆ B. Validity of the block list and coverage of the relevant selections are separate obligations.","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"How the Python producer obtains the matrices. The executable producer uses exact rational row reduction. It selects independent columns of D and extends them by standard basis vectors to a basis B of ℚ^m. The inverse B^-1 supplies the image coordinates and the complementary detector coordinates. The selected image coordinates are placed in the orig","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"0=-R. Two separating blocks suffice: B_0={0}, Y_0=(0,1,-1), B_1={1}, Y_1=(0,1,0). Both satisfy Y_iD=0, annihilate their masked candidate matrices, and give Y_iR=1.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"0=-R. Two separating blocks suffice: B_0={0}, Y_0=(0,1,-1), B_1={1}, Y_1=(0,1,0). Both satisfy Y_iD=0, annihilate their masked candidate matrices, and give Y_iR=1.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"Let b_1,b_2,b_3 be the standard basis of ℚ^3. Take D(u)=ub_1 and residual r=b_2+b_3. There are 24 candidate entries: a_0=b_2, a_1=b_3, a_j=b_1 (2≤ j<24). Their costs are two, three, and ten for each remaining entry. All subsets are admissible.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"Let b_1,b_2,b_3 be the standard basis of ℚ^3. Take D(u)=ub_1 and residual r=b_2+b_3. There are 24 candidate entries: a_0=b_2, a_1=b_3, a_j=b_1 (2≤ j<24). Their costs are two, three, and ten for each remaining entry. All subsets are admissible.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-5284f24ae8cda5b8","tex":"b_3","aliases":"b_3","display":"b_3","kind":"use","excerpt":"Let b_1,b_2,b_3 be the standard basis of ℚ^3. Take D(u)=ub_1 and residual r=b_2+b_3. There are 24 candidate entries: a_0=b_2, a_1=b_3, a_j=b_1 (2≤ j<24). Their costs are two, three, and ten for each remaining entry. All subsets are admissible.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c653e5b49f447e5d","tex":"\\operatorname{bestByCheck}","aliases":"bestByCheck operatorname","display":"\\operatorname{bestByCheck}","kind":"use","excerpt":"For a finite list L of candidates, a Boolean predicate h, and a natural cost c, define bestByCheck(h,c, L) to filter by h and then return an argument of minimum cost. It returns codenone exactly when the filtered list is empty. Ties need not define a mathematically unique optimum.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-9cae1166a50fd513","tex":"\\operatorname{bestRepairMenu}","aliases":"bestRepairMenu operatorname","display":"\\operatorname{bestRepairMenu}","kind":"use","excerpt":"The repair optimizer is bestRepairMenu =bestByCheck( S↦allowed(S)wedgecheck(S), c, L_J).","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"@rrrrr@ k · x_k · c_kbmod7 · t_k · x_k+1","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"At the quotient level, L induces an injection ℚ^m/im D→ℚ^d. If Lv=0, the split identity gives v=DPv, proving that its class is zero. The image is im L. Hence the required residual obstruction span is isomorphic to the column space of C=LR. This quotient interpretation is an ordinary consequence of the displayed identities.","count":6},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"A weighting makes the distinction sharper. Assigning positive costs c_0,c_1 gives optimum c_0+c_1 for this menu. A subspace containing e_0-e_1 does not charge those two entries separately unless its cost model explicitly does so.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"A weighting makes the distinction sharper. Assigning positive costs c_0,c_1 gives optimum c_0+c_1 for this menu. A subspace containing e_0-e_1 does not charge those two entries separately unless its cost model explicitly does so.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"Fix rational matrices D∈ℚ^m× n, E∈ℚ^m× q, R∈ℚ^m× s. The columns e_j of E are the available new directions. The finite family A⊆ P({0,…,q-1}) contains exactly the admissible selections. It need not be downward closed. Nonnegative integer weights c_j define the cost c(S)=∑_j∈ Sc_j. These matrices, selections, and costs are fixed original input. A cer","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-254051ed849e0a84","tex":"c_k","aliases":"c_k","display":"c_k","kind":"use","excerpt":"An exact seven-adic trace. For x^2=2 at p=7, start with x_1=3. At each step, let c_k=fracx_k^2-27^k, c_k+2x_kt_k≡0pmod7, x_k+1=x_k+7^kt_k. The following table records normalized choices for this example. Every displayed digit equation is an exact integer calculation.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-f56da37de96d4894","tex":"c_\\ast","aliases":"c_∗ ast","display":"c_\\ast","kind":"use","excerpt":"Nonnegative additive costs are monotone under inclusion: A⊆ T Longrightarrow c(A)≤ c(T). Therefore, when an incumbent repair has cost c_∗, the finite check c_∗≤ c(A) excludes every strictly cheaper repair in I(A,B). It does not say the interval is infeasible. It only says that it cannot improve the incumbent. formalKomorebiRepair.RationalCertificat","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-269cd99c3ea8db81","tex":"\\code","aliases":"code","display":null,"kind":"use","excerpt":"Verification and provenance. The evidence attached to the formal release. The v0.2.0 verification record reports successful compilation of ten proof modules, an axiom audit of all 49 named theorem declarations, and individual replay of those modules through Lean's kernel. The allowed transitive axiom set is {codepropext, codeClassical.choice, codeQ","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-073a71f814bdfa5f","tex":"\\coker","aliases":"coker operatorname","display":"{\\operatorname{coker}}","kind":"use","excerpt":"What the quotient remembers. Write Q_D=coker D=V/im D, q_D:V⟶ Q_D. The class q_D(r) records the part of r that cannot be changed by the original correction map. Since q_D(Du)=0, all residuals that differ by an original correction have the same class.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-ddc7e0f1c8697a47","tex":"\\mathsf D","aliases":"D mathsf","display":"\\mathsf D","kind":"use","excerpt":"[Rank difference in matrix coordinates] Let the columns of a matrix R be a finite residual family and let D represent D. Then dim O_D(r)=rank[ Dmid R]-rank D. Set M=im D+span{r_i}. The restriction q_Dvert_M has image O and kernel im D. Rank and nullity give dim M=dimim D+dim O. The columns of [ Dmid R] span M. sourcescopeThis coordinate corollary i","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-64c963165c866971","tex":"\\operatorname{decide}","aliases":"decide operatorname","display":"\\operatorname{decide}","kind":"use","excerpt":"For each finite subset S⊆ J, define the Boolean test check(S)=decide( ∀ i∈ I ∃ u∈ U ∃ a∈𝕜^S, Du+E_Sa=-r_i ). All quantified domains are finite. Decidable equality in V determines each candidate equation.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"0 · 0 · 1. Thus LD=0, DP=diag(1,0,0), and KL=diag(0,1,1). The residual coordinates are C=LR=1 · 2","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"[title=The central result] Let D:U→ V be linear over a field and let q:V→ V/im D, O=span{q(r_i):i∈ I}. If V is finite dimensional, the least added dimension that repairs every r_i is dim O. If an allowed subspace A≤ V satisfies O⊆ q(A), the minimum can be attained with all added directions in A. Selecting entries from a fixed menu is a different op","count":9},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-5f6b6465a649cdba","tex":"\\dim_{\\kk}","aliases":"dim_kk dim_𝕜 dim kk dim Bbbk","display":"\\dim_{{\\Bbbk}}","kind":"use","excerpt":"[Dimension lower bound] If E:W→ V repairs r and W is finite dimensional, then O=O_D(r) is finite dimensional and dim_𝕜O≤dim_𝕜W. By [source reference], O⊆im(q_DE). The image of a linear map from a finite dimensional space is finite dimensional and has dimension at most that of its domain. Hence dim O≤dimim(q_DE)≤dim W.qedhere formalrepair_dimension_","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c7c26f6b3b085b5c","tex":"E'","aliases":"E'","display":"E'","kind":"use","excerpt":"[Dimension lower certificate] Assume [source reference]. If an arbitrary competing augmentation E'∈ℚ^m× w has coefficients satisfying DU'+E'W'=-R, then t≤ w. Multiplication by H and Q gives HE'W'Q=-I_t, hence (HE')(-W'Q)=I_t. Thus HE':ℚ^w→ℚ^t has a right inverse. It is surjective, and consequently t≤ w. formalRationalCertificates.dimension_certific","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"One obstruction, two selected entries. [The difference between the two minima] Over ℚ, let D(u)=(u,0,0), r=(0,1,0), and make only the following directions available: e_0=(0,1,1), e_1=(0,0,1). The quotient by im D is the plane of the second and third coordinates. There q_D(r)=(1,0), q_D(e_0)=(1,1), and q_D(e_1)=(0,1).","count":7},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"One obstruction, two selected entries. [The difference between the two minima] Over ℚ, let D(u)=(u,0,0), r=(0,1,0), and make only the following directions available: e_0=(0,1,1), e_1=(0,0,1). The quotient by im D is the plane of the second and third coordinates. There q_D(r)=(1,0), q_D(e_0)=(1,1), and q_D(e_1)=(0,1).","count":7},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"A third candidate can make the cost distinction sharper. Add the direct direction e_0=(0,1,0) at cost five, and price e_1=(0,1,1) and e_2=(0,0,1) at one each. One menu entry can now repair the residual, but the cheapest repair selects two entries, {1,2}, at total cost two. Minimum free dimension, minimum menu cardinality, and minimum menu cost are ","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"Finite menus and conservation. A menu is an actual coefficient map. Return to a commutative ring 𝕜 and modules U,V. Let e:J→ V be a family of candidate directions and let S⊆ J be finite. A coefficient vector is a function a:S→𝕜. Define E_S:𝕜^S⟶ V, E_S(a)=∑_j∈ Sa_je_j. Finite addition and scalar multiplication show that E_S is linear. The coefficien","count":9},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"Nonlinear limits and native realizations. A linearized problem is a different equation. Suppose a nonlinear map F is differentiable at an approximation x_0. Its linearized correction equation is DF(x_0)u=-F(x_0). The cokernel criterion completely decides [source reference] once the derivative is fixed. It does not, by itself, decide whether there i","count":19},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F 𝔽 mathbb","display":"{\\mathbb{F}}","kind":"use","excerpt":"A fully evaluated example over two elements. Take the data of [source reference] over 𝔽_2. Negation is now the identity. The equation is (u,a,a+b)=(0,1,0). Thus the full menu succeeds with u=0, a=b=1. The two singleton menus still fail.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"[Attainment inside a legal subspace] If O⊆ q_D(A), there exists a linear map E:O→ V satisfying im E⊆ A, q_DE=ι_O, and repairing the entire family. If V is finite dimensional, its added dimension is optimal even among all unrestricted repairs. Consider the surjection f:A⟶ q_D(A), a⟼ q_D(a). Choose a linear section s_A:q_D(A)→ A. Let j:O↪ q_D(A) and ","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-101fbe8a1f9aa65c","tex":"\\F_2","aliases":"F_2 𝔽_2 F mathbb","display":"{\\mathbb{F}}_2","kind":"use","excerpt":"A fully evaluated example over two elements. Take the data of [source reference] over 𝔽_2. Negation is now the identity. The equation is (u,a,a+b)=(0,1,0). Thus the full menu succeeds with u=0, a=b=1. The two singleton menus still fail.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"[A nonlinear root with a nonzero first order obstruction] For a∈ℝ∖{0}, define F_a(x)=x^2-a^2. At zero, F_a'(0)=0, and the correction equation 0· u=a^2 has no solution. Nevertheless, F_a(a)=0. Differentiation gives F_a'(x)=2x, so the derivative at zero is zero. Since a≠0, its square is nonzero, while 0· u=0 for every u. Finally, F_a(a)=a^2-a^2=0. fo","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-676f97fc86bc8429","tex":"F_a'","aliases":"F_a'","display":"F_a'","kind":"use","excerpt":"[A nonlinear root with a nonzero first order obstruction] For a∈ℝ∖{0}, define F_a(x)=x^2-a^2. At zero, F_a'(0)=0, and the correction equation 0· u=a^2 has no solution. Nevertheless, F_a(a)=0. Differentiation gives F_a'(x)=2x, so the derivative at zero is zero. Since a≠0, its square is nonzero, while 0· u=0 for every u. Finally, F_a(a)=a^2-a^2=0. fo","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4658cf98f2ab845c","tex":"\\mathrm{false}","aliases":"false mathrm","display":"\\mathrm{false}","kind":"use","excerpt":"Additional admissibility is a supplied function allowed: P(J)→{false,true}. The cost is any supplied function c: P(J)→ℕ. The formal theorem does not require c to be additive in the selected entries.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"[A solvable linearization without a real root] Let G(x)=x^2+x+1. Then G'(0)=1, ∃ u∈ℝ (u=-1), so the linearized correction at zero is solvable. However, G has no real root. The derivative is 2x+1, which equals one at zero, and G(0)=1. Thus u=-1 solves the linearized equation. But G(x)=(x+frac12)^2+frac34>0 for every real x. formalNonlinear.linearize","count":13},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"[A solvable linearization without a real root] Let G(x)=x^2+x+1. Then G'(0)=1, ∃ u∈ℝ (u=-1), so the linearized correction at zero is solvable. However, G has no real root. The derivative is 2x+1, which equals one at zero, and G(0)=1. Thus u=-1 solves the linearized equation. But G(x)=(x+frac12)^2+frac34>0 for every real x. formalNonlinear.linearize","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"The quadratic lifting theorem. [One higher precision] Let p,x,a∈ℤ and k∈ℕ with k≥1. Assume p^kmid x^2-a, gcd(2x,p)=1. Then there exists t∈ℤ such that p^k+1mid (x+p^kt)^2-a. Choose c∈ℤ with x^2-a=p^kc. Coprimality supplies B ezout data for 2x and p, so [source reference] gives t with pmid c+2xt. Since k≥1, one has pmid p^k. Apply [source reference] ","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"Dimension certificates without a reported rank. Suppose an augmentation with t columns succeeds. Its optimality can be certified by two additional matrices H∈ℚ^t× m, Q∈ℚ^s× t with HD=0, HRQ=I_t. These identities make a lower bound on the required dimension explicit.","count":17},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"[column sep=large,row sep=large] O arrow[r,\"E\"] arrow[dr,hook,\"ι_O\"'] · V arrow[d,\"q_D\"] · Q_D The triangle records exactly what is preserved: a required quotient class is lifted to a representative of that same class. The lift does not manufacture an unrelated object with a similar invariant.","count":9},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-20d6646340ce20f0","tex":"I_d","aliases":"I_d","display":"I_d","kind":"use","excerpt":"Exact rank and menu cost. For A∈ℚ^a× b, a certificate of rank d consists of appropriately shaped rational matrices with A=FG, PAQ=I_d, where the inner dimension of FG is d.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-ad472b41377c2078","tex":"I_m","aliases":"I_m","display":"I_m","kind":"use","excerpt":"Constructive matrix attainment. A complete detector. Fix D∈ℚ^m× n and R∈ℚ^m× s. A split correction certificate consists of P∈ℚ^n× m, K∈ℚ^m× d, L∈ℚ^d× m with LD=0, DP+KL=I_m. The first identity makes L a detector that ignores original corrections. The second proves that no further information has been discarded: every target vector has the decomposi","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-34b5a08da5138c3a","tex":"I_t","aliases":"I_t","display":"I_t","kind":"use","excerpt":"Dimension certificates without a reported rank. Suppose an augmentation with t columns succeeds. Its optimality can be certified by two additional matrices H∈ℚ^t× m, Q∈ℚ^s× t with HD=0, HRQ=I_t. These identities make a lower bound on the required dimension explicit.","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id operatorname","display":"{\\operatorname{id}}","kind":"use","excerpt":"An attaining construction. [A section of the quotient] There is a linear map s:Q_D→ V with q_Ds=id_Q_D. Choose a basis of the vector space Q_D. Since q_D is surjective, choose a representative in V of each basis vector. Extend this assignment linearly. Its composite with q_D fixes the basis and therefore fixes all of Q_D. formalexists_obstruction_s","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-7d271959abb417d0","tex":"\\id_{\\Q}","aliases":"id_Q id_ℚ id Q operatorname mathbb","display":"{\\operatorname{id}}_{{\\mathbb{Q}}}","kind":"use","excerpt":"A transport that conceals failure. [Correct in the target, false in the source] Let D:ℚ→ℚ^2, D(u)=(u,0), D'=id_ℚ, a=id_ℚ, b(x,y)=x. Then bD=D'a. For r=(0,1) there is no u with (u,0)=(0,-1), so the source residual is not correctable. Its image b(r)=0 is correctable by D'.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-ae2553828d11012e","tex":"\\id_{Q_D}","aliases":"id_Q_D id operatorname","display":"{\\operatorname{id}}_{Q_D}","kind":"use","excerpt":"An attaining construction. [A section of the quotient] There is a linear map s:Q_D→ V with q_Ds=id_Q_D. Choose a basis of the vector space Q_D. Since q_D is surjective, choose a representative in V of each basis vector. Extend this assignment linearly. Its composite with q_D fixes the basis and therefore fixes all of Q_D. formalexists_obstruction_s","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"use","excerpt":"[title=The central result] Let D:U→ V be linear over a field and let q:V→ V/im D, O=span{q(r_i):i∈ I}. If V is finite dimensional, the least added dimension that repairs every r_i is dim O. If an allowed subspace A≤ V satisfies O⊆ q(A), the minimum can be attained with all added directions in A. Selecting entries from a fixed menu is a different op","count":32},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"Constructive matrix attainment. A complete detector. Fix D∈ℚ^m× n and R∈ℚ^m× s. A split correction certificate consists of P∈ℚ^n× m, K∈ℚ^m× d, L∈ℚ^d× m with LD=0, DP+KL=I_m. The first identity makes L a detector that ignores original corrections. The second proves that no further information has been discarded: every target vector has the decomposi","count":12},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"[column sep=large,row sep=large] O arrow[r,\"E\"] arrow[dr,hook,\"ι_O\"'] · V arrow[d,\"q_D\"] · Q_D The triangle records exactly what is preserved: a required quotient class is lifted to a representative of that same class. The lift does not manufacture an unrelated object with a similar invariant.","count":19},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"This condition is equivalent to ker B∩ O=0. It does not require injectivity on all of Q_D when only a smaller family of obstruction classes matters.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e6ebaeb35ff7fc8c","tex":"\\kk^S","aliases":"kk^S 𝕜^S kk Bbbk","display":"{\\Bbbk}^S","kind":"use","excerpt":"Finite menus and conservation. A menu is an actual coefficient map. Return to a commutative ring 𝕜 and modules U,V. Let e:J→ V be a family of candidate directions and let S⊆ J be finite. A coefficient vector is a function a:S→𝕜. Define E_S:𝕜^S⟶ V, E_S(a)=∑_j∈ Sa_je_j. Finite addition and scalar multiplication show that E_S is linear. The coefficien","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"The quotient also has a universal property. If L:V→ Z annihilates im D, it descends to a unique linear map overline L:Q_D→ Z with L=overline Lq_D. Thus every linear detector that ignores correctable directions factors through the same quotient. This is the classical cokernel construction used in Paper III~[source reference]. Canonicality is relativ","count":48},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"Enumerating menus and selecting an optimum. Enumerate J as a sorted list. Take every sublist and convert it to a finite set. The resulting list, denoted L_J, contains every subset of J.","count":9},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-9e7f2be77d295f01","tex":"\\mathcal L_J","aliases":"L_J mathcal","display":"\\mathcal L_J","kind":"use","excerpt":"Enumerating menus and selecting an optimum. Enumerate J as a sorted list. Take every sublist and convert it to a finite set. The resulting list, denoted L_J, contains every subset of J.","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Exact obstruction criterion] For every r∈ V, ∃ u∈ U (Du=-r) Longleftrightarrow q_D(r)=0. Applying q_D to Du=-r gives 0=-q_D(r), hence q_D(r)=0. Conversely, q_D(r)=0 means r∈im D. Choose v∈ U with Dv=r. Then D(-v)=-r. formalcorrectable_iff_obstruction_zero","count":4},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"[Rank difference in matrix coordinates] Let the columns of a matrix R be a finite residual family and let D represent D. Then dim O_D(r)=rank[ Dmid R]-rank D. Set M=im D+span{r_i}. The restriction q_Dvert_M has image O and kernel im D. Rank and nullity give dim M=dimim D+dim O. The columns of [ Dmid R] span M. sourcescopeThis coordinate corollary i","count":4},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"[Quadratic correction identity] Let p,m,x,a,c,t∈ℤ satisfy x^2-a=mc, pmid c+2xt, pmid m. Then mpmid (x+mt)^2-a. Write c+2xt=pd and m=ps. Direct expansion gives (x+mt)^2-a · =(x^2-a)+2mxt+m^2t^2 · =m(c+2xt)+m^2t^2 · =mp(d+st^2).qedhere The last expression displays the required quotient explicitly. formalArithmetic.quadratic_digit_correction","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"[Rank difference in matrix coordinates] Let the columns of a matrix R be a finite residual family and let D represent D. Then dim O_D(r)=rank[ Dmid R]-rank D. Set M=im D+span{r_i}. The restriction q_Dvert_M has image O and kernel im D. Rank and nullity give dim M=dimim D+dim O. The columns of [ Dmid R] span M. sourcescopeThis coordinate corollary i","count":12},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"[The attained free minimum] If V is finite dimensional, the minimum possible dimension of a finite dimensional added space that repairs the family is min{dim W: E:W→ V repairs r}=dim O_D(r). Every admissible term on the left is at least dim O by [source reference]. Because V is finite dimensional, its quotient and the subspace O are finite dimensio","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N ℕ mathbb","display":"{\\mathbb{N}}","kind":"use","excerpt":"Additional admissibility is a supplied function allowed: P(J)→{false,true}. The cost is any supplied function c: P(J)→ℕ. The formal theorem does not require c to be additive in the selected entries.","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-09061c29f4c104f3","tex":"\\nexists","aliases":"nexists","display":null,"kind":"use","excerpt":"If admissibility is specified by a list A, a terminal check establishes nexists T∈ A: A⊆ T⊆ B. It examines the original list, not the powerset of J. For the all-subsets rule this terminal certificate is unavailable. formalKomorebiRepair.RationalCertificates.intervalExcluded_sound","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","excerpt":"[Subspace impossibility] If Onot⊆ q_D(A), no added map E:W→ V with image contained in A repairs the family. Such a map satisfies im(q_DE)⊆ q_D(A). If it repaired the family, [source reference] would imply O⊆im(q_DE)⊆ q_D(A), contrary to the hypothesis. formaladmissible_subspace_no_go","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-d08d07be26360470","tex":"O_S","aliases":"O_S","display":"O_S","kind":"use","excerpt":"Write O_S=span_𝕜{q_D(e_j):j∈ S}.","count":6},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c6833411661004de","tex":"O_T","aliases":"O_T","display":"O_T","kind":"use","excerpt":"[Monotonicity of feasibility] If S⊆ T, then O_S⊆ O_T. Every family repairable with S is therefore repairable with T. Each generator defining O_S is also a generator defining O_T. Apply [source reference]. Equivalently, extend each coefficient vector by zero on T∖ S. formalrepairable_menu_mono","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e4b45cd66b9c8f4e","tex":"\\overline L","aliases":"overline L overline","display":"\\overline L","kind":"use","excerpt":"The quotient also has a universal property. If L:V→ Z annihilates im D, it descends to a unique linear map overline L:Q_D→ Z with L=overline Lq_D. Thus every linear detector that ignores correctable directions factors through the same quotient. This is the classical cokernel construction used in Paper III~[source reference]. Canonicality is relativ","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"Exact rank and menu cost. For A∈ℚ^a× b, a certificate of rank d consists of appropriately shaped rational matrices with A=FG, PAQ=I_d, where the inner dimension of FG is d.","count":15},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"Additional admissibility is a supplied function allowed: P(J)→{false,true}. The cost is any supplied function c: P(J)→ℕ. The formal theorem does not require c to be additive in the selected entries.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-b7f360bb2799a8d5","tex":"p^k","aliases":"p^k","display":"p^k","kind":"use","excerpt":"The quadratic lifting theorem. [One higher precision] Let p,x,a∈ℤ and k∈ℕ with k≥1. Assume p^kmid x^2-a, gcd(2x,p)=1. Then there exists t∈ℤ such that p^k+1mid (x+p^kt)^2-a. Choose c∈ℤ with x^2-a=p^kc. Coprimality supplies B ezout data for 2x and p, so [source reference] gives t with pmid c+2xt. Since k≥1, one has pmid p^k. Apply [source reference] ","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-f21691df4ee7e60b","tex":"p^{k+1}","aliases":"p^k+1","display":"p^{k+1}","kind":"use","excerpt":"The quadratic lifting theorem. [One higher precision] Let p,x,a∈ℤ and k∈ℕ with k≥1. Assume p^kmid x^2-a, gcd(2x,p)=1. Then there exists t∈ℤ such that p^k+1mid (x+p^kt)^2-a. Choose c∈ℤ with x^2-a=p^kc. Coprimality supplies B ezout data for 2x and p, so [source reference] gives t with pmid c+2xt. Since k≥1, one has pmid p^k. Apply [source reference] ","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"An exact seven-adic trace. For x^2=2 at p=7, start with x_1=3. At each step, let c_k=fracx_k^2-27^k, c_k+2x_kt_k≡0pmod7, x_k+1=x_k+7^kt_k. The following table records normalized choices for this example. Every displayed digit equation is an exact integer calculation.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-afeb63abf7eb7cda","tex":"\\Q^2","aliases":"Q^2 ℚ^2 Q mathbb","display":"{\\mathbb{Q}}^2","kind":"use","excerpt":"[A visible residual] Let D:ℚ→ℚ^2 be D(u)=(u,0) and let r=(2,3). The quotient is identified with the second coordinate. The correction u=-2 removes the first coordinate but leaves (0,3). The functional L(x,y)=y satisfies LD=0 and L(r)=3, so no original correction eliminates r.","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-1a8d4103c05783dd","tex":"\\Q^3","aliases":"Q^3 ℚ^3 Q mathbb","display":"{\\mathbb{Q}}^3","kind":"use","excerpt":"Let b_1,b_2,b_3 be the standard basis of ℚ^3. Take D(u)=ub_1 and residual r=b_2+b_3. There are 24 candidate entries: a_0=b_2, a_1=b_3, a_j=b_1 (2≤ j<24). Their costs are two, three, and ten for each remaining entry. All subsets are admissible.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-5ae47ba0f9ecc9f3","tex":"\\Q^{a\\times b}","aliases":"Q^a× b ℚ^a× b Q times mathbb times","display":"{\\mathbb{Q}}^{a\\times b}","kind":"use","excerpt":"Exact rank and menu cost. For A∈ℚ^a× b, a certificate of rank d consists of appropriately shaped rational matrices with A=FG, PAQ=I_d, where the inner dimension of FG is d.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-f9d22b30dcf659de","tex":"\\Q^d","aliases":"Q^d ℚ^d Q mathbb","display":"{\\mathbb{Q}}^d","kind":"use","excerpt":"At the quotient level, L induces an injection ℚ^m/im D→ℚ^d. If Lv=0, the split identity gives v=DPv, proving that its class is zero. The image is im L. Hence the required residual obstruction span is isomorphic to the column space of C=LR. This quotient interpretation is an ordinary consequence of the displayed identities.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-1417ba15b4b8b1d3","tex":"\\Q^{d\\times m}","aliases":"Q^d× m ℚ^d× m Q times mathbb times","display":"{\\mathbb{Q}}^{d\\times m}","kind":"use","excerpt":"Constructive matrix attainment. A complete detector. Fix D∈ℚ^m× n and R∈ℚ^m× s. A split correction certificate consists of P∈ℚ^n× m, K∈ℚ^m× d, L∈ℚ^d× m with LD=0, DP+KL=I_m. The first identity makes L a detector that ignores original corrections. The second proves that no further information has been discarded: every target vector has the decomposi","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-63eba5dd2251c6c9","tex":"\\Q^{d\\times t}","aliases":"Q^d× t ℚ^d× t Q times mathbb times","display":"{\\mathbb{Q}}^{d\\times t}","kind":"use","excerpt":"The attaining construction. Suppose the detected residual matrix C=LR has certificate data F∈ℚ^d× t, G∈ℚ^t× s, J∈ℚ^t× d, Q∈ℚ^s× t satisfying LR=FG, JLRQ=I_t. The new augmentation and its correction coefficients are not unspecified witnesses. They are E=KF, U=-PR, W=-G.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-dbcb7a433d1dbeaf","tex":"\\Q^m","aliases":"Q^m ℚ^m Q mathbb","display":"{\\mathbb{Q}}^m","kind":"use","excerpt":"For a fixed rational menu, either the correction matrices exist or a separating row exists, but not both. Exclusivity follows from the preceding theorem. If no correction matrices exist, some residual r_i lies outside the column space of [D widetilde E_S]. Choose a basis for that space, adjoin r_i, and extend to a basis of ℚ^m. The functional takin","count":4},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-eb841c358d0c39a0","tex":"\\Q^{m\\times d}","aliases":"Q^m× d ℚ^m× d Q times mathbb times","display":"{\\mathbb{Q}}^{m\\times d}","kind":"use","excerpt":"Constructive matrix attainment. A complete detector. Fix D∈ℚ^m× n and R∈ℚ^m× s. A split correction certificate consists of P∈ℚ^n× m, K∈ℚ^m× d, L∈ℚ^d× m with LD=0, DP+KL=I_m. The first identity makes L a detector that ignores original corrections. The second proves that no further information has been discarded: every target vector has the decomposi","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-cac4d5300cc950fc","tex":"\\Q^{m\\times n}","aliases":"Q^m× n ℚ^m× n Q times mathbb times","display":"{\\mathbb{Q}}^{m\\times n}","kind":"use","excerpt":"Fix matrices D∈ℚ^m× n, R∈ℚ^m× s, E∈ℚ^m× q. The residuals are the columns of R. For a selected menu S, let widetilde E_S retain the original candidate indexing and replace unselected columns by zero. The supplied coefficient matrices have shapes U∈ℚ^n× s and W∈ℚ^q× s. Their correction equation is DU+widetilde E_SW=-R. Keeping the original indexing a","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-d4a214c9340ff0ca","tex":"\\Q^{m\\times q}","aliases":"Q^m× q ℚ^m× q Q times mathbb times","display":"{\\mathbb{Q}}^{m\\times q}","kind":"use","excerpt":"Fix matrices D∈ℚ^m× n, R∈ℚ^m× s, E∈ℚ^m× q. The residuals are the columns of R. For a selected menu S, let widetilde E_S retain the original candidate indexing and replace unselected columns by zero. The supplied coefficient matrices have shapes U∈ℚ^n× s and W∈ℚ^q× s. Their correction equation is DU+widetilde E_SW=-R. Keeping the original indexing a","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-3ad33499ce250435","tex":"\\Q^{m\\times s}","aliases":"Q^m× s ℚ^m× s Q times mathbb times","display":"{\\mathbb{Q}}^{m\\times s}","kind":"use","excerpt":"Fix matrices D∈ℚ^m× n, R∈ℚ^m× s, E∈ℚ^m× q. The residuals are the columns of R. For a selected menu S, let widetilde E_S retain the original candidate indexing and replace unselected columns by zero. The supplied coefficient matrices have shapes U∈ℚ^n× s and W∈ℚ^q× s. Their correction equation is DU+widetilde E_SW=-R. Keeping the original indexing a","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a32a7007ed661040","tex":"\\Q^{m\\times w}","aliases":"Q^m× w ℚ^m× w Q times mathbb times","display":"{\\mathbb{Q}}^{m\\times w}","kind":"use","excerpt":"[Dimension lower certificate] Assume [source reference]. If an arbitrary competing augmentation E'∈ℚ^m× w has coefficients satisfying DU'+E'W'=-R, then t≤ w. Multiplication by H and Q gives HE'W'Q=-I_t, hence (HE')(-W'Q)=I_t. Thus HE':ℚ^w→ℚ^t has a right inverse. It is surjective, and consequently t≤ w. formalRationalCertificates.dimension_certific","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-cacd1c6ffacc4de6","tex":"\\Q^{n\\times m}","aliases":"Q^n× m ℚ^n× m Q times mathbb times","display":"{\\mathbb{Q}}^{n\\times m}","kind":"use","excerpt":"Constructive matrix attainment. A complete detector. Fix D∈ℚ^m× n and R∈ℚ^m× s. A split correction certificate consists of P∈ℚ^n× m, K∈ℚ^m× d, L∈ℚ^d× m with LD=0, DP+KL=I_m. The first identity makes L a detector that ignores original corrections. The second proves that no further information has been discarded: every target vector has the decomposi","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-7f2f638718b9ad00","tex":"\\Q^{n\\times s}","aliases":"Q^n× s ℚ^n× s Q times mathbb times","display":"{\\mathbb{Q}}^{n\\times s}","kind":"use","excerpt":"Fix matrices D∈ℚ^m× n, R∈ℚ^m× s, E∈ℚ^m× q. The residuals are the columns of R. For a selected menu S, let widetilde E_S retain the original candidate indexing and replace unselected columns by zero. The supplied coefficient matrices have shapes U∈ℚ^n× s and W∈ℚ^q× s. Their correction equation is DU+widetilde E_SW=-R. Keeping the original indexing a","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e51b51d3f94da9af","tex":"\\Q^{q\\times s}","aliases":"Q^q× s ℚ^q× s Q times mathbb times","display":"{\\mathbb{Q}}^{q\\times s}","kind":"use","excerpt":"Fix matrices D∈ℚ^m× n, R∈ℚ^m× s, E∈ℚ^m× q. The residuals are the columns of R. For a selected menu S, let widetilde E_S retain the original candidate indexing and replace unselected columns by zero. The supplied coefficient matrices have shapes U∈ℚ^n× s and W∈ℚ^q× s. Their correction equation is DU+widetilde E_SW=-R. Keeping the original indexing a","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-37f7ca1a9f082dea","tex":"\\Q^{s\\times t}","aliases":"Q^s× t ℚ^s× t Q times mathbb times","display":"{\\mathbb{Q}}^{s\\times t}","kind":"use","excerpt":"Dimension certificates without a reported rank. Suppose an augmentation with t columns succeeds. Its optimality can be certified by two additional matrices H∈ℚ^t× m, Q∈ℚ^s× t with HD=0, HRQ=I_t. These identities make a lower bound on the required dimension explicit.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-b7a93ad2a6dfb30b","tex":"\\Q^t","aliases":"Q^t ℚ^t Q mathbb","display":"{\\mathbb{Q}}^t","kind":"use","excerpt":"[Dimension lower certificate] Assume [source reference]. If an arbitrary competing augmentation E'∈ℚ^m× w has coefficients satisfying DU'+E'W'=-R, then t≤ w. Multiplication by H and Q gives HE'W'Q=-I_t, hence (HE')(-W'Q)=I_t. Thus HE':ℚ^w→ℚ^t has a right inverse. It is surjective, and consequently t≤ w. formalRationalCertificates.dimension_certific","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-5ee478b0f22901ca","tex":"\\Q^{t\\times d}","aliases":"Q^t× d ℚ^t× d Q times mathbb times","display":"{\\mathbb{Q}}^{t\\times d}","kind":"use","excerpt":"The attaining construction. Suppose the detected residual matrix C=LR has certificate data F∈ℚ^d× t, G∈ℚ^t× s, J∈ℚ^t× d, Q∈ℚ^s× t satisfying LR=FG, JLRQ=I_t. The new augmentation and its correction coefficients are not unspecified witnesses. They are E=KF, U=-PR, W=-G.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-45a465e3989677e4","tex":"\\Q^{t\\times m}","aliases":"Q^t× m ℚ^t× m Q times mathbb times","display":"{\\mathbb{Q}}^{t\\times m}","kind":"use","excerpt":"Dimension certificates without a reported rank. Suppose an augmentation with t columns succeeds. Its optimality can be certified by two additional matrices H∈ℚ^t× m, Q∈ℚ^s× t with HD=0, HRQ=I_t. These identities make a lower bound on the required dimension explicit.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-52273fcbbb7ddb82","tex":"\\Q^{t\\times s}","aliases":"Q^t× s ℚ^t× s Q times mathbb times","display":"{\\mathbb{Q}}^{t\\times s}","kind":"use","excerpt":"The attaining construction. Suppose the detected residual matrix C=LR has certificate data F∈ℚ^d× t, G∈ℚ^t× s, J∈ℚ^t× d, Q∈ℚ^s× t satisfying LR=FG, JLRQ=I_t. The new augmentation and its correction coefficients are not unspecified witnesses. They are E=KF, U=-PR, W=-G.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a0a98d8cc010de93","tex":"\\Q^w","aliases":"Q^w ℚ^w Q mathbb","display":"{\\mathbb{Q}}^w","kind":"use","excerpt":"[Dimension lower certificate] Assume [source reference]. If an arbitrary competing augmentation E'∈ℚ^m× w has coefficients satisfying DU'+E'W'=-R, then t≤ w. Multiplication by H and Q gives HE'W'Q=-I_t, hence (HE')(-W'Q)=I_t. Thus HE':ℚ^w→ℚ^t has a right inverse. It is surjective, and consequently t≤ w. formalRationalCertificates.dimension_certific","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a2db8f84dc58722f","tex":"Q_{D'}","aliases":"Q_D'","display":"Q_{D'}","kind":"use","excerpt":"It follows that there is a well defined linear map B:Q_D⟶ Q_D', B(q_D(v))=q_D'(b(v)). Indeed, changing v by an element of im D changes b(v) by an element of im D'. This is the quotient map constructed in the formal source, not an assumed function on unidentified obstruction spaces.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a2b27e6c1de4e544","tex":"\\qedhere","aliases":"qedhere","display":null,"kind":"use","excerpt":"[Dimension lower bound] If E:W→ V repairs r and W is finite dimensional, then O=O_D(r) is finite dimensional and dim_𝕜O≤dim_𝕜W. By [source reference], O⊆im(q_DE). The image of a linear map from a finite dimensional space is finite dimensional and has dimension at most that of its domain. Hence dim O≤dimim(q_DE)≤dim W.qedhere formalrepair_dimension_","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R ℝ mathbb","display":"{\\mathbb{R}}","kind":"use","excerpt":"[A nonlinear root with a nonzero first order obstruction] For a∈ℝ∖{0}, define F_a(x)=x^2-a^2. At zero, F_a'(0)=0, and the correction equation 0· u=a^2 has no solution. Nevertheless, F_a(a)=0. Differentiation gives F_a'(x)=2x, so the derivative at zero is zero. Since a≠0, its square is nonzero, while 0· u=0 for every u. Finally, F_a(a)=a^2-a^2=0. fo","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-02f6e210c4b44c83","tex":"\\mathsf R","aliases":"R mathsf","display":"\\mathsf R","kind":"use","excerpt":"[Rank difference in matrix coordinates] Let the columns of a matrix R be a finite residual family and let D represent D. Then dim O_D(r)=rank[ Dmid R]-rank D. Set M=im D+span{r_i}. The restriction q_Dvert_M has image O and kernel im D. Rank and nullity give dim M=dimim D+dim O. The columns of [ Dmid R] span M. sourcescopeThis coordinate corollary i","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank operatorname","display":"{\\operatorname{rank}}","kind":"use","excerpt":"[Rank difference in matrix coordinates] Let the columns of a matrix R be a finite residual family and let D represent D. Then dim O_D(r)=rank[ Dmid R]-rank D. Set M=im D+span{r_i}. The restriction q_Dvert_M has image O and kernel im D. Rank and nullity give dim M=dimim D+dim O. The columns of [ Dmid R] span M. sourcescopeThis coordinate corollary i","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c0bc731e693dd8ce","tex":"s_A","aliases":"s_A","display":"s_A","kind":"use","excerpt":"abelthm:legal If O⊆ q_D(A), there exists a linear map E:O→ V satisfying im E⊆ A, q_DE=ι_O, and repairing the entire family. If V is finite dimensional, its added dimension is optimal even among all unrestricted repairs. Consider the surjection f:A⟶ q_D(A), a⟼ q_D(a). Choose a linear section s_A:q_D(A)→ A. Let j:O↪ q_D(A) and ι_A:A↪ V be inclusions.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"[Exact branch partition] If j∈ B∖ A, then I(A,B)= I(A,B∖{j}) sqcup I(A∪{j},B). Every T∈ I(A,B) either excludes j or includes j. In the first case it belongs to the left interval, and in the second it belongs to the right interval. Conversely, each child interval is contained in the parent. Their membership conditions on j are incompatible, so the u","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"The fractional case uses D=(2/3,4/3)^ T and R=(1/2,5/6)^ T. It exercises nonintegral splitting and correction coefficients while retaining exact arithmetic. Empty and rank-deficient cases are included in the same exported corpus. Every case has its own original input file, certificate file, and formal optimality statement.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"What the algorithm does not promise. The reference solver is deliberately finite and exhaustive. With t=|J|, there are 2^t candidate subsets. For a subset of size s, each residual has at most |U| |𝕜|^s candidate coefficient and correction assignments to examine. Summing this upper bound over all subsets gives |I| |U|∑_s=0^tbinom ts|𝕜|^s =|I| |U|(1+","count":39},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"[Quadratic correction identity] Let p,m,x,a,c,t∈ℤ satisfy x^2-a=mc, pmid c+2xt, pmid m. Then mpmid (x+mt)^2-a. Write c+2xt=pd and m=ps. Direct expansion gives (x+mt)^2-a · =(x^2-a)+2mxt+m^2t^2 · =m(c+2xt)+m^2t^2 · =mp(d+st^2).qedhere The last expression displays the required quotient explicitly. formalArithmetic.quadratic_digit_correction","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-830c9c56b4f2bf21","tex":"t_k","aliases":"t_k","display":"t_k","kind":"use","excerpt":"An exact seven-adic trace. For x^2=2 at p=7, start with x_1=3. At each step, let c_k=fracx_k^2-27^k, c_k+2x_kt_k≡0pmod7, x_k+1=x_k+7^kt_k. The following table records normalized choices for this example. Every displayed digit equation is an exact integer calculation.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"[Exact finite menu feasibility] The selected menu S repairs the family exactly when O_D(r)⊆ O_S. Equivalently, ∀ i∈ I ∃ u_i∈ U ∃ a_i∈𝕜^S, Du_i+∑_j∈ Sa_ije_j=-r_i. Apply [source reference] to the actual map E_S and use [source reference]. Its reverse direction supplies the original correction witnesses and the selected coefficients, rather than only","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"[Exact obstruction criterion] For every r∈ V, ∃ u∈ U (Du=-r) Longleftrightarrow q_D(r)=0. Applying q_D to Du=-r gives 0=-q_D(r), hence q_D(r)=0. Conversely, q_D(r)=0 means r∈im D. Choose v∈ U with Dv=r. Then D(-v)=-r. formalcorrectable_iff_obstruction_zero","count":17},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c18a3ad17b546688","tex":"\\vert","aliases":"vert","display":"\\vert","kind":"use","excerpt":"Directions restricted to a subspace. An application usually prohibits some corrections. A first useful constraint is a subspace A≤ V containing every permitted added direction. Its quotient image is q_D(A)=im(q_Dvert_A:A→ Q_D).","count":4},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-68bd5e9811863038","tex":"\\vert_A","aliases":"vert_A vert","display":"\\vert_A","kind":"use","excerpt":"Directions restricted to a subspace. An application usually prohibits some corrections. A first useful constraint is a subspace A≤ V containing every permitted added direction. Its quotient image is q_D(A)=im(q_Dvert_A:A→ Q_D).","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c9515ecfe4178c4d","tex":"\\vert_M","aliases":"vert_M vert","display":"\\vert_M","kind":"use","excerpt":"[Rank difference in matrix coordinates] Let the columns of a matrix R be a finite residual family and let D represent D. Then dim O_D(r)=rank[ Dmid R]-rank D. Set M=im D+span{r_i}. The restriction q_Dvert_M has image O and kernel im D. Rank and nullity give dim M=dimim D+dim O. The columns of [ Dmid R] span M. sourcescopeThis coordinate corollary i","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-387c51e41585ea60","tex":"\\vert_O","aliases":"vert_O vert","display":"\\vert_O","kind":"use","excerpt":"[Equivalent injectivity criterion] The map B is faithful on O if and only if its restriction Bvert_O:O→ Q_D' is injective. If B is faithful and Bx=By for x,y∈ O, then x-y∈ O and B(x-y)=0. Faithfulness gives x=y. Conversely, if Bvert_O is injective and Bz=0=B0 for z∈ O, injectivity gives z=0. formalfaithfulOn_iff_injective","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-e9fe8ee5ada9c8bf","tex":"W'","aliases":"W'","display":"W'","kind":"use","excerpt":"[Dimension lower certificate] Assume [source reference]. If an arbitrary competing augmentation E'∈ℚ^m× w has coefficients satisfying DU'+E'W'=-R, then t≤ w. Multiplication by H and Q gives HE'W'Q=-I_t, hence (HE')(-W'Q)=I_t. Thus HE':ℚ^w→ℚ^t has a right inverse. It is surjective, and consequently t≤ w. formalRationalCertificates.dimension_certific","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-b99d2e5f37ae7e17","tex":"w'","aliases":"w'","display":"w'","kind":"use","excerpt":"Conversely, choose w with q_D(Ew)=q_D(r). Then q_D(r-Ew)=0, so there is v∈ U with Dv=r-Ew. Taking u=-v and w'=-w gives Du+Ew'=-(r-Ew)-Ew=-r. The witnesses solve the equation in V, not just in its quotient. formalrepairable_iff_obstruction_mem_range","count":4},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"The mathematical statements here use ∃ u,w with the same sign convention as the source. Families quantify ∀ i ∃ u_i,w_i. They do not assume one shared correction vector. The algebraic base is a commutative ring. The dimension and section results specialize to a field, and finiteness is stated when numerical dimensions or terminating enumeration are","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"Let J={0,…,q-1} be the original candidate index set. The matrices D,E,R and the original correction equation remain fixed: DU+widetilde E_S W=-R. Here widetilde E_S zeros unselected columns of E while retaining its original indexing. Costs are nonnegative integers attached to those same columns: c(S)=∑_j∈ Sw_j, w_j∈ℕ. The admissibility rule either ","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-81b57ad5b6166ce8","tex":"w_{ji}","aliases":"w_ji","display":"w_{ji}","kind":"use","excerpt":"[Successful certificate] If an admissible S and rational matrices U,W satisfy [source reference], they repair every original residual. Take the ith column of the displayed identity. It is the original equation Du_i+∑_j∈ Sw_jie_j=-r_i. formalRationalCertificates.checkRepair_sound","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"The repair optimizer is bestRepairMenu =bestByCheck( S↦allowed(S)wedgecheck(S), c, L_J).","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-864a514bc8558dfc","tex":"\\widetilde E_T","aliases":"widetilde E_T widetilde","display":"\\widetilde E_T","kind":"use","excerpt":"[Downward validity of a separator] If (B,Y) is a separating block and T⊆ B, then Y also separates T. In particular, T cannot satisfy [source reference]. For a selected column j∈ T, one also has j∈ B, so (YE)_0j=0 by the block identity. Every unselected column is zero after masking. Thus Ywidetilde E_T=0. The other two equations in [source reference","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"[Exact detection] Under [source reference], (∃ X, DX=-R) Longleftrightarrow LR=0. When LR=0, the explicit witness is X=-PR. A correction gives 0=LDX=-LR. Conversely, LR=0 gives R=(DP+KL)R=DPR, so D(-PR)=-R. formalRationalCertificates.split_correctable_iff","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"[A visible residual] Let D:ℚ→ℚ^2 be D(u)=(u,0) and let r=(2,3). The quotient is identified with the second coordinate. The correction u=-2 removes the first coordinate but leaves (0,3). The functional L(x,y)=y satisfies LD=0 and L(r)=3, so no original correction eliminates r.","count":47},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"[Quadratic correction identity] Let p,m,x,a,c,t∈ℤ satisfy x^2-a=mc, pmid c+2xt, pmid m. Then mpmid (x+mt)^2-a. Write c+2xt=pd and m=ps. Direct expansion gives (x+mt)^2-a · =(x^2-a)+2mxt+m^2t^2","count":10},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"Nonlinear limits and native realizations. A linearized problem is a different equation. Suppose a nonlinear map F is differentiable at an approximation x_0. Its linearized correction equation is DF(x_0)u=-F(x_0). The cokernel criterion completely decides [source reference] once the derivative is fixed. It does not, by itself, decide whether there i","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"An exact seven-adic trace. For x^2=2 at p=7, start with x_1=3. At each step, let c_k=fracx_k^2-27^k, c_k+2x_kt_k≡0pmod7, x_k+1=x_k+7^kt_k. The following table records normalized choices for this example. Every displayed digit equation is an exact integer calculation.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"use","excerpt":"An exact seven-adic trace. For x^2=2 at p=7, start with x_1=3. At each step, let c_k=fracx_k^2-27^k, c_k+2x_kt_k≡0pmod7, x_k+1=x_k+7^kt_k. The following table records normalized choices for this example. Every displayed digit equation is an exact integer calculation.","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-c7df6d1ed132831f","tex":"x_{k+1}","aliases":"x_k+1","display":"x_{k+1}","kind":"use","excerpt":"An exact seven-adic trace. For x^2=2 at p=7, start with x_1=3. At each step, let c_k=fracx_k^2-27^k, c_k+2x_kt_k≡0pmod7, x_k+1=x_k+7^kt_k. The following table records normalized choices for this example. Every displayed digit equation is an exact integer calculation.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-0c95817b579844a3","tex":"x_k^2","aliases":"x_k^2","display":"x_k^2","kind":"use","excerpt":"An exact seven-adic trace. For x^2=2 at p=7, start with x_1=3. At each step, let c_k=fracx_k^2-27^k, c_k+2x_kt_k≡0pmod7, x_k+1=x_k+7^kt_k. The following table records normalized choices for this example. Every displayed digit equation is an exact integer calculation.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[A visible residual] Let D:ℚ→ℚ^2 be D(u)=(u,0) and let r=(2,3). The quotient is identified with the second coordinate. The correction u=-2 removes the first coordinate but leaves (0,3). The functional L(x,y)=y satisfies LD=0 and L(r)=3, so no original correction eliminates r.","count":12},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-d842f0833fbfd1bc","tex":"Y_0","aliases":"Y_0","display":"Y_0","kind":"use","excerpt":"0=-R. Two separating blocks suffice: B_0={0}, Y_0=(0,1,-1), B_1={1}, Y_1=(0,1,0). Both satisfy Y_iD=0, annihilate their masked candidate matrices, and give Y_iR=1.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-7ef58ddaab7b5984","tex":"Y_1","aliases":"Y_1","display":"Y_1","kind":"use","excerpt":"0=-R. Two separating blocks suffice: B_0={0}, Y_0=(0,1,-1), B_1={1}, Y_1=(0,1,0). Both satisfy Y_iD=0, annihilate their masked candidate matrices, and give Y_iR=1.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-8e1b422661cf5616","tex":"Y_i","aliases":"Y_i","display":"Y_i","kind":"use","excerpt":"0=-R. Two separating blocks suffice: B_0={0}, Y_0=(0,1,-1), B_1={1}, Y_1=(0,1,0). Both satisfy Y_iD=0, annihilate their masked candidate matrices, and give Y_iR=1.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"The quotient also has a universal property. If L:V→ Z annihilates im D, it descends to a unique linear map overline L:Q_D→ Z with L=overline Lq_D. Thus every linear detector that ignores correctable directions factors through the same quotient. This is the classical cokernel construction used in Paper III~[source reference]. Canonicality is relativ","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z ℤ mathbb","display":"{\\mathbb{Z}}","kind":"use","excerpt":"The field hypothesis in the section argument is substantive. Over ℤ, take D:ℤ→ℤ to be multiplication by two. The quotient map q:ℤ→ℤ/2ℤ has no ℤ-linear section. A section would send the nonzero class to an integer n with 2n=0, forcing n=0 and contradicting q(n)=1. This elementary example explains why the module feasibility theorem does not automatic","count":16},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"What the algorithm does not promise. The reference solver is deliberately finite and exhaustive. With t=|J|, there are 2^t candidate subsets. For a subset of size s, each residual has at most |U| |𝕜|^s candidate coefficient and correction assignments to examine. Summing this upper bound over all subsets gives |I| |U|∑_s=0^tbinom ts|𝕜|^s =|I| |U|(1+","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-fa191ac2fb86a5b6","tex":"|^s","aliases":"|^s","display":"|^s","kind":"use","excerpt":"What the algorithm does not promise. The reference solver is deliberately finite and exhaustive. With t=|J|, there are 2^t candidate subsets. For a subset of size s, each residual has at most |U| |𝕜|^s candidate coefficient and correction assignments to examine. Summing this upper bound over all subsets gives |I| |U|∑_s=0^tbinom ts|𝕜|^s =|I| |U|(1+","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"For example, 10^2-2=98=7^2·2, and the next digit satisfies 2+2·10·2=42. Hence 108=10+7^2·2, 108^2-2=7^3·34. The new digit corrects the residual quotient at precisely the next required modulus.","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"[Attainment on the obstruction span] Let ι_O:O↪ Q_D be inclusion. There exists a linear map E:O→ V such that q_DE=ι_O and E repairs the family. Choose the section s from [source reference] and set E=sι_O. Then q_DE=q_Dsι_O=ι_O. Every z∈ O is therefore in im(q_DE). Family feasibility follows from [source reference]. formalexists_attaining_repair","count":10},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-700d5ce60eb6edd7","tex":"\\iota_A","aliases":"ι_A iota","display":"\\iota_A","kind":"use","excerpt":"linear map E:O→ V satisfying im E⊆ A, q_DE=ι_O, and repairing the entire family. If V is finite dimensional, its added dimension is optimal even among all unrestricted repairs. Consider the surjection f:A⟶ q_D(A), a⟼ q_D(a). Choose a linear section s_A:q_D(A)→ A. Let j:O↪ q_D(A) and ι_A:A↪ V be inclusions. Define E=ι_As_Aj. Its image is in A by con","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-adaa9f797481f128","tex":"\\iota_O","aliases":"ι_O iota","display":"\\iota_O","kind":"use","excerpt":"[Attainment on the obstruction span] Let ι_O:O↪ Q_D be inclusion. There exists a linear map E:O→ V such that q_DE=ι_O and E repairs the family. Choose the section s from [source reference] and set E=sι_O. Then q_DE=q_Dsι_O=ι_O. Every z∈ O is therefore in im(q_DE). Family feasibility follows from [source reference]. formalexists_attaining_repair","count":7},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"The repair optimizer is bestRepairMenu =bestByCheck( S↦allowed(S)wedgecheck(S), c, L_J).","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"[Attainment on the obstruction span] Let ι_O:O↪ Q_D be inclusion. There exists a linear map E:O→ V such that q_DE=ι_O and E repairs the family. Choose the section s from [source reference] and set E=sι_O. Then q_DE=q_Dsι_O=ι_O. Every z∈ O is therefore in im(q_DE). Family feasibility follows from [source reference]. formalexists_attaining_repair","count":3},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"[Exact obstruction criterion] For every r∈ V, ∃ u∈ U (Du=-r) Longleftrightarrow q_D(r)=0. Applying q_D to Du=-r gives 0=-q_D(r), hence q_D(r)=0. Conversely, q_D(r)=0 means r∈im D. Choose v∈ U with Dv=r. Then D(-v)=-r. formalcorrectable_iff_obstruction_zero","count":8},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"∅ · 0 · No","count":6},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"[Column-mask identity] For every row Y and column j, (Ywidetilde E_S)_0j= (YE)_0j, · j∈ S, 0, · j∉ S. The mask either retains the entire jth column or replaces it by zero. Expanding the finite matrix product gives the two cases. formalKomorebiRepair.RationalCertificates.mul_masked_entry","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Finite menus and conservation. A menu is an actual coefficient map. Return to a commutative ring 𝕜 and modules U,V. Let e:J→ V be a family of candidate directions and let S⊆ J be finite. A coefficient vector is a function a:S→𝕜. Define E_S:𝕜^S⟶ V, E_S(a)=∑_j∈ Sa_je_j. Finite addition and scalar multiplication show that E_S is linear. The coefficien","count":8},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-41acaa92cb8466b8","tex":"\\sum_{j\\in S}","aliases":"∑_j∈ S sum in","display":"\\sum_{j\\in S}","kind":"use","excerpt":"Finite menus and conservation. A menu is an actual coefficient map. Return to a commutative ring 𝕜 and modules U,V. Let e:J→ V be a family of candidate directions and let S⊆ J be finite. A coefficient vector is a function a:S→𝕜. Define E_S:𝕜^S⟶ V, E_S(a)=∑_j∈ Sa_je_j. Finite addition and scalar multiplication show that E_S is linear. The coefficien","count":7},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-0ff97622b5eb39b8","tex":"\\sum_{s=0}^t","aliases":"∑_s=0^t sum","display":"\\sum_{s=0}^t","kind":"use","excerpt":"What the algorithm does not promise. The reference solver is deliberately finite and exhaustive. With t=|J|, there are 2^t candidate subsets. For a subset of size s, each residual has at most |U| |𝕜|^s candidate coefficient and correction assignments to examine. Summing this upper bound over all subsets gives |I| |U|∑_s=0^tbinom ts|𝕜|^s =|I| |U|(1+","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-6f333436e7d549b7","tex":"\\ast","aliases":"∗ ast","display":"\\ast","kind":"use","excerpt":"Nonnegative additive costs are monotone under inclusion: A⊆ T Longrightarrow c(A)≤ c(T). Therefore, when an incumbent repair has cost c_∗, the finite check c_∗≤ c(A) excludes every strictly cheaper repair in I(A,B). It does not say the interval is infeasible. It only says that it cannot improve the incumbent. formalKomorebiRepair.RationalCertificat","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"This condition is equivalent to ker B∩ O=0. It does not require injectivity on all of Q_D when only a smaller family of obstruction classes matters.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"[Exact branch partition] If j∈ B∖ A, then I(A,B)= I(A,B∖{j}) sqcup I(A∪{j},B). Every T∈ I(A,B) either excludes j or includes j. In the first case it belongs to the left interval, and in the second it belongs to the right interval. Conversely, each child interval is contained in the parent. Their membership conditions on j are incompatible, so the u","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"[Availability of lower certificates] Let O be the span of the residual classes in ℚ^m/im D, and let t=dim O. Then matrices H,Q satisfying [source reference] exist. Choose coordinates O≅ℚ^t, extend that map to the whole quotient, and compose with the quotient map to obtain H. Then HD=0. The columns of HR span ℚ^t, so HR has a linear right inverse Q.","count":1},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"An exact seven-adic trace. For x^2=2 at p=7, start with x_1=3. At each step, let c_k=fracx_k^2-27^k, c_k+2x_kt_k≡0pmod7, x_k+1=x_k+7^kt_k. The following table records normalized choices for this example. Every displayed digit equation is an exact integer calculation.","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"The quadratic lifting theorem. [One higher precision] Let p,x,a∈ℤ and k∈ℕ with k≥1. Assume p^kmid x^2-a, gcd(2x,p)=1. Then there exists t∈ℤ such that p^k+1mid (x+p^kt)^2-a. Choose c∈ℤ with x^2-a=p^kc. Coprimality supplies B ezout data for 2x and p, so [source reference] gives t with pmid c+2xt. Since k≥1, one has pmid p^k. Apply [source reference] ","count":2},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"Let 𝕜 be a commutative ring. All modules are unital 𝕜-modules and all maps in this chapter are 𝕜-linear. Fix D:U⟶ V, and let r:I→ V be a family of residuals. The index set I need not be finite for the algebraic statements. When finite enumeration is used later, finiteness will be imposed explicitly.","count":5},{"work":"constructive-repair","url":"notation/constructive-repair.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[Attainment inside a legal subspace] If O⊆ q_D(A), there exists a linear map E:O→ V satisfying im E⊆ A, q_DE=ι_O, and repairing the entire family. If V is finite dimensional, its added dimension is optimal even among all unrestricted repairs. Consider the surjection f:A⟶ q_D(A), a⟼ q_D(a). Choose a linear section s_A:q_D(A)→ A. Let j:O↪ q_D(A) and ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"notation-table","excerpt":"midrule V,F_v,f^* · A declared view category, its fibers, and contravariant restrictions.","count":134},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.","count":1293},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.","count":243},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"notation-table","excerpt":"c_f,a_v · Coherent nonabelian transition labels and gauge values.","count":446},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-eae1dc41a7db82ef","tex":"a_v","aliases":"a_v","display":"a_v","kind":"notation-table","excerpt":"c_f,a_v · Coherent nonabelian transition labels and gauge values.","count":15},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-34b9981d517fb6b9","tex":"a_\\alpha","aliases":"a_α alpha","display":"a_\\alpha","kind":"notation-table","excerpt":"β_T,r(M),a_α · Minimum residual at correction budget M, and the quadratic regularized correction.","count":60},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"notation-table","excerpt":"χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.","count":242},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"notation-table","excerpt":"E_b(R) · Realizations equipped with an actual identification of their boundary with b.","count":179},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"notation-table","excerpt":"ω,Q_n,C_n,b_n · A residual, its prefix-product rate, movement coefficient, and summable step budget. Here Q_n is not the dyadic remainder operator Q_N.","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"notation-table","excerpt":"C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.","count":190},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"notation-table","excerpt":"c_f,a_v · Coherent nonabelian transition labels and gauge values.","count":170},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cdae72f972852da1","tex":"c_f","aliases":"c_f","display":"c_f","kind":"notation-table","excerpt":"c_f,a_v · Coherent nonabelian transition labels and gauge values.","count":23},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"notation-table","excerpt":"C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cfbf41dac773cbfc","tex":"C_n","aliases":"C_n","display":"C_n","kind":"notation-table","excerpt":"ω,Q_n,C_n,b_n · A residual, its prefix-product rate, movement coefficient, and summable step budget. 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Each target is declared where used.","count":69},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"notation-table","excerpt":"c(x,y),κ(x,y) · The two-bit pair phase and its antisymmetric scalar-gauge invariant.","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"notation-table","excerpt":"ν_B,d_W,d_W^sym · Residual floor, extended directed transport distance, and its symmetric maximum.","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6e8a05f74bb4671f","tex":"\\nu_B","aliases":"ν_B nu","display":"\\nu_B","kind":"notation-table","excerpt":"ν_B,d_W,d_W^sym · Residual floor, extended directed transport distance, and its symmetric maximum.","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"notation-table","excerpt":"π_W,U,H_U · The map forgetting part of the prescription and the corresponding invisible coherent ambiguity.","count":18},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-54a9ed6aa5df92f4","tex":"\\pi_{W,U}","aliases":"π_W,U pi","display":"\\pi_{W,U}","kind":"notation-table","excerpt":"π_W,U,H_U · The map forgetting part of the prescription and the corresponding invisible coherent ambiguity.","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"notation-table","excerpt":"ϕ,EventualHom · A monotone source schedule and transport classes modulo eventual equality.","count":86},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"notation-table","excerpt":"χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0bbd4413777b414e","tex":"\\chi_A","aliases":"χ_A chi","display":"\\chi_A","kind":"notation-table","excerpt":"χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"notation-table","excerpt":"ω,Q_n,C_n,b_n · A residual, its prefix-product rate, movement coefficient, and summable step budget. Here Q_n is not the dyadic remainder operator Q_N.","count":33},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"notation-table","excerpt":"δ,∂,d,D · Respectively the view differential, tower differential, a model residual, and a Hilbert defect operator. Each target is declared where used.","count":26},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"notation-table","excerpt":"≃,≃_L · A specified equivalence, and a continuous linear equivalence. Neither notation alone asserts an isometry.","count":17},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b3be7f4201ac0800","tex":"\\simeq_L","aliases":"≃_L simeq","display":"\\simeq_L","kind":"notation-table","excerpt":"≃,≃_L · A specified equivalence, and a continuous linear equivalence. Neither notation alone asserts an isometry.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"definition-passage","excerpt":"[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-06ddbea4003ffa9b","tex":"c_{1_v}","aliases":"c_1_v","display":"c_{1_v}","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-59651fe6df7d2f0f","tex":"c_g","aliases":"c_g","display":"c_g","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c04b92df09f891e3","tex":"c_{g\\circ f}","aliases":"c_g∘ f circ","display":"c_{g\\circ f}","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e84b1a98b5a23eb6","tex":"\\operatorname{ev}","aliases":"ev operatorname","display":"\\operatorname{ev}","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-db44ffded3c6918e","tex":"\\operatorname{ev}_v","aliases":"ev_v operatorname","display":"\\operatorname{ev}_v","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-29f81858f6292043","tex":"g^*","aliases":"g^*","display":"g^*","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8bea312cd06acfdb","tex":"G_v","aliases":"G_v","display":"G_v","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id","display":null,"kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob","display":null,"kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f814b60f967151c4","tex":"\\mathrm{op}","aliases":"op mathrm","display":"\\mathrm{op}","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2d2757f1a3838f81","tex":"\\Set","aliases":"Set","display":null,"kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0692c76d8494573e","tex":"T_c","aliases":"T_c","display":"T_c","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4ac0da71d791b586","tex":"\\V^{\\mathrm{op}}","aliases":"V^op V mathrm","display":null,"kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":96},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":18},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9d5ae088e190bcb8","tex":"x_w","aliases":"x_w","display":"x_w","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"definition-passage","excerpt":"[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"definition-passage","excerpt":"[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.","count":33},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":187},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":254},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ba20de831f2d9fe6","tex":"\\prod_{v\\in\\Ob\\V}","aliases":"∏_v∈ObV prod in Ob V","display":null,"kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":26},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":392},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0014cfc2591bdee6","tex":"a_*","aliases":"a_*","display":"a_*","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":13},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":47},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0083f442b19187bb","tex":"A_v","aliases":"A_v","display":"A_v","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8bed5507d096707b","tex":"B^{\\mathsf T}","aliases":"B^ T mathsf","display":"B^{\\mathsf T}","kind":"introduction-passage","excerpt":"The residual of the coordinate equation is a covector. In finite dimensions, use orthonormal coordinates in the ambient candidate and residual spaces. Let the columns of B form an arbitrary basis of H=ker p, not necessarily an orthonormal one. Write a=Bc, S=dB, and G=B^ TB. As before, G is positive definite. For an inexact coefficient vector c, the","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1a59e813e505c83b","tex":"B_U","aliases":"B_U","display":"B_U","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":15},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-69b376b6bbe4d3ac","tex":"b_v","aliases":"b_v","display":"b_v","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-edcd2529ad4f495e","tex":"B_X","aliases":"B_X","display":"B_X","kind":"introduction-passage","excerpt":"emma[Continuous inverse for Banach spaces] Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0df0443ecd4fe0fb","tex":"c^q","aliases":"c^q","display":"c^q","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":81},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-115ca4a8b4972e65","tex":"\\operatorname{ev}_n","aliases":"ev_n operatorname","display":"\\operatorname{ev}_n","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6d014ef8b92c530c","tex":"f^{(m)}","aliases":"f^(m)","display":"f^{(m)}","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e3f4ba757b6d2cf8","tex":"f_n","aliases":"f_n","display":"f_n","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":52},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"{0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lemma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆overlineS(nB_X). Here B(y_0,ρ) is an open ball. If normh<ρ, both y_0+h and y_0 b","count":145},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":99},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f18b3c49fd8055ae","tex":"i_0","aliases":"i_0","display":"i_0","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d529fadc23d1fc12","tex":"i_k","aliases":"i_k","display":"i_k","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-56e19bddcbcb5ec7","tex":"i_{k+1}","aliases":"i_k+1","display":"i_{k+1}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d822ff9312b40594","tex":"i_{k-1}","aliases":"i_k-1","display":"i_{k-1}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2ee8ac8c8d5446ba","tex":"i_m","aliases":"i_m","display":"i_m","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-45e9122f2e536e8b","tex":"I_{n,m}","aliases":"I_n,m","display":"I_{n,m}","kind":"introduction-passage","excerpt":"8 Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6a50e5637ed9a812","tex":"I_{n,m+1}","aliases":"I_n,m+1","display":"I_{n,m+1}","kind":"introduction-passage","excerpt":"nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4dab5e48ba9132bb","tex":"I_{n,s_n}","aliases":"I_n,s_n","display":"I_{n,s_n}","kind":"introduction-passage","excerpt":"roposition Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently, so only finitely many strict decreases are possible. There is an index s_n≥ n such that I_n,m=I_n,s_n for all m≥ s_n. Consequen","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im","display":null,"kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":147},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c4748b1ae1b9971c","tex":"L^m","aliases":"L^m","display":"L^m","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7054bd921fca19d4","tex":"L^\\perp","aliases":"L^⊥ perp","display":"L^\\perp","kind":"introduction-passage","excerpt":"For a closed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ed76d6b3d182baf7","tex":"L_n","aliases":"L_n","display":"L_n","kind":"introduction-passage","excerpt":"nly if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"introduction-passage","excerpt":"[Dyadic determination and finite coordinate recovery] In the actual dyadic system, all detail coordinates jointly determine the residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1 · =S_2M_1+M_2C_1. All terms map the first residual space into the last candidate space, in the written order. Finally, for fixed bounded ","count":13},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1 · =S_2M_1+M_2C_1. All terms map the first residual space into the last candidate space, in the written order. Finally, for fixed bounded ","count":14},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8200e811d73eda66","tex":"m_m^2","aliases":"m_m^2","display":"m_m^2","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ce78c2a469d42dfa","tex":"m_n","aliases":"m_n","display":"m_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a42932dc6aaac88d","tex":"m_n^2","aliases":"m_n^2","display":"m_n^2","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N","display":null,"kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":334},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-26e4083a07ea4a31","tex":"\\overline{S(nB_X)}","aliases":"overlineS(nB_X) overline","display":"\\overline{S(nB_X)}","kind":"introduction-passage","excerpt":":inverse-tool Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lem","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f21822f96efaf869","tex":"P_A","aliases":"P_A","display":"P_A","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":33},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-363d67fd271736f2","tex":"P_L","aliases":"P_L","display":"P_L","kind":"introduction-passage","excerpt":"For a closed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7970473ade3c6476","tex":"P_L^2","aliases":"P_L^2","display":"P_L^2","kind":"introduction-passage","excerpt":"sed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2=P_L. For","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ddc0eef4732461d5","tex":"Q_U","aliases":"Q_U","display":"Q_U","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e52a322b6d2467f2","tex":"q_v","aliases":"q_v","display":"q_v","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"introduction-passage","excerpt":"The topology is defined without assuming effectivity. For a composable pair q=(vxrightarrow f wxrightarrow g z), let its support be |q|={v,w,z}. A sieve R on W is covering when |q|⊆ W Longrightarrow (|q|↪ W)∈ R for every composable q. This condition depends only on the view category, not on the coefficient groups or on a promised global realization","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a03f214a14787ce8","tex":"R_j","aliases":"R_j","display":"R_j","kind":"introduction-passage","excerpt":"ery] In the actual dyadic system, all detail coordinates jointly determine the residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the res","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"introduction-passage","excerpt":"[Dyadic determination and finite coordinate recovery] In the actual dyadic system, all detail coordinates jointly determine the residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9993dc43cc574b8d","tex":"r_X","aliases":"r_X","display":"r_X","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ca43c3cdcb5f5a6f","tex":"r_Y","aliases":"r_Y","display":"r_Y","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fe39e36813ced2eb","tex":"S^{\\mathsf T}","aliases":"S^ T mathsf","display":"S^{\\mathsf T}","kind":"introduction-passage","excerpt":"rdinate equation is a covector In finite dimensions, use orthonormal coordinates in the ambient candidate and residual spaces. Let the columns of B form an arbitrary basis of H=ker p, not necessarily an orthonormal one. Write a=Bc, S=dB, and G=B^ TB. As before, G is positive definite. For an inexact coefficient vector c, the residual of the coordin","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-88d1f82215e67d59","tex":"s^{(j)}_{2m}","aliases":"s^(j)_2m","display":"s^{(j)}_{2m}","kind":"introduction-passage","excerpt":"residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0,","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-db9af401ac382039","tex":"S_2","aliases":"S_2","display":"S_2","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-91ec47101c58ab98","tex":"s_n","aliases":"s_n","display":"s_n","kind":"introduction-passage","excerpt":"ery later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently, so only finitely many strict decreases are possible. There is an index s_n≥ n such that I_n,m=I_n,s_n for all m≥ s_n. Con","count":17},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-38dbe99c73eeaafd","tex":"s_X","aliases":"s_X","display":"s_X","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-89069deb6219b9ac","tex":"s_Y","aliases":"s_Y","display":"s_Y","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3efb077b7b4a3c92","tex":"\\sup_n","aliases":"sup_n sup","display":"\\sup_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"introduction-passage","excerpt":"The residual of the coordinate equation is a covector. In finite dimensions, use orthonormal coordinates in the ambient candidate and residual spaces. Let the columns of B form an arbitrary basis of H=ker p, not necessarily an orthonormal one. Write a=Bc, S=dB, and G=B^ TB. As before, G is positive definite. For an inexact coefficient vector c, the","count":28},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"introduction-passage","excerpt":"t a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs t","count":134},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-10d6fa1215b924bf","tex":"T^*","aliases":"T^*","display":"T^*","kind":"introduction-passage","excerpt":"There are three different residuals. Keep H=ker p, T=d|_H, and a residual datum r. Fix α>0. Write a_α for the unique minimizer of Theorem~[source reference] and let a∈ H be any proposed correction. Three quantities now have different meanings: v=r-Ta, g=T^*(Ta-r)+α a, e=a-a_α. The vector v is the fit residual. The vector g is the normal-equation re","count":50},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"introduction-passage","excerpt":"orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norm","count":21},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-79b896ccd56f4035","tex":"T_q","aliases":"T_q","display":"T_q","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"introduction-passage","excerpt":"[The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-72b2bc9e12d0fa8e","tex":"U_k","aliases":"U_k","display":"U_k","kind":"introduction-passage","excerpt":"tion[The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from ","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f6951bc1a235a45d","tex":"U_{k+1}","aliases":"U_k+1","display":"U_{k+1}","kind":"introduction-passage","excerpt":". The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cd263f633c7dde0b","tex":"\\widetilde s","aliases":"widetilde s widetilde","display":"\\widetilde s","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":21},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":112},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2d9731b0ef10fdc5","tex":"x_{i_0}","aliases":"x_i_0","display":"x_{i_0}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e2a78aa8cf2a6130","tex":"x_{i_k}","aliases":"x_i_k","display":"x_{i_k}","kind":"introduction-passage","excerpt":"of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels eac","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a47c3bd800d8c2d7","tex":"x_{i_{k+1}}","aliases":"x_i_k+1","display":"x_{i_{k+1}}","kind":"introduction-passage","excerpt":"of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels eac","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fe618b7cfb9110a0","tex":"x_{i_m}","aliases":"x_i_m","display":"x_{i_m}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f3be737b42076753","tex":"X_m","aliases":"X_m","display":"X_m","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7a7064d4243ed8b7","tex":"X_{m+1}","aliases":"X_m+1","display":"X_{m+1}","kind":"introduction-passage","excerpt":"nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":16},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-98f50cf2afdfc449","tex":"X_{n+1}","aliases":"X_n+1","display":"X_{n+1}","kind":"introduction-passage","excerpt":"We next prove that restriction induces a surjection Y_n+1→ Y_n, even though X_n+1→ X_n need not be surjective. For membership, let z∈ Y_n+1. Given m≥ n+1, write z=ρ_n+1,m(w) for some w∈ X_m. Then ρ_n z=ρ_n,m(w). Together with the automatic condition at m=n, this shows ρ_nz∈ Y_n.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-caa3e2ac7047f01e","tex":"x_v^{-1}","aliases":"x_v^-1","display":"x_v^{-1}","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f9adc8efb1f46cfd","tex":"X_{\\phi(n)}","aliases":"X_ϕ(n) phi","display":"X_{\\phi(n)}","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":89},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"introduction-passage","excerpt":"with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lemma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f63c0b46cff9ca02","tex":"y_m","aliases":"y_m","display":"y_m","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3859ee776b04fc90","tex":"Y_n","aliases":"Y_n","display":"Y_n","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":16},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-00fe9b383f124db5","tex":"y_n","aliases":"y_n","display":"y_n","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3f18b642e61a8904","tex":"Y_{n+1}","aliases":"Y_n+1","display":"Y_{n+1}","kind":"introduction-passage","excerpt":"We next prove that restriction induces a surjection Y_n+1→ Y_n, even though X_n+1→ X_n need not be surjective. For membership, let z∈ Y_n+1. Given m≥ n+1, write z=ρ_n+1,m(w) for some w∈ X_m. Then ρ_n z=ρ_n,m(w). Together with the automatic condition at m=n, this shows ρ_nz∈ Y_n.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0,","count":128},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-42dba447a7ef210c","tex":"z_m","aliases":"z_m","display":"z_m","kind":"introduction-passage","excerpt":"residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-365f822c88edb476","tex":"|_L","aliases":"|_L","display":"|_L","kind":"introduction-passage","excerpt":"chapQuantitative descent in a common Hilbert model Local exactness and matching are separate operators. Let E be a real Hilbert space of raw local candidates. Let D:E→ R and C:E→ Y be continuous linear maps into Banach spaces, measuring local equation defects and overlap mismatch, respectively. Put L=ker D, B=C|_L, M=ker D∩ker C. The exact matched ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":13},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":17},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":80},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-61e9838ce05b2305","tex":"\\rho^X_{i,j}","aliases":"ρ^X_i,j rho","display":"\\rho^X_{i,j}","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3f6a68400896d8e4","tex":"\\rho^X_{\\phi(i),\\phi(j)}","aliases":"ρ^X_ϕ(i),ϕ(j) rho phi phi","display":"\\rho^X_{\\phi(i),\\phi(j)}","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-12f1c3dae45bea10","tex":"\\rho^Y_{i,j}","aliases":"ρ^Y_i,j rho","display":"\\rho^Y_{i,j}","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bcdc1fcd14dba16f","tex":"\\rho_{i,i}","aliases":"ρ_i,i rho","display":"\\rho_{i,i}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-919c94d29949aa9a","tex":"\\rho_{i,j}","aliases":"ρ_i,j rho","display":"\\rho_{i,j}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f9ea4d96db178cea","tex":"\\rho_{i,k}","aliases":"ρ_i,k rho","display":"\\rho_{i,k}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0019b82ccb43f3b4","tex":"\\rho_{j,k}","aliases":"ρ_j,k rho","display":"\\rho_{j,k}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-18406af98fd43046","tex":"\\rho_m","aliases":"ρ_m rho","display":"\\rho_m","kind":"introduction-passage","excerpt":"nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c03851211fa61002","tex":"\\rho_n","aliases":"ρ_n rho","display":"\\rho_n","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0da056617ca0ff1c","tex":"\\rho_{n+1,m}","aliases":"ρ_n+1,m rho","display":"\\rho_{n+1,m}","kind":"introduction-passage","excerpt":"We next prove that restriction induces a surjection Y_n+1→ Y_n, even though X_n+1→ X_n need not be surjective. For membership, let z∈ Y_n+1. Given m≥ n+1, write z=ρ_n+1,m(w) for some w∈ X_m. Then ρ_n z=ρ_n,m(w). Together with the automatic condition at m=n, this shows ρ_nz∈ Y_n.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-98944c463486b854","tex":"\\rho_{n,m}","aliases":"ρ_n,m rho","display":"\\rho_{n,m}","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-714e0157858a58aa","tex":"\\rho_{\\phi(n),\\phi(m)}","aliases":"ρ_ϕ(n),ϕ(m) rho phi phi","display":"\\rho_{\\phi(n),\\phi(m)}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":14},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f0f378bf6e9cc5de","tex":"\\tau_{i_0i_1}","aliases":"τ_i_0i_1 tau","display":"\\tau_{i_0i_1}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-744199ca7fcbd805","tex":"\\tau_{i_{k-1}i_k}","aliases":"τ_i_k-1i_k tau","display":"\\tau_{i_{k-1}i_k}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5ea924893eca4bcf","tex":"\\tau_{i_ki_{k+1}}","aliases":"τ_i_ki_k+1 tau","display":"\\tau_{i_ki_{k+1}}","kind":"introduction-passage","excerpt":". The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":13},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t","count":51},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"introduction-passage","excerpt":"orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norm","count":18},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":18},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"introduction-passage","excerpt":"The topology is defined without assuming effectivity. For a composable pair q=(vxrightarrow f wxrightarrow g z), let its support be |q|={v,w,z}. A sieve R on W is covering when |q|⊆ W Longrightarrow (|q|↪ W)∈ R for every composable q. This condition depends only on the view category, not on the coefficient groups or on a promised global realization","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4289d0aabfe27afc","tex":"\\prod_{v\\in U}","aliases":"∏_v∈ U prod in","display":"\\prod_{v\\in U}","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":46},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b745b62212003b48","tex":"\\sum_{k=0}^{m-1}","aliases":"∑_k=0^m-1 sum","display":"\\sum_{k=0}^{m-1}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":62},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"introduction-passage","excerpt":"chapQuantitative descent in a common Hilbert model Local exactness and matching are separate operators. Let E be a real Hilbert space of raw local candidates. Let D:E→ R and C:E→ Y be continuous linear maps into Banach spaces, measuring local equation defects and overlap mismatch, respectively. Put L=ker D, B=C|_L, M=ker D∩ker C. The exact matched ","count":13},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":287},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":91},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":24},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"introduction-passage","excerpt":"For a closed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2","count":33},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e519c34da7accd5d","tex":"\\bigcap_{m\\ge n}","aliases":"⋂_m≥ n bigcap ge","display":"\\bigcap_{m\\ge n}","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9c20590cf5841569","tex":"\\bigcap_n","aliases":"⋂_n bigcap","display":"\\bigcap_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"introduction-passage","excerpt":":inverse-tool Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lem","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ad4b0793f8d2d37d","tex":"\\bigcup_{n\\ge1}","aliases":"⋃_n≥1 bigcup ge","display":"\\bigcup_{n\\ge1}","kind":"introduction-passage","excerpt":":inverse-tool Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lem","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","excerpt":"orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norm","count":47},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","excerpt":"orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norm","count":47},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"Two boundary examples. On ℓ^∞(N,R), let C remove the first coordinate and let J prepend zero. Then Q_Nx is the tail of x, placed at its original coordinates with zeros before N. Every fixed coordinate is eventually zero in the remainder, and normQ_N≤1. Nevertheless the constant sequence 1 satisfies normQ_N1=1 for every N.","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-232f12a78d23a1e5","tex":"\\mathbf1^{\\mathsf T}","aliases":"1^ T mathbf mathsf","display":"\\mathbf1^{\\mathsf T}","kind":"use","excerpt":"-1 · 0 · -1. The ordinary cycle block satisfies A^ TA=3I-11^ T. It is zero on the constant line and equals 3I on its orthogonal complement. The reflected block satisfies B^ TB=I+vv^ T, v=(1,-1,1)^ T. Since norm v^2=3, this matrix has eigenvalue four in the direction of v and eigenvalue one on its orthogonal complement. In particular it is positive ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-03eeee33e0ea1dca","tex":"\\mathbf1_N","aliases":"1_N mathbf","display":"\\mathbf1_N","kind":"use","excerpt":"A complete calculation of the finite repairs. Let 1_N be one in coordinates 1,…,N and zero elsewhere. Put a_N=( 1_N,0), f_N=f_0-a_N=(- 1_N,1). Then p(f_N)=1, and the first N residual coordinates vanish exactly. The remaining residual and movement satisfy d(f_N)_n=0 · n≤ N,","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"Constructing the new certificate. Assume the target R is Banach and A:R→ E is a continuous linear right inverse of D, with DA=I, norm A≤ a, and aε<1. Set T=(D-D')A, Z=(I-T)^-1, A'=AZ. The inverse is supplied by the convergent Neumann series because norm T≤ aε<1.","count":14},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-92312bec7dc0bc70","tex":"a'_{ij}","aliases":"a'_ij","display":"a'_{ij}","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-54034542483dca3c","tex":"A^{\\mathsf T}","aliases":"A^ T mathsf","display":"A^{\\mathsf T}","kind":"use","excerpt":"-1 · 0 · -1. The ordinary cycle block satisfies A^ TA=3I-11^ T. It is zero on the constant line and equals 3I on its orthogonal complement. The reflected block satisfies B^ TB=I+vv^ T, v=(1,-1,1)^ T. Since norm v^2=3, this matrix has eigenvalue four in the direction of v and eigenvalue one on its orthogonal complement. In particular it is positive ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-40d113efdc774b45","tex":"a^{(m)}","aliases":"a^(m)","display":"a^{(m)}","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-13c13d5b2dd869b7","tex":"a^{(m)}_n","aliases":"a^(m)_n","display":"a^{(m)}_n","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"For the groupoid, an arrow a:c→ d means the equation a· c=d. If b:d→ e, then (ba)· c=b·(a· c)=e, giving composition. The identity label is 1 and the inverse label is a^-1. Associativity and the two inverse identities now follow from the corresponding multiplication laws in the product group. The label ba is therefore forced by the order of applicat","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"stable repair requires a stable matching problem Suppose ran D and ran B are closed, with residual gains a,b. Then norm u≤ anormDx, norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum no","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-004a43f65c127c71","tex":"a^\\dagger","aliases":"a^† dagger","display":"a^\\dagger","kind":"use","excerpt":"[The regularization limit alternative] As αdownarrow0, Ta_α⟶ b, normr-Ta_α⟶ q, αnorma_α^2⟶0. If b∈ran T, let a^† be its unique minimum norm preimage. Then a_α⟶ a^† in norm. If b∉ran T, then norma_α→∞. Step 1: identify the unavoidable component. Since r-b⊥overlineran T, the adjoint identity gives T^*(r-b)=0. Thus a_α(r)=a_α(b). For every correction ","count":20},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1db92fcd3339ffa0","tex":"a^\\dagger_n","aliases":"a^†_n dagger","display":"a^\\dagger_n","kind":"use","excerpt":"The same operator has an attainable datum. Change only the residual to b_n=1/n^2. Its minimum norm preimage is a^†_n=1/n∈ℓ^2. At α=m^-2, the regularized correction is c^(m)_n=fracm^2n(m^2+n^2), a^†_n-c^(m)_n=fracnm^2+n^2. The bias square is exactly the previously computed B_m. This links the two cases without conflating their existence hypotheses: ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The auto","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"Suppose now that A_1⊇ A_2⊇⋯ are nonempty closed affine subsets of a real Hilbert correction space H. A typical A_n consists of corrections satisfying the first n bounded linear observation equations. The sets are affine because the equations are linear, and closed because the observations are continuous.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The auto","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2941a7c766320c3c","tex":"a_1'","aliases":"a_1'","display":"a_1'","kind":"use","excerpt":"s a_1,a_2 at budgets M_1,M_2 and t∈[0,1]. The candidate a_t=ta_1+(1-t)a_2 satisfies norma_t≤ tM_1+(1-t)M_2, and its residual is t(r-Ta_1)+(1-t)(r-Ta_2). The triangle inequality therefore gives β(tM_1+(1-t)M_2)≤ tβ(M_1)+(1-t)β(M_2). Monotonicity follows from inclusion of balls. If 0≤ M_1≤ M_2 and M_2>0, choose a minimizer a_2 at M_2 and put a_1'=(M_","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bd409829ef36d868","tex":"a_1^{-1}","aliases":"a_1^-1","display":"a_1^{-1}","kind":"use","excerpt":"Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The auto","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"For convexity, take minimizers a_1,a_2 at budgets M_1,M_2 and t∈[0,1]. The candidate a_t=ta_1+(1-t)a_2 satisfies norma_t≤ tM_1+(1-t)M_2, and its residual is t(r-Ta_1)+(1-t)(r-Ta_2). The triangle inequality therefore gives β(tM_1+(1-t)M_2)≤ tβ(M_1)+(1-t)β(M_2). Monotonicity follows from inclusion of balls. If 0≤ M_1≤ M_2 and M_2>0, choose a minimize","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"2. A residual chart can change without changing the defect. Let d:R^2→R be d(u,r)=r. For a∈R, let A_a(r)=(ar,r). Compute the exact repair P_a, compare it with P_b, and decide when they commute.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6cfcb2dc290af84d","tex":"a_a","aliases":"a_a","display":"a_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-68421187e198542c","tex":"a_c","aliases":"a_c","display":"a_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-41ae46808180b866","tex":"a_e","aliases":"a_e","display":"a_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dc727ddbdff7e587","tex":"a_f","aliases":"a_f","display":"a_f","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-487f58499940e0c4","tex":"a_{ii}","aliases":"a_ii","display":"a_{ii}","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-daf800c8ae3d0681","tex":"a_{ij}","aliases":"a_ij","display":"a_{ij}","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1,","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9f6bca5c1c7a0354","tex":"a_{ij}^{-1}","aliases":"a_ij^-1","display":"a_{ij}^{-1}","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ae3cae372c62870b","tex":"a_{ik}","aliases":"a_ik","display":"a_{ik}","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1, a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-599f350727f7f626","tex":"a_{ji}","aliases":"a_ji","display":"a_{ji}","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0b50cb77343ec39b","tex":"a_{jk}","aliases":"a_jk","display":"a_{jk}","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1, a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B","count":16},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ba209639e02504e4","tex":"a_M","aliases":"a_M","display":"a_M","kind":"use","excerpt":"To prove sharpness, let a_M be an attaining correction from Theorem~[source reference] and put v=r-Ta_M. If v=0, then β(M)=0 and z=0 attains the dual bound. Assume henceforth v≠0 and set z=v/norm v.","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e66d0e178e1abc69","tex":"A_{m+1}","aliases":"A_m+1","display":"A_{m+1}","kind":"use","excerpt":"Step 3: move from regularization parameters to arbitrary budgets. Set M_m=sqrtA_m. Every positive coordinate m^2/(m^2+n^2) increases strictly with m, so A_m, and hence M_m, increases strictly. Since A_m/m→ c=π/4>0, these budgets tend to infinity and (fracM_m+1M_m)^2 =fracA_m+1/(m+1)A_m/mfracm+1m⟶1. Proposition~[source reference] identifies β(M_m)=s","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ffad8741cff39b12","tex":"A_{m,K}","aliases":"A_m,K","display":"A_{m,K}","kind":"use","excerpt":"s] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-758554b79826abfb","tex":"a_{m^{-2}}","aliases":"a_m^-2","display":"a_{m^{-2}}","kind":"use","excerpt":"lta with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By Proposition~[source reference], norma_m^-2(b)-a^†≤sqrt2/m. Theorem~[source reference] at the same ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-07112285aabb492d","tex":"a_N","aliases":"a_N","display":"a_N","kind":"use","excerpt":"A complete calculation of the finite repairs. Let 1_N be one in coordinates 1,…,N and zero elsewhere. Put a_N=( 1_N,0), f_N=f_0-a_N=(- 1_N,1). Then p(f_N)=1, and the first N residual coordinates vanish exactly. The remaining residual and movement satisfy d(f_N)_n=0 · n≤ N,","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0c376ef200c86635","tex":"a_n^{(m)}","aliases":"a_n^(m)","display":"a_n^{(m)}","kind":"use","excerpt":"fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and w=(1/n)∈ℓ^2, hence the residual sequence i","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cba9f3864736fa91","tex":"a_n^0","aliases":"a_n^0","display":"a_n^0","kind":"use","excerpt":"displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve the intended-data problem. All three corrections, including the pro","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-803b820dd09c733f","tex":"a_{n_k}","aliases":"a_n_k","display":"a_{n_k}","kind":"use","excerpt":"The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 b","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-db660538430f2d4b","tex":"A_p","aliases":"A_p","display":"A_p","kind":"use","excerpt":"Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a9457f85e2123199","tex":"a_p","aliases":"a_p","display":"a_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"For convexity, take minimizers a_1,a_2 at budgets M_1,M_2 and t∈[0,1]. The candidate a_t=ta_1+(1-t)a_2 satisfies norma_t≤ tM_1+(1-t)M_2, and its residual is t(r-Ta_1)+(1-t)(r-Ta_2). The triangle inequality therefore gives β(tM_1+(1-t)M_2)≤ tβ(M_1)+(1-t)β(M_2). Monotonicity follows from inclusion of balls. If 0≤ M_1≤ M_2 and M_2>0, choose a minimize","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4f1eee202dc651a9","tex":"A_u","aliases":"A_u","display":"A_u","kind":"use","excerpt":"Three maps that must not be confused. Keep the additive view system of Section~[source reference]. For a prescribed domain U, there are three distinct maps: p_U:E⟶∏_u∈ UA_u, r_U:Γ(A)⟶Γ_U(A), o_U:Γ_U(A)⟶ Q_U. The first restricts a raw family, whether or not it is coherent. The second restricts a global coherent family. The third detects whether an a","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-84e075bd0dff7e4d","tex":"a_u","aliases":"a_u","display":"a_u","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a6b72d71c89e03f5","tex":"a_v^{-1}","aliases":"a_v^-1","display":"a_v^{-1}","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1)","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-509b6f92fab6cf2c","tex":"a_w","aliases":"a_w","display":"a_w","kind":"use","excerpt":"a coordinate preserving equivalence Γ(B_a,b)≃ B(a,b). Under a transition cover, matching local bridge sections glue to a unique global bridge with both prescribed projections unchanged. We must first check that the displayed fibers really form a view system. Let m∈ B_a,b(w) and f:v→ w. Naturality of p and coherence of a give p_v(f_M^*m)=f_F^*p_w(m)","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9f9cdba8dad3a848","tex":"a_z","aliases":"a_z","display":"a_z","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitu","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-edac8f4090c64b8b","tex":"A_\\alpha","aliases":"A_α alpha","display":"A_\\alpha","kind":"use","excerpt":"gularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-753ee05230ab589b","tex":"a_{\\alpha(\\delta)}","aliases":"a_α(δ) alpha delta","display":"a_{\\alpha(\\delta)}","kind":"use","excerpt":"For the limiting statement, insert the exact-data correction at the selected parameter: norma_α(δ)(widetilde r)-a^† ≤fracδ2sqrtα(δ) +norma_α(δ)(r)-a^†. The first term tends to zero by the noise/parameter hypothesis. The second tends to zero by the attained-projection case of Theorem~[source reference], because the selected parameter tends to zero. ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b10de445db1a0c15","tex":"a_{\\alpha_n}","aliases":"a_α_n alpha","display":"a_{\\alpha_n}","kind":"use","excerpt":"Step 5: in the unattained case the norms tend to infinity. If b∉ran T and the norms failed to tend to infinity, there would exist M<∞ and positive α_n→0 with norma_α_n≤ M. Their images still tend to b by Step 2. Theorem~[source reference] would then put b in the actual range, a contradiction. This proves divergence of the norms, not just unboundedn","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-92645d9edf80cd55","tex":"a_{\\alpha_{n_k}}","aliases":"a_α_n_k alpha","display":"a_{\\alpha_{n_k}}","kind":"use","excerpt":"Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity giv","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7d7d4c82e2b393ad","tex":"A_\\gamma","aliases":"A_γ gamma","display":"A_\\gamma","kind":"use","excerpt":"a_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel ","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9fdf6a2a563742c0","tex":"a_\\gamma","aliases":"a_γ gamma","display":"a_\\gamma","kind":"use","excerpt":"Parameter monotonicity and the fixed-budget problem. [Regularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-75bb730778c58ee6","tex":"a_{\\kappa(v),i}","aliases":"a_κ(v),i kappa","display":"a_{\\kappa(v),i}","kind":"use","excerpt":"rded. Step 1: normalize the supplied identifications. The triple-overlap equation with all indices equal gives a_ii=a_iia_ii, hence a_ii=1. With indices i,j,i it gives a_jia_ij=1, hence a_ji=a_ij^-1. Composition coverage implies object coverage by inserting identity arrows. Choose a containing patch κ(v) for each view and put b_i(v)=a_κ(v),i(v) (v∈","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-01478c4149c7701c","tex":"\\Aut","aliases":"Aut","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] fill[KomorebiSoft!65,rounded corners=7pt] (-6.7,-.8) rectangle (-1.3,3.2); fill[KomorebiSoft!65,rounded corners=7pt] (.1,-.8) rectangle (6.7,3.2); node[box,circle,minimum size=1cm] (id) at (-4,1.15) 1; draw[arr] (id) to[loop above,min distance=12mm] (id); node[lab] at (-4,2.7) Aut(1)=S_3; node[lab] at (-4,-.25) identity cocycle; node[","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ea0b1d9a333c4d54","tex":"\\mathfrak B","aliases":"B mathfrak","display":"\\mathfrak B","kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e77f9eceea598e09","tex":"B'","aliases":"B'","display":"B'","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"use","excerpt":"Replacing b by an isomorphic boundary b' also gives an equivalence, by composing the identification with b→ b'. For R=id, the marked realization groupoid has center (b,1_b), and between (x,ϕ) and (y,ψ) there is exactly one compatible arrow ψ^-1ϕ. This is a useful example of contractible realization data, not a claim that the original groupoid has n","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7a0bd25f60ee2ba9","tex":"b'_i","aliases":"b'_i","display":"b'_i","kind":"use","excerpt":"The correct uniqueness statement. Suppose (c,b_i) and (c',b'_i) realize the same descent datum. The only possible compatible comparison is g_v=b'_i(v)^-1b_i(v), using any patch through v. The overlap identities make it independent of the patch, and b'_ig=b_i. It is therefore the unique global comparison respecting the fixed local identifications.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"The inverse therefore exists. Its linearity follows from uniqueness of preimages, and its bound is normB^-1y≤α^-1norm y. This construction does not assume closedness of ran T. It uses the positive term α I in the normal operator.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"stable repair requires a stable matching problem Suppose ran D and ran B are closed, with residual gains a,b. Then norm u≤ anormDx, norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum no","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"A failed triple. Let t≠1 be an involution and ask for frames b_0,b_1,b_2 satisfying b_1=tb_0, b_2=tb_1, b_2=tb_0. The first two give b_2=b_0, while the third forces t=1. Each displayed transition is individually invertible, but they cannot be the overlap identifications of a global frame. The defect lies in the triple relation.","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"[Baire's complete-space principle] A nonempty complete metric space cannot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open an","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"A failed triple. Let t≠1 be an involution and ask for frames b_0,b_1,b_2 satisfying b_1=tb_0, b_2=tb_1, b_2=tb_0. The first two give b_2=b_0, while the third forces t=1. Each displayed transition is individually invertible, but they cannot be the overlap identifications of a global frame. The defect lies in the triple relation.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"A failed triple. Let t≠1 be an involution and ask for frames b_0,b_1,b_2 satisfying b_1=tb_0, b_2=tb_1, b_2=tb_0. The first two give b_2=b_0, while the third forces t=1. Each displayed transition is individually invertible, but they cannot be the overlap identifications of a global frame. The defect lies in the triple relation.","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fe7473d969f990ef","tex":"B_{a,b}","aliases":"B_a,b","display":"B_{a,b}","kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"rded. Step 1: normalize the supplied identifications. The triple-overlap equation with all indices equal gives a_ii=a_iia_ii, hence a_ii=1. With indices i,j,i it gives a_jia_ij=1, hence a_ji=a_ij^-1. Composition coverage implies object coverage by inserting identity arrows. Choose a containing patch κ(v) for each view and put b_i(v)=a_κ(v),i(v) (v∈","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"s equal gives a_ii=a_iia_ii, hence a_ii=1. With indices i,j,i it gives a_jia_ij=1, hence a_ji=a_ij^-1. Composition coverage implies object coverage by inserting identity arrows. Choose a containing patch κ(v) for each view and put b_i(v)=a_κ(v),i(v) (v∈ U_i). At a point of U_i∩ U_j, the triple law, now on patches κ(v),i,j, gives b_j(v)=a_ij(v)b_i(v","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","excerpt":"[Summable movement gives exactification] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-28ef9ae6743fe0cd","tex":"b_l","aliases":"b_l","display":"b_l","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"use","excerpt":"[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B","count":13},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5df488c3f3b35a2b","tex":"B_{m,K}","aliases":"B_m,K","display":"B_{m,K}","kind":"use","excerpt":"s] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"use","excerpt":"orem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0","count":13},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4c553faf293d9e77","tex":"B_{n+1}","aliases":"B_n+1","display":"B_{n+1}","kind":"use","excerpt":"annot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open and meets every nonempty open set. Recursively choose a closed ball B_n","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-01ddcddf337569c9","tex":"B_n^2","aliases":"B_n^2","display":"B_n^2","kind":"use","excerpt":"p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really project","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9474dae99bcf0491","tex":"B_t","aliases":"B_t","display":"B_t","kind":"use","excerpt":"Subtract q_0=diag(0,1), obtaining the symmetric matrix B_t=q_t-q_0=frac11+t^2 t^2 · -t -t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5fe27b9bc4a36cd0","tex":"B_t^{\\mathsf T}","aliases":"B_t^ T mathsf","display":"B_t^{\\mathsf T}","kind":"use","excerpt":"-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives e","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1dbc5d4c102370ca","tex":"B_t^2","aliases":"B_t^2","display":"B_t^2","kind":"use","excerpt":"-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives e","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2dd50629910aaca0","tex":"b_v^{-1}","aliases":"b_v^-1","display":"b_v^{-1}","kind":"use","excerpt":"Suppose the orbit is neutral. Equality of orbit classes means there exists an actual based gauge b with b· c^q=1. At an arrow f:v→ w, this equation is b_vq_v^-1c_ff^*(q_w)[f^*(b_w)]^-1=1. Multiply on the left by q_vb_v^-1. Homomorphic preservation of products then gives c_ff^*(q_wb_w^-1)=q_vb_v^-1. Therefore x_v=q_vb_v^-1 is a twisted coherent fami","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ee397f54c2fce937","tex":"B_W","aliases":"B_W","display":"B_W","kind":"use","excerpt":"Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-44eebffdd43cc2d2","tex":"b_w","aliases":"b_w","display":"b_w","kind":"use","excerpt":"· =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitution gives (a·(b· c))_f =a_vb_vc_f[f^*(b_w)]^-1[f^*(a_w)]^-1 =((ab)· c)_f. The reversed order in the two inverse factors is exactly the formula for the inverse of a product. The identity gauge acts trivially.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9c8b5d749b8019c4","tex":"b_w^{-1}","aliases":"b_w^-1","display":"b_w^{-1}","kind":"use","excerpt":"Suppose the orbit is neutral. Equality of orbit classes means there exists an actual based gauge b with b· c^q=1. At an arrow f:v→ w, this equation is b_vq_v^-1c_ff^*(q_w)[f^*(b_w)]^-1=1. Multiply on the left by q_vb_v^-1. Homomorphic preservation of products then gives c_ff^*(q_wb_w^-1)=q_vb_v^-1. Therefore x_v=q_vb_v^-1 is a twisted coherent fami","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-86c749fe35cadfc1","tex":"B_Y","aliases":"B_Y","display":"B_Y","kind":"use","excerpt":"chapBanach repair and constructed split coordinates An actual complete candidate space. Let X_n,Y_n be Banach spaces over a complete nontrivially normed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formal","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.","count":15},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9f9c8470618c69e9","tex":"\\mathcal C'","aliases":"C' mathcal","display":"\\mathcal C'","kind":"use","excerpt":"Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","excerpt":"The correct uniqueness statement. Suppose (c,b_i) and (c',b'_i) realize the same descent datum. The only possible compatible comparison is g_v=b'_i(v)^-1b_i(v), using any patch through v. The overlap identities make it independent of the patch, and b'_ig=b_i. It is therefore the unique global comparison respecting the fixed local identifications.","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-97c15171699972db","tex":"C^{\\mathsf T}","aliases":"C^ T mathsf","display":"C^{\\mathsf T}","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5eda86663b484271","tex":"c^{\\mathsf T}","aliases":"c^ T mathsf","display":"c^{\\mathsf T}","kind":"use","excerpt":"covector-norm g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0. Thus G is positive definite and invertible. The matrix P=BG^-1B","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3f9e12aa522f08b3","tex":"c^{(m)}_n","aliases":"c^(m)_n","display":"c^{(m)}_n","kind":"use","excerpt":"The same operator has an attainable datum. Change only the residual to b_n=1/n^2. Its minimum norm preimage is a^†_n=1/n∈ℓ^2. At α=m^-2, the regularized correction is c^(m)_n=fracm^2n(m^2+n^2), a^†_n-c^(m)_n=fracnm^2+n^2. The bias square is exactly the previously computed B_m. This links the two cases without conflating their existence hypotheses: ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f076934b22cf32d0","tex":"c^q_f","aliases":"c^q_f","display":"c^q_f","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar s","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3134b412f41254b7","tex":"C_{2k}","aliases":"C_2k","display":"C_{2k}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7d25b1661025a3ae","tex":"C_{2k+1}","aliases":"C_2k+1","display":"C_{2k+1}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2dc3b3739f1aa480","tex":"C_{a,b}","aliases":"C_a,b","display":"C_{a,b}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-63159a30f6ea5ab4","tex":"c_f^{-1}","aliases":"c_f^-1","display":"c_f^{-1}","kind":"use","excerpt":"Automorphisms and ambiguity are not the same question. The stabilizer of a cocycle is the group of families with a_v=c_f f^*(a_w)c_f^-1. If one gauge a:c→ d is known, all gauges from c to d are uniquely of the form ah, with h∈Stab(c). Indeed a^-1b stabilizes c for every other gauge b. This is the ordinary Hom torsor principle in a groupoid.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7081746b55ffe251","tex":"c_f^q","aliases":"c_f^q","display":"c_f^q","kind":"use","excerpt":"relative extension] The obstruction is independent of the raw lift. It is neutral exactly when s extends to a global section of T_c. The normalized label c^q is the gauge transform of c by q^-1, so Proposition~[source reference] makes it a cocycle. If f:v→ w is internal to U, the partial-section equation says q_v=c_ff^*(q_w), and therefore c_f^q=1 ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d4c67adf201d51ce","tex":"c_f^t","aliases":"c_f^t","display":"c_f^t","kind":"use","excerpt":"If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1 · =t_v^-1c_ff^*(q_w) [f^*(q_w)]^-1f^*(t_w) =t_v^-1c_ff^*(t_w)=c_f^t. Thus changing the raw lift leaves the based orbit unchanged.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0ead68b043a941fd","tex":"C_{\\i}","aliases":"C_i i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8node[box,minimum width=1.5cm] (ri) at (x,1.7) x_i∈ R_i; foreach i/j in 0/1,1/2,2/3 draw[arr] (ri.east) -- node[lab,above] C_i (rj.west); draw[thinarr] (rj.north) to[out=130,in=50] node[lab,above] J_i (ri.north); foreach i/x in 0/-4.8,1/-1.6,2/1.6 node[box,minimum width=1.6cm] (hi) at (x,-.3) h_i","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1,","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1, a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cb563fdc61e28b56","tex":"c_j^2","aliases":"c_j^2","display":"c_j^2","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"[Variable-rate exactification] If Q_n→0 and ∑_n C_nQ_n<∞, then (f_n) has an exact limit and dist(f_n,x)≤ω(f_0)∑_k=n^∞C_kQ_k. For constant q_n=q∈[0,1) and constant C_n=C, the sharper restarted estimate is dist(f_n,x)≤fracC1-q ω(f_n). The estimate ω(f_n)≤ Q_nω(f_0) follows by induction. At n=0 it is equality. If it holds at n, nonnegativity of q_n pe","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-646fe15eb212b5b3","tex":"c_m","aliases":"c_m","display":"c_m","kind":"use","excerpt":"⟨ x-c,z-c⟩≤0 (z∈ C). If C=c_0+L is a closed affine subspace with direction space L, the condition is x-c⊥ L. Produce a Cauchy minimizing sequence. Put d=inf_z∈ Cnormx-z. Nonemptiness makes this a finite nonnegative real number. Choose c_n∈ C so that normx-c_n→ d. Convexity places (c_n+c_m)/2 in C. The parallelogram identity gives normc_n-c_m^2 =2no","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b08948d76a77d953","tex":"C_{M-1}","aliases":"C_M-1","display":"C_{M-1}","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1e15d2e27313753c","tex":"C_N","aliases":"C_N","display":"C_N","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bc287c0044452641","tex":"C_{N,M}","aliases":"C_N,M","display":"C_{N,M}","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3573f0e85f5731cd","tex":"C_{N-1}","aliases":"C_N-1","display":"C_{N-1}","kind":"use","excerpt":"egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-53faef096d1a0acc","tex":"c_v","aliases":"c_v","display":"c_v","kind":"use","excerpt":"[Joint determination and the universal property] The evaluations jointly determine a realization. If a set T carries maps c_v:T→ F_v satisfying f^*c_w=c_v, there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-06e1595497d8f5f2","tex":"c_w","aliases":"c_w","display":"c_w","kind":"use","excerpt":"[Joint determination and the universal property] The evaluations jointly determine a realization. If a set T carries maps c_v:T→ F_v satisfying f^*c_w=c_v, there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-31ddf16fe06684fb","tex":"C_\\varepsilon","aliases":"C_ε varepsilon","display":"C_\\varepsilon","kind":"use","excerpt":"It remains to prove exactness of the limit. Fix ε>0. Since ω(f_m)→0, an entire tail of the sequence belongs to the sublevel set C_ε={z:ω(z)≤ε}. Lower semicontinuity makes this set closed, so the limit of that tail belongs to it. Hence ω(x)≤ε for every positive ε. If ω(x)>0, choosing ε=ω(x)/2 contradicts this inequality. Since ω is nonnegative, ω(x)","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-78f0eff5d5a1c58a","tex":"C_\\phi","aliases":"C_ϕ phi","display":"C_\\phi","kind":"use","excerpt":"Actual tower resampling gives a defect map. Return to Banach towers with contractive source and target bonds. Let ϕ be a monotone schedule. Candidate sampling is (S_ϕ f)_n=f_ϕ(n). The sampled bonding map is the actual composite restriction between the selected levels. The residual map C_ϕ is not pointwise sampling of a scalar error score. It accumu","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4a21d8457df5b3c8","tex":"C_{\\phi(n),\\phi(n+1)}","aliases":"C_ϕ(n),ϕ(n+1) phi phi","display":"C_{\\phi(n),\\phi(n+1)}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-269cd99c3ea8db81","tex":"\\code","aliases":"code","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bb28e6eb35641c53","tex":"\\mathrm{comp}","aliases":"comp mathrm","display":"\\mathrm{comp}","kind":"use","excerpt":"[The composition topology] Condition~[source reference] defines a Grothendieck topology J_comp. The sieve generated by a family of inclusions U_i⊆ W is covering exactly when the family covers every composition support in W. In the inclusion poset, identify a sieve on W with a downward-closed collection of subsets of W. Pullback along Z⊆ W consists ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cef2f13a7966bbb0","tex":"\\coprod","aliases":"coprod","display":null,"kind":"use","excerpt":"Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coord","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8760e5ffe5f16a41","tex":"\\coprod_{\\substack{t\\in\\Gamma_W(A)\\\\t|_U=s}}","aliases":"coprod_substackt∈Γ_W(A) t|_U=s coprod substack in Gamma t","display":null,"kind":"use","excerpt":"Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coord","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8db6134243ea03f4","tex":"\\cos^2","aliases":"cos^2 cos","display":"\\cos^2","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"use","excerpt":"chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.","count":38},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c2a814548eab63ec","tex":"\\mathcal D'","aliases":"D' mathcal","display":"\\mathcal D'","kind":"use","excerpt":"Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6c038250227f91ca","tex":"d^{\\mathsf T}","aliases":"d^ T mathsf","display":"d^{\\mathsf T}","kind":"use","excerpt":"The restricted Hilbert normal residual is g=P d^ T(dBc-r)+α Bc. Substitute S=dB, G=B^ TB and the expression for P: g=BG^-1(S^ T(Sc-r)+α Gc)=BG^-1q. Equivalently q=B^ Tg records the pairings of g with the basis columns. Its squared Hilbert norm is g^ Tg =q^ TG^-1B^ TBG^-1q =q^ TG^-1q, using symmetry of G^-1. This proves both coordinate formulas.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d4ef1713d263e2f3","tex":"d^*","aliases":"d^*","display":"d^*","kind":"use","excerpt":"The adjoint below always means T^*:Y→ H, taken with the inherited Hilbert metric on H. In ambient coordinates this is the projection of d^* onto H. A coordinate basis of ker p need not be orthonormal. Its Gram matrix must then be included in any calculation of correction energy.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"If C=c_0+L is a closed affine subspace with direction space L, the condition is x-c⊥ L. Produce a Cauchy minimizing sequence. Put d=inf_z∈ Cnormx-z. Nonemptiness makes this a finite nonnegative real number. Choose c_n∈ C so that normx-c_n→ d. Convexity places (c_n+c_m)/2 in C. The parallelogram identity gives normc_n-c_m^2 =2normx-c_n^2+2normx-c_m^","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c4f88bd0811b1b2f","tex":"\\mathcal D_1","aliases":"D_1 mathcal","display":"\\mathcal D_1","kind":"use","excerpt":"The proposed objective is 11/16. Its actual excess over the optimum is 1/48. The residual witness gives D_1(3/4)=frac2132, J(a)- D_1(3/4)=frac132. The duality gap is an upper bound for suboptimality, not necessarily the exact suboptimality. Both repaired candidates preserve p(f)=1.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-943ac188116d74d8","tex":"d_a","aliases":"d_a","display":"d_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2b84d7d41efdc0b1","tex":"d_b","aliases":"d_b","display":"d_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b757b9d04b228415","tex":"d_c","aliases":"d_c","display":"d_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d3b4cd694374cac7","tex":"d_e","aliases":"d_e","display":"d_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fac4e28b7bf50eb9","tex":"d_f","aliases":"d_f","display":"d_f","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitu","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-59c0124e24487c12","tex":"d_g","aliases":"d_g","display":"d_g","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitu","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2b510e902537ebcf","tex":"d_{g\\circ f}","aliases":"d_g∘ f circ","display":"d_{g\\circ f}","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitu","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"If all h_j vanish, the finite telescope gives Q_Nx=x for every N. Conversely, if every Q_Nx=x, then L_jh_j=Q_jx-Q_j+1x=0. Apply the left inverse D_j of L_j to obtain h_j=0. The injectivity of the restoration composite is the step that recovers the original detail from its transported value. sourceKS26","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a6fe151e79c7d1a9","tex":"d_l","aliases":"d_l","display":"d_l","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f1be8e49b81c9b1e","tex":"D_M","aliases":"D_M","display":"D_M","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3fb8837a44922746","tex":"D_N","aliases":"D_N","display":"D_N","kind":"use","excerpt":"egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0efdddd015383ecb","tex":"d_p","aliases":"d_p","display":"d_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-acd8deecbfbf287e","tex":"D_s","aliases":"D_s","display":"D_s","kind":"use","excerpt":"-t · 1. The common right inverse r↦(r,0) supplies gain at most one for every t, and normD_s-D_t=|s-t|. The general theorem gives normq_s-q_t≤|s-t|.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"use","excerpt":"An all-real worked model. Let D_t(x,y)=x+ty. Its kernel is the line {(-tb,b):b∈R}, and its orthogonal projector is q_t=frac11+t^2 t^2 · -t","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b377ac1e0c05ff0f","tex":"d_t","aliases":"d_t","display":"d_t","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-30d2e95df9b4aa17","tex":"D_\\perp","aliases":"D_⊥ perp","display":"D_\\perp","kind":"use","excerpt":"such that every x has some exact y with normx-y≤ CnormDx. When the range is closed, the least admissible constant is the operator norm of the inverse of D:K^⊥→ran D. In the zero operator case that inverse has zero dimensional domain and its norm is zero. Let K=ker D and K^⊥ carry the induced Hilbert norm. Define D_⊥ u=Du, but regard its target as r","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1d2d51d62f51421c","tex":"D_\\perp^{-1}","aliases":"D_⊥^-1 perp","display":"D_\\perp^{-1}","kind":"use","excerpt":"Suppose first that the range is closed. Because R is Banach, ran D is then Banach. The domain K^⊥ is also complete. The bounded inverse theorem, proved in Appendix~[source reference], gives a bounded inverse A=D_⊥^-1. Applying it to Dx=D(x-qx) yields x-qx=A(Dx), normx-qx≤norm AnormDx. Hence condition 2 holds. If any C≥0 satisfies condition 2, then ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"[Repair of the identity family] The bounded candidate f_n=I is not exact. Its repair is (Pf)_n=diag(2^-n,2^-(n-1),…,2^-1,1). The new last output row is retained at every stage while inherited rows are propagated by the factor 1/2. At level zero the value is one. Exactness forces every inherited row at the next level to be one half of its previous v","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the positive case is dim H=n-rankA P. Set T=A","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-558c1dc971656de6","tex":"\\dist","aliases":"dist","display":null,"kind":"use","excerpt":"A small residual far from exactness. On ℓ^2(N), index the standard basis from one and define De_n=n^-1e_n. The map is bounded and injective, so its exact space is zero. Yet norme_n=1, normDe_n=1/n⟶0. No finite C can bound dist(e_n,ker D) by CnormDe_n. The range is not closed. For example finite truncations of (1/n)_n≥1 belong to the range and conve","count":22},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"C and b-Ta∈ C, every correction satisfies normr-Ta^2=q^2+normb-Ta^2. Thus β(M)≥ q. Given ϵ>0, range density in C supplies one correction a with normb-Ta<ϵ. For every M≥norm a it follows that q≤β(M)≤sqrtq^2+ϵ^2. First letting the budget increase and then letting ϵdownarrow0 proves the limit.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b87f37426ee94627","tex":"E^2","aliases":"E^2","display":"E^2","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-89d50d3d0bc30bea","tex":"e^2","aliases":"e^2","display":"e^2","kind":"use","excerpt":"0≤ J_α(a)-J_α(a_α) · ≤fracnorm g^2α. Let e=a-a_α and B=T^*T+α I. The normal equation of the constructed optimizer gives Be=(T^*T+α I)a-T^*r=g. Pairing with e yields the exact identity normTe^2+αnorm e^2=⟨ g,e⟩ ≤norm gnorm e. If e=0, all three conclusions hold with zero error. Otherwise drop the first nonnegative term and divide by αnorm e>0 to obta","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"Every expression a(x)+a(y)-a(x+y) is symmetric in x,y, because G is abelian. With e_1=(1,0) and e_2=(0,1), however, c(e_1,e_2)=0, c(e_2,e_1)=1. This contradicts the symmetry required by [source reference]. sourceKS80 The proof rules out all sixteen possible scalar functions a, not just linear functions or selected trial gauges. The source codeProje","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"Every expression a(x)+a(y)-a(x+y) is symmetric in x,y, because G is abelian. With e_1=(1,0) and e_2=(0,1), however, c(e_1,e_2)=0, c(e_2,e_1)=1. This contradicts the symmetry required by [source reference]. sourceKS80 The proof rules out all sixteen possible scalar functions a, not just linear functions or selected trial gauges. The source codeProje","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b6266893a62573e7","tex":"e_b","aliases":"e_b","display":"e_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6500ef20b6c4932a","tex":"e_c","aliases":"e_c","display":"e_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b49fd93911355d0a","tex":"e_e","aliases":"e_e","display":"e_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-420aa7a6632049fa","tex":"e_f","aliases":"e_f","display":"e_f","kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-53a44fa3ddda48c6","tex":"e_g","aliases":"e_g","display":"e_g","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"Flatness does not remove amplification. Consider a scalar chain with n+1 vertices, where n≥1, and coordinates e_i(z)=2^iz. Set τ_ij=e_ie_j^-1. This connection is flat, but reverse transport along one edge multiplies norms by two. Let x_i=8·2^i-1, y_i=7·2^i. The family y is exact and preserves x_o=7. On a forward edge the residual of x is 1/2, and o","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":", every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many vectors a_n. Finite rational linear combinations of those vectors form a countable dense set, so H_0 is separable. Apply Gram--Schmidt to a countable dense sequence, omitting zero remainders. This gives either a finite orthonormal","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-39cc49921fc88ad3","tex":"e_j^{-1}","aliases":"e_j^-1","display":"e_j^{-1}","kind":"use","excerpt":"Flatness does not remove amplification. Consider a scalar chain with n+1 vertices, where n≥1, and coordinates e_i(z)=2^iz. Set τ_ij=e_ie_j^-1. This connection is flat, but reverse transport along one edge multiplies norms by two. Let x_i=8·2^i-1, y_i=7·2^i. The family y is exact and preserves x_o=7. On a forward edge the residual of x is 1/2, and o","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-76853ffdc4938f3f","tex":"\\mathcal E_{Kb}","aliases":"E_Kb mathcal","display":"\\mathcal E_{Kb}","kind":"use","excerpt":"[Invariance of the marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-16230be50f47f21c","tex":"e_l","aliases":"e_l","display":"e_l","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","excerpt":"Existence does not make an arbitrary sequence converge strongly. Take H=Y=ℓ^2 and Te_n=e_n/n. The vectors a_n=e_n have norm one and satisfy Ta_n→0. An exact answer exists, namely zero. Yet norme_n-e_m=sqrt2 for n≠ m, so no subsequence converges in norm.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"A small residual far from exactness. On ℓ^2(N), index the standard basis from one and define De_n=n^-1e_n. The map is bounded and injective, so its exact space is zero. Yet norme_n=1, normDe_n=1/n⟶0. No finite C can bound dist(e_n,ker D) by CnormDe_n. The range is not closed. For example finite truncations of (1/n)_n≥1 belong to the range and conve","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-66c22a9ceb430f6a","tex":"e_o","aliases":"e_o","display":"e_o","kind":"use","excerpt":"For necessity, suppose a constant C satisfies dist(w,ker d)≤ Cnormdw_∞ for every vertex family w. Fix z∈ K^⊥ and take w=R_pz. Every exact family e has e_o∈ K, and root evaluation has norm at most one. Therefore norm z=dist(z,K)≤normz-e_o≤normR_pz-e_∞. Taking the infimum over e∈ker d is legitimate without assuming that this global nearest point is a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-caa961746837d921","tex":"e_s","aliases":"e_s","display":"e_s","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7cf754fef645cd67","tex":"\\endhead","aliases":"endhead","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-63474ae27b72cd77","tex":"\\operatorname{ev}_k","aliases":"ev_k operatorname","display":"\\operatorname{ev}_k","kind":"use","excerpt":"When global behavior determines the finite presentation. Say that X has surjective point projections when ∀ k ev_k:Point(X)→ X_k is onto. This means that every finite input state really occurs in some coherent global input. It is stronger than nonemptiness of the global space.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0cab149d070716dc","tex":"\\operatorname{ev}_w","aliases":"ev_w operatorname","display":"\\operatorname{ev}_w","kind":"use","excerpt":"The evaluations jointly determine a realization. If a set T carries maps c_v:T→ F_v satisfying f^*c_w=c_v, there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for x gives f^*(ev_w x)=f^*(x_w)=x_v=ev_v x. If x,y∈","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f1440080216cf2f2","tex":"\\operatorname{ev}_z","aliases":"ev_z operatorname","display":"\\operatorname{ev}_z","kind":"use","excerpt":"appearance x_z; draw[arr] (g.south west) -- node[lab,above left] ev_v (v.north); draw[arr] (g) -- node[lab,right] ev_w (w); draw[arr] (g.south east) -- node[lab,above right] ev_z (z.north); draw[arr] (w) -- node[lab,above] f^* (v); draw[arr] (z) -- node[lab,above] g^* (w); draw[arr] (z.south) to[out=-90,in=-90] node[lab,below] (g∘ f)^*=f^*g^* (v.so","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-564edd8700e5a5d8","tex":"\\Ext_{U'}^{H^*F}","aliases":"Ext_U'^H^*F Ext","display":null,"kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2c95751a9919cb1b","tex":"\\Ext_U^F","aliases":"Ext_U^F Ext","display":null,"kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-44cbdf7e5a5067f6","tex":"\\Ext_W","aliases":"Ext_W Ext","display":null,"kind":"use","excerpt":"Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coord","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"F_2 · Only zero · Two","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b6bb9a2af76d66fc","tex":"\\mathbf F","aliases":"F mathbf","display":"\\mathbf F","kind":"use","excerpt":"A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"Convergence has its own theorem. Call x norm recoverable when Q_Nx→0. This is equivalent to S_Nx→ x, but it does not follow merely from vanishing persistent subspace. Let F=⋃_Nker Q_N be the finite core.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2f4583c8b7c73ba0","tex":"f'","aliases":"f'","display":"f'","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5fefedf3e6c697a3","tex":"f'_j","aliases":"f'_j","display":"f'_j","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e5c12fbb78423d79","tex":"f'_{\\psi'(n)}","aliases":"f'_ψ'(n) psi","display":"f'_{\\psi'(n)}","kind":"use","excerpt":"nd g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7622cd705b527bdd","tex":"f^{(0)}","aliases":"f^(0)","display":"f^{(0)}","kind":"use","excerpt":"[Constructed geometric repair] Starting from f^(0)=f, define f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-51f77419951a1a0f","tex":"f^{(m+1)}","aliases":"f^(m+1)","display":"f^{(m+1)}","kind":"use","excerpt":"[Constructed geometric repair] Starting from f^(0)=f, define f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e1adf13ef9f0afde","tex":"f^{(m+j)}","aliases":"f^(m+j)","display":"f^{(m+j)}","kind":"use","excerpt":"nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-463fe71a6222bdb1","tex":"f^{(m+j+1)}","aliases":"f^(m+j+1)","display":"f^{(m+j+1)}","kind":"use","excerpt":"nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2715e8330c28c57a","tex":"f^{(n)}","aliases":"f^(n)","display":"f^{(n)}","kind":"use","excerpt":"Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","excerpt":"[Split tower correction] The recurrence defines a continuous linear map A:R→ E with ∂ A=I_R, norm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial co","count":23},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"[Baire's complete-space principle] A nonempty complete metric space cannot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open an","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"[Baire's complete-space principle] A nonempty complete metric space cannot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open an","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"F_2 · Only zero · Two","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-710b65b51795073d","tex":"\\mathbf F_2","aliases":"F_2 mathbf","display":"\\mathbf F_2","kind":"use","excerpt":"A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bb1d20c582e7f72b","tex":"\\mathbf F_2^2","aliases":"F_2^2 mathbf","display":"\\mathbf F_2^2","kind":"use","excerpt":"A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-54c7a07f66f1253f","tex":"\\mathbb F_3","aliases":"F_3 mathbb","display":"\\mathbb F_3","kind":"use","excerpt":"F_3 · Every element · One","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b314e2300364246a","tex":"f_a","aliases":"f_a","display":"f_a","kind":"use","excerpt":"a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[s","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0b8eb6ad7f8b7acf","tex":"f_b","aliases":"f_b","display":"f_b","kind":"use","excerpt":"a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[s","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-088f015dc54be4e4","tex":"f_F^*","aliases":"f_F^*","display":"f_F^*","kind":"use","excerpt":"Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ac9da4be99230127","tex":"f_G^*","aliases":"f_G^*","display":"f_G^*","kind":"use","excerpt":"Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c29f47f3dabc5355","tex":"f_{\\i}","aliases":"f_i i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8 node[box,minimum width=1.5cm] (xi) at (x,2) X_i; node[box,minimum width=1.5cm] (yi) at (x,0) Y_i; draw[arr] (xi) -- node[lab,right] f_i (yi); foreach j/i in 1/0,2/1,3/2 draw[arr] (xj) -- node[lab,above] ρ^X_i (xi); draw[arr] (yj) -- node[lab,below] ρ^Y_i (yi); node[lab,text width=12.2cm] at (0,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2f6b5c0f84584144","tex":"f_k","aliases":"f_k","display":"f_k","kind":"use","excerpt":"uad normC_ϕ≤ m, ∂_ϕ S_ϕ=C_ϕ∂. Exact candidates remain exact under every monotone common resampling, even without a gap bound. If A and B are bounded right inverses for the fine and sampled differentials, then normP_BS_ϕ f-S_ϕ P_Af ≤(norm A+mnorm B)norm∂ f. For a bounded candidate family, sup_nnormf_ϕ(n)≤sup_knormf_k, proving the sampling bound. For","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0be3a2ded0c0a234","tex":"f_{k+1}","aliases":"f_k+1","display":"f_{k+1}","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"use","excerpt":"on] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k=n^∞b_k. Given ε>0, convergence of th","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dc4a59cf35c9bb32","tex":"f_M^*","aliases":"f_M^*","display":"f_M^*","kind":"use","excerpt":"a coordinate preserving equivalence Γ(B_a,b)≃ B(a,b). Under a transition cover, matching local bridge sections glue to a unique global bridge with both prescribed projections unchanged. We must first check that the displayed fibers really form a view system. Let m∈ B_a,b(w) and f:v→ w. Naturality of p and coherence of a give p_v(f_M^*m)=f_F^*p_w(m)","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c57cafb013c6775f","tex":"F_n","aliases":"F_n","display":"F_n","kind":"use","excerpt":"The centers form a Cauchy sequence: all centers after stage n lie in B_n, whose diameter is at most 2^1-n. Completeness gives a limit. For each fixed n, a tail of centers lies in the closed set B_n, so the limit lies there. It therefore belongs to no F_n, contradicting that these sets cover the space.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-43135e362a6370f0","tex":"f_N","aliases":"f_N","display":"f_N","kind":"use","excerpt":"A complete calculation of the finite repairs. Let 1_N be one in coordinates 1,…,N and zero elsewhere. Put a_N=( 1_N,0), f_N=f_0-a_N=(- 1_N,1). Then p(f_N)=1, and the first N residual coordinates vanish exactly. The remaining residual and movement satisfy d(f_N)_n=0 · n≤ N,","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7817222f46df5e65","tex":"F_{n+1}","aliases":"F_n+1","display":"F_{n+1}","kind":"use","excerpt":"interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open and meets every nonempty open set. Recursively choose a closed ball B_n+1 of positive radius at most 2^-(n+1), contained in the open ball with the same center and radius as B_n, and","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7fc893a0531e77d4","tex":"f_{N+1}","aliases":"f_N+1","display":"f_{N+1}","kind":"use","excerpt":"The residual tends to zero, but the corrections grow like sqrt N. Successive candidates differ by a unit coordinate, so normf_N+1-f_N=1. The movement series diverges and the sequence is not Cauchy. This is exactly the hypothesis missing from the summable exactification theorem.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-af40e62da957c071","tex":"f_{n+1}","aliases":"f_n+1","display":"f_{n+1}","kind":"use","excerpt":"A candidate transport from X to Y consists of maps f_n:X_n→ Y_n. It is exact when ρ^Y_n f_n+1=f_nρ^X_n for all n.","count":19},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d2c04c9aaf92eb55","tex":"f_{n+j}","aliases":"f_n+j","display":"f_{n+j}","kind":"use","excerpt":"For constant rates, the global estimate can be sharpened by restarting at the current index. Fix n. A new induction on j gives ω(f_n+j)≤ q^jω(f_n), dist(f_n+j,f_n+j+1)≤ Cq^jω(f_n). Sum the finite chain from n to n+m, and then let m→∞ using the already established limit. The geometric series equals 1/(1-q), including q=0, where its first term is one","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-837249c7d9881c82","tex":"f_{n+j+1}","aliases":"f_n+j+1","display":"f_{n+j+1}","kind":"use","excerpt":"For constant rates, the global estimate can be sharpened by restarting at the current index. Fix n. A new induction on j gives ω(f_n+j)≤ q^jω(f_n), dist(f_n+j,f_n+j+1)≤ Cq^jω(f_n). Sum the finite chain from n to n+m, and then let m→∞ using the already established limit. The geometric series equals 1/(1-q), including q=0, where its first term is one","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b83ac00bc683ab3c","tex":"f_n^{(a)}","aliases":"f_n^(a)","display":"f_n^{(a)}","kind":"use","excerpt":"ct and protected value one, but their middle coordinates differ. They are distinct points of the same feasible fiber. Constant sequences at those points meet exactness and preservation but not a vanishing comparison bound, so these conditions alone cannot imply path independence. sourceKS74 A stronger worked example uses actual convergent paths fro","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-82848474eefdb909","tex":"f_n^{(b)}","aliases":"f_n^(b)","display":"f_n^{(b)}","kind":"use","excerpt":"eKS74 A stronger worked example uses actual convergent paths from the same initial input: f_n^(a)=(s,a(1-2^-n),2^-n) (a∈R). All paths begin at (s,0,1), keep p=s, and have residual 2^-n. Their maximum-norm movements are frac12max{|a|,1}2^-n, so every path satisfies geometric exactification. Their limits are (s,a,0). For different a,b, dist(f_n^(a),f","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","excerpt":"A global view and a useful warning. [Terminal evaluation] If t is terminal in V, evaluation induces an equivalence Γ(F)≃ F_t. For a∈ F_t, take its restriction along the unique arrow v→ t at each view. Uniqueness of the arrow to t, together with the functor laws, proves coherence. Evaluation at t recovers a. Conversely, any coherent family is the re","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a75d31ba6a020ef2","tex":"F_z","aliases":"F_z","display":"F_z","kind":"use","excerpt":"appearance x_w; node[box,minimum width=2.6cm] (z) at (4.7,0) F_z","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7ae800f554542e07","tex":"f_{\\phi(n)}","aliases":"f_ϕ(n) phi","display":"f_{\\phi(n)}","kind":"use","excerpt":"Actual tower resampling gives a defect map. Return to Banach towers with contractive source and target bonds. Let ϕ be a monotone schedule. Candidate sampling is (S_ϕ f)_n=f_ϕ(n). The sampled bonding map is the actual composite restriction between the selected levels. The residual map C_ϕ is not pointwise sampling of a scalar error score. It accumu","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c1c93a868bfd9d76","tex":"f_{\\psi(n)}","aliases":"f_ψ(n) psi","display":"f_{\\psi(n)}","kind":"use","excerpt":"The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9a7b27c4eeefb975","tex":"f_\\infty","aliases":"f_∞ infty","display":"f_\\infty","kind":"use","excerpt":"ed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formalization derives the required completeness from the operator spaces and their bounded product. The same differential satisfies norm∂ f_∞≤(B_X+B_Y)norm ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8096f24b23ced63d","tex":"\\fixedplate","aliases":"fixedplate","display":null,"kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"Motion of the actual descent repair. Compare pairs (D,C) and (D',C') on the same raw Hilbert space. Suppose each pair has closed local range and closed overlap range on its own locally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤ma","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-02b0b5ff2a25a5d9","tex":"g'","aliases":"g'","display":"g'","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7dd96346a453bb05","tex":"g'_n","aliases":"g'_n","display":"g'_n","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ccf2aef1e39540d4","tex":"g^{\\mathsf T}","aliases":"g^ T mathsf","display":"g^{\\mathsf T}","kind":"use","excerpt":"The restricted Hilbert normal residual is g=P d^ T(dBc-r)+α Bc. Substitute S=dB, G=B^ TB and the expression for P: g=BG^-1(S^ T(Sc-r)+α Gc)=BG^-1q. Equivalently q=B^ Tg records the pairings of g with the basis columns. Its squared Hilbert norm is g^ Tg =q^ TG^-1B^ TBG^-1q =q^ TG^-1q, using symmetry of G^-1. This proves both coordinate formulas.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-565b30cbcdc2ca2a","tex":"G^{-1}","aliases":"G^-1","display":"G^{-1}","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","excerpt":"A posteriori correction and fit bounds. [Normal residual as a solver certificate] For every proposed a∈ H, norma-a_α · ≤fracnorm gα, normT(a-a_α) · ≤fracnorm g2sqrtα, 0≤ J_α(a)-J_α(a_α) · ≤fracnorm g^2α. Let e=a-a_α and B=T^*T+α I. The normal equation of the constructed optimizer gives Be=(T^*T+α I)a-T^*r=g. Pairing with e yields the exact identity","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"use","excerpt":"uad norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum norm on R× Y, two sufficient gains are therefore G_2=sqrta^2+b^2(1+anorm C)^2 ≤ G_1=a(1+bnorm C)+b. The inequality follows from sq","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3b9dcaf46a46092c","tex":"G_2","aliases":"G_2","display":"G_2","kind":"use","excerpt":"uad norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum norm on R× Y, two sufficient gains are therefore G_2=sqrta^2+b^2(1+anorm C)^2 ≤ G_1=a(1+bnorm C)+b. The inequality follows from sq","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7ccf548c96ad41ac","tex":"g_d","aliases":"g_d","display":"g_d","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-15e892a32e1d61ee","tex":"g_n","aliases":"g_n","display":"g_n","kind":"use","excerpt":"[The exact tower category] Identity maps and composites of exact transports are exact. An exact transport induces a map of coherent point spaces by x↦(f_nx_n)_n, preserving identities and composition. For two exact transports, substitute ρ^Z_ng_n+1=g_nρ^Y_n and ρ^Y_nf_n+1=f_nρ^X_n into the composite square. For a coherent point, ρ^Y_n f_n+1x_n+1 =f","count":18},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dbbbaca70e55accd","tex":"g_{n+1}","aliases":"g_n+1","display":"g_{n+1}","kind":"use","excerpt":"[The exact tower category] Identity maps and composites of exact transports are exact. An exact transport induces a map of coherent point spaces by x↦(f_nx_n)_n, preserving identities and composition. For two exact transports, substitute ρ^Z_ng_n+1=g_nρ^Y_n and ρ^Y_nf_n+1=f_nρ^X_n into the composite square. For a coherent point, ρ^Y_n f_n+1x_n+1 =f","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-87a3fa4dd69238cf","tex":"g_r","aliases":"g_r","display":"g_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-caf9c919454201a8","tex":"\\mathcal G_U","aliases":"G_U mathcal","display":"\\mathcal G_U","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-99b7a1e1268a1b18","tex":"g_v","aliases":"g_v","display":"g_v","kind":"use","excerpt":"The correct uniqueness statement. Suppose (c,b_i) and (c',b'_i) realize the same descent datum. The only possible compatible comparison is g_v=b'_i(v)^-1b_i(v), using any patch through v. The overlap identities make it independent of the patch, and b'_ig=b_i. It is therefore the unique global comparison respecting the fixed local identifications.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6d584c042dd0c359","tex":"G_w","aliases":"G_w","display":"G_w","kind":"use","excerpt":"descentA set of solutions forgets the witnesses relating equivalent solutions. The next layer retains those witnesses, their composition, and their automorphisms. Gluing then has to respect the overlap identifications themselves. Nonabelian coherence and relative trivialization. Coherent transition labels. Fix groups G_v and restriction homomorphis","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-66d4cedfcc852439","tex":"G_z","aliases":"G_z","display":"G_z","kind":"use","excerpt":"groupoid] Equation~[source reference] defines an action on cocycles. A gauge carrying c to d induces an equivalence of twisted section sets by x_v↦ a_vx_v. The cocycles and actual gauges form a groupoid: an arrow a:c→ d followed by b:d→ e has label ba. First, T_c really is a view system. For f:v→ w, g:w→ z, and x∈ G_z, the homomorphism and cocycle ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c3d50e4b12d0681c","tex":"H^*","aliases":"H^*","display":"H^*","kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-daaa917c98a2902e","tex":"H^{-1}","aliases":"H^-1","display":"H^{-1}","kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"chapMultiscale residuals and the limits of reconstruction A split coarsener retains its missing detail. Let C:R→overline R and J:overline R→ R be linear maps with CJ=I. Put H=I-JC. Then r=JCr+Hr, CH=0, H^2=H. The two coordinates (Cr,Hr) retain all information. The coarse coordinate alone generally does not.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":". It satisfies T^*(r-Ta_α)=α a_α. For every a∈ H, J_α(a)-J_α(a_α) =normT(a-a_α)^2+αnorma-a_α^2. Step 1: the normal operator has an actual bounded inverse. Let B=T^*T+α I:H→ H. It is bounded and self-adjoint. For every h∈ H, ⟨ Bh,h⟩=normTh^2+αnorm h^2 ≥αnorm h^2. If h≠0, Cauchy--Schwarz and division by norm h give normBh≥αnorm h. The same inequality","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0b14fec69d50fa98","tex":"h_*","aliases":"h_*","display":"h_*","kind":"use","excerpt":"[Nearest feasible protected repair] Every feasible x has a unique closest exact answer y_* preserving p(x). Its correction h_*=x-y_* is the unique vector in the orthogonal complement of ker B inside M satisfying Bh_*=Dx. For every other feasible answer y, normx-y^2=normx-y_*^2+normy_*-y^2. Moreover, the following conditions are equivalent: ran B is","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"ent [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many ","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"ormx-y_*^2+normy_*-y^2. Moreover, the following conditions are equivalent: ran B is closed Longleftrightarrow ∃ C≥0 ∀ feasible x: normx-y_*≤ CnormDx. When these conditions hold, the least nonnegative constant is the transverse inverse norm of B, including the zero-range case. Feasibility gives some h_0=x-y_0∈ M with Bh_0=Dx. Conversely, every h∈ M ","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"The choice of x_0 only supplies an origin for the parameters. If x_1=x_0+h_1 is used instead, the parameter of x changes from x-x_0 to x-x_1=(x-x_0)-h_1. The affine set of answers itself has not changed. sourceKS10 An extension fiber is therefore an affine copy, or torsor, of the invisible coherent group. Selecting one solution gives coordinates on","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","excerpt":"0,in=50] node[lab,above] J_i (ri.north); foreach i/x in 0/-4.8,1/-1.6,2/1.6 node[box,minimum width=1.6cm] (hi) at (x,-.3) h_i; draw[arr] (ri) -- node[lab,right] detail (hi); node[box,draw=KomorebiOchre,fill=KomorebiWarm,minimum width=2.1cm] (q) at (4.8,-.3) Q_3x=L_3x_3; draw[arr] (r3)--node[lab,right] restore (q); node[lab,text width=13cm] at (0,-1","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-14fc04484b11ded4","tex":"h_{\\i}","aliases":"h_i i","display":null,"kind":"use","excerpt":"m] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8node[box,minimum width=1.5cm] (ri) at (x,1.7) x_i∈ R_i; foreach i/j in 0/1,1/2,2/3 draw[arr] (ri.east) -- node[lab,above] C_i (rj.west); draw[thinarr] (rj.north) to[out=130,in=50] node[lab,above] J_i (ri.north); foreach i/x in 0/-4.8,1/-1.6,2/1.6 node[box,minimum width=1.6cm] (hi) at (x,-.3) h_i; draw[arr]","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-da28811766777fc7","tex":"h_J","aliases":"h_J","display":"h_J","kind":"use","excerpt":"To verify weak convergence against h∈ H_0, choose a finite basis combination h_J close to h. Then |⟨ a_n_k-a,h⟩| ≤ |⟨ a_n_k-a,h_J⟩| +2Mnormh-h_J. For a prescribed tolerance, choose h_J first to control the second term and then k to control the first by finitely many coordinate limits. For a general h in the ambient space, write h=P_H_0h+h_⊥. The or","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-082db4c82cd55c3c","tex":"h_m","aliases":"h_m","display":"h_m","kind":"use","excerpt":"To prove closedness of its range, suppose Bh_n→ y. Applying the lower bound to h_n-h_m gives normh_n-h_m≤α^-1normBh_n-Bh_m. Thus h_n is Cauchy. Completeness of H supplies a limit h, and continuity gives Bh=y. If y is orthogonal to the range of B, then for every h, 0=⟨ Bh,y⟩=⟨ h,By⟩. Taking h=By gives By=0, and injectivity gives y=0. A closed subspa","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"use","excerpt":"Three hypotheses that cannot simply be dropped. Summable movement. In maximum-norm R^3, let f_n=(s,H_n,(n+1)^-1), H_n=∑_j=1^nfrac1j. The first coordinate is protected and the residual is the last coordinate. It tends to zero with the same ratios as the polynomial example. The movement equals 1/(n+1), however, and the unprotected coordinate diverges","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-785ae493560363c1","tex":"h_n","aliases":"h_n","display":"h_n","kind":"use","excerpt":"To prove closedness of its range, suppose Bh_n→ y. Applying the lower bound to h_n-h_m gives normh_n-h_m≤α^-1normBh_n-Bh_m. Thus h_n is Cauchy. Completeness of H supplies a limit h, and continuity gives Bh=y. If y is orthogonal to the range of B, then for every h, 0=⟨ Bh,y⟩=⟨ h,By⟩. Taking h=By gives By=0, and injectivity gives y=0. A closed subspa","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f823eeb6cd5f23d3","tex":"h_v","aliases":"h_v","display":"h_v","kind":"use","excerpt":"Let H_U={h:f^*(h_w)=h_v for every f, h|_U=1}.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-70dd52c54f29b990","tex":"H_W","aliases":"H_W","display":"H_W","kind":"use","excerpt":"Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f020f3626fdf00d6","tex":"h_w","aliases":"h_w","display":"h_w","kind":"use","excerpt":"Let H_U={h:f^*(h_w)=h_v for every f, h|_U=1}.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-87b87a125d1e88e4","tex":"h_\\perp","aliases":"h_⊥ perp","display":"h_\\perp","kind":"use","excerpt":"To verify weak convergence against h∈ H_0, choose a finite basis combination h_J close to h. Then |⟨ a_n_k-a,h⟩| ≤ |⟨ a_n_k-a,h_J⟩| +2Mnormh-h_J. For a prescribed tolerance, choose h_J first to control the second term and then k to control the first by finitely many coordinate limits. For a general h in the ambient space, write h=P_H_0h+h_⊥. The or","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"category Consider four vertices A,B,C,D, and the free path category generated by Axrightarrow0B, Bxrightarrow0A, Axrightarrow2C, Cxrightarrow5A. There are no arrows incident to D, apart from its identity. A path costs the sum of its edge labels. Concatenation is additive and empty paths cost zero, so all cost axioms hold. The directed distances are","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom","display":null,"kind":"use","excerpt":"The actual residual of a transport. Assume C is preadditive, so differences of maps make sense. The tower differential is (∂ f)_n=ρ^Y_nf_n+1-f_nρ^X_n. Its target has components Hom(X_n+1,Y_n), not Hom(X_n,Y_n).","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aa4ce940cc088be5","tex":"\\i","aliases":"i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8 node[box,minimum width=1.5cm] (xi) at (x,2) X_i; node[box,minimum width=1.5cm] (yi) at (x,0) Y_i; draw[arr] (xi) -- node[lab,right] f_i (yi); foreach j/i in 1/0,2/1,3/2 draw[arr] (xj) -- node[lab,above] ρ^X_i (xi); draw[arr] (yj) -- node[lab,below] ρ^Y_i (yi); node[lab,text width=12.2cm] at (0,","count":14},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a3dbd696a54cd3ed","tex":"I_E","aliases":"I_E","display":"I_E","kind":"use","excerpt":"A section is a choice of residual coordinates. Let d:E→ R be a continuous linear map between normed spaces over a nontrivially normed field. Suppose it has a continuous linear right inverse A:R→ E, so dA=I_R. Set K=ker d, P_A=I_E-Ad. Then dP_A=0, P_A fixes K, and P_A^2=P_A. The right inverse chooses where each residual is represented inside the can","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0625c4dc07eb960a","tex":"i_j","aliases":"i_j","display":"i_j","kind":"use","excerpt":"The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-33548fbd7d20a181","tex":"i_{j+1}","aliases":"i_j+1","display":"i_{j+1}","kind":"use","excerpt":"The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-714b73df80781bff","tex":"i_{m-1}","aliases":"i_m-1","display":"i_{m-1}","kind":"use","excerpt":"Distinct walks are retained as arrows of the view category. The product along a walk q=(i_0,…,i_m) is T_q=τ_i_0i_1⋯τ_i_m-1i_m. Concatenation gives the restriction composition law. An exact vertex family defines a coherent walk-view section, and a coherent walk-view section satisfies every edge equation. Thus this graph construction returns to the o","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5dc7ccbeace33e78","tex":"I_R","aliases":"I_R","display":"I_R","kind":"use","excerpt":"[Split tower correction] The recurrence defines a continuous linear map A:R→ E with ∂ A=I_R, norm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial co","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f3d67654dfc71a7c","tex":"I_{R_j}","aliases":"I_R_j","display":"I_{R_j}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3a19cea3d43d3b3d","tex":"I_{R_N}","aliases":"I_R_N","display":"I_{R_N}","kind":"use","excerpt":"j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent ma","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ad3476e646d4f9c7","tex":"I_\\V","aliases":"I_V V","display":null,"kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b1185bc920eea10a","tex":"\\id_A","aliases":"id_A id","display":null,"kind":"use","excerpt":"Extended directed distance. Let C be a category and let W assign each arrow a value in the extended nonnegative reals [0,∞]. Suppose W(id_A)=0, W(gf)≤ W(f)+W(g). The order of ordinary composition is first f, then g. Define d_W(A,B)=inf_f:A→ BW(f). The infimum of an empty family is ∞. No assumption of connectedness, finite cost, symmetry, or attainm","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-011988ed9bdfec11","tex":"\\id_F","aliases":"id_F id","display":null,"kind":"use","excerpt":"[Functorial transport] These induced maps preserve identities and composition: Γ(id_F)=id_Γ(F), Γ(βα)=Γ(β)Γ(α). Evaluate each side at an arbitrary view and apply the corresponding identity for the coordinate maps. Theorem~[source reference] then identifies the maps on the whole realization set. sourceKS02 An arbitrary collection of fiber maps need ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-974a5ed6832aaea3","tex":"\\id_\\W","aliases":"id_W id W","display":null,"kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-14fac349a823622b","tex":"\\id_{\\Gamma(F)}","aliases":"id_Γ(F) id Gamma","display":null,"kind":"use","excerpt":"[Functorial transport] These induced maps preserve identities and composition: Γ(id_F)=id_Γ(F), Γ(βα)=Γ(β)Γ(α). Evaluate each side at an arbitrary view and apply the corresponding identity for the coordinate maps. Theorem~[source reference] then identifies the maps on the whole realization set. sourceKS02 An arbitrary collection of fiber maps need ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"Extended directed distance. Let C be a category and let W assign each arrow a value in the extended nonnegative reals [0,∞]. Suppose W(id_A)=0, W(gf)≤ W(f)+W(g). The order of ordinary composition is first f, then g. Define d_W(A,B)=inf_f:A→ BW(f). The infimum of an empty family is ∞. No assumption of connectedness, finite cost, symmetry, or attainm","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f3d77613c0887bee","tex":"\\inf_{b\\in B}","aliases":"inf_b∈ B inf in","display":"\\inf_{b\\in B}","kind":"use","excerpt":"Distance to the actual correction image. Let R be a normed additive group and B≤ R an additive subgroup. In the protected linear model, B=d(ker p). For a residual Ω∈ R, define ν_B(Ω)=inf_b∈ BnormΩ-b. The subgroup contains zero, so this is a finite nonnegative real number. It is an infimum, not an assertion that a nearest admissible residual exists.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1f4466ecc1e7344a","tex":"\\inf_{f:A\\to B}","aliases":"inf_f:A→ B inf to","display":"\\inf_{f:A\\to B}","kind":"use","excerpt":"Extended directed distance. Let C be a category and let W assign each arrow a value in the extended nonnegative reals [0,∞]. Suppose W(id_A)=0, W(gf)≤ W(f)+W(g). The order of ordinary composition is first f, then g. Define d_W(A,B)=inf_f:A→ BW(f). The infimum of an empty family is ∞. No assumption of connectedness, finite cost, symmetry, or attainm","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-82ee963b450755be","tex":"\\inf_{\\norm a\\le M}","aliases":"inf_norm a≤ M inf norm le","display":null,"kind":"use","excerpt":"chapThe attainable residual at a fixed correction budget The unrestricted floor dist(r,ran T) forgets how large a correction must be. To retain that information, fix a nonnegative budget M and define β_T,r(M)=inf_norm a≤ Mnormr-Ta. The norm is the actual correction norm in H=ker p. It is not the norm of arbitrary coordinates used to represent that ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-99f24b2b87ddb9f1","tex":"\\inf_{z\\in C}","aliases":"inf_z∈ C inf in","display":"\\inf_{z\\in C}","kind":"use","excerpt":"d convex, and fix x∈ H. There is a unique c∈ C minimizing normx-c. It is characterized by ⟨ x-c,z-c⟩≤0 (z∈ C). If C=c_0+L is a closed affine subspace with direction space L, the condition is x-c⊥ L. Produce a Cauchy minimizing sequence. Put d=inf_z∈ Cnormx-z. Nonemptiness makes this a finite nonnegative real number. Choose c_n∈ C so that normx-c_n→","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-485db41ee683b54c","tex":"J_0","aliases":"J_0","display":"J_0","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-08e93c501f7d1688","tex":"J_1","aliases":"J_1","display":"J_1","kind":"use","excerpt":"0,in=50] node[lab,above] J_i (ri.north); foreach i/x in 0/-4.8,1/-1.6,2/1.6 node[box,minimum width=1.6cm] (hi) at (x,-.3) h_i; draw[arr] (ri) -- node[lab,right] detail (hi); node[box,draw=KomorebiOchre,fill=KomorebiWarm,minimum width=2.1cm] (q) at (4.8,-.3) Q_3x=L_3x_3; draw[arr] (r3)--node[lab,right] restore (q); node[lab,text width=13cm] at (0,-1","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-694a109c6881c1d9","tex":"J_{\\mathrm{comp}}","aliases":"J_comp mathrm","display":"J_{\\mathrm{comp}}","kind":"use","excerpt":"[The composition topology] Condition~[source reference] defines a Grothendieck topology J_comp. The sieve generated by a family of inclusions U_i⊆ W is covering exactly when the family covers every composition support in W. In the inclusion poset, identify a sieve on W with a downward-closed collection of subsets of W. Pullback along Z⊆ W consists ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-49737c591a8b7acf","tex":"J_{\\i}","aliases":"J_i i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8node[box,minimum width=1.5cm] (ri) at (x,1.7) x_i∈ R_i; foreach i/j in 0/1,1/2,2/3 draw[arr] (ri.east) -- node[lab,above] C_i (rj.west); draw[thinarr] (rj.north) to[out=130,in=50] node[lab,above] J_i (ri.north); foreach i/x in 0/-4.8,1/-1.6,2/1.6 node[box,minimum width=1.6cm] (hi) at (x,-.3) h_i","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-147f12c21f03b9ea","tex":"J_{M-1}","aliases":"J_M-1","display":"J_{M-1}","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6cca5b503f09502b","tex":"J_N","aliases":"J_N","display":"J_N","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2501a2cd4507c654","tex":"J_{N,M}","aliases":"J_N,M","display":"J_{N,M}","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9ec98c07d696ff5a","tex":"J_{N-1}","aliases":"J_N-1","display":"J_{N-1}","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5cbd3ac214345870","tex":"J_\\alpha","aliases":"J_α alpha","display":"J_\\alpha","kind":"use","excerpt":"The normal equation is derived in the protected space. For α>0, define on H=ker p J_α(a)=normTa-r^2+αnorm a^2. Both norms are the declared Hilbert norms. All minimizers below are corrections in H, so preservation is exact for every parameter, not just in the limit. This is ordinary quadratic Tikhonov regularization applied to the actual protected c","count":14},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d028b14938c1ba39","tex":"J_{\\alpha_n}","aliases":"J_α_n alpha","display":"J_{\\alpha_n}","kind":"use","excerpt":"Why a vanishing normal residual is not enough. Let T=0 on R, let r=0, and take α_n=n^-2 for n≥1. The exact optimizer is zero for every such n. The fixed proposal a_n=1 instead has v_n=0, g_n=α_n⟶0, J_α_n(a_n)=α_n⟶0, absa_n-a_α_n=1. Thus the fit residual, the normal residual, and even the absolute objective value can all tend to zero without converg","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a7e0c315ce3cf160","tex":"\\K","aliases":"K","display":null,"kind":"use","excerpt":"For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0,","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"Internal realization and intended objects. Suppose a separate category S contains the intended semantic objects of an application, and a realization construction gives a comparison functor R: S⟶ K, where K is the relevant category or groupoid of coherent data. Defining K does not establish that R is faithful, full, or essentially surjective. These ","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fc2228c8ece1f90d","tex":"\\K^{1\\times m}","aliases":"K^1× m K times","display":null,"kind":"use","excerpt":"[A finite certificate alternative] Exactly one of the following occurs: item There is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the po","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0834cc9214348f2c","tex":"\\K^{1\\times p}","aliases":"K^1× p K times","display":null,"kind":"use","excerpt":"[A finite certificate alternative] Exactly one of the following occurs: item There is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the po","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"Every other correction can be written uniquely as h_*+k with k∈ker B. Orthogonality gives normh_*+k^2=normh_*^2+norm k^2. Writing y_*=x-h_* and y=x-h_*-k, we have x-y=h_*+k and y_*-y=k. Substitution proves the stated Pythagorean identity. Its last term vanishes only when y=y_*, establishing the unique closest answer.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-16b09a3431d14563","tex":"K^3","aliases":"K^3","display":"K^3","kind":"use","excerpt":"is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9ead51bf85f85425","tex":"\\K^{m+p}","aliases":"K^m+p K","display":null,"kind":"use","excerpt":"m has zero left side and nonzero right side, assign arbitrary values to its free variables and solve for its pivot variables. This supplies a solution y_0. For any vector y, Ty=b Longleftrightarrow T(y-y_0)=0. Translation by y_0 is therefore a bijection from ker T to the answer set, with inverse y↦ y-y_0. Rank--nullity applied to the actual linear ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a16ce646a1aaff3e","tex":"\\K^{m\\times n}","aliases":"K^m× n K times","display":null,"kind":"use","excerpt":"For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fe0fe70909341737","tex":"\\K^n","aliases":"K^n K","display":null,"kind":"use","excerpt":"For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0,","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-473d859b7a40335f","tex":"\\K^{p\\times n}","aliases":"K^p× n K times","display":null,"kind":"use","excerpt":"For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3ff8bbec26df05be","tex":"K^\\perp","aliases":"K^⊥ perp","display":"K^\\perp","kind":"use","excerpt":"ntitative exactness] The following are equivalent: item ran D is closed. item There is C≥0 with normx-qx≤ CnormDx for all x. item There is C≥0 such that every x has some exact y with normx-y≤ CnormDx. When the range is closed, the least admissible constant is the operator norm of the inverse of D:K^⊥→ran D. In the zero operator case that inverse ha","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9817d114fe256b80","tex":"k_0","aliases":"k_0","display":"k_0","kind":"use","excerpt":"Let k_0=P^M_ker Bh_0 and h_*=h_0-k_0. Then Bh_*=Bh_0=Dx and h_*⊥ker B in M. If a different initial correction is chosen, it differs from h_0 by an element of ker B. Its orthogonal-complement component is therefore again h_*. This proves that the construction is independent of the feasibility witness.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b783228933c35b11","tex":"k_i","aliases":"k_i","display":"k_i","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b35f8ec2c58ce5a0","tex":"k_n","aliases":"k_n","display":"k_n","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"chapBanach repair and constructed split coordinates An actual complete candidate space. Let X_n,Y_n be Banach spaces over a complete nontrivially normed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formal","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-42ada035ee0b4eaf","tex":"L^3","aliases":"L^3","display":"L^3","kind":"use","excerpt":"The tails require a bound independent of sufficiently fine discretization. Since N≥ mL/2, the inequalities f(x)≤ x^-4 and g(x)≤ x^-2 give frac1m∑_n>Nf(n/m) ≤ m^3∑_n>Nn^-4≤fracm^33N^3 ≤frac83L^3, frac1m∑_n>Ng(n/m) ≤ m∑_n>Nn^-2≤frac mN≤frac2L. The integral tails are at most 1/(3L^3) and 1/L. Given an error tolerance, choose L first to control both di","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b3f55c1a3fc262cb","tex":"L_{0,N}","aliases":"L_0,N","display":"L_{0,N}","kind":"use","excerpt":"Now Q_Nr=L_0,Nx_N, so the support assertion applies to the restored remainder. Since S_Nr=r-Q_Nr, it gives the reconstruction equation. Finally, if two residuals have identical details at every scale, the finite telescope makes their S_N reconstructions equal for every N. At any fixed coordinate choose N with k+1<2^N. The reconstruction equation th","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c7f6796e75b9a514","tex":"L_3","aliases":"L_3","display":"L_3","kind":"use","excerpt":"draw[arr] (ri.east) -- node[lab,above] C_i (rj.west); draw[thinarr] (rj.north) to[out=130,in=50] node[lab,above] J_i (ri.north); foreach i/x in 0/-4.8,1/-1.6,2/1.6 node[box,minimum width=1.6cm] (hi) at (x,-.3) h_i; draw[arr] (ri) -- node[lab,right] detail (hi); node[box,draw=KomorebiOchre,fill=KomorebiWarm,minimum width=2.1cm] (q) at (4.8,-.3) Q_3x","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4808d25c7c0c8b6f","tex":"l_b","aliases":"l_b","display":"l_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ed4117db0a118834","tex":"l_c","aliases":"l_c","display":"l_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0672c5d4c101de32","tex":"l_d","aliases":"l_d","display":"l_d","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1459f22abbadb5f5","tex":"l_g","aliases":"l_g","display":"l_g","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a00df4a1263286e2","tex":"l_i","aliases":"l_i","display":"l_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a260dece327539f2","tex":"L_{j+1}","aliases":"L_j+1","display":"L_{j+1}","kind":"use","excerpt":"Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1e5f817a12a89680","tex":"L_{j+1,N}","aliases":"L_j+1,N","display":"L_{j+1,N}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c119997d7f38f38c","tex":"L_{j,0}","aliases":"L_j,0","display":"L_{j,0}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ce13126dfae5eb88","tex":"L_{j,N}","aliases":"L_j,N","display":"L_{j,N}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6ad58f1d489a5ca5","tex":"L_{j,N+1}","aliases":"L_j,N+1","display":"L_{j,N+1}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-06367b05ddac23e5","tex":"l_l","aliases":"l_l","display":"l_l","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a47d9642859cb109","tex":"L_M","aliases":"L_M","display":"L_M","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7e66b64743d52741","tex":"l_m","aliases":"l_m","display":"l_m","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c91dabfc8b700b3b","tex":"L_N","aliases":"L_N","display":"L_N","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a113110bb27da5fa","tex":"l_n","aliases":"l_n","display":"l_n","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b08b1edd5a95d60d","tex":"l_t","aliases":"l_t","display":"l_t","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4bd606a0f9ec9dda","tex":"l_w","aliases":"l_w","display":"l_w","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":")=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a mod","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"Continuity of d implies d(P_Af)=lim_m r_m=0. Protection holds before taking a limit: pA=0 gives pf^(m+1)=pf^(m), hence pf^(m)=pf for every m. Continuity of p gives p(P_Af)=pf. This explains why continuity of the observable belongs to this analytic statement.","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-538b1d43a9783a43","tex":"\\lim_k","aliases":"lim_k lim","display":"\\lim_k","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ccb8a376eaddc6e8","tex":"\\lim_m","aliases":"lim_m lim","display":"\\lim_m","kind":"use","excerpt":"Continuity of d implies d(P_Af)=lim_m r_m=0. Protection holds before taking a limit: pA=0 gives pf^(m+1)=pf^(m), hence pf^(m)=pf for every m. Continuity of p gives p(P_Af)=pf. This explains why continuity of the observable belongs to this analytic statement.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-76d061defd0810fd","tex":"\\lim_{M\\to\\infty}","aliases":"lim_M→∞ lim to infty","display":"\\lim_{M\\to\\infty}","kind":"use","excerpt":"sidual minimum exists [Attainment and shape of the budget profile] For a bounded linear map between real Hilbert spaces, the infimum defining β(M) is attained for every M≥0. The profile is nonincreasing and convex. For 0≤ M_1≤ M_2, 0≤β(M_1)-β(M_2)≤norm T(M_2-M_1). Writing C=overlineran T, b=P_Cr, and q=normr-b, one has lim_M→∞β(M)=q. The equality β","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-53f3164e2d42bc85","tex":"\\lim_n","aliases":"lim_n lim","display":"\\lim_n","kind":"use","excerpt":"Now assume condition 2. To prove closedness, take an arbitrary sequence r_n∈ran D with r_n→ r in R. Choose x_n with Dx_n=r_n and put u_n=x_n-qx_n. Then u_n∈ K^⊥ and Du_n=r_n. Applying the estimate to u_n-u_m gives normu_n-u_m≤ Cnormr_n-r_m. Thus (u_n) is Cauchy in the complete space K^⊥. Let its limit be u. Continuity gives Du=lim_n r_n=r, so r∈ran","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e5c19d75887aa542","tex":"\\liminf","aliases":"liminf","display":"\\liminf","kind":"use","excerpt":"The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 b","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bef3746ed99e451b","tex":"\\liminf_k","aliases":"liminf_k liminf","display":"\\liminf_k","kind":"use","excerpt":"The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 b","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-45e90aa85790529d","tex":"\\linewidth","aliases":"linewidth","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"Solution. The sampled value is η_ϕ(k)=2^-k. Recovering the original index gives k=log_2(n+1) on the sampled stages, so the value is still 1/(n+1). The schedule preserves vanishing but changes the meaning of the horizontal axis. No faster original process has been proved. Proposition~[source reference] is a convergence equivalence under a monotone c","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"Solution. The sampled value is η_ϕ(k)=2^-k. Recovering the original index gives k=log_2(n+1) on the sampled stages, so the value is still 1/(n+1). The schedule preserves vanishing but changes the meaning of the horizontal axis. No faster original process has been proved. Proposition~[source reference] is a convergence equivalence under a monotone c","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Extension criterion] For every coherent partial section, o_U(s)=0 Longleftrightarrow Ext_U(s)≠∅. Let π_U:R→ R/B_U be the quotient homomorphism. By the definition of a quotient by a subgroup, π_U(r)=0 Longleftrightarrow r∈ B_U. Here B_U is the image of ker p_U under δ. Consequently membership of δwidetilde s means the existence of a specific h∈ E s","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-768071a8b8aba296","tex":"m^{-1}","aliases":"m^-1","display":"m^{-1}","kind":"use","excerpt":"infinite problem, lim_M→∞Mβ_T,w(M)=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniform","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e615f08a31cd0f9c","tex":"m^{-1/2}","aliases":"m^-1/2","display":"m^{-1/2}","kind":"use","excerpt":"Thus the correction norm grows on the order of sqrt m, the residual norm decays on the order of m^-1/2, and the penalty α A_m=A_m/m^2 tends to zero. The smooth correction avoids the hard discontinuity at a prefix boundary, but it still approaches the forbidden constant sequence coordinate by coordinate.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2d9b06f7801188ec","tex":"m^{-2}","aliases":"m^-2","display":"m^{-2}","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9155222104384cb9","tex":"m^3","aliases":"m^3","display":"m^3","kind":"use","excerpt":"For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-af37a2ae9e4efb41","tex":"m^4","aliases":"m^4","display":"m^4","kind":"use","excerpt":"[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bb04371f38e2158f","tex":"M_{21}","aliases":"M_21","display":"M_{21}","kind":"use","excerpt":"For two consecutive compatible changes (S_1,C_1), (S_2,C_2), with compatible choices of sections at the three ends, their mismatches satisfy M_21=S_2M_1+M_2C_1. If dx_j→0 in norm for a fixed change and fixed sections, the repair discrepancy tends to zero. First DS=Cd sends ker d into ker D, and DM=DSA-DBC=CdA-C=C-C=0. Thus both terms in the propose","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bc4c24fa42db60db","tex":"m_a","aliases":"m_a","display":"m_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0f5cef98cfcf15fe","tex":"M_c","aliases":"M_c","display":"M_c","kind":"use","excerpt":"sectionA two-coordinate model in which the routes differ Use E=F=R^2, write candidates as (u,r), and take d(u,r)=D(u,r)=r, A(r)=B(r)=(0,r). The exact space is the horizontal axis and both repairs are P(u,r)=(u,0). For a real parameter c, set S_c(u,r)=(u+cr,r), C=I. The defect square commutes because the second coordinate is unchanged. The mismatch ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f1ff17ffbc6328a0","tex":"M_{c_1}","aliases":"M_c_1","display":"M_{c_1}","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4c6336cd396007b7","tex":"M_{c_2}","aliases":"M_c_2","display":"M_{c_2}","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4a4cfce379dd77c2","tex":"M_{c_2+c_1}","aliases":"M_c_2+c_1","display":"M_{c_2+c_1}","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3cc0ed3324bf0e59","tex":"M_m","aliases":"M_m","display":"M_m","kind":"use","excerpt":"Step 3: move from regularization parameters to arbitrary budgets. Set M_m=sqrtA_m. Every positive coordinate m^2/(m^2+n^2) increases strictly with m, so A_m, and hence M_m, increases strictly. Since A_m/m→ c=π/4>0, these budgets tend to infinity and (fracM_m+1M_m)^2 =fracA_m+1/(m+1)A_m/mfracm+1m⟶1. Proposition~[source reference] identifies β(M_m)=s","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3ba0c3b06de090fa","tex":"M_{m+1}","aliases":"M_m+1","display":"M_{m+1}","kind":"use","excerpt":"Step 3: move from regularization parameters to arbitrary budgets. Set M_m=sqrtA_m. Every positive coordinate m^2/(m^2+n^2) increases strictly with m, so A_m, and hence M_m, increases strictly. Since A_m/m→ c=π/4>0, these budgets tend to infinity and (fracM_m+1M_m)^2 =fracA_m+1/(m+1)A_m/mfracm+1m⟶1. Proposition~[source reference] identifies β(M_m)=s","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c1f72b92c8bb119d","tex":"m_N^2","aliases":"m_N^2","display":"m_N^2","kind":"use","excerpt":"Suppose these energies are bounded, and let L=lim_n m_n^2. Given ϵ>0, choose N with L-m_N^2<ϵ^2. For m≥ n≥ N, norma_m-a_n^2=m_m^2-m_n^2≤ L-m_N^2<ϵ^2. Thus (a_n) is Cauchy and has a norm limit a_*. For each fixed n, all terms from that index onward belong to the closed set A_n, so a_*∈ A_n. This places a_* in their entire intersection.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-19afb20993a1d995","tex":"m_r","aliases":"m_r","display":"m_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6f3ae7d6b37bdd61","tex":"m_t","aliases":"m_t","display":"m_t","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4ff979fe78820d20","tex":"M_v","aliases":"M_v","display":"M_v","kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cc2f5f4bfec5fb41","tex":"M_\\alpha","aliases":"M_α alpha","display":"M_\\alpha","kind":"use","excerpt":"Parameter monotonicity and the fixed-budget problem. [Regularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing ","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f7520c75e0aae3a9","tex":"\\Match","aliases":"Match","display":null,"kind":"use","excerpt":"[Strict gluing] Under [source reference], restriction induces an equivalence Γ(F)≃Match(F,(U_i)). Every fixed matching family has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an ex","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"ctions, two reindexed maps out of X are eventually equal exactly when they induce the same realized function. Surjective adjacent bonding maps are a sufficient condition for surjective point projections. Realized equality follows from eventual equality by the preceding theorem, so only its converse remains. Suppose |f|=|g|. Fix the output level n, ","count":15},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bae859577c210728","tex":"\\max_{0\\le i\\le n}","aliases":"max_0≤ i≤ n max le le","display":"\\max_{0\\le i\\le n}","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"use","excerpt":"The maps are linear. On the finite vertex product, normR_pz_∞≤(max_inormT_i^-1)norm z, normx_o≤norm x_∞. These bounds also hold on the subspaces K and ker d, proving continuity of both directions. No completeness is needed for these finite bounds.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e3e0c903e06afb04","tex":"\\max_{\\norm z\\le1}","aliases":"max_norm z≤1 max norm le","display":null,"kind":"use","excerpt":"A dual witness can rule out every correction in the budget. [A sharp budget certificate] For every M≥0, β_T,r(M)= max_norm z≤1(⟨ r,z⟩-MnormT^*z). Thus any single z with norm z≤1 and ⟨ r,z⟩-MnormT^*z>ε certifies that no correction of norm at most M has residual at most ε. For every norm a≤ M and norm z≤1, the adjoint identity and Cauchy--Schwarz giv","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"ositive integer m. The only possible unrecovered coordinates at depth N are 2^N-1, 2·2^N-1,quad3·2^N-1, …. Prove the assertion for L_j,N by induction, uniformly in j. At depth zero every positive integer is divisible by one, so there is nothing to prove. At depth N+1, even coordinates vanish. If k=2m+1, then 2^N+1mid(k+1) Longleftrightarrowquad2^Nm","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3f9db2ae5742966b","tex":"\\midrule","aliases":"midrule","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b75f3a902c5f7e60","tex":"n^{-1}","aliases":"n^-1","display":"n^{-1}","kind":"use","excerpt":"A small residual far from exactness. On ℓ^2(N), index the standard basis from one and define De_n=n^-1e_n. The map is bounded and injective, so its exact space is zero. Yet norme_n=1, normDe_n=1/n⟶0. No finite C can bound dist(e_n,ker D) by CnormDe_n. The range is not closed. For example finite truncations of (1/n)_n≥1 belong to the range and conve","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c0769f9271add232","tex":"n^{-2}","aliases":"n^-2","display":"n^{-2}","kind":"use","excerpt":"Finally, for K≥ m, both omitted tails consist of nonnegative terms. Bound the first by m^4∑_n>Kn^-4 and the second by ∑_n>Kn^-2. The same upper integral comparisons give the two certified remainders in [source reference]. No finite truncation is substituted for an equality of infinite energies. writtenWR10","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-076d94003a73a0df","tex":"n^{-3}","aliases":"n^-3","display":"n^{-3}","kind":"use","excerpt":"Solution. The noise term is at most 1/(2n), but the solver term is at most one. Thus the displayed tolerance does not provide the required guarantee. The scalar counterexample T=0, r=0, a_n=1 meets it and does not approach the exact optimizer. Replacing it by n^-3 makes the solver term at most 1/n, so the triangle inequality and the assumed vanishi","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-55d7cf56037490a0","tex":"n^{-4}","aliases":"n^-4","display":"n^{-4}","kind":"use","excerpt":"Finally, for K≥ m, both omitted tails consist of nonnegative terms. Bound the first by m^4∑_n>Kn^-4 and the second by ∑_n>Kn^-2. The same upper integral comparisons give the two certified remainders in [source reference]. No finite truncation is substituted for an equality of infinite energies. writtenWR10","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"Escape is not an accident of this particular algorithm. Suppose an arbitrary protected candidate (u,1) satisfies normd(u,1)≤ε. On its first N coordinates, ∑_n=1^N|u_n+1|^2 ≤ N^2∑_n=1^Nfrac|u_n+1|^2n^2 ≤ N^2ε^2. The reverse triangle inequality in R^N gives norm u≥sqrt N-Nε (N≥1). If a sequence of protected candidates has residual tending to zero, th","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"Unbounded gaps cause a different issue. With ϕ(n)=n^2, the bounded residual family r_k=1 has block sums 2n+1. These are unbounded. Therefore block accumulation cannot be a map from every bounded residual family to a bounded sampled residual family in this example. The constant residual is not claimed to be the differential of a bounded candidate on","count":24},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"The tails require a bound independent of sufficiently fine discretization. Since N≥ mL/2, the inequalities f(x)≤ x^-4 and g(x)≤ x^-2 give frac1m∑_n>Nf(n/m) ≤ m^3∑_n>Nn^-4≤fracm^33N^3 ≤frac83L^3, frac1m∑_n>Ng(n/m) ≤ m∑_n>Nn^-2≤frac mN≤frac2L. The integral tails are at most 1/(3L^3) and 1/L. Given an error tolerance, choose L first to control both di","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ac5e944c84a2e799","tex":"n^4","aliases":"n^4","display":"n^4","kind":"use","excerpt":"fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and w=(1/n)∈ℓ^2, hence the residual sequence i","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b18687907767a3b7","tex":"\\N_{>0}","aliases":"N_>0 N","display":null,"kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-99f3c225ef6f9c62","tex":"n_c","aliases":"n_c","display":"n_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b9588331116f694e","tex":"n_e","aliases":"n_e","display":"n_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cf9404d3f534ee7e","tex":"n_k","aliases":"n_k","display":"n_k","kind":"use","excerpt":"The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 b","count":15},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-35b602543437bbe8","tex":"N_n","aliases":"N_n","display":"N_n","kind":"use","excerpt":"Protected information beyond the seed. When the target grows, its new kernel directions are independent information. Define N_n(f)=(I-s_nρ^Y_n)f_n+1, (p_Lf)_0=f_0, (p_Lf)_n+1=N_n(f). Each N_n(f) lands in kerρ^Y_n. The observable p_L retains both the seed and every new output component, relative to the chosen sections.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aaec29026bff78cd","tex":"n_o","aliases":"n_o","display":"n_o","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-26511a3b0a02612d","tex":"N_p","aliases":"N_p","display":"N_p","kind":"use","excerpt":"Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5670a09f4b5b441b","tex":"N_p^{-1}","aliases":"N_p^-1","display":"N_p^{-1}","kind":"use","excerpt":"dist(w,ker d)≤ Cnormdw_∞ for every vertex family w. Fix z∈ K^⊥ and take w=R_pz. Every exact family e has e_o∈ K, and root evaluation has norm at most one. Therefore norm z=dist(z,K)≤normz-e_o≤normR_pz-e_∞. Taking the infimum over e∈ker d is legitimate without assuming that this global nearest point is attained. It gives norm z≤ CnormdR_pz_∞ ≤ Cnorm","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-232d662923754272","tex":"\\name","aliases":"name","display":null,"kind":"use","excerpt":"r] (a1)--(a2); draw[thinarr] (a0.north) to[out=60,in=120] (a2.north); foreach i/x in 0/1.4,1/2.8,2/4.2,3/5.6node[box,circle,inner sep=4pt] (bi) at (x,2.3) i; draw[arr](b0)--(b1);draw[arr](b1)--(b2);draw[arr](b2)--(b3); foreach i/x in 0/-5.4,1/-3.8,2/-2.2node[lab] at (x,1.25) i; foreach j/y/name in 0/.55/01,1/-.15/02,2/-.85/12node[lab] at (-6.25,y) ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-64824291a60cddb8","tex":"\\newline","aliases":"newline","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"[The dyadic support pattern] For the same lifts, 2^Nnmid(k+1) Longrightarrow (Q_Nr)_k=0. Thus coordinate k is certainly recovered once N>ν_2(k+1), where ν_2(m) is the largest power-of-two exponent dividing the positive integer m. The only possible unrecovered coordinates at depth N are 2^N-1, 2·2^N-1,quad3·2^N-1, …. Prove the assertion for L_j,N by","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-218f4a32e8adb732","tex":"o_a","aliases":"o_a","display":"o_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5e3f9866da12f022","tex":"o_f","aliases":"o_f","display":"o_f","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7fc03b5db3831c26","tex":"o_i","aliases":"o_i","display":"o_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d1a47c863ac7fc79","tex":"o_p","aliases":"o_p","display":"o_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-490c00130e07d0d8","tex":"\\mathfrak o_W","aliases":"o_W mathfrak","display":"\\mathfrak o_W","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0bb121817ecaf82e","tex":"o_z","aliases":"o_z","display":"o_z","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-54037e9a059bedc5","tex":"\\overline B","aliases":"overline B overline","display":"\\overline B","kind":"use","excerpt":"[The floor detects closure] For every b∈ B, 0≤ν_B(Ω)≤normΩ-b. Moreover, ν_B(Ω)=0 Longleftrightarrow Ω∈overline B. If B is closed, this is equivalent to Ω∈ B. The set of candidate distances is nonempty because 0∈ B, bounded below by zero, and contains the finite value normΩ. Its infimum is therefore a finite nonnegative real, and the defining lower-","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f5ab84da5f379ea9","tex":"\\overline{\\mathcal F}","aliases":"overline F overline mathcal","display":"\\overline{\\mathcal F}","kind":"use","excerpt":"[Density criterion for norm recovery] Suppose sup_NnormQ_N≤ K<∞. Then the norm recoverable vectors are exactly overline F. In particular all vectors are norm recoverable if and only if the finite core is dense. The nested identities first imply that the kernels increase: if Q_My=0 and N≥ M, then Q_Ny=Q_NQ_My=0. Thus the finite core really is an inc","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e4b45cd66b9c8f4e","tex":"\\overline L","aliases":"overline L overline","display":"\\overline L","kind":"use","excerpt":"Equivalently, if r-r'∈ d(ker p), then L(r-r')=0, so L(r)=L(r'). Hence overline L([r])=L(r) is a well-defined homomorphism on R/d(ker p). Its value on [dx] is nonzero. This supplies the quotient interpretation of the same contradiction, without assuming that every nonzero quotient class can be detected by a map into one fixed preferred coefficient g","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1a02ef2c814691f5","tex":"\\overline R","aliases":"overline R overline","display":"\\overline R","kind":"use","excerpt":"chapMultiscale residuals and the limits of reconstruction A split coarsener retains its missing detail. Let C:R→overline R and J:overline R→ R be linear maps with CJ=I. Put H=I-JC. Then r=JCr+Hr, CH=0, H^2=H. The two coordinates (Cr,Hr) retain all information. The coarse coordinate alone generally does not.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-553c7221d52abe12","tex":"\\overline{d(\\ker p)}","aliases":"overlined(ker p) overline ker","display":"\\overline{d(\\ker p)}","kind":"use","excerpt":"The bridge back to exactification. Proposition~[source reference] and Proposition~[source reference] fit together closely. If corrections produce a norm-convergent, protected path with vanishing residual, the actual quotient class vanishes. If only the corrected residuals tend to zero, all that follows is membership in overlined(ker p).","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cd341e70441d1406","tex":"\\overline{p_i}","aliases":"overlinep_i overline","display":"\\overline{p_i}","kind":"use","excerpt":"Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-67b35b32b305e3c7","tex":"\\overline{\\ran T}","aliases":"overlineran T overline ran","display":null,"kind":"use","excerpt":"An immediate contrapositive is useful. If r∈overlineran T∖ran T, then every sequence with Ta_n→ r satisfies norma_n→∞. Otherwise infinitely many terms would lie in one bounded ball, contradicting the theorem along that subsequence. This recovers the escape mechanism of Section~[source reference] without using the diagonal formula.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4af27a350f0c4ed1","tex":"\\overline{S(2nB_X)}","aliases":"overlineS(2nB_X) overline","display":"\\overline{S(2nB_X)}","kind":"use","excerpt":"mma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆overlineS(nB_X). Here B(y_0,ρ) is an open ball. If normh<ρ, both y_0+h and y_0 belong to this closure. Choose approximating image sequences for both. Subtracting their preimages shows h∈overlineS(2nB_X).","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6e36290a91c7877b","tex":"\\overline{S(B_X)}","aliases":"overlineS(B_X) overline","display":"\\overline{S(B_X)}","kind":"use","excerpt":"ere are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆overlineS(nB_X). Here B(y_0,ρ) is an open ball. If normh<ρ, both y_0+h and y_0 belong to this closure. Choose approximating image sequences for both. Subtracting their preimages shows h∈overlineS(2nB_X). Scaling by 2n gives a constant c=ρ/(2n)>0 with B(0,c)⊆overlineS(B_X). This only gives approximate ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"Set P=I-A∂. Then ∂ P=∂-∂ A∂=0, P^2=I-2A∂+A(∂ A)∂=I-A∂=P. The expression 2A∂ abbreviates addition of the same operator twice and uses no division by two. Every exact f is fixed because ∂ f=0. Since (Ar)_0=0, the seed of Pf is f_0. Finally f-Pf=A∂ f, and the bound on A gives the asserted residual estimate. The denominators above are justified exactly","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"For H=S_3, these elements are the identity and the three transpositions. There are exactly two orbit classes. The identity cocycle has automorphism group S_3, which is noncommutative. A nonidentity involution gives a coherent cocycle whose regular twisted section equation x=px has no solution. The only nontrivial cocycle relation is p^2=1. At a sin","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-26f9b4dce778eaea","tex":"P^M_{\\ker B}","aliases":"P^M_ker B ker","display":"P^M_{\\ker B}","kind":"use","excerpt":"Let k_0=P^M_ker Bh_0 and h_*=h_0-k_0. Then Bh_*=Bh_0=Dx and h_*⊥ker B in M. If a different initial correction is chosen, it differs from h_0 by an element of ker B. Its orthogonal-complement component is therefore again h_*. This proves that the construction is independent of the feasibility witness.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"2. A residual chart can change without changing the defect. Let d:R^2→R be d(u,r)=r. For a∈R, let A_a(r)=(ar,r). Compute the exact repair P_a, compare it with P_b, and decide when they commute.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e7d1a3be90d27945","tex":"p_A","aliases":"p_A","display":"p_A","kind":"use","excerpt":"Preservation changes the admissible arrows. One response to collapse is to restrict arrows to those satisfying a declared preservation contract. For objects equipped with maps p_A:A→ S, a morphism can be required to satisfy p_Bf=p_A. Identities and composition preserve this equation. Unless p_A=0, the zero arrow is not admissible. In other applicat","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-93f05451e0c22e43","tex":"P_A^2","aliases":"P_A^2","display":"P_A^2","kind":"use","excerpt":"n allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, every iterate equals f, hence P_Af=f. Its image is contained in ker d and it fixes that kernel, proving surjectivity onto its stated image and P_A^2=P_","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a5c35e870d97f9c4","tex":"P_B","aliases":"P_B","display":"P_B","kind":"use","excerpt":"[The split defect chart] The maps χ_A:E≃_L K× R, χ_A(x)=(P_Ax,dx), χ_A^-1(u,r)=u+Ar are inverse continuous linear equivalences. In these coordinates, P_A sends (u,r) to (u,0). If B is another right inverse, then A-B takes values in K, and P_Bx-P_Ax=(A-B)dx. The projections commute if and only if they are equal, and they are equal if and only if A=B","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","excerpt":"2. A residual chart can change without changing the defect. Let d:R^2→R be d(u,r)=r. For a∈R, let A_a(r)=(ar,r). Compute the exact repair P_a, compare it with P_b, and decide when they commute.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d072d8979775d522","tex":"p_B","aliases":"p_B","display":"p_B","kind":"use","excerpt":"Preservation changes the admissible arrows. One response to collapse is to restrict arrows to those satisfying a declared preservation contract. For objects equipped with maps p_A:A→ S, a morphism can be required to satisfy p_Bf=p_A. Identities and composition preserve this equation. Unless p_A=0, the zero arrow is not admissible. In other applicat","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d04253126b67c473","tex":"p_b","aliases":"p_b","display":"p_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0b67a2140e399c4e","tex":"P_C","aliases":"P_C","display":"P_C","kind":"use","excerpt":"ubsectionAt every finite budget the residual minimum exists [Attainment and shape of the budget profile] For a bounded linear map between real Hilbert spaces, the infimum defining β(M) is attained for every M≥0. The profile is nonincreasing and convex. For 0≤ M_1≤ M_2, 0≤β(M_1)-β(M_2)≤norm T(M_2-M_1). Writing C=overlineran T, b=P_Cr, and q=normr-b,","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4b003ef2553d3052","tex":"p_c","aliases":"p_c","display":"p_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-afaca47f8ebd7708","tex":"P_{D',C'}","aliases":"P_D',C'","display":"P_{D',C'}","kind":"use","excerpt":"ally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤max(ε_D,ε_C). Applying Theorem~[source reference] to the joint operators, and using the actual two-stage identity of Theorem~[source reference], gives normP_D,C-P_D',C' ≤max(G,G')max","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a5e7737709b11f98","tex":"P_{D,C}","aliases":"P_D,C","display":"P_{D,C}","kind":"use","excerpt":"ally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤max(ε_D,ε_C). Applying Theorem~[source reference] to the joint operators, and using the actual two-stage identity of Theorem~[source reference], gives normP_D,C-P_D',C' ≤max(G,G')max","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b9e70c157342a064","tex":"P_H","aliases":"P_H","display":"P_H","kind":"use","excerpt":"The adjoint in g is the restricted adjoint on H, not the unrestricted ambient adjoint. Equivalently, T^*v=P_Hd^*v. Thus the correction certificate must measure the part of the normal equation that belongs to the permitted correction space. A numerical program that forgets this projection can incorrectly reject an exact protected optimizer.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-237e766422ac8978","tex":"P_{H_0}","aliases":"P_H_0","display":"P_{H_0}","kind":"use","excerpt":"To verify weak convergence against h∈ H_0, choose a finite basis combination h_J close to h. Then |⟨ a_n_k-a,h⟩| ≤ |⟨ a_n_k-a,h_J⟩| +2Mnormh-h_J. For a prescribed tolerance, choose h_J first to control the second term and then k to control the first by finitely many coordinate limits. For a general h in the ambient space, write h=P_H_0h+h_⊥. The or","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"default. In a declared linear model, the compatibility equations become a bounded operator. Closed range, holonomy, and conditioning then determine what small residuals actually guarantee. Graph views, holonomy, and reconstruction. Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c5ca4f817be17bab","tex":"P_K","aliases":"P_K","display":"P_K","kind":"use","excerpt":"chapClosed range is the stability boundary Projection exists before a residual estimate. Let E be a real Hilbert space and R a real Banach space. Let D:E→ R be continuous linear. The kernel K=ker D is closed, so its orthogonal projection q=P_K exists~[source reference].","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f5e4a53d4c21d2f3","tex":"P_{\\ker D}","aliases":"P_ker D ker","display":"P_{\\ker D}","kind":"use","excerpt":"ready exact under permitted correction and the least correction is zero, so the optimal gain is zero. sourceKS56 The relevant stability boundary is therefore D(ker p), not automatically D(E). Restricting permitted changes can turn a well-conditioned unconstrained problem into an obstructed or poorly conditioned protected one. The nearest protected ","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-78436549de06ca5c","tex":"P_{\\ker D'}","aliases":"P_ker D' ker","display":"P_{\\ker D'}","kind":"use","excerpt":"chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8e5399faab2aeb9e","tex":"p_L^2","aliases":"p_L^2","display":"p_L^2","kind":"use","excerpt":"g p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zero of p_L is the identity, and every later coordinate applies B_n, so p_L^2=p_L and normp_L≤1+SB_Y. In particular the","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fe1b7ade23d1e9a1","tex":"P_M","aliases":"P_M","display":"P_M","kind":"use","excerpt":"), write ℓ=P_Lx and let Px be the projection of ℓ onto ker B inside L. Then Px∈ M, u=x-ℓ∈ L^⊥, and v=ℓ-Px∈ L is orthogonal to M. Since M⊆ L, the vector u is also orthogonal to M. Their sum x-Px=u+v is therefore orthogonal to M. A vector in M with that error is the unique orthogonal projection of x, so P=P_M.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ff4cecd8ff7a194d","tex":"p_o","aliases":"p_o","display":"p_o","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b0692a1c334c82d7","tex":"P_{\\rm anchor}","aliases":"P_rm anchor rm","display":"P_{\\rm anchor}","kind":"use","excerpt":"Preserving only the feasible projection of the anchor gives a different answer. Orthogonally project (2,1) to (2,0), then reconstruct along the tree. The result is P_rm anchorx=((2,0),(2,0),(2,0)). It is the unique exact family preserving the first component of the original anchor. It does not preserve its second component, and makes no claim to do","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1d00be659bf68fc1","tex":"P_{\\rm nearest}","aliases":"P_rm nearest rm","display":"P_{\\rm nearest}","kind":"use","excerpt":"For global nearest-point repair, declare the Euclidean product norm on (R^2)^3. The closest member of [source reference] uses the mean of the three first coordinates, namely three: P_rm nearestx=((3,0),(3,0),(3,0)). To verify this without appealing only to a general theorem, write the squared error at parameter a: (2-a)^2+(4-a)^2+(3-a)^2+1^2+(-1)^2","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b9cbd5c85d37e703","tex":"P_{T(\\{a:\\norm a\\le M\\})}","aliases":"P_T({a:norm a≤ M}) norm le","display":null,"kind":"use","excerpt":"The nearest reachable image y_M=P_T({a:norm a≤ M})r is unique, so the best residual vector r-y_M is unique as well. Corrections mapping to y_M can differ by vectors in ker T. Among those corrections there is a unique least norm correction. These two uniqueness statements should not be confused with injectivity of T.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-26a9ec94734fbef0","tex":"p_v","aliases":"p_v","display":"p_v","kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aacabfcb3613b64e","tex":"p_W","aliases":"p_W","display":"p_W","kind":"use","excerpt":"Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6634b2cbd59c3d1f","tex":"p_w","aliases":"p_w","display":"p_w","kind":"use","excerpt":"a coordinate preserving equivalence Γ(B_a,b)≃ B(a,b). Under a transition cover, matching local bridge sections glue to a unique global bridge with both prescribed projections unchanged. We must first check that the displayed fibers really form a view system. Let m∈ B_a,b(w) and f:v→ w. Naturality of p and coherence of a give p_v(f_M^*m)=f_F^*p_w(m)","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a9cab240c51624bb","tex":"\\mathcal P_\\infty","aliases":"P_∞ mathcal infty","display":"\\mathcal P_\\infty","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7eb28ed8746e003c","tex":"\\path","aliases":"path","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-74be1e0574f29a94","tex":"\\operatorname{Point}","aliases":"Point operatorname","display":"\\operatorname{Point}","kind":"use","excerpt":"The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"use","excerpt":"Q · Every rational · One","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.","count":23},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-34d43de423598186","tex":"q^{\\mathsf T}","aliases":"q^ T mathsf","display":"q^{\\mathsf T}","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6339057791df9dbc","tex":"q^{-1}","aliases":"q^-1","display":"q^{-1}","kind":"use","excerpt":"[Nonabelian relative extension] The obstruction is independent of the raw lift. It is neutral exactly when s extends to a global section of T_c. The normalized label c^q is the gauge transform of c by q^-1, so Proposition~[source reference] makes it a cocycle. If f:v→ w is internal to U, the partial-section equation says q_v=c_ff^*(q_w), and theref","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"[Exact orthogonal repair] For every x∈ E, D(qx)=0, q^2=q, normqx≤norm x, and qx is the unique closest point to x in K. Boundedness of D makes K=ker D closed. Lemma~[source reference] constructs q=P_K and gives the orthogonal decomposition x=qx+(x-qx). The first term belongs to K, so Dqx=0. Projection fixes every point of K, giving q(qx)=qx.","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-142e69180c7d1697","tex":"q^j","aliases":"q^j","display":"q^j","kind":"use","excerpt":"nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4a3a203830edc472","tex":"q^m","aliases":"q^m","display":"q^m","kind":"use","excerpt":"the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j ≤fracnorm A1-qnormr_m. Since normr_m≤ q^mnormdf tends to zero, thi","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1928485a8127cf57","tex":"q^N","aliases":"q^N","display":"q^N","kind":"use","excerpt":"[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar s","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-102bb645d8547d75","tex":"q^n","aliases":"q^n","display":"q^n","kind":"use","excerpt":"[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar s","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"use","excerpt":"Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","excerpt":"[Exact motion of the line] For every real t, norm(q_t-q_0)x^2=fract^21+t^2norm x^2, normq_t-q_0=frac|t|sqrt1+t^2. In the small perturbation interface, kernel transport sends (0,b) to (-tb,b). Orthogonal projection instead sends it to (-tb/(1+t^2),b/(1+t^2)). The vector v_t=(-t,1) spans ker D_t and has squared norm 1+t^2>0. Projection onto this line","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0306f7ccb079ea2e","tex":"Q_{2k}","aliases":"Q_2k","display":"Q_{2k}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8dfb1493f0f1cd4b","tex":"q_{2k}","aliases":"q_2k","display":"q_{2k}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aeeb84c561790b3a","tex":"Q_{2k+1}","aliases":"Q_2k+1","display":"Q_{2k+1}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4c1a7de923feba0c","tex":"q_{2k+1}","aliases":"q_2k+1","display":"q_{2k+1}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f5c3a1f5ac4f85f5","tex":"Q_3","aliases":"Q_3","display":"Q_3","kind":"use","excerpt":"draw[arr] (ri.east) -- node[lab,above] C_i (rj.west); draw[thinarr] (rj.north) to[out=130,in=50] node[lab,above] J_i (ri.north); foreach i/x in 0/-4.8,1/-1.6,2/1.6 node[box,minimum width=1.6cm] (hi) at (x,-.3) h_i; draw[arr] (ri) -- node[lab,right] detail (hi); node[box,draw=KomorebiOchre,fill=KomorebiWarm,minimum width=2.1cm] (q) at (4.8,-.3) Q_3x","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-345638a01538c367","tex":"q_f","aliases":"q_f","display":"q_f","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-01982e51585e4e66","tex":"Q_{j+1}","aliases":"Q_j+1","display":"Q_{j+1}","kind":"use","excerpt":"Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a9af1147d28b7af5","tex":"Q_k","aliases":"Q_k","display":"Q_k","kind":"use","excerpt":"[Variable-rate exactification] If Q_n→0 and ∑_n C_nQ_n<∞, then (f_n) has an exact limit and dist(f_n,x)≤ω(f_0)∑_k=n^∞C_kQ_k. For constant q_n=q∈[0,1) and constant C_n=C, the sharper restarted estimate is dist(f_n,x)≤fracC1-q ω(f_n). The estimate ω(f_n)≤ Q_nω(f_0) follows by induction. At n=0 it is equality. If it holds at n, nonnegativity of q_n pe","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cba1a0b5de3eccd8","tex":"Q_M","aliases":"Q_M","display":"Q_M","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4a77ffe628ff771a","tex":"q_n","aliases":"q_n","display":"q_n","kind":"use","excerpt":"Variable rates and a movement budget. Let q_n,C_n≥0, and define the prefix product Q_0=1, Q_n+1=q_nQ_n. Assume ω(f_n+1)≤ q_nω(f_n), dist(f_n,f_n+1)≤ C_nω(f_n).","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bf313ea6d2a73b75","tex":"Q_{n+1}","aliases":"Q_n+1","display":"Q_{n+1}","kind":"use","excerpt":"Variable rates and a movement budget. Let q_n,C_n≥0, and define the prefix product Q_0=1, Q_n+1=q_nQ_n. Assume ω(f_n+1)≤ q_nω(f_n), dist(f_n,f_n+1)≤ C_nω(f_n).","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5266bc3b7e8fe0cc","tex":"Q_N^2","aliases":"Q_N^2","display":"Q_N^2","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e11eab697b8866ee","tex":"q_s","aliases":"q_s","display":"q_s","kind":"use","excerpt":"-t · 1. The common right inverse r↦(r,0) supplies gain at most one for every t, and normD_s-D_t=|s-t|. The general theorem gives normq_s-q_t≤|s-t|.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"An all-real worked model. Let D_t(x,y)=x+ty. Its kernel is the line {(-tb,b):b∈R}, and its orthogonal projector is q_t=frac11+t^2 t^2 · -t -t · 1. The common right inverse r↦(r,0) supplies gain at most one for every t, and normD_s-D_t=|s-t|. The general theorem gives normq_s-q_t≤|s-t|.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8c29958522a9d167","tex":"q_v^{-1}","aliases":"q_v^-1","display":"q_v^{-1}","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ee3e599e7035de50","tex":"Q_W","aliases":"Q_W","display":"Q_W","kind":"use","excerpt":"Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2ac44abe0fa0cba1","tex":"q_w","aliases":"q_w","display":"q_w","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2eee74810f9c985d","tex":"q_z","aliases":"q_z","display":"q_z","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"A growing example and the gap boundary. Use X_n=Y_n=R^n+1 with maximum norms. Let π_n delete the last coordinate, take ρ^X_n=tfrac12π_n, ρ^Y_n=π_n, and let s_n append zero. Then S=B_Y=1, B_X=1/2, and K=2.","count":47},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"use","excerpt":"Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"Equivalently, if r-r'∈ d(ker p), then L(r-r')=0, so L(r)=L(r'). Hence overline L([r])=L(r) is a well-defined homomorphism on R/d(ker p). Its value on [dx] is nonzero. This supplies the quotient interpretation of the same contradiction, without assuming that every nonzero quotient class can be detected by a map into one fixed preferred coefficient g","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aff4a4490ea52474","tex":"r^{-1}","aliases":"r^-1","display":"r^{-1}","kind":"use","excerpt":"Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The auto","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cd6a51568c71e4ad","tex":"\\R^2","aliases":"R^2 R","display":null,"kind":"use","excerpt":"A two-coordinate model in which the routes differ. Use E=F=R^2, write candidates as (u,r), and take d(u,r)=D(u,r)=r, A(r)=B(r)=(0,r). The exact space is the horizontal axis and both repairs are P(u,r)=(u,0). For a real parameter c, set S_c(u,r)=(u+cr,r), C=I. The defect square commutes because the second coordinate is unchanged. The mismatch is M_c","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-78d6b9680d84cb1c","tex":"\\R^3","aliases":"R^3 R","display":null,"kind":"use","excerpt":"Three hypotheses that cannot simply be dropped. Summable movement. In maximum-norm R^3, let f_n=(s,H_n,(n+1)^-1), H_n=∑_j=1^nfrac1j. The first coordinate is protected and the residual is the last coordinate. It tends to zero with the same ratios as the polynomial example. The movement equals 1/(n+1), however, and the unprotected coordinate diverges","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-72b3b4ee79240b62","tex":"\\R^4","aliases":"R^4 R","display":null,"kind":"use","excerpt":"Here the general protected construction is completely explicit. With candidate space E=R^4, defect d=ℓ, and protected observable p(r)=(r_0,r_1,r_2), the allowed direction is the last coordinate axis. The restricted operator d|_ker p has gain one. Without protection, the least gain for distance to kerℓ is 1/2, since normℓ=2. The different constants ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b0a9532805ec4d05","tex":"\\R^N","aliases":"R^N R","display":null,"kind":"use","excerpt":"Escape is not an accident of this particular algorithm. Suppose an arbitrary protected candidate (u,1) satisfies normd(u,1)≤ε. On its first N coordinates, ∑_n=1^N|u_n+1|^2 ≤ N^2∑_n=1^Nfrac|u_n+1|^2n^2 ≤ N^2ε^2. The reverse triangle inequality in R^N gives norm u≥sqrt N-Nε (N≥1). If a sequence of protected candidates has residual tending to zero, th","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b74c15fecc6efa87","tex":"\\R^{n+1}","aliases":"R^n+1 R","display":null,"kind":"use","excerpt":"A growing example and the gap boundary. Use X_n=Y_n=R^n+1 with maximum norms. Let π_n delete the last coordinate, take ρ^X_n=tfrac12π_n, ρ^Y_n=π_n, and let s_n append zero. Then S=B_Y=1, B_X=1/2, and K=2.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8d86c98f6680ae2f","tex":"\\R^{\\N_{>0}}","aliases":"R^N_>0 R N","display":null,"kind":"use","excerpt":"A coherent inverse-limit point still exists in a larger space. At each finite level, the protected exact equations have the unique solution (-1,…,-1;1)∈R^N⊕R. Truncation maps the solution at level N+1 to the one at level N. These finite solution sets are singletons, so their inverse limit is also a singleton. Its coordinate description is (-1,-1,…;","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r","count":10},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0c95d3415f0ad112","tex":"r_{01}","aliases":"r_01","display":"r_{01}","kind":"use","excerpt":"0 · -1. For one chosen orientation of each edge, the residuals are r_01=x_1-x_0, r_12=x_2-x_1, r_20=Jx_0-x_2. The reverse transports are inverses, so this defines an inverse-consistent isometric connection. Exactness first forces x_1=x_0 and x_2=x_0, and the remaining edge then forces Jx_0=x_0. Hence every exact family is ((a,0),(a,0),(a,0)), a∈R. ","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"An equivalence between the moving kernels. [Explicit kernel transport] Under the hypotheses of Theorem~[source reference], J:ker D≃_Lker D', Jx=x-A'D'x, with inverse y↦ y-ADy. Moreover normJx-x≤fracaε1-aεnorm x. Let R_0=I-AD and R_1=I-A'D'. From DA=D'A'=I we obtain DR_0=0 and D'R_1=0. Each R_i fixes its corresponding kernel. Thus R_1 restricts to a","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-64ec1404fdffc06e","tex":"r_{12}","aliases":"r_12","display":"r_{12}","kind":"use","excerpt":"0 · -1. For one chosen orientation of each edge, the residuals are r_01=x_1-x_0, r_12=x_2-x_1, r_20=Jx_0-x_2. The reverse transports are inverses, so this defines an inverse-consistent isometric connection. Exactness first forces x_1=x_0 and x_2=x_0, and the remaining edge then forces Jx_0=x_0. Hence every exact family is ((a,0),(a,0),(a,0)), a∈R. ","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"use","excerpt":"nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-628aa4eb9fec0e5b","tex":"r_{20}","aliases":"r_20","display":"r_{20}","kind":"use","excerpt":"0 · -1. For one chosen orientation of each edge, the residuals are r_01=x_1-x_0, r_12=x_2-x_1, r_20=Jx_0-x_2. The reverse transports are inverses, so this defines an inverse-consistent isometric connection. Exactness first forces x_1=x_0 and x_2=x_0, and the remaining edge then forces Jx_0=x_0. Hence every exact family is ((a,0),(a,0),(a,0)), a∈R. ","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b95bc35897dc3f06","tex":"r_{2n}","aliases":"r_2n","display":"r_{2n}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-77c3d448c4fd635b","tex":"r_{2n+1}","aliases":"r_2n+1","display":"r_{2n+1}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d5c7051b56820f95","tex":"r_3","aliases":"r_3","display":"r_3","kind":"use","excerpt":"nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4bdf782a5897d64d","tex":"r_b","aliases":"r_b","display":"r_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-567e1106095a8a61","tex":"r_c","aliases":"r_c","display":"r_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"use","excerpt":"tween the moving kernels [Explicit kernel transport] Under the hypotheses of Theorem~[source reference], J:ker D≃_Lker D', Jx=x-A'D'x, with inverse y↦ y-ADy. Moreover normJx-x≤fracaε1-aεnorm x. Let R_0=I-AD and R_1=I-A'D'. From DA=D'A'=I we obtain DR_0=0 and D'R_1=0. Each R_i fixes its corresponding kernel. Thus R_1 restricts to a map from ker D to","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a1dc05f091a363a3","tex":"R_{\\i}","aliases":"R_i i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8node[box,minimum width=1.5cm] (ri) at (x,1.7) x_i∈ R_i; foreach i/j in 0/1,1/2,2/3 draw[arr] (ri.east) -- node[lab,above] C_i (rj.west); draw[thinarr] (rj.north) to[out=130,in=50] node[lab,above] J_i (ri.north); foreach i/x in 0/-4.8,1/-1.6,2/1.6 node[box,minimum width=1.6cm] (hi) at (x,-.3) h_i","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8505e2b9e062c506","tex":"r_{ij}","aliases":"r_ij","display":"r_{ij}","kind":"use","excerpt":"Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-53e3e9d1c2a6131f","tex":"R_{j+1}","aliases":"R_j+1","display":"R_{j+1}","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2a1d688e5f76248c","tex":"r_{j+1}","aliases":"r_j+1","display":"r_{j+1}","kind":"use","excerpt":"r a nonzero residual r∈ Y, set λ=2normr/c. The vector r/λ has norm c/2, so [source reference] supplies z∈ B_X with normSz-r/λ≤ c/4. Then u=λ z satisfies normu≤ 2normr/c, normr-Su≤normr/2. If the residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-41b9e630b25443f1","tex":"R_{j+N}","aliases":"R_j+N","display":"R_{j+N}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-272cb3ca57c00592","tex":"r_m","aliases":"r_m","display":"r_m","kind":"use","excerpt":"belKS19 Starting from f^(0)=f, define f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2b9d5b78881f1c09","tex":"r_{m+1}","aliases":"r_m+1","display":"r_{m+1}","kind":"use","excerpt":"fine f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-48ae5eb670db3065","tex":"r_{m+j}","aliases":"r_m+j","display":"r_{m+j}","kind":"use","excerpt":"nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4aa81ec436e2173c","tex":"R_N","aliases":"R_N","display":"R_N","kind":"use","excerpt":"egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8d214535a6b8b511","tex":"r_N","aliases":"r_N","display":"r_N","kind":"use","excerpt":"residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages converges in X by completeness, to a vector x with the displayed norm bound. Continuity of S, together with the finite telescoping equation S(∑_j=0^N-1u_j)=y-r_N, ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a4ff961cd74d1869","tex":"r_n","aliases":"r_n","display":"r_n","kind":"use","excerpt":"A right inverse from target splittings. Suppose each target bonding map has a continuous linear section s_n:Y_n→ Y_n+1, ρ^Y_ns_n=I, norms_n≤ S. Assume the propagation gap SB_X<1, K=fracS1-SB_X. For r∈ R, recursively set (Ar)_0=0, (Ar)_n+1=s_n(r_n+(Ar)_nρ^X_n).","count":15},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fd828431f1a05a6f","tex":"r_o","aliases":"r_o","display":"r_o","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-10d3f7c1355c5340","tex":"r_r","aliases":"r_r","display":"r_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-adea9ba150e9752c","tex":"r_U","aliases":"r_U","display":"r_U","kind":"use","excerpt":"The exact sequence behind the criterion. The additive structures make restriction a homomorphism r_U:Γ(A)→Γ_U(A). Using the zero extension, or lift independence, o_U is also a homomorphism.","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c16fbef0f6898c20","tex":"r_v","aliases":"r_v","display":"r_v","kind":"use","excerpt":"Step 4: retain every morphism. Let c,d be global cocycles. Their canonical restrictions have identity overlap gauges. A morphism between those restrictions is consequently a family r_i satisfying r_j(v)=r_i(v) on each overlap. Assemble r_v from any containing patch. For an arrow choose a patch containing both endpoints. Its gauge equation proves d_","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6c74d6c0c951e9a8","tex":"r_w","aliases":"r_w","display":"r_w","kind":"use","excerpt":"Step 4: retain every morphism. Let c,d be global cocycles. Their canonical restrictions have identity overlap gauges. A morphism between those restrictions is consequently a family r_i satisfying r_j(v)=r_i(v) on each overlap. Assemble r_v from any containing patch. For an arrow choose a patch containing both endpoints. Its gauge equation proves d_","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e81875b79f13e65e","tex":"R_\\alpha","aliases":"R_α alpha","display":"R_\\alpha","kind":"use","excerpt":"et frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ca0ed80fd48b48de","tex":"R_\\gamma","aliases":"R_γ gamma","display":"R_\\gamma","kind":"use","excerpt":"a_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f5cfdb40005f62a6","tex":"r_\\infty","aliases":"r_∞ infty","display":"r_\\infty","kind":"use","excerpt":"orm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial coordinate has norm zero. If norm(Ar)_n≤ Knorm r_∞, submultiplicativity of operator norms gives ","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3f0bb881a2e80b27","tex":"\\R_{\\ge0}^{\\infty}","aliases":"R_≥0^∞ R ge infty","display":null,"kind":"use","excerpt":"stances from A to C and from C to A is bounded by d_W^sym(A,B)+d_W^sym(B,C). Taking their maximum proves the symmetric triangle law. Zero diagonal and symmetry give reflexivity and symmetry of the zero-distance relation. The triangle inequality and nonnegativity give transitivity. sourceKS78 The formal source proves these laws directly in R_≥0^∞ an","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-512d411adf8ee1ff","tex":"\\ran","aliases":"ran","display":null,"kind":"use","excerpt":"ion gives norm x^2=normqx^2+normx-qx^2, hence normqx≤norm x. For any y∈ K, the vector qx-y lies in K, whereas x-qx lies in its orthogonal complement. Consequently normx-y^2=normx-qx^2+normqx-y^2≥normx-qx^2. Equality holds only when normqx-y=0, or y=qx. This proves nearest-point uniqueness without any closedness assumption on ran D. Closedness of th","count":39},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the positive case is dim H=n-rankA P. Set T=A","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":")=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a mod","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-46238367ef18be35","tex":"\\rightharpoonup","aliases":"rightharpoonup","display":"\\rightharpoonup","kind":"use","excerpt":"Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity giv","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"For the basis change B'=BC, the same ambient vector is represented by c'=C^-1c. Also S'=SC and G'=C^ TGC. Substitute these formulas into the new coordinate normal equation: q' · =((S')^ TS'+α G')c'-(S')^ Tr","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"In the previous example, B=(1,1)^ T, G=2, and the proposal c=1/4 gives q=-1/4. Thus q^ TG^-1q=1/32, as required. Replacing B by sB changes q to sq and G to s^2G. The Euclidean norm of the coordinate covector can be made arbitrarily small or large by changing the nonzero scalar s. The actual correction, normal residual, and certified error do not ch","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-913791bf1e0ffd22","tex":"s_{2n}","aliases":"s_2n","display":"s_{2n}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"A small noncommutative model. [A C_2 loop with S_3 coefficients] Take one view with a loop σ satisfying σ^2=1, and constant coefficients in a group H. A cocycle is determined by an element p∈ H whose square is the identity. Gauge equivalence is conjugacy, and its automorphisms form the centralizer of p.","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1dcf95b2a026d667","tex":"s_a","aliases":"s_a","display":"s_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5df9b8f58b7161eb","tex":"s_b","aliases":"s_b","display":"s_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8ec8b0faab9c7fd8","tex":"S_c","aliases":"S_c","display":"S_c","kind":"use","excerpt":"A two-coordinate model in which the routes differ. Use E=F=R^2, write candidates as (u,r), and take d(u,r)=D(u,r)=r, A(r)=B(r)=(0,r). The exact space is the horizontal axis and both repairs are P(u,r)=(u,0). For a real parameter c, set S_c(u,r)=(u+cr,r), C=I. The defect square commutes because the second coordinate is unchanged. The mismatch is M_c","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a64b77f6851063e5","tex":"s_c","aliases":"s_c","display":"s_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ae27dbb473a126cc","tex":"S_{c_2}","aliases":"S_c_2","display":"S_{c_2}","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7aa9c815c9ae3803","tex":"s_e","aliases":"s_e","display":"s_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1550303e0c5e431e","tex":"s_g","aliases":"s_g","display":"s_g","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e51f187cf7ad66bf","tex":"s_{HLw}","aliases":"s_HLw","display":"s_{HLw}","kind":"use","excerpt":"For the relative claim, pullback clearly sends an extension of s to an extension of H^*s on U'=H^-1U. Conversely suppose y extends H^*s. If w∈ U, the isomorphism w≅ HLw and isomorphism closure imply HLw∈ U. Hence Lw∈ U' and y_Lw=s_HLw. Both endpoints of t_w lie in U, so partial coherence gives (Ry)_w=F(t_w)s_HLw=s_w. Recovery therefore preserves th","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Let (U_i)_i∈ I be full patches. A matching family consists of s_i∈Γ_U_i(F) with equal values on every overlap. The relevant covering condition is for every f:v→ w, some U_i contains both v and w. Identities imply that this condition also covers every object.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-02cad8c82d44ed28","tex":"s_{i(v)}","aliases":"s_i(v)","display":"s_{i(v)}","kind":"use","excerpt":"rem[Strict gluing] Under [source reference], restriction induces an equivalence Γ(F)≃Match(F,(U_i)). Every fixed matching family has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"use","excerpt":"Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U_j, matching on the overlap U_i(v)∩ U_j gives x_v=(s_j)_v. t","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7b9f4293854c6f56","tex":"\\mathcal S_L","aliases":"S_L mathcal","display":"\\mathcal S_L","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ef445d6d03f47b90","tex":"s_n'","aliases":"s_n'","display":"s_n'","kind":"use","excerpt":"What is canonical. For fixed splitting and protected observable, the repair above is unique. Changing the splitting can change both the observable and the repaired answer. In the growing example, replace s_n(y)=(y,0) by s_n'(y)=(y,y_0). At the first positive level, repair of the identity becomes 1/2 · 0","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cc0051a685f15a7f","tex":"s_{n+1}","aliases":"s_n+1","display":"s_{n+1}","kind":"use","excerpt":"For surjectivity, fix y∈ Y_n and choose m≥max{s_n,s_n+1,n+1}. Since y∈ I_n,m, there is w∈ X_m with ρ_n,m(w)=y. Put z=ρ_n+1,m(w). Stabilization at level n+1 gives z∈ Y_n+1, and composition gives ρ_nz=y.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-efc33e545a71481e","tex":"s_s","aliases":"s_s","display":"s_s","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5dce6ea50bf9e787","tex":"s_u","aliases":"s_u","display":"s_u","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1d641eb69668a366","tex":"s_v","aliases":"s_v","display":"s_v","kind":"use","excerpt":"is neutral. Equality of orbit classes means there exists an actual based gauge b with b· c^q=1. At an arrow f:v→ w, this equation is b_vq_v^-1c_ff^*(q_w)[f^*(b_w)]^-1=1. Multiply on the left by q_vb_v^-1. Homomorphic preservation of products then gives c_ff^*(q_wb_w^-1)=q_vb_v^-1. Therefore x_v=q_vb_v^-1 is a twisted coherent family. On U, b_v=1 an","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7735513ecabe8703","tex":"s_w","aliases":"s_w","display":"s_w","kind":"use","excerpt":"For the relative claim, pullback clearly sends an extension of s to an extension of H^*s on U'=H^-1U. Conversely suppose y extends H^*s. If w∈ U, the isomorphism w≅ HLw and isomorphism closure imply HLw∈ U. Hence Lw∈ U' and y_Lw=s_HLw. Both endpoints of t_w lie in U, so partial coherence gives (Ry)_w=F(t_w)s_HLw=s_w. Recovery therefore preserves th","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f8039ab5de609643","tex":"S_\\phi","aliases":"S_ϕ phi","display":"S_\\phi","kind":"use","excerpt":"Actual tower resampling gives a defect map. Return to Banach towers with contractive source and target bonds. Let ϕ be a monotone schedule. Candidate sampling is (S_ϕ f)_n=f_ϕ(n). The sampled bonding map is the actual composite restriction between the selected levels. The residual map C_ϕ is not pointwise sampling of a scalar error score. It accumu","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-72398ff18d39e1f6","tex":"\\sec","aliases":"sec","display":"\\sec","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c3f001b6b563f398","tex":"\\sec^2","aliases":"sec^2 sec","display":"\\sec^2","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-87e5f1044d3860c9","tex":"\\operatorname{Set}","aliases":"Set operatorname","display":"\\operatorname{Set}","kind":"use","excerpt":"chapA genuine stack on the composition site The preceding theorem can be placed in a standard site, rather than called stack-like. We use the usual effective-descent meaning of a stack~[source reference]. The base of this site is the inclusion poset of full patches P(V)=(Set(ObV),⊆). This patch category is different from the original category of vi","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"In matrix form, the difference operator is D= -1 · 1 · 0 · 0 0 · -1 · 1 · 0 0 · 0 · -1 · 1 1 · 0 · 0 · -1 , ker D=span{(1,1,1,1)}. The explicit construction proves ran D=kerℓ, ℓ(r)=∑_i r_i. Thus ℓ is a complete feasibility test for this prescribed-difference problem. It annihilates every actual difference family, but takes a nonzero value on an inc","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-310b249a090cc66d","tex":"\\Stab","aliases":"Stab","display":null,"kind":"use","excerpt":"Automorphisms and ambiguity are not the same question. The stabilizer of a cocycle is the group of families with a_v=c_f f^*(a_w)c_f^-1. If one gauge a:c→ d is known, all gauges from c to d are uniquely of the form ah, with h∈Stab(c). Indeed a^-1b stabilizes c for every other gauge b. This is the ordinary Hom torsor principle in a groupoid.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-848c3c3d231100c2","tex":"\\subsection","aliases":"subsection","display":null,"kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a240bb4eb2bf25a3","tex":"\\sup_k","aliases":"sup_k sup","display":"\\sup_k","kind":"use","excerpt":"uad normC_ϕ≤ m, ∂_ϕ S_ϕ=C_ϕ∂. Exact candidates remain exact under every monotone common resampling, even without a gap bound. If A and B are bounded right inverses for the fine and sampled differentials, then normP_BS_ϕ f-S_ϕ P_Af ≤(norm A+mnorm B)norm∂ f. For a bounded candidate family, sup_nnormf_ϕ(n)≤sup_knormf_k, proving the sampling bound. For","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-61335044b48ba968","tex":"\\sup_N","aliases":"sup_N sup","display":"\\sup_N","kind":"use","excerpt":"[Density criterion for norm recovery] Suppose sup_NnormQ_N≤ K<∞. Then the norm recoverable vectors are exactly overline F. In particular all vectors are norm recoverable if and only if the finite core is dense. The nested identities first imply that the kernels increase: if Q_My=0 and N≥ M, then Q_Ny=Q_NQ_My=0. Thus the finite core really is an inc","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f07ba46881cb0326","tex":"\\sup_{\\norm{z}\\le1}","aliases":"sup_normz≤1 sup norm le","display":null,"kind":"use","excerpt":"linear map between real Hilbert spaces. For fixed z∈ R, the functional h↦⟨ Th,z⟩ is bounded by normTnormznormh. The lemma therefore defines a unique vector T^*z∈ H with ⟨ Th,z⟩=⟨ h,T^*z⟩. Uniqueness of representatives proves that T^* is linear. The same bound proves normT^*z≤normTnormz. Conversely, normTh=sup_normz≤1|⟨ Th,z⟩| ≤normT^*normh, where t","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d0f4101430d2716c","tex":"\\sup_{n\\ge N}","aliases":"sup_n≥ N sup ge","display":"\\sup_{n\\ge N}","kind":"use","excerpt":"[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar s","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","excerpt":"For transitivity, suppose R covers W, and that for every U∈ R the pullback of a sieve T to U covers U. Fix a composable pair q in W. Its support S=|q| belongs to R. Since patches are full, q is also a composable pair in S. The covering condition for the pullback of T to S therefore puts the identity inclusion S⊆ S in that pullback. By the definitio","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-104f4e0a72089cc5","tex":"T^0","aliases":"T^0","display":"T^0","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives e","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":"-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives e","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9ea7696d8ec4186b","tex":"T^j","aliases":"T^j","display":"T^j","kind":"use","excerpt":"Neumann inversion with both inverse identities. [A strict perturbation radius] Let W be a real Banach space and T:W→ W bounded linear, with normT≤θ<1 and θ≥0. Then Z=∑_j=0^∞T^j exists in operator norm and satisfies (I-T)Z=Z(I-T)=I, normZ≤frac11-θ. First recall why the space of bounded operators on W is complete. An operator-norm Cauchy sequence A_n","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-910092ae6df8d19a","tex":"T^n","aliases":"T^n","display":"T^n","kind":"use","excerpt":"The perturbed map is surjective, with D'A'=I, normA'≤fraca1-aε. Its exact projection satisfies normx-q'x≤fraca1-aεnormD'x, normq-q'≤fracaε1-aε. The map T=(D-D')A acts on the Banach target R and satisfies norm T≤ aε<1. The operator series Z=∑_n=0^∞ T^n converges in operator norm and obeys norm Z≤(1-aε)^-1. Multiplying its finite partial sums on eith","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-756b0e0aef75f3d4","tex":"T^{N+1}","aliases":"T^N+1","display":"T^{N+1}","kind":"use","excerpt":"1-aεnormD'x, normq-q'≤fracaε1-aε. The map T=(D-D')A acts on the Banach target R and satisfies norm T≤ aε<1. The operator series Z=∑_n=0^∞ T^n converges in operator norm and obeys norm Z≤(1-aε)^-1. Multiplying its finite partial sums on either side by I-T gives I-T^N+1. Passing to the operator-norm limit proves (I-T)Z=Z(I-T)=I. Appendix~[source refe","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4a9e32fb56cb7f6c","tex":"t_a","aliases":"t_a","display":"t_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bd730f6c2c2e8897","tex":"t_d","aliases":"t_d","display":"t_d","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c78b7188b7b592c0","tex":"t_e","aliases":"t_e","display":"t_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c377758d754d2ccd","tex":"t_{Hv}","aliases":"t_Hv","display":"t_{Hv}","kind":"use","excerpt":"If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-84eac95fe9a2a4fb","tex":"T_i^{-1}","aliases":"T_i^-1","display":"T_i^{-1}","kind":"use","excerpt":"Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"use","excerpt":"[x=1cm,y=1cm] node[box,circle] (o) at (-3.6,2.3) o; node[box,circle] (i) at (-5.5,-.7) i;node[box,circle] (j) at (-1.7,-.7) j; draw[arr] (i)--node[lab,left] T_i(o);draw[arr] (j)--node[lab,right] T_j(o); draw[arr,draw=KomorebiOchre] (j)--node[lab,below] τ_ij(i); draw[arr,dashed] (o) to[loop above,min distance=12mm] node[lab,above] K_ij=T_iτ_ijT_j^-1","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-95211c71e87133e2","tex":"T_j^{-1}","aliases":"T_j^-1","display":"T_j^{-1}","kind":"use","excerpt":"Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-733902ab0028bba1","tex":"t_m","aliases":"t_m","display":"t_m","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4505604ff161e9a8","tex":"T_N","aliases":"T_N","display":"T_N","kind":"use","excerpt":"Every finite operator is invertible, but the gains diverge. On R^N, let T_N=diag(1,1/2,…,1/N). It is invertible and normT_N^-1=N in the Euclidean norm. The upper bound follows by multiplying each coordinate by an integer at most N, and equality is attained at the last coordinate vector. Full finite rank therefore supplies no uniform gain as N→∞.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3aeff80b8378ac4b","tex":"T_N^{-1}","aliases":"T_N^-1","display":"T_N^{-1}","kind":"use","excerpt":"Every finite operator is invertible, but the gains diverge. On R^N, let T_N=diag(1,1/2,…,1/N). It is invertible and normT_N^-1=N in the Euclidean norm. The upper bound follows by multiplying each coordinate by an integer at most N, and equality is attained at the last coordinate vector. Full finite rank therefore supplies no uniform gain as N→∞.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dcee229e3aaf9526","tex":"t_o","aliases":"t_o","display":"t_o","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-81549140e4fc02bc","tex":"t_p","aliases":"t_p","display":"t_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dd4ff4382ddb04f0","tex":"T_{p_i}","aliases":"T_p_i","display":"T_{p_i}","kind":"use","excerpt":"Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-387970c466cde48f","tex":"t_v^{-1}","aliases":"t_v^-1","display":"t_v^{-1}","kind":"use","excerpt":"If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4ab411d9663afe22","tex":"t_w","aliases":"t_w","display":"t_w","kind":"use","excerpt":"If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1 · =t_v^-1c_ff^*(q_w) [f^*(q_w)]^-1f^*(t_w) =t_v^-1c_ff^*(t_w)=c_f^t. Thus changing the raw lift leaves the based orbit unchanged.","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b04a44cd7b0fa729","tex":"t_{w'}","aliases":"t_w'","display":"t_{w'}","kind":"use","excerpt":"ce and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Contravariance of F a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b00cb8546f0d5d6a","tex":"t_w^{-1}","aliases":"t_w^-1","display":"t_w^{-1}","kind":"use","excerpt":"If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1 · =t_v^-1c_ff^*(q_w) [f^*(q_w)]^-1f^*(t_w) =t_v^-1c_ff^*(t_w)=c_f^t. Thus changing the raw lift leaves the based orbit unchanged.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-75177911206476aa","tex":"\\tan","aliases":"tan","display":"\\tan","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3000c3bd47e8007c","tex":"\\toprule","aliases":"toprule","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Given u∈Γ(B_a,b), forget each fiber-membership condition and call the underlying family z_v∈ M_v. Coherence of u is precisely coherence of z, so z∈Γ(M). The equations p_vz_v=a_v hold at every coordinate. Joint determination in Γ(F) yields Γ(p)z=a, and the equations for q similarly yield Γ(q)z=b. Thus z is a global bridge.","count":119},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ec61c8382b2d3710","tex":"U'","aliases":"U'","display":"U'","kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ec7d40a789aa2d83","tex":"u'","aliases":"u'","display":"u'","kind":"use","excerpt":"For any h∈ H, the vector h-ℓ(h)v/ℓ(v) belongs to L. Pairing it with v therefore gives ⟨ h,v⟩=fracℓ(h)ℓ(v)normv^2. Consequently u=ℓ(v)v/normv^2 represents ℓ. If u' also represents it, pairing the difference of the two formulas with h=u-u' gives normu-u'^2=0, hence uniqueness.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6cf86c8bfef7f92f","tex":"U'_x","aliases":"U'_x","display":"U'_x","kind":"use","excerpt":"Changing lifts to U'_x=(-1)^a(x)U_x changes the phase to c'(x,y)=c(x,y)+a(x)+a(y)-a(x+y). No such change makes c' vanish identically. The antisymmetric part is unchanged by every scalar change of lift. In this model it is κ(x,y)=c(x,y)-c(y,x) =x_2y_1-y_2x_1. This is a directly observable obstruction certificate for the model. Its nonzero value at (","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"[The two-stage repair is the joint projection] The operator P is the orthogonal projection onto M, equivalently onto the kernel of Δ=(D,C). If u=x-P_Lx and v=P_Lx-Px, then u⊥ v, normx-Px^2=norm u^2+norm v^2. The subspace L=ker D is closed, hence Hilbert with the inherited inner product. The restriction B=C|_L is continuous, so its kernel is a close","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-305733ca1f18133f","tex":"U_{(p,q)}","aliases":"U_(p,q)","display":"U_{(p,q)}","kind":"use","excerpt":"The phase is visible in actual matrices. Over R, define X=0 · 1 1 · 0, Z=1 · 0 0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-652da1418f4adb94","tex":"U_{e_1}","aliases":"U_e_1","display":"U_{e_1}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a7dee972cf3034ae","tex":"U_{e_2}","aliases":"U_e_2","display":"U_{e_2}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"Let (U_i)_i∈ I be full patches. A matching family consists of s_i∈Γ_U_i(F) with equal values on every overlap. The relevant covering condition is for every f:v→ w, some U_i contains both v and w. Identities imply that this condition also covers every object.","count":18},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a1ef7ae339701690","tex":"U_{\\i}","aliases":"U_i i","display":null,"kind":"use","excerpt":"at (x,1.25) i; foreach j/y/name in 0/.55/01,1/-.15/02,2/-.85/12node[lab] at (-6.25,y) U_name; foreach x/y in -5.4/.55,-3.8/.55,-5.4/-.15,-2.2/-.15,-3.8/-.85,-2.2/-.85fill[KomorebiPine] (x,y) circle (2.5pt); foreach i/x in 0/1.4,1/2.8,2/4.2,3/5.6node[lab] at (x,1.25) i; foreach i/y in 0/.55,1/-.15,2/-.85,3/-1.55 node[lab] at (.45,y) U_i; foreach j/x","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2231f0f6ca7a3024","tex":"U_{i(v)}","aliases":"U_i(v)","display":"U_{i(v)}","kind":"use","excerpt":"has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U_j, matching on the overlap U_i(v)","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","excerpt":"_i)). Every fixed matching family has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"r a nonzero residual r∈ Y, set λ=2normr/c. The vector r/λ has norm c/2, so [source reference] supplies z∈ B_X with normSz-r/λ≤ c/4. Then u=λ z satisfies normu≤ 2normr/c, normr-Su≤normr/2. If the residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-25121c20156ac918","tex":"U_m","aliases":"U_m","display":"U_m","kind":"use","excerpt":"τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels each interior value once with each sign. The remaining terms are U_mx_i_m-U_0x_i_0=T_qx_i_m-x_i_0, pro","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8d86088ae482ad75","tex":"u_m","aliases":"u_m","display":"u_m","kind":"use","excerpt":"Now assume condition 2. To prove closedness, take an arbitrary sequence r_n∈ran D with r_n→ r in R. Choose x_n with Dx_n=r_n and put u_n=x_n-qx_n. Then u_n∈ K^⊥ and Du_n=r_n. Applying the estimate to u_n-u_m gives normu_n-u_m≤ Cnormr_n-r_m. Thus (u_n) is Cauchy in the complete space K^⊥. Let its limit be u. Continuity gives Du=lim_n r_n=r, so r∈ran","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"use","excerpt":"Now assume condition 2. To prove closedness, take an arbitrary sequence r_n∈ran D with r_n→ r in R. Choose x_n with Dx_n=r_n and put u_n=x_n-qx_n. Then u_n∈ K^⊥ and Du_n=r_n. Applying the estimate to u_n-u_m gives normu_n-u_m≤ Cnormr_n-r_m. Thus (u_n) is Cauchy in the complete space K^⊥. Let its limit be u. Continuity gives Du=lim_n r_n=r, so r∈ran","count":18},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8bd76b19257dd92a","tex":"U_{\\name}","aliases":"U_name name","display":null,"kind":"use","excerpt":"r] (a1)--(a2); draw[thinarr] (a0.north) to[out=60,in=120] (a2.north); foreach i/x in 0/1.4,1/2.8,2/4.2,3/5.6node[box,circle,inner sep=4pt] (bi) at (x,2.3) i; draw[arr](b0)--(b1);draw[arr](b1)--(b2);draw[arr](b2)--(b3); foreach i/x in 0/-5.4,1/-3.8,2/-2.2node[lab] at (x,1.25) i; foreach j/y/name in 0/.55/01,1/-.15/02,2/-.85/12node[lab] at (-6.25,y) ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6ae21ddbb7e958a2","tex":"U_t","aliases":"U_t","display":"U_t","kind":"use","excerpt":"Small holonomy can still be poorly conditioned. In R^2, the rational rotation U_t=frac11+t^2 1-t^2 · -2t 2t · 1-t^2 satisfies norm(U_t-I)z^2=frac4t^21+t^2norm z^2. For t≠0 its fixed space is zero and its residual gain is sqrt1+t^2/(2|t|), which diverges as t→0. At t=0, the entire fiber is fixed. Therefore small loop discrepancy does not supply a un","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-07ca84b023e0812f","tex":"U_x","aliases":"U_x","display":"U_x","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-931251200eb190e2","tex":"U_{x+y}","aliases":"U_x+y","display":"U_{x+y}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cf906e4bc6834869","tex":"U_{x+y+z}","aliases":"U_x+y+z","display":"U_{x+y+z}","kind":"use","excerpt":"Associativity of three lifted matrices requires the same total sign on both parenthesizations. Equation~[source reference] is exactly that requirement: (U_xU_y)U_z =(-1)^c(x,y)+c(x+y,z)U_x+y+z, U_x(U_yU_z) =(-1)^c(y,z)+c(x,y+z)U_x+y+z. Thus a nonzero phase is not automatically a failed associativity law. It can be the data that make the composition","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6df8bf7a9378fe0f","tex":"U_y","aliases":"U_y","display":"U_y","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-810dc694a1dc15a1","tex":"U_z","aliases":"U_z","display":"U_z","kind":"use","excerpt":"Associativity of three lifted matrices requires the same total sign on both parenthesizations. Equation~[source reference] is exactly that requirement: (U_xU_y)U_z =(-1)^c(x,y)+c(x+y,z)U_x+y+z, U_x(U_yU_z) =(-1)^c(y,z)+c(x,y+z)U_x+y+z. Thus a nonzero phase is not automatically a failed associativity law. It can be the data that make the composition","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"not come with a numerical distance by default. In a declared linear model, the compatibility equations become a bounded operator. Closed range, holonomy, and conditioning then determine what small residuals actually guarantee. Graph views, holonomy, and reconstruction. Actual transports on a finite graph. Let a finite connected simple graph have ve","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a8b2a9721419c0b6","tex":"v^{\\mathsf T}","aliases":"v^ T mathsf","display":"v^{\\mathsf T}","kind":"use","excerpt":"-1 · 0 · -1. The ordinary cycle block satisfies A^ TA=3I-11^ T. It is zero on the constant line and equals 3I on its orthogonal complement. The reflected block satisfies B^ TB=I+vv^ T, v=(1,-1,1)^ T. Since norm v^2=3, this matrix has eigenvalue four in the direction of v and eigenvalue one on its orthogonal complement. In particular it is positive ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ff15a0be2cf55ac8","tex":"v^{(m)}","aliases":"v^(m)","display":"v^{(m)}","kind":"use","excerpt":"[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6042bf2f76597e2e","tex":"v^{(m)}_n","aliases":"v^(m)_n","display":"v^{(m)}_n","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cafb796619c588dd","tex":"v^{-1}","aliases":"v^-1","display":"v^{-1}","kind":"use","excerpt":"Conversely, if x is a twisted extension of the same prescribed data, define h_v=(x_0)_v^-1x_v. On U its value is one. For every arrow, h_v · =[c_f f^*((x_0)_w)]^-1[c_f f^*(x_w)]","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"[The two-stage repair is the joint projection] The operator P is the orthogonal projection onto M, equivalently onto the kernel of Δ=(D,C). If u=x-P_Lx and v=P_Lx-Px, then u⊥ v, normx-Px^2=norm u^2+norm v^2. The subspace L=ker D is closed, hence Hilbert with the inherited inner product. The restriction B=C|_L is continuous, so its kernel is a close","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9c8da2b101444307","tex":"v^\\perp","aliases":"v^⊥ perp","display":"v^\\perp","kind":"use","excerpt":"These two identities show that the least nonnegative gain for the full defect-to-distance estimate in this Euclidean model is one. The lower bound is attained, for example, on a candidate with zero first components and second-component vector (1,1,0), which lies in v^⊥. The squared input distance and squared residual are both two.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"use","excerpt":"Why a vanishing normal residual is not enough. Let T=0 on R, let r=0, and take α_n=n^-2 for n≥1. The exact optimizer is zero for every such n. The fixed proposal a_n=1 instead has v_n=0, g_n=α_n⟶0, J_α_n(a_n)=α_n⟶0, absa_n-a_α_n=1. Thus the fit residual, the normal residual, and even the absolute objective value can all tend to zero without converg","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7be15d086b46f651","tex":"v_t","aliases":"v_t","display":"v_t","kind":"use","excerpt":"[Exact motion of the line] For every real t, norm(q_t-q_0)x^2=fract^21+t^2norm x^2, normq_t-q_0=frac|t|sqrt1+t^2. In the small perturbation interface, kernel transport sends (0,b) to (-tb,b). Orthogonal projection instead sends it to (-tb/(1+t^2),b/(1+t^2)). The vector v_t=(-t,1) spans ker D_t and has squared norm 1+t^2>0. Projection onto this line","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c3a76197560b59e1","tex":"v_\\alpha","aliases":"v_α alpha","display":"v_\\alpha","kind":"use","excerpt":"Parameter monotonicity and the fixed-budget problem. [Regularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bb5783d1e38c64af","tex":"\\vec E","aliases":"vec E vec","display":"\\vec E","kind":"use","excerpt":"The candidate and residual spaces are finite products with maximum norms: E=∏_i∈ VH, R=∏_(i,j)∈vec EH, (dx)_ij=τ_ijx_j-x_i. Both orientations of an edge are retained. If M bounds the norms of all measured edge maps, then norm d≤ M+1. The empty edge case has the usual zero residual.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c30dc958e19e4c13","tex":"\\W","aliases":"W","display":null,"kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b99d2e5f37ae7e17","tex":"w'","aliases":"w'","display":"w'","kind":"use","excerpt":"[ Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2bbe13df91f37646","tex":"w^{-1}","aliases":"w^-1","display":"w^{-1}","kind":"use","excerpt":"Conversely, if x is a twisted extension of the same prescribed data, define h_v=(x_0)_v^-1x_v. On U its value is one. For every arrow, h_v · =[c_f f^*((x_0)_w)]^-1[c_f f^*(x_w)] · =f^*((x_0)_w)^-1c_f^-1c_f f^*(x_w) · =f^*((x_0)_w^-1x_w)=f^*(h_w). The cancellation uses inverse multiplication in its actual order. No commuting factors have been moved ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"usR, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n has operator norm one. Since d(u,t)=Tu+tw, normd(u,t)≤norm u+|t|norm w ≤sqrt1+norm w^2 norm(u,t)_E. Thus bo","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8945947ef031a855","tex":"w_n","aliases":"w_n","display":"w_n","kind":"use","excerpt":"The obstruction class is nonzero while its floor is zero. The admissible correction space is ker p=ℓ^2⊕{0}, with correction image B=d(ker p)=ran T ={r∈ℓ^2:(nr_n)_n≥1∈ℓ^2}. It contains every finitely supported sequence, so it is dense in R. But w∉ B, because (nw_n)_n=(1,1,…) is not square summable. Therefore [w]≠0 in R/B, ν_B(w)=0. This is the middl","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-644e773747d7540b","tex":"\\widehat r","aliases":"widehat r widehat","display":"\\widehat r","kind":"use","excerpt":"To repair the residual, now regard the vector of edge data as a candidate in its own right. Its defect is ℓ(r). The exact subspace is the zero-sum hyperplane. With the ordinary Euclidean norm and no protected edge coordinates, the nearest compatible residual is widehat r=r-fracℓ(r)41 =(frac34,-frac14,-frac14,-frac14). The correction is perpendicula","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e8d6f1354dafb866","tex":"\\widehat x","aliases":"widehat x widehat","display":"\\widehat x","kind":"use","excerpt":"Anchor the repaired signal at zero. Its primitive is widehat x=(0,frac34,frac12,frac14). The next partial sum is zero, so it closes. If instead the desired representative has mean zero, subtract the mean 3/8: widehat x_rm mean =(-frac38,frac38,frac18,-frac18). Both primitives realize exactly widehat r. They differ by an element of ker D. The anchor","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e2110cbb1980b3b4","tex":"\\widehat x_{\\rm mean}","aliases":"widehat x_rm mean widehat rm","display":"\\widehat x_{\\rm mean}","kind":"use","excerpt":"Anchor the repaired signal at zero. Its primitive is widehat x=(0,frac34,frac12,frac14). The next partial sum is zero, so it closes. If instead the desired representative has mean zero, subtract the mean 3/8: widehat x_rm mean =(-frac38,frac38,frac18,-frac18). Both primitives realize exactly widehat r. They differ by an element of ker D. The anchor","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2b26d57eb5a9595f","tex":"\\widetilde a","aliases":"widetilde a widetilde","display":"\\widetilde a","kind":"use","excerpt":"arepsilon_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve th","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7a98ca23b7e037e5","tex":"\\widetilde a_n","aliases":"widetilde a_n widetilde","display":"\\widetilde a_n","kind":"use","excerpt":"arepsilon_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve th","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0a4e15e563cc5107","tex":"\\widetilde b","aliases":"widetilde b widetilde","display":"\\widetilde b","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2500eabb3d3f0489","tex":"\\widetilde c","aliases":"widetilde c widetilde","display":"\\widetilde c","kind":"use","excerpt":"Step 2: obtain literal matching labels. Define widetilde c_i(f)=b_i(v)^-1c_i(f)f^*(b_i(w)) (f:v→ w in U_i). This is a gauge transform and hence a cocycle on U_i. If the arrow lies also in U_j, substitute the original overlap equation and (*) at its two endpoints: widetilde c_j(f) · =b_i(v)^-1a_ij(v)^-1 a_ij(v)c_i(f)[f^*a_ij(w)]^-1 f^*(a_ij(w)b_i(w)","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c53d21b2c0934aea","tex":"\\widetilde c_i","aliases":"widetilde c_i widetilde","display":"\\widetilde c_i","kind":"use","excerpt":"Step 2: obtain literal matching labels. Define widetilde c_i(f)=b_i(v)^-1c_i(f)f^*(b_i(w)) (f:v→ w in U_i). This is a gauge transform and hence a cocycle on U_i. If the arrow lies also in U_j, substitute the original overlap equation and (*) at its two endpoints: widetilde c_j(f) · =b_i(v)^-1a_ij(v)^-1 a_ij(v)c_i(f)[f^*a_ij(w)]^-1 f^*(a_ij(w)b_i(w)","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-eff9a685778c43a7","tex":"\\widetilde c_j","aliases":"widetilde c_j widetilde","display":"\\widetilde c_j","kind":"use","excerpt":"Step 2: obtain literal matching labels. Define widetilde c_i(f)=b_i(v)^-1c_i(f)f^*(b_i(w)) (f:v→ w in U_i). This is a gauge transform and hence a cocycle on U_i. If the arrow lies also in U_j, substitute the original overlap equation and (*) at its two endpoints: widetilde c_j(f) · =b_i(v)^-1a_ij(v)^-1 a_ij(v)c_i(f)[f^*a_ij(w)]^-1 f^*(a_ij(w)b_i(w)","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d4a13a7052f06326","tex":"\\widetilde D","aliases":"widetilde D widetilde","display":"\\widetilde D","kind":"use","excerpt":"[Measurement and operator uncertainty] With the reference certificate above, normx-qx≤ C(norm y+η) if normDx-y≤η. More generally, let widetilde D:E→ R be the bounded linear operator actually used in the measurement. If normD-widetilde D≤ε, normwidetilde D x-y≤η, then normx-qx≤ C(norm y+η+εnorm x). The projection in this conclusion is still the proj","count":19},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-20e75cd544fc942d","tex":"\\widetilde D_t","aliases":"widetilde D_t widetilde","display":"\\widetilde D_t","kind":"use","excerpt":"plate22_gain_budgetFor the kernel-preserving family widetilde D_t(a,b)=(1-t)a, the exact residual gain is 1/(1-t). It deteriorates as the strict budget is exhausted. At the excluded endpoint the kernel changes, so the finite-radius estimate does not extend by substituting t=1.fig:gain-budget","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-929c170d8af8ade4","tex":"\\widetilde G","aliases":"widetilde G widetilde","display":"\\widetilde G","kind":"use","excerpt":"Retaining the phase restores an ordinary multiplication law. Instead of trying to discard the phase, include it in the state. On the eight-element set widetilde G= F_2× G, put (a,x)·(b,y)=(a+b+c(x,y),x+y). The first coordinate of the left-associated product of three elements is a+b+d+c(x,y)+c(x+y,z). The right-associated coordinate is a+b+d+c(y,z)+","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-818401732c1dd8d1","tex":"\\widetilde r","aliases":"widetilde r widetilde","display":"\\widetilde r","kind":"use","excerpt":"How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde","count":17},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2c8e01763b77b415","tex":"\\widetilde r_n","aliases":"widetilde r_n widetilde","display":"\\widetilde r_n","kind":"use","excerpt":"arepsilon_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve th","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-78ff0ae9d511f4cd","tex":"\\widetilde s'","aliases":"widetilde s' widetilde","display":"\\widetilde s'","kind":"use","excerpt":"The proof is independent of the selected raw lift. Indeed, if widetilde s' is another lift, k=widetilde s'-widetilde s lies in ker p_U, and δwidetilde s'-δwidetilde s=δ k∈ B_U. Thus the two quotient classes agree before either feasibility direction is invoked. A raw lift always exists in this additive model by using zero outside U. Constructing a r","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9fcf9ea0bdd21c78","tex":"\\widetilde T","aliases":"widetilde T widetilde","display":"\\widetilde T","kind":"use","excerpt":"How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e24a9d0c560ae65e","tex":"\\widetilde t","aliases":"widetilde t widetilde","display":"\\widetilde t","kind":"use","excerpt":"Γ(A)xrightarrowr_UΓ_U(A) xrightarrow o_UQ_U is exact at Γ_U(A). Thus ker o_U=im r_U, and restriction is onto precisely when o_U is the zero map. First verify that o_U is a homomorphism, since exactness is being asserted in an additive setting. Given s,t∈Γ_U(A), choose raw lifts widetilde s,widetilde t. Their sum lifts s+t. Lift independence and add","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6c8741428acf5118","tex":"\\widetilde T^*","aliases":"widetilde T^* widetilde","display":"\\widetilde T^*","kind":"use","excerpt":"ct the budget [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde T^*z -δ-η M. Fix a feasible correction a. ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c77170c848a2e8b4","tex":"\\widetilde x","aliases":"widetilde x widetilde","display":"\\widetilde x","kind":"use","excerpt":"Protecting three edges forces a different repair. Suppose the first three values (r_0,r_1,r_2)=(1,0,0) are required to remain fixed. The admissible corrections then affect only the final edge. Compatibility forces widetilde r=(1,0,0,-1), widetilde x=(0,1,1,1) for anchor zero. This is the unique repair preserving those three edges. Its squared corre","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-65e8ec76ae768d6c","tex":"x^{-1}","aliases":"x^-1","display":"x^{-1}","kind":"use","excerpt":"erent cocycle whose regular twisted section equation x=px has no solution. The only nontrivial cocycle relation is p^2=1. At a single view the gauge action is p↦ apa^-1. All three transpositions are conjugate and none is conjugate to the identity. The stabilizer is therefore the centralizer. Finally x=px implies p=1 after multiplication by x^-1. so","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d1d16f551dc7199c","tex":"x^{-2}","aliases":"x^-2","display":"x^{-2}","kind":"use","excerpt":"For the lower bound, n>m implies m^2+n^2≤2n^2, hence fracn^2(m^2+n^2)^2≥frac14n^2. The other direction of the decreasing-function integral comparison gives ∑_n>mfrac1n^2≥∫_m+1^∞ x^-2 dx=frac1m+1. This proves B_m≥1/(4(m+1)).","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-226a286fa7079023","tex":"x^{-4}","aliases":"x^-4","display":"x^{-4}","kind":"use","excerpt":"For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Pythagoras applied to that decomposition gives norm x^2=normqx^2+normx-qx^2, hence normqx≤norm x. For any y∈ K, the vector qx-y lies in K, whereas x-qx lies in its orthogonal complement. Consequently normx-y^2=normx-qx^2+normqx-y^2≥normx-qx^2. Equality holds only when normqx-y=0, or y=qx. This proves nearest-point uniqueness without any closedness ","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-baf58144b77f8bed","tex":"X^p","aliases":"X^p","display":"X^p","kind":"use","excerpt":"The phase is visible in actual matrices. Over R, define X=0 · 1 1 · 0, Z=1 · 0 0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3463f1fd3454cd7b","tex":"X^{y_1}","aliases":"X^y_1","display":"X^{y_1}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"[Classification of a nonempty extension fiber] If x_0∈Ext_U(s), translation gives an equivalence H_U≃Ext_U(s), h⟼ x_0+h. Consequently the extension is unique exactly when o_U(s)=0 and H_U=0. Define Θ:H_U→Ext_U(s) by Θ(h)=x_0+h. It lands in the extension set because δ(x_0+h)=δ x_0+δ h=0, p_U(x_0+h)=s+0=s. Define Ψ:Ext_U(s)→ H_U by Ψ(x)=x-x_0. Its di","count":38},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-68998c0c84fa2ca0","tex":"x_0^{-1}","aliases":"x_0^-1","display":"x_0^{-1}","kind":"use","excerpt":"· =f^*((x_0)_w^-1x_w)=f^*(h_w). The cancellation uses inverse multiplication in its actual order. No commuting factors have been moved past one another. Hence h∈ H_U. Pointwise group cancellation shows that h↦ x_0h and x↦ x_0^-1x are inverse maps.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"The choice of x_0 only supplies an origin for the parameters. If x_1=x_0+h_1 is used instead, the parameter of x changes from x-x_0 to x-x_1=(x-x_0)-h_1. The affine set of answers itself has not changed. sourceKS10 An extension fiber is therefore an affine copy, or torsor, of the invisible coherent group. Selecting one solution gives coordinates on","count":15},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"An initially free parameter can become an obstruction Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k,","count":18},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"For D(x_1,x_2,x_3)=x_1-x_2, p(x)=(x_1,x_3), and x=(2,0,5), the ordinary orthogonal answer is (1,1,5). The protected answer is (2,2,5). The respective squared errors are two and four. Both have zero residual, but only the latter preserves the declared data. The theorem is formalized in codeProtectedHilbertRepair.lean, using the actual operator on ke","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-73a4e2e74cee4766","tex":"X_b","aliases":"X_b","display":"X_b","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-702e017c46584077","tex":"x_c","aliases":"x_c","display":"x_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ebb507729655e93c","tex":"x_{HLw}","aliases":"x_HLw","display":"x_{HLw}","kind":"use","excerpt":"If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cd9406cbc1af71a5","tex":"x_{Hv}","aliases":"x_Hv","display":"x_{Hv}","kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-405cde31d8eff404","tex":"X_{\\i}","aliases":"X_i i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8 node[box,minimum width=1.5cm] (xi) at (x,2) X_i; node[box,minimum width=1.5cm] (yi) at (x,0) Y_i; draw[arr] (xi) -- node[lab,right] f_i (yi); foreach j/i in 1/0,2/1,3/2 draw[arr] (xj) -- node[lab,above] ρ^X_i (xi); draw[arr] (yj) -- node[lab,below] ρ^Y_i (yi); node[lab,text width=12.2cm] at (0,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Bitwise complementation is represented at level n by complementing the first n bits. The left shift F(x)_i=x_i+1 uses the schedule ϕ(n)=n+1. A nearest-neighbor difference over F_2, G(x)_i=x_i+x_i+1, also uses n+1 input bits for n output bits. Applying G twice uses n+2 bits, and the middle terms cancel: G(G(x))_i=x_i+x_i+2. The compatible finite map","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-eebbd1d2227e03a8","tex":"x_{\\i}","aliases":"x_i i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8node[box,minimum width=1.5cm] (ri) at (x,1.7) x_i∈ R_i; foreach i/j in 0/1,1/2,2/3 draw[arr] (ri.east) -- node[lab,above] C_i (rj.west); draw[thinarr] (rj.north) to[out=130,in=50] node[lab,above] J_i (ri.north); foreach i/x in 0/-4.8,1/-1.6,2/1.6 node[box,minimum width=1.6cm] (hi) at (x,-.3) h_i","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3eb6a4bc2837a67f","tex":"x_{i+1}","aliases":"x_i+1","display":"x_{i+1}","kind":"use","excerpt":"Bitwise complementation is represented at level n by complementing the first n bits. The left shift F(x)_i=x_i+1 uses the schedule ϕ(n)=n+1. A nearest-neighbor difference over F_2, G(x)_i=x_i+x_i+1, also uses n+1 input bits for n output bits. Applying G twice uses n+2 bits, and the middle terms cancel: G(G(x))_i=x_i+x_i+2. The compatible finite map","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4e65ecaade8e1cc1","tex":"x_{i+2}","aliases":"x_i+2","display":"x_{i+2}","kind":"use","excerpt":"Bitwise complementation is represented at level n by complementing the first n bits. The left shift F(x)_i=x_i+1 uses the schedule ϕ(n)=n+1. A nearest-neighbor difference over F_2, G(x)_i=x_i+x_i+1, also uses n+1 input bits for n output bits. Applying G twice uses n+2 bits, and the middle terms cancel: G(G(x))_i=x_i+x_i+2. The compatible finite map","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0430cf91ce15859e","tex":"x_{j+1}","aliases":"x_j+1","display":"x_{j+1}","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f88b14242652e8b9","tex":"X_k","aliases":"X_k","display":"X_k","kind":"use","excerpt":"Equality is witnessed at finite stages. Two different schedules can represent the same transformation. To compare (ϕ,f) and (ψ,g), require that for every output level n there be a finite level k≥ϕ(n),ψ(n) with f_nρ^X_ϕ(n),k=g_nρ^X_ψ(n),k as functions on all of X_k. Call this eventual equality. The witnessing k may depend on n. Equality is required ","count":11},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c7df6d1ed132831f","tex":"x_{k+1}","aliases":"x_k+1","display":"x_{k+1}","kind":"use","excerpt":"For the adjacent equation at k, choose one n with k+1≤ϕ(n). Then ρ_k,k+1x_k+1 =ρ_k,k+1ρ_k+1,ϕ(n)y_n =ρ_k,ϕ(n)y_n=x_k. The recovered family is therefore a point of the original tower.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f9c58e09df34b610","tex":"x_{k_n}","aliases":"x_k_n","display":"x_{k_n}","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","excerpt":"Finally, fix a level n. Every global point satisfies x_n=ρ_n,m(x_m) for each m≥ n, hence x_n∈ Y_n. Conversely, start with any y∈ Y_n, lift recursively above level n through the surjections just proved, and define the lower coordinates by x_i=ρ_i,n(y). The composite restriction laws verify coherence across and below level n. The resulting point eval","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2ce545975574982d","tex":"x_{m+1}","aliases":"x_m+1","display":"x_{m+1}","kind":"use","excerpt":"Suppose instead that x=(x_n) is a global point. Equality along each inclusion says that the underlying natural numbers of x_n+1 and x_n are equal. Induction therefore makes every underlying coordinate equal to the number m underlying x_0. Membership of x_m+1 in X_m+1 now gives m≥ m+1, a contradiction. Thus no global point exists.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ec84613b712fe298","tex":"x_N","aliases":"x_N","display":"x_N","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composi","count":21},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4715e03ab2461874","tex":"x_{n+1}","aliases":"x_n+1","display":"x_{n+1}","kind":"use","excerpt":"sidual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions tur","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-964b3165f4897ae1","tex":"x_o","aliases":"x_o","display":"x_o","kind":"use","excerpt":"Conversely, let x∈ker d. The path telescope along p_i has all zero summands, hence T_ix_i=x_o for every vertex. Applying T_i^-1 gives x_i=T_i^-1x_o, so x=R_p(x_o). The first part then places x_o in K. We have proved both inverse equations and uniqueness of an exact extension of a root value.","count":16},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a487d9d7f59851e2","tex":"x_z","aliases":"x_z","display":"x_z","kind":"use","excerpt":"[Differential identities] The map δ:E→ R is additive, Γ(A)=kerδ, and (δ x)_1_v=0, (δ x)_g∘ f=f^*((δ x)_g)+(δ x)_f. Additivity follows coordinate by coordinate. Vanishing is exactly the defining coherence equation. For composition, add and subtract f^*x_w: f^*g^*x_z-x_v =f^*(g^*x_z-x_w)+(f^*x_w-x_v). The identity equation uses (1_v)^*=id. sourceKS08","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-145fa231c2e954bf","tex":"X_\\theta","aliases":"X_θ theta","display":"X_\\theta","kind":"use","excerpt":"Cofinal changes also preserve the morphism calculus. A monotone unbounded schedule θ gives the sampled resolution X_θ. There is a transport from X to X_θ that reads the selected level. For the reverse direction choose, for each n, a level j(n) with n≤θ(j(n)), and restrict the available value down to X_n. Taking a least such index gives a monotone s","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-94f101af9ed70d0b","tex":"x_{\\phi(i)}","aliases":"x_ϕ(i) phi","display":"x_{\\phi(i)}","kind":"use","excerpt":"The remaining obligation is naturality on all finite states. Fix i≤ j and u∈ X_ϕ(j). By surjectivity of that point projection, choose x with x_ϕ(j)=u. Equation (*) and output coherence yield ρ^Y_i,jf_j(u) · =ρ^Y_i,jF(x)_j=F(x)_i, f_iρ^X_ϕ(i),ϕ(j)(u) · =f_i(x_ϕ(i))=F(x)_i. The state u was arbitrary. Hence the two finite-level functions are equal, wh","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4bf2d988c0ac047e","tex":"X_{\\phi(j)}","aliases":"X_ϕ(j) phi","display":"X_{\\phi(j)}","kind":"use","excerpt":"The remaining obligation is naturality on all finite states. Fix i≤ j and u∈ X_ϕ(j). By surjectivity of that point projection, choose x with x_ϕ(j)=u. Equation (*) and output coherence yield ρ^Y_i,jf_j(u) · =ρ^Y_i,jF(x)_j=F(x)_i,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-95e421cf839b657d","tex":"x_{\\phi(j)}","aliases":"x_ϕ(j) phi","display":"x_{\\phi(j)}","kind":"use","excerpt":"The remaining obligation is naturality on all finite states. Fix i≤ j and u∈ X_ϕ(j). By surjectivity of that point projection, choose x with x_ϕ(j)=u. Equation (*) and output coherence yield ρ^Y_i,jf_j(u) · =ρ^Y_i,jF(x)_j=F(x)_i,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c2bf72b185a8cab0","tex":"x_{\\phi(n)}","aliases":"x_ϕ(n) phi","display":"x_{\\phi(n)}","kind":"use","excerpt":"Sampling a full point and recovering its kth coordinate gives ρ_k,ϕ(n)x_ϕ(n)=x_k by its original coherence. Recovering a sampled point and evaluating at k=ϕ(n) allows the index n itself, giving ρ_ϕ(n),ϕ(n)y_n=y_n. These are both inverse identities. No surjectivity of the bonding maps or of the point evaluations was used. sourceKS18 This theorem is ","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a9479d7be2af7aae","tex":"x_{\\phi(\\psi(n))}","aliases":"x_ϕ(ψ(n)) phi psi","display":"x_{\\phi(\\psi(n))}","kind":"use","excerpt":"Finally, evaluate a witness equation at the kth coordinate of a coherent point x. Its two restricted arguments become x_ϕ(n) and x_ψ(n), proving equality of the realized outputs. Thus realization is well defined on the quotient. For a composite, direct evaluation gives |g∘ f|(x)_n=g_n(f_ψ(n)(x_ϕ(ψ(n)))) =(|g|(|f|(x)))_n. The identity law is equally","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-030c2588c09d3e01","tex":"x_{\\psi(n)}","aliases":"x_ψ(n) psi","display":"x_{\\psi(n)}","kind":"use","excerpt":"Finally, evaluate a witness equation at the kth coordinate of a coherent point x. Its two restricted arguments become x_ϕ(n) and x_ψ(n), proving equality of the realized outputs. Thus realization is well defined on the quotient. For a composite, direct evaluation gives |g∘ f|(x)_n=g_n(f_ψ(n)(x_ϕ(ψ(n)))) =(|g|(|f|(x)))_n. The identity law is equally","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b99185167014e4eb","tex":"x_\\infty","aliases":"x_∞ infty","display":"x_\\infty","kind":"use","excerpt":"The maps are linear. On the finite vertex product, normR_pz_∞≤(max_inormT_i^-1)norm z, normx_o≤norm x_∞. These bounds also hold on the subspaces K and ker d, proving continuity of both directions. No completeness is needed for these finite bounds.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-55abe5dbff13226b","tex":"\\xleftarrow","aliases":"xleftarrow","display":null,"kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2355745dec81ea53","tex":"\\xRightarrow","aliases":"xRightarrow","display":null,"kind":"use","excerpt":"Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e5f6938d2e45504d","tex":"y_*","aliases":"y_*","display":"y_*","kind":"use","excerpt":"[Nearest feasible protected repair] Every feasible x has a unique closest exact answer y_* preserving p(x). Its correction h_*=x-y_* is the unique vector in the orthogonal complement of ker B inside M satisfying Bh_*=Dx. For every other feasible answer y, normx-y^2=normx-y_*^2+normy_*-y^2. Moreover, the following conditions are equivalent: ran B is","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d842f0833fbfd1bc","tex":"Y_0","aliases":"Y_0","display":"Y_0","kind":"use","excerpt":"Choose x_0∈ Y_0 and recursively choose x_n+1∈ Y_n+1 above x_n. The preceding surjectivity justifies each step. Adjacent coherence implies every composite coherence equation, so this gives a point. This is an existence construction using countably many choices, not an asserted effective procedure for locating the stabilization indices.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dd412919846eece9","tex":"y_2","aliases":"y_2","display":"y_2","kind":"use","excerpt":"on c(x,y)+c(x+y,z)=c(y,z)+c(x,y+z). There is no function a:G→ F_2 such that c(x,y)=a(x)+a(y)-a(x+y) for every x,y∈ G. Normalization follows from multiplication by zero. Expanding the two sides of [source reference] gives, respectively, x_2y_1+(x_2+y_2)z_1, y_2z_1+x_2(y_1+z_1), which are equal by distributivity.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-536459ef72a0484d","tex":"Y_a","aliases":"Y_a","display":"Y_a","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-eccd6d835c836c19","tex":"Y_{\\i}","aliases":"Y_i i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8 node[box,minimum width=1.5cm] (xi) at (x,2) X_i; node[box,minimum width=1.5cm] (yi) at (x,0) Y_i; draw[arr] (xi) -- node[lab,right] f_i (yi); foreach j/i in 1/0,2/1,3/2 draw[arr] (xj) -- node[lab,above] ρ^X_i (xi); draw[arr] (yj) -- node[lab,below] ρ^Y_i (yi); node[lab,text width=12.2cm] at (0,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"Flatness does not remove amplification. Consider a scalar chain with n+1 vertices, where n≥1, and coordinates e_i(z)=2^iz. Set τ_ij=e_ie_j^-1. This connection is flat, but reverse transport along one edge multiplies norms by two. Let x_i=8·2^i-1, y_i=7·2^i. The family y is exact and preserves x_o=7. On a forward edge the residual of x is 1/2, and o","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-120d8dd974f4bf08","tex":"y_{LHv}","aliases":"y_LHv","display":"y_{LHv}","kind":"use","excerpt":"If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1f5b22a9aa4cdefc","tex":"y_{Lw}","aliases":"y_Lw","display":"y_{Lw}","kind":"use","excerpt":"sm and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturali","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ecfc6f094e651c7b","tex":"y_{Lw'}","aliases":"y_Lw'","display":"y_{Lw'}","kind":"use","excerpt":"sition Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Contravariance of F and coherence of y therefore give F(f)(Ry)_w' · =F(f)F(t_w')y_Lw' · =","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a916367bec9f924f","tex":"y_M","aliases":"y_M","display":"y_M","kind":"use","excerpt":"The nearest reachable image y_M=P_T({a:norm a≤ M})r is unique, so the best residual vector r-y_M is unique as well. Corrections mapping to y_M can differ by vectors in ker T. Among those corrections there is a unique least norm correction. These two uniqueness statements should not be confused with injectivity of T.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b3b03c916a5da762","tex":"y_N","aliases":"y_N","display":"y_N","kind":"use","excerpt":"Suppose Q_Nx→0. Put y_N=S_Nx=x-Q_Nx. Idempotence gives Q_Ny_N=Q_Nx-Q_N^2x=0, so y_N∈ F. Since x-y_N=Q_Nx→0, these are finite-core approximants converging to x. Hence x∈overline F.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7a4d74bf293c7c99","tex":"y_p","aliases":"y_p","display":"y_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1ecb58e66eedd43b","tex":"y_t","aliases":"y_t","display":"y_t","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-82ed34227c018c62","tex":"y_v","aliases":"y_v","display":"y_v","kind":"use","excerpt":"), there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for x gives f^*(ev_w x)=f^*(x_w)=x_v=ev_v x. If x,y∈Γ(F) have equal evaluations, then x_v=y_v for every v. Equality of functions into a dependent product gi","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c3e3ca27df8cd514","tex":"y_w","aliases":"y_w","display":"y_w","kind":"use","excerpt":"If d=a· c and x is a section of T_c, put y_v=a_vx_v. Then d_ff^*(y_w) =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(x_w) =a_vx_v=y_v. The inverse map is multiplication by a_v^-1 at each view, proving an equivalence of the actual section sets.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7286e65db353f4e6","tex":"Y_\\eta","aliases":"Y_η eta","display":"Y_\\eta","kind":"use","excerpt":"[Cofinal invariance of eventual transports] Sampling and recovery are inverse in the eventual transport category. Consequently, for monotone unbounded schedules θ and η, there is a bijection EventualHom(X,Y) ≃ EventualHom(X_θ,Y_η). On point spaces it agrees with the previously constructed cofinal realization equivalences. Let s:X→ X_θ be sampling, ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6b722730f1539238","tex":"Y_{\\psi(n)}","aliases":"Y_ψ(n) psi","display":"Y_{\\psi(n)}","kind":"use","excerpt":"The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0eb2c4237a529e5e","tex":"y_\\ell","aliases":"y_ℓ ell","display":"y_\\ell","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":41},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"use","excerpt":"[Parity and an independent ambiguity] For two views with f^*(m)=2m on Z, a prescribed value k at the coarse view extends exactly when k is even. The extension is (k,k/2) and is unique. In particular 1 is obstructed and 2 extends nontrivially. The parity statement is an instance of Theorem~[source reference]. If a separate, disconnected integer view","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-544ac8aed4a2c573","tex":"Z^1","aliases":"Z^1","display":"Z^1","kind":"use","excerpt":"to e, then (ba)· c=b·(a· c)=e, giving composition. The identity label is 1 and the inverse label is a^-1. Associativity and the two inverse identities now follow from the corresponding multiplication laws in the product group. The label ba is therefore forced by the order of application, not an arbitrary notational convention. sourceKS30 The action","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1a49e6bec04c7b56","tex":"Z^2","aliases":"Z^2","display":"Z^2","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ddecadd3df3a219e","tex":"\\Z^2","aliases":"Z^2 Z","display":null,"kind":"use","excerpt":"An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"2t · 1-t^2 satisfies norm(U_t-I)z^2=frac4t^21+t^2norm z^2. For t≠0 its fixed space is zero and its residual gain is sqrt1+t^2/(2|t|), which diverges as t→0. At t=0, the entire fiber is fixed. Therefore small loop discrepancy does not supply a uniform stable realization estimate. The fixed directions and their conditioning must be analyzed together.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c8d007212f447c06","tex":"Z^q","aliases":"Z^q","display":"Z^q","kind":"use","excerpt":"The phase is visible in actual matrices. Over R, define X=0 · 1 1 · 0, Z=1 · 0 0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d322c4a01ae30ded","tex":"Z^{x_2}","aliases":"Z^x_2","display":"Z^{x_2}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"on c(x,y)+c(x+y,z)=c(y,z)+c(x,y+z). There is no function a:G→ F_2 such that c(x,y)=a(x)+a(y)-a(x+y) for every x,y∈ G. Normalization follows from multiplication by zero. Expanding the two sides of [source reference] gives, respectively, x_2y_1+(x_2+y_2)z_1, y_2z_1+x_2(y_1+z_1), which are equal by distributivity.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5e32645afb213c74","tex":"z_2","aliases":"z_2","display":"z_2","kind":"use","excerpt":"fixed anchor vectors.; [shift=(3.2,.4)] draw[thinarr](-2.1,0)--(2.6,0) node[right,lab] z_1; draw[thinarr](0,-2)--(0,2.3) node[above,lab] z_2; draw[KomorebiPine,line width=2pt](-2,0)--(2.3,0); node[dot] (p) at (1.4,1.4) ;node[dot] (r) at (1.4,-1.4) ; draw[arr,dashed](p)--(r); node[lab,anchor=east] at (-.35,1.2) H=diag(1,-1); node[lab,left] at (p) (u","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9fdca6236d286603","tex":"Z_n","aliases":"Z_n","display":"Z_n","kind":"use","excerpt":"The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-53947a836b78402e","tex":"z_v","aliases":"z_v","display":"z_v","kind":"use","excerpt":"Given u∈Γ(B_a,b), forget each fiber-membership condition and call the underlying family z_v∈ M_v. Coherence of u is precisely coherence of z, so z∈Γ(M). The equations p_vz_v=a_v hold at every coordinate. Joint determination in Γ(F) yields Γ(p)z=a, and the equations for q similarly yield Γ(q)z=b. Thus z is a global bridge.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1011bf33a5864682","tex":"z_\\alpha","aliases":"z_α alpha","display":"z_\\alpha","kind":"use","excerpt":"[Exact regularized primal and dual gap] For every a∈ H and z∈ Y, J_α(a)- D_α(z) =normTa-r+z^2+ fracnormT^*z-α a^2α. Consequently D_α(z)≤ J_α(a) for every pair. At a=a_α and z=z_α=r-Ta_α, equality holds, so the dual maximum is attained and equals the primal minimum. Moreover, αnorma-a_α^2 ≤ J_α(a)- D_α(z). For the easily computed choice z=r-Ta, J_α(","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-02ad10b5fe941a06","tex":"|_{\\ker D}","aliases":"|_ker D ker","display":"|_{\\ker D}","kind":"use","excerpt":"[Closed range for the joint defect] Assume ran D is closed. Then ran(D,C) is closed Longleftrightarrow ran(C|_ker D) is closed. Under either condition, a finite joint residual-to-repair bound exists. Suppose the range of the joint map is closed. To test closedness of ran B, take a sequence Bk_n→ y with k_n∈ker D. The joint images are (Dk_n,Ck_n)=(0","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-00398672ab8c08af","tex":"|_{\\ker p}","aliases":"|_ker p ker","display":"|_{\\ker p}","kind":"use","excerpt":"Here the general protected construction is completely explicit. With candidate space E=R^4, defect d=ℓ, and protected observable p(r)=(r_0,r_1,r_2), the allowed direction is the last coordinate axis. The restricted operator d|_ker p has gain one. Without protection, the least gain for distance to kerℓ is 1/2, since normℓ=2. The different constants ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-89bc28b8775c3cb7","tex":"|_M","aliases":"|_M","display":"|_M","kind":"use","excerpt":"Stable repair with protected data. The preceding orthogonal projection does not generally preserve a separately declared observable. To join the analytic theory to Section~[source reference], let p:E→ S be continuous linear into a normed space. Define M=ker p, B=D|_M:M→ R, H=M∩ker D. The closed subspace M inherits its Hilbert structure from E. A ca","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-17be83227c76198b","tex":"|_U","aliases":"|_U","display":"|_U","kind":"use","excerpt":"chapPartial data, gluing, and correspondence fibers Prescribed views. For U⊆ObV, use the full subcategory on U. A partial section s∈Γ_U(F) satisfies all transition equations whose endpoints lie in U. Define Ext_U(s)={x∈Γ(F):x|_U=s}. This fiber can be empty, a singleton, or much larger. A partial section is already coherent on its prescribed domain.","count":9},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-62b2ba1a7f4ec5e9","tex":"|_{U_j}","aliases":"|_U_j","display":"|_{U_j}","kind":"use","excerpt":"Step 3: assembly and restriction are inverse. Formula (*) proves x|_U_j=s_j for every j. Conversely, start with y∈Γ(F) and assemble its restrictions. The resulting coordinate at v is y_v, regardless of which containing patch is chosen. Joint determination gives the original y. These two identities are exactly the inverse identities for the displaye","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2b9f0248ab6db655","tex":"|_W","aliases":"|_W","display":"|_W","kind":"use","excerpt":"The actual patch pseudofunctor. On a patch W, use K_G(W)=Z^1(V|_W;G|_W)//∏_v∈ WG_v. Restriction keeps the existing cocycle values and gauge values on the smaller patch. It first gives a strict category valued functor on the opposite patch poset. Its standard pseudofunctor form has identity comparison isomorphisms, with the required coherence laws.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"Projection (a,x)↦ x is a homomorphism onto G. Its kernel is {(0,0),(1,0)}, which commutes with every element. The map (a,x)⟼(-1)^aU_x is a homomorphism by [source reference]. Its eight matrices are distinct: they are ± I,± X,± Z,± XZ. It is therefore faithful. The phase extension is not an assertion that G itself has acquired an ordinary representa","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":13},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3f7f41ad8028cb36","tex":"\\alpha^{-1}","aliases":"α^-1 alpha","display":"\\alpha^{-1}","kind":"use","excerpt":"To prove closedness of its range, suppose Bh_n→ y. Applying the lower bound to h_n-h_m gives normh_n-h_m≤α^-1normBh_n-Bh_m. Thus h_n is Cauchy. Completeness of H supplies a limit h, and continuity gives Bh=y. If y is orthogonal to the range of B, then for every h, 0=⟨ Bh,y⟩=⟨ h,By⟩. Taking h=By gives By=0, and injectivity gives y=0. A closed subspa","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8bd745916dd5cb08","tex":"\\alpha^{-1/2}","aliases":"α^-1/2 alpha","display":"\\alpha^{-1/2}","kind":"use","excerpt":"fixedplate37_solver_amplificationA fixed normal-equation tolerance has different consequences at different regularization parameters. The curves are universal certified upper bounds for a declared normal-residual norm, not observed errors of a particular experiment. The correction estimate grows as α^-1 while the fitted-data estimate grows as α^-1/","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d830d9bc7e9c803c","tex":"\\alpha^2","aliases":"α^2 alpha","display":"\\alpha^2","kind":"use","excerpt":"To obtain the sharper fitted-data bound, retain the quadratic dependence on norm e instead of substituting the first estimate. Multiplying [source reference] by α and rearranging gives αnormTe^2 · ≤αnorm gnorm e-α^2norm e^2 · =fracnorm g^24 -(αnorm e-fracnorm g2)^2 ≤fracnorm g^24. Division by α and square roots give the factor 1/(2sqrtα).","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a823753ea2193233","tex":"\\alpha_n","aliases":"α_n alpha","display":"\\alpha_n","kind":"use","excerpt":"Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity giv","count":16},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-adddd1027194ff95","tex":"\\alpha_{n_k}","aliases":"α_n_k alpha","display":"\\alpha_{n_k}","kind":"use","excerpt":"Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity giv","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8d6e19f38149c347","tex":"\\alpha_v","aliases":"α_v alpha","display":"\\alpha_v","kind":"use","excerpt":"Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cda0d89220d5cb7b","tex":"\\alpha_w","aliases":"α_w alpha","display":"\\alpha_w","kind":"use","excerpt":"Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-17d9a794b7ae4113","tex":"\\beta_n","aliases":"β_n beta","display":"\\beta_n","kind":"use","excerpt":"When two repair paths select the same point. [Path independence requires a comparison] Let f_n→ x and g_n→ y in a metric space. If dist(f_n,g_n)≤β_n, β_n→0, then x=y. In contrast, exactness and preservation of a nonzero observable alone do not force a unique answer. For each n, the triangle inequality gives dist(x,y)≤dist(x,f_n)+dist(f_n,g_n)+dist(","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9d155779febbf3e2","tex":"\\beta_{T,w}","aliases":"β_T,w beta","display":"\\beta_{T,w}","kind":"use","excerpt":"[A sharp budget asymptotic for the diagonal datum] As m→∞, fracA_mm⟶fracπ4, mB_m⟶fracπ4. For the exact budget profile of this infinite problem, lim_M→∞Mβ_T,w(M)=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-44fa3cd067a59627","tex":"\\beta_{\\widetilde T,\\widetilde r}","aliases":"β_widetilde T,widetilde r beta widetilde widetilde","display":"\\beta_{\\widetilde T,\\widetilde r}","kind":"use","excerpt":"How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aed62ca958ae84bc","tex":"\\beta_x","aliases":"β_x beta","display":"\\beta_x","kind":"use","excerpt":"[Invariance of the marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c47cb5bddd3fc8d3","tex":"\\beta_y","aliases":"β_y beta","display":"\\beta_y","kind":"use","excerpt":"), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(ϕ)σ_x^-1=β_x. So E(a) is a marked arrow. Identity and compo","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8c55dcbd044066e1","tex":"\\Gamma_{U_i}","aliases":"Γ_U_i Gamma","display":"\\Gamma_{U_i}","kind":"use","excerpt":"Let (U_i)_i∈ I be full patches. A matching family consists of s_i∈Γ_U_i(F) with equal values on every overlap. The relevant covering condition is for every f:v→ w, some U_i contains both v and w. Identities imply that this condition also covers every object.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1d3aa5383bcccfb0","tex":"\\Gamma_W","aliases":"Γ_W Gamma","display":"\\Gamma_W","kind":"use","excerpt":"Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coord","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"[The two-stage repair is the joint projection] The operator P is the orthogonal projection onto M, equivalently onto the kernel of Δ=(D,C). If u=x-P_Lx and v=P_Lx-Px, then u⊥ v, normx-Px^2=norm u^2+norm v^2. The subspace L=ker D is closed, hence Hilbert with the inherited inner product. The restriction B=C|_L is continuous, so its kernel is a close","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1f70da82e50b4385","tex":"\\delta^{-2/3}","aliases":"δ^-2/3 delta","display":"\\delta^{-2/3}","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0b92c8f7e3920710","tex":"\\delta^{1/3}","aliases":"δ^1/3 delta","display":"\\delta^{1/3}","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"use","excerpt":"e=widetilde r-r. Subtract the two normal equations to obtain (T^*T+α I)h=T^*e. Pair with h and use the adjoint identity: normTh^2+αnorm h^2 =⟨ e,Th⟩≤δnormTh. Put t=normTh≥0. If t>0, dropping the nonnegative penalty and dividing by t gives t≤δ. If t=0, that inequality is already true. Completing the scalar square gives αnorm h^2≤δ t-t^2 =fracδ^24-(t","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b2b4dc127ad4bd0a","tex":"\\delta^{4/3}","aliases":"δ^4/3 delta","display":"\\delta^{4/3}","kind":"use","excerpt":"lceilδ^-2/3rceil and α=m^-2 as before. If an inexact solver satisfies ε≤καδ^1/3 (κ≥0), then the same argument gives norma-a^†≤(sqrt2+1+κ)δ^1/3. Thus the inner solve can retain the proven outer accuracy if its normal residual decreases on the scale αδ^1/3. Since α is on the scale δ^4/3, the corresponding normal-residual scale is δ^5/3, with the stat","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6b477b6ce1da12ae","tex":"\\delta^{5/3}","aliases":"δ^5/3 delta","display":"\\delta^{5/3}","kind":"use","excerpt":"an inexact solver satisfies ε≤καδ^1/3 (κ≥0), then the same argument gives norma-a^†≤(sqrt2+1+κ)δ^1/3. Thus the inner solve can retain the proven outer accuracy if its normal residual decreases on the scale αδ^1/3. Since α is on the scale δ^4/3, the corresponding normal-residual scale is δ^5/3, with the stated ceiling factors retained in the exact r","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ae5d1ee4ca82d29f","tex":"\\delta_n","aliases":"δ_n delta","display":"\\delta_n","kind":"use","excerpt":"[A consistent inexact stopping rule] Assume the projected intended datum belongs to the actual range of T, with minimum-norm preimage a^†. Along a sequence of positive parameters and measured problems, suppose α_n→0, fracδ_nsqrtα_n→0, fracε_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Conversely, fix x∈overline F and ϵ>0. If K=0, the common operator bound gives normQ_Nx=0 for every N, so convergence is immediate. Suppose K>0. Choose y∈ F satisfying normx-y<ϵ/(K+1). Some M has Q_My=0. For every N≥ M, nesting and the common operator bound give normQ_Nx=normQ_N(x-y)≤ Knormx-y <ϵfracKK+1≤ϵ. The approximation y is chosen before the d","count":17},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":79},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fb43c73461568514","tex":"\\epsilon^2","aliases":"ϵ^2 epsilon","display":"\\epsilon^2","kind":"use","excerpt":"Suppose these energies are bounded, and let L=lim_n m_n^2. Given ϵ>0, choose N with L-m_N^2<ϵ^2. For m≥ n≥ N, norma_m-a_n^2=m_m^2-m_n^2≤ L-m_N^2<ϵ^2. Thus (a_n) is Cauchy and has a norm limit a_*. For each fixed n, all terms from that index onward belong to the closed set A_n, so a_*∈ A_n. This places a_* in their entire intersection.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"Now write (q-q')x=q(I-q')x-(I-q)q'x. The two displayed terms lie in ker D and (ker D)^⊥, respectively, so they are orthogonal. The input components (I-q')x and q'x are also orthogonal. Applying the dual estimate to the first and the original cross estimate to the second gives norm(q-q')x^2 · =normq(I-q')x^2+norm(I-q)q'x^2 · ≤ (C'ε)^2norm(I-q')x^2 +","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5fafc40bd0085548","tex":"\\varepsilon_C","aliases":"ε_C varepsilon","display":"\\varepsilon_C","kind":"use","excerpt":"Motion of the actual descent repair. Compare pairs (D,C) and (D',C') on the same raw Hilbert space. Suppose each pair has closed local range and closed overlap range on its own locally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤ma","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7879135e402b1ed2","tex":"\\varepsilon_D","aliases":"ε_D varepsilon","display":"\\varepsilon_D","kind":"use","excerpt":"Motion of the actual descent repair. Compare pairs (D,C) and (D',C') on the same raw Hilbert space. Suppose each pair has closed local range and closed overlap range on its own locally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤ma","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dc06eebc88a8b163","tex":"\\varepsilon_n","aliases":"ε_n varepsilon","display":"\\varepsilon_n","kind":"use","excerpt":"[A consistent inexact stopping rule] Assume the projected intended datum belongs to the actual range of T, with minimum-norm preimage a^†. Along a sequence of positive parameters and measured problems, suppose α_n→0, fracδ_nsqrtα_n→0, fracε_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2289375dae5eef50","tex":"\\varepsilon_w^{-1}","aliases":"ε_w^-1 varepsilon","display":"\\varepsilon_w^{-1}","kind":"use","excerpt":"simeqΓ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"[Cofinal invariance of eventual transports] Sampling and recovery are inverse in the eventual transport category. Consequently, for monotone unbounded schedules θ and η, there is a bijection EventualHom(X,Y) ≃ EventualHom(X_θ,Y_η). On point spaces it agrees with the previously constructed cofinal realization equivalences. Let s:X→ X_θ be sampling, ","count":39},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-52f9cbbff7441c6e","tex":"\\eta_{2n}","aliases":"η_2n eta","display":"\\eta_{2n}","kind":"use","excerpt":"For failure without monotonicity, take η_2n=0, η_2n+1=1, and ϕ(n)=2n. All sampled values vanish, but infinitely many original values equal one. sourceKS75 Even in the decreasing case, the numerical rate changes. The profile η_n=1/(n+1), sampled at ϕ(n)=2^n-1, becomes 2^-n. This is exponential decay in the number of observations, not a proof of a fa","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6d059a7e178c6271","tex":"\\eta_{2n+1}","aliases":"η_2n+1 eta","display":"\\eta_{2n+1}","kind":"use","excerpt":"For failure without monotonicity, take η_2n=0, η_2n+1=1, and ϕ(n)=2n. All sampled values vanish, but infinitely many original values equal one. sourceKS75 Even in the decreasing case, the numerical rate changes. The profile η_n=1/(n+1), sampled at ϕ(n)=2^n-1, becomes 2^-n. This is exponential decay in the number of observations, not a proof of a fa","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5ed478059bcd3710","tex":"\\eta_n","aliases":"η_n eta","display":"\\eta_n","kind":"use","excerpt":"Changing the sampling schedule does not preserve a rate. A residual profile η_n indexed by resolution is not an iteration trajectory. Nevertheless a useful convergence property is invariant under a cofinal schedule when the profile is monotone.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fdf52ba77f1cb30f","tex":"\\eta_v","aliases":"η_v eta","display":"\\eta_v","kind":"use","excerpt":"If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4f321af5eb71fc0f","tex":"\\eta_{\\phi(k)}","aliases":"η_ϕ(k) eta phi","display":"\\eta_{\\phi(k)}","kind":"use","excerpt":"Solution. The sampled value is η_ϕ(k)=2^-k. Recovering the original index gives k=log_2(n+1) on the sampled stages, so the value is still 1/(n+1). The schedule preserves vanishing but changes the meaning of the horizontal axis. No faster original process has been proved. Proposition~[source reference] is a convergence equivalence under a monotone c","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-333f69862cebb794","tex":"\\eta_{\\phi(n)}","aliases":"η_ϕ(n) eta phi","display":"\\eta_{\\phi(n)}","kind":"use","excerpt":"[Cofinal sampling of a decreasing profile] Suppose η_n≥0 is nonincreasing and ϕ:N→N is monotone and unbounded. Then η_n→0 Longleftrightarrow η_ϕ(n)→0. Without the monotonicity of η, the reverse implication can fail even for a cofinal schedule. The schedule tends to infinity, so convergence of the full profile implies convergence of the sampled one.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"[Classification of a nonempty extension fiber] If x_0∈Ext_U(s), translation gives an equivalence H_U≃Ext_U(s), h⟼ x_0+h. Consequently the extension is unique exactly when o_U(s)=0 and H_U=0. Define Θ:H_U→Ext_U(s) by Θ(h)=x_0+h. It lands in the extension set because δ(x_0+h)=δ x_0+δ h=0, p_U(x_0+h)=s+0=s. Define Ψ:Ext_U(s)→ H_U by Ψ(x)=x-x_0. Its di","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"[Split tower correction] The recurrence defines a continuous linear map A:R→ E with ∂ A=I_R, norm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial co","count":33},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0c2176b912a83c9c","tex":"\\theta^j","aliases":"θ^j theta","display":"\\theta^j","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1c71e2a304d9a62a","tex":"\\theta^{N+1}","aliases":"θ^N+1 theta","display":"\\theta^{N+1}","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c381b1e9f2cf6895","tex":"\\nu_2","aliases":"ν_2 nu","display":"\\nu_2","kind":"use","excerpt":"[The dyadic support pattern] For the same lifts, 2^Nnmid(k+1) Longrightarrow (Q_Nr)_k=0. Thus coordinate k is certainly recovered once N>ν_2(k+1), where ν_2(m) is the largest power-of-two exponent dividing the positive integer m. The only possible unrecovered coordinates at depth N are 2^N-1, 2·2^N-1,quad3·2^N-1, …. Prove the assertion for L_j,N by","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6037193d07157d39","tex":"\\nu_{B_n}","aliases":"ν_B_n nu","display":"\\nu_{B_n}","kind":"use","excerpt":"[Uniformly positive floors obstruct asymptotic repair] Suppose c>0 and eventually c≤ν_B_n(Ω_n). For any choices b_n∈ B_n, it is impossible that normΩ_n-b_n⟶0. Every allowed correction satisfies normΩ_n-b_n≥ν_B_n(Ω_n). The corrected norms are therefore eventually at least c. They cannot converge to zero. sourceKS77 The quantifier covers all admissib","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-38449ee7b10fdb4d","tex":"\\pi_n","aliases":"π_n pi","display":"\\pi_n","kind":"use","excerpt":"A growing example and the gap boundary. Use X_n=Y_n=R^n+1 with maximum norms. Let π_n delete the last coordinate, take ρ^X_n=tfrac12π_n, ρ^Y_n=π_n, and let s_n append zero. Then S=B_Y=1, B_X=1/2, and K=2.","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-51f8c0dd58b64765","tex":"\\pi_U","aliases":"π_U pi","display":"\\pi_U","kind":"use","excerpt":"[Extension criterion] For every coherent partial section, o_U(s)=0 Longleftrightarrow Ext_U(s)≠∅. Let π_U:R→ R/B_U be the quotient homomorphism. By the definition of a quotient by a subgroup, π_U(r)=0 Longleftrightarrow r∈ B_U. Here B_U is the image of ker p_U under δ. Consequently membership of δwidetilde s means the existence of a specific h∈ E s","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cf29efe3e08463bd","tex":"\\pi_{U,U}","aliases":"π_U,U pi","display":"\\pi_{U,U}","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7377cb30d6304a53","tex":"\\pi_{Z,U}","aliases":"π_Z,U pi","display":"\\pi_{Z,U}","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-264156b9b1294150","tex":"\\pi_{Z,W}","aliases":"π_Z,W pi","display":"\\pi_{Z,W}","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b48f947c0ca70892","tex":"\\rho^X_{2n+1}","aliases":"ρ^X_2n+1 rho","display":"\\rho^X_{2n+1}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-755f7ceba9b9722a","tex":"\\rho^X_{a,b}","aliases":"ρ^X_a,b rho","display":"\\rho^X_{a,b}","kind":"use","excerpt":"a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[s","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3b30de3ed849f723","tex":"\\rho^X_{\\i}","aliases":"ρ^X_i rho i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8 node[box,minimum width=1.5cm] (xi) at (x,2) X_i; node[box,minimum width=1.5cm] (yi) at (x,0) Y_i; draw[arr] (xi) -- node[lab,right] f_i (yi); foreach j/i in 1/0,2/1,3/2 draw[arr] (xj) -- node[lab,above] ρ^X_i (xi); draw[arr] (yj) -- node[lab,below] ρ^Y_i (yi); node[lab,text width=12.2cm] at (0,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-73d72c2cfcbcc383","tex":"\\rho^X_k","aliases":"ρ^X_k rho","display":"\\rho^X_k","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b10f6fc6b14d2593","tex":"\\rho^X_{k+1,b}","aliases":"ρ^X_k+1,b rho","display":"\\rho^X_{k+1,b}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-779098801d242d63","tex":"\\rho^X_{k,b}","aliases":"ρ^X_k,b rho","display":"\\rho^X_{k,b}","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cd764dd5def48bd9","tex":"\\rho^X_{k,\\ell}","aliases":"ρ^X_k,ℓ rho ell","display":"\\rho^X_{k,\\ell}","kind":"use","excerpt":"ality is an equivalence relation and is preserved by composition. Its quotient therefore gives a category of resolutions and eventual transports. Realization descends to a functor into sets, preserving identities and composition. A finite-stage equality remains true after moving to a later stage. Indeed, if it holds on X_k and ℓ≥ k, precompose both","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4279f193653756cd","tex":"\\rho^X_{k_n,\\phi(n)}","aliases":"ρ^X_k_n,ϕ(n) rho phi","display":"\\rho^X_{k_n,\\phi(n)}","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9dcbb5c17aafa2b5","tex":"\\rho^X_n","aliases":"ρ^X_n rho","display":"\\rho^X_n","kind":"use","excerpt":"A candidate transport from X to Y consists of maps f_n:X_n→ Y_n. It is exact when ρ^Y_n f_n+1=f_nρ^X_n for all n.","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-19aa1ab1f48b5e8b","tex":"\\rho^X_{n,\\theta(j(n))}","aliases":"ρ^X_n,θ(j(n)) rho theta","display":"\\rho^X_{n,\\theta(j(n))}","kind":"use","excerpt":"with the previously constructed cofinal realization equivalences. Let s:X→ X_θ be sampling, with schedule θ and identity level maps. For recovery choose the least j(n) with n≤θ(j(n)). Unboundedness makes this set nonempty. Increasing n only strengthens the inequality, so j is monotone. The level map of r:X_θ→ X is ρ^X_n,θ(j(n)). Composite restricti","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f193474520572543","tex":"\\rho^X_{\\theta(n),\\theta(j(\\theta(n)))}","aliases":"ρ^X_θ(n),θ(j(θ(n))) rho theta theta theta","display":"\\rho^X_{\\theta(n),\\theta(j(\\theta(n)))}","kind":"use","excerpt":"For sr:X_θ→ X_θ, the schedule at output n is j(θ(n)), and the level map is ρ^X_θ(n),θ(j(θ(n))). Compare it with the identity at a sampled index k≥max{n,j(θ(n))}. Restricting either route from level k of the sampled system gives the same original restriction ρ^X_θ(n),θ(k). This proves eventual equality even when θ has repeated values. Hence the two ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-be807838c5137d76","tex":"\\rho^X_{\\theta(n),\\theta(k)}","aliases":"ρ^X_θ(n),θ(k) rho theta theta","display":"\\rho^X_{\\theta(n),\\theta(k)}","kind":"use","excerpt":"For sr:X_θ→ X_θ, the schedule at output n is j(θ(n)), and the level map is ρ^X_θ(n),θ(j(θ(n))). Compare it with the identity at a sampled index k≥max{n,j(θ(n))}. Restricting either route from level k of the sampled system gives the same original restriction ρ^X_θ(n),θ(k). This proves eventual equality even when θ has repeated values. Hence the two ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0507e0b0818d9ad2","tex":"\\rho^X_{\\phi'(j),k}","aliases":"ρ^X_ϕ'(j),k rho phi","display":"\\rho^X_{\\phi'(j),k}","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-eb28761e19c3c138","tex":"\\rho^X_{\\phi'(\\psi'(n)),k}","aliases":"ρ^X_ϕ'(ψ'(n)),k rho phi psi","display":"\\rho^X_{\\phi'(\\psi'(n)),k}","kind":"use","excerpt":"nd g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-28b1ed6f8781b936","tex":"\\rho^X_{\\phi(j),k}","aliases":"ρ^X_ϕ(j),k rho phi","display":"\\rho^X_{\\phi(j),k}","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8448b479f4cbb66a","tex":"\\rho^X_{\\phi(n),k}","aliases":"ρ^X_ϕ(n),k rho phi","display":"\\rho^X_{\\phi(n),k}","kind":"use","excerpt":"Equality is witnessed at finite stages. Two different schedules can represent the same transformation. To compare (ϕ,f) and (ψ,g), require that for every output level n there be a finite level k≥ϕ(n),ψ(n) with f_nρ^X_ϕ(n),k=g_nρ^X_ψ(n),k as functions on all of X_k. Call this eventual equality. The witnessing k may depend on n. Equality is required ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-000f6c73dd1d8b34","tex":"\\rho^X_{\\phi(\\psi(n)),k}","aliases":"ρ^X_ϕ(ψ(n)),k rho phi psi","display":"\\rho^X_{\\phi(\\psi(n)),k}","kind":"use","excerpt":"nd g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8bf3e7c4ceb8d89f","tex":"\\rho^X_{\\psi(n),k}","aliases":"ρ^X_ψ(n),k rho psi","display":"\\rho^X_{\\psi(n),k}","kind":"use","excerpt":"Equality is witnessed at finite stages. Two different schedules can represent the same transformation. To compare (ϕ,f) and (ψ,g), require that for every output level n there be a finite level k≥ϕ(n),ψ(n) with f_nρ^X_ϕ(n),k=g_nρ^X_ψ(n),k as functions on all of X_k. Call this eventual equality. The witnessing k may depend on n. Equality is required ","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-665c37b658f9761f","tex":"\\rho^Y_{2n}","aliases":"ρ^Y_2n rho","display":"\\rho^Y_{2n}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-00533f34433c2de0","tex":"\\rho^Y_{a,b}","aliases":"ρ^Y_a,b rho","display":"\\rho^Y_{a,b}","kind":"use","excerpt":"a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[s","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c2351fd63f9dbda2","tex":"\\rho^Y_{a,k}","aliases":"ρ^Y_a,k rho","display":"\\rho^Y_{a,k}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d7c357e7e2c7d8e9","tex":"\\rho^Y_{a,k+1}","aliases":"ρ^Y_a,k+1 rho","display":"\\rho^Y_{a,k+1}","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-240b8d2feea43872","tex":"\\rho^Y_{\\i}","aliases":"ρ^Y_i rho i","display":null,"kind":"use","excerpt":"[x=1cm,y=1cm] foreach i/x in 0/-4.8,1/-1.6,2/1.6,3/4.8 node[box,minimum width=1.5cm] (xi) at (x,2) X_i; node[box,minimum width=1.5cm] (yi) at (x,0) Y_i; draw[arr] (xi) -- node[lab,right] f_i (yi); foreach j/i in 1/0,2/1,3/2 draw[arr] (xj) -- node[lab,above] ρ^X_i (xi); draw[arr] (yj) -- node[lab,below] ρ^Y_i (yi); node[lab,text width=12.2cm] at (0,","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a2cac7880cf1ce17","tex":"\\rho^Y_k","aliases":"ρ^Y_k rho","display":"\\rho^Y_k","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-67648e382c5950ed","tex":"\\rho^Y_n","aliases":"ρ^Y_n rho","display":"\\rho^Y_n","kind":"use","excerpt":"A candidate transport from X to Y consists of maps f_n:X_n→ Y_n. It is exact when ρ^Y_n f_n+1=f_nρ^X_n for all n.","count":20},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d2886b332952430a","tex":"\\rho^Y_{\\psi'(n),j}","aliases":"ρ^Y_ψ'(n),j rho psi","display":"\\rho^Y_{\\psi'(n),j}","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-58ef887bb9f1247b","tex":"\\rho^Y_{\\psi(n),j}","aliases":"ρ^Y_ψ(n),j rho psi","display":"\\rho^Y_{\\psi(n),j}","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dc9254e83eae53d1","tex":"\\rho^Z_n","aliases":"ρ^Z_n rho","display":"\\rho^Z_n","kind":"use","excerpt":"[The exact tower category] Identity maps and composites of exact transports are exact. An exact transport induces a map of coherent point spaces by x↦(f_nx_n)_n, preserving identities and composition. For two exact transports, substitute ρ^Z_ng_n+1=g_nρ^Y_n and ρ^Y_nf_n+1=f_nρ^X_n into the composite square. For a coherent point, ρ^Y_n f_n+1x_n+1 =f","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a32d3695336eb34b","tex":"\\rho_{0,1}","aliases":"ρ_0,1 rho","display":"\\rho_{0,1}","kind":"use","excerpt":"An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3417c2f73c9f7d40","tex":"\\rho_{0,2}","aliases":"ρ_0,2 rho","display":"\\rho_{0,2}","kind":"use","excerpt":"An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-737cdd19f67dfe28","tex":"\\rho_1","aliases":"ρ_1 rho","display":"\\rho_1","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3f0529e276ae445a","tex":"\\rho_{1,2}","aliases":"ρ_1,2 rho","display":"\\rho_{1,2}","kind":"use","excerpt":"An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d7ba0ffb2518dd65","tex":"\\rho_2","aliases":"ρ_2 rho","display":"\\rho_2","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-14b329400c5c31fb","tex":"\\rho_{i,n}","aliases":"ρ_i,n rho","display":"\\rho_{i,n}","kind":"use","excerpt":"Finally, fix a level n. Every global point satisfies x_n=ρ_n,m(x_m) for each m≥ n, hence x_n∈ Y_n. Conversely, start with any y∈ Y_n, lift recursively above level n through the surjections just proved, and define the lower coordinates by x_i=ρ_i,n(y). The composite restriction laws verify coherence across and below level n. The resulting point eval","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d51449a1464f8b5e","tex":"\\rho_{k+1,\\phi(n)}","aliases":"ρ_k+1,ϕ(n) rho phi","display":"\\rho_{k+1,\\phi(n)}","kind":"use","excerpt":"For the adjacent equation at k, choose one n with k+1≤ϕ(n). Then ρ_k,k+1x_k+1 =ρ_k,k+1ρ_k+1,ϕ(n)y_n =ρ_k,ϕ(n)y_n=x_k. The recovered family is therefore a point of the original tower.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-bc61c5b129ff9df6","tex":"\\rho_{k,k+1}","aliases":"ρ_k,k+1 rho","display":"\\rho_{k,k+1}","kind":"use","excerpt":"For the adjacent equation at k, choose one n with k+1≤ϕ(n). Then ρ_k,k+1x_k+1 =ρ_k,k+1ρ_k+1,ϕ(n)y_n =ρ_k,ϕ(n)y_n=x_k. The recovered family is therefore a point of the original tower.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-beae91d16a7df6a7","tex":"\\rho_{k,\\phi(m)}","aliases":"ρ_k,ϕ(m) rho phi","display":"\\rho_{k,\\phi(m)}","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a2096330a2542786","tex":"\\rho_{k,\\phi(n)}","aliases":"ρ_k,ϕ(n) rho phi","display":"\\rho_{k,\\phi(n)}","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fb8f4b50aa209d1f","tex":"\\rho_{k,\\phi(\\ell)}","aliases":"ρ_k,ϕ(ℓ) rho phi ell","display":"\\rho_{k,\\phi(\\ell)}","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-997f0b9609eaf253","tex":"\\rho_{\\phi(n),\\phi(n)}","aliases":"ρ_ϕ(n),ϕ(n) rho phi phi","display":"\\rho_{\\phi(n),\\phi(n)}","kind":"use","excerpt":"Sampling a full point and recovering its kth coordinate gives ρ_k,ϕ(n)x_ϕ(n)=x_k by its original coherence. Recovering a sampled point and evaluating at k=ϕ(n) allows the index n itself, giving ρ_ϕ(n),ϕ(n)y_n=y_n. These are both inverse identities. No surjectivity of the bonding maps or of the point evaluations was used. sourceKS18 This theorem is ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aeb949f3ceb46809","tex":"\\rho_{\\phi(n),\\phi(\\ell)}","aliases":"ρ_ϕ(n),ϕ(ℓ) rho phi phi ell","display":"\\rho_{\\phi(n),\\phi(\\ell)}","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"A small noncommutative model. [A C_2 loop with S_3 coefficients] Take one view with a loop σ satisfying σ^2=1, and constant coefficients in a group H. A cocycle is determined by an element p∈ H whose square is the identity. Gauge equivalence is conjugacy, and its automorphisms form the centralizer of p.","count":12},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"A small noncommutative model. [A C_2 loop with S_3 coefficients] Take one view with a loop σ satisfying σ^2=1, and constant coefficients in a group H. A cocycle is determined by an element p∈ H whose square is the identity. Gauge equivalence is conjugacy, and its automorphisms form the centralizer of p.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7499d2052ab744b7","tex":"\\sigma_x","aliases":"σ_x sigma","display":"\\sigma_x","kind":"use","excerpt":"he marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-51ded5aa59c649df","tex":"\\sigma_x^{-1}","aliases":"σ_x^-1 sigma","display":"\\sigma_x^{-1}","kind":"use","excerpt":"[Invariance of the marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ae6135878257b347","tex":"\\sigma_y","aliases":"σ_y sigma","display":"\\sigma_y","kind":"use","excerpt":"he marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8bbe064464725f7e","tex":"\\sigma_y^{-1}","aliases":"σ_y^-1 sigma","display":"\\sigma_y^{-1}","kind":"use","excerpt":"he marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cd33bebc9c1e4ef9","tex":"\\tau_{i_ji_{j+1}}","aliases":"τ_i_ji_j+1 tau","display":"\\tau_{i_ji_{j+1}}","kind":"use","excerpt":"The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-92ea019ce8093a2c","tex":"\\tau_{i_{m-1}i_m}","aliases":"τ_i_m-1i_m tau","display":"\\tau_{i_{m-1}i_m}","kind":"use","excerpt":"Distinct walks are retained as arrows of the view category. The product along a walk q=(i_0,…,i_m) is T_q=τ_i_0i_1⋯τ_i_m-1i_m. Concatenation gives the restriction composition law. An exact vertex family defines a coherent walk-view section, and a coherent walk-view section satisfies every edge equation. Thus this graph construction returns to the o","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-14b801f95f5f5a95","tex":"\\tau_{ij}","aliases":"τ_ij tau","display":"\\tau_{ij}","kind":"use","excerpt":"lly guarantee. Graph views, holonomy, and reconstruction. Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear equivalences τ_ij, with τ_","count":8},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e3e46cde9b77d238","tex":"\\tau_{ij}^{-1}","aliases":"τ_ij^-1 tau","display":"\\tau_{ij}^{-1}","kind":"use","excerpt":"tionGraph views, holonomy, and reconstruction Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear equivalences τ_ij, with τ_ji=τ_ij^-1. ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-7370b2cce0bbcf1a","tex":"\\tau_{ji}","aliases":"τ_ji tau","display":"\\tau_{ji}","kind":"use","excerpt":"tionGraph views, holonomy, and reconstruction Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear equivalences τ_ij, with τ_ji=τ_ij^-1. ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-97267862925829aa","tex":"\\phi'","aliases":"ϕ' phi","display":"\\phi'","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-18f9ac23eff67597","tex":"\\Phi^{-1}","aliases":"Φ^-1 Phi","display":"\\Phi^{-1}","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-3dd15563f7adbbf0","tex":"\\chi_A^{-1}","aliases":"χ_A^-1 chi","display":"\\chi_A^{-1}","kind":"use","excerpt":"[The split defect chart] The maps χ_A:E≃_L K× R, χ_A(x)=(P_Ax,dx), χ_A^-1(u,r)=u+Ar are inverse continuous linear equivalences. In these coordinates, P_A sends (u,r) to (u,0). If B is another right inverse, then A-B takes values in K, and P_Bx-P_Ax=(A-B)dx. The projections commute if and only if they are equal, and they are equal if and only if A=B","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f3e9049b2d128d00","tex":"\\chi_B","aliases":"χ_B chi","display":"\\chi_B","kind":"use","excerpt":"[Triangular transport and repair mismatch] In split defect coordinates, the change of representation is χ_BSχ_A^-1(u,r)=(Su+Mr,Cr). The two repair routes differ by P_BSx-SP_Ax=M(dx). Consequently they agree on exact inputs, and normP_BSx-SP_Ax≤norm Mnormdx, norm M≤norm Snorm A+norm Bnorm C. They agree on all inputs if and only if SA=BC.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"10 If x_0∈Ext_U(s), translation gives an equivalence H_U≃Ext_U(s), h⟼ x_0+h. Consequently the extension is unique exactly when o_U(s)=0 and H_U=0. Define Θ:H_U→Ext_U(s) by Θ(h)=x_0+h. It lands in the extension set because δ(x_0+h)=δ x_0+δ h=0, p_U(x_0+h)=s+0=s. Define Ψ:Ext_U(s)→ H_U by Ψ(x)=x-x_0. Its differential is zero, and its restriction is s","count":3},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a1d181fcbd5bda98","tex":"\\psi'","aliases":"ψ' psi","display":"\\psi'","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-1ca12a002e4da1b8","tex":"\\psi^{-1}","aliases":"ψ^-1 psi","display":"\\psi^{-1}","kind":"use","excerpt":"Replacing b by an isomorphic boundary b' also gives an equivalence, by composing the identification with b→ b'. For R=id, the marked realization groupoid has center (b,1_b), and between (x,ϕ) and (y,ψ) there is exactly one compatible arrow ψ^-1ϕ. This is a useful example of contractible realization data, not a claim that the original groupoid has n","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"Distance to the actual correction image. Let R be a normed additive group and B≤ R an additive subgroup. In the protected linear model, B=d(ker p). For a residual Ω∈ R, define ν_B(Ω)=inf_b∈ BnormΩ-b. The subgroup contains zero, so this is a finite nonnegative real number. It is an infimum, not an assertion that a nearest admissible residual exists.","count":21},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-097ee86d56b24dd4","tex":"\\omega_f","aliases":"ω_f omega","display":"\\omega_f","kind":"use","excerpt":"stead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e9d6d9121cd11235","tex":"\\Omega_n","aliases":"Ω_n Omega","display":"\\Omega_n","kind":"use","excerpt":"A persistent floor blocks every correction sequence. The correction image can vary with the resolution or the admissibility rule. Let B_n≤ R and let Ω_n∈ R be given.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"use","excerpt":"[The regularization limit alternative] As αdownarrow0, Ta_α⟶ b, normr-Ta_α⟶ q, αnorma_α^2⟶0. If b∈ran T, let a^† be its unique minimum norm preimage. Then a_α⟶ a^† in norm. If b∉ran T, then norma_α→∞. Step 1: identify the unavoidable component. Since r-b⊥overlineran T, the adjoint identity gives T^*(r-b)=0. Thus a_α(r)=a_α(b). For every correction ","count":22},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-837406a230d80642","tex":"\\ell^\\infty","aliases":"ℓ^∞ ell infty","display":"\\ell^\\infty","kind":"use","excerpt":"Two boundary examples. On ℓ^∞(N,R), let C remove the first coordinate and let J prepend zero. Then Q_Nx is the tail of x, placed at its original coordinates with zeros before N. Every fixed coordinate is eventually zero in the remainder, and normQ_N≤1. Nevertheless the constant sequence 1 satisfies normQ_N1=1 for every N.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0d9104cdc1fe8aed","tex":"\\ell^\\infty_n","aliases":"ℓ^∞_n ell infty","display":"\\ell^\\infty_n","kind":"use","excerpt":"chapBanach repair and constructed split coordinates An actual complete candidate space. Let X_n,Y_n be Banach spaces over a complete nontrivially normed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formal","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"When global behavior determines the finite presentation. Say that X has surjective point projections when ∀ k ev_k:Point(X)→ X_k is onto. This means that every finite input state really occurs in some coherent global input. It is stronger than nonemptiness of the global space.","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-935f9c58c2ce6b38","tex":"\\partial_\\phi","aliases":"∂_ϕ partial phi","display":"\\partial_\\phi","kind":"use","excerpt":"[Bounded-gap resampling and selected repair] Suppose ϕ(n+1)-ϕ(n)≤ m for a fixed natural number m. Then the actual sampling maps are bounded and satisfy normS_ϕ≤1, normC_ϕ≤ m, ∂_ϕ S_ϕ=C_ϕ∂. Exact candidates remain exact under every monotone common resampling, even without a gap bound. If A and B are bounded right inverses for the fine and sampled di","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"[Semantics preserving correction] For x∈ E, the following conditions are equivalent: · [dx]=0 in R/d(ker p), · ∃ h∈ E: p(h)=0, d(h)=d(x), · ∃ y∈ E: d(y)=0, p(y)=p(x). The class of dx is unchanged when an admissible correction is subtracted from x. The image d(ker p) is a subgroup of R. The defining property of the quotient says [dx]=0 Longleftright","count":7},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"[Extension criterion] For every coherent partial section, o_U(s)=0 Longleftrightarrow Ext_U(s)≠∅. Let π_U:R→ R/B_U be the quotient homomorphism. By the definition of a quotient by a subgroup, π_U(r)=0 Longleftrightarrow r∈ B_U. Here B_U is the image of ker p_U under δ. Consequently membership of δwidetilde s means the existence of a specific h∈ E s","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"For route independence, the proved equivalence identifies K with the image of root evaluation from the intrinsically defined space ker d. That image does not refer to any chosen routes, so every route system yields the same K. For z∈ K, reconstruction using either system is an exact family with root z, and uniqueness makes the families equal. For z","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4e78af967b166a06","tex":"\\prod_{(i,j)\\in\\vec E}","aliases":"∏_(i,j)∈vec E prod in vec","display":"\\prod_{(i,j)\\in\\vec E}","kind":"use","excerpt":"The candidate and residual spaces are finite products with maximum norms: E=∏_i∈ VH, R=∏_(i,j)∈vec EH, (dx)_ij=τ_ijx_j-x_i. Both orientations of an edge are retained. If M bounds the norms of all measured edge maps, then norm d≤ M+1. The empty edge case has the usual zero residual.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-aaf9cd5f4f9933fc","tex":"\\prod_{f:v\\to w}","aliases":"∏_f:v→ w prod to","display":"\\prod_{f:v\\to w}","kind":"use","excerpt":"The compatibility differential. Assume each fiber A_v is an abelian group and each restriction is additive. Set E=∏_v A_v, R=∏_f:v→ wA_v, (δ x)_f=f^*(x_w)-x_v. The group R retains an entry for every arrow. It is not silently identified with a cohomology group.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4dd9169b011221a2","tex":"\\prod_{i\\in V}","aliases":"∏_i∈ V prod in","display":"\\prod_{i\\in V}","kind":"use","excerpt":"The candidate and residual spaces are finite products with maximum norms: E=∏_i∈ VH, R=∏_(i,j)∈vec EH, (dx)_ij=τ_ijx_j-x_i. Both orientations of an edge are retained. If M bounds the norms of all measured edge maps, then norm d≤ M+1. The empty edge case has the usual zero residual.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-badead017d25968f","tex":"\\prod_{j<N}","aliases":"∏_j<N prod","display":"\\prod_{j<N}","kind":"use","excerpt":"nce. This proves equality of the two sets. Applying it to every x∈ R_0 gives the density criterion. sourceKS27 There are other sufficient conditions. If the transported details are absolutely summable in the Banach space R_0, their sum exists and the remaining boundary lies in P_∞. If that persistent subspace is zero, the sum equals x. Alternativel","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-709a1d5e851efb47","tex":"\\prod_{j=0}^{k-1}","aliases":"∏_j=0^k-1 prod","display":"\\prod_{j=0}^{k-1}","kind":"use","excerpt":"The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-37de85f98def7f5b","tex":"\\prod_{u\\in U}","aliases":"∏_u∈ U prod in","display":"\\prod_{u\\in U}","kind":"use","excerpt":"Three maps that must not be confused. Keep the additive view system of Section~[source reference]. For a prescribed domain U, there are three distinct maps: p_U:E⟶∏_u∈ UA_u, r_U:Γ(A)⟶Γ_U(A), o_U:Γ_U(A)⟶ Q_U. The first restricts a raw family, whether or not it is coherent. The second restricts a global coherent family. The third detects whether an a","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d212f3e24adbf170","tex":"\\prod_v","aliases":"∏_v prod","display":"\\prod_v","kind":"use","excerpt":"The compatibility differential. Assume each fiber A_v is an abelian group and each restriction is additive. Set E=∏_v A_v, R=∏_f:v→ wA_v, (δ x)_f=f^*(x_w)-x_v. The group R retains an entry for every arrow. It is not silently identified with a cohomology group.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-70c07b4410f6643a","tex":"\\prod_{v\\in W}","aliases":"∏_v∈ W prod in","display":"\\prod_{v\\in W}","kind":"use","excerpt":"The actual patch pseudofunctor. On a patch W, use K_G(W)=Z^1(V|_W;G|_W)//∏_v∈ WG_v. Restriction keeps the existing cocycle values and gauge values on the smaller patch. It first gives a strict category valued functor on the opposite patch poset. Its standard pseudofunctor form has identity comparison isomorphisms, with the required coherence laws.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Every edge map is bijective. Around the cycle their composite is translation by -∑_i r_i. A global realization must be fixed by that translation. Thus the example makes local invertibility and global compatibility visibly different without needing a noncommutative group.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6cb04d84bd2adbc3","tex":"\\sum_{j<N}","aliases":"∑_j<N sum","display":"\\sum_{j<N}","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b9a7896e38144f2e","tex":"\\sum_{j=0}^{N}","aliases":"∑_j=0^N sum","display":"\\sum_{j=0}^{N}","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-586bc3571b381e64","tex":"\\sum_{j=0}^{N-1}","aliases":"∑_j=0^N-1 sum","display":"\\sum_{j=0}^{N-1}","kind":"use","excerpt":"residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages converges in X by completeness, to a vector x with the displayed norm bound. Continuity of S, together with the finite telescoping equation S(∑_j=0^N-1u_j)=y-r_N, ","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-b8194ba746cad45a","tex":"\\sum_{j=0}^{n-m-1}","aliases":"∑_j=0^n-m-1 sum","display":"\\sum_{j=0}^{n-m-1}","kind":"use","excerpt":"Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-c579ae579c844347","tex":"\\sum_{j=0}^{\\infty}","aliases":"∑_j=0^∞ sum infty","display":"\\sum_{j=0}^{\\infty}","kind":"use","excerpt":"Neumann inversion with both inverse identities. [A strict perturbation radius] Let W be a real Banach space and T:W→ W bounded linear, with normT≤θ<1 and θ≥0. Then Z=∑_j=0^∞T^j exists in operator norm and satisfies (I-T)Z=Z(I-T)=I, normZ≤frac11-θ. First recall why the space of bounded operators on W is complete. An operator-norm Cauchy sequence A_n","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-16d89af80aea902e","tex":"\\sum_{j=1}^J","aliases":"∑_j=1^J sum","display":"\\sum_{j=1}^J","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-ef75e15ec0684e46","tex":"\\sum_{j=1}^{n}","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^{n}","kind":"use","excerpt":"Three hypotheses that cannot simply be dropped. Summable movement. In maximum-norm R^3, let f_n=(s,H_n,(n+1)^-1), H_n=∑_j=1^nfrac1j. The first coordinate is protected and the residual is the last coordinate. It tends to zero with the same ratios as the polynomial example. The movement equals 1/(n+1), however, and the unprotected coordinate diverges","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-71e4a7d709b5778b","tex":"\\sum_{j=N+1}^{M}","aliases":"∑_j=N+1^M sum","display":"\\sum_{j=N+1}^{M}","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6d14accaece49594","tex":"\\sum_{j\\le J}","aliases":"∑_j≤ J sum le","display":"\\sum_{j\\le J}","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-49afb54019149b8f","tex":"\\sum_{j\\ge0}","aliases":"∑_j≥0 sum ge","display":"\\sum_{j\\ge0}","kind":"use","excerpt":"λ=2normr/c. The vector r/λ has norm c/2, so [source reference] supplies z∈ B_X with normSz-r/λ≤ c/4. Then u=λ z satisfies normu≤ 2normr/c, normr-Su≤normr/2. If the residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f1c2ff812e2b7828","tex":"\\sum_{k=0}^{\\infty}","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^{\\infty}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4a94609cba901bff","tex":"\\sum_{k=a}^{b-1}","aliases":"∑_k=a^b-1 sum","display":"\\sum_{k=a}^{b-1}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-baa8f7877711f9a7","tex":"\\sum_{k=n}^{m-1}","aliases":"∑_k=n^m-1 sum","display":"\\sum_{k=n}^{m-1}","kind":"use","excerpt":"∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k=n^∞b_k. Given ε>0, convergence of the nonnegative serie","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f26710096ba8e9ae","tex":"\\sum_{k=n}^{\\infty}","aliases":"∑_k=n^∞ sum infty","display":"\\sum_{k=n}^{\\infty}","kind":"use","excerpt":"[Summable movement gives exactification] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k","count":4},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","excerpt":"[Variable-rate exactification] If Q_n→0 and ∑_n C_nQ_n<∞, then (f_n) has an exact limit and dist(f_n,x)≤ω(f_0)∑_k=n^∞C_kQ_k. For constant q_n=q∈[0,1) and constant C_n=C, the sharper restarted estimate is dist(f_n,x)≤fracC1-q ω(f_n). The estimate ω(f_n)≤ Q_nω(f_0) follows by induction. At n=0 it is equality. If it holds at n, nonnegativity of q_n pe","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-8773b980988ccc84","tex":"\\sum_{n=0}^\\infty","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^\\infty","kind":"use","excerpt":"The perturbed map is surjective, with D'A'=I, normA'≤fraca1-aε. Its exact projection satisfies normx-q'x≤fraca1-aεnormD'x, normq-q'≤fracaε1-aε. The map T=(D-D')A acts on the Banach target R and satisfies norm T≤ aε<1. The operator series Z=∑_n=0^∞ T^n converges in operator norm and obeys norm Z≤(1-aε)^-1. Multiplying its finite partial sums on eith","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a7626aeba75a509f","tex":"\\sum_{n=0}^{\\infty}","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^{\\infty}","kind":"use","excerpt":"[Summable movement gives exactification] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-cdd7cb47a2dff84a","tex":"\\sum_{n=1}^m","aliases":"∑_n=1^m sum","display":"\\sum_{n=1}^m","kind":"use","excerpt":"For n≤ m, the residual square is at most n^2/m^4. The sum-of-squares estimate ∑_n=1^m n^2=fracm(m+1)(2m+1)6≤ m^3 holds for m≥1, so this part contributes at most 1/m. For n>m, each residual square is at most 1/n^2, whose tail is at most 1/m. This proves B_m≤2/m.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-a3ec24382fc8477a","tex":"\\sum_{n=1}^N","aliases":"∑_n=1^N sum","display":"\\sum_{n=1}^N","kind":"use","excerpt":"1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-afbea6d048f6cfef","tex":"\\sum_{n=1}^{N}","aliases":"∑_n=1^N sum","display":"\\sum_{n=1}^{N}","kind":"use","excerpt":"Escape is not an accident of this particular algorithm. Suppose an arbitrary protected candidate (u,1) satisfies normd(u,1)≤ε. On its first N coordinates, ∑_n=1^N|u_n+1|^2 ≤ N^2∑_n=1^Nfrac|u_n+1|^2n^2 ≤ N^2ε^2. The reverse triangle inequality in R^N gives norm u≥sqrt N-Nε (N≥1). If a sequence of protected candidates has residual tending to zero, th","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-67e8088ae0f488e2","tex":"\\sum_{n>K}","aliases":"∑_n>K sum","display":"\\sum_{n>K}","kind":"use","excerpt":"Finally, for K≥ m, both omitted tails consist of nonnegative terms. Bound the first by m^4∑_n>Kn^-4 and the second by ∑_n>Kn^-2. The same upper integral comparisons give the two certified remainders in [source reference]. No finite truncation is substituted for an equality of infinite energies. writtenWR10","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6c1ff4839728583e","tex":"\\sum_{n>m}","aliases":"∑_n>m sum","display":"\\sum_{n>m}","kind":"use","excerpt":"For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9b846ee3188af45a","tex":"\\sum_{n>N}","aliases":"∑_n>N sum","display":"\\sum_{n>N}","kind":"use","excerpt":"1/n · n>N, norma_N^2=N, normf_N^2=N+1. For every N≥1, telescoping comparisons give frac1N+1≤normd(f_N)^2 =∑_n>Nfrac1n^2≤frac1N. Indeed 1/[n(n+1)]≤1/n^2≤1/[n(n-1)] for n≥2. Summing the two telescoping bounds proves [source reference] without numerically evaluating an infinite series.","count":6},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-069c58c03dba2dbd","tex":"\\sum_{n\\ge2}","aliases":"∑_n≥2 sum ge","display":"\\sum_{n\\ge2}","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Ω∈overline B∖ B · Nonzero · Arbitrarily small corrected residuals exist, but no exact cancellation exists.","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"The category of elements ∫_V F and Γ(F) also have different roles. The former organizes individual appearances and their transitions. The latter selects coherent families across all views. The category of elements serves here as explanatory notation, while the Lean formalization directly represents families and their coherence.","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-f21fa71dce8d6708","tex":"\\int_0^{N/m}","aliases":"∫_0^N/m int","display":"\\int_0^{N/m}","kind":"use","excerpt":"1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dad46397774b52e2","tex":"\\int_0^{\\pi/2}","aliases":"∫_0^π/2 int pi","display":"\\int_0^{\\pi/2}","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-5f9323382a61523c","tex":"\\int_{m+1}^\\infty","aliases":"∫_m+1^∞ int infty","display":"\\int_{m+1}^\\infty","kind":"use","excerpt":"For the lower bound, n>m implies m^2+n^2≤2n^2, hence fracn^2(m^2+n^2)^2≥frac14n^2. The other direction of the decreasing-function integral comparison gives ∑_n>mfrac1n^2≥∫_m+1^∞ x^-2 dx=frac1m+1. This proves B_m≥1/(4(m+1)).","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-fb3ba7923792333d","tex":"\\int_m^\\infty","aliases":"∫_m^∞ int infty","display":"\\int_m^\\infty","kind":"use","excerpt":"For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-dd7b189feddbc76a","tex":"\\int_\\V","aliases":"∫_V int V","display":null,"kind":"use","excerpt":"The category of elements ∫_V F and Γ(F) also have different roles. The former organizes individual appearances and their transitions. The latter selects coherent families across all views. The category of elements serves here as explanatory notation, while the Lean formalization directly represents families and their coherence.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"For the relative claim, pullback clearly sends an extension of s to an extension of H^*s on U'=H^-1U. Conversely suppose y extends H^*s. If w∈ U, the isomorphism w≅ HLw and isomorphism closure imply HLw∈ U. Hence Lw∈ U' and y_Lw=s_HLw. Both endpoints of t_w lie in U, so partial coherence gives (Ry)_w=F(t_w)s_HLw=s_w. Recovery therefore preserves th","count":2},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"[Extension criterion] For every coherent partial section, o_U(s)=0 Longleftrightarrow Ext_U(s)≠∅. Let π_U:R→ R/B_U be the quotient homomorphism. By the definition of a quotient by a subgroup, π_U(r)=0 Longleftrightarrow r∈ B_U. Here B_U is the image of ker p_U under δ. Consequently membership of δwidetilde s means the existence of a specific h∈ E s","count":23},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"[Bounded approximate feasibility] For a bounded linear map T:H→ Y between real Hilbert spaces and r∈ Y, the following are equivalent: · r∈ran T, · there is a bounded sequence (a_n)⊂ H with normTa_n-r⟶0. If all norma_n≤ M, an exact solution exists with norm at most M. If Ta=r, the constant sequence a_n=a is bounded and has zero residual. For the con","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9f55fb1c51ddcd12","tex":"\\supseteq","aliases":"⊇ supseteq","display":"\\supseteq","kind":"use","excerpt":"Suppose now that A_1⊇ A_2⊇⋯ are nonempty closed affine subsets of a real Hilbert correction space H. A typical A_n consists of corrections satisfying the first n bounded linear observation equations. The sets are affine because the equations are linear, and closed because the observations are continuous.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":5},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-652adc7f79a6e08d","tex":"\\perp_M","aliases":"⊥_M perp","display":"\\perp_M","kind":"use","excerpt":"If ran B is closed, Theorem~[source reference] applied on M gives a bounded inverse of B on (ker B)^⊥_M. The vector h_* is exactly that inverse applied to Dx, so its norm obeys the required bound.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-9a4555c51b02f0a3","tex":"\\bigcap_{(i,j)\\in\\vec E}","aliases":"⋂_(i,j)∈vec E bigcap in vec","display":"\\bigcap_{(i,j)\\in\\vec E}","kind":"use","excerpt":"Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-d24cfe84a3e98784","tex":"\\bigcup_N","aliases":"⋃_N bigcup","display":"\\bigcup_N","kind":"use","excerpt":"Convergence has its own theorem. Call x norm recoverable when Q_Nx→0. This is equivalent to S_Nx→ x, but it does not follow merely from vanishing persistent subspace. Let F=⋃_Nker Q_N be the finite core.","count":1},{"work":"komorebi-space","url":"notation/komorebi-space.html#symbol-6ef7ee64cfa071c4","tex":"\\rangle^2","aliases":"⟩^2 rangle","display":"\\rangle^2","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"padic-canonical-realization","url":"notation/padic-canonical-realization.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"export RUNTIME=/absolute/path/to/Komorebi_Portable_Lean_4.34.0-rc2 export PATH=\"RUNTIME/toolchain/bin:PATH\" python verification/prepare_offline.py \"RUNTIME\" lean --version lake --version PYTHONDONTWRITEBYTECODE=1 python verification/run_final.py","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"notation-table","excerpt":"S_A · The prime set {p_2n+one_A(n):n∈N}.","count":48},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"midrule C_p=F_p · The additive cyclic group of prime order p.","count":152},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"S_A · The prime set {p_2n+one_A(n):n∈N}.","count":261},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"notation-table","excerpt":"D_B · The subgroup supported on prime indices in B⊆N.","count":100},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"notation-table","excerpt":"midrule C_p=F_p · The additive cyclic group of prime order p.","count":65},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"notation-table","excerpt":"c_A(r,t) · The finite cocycle s_A(r)+s_A(t)-s_A(r+t).","count":10},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-9b74ac7e4a926fd8","tex":"c_A","aliases":"c_A","display":"c_A","kind":"notation-table","excerpt":"c_A(r,t) · The finite cocycle s_A(r)+s_A(t)-s_A(r+t).","count":9},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-374d4fe4488ce15e","tex":"C_p","aliases":"C_p","display":"C_p","kind":"notation-table","excerpt":"midrule C_p=F_p · The additive cyclic group of prime order p.","count":23},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"notation-table","excerpt":"D=⊕_p C_p · Finite-support prime-coordinate vectors.","count":128},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-ee5bb7d36741efec","tex":"D_B","aliases":"D_B","display":"D_B","kind":"notation-table","excerpt":"D_B · The subgroup supported on prime indices in B⊆N.","count":20},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"notation-table","excerpt":"e_p · The vector supported at p with value one.","count":35},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1b09d4efdcdc1f39","tex":"e_p","aliases":"e_p","display":"e_p","kind":"notation-table","excerpt":"e_p · The vector supported at p with value one.","count":10},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0a108e0b17fae60a","tex":"\\eqfin","aliases":"eqfin","display":null,"kind":"notation-table","excerpt":"Aeqfin B · The symmetric difference Atriangle B is finite.","count":11},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F","display":null,"kind":"notation-table","excerpt":"midrule C_p=F_p · The additive cyclic group of prime order p.","count":7},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-fda40c00a5457366","tex":"\\F_p","aliases":"F_p F","display":null,"kind":"notation-table","excerpt":"midrule C_p=F_p · The additive cyclic group of prime order p.","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"notation-table","excerpt":"G_A · The group q^-1(Q q(u_A)).","count":119},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-3a9e648245c571b5","tex":"\\mathfrak G","aliases":"G mathfrak","display":"\\mathfrak G","kind":"notation-table","excerpt":"K_ G(G) · Least admissible finite description length, or +∞.","count":13},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"notation-table","excerpt":"pmid_G g · There exists y∈ G with py=g.","count":23},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-28a0f54c10c0e63f","tex":"G_A","aliases":"G_A","display":"G_A","kind":"notation-table","excerpt":"G_A · The group q^-1(Q q(u_A)).","count":93},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"notation-table","excerpt":"H_W=q^-1(W) · The intermediate group associated to a rational subspace W≤ V.","count":55},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-70dd52c54f29b990","tex":"H_W","aliases":"H_W","display":"H_W","kind":"notation-table","excerpt":"H_W=q^-1(W) · The intermediate group associated to a rational subspace W≤ V.","count":27},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","excerpt":"K_ G(G) · Least admissible finite description length, or +∞.","count":15},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8bb7d51a601e1a38","tex":"K_{\\mathfrak G}","aliases":"K_ G mathfrak","display":"K_{\\mathfrak G}","kind":"notation-table","excerpt":"K_ G(G) · Least admissible finite description length, or +∞.","count":6},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"notation-table","excerpt":"rad(m) · The product of the distinct prime divisors of m, with rad(1)=1.","count":122},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"notation-table","excerpt":"pmid_G g · There exists y∈ G with py=g.","count":23},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-27d124b3cf06bc76","tex":"\\mid_G","aliases":"mid_G mid","display":null,"kind":"notation-table","excerpt":"pmid_G g · There exists y∈ G with py=g.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N","display":null,"kind":"notation-table","excerpt":"D_B · The subgroup supported on prime indices in B⊆N.","count":26},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"notation-table","excerpt":"S_A · The prime set {p_2n+one_A(n):n∈N}.","count":31},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-7d905bc6806c132f","tex":"\\one","aliases":"one","display":null,"kind":"notation-table","excerpt":"S_A · The prime set {p_2n+one_A(n):n∈N}.","count":6},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-05cfecec361e4193","tex":"\\one_A","aliases":"one_A one","display":null,"kind":"notation-table","excerpt":"S_A · The prime set {p_2n+one_A(n):n∈N}.","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"P=∏_p C_p · All prime-coordinate vectors, with product topology.","count":97},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"notation-table","excerpt":"midrule C_p=F_p · The additive cyclic group of prime order p.","count":124},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-3b4c24291c35777c","tex":"p_{2n+\\one_A(n)}","aliases":"p_2n+one_A(n) one","display":null,"kind":"notation-table","excerpt":"S_A · The prime set {p_2n+one_A(n):n∈N}.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"notation-table","excerpt":"G_A · The group q^-1(Q q(u_A)).","count":34},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"notation-table","excerpt":"q:P→ V · The quotient map to V=P/D.","count":27},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-6339057791df9dbc","tex":"q^{-1}","aliases":"q^-1","display":"q^{-1}","kind":"notation-table","excerpt":"H_W=q^-1(W) · The intermediate group associated to a rational subspace W≤ V.","count":8},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"notation-table","excerpt":"s_A(r) · The specified rational lift of r q(u_A) to P.","count":60},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-650ae3274f788b89","tex":"\\rad","aliases":"rad","display":null,"kind":"notation-table","excerpt":"rad(m) · The product of the distinct prime divisors of m, with rad(1)=1.","count":13},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"notation-table","excerpt":"S_A · The prime set {p_2n+one_A(n):n∈N}.","count":28},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"notation-table","excerpt":"s_A(r) · The specified rational lift of r q(u_A) to P.","count":37},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2badf92439cc7951","tex":"S_A","aliases":"S_A","display":"S_A","kind":"notation-table","excerpt":"S_A · The prime set {p_2n+one_A(n):n∈N}.","count":20},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-c0bc731e693dd8ce","tex":"s_A","aliases":"s_A","display":"s_A","kind":"notation-table","excerpt":"s_A(r) · The specified rational lift of r q(u_A) to P.","count":32},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"notation-table","excerpt":"A≤_T X · Membership in A is computable using oracle X.","count":14},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"notation-table","excerpt":"c_A(r,t) · The finite cocycle s_A(r)+s_A(t)-s_A(r+t).","count":18},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-38a6461e5f88ab3d","tex":"\\triangle","aliases":"triangle","display":"\\triangle","kind":"notation-table","excerpt":"Aeqfin B · The symmetric difference Atriangle B is finite.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"notation-table","excerpt":"u_A · The coordinate indicator of S_A in P.","count":49},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-432c14b23419ac6d","tex":"u_A","aliases":"u_A","display":"u_A","kind":"notation-table","excerpt":"u_A · The coordinate indicator of S_A in P.","count":42},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"notation-table","excerpt":"q:P→ V · The quotient map to V=P/D.","count":18},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"notation-table","excerpt":"H_W=q^-1(W) · The intermediate group associated to a rational subspace W≤ V.","count":47},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4ae2c725a0324b8f","tex":"\\widehat G","aliases":"widehat G widehat","display":"\\widehat G","kind":"notation-table","excerpt":"widehat G · The profinite completion, not the Pontryagin dual.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"notation-table","excerpt":"A≤_T X · Membership in A is computable using oracle X.","count":25},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"notation-table","excerpt":"pmid_G g · There exists y∈ G with py=g.","count":18},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"notation-table","excerpt":"q:P→ V · The quotient map to V=P/D.","count":25},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"notation-table","excerpt":"S_A · The prime set {p_2n+one_A(n):n∈N}.","count":52},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"notation-table","excerpt":"P=∏_p C_p · All prime-coordinate vectors, with product topology.","count":9},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-b9d5148b142926aa","tex":"\\prod_p","aliases":"∏_p prod","display":"\\prod_p","kind":"notation-table","excerpt":"P=∏_p C_p · All prime-coordinate vectors, with product topology.","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"notation-table","excerpt":"K_ G(G) · Least admissible finite description length, or +∞.","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"notation-table","excerpt":"H_W=q^-1(W) · The intermediate group associated to a rational subspace W≤ V.","count":17},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-faac5de51f69f1dd","tex":"\\le_T","aliases":"≤_T le","display":"\\le_T","kind":"notation-table","excerpt":"A≤_T X · Membership in A is computable using oracle X.","count":6},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"notation-table","excerpt":"D_B · The subgroup supported on prime indices in B⊆N.","count":27},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"notation-table","excerpt":"D=⊕_p C_p · Finite-support prime-coordinate vectors.","count":7},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-ce6b462bda62b030","tex":"\\bigoplus_p","aliases":"⊕_p bigoplus","display":"\\bigoplus_p","kind":"notation-table","excerpt":"D=⊕_p C_p · Finite-support prime-coordinate vectors.","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":37},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End","display":null,"kind":"definition-passage","excerpt":"[Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed add","count":6},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.","count":29},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1633dfbf62528f10","tex":"\\lvert","aliases":"lvert","display":"\\lvert","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":9},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"definition-passage","excerpt":"[Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed add","count":21},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"definition-passage","excerpt":"[Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed add","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1b983f28fc05d3a0","tex":"\\operatorname{Realize}","aliases":"Realize operatorname","display":"\\operatorname{Realize}","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-d2e3ad5bca4f700e","tex":"\\operatorname{Realize}_{\\mathfrak G}","aliases":"Realize_ G operatorname mathfrak","display":"\\operatorname{Realize}_{\\mathfrak G}","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-226626f51f7539ed","tex":"\\rvert","aliases":"rvert","display":"\\rvert","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":9},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0dcc526e2e33457d","tex":"\\trank","aliases":"trank","display":null,"kind":"definition-passage","excerpt":"[Topological generation] For a topological group M, a subset S topologically generates M when the closure of the subgroup generated by S is all of M. Its least possible size is denoted trank(M). An operator version takes the closure of the operator hull instead. The closure is taken in the stated topology. A change of topology is a change of contra","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"definition-passage","excerpt":"[Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.","count":5},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":20},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"[Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.","count":12},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":7},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":13},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-16636769fc160a5a","tex":"b^{-1}","aliases":"b^-1","display":"b^{-1}","kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"introduction-passage","excerpt":"For each j whose output lies in the family, choose B_j⊆N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.","count":8},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-ae1726ace684b30c","tex":"G_{B_j}","aliases":"G_B_j","display":"G_{B_j}","kind":"introduction-passage","excerpt":"For each j whose output lies in the family, choose B_j⊆N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":30},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"introduction-passage","excerpt":"[Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonze","count":9},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-43d85ec54b4d02c3","tex":"\\mid_{G_A}","aliases":"mid_G_A mid","display":null,"kind":"introduction-passage","excerpt":"[Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonze","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-e942b1c655fe6bde","tex":"\\Obs","aliases":"Obs","display":null,"kind":"introduction-passage","excerpt":"The observation contract. Let Obs(G) denote the unmarked inverse system of finite quotient groups and their transition maps, considered up to isomorphism as a pro-object. The source group G is not supplied again as part of the input. Nor do we retain evaluations of named source elements. Supplying the source itself would make a reconstruction probl","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-41dacfd07761cf94","tex":"\\overline r","aliases":"overline r overline","display":"\\overline r","kind":"introduction-passage","excerpt":"The endomorphism ring and its units. Write overline r_p for reduction of a rational r at a prime not dividing its denominator. The phrase ``for almost every selected prime'' below means outside a finite set, which may include those denominator primes.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-53f019d1026f6086","tex":"\\overline r_p","aliases":"overline r_p overline","display":"\\overline r_p","kind":"introduction-passage","excerpt":"The endomorphism ring and its units. Write overline r_p for reduction of a rational r at a prime not dividing its denominator. The phrase ``for almost every selected prime'' below means outside a finite set, which may include those denominator primes.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":6},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a03f214a14787ce8","tex":"R_j","aliases":"R_j","display":"R_j","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0ef3dbe5a13282d7","tex":"\\srank","aliases":"srank","display":null,"kind":"introduction-passage","excerpt":"If all endomorphisms are allowed, this is module generation over the unital ring End(M). We write srank_End(M)(M) for its least generating cardinal. An infinite value can therefore express a strong obstruction even when the operator collection itself is unrestricted.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-9567a4e9a9c809fe","tex":"\\srank_{\\End(M)}","aliases":"srank_End(M) srank End","display":null,"kind":"introduction-passage","excerpt":"If all endomorphisms are allowed, this is module generation over the unital ring End(M). We write srank_End(M)(M) for its least generating cardinal. An infinite value can therefore express a strong obstruction even when the operator collection itself is unrestricted.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1084f45e6b3e947e","tex":"\\supp","aliases":"supp","display":null,"kind":"introduction-passage","excerpt":"[Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonze","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":5},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"introduction-passage","excerpt":"For each j whose output lies in the family, choose B_j⊆N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.","count":5},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":8},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1e403224a9d979ef","tex":"\\mathbf0","aliases":"0 mathbf","display":"\\mathbf0","kind":"use","excerpt":"ystem presentation of Chapter~[source reference]. Choose any set A of the desired degree and apply Theorem~[source reference]. All the structural and completion properties were proved in Chapters~[source reference]--[source reference]. For A=∅, the copy is computable. For A=K, the group has no computable copy and its least degree is the halting deg","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0b5274a74be9e8df","tex":"\\mathbf0'","aliases":"0' mathbf","display":"\\mathbf0'","kind":"use","excerpt":"ystem presentation of Chapter~[source reference]. Choose any set A of the desired degree and apply Theorem~[source reference]. All the structural and completion properties were proved in Chapters~[source reference]--[source reference]. For A=∅, the copy is computable. For A=K, the group has no computable copy and its least degree is the halting deg","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-66be131002b8b5af","tex":"A_C","aliases":"A_C","display":"A_C","kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩),","count":9},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-65235a7c0d0ee96d","tex":"A_{C'}","aliases":"A_C'","display":"A_{C'}","kind":"use","excerpt":"If C≠ C', choose j at which they differ. Rule [source reference] makes A_C and A_C' differ at infinitely many odd indices. Theorems~[source reference] and [source reference] then give nonisomorphism and nonembeddability. The remaining properties hold for every paired-prime group.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":7},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"Choosing a rational line. Let A⊆N, write a_n=one_A(n), and define S_A={p_2n+a_n:n∈N}. Exactly one prime from each pair is selected. Both S_A and its complement are infinite. Let u_A∈ P be its indicator: (u_A)_p=1, · p∈ S_A,","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-99d30db923da3b41","tex":"A_{N+1}","aliases":"A_N+1","display":"A_{N+1}","kind":"use","excerpt":"Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-b645d837ca596c1a","tex":"\\aleph","aliases":"aleph","display":null,"kind":"use","excerpt":"[Countable-code bound] Let G be a finite-description grammar as above. Then its class of realized group isomorphism types has cardinal at most aleph_0. For each length n, the set of strings of length n is countable. The union over n∈N is countable. The valid strings form a subset, and a single-valued image of a countable set is countable. This is a","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8cafa9e3d0a185fe","tex":"\\aleph_0","aliases":"aleph_0 aleph","display":null,"kind":"use","excerpt":"[Countable-code bound] Let G be a finite-description grammar as above. Then its class of realized group isomorphism types has cardinal at most aleph_0. For each length n, the set of strings of length n is countable. The union over n∈N is countable. The valid strings form a subset, and a single-valued image of a countable set is countable. This is a","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finit","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finit","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finit","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-87f4e4b9b8d7159b","tex":"B_K","aliases":"B_K","display":"B_K","kind":"use","excerpt":"Define B_K={2n:n∈N}∪{2n+1:n∉ K}, H=D_B_K. This is an explicit mathematical definition of a countably infinite, locally cyclic abelian group.","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-012fc6190ee8df8e","tex":"\\bits","aliases":"bits","display":null,"kind":"use","excerpt":"Give Sub(D) the topology inherited from {0,1}^D by its membership functions. Membership of a vector with finite support depends on finitely many selector bits. Conversely, membership of e_p_i recovers the ith bit. Consequently the order isomorphism above is also a homeomorphism from Cantor space bits to Sub(D).","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"If C≠ C', choose j at which they differ. Rule [source reference] makes A_C and A_C' differ at infinitely many odd indices. Theorems~[source reference] and [source reference] then give nonisomorphism and nonembeddability. The remaining properties hold for every paired-prime group.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"[Finitely generated subgroups] Every finitely generated subgroup of G_A is either finite cyclic or isomorphic to Z⊕ C_m for a squarefree positive integer m, with C_1 trivial. Let L≤ G_A be generated by finitely many elements. Its image in Q is a finitely generated subgroup of Q, hence either zero or infinite cyclic. If it is zero, then L≤ D and its","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-6e40ae85b89b68e6","tex":"C_{12}","aliases":"C_12","display":"C_{12}","kind":"use","excerpt":"For m=12, only coordinates 2 and 3 matter. The six representatives are 0, e_2, e_3, 2e_3, e_2+e_3, e_2+2e_3. The vector e_2+e_3 generates this quotient, which is C_6, not C_12. Repeated prime powers in the multiplier do not create additional coordinates.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"midrule 1 · 2 · 2 · C_2","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0413afc6a75dd0cb","tex":"C_{210}","aliases":"C_210","display":"C_{210}","kind":"use","excerpt":"4 · 2,3,5,7 · 210 · C_210","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-155e8a6991cddb21","tex":"C_{2310}","aliases":"C_2310","display":"C_{2310}","kind":"use","excerpt":"5 · 2,3,5,7,11 · 2310 · C_2310","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"At the unselected prime 3, the coordinate of u_A is zero. In this case 3(1/3,0)=(1,0). The three roots are (1/3,0), (1/3,e_3), (1/3,2e_3). Their differences form G_A[3]=C_3, as predicted by Proposition~[source reference].","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2341bcb68032a02e","tex":"C_{30}","aliases":"C_30","display":"C_{30}","kind":"use","excerpt":"3 · 2,3,5 · 30 · C_30","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-dcb4e0ba5e2088bf","tex":"C_6","aliases":"C_6","display":"C_6","kind":"use","excerpt":"2 · 2,3 · 6 · C_6","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"[Finitely generated subgroups] Every finitely generated subgroup of G_A is either finite cyclic or isomorphic to Z⊕ C_m for a squarefree positive integer m, with C_1 trivial. Let L≤ G_A be generated by finitely many elements. Its image in Q is a finitely generated subgroup of Q, hence either zero or infinite cyclic. If it is zero, then L≤ D and its","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-3d0472df3e9d0eb7","tex":"C_{p_i}","aliases":"C_p_i","display":"C_{p_i}","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-5281a9b0294e8bc6","tex":"C_{p_n}","aliases":"C_p_n","display":"C_{p_n}","kind":"use","excerpt":"A purely finite-state boundary. Consider a restricted grammar in which a finite automaton reads the unary word 1^n and selects the factor C_p_n exactly when it accepts.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-e1111c7a3d137309","tex":"C_{\\rad(m)}","aliases":"C_rad(m) rad","display":null,"kind":"use","excerpt":"The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff Aeqfin B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with ","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-e5cba341fdb2a255","tex":"D_{B_K}","aliases":"D_B_K","display":"D_{B_K}","kind":"use","excerpt":"Define B_K={2n:n∈N}∪{2n+1:n∉ K}, H=D_B_K. This is an explicit mathematical definition of a countably infinite, locally cyclic abelian group.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-13f2e328ffd83191","tex":"D_{B\\cap C}","aliases":"D_B∩ C cap","display":"D_{B\\cap C}","kind":"use","excerpt":"e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic d","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-30566182622f13f1","tex":"D_{B\\cup C}","aliases":"D_B∪ C cup","display":"D_{B\\cup C}","kind":"use","excerpt":"e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic d","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-f7fb93f3c3255dcc","tex":"D_C","aliases":"D_C","display":"D_C","kind":"use","excerpt":"[Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P(N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-688ebedc522ae64c","tex":"d_m","aliases":"d_m","display":"d_m","kind":"use","excerpt":"Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-44c6bb8c38959e5b","tex":"D_{\\N\\setminus B}","aliases":"D_N∖ B N setminus","display":null,"kind":"use","excerpt":"e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic d","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0efdddd015383ecb","tex":"d_p","aliases":"d_p","display":"d_p","kind":"use","excerpt":"bijection between rational subspaces of V and pure subgroups H of P containing D. First let W be rational and suppose h∈ H_W∩ mP. Because W is closed under division by m, choose x∈ H_W with q(x)=q(h)/m. Then d=h-mx belongs to D. At a prime dividing m, the coordinate of h is zero because h∈ mP, and that of mx is also zero. Thus d_p=0 whenever pmid m","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"[Uniform construction relative to the selector] If A≤_T X, then the normal-form construction gives an X-computable copy of G_A. In its canonical normal-form presentation, the group operation has Turing degree exactly deg_T(A). To add (r,d) and (t,e), compute the prime divisors of den(r)den(t). At each of these finitely many primes, query the corres","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0d469a888ac8840b","tex":"\\deg_T","aliases":"deg_T deg","display":"\\deg_T","kind":"use","excerpt":"[Uniform construction relative to the selector] If A≤_T X, then the normal-form construction gives an X-computable copy of G_A. In its canonical normal-form presentation, the group operation has Turing degree exactly deg_T(A). To add (r,d) and (t,e), compute the prime divisors of den(r)den(t). At each of these finitely many primes, query the corres","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-fbaa37d22c49a1eb","tex":"\\den","aliases":"den","display":null,"kind":"use","excerpt":"[A finite support bound] For all r,t∈Q, supp c_A(r,t)⊆ {p:pmidden(r)den(t)}. At a prime not dividing either denominator, all three rationals have well-defined reductions. On a selected coordinate, reduction respects addition, so the defect is zero. On an unselected coordinate, all section values are zero. The reduced denominator of r+t has no prime","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"A splitting would embed the nonzero divisible group W into P. Theorem~[source reference] excludes every such subgroup. The rank of the torsion-free quotient is exactly dim_QW. This gives countable examples of every finite rank and of countably infinite rank with the same finite-quotient system. The rank-one family studied next is chosen because it ","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a4b7e80acc67bb28","tex":"\\dim_{\\Q}","aliases":"dim_Q dim Q","display":null,"kind":"use","excerpt":"A splitting would embed the nonzero divisible group W into P. Theorem~[source reference] excludes every such subgroup. The rank of the torsion-free quotient is exactly dim_QW. This gives countable examples of every finite rank and of countably infinite rank with the same finite-quotient system. The rank-one family studied next is chosen because it ","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":6},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"[The same group with a different marked point] Let A=∅ and B={0}. Only the first prime pair changes. Then G_A=G_B inside P, but u_A-u_B=e_2-e_3. The distinguished element is not preserved. In particular, the statement G_A≅ G_Biff A=B would be false for unmarked groups.","count":7},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"[The same group with a different marked point] Let A=∅ and B={0}. Only the first prime pair changes. Then G_A=G_B inside P, but u_A-u_B=e_2-e_3. The distinguished element is not preserved. In particular, the statement G_A≅ G_Biff A=B would be false for unmarked groups.","count":9},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-384a7e923a8d2319","tex":"e_7","aliases":"e_7","display":"e_7","kind":"use","excerpt":"A finite tail exception. Consider an infinite-order element x=s_A(5/6)+2e_7. An exception set for its signature is {2,3,5,7}. Outside these primes, the denominator is invertible, the numerator is nonzero, and the torsion correction vanishes. Thus the root pattern of x agrees with that of u_A in every later prime pair.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8db0cea46fdd60f3","tex":"E_A","aliases":"E_A","display":"E_A","kind":"use","excerpt":"[Endomorphism ring] Under coordinate multiplication, End(G_A)≅ E_A={λ∈ P:∃ r∈Q, λ_p=overline r_p for almost every p∈ S_A}. The rational r is unique. It is the scalar induced on G_A/D, and the map E_A→Q sending λ to r is a surjective ring homomorphism. Apply Theorem~[source reference] with B=A. At unselected coordinates, both λ u_A and s_A(r) vanish","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-86ea2a47e4e2f4df","tex":"e_{p_i}","aliases":"e_p_i","display":"e_{p_i}","kind":"use","excerpt":"Let B be the set of indices of these factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice oper","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":20},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"The complete finite-quotient system. [Profinite completion of the direct sum] The canonical embedding of D into P identifies P with widehat D. Let f:D→ F be a homomorphism to a finite group. The image of a prime-order factor C_p is trivial unless p divides lvert Frvert. Thus f vanishes on all but finitely many prime components and factors through a","count":18},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-08c122811477fa77","tex":"\\F_3","aliases":"F_3 F","display":null,"kind":"use","excerpt":"A torsion correction that cannot be discarded. Take the purely torsion target e_3 and multiplier 2. The root criterion succeeds because the target has zero 2-coordinate. Its rational part is zero, but the answer is not the zero element: 2(0,2e_3)=(0,e_3). In F_3, multiplication by 2 sends 2 to 1. This is the simplest example showing why the finite ","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0a27cb068825ed9a","tex":"\\F_5","aliases":"F_5 F","display":null,"kind":"use","excerpt":"bottomrule Each residue is exact. At p=2, the lift of 1/2 is set to zero because the denominator is not invertible. At p=5, its value is 3 because 2·3=1 in F_5.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2bbdafafd965559e","tex":"\\F_{p_{b_j}}","aliases":"F_p_b_j F","display":null,"kind":"use","excerpt":"Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finit","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-3806cb12e1e3e734","tex":"G_{A_C}","aliases":"G_A_C","display":"G_{A_C}","kind":"use","excerpt":"[Escape within one profinite-completion class] For every C⊆N, the group G_A_C is not isomorphic to any output of G. Distinct C give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion P. Fix a description index j. If R_j is outside the family, it c","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-74c538d30573ec20","tex":"G_B","aliases":"G_B","display":"G_B","kind":"use","excerpt":"G_A≅ G_Biff Atriangle B is finite, G_A has an X-computable copyiff A≤_T X.","count":19},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-82170a73bb5aa973","tex":"G_K","aliases":"G_K","display":"G_K","kind":"use","excerpt":"The group G_K still has a finite mathematical definition. Its noncomputability is not a proof that it cannot be named in a finite semantic language. The theorem concerns effective copies with computable operations and equality.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-f046713e329fd428","tex":"g_p","aliases":"g_p","display":"g_p","kind":"use","excerpt":"[Root or obstruction] Let g=s_A(r)+d∈ G_A and let m≥1. Then g∈ mG_A Longleftrightarrow g_p=0 for every pmid m. If the condition fails, a prime pmid m with g_p≠0 is a nonexistence certificate. If it holds, a root is obtained by the finite procedure in the proof. Necessity follows because multiplication by m is zero in every coordinate C_p with pmid ","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1bf12537a68f2d94","tex":"\\mathfrak H","aliases":"H mathfrak","display":"\\mathfrak H","kind":"use","excerpt":"Monotonicity under grammar extension. [Admissible expansion] Suppose every valid description in G translates to a valid description in H with the same semantic group. Then every G-realizable group is H-realizable. If every translated description has length at most f(lvert drvert), then K_ H(G)≤ f(K_ G(G)) whenever K_ G(G) is finite. Translate a sho","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-d54b1a6be7c911b2","tex":"h_p","aliases":"h_p","display":"h_p","kind":"use","excerpt":"Root conditions do not depend on the subspace. [Ambient root criterion] For every rational subspace W, every h∈ H_W, and every m≥1, h∈ mH_W Longleftrightarrow h_p=0 for all primes pmid m. Multiplication by m on P is zero at primes dividing m and invertible on every other coordinate. Thus the right-hand condition is equivalent to h∈ mP. Apply purity","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-3ea834c81d3dfd1d","tex":"H_V","aliases":"H_V","display":"H_V","kind":"use","excerpt":"D agree with h on the primes dividing m and vanish elsewhere. Then h-d∈ mP∩ H. Purity gives h-d=mx for some x∈ H, so q(x)=q(h)/m. Thus q(H) is closed under rational scalars. Since D⊆ H, the equality H=q^-1(q(H)) holds. This proves that the constructions are inverse. Both preserve inclusion. The bottom object is H_{0}=D, and the top object is H_V=P.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2bf812a56567a3f8","tex":"H_{\\{0\\}}","aliases":"H_{0}","display":"H_{\\{0\\}}","kind":"use","excerpt":"Given h∈ H and m≥1, let d∈ D agree with h on the primes dividing m and vanish elsewhere. Then h-d∈ mP∩ H. Purity gives h-d=mx for some x∈ H, so q(x)=q(h)/m. Thus q(H) is closed under rational scalars. Since D⊆ H, the equality H=q^-1(q(H)) holds. This proves that the constructions are inverse. Both preserve inclusion. The bottom object is H_{0}=D, a","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom","display":null,"kind":"use","excerpt":"[Finite-target equivalence] For every finite group F, restriction along D↪ H_W gives a natural bijection Hom(H_W,F)⟶Hom(D,F). Put m=lvert Frvert. Every h∈ H_W agrees modulo mH_W with an element of D, namely its finite coordinate vector on primes dividing m. Every homomorphism to F kills mH_W. Thus two maps agreeing on D agree everywhere.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-996aae2a581f4236","tex":"\\Hull","aliases":"Hull","display":null,"kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-b0ad653a11ea5542","tex":"\\Hull_{\\mathcal O}","aliases":"Hull_ O Hull mathcal","display":null,"kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":17},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"G_A≅ G_Biff Atriangle B is finite, G_A has an X-computable copyiff A≤_T X.","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":16},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-01259f2441a815dd","tex":"K_{\\mathfrak H}","aliases":"K_ H mathfrak","display":"K_{\\mathfrak H}","kind":"use","excerpt":"Monotonicity under grammar extension. [Admissible expansion] Suppose every valid description in G translates to a valid description in H with the same semantic group. Then every G-realizable group is H-realizable. If every translated description has length at most f(lvert drvert), then K_ H(G)≤ f(K_ G(G)) whenever K_ G(G) is finite. Translate a sho","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"[Finitely generated subgroups] Every finitely generated subgroup of G_A is either finite cyclic or isomorphic to Z⊕ C_m for a squarefree positive integer m, with C_1 trivial. Let L≤ G_A be generated by finitely many elements. Its image in Q is a finitely generated subgroup of Q, hence either zero or infinite cyclic. If it is zero, then L≤ D and its","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-768071a8b8aba296","tex":"m^{-1}","aliases":"m^-1","display":"m^{-1}","kind":"use","excerpt":"[Unique division in the ambient quotient] Multiplication by every positive integer is bijective on V. Thus V has a unique rational vector-space structure extending its additive group structure. The product P has no nonzero divisible subgroup. Fix m≥1 and x∈ P. Define y_p=m^-1x_p when pnmid m, and y_p=0 when pmid m. Then x-my has finite support. Thu","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-df92634da3a3e068","tex":"M_i","aliases":"M_i","display":"M_i","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-bf2227566ef624f5","tex":"M_j","aliases":"M_j","display":"M_j","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-ea260e96669ea599","tex":"m_N","aliases":"m_N","display":"m_N","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"The distinction is easiest to express in terms of an embedding G⟶ varprojlim G/N. Residual finiteness makes this map injective. The inverse limit includes all compatible tuples. The image of the original G can nevertheless be a proper dense subgroup. Forgetting that image can forget the object even while retaining enough coordinates to distinguish ","count":18},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-6ac2b834a040f058","tex":"\\N^2","aliases":"N^2 N","display":null,"kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩),","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-bc7de739b4277782","tex":"\\neqfin","aliases":"neqfin","display":null,"kind":"use","excerpt":"[Torsion-image obstruction] If Aneqfin B, every homomorphism G_A→ G_B has image contained in D. Consequently neither group embeds in the other, and neither is a quotient of the other. A homomorphism preserves torsion and induces a map Q→Q on the quotients, necessarily multiplication by a rational r. If r=0, its image lies in D.","count":5},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4761badfcfad5d29","tex":"\\one_{A_C}","aliases":"one_A_C one","display":null,"kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩), one_A_C(2⟨ j,k⟩+1) · =one_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd posit","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-af8729ce8553c931","tex":"\\one_{B_j}","aliases":"one_B_j one","display":null,"kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩), one_A_C(2⟨ j,k⟩+1) · =one_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd posit","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-e49bb6af86c95a09","tex":"\\one_C","aliases":"one_C one","display":null,"kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩:N^2→N. For a freely chosen set C⊆N, define a new selector A_C by one_A_C(2⟨ j,k⟩) · =1-one_B_j(2⟨ j,k⟩), one_A_C(2⟨ j,k⟩+1) · =one_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd posit","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-ca3ea4943ff504c7","tex":"\\one_{S_A}","aliases":"one_S_A one","display":null,"kind":"use","excerpt":"[One cocycle value reads one coordinate] For every prime p, c_A(1/p,(p-1)/p)=-one_S_A(p)e_p. The support bound leaves only the coordinate p. At that coordinate both fractional lifts are zero, and the lift of their sum, which is one, is one_S_A(p). Dropping the cocycle is therefore not a harmless simplification. The direct product operation on Q× D ","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-9d8a1921f2a37e0a","tex":"\\overline M","aliases":"overline M overline","display":"\\overline M","kind":"use","excerpt":"All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4f9e74405a47101e","tex":"\\overline M_\\lambda","aliases":"overline M_λ overline lambda","display":"\\overline M_\\lambda","kind":"use","excerpt":"All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-efa8224a327a3d68","tex":"\\overline{\\langle u\\rangle}","aliases":"overline⟨ u⟩ overline langle rangle","display":"\\overline{\\langle u\\rangle}","kind":"use","excerpt":"A single topological generator after completion. [Topological cyclicity] Let u=(1,1,1,…)∈ P. Then overline⟨ u⟩=P. In particular, trank(P)=1. A basic open set prescribes a tuple on finitely many distinct prime coordinates. The Chinese remainder theorem supplies an integer k with the specified residues. The element ku belongs to the given open set. T","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"[Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P(N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ S","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ S","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ S","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-f5af782990149da7","tex":"p_{2n}","aliases":"p_2n","display":"p_{2n}","kind":"use","excerpt":"The information retained by a marked element. For the distinguished element u_A=s_A(1), the coordinate criterion simplifies to pmid_G_Au_A Longleftrightarrow p∉ S_A. Consequently, for each n, p_2nmid_G_Au_A · Longleftrightarrow n∈ A, p_2n+1mid_G_Au_A · Longleftrightarrow n∉ A. Exactly one root equation is solvable. This binary pattern is the select","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-06182cfbeb52a59d","tex":"p_{2n+1}","aliases":"p_2n+1","display":"p_{2n+1}","kind":"use","excerpt":"The information retained by a marked element. For the distinguished element u_A=s_A(1), the coordinate criterion simplifies to pmid_G_Au_A Longleftrightarrow p∉ S_A. Consequently, for each n, p_2nmid_G_Au_A · Longleftrightarrow n∈ A, p_2n+1mid_G_Au_A · Longleftrightarrow n∉ A. Exactly one root equation is solvable. This binary pattern is the select","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-77a74a36475e5eda","tex":"p_{2n+a_n}","aliases":"p_2n+a_n","display":"p_{2n+a_n}","kind":"use","excerpt":"Choosing a rational line. Let A⊆N, write a_n=one_A(n), and define S_A={p_2n+a_n:n∈N}. Exactly one prime from each pair is selected. Both S_A and its complement are infinite. Let u_A∈ P be its indicator: (u_A)_p=1, · p∈ S_A,","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"use","excerpt":"Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ S","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-24b885024a7a3479","tex":"p_{b_j}","aliases":"p_b_j","display":"p_{b_j}","kind":"use","excerpt":"Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finit","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":8},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-346cf456a77b1ce5","tex":"p_n","aliases":"p_n","display":"p_n","kind":"use","excerpt":"A purely finite-state boundary. Consider a restricted grammar in which a finite automaton reads the unary word 1^n and selects the factor C_p_n exactly when it accepts.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"normal form] Every element of G_A has a unique expression g=s_A(r)+d, (r,d)∈Q× D. Consequently G_A is countably infinite, with a countable normal-form domain that does not depend on the selector. Existence follows from the preimage definition and the displayed quotient identity. If two forms are equal, applying q gives (r-r')q(u_A)=0. The vector q(","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-bf4c43a79117c6ae","tex":"\\operatorname{Reach}","aliases":"Reach operatorname","display":"\\operatorname{Reach}","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-ea9d64aac68c5482","tex":"\\operatorname{Reach}_E","aliases":"Reach_E operatorname","display":"\\operatorname{Reach}_E","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-450f6a43de6765bd","tex":"S_B","aliases":"S_B","display":"S_B","kind":"use","excerpt":"ghtarrow Aeqfin B. When Aeqfin B, the two groups are equal as subgroups of the fixed product P. Suppose f:G_A→ G_B is an isomorphism. The image of u_A has infinite order. An isomorphism preserves the existence of a pth root for every prime p. Combining the exact signature [source reference] with Lemma~[source reference] for f(u_A) gives S_Aeqfin S_","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-fb3ff7699e0538a9","tex":"s_B","aliases":"s_B","display":"s_B","kind":"use","excerpt":"Conversely, suppose Aeqfin B. For every rational r, the vectors s_A(r) and s_B(r) differ only on the finitely many prime pairs where the selectors differ. Thus their difference belongs to D. The descriptions G_A=s_A(Q)+D and G_B=s_B(Q)+D define the same subset of P. The normal forms need not be identical. Their coordinate change is (r,d)⟼ (r,d+s_A(","count":5},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-d53379e798f70907","tex":"\\srank_{\\End(G_A)}","aliases":"srank_End(G_A) srank End","display":null,"kind":"use","excerpt":"Infinite seed rank before completion. [Seed-rank separation] For every A, srank_End(G_A)(G_A)=aleph_0. No finite family topologically generates G_A, even after arbitrary endomorphisms are admitted. Yet its completion has ordinary topological generating rank one. Each element s_A(r)+d has only finitely many nonzero coordinates outside S_A. A finite ","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-5f9d50c526ca5e23","tex":"\\Sub","aliases":"Sub","display":null,"kind":"use","excerpt":"[Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P(N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-d97b932ee16cbf3d","tex":"\\Tor","aliases":"Tor","display":null,"kind":"use","excerpt":"The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff Aeqfin B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with ","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-924579a9b7ed33d9","tex":"u_B","aliases":"u_B","display":"u_B","kind":"use","excerpt":"[Marked classification] An isomorphism (G_A,u_A)≅(G_B,u_B) exists if and only if A=B. A marked isomorphism preserves the exact two-prime root test [source reference]--[source reference] at every index. Therefore every bit agrees. The converse is the identity map.","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"For injectivity, suppose mx∈ D. Outside the union of the finite support of mx and the prime divisors of m, multiplication by m is invertible and x_p=0. Hence x∈ D and q(x)=0. Unique division defines (a/b)v as the unique solution w of bw=av. The vector-space laws follow by clearing positive denominators and using injectivity of integer multiplicatio","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-27aee75937940a51","tex":"\\varinjlim","aliases":"varinjlim","display":null,"kind":"use","excerpt":"Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2e2fc35414ba9c6b","tex":"\\varinjlim_N","aliases":"varinjlim_N varinjlim","display":null,"kind":"use","excerpt":"Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"use","excerpt":"The distinction is easiest to express in terms of an embedding G⟶ varprojlim G/N. Residual finiteness makes this map injective. The inverse limit includes all compatible tuples. The image of the original G can nevertheless be a proper dense subgroup. Forgetting that image can forget the object even while retaining enough coordinates to distinguish ","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-d8963e7f8e8a07d5","tex":"\\varprojlim_N","aliases":"varprojlim_N varprojlim","display":null,"kind":"use","excerpt":"Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"For injectivity, suppose mx∈ D. Outside the union of the finite support of mx and the prime divisors of m, multiplication by m is invertible and x_p=0. Hence x∈ D and q(x)=0. Unique division defines (a/b)v as the unique solution w of bw=av. The vector-space laws follow by clearing positive denominators and using injectivity of integer multiplicatio","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-bd5ef9a5ba57a080","tex":"W_A","aliases":"W_A","display":"W_A","kind":"use","excerpt":"0, · p∉ S_A. Because u_A has infinite support, q(u_A)≠0. We can therefore define a rational line and its preimage by W_A=Q q(u_A), G_A=q^-1(W_A). This definition produces an actual group for every set A. It does not claim an effective method for obtaining the selector bits of an arbitrary A.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-d1655f35bb316a56","tex":"\\widehat D","aliases":"widehat D widehat","display":"\\widehat D","kind":"use","excerpt":"The complete finite-quotient system. [Profinite completion of the direct sum] The canonical embedding of D into P identifies P with widehat D. Let f:D→ F be a homomorphism to a finite group. The image of a prime-order factor C_p is trivial unless p divides lvert Frvert. Thus f vanishes on all but finitely many prime components and factors through a","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"[Reflection within the paired-prime family] For a homomorphism f:G_A→ G_B, the induced completion map widehat f is an isomorphism if and only if f is an isomorphism. One implication follows from functoriality. Conversely, suppose widehat f is an isomorphism. By Theorem~[source reference], its coordinate action is multiplication by a diagonal sequen","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-70dd49fe06536922","tex":"\\widehat{G_A}","aliases":"widehatG_A widehat","display":"\\widehat{G_A}","kind":"use","excerpt":"The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff Aeqfin B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with ","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-9fb7a711460d4e2d","tex":"\\widehat{H_W}","aliases":"widehatH_W widehat","display":"\\widehat{H_W}","kind":"use","excerpt":"em[One finite-quotient system for every preimage] For every rational subspace W and every m≥1, the coordinate map induces an isomorphism H_W/mH_W ≅ ∏_pmid mC_p. For mmid n, the quotient transition map is the coordinate restriction from primes dividing n to primes dividing m. These quotients are cofinal among all finite quotients, and widehatH_W≅ P.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"Conversely, suppose mx=0 for a positive integer m. Whenever pnmid m, multiplication by m is invertible in C_p, and hence x_p=0. The support of x is contained in the finite set of prime divisors of m. Thus x∈ D. The all-one vector has infinite support, so it belongs to P∖ D.","count":39},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-af6e3ccbb4a25be4","tex":"x_{2n+1}","aliases":"x_2n+1","display":"x_{2n+1}","kind":"use","excerpt":"For the relator description, introduce a commuting generator x_i for each prime index, impose p_i x_i=0, and impose x_2n+1=0 whenever program n is observed to halt on input n. The resulting quotient retains exactly the prime factors indexed by B_K. All relators are effectively enumerable. Deciding whether an odd generator has become trivial is not ","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"For the relator description, introduce a commuting generator x_i for each prime index, impose p_i x_i=0, and impose x_2n+1=0 whenever program n is observed to halt on input n. The resulting quotient retains exactly the prime factors indexed by B_K. All relators are effectively enumerable. Deciding whether an odd generator has become trivial is not ","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-53687bf254c09aec","tex":"x_p","aliases":"x_p","display":"x_p","kind":"use","excerpt":"Conversely, suppose mx=0 for a positive integer m. Whenever pnmid m, multiplication by m is invertible in C_p, and hence x_p=0. The support of x is contained in the finite set of prime divisors of m. Thus x∈ D. The all-one vector has infinite support, so it belongs to P∖ D.","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"Structure of the extension. By Chapter~[source reference], the group is pure in P, reduced, residually finite, and dense. Its torsion is D, and the rational coordinate gives the exact sequence 0⟶ D⟶ G_Axrightarrowρ_AQ⟶0, ρ_A(s_A(r)+d)=r. It does not split.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"[All roots and their number] If my=g is solvable, its solution set is the coset y_0+G_A[m], G_A[m]=⊕_pmid mC_p. It has exactly rad(m) elements. Two solutions differ by an element annihilated by m, and adding such an element to a solution gives another. The ambient product has a nonzero m-torsion coordinate only when pmid m. All those finite coordin","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-7a4d74bf293c7c99","tex":"y_p","aliases":"y_p","display":"y_p","kind":"use","excerpt":"[Unique division in the ambient quotient] Multiplication by every positive integer is bijective on V. Thus V has a unique rational vector-space structure extending its additive group structure. The product P has no nonzero divisible subgroup. Fix m≥1 and x∈ P. Define y_p=m^-1x_p when pnmid m, and y_p=0 when pmid m. Then x-my has finite support. Thu","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"use","excerpt":"The integer group Z illustrates only the first distinction. One generator and its integer multiples produce infinitely many elements. The groups developed below separate the other distinctions as well. Their purpose is not to make infinite objects mysterious. It is to account for the information that a construction actually consumes.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"Decoding a marked copy. [Uniform marked decoding] Given an X-computable copy of G_A together with the name of u_A, one can uniformly compute A in X. For input n, search the domain simultaneously for solutions of p_2ny=u_A, p_2n+1z=u_A. Each candidate test is decidable in X. Exactly one equation has a solution by [source reference]--[source referenc","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"bottomrule Each residue is exact. At p=2, the lift of 1/2 is set to zero because the denominator is not invertible. At p=5, its value is 3 because 2·3=1 in F_5.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"[Unique normal form] Every element of G_A has a unique expression g=s_A(r)+d, (r,d)∈Q× D. Consequently G_A is countably infinite, with a countable normal-form domain that does not depend on the selector. Existence follows from the preimage definition and the displayed quotient identity. If two forms are equal, applying q gives (r-r')q(u_A)=0. The v","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.","count":18},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-fdc2dc9f687c22a4","tex":"\\lambda_p","aliases":"λ_p lambda","display":"\\lambda_p","kind":"use","excerpt":"[Complete homomorphism description] Every homomorphism f:G_A→ G_B has the form f(g)_p=λ_p g_p (p prime) for a unique λ∈ P. Conversely, coordinate multiplication by λ defines such a homomorphism exactly when q(λ u_A)∈Q q(u_B). Equivalently, there is a rational r for which λ u_A-s_B(r) has finite support. For a homomorphism f, the composite G_A→ G_B→","count":6},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-831c58f6031d9f05","tex":"\\lambda_p^{-1}","aliases":"λ_p^-1 lambda","display":"\\lambda_p^{-1}","kind":"use","excerpt":"[Automorphisms] An endomorphism with diagonal λ∈ E_A is an automorphism if and only if every coordinate λ_p is nonzero. In that case its rational quotient scalar is nonzero, and its inverse has diagonal λ_p^-1. An automorphism restricts to an automorphism of each torsion factor, so every scalar is nonzero. Conversely, if every scalar is nonzero, th","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"The quotient formula. [All integer quotients] For m≥1, the coordinate map π_m:G_A⟶∏_pmid mC_p is surjective with kernel mG_A. Thus G_A/mG_A≅∏_pmid mC_p≅ C_rad(m). For mmid n, the transition map is the corresponding coordinate projection. A vector supported on the prime divisors of m realizes any tuple in the target, proving surjectivity. The root c","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-804bed66bc586497","tex":"\\pi_m","aliases":"π_m pi","display":"\\pi_m","kind":"use","excerpt":"The quotient formula. [All integer quotients] For m≥1, the coordinate map π_m:G_A⟶∏_pmid mC_p is surjective with kernel mG_A. Thus G_A/mG_A≅∏_pmid mC_p≅ C_rad(m). For mmid n, the transition map is the corresponding coordinate projection. A vector supported on the prime divisors of m realizes any tuple in the target, proving surjectivity. The root c","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"Structure of the extension. By Chapter~[source reference], the group is pure in P, reduced, residually finite, and dense. Its torsion is D, and the rational coordinate gives the exact sequence 0⟶ D⟶ G_Axrightarrowρ_AQ⟶0, ρ_A(s_A(r)+d)=r. It does not split.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-362dfb6c4b153cec","tex":"\\rho_A","aliases":"ρ_A rho","display":"\\rho_A","kind":"use","excerpt":"Structure of the extension. By Chapter~[source reference], the group is pure in P, reduced, residually finite, and dense. Its torsion is D, and the rational coordinate gives the exact sequence 0⟶ D⟶ G_Axrightarrowρ_AQ⟶0, ρ_A(s_A(r)+d)=r. It does not split.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"The entire subgroup lattice. For B⊆N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2792c54819372788","tex":"\\sigma_p","aliases":"σ_p sigma","display":"\\sigma_p","kind":"use","excerpt":"The entire subgroup lattice. For B⊆N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-8939774e8d5b06af","tex":"\\sigma_{p_i}","aliases":"σ_p_i sigma","display":"\\sigma_{p_i}","kind":"use","excerpt":"Let B be the set of indices of these factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice oper","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P(N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"If H_W→ F is a homomorphism to a finite group, then the image is abelian and is killed by lvert Frvert. Its kernel therefore contains lvert Frvert H_W. Hence the displayed kernels are cofinal. Their inverse limit is the product of all prime coordinates, which is P. Under this identification the completion map is the original inclusion H_W↪ P. There","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"[Endomorphism ring] Under coordinate multiplication, End(G_A)≅ E_A={λ∈ P:∃ r∈Q, λ_p=overline r_p for almost every p∈ S_A}. The rational r is unique. It is the scalar induced on G_A/D, and the map E_A→Q sending λ to r is a surjective ring homomorphism. Apply Theorem~[source reference] with B=A. At unselected coordinates, both λ u_A and s_A(r) vanish","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-e3684783b225d098","tex":"\\prod_{i<N}","aliases":"∏_i<N prod","display":"\\prod_{i<N}","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-92351115e501ee93","tex":"\\prod_{p\\mid m}","aliases":"∏_pmid m prod mid","display":null,"kind":"use","excerpt":"[One finite-quotient system for every preimage] For every rational subspace W and every m≥1, the coordinate map induces an isomorphism H_W/mH_W ≅ ∏_pmid mC_p. For mmid n, the quotient transition map is the coordinate restriction from primes dividing n to primes dividing m. These quotients are cofinal among all finite quotients, and widehatH_W≅ P. T","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0c8a7313d6794de5","tex":"\\sum_{p\\mid m}","aliases":"∑_pmid m sum mid","display":null,"kind":"use","excerpt":"Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Conversely, suppose mx=0 for a positive integer m. Whenever pnmid m, multiplication by m is invertible in C_p, and hence x_p=0. The support of x is contained in the finite set of prime divisors of m. Thus x∈ D. The all-one vector has infinite support, so it belongs to P∖ D.","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic d","count":2},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_","count":9},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic d","count":3},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-e8a1aa147b4b8b89","tex":"\\bigoplus_{i\\in B}","aliases":"⊕_i∈ B bigoplus in","display":"\\bigoplus_{i\\in B}","kind":"use","excerpt":"The entire subgroup lattice. For B⊆N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-fa0b3e595fc9f5b4","tex":"\\bigoplus_{i\\in I}","aliases":"⊕_i∈ I bigoplus in","display":"\\bigoplus_{i\\in I}","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-2610e51a095ab37e","tex":"\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}","aliases":"⊕_j∈Reach_E(F) bigoplus in operatorname","display":"\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-e2983c1eebc209b8","tex":"\\bigoplus_{p\\mid m}","aliases":"⊕_pmid m bigoplus mid","display":null,"kind":"use","excerpt":"[All roots and their number] If my=g is solvable, its solution set is the coset y_0+G_A[m], G_A[m]=⊕_pmid mC_p. It has exactly rad(m) elements. Two solutions differ by an element annihilated by m, and adding such an element to a solution gives another. The ambient product has a nonzero m-torsion coordinate only when pmid m. All those finite coordin","count":1},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"Otherwise choose a lift x∈ L of a generator of its rational image. Subtracting suitable integer multiples of x from the given generators produces finitely many elements of D. They generate the kernel of L→ρ_A(L), which is finite cyclic of squarefree order. The subgroup ⟨ x⟩ intersects that torsion kernel trivially, giving the direct sum. The extens","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"Otherwise choose a lift x∈ L of a generator of its rational image. Subtracting suitable integer multiples of x from the given generators produces finitely many elements of D. They generate the kernel of L→ρ_A(L), which is finite cyclic of squarefree order. The subgroup ⟨ x⟩ intersects that torsion kernel trivially, giving the direct sum. The extens","count":4},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"The distinction is easiest to express in terms of an embedding G⟶ varprojlim G/N. Residual finiteness makes this map injective. The inverse limit includes all compatible tuples. The image of the original G can nevertheless be a proper dense subgroup. Forgetting that image can forget the object even while retaining enough coordinates to distinguish ","count":5},{"work":"realization-boundaries","url":"notation/realization-boundaries.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[Pure-preimage correspondence] The map W⟼ H_W=q^-1(W) is an inclusion-preserving bijection between rational subspaces of V and pure subgroups H of P containing D. First let W be rational and suppose h∈ H_W∩ mP. Because W is closed under division by m, choose x∈ H_W with q(x)=q(h)/m. Then d=h-mx belongs to D. At a prime dividing m, the coordinate of","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":492},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"Multiplying δ≤γη by nn p≥0 and the strict margin γnn p<τ-1 by η>0 proves the ratio specialization. The chosen source is defined by classical choice from the existence theorem. Its raw definition takes A,p and the truth/feasibility proofs as arguments; it is a theorem-level witness rather than an extracted algorithm using only available measurements","count":337},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5761fa2adf61c496","tex":"\\cl","aliases":"cl operatorname","display":"{\\operatorname{cl}}","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":8},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-558c1dc971656de6","tex":"\\dist","aliases":"dist operatorname","display":"{\\operatorname{dist}}","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":12},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ad49f2063354d25b","tex":"\\inf_{\\ell\\in L}","aliases":"inf_ℓ∈ L inf ell in","display":"\\inf_{\\ell\\in L}","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":51},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-96bff98fdef7a4f7","tex":"\\nn","aliases":"nn","display":null,"kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":267},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":66},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Multiplying δ≤γη by nn p≥0 and the strict margin γnn p<τ-1 by η>0 proves the ratio specialization. The chosen source is defined by classical choice from the existence theorem. Its raw definition takes A,p and the truth/feasibility proofs as arguments; it is a theorem-level witness rather than an extracted algorithm using only available measurements","count":118},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-363d67fd271736f2","tex":"P_L","aliases":"P_L","display":"P_L","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":216},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"introduction-passage","excerpt":"Multiplying δ≤γη by nn p≥0 and the strict margin γnn p<τ-1 by η>0 proves the ratio specialization. The chosen source is defined by classical choice from the existence theorem. Its raw definition takes A,p and the truth/feasibility proofs as arguments; it is a theorem-level witness rather than an extracted algorithm using only available measurements","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"introduction-passage","excerpt":"Multiplying δ≤γη by nn p≥0 and the strict margin γnn p<τ-1 by η>0 proves the ratio specialization. The chosen source is defined by classical choice from the existence theorem. Its raw definition takes A,p and the truth/feasibility proofs as arguments; it is a theorem-level witness rather than an extracted algorithm using only available measurements","count":100},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"introduction-passage","excerpt":"Multiplying δ≤γη by nn p≥0 and the strict margin γnn p<τ-1 by η>0 proves the ratio specialization. The chosen source is defined by classical choice from the existence theorem. Its raw definition takes A,p and the truth/feasibility proofs as arguments; it is a theorem-level witness rather than an extracted algorithm using only available measurements","count":95},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"Multiplying δ≤γη by nn p≥0 and the strict margin γnn p<τ-1 by η>0 proves the ratio specialization. The chosen source is defined by classical choice from the existence theorem. Its raw definition takes A,p and the truth/feasibility proofs as arguments; it is a theorem-level witness rather than an extracted algorithm using only available measurements","count":47},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"For a subspace L admitting orthogonal projection, P_L denotes that actual projection. On a Hilbert space, closed subspaces admit it. The notation dist(v,L)=inf_ℓ∈ Lnnv-ℓ always refers to the actual subspace, not a replacement chosen to make a theorem true. Closure is written cl L.","count":51},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"Multiplying δ≤γη by nn p≥0 and the strict margin γnn p<τ-1 by η>0 proves the ratio specialization. The chosen source is defined by classical choice from the existence theorem. Its raw definition takes A,p and the truth/feasibility proofs as arguments; it is a theorem-level witness rather than an extracted algorithm using only available measurements","count":215},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"Multiplying δ≤γη by nn p≥0 and the strict margin γnn p<τ-1 by η>0 proves the ratio specialization. The chosen source is defined by classical choice from the existence theorem. Its raw definition takes A,p and the truth/feasibility proofs as arguments; it is a theorem-level witness rather than an extracted algorithm using only available measurements","count":31},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Fix a bounded operator A:G→ V between real Hilbert spaces, without assuming closed range. For λ>0, put N_λ=A^*A+λ I, R_λ=N_λ^-1, B_λ=R_λ A^*. The inverse is constructed from coercivity. It is not supplied as a certificate premise. Here A^* is the actual Hilbert adjoint.","count":52},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":181},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":60},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"Commuting presentation maps LA=A'F and δ'L=Mδ induce a map overline L: H→ H' by [v]↦[Lv]. These maps compose and preserve zero classes. Reflection of zero follows when overline L is injective.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e0c9947b1daf0522","tex":"A^*","aliases":"A^*","display":"A^*","kind":"use","excerpt":"Fix a bounded operator A:G→ V between real Hilbert spaces, without assuming closed range. For λ>0, put N_λ=A^*A+λ I, R_λ=N_λ^-1, B_λ=R_λ A^*. The inverse is constructed from coercivity. It is not supplied as a certificate premise. Here A^* is the actual Hilbert adjoint.","count":52},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0db91bbb5873c5b3","tex":"a^0","aliases":"a^0","display":"a^0","kind":"use","excerpt":"Induction proves the finite-sum formula: multiplying the old sum by a and adjoining b^n generates every term of the new sum. Multiplication by a-b telescopes, leaving a^n-b^n. Division is used only when the rates are distinct. If they coincide, all n+1 terms of H_n+1 equal a^n, including the zero-rate boundary with the usual a^0=1. For uniform boun","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"The resolvent parameter identity gives, for s,t≥ a>0, |r_v(s)-r_v(t)|≤ fracnn AnnA^*va^2|s-t|, proving continuity on positive compact intervals. The strict floor permits a comparator g with nnAg-v<ρ; objective minimization at sufficiently small positive λ gives a lower residual bracket. For large λ, nnB_λ v≤nnA^*v/λ, so the residual approaches nn v","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-61bf579da1b5c924","tex":"a^j","aliases":"a^j","display":"a^j","kind":"use","excerpt":"For H_0(a,b)=0 and H_n+1(a,b)=aH_n(a,b)+b^n, H_n(a,b)=∑_j=0^n-1a^jb^n-1-j, (a-b)H_n=a^n-b^n. If a≠ b, H_n=(a^n-b^n)/(a-b); for equal rates, H_n+1(a,a)=(n+1)a^n. Uniform step bounds give nnE_w≤ϵ H_n(a,b), nnOT_w-VU_wq≤η a^n+nn Vϵ H_n(a,b).","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a472735db532714b","tex":"A^{k+2}","aliases":"A^k+2","display":"A^{k+2}","kind":"use","excerpt":"The following conditions are equivalent: E_n=A^n (∀ n), M_k=0 (∀ k), E_m+n=E_mE_n (∀ m,n). More precisely, E_n=A^n (n≤ N+1)iff M_k=0 (k<N). If M_j=0 for j<k, then E_k+2-A^k+2=M_k. Closure for every initial hidden state is equivalent to B=0.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9a52b1bc77ae2b6d","tex":"A^n","aliases":"A^n","display":"A^n","kind":"use","excerpt":"The following conditions are equivalent: E_n=A^n (∀ n), M_k=0 (∀ k), E_m+n=E_mE_n (∀ m,n). More precisely, E_n=A^n (n≤ N+1)iff M_k=0 (k<N). If M_j=0 for j<k, then E_k+2-A^k+2=M_k. Closure for every initial hidden state is equivalent to B=0.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c60967b39bed49c7","tex":"a^n","aliases":"a^n","display":"a^n","kind":"use","excerpt":"For H_0(a,b)=0 and H_n+1(a,b)=aH_n(a,b)+b^n, H_n(a,b)=∑_j=0^n-1a^jb^n-1-j, (a-b)H_n=a^n-b^n. If a≠ b, H_n=(a^n-b^n)/(a-b); for equal rates, H_n+1(a,a)=(n+1)a^n. Uniform step bounds give nnE_w≤ϵ H_n(a,b), nnOT_w-VU_wq≤η a^n+nn Vϵ H_n(a,b).","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5494a82a5f60bced","tex":"A_c","aliases":"A_c","display":"A_c","kind":"use","excerpt":"changing its image. Coercivity makes this source sequence Cauchy. Completeness gives a limit g∈ S, and continuity gives the image limit Ag∈Ran A. Hence the range is closed. If bounded B satisfies BA=P_S, then for g∈ S, nn g=nnBAg≤nn BnnAg. The same positive choice 1/(nn B+1) gives coercivity. Finite-dimensional image subspaces are closed in a norme","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f78009c8b1487c7d","tex":"A_S","aliases":"A_S","display":"A_S","kind":"use","excerpt":"Restriction A_S:S→ R is a bounded linear bijection with bounded inverse. Define B=ι_S A_S^-1P_R. Then AB=P_R, BA=P_S, ABA=A, and BAB=B. Its two projections are symmetric, ker B=R^⊥, and Ran B=S. The best-image identity AB=P_R and the condition Ran B⊆ S uniquely determine B.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-784a3b525d0e2e98","tex":"A_S^{-1}","aliases":"A_S^-1","display":"A_S^{-1}","kind":"use","excerpt":"Restriction A_S:S→ R is a bounded linear bijection with bounded inverse. Define B=ι_S A_S^-1P_R. Then AB=P_R, BA=P_S, ABA=A, and BAB=B. Its two projections are symmetric, ker B=R^⊥, and Ran B=S. The best-image identity AB=P_R and the condition Ran B⊆ S uniquely determine B.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4f1eee202dc651a9","tex":"A_u","aliases":"A_u","display":"A_u","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9771bd16cd82b5a0","tex":"a_X","aliases":"a_X","display":"a_X","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":10},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-be9fbced69e42da2","tex":"a_Y","aliases":"a_Y","display":"a_Y","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-08a584d38018c160","tex":"a_Z","aliases":"a_Z","display":"a_Z","kind":"use","excerpt":"The splitting gives E_X^2=a_X(s_Xa_X)s_X=a_Xs_X. If E_Xv=v, then v=a_X(s_Xv); conversely E_Xa_Xx=a_Xx. Substituting the definitions yields s_Y(a_YTs_X)a_X=T, a_Y(s_YKa_X)s_X=E_YKE_X. For compatible K, take T= D(K). If another T' encodes to K, decoding proves T'= D(K). Finally, an intermediate resolution of Y has s_Ya_Y=I_Y, so K(S) K(T)=a_ZS(s_Ya_Y","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"Let V be visible state and H hidden state. Over a commutative ring, specify four maps and form the actual full dynamics T(y,z)=(Ay+Bz,Gy+Dz). Put C(y,z)=y, Ry=(y,0), E_n=CT^nR, and M_k=BD^kG. The return M_k contains entry into H, k hidden transitions, and return to V. Its first effect occurs after k+2 full transitions.","count":103},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":39},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e77f9eceea598e09","tex":"B'","aliases":"B'","display":"B'","kind":"use","excerpt":"orem supplies A_S^-1 as an actual continuous linear inverse. The definition then gives AB=P_R. For g, BAg is the unique element of S with image Ag, namely P_Sg. The two inverse identities imply the generalized inverse laws. Orthogonal projections are symmetric, proving the symmetry conditions. The definition and injectivity of A_S^-1 identify the k","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"For the source-condition rate, norm domination and p=A^*w give nny-p^2≤2ippp-y≤2nn w nnA(p-y) for an exact comparison y. Bound its image by the first conclusion. Then nnx-p≤nnx-y+nny-p and (a+b)^2≤2a^2+2b^2 prove the stated inexact rate.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-408547371092547a","tex":"b^n","aliases":"b^n","display":"b^n","kind":"use","excerpt":"For H_0(a,b)=0 and H_n+1(a,b)=aH_n(a,b)+b^n, H_n(a,b)=∑_j=0^n-1a^jb^n-1-j, (a-b)H_n=a^n-b^n. If a≠ b, H_n=(a^n-b^n)/(a-b); for equal rates, H_n+1(a,a)=(n+1)a^n. Uniform step bounds give nnE_w≤ϵ H_n(a,b), nnOT_w-VU_wq≤η a^n+nn Vϵ H_n(a,b).","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d518a5f110d0ab48","tex":"b^{n-1}","aliases":"b^n-1","display":"b^{n-1}","kind":"use","excerpt":"For a=b=1, the response sum has n unit terms. For a=1 and b<1, it is 1+b+⋯+b^n-1≤1/(1-b). SR-15 gives the operator estimates. For initialization, add VU_w(qx-z) to the operator output error applied to x, and use nnU_w≤ b^n. The common bound concerns every allowed operation in the chosen norm. Individual spectral radii do not imply a common product ","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e466f81dc36eab90","tex":"b^{n-1-j}","aliases":"b^n-1-j","display":"b^{n-1-j}","kind":"use","excerpt":"For H_0(a,b)=0 and H_n+1(a,b)=aH_n(a,b)+b^n, H_n(a,b)=∑_j=0^n-1a^jb^n-1-j, (a-b)H_n=a^n-b^n. If a≠ b, H_n=(a^n-b^n)/(a-b); for equal rates, H_n+1(a,a)=(n+1)a^n. Uniform step bounds give nnE_w≤ϵ H_n(a,b), nnOT_w-VU_wq≤η a^n+nn Vϵ H_n(a,b).","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-7e782449bb70aa30","tex":"b^{|w|}","aliases":"b^|w|","display":"b^{|w|}","kind":"use","excerpt":"ion proves the finite-sum formula: multiplying the old sum by a and adjoining b^n generates every term of the new sum. Multiplication by a-b telescopes, leaving a^n-b^n. Division is used only when the rates are distinct. If they coincide, all n+1 terms of H_n+1 equal a^n, including the zero-rate boundary with the usual a^0=1. For uniform bounds, th","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"[A metric must survive its coordinate change] For K={(x,y,0)} and R=span{(1,0,0)}, take the compatible basis b_1=(1,0,0), b_2=(1,2,0). Its Gram matrix is (1 · 1","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"[A metric must survive its coordinate change] For K={(x,y,0)} and R=span{(1,0,0)}, take the compatible basis b_1=(1,0,0), b_2=(1,2,0). Its Gram matrix is (1 · 1","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e0205e635cc74e4e","tex":"B_{A,\\lambda}","aliases":"B_A,λ lambda","display":"B_{A,\\lambda}","kind":"use","excerpt":"For two operators and parameters, R_A,λ-R_C,μ =R_A,λ(N_C,μ-N_A,λ)R_C,μ. At fixed positive λ, with ϵ=nnA-C, nnB_A,λ-B_C,λ ≤fracϵλ+ frac(nn A+nn C)ϵnn Cλ^2. If A has closed range and canonical inverse B, then B_λ v-Bv=-λ R_λ Bv. For δ A=0, nnv-AB_λ v^2 =nnc(v)^2+nni(v)^2+nnA(B_λ v-Bv)^2.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d41402bfd3b150fb","tex":"B_c","aliases":"B_c","display":"B_c","kind":"use","excerpt":"s source sequence Cauchy. Completeness gives a limit g∈ S, and continuity gives the image limit Ag∈Ran A. Hence the range is closed. If bounded B satisfies BA=P_S, then for g∈ S, nn g=nnBAg≤nn BnnAg. The same positive choice 1/(nn B+1) gives coercivity. Finite-dimensional image subspaces are closed in a normed space. Finally A_cg=cg, c>0, has canon","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ce5af6a2829bb0bb","tex":"B_{C,\\lambda}","aliases":"B_C,λ lambda","display":"B_{C,\\lambda}","kind":"use","excerpt":"For two operators and parameters, R_A,λ-R_C,μ =R_A,λ(N_C,μ-N_A,λ)R_C,μ. At fixed positive λ, with ϵ=nnA-C, nnB_A,λ-B_C,λ ≤fracϵλ+ frac(nn A+nn C)ϵnn Cλ^2. If A has closed range and canonical inverse B, then B_λ v-Bv=-λ R_λ Bv. For δ A=0, nnv-AB_λ v^2 =nnc(v)^2+nni(v)^2+nnA(B_λ v-Bv)^2.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-aba34f1446cfa879","tex":"B_v","aliases":"B_v","display":"B_v","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5902c64cbd2a059e","tex":"B_w","aliases":"B_w","display":"B_w","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f8dd14ae875161a4","tex":"B_\\lambda","aliases":"B_λ lambda","display":"B_\\lambda","kind":"use","excerpt":"Fix a bounded operator A:G→ V between real Hilbert spaces, without assuming closed range. For λ>0, put N_λ=A^*A+λ I, R_λ=N_λ^-1, B_λ=R_λ A^*. The inverse is constructed from coercivity. It is not supplied as a certificate premise. Here A^* is the actual Hilbert adjoint.","count":22},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3a0b3bc889c93357","tex":"B_\\lambda^A","aliases":"B_λ^A lambda","display":"B_\\lambda^A","kind":"use","excerpt":"For arbitrary data v, AB_λ^Av=B_λ^A^*A^*v ⟶ P_clRan Av. Adding data noise e_i→0 preserves this fitted-data limit. If v∈clRan A∖Ran A, fitted data converge to v but the corresponding reconstructed source family has no norm limit in G.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3e9f102a12c3ffa1","tex":"B_\\lambda^{A^*}","aliases":"B_λ^A^* lambda","display":"B_\\lambda^{A^*}","kind":"use","excerpt":"For arbitrary data v, AB_λ^Av=B_λ^A^*A^*v ⟶ P_clRan Av. Adding data noise e_i→0 preserves this fitted-data limit. If v∈clRan A∖Ran A, fitted data converge to v but the corresponding reconstructed source family has no norm limit in G.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-de801c8302f701cf","tex":"B_{\\lambda_i}","aliases":"B_λ_i lambda","display":"B_{\\lambda_i}","kind":"use","excerpt":"If λ_i>0 tends to zero and nne_i/sqrtλ_i→0, then B_λ_i(Ag+e_i)⟶ P_(ker A)^⊥g. The same limit holds for computed x_i when additionally fracnn(A^*A+λ_iI)x_i-A^*(Ag+e_i)λ_i→0. For example, λ_i=s_i^2, s_i>0→0, and nne_i≤ s_i^2 give the exact-solve consistency condition.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"smallHypotheses. Commutative-ring modules; linear C:X→ Z, R:Z→ X with CR=I_Z; arbitrary endomorphisms S,T,U of X.","count":106},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"[Redundant coordinates] For X= R and V_X= R^2, take a(x)=(x,x) and s(u,v)=(u+v)/2. Then E(u,v)=((u+v)/2,(u+v)/2). The encoded scalar transformation x↦ cx has matrix frac c2(1 · 1","count":50},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"use","excerpt":"Multiply the difference of inverses on the left and right by their normal operators, or insert their inverse laws, to obtain the resolvent identity. SR-38 bounds its norm by nnN_C,μ-N_A,λ/(λμ). At equal parameters, C^*C-A^*A=(C^*-A^*)C+A^*(C-A), whose norm is at most (nn A+nn C)ϵ. Splitting R_AA^*-R_CC^*=R_A(A^*-C^*)+(R_A-R_C)C^* proves the stated ","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","excerpt":"[Exact scalar response and a two-coordinate example] For A_cx=cx, B_λ v=cv/(c^2+λ). At c>0, λ=c^2, the value is v/(2c), attaining the noise bound. For A(g_1,g_2)=(g_1+g_2,0,0), v=(3,4,5), and λ=1, the source is (1,1). Its squared fit is 42=16+25+1, separating obstruction, inconsistency, and regularization image bias. Its objective is 44, whereas th","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"Suppose p∈ S_A, nn p≤ M, and nnC_i-A≤δ_i, nnAp-v_i≤η_i . If x_i lies within κ_i of an exact accepted source for C_i,v_i with allowance ϵ_i=η_i+δ_iM, and η_i,δ_i,κ_i→0, then x_i→ p. The model kernels and ranks may change. For arbitrary g, using p=P_S_Ag yields nnx_i-g→nnP_N_Ag and P_N_Ax_i→0.","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-307d2c7e46d5ba73","tex":"C_s","aliases":"C_s","display":"C_s","kind":"use","excerpt":"Let U(t)=(t,0) and C_s(x,y)=sy. Then C_sU=0 and nnC_s=|s|. For every real M there is an operator agreeing perfectly with zero on the entire calibration subspace yet satisfying nnC_s>M. The calibration synthesis is not surjective. Thus perfect agreement on a proper source subspace does not control whole-source operator accuracy.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4788720426dbb81f","tex":"c_{\\eta,\\delta}","aliases":"c_η,δ eta delta","display":"c_{\\eta,\\delta}","kind":"use","excerpt":"The acceptance rule can avoid an externally supplied number M by using the candidate itself in the uncertainty allowance. Put c_η,δ(x)=η+δnn x . A generalized accepted source is either zero in the small-data branch nn v≤τη, or an actual Tikhonov source whose residual lies between c_η,δ(x) and τ c_η,δ(x). This definition contains C,v,η,δ,τ,x and the","count":12},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"Let V be visible state and H hidden state. Over a commutative ring, specify four maps and form the actual full dynamics T(y,z)=(Ay+Bz,Gy+Dz). Put C(y,z)=y, Ry=(y,0), E_n=CT^nR, and M_k=BD^kG. The return M_k contains entry into H, k hidden transitions, and return to V. Its first effect occurs after k+2 full transitions.","count":51},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"If a monic polynomial p annihilates D, vanishing of M_k for k<deg p implies vanishing for all k. If dim H=d<∞, then E_n=A^n (∀ n) iff M_k=0 (0≤ k<d) iff E_n=A^n (0≤ n≤ d+1). The visible space need not be finite-dimensional. The case d=0 is included.","count":22},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","excerpt":"[A linear continuous memory kernel] Take V= R, H= R^2, A=0, B(z_1,z_2)=z_1, G(y)=(0,y), and D(z_1,z_2)=(z_2,0). Since D^2=0, e^tD(z_1,z_2)=(z_1+tz_2,z_2) and M(t)=tI. The first two prepared jets agree with the reduced generator, while the third derivative differs by I. Changing D to zero leaves both boundary couplings nonzero but makes the memory k","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2f80fcde2a727a6d","tex":"D^k","aliases":"D^k","display":"D^k","kind":"use","excerpt":"Let V be visible state and H hidden state. Over a commutative ring, specify four maps and form the actual full dynamics T(y,z)=(Ay+Bz,Gy+Dz). Put C(y,z)=y, Ry=(y,0), E_n=CT^nR, and M_k=BD^kG. The return M_k contains entry into H, k hidden transitions, and return to V. Its first effect occurs after k+2 full transitions.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-76b7929465487fe2","tex":"D^n","aliases":"D^n","display":"D^n","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c8eca49f4ae26f1e","tex":"D^{n-1-j}","aliases":"D^n-1-j","display":"D^{n-1-j}","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-85e05bb77235fdc9","tex":"D_{i::w}","aliases":"D_i::w","display":"D_{i::w}","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ee1684d3a8f6d798","tex":"D_u","aliases":"D_u","display":"D_u","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5f210917e7185dac","tex":"D_{u\\mathbin{+\\!+}v}","aliases":"D_umathbin+ +v mathbin","display":null,"kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-960a9e4480b08df9","tex":"D_w","aliases":"D_w","display":"D_w","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c1198614ad304db3","tex":"D_\\varnothing","aliases":"D_∅ varnothing","display":"D_\\varnothing","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"If a monic polynomial p annihilates D, vanishing of M_k for k<deg p implies vanishing for all k. If dim H=d<∞, then E_n=A^n (∀ n) iff M_k=0 (0≤ k<d) iff E_n=A^n (0≤ n≤ d+1). The visible space need not be finite-dimensional. The case d=0 is included.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"If a monic polynomial p annihilates D, vanishing of M_k for k<deg p implies vanishing for all k. If dim H=d<∞, then E_n=A^n (∀ n) iff M_k=0 (0≤ k<d) iff E_n=A^n (0≤ n≤ d+1). The visible space need not be finite-dimensional. The case d=0 is included.","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"the range is closed. If bounded B satisfies BA=P_S, then for g∈ S, nn g=nnBAg≤nn BnnAg. The same positive choice 1/(nn B+1) gives coercivity. Finite-dimensional image subspaces are closed in a normed space. Finally A_cg=cg, c>0, has canonical inverse B_cv=v/c. At v=c, data tend to zero with c while the recovered source remains one, ruling out a com","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":77},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"Let L be a bounded real operator on a Banach space. Its flow is the actual operator exponential e^tL. Boundedness allows differentiation of the norm-convergent exponential series and integration of continuous operator curves. Equalities in this section hold for all real t, using oriented interval integrals when t<0.","count":65},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c7c26f6b3b085b5c","tex":"E'","aliases":"E'","display":"E'","kind":"use","excerpt":"SR-18 converts prepared flow equality into prepared power equality and memory flow vanishing into memory moment vanishing. SR-06 equates the corresponding algebraic conditions, and SR-07 supplies the finite moment criterion. Exact prepared flow gives its semigroup law. Conversely, if E is a semigroup, differentiating E(s+t)=E(s)E(t) in s at zero gi","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8ea99e628b8bd991","tex":"e^{(a-b)s}","aliases":"e^(a-b)s","display":"e^{(a-b)s}","kind":"use","excerpt":"Apply the integral norm inequality and submultiplicativity to SR-21. For 0≤ s≤ t, the integrand norm is at most M_Ae^b(t-s)ϵ M_Le^as. Integration gives the first estimate. Factoring e^bt and integrating e^(a-b)s gives the distinct-rate formula; for equal rates the integrand is constant e^at. The output decomposition is F(t)=(O-Vq)e^tL+VE(t), giving","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6e1e5f56fa519967","tex":"E^{(k+2)}","aliases":"E^(k+2)","display":"E^{(k+2)}","kind":"use","excerpt":"Let E(t)=Ce^tLR. The following are equivalent: E(t)=e^tA (∀ t), M(t)=0 (∀ t), BD^kG=0 (∀ k), E(s+t)=E(s)E(t) (∀ s,t). If dim H=d, each is equivalent to BD^kG=0 for k<d, or to E^(n)(0)=A^n for n≤ d+1. If the first k return moments vanish, E^(k+2)(0)-A^k+2=BD^kG.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-59aa01a9eeebb73a","tex":"E^{(n)}","aliases":"E^(n)","display":"E^{(n)}","kind":"use","excerpt":"Let E(t)=Ce^tLR. The following are equivalent: E(t)=e^tA (∀ t), M(t)=0 (∀ t), BD^kG=0 (∀ k), E(s+t)=E(s)E(t) (∀ s,t). If dim H=d, each is equivalent to BD^kG=0 for k<d, or to E^(n)(0)=A^n for n≤ d+1. If the first k return moments vanish, E^(k+2)(0)-A^k+2=BD^kG.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-51d61932ab8dcf5e","tex":"e^{(s+t)D}","aliases":"e^(s+t)D","display":"e^{(s+t)D}","kind":"use","excerpt":"frac ddte^tD=De^tD=e^tDD, e^(s+t)D=e^sDe^tD, and e^-tD is the inverse flow. Every actual curve satisfying z'=Dz+f with continuous f obeys z(t)=e^tDz(0)+∫_0^t e^(t-s)Df(s) ds. The initial value uniquely determines the homogeneous solution; two forced solutions with the same forcing and initial value also agree.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ca4a7e885b45520d","tex":"e^{(s+t)L}","aliases":"e^(s+t)L","display":"e^{(s+t)L}","kind":"use","excerpt":"For bounded linear observation maps K,J on bounded generator algebras, K(e^tL)=J(e^tD) (∀ t) iff K(L^n)=J(D^n) (∀ n). In particular, Be^tDG=0 for every t exactly when BD^nG=0 for every n. For CR=I, A=CLR, one has Ce^tLR=e^tA (∀ t) iff CL^nR=A^n (∀ n), and, with Q=I-RC, Ce^(s+t)LR-(Ce^sLR)(Ce^tLR)=Ce^sLQe^tLR.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-150c83a53ca293d8","tex":"e^{(t-s)A}","aliases":"e^(t-s)A","display":"e^{(t-s)A}","kind":"use","excerpt":"One has the operator identity E(t)=∫_0^t e^(t-s)Aδ e^sL ds, and δ=0 exactly when E(t)=0 for every t. Also E(s+t)=E(s)e^tL+e^sAE(t). For a second representation p:Z→ W with generator B, δ_pq=pδ_q+δ_pq, E_pq(t)=pE_q(t)+E_p(t)q.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-7cbbd3a218210e8d","tex":"e^{(t-s)D}","aliases":"e^(t-s)D","display":"e^{(t-s)D}","kind":"use","excerpt":"frac ddte^tD=De^tD=e^tDD, e^(s+t)D=e^sDe^tD, and e^-tD is the inverse flow. Every actual curve satisfying z'=Dz+f with continuous f obeys z(t)=e^tDz(0)+∫_0^t e^(t-s)Df(s) ds. The initial value uniquely determines the homogeneous solution; two forced solutions with the same forcing and initial value also agree.","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f7ad3235ab5e2717","tex":"e^{-rt}","aliases":"e^-rt","display":"e^{-rt}","kind":"use","excerpt":"If nne^tL≤ M_L and nne^tA≤ M_Ae^-rt with r>0, then nnE(t)≤fracM_LM_Aϵr, nnF(t)≤η M_L+nn VfracM_LM_Aϵr. For arbitrary x,z, nnOe^tLx-Ve^tAz · ≤(η M_L+nn VfracM_LM_Aϵr)nn x","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c4a97627ee2a139a","tex":"e^{-tD}","aliases":"e^-tD","display":"e^{-tD}","kind":"use","excerpt":"frac ddte^tD=De^tD=e^tDD, e^(s+t)D=e^sDe^tD, and e^-tD is the inverse flow. Every actual curve satisfying z'=Dz+f with continuous f obeys z(t)=e^tDz(0)+∫_0^t e^(t-s)Df(s) ds. The initial value uniquely determines the homogeneous solution; two forced solutions with the same forcing and initial value also agree.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-7f59d6c54c6ca3e9","tex":"e^{as}","aliases":"e^as","display":"e^{as}","kind":"use","excerpt":"smallHypotheses. Real Banach spaces X,Z; bounded autonomous real generators L,A and bounded representation/readouts q,O,V; real rates a,b of either sign; t≥0 and M_L,M_A,ϵ,η≥0. For every s≥0, nne^sL≤ M_Le^as and nne^sA≤ M_Ae^bs; also nnqL-Aq≤ϵ and nnO-Vq≤η.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4a1d25b2a3b14026","tex":"e^{at}","aliases":"e^at","display":"e^{at}","kind":"use","excerpt":"For nnδ≤ϵ, put R(a,b,t)=∫_0^t e^b(t-s)e^as ds =(e^at-e^bt)/(a-b), · a≠ b, te^at, · a=b. Then nnE(t)≤ M_LM_Aϵ R(a,b,t). If nnO-Vq≤η, the output defect F(t)=Oe^tL-Ve^tAq obeys nnF(t)≤η M_Le^at+nn V M_LM_Aϵ R(a,b,t).","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f137872a071bd805","tex":"e^{b(t-s)}","aliases":"e^b(t-s)","display":"e^{b(t-s)}","kind":"use","excerpt":"For nnδ≤ϵ, put R(a,b,t)=∫_0^t e^b(t-s)e^as ds =(e^at-e^bt)/(a-b), · a≠ b, te^at, · a=b. Then nnE(t)≤ M_LM_Aϵ R(a,b,t). If nnO-Vq≤η, the output defect F(t)=Oe^tL-Ve^tAq obeys nnF(t)≤η M_Le^at+nn V M_LM_Aϵ R(a,b,t).","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9a6e3b1236136b41","tex":"e^{bs}","aliases":"e^bs","display":"e^{bs}","kind":"use","excerpt":"smallHypotheses. Real Banach spaces X,Z; bounded autonomous real generators L,A and bounded representation/readouts q,O,V; real rates a,b of either sign; t≥0 and M_L,M_A,ϵ,η≥0. For every s≥0, nne^sL≤ M_Le^as and nne^sA≤ M_Ae^bs; also nnqL-Aq≤ϵ and nnO-Vq≤η.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-25ed668f8cf6d047","tex":"e^{bt}","aliases":"e^bt","display":"e^{bt}","kind":"use","excerpt":"For nnδ≤ϵ, put R(a,b,t)=∫_0^t e^b(t-s)e^as ds =(e^at-e^bt)/(a-b), · a≠ b, te^at, · a=b. Then nnE(t)≤ M_LM_Aϵ R(a,b,t). If nnO-Vq≤η, the output defect F(t)=Oe^tL-Ve^tAq obeys nnF(t)≤η M_Le^at+nn V M_LM_Aϵ R(a,b,t).","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-21367aaf2687869b","tex":"e^{sA}","aliases":"e^sA","display":"e^{sA}","kind":"use","excerpt":"One has the operator identity E(t)=∫_0^t e^(t-s)Aδ e^sL ds, and δ=0 exactly when E(t)=0 for every t. Also E(s+t)=E(s)e^tL+e^sAE(t). For a second representation p:Z→ W with generator B, δ_pq=pδ_q+δ_pq, E_pq(t)=pE_q(t)+E_p(t)q.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5bff803934ea9e7a","tex":"e^{sD}","aliases":"e^sD","display":"e^{sD}","kind":"use","excerpt":"frac ddte^tD=De^tD=e^tDD, e^(s+t)D=e^sDe^tD, and e^-tD is the inverse flow. Every actual curve satisfying z'=Dz+f with continuous f obeys z(t)=e^tDz(0)+∫_0^t e^(t-s)Df(s) ds. The initial value uniquely determines the homogeneous solution; two forced solutions with the same forcing and initial value also agree.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-cca7a466f6552b96","tex":"e^{sL}","aliases":"e^sL","display":"e^{sL}","kind":"use","excerpt":"For bounded linear observation maps K,J on bounded generator algebras, K(e^tL)=J(e^tD) (∀ t) iff K(L^n)=J(D^n) (∀ n). In particular, Be^tDG=0 for every t exactly when BD^nG=0 for every n. For CR=I, A=CLR, one has Ce^tLR=e^tA (∀ t) iff CL^nR=A^n (∀ n), and, with Q=I-RC, Ce^(s+t)LR-(Ce^sLR)(Ce^tLR)=Ce^sLQe^tLR.","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0bc0eec613816bce","tex":"e^{tA}","aliases":"e^tA","display":"e^{tA}","kind":"use","excerpt":"For bounded linear observation maps K,J on bounded generator algebras, K(e^tL)=J(e^tD) (∀ t) iff K(L^n)=J(D^n) (∀ n). In particular, Be^tDG=0 for every t exactly when BD^nG=0 for every n. For CR=I, A=CLR, one has Ce^tLR=e^tA (∀ t) iff CL^nR=A^n (∀ n), and, with Q=I-RC, Ce^(s+t)LR-(Ce^sLR)(Ce^tLR)=Ce^sLQe^tLR.","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9b6d0b4fad11116f","tex":"e^{tD}","aliases":"e^tD","display":"e^{tD}","kind":"use","excerpt":"frac ddte^tD=De^tD=e^tDD, e^(s+t)D=e^sDe^tD, and e^-tD is the inverse flow. Every actual curve satisfying z'=Dz+f with continuous f obeys z(t)=e^tDz(0)+∫_0^t e^(t-s)Df(s) ds. The initial value uniquely determines the homogeneous solution; two forced solutions with the same forcing and initial value also agree.","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e7ca070f23ef9e5b","tex":"e^{tL}","aliases":"e^tL","display":"e^{tL}","kind":"use","excerpt":"Let L be a bounded real operator on a Banach space. Its flow is the actual operator exponential e^tL. Boundedness allows differentiation of the norm-convergent exponential series and integration of continuous operator curves. Equalities in this section hold for all real t, using oriented interval integrals when t<0.","count":16},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-84d9cf7fe723f14a","tex":"e^u","aliases":"e^u","display":"e^u","kind":"use","excerpt":"The scalar exponential is the actual flow by the homogeneous ODE uniqueness theorem. For a>b, the response formula immediately gives the equality; for a=b, both sides vanish. In the second example, the generator defect is multiplication by ϵ, while the evolution defect is multiplication by e^ϵ t-1. The elementary exponential inequality e^u≥1+u for ","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d4dca010cab0ce47","tex":"e^{\\epsilon t}","aliases":"e^ϵ t epsilon","display":"e^{\\epsilon t}","kind":"use","excerpt":"For scalar generators a≥ b, |e^at-e^bt|=(a-b)R(a,b,t), so the response estimate is attained. For every ϵ>0, choosing L=ϵ I, A=0, q=I gives nnδ=ϵ, nnE(t)=e^ϵ t-1≥ϵ t. Its evolution defect exceeds every fixed bound at some nonnegative time.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"[Delayed return and disconnected return] On Q^3, set T(y,z_1,z_2)=(z_1,z_2,y) and retain y. Then A=0, M_0=0, M_1=I, and E_1=E_2=0 while E_3=I. Two shallow exact checks cannot certify all depths without the dimension budget. For another system, let A=D=I, B(z_1,z_2)=z_1, and G(y)=(0,y). Both couplings are nonzero but every BD^kG vanishes. Prepared c","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-82c2d872ef715d1f","tex":"E_3","aliases":"E_3","display":"E_3","kind":"use","excerpt":"[Delayed return and disconnected return] On Q^3, set T(y,z_1,z_2)=(z_1,z_2,y) and retain y. Then A=0, M_0=0, M_1=I, and E_1=E_2=0 while E_3=I. Two shallow exact checks cannot certify all depths without the dimension budget. For another system, let A=D=I, B(z_1,z_2)=z_1, and G(y)=(0,y). Both couplings are nonzero but every BD^kG vanishes. Prepared c","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"If λ_i>0 tends to zero and nne_i/sqrtλ_i→0, then B_λ_i(Ag+e_i)⟶ P_(ker A)^⊥g. The same limit holds for computed x_i when additionally fracnn(A^*A+λ_iI)x_i-A^*(Ag+e_i)λ_i→0. For example, λ_i=s_i^2, s_i>0→0, and nne_i≤ s_i^2 give the exact-solve consistency condition.","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3322a5f94182b5f5","tex":"E_{i::w}","aliases":"E_i::w","display":"E_{i::w}","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-69b76e4f5fee61c8","tex":"E_{k+1}","aliases":"E_k+1","display":"E_{k+1}","kind":"use","excerpt":"If all memory maps vanish, the recurrence in SR-05 reduces to E_n+1=AE_n, proving E_n=A^n by induction. Conversely, suppose exact powers hold. Induct on k. In the recurrence for E_k+2, every term except M_kE_0 involves a previously vanishing memory map. Thus E_k+2=AE_k+1+M_k, forcing M_k=0. The same induction restricted to k<N proves the finite-pre","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-643a0628c5800ac8","tex":"E_{k+2}","aliases":"E_k+2","display":"E_{k+2}","kind":"use","excerpt":"The following conditions are equivalent: E_n=A^n (∀ n), M_k=0 (∀ k), E_m+n=E_mE_n (∀ m,n). More precisely, E_n=A^n (n≤ N+1)iff M_k=0 (k<N). If M_j=0 for j<k, then E_k+2-A^k+2=M_k. Closure for every initial hidden state is equivalent to B=0.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e749405022a66e6b","tex":"E_m","aliases":"E_m","display":"E_m","kind":"use","excerpt":"The following conditions are equivalent: E_n=A^n (∀ n), M_k=0 (∀ k), E_m+n=E_mE_n (∀ m,n). More precisely, E_n=A^n (n≤ N+1)iff M_k=0 (k<N). If M_j=0 for j<k, then E_k+2-A^k+2=M_k. Closure for every initial hidden state is equivalent to B=0.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-93d8ef4727e39bde","tex":"E_{m+n}","aliases":"E_m+n","display":"E_{m+n}","kind":"use","excerpt":"The following conditions are equivalent: E_n=A^n (∀ n), M_k=0 (∀ k), E_m+n=E_mE_n (∀ m,n). More precisely, E_n=A^n (n≤ N+1)iff M_k=0 (k<N). If M_j=0 for j<k, then E_k+2-A^k+2=M_k. Closure for every initial hidden state is equivalent to B=0.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-da2ce42dc70bb3da","tex":"E_n","aliases":"E_n","display":"E_n","kind":"use","excerpt":"Let V be visible state and H hidden state. Over a commutative ring, specify four maps and form the actual full dynamics T(y,z)=(Ay+Bz,Gy+Dz). Put C(y,z)=y, Ry=(y,0), E_n=CT^nR, and M_k=BD^kG. The return M_k contains entry into H, k hidden transitions, and return to V. Its first effect occurs after k+2 full transitions.","count":8},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8448bff242e062e8","tex":"E_{n+1}","aliases":"E_n+1","display":"E_{n+1}","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0dd6372c1de2074b","tex":"E_p","aliases":"E_p","display":"E_p","kind":"use","excerpt":"One has the operator identity E(t)=∫_0^t e^(t-s)Aδ e^sL ds, and δ=0 exactly when E(t)=0 for every t. Also E(s+t)=E(s)e^tL+e^sAE(t). For a second representation p:Z→ W with generator B, δ_pq=pδ_q+δ_pq, E_pq(t)=pE_q(t)+E_p(t)q.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-97ee1bda129eb998","tex":"E_{pq}","aliases":"E_pq","display":"E_{pq}","kind":"use","excerpt":"One has the operator identity E(t)=∫_0^t e^(t-s)Aδ e^sL ds, and δ=0 exactly when E(t)=0 for every t. Also E(s+t)=E(s)e^tL+e^sAE(t). For a second representation p:Z→ W with generator B, δ_pq=pδ_q+δ_pq, E_pq(t)=pE_q(t)+E_p(t)q.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b3f1fc86e79071cb","tex":"E_q","aliases":"E_q","display":"E_q","kind":"use","excerpt":"One has the operator identity E(t)=∫_0^t e^(t-s)Aδ e^sL ds, and δ=0 exactly when E(t)=0 for every t. Also E(s+t)=E(s)e^tL+e^sAE(t). For a second representation p:Z→ W with generator B, δ_pq=pδ_q+δ_pq, E_pq(t)=pE_q(t)+E_p(t)q.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9438a38b00614be2","tex":"E_u","aliases":"E_u","display":"E_u","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-37fb925f37df4d2b","tex":"E_{u\\mathbin{+\\!+}v}","aliases":"E_umathbin+ +v mathbin","display":null,"kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0796f4b2ba69fba8","tex":"E_v","aliases":"E_v","display":"E_v","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4c336ec08d156ec7","tex":"E_w","aliases":"E_w","display":"E_w","kind":"use","excerpt":"Let q:X→ Z be a proposed representation, with true operations T_i, reduced operations U_i, and readouts O:X→ Y, V:Z→ Y. Define the step and word defects δ_i=qT_i-U_iq, E_w=qT_w-U_wq. Algebraic identities require only the stated commutative-ring structure. Quantitative estimates below use continuous linear maps over a nontrivially normed field and t","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-44985fac8d8e1583","tex":"E_X","aliases":"E_X","display":"E_X","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":10},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-00a67cdd394dbd2e","tex":"E_X^2","aliases":"E_X^2","display":"E_X^2","kind":"use","excerpt":"E_X^2=E_X, and E_Xv=v exactly when v∈Ran a_X. Moreover D K(T)=T, K D(K)=E_YKE_X. Consequently a compatible K is the encoding of a unique map T:X→ Y. Encoding preserves composition. Its identity on resolved coordinates is E_X.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-737e7b2e93461476","tex":"E_Y","aliases":"E_Y","display":"E_Y","kind":"use","excerpt":"contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator compatible whe","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-74e2d51d64f7fad4","tex":"E_\\lambda","aliases":"E_λ lambda","display":"E_\\lambda","kind":"use","excerpt":"Let E_λ=B_λ A-P_(ker A)^⊥. Always nnE_λ≤1. If Ran A is nonclosed, then nnE_λ=1 for every positive parameter. For a positive family tending to zero along a nontrivial filter, E_λ→0 in operator normiffRan A is closed. Under source coercivity α>0, nnE_λ≤λ/(α^2+λ).","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3724a04b48e23949","tex":"E_\\varnothing","aliases":"E_∅ varnothing","display":"E_\\varnothing","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"Commuting presentation maps LA=A'F and δ'L=Mδ induce a map overline L: H→ H' by [v]↦[Lv]. These maps compose and preserve zero classes. Reflection of zero follows when overline L is injective.","count":26},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"frac ddte^tD=De^tD=e^tDD, e^(s+t)D=e^sDe^tD, and e^-tD is the inverse flow. Every actual curve satisfying z'=Dz+f with continuous f obeys z(t)=e^tDz(0)+∫_0^t e^(t-s)Df(s) ds. The initial value uniquely determines the homogeneous solution; two forced solutions with the same forcing and initial value also agree.","count":16},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"use","excerpt":"For recovery assume G,V are real Hilbert spaces, A is a fixed bounded true operator, τ≥0 is fixed, and p∈(ker A)^⊥ is fixed. Retain the whole calibration bounds and the true data bounds nnAp-v_i≤η_i. Put δ_i=(η_rm cal,i+f_i)nn R. If the true data, calibration-truth and model-fit allowances tend to zero, then δ_i→0. Generalized accepted sources usin","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-832e56d8bd1b7868","tex":"F_\\lambda","aliases":"F_λ lambda","display":"F_\\lambda","kind":"use","excerpt":"For F_λ(v,x)=nnAx-v^2+λnn x^2, F_λ(v,x)=F_λ(v,x_λ) +nnA(x-x_λ)^2+λnnx-x_λ^2. Thus x_λ is the unique minimizer and lies in (ker A)^⊥. The whole-map bounds are nnR_λ≤1/λ, nnB_λ≤frac12sqrtλ, nnAB_λ v≤nn v, nnAB_λ v-v≤nn v. For r=N_λ x-A^*v, one has x-x_λ=R_λ r and nnx-x_λ≤nn r/λ.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"smallHypotheses. Commutative-ring modules and a linear presentation GxrightarrowAVxrightarrowδW with δ A=0; a compatible vector v∈kerδ.","count":34},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"For δ A=0, define K=kerδ, R=Ran A⊆ K, and H=K/R. A compatible v has class zero exactly when v=Ag for some g. All classes vanish exactly when R=K. If A is injective, a zero class has a unique source.","count":103},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","excerpt":"Objective minimality gives r_v(λ)^2+λnnB_λ v^2 ≤nnAg-v^2+λnn g^2. The residual ordering and positive λ imply norm domination. For the source estimate, expand nnx-p^2 and use nn x≤nn p to obtain nnx-p^2≤2ip pp-x. Write p=(p-A^*w)+A^*w. The first pairing is at most nnp-A^*wnnp-x≤2nn pnnp-A^*w; the adjoint identity bounds the second by nn wnnAp-Ax. Th","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"[Source and image are different optima] Let A(g_1,g_2)=(g_1+g_2,0,0) and v=(3,4,5). The best image is (3,0,0), with the same squared local error 41. The minimum-norm source is (3/2,3/2) of squared norm 9/2. The source (3,0) gives the same best image but has squared norm 9. Their difference (3/2,-3/2) is kernel ambiguity, not local obstruction.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e6df94744f7ade62","tex":"g_2","aliases":"g_2","display":"g_2","kind":"use","excerpt":"[Source and image are different optima] Let A(g_1,g_2)=(g_1+g_2,0,0) and v=(3,4,5). The best image is (3,0,0), with the same squared local error 41. The minimum-norm source is (3/2,3/2) of squared norm 9/2. The source (3,0) gives the same best image but has squared norm 9. Their difference (3/2,-3/2) is kernel ambiguity, not local obstruction.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-15e892a32e1d61ee","tex":"g_n","aliases":"g_n","display":"g_n","kind":"use","excerpt":"Closed range gives the bounded constructor and coercivity by SR-29--SR-31. Conversely suppose coercivity holds. If Ag_n converges in V, replace g_n by P_Sg_n without changing its image. Coercivity makes this source sequence Cauchy. Completeness gives a limit g∈ S, and continuity gives the image limit Ag∈Ran A. Hence the range is closed. If bounded ","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"Let V be visible state and H hidden state. Over a commutative ring, specify four maps and form the actual full dynamics T(y,z)=(Ay+Bz,Gy+Dz). Put C(y,z)=y, Ry=(y,0), E_n=CT^nR, and M_k=BD^kG. The return M_k contains entry into H, k hidden transitions, and return to V. Its first effect occurs after k+2 full transitions.","count":32},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"For δ A=0, define K=kerδ, R=Ran A⊆ K, and H=K/R. A compatible v has class zero exactly when v=Ag for some g. All classes vanish exactly when R=K. If A is injective, a zero class has a unique source.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"[A noncommuting minimal model] On Q^3, let O(x,y,h)=x, T_A(x,y,h)=(y,0,h), and T_B(x,y,h)=(0,x,h). The two operations do not commute: on (1,0,0) their two compositions give (1,0,0) and (0,0,0). The map q(x,y,h)=(x,y) is generated by O and OT_A, intertwines both operations, and factors O. It therefore has a depth-one certificate. The hidden coordina","count":20},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-936b3a65aaba44d8","tex":"\\mathcal H'","aliases":"H' mathcal","display":"\\mathcal H'","kind":"use","excerpt":"Commuting presentation maps LA=A'F and δ'L=Mδ induce a map overline L: H→ H' by [v]↦[Lv]. These maps compose and preserve zero classes. Reflection of zero follows when overline L is injective.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"For H_0(a,b)=0 and H_n+1(a,b)=aH_n(a,b)+b^n, H_n(a,b)=∑_j=0^n-1a^jb^n-1-j, (a-b)H_n=a^n-b^n. If a≠ b, H_n=(a^n-b^n)/(a-b); for equal rates, H_n+1(a,a)=(n+1)a^n. Uniform step bounds give nnE_w≤ϵ H_n(a,b), nnOT_w-VU_wq≤η a^n+nn Vϵ H_n(a,b).","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0231a655bf13a98d","tex":"H_k","aliases":"H_k","display":"H_k","kind":"use","excerpt":"Let H_kx=(OT_wx)_|w|≤ k and Z_k=Ran H_k. Then ker H_k=N and X/N≅ Z_k. The formulas U_i(H_kx)=H_kT_ix, V(H_kx)=Ox define actual linear maps on Z_k and preserve every finite future word. This model has minimum exact dimension.","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"use","excerpt":"For H_0(a,b)=0 and H_n+1(a,b)=aH_n(a,b)+b^n, H_n(a,b)=∑_j=0^n-1a^jb^n-1-j, (a-b)H_n=a^n-b^n. If a≠ b, H_n=(a^n-b^n)/(a-b); for equal rates, H_n+1(a,a)=(n+1)a^n. Uniform step bounds give nnE_w≤ϵ H_n(a,b), nnOT_w-VU_wq≤η a^n+nn Vϵ H_n(a,b).","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8ff15c0817c304f2","tex":"H_{n+1}","aliases":"H_n+1","display":"H_{n+1}","kind":"use","excerpt":"For H_0(a,b)=0 and H_n+1(a,b)=aH_n(a,b)+b^n, H_n(a,b)=∑_j=0^n-1a^jb^n-1-j, (a-b)H_n=a^n-b^n. If a≠ b, H_n=(a^n-b^n)/(a-b); for equal rates, H_n+1(a,a)=(n+1)a^n. Uniform step bounds give nnE_w≤ϵ H_n(a,b), nnOT_w-VU_wq≤η a^n+nn Vϵ H_n(a,b).","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2d64df6fa5868f4c","tex":"H_r","aliases":"H_r","display":"H_r","kind":"use","excerpt":"For a finite operation alphabet, suppose a proposed representation satisfies q=F H_r, qT_i=U_iq, Vq=O. Then ker q=N, regardless of whether r reaches the a priori horizon. If q is surjective, it gives an equivalence Z≅ X/N preserving the observation and intertwining every operation. Without surjectivity, the equivalence applies to Ran q.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-82c1b691fedf0351","tex":"H_{|w|}","aliases":"H_|w|","display":"H_{|w|}","kind":"use","excerpt":"ld sum by a and adjoining b^n generates every term of the new sum. Multiplication by a-b telescopes, leaving a^n-b^n. Division is used only when the rates are distinct. If they coincide, all n+1 terms of H_n+1 equal a^n, including the zero-rate boundary with the usual a^0=1. For uniform bounds, the heterogeneous budget becomes D_i::w=aD_w+b^|w|ϵ, s","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":40},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":110},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-fc66c01638c40151","tex":"i^2","aliases":"i^2","display":"i^2","kind":"use","excerpt":"The components lie in R, K∩ R^⊥, and K^⊥, respectively, are pairwise orthogonal, and give the unique decomposition v=r(v)+o(v)+i(v). Furthermore, nn v^2=nn r^2+nn o^2+nn i^2, dist(v,R)^2=nn o^2+nn i^2. For any u∈ R, nnv-u^2=dist(v,R)^2+nnr(v)-u^2. Thus r(v) is the unique nearest represented vector.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-fa7f3bce1a687b82","tex":"I_G","aliases":"I_G","display":"I_G","kind":"use","excerpt":"The final link connects justified calibration measurements with the whole-operator premise used above. It does not infer the calibration truth allowance from the model's own fit. Let K be a real normed calibration coordinate space, G a real normed source space and V a real normed response space. A split frame has bounded synthesis U:K→ G and analys","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0625c4dc07eb960a","tex":"i_j","aliases":"i_j","display":"i_j","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-33548fbd7d20a181","tex":"i_{j+1}","aliases":"i_j+1","display":"i_{j+1}","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a2a252b49f986e1b","tex":"i_{j-1}","aliases":"i_j-1","display":"i_{j-1}","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-cf48159a73d856b5","tex":"i_n","aliases":"i_n","display":"i_n","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2d1fd7dc4ba460e1","tex":"I_{\\RR^2}","aliases":"I_RR^2 I_ R^2 RR mathbb","display":"I_{{\\mathbb R}^2}","kind":"use","excerpt":"The vector (0,1) is outside the range of U, proving nonsurjectivity. A split reconstruction UR=I_ R^2 would make every source a range vector, so no such reconstruction exists here. Nor can any approximate reconstruction for this synthesis have whole defect below one: the range of UR has second coordinate zero, and therefore (I-UR)(0,1) has second c","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ab3ef27be5e3d7f7","tex":"I_Y","aliases":"I_Y","display":"I_Y","kind":"use","excerpt":"The splitting gives E_X^2=a_X(s_Xa_X)s_X=a_Xs_X. If E_Xv=v, then v=a_X(s_Xv); conversely E_Xa_Xx=a_Xx. Substituting the definitions yields s_Y(a_YTs_X)a_X=T, a_Y(s_YKa_X)s_X=E_YKE_X. For compatible K, take T= D(K). If another T' encodes to K, decoding proves T'= D(K). Finally, an intermediate resolution of Y has s_Ya_Y=I_Y, so K(S) K(T)=a_ZS(s_Ya_Y","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-bdedc0251851d395","tex":"I_Z","aliases":"I_Z","display":"I_Z","kind":"use","excerpt":"smallHypotheses. Commutative-ring modules; linear C:X→ Z, R:Z→ X with CR=I_Z; arbitrary endomorphisms S,T,U of X.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"The chain condition puts the actual range of A inside K. The zero class criterion for a quotient is membership in the submodule divided out, giving [v]=0iff v∈ Riff∃ g,Ag=v. Every compatible class is zero exactly when K⊆ R; the reverse inclusion is already known. Injectivity of A gives uniqueness of any preimage.","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c0031bdf2737d3ca","tex":"\\ip","aliases":"ip","display":null,"kind":"use","excerpt":"N_λ is a bounded bijection with bounded inverse R_λ. For x_λ=B_λ v, N_λ x_λ=A^*v, A^*(Ax_λ-v)+λ x_λ=0, and nnAx_λ^2+λnnx_λ^2 =ip vAx_λ.","count":10},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"For bounded linear observation maps K,J on bounded generator algebras, K(e^tL)=J(e^tD) (∀ t) iff K(L^n)=J(D^n) (∀ n). In particular, Be^tDG=0 for every t exactly when BD^nG=0 for every n. For CR=I, A=CLR, one has Ce^tLR=e^tA (∀ t) iff CL^nR=A^n (∀ n), and, with Q=I-RC, Ce^(s+t)LR-(Ce^sLR)(Ce^tLR)=Ce^sLQe^tLR.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":64},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"Let V be visible state and H hidden state. Over a commutative ring, specify four maps and form the actual full dynamics T(y,z)=(Ay+Bz,Gy+Dz). Put C(y,z)=y, Ry=(y,0), E_n=CT^nR, and M_k=BD^kG. The return M_k contains entry into H, k hidden transitions, and return to V. Its first effect occurs after k+2 full transitions.","count":61},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"ABv=P_Rv, so SR-25's candidate identity with u=Ag proves the first formula. Equality to the minimum image error is equivalent to A(g-Bv)=0, yielding exactly the affine family of minimizers. Since Bv∈ S and k∈ N_A, Pythagoras gives nnBv+k^2=nnBv^2+nn k^2. It proves minimum norm and uniqueness. The identity BA=P_S proves the exact recovery criterion.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3ff8bbec26df05be","tex":"K^\\perp","aliases":"K^⊥ perp","display":"K^\\perp","kind":"use","excerpt":"The components lie in R, K∩ R^⊥, and K^⊥, respectively, are pairwise orthogonal, and give the unique decomposition v=r(v)+o(v)+i(v). Furthermore, nn v^2=nn r^2+nn o^2+nn i^2, dist(v,R)^2=nn o^2+nn i^2. For any u∈ R, nnv-u^2=dist(v,R)^2+nnr(v)-u^2. Thus r(v) is the unique nearest represented vector.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1bdfdabfb7ccb3ec","tex":"k_1","aliases":"k_1","display":"k_1","kind":"use","excerpt":"Redundancy causes no logical problem. For instance, let K= R^2, G= R, U(k_1,k_2)=k_1 and Rg=(g,0), with the standard product norm. Then UR=I, nn U=nn R=1, but U has a nontrivial kernel. The frame spans the source even though it has an unused coordinate. A true scalar map A=a and model C=c with measured row Y(k_1,k_2)=yk_1 give |y-a|≤η_rm cal and |y","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a8cd867fa822df3a","tex":"k_2","aliases":"k_2","display":"k_2","kind":"use","excerpt":"Redundancy causes no logical problem. For instance, let K= R^2, G= R, U(k_1,k_2)=k_1 and Rg=(g,0), with the standard product norm. Then UR=I, nn U=nn R=1, but U has a nontrivial kernel. The frame spans the source even though it has an unused coordinate. A true scalar map A=a and model C=c with measured row Y(k_1,k_2)=yk_1 give |y-a|≤η_rm cal and |y","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"smallHypotheses. Commutative-ring modules and a linear presentation GxrightarrowAVxrightarrowδW with δ A=0; a compatible vector v∈kerδ.","count":49},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"The observed response Y:K→ V is an operator, and the assumptions nnY-AU≤η_rm cal and nnY-CU≤ f are operator-norm enclosures. Individual calibration samples do not automatically give these enclosures. A mathematical or experimental justification of η_rm cal remains a separate input.","count":12},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-478acc5f340c0573","tex":"L^n","aliases":"L^n","display":"L^n","kind":"use","excerpt":"For bounded linear observation maps K,J on bounded generator algebras, K(e^tL)=J(e^tD) (∀ t) iff K(L^n)=J(D^n) (∀ n). In particular, Be^tDG=0 for every t exactly when BD^nG=0 for every n. For CR=I, A=CLR, one has Ce^tLR=e^tA (∀ t) iff CL^nR=A^n (∀ n), and, with Q=I-RC, Ce^(s+t)LR-(Ce^sLR)(Ce^tLR)=Ce^sLQe^tLR.","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"Commuting presentation maps LA=A'F and δ'L=Mδ induce a map overline L: H→ H' by [v]↦[Lv]. These maps compose and preserve zero classes. Reflection of zero follows when overline L is injective.","count":55},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"The following conditions are equivalent: E_n=A^n (∀ n), M_k=0 (∀ k), E_m+n=E_mE_n (∀ m,n). More precisely, E_n=A^n (n≤ N+1)iff M_k=0 (k<N). If M_j=0 for j<k, then E_k+2-A^k+2=M_k. Closure for every initial hidden state is equivalent to B=0.","count":12},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","excerpt":"[Delayed return and disconnected return] On Q^3, set T(y,z_1,z_2)=(z_1,z_2,y) and retain y. Then A=0, M_0=0, M_1=I, and E_1=E_2=0 while E_3=I. Two shallow exact checks cannot certify all depths without the dimension budget. For another system, let A=D=I, B(z_1,z_2)=z_1, and G(y)=(0,y). Both couplings are nonzero but every BD^kG vanishes. Prepared c","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"[Delayed return and disconnected return] On Q^3, set T(y,z_1,z_2)=(z_1,z_2,y) and retain y. Then A=0, M_0=0, M_1=I, and E_1=E_2=0 while E_3=I. Two shallow exact checks cannot certify all depths without the dimension budget. For another system, let A=D=I, B(z_1,z_2)=z_1, and G(y)=(0,y). Both couplings are nonzero but every BD^kG vanishes. Prepared c","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-63019db73ab92a7c","tex":"M_A","aliases":"M_A","display":"M_A","kind":"use","excerpt":"smallHypotheses. Real Banach spaces X,Z; bounded autonomous real generators L,A and bounded representation/readouts q,O,V; real rates a,b of either sign; t≥0 and M_L,M_A,ϵ,η≥0. For every s≥0, nne^sL≤ M_Le^as and nne^sA≤ M_Ae^bs; also nnqL-Aq≤ϵ and nnO-Vq≤η.","count":8},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-bf2227566ef624f5","tex":"M_j","aliases":"M_j","display":"M_j","kind":"use","excerpt":"The following conditions are equivalent: E_n=A^n (∀ n), M_k=0 (∀ k), E_m+n=E_mE_n (∀ m,n). More precisely, E_n=A^n (n≤ N+1)iff M_k=0 (k<N). If M_j=0 for j<k, then E_k+2-A^k+2=M_k. Closure for every initial hidden state is equivalent to B=0.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c9f29e7d9e7a175b","tex":"M_k","aliases":"M_k","display":"M_k","kind":"use","excerpt":"Let V be visible state and H hidden state. Over a commutative ring, specify four maps and form the actual full dynamics T(y,z)=(Ay+Bz,Gy+Dz). Put C(y,z)=y, Ry=(y,0), E_n=CT^nR, and M_k=BD^kG. The return M_k contains entry into H, k hidden transitions, and return to V. Its first effect occurs after k+2 full transitions.","count":10},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b600b58f986c2898","tex":"M_L","aliases":"M_L","display":"M_L","kind":"use","excerpt":"smallHypotheses. Real Banach spaces X,Z; bounded autonomous real generators L,A and bounded representation/readouts q,O,V; real rates a,b of either sign; t≥0 and M_L,M_A,ϵ,η≥0. For every s≥0, nne^sL≤ M_Le^as and nne^sA≤ M_Ae^bs; also nnqL-Aq≤ϵ and nnO-Vq≤η.","count":8},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-31810b4520930800","tex":"M_{n-1-j}","aliases":"M_n-1-j","display":"M_{n-1-j}","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"Suppose a record supplies λ>0, nonnegative ρ,ξ,κ,m,ω, and nnr_λ(x)^2≤ρ^2, ρ^2≤(λκ)^2, ρ^2≤4λξ^2, max(0,m-ω)^2≤nnAx-v^2≤(m+ω)^2, η+ω+ξ≤ m≤τη-ω-ξ . Then x is near accepted with source budget κ. Families of such records give SR-51 when the genuine data allowances and source budgets tend to zero.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Let N=⋂_wker(OT_w). It is the largest submodule contained in ker O and invariant under every T_i. The operations and observation descend to X/N, preserving every future observation. In particular, [x]=[y]iff OT_wx=OT_wy for every finite word w. If q:X→ Z satisfies qT_i=U_iq and Vq=O, then ker q⊆ N.","count":71},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":57},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ebc65dd86b43b2d4","tex":"N_0","aliases":"N_0","display":"N_0","kind":"use","excerpt":"N_0=ker O and N_r+1=ker O∩⋂_i T_i^-1N_r⊆ N_r. One plateau N_r+1=N_r implies N_r=N. In all cases N_k=N. Thus equality of future observations through word length k is equivalent to equality for every finite word.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-48a2e73ce7f173e0","tex":"N_A","aliases":"N_A","display":"N_A","kind":"use","excerpt":"Let A:G→ V be bounded between real Hilbert spaces, with actual closed range R=Ran A. Write N_A=ker A and S=N_A^⊥. Closedness is a substantive hypothesis for a bounded exact inverse. It will be removed for regularized recovery in the following sections.","count":17},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a7163b5efb8753a2","tex":"N_{A,\\lambda}","aliases":"N_A,λ lambda","display":"N_{A,\\lambda}","kind":"use","excerpt":"For two operators and parameters, R_A,λ-R_C,μ =R_A,λ(N_C,μ-N_A,λ)R_C,μ. At fixed positive λ, with ϵ=nnA-C, nnB_A,λ-B_C,λ ≤fracϵλ+ frac(nn A+nn C)ϵnn Cλ^2. If A has closed range and canonical inverse B, then B_λ v-Bv=-λ R_λ Bv. For δ A=0, nnv-AB_λ v^2 =nnc(v)^2+nni(v)^2+nnA(B_λ v-Bv)^2.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9103bc76a228e554","tex":"N_A^\\perp","aliases":"N_A^⊥ perp","display":"N_A^\\perp","kind":"use","excerpt":"Let A:G→ V be bounded between real Hilbert spaces, with actual closed range R=Ran A. Write N_A=ker A and S=N_A^⊥. Closedness is a substantive hypothesis for a bounded exact inverse. It will be removed for regularized recovery in the following sections.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4c0f6e46a06c4ce1","tex":"N_{C,\\mu}","aliases":"N_C,μ mu","display":"N_{C,\\mu}","kind":"use","excerpt":"For two operators and parameters, R_A,λ-R_C,μ =R_A,λ(N_C,μ-N_A,λ)R_C,μ. At fixed positive λ, with ϵ=nnA-C, nnB_A,λ-B_C,λ ≤fracϵλ+ frac(nn A+nn C)ϵnn Cλ^2. If A has closed range and canonical inverse B, then B_λ v-Bv=-λ R_λ Bv. For δ A=0, nnv-AB_λ v^2 =nnc(v)^2+nni(v)^2+nnA(B_λ v-Bv)^2.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9cc3cd93a260a3e9","tex":"N_k","aliases":"N_k","display":"N_k","kind":"use","excerpt":"N_0=ker O and N_r+1=ker O∩⋂_i T_i^-1N_r⊆ N_r. One plateau N_r+1=N_r implies N_r=N. In all cases N_k=N. Thus equality of future observations through word length k is equivalent to equality for every finite word.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-75599878a7100180","tex":"N_r","aliases":"N_r","display":"N_r","kind":"use","excerpt":"Now work over a field and, for dimension statements, assume dim X<∞. The allowed operations need not commute. Define N_r=⋂_|w|≤ rker(OT_w), k=dimker O. A finite horizon means bounded word length. It gives a finite enumeration of constraints only when the operation alphabet is finite.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-13a0a5c179de09fe","tex":"N_{r+1}","aliases":"N_r+1","display":"N_{r+1}","kind":"use","excerpt":"N_0=ker O and N_r+1=ker O∩⋂_i T_i^-1N_r⊆ N_r. One plateau N_r+1=N_r implies N_r=N. In all cases N_k=N. Thus equality of future observations through word length k is equivalent to equality for every finite word.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c42719b2f08b8d28","tex":"N_\\lambda","aliases":"N_λ lambda","display":"N_\\lambda","kind":"use","excerpt":"Fix a bounded operator A:G→ V between real Hilbert spaces, without assuming closed range. For λ>0, put N_λ=A^*A+λ I, R_λ=N_λ^-1, B_λ=R_λ A^*. The inverse is constructed from coercivity. It is not supplied as a certificate premise. Here A^* is the actual Hilbert adjoint.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-122c6c6d12ed19d3","tex":"N_\\lambda^{-1}","aliases":"N_λ^-1 lambda","display":"N_\\lambda^{-1}","kind":"use","excerpt":"Fix a bounded operator A:G→ V between real Hilbert spaces, without assuming closed range. For λ>0, put N_λ=A^*A+λ I, R_λ=N_λ^-1, B_λ=R_λ A^*. The inverse is constructed from coercivity. It is not supplied as a certificate premise. Here A^* is the actual Hilbert adjoint.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"smallHypotheses. Commutative-ring modules; a linear observation O:X→ Y and an arbitrary family of allowed endomorphisms T_i:X→ X; every finite chronological word is included.","count":42},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"Let R⊆ K be subspaces of a real inner-product space V, both admitting orthogonal projections. Define the actual maps r(v)=P_Rv, o(v)=(P_K-P_R)v, i(v)=(I-P_K)v. They distinguish realized data, compatible obstruction, and incompatibility. In a globalization presentation use R=Ran A, K=kerδ with δ A=0.","count":11},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a45c9d3015875693","tex":"o^2","aliases":"o^2","display":"o^2","kind":"use","excerpt":"The components lie in R, K∩ R^⊥, and K^⊥, respectively, are pairwise orthogonal, and give the unique decomposition v=r(v)+o(v)+i(v). Furthermore, nn v^2=nn r^2+nn o^2+nn i^2, dist(v,R)^2=nn o^2+nn i^2. For any u∈ R, nnv-u^2=dist(v,R)^2+nnr(v)-u^2. Thus r(v) is the unique nearest represented vector.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-cc45e09328b547a1","tex":"o_0","aliases":"o_0","display":"o_0","kind":"use","excerpt":"In finite dimension some σ>0 satisfies SR-27. Under δ A=0, ∃ c>0 ∀ v:dist(v,Ran A)≤ cnnδ v iffRan A=kerδ. For projected R_0⊆ R_1⊆ K in a fixed metric, put g(v)=(P_R_1-P_R_0)v. Then nno_0(v)^2=nno_1(v)^2+nng(v)^2, dist(v,R_0)^2=dist(v,R_1)^2+nng(v)^2. No uniform conditioning constant exists across all nonzero scalar compatibility maps.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e1d6eed537fd5981","tex":"o_1","aliases":"o_1","display":"o_1","kind":"use","excerpt":"In finite dimension some σ>0 satisfies SR-27. Under δ A=0, ∃ c>0 ∀ v:dist(v,Ran A)≤ cnnδ v iffRan A=kerδ. For projected R_0⊆ R_1⊆ K in a fixed metric, put g(v)=(P_R_1-P_R_0)v. Then nno_0(v)^2=nno_1(v)^2+nng(v)^2, dist(v,R_0)^2=dist(v,R_1)^2+nng(v)^2. No uniform conditioning constant exists across all nonzero scalar compatibility maps.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e4b45cd66b9c8f4e","tex":"\\overline L","aliases":"overline L overline","display":"\\overline L","kind":"use","excerpt":"Commuting presentation maps LA=A'F and δ'L=Mδ induce a map overline L: H→ H' by [v]↦[Lv]. These maps compose and preserve zero classes. Reflection of zero follows when overline L is injective.","count":8},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-000f4f5a235b04fa","tex":"P_{(\\ker A)^\\perp}","aliases":"P_(ker A)^⊥ ker perp","display":"P_{(\\ker A)^\\perp}","kind":"use","excerpt":"Write p=P_(ker A)^⊥g, k=P_ker Ag. Then B_λ Ag+λ R_λ g=g, B_λ(Ag+e)-g=-λ R_λ g+B_λ e, and nnB_λ(Ag+e)-g^2 =nn-λ R_λ p+B_λ e^2+nn k^2. If A has source coercivity α>0, this is at most (fracλα^2+λnn p +fracnn e2sqrtλ)^2+nn k^2. For a supplied source condition g=A^*Aw, nnB_λ Ag-g≤λnn w without requiring closed range.","count":8},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b60e29fadb7e1879","tex":"P_{\\cl\\Ran A}","aliases":"P_clRan A cl Ran operatorname operatorname","display":"P_{{\\operatorname{cl}}{\\operatorname{Ran}} A}","kind":"use","excerpt":"For arbitrary data v, AB_λ^Av=B_λ^A^*A^*v ⟶ P_clRan Av. Adding data noise e_i→0 preserves this fitted-data limit. If v∈clRan A∖Ran A, fitted data converge to v but the corresponding reconstructed source family has no norm limit in G.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c5ca4f817be17bab","tex":"P_K","aliases":"P_K","display":"P_K","kind":"use","excerpt":"Let R⊆ K be subspaces of a real inner-product space V, both admitting orthogonal projections. Define the actual maps r(v)=P_Rv, o(v)=(P_K-P_R)v, i(v)=(I-P_K)v. They distinguish realized data, compatible obstruction, and incompatibility. In a globalization presentation use R=Ran A, K=kerδ with δ A=0.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2d9d3fbb73ac4b3d","tex":"P_{\\ker A}","aliases":"P_ker A ker","display":"P_{\\ker A}","kind":"use","excerpt":"Write p=P_(ker A)^⊥g, k=P_ker Ag. Then B_λ Ag+λ R_λ g=g, B_λ(Ag+e)-g=-λ R_λ g+B_λ e, and nnB_λ(Ag+e)-g^2 =nn-λ R_λ p+B_λ e^2+nn k^2. If A has source coercivity α>0, this is at most (fracλα^2+λnn p +fracnn e2sqrtλ)^2+nn k^2. For a supplied source condition g=A^*Aw, nnB_λ Ag-g≤λnn w without requiring closed range.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f3be644b8cc93602","tex":"P_{N_A}","aliases":"P_N_A","display":"P_{N_A}","kind":"use","excerpt":"For every source g, nnv-Ag^2=dist(v,R)^2+nnA(g-Bv)^2. All least-squares minimizers are Bv+k with k∈ker A. Among them Bv is the unique minimum-norm source. Exact source recovery BAg=g holds exactly for g∈ S. For data Ag+e, nnB(Ag+e)-g^2=nnBe^2+nnP_N_Ag^2.","count":13},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6437e41273729f36","tex":"P_R","aliases":"P_R","display":"P_R","kind":"use","excerpt":"Let R⊆ K be subspaces of a real inner-product space V, both admitting orthogonal projections. Define the actual maps r(v)=P_Rv, o(v)=(P_K-P_R)v, i(v)=(I-P_K)v. They distinguish realized data, compatible obstruction, and incompatibility. In a globalization presentation use R=Ran A, K=kerδ with δ A=0.","count":18},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ea39b7400542f2b0","tex":"P_{R_0}","aliases":"P_R_0","display":"P_{R_0}","kind":"use","excerpt":"In finite dimension some σ>0 satisfies SR-27. Under δ A=0, ∃ c>0 ∀ v:dist(v,Ran A)≤ cnnδ v iffRan A=kerδ. For projected R_0⊆ R_1⊆ K in a fixed metric, put g(v)=(P_R_1-P_R_0)v. Then nno_0(v)^2=nno_1(v)^2+nng(v)^2, dist(v,R_0)^2=dist(v,R_1)^2+nng(v)^2. No uniform conditioning constant exists across all nonzero scalar compatibility maps.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-cc193fd9124b3188","tex":"P_{R_1}","aliases":"P_R_1","display":"P_{R_1}","kind":"use","excerpt":"In finite dimension some σ>0 satisfies SR-27. Under δ A=0, ∃ c>0 ∀ v:dist(v,Ran A)≤ cnnδ v iffRan A=kerδ. For projected R_0⊆ R_1⊆ K in a fixed metric, put g(v)=(P_R_1-P_R_0)v. Then nno_0(v)^2=nno_1(v)^2+nng(v)^2, dist(v,R_0)^2=dist(v,R_1)^2+nng(v)^2. No uniform conditioning constant exists across all nonzero scalar compatibility maps.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-010f131ed455d9fc","tex":"P_S","aliases":"P_S","display":"P_S","kind":"use","excerpt":"Restriction A_S:S→ R is a bounded linear bijection with bounded inverse. Define B=ι_S A_S^-1P_R. Then AB=P_R, BA=P_S, ABA=A, and BAB=B. Its two projections are symmetric, ker B=R^⊥, and Ran B=S. The best-image identity AB=P_R and the condition Ran B⊆ S uniquely determine B.","count":10},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-be612c9650c6dc84","tex":"P_{S_A}","aliases":"P_S_A","display":"P_{S_A}","kind":"use","excerpt":"Call x near accepted with budget κ when some exact accepted source y satisfies nnx-y≤κ. If nnAg-v_i≤η_i, η_i→0, and near-accepted x_i have κ_i→0, then x_i⟶ P_S_Ag, nnx_i-g⟶nnP_N_Ag. At every index nnP_N_Ax_i≤κ_i; therefore the computed true-kernel component tends to zero. No separate parameter schedule is assumed.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3f1b6dc519af96a7","tex":"\\prime","aliases":"prime","display":"\\prime","kind":"use","excerpt":"smallHypotheses. Real Banach space; bounded autonomous D; continuous forcing f; differentiable actual z with z^prime=Dz+f at every real time.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Set P=RC, Q=I-P, Φ(T)=CTR, and Δ(S,T)=Φ(ST)-Φ(S)Φ(T). Then Δ(S,T)=CSQTR, and associativity gives Δ(S,T)Φ(U)+Δ(ST,U) =Φ(S)Δ(T,U)+Δ(S,TU). For two successive split compressions, Δ_21(S,T)=Φ_2(Δ_1(S,T)) +Δ_2(Φ_1(S),Φ_1(T)).","count":18},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"Let N=⋂_wker(OT_w). It is the largest submodule contained in ker O and invariant under every T_i. The operations and observation descend to X/N, preserving every future observation. In particular, [x]=[y]iff OT_wx=OT_wy for every finite word w. If q:X→ Z satisfies qT_i=U_iq and Vq=O, then ker q⊆ N.","count":67},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4a77ffe628ff771a","tex":"q_n","aliases":"q_n","display":"q_n","kind":"use","excerpt":"Divide X^n by the monic p to obtain a remainder q_n of degree below deg p. Evaluation at D gives D^n=q_n(D) because p(D)=0. Writing q_n(D) as a finite linear combination of the powers D^k below deg p shows BD^nG=0. For finite-dimensional H, Cayley--Hamilton provides the monic characteristic polynomial of D with degree d. SR-06 then yields both disp","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ Q mathbb","display":"{\\mathbb Q}","kind":"use","excerpt":"[Delayed return and disconnected return] On Q^3, set T(y,z_1,z_2)=(z_1,z_2,y) and retain y. Then A=0, M_0=0, M_1=I, and E_1=E_2=0 while E_3=I. Two shallow exact checks cannot certify all depths without the dimension budget. For another system, let A=D=I, B(z_1,z_2)=z_1, and G(y)=(0,y). Both couplings are nonzero but every BD^kG vanishes. Prepared c","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-279caa0ce5bda7c4","tex":"\\QQ^3","aliases":"QQ^3 Q^3 QQ mathbb","display":"{\\mathbb Q}^3","kind":"use","excerpt":"[Delayed return and disconnected return] On Q^3, set T(y,z_1,z_2)=(z_1,z_2,y) and retain y. Then A=0, M_0=0, M_1=I, and E_1=E_2=0 while E_3=I. Two shallow exact checks cannot certify all depths without the dimension budget. For another system, let A=D=I, B(z_1,z_2)=z_1, and G(y)=(0,y). Both couplings are nonzero but every BD^kG vanishes. Prepared c","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"smallHypotheses. Commutative-ring modules; linear C:X→ Z, R:Z→ X with CR=I_Z; arbitrary endomorphisms S,T,U of X.","count":144},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"Now work over a field and, for dimension statements, assume dim X<∞. The allowed operations need not commute. Define N_r=⋂_|w|≤ rker(OT_w), k=dimker O. A finite horizon means bounded word length. It gives a finite enumeration of constraints only when the operation alphabet is finite.","count":58},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"use","excerpt":"For transport, if v∈ K, then δ'Lv=Mδ v=0. If v=Ag, then Lv=A'Fg lies in R'. Hence L descends to the actual quotients. Evaluating on representatives proves the composition law. A linear map sends zero to zero. If its induced quotient map is injective, overline L[v]=0=overline L[0] implies [v]=0. The commuting squares alone provide no such injectivit","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"The components lie in R, K∩ R^⊥, and K^⊥, respectively, are pairwise orthogonal, and give the unique decomposition v=r(v)+o(v)+i(v). Furthermore, nn v^2=nn r^2+nn o^2+nn i^2, dist(v,R)^2=nn o^2+nn i^2. For any u∈ R, nnv-u^2=dist(v,R)^2+nnr(v)-u^2. Thus r(v) is the unique nearest represented vector.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3582d47c95ec52c7","tex":"r^A","aliases":"r^A","display":"r^A","kind":"use","excerpt":"s cancel. Insert and subtract A^*Cx to obtain the displayed identity. Adjoint isometry gives nnA^*-C^*=nnA-C≤δ and nnA^*=nn A. Submultiplicativity therefore bounds its two terms by δnnCx-v and nn A δnn x. The triangle inequality gives nn A≤nn C+δ, yielding a bound written in terms of available-model quantities and the allowance. Finally apply nnr^A","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-536fcb6cfe63ade0","tex":"r^A_\\lambda","aliases":"r^A_λ lambda","display":"r^A_\\lambda","kind":"use","excerpt":"For the normal residuals formed with the same parameter, r^A_λ(x)-r^C_λ(x) =(A^*-C^*)(Cx-v)+A^*(A-C)x . If nnC-A≤δ, then nnr^A_λ(x)-r^C_λ(x) ≤δ(nnCx-v+(nn C+δ)nn x). Hence a computed-model normal allowance nnr^C_λ(x)≤ρ gives the corresponding true normal allowance by adding this expression.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2b687d85de872606","tex":"r^C","aliases":"r^C","display":"r^C","kind":"use","excerpt":"s cancel. Insert and subtract A^*Cx to obtain the displayed identity. Adjoint isometry gives nnA^*-C^*=nnA-C≤δ and nnA^*=nn A. Submultiplicativity therefore bounds its two terms by δnnCx-v and nn A δnn x. The triangle inequality gives nn A≤nn C+δ, yielding a bound written in terms of available-model quantities and the allowance. Finally apply nnr^A","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b74e686f4b56f2e5","tex":"r^C_\\lambda","aliases":"r^C_λ lambda","display":"r^C_\\lambda","kind":"use","excerpt":"For the normal residuals formed with the same parameter, r^A_λ(x)-r^C_λ(x) =(A^*-C^*)(Cx-v)+A^*(A-C)x . If nnC-A≤δ, then nnr^A_λ(x)-r^C_λ(x) ≤δ(nnCx-v+(nn C+δ)nn x). Hence a computed-model normal allowance nnr^C_λ(x)≤ρ gives the corresponding true normal allowance by adding this expression.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-34f304974ae3a8a9","tex":"R^\\perp","aliases":"R^⊥ perp","display":"R^\\perp","kind":"use","excerpt":"The components lie in R, K∩ R^⊥, and K^⊥, respectively, are pairwise orthogonal, and give the unique decomposition v=r(v)+o(v)+i(v). Furthermore, nn v^2=nn r^2+nn o^2+nn i^2, dist(v,R)^2=nn o^2+nn i^2. For any u∈ R, nnv-u^2=dist(v,R)^2+nnr(v)-u^2. Thus r(v) is the unique nearest represented vector.","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"In finite dimension some σ>0 satisfies SR-27. Under δ A=0, ∃ c>0 ∀ v:dist(v,Ran A)≤ cnnδ v iffRan A=kerδ. For projected R_0⊆ R_1⊆ K in a fixed metric, put g(v)=(P_R_1-P_R_0)v. Then nno_0(v)^2=nno_1(v)^2+nng(v)^2, dist(v,R_0)^2=dist(v,R_1)^2+nng(v)^2. No uniform conditioning constant exists across all nonzero scalar compatibility maps.","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"In finite dimension some σ>0 satisfies SR-27. Under δ A=0, ∃ c>0 ∀ v:dist(v,Ran A)≤ cnnδ v iffRan A=kerδ. For projected R_0⊆ R_1⊆ K in a fixed metric, put g(v)=(P_R_1-P_R_0)v. Then nno_0(v)^2=nno_1(v)^2+nng(v)^2, dist(v,R_0)^2=dist(v,R_1)^2+nng(v)^2. No uniform conditioning constant exists across all nonzero scalar compatibility maps.","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ac27582ad7afcb50","tex":"R_A","aliases":"R_A","display":"R_A","kind":"use","excerpt":"Multiply the difference of inverses on the left and right by their normal operators, or insert their inverse laws, to obtain the resolvent identity. SR-38 bounds its norm by nnN_C,μ-N_A,λ/(λμ). At equal parameters, C^*C-A^*A=(C^*-A^*)C+A^*(C-A), whose norm is at most (nn A+nn C)ϵ. Splitting R_AA^*-R_CC^*=R_A(A^*-C^*)+(R_A-R_C)C^* proves the stated ","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e9909e5a7891fe92","tex":"R_{A,\\lambda}","aliases":"R_A,λ lambda","display":"R_{A,\\lambda}","kind":"use","excerpt":"For two operators and parameters, R_A,λ-R_C,μ =R_A,λ(N_C,μ-N_A,λ)R_C,μ. At fixed positive λ, with ϵ=nnA-C, nnB_A,λ-B_C,λ ≤fracϵλ+ frac(nn A+nn C)ϵnn Cλ^2. If A has closed range and canonical inverse B, then B_λ v-Bv=-λ R_λ Bv. For δ A=0, nnv-AB_λ v^2 =nnc(v)^2+nni(v)^2+nnA(B_λ v-Bv)^2.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0f7be7718d9eb684","tex":"R_C","aliases":"R_C","display":"R_C","kind":"use","excerpt":"Multiply the difference of inverses on the left and right by their normal operators, or insert their inverse laws, to obtain the resolvent identity. SR-38 bounds its norm by nnN_C,μ-N_A,λ/(λμ). At equal parameters, C^*C-A^*A=(C^*-A^*)C+A^*(C-A), whose norm is at most (nn A+nn C)ϵ. Splitting R_AA^*-R_CC^*=R_A(A^*-C^*)+(R_A-R_C)C^* proves the stated ","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2df5125174d79186","tex":"R_{C,\\mu}","aliases":"R_C,μ mu","display":"R_{C,\\mu}","kind":"use","excerpt":"For two operators and parameters, R_A,λ-R_C,μ =R_A,λ(N_C,μ-N_A,λ)R_C,μ. At fixed positive λ, with ϵ=nnA-C, nnB_A,λ-B_C,λ ≤fracϵλ+ frac(nn A+nn C)ϵnn Cλ^2. If A has closed range and canonical inverse B, then B_λ v-Bv=-λ R_λ Bv. For δ A=0, nnv-AB_λ v^2 =nnc(v)^2+nni(v)^2+nnA(B_λ v-Bv)^2.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8acad0d31da451a8","tex":"R_E","aliases":"R_E","display":"R_E","kind":"use","excerpt":"For split frames (U_E,R_E):Jrightleftarrows K and (U_F,R_F):Krightleftarrows G, the composite is split, with analysis R_ER_F and nnR_ER_F≤nnR_EnnR_F. The corresponding calibration allowance is at most (η_rm cal+f)nnR_EnnR_F.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c9377ce9ef801522","tex":"R_F","aliases":"R_F","display":"R_F","kind":"use","excerpt":"For split frames (U_E,R_E):Jrightleftarrows K and (U_F,R_F):Krightleftarrows G, the composite is split, with analysis R_ER_F and nnR_ER_F≤nnR_EnnR_F. The corresponding calibration allowance is at most (η_rm cal+f)nnR_EnnR_F.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c16fbef0f6898c20","tex":"r_v","aliases":"r_v","display":"r_v","kind":"use","excerpt":"Still fix a known bounded A between real Hilbert spaces. Define the actual residual curve on positive parameters r_v(λ)=nnAB_λ v-v. Its parameter is mathematical data, not an assumed noise estimate. Let η be a justified true-data allowance and τ>1 a band multiplier. A total rule must include a small-data branch.","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-59ea2d68a5ddd62a","tex":"R_\\lambda","aliases":"R_λ lambda","display":"R_\\lambda","kind":"use","excerpt":"Fix a bounded operator A:G→ V between real Hilbert spaces, without assuming closed range. For λ>0, put N_λ=A^*A+λ I, R_λ=N_λ^-1, B_λ=R_λ A^*. The inverse is constructed from coercivity. It is not supplied as a certificate premise. Here A^* is the actual Hilbert adjoint.","count":13},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-78422f2967c187e9","tex":"r_\\lambda","aliases":"r_λ lambda","display":"r_\\lambda","kind":"use","excerpt":"Throughout this section A:G→ V is bounded between real Hilbert spaces, N_A=ker A, S_A=N_A^⊥, and λ>0. Write N_λ=A^*A+λ I, x_λ=N_λ^-1A^*v, r_λ(x)=N_λ x-A^*v . The inverse is the actual positive-parameter constructor already established. The vector r_λ(x) is the normal-equation residual of the computed source x. Its size measures source correction an","count":11},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b11c81e6903fa9bf","tex":"\\Ran","aliases":"Ran operatorname","display":"{\\operatorname{Ran}}","kind":"use","excerpt":"E_X^2=E_X, and E_Xv=v exactly when v∈Ran a_X. Moreover D K(T)=T, K D(K)=E_YKE_X. Consequently a compatible K is the encoding of a unique map T:X→ Y. Encoding preserves composition. Its identity on resolved coordinates is E_X.","count":33},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5c93e964f5318448","tex":"\\rightleftarrows","aliases":"rightleftarrows","display":null,"kind":"use","excerpt":"For split frames (U_E,R_E):Jrightleftarrows K and (U_F,R_F):Krightleftarrows G, the composite is split, with analysis R_ER_F and nnR_ER_F≤nnR_EnnR_F. The corresponding calibration allowance is at most (η_rm cal+f)nnR_EnnR_F.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"The observed response Y:K→ V is an operator, and the assumptions nnY-AU≤η_rm cal and nnY-CU≤ f are operator-norm enclosures. Individual calibration samples do not automatically give these enclosures. A mathematical or experimental justification of η_rm cal remains a separate input.","count":12},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a62a596621079394","tex":"\\RR","aliases":"RR R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"[Redundant coordinates] For X= R and V_X= R^2, take a(x)=(x,x) and s(u,v)=(u+v)/2. Then E(u,v)=((u+v)/2,(u+v)/2). The encoded scalar transformation x↦ cx has matrix frac c2(1 · 1","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-78ecfb9af1021553","tex":"\\RR^2","aliases":"RR^2 R^2 RR mathbb","display":"{\\mathbb R}^2","kind":"use","excerpt":"[Redundant coordinates] For X= R and V_X= R^2, take a(x)=(x,x) and s(u,v)=(u+v)/2. Then E(u,v)=((u+v)/2,(u+v)/2). The encoded scalar transformation x↦ cx has matrix frac c2(1 · 1","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":58},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":61},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"Range enlargement induces a surjective contraction F:K/R_0→ K/R_1. For the old minimum representative s_0(z), nn z^2=nnFz^2+nn(P_R_1-P_R_0)s_0(z)^2. Without a closedness or projection hypothesis, a general quotient instead has nn[v]=0iff v∈cl R, [v]=0iff v∈ R. Thus its seminorm detects zero exactly when R is closed. Nonclosed range can leave a nonz","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2badf92439cc7951","tex":"S_A","aliases":"S_A","display":"S_A","kind":"use","excerpt":"Throughout this section A:G→ V is bounded between real Hilbert spaces, N_A=ker A, S_A=N_A^⊥, and λ>0. Write N_λ=A^*A+λ I, x_λ=N_λ^-1A^*v, r_λ(x)=N_λ x-A^*v . The inverse is the actual positive-parameter constructor already established. The vector r_λ(x) is the normal-equation residual of the computed source x. Its size measures source correction an","count":13},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"If λ_i>0 tends to zero and nne_i/sqrtλ_i→0, then B_λ_i(Ag+e_i)⟶ P_(ker A)^⊥g. The same limit holds for computed x_i when additionally fracnn(A^*A+λ_iI)x_i-A^*(Ag+e_i)λ_i→0. For example, λ_i=s_i^2, s_i>0→0, and nne_i≤ s_i^2 give the exact-solve consistency condition.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5bec34c0ded0e2d5","tex":"s_i^2","aliases":"s_i^2","display":"s_i^2","kind":"use","excerpt":"If λ_i>0 tends to zero and nne_i/sqrtλ_i→0, then B_λ_i(Ag+e_i)⟶ P_(ker A)^⊥g. The same limit holds for computed x_i when additionally fracnn(A^*A+λ_iI)x_i-A^*(Ag+e_i)λ_i→0. For example, λ_i=s_i^2, s_i>0→0, and nne_i≤ s_i^2 give the exact-solve consistency condition.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-38dbe99c73eeaafd","tex":"s_X","aliases":"s_X","display":"s_X","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-89069deb6219b9ac","tex":"s_Y","aliases":"s_Y","display":"s_Y","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5945fa8a59c08e31","tex":"\\spanof","aliases":"spanof span operatorname","display":"{\\operatorname{span}}","kind":"use","excerpt":"[A metric must survive its coordinate change] For K={(x,y,0)} and R=span{(1,0,0)}, take the compatible basis b_1=(1,0,0), b_2=(1,2,0). Its Gram matrix is (1 · 1","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":88},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"Let L be a bounded real operator on a Banach space. Its flow is the actual operator exponential e^tL. Boundedness allows differentiation of the norm-convergent exponential series and integration of continuous operator curves. Equalities in this section hold for all real t, using oriented interval integrals when t<0.","count":71},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6da3583735a2a7a0","tex":"T'","aliases":"T'","display":"T'","kind":"use","excerpt":"The splitting gives E_X^2=a_X(s_Xa_X)s_X=a_Xs_X. If E_Xv=v, then v=a_X(s_Xv); conversely E_Xa_Xx=a_Xx. Substituting the definitions yields s_Y(a_YTs_X)a_X=T, a_Y(s_YKa_X)s_X=E_YKE_X. For compatible K, take T= D(K). If another T' encodes to K, decoding proves T'= D(K). Finally, an intermediate resolution of Y has s_Ya_Y=I_Y, so K(S) K(T)=a_ZS(s_Ya_Y","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-910092ae6df8d19a","tex":"T^n","aliases":"T^n","display":"T^n","kind":"use","excerpt":"Let V be visible state and H hidden state. Over a commutative ring, specify four maps and form the actual full dynamics T(y,z)=(Ay+Bz,Gy+Dz). Put C(y,z)=y, Ry=(y,0), E_n=CT^nR, and M_k=BD^kG. The return M_k contains entry into H, k hidden transitions, and return to V. Its first effect occurs after k+2 full transitions.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"use","excerpt":"The nth derivative at zero of e^tL is L^n. Bounded linear maps commute with these derivatives, so equality for all times implies equality of every observed power. Conversely bounded linear maps commute with the sum of the norm-convergent exponential series: K(e^tL)=∑_n(t^n/n!)K(L^n). Equality term by term gives equality of the sums. Sandwiching and","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-41ed6a0ad0f8e525","tex":"T_{[i_1,\\ldots,i_{j-1}]}","aliases":"T_[i_1,…,i_j-1] ldots","display":"T_{[i_1,\\ldots,i_{j-1}]}","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-7f7abcf08b265257","tex":"T_A","aliases":"T_A","display":"T_A","kind":"use","excerpt":"[A noncommuting minimal model] On Q^3, let O(x,y,h)=x, T_A(x,y,h)=(y,0,h), and T_B(x,y,h)=(0,x,h). The two operations do not commute: on (1,0,0) their two compositions give (1,0,0) and (0,0,0). The map q(x,y,h)=(x,y) is generated by O and OT_A, intertwines both operations, and factors O. It therefore has a depth-one certificate. The hidden coordina","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b037fe8890ca7d73","tex":"T_B","aliases":"T_B","display":"T_B","kind":"use","excerpt":"[A noncommuting minimal model] On Q^3, let O(x,y,h)=x, T_A(x,y,h)=(y,0,h), and T_B(x,y,h)=(0,x,h). The two operations do not commute: on (1,0,0) their two compositions give (1,0,0) and (0,0,0). The map q(x,y,h)=(x,y) is generated by O and OT_A, intertwines both operations, and factors O. It therefore has a depth-one certificate. The hidden coordina","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-219b7412a89e1904","tex":"T_i","aliases":"T_i","display":"T_i","kind":"use","excerpt":"smallHypotheses. Commutative-ring modules; a linear observation O:X→ Y and an arbitrary family of allowed endomorphisms T_i:X→ X; every finite chronological word is included.","count":19},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b907416b17e39006","tex":"T_{i::w}","aliases":"T_i::w","display":"T_{i::w}","kind":"use","excerpt":"An intersection of kernels is a submodule. If x∈ N, then OT_wT_ix=OT_i::wx=0, so T_ix∈ N. If an invariant submodule L⊆ker O contains x, induction on word length gives T_wx∈ L; therefore x∈ N. This establishes maximality.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-84eac95fe9a2a4fb","tex":"T_i^{-1}","aliases":"T_i^-1","display":"T_i^{-1}","kind":"use","excerpt":"N_0=ker O and N_r+1=ker O∩⋂_i T_i^-1N_r⊆ N_r. One plateau N_r+1=N_r implies N_r=N. In all cases N_k=N. Thus equality of future observations through word length k is equivalent to equality for every finite word.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-bd9b3a0387d045a0","tex":"T_{i_1}","aliases":"T_i_1","display":"T_{i_1}","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-29a942fbb2721151","tex":"T_{i_n}","aliases":"T_i_n","display":"T_{i_n}","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-36236f8bca177d26","tex":"T_u","aliases":"T_u","display":"T_u","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-72c334562393242f","tex":"T_{u\\mathbin{+\\!+}v}","aliases":"T_umathbin+ +v mathbin","display":null,"kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1b2417ec3f1c4705","tex":"T_v","aliases":"T_v","display":"T_v","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d1272d40e795db7e","tex":"T_w","aliases":"T_w","display":"T_w","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":24},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-03604253eae01f52","tex":"T_\\varnothing","aliases":"T_∅ varnothing","display":"T_\\varnothing","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"smallHypotheses. Commutative-ring modules; linear C:X→ Z, R:Z→ X with CR=I_Z; arbitrary endomorphisms S,T,U of X.","count":67},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":29},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"For y=R_λ r, its energy gives λnn y^2≤ip ry≤nn rnn y, hence the resolvent bound. For x_λ, set u=nnAx_λ. The energy gives u^2+λnnx_λ^2≤nn v u. Completing the square in u yields 4λnnx_λ^2≤nn v^2 and also u≤nn v. Comparing the minimum objective with F_λ(v,0)=nn v^2 bounds the fitting residual. Subtract the exact normal equation from the computed one a","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9c8d9aa4182b4194","tex":"U_{[i_{j+1},\\ldots,i_n]}","aliases":"U_[i_j+1,…,i_n] ldots","display":"U_{[i_{j+1},\\ldots,i_n]}","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-87e8d8c77a91175e","tex":"U_E","aliases":"U_E","display":"U_E","kind":"use","excerpt":"For split frames (U_E,R_E):Jrightleftarrows K and (U_F,R_F):Krightleftarrows G, the composite is split, with analysis R_ER_F and nnR_ER_F≤nnR_EnnR_F. The corresponding calibration allowance is at most (η_rm cal+f)nnR_EnnR_F.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-393db2f05e282178","tex":"U_F","aliases":"U_F","display":"U_F","kind":"use","excerpt":"For split frames (U_E,R_E):Jrightleftarrows K and (U_F,R_F):Krightleftarrows G, the composite is split, with analysis R_ER_F and nnR_ER_F≤nnR_EnnR_F. The corresponding calibration allowance is at most (η_rm cal+f)nnR_EnnR_F.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"Let N=⋂_wker(OT_w). It is the largest submodule contained in ker O and invariant under every T_i. The operations and observation descend to X/N, preserving every future observation. In particular, [x]=[y]iff OT_wx=OT_wy for every finite word w. If q:X→ Z satisfies qT_i=U_iq and Vq=O, then ker q⊆ N.","count":11},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1b6b2cac1ed2c1b6","tex":"U_v","aliases":"U_v","display":"U_v","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b617c2317bcc4406","tex":"U_w","aliases":"U_w","display":"U_w","kind":"use","excerpt":"Invariance defines quotient operations [x]↦[T_ix]; containment in ker O defines [x]↦ Ox. Induction gives their all-word fidelity. Equality [x]=[y] means x-y∈ N, which is precisely equality of the displayed observations by linearity. In an exact competing model, induction gives qT_w=U_wq and hence OT_w=VU_wq. If qx=0, every future observation of x i","count":18},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":43},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"The components lie in R, K∩ R^⊥, and K^⊥, respectively, are pairwise orthogonal, and give the unique decomposition v=r(v)+o(v)+i(v). Furthermore, nn v^2=nn r^2+nn o^2+nn i^2, dist(v,R)^2=nn o^2+nn i^2. For any u∈ R, nnv-u^2=dist(v,R)^2+nnr(v)-u^2. Thus r(v) is the unique nearest represented vector.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"smallHypotheses. Known bounded A:G→ V between real Hilbert spaces and a fixed true source g; a valid pointwise allowance nnAg-v≤η. Accepted-source consistency uses a fixed τ≥1 and genuine bounds nnAg-v_i≤η_i with η_i→0. Total selector existence additionally requires η>0 and τ>1.","count":8},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5819b9247c5027fb","tex":"V_X","aliases":"V_X","display":"V_X","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-50ce1a62663b77d9","tex":"\\Vert","aliases":"Vert","display":"\\Vert","kind":"use","excerpt":"smallHypotheses. Normed spaces over nontrivially normed field; bounded maps; nonnegative letter bounds a_i,b_i,epsilon_i and eta≥VertO-VqVert.","count":19},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"smallHypotheses. Commutative-ring modules and a linear presentation GxrightarrowAVxrightarrowδW with δ A=0; a compatible vector v∈kerδ.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":75},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-644e773747d7540b","tex":"\\widehat r","aliases":"widehat r widehat","display":"\\widehat r","kind":"use","excerpt":"If the normal residual itself is evaluated approximately as widehat r, the distinct assumptions nnr_λ(x)-widehat r≤ν and nnwidehat r≤widehatρ give the allowance widehatρ+ν. Substituting that sum into both correction tests retains the evaluation error rather than silently identifying it with zero.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-80b3be1647ac2fcb","tex":"\\widehat\\rho","aliases":"widehatρ widehat rho","display":"\\widehat\\rho","kind":"use","excerpt":"If the normal residual itself is evaluated approximately as widehat r, the distinct assumptions nnr_λ(x)-widehat r≤ν and nnwidehat r≤widehatρ give the allowance widehatρ+ν. Substituting that sum into both correction tests retains the evaluation error rather than silently identifying it with zero.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":52},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":206},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"The bilinear form b(x,y)=ipAxAy+λip xy is bounded and satisfies b(x,x)≥λnn x^2. Lax--Milgram gives a continuous linear equivalence from the source to itself under the Riesz identification. The adjoint identity identifies its map with A^*A+λ I. The two inverse laws give an actual resolvent. Applying one inverse law to A^*v yields the normal equation","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e189c65b70ed9178","tex":"X^n","aliases":"X^n","display":"X^n","kind":"use","excerpt":"Divide X^n by the monic p to obtain a remainder q_n of degree below deg p. Evaluation at D gives D^n=q_n(D) because p(D)=0. Writing q_n(D) as a finite linear combination of the powers D^k below deg p shows BD^nG=0. For finite-dimensional H, Cayley--Hamilton provides the monic characteristic polynomial of D with degree d. SR-06 then yields both disp","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"If λ_i>0 tends to zero and nne_i/sqrtλ_i→0, then B_λ_i(Ag+e_i)⟶ P_(ker A)^⊥g. The same limit holds for computed x_i when additionally fracnn(A^*A+λ_iI)x_i-A^*(Ag+e_i)λ_i→0. For example, λ_i=s_i^2, s_i>0→0, and nne_i≤ s_i^2 give the exact-solve consistency condition.","count":19},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1a80ff6e24b3fc04","tex":"x_\\lambda","aliases":"x_λ lambda","display":"x_\\lambda","kind":"use","excerpt":"N_λ is a bounded bijection with bounded inverse R_λ. For x_λ=B_λ v, N_λ x_λ=A^*v, A^*(Ax_λ-v)+λ x_λ=0, and nnAx_λ^2+λnnx_λ^2 =ip vAx_λ.","count":36},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-52a79f802179be19","tex":"x_\\mu","aliases":"x_μ mu","display":"x_\\mu","kind":"use","excerpt":"Compare the minimum objectives at 0<λ<μ using each minimizer as a competitor for the other. Adding the two inequalities gives (μ-λ)(nnx_μ^2-nnx_λ^2)≤0, so source norms are antitone. Substitution back into the smaller-parameter comparison shows residual monotonicity. If the two residuals were equal, both norms and objective values would be equal, an","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"smallHypotheses. Commutative-ring modules and a linear presentation GxrightarrowAVxrightarrowδW with δ A=0; a compatible vector v∈kerδ.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":19},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"Let N=⋂_wker(OT_w). It is the largest submodule contained in ker O and invariant under every T_i. The operations and observation descend to X/N, preserving every future observation. In particular, [x]=[y]iff OT_wx=OT_wy for every finite word w. If q:X→ Z satisfies qT_i=U_iq and Vq=O, then ker q⊆ N.","count":78},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"use","excerpt":"For L(y,z)=(Ay+Bz,Gy+Dz) and (y(t),z(t))=e^tL(y_0,z_0), z(t)=e^tDz_0+∫_0^t e^(t-s)DGy(s) ds. Writing M(t)=Be^tDG, the visible dynamics satisfy y'(t)=Ay(t)+Be^tDz_0+∫_0^t M(t-s)y(s) ds. Prepared initialization removes the term involving z_0.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"For y=R_λ r, its energy gives λnn y^2≤ip ry≤nn rnn y, hence the resolvent bound. For x_λ, set u=nnAx_λ. The energy gives u^2+λnnx_λ^2≤nn v u. Completing the square in u yields 4λnnx_λ^2≤nn v^2 and also u≤nn v. Comparing the minimum objective with F_λ(v,0)=nn v^2 bounds the fitting residual. Subtract the exact normal equation from the computed one a","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"Choose accepted comparison sources y_i from the definition. Exact accepted consistency gives y_i→ p=P_S_Ag, while nnx_i-y_i≤κ_i→0 gives x_i-y_i→0. Adding the two limits proves x_i→ p. Continuity of the norm gives the source-error limit, because p-g=-P_N_Ag.","count":17},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e3d4192e022af00a","tex":"y_j","aliases":"y_j","display":"y_j","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-00fe9b383f124db5","tex":"y_n","aliases":"y_n","display":"y_n","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b45994edf90285b5","tex":"y_{n+1}","aliases":"y_n+1","display":"y_{n+1}","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"The splitting gives E_X^2=a_X(s_Xa_X)s_X=a_Xs_X. If E_Xv=v, then v=a_X(s_Xv); conversely E_Xa_Xx=a_Xx. Substituting the definitions yields s_Y(a_YTs_X)a_X=T, a_Y(s_YKa_X)s_X=E_YKE_X. For compatible K, take T= D(K). If another T' encodes to K, decoding proves T'= D(K). Finally, an intermediate resolution of Y has s_Ya_Y=I_Y, so K(S) K(T)=a_ZS(s_Ya_Y","count":23},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"Let V be visible state and H hidden state. Over a commutative ring, specify four maps and form the actual full dynamics T(y,z)=(Ay+Bz,Gy+Dz). Put C(y,z)=y, Ry=(y,0), E_n=CT^nR, and M_k=BD^kG. The return M_k contains entry into H, k hidden transitions, and return to V. Its first effect occurs after k+2 full transitions.","count":64},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"frac ddte^tD=De^tD=e^tDD, e^(s+t)D=e^sDe^tD, and e^-tD is the inverse flow. Every actual curve satisfying z'=Dz+f with continuous f obeys z(t)=e^tDz(0)+∫_0^t e^(t-s)Df(s) ds. The initial value uniquely determines the homogeneous solution; two forced solutions with the same forcing and initial value also agree.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"Every class z has a unique compatible minimum-norm representative s(z), and every compatible representative v satisfies nn v^2=nn z^2+nnv-s(z)^2. For arbitrary data define c(v)=[P_Kv]. This is nonexpansive and dist(v,R)^2=nnc(v)^2+nni(v)^2. The datum is realizable exactly when both c(v)=0 and δ v=0. When the canonical inverse exists, nnv-Ag^2=nnc(v","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":12},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"[Delayed return and disconnected return] On Q^3, set T(y,z_1,z_2)=(z_1,z_2,y) and retain y. Then A=0, M_0=0, M_1=I, and E_1=E_2=0 while E_3=I. Two shallow exact checks cannot certify all depths without the dimension budget. For another system, let A=D=I, B(z_1,z_2)=z_1, and G(y)=(0,y). Both couplings are nonzero but every BD^kG vanishes. Prepared c","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5e32645afb213c74","tex":"z_2","aliases":"z_2","display":"z_2","kind":"use","excerpt":"[Delayed return and disconnected return] On Q^3, set T(y,z_1,z_2)=(z_1,z_2,y) and retain y. Then A=0, M_0=0, M_1=I, and E_1=E_2=0 while E_3=I. Two shallow exact checks cannot certify all depths without the dimension budget. For another system, let A=D=I, B(z_1,z_2)=z_1, and G(y)=(0,y). Both couplings are nonzero but every BD^kG vanishes. Prepared c","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d533223c5565c514","tex":"Z_k","aliases":"Z_k","display":"Z_k","kind":"use","excerpt":"Let H_kx=(OT_wx)_|w|≤ k and Z_k=Ran H_k. Then ker H_k=N and X/N≅ Z_k. The formulas U_i(H_kx)=H_kT_ix, V(H_kx)=Ox define actual linear maps on Z_k and preserve every finite future word. This model has minimum exact dimension.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-bdc1562155501d06","tex":"z_n","aliases":"z_n","display":"z_n","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a1bb8bb52da2ab42","tex":"z_{n+1}","aliases":"z_n+1","display":"z_{n+1}","kind":"use","excerpt":"At n=0, the first formula is z_0=z_0 with an empty sum. If it holds at n, the actual recurrence z_n+1=Gy_n+Dz_n gives the same formula at n+1 after distributing D and appending the newest forcing term. Substituting this expression into y_n+1=Ay_n+Bz_n proves the visible memory equation. For a prepared input y_0, each y_j equals E_jy_0. The pointwis","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"Now work over a field and, for dimension statements, assume dim X<∞. The allowed operations need not commute. Define N_r=⋂_|w|≤ rker(OT_w), k=dimker O. A finite horizon means bounded word length. It gives a finite enumeration of constraints only when the operation alphabet is finite.","count":22},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"For the scalar boundary, the true data error is zero and nnC-A=1≤δ, so both stated truth allowances are valid. The small-data branch would require 1≤2/10, which is false. For every positive parameter, the actual regularized source of C=0 is zero and its fitting residual is one. Its candidate noise is 1/10+1·0=1/10, so the upper generalized band wou","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"For C:X→ Z, R:Z→ X, CR=I, let Q=I-RC and H=ker C. The maps x⟼(Cx,Qx), (y,z)⟼ Ry+z are inverse linear maps between X and Z× H. An endomorphism T is conjugate to block dynamics with A=CTR, B=CTι_H, G=QTR, D=QTι_H. Their prepared maps equal CT^nR. Hence exact compressed powers are equivalent to zero canonical memory. If dimker C=d, complete compressed","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"Some α>0 satisfies αnn g≤nnAg on S, and any supplied such constant gives nn B≤1/α. Therefore nnB(Ag+e)-g^2≤(nn e/α)^2+nnP_N_Ag^2. For an arbitrary computed source h, nnh-Bv^2 ≤(fracnnAh-P_Rvα)^2+nnP_N_Ah^2. If also δ A=0, a compatibility constant σ bounds the local error as in SR-27. The constants α and σ have separate roles.","count":16},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d830d9bc7e9c803c","tex":"\\alpha^2","aliases":"α^2 alpha","display":"\\alpha^2","kind":"use","excerpt":"Write p=P_(ker A)^⊥g, k=P_ker Ag. Then B_λ Ag+λ R_λ g=g, B_λ(Ag+e)-g=-λ R_λ g+B_λ e, and nnB_λ(Ag+e)-g^2 =nn-λ R_λ p+B_λ e^2+nn k^2. If A has source coercivity α>0, this is at most (fracλα^2+λnn p +fracnn e2sqrtλ)^2+nn k^2. For a supplied source condition g=A^*Aw, nnB_λ Ag-g≤λnn w without requiring closed range.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Set P=RC, Q=I-P, Φ(T)=CTR, and Δ(S,T)=Φ(ST)-Φ(S)Φ(T). Then Δ(S,T)=CSQTR, and associativity gives Δ(S,T)Φ(U)+Δ(ST,U) =Φ(S)Δ(T,U)+Δ(S,TU). For two successive split compressions, Δ_21(S,T)=Φ_2(Δ_1(S,T)) +Δ_2(Φ_1(S),Φ_1(T)).","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2d8fd38ef1b9fe4d","tex":"\\delta'","aliases":"δ' delta","display":"\\delta'","kind":"use","excerpt":"Commuting presentation maps LA=A'F and δ'L=Mδ induce a map overline L: H→ H' by [v]↦[Lv]. These maps compose and preserve zero classes. Reflection of zero follows when overline L is injective.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-2bb168c166c79f1d","tex":"\\Delta_1","aliases":"Δ_1 Delta","display":"\\Delta_1","kind":"use","excerpt":"Set P=RC, Q=I-P, Φ(T)=CTR, and Δ(S,T)=Φ(ST)-Φ(S)Φ(T). Then Δ(S,T)=CSQTR, and associativity gives Δ(S,T)Φ(U)+Δ(ST,U) =Φ(S)Δ(T,U)+Δ(S,TU). For two successive split compressions, Δ_21(S,T)=Φ_2(Δ_1(S,T)) +Δ_2(Φ_1(S),Φ_1(T)).","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-6c0b4ab8dd399bba","tex":"\\Delta_2","aliases":"Δ_2 Delta","display":"\\Delta_2","kind":"use","excerpt":"Set P=RC, Q=I-P, Φ(T)=CTR, and Δ(S,T)=Φ(ST)-Φ(S)Φ(T). Then Δ(S,T)=CSQTR, and associativity gives Δ(S,T)Φ(U)+Δ(ST,U) =Φ(S)Δ(T,U)+Δ(S,TU). For two successive split compressions, Δ_21(S,T)=Φ_2(Δ_1(S,T)) +Δ_2(Φ_1(S),Φ_1(T)).","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-aef985bdcdd75d2b","tex":"\\Delta_{21}","aliases":"Δ_21 Delta","display":"\\Delta_{21}","kind":"use","excerpt":"Set P=RC, Q=I-P, Φ(T)=CTR, and Δ(S,T)=Φ(ST)-Φ(S)Φ(T). Then Δ(S,T)=CSQTR, and associativity gives Δ(S,T)Φ(U)+Δ(ST,U) =Φ(S)Δ(T,U)+Δ(S,TU). For two successive split compressions, Δ_21(S,T)=Φ_2(Δ_1(S,T)) +Δ_2(Φ_1(S),Φ_1(T)).","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a104cc9eebe89629","tex":"\\delta_i","aliases":"δ_i delta","display":"\\delta_i","kind":"use","excerpt":"Let q:X→ Z be a proposed representation, with true operations T_i, reduced operations U_i, and readouts O:X→ Y, V:Z→ Y. Define the step and word defects δ_i=qT_i-U_iq, E_w=qT_w-U_wq. Algebraic identities require only the stated commutative-ring structure. Quantitative estimates below use continuous linear maps over a nontrivially normed field and t","count":14},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-075418209d63c029","tex":"\\delta_{i_j}","aliases":"δ_i_j delta","display":"\\delta_{i_j}","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-489fe5f71d54d77a","tex":"\\delta_p","aliases":"δ_p delta","display":"\\delta_p","kind":"use","excerpt":"One has the operator identity E(t)=∫_0^t e^(t-s)Aδ e^sL ds, and δ=0 exactly when E(t)=0 for every t. Also E(s+t)=E(s)e^tL+e^sAE(t). For a second representation p:Z→ W with generator B, δ_pq=pδ_q+δ_pq, E_pq(t)=pE_q(t)+E_p(t)q.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-41af0b6beae368c8","tex":"\\delta_{pq}","aliases":"δ_pq delta","display":"\\delta_{pq}","kind":"use","excerpt":"One has the operator identity E(t)=∫_0^t e^(t-s)Aδ e^sL ds, and δ=0 exactly when E(t)=0 for every t. Also E(s+t)=E(s)e^tL+e^sAE(t). For a second representation p:Z→ W with generator B, δ_pq=pδ_q+δ_pq, E_pq(t)=pE_q(t)+E_p(t)q.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-53532b4a4b2a0313","tex":"\\delta_q","aliases":"δ_q delta","display":"\\delta_q","kind":"use","excerpt":"One has the operator identity E(t)=∫_0^t e^(t-s)Aδ e^sL ds, and δ=0 exactly when E(t)=0 for every t. Also E(s+t)=E(s)e^tL+e^sAE(t). For a second representation p:Z→ W with generator B, δ_pq=pδ_q+δ_pq, E_pq(t)=pE_q(t)+E_p(t)q.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-df62fbeae71e37b5","tex":"\\delta_s","aliases":"δ_s delta","display":"\\delta_s","kind":"use","excerpt":"Nested orthogonal projections split o_0(v)=o_1(v)+g(v) into orthogonal sectors, and the inconsistency component stays fixed. Pythagoras gives both refinement identities. For the scalar family δ_s(x)=sx, s>0, take R=0 and v=1. The reconstruction distance is one and the residual is s. For any proposed common c, choosing s sufficiently small violates ","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":37},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0773a19734ee0848","tex":"\\epsilon_i","aliases":"ϵ_i epsilon","display":"\\epsilon_i","kind":"use","excerpt":"Suppose nnT_i≤ a_i, nnU_i≤ b_i, nnδ_i≤ϵ_i. Let A_w,B_w be the products of the respective letter bounds and define D_∅=0, D_i::w=D_wa_i+B_wϵ_i. Then nnT_w≤ A_w, nnU_w≤ B_w, nnE_w≤ D_w, and D_umathbin+ +v=D_vA_u+B_vD_u. If nnO-Vq≤η, then nnOT_w-VU_wq≤η A_w+nn V D_w. For arbitrary initial x∈ X, z∈ Z, nnqT_wx-U_wz≤ D_wnn x+B_wnnqx-z.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-412b907de59c5fc0","tex":"\\eta_i","aliases":"η_i eta","display":"\\eta_i","kind":"use","excerpt":"smallHypotheses. Known bounded A:G→ V between real Hilbert spaces and a fixed true source g; a valid pointwise allowance nnAg-v≤η. Accepted-source consistency uses a fixed τ≥1 and genuine bounds nnAg-v_i≤η_i with η_i→0. Total selector existence additionally requires η>0 and τ>1.","count":15},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-5cbc585d5396c95e","tex":"\\eta_{\\rm cal}","aliases":"η_rm cal eta rm","display":"\\eta_{\\rm cal}","kind":"use","excerpt":"The observed response Y:K→ V is an operator, and the assumptions nnY-AU≤η_rm cal and nnY-CU≤ f are operator-norm enclosures. Individual calibration samples do not automatically give these enclosures. A mathematical or experimental justification of η_rm cal remains a separate input.","count":11},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-cb6432bc124f783a","tex":"\\eta_{{\\rm cal},i}","aliases":"η_rm cal,i eta rm","display":"\\eta_{{\\rm cal},i}","kind":"use","excerpt":"For recovery assume G,V are real Hilbert spaces, A is a fixed bounded true operator, τ≥0 is fixed, and p∈(ker A)^⊥ is fixed. Retain the whole calibration bounds and the true data bounds nnAp-v_i≤η_i. Put δ_i=(η_rm cal,i+f_i)nn R. If the true data, calibration-truth and model-fit allowances tend to zero, then δ_i→0. Generalized accepted sources usin","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"For C:X→ Z, R:Z→ X, CR=I, let Q=I-RC and H=ker C. The maps x⟼(Cx,Qx), (y,z)⟼ Ry+z are inverse linear maps between X and Z× H. An endomorphism T is conjugate to block dynamics with A=CTR, B=CTι_H, G=QTR, D=QTι_H. Their prepared maps equal CT^nR. Hence exact compressed powers are equivalent to zero canonical memory. If dimker C=d, complete compressed","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-50b831621b666dcd","tex":"\\iota_H","aliases":"ι_H iota","display":"\\iota_H","kind":"use","excerpt":"For C:X→ Z, R:Z→ X, CR=I, let Q=I-RC and H=ker C. The maps x⟼(Cx,Qx), (y,z)⟼ Ry+z are inverse linear maps between X and Z× H. An endomorphism T is conjugate to block dynamics with A=CTR, B=CTι_H, G=QTR, D=QTι_H. Their prepared maps equal CT^nR. Hence exact compressed powers are equivalent to zero canonical memory. If dimker C=d, complete compressed","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-3a3070a4a5f283b5","tex":"\\iota_S","aliases":"ι_S iota","display":"\\iota_S","kind":"use","excerpt":"Restriction A_S:S→ R is a bounded linear bijection with bounded inverse. Define B=ι_S A_S^-1P_R. Then AB=P_R, BA=P_S, ABA=A, and BAB=B. Its two projections are symmetric, ker B=R^⊥, and Ran B=S. The best-image identity AB=P_R and the condition Ran B⊆ S uniquely determine B.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"The distinction matters when λ is small. A normal residual sufficient for a prescribed image tolerance may be insufficient for a prescribed source tolerance. If source correction must be at most κ and image correction at most ξ, a sound pair of conditions is nnr_λ(x)≤λκ and nnr_λ(x)≤2sqrtλ ξ. These are parameter-dependent requirements, not a unifor","count":26},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0fc1259c08b990e9","tex":"\\kappa^2","aliases":"κ^2 kappa","display":"\\kappa^2","kind":"use","excerpt":"A generalized accepted x obeys nnAx-Ap≤(τ+1)c_η,δ(p). A source within κ of such an x obeys the same bound with nn A κ added. For p∈ S_A, vanishing data/operator allowances and near-source budgets imply strong convergence to p. If p=A^*w, an inexact accepted candidate also satisfies nnx-p^2≤ 2κ^2+4nn w(τ+1)c_η,δ(p).","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c818d31da156fd8b","tex":"\\kappa_i","aliases":"κ_i kappa","display":"\\kappa_i","kind":"use","excerpt":"Call x near accepted with budget κ when some exact accepted source y satisfies nnx-y≤κ. If nnAg-v_i≤η_i, η_i→0, and near-accepted x_i have κ_i→0, then x_i⟶ P_S_Ag, nnx_i-g⟶nnP_N_Ag. At every index nnP_N_Ax_i≤κ_i; therefore the computed true-kernel component tends to zero. No separate parameter schedule is assumed.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Fix a bounded operator A:G→ V between real Hilbert spaces, without assuming closed range. For λ>0, put N_λ=A^*A+λ I, R_λ=N_λ^-1, B_λ=R_λ A^*. The inverse is constructed from coercivity. It is not supplied as a certificate premise. Here A^* is the actual Hilbert adjoint.","count":143},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-308a821eda6e7229","tex":"\\lambda^2","aliases":"λ^2 lambda","display":"\\lambda^2","kind":"use","excerpt":"For two operators and parameters, R_A,λ-R_C,μ =R_A,λ(N_C,μ-N_A,λ)R_C,μ. At fixed positive λ, with ϵ=nnA-C, nnB_A,λ-B_C,λ ≤fracϵλ+ frac(nn A+nn C)ϵnn Cλ^2. If A has closed range and canonical inverse B, then B_λ v-Bv=-λ R_λ Bv. For δ A=0, nnv-AB_λ v^2 =nnc(v)^2+nni(v)^2+nnA(B_λ v-Bv)^2.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"use","excerpt":"If λ_i>0 tends to zero and nne_i/sqrtλ_i→0, then B_λ_i(Ag+e_i)⟶ P_(ker A)^⊥g. The same limit holds for computed x_i when additionally fracnn(A^*A+λ_iI)x_i-A^*(Ag+e_i)λ_i→0. For example, λ_i=s_i^2, s_i>0→0, and nne_i≤ s_i^2 give the exact-solve consistency condition.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"For two operators and parameters, R_A,λ-R_C,μ =R_A,λ(N_C,μ-N_A,λ)R_C,μ. At fixed positive λ, with ϵ=nnA-C, nnB_A,λ-B_C,λ ≤fracϵλ+ frac(nn A+nn C)ϵnn Cλ^2. If A has closed range and canonical inverse B, then B_λ v-Bv=-λ R_λ Bv. For δ A=0, nnv-AB_λ v^2 =nnc(v)^2+nni(v)^2+nnA(B_λ v-Bv)^2.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"If the normal residual itself is evaluated approximately as widehat r, the distinct assumptions nnr_λ(x)-widehat r≤ν and nnwidehat r≤widehatρ give the allowance widehatρ+ν. Substituting that sum into both correction tests retains the evaluation error rather than silently identifying it with zero.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"The distinction matters when λ is small. A normal residual sufficient for a prescribed image tolerance may be insufficient for a prescribed source tolerance. If source correction must be at most κ and image correction at most ξ, a sound pair of conditions is nnr_λ(x)≤λκ and nnr_λ(x)≤2sqrtλ ξ. These are parameter-dependent requirements, not a unifor","count":12},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e8149ec697a9ee64","tex":"\\xi^2","aliases":"ξ^2 xi","display":"\\xi^2","kind":"use","excerpt":"Suppose a record supplies λ>0, nonnegative ρ,ξ,κ,m,ω, and nnr_λ(x)^2≤ρ^2, ρ^2≤(λκ)^2, ρ^2≤4λξ^2, max(0,m-ω)^2≤nnAx-v^2≤(m+ω)^2, η+ω+ξ≤ m≤τη-ω-ξ . Then x is near accepted with source budget κ. Families of such records give SR-51 when the genuine data allowances and source budgets tend to zero.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"The residual is continuous and nondecreasing on positive parameters; the source norm is nonincreasing. It is strictly increasing when A^*v≠0, and constant at nn v when A^*v=0. For dist(v,Ran A)<ρ<nn v, there is exactly one λ>0 with r_v(λ)=ρ. The resulting exact real parameter is a noncomputable choice with a proved specification.","count":18},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-bd1aac0f414abebd","tex":"\\rho^2","aliases":"ρ^2 rho","display":"\\rho^2","kind":"use","excerpt":"Suppose a record supplies λ>0, nonnegative ρ,ξ,κ,m,ω, and nnr_λ(x)^2≤ρ^2, ρ^2≤(λκ)^2, ρ^2≤4λξ^2, max(0,m-ω)^2≤nnAx-v^2≤(m+ω)^2, η+ω+ξ≤ m≤τη-ω-ξ . Then x is near accepted with source budget κ. Families of such records give SR-51 when the genuine data allowances and source budgets tend to zero.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"If σnn x≤nnδ x for x∈ K^⊥ and σ>0, then δ i(v)=δ v, nni(v)≤fracnnδ vσ, and nno(v)≤dist(v,R), dist(v,R)^2≤nno(v)^2+(fracnnδ vσ)^2. For compatible data, dist(v,R)=nno(v).","count":11},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"Set P=RC, Q=I-P, Φ(T)=CTR, and Δ(S,T)=Φ(ST)-Φ(S)Φ(T). Then Δ(S,T)=CSQTR, and associativity gives Δ(S,T)Φ(U)+Δ(ST,U) =Φ(S)Δ(T,U)+Δ(S,TU). For two successive split compressions, Δ_21(S,T)=Φ_2(Δ_1(S,T)) +Δ_2(Φ_1(S),Φ_1(T)).","count":9},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0dd8e14cc9a785dc","tex":"\\Phi_1","aliases":"Φ_1 Phi","display":"\\Phi_1","kind":"use","excerpt":"Set P=RC, Q=I-P, Φ(T)=CTR, and Δ(S,T)=Φ(ST)-Φ(S)Φ(T). Then Δ(S,T)=CSQTR, and associativity gives Δ(S,T)Φ(U)+Δ(ST,U) =Φ(S)Δ(T,U)+Δ(S,TU). For two successive split compressions, Δ_21(S,T)=Φ_2(Δ_1(S,T)) +Δ_2(Φ_1(S),Φ_1(T)).","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-d33036383a860124","tex":"\\Phi_2","aliases":"Φ_2 Phi","display":"\\Phi_2","kind":"use","excerpt":"Set P=RC, Q=I-P, Φ(T)=CTR, and Δ(S,T)=Φ(ST)-Φ(S)Φ(T). Then Δ(S,T)=CSQTR, and associativity gives Δ(S,T)Φ(U)+Δ(ST,U) =Φ(S)Δ(T,U)+Δ(S,TU). For two successive split compressions, Δ_21(S,T)=Φ_2(Δ_1(S,T)) +Δ_2(Φ_1(S),Φ_1(T)).","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Suppose m is a measured fitting norm with |m-nnAx-v|≤ω, nnr_λ(x)≤λκ, nnr_λ(x)≤2sqrtλ ξ . If η+ω+ξ≤ m≤τη-ω-ξ, then the actual exact source x_λ has discrepancy in [η,τη], and nnx-x_λ≤κ. A protected scalar window exists exactly when 2(ω+ξ)≤(τ-1)η . Its midpoint (τ+1)η/2 is then admissible.","count":12},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"The first contract supplies two maps rather than identifying a coordinate space with the object by convention. Let a_X:X→ V_X and s_X:V_X→ X be linear with s_Xa_X=I_X. The ambient space V_X may have redundant coordinates. Define E_X=a_Xs_X. For another such resolution of Y, put K(T)=a_YTs_X, D(K)=s_YKa_X. We call a proposed coordinate operator comp","count":60},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Redundant coordinates] For X= R and V_X= R^2, take a(x)=(x,x) and s(u,v)=(u+v)/2. Then E(u,v)=((u+v)/2,(u+v)/2). The encoded scalar transformation x↦ cx has matrix frac c2(1 · 1","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"The following conditions are equivalent: E_n=A^n (∀ n), M_k=0 (∀ k), E_m+n=E_mE_n (∀ m,n). More precisely, E_n=A^n (n≤ N+1)iff M_k=0 (k<N). If M_j=0 for j<k, then E_k+2-A^k+2=M_k. Closure for every initial hidden state is equivalent to B=0.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"The chain condition puts the actual range of A inside K. The zero class criterion for a quotient is membership in the submodule divided out, giving [v]=0iff v∈ Riff∃ g,Ag=v. Every compatible class is zero exactly when K⊆ R; the reverse inclusion is already known. Injectivity of A gives uniqueness of any preimage.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"iable space is (ker A^*)^⊥=clRan A, yielding the fitted limit. SR-38 shows AB_λ is contractive, so changing data by e_i changes the fitted value by at most nne_i. If reconstructed sources converged in norm to some g, continuity of A would make their images converge to Ag. On a nontrivial filter, the already established image limit v would force Ag=","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-f7b37e8032ebe855","tex":"\\sum_{j=0}^{n-1}","aliases":"∑_j=0^n-1 sum","display":"\\sum_{j=0}^{n-1}","kind":"use","excerpt":"For the trajectory (y_n,z_n)=T^n(y_0,z_0), z_n=D^nz_0+∑_j=0^n-1D^n-1-jGy_j, and y_n+1=Ay_n+BD^nz_0+∑_j=0^n-1M_n-1-jy_j. Prepared evolution (z_0=0) satisfies E_0=I, E_1=A, and E_n+1=AE_n+∑_j=0^n-1M_n-1-jE_j.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-db50c77870a75686","tex":"\\sum_{j=1}^n","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^n","kind":"use","excerpt":"One has E_∅=0, E_i::w=E_wT_i+U_wδ_i, E_umathbin+ +v=E_vT_u+U_vE_u. For w=[i_1,…,i_n] this yields E_w=∑_j=1^n U_[i_j+1,…,i_n]δ_i_jT_[i_1,…,i_j-1]. The output defect is (O-Vq)T_w+VE_w.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","excerpt":"The nth derivative at zero of e^tL is L^n. Bounded linear maps commute with these derivatives, so equality for all times implies equality of every observed power. Conversely bounded linear maps commute with the sum of the norm-convergent exponential series: K(e^tL)=∑_n(t^n/n!)K(L^n). Equality term by term gives equality of the sums. Sandwiching and","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"For any submodule, the native quotient seminorm equals distance to that submodule. Zero distance is equivalent to membership in its closure by the metric closure criterion; algebraic quotient zero remains actual membership. If R is nonclosed, choose v∈cl R∖ R. Its class is nonzero with zero seminorm. Conversely closedness makes zero distance imply ","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"If a monic polynomial p annihilates D, vanishing of M_k for k<deg p implies vanishing for all k. If dim H=d<∞, then E_n=A^n (∀ n) iff M_k=0 (0≤ k<d) iff E_n=A^n (0≤ n≤ d+1). The visible space need not be finite-dimensional. The case d=0 is included.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"N_0=ker O and N_r+1=ker O∩⋂_i T_i^-1N_r⊆ N_r. One plateau N_r+1=N_r implies N_r=N. In all cases N_k=N. Thus equality of future observations through word length k is equivalent to equality for every finite word.","count":7},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"frac ddte^tD=De^tD=e^tDD, e^(s+t)D=e^sDe^tD, and e^-tD is the inverse flow. Every actual curve satisfying z'=Dz+f with continuous f obeys z(t)=e^tDz(0)+∫_0^t e^(t-s)Df(s) ds. The initial value uniquely determines the homogeneous solution; two forced solutions with the same forcing and initial value also agree.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-a2972272b24e9c45","tex":"\\int_0^t","aliases":"∫_0^t int","display":"\\int_0^t","kind":"use","excerpt":"frac ddte^tD=De^tD=e^tDD, e^(s+t)D=e^sDe^tD, and e^-tD is the inverse flow. Every actual curve satisfying z'=Dz+f with continuous f obeys z(t)=e^tDz(0)+∫_0^t e^(t-s)Df(s) ds. The initial value uniquely determines the homogeneous solution; two forced solutions with the same forcing and initial value also agree.","count":6},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"The model X/N is observable. Every exact competitor induces a unique surjection Ran q→ X/N, taking qx to [x]. Therefore dim(X/N)≤dimRan q, with equality exactly when ker q=N. In that case Ran q≅ X/N. If Z is finite-dimensional, also dimRan q≤dim Z. Surjectivity of q onto Z is not required.","count":5},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"For H_0(a,b)=0 and H_n+1(a,b)=aH_n(a,b)+b^n, H_n(a,b)=∑_j=0^n-1a^jb^n-1-j, (a-b)H_n=a^n-b^n. If a≠ b, H_n=(a^n-b^n)/(a-b); for equal rates, H_n+1(a,a)=(n+1)a^n. Uniform step bounds give nnE_w≤ϵ H_n(a,b), nnOT_w-VU_wq≤η a^n+nn Vϵ H_n(a,b).","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"For δ A=0, define K=kerδ, R=Ran A⊆ K, and H=K/R. A compatible v has class zero exactly when v=Ag for some g. All classes vanish exactly when R=K. If A is injective, a zero class has a unique source.","count":14},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"The components lie in R, K∩ R^⊥, and K^⊥, respectively, are pairwise orthogonal, and give the unique decomposition v=r(v)+o(v)+i(v). Furthermore, nn v^2=nn r^2+nn o^2+nn i^2, dist(v,R)^2=nn o^2+nn i^2. For any u∈ R, nnv-u^2=dist(v,R)^2+nnr(v)-u^2. Thus r(v) is the unique nearest represented vector.","count":31},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"Let N=⋂_wker(OT_w). It is the largest submodule contained in ker O and invariant under every T_i. The operations and observation descend to X/N, preserving every future observation. In particular, [x]=[y]iff OT_wx=OT_wy for every finite word w. If q:X→ Z satisfies qT_i=U_iq and Vq=O, then ker q⊆ N.","count":3},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-892cecbf386dea45","tex":"\\bigcap_i","aliases":"⋂_i bigcap","display":"\\bigcap_i","kind":"use","excerpt":"N_0=ker O and N_r+1=ker O∩⋂_i T_i^-1N_r⊆ N_r. One plateau N_r+1=N_r implies N_r=N. In all cases N_k=N. Thus equality of future observations through word length k is equivalent to equality for every finite word.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-27b1d2fc9bc15338","tex":"\\bigcap_w","aliases":"⋂_w bigcap","display":"\\bigcap_w","kind":"use","excerpt":"Let N=⋂_wker(OT_w). It is the largest submodule contained in ker O and invariant under every T_i. The operations and observation descend to X/N, preserving every future observation. In particular, [x]=[y]iff OT_wx=OT_wy for every finite word w. If q:X→ Z satisfies qT_i=U_iq and Vq=O, then ker q⊆ N.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-930bb050c29b1bb0","tex":"\\bigcap_{|w|\\le r}","aliases":"⋂_|w|≤ r bigcap le","display":"\\bigcap_{|w|\\le r}","kind":"use","excerpt":"Now work over a field and, for dimension statements, assume dim X<∞. The allowed operations need not commute. Define N_r=⋂_|w|≤ rker(OT_w), k=dimker O. A finite horizon means bounded word length. It gives a finite enumeration of constraints only when the operation alphabet is finite.","count":1},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"Conventions and domains. Operator multiplication applies the right factor first: STx=S(Tx). A word w=[i_1,…,i_n] is chronological, so T_w=T_i_n⋯ T_i_1; T_∅=I. Concatenating u then v gives T_umathbin+ +v=T_vT_u. Empty sums are zero and empty products are one. All equalities of operators are equalities on the entire stated domain.","count":2},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"One has clRan(A^*A)=(ker A)^⊥, and for every g,w, nnB_λ Ag-g≤nng-A^*Aw+λnn w. As λdownarrow0, B_λ Ag⟶ P_(ker A)^⊥g, nnB_λ Ag-g⟶nnP_ker Ag. For a nontrivial parameter filter, convergence to g itself holds exactly when g∈(ker A)^⊥.","count":4},{"work":"sufficient-resolution","url":"notation/sufficient-resolution.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"For C:X→ Z, R:Z→ X, CR=I, let Q=I-RC and H=ker C. The maps x⟼(Cx,Qx), (y,z)⟼ Ry+z are inverse linear maps between X and Z× H. An endomorphism T is conjugate to block dynamics with A=CTR, B=CTι_H, G=QTR, D=QTι_H. Their prepared maps equal CT^nR. Hence exact compressed powers are equivalent to zero canonical memory. If dimker C=d, complete compressed","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":52},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":266},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":81},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":155},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-bb3cdc779ccbade8","tex":"\\bar J","aliases":"bar J bar","display":"\\bar J","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-446fba0b12b9aac6","tex":"\\bar J_0","aliases":"bar J_0 bar","display":"\\bar J_0","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-58d2d6ac0d680d9f","tex":"\\bar J_h","aliases":"bar J_h bar","display":"\\bar J_h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-32da9c12730e8ae8","tex":"\\bar J_{h-1}","aliases":"bar J_h-1 bar","display":"\\bar J_{h-1}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c0435683f947250d","tex":"\\bar z","aliases":"bar z bar","display":"\\bar z","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f68011f6686ac355","tex":"\\bar z_{a,o}","aliases":"bar z_a,o bar","display":"\\bar z_{a,o}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4415f58b83fbd8c6","tex":"\\bar\\Psi","aliases":"barΨ bar Psi","display":"\\bar\\Psi","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5c18dd7e6706bb6e","tex":"\\bar\\Psi_{a,o}","aliases":"barΨ_a,o bar Psi","display":"\\bar\\Psi_{a,o}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":22},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":51},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":68},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":24},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":12},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-24ceed0c3b629a6f","tex":"F_{a_1}","aliases":"F_a_1","display":"F_{a_1}","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d6d2fa627b1c221a","tex":"F_{a_k}","aliases":"F_a_k","display":"F_{a_k}","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":48},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":16},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6e30319bc4adb843","tex":"J_h","aliases":"J_h","display":"J_h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":22},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":18},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a20d1b2c5e09f23a","tex":"\\min_{a\\in A(q)}","aliases":"min_a∈ A(q) min in","display":"\\min_{a\\in A(q)}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-73ed63492f6711c5","tex":"\\min_{d\\in D(q)}","aliases":"min_d∈ D(q) min in","display":"\\min_{d\\in D(q)}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":91},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":72},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":36},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-bbf62fa0f1dd5f2f","tex":"\\mathcal Q_h","aliases":"Q_h mathcal","display":"\\mathcal Q_h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":42},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":31},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":31},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-7151afe2b5220a1a","tex":"\\ell_d","aliases":"ℓ_d ell","display":"\\ell_d","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":7},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":36},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":37},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":39},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0876d820c228b469","tex":"\\sum_{o:\\bar z_{a,o}(q)>0}","aliases":"∑_o:bar z_a,o(q)>0 sum bar","display":"\\sum_{o:\\bar z_{a,o}(q)>0}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":44},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2876747ca029362d","tex":"\\mathbf1^\\top","aliases":"1^top mathbf top","display":"\\mathbf1^\\top","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ca30d7bde7b966d9","tex":"\\mathbf1_X^\\top","aliases":"1_X^top mathbf top","display":"\\mathbf1_X^\\top","kind":"use","excerpt":"Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point m","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c907afa1681c3705","tex":"\\mathbf1_Z^\\top","aliases":"1_Z^top mathbf top","display":"\\mathbf1_Z^\\top","kind":"use","excerpt":"Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point m","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0014cfc2591bdee6","tex":"a_*","aliases":"a_*","display":"a_*","kind":"use","excerpt":"For a true minimizer a_*, R(widehat a)≤widehat R(widehat a)+ξ ≤widehat R(a_*)+ξ≤ R(a_*)+2ξ. The loss range supplies the cap one. Finally, for any other a, subtract the two error bounds to obtain R(a)-R(widehat a)≥widehat R(a)-widehat R(widehat a)-2ξ>0. source[GA], ``A common quantitative grounding certificate''; [MU], ``A joint posterior and recipi","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ee19b85346d2e7d2","tex":"A_A","aliases":"A_A","display":"A_A","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be finite dimensional, let T_a^θ be finitely many candidate linear operators, and let A_a be one complete candidate system chosen as reference. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear rea","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6c1908f93f8dee44","tex":"A_B","aliases":"A_B","display":"A_B","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5a8e57a002dc32f4","tex":"a_d","aliases":"a_d","display":"a_d","kind":"use","excerpt":"Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined b","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"E_μ[1_A_jmid H_j]≤ d_j μ-almost surely, for every j.","count":9},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5b950cf09bc94cd2","tex":"A_{\\rm end}","aliases":"A_rm end rm","display":"A_{\\rm end}","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4f6eb91cb1eca840","tex":"A_{\\rm run}","aliases":"A_rm run rm","display":"A_{\\rm run}","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-82821bc2124f1658","tex":"A_t","aliases":"A_t","display":"A_t","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-45d4500122db1f19","tex":"\\addcontentsline","aliases":"addcontentsline","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4afb314fdcd4106f","tex":"\\arraybackslash","aliases":"arraybackslash","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"J_m(o,z)=L_m(o)B(o,z), Q_m(z)=∑_o J_m(o,z).","count":39},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"J_c≤widetilde J_c+Bd ≤widetilde J_b+Bd ≤ J_b+2Bd.","count":63},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-eebc9b85b3d9979a","tex":"\\bar D","aliases":"bar D bar","display":"\\bar D","kind":"use","excerpt":"[A quotient that preserves legal operations] For a surjection π:X→ Z, quotient labels, enabled sets, and successors satisfying ℓ=barℓπ, D=bar Dπ, π F(x,a)=bar F(π x,a) exist exactly when equal fibres have equal labels, equal enabled sets, and equal projected successors for every enabled action. The quotient data are unique and preserve every legal ","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-52555ee0fe55f126","tex":"\\bar F","aliases":"bar F bar","display":"\\bar F","kind":"use","excerpt":"[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a(","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0ddd6936013fb39a","tex":"\\bar F_a","aliases":"bar F_a bar","display":"\\bar F_a","kind":"use","excerpt":"[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a(","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-fad6ae0fd6c0acb8","tex":"\\bar K","aliases":"bar K bar","display":"\\bar K","kind":"use","excerpt":"C K_a,o=bar K_a,oC.","count":11},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d9966b1737faabea","tex":"\\bar K_{a,o}","aliases":"bar K_a,o bar","display":"\\bar K_{a,o}","kind":"use","excerpt":"C K_a,o=bar K_a,oC.","count":9},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-742fe606e2cdb4d4","tex":"\\bar K_a^\\top","aliases":"bar K_a^top bar top","display":"\\bar K_a^\\top","kind":"use","excerpt":"nique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly positive coarse likelihoods for every prior. At a stable partition the probability of reaching each coarse block depends only on the current block. These sums define a stochastic coarse tran","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8f8a4d0a063c4065","tex":"\\bar Q","aliases":"bar Q bar","display":"\\bar Q","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also pre","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"use","excerpt":"havioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a([x])=[F_ax] and its read","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2311bbf2ac9fa5af","tex":"\\bar u","aliases":"bar u bar","display":"\\bar u","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-38f0a5c7c65b9678","tex":"\\bar u_h","aliases":"bar u_h bar","display":"\\bar u_h","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-028d200d0ea01a8b","tex":"\\bar u_\\varnothing","aliases":"bar u_∅ bar varnothing","display":"\\bar u_\\varnothing","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d97fba459e62ed0e","tex":"\\bar\\ell","aliases":"barℓ bar ell","display":"\\bar\\ell","kind":"use","excerpt":"For minimality, take any admissible refinement η of π_0. It has at least two blocks. Fix a strictly positive fine likelihood ℓ and a strictly positive coarse likelihood barℓ inducing its assumed update. For states x,y in one block and z in a different block, compare priors assigning half their mass to x,z and to y,z. Their coarse priors agree. Equa","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"An agent given only y can predict every future display under repeated flips: after k flips it is y⊕(kbmod2). It cannot determine the current physical bit. Worlds (0,0,0) and (1,1,0) share display zero and have different targets. The obstruction is the fibre of the available record, not a shortage of computational effort.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"∑_x p(x)ℓ_d(π x)=∑_z(Cp)(z)ℓ_d(z).","count":67},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":14},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"TV(C'Kp,Lq) ≤ TV(Cp,q)+ε.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0; in the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0; in the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"use","excerpt":"tion, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegativ","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-efafb53fff3f1cb9","tex":"\\mathcal C_h","aliases":"C_h mathcal","display":"\\mathcal C_h","kind":"use","excerpt":"Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c591dfc771720384","tex":"c_{\\max}","aliases":"c_max max","display":"c_{\\max}","kind":"use","excerpt":"te planning must not share one unqualified guarantee. Fix a finite horizon H and one finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-state and action/report path in both models, taking values in [0,B]. For example, a terminal loss in [0,1] and ","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-42eb4bc96455bb91","tex":"C_\\pi","aliases":"C_π pi","display":"C_\\pi","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2949436ae89cb480","tex":"\\chapter","aliases":"chapter","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6de3eb080886bbf6","tex":"D_a^\\theta","aliases":"D_a^θ theta","display":"D_a^\\theta","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3b858f3e8665ea65","tex":"D_e","aliases":"D_e","display":"D_e","kind":"use","excerpt":"One investigation, one resource budget. questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X,","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5341c066ee58261e","tex":"d_H","aliases":"d_H","display":"d_H","kind":"use","excerpt":"d_H=min(1,δ+∑_t=0^H-1ε_t).","count":23},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"E_μ[1_A_jmid H_j]≤ d_j μ-almost surely, for every j.","count":3},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"use","excerpt":"Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined b","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"TV(P(·mid E),Q(·mid E)) ≤min(1,fracTV(P,Q)max(p,q)).","count":62},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"use","excerpt":"E_μ[1_A_jmid H_j]≤ d_j μ-almost surely, for every j.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"V_e,o={x∈ V:o_e(x)=o}.","count":31},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"[Minimal exact memory for a correction contract] Equality of Σ is a congruence for every update V↦ V∩ E_0. The signature supports the current target set and its exact update after every finite evidence sequence. Any deterministic retained state supporting the same readouts and updates on all candidate subsets must distinguish every two distinct sig","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e963eb97b7c59e43","tex":"\\mathbb E_\\mu","aliases":"E_μ mathbb mu","display":"\\mathbb E_\\mu","kind":"use","excerpt":"E_μ[1_A_jmid H_j]≤ d_j μ-almost surely, for every j.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"f(s_a,o)=f(s)_a,o","count":9},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-48dc92a98cff75ec","tex":"f_e","aliases":"f_e","display":"f_e","kind":"use","excerpt":", one resource budget questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a determinist","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f063dc7c1c3b3019","tex":"\\fontsize","aliases":"fontsize","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.","count":28},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that ","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also pre","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"A terminal artifact a is acceptable only if every fixed recipient succeeds in every state of V. Its worst-case terminal cost is g_a(V), computed componentwise. A controller is a literal finite tree with one record per experiment or terminal node. Stored memory counts all branches, including those not visited in one execution.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5f29bf5e40e3c97b","tex":"G_{\\tau(a)}","aliases":"G_τ(a) tau","display":"G_{\\tau(a)}","kind":"use","excerpt":"A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a5620dda8c46dbb1","tex":"G_{\\tau(w)}","aliases":"G_τ(w) tau","display":"G_{\\tau(w)}","kind":"use","excerpt":"[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"d_H=min(1,δ+∑_t=0^H-1ε_t).","count":41},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8c7da95775a0626f","tex":"H_j","aliases":"H_j","display":"H_j","kind":"use","excerpt":"E_μ[1_A_jmid H_j]≤ d_j μ-almost surely, for every j.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5bc59e2e502080d9","tex":"\\mathcal H_{j-1}","aliases":"H_j-1 mathcal","display":"\\mathcal H_{j-1}","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4936b070e854f399","tex":"\\hfill","aliases":"hfill","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-7a664dd94a0911d8","tex":"\\hrule","aliases":"hrule","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-387f38a221f45ced","tex":"\\hypersetup","aliases":"hypersetup","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"I=P_0∩ker(x↦(l_o(x))_o).","count":26},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"|J_i-widetilde J_i|≤ BTV(P_i,Q_i).","count":51},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of ","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0475b2ac7d225af5","tex":"i_2","aliases":"i_2","display":"i_2","kind":"use","excerpt":"Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of ","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-34af8cf33f9487de","tex":"I_U","aliases":"I_U","display":"I_U","kind":"use","excerpt":"[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f5bae71a192fc64d","tex":"I_V","aliases":"I_V","display":"I_V","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-bdedc0251851d395","tex":"I_Z","aliases":"I_Z","display":"I_Z","kind":"use","excerpt":"be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"use","excerpt":":reconstruction There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv on im A. If Av=Aw, then","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"E_μ[1_A_jmid H_j]≤ d_j μ-almost surely, for every j.","count":19},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-88fbeb34dbc66890","tex":"J_b","aliases":"J_b","display":"J_b","kind":"use","excerpt":"J_c≤widetilde J_c+Bd ≤widetilde J_b+Bd ≤ J_b+2Bd.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-365e4afde3f07c22","tex":"J_c","aliases":"J_c","display":"J_c","kind":"use","excerpt":"J_c≤widetilde J_c+Bd ≤widetilde J_b+Bd ≤ J_b+2Bd.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-11d88ad682117768","tex":"J_i","aliases":"J_i","display":"J_i","kind":"use","excerpt":"|J_i-widetilde J_i|≤ BTV(P_i,Q_i).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-88e96a5d290eef2b","tex":"J_m","aliases":"J_m","display":"J_m","kind":"use","excerpt":"J_m(o,z)=L_m(o)B(o,z), Q_m(z)=∑_o J_m(o,z).","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"C K_a,o=bar K_a,oC.","count":38},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state n","count":16},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6cc92a51c6969e2c","tex":"K_a","aliases":"K_a","display":"K_a","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-16b5593bbf90ad19","tex":"K_{a,o}","aliases":"K_a,o","display":"K_{a,o}","kind":"use","excerpt":"C K_a,o=bar K_a,oC.","count":17},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0a0b4681c427fb78","tex":"K_a^\\top","aliases":"K_a^top top","display":"K_a^\\top","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"I=P_0∩ker(x↦(l_o(x))_o).","count":13},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"J_m(o,z)=L_m(o)B(o,z), Q_m(z)=∑_o J_m(o,z).","count":37},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"fracl(x)l(x)+l(z)=fracl(y)l(y)+l(z).","count":12},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-73f17951391086e7","tex":"L_*","aliases":"L_*","display":"L_*","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":14},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a716b0894950a10a","tex":"L_i","aliases":"L_i","display":"L_i","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-09eb22fcb1ad8180","tex":"L_{k+1}","aliases":"L_k+1","display":"L_{k+1}","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"use","excerpt":"J_m(o,z)=L_m(o)B(o,z), Q_m(z)=∑_o J_m(o,z).","count":11},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2eaab2297ad09dd7","tex":"l_o","aliases":"l_o","display":"l_o","kind":"use","excerpt":"I=P_0∩ker(x↦(l_o(x))_o).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-987eab22f86d841c","tex":"\\LARGE","aliases":"LARGE","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-bda22e23ac42a073","tex":"\\Large","aliases":"Large","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3b3d7f267bfbab0b","tex":"\\large","aliases":"large","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"W(T)≥ 1+lceillog_2 |V|rceil, records(T)≥ 2|V|-1.","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-45e90aa85790529d","tex":"\\linewidth","aliases":"linewidth","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"W(T)≥ 1+lceillog_2 |V|rceil, records(T)≥ 2|V|-1.","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"W(T)≥ 1+lceillog_2 |V|rceil, records(T)≥ 2|V|-1.","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"M.solve(h,V)={p_|V|}.","count":5},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"J_m(o,z)=L_m(o)B(o,z), Q_m(z)=∑_o J_m(o,z).","count":42},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-7f54f92432b51276","tex":"m_o","aliases":"m_o","display":"m_o","kind":"use","excerpt":"athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasib","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-dc6fa4cc1dd82d35","tex":"\\mainmatter","aliases":"mainmatter","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"TV(P(·mid E),Q(·mid E)) ≤min(1,fracTV(P,Q)max(p,q)).","count":15},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d441c86815baa4be","tex":"\\max_o","aliases":"max_o max","display":"\\max_o","kind":"use","excerpt":"m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty ","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"ρ(omid z)=frac∑_m p(m)J_m(o,z)∑_m p(m)Q_m(z)","count":33},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3f9db2ae5742966b","tex":"\\midrule","aliases":"midrule","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-1e2deab3af39b182","tex":"\\min_a","aliases":"min_a min","display":"\\min_a","kind":"use","excerpt":"[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add a","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-13d3a20824a03360","tex":"\\min_{w\\in W}","aliases":"min_w∈ W min in","display":"\\min_{w\\in W}","kind":"use","excerpt":"[Which gains survive every admitted price?] The saving w·Δ is nonnegative for every nonnegative price vector if and only if every coordinate of Δ is nonnegative. If prices lie in the convex hull of a finite nonempty set W, their minimum saving equals min_w∈ Ww·Δ. A componentwise nonnegative vector has nonnegative inner product with every nonnegativ","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that ","count":7},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"p_n=(1+lceillog_2 nrceil,2n-1).","count":45},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8bbbd92a94033296","tex":"n'","aliases":"n'","display":"n'","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive me","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3554f05e6897e490","tex":"O_e","aliases":"O_e","display":"O_e","kind":"use","excerpt":"belch:inquiry questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic succe","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-b2bdbcfb9c9cfc41","tex":"o_e","aliases":"o_e","display":"o_e","kind":"use","excerpt":"V_e,o={x∈ V:o_e(x)=o}.","count":5},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2a82579e1e2588f5","tex":"O_{t+1}","aliases":"O_t+1","display":"O_{t+1}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"I=P_0∩ker(x↦(l_o(x))_o).","count":40},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitte","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"use","excerpt":"An approximate model supports a different, quantitative claim. Exact policy transport and approximate planning must not share one unqualified guarantee. Fix a finite horizon H and one finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-state and ac","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","excerpt":"The theorem permits removal of an independent nuisance flag. It does not permit removal of every association. Consider two priors on (b,s), with rows indexed by b and columns by s: P=1/2 · 0 0 · 1/2, P'=0 · 1/2 1/2 · 0. Both individual marginals are uniform. A noisy calibration reports s correctly with probability 3/4, independently conditional on ","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"I=P_0∩ker(x↦(l_o(x))_o).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the report probabilities and the aggregated positive-probability posteriors: z_a,o(p)=bar z_a,o(Cp), CΨ_a,o(p)=barΨ_a,o(Cp). Starting from Cp_0","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5c39c7933673208b","tex":"p_E","aliases":"p_E","display":"p_E","kind":"use","excerpt":"d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, th","count":14},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-b9e70c157342a064","tex":"P_H","aliases":"P_H","display":"P_H","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5627e8b6e58a3075","tex":"\\mathcal P_h","aliases":"P_h mathcal","display":"\\mathcal P_h","kind":"use","excerpt":"The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitte","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"|J_i-widetilde J_i|≤ BTV(P_i,Q_i).","count":11},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-346cf456a77b1ce5","tex":"p_n","aliases":"p_n","display":"p_n","kind":"use","excerpt":"p_n=(1+lceillog_2 nrceil,2n-1).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-cc9e0dabacaadc8b","tex":"p_{N-1}","aliases":"p_N-1","display":"p_{N-1}","kind":"use","excerpt":"Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that ","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-fe0059a8d2b46448","tex":"P_{\\rm av}","aliases":"P_rm av rm","display":"P_{\\rm av}","kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-43031e8fba4068f2","tex":"P_{\\rm det}","aliases":"P_rm det rm","display":"P_{\\rm det}","kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-18a36bddef523d4c","tex":"p_{|V|}","aliases":"p_|V|","display":"p_{|V|}","kind":"use","excerpt":"M.solve(h,V)={p_|V|}.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4bc9c56157a62dc5","tex":"\\operatorname{pa}","aliases":"pa operatorname","display":"\\operatorname{pa}","kind":"use","excerpt":"Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined b","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-bc660fe2b7285e52","tex":"\\par","aliases":"par","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-bfd0b07cc3c57995","tex":"\\operatorname{Pareto}","aliases":"Pareto operatorname","display":"\\operatorname{Pareto}","kind":"use","excerpt":"The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitte","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e695ee90e9b92c62","tex":"\\operatorname{Post}","aliases":"Post operatorname","display":"\\operatorname{Post}","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d692f9ac1c18626a","tex":"\\Pow","aliases":"Pow P mathcal","display":"{\\mathcal P}","kind":"use","excerpt":"Let E⊆ P(Ω) contain Ω and be closed under intersection. In this chapter the world does not change when an evidence event is received. The update is V↦ V∩ E. Physical actions and their effects will return in Chapter~[source reference].","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive","count":12},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"J_m(o,z)=L_m(o)B(o,z), Q_m(z)=∑_o J_m(o,z).","count":51},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5950b0378700bfba","tex":"Q_i","aliases":"Q_i","display":"Q_i","kind":"use","excerpt":"|J_i-widetilde J_i|≤ BTV(P_i,Q_i).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3406acd3f0f64ef4","tex":"Q_m","aliases":"Q_m","display":"Q_m","kind":"use","excerpt":"J_m(o,z)=L_m(o)B(o,z), Q_m(z)=∑_o J_m(o,z).","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8af81721e09a3499","tex":"Q_V","aliases":"Q_V","display":"Q_V","kind":"use","excerpt":"Q_V(s,a)=c(s,a)+∑_o:P(omid s,a)>0P(omid s,a)V(s_a,o).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6e3edc954e0d904a","tex":"\\question","aliases":"question","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":28},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":26},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a3d472d51734fedb","tex":"\\mathbb R^2","aliases":"R^2 mathbb","display":"\\mathbb R^2","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2f01b4d5a5d6a367","tex":"\\mathbb R^d_{\\ge0}","aliases":"R^d_≥0 mathbb ge","display":"\\mathbb R^d_{\\ge0}","kind":"use","excerpt":"beginproposition[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be ","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e13c04e5f7fac1fc","tex":"\\raggedright","aliases":"raggedright","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"W(T)≥ 1+lceillog_2 |V|rceil, records(T)≥ 2|V|-1.","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c822b493afd80c09","tex":"\\operatorname{records}","aliases":"records operatorname","display":"\\operatorname{records}","kind":"use","excerpt":"records(T)+1=2L(T)+u(T),","count":3},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero.","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2f2fae6efa7605a9","tex":"\\rule","aliases":"rule","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":23},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"Q_V(s,a)=c(s,a)+∑_o:P(omid s,a)>0P(omid s,a)V(s_a,o).","count":70},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-46fc7c1e02ec2d2e","tex":"s_{a,o}","aliases":"s_a,o","display":"s_{a,o}","kind":"use","excerpt":"Q_V(s,a)=c(s,a)+∑_o:P(omid s,a)>0P(omid s,a)V(s_a,o).","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive me","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-7c6199f1c7d32ba5","tex":"\\selectfont","aliases":"selectfont","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-9bd6563a665b1ad1","tex":"\\small","aliases":"small","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0d20e3c4a33762b5","tex":"\\operatorname{solve}","aliases":"solve operatorname","display":"\\operatorname{solve}","kind":"use","excerpt":"M.solve(h,V)={p_|V|}.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span span operatorname","display":"{\\operatorname{span}}","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-7deace183d0a867f","tex":"\\sup_\\omega","aliases":"sup_ω sup omega","display":"\\sup_\\omega","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"|V|=L(T),","count":86},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"d_H=min(1,δ+∑_t=0^H-1ε_t).","count":41},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5d42311231194d5c","tex":"t^{-1}","aliases":"t^-1","display":"t^{-1}","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0bc6dba302face63","tex":"T^\\theta","aliases":"T^θ theta","display":"T^\\theta","kind":"use","excerpt":"For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f90055131c0edf2c","tex":"T_*","aliases":"T_*","display":"T_*","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":18},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-1a4a5aadced24804","tex":"T_a^\\theta","aliases":"T_a^θ theta","display":"T_a^\\theta","kind":"use","excerpt":"One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be finite dimensional, let T_a^θ be finitely many candidate linear operators, and let A_a be one complete candidate system chosen as reference. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear rea","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-b6d1c35d86f0d39d","tex":"T_k","aliases":"T_k","display":"T_k","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4e56fe1bda71e196","tex":"T_{k+1}","aliases":"T_k+1","display":"T_{k+1}","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ed0619d26eb6427a","tex":"T_w^\\theta","aliases":"T_w^θ theta","display":"T_w^\\theta","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also pre","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a9a485c98291d899","tex":"\\tableofcontents","aliases":"tableofcontents","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1","count":16},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3000c3bd47e8007c","tex":"\\toprule","aliases":"toprule","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV d_TV mathrm","display":"{d_{\\mathrm{TV}}}","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-17aada5f90193f08","tex":"\\operatorname{TV}","aliases":"TV operatorname","display":"\\operatorname{TV}","kind":"use","excerpt":"TV(C'Kp,Lq) ≤ TV(Cp,q)+ε.","count":5},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":16},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"u-v _1+|p-q|≤ 2TV(P,Q).","count":33},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6fdb23dc8f442414","tex":"u_h","aliases":"u_h","display":"u_h","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-9eff72e097fd075b","tex":"u_\\varnothing","aliases":"u_∅ varnothing","display":"u_\\varnothing","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"Q_V(s,a)=c(s,a)+∑_o:P(omid s,a)>0P(omid s,a)V(s_a,o).","count":80},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"u-v _1+|p-q|≤ 2TV(P,Q).","count":67},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-9bacaf6e1626666c","tex":"V_{e,o}","aliases":"V_e,o","display":"V_{e,o}","kind":"use","excerpt":"V_e,o={x∈ V:o_e(x)=o}.","count":5},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8e7676bd93a7edb5","tex":"v_o","aliases":"v_o","display":"v_o","kind":"use","excerpt":"athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasib","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-786a3c53c74db475","tex":"v_\\omega","aliases":"v_ω omega","display":"v_\\omega","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0fef838539965a6b","tex":"\\vfill","aliases":"vfill","display":null,"kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"W(T)=1+d(T).","count":14},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add a","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-967c63ebf4cd78be","tex":"\\widehat R","aliases":"widehat R widehat","display":"\\widehat R","kind":"use","excerpt":"widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f207fa5d22c484f6","tex":"\\widehat v","aliases":"widehat v widehat","display":"\\widehat v","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8307c5f38fc587f5","tex":"\\widehat v_\\omega","aliases":"widehat v_ω widehat omega","display":"\\widehat v_\\omega","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3f476ecaf7bf4a0d","tex":"\\widetilde J","aliases":"widetilde J widetilde","display":"\\widetilde J","kind":"use","excerpt":"|J_i-widetilde J_i|≤ BTV(P_i,Q_i).","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e12bbf073a97ae67","tex":"\\widetilde J_b","aliases":"widetilde J_b widetilde","display":"\\widetilde J_b","kind":"use","excerpt":"J_c≤widetilde J_c+Bd ≤widetilde J_b+Bd ≤ J_b+2Bd.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-c039364200b22983","tex":"\\widetilde J_c","aliases":"widetilde J_c widetilde","display":"\\widetilde J_c","kind":"use","excerpt":"J_c≤widetilde J_c+Bd ≤widetilde J_b+Bd ≤ J_b+2Bd.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-7bf755c5954f8109","tex":"\\widetilde J_i","aliases":"widetilde J_i widetilde","display":"\\widetilde J_i","kind":"use","excerpt":"|J_i-widetilde J_i|≤ BTV(P_i,Q_i).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-9117c60fc5e9378b","tex":"\\widetilde K","aliases":"widetilde K widetilde","display":"\\widetilde K","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8da03ee77856ac56","tex":"\\widetilde K_{a,o}","aliases":"widetilde K_a,o widetilde","display":"\\widetilde K_{a,o}","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-649a6c9973f355dd","tex":"\\widetilde p","aliases":"widetilde p widetilde","display":"\\widetilde p","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_","count":14},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4741526210fb0d8d","tex":"\\widetilde p_0","aliases":"widetilde p_0 widetilde","display":"\\widetilde p_0","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-47fb519e5af1c253","tex":"\\widetilde p_E","aliases":"widetilde p_E widetilde","display":"\\widetilde p_E","kind":"use","excerpt":"d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, th","count":12},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-26f82a5d2a875a6e","tex":"\\widetilde P_H","aliases":"widetilde P_H widetilde","display":"\\widetilde P_H","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a25d72cc47e636ba","tex":"\\widetilde\\pi","aliases":"widetildeπ widetilde pi","display":"\\widetilde\\pi","kind":"use","excerpt":"[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certi","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"fracl(x)l(x)+l(z)=fracl(y)l(y)+l(z).","count":99},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.","count":16},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"On the nonnegative coordinate cone, preservation is equivalent to nonnegative matrix entries, as is seen by applying a matrix to every standard basis vector. Because the matrices sum to I, their nonnegative off-diagonal entries must all vanish. An idempotent diagonal entry satisfies x^2=x, hence is zero or one. The diagonal sums are one, so each co","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-1e4fb4e21dd5aa54","tex":"X_{t+1}","aliases":"X_t+1","display":"X_{t+1}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"fracl(x)l(x)+l(z)=fracl(y)l(y)+l(z).","count":47},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":40},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"J_m(o,z)=L_m(o)B(o,z), Q_m(z)=∑_o J_m(o,z).","count":74},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the repor","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-9a75d558418afc42","tex":"z_{a,o}","aliases":"z_a,o","display":"z_{a,o}","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-45eb5061d9b39786","tex":"Z_H","aliases":"Z_H","display":"Z_H","kind":"use","excerpt":"e distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"|X|-|X/I|.","count":23},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"TV(P(·mid E),Q(·mid E)) ≤min(1,fracTV(P,Q)max(p,q)).","count":9},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6cb4d369799123ce","tex":"\\beta_i","aliases":"β_i beta","display":"\\beta_i","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.","count":14},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"d_H=min(1,δ+∑_t=0^H-1ε_t).","count":21},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":12},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"TV(C'Kp,Lq) ≤ TV(Cp,q)+ε.","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d08739f367199333","tex":"\\epsilon_t","aliases":"ϵ_t epsilon","display":"\\epsilon_t","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":12},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-76ab81e6b93d8a15","tex":"\\varepsilon_t","aliases":"ε_t varepsilon","display":"\\varepsilon_t","kind":"use","excerpt":"d_H=min(1,δ+∑_t=0^H-1ε_t).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"For minimality, take any admissible refinement η of π_0. It has at least two blocks. Fix a strictly positive fine likelihood ℓ and a strictly positive coarse likelihood barℓ inducing its assumed update. For states x,y in one block and z in a different block, compare priors assigning half their mass to x,z and to y,z. Their coarse priors agree. Equa","count":16},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be finite dimensional, let T_a^θ be finitely many candidate linear operators, and let A_a be one complete candidate system chosen as reference. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear rea","count":30},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state n","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"E_μ[1_A_jmid H_j]≤ d_j μ-almost surely, for every j.","count":3},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add a","count":12},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"∑_x p(x)ℓ_d(π x)=∑_z(Cp)(z)ℓ_d(z).","count":43},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-6d2412f84c728949","tex":"\\pi_*","aliases":"π_* pi","display":"\\pi_*","kind":"use","excerpt":"[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certi","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-2d28a27ac7821632","tex":"\\pi_0","aliases":"π_0 pi","display":"\\pi_0","kind":"use","excerpt":"A mandatory partition π_0 with at least two blocks protects the target distinctions. We seek a refinement that supports every prediction and each separate Bayesian update for every prior. The observation is an operation available in its own right. This is stronger than inspecting only a chosen combined transition-and-report schedule.","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-560c891f5e585ac2","tex":"\\pi_{\\rm init}","aliases":"π_rm init pi rm","display":"\\pi_{\\rm init}","kind":"use","excerpt":"[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest ","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"ρ(omid z)=frac∑_m p(m)J_m(o,z)∑_m p(m)Q_m(z)","count":11},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"A state sufficient for every later filter. Define the correction signature Σ(V)=(t(V∩ E))_E∈ E. It records what each admitted future filter would leave of the possible target values.","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"For a compression, reverse the splitting. Let C:V→ Z and R:Z→ V satisfy CR=I_Z, and put P=RC. The compressed operator is Φ(T)=CTR.","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-437405ce641f5157","tex":"\\Psi_{a,o}","aliases":"Ψ_a,o Psi","display":"\\Psi_{a,o}","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":18},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":20},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-efdf8990a98fa39d","tex":"\\omega'","aliases":"ω' omega","display":"\\omega'","kind":"use","excerpt":"[3mm]Reasoning fontsize2027selectfont Evidence, Methods, and Accountable Revision colorAccentrule1.9in2pt Large Komorebi P. vfill Large Reviewed Mathematical Core large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. Only the selected mathematical statements d","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":10},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"For conditioning, let u,v be the restrictions of the two history laws to E, with masses p_E,widetilde p_E. The difference on the complement has ℓ^1 norm at least |p_E-widetilde p_E|. Thus u-v _1+|p_E-widetilde p_E|≤2d_H. Adding and subtracting v/p_E bounds the conditional distance by d_H/p_E. Interchanging the laws gives d_H/widetilde p_E, so the s","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-313774f1667bd9ea","tex":"\\ell_o","aliases":"ℓ_o ell","display":"\\ell_o","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"I=P_0∩ker(x↦(l_o(x))_o).","count":11},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ff1f52005cd89e31","tex":"\\sum_{i:w_i>0}","aliases":"∑_i:w_i>0 sum","display":"\\sum_{i:w_i>0}","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-49afb54019149b8f","tex":"\\sum_{j\\ge0}","aliases":"∑_j≥0 sum ge","display":"\\sum_{j\\ge0}","kind":"use","excerpt":"μ(⋃_j≥0A_j)≤∑_j≥0d_j≤ D.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0c18bcf01edb7e76","tex":"\\sum_{j\\ge1}","aliases":"∑_j≥1 sum ge","display":"\\sum_{j\\ge1}","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-7dd1f8372600c494","tex":"\\sum_m","aliases":"∑_m sum","display":"\\sum_m","kind":"use","excerpt":"ρ(omid z)=frac∑_m p(m)J_m(o,z)∑_m p(m)Q_m(z)","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ccbd66bdaed18a2b","tex":"\\sum_o","aliases":"∑_o sum","display":"\\sum_o","kind":"use","excerpt":"J_m(o,z)=L_m(o)B(o,z), Q_m(z)=∑_o J_m(o,z).","count":7},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-3abe3b490f0f125b","tex":"\\sum_{o,x'}","aliases":"∑_o,x' sum","display":"\\sum_{o,x'}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d984258d08d344d9","tex":"\\sum_{o:P(o\\mid s,a)>0}","aliases":"∑_o:P(omid s,a)>0 sum mid","display":null,"kind":"use","excerpt":"Q_V(s,a)=c(s,a)+∑_o:P(omid s,a)>0P(omid s,a)V(s_a,o).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f1e914837dbf8c89","tex":"\\sum_{t=0}^{H-1}","aliases":"∑_t=0^H-1 sum","display":"\\sum_{t=0}^{H-1}","kind":"use","excerpt":"d_H=min(1,δ+∑_t=0^H-1ε_t).","count":3},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0e44b9be25437975","tex":"\\sum_x","aliases":"∑_x sum","display":"\\sum_x","kind":"use","excerpt":"∑_x p(x)ℓ_d(π x)=∑_z(Cp)(z)ℓ_d(z).","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-31e59f15403939ba","tex":"\\sum_{x':\\pi(x')=z'}","aliases":"∑_x':π(x')=z' sum pi","display":"\\sum_{x':\\pi(x')=z'}","kind":"use","excerpt":"[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the repor","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-f05731197cd413ab","tex":"\\sum_{x'\\in B}","aliases":"∑_x'∈ B sum in","display":"\\sum_{x'\\in B}","kind":"use","excerpt":"[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest ","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-5966b9097878350e","tex":"\\sum_z","aliases":"∑_z sum","display":"\\sum_z","kind":"use","excerpt":"∑_x p(x)ℓ_d(π x)=∑_z(Cp)(z)ℓ_d(z).","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-ea8a48c6147135b3","tex":"\\sum_\\omega","aliases":"∑_ω sum omega","display":"\\sum_\\omega","kind":"use","excerpt":"widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"I=P_0∩ker(x↦(l_o(x))_o).","count":15},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"μ(A_j)=∫ 1_A_j dμ =∫ E_μ[1_A_jmid H_j] dμ ≤∫ d_j dμ=d_j.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":8},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"L(T)≤ 2^d(T), d(T)≤ h.","count":36},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"TV(C'Kp,Lq) ≤ TV(Cp,q)+ε.","count":6},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"W(T)≥ 1+lceillog_2 |V|rceil, records(T)≥ 2|V|-1.","count":20},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also pre","count":26},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"The sensor used throughout the book. Write a world as ω=(b,s,n)∈{0,1}^3. The physical target bit is b. The sensor convention is s. The nuisance bit n never affects the operations or outputs of the initial contract. The display is y=b⊕ s, where ⊕ is addition modulo two. A permitted control f flips the physical bit and leaves s,n unchanged. Consequen","count":12},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).","count":4},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-892cecbf386dea45","tex":"\\bigcap_i","aliases":"⋂_i bigcap","display":"\\bigcap_i","kind":"use","excerpt":"[Target-relative joint reconstruction] There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv ","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-9d31d41a627eed08","tex":"\\bigcap_{x:\\alpha(x)=h}","aliases":"⋂_x:α(x)=h bigcap alpha","display":"\\bigcap_{x:\\alpha(x)=h}","kind":"use","excerpt":"Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).","count":2},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"μ(⋃_j≥0A_j)≤∑_j≥0d_j≤ D.","count":1},{"work":"monographs-machine-native-reasoning","url":"notation/monographs-machine-native-reasoning.html#symbol-bc606610d429e31c","tex":"\\bigcup_{j\\ge0}","aliases":"⋃_j≥0 bigcup ge","display":"\\bigcup_{j\\ge0}","kind":"use","excerpt":"μ(⋃_j≥0A_j)≤∑_j≥0d_j≤ D.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"notation-table","excerpt":"β(f),β^*(f) · Complete-cover degree and its affine-output optimization.","count":62},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"notation-table","excerpt":"midrule C(f) · Minimum fan-in-two AND/OR/NOT gate count, allowing fan-out.","count":76},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"notation-table","excerpt":"midrule C(f) · Minimum fan-in-two AND/OR/NOT gate count, allowing fan-out.","count":142},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"notation-table","excerpt":"I_f · Boolean functions supported only on positive inputs of f.","count":113},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-997181a4d855ed8f","tex":"I_f","aliases":"I_f","display":"I_f","kind":"notation-table","excerpt":"I_f · Boolean functions supported only on positive inputs of f.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-155728639f769fe0","tex":"\\KTrest","aliases":"KTrest","display":null,"kind":"notation-table","excerpt":"KTrest(f) · Minimum such Toffoli count, allowing any finite zero workspace.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","excerpt":"L(f) · Minimum total NOT/CNOT/Toffoli count of a restored implementation.","count":154},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"notation-table","excerpt":"M(f) · Minimum AND count in an XOR--AND computation, with affine operations uncharged.","count":89},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"notation-table","excerpt":"Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|","count":616},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"[P]_2 · Homogeneous degree-two component of the multilinear polynomial P in the specified coordinates.","count":306},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"notation-table","excerpt":"Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-793e8333d34ceb82","tex":"\\Sp","aliases":"Sp","display":null,"kind":"notation-table","excerpt":"Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|","count":13},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"notation-table","excerpt":"T · Number of Toffoli gates in a specified implementation.","count":123},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"notation-table","excerpt":"β_v(f) · Minimum degree of a member of I_f that is one at v.","count":211},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"notation-table","excerpt":"Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|","count":119},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"notation-table","excerpt":"α_1(f) · Minimum degree of one nonzero member of I_f.","count":57},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"notation-table","excerpt":"α_1(f) · Minimum degree of one nonzero member of I_f.","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"notation-table","excerpt":"β_v(f) · Minimum degree of a member of I_f that is one at v.","count":86},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bd66c98b32c8ec9d","tex":"\\beta^*","aliases":"β^* beta","display":"\\beta^*","kind":"notation-table","excerpt":"β(f),β^*(f) · Complete-cover degree and its affine-output optimization.","count":28},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-68aa4544acddd88f","tex":"\\beta_v","aliases":"β_v beta","display":"\\beta_v","kind":"notation-table","excerpt":"β_v(f) · Minimum degree of a member of I_f that is one at v.","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":",c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.","count":426},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":1108},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":280},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":216},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-74cca85a5b12f4be","tex":"\\mathrm{amb}","aliases":"amb mathrm","display":"\\mathrm{amb}","kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":17},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":126},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0eaa1204dfcfa445","tex":"\\mathbf c","aliases":"c mathbf","display":"\\mathbf c","kind":"definition-passage","excerpt":"[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":171},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"definition-passage","excerpt":"[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"definition-passage","excerpt":"[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5a8099e14b561f14","tex":"\\CX","aliases":"CX","display":null,"kind":"definition-passage","excerpt":"[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.","count":19},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-11ceeb994a2dc6d2","tex":"\\CX_{i\\to j}","aliases":"CX_i→ j CX to","display":null,"kind":"definition-passage","excerpt":"[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-278103e84fa8e88e","tex":"\\CZ","aliases":"CZ","display":null,"kind":"definition-passage","excerpt":"[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-82247e11a0d4696e","tex":"\\CZ_{ij}","aliases":"CZ_ij CZ","display":null,"kind":"definition-passage","excerpt":"[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.","count":111},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a","count":220},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d3405c1cc6903854","tex":"D_n","aliases":"D_n","display":"D_n","kind":"definition-passage","excerpt":"Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.","count":15},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ad4185b7fb435e1b","tex":"\\DeltaDef","aliases":"DeltaDef","display":null,"kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":18},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"definition-passage","excerpt":"[Catalytic excess] For a normal form with r matched excursions and C useful cross interactions, define E=C-2r when each consumer contributes one unit and each opening or closing costs one unit. Positive excess is necessary in the simplest transvection model, but it is not sufficient. An all-cross configuration may admit a mirror word of equal restr","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-586662d51e814227","tex":"\\ellAll","aliases":"ellAll","display":null,"kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":62},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d0dc07d3d401efa0","tex":"\\ellAmb","aliases":"ellAmb","display":null,"kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":15},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0175f94ec32f9fd7","tex":"\\ellBudget","aliases":"ellBudget","display":null,"kind":"definition-passage","excerpt":"[Upper-gate budget] For q≥0 and A∈U_n(F), define ellBudgetq(A) = min{|w|:[w]=A, u(w)≤ q}. Thus ellBudget0(A)=ellDown(A), ellBudgetq+1(A)≤ellBudgetq(A), ellAll(A)≤ellBudgetq(A).","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c68a13f648d86b97","tex":"\\ellDown","aliases":"ellDown","display":null,"kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":64},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-159df96fef451807","tex":"\\ellRes","aliases":"ellRes","display":null,"kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":17},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F","display":null,"kind":"definition-passage","excerpt":"[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).","count":131},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":105},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":12},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-09f3e72028e864cd","tex":"G_m","aliases":"G_m","display":"G_m","kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b4f9811bf3c61e66","tex":"G_t","aliases":"G_t","display":"G_t","kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":18},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a","count":124},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":82},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d3cba85783189f4d","tex":"H_i","aliases":"H_i","display":"H_i","kind":"definition-passage","excerpt":"[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.","count":18},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-333bf23be8bc780a","tex":"h_i","aliases":"h_i","display":"h_i","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-082db4c82cd55c3c","tex":"h_m","aliases":"h_m","display":"h_m","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f5b2a3780f3d2aaf","tex":"\\Hh","aliases":"Hh","display":null,"kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":23},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Flagged matroid tour] Let M be a representable matroid with designated initial basis, target basis, and consumer states. A flagged tour is a sequence of basis exchanges or transvections in which the ith carrier remains inside a specified filtration layer. An ambient tour drops this constraint. The six-row atom realizes the graphic regular matroid ","count":190},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"definition-passage","excerpt":"te model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realiza","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cd60629dee2b50e9","tex":"\\operatorname{Int}","aliases":"Int operatorname","display":"\\operatorname{Int}","kind":"definition-passage","excerpt":"[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":100},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b532789d6ddb0be6","tex":"L_v","aliases":"L_v","display":"L_v","kind":"definition-passage","excerpt":"[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"definition-passage","excerpt":"[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a7109fcc0efd3c8f","tex":"\\lim_{r\\to\\infty}","aliases":"lim_r→∞ lim to infty","display":"\\lim_{r\\to\\infty}","kind":"definition-passage","excerpt":"[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f0042c6cf9ee0b72","tex":"\\mathfrak M","aliases":"M mathfrak","display":"\\mathfrak M","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Tensor synergy] Define Syn(U,V) = nellAmb(U)+mellAmb(V)-ellAmb(U⊗ V). Positive synergy means that the joint ambient realization improves on independent lifted circuits. Restricted tree-tight metrics have zero tensor synergy. Ambient synergy is a possible source of collective activation.","count":276},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ae8d8fa77a517e25","tex":"m^{r-1}","aliases":"m^r-1","display":"m^{r-1}","kind":"definition-passage","excerpt":"[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d964d3e99666ab79","tex":"\\mathfrak M_{\\mathrm{amb}}","aliases":"M_amb mathfrak mathrm","display":"\\mathfrak M_{\\mathrm{amb}}","kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4d169601583f0ad4","tex":"\\mathfrak M_{\\mathrm{res}}","aliases":"M_res mathfrak mathrm","display":"\\mathfrak M_{\\mathrm{res}}","kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"definition-passage","excerpt":"Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.","count":25},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5d77bde68920bec8","tex":"\\overline{\\mathcal W}","aliases":"overline W overline mathcal","display":"\\overline{\\mathcal W}","kind":"definition-passage","excerpt":"Approximate synthesis and border defects. For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c78f6cd66c4bff5a","tex":"\\overline{\\mathcal W}_{L}","aliases":"overline W_L overline mathcal","display":"\\overline{\\mathcal W}_{L}","kind":"definition-passage","excerpt":"Approximate synthesis and border defects. For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"definition-passage","excerpt":"[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":102},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3d9161e8faa2bd53","tex":"\\mathcal P_{\\mathfrak M}","aliases":"P_ M mathcal mathfrak","display":"\\mathcal P_{\\mathfrak M}","kind":"definition-passage","excerpt":"[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":18},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-12b33428971a165a","tex":"P_t","aliases":"P_t","display":"P_t","kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":13},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-71a2a7c8e27d6152","tex":"P_{t-1}","aliases":"P_t-1","display":"P_{t-1}","kind":"definition-passage","excerpt":"[Packet schedule] For a circuit U=G_m⋯ G_1 on a Hilbert space Hh, a packet schedule is a sequence of orthogonal projectors P_0,P_1,…,P_m. It is exact when G_tP_t-1=P_tG_tP_t-1 (1≤ t≤ m).","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a","count":186},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.","count":147},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.","count":193},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"definition-passage","excerpt":"Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-257791562a17d424","tex":"R_v","aliases":"R_v","display":"R_v","kind":"definition-passage","excerpt":"[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-944be825dd3a726a","tex":"\\R_{\\ge0}","aliases":"R_≥0 R ge","display":null,"kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-71197176f19316fe","tex":"\\R_{\\ge0}^r","aliases":"R_≥0^r R ge","display":null,"kind":"definition-passage","excerpt":"[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank","display":null,"kind":"definition-passage","excerpt":"[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3695272564a51a9c","tex":"\\mathrm{res}","aliases":"res mathrm","display":"\\mathrm{res}","kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":17},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":143},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":13},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":136},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":26},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-430721124e951232","tex":"\\mathcal S_{\\mathrm{amb}}","aliases":"S_amb mathcal mathrm","display":"\\mathcal S_{\\mathrm{amb}}","kind":"definition-passage","excerpt":"[Cost-preserving equivalence] Targets U and V are equivalent if an automorphism φ of the ambient realization system satisfies φ(U)=V, φ( S_res)= S_res, φ( S_amb)= S_amb, and preserves the gate costs. Under this equivalence, both lengths and the defect are orbit invariants. A useful threshold theorem should classify minimal defective orbits, not mer","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e2272e31d4eaf8ee","tex":"\\mathcal S_C","aliases":"S_C mathcal","display":"\\mathcal S_C","kind":"definition-passage","excerpt":"[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-698cd4222a287d75","tex":"S_i","aliases":"S_i","display":"S_i","kind":"definition-passage","excerpt":"[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-17d9f30419ac2010","tex":"s_L","aliases":"s_L","display":"s_L","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2f60b290dee7f333","tex":"S_n^*","aliases":"S_n^*","display":"S_n^*","kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7d3276f6eb859da7","tex":"S_n^\\downarrow","aliases":"S_n^downarrow downarrow","display":"S_n^\\downarrow","kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a38eec2859897799","tex":"\\mathcal S_P","aliases":"S_P mathcal","display":"\\mathcal S_P","kind":"definition-passage","excerpt":"[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-649ae432b13c7bd3","tex":"\\mathcal S_{\\mathrm{res}}","aliases":"S_res mathcal mathrm","display":"\\mathcal S_{\\mathrm{res}}","kind":"definition-passage","excerpt":"[Cost-preserving equivalence] Targets U and V are equivalent if an automorphism φ of the ambient realization system satisfies φ(U)=V, φ( S_res)= S_res, φ( S_amb)= S_amb, and preserves the gate costs. Under this equivalence, both lengths and the defect are orbit invariants. A useful threshold theorem should classify minimal defective orbits, not mer","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"definition-passage","excerpt":"Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-020c17c2ab7cbabb","tex":"\\sup_{U\\in\\mathcal T_n}","aliases":"sup_U∈ T_n sup in mathcal","display":"\\sup_{U\\in\\mathcal T_n}","kind":"definition-passage","excerpt":"Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-17b391d057c23c7b","tex":"\\Syn","aliases":"Syn","display":null,"kind":"definition-passage","excerpt":"[Tensor synergy] Define Syn(U,V) = nellAmb(U)+mellAmb(V)-ellAmb(U⊗ V). Positive synergy means that the joint ambient realization improves on independent lifted circuits. Restricted tree-tight metrics have zero tensor synergy. Ambient synergy is a possible source of collective activation.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a","count":191},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bb69253aa74e4ac4","tex":"\\mathcal T_n","aliases":"T_n mathcal","display":"\\mathcal T_n","kind":"definition-passage","excerpt":"Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"begindefinition[Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The val","count":161},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-300f218594b1347a","tex":"\\U","aliases":"U","display":null,"kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":66},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Upper-gate budget] For q≥0 and A∈U_n(F), define ellBudgetq(A) = min{|w|:[w]=A, u(w)≤ q}. Thus ellBudget0(A)=ellDown(A), ellBudgetq+1(A)≤ellBudgetq(A), ellAll(A)≤ellBudgetq(A).","count":126},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-af5759fc501d02ce","tex":"U^{\\otimes r}","aliases":"U^⊗ r otimes","display":"U^{\\otimes r}","kind":"definition-passage","excerpt":"[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-167a8b5df4139bcf","tex":"U_n","aliases":"U_n","display":"U_n","kind":"definition-passage","excerpt":"[Critical catalyst penalty] For a target family U_n, define λ_c(U_n) = sup{λ:ℓ_λ,amb(U_n)<ellRes(U_n)}. Determining λ_c exactly is a refined synthesis problem. It can be more physically meaningful than unweighted gate count.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-51060ad800483114","tex":"\\U_n","aliases":"U_n U","display":null,"kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":46},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bd11ffa7d66071e4","tex":"\\underline\\Delta","aliases":"underlineΔ underline Delta","display":"\\underline\\Delta","kind":"definition-passage","excerpt":"pproximate synthesis and border defects For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= un","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d1536a015093f505","tex":"\\underline\\ell","aliases":"underlineℓ underline ell","display":"\\underline\\ell","kind":"definition-passage","excerpt":"Approximate synthesis and border defects. For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-166db2fe610198ca","tex":"\\underline\\ell_{\\mathrm{amb}}","aliases":"underlineℓ_amb underline ell mathrm","display":"\\underline\\ell_{\\mathrm{amb}}","kind":"definition-passage","excerpt":"pproximate synthesis and border defects For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= un","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6d810bcd90e0c723","tex":"\\underline\\ell_{\\mathrm{res}}","aliases":"underlineℓ_res underline ell mathrm","display":"\\underline\\ell_{\\mathrm{res}}","kind":"definition-passage","excerpt":"pproximate synthesis and border defects For continuous gate models, exact word length can differ from the minimum length achievable as a limit. [Border cost] Let overline W_L be the topological or Zariski closure of targets realizable with cost at most L. The border cost is underlineℓ(U)=inf{L:U∈overline W_L}. The border defect is underlineΔ(U)= un","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Cost-preserving equivalence] Targets U and V are equivalent if an automorphism φ of the ambient realization system satisfies φ(U)=V, φ( S_res)= S_res, φ( S_amb)= S_amb, and preserves the gate costs. Under this equivalence, both lengths and the defect are orbit invariants. A useful threshold theorem should classify minimal defective orbits, not mer","count":47},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":87},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a","count":83},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-09868efb784faa54","tex":"\\mathcal X","aliases":"X mathcal","display":"\\mathcal X","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":242},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"definition-passage","excerpt":"Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"definition-passage","excerpt":"Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8f1384780cc6224d","tex":"X_L","aliases":"X_L","display":"X_L","kind":"definition-passage","excerpt":"Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"definition-passage","excerpt":"Excursion budgets. Counting ambient-only gates is only one measure of how far a circuit leaves its restricted model. [Excursion profile] For a circuit path X_0,X_1,…,X_L, an excursion profile may include item the number q of ambient-only gates, item the maximum Schreier distance d(X_tH,H), item the rank of a forbidden block, item the bandwidth of a","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":35},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aa73a0d73b15269b","tex":"|_{S_n^*(R)}","aliases":"|_S_n^*(R)","display":"|_{S_n^*(R)}","kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-647865d8d642d037","tex":"|_{S_n^\\downarrow(R)}","aliases":"|_S_n^downarrow(R) downarrow","display":"|_{S_n^\\downarrow(R)}","kind":"definition-passage","excerpt":"[Intrinsic and ambient lengths] For A∈U_n(R), define ellDown(A)=|A|_S_n^downarrow(R), ellAll(A)=|A|_S_n^*(R). The catalytic defect is DeltaDef(A)=ellDown(A)-ellAll(A)≥0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"definition-passage","excerpt":"ed realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":22},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"definition-passage","excerpt":"[Approximate symmetry collapse] A uniform circuit family exhibits an approximate symmetry collapse to error ε(n) when a polynomial-size classical representation reproduces its acceptance probability to additive error ε(n), with the error below the promise gap required by the decision problem.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"definition-passage","excerpt":"[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cdc1d07dfd6822eb","tex":"\\kappa_\\otimes","aliases":"κ_⊗ kappa otimes","display":"\\kappa_\\otimes","kind":"definition-passage","excerpt":"[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"[Critical catalyst penalty] For a target family U_n, define λ_c(U_n) = sup{λ:ℓ_λ,amb(U_n)<ellRes(U_n)}. Determining λ_c exactly is a refined synthesis problem. It can be more physically meaningful than unweighted gate count.","count":76},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-65d44243a226ac8f","tex":"\\lambda_c","aliases":"λ_c lambda","display":"\\lambda_c","kind":"definition-passage","excerpt":"[Critical catalyst penalty] For a target family U_n, define λ_c(U_n) = sup{λ:ℓ_λ,amb(U_n)<ellRes(U_n)}. Determining λ_c exactly is a refined synthesis problem. It can be more physically meaningful than unweighted gate count.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"Worst-case and threshold functions. For a size parameter n, let T_n denote the target family. [Extremal distortion] Define D_n=sup_U∈ T_nΔ(U) and, when 0<ellAmb(U)<∞, R_n=sup_U∈ T_n fracellRes(U)ellAmb(U). The first defective size is τ=min{n:D_n>0}. We set a threshold to +∞ when its defining set is empty.","count":45},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-273b5cbc261575f0","tex":"\\tau_{p,q}","aliases":"τ_p,q tau","display":"\\tau_{p,q}","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-831204a426bd149e","tex":"\\tau_{p,x}","aliases":"τ_p,x tau","display":"\\tau_{p,x}","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-79ad64058fda49c4","tex":"\\tau_{q,x}","aliases":"τ_q,x tau","display":"\\tau_{q,x}","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-94b9ce3e9e7091fb","tex":"\\tau_{y,p}","aliases":"τ_y,p tau","display":"\\tau_{y,p}","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0e084704755e4b52","tex":"\\tau_{y,q}","aliases":"τ_y,q tau","display":"\\tau_{y,q}","kind":"definition-passage","excerpt":"[Simple one-catalyst word] A simple one-catalyst word has the form τ_p,q(-a) h_m⋯ h_1 τ_p,q(a), p<q, where every h_i is lower directed and is one of τ_q,x(b), τ_p,x(b), τ_y,p(b), τ_y,q(b) with x<p<q<y. Auxiliary rows are distinct unless stated otherwise.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e809ce7dd3704c0d","tex":"\\tau_\\otimes","aliases":"τ_⊗ tau otimes","display":"\\tau_\\otimes","kind":"definition-passage","excerpt":"[Stable tensor cost] For a size-m target U, define τ_⊗(U) = lim_r→∞ fracellAmb(U^⊗ r)r m^r-1, when the subadditivity argument applies. Define κ_⊗(U) = ellRes(U)-τ_⊗(U). Fekete's lemma gives the limit from the tensor subadditivity inequality.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).","count":30},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"definition-passage","excerpt":"[Cost-preserving equivalence] Targets U and V are equivalent if an automorphism φ of the ambient realization system satisfies φ(U)=V, φ( S_res)= S_res, φ( S_amb)= S_amb, and preserves the gate costs. Under this equivalence, both lengths and the defect are orbit invariants. A useful threshold theorem should classify minimal defective orbits, not mer","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-80503d26854475ff","tex":"\\Phi_T","aliases":"Φ_T Phi","display":"\\Phi_T","kind":"definition-passage","excerpt":"[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Resource vector] Let c(s)=(c_1(s),…,c_r(s))∈R_≥0^r be the resource vector of a gate. The achievable resource region of a target is P_ M(U) = { ∑_j=1^L c(s_j):s_L⋯ s_1=U }. Its nondominated points form the Pareto boundary.","count":82},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"definition-passage","excerpt":"te model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realiza","count":64},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5b39806e93f84fd5","tex":"\\ell_{\\mathfrak M}","aliases":"ℓ_ M ell mathfrak","display":"\\ell_{\\mathfrak M}","kind":"definition-passage","excerpt":"te model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realiza","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b94197b63517cbf5","tex":"\\ell_{\\mathfrak M_{\\mathrm{amb}}}","aliases":"ℓ_ M_amb ell mathfrak mathrm","display":"\\ell_{\\mathfrak M_{\\mathrm{amb}}}","kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-977459baa8337ae2","tex":"\\ell_{\\mathfrak M_{\\mathrm{res}}}","aliases":"ℓ_ M_res ell mathfrak mathrm","display":"\\ell_{\\mathfrak M_{\\mathrm{res}}}","kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4b3a0d2b201fef79","tex":"\\ell_{\\lambda,\\mathrm{amb}}","aliases":"ℓ_λ,amb ell lambda mathrm","display":"\\ell_{\\lambda,\\mathrm{amb}}","kind":"definition-passage","excerpt":"[Critical catalyst penalty] For a target family U_n, define λ_c(U_n) = sup{λ:ℓ_λ,amb(U_n)<ellRes(U_n)}. Determining λ_c exactly is a refined synthesis problem. It can be more physically meaningful than unweighted gate count.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":74},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"begindefinition[Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The val","count":215},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":160},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-25b387c951fb9265","tex":"\\sum_{j=1}^L","aliases":"∑_j=1^L sum","display":"\\sum_{j=1}^L","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9e0ed1ad28d5a289","tex":"\\sum_{v\\in\\operatorname{Int}(T)}","aliases":"∑_v∈Int(T) sum in operatorname","display":"\\sum_{v\\in\\operatorname{Int}(T)}","kind":"definition-passage","excerpt":"[Interval tree] An interval tree on the ordered coordinates 0,1,…,n-1 is a rooted binary tree whose leaves occur in that order and whose every node is a contiguous interval. If an internal node v splits its interval into consecutive left and right children L_v and R_v, define Φ_T(A) = ∑_v∈Int(T) rank A[R_v,L_v], A∈U_n(F).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":",c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The value is +∞ when the target has no realization.","count":18},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"definition-passage","excerpt":"[Parabolic and ambient Clifford metrics] The restricted alphabet is S_P = {S_i,CZ_ij,CX_i→ j}, and the ambient alphabet is S_C = S_P∪{H_i}. Targets lie in P_n. We study both total primitive count and entangling count, where H_i and S_i have cost zero while CX and CZ have cost one.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"[Upper-gate budget] For q≥0 and A∈U_n(F), define ellBudgetq(A) = min{|w|:[w]=A, u(w)≤ q}. Thus ellBudget0(A)=ellDown(A), ellBudgetq+1(A)≤ellBudgetq(A), ellAll(A)≤ellBudgetq(A).","count":183},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":128},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Nested realization pair] A nested realization pair consists of restricted and ambient models M_res⊆ M_amb with a common target class T of finite ambient cost and the same cost on common generators. For U∈ T, define ellRes(U)=ℓ_ M_res(U), ellAmb(U)=ℓ_ M_amb(U). The additive defect is Δ(U)=ellRes(U)-ellAmb(U).","count":25},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"definition-passage","excerpt":"[Tensor synergy] Define Syn(U,V) = nellAmb(U)+mellAmb(V)-ellAmb(U⊗ V). Positive synergy means that the joint ambient realization improves on independent lifted circuits. Restricted tree-tight metrics have zero tensor synergy. Ambient synergy is a possible source of collective activation.","count":54},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"Exact gate models. [Gate model] An exact gate model is a tuple M=( X, S,c,∘), where X is a set of states or morphisms, S is a specified family of elementary generators, c: S→R_≥0 is a cost function, and ∘ is the composition law. A word w=s_L⋯ s_1 has cost c(w)=∑_j=1^L c(s_j). For a target U∈ X, the exact synthesis cost is ℓ_ M(U)=inf{c(w):w=U}. The","count":46},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"introduction-passage","excerpt":"aths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be c","count":20},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":104},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9af3c6409bcf85f1","tex":"b^+","aliases":"b^+","display":"b^+","kind":"introduction-passage","excerpt":"The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected famil","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"introduction-passage","excerpt":"A signal is a vector x:I_q→ C. We write Id_q for the identity matrix of dimension q, and I for an identity matrix when its dimension is clear. Our inner product is conjugate-linear in the first argument:","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d6e0f61000c3e183","tex":"\\Deg","aliases":"Deg","display":null,"kind":"introduction-passage","excerpt":"ocsectionThe formal endpoint Let c be a finite legal classical NOT/CNOT/Toffoli circuit. For n input bits and a zero-initialized ancillas, assume for every x and both initial target values b that run(c)(x,0^a,b)=(x,0^a,bx f(x)). Write T=TC(c) for its number of Toffoli gates. The Block 02 joint endpoint is Degree and complete-cover bound Deg(T+1,f) ","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"tionExtracting the quadratic grade [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient i","count":134},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7d9b5abc6519e56b","tex":"e_u","aliases":"e_u","display":"e_u","kind":"introduction-passage","excerpt":"labelc:thm:grade-cost Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-042162b6d9ca555d","tex":"e_v","aliases":"e_v","display":"e_v","kind":"introduction-passage","excerpt":"tionExtracting the quadratic grade [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient i","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b9e4d09fcc96b0b9","tex":"\\ellExc","aliases":"ellExc","display":null,"kind":"introduction-passage","excerpt":"The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected famil","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"Here D is a diagonal zero-or-one matrix, F is unitary, and † denotes conjugate transpose. Consequently P is an orthogonal projector. The transformation F is reversible. The selection P need not be. A complete decomposition consists of both Px and (I-P)x, whose sum recovers x.","count":88},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6511f45686c84845","tex":"\\F_2^m","aliases":"F_2^m F","display":null,"kind":"introduction-passage","excerpt":"guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. I","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"introduction-passage","excerpt":"Write widehatx_r for the unscaled M-point transform of lane r. Then for k∈ I_N,","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-69a4780a0be9987e","tex":"I_q","aliases":"I_q","display":"I_q","kind":"introduction-passage","excerpt":"A signal is a vector x:I_q→ C. We write Id_q for the identity matrix of dimension q, and I for an identity matrix when its dimension is clear. Our inner product is conjugate-linear in the first argument:","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5552078a68453cbe","tex":"\\operatorname{Id}","aliases":"Id operatorname","display":"\\operatorname{Id}","kind":"introduction-passage","excerpt":"A signal is a vector x:I_q→ C. We write Id_q for the identity matrix of dimension q, and I for an identity matrix when its dimension is clear. Our inner product is conjugate-linear in the first argument:","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a82555acb6a3be97","tex":"\\operatorname{Id}_q","aliases":"Id_q operatorname","display":"\\operatorname{Id}_q","kind":"introduction-passage","excerpt":"A signal is a vector x:I_q→ C. We write Id_q for the identity matrix of dimension q, and I for an identity matrix when its dimension is clear. Our inner product is conjugate-linear in the first argument:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"Write widehatx_r for the unscaled M-point transform of lane r. Then for k∈ I_N,","count":223},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"introduction-passage","excerpt":"guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. I","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"introduction-passage","excerpt":"aths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be c","count":20},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9dfa6b6b1427219a","tex":"\\min_{a:\\,P_a=P}","aliases":"min_a: P_a=P min","display":"\\min_{a:\\,P_a=P}","kind":"introduction-passage","excerpt":"Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"introduction-passage","excerpt":"Write widehatx_r for the unscaled M-point transform of lane r. Then for k∈ I_N,","count":141},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e0d92ca5df54dffc","tex":"\\operatorname{odd}","aliases":"odd operatorname","display":"\\operatorname{odd}","kind":"introduction-passage","excerpt":"Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"introduction-passage","excerpt":"Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"introduction-passage","excerpt":"Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inve","count":64},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3406acd3f0f64ef4","tex":"Q_m","aliases":"Q_m","display":"Q_m","kind":"introduction-passage","excerpt":"Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inve","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-14c10a0962d6f3d5","tex":"r^+","aliases":"r^+","display":"r^+","kind":"introduction-passage","excerpt":"The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected famil","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2ee1f012ae4017c4","tex":"\\operatorname{Real}","aliases":"Real operatorname","display":"\\operatorname{Real}","kind":"introduction-passage","excerpt":"Realization fibers. Let P denote packet data and let Real_res(P) be the space of restricted realizations compatible with P. The packet may completely classify the discrete object while the realization fiber remains empty, positive dimensional, disconnected, or singular.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-60dd403987d16995","tex":"\\operatorname{Real}_{\\mathrm{res}}","aliases":"Real_res operatorname mathrm","display":"\\operatorname{Real}_{\\mathrm{res}}","kind":"introduction-passage","excerpt":"Realization fibers. Let P denote packet data and let Real_res(P) be the space of restricted realizations compatible with P. The packet may completely classify the discrete object while the realization fiber remains empty, positive dimensional, disconnected, or singular.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"introduction-passage","excerpt":"aths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected family can be c","count":20},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0bfd0939e0882ca4","tex":"\\operatorname{run}","aliases":"run operatorname","display":"\\operatorname{run}","kind":"introduction-passage","excerpt":"the original rights record and the original release files. There is no new publication-novelty or unrestricted hardness claim. The formal endpoint. addcontentslinetocsectionThe formal endpoint Let c be a finite legal classical NOT/CNOT/Toffoli circuit. For n input bits and a zero-initialized ancillas, assume for every x and both initial target valu","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1d641eb69668a366","tex":"s_v","aliases":"s_v","display":"s_v","kind":"introduction-passage","excerpt":"Connected higher-dimensional qudit families. The snapshot target A(u;s_1,…,s_v) from the core chapter is defined over any coefficient ring for which the displayed additions make sense. Over Z_d, it becomes a modular SUM circuit. Its explicit ambient word uses exactly two reverse-directed SUM gates and has length u+v+2. The interval-tree proof of ex","count":15},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b1bfe89c1a60d4ba","tex":"T_{ab\\to t}","aliases":"T_ab→ t to","display":"T_{ab\\to t}","kind":"introduction-passage","excerpt":"Conjugation formulas. Let T_ab→ t denote Toffoli. Conjugating an affine gate by T can produce nonlinear permutations. For example, if X_t flips the target, T_ab→ tX_tT_ab→ t=X_t, so no gain occurs. If a CNOT changes a control or exports the target, the conjugated map can contain several quadratic terms. The exact formula is easiest in algebraic nor","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cfd3c90bc1885140","tex":"\\TC","aliases":"TC","display":null,"kind":"introduction-passage","excerpt":"here is no new publication-novelty or unrestricted hardness claim. The formal endpoint. addcontentslinetocsectionThe formal endpoint Let c be a finite legal classical NOT/CNOT/Toffoli circuit. For n input bits and a zero-initialized ancillas, assume for every x and both initial target values b that run(c)(x,0^a,b)=(x,0^a,bx f(x)). Write T=TC(c) for","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"introduction-passage","excerpt":"[Proof of [source reference]] For n≥7, put v_n=lfloorfracn-12rfloor, u_n=n-v_n-2, and take the phases 0,1,…,v_n-1. The inequality v_n≤ u_n+1 holds, so the snapshot target is defined. By [source reference], ellDown(A_n)=u_n+2v_n=n+v_n-2, while ellAll(A_n)≤ u_n+v_n+2=n. Hence ellDown(A_n)-ellAll(A_n) ≥ v_n-2 = lfloorfracn-52rfloor. Connectivity follo","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"introduction-passage","excerpt":"guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. I","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"introduction-passage","excerpt":"[Proof of [source reference]] For n≥7, put v_n=lfloorfracn-12rfloor, u_n=n-v_n-2, and take the phases 0,1,…,v_n-1. The inequality v_n≤ u_n+1 holds, so the snapshot target is defined. By [source reference], ellDown(A_n)=u_n+2v_n=n+v_n-2, while ellAll(A_n)≤ u_n+v_n+2=n. Hence ellDown(A_n)-ellAll(A_n) ≥ v_n-2 = lfloorfracn-52rfloor. Connectivity follo","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-14171564f9ff64a8","tex":"\\widehat{x_r}","aliases":"widehatx_r widehat","display":"\\widehat{x_r}","kind":"introduction-passage","excerpt":"Write widehatx_r for the unscaled M-point transform of lane r. Then for k∈ I_N,","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-48214699dd0a817f","tex":"\\operatorname{wt}","aliases":"wt operatorname","display":"\\operatorname{wt}","kind":"introduction-passage","excerpt":"Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a86d6dedf5bd962e","tex":"\\x","aliases":"x","display":null,"kind":"introduction-passage","excerpt":"the original rights record and the original release files. There is no new publication-novelty or unrestricted hardness claim. The formal endpoint. addcontentslinetocsectionThe formal endpoint Let c be a finite legal classical NOT/CNOT/Toffoli circuit. For n input bits and a zero-initialized ancillas, assume for every x and both initial target valu","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ad57b606030fc799","tex":"X^+","aliases":"X^+","display":"X^+","kind":"introduction-passage","excerpt":"The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected famil","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0298cc60bfc2d7e5","tex":"X_{ij}","aliases":"X_ij","display":"X_{ij}","kind":"introduction-passage","excerpt":"The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected famil","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2ee20c14455ce834","tex":"X_s","aliases":"X_s","display":"X_s","kind":"introduction-passage","excerpt":"Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency ch","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e971db48ca9b367c","tex":"X_t^a","aliases":"X_t^a","display":"X_t^a","kind":"introduction-passage","excerpt":"Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency ch","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6590090cc9a70e3b","tex":"x_u","aliases":"x_u","display":"x_u","kind":"introduction-passage","excerpt":"labelc:thm:grade-cost Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inversion therefore gives [x_ux_v]P_m=P_m(e_u+e_v)","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"introduction-passage","excerpt":"Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inve","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e86d713b2da9186d","tex":"x_{v+a}","aliases":"x_v+a","display":"x_{v+a}","kind":"introduction-passage","excerpt":"Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inve","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"introduction-passage","excerpt":"Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t,","count":17},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"introduction-passage","excerpt":"Connected higher-dimensional qudit families. The snapshot target A(u;s_1,…,s_v) from the core chapter is defined over any coefficient ring for which the displayed additions make sense. Over Z_d, it becomes a modular SUM circuit. Its explicit ambient word uses exactly two reverse-directed SUM gates and has length u+v+2. The interval-tree proof of ex","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":109},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f6fd9fa43213fd6e","tex":"z^{c(s)}","aliases":"z^c(s)","display":"z^{c(s)}","kind":"introduction-passage","excerpt":"For strictly positive integer costs, replace each generator s by the monomial z^c(s)ρ(s). Every coefficient then counts finitely many words of a given total cost, and the same formal inverse is defined. Zero-cost cycles require separate treatment: they can create infinitely many words of cost zero, so the ordinary counting series need not exist. Th","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-788fb106ab10a5c7","tex":"\\Z_8^{2^n-1}","aliases":"Z_8^2^n-1 Z","display":null,"kind":"introduction-passage","excerpt":"Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-505d159cef1afbcb","tex":"\\Z_d","aliases":"Z_d Z","display":null,"kind":"introduction-passage","excerpt":"Connected higher-dimensional qudit families. The snapshot target A(u;s_1,…,s_v) from the core chapter is defined over any coefficient ring for which the displayed additions make sense. Over Z_d, it becomes a modular SUM circuit. Its explicit ambient word uses exactly two reverse-directed SUM gates and has length u+v+2. The interval-tree proof of ex","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9cb094fff0fdadf7","tex":"Z_s","aliases":"Z_s","display":"Z_s","kind":"introduction-passage","excerpt":"Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency ch","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a48b492ab6d052c6","tex":"z_S","aliases":"z_S","display":"z_S","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-46fed9b3872ac70f","tex":"Z_s^{-a}","aliases":"Z_s^-a","display":"Z_s^{-a}","kind":"introduction-passage","excerpt":"Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency ch","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-77149717f7fe9972","tex":"z_S^s","aliases":"z_S^s","display":"z_S^s","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3480de748bc9723a","tex":"Z_t","aliases":"Z_t","display":"Z_t","kind":"introduction-passage","excerpt":"Pauli action for prime qudits. For odd prime d, let X and Z denote generalized Pauli operators. The SUM gate acts by X_s · ↦ X_sX_t^a, · X_t · ↦ X_t, Z_s · ↦ Z_s, · Z_t · ↦ Z_s^-aZ_t. Thus the modular linear action embeds into a symplectic transformation. The catalytic row excursion has a dual Pauli interpretation: a temporary reverse dependency ch","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-718a1696dd5c68b7","tex":"z_T^t","aliases":"z_T^t","display":"z_T^t","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"For strictly positive integer costs, replace each generator s by the monomial z^c(s)ρ(s). Every coefficient then counts finitely many words of a given total cost, and the same formal inverse is defined. Zero-cost cycles require separate treatment: they can create infinitely many words of cost zero, so the ordinary counting series need not exist. Th","count":36},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2a1b7ee96a9c377c","tex":"\\tau_w","aliases":"τ_w tau","display":"\\tau_w","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-62ae215fde9a7304","tex":"\\tau_{z_S}","aliases":"τ_z_S tau","display":"\\tau_{z_S}","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"introduction-passage","excerpt":"Here D is a diagonal zero-or-one matrix, F is unitary, and † denotes conjugate transpose. Consequently P is an orthogonal projector. The transformation F is reversible. The selection P need not be. A complete decomposition consists of both Px and (I-P)x, whose sum recovers x.","count":56},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9bb76059a9d9aed1","tex":"\\ell_T","aliases":"ℓ_T ell","display":"\\ell_T","kind":"introduction-passage","excerpt":"Let a∈Z_8^2^n-1 index nonzero linear forms and let odd(a) denote the binary vector recording which coefficients are odd. An abstract restricted T-count is ℓ_T(P)= min_a: P_a=P wt(odd(a)). An ambient Clifford excursion changes not only the coefficient vector but also the Pauli frame in which the rotations are represented. A lower bound that survives","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"introduction-passage","excerpt":"guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. I","count":20},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"introduction-passage","excerpt":"guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. I","count":24},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b8080a407bf6180c","tex":"\\prod_{j=1}^m","aliases":"∏_j=1^m prod","display":"\\prod_{j=1}^m","kind":"introduction-passage","excerpt":"guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. I","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7d98fe9de6bebd3c","tex":"\\sum_s","aliases":"∑_s sum","display":"\\sum_s","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-05725bd2c82a5e24","tex":"\\sum_{s,t}","aliases":"∑_s,t sum","display":"\\sum_{s,t}","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-90c396c1f06e154a","tex":"\\sum_t","aliases":"∑_t sum","display":"\\sum_t","kind":"introduction-passage","excerpt":"guidepageDisjoint-variable factorization uses each predicate's own block Suppose A depends on witness variables in block S and B on a disjoint block T. Write τ_z_S(A)=∑_s A(s)z_S^s for the weighted trace on S, distinct from the unweighted top-coefficient notation τ_w. Weighted sum over the union factors as ∑_s,t A(s)B(t)z_S^s z_T^t =(∑_s A(s)z_S^s)","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2d63a79aa697bf49","tex":"\\sum_{v:v_j=1}","aliases":"∑_v:v_j=1 sum","display":"\\sum_{v:v_j=1}","kind":"introduction-passage","excerpt":"guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. I","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7abf820cf7b469f1","tex":"\\sum_{\\{v,v+a\\}\\subseteq V}","aliases":"∑_{v,v+a}⊆ V sum subseteq","display":"\\sum_{\\{v,v+a\\}\\subseteq V}","kind":"introduction-passage","excerpt":"Extracting the quadratic grade. [Exact cost of the extracted grade] Let Q_m=[P_m]_2 denote the homogeneous quadratic part in the original x_v coordinates. Then Q_m=∑_{v,v+a}⊆ Vx_vx_v+a, where every unordered pair is used once, and M(Q_m)=KTrest(Q_m)=n/2=2^m-1-1. The excluded columns give P_m(0)=P_m(e_v)=0. For distinct u,v, Boolean coefficient inve","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"introduction-passage","excerpt":"guidepageProjection is defined in the original coordinates For m≥2, put a=(1,…,1) and V=F_2^m∖{0,a}, n=2^m-2. There is one input bit x_v for each v∈ V. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^m L_j(x). The linear map x↦ L(x) has rank m, because the standard basis columns are present. In the L coordinates, P_m is a monomial in m independent bits. I","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"The same paths remain extremely close to the restricted subgroup. For a matrix X over a field, let X^+ be its strict upper part, let r^+(X)=rank X^+, and let b^+(X) be the largest upper displacement j-i with X_ij≠0. If ellExcrb denotes the minimum length among circuits satisfying r^+(X_t)≤ r and b^+(X_t)≤ b at every prefix, then the connected famil","count":37},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"introduction-passage","excerpt":"Tensor and direct-sum conventions. For linear maps, A⊕ B denotes a block-diagonal direct sum. The Kronecker product A⊗ B uses lexicographic ordering. In quantum chapters, U⊗ V denotes the physical tensor product on disjoint systems. These operations have different dimension and cost normalizations and are not interchanged silently.","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d).","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bd88e31dacd4a68e","tex":"\\mathbf1_{12\\mid n}","aliases":"1_12mid n mathbf mid","display":null,"kind":"use","excerpt":"c(n)=21_2mid n-41_4mid n -61_6mid n+121_12mid n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-30230a427fb70421","tex":"\\mathbf1_{2\\mid n}","aliases":"1_2mid n mathbf mid","display":null,"kind":"use","excerpt":"c(n)=21_2mid n-41_4mid n -61_6mid n+121_12mid n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dcc875a8c9c85480","tex":"\\mathbf1_{4\\mid n}","aliases":"1_4mid n mathbf mid","display":null,"kind":"use","excerpt":"c(n)=21_2mid n-41_4mid n -61_6mid n+121_12mid n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2d7fc4f624d255ac","tex":"\\mathbf1_{6\\mid n}","aliases":"1_6mid n mathbf mid","display":null,"kind":"use","excerpt":"c(n)=21_2mid n-41_4mid n -61_6mid n+121_12mid n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d045a8e823f610f0","tex":"\\mathbf1_{a\\in S_d}","aliases":"1_a∈ S_d mathbf in","display":"\\mathbf1_{a\\in S_d}","kind":"use","excerpt":"The set S_d contains the labels of characters of exact order d. Put D_d=diag(1_a∈ S_d) and","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6bdb212b6060b205","tex":"\\mathbf1_{a\\in U_{12}}","aliases":"1_a∈ U_12 mathbf in","display":"\\mathbf1_{a\\in U_{12}}","kind":"use","excerpt":"With D=diag(1_a∈ U_12) and F=F_12,","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-feebc73f1c32934e","tex":"\\mathbf1_{\\gcd(a,q)=1}","aliases":"1_gcd(a,q)=1 mathbf gcd","display":"\\mathbf1_{\\gcd(a,q)=1}","kind":"use","excerpt":"1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8e8f6ac22ef91590","tex":"\\mathbf1_{\\gcd(k,q)=1}","aliases":"1_gcd(k,q)=1 mathbf gcd","display":"\\mathbf1_{\\gcd(k,q)=1}","kind":"use","excerpt":"integer waveform changes with the modulus, while the same arithmetic identity supplies it. fig19_other_arithmetic_signals1.0Four exact Ramanujan signals constructed by the divisor formula. Their periods and amplitudes differ. These examples illustrate the general formulas; their finite verification does not prove those formulas universally. For gen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0c9e28aea8cddc7a","tex":"\\mathbf1_{k\\in U_{12}}","aliases":"1_k∈ U_12 mathbf in","display":"\\mathbf1_{k\\in U_{12}}","kind":"use","excerpt":"s the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 and exactly zero elsewhere. This is an algebraic identity, not a spectral threshold","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-137aaf4a58ff8d6b","tex":"\\mathbf1_{k\\in U_q}","aliases":"1_k∈ U_q mathbf in","display":"\\mathbf1_{k\\in U_q}","kind":"use","excerpt":"widehatc_q(k)=q 1_k∈ U_q, k∈ I_q.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-104c1b3f1f4ff327","tex":"\\mathrm{2q-depth}","aliases":"2q-depth mathrm","display":"\\mathrm{2q-depth}","kind":"use","excerpt":"Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"Metric collapse and existence collapse. The simulation theorems compare costs of two realizations that both exist. A broader restricted-realization framework also allows the restricted object to be absent. If R(x)=∅, A(x)≠∅, then enlarging the realization category changes feasibility rather than merely runtime.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"Proof. The equation (ζ_q^a)^t=1 holds exactly when q divides at. Write g=gcd(a,q), q=gq', and a=ga', with gcd(a',q')=1. The divisibility condition is q'mid a't, which is equivalent to q'mid t. Its smallest positive solution is t=q'. This includes a=0: then g=q and the order is one. square","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"If the unique gate is R_t← R_t+aR_s, then aR_s=w. Choose a coordinate j for which w_j is a unit. Since a(R_s)_j=w_j, the factor a is a unit in the commutative ring R. Hence R_s=a^-1w.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2382d8c5b209022e","tex":"A^{-T}","aliases":"A^-T","display":"A^{-T}","kind":"use","excerpt":"Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-45bd8af79b3ac47d","tex":"A^{[0]}","aliases":"A^[0]","display":"A^{[0]}","kind":"use","excerpt":"Let A^[0]=A and A^[r+1]= Z(A^[r]). If A has size m, restricted length a, and ambient length b, then ellDown(A^[r]) = 2^r a+r m2^r-1, and ellAll(A^[r]) ≤ 2^r b+r m2^r-1. Thus DeltaDef(A^[r])≥2^rDeltaDef(A). For the canonical atom C, the first values are @rrrrr@ r · dimension · restricted exact · ambient upper bound · certified gap","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4ab0a490a7450590","tex":"A^{[r+1]}","aliases":"A^[r+1]","display":"A^{[r+1]}","kind":"use","excerpt":"Let A^[0]=A and A^[r+1]= Z(A^[r]). If A has size m, restricted length a, and ambient length b, then ellDown(A^[r]) = 2^r a+r m2^r-1, and ellAll(A^[r]) ≤ 2^r b+r m2^r-1. Thus DeltaDef(A^[r])≥2^rDeltaDef(A). For the canonical atom C, the first values are @rrrrr@ r · dimension · restricted exact · ambient upper bound · certified gap","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5675cae831d23ecb","tex":"A^{[r]}","aliases":"A^[r]","display":"A^{[r]}","kind":"use","excerpt":"Let A^[0]=A and A^[r+1]= Z(A^[r]). If A has size m, restricted length a, and ambient length b, then ellDown(A^[r]) = 2^r a+r m2^r-1, and ellAll(A^[r]) ≤ 2^r b+r m2^r-1. Thus DeltaDef(A^[r])≥2^rDeltaDef(A). For the canonical atom C, the first values are @rrrrr@ r · dimension · restricted exact · ambient upper bound · certified gap","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-943bc84d09f5dc3b","tex":"a^k","aliases":"a^k","display":"a^k","kind":"use","excerpt":"The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c313f2fab2d77584","tex":"A^\\ell","aliases":"A^ℓ ell","display":"A^\\ell","kind":"use","excerpt":"E_a=frac1N∑_ℓ=0^N-1ζ_N^aℓA^ℓ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e8b73dca4a9bb810","tex":"A^{\\otimes r}","aliases":"A^⊗ r otimes","display":"A^{\\otimes r}","kind":"use","excerpt":"[Tensor-amplified snapshot family] Let A=A(u;s_1,…,s_v) have dimension m=u+v+2. For N=m^r, ellDown(A^⊗ r) =r m^r-1(u+2v), while ellAll(A^⊗ r) ≤ r m^r. Consequently, ellDown(A^⊗ r)-ellAll(A^⊗ r) ≥ fracv-2mlog_2mNlog_2N.","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9904992b50efc27c","tex":"A^{\\otimes s}","aliases":"A^⊗ s otimes","display":"A^{\\otimes s}","kind":"use","excerpt":"The tensor inequality also defines a regularized ambient invariant. Let A∈U_m(F) and put b_r(A) = fracellAll(A^⊗ r)m^r-1. Applying [source reference] to A^⊗ r⊗ A^⊗ s gives b_r+s(A)≤ b_r(A)+b_s(A).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"Suggested sequence: A_2→ B_2=C_2→ G_2→ A_r. Low-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-89b2e69cbc9c02e9","tex":"A_{21}","aliases":"A_21","display":"A_{21}","kind":"use","excerpt":"Within the snapshot family, the best coefficient is obtained by the twenty-one-dimensional seed A_21=A(9;0,1,…,9), for which ellDown(A_21)=29, ellAll(A_21)≤21, DeltaDef(A_21)≥8.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ffff5e7ed44d00db","tex":"A_{21}^{\\otimes j}","aliases":"A_21^⊗ j otimes","display":"A_{21}^{\\otimes j}","kind":"use","excerpt":"[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7b8dbe5d7c4584e8","tex":"A_{21}^{\\otimes0}","aliases":"A_21^⊗0 otimes","display":"A_{21}^{\\otimes0}","kind":"use","excerpt":"[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f8255ddcd673535f","tex":"A_{2^n}","aliases":"A_2^n","display":"A_{2^n}","kind":"use","excerpt":"tring x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained in A_2^n and is not the full symmetric g","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a991039ddc6de3af","tex":"A_5","aliases":"A_5","display":"A_5","kind":"use","excerpt":"Type texorpdfstringAA and beyond. Row transvections are root-group elements of type A. The established unitriangular atom occurs in A_5. In nonsimply laced systems, root strings can have length greater than two. This raises the possibility that one ambient root can create multiple restricted effects per consumer.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"erage] Let a restored oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-19d7b7e300daef9b","tex":"a_{i,S}","aliases":"a_i,S","display":"a_{i,S}","kind":"use","excerpt":"Algebraic normal form. Every Boolean coordinate function has a unique algebraic normal form f_i(x) = ⊕_S⊆[n]a_i,S∏_j∈ Sx_j. Affine maps have degree at most one. A Toffoli gate targeting t with controls a,b performs x_t⟼ x_t+x_ax_b, which can raise the degree of later coordinates. The final affine target requires every higher-degree coefficient to c","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d7b2e04d502da719","tex":"a_i^\\dagger","aliases":"a_i^† dagger","display":"a_i^\\dagger","kind":"use","excerpt":"Gate updates. A number-preserving one-body unitary acts on mode creation operators by a_i^† ⟼ ∑_j U_jia_j^†. A generic dense mode transformation can induce many occupation-basis transitions, but for fixed N the basis dimension remains polynomial. Local hopping and phase gates are sparse. Few-body interactions remain polynomial to apply for fixed N,","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c9f16574b6182941","tex":"A_{ij}","aliases":"A_ij","display":"A_{ij}","kind":"use","excerpt":"Pattern groups and architecture constraints. For a poset P, define the pattern group U_P(K) = {I+A:A_ij=0 unless j<_P i}. When the allowed positions are transitively closed, U_P is a group. Compare the intrinsic metric generated by P-allowed transvections with the metric induced by all transvections.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c727b0e4770ce93c","tex":"a_j^\\dagger","aliases":"a_j^† dagger","display":"a_j^\\dagger","kind":"use","excerpt":"Gate updates. A number-preserving one-body unitary acts on mode creation operators by a_i^† ⟼ ∑_j U_jia_j^†. A generic dense mode transformation can induce many occupation-basis transitions, but for fixed N the basis dimension remains polynomial. Local hopping and phase gates are sparse. Few-body interactions remain polynomial to apply for fixed N,","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7d8e3b8668ae2ddd","tex":"A_k","aliases":"A_k","display":"A_k","kind":"use","excerpt":"[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c558af70c2e8eb08","tex":"a_L","aliases":"a_L","display":"a_L","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"Let L≥1 and let f(z)=∑_n≥0a_nz^n be a formal power series over a field containing a primitive Lth root ω and 1/L. For 0≤ r<L, define","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"use","excerpt":"Suggested sequence: A_2→ B_2=C_2→ G_2→ A_r. Low-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"use","excerpt":"Template XI: stable capacity. If a_r+s≤ m^s a_r+m^r a_s, then b_r=fraca_rm^r-1 is subadditive and lim_r→∞fraca_rr m^r-1 exists by Fekete's lemma. This defines a regularized ambient cost and exposes possible tensor activation.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6c74ebcb7e75b37a","tex":"a_{r+Lm}","aliases":"a_r+Lm","display":"a_{r+Lm}","kind":"use","excerpt":"E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8b81abe263690665","tex":"a_{r+s}","aliases":"a_r+s","display":"a_{r+s}","kind":"use","excerpt":"Template XI: stable capacity. If a_r+s≤ m^s a_r+m^r a_s, then b_r=fraca_rm^r-1 is subadditive and lim_r→∞fraca_rr m^r-1 exists by Fekete's lemma. This defines a regularized ambient cost and exposes possible tensor activation.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f78009c8b1487c7d","tex":"A_S","aliases":"A_S","display":"A_S","kind":"use","excerpt":"Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.","count":12},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bb2614930ea303a4","tex":"a_s","aliases":"a_s","display":"a_s","kind":"use","excerpt":"Template XI: stable capacity. If a_r+s≤ m^s a_r+m^r a_s, then b_r=fraca_rm^r-1 is subadditive and lim_r→∞fraca_rr m^r-1 exists by Fekete's lemma. This defines a regularized ambient cost and exposes possible tensor activation.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cf5deb93afba38c3","tex":"A_S^k","aliases":"A_S^k","display":"A_S^k","kind":"use","excerpt":"Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-82821bc2124f1658","tex":"A_t","aliases":"A_t","display":"A_t","kind":"use","excerpt":"Adaptive packet enlargement. A practical simulator need not fix one packet forever. It can maintain an active set A_t of packets, estimate transfers P_α G_tP_β, and add a packet when its predicted mass exceeds a tolerance. The proof obligation becomes a global accounting identity: total error ≤ discarded transfer + numerical error + measurement app","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1288c776dbe9d9fe","tex":"A_\\lambda","aliases":"A_λ lambda","display":"A_\\lambda","kind":"use","excerpt":"Decomposition For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e13a855d2f8c887b","tex":"\\Aff","aliases":"Aff","display":null,"kind":"use","excerpt":"For a positive input v, define instead β_v(f)=min{deg h:h∈ I_f, h(v)=1}, β(f)=max_v:f(v)=1β_v(f). Set β(0)=0. Thus β(f) is the degree needed before safe polynomials collectively detect every positive input. Finally, put β^*(f)=max_ℓ∈Aff_nβ(f+ℓ), where Aff_n is the set of affine Boolean functions on the logical inputs.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aed58fd23b735fcb","tex":"\\Aff^L","aliases":"Aff^L Aff","display":null,"kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ce34f97dc65f139e","tex":"\\Aff^N","aliases":"Aff^N Aff","display":null,"kind":"use","excerpt":"After base change to an algebraic closure overline K, [source reference] gives dimoverline R^*_n,≤ N-1≤ N-1<N=dimU_n. Because the word maps are defined over K, this proper closed set descends to a proper closed subset of U_n over K. Its complement is a nonempty Zariski-open set Ω_n. Since K is infinite and U_n≅Aff^N, the set Ω_n(K) is nonempty and ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5efea09f75bd83b5","tex":"\\Aff_n","aliases":"Aff_n Aff","display":null,"kind":"use","excerpt":"For a positive input v, define instead β_v(f)=min{deg h:h∈ I_f, h(v)=1}, β(f)=max_v:f(v)=1β_v(f). Set β(0)=0. Thus β(f) is the degree needed before safe polynomials collectively detect every positive input. Finally, put β^*(f)=max_ℓ∈Aff_nβ(f+ℓ), where Aff_n is the set of affine Boolean functions on the logical inputs.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c0428bd35161a74c","tex":"\\operatorname{AGL}","aliases":"AGL operatorname","display":"\\operatorname{AGL}","kind":"use","excerpt":"Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained i","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ee9fbd7c37205e3a","tex":"\\mathrm{AND}","aliases":"AND mathrm","display":"\\mathrm{AND}","kind":"use","excerpt":"A degree-one complete safe cover for AND would include a detector of its sole accepting input. Safety forces that detector to be AND itself, contradicting the corner-parity lemma. The degree-two monomial is a valid detector. The source proves β(AND_2)=β^*(AND_2)=2 and therefore every restored implementation of AND_2 uses at least one Toffoli.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8c224d7c7cbaebe8","tex":"\\mathrm{AND}_2","aliases":"AND_2 mathrm","display":"\\mathrm{AND}_2","kind":"use","excerpt":"A degree-one complete safe cover for AND would include a detector of its sole accepting input. Safety forces that detector to be AND itself, contradicting the corner-parity lemma. The degree-two monomial is a valid detector. The source proves β(AND_2)=β^*(AND_2)=2 and therefore every restored implementation of AND_2 uses at least one Toffoli.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1cad61c08170218e","tex":"\\B","aliases":"B","display":null,"kind":"use","excerpt":"Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies ","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1267a67d957e703c","tex":"\\mathrm B","aliases":"B mathrm","display":"\\mathrm B","kind":"use","excerpt":"Bosonic and fermionic analogues. For N fermions in m modes, D_ F(m,N)=binom mN. For N bosons in m modes, D_ B(m,N)=binomm+N-1N. The same binomial analysis gives:","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-16636769fc160a5a","tex":"b^{-1}","aliases":"b^-1","display":"b^{-1}","kind":"use","excerpt":"Let B_4={g∈GL_6(F_2):ellAll(g)≤4}. Exact breadth-first search gives |B_4|=110626. Group B_4 by right cosets of U_6(F_2). If a,b lie in the same right coset, then b^-1a∈U_6(F_2), and ellAll(b^-1a)≤ellAll(b)+ellAll(a)≤8. The implementation uses a canonical right-coset key obtained by reducing matrix columns from right to left. Right multiplication by","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c4af5f5e5ed160bb","tex":"\\B^m","aliases":"B^m B","display":null,"kind":"use","excerpt":"The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c8d6e62a2ef2fb37","tex":"\\B^n","aliases":"B^n B","display":null,"kind":"use","excerpt":"Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"Define Q_R=fracI+Rn, Q_C=fracI+Cn, Q_S=fracI+Sn, B_0=fracJn^2. Then the five joint projectors are B_0, Q_R-B_0, Q_C-B_0, Q_S-B_0, I-Q_R-Q_C-Q_S+2B_0. Their multiplicities are 1, n-1, n-1, n-1, (n-1)(n-2). The Latin-square graph R+C+S has universal spectrum 3(n-1)^1, (n-3)^3(n-1), (-3)^(n-1)(n-2). At n=6 this becomes 15^1,qquad3^15, (-3)^20. Thus th","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c8004fef7e4fe811","tex":"B_0^*","aliases":"B_0^*","display":"B_0^*","kind":"use","excerpt":"[Exact dynamic-packet theorem] Suppose P_0ketψ_0=ketψ_0 and [source reference] holds for every gate. Put z_0=B_0^*ketψ_0, z_t=M_tz_t-1. Then G_t⋯ G_1ketψ_0=B_tz_t for every t.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"The packet access model. For each input length, suppose there is a packet projector P_n and a basis B_n= {ketb_1,…,ketb_D_n} of P_nHh_n. The simulation interface has four routines.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","excerpt":"Suggested sequence: A_2→ B_2=C_2→ G_2→ A_r. Low-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4b94fddf75c4f060","tex":"B_4","aliases":"B_4","display":"B_4","kind":"use","excerpt":"Let B_4={g∈GL_6(F_2):ellAll(g)≤4}. Exact breadth-first search gives |B_4|=110626. Group B_4 by right cosets of U_6(F_2). If a,b lie in the same right coset, then b^-1a∈U_6(F_2), and ellAll(b^-1a)≤ellAll(b)+ellAll(a)≤8. The implementation uses a canonical right-coset key obtained by reducing matrix columns from right to left. Right multiplication by","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-df9249aec3709c3c","tex":"b_{D_n}","aliases":"b_D_n","display":"b_{D_n}","kind":"use","excerpt":"The packet access model. For each input length, suppose there is a packet projector P_n and a basis B_n= {ketb_1,…,ketb_D_n} of P_nHh_n. The simulation interface has four routines.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"use","excerpt":"erage] Let a restored oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-07b578d231cf4bc6","tex":"\\B_n","aliases":"B_n B","display":null,"kind":"use","excerpt":"The packet access model. For each input length, suppose there is a packet projector P_n and a basis B_n= {ketb_1,…,ketb_D_n} of P_nHh_n. The simulation interface has four routines.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ac0d31a2615b8b66","tex":"b_r","aliases":"b_r","display":"b_r","kind":"use","excerpt":"The tensor inequality also defines a regularized ambient invariant. Let A∈U_m(F) and put b_r(A) = fracellAll(A^⊗ r)m^r-1. Applying [source reference] to A^⊗ r⊗ A^⊗ s gives b_r+s(A)≤ b_r(A)+b_s(A).","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3dafff157450459a","tex":"b_{r+s}","aliases":"b_r+s","display":"b_{r+s}","kind":"use","excerpt":"The tensor inequality also defines a regularized ambient invariant. Let A∈U_m(F) and put b_r(A) = fracellAll(A^⊗ r)m^r-1. Applying [source reference] to A^⊗ r⊗ A^⊗ s gives b_r+s(A)≤ b_r(A)+b_s(A).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9a8f3582375c42a4","tex":"B_r^{(M)}","aliases":"B_r^(M)","display":"B_r^{(M)}","kind":"use","excerpt":"Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis a","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-10c299c3911c0c85","tex":"b_s","aliases":"b_s","display":"b_s","kind":"use","excerpt":"The tensor inequality also defines a regularized ambient invariant. Let A∈U_m(F) and put b_r(A) = fracellAll(A^⊗ r)m^r-1. Applying [source reference] to A^⊗ r⊗ A^⊗ s gives b_r+s(A)≤ b_r(A)+b_s(A).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9474dae99bcf0491","tex":"B_t","aliases":"B_t","display":"B_t","kind":"use","excerpt":"Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ecaf6432b2bc8af5","tex":"B_{t-1}","aliases":"B_t-1","display":"B_{t-1}","kind":"use","excerpt":"Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a21e613af2f0157f","tex":"B_t^*","aliases":"B_t^*","display":"B_t^*","kind":"use","excerpt":"Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bec856d92c5ab12c","tex":"b_x","aliases":"b_x","display":"b_x","kind":"use","excerpt":"Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-a","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f8dd14ae875161a4","tex":"B_\\lambda","aliases":"B_λ lambda","display":"B_\\lambda","kind":"use","excerpt":"Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-58af0f264fe7ee38","tex":"\\backslash","aliases":"backslash","display":null,"kind":"use","excerpt":"Set G=Egrp_n(R), H=U_n(R). Consider the right Schreier graph on the left cosets Hbackslash G: a generator s∈ S_n^*(R) gives an edge Hg⟶ Hgs. A word representing h∈ H traces a based loop at the coset H. Its label product is the holonomy h.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b61c22b86277b610","tex":"\\bigvee","aliases":"bigvee","display":"\\bigvee","kind":"use","excerpt":"d oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9120115ca46b0941","tex":"\\bigvee_i","aliases":"bigvee_i bigvee","display":"\\bigvee_i","kind":"use","excerpt":"d oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e90bcaa8fd887e29","tex":"\\mathrm{Bool}","aliases":"Bool mathrm","display":"\\mathrm{Bool}","kind":"use","excerpt":"pageheadModel and definitionsTHE ASSUMPTIONS ARE PART OF THE THEOREM A general register state is a Boolean function W→Bool on any wire type with decidable equality. C01 does not assume that W is finite. The canonical degree model has a finite wire set with n input wires, a ancillas, and one target. A NOT complements a bit. A CNOT adds one control t","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-43fa8bad7cce3005","tex":"\\bra","aliases":"bra","display":null,"kind":"use","excerpt":"Hypercube quotient coupling. Let ket k = binom nk^-1/2 ∑_|x|=kket x. Every weight-k string has n-k neighbors of weight k+1, while every weight-(k+1) string has k+1 neighbors of weight k. Counting the bipartite edges yields brak+1Aket k = fracbinom nk(n-k) sqrtbinom nkbinom nk+1 = sqrt(k+1)(n-k), proving [source reference].","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C","display":null,"kind":"use","excerpt":"Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"Exact cut comparison and safe discrepancies. claimC01 [Cut-comparison identity and complete coverage] Let a restored oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ab8bde9a479f56a0","tex":"\\mathfrak C","aliases":"C mathfrak","display":"\\mathfrak C","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a620c9c352cae951","tex":"\\C^2","aliases":"C^2 C","display":null,"kind":"use","excerpt":"Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5d2ac80c37b71f32","tex":"\\C^d","aliases":"C^d C","display":null,"kind":"use","excerpt":"Symmetric qudits. For local dimension d, the symmetric subspace has occupation-number basis (n_1,…,n_d), n_1+⋯+n_d=n, and dimension dimSym^n(C^d) = binomn+d-1d-1. For fixed d, this is polynomial in n.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2f1bee7bf2d4c335","tex":"\\C^{D_t}","aliases":"C^D_t C","display":null,"kind":"use","excerpt":"Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-054405a42ce6c01d","tex":"\\mathbb C^N","aliases":"C^N mathbb","display":"\\mathbb C^N","kind":"use","excerpt":"For every x∈ C^N,","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bbccad0aecc9be25","tex":"C^\\dagger","aliases":"C^† dagger","display":"C^\\dagger","kind":"use","excerpt":"Pauli rotations. For a Hermitian Pauli P, R_P(θ)=e^-iθ P/2. Clifford conjugation gives C R_P(θ) C^†=R_CPC^†(θ). The sign of CPC^† can be absorbed by changing θ to -θ.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1f5d67fa8f4b0cf1","tex":"C^{\\oplus k}","aliases":"C^⊕ k oplus","display":"C^{\\oplus k}","kind":"use","excerpt":"[Block-sum amplification] For every field and every n≥6, there is an A_n∈U_n(F) with ellDown(A_n)-ellAll(A_n) ≥ lfloorfrac n6rfloor. For n=6k, the target C^⊕ k satisfies ellDown(C^⊕ k)=7k, ellAll(C^⊕ k)≤6k.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2f5a4fbf8658bbe7","tex":"c_{(n-m)\\bmod12}","aliases":"c_(n-m)bmod12 bmod","display":"c_{(n-m)\\bmod12}","kind":"use","excerpt":"The cyclic matrix has entry c_(n-m)bmod12/12. The original layer proves its symmetry, idempotence, preservation of c, rejection of the chosen perturbation, and exact recovery from the displayed contaminated signal. The exact energy contract is 150=48+102.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"Any affine polynomial is a coefficient covector c=(c_0,c_a,c_b). Safety means that it pairs to zero with each negative row. The displayed dependence therefore forces its value at (1,1) to be zero. No safe affine packet detects the positive point.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c770fedcda9c0635","tex":"c_{12}","aliases":"c_12","display":"c_{12}","kind":"use","excerpt":"twelve will become an integer signal, an exact rational filter, and a normalized quantum state. Every bridge will be written down. The guide follows one object through three coordinate systems. The starting condition is coprimality. The signal realization adds its selected waves. The quantum realization normalizes the amplitude vector and distingui","count":16},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"Suggested sequence: A_2→ B_2=C_2→ G_2→ A_r. Low-rank nonsimply laced systems are attractive because longer root strings may create more than one useful effect per consumer.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"c_0 · =(4,0,-4,0), c_1 · =(0,-2,0,2), c_2 · =(2,0,-2,0).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"use","excerpt":"Any affine polynomial is a coefficient covector c=(c_0,c_a,c_b). Safety means that it pairs to zero with each negative row. The displayed dependence therefore forces its value at (1,1) to be zero. No safe affine packet detects the positive point.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c0ef80510cceb437","tex":"\\mathcal C_{\\mathrm{amb}}","aliases":"C_amb mathcal mathrm","display":"\\mathcal C_{\\mathrm{amb}}","kind":"use","excerpt":"Metric and existence defects in one category. Let C_res⊆ C_amb be categories with weighted generating morphisms. For a morphism f in the restricted target class, the same definition Δ(f)=c_res(f)-c_amb(f) includes group-word metrics, adaptive quantum channels, tensor-network realizations, and orthogonality designs. This categorical form is essentia","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1d40e962ef49cbf1","tex":"c_{\\mathrm{amb}}","aliases":"c_amb mathrm","display":"c_{\\mathrm{amb}}","kind":"use","excerpt":"Metric and existence defects in one category. Let C_res⊆ C_amb be categories with weighted generating morphisms. For a morphism f in the restricted target class, the same definition Δ(f)=c_res(f)-c_amb(f) includes group-word metrics, adaptive quantum channels, tensor-network realizations, and orthogonality designs. This categorical form is essentia","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e78a957e4b60188f","tex":"c_b","aliases":"c_b","display":"c_b","kind":"use","excerpt":"Any affine polynomial is a coefficient covector c=(c_0,c_a,c_b). Safety means that it pairs to zero with each negative row. The displayed dependence therefore forces its value at (1,1) to be zero. No safe affine packet detects the positive point.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cc6784245cabc541","tex":"c_{\\mathrm{close}}","aliases":"c_close mathrm","display":"c_{\\mathrm{close}}","kind":"use","excerpt":"Template II: amortized consumers. Suppose a catalyst costs c_open+c_close and each of m consumers saves at least s_i restricted cost. Then Δ(U) ≥ ∑_i=1^m s_i -c_open-c_close-r, where r is repair or interference cost. A threshold occurs when the right side first becomes positive.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-43270a4fe1c36b14","tex":"C_{\\mathrm{compile}}","aliases":"C_compile mathrm","display":"C_{\\mathrm{compile}}","kind":"use","excerpt":"State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectoriza","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-227ceb7baf501818","tex":"c_d","aliases":"c_d","display":"c_d","kind":"use","excerpt":"(P_d^(N))_nm=frac1N c_d(n-m).","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4a8f37541acc8491","tex":"C_{\\mathrm{err}}","aliases":"C_err mathrm","display":"C_{\\mathrm{err}}","kind":"use","excerpt":"For stochastic error rates p_g, a first-order weighted cost is C_err(W)=∑_g∈ W-log(1-p_g). This additive approximation can be inserted directly into Dijkstra search. It does not capture coherent or correlated errors, but it produces a declared and reproducible metric.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b78e78268ca8449a","tex":"C_{\\mathrm{full}}","aliases":"C_full mathrm","display":"C_{\\mathrm{full}}","kind":"use","excerpt":"State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectoriza","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"use","excerpt":"A cubic on three variables has the form x_1x_2x_3+q(x)+ℓ(x), with q quadratic and ℓ affine. Translating x_i↦ x_i+c_i adjusts independently the coefficients of x_jx_k, so all quadratic terms can be cancelled. The resulting function is a three-variable monomial plus an affine function. Affine input changes and affine output additions cost no Toffolis","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"For an invertible binary matrix A, let c_j be column j. Starting at j=n-1 and moving left, reduce c_j by the already canonicalized columns c_j+1,…,c_n-1, using their pivot coordinates. The resulting ordered reduced columns form the key κ(A).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-771ded1e1e280422","tex":"c_{j+1}","aliases":"c_j+1","display":"c_{j+1}","kind":"use","excerpt":"For an invertible binary matrix A, let c_j be column j. Starting at j=n-1 and moving left, reduce c_j by the already canonicalized columns c_j+1,…,c_n-1, using their pivot coordinates. The resulting ordered reduced columns form the key κ(A).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3ef0835ad712055f","tex":"c_j^\\dagger","aliases":"c_j^† dagger","display":"c_j^\\dagger","kind":"use","excerpt":"Fermionic Gaussian and matchgate circuits. Let c_j,c_j^† be fermionic creation and annihilation operators. Quadratic Hamiltonians generate Gaussian unitaries, and their action is determined by an orthogonal transformation on Majoranas. Number-conserving Gaussian operations correspond to a unitary action on modes, while general Gaussian operations a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-254051ed849e0a84","tex":"c_k","aliases":"c_k","display":"c_k","kind":"use","excerpt":"The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5c6080ace6a6a36a","tex":"\\mathfrak C_n","aliases":"C_n mathfrak","display":"\\mathfrak C_n","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b7e8c8a0a1fa875e","tex":"c_{n-1}","aliases":"c_n-1","display":"c_{n-1}","kind":"use","excerpt":"For an invertible binary matrix A, let c_j be column j. Starting at j=n-1 and moving left, reduce c_j by the already canonicalized columns c_j+1,…,c_n-1, using their pivot coordinates. The resulting ordered reduced columns form the key κ(A).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-56c07fef600c26a6","tex":"c_{\\mathrm{open}}","aliases":"c_open mathrm","display":"c_{\\mathrm{open}}","kind":"use","excerpt":"Template II: amortized consumers. Suppose a catalyst costs c_open+c_close and each of m consumers saves at least s_i restricted cost. Then Δ(U) ≥ ∑_i=1^m s_i -c_open-c_close-r, where r is repair or interference cost. A threshold occurs when the right side first becomes positive.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c34fe880990b4de6","tex":"C_{\\mathrm{packet}}","aliases":"C_packet mathrm","display":"C_{\\mathrm{packet}}","kind":"use","excerpt":"State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectoriza","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e8b9f48bd6d09aed","tex":"c_q","aliases":"c_q","display":"c_q","kind":"use","excerpt":"c_q(n)=∑_a∈ U_qζ_q^an, n∈ Z.","count":12},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-607644686572680d","tex":"\\mathcal C_{\\mathrm{res}}","aliases":"C_res mathcal mathrm","display":"\\mathcal C_{\\mathrm{res}}","kind":"use","excerpt":"Metric and existence defects in one category. Let C_res⊆ C_amb be categories with weighted generating morphisms. For a morphism f in the restricted target class, the same definition Δ(f)=c_res(f)-c_amb(f) includes group-word metrics, adaptive quantum channels, tensor-network realizations, and orthogonality designs. This categorical form is essentia","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-338103e14c66c689","tex":"c_{\\mathrm{res}}","aliases":"c_res mathrm","display":"c_{\\mathrm{res}}","kind":"use","excerpt":"Metric and existence defects in one category. Let C_res⊆ C_amb be categories with weighted generating morphisms. For a morphism f in the restricted target class, the same definition Δ(f)=c_res(f)-c_amb(f) includes group-word metrics, adaptive quantum channels, tensor-network realizations, and orthogonality designs. This categorical form is essentia","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-10c5f167f7dd263f","tex":"c_{\\rm amb}","aliases":"c_rm amb rm","display":"c_{\\rm amb}","kind":"use","excerpt":"Unified terminology. For a target x, let c_rm res(x) and c_rm amb(x) be minimum costs, with +∞ assigned when a realization does not exist. Then Δ(x) = c_rm res(x)-c_rm amb(x) includes both finite metric defects and infinite existence defects. Symmetry collapse concerns the special case where the ambient quantum description has a cheaper classical r","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aed911f24cc41ff6","tex":"C_{\\rm apply}","aliases":"C_rm apply rm","display":"C_{\\rm apply}","kind":"use","excerpt":"Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1dd324d85a939053","tex":"C_{\\rm compile}","aliases":"C_rm compile rm","display":"C_{\\rm compile}","kind":"use","excerpt":"Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1fcfdece5694cf71","tex":"C_{\\rm full}","aliases":"C_rm full rm","display":"C_{\\rm full}","kind":"use","excerpt":"Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e6e40759f4d8fa4b","tex":"C_{\\rm packet}","aliases":"C_rm packet rm","display":"C_{\\rm packet}","kind":"use","excerpt":"Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b4d4256e9c9e09eb","tex":"c_{\\rm res}","aliases":"c_rm res rm","display":"c_{\\rm res}","kind":"use","excerpt":"Unified terminology. For a target x, let c_rm res(x) and c_rm amb(x) be minimum costs, with +∞ assigned when a realization does not exist. Then Δ(x) = c_rm res(x)-c_rm amb(x) includes both finite metric defects and infinite existence defects. Symmetry collapse concerns the special case where the ambient quantum description has a cheaper classical r","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c9c457fd3fefb6bf","tex":"c_u","aliases":"c_u","display":"c_u","kind":"use","excerpt":"ltiplicative map into a K-algebra. Suppose Φ(δ_v)≠0 for every v∈ W. Then dim_K A≥2^m. No unital hypothesis on Φ is required. Write e_v=Φ(δ_v). Multiplicativity makes these nonzero orthogonal idempotents. If ∑_v∈ Wc_ve_v=0, multiply the equality by e_u. Orthogonality annihilates every term except c_ue_u, which must be zero. Since K is a field and e_","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-53faef096d1a0acc","tex":"c_v","aliases":"c_v","display":"c_v","kind":"use","excerpt":"a faithful multiplicative summary] Let Φ: B→ A be a K-linear multiplicative map into a K-algebra. Suppose Φ(δ_v)≠0 for every v∈ W. Then dim_K A≥2^m. No unital hypothesis on Φ is required. Write e_v=Φ(δ_v). Multiplicativity makes these nonzero orthogonal idempotents. If ∑_v∈ Wc_ve_v=0, multiply the equality by e_u. Orthogonality annihilates every te","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a6ff7660eca84d81","tex":"C_x","aliases":"C_x","display":"C_x","kind":"use","excerpt":"Circuit families. A quantum decision procedure is a polynomial-time uniform family of circuits {C_x}, where the input string x determines a circuit on n=n(|x|) qubits with at most poly(|x|) gates. Let C_x=G_m⋯ G_1. A classical simulation theorem must describe an algorithm receiving the same succinct description of x and the circuit. It is not enoug","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c5d2dcf9c6cf601e","tex":"c_y","aliases":"c_y","display":"c_y","kind":"use","excerpt":"Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-a","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d3464ab969722596","tex":"\\mathrm{cat}","aliases":"cat mathrm","display":"\\mathrm{cat}","kind":"use","excerpt":"Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-780075647c19479b","tex":"\\mathrm{clean}","aliases":"clean mathrm","display":"\\mathrm{clean}","kind":"use","excerpt":"Ancillas define a family of defects Δ_a,clean(U), Δ_a,dirty(U), where a is the number of ancillary qubits or modes. Stable ancilla capacity asks whether a fixed number of reusable ancillas can yield an unbounded saving as the data size grows.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-04eb0e105b818917","tex":"\\mathrm{close}","aliases":"close mathrm","display":"\\mathrm{close}","kind":"use","excerpt":"Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2c2388b162846f8a","tex":"\\operatorname{code}","aliases":"code operatorname","display":"\\operatorname{code}","kind":"use","excerpt":"A lower-unitriangular binary matrix is encoded by reading the strict lower entries row by row: (1,0), (2,0),(2,1), …, (5,0),…,(5,4). The resulting fifteen-bit integer is denoted by code(A). A row is displayed as the integer whose binary expansion is its support vector. [source reference] lists all defects. Every row has ellDown=7 and ellAll=6.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-629697d6f7abf02a","tex":"\\codim","aliases":"codim","display":null,"kind":"use","excerpt":"Defect codimension. If every target has restricted cost at most N, then defect at least d forces ambient cost at most N-d. The dimension-per-gate theorem gives codim_Voverline{U:Δ(U)≥ d} ≥ d when each ambient gate carries one parameter.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4bff559ded2e71c1","tex":"\\codim_{(\\U_n)_{\\overline K}}","aliases":"codim_(U_n)_overline K codim U overline","display":null,"kind":"use","excerpt":"-rigidity-intro Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline K overline D_n,≥ d(overlin","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fdfb8367b857a24b","tex":"\\codim_{\\U_n}","aliases":"codim_U_n codim U","display":null,"kind":"use","excerpt":"[Defect--codimension bound] Let K be algebraically closed. For every 1≤ d≤ N, dimoverline D_n,≥ d≤ N-d. Equivalently, codim_U_noverline D_n,≥ d≥ d. More finely, on the stratum where ellDown(A)=L and DeltaDef(A)≥ d, the closure has dimension at most L-d.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ae512d9031755044","tex":"\\codim_V","aliases":"codim_V codim","display":null,"kind":"use","excerpt":"Defect codimension. If every target has restricted cost at most N, then defect at least d forces ambient cost at most N-d. The dimension-per-gate theorem gives codim_Voverline{U:Δ(U)≥ d} ≥ d when each ambient gate carries one parameter.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8b5b78faba460a79","tex":"\\mathrm{compile}","aliases":"compile mathrm","display":"\\mathrm{compile}","kind":"use","excerpt":"State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectoriza","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-99d382e0d36b1155","tex":"\\mathrm{conn}","aliases":"conn mathrm","display":"\\mathrm{conn}","kind":"use","excerpt":"Architecture and routing. On a line or grid, Gaussian and matchgate circuits may be dominated by routing. Let G_conn be the hardware graph. A mode permutation π can be treated as a catalyst if it is opened, many interactions are executed in the reordered frame, and π^-1 restores the logical labeling. The benefit is routing saved across consumers - ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6a6011fe2503acfc","tex":"\\mathrm{controls}","aliases":"controls mathrm","display":"\\mathrm{controls}","kind":"use","excerpt":"Suppose the next gate toggles target u by the product of a control list of length q. A product that omits u is unchanged. If u occurs, idempotence removes all its repetitions. With R the remaining list, the updated product is (s_u∏_i∈ Rs_i) x (∏_j∈controlss_j∏_i∈ Rs_i). The two old-state products have certified degrees at most t+1+|R| and t+q+|R|. ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"The rational signal and its filter. The first layer uses literal rational lists. The chosen perturbation is r_n=3cos(π n/2)+2, whose displayed rational entries can be certified without numerical trigonometric evaluation. Its sum with c is the test signal.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e1a4d4510c259fa0","tex":"\\cost","aliases":"cost","display":null,"kind":"use","excerpt":"For a target channel Φ in the restricted category, define c_res(Φ) = inf{cost(W):W realizes Φ in C_res} and similarly c_amb(Φ). The value +∞ is allowed. A finite positive difference is a metric defect; an infinite difference is an existence defect.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dadbefeac6b8f90a","tex":"\\operatorname{cost}","aliases":"cost operatorname","display":"\\operatorname{cost}","kind":"use","excerpt":"A hardware or fault model may penalize an upper-directed interaction more heavily than a lower-directed one. Fix λ>0. If w has d(w) lower-directed gates and u(w) upper-directed gates, put cost_λ(w)=d(w)+λ u(w), and define ellWeightλ(A) = inf{cost_λ(w):[w]=A}. At λ=1 this is the ordinary ambient length, while large λ increasingly penalizes flag excu","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8b2b671644194ae8","tex":"\\operatorname{cost}_{\\lambda}","aliases":"cost_λ operatorname lambda","display":"\\operatorname{cost}_{\\lambda}","kind":"use","excerpt":"A hardware or fault model may penalize an upper-directed interaction more heavily than a lower-directed one. Fix λ>0. If w has d(w) lower-directed gates and u(w) upper-directed gates, put cost_λ(w)=d(w)+λ u(w), and define ellWeightλ(A) = inf{cost_λ(w):[w]=A}. At λ=1 this is the ordinary ambient length, while large λ increasingly penalizes flag excu","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-94fb4fe48aaaaf97","tex":"\\CX_{0\\to1}","aliases":"CX_0→1 CX to","display":null,"kind":"use","excerpt":"1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f3d5c28121876f74","tex":"\\CX_{0\\to2}","aliases":"CX_0→2 CX to","display":null,"kind":"use","excerpt":"0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7727c2da277bccb7","tex":"\\CX_{1\\to0}","aliases":"CX_1→0 CX to","display":null,"kind":"use","excerpt":"1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-deb5279fd8b2545e","tex":"\\CX_{1\\to2}","aliases":"CX_1→2 CX to","display":null,"kind":"use","excerpt":"0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-61d8ac7abd099509","tex":"\\CX_{2\\to0}","aliases":"CX_2→0 CX to","display":null,"kind":"use","excerpt":"0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3adf45158b6a5dda","tex":"\\CX_{2\\to1}","aliases":"CX_2→1 CX to","display":null,"kind":"use","excerpt":"0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4addab54417d4899","tex":"\\CX_{c\\to t}","aliases":"CX_c→ t CX to","display":null,"kind":"use","excerpt":"Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained i","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-61b2e836c52d349d","tex":"\\CZ_{1,2}","aliases":"CZ_1,2 CZ","display":null,"kind":"use","excerpt":"0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-31aff3260d963816","tex":"\\CZ_{12}","aliases":"CZ_12 CZ","display":null,"kind":"use","excerpt":"0 · 1 · 0 · 1 · 0 · 0 . A restricted word of length seven is CX_0→2, CZ_12, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1. An ambient word of length six is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2, H_0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"[Generic rigidity and defect codimension] Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline ","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","excerpt":"With the positive-sign unitary matrix F, the negative-sign DFT satisfies F^† x=widehat x/sqrt12. fig05_transported_projector1.0Left: the frequency selector D. Right: its transported sample-basis matrix P. The right colorbar displays its five possible rational entries. The two panels use different scales. P=FDF^†, P_nm=fracc_12(n-m)12. Because D^2=D","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"umber of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-96cd702de2890fbe","tex":"D^\\dagger","aliases":"D^† dagger","display":"D^\\dagger","kind":"use","excerpt":"-sign DFT satisfies F^† x=widehat x/sqrt12. fig05_transported_projector1.0Left: the frequency selector D. Right: its transported sample-basis matrix P. The right colorbar displays its five possible rational entries. The two panels use different scales. P=FDF^†, P_nm=fracc_12(n-m)12. Because D^2=D and F^† F=I, we obtain P^2=P. Because D^†=D, we also","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d4781bd7c2512100","tex":"D_{\\mathrm B}","aliases":"D_ B mathrm","display":"D_{\\mathrm B}","kind":"use","excerpt":"Bosonic and fermionic analogues. For N fermions in m modes, D_ F(m,N)=binom mN. For N bosons in m modes, D_ B(m,N)=binomm+N-1N. The same binomial analysis gives:","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-00f91106b48ae9fb","tex":"D_{\\mathrm F}","aliases":"D_ F mathrm","display":"D_{\\mathrm F}","kind":"use","excerpt":"Bosonic and fermionic analogues. For N fermions in m modes, D_ F(m,N)=binom mN. For N bosons in m modes, D_ B(m,N)=binomm+N-1N. The same binomial analysis gives:","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e9d8746172823369","tex":"d_*","aliases":"d_*","display":"d_*","kind":"use","excerpt":"[Radius-four right-coset meet in the middle] Compute the exact radius-four ball B_4 in GL_6(F_2). Bucket its elements by κ. For each bucket and every ordered pair (b,a) in that bucket, compute u=b^-1a∈U_6 and update d_*(u)← min(d_*(u),d(b)+d(a)).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"The four exceptional matrices admit one common lower-bound proof. Work in the declared symplectic quotient and entangling metric, with H_i and S_i free and SWAP excluded as a primitive. Put J_2=0 · 1 1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"use","excerpt":"The four exceptional matrices admit one common lower-bound proof. Work in the declared symplectic quotient and entangling metric, with H_i and S_i free and SWAP excluded as a primitive. Put J_2=0 · 1 1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0ec57d816513bae2","tex":"d_A","aliases":"d_A","display":"d_A","kind":"use","excerpt":"[Breadth-first defect census] Given restricted generators S_R, ambient generators S_A, and identity e: item compute d_R on ⟨ S_R⟩; item compute d_A on ⟨ S_A⟩; item for every restricted state x, compare d_R(x) and d_A(x); item record the pair (d_R(x),d_A(x)) and reconstruct witnesses for strict inequalities.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-702eb05bdd1706ca","tex":"D_{aa}","aliases":"D_aa","display":"D_{aa}","kind":"use","excerpt":"A diagonal selector becomes a rational filter. addcontentslinetocsectionA diagonal selector becomes a rational filter leadThe filter is diagonal in one basis and a rational convolution in the other. Let F_na=ζ^na/sqrt12 and D_aa=1_a∈ U_12. With the positive-sign unitary matrix F, the negative-sign DFT satisfies F^† x=widehat x/sqrt12. fig05_transpo","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-18b312b01ad13df4","tex":"D_d","aliases":"D_d","display":"D_d","kind":"use","excerpt":"The set S_d contains the labels of characters of exact order d. Put D_d=diag(1_a∈ S_d) and","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3b858f3e8665ea65","tex":"D_e","aliases":"D_e","display":"D_e","kind":"use","excerpt":"QA04 proves rationality. Each D_d is a self-adjoint diagonal projector. The sets S_d partition I_N, because every character has exactly one order dividing N. Hence D_dD_e=δ_deD_d and ∑_dmid ND_d=Id_N. Conjugating these identities by the unitary F_N proves the projector identities. The rank is unchanged under multiplication by invertible matrices, s","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"use","excerpt":"Cut transport · Inverse suffix meets the appropriate forward target branch; H_i=fD_i gives safety and pointwise completeness","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5f5c77bbe3f93916","tex":"D_{\\mathrm{internal}}","aliases":"D_internal mathrm","display":"D_{\\mathrm{internal}}","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-753f6f7906cf9038","tex":"D_{n,k}","aliases":"D_n,k","display":"D_{n,k}","kind":"use","excerpt":"Exact sector size is a complexity phase variable. The statement ``fixed particle number is polynomial'' has a sharp extension. Let the particle number depend on the system size: k=k(n). The explicit-amplitude simulation cost of one fermionic or qubit weight packet is governed by D_n,k=binom nk. For 1≤ k≤ n, the standard bounds (frac nk)^k ≤ binom n","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3b21fae5fce3d6d2","tex":"\\mathcal D_{n,\\ge d}","aliases":"D_n,≥ d mathcal ge","display":"\\mathcal D_{n,\\ge d}","kind":"use","excerpt":"[Generic rigidity and defect codimension] Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6610be0b6d794c3f","tex":"D_n^{(1)}","aliases":"D_n^(1)","display":"D_n^{(1)}","kind":"use","excerpt":"Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-daddf1899dad021e","tex":"D_n^{(w)}","aliases":"D_n^(w)","display":"D_n^{(w)}","kind":"use","excerpt":"Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-72ffc1ce840ae45d","tex":"D_n^k","aliases":"D_n^k","display":"D_n^k","kind":"use","excerpt":"The symmetric subspace. Let the symmetric group S_n act by permuting the n tensor factors. The fully symmetric qubit subspace is spanned by the Dicke states ketD_n^k = binom nk^-1/2 ∑_|x|=kket x, 0≤ k≤ n. Its dimension is n+1.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d6fed9dcde083506","tex":"D_n^{k+1}","aliases":"D_n^k+1","display":"D_n^{k+1}","kind":"use","excerpt":"uad a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum numb","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7e976dea002f3e70","tex":"D_n^{k-1}","aliases":"D_n^k-1","display":"D_n^{k-1}","kind":"use","excerpt":"mmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decreasing excitation number rather than magnetic quantum number. Then J_+ketD_n^k = sqrt(n-k)(k+1)ketD_n^k+1,","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-79e4e0f34503c5b8","tex":"d_R","aliases":"d_R","display":"d_R","kind":"use","excerpt":"[Breadth-first defect census] Given restricted generators S_R, ambient generators S_A, and identity e: item compute d_R on ⟨ S_R⟩; item compute d_A on ⟨ S_A⟩; item for every restricted state x, compare d_R(x) and d_A(x); item record the pair (d_R(x),d_A(x)) and reconstruct witnesses for strict inequalities.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-891deb76dd6cd499","tex":"D_{\\rm boson}","aliases":"D_rm boson rm","display":"D_{\\rm boson}","kind":"use","excerpt":"Bosonic occupation space. For N indistinguishable bosons in m modes, a Fock basis state is an occupation vector (n_1,…,n_m), n_1+⋯+n_m=N. The dimension is D_rm boson(m,N) = binomN+m-1N. For fixed N, this is O(m^N).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-372f7844af966f00","tex":"D_{\\rm fermion}","aliases":"D_rm fermion rm","display":"D_{\\rm fermion}","kind":"use","excerpt":"Fermionic occupation space. For N fermions in m modes, each mode is occupied at most once and D_rm fermion(m,N) = binom mN. For fixed N, this is also polynomial in m.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-530475fd78b2a838","tex":"D_{\\rm internal}","aliases":"D_rm internal rm","display":"D_{\\rm internal}","kind":"use","excerpt":"The core research invariant is the complete classical representation record ( discovery cost, N_rm packets, D_rm internal, S_rm transfer, L_rm leakage, measurement cost ). A future compiler that computes and certifies this record can decide when the full state vector is unnecessary, when a hybrid representation is preferable, and when symmetry alon","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-beffdc290038be56","tex":"D_{\\rm packet}","aliases":"D_rm packet rm","display":"D_{\\rm packet}","kind":"use","excerpt":"Entanglement inside a packet. A small number of symmetry labels does not imply low entanglement. The half-filled packet can support volume-law entanglement. Tensor-network bond dimension and packet dimension are independent resources. A robust simulator should therefore track both (D_rm packet,χ_rm internal), where χ_rm internal measures the cost o","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6ea3eb665bcad800","tex":"D_S","aliases":"D_S","display":"D_S","kind":"use","excerpt":"For any set S⊆ I_N, let D_S be its diagonal indicator and P_S=F_ND_SF_N^†. This construction does not require S to be an order class. Rationality is special, but orthogonal projection is general.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4ffc869e223f4567","tex":"d_S","aliases":"d_S","display":"d_S","kind":"use","excerpt":"[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"use","excerpt":"Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b377ac1e0c05ff0f","tex":"d_t","aliases":"d_t","display":"d_t","kind":"use","excerpt":"Fix the projectors chosen during the particular run. Put d_t=(I-P_t)G_tϕ_t-1. Subtraction gives the exact recurrence ψ_t-ϕ_t=G_t(ψ_t-1-ϕ_t-1)+d_t. Since ψ_0-ϕ_0=0, iteration yields ψ_m-ϕ_m=∑_t=1^mG_m⋯ G_t+1d_t. Every unitary product preserves norm, so the triangle inequality proves the state estimate. Also ψ_m =1 and ϕ_m ≤1 because each projector i","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ee1684d3a8f6d798","tex":"D_u","aliases":"D_u","display":"D_u","kind":"use","excerpt":"A candidate lower-bound invariant must do more than track maximum degree. One Toffoli creates degree two and a second can remove it. Useful refinements include: item the dimension of the quadratic monomial span, item directional derivatives D_u f(x)=f(x+u)+f(x), item the rank of the bilinear polarization, item the number of outputs consuming a nonl","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1f87edb2dfa256d3","tex":"d_\\chi","aliases":"d_χ chi","display":"d_\\chi","kind":"use","excerpt":"Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-593fbff05f5c23a7","tex":"D_{\\le K}","aliases":"D_≤ K le","display":"D_{\\le K}","kind":"use","excerpt":"A bounded-excitation hierarchy. The union of the first K weight sectors has dimension D_≤ K(n) = ∑_k=0^Kbinom nk. For fixed K, D_≤ K(n)=O(n^K). If K=O(log n), the same union is quasipolynomial. This refines [source reference]: bounded symmetry breaking preserves polynomial simulation, while logarithmically growing excitation gives a controlled quas","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1e31f1e54430399f","tex":"\\operatorname{decode}","aliases":"decode operatorname","display":"\\operatorname{decode}","kind":"use","excerpt":"Let R_n be a class of classical descriptions, and let decode: R_n→ P_nHh_n associate a quantum state with each description.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.","count":15},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a674cead272def3d","tex":"\\mathrm{depth}","aliases":"depth mathrm","display":"\\mathrm{depth}","kind":"use","excerpt":"Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag","display":null,"kind":"use","excerpt":"Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ffa9d7f17ebd6a0c","tex":"\\mathrm{diag}","aliases":"diag mathrm","display":"\\mathrm{diag}","kind":"use","excerpt":"ρ_diag=diag(c(n)^2/48)_n=0^11.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0ebaf71d15dcdb62","tex":"\\operatorname{diam}","aliases":"diam operatorname","display":"\\operatorname{diam}","kind":"use","excerpt":"Clifford census summary. @ccccrrrr@ Qubits · Metric · SWAP · Restricted size · Ambient size · diam_R · diam_A · Defects","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a275a5b0db0035e1","tex":"\\operatorname{diam}_A","aliases":"diam_A operatorname","display":"\\operatorname{diam}_A","kind":"use","excerpt":"Clifford census summary. @ccccrrrr@ Qubits · Metric · SWAP · Restricted size · Ambient size · diam_R · diam_A · Defects","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5f6e237585322826","tex":"\\operatorname{diam}_R","aliases":"diam_R operatorname","display":"\\operatorname{diam}_R","kind":"use","excerpt":"Clifford census summary. @ccccrrrr@ Qubits · Metric · SWAP · Restricted size · Ambient size · diam_R · diam_A · Defects","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"ratic-cost For even n, let Q(x)=x_1x_2+x_3x_4+⋯+x_n-1x_n. Then M(Q)=KTrest(Q)=n/2. The polar bilinear form of Q is the full-rank alternating form made of n/2 disjoint 2×2 blocks. If Q is affine on an affine subspace with direction space U, its second differences vanish on U, so U is totally isotropic. Nondegeneracy gives dim U≤ n/2.","count":15},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-98d940b7f3ffe088","tex":"\\dim_K","aliases":"dim_K dim","display":"\\dim_K","kind":"use","excerpt":"[Dimension of a faithful multiplicative summary] Let Φ: B→ A be a K-linear multiplicative map into a K-algebra. Suppose Φ(δ_v)≠0 for every v∈ W. Then dim_K A≥2^m. No unital hypothesis on Φ is required. Write e_v=Φ(δ_v). Multiplicativity makes these nonzero orthogonal idempotents. If ∑_v∈ Wc_ve_v=0, multiply the equality by e_u. Orthogonality annihi","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fe7112a7c8a12719","tex":"\\mathrm{dirty}","aliases":"dirty mathrm","display":"\\mathrm{dirty}","kind":"use","excerpt":"Ancillas define a family of defects Δ_a,clean(U), Δ_a,dirty(U), where a is the number of ancillary qubits or modes. Stable ancilla capacity asks whether a fixed number of reusable ancillas can yield an unbounded saving as the data size grows.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.","count":17},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3f0565f465b345a1","tex":"e^{-2\\pi ikn/12}","aliases":"e^-2π ikn/12 pi","display":"e^{-2\\pi ikn/12}","kind":"use","excerpt":"The spectrum recovers the selector. addcontentslinetocsectionThe spectrum recovers the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 an","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1fe74a04db9b8f15","tex":"e^{-2\\pi ikr/12}","aliases":"e^-2π ikr/12 pi","display":"e^{-2\\pi ikr/12}","kind":"use","excerpt":"Phase factors connect the lane spectra. addcontentslinetocsectionPhase factors connect the lane spectra leadEach local Fourier transform contributes to every appropriate global frequency. With four-point negative-sign DFTs in the lanes, widehat x(k)=∑_r=0^2e^-2π ikr/12 widehatE_rx(kbmod4). fig18_polyphase_alias_assembly1.0Left: magnitudes of the th","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bc0281d897aa4a39","tex":"e^{-2\\pi i\\ell/3}","aliases":"e^-2π iℓ/3 pi ell","display":"e^{-2\\pi i\\ell/3}","kind":"use","excerpt":"2(4+2e^-2π iℓ/3+2e^-4π iℓ/3).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-14d42a489603418a","tex":"e^{-4\\pi i\\ell/3}","aliases":"e^-4π iℓ/3 pi ell","display":"e^{-4\\pi i\\ell/3}","kind":"use","excerpt":"2(4+2e^-2π iℓ/3+2e^-4π iℓ/3).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5a1e1b0a1da34e35","tex":"e^{-i(t-s)H}","aliases":"e^-i(t-s)H","display":"e^{-i(t-s)H}","kind":"use","excerpt":"Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-390457a95f8f6d57","tex":"e^{-isH}","aliases":"e^-isH","display":"e^{-isH}","kind":"use","excerpt":"Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9eaf8275847a72c8","tex":"e^{-itA}","aliases":"e^-itA","display":"e^{-itA}","kind":"use","excerpt":"Fourier packets. On a cycle, translation characters diagonalize the adjacency operator. The continuous-time walk ketψ(t)=e^-itAket0 is a sum of Fourier-character packets. This is the same discrete Fourier transform that appears as a signal-processing coordinate change, a group-character decomposition, and a quantum basis transformation.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2637b6dd9fae991f","tex":"e^{-itH}","aliases":"e^-itH","display":"e^{-itH}","kind":"use","excerpt":"Hamiltonian leakage. Exact conservation is not the only useful regime. Let P be an orthogonal projector and let U(t)=e^-itH for a bounded self-adjoint Hamiltonian H. The leakage operator from the packet is L_P(t)=(I-P)e^-itHP.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-30cc66e2c5389e47","tex":"e^{-i\\theta P/2}","aliases":"e^-iθ P/2 theta","display":"e^{-i\\theta P/2}","kind":"use","excerpt":"Pauli rotations. For a Hermitian Pauli P, R_P(θ)=e^-iθ P/2. Clifford conjugation gives C R_P(θ) C^†=R_CPC^†(θ). The sign of CPC^† can be absorbed by changing θ to -θ.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f0200738329f1979","tex":"e^{-i\\phi}","aliases":"e^-iϕ phi","display":"e^{-i\\phi}","kind":"use","excerpt":"Number-preserving gates. Single-qubit R_Z gates commute with widehat N. A two-qubit exchange rotation acts in the ordered basis ket00,ket01,ket10,ket11 as U_XY(θ,ϕ) = 1 · 0 · 0 · 0 0 · cosθ · -ie^-iϕsinθ · 0 0 · -ie^iϕsinθ · cosθ · 0 0 · 0 · 0 · 1 . It rotates ket01 and ket10 while fixing ket00 and ket11, so it preserves exact Hamming weight.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1933b821b71bc9d3","tex":"e^{-\\varepsilon X}","aliases":"e^-ε X varepsilon","display":"e^{-\\varepsilon X}","kind":"use","excerpt":"Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-037c6d05704c1045","tex":"e^{-\\varepsilon Y}","aliases":"e^-ε Y varepsilon","display":"e^{-\\varepsilon Y}","kind":"use","excerpt":"Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b7171450b0b2fd1d","tex":"e^{2\\pi i/12}","aliases":"e^2π i/12 pi","display":"e^{2\\pi i/12}","kind":"use","excerpt":"Coprimality selects primitive roots. addcontentslinetocsectionCoprimality selects primitive roots leadA selected residue is a label for a wave whose phase has exact order twelve. Set ζ=e^2π i/12. The phase ζ^a first returns to 1 after 12/gcd(a,12) powers. Thus the coprime labels 1,5,7,11 select exactly the primitive twelfth roots. fig01_primitive_r","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-47852f9fe0da341d","tex":"e^{2\\pi i/M}","aliases":"e^2π i/M pi","display":"e^{2\\pi i/M}","kind":"use","excerpt":"Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a9b3a26c9c59d042","tex":"e^{2\\pi i/q}","aliases":"e^2π i/q pi","display":"e^{2\\pi i/q}","kind":"use","excerpt":"ζ_q=e^2π i/q.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d059d359ebd2974f","tex":"e^{2\\pi ian/12}","aliases":"e^2π ian/12 pi","display":"e^{2\\pi ian/12}","kind":"use","excerpt":"twelve will become an integer signal, an exact rational filter, and a normalized quantum state. Every bridge will be written down. The guide follows one object through three coordinate systems. The starting condition is coprimality. The signal realization adds its selected waves. The quantum realization normalizes the amplitude vector and distingui","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1b0fcc15a66dea8a","tex":"e^{2\\pi ibn/(q/d)}","aliases":"e^2π ibn/(q/d) pi","display":"e^{2\\pi ibn/(q/d)}","kind":"use","excerpt":"c_q(n) · =∑_dmid qμ(d) ∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q (q/d)mid nμ(d)frac qd.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2f1d2e66e9a3c7f5","tex":"e^{2\\pi i\\ell Q/M}","aliases":"e^2π iℓ Q/M pi ell","display":"e^{2\\pi i\\ell Q/M}","kind":"use","excerpt":"Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-682a47b2eea70d50","tex":"e^{i\\theta}","aliases":"e^iθ theta","display":"e^{i\\theta}","kind":"use","excerpt":"χ_θ=v_++e^iθv_-.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-86b671739d5b7f33","tex":"e^{i\\pi/4}","aliases":"e^iπ/4 pi","display":"e^{i\\pi/4}","kind":"use","excerpt":"Let ω=e^iπ/4. A phase-polynomial circuit on n qubits has the form [source reference] Uketx=ω^P(x)ketAx+b, where A∈GL(n,2), b∈F_2^n, and P:F_2^n→Z_8 is represented as a sum of binary linear forms with coefficients in Z_8. The diagonal case has A=I and b=0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f602ae83d3f7d022","tex":"e^{i\\phi}","aliases":"e^iϕ phi","display":"e^{i\\phi}","kind":"use","excerpt":"Number-preserving gates. Single-qubit R_Z gates commute with widehat N. A two-qubit exchange rotation acts in the ordered basis ket00,ket01,ket10,ket11 as U_XY(θ,ϕ) = 1 · 0 · 0 · 0 0 · cosθ · -ie^-iϕsinθ · 0 0 · -ie^iϕsinθ · cosθ · 0 0 · 0 · 0 · 1 . It rotates ket01 and ket10 while fixing ket00 and ket11, so it preserves exact Hamming weight.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-eb408ac1c5514170","tex":"e^{\\varepsilon X}","aliases":"e^ε X varepsilon","display":"e^{\\varepsilon X}","kind":"use","excerpt":"Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-df3e226f91cc1e0c","tex":"e^{\\varepsilon Y}","aliases":"e^ε Y varepsilon","display":"e^{\\varepsilon Y}","kind":"use","excerpt":"Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2fe526a56a9dcea8","tex":"e^{\\varepsilon^2[X,Y]+O(\\varepsilon^3)}","aliases":"e^ε^2[X,Y]+O(ε^3) varepsilon varepsilon","display":"e^{\\varepsilon^2[X,Y]+O(\\varepsilon^3)}","kind":"use","excerpt":"Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ed3905e7c0b0f722","tex":"e^{\\pi i/6}","aliases":"e^π i/6 pi","display":"e^{\\pi i/6}","kind":"use","excerpt":"U_12={1,5,7,11}, φ(12)=4, ζ=e^π i/6=fracsqrt32+frac i2.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"The six-gate ambient circuit uses row 1 as a dirty workspace. It first adds row 2 into row 1, exports the carrier e_1+e_2, changes the carrier to e_0+e_1+e_2, exports it twice, and finally removes the temporary e_2 component. No extra wire is introduced. The workspace is one of the target rows and is restored to its prescribed value.","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"The six-gate ambient circuit uses row 1 as a dirty workspace. It first adds row 2 into row 1, exports the carrier e_1+e_2, changes the carrier to e_0+e_1+e_2, exports it twice, and finally removes the temporary e_2 component. No extra wire is introduced. The workspace is one of the target rows and is restored to its prescribed value.","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"The six-gate ambient circuit uses row 1 as a dirty workspace. It first adds row 2 into row 1, exports the carrier e_1+e_2, changes the carrier to e_0+e_1+e_2, exports it twice, and finally removes the temporary e_2 component. No extra wire is introduced. The workspace is one of the target rows and is restored to its prescribed value.","count":17},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-95cc755c1979e96c","tex":"e_{2n-1}","aliases":"e_2n-1","display":"e_{2n-1}","kind":"use","excerpt":"The parabolic subgroup used in the census is generated by S_i, CZ_ij, and CX_i→ j. It stabilizes the Z Lagrangian L_Z=Span(e_n,…,e_2n-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-67ab753297c61ce5","tex":"e_4","aliases":"e_4","display":"e_4","kind":"use","excerpt":"Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a64c5d974442c22a","tex":"e_5","aliases":"e_5","display":"e_5","kind":"use","excerpt":"Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d9b6193b1833f637","tex":"E_a","aliases":"E_a","display":"E_a","kind":"use","excerpt":"E_a=frac1N∑_ℓ=0^N-1ζ_N^aℓA^ℓ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-19f85b8b87f42d0c","tex":"e_{i_r}","aliases":"e_i_r","display":"e_{i_r}","kind":"use","excerpt":"C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ca7e75161b6bbf35","tex":"E_{ij}","aliases":"E_ij","display":"E_{ij}","kind":"use","excerpt":"Under a Gaussian gate alphabet in which each embedded elementary transvection ι(I+aE_ij) is one primitive and direction is the only restriction, the universal six-row catalytic atom embeds into a twelve-quadrature Gaussian atom with restricted length seven and ambient length six.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a0f2ded25108c22b","tex":"e_{j_r}^{T}","aliases":"e_j_r^T","display":"e_{j_r}^{T}","kind":"use","excerpt":"C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"The parabolic subgroup used in the census is generated by S_i, CZ_ij, and CX_i→ j. It stabilizes the Z Lagrangian L_Z=Span(e_n,…,e_2n-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fdcecacb4b61f1a1","tex":"e_{n+i}","aliases":"e_n+i","display":"e_{n+i}","kind":"use","excerpt":"Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1b09d4efdcdc1f39","tex":"e_p","aliases":"e_p","display":"e_p","kind":"use","excerpt":"inF^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_y","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-244f69b2cbc609aa","tex":"e_q","aliases":"e_q","display":"e_q","kind":"use","excerpt":"inF^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_y","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3a840ddc04edad8c","tex":"E_r","aliases":"E_r","display":"E_r","kind":"use","excerpt":"E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"Suppose instead that an output row, say row r∈{3,4,5}, is targeted twice. The other two outputs and row 1 are targeted once. We claim that row r cannot lie in W between its two targetings. To cancel its unique quotient pivot e_r at the first targeting, the source would already have to contain an e_r component. Before row r is first targeted, the on","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-caa961746837d921","tex":"e_s","aliases":"e_s","display":"e_s","kind":"use","excerpt":"Projection and unitary change are different operations. addcontentslinetocsectionProjection and unitary change are different operations leadA basis change preserves the whole vector. Projection keeps only one orthogonal component. Let e_s=c_12/sqrt48 and e_r=(I-P)x/sqrt102. They are orthonormal, so the normalized input has the exact two-dimensional","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2775a21153e9ebc4","tex":"e_t","aliases":"e_t","display":"e_t","kind":"use","excerpt":"[One-target source condition] Let W≤ R^n. Suppose row t starts at e_t, is targeted exactly once, and finishes at e_t+w, w∈ W, where at least one coordinate of w is a unit of R. At the moment of its unique targeting, the gate coefficient is a unit and the source row is a unit multiple of w.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a6847984f15b8468","tex":"E_{t,s}","aliases":"E_t,s","display":"E_{t,s}","kind":"use","excerpt":"Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d6ed61024ab0139e","tex":"E_{t_1,s_1}","aliases":"E_t_1,s_1","display":"E_{t_1,s_1}","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-be42e928df106422","tex":"e_{t_j}","aliases":"e_t_j","display":"e_{t_j}","kind":"use","excerpt":"[ 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4d5522763d037924","tex":"E_{t_L,s_L}","aliases":"E_t_L,s_L","display":"E_{t_L,s_L}","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d4c39e5e1f0a0028","tex":"e_x","aliases":"e_x","display":"e_x","kind":"use","excerpt":"Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9a47be0aefdd6137","tex":"e_{x_i}","aliases":"e_x_i","display":"e_{x_i}","kind":"use","excerpt":"Fix integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6cb49e563b7d9a67","tex":"e_y","aliases":"e_y","display":"e_y","kind":"use","excerpt":"inF^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-aB, R_q=e_q+B, R_y=e_y+c_ye_p+ac_y","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-424615e27a7fb7da","tex":"\\Egrp","aliases":"Egrp","display":null,"kind":"use","excerpt":"rected when s<t. The lower-directed gates generate U_n(R). We compare ellDown(A) =min{number of lower-directed transvections producing A} with ellAll(A) =min{number of arbitrary row transvections producing A}, A∈U_n(R). For a field F, the first metric is intrinsic to the unipotent radical and the second is induced from the elementary group Egrp_n(F","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9232d5f8f9f4a3b5","tex":"\\Egrp_n","aliases":"Egrp_n Egrp","display":null,"kind":"use","excerpt":"rected when s<t. The lower-directed gates generate U_n(R). We compare ellDown(A) =min{number of lower-directed transvections producing A} with ellAll(A) =min{number of arbitrary row transvections producing A}, A∈U_n(R). For a field F, the first metric is intrinsic to the unipotent radical and the second is induced from the elementary group Egrp_n(F","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2437637ee637961e","tex":"\\ellAmb^{(w)}","aliases":"ellAmb^(w) ellAmb","display":null,"kind":"use","excerpt":"Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-808df610e2444e78","tex":"\\ellWeight","aliases":"ellWeight","display":null,"kind":"use","excerpt":"en gates and weighted penalties] For every field and every n≥7, there is a connected target A_n∈U_n(F) such that ellBudget0(A_n)-ellBudget2(A_n) ≥ lfloorfracn-52rfloor, where ellBudgetq permits at most q upper-directed gates. More generally, if lower gates have cost 1 and upper gates have cost λ>0, then ellDown(A_n)-ellWeightλ(A_n) ≥ lfloorfracn-12","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b7739c0b6665ad79","tex":"\\mathrm{encode}","aliases":"encode mathrm","display":"\\mathrm{encode}","kind":"use","excerpt":"pageheadReading the formal model literallyQUANTIFIERS AND ENCODING CARRY THE HYPOTHESES The generic cut theory uses a logical-input type X, a wire type W with decidable equality, and a boundary structure. Its essential fields are an encoding encode:X→Bool→(W→Bool), a target wire, and the proof encode(x,b)(target)=b. This target equation separates t","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-24553baae83bc4c0","tex":"\\mathrm{entangled}","aliases":"entangled mathrm","display":"\\mathrm{entangled}","kind":"use","excerpt":"This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8614d1502b190274","tex":"\\mathrm{err}","aliases":"err mathrm","display":"\\mathrm{err}","kind":"use","excerpt":"For stochastic error rates p_g, a first-order weighted cost is C_err(W)=∑_g∈ W-log(1-p_g). This additive approximation can be inserted directly into Dijkstra search. It does not capture coherent or correlated errors, but it produces a declared and reproducible metric.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b587845b9aeba50f","tex":"\\mathrm{exist}","aliases":"exist mathrm","display":"\\mathrm{exist}","kind":"use","excerpt":"Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-39ea6df1a878e2c7","tex":"\\Exp","aliases":"Exp","display":null,"kind":"use","excerpt":"[Finite-field defect density] For 1≤ d≤ N, Prob_A∈U_n(F_q)[DeltaDef(A)≥ d] ≤ min{1,M^N-dq^-d}. In particular, for fixed n, Prob[DeltaDef(A)>0]=O_n(q^-1), Exp[DeltaDef(A)]=O_n(q^-1) as q→∞.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"The typed mathematical target. The complex matrices act on vectors indexed by Fin(12). The Fourier convention is F_nk=frac1sqrt12exp (frac2π i nk12), n,k∈{0,…,11}. The adjoint F^* therefore carries the negative exponential sign. The selector D is diagonal with value one precisely at the residues U={1,5,7,11}. The transported operator is P=FDF^*.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"All arithmetic is over F_2, implemented using Boolean XOR and AND. Each gate is its own inverse. Circuit reversal, restoration, and every cut are inherited from the unchanged Block 01 source.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-484de1c254ac2bb8","tex":"\\mathrm F","aliases":"F mathrm","display":"\\mathrm F","kind":"use","excerpt":"Bosonic and fermionic analogues. For N fermions in m modes, D_ F(m,N)=binom mN. For N bosons in m modes, D_ B(m,N)=binomm+N-1N. The same binomial analysis gives:","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7622cd705b527bdd","tex":"f^{(0)}","aliases":"f^(0)","display":"f^{(0)}","kind":"use","excerpt":"Higher-order Möbius coefficients. The ANF coefficients are the Möbius transform of the truth table. Projectors onto degree layers give an exact packet decomposition f=f^(0)+f^(1)+f^(2)+⋯. A restricted affine circuit remains in degrees zero and one. The ambient circuit can populate higher packets temporarily. A successful lower bound must charge the","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cd92a6255ed1ce32","tex":"f^{(1)}","aliases":"f^(1)","display":"f^{(1)}","kind":"use","excerpt":"Higher-order Möbius coefficients. The ANF coefficients are the Möbius transform of the truth table. Projectors onto degree layers give an exact packet decomposition f=f^(0)+f^(1)+f^(2)+⋯. A restricted affine circuit remains in degrees zero and one. The ambient circuit can populate higher packets temporarily. A successful lower bound must charge the","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5aab3187abf7c305","tex":"f^{(2)}","aliases":"f^(2)","display":"f^{(2)}","kind":"use","excerpt":"Higher-order Möbius coefficients. The ANF coefficients are the Möbius transform of the truth table. Projectors onto degree layers give an exact packet decomposition f=f^(0)+f^(1)+f^(2)+⋯. A restricted affine circuit remains in degrees zero and one. The ambient circuit can populate higher packets temporarily. A successful lower bound must charge the","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a3382034c436fa8d","tex":"F^*","aliases":"F^*","display":"F^*","kind":"use","excerpt":"The typed mathematical target. The complex matrices act on vectors indexed by Fin(12). The Fourier convention is F_nk=frac1sqrt12exp (frac2π i nk12), n,k∈{0,…,11}. The adjoint F^* therefore carries the negative exponential sign. The selector D is diagonal with value one precisely at the residues U={1,5,7,11}. The transported operator is P=FDF^*.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-49ddaa56c96d4e05","tex":"f^{-1}","aliases":"f^-1","display":"f^{-1}","kind":"use","excerpt":"Exact evaluation-span certificates. For an integer d≥0, form the evaluation vector v_d(x)=(x^S)_S⊆[n], |S|≤ d, and write U=f^-1(0).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9ee8247244d03326","tex":"\\F^m","aliases":"F^m F","display":null,"kind":"use","excerpt":"A compact whole function. Let m≥2, let a=(1,…,1)∈F^m, and introduce one logical bit x_v for each v∈ V=F^m∖{0,a}. There are n=2^m-2 such bits. Define L_j(x)=∑_v:v_j=1x_v, P_m(x)=∏_j=1^mL_j(x).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-97b85bcfaa51fa20","tex":"\\F^n","aliases":"F^n F","display":null,"kind":"use","excerpt":"For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among fr","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b37ef33aef07505b","tex":"\\F^\\times","aliases":"F^× F times","display":null,"kind":"use","excerpt":"Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-27e0e37e6fd58273","tex":"F^\\dagger","aliases":"F^† dagger","display":"F^\\dagger","kind":"use","excerpt":"The later source explicitly supplies FDF^†=P, both normalized state-action identities, the conventional exponential formula for F, identical initial probabilities, orthogonality, exact selected probability one versus zero, histogram nonreconstruction, and projector noninjectivity. It therefore closes the representation-transport obligations that th","count":28},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-80e8e28481341311","tex":"F_{12}","aliases":"F_12","display":"F_{12}","kind":"use","excerpt":"With D=diag(1_a∈ U_12) and F=F_12,","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-70149f0cbcab288a","tex":"\\F_{16}","aliases":"F_16 F","display":null,"kind":"use","excerpt":"The field is F_16=F_2[θ]/(θ^4+θ+1). The checker stores field elements as four-bit coefficient vectors: θ is integer 2 and θ^2 is integer 4. These integers encode field elements; they are not ordinary integer multiplication. Exact field multiplication gives the reported mismatch.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-101fbe8a1f9aa65c","tex":"\\F_2","aliases":"F_2 F","display":null,"kind":"use","excerpt":"All polynomial arithmetic in the Boolean state model is over F_2, with x_i^2=x_i. Every function has a unique multilinear algebraic normal form. Its degree is the largest support size with a nonzero coefficient; the zero function has mathematical degree -1 in the Python helpers. A formal degree bound uses a natural number, so zero has a degree-zero","count":38},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"All arithmetic is over F_2, implemented using Boolean XOR and AND. Each gate is its own inverse. Circuit reversal, restoration, and every cut are inherited from the unchanged Block 01 source.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-759cd3fb7e585adb","tex":"\\F_2^{2n}","aliases":"F_2^2n F","display":null,"kind":"use","excerpt":"Binary symplectic representation. Modulo global phase and Pauli translations, an n-qubit Clifford operation acts on Pauli exponent vectors v=(x_0,…,x_n-1mid z_0,…,z_n-1)^T∈F_2^2n through a symplectic matrix M∈Sp(2n,2). The elementary gates used here act as follows: H_i · : (x_i,z_i)↦(z_i,x_i),","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-62198a97093af5d0","tex":"\\F_2^3","aliases":"F_2^3 F","display":null,"kind":"use","excerpt":"Over F_2, write x=e_0, p=e_1, q=e_2. The canonical ambient word uses the six nonzero vectors x, p, q, p+q, p+q+x, p+x inside Span(x,p,q)≅F_2^3. The only missing nonzero vector is x+q.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5a5c9b41f925e2ad","tex":"\\F_2^n","aliases":"F_2^n F","display":null,"kind":"use","excerpt":"Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained i","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-74317064bf2c9a4d","tex":"F_3","aliases":"F_3","display":"F_3","kind":"use","excerpt":"guidepageOne safe packet need not cover every accepting point For k=3 consider F_3(a,b,z_1,z_2,z_3)=ab OR z_1z_2z_3. The packet ab is safe and has degree two. It detects every teal point below. It misses three additional positives, which require the other packet z_1z_2z_3.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-08c122811477fa77","tex":"\\F_3","aliases":"F_3 F","display":null,"kind":"use","excerpt":"The equality ellDown(A)=ellAll(A) is tempting because the target and the obvious Gaussian-elimination paths are lower unitriangular. It is false. The failure first occurs at six wires over F_2 and F_3, and it is caused by a reusable temporary carrier rather than by a better ordering of the same lower gates.","count":13},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0a27cb068825ed9a","tex":"\\F_5","aliases":"F_5 F","display":null,"kind":"use","excerpt":"The n=5 ternary computation uses radius-three and radius-four balls, checks 38415205 right-coset pairs, and recovers all 59049 targets. It finds no defect. The field-independent [source reference] supplies a defect at n=6, proving [source reference]. Separate exact searches over F_3 and F_5 verify the distances of the canonical atom directly. The F","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b314e2300364246a","tex":"f_a","aliases":"f_a","display":"f_a","kind":"use","excerpt":"Let A be the cyclic shift (Ax)(n)=x(n-1). A Fourier column f_a(n)=N^-1/2ζ_N^na satisfies Af_a=ζ_N^-af_a. Therefore","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"use","excerpt":"Algebraic normal form. Every Boolean coordinate function has a unique algebraic normal form f_i(x) = ⊕_S⊆[n]a_i,S∏_j∈ Sx_j. Affine maps have degree at most one. A Toffoli gate targeting t with controls a,b performs x_t⟼ x_t+x_ax_b, which can raise the degree of later coordinates. The final affine target requires every higher-degree coefficient to c","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7e04443bf0b17f7b","tex":"F_k","aliases":"F_k","display":"F_k","kind":"use","excerpt":"Calibration families and exact small oracles. claimC03 and C04 An unbalanced disjunction. For k≥2 and n=k+2, define B=z_1⋯ z_k, F_k(a,b,z)=(ab)lor B=ab+B+abB, G_k=F_k+a. This is a deliberately easy family with two differently sized accepting regions. It separates the cost of finding one safe packet from the degree needed for complete coverage.","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0fe9f993f4ff4310","tex":"F_N","aliases":"F_N","display":"F_N","kind":"use","excerpt":"P_d^(N)=F_ND_dF_N^†.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3a1a7da23fedbabf","tex":"F_N^\\dagger","aliases":"F_N^† dagger","display":"F_N^\\dagger","kind":"use","excerpt":"P_d^(N)=F_ND_dF_N^†.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8883f5f802462156","tex":"F_{na}","aliases":"F_na","display":"F_{na}","kind":"use","excerpt":"A diagonal selector becomes a rational filter. addcontentslinetocsectionA diagonal selector becomes a rational filter leadThe filter is diagonal in one basis and a rational convolution in the other. Let F_na=ζ^na/sqrt12 and D_aa=1_a∈ U_12. With the positive-sign unitary matrix F, the negative-sign DFT satisfies F^† x=widehat x/sqrt12. fig05_transpo","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8de2e0374868ba51","tex":"F_{nk}","aliases":"F_nk","display":"F_{nk}","kind":"use","excerpt":"The typed mathematical target. The complex matrices act on vectors indexed by Fin(12). The Fourier convention is F_nk=frac1sqrt12exp (frac2π i nk12), n,k∈{0,…,11}. The adjoint F^* therefore carries the negative exponential sign. The selector D is diagonal with value one precisely at the residues U={1,5,7,11}. The transported operator is P=FDF^*.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-666cac5565f15579","tex":"F_q","aliases":"F_q","display":"F_q","kind":"use","excerpt":"(F_q)_na=q^-1/2ζ_q^na.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f9d082b0820d2930","tex":"\\F_q","aliases":"F_q F","display":null,"kind":"use","excerpt":"Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estima","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-83eec918d09bb7e3","tex":"F_q^\\dagger","aliases":"F_q^† dagger","display":"F_q^\\dagger","kind":"use","excerpt":"Thus F_q synthesizes a signal from normalized frequency coordinates, while F_q^† x=widehatx/sqrt q analyzes it. This convention explains all signs and all square-root factors below. The negative-sign transform and the positive-sign synthesis are adjoints of one another.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aafff6a8970aae8f","tex":"f_\\lambda","aliases":"f_λ lambda","display":"f_\\lambda","kind":"use","excerpt":"Exact commutant record. Let u_λ=dim U_λ, f_λ=dim V_λ. Schur's lemma gives the following normal form.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-280dad33f94a39d1","tex":"\\operatorname{Fin}","aliases":"Fin operatorname","display":"\\operatorname{Fin}","kind":"use","excerpt":"The typed mathematical target. The complex matrices act on vectors indexed by Fin(12). The Fourier convention is F_nk=frac1sqrt12exp (frac2π i nk12), n,k∈{0,…,11}. The adjoint F^* therefore carries the negative exponential sign. The selector D is diagonal with value one precisely at the residues U={1,5,7,11}. The transported operator is P=FDF^*.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-91dc582adffb9b2f","tex":"\\mathrm{final}","aliases":"final mathrm","display":"\\mathrm{final}","kind":"use","excerpt":"endpmatrix. A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-00c37c61b4205056","tex":"\\mathrm{full}","aliases":"full mathrm","display":"\\mathrm{full}","kind":"use","excerpt":"State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectoriza","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.","count":12},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"β^*(f) · Least upper bound on β(fx g) over all degree-one g.","count":78},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"[Retraction inheritance] Suppose a smaller nested pair H≤ G embeds into H'≤ G' by homomorphisms preserving the inclusion, the ambient embedding does not increase gate cost, and there is a homomorphic retraction π:H'→ H onto the embedded copy. Require ℓ_H(π(s))≤ c(s) for every restricted generator s of H'. Then every defect in H≤ G gives a defect at","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aa45a7ad376e6ef9","tex":"g^{-1}","aliases":"g^-1","display":"g^{-1}","kind":"use","excerpt":"The basic conjugation inequality. Let g∈ G and k∈ H satisfy gkg^-1∈ H. Then ℓ_G(gkg^-1) ≤ 2ℓ_G(g)+ℓ_H(k). Define the intrinsic conjugation stretch Stretch_g(k) = ℓ_H(gkg^-1)-ℓ_H(k).","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f1ad004ee5a8f063","tex":"G^L","aliases":"G^L","display":"G^L","kind":"use","excerpt":"Every irreducible component of the Cartesian power X^× L has dimension at most rL. Multiplication G^L→ G is a morphism, so the closure of the image of each component has dimension at most rL. There are finitely many components, and intersecting the resulting finite union with V cannot increase dimension. Since the identity belongs to X, products of","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d89460d4612fc422","tex":"\\mathcal G_{*,C}","aliases":"G_*,C mathcal","display":"\\mathcal G_{*,C}","kind":"use","excerpt":"For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3b9dcaf46a46092c","tex":"G_2","aliases":"G_2","display":"G_2","kind":"use","excerpt":"[Long-root catalytic gain] There exist gate alphabets in types B, C, or G_2 for which a clean one-catalyst multiplicative distortion exceeds the type-A snapshot limit of 3/2. No lower-bound certificate is presently supplied. The conjecture identifies a concrete reason to test nonsimply laced root systems rather than merely repeating type A computat","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2b23d0ba1d3b9477","tex":"G_{\\mathrm{conn}}","aliases":"G_conn mathrm","display":"G_{\\mathrm{conn}}","kind":"use","excerpt":"Architecture and routing. On a line or grid, Gaussian and matchgate circuits may be dominated by routing. Let G_conn be the hardware graph. A mode permutation π can be treated as a catalyst if it is opened, many interactions are executed in the reordered frame, and π^-1 restores the logical labeling. The benefit is routing saved across consumers - ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f5227f14a17f2457","tex":"\\mathcal G_{\\downarrow,C}","aliases":"G_downarrow,C mathcal downarrow","display":"\\mathcal G_{\\downarrow,C}","kind":"use","excerpt":"For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-96fd9511ef84bb66","tex":"G_{\\mathrm{Gauss}}","aliases":"G_Gauss mathrm","display":"G_{\\mathrm{Gauss}}","kind":"use","excerpt":"Temporary non-Gaussianity. Suppose a non-Gaussian primitive g is introduced and later removed: U=g h_m⋯ h_1 g^-1, U∈ G_Gauss. If g h_i g^-1 generates several useful Gaussian effects, the excursion may be catalytic. However, many standard non-Gaussianity monotones vanish only on states rather than unitaries, and cancellation at the endpoint complica","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ec1c980979f73c29","tex":"G_j","aliases":"G_j","display":"G_j","kind":"use","excerpt":"Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-103f7de3482048f3","tex":"G_k","aliases":"G_k","display":"G_k","kind":"use","excerpt":"Calibration families and exact small oracles. claimC03 and C04 An unbalanced disjunction. For k≥2 and n=k+2, define B=z_1⋯ z_k, F_k(a,b,z)=(ab)lor B=ab+B+abB, G_k=F_k+a. This is a deliberately easy family with two differently sized accepting regions. It separates the cost of finding one safe packet from the degree needed for complete coverage.","count":12},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-80e1562860aa6aaf","tex":"G_{k+1}","aliases":"G_k+1","display":"G_{k+1}","kind":"use","excerpt":"Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-46dfc77dfada741b","tex":"\\mathcal G_{S,g}","aliases":"G_S,g mathcal","display":"\\mathcal G_{S,g}","kind":"use","excerpt":"Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4d3d077685f05b94","tex":"\\mathcal G_{S_A,g}","aliases":"G_S_A,g mathcal","display":"\\mathcal G_{S_A,g}","kind":"use","excerpt":"[First-arrival template] If word multiplicities are nonnegative, then d_S(e,g)=ord_z=0 G_S,g(z). For restricted and ambient sets, Δ(g)=ord_0 G_S_R,g-ord_0 G_S_A,g.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6e8acd5b0c81ff07","tex":"\\mathcal G_{S_R,g}","aliases":"G_S_R,g mathcal","display":"\\mathcal G_{S_R,g}","kind":"use","excerpt":"[First-arrival template] If word multiplicities are nonnegative, then d_S(e,g)=ord_z=0 G_S,g(z). For restricted and ambient sets, Δ(g)=ord_0 G_S_R,g-ord_0 G_S_A,g.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1481d5cc6f09169d","tex":"g_t","aliases":"g_t","display":"g_t","kind":"use","excerpt":"Depth, count, and space-time volume. Let g_t be the set of gates scheduled in layer t. Common scalar costs are ∑_t |g_t|, #{t:g_t≠∅}, ∑_t #{active qubits in layer t}. A catalyst may reduce count but increase depth, or vice versa. Space-time volume is often the more stable hardware quantity.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d8313a7d60fe037e","tex":"G_{t+1}","aliases":"G_t+1","display":"G_{t+1}","kind":"use","excerpt":"Fix the projectors chosen during the particular run. Put d_t=(I-P_t)G_tϕ_t-1. Subtraction gives the exact recurrence ψ_t-ϕ_t=G_t(ψ_t-1-ϕ_t-1)+d_t. Since ψ_0-ϕ_0=0, iteration yields ψ_m-ϕ_m=∑_t=1^mG_m⋯ G_t+1d_t. Every unitary product preserves norm, so the triangle inequality proves the state estimate. Also ψ_m =1 and ϕ_m ≤1 because each projector i","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f2604befea683337","tex":"G_{t;\\beta\\alpha}","aliases":"G_t;βα beta alpha","display":"G_{t;\\beta\\alpha}","kind":"use","excerpt":"Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-12a4192e16f68b07","tex":"\\mathrm{Gauss}","aliases":"Gauss mathrm","display":"\\mathrm{Gauss}","kind":"use","excerpt":"Temporary non-Gaussianity. Suppose a non-Gaussian primitive g is introduced and later removed: U=g h_m⋯ h_1 g^-1, U∈ G_Gauss. If g h_i g^-1 generates several useful Gaussian effects, the excursion may be catalytic. However, many standard non-Gaussianity monotones vanish only on states rather than unitaries, and cancellation at the endpoint complica","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"frac qgcd(a,q).","count":16},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-be87b2234ab9d39e","tex":"\\mathrm{GF}","aliases":"GF mathrm","display":"\\mathrm{GF}","kind":"use","excerpt":"The finite scalar controls enumerate 256 linear maps from a two-point witness algebra to GF(16), 65,536 maps for four points, and 256 maps from an eight-point algebra to F. The respective counts of unital algebra homomorphisms are two, four, and eight, matching the evaluation classification. The all-size result rests on the proof above.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f9b55a29a191e028","tex":"\\GL","aliases":"GL","display":null,"kind":"use","excerpt":"Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).","count":13},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-05d2a88242008ca3","tex":"\\GL_6","aliases":"GL_6 GL","display":null,"kind":"use","excerpt":"Let B_4={g∈GL_6(F_2):ellAll(g)≤4}. Exact breadth-first search gives |B_4|=110626. Group B_4 by right cosets of U_6(F_2). If a,b lie in the same right coset, then b^-1a∈U_6(F_2), and ellAll(b^-1a)≤ellAll(b)+ellAll(a)≤8. The implementation uses a canonical right-coset key obtained by reducing matrix columns from right to left. Right multiplication by","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5e917386367b6fdc","tex":"\\GL_n","aliases":"GL_n GL","display":null,"kind":"use","excerpt":"A CNOT circuit on n labeled wires represents an element of GL_n(F_2)=SL_n(F_2), and minimizing its gate count is the word-metric problem for transvection generators. This formulation supports asymptotically efficient synthesis, architecture-aware elimination, computable lower bounds, and exact small-width optimization [source reference]. Recent wor","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"Bosonic gate hierarchies. Natural nested models include triangular shears ⊂ all Gaussian unitaries ⊂ Gaussian plus a non-Gaussian primitive ⊂ U( H). Typical primitive resources are: item SUM or controlled displacement gates, item beam splitters, item single-mode and two-mode squeezers, item Fourier transforms, item cubic phase gates, item photon co","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b8bdb1c02444ca91","tex":"H'","aliases":"H'","display":"H'","kind":"use","excerpt":"[Retraction inheritance] Suppose a smaller nested pair H≤ G embeds into H'≤ G' by homomorphisms preserving the inclusion, the ambient embedding does not increase gate cost, and there is a homomorphic retraction π:H'→ H onto the embedded copy. Require ℓ_H(π(s))≤ c(s) for every restricted generator s of H'. Then every defect in H≤ G gives a defect at","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","excerpt":"Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"use","excerpt":"t dimension is polynomial in n if r(n)=O(1); item the packet dimension is superpolynomial whenever r(n)→∞; item if r(n)=O(log n), then D_n,k=2^O((log n)^2), so the explicit packet simulation is quasipolynomial; item if r(n)=O(n^α) for a fixed 0<α<1, then D_n,k=2^O(n^αlog n), which is subexponential; item if k(n)/n→ p∈(0,1), then log_2 D_n,k = nH_2(","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6144cd1a30920138","tex":"H_7","aliases":"H_7","display":"H_7","kind":"use","excerpt":"[Exact binary cases in dimensions seven through nine] Over F_2, let H_7=A(2;0,1,2), H_8=A(3;0,1,2), H_9=A(3;0,1,2,3). Then (ellDown(H_7),ellAll(H_7))=(8,7), (ellDown(H_8),ellAll(H_8))=(9,8), and (ellDown(H_9),ellAll(H_9))=(11,9).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-902dedcbd6a5754e","tex":"H_8","aliases":"H_8","display":"H_8","kind":"use","excerpt":"[Exact binary cases in dimensions seven through nine] Over F_2, let H_7=A(2;0,1,2), H_8=A(3;0,1,2), H_9=A(3;0,1,2,3). Then (ellDown(H_7),ellAll(H_7))=(8,7), (ellDown(H_8),ellAll(H_8))=(9,8), and (ellDown(H_9),ellAll(H_9))=(11,9).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c80fb3a69306b1a1","tex":"H_9","aliases":"H_9","display":"H_9","kind":"use","excerpt":"[Exact binary cases in dimensions seven through nine] Over F_2, let H_7=A(2;0,1,2), H_8=A(3;0,1,2), H_9=A(3;0,1,2,3). Then (ellDown(H_7),ellAll(H_7))=(8,7), (ellDown(H_8),ellAll(H_8))=(9,8), and (ellDown(H_9),ellAll(H_9))=(11,9).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a6ccc049e4b6854f","tex":"h_{i_1j_1}","aliases":"h_i_1j_1","display":"h_{i_1j_1}","kind":"use","excerpt":"itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b3833027ffab4b2f","tex":"h_{i_kj_k}","aliases":"h_i_kj_k","display":"h_{i_kj_k}","kind":"use","excerpt":"itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-46e2d2e905a9d0ff","tex":"h_{ij}","aliases":"h_ij","display":"h_{ij}","kind":"use","excerpt":"Holonomy packets. Let a family of local charts or rows be represented by permutations ρ_i. Define normalized transitions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partit","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d83201f3b36e1284","tex":"H_Q","aliases":"H_Q","display":"H_Q","kind":"use","excerpt":"Symmetry-preserving circuits. Let Q be a conserved charge and let H_Q={U:[U,Q]=0}. Restricted circuits use gates commuting with Q. Ambient circuits may temporarily leave H_Q but must return to a charge-preserving endpoint.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-348b8b1ce3cec4b3","tex":"\\mathcal H_q","aliases":"H_q mathcal","display":"\\mathcal H_q","kind":"use","excerpt":"The Hilbert space decomposes as H=⊕_q H_q. A restricted circuit is block diagonal. An ambient catalyst may couple sectors, use the enlarged route, and decouple them at the endpoint.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2d64df6fa5868f4c","tex":"H_r","aliases":"H_r","display":"H_r","kind":"use","excerpt":"Holonomy packets. Let a family of local charts or rows be represented by permutations ρ_i. Define normalized transitions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partit","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5e073754a563e30e","tex":"\\hbox","aliases":"hbox","display":null,"kind":"use","excerpt":"d oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b894251d06346420","tex":"\\Hh_n","aliases":"Hh_n Hh","display":null,"kind":"use","excerpt":"The packet access model. For each input length, suppose there is a packet projector P_n and a basis B_n= {ketb_1,…,ketb_D_n} of P_nHh_n. The simulation interface has four routines.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1d6e81930625f21d","tex":"\\Hh_{n,d}","aliases":"Hh_n,d Hh","display":null,"kind":"use","excerpt":"Let S_n act on Hh_n,d=(C^d)^⊗ n by permuting tensor factors. Schur--Weyl duality gives Hh_n,d ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where S_n acts trivially on the first tensor factor and irreducibly on V_λ.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dbea7c4d3c725e32","tex":"\\Hh_{n,k}","aliases":"Hh_n,k Hh","display":null,"kind":"use","excerpt":"[Bounded-excitation collapse] Consider n qubits and suppose the initial state and every circuit prefix are certified to lie in Hh_≤ K = ⊕_k=0^KHh_n,k for a fixed constant K. If initialization, the gate actions, and the measurement are computable in polynomial time in the occupation basis, then the circuit family lies in P, even when individual gate","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e4c34c007bad7ce7","tex":"\\Hh_{\\le K}","aliases":"Hh_≤ K Hh le","display":null,"kind":"use","excerpt":"[Bounded-excitation collapse] Consider n qubits and suppose the initial state and every circuit prefix are certified to lie in Hh_≤ K = ⊕_k=0^KHh_n,k for a fixed constant K. If initialization, the gate actions, and the measurement are computable in polynomial time in the occupation basis, then the circuit family lies in P, even when individual gate","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"There are 4+16+256=276 functions over the three input sizes. The distribution is c|rrr Optimum Toffolis · 0 · 1 · 3 Number of functions · 28 · 120 · 128. The source constructs and checks the actual circuits. Optimality is obtained from the written lower bounds and matching constructions, not from exhaustive enumeration of all reversible circuits.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fc66c01638c40151","tex":"i^2","aliases":"i^2","display":"i^2","kind":"use","excerpt":"The exact phase arithmetic. The structure Cyclo12 represents (a,b,c,d)⟼ a+bsqrt3+i(c+dsqrt3), a,b,c,d∈ Q. Its multiplication explicitly incorporates (sqrt3)^2=3 and i^2=-1. It is a concrete arithmetic representation. The development does not need or install a global field instance to certify the finite example.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-51bc4bff81cf0295","tex":"I_{\\mathcal V_\\lambda}","aliases":"I_ V_λ mathcal lambda","display":"I_{\\mathcal V_\\lambda}","kind":"use","excerpt":"Decomposition For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-37475f8aeaec6cce","tex":"I_1","aliases":"I_1","display":"I_1","kind":"use","excerpt":"[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0 I_2+DJ_2 · J_2.","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-78fd137c582a5e5f","tex":"I_4","aliases":"I_4","display":"I_4","kind":"use","excerpt":"This is a mathematical quantum-state construction. An actual realization would additionally require state preparation, a twelve-dimensional unitary implementation, coherence control, and sampling. A twelve-dimensional Fourier matrix is not the ordinary sixteen-dimensional Fourier transformation on four qubits. The unitary direct sum F_12⊕ I_4 suppl","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9d9060f8fe5d796b","tex":"I_A","aliases":"I_A","display":"I_A","kind":"use","excerpt":"Template XIII: ancilla restoration. Let W act on data D and ancilla A. To certify a dirty catalyst, prove the operator identity W=U_D⊗ I_A on the full joint space, not only on one ancilla input. For a clean catalyst, prove W(ketψ⊗ket0) =(Uketψ)⊗ket0 for all ketψ. The distinction changes both the target equations and lower bounds.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dacb44b03addfbd2","tex":"I_{D_t}","aliases":"I_D_t","display":"I_{D_t}","kind":"use","excerpt":"Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4925b1b0231c7842","tex":"I_j","aliases":"I_j","display":"I_j","kind":"use","excerpt":"Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d529fadc23d1fc12","tex":"i_k","aliases":"i_k","display":"i_k","kind":"use","excerpt":"itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a3ecf779d0b53571","tex":"I_M","aliases":"I_M","display":"I_M","kind":"use","excerpt":"Conversely, for ℓ∈ I_M,","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ad472b41377c2078","tex":"I_m","aliases":"I_m","display":"I_m","kind":"use","excerpt":"For the matching lower construction, first realize I_m⊗ B using m copies of a circuit for B. Then realize A⊗ I_n. Each gate of a circuit for A lifts to n coordinatewise gates. Since (A⊗ I_n)(I_m⊗ B)=A⊗ B, the cost is the right side of [source reference]. The same construction with ambient circuits proves [source reference].","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-95d680fad2ccb816","tex":"I_n","aliases":"I_n","display":"I_n","kind":"use","excerpt":"Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n I_n · 0 over F_2. A matrix M is symplectic when M^TJM=J.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-05de3a72487098dc","tex":"I_{n-6}","aliases":"I_n-6","display":"I_{n-6}","kind":"use","excerpt":"Embed C as C⊕ I_n-6. A six-gate ambient word acts on the first six rows. In a lower-directed circuit, rows with index at least six cannot be used as sources for targets among the first six rows, so the restricted lower bound of seven persists.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e1b2255977cf2b95","tex":"I_{P_3}","aliases":"I_P_3","display":"I_{P_3}","kind":"use","excerpt":"The three linear forms evaluate to L=(0,1,1), so P_3=L_1L_2L_3=0. The quadratic grade has pairs (1,6), (2,5), (3,4), so Q_3=x_1x_6+x_2x_5+x_3x_4=0+0+1=1. Thus Q_3∉ I_P_3: homogeneous extraction does not preserve the positive-support ideal. The whole polynomial is supported on its own accepting set, but its isolated homogeneous components need not b","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e9855187f190ff70","tex":"I_r","aliases":"I_r","display":"I_r","kind":"use","excerpt":"For n=6k+r, take C^⊕ k⊕ I_r and apply block additivity together with k copies of the ambient six-gate word.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-afccf775c866af1f","tex":"i_r","aliases":"i_r","display":"i_r","kind":"use","excerpt":"C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-111b6ee2c5f8dbd2","tex":"I_s","aliases":"I_s","display":"I_s","kind":"use","excerpt":"Dynamic rank-one factorization. Consider a height-two block shear E(C)= I_s · 0 C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final blo","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-34b5a08da5138c3a","tex":"I_t","aliases":"I_t","display":"I_t","kind":"use","excerpt":"Dynamic rank-one factorization. Consider a height-two block shear E(C)= I_s · 0 C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final blo","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b09c3d3928fe2c8b","tex":"\\operatorname{id}","aliases":"id operatorname","display":"\\operatorname{id}","kind":"use","excerpt":"When defects are impossible. A useful positive theorem is a criterion guaranteeing geodesic convexity. [One-Lipschitz retraction criterion] Suppose there is a group homomorphism π:G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambien","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0139f1bb81d79a32","tex":"\\operatorname{id}_H","aliases":"id_H operatorname","display":"\\operatorname{id}_H","kind":"use","excerpt":"When defects are impossible. A useful positive theorem is a criterion guaranteeing geodesic convexity. [One-Lipschitz retraction criterion] Suppose there is a group homomorphism π:G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambien","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f5a816f703a8485b","tex":"\\operatorname{Id}_N","aliases":"Id_N operatorname","display":"\\operatorname{Id}_N","kind":"use","excerpt":"(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","excerpt":"A real three-dimensional view of one amplitude. addcontentslinetocsectionA real three-dimensional view of one amplitude leadThe plot follows an actual complex amplitude as the relative phase advances. At sample n=6, the phase-family amplitude is χ_θ(6)=frace^iθsqrt3. fig16_complex_amplitude_trajectory0.84Coordinates are (θ,Reχ_θ(6),Imχ_θ(6)). In th","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-96eb28da82a0610a","tex":"\\inf_{r\\ge1}","aliases":"inf_r≥1 inf ge","display":"\\inf_{r\\ge1}","kind":"use","excerpt":"[Stable tensor length] The limit τ_*(A) = lim_r→∞ fracellAll(A^⊗ r)r m^r-1 = inf_r≥1 fracellAll(A^⊗ r)r m^r-1 exists. If A is tree tight, then the stable catalytic capacity κ_⊗(A) = ellDown(A)-τ_*(A) satisfies κ_⊗(A)≥DeltaDef(A).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6923dc2eea348b1d","tex":"\\inf_{x\\in\\mathcal P_{\\mathfrak M}(U)}","aliases":"inf_x∈ P_ M(U) inf in mathcal mathfrak","display":"\\inf_{x\\in\\mathcal P_{\\mathfrak M}(U)}","kind":"use","excerpt":"For a weight vector λ∈R_≥0^r, scalarization gives ℓ_ M^λ(U) = inf_x∈ P_ M(U)λ· x. The scalar defect Δ^λ(U) can vary with λ even when the target and gate alphabets are fixed.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cda9070e167070fb","tex":"\\mathrm{internal}","aliases":"internal mathrm","display":"\\mathrm{internal}","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n I_n · 0 over F_2. A matrix M is symplectic when M^TJM=J.","count":20},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0a4bf791b6b514b4","tex":"J_+","aliases":"J_+","display":"J_+","kind":"use","excerpt":"Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decrea","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c94b9cc5a134bb16","tex":"J_-","aliases":"J_-","display":"J_-","kind":"use","excerpt":"Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decrea","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0e97a80aa927d24d","tex":"j_1","aliases":"j_1","display":"j_1","kind":"use","excerpt":"itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-29ac64298abba628","tex":"J_2","aliases":"J_2","display":"J_2","kind":"use","excerpt":"The four exceptional matrices admit one common lower-bound proof. Work in the declared symplectic quotient and entangling metric, with H_i and S_i free and SWAP excluded as a primitive. Put J_2=0 · 1 1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f63cab9c738188a7","tex":"J_a","aliases":"J_a","display":"J_a","kind":"use","excerpt":"Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decrea","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-50c53edb72dca2a6","tex":"j_k","aliases":"j_k","display":"j_k","kind":"use","excerpt":"itions h_ij=ρ_iρ_j^-1 and the transition group H_r(L)=⟨ h_ij:i,j⟩. A holonomy packet may include: item the isomorphism type or order of H_r(L), item the cycle-type inventory of the distinguished h_ij, item block systems and orbit partitions, item characters or matrix coefficients of words in the h_ij, item noncommutative moments tr(h_i_1j_1⋯ h_i_kj","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-456e17f40199861f","tex":"j_r","aliases":"j_r","display":"j_r","kind":"use","excerpt":"C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8af4e249618e23ea","tex":"J_x","aliases":"J_x","display":"J_x","kind":"use","excerpt":"Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decrea","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ab5e8580e0cfc647","tex":"J_y","aliases":"J_y","display":"J_y","kind":"use","excerpt":"Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decrea","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2c8ffdbbdca4c726","tex":"J_z","aliases":"J_z","display":"J_z","kind":"use","excerpt":"Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decrea","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/","count":83},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cd503df35d57e6f5","tex":"K^{-1}","aliases":"K^-1","display":"K^{-1}","kind":"use","excerpt":"A Gaussian catalytic theorem requires: item a declared primitive library and cost vector; item a target subgroup, such as triangular, number conserving, or local; item an explicit sandwich KHK^-1 returning to the target subgroup; item a lower bound based on symplectic rank, singular values, graph cuts, or mode support; item treatment of approximati","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-438178e7f0b470d8","tex":"k^{-1/2}","aliases":"k^-1/2","display":"k^{-1/2}","kind":"use","excerpt":"Hypercube quotient coupling. Let ket k = binom nk^-1/2 ∑_|x|=kket x. Every weight-k string has n-k neighbors of weight k+1, while every weight-(k+1) string has k+1 neighbors of weight k. Counting the bipartite edges yields brak+1Aket k = fracbinom nk(n-k) sqrtbinom nkbinom nk+1 = sqrt(k+1)(n-k), proving [source reference].","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e27d2847281801ab","tex":"K^4","aliases":"K^4","display":"K^4","kind":"use","excerpt":"The four indicator functions form a basis of K^4. General m has 2^m such basis elements. A verifier is the sum of the indicators of its accepted witnesses, and pointwise multiplication models intersection of accepting sets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9d2b8e9b32c3f214","tex":"K^m","aliases":"K^m","display":"K^m","kind":"use","excerpt":"The conditional algorithmic consequence. [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9724859fee3b9e2b","tex":"K^W","aliases":"K^W","display":"K^W","kind":"use","excerpt":"The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cf5fcf757b255a84","tex":"K_1","aliases":"K_1","display":"K_1","kind":"use","excerpt":"1 · 0) and has rank one. In either directed CNOT or CZ, both diagonal blocks are I_2 and have rank two. Left and right multiplication by local Cliffords preserve these diagonal-block ranks. A circuit with one charged gate has the form K_1 E K_2 with K_1,K_2 local and E a CNOT or CZ, so it cannot equal M_D. A circuit with no charged gate is local an","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-173833828c3529a3","tex":"K_2","aliases":"K_2","display":"K_2","kind":"use","excerpt":"1 · 0) and has rank one. In either directed CNOT or CZ, both diagonal blocks are I_2 and have rank two. Left and right multiplication by local Cliffords preserve these diagonal-block ranks. A circuit with one charged gate has the form K_1 E K_2 with K_1,K_2 local and E a CNOT or CZ, so it cannot equal M_D. A circuit with no charged gate is local an","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-15201584ff766791","tex":"K_4","aliases":"K_4","display":"K_4","kind":"use","excerpt":"[Flagged matroid tour] Let M be a representable matroid with designated initial basis, target basis, and consumer states. A flagged tour is a sequence of basis exchanges or transvections in which the ith carrier remains inside a specified filtration layer. An ambient tour drops this constraint. The six-row atom realizes the graphic regular matroid ","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8e6a451de88cfbb0","tex":"K_\\mu","aliases":"K_μ mu","display":"K_\\mu","kind":"use","excerpt":"Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7414a80ba006680c","tex":"K_\\mu^\\dagger","aliases":"K_μ^† mu dagger","display":"K_\\mu^\\dagger","kind":"use","excerpt":"Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-44819be810b8a1c8","tex":"\\ket","aliases":"ket","display":null,"kind":"use","excerpt":"Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.","count":55},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-18563c4c5939a48e","tex":"\\KTop","aliases":"KTop","display":null,"kind":"use","excerpt":"[Weighted top-coefficient identity] Let KTop_w extract the coefficient of w_1⋯ w_m after reduction in the Boolean w algebra. Then P_R(z)=KTop_w[ R(w)∏_i=1^m(1+(1+z_i)w_i) ]. At w_i=0, the ith factor is one. At w_i=1, it is z_i. Lemma~[source reference] therefore converts the right side to exactly the weighted sum defining P_R.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-849240f6b4987eb4","tex":"\\KTop_w","aliases":"KTop_w KTop","display":null,"kind":"use","excerpt":"[Weighted top-coefficient identity] Let KTop_w extract the coefficient of w_1⋯ w_m after reduction in the Boolean w algebra. Then P_R(z)=KTop_w[ R(w)∏_i=1^m(1+(1+z_i)w_i) ]. At w_i=0, the ith factor is one. At w_i=1, it is z_i. Lemma~[source reference] therefore converts the right side to exactly the weighted sum defining P_R.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cb47427f88aad3bc","tex":"\\KTr","aliases":"KTr","display":null,"kind":"use","excerpt":"[Shared-variable multiplication fails] For a single witness bit, define KTr_z(A)=A(0)+zA(1). Then in the Boolean w algebra, KTr_z(w)=z, KTr_z(1+w)=1, but KTr_z(w(1+w))=0 ≠ z=KTr_z(w)KTr_z(1+w) whenever z≠0. Also, KTr_z(w^2)=z, KTr_z(w)^2=z^2, which differ for field elements z with z^2≠ z. Evaluate w and 1+w at zero and one. In the Boolean quotient ","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0f21f9e1c1ac6cf0","tex":"\\KTr_z","aliases":"KTr_z KTr","display":null,"kind":"use","excerpt":"[Shared-variable multiplication fails] For a single witness bit, define KTr_z(A)=A(0)+zA(1). Then in the Boolean w algebra, KTr_z(w)=z, KTr_z(1+w)=1, but KTr_z(w(1+w))=0 ≠ z=KTr_z(w)KTr_z(1+w) whenever z≠0. Also, KTr_z(w^2)=z, KTr_z(w)^2=z^2, which differ for field elements z with z^2≠ z. Evaluate w and 1+w at zero and one. In the Boolean quotient ","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"Lindblad dynamics. A Markovian open system obeys fracdρdt = L(ρ) = -i[H,ρ] + ∑_a ( L_aρ L_a^† - frac12{L_a^† L_a,ρ} ). Packet selection rules of H and L_a determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"The detailed defect distributions are: c|l model · restricted→ambient pairs n=2 entangling · 3→2:4 n=3 total · 7→6:6,quad8→7:18 n=3 entangling · 3→2:24, 4→3:432, 5→4:1056, 6→4:48, 6→5:64.","count":12},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"[Exact nonlinear cost of the whole function] For the family above, M(P_m)=m-1, KTrest(P_m)=2m-3. The linear map x↦(L_1(x),…,L_m(x)) has rank m, since its columns include the standard basis vectors. Extend these forms to an invertible system of linear coordinates. In those coordinates P_m is the product of the first m variables, with the other n-m v","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"The three linear forms evaluate to L=(0,1,1), so P_3=L_1L_2L_3=0. The quadratic grade has pairs (1,6), (2,5), (3,4), so Q_3=x_1x_6+x_2x_5+x_3x_4=0+0+1=1. Thus Q_3∉ I_P_3: homogeneous extraction does not preserve the positive-support ideal. The whole polynomial is supported on its own accepting set, but its isolated homogeneous components need not b","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c7f6796e75b9a514","tex":"L_3","aliases":"L_3","display":"L_3","kind":"use","excerpt":"The three linear forms evaluate to L=(0,1,1), so P_3=L_1L_2L_3=0. The quadratic grade has pairs (1,6), (2,5), (3,4), so Q_3=x_1x_6+x_2x_5+x_3x_4=0+0+1=1. Thus Q_3∉ I_P_3: homogeneous extraction does not preserve the positive-support ideal. The whole polynomial is supported on its own accepting set, but its isolated homogeneous components need not b","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a11ee0ad40de6dac","tex":"L_a","aliases":"L_a","display":"L_a","kind":"use","excerpt":"Lindblad dynamics. A Markovian open system obeys fracdρdt = L(ρ) = -i[H,ρ] + ∑_a ( L_aρ L_a^† - frac12{L_a^† L_a,ρ} ). Packet selection rules of H and L_a determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-deaa1521a6e1fbc4","tex":"\\mathcal L_A","aliases":"L_A mathcal","display":"\\mathcal L_A","kind":"use","excerpt":"Error accounting for truncated channels. Let Π_A project operator space onto an active set of coherence blocks. A reduced evolution uses L_A = Π_A LΠ_A. The discarded transfer is (I-Π_A) LΠ_A. Duhamel's formula yields an error bound analogous to the Hamiltonian leakage theorem. The resulting estimate depends on the induced norm of the discarded sup","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-42dc4827975a3189","tex":"L_a^\\dagger","aliases":"L_a^† dagger","display":"L_a^\\dagger","kind":"use","excerpt":"Lindblad dynamics. A Markovian open system obeys fracdρdt = L(ρ) = -i[H,ρ] + ∑_a ( L_aρ L_a^† - frac12{L_a^† L_a,ρ} ). Packet selection rules of H and L_a determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c759e173e558f4b0","tex":"L_{\\mathrm{leakage}}","aliases":"L_leakage mathrm","display":"L_{\\mathrm{leakage}}","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"use","excerpt":"[Exact nonlinear cost of the whole function] For the family above, M(P_m)=m-1, KTrest(P_m)=2m-3. The linear map x↦(L_1(x),…,L_m(x)) has rank m, since its columns include the standard basis vectors. Extend these forms to an invertible system of linear coordinates. In those coordinates P_m is the product of the first m variables, with the other n-m v","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b450d7bbb1f72fc4","tex":"L_P","aliases":"L_P","display":"L_P","kind":"use","excerpt":"Hamiltonian leakage. Exact conservation is not the only useful regime. Let P be an orthogonal projector and let U(t)=e^-itH for a bounded self-adjoint Hamiltonian H. The leakage operator from the packet is L_P(t)=(I-P)e^-itHP.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-405ec28531eabb81","tex":"L_{\\rm leakage}","aliases":"L_rm leakage rm","display":"L_{\\rm leakage}","kind":"use","excerpt":"The core research invariant is the complete classical representation record ( discovery cost, N_rm packets, D_rm internal, S_rm transfer, L_rm leakage, measurement cost ). A future compiler that computes and certifies this record can decide when the full state vector is unnecessary, when a hybrid representation is preferable, and when symmetry alon","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d3ef79a3b07a0aac","tex":"L_{v_0}","aliases":"L_v_0","display":"L_{v_0}","kind":"use","excerpt":"Suppose the gate is s→ t with s<t. Let v_0 be the lowest node of T whose two children separate s and t. The order of the leaves forces s∈ L_v_0 and t∈ R_v_0. At this one node, a single row of the cross block X[R_v_0,L_v_0] changes, so its rank changes by at most one.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-81cc2032770c59f1","tex":"L_Z","aliases":"L_Z","display":"L_Z","kind":"use","excerpt":"The parabolic subgroup used in the census is generated by S_i, CZ_ij, and CX_i→ j. It stabilizes the Z Lagrangian L_Z=Span(e_n,…,e_2n-1).","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-01208e159d2aec8c","tex":"\\land","aliases":"land","display":"\\land","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ac11752100d57466","tex":"\\mathrm{large}","aliases":"large mathrm","display":"\\mathrm{large}","kind":"use","excerpt":"a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). T","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a5ad71655c3b7e97","tex":"\\mathrm{large,res}","aliases":"large,res mathrm","display":"\\mathrm{large,res}","kind":"use","excerpt":"Template VIII: block retraction. Let a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves ga","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"The balanced cut is constructed. Because each gate has cost zero or one, a prefix can realize every Toffoli count from zero to T. The source proves this fact and extracts a cut with max{TC(p),TC(q)}≤lceil T/2rceil. The complete safe cut cover then proves β(f)≤lceil T/2rceil+1, 2β(f)≤ T+3. Natural-number arithmetic retains the floor/ceiling and smal","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-973cb69b1393624f","tex":"\\mathrm{leakage}","aliases":"leakage mathrm","display":"\\mathrm{leakage}","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e5c19d75887aa542","tex":"\\liminf","aliases":"liminf","display":"\\liminf","kind":"use","excerpt":"Tensor and repeated-module questions. For targets U and V, independent synthesis gives ℓ_T(U⊗ V) ≤ ℓ_T(U)+ℓ_T(V) when the systems are disjoint and local Clifford cost is ignored. Joint synthesis may be better because phase identities can couple terms across modules. This motivates a stable regularization τ_T(U)= liminf_r→∞ fracℓ_T(U^⊗ r)r. A one-co","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2468891cd8895610","tex":"\\liminf_{r\\to\\infty}","aliases":"liminf_r→∞ liminf to infty","display":"\\liminf_{r\\to\\infty}","kind":"use","excerpt":"Tensor and repeated-module questions. For targets U and V, independent synthesis gives ℓ_T(U⊗ V) ≤ ℓ_T(U)+ℓ_T(V) when the systems are disjoint and local Clifford cost is ignored. Joint synthesis may be better because phase identities can couple terms across modules. This motivates a stable regularization τ_T(U)= liminf_r→∞ fracℓ_T(U^⊗ r)r. A one-co","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"use","excerpt":"[Asymptotic multiplicative distortion] For every field, limsup_n→∞ sup_A∈U_n(F), A≠ I fracellDown(A)ellAll(A) ≥frac32.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3e2f995f3f54e413","tex":"\\limsup_{n\\to\\infty}","aliases":"limsup_n→∞ limsup to infty","display":"\\limsup_{n\\to\\infty}","kind":"use","excerpt":"[Asymptotic multiplicative distortion] For every field, limsup_n→∞ sup_A∈U_n(F), A≠ I fracellDown(A)ellAll(A) ≥frac32.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"These are exact cost formulas. The large-input entries were not obtained by searching for optimal circuits. The full construction uses 2m(2^m-1-1) CNOTs as well as 2m-3 Toffolis, giving total count O(nlog n). The projected quadratic has n/2 total gates.","count":49},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"[Tensor-amplified snapshot family] Let A=A(u;s_1,…,s_v) have dimension m=u+v+2. For N=m^r, ellDown(A^⊗ r) =r m^r-1(u+2v), while ellAll(A^⊗ r) ≤ r m^r. Consequently, ellDown(A^⊗ r)-ellAll(A^⊗ r) ≥ fracv-2mlog_2mNlog_2N.","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-13b051ca8964d665","tex":"\\log_{21}","aliases":"log_21 log","display":"\\log_{21}","kind":"use","excerpt":"[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-68a67f309b99ec73","tex":"\\longleftarrow","aliases":"longleftarrow","display":null,"kind":"use","excerpt":"Let R be a nonzero commutative ring with identity. A row transvection is τ_t,s(a)=I+aE_t,s, t≠ s, a∈ R∖{0}, acting by R_tlongleftarrow R_t+aR_s. The transvection is lower directed when s<t. The lower-directed gates generate U_n(R). We compare ellDown(A) =min{number of lower-directed transvections producing A} with ellAll(A) =min{number of arbitrary","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"In the generic C01 theorem, completeness is the existential statement f(x)=1 Longleftrightarrow ∃ i:W, H_i(x)=1. No finite disjunction is needed there. Specializing to the canonical finite wires yields the finite complete family used in the main paper. Completeness does not say that the XOR of all packets reconstructs f; detecting packets can cance","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"Dependency structure. The main proof dependencies are: restored oracle convention Longrightarrow cut comparison, cut comparison+degree per multiplication Longrightarrow complete-cover lower bound, safe ideal+finite linear duality Longrightarrow evaluation-span certificates, monotone ideal+affine input/output changes Longrightarrow calibration formu","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-529ad2daacd7efcb","tex":"\\lor","aliases":"lor","display":"\\lor","kind":"use","excerpt":"rScope, notation, and computational model Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dd221cbf2cb1954c","tex":"m^j","aliases":"m^j","display":"m^j","kind":"use","excerpt":"[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-09a4e391e923d287","tex":"m^N","aliases":"m^N","display":"m^N","kind":"use","excerpt":"Bosonic occupation space. For N indistinguishable bosons in m modes, a Fock basis state is an occupation vector (n_1,…,n_m), n_1+⋯+n_m=N. The dimension is D_rm boson(m,N) = binomN+m-1N. For fixed N, this is O(m^N).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8004570aa4bfea5e","tex":"M^{N-1}","aliases":"M^N-1","display":"M^{N-1}","kind":"use","excerpt":"The same counting argument also gives Prob_A∈U_n(F_q)[ellAll(A)=ellDown(A)=N] ≥ 1-M^N-1q^-1, whenever the right side is positive. Thus maximal ambient and intrinsic length is not only Zariski generic over infinite fields but asymptotically typical over growing finite fields at fixed dimension.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ff58bb89d2493d56","tex":"M^{N-d}","aliases":"M^N-d","display":"M^{N-d}","kind":"use","excerpt":"[Finite-field sparsity] Let N=binom n2 and M=n(n-1). For 1≤ d≤ N, Prob_A∈U_n(F_q)[DeltaDef(A)≥ d] ≤ min{1,M^N-dq^-d}. For fixed n, a uniformly random target therefore has defect zero with probability 1-O_n(q^-1) as q→∞.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6b262273fadfeae1","tex":"m^r","aliases":"m^r","display":"m^r","kind":"use","excerpt":"[Tensor-amplified snapshot family] Let A=A(u;s_1,…,s_v) have dimension m=u+v+2. For N=m^r, ellDown(A^⊗ r) =r m^r-1(u+2v), while ellAll(A^⊗ r) ≤ r m^r. Consequently, ellDown(A^⊗ r)-ellAll(A^⊗ r) ≥ fracv-2mlog_2mNlog_2N.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e3cd89a7aee64fd2","tex":"m^s","aliases":"m^s","display":"m^s","kind":"use","excerpt":"Template XI: stable capacity. If a_r+s≤ m^s a_r+m^r a_s, then b_r=fraca_rm^r-1 is subadditive and lim_r→∞fraca_rr m^r-1 exists by Fekete's lemma. This defines a regularized ambient cost and exposes possible tensor activation.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8868d63653f7b040","tex":"M^T","aliases":"M^T","display":"M^T","kind":"use","excerpt":"I_n · 0 over F_2. A matrix M is symplectic when M^TJM=J.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7615cae6d0b78c4b","tex":"m^\\alpha","aliases":"m^α alpha","display":"m^\\alpha","kind":"use","excerpt":"[Few-particle scaling] With the number of modes as the input-size parameter: item fixed N gives polynomial explicit-amplitude simulation; item N=O(log m) gives quasipolynomial sector size; item N=O(m^α), 0<α<1, gives subexponential sector size; item finite-density fermionic sectors remain exponential. For bosons, the statements hold in the dilute r","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","excerpt":"ubit orbit and exact costs] Each of the four distinct matrices M_D lies in P_2 and has restricted entangling cost three and ambient entangling cost two. They form one orbit under left multiplication by the free restricted subgroup generated by S_0,S_1. The ambient sequence H_1, CX_1→0, CX_0→1, H_0 has matrix M_0=(J_2 · 0 I_2 · J_2). Applying S_0^d_","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"Two-qubit Clifford witness. The target symplectic matrix is M_2= 0 · 1 · 0 · 0 1 · 0 · 0 · 0 1 · 1 · 0 · 1 1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fa02e26ac15b5f1d","tex":"M_3","aliases":"M_3","display":"M_3","kind":"use","excerpt":"Three-qubit Clifford witness. The target symplectic matrix is M_3= 0 · 1 · 0 · 0 · 0 · 0 0 · 0 · 1 · 0 · 0 · 0 1 · 0 · 0 · 0 · 0 · 0 1 · 0 · 1 · 0 · 1 · 0 0 · 1 · 0 · 0 · 0 · 1 0 · 1 · 0 · 1 · 0 · 0 . A restricted total-count path is CX_0→2, CZ_1,2, CX_2→0, CX_0→2, CX_0→1, CX_1→0, CX_0→1, while an ambient path is H_1, CX_1→2, CX_2→1, CX_2→0, CX_0→2","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6b34051ea3264f33","tex":"M_D","aliases":"M_D","display":"M_D","kind":"use","excerpt":"1 · 0, D=diag(d_0,d_1), d_0,d_1∈F_2, and define M_D=J_2 · 0 I_2+DJ_2 · J_2.","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"use","excerpt":"_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossing transvection projects to the identity in","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0442e5a2d693c5a9","tex":"M_{I_2}","aliases":"M_I_2","display":"M_{I_2}","kind":"use","excerpt":"D · I_2) and gives M_D. These matrices are distinct because J_2 is invertible. This also proves their orbit statement and the ambient upper bound two. The displayed restricted sequence for M_2 above gives M_I_2 using three CNOT gates and one free S_1. Free final phase gates then give every M_D, proving the restricted upper bound three.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9de6a9970cb6757b","tex":"M_t","aliases":"M_t","display":"M_t","kind":"use","excerpt":"Exact dynamic-packet simulation. Choose isometries B_t:C^D_t⟶Hh, B_tB_t^*=P_t, B_t^*B_t=I_D_t, where D_t=rank P_t. Define the rectangular coordinate update M_t=B_t^*G_tB_t-1.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ad86bcffe5ce1d96","tex":"M_\\lambda","aliases":"M_λ lambda","display":"M_\\lambda","kind":"use","excerpt":"A permutation-invariant unitary has the form U = ⊕_λ U_λ⊗ I_ V_λ, and a permutation-invariant measurement operator has the analogous form M = ⊕_λ M_λ⊗ I_ V_λ. The output probability is Tr(MUρ U^*) = ∑_λ p_λ Tr(M_λ U_λρ_λ U_λ^*).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"use","excerpt":"pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"Degree and complete-cover bound Deg(T+1,f) and 2β^*(f)≤ T+3. pathrefCircuitObstructions.restored_degree_and_cover_bound The first conclusion is an explicit certificate that f has a Boolean polynomial representation of degree at most T+1. The second is the affine-optimized complete safe-cover lower bound, equivalently T≥max{0,2β^*(f)-3}.","count":24},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-71f01340a1f6ac14","tex":"\\max_{q\\ne r}","aliases":"max_q≠ r max ne","display":"\\max_{q\\ne r}","kind":"use","excerpt":"Sector-flow invariants. Let P_q be the projector onto sector q. For an intermediate unitary U_t, define transfer blocks T_q,r(t)=P_q U_t P_r. A restricted path has T_q,r=0 for q≠ r. Ambient paths may populate off-diagonal blocks. Possible excursion measures include ∑_q≠ rrank T_q,r, max_q≠ r T_q,r , bandwidth in |q-r|. These are analogues of strict","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cd7ba0f5e9cf0737","tex":"\\max_t","aliases":"max_t max","display":"\\max_t","kind":"use","excerpt":"Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-01913498f102729b","tex":"\\max_{U}","aliases":"max_U max","display":"\\max_{U}","kind":"use","excerpt":"Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4876d7ae47f16ea5","tex":"\\max_{v:f(v)=1}","aliases":"max_v:f(v)=1 max","display":"\\max_{v:f(v)=1}","kind":"use","excerpt":"For a positive input v, define instead β_v(f)=min{deg h:h∈ I_f, h(v)=1}, β(f)=max_v:f(v)=1β_v(f). Set β(0)=0. Thus β(f) is the degree needed before safe polynomials collectively detect every positive input. Finally, put β^*(f)=max_ℓ∈Aff_nβ(f+ℓ), where Aff_n is the set of affine Boolean functions on the logical inputs.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-416d9ed969d1b79d","tex":"\\max_\\ell","aliases":"max_ℓ max ell","display":"\\max_\\ell","kind":"use","excerpt":"@rrrrr@ Inputs n · max_ℓα_1 · β^* · One-packet bound · Complete-cover bound","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aa0979c4da4360df","tex":"\\max_{\\ell}","aliases":"max_ℓ max ell","display":"\\max_{\\ell}","kind":"use","excerpt":"The preceding combined bound max{0,deg F_k-1,2max_ℓα_1(F_k+ℓ)-3} gives n-1 for this family; its immunity-only term is the constant 3. The complete-cover bound gives 2n-5. The stronger estimate can also be recognized by restriction to b=1 followed by a free affine output adjustment. No exact optimum is claimed when the displayed endpoints differ.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-800c1a6f96cf2186","tex":"\\max_{\\ell\\text{ affine}}","aliases":"max_ℓ affine max ell text","display":"\\max_{\\ell\\text{ affine}}","kind":"use","excerpt":"An affine output mask ℓ(x) costs zero additional Toffolis: append CNOTs from restored inputs and, if needed, a target NOT. This constructs a restored oracle for f+ℓ, so the bound applies to every affine mask. Define β^*(f)=max_ℓ affineβ(f+ℓ). Together with the output degree bound, T≥max{0,deg f-1,2β^*(f)-3}.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7bc191c668c93c7b","tex":"\\max_{\\ell\\in\\Aff_n}","aliases":"max_ℓ∈Aff_n max ell in Aff","display":null,"kind":"use","excerpt":"For a positive input v, define instead β_v(f)=min{deg h:h∈ I_f, h(v)=1}, β(f)=max_v:f(v)=1β_v(f). Set β(0)=0. Thus β(f) is the degree needed before safe polynomials collectively detect every positive input. Finally, put β^*(f)=max_ℓ∈Aff_nβ(f+ℓ), where Aff_n is the set of affine Boolean functions on the logical inputs.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-efd6ea68821c6cf7","tex":"\\mathrm{metric}","aliases":"metric mathrm","display":"\\mathrm{metric}","kind":"use","excerpt":"Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"Binary symplectic representation. Modulo global phase and Pauli translations, an n-qubit Clifford operation acts on Pauli exponent vectors v=(x_0,…,x_n-1mid z_0,…,z_n-1)^T∈F_2^2n through a symplectic matrix M∈Sp(2n,2). The elementary gates used here act as follows: H_i · : (x_i,z_i)↦(z_i,x_i),","count":22},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e4d292a6927cbbf8","tex":"\\min_{c\\in\\mathcal C_{\\mathrm{amb}}(U)}","aliases":"min_c∈ C_amb(U) min in mathcal mathrm","display":"\\min_{c\\in\\mathcal C_{\\mathrm{amb}}(U)}","kind":"use","excerpt":"Scalarization by positive weights w gives Δ_w(U)= min_c∈ C_res(U)w· c - min_c∈ C_amb(U)w· c. Different w can change the magnitude of the defect and whether it is zero or positive. It remains nonnegative because every restricted realization is also an ambient realization with the same resource vector.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0305ff10d4bfd523","tex":"\\min_{c\\in\\mathcal C_{\\mathrm{res}}(U)}","aliases":"min_c∈ C_res(U) min in mathcal mathrm","display":"\\min_{c\\in\\mathcal C_{\\mathrm{res}}(U)}","kind":"use","excerpt":"Scalarization by positive weights w gives Δ_w(U)= min_c∈ C_res(U)w· c - min_c∈ C_amb(U)w· c. Different w can change the magnitude of the defect and whether it is zero or positive. It remains nonnegative because every restricted realization is also an ambient realization with the same resource vector.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N","display":null,"kind":"use","excerpt":"Multiobjective search. Let each gate have a vector cost in N^r. At each state, retain only nondominated labels. The algorithm terminates for finite state spaces with a bounded search region, but the frontier can grow exponentially.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5a1fe216a765aa7c","tex":"N^{-1/2}","aliases":"N^-1/2","display":"N^{-1/2}","kind":"use","excerpt":"Let A be the cyclic shift (Ax)(n)=x(n-1). A Fourier column f_a(n)=N^-1/2ζ_N^na satisfies Af_a=ζ_N^-af_a. Therefore","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"Forrelation amplitude. Let f,g:{0,1}^n→{-1,1} and N=2^n. The circuit with alternating Hadamards and phase oracles has zero-state amplitude Φ(f,g) = frac1N^3/2 ∑_x,y f(x)(-1)^x· yg(y). The circuit does not evaluate the N^2 terms separately. It preserves phases and lets them interfere.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"[Correct growth of the maximum defect] Is D_n(F) of order nlog n, or can interacting catalysts produce a larger power of n? The current bounds are Ω(nlog n) ≤ D_n(F) ≤ O(n^2).","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-683c44c732aa9aff","tex":"n^{20}","aliases":"n^20","display":"n^{20}","kind":"use","excerpt":"Complexity class versus practical runtime. A polynomial sector can still have a large exponent. Fixed k=20 gives O(n^20) explicit amplitudes, which is in P but may be impractical. Conversely, a subexponential sector may be useful at moderate size without changing the formal complexity class.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0969e62c250901c5","tex":"n^3","aliases":"n^3","display":"n^3","kind":"use","excerpt":"Qubit corollary. For qubits, Schur labels are total spins j. The carrier dimension is u_j=2j+1≤ n+1, and there are at most lfloor n/2rfloor+1 labels. Hence ∑_j(2j+1)^2=O(n^3).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dac29c73b980bfce","tex":"N^{3/2}","aliases":"N^3/2","display":"N^{3/2}","kind":"use","excerpt":"Forrelation amplitude. Let f,g:{0,1}^n→{-1,1} and N=2^n. The circuit with alternating Hadamards and phase oracles has zero-state amplitude Φ(f,g) = frac1N^3/2 ∑_x,y f(x)(-1)^x· yg(y). The circuit does not evaluate the N^2 terms separately. It preserves phases and lets them interfere.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-94e0ebe962e11137","tex":"n^{d(d-1)/2}","aliases":"n^d(d-1)/2","display":"n^{d(d-1)/2}","kind":"use","excerpt":"Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-936a6bfdc7b6fddb","tex":"n^{d-1}","aliases":"n^d-1","display":"n^{d-1}","kind":"use","excerpt":"Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-58abfdcc71452677","tex":"n^{d^2-1}","aliases":"n^d^2-1","display":"n^{d^2-1}","kind":"use","excerpt":"umber of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9f138cd944c86429","tex":"n^K","aliases":"n^K","display":"n^K","kind":"use","excerpt":"The active dimension is ∑_k=0^Kbinom nk =O(n^K). Let P project onto Hh_≤ K. By the hypothesis on the actual circuit prefixes, Pψ_t=ψ_t at every step. Consequently PG_tPψ_t-1=G_tψ_t-1 on the evolving state, even if the gate does not preserve all of PHh. The restricted update therefore reproduces that trajectory exactly. The hypothesis supplies effic","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e97306133ed6eb8c","tex":"n^k","aliases":"n^k","display":"n^k","kind":"use","excerpt":"For fixed k, dimHh_n,k=binom nk=O(n^k). The weight-k basis is efficiently indexed by k-element subsets. A local number-preserving gate changes only basis states containing the affected occupation pattern. Apply [source reference].","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4877fade481241a6","tex":"\\N^r","aliases":"N^r N","display":null,"kind":"use","excerpt":"Multiobjective search. Let each gate have a vector cost in N^r. At each state, retain only nondominated labels. The algorithm terminates for finite state spaces with a bounded search region, but the frontier can grow exponentially.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f652639badcf7db2","tex":"n^r","aliases":"n^r","display":"n^r","kind":"use","excerpt":"The upper bounds follow from [source reference]. If r=O(1), then binom nr=O(n^r). Conversely, suppose r→∞ and r≤ n/2. If r≤sqrt n, then logbinom nr ≥ rlog(n/r) ≥ frac r2log n, whose ratio to log n diverges. If r>sqrt n, then logbinom nr ≥ rlog2 ≥ sqrt nlog2, which also dominates every constant multiple of log n. For the final assertion, p_n=k(n)/n ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-526a976a5ea4910d","tex":"n^\\alpha","aliases":"n^α alpha","display":"n^\\alpha","kind":"use","excerpt":"d-charge phase diagram] For explicit amplitude storage in a weight-k(n) packet: item the packet dimension is polynomial in n if r(n)=O(1); item the packet dimension is superpolynomial whenever r(n)→∞; item if r(n)=O(log n), then D_n,k=2^O((log n)^2), so the explicit packet simulation is quasipolynomial; item if r(n)=O(n^α) for a fixed 0<α<1, then D","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","excerpt":"Template XV: exhaustive threshold classification. A finite first-threshold theorem should contain: item no defects below size n_0; item at least one defect at n_0; item exact classification of every defect at n_0 modulo declared equivalence; item analytic generation of the exceptional motifs; item equality between the analytic set and computational","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossi","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bd8030972b621b0e","tex":"N_{1,3}","aliases":"N_1,3","display":"N_{1,3}","kind":"use","excerpt":"[Motif count] For a split (L,R)=(|L|,|R|) with L+R=4, the number of minimal phase-separated targets is N_L,R =L(2^R-2)+R(2^L-2). For the three possible splits, N_1,3=6, N_2,2=8, N_3,1=6. Hence there are twenty targets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ec18fff607e7dd81","tex":"N_{2,2}","aliases":"N_2,2","display":"N_{2,2}","kind":"use","excerpt":"[Motif count] For a split (L,R)=(|L|,|R|) with L+R=4, the number of minimal phase-separated targets is N_L,R =L(2^R-2)+R(2^L-2). For the three possible splits, N_1,3=6, N_2,2=8, N_3,1=6. Hence there are twenty targets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-06f72a764986a5f6","tex":"N_{3,1}","aliases":"N_3,1","display":"N_{3,1}","kind":"use","excerpt":"[Motif count] For a split (L,R)=(|L|,|R|) with L+R=4, the number of minimal phase-separated targets is N_L,R =L(2^R-2)+R(2^L-2). For the three possible splits, N_1,3=6, N_2,2=8, N_3,1=6. Hence there are twenty targets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4d2462ef3372559a","tex":"n_d","aliases":"n_d","display":"n_d","kind":"use","excerpt":"Symmetric-subspace dimension. A symmetric basis state of n qudits is indexed by occupation numbers (n_1,…,n_d), n_1+⋯+n_d=n. The stars-and-bars count is binomn+d-1d-1, proving [source reference].","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cf9404d3f534ee7e","tex":"n_k","aliases":"n_k","display":"n_k","kind":"use","excerpt":"For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossi","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5633c34972b23288","tex":"N_{L,R}","aliases":"N_L,R","display":"N_{L,R}","kind":"use","excerpt":"[Motif count] For a split (L,R)=(|L|,|R|) with L+R=4, the number of minimal phase-separated targets is N_L,R =L(2^R-2)+R(2^L-2). For the three possible splits, N_1,3=6, N_2,2=8, N_3,1=6. Hence there are twenty targets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d7c0e2c4594eec1b","tex":"n_m","aliases":"n_m","display":"n_m","kind":"use","excerpt":"Bosonic occupation space. For N indistinguishable bosons in m modes, a Fock basis state is an occupation vector (n_1,…,n_m), n_1+⋯+n_m=N. The dimension is D_rm boson(m,N) = binomN+m-1N. For fixed N, this is O(m^N).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9a0961b40194a985","tex":"N_{\\mathrm{packets}}","aliases":"N_packets mathrm","display":"N_{\\mathrm{packets}}","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-049e49880b9ef7e5","tex":"N_{\\rm packets}","aliases":"N_rm packets rm","display":"N_{\\rm packets}","kind":"use","excerpt":"The core research invariant is the complete classical representation record ( discovery cost, N_rm packets, D_rm internal, S_rm transfer, L_rm leakage, measurement cost ). A future compiler that computes and certifies this record can decide when the full state vector is unnecessary, when a hybrid representation is preferable, and when symmetry alon","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"∑_n=0^q-1ζ_q^hn =q, · qmid h, 0, · qnmid h.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"These are exact cost formulas. The large-input entries were not obtained by searching for optimal circuits. The full construction uses 2m(2^m-1-1) CNOTs as well as 2m-3 Toffolis, giving total count O(nlog n). The projected quadratic has n/2 total gates.","count":50},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"d oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently, the coordinate discrepancies H_i=A_i+B_i satisfy H_i≤ f hboxfor every i, bigvee_iH_i=f. For a boundary state with","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-429f398b1a83839f","tex":"O^T","aliases":"O^T","display":"O^T","kind":"use","excerpt":"Covariance-matrix lower bounds. For a Gaussian state, the covariance matrix Γ transforms as Γ⟼ SΓ S^T in the bosonic case and Γ⟼ OΓ O^T in the fermionic case. Possible restricted lower bounds include: item rank change of off-diagonal covariance blocks, item symplectic singular-value obligations, item growth of mode support, item graph bandwidth and","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-728136f7294269e0","tex":"\\mathcal O_1","aliases":"O_1 mathcal","display":"\\mathcal O_1","kind":"use","excerpt":"Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6f0c09ddcc25554c","tex":"\\mathcal O_a","aliases":"O_a mathcal","display":"\\mathcal O_a","kind":"use","excerpt":"Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9bcd40cc6f81de81","tex":"\\mathcal O_b","aliases":"O_b mathcal","display":"\\mathcal O_b","kind":"use","excerpt":"The quotient matrix is widetilde H_ab = bra O_aHket O_b.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-60614ca2902b889f","tex":"O_d","aliases":"O_d","display":"O_d","kind":"use","excerpt":"Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-93f77af3d4162a81","tex":"O_n","aliases":"O_n","display":"O_n","kind":"use","excerpt":"[Finite-field sparsity] Let N=binom n2 and M=n(n-1). For 1≤ d≤ N, Prob_A∈U_n(F_q)[DeltaDef(A)≥ d] ≤ min{1,M^N-dq^-d}. For fixed n, a uniformly random target therefore has defect zero with probability 1-O_n(q^-1) as q→∞.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c925667d3a1d57c0","tex":"O_{N,M}","aliases":"O_N,M","display":"O_{N,M}","kind":"use","excerpt":"Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estima","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d3ef455391321bdb","tex":"\\mathcal O_r","aliases":"O_r mathcal","display":"\\mathcal O_r","kind":"use","excerpt":"Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-106a86b45be386c3","tex":"\\mathrm{open}","aliases":"open mathrm","display":"\\mathrm{open}","kind":"use","excerpt":"Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a69f9d4b1d750b15","tex":"\\mathrm{OR}","aliases":"OR mathrm","display":"\\mathrm{OR}","kind":"use","excerpt":"guidepageOne safe packet need not cover every accepting point For k=3 consider F_3(a,b,z_1,z_2,z_3)=ab OR z_1z_2z_3. The packet ab is safe and has degree two. It detects every teal point below. It misses three additional positives, which require the other packet z_1z_2z_3.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fd71dfd389a79ac5","tex":"\\operatorname{ord}","aliases":"ord operatorname","display":"\\operatorname{ord}","kind":"use","excerpt":"exact order twelve. Set ζ=e^2π i/12. The phase ζ^a first returns to 1 after 12/gcd(a,12) powers. Thus the coprime labels 1,5,7,11 select exactly the primitive twelfth roots. fig01_primitive_roots0.91The circle is the complex unit circle. Teal points are the four primitive roots; pale points are the other eight roots. Labels are residues a, not meas","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ab3321d6961879fc","tex":"\\ord","aliases":"ord","display":null,"kind":"use","excerpt":"[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7f0c4545d3e4be82","tex":"\\ord_0","aliases":"ord_0 ord","display":null,"kind":"use","excerpt":"G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-43b46d27e623d852","tex":"\\ord_{z=0}","aliases":"ord_z=0 ord","display":null,"kind":"use","excerpt":"[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b1e4fb989c1ebd63","tex":"\\overline{\\mathcal D_{n,\\ge d}}","aliases":"overline D_n,≥ d overline mathcal ge","display":"\\overline{\\mathcal D_{n,\\ge d}}","kind":"use","excerpt":"[Defect--codimension bound] Let K be algebraically closed. For every 1≤ d≤ N, dimoverline D_n,≥ d≤ N-d. Equivalently, codim_U_noverline D_n,≥ d≥ d. More finely, on the stratum where ellDown(A)=L and DeltaDef(A)≥ d, the closure has dimension at most L-d.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-476bf0ed60071234","tex":"\\overline{\\mathcal D_{n,\\ge d}(\\overline K)}","aliases":"overline D_n,≥ d(overline K) overline mathcal ge overline","display":"\\overline{\\mathcal D_{n,\\ge d}(\\overline K)}","kind":"use","excerpt":"-rigidity-intro Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline K overline D_n,≥ d(overlin","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d27738e1133b7635","tex":"\\overline{\\mathcal D_{\\ge d}}","aliases":"overline D_≥ d overline mathcal ge","display":"\\overline{\\mathcal D_{\\ge d}}","kind":"use","excerpt":"Template VI: defect codimension. Assume every target has restricted cost at most M. A defect of size at least d implies ambient cost at most M-d. If the (M-d)-gate ambient locus has dimension at most N-d, then codimoverline D_≥ d≥ d. This is the general mechanism behind algebraic rarity.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b7a5b96653c1dfce","tex":"\\overline K","aliases":"overline K overline","display":"\\overline K","kind":"use","excerpt":"[Generic rigidity and defect codimension] Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline ","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3525bac8c8407c33","tex":"\\overline{\\mathcal R^*_{n,\\le L}}","aliases":"overline R^*_n,≤ L overline mathcal le","display":"\\overline{\\mathcal R^*_{n,\\le L}}","kind":"use","excerpt":"[Dimension of bounded-length loci] For every L≥0, the sets R^*_n,≤ L and R^downarrow_n,≤ L are constructible and dimoverline R^*_n,≤ L ≤ L, dimoverline R^downarrow_n,≤ L ≤ L.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1b06286f0d97f645","tex":"\\overline{\\mathcal R^*_{n,\\le N-1}}","aliases":"overline R^*_n,≤ N-1 overline mathcal le","display":"\\overline{\\mathcal R^*_{n,\\le N-1}}","kind":"use","excerpt":"After base change to an algebraic closure overline K, [source reference] gives dimoverline R^*_n,≤ N-1≤ N-1<N=dimU_n. Because the word maps are defined over K, this proper closed set descends to a proper closed subset of U_n over K. Its complement is a nonempty Zariski-open set Ω_n. Since K is infinite and U_n≅Aff^N, the set Ω_n(K) is nonempty and ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1dea600ec3ebb803","tex":"\\overline{\\mathcal R^\\downarrow_{n,\\le L}}","aliases":"overline R^downarrow_n,≤ L overline mathcal downarrow le","display":"\\overline{\\mathcal R^\\downarrow_{n,\\le L}}","kind":"use","excerpt":"[Dimension of bounded-length loci] For every L≥0, the sets R^*_n,≤ L and R^downarrow_n,≤ L are constructible and dimoverline R^*_n,≤ L ≤ L, dimoverline R^downarrow_n,≤ L ≤ L.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ac62936f96a7dcea","tex":"\\overline{V\\cap X^{(L)}}","aliases":"overlineV∩ X^(L) overline cap","display":"\\overline{V\\cap X^{(L)}}","kind":"use","excerpt":"[Dimension per gate] Let G be an algebraic group, let V⊆ G be an irreducible subvariety of dimension D, and let X⊆ G be a finite union of locally closed subvarieties, each of dimension at most r, with the identity in X. Write X^(L) = {x_L⋯ x_1:x_1,…,x_L∈ X}. Then dimoverlineV∩ X^(L)≤ rL. In particular, if rL<D, a Zariski-generic point of V has X-pr","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6a505014569e9196","tex":"\\overline{x(n)}","aliases":"overlinex(n) overline","display":"\\overline{x(n)}","kind":"use","excerpt":"⟨ x,y⟩=∑_n=0^q-1overlinex(n)y(n), x ^2=⟨ x,x⟩.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-59444bb791725ca3","tex":"\\overline{X^L}","aliases":"overlineX^L overline","display":"\\overline{X^L}","kind":"use","excerpt":"Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but t","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c8495553d71183bf","tex":"\\overline{\\{U:\\Delta(U)\\ge d\\}}","aliases":"overline{U:Δ(U)≥ d} overline Delta ge","display":"\\overline{\\{U:\\Delta(U)\\ge d\\}}","kind":"use","excerpt":"Defect codimension. If every target has restricted cost at most N, then defect at least d forces ambient cost at most N-d. The dimension-per-gate theorem gives codim_Voverline{U:Δ(U)≥ d} ≥ d when each ambient gate carries one parameter.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-87d3c0d8c6a9d614","tex":"\\overline{\\chi(g)}","aliases":"overlineχ(g) overline chi","display":"\\overline{\\chi(g)}","kind":"use","excerpt":"Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-794a001c7c3f08ff","tex":"P^{(1)}_{\\alpha_1}","aliases":"P^(1)_α_1 alpha","display":"P^{(1)}_{\\alpha_1}","kind":"use","excerpt":"Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential pre","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bf80395c8ca8a826","tex":"P^{(s)}_{\\alpha_s}","aliases":"P^(s)_α_s alpha","display":"P^{(s)}_{\\alpha_s}","kind":"use","excerpt":"Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential pre","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"projector12_idempotent · Exact rational multiplication gives P^2=P","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d9f720eba6b7fab0","tex":"P^\\dagger","aliases":"P^† dagger","display":"P^\\dagger","kind":"use","excerpt":"P=FDF^†, P^2=P, P^†=P, rankP=trP=4.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1a49ce15b81ac349","tex":"P_{\\mathcal D}","aliases":"P_ D mathcal","display":"P_{\\mathcal D}","kind":"use","excerpt":"P_ D=∑_d∈ DP_d^(N)","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-830e190d560317dd","tex":"P_{10}","aliases":"P_10","display":"P_{10}","kind":"use","excerpt":"visualfigureprojection_m10Two different resource comparisons at the same input length. The right panel reports the totals of the stated constructions, not an asserted optimal total for P_10.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7a684494a361857c","tex":"P_{12}^{(12)}","aliases":"P_12^(12)","display":"P_{12}^{(12)}","kind":"use","excerpt":"Put P=P_12^(12). Its entries are","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"[Two-qubit orbit and exact costs] Each of the four distinct matrices M_D lies in P_2 and has restricted entangling cost three and ambient entangling cost two. They form one orbit under left multiplication by the free restricted subgroup generated by S_0,S_1. The ambient sequence H_1, CX_1→0, CX_0→1, H_0 has matrix M_0=(J_2 · 0","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f0242a419961441b","tex":"P_3","aliases":"P_3","display":"P_3","kind":"use","excerpt":"[Projection does not preserve safe support] The quadratic extraction [P_m]_2 need not be supported on the positive set of P_m. Take m=3, identify the columns with the integers 1,…,6 using their three-bit expansions, and set x_1=x_3=x_4=1 with all other inputs zero. Their column sum is 1+3+4=6 under bitwise XOR, not a=7, so P_3=0. Among the compleme","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a5377f21b2465a55","tex":"P_6","aliases":"P_6","display":"P_6","kind":"use","excerpt":"This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-472d75719eeab466","tex":"P_{<r}","aliases":"P_<r","display":"P_{<r}","kind":"use","excerpt":"C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f21822f96efaf869","tex":"P_A","aliases":"P_A","display":"P_A","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f6ef94023fe2dd2f","tex":"P_{A\\land B}","aliases":"P_Aland B land","display":"P_{A\\land B}","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a5c35e870d97f9c4","tex":"P_B","aliases":"P_B","display":"P_B","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8a232427a9b65168","tex":"p_C","aliases":"p_C","display":"p_C","kind":"use","excerpt":"A representation-relative invariant. For a declared classical representation class R, define TC^ R_ε(C) = min{ cost(R): R∈ R, |R(C)-p_C|≤ε }, where p_C is the target acceptance probability. This permits a disciplined ladder:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-14f8900152d6716a","tex":"P_d","aliases":"P_d","display":"P_d","kind":"use","excerpt":"six order classes leadThe primitive-twelve projector is one packet in a complete orthogonal decomposition. A frequency label k has root order d=12/gcd(k,12). The possible orders are the divisors 1,2,3,4,6,12. fig08_order_partition1.0The number inside each tile is its root order d. The horizontal label is frequency k. Every label belongs to one clas","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0ad5d4b1b4588901","tex":"P_d^{(N)}","aliases":"P_d^(N)","display":"P_d^{(N)}","kind":"use","excerpt":"P_d^(N)=F_ND_dF_N^†.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2fd63ee4a5169ac2","tex":"P_e","aliases":"P_e","display":"P_e","kind":"use","excerpt":"six order classes leadThe primitive-twelve projector is one packet in a complete orthogonal decomposition. A frequency label k has root order d=12/gcd(k,12). The possible orders are the divisors 1,2,3,4,6,12. fig08_order_partition1.0The number inside each tile is its root order d. The horizontal label is frequency k. Every label belongs to one clas","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ece7daf927df3f9e","tex":"P_e^{(N)}","aliases":"P_e^(N)","display":"P_e^{(N)}","kind":"use","excerpt":"(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-67970ebd56f6d184","tex":"P_{\\mathrm{final}}","aliases":"P_final mathrm","display":"P_{\\mathrm{final}}","kind":"use","excerpt":"endpmatrix. A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2dd4810c8ecd9090","tex":"p_g","aliases":"p_g","display":"p_g","kind":"use","excerpt":"For stochastic error rates p_g, a first-order weighted cost is C_err(W)=∑_g∈ W-log(1-p_g). This additive approximation can be inserted directly into Dijkstra search. It does not capture coherent or correlated errors, but it produces a declared and reproducible metric.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"[Invariant quantum packets] If a unitary U satisfies [U,Q]=0, then P_iUP_j=0 (i≠ j), and U=⊕_i U_i, U_i=P_iUP_i|_P_iHh. If P_kketψ=ketψ, then P_kUketψ=Uketψ.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The le","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"Conserved observables. Let Q=∑_k q_kP_k be a self-adjoint observable with spectral projectors P_k.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-346cf456a77b1ce5","tex":"p_n","aliases":"p_n","display":"p_n","kind":"use","excerpt":"Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b159343d7051969d","tex":"P_{n,k}","aliases":"P_n,k","display":"P_{n,k}","kind":"use","excerpt":"Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-db3c59d3185e4a23","tex":"P_{nm}","aliases":"P_nm","display":"P_{nm}","kind":"use","excerpt":"P_nm=fracc(n-m)12.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7559923eaf5c65b9","tex":"P_{nn}","aliases":"P_nn","display":"P_{nn}","kind":"use","excerpt":"tr(Pρ_diag) =∑_n P_nnfracc(n)^248 =frac13∑_nfracc(n)^248 =frac13. square","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d61b96062b482c72","tex":"P_q","aliases":"P_q","display":"P_q","kind":"use","excerpt":"Sector-flow invariants. Let P_q be the projector onto sector q. For an intermediate unitary U_t, define transfer blocks T_q,r(t)=P_q U_t P_r. A restricted path has T_q,r=0 for q≠ r. Ambient paths may populate off-diagonal blocks. Possible excursion measures include ∑_q≠ rrank T_q,r, max_q≠ r T_q,r , bandwidth in |q-r|. These are analogues of strict","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6437e41273729f36","tex":"P_R","aliases":"P_R","display":"P_R","kind":"use","excerpt":"Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4d3243980b9ae21f","tex":"P_r","aliases":"P_r","display":"P_r","kind":"use","excerpt":"tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The le","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-010f131ed455d9fc","tex":"P_S","aliases":"P_S","display":"P_S","kind":"use","excerpt":"For any set S⊆ I_N, let D_S be its diagonal indicator and P_S=F_ND_SF_N^†. This construction does not require S to be an order class. Rationality is special, but orthogonal projection is general.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-78aa8f72acca2b03","tex":"P_s","aliases":"P_s","display":"P_s","kind":"use","excerpt":"Syndrome projectors. Let S_1,…,S_m be commuting Pauli stabilizers with eigenvalues ±1. A syndrome s∈{0,1}^m defines P_s = ∏_j=1^m fracI+(-1)^s_jS_j2. These projectors are orthogonal and sum to the identity. Syndrome measurement selects one packet, while recovery applies a syndrome-dependent transfer toward the code packet.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e3d9206a2554a848","tex":"P_S^2","aliases":"P_S^2","display":"P_S^2","kind":"use","excerpt":"Proof. The first equality is distributivity. Since P_S^†=P_S and P_S^2=P_S,","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b1ce984f17e4daf7","tex":"P_S^\\dagger","aliases":"P_S^† dagger","display":"P_S^\\dagger","kind":"use","excerpt":"Proof. The first equality is distributivity. Since P_S^†=P_S and P_S^2=P_S,","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"use","excerpt":"Coherence block · P_αρ P_β with possibly distinct left and right packet labels.","count":24},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8a8ef493645dafcb","tex":"P_{\\bm\\alpha}","aliases":"P_α bm alpha","display":"P_{\\bm\\alpha}","kind":"use","excerpt":"Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential pre","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f29b3a10d396763d","tex":"p_\\alpha","aliases":"p_α alpha","display":"p_\\alpha","kind":"use","excerpt":"Projective measurement. For projectors {P_α}, the probability of outcome α is p_α = Tr(P_αρ), and the conditioned state is ρ_α = fracP_αρ P_αp_α when p_α>0. Measurement is therefore packet selection followed by normalization.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2a1f43875f2877e0","tex":"\\mathcal P_{\\alpha\\beta}","aliases":"P_αβ mathcal alpha beta","display":"\\mathcal P_{\\alpha\\beta}","kind":"use","excerpt":"Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aad7607e608d0052","tex":"P_\\beta","aliases":"P_β beta","display":"P_\\beta","kind":"use","excerpt":"Coherence block · P_αρ P_β with possibly distinct left and right packet labels.","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6d16112b6efeea58","tex":"\\mathcal P_{\\gamma\\delta}","aliases":"P_γδ mathcal gamma delta","display":"\\mathcal P_{\\gamma\\delta}","kind":"use","excerpt":"Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-846abc4e6a928e61","tex":"p_\\lambda","aliases":"p_λ lambda","display":"p_\\lambda","kind":"use","excerpt":"[Permutation-invariant state normal form] A density operator ρ commutes with every permutation if and only if it can be written as ρ = ⊕_λ p_λ ρ_λ⊗ fracI_ V_λf_λ, where p_λ≥0, ∑_λ p_λ=1, and each ρ_λ is a density operator on U_λ whenever p_λ>0. The weights p_λ and the unnormalized blocks p_λρ_λ are unique. The normalized block ρ_λ is unique only wh","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-de2246ea020bc17b","tex":"P_{\\rho_i\\rho_j^{-1}}","aliases":"P_ρ_iρ_j^-1 rho rho","display":"P_{\\rho_i\\rho_j^{-1}}","kind":"use","excerpt":"Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8df653791fb7bee7","tex":"P_{\\rho_j\\rho_i^{-1}}","aliases":"P_ρ_jρ_i^-1 rho rho","display":"P_{\\rho_j\\rho_i^{-1}}","kind":"use","excerpt":"Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-43ec088454bf05d8","tex":"P_\\chi","aliases":"P_χ chi","display":"P_\\chi","kind":"use","excerpt":"Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1039ef467054d89f","tex":"P_\\psi","aliases":"P_ψ psi","display":"P_\\psi","kind":"use","excerpt":"Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-48376cdc9be29993","tex":"\\mathrm{packet}","aliases":"packet mathrm","display":"\\mathrm{packet}","kind":"use","excerpt":"State-vector cost inside a packet. If the simulator stores D amplitudes and a local gate can be applied in O(nD) arithmetic operations, a depth-d packet simulation has schematic cost C_packet=O(dnD)+C_compile. The corresponding full state-vector path is C_full=O(dn2^n). These formulas are intentionally schematic. Cache behavior, indexing, vectoriza","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a156778c1d9d8562","tex":"\\mathrm{packets}","aliases":"packets mathrm","display":"\\mathrm{packets}","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bfd0b07cc3c57995","tex":"\\operatorname{Pareto}","aliases":"Pareto operatorname","display":"\\operatorname{Pareto}","kind":"use","excerpt":"A vector-valued defect. For a target U in a restricted phase model, define C_res(U) = { c( W): W is a restricted circuit for U} and similarly C_amb(U). Their Pareto frontiers are denoted Pareto_res(U) and Pareto_amb(U).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-35495a296f9f4211","tex":"\\operatorname{Pareto}_{\\mathrm{amb}}","aliases":"Pareto_amb operatorname mathrm","display":"\\operatorname{Pareto}_{\\mathrm{amb}}","kind":"use","excerpt":"A vector-valued defect. For a target U in a restricted phase model, define C_res(U) = { c( W): W is a restricted circuit for U} and similarly C_amb(U). Their Pareto frontiers are denoted Pareto_res(U) and Pareto_amb(U).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-961930c638d32a85","tex":"\\operatorname{Pareto}_{\\mathrm{res}}","aliases":"Pareto_res operatorname mathrm","display":"\\operatorname{Pareto}_{\\mathrm{res}}","kind":"use","excerpt":"A vector-valued defect. For a target U in a restricted phase model, define C_res(U) = { c( W): W is a restricted circuit for U} and similarly C_amb(U). Their Pareto frontiers are denoted Pareto_res(U) and Pareto_amb(U).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9db0dae5ed6cdfcc","tex":"\\operatorname{PG}","aliases":"PG operatorname","display":"\\operatorname{PG}","kind":"use","excerpt":"Over F_2, the carrier states used by the canonical word form six of the seven points of PG(2,2). The four projective lines supported on those six points form a Pasch configuration. The same six vectors represent the graphic matroid M(K_4) over every field; this regular-matroid structure isolates the characteristic-independent core of the atom. At t","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"Moment and derivative packets. For a phase function P, finite differences D_uP(x)=P(x+u)-P(x)pmod8 separate affine, quadratic, and higher interactions. A hierarchy of derivative tensors can record temporary flow into nonlinear phase packets.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6fdfdc94488996a0","tex":"\\poly","aliases":"poly","display":null,"kind":"use","excerpt":"Circuit families. A quantum decision procedure is a polynomial-time uniform family of circuits {C_x}, where the input string x determines a circuit on n=n(|x|) qubits with at most poly(|x|) gates. Let C_x=G_m⋯ G_1. A classical simulation theorem must describe an algorithm receiving the same succinct description of x and the circuit. It is not enoug","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-137958caf657ae18","tex":"\\operatorname{polyEval}","aliases":"polyEval operatorname","display":"\\operatorname{polyEval}","kind":"use","excerpt":"Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"The conditional algorithmic consequence. [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0463a5c7d3865eaf","tex":"\\Pr_{F^\\dagger\\chi}","aliases":"Pr_F^†χ Pr dagger chi","display":"\\Pr_{F^\\dagger\\chi}","kind":"use","excerpt":"Pr_F^†ψ(k∈ U_12)=1, Pr_F^†χ(k∈ U_12)=0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-017471d31c68196e","tex":"\\Pr_{F^\\dagger\\chi_\\theta}","aliases":"Pr_F^†χ_θ Pr dagger chi theta","display":"\\Pr_{F^\\dagger\\chi_\\theta}","kind":"use","excerpt":"Pr_F^†χ_θ(k∈ U_12) = Pχ_θ ^2=sin^2(θ/2).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6fd769d38fe9dc8e","tex":"\\Pr_{F^\\dagger\\psi}","aliases":"Pr_F^†ψ Pr dagger psi","display":"\\Pr_{F^\\dagger\\psi}","kind":"use","excerpt":"Pr_F^†ψ(k∈ U_12)=1, Pr_F^†χ(k∈ U_12)=0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a44c0eff3ae23ff3","tex":"\\Pr_{z\\text{ uniform in }K^m}","aliases":"Pr_z uniform in K^m Pr text","display":"\\Pr_{z\\text{ uniform in }K^m}","kind":"use","excerpt":"The conditional algorithmic consequence. [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-003001ebf5867324","tex":"\\prec","aliases":"prec","display":null,"kind":"use","excerpt":"The trivial representation records only the total number of loops returning to the subgroup. Nontrivial H-types retain their holonomy. This yields a hierarchy: spherical return data prec type-resolved holonomy packets prec exact endpoint realization.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-71a98ec1e2bb9f0a","tex":"\\preceq","aliases":"preceq","display":"\\preceq","kind":"use","excerpt":"Template XII: vector costs. Let gates carry costs in R_+^k. A lower bound may be a vector inequality F(U)preceq∑_g∈ Wc(g) or a family of scalar inequalities indexed by dual weights w∈R_+^k: w· F(U)≤ w· c(W). A complete Pareto proof may require enough supporting weights to separate the restricted and ambient convex lower hulls.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-80b4b6d1ff3db017","tex":"\\Prob","aliases":"Prob","display":null,"kind":"use","excerpt":"Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estima","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8800c592d390fda6","tex":"\\Prob_{A\\in\\U_n(\\F_q)}","aliases":"Prob_A∈U_n(F_q) Prob in U F","display":null,"kind":"use","excerpt":"[Finite-field sparsity] Let N=binom n2 and M=n(n-1). For 1≤ d≤ N, Prob_A∈U_n(F_q)[DeltaDef(A)≥ d] ≤ min{1,M^N-dq^-d}. For fixed n, a uniformly random target therefore has defect zero with probability 1-O_n(q^-1) as q→∞.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bd1f83741366f036","tex":"\\mathrm{product}","aliases":"product mathrm","display":"\\mathrm{product}","kind":"use","excerpt":"This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"The exact phase arithmetic. The structure Cyclo12 represents (a,b,c,d)⟼ a+bsqrt3+i(c+dsqrt3), a,b,c,d∈ Q. Its multiplication explicitly incorporates (sqrt3)^2=3 and i^2=-1. It is a concrete arithmetic representation. The development does not need or install a global field instance to certify the finite example.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"Proof. The equation (ζ_q^a)^t=1 holds exactly when q divides at. Write g=gcd(a,q), q=gq', and a=ga', with gcd(a',q')=1. The divisibility condition is q'mid a't, which is equivalent to q'mid t. Its smallest positive solution is t=q'. This includes a=0: then g=q and the order is one. square","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f3003d8698b757c1","tex":"Q^{-1}","aliases":"Q^-1","display":"Q^{-1}","kind":"use","excerpt":"Exact cut comparison and safe discrepancies. claimC01 [Cut-comparison identity and complete coverage] Let a restored oracle be written C=Q∘ P, and put R=Q^-1. Suppress zero ancillas in boundary arguments and define A(x)=P(x,0), B(x)=R(x,0), D(x)=P(x,1)+P(x,0). These are vectors indexed by all registers. Then A(x)+B(x)=f(x)D(x), D(x)≠0. Consequently","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6339057791df9dbc","tex":"q^{-1}","aliases":"q^-1","display":"q^{-1}","kind":"use","excerpt":"pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d9c016afb8932679","tex":"q^{-1/2}","aliases":"q^-1/2","display":"q^{-1/2}","kind":"use","excerpt":"(F_q)_na=q^-1/2ζ_q^na.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a121f605bcf8a91d","tex":"q^{-d}","aliases":"q^-d","display":"q^{-d}","kind":"use","excerpt":"Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estima","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"c_q ^2 =frac1q∑_k|widehatc_q(k)|^2 =frac1qφ(q)q^2. square","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5fa217a1f05c0a4d","tex":"\\mathbb Q^4","aliases":"Q^4 mathbb","display":"\\mathbb Q^4","kind":"use","excerpt":"(a,b,c,d)∈ Q^4","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1928485a8127cf57","tex":"q^N","aliases":"q^N","display":"q^N","kind":"use","excerpt":"Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estima","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-50cffe607a1e0f78","tex":"q^r","aliases":"q^r","display":"q^r","kind":"use","excerpt":"Template VII: finite-field sparsity. Suppose there are at most B gate directions and at most q^r parameter choices per gate over F_q. Then the number of words of length at most L is at most ∑_j=0^L(Bq^r)^j. If the target family has q^N points and defect d forces L≤ (N-d)/r, this yields a density bound of order q^-d up to a fixed combinatorial facto","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"use","excerpt":"Quiver gate models. Let Q=(Q_0,Q_1) be a quiver. Associate a vector space V_v to every vertex. Internal gates change bases inside V_v, while edge gates transmit low-rank maps along arrows. A target is a representation of the path algebra or an automorphism of a filtered quiver module.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"use","excerpt":"Quiver gate models. Let Q=(Q_0,Q_1) be a quiver. Associate a vector space V_v to every vertex. Internal gates change bases inside V_v, while edge gates transmit low-rank maps along arrows. A target is a representation of the path algebra or an automorphism of a filtered quiver module.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"guidepageDegree bounds must retain shared register products Take four independent input bits. Compute q_1=x_1x_2, q_2=x_3x_4, q_3=q_1q_2. After two ANDs, each nonlinear register has degree two, but their product has degree four. After the third AND, a coordinate of degree four appears.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"guidepageDegree bounds must retain shared register products Take four independent input bits. Compute q_1=x_1x_2, q_2=x_3x_4, q_3=q_1q_2. After two ANDs, each nonlinear register has degree two, but their product has degree four. After the third AND, a coordinate of degree four appears.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f5c3a1f5ac4f85f5","tex":"Q_3","aliases":"Q_3","display":"Q_3","kind":"use","excerpt":"d not be supported on the positive set of P_m. Take m=3, identify the columns with the integers 1,…,6 using their three-bit expansions, and set x_1=x_3=x_4=1 with all other inputs zero. Their column sum is 1+3+4=6 under bitwise XOR, not a=7, so P_3=0. Among the complementary pairs (1,6),(2,5),(3,4), exactly (3,4) has both inputs one. Thus Q_3=1. In","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8ae9808cb0f6e77d","tex":"q_3","aliases":"q_3","display":"q_3","kind":"use","excerpt":"guidepageDegree bounds must retain shared register products Take four independent input bits. Compute q_1=x_1x_2, q_2=x_3x_4, q_3=q_1q_2. After two ANDs, each nonlinear register has degree two, but their product has degree four. After the third AND, a coordinate of degree four appears.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-51841caabeb61333","tex":"Q_{>r}","aliases":"Q_>r","display":"Q_{>r}","kind":"use","excerpt":"C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-72684694ad1ea082","tex":"Q_C","aliases":"Q_C","display":"Q_C","kind":"use","excerpt":"Define Q_R=fracI+Rn, Q_C=fracI+Cn, Q_S=fracI+Sn, B_0=fracJn^2. Then the five joint projectors are B_0, Q_R-B_0, Q_C-B_0, Q_S-B_0, I-Q_R-Q_C-Q_S+2B_0. Their multiplicities are 1, n-1, n-1, n-1, (n-1)(n-2). The Latin-square graph R+C+S has universal spectrum 3(n-1)^1, (n-3)^3(n-1), (-3)^(n-1)(n-2). At n=6 this becomes 15^1,qquad3^15, (-3)^20. Thus th","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-34f554df01422a56","tex":"Q_{\\mathrm{final}}","aliases":"Q_final mathrm","display":"Q_{\\mathrm{final}}","kind":"use","excerpt":"endpmatrix. A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e15bd658e05b8d9b","tex":"q_k","aliases":"q_k","display":"q_k","kind":"use","excerpt":"Conserved observables. Let Q=∑_k q_kP_k be a self-adjoint observable with spectral projectors P_k.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4a77ffe628ff771a","tex":"q_n","aliases":"q_n","display":"q_n","kind":"use","excerpt":"Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-808125305b3a9f1c","tex":"Q_R","aliases":"Q_R","display":"Q_R","kind":"use","excerpt":"Define Q_R=fracI+Rn, Q_C=fracI+Cn, Q_S=fracI+Sn, B_0=fracJn^2. Then the five joint projectors are B_0, Q_R-B_0, Q_C-B_0, Q_S-B_0, I-Q_R-Q_C-Q_S+2B_0. Their multiplicities are 1, n-1, n-1, n-1, (n-1)(n-2). The Latin-square graph R+C+S has universal spectrum 3(n-1)^1, (n-3)^3(n-1), (-3)^(n-1)(n-2). At n=6 this becomes 15^1,qquad3^15, (-3)^20. Thus th","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-430a2da3291a2845","tex":"Q_S","aliases":"Q_S","display":"Q_S","kind":"use","excerpt":"Define Q_R=fracI+Rn, Q_C=fracI+Cn, Q_S=fracI+Sn, B_0=fracJn^2. Then the five joint projectors are B_0, Q_R-B_0, Q_C-B_0, Q_S-B_0, I-Q_R-Q_C-Q_S+2B_0. Their multiplicities are 1, n-1, n-1, n-1, (n-1)(n-2). The Latin-square graph R+C+S has universal spectrum 3(n-1)^1, (n-3)^3(n-1), (-3)^(n-1)(n-2). At n=6 this becomes 15^1,qquad3^15, (-3)^20. Thus th","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d70d66d45c49fb6a","tex":"Q_s","aliases":"Q_s","display":"Q_s","kind":"use","excerpt":"Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential pre","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8ac168ec1a0ac655","tex":"\\mathbf r","aliases":"r mathbf","display":"\\mathbf r","kind":"use","excerpt":"For n bosonic modes, a Gaussian unitary acts on quadratures r=(q_1,…,q_n,p_1,…,p_n)^T by an affine symplectic transformation r⟼ S r+d, S∈Sp(2n,R). For n fermionic modes, a fermionic Gaussian unitary acts on 2n Majorana operators by γ⟼ Oγ, O∈ SO(2n). Nearest-neighbor matchgates give a circuit presentation of a fermionic Gaussian subgroup on qubits.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies ","count":16},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c3b10d2685a65f34","tex":"\\mathcal R^*_{n,\\le L}","aliases":"R^*_n,≤ L mathcal le","display":"\\mathcal R^*_{n,\\le L}","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0d336d0befb26bc2","tex":"\\mathcal R^*_{n,\\le N-d}","aliases":"R^*_n,≤ N-d mathcal le","display":"\\mathcal R^*_{n,\\le N-d}","kind":"use","excerpt":"If DeltaDef(A)≥ d, then ellAll(A) ≤ ellDown(A)-d ≤ N-d. Thus D_n,≥ d⊆ R^*_n,≤ N-d, and [source reference] gives the first assertion. On the intrinsic-length-L stratum one instead has ellAll(A)≤ L-d, proving the refinement.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-37e1261ad0123242","tex":"R^6","aliases":"R^6","display":"R^6","kind":"use","excerpt":"Project the rows to the quotient module R^6/W. Rows 0,1,2 have zero image, while rows 3,4,5 begin and end at the three distinct quotient basis vectors of the free quotient module. An output row targeted exactly once must therefore use a source lying in W, by [source reference].","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aeff67e9ace2cff5","tex":"\\mathcal R^\\downarrow_{n,\\le L}","aliases":"R^downarrow_n,≤ L mathcal downarrow le","display":"\\mathcal R^\\downarrow_{n,\\le L}","kind":"use","excerpt":"mial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^downarrow_n,≤ L analogously using only lower directions.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4415f2a3fc7b65ea","tex":"R^n","aliases":"R^n","display":"R^n","kind":"use","excerpt":"[One-target source condition] Let W≤ R^n. Suppose row t starts at e_t, is targeted exactly once, and finishes at e_t+w, w∈ W, where at least one coordinate of w is a unit of R. At the moment of its unique targeting, the gate coefficient is a unit and the source row is a unit multiple of w.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-94f941f054a1b775","tex":"r^q","aliases":"r^q","display":"r^q","kind":"use","excerpt":"Proof. If qmid h, every summand is one. Otherwise r=ζ_q^h≠1 and r^q=1. Multiplying the sum by 1-r gives 1-r^q=0, so the sum is zero. This argument also covers q=1, where only the first case occurs. square","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-53df6e3f42cc9daa","tex":"\\R_+^k","aliases":"R_+^k R","display":null,"kind":"use","excerpt":"Template XII: vector costs. Let gates carry costs in R_+^k. A lower bound may be a vector inequality F(U)preceq∑_g∈ Wc(g) or a family of scalar inequalities indexed by dual weights w∈R_+^k: w· F(U)≤ w· c(W). A complete Pareto proof may require enough supporting weights to separate the restricted and ambient convex lower hulls.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among fr","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"State models. Binary matrices. An m× m matrix over F_2 is stored as (r_0,…,r_m-1), where bit j of r_i is the (i,j) entry. Matrix multiplication is exact XOR accumulation.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d0aa3589089d58ba","tex":"R_3","aliases":"R_3","display":"R_3","kind":"use","excerpt":"Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0fe85010c113a2aa","tex":"R_4","aliases":"R_4","display":"R_4","kind":"use","excerpt":"Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5dbe454ca813d1d3","tex":"R_5","aliases":"R_5","display":"R_5","kind":"use","excerpt":"Let R be a nonzero commutative ring with identity, and let v=e_1+e_2, w=e_0+e_1+e_2, W=Span(e_0,e_1,e_2). The target C of [source reference] has rows R_0 · =e_0, · R_1 · =e_1+e_0, · R_2 · =e_2, R_3 · =e_3+v, · R_4 · =e_4+w, · R_5 · =e_5+w.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9067584bd8dfe06a","tex":"R_{CP_jC^\\dagger}","aliases":"R_CP_jC^† dagger","display":"R_{CP_jC^\\dagger}","kind":"use","excerpt":"The simplest symbolic sandwich is C(∏_j=1^m R_P_j(θ_j))C^† = ∏_j=1^m R_CP_jC^†(θ_j). If the P_j commute, their conjugates commute. A catalyst can therefore change the geometry of the support without changing the algebra of the angles.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7c5aabb43589552c","tex":"R_{CPC^\\dagger}","aliases":"R_CPC^† dagger","display":"R_{CPC^\\dagger}","kind":"use","excerpt":"Pauli rotations. For a Hermitian Pauli P, R_P(θ)=e^-iθ P/2. Clifford conjugation gives C R_P(θ) C^†=R_CPC^†(θ). The sign of CPC^† can be absorbed by changing θ to -θ.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"use","excerpt":"For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among fr","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"Many consumers and catalytic excess. Suppose the middle word is k=h_m⋯ h_1, h_i∈ H. The ambient word gh_m⋯ h_1g^-1 has cost at most 2c(g)+∑_i c(h_i). If the restricted images gh_ig^-1 require costs r_i and a lower-bound invariant proves that their obligations cannot be shared, then ℓ_H(gkg^-1)≥∑_i r_i. A sufficient inequality is ∑_i(r_i-c(h_i))>2c(","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5735cc8d934d4beb","tex":"r_{m-1}","aliases":"r_m-1","display":"r_{m-1}","kind":"use","excerpt":"State models. Binary matrices. An m× m matrix over F_2 is stored as (r_0,…,r_m-1), where bit j of r_i is the (i,j) entry. Matrix multiplication is exact XOR accumulation.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8e64bf438310da1c","tex":"\\mathcal R_n","aliases":"R_n mathcal","display":"\\mathcal R_n","kind":"use","excerpt":"Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies ","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a4ff961cd74d1869","tex":"r_n","aliases":"r_n","display":"r_n","kind":"use","excerpt":"The rational signal and its filter. The first layer uses literal rational lists. The chosen perturbation is r_n=3cos(π n/2)+2, whose displayed rational entries can be certified without numerical trigonometric evaluation. Its sum with c is the test signal.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b8a8516575d862bd","tex":"R_{n-1}","aliases":"R_n-1","display":"R_{n-1}","kind":"use","excerpt":"For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among fr","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1b85fe58dfd4105d","tex":"R_P","aliases":"R_P","display":"R_P","kind":"use","excerpt":"Pauli rotations. For a Hermitian Pauli P, R_P(θ)=e^-iθ P/2. Clifford conjugation gives C R_P(θ) C^†=R_CPC^†(θ). The sign of CPC^† can be absorbed by changing θ to -θ.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b5c9a939784c608f","tex":"R_p","aliases":"R_p","display":"R_p","kind":"use","excerpt":"Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-a","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-25dded27e9cb56fe","tex":"R_{P_j}","aliases":"R_P_j","display":"R_{P_j}","kind":"use","excerpt":"The simplest symbolic sandwich is C(∏_j=1^m R_P_j(θ_j))C^† = ∏_j=1^m R_CP_jC^†(θ_j). If the P_j commute, their conjugates commute. A catalyst can therefore change the geometry of the support without changing the algebra of the angles.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c2d59bfcf91e5807","tex":"R_q","aliases":"R_q","display":"R_q","kind":"use","excerpt":"Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-a","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1e4761712190b8fb","tex":"R_r","aliases":"R_r","display":"R_r","kind":"use","excerpt":"E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f7b985b4cff51815","tex":"R_s","aliases":"R_s","display":"R_s","kind":"use","excerpt":"Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-49121c793f36cbb5","tex":"R_t","aliases":"R_t","display":"R_t","kind":"use","excerpt":"Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9fb33ac102cd9b2c","tex":"R_{t_j}","aliases":"R_t_j","display":"R_{t_j}","kind":"use","excerpt":"[ 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d47df91897a0edf1","tex":"R_{v_0}","aliases":"R_v_0","display":"R_{v_0}","kind":"use","excerpt":"Suppose the gate is s→ t with s<t. Let v_0 be the lowest node of T whose two children separate s and t. The order of the leaves forces s∈ L_v_0 and t∈ R_v_0. At this one node, a single row of the cross block X[R_v_0,L_v_0] changes, so its rank changes by at most one.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3088a38f12fda26d","tex":"R_x","aliases":"R_x","display":"R_x","kind":"use","excerpt":"Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-a","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fe6cb9818c90a480","tex":"R_{x_i}","aliases":"R_x_i","display":"R_{x_i}","kind":"use","excerpt":"integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a3c72cb58f67a4d1","tex":"R_y","aliases":"R_y","display":"R_y","kind":"use","excerpt":"Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-a","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-619e2ba07104f75c","tex":"R_Z","aliases":"R_Z","display":"R_Z","kind":"use","excerpt":"Number-preserving gates. Single-qubit R_Z gates commute with widehat N. A two-qubit exchange rotation acts in the ordered basis ket00,ket01,ket10,ket11 as U_XY(θ,ϕ) = 1 · 0 · 0 · 0","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-68df6b2bfb142937","tex":"r_\\alpha","aliases":"r_α alpha","display":"r_\\alpha","kind":"use","excerpt":"Hybrid transfer graphs. Suppose packets are nodes of a sparse graph and each node carries an internal representation. A gate may update a node internally or transfer a compact component to neighboring nodes. The simulation cost depends on ∑_α∈ A_tsize(r_α) + #{nonzero transfers at time t}. This model combines symmetry selection rules with compact s","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"The balanced cut is constructed. Because each gate has cost zero or one, a prefix can realize every Toffoli count from zero to T. The source proves this fact and extracts a cut with max{TC(p),TC(q)}≤lceil T/2rceil. The complete safe cut cover then proves β(f)≤lceil T/2rceil+1, 2β(f)≤ T+3. Natural-number arithmetic retains the floor/ceiling and smal","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"use","excerpt":"A real three-dimensional view of one amplitude. addcontentslinetocsectionA real three-dimensional view of one amplitude leadThe plot follows an actual complex amplitude as the relative phase advances. At sample n=6, the phase-family amplitude is χ_θ(6)=frace^iθsqrt3. fig16_complex_amplitude_trajectory0.84Coordinates are (θ,Reχ_θ(6),Imχ_θ(6)). In th","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-73947eeba72e530f","tex":"\\operatorname{Real}_{\\mathrm{entangled}}","aliases":"Real_entangled operatorname mathrm","display":"\\operatorname{Real}_{\\mathrm{entangled}}","kind":"use","excerpt":"This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d614b629b19d55d6","tex":"\\operatorname{Real}_{\\mathrm{product}}","aliases":"Real_product operatorname mathrm","display":"\\operatorname{Real}_{\\mathrm{product}}","kind":"use","excerpt":"This is naturally expressed as Real_product(P_6)=∅, Real_entangled(P_6)≠∅. It is an existence defect, not a gate-count defect.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7dbae09ebccab752","tex":"\\mathrm{rest}","aliases":"rest mathrm","display":"\\mathrm{rest}","kind":"use","excerpt":"guidepageExact nonlinear projection costs versus actual input length The source proves the all-size formulas M(P_m)=m-1, T_rest(P_m)=2m-3, M(Q_m)=T_rest(Q_m)=n/2. The chart uses actual input length n=2^m-2 on its horizontal axis.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ecd7e56200e659e5","tex":"\\operatorname{rk}","aliases":"rk operatorname","display":"\\operatorname{rk}","kind":"use","excerpt":"Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but t","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-db5fe6198af8dba2","tex":"\\operatorname{rk}_{X}","aliases":"rk_X operatorname","display":"\\operatorname{rk}_{X}","kind":"use","excerpt":"Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but t","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"Repeated application versus one shot. Let C_rm compile be the cost of constructing the packet basis and restricted gate plans, and let C_rm apply be one packet application. Reusing the compiled plan R times costs C_rm packet(R) = C_rm compile+RC_rm apply. If the full path costs RC_rm full, the packet method breaks even when R> fracC_rm compile C_rm","count":19},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c2b4530a730b06f1","tex":"\\mathrm{run}","aliases":"run mathrm","display":"\\mathrm{run}","kind":"use","excerpt":"The restoration predicate is ∀ x b, run(c)(encode(x,b)) =encode(x,b+f(x)). It is a hypothesis of the conditional theorem. The code does not assert that every gate list realizes a restored oracle. The canonical encoding sets each input wire to its input bit, each ancillary wire to zero, and the target to b. Its wire type is Sum (Fin n) (Sum (Fin a) ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","excerpt":"Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-62b006e8dceb3bc0","tex":"S^T","aliases":"S^T","display":"S^T","kind":"use","excerpt":"Covariance-matrix lower bounds. For a Gaussian state, the covariance matrix Γ transforms as Γ⟼ SΓ S^T in the bosonic case and Γ⟼ OΓ O^T in the fermionic case. Possible restricted lower bounds include: item rank change of off-diagonal covariance blocks, item symplectic singular-value obligations, item growth of mode support, item graph bandwidth and","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"use","excerpt":"1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4c67250f901a56f7","tex":"S_0^{d_0}","aliases":"S_0^d_0","display":"S_0^{d_0}","kind":"use","excerpt":"I_2 · J_2). Applying S_0^d_0S_1^d_1 afterwards left multiplies by (I_2 · 0","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"use","excerpt":"1 · 1 · 1 · 0 . A restricted path is CX_0→1, S_1, CX_1→0, CX_0→1, with three charged entangling gates. An ambient path is H_1, CX_1→0, CX_0→1, H_0, S_1, S_0, with two charged entangling gates when H,S are free.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1616cfa7e151ed72","tex":"S_1^{d_1}","aliases":"S_1^d_1","display":"S_1^{d_1}","kind":"use","excerpt":"I_2 · J_2). Applying S_0^d_0S_1^d_1 afterwards left multiplies by (I_2 · 0","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"use","excerpt":"[Holonomy-compression identity] Let every generator have unit cost. For h∈ H, ℓ_G(h) is the minimum length of a based loop in G/H whose lifted word has holonomy h. The intrinsic length ℓ_H(h) is the minimum length among loops that remain in the base fiber. An ambient word s_k⋯ s_1=h projects to the loop H, s_1H, s_2s_1H, …, hH=H. Conversely, a base","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-71a07bf98e34b5f3","tex":"S_8","aliases":"S_8","display":"S_8","kind":"use","excerpt":"Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained i","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2badf92439cc7951","tex":"S_A","aliases":"S_A","display":"S_A","kind":"use","excerpt":"[Breadth-first defect census] Given restricted generators S_R, ambient generators S_A, and identity e: item compute d_R on ⟨ S_R⟩; item compute d_A on ⟨ S_A⟩; item for every restricted state x, compare d_R(x) and d_A(x); item record the pair (d_R(x),d_A(x)) and reconstruct witnesses for strict inequalities.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-508e9a069e81ec8b","tex":"S_d","aliases":"S_d","display":"S_d","kind":"use","excerpt":"S_d={a∈ I_N:N/gcd(a,N)=d}.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-20ce4b1ef4086ef3","tex":"\\mathcal S_G","aliases":"S_G mathcal","display":"\\mathcal S_G","kind":"use","excerpt":"Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry ","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e80bf6a0d513c10b","tex":"\\mathcal S_H","aliases":"S_H mathcal","display":"\\mathcal S_H","kind":"use","excerpt":"Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry ","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"pageheadWhy sharing does not break the degree boundA STRONGER INVARIANT THAN SINGLE-REGISTER DEGREE For a current state s(x) after t Toffolis, the invariant is ∀ L, Deg (t+|L|, x↦∏_i∈ Ls_i(x)). L may be any finite list, including repeated registers. Initially each coordinate is a variable or a constant, so a product of r coordinates has degree at m","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dfbf9a1694a645b8","tex":"s_i'","aliases":"s_i'","display":"s_i'","kind":"use","excerpt":"guidepageThe invariant handles duplicates, NOT, and CNOT Suppose a gate toggles target u by a product of c controls. In a product list L that contains u, remove all occurrences of u to obtain R. Boolean idempotence leaves one updated target factor, and ∏_i∈ Ls_i'= s_u∏_i∈ Rs_i+ (∏_j∈controlss_j)∏_i∈ Rs_i. The old invariant bounds these terms by t+1","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bbf4aa0bb7699587","tex":"S_j","aliases":"S_j","display":"S_j","kind":"use","excerpt":"Syndrome projectors. Let S_1,…,S_m be commuting Pauli stabilizers with eigenvalues ±1. A syndrome s∈{0,1}^m defines P_s = ∏_j=1^m fracI+(-1)^s_jS_j2. These projectors are orthogonal and sum to the identity. Syndrome measurement selects one packet, while recovery applies a syndrome-dependent transfer toward the code packet.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b4ebcce0dc1a5c39","tex":"s_k","aliases":"s_k","display":"s_k","kind":"use","excerpt":"[Holonomy-compression identity] Let every generator have unit cost. For h∈ H, ℓ_G(h) is the minimum length of a based loop in G/H whose lifted word has holonomy h. The intrinsic length ℓ_H(h) is the minimum length among loops that remain in the base fiber. An ambient word s_k⋯ s_1=h projects to the loop H, s_1H, s_2s_1H, …, hH=H. Conversely, a base","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-57c446e8e6d068a3","tex":"S_m","aliases":"S_m","display":"S_m","kind":"use","excerpt":"Syndrome projectors. Let S_1,…,S_m be commuting Pauli stabilizers with eigenvalues ±1. A syndrome s∈{0,1}^m defines P_s = ∏_j=1^m fracI+(-1)^s_jS_j2. These projectors are orthogonal and sum to the identity. Syndrome measurement selects one packet, while recovery applies a syndrome-dependent transfer toward the code packet.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b9746e4f6fc18b24","tex":"s_m","aliases":"s_m","display":"s_m","kind":"use","excerpt":"Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8c0785670c74cfb3","tex":"S_n","aliases":"S_n","display":"S_n","kind":"use","excerpt":"The symmetric subspace. Let the symmetric group S_n act by permuting the n tensor factors. The fully symmetric qubit subspace is spanned by the Dicke states ketD_n^k = binom nk^-1/2 ∑_|x|=kket x, 0≤ k≤ n. Its dimension is n+1.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f7af78049dcddf01","tex":"S_R","aliases":"S_R","display":"S_R","kind":"use","excerpt":"[Breadth-first defect census] Given restricted generators S_R, ambient generators S_A, and identity e: item compute d_R on ⟨ S_R⟩; item compute d_A on ⟨ S_A⟩; item for every restricted state x, compare d_R(x) and d_A(x); item record the pair (d_R(x),d_A(x)) and reconstruct witnesses for strict inequalities.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8fa000bc9870ade8","tex":"S_{\\rm transfer}","aliases":"S_rm transfer rm","display":"S_{\\rm transfer}","kind":"use","excerpt":"The core research invariant is the complete classical representation record ( discovery cost, N_rm packets, D_rm internal, S_rm transfer, L_rm leakage, measurement cost ). A future compiler that computes and certifies this record can decide when the full state vector is unnecessary, when a hybrid representation is preferable, and when symmetry alon","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-87807c21edc049cd","tex":"S_t","aliases":"S_t","display":"S_t","kind":"use","excerpt":"Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9049496dbf9180c9","tex":"S_{t-1}","aliases":"S_t-1","display":"S_{t-1}","kind":"use","excerpt":"Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-54fa66f2e8084241","tex":"S_{\\mathrm{transfer}}","aliases":"S_transfer mathrm","display":"S_{\\mathrm{transfer}}","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5dce6ea50bf9e787","tex":"s_u","aliases":"s_u","display":"s_u","kind":"use","excerpt":"Suppose the next gate toggles target u by the product of a control list of length q. A product that omits u is unchanged. If u occurs, idempotence removes all its repetitions. With R the remaining list, the updated product is (s_u∏_i∈ Rs_i) x (∏_j∈controlss_j∏_i∈ Rs_i). The two old-state products have certified degrees at most t+1+|R| and t+q+|R|. ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"QA19 · Relative-phase family · Prose extension sin^2(θ/2); not an inherited Lean theorem.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"use","excerpt":"QA19 · Relative-phase family · Prose extension sin^2(θ/2); not an inherited Lean theorem.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-323bf8e2b35a8f67","tex":"\\operatorname{size}","aliases":"size operatorname","display":"\\operatorname{size}","kind":"use","excerpt":"Hybrid transfer graphs. Suppose packets are nodes of a sparse graph and each node carries an internal representation. A gate may update a node internally or transfer a compact component to neighboring nodes. The simulation cost depends on ∑_α∈ A_tsize(r_α) + #{nonzero transfers at time t}. This model combines symmetry selection rules with compact s","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-40de2b59394cdbb4","tex":"\\SL","aliases":"SL","display":null,"kind":"use","excerpt":"A CNOT circuit on n labeled wires represents an element of GL_n(F_2)=SL_n(F_2), and minimizing its gate count is the word-metric problem for transvection generators. This formulation supports asymptotically efficient synthesis, architecture-aware elimination, computable lower bounds, and exact small-width optimization [source reference]. Recent wor","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-25e592c757881df5","tex":"\\SL_n","aliases":"SL_n SL","display":null,"kind":"use","excerpt":"A CNOT circuit on n labeled wires represents an element of GL_n(F_2)=SL_n(F_2), and minimizing its gate count is the word-metric problem for transvection generators. This formulation supports asymptotically efficient synthesis, architecture-aware elimination, computable lower bounds, and exact small-width optimization [source reference]. Recent wor","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7eb575280d7a6bca","tex":"\\mathrm{small}","aliases":"small mathrm","display":"\\mathrm{small}","kind":"use","excerpt":"a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). T","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-44f6ac12b2045864","tex":"\\mathrm{small,res}","aliases":"small,res mathrm","display":"\\mathrm{small,res}","kind":"use","excerpt":"Template VIII: block retraction. Let a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves ga","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c5fe8318028f5a40","tex":"\\mathrm{space-time}","aliases":"space-time mathrm","display":"\\mathrm{space-time}","kind":"use","excerpt":"Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span","display":null,"kind":"use","excerpt":"[Dual certificate for failure of safe detection] For every positive input v, β_v(f)>d Longleftrightarrow v_d(v)∈Span_F{v_d(u):u∈ U}. A degree-at-most-d polynomial is a coefficient covector on the evaluation space. It is safe exactly when it annihilates every negative evaluation vector, equivalently their span. Such a covector can equal one on v_d(v","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0801818e36621f46","tex":"\\Span_{\\F}","aliases":"Span_F Span F","display":null,"kind":"use","excerpt":"[Dual certificate for failure of safe detection] For every positive input v, β_v(f)>d Longleftrightarrow v_d(v)∈Span_F{v_d(u):u∈ U}. A degree-at-most-d polynomial is a coefficient covector on the evaluation space. It is safe exactly when it annihilates every negative evaluation vector, equivalently their span. Such a covector can equal one on v_d(v","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-649e98aecfe8e810","tex":"\\Span_{\\F_2}","aliases":"Span_F_2 Span F","display":null,"kind":"use","excerpt":"guidepageThe finite evaluation-span dual in full Fix a positive v and integer d≥0. Let v_d(x)=(x^S)_S⊆[n], |S|≤ d, U=f^-1(0), N=Span_F_2{v_d(u):u∈ U}. The constant monomial S=∅ is included. Degree-d coefficient covectors form the finite dual of this evaluation space. A covector is safe exactly when it annihilates N.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"use","excerpt":"Proof. The equation (ζ_q^a)^t=1 holds exactly when q divides at. Write g=gcd(a,q), q=gq', and a=ga', with gcd(a',q')=1. The divisibility condition is q'mid a't, which is equivalent to q'mid t. Its smallest positive solution is t=q'. This includes a=0: then g=q and the order is one. square","count":20},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2f85dcc86f67bb96","tex":"\\operatorname{Stretch}","aliases":"Stretch operatorname","display":"\\operatorname{Stretch}","kind":"use","excerpt":"The basic conjugation inequality. Let g∈ G and k∈ H satisfy gkg^-1∈ H. Then ℓ_G(gkg^-1) ≤ 2ℓ_G(g)+ℓ_H(k). Define the intrinsic conjugation stretch Stretch_g(k) = ℓ_H(gkg^-1)-ℓ_H(k).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4bc4928f8b82b26a","tex":"\\operatorname{Stretch}_g","aliases":"Stretch_g operatorname","display":"\\operatorname{Stretch}_g","kind":"use","excerpt":"The basic conjugation inequality. Let g∈ G and k∈ H satisfy gkg^-1∈ H. Then ℓ_G(gkg^-1) ≤ 2ℓ_G(g)+ℓ_H(k). Define the intrinsic conjugation stretch Stretch_g(k) = ℓ_H(gkg^-1)-ℓ_H(k).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-26ea29c87e6756b5","tex":"\\SUM","aliases":"SUM","display":null,"kind":"use","excerpt":"Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8455408a8dae9f51","tex":"\\SUM^{(-1)}_{q\\to p}","aliases":"SUM^(-1)_q→ p SUM to","display":null,"kind":"use","excerpt":"t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d990d248655a3919","tex":"\\SUM^{(a)}_{s\\to t}","aliases":"SUM^(a)_s→ t SUM to","display":null,"kind":"use","excerpt":"Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ba46169cfecbd468","tex":"\\SUM_{p\\to t_0}","aliases":"SUM_p→ t_0 SUM to","display":null,"kind":"use","excerpt":"t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-84bca647e658ac9f","tex":"\\SUM_{p\\to t_1}","aliases":"SUM_p→ t_1 SUM to","display":null,"kind":"use","excerpt":"t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-03514378c56b283a","tex":"\\SUM_{p\\to t_2}","aliases":"SUM_p→ t_2 SUM to","display":null,"kind":"use","excerpt":"t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e0c0a0aca4362d9f","tex":"\\SUM_{q\\to p}","aliases":"SUM_q→ p SUM to","display":null,"kind":"use","excerpt":"t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e69c92a0ec06a901","tex":"\\SUM_{x\\to p}","aliases":"SUM_x→ p SUM to","display":null,"kind":"use","excerpt":"t_2 · ↦ t_2+x+p+q, with x and q unchanged. An ambient word is SUM_q→ p, SUM_p→ t_0, SUM_x→ p, SUM_p→ t_1, SUM_p→ t_2, SUM^(-1)_q→ p. The first gate adds the dirty q component to p. The middle gates export two different carrier states. The final inverse gate restores p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9a1c9a99f33f46d1","tex":"\\sup_{A\\in\\U_n(\\F)}","aliases":"sup_A∈U_n(F) sup in U F","display":null,"kind":"use","excerpt":"[Explicit nlog n distortion] For D_n(F) = sup_A∈U_n(F) (ellDown(A)-ellAll(A)), one has, for every field and every n, D_n(F) ≥ frac821log_2 21nlog_2n - frac82105n. The coefficient is attained within the snapshot family by the twenty-one-dimensional seed A(9;0,1,…,9).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1b5f3ed90927c97c","tex":"\\sup_{A\\in\\U_n(\\F),\\,A\\ne I}","aliases":"sup_A∈U_n(F), A≠ I sup in U F ne","display":null,"kind":"use","excerpt":"[Asymptotic multiplicative distortion] For every field, limsup_n→∞ sup_A∈U_n(F), A≠ I fracellDown(A)ellAll(A) ≥frac32.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9878a88bb8c17e85","tex":"\\sup_{n,A\\in\\U_n(\\F)}","aliases":"sup_n,A∈U_n(F) sup in U F","display":null,"kind":"use","excerpt":"[Multiplicative distortion] Is sup_n,A∈U_n(F) fracellDown(A)ellAll(A) bounded? The connected snapshot family gives an asymptotic lower bound of 3/2. Exceeding 3/2 appears to require a mechanism beyond one clean carrier exporting independent snapshots.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1084f45e6b3e947e","tex":"\\supp","aliases":"supp","display":null,"kind":"use","excerpt":"Monotone specialization. [Safe monomial ideal for a monotone function] If f is monotone, I_f has a monomial basis indexed by the positive supports of f. For each positive v, β_v(f)=min{|S|: S⊆supp(v), f(1_S)=1}. Consequently, for nonzero f, α_1(f)=minimum minterm size, β(f)=maximum minterm size. For any Boolean polynomial h, its coefficient on x^S ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-46a32ae3908732ce","tex":"\\SWAP","aliases":"SWAP","display":null,"kind":"use","excerpt":"SWAP sensitivity. Add SWAP_ij to both alphabets as a unit-cost primitive. The target groups remain unchanged, but the metric changes.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-00b28a6e5f29ae66","tex":"\\SWAP_{ij}","aliases":"SWAP_ij SWAP","display":null,"kind":"use","excerpt":"SWAP sensitivity. Add SWAP_ij to both alphabets as a unit-cost primitive. The target groups remain unchanged, but the metric changes.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-220cf30acb4b6152","tex":"\\Sym","aliases":"Sym","display":null,"kind":"use","excerpt":"Symmetric qudits. For local dimension d, the symmetric subspace has occupation-number basis (n_1,…,n_d), n_1+⋯+n_d=n, and dimension dimSym^n(C^d) = binomn+d-1d-1. For fixed d, this is polynomial in n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-39703f0cb2aaea8a","tex":"\\Sym^n","aliases":"Sym^n Sym","display":null,"kind":"use","excerpt":"Symmetric qudits. For local dimension d, the symmetric subspace has occupation-number basis (n_1,…,n_d), n_1+⋯+n_d=n, and dimension dimSym^n(C^d) = binomn+d-1d-1. For fixed d, this is polynomial in n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"use","excerpt":"A formal nonlinear catalyst. A clean one-workspace template is T h_m⋯ h_1 T^-1, where T is Toffoli and the middle gates use the quadratic term created in the workspace. Since Toffoli is self-inverse, the displayed circuit uses two nonlinear gates.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":"The exact tests use GF(16)=F[t]/(t^4+t+1). They evaluate every specified verifier through three witness bits at every field point, while the independent top-coefficient check uses all field points through two witness bits and the declared smaller coordinate grid at three bits. These domains must not be conflated.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8eaeac84691b5c3b","tex":"T^k","aliases":"T^k","display":"T^k","kind":"use","excerpt":"Phase polynomials and parity carriers. A CNOT network computes linear parities λ(x)=a· x∈F_2. Applying T^k to a wire carrying λ(x) contributes ω^kλ(x) to the phase polynomial. A parity network must create each required parity, expose it to a phase gate, and restore the linear map required at the endpoint.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aea3af576abad800","tex":"t^m","aliases":"t^m","display":"t^m","kind":"use","excerpt":"E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","excerpt":"The universal six-qudit circuit. Use the coordinate names x,p,q,t_0,t_1,t_2. The target acts on basis labels by p · ↦ p+x,","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5d08324acb961fcf","tex":"t_2","aliases":"t_2","display":"t_2","kind":"use","excerpt":"The universal six-qudit circuit. Use the coordinate names x,p,q,t_0,t_1,t_2. The target acts on basis labels by p · ↦ p+x,","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"use","excerpt":"Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-25bdedf7497647d3","tex":"t_j","aliases":"t_j","display":"t_j","kind":"use","excerpt":"[ 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-576234d87c792d2f","tex":"t_j^{\\mathrm{close}}","aliases":"t_j^close mathrm","display":"t_j^{\\mathrm{close}}","kind":"use","excerpt":"Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9742de8e88997f45","tex":"t_j^{\\mathrm{open}}","aliases":"t_j^open mathrm","display":"t_j^{\\mathrm{open}}","kind":"use","excerpt":"Catalytic interval geometry. Associate to every matched opening and closing operation an interval I_j=[t_j^open,t_j^close]. The family of intervals can be disjoint, nested, laminar, or crossing.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9caf405f71847e16","tex":"t_L","aliases":"t_L","display":"t_L","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b8e003e6dbf7e04c","tex":"T_m","aliases":"T_m","display":"T_m","kind":"use","excerpt":"Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-733902ab0028bba1","tex":"t_m","aliases":"t_m","display":"t_m","kind":"use","excerpt":"Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-072176a16581e3eb","tex":"T_{q,r}","aliases":"T_q,r","display":"T_{q,r}","kind":"use","excerpt":"Sector-flow invariants. Let P_q be the projector onto sector q. For an intermediate unitary U_t, define transfer blocks T_q,r(t)=P_q U_t P_r. A restricted path has T_q,r=0 for q≠ r. Ambient paths may populate off-diagonal blocks. Possible excursion measures include ∑_q≠ rrank T_q,r, max_q≠ r T_q,r , bandwidth in |q-r|. These are analogues of strict","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-70319a4f329d90c9","tex":"T_{qr}","aliases":"T_qr","display":"T_{qr}","kind":"use","excerpt":"Template XIV: symmetry-sector escape. Given projectors P_q, define off-sector transfer blocks T_qr=P_qUP_r. A restricted gate preserves all diagonal sectors. An ambient gate changes a controlled collection of blocks. A candidate potential is a weighted sum of ranks or norms of obligations relative to the final target. The proof must charge both cre","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5864a811907b58da","tex":"T_{\\mathrm{rest}}","aliases":"T_rest mathrm","display":"T_{\\mathrm{rest}}","kind":"use","excerpt":"guidepageExact nonlinear projection costs versus actual input length The source proves the all-size formulas M(P_m)=m-1, T_rest(P_m)=2m-3, M(Q_m)=T_rest(Q_m)=n/2. The chart uses actual input length n=2^m-2 on its horizontal axis.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0fa28f555a8ae981","tex":"t_v","aliases":"t_v","display":"t_v","kind":"use","excerpt":"Fix integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3c0ebebd59054caf","tex":"\\mathrm{target}","aliases":"target mathrm","display":"\\mathrm{target}","kind":"use","excerpt":"pageheadReading the formal model literallyQUANTIFIERS AND ENCODING CARRY THE HYPOTHESES The generic cut theory uses a logical-input type X, a wire type W with decidable equality, and a boundary structure. Its essential fields are an encoding encode:X→Bool→(W→Bool), a target wire, and the proof encode(x,b)(target)=b. This target equation separates t","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-18df9116a49563bb","tex":"\\operatorname{TC}","aliases":"TC operatorname","display":"\\operatorname{TC}","kind":"use","excerpt":"A representation-relative invariant. For a declared classical representation class R, define TC^ R_ε(C) = min{ cost(R): R∈ R, |R(C)-p_C|≤ε }, where p_C is the target acceptance probability. This permits a disciplined ladder:","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c2ab9fbd602eb992","tex":"\\operatorname{TC}^{\\mathcal R}_\\varepsilon","aliases":"TC^ R_ε operatorname mathcal varepsilon","display":"\\operatorname{TC}^{\\mathcal R}_\\varepsilon","kind":"use","excerpt":"A representation-relative invariant. For a declared classical representation class R, define TC^ R_ε(C) = min{ cost(R): R∈ R, |R(C)-p_C|≤ε }, where p_C is the target acceptance probability. This permits a disciplined ladder:","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5a8aa6bed28fce0e","tex":"\\TOF","aliases":"TOF","display":null,"kind":"use","excerpt":"Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained i","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9bda6b31a43171bb","tex":"\\TOF_{a,b\\to t}","aliases":"TOF_a,b→ t TOF to","display":null,"kind":"use","excerpt":"Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained i","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr","display":null,"kind":"use","excerpt":"[Invariant observables see only the twirl] If A commutes with every U_π, then Tr(Aρ) = Tr(A T_n(ρ)) for every density operator ρ.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehat","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-06cd3fcbb39111f5","tex":"\\mathrm{transfer}","aliases":"transfer mathrm","display":"\\mathrm{transfer}","kind":"use","excerpt":"The four audit quantities. For a proposed reduction, define the schematic complexity record C_n= ( N_packets, D_internal, S_transfer, L_leakage ). Here:","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"use","excerpt":"Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1296a5898d2e7d5f","tex":"U^*","aliases":"U^*","display":"U^*","kind":"use","excerpt":"If a unitary U and acceptance operator M are permutation invariant, then U^*MU is invariant. Therefore the final probability depends on the input only through its twirl. This permits an observable-specific collapse even when the original pure input is not itself symmetric.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ed2a9915e2620c3c","tex":"U^\\dagger","aliases":"U^† dagger","display":"U^\\dagger","kind":"use","excerpt":"Symmetry-preserving unitary dynamics. If U commutes with every packet projector, then P_α Uρ U^† P_β = U(P_αρ P_β)U^†. The left and right packet labels are preserved. Populations and coherences evolve independently by their declared labels.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"At q=1, U_1={0} because gcd(0,1)=1. The phase sum and the divisor formula both give c_1(n)=1. This boundary case is included in the proofs even though the inherited arithmetic regression range begins at modulus two.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a3ce806ceefcfc1b","tex":"U_{12}","aliases":"U_12","display":"U_{12}","kind":"use","excerpt":"U_12={1,5,7,11}, φ(12)=4, ζ=e^π i/6=fracsqrt32+frac i2.","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b7ee86b45f7cbd54","tex":"\\U_{2m}","aliases":"U_2m U","display":null,"kind":"use","excerpt":"For A∈U_m(F), define Z(A) = A · 0 A · A ∈U_2m(F).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6fa4afab67a8d383","tex":"U_6","aliases":"U_6","display":"U_6","kind":"use","excerpt":"Core binary six-dimensional census. For U_6(F_2) there are 2^15=32,768 targets. The complete comparison is: @rrr@ Restricted distance · Ambient distance · Number of targets","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f8bde68cb288a80d","tex":"\\U_6","aliases":"U_6 U","display":null,"kind":"use","excerpt":"[Universal catalytic atom over rings] For every nonzero commutative ring R with identity, let C= 1 · 0 · 0 · 0 · 0 · 0 1 · 1 · 0 · 0 · 0 · 0 0 · 0 · 1 · 0 · 0 · 0 0 · 1 · 1 · 1 · 0 · 0 1 · 1 · 1 · 0 · 1 · 0 1 · 1 · 1 · 0 · 0 · 1 ∈U_6(R). Then ellDown(C)=7, ellAll(C)=6.","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9f091aaabc0f17cd","tex":"U_D","aliases":"U_D","display":"U_D","kind":"use","excerpt":"Template XIII: ancilla restoration. Let W act on data D and ancilla A. To certify a dirty catalyst, prove the operator identity W=U_D⊗ I_A on the full joint space, not only on one ancilla input. For a clean catalyst, prove W(ketψ⊗ket0) =(Uketψ)⊗ket0 for all ketψ. The distinction changes both the target equations and lower bounds.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0b3db69758fee0b4","tex":"U_d","aliases":"U_d","display":"U_d","kind":"use","excerpt":"(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9f5988962b4f6877","tex":"U_g","aliases":"U_g","display":"U_g","kind":"use","excerpt":"Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"[Invariant quantum packets] If a unitary U satisfies [U,Q]=0, then P_iUP_j=0 (i≠ j), and U=⊕_i U_i, U_i=P_iUP_i|_P_iHh. If P_kketψ=ketψ, then P_kUketψ=Uketψ.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"Qubit corollary. For qubits, Schur labels are total spins j. The carrier dimension is u_j=2j+1≤ n+1, and there are at most lfloor n/2rfloor+1 labels. Hence ∑_j(2j+1)^2=O(n^3).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7e82f1d174290b01","tex":"U_{ji}","aliases":"U_ji","display":"U_{ji}","kind":"use","excerpt":"Gate updates. A number-preserving one-body unitary acts on mode creation operators by a_i^† ⟼ ∑_j U_jia_j^†. A generic dense mode transformation can induce many occupation-basis transitions, but for fixed N the basis dimension remains polynomial. Local hopping and phase gates are sparse. Few-body interactions remain polynomial to apply for fixed N,","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-836830fc78b794aa","tex":"u_k","aliases":"u_k","display":"u_k","kind":"use","excerpt":"The arithmetic signal is a coherent amplitude vector. addcontentslinetocsectionThe arithmetic signal is a coherent amplitude vector leadNormalization makes the signal a unit vector without changing its relative phases. Define u_k=1/2 for k∈ U_12 and zero otherwise. Then u =1, and ψ=Fu, ψ_n=fracc_12(n)sqrt48, F^†ψ=u. fig11_same_state_two_bases1.0Sep","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3fd45350d19e2589","tex":"\\U_m","aliases":"U_m U","display":null,"kind":"use","excerpt":"For A∈U_m(F), define Z(A) = A · 0","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8495e8c1fbfb2f8d","tex":"\\U_{n_1+\\cdots+n_k}","aliases":"U_n_1+⋯+n_k U cdots","display":null,"kind":"use","excerpt":"For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossi","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-462d4efdbcdb34a3","tex":"\\U_{n_i}","aliases":"U_n_i U","display":null,"kind":"use","excerpt":"[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6a6f2c5ce2aaa4a4","tex":"U_P","aliases":"U_P","display":"U_P","kind":"use","excerpt":"Pattern groups and architecture constraints. For a poset P, define the pattern group U_P(K) = {I+A:A_ij=0 unless j<_P i}. When the allowed positions are transitively closed, U_P is a group. Compare the intrinsic metric generated by P-allowed transvections with the metric induced by all transvections.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d42ca9e4eff519f2","tex":"U_q","aliases":"U_q","display":"U_q","kind":"use","excerpt":"U_q={a∈ I_q:gcd(a,q)=1}, φ(q)=|U_q|.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-44229cca67dc816c","tex":"U_s","aliases":"U_s","display":"U_s","kind":"use","excerpt":"C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6ae21ddbb7e958a2","tex":"U_t","aliases":"U_t","display":"U_t","kind":"use","excerpt":"C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-16d258d900472e2b","tex":"\\U_{u+v+2}","aliases":"U_u+v+2 U","display":null,"kind":"use","excerpt":"[Connected snapshot family] Let u≥1, let 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<⋯<s_v≤ u. There is a connected target A(u;s_1,…,s_v)∈U_u+v+2(F) with ellDown(A(u;s_1,…,s_v))=u+2v and ellAll(A(u;s_1,…,s_v))≤ u+v+2. Hence its certified defect is at least v-2.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3ec0a98ad4a6fba0","tex":"U_{XY}","aliases":"U_XY","display":"U_{XY}","kind":"use","excerpt":"Number-preserving gates. Single-qubit R_Z gates commute with widehat N. A two-qubit exchange rotation acts in the ordered basis ket00,ket01,ket10,ket11 as U_XY(θ,ϕ) = 1 · 0 · 0 · 0 0 · cosθ · -ie^-iϕsinθ · 0 0 · -ie^iϕsinθ · cosθ · 0 0 · 0 · 0 · 1 . It rotates ket01 and ket10 while fixing ket00 and ket11, so it preserves exact Hamming weight.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cb6e7943a238bc29","tex":"U_\\lambda","aliases":"U_λ lambda","display":"U_\\lambda","kind":"use","excerpt":"A permutation-invariant unitary has the form U = ⊕_λ U_λ⊗ I_ V_λ, and a permutation-invariant measurement operator has the analogous form M = ⊕_λ M_λ⊗ I_ V_λ. The output probability is Tr(MUρ U^*) = ∑_λ p_λ Tr(M_λ U_λρ_λ U_λ^*).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a0b86a7cfbe87473","tex":"\\mathcal U_\\lambda","aliases":"U_λ mathcal lambda","display":"\\mathcal U_\\lambda","kind":"use","excerpt":"Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-42fb4947708961ca","tex":"u_\\lambda","aliases":"u_λ lambda","display":"u_\\lambda","kind":"use","excerpt":"Exact commutant record. Let u_λ=dim U_λ, f_λ=dim V_λ. Schur's lemma gives the following normal form.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6bded57f80fecf4a","tex":"U_\\lambda^*","aliases":"U_λ^* lambda","display":"U_\\lambda^*","kind":"use","excerpt":"A permutation-invariant unitary has the form U = ⊕_λ U_λ⊗ I_ V_λ, and a permutation-invariant measurement operator has the analogous form M = ⊕_λ M_λ⊗ I_ V_λ. The output probability is Tr(MUρ U^*) = ∑_λ p_λ Tr(M_λ U_λρ_λ U_λ^*).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4573c61aa1333f01","tex":"u_\\lambda^2","aliases":"u_λ^2 lambda","display":"u_\\lambda^2","kind":"use","excerpt":"umber of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5eada3bebae789bc","tex":"U_\\pi","aliases":"U_π pi","display":"U_\\pi","kind":"use","excerpt":"The permutation twirl. Define the twirling channel T_n(X) = frac1n! ∑_π∈ S_n U_π XU_π^*. Its image is the commutant of the permutation action.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d512c1190d6e9e75","tex":"U_\\pi^*","aliases":"U_π^* pi","display":"U_\\pi^*","kind":"use","excerpt":"The permutation twirl. Define the twirling channel T_n(X) = frac1n! ∑_π∈ S_n U_π XU_π^*. Its image is the commutant of the permutation action.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-370b50560dbc1db1","tex":"U_\\Psi","aliases":"U_Ψ Psi","display":"U_\\Psi","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fa632165f7ea3097","tex":"\\underline{\\operatorname{rk}}","aliases":"underlinerk underline operatorname","display":"\\underline{\\operatorname{rk}}","kind":"use","excerpt":"Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but t","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-758d511aeb7329e7","tex":"\\underline{\\operatorname{rk}}_{X}","aliases":"underlinerk_X underline operatorname","display":"\\underline{\\operatorname{rk}}_{X}","kind":"use","excerpt":"Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but t","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d7b46859277c8004","tex":"\\underline\\ell_X","aliases":"underlineℓ_X underline ell","display":"\\underline\\ell_X","kind":"use","excerpt":"For an algebraic generating set X, define the border product length underlineℓ_X(A) = min{L:A∈overlineX^L}. This is analogous to the passage from tensor rank to border rank [source reference]. The generic value is still N by [source reference], but exact and border defects can differ on exceptional strata. It is presently open whether the canonical","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"use","excerpt":"Intermediate states as a representable matroid. A linear circuit visits vectors that satisfy exact dependence relations. If V is the set of all row states appearing during a computation, then V determines a representable matroid. A transvection u⟼ u+a v creates a three-point projective line [u], [v], [u+av]. The ambient circuit chooses a tour throu","count":12},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-34ed924ec8641ec0","tex":"v^\\dagger","aliases":"v^† dagger","display":"v^\\dagger","kind":"use","excerpt":"For a normalized vector v, its density matrix is vv^†. Complete dephasing in the sample basis keeps only its diagonal entries. In the present example, dephasing either ψ or χ gives the same diagonal matrix","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-511e66af8de7448d","tex":"v_+","aliases":"v_+","display":"v_+","kind":"use","excerpt":"The two endpoint states are especially simple, but the mechanism is continuous. Define the nonnegative vectors v_+ and v_- by placing |c(n)|/sqrt48 respectively on the positive and negative entries of c, with zero elsewhere. Thus","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-decc309ddf1eace0","tex":"v_-","aliases":"v_-","display":"v_-","kind":"use","excerpt":"The two endpoint states are especially simple, but the mechanism is continuous. Define the nonnegative vectors v_+ and v_- by placing |c(n)|/sqrt48 respectively on the positive and negative entries of c, with zero elsewhere. Thus","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":", assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"[Scalar unital summaries are evaluations] Every unital K-linear multiplicative map Φ:K^W→ K is evaluation at exactly one witness. An idempotent scalar in a field is either zero or one. Orthogonality allows at most one Φ(δ_v) to be one, and preservation of the unit requires exactly one. If it occurs at v_0, linearity gives Φ(g)=Φ(∑_v g(v)δ_v)=g(v_0)","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":", assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"guidepageA degree-one dual certificate for two-input AND Let f(a,b)=ab, with unique positive point (1,1). For degree at most one, the evaluation vector is v_1(a,b)=(1,a,b). The three negative points already span the positive vector.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-47a39d32d4328cbf","tex":"v_d","aliases":"v_d","display":"v_d","kind":"use","excerpt":"Exact evaluation-span certificates. For an integer d≥0, form the evaluation vector v_d(x)=(x^S)_S⊆[n], |S|≤ d, and write U=f^-1(0).","count":13},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-91a52a41351c928c","tex":"V_i","aliases":"V_i","display":"V_i","kind":"use","excerpt":"For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among fr","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"The least safe degree exists. For a positive v, take the point indicator δ_v(x)=∏_i=1^n x_i, · v_i=1, 1+x_i, · v_i=0. It is safe, detects v, and has a degree-n certificate. Hence complete safe coverage exists at degree n for every function. Strong induction proves the existence of a least admissible natural degree. The definition pathrefbeta choose","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1fc9a341ce34ffe6","tex":"V_{i-1}","aliases":"V_i-1","display":"V_{i-1}","kind":"use","excerpt":"For the geometric interpretation in this subsection, assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among fr","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"use","excerpt":"Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fa886686fadc3c64","tex":"V_k","aliases":"V_k","display":"V_k","kind":"use","excerpt":"Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0a435f82d4918621","tex":"v_k","aliases":"v_k","display":"v_k","kind":"use","excerpt":"An explicit moment certificate. On the face b=1, every point except v=(a=0,z=(1,…,1)) is negative for G_k. A polynomial of degree at most k sums to zero over the (k+1)-dimensional cube: each monomial omits some coordinate and contributes an even number of ones. Therefore v_k(v)=∑_u≠ v on the facev_k(u). This is the certificate from Theorem~[source ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0eec75f74f8b5fe0","tex":"V_{k-1}","aliases":"V_k-1","display":"V_{k-1}","kind":"use","excerpt":"Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-43c15d242a46a51d","tex":"V_M","aliases":"V_M","display":"V_M","kind":"use","excerpt":"Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8b86c046f5cd21ee","tex":"V_m","aliases":"V_m","display":"V_m","kind":"use","excerpt":"Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-278526e9690119fe","tex":"V_M^\\ell","aliases":"V_M^ℓ ell","display":"V_M^\\ell","kind":"use","excerpt":"Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-93d342a2e6bb38b4","tex":"V_{n-1}","aliases":"V_n-1","display":"V_{n-1}","kind":"use","excerpt":", assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b8d2185f95f25265","tex":"V_v","aliases":"V_v","display":"V_v","kind":"use","excerpt":"Quiver gate models. Let Q=(Q_0,Q_1) be a quiver. Associate a vector space V_v to every vertex. Internal gates change bases inside V_v, while edge gates transmit low-rank maps along arrows. A target is a representation of the path algebra or an automorphism of a filtered quiver module.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d16dd8144c2ba548","tex":"\\mathcal V_\\lambda","aliases":"V_λ mathcal lambda","display":"\\mathcal V_\\lambda","kind":"use","excerpt":"Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d42e4596a0b89189","tex":"\\vdash","aliases":"vdash","display":null,"kind":"use","excerpt":"Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"pageheadModel and definitionsTHE ASSUMPTIONS ARE PART OF THE THEOREM A general register state is a Boolean function W→Bool on any wire type with decidable equality. C01 does not assume that W is finite. The canonical degree model has a finite wire set with n input wires, a ancillas, and one target. A NOT complements a bit. A CNOT adds one control t","count":55},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e83c58309eb2060b","tex":"\\mathcal W","aliases":"W mathcal","display":"\\mathcal W","kind":"use","excerpt":"A vector-valued defect. For a target U in a restricted phase model, define C_res(U) = { c( W): W is a restricted circuit for U} and similarly C_amb(U). Their Pareto frontiers are denoted Pareto_res(U) and Pareto_amb(U).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"[Shared-variable multiplication fails] For a single witness bit, define KTr_z(A)=A(0)+zA(1). Then in the Boolean w algebra, KTr_z(w)=z, KTr_z(1+w)=1, but KTr_z(w(1+w))=0 ≠ z=KTr_z(w)KTr_z(1+w) whenever z≠0. Also, KTr_z(w^2)=z, KTr_z(w)^2=z^2, which differ for field elements z with z^2≠ z. Evaluate w and 1+w at zero and one. In the Boolean quotient ","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"[Moving-frame collapse] Suppose a polynomial-time uniform circuit admits efficiently described frames W_t for which the transformed gates in [source reference] preserve one fixed polynomial-dimensional packet, the initial state has efficiently computable coordinates in W_0PHh, and their restricted matrices and the transformed final measurement are ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3106fac3e3c8f992","tex":"w_2","aliases":"w_2","display":"w_2","kind":"use","excerpt":"For A=w_1 and B=w_2, τ_z_1(A)=z_1, τ_z_2(B)=z_2, τ_z_1,z_2(AB)=z_1z_2. Tracing both factors over the full two-bit ambient domain introduces unused variable sums: τ_z_1,z_2(A)=z_1(1+z_2), τ_z_1,z_2(B)=z_2(1+z_1). Their product generally differs from the union trace. At z_1=z_2=1, each full ambient factor is zero, while the trace of AB is one. Disjoi","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-35dec78d77d191fb","tex":"w_{\\mathrm{cat}}","aliases":"w_cat mathrm","display":"w_{\\mathrm{cat}}","kind":"use","excerpt":"Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4891d65e523c7c18","tex":"w_i^2","aliases":"w_i^2","display":"w_i^2","kind":"use","excerpt":"and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"If the unique gate is R_t← R_t+aR_s, then aR_s=w. Choose a coordinate j for which w_j is a unit. Since a(R_s)_j=w_j, the factor a is a unit in the commutative ring R. Hence R_s=a^-1w.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a6dc04e23090a2a5","tex":"w_m","aliases":"w_m","display":"w_m","kind":"use","excerpt":"and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/(w_i^2-w_i:1≤ i≤ m).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-04fbfe585bfb125e","tex":"w_s","aliases":"w_s","display":"w_s","kind":"use","excerpt":"Fix integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-232dbeb170b28aa8","tex":"w_{s_1}","aliases":"w_s_1","display":"w_{s_1}","kind":"use","excerpt":"It remains to prove optimality in the lower model. At the root, split {x_1,…,x_u,p} | {q,t_1,…,t_v}. The nonzero rows of the root cross block are w_s_1,…,w_s_v. Distinct prefix vectors are linearly independent: in a linear relation, inspect successively a coordinate occurring in w_s_j but in none of the earlier prefixes, starting from the largest p","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ca7da45f88604a96","tex":"w_{s_j}","aliases":"w_s_j","display":"w_{s_j}","kind":"use","excerpt":"[ 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-842617fdf356a5d3","tex":"w_{s_v}","aliases":"w_s_v","display":"w_{s_v}","kind":"use","excerpt":"It remains to prove optimality in the lower model. At the root, split {x_1,…,x_u,p} | {q,t_1,…,t_v}. The nonzero rows of the root cross block are w_s_1,…,w_s_v. Distinct prefix vectors are linearly independent: in a linear relation, inspect successively a coordinate occurring in w_s_j but in none of the earlier prefixes, starting from the largest p","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ae58458e74e50c6f","tex":"W_t","aliases":"W_t","display":"W_t","kind":"use","excerpt":"Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ce6a04ea561dd517","tex":"W_{t-1}","aliases":"W_t-1","display":"W_{t-1}","kind":"use","excerpt":"Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e6e18cfffab1d167","tex":"W_t^*","aliases":"W_t^*","display":"W_t^*","kind":"use","excerpt":"Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1818278ca4f147b9","tex":"w_u","aliases":"w_u","display":"w_u","kind":"use","excerpt":"integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"widehat a=(16,0,4,0,4,0,16,0,4,0,4,0).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ea83107951d84fac","tex":"\\widehat b","aliases":"widehat b widehat","display":"\\widehat b","kind":"use","excerpt":"Thus widehat b equals twenty-four at zero, eighteen at three and nine, and zero elsewhere. Multiplication by D fixes the spectrum of c and kills the spectrum of b, proving the filter identities. Their supports are disjoint, so Parseval for inner products gives ⟨ c,b⟩=0. QA05 gives c ^2=12·4=48, and","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e7a695b8f5409721","tex":"\\widehat c","aliases":"widehat c widehat","display":"\\widehat c","kind":"use","excerpt":"widehat c=(0,12,0,0,0,12,0,12,0,0,0,12).","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8cab6bcb80e1080d","tex":"\\widehat c_0","aliases":"widehat c_0 widehat","display":"\\widehat c_0","kind":"use","excerpt":"widehat c_0=(0,8,0,8), widehat c_1=(0,4i,0,-4i), widehat c_2=(0,4,0,4).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-be547fad0a087d79","tex":"\\widehat c_1","aliases":"widehat c_1 widehat","display":"\\widehat c_1","kind":"use","excerpt":"widehat c_0=(0,8,0,8), widehat c_1=(0,4i,0,-4i), widehat c_2=(0,4,0,4).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a12d8ebfe6a581b4","tex":"\\widehat c_2","aliases":"widehat c_2 widehat","display":"\\widehat c_2","kind":"use","excerpt":"widehat c_0=(0,8,0,8), widehat c_1=(0,4i,0,-4i), widehat c_2=(0,4,0,4).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4ae2c725a0324b8f","tex":"\\widehat G","aliases":"widehat G widehat","display":"\\widehat G","kind":"use","excerpt":"Approximate dynamic packets. Define the one-step leakage from the scheduled packet by δ_t = (I-P_t)G_tP_t-1 . Let widehat G_t=P_tG_tP_t-1.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e03a626dda42047a","tex":"\\widehat{\\mathcal G}","aliases":"widehat G widehat mathcal","display":"\\widehat{\\mathcal G}","kind":"use","excerpt":"Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint s","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4e3fd513853783c2","tex":"\\widehat G_1","aliases":"widehat G_1 widehat","display":"\\widehat G_1","kind":"use","excerpt":"[Dynamic sum-of-leakages bound] For an initial unit vector in P_0Hh, G_m⋯ G_1ketψ_0 - widehat G_m⋯widehat G_1ketψ_0 ≤ ∑_t=1^mδ_t. For any acceptance operator 0≤ M≤ I, the corresponding acceptance probabilities differ by at most 2∑_t=1^mδ_t.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-24475bfe768b87e8","tex":"\\widehat G_m","aliases":"widehat G_m widehat","display":"\\widehat G_m","kind":"use","excerpt":"[Dynamic sum-of-leakages bound] For an initial unit vector in P_0Hh, G_m⋯ G_1ketψ_0 - widehat G_m⋯widehat G_1ketψ_0 ≤ ∑_t=1^mδ_t. For any acceptance operator 0≤ M≤ I, the corresponding acceptance probabilities differ by at most 2∑_t=1^mδ_t.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ec1657c3123651d2","tex":"\\widehat G_t","aliases":"widehat G_t widehat","display":"\\widehat G_t","kind":"use","excerpt":"Approximate dynamic packets. Define the one-step leakage from the scheduled packet by δ_t = (I-P_t)G_tP_t-1 . Let widehat G_t=P_tG_tP_t-1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dcef84002a724d4c","tex":"\\widehat{\\mathcal G}_\\tau","aliases":"widehat G_τ widehat mathcal tau","display":"\\widehat{\\mathcal G}_\\tau","kind":"use","excerpt":"Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehatμ_k(τ)z^k recovers every endpoint s","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5c577d5457da31ff","tex":"\\widehat H","aliases":"widehat H widehat","display":"\\widehat H","kind":"use","excerpt":"Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehat","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4a6775720ae1e169","tex":"\\widehat N","aliases":"widehat N widehat","display":"\\widehat N","kind":"use","excerpt":"Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e8d6f1354dafb866","tex":"\\widehat x","aliases":"widehat x widehat","display":"\\widehat x","kind":"use","excerpt":"The spectrum recovers the selector. addcontentslinetocsectionThe spectrum recovers the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 an","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b0112acb63a62fcf","tex":"\\widehat{c_{12}}","aliases":"widehatc_12 widehat","display":"\\widehat{c_{12}}","kind":"use","excerpt":"s the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 and exactly zero elsewhere. This is an algebraic identity, not a spectral threshold","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f9cfbc84329efb1c","tex":"\\widehat{c_q}","aliases":"widehatc_q widehat","display":"\\widehat{c_q}","kind":"use","excerpt":"widehatc_q(k)=q 1_k∈ U_q, k∈ I_q.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-79fc3eda55ed8982","tex":"\\widehat{E_rx}","aliases":"widehatE_rx widehat","display":"\\widehat{E_rx}","kind":"use","excerpt":"Phase factors connect the lane spectra. addcontentslinetocsectionPhase factors connect the lane spectra leadEach local Fourier transform contributes to every appropriate global frequency. With four-point negative-sign DFTs in the lanes, widehat x(k)=∑_r=0^2e^-2π ikr/12 widehatE_rx(kbmod4). fig18_polyphase_alias_assembly1.0Left: magnitudes of the th","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-723998d1f368812a","tex":"\\widehat{x}","aliases":"widehatx widehat","display":"\\widehat{x}","kind":"use","excerpt":"widehatx(k)=∑_n=0^q-1x(n)ζ_q^-kn.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b186d96cd6172b72","tex":"\\widehat\\mu","aliases":"widehatμ widehat mu","display":"\\widehat\\mu","kind":"use","excerpt":"Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehat","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6571f4fa27e30341","tex":"\\widehat\\mu_k","aliases":"widehatμ_k widehat mu","display":"\\widehat\\mu_k","kind":"use","excerpt":"Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehat","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-929c170d8af8ade4","tex":"\\widetilde G","aliases":"widetilde G widetilde","display":"\\widetilde G","kind":"use","excerpt":"Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3e18c25679c99456","tex":"\\widetilde G_t","aliases":"widetilde G_t widetilde","display":"\\widetilde G_t","kind":"use","excerpt":"Moving frames and hidden symmetry. The packet schedule has a useful coordinate form. Let P be one fixed projector and let W_t be unitaries with P_t=W_tPW_t^*. In the moving frame, the transformed gate is widetilde G_t =W_t^*G_tW_t-1. The schedule condition is equivalent to widetilde G_tP=Pwidetilde G_tP.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-70fad85f996380ab","tex":"\\widetilde H","aliases":"widetilde H widetilde","display":"\\widetilde H","kind":"use","excerpt":"The quotient matrix is widetilde H_ab = bra O_aHket O_b.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0908e5ac8db9724e","tex":"\\widetilde H_{ab}","aliases":"widetilde H_ab widetilde","display":"\\widetilde H_{ab}","kind":"use","excerpt":"The quotient matrix is widetilde H_ab = bra O_aHket O_b.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f5857cac0ddcea35","tex":"X'","aliases":"X'","display":"X'","kind":"use","excerpt":"[One-cut locality] Let X∈U_n(F) and let X' be obtained from X by one lower-directed transvection. Then |Φ_T(X')-Φ_T(X)|≤1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b3d1765f411a08bb","tex":"X^{(L)}","aliases":"X^(L)","display":"X^{(L)}","kind":"use","excerpt":"[Dimension per gate] Let G be an algebraic group, let V⊆ G be an irreducible subvariety of dimension D, and let X⊆ G be a finite union of locally closed subvarieties, each of dimension at most r, with the identity in X. Write X^(L) = {x_L⋯ x_1:x_1,…,x_L∈ X}. Then dimoverlineV∩ X^(L)≤ rL. In particular, if rL<D, a Zariski-generic point of V has X-pr","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9bd18fad09263723","tex":"X^L","aliases":"X^L","display":"X^L","kind":"use","excerpt":"Word maps and product loci. Let K be a field and let X⊆ G be an algebraic gate family. For a fixed length L, multiplication defines μ_L:X^L→ G, (x_L,…,x_1)↦ x_L⋯ x_1. The exact length-L locus is μ_L(X^L), and the length-at-most-L locus is a finite union of such images when the gate family has finitely many components.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5d4bf85504a30a46","tex":"x^S","aliases":"x^S","display":"x^S","kind":"use","excerpt":"For S⊆[n], write x^S=∏_i∈ Sx_i and let 1_S be the input with support S. A minterm of a monotone function means an inclusion-minimal positive support, not an arbitrary positive input.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b899bd7a7bbae605","tex":"X^{\\times L}","aliases":"X^× L times","display":"X^{\\times L}","kind":"use","excerpt":"Every irreducible component of the Cartesian power X^× L has dimension at most rL. Multiplication G^L→ G is a morphism, so the closure of the image of each component has dimension at most rL. There are finitely many components, and intersecting the resulting finite union with V cannot increase dimension. Since the identity belongs to X, products of","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-efa2945595153d6e","tex":"x^\\dagger","aliases":"x^† dagger","display":"x^\\dagger","kind":"use","excerpt":"F_q^† x ^2 =x^† F_qF_q^† x =x^† x.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4be2f139ee94332e","tex":"X_*","aliases":"X_*","display":"X_*","kind":"use","excerpt":"Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5bf6fa577d25bb9b","tex":"X_*^L","aliases":"X_*^L","display":"X_*^L","kind":"use","excerpt":"Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies ","count":17},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f17c0dba15962adb","tex":"X_{11}","aliases":"X_11","display":"X_{11}","kind":"use","excerpt":"For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossi","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[Exact monomial oracle cost] For k≥2, KTrest(x_1x_2⋯ x_k)=2k-3. The unique positive input of the k-variable monomial needs a safe polynomial of degree k. The complete-cover lower bound gives 2k-3. For the matching construction, successively compute x_1x_2, then x_1x_2x_3, and so on up to a (k-1)-fold product on clean work wires. Toggle the output u","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"[Exact monomial oracle cost] For k≥2, KTrest(x_1x_2⋯ x_k)=2k-3. The unique positive input of the k-variable monomial needs a safe polynomial of degree k. The complete-cover lower bound gives 2k-3. For the matching construction, successively compute x_1x_2, then x_1x_2x_3, and so on up to a (k-1)-fold product on clean work wires. Toggle the output u","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ca7008ae447ee9dc","tex":"x_4","aliases":"x_4","display":"x_4","kind":"use","excerpt":"[Exact cost of a nondegenerate paired quadratic] For even n, let Q(x)=x_1x_2+x_3x_4+⋯+x_n-1x_n. Then M(Q)=KTrest(Q)=n/2. The polar bilinear form of Q is the full-rank alternating form made of n/2 disjoint 2×2 blocks. If Q is affine on an affine subspace with direction space U, its second differences vanish on U, so U is totally isotropic. Nondegene","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-98391915cbb7346b","tex":"x_5","aliases":"x_5","display":"x_5","kind":"use","excerpt":"guidepageQuadratic projection can destroy safety At m=3, variables are ordered by source labels v=1,2,3,4,5,6. Assignment index 13 is read least-significant-bit first: (x_1,x_2,x_3,x_4,x_5,x_6)=(1,0,1,1,0,0). The active labels have syndrome 1+3+4=6, not a=7.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3864fe3cacc7c90d","tex":"x_6","aliases":"x_6","display":"x_6","kind":"use","excerpt":"guidepageQuadratic projection can destroy safety At m=3, variables are ordered by source labels v=1,2,3,4,5,6. Assignment index 13 is read least-significant-bit first: (x_1,x_2,x_3,x_4,x_5,x_6)=(1,0,1,1,0,0). The active labels have syndrome 1+3+4=6, not a=7.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9067dcf296a5b175","tex":"x_a","aliases":"x_a","display":"x_a","kind":"use","excerpt":"Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained i","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0553948e71973b1c","tex":"X_{\\mathrm{amb}}","aliases":"X_amb mathrm","display":"X_{\\mathrm{amb}}","kind":"use","excerpt":"Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but t","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-14c8085d57b478d9","tex":"x_b","aliases":"x_b","display":"x_b","kind":"use","excerpt":"Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained i","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-702e017c46584077","tex":"x_c","aliases":"x_c","display":"x_c","kind":"use","excerpt":"Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-668d82d1fa66516b","tex":"X_\\downarrow","aliases":"X_downarrow downarrow","display":"X_\\downarrow","kind":"use","excerpt":"Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e909423b75101a7c","tex":"X_\\downarrow^L","aliases":"X_downarrow^L downarrow","display":"X_\\downarrow^L","kind":"use","excerpt":"Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"use","excerpt":"Classical reversible gates. For a bit string x∈F_2^n: X_i:x_i← x_i+1, CX_c→ t:x_t← x_t+x_c, TOF_a,b→ t:x_t← x_t+x_ax_b. All other coordinates are unchanged. The first two generate AGL(n,2). On three bits, all three generate the full permutation group S_8. For n≥4 without ancillas they are all even permutations, so the generated group is contained i","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a ","count":17},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0597cb543b16b78f","tex":"x_i^2","aliases":"x_i^2","display":"x_i^2","kind":"use","excerpt":"Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"A cubic on three variables has the form x_1x_2x_3+q(x)+ℓ(x), with q quadratic and ℓ affine. Translating x_i↦ x_i+c_i adjusts independently the coefficients of x_jx_k, so all quadratic terms can be cancelled. The resulting function is a three-variable monomial plus an affine function. Affine input changes and affine output additions cost no Toffolis","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"use","excerpt":"[Exact monomial oracle cost] For k≥2, KTrest(x_1x_2⋯ x_k)=2k-3. The unique positive input of the k-variable monomial needs a safe polynomial of degree k. The complete-cover lower bound gives 2k-3. For the matching construction, successively compute x_1x_2, then x_1x_2x_3, and so on up to a (k-1)-fold product on clean work wires. Toggle the output u","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5802d1845302b6e3","tex":"X_{kk}","aliases":"X_kk","display":"X_{kk}","kind":"use","excerpt":"For the lower bound, view an element of U_n_1+⋯+n_k(F) as a lower block-triangular matrix and project to its diagonal blocks: π(X)=(X_11,…,X_kk). Because diagonal blocks multiply independently in a lower block-triangular product, π is a homomorphism. A lower transvection internal to one block projects to one generator in that factor, while a crossi","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-72c04885d210371b","tex":"x_L","aliases":"x_L","display":"x_L","kind":"use","excerpt":"Word maps and product loci. Let K be a field and let X⊆ G be an algebraic gate family. For a fixed length L, multiplication defines μ_L:X^L→ G, (x_L,…,x_1)↦ x_L⋯ x_1. The exact length-L locus is μ_L(X^L), and the length-at-most-L locus is a finite union of such images when the gate family has finitely many components.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7ba25f8ea2f76a5f","tex":"x_{L-1}","aliases":"x_L-1","display":"x_{L-1}","kind":"use","excerpt":"The addressing map T:x↦(x_0,…,x_L-1) is bijective. Interleaving the lanes reconstructs x, and","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"Scope, notation, and computational model. Boolean and algebraic conventions. A Boolean function is a map f:B^n→B. Unless a different field is explicitly stated, polynomials are the unique multilinear representatives in R_n=F[x_1,…,x_n]/(x_i^2-x_i:1≤ i≤ n). Addition in this ring is XOR. Multiplication is AND. Boolean OR is written lor and satisfies ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6f09ac0a5ce1f254","tex":"x_{n-1}","aliases":"x_n-1","display":"x_{n-1}","kind":"use","excerpt":"[Exact cost of a nondegenerate paired quadratic] For even n, let Q(x)=x_1x_2+x_3x_4+⋯+x_n-1x_n. Then M(Q)=KTrest(Q)=n/2. The polar bilinear form of Q is the full-rank alternating form made of n/2 disjoint 2×2 blocks. If Q is affine on an affine subspace with direction space U, its second differences vanish on U, so U is totally isotropic. Nondegene","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"x_r(m)=x(r+Lm), 0≤ r<L, 0≤ m<M.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9043fc6eac4d4371","tex":"X_{\\mathrm{res}}","aliases":"X_res mathrm","display":"X_{\\mathrm{res}}","kind":"use","excerpt":"Border catalysis. Let X_res and X_amb be algebraic gate varieties. The exact product rank is rk_X(U) = min{L:U∈ X^L}, while border product rank is underlinerk_X(U) = min{L:U∈overlineX^L}. The four possible comparisons are: item exact and border defects both positive, item exact defect positive but border defect zero, item one-copy defect zero but t","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e8786e121e9dc2d7","tex":"x_s","aliases":"x_s","display":"x_s","kind":"use","excerpt":"Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-81873d6b22d5ae0c","tex":"X_{s,s}","aliases":"X_s,s","display":"X_{s,s}","kind":"use","excerpt":"A lower-directed gate preserves lower triangularity. Conversely, if X is lower unitriangular and an upper-directed gate R_t← R_t+aR_s with t<s is applied, then the new (t,s) entry equals X_t,s+aX_s,s=a≠0. The next state is therefore not lower triangular. A circuit whose every prefix has zero strict upper part uses only lower-directed gates.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1029cb8ad51ed272","tex":"x_t","aliases":"x_t","display":"x_t","kind":"use","excerpt":"Qudit SUM gate. On computational basis labels in Z_d, SUM^(a)_s→ t ketx_s,x_t = ketx_s,x_t+a x_sbmod d. Its classical linear action is the transvection I+aE_t,s over Z_d.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7ac71fefeedcc8d1","tex":"X_{t,s}","aliases":"X_t,s","display":"X_{t,s}","kind":"use","excerpt":"A lower-directed gate preserves lower triangularity. Conversely, if X is lower unitriangular and an upper-directed gate R_t← R_t+aR_s with t<s is applied, then the new (t,s) entry equals X_t,s+aX_s,s=a≠0. The next state is therefore not lower triangular. A circuit whose every prefix has zero strict upper part uses only lower-directed gates.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-54ac43616fbfd36c","tex":"x_\\alpha","aliases":"x_α alpha","display":"x_\\alpha","kind":"use","excerpt":"The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bcf7586068a6a313","tex":"x_\\beta","aliases":"x_β beta","display":"x_\\beta","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-568686672cb5f519","tex":"x_{\\beta+k\\alpha}","aliases":"x_β+kα beta alpha","display":"x_{\\beta+k\\alpha}","kind":"use","excerpt":"The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4dae69c87b9fa50a","tex":"x_{\\beta_1}","aliases":"x_β_1 beta","display":"x_{\\beta_1}","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-77aa6d6c1e6eb64b","tex":"x_{\\beta_L}","aliases":"x_β_L beta","display":"x_{\\beta_L}","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"guidepageA dirty scratch wire can be fully restored Consider the four-wire oracle in the Lean examples. First compute ab into scratch s, copy s into the target, and uncompute s: (a,b,s,y)xrightarrowT(a,b;s)(a,b,s+ab,y) xrightarrowC(s;y)(a,b,s+ab,y+s+ab) xrightarrowT(a,b;s)(a,b,s,y+s+ab). The displayed final expression specializes to the required or","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"The program further_generalization_tests.py verifies the universal circuits over Z/4 Z, Z/6 Z, and Z/8 Z, audits the rank-one bandwidth-one excursion, checks the finite-field counting bounds symbolically, and verifies the stable tensor subadditivity identities.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c656dc90ffbe53cd","tex":"\\mathcal Z","aliases":"Z mathcal","display":"\\mathcal Z","kind":"use","excerpt":"For A∈U_m(F), define Z(A) = A · 0 A · A ∈U_2m(F).","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-69ffc71de82ab37e","tex":"z^{10}","aliases":"z^10","display":"z^{10}","kind":"use","excerpt":"For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"[Shared-variable multiplication fails] For a single witness bit, define KTr_z(A)=A(0)+zA(1). Then in the Boolean w algebra, KTr_z(w)=z, KTr_z(1+w)=1, but KTr_z(w(1+w))=0 ≠ z=KTr_z(w)KTr_z(1+w) whenever z≠0. Also, KTr_z(w^2)=z, KTr_z(w)^2=z^2, which differ for field elements z with z^2≠ z. Evaluate w and 1+w at zero and one. In the Boolean quotient ","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-34d4b252430fe95b","tex":"z^4","aliases":"z^4","display":"z^4","kind":"use","excerpt":"bottomrule noteThe finite verifier also evaluates the twelve phase sums in the exact ring Z[z]/(z^4-z^2+1). The plots use floating coordinates for rendering, while the sample identities use exact arithmetic.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6dedd41de3fd2c9b","tex":"z^6","aliases":"z^6","display":"z^6","kind":"use","excerpt":"For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-82a6f2d2c744dfea","tex":"z^7","aliases":"z^7","display":"z^7","kind":"use","excerpt":"For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b5c7ddf77cf12cf9","tex":"z^8","aliases":"z^8","display":"z^8","kind":"use","excerpt":"For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-11c6aa222a0c3d6c","tex":"z^9","aliases":"z^9","display":"z^9","kind":"use","excerpt":"For the canonical target C, exact word counts begin G_downarrow,C(z) · = 543z^7+25080z^8+510703z^9+10824940z^10+⋯, G_*,C(z) · = 2z^6+776z^7+39454z^8+⋯. The metric defect is the valuation drop ord_0 G_downarrow,C - ord_0 G_*,C =1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","excerpt":"[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4e5172da9206de94","tex":"z^L","aliases":"z^L","display":"z^L","kind":"use","excerpt":"E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"use","excerpt":"Let L≥1 and let f(z)=∑_n≥0a_nz^n be a formal power series over a field containing a primitive Lth root ω and 1/L. For 0≤ r<L, define","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-10d0329c35407a66","tex":"z^r","aliases":"z^r","display":"z^r","kind":"use","excerpt":"E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"Calibration families and exact small oracles. claimC03 and C04 An unbalanced disjunction. For k≥2 and n=k+2, define B=z_1⋯ z_k, F_k(a,b,z)=(ab)lor B=ab+B+abB, G_k=F_k+a. This is a deliberately easy family with two differently sized accepting regions. It separates the cost of finding one safe packet from the degree needed for complete coverage.","count":19},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-370c87df70534fa4","tex":"\\mathbb Z_2","aliases":"Z_2 mathbb","display":"\\mathbb Z_2","kind":"use","excerpt":"bibitemg:GongYu2026 S.~Gong and A.~Yu, Explicit matrices over Z_2 with CNOT and row complexity 4n-o(n) and local logic gates, arXiv:2607.28598, 2026.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5e32645afb213c74","tex":"z_2","aliases":"z_2","display":"z_2","kind":"use","excerpt":"guidepageOne safe packet need not cover every accepting point For k=3 consider F_3(a,b,z_1,z_2,z_3)=ab OR z_1z_2z_3. The packet ab is safe and has degree two. It detects every teal point below. It misses three additional positives, which require the other packet z_1z_2z_3.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0702dc64afa5c9d8","tex":"z_3","aliases":"z_3","display":"z_3","kind":"use","excerpt":"guidepageOne safe packet need not cover every accepting point For k=3 consider F_3(a,b,z_1,z_2,z_3)=ab OR z_1z_2z_3. The packet ab is safe and has degree two. It detects every teal point below. It misses three additional positives, which require the other packet z_1z_2z_3.","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-04228e62f1e4bbef","tex":"\\Z_8","aliases":"Z_8 Z","display":null,"kind":"use","excerpt":"Let ω=e^iπ/4. A phase-polynomial circuit on n qubits has the form [source reference] Uketx=ω^P(x)ketAx+b, where A∈GL(n,2), b∈F_2^n, and P:F_2^n→Z_8 is represented as a sum of binary linear forms with coefficients in Z_8. The diagonal case has A=I and b=0.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-64e1ce5490d9a00b","tex":"z_c","aliases":"z_c","display":"z_c","kind":"use","excerpt":"Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-eabd489a99110dec","tex":"\\Z_d^n","aliases":"Z_d^n Z","display":null,"kind":"use","excerpt":"From transvections to quantum gates. Let d≥2. On n qudits with computational basis labeled by Z_d^n, define the scaled SUM gate SUM^(a)_s→ t ketx_0,…,x_n-1 = ketx_0,…,x_t+a x_s,…,x_n-1, a∈Z_d. It is a permutation unitary. Its action on basis labels is the row transvection I+aE_t,s. For d=2 and a=1, it is CNOT. For prime d, these gates lie in the st","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-86298b26f7dd1fb4","tex":"z_i","aliases":"z_i","display":"z_i","kind":"use","excerpt":"Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d529741b0a991b24","tex":"z_i^2","aliases":"z_i^2","display":"z_i^2","kind":"use","excerpt":"fingerprints claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c0af3e0d23ead917","tex":"z_i^{w_i}","aliases":"z_i^w_i","display":"z_i^{w_i}","kind":"use","excerpt":"Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ae33ff0c9a9cfbb4","tex":"z_j","aliases":"z_j","display":"z_j","kind":"use","excerpt":"Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-74e0eb883b673576","tex":"z_k","aliases":"z_k","display":"z_k","kind":"use","excerpt":"Calibration families and exact small oracles. claimC03 and C04 An unbalanced disjunction. For k≥2 and n=k+2, define B=z_1⋯ z_k, F_k(a,b,z)=(ab)lor B=ab+B+abB, G_k=F_k+a. This is a deliberately easy family with two differently sized accepting regions. It separates the cost of finding one safe packet from the degree needed for complete coverage.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-42dba447a7ef210c","tex":"z_m","aliases":"z_m","display":"z_m","kind":"use","excerpt":"Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c6e38018e6d2dbf7","tex":"z_{m-1}","aliases":"z_m-1","display":"z_{m-1}","kind":"use","excerpt":"tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The le","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d8fbdcb982fab731","tex":"z_m^j","aliases":"z_m^j","display":"z_m^j","kind":"use","excerpt":"tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The le","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f3c2576851dd5e4c","tex":"z_{n-1}","aliases":"z_n-1","display":"z_{n-1}","kind":"use","excerpt":"Binary symplectic form. Pauli exponents are ordered as (x_0,…,x_n-1,z_0,…,z_n-1). The symplectic form is J= 0 · I_n","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f0fa5f2a39dbffe4","tex":"Z_q","aliases":"Z_q","display":"Z_q","kind":"use","excerpt":"Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9f2c528ad5400157","tex":"z_{t-1}","aliases":"z_t-1","display":"z_{t-1}","kind":"use","excerpt":"[Exact dynamic-packet theorem] Suppose P_0ketψ_0=ketψ_0 and [source reference] holds for every gate. Put z_0=B_0^*ketψ_0, z_t=M_tz_t-1. Then G_t⋯ G_1ketψ_0=B_tz_t for every t.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9deac6e22bedbcb9","tex":"z_u","aliases":"z_u","display":"z_u","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-43f419d43365fb39","tex":"z_u^u","aliases":"z_u^u","display":"z_u^u","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-53947a836b78402e","tex":"z_v","aliases":"z_v","display":"z_v","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1ec89745a1e0cc3f","tex":"z_v^v","aliases":"z_v^v","display":"z_v^v","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.","count":13},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-806a9d71892a199f","tex":"|_{\\mathcal S_G}","aliases":"|_ S_G mathcal","display":"|_{\\mathcal S_G}","kind":"use","excerpt":"Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c861ba7ebbc3a5b7","tex":"|_{\\mathcal S_H}","aliases":"|_ S_H mathcal","display":"|_{\\mathcal S_H}","kind":"use","excerpt":"Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d48be818bc13aef1","tex":"|_{a=0}","aliases":"|_a=0","display":"|_{a=0}","kind":"use","excerpt":"For equality, take ℓ=a. The degree-three polynomial a(1+b)(1+z_1) is safe for G_k. Suppose a safe h of degree at most two existed. On a=0, acceptance requires B=1. For k≥3, the monotone-ideal argument forces h|_a=0=0, hence h=aL with L affine in the remaining variables. The negative face a=b=1 forces L(1,z)=0, so L=c(1+b). The negative point a=1,b=","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6243cb402ad5c55f","tex":"|_{b=1}","aliases":"|_b=1","display":"|_{b=1}","kind":"use","excerpt":"[Affine optimization of complete coverage] For every k≥2, β^*(F_k)=k+1. Restrict G_k=F_k+a to b=1. Then G_k|_b=1=(1+a)B. This is the indicator of one point on k+1 variables. At the positive input a=0,b=1,z=(1,…,1), a safe detecting polynomial must restrict to that singleton indicator and therefore have degree at least k+1. Thus β(G_k)≥ k+1.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3d7f36918ca613a9","tex":"|_H","aliases":"|_H","display":"|_H","kind":"use","excerpt":"When defects are impossible. A useful positive theorem is a criterion guaranteeing geodesic convexity. [One-Lipschitz retraction criterion] Suppose there is a group homomorphism π:G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambien","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a33bcd012d3f8ff2","tex":"|_{P_i\\Hh}","aliases":"|_P_iHh Hh","display":null,"kind":"use","excerpt":"[Invariant quantum packets] If a unitary U satisfies [U,Q]=0, then P_iUP_j=0 (i≠ j), and U=⊕_i U_i, U_i=P_iUP_i|_P_iHh. If P_kketψ=ketψ, then P_kUketψ=Uketψ.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-47cf7947cf568f1e","tex":"|_{z_1=0}","aliases":"|_z_1=0","display":"|_{z_1=0}","kind":"use","excerpt":"[Affine optimization of the one-packet parameter] For every k≥2, max_ℓ∈Aff_nα_1(F_k+ℓ)=3. First restrict to the face z_1=0. There F_k+ℓ becomes ab+ℓ|_z_1=0, a nonzero quadratic. If the restricted affine term contains a remaining z variable, set a=0 and set the remaining affine form to one. This gives an accepting affine subspace of codimension at m","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"[Regular-matroid core] Every square minor of B belongs to {0,±1}, so B is totally unimodular. Hence the column-dependence matroid is the same over every field. Under the edge labeling c|cccccc state · p · x · x+p · p+q · q · x+p+q","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"guidepageAt m=10, the total-gate accounting points the other way Here n=1,022. The full function uses 17 Toffolis; its quadratic grade needs 511. However, computing and uncomputing the m parities also uses 2m(2^m-1-1)=20·511=10,220 CNOTs. The explicit full construction has 10,237 total gates. The quadratic accumulation uses only 511 total gates.","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"[Exact cost of a nondegenerate paired quadratic] For even n, let Q(x)=x_1x_2+x_3x_4+⋯+x_n-1x_n. Then M(Q)=KTrest(Q)=n/2. The polar bilinear form of Q is the full-rank alternating form made of n/2 disjoint 2×2 blocks. If Q is affine on an affine subspace with direction space U, its second differences vanish on U, so U is totally isotropic. Nondegene","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fc85b00eb1c7e18c","tex":"\\bm\\alpha","aliases":"α bm alpha","display":"\\bm\\alpha","kind":"use","excerpt":"Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential pre","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cd53b89eabc58b71","tex":"\\alpha_s","aliases":"α_s alpha","display":"\\alpha_s","kind":"use","excerpt":"Nested refinements. If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables Q_1,…,Q_s, a joint packet has the form P_α = P^(1)_α_1⋯ P^(s)_α_s. The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential pre","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1bac3c7ce6eb2507","tex":"\\beta_1","aliases":"β_1 beta","display":"\\beta_1","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4c15ec1f2aa21093","tex":"\\beta_{11}","aliases":"β_11 beta","display":"\\beta_{11}","kind":"use","excerpt":"The degree-two polynomial ab does detect it, so β_11(f)=β(f)=2. Since β^*≤ n=2 and β^*≥β, β^*=2. The complete bound gives T≥1, attained by one direct Toffoli to the target.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6cb4d369799123ce","tex":"\\beta_i","aliases":"β_i beta","display":"\\beta_i","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bc2581b62d7df219","tex":"\\beta_L","aliases":"β_L beta","display":"\\beta_L","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"texorpdfstringTT-depth and commuting layers. Two phase gadgets may share a T layer when a suitable Clifford network maps their parities to distinct wires or when ancilla resources permit parallel extraction. Define the conflict graph Γ_P whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the ","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"For n bosonic modes, a Gaussian unitary acts on quadratures r=(q_1,…,q_n,p_1,…,p_n)^T by an affine symplectic transformation r⟼ S r+d, S∈Sp(2n,R). For n fermionic modes, a fermionic Gaussian unitary acts on 2n Majorana operators by γ⟼ Oγ, O∈ SO(2n). Nearest-neighbor matchgates give a circuit presentation of a fermionic Gaussian subgroup on qubits.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9c815a51ca56e21f","tex":"\\Gamma'","aliases":"Γ' Gamma","display":"\\Gamma'","kind":"use","excerpt":"For a target U, define ℓ_Γ(U) = minimum hardware-respecting cost, and compare it with an enlarged graph Γ'. A matched routing excursion has the form π W π^-1, where π is a logical permutation. The excursion is profitable when many gates in W become local in the permuted frame.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d947132b6f8a7176","tex":"\\Gamma_P","aliases":"Γ_P Gamma","display":"\\Gamma_P","kind":"use","excerpt":"texorpdfstringTT-depth and commuting layers. Two phase gadgets may share a T layer when a suitable Clifford network maps their parities to distinct wires or when ancilla resources permit parallel extraction. Define the conflict graph Γ_P whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"The least safe degree exists. For a positive v, take the point indicator δ_v(x)=∏_i=1^n x_i, · v_i=1, 1+x_i, · v_i=0. It is safe, detects v, and has a degree-n certificate. Hence complete safe coverage exists at degree n for every function. Strong induction proves the existence of a least admissible natural degree. The definition pathrefbeta choose","count":39},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6b281769709306f6","tex":"\\Delta^\\lambda","aliases":"Δ^λ Delta lambda","display":"\\Delta^\\lambda","kind":"use","excerpt":"For a weight vector λ∈R_≥0^r, scalarization gives ℓ_ M^λ(U) = inf_x∈ P_ M(U)λ· x. The scalar defect Δ^λ(U) can vary with λ even when the target and gate alphabets are fixed.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0f784679c5c0c208","tex":"\\Delta_{\\mathrm{2q-depth}}","aliases":"Δ_2q-depth Delta mathrm","display":"\\Delta_{\\mathrm{2q-depth}}","kind":"use","excerpt":"Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2455bd5a0b57602a","tex":"\\Delta_{a,\\mathrm{clean}}","aliases":"Δ_a,clean Delta mathrm","display":"\\Delta_{a,\\mathrm{clean}}","kind":"use","excerpt":"Ancillas define a family of defects Δ_a,clean(U), Δ_a,dirty(U), where a is the number of ancillary qubits or modes. Stable ancilla capacity asks whether a fixed number of reusable ancillas can yield an unbounded saving as the data size grows.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c3ff7c9996ec9334","tex":"\\Delta_{a,\\mathrm{dirty}}","aliases":"Δ_a,dirty Delta mathrm","display":"\\Delta_{a,\\mathrm{dirty}}","kind":"use","excerpt":"Ancillas define a family of defects Δ_a,clean(U), Δ_a,dirty(U), where a is the number of ancillary qubits or modes. Stable ancilla capacity asks whether a fixed number of reusable ancillas can yield an unbounded saving as the data size grows.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9437cea347b38d28","tex":"\\delta_{de}","aliases":"δ_de delta","display":"\\delta_{de}","kind":"use","excerpt":"(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0c8a34e3dd176f15","tex":"\\Delta_{\\mathrm{depth}}","aliases":"Δ_depth Delta mathrm","display":"\\Delta_{\\mathrm{depth}}","kind":"use","excerpt":"Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3069cf1843723de2","tex":"\\delta_e","aliases":"δ_e delta","display":"\\delta_e","kind":"use","excerpt":"Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4da675801bf99258","tex":"\\delta_g","aliases":"δ_g delta","display":"\\delta_g","kind":"use","excerpt":"Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-45769bb69d61aef4","tex":"\\delta_h","aliases":"δ_h delta","display":"\\delta_h","kind":"use","excerpt":"Let G be a finite group and S=S^-1 a symmetric generating set. On ℓ^2(G), let λ be the left regular representation, λ(s)δ_h=δ_sh, and define A_S=∑_s∈ Sλ(s). For g∈ G, define the endpoint Green series G_S,g(z) = ⟨ δ_g, (I-zA_S)^-1δ_e ⟩.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2df1a157779fc64c","tex":"\\Delta_{H\\hookrightarrow G}","aliases":"Δ_H↪ G Delta hookrightarrow","display":"\\Delta_{H\\hookrightarrow G}","kind":"use","excerpt":"Thus Δ_H↪ G(h) = intrinsic fiber cost - shortest holonomy loop cost. This is the geometric content of a dirty-workspace shortcut.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"Circuit leakage. For a gate G_j, define δ_j= (I-P)G_jP . Let V_j=PG_jP. A telescoping expansion gives G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Taking norms yields the following.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d9ae4b7f092de1a7","tex":"\\Delta_{\\mathrm{large}}","aliases":"Δ_large Delta mathrm","display":"\\Delta_{\\mathrm{large}}","kind":"use","excerpt":"a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). T","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-51a87ccbd55b1aa4","tex":"\\delta_{rs}","aliases":"δ_rs delta","display":"\\delta_{rs}","kind":"use","excerpt":"R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d536d1106b2df83f","tex":"\\delta_{sh}","aliases":"δ_sh delta","display":"\\delta_{sh}","kind":"use","excerpt":"Let G be a finite group and S=S^-1 a symmetric generating set. On ℓ^2(G), let λ be the left regular representation, λ(s)δ_h=δ_sh, and define A_S=∑_s∈ Sλ(s). For g∈ G, define the endpoint Green series G_S,g(z) = ⟨ δ_g, (I-zA_S)^-1δ_e ⟩.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3b5512cdb2a2af3b","tex":"\\Delta_{\\mathrm{small}}","aliases":"Δ_small Delta mathrm","display":"\\Delta_{\\mathrm{small}}","kind":"use","excerpt":"a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves gate cost, Δ_large(ι U) ≥ Δ_small(U). T","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-63a5ca1c8e8a4e9a","tex":"\\Delta_{\\mathrm{space-time}}","aliases":"Δ_space-time Delta mathrm","display":"\\Delta_{\\mathrm{space-time}}","kind":"use","excerpt":"Depth and parallelism. Gate-count defects do not automatically imply depth defects. Given a gate sequence, define its dependency graph and schedule commuting or disjoint gates in parallel. The appropriate quantities are Δ_depth(U), Δ_2q-depth(U), Δ_space-time(U). A basis change applied transversally to many qubits may have small depth even when its","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4abc637c4be411ae","tex":"\\delta_t","aliases":"δ_t delta","display":"\\delta_t","kind":"use","excerpt":"Approximate dynamic packets. Define the one-step leakage from the scheduled packet by δ_t = (I-P_t)G_tP_t-1 . Let widehat G_t=P_tG_tP_t-1.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b35f037ae1ca06a1","tex":"\\delta_u","aliases":"δ_u delta","display":"\\delta_u","kind":"use","excerpt":"The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c840de7a2da7dc96","tex":"\\delta_v","aliases":"δ_v delta","display":"\\delta_v","kind":"use","excerpt":"The least safe degree exists. For a positive v, take the point indicator δ_v(x)=∏_i=1^n x_i, · v_i=1, 1+x_i, · v_i=0. It is safe, detects v, and has a degree-n certificate. Hence complete safe coverage exists at degree n for every function. Strong induction proves the existence of a least admissible natural degree. The definition pathrefbeta choose","count":14},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-761417b4c6aba833","tex":"\\delta_v^2","aliases":"δ_v^2 delta","display":"\\delta_v^2","kind":"use","excerpt":"The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5c105ddb18df42d7","tex":"\\Delta_w","aliases":"Δ_w Delta","display":"\\Delta_w","kind":"use","excerpt":"Scalarization by positive weights w gives Δ_w(U)= min_c∈ C_res(U)w· c - min_c∈ C_amb(U)w· c. Different w can change the magnitude of the defect and whether it is zero or positive. It remains nonnegative because every restricted realization is also an ambient realization with the same resource vector.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f83eb0c8a84cbdce","tex":"\\delta_w","aliases":"δ_w delta","display":"\\delta_w","kind":"use","excerpt":"If v and w are distinct positive inputs, δ_v+δ_w is safe and equals one at v. The degree-n coefficients of its two summands are both one, so they cancel. Its degree is at most n-1. Do this for each positive v. The last assertion applies the general bound to every affine adjustment, with the stated zero convention.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0fcb1c3cf5e3bf16","tex":"\\delta_{\\alpha\\beta}","aliases":"δ_αβ delta alpha beta","display":"\\delta_{\\alpha\\beta}","kind":"use","excerpt":"Orthogonal packet decompositions. Let Hh be a finite-dimensional Hilbert space and let {P_α}_α∈ A be orthogonal projectors satisfying P_α P_β=δ_αβP_α, ∑_α∈ AP_α=I. Every state and operator decompose as ketψ=∑_α P_αketψ, X=∑_α,βP_α X P_β. The diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8bdd397b1020fafd","tex":"\\delta_{\\chi\\psi}","aliases":"δ_χψ delta chi psi","display":"\\delta_{\\chi\\psi}","kind":"use","excerpt":"Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3bbbf66ed7bfacbd","tex":"\\varepsilon^3","aliases":"ε^3 varepsilon","display":"\\varepsilon^3","kind":"use","excerpt":"Commutator catalysis. An ambient commutator creates a new direction: [g,h]=ghg^-1h^-1. For continuous controls, e^ε Xe^ε Ye^-ε Xe^-ε Y = e^ε^2[X,Y]+O(ε^3). The desired effective generator may lie in the restricted algebra even though the path does not.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"The phase ζ=(0,1/2,1/2,0) has complex meaning sqrt3/2+i/2. Exact finite certificates establish its twelfth power, inverse, and absence of a smaller positive power below twelve. Later the convention module identifies it with exp(π i/6).","count":33},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d8970cd3190ee797","tex":"\\zeta^{(a-k)n}","aliases":"ζ^(a-k)n zeta","display":"\\zeta^{(a-k)n}","kind":"use","excerpt":"s the selector leadThe Fourier transform returns exactly the four labels selected by coprimality. Use the negative-sign, unnormalized discrete Fourier transform widehat x(k)=∑_n=0^11x(n)e^-2π ikn/12. fig04_exact_spectrum1.0The coefficient is exactly 12 at k=1,5,7,11 and exactly zero elsewhere. This is an algebraic identity, not a spectral threshold","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a1810dd5bbc6aace","tex":"\\zeta^{-1}","aliases":"ζ^-1 zeta","display":"\\zeta^{-1}","kind":"use","excerpt":"widehat c(1)=8+4iζ^-1+4ζ^-2=12.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-127d9b43b80b25fa","tex":"\\zeta^{-18}","aliases":"ζ^-18 zeta","display":"\\zeta^{-18}","kind":"use","excerpt":"widehat c(9)=8+4iζ^-9+4ζ^-18=8-4-4=0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9850f4d0232b97b6","tex":"\\zeta^{-2}","aliases":"ζ^-2 zeta","display":"\\zeta^{-2}","kind":"use","excerpt":"widehat c(1)=8+4iζ^-1+4ζ^-2=12.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3cca9850830d1525","tex":"\\zeta^{-9}","aliases":"ζ^-9 zeta","display":"\\zeta^{-9}","kind":"use","excerpt":"widehat c(9)=8+4iζ^-9+4ζ^-18=8-4-4=0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1a2126a079277e24","tex":"\\zeta^{11n}","aliases":"ζ^11n zeta","display":"\\zeta^{11n}","kind":"use","excerpt":"c(n)=ζ^n+ζ^5n+ζ^7n+ζ^11n =2cos(π n/6)+2cos(5π n/6).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0c1bf496c0496149","tex":"\\zeta^{3n}","aliases":"ζ^3n zeta","display":"\\zeta^{3n}","kind":"use","excerpt":"3cos(π n/2)=frac32(ζ^3n+ζ^9n).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4e594232609bbaa3","tex":"\\zeta^{5n}","aliases":"ζ^5n zeta","display":"\\zeta^{5n}","kind":"use","excerpt":"c(n)=ζ^n+ζ^5n+ζ^7n+ζ^11n =2cos(π n/6)+2cos(5π n/6).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7185f593ef543339","tex":"\\zeta^{7n}","aliases":"ζ^7n zeta","display":"\\zeta^{7n}","kind":"use","excerpt":"c(n)=ζ^n+ζ^5n+ζ^7n+ζ^11n =2cos(π n/6)+2cos(5π n/6).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-01467fbdfed4b463","tex":"\\zeta^{9n}","aliases":"ζ^9n zeta","display":"\\zeta^{9n}","kind":"use","excerpt":"3cos(π n/2)=frac32(ζ^3n+ζ^9n).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5897adbee959fca1","tex":"\\zeta^a","aliases":"ζ^a zeta","display":"\\zeta^a","kind":"use","excerpt":"Coprimality selects primitive roots. addcontentslinetocsectionCoprimality selects primitive roots leadA selected residue is a label for a wave whose phase has exact order twelve. Set ζ=e^2π i/12. The phase ζ^a first returns to 1 after 12/gcd(a,12) powers. Thus the coprime labels 1,5,7,11 select exactly the primitive twelfth roots. fig01_primitive_r","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1341203ef6755f41","tex":"\\zeta^{an}","aliases":"ζ^an zeta","display":"\\zeta^{an}","kind":"use","excerpt":"The selected phases add and cancel. addcontentslinetocsectionThe selected phases add and cancel leadThe same four waves can reinforce, cancel, or reinforce with the opposite sign. At each sample n, add the four numbers ζ^an with a∈ U_12. Each has magnitude one. Their sum need not have magnitude one. fig02_phase_addition1.0At n=0 the sum is 4. At n=","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6d3331bda5586731","tex":"\\zeta^n","aliases":"ζ^n zeta","display":"\\zeta^n","kind":"use","excerpt":"c(n)=ζ^n+ζ^5n+ζ^7n+ζ^11n =2cos(π n/6)+2cos(5π n/6).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9918735dac797be3","tex":"\\zeta^{na}","aliases":"ζ^na zeta","display":"\\zeta^{na}","kind":"use","excerpt":"frac12sqrt12∑_a∈ U_12ζ^na =fracc(n)sqrt48.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b4497751a0adc1f3","tex":"\\zeta_d^{b(n-m)}","aliases":"ζ_d^b(n-m) zeta","display":"\\zeta_d^{b(n-m)}","kind":"use","excerpt":"(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3ac5aed7cba8ac35","tex":"\\zeta_M^{(k-\\ell)m}","aliases":"ζ_M^(k-ℓ)m zeta ell","display":"\\zeta_M^{(k-\\ell)m}","kind":"use","excerpt":"widehatx_r(ℓ) · =∑_m=0^M-1x(r+Lm)ζ_M^-ℓ m · =frac1N∑_k=0^N-1widehatx(k)ζ_N^kr ∑_m=0^M-1ζ_M^(k-ℓ)m.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9a2c752b909f2437","tex":"\\zeta_M^{-km}","aliases":"ζ_M^-km zeta","display":"\\zeta_M^{-km}","kind":"use","excerpt":"widehatx(k) · =∑_r=0^L-1∑_m=0^M-1 x_r(m)ζ_N^-k(r+Lm) · =∑_rζ_N^-kr∑_mx_r(m)ζ_M^-km.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a12f683b30977209","tex":"\\zeta_M^{-\\ell m}","aliases":"ζ_M^-ℓ m zeta ell","display":"\\zeta_M^{-\\ell m}","kind":"use","excerpt":"widehatx_r(ℓ) · =∑_m=0^M-1x(r+Lm)ζ_M^-ℓ m · =frac1N∑_k=0^N-1widehatx(k)ζ_N^kr ∑_m=0^M-1ζ_M^(k-ℓ)m.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d2b6ee6a5c1dbac2","tex":"\\zeta_N^{-a}","aliases":"ζ_N^-a zeta","display":"\\zeta_N^{-a}","kind":"use","excerpt":"Let A be the cyclic shift (Ax)(n)=x(n-1). A Fourier column f_a(n)=N^-1/2ζ_N^na satisfies Af_a=ζ_N^-af_a. Therefore","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6ff715006f04845d","tex":"\\zeta_N^{-k(r+Lm)}","aliases":"ζ_N^-k(r+Lm) zeta","display":"\\zeta_N^{-k(r+Lm)}","kind":"use","excerpt":"widehatx(k) · =∑_r=0^L-1∑_m=0^M-1 x_r(m)ζ_N^-k(r+Lm) · =∑_rζ_N^-kr∑_mx_r(m)ζ_M^-km.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-33845af6e74ead05","tex":"\\zeta_N^{-kr}","aliases":"ζ_N^-kr zeta","display":"\\zeta_N^{-kr}","kind":"use","excerpt":"widehatx(k) =∑_r=0^L-1ζ_N^-krwidehatx_r(kbmod M).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-693d85c1e16c4a0b","tex":"\\zeta_N^{a(n-m)}","aliases":"ζ_N^a(n-m) zeta","display":"\\zeta_N^{a(n-m)}","kind":"use","excerpt":"(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-065ecc0ecf3ed319","tex":"\\zeta_N^{a\\ell}","aliases":"ζ_N^aℓ zeta ell","display":"\\zeta_N^{a\\ell}","kind":"use","excerpt":"E_a=frac1N∑_ℓ=0^N-1ζ_N^aℓA^ℓ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9e4f3f44008b6e40","tex":"\\zeta_N^{kr}","aliases":"ζ_N^kr zeta","display":"\\zeta_N^{kr}","kind":"use","excerpt":"widehatx_r(ℓ) · =∑_m=0^M-1x(r+Lm)ζ_M^-ℓ m · =frac1N∑_k=0^N-1widehatx(k)ζ_N^kr ∑_m=0^M-1ζ_M^(k-ℓ)m.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c01a289b0f3b189f","tex":"\\zeta_N^{na}","aliases":"ζ_N^na zeta","display":"\\zeta_N^{na}","kind":"use","excerpt":"Let A be the cyclic shift (Ax)(n)=x(n-1). A Fourier column f_a(n)=N^-1/2ζ_N^na satisfies Af_a=ζ_N^-af_a. Therefore","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-01eafec7a00c20ee","tex":"\\zeta_N^{r(\\ell+jM)}","aliases":"ζ_N^r(ℓ+jM) zeta ell","display":"\\zeta_N^{r(\\ell+jM)}","kind":"use","excerpt":"widehatx_r(ℓ) =frac1L∑_j=0^L-1 ζ_N^r(ℓ+jM)widehatx(ℓ+jM).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-06a4aa853ca66102","tex":"\\zeta_N^{r\\ell}","aliases":"ζ_N^rℓ zeta ell","display":"\\zeta_N^{r\\ell}","kind":"use","excerpt":"Every lane transform combines an alias class of L full-spectrum coefficients. One decimated lane generally cannot distinguish those coefficients. All L offsets together retain the original signal. For a fixed alias class ℓ, the equations across r are an invertible Fourier mixing system after the nonzero phases ζ_N^rℓ are removed. The apparent colla","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f362658753c0c7bf","tex":"\\zeta_q","aliases":"ζ_q zeta","display":"\\zeta_q","kind":"use","excerpt":"ζ_q=e^2π i/q.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-da6c78158199d6c3","tex":"\\zeta_q^{(a-k)n}","aliases":"ζ_q^(a-k)n zeta","display":"\\zeta_q^{(a-k)n}","kind":"use","excerpt":"widehatc_q(k) =∑_a∈ U_q∑_n=0^q-1ζ_q^(a-k)n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e55dc3af76bfc442","tex":"\\zeta_q^{-kn}","aliases":"ζ_q^-kn zeta","display":"\\zeta_q^{-kn}","kind":"use","excerpt":"widehatx(k)=∑_n=0^q-1x(n)ζ_q^-kn.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7b3bf17dd818a6cf","tex":"\\zeta_q^a","aliases":"ζ_q^a zeta","display":"\\zeta_q^a","kind":"use","excerpt":"For a∈ I_q, the multiplicative order of ζ_q^a is","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7541e1841af403eb","tex":"\\zeta_q^{an}","aliases":"ζ_q^an zeta","display":"\\zeta_q^{an}","kind":"use","excerpt":"c_q(n)=∑_a∈ U_qζ_q^an, n∈ Z.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-22ca22169bba828d","tex":"\\zeta_q^h","aliases":"ζ_q^h zeta","display":"\\zeta_q^h","kind":"use","excerpt":"Proof. If qmid h, every summand is one. Otherwise r=ζ_q^h≠1 and r^q=1. Multiplying the sum by 1-r gives 1-r^q=0, so the sum is zero. This argument also covers q=1, where only the first case occurs. square","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8783c6712ac0ff59","tex":"\\zeta_q^{hn}","aliases":"ζ_q^hn zeta","display":"\\zeta_q^{hn}","kind":"use","excerpt":"∑_n=0^q-1ζ_q^hn =q, · qmid h, 0, · qnmid h.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-90a61c77d63274df","tex":"\\zeta_q^{kn}","aliases":"ζ_q^kn zeta","display":"\\zeta_q^{kn}","kind":"use","excerpt":"x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9a66eb3024520f72","tex":"\\zeta_q^{n(b-a)}","aliases":"ζ_q^n(b-a) zeta","display":"\\zeta_q^{n(b-a)}","kind":"use","excerpt":"frac1q∑_n=0^q-1ζ_q^n(b-a).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-199f6e3c0bef300d","tex":"\\zeta_q^{na}","aliases":"ζ_q^na zeta","display":"\\zeta_q^{na}","kind":"use","excerpt":"(F_q)_na=q^-1/2ζ_q^na.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7b93f1be7412facb","tex":"\\eta_1","aliases":"η_1 eta","display":"\\eta_1","kind":"use","excerpt":"0. Using G_1=G_2=G and P_1=P_2=P gives ϕ_2=9/25 0, ψ_2=-7/25 24/25, η_1=4/5, η_2=12/25. The squared state error is 832/625, below (η_1+η_2)^2=1024/625. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2cb721f0ce1499b1","tex":"\\eta_2","aliases":"η_2 eta","display":"\\eta_2","kind":"use","excerpt":"0. Using G_1=G_2=G and P_1=P_2=P gives ϕ_2=9/25 0, ψ_2=-7/25 24/25, η_1=4/5, η_2=12/25. The squared state error is 832/625, below (η_1+η_2)^2=1024/625. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-12235ebd9b8ec538","tex":"\\eta_t","aliases":"η_t eta","display":"\\eta_t","kind":"use","excerpt":"[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"L3 · Maximum unitriangular defect is Θ(nlog n). · Open · Matching upper bound or larger interacting-catalyst family.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"visualfigureoverlap_gf16A concrete field point z=θ with θ≠0,1. The two columns are unequal in both rows.","count":30},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","excerpt":"The field is F_16=F_2[θ]/(θ^4+θ+1). The checker stores field elements as four-bit coefficient vectors: θ is integer 2 and θ^2 is integer 4. These integers encode field elements; they are not ordinary integer multiplication. Exact field multiplication gives the reported mismatch.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c112a1cdbb689680","tex":"\\theta^4","aliases":"θ^4 theta","display":"\\theta^4","kind":"use","excerpt":"The field is F_16=F_2[θ]/(θ^4+θ+1). The checker stores field elements as four-bit coefficient vectors: θ is integer 2 and θ^2 is integer 4. These integers encode field elements; they are not ordinary integer multiplication. Exact field multiplication gives the reported mismatch.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c8082d88b8040556","tex":"\\theta_j","aliases":"θ_j theta","display":"\\theta_j","kind":"use","excerpt":"The simplest symbolic sandwich is C(∏_j=1^m R_P_j(θ_j))C^† = ∏_j=1^m R_CP_jC^†(θ_j). If the P_j commute, their conjugates commute. A catalyst can therefore change the geometry of the support without changing the algebra of the angles.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"Bosonic symplectic embedding. For A∈GL(n,R), ι(A)=diag(A,A^-T)∈Sp(2n,R). This convention orders quadratures as (q_1,…,q_n,p_1,…,p_n).","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7238fc50817cce55","tex":"\\lambda_0","aliases":"λ_0 lambda","display":"\\lambda_0","kind":"use","excerpt":"This makes phase-polynomial synthesis a close relative of the snapshot families in the linear core. A parity row may evolve through λ_0,λ_1,…,λ_m, with selected states consumed by phase gates. The difference is that consumption need not copy the parity to another row; it can deposit phase while leaving the carrier unchanged.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"This makes phase-polynomial synthesis a close relative of the snapshot families in the linear core. A parity row may evolve through λ_0,λ_1,…,λ_m, with selected states consumed by phase gates. The difference is that consumption need not copy the parity to another row; it can deposit phase while leaving the carrier unchanged.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"use","excerpt":"Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"use","excerpt":"Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-129bdd0acbdea966","tex":"\\lambda_m","aliases":"λ_m lambda","display":"\\lambda_m","kind":"use","excerpt":"This makes phase-polynomial synthesis a close relative of the snapshot families in the linear core. A parity row may evolve through λ_0,λ_1,…,λ_m, with selected states consumed by phase gates. The difference is that consumption need not copy the parity to another row; it can deposit phase while leaving the carrier unchanged.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehat","count":23},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-708a6fe1c255bc51","tex":"\\mu_k","aliases":"μ_k mu","display":"\\mu_k","kind":"use","excerpt":"Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehat","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-247bec05dfd02479","tex":"\\mu_L","aliases":"μ_L mu","display":"\\mu_L","kind":"use","excerpt":"Word maps and product loci. Let K be a field and let X⊆ G be an algebraic gate family. For a fixed length L, multiplication defines μ_L:X^L→ G, (x_L,…,x_1)↦ x_L⋯ x_1. The exact length-L locus is μ_L(X^L), and the length-at-most-L locus is a finite union of such images when the gate family has finitely many components.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"When defects are impossible. A useful positive theorem is a criterion guaranteeing geodesic convexity. [One-Lipschitz retraction criterion] Suppose there is a group homomorphism π:G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambien","count":45},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6a331de74ff475c9","tex":"\\pi^{-1}","aliases":"π^-1 pi","display":"\\pi^{-1}","kind":"use","excerpt":"Architecture and routing. On a line or grid, Gaussian and matchgate circuits may be dominated by routing. Let G_conn be the hardware graph. A mode permutation π can be treated as a catalyst if it is opened, many interactions are executed in the reordered frame, and π^-1 restores the logical labeling. The benefit is routing saved across consumers - ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-222fb49e971fef73","tex":"\\pi_1","aliases":"π_1 pi","display":"\\pi_1","kind":"use","excerpt":"Direct sums and monoidal amplification. Suppose targets admit a direct sum U⊕ V and a homomorphism from the large restricted group onto the two factor groups sends U⊕ V to (U,V). Require the aggregate generator bound ellRes(π_1(s))+ellRes(π_2(s))≤ c(s) for every large restricted generator s. Assume that both restricted and ambient factor circuits e","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a539d0a4a394483f","tex":"\\pi_2","aliases":"π_2 pi","display":"\\pi_2","kind":"use","excerpt":"Direct sums and monoidal amplification. Suppose targets admit a direct sum U⊕ V and a homomorphism from the large restricted group onto the two factor groups sends U⊕ V to (U,V). Require the aggregate generator bound ellRes(π_1(s))+ellRes(π_2(s))≤ c(s) for every large restricted generator s. Assume that both restricted and ambient factor circuits e","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6f490323eba8fe3b","tex":"\\Pi_A","aliases":"Π_A Pi","display":"\\Pi_A","kind":"use","excerpt":"Error accounting for truncated channels. Let Π_A project operator space onto an active set of coherence blocks. A reduced evolution uses L_A = Π_A LΠ_A. The discarded transfer is (I-Π_A) LΠ_A. Duhamel's formula yields an error bound analogous to the Hamiltonian leakage theorem. The resulting estimate depends on the induced norm of the discarded sup","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-212c11c326b42e7c","tex":"\\Pi_\\lambda","aliases":"Π_λ Pi lambda","display":"\\Pi_\\lambda","kind":"use","excerpt":"Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-13f955c78f292d4b","tex":"\\rho_a^{\\rm Gauss}","aliases":"ρ_a^rm Gauss rho rm","display":"\\rho_a^{\\rm Gauss}","kind":"use","excerpt":"Low-rank non-Gaussian corrections. One research direction is to represent a weakly non-Gaussian state as ρ = ∑_a=1^R c_aρ_a^rm Gauss with small Gaussian rank R, or as a Gaussian reference plus sparse packet corrections. A polynomial collapse requires R and the update cost to remain polynomial.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-10f1368e507fb58b","tex":"\\rho_{\\mathrm{diag}}","aliases":"ρ_diag rho mathrm","display":"\\rho_{\\mathrm{diag}}","kind":"use","excerpt":"ρ_diag=diag(c(n)^2/48)_n=0^11.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-936d7991f33298f9","tex":"\\rho_i","aliases":"ρ_i rho","display":"\\rho_i","kind":"use","excerpt":"Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-321756df015e4e7f","tex":"\\rho_i^{-1}","aliases":"ρ_i^-1 rho","display":"\\rho_i^{-1}","kind":"use","excerpt":"Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c86a36ff5a44dcd8","tex":"\\rho_j","aliases":"ρ_j rho","display":"\\rho_j","kind":"use","excerpt":"Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bf4294b6f77236f4","tex":"\\rho_j^{-1}","aliases":"ρ_j^-1 rho","display":"\\rho_j^{-1}","kind":"use","excerpt":"Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8ca4d7e240aeeb8a","tex":"\\rho_\\alpha","aliases":"ρ_α rho alpha","display":"\\rho_\\alpha","kind":"use","excerpt":"Projective measurement. For projectors {P_α}, the probability of outcome α is p_α = Tr(P_αρ), and the conditioned state is ρ_α = fracP_αρ P_αp_α when p_α>0. Measurement is therefore packet selection followed by normalization.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fc125fdd0aa28df4","tex":"\\rho_\\lambda","aliases":"ρ_λ rho lambda","display":"\\rho_\\lambda","kind":"use","excerpt":"[Permutation-invariant state normal form] A density operator ρ commutes with every permutation if and only if it can be written as ρ = ⊕_λ p_λ ρ_λ⊗ fracI_ V_λf_λ, where p_λ≥0, ∑_λ p_λ=1, and each ρ_λ is a density operator on U_λ whenever p_λ>0. The weights p_λ and the unnormalized blocks p_λρ_λ are unique. The normalized block ρ_λ is unique only wh","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-afc4088500ea8a3f","tex":"\\sigma_a^{(q)}","aliases":"σ_a^(q) sigma","display":"\\sigma_a^{(q)}","kind":"use","excerpt":"Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decrea","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3fa147d5109a56b1","tex":"\\sigma_z","aliases":"σ_z sigma","display":"\\sigma_z","kind":"use","excerpt":"Collective spin operators. Define J_a=frac12∑_q=1^nσ_a^(q), a∈{x,y,z}. The operators J_a preserve the symmetric subspace. We use the computational convention σ_zket0=ket0 and σ_zket1=-ket1. Thus J_zketD_n^k=(n/2-k)ketD_n^k. For the following occupation formulas define J_+=J_x-iJ_y and J_-=J_x+iJ_y, so that the subscript records increasing or decrea","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-73e64e0e3513f2e3","tex":"\\tau_*","aliases":"τ_* tau","display":"\\tau_*","kind":"use","excerpt":"[Stable tensor length] The limit τ_*(A) = lim_r→∞ fracellAll(A^⊗ r)r m^r-1 = inf_r≥1 fracellAll(A^⊗ r)r m^r-1 exists. If A is tree tight, then the stable catalytic capacity κ_⊗(A) = ellDown(A)-τ_*(A) satisfies κ_⊗(A)≥DeltaDef(A).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6a7d24e503908c92","tex":"\\tau_{1,0}","aliases":"τ_1,0 tau","display":"\\tau_{1,0}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-34ac135663e84b81","tex":"\\tau_{1,2}","aliases":"τ_1,2 tau","display":"\\tau_{1,2}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fa5cfaba17dc6591","tex":"\\tau_{2,1}","aliases":"τ_2,1 tau","display":"\\tau_{2,1}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-fdb2d095b882422d","tex":"\\tau_{3,0}","aliases":"τ_3,0 tau","display":"\\tau_{3,0}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5bdee6934da21f68","tex":"\\tau_{3,1}","aliases":"τ_3,1 tau","display":"\\tau_{3,1}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-feee45b56f2f7cf8","tex":"\\tau_{3,2}","aliases":"τ_3,2 tau","display":"\\tau_{3,2}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cc7cc7429dfe7799","tex":"\\tau_{4,1}","aliases":"τ_4,1 tau","display":"\\tau_{4,1}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b9ed6d9ebb589344","tex":"\\tau_{4,2}","aliases":"τ_4,2 tau","display":"\\tau_{4,2}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-323c5dfd8fc2f020","tex":"\\tau_{5,1}","aliases":"τ_5,1 tau","display":"\\tau_{5,1}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6299e1a598829e60","tex":"\\tau_{5,2}","aliases":"τ_5,2 tau","display":"\\tau_{5,2}","kind":"use","excerpt":"1 · 1 · 1 · 0 · 0 · 1 . One ambient word is τ_1,2(1), τ_3,1(1), τ_1,0(1), τ_4,1(1), τ_5,1(1), τ_1,2(-1). A restricted word of length seven is τ_1,0(1), τ_2,1(1), τ_3,0(-1), τ_3,2(1), τ_4,2(1), τ_5,2(1), τ_2,1(-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cfa3fe1387473802","tex":"\\tau_{\\mathrm{exist}}","aliases":"τ_exist tau mathrm","display":"\\tau_{\\mathrm{exist}}","kind":"use","excerpt":"Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a6b9c16123520021","tex":"\\tau_{\\mathrm{metric}}","aliases":"τ_metric tau mathrm","display":"\\tau_{\\mathrm{metric}}","kind":"use","excerpt":"Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bc7af58eacec64a8","tex":"\\tau_{q}","aliases":"τ_q tau","display":"\\tau_{q}","kind":"use","excerpt":"Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f37d41af5076d525","tex":"\\tau_T","aliases":"τ_T tau","display":"\\tau_T","kind":"use","excerpt":"Tensor and repeated-module questions. For targets U and V, independent synthesis gives ℓ_T(U⊗ V) ≤ ℓ_T(U)+ℓ_T(V) when the systems are disjoint and local Clifford cost is ignored. Joint synthesis may be better because phase identities can couple terms across modules. This motivates a stable regularization τ_T(U)= liminf_r→∞ fracℓ_T(U^⊗ r)r. A one-co","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d439f4e3c6e0a10f","tex":"\\tau_{t,s}","aliases":"τ_t,s tau","display":"\\tau_{t,s}","kind":"use","excerpt":"Row-transvection convention. Matrices act on column coordinate vectors, while row operations left multiply the current matrix. A row transvection τ_t,s(a)=I+aE_t,s performs R_t← R_t+aR_s. In the binary CNOT interpretation, s is the control and t is the target.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1d464e1defd05d81","tex":"\\tau_{t_1,s_1}","aliases":"τ_t_1,s_1 tau","display":"\\tau_{t_1,s_1}","kind":"use","excerpt":"Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3b4e00f833573dcd","tex":"\\tau_{t_m,s_m}","aliases":"τ_t_m,s_m tau","display":"\\tau_{t_m,s_m}","kind":"use","excerpt":"Fix a nonzero commutative ring R with identity and an integer n≥2. Rows and columns are indexed by 0,1,…,n-1. For s≠ t and a∈ R∖{0}, set τ_t,s(a)=I+aE_t,s. Left multiplication by τ_t,s(a) performs R_tlongleftarrow R_t+aR_s. We call s the source and t the target. A circuit word w=(τ_t_m,s_m(a_m),…,τ_t_1,s_1(a_1)) produces [w]=τ_t_m,s_m(a_m)⋯τ_t_1,s_","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7f8e856d3acef855","tex":"\\tau_z","aliases":"τ_z tau","display":"\\tau_z","kind":"use","excerpt":"guidepageOverlap breaks the attempted context-free product rule One shared witness bit already defeats multiplicativity. With τ_z(P)=P(0)+zP(1), τ_z(w)=z, τ_z(1+w)=1, τ_z(w(1+w))=0. The product of the separate summaries is z. Also w^2=w in the Boolean ring, so τ_z(w^2)=z, whereas τ_z(w)^2=z^2.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a56a5e943115ea47","tex":"\\tau_{z_1}","aliases":"τ_z_1 tau","display":"\\tau_{z_1}","kind":"use","excerpt":"For A=w_1 and B=w_2, τ_z_1(A)=z_1, τ_z_2(B)=z_2, τ_z_1,z_2(AB)=z_1z_2. Tracing both factors over the full two-bit ambient domain introduces unused variable sums: τ_z_1,z_2(A)=z_1(1+z_2), τ_z_1,z_2(B)=z_2(1+z_1). Their product generally differs from the union trace. At z_1=z_2=1, each full ambient factor is zero, while the trace of AB is one. Disjoi","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4c76a44c6f6712d6","tex":"\\tau_{z_1,z_2}","aliases":"τ_z_1,z_2 tau","display":"\\tau_{z_1,z_2}","kind":"use","excerpt":"For A=w_1 and B=w_2, τ_z_1(A)=z_1, τ_z_2(B)=z_2, τ_z_1,z_2(AB)=z_1z_2. Tracing both factors over the full two-bit ambient domain introduces unused variable sums: τ_z_1,z_2(A)=z_1(1+z_2), τ_z_1,z_2(B)=z_2(1+z_1). Their product generally differs from the union trace. At z_1=z_2=1, each full ambient factor is zero, while the trace of AB is one. Disjoi","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e611fee4ddbe51e3","tex":"\\tau_{z_2}","aliases":"τ_z_2 tau","display":"\\tau_{z_2}","kind":"use","excerpt":"For A=w_1 and B=w_2, τ_z_1(A)=z_1, τ_z_2(B)=z_2, τ_z_1,z_2(AB)=z_1z_2. Tracing both factors over the full two-bit ambient domain introduces unused variable sums: τ_z_1,z_2(A)=z_1(1+z_2), τ_z_1,z_2(B)=z_2(1+z_1). Their product generally differs from the union trace. At z_1=z_2=1, each full ambient factor is zero, while the trace of AB is one. Disjoi","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"Let U act on the data register. A clean-ancilla realization is a unitary W satisfying W(ketψ⊗ket0) =(Uketψ)⊗ket0 for every ketψ. A dirty-ancilla realization requires W(ketψ⊗ketϕ) =(Uketψ)⊗ketϕ for every ketψ,ketϕ. The second condition is strictly stronger.","count":20},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-28b947267f902868","tex":"\\phi_0","aliases":"ϕ_0 phi","display":"\\phi_0","kind":"use","excerpt":"[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-69c13435fecd680e","tex":"\\phi_2","aliases":"ϕ_2 phi","display":"\\phi_2","kind":"use","excerpt":"0. Using G_1=G_2=G and P_1=P_2=P gives ϕ_2=9/25 0, ψ_2=-7/25 24/25, η_1=4/5, η_2=12/25. The squared state error is 832/625, below (η_1+η_2)^2=1024/625. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c7452277b5b2947b","tex":"\\Phi_m","aliases":"Φ_m Phi","display":"\\Phi_m","kind":"use","excerpt":"Measurement and feedforward. A measurement-assisted realization is described by an instrument with classical outcomes m, conditional channels Φ_m, and feedforward operations. Exact deterministic implementation of a unitary target requires every branch, after correction, to induce the same channel.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-decd81e7725d5b31","tex":"\\phi_m","aliases":"ϕ_m phi","display":"\\phi_m","kind":"use","excerpt":"ate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7016cbae921a9f68","tex":"\\phi_t","aliases":"ϕ_t phi","display":"\\phi_t","kind":"use","excerpt":"[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c87ae920dfc1cb95","tex":"\\phi_{t-1}","aliases":"ϕ_t-1 phi","display":"\\phi_{t-1}","kind":"use","excerpt":"[Adaptive discarded-state certificate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2d55166cd7b4fa9a","tex":"\\Phi_{\\gamma\\delta,\\alpha\\beta}","aliases":"Φ_γδ,αβ Phi gamma delta alpha beta","display":"\\Phi_{\\gamma\\delta,\\alpha\\beta}","kind":"use","excerpt":"Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-17b6bc4f52685e61","tex":"\\Phi_\\sigma","aliases":"Φ_σ Phi sigma","display":"\\Phi_\\sigma","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"gTT-depth and commuting layers Two phase gadgets may share a T layer when a suitable Clifford network maps their parities to distinct wires or when ancilla resources permit parallel extraction. Define the conflict graph Γ_P whose vertices are odd phase terms and whose edges encode inability to realize two terms in one layer under the chosen ancilla","count":59},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c5fb43cac0e529ac","tex":"\\chi_0","aliases":"χ_0 chi","display":"\\chi_0","kind":"use","excerpt":"It is a normalized state with the same original-basis probabilities for every θ. The endpoints are χ_0=χ and χ_π=ψ.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c8918cf9369cdef5","tex":"\\chi_n","aliases":"χ_n chi","display":"\\chi_n","kind":"use","excerpt":"Identical histograms can hide orthogonal states. addcontentslinetocsectionIdentical histograms can hide orthogonal states leadErasing the signs leaves sample probabilities unchanged and changes the state. Let χ_n=|c_12(n)|/sqrt48. Both states are normalized and |ψ_n|^2=|χ_n|^2 for every n, ⟨ψ,χ⟩=0. fig13_same_histogram_distinct_states1.0The top pan","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-04ef75ecd4489dd7","tex":"\\chi_{\\rm internal}","aliases":"χ_rm internal chi rm","display":"\\chi_{\\rm internal}","kind":"use","excerpt":"Entanglement inside a packet. A small number of symmetry labels does not imply low entanglement. The half-filled packet can support volume-law entanglement. Tensor-network bond dimension and packet dimension are independent resources. A robust simulator should therefore track both (D_rm packet,χ_rm internal), where χ_rm internal measures the cost o","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f9093cf6c7bb37fd","tex":"\\chi_\\theta","aliases":"χ_θ chi theta","display":"\\chi_\\theta","kind":"use","excerpt":"χ_θ=v_++e^iθv_-.","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6385a0ef712fbb38","tex":"\\chi_\\pi","aliases":"χ_π chi pi","display":"\\chi_\\pi","kind":"use","excerpt":"It is a normalized state with the same original-basis probabilities for every θ. The endpoints are χ_0=χ and χ_π=ψ.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Let U act on the data register. A clean-ancilla realization is a unitary W satisfying W(ketψ⊗ket0) =(Uketψ)⊗ket0 for every ketψ. A dirty-ancilla realization requires W(ketψ⊗ketϕ) =(Uketψ)⊗ketϕ for every ketψ,ketϕ. The second condition is strictly stronger.","count":86},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c1b88ae641d78c35","tex":"\\psi_0","aliases":"ψ_0 psi","display":"\\psi_0","kind":"use","excerpt":"Gatewise invariance. Let P be an orthogonal projector and suppose the initial state obeys Pketψ_0=ketψ_0. If every gate satisfies G_jP=PG_jP, then G_j maps PHh into itself. For a general linear map, this equation asserts only invariance of the selected subspace. For the finite-dimensional unitary gates considered here, invariance also implies invar","count":11},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bb23e1f1b9335e87","tex":"\\psi_2","aliases":"ψ_2 psi","display":"\\psi_2","kind":"use","excerpt":"0. Using G_1=G_2=G and P_1=P_2=P gives ϕ_2=9/25 0, ψ_2=-7/25 24/25, η_1=4/5, η_2=12/25. The squared state error is 832/625, below (η_1+η_2)^2=1024/625. The rational validation script checks these values exactly. The deliberately large tail illustrates a valid but loose certificate, not an efficient approximation.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d242c66daa18da63","tex":"\\psi_{j-1}","aliases":"ψ_j-1 psi","display":"\\psi_{j-1}","kind":"use","excerpt":"Inductively, if Pketψ_j-1=ketψ_j-1, then G_jketψ_j-1 = G_jPketψ_j-1 = PG_jPketψ_j-1, so the next state remains in PHh. Iterating proves the identity.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-631a8dfac98b7969","tex":"\\psi_k","aliases":"ψ_k psi","display":"\\psi_k","kind":"use","excerpt":"Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ec70e224c49c0da3","tex":"\\psi_m","aliases":"ψ_m psi","display":"\\psi_m","kind":"use","excerpt":"ate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d9e406735fd906a4","tex":"\\psi_n","aliases":"ψ_n psi","display":"\\psi_n","kind":"use","excerpt":"The arithmetic signal is a coherent amplitude vector. addcontentslinetocsectionThe arithmetic signal is a coherent amplitude vector leadNormalization makes the signal a unit vector without changing its relative phases. Define u_k=1/2 for k∈ U_12 and zero otherwise. Then u =1, and ψ=Fu, ψ_n=fracc_12(n)sqrt48, F^†ψ=u. fig11_same_state_two_bases1.0Sep","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-db624dbd058b53b6","tex":"\\psi_t","aliases":"ψ_t psi","display":"\\psi_t","kind":"use","excerpt":"The active dimension is ∑_k=0^Kbinom nk =O(n^K). Let P project onto Hh_≤ K. By the hypothesis on the actual circuit prefixes, Pψ_t=ψ_t at every step. Consequently PG_tPψ_t-1=G_tψ_t-1 on the evolving state, even if the gate does not preserve all of PHh. The restricted update therefore reproduces that trajectory exactly. The hypothesis supplies effic","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6786e5fadf9b4295","tex":"\\psi_{t-1}","aliases":"ψ_t-1 psi","display":"\\psi_{t-1}","kind":"use","excerpt":"The active dimension is ∑_k=0^Kbinom nk =O(n^K). Let P project onto Hh_≤ K. By the hypothesis on the actual circuit prefixes, Pψ_t=ψ_t at every step. Consequently PG_tPψ_t-1=G_tψ_t-1 on the evolving state, even if the gate does not preserve all of PHh. The restricted update therefore reproduces that trajectory exactly. The hypothesis supplies effic","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"Define the catalytic width w_cat= max_t #{j:t∈ I_j}. One may then study D_n^(w) = max_U ( ellRes(U)-ellAmb^(w)(U) ), where the ambient circuit has width at most w. The unitriangular snapshot family gives D_n^(1)=Ω(n) in its corresponding interval model.","count":15},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Let ω=e^iπ/4. A phase-polynomial circuit on n qubits has the form [source reference] Uketx=ω^P(x)ketAx+b, where A∈GL(n,2), b∈F_2^n, and P:F_2^n→Z_8 is represented as a sum of binary linear forms with coefficients in Z_8. The diagonal case has A=I and b=0.","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-386e34e7a2b2e959","tex":"\\omega^{-rj}","aliases":"ω^-rj omega","display":"\\omega^{-rj}","kind":"use","excerpt":"R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a27b633daa590319","tex":"\\omega^j","aliases":"ω^j omega","display":"\\omega^j","kind":"use","excerpt":"R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7df9e0c332c1354e","tex":"\\omega^{j(n-r)}","aliases":"ω^j(n-r) omega","display":"\\omega^{j(n-r)}","kind":"use","excerpt":"fraca_nL∑_j=0^L-1ω^j(n-r).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0e5cf3e86d083b11","tex":"\\omega^{k\\lambda(x)}","aliases":"ω^kλ(x) omega lambda","display":"\\omega^{k\\lambda(x)}","kind":"use","excerpt":"Phase polynomials and parity carriers. A CNOT network computes linear parities λ(x)=a· x∈F_2. Applying T^k to a wire carrying λ(x) contributes ω^kλ(x) to the phase polynomial. A parity network must create each required parity, expose it to a phase gate, and restore the linear map required at the endpoint.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-30386664da55d0e7","tex":"\\omega^{P(x)}","aliases":"ω^P(x) omega","display":"\\omega^{P(x)}","kind":"use","excerpt":"Let ω=e^iπ/4. A phase-polynomial circuit on n qubits has the form [source reference] Uketx=ω^P(x)ketAx+b, where A∈GL(n,2), b∈F_2^n, and P:F_2^n→Z_8 is represented as a sum of binary linear forms with coefficients in Z_8. The diagonal case has A=I and b=0.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f5c088733d7ceedc","tex":"\\omega_M","aliases":"ω_M omega","display":"\\omega_M","kind":"use","excerpt":"Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b4578acf9f0f6fc0","tex":"\\omega_M^{(k-r)\\ell}","aliases":"ω_M^(k-r)ℓ omega ell","display":"\\omega_M^{(k-r)\\ell}","kind":"use","excerpt":"Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6b1427c9539ef3d3","tex":"\\omega_M^{-r\\ell}","aliases":"ω_M^-rℓ omega ell","display":"\\omega_M^{-r\\ell}","kind":"use","excerpt":"Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e9d6d9121cd11235","tex":"\\Omega_n","aliases":"Ω_n Omega","display":"\\Omega_n","kind":"use","excerpt":"[Generic rigidity and defect codimension] Let K be an infinite field and put N=binom n2. There is a nonempty Zariski-open subset Ω_n⊆U_n with Ω_n(K)≠∅ such that ellDown(A)=ellAll(A)=N (A∈Ω_n(K)). Moreover, after base change to an algebraic closure overline K, for D_n,≥ d(overline K) ={A∈U_n(overline K):DeltaDef(A)≥ d}, one has codim_(U_n)_overline ","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"Let G be a finite group and S=S^-1 a symmetric generating set. On ℓ^2(G), let λ be the left regular representation, λ(s)δ_h=δ_sh, and define A_S=∑_s∈ Sλ(s). For g∈ G, define the endpoint Green series G_S,g(z) = ⟨ δ_g, (I-zA_S)^-1δ_e ⟩.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7361e32e32198703","tex":"\\ell_{\\mathfrak M}^{\\lambda}","aliases":"ℓ_ M^λ ell mathfrak lambda","display":"\\ell_{\\mathfrak M}^{\\lambda}","kind":"use","excerpt":"For a weight vector λ∈R_≥0^r, scalarization gives ℓ_ M^λ(U) = inf_x∈ P_ M(U)λ· x. The scalar defect Δ^λ(U) can vary with λ even when the target and gate alphabets are fixed.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2b95471cbd4330f4","tex":"\\ell_G","aliases":"ℓ_G ell","display":"\\ell_G","kind":"use","excerpt":"Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry ","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-491b29670b68e8f4","tex":"\\ell_H","aliases":"ℓ_H ell","display":"\\ell_H","kind":"use","excerpt":"Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry ","count":9},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dc076941cda3895f","tex":"\\ell_{\\mathrm{large,res}}","aliases":"ℓ_large,res ell mathrm","display":"\\ell_{\\mathrm{large,res}}","kind":"use","excerpt":"Template VIII: block retraction. Let a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves ga","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d67af754c5189c6e","tex":"\\ell_{\\mathrm{res},\\varepsilon}","aliases":"ℓ_res,ε ell mathrm varepsilon","display":"\\ell_{\\mathrm{res},\\varepsilon}","kind":"use","excerpt":"Approximate synthesis. For continuous gate sets, define ℓ_res,ε(U) = min{cost(W): W-U ≤ε}. There are two distinct border phenomena: item algebraic border length, where a target lies in the closure of a bounded exact word locus; item analytic approximation length, where an error tolerance is explicitly permitted. They should not be conflated.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bda084b13b1e44a7","tex":"\\ell_{\\mathrm{small,res}}","aliases":"ℓ_small,res ell mathrm","display":"\\ell_{\\mathrm{small,res}}","kind":"use","excerpt":"Template VIII: block retraction. Let a large restricted group contain a smaller one through a homomorphic embedding ι, and suppose a homomorphic retraction π satisfies πι=id. Require that every large restricted generator maps to a small restricted word of no greater cost. Then ℓ_large,res(ι U) ≥ ℓ_small,res(U). If the ambient embedding preserves ga","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8d059dc232277f9e","tex":"\\ell_\\Gamma","aliases":"ℓ_Γ ell Gamma","display":"\\ell_\\Gamma","kind":"use","excerpt":"For a target U, define ℓ_Γ(U) = minimum hardware-respecting cost, and compare it with an enlarged graph Γ'. A matched routing excursion has the form π W π^-1, where π is a logical permutation. The excursion is profitable when many gates in W become local in the permuted frame.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-27fd909ba18fb394","tex":"\\ell_\\Phi","aliases":"ℓ_Φ ell Phi","display":"\\ell_\\Phi","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4e8aebcf3644f2cd","tex":"\\ell_\\Psi","aliases":"ℓ_Ψ ell Psi","display":"\\ell_\\Psi","kind":"use","excerpt":"Root subgroups and Chevalley relations. Let G be a split reductive group [source reference] with root system Φ. Let Ψ⊆Φ be a closed set of allowed roots and let U_Ψ be generated by the root groups x_β(t) for β∈Ψ. Define ℓ_Ψ(u) = min{L:u=x_β_L(t_L)⋯ x_β_1(t_1), β_i∈Ψ}, and let ℓ_Φ permit all roots.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-956bf6a9695a20b1","tex":"\\ell_{\\le q}","aliases":"ℓ_≤ q ell le","display":"\\ell_{\\le q}","kind":"use","excerpt":"Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2d112476c7188527","tex":"\\ell_{\\le0}","aliases":"ℓ_≤0 ell le","display":"\\ell_{\\le0}","kind":"use","excerpt":"Several threshold notions should not be conflated: τ_metric · =min{n:∃ U, 0<Δ(U)<∞}, τ_exist · =min{n:∃ U, ellRes(U)=∞, ellAmb(U)<∞}, τ_q · =min{n:∃ U, ℓ_≤ q(U)<ℓ_≤0(U)}, where ℓ_≤ q permits at most q ambient-only gates. A model may have an early metric threshold and a later existence threshold, or no threshold at all.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"use","excerpt":"gical input wires x, zero-initialized ancillary wires, and a target bit y. A restored oracle is a legal classical circuit satisfying C(x,0^a,y)=(x,0^a,y+f(x)) for every x and both y. The input and scratch restoration conditions hold at the boundary. Wires may change internally, and the target may control an intermediate gate. Each primitive gate is","count":15},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"Clifford generator matrices. For qubit i, the Hadamard swaps x_i and z_i: H_i:e_i↔ e_n+i. The phase gate has z_i← z_i+x_i. For CNOT with control c and target t, x_t← x_t+x_c, z_c← z_c+z_t. For CZ on i,j, z_i← z_i+x_j, z_j← z_j+x_i. SWAP exchanges both x and z coordinates of the two qubits.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"Thus Δ_H↪ G(h) = intrinsic fiber cost - shortest holonomy loop cost. This is the geometric content of a dirty-workspace shortcut.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"Safe detector · A Boolean h with h(x)=1⇒ f(x)=1.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"pageheadWhy sharing does not break the degree boundA STRONGER INVARIANT THAN SINGLE-REGISTER DEGREE For a current state s(x) after t Toffolis, the invariant is ∀ L, Deg (t+|L|, x↦∏_i∈ Ls_i(x)). L may be any finite list, including repeated registers. Initially each coordinate is a variable or a constant, so a product of r coordinates has degree at m","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"pageheadFrom the cut to a numerical lower boundEXISTENCE, BALANCE, AND COMPLETE COVERAGE Write c=pmathbin++q. At this cut, Block 01 defines H_i(x)=run(p)(x,0^a,0)_ix run(q^-1)(x,0^a,0)_i and proves H_i≤ f, f(x)=1iff∃ i H_i(x)=1. Reversal does not change Toffoli count. The new coordinate degree theorem therefore gives deg(H_i)≤max{TC(p),TC(q)}+1.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"guidepageThe finite evaluation-span dual in full Fix a positive v and integer d≥0. Let v_d(x)=(x^S)_S⊆[n], |S|≤ d, U=f^-1(0), N=Span_F_2{v_d(u):u∈ U}. The constant monomial S=∅ is included. Degree-d coefficient covectors form the finite dual of this evaluation space. A covector is safe exactly when it annihilates N.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"If v_d(v)∈ N, every safe covector vanishes at v. If v_d(v)∉ N, extend a basis of N by v_d(v) and then to a basis of the full space. Assign zero to the N basis and one to v_d(v). The resulting covector is a safe detector. Hence β_v(f)>d Longleftrightarrow v_d(v)∈ N. Choose a basis of N from negative evaluation rows. A span representation of the posi","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-cf4624001975fb1d","tex":"\\prod_{1\\le i<j\\le d}","aliases":"∏_1≤ i<j≤ d prod le le","display":"\\prod_{1\\le i<j\\le d}","kind":"use","excerpt":"Polynomial parameter bound. The number of partitions of n with at most d parts is bounded by the number of weak compositions: #{λ:ℓ(λ)≤ d} ≤ binomn+d-1d-1 =O_d(n^d-1). The Weyl dimension formula gives u_λ = ∏_1≤ i<j≤ d fracλ_i-λ_j+j-ij-i =O_d(n^d(d-1)/2). Therefore ∑_λ u_λ^2 =O_d(n^d^2-1).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"guidepageDistinct witness monomials prevent formal cancellation Over a characteristic-two field, define P_R(z)=∑_w∈{0,1}^mR(w)∏_i z_i^w_i. The z_i are formal indeterminates, not Boolean variables. Different witnesses produce different monomials, so P_R=0 exactly when there is no accepting witness.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-446591f1c729b814","tex":"\\prod_{i:v_i=0}","aliases":"∏_i:v_i=0 prod","display":"\\prod_{i:v_i=0}","kind":"use","excerpt":"[Basic size bounds] For nonzero f on n inputs, α_1(f)≤β(f)≤ n. If f has at least two positive inputs, then β(f)≤ n-1. In particular β^*(f)≤ n. Any polynomial detecting a specified positive input is nonzero, giving the first inequality. For v∈B^n, its point indicator is δ_v(x)=∏_i:v_i=1x_i∏_i:v_i=0(1+x_i). If f(v)=1, then δ_v∈ I_f and has degree n, ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1024eebaaacc32f2","tex":"\\prod_{i:v_i=1}","aliases":"∏_i:v_i=1 prod","display":"\\prod_{i:v_i=1}","kind":"use","excerpt":"[Basic size bounds] For nonzero f on n inputs, α_1(f)≤β(f)≤ n. If f has at least two positive inputs, then β(f)≤ n-1. In particular β^*(f)≤ n. Any polynomial detecting a specified positive input is nonzero, giving the first inequality. For v∈B^n, its point indicator is δ_v(x)=∏_i:v_i=1x_i∏_i:v_i=0(1+x_i). If f(v)=1, then δ_v∈ I_f and has degree n, ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0d6d3de154e219e1","tex":"\\prod_{i=1}^m","aliases":"∏_i=1^m prod","display":"\\prod_{i=1}^m","kind":"use","excerpt":"Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f486d8e45d3686cb","tex":"\\prod_{i=1}^{n}","aliases":"∏_i=1^n prod","display":"\\prod_{i=1}^{n}","kind":"use","excerpt":"The least safe degree exists. For a positive v, take the point indicator δ_v(x)=∏_i=1^n x_i, · v_i=1, 1+x_i, · v_i=0. It is safe, detects v, and has a degree-n certificate. Hence complete safe coverage exists at degree n for every function. Strong induction proves the existence of a least admissible natural degree. The definition pathrefbeta choose","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f3161dc88a11fbb0","tex":"\\prod_{i\\in L}","aliases":"∏_i∈ L prod in","display":"\\prod_{i\\in L}","kind":"use","excerpt":"Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a ","count":6},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-62e1f79a28839407","tex":"\\prod_{i\\in R}","aliases":"∏_i∈ R prod in","display":"\\prod_{i\\in R}","kind":"use","excerpt":"Suppose the next gate toggles target u by the product of a control list of length q. A product that omits u is unchanged. If u occurs, idempotence removes all its repetitions. With R the remaining list, the updated product is (s_u∏_i∈ Rs_i) x (∏_j∈controlss_j∏_i∈ Rs_i). The two old-state products have certified degrees at most t+1+|R| and t+q+|R|. ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e126cbffee739a8e","tex":"\\prod_{i\\in S}","aliases":"∏_i∈ S prod in","display":"\\prod_{i\\in S}","kind":"use","excerpt":"For S⊆[n], write x^S=∏_i∈ Sx_i and let 1_S be the input with support S. A minterm of a monotone function means an inclusion-minimal positive support, not an arbitrary positive input.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6f7d5c597bf5d07a","tex":"\\prod_{j=1}^n","aliases":"∏_j=1^n prod","display":"\\prod_{j=1}^n","kind":"use","excerpt":"Exact group sizes. The finite state spaces are |P_n| = 2^n(n+1)/2|GL(n,2)|, and |Sp(2n,2)| = 2^n^2∏_j=1^n(2^2j-1). For n=1,2,3 this gives: @rcc@ n · |P_n| · |Sp(2n,2)|","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-94402a300682c961","tex":"\\prod_{j\\in S}","aliases":"∏_j∈ S prod in","display":"\\prod_{j\\in S}","kind":"use","excerpt":"Algebraic normal form. Every Boolean coordinate function has a unique algebraic normal form f_i(x) = ⊕_S⊆[n]a_i,S∏_j∈ Sx_j. Affine maps have degree at most one. A Toffoli gate targeting t with controls a,b performs x_t⟼ x_t+x_ax_b, which can raise the degree of later coordinates. The final affine target requires every higher-degree coefficient to c","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e1ed43159793540b","tex":"\\prod_{j\\in\\mathrm{controls}}","aliases":"∏_j∈controls prod in mathrm","display":"\\prod_{j\\in\\mathrm{controls}}","kind":"use","excerpt":"Suppose the next gate toggles target u by the product of a control list of length q. A product that omits u is unchanged. If u occurs, idempotence removes all its repetitions. With R the remaining list, the updated product is (s_u∏_i∈ Rs_i) x (∏_j∈controlss_j∏_i∈ Rs_i). The two old-state products have certified degrees at most t+1+|R| and t+q+|R|. ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4a38df6498519a9e","tex":"\\prod_{k>0}","aliases":"∏_k>0 prod","display":"\\prod_{k>0}","kind":"use","excerpt":"The Chevalley conjugation formula has the schematic form x_α(a)x_β(b)x_α(a)^-1 = x_β(b) ∏_k>0 x_β+kα(c_k a^k b). A forbidden root α can therefore create several allowed root effects before being removed.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"Lindblad dynamics. A Markovian open system obeys fracdρdt = L(ρ) = -i[H,ρ] + ∑_a ( L_aρ L_a^† - frac12{L_a^† L_a,ρ} ). Packet selection rules of H and L_a determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d80abc02ff3109ba","tex":"\\sum_{a=1}^R","aliases":"∑_a=1^R sum","display":"\\sum_{a=1}^R","kind":"use","excerpt":"Low-rank non-Gaussian corrections. One research direction is to represent a weakly non-Gaussian state as ρ = ∑_a=1^R c_aρ_a^rm Gauss with small Gaussian rank R, or as a Gaussian reference plus sparse packet corrections. A polynomial collapse requires R and the update cost to remain polynomial.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-44ec4c5e67a66fd7","tex":"\\sum_{a\\in S_d}","aliases":"∑_a∈ S_d sum in","display":"\\sum_{a\\in S_d}","kind":"use","excerpt":"(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bab724d9c8d3bbdf","tex":"\\sum_{a\\in U_{12}}","aliases":"∑_a∈ U_12 sum in","display":"\\sum_{a\\in U_{12}}","kind":"use","excerpt":"frac12sqrt12∑_a∈ U_12ζ^na =fracc(n)sqrt48.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-33a14c19886408f0","tex":"\\sum_{a\\in U_q}","aliases":"∑_a∈ U_q sum in","display":"\\sum_{a\\in U_q}","kind":"use","excerpt":"c_q(n)=∑_a∈ U_qζ_q^an, n∈ Z.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-aaf3d937436f3061","tex":"\\sum_{b=0}^{q/d-1}","aliases":"∑_b=0^q/d-1 sum","display":"\\sum_{b=0}^{q/d-1}","kind":"use","excerpt":"c_q(n) · =∑_dmid qμ(d) ∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q (q/d)mid nμ(d)frac qd.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4382053c4de0a1e5","tex":"\\sum_{b\\in U_d}","aliases":"∑_b∈ U_d sum in","display":"\\sum_{b\\in U_d}","kind":"use","excerpt":"(F_ND_dF_N^†)_nm · =frac1N∑_a∈ S_dζ_N^a(n-m) · =frac1N∑_b∈ U_dζ_d^b(n-m) =frac1N c_d(n-m).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d08f0b8b027f6325","tex":"\\sum_{d=1}^{N}","aliases":"∑_d=1^N sum","display":"\\sum_{d=1}^{N}","kind":"use","excerpt":"The defect condition implies ambient length at most N-d, so the number of such targets is at most (Mq)^N-d. Divide by |U_n(F_q)|=q^N. The expectation bound follows from Exp[DeltaDef]=∑_d=1^NProb[DeltaDef≥ d].","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7111f6c0d5cc1eeb","tex":"\\sum_{d\\mid g}","aliases":"∑_dmid g sum mid","display":null,"kind":"use","excerpt":"∑_dmid gμ(d) =1, · g=1, 0, · g>1.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6ed9ea0a78d4e4c9","tex":"\\sum_{d\\mid N}","aliases":"∑_dmid N sum mid","display":null,"kind":"use","excerpt":"(P_d^(N))^† · =P_d^(N), P_d^(N)P_e^(N) · =δ_deP_d^(N), ∑_dmid NP_d^(N) · =Id_N, rankP_d^(N) · =trP_d^(N)=φ(d).","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-462c540990e16831","tex":"\\sum_{d\\mid q}","aliases":"∑_dmid q sum mid","display":null,"kind":"use","excerpt":"c_q(n) · =∑_dmid qμ(d) ∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q (q/d)mid nμ(d)frac qd.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-106cc05d7ac21b42","tex":"\\sum_{d\\mid12}","aliases":"∑_dmid12 sum mid","display":null,"kind":"use","excerpt":"six order classes leadThe primitive-twelve projector is one packet in a complete orthogonal decomposition. A frequency label k has root order d=12/gcd(k,12). The possible orders are the divisors 1,2,3,4,6,12. fig08_order_partition1.0The number inside each tile is its root order d. The horizontal label is frequency k. Every label belongs to one clas","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a7098fdc2c2e88c9","tex":"\\sum_{d\\mid\\gcd(a,q)}","aliases":"∑_dmidgcd(a,q) sum mid gcd","display":null,"kind":"use","excerpt":"1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d).","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-23dde09cdca24e06","tex":"\\sum_{d\\mid\\gcd(q,n)}","aliases":"∑_dmidgcd(q,n) sum mid gcd","display":null,"kind":"use","excerpt":"c_q(n)=∑_dmidgcd(q,n)d μ(q/d)∈ Z.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-279edb1f840a40a8","tex":"\\sum_{d\\in\\mathcal D}","aliases":"∑_d∈ D sum in mathcal","display":"\\sum_{d\\in\\mathcal D}","kind":"use","excerpt":"P_ D=∑_d∈ DP_d^(N)","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7cf33a36a09fa87d","tex":"\\sum_{e\\mid\\gcd(q,n)}","aliases":"∑_emidgcd(q,n) sum mid gcd","display":null,"kind":"use","excerpt":"ntentslinetocsectionWhy the phase sum is an integer leadA divisor identity certifies integrality without rounding trigonometric values. For q≥1, insert the M obius coprimality identity into the finite phase sum: 1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d). Write a=db after interchanging the finite sums. Character orthogonality evaluates the inner geometric ser","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6d39fe36bb310cf9","tex":"\\sum_{g\\in G}","aliases":"∑_g∈ G sum in","display":"\\sum_{g\\in G}","kind":"use","excerpt":"Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6693c796a8ea30e1","tex":"\\sum_{g\\in W}","aliases":"∑_g∈ W sum in","display":"\\sum_{g\\in W}","kind":"use","excerpt":"For stochastic error rates p_g, a first-order weighted cost is C_err(W)=∑_g∈ W-log(1-p_g). This additive approximation can be inserted directly into Dijkstra search. It does not capture coherent or correlated errors, but it produces a declared and reproducible metric.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e177bb3bef4abaec","tex":"\\sum_{h\\in H}","aliases":"∑_h∈ H sum in","display":"\\sum_{h\\in H}","kind":"use","excerpt":"Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehat","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dd79b3b4cac0915f","tex":"\\sum_{h\\ge1}","aliases":"∑_h≥1 sum ge","display":"\\sum_{h\\ge1}","kind":"use","excerpt":"inproof[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Many consumers and catalytic excess. Suppose the middle word is k=h_m⋯ h_1, h_i∈ H. The ambient word gh_m⋯ h_1g^-1 has cost at most 2c(g)+∑_i c(h_i). If the restricted images gh_ig^-1 require costs r_i and a lower-bound invariant proves that their obligations cannot be shared, then ℓ_H(gkg^-1)≥∑_i r_i. A sufficient inequality is ∑_i(r_i-c(h_i))>2c(","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8524806a0ba8de34","tex":"\\sum_{i<j}","aliases":"∑_i<j sum","display":"\\sum_{i<j}","kind":"use","excerpt":"Transition fusion also collapses. Let ρ_i be the row permutations of a Latin square. The symmetrized aggregate of all pair transitions satisfies ∑_i<j ( P_ρ_iρ_j^-1 + P_ρ_jρ_i^-1 ) =n(J-I). Hence first-order fusion destroys the class information as well.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7b615a361efd7684","tex":"\\sum_{i=1}^k","aliases":"∑_i=1^k sum","display":"\\sum_{i=1}^k","kind":"use","excerpt":"[Block additivity] Let A_i∈U_n_i(F) for 1≤ i≤ k. Then ellDown(diag(A_1,…,A_k)) = ∑_i=1^kellDown(A_i).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7a21c7919adfbd23","tex":"\\sum_{i=1}^m","aliases":"∑_i=1^m sum","display":"\\sum_{i=1}^m","kind":"use","excerpt":"Template II: amortized consumers. Suppose a catalyst costs c_open+c_close and each of m consumers saves at least s_i restricted cost. Then Δ(U) ≥ ∑_i=1^m s_i -c_open-c_close-r, where r is repair or interference cost. A threshold occurs when the right side first becomes positive.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1ff9a92fe833dc54","tex":"\\sum_{i=1}^{s}","aliases":"∑_i=1^s sum","display":"\\sum_{i=1}^{s}","kind":"use","excerpt":"Fix integers u≥1, 1≤ v≤ u+1, and choose distinct phases 0≤ s_1<s_2<⋯<s_v≤ u. Order the coordinates as x_1,…,x_u, p, q, t_1,…,t_v. For 0≤ s≤ u, put w_s=e_p+∑_i=1^se_x_i. Define A(u;s_1,…,s_v)∈U_u+v+2(F) by R_x_i=e_x_i, R_p=w_u, R_q=e_q, and R_t_j=e_t_j+e_q+w_s_j (1≤ j≤ v). The output rows record selected intermediate versions of the carrier p.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":":G→ H with π|_H=id_H such that every ambient generator s∈ S_G satisfies ℓ_H(π(s))≤ c(s). Then ℓ_H(h)≤ℓ_G(h) for every h∈ H, hence the two metrics agree on H. For an ambient word h=s_L⋯ s_1, multiplicativity gives h=π(h)=π(s_L)⋯π(s_1). Subadditivity and the generator bounds give ℓ_H(h)≤∑_j c(s_j). Taking the infimum over ambient words proves the cla","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7858280d9cf575eb","tex":"\\sum_{j<r}","aliases":"∑_j<r sum","display":"\\sum_{j<r}","kind":"use","excerpt":"inproof[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f174ef085cd02c56","tex":"\\sum_{j=0}^L","aliases":"∑_j=0^L sum","display":"\\sum_{j=0}^L","kind":"use","excerpt":"Finite-field sparsity. A word-counting analogue applies over F_q. If the ambient alphabet has at most M(q-1) nontrivial scaled gates, then the number of targets reached in L gates is at most ∑_j=0^L[M(q-1)]^j. When the target space has q^N points and defect d implies ambient length at most N-d, the crude bound is Prob[Δ≥ d] =O_N,M(q^-d). The estima","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c57a71fec0503f9d","tex":"\\sum_{j=0}^{L-1}","aliases":"∑_j=0^L-1 sum","display":"\\sum_{j=0}^{L-1}","kind":"use","excerpt":"R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2927e2a188cda507","tex":"\\sum_{j=0}^{\\min(d,n)}","aliases":"∑_j=0^min(d,n) sum min","display":"\\sum_{j=0}^{\\min(d,n)}","kind":"use","excerpt":"[Finite support of a dual witness] When the right side of [source reference] holds, there is W⊆ U with v_d(v)=∑_u∈ Wv_d(u), |W|≤∑_j=0^min(d,n)binom nj. The cardinality |W| is odd. Choose a basis of the span from negative evaluation vectors and express v_d(v) in that basis. Over F, the nonzero coefficients select a subset W of size at most the ambie","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9098bd3c42e16d5a","tex":"\\sum_{j=0}^r","aliases":"∑_j=0^r sum","display":"\\sum_{j=0}^r","kind":"use","excerpt":"tional algorithmic consequence [Finite-field root bound] If a nonzero polynomial P∈ K[z_1,…,z_m] has total degree at most d, where K is finite, then Pr_z uniform in K^m[P(z)=0]≤frac d|K|. Induct on the number of variables. For one variable this is the elementary root bound. In the inductive step, write P=∑_j=0^r P_j(z_1,…,z_m-1)z_m^j, P_r≠0. The le","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-47306c517d6a1276","tex":"\\sum_{j=0}^{r}","aliases":"∑_j=0^r sum","display":"\\sum_{j=0}^{r}","kind":"use","excerpt":"[Proof of [source reference]] Let m=21 and write n=∑_j=0^rd_jm^j, 0≤ d_j<m. Take the block sum of d_j copies of A_21^⊗ j for every j, with the convention A_21^⊗0=I_1. Block additivity and the tensor estimate give D_n(F) ≥ frac821 ∑_j=0^rj d_j21^j. Put S=∑_j j d_j21^j. Since r=lfloorlog_21nrfloor, r n-S = ∑_j<r(r-j)d_j21^j ≤ 20∑_h≥1h21^r-h ≤ frac21^","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-24dfb51124d8ae91","tex":"\\sum_{j=1}^m","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^m","kind":"use","excerpt":"[Sum-of-leakages bound] For an initial state in PHh, G_m⋯ G_1ketψ_0 - V_m⋯ V_1ketψ_0 ≤ ∑_j=1^mδ_j.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7c520b68dc16ea6b","tex":"\\sum_{k=0}^{K}","aliases":"∑_k=0^K sum","display":"\\sum_{k=0}^{K}","kind":"use","excerpt":"The active dimension is ∑_k=0^Kbinom nk =O(n^K). Let P project onto Hh_≤ K. By the hypothesis on the actual circuit prefixes, Pψ_t=ψ_t at every step. Consequently PG_tPψ_t-1=G_tψ_t-1 on the evolving state, even if the gate does not preserve all of PHh. The restricted update therefore reproduces that trajectory exactly. The hypothesis supplies effic","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b7f7abb917a0d195","tex":"\\sum_{k=0}^n","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^n","kind":"use","excerpt":"Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7796ee229bc8c35a","tex":"\\sum_{k=0}^{N-1}","aliases":"∑_k=0^N-1 sum","display":"\\sum_{k=0}^{N-1}","kind":"use","excerpt":"widehatx_r(ℓ) · =∑_m=0^M-1x(r+Lm)ζ_M^-ℓ m · =frac1N∑_k=0^N-1widehatx(k)ζ_N^kr ∑_m=0^M-1ζ_M^(k-ℓ)m.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b2a479cd3da28c74","tex":"\\sum_{k=0}^{q-1}","aliases":"∑_k=0^q-1 sum","display":"\\sum_{k=0}^{q-1}","kind":"use","excerpt":"x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-374ba43ef897c062","tex":"\\sum_{k=1}^m","aliases":"∑_k=1^m sum","display":"\\sum_{k=1}^m","kind":"use","excerpt":"Circuit leakage telescoping. Let V_j=PG_jP. Insert and subtract one factor at a time: G_m⋯ G_1P-V_m⋯ V_1P = ∑_k=1^m G_m⋯ G_k+1(G_k-V_k)V_k-1⋯ V_1P. Every G_j has norm one and every V_j has norm at most one. On PHh, (G_k-V_k)P=(I-P)G_kP. Taking norms gives [source reference].","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-13ca266a2ea1f441","tex":"\\sum_{k\\equiv r\\pmod M}","aliases":"∑_k≡ rpmod M sum equiv pmod","display":"\\sum_{k\\equiv r\\pmod M}","kind":"use","excerpt":"Residue and character packets. Suppose Q has integer spectrum. For a modulus M, define B_r^(M)= ∑_k≡ rpmod MP_k. Let V_M=exp(frac2π iMQ), ω_M=e^2π i/M. Then B_r^(M) = frac1M∑_ℓ=0^M-1 ω_M^-rℓV_M^ℓ. On a Q-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-1ebed7c5a175f44d","tex":"\\sum_{k\\ge0}","aliases":"∑_k≥0 sum ge","display":"\\sum_{k\\ge0}","kind":"use","excerpt":"[Endpoint Green valuation] The word distance is d_S(e,g) = ord_z=0 ⟨ δ_g,(I-zA_S)^-1δ_e ⟩. Expand the resolvent as a formal power series: (I-zA_S)^-1=∑_k≥0z^kA_S^k. The first nonzero endpoint coefficient occurs at the shortest word length.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7dd1f8372600c494","tex":"\\sum_m","aliases":"∑_m sum","display":"\\sum_m","kind":"use","excerpt":"widehatx(k) · =∑_r=0^L-1∑_m=0^M-1 x_r(m)ζ_N^-k(r+Lm) · =∑_rζ_N^-kr∑_mx_r(m)ζ_M^-km.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8ff4cb52f1cd00da","tex":"\\sum_{m=0}^{M-1}","aliases":"∑_m=0^M-1 sum","display":"\\sum_{m=0}^{M-1}","kind":"use","excerpt":"∑_n=0^N-1|x(n)|^2 =∑_r=0^L-1∑_m=0^M-1|x_r(m)|^2.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d328b95fa488fa30","tex":"\\sum_{m\\ge0}","aliases":"∑_m≥0 sum ge","display":"\\sum_{m\\ge0}","kind":"use","excerpt":"E_rf(t)=∑_m≥0a_r+Lmt^m, R_rf(z)=z^rE_rf(z^L).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","excerpt":"x(n)=frac1q∑_k=0^q-1widehatx(k)ζ_q^kn, ∑_n|x(n)|^2=frac1q∑_k|widehatx(k)|^2.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-13c477d68290314d","tex":"\\sum_{n=0}^{11}","aliases":"∑_n=0^11 sum","display":"\\sum_{n=0}^{11}","kind":"use","excerpt":"|ψ⟩=frac1sqrt48∑_n=0^11c(n)|n⟩, |χ⟩=frac1sqrt48∑_n=0^11|c(n)| |n⟩.","count":5},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0fbc71e3cef7af4e","tex":"\\sum_{n=0}^{N-1}","aliases":"∑_n=0^N-1 sum","display":"\\sum_{n=0}^{N-1}","kind":"use","excerpt":"∑_n=0^N-1|x(n)|^2 =∑_r=0^L-1∑_m=0^M-1|x_r(m)|^2.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b1606deb82bb8d8b","tex":"\\sum_{n=0}^{q-1}","aliases":"∑_n=0^q-1 sum","display":"\\sum_{n=0}^{q-1}","kind":"use","excerpt":"⟨ x,y⟩=∑_n=0^q-1overlinex(n)y(n), x ^2=⟨ x,x⟩.","count":7},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"use","excerpt":"Let L≥1 and let f(z)=∑_n≥0a_nz^n be a formal power series over a field containing a primitive Lth root ω and 1/L. For 0≤ r<L, define","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-18f531fd7c19a676","tex":"\\sum_{q=1}^n","aliases":"∑_q=1^n sum","display":"\\sum_{q=1}^n","kind":"use","excerpt":"Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3eecdf1ce19e226f","tex":"\\sum_{q\\ne r}","aliases":"∑_q≠ r sum ne","display":"\\sum_{q\\ne r}","kind":"use","excerpt":"Sector-flow invariants. Let P_q be the projector onto sector q. For an intermediate unitary U_t, define transfer blocks T_q,r(t)=P_q U_t P_r. A restricted path has T_q,r=0 for q≠ r. Ambient paths may populate off-diagonal blocks. Possible excursion measures include ∑_q≠ rrank T_q,r, max_q≠ r T_q,r , bandwidth in |q-r|. These are analogues of strict","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"widehatx(k) · =∑_r=0^L-1∑_m=0^M-1 x_r(m)ζ_N^-k(r+Lm) · =∑_rζ_N^-kr∑_mx_r(m)ζ_M^-km.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-431d9cc5c3f4d6bb","tex":"\\sum_{r=0}^{2}","aliases":"∑_r=0^2 sum","display":"\\sum_{r=0}^{2}","kind":"use","excerpt":"Phase factors connect the lane spectra. addcontentslinetocsectionPhase factors connect the lane spectra leadEach local Fourier transform contributes to every appropriate global frequency. With four-point negative-sign DFTs in the lanes, widehat x(k)=∑_r=0^2e^-2π ikr/12 widehatE_rx(kbmod4). fig18_polyphase_alias_assembly1.0Left: magnitudes of the th","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a977c1aed3ee7d23","tex":"\\sum_{r=0}^{L-1}","aliases":"∑_r=0^L-1 sum","display":"\\sum_{r=0}^{L-1}","kind":"use","excerpt":"R_rf(z) · =frac1L∑_j=0^L-1ω^-rjf(ω^jz), R_rR_s · =δ_rsR_r, ∑_r=0^L-1R_rf · =f.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f4c08c8ec2f1f0e1","tex":"\\sum_{r=1}^{c}","aliases":"∑_r=1^c sum","display":"\\sum_{r=1}^{c}","kind":"use","excerpt":"C · I_t . A restricted circuit may change source and output bases while transmitting rank-one terms. Let P_r∈ U_s be the source basis before the rth transmission and let Q_>r∈ U_t be the future output basis. A cross gate contributes Q_>re_i_re_j_r^TP_<r to the final block. Hence C= ∑_r=1^c Q_>re_i_re_j_r^TP_<r, with closed basis paths P_final=I_s, ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-706a421df4de6105","tex":"\\sum_{\\substack{d\\mid q\\\\(q/d)\\mid n}}","aliases":"∑_substackdmid q (q/d)mid n sum substack mid mid","display":null,"kind":"use","excerpt":"c_q(n) · =∑_dmid qμ(d) ∑_b=0^q/d-1e^2π ibn/(q/d) · =∑_substackdmid q (q/d)mid nμ(d)frac qd.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-92eeaf2911f8ec48","tex":"\\sum_{\\substack{d\\mid q\\\\q/d\\mid n}}","aliases":"∑_substackdmid q q/dmid n sum substack mid q mid","display":null,"kind":"use","excerpt":"ntentslinetocsectionWhy the phase sum is an integer leadA divisor identity certifies integrality without rounding trigonometric values. For q≥1, insert the M obius coprimality identity into the finite phase sum: 1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d). Write a=db after interchanging the finite sums. Character orthogonality evaluates the inner geometric ser","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7a26fb0b26e32548","tex":"\\sum_{s\\in S}","aliases":"∑_s∈ S sum in","display":"\\sum_{s\\in S}","kind":"use","excerpt":"Word distance as first arrival. Let G be finite and let S=S^-1 be a symmetric generating multiset. In the right regular representation, define A_S=∑_s∈ Sρ(s). Then ⟨δ_g,A_S^kδ_e⟩ counts length-k words ending at g.","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4d8f038bb55517d6","tex":"\\sum_{t=1}^m","aliases":"∑_t=1^m sum","display":"\\sum_{t=1}^m","kind":"use","excerpt":"ate] Let G_1,…,G_m be unitary operators on a finite-dimensional Hilbert space, and let ψ_0=ϕ_0 be a unit vector. At each step choose an orthogonal projector P_t, possibly as a function of the previously computed approximate states. Define ψ_t=G_tψ_t-1, ϕ_t=P_tG_tϕ_t-1, η_t= (I-P_t)G_tϕ_t-1 . Then ψ_m-ϕ_m ≤∑_t=1^mη_t. For an effect 0≤ M≤ I, |⟨ψ_m,Mψ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ac4c06b90cfdee40","tex":"\\sum_{t=1}^{m}","aliases":"∑_t=1^m sum","display":"\\sum_{t=1}^{m}","kind":"use","excerpt":"[Dynamic sum-of-leakages bound] For an initial unit vector in P_0Hh, G_m⋯ G_1ketψ_0 - widehat G_m⋯widehat G_1ketψ_0 ≤ ∑_t=1^mδ_t. For any acceptance operator 0≤ M≤ I, the corresponding acceptance probabilities differ by at most 2∑_t=1^mδ_t.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7aa48f2333e4e738","tex":"\\sum_{T\\subseteq S}","aliases":"∑_T⊆ S sum subseteq","display":"\\sum_{T\\subseteq S}","kind":"use","excerpt":"hm:monotone If f is monotone, I_f has a monomial basis indexed by the positive supports of f. For each positive v, β_v(f)=min{|S|: S⊆supp(v), f(1_S)=1}. Consequently, for nonzero f, α_1(f)=minimum minterm size, β(f)=maximum minterm size. For any Boolean polynomial h, its coefficient on x^S is [x^S]h=∑_T⊆ Sh(1_T) over F. If S is negative for monoton","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d96e96aadd273752","tex":"\\sum_u","aliases":"∑_u sum","display":"\\sum_u","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a45293ecf74a34d1","tex":"\\sum_{u,v}","aliases":"∑_u,v sum","display":"\\sum_{u,v}","kind":"use","excerpt":"[Disjoint-variable factorization] If predicates A(u) and B(v) use disjoint witness-variable sets, then their weighted fingerprints satisfy P_Aland B(z_u,z_v)=P_A(z_u)P_B(z_v). Expand the left side as a sum over the Cartesian product of witnesses: ∑_u,vA(u)B(v)z_u^uz_v^v =(∑_uA(u)z_u^u) (∑_vB(v)z_v^v). Disjointness makes the exponent data independen","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-58c4e875861cb6bd","tex":"\\sum_{u\\in W}","aliases":"∑_u∈ W sum in","display":"\\sum_{u\\in W}","kind":"use","excerpt":"[Finite support of a dual witness] When the right side of [source reference] holds, there is W⊆ U with v_d(v)=∑_u∈ Wv_d(u), |W|≤∑_j=0^min(d,n)binom nj. The cardinality |W| is odd. Choose a basis of the span from negative evaluation vectors and express v_d(v) in that basis. Over F, the nonzero coefficients select a subset W of size at most the ambie","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-67dea2a8af46b4ec","tex":"\\sum_{u\\ne v\\text{ on the face}}","aliases":"∑_u≠ v on the face sum ne text","display":"\\sum_{u\\ne v\\text{ on the face}}","kind":"use","excerpt":"An explicit moment certificate. On the face b=1, every point except v=(a=0,z=(1,…,1)) is negative for G_k. A polynomial of degree at most k sums to zero over the (k+1)-dimensional cube: each monomial omits some coordinate and contributes an even number of ones. Therefore v_k(v)=∑_u≠ v on the facev_k(u). This is the certificate from Theorem~[source ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-53c07efe2188e4b0","tex":"\\sum_v","aliases":"∑_v sum","display":"\\sum_v","kind":"use","excerpt":"calar unital summaries are evaluations] Every unital K-linear multiplicative map Φ:K^W→ K is evaluation at exactly one witness. An idempotent scalar in a field is either zero or one. Orthogonality allows at most one Φ(δ_v) to be one, and preservation of the unit requires exactly one. If it occurs at v_0, linearity gives Φ(g)=Φ(∑_v g(v)δ_v)=g(v_0). ","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5e7068b070ecec3f","tex":"\\sum_{v\\in W}","aliases":"∑_v∈ W sum in","display":"\\sum_{v\\in W}","kind":"use","excerpt":"The witness algebra. Let K be a field and let W=B^m. The full witness algebra is B=K^W, with pointwise addition and multiplication. For each witness v, let δ_v be its indicator function. These form a basis of B and satisfy δ_v^2=δ_v, δ_vδ_u=0 (u≠ v), ∑_v∈ Wδ_v=1. In characteristic two, this is equivalently the Boolean quotient algebra K[w_1,…,w_m]/","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-5cf055eb2e581dff","tex":"\\sum_{v\\in\\B^m}","aliases":"∑_v∈B^m sum in B","display":null,"kind":"use","excerpt":"Exact top-coefficient extraction. [Top coefficient as a Boolean-cube sum] Let G(w) be a multilinear polynomial over a characteristic-two field. Then [w_1⋯ w_m]G=∑_v∈B^mG(v). It is enough to check a monomial. A monomial of proper support is one on an even number 2^m-|S| of Boolean inputs, so its cube sum is zero in characteristic two. The full-suppo","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ae10112f63c03182","tex":"\\sum_{v\\in\\{0,1\\}^m}","aliases":"∑_v∈{0,1}^m sum in","display":"\\sum_{v\\in\\{0,1\\}^m}","kind":"use","excerpt":"guidepageA short expression inside a trace is not a computed trace Let τ_w extract the top multilinear coefficient in the Boolean witness ring. For any G(w) over a characteristic-two field, τ_w(G)=∑_v∈{0,1}^mG(v). A proper-support monomial evaluates to one an even number of times; the full-support monomial does so once. This proves the identity dir","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-731f2fca5560b35b","tex":"\\sum_{w\\in\\B^m}","aliases":"∑_w∈B^m sum in B","display":null,"kind":"use","excerpt":"Weighted witness fingerprints. claimC07 A polynomial that distinguishes witnesses. Let R:B^m→B be a Boolean verifier, and let K be a characteristic-two field. Define the formal polynomial P_R(z_1,…,z_m)= ∑_w∈B^mR(w)∏_i=1^m z_i^w_i∈ K[z_1,…,z_m]. The variables z_i in this expression are ordinary polynomial variables, not reduced by z_i^2=z_i.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7b50cf402e41f565","tex":"\\sum_{w\\in\\{0,1\\}^m}","aliases":"∑_w∈{0,1}^m sum in","display":"\\sum_{w\\in\\{0,1\\}^m}","kind":"use","excerpt":"guidepageDistinct witness monomials prevent formal cancellation Over a characteristic-two field, define P_R(z)=∑_w∈{0,1}^mR(w)∏_i z_i^w_i. The z_i are formal indeterminates, not Boolean variables. Different witnesses produce different monomials, so P_R=0 exactly when there is no accepting witness.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c15f8eeeaa0c6aef","tex":"\\sum_{x,y}","aliases":"∑_x,y sum","display":"\\sum_{x,y}","kind":"use","excerpt":"Forrelation amplitude. Let f,g:{0,1}^n→{-1,1} and N=2^n. The circuit with alternating Hadamards and phase oracles has zero-state amplitude Φ(f,g) = frac1N^3/2 ∑_x,y f(x)(-1)^x· yg(y). The circuit does not evaluate the N^2 terms separately. It preserves phases and lets them interfere.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0364d5e38876f99a","tex":"\\sum_{x\\in\\mathcal O_a}","aliases":"∑_x∈ O_a sum in mathcal","display":"\\sum_{x\\in\\mathcal O_a}","kind":"use","excerpt":"Orbit states. Let a group G act by automorphisms on a finite graph with vertex set V. The orbits are O_1,…, O_r. Define normalized orbit states ket O_a = frac1sqrt| O_a| ∑_x∈ O_aket x. If the adjacency or Hamiltonian H commutes with the group action, the span of these states is invariant whenever the partition is equitable.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-32572785b48f6bd4","tex":"\\sum_{x\\in X}","aliases":"∑_x∈ X sum in","display":"\\sum_{x\\in X}","kind":"use","excerpt":"Write the opening coefficient as a∈F^×, the left-update coefficients as b_x, and the right-read coefficients as c_y. Since the all-cross middle gates commute past one another, the ambient gate list may be written R_p← R_p+aR_q; [R_q← R_q+b_xR_x]_x∈ X; [R_y← R_y+c_yR_p]_y∈ Y; R_p← R_p-aR_q. Put B=∑_x∈ Xb_xe_x. Its final nontrivial rows are R_p=e_p-a","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c74d43920893ab65","tex":"\\sum_{\\{v,v+a\\}}","aliases":"∑_{v,v+a} sum","display":"\\sum_{\\{v,v+a\\}}","kind":"use","excerpt":"Now extract the quadratic homogeneous ANF component Q_m in the original x_v coordinates. Since P_m(0)=P_m(e_v)=0, for distinct u,v its quadratic coefficient is [x_ux_v]P_m=P_m(e_u+e_v)=[u+v=a]. The involution v↦ v+a pairs every variable with one other variable: Q_m=∑_{v,v+a}x_vx_v+a. It is a quadratic inner product on n/2 disjoint pairs. Its altern","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-0ed8983883cf9b1c","tex":"\\sum_{|x|=k}","aliases":"∑_|x|=k sum","display":"\\sum_{|x|=k}","kind":"use","excerpt":"Hypercube quotient coupling. Let ket k = binom nk^-1/2 ∑_|x|=kket x. Every weight-k string has n-k neighbors of weight k+1, while every weight-(k+1) string has k+1 neighbors of weight k. Counting the bipartite edges yields brak+1Aket k = fracbinom nk(n-k) sqrtbinom nkbinom nk+1 = sqrt(k+1)(n-k), proving [source reference].","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"use","excerpt":"Orthogonal packet decompositions. Let Hh be a finite-dimensional Hilbert space and let {P_α}_α∈ A be orthogonal projectors satisfying P_α P_β=δ_αβP_α, ∑_α∈ AP_α=I. Every state and operator decompose as ketψ=∑_α P_αketψ, X=∑_α,βP_α X P_β. The diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","excerpt":"Orthogonal packet decompositions. Let Hh be a finite-dimensional Hilbert space and let {P_α}_α∈ A be orthogonal projectors satisfying P_α P_β=δ_αβP_α, ∑_α∈ AP_α=I. Every state and operator decompose as ketψ=∑_α P_αketψ, X=∑_α,βP_α X P_β. The diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-65bbcf65283d2b6f","tex":"\\sum_{\\alpha\\in A}","aliases":"∑_α∈ A sum alpha in","display":"\\sum_{\\alpha\\in A}","kind":"use","excerpt":"Orthogonal packet decompositions. Let Hh be a finite-dimensional Hilbert space and let {P_α}_α∈ A be orthogonal projectors satisfying P_α P_β=δ_αβP_α, ∑_α∈ AP_α=I. Every state and operator decompose as ketψ=∑_α P_αketψ, X=∑_α,βP_α X P_β. The diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-bdca70a598b083c8","tex":"\\sum_{\\alpha\\in A_t}","aliases":"∑_α∈ A_t sum alpha in","display":"\\sum_{\\alpha\\in A_t}","kind":"use","excerpt":"Hybrid transfer graphs. Suppose packets are nodes of a sparse graph and each node carries an internal representation. A gate may update a node internally or transfer a compact component to neighboring nodes. The simulation cost depends on ∑_α∈ A_tsize(r_α) + #{nonzero transfers at time t}. This model combines symmetry selection rules with compact s","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a97633ad5296debd","tex":"\\sum_{\\alpha\\in S_t}","aliases":"∑_α∈ S_t sum alpha in","display":"\\sum_{\\alpha\\in S_t}","kind":"use","excerpt":"[Reachable packet-graph collapse] Suppose the packet sets S_t and nonzero transfer blocks are computable in polynomial time, and D_t = ∑_α∈ S_trank P_α ≤poly(n) for every t. If all active block updates and the final measurement are efficiently computable, then the circuit family is classically polynomial-time simulable.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f93f7a7815ae0b42","tex":"\\sum_{\\alpha\\in\\Lambda}","aliases":"∑_α∈Λ sum alpha in Lambda","display":"\\sum_{\\alpha\\in\\Lambda}","kind":"use","excerpt":"Sparse packet-transfer graphs. Let I=∑_α∈ΛP_α be an orthogonal packet decomposition. Write the gate blocks as G_t;βα=P_β G_tP_α. Starting from an active packet set S_0, define recursively S_t = { β:∃α∈ S_t-1 with G_t;βα≠0 }.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ade7e6465dee6e86","tex":"\\sum_\\lambda","aliases":"∑_λ sum lambda","display":"\\sum_\\lambda","kind":"use","excerpt":"Why eigenvalues alone are insufficient. If A_S=∑_λλΠ_λ, then G_S,g(z) = ∑_λ frac⟨δ_g,Π_λδ_e⟩1-λ z. The eigenvalues and multiplicities do not determine the endpoint coefficients ⟨δ_g,Π_λδ_e⟩. Two Cayley or Schreier systems can therefore be cospectral and still have different first-arrival behavior at distinguished targets.","count":10},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-614ab5cb07fb5cef","tex":"\\sum_{\\lambda}","aliases":"∑_λ sum lambda","display":"\\sum_{\\lambda}","kind":"use","excerpt":"Spherical packets. Let T_1,…,T_m be commuting normal operators on a finite-dimensional Hilbert space. Their unique coarsest joint spectral resolution is T_j=∑_λt_j(λ)B_λ, ∑_λ B_λ=I. The tuple λ=(t_1(λ),…,t_m(λ)) is the spherical label. The abstract spectral packet records these labels and the ranks of B_λ.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f85a9d219ed81c46","tex":"\\sum_\\mu","aliases":"∑_μ sum mu","display":"\\sum_\\mu","kind":"use","excerpt":"Four-index transfer. A quantum channel with Kraus operators K_μ acts by Φ(ρ) = ∑_μ K_μρ K_μ^†. Relative to operator packets P_αβ(X) = P_α X P_β, the channel has blocks Φ_γδ,αβ = P_γδ Φ P_αβ. Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f29c4345909a9314","tex":"\\sum_\\pi","aliases":"∑_π sum pi","display":"\\sum_\\pi","kind":"use","excerpt":"Using cyclicity of the trace and U_π^*AU_π=A, Tr(A T_n(ρ)) = frac1n!∑_π Tr(AU_πρ U_π^*) = frac1n!∑_π Tr(U_π^*AU_πρ) = Tr(Aρ).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9bb019c9a0efe97b","tex":"\\sum_{\\pi\\in S_n}","aliases":"∑_π∈ S_n sum pi in","display":"\\sum_{\\pi\\in S_n}","kind":"use","excerpt":"The permutation twirl. Define the twirling channel T_n(X) = frac1n! ∑_π∈ S_n U_π XU_π^*. Its image is the commutant of the permutation action.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-f3a43e1e756dd849","tex":"\\sum_{\\tau\\in\\widehat H}","aliases":"∑_τ∈widehat H sum tau in widehat","display":"\\sum_{\\tau\\in\\widehat H}","kind":"use","excerpt":"Subgroup-holonomy Fourier packets. Let H≤ G. Let μ_k(h) count ambient words of length k with product h∈ H. For every irreducible representation τ∈widehat H, define widehatμ_k(τ) = ∑_h∈ Hμ_k(h)τ(h). Fourier inversion gives μ_k(h) = frac1|H| ∑_τ∈widehat H dim(τ) tr ( τ(h^-1)widehatμ_k(τ) ). The matrix-valued Green packet widehat G_τ(z) = ∑_k≥0widehat","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c37d85c369fd4aad","tex":"\\sum_\\chi","aliases":"∑_χ sum chi","display":"\\sum_\\chi","kind":"use","excerpt":"Finite-group projectors. Let a finite group G act unitarily by U_g. For an irreducible character χ of degree d_χ, the central isotypic projector is P_χ = fracd_χ|G| ∑_g∈ G overlineχ(g)U_g. Character orthogonality gives P_χ P_ψ=δ_χψP_χ, ∑_χ P_χ=I on the represented space. If an operator A commutes with every U_g, then P_χ A P_ψ=0 (χ≠ψ).","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-4e96d3bac74365f7","tex":"\\sum_{\\ell=0}^{M-1}","aliases":"∑_ℓ=0^M-1 sum ell","display":"\\sum_{\\ell=0}^{M-1}","kind":"use","excerpt":"Residue projector formula. Let Qketψ_k=kketψ_k. Then frac1M∑_ℓ=0^M-1 ω_M^-rℓe^2π iℓ Q/Mketψ_k = ( frac1M∑_ℓ=0^M-1 ω_M^(k-r)ℓ )ketψ_k. The bracket equals one when k≡ rpmod M and zero otherwise. This proves [source reference] and the orthogonality of the residue packets.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b98eb4d33c4dd25d","tex":"\\sum_{\\ell=0}^{N-1}","aliases":"∑_ℓ=0^N-1 sum ell","display":"\\sum_{\\ell=0}^{N-1}","kind":"use","excerpt":"E_a=frac1N∑_ℓ=0^N-1ζ_N^aℓA^ℓ","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"rangle. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry is the left Schreier graph G/H, with edges gH⟼ sgH. For the base-fiber assertion below, assume also S_H= S_G∩ H. ","count":3},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-a2972272b24e9c45","tex":"\\int_0^t","aliases":"∫_0^t int","display":"\\int_0^t","kind":"use","excerpt":"Define F(s)=e^-i(t-s)HPe^-isHP. Then F(0)=e^-itHP and F(t)=Pe^-itHP. Differentiation gives F'(s)=ie^-i(t-s)H[H,P]e^-isHP. Therefore (I-P)e^-itHP =-(I-P)∫_0^tF'(s) ds. The unitary factors and projectors have norm at most one, so the integral is bounded by |t| [H,P] .","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"Half filling. At half filling, binom nlfloor n/2rfloor ∼ frac2^nsqrtπ n/2. The packet is smaller by a polynomial factor, but it remains exponential. At n=20, the half-filled packet contains 184,756 amplitudes. The archived repeated-application gain is approximately 7.396, while the one-shot gain is 0.6529, meaning the packet path was slower in that","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"Over F_2, write x=e_0, p=e_1, q=e_2. The canonical ambient word uses the six nonzero vectors x, p, q, p+q, p+q+x, p+x inside Span(x,p,q)≅F_2^3. The only missing nonzero vector is x+q.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"Define the packet-transfer graph of an operator X by placing an edge β→α whenever P_α X P_β≠0. An exactly symmetry-preserving operator has only diagonal edges. A sparse transfer graph may still permit efficient simulation if the active packets and internal states remain compact.","count":8},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"Character orthogonality makes it a_n when n≡ rpmod L and zero otherwise. This is precisely the coefficient of R_rf. Each coefficient position belongs to a unique lane, so the projector and completeness identities follow coefficientwise. The equality is formal, with no analytic convergence hypothesis. square","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"ove those formulas universally. For general q, character orthogonality gives widehatc_q(k)=q1_gcd(k,q)=1, ∑_n=0^q-1|c_q(n)|^2=qφ(q). The associated primitive-order projector has rank φ(q). The general mathematical conclusions follow from the finite-sum derivations in the main paper. The source verifier separately checks its declared finite range 2≤","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"Why the phase sum is an integer. addcontentslinetocsectionWhy the phase sum is an integer leadA divisor identity certifies integrality without rounding trigonometric values. For q≥1, insert the M obius coprimality identity into the finite phase sum: 1_gcd(a,q)=1=∑_dmidgcd(a,q)μ(d). Write a=db after interchanging the finite sums. Character orthogona","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":", assume R=F is a field. An invertible matrix is an ordered basis (R_0,…,R_n-1) of F^n. A transvection replaces one basis vector by its sum with a nonzero multiple of another. Let V_i=Span(e_0,…,e_i). A matrix lies in U_n(F) exactly when R_i∈ e_i+V_i-1 for every i. Thus lower circuits move among frames compatible with the standard complete flag V_0","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a ","count":12},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","excerpt":"[Invariant quantum packets] If a unitary U satisfies [U,Q]=0, then P_iUP_j=0 (i≠ j), and U=⊕_i U_i, U_i=P_iUP_i|_P_iHh. If P_kketψ=ketψ, then P_kUketψ=Uketψ.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-8817e97bab58b0ec","tex":"\\bigoplus_{k=0}^{K}","aliases":"⊕_k=0^K bigoplus","display":"\\bigoplus_{k=0}^{K}","kind":"use","excerpt":"[Bounded-excitation collapse] Consider n qubits and suppose the initial state and every circuit prefix are certified to lie in Hh_≤ K = ⊕_k=0^KHh_n,k for a fixed constant K. If initialization, the gate actions, and the measurement are computable in polynomial time in the occupation basis, then the circuit family lies in P, even when individual gate","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-dde487846d5c46d3","tex":"\\bigoplus_{k=0}^n","aliases":"⊕_k=0^n bigoplus","display":"\\bigoplus_{k=0}^n","kind":"use","excerpt":"Excitation-number packets. For n qubits, define widehat N = ∑_q=1^nfracI-Z_q2. Its eigenvalues are k=0,1,…,n. The projector onto Hamming weight k is P_n,k = ∑_|x|=kket xbra x, and dim P_n,kHh = binom nk. The decomposition (C^2)^⊗ n = ⊕_k=0^nHh_n,k recovers the full dimension through ∑_k=0^nbinom nk=2^n.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-d98af3ffc3344961","tex":"\\bigoplus_{L\\in p}","aliases":"⊕_L∈ p bigoplus in","display":"\\bigoplus_{L\\in p}","kind":"use","excerpt":"Polynomial certificates are not postulated degree oracles. The inductive certificate has three constructors: zero, a bounded-length monomial, and XOR of two certified functions. It is tied to the concrete evaluator polyEval(p,x)=⊕_L∈ p∏_i∈ Lx_i by a proved equivalence. A certificate yields an explicit finite monomial list, and such a list yields a ","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-73d4349c6a5148b3","tex":"\\bigoplus_q","aliases":"⊕_q bigoplus","display":"\\bigoplus_q","kind":"use","excerpt":"The Hilbert space decomposes as H=⊕_q H_q. A restricted circuit is block diagonal. An ambient catalyst may couple sectors, use the enlarged route, and decouple them at the endpoint.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-6e42b29f036c3ac2","tex":"\\bigoplus_{S\\subseteq[n]}","aliases":"⊕_S⊆[n] bigoplus subseteq","display":"\\bigoplus_{S\\subseteq[n]}","kind":"use","excerpt":"Algebraic normal form. Every Boolean coordinate function has a unique algebraic normal form f_i(x) = ⊕_S⊆[n]a_i,S∏_j∈ Sx_j. Affine maps have degree at most one. A Toffoli gate targeting t with controls a,b performs x_t⟼ x_t+x_ax_b, which can raise the degree of later coordinates. The final affine target requires every higher-degree coefficient to c","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-ca12baf923be071e","tex":"\\bigoplus_\\lambda","aliases":"⊕_λ bigoplus lambda","display":"\\bigoplus_\\lambda","kind":"use","excerpt":"Decomposition For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-9134e99ccc88c15b","tex":"\\bigoplus_{\\lambda\\vdash n,\\,\\ell(\\lambda)\\le d}","aliases":"⊕_λvdash n, ℓ(λ)≤ d bigoplus lambda vdash ell lambda le","display":null,"kind":"use","excerpt":"Decomposition. For n qudits of local dimension d, Schur--Weyl duality decomposes (C^d)^⊗ n ≅ ⊕_λvdash n, ℓ(λ)≤ d U_λ⊗ V_λ, where U_λ carries an irreducible unitary-group representation and V_λ carries a symmetric-group representation. A permutation-invariant operator acts as A = ⊕_λ A_λ⊗ I_ V_λ.","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":2},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-869d7204e96d9910","tex":"\\bigcup_{s<t}","aliases":"⋃_s<t bigcup","display":"\\bigcup_{s<t}","kind":"use","excerpt":"Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-89033592672e0222","tex":"\\bigcup_{t\\ne s}","aliases":"⋃_t≠ s bigcup ne","display":"\\bigcup_{t\\ne s}","kind":"use","excerpt":"Equivalently, put X_* = ⋃_t≠ s{I+aE_t,s:a∈ K}, X_downarrow = ⋃_s<t{I+aE_t,s:a∈ K}. Then ellAll(A) and ellDown(A) are the corresponding transvection product lengths: the least L for which A∈ X_*^L or A∈ X_downarrow^L.","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-51f354fe46d1388b","tex":"\\bigcup_{\\sigma\\in\\{(t,s):t\\ne s\\}^L}","aliases":"⋃_σ∈{(t,s):t≠ s}^L bigcup sigma in ne","display":"\\bigcup_{\\sigma\\in\\{(t,s):t\\ne s\\}^L}","kind":"use","excerpt":"For a direction skeleton σ=((t_1,s_1),…,(t_L,s_L)), t_i≠ s_i, define the polynomial word map Φ_σ:Aff^L⟶SL_n, (a_1,…,a_L) ⟼ (I+a_LE_t_L,s_L)⋯(I+a_1E_t_1,s_1). The coefficients are allowed to vanish, so a word of length at most L can be padded to length L. Let R^*_n,≤ L = U_n∩ ⋃_σ∈{(t,s):t≠ s}^L imΦ_σ be the ambient reachability locus, and define R^d","count":1},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry ","count":27},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"Subgroup metrics and Schreier loops. Assume first that the restricted and ambient models are groups H=⟨ S_H⟩ ≤ G=⟨ S_G⟩. For h∈ H, the intrinsic and ambient word metrics are ℓ_H(h)=|h|_ S_H, ℓ_G(h)=|h|_ S_G. In the group-metric statements, the generating alphabets are symmetric and a generator and its inverse have equal cost. The quotient geometry ","count":32},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"Set G=Egrp_n(R), H=U_n(R). Consider the right Schreier graph on the left cosets Hbackslash G: a generator s∈ S_n^*(R) gives an edge Hg⟶ Hgs. A word representing h∈ H traces a based loop at the coset H. Its label product is the holonomy h.","count":4},{"work":"monographs-quantum-and-circuits","url":"notation/monographs-quantum-and-circuits.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"A restored oracle is a circuit satisfying (x,0^a,y)⟼(x,0^a,y+f(x)) for every logical input x and both initial target values y∈B. The zero-workspace condition is imposed at the boundary, not at intermediate times. The circuit may disturb logical inputs, use output feedback, reuse intermediate values, and make arbitrary reversible excursions before r","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"notation-table","excerpt":"Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.","count":180},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.","count":1774},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"notation-table","excerpt":"Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.","count":459},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"notation-table","excerpt":"Transfer block · L_ba=P_bLP_a takes a component in packet a to packet b.","count":218},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-598f62db34f4eb27","tex":"\\Dom","aliases":"Dom","display":null,"kind":"notation-table","excerpt":"Unbounded action · An expression such as Hψ includes the condition ψ∈Dom(H).","count":23},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2dfbbad49009071f","tex":"\\dot p","aliases":"dot p dot","display":"\\dot p","kind":"notation-table","excerpt":"Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-81596f18cb9ce128","tex":"\\dot p_b","aliases":"dot p_b dot","display":"\\dot p_b","kind":"notation-table","excerpt":"Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-381aa4b32306117e","tex":"\\dot\\psi","aliases":"dotψ dot psi","display":"\\dot\\psi","kind":"notation-table","excerpt":"State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"notation-table","excerpt":"State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.","count":359},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"notation-table","excerpt":"Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.","count":468},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"notation-table","excerpt":"State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.","count":298},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d4faea86910a340e","tex":"\\inner","aliases":"inner","display":null,"kind":"notation-table","excerpt":"midrule Inner product · innerxy is conjugate-linear in x and linear in y.","count":55},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"notation-table","excerpt":"Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.","count":128},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"notation-table","excerpt":"Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.","count":293},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1a670656747a1061","tex":"J_{b\\leftarrow a}","aliases":"J_b← a leftarrow","display":"J_{b\\leftarrow a}","kind":"notation-table","excerpt":"Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","excerpt":"State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.","count":116},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-997c382b71c367a6","tex":"K^*","aliases":"K^*","display":"K^*","kind":"notation-table","excerpt":"State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","excerpt":"Transfer block · L_ba=P_bLP_a takes a component in packet a to packet b.","count":170},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-03713cede48e6adf","tex":"L_{ba}","aliases":"L_ba","display":"L_{ba}","kind":"notation-table","excerpt":"Transfer block · L_ba=P_bLP_a takes a component in packet a to packet b.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"notation-table","excerpt":"Packet weight · p_a=normP_aψ^2. It is a probability for a normalized state and a complete orthogonal resolution. It is not an energy expectation by definition.","count":158},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.","count":246},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"notation-table","excerpt":"Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.","count":543},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"notation-table","excerpt":"Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.","count":59},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c261d0404352ff34","tex":"p_a","aliases":"p_a","display":"p_a","kind":"notation-table","excerpt":"Packet weight · p_a=normP_aψ^2. It is a probability for a normalized state and a complete orthogonal resolution. It is not an energy expectation by definition.","count":26},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a7abbe02c9f0ed6a","tex":"P_a^*","aliases":"P_a^*","display":"P_a^*","kind":"notation-table","excerpt":"Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-72bf1da84f9b9c1d","tex":"P_a^2","aliases":"P_a^2","display":"P_a^2","kind":"notation-table","excerpt":"Projector · P_a=P_a^*=P_a^2, with orthogonality between distinct packet labels.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"notation-table","excerpt":"Transfer block · L_ba=P_bLP_a takes a component in packet a to packet b.","count":36},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"notation-table","excerpt":"Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.","count":17},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"notation-table","excerpt":"Centered approximation · The multiplier with kinetic symbol sin(hp_j)/h on the same integer mode index set.","count":47},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"notation-table","excerpt":"Mixed state · ρ≥0, trρ=1 in the finite-dimensional sections.","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"notation-table","excerpt":"midrule Inner product · innerxy is conjugate-linear in x and linear in y.","count":321},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"notation-table","excerpt":"midrule Inner product · innerxy is conjugate-linear in x and linear in y.","count":100},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"notation-table","excerpt":"Mixed state · ρ≥0, trρ=1 in the finite-dimensional sections.","count":61},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"notation-table","excerpt":"State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.","count":180},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a86ef458cc61e377","tex":"\\hbar","aliases":"ℏ hbar","display":"\\hbar","kind":"notation-table","excerpt":"State equation · dotψ=Kψ, with K^*=-K. The Hamiltonian convention is K=-iH in units ℏ=1.","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"notation-table","excerpt":"Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.","count":29},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"notation-table","excerpt":"Unbounded action · An expression such as Hψ includes the condition ψ∈Dom(H).","count":161},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"notation-table","excerpt":"Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.","count":278},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"notation-table","excerpt":"Current orientation · J_b← a is inflow into packet b from packet a, so dot p_b=∑_aJ_b← a in the finite closed setting.","count":39},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"notation-table","excerpt":"Mixed state · ρ≥0, trρ=1 in the finite-dimensional sections.","count":72},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.","count":559},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.","count":240},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1aa9a70355d1f987","tex":"b^{m+1}","aliases":"b^m+1","display":"b^{m+1}","kind":"definition-passage","excerpt":"[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.","count":167},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F","display":null,"kind":"definition-passage","excerpt":"[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to e","count":14},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.","count":205},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f1ceb5c186bbf16a","tex":"\\F^{-1}","aliases":"F^-1 F","display":null,"kind":"definition-passage","excerpt":"[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to e","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"definition-passage","excerpt":"[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a67dba11d23b584e","tex":"F_a^*","aliases":"F_a^*","display":"F_a^*","kind":"definition-passage","excerpt":"[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7937103467a5b8d0","tex":"\\HH","aliases":"HH","display":null,"kind":"definition-passage","excerpt":"[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.","count":59},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.","count":235},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id","display":null,"kind":"definition-passage","excerpt":"[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"definition-passage","excerpt":"[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.","count":273},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5ab8627907d416c2","tex":"P_\\Lambda","aliases":"P_Λ Lambda","display":"P_\\Lambda","kind":"definition-passage","excerpt":"[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"definition-passage","excerpt":"[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.","count":140},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"definition-passage","excerpt":"[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.","count":25},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.","count":170},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"definition-passage","excerpt":"[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.","count":85},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to e","count":305},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"definition-passage","excerpt":"[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b8e003e6dbf7e04c","tex":"T_m","aliases":"T_m","display":"T_m","kind":"definition-passage","excerpt":"[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db2bda42ac0870c5","tex":"T_{m+1}","aliases":"T_m+1","display":"T_{m+1}","kind":"definition-passage","excerpt":"[The actual refinement tower] Set T_0f(h)=f(h) and, for m≥0, define T_m+1f(h)=fracb^m+1T_mf(h/b)-T_mf(h)b^m+1-1.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to e","count":184},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"definition-passage","excerpt":"[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to e","count":14},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8a830c0eacd44ea9","tex":"U_h","aliases":"U_h","display":"U_h","kind":"definition-passage","excerpt":"[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to e","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6908b76a39ebf92a","tex":"\\mathcal U_h","aliases":"U_h mathcal","display":"\\mathcal U_h","kind":"definition-passage","excerpt":"[Physical propagators] Define U(t)=F^-1U(t)F, U_h(t)=F^-1U_h(t)F. Both preserve the physical L^2 norm and inner product. Their group laws follow by inserting FF^-1=I between consecutive factors. Their residual satisfies the exact equality norm U_h(t)f- U(t)f_HH =normU_h(t)F f-U(t)F f_KK. The compact-time convergence theorem therefore transfers to e","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"definition-passage","excerpt":"[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.","count":125},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1525b57dc165eb8a","tex":"\\delta_{ab}","aliases":"δ_ab delta","display":"\\delta_{ab}","kind":"definition-passage","excerpt":"[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"definition-passage","excerpt":"[Charged mass coordinate] The charged Misner--Sharp coordinate in this canonical chart is M=frac R2[1-(fracR'Λ)^2 +(fracP_ΛR)^2] +fracQ^22R. The charge contribution is included in M so that it is constant throughout an electrovac Reissner--Nordstr om solution.","count":55},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ef65869303e9087f","tex":"\\psi^2","aliases":"ψ^2 psi","display":"\\psi^2","kind":"definition-passage","excerpt":"[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d80f8057d7f2fe60","tex":"\\psi_a","aliases":"ψ_a psi","display":"\\psi_a","kind":"definition-passage","excerpt":"[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ad151dded7ef7d37","tex":"\\sum_{a\\in A}","aliases":"∑_a∈ A sum in","display":"\\sum_{a\\in A}","kind":"definition-passage","excerpt":"[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2b67385de1e4af35","tex":"\\sum_{a\\in I}","aliases":"∑_a∈ I sum in","display":"\\sum_{a\\in I}","kind":"definition-passage","excerpt":"[Finite orthogonal atlas] A finite orthogonal atlas is a family (P_a)_a∈ I satisfying P_a^*=P_a, P_aP_b=δ_abP_a, ∑_a∈ IP_a=id. Its packet of a vector ψ is ψ_a=P_aψ. Its packet weight is p_a=normψ_a^2. Zero projectors are allowed, so a label can describe an empty sector.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"[Finite effect atlas] Let HH be a complex Hilbert space and let A be a finite label set. An effect atlas is a family of bounded operators satisfying F_a=F_a^*≥0, ∑_a∈ AF_a=I. Its weight in a state ψ is p_a=innerψF_aψ. The partition identity implies 0≤ F_a≤ I, so p_a≥0 and ∑_a p_a=normψ^2.","count":86},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"introduction-passage","excerpt":"· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b","count":25},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e0c9947b1daf0522","tex":"A^*","aliases":"A^*","display":"A^*","kind":"introduction-passage","excerpt":"use the forward convention of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert n","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"introduction-passage","excerpt":"forward convention of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm stat","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e7851cacbe3babd8","tex":"\\mathcal A_{3,1}","aliases":"A_3,1 mathcal","display":"\\mathcal A_{3,1}","kind":"introduction-passage","excerpt":"LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a0e924755e68f546","tex":"\\mathcal A_{L,j}","aliases":"A_L,j mathcal","display":"\\mathcal A_{L,j}","kind":"introduction-passage","excerpt":"· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"introduction-passage","excerpt":"Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+","count":47},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4d8140b92561cfb1","tex":"\\mathrm{arc}","aliases":"arc mathrm","display":"\\mathrm{arc}","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,","count":164},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f673c03c77f9ac50","tex":"B^*","aliases":"B^*","display":"B^*","kind":"introduction-passage","excerpt":"Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,","count":18},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"introduction-passage","excerpt":"mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ac0d31a2615b8b66","tex":"b_r","aliases":"b_r","display":"b_r","kind":"introduction-passage","excerpt":"mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-10c299c3911c0c85","tex":"b_s","aliases":"b_s","display":"b_s","kind":"introduction-passage","excerpt":"mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d7d42b2ecde077f4","tex":"\\bar s","aliases":"bar s bar","display":"\\bar s","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-87909045a7273a12","tex":"\\bar s_C","aliases":"bar s_C bar","display":"\\bar s_C","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8aba1c70aa95f18d","tex":"\\bar s_C^2","aliases":"bar s_C^2 bar","display":"\\bar s_C^2","kind":"introduction-passage","excerpt":"raded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The lost path starts in retained data, enters a discard","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1dd2fc5c9ef2e084","tex":"\\mathrm{bright}","aliases":"bright mathrm","display":"\\mathrm{bright}","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":153},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C","display":null,"kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":46},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":". Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when","count":160},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-da5f510477c944c3","tex":"\\C^4","aliases":"C^4 C","display":null,"kind":"introduction-passage","excerpt":"on of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ace7df14d6591de3","tex":"\\C^{\\tau_p}","aliases":"C^τ_p C tau","display":null,"kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c8691383f78796e0","tex":"\\mathcal C_C","aliases":"C_C mathcal","display":"\\mathcal C_C","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6bb415dfb5cb7b14","tex":"\\mathcal C_F","aliases":"C_F mathcal","display":"\\mathcal C_F","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1e15d2e27313753c","tex":"C_N","aliases":"C_N","display":"C_N","kind":"introduction-passage","excerpt":"[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":43},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Now let H:D(H)⊂HH→HH be a densely defined symmetric operator. To write HP_aψ for every domain state, require P_aD(H)⊂ D(H). Define norm x_H^2=norm x^2+normHx^2. A domain-compatible atlas is graph compatible when S_Fx⟶ x in the graph norm for every x∈ D(H). Equivalently, both S_Fx→ x and HS_Fx→ Hx hold in HH. The second convergence does not follow f","count":191},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-62bae077a0709320","tex":"\\mathrm D","aliases":"D mathrm","display":"\\mathrm D","kind":"introduction-passage","excerpt":"Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implic","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.","count":141},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"introduction-passage","excerpt":"la-spectral-path-inverse For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the rec","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"introduction-passage","excerpt":"-path-inverse For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"introduction-passage","excerpt":"t and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recu","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4144475705f1d2e6","tex":"d_{n-1}","aliases":"d_n-1","display":"d_{n-1}","kind":"introduction-passage","excerpt":"t and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recu","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-30c3820e382d42d4","tex":"d_{n-2}","aliases":"d_n-2","display":"d_{n-2}","kind":"introduction-passage","excerpt":"t and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recu","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d3b6e2eaef5fb02a","tex":"d_v","aliases":"d_v","display":"d_v","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-052240bfc014845d","tex":"d_v^{-1/2}","aliases":"d_v^-1/2","display":"d_v^{-1/2}","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"introduction-passage","excerpt":"[Dirichlet path determinant and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-35679b068dabbae2","tex":"\\dot P","aliases":"dot P dot","display":"\\dot P","kind":"introduction-passage","excerpt":"The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and complete","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e4b951c58979abc7","tex":"\\dot P_a","aliases":"dot P_a dot","display":"\\dot P_a","kind":"introduction-passage","excerpt":"The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and complete","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d1997e4b5359f675","tex":"\\dot U","aliases":"dot U dot","display":"\\dot U","kind":"introduction-passage","excerpt":"The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and complete","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-913b6583a9281098","tex":"\\dot X","aliases":"dot X dot","display":"\\dot X","kind":"introduction-passage","excerpt":"The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and complete","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"tood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^","count":149},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5a3fc062502fca22","tex":"\\mathsf E","aliases":"E mathsf","display":"\\mathsf E","kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":38},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":286},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cc613c8c38547509","tex":"e^{-\\mathrm{i}tH_{\\mathrm{bright}}}","aliases":"e^-itH_bright mathrm mathrm","display":"e^{-\\mathrm{i}tH_{\\mathrm{bright}}}","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7abee99367ff24ec","tex":"e^{-itH_{h,n}}","aliases":"e^-itH_h,n","display":"e^{-itH_{h,n}}","kind":"introduction-passage","excerpt":"Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-41db5006fbb2b3f4","tex":"e^{-itH_n}","aliases":"e^-itH_n","display":"e^{-itH_n}","kind":"introduction-passage","excerpt":"Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-acc086f034dde581","tex":"e^{-\\mathrm{i}t\\overline E}","aliases":"e^-itoverline E mathrm overline","display":"e^{-\\mathrm{i}t\\overline E}","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"introduction-passage","excerpt":"Let e_0=(1,0,0,0)^T denote the first spinor basis vector, and choose","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-58f890b6f450ab6a","tex":"E_{3,1}","aliases":"E_3,1","display":"E_{3,1}","kind":"introduction-passage","excerpt":"hypergeometric-lane Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27)","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ad3caf981ac30812","tex":"E_c","aliases":"E_c","display":"E_c","kind":"introduction-passage","excerpt":"tood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"introduction-passage","excerpt":"it-walk-refinement-counterexample Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p. For an old outgoi","count":14},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bb785fdcd708a453","tex":"E_{L,j}","aliases":"E_L,j","display":"E_{L,j}","kind":"introduction-passage","excerpt":"· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d8c9f25b4e1a222a","tex":"\\mathsf E_m","aliases":"E_m mathsf","display":"\\mathsf E_m","kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":14},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d4c39e5e1f0a0028","tex":"e_x","aliases":"e_x","display":"e_x","kind":"introduction-passage","excerpt":"Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing t","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28d5a6a46f63be54","tex":"e_{\\tau x}","aliases":"e_τ x tau","display":"e_{\\tau x}","kind":"introduction-passage","excerpt":"Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing t","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"introduction-passage","excerpt":"hypergeometric-lane Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27)","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f43c89d5480f3fa1","tex":"F_{L-1}","aliases":"F_L-1","display":"F_{L-1}","kind":"introduction-passage","excerpt":"· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-30ade5d863641faa","tex":"F_s","aliases":"F_s","display":"F_s","kind":"introduction-passage","excerpt":"mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5ac97b405fd2d8eb","tex":"F_{u\\to v}^n","aliases":"F_u→ v^n to","display":"F_{u\\to v}^n","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5b2cee62bbda2690","tex":"F_{v\\to u}^n","aliases":"F_v→ u^n to","display":"F_{v\\to u}^n","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":73},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"We use the forward convention of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilber","count":118},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-81a99880d3bfaeb5","tex":"G_{p,w}","aliases":"G_p,w","display":"G_{p,w}","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8bea312cd06acfdb","tex":"G_v","aliases":"G_v","display":"G_v","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ec2d03508e15ce2","tex":"H_{\\mathrm{bright}}","aliases":"H_bright mathrm","display":"H_{\\mathrm{bright}}","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-51a7901d5bb6ee54","tex":"H_{h,n}","aliases":"H_h,n","display":"H_{h,n}","kind":"introduction-passage","excerpt":"Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e246b7f4f9bbfa9","tex":"H_{h,S}","aliases":"H_h,S","display":"H_{h,S}","kind":"introduction-passage","excerpt":"Let H_S,H_h,S denote the finite block diagonal Hamiltonians. In coefficient coordinates, squared norm and continuum Hamiltonian expectation are","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"introduction-passage","excerpt":"Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+","count":33},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2e77d5fad40985b0","tex":"H_S","aliases":"H_S","display":"H_S","kind":"introduction-passage","excerpt":"Let H_S,H_h,S denote the finite block diagonal Hamiltonians. In coefficient coordinates, squared norm and continuum Hamiltonian expectation are","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-32945ba3eade4c63","tex":"\\HH_{\\mathrm{arc}}","aliases":"HH_arc HH mathrm","display":null,"kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-45418eea5d6e493e","tex":"\\HH_p","aliases":"HH_p HH","display":null,"kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-510e2db8852c553e","tex":"\\mathrm{i}","aliases":"i mathrm","display":"\\mathrm{i}","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0f6f9e87439c63ef","tex":"\\mathrm{id}","aliases":"id mathrm","display":"\\mathrm{id}","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8a8b98766d675837","tex":"\\mathrm{id}_C","aliases":"id_C mathrm","display":"\\mathrm{id}_C","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-71b9c5b82b8f477d","tex":"\\mathrm{id}_F","aliases":"id_F mathrm","display":"\\mathrm{id}_F","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bb9a538f5e406f30","tex":"K_n","aliases":"K_n","display":"K_n","kind":"introduction-passage","excerpt":"[Dirichlet path determinant and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3ab3f094fe0f10d8","tex":"K_{n-1}","aliases":"K_n-1","display":"K_{n-1}","kind":"introduction-passage","excerpt":"nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x. This is exactly the recurrence and initial data for U_n(x/2). Deleting the first or last row and column leaves K_n-1, proving the diagona","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-309607a909e10c97","tex":"\\KK","aliases":"KK","display":null,"kind":"introduction-passage","excerpt":"Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+","count":33},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9d0fbf856fe55b03","tex":"L^1","aliases":"L^1","display":"L^1","kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":57},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-318bb55567cfc011","tex":"L^L","aliases":"L^L","display":"L^L","kind":"introduction-passage","excerpt":"· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,","count":187},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0901f1f56ba2bc5e","tex":"M_N","aliases":"M_N","display":"M_N","kind":"introduction-passage","excerpt":"Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e16a4907811a989","tex":"m_x","aliases":"m_x","display":"m_x","kind":"introduction-passage","excerpt":"Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing t","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a6a28ba806797f76","tex":"m_{\\tau x}","aliases":"m_τ x tau","display":"m_{\\tau x}","kind":"introduction-passage","excerpt":"Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing t","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"introduction-passage","excerpt":"[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p","count":142},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":374},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ebcd9b4bdca5cfcd","tex":"N^r","aliases":"N^r","display":"N^r","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0ff3814053977f1f","tex":"\\overline E","aliases":"overline E overline","display":"\\overline E","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a71d099153a6ba75","tex":"p^{-(h-r+1)/2}","aliases":"p^-(h-r+1)/2","display":"p^{-(h-r+1)/2}","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7441b5cfbaa2c0c0","tex":"p^m","aliases":"p^m","display":"p^m","kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-da8a2616906ad75b","tex":"p^{r-1}","aliases":"p^r-1","display":"p^{r-1}","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,","count":14},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-95c96988a1a8fc97","tex":"p^w","aliases":"p^w","display":"p^w","kind":"introduction-passage","excerpt":"inequation Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^w when the hidden amplitude stays radial.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a825e050f8755dad","tex":"P_+","aliases":"P_+","display":"P_+","kind":"introduction-passage","excerpt":"x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is ","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cec3c913d3e4fd13","tex":"P_-","aliases":"P_-","display":"P_-","kind":"introduction-passage","excerpt":"x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is ","count":17},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4f0aaf3fe5bb5cd6","tex":"P_\\pm","aliases":"P_± pm","display":"P_\\pm","kind":"introduction-passage","excerpt":"refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"introduction-passage","excerpt":"e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanis","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,","count":135},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,","count":307},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a03f214a14787ce8","tex":"R_j","aliases":"R_j","display":"R_j","kind":"introduction-passage","excerpt":"j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right p","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b5c9a939784c608f","tex":"R_p","aliases":"R_p","display":"R_p","kind":"introduction-passage","excerpt":"The polynomial map R_p is defined at every x_j, while the Schur coefficients in [source reference] are not. Cancelling denominators in a determinant can cross this chart boundary; dividing by them in an eigenvector reconstruction cannot. A correct description there keeps the resonant cell modes or solves the full finite system. The fixed poles of t","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-76599921b0fd3fb7","tex":"R_{p,w}","aliases":"R_p,w","display":"R_{p,w}","kind":"introduction-passage","excerpt":"e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanis","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7fdf44047d9358ee","tex":"\\mathrm{red}","aliases":"red mathrm","display":"\\mathrm{red}","kind":"introduction-passage","excerpt":"Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implic","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"introduction-passage","excerpt":"Let H_S,H_h,S denote the finite block diagonal Hamiltonians. In coefficient coordinates, squared norm and continuum Hamiltonian expectation are","count":156},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":140},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8fb5b936111102da","tex":"S_{A,\\tau,u}","aliases":"S_A,τ,u tau","display":"S_{A,\\tau,u}","kind":"introduction-passage","excerpt":"x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is ","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a9e11db6243bd912","tex":"S_F","aliases":"S_F","display":"S_F","kind":"introduction-passage","excerpt":"Let (P_a)_a∈ A be a countable orthogonal family of bounded projections with strong sum I. For finite F⊂ A, write S_F=∑_a∈ FP_a. Strong completeness means S_Fx→ x for each fixed x∈HH, over the directed set of finite subsets. Orthogonality gives normS_Fx^2=∑_a∈ FnormP_ax^2, ∑_anormP_ax^2=norm x^2. These are Hilbert-space statements. They contain no u","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-26736bc4d2f28e27","tex":"s_F","aliases":"s_F","display":"s_F","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a8c7172f7f354bea","tex":"s_F^2","aliases":"s_F^2","display":"s_F^2","kind":"introduction-passage","excerpt":"What direct projection loses. Let ( C_F,s_F) be a graded fine complex with s_F^2=0. Let P: C_F→ C_C and I: C_C→ C_F be degree-zero maps satisfying PI=id_C, and set Q=id_F-IP, bar s_C=Ps_FI. Here I is the inclusion map, whereas id denotes an identity operator. Insertion of IP=id_F-Q proves the exact identity bar s_C^2 =Ps_F(IP)s_FI =-Ps_FQs_FI. The ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1d641eb69668a366","tex":"s_v","aliases":"s_v","display":"s_v","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"introduction-passage","excerpt":"e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanis","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"introduction-passage","excerpt":"[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p","count":19},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"introduction-passage","excerpt":"The source's transported Clifford action is defined by v↦ T(A(T^-1v)). The same is true for the displayed diagonal beta action. Conjugation preserves real coefficients and finite sums, proving the Hamiltonian equality. Apply the exponential identity with z=-it. Substituting each coefficient equality into the finite Fourier sum then proves equality ","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1e83927f5099f8d9","tex":"T_{p,h}","aliases":"T_p,h","display":"T_{p,h}","kind":"introduction-passage","excerpt":"[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b1dcc31cf92718a0","tex":"T_{p,h+1}","aliases":"T_p,h+1","display":"T_{p,h+1}","kind":"introduction-passage","excerpt":"[A boundary coin breaks the canonical arc embedding] Take h≥1 and N≥3, and embed the old arcs of C_Nsquare T_p,h in C_Nsquare T_p,h+1 by identical labels, with zero amplitude on new arcs. Denote this isometry by J. At an old leaf, the old degree is d=3: two cycle neighbors and one parent. The larger graph adds p children, changing the degree to d+p","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"introduction-passage","excerpt":"Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY, (δ_BX)^* · =δ_B(X^*), Tr((δ_BX)Y) · =-Tr(Xδ_BY). For the first identity, the middle terms -iXBY and iXBY cancel. For th","count":23},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":144},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1296a5898d2e7d5f","tex":"U^*","aliases":"U^*","display":"U^*","kind":"introduction-passage","excerpt":"The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and complete","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-35ecd39772b3647e","tex":"u^g","aliases":"u^g","display":"u^g","kind":"introduction-passage","excerpt":"We use the forward convention of the formal refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilber","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-167a8b5df4139bcf","tex":"U_n","aliases":"U_n","display":"U_n","kind":"introduction-passage","excerpt":"[Dirichlet path determinant and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f221f487fed33512","tex":"U_{n-1}","aliases":"U_n-1","display":"U_{n-1}","kind":"introduction-passage","excerpt":"[Dirichlet path determinant and endpoint entries] For n≥1, det K_n(x)=U_n(x/2). If this determinant is nonzero, (K_n(x)^-1)_11=(K_n(x)^-1)_nn =fracU_n-1(x/2)U_n(x/2), (K_n(x)^-1)_1n=frac1U_n(x/2). For n=1 these entries all denote the same value 1/x. Expansion along the last row gives the determinant recurrence d_n=xd_n-1-d_n-2, with d_0=1 and d_1=x","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+","count":101},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":171},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a0402d5bad7ad318","tex":"V_{h,n}","aliases":"V_h,n","display":"V_{h,n}","kind":"introduction-passage","excerpt":"Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a6735e9f71ad2ebf","tex":"V_n","aliases":"V_n","display":"V_n","kind":"introduction-passage","excerpt":"Apply this construction to V_n(t)=e^-itH_n, V_h,n(t)=e^-itH_h,n. The finite-dimensional norm preservation already proved supplies its hypothesis. Define (U(t)a)_n=V_n(t)a_n, (U_h(t)a)_n=V_h,n(t)a_n. Both maps are everywhere-defined unitary maps on KK. Their inverses are obtained at time -t. The modewise time-group laws imply U(s+t)=U(s)U(t), U_h(s+","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bb5783d1e38c64af","tex":"\\vec E","aliases":"vec E vec","display":"\\vec E","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","excerpt":"Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at ","count":35},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":97},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-65d03882bd3591c3","tex":"w^q","aliases":"w^q","display":"w^q","kind":"introduction-passage","excerpt":"mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6b00da79051237b2","tex":"W_{0j}","aliases":"W_0j","display":"W_{0j}","kind":"introduction-passage","excerpt":"j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right p","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-107faad68a0eb5b2","tex":"W_{1j}","aliases":"W_1j","display":"W_{1j}","kind":"introduction-passage","excerpt":"j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right p","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f504f709b964471b","tex":"W_{ij}","aliases":"W_ij","display":"W_{ij}","kind":"introduction-passage","excerpt":"Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"The same principle governs a time-dependent operator block. Write V=dot UU^* for the frame velocity in laboratory coordinates. Then dot P_a=[V,P_a], fracddt(P_bXP_a) =[V,P_b]XP_a+P_bdot XP_a+P_bX[V,P_a]. Both packet boundaries contribute, as does the operator itself. Conjugation by a general invertible family also preserves idempotence and complete","count":103},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-71a31718ae729ffa","tex":"X^*","aliases":"X^*","display":"X^*","kind":"introduction-passage","excerpt":"Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY, (δ_BX)^* · =δ_B(X^*), Tr((δ_BX)Y) · =-Tr(Xδ_BY). For the first identity, the middle terms -iXBY and iXBY cancel. For th","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"introduction-passage","excerpt":"Let (P_a)_a∈ A be a countable orthogonal family of bounded projections with strong sum I. For finite F⊂ A, write S_F=∑_a∈ FP_a. Strong completeness means S_Fx→ x for each fixed x∈HH, over the directed set of finite subsets. Orthogonality gives normS_Fx^2=∑_a∈ FnormP_ax^2, ∑_anormP_ax^2=norm x^2. These are Hilbert-space statements. They contain no u","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b16b8b5c9ec900ff","tex":"X_{0j}","aliases":"X_0j","display":"X_{0j}","kind":"introduction-passage","excerpt":"j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right p","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f26f4a25faf10ece","tex":"X_{0j}^{-1}","aliases":"X_0j^-1","display":"X_{0j}^{-1}","kind":"introduction-passage","excerpt":"j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at (0,j), are R_j=(X_0jW_1jX_0j^-1)W_0j, j=0,1. The conjugation transports the right p","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-61836166435158f9","tex":"x_H^2","aliases":"x_H^2","display":"x_H^2","kind":"introduction-passage","excerpt":"Now let H:D(H)⊂HH→HH be a densely defined symmetric operator. To write HP_aψ for every domain state, require P_aD(H)⊂ D(H). Define norm x_H^2=norm x^2+normHx^2. A domain-compatible atlas is graph compatible when S_Fx⟶ x in the graph norm for every x∈ D(H). Equivalently, both S_Fx→ x and HS_Fx→ Hx hold in HH. The second convergence does not follow f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b93e02a435f89a0","tex":"X_{i,j+1}^{-1}","aliases":"X_i,j+1^-1","display":"X_{i,j+1}^{-1}","kind":"introduction-passage","excerpt":"Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0298cc60bfc2d7e5","tex":"X_{ij}","aliases":"X_ij","display":"X_{ij}","kind":"introduction-passage","excerpt":"Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"introduction-passage","excerpt":"The polynomial map R_p is defined at every x_j, while the Schur coefficients in [source reference] are not. Cancelling denominators in a determinant can cross this chart boundary; dividing by them in an eigenvector reconstruction cannot. A correct description there keeps the resonant cell modes or solves the full finite system. The fixed poles of t","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY, (δ_BX)^* · =δ_B(X^*), Tr((δ_BX)Y) · =-Tr(Xδ_BY). For the first identity, the middle terms -iXBY and iXBY cancel. For th","count":49},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3e697fdc5343c7a2","tex":"Y_{i+1,j}","aliases":"Y_i+1,j","display":"Y_{i+1,j}","kind":"introduction-passage","excerpt":"Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-25a00e69dec9ace2","tex":"Y_{ij}","aliases":"Y_ij","display":"Y_{ij}","kind":"introduction-passage","excerpt":"Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8a25f3ab0c5819b4","tex":"Y_{ij}^{-1}","aliases":"Y_ij^-1","display":"Y_{ij}^{-1}","kind":"introduction-passage","excerpt":"Write X_ij for the backward horizontal link from (i+1,j) to (i,j), where i=0,1 and j=0,1,2. Write Y_ij for the backward vertical link from (i,j+1) to (i,j), where i=0,1,2 and j=0,1. Define the four fine plaquettes by W_ij=X_ijY_i+1,jX_i,j+1^-1Y_ij^-1, i,j∈{0,1}. Each W_ij acts at its own lower-left basepoint (i,j). The two row holonomies, based at ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":25},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b","count":196},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4140de75758a272d","tex":"z^3","aliases":"z^3","display":"z^3","kind":"introduction-passage","excerpt":"hypergeometric-lane Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3;z^3/27)","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"introduction-passage","excerpt":"· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4e5172da9206de94","tex":"z^L","aliases":"z^L","display":"z^L","kind":"introduction-passage","excerpt":"· =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every member is positive, so none is a forbidden nonpositive integer in a hypergeometric denominator. Since (1)_q=q!, E_L,j(z)=fracz^jj! _0F_L-1 (- A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b910972bdb78c268","tex":"\\Z_p","aliases":"Z_p Z","display":null,"kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":243},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":61},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"introduction-passage","excerpt":"refinement modules. A link u(x) transports from x to τ x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implic","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8474abafe881bcf4","tex":"\\delta_B","aliases":"δ_B delta","display":"\\delta_B","kind":"introduction-passage","excerpt":"Move now to M_N(C). To avoid confusing a Hermitian observable with the skew-adjoint state generator K, denote a Hermitian matrix by B=B^* and define δ_BX=i[B,X]=i(BX-XB). Then direct multiplication proves δ_B(XY) · =(δ_BX)Y+Xδ_BY,","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"introduction-passage","excerpt":"tood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when Ω=0, equivalently when both δ and G_p,w vanish. On resonance E_c=ϵ it reduces to sin^2(|G_p,w|t). The individual descendant probabilities are Pr(R_p,w;t)/p^","count":93},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-890e777d8bea7959","tex":"\\epsilon_0","aliases":"ϵ_0 epsilon","display":"\\epsilon_0","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"introduction-passage","excerpt":"Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implic","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca62cb8c93dad62f","tex":"\\iota^*","aliases":"ι^* iota","display":"\\iota^*","kind":"introduction-passage","excerpt":"Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implic","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-03cd5a90c4468e52","tex":"\\Lambda^2","aliases":"Λ^2 Lambda","display":"\\Lambda^2","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":70},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,","count":100},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-21e99a4d2b6a614c","tex":"\\rho_v^n","aliases":"ρ_v^n rho","display":"\\rho_v^n","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3628d2a1d0d8cb5d","tex":"\\rho_v^{n+1}","aliases":"ρ_v^n+1 rho","display":"\\rho_v^{n+1}","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"introduction-passage","excerpt":"Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implic","count":40},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","excerpt":"Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing t","count":67},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"introduction-passage","excerpt":"Midpoint geometry and independent gauges. Write e_x for the retained endpoint and m_x for the midpoint of the edge x→τ x. The fine translation is the bijection σ e_x=m_x, σ m_x=e_τ x. Consequently σ^2e_x=e_τ x and σ^2m_x=m_τ x. This is the precise sense in which two half translations produce one coarse translation. Assigning half the edge spacing t","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":71},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cff2c5f7d8b191ec","tex":"\\tau^{-1}","aliases":"τ^-1 tau","display":"\\tau^{-1}","kind":"introduction-passage","excerpt":"x, so u^g(x)=g(τ x)u(x)g(x)^-1. Thus u(x)=U(x)^-1 when U(x) denotes the backward link used in the covariant difference stencil. Products act from right to left. All links in this part belong to U(1); the unit modulus assumption is used in every Hilbert norm statement. Let A=A^*, A^2=1, act on C^4, and put P_±=(1± A)/2. The directional transport is ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e675956848d7751d","tex":"\\tau_2","aliases":"τ_2 tau","display":"\\tau_2","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-96a7e03674728790","tex":"\\tau_p","aliases":"τ_p tau","display":"\\tau_p","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":50},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":36},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea3431124039103d","tex":"\\phi^n","aliases":"ϕ^n phi","display":"\\phi^n","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-12e16dc509b4020a","tex":"\\Phi_e","aliases":"Φ_e Phi","display":"\\Phi_e","kind":"introduction-passage","excerpt":"The electrovac sector sets the Dirac field to zero and imposes spherical symmetry. It retains a real canonical Einstein--Maxwell system, but it has no nonspherical gravitational radiation. Work in units G=c=4πϵ_0=1. A prime denotes a radial derivative and a dot denotes a time derivative. On a radial interval where R>0 and Λ>0, write ds^2=-N^2dt^2+Λ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d28947f5acd9f188","tex":"\\psi^n","aliases":"ψ^n psi","display":"\\psi^n","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c498202a7960764","tex":"\\psi_{r,a,q}^{(h)}","aliases":"ψ_r,a,q^(h) psi","display":"\\psi_{r,a,q}^{(h)}","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":45},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,","count":37},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-23a05885bf7340b9","tex":"\\Omega^2","aliases":"Ω^2 Omega","display":"\\Omega^2","kind":"introduction-passage","excerpt":"G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc7236f880f8be15","tex":"\\omega_p","aliases":"ω_p omega","display":"\\omega_p","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a791b3585eff1a50","tex":"\\omega_p^{q d_r(x)}","aliases":"ω_p^q d_r(x) omega","display":"\\omega_p^{q d_r(x)}","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dadabf5f8288039e","tex":"\\Omega_{\\mathrm{red}}","aliases":"Ω_red Omega mathrm","display":"\\Omega_{\\mathrm{red}}","kind":"introduction-passage","excerpt":"Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implic","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d7aecc42f2bbbfbb","tex":"\\Omega_\\Sigma","aliases":"Ω_Σ Omega Sigma","display":"\\Omega_\\Sigma","kind":"introduction-passage","excerpt":"Descent and the boundary of the chart. Let ι denote the local solution map for the second-class relations and let {·,·}_ D be its regular Dirac bracket. If an already constructed boundary BFV vector field is tangent to that second-class surface, and its charge descends through the regular reduction, then Ω_red=ι^*Ω_Σ, {Ω_red,Ω_red}_ D=0. The implic","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1,","count":25},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":50},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":26},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":75},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"introduction-passage","excerpt":"The source's transported Clifford action is defined by v↦ T(A(T^-1v)). The same is true for the displayed diagonal beta action. Conjugation preserves real coefficients and finite sums, proving the Hamiltonian equality. Apply the exponential identity with z=-it. Substituting each coefficient equality into the finite Fourier sum then proves equality ","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"introduction-passage","excerpt":"mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0fcd1056f96dbe36","tex":"\\prod_r","aliases":"∏_r prod","display":"\\prod_r","kind":"introduction-passage","excerpt":"mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-07961363fdf94e98","tex":"\\sum_{a\\in F}","aliases":"∑_a∈ F sum in","display":"\\sum_{a\\in F}","kind":"introduction-passage","excerpt":"Let (P_a)_a∈ A be a countable orthogonal family of bounded projections with strong sum I. For finite F⊂ A, write S_F=∑_a∈ FP_a. Strong completeness means S_Fx→ x for each fixed x∈HH, over the directed set of finite subsets. Orthogonality gives normS_Fx^2=∑_a∈ FnormP_ax^2, ∑_anormP_ax^2=norm x^2. These are Hilbert-space statements. They contain no u","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eb07f8d7b26ea999","tex":"\\sum_{q\\geq0}","aliases":"∑_q≥0 sum geq","display":"\\sum_{q\\geq0}","kind":"introduction-passage","excerpt":"mathcal A_L,j; fracz^LL^L). Here _0F_s(;b_1,…,b_s;w) is defined by ∑_q≥0w^q/[q!∏_r(b_r)_q]. The ratio of successive terms has magnitude |w|/[(q+1)∏_r|q+b_r|], which tends to zero. Its series is entire in w, so [source reference] is an everywhere convergent identity with the factorial series. For example A_3,1={2/3,4/3} and E_3,1(z)=z _0F_2(;2/3,4/3","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d96e96aadd273752","tex":"\\sum_u","aliases":"∑_u sum","display":"\\sum_u","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-93b73f4100ccda1a","tex":"\\sum_{u\\sim v}","aliases":"∑_u∼ v sum sim","display":"\\sum_{u\\sim v}","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":56},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-edf83e015e4dd5ae","tex":"\\int_{x+p^m\\Z_p}","aliases":"∫_x+p^mZ_p int Z","display":null,"kind":"introduction-passage","excerpt":"For f∈HH_p, define ( E_m f)(x) =p^m∫_x+p^mZ_p f(y) dμ_p(y). The integral is defined since L^2 is contained in L^1 on this probability space. It produces a function constant on each depth-m cylinder.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"introduction-passage","excerpt":"Write ψ^n(v,u) for the internal amplitude on an arc leaving v before the coin, and ϕ^n=Cψ^n for its post-coin value. Define ρ_v^n=∑_u∼ vnormψ^n(v,u)^2, F_v→ u^n=normϕ^n(v,u)^2. The local unitary coin gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"introduction-passage","excerpt":"Write ω_p=exp(2πi/p). For 1≤ r≤ h, parent prefix a∈{0,…,p^r-1-1}, and character q∈{1,…,p-1}, define ψ_r,a,q^(h)(x)= p^-(h-r+1)/2ω_p^q d_r(x), · x≡ apmodp^r-1, 0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"introduction-passage","excerpt":"Let (P_a)_a∈ A be a countable orthogonal family of bounded projections with strong sum I. For finite F⊂ A, write S_F=∑_a∈ FP_a. Strong completeness means S_Fx→ x for each fixed x∈HH, over the directed set of finite subsets. Orthogonality gives normS_Fx^2=∑_a∈ FnormP_ax^2, ∑_anormP_ax^2=norm x^2. These are Hilbert-space statements. They contain no u","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":37},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","excerpt":"Let vec E denote the directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion","count":49},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","excerpt":". Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_bright-overline E IΩ sin(Ω t)]. At Ω=0 this formula is understood by continuous extension. Starting in |c⟩, the radial probability is therefore Pr(R_p,w;t)= frac|G_p,w|^2δ^2+|G_p,w|^2 sin^2 (tsqrtδ^2+|G_p,w|^2). The probability is defined to be zero when","count":93},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"introduction-passage","excerpt":"Now let H:D(H)⊂HH→HH be a densely defined symmetric operator. To write HP_aψ for every domain state, require P_aD(H)⊂ D(H). Define norm x_H^2=norm x^2+normHx^2. A domain-compatible atlas is graph compatible when S_Fx⟶ x in the graph norm for every x∈ D(H). Equivalently, both S_Fx→ x and HS_Fx→ Hx hold in HH. The second convergence does not follow f","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2a3551ffdc934706","tex":"+_p","aliases":"+_p","display":"+_p","kind":"use","excerpt":"One more digit supplies p children for every address. The uniform refinement map is (V_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b0fd131ff163229e","tex":"+_x","aliases":"+_x","display":"+_x","kind":"use","excerpt":"Z_sign have eigenvalues +1,-1 on a dyadic sign qubit, and let B=B^* act on the remaining factors. Assume there are orthonormal states |b_0⟩,|b_1⟩ with B|b_0⟩=0 and B|b_1⟩=|b_1⟩. One concrete choice is an odd-prime torsion involution times a fixed odd-prime detail projector, with the torsion state in its +1 eigenspace. For real g≠0, use H_int=gZ_sig","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-23aa8a462b6e6789","tex":"\\boldsymbol1","aliases":"1 boldsymbol","display":"\\boldsymbol1","kind":"use","excerpt":"For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-256011eff0fc0b21","tex":"\\boldsymbol1_{\\mathrm c}","aliases":"1_ c boldsymbol mathrm","display":"\\boldsymbol1_{\\mathrm c}","kind":"use","excerpt":"For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9022dcc510c5116d","tex":"\\boldsymbol1_{\\mathrm f}","aliases":"1_ f boldsymbol mathrm","display":"\\boldsymbol1_{\\mathrm f}","kind":"use","excerpt":"For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a107946ab1522f43","tex":"\\mathbf1_{B_{m,a}}","aliases":"1_B_m,a mathbf","display":"\\mathbf1_{B_{m,a}}","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df2fdc1df922b224","tex":"\\mathbf1_{B_{m,a+d p^{m-1}}}","aliases":"1_B_m,a+d p^m-1 mathbf","display":"\\mathbf1_{B_{m,a+d p^{m-1}}}","kind":"use","excerpt":"For an explicit basis, put ω_p=exp(2π i/p) and define ψ_m,a,q=p^(m-1)/2∑_d=0^p-1 ω_p^qd1_B_m,a+d p^m-1, m≥1, 0≤ a<p^m-1, 1≤ q<p. Each has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ee4af272d467ef2b","tex":"\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^r)}","aliases":"1_x≡ y (mod p^r) mathbf equiv mathrm","display":"\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^r)}","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determine","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-572d3cc9f9fb28e2","tex":"\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}","aliases":"1_x≡ y (mod p^r-1) mathbf equiv mathrm","display":"\\mathbf1_{x\\equiv y\\,(\\mathrm{mod}\\,p^{r-1})}","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determine","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-033181fac41ece68","tex":"\\mathbf1_{\\{x\\equiv y\\, (\\mathrm{mod}\\,p^r)\\}}","aliases":"1_{x≡ y (mod p^r)} mathbf equiv mathrm","display":"\\mathbf1_{\\{x\\equiv y\\, (\\mathrm{mod}\\,p^r)\\}}","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-02cdc56fb77a61c8","tex":"A^{\\mathcal H}","aliases":"A^ H mathcal","display":"A^{\\mathcal H}","kind":"use","excerpt":"Proposition PH09 (coordinate conjugacy of generators and exponentials). Let A be a complex-linear endomorphism of A, and let A^ H=TAT^-1. For every complex z,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-44f43aa87edf0679","tex":"a^{(r)}","aliases":"a^(r)","display":"a^{(r)}","kind":"use","excerpt":"Closedness follows from self-adjointness. It can also be seen directly for this multiplier. If a^(r)→ a and widehat H_ma^(r)→ b in KK, then convergence holds at each coordinate. Since the finite matrix H_n is continuous, b_n=lim_rH_na_n^(r)=H_na_n. The output sequence equals b∈KK, so a belongs to the maximal domain and widehat H_ma=b. This direct a","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-00ae19a4939e8cce","tex":"\\mathcal A^*","aliases":"A^* mathcal","display":"\\mathcal A^*","kind":"use","excerpt":"section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality f","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify th","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a cor","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7b48479868582d73","tex":"a^3","aliases":"a^3","display":"a^3","kind":"use","excerpt":"Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a corner instead generally leaves an O(a^3) phase error and hence an O(a^5) action-density error. These ord","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e046b0ebe8054038","tex":"a^4","aliases":"a^4","display":"a^4","kind":"use","excerpt":"^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-09acaea0cd848de9","tex":"a^5","aliases":"a^5","display":"a^5","kind":"use","excerpt":"abelg:wilson-expansion 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a corner instead generally leaves an O(a^3) phase error and hence an O(a^5) action-density error. ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-79f1bb0d0b74b664","tex":"a^6","aliases":"a^6","display":"a^6","kind":"use","excerpt":"u is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b51192cce36ea46d","tex":"a^8","aliases":"a^8","display":"a^8","kind":"use","excerpt":"ered-plaquette θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(a^8). Evaluating the curvature at a corner instead generally leaves an O(a^3) phase error and he","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c44dac86d60fd4be","tex":"\\mathcal A^{\\mathrm{ad}}_{p,h}","aliases":"A^ad_p,h mathcal mathrm","display":"\\mathcal A^{\\mathrm{ad}}_{p,h}","kind":"use","excerpt":"For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-be96b81185b4c8b5","tex":"a^g","aliases":"a^g","display":"a^g","kind":"use","excerpt":"Coarse endpoint gauges g(x) and midpoint gauges q(x) may be chosen independently. The two sublinks transform by a^g(x)=q(x)a(x)g(x)^-1, b^g(x)=g(τ x)b(x)q(x)^-1. The midpoint factors cancel in b^ga^g, leaving the required coarse transformation. At the midpoint, a^g(x)g(x)ψ(x)=q(x)a(x)ψ(x), while endpoint covariance is immediate. Hence interpolation","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e0424812d1c195ce","tex":"A^k","aliases":"A^k","display":"A^k","kind":"use","excerpt":"Proof. Induction gives (TAT^-1)^k=TA^kT^-1. For k=0, use TT^-1=I; for the induction step, cancel the adjacent T^-1T. Every exponential partial sum therefore satisfies","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-943bc84d09f5dc3b","tex":"a^k","aliases":"a^k","display":"a^k","kind":"use","excerpt":"is sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of the same operator, so there is no commuting-operator assumption. For the tail, bound a lane ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c60967b39bed49c7","tex":"a^n","aliases":"a^n","display":"a^n","kind":"use","excerpt":"is sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of the same operator, so there is no commuting-operator assumption. For the tail, bound a lane ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1d7c8ce7a043e522","tex":"a^{Q+1}","aliases":"a^Q+1","display":"a^{Q+1}","kind":"use","excerpt":"∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lane, the omitted operator norm is at most e^afraca^Q+1(Q+1)!, a=|t|normA_0. The sum of norms of all exponential terms is at most e^a. A lane is a subseries of this sum, so it converges absolutely. The character calculation and factorial split can therefore be applied i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b2c8f1aafb4a6058","tex":"\\mathcal A^{\\mathrm{std}}_{p,h}","aliases":"A^std_p,h mathcal mathrm","display":"\\mathcal A^{\\mathrm{std}}_{p,h}","kind":"use","excerpt":"For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-004a43f65c127c71","tex":"a^\\dagger","aliases":"a^† dagger","display":"a^\\dagger","kind":"use","excerpt":"Finally the annihilation operator routes adjacent residues: Q_r^(L)aQ_s^(L)=0 unless r≡ s-1pmod L, Q_r^(L)a^† Q_s^(L)=0 unless r≡ s+1pmod L. On a finite number-state combination this follows from a|n⟩=sqrt n |n-1⟩ and a^†|n⟩=sqrtn+1 |n+1⟩. Each residue projector preserves Dom(a) and Dom(a^†), because it only removes terms from their weighted number","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-60168b3ae1d1b815","tex":"a_{\\mathbf d}","aliases":"a_ d mathbf","display":"a_{\\mathbf d}","kind":"use","excerpt":"). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt n γ_p,w. Then H_int|c⟩=G_p,w|R_p,w⟩ and H_int|R_p,w⟩= overlineG_p,w|c⟩. If |A⟩=∑_ da_ d| d⟩ is orthogonal to |R_p,w⟩, its coefficients satisfy ∑_ da_ d=0. Hence H_int|A⟩=0. Only the radial vector is bright under this equal-leaf int","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8200d9e2af5f6cdb","tex":"A_{\\mathbf r}","aliases":"A_ r mathbf","display":"A_{\\mathbf r}","kind":"use","excerpt":"[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"Summable refinement and what a limit still requires. For the dyadic mesh h_n=a_0 2^-n, the regular estimate produces the nonnegative sequence ε_n=C4^-n, C=a_0^2 A_∂. The actual generated operator norms are bounded by this sequence and hence are summable. This is more information than merely ε_n→0. A sequence with adjacent errors tending to zero nee","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-826262fb5a1181ed","tex":"a_0^2","aliases":"a_0^2","display":"a_0^2","kind":"use","excerpt":"Summable refinement and what a limit still requires. For the dyadic mesh h_n=a_0 2^-n, the regular estimate produces the nonnegative sequence ε_n=C4^-n, C=a_0^2 A_∂. The actual generated operator norms are bounded by this sequence and hence are summable. This is more information than merely ε_n→0. A sequence with adjacent errors tending to zero nee","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ed1a1979512421da","tex":"A_0^j","aliases":"A_0^j","display":"A_0^j","kind":"use","excerpt":"If A_0^L=α I, the hypergeometric expression reduces every lane to a scalar function multiplying A_0^j. The propagator lies in the finite span of I,A_0,…,A_0^L-1. A more general polynomial identity gives the corresponding finite algebra of powers. These statements follow from the supplied algebraic relation; they do not assert one for an arbitrary H","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6906ba14ccd2485d","tex":"A_0^L","aliases":"A_0^L","display":"A_0^L","kind":"use","excerpt":"If A_0^L=α I, the hypergeometric expression reduces every lane to a scalar function multiplying A_0^j. The propagator lies in the finite span of I,A_0,…,A_0^L-1. A more general polynomial identity gives the corresponding finite algebra of powers. These statements follow from the supplied algebraic relation; they do not assert one for an arbitrary H","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-93103c835d52db50","tex":"A_0^{L-1}","aliases":"A_0^L-1","display":"A_0^{L-1}","kind":"use","excerpt":"If A_0^L=α I, the hypergeometric expression reduces every lane to a scalar function multiplying A_0^j. The propagator lies in the finite span of I,A_0,…,A_0^L-1. A more general polynomial identity gives the corresponding finite algebra of powers. These statements follow from the supplied algebraic relation; they do not assert one for an arbitrary H","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e95e07807750445a","tex":"a_{2n+1,2n}","aliases":"a_2n+1,2n","display":"a_{2n+1,2n}","kind":"use","excerpt":"For a simple warning, the antisymmetric scalar array with a_2n,2n+1=1/n, a_2n+1,2n=-1/n for n≥1, and all other entries zero has finite support in each row but infinite total absolute sum. This array is only a summation example, not an asserted physical realization of the current. It shows why rowwise convergence alone cannot authorize arbitrary rea","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-126841ea628e2845","tex":"a_{2n,2n+1}","aliases":"a_2n,2n+1","display":"a_{2n,2n+1}","kind":"use","excerpt":"For a simple warning, the antisymmetric scalar array with a_2n,2n+1=1/n, a_2n+1,2n=-1/n for n≥1, and all other entries zero has finite support in each row but infinite total absolute sum. This array is only a summation example, not an asserted physical realization of the current. It shows why rowwise convergence alone cannot authorize arbitrary rea","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"[The resolved algebraic transfer step] Assume the continuum polynomial constraint identities in this convention and 16K<N. Their full-grid sampled evaluations obey {G[λ],G[μ]}_N · =G[λ×μ], {D[ξ],G[λ]}_N · =G[ξ^aδ_aλ], {D[ξ],D[η]}_N · =D[[ξ,η]], {G[λ],H[n]}_N · =0, {D[ξ],H[n]}_N · =H[ L_ξ^(-1)n], {H[n],H[m]}_N · =V[S(n,m)] =D[S(n,m)]+G[S^a(n,m)A_a].","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c57dad5f04c4ba4a","tex":"A_a^i","aliases":"A_a^i","display":"A_a^i","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c2c0df3352740364","tex":"A_a^j","aliases":"A_a^j","display":"A_a^j","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ac0ad444a337d8a7","tex":"A_b^i","aliases":"A_b^i","display":"A_b^i","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-69ddbdb9d32d9cf1","tex":"A_b^k","aliases":"A_b^k","display":"A_b^k","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8da2f1b2bafe729c","tex":"A_h","aliases":"A_h","display":"A_h","kind":"use","excerpt":"Direct and split radius evolution. On the positive branch, restriction of the radius equation to the reconstruction manifold gives dot R_j=g(R_j)+b_j(D_4R)_j, g(R)=-fracsqrt2MR-Q^2R, where b=N^r is prescribed in the electrovac benchmark. A split alternative is dot R=g(R)+A_h[b]R, A_h[b]u=frac12(bD_4u+D_4(bu)-(D_4b)u). On smooth data these formulas ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-daf800c8ae3d0681","tex":"a_{ij}","aliases":"a_ij","display":"a_{ij}","kind":"use","excerpt":"· -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are neede","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-622bd8519fe153e7","tex":"a_{j+1/2}","aliases":"a_j+1/2","display":"a_{j+1/2}","kind":"use","excerpt":"Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are neede","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28ff8ef535373c67","tex":"a_{Lq+j}","aliases":"a_Lq+j","display":"a_{Lq+j}","kind":"use","excerpt":", and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f327e0098ac5621a","tex":"A_{m,0}","aliases":"A_m,0","display":"A_{m,0}","kind":"use","excerpt":"[A quantified remainder and perturbation bound] Let p be a nonnegative integer and suppose normR(h)≤ C|h|^p, normg(h)-f(h)≤ε (0<|h|<δ), where C,ε≥0. Define A_m,p=∏_k=1^mfracb^k-p+1b^k-1. Then, throughout the same punctured interval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-290d99ad3e8de4dc","tex":"A_{m,p}","aliases":"A_m,p","display":"A_{m,p}","kind":"use","excerpt":"[A quantified remainder and perturbation bound] Let p be a nonnegative integer and suppose normR(h)≤ C|h|^p, normg(h)-f(h)≤ε (0<|h|<δ), where C,ε≥0. Define A_m,p=∏_k=1^mfracb^k-p+1b^k-1. Then, throughout the same punctured interval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"Two other bounded multipliers approach one along this positive spectrum: b(n)=fracnsqrt1+n^2, q(n)=fracn-in+i. Their correct anchored cutoff is A_N(f)=S_Nf(H_λ)S_N+(I-S_N). The errors are then normb(H_λ)-A_N(b) =sup_n>N|fracnsqrt1+n^2-1|⟶0, and normq(H_λ)-A_N(q) =sup_n>Nfrac2sqrtn^2+1⟶0. The identity tail is an analytic choice justified by these as","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1dc1a9b01a1e0c65","tex":"a_n^{(r)}","aliases":"a_n^(r)","display":"a_n^{(r)}","kind":"use","excerpt":"Closedness follows from self-adjointness. It can also be seen directly for this multiplier. If a^(r)→ a and widehat H_ma^(r)→ b in KK, then convergence holds at each coordinate. Since the finite matrix H_n is continuous, b_n=lim_rH_na_n^(r)=H_na_n. The output sequence equals b∈KK, so a belongs to the maximal domain and widehat H_ma=b. This direct a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-612a17fd14dd2159","tex":"a_{n_0}","aliases":"a_n_0","display":"a_{n_0}","kind":"use","excerpt":"a_n_0=frac1sqrt2e_0, a_n_A=frac12e_0, a_n_D=frac12e_0.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2fc3b3c3f3fa024a","tex":"a_{n_A}","aliases":"a_n_A","display":"a_{n_A}","kind":"use","excerpt":"a_n_0=frac1sqrt2e_0, a_n_A=frac12e_0, a_n_D=frac12e_0.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a7522c9b01a1bd98","tex":"a_{n_D}","aliases":"a_n_D","display":"a_{n_D}","kind":"use","excerpt":"a_n_0=frac1sqrt2e_0, a_n_A=frac12e_0, a_n_D=frac12e_0.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"use","excerpt":"Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-073c74444893e767","tex":"a_{\\rm tr}","aliases":"a_rm tr rm","display":"a_{\\rm tr}","kind":"use","excerpt":"· -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h · -innerA(a)b_rm tr_h -innera_rm trA(b)_h -∑_kinnerδ_ka_rm trδ_kb_rm tr_h. This expression is symmetric in a,b.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-895b4c69e53e4df6","tex":"A_x","aliases":"A_x","display":"A_x","kind":"use","excerpt":"Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a9f66cfc2078279a","tex":"A_y","aliases":"A_y","display":"A_y","kind":"use","excerpt":"Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2e74d3cc58dddca4","tex":"A_z","aliases":"A_z","display":"A_z","kind":"use","excerpt":"Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-06b5cc3e240ef745","tex":"a_\\mu","aliases":"a_μ mu","display":"a_\\mu","kind":"use","excerpt":"Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivati","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4d9af95cd8418d57","tex":"a_\\mu^2","aliases":"a_μ^2 mu","display":"a_\\mu^2","kind":"use","excerpt":"u is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-39dbcce24a8d9bcb","tex":"a_\\nu","aliases":"a_ν nu","display":"a_\\nu","kind":"use","excerpt":"Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivati","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1c9cbd53094797df","tex":"a_\\nu^2","aliases":"a_ν^2 nu","display":"a_\\nu^2","kind":"use","excerpt":"u is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5ac775657d62befa","tex":"\\mathcal A_{\\partial}","aliases":"A_∂ mathcal partial","display":"\\mathcal A_{\\partial}","kind":"use","excerpt":"[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f6c4b36b119e9415","tex":"a_\\perp","aliases":"a_⊥ perp","display":"a_\\perp","kind":"use","excerpt":"An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-357745da2a118b68","tex":"\\mathrm{ad}","aliases":"ad mathrm","display":"\\mathrm{ad}","kind":"use","excerpt":"For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ea43fbb8cbfe05b","tex":"\\operatorname{area}","aliases":"area operatorname","display":"\\operatorname{area}","kind":"use","excerpt":"[Constant curvature density] For ϕ(t,x)=b(t^2-x^2)/2, one has ϕ_tt-ϕ_xx=2b, R=4b e^-2ϕ, W_square=e^2b area(square)K. The curvature density e^2ϕR/2 is constant although R varies with position. Adding oriented subrectangle integrals reconstructs the coarse rapidity exactly.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9544dc0281e02973","tex":"\\operatorname{asinh}","aliases":"asinh operatorname","display":"\\operatorname{asinh}","kind":"use","excerpt":"N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb), α(L,b","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-16636769fc160a5a","tex":"b^{-1}","aliases":"b^-1","display":"b^{-1}","kind":"use","excerpt":"The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify th","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a80a840f9ada088f","tex":"b^{-r}","aliases":"b^-r","display":"b^{-r}","kind":"use","excerpt":"ition[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"+(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro ·","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a7ffa2898efa8288","tex":"\\mathcal B^a","aliases":"B^a mathcal","display":"\\mathcal B^a","kind":"use","excerpt":"Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ad0ae9f47e95e104","tex":"\\mathcal B^b","aliases":"B^b mathcal","display":"\\mathcal B^b","kind":"use","excerpt":"Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-97a0df13c9cd3826","tex":"b^g","aliases":"b^g","display":"b^g","kind":"use","excerpt":"Coarse endpoint gauges g(x) and midpoint gauges q(x) may be chosen independently. The two sublinks transform by a^g(x)=q(x)a(x)g(x)^-1, b^g(x)=g(τ x)b(x)q(x)^-1. The midpoint factors cancel in b^ga^g, leaving the required coarse transformation. At the midpoint, a^g(x)g(x)ψ(x)=q(x)a(x)ψ(x), while endpoint covariance is immediate. Hence interpolation","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c4a00e5b81eeab04","tex":"b^k","aliases":"b^k","display":"b^k","kind":"use","excerpt":"[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next fact","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d260ef111195a9e3","tex":"b^{k-p}","aliases":"b^k-p","display":"b^{k-p}","kind":"use","excerpt":"[A quantified remainder and perturbation bound] Let p be a nonnegative integer and suppose normR(h)≤ C|h|^p, normg(h)-f(h)≤ε (0<|h|<δ), where C,ε≥0. Define A_m,p=∏_k=1^mfracb^k-p+1b^k-1. Then, throughout the same punctured interval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8b15ec8eb6abf2f6","tex":"b^{k-r}","aliases":"b^k-r","display":"b^{k-r}","kind":"use","excerpt":"[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next fact","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cea555c0ffd5ad40","tex":"b^{m+1-p}","aliases":"b^m+1-p","display":"b^{m+1-p}","kind":"use","excerpt":"nterval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower step gives normT_m+1R(h)≤ fracb^m+1normT_mR(h/b)+normT_mR(h)b^m+1-1. Starting with the stated remainder bound and proceeding by induction multiplies its constant at step m+1 by (b^m+1-p+1)/(b^m+1-1). Hence normT_mR(h)≤ A_m,p","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"Let Z_sign have eigenvalues +1,-1 on a dyadic sign qubit, and let B=B^* act on the remaining factors. Assume there are orthonormal states |b_0⟩,|b_1⟩ with B|b_0⟩=0 and B|b_1⟩=|b_1⟩. One concrete choice is an odd-prime torsion involution times a fixed odd-prime detail projector, with the torsion state in its +1 eigenspace. For real g≠0, use H_int=gZ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf1be6e890041939","tex":"B_{0i}","aliases":"B_0i","display":"B_{0i}","kind":"use","excerpt":"c:constraints:regular-triad The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fa1a68eae3317067","tex":"B_{ba}","aliases":"B_ba","display":"B_{ba}","kind":"use","excerpt":"Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the ca","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ffbdc00380cf8d6d","tex":"B_{bc}","aliases":"B_bc","display":"B_{bc}","kind":"use","excerpt":"Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the ca","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b5384093c6e0cfdb","tex":"B_{ca}","aliases":"B_ca","display":"B_{ca}","kind":"use","excerpt":"Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the ca","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"use","excerpt":"For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely wh","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28c8ff44f8fbfe84","tex":"B_{IJ}","aliases":"B_IJ","display":"B_{IJ}","kind":"use","excerpt":"IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ff1187fcdc685de","tex":"B_{ij}","aliases":"B_ij","display":"B_{ij}","kind":"use","excerpt":"c:constraints:regular-triad The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","excerpt":"For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely wh","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"Direct and split radius evolution. On the positive branch, restriction of the radius equation to the reconstruction manifold gives dot R_j=g(R_j)+b_j(D_4R)_j, g(R)=-fracsqrt2MR-Q^2R, where b=N^r is prescribed in the electrovac benchmark. A split alternative is dot R=g(R)+A_h[b]R, A_h[b]u=frac12(bD_4u+D_4(bu)-(D_4b)u). On smooth data these formulas ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-89304b2594ebe322","tex":"B_{j-1}","aliases":"B_j-1","display":"B_{j-1}","kind":"use","excerpt":"Put B_j=c_j+1/2Δ_j. Using the equation of motion, the derivative of the kinetic and mass terms at site j is fracdotφ_jh(B_j-B_j-1) -frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2. The right edge contribution differentiates to fracdot c_j+1/2Δ_j^24h +fracB_j2h(dotφ_j+1-dotφ_j), and the left edge has the analogous expression. Adding these terms leaves B","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2fc9f3d3fd680082","tex":"B_{j\\mid i}","aliases":"B_jmid i mid","display":null,"kind":"use","excerpt":"formal connection structure records the three components, the six cross derivatives ∂_iA_j for i≠ j, and their one-dimensional derivative identities along the relevant coordinate lines. Each component is continuous along its own integration direction. Each cross derivative is continuous along the line in its differentiation direction. Finally there","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8184a87901449233","tex":"B_{L,j}","aliases":"B_L,j","display":"B_{L,j}","kind":"use","excerpt":"The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-be137cd5e9ecea3a","tex":"B_{L,j-1\\bmod L}","aliases":"B_L,j-1bmod L bmod","display":"B_{L,j-1\\bmod L}","kind":"use","excerpt":"One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters are positive and the ratio test again gives entire convergence. Character filtering reconstructs B=∑_jB_L,j. With the Euler derivative Θ=y d/dy, termwise differentiation also gives the cyclic system Θ^","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9277d882004917b6","tex":"B_{m+1,a+d p^m}","aliases":"B_m+1,a+d p^m","display":"B_{m+1,a+d p^m}","kind":"use","excerpt":"The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-76b84f0a7edc903f","tex":"B_{m,a}","aliases":"B_m,a","display":"B_{m,a}","kind":"use","excerpt":"Fix a prime p. The compact additive group Z_p has normalized Haar measure μ_p, with B_m,a=a+p^mZ_p, 0≤ a<p^m, μ_p(B_m,a)=p^-m. Write HH_p=L^2(Z_p,μ_p;C). We use the complex inner product innerfg=∫overline f g dμ_p, linear in its second argument. The p-adic coordinate belongs to the measurable space. Amplitudes and the Born probabilities use ordinar","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3b4beaedece992ae","tex":"B_{m,a+d p^{m-1}}","aliases":"B_m,a+d p^m-1","display":"B_{m,a+d p^{m-1}}","kind":"use","excerpt":"For an explicit basis, put ω_p=exp(2π i/p) and define ψ_m,a,q=p^(m-1)/2∑_d=0^p-1 ω_p^qd1_B_m,a+d p^m-1, m≥1, 0≤ a<p^m-1, 1≤ q<p. Each has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f7dc3c6a96166d73","tex":"B_{m,b}","aliases":"B_m,b","display":"B_{m,b}","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","excerpt":"Completion and continuum evolution. The complete coefficient Hilbert space. Let KK=ℓ^2(Z^3;C^4) ={a=(a_n):∑_n∈Z^3norma_n^2<∞}. Its inner product is innerab=∑_ninnera_nb_n. Absolute convergence follows from the scalar Cauchy--Schwarz inequality. The norm therefore measures the total squared amplitude across all Fourier packets. A finite family of am","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-82d5eaeb539c2f07","tex":"\\mathcal B_{r,j}","aliases":"B_r,j mathcal","display":"\\mathcal B_{r,j}","kind":"use","excerpt":"The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the se","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8cdcafa2076b5d63","tex":"\\mathcal B_{r,j-1}","aliases":"B_r,j-1 mathcal","display":"\\mathcal B_{r,j-1}","kind":"use","excerpt":"The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the se","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b94a4e9d2f773e0","tex":"b_{\\rm tr}","aliases":"b_rm tr rm","display":"b_{\\rm tr}","kind":"use","excerpt":"· -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h · -innerA(a)b_rm tr_h -innera_rm trA(b)_h -∑_kinnerδ_ka_rm trδ_kb_rm tr_h. This expression is symmetric in a,b.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4a3dd5cb1c2ae296","tex":"B_{x\\mid y}","aliases":"B_xmid y mid","display":null,"kind":"use","excerpt":"· -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a s","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d93a18a8f7a52a97","tex":"B_{x\\mid z}","aliases":"B_xmid z mid","display":null,"kind":"use","excerpt":"[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-123a2e3b9d146b7e","tex":"B_{y\\mid x}","aliases":"B_ymid x mid","display":null,"kind":"use","excerpt":"oss derivatives ∂_iA_j for i≠ j, and their one-dimensional derivative identities along the relevant coordinate lines. Each component is continuous along its own integration direction. Each cross derivative is continuous along the line in its differentiation direction. Finally there are global nonnegative constants |∂_iA_j(p)|≤ B_jmid i (i≠ j). For ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cb32484d4f016b34","tex":"B_{y\\mid z}","aliases":"B_ymid z mid","display":null,"kind":"use","excerpt":"[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-75c7c82addb66e22","tex":"B_{z\\mid x}","aliases":"B_zmid x mid","display":null,"kind":"use","excerpt":"[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a0327f04e760f963","tex":"B_{z\\mid y}","aliases":"B_zmid y mid","display":null,"kind":"use","excerpt":"[Regular background estimate] Suppose the coordinate regularity and derivative bounds above hold. Set A_∂= · B_ymid x+B_xmid y +2(B_zmid x+B_xmid z)notag · +2(B_zmid y+B_ymid z). Generate the reconstructed and prescribed finite-site links from the exact integrals [source reference], and generate the twelve links of the comparison cube from the same","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-093efc86fbc935a9","tex":"B_\\partial","aliases":"B_∂ partial","display":"B_\\partial","kind":"use","excerpt":"da>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift. The Maxwell multiplier is denoted Φ_e to distinguish it from the scalar gradient used later. The bulk canonical action is S=∫ dt∫ dr (P_ΛdotΛ+P_Rdot R+Qdot A_r -N H-N^r D-Φ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf72cfc588b3df8b","tex":"\\bar p","aliases":"bar p bar","display":"\\bar p","kind":"use","excerpt":"-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpo","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"use","excerpt":"-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpo","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db1d2ae53aa18234","tex":"\\bar\\rho","aliases":"barρ bar rho","display":"\\bar\\rho","kind":"use","excerpt":"[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a09b010e18ba8e9a","tex":"\\bar\\chi","aliases":"barχ bar chi","display":"\\bar\\chi","kind":"use","excerpt":"the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Different","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-40c6a191c91db0c5","tex":"\\bar\\chi_B","aliases":"barχ_B bar chi","display":"\\bar\\chi_B","kind":"use","excerpt":"the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Different","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9c3ecc4b70c12695","tex":"\\bar\\psi","aliases":"barψ bar psi","display":"\\bar\\psi","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5bd492e4ea78c5fe","tex":"\\bigsqcup","aliases":"bigsqcup","display":null,"kind":"use","excerpt":"The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-15ef91d0a0f35fa0","tex":"\\bigsqcup_d","aliases":"bigsqcup_d bigsqcup","display":null,"kind":"use","excerpt":"The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f805aa5d780b628b","tex":"\\bigsqcup_{\\alpha}","aliases":"bigsqcup_α bigsqcup alpha","display":null,"kind":"use","excerpt":"For a partition A=bigsqcup_αC_α, the strong sum Q_α=∑_a∈ C_αP_a is a bounded orthogonal projection. Its weight is always normQ_αψ^2=∑_a∈ C_αnormP_aψ^2. To define a coarse current with HQ_αψ, additionally require Q_α D(H)⊂ D(H). Fine domain preservation alone does not state this infinite-cell property.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify th","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f157f71e0163a897","tex":"\\Box","aliases":"Box","display":"\\Box","kind":"use","excerpt":"The geometric fields are real, but the physical spinor convention needs a separate check. The displayed action retains the inherited Dirac-type polynomial gauge model. With the stated Clifford sign and kinetic factor i, the flat-space operators iγ^I∂_I± m have product (iγ^I∂_I+m)(iγ^J∂_J-m) =-Box-m^2, Box=η^IJ∂_I∂_J. Thus its squared field equation","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c0d312a637b5f115","tex":"\\mathrm{BV}","aliases":"BV mathrm","display":"\\mathrm{BV}","kind":"use","excerpt":"The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or eq","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"Components of a spinor are numbered 0,1,2,3, in the exact order of the source representation. For v,w∈ C^4,","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ac286c55ce5c92f8","tex":"\\mathrm c","aliases":"c mathrm","display":"\\mathrm c","kind":"use","excerpt":"[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","excerpt":"in and all the chains [source reference] form an orthogonal decomposition of ℓ^2(T_p,h). A chain starting at depth r has length h-r+1 and appears with multiplicity (p-1)p^r-1, for 1≤ r≤ h. Within one chain, different s have disjoint level supports. Chains based at the same u and different contrast vectors have inner product ⟨ c,c'⟩=0 at a common le","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"use","excerpt":"|s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk U_p,h,N=SC is unitary because U_p,h,N^*U_p,h,N=C^*S^*SC=C^*C=I. Here locality concerns the graph ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"use","excerpt":"det(z-H) · =det(z-D)det S(z). For E∈R∖σ(D), an eigenvector of H is equivalent to a nonzero solution of S(E)u=0, with hidden component v=(E-D)^-1B^*u. Put C=z-D. Direct block multiplication gives z-H= I · -BC^-1 0 · I S(z) · 0 0 · C I · 0 -C^-1B^* · I. Both outer factors are invertible and have determinant one. This proves the invertibility equivale","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","excerpt":"For the faster normalized tail c_n=C/n^2, both errors are controlled: normc-S_Nc^2=C^2∑_n>Nn^-4, normH_λ c-S_NH_λ c^2=C^2∑_n>Nn^-2. The state tail is O(N^-3/2) in norm, while the velocity tail is O(N^-1/2). A small state error can therefore coexist with a much larger derivative error even inside the true domain.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a620c9c352cae951","tex":"\\C^2","aliases":"C^2 C","display":null,"kind":"use","excerpt":"[An exactly solved exchange] On C^2, take the coordinate atlas, H=g0 · 1","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","excerpt":"e⊗|b_1⟩). Tracing out the orthogonal b states yields the equally weighted mixture of the two displayed sign states. Their overlap is cos(gt). A mixture of two unit vectors with equal weights and overlap magnitude c has eigenvalues (1+c)/2 and (1-c)/2: its trace is one and its determinant on their two-dimensional span is (1-c^2)/4. Thus the sign ent","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-da615aa5778187eb","tex":"\\C^2_A","aliases":"C^2_A C","display":null,"kind":"use","excerpt":"2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-071a19e037d78134","tex":"\\C^2_B","aliases":"C^2_B C","display":null,"kind":"use","excerpt":"2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7de172a5e1aa4414","tex":"\\C^2_{\\mathrm{sign}}","aliases":"C^2_sign C mathrm","display":null,"kind":"use","excerpt":"In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They d","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4a4fb977dca130a2","tex":"C^3","aliases":"C^3","display":"C^3","kind":"use","excerpt":"The exact finite identities did not assume a smooth field in the background. A continuum comparison does. Let A be a real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains t","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5a1581d983eceed7","tex":"\\mathbb C^4","aliases":"C^4 mathbb","display":"\\mathbb C^4","kind":"use","excerpt":"Components of a spinor are numbered 0,1,2,3, in the exact order of the source representation. For v,w∈ C^4,","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-082b7a73628d3cfa","tex":"c^A","aliases":"c^A","display":"c^A","kind":"use","excerpt":"There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total gh","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0330172c3e4700bc","tex":"c^B","aliases":"c^B","display":"c^B","kind":"use","excerpt":"There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total gh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-84321f2846a02874","tex":"\\C^{h+1}","aliases":"C^h+1 C","display":null,"kind":"use","excerpt":"Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · ·","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3740fcff7b45731","tex":"C^{IJ}","aliases":"C^IJ","display":"C^{IJ}","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d8416f1ffda0d03b","tex":"\\C^M","aliases":"C^M C","display":null,"kind":"use","excerpt":"Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ff8af0060364d2c","tex":"\\C^N","aliases":"C^N C","display":null,"kind":"use","excerpt":"Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-06e2221a9a4e2168","tex":"\\C^n","aliases":"C^n C","display":null,"kind":"use","excerpt":"For this section take HH=C^n. A density matrix is a positive semidefinite Hermitian matrix ρ with Trρ=1. Its probability in a sharp packet is p_a=Tr(P_aρ). Hermiticity makes this trace real. Positivity makes it nonnegative, since cyclicity and P_a^2=P_a give Tr(P_aρ)=Tr(P_aρ P_a)≥0. Completeness gives ∑_ap_a=Trρ=1. These are separate uses of the de","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d14ca8f6e8828e5e","tex":"\\C^p","aliases":"C^p C","display":null,"kind":"use","excerpt":"There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite en","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1756adec424805d9","tex":"\\C^{p-1}_{\\mathrm{tors}}","aliases":"C^p-1_tors C mathrm","display":null,"kind":"use","excerpt":"In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They d","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fcd04e249acdc68d","tex":"\\C^r","aliases":"C^r C","display":null,"kind":"use","excerpt":"Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have t","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ab8119bd804b07e","tex":"\\C^s","aliases":"C^s C","display":null,"kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0fa59d916aa991b8","tex":"\\C^\\times","aliases":"C^× C times","display":null,"kind":"use","excerpt":"The genuine unit-circle hypothesis is essential. A nonzero complex scalar is invertible, but multiplication by 2 multiplies squared norm by 4. The phase must therefore lie in the unit circle, not merely in C^×. Norm preservation depends on that mathematical hypothesis. The formalization companion identifies its precise type and transport convention","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7ce18b1e2aa002e0","tex":"c^\\mu","aliases":"c^μ mu","display":"c^\\mu","kind":"use","excerpt":"Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-16df97f80ced5579","tex":"c^\\nu","aliases":"c^ν nu","display":"c^\\nu","kind":"use","excerpt":"Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-78a649bb99dbf34d","tex":"c_+","aliases":"c_+","display":"c_+","kind":"use","excerpt":"dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-83dae4cbc4840c66","tex":"c_-","aliases":"c_-","display":"c_-","kind":"use","excerpt":"dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite en","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-936cd180d8ae5af5","tex":"c_0^2","aliases":"c_0^2","display":"c_0^2","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf2b03123f263521","tex":"c_0^3","aliases":"c_0^3","display":"c_0^3","kind":"use","excerpt":"N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb), α(L,b","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-158ec65eae30561d","tex":"c_0^4","aliases":"c_0^4","display":"c_0^4","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-94104e5692ada178","tex":"C_{1,j}^n","aliases":"C_1,j^n","display":"C_{1,j}^n","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3080f350b29a38e7","tex":"C_{2,j}^n","aliases":"C_2,j^n","display":"C_{2,j}^n","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"[Local logarithmic refinement] On the common analytic logarithm branch, in any fixed matrix norm, F_h · =[A,B]+hC_3(A,B)+O(h^2), C_3(A,B) · =tfrac12[A+B,[A,B]], F_h^[1] · :=2F_h/2-F_h=[A,B]+O(h^2). For constant G∈ SU(2), simultaneous conjugation of A,B conjugates both F_h and F_h^[1] by G.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-99dbb247eeb251a3","tex":"C_A","aliases":"C_A","display":"C_A","kind":"use","excerpt":"Let A be a finite set of coarse labels and let f:I→ A assign every fine packet to exactly one coarse label. Write C_A=f^-1(A) and define Q_A=∑_a∈ C_AP_a. The fibers are disjoint and cover I, so (Q_A) is another orthogonal atlas. Empty fibers give zero projectors. This is a grouping of an existing resolution. It need not be an approximation to a dif","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"use","excerpt":"Cell values and normalized coefficients. For a measurable finite partition into cells of equal volume v>0, distinguish the physical constant value q_a∈C^4 on cell C_a from the normalized coefficient ψ_a=sqrt v q_a. The step interpolation is (Jψ)(x)=v^-1/2ψ_a (x∈ C_a), with equality understood almost everywhere. Direct integration gives normJψ_L^2^2","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"use","excerpt":"_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probabili","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-89de6f3841051f82","tex":"c_{ab}","aliases":"c_ab","display":"c_{ab}","kind":"use","excerpt":"For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a717968c751b4646","tex":"C_B","aliases":"C_B","display":"C_B","kind":"use","excerpt":"[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-389556078e08449f","tex":"c_{ba}","aliases":"c_ba","display":"c_{ba}","kind":"use","excerpt":"For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-227ceb7baf501818","tex":"c_d","aliases":"c_d","display":"c_d","kind":"use","excerpt":"There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite en","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fd685bf246955178","tex":"C_h","aliases":"C_h","display":"C_h","kind":"use","excerpt":"dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity variables when the configuration P,Q is fixed.","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d17bd79161d5e884","tex":"\\mathcal C_I","aliases":"C_I mathcal","display":"\\mathcal C_I","kind":"use","excerpt":"The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-de72a57fe0143ff0","tex":"C_{i,x}","aliases":"C_i,x","display":"C_{i,x}","kind":"use","excerpt":"For a partition of fine spatial sites into coarse regions A, the nonnegative diagnostic E_A^rm con=∑_x∈ Ah^3 (|C_x|^2+∑_i|C_i,x|^2) adds exactly to the full fine diagnostic. This is an exact partition identity. It does not assert that differentiating a coarse field reconstructs its fine constraint values.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ba0b2b1b046df4d1","tex":"\\mathcal C_I^{\\mathrm D}","aliases":"C_I^ D mathcal mathrm","display":"\\mathcal C_I^{\\mathrm D}","kind":"use","excerpt":"The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-94e1dd125c9c41b7","tex":"\\mathcal C_I^{\\mathrm G}","aliases":"C_I^ G mathcal mathrm","display":"\\mathcal C_I^{\\mathrm G}","kind":"use","excerpt":"The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-56824ebc9521ba33","tex":"\\mathcal C_I^{\\mathrm M}","aliases":"C_I^ M mathcal mathrm","display":"\\mathcal C_I^{\\mathrm M}","kind":"use","excerpt":"The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-24211da935b930d3","tex":"c_{j+1/2}","aliases":"c_j+1/2","display":"c_{j+1/2}","kind":"use","excerpt":"Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are neede","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-88f8e11606761017","tex":"C_{j-1}","aliases":"C_j-1","display":"C_{j-1}","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-120b20c73e5f2961","tex":"c_{j-1/2}","aliases":"c_j-1/2","display":"c_{j-1/2}","kind":"use","excerpt":"equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a8e8fbaa4e63d973","tex":"C_L","aliases":"C_L","display":"C_L","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-56b847776dc4e558","tex":"C_L^L","aliases":"C_L^L","display":"C_L^L","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-93c7b645d58c2453","tex":"C_L^n","aliases":"C_L^n","display":"C_L^n","kind":"use","excerpt":"gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aa77d76fccb29372","tex":"c_{m,0}","aliases":"c_m,0","display":"c_{m,0}","kind":"use","excerpt":"[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next fact","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-127e9660aac8b912","tex":"C_{M,j}","aliases":"C_M,j","display":"C_{M,j}","kind":"use","excerpt":"[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-542cae4028b5c3e4","tex":"c_{m,r}","aliases":"c_m,r","display":"c_{m,r}","kind":"use","excerpt":"[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next fact","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"use","excerpt":"equals its Hilbert adjoint. Let b∈Dom(widehat H_m^*) and write c=widehat H_m^*b. By the adjoint relation, innerwidehat H_mab=innerac (a∈Dom(widehat H_m)). Fix a mode n and choose a with the single nonzero coefficient a_n=v, where v∈C^4 is arbitrary. Such a field belongs to the maximal domain. The adjoint identity becomes innerH_nvb_n=innervc_n. Her","count":17},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3a505891c8ed892e","tex":"C_{P,j}","aliases":"C_P,j","display":"C_{P,j}","kind":"use","excerpt":"[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c6bc925728d707f0","tex":"c_{p-1}","aliases":"c_p-1","display":"c_{p-1}","kind":"use","excerpt":"There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite en","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9980fbd02a4085b1","tex":"C_{\\mathrm{phase}}","aliases":"C_phase mathrm","display":"C_{\\mathrm{phase}}","kind":"use","excerpt":"The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e8b9f48bd6d09aed","tex":"c_q","aliases":"c_q","display":"c_q","kind":"use","excerpt":"The error also has an exact expression. For fixed mode, let r=p(n), q=q_h(n), with the corresponding c_r,s_r,c_q,s_q. Coordinate subtraction gives","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1e2542ffdcecdf00","tex":"C_{Q,j}","aliases":"C_Q,j","display":"C_{Q,j}","kind":"use","excerpt":"[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"use","excerpt":"Set E_r=sqrt1+|r|^2, c_r(t)=cos(E_rt), and s_r(t)=sin(E_rt)/E_r. PH03 gives","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a6ff7660eca84d81","tex":"C_x","aliases":"C_x","display":"C_x","kind":"use","excerpt":"For a partition of fine spatial sites into coarse regions A, the nonnegative diagnostic E_A^rm con=∑_x∈ Ah^3 (|C_x|^2+∑_i|C_i,x|^2) adds exactly to the full fine diagnostic. This is an exact partition identity. It does not assert that differentiating a coarse field reconstructs its fine constraint values.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-637c0e5aa42a2ea2","tex":"C_\\alpha","aliases":"C_α alpha","display":"C_\\alpha","kind":"use","excerpt":"For a partition A=bigsqcup_αC_α, the strong sum Q_α=∑_a∈ C_αP_a is a bounded orthogonal projection. Its weight is always normQ_αψ^2=∑_a∈ C_αnormP_aψ^2. To define a coarse current with HQ_αψ, additionally require Q_α D(H)⊂ D(H). Fine domain preservation alone does not state this infinite-cell property.","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-581321bc41f1865f","tex":"C_\\beta","aliases":"C_β beta","display":"C_\\beta","kind":"use","excerpt":"Under this coarse-domain hypothesis, symmetry and the strong cell sums justify both iterated sums separately: J^rm c_βα · =2IminnerQ_βψHQ_αψ · =∑_b∈ C_β(∑_a∈ C_αJ_ba) =∑_a∈ C_α(∑_b∈ C_βJ_ba). For example, first summing a moves H onto P_bψ, then the outer sum pairs ∑_bP_bψ with the fixed vector HQ_αψ. The reverse order has its own symmetric argument","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1b598868f1055551","tex":"C_{\\Lambda,j}","aliases":"C_Λ,j Lambda","display":"C_{\\Lambda,j}","kind":"use","excerpt":"[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8ec5cf28b1acacd8","tex":"c_\\ell","aliases":"c_ℓ ell","display":"c_\\ell","kind":"use","excerpt":"The same construction works for any circulant polynomial H=∑_ℓ c_ℓ S^ℓ, with symbol ∑_ℓ c_ℓ P_p(θ)^ℓ. It preserves all bands, whether or not their energies coincide. For N=p^h, digit reversal relates this standard residue split to the newest-digit adic split developed earlier. It is still necessary to distinguish the retained residue-zero sites fro","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-67c1622d52b0e019","tex":"\\mathrm{coarse}","aliases":"coarse mathrm","display":"\\mathrm{coarse}","kind":"use","excerpt":"lation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of [source reference]: S_i,fine^2 I_r=I_rS_i,coarse. If the mass sub","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8db6134243ea03f4","tex":"\\cos^2","aliases":"cos^2 cos","display":"\\cos^2","kind":"use","excerpt":"u_r(t) _2^2 =c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2) =cos^2(E_rt)+sin^2(E_rt)=1.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"use","excerpt":"The coefficient maps R_L,j in [source reference] are idempotents on analytic functions. The evaluated operators E_L,j(tA_0) generally are not. For L=2, E_2,0(tA_0)=cosh(tA_0) and E_2,1(tA_0)=sinh(tA_0), which are neither an orthogonal projector family nor a pair of independent unitary evolutions. The sum reconstructs an evolution; an individual lan","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7c63d9340937698","tex":"\\mathrm{CRT}","aliases":"CRT mathrm","display":"\\mathrm{CRT}","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eae54bb0ec4239ae","tex":"\\mathbf d","aliases":"d mathbf","display":"\\mathbf d","kind":"use","excerpt":"D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"Coherent exchange is one mechanism for changing packet weights. A second comes from an explicitly chosen family of dissipative transitions. Fix an orthonormal basis (e_a)_a∈ I of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(","count":21},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e63d808025fba917","tex":"\\mathrm d","aliases":"d mathrm","display":"\\mathrm d","kind":"use","excerpt":"If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j), τ_N(∏_jS_Nf_j) · =(2π)^-3∫_ T^3∏_jf_j d^3x. The Leibniz rule and trace integration by parts consequently hold for the declared products. They do not hold for arbitrary sampled poi","count":21},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ddc7e0f1c8697a47","tex":"\\mathsf D","aliases":"D mathsf","display":"\\mathsf D","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf5b0623c1cc32bf","tex":"D^{\\mathsf T}","aliases":"D^ T mathsf","display":"D^{\\mathsf T}","kind":"use","excerpt":"Coordinate differentiation gives ∂_p D_h[ξ]=hξ Dq and ∂_q D_h[ξ]=-hD(ξ p). Insert these in the canonical bracket and use D^ T=-D: { D_h[ξ], D_h[η]}_h =h∑_ap_a[ξ_aD(η Dq)_a -η_aD(ξ Dq)_a]. Expanding each derivative using the definition of L_D cancels the two ξη D^2q terms. The remaining product-rule terms give the first term of [source reference], a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7e8acadc439b10d5","tex":"D^*","aliases":"D^*","display":"D^*","kind":"use","excerpt":"Let a finite Hilbert space be the orthogonal sum HH_L⊕HH_D, and write a self-adjoint Hamiltonian as H=A · B B^* · D, A=A^*, D=D^*. Here B:HH_D→HH_L. Let ι_L embed HH_L into the full space. This notation fixes the coordinate type of every compression.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d4ef1713d263e2f3","tex":"d^*","aliases":"d^*","display":"d^*","kind":"use","excerpt":"A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5a84c9ff6c23f102","tex":"D^+_{\\mu,U}","aliases":"D^+_μ,U mu","display":"D^+_{\\mu,U}","kind":"use","excerpt":"[Covariance of linked differences] For spacing h>0, define (D^+_μ,Uψ)_x · =fracU_x,μψ_x+widehatμ-ψ_xh, (Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-96f254806d6d6b55","tex":"D^+_{\\mu,U^g}","aliases":"D^+_μ,U^g mu","display":"D^+_{\\mu,U^g}","kind":"use","excerpt":"(Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6311982fa9099aea","tex":"D^{-1/2}","aliases":"D^-1/2","display":"D^{-1/2}","kind":"use","excerpt":"x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies d","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a182cf4276219221","tex":"d^{-1/2}","aliases":"d^-1/2","display":"d^{-1/2}","kind":"use","excerpt":"Eliminating z first gives the effective coefficient μ for y^2 and the factor (m+n)^-1/2. Eliminating y next gives d^-1/2. Their product is D^-1/2, exactly as in direct elimination. For the Euclidean weight e^-S/ℏ, using a flat coarse measure instead records this determinant as S_C,ℏ=S_C+tfracℏ2log D, up to an additive constant. A nonlinear symmetry","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","excerpt":"f Coordinate differentiation gives ∂_p D_h[ξ]=hξ Dq and ∂_q D_h[ξ]=-hD(ξ p). Insert these in the canonical bracket and use D^ T=-D: { D_h[ξ], D_h[η]}_h =h∑_ap_a[ξ_aD(η Dq)_a -η_aD(ξ Dq)_a]. Expanding each derivative using the definition of L_D cancels the two ξη D^2q terms. The remaining product-rule terms give the first term of [source reference],","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"(tfrac14d^2+tfrac14 3d^2)^1/2=d,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7b5634dbd71abc53","tex":"\\mathrm d^3","aliases":"d^3 mathrm","display":"\\mathrm d^3","kind":"use","excerpt":"If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j), τ_N(∏_jS_Nf_j) · =(2π)^-3∫_ T^3∏_jf_j d^3x. The Leibniz rule and trace integration by parts consequently hold for the declared products. They do not hold for arbitrary sampled poi","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d8ae000fbd2c1692","tex":"D^T","aliases":"D^T","display":"D^T","kind":"use","excerpt":"For an interval summation-by-parts derivative with a diagonal positive norm matrix H and HD+D^TH=B, multiplication by b commutes with H. The uncancelled term is instead inneruA_h[b]u_H =frac12u^TB diag(b)u -frac12inneru(Db)u_H. Its boundary flux must be controlled using an actual boundary closure and appropriate boundary data. Saying that a stencil","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9d8746172823369","tex":"d_*","aliases":"d_*","display":"d_*","kind":"use","excerpt":"A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-70ffa9c8bbca2ebd","tex":"D_+","aliases":"D_+","display":"D_+","kind":"use","excerpt":"The characteristic evolution uses the second-order one-sided derivatives (D_+u)_j=frac-3u_j+4u_j+1-u_j+22h, (D_-u)_j=frac3u_j-4u_j-1+u_j-22h. The inward channel uses D_+ and the outgoing channel uses D_- outside a horizon, switching to D_+ inside. These interior matter stencils have radius two. The source also uses a radius-two fourth-order interio","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ff76dfe8b26b12f2","tex":"D_-","aliases":"D_-","display":"D_-","kind":"use","excerpt":"The characteristic evolution uses the second-order one-sided derivatives (D_+u)_j=frac-3u_j+4u_j+1-u_j+22h, (D_-u)_j=frac3u_j-4u_j-1+u_j-22h. The inward channel uses D_+ and the outgoing channel uses D_- outside a horizon, switching to D_+ inside. These interior matter stencils have radius two. The source also uses a radius-two fourth-order interio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18df6123ecfe74ae","tex":"\\mathbf d_0","aliases":"d_0 mathbf","display":"\\mathbf d_0","kind":"use","excerpt":"[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-589f69b8f3dcb451","tex":"D_0^{(h)}","aliases":"D_0^(h)","display":"D_0^{(h)}","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca8301964f9bd376","tex":"D_4","aliases":"D_4","display":"D_4","kind":"use","excerpt":"On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,","count":14},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-42dbc6c2397b687f","tex":"\\mathcal D_a","aliases":"D_a mathcal","display":"\\mathcal D_a","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9bf608c22091e074","tex":"d_{C_N}","aliases":"d_C_N","display":"d_{C_N}","kind":"use","excerpt":"[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fdadfd14bd665b99","tex":"D_h","aliases":"D_h","display":"D_h","kind":"use","excerpt":"[The precise defect of a forward difference] On a cyclic grid, let D_hf_j=(f_j+1-f_j)/h, with h≠0. Expanding the product gives D_h(fg)_j-(D_hf)_jg_j-f_j(D_hg)_j =h(D_hf)_j(D_hg)_j. Thus the ordinary Leibniz identity fails unless the displayed product vanishes. A small defect on sufficiently regular sampled functions is compatible with the no-go the","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a7b7b693578bf211","tex":"\\mathcal D_h","aliases":"D_h mathcal","display":"\\mathcal D_h","kind":"use","excerpt":"Let a periodic real lattice have canonical brackets {q_a,p_b}_h=δ_ab/h and a skew matrix D. Define D_h[ξ]=h∑_aξ_ap_a(Dq)_a, L_D(f,g)=D(fg)-(Df)g-f(Dg). All products on this lattice are pointwise.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bf1b37e9c699a1d1","tex":"d_h","aliases":"d_h","display":"d_h","kind":"use","excerpt":"g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keep","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0bac9a55ad46a04a","tex":"D_{h+1}^{(h+1)}","aliases":"D_h+1^(h+1)","display":"D_{h+1}^{(h+1)}","kind":"use","excerpt":"is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(h+1), I-V_p,hV_p,h^*=D_h+1^(h+1). The first composition is an identity on ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fa19135e19c1fe52","tex":"D_{h,j}","aliases":"D_h,j","display":"D_{h,j}","kind":"use","excerpt":"(D_h,jf)(x)=fracf(x+he_j)-f(x-he_j)2h.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-15102a51e263a437","tex":"d_{h-1}","aliases":"d_h-1","display":"d_{h-1}","kind":"use","excerpt":"ue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-843cae1be4629d0b","tex":"d_{h-2}","aliases":"d_h-2","display":"d_{h-2}","kind":"use","excerpt":"-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores the prefix a=d_0+⋯+d_h-2p^h-2. This proves the identity on an orthonormal basis.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cd12d05aa24858ef","tex":"D_h^{\\mathsf T}","aliases":"D_h^ T mathsf","display":"D_h^{\\mathsf T}","kind":"use","excerpt":"Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-88af6b8b39907d56","tex":"D_h^{(h)}","aliases":"D_h^(h)","display":"D_h^{(h)}","kind":"use","excerpt":"Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*, W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-243cb287b7b23ea4","tex":"D_h^A","aliases":"D_h^A","display":"D_h^A","kind":"use","excerpt":"The exact finite identities did not assume a smooth field in the background. A continuum comparison does. Let A be a real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains t","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-476e3927be5169c0","tex":"\\mathcal D_L","aliases":"D_L mathcal","display":"\\mathcal D_L","kind":"use","excerpt":"Coherent exchange is one mechanism for changing packet weights. A second comes from an explicitly chosen family of dissipative transitions. Fix an orthonormal basis (e_a)_a∈ I of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7c4ec459e6d82d41","tex":"\\mathcal D_{L_{ba}}","aliases":"D_L_ba mathcal","display":"\\mathcal D_{L_{ba}}","kind":"use","excerpt":"of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(L^*Lρ+ρ L^*L). Consider a supplied differentiable matrix trajectory satisfying dotρ=∑_b≠ a D_L_ba(ρ). This choice of jumps is tied to the selected basis. A general dissipative ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f738a79642fbf617","tex":"\\mathsf D_m","aliases":"D_m mathsf","display":"\\mathsf D_m","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-688ebedc522ae64c","tex":"d_m","aliases":"d_m","display":"d_m","kind":"use","excerpt":"Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5f2dbbf05d954b28","tex":"D_m^{(M)}","aliases":"D_m^(M)","display":"D_m^{(M)}","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-237fc9015f157bd3","tex":"D_r","aliases":"D_r","display":"D_r","kind":"use","excerpt":"For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-505efcf9f8d360c3","tex":"D_r^{(h)}","aliases":"D_r^(h)","display":"D_r^{(h)}","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d260dde0cb637baf","tex":"D_r^{(h+1)}","aliases":"D_r^(h+1)","display":"D_r^{(h+1)}","kind":"use","excerpt":"Replicated vectors are constant in the new digit. All old cylinder averages commute with replication, giving E_r^(h+1)V_p,h · =V_p,hE_r^(h) (0≤ r≤ h), nonumber D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a90caebfafd4c4a6","tex":"D_{\\rm fine}","aliases":"D_rm fine rm","display":"D_{\\rm fine}","kind":"use","excerpt":"A flat cube need not reconstruct the prescribed edge. Conversely, take an arbitrary ordered half-link pair (a,b) and call its actual product v=ba. It is always a subdivision of v, even when the separately prescribed coarse link is u≠ v. The exact directional defect is D_rm fineψ · :=S_A,σ,(a,b)^2I_aψ-I_aS_A,τ,uψnotag · =I_a(S_A,τ,v-S_A,τ,u)ψ. The e","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eb0eb9d6a46d10ef","tex":"\\mathcal D_{\\rm matt}","aliases":"D_rm matt mathcal rm","display":"\\mathcal D_{\\rm matt}","kind":"use","excerpt":"To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source refe","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1864dd3f4ea54a01","tex":"D_{\\rm total}","aliases":"D_rm total rm","display":"D_{\\rm total}","kind":"use","excerpt":"[Combined finite Hilbert estimate] With the common-space walks and contractive mass factors from [source reference], define D_rm total=Q_fW_v-Q_cW_u. Then normD_rm total ≤ 2δ_x+2δ_y+δ_z+κ(ℓ). If v_i=u_i for all three directions and the cube is flat, the defect is exactly zero. Insert and subtract Q_fW_u: D_rm total=Q_f(W_v-W_u)+(Q_f-Q_c)W_u. All ph","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3393ff249d1d0525","tex":"D_s^{(h)}","aliases":"D_s^(h)","display":"D_s^{(h)}","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber rankD_0^(h) · =1, · rankD_r^(h) · =(p-1)p^r-1 (r≥1).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d57ec070569225e9","tex":"d_T","aliases":"d_T","display":"d_T","kind":"use","excerpt":"[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cbc718b2c6c7ed2a","tex":"\\mathcal D_\\Sigma","aliases":"D_Σ mathcal Sigma","display":"\\mathcal D_\\Sigma","kind":"use","excerpt":"The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gau","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-84f59126b5493abd","tex":"\\mathrm d_\\Sigma","aliases":"d_Σ mathrm Sigma","display":"\\mathrm d_\\Sigma","kind":"use","excerpt":"An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-525495744ec84b4c","tex":"D_\\Omega","aliases":"D_Ω Omega","display":"D_\\Omega","kind":"use","excerpt":"lit On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28127f9473d827e7","tex":"D_\\omega","aliases":"D_ω omega","display":"D_\\omega","kind":"use","excerpt":"Gauge transformations and graded conventions. For even parameters (ξ,λ,u), the coframe transforms by its Lie derivative and Lorentz rotation, the connection by L_ξω-D_ωλ, and the Abelian potential by L_ξ a+ du. The spinor transforms by L_ξψ+tfrac14λ^IJγ_IJψ+iquψ, with the dual action on barψ. All internal contractions in~[source reference] are inva","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fa5a09d88f21bc1a","tex":"\\ddagger","aliases":"ddagger","display":null,"kind":"use","excerpt":"The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or eq","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-35b0e63593ef49a1","tex":"\\ddot h","aliases":"ddot h ddot","display":"\\ddot h","kind":"use","excerpt":"Choose n=n_i=0 and impose the transverse and traceless conditions δ_jh_ij=0, h_rm tr=0, δ_jπ_ij=0, π_rm tr=0. They are preserved by the evolution because G_ij(h)=-tfrac12δ^2h_ij, dot h=2π, dotπ=tfrac12δ^2h, ddot h=δ^2h. On this reduced sector the energy is H_rm TT=normπ_h^2 +frac14∑_knormδ_kh_h^2≥0. Its derivative is zero: substituting the two firs","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0d09000efb054df8","tex":"\\ddots","aliases":"ddots","display":"\\ddots","kind":"use","excerpt":"Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decompo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-237a2ee814c86660","tex":"\\ddot\\varphi","aliases":"ddotφ ddot varphi","display":"\\ddot\\varphi","kind":"use","excerpt":"equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a83e49235f2fc701","tex":"\\ddot\\varphi_j","aliases":"ddotφ_j ddot varphi","display":"\\ddot\\varphi_j","kind":"use","excerpt":"equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","excerpt":"normQe^-itH_λP · ≤min{1,|tλ|normQBP}. For z∉σ(QH_λ Q|_QHH), the coarse self-energy is Σ_P(z)=PH_λ Q (z-QH_λ Q|_QHH)^-1QH_λ P, where the inverse acts only on QHH, and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8f42a8e27ac6c866","tex":"\\operatorname{div}","aliases":"div operatorname","display":"\\operatorname{div}","kind":"use","excerpt":"stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies divR_C=fracx_1d, R_Clogρ_C=-fracx_1d. Their sum van","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4e181058d0a68294","tex":"\\dot A","aliases":"dot A dot","display":"\\dot A","kind":"use","excerpt":"βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_j","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6f9f83efac22f7f6","tex":"\\dot a","aliases":"dot a dot","display":"\\dot a","kind":"use","excerpt":"[Time dependent expansion] Let the lapse equal one, let a(t)>0 be spatially uniform, and let m=0. Then w=a, c=1/a, and dot E_h=-frac h2dot a∑_jdotφ_j^2 -fracdot a2ha^2∑_jΔ_j^2. An increasing a transfers energy out of this coordinate-time field Hamiltonian. The supplied background receives no evolution equation from this calculation. A self-consiste","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fd4704b404cc6811","tex":"\\dot A_j","aliases":"dot A_j dot","display":"\\dot A_j","kind":"use","excerpt":"βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_j","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-30f684141cc7fd3b","tex":"\\dot A_r","aliases":"dot A_r dot","display":"\\dot A_r","kind":"use","excerpt":"da>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift. The Maxwell multiplier is denoted Φ_e to distinguish it from the scalar gradient used later. The bulk canonical action is S=∫ dt∫ dr (P_ΛdotΛ+P_Rdot R+Qdot A_r -N H-N^r D-Φ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e25dc6c80c9dc8ef","tex":"\\dot C","aliases":"dot C dot","display":"\\dot C","kind":"use","excerpt":"[Exact finite linearized closure] For every solution of [source reference], dot C=-δ_iC_i, dot C_i=0. Thus initially vanishing constraints remain zero. With the canonical Frobenius bracket and smearings C[n]=⟨ n,C⟩_h, D[ξ]=⟨ξ_i,C_i⟩_h, all three brackets {C[n],C[m]}, {D[ξ],D[η]}, and {C[n],D[ξ]} vanish identically.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c6ef1a371ad0858a","tex":"\\dot c","aliases":"dot c dot","display":"\\dot c","kind":"use","excerpt":"[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8b8fe0baf483f067","tex":"\\dot C_i","aliases":"dot C_i dot","display":"\\dot C_i","kind":"use","excerpt":"[Exact finite linearized closure] For every solution of [source reference], dot C=-δ_iC_i, dot C_i=0. Thus initially vanishing constraints remain zero. With the canonical Frobenius bracket and smearings C[n]=⟨ n,C⟩_h, D[ξ]=⟨ξ_i,C_i⟩_h, all three brackets {C[n],C[m]}, {D[ξ],D[η]}, and {C[n],D[ξ]} vanish identically.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc603732425b2013","tex":"\\dot c_{j+1/2}","aliases":"dot c_j+1/2 dot","display":"\\dot c_{j+1/2}","kind":"use","excerpt":"[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3ea553ff75be986b","tex":"\\dot c_{j-1/2}","aliases":"dot c_j-1/2 dot","display":"\\dot c_{j-1/2}","kind":"use","excerpt":"[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca7356d8caa1a365","tex":"\\dot E","aliases":"dot E dot","display":"\\dot E","kind":"use","excerpt":"y conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0f5f8f38d148f71b","tex":"\\dot e","aliases":"dot e dot","display":"\\dot e","kind":"use","excerpt":"s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-67df9db4a9358f3a","tex":"\\dot E_h","aliases":"dot E_h dot","display":"\\dot E_h","kind":"use","excerpt":"s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5cf1005dd53a7fe5","tex":"\\dot E_j","aliases":"dot E_j dot","display":"\\dot E_j","kind":"use","excerpt":"y conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6d96070594af5331","tex":"\\dot e_j","aliases":"dot e_j dot","display":"\\dot e_j","kind":"use","excerpt":"s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f40edbe0536a4283","tex":"\\dot E_{j-1}","aliases":"dot E_j-1 dot","display":"\\dot E_{j-1}","kind":"use","excerpt":"2Re(ψ_j^*dotψ_j). The mass term contributes zero, since ψ_j^*βψ_j is real. The forward hopping term contributes -J_j/h. The backward scalar is the conjugate of ψ_j-1^*α U_j-1ψ_j, using α=α^* and the scalar nature of U. Its contribution is J_j-1/h. This proves continuity. Periodic summation gives dotbarρ=0. Finally, dot G_j =fracdot E_j-dot E_j-1h+d","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c59246a191bf113","tex":"\\dot F","aliases":"dot F dot","display":"\\dot F","kind":"use","excerpt":"[Effect continuity] Under these hypotheses, dot p_b=∑_a J^F_ba, ∑_bdot p_b=0. If F_b=F_b(t) is differentiable in operator norm, the first identity becomes dot p_b=s_b+∑_a J^F_ba, s_b=innerψdot F_bψ, ∑_b s_b=0. Self-adjointness of F_b and differentiation give dot p_b=2ReinnerF_bψKψ for a fixed effect. Summing the two terms of [source reference] over","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7deecd1c9f42c06a","tex":"\\dot F_b","aliases":"dot F_b dot","display":"\\dot F_b","kind":"use","excerpt":"[Effect continuity] Under these hypotheses, dot p_b=∑_a J^F_ba, ∑_bdot p_b=0. If F_b=F_b(t) is differentiable in operator norm, the first identity becomes dot p_b=s_b+∑_a J^F_ba, s_b=innerψdot F_bψ, ∑_b s_b=0. Self-adjointness of F_b and differentiation give dot p_b=2ReinnerF_bψKψ for a fixed effect. Summing the two terms of [source reference] over","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0c809511415d8469","tex":"\\dot{\\mathcal G}","aliases":"dot G dot mathcal","display":"\\dot{\\mathcal G}","kind":"use","excerpt":"[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3f95e562bedcba2c","tex":"\\dot{\\mathcal G}_j","aliases":"dot G_j dot mathcal","display":"\\dot{\\mathcal G}_j","kind":"use","excerpt":"[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d576368deebdc90f","tex":"\\dot H","aliases":"dot H dot","display":"\\dot H","kind":"use","excerpt":"equation dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f0d1701d0d18bb49","tex":"\\dot h","aliases":"dot h dot","display":"\\dot h","kind":"use","excerpt":"Set T_h=innerππ_h-tfrac12innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-217fb857dc45993a","tex":"\\dot H_{\\mathrm m}","aliases":"dot H_ m dot mathrm","display":"\\dot H_{\\mathrm m}","kind":"use","excerpt":"[Electric work and total energy] Along the same solutions, dot H_E=h∑_jE_jJ_j, dot H_ m=-h∑_jE_jJ_j, dot H_h=0.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2d1b19f5284ca29c","tex":"\\dot H_E","aliases":"dot H_E dot","display":"\\dot H_E","kind":"use","excerpt":"[Electric work and total energy] Along the same solutions, dot H_E=h∑_jE_jJ_j, dot H_ m=-h∑_jE_jJ_j, dot H_h=0.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cfe2edb8994f8ac8","tex":"\\dot H_h","aliases":"dot H_h dot","display":"\\dot H_h","kind":"use","excerpt":"[Electric work and total energy] Along the same solutions, dot H_E=h∑_jE_jJ_j, dot H_ m=-h∑_jE_jJ_j, dot H_h=0.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9b6d09980d703f8f","tex":"\\dot h_{ij}","aliases":"dot h_ij dot","display":"\\dot h_{ij}","kind":"use","excerpt":"innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca30477e01c3f93b","tex":"\\dot H_q","aliases":"dot H_q dot","display":"\\dot H_q","kind":"use","excerpt":"equation dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positivity makes the powers","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-efe523b3236f6750","tex":"\\dot M","aliases":"dot M dot","display":"\\dot M","kind":"use","excerpt":"For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b48a467a636fcefb","tex":"\\dot m","aliases":"dot m dot","display":"\\dot m","kind":"use","excerpt":"Exact mass flux and compatibility. Put A=Π+vΦ and B=Φ+vΠ. The remaining Einstein momentum evolution gives dot m=r^2 J, J=σ AB. One can verify its origin directly from the P_Λ equation. In the flat-slice gauge, the gravitational terms give σ vP_Λ'-σ v^2/2-rσ'. The lapse-gradient term is essential. The matter variation contributes σ r^2(Π^2+Φ^2)/2. A","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ffe398ee826021a","tex":"\\dot M_q","aliases":"dot M_q dot","display":"\\dot M_q","kind":"use","excerpt":"For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8e546668fa5b487a","tex":"\\dot p_1","aliases":"dot p_1 dot","display":"\\dot p_1","kind":"use","excerpt":"[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-974902f9939d7ec2","tex":"\\dot p_2","aliases":"dot p_2 dot","display":"\\dot p_2","kind":"use","excerpt":"-isin(gt), p_1=cos^2(gt), p_2=sin^2(gt). The current is J_2←1 =2Im(i gsin(gt)cos(gt)) =gsin(2gt)=dot p_2. It reverses sign during the cycle. At t=0 the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at second order in time. This is why coupling strength and instan","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8970ef006a53c2b6","tex":"\\dot p_A^{\\mathrm c}","aliases":"dot p_A^ c dot mathrm","display":"\\dot p_A^{\\mathrm c}","kind":"use","excerpt":"J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). For A={1,2} and B={3,4}, the internal currents 3 and 1 cancel, leaving J^ c_B← A=2-1/2=3/2. Hence dot p_A^ c=-3/2 and","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ae88b8db41308a09","tex":"\\dot p_B^{\\mathrm c}","aliases":"dot p_B^ c dot mathrm","display":"\\dot p_B^{\\mathrm c}","kind":"use","excerpt":"J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a4a1cd55f634f90d","tex":"\\dot P_R","aliases":"dot P_R dot","display":"\\dot P_R","kind":"use","excerpt":"ginequation dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2], dot P_R · =fracΛ(P_Λ^2+Q^2)R^3 -fracP_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7ab430cdfa9f3016","tex":"\\dot P_\\Lambda","aliases":"dot P_Λ dot Lambda","display":"\\dot P_\\Lambda","kind":"use","excerpt":"ginequation dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2], dot P_R · =fracΛ(P_Λ^2+Q^2)R^3 -fracP_","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db307e9343ea80e8","tex":"\\dot Q","aliases":"dot Q dot","display":"\\dot Q","kind":"use","excerpt":"Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c0667a78ae67ed93","tex":"\\dot R","aliases":"dot R dot","display":"\\dot R","kind":"use","excerpt":"da>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift. The Maxwell multiplier is denoted Φ_e to distinguish it from the scalar gradient used later. The bulk canonical action is S=∫ dt∫ dr (P_ΛdotΛ+P_Rdot R+Qdot A_r -N H-N^r D-Φ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea1e67564418c94f","tex":"\\dot R_j","aliases":"dot R_j dot","display":"\\dot R_j","kind":"use","excerpt":"Direct and split radius evolution. On the positive branch, restriction of the radius equation to the reconstruction manifold gives dot R_j=g(R_j)+b_j(D_4R)_j, g(R)=-fracsqrt2MR-Q^2R, where b=N^r is prescribed in the electrovac benchmark. A split alternative is dot R=g(R)+A_h[b]R, A_h[b]u=frac12(bD_4u+D_4(bu)-(D_4b)u). On smooth data these formulas ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aacc3fef7c69d461","tex":"\\dot S","aliases":"dot S dot","display":"\\dot S","kind":"use","excerpt":"For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-51f3bbd1503295c5","tex":"\\dot u","aliases":"dot u dot","display":"\\dot u","kind":"use","excerpt":"[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bddb44ee5b7358ac","tex":"\\dot U^*","aliases":"dot U^* dot","display":"\\dot U^*","kind":"use","excerpt":"Differentiate U^*U=id to obtain dot U^*U+U^*dot U=0. This proves Γ^*=-Γ. Multiplication on the right by U^* gives dot U^*=-Γ U^*. Therefore dotϕ=dot U^*ψ+U^*dotψ =-Γϕ+U^*KUϕ. Unitary conjugation preserves skew-adjointness. The difference of the two skew-adjoint operators in [source reference] is skew-adjoint, so the transfer theorem applies.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c63fc0870cbf6ad7","tex":"\\dot U_{\\rm raw}","aliases":"dot U_rm raw dot rm","display":"\\dot U_{\\rm raw}","kind":"use","excerpt":"The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservatio","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ae6eb7cbea0e8357","tex":"\\dot V","aliases":"dot V dot","display":"\\dot V","kind":"use","excerpt":"Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber dot U · =γΔ_hP+e^2P[V^2-γ(D_hQ)^2],nonumber dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and expone","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3ec2563613dd0ed8","tex":"\\dot v","aliases":"dot v dot","display":"\\dot v","kind":"use","excerpt":"[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-458452ed36c03a28","tex":"\\dot v_j","aliases":"dot v_j dot","display":"\\dot v_j","kind":"use","excerpt":"[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ef190f09437c0f51","tex":"\\dot V_{\\rm raw}","aliases":"dot V_rm raw dot rm","display":"\\dot V_{\\rm raw}","kind":"use","excerpt":"The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservatio","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-60e5e5cbe336ff2a","tex":"\\dot w","aliases":"dot w dot","display":"\\dot w","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cb751fbbac757a0e","tex":"\\dot w_+","aliases":"dot w_+ dot","display":"\\dot w_+","kind":"use","excerpt":"Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-], dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-121f6cefc8878e5a","tex":"\\dot w_-","aliases":"dot w_- dot","display":"\\dot w_-","kind":"use","excerpt":"Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-], dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-caed6451c7e16f37","tex":"\\dot w_j","aliases":"dot w_j dot","display":"\\dot w_j","kind":"use","excerpt":"equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms i","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-30ea3aa7299df5c5","tex":"\\dot w_r","aliases":"dot w_r dot","display":"\\dot w_r","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-825f9bf753046ea5","tex":"\\dot z","aliases":"dot z dot","display":"\\dot z","kind":"use","excerpt":"[Fixed-mesh second-order time accuracy] Fix h. Suppose a smooth exact solution of dot z=widehat f_h(τ,z) and its numerical steps remain in a compact regular neighborhood where d_h≥ d_0>0. Assume the mixed time and state derivatives of widehat f_h through order three and the state derivatives of C_h through order four are bounded there. Apply the en","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c9e52621fb9f724d","tex":"\\dot{\\bar\\rho}","aliases":"dotbarρ dot bar rho","display":"\\dot{\\bar\\rho}","kind":"use","excerpt":"[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-74d7eef59b015f03","tex":"\\dot\\theta","aliases":"dotθ dot theta","display":"\\dot\\theta","kind":"use","excerpt":"sinθ(t) · cosθ(t). With the coordinate projectors as P_a^(0), ϕ=cosθ -sinθ, Γ=dotθ0 · -1 1 · 0, K_eff=-Γ. The moving weights are p_1=cos^2θ and p_2=sin^2θ, with J_2←1=dotθsin(2θ)=dot p_2. The state has not moved in the original Hilbert space. The measured sectors have moved around it. The connection term accounts for the entire change.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a741df871091194a","tex":"\\dot\\Lambda","aliases":"dotΛ dot Lambda","display":"\\dot\\Lambda","kind":"use","excerpt":"da>0, write ds^2=-N^2dt^2+Λ^2(dr+N^r dt)^2+R^2dΩ^2, A=A_rdr+Φ_e dt. Here R is the areal radius, Λ determines proper radial distance, N is the lapse, and N^r is the radial shift. The Maxwell multiplier is denoted Φ_e to distinguish it from the scalar gradient used later. The bulk canonical action is S=∫ dt∫ dr (P_ΛdotΛ+P_Rdot R+Qdot A_r -N H-N^r D-Φ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-93c44eb7ba707095","tex":"\\dot\\Pi","aliases":"dotΠ dot Pi","display":"\\dot\\Pi","kind":"use","excerpt":"E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ), dotΦ · =∂_r[σ(Π+vΦ)], dotΠ · =r^-2∂_r[r^2σ(Φ+vΠ)]. If Φ=φ' initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ba309cf054813f32","tex":"\\dot\\pi","aliases":"dotπ dot pi","display":"\\dot\\pi","kind":"use","excerpt":"innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic g","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f3cfdbc357b7c1cb","tex":"\\dot\\pi_{ij}","aliases":"dotπ_ij dot pi","display":"\\dot\\pi_{ij}","kind":"use","excerpt":"innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-826d719bf280601a","tex":"\\dot\\rho","aliases":"dotρ dot rho","display":"\\dot\\rho","kind":"use","excerpt":"Suppose a differentiable matrix trajectory obeys the von Neumann equation dotρ=-i[H,ρ], H=H^*. For its coherent transfer define J^(ρ)_b← a =2ImTr(P_bHP_aρ).","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b72c01f71a7b828c","tex":"\\dot\\rho_{ij}","aliases":"dotρ_ij dot rho","display":"\\dot\\rho_{ij}","kind":"use","excerpt":"In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3d8a0fef306e4e52","tex":"\\dot\\rho_j","aliases":"dotρ_j dot rho","display":"\\dot\\rho_j","kind":"use","excerpt":"[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-befcf78b7c16b7f6","tex":"\\dot\\Phi","aliases":"dotΦ dot Phi","display":"\\dot\\Phi","kind":"use","excerpt":"E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ), dotΦ · =∂_r[σ(Π+vΦ)], dotΠ · =r^-2∂_r[r^2σ(Φ+vΠ)]. If Φ=φ' initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c8b9d31659a8bf5","tex":"\\dot\\phi","aliases":"dotϕ dot phi","display":"\\dot\\phi","kind":"use","excerpt":"[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4512c0db0b524163","tex":"\\dot\\varphi","aliases":"dotφ dot varphi","display":"\\dot\\varphi","kind":"use","excerpt":"w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9a8c0cc0f1e0e9e6","tex":"\\dot\\varphi_j","aliases":"dotφ_j dot varphi","display":"\\dot\\varphi_j","kind":"use","excerpt":"equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms i","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d706b990909bd88e","tex":"\\dot\\varphi_{j+1}","aliases":"dotφ_j+1 dot varphi","display":"\\dot\\varphi_{j+1}","kind":"use","excerpt":"[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cd1a0c9fdd3cf81a","tex":"\\dot\\varphi_j^2","aliases":"dotφ_j^2 dot varphi","display":"\\dot\\varphi_j^2","kind":"use","excerpt":"w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-302ba2bfdf019cf6","tex":"\\dot\\chi","aliases":"dotχ dot chi","display":"\\dot\\chi","kind":"use","excerpt":"This is the residual symmetry of the stated temporal-gauge model. If χ_j depends on time, the matter time derivative acquires idotχ_jψ_j, while dot A_j acquires (dotχ_j+1-dotχ_j)/h. With E_j fixed and no scalar potential in [source reference], these changes do not preserve the displayed equations. The fully time-dependent covariance of the discrete","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0bcb61993d85cfe8","tex":"\\dot\\chi_j","aliases":"dotχ_j dot chi","display":"\\dot\\chi_j","kind":"use","excerpt":"This is the residual symmetry of the stated temporal-gauge model. If χ_j depends on time, the matter time derivative acquires idotχ_jψ_j, while dot A_j acquires (dotχ_j+1-dotχ_j)/h. With E_j fixed and no scalar potential in [source reference], these changes do not preserve the displayed equations. The fully time-dependent covariance of the discrete","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-56ba2edb3284c438","tex":"\\dot\\chi_{j+1}","aliases":"dotχ_j+1 dot chi","display":"\\dot\\chi_{j+1}","kind":"use","excerpt":"This is the residual symmetry of the stated temporal-gauge model. If χ_j depends on time, the matter time derivative acquires idotχ_jψ_j, while dot A_j acquires (dotχ_j+1-dotχ_j)/h. With E_j fixed and no scalar potential in [source reference], these changes do not preserve the displayed equations. The fully time-dependent covariance of the discrete","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-47f08e6aab43dd03","tex":"\\dot\\psi_j","aliases":"dotψ_j dot psi","display":"\\dot\\psi_j","kind":"use","excerpt":"βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_j","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6038007d6ae6dd28","tex":"\\dot\\psi_j^*","aliases":"dotψ_j^* dot psi","display":"\\dot\\psi_j^*","kind":"use","excerpt":"βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_j","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bb79f280e36ef7a2","tex":"\\dot\\psi_r","aliases":"dotψ_r dot psi","display":"\\dot\\psi_r","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8052e1ac0acaea2d","tex":"\\dot\\Omega","aliases":"dotΩ dot Omega","display":"\\dot\\Omega","kind":"use","excerpt":"lit On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7015f1f6989ac898","tex":"\\mathbf E","aliases":"E mathbf","display":"\\mathbf E","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"N(a)=∑_n∈ S a_n _2^2, E(a)=∑_n∈ S⟨ a_n,H_na_n⟩,","count":28},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-303b235f22c45f55","tex":"E'_{L,0}","aliases":"E'_L,0","display":"E'_{L,0}","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2099c82204eda1e6","tex":"E'_{L,j}","aliases":"E'_L,j","display":"E'_{L,j}","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-67b08d219bc9d5a2","tex":"e^{(L_g+S_1/2)t}","aliases":"e^(L_g+S_1/2)t","display":"e^{(L_g+S_1/2)t}","kind":"use","excerpt":"djoint for the grid inner product inneruw_h, and let multiplication by the real array b be self-adjoint. Then inneruA_h[b]u_h =-frac12inneru(Db)u_h. For two solutions of the split equation with the same prescribed b, remaining in a positive-radicand interval where |g'|≤ L_g, one obtains normR(t)-widetilde R(t)_h ≤ e^(L_g+S_1/2)tnormR(0)-widetilde R","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dbf30aa11f3ef3cc","tex":"e^{-(r-10)^2}","aliases":"e^-(r-10)^2","display":"e^{-(r-10)^2}","kind":"use","excerpt":"The initial scalar packet is φ(0,r)=A e^-(r-10)^2, Φ(0,r)=∂_rφ(0,r), Π(0,r)=Φ(0,r)+fracφ(0,r)r. The last equation is the flat-space ingoing condition for χ=rφ, used to initialize the scalar. The nonlinear gravitational constraints are then solved. It is therefore an ingoing packet prescription, rather than exactly freezing the geometry to Minkowski","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d7344e7d866b8403","tex":"e^{-(r_i+r_j)t/2}","aliases":"e^-(r_i+r_j)t/2","display":"e^{-(r_i+r_j)t/2}","kind":"use","excerpt":"on closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=e^-(r_i+r_j)t/2ρ_ij(0). This decay ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c2349512e48c7e97","tex":"e^{-1}","aliases":"e^-1","display":"e^{-1}","kind":"use","excerpt":"There is also a spectral boundary. Even when e_a^i are trigonometric polynomials, e^-1, e^1/2, and the torsion-free connection are generally not finite Fourier polynomials. The unreduced degree rules 16K<N and 32K<N therefore cannot simply be reapplied to the reduced chart. One may evaluate its finite-dimensional regular functions directly, or reta","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fbb1629ff78a5896","tex":"e^{-2kt}","aliases":"e^-2kt","display":"e^{-2kt}","kind":"use","excerpt":"[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a1b4272a33792a85","tex":"e^{-2P}","aliases":"e^-2P","display":"e^{-2P}","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e4256bd4d37cf206","tex":"e^{-2\\beta\\epsilon_m}","aliases":"e^-2βϵ_m beta epsilon","display":"e^{-2\\beta\\epsilon_m}","kind":"use","excerpt":"For β>0, the multiplier e^-β H_ϵ is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the posi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-462eff40a0d76a62","tex":"e^{-2\\pi ijk/M}","aliases":"e^-2π ijk/M pi","display":"e^{-2\\pi ijk/M}","kind":"use","excerpt":"Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and se","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6332bad66e756810","tex":"e^{-2\\tau}","aliases":"e^-2τ tau","display":"e^{-2\\tau}","kind":"use","excerpt":"Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-414a8cb533bd9054","tex":"e^{-2\\phi}","aliases":"e^-2ϕ phi","display":"e^{-2\\phi}","kind":"use","excerpt":"1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. Therefore Ω=dω=(ϕ_tt-ϕ_xx)K dtwedge dx, R=2e^-2ϕ(ϕ_tt-ϕ_xx). This fixes the scalar-curvature sign convention used in this monograph.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6aa40be28a41f8ce","tex":"e^{-hA}","aliases":"e^-hA","display":"e^{-hA}","kind":"use","excerpt":"Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all suffi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bfb1cd35e1e6c75a","tex":"e^{-hB}","aliases":"e^-hB","display":"e^{-hB}","kind":"use","excerpt":"Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all suffi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-46080bafb5da7953","tex":"e^{-i(s+t)E}","aliases":"e^-i(s+t)E","display":"e^{-i(s+t)E}","kind":"use","excerpt":"U_H(s)U_H(t) =e^-i(s+t)EP_++e^i(s+t)EP_-=U_H(s+t).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-515496daf359dcbc","tex":"e^{-i(t-s)A}","aliases":"e^-i(t-s)A","display":"e^{-i(t-s)A}","kind":"use","excerpt":"The same holds for A. Since A commutes with every power of itself, it commutes with e^-iuA. Fix t and define F(s)=e^-i(t-s)Ae^-isB. The product rule gives","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b62117803ef27c0d","tex":"e^{-i(t-s)D}","aliases":"e^-i(t-s)D","display":"e^{-i(t-s)D}","kind":"use","excerpt":"K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conv","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ea38b9782a5a3dd","tex":"e^{-i(t-s)H_\\lambda}","aliases":"e^-i(t-s)H_λ lambda","display":"e^{-i(t-s)H_\\lambda}","kind":"use","excerpt":"and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-diagonal assumption and reduction by H_0 give B e^-isH_0P=QBP","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7e78dacd05eb2880","tex":"e^{-igtZ_{\\mathrm{sign}}}","aliases":"e^-igtZ_sign mathrm","display":"e^{-igtZ_{\\mathrm{sign}}}","kind":"use","excerpt":"b_1⟩=|b_1⟩. One concrete choice is an odd-prime torsion involution times a fixed odd-prime detail projector, with the torsion state in its +1 eigenspace. For real g≠0, use H_int=gZ_sign⊗ B, |Ψ(0)⟩=|+_x⟩⊗ frac|b_0⟩+|b_1⟩sqrt2. Evolution under this explicit interaction alone gives |Ψ(t)⟩=frac1sqrt2 (|+_x⟩⊗|b_0⟩ +e^-igtZ_sign|+_x⟩⊗|b_1⟩). Tracing out ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-426af39a89a86187","tex":"e^{-ih^2}","aliases":"e^-ih^2","display":"e^{-ih^2}","kind":"use","excerpt":"[Left-endpoint sampling can lose an order] Take A(x)=x near the origin and ψ(x)=1. With the left-endpoint prescription widehat U_h(x)=e^-ihA(x), the stencil at zero is fracwidehat U_h(0)-widehat U_h(-h)^-12h =frac1-e^-ih^22h=fraci2h+O(h^3). The continuum value (∂_x-iA(0))ψ(0) is zero. Thus this perfectly smooth input has a first-order error. The fo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca72e42f27fac12f","tex":"e^{-ihA(x)}","aliases":"e^-ihA(x)","display":"e^{-ihA(x)}","kind":"use","excerpt":"[Left-endpoint sampling can lose an order] Take A(x)=x near the origin and ψ(x)=1. With the left-endpoint prescription widehat U_h(x)=e^-ihA(x), the stencil at zero is fracwidehat U_h(0)-widehat U_h(-h)^-12h =frac1-e^-ih^22h=fraci2h+O(h^3). The continuum value (∂_x-iA(0))ψ(0) is zero. Thus this perfectly smooth input has a first-order error. The fo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5168a10b25fab030","tex":"e^{-ihA(x+h/2)}","aliases":"e^-ihA(x+h/2)","display":"e^{-ihA(x+h/2)}","kind":"use","excerpt":"A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-06c04b308af64a05","tex":"e^{-ihA_j}","aliases":"e^-ihA_j","display":"e^{-ihA_j}","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ba26c168a779953c","tex":"e^{-ihp_j(n)}","aliases":"e^-ihp_j(n)","display":"e^{-ihp_j(n)}","kind":"use","excerpt":"D_h,je_n =e_nfrace^ihp_j(n)-e^-ihp_j(n)2h =ifracsin(hp_j(n))he_n.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-938d1b8f1712512f","tex":"e^{-\\operatorname{Im}z\\,\\tau}","aliases":"e^-Imz τ operatorname tau","display":"e^{-\\operatorname{Im}z\\,\\tau}","kind":"use","excerpt":"itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conversely the reconstruction differentiates to the hidden equation and supplies the prescribed initial value, so it recovers the full evolution. Since D is self-adjoint, normK(τ)≤normB^2. The factor e^-Imz ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3fbb729a001bbbc","tex":"e^{-isB}","aliases":"e^-isB","display":"e^{-isB}","kind":"use","excerpt":"Proof. First justify differentiation. On |s|≤ R, the series for e^-isB converges uniformly, and its derivative series is norm-dominated by","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3cd2d38b3e555da","tex":"e^{-isH_0}","aliases":"e^-isH_0","display":"e^{-isH_0}","kind":"use","excerpt":"and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-diagonal assumption and reduction by H_0 give B e^-isH_0P=QBP","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a01743648714134d","tex":"e^{-it}","aliases":"e^-it","display":"e^{-it}","kind":"use","excerpt":"In particular, u_0(t)=e^-ite_0. For the centered axis and diagonal modes,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4a353a46a2b6c1b4","tex":"e^{-it\\mathcal A H\\mathcal A^*}","aliases":"e^-it A H A^* mathcal mathcal","display":"e^{-it\\mathcal A H\\mathcal A^*}","kind":"use","excerpt":"For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bd6eab3f096ffab4","tex":"e^{-it_0\\epsilon_m}","aliases":"e^-it_0ϵ_m epsilon","display":"e^{-it_0\\epsilon_m}","kind":"use","excerpt":"The modulus-one multipliers give U(t)^*U(t)=I and U(t+s)=U(t)U(s). They also give strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9eaf8275847a72c8","tex":"e^{-itA}","aliases":"e^-itA","display":"e^{-itA}","kind":"use","excerpt":"e^-itA-e^-itB ≤ |t| A-B .","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e39fb67a6a73c3c4","tex":"e^{-itB}","aliases":"e^-itB","display":"e^{-itB}","kind":"use","excerpt":"e^-itA-e^-itB ≤ |t| A-B .","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ac8c969ea731f537","tex":"e^{-itD}","aliases":"e^-itD","display":"e^{-itD}","kind":"use","excerpt":"[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-side","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fae7499816e18e18","tex":"e^{-itE}","aliases":"e^-itE","display":"e^{-itE}","kind":"use","excerpt":"U_H(t)=e^-itEP_++e^itEP_- =cos(tE)I_4-ifracsin(tE)EH.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2637b6dd9fae991f","tex":"e^{-itH}","aliases":"e^-itH","display":"e^{-itH}","kind":"use","excerpt":"For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-71404fd53acc476a","tex":"e^{-itH_0}","aliases":"e^-itH_0","display":"e^{-itH_0}","kind":"use","excerpt":"quation where the inverse acts only on QHH, and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-diagonal assumpti","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-33b5b515ddb5e759","tex":"e^{-itH_A}","aliases":"e^-itH_A","display":"e^{-itH_A}","kind":"use","excerpt":"ge reduces H_A, and its finite restriction H_A^( h) satisfies exact embedding intertwining with every larger rectangular restriction. For bounded Borel F, F(H_A)f=∑_ rF(A_ r) F_ rf, F(H_A) E_ h→ F(H_A) strongly. In particular, norme^-itH_A(I- E_ h)f =norm(I- E_ h)f. Finite rectangular sums belong to the domain and are dense. Symmetry holds by symme","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-46067ecf745d5b8c","tex":"e^{-\\mathrm{i}tH_c}","aliases":"e^-itH_c mathrm","display":"e^{-\\mathrm{i}tH_c}","kind":"use","excerpt":"[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber e^-β H_fV · =Ve^-β H_c,nonumber (z-H_f)^-1V · =V(z-H_c)^-1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7d3c46f1ac08755a","tex":"e^{-\\mathrm{i}tH_f}","aliases":"e^-itH_f mathrm","display":"e^{-\\mathrm{i}tH_f}","kind":"use","excerpt":"[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber e^-β H_fV · =Ve^-β H_c,nonumber (z-H_f)^-1V · =V(z-H_c)^-1.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-77fd83ae503c187e","tex":"e^{-itH_m^{\\mathrm{phys}}}","aliases":"e^-itH_m^phys mathrm","display":"e^{-itH_m^{\\mathrm{phys}}}","kind":"use","excerpt":"Connecting the group to its generator. The next chapter proves that the constructed group is the spectral evolution of this Hamiltonian: U(t)=e^-itH_m^phys. The proof establishes strong continuity and differentiates on the full Hamiltonian domain with control of the Fourier tail. The reverse domain inclusion then shows that the infinitesimal genera","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc773578bf2509d3","tex":"e^{-\\mathrm{i}tH_{p,h}^{\\mathrm{Haar}}}","aliases":"e^-itH_p,h^Haar mathrm mathrm","display":"e^{-\\mathrm{i}tH_{p,h}^{\\mathrm{Haar}}}","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4ce53f9cf88dacbf","tex":"e^{-itH_\\epsilon}","aliases":"e^-itH_ϵ epsilon","display":"e^{-itH_\\epsilon}","kind":"use","excerpt":"For bounded Borel F:R→C, define F(H_ϵ)f=∑_m≥0F(ϵ_m) F_mf. The series converges by Parseval and normF(H_ϵ)f_2≤sup_m|F(ϵ_m)|norm f_2. The same level decomposition verifies sums, products, and adjoints of these multipliers directly. In particular, U(t) · =e^-itH_ϵ =∑_m≥0e^-itϵ_m F_m, (z-H_ϵ)^-1 · =∑_m≥0(z-ϵ_m)^-1 F_m, z∉σ(H_ϵ). The projections in thes","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ee9845200fd38a4","tex":"e^{-itH_\\lambda}","aliases":"e^-itH_λ lambda","display":"e^{-itH_\\lambda}","kind":"use","excerpt":"[Defect, leakage, and a second-order self-energy] Under these hypotheses, normH_λ V-VH_c · =|λ|normBV, normQe^-itH_λP · ≤min{1,|tλ|normQBP}. For z∉σ(QH_λ Q|_QHH), the coarse self-energy is Σ_P(z)=PH_λ Q (z-QH_λ Q|_QHH)^-1QH_λ P, where the inverse acts only on QHH, and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2f444d0d93addeee","tex":"e^{-itH_\\infty}","aliases":"e^-itH_∞ infty","display":"e^{-itH_\\infty}","kind":"use","excerpt":"E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It does not say th","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f75d36d14a6c2f17","tex":"e^{-it\\widetilde H_M}","aliases":"e^-itwidetilde H_M widetilde","display":"e^{-it\\widetilde H_M}","kind":"use","excerpt":"An alternative is to extend the finite Hamiltonian by the zero Hamiltonian on the omitted subspace: widetilde H_M=H_ϵ E_M =∑_m≤ Mϵ_m F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c8ba533552f605b6","tex":"e^{-it\\epsilon}","aliases":"e^-itϵ epsilon","display":"e^{-it\\epsilon}","kind":"use","excerpt":"U(t)^*U(t)=I and U(t+s)=U(t)U(s). They also give strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ-1)/t|≤|ϵ|, proves lim_t→0fracU(t","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5ce269a7f5673f0b","tex":"e^{-it\\epsilon_m}","aliases":"e^-itϵ_m epsilon","display":"e^{-it\\epsilon_m}","kind":"use","excerpt":"For bounded Borel F:R→C, define F(H_ϵ)f=∑_m≥0F(ϵ_m) F_mf. The series converges by Parseval and normF(H_ϵ)f_2≤sup_m|F(ϵ_m)|norm f_2. The same level decomposition verifies sums, products, and adjoints of these multipliers directly. In particular, U(t) · =e^-itH_ϵ =∑_m≥0e^-itϵ_m F_m, (z-H_ϵ)^-1 · =∑_m≥0(z-ϵ_m)^-1 F_m, z∉σ(H_ϵ). The projections in thes","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3585bee2603724b","tex":"e^{-\\mathrm{i}t\\epsilon_{p,0}}","aliases":"e^-itϵ_p,0 mathrm epsilon","display":"e^{-\\mathrm{i}t\\epsilon_{p,0}}","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determine","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b9adfb35026d06fa","tex":"e^{-\\mathrm{i}t\\epsilon_{p,r}}","aliases":"e^-itϵ_p,r mathrm epsilon","display":"e^{-\\mathrm{i}t\\epsilon_{p,r}}","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ce034acd93c2c8ef","tex":"e^{-it\\lambda}","aliases":"e^-itλ lambda","display":"e^{-it\\lambda}","kind":"use","excerpt":"Let H_∞ be self-adjoint with spectral measure E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum o","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9638a772c36bf7ae","tex":"e^{-it\\lambda_n}","aliases":"e^-itλ_n lambda","display":"e^{-it\\lambda_n}","kind":"use","excerpt":"For real (λ_n) define (U(t)c)_n=e^-itλ_nc_n, c∈ℓ^2. The modulus of every multiplier is one. Hence U(t) is a unitary operator, U(t+s)=U(t)U(s), and U(-t)=U(t)^*. For each fixed state it is strongly continuous. To prove the latter claim, split c into a finite head and a small ℓ^2 tail. The finite head is continuous coordinatewise, while the norm of t","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-53b868ab17bc229d","tex":"e^{-iu}","aliases":"e^-iu","display":"e^{-iu}","kind":"use","excerpt":"ability domain] The orbit t↦ U(t)c is differentiable in ℓ^2 at one, equivalently every, time precisely when c∈ D(H_λ). On that domain, fracddtU(t)c=-iH_λ U(t)c, H_λ U(t)c=U(t)H_λ c. If c∈ D(H_λ), each coordinate difference quotient tends to -iλ_ne^-itλ_nc_n. The elementary bound |e^-iu-1|≤|u| bounds the quotient magnitude by |λ_nc_n|. The squared d","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5a7fb061cea5b411","tex":"e^{-iuA}","aliases":"e^-iuA","display":"e^{-iuA}","kind":"use","excerpt":"The same holds for A. Since A commutes with every power of itself, it commutes with e^-iuA. Fix t and define F(s)=e^-i(t-s)Ae^-isB. The product rule gives","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-47ca6386cf2c63a4","tex":"e^{-iuK}","aliases":"e^-iuK","display":"e^{-iuK}","kind":"use","excerpt":"The small-time leading term is -itλ QBP. More precisely, normQe^-itH_λP+itλ QBP ≤frac|λ|t^22 (normH_λ+normH_0)normQBP. To verify this, subtract the constant integrand QBP in Duhamel's formula and use norme^-iuK-I≤|u|norm K for a finite self-adjoint K. The two deviations are bounded by (|t-s|normH_λ+|s|normH_0)normQBP on the oriented integration int","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-58d59de3254bae16","tex":"e^{-i\\alpha_q}","aliases":"e^-iα_q alpha","display":"e^{-i\\alpha_q}","kind":"use","excerpt":"pa cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f9083e6f281158cc","tex":"e^{-i\\theta}","aliases":"e^-iθ theta","display":"e^{-i\\theta}","kind":"use","excerpt":"M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d91f717734c65b3e","tex":"e^{-i\\theta_j}","aliases":"e^-iθ_j theta","display":"e^{-i\\theta_j}","kind":"use","excerpt":"rier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-09975dc298fd8d03","tex":"e^{-i\\tau D}","aliases":"e^-iτ D tau","display":"e^{-i\\tau D}","kind":"use","excerpt":"[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-side","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-032fc47c4d958027","tex":"e^{-i\\tau J_{1:h}}","aliases":"e^-iτ J_1:h tau","display":"e^{-i\\tau J_{1:h}}","kind":"use","excerpt":"The scalar causal kernel is K(τ)=γ_0^2⟨1,e^-iτ J_1:h1⟩, multiplying I_2. It uses the same radial coefficients as the rational self-energy. Its exact causal relation is [source reference]. adicfigure08_radial_self_energyA branch-normalized finite tree has the same root response as its radial Jacobi chain. The complex-energy resolvent avoids real pol","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8c46d727c71945e9","tex":"e^{-kt}","aliases":"e^-kt","display":"e^{-kt}","kind":"use","excerpt":"[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bd14305f0c1887f7","tex":"e^{-r^2/(2\\sigma^2)}","aliases":"e^-r^2/(2σ^2) sigma","display":"e^{-r^2/(2\\sigma^2)}","kind":"use","excerpt":"For the normalized Gaussian density ρ(r)=fracM_0(2π)^3/2σ^3e^-r^2/(2σ^2), σ>0, direct differentiation verifies M_rm enc(r) · =M_0[ erf(frac rsqrt2σ) -sqrtfrac2πfrac rσ e^-r^2/(2σ^2)],","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-543348375bb29268","tex":"e^{-S/\\hbar}","aliases":"e^-S/ℏ hbar","display":"e^{-S/\\hbar}","kind":"use","excerpt":"Eliminating z first gives the effective coefficient μ for y^2 and the factor (m+n)^-1/2. Eliminating y next gives d^-1/2. Their product is D^-1/2, exactly as in direct elimination. For the Euclidean weight e^-S/ℏ, using a flat coarse measure instead records this determinant as S_C,ℏ=S_C+tfracℏ2log D, up to an additive constant. A nonlinear symmetry","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e7a07fecc096889f","tex":"e^{-t(N+1)}","aliases":"e^-t(N+1)","display":"e^{-t(N+1)}","kind":"use","excerpt":"For λ_n=n, a resolvent at iη, η≠0, has zero-padded error sup_n>Nfrac1sqrtn^2+η^2 =frac1sqrt(N+1)^2+η^2⟶0. The heat multiplier e^-tn has error e^-t(N+1) for fixed t>0. These tails decay because the operators are compact. At t=0 the heat operator is the identity, so the same norm-convergence assertion is false.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9524e9b54a42d871","tex":"e^{-tn}","aliases":"e^-tn","display":"e^{-tn}","kind":"use","excerpt":"For λ_n=n, a resolvent at iη, η≠0, has zero-padded error sup_n>Nfrac1sqrtn^2+η^2 =frac1sqrt(N+1)^2+η^2⟶0. The heat multiplier e^-tn has error e^-t(N+1) for fixed t>0. These tails decay because the operators are compact. At t=0 the heat operator is the identity, so the same norm-convergence assertion is false.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-04c554b9071d41ff","tex":"e^{-|\\alpha|^2/2}","aliases":"e^-|α|^2/2 alpha","display":"e^{-|\\alpha|^2/2}","kind":"use","excerpt":"For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea459297b3497aa4","tex":"e^{-\\beta H}","aliases":"e^-β H beta","display":"e^{-\\beta H}","kind":"use","excerpt":"The Euclidean operator e^-β H in this corollary is a spectral multiplier. It is not generally unitary, and a normalized state after Euclidean evolution requires its own normalization. The identity does not compare normalized Gibbs states of the entire fine and coarse systems, since the fine partition function also counts the detail block. For a com","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5b952d109b96b1f8","tex":"e^{-\\beta H_c}","aliases":"e^-β H_c beta","display":"e^{-\\beta H_c}","kind":"use","excerpt":"[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber e^-β H_fV · =Ve^-β H_c,nonumber (z-H_f)^-1V · =V(z-H_c)^-1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ec933d1e365cdeb7","tex":"e^{-\\beta H_f}","aliases":"e^-β H_f beta","display":"e^{-\\beta H_f}","kind":"use","excerpt":"[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber e^-β H_fV · =Ve^-β H_c,nonumber (z-H_f)^-1V · =V(z-H_c)^-1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a8526e6c4580f2c3","tex":"e^{-\\beta H_\\epsilon}","aliases":"e^-β H_ϵ beta epsilon","display":"e^{-\\beta H_\\epsilon}","kind":"use","excerpt":"For β>0, the multiplier e^-β H_ϵ is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the posi","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a5580f2eb87c1778","tex":"e^{-\\beta\\epsilon_0}","aliases":"e^-βϵ_0 beta epsilon","display":"e^{-\\beta\\epsilon_0}","kind":"use","excerpt":"silon is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the positive operator has trace Tr(","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5961b144710322df","tex":"e^{-\\beta\\epsilon_m}","aliases":"e^-βϵ_m beta epsilon","display":"e^{-\\beta\\epsilon_m}","kind":"use","excerpt":"silon is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the positive operator has trace Tr(","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3f49e8cdefb83b41","tex":"e^{-\\mu}","aliases":"e^-μ mu","display":"e^{-\\mu}","kind":"use","excerpt":"For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^μ(ω_L^ℓ-1). The factorial series proves nonnegativity, including the limiting values p_L,0(0)=1 and p_L,j(0)=0 for j>0. Partition of the Poisson series p","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-53b84cd8e738559b","tex":"e^{-\\mu(1-\\cos(2\\pi/L))}","aliases":"e^-μ(1-cos(2π/L)) mu cos pi","display":"e^{-\\mu(1-\\cos(2\\pi/L))}","kind":"use","excerpt":"[Approach to uniform residue probability] For fixed L≥2 and μ≥0, |p_L,j(μ)-frac1L| ≤frac1L∑_ℓ=1^L-1 e^-μ(1-cos(2πℓ/L)) ≤fracL-1L e^-μ(1-cos(2π/L)). In [source reference], the ℓ=0 term is 1/L. The modulus of every other term is e^μ(Reω_L^ℓ-1)/L. The triangle inequality gives the first bound. Among the nonzero residues, the largest cosine is cos(2π/L","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2af989611d84b3ff","tex":"e^{-\\mu(1-\\cos(2\\pi\\ell/L))}","aliases":"e^-μ(1-cos(2πℓ/L)) mu cos pi ell","display":"e^{-\\mu(1-\\cos(2\\pi\\ell/L))}","kind":"use","excerpt":"[Approach to uniform residue probability] For fixed L≥2 and μ≥0, |p_L,j(μ)-frac1L| ≤frac1L∑_ℓ=1^L-1 e^-μ(1-cos(2πℓ/L)) ≤fracL-1L e^-μ(1-cos(2π/L)). In [source reference], the ℓ=0 term is 1/L. The modulus of every other term is e^μ(Reω_L^ℓ-1)/L. The triangle inequality gives the first bound. Among the nonzero residues, the largest cosine is cos(2π/L","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8e4608509fba303e","tex":"e^{-\\tau}","aliases":"e^-τ tau","display":"e^{-\\tau}","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-15348357d90941e2","tex":"e^{-\\chi}","aliases":"e^-χ chi","display":"e^{-\\chi}","kind":"use","excerpt":"If the integrated coefficient is χ, then e^χ K=coshχ I+sinhχ K. This preserves the Lorentz form diag(-1,1) and has positive eigenvalues e^χ and e^-χ. Its real principal matrix logarithm is χ K. These are Lorentz boosts. The positive Hilbert norm used for unitary color transport in the gauge monograph is a different structure.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-99e4c97cf0f019b5","tex":"E^0","aliases":"E^0","display":"E^0","kind":"use","excerpt":"Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1feddf7f1342e789","tex":"e^0","aliases":"e^0","display":"e^0","kind":"use","excerpt":"n of φ gives the unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ec1ff447b7876d8d","tex":"e^1","aliases":"e^1","display":"e^1","kind":"use","excerpt":"n of φ gives the unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-26deaea0ab56be9e","tex":"e^{1/2}","aliases":"e^1/2","display":"e^{1/2}","kind":"use","excerpt":"al becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-89d50d3d0bc30bea","tex":"e^2","aliases":"e^2","display":"e^2","kind":"use","excerpt":"unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2cc4e438f644083b","tex":"e^{2b\\,\\operatorname{area}(\\square)K}","aliases":"e^2b area(square)K operatorname square","display":"e^{2b\\,\\operatorname{area}(\\square)K}","kind":"use","excerpt":"[Constant curvature density] For ϕ(t,x)=b(t^2-x^2)/2, one has ϕ_tt-ϕ_xx=2b, R=4b e^-2ϕ, W_square=e^2b area(square)K. The curvature density e^2ϕR/2 is constant although R varies with position. Adding oriented subrectangle integrals reconstructs the coarse rapidity exactly.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7dc913e427c2b1e7","tex":"e^{2P}","aliases":"e^2P","display":"e^{2P}","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9aa10317885fcb2","tex":"e^{2P_j}","aliases":"e^2P_j","display":"e^{2P_j}","kind":"use","excerpt":"dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity variables when the configuration P,Q is fixed.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ea4a26d6415bea6","tex":"e^{2|t|E_n}","aliases":"e^2|t|E_n","display":"e^{2|t|E_n}","kind":"use","excerpt":"f_h(t)-f(t) _L^2 ≤ h^2( ∑_n∈ S|t|^2e^2|t|E_nC(n)^2 a_n _2^2 )^1/2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a7ea9b34c288dc20","tex":"e^{2|t||\\lambda|}","aliases":"e^2|t||λ| lambda","display":"e^{2|t||\\lambda|}","kind":"use","excerpt":"A sufficient common-domain condition is ∫_R e^2|t||λ| d⟨ψ,E(λ)ψ⟩<∞, since |f_j(λ)|≤ e^|t||λ|. Spectral cutoffs P_R=E([-R,R]) supply a dense set satisfying this condition. On Ran P_R, the restricted Hamiltonian is bounded, the norm-series identities apply, and ∑_jf_j(H_∞)P_Rψ=e^-itH_∞P_Rψ ⟶ e^-itH_∞ψ. On the common domain of the individual lanes, ea","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-23fccd81348570f0","tex":"e^{2\\pi i n\\cdot x}","aliases":"e^2π i n· x pi cdot","display":"e^{2\\pi i n\\cdot x}","kind":"use","excerpt":"Fourier synthesis on the physical torus. The scalar Fourier characters e_n(x)=e^2π i n· x are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is F:HH⟶KK, HH=L^2(T^3;C^4). The four-spinor Parseval identity and surjectivity of s","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ede1673d5c208439","tex":"e^{2\\pi i(k-n)\\cdot x}","aliases":"e^2π i(k-n)· x pi cdot","display":"e^{2\\pi i(k-n)\\cdot x}","kind":"use","excerpt":"⟨ J_nv,J_kw⟩_L^2 =⟨ v,w⟩∫_ T^3e^2π i(k-n)· x dx.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f81e7a1f8753f574","tex":"e^{2\\pi i(x_1+x_2+x_3)}","aliases":"e^2π i(x_1+x_2+x_3) pi","display":"e^{2\\pi i(x_1+x_2+x_3)}","kind":"use","excerpt":"f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-11bafad14ed5d97c","tex":"e^{2\\pi i/L}","aliases":"e^2π i/L pi","display":"e^{2\\pi i/L}","kind":"use","excerpt":"Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-450272f382dba42d","tex":"e^{2\\pi ir}","aliases":"e^2π ir pi","display":"e^{2\\pi ir}","kind":"use","excerpt":"The integral is the product of three one-dimensional integrals. For integer r, the integral of e^2π irx over [0,1) is 1 when r=0; for r≠0, it is (e^2π ir-1)/(2π ir)=0. The product is therefore 1 exactly when n=k. Linearity is immediate, and setting k=n,w=v proves isometry. Expansion of the squared norm of the finite sum gives a double sum of pairin","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-212c2fd58b825ca3","tex":"e^{2\\pi irx}","aliases":"e^2π irx pi","display":"e^{2\\pi irx}","kind":"use","excerpt":"The integral is the product of three one-dimensional integrals. For integer r, the integral of e^2π irx over [0,1) is 1 when r=0; for r≠0, it is (e^2π ir-1)/(2π ir)=0. The product is therefore 1 exactly when n=k. Linearity is immediate, and setting k=n,w=v proves isometry. Expansion of the squared norm of the finite sum gives a double sum of pairin","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-014cd8dee73ba62b","tex":"e^{2\\pi i\\widehat N/L}","aliases":"e^2π iwidehat N/L pi widehat","display":"e^{2\\pi i\\widehat N/L}","kind":"use","excerpt":"On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f430d97b17e0353a","tex":"e^{2\\pi ix_1}","aliases":"e^2π ix_1 pi","display":"e^{2\\pi ix_1}","kind":"use","excerpt":"f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0dedbe3b64a4ae5e","tex":"e^{2\\phi}","aliases":"e^2ϕ phi","display":"e^{2\\phi}","kind":"use","excerpt":"Consider a twice continuously differentiable real function ϕ on an oriented (t,x) rectangle, and the coframe e^0=e^ϕ dt, e^1=e^ϕ dx, ds^2=e^2ϕ(-dt^2+dx^2). Write K=0 · 1","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-22e244f5761b2581","tex":"e^3","aliases":"e^3","display":"e^3","kind":"use","excerpt":"unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f79ae30615c4831b","tex":"\\mathcal E^a","aliases":"E^a mathcal","display":"\\mathcal E^a","kind":"use","excerpt":"Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b56e4649574b8fa0","tex":"e^a","aliases":"e^a","display":"e^a","kind":"use","excerpt":"∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lane, the omitted operator norm is at most e^afraca^Q+1(Q+1)!, a=|t|normA_0. The sum of norms of all exponential terms is at most e^a. A lane is a subseries of this sum, so it converges absolutely. The character calculation and factorial split can therefore be applied i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ba0a0499ab6712e5","tex":"E^a_i","aliases":"E^a_i","display":"E^a_i","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k,","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bd4227990f2f867b","tex":"E^a_k","aliases":"E^a_k","display":"E^a_k","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-42f98c4694f18001","tex":"\\mathcal E^b","aliases":"E^b mathcal","display":"\\mathcal E^b","kind":"use","excerpt":"Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dd7eab8cdf3e9482","tex":"E^b_i","aliases":"E^b_i","display":"E^b_i","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-66eb38162a08955e","tex":"E^b_j","aliases":"E^b_j","display":"E^b_j","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4ac68a8f5d0d5665","tex":"E^{bi}","aliases":"E^bi","display":"E^{bi}","kind":"use","excerpt":"Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2b1300a4ddb90178","tex":"E^c_k","aliases":"E^c_k","display":"E^c_k","kind":"use","excerpt":"H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-65082530d34bd1ec","tex":"e^{hA}","aliases":"e^hA","display":"e^{hA}","kind":"use","excerpt":"Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all suffi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf13ff3dd80732f9","tex":"e^{hB}","aliases":"e^hB","display":"e^{hB}","kind":"use","excerpt":"Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all suffi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e0f8b8099b1a4d62","tex":"E^I","aliases":"E^I","display":"E^I","kind":"use","excerpt":"An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9e2a2077329c507","tex":"E^i","aliases":"E^i","display":"E^i","kind":"use","excerpt":"Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dfd4f967373eb983","tex":"e^I","aliases":"e^I","display":"e^I","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4bd6cf0e6c9e8055","tex":"e^{i p(n)\\cdot x}","aliases":"e^i p(n)· x cdot","display":"e^{i p(n)\\cdot x}","kind":"use","excerpt":"e_n(x)=e^2π i n· x=e^i p(n)· x.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6711f41ccb2a9da4","tex":"e^{i(s+t)E}","aliases":"e^i(s+t)E","display":"e^{i(s+t)E}","kind":"use","excerpt":"U_H(s)U_H(t) =e^-i(s+t)EP_++e^i(s+t)EP_-=U_H(s+t).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3d7d478f333b07a6","tex":"e^{i(z-\\lambda)\\tau}","aliases":"e^i(z-λ)τ lambda tau","display":"e^{i(z-\\lambda)\\tau}","kind":"use","excerpt":"titution in the visible equation gives [source reference]. Conversely the reconstruction differentiates to the hidden equation and supplies the prescribed initial value, so it recovers the full evolution. Since D is self-adjoint, normK(τ)≤normB^2. The factor e^-Imz τ makes the transform absolutely convergent. On a spectral eigenvalue of D, ∫_0^∞ e^","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-85b4fb052ac49250","tex":"e^{ia}","aliases":"e^ia","display":"e^{ia}","kind":"use","excerpt":"A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f4ea3d6c305c710f","tex":"e^{ib}","aliases":"e^ib","display":"e^{ib}","kind":"use","excerpt":"A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e94881ab2b7c428e","tex":"e^{ih^2F_{ij}}","aliases":"e^ih^2F_ij","display":"e^{ih^2F_{ij}}","kind":"use","excerpt":"For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argu","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-55f51293b683c65d","tex":"e^{ihp_j(n)}","aliases":"e^ihp_j(n)","display":"e^{ihp_j(n)}","kind":"use","excerpt":"D_h,je_n =e_nfrace^ihp_j(n)-e^-ihp_j(n)2h =ifracsin(hp_j(n))he_n.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ed7844fbf82ff3f5","tex":"e^{ik\\cdot x}","aliases":"e^ik· x cdot","display":"e^{ik\\cdot x}","kind":"use","excerpt":"A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normaliz","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-24d92c2ffc4b376f","tex":"e^{ip\\alpha_q}","aliases":"e^ipα_q alpha","display":"e^{ip\\alpha_q}","kind":"use","excerpt":"At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7663ef411b37c4f9","tex":"e^{is}","aliases":"e^is","display":"e^{is}","kind":"use","excerpt":"· -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cross derivatives. This proof bounds absolute derivatives separately; it does not prove a s","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1d3442f53084db4f","tex":"e^{iS_C/\\hbar}","aliases":"e^iS_C/ℏ hbar","display":"e^{iS_C/\\hbar}","kind":"use","excerpt":"regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0d43ec957b56df25","tex":"e^{iS_F/\\hbar}","aliases":"e^iS_F/ℏ hbar","display":"e^{iS_F/\\hbar}","kind":"use","excerpt":"regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9fd148b730c54fcd","tex":"e^{it}","aliases":"e^it","display":"e^{it}","kind":"use","excerpt":"For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argu","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8499e0c2a6028581","tex":"e^{itD}","aliases":"e^itD","display":"e^{itD}","kind":"use","excerpt":"together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conversely the reconstruction d","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-97c26b61dd48806a","tex":"e^{itE}","aliases":"e^itE","display":"e^{itE}","kind":"use","excerpt":"U_H(t)=e^-itEP_++e^itEP_- =cos(tE)I_4-ifracsin(tE)EH.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4ce7d415045642ef","tex":"e^{iuA}","aliases":"e^iuA","display":"e^{iuA}","kind":"use","excerpt":"For Hermitian matrices the two exponentials in the integrand are unitary. This does not rely on the special Clifford algebra: the series gives (e^-iuA)^*=e^iuA, and the Cauchy product of the two absolutely convergent series gives e^iuAe^-iuA=I. Indeed, the coefficient of A^k for k>0 is (iu)^k∑_r=0^kbinomkr(-1)^r/k!=0, and the constant coefficient i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a50b74d94a7f721f","tex":"e^{iz\\tau}","aliases":"e^izτ tau","display":"e^{iz\\tau}","kind":"use","excerpt":"K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conv","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d1901d33f29f2a30","tex":"e^{i\\alpha_qr}","aliases":"e^iα_qr alpha","display":"e^{i\\alpha_qr}","kind":"use","excerpt":"p(θ) are E_q(θ)=μ+2κ cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-682a47b2eea70d50","tex":"e^{i\\theta}","aliases":"e^iθ theta","display":"e^{i\\theta}","kind":"use","excerpt":"At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-add9683f9060c3e6","tex":"e^{i\\theta_f}","aliases":"e^iθ_f theta","display":"e^{i\\theta_f}","kind":"use","excerpt":"six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If W_f=e^iθ_f with each θ_f chosen as a princ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f2fd45f05bfc81bb","tex":"e^{i\\theta_i(p,h)}","aliases":"e^iθ_i(p,h) theta","display":"e^{i\\theta_i(p,h)}","kind":"use","excerpt":"Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-01b838c630a22dad","tex":"e^{i\\chi_j}","aliases":"e^iχ_j chi","display":"e^{i\\chi_j}","kind":"use","excerpt":"For time-independent real χ_j, the transformation ψ_j↦ e^iχ_jψ_j, A_j↦ A_j+fracχ_j+1-χ_jh, E_j↦ E_j leaves the action and equations invariant. In fact U_j then has precisely the backward-link law [source reference], every hopping scalar z_j is unchanged, and the time derivatives of the gauge parameters vanish.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c115057b598e60d1","tex":"E^J","aliases":"E^J","display":"E^J","kind":"use","excerpt":"IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5d361bdf2478b59b","tex":"E^j","aliases":"E^j","display":"E^j","kind":"use","excerpt":"c:constraints:regular-triad The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c77a77855f0f7f8b","tex":"e^J","aliases":"e^J","display":"e^J","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0c9eb1acc19f1994","tex":"E^K","aliases":"E^K","display":"E^K","kind":"use","excerpt":"IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b607a74a7267e802","tex":"E^k","aliases":"E^k","display":"E^k","kind":"use","excerpt":"The identity I_4=P_++P_- and the two eigenvalue identities give, by induction, H^k=E^kP_++(-E)^kP_- for every k≥0. Substitute into the absolutely convergent series to obtain the spectral expression. Expanding its projectors gives the cosine and sine expression.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-420daa57bb2be484","tex":"e^K","aliases":"e^K","display":"e^K","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-52ee39a17082ee58","tex":"E^L","aliases":"E^L","display":"E^L","kind":"use","excerpt":"IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e0cff6c1252f56cd","tex":"e^L","aliases":"e^L","display":"e^L","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bb582ca6960394af","tex":"e^{LT}","aliases":"e^LT","display":"e^{LT}","kind":"use","excerpt":"nd on g make the correction O(Δτ^3). Thus the corrected local error has the same order. Under the stated derivative bounds the corrected step map has Lipschitz constant at most 1+LΔτ for small steps. If e_n is its error against the exact solution, then e_n+1≤(1+LΔτ)e_n+MΔτ^3. Iterating this inequality for nΔτ≤ T bounds the error by MT e^LTΔτ^2 when","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ed38d997f7481198","tex":"e^R","aliases":"e^R","display":"e^R","kind":"use","excerpt":"e^X-e^Y ≤ X-Y ∑_k=1^∞frackR^k-1k! =e^R X-Y .","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b68cf7b255e2e9f","tex":"e^{R\\|B\\|}","aliases":"e^R B","display":"e^{R\\|B\\|}","kind":"use","excerpt":"∑_k=1^∞frackR^k-1 B ^kk! = B e^R B .","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3dd973573165efc7","tex":"e^{tA_0}","aliases":"e^tA_0","display":"e^{tA_0}","kind":"use","excerpt":"[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber · =frac(tA_0)^jj! _0F_L-1 (- A_L,j; frac(tA_0)^LL^L),nonumber ∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eb1d3777cb0c1a31","tex":"e^X","aliases":"e^X","display":"e^X","kind":"use","excerpt":"e^X-e^Y ≤ X-Y ∑_k=1^∞frackR^k-1k! =e^R X-Y .","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1df81a064aee7511","tex":"e^Y","aliases":"e^Y","display":"e^Y","kind":"use","excerpt":"e^X-e^Y ≤ X-Y ∑_k=1^∞frackR^k-1k! =e^R X-Y .","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8611063e24d14e09","tex":"e^z","aliases":"e^z","display":"e^z","kind":"use","excerpt":"Apply this filter to e^z and put E_L,j(z)=∑_q=0^∞fracz^Lq+j(Lq+j)!. For α∈C, write (α)_q=α(α+1)⋯(α+q-1) and (α)_0=1. Splitting the factorial into step-L factors gives frac(Lq+j)!j! · =∏_r=1^L∏_k=0^q-1(j+r+Lk)nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-46b65184169dd9a1","tex":"e^{zA}","aliases":"e^zA","display":"e^{zA}","kind":"use","excerpt":"e^zA^ H=Te^zAT^-1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cecf606487256a5e","tex":"e^{zA^{\\mathcal H}}","aliases":"e^zA^ H mathcal","display":"e^{zA^{\\mathcal H}}","kind":"use","excerpt":"e^zA^ H=Te^zAT^-1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4eac7219693d8601","tex":"e^{zC_L}","aliases":"e^zC_L","display":"e^{zC_L}","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-780a43111d19c036","tex":"e^{|t|E_n}","aliases":"e^|t|E_n","display":"e^{|t|E_n}","kind":"use","excerpt":"U_H_h,n(t)-U_H_n(t) ≤ |t|e^|t|E_nh^2C(n).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cce7c34a27a2a30a","tex":"e^{|t||\\lambda|}","aliases":"e^|t||λ| lambda","display":"e^{|t||\\lambda|}","kind":"use","excerpt":"A sufficient common-domain condition is ∫_R e^2|t||λ| d⟨ψ,E(λ)ψ⟩<∞, since |f_j(λ)|≤ e^|t||λ|. Spectral cutoffs P_R=E([-R,R]) supply a dense set satisfying this condition. On Ran P_R, the restricted Hamiltonian is bounded, the norm-series identities apply, and ∑_jf_j(H_∞)P_Rψ=e^-itH_∞P_Rψ ⟶ e^-itH_∞ψ. On the common domain of the individual lanes, ea","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c99f921ba3d45581","tex":"e^{\\pm itE_{h,n}}","aliases":"e^± itE_h,n pm","display":"e^{\\pm itE_{h,n}}","kind":"use","excerpt":"|e^± itE_h,n-e^± itE_n| ≤ |t| |E_h,n-E_n|,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d9d6541bc9c84d0f","tex":"e^{\\pm itE_n}","aliases":"e^± itE_n pm","display":"e^{\\pm itE_n}","kind":"use","excerpt":"|e^± itE_h,n-e^± itE_n| ≤ |t| |E_h,n-E_n|,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e19ad9bc486a5d76","tex":"e^{\\pm its}","aliases":"e^± its pm","display":"e^{\\pm its}","kind":"use","excerpt":"by integrating the derivative of s↦ e^± its between the two real energies. Phase comparison alone does not compare full spinor evolution: the eigenspaces also depend on the kinetic vector. The operator argument below controls both effects.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-356289f0a237ed70","tex":"e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}","aliases":"e^μ(Reω_L^ℓ-1) mu operatorname omega ell","display":"e^{\\mu(\\operatorname{Re}\\omega_L^\\ell-1)}","kind":"use","excerpt":"For fixed L≥2 and μ≥0, |p_L,j(μ)-frac1L| ≤frac1L∑_ℓ=1^L-1 e^-μ(1-cos(2πℓ/L)) ≤fracL-1L e^-μ(1-cos(2π/L)). In [source reference], the ℓ=0 term is 1/L. The modulus of every other term is e^μ(Reω_L^ℓ-1)/L. The triangle inequality gives the first bound. Among the nonzero residues, the largest cosine is cos(2π/L), giving the second.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf514f58b73ee1a0","tex":"e^{\\mu(\\omega_L^\\ell-1)}","aliases":"e^μ(ω_L^ℓ-1) mu omega ell","display":"e^{\\mu(\\omega_L^\\ell-1)}","kind":"use","excerpt":"For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^μ(ω_L^ℓ-1). The factorial series proves nonnegativity, including the limiting values p_L,0(0)=1 and p_L,j(0)=0 for j>0. Partition of the Poisson series p","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b70233ddf00e967e","tex":"e^\\phi","aliases":"e^ϕ phi","display":"e^\\phi","kind":"use","excerpt":"Consider a twice continuously differentiable real function ϕ on an oriented (t,x) rectangle, and the coframe e^0=e^ϕ dt, e^1=e^ϕ dx, ds^2=e^2ϕ(-dt^2+dx^2). Write K=0 · 1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3980ccd55c0d7847","tex":"e^{\\chi}","aliases":"e^χ chi","display":"e^{\\chi}","kind":"use","excerpt":"If the integrated coefficient is χ, then e^χ K=coshχ I+sinhχ K. This preserves the Lorentz form diag(-1,1) and has positive eigenvalues e^χ and e^-χ. Its real principal matrix logarithm is χ K. These are Lorentz boosts. The positive Hilbert norm used for unitary color transport in the gauge monograph is a different structure.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0073819703c9eac7","tex":"e^{\\chi K}","aliases":"e^χ K chi","display":"e^{\\chi K}","kind":"use","excerpt":"If the integrated coefficient is χ, then e^χ K=coshχ I+sinhχ K. This preserves the Lorentz form diag(-1,1) and has positive eigenvalues e^χ and e^-χ. Its real principal matrix logarithm is χ K. These are Lorentz boosts. The positive Hilbert norm used for unitary color transport in the gauge monograph is a different structure.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28c070e65388214e","tex":"e^{\\chi(t,x)K}","aliases":"e^χ(t,x)K chi","display":"e^{\\chi(t,x)K}","kind":"use","excerpt":"The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and pa","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6956e1ff41ab0d60","tex":"e^{\\omega_L^\\ell tA_0}","aliases":"e^ω_L^ℓ tA_0 omega ell","display":"e^{\\omega_L^\\ell tA_0}","kind":"use","excerpt":"[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber · =frac(tA_0)^jj! _0F_L-1 (- A_L,j; frac(tA_0)^LL^L),nonumber ∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e6baee59f187d10","tex":"e^{\\omega_L^\\ell z}","aliases":"e^ω_L^ℓ z omega ell","display":"e^{\\omega_L^\\ell z}","kind":"use","excerpt":"j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z)=frac1L∑_ℓ=0^L-1 ω_L^-jℓe^ω_L^ℓ z. Thus factorial, hypergeometric, companion-state, and root-filter forms descri","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a48ab850207c4126","tex":"\\mathsf E_{\\mathbf h}","aliases":"E_ h mathsf mathbf","display":"\\mathsf E_{\\mathbf h}","kind":"use","excerpt":"imensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db8e0e8ced5b248a","tex":"\\mathsf E_{\\mathbf k}","aliases":"E_ k mathsf mathbf","display":"\\mathsf E_{\\mathbf k}","kind":"use","excerpt":"⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections and ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d8be2ba7e790af78","tex":"E_+","aliases":"E_+","display":"E_+","kind":"use","excerpt":"0 · -1, A^2=I, E_±=fracI± A2. The identities E_±^2=E_±, E_+E_-=0, and E_++E_-=I follow directly from A^2=I. Both idempotents contain a denominator two, and the resultant of x-1 and x+1 has dyadic valuation one. Modulo two the two roots agree and (A-I)^2=0, while A-I is nonzero. Thus the residual matrix can retain a nilpotent extension even though i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca57caa14f72ba3b","tex":"E_-","aliases":"E_-","display":"E_-","kind":"use","excerpt":"0 · -1, A^2=I, E_±=fracI± A2. The identities E_±^2=E_±, E_+E_-=0, and E_++E_-=I follow directly from A^2=I. Both idempotents contain a denominator two, and the resultant of x-1 and x+1 has dyadic valuation one. Modulo two the two roots agree and (A-I)^2=0, while A-I is nonzero. Thus the residual matrix can retain a nilpotent extension even though i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f912500efb3ee692","tex":"\\mathsf E_0","aliases":"E_0 mathsf","display":"\\mathsf E_0","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3536644415c7e956","tex":"E_0^{(h)}","aliases":"E_0^(h)","display":"E_0^{(h)}","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"[A stationary vector viewed through a rotating split] Let ψ(t)=e_1, K=0, and U(t)=cosθ(t) · -sinθ(t)","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"[An exact bracket failure] Use N=5, h=1, and (Df)_a=(f_a+1-f_a-1)/2. Take q=e_0, p=ξ=e_1, and η=e_2. Then (Dq)_1=-1/2, (Dη)_1=1/2, and η Dq=0. The direct bracket in the preceding proof is zero, while D_h[ξ Dη-η Dξ]=-tfrac14. The defect is therefore exactly +1/4. This finite witness shows that the naive centered scalar constraint does fail to close ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-75efec2c359421e6","tex":"E_{2,0}","aliases":"E_2,0","display":"E_{2,0}","kind":"use","excerpt":"The coefficient maps R_L,j in [source reference] are idempotents on analytic functions. The evaluated operators E_L,j(tA_0) generally are not. For L=2, E_2,0(tA_0)=cosh(tA_0) and E_2,1(tA_0)=sinh(tA_0), which are neither an orthogonal projector family nor a pair of independent unitary evolutions. The sum reconstructs an evolution; an individual lan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fdddb5e6146d680d","tex":"E_{2,1}","aliases":"E_2,1","display":"E_{2,1}","kind":"use","excerpt":"The coefficient maps R_L,j in [source reference] are idempotents on analytic functions. The evaluated operators E_L,j(tA_0) generally are not. For L=2, E_2,0(tA_0)=cosh(tA_0) and E_2,1(tA_0)=sinh(tA_0), which are neither an orthogonal projector family nor a pair of independent unitary evolutions. The sum reconstructs an evolution; an individual lan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9aaef4e6308432c","tex":"e_a","aliases":"e_a","display":"e_a","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":19},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ce80e19252178918","tex":"e_{a+d p^m}","aliases":"e_a+d p^m","display":"e_{a+d p^m}","kind":"use","excerpt":"The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-274edfe3f1c1c948","tex":"e_a^2","aliases":"e_a^2","display":"e_a^2","kind":"use","excerpt":"[No nonzero derivation of the finite pointwise algebra] Let D:C^N→C^N be complex linear and satisfy D(fg)=D(f)g+fD(g) for every pair of vectors, where multiplication is coordinatewise. Then D=0. For the coordinate idempotent e_a, the identity e_a^2=e_a gives D(e_a)=2e_aD(e_a). At a coordinate b≠ a, the right side is zero. At b=a, the equality reads","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b03eb1832ae32d9a","tex":"E_a^I","aliases":"E_a^I","display":"E_a^I","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b207391fd280ff7e","tex":"E_a^J","aliases":"E_a^J","display":"E_a^J","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fc6e8ae822395078","tex":"E_A^{\\rm con}","aliases":"E_A^rm con rm","display":"E_A^{\\rm con}","kind":"use","excerpt":"For a partition of fine spatial sites into coarse regions A, the nonnegative diagnostic E_A^rm con=∑_x∈ Ah^3 (|C_x|^2+∑_i|C_i,x|^2) adds exactly to the full fine diagnostic. This is an exact partition identity. It does not assert that differentiating a coarse field reconstructs its fine constraint values.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b6266893a62573e7","tex":"e_b","aliases":"e_b","display":"e_b","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eb1e556f8b1d6a8e","tex":"E_b^J","aliases":"E_b^J","display":"E_b^J","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a8691c09c2b87496","tex":"E_b^K","aliases":"E_b^K","display":"E_b^K","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6500ef20b6c4932a","tex":"e_c","aliases":"e_c","display":"e_c","kind":"use","excerpt":"Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3174fd0f13b3655d","tex":"E_c^L","aliases":"E_c^L","display":"E_c^L","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dfb98ad8aedbb4dd","tex":"E_h","aliases":"E_h","display":"E_h","kind":"use","excerpt":"s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e8d4f703100b5fef","tex":"\\mathcal E_h","aliases":"E_h mathcal","display":"\\mathcal E_h","kind":"use","excerpt":"with centered expectation E_h(a) obtained by replacing H_n with H_h,n. Hermiticity makes each expectation real: its complex conjugate is ⟨ H_na_n,a_n⟩=⟨ a_n,H_na_n⟩.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-567fce9b1e56d505","tex":"E_{h,n}","aliases":"E_h,n","display":"E_{h,n}","kind":"use","excerpt":"H_h,n^2=E_h,n^2I_4, E_h,n=sqrtm^2+|q_h(n)|^2.","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-63ddef0ed4a25d17","tex":"E_{h,n}^2","aliases":"E_h,n^2","display":"E_{h,n}^2","kind":"use","excerpt":"H_h,n^2=E_h,n^2I_4, E_h,n=sqrtm^2+|q_h(n)|^2.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5479790a579a7cad","tex":"E_h^{(h)}","aliases":"E_h^(h)","display":"E_h^{(h)}","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-81545aa21397b4c1","tex":"E_h^{(h+1)}","aliases":"E_h^(h+1)","display":"E_h^{(h+1)}","kind":"use","excerpt":"is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(h+1), I-V_p,hV_p,h^*=D_h+1^(h+1). The first composition is an identity on ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-09f94d8cb8d59698","tex":"e_h^2","aliases":"e_h^2","display":"e_h^2","kind":"use","excerpt":"y≤ S_1. Skew adjointness cancels innerubDu_h+inneruD(bu)_h. The remaining term gives [source reference]. With e=R-widetilde R, the nonlinear difference satisfies |g(R_j)-g(widetilde R_j)|≤ L_g|e_j|, where g'(R)=fracMR-Q^2R^2sqrt2MR-Q^2. Thus tfrac12∂_tnorm e_h^2≤(L_g+S_1/2)norm e_h^2, and integration gives the claim.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3296004731e3487c","tex":"E_{j-1}","aliases":"E_j-1","display":"E_{j-1}","kind":"use","excerpt":"[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-204f9ab8a5b56738","tex":"E_j^2","aliases":"E_j^2","display":"E_j^2","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations spec","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-542812c54fe5f147","tex":"e_k","aliases":"e_k","display":"e_k","kind":"use","excerpt":"l p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, with (e_k)_k indexing the basis of HH_B. In the resulting product basis the partial trace has entries (ρ_A)_ij=∑_k(ρ_AB)_(i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fcbad8d16f70335c","tex":"\\mathbf E_L","aliases":"E_L mathbf","display":"\\mathbf E_L","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-568316c37cf711b8","tex":"\\mathbf E_L'","aliases":"E_L' mathbf","display":"\\mathbf E_L'","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b45e7eb6ee17b6f3","tex":"E_{L,0}","aliases":"E_L,0","display":"E_{L,0}","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0f77f254fddb0f70","tex":"E_{L,j-1}","aliases":"E_L,j-1","display":"E_{L,j-1}","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b6b4a7ea5864023d","tex":"E_{L,L-1}","aliases":"E_L,L-1","display":"E_{L,L-1}","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c7c5122c570797bd","tex":"\\mathsf E_M","aliases":"E_M mathsf","display":"\\mathsf E_M","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0972d4ba92947985","tex":"\\mathsf E_{m-1}","aliases":"E_m-1 mathsf","display":"\\mathsf E_{m-1}","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bc1cb9d9d07da221","tex":"E_m^{(h)}","aliases":"E_m^(h)","display":"E_m^{(h)}","kind":"use","excerpt":"The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an addition","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6fe2642a9fd5fc0e","tex":"\\mathsf E_m^*","aliases":"E_m^* mathsf","display":"\\mathsf E_m^*","kind":"use","excerpt":"[Nested cylinder projections] The operator E_m is the orthogonal projection onto V_m. For m,n≥0, E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows inner","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4d74754653932380","tex":"\\mathsf E_m^2","aliases":"E_m^2 mathsf","display":"\\mathsf E_m^2","kind":"use","excerpt":"[Nested cylinder projections] The operator E_m is the orthogonal projection onto V_m. For m,n≥0, E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows inner","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5ee8d3321f36c338","tex":"\\mathsf E_{\\min(\\mathbf h,\\mathbf k)}","aliases":"E_min( h, k) mathsf min mathbf mathbf","display":"\\mathsf E_{\\min(\\mathbf h,\\mathbf k)}","kind":"use","excerpt":"⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections and ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-30864bbcd81fdb76","tex":"\\mathsf E_{\\min(m,n)}","aliases":"E_min(m,n) mathsf min","display":"\\mathsf E_{\\min(m,n)}","kind":"use","excerpt":"[Nested cylinder projections] The operator E_m is the orthogonal projection onto V_m. For m,n≥0, E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows inner","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-253619f7a5d93163","tex":"E_{\\min(r,s)}^{(h)}","aliases":"E_min(r,s)^(h) min","display":"E_{\\min(r,s)}^{(h)}","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-da2ce42dc70bb3da","tex":"E_n","aliases":"E_n","display":"E_n","kind":"use","excerpt":"H_n^2=E_n^2I_4, E_n=sqrtm^2+|p(n)|^2,","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3d0abe95633dd7c9","tex":"\\mathsf E_n","aliases":"E_n mathsf","display":"\\mathsf E_n","kind":"use","excerpt":"[Nested cylinder projections] The operator E_m is the orthogonal projection onto V_m. For m,n≥0, E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows inner","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bade16369687b04f","tex":"e_N","aliases":"e_N","display":"e_N","kind":"use","excerpt":"[A small state can have a large generator image] On ℓ^2(N_≥1), let He_n=ne_n on its maximal domain. The finitely supported vectors x_N=e_N/N satisfy normx_N=1/N→0, while normHx_N=1. This gives a direct reason to track the generator tail in addition to the state tail. It does not contradict closedness, because Hx_N=e_N has no norm limit.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"Fourier synthesis on the physical torus. The scalar Fourier characters e_n(x)=e^2π i n· x are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is F:HH⟶KK, HH=L^2(T^3;C^4). The four-spinor Parseval identity and surjectivity of s","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bd332fc3db09af0f","tex":"e_{N+1}","aliases":"e_N+1","display":"e_{N+1}","kind":"use","excerpt":"[The identity never has a small omitted tail in operator norm] On an infinite-dimensional coordinate space, let S_N retain the first N coordinates. For X=I, normS_NIS_N-I=normS_N-I=1. A unit vector e_N+1 witnesses the lower bound. Nevertheless, S_Nx→ x for every fixed x∈ℓ^2. This is the difference between pointwise control over states and uniform c","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d3e1f9162e4702a9","tex":"e_{n+1}","aliases":"e_n+1","display":"e_{n+1}","kind":"use","excerpt":"atives, the provisional constraint defect is also O(Δτ^3). The lower bound on the denominator and the upper bound on g make the correction O(Δτ^3). Thus the corrected local error has the same order. Under the stated derivative bounds the corrected step map has Lipschitz constant at most 1+LΔτ for small steps. If e_n is its error against the exact s","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7bfa75c99a6ca248","tex":"\\mathsf E_{n-1}","aliases":"E_n-1 mathsf","display":"\\mathsf E_{n-1}","kind":"use","excerpt":"depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 and cancel. For m=n the same expansion is E_m- E_m-1. The cases in","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5d6f034bf6418f2e","tex":"E_n^2","aliases":"E_n^2","display":"E_n^2","kind":"use","excerpt":"H_n^2=E_n^2I_4, E_n=sqrtm^2+|p(n)|^2,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a48064d29c441e34","tex":"e_{n\\bmod L}","aliases":"e_nbmod L bmod","display":"e_{n\\bmod L}","kind":"use","excerpt":"gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the last equality directly: C_L^ne_0=e_nbmod L, so its jth exponential coordinate is exactly [source reference]. Character filtering also gives E_L,j(z","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a06214110d9137a5","tex":"\\mathcal E_P","aliases":"E_P mathcal","display":"\\mathcal E_P","kind":"use","excerpt":"∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf5bac72f2b1a2ac","tex":"\\mathcal E_{P,h}","aliases":"E_P,h mathcal","display":"\\mathcal E_{P,h}","kind":"use","excerpt":"The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservatio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b6b70465057ce11c","tex":"\\mathsf E_{p,h_p}","aliases":"E_p,h_p mathsf","display":"\\mathsf E_{p,h_p}","kind":"use","excerpt":"imensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-98f6149adeb54f57","tex":"E_{pL,j+aL}","aliases":"E_pL,j+aL","display":"E_{pL,j+aL}","kind":"use","excerpt":"Analytic refinement of a factorial lane is independent of this unit condition. Splitting q=pr+a, 0≤ a<p, gives the exact identity E_L,j(z)=∑_a=0^p-1E_pL,j+aL(z), 0≤ j<L. Absolute convergence permits this partition of the summation index. The labels j+aL all lie in [0,pL), but need not lie in the parent's short unit window. Exact complex analytic re","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b3f1fc86e79071cb","tex":"E_q","aliases":"E_q","display":"E_q","kind":"use","excerpt":"[Exact folded cyclic spectrum] For q=0,…,p-1, the eigenvalues of H_p(θ) are E_q(θ)=μ+2κ cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrap","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f7f8687050d24b46","tex":"\\mathcal E_Q","aliases":"E_Q mathcal","display":"\\mathcal E_Q","kind":"use","excerpt":"∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ec5435df861a0c30","tex":"\\mathcal E_{Q,h}","aliases":"E_Q,h mathcal","display":"\\mathcal E_{Q,h}","kind":"use","excerpt":"The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservatio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3a840ddc04edad8c","tex":"E_r","aliases":"E_r","display":"E_r","kind":"use","excerpt":"If r≤ s, the level-s cylinders partition each level-r cylinder into equally sized groups. Averaging over the smaller groups and then over the whole group returns the full average, giving E_rE_s=E_r. The self-adjointness of both sides also gives E_sE_r=E_r. This proves the minimum rule.","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7b243e274b6d1702","tex":"e_{r+1}","aliases":"e_r+1","display":"e_{r+1}","kind":"use","excerpt":"M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-47afbcdfac96063a","tex":"E_{r-1}","aliases":"E_r-1","display":"E_{r-1}","kind":"use","excerpt":"For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aa3a317d7ce08707","tex":"E_{r-1}^{(h)}","aliases":"E_r-1^(h)","display":"E_{r-1}^{(h)}","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a7d70bc7f1f74924","tex":"E_r^{(h)}","aliases":"E_r^(h)","display":"E_r^{(h)}","kind":"use","excerpt":"For 0≤ r≤ h, define the cylinder average (E_r^(h)f)(x)=p^-(h-r) ∑_y≡ x (mod p^r)f(y). Its matrix is real and symmetric. Averaging twice over the same cylinder has the same effect as averaging once. Thus E_r^(h) is the orthogonal projector onto functions constant on level-r cylinders. The range has dimension p^r.","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-95f29086cb5fde1d","tex":"E_r^{(h+1)}","aliases":"E_r^(h+1)","display":"E_r^{(h+1)}","kind":"use","excerpt":"Replicated vectors are constant in the new digit. All old cylinder averages commute with replication, giving E_r^(h+1)V_p,h · =V_p,hE_r^(h) (0≤ r≤ h), nonumber D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9a38932300c921be","tex":"E_s","aliases":"E_s","display":"E_s","kind":"use","excerpt":"If r≤ s, the level-s cylinders partition each level-r cylinder into equally sized groups. Averaging over the smaller groups and then over the whole group returns the full average, giving E_rE_s=E_r. The self-adjointness of both sides also gives E_sE_r=E_r. This proves the minimum rule.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cfc2ed00b0c304ae","tex":"E_{s-1}","aliases":"E_s-1","display":"E_{s-1}","kind":"use","excerpt":"For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e7970e33de410d2","tex":"E_s^{(h)}","aliases":"E_s^(h)","display":"E_s^{(h)}","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6cb49e563b7d9a67","tex":"e_y","aliases":"e_y","display":"e_y","kind":"use","excerpt":"For the xy face, the two-path angle difference is Θ_xy(p,h) = · θ_y(p+h e_x,h)-θ_y(p,h) · -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cro","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-23943c7271363293","tex":"E_\\pm","aliases":"E_± pm","display":"E_\\pm","kind":"use","excerpt":"A separate arithmetic example shows why integral residue selectors and rational spectral idempotents should be distinguished. Let A=1 · 1 0 · -1, A^2=I, E_±=fracI± A2. The identities E_±^2=E_±, E_+E_-=0, and E_++E_-=I follow directly from A^2=I. Both idempotents contain a denominator two, and the resultant of x-1 and x+1 has dyadic valuation one. M","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eefa259b77b7a2fe","tex":"E_\\pm^2","aliases":"E_±^2 pm","display":"E_\\pm^2","kind":"use","excerpt":"0 · -1, A^2=I, E_±=fracI± A2. The identities E_±^2=E_±, E_+E_-=0, and E_++E_-=I follow directly from A^2=I. Both idempotents contain a denominator two, and the resultant of x-1 and x+1 has dyadic valuation one. Modulo two the two roots agree and (A-I)^2=0, while A-I is nonzero. Thus the residual matrix can retain a nilpotent extension even though i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-74e2d51d64f7fad4","tex":"E_\\lambda","aliases":"E_λ lambda","display":"E_\\lambda","kind":"use","excerpt":"Let D=∑_λλ E_λ be its spectral resolution. Then Σ(z)=∑_λfracB E_λ B^*z-λ. Every residue B E_λ B^* is positive semidefinite. Its rank records which directions of the hidden eigenspace reach the visible space. A vanishing residue removes the corresponding pole of Σ, even though z-D is still singular there.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9bd68f1d61a4d151","tex":"e_\\perp^I","aliases":"e_⊥^I perp","display":"e_\\perp^I","kind":"use","excerpt":"An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f8483dce1203449b","tex":"\\mathrm{eff}","aliases":"eff mathrm","display":"\\mathrm{eff}","kind":"use","excerpt":"[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a66b2c302d0c8a7d","tex":"\\mathrm{emb}","aliases":"emb mathrm","display":"\\mathrm{emb}","kind":"use","excerpt":"Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0bff408d7104cbba","tex":"\\operatorname{erf}","aliases":"erf operatorname","display":"\\operatorname{erf}","kind":"use","excerpt":"For the normalized Gaussian density ρ(r)=fracM_0(2π)^3/2σ^3e^-r^2/(2σ^2), σ>0, direct differentiation verifies M_rm enc(r) · =M_0[ erf(frac rsqrt2σ) -sqrtfrac2πfrac rσ e^-r^2/(2σ^2)], Φ(r) · =-fracGM_0r erf(frac rsqrt2σ). The potential tends to zero at infinity and has the finite central value -sqrt2/π GM_0/σ. This removes the point-source singular","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"The packet space F_S={f_a:a∈ K_S} will be identified isometrically with K_S. Its elements are smooth periodic functions, so the differential and centered-translation calculations below raise no regularity question. Empty support is allowed: both spaces then contain only zero and all packet identities reduce to 0=0.","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28240dde4f98a443","tex":"\\mathrm f","aliases":"f mathrm","display":"\\mathrm f","kind":"use","excerpt":"For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aaec5c835f65f308","tex":"\\mathsf F","aliases":"F mathsf","display":"\\mathsf F","kind":"use","excerpt":"adicfigure02_haar_channelsFor p=3, the normalized child characters contain one average and two orthogonal complex details. Solid curves show real parts and dashed curves show imaginary parts. The level rank (p-1)p^r-1 grows by the branching factor. The atlas uses F_r for the corresponding infinite cylinder detail; its finite restriction is D_r^(h) ","count":27},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"F'(s)=i e^-i(t-s)A(A-B)e^-isB.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2f4583c8b7c73ba0","tex":"f'","aliases":"f'","display":"f'","kind":"use","excerpt":"An antisymmetric current also organizes derivatives of scalar functionals of the packet distribution. Let p_a(t)>0 be differentiable, normalized, and obey dot p_b=∑_aJ_b← a with J antisymmetric. For a differentiable real function f on the positive half-line, fracddt∑_b f(p_b) =frac12∑_b,aJ_b← a (f'(p_b)-f'(p_a)). To prove it, apply the ordinary cha","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9337f69b3f162423","tex":"f''","aliases":"f''","display":"f''","kind":"use","excerpt":"For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to co","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-058246411cc5b595","tex":"f'''","aliases":"f'''","display":"f'''","kind":"use","excerpt":"For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to co","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-49ddaa56c96d4e05","tex":"f^{-1}","aliases":"f^-1","display":"f^{-1}","kind":"use","excerpt":"Let A be a finite set of coarse labels and let f:I→ A assign every fine packet to exactly one coarse label. Write C_A=f^-1(A) and define Q_A=∑_a∈ C_AP_a. The fibers are disjoint and cover I, so (Q_A) is another orthogonal atlas. Empty fibers give zero projectors. This is a grouping of an existing resolution. It need not be an approximation to a dif","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","excerpt":"ambda^2+Q^2)R^3 -fracP_Λ P_RR^2+(N^rP_R)' -fracR''Λ+fracΛ'R'Λ^2. Substituting the data and differentiating p(F)^2=2MF-Q^2 reduces their right sides to fracMF_tp(F), M[fracF_trp(F) -fracMF_rF_tp(F)^3], which are the direct time derivatives of the two momenta. Finally, Φ_e'=QF_r/F^2 cancels NΛ Q/R^2 in dot A_r=NΛ Q/R^2-Φ_e'.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8c4caf5c79a35c9c","tex":"f^2","aliases":"f^2","display":"f^2","kind":"use","excerpt":"The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an addition","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c9a3c3f9d72a35e1","tex":"F^{IJ}","aliases":"F^IJ","display":"F^{IJ}","kind":"use","excerpt":"ta⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b9284f7a50c1ed02","tex":"f_{\\mathfrak H_p^s}","aliases":"f_ H_p^s mathfrak","display":"f_{\\mathfrak H_p^s}","kind":"use","excerpt":"[Exact state error for a compatible cutoff] For every f∈HH_p, normU(t)f-U_M^emb(t)f_2 =norm(I- E_M)f_2. This equality is uniform for all real t. For f∈ H_p^s, the right side is bounded by p^-s(M+1)norm f_ H_p^s. For f∈Dom(H_ϵ) there is also the exact graph-norm identity · normU(t)f-U_M^emb(t)f_2^2 +normH_ϵ(U(t)f-U_M^emb(t)f)_2^2 nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-71bf007f614b8921","tex":"\\mathsf F_{\\mathbf r}","aliases":"F_ r mathsf mathbf","display":"\\mathsf F_{\\mathbf r}","kind":"use","excerpt":"Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0e24ae7f63f35dfa","tex":"f_{\\mathbf r}","aliases":"f_ r mathbf","display":"f_{\\mathbf r}","kind":"use","excerpt":"domain and are dense. Symmetry holds by symmetry on every block. If g belongs to the adjoint domain and k=H_A^*g, testing against an arbitrary vector in KK_ r gives F_ rk=A_ r F_ rg. Square summability of k is precisely the domain condition in [source reference]. Thus H_A^*=H_A. Truncating the two square-summable sequences f_ r and A_ rf_ r proves ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dcde29ad44d73958","tex":"F_+","aliases":"F_+","display":"F_+","kind":"use","excerpt":"[Overlapping two-channel weights] On C^2, take F_+=4/5 · 0 0 · 1/5, F_-=1/5 · 0 0 · 4/5, ψ=binom10. The true weights are 4/5 and 1/5. Squared effect-vector norms would instead give 16/25 and 1/25, whose sum is 17/25. The state still satisfies F_+ψ+F_-ψ=ψ, but its two summands are not orthogonal. Reconstruction and an orthogonal decomposition are di","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1cef4fb299762615","tex":"F_-","aliases":"F_-","display":"F_-","kind":"use","excerpt":"[Overlapping two-channel weights] On C^2, take F_+=4/5 · 0 0 · 1/5, F_-=1/5 · 0 0 · 4/5, ψ=binom10. The true weights are 4/5 and 1/5. Squared effect-vector norms would instead give 16/25 and 1/25, whose sum is 17/25. The state still satisfies F_+ψ+F_-ψ=ψ, but its two summands are not orthogonal. Reconstruction and an orthogonal decomposition are di","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b910d55bebfcc906","tex":"\\mathsf F_0","aliases":"F_0 mathsf","display":"\\mathsf F_0","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","excerpt":"For an explicit example, take L=4. The scalar formula gives f_0(λ)=tfrac12(cosh(tλ)+cos(tλ)), f_2(λ)=tfrac12(cos(tλ)-cosh(tλ)). If H_∞ is multiplication by λ on L^2(R) and t≠0, these functions define unbounded operators. Their sum cancels the growing hyperbolic contribution. The full propagator e^-itH_∞ is nevertheless unitary on all of L^2(R). For","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3432ebe25e9aabbb","tex":"f_{1/12}","aliases":"f_1/12","display":"f_{1/12}","kind":"use","excerpt":"f_1/12(t)-f(t) _L^2 ≤(2π-6)|t|≤fracπ^3108|t|.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","excerpt":"For f∈Dom(H_ϵ), the cylinder truncation E_M f satisfies both normf- E_M f_2→0 and normH_ϵ(f- E_M f)_2→0, because the two squared norms are the tails of convergent nonnegative series. This is precisely density in the graph norm and proves the core assertion. If the energies have supremum C<∞, Parseval gives normH_ϵ f_2≤ Cnorm f_2. Conversely, a unit","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eea600dc4f325e8a","tex":"F_{2^{h_2}}","aliases":"F_2^h_2","display":"F_{2^{h_2}}","kind":"use","excerpt":"We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e38e6f593dba41e","tex":"F_{2L-1}","aliases":"F_2L-1","display":"F_{2L-1}","kind":"use","excerpt":"The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( - 1, A_L,j, A_L,j; frac(-1)^Ly^LL^2L). O","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b314e2300364246a","tex":"f_a","aliases":"f_a","display":"f_a","kind":"use","excerpt":"f_a(x)=∑_n∈ Se_n(x)a_n.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1438a4d487c7b22d","tex":"f_{a+1}","aliases":"f_a+1","display":"f_{a+1}","kind":"use","excerpt":"[An exact bracket failure] Use N=5, h=1, and (Df)_a=(f_a+1-f_a-1)/2. Take q=e_0, p=ξ=e_1, and η=e_2. Then (Dq)_1=-1/2, (Dη)_1=1/2, and η Dq=0. The direct bracket in the preceding proof is zero, while D_h[ξ Dη-η Dξ]=-tfrac14. The defect is therefore exactly +1/4. This finite witness shows that the naive centered scalar constraint does fail to close ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cacee2cb9dea5558","tex":"f_{a-1}","aliases":"f_a-1","display":"f_{a-1}","kind":"use","excerpt":"[An exact bracket failure] Use N=5, h=1, and (Df)_a=(f_a+1-f_a-1)/2. Take q=e_0, p=ξ=e_1, and η=e_2. Then (Dq)_1=-1/2, (Dη)_1=1/2, and η Dq=0. The direct bracket in the preceding proof is zero, while D_h[ξ Dη-η Dξ]=-tfrac14. The defect is therefore exactly +1/4. This finite witness shows that the naive centered scalar constraint does fail to close ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6f98938f2b1b013e","tex":"F_a^{1/2}","aliases":"F_a^1/2","display":"F_a^{1/2}","kind":"use","excerpt":"*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, the compressed effect is F_a=V^*Π_aV=V_a^*V_a, p_a=normV_aψ^2=normΠ_aVψ^2. Every finite effect atlas admits this construction by taking KK_a=HH and V_a=F_a^1/2. Positi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0fb521484b5b5711","tex":"F_a^2","aliases":"F_a^2","display":"F_a^2","kind":"use","excerpt":"For a sharp atlas, F_a=P_a and p_a=normP_aψ^2. In general, p_a is not normF_aψ^2. The latter quantity is innerψF_a^2ψ and therefore loses precisely the positive defect F_a-F_a^2.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dd0b7a11832442ee","tex":"F_{ab}^i","aliases":"F_ab^i","display":"F_{ab}^i","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-08d88035e5bf8864","tex":"F_{ab}^k","aliases":"F_ab^k","display":"F_{ab}^k","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-36f0dfbeaf4bfcdb","tex":"F_{a\\leftarrow b}","aliases":"F_a← b leftarrow","display":"F_{a\\leftarrow b}","kind":"use","excerpt":"[Population gain and loss] The diagonal populations p_b=ρ_bb of a Hermitian solution satisfy dot p_b=∑_aF_b← a, F_b← a=γ_bap_a-γ_abp_b, F_b← a=-F_a← b. The total population is constant, and every coarse population has the sum of the boundary jump currents as its derivative.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-54b44dec26d2ba54","tex":"F_b","aliases":"F_b","display":"F_b","kind":"use","excerpt":"P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is c","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9e7bf13ec229e895","tex":"\\mathcal F_{bc}","aliases":"F_bc mathcal","display":"\\mathcal F_{bc}","kind":"use","excerpt":"Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fb505b26292653fe","tex":"F_{b\\leftarrow a}","aliases":"F_b← a leftarrow","display":"F_{b\\leftarrow a}","kind":"use","excerpt":"[Population gain and loss] The diagonal populations p_b=ρ_bb of a Hermitian solution satisfy dot p_b=∑_aF_b← a, F_b← a=γ_bap_a-γ_abp_b, F_b← a=-F_a← b. The total population is constant, and every coarse population has the sum of the boundary jump currents as its derivative.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e982a068c0aac4ff","tex":"F_h","aliases":"F_h","display":"F_h","kind":"use","excerpt":"ations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all sufficiently small |h|. Define F_h=h^-2log W_h, h≠0. This construction describes a local","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1551bc759aa860fc","tex":"f_h","aliases":"f_h","display":"f_h","kind":"use","excerpt":"f(t,x)=∑_n∈ Se_n(x)U_H_n(t)a_n, f_h(t,x)=∑_n∈ Se_n(x)U_H_h,n(t)a_n.","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fc9d2a85cc280a7e","tex":"f_{h,a}","aliases":"f_h,a","display":"f_{h,a}","kind":"use","excerpt":"⟨ f_h,a(t),f_h,b(t)⟩_L^2 =∑_n∈ S ⟨ U_H_h,n(t)a_n,U_H_h,n(t)b_n⟩ =∑_n∈ S⟨ a_n,b_n⟩.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4568892b4cd9f787","tex":"f_{h,b}","aliases":"f_h,b","display":"f_{h,b}","kind":"use","excerpt":"⟨ f_h,a(t),f_h,b(t)⟩_L^2 =∑_n∈ S ⟨ U_H_h,n(t)a_n,U_H_h,n(t)b_n⟩ =∑_n∈ S⟨ a_n,b_n⟩.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e8a91baf437e5b29","tex":"F_{h/2}","aliases":"F_h/2","display":"F_{h/2}","kind":"use","excerpt":"[Local logarithmic refinement] On the common analytic logarithm branch, in any fixed matrix norm, F_h · =[A,B]+hC_3(A,B)+O(h^2), C_3(A,B) · =tfrac12[A+B,[A,B]], F_h^[1] · :=2F_h/2-F_h=[A,B]+O(h^2). For constant G∈ SU(2), simultaneous conjugation of A,B conjugates both F_h and F_h^[1] by G.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8a63a9f5145880e2","tex":"F_h^{[1]}","aliases":"F_h^[1]","display":"F_h^{[1]}","kind":"use","excerpt":"[Local logarithmic refinement] On the common analytic logarithm branch, in any fixed matrix norm, F_h · =[A,B]+hC_3(A,B)+O(h^2), C_3(A,B) · =tfrac12[A+B,[A,B]], F_h^[1] · :=2F_h/2-F_h=[A,B]+O(h^2). For constant G∈ SU(2), simultaneous conjugation of A,B conjugates both F_h and F_h^[1] by G.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-74eec0b63ac7044d","tex":"F_{IJ}","aliases":"F_IJ","display":"F_{IJ}","kind":"use","excerpt":"ta⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-09888c9e8426e6e3","tex":"F_{ij}","aliases":"F_ij","display":"F_{ij}","kind":"use","excerpt":"For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argu","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-967ebb9573a3e321","tex":"\\mathcal F_{ij}","aliases":"F_ij mathcal","display":"\\mathcal F_{ij}","kind":"use","excerpt":"For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely wh","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"use","excerpt":"Let H_∞ be self-adjoint with spectral measure E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum o","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8576ad4cc4bc1341","tex":"F_{j+1}","aliases":"F_j+1","display":"F_{j+1}","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0fe97be4e95beb8d","tex":"f_{j+1}","aliases":"f_j+1","display":"f_{j+1}","kind":"use","excerpt":"[The precise defect of a forward difference] On a cyclic grid, let D_hf_j=(f_j+1-f_j)/h, with h≠0. Expanding the product gives D_h(fg)_j-(D_hf)_jg_j-f_j(D_hg)_j =h(D_hf)_j(D_hg)_j. Thus the ordinary Leibniz identity fails unless the displayed product vanishes. A small defect on sufficiently regular sampled functions is compatible with the no-go the","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-44073352c09346c9","tex":"F_{j+1/2}","aliases":"F_j+1/2","display":"F_{j+1/2}","kind":"use","excerpt":"[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d54d63626059c12d","tex":"f_{j-1}","aliases":"f_j-1","display":"f_{j-1}","kind":"use","excerpt":"Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d63682be513a74f9","tex":"F_{j-1/2}","aliases":"F_j-1/2","display":"F_{j-1/2}","kind":"use","excerpt":"s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f0160f5b8a57192c","tex":"F_M","aliases":"F_M","display":"F_M","kind":"use","excerpt":"Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and se","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3b20c59729d1aa1f","tex":"\\mathsf F_m","aliases":"F_m mathsf","display":"\\mathsf F_m","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":19},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"use","excerpt":"losed, and self-adjoint. The finite cylinder functions form an operator core. It is bounded if and only if sup_m|ϵ_m|<∞, and its spectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8b7438119014ae74","tex":"\\mathsf F_{M+1}","aliases":"F_M+1 mathsf","display":"\\mathsf F_{M+1}","kind":"use","excerpt":"ence E_M→ I is strong and is not convergence in operator norm. The omitted details are orthogonal, so Parseval gives the equality. For m>M, p^2sm≥ p^2s(M+1). Multiplying the omitted summands by this lower bound proves the squared estimate, and taking square roots gives [source reference]. At every finite M there is a unit vector in Ran F_M+1, on wh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0fe9f993f4ff4310","tex":"F_N","aliases":"F_N","display":"F_N","kind":"use","excerpt":"P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is c","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c57cafb013c6775f","tex":"F_n","aliases":"F_n","display":"F_n","kind":"use","excerpt":"We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e213f3328b138478","tex":"\\mathsf F_n","aliases":"F_n mathsf","display":"\\mathsf F_n","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e3f4ba757b6d2cf8","tex":"f_n","aliases":"f_n","display":"f_n","kind":"use","excerpt":"[A complete-space limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence i","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-af40e62da957c071","tex":"f_{n+1}","aliases":"f_n+1","display":"f_{n+1}","kind":"use","excerpt":"[A complete-space limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6a27312454be2902","tex":"F_p","aliases":"F_p","display":"F_p","kind":"use","excerpt":"Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3662907c3ec44b16","tex":"\\mathsf F_{p,r_p}","aliases":"F_p,r_p mathsf","display":"\\mathsf F_{p,r_p}","kind":"use","excerpt":"Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c05872ad0a26184c","tex":"F_{p-1}","aliases":"F_p-1","display":"F_{p-1}","kind":"use","excerpt":"We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fb50aad1fbf28b7f","tex":"F_{p^{h_p}}","aliases":"F_p^h_p","display":"F_{p^{h_p}}","kind":"use","excerpt":"We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7b7f016ef3a20773","tex":"F_{p_a}","aliases":"F_p_a","display":"F_{p_a}","kind":"use","excerpt":"midrule Canonical brackets · Each sector declares its variables and sign. In standard coordinates, {F,G}=∑_a(F_q_aG_p_a-F_p_aG_q_a), so dot F={F,H}.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ba2ac5456d322b89","tex":"f_q","aliases":"f_q","display":"f_q","kind":"use","excerpt":"[Convergent states and finite energy derivatives] Choose one unit vector u_m∈Ran F_m at every level, let 0<q<1, and set f_q=∑_m≥0sqrt(1-q)q^m u_m, ϵ_m=2^m. Then normf_q_2=1 and norm(I- E_M)f_q_2^2=q^M+1. However, f_q∈Dom(H_ϵ) exactly when 4q<1: normH_ϵ f_q_2^2 =(1-q)∑_m≥0(4q)^m. For q<1/4, its graph tail is exactly q^M+1+frac(1-q)(4q)^M+11-4q. For ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b2fb504bd6f0611c","tex":"F_{q_a}","aliases":"F_q_a","display":"F_{q_a}","kind":"use","excerpt":"midrule Canonical brackets · Each sector declares its variables and sign. In standard coordinates, {F,G}=∑_a(F_q_aG_p_a-F_p_aG_q_a), so dot F={F,H}.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f7110079fe13a85a","tex":"F_r","aliases":"F_r","display":"F_r","kind":"use","excerpt":"Take a smooth radial map R=F(t,r) with F_r>0 and identify t_P=t. Expanding dR=F_tdt+F_rdr identifies N=1, Λ=F_r, N^r=fracF_t+v(F)F_r. The corresponding canonical data are P_Λ=p(F), P_R=fracMF_rp(F), A_r=0, Φ_e=-frac QF, Q'=0. The sign in P_R is contained in p(F).","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-92f4bda95e0af13d","tex":"\\mathsf F_r","aliases":"F_r mathsf","display":"\\mathsf F_r","kind":"use","excerpt":"adicfigure02_haar_channelsFor p=3, the normalized child characters contain one average and two orthogonal complex details. Solid curves show real parts and dashed curves show imaginary parts. The level rank (p-1)p^r-1 grows by the branching factor. The atlas uses F_r for the corresponding infinite cylinder detail; its finite restriction is D_r^(h) ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-142a68e175e6731e","tex":"F_{r-1/2}","aliases":"F_r-1/2","display":"F_{r-1/2}","kind":"use","excerpt":"Every e_j is nonnegative under the stated coefficient assumptions. An interval of cells r,…,s obeys the exact coarse balance fracddt∑_j=r^se_j +F_s+1/2-F_r-1/2=∑_j=r^ss_j. Thus coarse energy is obtained by adding fine cell energies and retaining only the exposed boundary fluxes. This is the same cancellation mechanism as projector transfer, applied","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a46e3f3e2645bd3d","tex":"\\mathcal F_S","aliases":"F_S mathcal","display":"\\mathcal F_S","kind":"use","excerpt":"The packet space F_S={f_a:a∈ K_S} will be identified isometrically with K_S. Its elements are smooth periodic functions, so the differential and centered-translation calculations below raise no regularity question. Empty support is allowed: both spaces then contain only zero and all packet identities reduce to 0=0.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ef3bbb91e99e25c","tex":"F_{s+1/2}","aliases":"F_s+1/2","display":"F_{s+1/2}","kind":"use","excerpt":"Every e_j is nonnegative under the stated coefficient assumptions. An interval of cells r,…,s obeys the exact coarse balance fracddt∑_j=r^se_j +F_s+1/2-F_r-1/2=∑_j=r^ss_j. Thus coarse energy is obtained by adding fine cell energies and retaining only the exposed boundary fluxes. This is the same cancellation mechanism as projector transfer, applied","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","excerpt":"Take a smooth radial map R=F(t,r) with F_r>0 and identify t_P=t. Expanding dR=F_tdt+F_rdr identifies N=1, Λ=F_r, N^r=fracF_t+v(F)F_r. The corresponding canonical data are P_Λ=p(F), P_R=fracMF_rp(F), A_r=0, Φ_e=-frac QF, Q'=0. The sign in P_R is contained in p(F).","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-392398bb2bd705e1","tex":"F_{tr}","aliases":"F_tr","display":"F_{tr}","kind":"use","excerpt":"For evolution, the radius equation gives dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2],","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0ddd8046eaa27be9","tex":"F_{U_M}","aliases":"F_U_M","display":"F_{U_M}","kind":"use","excerpt":"We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-308fda5bda779517","tex":"f_x","aliases":"f_x","display":"f_x","kind":"use","excerpt":"For fixed x, put f_x(s)=exp(-i∫_x^x+sA(r) dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bo","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df08b9f9771e4e17","tex":"f_x'","aliases":"f_x'","display":"f_x'","kind":"use","excerpt":"For fixed x, put f_x(s)=exp(-i∫_x^x+sA(r) dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bo","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-faca7756d6387453","tex":"f_x'''","aliases":"f_x'''","display":"f_x'''","kind":"use","excerpt":") dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bounds for A and ψ give the required unifo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7a3e381615a63515","tex":"F_{xy}","aliases":"F_xy","display":"F_{xy}","kind":"use","excerpt":"For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argu","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c347f30d5a08d9fc","tex":"F_{xz}","aliases":"F_xz","display":"F_{xz}","kind":"use","excerpt":"For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argu","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b8a2189d0ac907c","tex":"F_{yz}","aliases":"F_yz","display":"F_{yz}","kind":"use","excerpt":"For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argu","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3aeda5cb74c7b3c2","tex":"F_{\\mu\\nu}","aliases":"F_μν mu nu","display":"F_{\\mu\\nu}","kind":"use","excerpt":"Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivati","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf0883ad1e961a59","tex":"F_\\infty","aliases":"F_∞ infty","display":"F_\\infty","kind":"use","excerpt":"rectangular projection selects a collection of whole blocks, so it commutes with H_A on the domain and reduces the operator. The restriction to each retained block is literally the same matrix at every subsequent depth. This proves the finite intertwining, and finite-dimensional spectral calculus on each block gives [source reference]. Boundedness ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9a7b27c4eeefb975","tex":"f_\\infty","aliases":"f_∞ infty","display":"f_\\infty","kind":"use","excerpt":"[A complete-space limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ed487b1e5f57fec8","tex":"\\mathrm{fine}","aliases":"fine mathrm","display":"\\mathrm{fine}","kind":"use","excerpt":"lation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of [source reference]: S_i,fine^2 I_r=I_rS_i,coarse. If the mass sub","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ff9bce76b90458b5","tex":"\\mathrm{fld}","aliases":"fld mathrm","display":"\\mathrm{fld}","kind":"use","excerpt":"The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"Let N≥3 and h≥0. Let C_N be the physical cycle and let T_p,h be the rooted finite p-ary tree of prefixes of length at most h. The product graph is G=C_Nsquare T_p,h. A vertex is v=(x, d), with x∈Z/NZ and d a prefix. An edge changes either x by one on the cycle or the prefix by one parent-child step. The graph has N∑_r=0^h p^r vertices. It includes ","count":23},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7b55fe92541ca0f","tex":"\\mathrm G","aliases":"G mathrm","display":"\\mathrm G","kind":"use","excerpt":"The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-02b0b5ff2a25a5d9","tex":"g'","aliases":"g'","display":"g'","kind":"use","excerpt":"For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to co","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eb695a8d9657252e","tex":"g''","aliases":"g''","display":"g''","kind":"use","excerpt":"For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to co","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","excerpt":"Nilpotency follows from the graded Jacobi identity for the parameter algebra and its action on each representation. A simple internal check makes the sign visible. Write the Lorentz representation-valued ghost as g=tfrac14χ^IJγ_IJ and suppress diffeomorphisms and the Abelian factor. Then sg=g^2, sψ=gψ, and the dual rule is sbarψ=barψ g when the odd","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1dd057a76e8bf9c5","tex":"G^n","aliases":"G^n","display":"G^n","kind":"use","excerpt":"For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the s","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8247fba132aaeb48","tex":"G^{n+1}","aliases":"G^n+1","display":"G^{n+1}","kind":"use","excerpt":"For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the s","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ac1e4b86b2063762","tex":"G^T","aliases":"G^T","display":"G^T","kind":"use","excerpt":"equation overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. Direct multiplication proves GJ_ D^Δ=0. Since","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-531a7fd60afe23fd","tex":"G_{\\mathrm L}","aliases":"G_ L mathrm","display":"G_{\\mathrm L}","kind":"use","excerpt":"What is explicit in the internal BFV sector. On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation ca","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-57321f6d631ce0e0","tex":"G_{\\mathrm U}","aliases":"G_ U mathrm","display":"G_{\\mathrm U}","kind":"use","excerpt":"What is explicit in the internal BFV sector. On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation ca","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f28a064eaf92ff39","tex":"\\mathcal G_{\\mathrm U}","aliases":"G_ U mathcal mathrm","display":"\\mathcal G_{\\mathrm U}","kind":"use","excerpt":"C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The au","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-30418dea0b4dc62f","tex":"\\mathcal G_{\\mathrm U}^{\\mathrm{red}}","aliases":"G_ U^red mathcal mathrm mathrm","display":"\\mathcal G_{\\mathrm U}^{\\mathrm{red}}","kind":"use","excerpt":"For example, the residual electric constraint is G_ U^red=∂_a E^a+qΨ^†Ψ, Q[u]=-∫_Σ u G_ U^red d^3x. It generates δ_uA_a=∂_au and δ_uΨ=iquΨ. The half-density Lie derivative in the spatial generator is L_η^(1/2)Ψ =η^a∂_aΨ+tfrac12(∂_aη^a)Ψ. This term is forced by the density weight. Treating Ψ as a scalar would give a different moment map.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0a566b1db5acde7f","tex":"g_0","aliases":"g_0","display":"g_0","kind":"use","excerpt":"g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a43ad99289f696af","tex":"G_g","aliases":"G_g","display":"G_g","kind":"use","excerpt":"tution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing x through τ in each channel, gives ∑_xinnerSψ(x)Sϕ(x) =∑_xinnerψ(x)ϕ(x). For u^g(x)=g(τ x)u(x)g(x)^-1, direct endpoint cancellation gives S_A,τ,u^gG_g=G_gS_A,τ,u. Products of these directio","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-624b8e0e8ca99130","tex":"G_h","aliases":"G_h","display":"G_h","kind":"use","excerpt":"g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keep","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ba4cd8d81889283","tex":"g_h","aliases":"g_h","display":"g_h","kind":"use","excerpt":"For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomp","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d5c5a1afa19b22e1","tex":"\\mathcal G_i","aliases":"G_i mathcal","display":"\\mathcal G_i","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-967e5345c58b31c4","tex":"\\mathcal G_{IJ}","aliases":"G_IJ mathcal","display":"\\mathcal G_{IJ}","kind":"use","excerpt":"C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The au","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4e1d92ceb75803b7","tex":"\\mathcal G_{ij}","aliases":"G_ij mathcal","display":"\\mathcal G_{ij}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9ef43ece144c5d2a","tex":"\\mathcal G_j","aliases":"G_j mathcal","display":"\\mathcal G_j","kind":"use","excerpt":"[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f7288fb9331c8dd5","tex":"g_j","aliases":"g_j","display":"g_j","kind":"use","excerpt":"[The precise defect of a forward difference] On a cyclic grid, let D_hf_j=(f_j+1-f_j)/h, with h≠0. Expanding the product gives D_h(fg)_j-(D_hf)_jg_j-f_j(D_hg)_j =h(D_hf)_j(D_hg)_j. Thus the ordinary Leibniz identity fails unless the displayed product vanishes. A small defect on sufficiently regular sampled functions is compatible with the no-go the","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b371104368ef3d3d","tex":"g_{j+1}^{-1}","aliases":"g_j+1^-1","display":"g_{j+1}^{-1}","kind":"use","excerpt":"Linked shifts · A coefficient U_j in U_jψ_j+1 transports from j+1 back to j. Thus U_j'=g_jU_jg_j+1^-1. An outward link uses the opposite endpoint convention.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-89aa350d075d9b5d","tex":"g_{j+1}^n","aliases":"g_j+1^n","display":"g_{j+1}^n","kind":"use","excerpt":"For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the s","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c7ba05d541dd915","tex":"g_j^n","aliases":"g_j^n","display":"g_j^n","kind":"use","excerpt":"For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the s","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-65205833b6a4bc33","tex":"g_j^{n+1}","aliases":"g_j^n+1","display":"g_j^{n+1}","kind":"use","excerpt":"For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the s","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1a4a7f45555ebe7e","tex":"g_m","aliases":"g_m","display":"g_m","kind":"use","excerpt":"ginproof The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, suppose g∈Dom(H_ϵ^*) and write k=H_ϵ^*g. Testing the adjoint identity against an arbitrary vector supported on level m","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3d27b1546a14b2e","tex":"G_{p,h}","aliases":"G_p,h","display":"G_{p,h}","kind":"use","excerpt":"The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least si","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bbd4724f38071f66","tex":"G_{p_a}","aliases":"G_p_a","display":"G_{p_a}","kind":"use","excerpt":"midrule Canonical brackets · Each sector declares its variables and sign. In standard coordinates, {F,G}=∑_a(F_q_aG_p_a-F_p_aG_q_a), so dot F={F,H}.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7eadc3987326373a","tex":"g_q","aliases":"g_q","display":"g_q","kind":"use","excerpt":"Let p be prime, let L=mp for a positive integer m, and take 0≤ j<p. For every q≥0, g_q=fracj!(Lq)!(Lq+j)! =binomLq+jj^-1∈Z_p^×. Indeed binomLq+jj=∏_r=1^j(Lq+r)/r. Each numerator and denominator is a p-adic unit because Lq≡0pmod p and 1≤ r<p. For j=0 the empty product is one. The proof also covers p=2 in its short window. Outside this window, a fact","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bf4f844ef337a256","tex":"G_{q_a}","aliases":"G_q_a","display":"G_{q_a}","kind":"use","excerpt":"midrule Canonical brackets · Each sector declares its variables and sign. In standard coordinates, {F,G}=∑_a(F_q_aG_p_a-F_p_aG_q_a), so dot F={F,H}.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-87a3fa4dd69238cf","tex":"g_r","aliases":"g_r","display":"g_r","kind":"use","excerpt":"For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c9668d016de81fd0","tex":"g_{r+1}","aliases":"g_r+1","display":"g_{r+1}","kind":"use","excerpt":"For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomp","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-169c77a4f93260fb","tex":"g_U","aliases":"g_U","display":"g_U","kind":"use","excerpt":"For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tan","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-08d86714390ee3fe","tex":"g_V","aliases":"g_V","display":"g_V","kind":"use","excerpt":"For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tan","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-310a48051f387508","tex":"g_x","aliases":"g_x","display":"g_x","kind":"use","excerpt":"Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have t","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7b7b7cb3a0e5cda8","tex":"g_{x+\\widehat\\mu}","aliases":"g_x+widehatμ widehat mu","display":"g_{x+\\widehat\\mu}","kind":"use","excerpt":"The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particula","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c1bd1851814a55bc","tex":"g_{x+\\widehat\\mu}^{-1}","aliases":"g_x+widehatμ^-1 widehat mu","display":"g_{x+\\widehat\\mu}^{-1}","kind":"use","excerpt":"ing on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Product","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-badd526519f7d4c1","tex":"g_{x-\\widehat\\mu}","aliases":"g_x-widehatμ widehat mu","display":"g_{x-\\widehat\\mu}","kind":"use","excerpt":"For the positive edge, substitute [source reference]. The two gauge factors at x+widehatμ cancel, leaving g_x on the left. For the negative edge, (U_x-widehatμ,μ^g)^-1g_x-widehatμ =g_xU_x-widehatμ,μ^-1. Every term in either difference therefore has the common factor g_x.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5feaf61feae29f5f","tex":"g_x^*","aliases":"g_x^*","display":"g_x^*","kind":"use","excerpt":"Now take ψ_x∈C^2, U_x,μ∈ SU(2), and local gauges g_x∈ SU(2), as in the legacy non-Abelian chapter. The linked scalar z_x,μ=ψ_x^*U_x,μψ_x+widehatμ is gauge invariant: the adjoint factor g_x^* cancels g_x, and the factors at x+widehatμ also cancel. It follows that the source's scalar link observable j_x,μ=2Imz_x,μ is invariant. To obtain a continuity","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c4b741b3ca6387a1","tex":"g_x^{-1}","aliases":"g_x^-1","display":"g_x^{-1}","kind":"use","excerpt":"x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to left. More generally, a transporter T_y← x has the transformation g_yT_y← xg_x^-1. This notation checks both the input and output of every ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-889e84d345b27e17","tex":"g_y","aliases":"g_y","display":"g_y","kind":"use","excerpt":"x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to left. More generally, a transporter T_y← x has the transformation g_yT_y← xg_x^-1. This notation checks both the input and output of every ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e34f9d3fec41ee83","tex":"g_{\\mathrm{YM}}","aliases":"g_YM mathrm","display":"g_{\\mathrm{YM}}","kind":"use","excerpt":"For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the anal","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c445feaff50474e5","tex":"g_{\\mathrm{YM}}^2","aliases":"g_YM^2 mathrm","display":"g_{\\mathrm{YM}}^2","kind":"use","excerpt":"For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the anal","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-876a72a7054676bb","tex":"G_\\beta","aliases":"G_β beta","display":"G_\\beta","kind":"use","excerpt":"If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f907805278b68599","tex":"\\operatorname{gh}","aliases":"gh operatorname","display":"\\operatorname{gh}","kind":"use","excerpt":"The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or eq","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1725e2bf2fce3bf7","tex":"\\mathrm{graph}","aliases":"graph mathrm","display":"\\mathrm{graph}","kind":"use","excerpt":"widehat H_ma are square summable. Given ε>0, choose finite sets that make each of their tails small and take their union. The resulting one finite set S satisfies norma-P_Sa<ε, normwidehat H_ma-widehat H_mP_Sa<ε. This graph-approximation statement is represented in the formal source. The finite Fourier fields are dense for the graph norm norma_grap","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b75cc7bdfa6f5f94","tex":"\\mathbf h","aliases":"h mathbf","display":"\\mathbf h","kind":"use","excerpt":"mathrmtor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least si","count":19},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1bf12537a68f2d94","tex":"\\mathfrak H","aliases":"H mathfrak","display":"\\mathfrak H","kind":"use","excerpt":"For s>0, define the Haar regularity norm normf_ H_p^s^2 =∑_m≥0p^2smnorm F_mf_2^2, H_p^s={f∈HH_p:normf_ H_p^s<∞}. No identification with another Sobolev convention is required here. The definition itself gives the quantitative cutoff estimate.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-823170eb09d88517","tex":"\\mathsf H","aliases":"H mathsf","display":"\\mathsf H","kind":"use","excerpt":"M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b8bdb1c02444ca91","tex":"H'","aliases":"H'","display":"H'","kind":"use","excerpt":"Refinement · An isometry V has V^*V=I. Its old-range projector is P=VV^* and its new-detail projector is Q=I-P. Exact dynamical compatibility is a separate identity, H'V=VH.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c3d50e4b12d0681c","tex":"H^*","aliases":"H^*","display":"H^*","kind":"use","excerpt":"+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","excerpt":"real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking A_j to be its edge average h^-1∫_x_j^x_j+hA(s) ds. This is an additional sampling prescription","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fa0621c04438e85a","tex":"h^{-2}","aliases":"h^-2","display":"h^{-2}","kind":"use","excerpt":"The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm ter","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"The inhomogeneous weight retains the zero mode even when m=0. There is no loss of the ambient square-summability requirement in the massless case. The phrase first-order Fourier domain refers to this exact condition. It agrees with the conventional Fourier definition of periodic H^1 regularity; this manuscript does not claim a new checked equivalen","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"A precise conditional transfer theorem. Split s_F=s_0+δ, where s_0^2=0 and δ has degree one. Suppose ( C_C,s_0,C) is a coarse complex and the maps are chain maps: s_0I=Is_0,C, Ps_0=s_0,CP. Assume a degree-minus-one homotopy H satisfies PI=id_C, id_F-IP=s_0H+Hs_0, PH=0, HI=0, H^2=0. Finally, require (s_0+δ)^2=0 and an inverse of id_F+Hδ. The inverse","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":"Take V=U_h(t), W=U(t), and b=P_Sa. For a fixed finite S, the compact-time packet estimate gives normU_h(t)P_Sa-U(t)P_Sa≤ K_S,a,Th^2 for all |t|≤ T and sufficiently small positive h. The constant is allowed to depend on the support, the amplitudes, and the time window. Combining the estimates gives normU_h(t)a-U(t)a≤ 2norma-P_Sa+K_S,a,Th^2.","count":54},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9217de3017a2652b","tex":"h^{2.084394965}","aliases":"h^2.084394965","display":"h^{2.084394965}","kind":"use","excerpt":"The coarsest two event coordinates are not monotone with refinement. On the six finest resolutions the retained nonlinear fits are t_H(h) · ≃4.6893522370+33.88577338 h^2.173853282, r_H(h) · ≃3.1204965744-65.59310603 h^2.084394965. These fit exponents are consistent with second-order event convergence on this bounded family. They are not a convergen","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-92940262b09349da","tex":"h^{2.173853282}","aliases":"h^2.173853282","display":"h^{2.173853282}","kind":"use","excerpt":"The coarsest two event coordinates are not monotone with refinement. On the six finest resolutions the retained nonlinear fits are t_H(h) · ≃4.6893522370+33.88577338 h^2.173853282, r_H(h) · ≃3.1204965744-65.59310603 h^2.084394965. These fit exponents are consistent with second-order event convergence on this bounded family. They are not a convergen","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fc3c9dc938c10528","tex":"h^3","aliases":"h^3","display":"h^3","kind":"use","excerpt":"T_3f(h) · =frac64f(h/8)-56f(h/4)+14f(h/2)-f(h)21. Take the exact scalar polynomial f(h)=1+2h-3h^2+4h^3. The responses are T_1f(h)=1+frac32h^2-3h^3, T_2f(h)=1+frac12h^3, T_3f(h)=1. These are polynomial identities for every real h. If each of the three samples used by T_2 has absolute error at most ε, its output error is at most frac8+6+13ε=5ε. This ","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fd80ec0462f00d62","tex":"h^4","aliases":"h^4","display":"h^4","kind":"use","excerpt":"q_j(h,p(n))=p_j(n)-frach^2p_j(n)^36+O(h^4),","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-425652a73ca954b2","tex":"H^k","aliases":"H^k","display":"H^k","kind":"use","excerpt":"The identity I_4=P_++P_- and the two eigenvalue identities give, by induction, H^k=E^kP_++(-E)^kP_- for every k≥0. Substitute into the absolutely convergent series to obtain the spectral expression. Expanding its projectors gives the cosine and sine expression.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7c6fa035c4defd46","tex":"H^n","aliases":"H^n","display":"H^n","kind":"use","excerpt":"For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-858ebfe3a6d1e027","tex":"h^r","aliases":"h^r","display":"h^r","kind":"use","excerpt":"[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next fact","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1fe35faf74f825f6","tex":"H^T","aliases":"H^T","display":"H^T","kind":"use","excerpt":"overline∇ H(z_0,z_1) with invertible Δ_Δ satisfies Φ(z_1)=Φ(z_0), H(z_1)=H(z_0). The componentwise chain rule gives Φ(z_1)-Φ(z_0)=G(z_1-z_0), which vanishes because GJ_ D^Δ=0. For the Hamiltonian, H(z_1)-H(z_0) =Δ t overline∇ H^T J_ D^Δoverline∇ H=0 by skewness.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7e488733e11e45b8","tex":"H_{\\mathrm m}","aliases":"H_ m mathrm","display":"H_{\\mathrm m}","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations spec","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-16ef1e898cbf3ee0","tex":"\\mathcal H_{\\mathrm M}","aliases":"H_ M mathcal mathrm","display":"\\mathcal H_{\\mathrm M}","kind":"use","excerpt":"Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"Let V be an isometry from a coarse space to a finite fine space, set P=VV^* and Q=I-P, and let H_0=H_0^* satisfy H_0V=VH_c. Let B=B^* be purely off diagonal: PBP=0, QBQ=0. For a real λ, define H_λ=H_0+λ B. In the source model λ=p^-w for an assigned weight w≥1.","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-49ecf53502f364f0","tex":"H_0^*","aliases":"H_0^*","display":"H_0^*","kind":"use","excerpt":"Let V be an isometry from a coarse space to a finite fine space, set P=VV^* and Q=I-P, and let H_0=H_0^* satisfy H_0V=VH_c. Let B=B^* be purely off diagonal: PBP=0, QBQ=0. For a real λ, define H_λ=H_0+λ B. In the source model λ=p^-w for an assigned weight w≥1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bea5d0e7eb485d1c","tex":"H_A","aliases":"H_A","display":"H_A","kind":"use","excerpt":"[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber H_A f · =∑_ rA_ r F_ rf. Then H_A is self-adjoint, and finite rectangular packet sums form a core. Every rectangula","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e337b6fe26be0666","tex":"H_A^{(\\mathbf h)}","aliases":"H_A^( h) mathbf","display":"H_A^{(\\mathbf h)}","kind":"use","excerpt":"H_A f · =∑_ rA_ r F_ rf. Then H_A is self-adjoint, and finite rectangular packet sums form a core. Every rectangular range reduces H_A, and its finite restriction H_A^( h) satisfies exact embedding intertwining with every larger rectangular restriction. For bounded Borel F, F(H_A)f=∑_ rF(A_ r) F_ rf, F(H_A) E_ h→ F(H_A) strongly. In particular, nor","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c54ddda591ca604e","tex":"H_A^*","aliases":"H_A^*","display":"H_A^*","kind":"use","excerpt":"xtstrongly. In particular, norme^-itH_A(I- E_ h)f =norm(I- E_ h)f. Finite rectangular sums belong to the domain and are dense. Symmetry holds by symmetry on every block. If g belongs to the adjoint domain and k=H_A^*g, testing against an arbitrary vector in KK_ r gives F_ rk=A_ r F_ rg. Square summability of k is precisely the domain condition in [","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5294eca4b0cb55fe","tex":"H_{AA}","aliases":"H_AA","display":"H_{AA}","kind":"use","excerpt":"2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depen","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c9c78162ab0afe1d","tex":"H_{AB}","aliases":"H_AB","display":"H_{AB}","kind":"use","excerpt":"2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depen","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-587cf05864943581","tex":"H_{BA}","aliases":"H_BA","display":"H_{BA}","kind":"use","excerpt":"2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depen","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8b69ca3ee83e3f48","tex":"H_{ba}","aliases":"H_ba","display":"H_{ba}","kind":"use","excerpt":"In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7cee56da1c8c7844","tex":"H_{BA}^*","aliases":"H_BA^*","display":"H_{BA}^*","kind":"use","excerpt":"2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depen","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cca8f5fa2a64839a","tex":"H_{BB}","aliases":"H_BB","display":"H_{BB}","kind":"use","excerpt":"2 · 0 · 1 · 0 . Relative to C^4=C^2_A⊕C^2_B, the four blocks are H_AA=H_BB=0 · 1 1 · 0, H_BA=0 · -1 2 · 0, H_AB=H_BA^*. The block from A to B contains two distinct coordinate channels. Neither has to vanish for their net probability current to vanish at a particular state. The operator block describes available coupling. The current will also depen","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b8541d1f64196274","tex":"H_c","aliases":"H_c","display":"H_c","kind":"use","excerpt":"[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2b355180d8491077","tex":"\\mathcal H_c","aliases":"H_c mathcal","display":"\\mathcal H_c","kind":"use","excerpt":"[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-559e90a1829db889","tex":"H_D","aliases":"H_D","display":"H_D","kind":"use","excerpt":"As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0028a7a569a1dcc9","tex":"H_E","aliases":"H_E","display":"H_E","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations spec","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7b786d951bfb29bf","tex":"H_{\\mathrm{eff}}","aliases":"H_eff mathrm","display":"H_{\\mathrm{eff}}","kind":"use","excerpt":"The connection Γ records the velocity of the frame. Omitting it would interpret a change of packet coordinates as if it were entirely a change of the physical state. In Hamiltonian notation the corresponding effective Hamiltonian is H_eff=U^*HU-iΓ, which is self-adjoint because Γ is skew-adjoint.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-399afbaa0e9a8610","tex":"H_f","aliases":"H_f","display":"H_f","kind":"use","excerpt":"[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,","count":18},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4bc955a2f0ca796d","tex":"\\mathcal H_f","aliases":"H_f mathcal","display":"\\mathcal H_f","kind":"use","excerpt":"[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c2c11852c7f707dc","tex":"H_G","aliases":"H_G","display":"H_G","kind":"use","excerpt":"Define the densities H_G · =P_τ^2+γ P_θ^2 +e^2P(Q_τ^2+γ Q_θ^2),nonumber M_G · =2(P_τ P_θ+e^2PQ_τ Q_θ). The remaining reconstruction equations are λ_τ=H_G and λ_θ=M_G.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-102402835dbe0bc2","tex":"H_{G,h}","aliases":"H_G,h","display":"H_{G,h}","kind":"use","excerpt":"The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservatio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-820491df7b55e699","tex":"H_h","aliases":"H_h","display":"H_h","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations spec","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-027276bb71e82248","tex":"H_{h,n}^2","aliases":"H_h,n^2","display":"H_{h,n}^2","kind":"use","excerpt":"H_h,n^2=E_h,n^2I_4, E_h,n=sqrtm^2+|q_h(n)|^2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-46e2d2e905a9d0ff","tex":"h_{ij}","aliases":"h_ij","display":"h_{ij}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b90556c898d304c9","tex":"h_{ik}","aliases":"h_ik","display":"h_{ik}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-60f212957626b8f2","tex":"H_{\\mathrm{int}}","aliases":"H_int mathrm","display":"H_{\\mathrm{int}}","kind":"use","excerpt":"Introduce a visible address |c⟩ orthogonal to all descendants. Suppose every descendant has the same coupling from |c⟩, with complex matrix element γ_p,w. The Hermitian interaction is H_int= ∑_ d (γ_p,w| d⟩⟨ c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e6e1d7ad2fc5bdfa","tex":"h_{ji}","aliases":"h_ji","display":"h_{ji}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-957daf505ba79710","tex":"h_{jk}","aliases":"h_jk","display":"h_{jk}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-95273a360d135670","tex":"h_{kk}","aliases":"h_kk","display":"h_{kk}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df3ba0db246bbd1a","tex":"h_{kl}","aliases":"h_kl","display":"h_{kl}","kind":"use","excerpt":"Twice its divergence expands as 2δ_j G_ij = · δ_iδ_jδ_kh_jk +δ^2δ_kh_ik-δ^2δ_jh_ij -δ_iδ^2h_rm tr · -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0239d5e810818427","tex":"H_m^{\\mathrm{phys}}","aliases":"H_m^phys mathrm","display":"H_m^{\\mathrm{phys}}","kind":"use","excerpt":"Connecting the group to its generator. The next chapter proves that the constructed group is the spectral evolution of this Hamiltonian: U(t)=e^-itH_m^phys. The proof establishes strong continuity and differentiates on the full Hamiltonian domain with control of the Fourier tail. The reverse domain inclusion then shows that the infinitesimal genera","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-785ae493560363c1","tex":"h_n","aliases":"h_n","display":"h_n","kind":"use","excerpt":"Summable refinement and what a limit still requires. For the dyadic mesh h_n=a_0 2^-n, the regular estimate produces the nonnegative sequence ε_n=C4^-n, C=a_0^2 A_∂. The actual generated operator norms are bounded by this sequence and hence are summable. This is more information than merely ε_n→0. A sequence with adjacent errors tending to zero nee","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0ced436a17ff0e3c","tex":"H_n^*","aliases":"H_n^*","display":"H_n^*","kind":"use","excerpt":"The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass ter","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-921fd2ec6756cc1a","tex":"H_n^2","aliases":"H_n^2","display":"H_n^2","kind":"use","excerpt":"The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass ter","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0614636d0bfeab26","tex":"\\mathsf H_p","aliases":"H_p mathsf","display":"\\mathsf H_p","kind":"use","excerpt":"M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d54b1a6be7c911b2","tex":"h_p","aliases":"h_p","display":"h_p","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-263fca6973f674db","tex":"H_{p,h}","aliases":"H_p,h","display":"H_{p,h}","kind":"use","excerpt":"Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5d63a92771aeec1f","tex":"\\mathcal H_{p,h}","aliases":"H_p,h mathcal","display":"\\mathcal H_{p,h}","kind":"use","excerpt":"The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least si","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-44144103f9354925","tex":"H_{p,h+1}^{\\mathrm{Haar}}","aliases":"H_p,h+1^Haar mathrm","display":"H_{p,h+1}^{\\mathrm{Haar}}","kind":"use","excerpt":"Choose a real sequence of level energies ϵ_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonaliz","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8032d1774cbfdf13","tex":"H_{p,h}^{\\mathrm{Haar}}","aliases":"H_p,h^Haar mathrm","display":"H_{p,h}^{\\mathrm{Haar}}","kind":"use","excerpt":"Choose a real sequence of level energies ϵ_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonaliz","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3a2ec2244c17754a","tex":"\\mathfrak H_p^s","aliases":"H_p^s mathfrak","display":"\\mathfrak H_p^s","kind":"use","excerpt":"For s>0, define the Haar regularity norm normf_ H_p^s^2 =∑_m≥0p^2smnorm F_mf_2^2, H_p^s={f∈HH_p:normf_ H_p^s<∞}. No identification with another Sobolev convention is required here. The definition itself gives the quantitative cutoff estimate.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d83201f3b36e1284","tex":"H_Q","aliases":"H_Q","display":"H_Q","kind":"use","excerpt":"QH_fP · =0, PH_fP=VH_cV^*. They hold exactly when, on Ran P⊕Ran Q, H_f=(VH_cV^*)|_Ran P ⊕ H_Q, H_Q=(QH_fQ)|_Ran Q. The detail operator H_Q may be any self-adjoint operator on Ran Q.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a06593e7c86b3be5","tex":"H_q","aliases":"H_q","display":"H_q","kind":"use","excerpt":")=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1-q)M_q). Strict positiv","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e3b81d1ac3c985ef","tex":"H_{\\rm eff}^{(2)}","aliases":"H_rm eff^(2) rm","display":"H_{\\rm eff}^{(2)}","kind":"use","excerpt":"The normalized effective energy is (z-μ_p(z))/κ_p(z)=R_p(x). The parameter z remains fixed through the elimination. Since μ_p and κ_p depend on it, this is an energy-dependent spectral closure, not an energy-independent Hamiltonian flow. For p=2, R_2(x)=x^2-2 and U_1(x/2)=x, giving H_rm eff^(2)(z) =μ I+fracκ^2z-μ(2I+T+T^*), z≠μ. This decimation ret","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b60b9efbddaafac9","tex":"H_{\\rm eff}^{(p)}","aliases":"H_rm eff^(p) rm","display":"H_{\\rm eff}^{(p)}","kind":"use","excerpt":"[Exact free-cycle decimation] For U_p-1(x/2)≠0, the Schur operator on the retained sublattice is S_p(z)=fracκU_p-1(x/2) [R_p(x)I_M-(T+T^*)]. Equivalently, S_p(z)=zI-H_rm eff^(p)(z), where H_rm eff^(p)(z) · =μ_p(z)I+κ_p(z)(T+T^*),nonumber","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-da0dbd41a50083a7","tex":"H_{\\rm full}","aliases":"H_rm full rm","display":"H_{\\rm full}","kind":"use","excerpt":"=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenv","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3001aa6263f9409","tex":"\\mathcal H_{\\rm matt}","aliases":"H_rm matt mathcal rm","display":"\\mathcal H_{\\rm matt}","kind":"use","excerpt":"To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source refe","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3d87103c514a144c","tex":"h_{\\rm tr}","aliases":"h_rm tr rm","display":"h_{\\rm tr}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b6bc647753035f9f","tex":"H_{\\rm TT}","aliases":"H_rm TT rm","display":"H_{\\rm TT}","kind":"use","excerpt":"Choose n=n_i=0 and impose the transverse and traceless conditions δ_jh_ij=0, h_rm tr=0, δ_jπ_ij=0, π_rm tr=0. They are preserved by the evolution because G_ij(h)=-tfrac12δ^2h_ij, dot h=2π, dotπ=tfrac12δ^2h, ddot h=δ^2h. On this reduced sector the energy is H_rm TT=normπ_h^2 +frac14∑_knormδ_kh_h^2≥0. Its derivative is zero: substituting the two firs","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-00e5cb4d6d94dccd","tex":"H_{rs}","aliases":"H_rs","display":"H_{rs}","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-668caf8df5eeb048","tex":"H_{rs}^*","aliases":"H_rs^*","display":"H_{rs}^*","kind":"use","excerpt":"mes|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f2b2331953e04c5c","tex":"H_{sr}","aliases":"H_sr","display":"H_{sr}","kind":"use","excerpt":"mes|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ea1daba7a0d5088","tex":"\\mathcal H_t","aliases":"H_t mathcal","display":"\\mathcal H_t","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-95f7cb457015e1aa","tex":"H_\\epsilon","aliases":"H_ϵ epsilon","display":"H_\\epsilon","kind":"use","excerpt":"Let (ϵ_m)_m≥0 be any sequence of real numbers. Define Dom(H_ϵ) · ={f∈HH_p: ∑_m≥0|ϵ_m|^2norm F_mf_2^2<∞}, nonumber H_ϵ f · =∑_m≥0ϵ_m F_mf. The series converges in L^2 exactly on this domain. It is not necessary to bound the energies uniformly, but it is necessary to state the domain.","count":28},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-49ad7821cf07d564","tex":"H_\\epsilon^{(M)}","aliases":"H_ϵ^(M) epsilon","display":"H_\\epsilon^{(M)}","kind":"use","excerpt":"Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-15016e5fa808b1d5","tex":"H_\\epsilon^*","aliases":"H_ϵ^* epsilon","display":"H_\\epsilon^*","kind":"use","excerpt":"pectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, suppose g∈Dom(H_ϵ^*) and write k=H_ϵ^*g. Testing the adjoint identity against an arbitrary","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18adc48f226810ce","tex":"H_\\lambda","aliases":"H_λ lambda","display":"H_\\lambda","kind":"use","excerpt":"Let V be an isometry from a coarse space to a finite fine space, set P=VV^* and Q=I-P, and let H_0=H_0^* satisfy H_0V=VH_c. Let B=B^* be purely off diagonal: PBP=0, QBQ=0. For a real λ, define H_λ=H_0+λ B. In the source model λ=p^-w for an assigned weight w≥1.","count":33},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4f4f51cfd6f3785b","tex":"H_\\lambda^*","aliases":"H_λ^* lambda","display":"H_\\lambda^*","kind":"use","excerpt":"[Self-adjointness of the diagonal realization] The operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e1ec9ac41ad6dd0e","tex":"H_\\lambda^{1/2}","aliases":"H_λ^1/2 lambda","display":"H_\\lambda^{1/2}","kind":"use","excerpt":"For positive λ_n, the quadratic-form domain instead uses ∑_nλ_n|c_n|^2<∞. This is the domain of H_λ^1/2, not generally the domain of H_λ. With λ_n=n and c_n=n^-s, the three thresholds are c|c membership · condition","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-743400708b98042d","tex":"H_\\infty","aliases":"H_∞ infty","display":"H_\\infty","kind":"use","excerpt":"Let H_∞ be self-adjoint with spectral measure E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum o","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7fa5c4539bb70667","tex":"\\mathrm{Haar}","aliases":"Haar mathrm","display":"\\mathrm{Haar}","kind":"use","excerpt":"Choose a real sequence of level energies ϵ_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonaliz","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e073754a563e30e","tex":"\\hbox","aliases":"hbox","display":null,"kind":"use","excerpt":"f_h(t)-f(t)=O(h^2) hboxin L^2( T^3; C^4),","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-04e8653f90df6217","tex":"\\HH_{\\mathcal P}^{\\infty}","aliases":"HH_ P^∞ HH mathcal infty","display":null,"kind":"use","excerpt":"Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7f4a69bec175bb06","tex":"\\HH_A","aliases":"HH_A HH","display":null,"kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e68521b49f17a05c","tex":"\\HH_B","aliases":"HH_B HH","display":null,"kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-02afd49e2184a454","tex":"\\HH_D","aliases":"HH_D HH","display":null,"kind":"use","excerpt":"Let a finite Hilbert space be the orthogonal sum HH_L⊕HH_D, and write a self-adjoint Hamiltonian as H=A · B","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-169da2322bee3862","tex":"\\HH_L","aliases":"HH_L HH","display":null,"kind":"use","excerpt":"Let a finite Hilbert space be the orthogonal sum HH_L⊕HH_D, and write a self-adjoint Hamiltonian as H=A · B","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e1e857460125b3b5","tex":"\\HH_{p,m}","aliases":"HH_p,m HH","display":null,"kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-540b985016256f81","tex":"\\HH_S","aliases":"HH_S HH","display":null,"kind":"use","excerpt":"Hilbert estimates with explicit constants. Finite-dimensional continuity is not enough to identify an operator norm with physical ℓ^2 energy. The earlier continuous-linear packaging retains the ambient norm on a function space. The Hilbert scalar layer instead realizes a finite four-spinor field as HH_S=ℓ^2(S×{0,1,2,3};C), normψ^2=∑_x∈ Snormψ(x)_C^","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-26ff68a22dbb0da1","tex":"\\HH_x","aliases":"HH_x HH","display":null,"kind":"use","excerpt":"Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have t","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b70e977f58fbef0b","tex":"\\HH_{x+\\widehat\\mu}","aliases":"HH_x+widehatμ HH widehat mu","display":null,"kind":"use","excerpt":"Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have t","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"For positive λ_n, the quadratic-form domain instead uses ∑_nλ_n|c_n|^2<∞. This is the domain of H_λ^1/2, not generally the domain of H_λ. With λ_n=n and c_n=n^-s, the three thresholds are c|c membership · condition ℓ^2 · s>1/2 D(H_λ^1/2) · s>1 D(H_λ) · s>3/2. Each equality case diverges harmonically. Thus a state with finite mean positive energy ca","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"use","excerpt":"The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservatio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-78fd137c582a5e5f","tex":"I_4","aliases":"I_4","display":"I_4","kind":"use","excerpt":"The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass ter","count":25},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eff6aa27c552c5d2","tex":"I_a","aliases":"I_a","display":"I_a","kind":"use","excerpt":"Choose unit links a(x) and b(x) on the first and second half-edge with b(x)a(x)=u(x). The associated fine link is v(e_x)=a(x), v(m_x)=b(x). The interpolation and reconstruction maps are (I_aψ)(e_x) · =ψ(x), · (I_aψ)(m_x) · =a(x)ψ(x), (R_aΨ)(x) · =tfrac12(Ψ(e_x)+a(x)^-1Ψ(m_x)). The midpoint is a transported value, not an independently sampled field ","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d971406bf59d1ecc","tex":"\\mathcal I_h","aliases":"I_h mathcal","display":"\\mathcal I_h","kind":"use","excerpt":"The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservatio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1838bc7ba65331a7","tex":"I_{\\KK}","aliases":"I_KK KK","display":null,"kind":"use","excerpt":"For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-136567efa8d010b5","tex":"I_{\\KK_{\\mathrm{tor}}}","aliases":"I_KK_tor KK mathrm","display":null,"kind":"use","excerpt":"Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3685c5d2d3894445","tex":"I_{K\\to L}","aliases":"I_K→ L to","display":"I_{K\\to L}","kind":"use","excerpt":"What the closure tower does and does not preserve. At a resolved point, a Hamiltonian vector field can immediately generate higher bands. In particular, the fixed space T_K is not generally invariant under nonlinear evolution. The valid procedure enlarges the representation before the newly generated support aliases. Fourier zero padding satisfies ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-acc489cb07feb1b4","tex":"I_{K\\to M}","aliases":"I_K→ M to","display":"I_{K\\to M}","kind":"use","excerpt":"What the closure tower does and does not preserve. At a resolved point, a Hamiltonian vector field can immediately generate higher bands. In particular, the fixed space T_K is not generally invariant under nonlinear evolution. The valid procedure enlarges the representation before the newly generated support aliases. Fourier zero padding satisfies ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-03e4b06b60f9a7dc","tex":"I_{L\\to M}","aliases":"I_L→ M to","display":"I_{L\\to M}","kind":"use","excerpt":"What the closure tower does and does not preserve. At a resolved point, a Hamiltonian vector field can immediately generate higher bands. In particular, the fixed space T_K is not generally invariant under nonlinear evolution. The valid procedure enlarges the representation before the newly generated support aliases. Fourier zero padding satisfies ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3ecf779d0b53571","tex":"I_M","aliases":"I_M","display":"I_M","kind":"use","excerpt":"Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ad472b41377c2078","tex":"I_m","aliases":"I_m","display":"I_m","kind":"use","excerpt":"Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"use","excerpt":"Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c7b33cae672307e6","tex":"I_p","aliases":"I_p","display":"I_p","kind":"use","excerpt":"p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-89e0ceb2ea34de35","tex":"I_{\\mathrm{phase},\\mathbf r}","aliases":"I_phase, r mathrm mathbf","display":"I_{\\mathrm{phase},\\mathbf r}","kind":"use","excerpt":"For example, on a joint depth packet one may take A_ r =(∑_aϵ_a, rT_a) ⊗ I_phase, r, ϵ_a, r∈R, T_a=T_a^*. The matrices T_a act on the sign and torsion factors. Their mutual commutation is unnecessary because their sum is self-adjoint on the finite block. The coefficients assigned to an existing packet are fixed under refinement. An extension acting","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9855187f190ff70","tex":"I_r","aliases":"I_r","display":"I_r","kind":"use","excerpt":"ndent The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phase frame r(s,c)∈ U(1) defines (I_rψ)(s,c)=r(s,c)ψ(s)","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fad10e84b7053877","tex":"I_\\delta","aliases":"I_δ delta","display":"I_\\delta","kind":"use","excerpt":"[Transfer of a differential] Under these assumptions, define I_δ=(id_F+Hδ)^-1I, s_C=s_0,C+Pδ I_δ. Then PI_δ=id_C, (s_0+δ)I_δ=I_δ s_C, and s_C^2=0. The inverse equation is I_δ=I-Hδ I_δ. Application of P proves PI_δ=PI=id_C. Let K=(s_0+δ)I_δ-I_δ s_C. Using the chain-map identity and id_F-IP=s_0H+Hs_0 in this expression gives K=H(s_0δ I_δ+ δ I_δ s_0,C","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f2a427dbf773b62b","tex":"I_\\ell","aliases":"I_ℓ ell","display":"I_\\ell","kind":"use","excerpt":"Interpolation is possible even when it depends on a path. Choose one reference path from the origin to each vertex and let r(v) be its transport. Then (I_ℓψ)(v)=r(v)ψ, innerΨΦ_rm cube =tfrac18∑_v∈{0,1}^3innerΨ(v)Φ(v). Every r(v) is unitary, so this map is a Hilbert isometry from one spinor to the eight-child cell. Its gauge covariance follows from ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b1185bc920eea10a","tex":"\\id_A","aliases":"id_A id","display":null,"kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b6db13550ab9effd","tex":"\\id_B","aliases":"id_B id","display":null,"kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","excerpt":"athsf F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber norm((z-widetilde H_M)^-1-(z-H_ϵ)^-1)","count":19},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ef28ee4048cca398","tex":"\\mathrm{in}","aliases":"in mathrm","display":"\\mathrm{in}","kind":"use","excerpt":"The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify th","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-55466b76f7f0c090","tex":"\\mathrm{int}","aliases":"int mathrm","display":"\\mathrm{int}","kind":"use","excerpt":"Introduce a visible address |c⟩ orthogonal to all descendants. Suppose every descendant has the same coupling from |c⟩, with complex matrix element γ_p,w. The Hermitian interaction is H_int= ∑_ d (γ_p,w| d⟩⟨ c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-146cbf287aa37a5f","tex":"\\mathcal J","aliases":"J mathcal","display":"\\mathcal J","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-41076cb14175109c","tex":"J^{\\mathrm c}","aliases":"J^ c mathrm","display":"J^{\\mathrm c}","kind":"use","excerpt":"For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eda0982baceee7a4","tex":"J^{\\mathrm c}_{B\\leftarrow A}","aliases":"J^ c_B← A mathrm leftarrow","display":"J^{\\mathrm c}_{B\\leftarrow A}","kind":"use","excerpt":"_2←1=3, J_3←2=2, J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). For A={1,2} and B={3,4}, the internal currents 3 and 1 cancel, leaving J^ c_B← A=2-1/2=3/2. Hence do","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6d08297f6303506d","tex":"J^{(\\rho)}_{a\\leftarrow b}","aliases":"J^(ρ)_a← b rho leftarrow","display":"J^{(\\rho)}_{a\\leftarrow b}","kind":"use","excerpt":"[Trace-current balance] For a Hermitian ρ satisfying [source reference], dot p_b=∑_aJ^(ρ)_b← a, J^(ρ)_b← a=-J^(ρ)_a← b. The subset boundary formula and exact coarse-current summation remain valid.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e17cbcbf5402efc6","tex":"J^{(\\rho)}_{b\\leftarrow a}","aliases":"J^(ρ)_b← a rho leftarrow","display":"J^{(\\rho)}_{b\\leftarrow a}","kind":"use","excerpt":"Suppose a differentiable matrix trajectory obeys the von Neumann equation dotρ=-i[H,ρ], H=H^*. For its coherent transfer define J^(ρ)_b← a =2ImTr(P_bHP_aρ).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-35b3ffd4dcaf66dc","tex":"J^*","aliases":"J^*","display":"J^*","kind":"use","excerpt":"ds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary preserves its internal norm. The subsequent reversal shift sends these amplitudes to new arcs at the children, outside the old arc range. Therefore, for a unit","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cd5b3dbd78a34d9d","tex":"J^F","aliases":"J^F","display":"J^F","kind":"use","excerpt":"r(a)=αJ^F_ba. No orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to J^F without changing its row sums. Thus the balance law determines the change in weight, while the generator-induced formula selects a particular representative of the current.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e0b1d825c322ca4","tex":"J^F_{ba}","aliases":"J^F_ba","display":"J^F_{ba}","kind":"use","excerpt":"Suppose that a differentiable state curve satisfies dotψ=Kψ. Define J^F_ba=ReinnerF_bψKF_aψ -ReinnerF_aψKF_bψ. This current is antisymmetric directly from its definition. Skew-adjointness also gives J^F_ba=2ReinnerF_bψKF_aψ. The same bilinear expression occurs for sharp packets, but its interpretation now concerns expectation weights.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-34af18168642b919","tex":"J^{F}_{ba}","aliases":"J^F_ba","display":"J^{F}_{ba}","kind":"use","excerpt":"Suppose that a differentiable state curve satisfies dotψ=Kψ. Define J^F_ba=ReinnerF_bψKF_aψ -ReinnerF_aψKF_bψ. This current is antisymmetric directly from its definition. Skew-adjointness also gives J^F_ba=2ReinnerF_bψKF_aψ. The same bilinear expression occurs for sharp packets, but its interpretation now concerns expectation weights.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ae0b0d392d2d752f","tex":"J^{\\rm c}_{\\beta\\alpha}","aliases":"J^rm c_βα rm beta alpha","display":"J^{\\rm c}_{\\beta\\alpha}","kind":"use","excerpt":"If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β r(a)=αJ^F_ba. No orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to J^F without changing its row sums. Thus the balance law determines the chan","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9d5a89b8a95b615","tex":"J_{\\mathrm D}^{\\Delta}","aliases":"J_ D^Δ mathrm Delta","display":"J_{\\mathrm D}^{\\Delta}","kind":"use","excerpt":"labelc:constraints:discrete-gradient The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. Direct multiplication proves GJ_ D^Δ=0. Since Δ_Δ and its inverse ","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-485db41ee683b54c","tex":"J_0","aliases":"J_0","display":"J_0","kind":"use","excerpt":"The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-67957011b9286dc7","tex":"J_0'","aliases":"J_0'","display":"J_0'","kind":"use","excerpt":"Set Q=0 and take an integer n≥1. Define the entire function J_0(x)=∑_m=0^∞frac(-1)^m(x^2/4)^m(m!)^2. The series and its differentiated series converge uniformly on compact sets. Coefficient comparison gives x^2J_0''+xJ_0'+x^2J_0=0. Thus P(τ,θ)=A J_0(ne^-τ)cos(nθ) is an exact polarized solution. Indeed, for x=ne^-τ, ∂_τ^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-351696876f533574","tex":"J_0''","aliases":"J_0''","display":"J_0''","kind":"use","excerpt":"Set Q=0 and take an integer n≥1. Define the entire function J_0(x)=∑_m=0^∞frac(-1)^m(x^2/4)^m(m!)^2. The series and its differentiated series converge uniformly on compact sets. Coefficient comparison gives x^2J_0''+xJ_0'+x^2J_0=0. Thus P(τ,θ)=A J_0(ne^-τ)cos(nθ) is an exact polarized solution. Indeed, for x=ne^-τ, ∂_τ^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1048fbee73eb4c54","tex":"J_{1:h}","aliases":"J_1:h","display":"J_{1:h}","kind":"use","excerpt":"As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ac431dc5751a954","tex":"J_{1\\leftarrow4}","aliases":"J_1←4 leftarrow","display":"J_{1\\leftarrow4}","kind":"use","excerpt":"[Internal exchange disappears from the coarse boundary] Consider four labels with nonzero currents J_2←1=3, J_3←2=2, J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-02186f046775278e","tex":"J_{2\\leftarrow1}","aliases":"J_2←1 leftarrow","display":"J_{2\\leftarrow1}","kind":"use","excerpt":"-isin(gt), p_1=cos^2(gt), p_2=sin^2(gt). The current is J_2←1 =2Im(i gsin(gt)cos(gt)) =gsin(2gt)=dot p_2. It reverses sign during the cycle. At t=0 the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at second order in time. This is why coupling strength and instan","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-86b61ad61a12e40d","tex":"J_{3\\leftarrow2}","aliases":"J_3←2 leftarrow","display":"J_{3\\leftarrow2}","kind":"use","excerpt":"[Internal exchange disappears from the coarse boundary] Consider four labels with nonzero currents J_2←1=3, J_3←2=2, J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2e4348540a1ad514","tex":"J_{4\\leftarrow3}","aliases":"J_4←3 leftarrow","display":"J_{4\\leftarrow3}","kind":"use","excerpt":"[Internal exchange disappears from the coarse boundary] Consider four labels with nonzero currents J_2←1=3, J_3←2=2, J_4←3=1, J_1←4=tfrac12, their negatives in the reverse directions, and all other currents zero. This is an example of the balance algebra, without an asserted underlying quantum trajectory. The weight derivatives are (-5/2,1,1,1/2). ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9448a796bda79d54","tex":"J_5^0","aliases":"J_5^0","display":"J_5^0","kind":"use","excerpt":"a^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the sub","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-07efa069ab951ea9","tex":"J_5^I","aliases":"J_5^I","display":"J_5^I","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aa1c6e5f9fe8f17e","tex":"J_5^J","aliases":"J_5^J","display":"J_5^J","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7fbcde5169efaf2d","tex":"J_5^K","aliases":"J_5^K","display":"J_5^K","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a7ed374167b55075","tex":"J_{5I}","aliases":"J_5I","display":"J_{5I}","kind":"use","excerpt":"tion is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the substituted connection, or use a torsion-free representation with the induced local four-spinor interaction. In the declared action convention the latter action contribution is -tfrac3κ16∫ J_5IJ_5^Ivol_e. It","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4740cf6406194851","tex":"J_{a-1}","aliases":"J_a-1","display":"J_{a-1}","kind":"use","excerpt":"rge without compensating background would be inconsistent. Equation [source reference] supplies that neutralizing mean. Conservation of G_j is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over a≤ j≤ b gives the physical-l","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e268ef5e9ac6d864","tex":"J_{ab}","aliases":"J_ab","display":"J_{ab}","kind":"use","excerpt":"Suppose a supplied curve ψ(t)∈ D(H) is differentiable as a curve in HH and satisfies dotψ=-iHψ. For a domain-preserving atlas, put p_b=normP_bψ^2, J_ba=2IminnerP_bψHP_aψ. Symmetry gives J_ba=-J_ab. Differentiating the bounded projected norm gives dot p_b=2IminnerP_bψHψ. Graph compatibility permits direct insertion of HS_Fψ→ Hψ. The formal developme","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8a5d3aa52b0f5bb8","tex":"J_{a\\leftarrow a}","aliases":"J_a← a leftarrow","display":"J_{a\\leftarrow a}","kind":"use","excerpt":"[Finite packet transfer] For the trajectory in [source reference], dot p_b · =∑_aJ_b← a, J_b← a · =-J_a← b, J_a← a=0, fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-777ddad06f846983","tex":"J_{a\\leftarrow b}","aliases":"J_a← b leftarrow","display":"J_{a\\leftarrow b}","kind":"use","excerpt":"[Finite packet transfer] For the trajectory in [source reference], dot p_b · =∑_aJ_b← a, J_b← a · =-J_a← b, J_a← a=0, fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-03acd6e46a6b3e27","tex":"J_{A\\leftarrow B}^{\\mathrm c}","aliases":"J_A← B^ c leftarrow mathrm","display":"J_{A\\leftarrow B}^{\\mathrm c}","kind":"use","excerpt":"J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-88fbeb34dbc66890","tex":"J_b","aliases":"J_b","display":"J_b","kind":"use","excerpt":"rge without compensating background would be inconsistent. Equation [source reference] supplies that neutralizing mean. Conservation of G_j is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over a≤ j≤ b gives the physical-l","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b1866826ed9899dc","tex":"J_{ba}","aliases":"J_ba","display":"J_{ba}","kind":"use","excerpt":"Throughout this chapter, inner products are conjugate linear in the first variable. A bounded state generator is denoted by K, with K^*=-K. When a Hamiltonian is present, K=-iH. The receiving label is written first in J_ba.","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8c11d396019c9a67","tex":"J_{B\\leftarrow A}^{\\mathrm c}","aliases":"J_B← A^ c leftarrow mathrm","display":"J_{B\\leftarrow A}^{\\mathrm c}","kind":"use","excerpt":"[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e30319bc4adb843","tex":"J_h","aliases":"J_h","display":"J_h","kind":"use","excerpt":"Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · ·","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2669ee857bf534f7","tex":"J_j","aliases":"J_j","display":"J_j","kind":"use","excerpt":"sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1, dot A_j · =E_j, · dot E_j · =J_j, · J_j · =Re(ψ_j^*α U_jψ_j+1)","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d7945cdcf60e1077","tex":"J_{j+1/2}^n","aliases":"J_j+1/2^n","display":"J_{j+1/2}^n","kind":"use","excerpt":"[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-14e9327ada8965fb","tex":"J_{j-1}","aliases":"J_j-1","display":"J_{j-1}","kind":"use","excerpt":"[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e866d0c3b0e9c007","tex":"J_{j-1/2}^n","aliases":"J_j-1/2^n","display":"J_{j-1/2}^n","kind":"use","excerpt":"[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b78dc9dea4a072f6","tex":"J_k","aliases":"J_k","display":"J_k","kind":"use","excerpt":"⟨ J_nv,J_kw⟩_L^2 =⟨ v,w⟩, · n=k, 0, · n≠ k.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a620020ba505249b","tex":"J_n","aliases":"J_n","display":"J_n","kind":"use","excerpt":"Let J_n: C^4→ L^2( T^3; C^4) be J_nv=e_nv.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fe88e020806e1739","tex":"J_r","aliases":"J_r","display":"J_r","kind":"use","excerpt":"Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e572acf19e0e35d0","tex":"J_{r:h}","aliases":"J_r:h","display":"J_{r:h}","kind":"use","excerpt":"For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3a4b59b7e16443be","tex":"J_r^*","aliases":"J_r^*","display":"J_r^*","kind":"use","excerpt":"Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dacf8aa7e30755b8","tex":"\\mathcal J_{r\\leftarrow s}","aliases":"J_r← s mathcal leftarrow","display":"\\mathcal J_{r\\leftarrow s}","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7b5625b6d739181","tex":"J_s","aliases":"J_s","display":"J_s","kind":"use","excerpt":"Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e7616576e5283b5","tex":"J_s^*","aliases":"J_s^*","display":"J_s^*","kind":"use","excerpt":"section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality f","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-faef0668e8df6422","tex":"\\mathcal J_{s\\leftarrow r}","aliases":"J_s← r mathcal leftarrow","display":"\\mathcal J_{s\\leftarrow r}","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-07b6123d41bb6991","tex":"J_t","aliases":"J_t","display":"J_t","kind":"use","excerpt":", its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f642708102f36295","tex":"j_{x,\\mu}","aliases":"j_x,μ mu","display":"j_{x,\\mu}","kind":"use","excerpt":"C^2, U_x,μ∈ SU(2), and local gauges g_x∈ SU(2), as in the legacy non-Abelian chapter. The linked scalar z_x,μ=ψ_x^*U_x,μψ_x+widehatμ is gauge invariant: the adjoint factor g_x^* cancels g_x, and the factors at x+widehatμ also cancel. It follows that the source's scalar link observable j_x,μ=2Imz_x,μ is invariant. To obtain a continuity equation wit","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-be10909ba1926f76","tex":"J_\\Sigma","aliases":"J_Σ Sigma","display":"J_\\Sigma","kind":"use","excerpt":", the Cartan identities give L_ηΩ=ι_η R+D_Ω(ι_ηΩ), L_η A=ι_η F+ d_Σ(ι_η A). Accordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression J_Σ[c,χ,u]=C[ζ(c)] +G_ L[χ-ι_cΩ] +G_ U[u+ι_cA]. For tangential c, ζ(c)=ι_cE is a spatial contraction. Fo","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-39da9152b74aaf7c","tex":"J_{\\partial_+A}^n","aliases":"J_∂_+A^n partial","display":"J_{\\partial_+A}^n","kind":"use","excerpt":"i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-175fcc560bff2509","tex":"J_{\\partial_-A}^n","aliases":"J_∂_-A^n partial","display":"J_{\\partial_-A}^n","kind":"use","excerpt":"i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-655e6df9493408f0","tex":"\\mathbf k","aliases":"k mathbf","display":"\\mathbf k","kind":"use","excerpt":"⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k = E_min( h, k), E_ hf→ f as every h_p→∞. To justify the second statement, first take an elementary tensor. The difference of the tensor projections and ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup gene","count":109},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1406e790403afdc4","tex":"k^{-2s}","aliases":"k^-2s","display":"k^{-2s}","kind":"use","excerpt":"A concrete domain boundary. As an ordinary example, put nonzero coefficients only on the positive axis modes n=(k,0,0), k≥1, and set a_n=k^-sv for a fixed unit spinor v. All remaining coefficients are zero. Then ∑_nnorma_n^2=∑_k≥1k^-2s, which is finite exactly when s>1/2. The additional momentum series is 4π^2∑_k≥1k^2-2s, which is finite exactly wh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e47a0b4ba7b7781f","tex":"k^{-s}","aliases":"k^-s","display":"k^{-s}","kind":"use","excerpt":"A concrete domain boundary. As an ordinary example, put nonzero coefficients only on the positive axis modes n=(k,0,0), k≥1, and set a_n=k^-sv for a fixed unit spinor v. All remaining coefficients are zero. Then ∑_nnorma_n^2=∑_k≥1k^-2s, which is finite exactly when s>1/2. The additional momentum series is 4π^2∑_k≥1k^2-2s, which is finite exactly wh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b779dcac03cb4e5","tex":"K^2","aliases":"K^2","display":"K^2","kind":"use","excerpt":"Proof. For E>0, the Hermitian matrix K=H/E satisfies K^2=I_4. Thus","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenv","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-479662d70e817846","tex":"k^{2-2s}","aliases":"k^2-2s","display":"k^{2-2s}","kind":"use","excerpt":"A concrete domain boundary. As an ordinary example, put nonzero coefficients only on the positive axis modes n=(k,0,0), k≥1, and set a_n=k^-sv for a fixed unit spinor v. All remaining coefficients are zero. Then ∑_nnorma_n^2=∑_k≥1k^-2s, which is finite exactly when s>1/2. The additional momentum series is 4π^2∑_k≥1k^2-2s, which is finite exactly wh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ee47a0653a7c8273","tex":"K^i","aliases":"K^i","display":"K^i","kind":"use","excerpt":"ented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6cc92a51c6969e2c","tex":"K_a","aliases":"K_a","display":"K_a","kind":"use","excerpt":"c:constraints:regular-triad The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_iδ K_a^i d^3x","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-de6eed3cf1d7e144","tex":"k_a","aliases":"k_a","display":"k_a","kind":"use","excerpt":"al P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b i","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f7f5fb972f2ae455","tex":"k_b","aliases":"k_b","display":"k_b","kind":"use","excerpt":"gle · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congru","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5203f1c6488200ed","tex":"K_{ba}","aliases":"K_ba","display":"K_{ba}","kind":"use","excerpt":"Define the receiving-first current by J_b← a(t) =2Reinnerψ_b(t)K(t)ψ_a(t) =2Reinnerψ_b(t)K_ba(t)ψ_a(t). The second equality follows because P_bψ_b=ψ_b. A positive value means that the contribution assigned to the pair increases the weight in b. The current is a real number, and its sign convention is fixed by the derivative calculation below.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7346537331b57441","tex":"K_{\\mathrm{eff}}","aliases":"K_eff mathrm","display":"K_{\\mathrm{eff}}","kind":"use","excerpt":"[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2e49264ff683642e","tex":"K_{\\mathrm{eff}}^*","aliases":"K_eff^* mathrm","display":"K_{\\mathrm{eff}}^*","kind":"use","excerpt":"[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b783228933c35b11","tex":"k_i","aliases":"k_i","display":"k_i","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-69e98d4d896d2853","tex":"k_i^4","aliases":"k_i^4","display":"k_i^4","kind":"use","excerpt":"The resulting frequency obeys ω_h^2=∑_ifracsin^2(k_ih)h^2. For fixed k≠0 as h→0, ω_h^2=| k|^2-frach^23∑_i k_i^4+O(h^4), ω_h=| k| -frach^26| k|∑_i k_i^4+O(h^4). These follow by expanding the sine and then the positive square root. The regime keeps k fixed. It is not uniform up to the Nyquist boundary.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca931e9dd55037b6","tex":"K_j","aliases":"K_j","display":"K_j","kind":"use","excerpt":"If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j),","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d383fcbbf03353b5","tex":"K_M","aliases":"K_M","display":"K_M","kind":"use","excerpt":"There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-792ddfc072fc3946","tex":"k_m","aliases":"k_m","display":"k_m","kind":"use","excerpt":"ginproof The orthogonal decomposition [source reference] identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, suppose g∈Dom(H_ϵ^*) and write k=H_ϵ^*g. Testing the adjoint identity against an arbitrary vector supported on level m","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-76c882ff3b5f883b","tex":"K_{M+1}","aliases":"K_M+1","display":"K_{M+1}","kind":"use","excerpt":"There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9a37605986fa4ad","tex":"\\mathcal K_p","aliases":"K_p mathcal","display":"\\mathcal K_p","kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3b86edec3341206","tex":"K_{p-1}","aliases":"K_p-1","display":"K_{p-1}","kind":"use","excerpt":"The stationary equation on an interior site is ψ_n+1=xψ_n-ψ_n-1. The eliminated block of z-H in one cell is κ K_p-1(x), where K_n(x) has diagonal x and neighboring off-diagonal entries -1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d2422ad0559ddeb1","tex":"\\mathcal K_p^*","aliases":"K_p^* mathcal","display":"\\mathcal K_p^*","kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1587c181ffea0832","tex":"\\mathcal K_S","aliases":"K_S mathcal","display":"\\mathcal K_S","kind":"use","excerpt":"K_S=⊕_n∈ S C^4, a _ K_S^2=∑_n∈ S a_n _2^2.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e45ff3c8c2a5a13f","tex":"K_{S,a,T}","aliases":"K_S,a,T","display":"K_{S,a,T}","kind":"use","excerpt":"Take V=U_h(t), W=U(t), and b=P_Sa. For a fixed finite S, the compact-time packet estimate gives normU_h(t)P_Sa-U(t)P_Sa≤ K_S,a,Th^2 for all |t|≤ T and sufficiently small positive h. The constant is allowed to depend on the support, the amplitudes, and the time window. Combining the estimates gives normU_h(t)a-U(t)a≤ 2norma-P_Sa+K_S,a,Th^2.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8e6a451de88cfbb0","tex":"K_\\mu","aliases":"K_μ mu","display":"K_\\mu","kind":"use","excerpt":"For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8929221b063811cf","tex":"K_\\mu^*","aliases":"K_μ^* mu","display":"K_\\mu^*","kind":"use","excerpt":"For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-797b703ad526a53a","tex":"K_\\mu^{\\rm poly}","aliases":"K_μ^rm poly mu rm","display":"K_\\mu^{\\rm poly}","kind":"use","excerpt":"xpansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4bd54056b2ab6858","tex":"\\KK_{\\mathbf r}","aliases":"KK_ r KK mathbf","display":null,"kind":"use","excerpt":"[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-563d5bb39d3ff859","tex":"\\KK_a","aliases":"KK_a KK","display":null,"kind":"use","excerpt":"Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1d66668754c625dc","tex":"\\KK_{\\mathrm{tor}}","aliases":"KK_tor KK mathrm","display":null,"kind":"use","excerpt":"Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","count":13},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e3d8f1ac9931ea94","tex":"\\mathfrak L","aliases":"L mathfrak","display":"\\mathfrak L","kind":"use","excerpt":"Let a periodic real lattice have canonical brackets {q_a,p_b}_h=δ_ab/h and a skew matrix D. Define D_h[ξ]=h∑_aξ_ap_a(Dq)_a, L_D(f,g)=D(fg)-(Df)g-f(Dg). All products on this lattice are pointwise.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c373cbbb5a44e89","tex":"\\mathrm L","aliases":"L mathrm","display":"\\mathrm L","kind":"use","excerpt":"What is explicit in the internal BFV sector. On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation ca","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"xpansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This i","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a44e55170cadffdb","tex":"L^*","aliases":"L^*","display":"L^*","kind":"use","excerpt":"Coherent exchange is one mechanism for changing packet weights. A second comes from an explicitly chosen family of dissipative transitions. Fix an orthonormal basis (e_a)_a∈ I of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cc65d4ac028d8c8c","tex":"L^{-1}","aliases":"L^-1","display":"L^{-1}","kind":"use","excerpt":"vergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follow by filtering these coefficients agai","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d4d9318beb0bea26","tex":"L^{2L}","aliases":"L^2L","display":"L^{2L}","kind":"use","excerpt":"The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( - 1, A_L,j, A_L,j; frac(-1)^Ly^LL^2L). O","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0387f51f80abc39f","tex":"L^{Lq}","aliases":"L^Lq","display":"L^{Lq}","kind":"use","excerpt":"Apply this filter to e^z and put E_L,j(z)=∑_q=0^∞fracz^Lq+j(Lq+j)!. For α∈C, write (α)_q=α(α+1)⋯(α+q-1) and (α)_0=1. Splitting the factorial into step-L factors gives frac(Lq+j)!j! · =∏_r=1^L∏_k=0^q-1(j+r+Lk)nonumber · =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every m","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"Let M_0∈ SL_2(C) and x=trM_0. No diagonalizability assumption is needed. Cayley--Hamilton gives M_0^2-xM_0+I=0, hence M_0^n=U_n-1(x/2)M_0-U_n-2(x/2)I, n≥1, with U_-1=0. For n=1 it is the identity M_0=M_0, and for n=2 it is Cayley--Hamilton. Multiplying by M_0 and replacing M_0^2 by xM_0-I proves the induction step. Taking traces yields tr(M_0^n)=R_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-63e820edf4b8f186","tex":"L_{ba}^*","aliases":"L_ba^*","display":"L_{ba}^*","kind":"use","excerpt":"For the one transition a→ b, L_baρ L_ba^*=γ_bap_aP_b, L_ba^*L_ba=γ_baP_a. Its diagonal gain is γ_bap_a at b, and the two halves of the anticommutator remove that same amount at a. No other diagonal entry changes from this one jump. Sum over ordered transitions to obtain [source reference]. Antisymmetry is immediate, and the finite boundary cancella","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f44072e22348fc64","tex":"\\mathcal L_c","aliases":"L_c mathcal","display":"\\mathcal L_c","kind":"use","excerpt":"Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c11b0603a60a2a4c","tex":"L_{CC}","aliases":"L_CC","display":"L_{CC}","kind":"use","excerpt":"Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local c","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3735436a6c30f5d0","tex":"L_{CD}","aliases":"L_CD","display":"L_{CD}","kind":"use","excerpt":"Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local c","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d3cc0ec7c5debf23","tex":"\\mathcal L_D","aliases":"L_D mathcal","display":"\\mathcal L_D","kind":"use","excerpt":"ectionThe conditional BV pushforward statement Suppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f73fafaae8f13dd0","tex":"\\mathfrak L_D","aliases":"L_D mathfrak","display":"\\mathfrak L_D","kind":"use","excerpt":"Let a periodic real lattice have canonical brackets {q_a,p_b}_h=δ_ab/h and a skew matrix D. Define D_h[ξ]=h∑_aξ_ap_a(Dq)_a, L_D(f,g)=D(fg)-(Df)g-f(Dg). All products on this lattice are pointwise.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c582ec52f23a0898","tex":"\\mathfrak L_{D_h}","aliases":"L_D_h mathfrak","display":"\\mathfrak L_{D_h}","kind":"use","excerpt":"For samples of sufficiently smooth periodic functions, the centered derivative obeys D_hf=f'+frach^26f'''+O(h^4). If f,g have uniformly bounded derivatives through order five, its Leibniz defect consequently satisfies L_D_h(f,g) =frach^22(f''g'+f'g'')+O(h^4). For fixed smooth p,ξ,η,q, with sufficiently many additional bounded derivatives of q to co","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bf1c6a9d4287ca7b","tex":"L_{DC}","aliases":"L_DC","display":"L_{DC}","kind":"use","excerpt":"Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local c","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18907e293d8ed85e","tex":"L_{DD}^{-1}","aliases":"L_DD^-1","display":"L_{DD}^{-1}","kind":"use","excerpt":"Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local c","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7ef8c7cca2b22b6a","tex":"L_{\\mathrm{eff}}","aliases":"L_eff mathrm","display":"L_{\\mathrm{eff}}","kind":"use","excerpt":"Similarly, direct and iterated elimination agree when the integrals and product gauge fixings admit the relevant Fubini or formal pushforward composition. The quadratic special case is Schur elimination, L_eff=L_CC-L_CDL_DD^-1L_DC, with the required inverses. A semigroup of elimination maps under these hypotheses does not imply a physically local c","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ff93f51daddcc717","tex":"L_g","aliases":"L_g","display":"L_g","kind":"use","excerpt":"it-form energy identity] Let D be skew adjoint for the grid inner product inneruw_h, and let multiplication by the real array b be self-adjoint. Then inneruA_h[b]u_h =-frac12inneru(Db)u_h. For two solutions of the split equation with the same prescribed b, remaining in a positive-radicand interval where |g'|≤ L_g, one obtains normR(t)-widetilde R(t","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-743ff0ea070acbd8","tex":"L_h","aliases":"L_h","display":"L_h","kind":"use","excerpt":"w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"use","excerpt":"The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particula","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-777bcedff418f3fc","tex":"L_{p,h}","aliases":"L_p,h","display":"L_{p,h}","kind":"use","excerpt":"Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-320283b00a6d4f02","tex":"\\mathcal L_\\eta","aliases":"L_η mathcal eta","display":"\\mathcal L_\\eta","kind":"use","excerpt":"The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d760990430efe993","tex":"\\mathcal L_\\eta^{(1/2)}","aliases":"L_η^(1/2) mathcal eta","display":"\\mathcal L_\\eta^{(1/2)}","kind":"use","excerpt":"For example, the residual electric constraint is G_ U^red=∂_a E^a+qΨ^†Ψ, Q[u]=-∫_Σ u G_ U^red d^3x. It generates δ_uA_a=∂_au and δ_uΨ=iquΨ. The half-density Lie derivative in the spatial generator is L_η^(1/2)Ψ =η^a∂_aΨ+tfrac12(∂_aη^a)Ψ. This term is forced by the density weight. Treating Ψ as a scalar would give a different moment map.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-48918c8f911d3438","tex":"\\mathcal L_\\xi","aliases":"L_ξ mathcal xi","display":"\\mathcal L_\\xi","kind":"use","excerpt":"Gauge transformations and graded conventions. For even parameters (ξ,λ,u), the coframe transforms by its Lie derivative and Lorentz rotation, the connection by L_ξω-D_ωλ, and the Abelian potential by L_ξ a+ du. The spinor transforms by L_ξψ+tfrac14λ^IJγ_IJψ+iquψ, with the dual action on barψ. All internal contractions in~[source reference] are inva","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d1f5eb93d8b633b4","tex":"\\mathcal L_\\xi^{(-1)}","aliases":"L_ξ^(-1) mathcal xi","display":"\\mathcal L_\\xi^{(-1)}","kind":"use","excerpt":"Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f52ec1f6f7fb5fa2","tex":"\\operatorname{lcp}","aliases":"lcp operatorname","display":"\\operatorname{lcp}","kind":"use","excerpt":"[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"ive strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ-1)/t|≤|ϵ|, proves lim_t→0fracU(t)f-ft=-iH_ϵ f. Thus the declared self-adjoint","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-436d6e601e9eab47","tex":"\\lim_r","aliases":"lim_r lim","display":"\\lim_r","kind":"use","excerpt":"Closedness follows from self-adjointness. It can also be seen directly for this multiplier. If a^(r)→ a and widehat H_ma^(r)→ b in KK, then convergence holds at each coordinate. Since the finite matrix H_n is continuous, b_n=lim_rH_na_n^(r)=H_na_n. The output sequence equals b∈KK, so a belongs to the maximal domain and widehat H_ma=b. This direct a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-244cdcc18a0e405f","tex":"\\lim_{r\\downarrow0}","aliases":"lim_rdownarrow0 lim downarrow","display":"\\lim_{r\\downarrow0}","kind":"use","excerpt":"Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1416c21d48d701c1","tex":"\\lim_{t\\to0}","aliases":"lim_t→0 lim to","display":"\\lim_{t\\to0}","kind":"use","excerpt":"ive strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ-1)/t|≤|ϵ|, proves lim_t→0fracU(t)f-ft=-iH_ϵ f. Thus the declared self-adjoint","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2a707ccad5a125ef","tex":"\\ll","aliases":"ll","display":null,"kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"tor has trace Tr(e^-β H_ϵ) =e^-βϵ_0 +∑_m≥1(p-1)p^m-1e^-βϵ_m, possibly infinite. This follows by summing its nonnegative diagonal entries in the complete Haar basis. It is trace class exactly when this series converges. For ϵ_m=c m with c>0, the condition is β c>log p, by the geometric-series test. For ϵ_m=p^α m with α>0, it converges for every β>0,","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"Tracing out the orthogonal b states yields the equally weighted mixture of the two displayed sign states. Their overlap is cos(gt). A mixture of two unit vectors with equal weights and overlap magnitude c has eigenvalues (1+c)/2 and (1-c)/2: its trace is one and its determinant on their two-dimensional span is (1-c^2)/4. Thus the sign entropy in bi","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass ter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-25d467671d09a670","tex":"\\lVert","aliases":"lVert","display":"\\lVert","kind":"use","excerpt":"The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm ter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ab13851c6dc5f1f","tex":"\\mathrm M","aliases":"M mathrm","display":"\\mathrm M","kind":"use","excerpt":"The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4977ee063fbe9a0d","tex":"\\mathrm m","aliases":"m mathrm","display":"\\mathrm m","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations spec","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ee5820332effab6","tex":"\\mathcal M'","aliases":"M' mathcal","display":"\\mathcal M'","kind":"use","excerpt":"[Exact triangular mass identity] For differentiable canonical data in the regular chart, M'=-fracR'Λ H -fracP_ΛRΛ D +frac QR G. Consequently, on a patch where R'/Λ≠0, the systems ( H, D, G)=0 and ( M', D, G)=0 have the same zero set. Differentiating [source reference] gives M'= · fracR'2-fracR'^32Λ^2 -fracRR'R''Λ^2+fracRR'^2Λ'Λ^3 +fracP_Λ P_Λ'R-fra","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a4c7fd7f94be16fe","tex":"m'","aliases":"m'","display":"m'","kind":"use","excerpt":"The momentum constraint removes P_R algebraically: P_R=P_Λ'-P_φΦ. Since m'=P_Λ P_Λ'/r-P_Λ^2/(2r^2), inserting this relation into the Hamiltonian constraint gives m'=r^2 E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ),","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca52a161c084be9c","tex":"M^{-1/2}","aliases":"M^-1/2","display":"M^{-1/2}","kind":"use","excerpt":"Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and se","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","excerpt":"Charged horizons and the turning surface. Write f(R)=1-frac2MR+fracQ^2R^2. For 0<|Q|<M, its two positive roots are r_±=M±sqrtM^2-Q^2. At either root p(R)^2=R^2, so the canonical quantities in [source reference] remain finite if F_r does. For |Q|=M, the two roots coincide at R=M. For |Q|>M there is no real root. At Q=0, the outer root is 2M and the ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"The exact first-order Fourier domain. The Clifford square identity sharpens the definition of the maximal domain. Since H_n^*=H_n and H_n^2=(m^2+|p(n)|^2)I_4, normH_nv^2 =innervH_n^*H_nv =(m^2+|p(n)|^2)normv^2. For the coefficient field this gives the exact equivalence a∈Dom(widehat H_m) Longleftrightarrow ∑_n(m^2+|p(n)|^2)norma_n^2<∞. The mass ter","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","excerpt":"Let M_0∈ SL_2(C) and x=trM_0. No diagonalizability assumption is needed. Cayley--Hamilton gives M_0^2-xM_0+I=0, hence M_0^n=U_n-1(x/2)M_0-U_n-2(x/2)I, n≥1, with U_-1=0. For n=1 it is the identity M_0=M_0, and for n=2 it is Cayley--Hamilton. Multiplying by M_0 and replacing M_0^2 by xM_0-I proves the induction step. Taking traces yields tr(M_0^n)=R_","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":"As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a0f711a1afb04d3f","tex":"M_0^2","aliases":"M_0^2","display":"M_0^2","kind":"use","excerpt":"Let M_0∈ SL_2(C) and x=trM_0. No diagonalizability assumption is needed. Cayley--Hamilton gives M_0^2-xM_0+I=0, hence M_0^n=U_n-1(x/2)M_0-U_n-2(x/2)I, n≥1, with U_-1=0. For n=1 it is the identity M_0=M_0, and for n=2 it is Cayley--Hamilton. Multiplying by M_0 and replacing M_0^2 by xM_0-I proves the induction step. Taking traces yields tr(M_0^n)=R_","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5fa7968cb2b9c241","tex":"m_0^2","aliases":"m_0^2","display":"m_0^2","kind":"use","excerpt":"=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z-σ(z))^2-k^2-m_0^2). Consequently any eigenv","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca9e6b144b03a15e","tex":"M_0^j","aliases":"M_0^j","display":"M_0^j","kind":"use","excerpt":"If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the requ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-af2fd5cefc0c8b88","tex":"M_0^n","aliases":"M_0^n","display":"M_0^n","kind":"use","excerpt":"Let M_0∈ SL_2(C) and x=trM_0. No diagonalizability assumption is needed. Cayley--Hamilton gives M_0^2-xM_0+I=0, hence M_0^n=U_n-1(x/2)M_0-U_n-2(x/2)I, n≥1, with U_-1=0. For n=1 it is the identity M_0=M_0, and for n=2 it is Cayley--Hamilton. Multiplying by M_0 and replacing M_0^2 by xM_0-I proves the induction step. Taking traces yields tr(M_0^n)=R_","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"For a general finite transfer matrix M_1, the residue-state identity v_Lq+j=M_1^j(M_1^L)^qv_0 follows by adding exponents. It needs no hypergeometric coefficients. A factorial lane provides a particular analytic coordinate system; it does not impose a unique microscopic Hamiltonian on every such transfer problem.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"[Weighted domain and mass independence] For every real m, Dom(widehat H_m) ={a∈KK: ∑_n(1+|p(n)|^2)norma_n^2<∞}. Consequently Dom(widehat H_m_1)=Dom(widehat H_m_2) for every two real masses. The weighted energy series is the sum of the nonnegative momentum series and m^2∑_nnorma_n^2. The second series is finite. Thus the energy series is finite exac","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b9396bdc88291f43","tex":"M_1^j","aliases":"M_1^j","display":"M_1^j","kind":"use","excerpt":"For a general finite transfer matrix M_1, the residue-state identity v_Lq+j=M_1^j(M_1^L)^qv_0 follows by adding exponents. It needs no hypergeometric coefficients. A factorial lane provides a particular analytic coordinate system; it does not impose a unique microscopic Hamiltonian on every such transfer problem.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c6d357730a0838d5","tex":"M_1^L","aliases":"M_1^L","display":"M_1^L","kind":"use","excerpt":"For a general finite transfer matrix M_1, the residue-state identity v_Lq+j=M_1^j(M_1^L)^qv_0 follows by adding exponents. It needs no hypergeometric coefficients. A factorial lane provides a particular analytic coordinate system; it does not impose a unique microscopic Hamiltonian on every such transfer problem.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-651ec01c3f6e67cb","tex":"m_2","aliases":"m_2","display":"m_2","kind":"use","excerpt":"[Weighted domain and mass independence] For every real m, Dom(widehat H_m) ={a∈KK: ∑_n(1+|p(n)|^2)norma_n^2<∞}. Consequently Dom(widehat H_m_1)=Dom(widehat H_m_2) for every two real masses. The weighted energy series is the sum of the nonnegative momentum series and m^2∑_nnorma_n^2. The second series is finite. Thus the energy series is finite exac","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6b34051ea3264f33","tex":"M_D","aliases":"M_D","display":"M_D","kind":"use","excerpt":"nsfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary divergence. The detail Hessian is M_D=x_1^2+m · -m -m · m+n, D=(m+n)x_1^2+mn>0. Writi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7d6064a1e2fd6d56","tex":"M_G","aliases":"M_G","display":"M_G","kind":"use","excerpt":"Define the densities H_G · =P_τ^2+γ P_θ^2 +e^2P(Q_τ^2+γ Q_θ^2),nonumber M_G · =2(P_τ P_θ+e^2PQ_τ Q_θ). The remaining reconstruction equations are λ_τ=H_G and λ_θ=M_G.","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6f65dba0db58365e","tex":"M_{G,h}","aliases":"M_G,h","display":"M_{G,h}","kind":"use","excerpt":"The local compatibility diagnostic is a separate quantity. Define H_G,h,M_G,h by replacing spatial derivatives in [source reference] with D_h. If E_P,h=dot U-dot U_rm raw and E_Q,h=dot V-dot V_rm raw, then I_h=∂_τ M_G,h-D_hH_G,h -2(D_hP) E_P,h -2e^2P(D_hQ) E_Q,h subtracts the explicit constraint-force contribution. Exact global momentum preservatio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-29c7b0251fbf7073","tex":"m_H","aliases":"m_H","display":"m_H","kind":"use","excerpt":"For N_r=2400, the interpolated event has t_H=4.69182094580, r_H=3.11341154797, m_H=r_H/2=1.55670577399. The relation m_H=r_H/2 is imposed by the interpolated marginal-sphere criterion. It is not an independent numerical conservation test.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c879cfab2bd64d1b","tex":"m_j","aliases":"m_j","display":"m_j","kind":"use","excerpt":"limit-strong-resolvent. If the absolute energies tend to infinity, the right side of [source reference] tends to zero. The retained part of the resolvent has finite rank, so the resolvent is a norm limit of finite-rank operators and is compact. Conversely, if the absolute energies do not tend to infinity, there are infinitely many distinct levels m","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c47f1cec87420595","tex":"M_{p,h}","aliases":"M_p,h","display":"M_{p,h}","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8640417ce531dc32","tex":"M_{p,h}^{\\ell}","aliases":"M_p,h^ℓ ell","display":"M_{p,h}^{\\ell}","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dbd15cd98c87c0e2","tex":"M_q","aliases":"M_q","display":"M_q","kind":"use","excerpt":"For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-986d5a7882d8f884","tex":"M_{\\rm enc}","aliases":"M_rm enc rm","display":"M_{\\rm enc}","kind":"use","excerpt":"Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ca9432f255742492","tex":"\\mathring","aliases":"mathring","display":null,"kind":"use","excerpt":"ebraic torsion and the reduced phase space For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. I","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"Let A be the coordinate space {0,1,2,3}→ C with maximum norm v _∞=max_r|v_r|, and let H be the same coordinates with the Euclidean inner product. The map","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c52aaa85647b7be3","tex":"\\max_r","aliases":"max_r max","display":"\\max_r","kind":"use","excerpt":"Let A be the coordinate space {0,1,2,3}→ C with maximum norm v _∞=max_r|v_r|, and let H be the same coordinates with the Euclidean inner product. The map","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8c09df9956ae55f5","tex":"\\max_{x\\in S}","aliases":"max_x∈ S max in","display":"\\max_{x\\in S}","kind":"use","excerpt":"[Constant-one directional mismatch bound] Let A be a Hermitian Clifford involution, τ a site bijection, and u,v:S→ U(1). Then normS_A,τ,v-S_A,τ,u ≤δ(u,v):=max_x∈ S|v(x)-u(x)| =max_x∈ S|v(x)u(x)^-1-1|. The constant is independent of the number of sites. The actual half-grid defect in [source reference], with half-cell Hilbert measure, obeys the same","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f7ed41372efc24c8","tex":"\\mathrm{mod}","aliases":"mod mathrm","display":"\\mathrm{mod}","kind":"use","excerpt":"For 0≤ r≤ h, define the cylinder average (E_r^(h)f)(x)=p^-(h-r) ∑_y≡ x (mod p^r)f(y). Its matrix is real and symmetric. Averaging twice over the same cylinder has the same effect as averaging once. Thus E_r^(h) is the orthogonal projector onto functions constant on level-r cylinders. The range has dimension p^r.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N","display":null,"kind":"use","excerpt":"Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"use","excerpt":"regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c947cbb5e27a6a1d","tex":"\\N^{\\mathcal P}","aliases":"N^ P N mathcal","display":null,"kind":"use","excerpt":"Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f40bbcca7096796b","tex":"\\mathcal N^{-1}","aliases":"N^-1 mathcal","display":"\\mathcal N^{-1}","kind":"use","excerpt":"regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b75f3a902c5f7e60","tex":"n^{-1}","aliases":"n^-1","display":"n^{-1}","kind":"use","excerpt":"After w refinements a packet has n=p^w orthonormal descendants. Its normalized radial vector is |R_p,w⟩=n^-1/2∑_ d| d⟩. The norm is n^-1∑_ d1=1, so each descendant has amplitude modulus p^-w/2 and Born probability p^-w. This reproduces [source reference]. The words ``radial'' and ``angular'' here refer to the uniform vector and its orthogonal compl","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5a1fe216a765aa7c","tex":"N^{-1/2}","aliases":"N^-1/2","display":"N^{-1/2}","kind":"use","excerpt":"bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is congruent to k modulo both a and b,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-79616ac402fc2170","tex":"n^{-1/2}","aliases":"n^-1/2","display":"n^{-1/2}","kind":"use","excerpt":"We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit g","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c0769f9271add232","tex":"n^{-2}","aliases":"n^-2","display":"n^{-2}","kind":"use","excerpt":"foundationfigure[H]11_graph_tailsFor the normalized state proportional to n^-2 and He_n=ne_n, the state tail and generator tail decay at different rates. Graph control retains both contributions.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2390514d28436a26","tex":"N^{-3}","aliases":"N^-3","display":"N^{-3}","kind":"use","excerpt":"e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-acb2c967f2e3c2e2","tex":"N^{-3/2}","aliases":"N^-3/2","display":"N^{-3/2}","kind":"use","excerpt":"For the faster normalized tail c_n=C/n^2, both errors are controlled: normc-S_Nc^2=C^2∑_n>Nn^-4, normH_λ c-S_NH_λ c^2=C^2∑_n>Nn^-2. The state tail is O(N^-3/2) in norm, while the velocity tail is O(N^-1/2). A small state error can therefore coexist with a much larger derivative error even inside the true domain.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-55d7cf56037490a0","tex":"n^{-4}","aliases":"n^-4","display":"n^{-4}","kind":"use","excerpt":"For the faster normalized tail c_n=C/n^2, both errors are controlled: normc-S_Nc^2=C^2∑_n>Nn^-4, normH_λ c-S_NH_λ c^2=C^2∑_n>Nn^-2. The state tail is O(N^-3/2) in norm, while the velocity tail is O(N^-1/2). A small state error can therefore coexist with a much larger derivative error even inside the true domain.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-38855ce590112f03","tex":"n^{-s}","aliases":"n^-s","display":"n^{-s}","kind":"use","excerpt":"For positive λ_n, the quadratic-form domain instead uses ∑_nλ_n|c_n|^2<∞. This is the domain of H_λ^1/2, not generally the domain of H_λ. With λ_n=n and c_n=n^-s, the three thresholds are c|c membership · condition","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"bottom parameters are positive and the ratio test again gives entire convergence. Character filtering reconstructs B=∑_jB_L,j. With the Euler derivative Θ=y d/dy, termwise differentiation also gives the cyclic system Θ^2B_L,j(y)=-yB_L,j-1bmod L(y). For a term of positive degree n, its new coefficient is n^2(-1)^n/(n!)^2=-(-1)^n-1/[(n-1)!]^2. The de","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","excerpt":"S={n_0,n_A,n_D}, n_0=(0,0,0), n_A=(1,0,0), n_D=(1,1,1).","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b6b337ece4924770","tex":"n_2","aliases":"n_2","display":"n_2","kind":"use","excerpt":"equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b69db358ab95952d","tex":"n_3","aliases":"n_3","display":"n_3","kind":"use","excerpt":"equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b9de340bc0b2a993","tex":"n_A","aliases":"n_A","display":"n_A","kind":"use","excerpt":"S={n_0,n_A,n_D}, n_0=(0,0,0), n_A=(1,0,0), n_D=(1,1,1).","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-64d3fd3e91a16874","tex":"n_a","aliases":"n_a","display":"n_a","kind":"use","excerpt":"P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is c","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-de5a861749813249","tex":"n_b","aliases":"n_b","display":"n_b","kind":"use","excerpt":"P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is c","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-71f77540e0e9914c","tex":"n_D","aliases":"n_D","display":"n_D","kind":"use","excerpt":"S={n_0,n_A,n_D}, n_0=(0,0,0), n_A=(1,0,0), n_D=(1,1,1).","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"Set T_h=innerππ_h-tfrac12innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e1613ab8834072c1","tex":"N_j","aliases":"N_j","display":"N_j","kind":"use","excerpt":"Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are neede","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic g","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-946f0d537d4b4f30","tex":"N_{j+1/2}","aliases":"N_j+1/2","display":"N_{j+1/2}","kind":"use","excerpt":"Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are neede","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf9404d3f534ee7e","tex":"n_k","aliases":"n_k","display":"n_k","kind":"use","excerpt":"The trace of the momentum part of dot h is -π_rm tr. Its contribution to dot C is therefore 2δ_iδ_jπ_ij=-δ_iC_i. The shift contribution is δ_iδ_j(δ_in_j+δ_jn_i) -δ^2(2δ_kn_k)=0. For dot C_i, the divergence of G vanishes by Lemma~[source reference], and δ_j(δ_iδ_jn-δ_ijδ^2n)=0. This proves the propagation equations and their stated implication.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-75599878a7100180","tex":"N_r","aliases":"N_r","display":"N_r","kind":"use","excerpt":"Recorded first marginally trapped sphere. The event finder takes a quadratic interpolation of the spatial maximum of 2m/r and a linear interpolation between consecutive time steps bracketing its first crossing of one. The archived event table is small rrrrr N_r · h · t_H · r_H · Δ M/M_0","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1dad9952c8e86d69","tex":"\\mathcal N_{\\rm emit}","aliases":"N_rm emit mathcal rm","display":"\\mathcal N_{\\rm emit}","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8e74a14211720a9d","tex":"\\mathcal N_{\\rm obs}","aliases":"N_rm obs mathcal rm","display":"\\mathcal N_{\\rm obs}","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0ea5b5c0e57e64fb","tex":"\\N_{\\geq1}","aliases":"N_≥1 N geq","display":null,"kind":"use","excerpt":"[A small state can have a large generator image] On ℓ^2(N_≥1), let He_n=ne_n on its maximal domain. The finitely supported vectors x_N=e_N/N satisfy normx_N=1/N→0, while normHx_N=1. This gives a direct reason to track the generator tail in addition to the state tail. It does not contradict closedness, because Hx_N=e_N has no norm limit.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup gene","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"For a self-adjoint fine Hamiltonian with coarse projector P, set B=QH_fP. If a unit vector ψ lies in Ran P, its initial detail amplitude is Qe^-itH_fψ=-itBψ+O(t^2), normQe^-itH_fψ^2 =t^2normBψ^2+O(|t|^3). These are finite Taylor expansions for fixed H_f and ψ. Thus a nonzero B supplies a state with nonzero short-time leakage. The two-dimensional ex","count":48},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50cad36890440685","tex":"\\mathrm o","aliases":"o mathrm","display":"\\mathrm o","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"xpansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This i","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b77c3e9c22499671","tex":"\\oint","aliases":"oint","display":null,"kind":"use","excerpt":"We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt alon","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-95ca3968ddfe525b","tex":"\\oint_{\\partial\\square}","aliases":"oint_∂square oint partial square","display":null,"kind":"use","excerpt":"We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt alon","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-754626c55688277f","tex":"\\operatorname{or}","aliases":"or operatorname","display":"\\operatorname{or}","kind":"use","excerpt":"For scalar links, multiply the oriented plaquettes on the six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact gro","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-44a55b48f8dea448","tex":"\\mathrm{out}","aliases":"out mathrm","display":"\\mathrm{out}","kind":"use","excerpt":"The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify th","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-01eecdf5cc4bca8b","tex":"\\overline A","aliases":"overline A overline","display":"\\overline A","kind":"use","excerpt":"The Lorentzian limitation of this self-dual chart is also explicit. A single Lorentzian self-dual connection is complex, with a physical sector requiring real E and A+overline A=2Γ(E). The source chapter does not construct a finite step preserving that sector. The real parent below addresses reality by changing variables and retaining both chiral s","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e83fbd63131abb69","tex":"\\overline f","aliases":"overline f overline","display":"\\overline f","kind":"use","excerpt":"Fix a prime p. The compact additive group Z_p has normalized Haar measure μ_p, with B_m,a=a+p^mZ_p, 0≤ a<p^m, μ_p(B_m,a)=p^-m. Write HH_p=L^2(Z_p,μ_p;C). We use the complex inner product innerfg=∫overline f g dμ_p, linear in its second argument. The p-adic coordinate belongs to the measurable space. Amplitudes and the Born probabilities use ordinar","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9b0ad0f92e6daea0","tex":"\\overline G","aliases":"overline G overline","display":"\\overline G","kind":"use","excerpt":"Assume those invariance conditions and order the basis as (|c⟩,|R_p,w⟩). With visible energy E_c and radial energy ϵ, the exact bright block is H_bright= E_c · overline G_p,w G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_brig","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5f51cdcfd251a311","tex":"\\overline g","aliases":"overline g overline","display":"\\overline g","kind":"use","excerpt":"E_m^*= E_m, E_m^2= E_m, E_m E_n= E_min(m,n). Moreover E_m f→ f in L^2 for every f∈HH_p. For a cylinder-constant g, the integral of f- E_m f over each depth-m cylinder is zero. Multiplying by the constant value of overline g there and summing shows innergf- E_mf=0. The image is in V_m, and averaging a function in V_m returns the function. These stat","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-69a71f8a9a72cb1e","tex":"\\overline G_{p,w}","aliases":"overline G_p,w overline","display":"\\overline G_{p,w}","kind":"use","excerpt":"Assume those invariance conditions and order the basis as (|c⟩,|R_p,w⟩). With visible energy E_c and radial energy ϵ, the exact bright block is H_bright= E_c · overline G_p,w G_p,w · ϵ. Set overline E=(E_c+ϵ)/2, δ=(E_c-ϵ)/2, and Ω=(δ^2+|G_p,w|^2)^1/2. Then (H_bright-overline E I)^2=Ω^2I, giving e^-itH_bright =e^-itoverline E [Icos(Ω t) -ifracH_brig","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc6c360a7accd337","tex":"\\overline{G_{p,w}}","aliases":"overlineG_p,w overline","display":"\\overline{G_{p,w}}","kind":"use","excerpt":"c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt n γ_p,w. Then H_int|c⟩=G_p,w|R_p,w⟩ and H_int|R_p,w⟩= overlineG_p,w|c⟩. If |A⟩=∑_ da_ d| d⟩ is orthogonal to |R_p,w⟩, its coefficients satisfy ∑_ da_ d=0. Hence H_int|A⟩=0. Only the radial vector is bright u","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0a76f689d811f120","tex":"\\overline{\\inner{\\psi_b}{K\\psi_a}}","aliases":"overlineinnerψ_bKψ_a overline inner psi psi","display":null,"kind":"use","excerpt":"· =∑_a2Reinnerψ_bKψ_a. The reconstruction of ψ justifies the last finite sum. Skew-adjointness gives innerψ_aKψ_b =-innerKψ_aψ_b =-overlineinnerψ_bKψ_a. Taking real parts proves antisymmetry. Putting a=b gives zero diagonal current. Every internal ordered pair in C× C cancels with its reverse. Splitting the sum over all sources into C and its compl","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ec76d89d14f0c305","tex":"\\overline{P_+X}","aliases":"overlineP_+X overline","display":"\\overline{P_+X}","kind":"use","excerpt":"Reality is built into e and ω. For real Lorentz bivectors, the internal dual obeys ⋆^2=-id and the projectors P_±=tfrac12(id∓ i⋆) satisfy overlineP_+X=P_-X. Both chiral blocks are retained and related by conjugation. A real vector field with an existing unique flow preserves this real locus. This parent-level statement supplies a real classical sta","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0a2674f534bda381","tex":"\\overline{v_r}","aliases":"overlinev_r overline","display":"\\overline{v_r}","kind":"use","excerpt":"⟨ v,w⟩=∑_r=0^3overlinev_rw_r, v _2^2=∑_r=0^3|v_r|^2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-739fbcd12238ffb3","tex":"\\overline{z_{ba}}","aliases":"overlinez_ba overline","display":"\\overline{z_{ba}}","kind":"use","excerpt":"Let z_ba=Tr(P_bHP_aρ). Taking the conjugate trace and cyclically moving the first factor to the end yields overlinez_ba=Tr(ρ P_aHP_b) =Tr(P_aHP_bρ)=z_ab. Completeness and cyclicity resolve the derivative as dot p_b=-i∑_a(z_ba-z_ab) =∑_a2Imz_ba. The conjugate relation gives antisymmetry. Finite sums then prove cancellation and coarse aggregation exa","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0b2f408b058396fe","tex":"\\overline{\\{\\epsilon_m:m\\geq0\\}}","aliases":"overline{ϵ_m:m≥0} overline epsilon geq","display":"\\overline{\\{\\epsilon_m:m\\geq0\\}}","kind":"use","excerpt":"[Real level energies define a self-adjoint operator] The operator in [source reference] is densely defined, closed, and self-adjoint. The finite cylinder functions form an operator core. It is bounded if and only if sup_m|ϵ_m|<∞, and its spectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the squa","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c3a4a38379b7068","tex":"\\overline\\gamma","aliases":"overlineγ overline gamma","display":"\\overline\\gamma","kind":"use","excerpt":"se projector P, set B=QH_fP. If a unit vector ψ lies in Ran P, its initial detail amplitude is Qe^-itH_fψ=-itBψ+O(t^2), normQe^-itH_fψ^2 =t^2normBψ^2+O(|t|^3). These are finite Taylor expansions for fixed H_f and ψ. Thus a nonzero B supplies a state with nonzero short-time leakage. The two-dimensional example H_f=0 · overlineγ γ · 0, P=1 · 0 0 · 0 ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e4c3e0b2110e648","tex":"\\overline{\\gamma_{p,w}}","aliases":"overlineγ_p,w overline gamma","display":"\\overline{\\gamma_{p,w}}","kind":"use","excerpt":"Introduce a visible address |c⟩ orthogonal to all descendants. Suppose every descendant has the same coupling from |c⟩, with complex matrix element γ_p,w. The Hermitian interaction is H_int= ∑_ d (γ_p,w| d⟩⟨ c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-55dc76c431a12088","tex":"\\overline\\psi","aliases":"overlineψ overline psi","display":"\\overline\\psi","kind":"use","excerpt":"al Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-74afe80986b7b88b","tex":"\\overline\\psi_j","aliases":"overlineψ_j overline psi","display":"\\overline\\psi_j","kind":"use","excerpt":"al Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aab687f6768e37ac","tex":"\\overline\\nabla","aliases":"overline∇ overline nabla","display":"\\overline\\nabla","kind":"use","excerpt":"hart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient o","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-57fcee80b46c2130","tex":"\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}","aliases":"overline⟨ψ_r,H_rsψ_s⟩ overline langle psi psi rangle","display":"\\overline{\\langle\\psi_r,H_{rs}\\psi_s\\rangle}","kind":"use","excerpt":"al J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. The currents therefore cancel in pairs, including a zero diagonal current. Here w_r is a pr","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-627237205a8ef32d","tex":"\\mathrm P","aliases":"P mathrm","display":"\\mathrm P","kind":"use","excerpt":"the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Different","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-394515d30df05582","tex":"p^{\\mathrm c}","aliases":"p^ c mathrm","display":"p^{\\mathrm c}","kind":"use","excerpt":"For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9b5512f13515252c","tex":"p^{(1-2\\alpha)w}","aliases":"p^(1-2α)w alpha","display":"p^{(1-2\\alpha)w}","kind":"use","excerpt":"Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf inter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e3c49b631652076","tex":"p^{(1/2-\\alpha)w}","aliases":"p^(1/2-α)w alpha","display":"p^{(1/2-\\alpha)w}","kind":"use","excerpt":"Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf inter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d0b51e771199536c","tex":"p^{(m+1)/2}","aliases":"p^(m+1)/2","display":"p^{(m+1)/2}","kind":"use","excerpt":"The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cfe2afc69712b120","tex":"p^{(m-1)/2}","aliases":"p^(m-1)/2","display":"p^{(m-1)/2}","kind":"use","excerpt":"For an explicit basis, put ω_p=exp(2π i/p) and define ψ_m,a,q=p^(m-1)/2∑_d=0^p-1 ω_p^qd1_B_m,a+d p^m-1, m≥1, 0≤ a<p^m-1, 1≤ q<p. Each has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6feb1c26b70e6413","tex":"P^+_A","aliases":"P^+_A","display":"P^+_A","kind":"use","excerpt":"The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let A=A^* and A^2=I on a finite spinor space. Then P^+_A=tfrac12(I+A), P^-_A=tfrac12(I-A) are self-adjoint complementary orthogonal projections. Idempotence and P_A^+P_A^-=0 follow by expanding the products and using A^2=I; self-adjointness ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d61400e021ea8cb0","tex":"p^{-(h-r)}","aliases":"p^-(h-r)","display":"p^{-(h-r)}","kind":"use","excerpt":"For 0≤ r≤ h, define the cylinder average (E_r^(h)f)(x)=p^-(h-r) ∑_y≡ x (mod p^r)f(y). Its matrix is real and symmetric. Averaging twice over the same cylinder has the same effect as averaging once. Thus E_r^(h) is the orthogonal projector onto functions constant on level-r cylinders. The range has dimension p^r.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0e19dd0107059fb7","tex":"p^{-(h-r+1)}","aliases":"p^-(h-r+1)","display":"p^{-(h-r+1)}","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determine","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-050b78fe0bbad3ec","tex":"p^{-(r+1)/2}","aliases":"p^-(r+1)/2","display":"p^{-(r+1)/2}","kind":"use","excerpt":"dots,h. They are orthonormal because different levels are disjoint and level r contains p^r vertices. A direct action calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/2. At each parent at level r-1, the p child contributi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aedd6029b023e559","tex":"p^{-(r-1)/2}","aliases":"p^-(r-1)/2","display":"p^{-(r-1)/2}","kind":"use","excerpt":"on calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/2. At each parent at level r-1, the p child contributions instead give p p^-r/2γ_r-1/sqrt p =γ_r-1p^-(r-1)/2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e57276c69cbfb04f","tex":"p^{-1}","aliases":"p^-1","display":"p^{-1}","kind":"use","excerpt":"Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d718e07c40dc0d2e","tex":"p^{-1/2}","aliases":"p^-1/2","display":"p^{-1/2}","kind":"use","excerpt":"One more digit supplies p children for every address. The uniform refinement map is (V_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-054866e9b11e6acb","tex":"p^{-2w}","aliases":"p^-2w","display":"p^{-2w}","kind":"use","excerpt":"A model that directly couples |c⟩ to one normalized hidden packet with matrix element g p^-w instead has a collective coefficient of that same size and a self-energy numerator |g|^2p^-2w. It is a valid two-level model, but it is not the equal-leaf model [source reference]. The number of descendant addresses alone does not select between these inter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8dec76303c90ddfc","tex":"P^-_A","aliases":"P^-_A","display":"P^-_A","kind":"use","excerpt":"The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let A=A^* and A^2=I on a finite spinor space. Then P^+_A=tfrac12(I+A), P^-_A=tfrac12(I-A) are self-adjoint complementary orthogonal projections. Idempotence and P_A^+P_A^-=0 follow by expanding the products and using A^2=I; self-adjointness ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-570d98a38290a283","tex":"p^{-h}","aliases":"p^-h","display":"p^{-h}","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determine","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d044418aad4b95c8","tex":"p^{-h/2}","aliases":"p^-h/2","display":"p^{-h/2}","kind":"use","excerpt":"0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8a1b7e2cb35dff5a","tex":"p^{-m}","aliases":"p^-m","display":"p^{-m}","kind":"use","excerpt":"Fix a prime p. The compact additive group Z_p has normalized Haar measure μ_p, with B_m,a=a+p^mZ_p, 0≤ a<p^m, μ_p(B_m,a)=p^-m. Write HH_p=L^2(Z_p,μ_p;C). We use the complex inner product innerfg=∫overline f g dμ_p, linear in its second argument. The p-adic coordinate belongs to the measurable space. Amplitudes and the Born probabilities use ordinar","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cfebc46ff80f9645","tex":"p^{-r/2}","aliases":"p^-r/2","display":"p^{-r/2}","kind":"use","excerpt":"Define normalized level states |r⟩=p^-r/2∑_|v|=r|v⟩, r=0,…,h. They are orthonormal because different levels are disjoint and level r contains p^r vertices. A direct action calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-46695d5ac294b8b0","tex":"p^{-s(M+1)}","aliases":"p^-s(M+1)","display":"p^{-s(M+1)}","kind":"use","excerpt":"[Exact tail and regularity bound] For f∈HH_p and every M≥0, norm(I- E_M)f_2^2 =∑_m>Mnorm F_mf_2^2. If f∈ H_p^s, then norm(I- E_M)f_2 ≤ p^-s(M+1)normf_ H_p^s. The convergence E_M→ I is strong and is not convergence in operator norm. The omitted details are orthogonal, so Parseval gives the equality. For m>M, p^2sm≥ p^2s(M+1). Multiplying the omitted","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fd6c590f56fc7427","tex":"p^{-s/2}","aliases":"p^-s/2","display":"p^{-s/2}","kind":"use","excerpt":"There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite en","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d2c543a54857b35f","tex":"p^{-w}","aliases":"p^-w","display":"p^{-w}","kind":"use","excerpt":"After w refinements a packet has n=p^w orthonormal descendants. Its normalized radial vector is |R_p,w⟩=n^-1/2∑_ d| d⟩. The norm is n^-1∑_ d1=1, so each descendant has amplitude modulus p^-w/2 and Born probability p^-w. This reproduces [source reference]. The words ``radial'' and ``angular'' here refer to the uniform vector and its orthogonal compl","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4618987e95ac91f8","tex":"p^{-w/2}","aliases":"p^-w/2","display":"p^{-w/2}","kind":"use","excerpt":"D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-40ea66b34b55a33f","tex":"p^{-\\alpha w}","aliases":"p^-α w alpha","display":"p^{-\\alpha w}","kind":"use","excerpt":"Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf inter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3100e6c13d0c66d3","tex":"p^{2s(M+1)}","aliases":"p^2s(M+1)","display":"p^{2s(M+1)}","kind":"use","excerpt":"a-limit-tail-identity If f∈ H_p^s, then norm(I- E_M)f_2 ≤ p^-s(M+1)normf_ H_p^s. The convergence E_M→ I is strong and is not convergence in operator norm. The omitted details are orthogonal, so Parseval gives the equality. For m>M, p^2sm≥ p^2s(M+1). Multiplying the omitted summands by this lower bound proves the squared estimate, and taking square ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0000c447057fb504","tex":"p^{2sm}","aliases":"p^2sm","display":"p^{2sm}","kind":"use","excerpt":"For s>0, define the Haar regularity norm normf_ H_p^s^2 =∑_m≥0p^2smnorm F_mf_2^2, H_p^s={f∈HH_p:normf_ H_p^s<∞}. No identification with another Sobolev convention is required here. The definition itself gives the quantitative cutoff estimate.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df7ab5771f41cb6f","tex":"p^A","aliases":"p^A","display":"p^A","kind":"use","excerpt":"For a product state ρ_AB=σ_A⊗τ_B, the trace factorizes and p_aμ=p_a^Ap_μ^B. General states need not have that property. The normalized vector Ψ=(e_0⊗ e_0+e_1⊗ e_1)/sqrt2 has joint probabilities (p_aμ)=1/2 · 0 0 · 1/2, p^A=p^B=(1/2,1/2). The product density matrix (id_A/2)⊗(id_B/2) has the same marginals but every joint entry is 1/4. Marginals there","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6c2132e901148066","tex":"p^B","aliases":"p^B","display":"p^B","kind":"use","excerpt":"For a product state ρ_AB=σ_A⊗τ_B, the trace factorizes and p_aμ=p_a^Ap_μ^B. General states need not have that property. The normalized vector Ψ=(e_0⊗ e_0+e_1⊗ e_1)/sqrt2 has joint probabilities (p_aμ)=1/2 · 0 0 · 1/2, p^A=p^B=(1/2,1/2). The product density matrix (id_A/2)⊗(id_B/2) has the same marginals but every joint entry is 1/4. Marginals there","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7f4968674b99b5fb","tex":"p^b","aliases":"p^b","display":"p^b","kind":"use","excerpt":"its fixed ancestral coefficient, on all children below the lower vertex at a common level. The lower chain has zero sum over its own child contrasts, so their inner product is zero. The same zero-sum calculation makes every angular vector orthogonal to the uniform radial level vector. At a fixed depth b, the radial vector contributes one dimension.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0edaee8e5b48a0f0","tex":"p^h","aliases":"p^h","display":"p^h","kind":"use","excerpt":"[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ, (s,a) · ⟼(-1)^s5^apmod2^h+2, · (s,a) · ∈Z/2Z×Z/2^hZ are group isomorphisms onto U_p,h and U_2,h respectively.","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c872c2116fdfaa92","tex":"p^{h+1}","aliases":"p^h+1","display":"p^{h+1}","kind":"use","excerpt":"The first coordinate issue is specific to units. For an odd prime p and an integer h≥0, set U_p,h=(Z/p^h+1Z)^×. For p=2 the useful indexing starts one digit later: U_2,h=(Z/2^h+2Z)^×. The shifted dyadic index retains the sign even when the principal phase register has dimension one. It prevents the two units modulo four from being collapsed into on","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9952deedd172b29f","tex":"p^{h-1}","aliases":"p^h-1","display":"p^{h-1}","kind":"use","excerpt":"For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8afb4c38411cb8b1","tex":"p^{h-2}","aliases":"p^h-2","display":"p^{h-2}","kind":"use","excerpt":"-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h-1p^h-1 makes d_h-1 the least significant digit. Standard analysis removes that digit, and reversing the remaining h-1 digits restores the prefix a=d_0+⋯+d_h-2p^h-2. This proves the identity on an orthonormal basis.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-220bbebb58967375","tex":"p^{h-k-1}","aliases":"p^h-k-1","display":"p^{h-k-1}","kind":"use","excerpt":"x≡ y (mod p^r-1) ]. Every indicator is determined by the truncated valuation min{v_p(x-y),h}, with the diagonal assigned value h. The transition amplitude, and therefore the individual-address Born probability, is constant on each such valuation shell. At a fixed initial address y, the shell of valuation k<h contains (p-1)p^h-k-1 final addresses. I","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-be94f0538ad303b4","tex":"p^{h-r}","aliases":"p^h-r","display":"p^{h-r}","kind":"use","excerpt":"The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least si","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f19099897aa002b9","tex":"p^{h-r+1}","aliases":"p^h-r+1","display":"p^{h-r+1}","kind":"use","excerpt":"Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-351cda8cd86b95b8","tex":"p^{h_p}","aliases":"p^h_p","display":"p^{h_p}","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b9a485297aa071aa","tex":"p^{h_p+1}","aliases":"p^h_p+1","display":"p^{h_p+1}","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7f360bb2799a8d5","tex":"p^k","aliases":"p^k","display":"p^k","kind":"use","excerpt":"For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup gene","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f21691df4ee7e60b","tex":"p^{k+1}","aliases":"p^k+1","display":"p^{k+1}","kind":"use","excerpt":"For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup gene","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc82ec356cf9d9b6","tex":"p^{k+2}","aliases":"p^k+2","display":"p^{k+2}","kind":"use","excerpt":"For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup gene","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4927a5c0083e2327","tex":"p^{m-1}","aliases":"p^m-1","display":"p^{m-1}","kind":"use","excerpt":"Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ab24a79a4655574e","tex":"p^{m/2}","aliases":"p^m/2","display":"p^{m/2}","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d3cdc9ecc001d6c4","tex":"p^n","aliases":"p^n","display":"p^n","kind":"use","excerpt":"For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the requir","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c020cb0886702b18","tex":"p^{n+1}","aliases":"p^n+1","display":"p^{n+1}","kind":"use","excerpt":"For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the requir","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6f0ea21bbfbf69fb","tex":"p^r","aliases":"p^r","display":"p^r","kind":"use","excerpt":"The principal phase register has a canonical finite filtration. For a prime p and depth h≥0, set G_p,h=Z/p^hZ, H_p,h=ℓ^2(G_p,h;C). At depth zero this is a one-dimensional space. A level-r cylinder contains all residues with a prescribed prefix modulo p^r, and has p^h-r members. Throughout this section depth refers to the number of retained least si","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1fc0b92fec83c631","tex":"p^{\\rm c}_\\beta","aliases":"p^rm c_β rm beta","display":"p^{\\rm c}_\\beta","kind":"use","excerpt":"If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β r(a)=αJ^F_ba. No orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to J^F without changing its row sums. Thus the balance law determines the chan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-878e0df7265da180","tex":"p^{w/2}","aliases":"p^w/2","display":"p^{w/2}","kind":"use","excerpt":"Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf inter","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9377ed70d686e166","tex":"p^{\\alpha m}","aliases":"p^α m alpha","display":"p^{\\alpha m}","kind":"use","excerpt":"-β H_ϵ) =e^-βϵ_0 +∑_m≥1(p-1)p^m-1e^-βϵ_m, possibly infinite. This follows by summing its nonnegative diagonal entries in the complete Haar basis. It is trace class exactly when this series converges. For ϵ_m=c m with c>0, the condition is β c>log p, by the geometric-series test. For ϵ_m=p^α m with α>0, it converges for every β>0, since the exponent","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-832eb479480de132","tex":"\\mathcal P_{\\mathrm o}","aliases":"P_ o mathcal mathrm","display":"\\mathcal P_{\\mathrm o}","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7cd48fb48ab38484","tex":"p_*","aliases":"p_*","display":"p_*","kind":"use","excerpt":"A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞, normδ P_Λ_∞ · ≤frac Mp_*normδ R_∞, normδ P_R_∞ · ≤frac Mp_*normD_4δ R_∞ +fracM^2d^*p_*^3normδ ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-25581a5c9fb5bdcb","tex":"p_*^2","aliases":"p_*^2","display":"p_*^2","kind":"use","excerpt":"A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-83f42125c2adbdfd","tex":"p_*^3","aliases":"p_*^3","display":"p_*^3","kind":"use","excerpt":"A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞, normδ P_Λ_∞ · ≤frac Mp_*normδ R_∞, normδ P_R_∞ · ≤frac Mp_*normD_4δ R_∞ +fracM^2d^*p_*^3normδ ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"0. Direct differentiation verifies the solution ψ(t)=cos(gt) -isin(gt), p_1=cos^2(gt), p_2=sin^2(gt). The current is J_2←1 =2Im(i gsin(gt)cos(gt)) =gsin(2gt)=dot p_2. It reverses sign during the cycle. At t=0 the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at s","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"0. Direct differentiation verifies the solution ψ(t)=cos(gt) -isin(gt), p_1=cos^2(gt), p_2=sin^2(gt). The current is J_2←1 =2Im(i gsin(gt)cos(gt)) =gsin(2gt)=dot p_2. It reverses sign during the cycle. At t=0 the coupling is nonzero but the instantaneous current is zero, since the destination amplitude vanishes. The population begins to change at s","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f21822f96efaf869","tex":"P_A","aliases":"P_A","display":"P_A","kind":"use","excerpt":"Suppose P_A=∑_a∈ C(A)P_a and P_B=∑_b∈ C(B)P_b. Then P_BLP_A=∑_b∈ C(B)∑_a∈ C(A)P_bLP_a. The identity follows by distributing two finite sums. It says exactly which fine transfers a coarse observation retains. A scalar summary or a diagonal-only summary may retain less information and therefore needs its own loss statement.","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e7d1a3be90d27945","tex":"p_A","aliases":"p_A","display":"p_A","kind":"use","excerpt":"[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fc3eca2c0cc0fb84","tex":"p_A^{\\mathrm c}","aliases":"p_A^ c mathrm","display":"p_A^{\\mathrm c}","kind":"use","excerpt":"[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4338c025955f4745","tex":"P_a^{(0)}","aliases":"P_a^(0)","display":"P_a^{(0)}","kind":"use","excerpt":"A resolution can change with time even when the state is stationary. To retain orthogonality, transport the entire atlas by a unitary family. Let U(t) be differentiable in operator norm and satisfy both U^*U=id and UU^*=id. Set P_a(t)=U(t)P_a^(0)U(t)^*, ϕ(t)=U(t)^*ψ(t), Γ(t)=U(t)^*dot U(t). The coordinates ϕ see fixed projectors P_a^(0). The physic","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50041424806fdeac","tex":"P_A^+","aliases":"P_A^+","display":"P_A^+","kind":"use","excerpt":"The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let A=A^* and A^2=I on a finite spinor space. Then P^+_A=tfrac12(I+A), P^-_A=tfrac12(I-A) are self-adjoint complementary orthogonal projections. Idempotence and P_A^+P_A^-=0 follow by expanding the products and using A^2=I; self-adjointness ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7931d33253166dfc","tex":"P_A^-","aliases":"P_A^-","display":"P_A^-","kind":"use","excerpt":"The later formal development makes the directional channels depend on the Clifford matrix of that direction. Let A=A^* and A^2=I on a finite spinor space. Then P^+_A=tfrac12(I+A), P^-_A=tfrac12(I-A) are self-adjoint complementary orthogonal projections. Idempotence and P_A^+P_A^-=0 follow by expanding the products and using A^2=I; self-adjointness ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d2ca5ff6fec3e166","tex":"p_a^A","aliases":"p_a^A","display":"p_a^A","kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8b2da815d2e052f3","tex":"p_a^q","aliases":"p_a^q","display":"p_a^q","kind":"use","excerpt":"For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1ff79702d0f06146","tex":"p_a^{q-1}","aliases":"p_a^q-1","display":"p_a^{q-1}","kind":"use","excerpt":"For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-484d493d93b99218","tex":"p_{a\\mu}","aliases":"p_aμ mu","display":"p_{a\\mu}","kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a5c35e870d97f9c4","tex":"P_B","aliases":"P_B","display":"P_B","kind":"use","excerpt":"Suppose P_A=∑_a∈ C(A)P_a and P_B=∑_b∈ C(B)P_b. Then P_BLP_A=∑_b∈ C(B)∑_a∈ C(A)P_bLP_a. The identity follows by distributing two finite sums. It says exactly which fine transfers a coarse observation retains. A scalar summary or a diagonal-only summary may retain less information and therefore needs its own loss statement.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d072d8979775d522","tex":"p_B","aliases":"p_B","display":"p_B","kind":"use","excerpt":"[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d04253126b67c473","tex":"p_b","aliases":"p_b","display":"p_b","kind":"use","excerpt":"fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-43452d7cc9435d59","tex":"p_B^{\\mathrm c}","aliases":"p_B^ c mathrm","display":"p_B^{\\mathrm c}","kind":"use","excerpt":"The coarse jump balance is always exact. For it to close as a time-homogeneous Markov equation for coarse weights alone, a further compatibility condition suffices. For distinct coarse cells A,B, suppose ∑_b∈ C_Bγ_ba=q_BA for every a∈ C_A. Then summing the fine gains gives q_BAp_A^ c, so dot p_B^ c=∑_A≠ B(q_BAp_A^ c-q_ABp_B^ c). Without this equali","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bb071e0da3d1f51a","tex":"P_b^2","aliases":"P_b^2","display":"P_b^2","kind":"use","excerpt":"Insert ∑_bP_b=id to the left of X and ∑_aP_a=id to its right. This gives the first formula. Evaluate the right resolution on ψ for the second. For the product, P_cXYP_a=P_cX(∑_bP_b)YP_a =∑_b(P_cXP_b)(P_bYP_a), where P_b^2=P_b supplies the repeated middle projector. Finally, (P_bXP_a)^*=P_aX^*P_b.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0f1c833f254f57e3","tex":"p_b^{q-1}","aliases":"p_b^q-1","display":"p_b^{q-1}","kind":"use","excerpt":"For Shannon entropy S(p)=-∑_ap_alog p_a, this gives dot S=-frac12∑_b,aJ_b← alogfracp_bp_a. The constant derivative term cancels because the total current is zero. For a real q, the power sum M_q=∑_ap_a^q has derivative dot M_q=frac q2∑_b,aJ_b← a (p_b^q-1-p_a^q-1). For q≠1, the R enyi entropy H_q=log M_q/(1-q) therefore satisfies dot H_q=dot M_q/((1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3270e71bf1ae54c5","tex":"P_c","aliases":"P_c","display":"P_c","kind":"use","excerpt":"For two operators L and M, insert the resolution between them: P_cMLP_a=∑_b∈ A(P_cMP_b)(P_bLP_a). This is block multiplication written without a choice of basis inside each packet. The intermediate index records the interface through which a composition passes. It is also the reason that a change of resolution must specify how fine packets assemble","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-325fba8fb10630f7","tex":"\\mathcal P_{\\mathrm{in}}","aliases":"P_in mathcal mathrm","display":"\\mathcal P_{\\mathrm{in}}","kind":"use","excerpt":"The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify th","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity variables when the configuration P,Q is fixed.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-045a1b8959b0bef9","tex":"p_j^2","aliases":"p_j^2","display":"p_j^2","kind":"use","excerpt":"Proof. Adjoint is additive, and a real multiple of a Hermitian matrix is Hermitian. PH01 therefore proves the first identity. Expanding the square gives the square terms m^2β^2+∑_jp_j^2α_j^2, the terms mp_j(βα_j+α_jβ), and the terms p_jp_k(α_jα_k+α_kα_j) for j<k. The anticommutators vanish and all four squares are I_4, giving (m^2+∑_jp_j^2)I_4. Sub","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc7ba1c49968c37c","tex":"p_k","aliases":"p_k","display":"p_k","kind":"use","excerpt":"Proof. Adjoint is additive, and a real multiple of a Hermitian matrix is Hermitian. PH01 therefore proves the first identity. Expanding the square gives the square terms m^2β^2+∑_jp_j^2α_j^2, the terms mp_j(βα_j+α_jβ), and the terms p_jp_k(α_jα_k+α_kα_j) for j<k. The anticommutators vanish and all four squares are I_4, giving (m^2+∑_jp_j^2)I_4. Sub","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-363d67fd271736f2","tex":"P_L","aliases":"P_L","display":"P_L","kind":"use","excerpt":"If P_L=ι_Lι_L^*, the full space compression is P_L(z-H)^-1P_L=ι_L S(z)^-1ι_L^*. When a Haar transform W first exposes low and detail channels, the embedding in the original coordinates is ι_L= W^*(|0⟩⊗ I). The inverse on HH_L must be embedded in this way before it is compared with a full space matrix.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-82a6b3a669544b39","tex":"p_{L,0}","aliases":"p_L,0","display":"p_{L,0}","kind":"use","excerpt":"· =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^μ(ω_L^ℓ-1). The factorial series proves nonnegativity, including the limiting values p_L,0(0)=1 and p_L,j(0)=0 for j>0. Partition of the Poisson series proves ∑_jp_L,j=1. Their equivalent hypergeometric form is e^-μ times [source reference] at z=μ.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f3065e30b0c8c4bc","tex":"p_{L,j}","aliases":"p_L,j","display":"p_{L,j}","kind":"use","excerpt":"For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^μ(ω_L^ℓ-1). The factorial series proves nonnegativity, including the limiting values p_L,0(0)=1 and p_L,j(0)=0 for j>0. Partition of the Poisson series p","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"use","excerpt":"The transported and concrete closed levels are then proved equal. Their union is dense, and the orthogonal projections P_n satisfy normP_nf-f_L^2⟶0 for every f∈ L^2([0,1]^3;C^4). This is a state-space approximation theorem. Strong convergence of projections does not control an unbounded Dirac operator, establish graph convergence, or identify the l","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-081e137663ec699f","tex":"\\mathcal P_{\\mathrm{out}}","aliases":"P_out mathcal mathrm","display":"\\mathcal P_{\\mathrm{out}}","kind":"use","excerpt":"The additive transform has a related factorization, with different permutations on its input and its dual output. Let a,b be positive coprime integers and N=ab. Define P_in|n⟩ · =|nbmod a⟩⊗|nbmod b⟩, P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0f540fdbe066e4cf","tex":"\\mathcal P_{\\mathrm{out}}^*","aliases":"P_out^* mathcal mathrm","display":"\\mathcal P_{\\mathrm{out}}^*","kind":"use","excerpt":"P_out|k⟩ · =|k b^-1bmod a⟩⊗|k a^-1bmod b⟩, where the inverses are taken in the indicated residue rings. Then F_N= P_out^*(F_a⊗ F_b) P_in. To verify the phase, let n_a=nbmod a and n_b=nbmod b. At the dual coordinates k_a=kb^-1bmod a and k_b=ka^-1bmod b, the product kernel is N^-1/2exp [2πi (frack_a n_aa+frack_b n_bb)]. The numerator b k_a+a k_b is c","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d4876c09442ebddb","tex":"P_p","aliases":"P_p","display":"P_p","kind":"use","excerpt":"M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and sends (p-1,k) to (0,k+1). The coarse shift has Fourier eigenvalue e^-iθ_j. Thus define P_p(θ)e_r= e_r+1, ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e83884e0b8fb5b18","tex":"P_p^*","aliases":"P_p^*","display":"P_p^*","kind":"use","excerpt":"y, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The v_q are orthonormal by the finite character sum. Applying μ I+κ(P_p+P_p^*) gives [source reference]. Every fine F","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d61b96062b482c72","tex":"P_q","aliases":"P_q","display":"P_q","kind":"use","excerpt":"A two-stage edge and path estimate. The cube phase discrepancies can be made into bounded operators on the same finite Hilbert state space. For z∈ U(1), let P_zψ=zψ uniformly at every site. If p=Ω q, then P_p-P_q=q(Ω-1)id, normP_p-P_q=|Ω-1|. The norm equality uses a nonzero Hilbert space. It is the operator form of [source reference].","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6437e41273729f36","tex":"P_R","aliases":"P_R","display":"P_R","kind":"use","excerpt":"A sufficient common-domain condition is ∫_R e^2|t||λ| d⟨ψ,E(λ)ψ⟩<∞, since |f_j(λ)|≤ e^|t||λ|. Spectral cutoffs P_R=E([-R,R]) supply a dense set satisfying this condition. On Ran P_R, the restricted Hamiltonian is bounded, the norm-series identities apply, and ∑_jf_j(H_∞)P_Rψ=e^-itH_∞P_Rψ ⟶ e^-itH_∞ψ. On the common domain of the individual lanes, ea","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-824f428da635dd8f","tex":"p_R","aliases":"p_R","display":"p_R","kind":"use","excerpt":"For evolution, the radius equation gives dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2],","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b35e522aef4bea0d","tex":"P_{R,j}","aliases":"P_R,j","display":"P_{R,j}","kind":"use","excerpt":"[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-010f131ed455d9fc","tex":"P_S","aliases":"P_S","display":"P_S","kind":"use","excerpt":"For a finite set S⊂Z^3, define the truncation P_Sa by retaining a_n when n∈ S and assigning zero otherwise. Then norma-P_Sa^2=∑_n∉ Snorma_n^2. The sum on the right can be made arbitrarily small because the full nonnegative series converges. Thus finite packets are dense in KK. The statement concerns the norm of the omitted amplitudes and does not d","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-27ac1d6f9a948981","tex":"p_x","aliases":"p_x","display":"p_x","kind":"use","excerpt":"H(m,p)=mβ+p_xα_x+p_yα_y+p_zα_z.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-398dcf493d87760a","tex":"p_{xy}","aliases":"p_xy","display":"p_{xy}","kind":"use","excerpt":"For example, the two positive paths across an xy face at height z are p_xy(z)=ℓ_y(1,z)ℓ_x(0,z), p_yx(z)=ℓ_x(1,z)ℓ_y(0,z). Define Ω_xy(z)=p_xy(z)p_yx(z)^-1, and define the xz and yz face phases analogously. Both paths have the same endpoints, so their gauge factors cancel in this abelian ratio. Thus each Ω is gauge invariant, and Ω=1 precisely when ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a1a9970aad3792ab","tex":"p_{XYZ}","aliases":"p_XYZ","display":"p_{XYZ}","kind":"use","excerpt":"Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0ba60bcf1db4d4e8","tex":"p_{XZY}","aliases":"p_XZY","display":"p_{XZY}","kind":"use","excerpt":"Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-17584b33b8ab7132","tex":"p_y","aliases":"p_y","display":"p_y","kind":"use","excerpt":"H(m,p)=mβ+p_xα_x+p_yα_y+p_zα_z.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eecfb231ea3c5eb1","tex":"p_{yx}","aliases":"p_yx","display":"p_{yx}","kind":"use","excerpt":"For example, the two positive paths across an xy face at height z are p_xy(z)=ℓ_y(1,z)ℓ_x(0,z), p_yx(z)=ℓ_x(1,z)ℓ_y(0,z). Define Ω_xy(z)=p_xy(z)p_yx(z)^-1, and define the xz and yz face phases analogously. Both paths have the same endpoints, so their gauge factors cancel in this abelian ratio. Thus each Ω is gauge invariant, and Ω=1 precisely when ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-611887b9ab141940","tex":"p_{YXZ}","aliases":"p_YXZ","display":"p_{YXZ}","kind":"use","excerpt":"Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-567a8517d9d15c5f","tex":"p_{YZX}","aliases":"p_YZX","display":"p_{YZX}","kind":"use","excerpt":"Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-91513c11f34221ca","tex":"P_z","aliases":"P_z","display":"P_z","kind":"use","excerpt":"A two-stage edge and path estimate. The cube phase discrepancies can be made into bounded operators on the same finite Hilbert state space. For z∈ U(1), let P_zψ=zψ uniformly at every site. If p=Ω q, then P_p-P_q=q(Ω-1)id, normP_p-P_q=|Ω-1|. The norm equality uses a nonzero Hilbert space. It is the operator form of [source reference].","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-60af386a48855ec5","tex":"p_z","aliases":"p_z","display":"p_z","kind":"use","excerpt":"H(m,p)=mβ+p_xα_x+p_yα_y+p_zα_z.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f61bdb6306d6c2c5","tex":"p_{ZXY}","aliases":"p_ZXY","display":"p_{ZXY}","kind":"use","excerpt":"Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-783bdc5dbbc88b2c","tex":"p_{ZYX}","aliases":"p_ZYX","display":"p_{ZYX}","kind":"use","excerpt":"Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3298ad8090debb2c","tex":"P_\\pm^*","aliases":"P_±^* pm","display":"P_\\pm^*","kind":"use","excerpt":"Then P_±^*=P_±, P_±^2=P_±, P_+P_-=P_-P_+=0, P_++P_-=I_4, and HP_±=± EP_±. Both projectors have rank two, and","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-66044690d086e578","tex":"P_\\pm^2","aliases":"P_±^2 pm","display":"P_\\pm^2","kind":"use","excerpt":"Then P_±^*=P_±, P_±^2=P_±, P_+P_-=P_-P_+=0, P_++P_-=I_4, and HP_±=± EP_±. Both projectors have rank two, and","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-58fb42807c0a2ca7","tex":"P_\\theta","aliases":"P_θ theta","display":"P_\\theta","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1a282e80877df43d","tex":"P_\\theta^2","aliases":"P_θ^2 theta","display":"P_\\theta^2","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b0351f4b03177b3e","tex":"P_{\\theta\\theta}","aliases":"P_θθ theta theta","display":"P_{\\theta\\theta}","kind":"use","excerpt":"∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-68c3342b398c78bd","tex":"P_\\Lambda'","aliases":"P_Λ' Lambda","display":"P_\\Lambda'","kind":"use","excerpt":"The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0a2084aa04022a3d","tex":"P_{\\Lambda,j}","aliases":"P_Λ,j Lambda","display":"P_{\\Lambda,j}","kind":"use","excerpt":"[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2b4b53c02eca0902","tex":"P_{\\Lambda,j}^2","aliases":"P_Λ,j^2 Lambda","display":"P_{\\Lambda,j}^2","kind":"use","excerpt":"[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6aaf97ce7bc4622c","tex":"P_\\Lambda^2","aliases":"P_Λ^2 Lambda","display":"P_\\Lambda^2","kind":"use","excerpt":"The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df000ea46aaa7e45","tex":"p_\\mu^B","aliases":"p_μ^B mu","display":"p_\\mu^B","kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c17f246de5f52801","tex":"P_\\tau","aliases":"P_τ tau","display":"P_\\tau","kind":"use","excerpt":"∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db675a504cfbee47","tex":"P_\\tau^2","aliases":"P_τ^2 tau","display":"P_\\tau^2","kind":"use","excerpt":"Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0aa9f14ee49222c6","tex":"P_{\\tau\\theta}","aliases":"P_τθ tau theta","display":"P_{\\tau\\theta}","kind":"use","excerpt":"Differentiate the two densities. The terms with P_τ P_τθ and Q_τ Q_τθ cancel, leaving ∂_τ M_G-∂_θ H_G = · 2P_θ(P_ττ-γ P_θθ)","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-308d77c0d16ee668","tex":"P_{\\tau\\tau}","aliases":"P_ττ tau tau","display":"P_{\\tau\\tau}","kind":"use","excerpt":"∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cfc2554aee2ed59e","tex":"P_\\varphi","aliases":"P_φ varphi","display":"P_\\varphi","kind":"use","excerpt":"To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source refe","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-519ea6b75332fc0d","tex":"P_\\varphi^2","aliases":"P_φ^2 varphi","display":"P_\\varphi^2","kind":"use","excerpt":"To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source refe","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8a243f3b87b7dca3","tex":"\\mathrm{par}","aliases":"par mathrm","display":"\\mathrm{par}","kind":"use","excerpt":"The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fe8f749b378cd86e","tex":"\\mathrm{phase}","aliases":"phase mathrm","display":"\\mathrm{phase}","kind":"use","excerpt":"The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-53743af9d1feb8d1","tex":"\\mathrm{phys}","aliases":"phys mathrm","display":"\\mathrm{phys}","kind":"use","excerpt":"Connecting the group to its generator. The next chapter proves that the constructed group is the spectral evolution of this Hamiltonian: U(t)=e^-itH_m^phys. The proof establishes strong continuity and differentiates on the full Hamiltonian domain with control of the Fourier tail. The reverse domain inclusion then shows that the infinitesimal genera","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eda0fef223ad56b8","tex":"\\propto","aliases":"propto","display":null,"kind":"use","excerpt":"x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies d","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"item S. V. Kozyrev, ``Wavelet theory as p-adic spectral analysis,'' Izvestiya: Mathematics 66(2), 367--376 (2002). https://doi.org/10.1070/IM2002v066n02ABEH000381doi:10.1070/IM2002v066n02ABEH000381. Published English Theorems 2 and 7 inspected. The Q_p Vladimirov setting supplies wavelet context; the present Z_p level-diagonal construction is prove","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0c370aa793b02aac","tex":"Q'","aliases":"Q'","display":"Q'","kind":"use","excerpt":"The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"u is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e4b91cbee49fa788","tex":"Q^4","aliases":"Q^4","display":"Q^4","kind":"use","excerpt":"The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this inter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d014cc8d6971e544","tex":"q^{ab}","aliases":"q^ab","display":"q^{ab}","kind":"use","excerpt":"nequation widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4a3a203830edc472","tex":"q^m","aliases":"q^m","display":"q^m","kind":"use","excerpt":"[Convergent states and finite energy derivatives] Choose one unit vector u_m∈Ran F_m at every level, let 0<q<1, and set f_q=∑_m≥0sqrt(1-q)q^m u_m, ϵ_m=2^m. Then normf_q_2=1 and norm(I- E_M)f_q_2^2=q^M+1. However, f_q∈Dom(H_ϵ) exactly when 4q<1: normH_ϵ f_q_2^2 =(1-q)∑_m≥0(4q)^m. For q<1/4, its graph tail is exactly q^M+1+frac(1-q)(4q)^M+11-4q. For ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8622e3ac0dd5438d","tex":"q^{M+1}","aliases":"q^M+1","display":"q^{M+1}","kind":"use","excerpt":"[Convergent states and finite energy derivatives] Choose one unit vector u_m∈Ran F_m at every level, let 0<q<1, and set f_q=∑_m≥0sqrt(1-q)q^m u_m, ϵ_m=2^m. Then normf_q_2=1 and norm(I- E_M)f_q_2^2=q^M+1. However, f_q∈Dom(H_ϵ) exactly when 4q<1: normH_ϵ f_q_2^2 =(1-q)∑_m≥0(4q)^m. For q<1/4, its graph tail is exactly q^M+1+frac(1-q)(4q)^M+11-4q. For ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","excerpt":"-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"-2 q ^2 · 0. It is invertible whenever q≠0. On the intended constraint surface, q lies on the unit sphere and p is tangent to that sphere. Writing bar q=(q_0+q_1)/2 and bar p=(p_0+p_1)/2, the exact discrete gradients are overline∇Φ_1=(2bar q,0), overline∇Φ_2=(bar p,bar q). Consequently Δ_Δ has the same displayed form with q replaced by bar q. Endpo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-506afac37903bdfd","tex":"Q_A","aliases":"Q_A","display":"Q_A","kind":"use","excerpt":"Let A be a finite set of coarse labels and let f:I→ A assign every fine packet to exactly one coarse label. Write C_A=f^-1(A) and define Q_A=∑_a∈ C_AP_a. The fibers are disjoint and cover I, so (Q_A) is another orthogonal atlas. Empty fibers give zero projectors. This is a grouping of an existing resolution. It need not be an approximation to a dif","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-55d4600438de195a","tex":"q_a","aliases":"q_a","display":"q_a","kind":"use","excerpt":"Cell values and normalized coefficients. For a measurable finite partition into cells of equal volume v>0, distinguish the physical constant value q_a∈C^4 on cell C_a from the normalized coefficient ψ_a=sqrt v q_a. The step interpolation is (Jψ)(x)=v^-1/2ψ_a (x∈ C_a), with equality understood almost everywhere. Direct integration gives normJψ_L^2^2","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d8eb43a9c9648747","tex":"Q_A^n","aliases":"Q_A^n","display":"Q_A^n","kind":"use","excerpt":"i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-975da5ad2f8f771b","tex":"Q_A^{n+1}","aliases":"Q_A^n+1","display":"Q_A^{n+1}","kind":"use","excerpt":"i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4ac081124caea3b8","tex":"q_{AB}","aliases":"q_AB","display":"q_{AB}","kind":"use","excerpt":"The coarse jump balance is always exact. For it to close as a time-homogeneous Markov equation for coarse weights alone, a further compatibility condition suffices. For distinct coarse cells A,B, suppose ∑_b∈ C_Bγ_ba=q_BA for every a∈ C_A. Then summing the fine gains gives q_BAp_A^ c, so dot p_B^ c=∑_A≠ B(q_BAp_A^ c-q_ABp_B^ c). Without this equali","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-44b1f051975da19b","tex":"q_{ab}","aliases":"q_ab","display":"q_{ab}","kind":"use","excerpt":"Dbarψ · = dbarψ-tfrac14barψω^IJγ_IJ+iq abarψ. The parameter q here is electric charge. The determinant of a spatial metric below will be written det(q_ab) to avoid confusing the two uses.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b6cbebd4328bd504","tex":"Q_B","aliases":"Q_B","display":"Q_B","kind":"use","excerpt":"[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3653386b19dd38cf","tex":"q_{BA}","aliases":"q_BA","display":"q_{BA}","kind":"use","excerpt":"The coarse jump balance is always exact. For it to close as a time-homogeneous Markov equation for coarse weights alone, a further compatibility condition suffices. For distinct coarse cells A,B, suppose ∑_b∈ C_Bγ_ba=q_BA for every a∈ C_A. Then summing the fine gains gives q_BAp_A^ c, so dot p_B^ c=∑_A≠ B(q_BAp_A^ c-q_ABp_B^ c). Without this equali","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a4591bb144ad9c3f","tex":"Q_c","aliases":"Q_c","display":"Q_c","kind":"use","excerpt":"The formal construction selects five ordered phase pairs, in this order: (p_XYZ,p_YXZ), (p_YXZ,p_YZX), (p_YZX,p_ZYX), (p_XYZ,p_XZY), (p_XZY,p_ZXY). Let Q_f be the execution-order product of the first phase operators in these pairs, and Q_c the product of the second ones. Identity factors can be placed at the ends to fit the seven-factor interface. ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ae104bef53b5518c","tex":"q_{cd}","aliases":"q_cd","display":"q_{cd}","kind":"use","excerpt":"nequation widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-688d5aa3a341b160","tex":"Q_f","aliases":"Q_f","display":"Q_f","kind":"use","excerpt":"The formal construction selects five ordered phase pairs, in this order: (p_XYZ,p_YXZ), (p_YXZ,p_YZX), (p_YZX,p_ZYX), (p_XYZ,p_XZY), (p_XZY,p_ZXY). Let Q_f be the execution-order product of the first phase operators in these pairs, and Q_c the product of the second ones. Identity factors can be placed at the ends to fit the seven-factor interface. ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-393f3636d9c3ffef","tex":"q_h","aliases":"q_h","display":"q_h","kind":"use","excerpt":"Write q_h(n)=q(h,p(n)) and H_h,n=H(m,q_h(n)). The function q_j is even in h, and its value at zero is justified by sin z/z→1. For h≠0, a centered translation difference on a periodic smooth function is","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_jQ_j=I. No exponential norm series for the unbounded number operator is nee","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ecfc6d63a3b68fb8","tex":"q_j","aliases":"q_j","display":"q_j","kind":"use","excerpt":"q_j(h,p)= sin(hp_j)/h, · h≠0, p_j, · h=0.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-acc408ad72dcbd3f","tex":"Q_j^{(L)}","aliases":"Q_j^(L)","display":"Q_j^{(L)}","kind":"use","excerpt":"On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e88cbadb72581711","tex":"q_j^*","aliases":"q_j^*","display":"q_j^*","kind":"use","excerpt":"The first formula follows from dot E_j=J_j. For the matter contribution, let q_j=∂ H_ m/∂overlineψ_j=ihdotψ_j. The state-dependent part of the real derivative is 2Re∑_jq_j^*dotψ_j=0, because q_j^*dotψ_j=-ihnormdotψ_j^2 is purely imaginary. The remaining explicit link derivative is ∑_jfrac∂ H_ m∂ A_jdot A_j =-h∑_jJ_jE_j. Adding the two sectors prove","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a9af1147d28b7af5","tex":"Q_k","aliases":"Q_k","display":"Q_k","kind":"use","excerpt":"unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_jQ_j=I. No exponential norm series for the unbounded number operator is nee","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cd98a9c47ecfa404","tex":"\\mathbb Q_p","aliases":"Q_p mathbb","display":"\\mathbb Q_p","kind":"use","excerpt":"item S. V. Kozyrev, ``Wavelet theory as p-adic spectral analysis,'' Izvestiya: Mathematics 66(2), 367--376 (2002). https://doi.org/10.1070/IM2002v066n02ABEH000381doi:10.1070/IM2002v066n02ABEH000381. Published English Theorems 2 and 7 inspected. The Q_p Vladimirov setting supplies wavelet context; the present Z_p level-diagonal construction is prove","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-237bdbcaf2851c9d","tex":"Q_r^{(L)}","aliases":"Q_r^(L)","display":"Q_r^{(L)}","kind":"use","excerpt":"Finally the annihilation operator routes adjacent residues: Q_r^(L)aQ_s^(L)=0 unless r≡ s-1pmod L, Q_r^(L)a^† Q_s^(L)=0 unless r≡ s+1pmod L. On a finite number-state combination this follows from a|n⟩=sqrt n |n-1⟩ and a^†|n⟩=sqrtn+1 |n+1⟩. Each residue projector preserves Dom(a) and Dom(a^†), because it only removes terms from their weighted number","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c07593a62404ec46","tex":"Q_s^{(L)}","aliases":"Q_s^(L)","display":"Q_s^{(L)}","kind":"use","excerpt":"Finally the annihilation operator routes adjacent residues: Q_r^(L)aQ_s^(L)=0 unless r≡ s-1pmod L, Q_r^(L)a^† Q_s^(L)=0 unless r≡ s+1pmod L. On a finite number-state combination this follows from a|n⟩=sqrt n |n-1⟩ and a^†|n⟩=sqrtn+1 |n+1⟩. Each residue projector preserves Dom(a) and Dom(a^†), because it only removes terms from their weighted number","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-65fa07eabd1080a1","tex":"q_x^a","aliases":"q_x^a","display":"q_x^a","kind":"use","excerpt":"Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e6c82058fa3275d","tex":"q_x^b","aliases":"q_x^b","display":"q_x^b","kind":"use","excerpt":"Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-629e3cebbe7000f8","tex":"q_x^{g,a}","aliases":"q_x^g,a","display":"q_x^{g,a}","kind":"use","excerpt":"Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea3d9f514fcaceb6","tex":"Q_\\alpha","aliases":"Q_α alpha","display":"Q_\\alpha","kind":"use","excerpt":"For a partition A=bigsqcup_αC_α, the strong sum Q_α=∑_a∈ C_αP_a is a bounded orthogonal projection. Its weight is always normQ_αψ^2=∑_a∈ C_αnormP_aψ^2. To define a coarse current with HQ_αψ, additionally require Q_α D(H)⊂ D(H). Fine domain preservation alone does not state this infinite-cell property.","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d2f8225af58ea4bd","tex":"Q_\\beta","aliases":"Q_β beta","display":"Q_\\beta","kind":"use","excerpt":"Under this coarse-domain hypothesis, symmetry and the strong cell sums justify both iterated sums separately: J^rm c_βα · =2IminnerQ_βψHQ_αψ · =∑_b∈ C_β(∑_a∈ C_αJ_ba) =∑_a∈ C_α(∑_b∈ C_βJ_ba). For example, first summing a moves H onto P_bψ, then the outer sum pairs ∑_bP_bψ with the fixed vector HQ_αψ. The reverse order has its own symmetric argument","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fbcb007cb50c0f2b","tex":"Q_\\theta","aliases":"Q_θ theta","display":"Q_\\theta","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9310f52f93902981","tex":"Q_\\theta^2","aliases":"Q_θ^2 theta","display":"Q_\\theta^2","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0b28e92c9075b032","tex":"Q_{\\theta\\theta}","aliases":"Q_θθ theta theta","display":"Q_{\\theta\\theta}","kind":"use","excerpt":"∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6f7058a34b152234","tex":"Q_\\mu","aliases":"Q_μ mu","display":"Q_\\mu","kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9e65ea2a28eb600e","tex":"Q_\\tau","aliases":"Q_τ tau","display":"Q_\\tau","kind":"use","excerpt":"∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-76f99c7b5823d720","tex":"Q_\\tau^2","aliases":"Q_τ^2 tau","display":"Q_\\tau^2","kind":"use","excerpt":"Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c341e0f3ba60fc73","tex":"Q_{\\tau\\theta}","aliases":"Q_τθ tau theta","display":"Q_{\\tau\\theta}","kind":"use","excerpt":"Differentiate the two densities. The terms with P_τ P_τθ and Q_τ Q_τθ cancel, leaving ∂_τ M_G-∂_θ H_G = · 2P_θ(P_ττ-γ P_θθ)","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1fc923810ce585fe","tex":"Q_{\\tau\\tau}","aliases":"Q_ττ tau tau","display":"Q_{\\tau\\tau}","kind":"use","excerpt":"∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"For bounded Borel F:R→C, define F(H_ϵ)f=∑_m≥0F(ϵ_m) F_mf. The series converges by Parseval and normF(H_ϵ)f_2≤sup_m|F(ϵ_m)|norm f_2. The same level decomposition verifies sums, products, and adjoints of these multipliers directly. In particular, U(t) · =e^-itH_ϵ =∑_m≥0e^-itϵ_m F_m,","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"T^3=( R/ Z)^3,","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8ac168ec1a0ac655","tex":"\\mathbf r","aliases":"r mathbf","display":"\\mathbf r","kind":"use","excerpt":"Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-32fe36360874826c","tex":"\\mathfrak R","aliases":"R mathfrak","display":"\\mathfrak R","kind":"use","excerpt":"Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0c8820d81c53436b","tex":"R''","aliases":"R''","display":"R''","kind":"use","excerpt":"The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cdb814bc7d9fc317","tex":"R'^2","aliases":"R'^2","display":"R'^2","kind":"use","excerpt":"The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-37387465eb47206f","tex":"R'^3","aliases":"R'^3","display":"R'^3","kind":"use","excerpt":"-fracP_ΛRΛ D +frac QR G. Consequently, on a patch where R'/Λ≠0, the systems ( H, D, G)=0 and ( M', D, G)=0 have the same zero set. Differentiating [source reference] gives M'= · fracR'2-fracR'^32Λ^2 -fracRR'R''Λ^2+fracRR'^2Λ'Λ^3 +fracP_Λ P_Λ'R-fracP_Λ^2R'2R^2 · +fracQQ'R-fracQ^2R'2R^2. Expanding the right side of [source reference] cancels the two ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e672c832eb007691","tex":"R^{\\mathsf T}","aliases":"R^ T mathsf","display":"R^{\\mathsf T}","kind":"use","excerpt":"For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-055d5ee422435b80","tex":"R^*","aliases":"R^*","display":"R^*","kind":"use","excerpt":"Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-072e7fe97b551a6a","tex":"r^{-2}","aliases":"r^-2","display":"r^{-2}","kind":"use","excerpt":"E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ), dotΦ · =∂_r[σ(Π+vΦ)], dotΠ · =r^-2∂_r[r^2σ(Φ+vΠ)]. If Φ=φ' initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.","count":19},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-061c21e92840dde5","tex":"\\mathbb R^{2n}","aliases":"R^2n mathbb","display":"\\mathbb R^{2n}","kind":"use","excerpt":"Endpoint Dirac projection. Consider a finite-dimensional real canonical chart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. F","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-508222661b47a325","tex":"R^3","aliases":"R^3","display":"R^3","kind":"use","excerpt":"ginequation dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2], dot P_R · =fracΛ(P_Λ^2+Q^2)R^3 -fracP_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-15bd74f3637fba8d","tex":"\\mathbb R^3","aliases":"R^3 mathbb","display":"\\mathbb R^3","kind":"use","excerpt":"For p∈ R^3, define","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c6163472e649a229","tex":"\\mathbb R^4","aliases":"R^4 mathbb","display":"\\mathbb R^4","kind":"use","excerpt":"The reverse triangle inequality in R^4, applied to (m,q_h(n)) and (m,p(n)), gives","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-37e1261ad0123242","tex":"R^6","aliases":"R^6","display":"R^6","kind":"use","excerpt":"The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this inter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3a803a9bd1627024","tex":"\\mathbb R^6","aliases":"R^6 mathbb","display":"\\mathbb R^6","kind":"use","excerpt":"A worked particle constraint. In R^6 with z=(q,p) and canonical J, set Φ_1=q· q-1, Φ_2=q· p. At one point the ordinary Dirac matrix is Δ(z)=0 · 2 q ^2","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d6d881fd83e7a2b0","tex":"R^7","aliases":"R^7","display":"R^7","kind":"use","excerpt":"The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this inter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-be22a5b4e4f47e1b","tex":"R^8","aliases":"R^8","display":"R^8","kind":"use","excerpt":"The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this inter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-89e5eada6032fa59","tex":"R^{IJ}","aliases":"R^IJ","display":"R^{IJ}","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-92a95a3f59d5b018","tex":"R^{k-1}","aliases":"R^k-1","display":"R^{k-1}","kind":"use","excerpt":"X^k-Y^k ≤ kR^k-1 X-Y .","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-19ee12c051e5e943","tex":"R^{KL}","aliases":"R^KL","display":"R^{KL}","kind":"use","excerpt":"The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e, S_ M^(1) · =∫(tfrac12φ_IJFwedge e^Iwedge e^J -tfrac14φ_IJφ^IJvol_e), S_ D · =frac i2∫Σ_Iwedge (barψγ^I Dψ -( Dbarψ)γ^Iψ) -m∫vol_ebarψψ. This formulation uses no inverse tetrad in its defining action. Its largest monomials have six field fac","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f1a40c4569184e2d","tex":"\\mathcal R^{KL}","aliases":"R^KL mathcal","display":"\\mathcal R^{KL}","kind":"use","excerpt":"The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gau","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-236f80d54e55c576","tex":"R_*","aliases":"R_*","display":"R_*","kind":"use","excerpt":"A level-independent reconstruction estimate. Assume a uniform regular set R_j≥ R_*>0, 2MR_j-Q^2≥ p_*^2>0, 0<d_*≤ D_4R_j≤ d^*. For two arrays in this set, with the same (M,Q,ε), differentiation along the segment between them gives normδΛ_∞ · ≤normD_4δ R_∞,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7f4928b93fd91121","tex":"r_+","aliases":"r_+","display":"r_+","kind":"use","excerpt":"The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this inter","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7defeb26f73dee49","tex":"r_{+,h}","aliases":"r_+,h","display":"r_{+,h}","kind":"use","excerpt":"The outer horizon remains at the fixed areal radius 1.6. Its computational coordinate moves because F changes. The retained final coordinate comparison at N=256 gives |r_+,h-r_+^rm exact|=1.89262 10^-11. This measures tracking of a known horizon under coordinate deformation. It does not measure the formation or growth of a black hole.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2d4a4709819c91da","tex":"r_+^2","aliases":"r_+^2","display":"r_+^2","kind":"use","excerpt":"Horizon conditioning. For a subextremal outer horizon, put d=sqrtM^2-Q^2>0. Then f'(r_+)=frac2dr_+^2. If a nearby continuous numerical profile has a selected root near this simple root, the local inverse estimate gives |δ r_+|≤fracr_+^22d normδ f_∞ +O(normδ f_∞^2). The estimate describes the nearby root and presupposes its localization to a region ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bcc60ff453e6985c","tex":"r_+^{\\rm exact}","aliases":"r_+^rm exact rm","display":"r_+^{\\rm exact}","kind":"use","excerpt":"The outer horizon remains at the fixed areal radius 1.6. Its computational coordinate moves because F changes. The retained final coordinate comparison at N=256 gives |r_+,h-r_+^rm exact|=1.89262 10^-11. This measures tracking of a known horizon under coordinate deformation. It does not measure the formation or growth of a black hole.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7335b8673a12eddc","tex":"r_-","aliases":"r_-","display":"r_-","kind":"use","excerpt":"The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this inter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"The normalized effective energy is (z-μ_p(z))/κ_p(z)=R_p(x). The parameter z remains fixed through the elimination. Since μ_p and κ_p depend on it, this is an energy-dependent spectral closure, not an energy-independent Hamiltonian flow. For p=2, R_2(x)=x^2-2 and U_1(x/2)=x, giving H_rm eff^(2)(z) =μ I+fracκ^2z-μ(2I+T+T^*), z≠μ. This decimation ret","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d0aa3589089d58ba","tex":"R_3","aliases":"R_3","display":"R_3","kind":"use","excerpt":"adicfigure11_prime_renormalizationThe normalized maps R_2, R_3, and R_5 preserve the same arcsine spectral measure. The accompanying bounded moment checks illustrate the exact measure identity, while each Schur realization retains its own excluded poles.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5dbe454ca813d1d3","tex":"R_5","aliases":"R_5","display":"R_5","kind":"use","excerpt":"adicfigure11_prime_renormalizationThe normalized maps R_2, R_3, and R_5 preserve the same arcsine spectral measure. The accompanying bounded moment checks illustrate the exact measure identity, while each Schur realization retains its own excluded poles.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-33462a02fcce1a84","tex":"R_a","aliases":"R_a","display":"R_a","kind":"use","excerpt":"Choose unit links a(x) and b(x) on the first and second half-edge with b(x)a(x)=u(x). The associated fine link is v(e_x)=a(x), v(m_x)=b(x). The interpolation and reconstruction maps are (I_aψ)(e_x) · =ψ(x), · (I_aψ)(m_x) · =a(x)ψ(x), (R_aΨ)(x) · =tfrac12(Ψ(e_x)+a(x)^-1Ψ(m_x)). The midpoint is a transported value, not an independently sampled field ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f79643ae14df8c41","tex":"R_{Aa}","aliases":"R_Aa","display":"R_{Aa}","kind":"use","excerpt":"For an incidence matrix R_Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-12288fef14aab090","tex":"R_{a\\mu}","aliases":"R_aμ mu","display":"R_{a\\mu}","kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-849a3daf5b3e8a7a","tex":"R_{b\\nu}","aliases":"R_bν nu","display":"R_{b\\nu}","kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0f7be7718d9eb684","tex":"R_C","aliases":"R_C","display":"R_C","kind":"use","excerpt":"-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constan","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-71e0f18ed7179e19","tex":"R_d","aliases":"R_d","display":"R_d","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bfb439ae05415aec","tex":"R_d^*","aliases":"R_d^*","display":"R_d^*","kind":"use","excerpt":"dulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from the projectors onto ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-55c658da2911c141","tex":"R_e","aliases":"R_e","display":"R_e","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c9377ce9ef801522","tex":"R_F","aliases":"R_F","display":"R_F","kind":"use","excerpt":"A nonlinear Gaussian reduction in closed form. The small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3a7065b026fde28d","tex":"r_H","aliases":"r_H","display":"r_H","kind":"use","excerpt":"Recorded first marginally trapped sphere. The event finder takes a quadratic interpolation of the spatial maximum of 2m/r and a linear interpolation between consecutive time steps bracketing its first crossing of one. The archived event table is small rrrrr N_r · h · t_H · r_H · Δ M/M_0","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d0a12b6c6c489d99","tex":"r_i^2","aliases":"r_i^2","display":"r_i^2","kind":"use","excerpt":"On a fixed radial interval [r_i,r_o], the exact balance is fracddt(m(t,r_o)-m(t,r_i)) =r_o^2 J(t,r_o)-r_i^2 J(t,r_i). Boundary fluxes, rather than a statement that a finite-domain mass must be exactly constant, are the appropriate continuum reference. The numerical prefix solve below does not satisfy this identity algebraically at every finite grid","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-229e4596f8cda462","tex":"R_{i\\eta}","aliases":"R_iη eta","display":"R_{i\\eta}","kind":"use","excerpt":"projection preserving D(H), and [H,P] on D(H) extends to a bounded operator B. For nonreal z, let R_z=(H-z)^-1. It maps HH into D(H), so all products used below have their stated meaning. Multiplying the commutator identity on the domain by the resolvents gives [R_z,P]=-R_zBR_z. Indeed, for x∈HH, apply H-z to PR_zx to obtain Px+BR_zx, then apply R_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"u_q(t)-u_r(t) _2^2 =(c_q-c_r)^2+(s_q-s_r)^2 +∑_j=x,y,z(q_js_q-r_js_r)^2.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1e373259488dec7c","tex":"\\mathfrak R_{L,j}","aliases":"R_L,j mathfrak","display":"\\mathfrak R_{L,j}","kind":"use","excerpt":"Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a951830e0ee3b4ef","tex":"\\mathfrak R_{L,k}","aliases":"R_L,k mathfrak","display":"\\mathfrak R_{L,k}","kind":"use","excerpt":", and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-25ee15775b880083","tex":"R_m","aliases":"R_m","display":"R_m","kind":"use","excerpt":"[Chebyshev composition and Schur composition] For all positive integers m,n, the polynomial identities R_n(R_m(x)) · =R_mn(x)=R_m(R_n(x)),nonumber U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur i","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-091fb9d0a9551219","tex":"r_{\\max}","aliases":"r_max max","display":"r_{\\max}","kind":"use","excerpt":"What is local in the algorithm. At each RK4 stage the matter channels determine the geometry through two radial solves. The mass constraint is integrated outward with m(r_rm exc)=0. The lapse derivative is integrated and the result normalized by σ(r_max)=1. These are triangular prefix operations. A matter value near the inner end can affect the rec","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a75d4101f5ae9880","tex":"R_{mn}","aliases":"R_mn","display":"R_{mn}","kind":"use","excerpt":"[Chebyshev composition and Schur composition] For all positive integers m,n, the polynomial identities R_n(R_m(x)) · =R_mn(x)=R_m(R_n(x)),nonumber U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur i","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"use","excerpt":"We now eliminate intermediate residue sites in [source reference], assuming κ≠0. The retained sites are C={0,p,…,(M-1)p} and the eliminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fd828431f1a05a6f","tex":"r_o","aliases":"r_o","display":"r_o","kind":"use","excerpt":"On a fixed radial interval [r_i,r_o], the exact balance is fracddt(m(t,r_o)-m(t,r_i)) =r_o^2 J(t,r_o)-r_i^2 J(t,r_i). Boundary fluxes, rather than a statement that a finite-domain mass must be exactly constant, are the appropriate continuum reference. The numerical prefix solve below does not satisfy this identity algebraically at every finite grid","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5a71c171d2d1861c","tex":"r_o^2","aliases":"r_o^2","display":"r_o^2","kind":"use","excerpt":"On a fixed radial interval [r_i,r_o], the exact balance is fracddt(m(t,r_o)-m(t,r_i)) =r_o^2 J(t,r_o)-r_i^2 J(t,r_i). Boundary fluxes, rather than a statement that a finite-domain mass must be exactly constant, are the appropriate continuum reference. The numerical prefix solve below does not satisfy this identity algebraically at every finite grid","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0d5a58c436e4715b","tex":"r_p","aliases":"r_p","display":"r_p","kind":"use","excerpt":"Define the joint detail projections F_ r =I_KK_tor⊗ ⊗_p∈ P F_p,r_p, r∈N^ P. They are pairwise orthogonal and have finite-dimensional ranges. Rectangular sums equal E_ h, so their strong sum is the identity. This permits interactions between primes while preserving the exact depth filtration.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18f41d2d86df50af","tex":"R_p'","aliases":"R_p'","display":"R_p'","kind":"use","excerpt":"The mean logarithmic expansion of this map is also exact: ∫_-2^2log|R_p'(x)| dν(x)=log p. Indeed for x=2cosθ, R_p'(x)=psin(pθ)/sinθ. Logarithms at its finitely many zeros are integrable, since they have the local form log|θ-θ_0| plus a bounded term. Hence the integral is log p plus the difference of the integrals of log|sin(pθ)| and logsinθ over [0","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-473eb620029bf88c","tex":"\\mathcal R_{p,0}","aliases":"R_p,0 mathcal","display":"\\mathcal R_{p,0}","kind":"use","excerpt":"A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-646f864dabf7bf1e","tex":"\\mathcal R_{p,h}","aliases":"R_p,h mathcal","display":"\\mathcal R_{p,h}","kind":"use","excerpt":"A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1a4a0bc6234fa1f9","tex":"\\mathcal R_{p,h-1}","aliases":"R_p,h-1 mathcal","display":"\\mathcal R_{p,h-1}","kind":"use","excerpt":"p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R_p,h reverse the h base-p digits, with R_p,0=I on the one-dimensional space. Then A^ad_p,h= (I_p⊗ R_p,h-1) A^std_p,h R_p,h. Indeed reversing d_0+d_1p+⋯+d_h","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-39702065bb040b60","tex":"r_{\\rm exc}","aliases":"r_rm exc rm","display":"r_{\\rm exc}","kind":"use","excerpt":"What is local in the algorithm. At each RK4 stage the matter channels determine the geometry through two radial solves. The mass constraint is integrated outward with m(r_rm exc)=0. The lapse derivative is integrated and the result normalized by σ(r_max)=1. These are triangular prefix operations. A matter value near the inner end can affect the rec","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-59eb0e6ce7c7f455","tex":"R_{\\rm turn}","aliases":"R_rm turn rm","display":"R_{\\rm turn}","kind":"use","excerpt":"The square-root chart instead meets a turning surface at R_rm turn=fracQ^22M. There p=0, and P_R=MF_r/p becomes unbounded for nonzero F_r. This is a limitation of the chosen flat spatial slice and canonical chart. Horizon penetration does not remove the need for a positive distance from this turning surface. It also does not remove the physical cur","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-257791562a17d424","tex":"R_v","aliases":"R_v","display":"R_v","kind":"use","excerpt":"directed arcs and let τ_2=2, τ_p=p-1 for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and p","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3088a38f12fda26d","tex":"R_x","aliases":"R_x","display":"R_x","kind":"use","excerpt":"Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basi","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f0e893f7b0a8cb70","tex":"r_x","aliases":"r_x","display":"r_x","kind":"use","excerpt":"H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-84d9e04e97e949b2","tex":"R_x^{\\mathsf T}","aliases":"R_x^ T mathsf","display":"R_x^{\\mathsf T}","kind":"use","excerpt":"pand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basis is orthonormal for it; hence R_xR_x^ T=I. Thus the color vector rotates while its Euclid","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-06cb8b88ad7472f6","tex":"r_x^2","aliases":"r_x^2","display":"r_x^2","kind":"use","excerpt":"u_r(t) _2^2 =c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2) =cos^2(E_rt)+sin^2(E_rt)=1.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5d6e45e73e872e8e","tex":"r_y","aliases":"r_y","display":"r_y","kind":"use","excerpt":"H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-607cd9691e7de5b0","tex":"r_y^2","aliases":"r_y^2","display":"r_y^2","kind":"use","excerpt":"u_r(t) _2^2 =c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2) =cos^2(E_rt)+sin^2(E_rt)=1.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e4805d2243d9f7ec","tex":"R_z","aliases":"R_z","display":"R_z","kind":"use","excerpt":"A packet need not commute with an unbounded Hamiltonian. Suppose now that H is self-adjoint, P is a bounded orthogonal projection preserving D(H), and [H,P] on D(H) extends to a bounded operator B. For nonreal z, let R_z=(H-z)^-1. It maps HH into D(H), so all products used below have their stated meaning. Multiplying the commutator identity on the ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-22b0ec1ebb0d2020","tex":"r_z","aliases":"r_z","display":"r_z","kind":"use","excerpt":"H(1,r)e_0=(1,0,r_z,r_x+ir_y)^T.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-98b9fa48a3b1a5b4","tex":"r_z^2","aliases":"r_z^2","display":"r_z^2","kind":"use","excerpt":"u_r(t) _2^2 =c_r(t)^2+s_r(t)^2(1+r_z^2+r_x^2+r_y^2) =cos^2(E_rt)+sin^2(E_rt)=1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8c6f7b1421c8035e","tex":"r_\\pm","aliases":"r_± pm","display":"r_\\pm","kind":"use","excerpt":"Charged horizons and the turning surface. Write f(R)=1-frac2MR+fracQ^2R^2. For 0<|Q|<M, its two positive roots are r_±=M±sqrtM^2-Q^2. At either root p(R)^2=R^2, so the canonical quantities in [source reference] remain finite if F_r does. For |Q|=M, the two roots coincide at R=M. For |Q|>M there is no real root. At Q=0, the outer root is 2M and the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b11c81e6903fa9bf","tex":"\\Ran","aliases":"Ran","display":null,"kind":"use","excerpt":"For r≥1, (E_r-E_r-1)^2=E_r-2E_r-1+E_r-1=D_r. For 1≤ r<s, expansion of (E_r-E_r-1)(E_s-E_s-1) gives zero by the same minimum rule. The products with D_0 vanish as well. Telescoping gives the resolution of the identity. Finally Ran E_r-1⊂Ran E_r, so the detail rank is p^r-p^r-1, while E_0 has rank one.","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-040c923c57f7a5f9","tex":"\\operatorname{Ran}","aliases":"Ran operatorname","display":"\\operatorname{Ran}","kind":"use","excerpt":"Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"The linear address-forgetting map C_phase|s,a⟩=|a⟩ maps a 2^h+1-dimensional dyadic unit register onto a 2^h-dimensional principal phase register. Its kernel is the span of the 2^h vectors |0,a⟩-|1,a⟩. Thus rankC_phase=2^h, dimker C_phase=2^h. This is a rank-deficient linear sum of amplitudes. It is not an isometry, and it is not the quantum channel","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"use","excerpt":"um_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint block symmetry implies ⟨ψ_s,H_srψ_r⟩ =overline⟨ψ_r,H_rsψ_s⟩. Th","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"xpansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm poly=I. This i","count":43},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a58e4c2e82e7cd1b","tex":"\\mathrm{root}","aliases":"root mathrm","display":"\\mathrm{root}","kind":"use","excerpt":"g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-61f9609b70de73a0","tex":"\\rVert","aliases":"rVert","display":"\\rVert","kind":"use","excerpt":"The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm ter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-78e04f6069787e74","tex":"\\rVert^2","aliases":"rVert^2 rVert","display":"\\rVert^2","kind":"use","excerpt":"The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm ter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"[Finite Feshbach identity] If z∉σ(D), define Σ(z)=B(z-D)^-1B^*, S(z)=z-A-Σ(z). Then z-H is invertible exactly when S(z) is invertible. On this common regular set, ι_L^*(z-H)^-1ι_L · = S(z)^-1,nonumber","count":18},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fe39e36813ced2eb","tex":"S^{\\mathsf T}","aliases":"S^ T mathsf","display":"S^{\\mathsf T}","kind":"use","excerpt":"Aa=1 when f(a)=A and zero otherwise, these formulas read p^ c=Rp, J^ c=RJR^ T, R^ T1_ c=1_ f. The last identity expresses exactly one coarse assignment per fine index. An arbitrary binary matrix does not suffice. If S performs a second grouping, then S(Rp)=(SR)p, S(RJR^ T)S^ T =(SR)J(SR)^ T. Thus a tower of exact groupings is associative. There is ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d7305573ca2e697d","tex":"S^*","aliases":"S^*","display":"S^*","kind":"use","excerpt":"malized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The on","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c8d0b37b4af3bfcc","tex":"S^2","aliases":"S^2","display":"S^2","kind":"use","excerpt":"oing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bc919570ec8558a8","tex":"S^a","aliases":"S^a","display":"S^a","kind":"use","excerpt":"Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dcc980c36e2b8074","tex":"S^\\ell","aliases":"S^ℓ ell","display":"S^\\ell","kind":"use","excerpt":"The same construction works for any circulant polynomial H=∑_ℓ c_ℓ S^ℓ, with symbol ∑_ℓ c_ℓ P_p(θ)^ℓ. It preserves all bands, whether or not their energies coincide. For N=p^h, digit reversal relates this standard residue split to the newest-digit adic split developed earlier. It is still necessary to distinguish the retained residue-zero sites fro","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c7fcb94025e0dcdc","tex":"S_{\\mathrm D}","aliases":"S_ D mathrm","display":"S_{\\mathrm D}","kind":"use","excerpt":"The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d510ede8513110af","tex":"S_{\\mathrm G}","aliases":"S_ G mathrm","display":"S_{\\mathrm G}","kind":"use","excerpt":"The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b14c6b717dca7bbf","tex":"S_{\\mathrm M}^{(1)}","aliases":"S_ M^(1) mathrm","display":"S_{\\mathrm M}^{(1)}","kind":"use","excerpt":"The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"use","excerpt":"The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e,","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"A precise conditional transfer theorem. Split s_F=s_0+δ, where s_0^2=0 and δ has degree one. Suppose ( C_C,s_0,C) is a coarse complex and the maps are chain maps: s_0I=Is_0,C, Ps_0=s_0,CP. Assume a degree-minus-one homotopy H satisfies PI=id_C, id_F-IP=s_0H+Hs_0,","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7cad08312c1a5a2b","tex":"s_{0,C}","aliases":"s_0,C","display":"s_{0,C}","kind":"use","excerpt":"A precise conditional transfer theorem. Split s_F=s_0+δ, where s_0^2=0 and δ has degree one. Suppose ( C_C,s_0,C) is a coarse complex and the maps are chain maps: s_0I=Is_0,C, Ps_0=s_0,CP. Assume a degree-minus-one homotopy H satisfies PI=id_C, id_F-IP=s_0H+Hs_0,","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1782caccc889225f","tex":"s_0^2","aliases":"s_0^2","display":"s_0^2","kind":"use","excerpt":"A precise conditional transfer theorem. Split s_F=s_0+δ, where s_0^2=0 and δ has degree one. Suppose ( C_C,s_0,C) is a coarse complex and the maps are chain maps: s_0I=Is_0,C, Ps_0=s_0,CP. Assume a degree-minus-one homotopy H satisfies PI=id_C, id_F-IP=s_0H+Hs_0,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"use","excerpt":"djoint for the grid inner product inneruw_h, and let multiplication by the real array b be self-adjoint. Then inneruA_h[b]u_h =-frac12inneru(Db)u_h. For two solutions of the split equation with the same prescribed b, remaining in a positive-radicand interval where |g'|≤ L_g, one obtains normR(t)-widetilde R(t)_h ≤ e^(L_g+S_1/2)tnormR(0)-widetilde R","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-92f500208fc7568a","tex":"S_{A,1,-1}","aliases":"S_A,1,-1","display":"S_{A,1,-1}","kind":"use","excerpt":"A concrete counterexample removes any possible inference from flatness alone. Let both half-links be i, and prescribe u=1. Their product is v=-1. A cube whose twelve positive links are all i has every face plaquette equal to one. On a one-site periodic lattice, however, τ=1 and S_A,1,1=P_++P_-=1, S_A,1,-1=-P_+-P_-=-1. Thus the coarse mismatch is -2","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-241b8b95f6be5201","tex":"S_{A,1,1}","aliases":"S_A,1,1","display":"S_{A,1,1}","kind":"use","excerpt":"A concrete counterexample removes any possible inference from flatness alone. Let both half-links be i, and prescribe u=1. Their product is v=-1. A cube whose twelve positive links are all i has every face plaquette equal to one. On a one-site periodic lattice, however, τ=1 and S_A,1,1=P_++P_-=1, S_A,1,-1=-P_+-P_-=-1. Thus the coarse mismatch is -2","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-860b396d57115c52","tex":"S_{A,\\sigma,(a,b)}^2","aliases":"S_A,σ,(a,b)^2 sigma","display":"S_{A,\\sigma,(a,b)}^2","kind":"use","excerpt":"A flat cube need not reconstruct the prescribed edge. Conversely, take an arbitrary ordered half-link pair (a,b) and call its actual product v=ba. It is always a subdivision of v, even when the separately prescribed coarse link is u≠ v. The exact directional defect is D_rm fineψ · :=S_A,σ,(a,b)^2I_aψ-I_aS_A,τ,uψnotag · =I_a(S_A,τ,v-S_A,τ,u)ψ. The e","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-da78c73c79172728","tex":"S_{A,\\sigma,v}","aliases":"S_A,σ,v sigma","display":"S_{A,\\sigma,v}","kind":"use","excerpt":"s step can be refined factor by factor. If a local linear map N satisfies N^2=M, then N commutes with the scalar interpolation. Consequently N^2S_A,σ,v^ 2N^2I_a =I_aMS_A,τ,uM. Unitary spectral phases provide explicit choices of N. The refined expression replaces each coarse factor by its two half factors. It does not assert that two repetitions of ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1832780214718152","tex":"S_{A,\\sigma,v}^{\\,2}","aliases":"S_A,σ,v^ 2 sigma","display":"S_{A,\\sigma,v}^{\\,2}","kind":"use","excerpt":"[Exact two-half-step refinement] For every coarse state ψ and subdivision satisfying [source reference], S_A,σ,v^ 2I_aψ=I_aS_A,τ,uψ. No flatness hypothesis is needed. When the fine shift is applied twice, the mixed projector terms vanish because P_+P_-=P_-P_+=0. The remaining positive channel travels two half-edges forward, and the negative channel","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-19bf8b5638e7fe88","tex":"S_{A,\\tau,u}^{-1}","aliases":"S_A,τ,u^-1 tau","display":"S_{A,\\tau,u}^{-1}","kind":"use","excerpt":"For a bijection τ:V→ V and a forward scalar link u:V→ U(1), define (S_A,τ,uψ)(x) =u(τ^-1x)P_A^+ψ(τ^-1x) +u(x)^-1P_A^-ψ(τ x). This agrees with the orientation dictionary [source reference]. Its inverse is (S_A,τ,u^-1ϕ)(x) =u(x)^-1P_A^+ϕ(τ x) +u(τ^-1x)P_A^-ϕ(τ^-1x). Substitution proves both inverse identities: same-channel terms cancel the link with ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f2070054f7f973f8","tex":"S_{A,\\tau,u^g}","aliases":"S_A,τ,u^g tau","display":"S_{A,\\tau,u^g}","kind":"use","excerpt":"tution proves both inverse identities: same-channel terms cancel the link with its inverse and mixed-channel terms vanish. Orthogonality, followed by reindexing x through τ in each channel, gives ∑_xinnerSψ(x)Sϕ(x) =∑_xinnerψ(x)ϕ(x). For u^g(x)=g(τ x)u(x)g(x)^-1, direct endpoint cancellation gives S_A,τ,u^gG_g=G_gS_A,τ,u. Products of these directio","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-300f3cadafb9f8db","tex":"S_{A,\\tau,v}","aliases":"S_A,τ,v tau","display":"S_{A,\\tau,v}","kind":"use","excerpt":"A flat cube need not reconstruct the prescribed edge. Conversely, take an arbitrary ordered half-link pair (a,b) and call its actual product v=ba. It is always a subdivision of v, even when the separately prescribed coarse link is u≠ v. The exact directional defect is D_rm fineψ · :=S_A,σ,(a,b)^2I_aψ-I_aS_A,τ,uψnotag · =I_a(S_A,τ,v-S_A,τ,u)ψ. The e","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5df9b8f58b7161eb","tex":"s_b","aliases":"s_b","display":"s_b","kind":"use","excerpt":"[Effect continuity] Under these hypotheses, dot p_b=∑_a J^F_ba, ∑_bdot p_b=0. If F_b=F_b(t) is differentiable in operator norm, the first identity becomes dot p_b=s_b+∑_a J^F_ba, s_b=innerψdot F_bψ, ∑_b s_b=0. Self-adjointness of F_b and differentiation give dot p_b=2ReinnerF_bψKψ for a fixed effect. Summing the two terms of [source reference] over","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-971eac196f4fcd05","tex":"S_{\\mathrm{BV}}","aliases":"S_BV mathrm","display":"S_{\\mathrm{BV}}","kind":"use","excerpt":"The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or eq","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-92045ce8b164570b","tex":"S_C","aliases":"S_C","display":"S_C","kind":"use","excerpt":"-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constan","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a04e13a6c34c8c88","tex":"s_C","aliases":"s_C","display":"s_C","kind":"use","excerpt":"0 · 1/25 · 0 . Its square is nonzero, as its (2,2) entry is -1/25. The transferred block in that same test is s_C=0 · 0 · 0 1 · 0 · 1/25 0 · 0 · 0, s_C^2=0. This small rational example illustrates the distinction between a projection and a transfer. The matrices by themselves are an algebraic diagnostic. They do not define gravitational constraints","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-59f247a519a27877","tex":"S_{C,\\hbar}","aliases":"S_C,ℏ hbar","display":"S_{C,\\hbar}","kind":"use","excerpt":"Eliminating z first gives the effective coefficient μ for y^2 and the factor (m+n)^-1/2. Eliminating y next gives d^-1/2. Their product is D^-1/2, exactly as in direct elimination. For the Euclidean weight e^-S/ℏ, using a flat coarse measure instead records this determinant as S_C,ℏ=S_C+tfracℏ2log D, up to an additive constant. A nonlinear symmetry","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4d4661ecebad12fb","tex":"s_C^2","aliases":"s_C^2","display":"s_C^2","kind":"use","excerpt":"0 · 1/25 · 0 . Its square is nonzero, as its (2,2) entry is -1/25. The transferred block in that same test is s_C=0 · 0 · 0 1 · 0 · 1/25 0 · 0 · 0, s_C^2=0. This small rational example illustrates the distinction between a projection and a transfer. The matrices by themselves are an algebraic diagnostic. They do not define gravitational constraints","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a1170f96f1f38668","tex":"S_G","aliases":"S_G","display":"S_G","kind":"use","excerpt":"Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e8fe4e26c95819b","tex":"S_h","aliases":"S_h","display":"S_h","kind":"use","excerpt":"Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-64eeff7a0fb8b5f2","tex":"\\mathcal S_h","aliases":"S_h mathcal","display":"\\mathcal S_h","kind":"use","excerpt":"βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations specify a Dirac representation but are not needed to prove charge or energy conservation in this finite model. The action fixes all normalization and sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_j","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-612081e3dc461c18","tex":"S_h^{(2)}","aliases":"S_h^(2)","display":"S_h^{(2)}","kind":"use","excerpt":"Set T_h=innerππ_h-tfrac12innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c6c3417bdce5077d","tex":"S_{i,\\mathrm{coarse}}","aliases":"S_i,coarse mathrm","display":"S_{i,\\mathrm{coarse}}","kind":"use","excerpt":"lation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of [source reference]: S_i,fine^2 I_r=I_rS_i,coarse. If the mass sub","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d71adb22cfa23dea","tex":"S_{i,\\mathrm{fine}}^2","aliases":"S_i,fine^2 mathrm","display":"S_{i,\\mathrm{fine}}^2","kind":"use","excerpt":"lation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source reference] is the explicit straight compatibility hypothesis. It is gauge covariant, and it is exactly the hypothesis needed to repeat the projector argument of [source reference]: S_i,fine^2 I_r=I_rS_i,coarse. If the mass sub","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"use","excerpt":"[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-22cf9853e0e53ac8","tex":"S_N","aliases":"S_N","display":"S_N","kind":"use","excerpt":"operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there. For S_Nc=∑_n≤ Nc_ne_n, normc-S_Nc_H_λ^2 =∑_n>N(1+","count":26},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-74d9bae07e421353","tex":"S_N^*","aliases":"S_N^*","display":"S_N^*","kind":"use","excerpt":"For the last assertion, fix an orthonormal basis (e_j) and define norm X_2^2=∑_jnormXe_j^2<∞. Each column (S_N-I)Xe_j tends to zero, and its squared norm is bounded by 4normXe_j^2. Dominated convergence gives norm(S_N-I)X_2→0. The right error follows by taking adjoints, using S_N^*=S_N and normX^*_2=norm X_2. The ideal inequality normAY_2≤norm Anor","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-25afd0ab9b73955e","tex":"\\mathcal S_p","aliases":"S_p mathcal","display":"\\mathcal S_p","kind":"use","excerpt":"[Exact free-cycle decimation] For U_p-1(x/2)≠0, the Schur operator on the retained sublattice is S_p(z)=fracκU_p-1(x/2) [R_p(x)I_M-(T+T^*)]. Equivalently, S_p(z)=zI-H_rm eff^(p)(z), where H_rm eff^(p)(z) · =μ_p(z)I+κ_p(z)(T+T^*),nonumber","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7994b21f9b979579","tex":"S_{p,h}","aliases":"S_p,h","display":"S_{p,h}","kind":"use","excerpt":"Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ee7713fc89d10bf","tex":"S_{p,h+1}","aliases":"S_p,h+1","display":"S_{p,h+1}","kind":"use","excerpt":"Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f12cd5af3f8bce51","tex":"S_{p,h}^*","aliases":"S_p,h^*","display":"S_{p,h}^*","kind":"use","excerpt":"Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5b939a36dfaae506","tex":"s_q","aliases":"s_q","display":"s_q","kind":"use","excerpt":"The error also has an exact expression. For fixed mode, let r=p(n), q=q_h(n), with the corresponding c_r,s_r,c_q,s_q. Coordinate subtraction gives","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a6bf5eb7ecfc145e","tex":"s_r","aliases":"s_r","display":"s_r","kind":"use","excerpt":"Set E_r=sqrt1+|r|^2, c_r(t)=cos(E_rt), and s_r(t)=sin(E_rt)/E_r. PH03 gives","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c1c5554e3132b956","tex":"S_{\\mathrm{sign}}","aliases":"S_sign mathrm","display":"S_{\\mathrm{sign}}","kind":"use","excerpt":"Tracing out the orthogonal b states yields the equally weighted mixture of the two displayed sign states. Their overlap is cos(gt). A mixture of two unit vectors with equal weights and overlap magnitude c has eigenvalues (1+c)/2 and (1-c)/2: its trace is one and its determinant on their two-dimensional span is (1-c^2)/4. Thus the sign entropy in bi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b387b1687e20541b","tex":"S_u","aliases":"S_u","display":"S_u","kind":"use","excerpt":"1|. The constant is independent of the number of sites. The actual half-grid defect in [source reference], with half-cell Hilbert measure, obeys the same bound. At each output site the difference of the shifts has a positive-channel term and a negative-channel term. Orthogonality gives the exact energy identity norm(S_v-S_u)ψ^2 =∑_x[ · |v(τ^-1x)-u(","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2f57d1ab5cc8f2ce","tex":"S_v","aliases":"S_v","display":"S_v","kind":"use","excerpt":"1|. The constant is independent of the number of sites. The actual half-grid defect in [source reference], with half-cell Hilbert measure, obeys the same bound. At each output site the difference of the shifts has a positive-channel term and a negative-channel term. Orthogonality gives the exact energy identity norm(S_v-S_u)ψ^2 =∑_x[ · |v(τ^-1x)-u(","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50b3713c204305be","tex":"S_W","aliases":"S_W","display":"S_W","kind":"use","excerpt":"For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the anal","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6a81d4e3e02ea338","tex":"\\mathrm{sign}","aliases":"sign mathrm","display":"\\mathrm{sign}","kind":"use","excerpt":"In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They d","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-21670bd4e275805e","tex":"\\sinh","aliases":"sinh","display":"\\sinh","kind":"use","excerpt":"The coefficient maps R_L,j in [source reference] are idempotents on analytic functions. The evaluated operators E_L,j(tA_0) generally are not. For L=2, E_2,0(tA_0)=cosh(tA_0) and E_2,1(tA_0)=sinh(tA_0), which are neither an orthogonal projector family nor a pair of independent unitary evolutions. The sum reconstructs an evolution; an individual lan","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e02ae8115a0a024c","tex":"\\mathrm{Site}","aliases":"Site mathrm","display":"\\mathrm{Site}","kind":"use","excerpt":"Exact straight refinement and curvature are independent. The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phas","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-72d39f9d2160995b","tex":"\\mathrm{std}","aliases":"std mathrm","display":"\\mathrm{std}","kind":"use","excerpt":"For h≥1, put N=p^h and m=p^h-1. Two permutation unitaries separate different digits: A^std_p,h|pk+r⟩ · =|r⟩⊗|k⟩, · 0≤ r<p,quad0≤ k<m,nonumber A^ad_p,h|a+d p^h-1⟩ · =|d⟩⊗|a⟩, · 0≤ d<p,quad0≤ a<m. The standard polyphase split extracts the least significant residue. The adic split extracts the newest, most significant digit at the current depth. Let R","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f90138132573de84","tex":"\\mathfrak{su}","aliases":"su mathfrak","display":"\\mathfrak{su}","kind":"use","excerpt":"Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all suffi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"[Real level energies define a self-adjoint operator] The operator in [source reference] is densely defined, closed, and self-adjoint. The finite cylinder functions form an operator core. It is bounded if and only if sup_m|ϵ_m|<∞, and its spectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the squa","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0341617620bd99b1","tex":"\\sup_m","aliases":"sup_m sup","display":"\\sup_m","kind":"use","excerpt":"[Real level energies define a self-adjoint operator] The operator in [source reference] is densely defined, closed, and self-adjoint. The finite cylinder functions form an operator core. It is bounded if and only if sup_m|ϵ_m|<∞, and its spectrum is σ(H_ϵ)=overline{ϵ_m:m≥0}. The orthogonal decomposition [source reference] identifies f with the squa","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7e2fcded366f165","tex":"\\sup_{m>M}","aliases":"sup_m>M sup","display":"\\sup_{m>M}","kind":"use","excerpt":"The omitted embedded resolvent has the exact norm norm(I- E_M)(z-H_ϵ)^-1 =sup_m>M|z-ϵ_m|^-1. The upper bound follows by Parseval. For the lower bound, test on a unit vector at any omitted level and take the supremum. Since every level is nonzero and finite dimensional, this proves the following equivalence.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9ee1f451cebb509d","tex":"\\sup_{n>N}","aliases":"sup_n>N sup","display":"\\sup_{n>N}","kind":"use","excerpt":"For a diagonal bounded multiplier f(H_λ), zero padding gives the exact formula normf(H_λ)-S_Nf(H_λ)S_N =sup_n>N|f(λ_n)|. For the unitary multiplier f(λ)=e^-itλ, this norm is one at every finite cutoff. Equation [source reference] remains valid. Uniformity in time for a fixed state and failure of operator-norm convergence are therefore compatible.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8fdddc59d2c468c6","tex":"\\sup_{|t|\\le T}","aliases":"sup_|t|≤ T sup le","display":"\\sup_{|t|\\le T}","kind":"use","excerpt":"sup_|t|≤ T f_h(t)-f(t) _L^2 ≤ Th^2(∑_n∈ SC(n)^2 a_n _2^2)^1/2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c161d17e61a95e63","tex":"\\T","aliases":"T","display":null,"kind":"use","excerpt":"Fourier synthesis on the physical torus. The scalar Fourier characters e_n(x)=e^2π i n· x are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is F:HH⟶KK, HH=L^2(T^3;C^4). The four-spinor Parseval identity and surjectivity of s","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-adac703e9d252af9","tex":"\\mathbb T","aliases":"T mathbb","display":"\\mathbb T","kind":"use","excerpt":"T^3=( R/ Z)^3,","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","excerpt":"W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"-C^-1B^* · I. Both outer factors are invertible and have determinant one. This proves the invertibility equivalence and determinant formula. Inverting the factors gives the stated upper left block. Alternatively, in (z-H)(u,v)^ T=(f,0)^ T, the lower equation gives v=C^-1B^*u and the upper equation becomes S(z)u=f. The same elimination with f=0 prov","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-10d6fa1215b924bf","tex":"T^*","aliases":"T^*","display":"T^*","kind":"use","excerpt":"[Exact free-cycle decimation] For U_p-1(x/2)≠0, the Schur operator on the retained sublattice is S_p(z)=fracκU_p-1(x/2) [R_p(x)I_M-(T+T^*)]. Equivalently, S_p(z)=zI-H_rm eff^(p)(z), where H_rm eff^(p)(z) · =μ_p(z)I+κ_p(z)(T+T^*),nonumber","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-27360346a415cf2f","tex":"\\T^3","aliases":"T^3 T","display":null,"kind":"use","excerpt":"Fourier synthesis on the physical torus. The scalar Fourier characters e_n(x)=e^2π i n· x are normalized relative to Haar measure of total mass one. For a four-spinor field, apply the scalar Fourier coefficient map to each of its four coordinates. The resulting map is F:HH⟶KK, HH=L^2(T^3;C^4). The four-spinor Parseval identity and surjectivity of s","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b1c3e8d8054aff08","tex":"\\mathbb T^3","aliases":"T^3 mathbb","display":"\\mathbb T^3","kind":"use","excerpt":"T^3=( R/ Z)^3,","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-78de04bec91c6824","tex":"T^g","aliases":"T^g","display":"T^g","kind":"use","excerpt":"For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the s","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ceeef154f3220be7","tex":"T^I","aliases":"T^I","display":"T^I","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4ff31b43de1984cc","tex":"t_*","aliases":"t_*","display":"t_*","kind":"use","excerpt":"th equal weights and overlap magnitude c has eigenvalues (1+c)/2 and (1-c)/2: its trace is one and its determinant on their two-dimensional span is (1-c^2)/4. Thus the sign entropy in bits is S_sign(t) =h_2(frac1+|cos(gt)|2), h_2(u)=-ulog_2u-(1-u)log_2(1-u), with 0log_2 0=0. At t_* =π/(2|g|) the two sign states are orthogonal and the entropy is one","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","excerpt":"The modulus-one multipliers give U(t)^*U(t)=I and U(t+s)=U(t)U(s). They also give strong continuity: norm(U(t)-U(t_0))f_2^2 =∑_m|e^-itϵ_m-e^-it_0ϵ_m|^2 norm F_mf_2^2⟶0. To justify this limit, first bound the omitted summands by 4∑_m>Mnorm F_mf_2^2, then use ordinary continuity in the finite sum up to M. On Dom(H_ϵ) the same argument, using |(e^-itϵ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"A complete three-level example. For b=2, direct substitution gives T_1f(h) · =2f(h/2)-f(h), T_2f(h) · =frac8f(h/4)-6f(h/2)+f(h)3, T_3f(h) · =frac64f(h/8)-56f(h/4)+14f(h/2)-f(h)21. Take the exact scalar polynomial f(h)=1+2h-3h^2+4h^3. The responses are T_1f(h)=1+frac32h^2-3h^3, T_2f(h)=1+frac12h^3, T_3f(h)=1. These are polynomial identities for ever","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"A complete three-level example. For b=2, direct substitution gives T_1f(h) · =2f(h/2)-f(h), T_2f(h) · =frac8f(h/4)-6f(h/2)+f(h)3, T_3f(h) · =frac64f(h/8)-56f(h/4)+14f(h/2)-f(h)21. Take the exact scalar polynomial f(h)=1+2h-3h^2+4h^3. The responses are T_1f(h)=1+frac32h^2-3h^3, T_2f(h)=1+frac12h^3, T_3f(h)=1. These are polynomial identities for ever","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","excerpt":"A complete three-level example. For b=2, direct substitution gives T_1f(h) · =2f(h/2)-f(h), T_2f(h) · =frac8f(h/4)-6f(h/2)+f(h)3, T_3f(h) · =frac64f(h/8)-56f(h/4)+14f(h/2)-f(h)21. Take the exact scalar polynomial f(h)=1+2h-3h^2+4h^3. The responses are T_1f(h)=1+frac32h^2-3h^3, T_2f(h)=1+frac12h^3, T_3f(h)=1. These are polynomial identities for ever","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"use","excerpt":"For example, on a joint depth packet one may take A_ r =(∑_aϵ_a, rT_a) ⊗ I_phase, r, ϵ_a, r∈R, T_a=T_a^*. The matrices T_a act on the sign and torsion factors. Their mutual commutation is unnecessary because their sum is self-adjoint on the finite block. The coefficients assigned to an existing packet are fixed under refinement. An extension acting","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50b87ddd607eac6e","tex":"T_a^*","aliases":"T_a^*","display":"T_a^*","kind":"use","excerpt":"For example, on a joint depth packet one may take A_ r =(∑_aϵ_a, rT_a) ⊗ I_phase, r, ϵ_a, r∈R, T_a=T_a^*. The matrices T_a act on the sign and torsion factors. Their mutual commutation is unnecessary because their sum is self-adjoint on the finite block. The coefficients assigned to an existing packet are fixed under refinement. An extension acting","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0b61846000086dfe","tex":"T_h","aliases":"T_h","display":"T_h","kind":"use","excerpt":"Set T_h=innerππ_h-tfrac12innerπ_rm trπ_rm tr_h, U_h=tfrac12innerh G(h)_h. With lapse perturbation n and shift perturbation n_i, the action is S_h^(2)=∫[ innerπdot h_h-T_h-U_h +innernC_h-innern_iC_i_h]dt. Variation of n,n_i imposes C=C_i=0. The remaining equations are dot h_ij · =2π_ij-δ_ijπ_rm tr +δ_in_j+δ_jn_i,nonumber","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8428b354040d2207","tex":"t_H","aliases":"t_H","display":"t_H","kind":"use","excerpt":"Recorded first marginally trapped sphere. The event finder takes a quadratic interpolation of the spatial maximum of 2m/r and a linear interpolation between consecutive time steps bracketing its first crossing of one. The archived event table is small rrrrr N_r · h · t_H · r_H · Δ M/M_0","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c6f45d72806f285","tex":"T_j^n","aliases":"T_j^n","display":"T_j^n","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d9a3064def648911","tex":"\\mathcal T_K","aliases":"T_K mathcal","display":"\\mathcal T_K","kind":"use","excerpt":"A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normaliz","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-27d2210af02484ac","tex":"\\mathcal T_{K+L}","aliases":"T_K+L mathcal","display":"\\mathcal T_{K+L}","kind":"use","excerpt":"A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normaliz","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0874c28c34a14314","tex":"\\mathcal T_K^{(4)}","aliases":"T_K^(4) mathcal","display":"\\mathcal T_K^{(4)}","kind":"use","excerpt":"For coefficients in T_K^(4) and a finite Grassmann algebra, the parent action has degree at most six, and the displayed antifield couplings have degree at most three. The master-equation and invariance coefficients have at most seven band-K factors. The sufficient rule 16K<N places them below Nyquist and transfers the classical identities coefficie","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6fc5e3fb71882a80","tex":"\\mathcal T_{K_j}","aliases":"T_K_j mathcal","display":"\\mathcal T_{K_j}","kind":"use","excerpt":"If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j),","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fae727c1986a152d","tex":"\\mathcal T_L","aliases":"T_L mathcal","display":"\\mathcal T_L","kind":"use","excerpt":"A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normaliz","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-64afaf0ed6cbcadc","tex":"T_n","aliases":"T_n","display":"T_n","kind":"use","excerpt":"We now eliminate intermediate residue sites in [source reference], assuming κ≠0. The retained sites are C={0,p,…,(M-1)p} and the eliminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-807e619b0b36adc6","tex":"T_p","aliases":"T_p","display":"T_p","kind":"use","excerpt":"eighboring paths. Each path contributes the onsite self-energy κ U_p-2/U_p-1 by Lemma~[source reference]; the two contributions add. The path joining two adjacent retained sites contributes off-diagonal self-energy κ/U_p-1. Thus S_p(z) =κ(x-frac2U_p-2(x/2)U_p-1(x/2))I -fracκU_p-1(x/2)(T+T^*). The identity xU_p-1(x/2)-2U_p-2(x/2)=2T_p(x/2) follows f","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1a4b381b4644d7ce","tex":"t_P","aliases":"t_P","display":"t_P","kind":"use","excerpt":"Fix M>0, a real charge Q, and a branch ε∈{+1,-1}. On 2MR-Q^2>0, define p(R)=εsqrt2MR-Q^2, v(R)=fracp(R)R. The Painlev e--Gullstrand metric takes the form ds^2=-dt_P^2+(dR+v(R)dt_P)^2+R^2dΩ^2. No inverse factor of 1-2M/R+Q^2/R^2 appears. The metric can therefore cross a horizon while the lapse and radial spatial metric remain regular.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-efd63624efc4bcf3","tex":"\\mathcal T_{p,h}","aliases":"T_p,h mathcal","display":"\\mathcal T_{p,h}","kind":"use","excerpt":"W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-40affe5513361680","tex":"\\mathcal T_{p,h}^*","aliases":"T_p,h^* mathcal","display":"\\mathcal T_{p,h}^*","kind":"use","excerpt":"d The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). With our positive Fourier sign, th","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9ef06c1f67b393ad","tex":"t_P^2","aliases":"t_P^2","display":"t_P^2","kind":"use","excerpt":"Fix M>0, a real charge Q, and a branch ε∈{+1,-1}. On 2MR-Q^2>0, define p(R)=εsqrt2MR-Q^2, v(R)=fracp(R)R. The Painlev e--Gullstrand metric takes the form ds^2=-dt_P^2+(dR+v(R)dt_P)^2+R^2dΩ^2. No inverse factor of 1-2M/R+Q^2/R^2 appears. The metric can therefore cross a horizon while the lapse and radial spatial metric remain regular.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bcbfdf0d1b59ea6b","tex":"t_{\\rm Shapiro}","aliases":"t_rm Shapiro rm","display":"t_{\\rm Shapiro}","kind":"use","excerpt":"N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb), α(L,b","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e88803809ed376b6","tex":"T_{y\\leftarrow x}","aliases":"T_y← x leftarrow","display":"T_{y\\leftarrow x}","kind":"use","excerpt":"_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Products act on column vectors from right to left. More generally, a transporter T_y← x has the transformation g_yT_y← xg_x^-1. This notation checks b","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d52dba8cafc2212f","tex":"T_{z\\leftarrow y}","aliases":"T_z← y leftarrow","display":"T_{z\\leftarrow y}","kind":"use","excerpt":"Matrix order · T_z← yT_y← x transports first from x to y, then from y to z. The rightmost factor acts first.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a084b0e913e1579b","tex":"\\mathrm{tor}","aliases":"tor mathrm","display":"\\mathrm{tor}","kind":"use","excerpt":"Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8dd594b4bce51346","tex":"\\mathrm{tors}","aliases":"tors mathrm","display":"\\mathrm{tors}","kind":"use","excerpt":"In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They d","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-89161acad9d67f8d","tex":"\\operatorname{Tr}_B","aliases":"Tr_B operatorname","display":"\\operatorname{Tr}_B","kind":"use","excerpt":"al because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, with (e","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7ded009c618f2421","tex":"\\operatorname{Tr}_{\\mathrm{sign}}","aliases":"Tr_sign operatorname mathrm","display":"\\operatorname{Tr}_{\\mathrm{sign}}","kind":"use","excerpt":"Discarding the sign as a quantum operation instead takes the partial trace ρ↦Tr_signρ. Both pure states |0,a⟩ and |1,a⟩ then produce the same reduced phase state |a⟩⟨ a|. Yet the sign observable Z_sign=(|0⟩⟨0|-|1⟩⟨1|)⊗ I has expectations +1 and -1 on those states. No measurement on the retained phase state can reconstruct this difference. The obstr","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e37ab0f79926296","tex":"\\mathrm U","aliases":"U mathrm","display":"\\mathrm U","kind":"use","excerpt":"C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The au","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ec7d40a789aa2d83","tex":"u'","aliases":"u'","display":"u'","kind":"use","excerpt":"Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-11b41534cdf61584","tex":"u''","aliases":"u''","display":"u''","kind":"use","excerpt":"Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df5c58f22a65049f","tex":"u^{-1}","aliases":"u^-1","display":"u^{-1}","kind":"use","excerpt":"· +|v(x)^-1-u(x)^-1|^2 normP_-ψ(τ x)^2]. There is no mixed term, even though the two spinors come from different sites: they lie in orthogonal fixed channel ranges. Unit modulus implies |v^-1-u^-1|=|v-u|. Bound every coefficient by δ^2, reindex the two sums through τ, and use normP_+η^2+normP_-η^2=normη^2. The result is at most δ^2normψ^2. Taking s","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"|1-cos u|=2sin^2(u/2)≤ u^2/2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-12b5175db95756e4","tex":"U^g","aliases":"U^g","display":"U^g","kind":"use","excerpt":"(Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-da5e436b748b454f","tex":"u^p","aliases":"u^p","display":"u^p","kind":"use","excerpt":"For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup gene","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7c05ecd0a14b319f","tex":"u^T","aliases":"u^T","display":"u^T","kind":"use","excerpt":"For an interval summation-by-parts derivative with a diagonal positive norm matrix H and HD+D^TH=B, multiplication by b commutes with H. The uncancelled term is instead inneruA_h[b]u_H =frac12u^TB diag(b)u -frac12inneru(Db)u_H. Its boundary flux must be controlled using an actual boundary closure and appropriate boundary data. Saying that a stencil","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7d05da6d886b7035","tex":"u_{(2\\pi,0,0)}","aliases":"u_(2π,0,0) pi","display":"u_{(2\\pi,0,0)}","kind":"use","excerpt":"The continuum packet has the same form with u_A,u_D replaced by u_(2π,0,0) and u_(2π,2π,2π). At t=0, both equal","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5274c3aacd45a0cc","tex":"u_{(2\\pi,2\\pi,2\\pi)}","aliases":"u_(2π,2π,2π) pi pi pi","display":"u_{(2\\pi,2\\pi,2\\pi)}","kind":"use","excerpt":"The continuum packet has the same form with u_A,u_D replaced by u_(2π,0,0) and u_(2π,2π,2π). At t=0, both equal","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b0b57f70ac14529","tex":"u_{(6,0,0)}","aliases":"u_(6,0,0)","display":"u_{(6,0,0)}","kind":"use","excerpt":"f_h(t)-f(t) _L^2^2 = · frac14 u_(6,0,0)(t)-u_(2π,0,0)(t) _2^2 · +frac14 u_(6,6,6)(t)-u_(2π,2π,2π)(t) _2^2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8b335f28186fa8e0","tex":"u_{(6,6,6)}","aliases":"u_(6,6,6)","display":"u_{(6,6,6)}","kind":"use","excerpt":"f_h(t)-f(t) _L^2^2 = · frac14 u_(6,0,0)(t)-u_(2π,0,0)(t) _2^2 · +frac14 u_(6,6,6)(t)-u_(2π,2π,2π)(t) _2^2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df73ec13d71ee514","tex":"u_*","aliases":"u_*","display":"u_*","kind":"use","excerpt":"For a nondegenerate visible spin direction with normalized low-space direction u_*, the visible weight of its normalized full eigenstate is [1-σ'(E)]^-1. If k=m_0=0, the spin eigenspace is degenerate, but the same normalization relation holds for each chosen unit vector in that eigenspace.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e124bcd45f61b489","tex":"U_{-1}","aliases":"U_-1","display":"U_{-1}","kind":"use","excerpt":"dots,(M-1)p} and the eliminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The second-kind polynomials U_n satisfy U_-1=0, U_0=1 and U_n+1(t)=2tU_n(t)-U_n-1(t).","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","excerpt":"and the eliminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The second-kind polynomials U_n satisfy U_-1=0, U_0=1 and U_n+1(t)=2tU_n(t)-U_n-1(t).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"[Causal memory with the initial hidden forcing] The finite system idot u=Au+Bv, idot v=Dv+B^*u, with initial data (u_0,v_0), is equivalent to idot u(t) · =Au(t)+B e^-itDv_0 -i∫_0^tK(t-s)u(s) ds,nonumber","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"The normalized effective energy is (z-μ_p(z))/κ_p(z)=R_p(x). The parameter z remains fixed through the elimination. Since μ_p and κ_p depend on it, this is an energy-dependent spectral closure, not an energy-independent Hamiltonian flow. For p=2, R_2(x)=x^2-2 and U_1(x/2)=x, giving H_rm eff^(2)(z) =μ I+fracκ^2z-μ(2I+T+T^*), z≠μ. This decimation ret","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e94ce7ed00675a9a","tex":"U_{2,h}","aliases":"U_2,h","display":"U_{2,h}","kind":"use","excerpt":"The first coordinate issue is specific to units. For an odd prime p and an integer h≥0, set U_p,h=(Z/p^h+1Z)^×. For p=2 the useful indexing starts one digit later: U_2,h=(Z/2^h+2Z)^×. The shifted dyadic index retains the sign even when the principal phase register has dimension one. It prevents the two units modulo four from being collapsed into on","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-432c14b23419ac6d","tex":"u_A","aliases":"u_A","display":"u_A","kind":"use","excerpt":"u_A(t)= cos(sqrt37t)-isin(sqrt37t)/sqrt37 0 0 -6isin(sqrt37t)/sqrt37 ,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-91c677571574a0b9","tex":"u_a","aliases":"u_a","display":"u_a","kind":"use","excerpt":"For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-77eae673cef2716c","tex":"U_{AB}","aliases":"U_AB","display":"U_{AB}","kind":"use","excerpt":"There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total gh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d776df64b15c1825","tex":"U_{a\\to b}","aliases":"U_a→ b to","display":"U_{a\\to b}","kind":"use","excerpt":"The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and pa","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-05e45e2c62b065b7","tex":"U_{a\\to b}'","aliases":"U_a→ b' to","display":"U_{a\\to b}'","kind":"use","excerpt":"The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and pa","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e21dd43c611f6c4e","tex":"u_b","aliases":"u_b","display":"u_b","kind":"use","excerpt":"For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1f20e1d7a367ca68","tex":"\\mathcal U_{\\mathrm{CRT}}","aliases":"U_CRT mathcal mathrm","display":"\\mathcal U_{\\mathrm{CRT}}","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-344059846d20b215","tex":"\\mathcal U_{\\mathrm{CRT}}^*","aliases":"U_CRT^* mathcal mathrm","display":"\\mathcal U_{\\mathrm{CRT}}^*","kind":"use","excerpt":"We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7a598e95d01e562","tex":"u_D","aliases":"u_D","display":"u_D","kind":"use","excerpt":"u_D(t)= cos(sqrt109t)-isin(sqrt109t)/sqrt109 0 -6isin(sqrt109t)/sqrt109 6(1-i)sin(sqrt109t)/sqrt109 .","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cb69d506a7bd35e5","tex":"U_e","aliases":"U_e","display":"U_e","kind":"use","excerpt":"We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt alon","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f72fa924cc22eb23","tex":"U_H","aliases":"U_H","display":"U_H","kind":"use","excerpt":"U_H(t)=e^-itH=∑_k=0^∞frac(-itH)^kk!.","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f59a4109ae21c00d","tex":"U_{H(1,r)}","aliases":"U_H(1,r)","display":"U_{H(1,r)}","kind":"use","excerpt":"u_r(t)=U_H(1,r)(t)e_0= c_r(t)-is_r(t) 0 -ir_zs_r(t) (r_y-ir_x)s_r(t) .","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6387f759e35b4e7d","tex":"U_{h,S}","aliases":"U_h,S","display":"U_{h,S}","kind":"use","excerpt":"Both remain in F_S. Their operators U_S(t),U_h,S(t) have the finite block diagonal coefficient actions with blocks U_H_n(t),U_H_h,n(t).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f51e0c71f0f6f66a","tex":"U_{H_{h,n}}","aliases":"U_H_h,n","display":"U_{H_{h,n}}","kind":"use","excerpt":"f(t,x)=∑_n∈ Se_n(x)U_H_n(t)a_n, f_h(t,x)=∑_n∈ Se_n(x)U_H_h,n(t)a_n.","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1e4c058a9b28e970","tex":"U_{H_n}","aliases":"U_H_n","display":"U_{H_n}","kind":"use","excerpt":"f(t,x)=∑_n∈ Se_n(x)U_H_n(t)a_n, f_h(t,x)=∑_n∈ Se_n(x)U_H_h,n(t)a_n.","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"The straight compatibility contract. For each direction, canonical fine links can be defined by v_i(p)=r(σ_i p) γ_i(p) r(p)^-1, where γ_i(p)=1 on a noncarrying half step and γ_i(p)=u_i(s) on a carrying half step. Cancellation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source referen","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-423beb526b282cb9","tex":"u_{i+1}","aliases":"u_i+1","display":"u_{i+1}","kind":"use","excerpt":"Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-37b4eb196aed5935","tex":"u_{i-1}","aliases":"u_i-1","display":"u_{i-1}","kind":"use","excerpt":"[Unique finite radial solution] With u_0,u_M fixed, S_h has a unique minimizer. Its interior values satisfy fracu_i+1-2u_i+u_i-1h^2=4π G r_iρ_i.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"If z is outside the closed energy set, the sequence (ϵ_m-z)^-1 is bounded and defines an inverse for H_ϵ-z. Its image lies in the domain because ϵ_m/(ϵ_m-z)=1+z/(ϵ_m-z) is bounded as well. This proves that such z is in the resolvent set. An energy ϵ_m is an eigenvalue, while a limit point z has unit level vectors u_j with norm(H_ϵ-z)u_j_2→0. A boun","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d776fc575cc5531d","tex":"U_j'","aliases":"U_j'","display":"U_j'","kind":"use","excerpt":"Linked shifts · A coefficient U_j in U_jψ_j+1 transports from j+1 back to j. Thus U_j'=g_jU_jg_j+1^-1. An outward link uses the opposite endpoint convention.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-79c73d373618f65a","tex":"u_{j+1}","aliases":"u_j+1","display":"u_{j+1}","kind":"use","excerpt":"On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3ea98bddbefa3e22","tex":"u_{j+2}","aliases":"u_j+2","display":"u_{j+2}","kind":"use","excerpt":"On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ef4170d2ceda737","tex":"U_{j-1}","aliases":"U_j-1","display":"U_{j-1}","kind":"use","excerpt":"Differentiate ρ_j and use dotρ_j=2Re(ψ_j^*dotψ_j). The mass term contributes zero, since ψ_j^*βψ_j is real. The forward hopping term contributes -J_j/h. The backward scalar is the conjugate of ψ_j-1^*α U_j-1ψ_j, using α=α^* and the scalar nature of U. Its contribution is J_j-1/h. This proves continuity. Periodic summation gives dotbarρ=0. Finally, ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-276fce3b16629317","tex":"u_{j-1}","aliases":"u_j-1","display":"u_{j-1}","kind":"use","excerpt":"On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e4ddf66f1da3ef2c","tex":"U_{j-1}^*","aliases":"U_j-1^*","display":"U_{j-1}^*","kind":"use","excerpt":"sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1, dot A_j · =E_j, · dot E_j · =J_j, · J_j · =Re(ψ_j^*α U_jψ_j+1)","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ae1b72e4bc4c00c0","tex":"U_{j-1}^n","aliases":"U_j-1^n","display":"U_{j-1}^n","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-969a2f5751eb71e7","tex":"u_{j-2}","aliases":"u_j-2","display":"u_{j-2}","kind":"use","excerpt":"On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b2a7e488064e7e3e","tex":"U_j^n","aliases":"U_j^n","display":"U_j^n","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c8e7ba9b6fa73cc6","tex":"U_L","aliases":"U_L","display":"U_L","kind":"use","excerpt":"For an embedding I_K→ L and a level-L step U_L, the adaptive map is U_L∘ I_K→ L. The level-L step preserves the invariants of its embedded initial state. To preserve the intended old data, the embedding must also satisfy the appropriate pullback relations for the Hamiltonian, constraints, gauge conditions, and symplectic form. A generic zero paddin","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a49e958a93a3404a","tex":"U_M","aliases":"U_M","display":"U_M","kind":"use","excerpt":"We use the positive-phase Fourier convention (F_n)_k x=n^-1/2exp(2πikx/n). The character transform of the unit group consequently factors as F_U_M= U_CRT^* [F_2⊗ F_2^h_2⊗ ⊗_p∈ P_ o(F_p-1⊗ F_p^h_p) ] U_CRT. Here the unit-group character labels have been identified with the same cyclic coordinate sets. This is a transform on the multiplicative unit g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c05f0648abbf982d","tex":"u_M","aliases":"u_M","display":"u_M","kind":"use","excerpt":"[Unique finite radial solution] With u_0,u_M fixed, S_h has a unique minimizer. Its interior values satisfy fracu_i+1-2u_i+u_i-1h^2=4π G r_iρ_i.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8d86088ae482ad75","tex":"u_m","aliases":"u_m","display":"u_m","kind":"use","excerpt":"[Convergent states and finite energy derivatives] Choose one unit vector u_m∈Ran F_m at every level, let 0<q<1, and set f_q=∑_m≥0sqrt(1-q)q^m u_m, ϵ_m=2^m. Then normf_q_2=1 and norm(I- E_M)f_q_2^2=q^M+1. However, f_q∈Dom(H_ϵ) exactly when 4q<1: normH_ϵ f_q_2^2 =(1-q)∑_m≥0(4q)^m. For q<1/4, its graph tail is exactly q^M+1+frac(1-q)(4q)^M+11-4q. For ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-65a04898bfc53b4d","tex":"U_{m-1}","aliases":"U_m-1","display":"U_{m-1}","kind":"use","excerpt":"[Chebyshev composition and Schur composition] For all positive integers m,n, the polynomial identities R_n(R_m(x)) · =R_mn(x)=R_m(R_n(x)),nonumber U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur i","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-139443253ce41588","tex":"U_M^{\\mathrm{emb}}","aliases":"U_M^emb mathrm","display":"U_M^{\\mathrm{emb}}","kind":"use","excerpt":"Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-15cdb597eea20cc8","tex":"U_{mn-1}","aliases":"U_mn-1","display":"U_{mn-1}","kind":"use","excerpt":"[Chebyshev composition and Schur composition] For all positive integers m,n, the polynomial identities R_n(R_m(x)) · =R_mn(x)=R_m(R_n(x)),nonumber U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur i","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2b181e5223e87432","tex":"U_{n+1}","aliases":"U_n+1","display":"U_{n+1}","kind":"use","excerpt":"liminated sites comprise the p-1 interior sites between adjacent retained sites. Identify ℓ^2(C) with C^M and let T|j⟩=|j+1pmod M⟩. Set x=fracz-μκ, R_n(x)=2T_n(x/2), where T_n is the first-kind Chebyshev polynomial. The second-kind polynomials U_n satisfy U_-1=0, U_0=1 and U_n+1(t)=2tU_n(t)-U_n-1(t).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8acdf74b89e87442","tex":"U_{n-2}","aliases":"U_n-2","display":"U_{n-2}","kind":"use","excerpt":"1 · 0. Its determinant is one and induction gives A(x)^n= U_n(x/2) · -U_n-1(x/2) U_n-1(x/2) · -U_n-2(x/2) , n≥1. For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps ther","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d17a28f5e5468c9f","tex":"U_p","aliases":"U_p","display":"U_p","kind":"use","excerpt":"-1 by Lemma~[source reference]; the two contributions add. The path joining two adjacent retained sites contributes off-diagonal self-energy κ/U_p-1. Thus S_p(z) =κ(x-frac2U_p-2(x/2)U_p-1(x/2))I -fracκU_p-1(x/2)(T+T^*). The identity xU_p-1(x/2)-2U_p-2(x/2)=2T_p(x/2) follows from the three-term recurrence and U_p-U_p-2=2T_p. For the latter identity,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-279939f80579ae9e","tex":"U_{p,h}","aliases":"U_p,h","display":"U_{p,h}","kind":"use","excerpt":"The first coordinate issue is specific to units. For an odd prime p and an integer h≥0, set U_p,h=(Z/p^h+1Z)^×. For p=2 the useful indexing starts one digit later: U_2,h=(Z/2^h+2Z)^×. The shifted dyadic index retains the sign even when the principal phase register has dimension one. It prevents the two units modulo four from being collapsed into on","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9fa59f1be311c29b","tex":"U_{p,h+1,N}","aliases":"U_p,h+1,N","display":"U_{p,h+1,N}","kind":"use","excerpt":"ds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary preserves its internal norm. The subsequent reversal shift sends these amplitudes to new arcs at the children, outside the old arc range. Therefore, for a unit","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-600ccddcc17ae38b","tex":"U_{p,h,N}","aliases":"U_p,h,N","display":"U_{p,h,N}","kind":"use","excerpt":"nd the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk U_p,h,N=SC is unitary because U_p","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c62e6bd466c701a3","tex":"U_{p,h,N}^*","aliases":"U_p,h,N^*","display":"U_{p,h,N}^*","kind":"use","excerpt":"|s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shift S sends the arc v→ u to u→ v, retaining the internal register. It is a permutation with S^*=S and S^2=I. The one-tick walk U_p,h,N=SC is unitary because U_p,h,N^*U_p,h,N=C^*S^*SC=C^*C=I. Here locality concerns the graph ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-15c0f31655cd03d4","tex":"\\mathcal U_{p,h_p}","aliases":"U_p,h_p mathcal","display":"\\mathcal U_{p,h_p}","kind":"use","excerpt":"Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3e789f6b6ba49589","tex":"\\mathcal U_{p,M}","aliases":"U_p,M mathcal","display":"\\mathcal U_{p,M}","kind":"use","excerpt":"The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projectio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4a0208673d51f84e","tex":"\\mathcal U_{p,m}","aliases":"U_p,m mathcal","display":"\\mathcal U_{p,m}","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6001a276c34996c2","tex":"\\mathcal U_{p,m+1}","aliases":"U_p,m+1 mathcal","display":"\\mathcal U_{p,m+1}","kind":"use","excerpt":"The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-63cfb5dda4ee754e","tex":"\\mathcal U_{p,M}^*","aliases":"U_p,M^* mathcal","display":"\\mathcal U_{p,M}^*","kind":"use","excerpt":"The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projectio","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e1387ffa6db2b83b","tex":"U_{p-1}","aliases":"U_p-1","display":"U_{p-1}","kind":"use","excerpt":"[Exact free-cycle decimation] For U_p-1(x/2)≠0, the Schur operator on the retained sublattice is S_p(z)=fracκU_p-1(x/2) [R_p(x)I_M-(T+T^*)]. Equivalently, S_p(z)=zI-H_rm eff^(p)(z), where H_rm eff^(p)(z) · =μ_p(z)I+κ_p(z)(T+T^*),nonumber","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d340f681877ced98","tex":"U_{p-2}","aliases":"U_p-2","display":"U_{p-2}","kind":"use","excerpt":", μ_p(z)=z-fracκ R_p(x)U_p-1(x/2). The eliminated sites split into M disjoint Dirichlet paths of length p-1. Hence their resolvent exists exactly when U_p-1(x/2)≠0. A retained site touches one endpoint on each of its two neighboring paths. Each path contributes the onsite self-energy κ U_p-2/U_p-1 by Lemma~[source reference]; the two contributions ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0835809a6850acb4","tex":"u_q","aliases":"u_q","display":"u_q","kind":"use","excerpt":"u_q(t)-u_r(t) _2^2 =(c_q-c_r)^2+(s_q-s_r)^2 +∑_j=x,y,z(q_js_q-r_js_r)^2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a822fb370eb1244d","tex":"u_r","aliases":"u_r","display":"u_r","kind":"use","excerpt":"u_r(t)=U_H(1,r)(t)e_0= c_r(t)-is_r(t) 0 -ir_zs_r(t) (r_y-ir_x)s_r(t) .","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2c0c595b21c3bb2f","tex":"U_{rs}","aliases":"U_rs","display":"U_{rs}","kind":"use","excerpt":", its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ec1736aa4eb4331a","tex":"U_{rs}^*","aliases":"U_rs^*","display":"U_{rs}^*","kind":"use","excerpt":", its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-27d332518a7e4b64","tex":"U_{rt}","aliases":"U_rt","display":"U_{rt}","kind":"use","excerpt":", its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0b4d1c17b7c34773","tex":"U_S","aliases":"U_S","display":"U_S","kind":"use","excerpt":"Both remain in F_S. Their operators U_S(t),U_h,S(t) have the finite block diagonal coefficient actions with blocks U_H_n(t),U_H_h,n(t).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-737d443707dfaa85","tex":"U_{ts}^*","aliases":"U_ts^*","display":"U_{ts}^*","kind":"use","excerpt":", its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ad63d401332166ba","tex":"u_x","aliases":"u_x","display":"u_x","kind":"use","excerpt":"For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [sourc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-72ae5158c8aa8c59","tex":"U_{x+\\widehat\\mu,\\nu}","aliases":"U_x+widehatμ,ν widehat mu nu","display":"U_{x+\\widehat\\mu,\\nu}","kind":"use","excerpt":"On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-66ca98fe046c552c","tex":"U_{x+\\widehat\\nu,\\mu}^{-1}","aliases":"U_x+widehatν,μ^-1 widehat nu mu","display":"U_{x+\\widehat\\nu,\\mu}^{-1}","kind":"use","excerpt":"On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ca8103aa551294d","tex":"U_{x,\\mu}","aliases":"U_x,μ mu","display":"U_{x,\\mu}","kind":"use","excerpt":"Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have t","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3127b0bda4adc1a6","tex":"u_{x,\\mu}","aliases":"u_x,μ mu","display":"u_{x,\\mu}","kind":"use","excerpt":"The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particula","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9681ef253d45ed2b","tex":"U_{x,\\mu}^*","aliases":"U_x,μ^* mu","display":"U_{x,\\mu}^*","kind":"use","excerpt":"Let V be a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have t","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ed02f96136bb7a27","tex":"U_{x,\\mu}^{-1}","aliases":"U_x,μ^-1 mu","display":"U_{x,\\mu}^{-1}","kind":"use","excerpt":"The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particula","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e53279255a429197","tex":"U_{x,\\mu}^g","aliases":"U_x,μ^g mu","display":"U_{x,\\mu}^g","kind":"use","excerpt":"ing on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Product","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-776573e0476b4656","tex":"u_{x,\\mu}^g","aliases":"u_x,μ^g mu","display":"u_{x,\\mu}^g","kind":"use","excerpt":"The forward link is u_x,μ=U_x,μ^-1:HH_x⟶HH_x+widehatμ, u_x,μ^g=g_x+widehatμu_x,μg_x^-1. The older time-dependent Abelian walk writes this forward link as L_j. The later geometric Lean sources use u. The non-Abelian stencil and the semidiscrete Hamiltonian use the backward U. These formulas describe the same edge in opposite directions. In particula","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-643d4c9f3c093afb","tex":"U_{x,\\nu}^{-1}","aliases":"U_x,ν^-1 nu","display":"U_{x,\\nu}^{-1}","kind":"use","excerpt":"On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1174cbd293ed0592","tex":"U_{x-\\widehat\\mu,\\mu}^{-1}","aliases":"U_x-widehatμ,μ^-1 widehat mu mu","display":"U_{x-\\widehat\\mu,\\mu}^{-1}","kind":"use","excerpt":"[Covariance of linked differences] For spacing h>0, define (D^+_μ,Uψ)_x · =fracU_x,μψ_x+widehatμ-ψ_xh, (Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7dded304aadbda87","tex":"U_{x-\\widehat\\mu,\\mu}^g","aliases":"U_x-widehatμ,μ^g widehat mu mu","display":"U_{x-\\widehat\\mu,\\mu}^g","kind":"use","excerpt":"For the positive edge, substitute [source reference]. The two gauge factors at x+widehatμ cancel, leaving g_x on the left. For the negative edge, (U_x-widehatμ,μ^g)^-1g_x-widehatμ =g_xU_x-widehatμ,μ^-1. Every term in either difference therefore has the common factor g_x.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c6df7a94e294a06b","tex":"u_y","aliases":"u_y","display":"u_y","kind":"use","excerpt":"For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [sourc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-07a46937ac588412","tex":"u_z","aliases":"u_z","display":"u_z","kind":"use","excerpt":"For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [sourc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":23},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4e7142ad42e2513e","tex":"v'","aliases":"v'","display":"v'","kind":"use","excerpt":"partial_t(m'-r^2 E)=0. A radial mass constraint imposed initially propagates under this continuum system. Expanding [source reference] gives dotΦ=(σ A)' and dotΠ=(σ B)'+2σ B/r. Differentiate E and subtract the radial derivative of the flux. Terms involving Π', Φ', and v' cancel, leaving ∂_t(r^2 E)-∂_r(r^2 J) =r^2[σ'AB+ΠΦdot v]. Since m=rv^2/2, the ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"use","excerpt":"[Finite refinement classification] Let V: H_c→ H_f be an isometry between finite Hilbert spaces. Set P=VV^* and Q=I-P. For self-adjoint operators H_c,H_f, the following conditions are equivalent: H_fV · =VH_c,","count":18},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cafb796619c588dd","tex":"v^{-1}","aliases":"v^-1","display":"v^{-1}","kind":"use","excerpt":"· +|v(x)^-1-u(x)^-1|^2 normP_-ψ(τ x)^2]. There is no mixed term, even though the two spinors come from different sites: they lie in orthogonal fixed channel ranges. Unit modulus implies |v^-1-u^-1|=|v-u|. Bound every coefficient by δ^2, reindex the two sums through τ, and use normP_+η^2+normP_-η^2=normη^2. The result is at most δ^2normψ^2. Taking s","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d4bbf0f17c40b804","tex":"v^{-1/2}","aliases":"v^-1/2","display":"v^{-1/2}","kind":"use","excerpt":"Cell values and normalized coefficients. For a measurable finite partition into cells of equal volume v>0, distinguish the physical constant value q_a∈C^4 on cell C_a from the normalized coefficient ψ_a=sqrt v q_a. The step interpolation is (Jψ)(x)=v^-1/2ψ_a (x∈ C_a), with equality understood almost everywhere. Direct integration gives normJψ_L^2^2","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-759397de55065787","tex":"V^2","aliases":"V^2","display":"V^2","kind":"use","excerpt":"Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber dot U · =γΔ_hP+e^2P[V^2-γ(D_hQ)^2],nonumber dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and expone","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"Impose R=r and Λ=1, and set N=σ>0, N^r=σ v, P_Λ=rv, P_φ=r^2Π, Φ=φ'. Here Π is the scalar momentum per areal area, not the reconstruction map Π_h. Define m=frac r2v^2, v=sqrtfrac2mr. The inward branch gives ds^2=-σ^2dt^2+(dr+σ vdt)^2+r^2dΩ^2. The following derivation applies on smooth patches with r>0, σ>0, and v>0. These hypotheses keep every divis","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-baf0eae4627f5125","tex":"V^\\ell","aliases":"V^ℓ ell","display":"V^\\ell","kind":"use","excerpt":"On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1edd83621988d298","tex":"\\mathcal V_0","aliases":"V_0 mathcal","display":"\\mathcal V_0","kind":"use","excerpt":"There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7f891d9184f7892b","tex":"\\mathcal V_1","aliases":"V_1 mathcal","display":"\\mathcal V_1","kind":"use","excerpt":"There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"In particular, α_yv=(-iv_3,iv_2,-iv_1,iv_0); this fixes a sign that cannot be changed silently in a worked example. The matrices act on column spinors. Grouping the first two and last two components gives their block form without any coordinate permutation:","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"For the dyadic case, successive squaring gives v_2(5^2^k-1)=k+2. Indeed 5-1 has valuation two, and for every such power the factor 5^2^k+1 has valuation one. Thus 5 has order 2^h modulo 2^h+2. Its powers are exactly the 2^h units congruent to one modulo four. Multiplication by -1 supplies the units congruent to three modulo four. These two classes ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"In particular, α_yv=(-iv_3,iv_2,-iv_1,iv_0); this fixes a sign that cannot be changed silently in a worked example. The matrices act on column spinors. Grouping the first two and last two components gives their block form without any coordinate permutation:","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a478c8ae160947ff","tex":"V_a","aliases":"V_a","display":"V_a","kind":"use","excerpt":"Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a23b8f8fe4c89a4d","tex":"\\mathcal V_a","aliases":"V_a mathcal","display":"\\mathcal V_a","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-56405f2e425e038c","tex":"V_a^*","aliases":"V_a^*","display":"V_a^*","kind":"use","excerpt":"Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2d8b529eb2a0a517","tex":"V_b","aliases":"V_b","display":"V_b","kind":"use","excerpt":"For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3c64cfa0416300f6","tex":"V_b^*","aliases":"V_b^*","display":"V_b^*","kind":"use","excerpt":"For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0b837e89a1e8104e","tex":"V_h","aliases":"V_h","display":"V_h","kind":"use","excerpt":"Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"The straight compatibility contract. For each direction, canonical fine links can be defined by v_i(p)=r(σ_i p) γ_i(p) r(p)^-1, where γ_i(p)=1 on a noncarrying half step and γ_i(p)=u_i(s) on a carrying half step. Cancellation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source referen","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-283debf6bf3b04eb","tex":"v_{i+1}","aliases":"v_i+1","display":"v_{i+1}","kind":"use","excerpt":"Differentiation with respect to u_i gives (2u_i-u_i-1-u_i+1)/(Gh)+4π h r_iρ_i. Setting it to zero gives the equation. For a variation v with v_0=v_M=0, the Hessian quadratic form is frac1Gh∑_i=0^M-1(v_i+1-v_i)^2. Its vanishing forces v to be constant and hence zero. The finite Hessian is positive definite, so the quadratic action has one stationary","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"use","excerpt":"dot V · =γΔ_hQ-2[UV-γ(D_hP)(D_hQ)]. Products and exponentials in these formulas are componentwise. The finite global momentum is C_h(z)=2h∑_j[U_j(D_hP)_j+e^2P_jV_j(D_hQ)_j]. It is linear in the velocity variables when the configuration P,Q is fixed.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are neede","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7b6a271c4eb9d3c3","tex":"v_{j-1}","aliases":"v_j-1","display":"v_{j-1}","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9381c9616ec2e3a3","tex":"\\mathcal V_L","aliases":"V_L mathcal","display":"\\mathcal V_L","kind":"use","excerpt":"The rate is a bound on states in the displayed regularity class. For an arbitrary L^2 state the tail still tends to zero, but no common geometric rate follows. A finite array already in V_L has zero tail once M≥ L, which is a different statement from testing an infinite tail.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-05a897b8ae55039b","tex":"v_{L-1}","aliases":"v_L-1","display":"v_{L-1}","kind":"use","excerpt":"Termwise differentiation on compact subsets gives E'_L,j=E_L,j-1 for j≥1. For j=0, the constant term vanishes under differentiation and reindexing gives E'_L,0=E_L,L-1. Let E_L=(E_L,0,…,E_L,L-1)^ T and define the cyclic matrix by (C_Lv)_0=v_L-1 and (C_Lv)_j=v_j-1 for j≥1. Then E_L'=C_L E_L, E_L(0)=e_0, C_L^L=I, E_L(z)=e^zC_Le_0. One can verify the ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f960dcf025809c6a","tex":"v_{Lq+j}","aliases":"v_Lq+j","display":"v_{Lq+j}","kind":"use","excerpt":"For a general finite transfer matrix M_1, the residue-state identity v_Lq+j=M_1^j(M_1^L)^qv_0 follows by adding exponents. It needs no hypergeometric coefficients. A factorial lane provides a particular analytic coordinate system; it does not impose a unique microscopic Hamiltonian on every such transfer problem.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db3b5f8ecfc5c837","tex":"\\mathcal V_M","aliases":"V_M mathcal","display":"\\mathcal V_M","kind":"use","excerpt":"Given ε>0, choose g∈ V_M with normf-g_2<ε. For m≥ M, the projection fixes g, so normf- E_m f_2 =norm(I- E_m)(f-g)_2≤normf-g_2<ε. This proves strong convergence without exchanging an infinite integral with a limit.","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eb7cc70f89201b8d","tex":"\\mathcal V_m","aliases":"V_m mathcal","display":"\\mathcal V_m","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-20cd50e2bdb5b105","tex":"v_M","aliases":"v_M","display":"v_M","kind":"use","excerpt":"Differentiation with respect to u_i gives (2u_i-u_i-1-u_i+1)/(Gh)+4π h r_iρ_i. Setting it to zero gives the equation. For a variation v with v_0=v_M=0, the Hessian quadratic form is frac1Gh∑_i=0^M-1(v_i+1-v_i)^2. Its vanishing forces v to be constant and hence zero. The finite Hessian is positive definite, so the quadratic action has one stationary","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c5e6ae3f2110675e","tex":"\\mathcal V_{M+1}","aliases":"V_M+1 mathcal","display":"\\mathcal V_{M+1}","kind":"use","excerpt":"There is also an exact construction for any specified self-adjoint compatible finite family along a cofinal nested sequence. Suppose finite-dimensional spaces V_0⊂ V_1 ⊂⋯ have dense union, and self-adjoint K_M satisfy K_M+1f=K_Mf for f∈ V_M. Self-adjointness implies that V_M^⊥∩ V_M+1 is invariant: for g there and f∈ V_M, innerfK_M+1g=innerK_M+1fg=0","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-24a238a771b3be94","tex":"\\mathcal V_M^\\perp","aliases":"V_M^⊥ mathcal perp","display":"\\mathcal V_M^\\perp","kind":"use","excerpt":"Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ac8b12063b2b1211","tex":"V_p","aliases":"V_p","display":"V_p","kind":"use","excerpt":"Refinement in distinct coordinates is path independent because (V_p⊗ I)(I⊗ V_q) =(I⊗ V_q)(V_p⊗ I)=V_p⊗ V_q. The identity on KK_tor leaves the separately retained sign channel untouched by the averaging. This is a finite set of primes. An infinite tensor product, a restricted adelic product, and a model including the Archimedean place each require f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"use","excerpt":"For odd p the valuation identity v_p((1+p)^p^k-1)=k+1 (k≥0) shows that 1+p has order p^h modulo p^h+1. To check the identity, the case k=0 is immediate. If u=1+c p^k+1 with pnmid c, the binomial expansion of u^p-1 has first term c p^k+2. Every remaining term has strictly larger p-valuation when p is odd. This proves the induction. The subgroup gene","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ecc628efee7ceee2","tex":"V_{p,h}","aliases":"V_p,h","display":"V_{p,h}","kind":"use","excerpt":"One more digit supplies p children for every address. The uniform refinement map is (V_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8da5c99bd0752f70","tex":"V_{p,h+w-1}","aliases":"V_p,h+w-1","display":"V_{p,h+w-1}","kind":"use","excerpt":"D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1ce4eff1892c6bcc","tex":"V_{p,h-1}^*","aliases":"V_p,h-1^*","display":"V_{p,h-1}^*","kind":"use","excerpt":"Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*, W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f8bbef892e55e829","tex":"V_{p,h}^*","aliases":"V_p,h^*","display":"V_{p,h}^*","kind":"use","excerpt":"_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V_p,h^*g)(a)=p^-1/2∑_d=0^p-1g(a+d p^h). Consequently V_p,h^*V_p,h=I, V_p,hV_p,h^*=E_h^(","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ada632c6fceb6663","tex":"V_{p,m}","aliases":"V_p,m","display":"V_{p,m}","kind":"use","excerpt":"The finite refinement map is V_p,me_a=frac1sqrt p ∑_d=0^p-1e_a+d p^m. Since B_m,a=bigsqcup_d B_m+1,a+d p^m, it satisfies U_p,m+1V_p,m= U_p,m. Indeed, the coefficient on each child indicator is p^-1/2p^(m+1)/2=p^m/2, precisely the coefficient on the parent. Thus the refinement changes the address resolution of an embedded state without changing the ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-310efc759ac5294c","tex":"V_{p,M-1}","aliases":"V_p,M-1","display":"V_{p,M-1}","kind":"use","excerpt":"The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projectio","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f0b72f4f5ef32b81","tex":"V_{p,m\\to M}","aliases":"V_p,m→ M to","display":"V_{p,m\\to M}","kind":"use","excerpt":"The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projectio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e96f90baddd90c65","tex":"V_{p,m\\to M}^*","aliases":"V_p,m→ M^* to","display":"V_{p,m\\to M}^*","kind":"use","excerpt":"The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projectio","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b70c2d293c03377d","tex":"V_q","aliases":"V_q","display":"V_q","kind":"use","excerpt":"Refinement in distinct coordinates is path independent because (V_p⊗ I)(I⊗ V_q) =(I⊗ V_q)(V_p⊗ I)=V_p⊗ V_q. The identity on KK_tor leaves the separately retained sign channel untouched by the averaging. This is a finite set of primes. An infinite tensor product, a restricted adelic product, and a model including the Archimedean place each require f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-724afc19e77e7bfd","tex":"v_q","aliases":"v_q","display":"v_q","kind":"use","excerpt":"mathsf H_p(θ) are E_q(θ)=μ+2κ cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component has the same value because e^ipα_q=e^iθ. The","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2e67d412be824541","tex":"v_R","aliases":"v_R","display":"v_R","kind":"use","excerpt":"For evolution, the radius equation gives dot R=-fracP_ΛR+N^rR' =-v(F)+F_t+v(F)=F_t. The radial-metric equation is dotΛ=fracΛ P_ΛR^2 -fracP_RR+(N^rΛ)'. Use p_R=M/p and v_R=M/(pR)-p/R^2. The non-F_tr terms cancel, leaving dotΛ=F_tr. The momentum equations in the N=1 convention are dot P_Λ · =N^rP_Λ' +frac12[1-fracP_Λ^2+Q^2R^2 -(fracR'Λ)^2],","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-51fd0818257204a0","tex":"v_r","aliases":"v_r","display":"v_r","kind":"use","excerpt":"⟨ v,w⟩=∑_r=0^3overlinev_rw_r, v _2^2=∑_r=0^3|v_r|^2.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c580baf0833ca721","tex":"v_x","aliases":"v_x","display":"v_x","kind":"use","excerpt":"For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [sourc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5b59d34e2b2271f4","tex":"v_y","aliases":"v_y","display":"v_y","kind":"use","excerpt":"For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [sourc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f88941ebb3c7a6bd","tex":"v_z","aliases":"v_z","display":"v_z","kind":"use","excerpt":"For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [sourc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50ce1a62663b77d9","tex":"\\Vert","aliases":"Vert","display":"\\Vert","kind":"use","excerpt":"[Relative-entropy dissipation] For a strictly positive solution of the finite jump balance and a fixed positive reference satisfying [source reference], D(pVertπ) · =∑_ap_alogfracp_aπ_a, fracddtD(pVertπ) · =-frac12∑_b,ac_ba(u_b-u_a)(log u_b-log u_a)≤0. notag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e355f29d98c97c06","tex":"\\operatorname{vol}","aliases":"vol operatorname","display":"\\operatorname{vol}","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c6a2a0e38dc26bdf","tex":"\\operatorname{vol}_e","aliases":"vol_e operatorname","display":"\\operatorname{vol}_e","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e83c58309eb2060b","tex":"\\mathcal W","aliases":"W mathcal","display":"\\mathcal W","kind":"use","excerpt":"Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*,","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b99d2e5f37ae7e17","tex":"w'","aliases":"w'","display":"w'","kind":"use","excerpt":"dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=σ(1-v). The plus channel is inward on the positive branch. The minus channel is outward when v<1, stationary at v=1, and inward when v>1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5638fa9691fa3e47","tex":"W^{(2)}","aliases":"W^(2)","display":"W^{(2)}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f6df49aeb91e7b72","tex":"\\mathcal W^*","aliases":"W^* mathcal","display":"\\mathcal W^*","kind":"use","excerpt":"If P_L=ι_Lι_L^*, the full space compression is P_L(z-H)^-1P_L=ι_L S(z)^-1ι_L^*. When a Haar transform W first exposes low and detail channels, the embedding in the original coordinates is ι_L= W^*(|0⟩⊗ I). The inverse on HH_L must be embedded in this way before it is compared with a full space matrix.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-11a0326400d8d7a5","tex":"W^{1,\\infty}","aliases":"W^1,∞ infty","display":"W^{1,\\infty}","kind":"use","excerpt":"The input norm contains a radial derivative. A small pointwise radius error alone therefore does not establish a small error in every reconstructed canonical field. This estimate transfers an already controlled discrete W^1,∞ radius error to the reconstructed state. It does not create that derivative control from an L^2 estimate.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"The constructed orbit is the unique global domain-valued solution with a given domain initial state. If two supplied solutions have the same initial value, their difference w satisfies dot w=-iH_λ w. Symmetry gives fracddtnorm w^2 =2Reinnerw-iH_λ w=0. The initial zero norm remains zero. This uniqueness proof extends to supplied solutions of a symme","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b2aee1add211ab2b","tex":"w_+","aliases":"w_+","display":"w_+","kind":"use","excerpt":"Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-],","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7bb383dea36e9371","tex":"w_+'","aliases":"w_+'","display":"w_+'","kind":"use","excerpt":"Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-], dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5d0598ac5861dddc","tex":"w_+^2","aliases":"w_+^2","display":"w_+^2","kind":"use","excerpt":"The same channels give the exact algebraic identities E · =frac14[(1+v)w_+^2+(1-v)w_-^2], J · =fracσ4[(1+v)^2w_+^2-(1-v)^2w_-^2]. For 0≤ v<1, E≥(1-v)(w_+^2+w_-^2)/4. This lower bound degenerates at a horizon. For v>1, the outward-channel coefficient in this coordinate mass density is negative. The scalar's local normal-frame quadratic density (Π^2+","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9182875dbc779918","tex":"w_-","aliases":"w_-","display":"w_-","kind":"use","excerpt":"Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-],","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c0eb6b90de5e504c","tex":"w_-'","aliases":"w_-'","display":"w_-'","kind":"use","excerpt":"Characteristic channels and the loss of a coordinate energy bound. Define w_+=Π+Φ and w_-=Π-Φ. Their equations are dot w_+= · σ(v+1)w_+' +[σ v'+σ'(v+1)]w_+ +fracσ r[(v+1)w_++(v-1)w_-], dot w_-= · σ(v-1)w_-' +[σ v'+σ'(v-1)]w_- +fracσ r[(v-1)w_-+(v+1)w_+]. Written as transport equations dot w+c w'=source, their coordinate speeds are c_+=-σ(v+1), c_-=","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e48a71f06564da0e","tex":"w_-^2","aliases":"w_-^2","display":"w_-^2","kind":"use","excerpt":"The same channels give the exact algebraic identities E · =frac14[(1+v)w_+^2+(1-v)w_-^2], J · =fracσ4[(1+v)^2w_+^2-(1-v)^2w_-^2]. For 0≤ v<1, E≥(1-v)(w_+^2+w_-^2)/4. This lower bound degenerates at a horizon. For v>1, the outward-channel coefficient in this coordinate mass density is negative. The scalar's local normal-frame quadratic density (Π^2+","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all suffi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-59e63012bc6d4f87","tex":"w_0","aliases":"w_0","display":"w_0","kind":"use","excerpt":"If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the requ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-84ca84b36e79ec9b","tex":"W_f","aliases":"W_f","display":"W_f","kind":"use","excerpt":"six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If W_f=e^iθ_f with each θ_f chosen as a princ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-057eae80a542fec9","tex":"W_f^{\\operatorname{or}(f,c)}","aliases":"W_f^or(f,c) operatorname","display":"W_f^{\\operatorname{or}(f,c)}","kind":"use","excerpt":"For scalar links, multiply the oriented plaquettes on the six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact gro","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db94573662ebfd05","tex":"W_h","aliases":"W_h","display":"W_h","kind":"use","excerpt":"Exact group multiplication and approximate Lie-algebra curvature are different operations. For fixed A,B∈su(2) define the commutator loop W_h=e^hAe^hBe^-hAe^-hB. Here A and B are constant skew-Hermitian traceless matrices, and h is a real scalar near zero. Because W_0=I, a single analytic matrix logarithm near the identity can be used for all suffi","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"Take a periodic chain with N≥3 sites, spacing h>0, and indices modulo N. The continuum metric motivating its coefficients is ds^2=-N(t,x)^2dt^2+a(t,x)^2dx^2, N(t,x)>0, a(t,x)>0. The repeated letter N in this metric is the lapse, whereas the integer N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are neede","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ebdf9f205f201a52","tex":"w_k","aliases":"w_k","display":"w_k","kind":"use","excerpt":"If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the requ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1d116e67893836b3","tex":"w_{k+1}","aliases":"w_k+1","display":"w_{k+1}","kind":"use","excerpt":"If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the requ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-466a54f902244928","tex":"W_n","aliases":"W_n","display":"W_n","kind":"use","excerpt":"[Completion of a modewise isometry] Suppose that for every mode n, V_n:C^4→C^4 is a complex linear isometry. The coordinate rule (Va)_n=V_na_n defines a complex linear isometry on KK. If each V_n has an inverse isometry W_n, the induced V and W are inverse maps on KK. For each n, normV_na_n^2=norma_n^2. Summability of the latter terms gives members","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9b94508bc7508bf1","tex":"\\mathcal W_n","aliases":"W_n mathcal","display":"\\mathcal W_n","kind":"use","excerpt":"For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the s","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-850addc7bcb89021","tex":"w_{nq+j}","aliases":"w_nq+j","display":"w_{nq+j}","kind":"use","excerpt":"If w_k+1=M_0w_k, each residue subsequence satisfies w_nq+j=M_0^j(M_0^n)^q w_0, 0≤ j<n. Thus ordinary matrix powering produces the normalized trace semigroup, including matrices with repeated eigenvalues at trace ±2. The trace alone does not determine M_0, its eigenvectors, or a local Hamiltonian realization. In the free chain, M_0=A(x) has the requ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-05ee5dd14eda9a8e","tex":"\\mathcal W_{p,h}","aliases":"W_p,h mathcal","display":"\\mathcal W_{p,h}","kind":"use","excerpt":"Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-42b5720616e472f2","tex":"\\mathcal W_{p,h}^*","aliases":"W_p,h^* mathcal","display":"\\mathcal W_{p,h}^*","kind":"use","excerpt":"Apply the child Fourier transform after adic analysis: W_p,h=(F_p⊗ I_m) A^ad_p,h. The zero child-frequency row of F_p is p^-1/2(1,…,1). Hence (⟨0|⊗ I_m) W_p,h · =V_p,h-1^*, W_p,h^* [(I_p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dfa15e280d3611a8","tex":"w_r","aliases":"w_r","display":"w_r","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-75d29d71343727f9","tex":"W_\\square","aliases":"W_square square","display":"W_\\square","kind":"use","excerpt":"We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt alon","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28b939ffc4f05e71","tex":"W_u","aliases":"W_u","display":"W_u","kind":"use","excerpt":"both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] pr","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-377769d6a9c94ee0","tex":"W_v","aliases":"W_v","display":"W_v","kind":"use","excerpt":"both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] pr","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a1354be5a54b8730","tex":"W_{x,\\mu\\nu}","aliases":"W_x,μν mu nu","display":"W_{x,\\mu\\nu}","kind":"use","excerpt":"On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-782a1381512fbead","tex":"W_{x,\\mu\\nu}^g","aliases":"W_x,μν^g mu nu","display":"W_{x,\\mu\\nu}^g","kind":"use","excerpt":"ν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber. Each intermediate gauge factor cancels in its written order, giving W_x,μν^g=g_xW_x,μνg_x","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":18},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d1655f35bb316a56","tex":"\\widehat D","aliases":"widehat D widehat","display":"\\widehat D","kind":"use","excerpt":"The periodic Fourier symbol is widehat D_4(θ)=fracihfrac8sinθ-sin2θ6. Its maximum dimensionless magnitude is 1.3722219798…. Applying the RK4 imaginary-axis stability interval to frozen constant advection gives frac|b|Δ th≤2.0612023174…. This is a frozen-coefficient diagnostic. A full variable-coefficient nonlinear RK4 convergence theorem still requ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-35728400f785376e","tex":"\\widehat D_4","aliases":"widehat D_4 widehat","display":"\\widehat D_4","kind":"use","excerpt":"The periodic Fourier symbol is widehat D_4(θ)=fracihfrac8sinθ-sin2θ6. Its maximum dimensionless magnitude is 1.3722219798…. Applying the RK4 imaginary-axis stability interval to frozen constant advection gives frac|b|Δ th≤2.0612023174…. This is a frozen-coefficient diagnostic. A full variable-coefficient nonlinear RK4 convergence theorem still requ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"[Physical torus Hamiltonian] The operator H_m^phys is densely defined, symmetric, self-adjoint, and closed on HH=L^2(T^3;C^4). Its domain consists exactly of fields satisfying ∑_n∈Z^3(1+|2π n|^2)normwidehat f(n)^2<∞, and for every domain vector, normH_m^physf_HH^2 =∑_n(m^2+|2π n|^2)normwidehat f(n)^2. The domain characterization is the coefficient ","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9f20b388cdc4505b","tex":"\\widehat f_h","aliases":"widehat f_h widehat","display":"\\widehat f_h","kind":"use","excerpt":"g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tangency identity ∇ C_h·widehat f_h =∇ C_h· f_h-Λ_hd_h=0. Every differentiable solution that remains in this region therefore keep","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4aae4b9331147cb6","tex":"\\widehat f_k","aliases":"widehat f_k widehat","display":"\\widehat f_k","kind":"use","excerpt":"A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normaliz","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c577d5457da31ff","tex":"\\widehat H","aliases":"widehat H widehat","display":"\\widehat H","kind":"use","excerpt":"Define Dom(widehat H_m) ={a∈KK:∑_nnormH_na_n^2<∞}, (widehat H_ma)_n=H_na_n. The domain is maximal for this coefficient action: it contains precisely those coefficient fields for which the stated output belongs to the ambient Hilbert space. It is a vector subspace. For example, normx+y^2≤2normx^2+2normy^2 and linearity of each H_n prove closure unde","count":25},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-49b14a998ffa2541","tex":"\\widehat H_m","aliases":"widehat H_m widehat","display":"\\widehat H_m","kind":"use","excerpt":"Define Dom(widehat H_m) ={a∈KK:∑_nnormH_na_n^2<∞}, (widehat H_ma)_n=H_na_n. The domain is maximal for this coefficient action: it contains precisely those coefficient fields for which the stated output belongs to the ambient Hilbert space. It is a vector subspace. For example, normx+y^2≤2normx^2+2normy^2 and linearity of each H_n prove closure unde","count":22},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df544400edfb9943","tex":"\\widehat H_m^*","aliases":"widehat H_m^* widehat","display":"\\widehat H_m^*","kind":"use","excerpt":"The adjoint domain and self-adjointness. Modewise Hermiticity gives symmetry on the maximal domain. If a,b belong to that domain, the two series in innerwidehat H_mab =∑_ninnerH_na_nb_n =∑_ninnera_nH_nb_n =innerawidehat H_mb converge absolutely by Cauchy--Schwarz. This establishes the inclusion widehat H_m⊆widehat H_m^*, including the corresponding","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3546643ac017f642","tex":"\\widehat H_{m_1}","aliases":"widehat H_m_1 widehat","display":"\\widehat H_{m_1}","kind":"use","excerpt":"[Weighted domain and mass independence] For every real m, Dom(widehat H_m) ={a∈KK: ∑_n(1+|p(n)|^2)norma_n^2<∞}. Consequently Dom(widehat H_m_1)=Dom(widehat H_m_2) for every two real masses. The weighted energy series is the sum of the nonnegative momentum series and m^2∑_nnorma_n^2. The second series is finite. Thus the energy series is finite exac","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-937384cfab5dedd5","tex":"\\widehat H_{m_2}","aliases":"widehat H_m_2 widehat","display":"\\widehat H_{m_2}","kind":"use","excerpt":"[Weighted domain and mass independence] For every real m, Dom(widehat H_m) ={a∈KK: ∑_n(1+|p(n)|^2)norma_n^2<∞}. Consequently Dom(widehat H_m_1)=Dom(widehat H_m_2) for every two real masses. The weighted energy series is the sum of the nonnegative momentum series and m^2∑_nnorma_n^2. The second series is finite. Thus the energy series is finite exac","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4a6775720ae1e169","tex":"\\widehat N","aliases":"widehat N widehat","display":"\\widehat N","kind":"use","excerpt":"On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5ec76de4222f5aa9","tex":"\\widehat U","aliases":"widehat U widehat","display":"\\widehat U","kind":"use","excerpt":"[Left-endpoint sampling can lose an order] Take A(x)=x near the origin and ψ(x)=1. With the left-endpoint prescription widehat U_h(x)=e^-ihA(x), the stencil at zero is fracwidehat U_h(0)-widehat U_h(-h)^-12h =frac1-e^-ih^22h=fraci2h+O(h^3). The continuum value (∂_x-iA(0))ψ(0) is zero. Thus this perfectly smooth input has a first-order error. The fo","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bf98495fc78739ad","tex":"\\widehat U_h","aliases":"widehat U_h widehat","display":"\\widehat U_h","kind":"use","excerpt":"[Left-endpoint sampling can lose an order] Take A(x)=x near the origin and ψ(x)=1. With the left-endpoint prescription widehat U_h(x)=e^-ihA(x), the stencil at zero is fracwidehat U_h(0)-widehat U_h(-h)^-12h =frac1-e^-ih^22h=fraci2h+O(h^3). The continuum value (∂_x-iA(0))ψ(0) is zero. Thus this perfectly smooth input has a first-order error. The fo","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b186d96cd6172b72","tex":"\\widehat\\mu","aliases":"widehatμ widehat mu","display":"\\widehat\\mu","kind":"use","excerpt":"A field assigns a vector to each site, but a local change of frame can be chosen independently at every site. A difference such as ψ_x+widehatμ-ψ_x therefore needs a rule for comparing its two entries. A link supplies that rule. Its orientation is part of its definition, rather than a sign convention to be inferred after a calculation.","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-81743d091025fb84","tex":"\\widehat\\nu","aliases":"widehatν widehat nu","display":"\\widehat\\nu","kind":"use","excerpt":"On a two-dimensional coordinate plane, the backward links define the based plaquette W_x,μν =U_x,μU_x+widehatμ,ν U_x+widehatν,μ^-1U_x,ν^-1. Starting with a vector at x, the rightmost factor moves it to x+widehatν; the next moves it to the far corner; the last two bring it back to x. The matrix product is therefore an endomorphism of the base fiber.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8d678effd8898b69","tex":"\\widetilde d","aliases":"widetilde d widetilde","display":"\\widetilde d","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-44074f8fd9fb43af","tex":"\\widetilde d_h","aliases":"widetilde d_h widetilde","display":"\\widetilde d_h","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-20f230bba299446b","tex":"\\widetilde E","aliases":"widetilde E widetilde","display":"\\widetilde E","kind":"use","excerpt":"Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5cb9657258ebae5f","tex":"\\widetilde E^a","aliases":"widetilde E^a widetilde","display":"\\widetilde E^a","kind":"use","excerpt":"Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-089f71382a559dc8","tex":"\\widetilde g","aliases":"widetilde g widetilde","display":"\\widetilde g","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-434fbfd6e1ff4264","tex":"\\widetilde g_U","aliases":"widetilde g_U widetilde","display":"\\widetilde g_U","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-34052e77aaaf9a0d","tex":"\\widetilde g_V","aliases":"widetilde g_V widetilde","display":"\\widetilde g_V","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-70fad85f996380ab","tex":"\\widetilde H","aliases":"widetilde H widetilde","display":"\\widetilde H","kind":"use","excerpt":"An alternative is to extend the finite Hamiltonian by the zero Hamiltonian on the omitted subspace: widetilde H_M=H_ϵ E_M =∑_m≤ Mϵ_m F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c74e69495f6cc411","tex":"\\widetilde H_M","aliases":"widetilde H_M widetilde","display":"\\widetilde H_M","kind":"use","excerpt":"An alternative is to extend the finite Hamiltonian by the zero Hamiltonian on the omitted subspace: widetilde H_M=H_ϵ E_M =∑_m≤ Mϵ_m F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9117c60fc5e9378b","tex":"\\widetilde K","aliases":"widetilde K widetilde","display":"\\widetilde K","kind":"use","excerpt":"The dynamics respects the same construction. Put widetilde K=VKV^*. It is skew-adjoint, and fracddtVψ=widetilde K Vψ. Moving adjoints across the inner product shows 2ReinnerΠ_bVψwidetilde KΠ_aVψ =2ReinnerF_bψKF_aψ=J^F_ba. The positive-effect current is therefore exactly the sharp current of this lifted flow evaluated on the embedded state. It is no","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-152380ca5fee3168","tex":"\\widetilde k","aliases":"widetilde k widetilde","display":"\\widetilde k","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3f611dfe324685e","tex":"\\widetilde{\\mathbf k}","aliases":"widetilde k widetilde mathbf","display":"\\widetilde{\\mathbf k}","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4e03157a81c6cd1b","tex":"\\widetilde k_i","aliases":"widetilde k_i widetilde","display":"\\widetilde k_i","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7d1e1357771f82a","tex":"\\widetilde k_j","aliases":"widetilde k_j widetilde","display":"\\widetilde k_j","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-939a02f74ff6b85e","tex":"\\widetilde n","aliases":"widetilde n widetilde","display":"\\widetilde n","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-271a4979a4fd2464","tex":"\\widetilde n^I","aliases":"widetilde n^I widetilde","display":"\\widetilde n^I","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e250d9bc1a41414","tex":"\\widetilde n_I","aliases":"widetilde n_I widetilde","display":"\\widetilde n_I","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c51909f438842cb3","tex":"\\widetilde n_J","aliases":"widetilde n_J widetilde","display":"\\widetilde n_J","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ed8c63057cfee2e2","tex":"\\widetilde Q","aliases":"widetilde Q widetilde","display":"\\widetilde Q","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a1dfb1836c1fd75b","tex":"\\widetilde R","aliases":"widetilde R widetilde","display":"\\widetilde R","kind":"use","excerpt":"djoint for the grid inner product inneruw_h, and let multiplication by the real array b be self-adjoint. Then inneruA_h[b]u_h =-frac12inneru(Db)u_h. For two solutions of the split equation with the same prescribed b, remaining in a positive-radicand interval where |g'|≤ L_g, one obtains normR(t)-widetilde R(t)_h ≤ e^(L_g+S_1/2)tnormR(0)-widetilde R","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-23b8f5bb7ff61960","tex":"\\widetilde R_j","aliases":"widetilde R_j widetilde","display":"\\widetilde R_j","kind":"use","excerpt":"ns normR(t)-widetilde R(t)_h ≤ e^(L_g+S_1/2)tnormR(0)-widetilde R(0)_h, normDb_∞≤ S_1. Skew adjointness cancels innerubDu_h+inneruD(bu)_h. The remaining term gives [source reference]. With e=R-widetilde R, the nonlinear difference satisfies |g(R_j)-g(widetilde R_j)|≤ L_g|e_j|, where g'(R)=fracMR-Q^2R^2sqrt2MR-Q^2. Thus tfrac12∂_tnorm e_h^2≤(L_g+S_1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aba1aecba944a16d","tex":"\\widetilde U","aliases":"widetilde U widetilde","display":"\\widetilde U","kind":"use","excerpt":"A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc68d486952395d8","tex":"\\widetilde U_h","aliases":"widetilde U_h widetilde","display":"\\widetilde U_h","kind":"use","excerpt":"A midpoint link widetilde U_h(x)=e^-ihA(x+h/2) has the same conclusion under these bounds. The midpoint quadrature error in the integral is O(h^3), and |e^ia-e^ib|≤|a-b| for real a,b transfers that error to the link. The two numerator perturbations are each O(h^3) for bounded ψ, hence the stencil perturbation is O(h^2).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-60cd197d59c08b6e","tex":"\\widetilde V","aliases":"widetilde V widetilde","display":"\\widetilde V","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea2a8d1df2130cc8","tex":"\\widetilde X","aliases":"widetilde X widetilde","display":"\\widetilde X","kind":"use","excerpt":"For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-86c1d03c1a864e81","tex":"\\widetilde z","aliases":"widetilde z widetilde","display":"\\widetilde z","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2e6d3d996f44d01d","tex":"\\mathbf x","aliases":"x mathbf","display":"\\mathbf x","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8aac0daabbebf28a","tex":"\\mathbf x^2","aliases":"x^2 mathbf","display":"\\mathbf x^2","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f3522098af4f1d38","tex":"x^a","aliases":"x^a","display":"x^a","kind":"use","excerpt":"Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bad8ba0dd09b4f7d","tex":"X^k","aliases":"X^k","display":"X^k","kind":"use","excerpt":"X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f15ad8081e22a9d1","tex":"X^{k-1-r}","aliases":"X^k-1-r","display":"X^{k-1-r}","kind":"use","excerpt":"X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7c235d1ebb9c7923","tex":"X^{k-r}","aliases":"X^k-r","display":"X^{k-r}","kind":"use","excerpt":"No commutation is assumed. Expanding the summands gives successive differences X^k-rY^r-X^k-1-rY^r+1, which cancel. If X , Y ≤ R, then","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0ed7bdf4a651c709","tex":"x^m","aliases":"x^m","display":"x^m","kind":"use","excerpt":"lows from the elementary fact that a degree-d polynomial over a field has at most d roots: in a finite subgroup of the multiplicative group, the least common multiple m of the element orders is realized by an element. To see this, take for each prime divisor of m a component having the maximal prime-power order and multiply those commuting componen","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-efe34588925c4a06","tex":"x^{p-1}","aliases":"x^p-1","display":"x^{p-1}","kind":"use","excerpt":"For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the requir","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f8239c285a2b08c9","tex":"x^{p-2}","aliases":"x^p-2","display":"x^{p-2}","kind":"use","excerpt":"For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the requir","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ff79273075ae7ad3","tex":"X_{0,j+1}^{-1}","aliases":"X_0,j+1^-1","display":"X_{0,j+1}^{-1}","kind":"use","excerpt":"First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c4a48827c05d4509","tex":"X_{00}","aliases":"X_00","display":"X_{00}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3f15eebc2170872f","tex":"X_{01}","aliases":"X_01","display":"X_{01}","kind":"use","excerpt":"· =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I · · X_01X_11Y_21X_12^-1X_02^-1Y_01^-1Y_00^-1. Now X_01^-1X_01 cancels, followed by X_11^-1X_11. The remaining factors are exactly [source","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5d877f50bbc2ba84","tex":"X_{01}^{-1}","aliases":"X_01^-1","display":"X_{01}^{-1}","kind":"use","excerpt":"· =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I · · X_01X_11Y_21X_12^-1X_02^-1Y_01^-1Y_00^-1. Now X_01^-1X_01 cancels, followed by X_11^-1X_11. The remaining factors are exactly [source","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2b1877e0f569623b","tex":"X_{02}^{-1}","aliases":"X_02^-1","display":"X_{02}^{-1}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bd1d17514be24765","tex":"X_{1,j+1}^{-1}","aliases":"X_1,j+1^-1","display":"X_{1,j+1}^{-1}","kind":"use","excerpt":"First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-805649c1b3a8b1e8","tex":"X_{10}","aliases":"X_10","display":"X_{10}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f17c0dba15962adb","tex":"X_{11}","aliases":"X_11","display":"X_{11}","kind":"use","excerpt":"· =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I · · X_01X_11Y_21X_12^-1X_02^-1Y_01^-1Y_00^-1. Now X_01^-1X_01 cancels, followed by X_11^-1X_11. The remaining factors are exactly [source","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5ec4a4d89f255491","tex":"X_{11}^{-1}","aliases":"X_11^-1","display":"X_{11}^{-1}","kind":"use","excerpt":"· =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I · · X_01X_11Y_21X_12^-1X_02^-1Y_01^-1Y_00^-1. Now X_01^-1X_01 cancels, followed by X_11^-1X_11. The remaining factors are exactly [source","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7a61a0fe64865421","tex":"X_{12}^{-1}","aliases":"X_12^-1","display":"X_{12}^{-1}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3674278fc192e035","tex":"x_1^2","aliases":"x_1^2","display":"x_1^2","kind":"use","excerpt":"nsfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary divergence. The detail Hessian is M_D=x_1^2+m · -m -m · m+n, D=(m+n)x_1^2+mn>0. Writi","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-48fbe887931fba9d","tex":"X_{1j}","aliases":"X_1j","display":"X_{1j}","kind":"use","excerpt":"First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"For the last assertion, fix an orthonormal basis (e_j) and define norm X_2^2=∑_jnormXe_j^2<∞. Each column (S_N-I)Xe_j tends to zero, and its squared norm is bounded by 4normXe_j^2. Dominated convergence gives norm(S_N-I)X_2→0. The right error follows by taking adjoints, using S_N^*=S_N and normX^*_2=norm X_2. The ideal inequality normAY_2≤norm Anor","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6f4fed7ff48f6423","tex":"X_2^2","aliases":"X_2^2","display":"X_2^2","kind":"use","excerpt":"For the last assertion, fix an orthonormal basis (e_j) and define norm X_2^2=∑_jnormXe_j^2<∞. Each column (S_N-I)Xe_j tends to zero, and its squared norm is bounded by 4normXe_j^2. Dominated convergence gives norm(S_N-I)X_2→0. The right error follows by taking adjoints, using S_N^*=S_N and normX^*_2=norm X_2. The ideal inequality normAY_2≤norm Anor","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8d0a8687ec2ae46a","tex":"x_2^2","aliases":"x_2^2","display":"x_2^2","kind":"use","excerpt":"-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"f_h(t,x)=frace^-itsqrt2e_0 +frac12e^2π ix_1u_A(t) +frac12e^2π i(x_1+x_2+x_3)u_D(t).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-77b2927cea561ce1","tex":"X_{ba}","aliases":"X_ba","display":"X_{ba}","kind":"use","excerpt":"The same atlas resolves an operator on both its input and output sides. For X∈ B(HH), define X_ba=P_bXP_a. The first index names the destination and the second names the source. As a map between packet spaces, this is X_ba:Ran P_a→Ran P_b. As a bounded operator on HH, it is extended by zero on the orthogonal complement of Ran P_a. Keeping these two","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7a2d99f30c0cc9af","tex":"X_{ba}^{V}","aliases":"X_ba^V","display":"X_{ba}^{V}","kind":"use","excerpt":"For X∈ B(HH), its channel block is X_ba^V=V_bXV_a^*. Inserting the partition on both sides proves X=∑_a,bV_b^*X_ba^VV_a. The lifted operator widetilde X=VXV^* is multiplicative on lifted operators, since V^*V=I. Its identity is R, not I_KK. Conversely, compressing arbitrary ambient operators need not preserve multiplication: V^*ABV-(V^*AV)(V^*BV)=V","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e3e9d53e5f23ac1e","tex":"X_{bc}","aliases":"X_bc","display":"X_{bc}","kind":"use","excerpt":"Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the ca","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-702e017c46584077","tex":"x_c","aliases":"x_c","display":"x_c","kind":"use","excerpt":"ampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivatives of F. If the rectangle is centered a","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-24f9a3fe39a3dae6","tex":"X_{ca}","aliases":"X_ca","display":"X_{ca}","kind":"use","excerpt":"Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the ca","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-14a3d06f2604bc2a","tex":"X_{cb}","aliases":"X_cb","display":"X_{cb}","kind":"use","excerpt":"[Block calculus] For bounded X,Y, X · =∑_b,aX_ba, · P_bXψ · =∑_aX_baψ_a, (XY)_ca · =∑_bX_cbY_ba, · (X^*)_ab · =(X_ba)^*. The identity has diagonal blocks P_a and zero off-diagonal blocks. If X=X^*, its opposite blocks are adjoints.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"use","excerpt":"For the fine cycle, x_k=2cos(2π k/(pM)) maps to 2cos(2π k/M). Each coarse Fourier label has exactly p fine preimages. If two labels already have the same numerical energy, their multiplicities add as usual. Band folding retains the branch label and its vector, whereas normalized decimation identifies all of these labels at the same coarse energy. T","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","excerpt":"an interval and is therefore an identity of polynomials, including complex x. For t=cosθ away from zeros of the sine denominators, U_n-1(T_m(t))U_m-1(t) =fracsin(mnθ)sin(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ec84613b712fe298","tex":"x_N","aliases":"x_N","display":"x_N","kind":"use","excerpt":"[A small state can have a large generator image] On ℓ^2(N_≥1), let He_n=ne_n on its maximal domain. The finitely supported vectors x_N=e_N/N satisfy normx_N=1/N→0, while normHx_N=1. This gives a direct reason to track the generator tail in addition to the state tail. It does not contradict closedness, because Hx_N=e_N has no norm limit.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"For completeness, an order-p-1 lift of any nonzero residue can be chosen coherently. The root equation x^p-1=1 has nonzero derivative (p-1)x^p-2 modulo p. If x_n is a root modulo p^n, substituting x_n+c p^n modulo p^n+1 determines one and only one residue c that cancels the error. Starting with a generator of (Z/pZ)^× produces ζ_p,h with the requir","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-77180a1ffcf8a2ca","tex":"x_{p,q}","aliases":"x_p,q","display":"x_{p,q}","kind":"use","excerpt":"The folded branches from Proposition~[source reference], in normalized coordinates, satisfy x_p,q(θ)=2cos (fracθ+2π qp), R_p(x_p,q(θ))=2cosθ. The second equality is just T_p(cosα)=cos(pα). For a coarse normalized energy in the open interval (-2,2), choose its angle in (0,π). These formulas supply the p distinct real roots of R_p(x)=2cosθ. At the en","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-facd8502ae1cad01","tex":"X_{rs}","aliases":"X_rs","display":"X_{rs}","kind":"use","excerpt":"rimality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-58812c8c592a182f","tex":"Y^*","aliases":"Y^*","display":"Y^*","kind":"use","excerpt":"For compact X, choose finite-rank Y with normX-Y<ε. Strong convergence is uniform on each fixed finite-dimensional subspace, and self-adjointness of S_N controls the right error by the corresponding left error for Y^*. Thus normS_NYS_N-Y→0. The difference between the two reconstruction errors is at most 2ε, proving the claim.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-61c288ea4029d6a5","tex":"y^{-1}","aliases":"y^-1","display":"y^{-1}","kind":"use","excerpt":"[Upper half-plane sign and the exact visible norm] For z=x+iy with y>0, ImΣ(z) =-yB((x-D)^2+y^2I)^-1B^*≤0, Im S(z)≥ yI. In particular S(z) is invertible and norm S(z)^-1≤ y^-1. For real E∉σ(D), Σ'(E)=-B(E-D)^-2B^*. If (u,v) is a normalized full eigenvector at E, then 1=⟨ u,(I-Σ'(E))u⟩. For real λ, Im(z-λ)^-1=-y/((x-λ)^2+y^2). Applying this identity","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"[Upper half-plane sign and the exact visible norm] For z=x+iy with y>0, ImΣ(z) =-yB((x-D)^2+y^2I)^-1B^*≤0, Im S(z)≥ yI. In particular S(z) is invertible and norm S(z)^-1≤ y^-1. For real E∉σ(D), Σ'(E)=-B(E-D)^-2B^*. If (u,v) is a normalized full eigenvector at E, then 1=⟨ u,(I-Σ'(E))u⟩. For real λ, Im(z-λ)^-1=-y/((x-λ)^2+y^2). Applying this identity","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-827a2f52108af912","tex":"y^j","aliases":"y^j","display":"y^j","kind":"use","excerpt":"The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( - 1, A_L,j, A_L,j; frac(-1)^Ly^LL^2L). O","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-13920cdfdee3eee1","tex":"Y^k","aliases":"Y^k","display":"Y^k","kind":"use","excerpt":"X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-10e9863564d53a0d","tex":"y^L","aliases":"y^L","display":"y^L","kind":"use","excerpt":"The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( - 1, A_L,j, A_L,j; frac(-1)^Ly^LL^2L). O","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d0d8e30ac786064e","tex":"y^{Lq+j}","aliases":"y^Lq+j","display":"y^{Lq+j}","kind":"use","excerpt":"The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-de7ba1dbdac1562b","tex":"y^n","aliases":"y^n","display":"y^n","kind":"use","excerpt":"The same separation gives the entire function B(y)=∑_n=0^∞frac(-1)^ny^n(n!)^2 =_0F_1(;1;-y). For y≥0, this is J_0(2sqrt y); the entire series itself does not need a square-root branch. Define B_L,j(y)=∑_q≥0(-1)^Lq+jy^Lq+j/[(Lq+j)!]^2. Squaring [source reference] gives B_L,j(y) =frac(-1)^jy^j(j!)^2 _0F_2L-1 ( -","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-90d98d4bd026d21c","tex":"Y^r","aliases":"Y^r","display":"Y^r","kind":"use","excerpt":"X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-95314123831de1c4","tex":"Y^{r+1}","aliases":"Y^r+1","display":"Y^{r+1}","kind":"use","excerpt":"No commutation is assumed. Expanding the summands gives successive differences X^k-rY^r-X^k-1-rY^r+1, which cancel. If X , Y ≤ R, then","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-211b18dbfa6b49f3","tex":"y^{rq+j}","aliases":"y^rq+j","display":"y^{rq+j}","kind":"use","excerpt":"The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the se","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e5f6938d2e45504d","tex":"y_*","aliases":"y_*","display":"y_*","kind":"use","excerpt":"-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constan","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d50f75767567fffe","tex":"Y_{00}","aliases":"Y_00","display":"Y_{00}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dadd0c0a7ac5ceef","tex":"Y_{00}^{-1}","aliases":"Y_00^-1","display":"Y_{00}^{-1}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8c837e3015731152","tex":"Y_{01}^{-1}","aliases":"Y_01^-1","display":"Y_{01}^{-1}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-344ce98a533836a9","tex":"Y_{0j}^{-1}","aliases":"Y_0j^-1","display":"Y_{0j}^{-1}","kind":"use","excerpt":"First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9ece1fcef8e48fb6","tex":"Y_{1j}","aliases":"Y_1j","display":"Y_{1j}","kind":"use","excerpt":"First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-00758ebfad58b574","tex":"Y_{1j}^{-1}","aliases":"Y_1j^-1","display":"Y_{1j}^{-1}","kind":"use","excerpt":"First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-57080b7d6f26d5e6","tex":"Y_2","aliases":"Y_2","display":"Y_2","kind":"use","excerpt":"For the last assertion, fix an orthonormal basis (e_j) and define norm X_2^2=∑_jnormXe_j^2<∞. Each column (S_N-I)Xe_j tends to zero, and its squared norm is bounded by 4normXe_j^2. Dominated convergence gives norm(S_N-I)X_2→0. The right error follows by taking adjoints, using S_N^*=S_N and normX^*_2=norm X_2. The ideal inequality normAY_2≤norm Anor","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f5ca61ea6e5f00f5","tex":"Y_{20}","aliases":"Y_20","display":"Y_{20}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9e5725576b0ba70d","tex":"Y_{21}","aliases":"Y_21","display":"Y_{21}","kind":"use","excerpt":"[Ordered patch identity] For invertible links in any matrix group, the coarse boundary holonomy satisfies W^(2) · :=X_00X_10Y_20Y_21 X_12^-1X_02^-1Y_01^-1Y_00^-1 · =R_0Y_00R_1Y_00^-1. This identity requires neither small links nor commuting links.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9597b8a338d4d116","tex":"Y_{2j}","aliases":"Y_2j","display":"Y_{2j}","kind":"use","excerpt":"First expand one row while retaining the product order: R_j · =X_0jX_1jY_2jX_1,j+1^-1Y_1j^-1 X_0j^-1X_0jY_1jX_0,j+1^-1Y_0j^-1 · =X_0jX_1jY_2jX_1,j+1^-1X_0,j+1^-1Y_0j^-1. The adjacent pairs X_0j^-1X_0j and Y_1j^-1Y_1j cancel. Substituting these row expressions gives R_0Y_00R_1Y_00^-1 = · X_00X_10Y_20X_11^-1X_01^-1 underbraceY_00^-1Y_00_I","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7be44d74687569ce","tex":"y_a","aliases":"y_a","display":"y_a","kind":"use","excerpt":"Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4c1e8c1c8cca56ba","tex":"Y_{ba}","aliases":"Y_ba","display":"Y_{ba}","kind":"use","excerpt":"[Block calculus] For bounded X,Y, X · =∑_b,aX_ba, · P_bXψ · =∑_aX_baψ_a, (XY)_ca · =∑_bX_cbY_ba, · (X^*)_ab · =(X_ba)^*. The identity has diagonal blocks P_a and zero off-diagonal blocks. If X=X^*, its opposite blocks are adjoints.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-00fe9b383f124db5","tex":"y_n","aliases":"y_n","display":"y_n","kind":"use","excerpt":"[Self-adjointness of the diagonal realization] The operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bbfc803998827b8a","tex":"\\mathrm{YM}","aliases":"YM mathrm","display":"\\mathrm{YM}","kind":"use","excerpt":"For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the anal","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Discarding the sign as a quantum operation instead takes the partial trace ρ↦Tr_signρ. Both pure states |0,a⟩ and |1,a⟩ then produce the same reduced phase state |a⟩⟨ a|. Yet the sign observable Z_sign=(|0⟩⟨0|-|1⟩⟨1|)⊗ I has expectations +1 and -1 on those states. No measurement on the retained phase state can reconstruct this difference. The obstr","count":31},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"item S. V. Kozyrev, ``Wavelet theory as p-adic spectral analysis,'' Izvestiya: Mathematics 66(2), 367--376 (2002). https://doi.org/10.1070/IM2002v066n02ABEH000381doi:10.1070/IM2002v066n02ABEH000381. Published English Theorems 2 and 7 inspected. The Q_p Vladimirov setting supplies wavelet context; the present Z_p level-diagonal construction is prove","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-449753faf1f7a481","tex":"z^{-1}","aliases":"z^-1","display":"z^{-1}","kind":"use","excerpt":"athsf F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber norm((z-widetilde H_M)^-1-(z-H_ϵ)^-1)","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1a49e6bec04c7b56","tex":"Z^2","aliases":"Z^2","display":"Z^2","kind":"use","excerpt":"Multiplying the four exponential power series through degree three gives W_h=I+h^2[A,B]+frach^32[A+B,[A,B]]+O(h^4). The analytic logarithm has log(I+Z)=Z+O(norm Z^2) near zero. Since Z=W_h-I=O(h^2), its correction begins at order h^4. Dividing by h^2 proves [source reference]. At half spacing the linear term is hC_3/2, so 2F_h/2-F_h removes it exac","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"A nonlinear Gaussian reduction in closed form. The small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d425233dded4b47a","tex":"\\Z^3","aliases":"Z^3 Z","display":null,"kind":"use","excerpt":"Completion and continuum evolution. The complete coefficient Hilbert space. Let KK=ℓ^2(Z^3;C^4) ={a=(a_n):∑_n∈Z^3norma_n^2<∞}. Its inner product is innerab=∑_ninnera_nb_n. Absolute convergence follows from the scalar Cauchy--Schwarz inequality. The norm therefore measures the total squared amplitude across all Fourier packets. A finite family of am","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7fb250c40766c45f","tex":"\\mathbb Z^3","aliases":"Z^3 mathbb","display":"\\mathbb Z^3","kind":"use","excerpt":"equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5b75520135285cf1","tex":"Z^A","aliases":"Z^A","display":"Z^A","kind":"use","excerpt":"An exact matrix momentum sector. Let the fine canonical coordinates be Hermitian matrix pairs (Z^A,P_A), with real components and the trace bracket {F,G}=∑_ATr( ∇_Z^AF∇_P_AG- ∇_P_AF∇_Z^AG). The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjug","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","excerpt":"∑_k=0^Nfracz^k(A^ H)^kk! =T(∑_k=0^Nfracz^kA^kk!)T^-1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0bb5e126e766a7f8","tex":"z^{Lq+j}","aliases":"z^Lq+j","display":"z^{Lq+j}","kind":"use","excerpt":", and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, th","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"use","excerpt":"Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-10c7caeba58a1aa9","tex":"z_*","aliases":"z_*","display":"z_*","kind":"use","excerpt":"-m · m+n, D=(m+n)x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"hart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient o","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"hart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient o","count":12},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-642b65364417d9f0","tex":"z_{ab}","aliases":"z_ab","display":"z_{ab}","kind":"use","excerpt":"Let z_ba=Tr(P_bHP_aρ). Taking the conjugate trace and cyclically moving the first factor to the end yields overlinez_ba=Tr(ρ P_aHP_b) =Tr(P_aHP_bρ)=z_ab. Completeness and cyclicity resolve the derivative as dot p_b=-i∑_a(z_ba-z_ab) =∑_a2Imz_ba. The conjugate relation gives antisymmetry. Finite sums then prove cancellation and coarse aggregation exa","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-32babd86fec042f0","tex":"z_{ba}","aliases":"z_ba","display":"z_{ba}","kind":"use","excerpt":"Let z_ba=Tr(P_bHP_aρ). Taking the conjugate trace and cyclically moving the first factor to the end yields overlinez_ba=Tr(ρ P_aHP_b) =Tr(P_aHP_bρ)=z_ab. Completeness and cyclicity resolve the derivative as dot p_b=-i∑_a(z_ba-z_ab) =∑_a2Imz_ba. The conjugate relation gives antisymmetry. Finite sums then prove cancellation and coarse aggregation exa","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ae33ff0c9a9cfbb4","tex":"z_j","aliases":"z_j","display":"z_j","kind":"use","excerpt":"The sine formula U_p-1(cosθ)=sin(pθ)/sinθ shows that the excluded normalized energies are x_j=2cos(jπ/p), z_j=μ+2κcos(jπ/p), 1≤ j<p. These are the energies of one eliminated path. Indeed its jth eigenvector has interior components sin(jπ r/p), r=1,…,p-1, and the nearest-neighbor recurrence gives precisely z_j. Both boundary values at r=0,p vanish. ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bdc1562155501d06","tex":"z_n","aliases":"z_n","display":"z_n","kind":"use","excerpt":"[Self-adjointness of the diagonal realization] The operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a1bb8bb52da2ab42","tex":"z_{n+1}","aliases":"z_n+1","display":"z_{n+1}","kind":"use","excerpt":"[Exact endpoint momentum] Start from z_n and let a provisional step produce widetilde z=(widetilde P,widetilde Q,widetilde U,widetilde V). Form widetilde g_U,widetilde g_V at its configuration and assume widetilde d_h>0. The correction μ=fracC_h(widetilde z)-C_h(z_n)widetilde d_h, z_n+1=(widetilde P,widetilde Q, widetilde U-μwidetilde g_U, widetild","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8723c894d5150ea2","tex":"\\mathbb Z_p","aliases":"Z_p mathbb","display":"\\mathbb Z_p","kind":"use","excerpt":"item S. V. Kozyrev, ``Wavelet theory as p-adic spectral analysis,'' Izvestiya: Mathematics 66(2), 367--376 (2002). https://doi.org/10.1070/IM2002v066n02ABEH000381doi:10.1070/IM2002v066n02ABEH000381. Published English Theorems 2 and 7 inspected. The Q_p Vladimirov setting supplies wavelet context; the present Z_p level-diagonal construction is prove","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-31d760e606ff3dad","tex":"\\Z_p^\\times","aliases":"Z_p^× Z times","display":null,"kind":"use","excerpt":"Let p be prime, let L=mp for a positive integer m, and take 0≤ j<p. For every q≥0, g_q=fracj!(Lq)!(Lq+j)! =binomLq+jj^-1∈Z_p^×. Indeed binomLq+jj=∏_r=1^j(Lq+r)/r. Each numerator and denominator is a p-adic unit because Lq≡0pmod p and 1≤ r<p. For j=0 the empty product is one. The proof also covers p=2 in its short window. Outside this window, a fact","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-99e9f1953d7e02c1","tex":"Z_{\\mathrm{sign}}","aliases":"Z_sign mathrm","display":"Z_{\\mathrm{sign}}","kind":"use","excerpt":"Discarding the sign as a quantum operation instead takes the partial trace ρ↦Tr_signρ. Both pure states |0,a⟩ and |1,a⟩ then produce the same reduced phase state |a⟩⟨ a|. Yet the sign observable Z_sign=(|0⟩⟨0|-|1⟩⟨1|)⊗ I has expectations +1 and -1 on those states. No measurement on the retained phase state can reconstruct this difference. The obstr","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3338e0334cca7109","tex":"z_{x,\\mu}","aliases":"z_x,μ mu","display":"z_{x,\\mu}","kind":"use","excerpt":"Now take ψ_x∈C^2, U_x,μ∈ SU(2), and local gauges g_x∈ SU(2), as in the legacy non-Abelian chapter. The linked scalar z_x,μ=ψ_x^*U_x,μψ_x+widehatμ is gauge invariant: the adjoint factor g_x^* cancels g_x, and the factors at x+widehatμ also cancel. It follows that the source's scalar link observable j_x,μ=2Imz_x,μ is invariant. To obtain a continuity","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6820374f59f0b332","tex":"|^{-1}","aliases":"|^-1","display":"|^{-1}","kind":"use","excerpt":"athsf F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_2. Likewise, for nonreal z, (z-widetilde H_M)^-1 · = E_M(z-H_ϵ)^-1 +z^-1(I- E_M),nonumber norm((z-widetilde H_M)^-1-(z-H_ϵ)^-1)","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3e3225555f03b1a7","tex":"|^3","aliases":"|^3","display":"|^3","kind":"use","excerpt":"For a self-adjoint fine Hamiltonian with coarse projector P, set B=QH_fP. If a unit vector ψ lies in Ran P, its initial detail amplitude is Qe^-itH_fψ=-itBψ+O(t^2), normQe^-itH_fψ^2 =t^2normBψ^2+O(|t|^3). These are finite Taylor expansions for fixed H_f and ψ. Thus a nonzero B supplies a state with nonzero short-time leakage. The two-dimensional ex","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-66f78e7a5c255442","tex":"|^5","aliases":"|^5","display":"|^5","kind":"use","excerpt":"For clarity, Taylor's theorem through degree four has zero even coefficients for sine, and its fifth-derivative remainder is at most |hp_j|^5/120. Division by |h| gives O(h^4); squaring the three expansions and summing preserves that order for a fixed mode. If E_n>0, the square root is smooth near E_n^2, so","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a2e3fd70ad04e933","tex":"|^6","aliases":"|^6","display":"|^6","kind":"use","excerpt":"C(n)=frac16(∑_j=x,y,z|p_j(n)|^6)^1/2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-06b81a055469dbd1","tex":"|^p","aliases":"|^p","display":"|^p","kind":"use","excerpt":"[A quantified remainder and perturbation bound] Let p be a nonnegative integer and suppose normR(h)≤ C|h|^p, normg(h)-f(h)≤ε (0<|h|<δ), where C,ε≥0. Define A_m,p=∏_k=1^mfracb^k-p+1b^k-1. Then, throughout the same punctured interval, normT_mg(h)-F≤ A_m,pC|h|^p+A_m,0ε. The interval is stable under h↦ h/b. The triangle inequality applied to one tower ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18ec6838a96d8568","tex":"|_{\\mathcal V_M}","aliases":"|_ V_M mathcal","display":"|_{\\mathcal V_M}","kind":"use","excerpt":"The completion of the isometric direct system (HH_p,m,V_p,m) is therefore HH_p: equation [source reference] gives a single isometric map on the algebraic direct limit, and its image is the dense union of the V_m. For m≤ M, with V_p,m→ M=V_p,M-1⋯ V_p,m, one also has E_m|_ V_M = U_p,MV_p,m→ MV_p,m→ M^* U_p,M^*. Both sides are the orthogonal projectio","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ffbdd0e2e2408112","tex":"|_{Q\\HH}","aliases":"|_QHH HH","display":null,"kind":"use","excerpt":"normQe^-itH_λP · ≤min{1,|tλ|normQBP}. For z∉σ(QH_λ Q|_QHH), the coarse self-energy is Σ_P(z)=PH_λ Q (z-QH_λ Q|_QHH)^-1QH_λ P, where the inverse acts only on QHH, and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2db86546abe159fd","tex":"|_{\\Ran P}","aliases":"|_Ran P Ran","display":null,"kind":"use","excerpt":"QH_fP · =0, PH_fP=VH_cV^*. They hold exactly when, on Ran P⊕Ran Q, H_f=(VH_cV^*)|_Ran P ⊕ H_Q, H_Q=(QH_fQ)|_Ran Q. The detail operator H_Q may be any self-adjoint operator on Ran Q.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7302e1b720e88ef","tex":"|_{\\Ran Q}","aliases":"|_Ran Q Ran","display":null,"kind":"use","excerpt":"QH_fP · =0, PH_fP=VH_cV^*. They hold exactly when, on Ran P⊕Ran Q, H_f=(VH_cV^*)|_Ran P ⊕ H_Q, H_Q=(QH_fQ)|_Ran Q. The detail operator H_Q may be any self-adjoint operator on Ran Q.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9152fcee04370f9c","tex":"|_{\\varphi}","aliases":"|_φ varphi","display":"|_{\\varphi}","kind":"use","excerpt":"ta⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into the action gives the conventional result S_ M^(1)|_φ =-tfrac14∫ F_IJF^IJvol","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"The first coordinate issue is specific to units. For an odd prime p and an integer h≥0, set U_p,h=(Z/p^h+1Z)^×. For p=2 the useful indexing starts one digit later: U_2,h=(Z/2^h+2Z)^×. The shifted dyadic index retains the sign even when the principal phase register has dimension one. It prevents the two units modulo four from being collapsed into on","count":24},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"Let g∈C be fixed and choose γ_p,w=g p^-α w for a real exponent α. Then G_p,w=g p^(1/2-α)w, |G_p,w|^2=|g|^2p^(1-2α)w. The exact branching multiplicity produces the factor p^w/2. Thus the collective coefficient grows with w for α<1/2, is independent of w for α=1/2, and decays for α>1/2. The same cases govern the operator norm of this equal-leaf inter","count":59},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6846e04eb85319e1","tex":"\\alpha^*","aliases":"α^* alpha","display":"\\alpha^*","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-34a0d2bb0218af80","tex":"\\alpha^n","aliases":"α^n alpha","display":"\\alpha^n","kind":"use","excerpt":"For a coherent state |α⟩=e^-|α|^2/2 ∑_n=0^∞fracα^nsqrtn!|n⟩, μ=|α|^2≥0, the packet probability is the Poisson residue sum p_L,j(μ) · =⟨α,Q_j^(L)α⟩ =e^-μE_L,j(μ)nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","excerpt":"On finite Fourier polynomials, the operator agrees with the matrix differential expression -i∑_jα_j∂_j+mβ already calculated. Its extension is specified by Fourier transport and the maximal domain, so no pointwise differentiability assumption is imposed on every L^2 representative. Equality with a separately constructed weak differential operator i","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-70124300c10527d0","tex":"\\alpha_j^*","aliases":"α_j^* alpha","display":"\\alpha_j^*","kind":"use","excerpt":"α_j^*=α_j, β^*=β, α_j^2=β^2=I_4,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8ae91c89408a2a13","tex":"\\alpha_j^2","aliases":"α_j^2 alpha","display":"\\alpha_j^2","kind":"use","excerpt":"α_j^*=α_j, β^*=β, α_j^2=β^2=I_4,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-993a80257a1ce549","tex":"\\alpha_k","aliases":"α_k alpha","display":"\\alpha_k","kind":"use","excerpt":"α_jα_k+α_kα_j=0 (j≠ k), α_jβ+βα_j=0.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-48b4dca7f4ffcdf4","tex":"\\alpha_q","aliases":"α_q alpha","display":"\\alpha_q","kind":"use","excerpt":"al-folded-bands For q=0,…,p-1, the eigenvalues of H_p(θ) are E_q(θ)=μ+2κ cos (fracθ+2π qp). At θ=θ_j, the union of these pM eigenvalues, counting multiplicity, is exactly the spectrum of [source reference]. Put α_q=(θ+2π q)/p and let the rth component of v_q be p^-1/2e^iα_qr. On the interior components, P_p(θ)v_q=e^-iα_qv_q. The wrapped component h","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c26b0386db57f0c4","tex":"\\alpha_x","aliases":"α_x alpha","display":"\\alpha_x","kind":"use","excerpt":"α_x= 0 · 0 · 0 · 1 0 · 0 · 1 · 0 0 · 1 · 0 · 0 1 · 0 · 0 · 0 , α_y= 0 · 0 · 0 · -i 0 · 0 · i · 0 0 · -i · 0 · 0 i · 0 · 0 · 0 ,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-58f0a9598ff4ff43","tex":"\\alpha_y","aliases":"α_y alpha","display":"\\alpha_y","kind":"use","excerpt":"α_x= 0 · 0 · 0 · 1 0 · 0 · 1 · 0 0 · 1 · 0 · 0 1 · 0 · 0 · 0 , α_y= 0 · 0 · 0 · -i 0 · 0 · i · 0 0 · -i · 0 · 0 i · 0 · 0 · 0 ,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f60ac066d843ff0e","tex":"\\alpha_z","aliases":"α_z alpha","display":"\\alpha_z","kind":"use","excerpt":"α_z= 0 · 0 · 1 · 0 0 · 0 · 0 · -1 1 · 0 · 0 · 0 0 · -1 · 0 · 0 , β= 1 · 0 · 0 · 0 0 · 1 · 0 · 0 0 · 0 · -1 · 0 0 · 0 · 0 · -1 .","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber","count":33},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bd66c98b32c8ec9d","tex":"\\beta^*","aliases":"β^* beta","display":"\\beta^*","kind":"use","excerpt":"α_j^*=α_j, β^*=β, α_j^2=β^2=I_4,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ab15d1c00936ee24","tex":"\\beta^2","aliases":"β^2 beta","display":"\\beta^2","kind":"use","excerpt":"α_j^*=α_j, β^*=β, α_j^2=β^2=I_4,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"A resolution can change with time even when the state is stationary. To retain orthogonality, transport the entire atlas by a unitary family. Let U(t) be differentiable in operator norm and satisfy both U^*U=id and UU^*=id. Set P_a(t)=U(t)P_a^(0)U(t)^*, ϕ(t)=U(t)^*ψ(t), Γ(t)=U(t)^*dot U(t). The coordinates ϕ see fixed projectors P_a^(0). The physic","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"se projector P, set B=QH_fP. If a unit vector ψ lies in Ran P, its initial detail amplitude is Qe^-itH_fψ=-itBψ+O(t^2), normQe^-itH_fψ^2 =t^2normBψ^2+O(|t|^3). These are finite Taylor expansions for fixed H_f and ψ. Thus a nonzero B supplies a state with nonzero short-time leakage. The two-dimensional example H_f=0 · overlineγ γ · 0, P=1 · 0 0 · 0 ","count":59},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-03569cbd604e9e67","tex":"\\Gamma^*","aliases":"Γ^* Gamma","display":"\\Gamma^*","kind":"use","excerpt":"[The transformed generator] If dotψ=Kψ and K^*=-K, then dotϕ=K_effϕ, K_eff=U^*KU-Γ, Γ^*=-Γ, K_eff^*=-K_eff. Consequently the fixed-atlas current formula, applied to ϕ and K_eff, gives the derivative of the moving physical packet weights.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-628f274efa0c1a48","tex":"\\gamma^0","aliases":"γ^0 gamma","display":"\\gamma^0","kind":"use","excerpt":"Fermionic primary constraints and half densities. Impose the inherited algebraic spinor relation barψ=ψ^†γ^0, where (γ^0)^2=-id. Its phase convention must be included in the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-660830aacf9a8a38","tex":"\\gamma^1","aliases":"γ^1 gamma","display":"\\gamma^1","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4569df46706329ae","tex":"\\gamma^2","aliases":"γ^2 gamma","display":"\\gamma^2","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-03799c3dc7cb9b3c","tex":"\\gamma^3","aliases":"γ^3 gamma","display":"\\gamma^3","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f455ccf4bfbfc11e","tex":"\\Gamma^i","aliases":"Γ^i Gamma","display":"\\Gamma^i","kind":"use","excerpt":"E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-768b2242190301f3","tex":"\\gamma^I","aliases":"γ^I gamma","display":"\\gamma^I","kind":"use","excerpt":"vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-23236d6d32bb1087","tex":"\\gamma^J","aliases":"γ^J gamma","display":"\\gamma^J","kind":"use","excerpt":"vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e26c0ed5d9be65cf","tex":"\\gamma^K","aliases":"γ^K gamma","display":"\\gamma^K","kind":"use","excerpt":"C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The au","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2771b61f3f3f5974","tex":"\\gamma_0","aliases":"γ_0 gamma","display":"\\gamma_0","kind":"use","excerpt":"Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decompo","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fd17a77e6abbf292","tex":"\\gamma_0^2","aliases":"γ_0^2 gamma","display":"\\gamma_0^2","kind":"use","excerpt":"As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d2e737aa54abf669","tex":"\\gamma_1","aliases":"γ_1 gamma","display":"\\gamma_1","kind":"use","excerpt":"Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decompo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e27dee4a4efd5ee2","tex":"\\gamma_{12}","aliases":"γ_12 gamma","display":"\\gamma_{12}","kind":"use","excerpt":"[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e896a0b8b00bb35e","tex":"\\gamma_{21}","aliases":"γ_21 gamma","display":"\\gamma_{21}","kind":"use","excerpt":"[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7fd85439665ac62c","tex":"\\gamma_5","aliases":"γ_5 gamma","display":"\\gamma_5","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea15a4cf61a5a234","tex":"\\Gamma_a","aliases":"Γ_a Gamma","display":"\\Gamma_a","kind":"use","excerpt":"a^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe sector. It fixes Γ_a^i=mathringΓ_a^i(E) -tfracκ4J_5^0e_a^i. One may retain torsion in K and in the sub","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1c4421c45efc7807","tex":"\\gamma_{a+1}","aliases":"γ_a+1 gamma","display":"\\gamma_{a+1}","kind":"use","excerpt":"ldots,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportional to ∑_dc_d, so it vanishes. At s>0 the sa","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0e0f0609597bb3e8","tex":"\\gamma_{ab}","aliases":"γ_ab gamma","display":"\\gamma_{ab}","kind":"use","excerpt":"[Population gain and loss] The diagonal populations p_b=ρ_bb of a Hermitian solution satisfy dot p_b=∑_aF_b← a, F_b← a=γ_bap_a-γ_abp_b, F_b← a=-F_a← b. The total population is constant, and every coarse population has the sum of the boundary jump currents as its derivative.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e593d31e966a0e06","tex":"\\gamma_{ba}","aliases":"γ_ba gamma","display":"\\gamma_{ba}","kind":"use","excerpt":"Coherent exchange is one mechanism for changing packet weights. A second comes from an explicitly chosen family of dissipative transitions. Fix an orthonormal basis (e_a)_a∈ I of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-54218a765dc34dd7","tex":"\\gamma_{bi}","aliases":"γ_bi gamma","display":"\\gamma_{bi}","kind":"use","excerpt":"In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9e99730a6fe995c5","tex":"\\gamma_{h-1}","aliases":"γ_h-1 gamma","display":"\\gamma_{h-1}","kind":"use","excerpt":"Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decompo","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8e6227ccf23c3c25","tex":"\\gamma_I","aliases":"γ_I gamma","display":"\\gamma_I","kind":"use","excerpt":"vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ae2e30306bc89e00","tex":"\\gamma_i","aliases":"γ_i gamma","display":"\\gamma_i","kind":"use","excerpt":"The straight compatibility contract. For each direction, canonical fine links can be defined by v_i(p)=r(σ_i p) γ_i(p) r(p)^-1, where γ_i(p)=1 on a noncarrying half step and γ_i(p)=u_i(s) on a carrying half step. Cancellation of the intermediate frame gives v_i(σ_i p)v_i(p) =r(σ_i^2p)u_i(s)r(p)^-1. For arbitrary fine links, equation [source referen","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eb6169646c2f7354","tex":"\\gamma_{IJ}","aliases":"γ_IJ gamma","display":"\\gamma_{IJ}","kind":"use","excerpt":"vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-322e5c7da80d563a","tex":"\\gamma_J","aliases":"γ_J gamma","display":"\\gamma_J","kind":"use","excerpt":"vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b8f615ebae18b9ca","tex":"\\gamma_{p,w}","aliases":"γ_p,w gamma","display":"\\gamma_{p,w}","kind":"use","excerpt":"Introduce a visible address |c⟩ orthogonal to all descendants. Suppose every descendant has the same coupling from |c⟩, with complex matrix element γ_p,w. The Hermitian interaction is H_int= ∑_ d (γ_p,w| d⟩⟨ c| +overlineγ_p,w|c⟩⟨ d|). The conjugate reverse coefficient is necessary when γ_p,w is not real. Define the collective coefficient G_p,w=sqrt","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-788db25068d804f5","tex":"\\gamma_r","aliases":"γ_r gamma","display":"\\gamma_r","kind":"use","excerpt":"Let T_p,h be a rooted p-ary tree of depth h≥1. Vertices at depth r have onsite energy ϵ_r∈R. Every parent-to-child edge from level r to r+1 has real coupling γ_r/sqrt p, where γ_r∈R. The Hamiltonian is the corresponding real symmetric adjacency matrix with onsite diagonal. The level symmetry and this branch normalization are structural assumptions ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e850f2bbe26688ef","tex":"\\gamma_{r-1}","aliases":"γ_r-1 gamma","display":"\\gamma_{r-1}","kind":"use","excerpt":"Define normalized level states |r⟩=p^-r/2∑_|v|=r|v⟩, r=0,…,h. They are orthonormal because different levels are disjoint and level r contains p^r vertices. A direct action calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-311a62961865d104","tex":"\\gamma_r^2","aliases":"γ_r^2 gamma","display":"\\gamma_r^2","kind":"use","excerpt":"For a tail J_r:h, put g_r(z)=⟨ r,(z-J_r:h)^-1r⟩. On a common regular domain, scalar Schur elimination gives g_h(z) · =(z-ϵ_h)^-1,nonumber g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomp","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Δ_h,n:=H_h,n-H_n =∑_j=x,y,zδ_j(h,n)α_j, δ_j(h,n)=q_j(h,p(n))-p_j(n).","count":53},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-02c68d637b81b90c","tex":"\\delta^3","aliases":"δ^3 delta","display":"\\delta^3","kind":"use","excerpt":"the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Different","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8d634cbf29cf1a83","tex":"\\delta_{\\mathcal F_{ij}}","aliases":"δ_ F_ij delta mathcal","display":"\\delta_{\\mathcal F_{ij}}","kind":"use","excerpt":"For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely wh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9adbb02c2ad54bb0","tex":"\\delta_1","aliases":"δ_1 delta","display":"\\delta_1","kind":"use","excerpt":"Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dac3f880e6e3670a","tex":"\\delta_2","aliases":"δ_2 delta","display":"\\delta_2","kind":"use","excerpt":"Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-020f6e328243f9c3","tex":"\\delta_3","aliases":"δ_3 delta","display":"\\delta_3","kind":"use","excerpt":"Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-08ab92c996841336","tex":"\\delta_{\\left[U,V\\right]_{\\mathrm{par}}}","aliases":"δ_[U,V]_par delta left right mathrm","display":"\\delta_{\\left[U,V\\right]_{\\mathrm{par}}}","kind":"use","excerpt":"The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","excerpt":"|_∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3ff78dbd4ffd8b65","tex":"\\delta_{AB}","aliases":"δ_AB delta","display":"\\delta_{AB}","kind":"use","excerpt":"the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Different","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-845f98ac1e4d0fbd","tex":"\\delta_b","aliases":"δ_b delta","display":"\\delta_b","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8510827ccfbd978d","tex":"\\delta_{B_i}","aliases":"δ_B_i delta","display":"\\delta_{B_i}","kind":"use","excerpt":"For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely wh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0e4773cb2ae1266d","tex":"\\delta_{B_j}","aliases":"δ_B_j delta","display":"\\delta_{B_j}","kind":"use","excerpt":"For Hermitian generators B_i,B_j, put F_ij=i[B_i,B_j]. This curvature matrix is Hermitian, and the Jacobi identity gives [δ_B_i,δ_B_j]X =-[[B_i,B_j],X] =i[ F_ij,X]=δ_ F_ijX. Commuting generators produce commuting derivations. A nonzero curvature matrix need not act nontrivially on every observable: its action on a particular X vanishes precisely wh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-60edcde84b590e7e","tex":"\\Delta_C","aliases":"Δ_C Delta","display":"\\Delta_C","kind":"use","excerpt":"The conditional BV pushforward statement. Suppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c6a395594f216319","tex":"\\Delta_D","aliases":"Δ_D Delta","display":"\\Delta_D","kind":"use","excerpt":"The conditional BV pushforward statement. Suppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9437cea347b38d28","tex":"\\delta_{de}","aliases":"δ_de delta","display":"\\delta_{de}","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9e14435a4d85c517","tex":"\\Delta_F","aliases":"Δ_F Delta","display":"\\Delta_F","kind":"use","excerpt":"The conditional BV pushforward statement. Suppose a regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-608581f0be0c4b00","tex":"\\Delta_h","aliases":"Δ_h Delta","display":"\\Delta_h","kind":"use","excerpt":"Let h=2π/N and use the periodic centered derivative (D_hf)_j=fracf_j+1-f_j-12h, (Δ_hf)_j=fracf_j+1-2f_j+f_j-1h^2. Here D_h^ T=-D_h. With U=dot P, V=dot Q, the raw semidiscrete vector field f_h(τ,z) for z=(P,Q,U,V) is dot P · =U, · dot Q · =V,nonumber","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-990016db1ae37714","tex":"\\Delta_{h,n}","aliases":"Δ_h,n Delta","display":"\\Delta_{h,n}","kind":"use","excerpt":"Δ_h,n:=H_h,n-H_n =∑_j=x,y,zδ_j(h,n)α_j, δ_j(h,n)=q_j(h,p(n))-p_j(n).","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eaf44d5b23935330","tex":"\\Delta_{h,n}^*","aliases":"Δ_h,n^* Delta","display":"\\Delta_{h,n}^*","kind":"use","excerpt":"Δ_h,nv _2^2 =⟨ v,Δ_h,n^*Δ_h,nv⟩ =|δ(h,n)|^2 v _2^2.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-83b61b3d00ad9b6e","tex":"\\Delta_{h,n}^2","aliases":"Δ_h,n^2 Delta","display":"\\Delta_{h,n}^2","kind":"use","excerpt":"The Hermitian matrix Δ_h,n satisfies Δ_h,n^2=|δ(h,n)|^2I_4 by PH01. Hence","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a104cc9eebe89629","tex":"\\delta_i","aliases":"δ_i delta","display":"\\delta_i","kind":"use","excerpt":"For the scalar seven-factor estimate, both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [sourc","count":14},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d900de86d0f4ee16","tex":"\\delta_i^2","aliases":"δ_i^2 delta","display":"\\delta_i^2","kind":"use","excerpt":"Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-288a046779bf284f","tex":"\\Delta_j","aliases":"Δ_j Delta","display":"\\Delta_j","kind":"use","excerpt":"ites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"Δ_h,n:=H_h,n-H_n =∑_j=x,y,zδ_j(h,n)α_j, δ_j(h,n)=q_j(h,p(n))-p_j(n).","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-147b73e82dc15b8b","tex":"\\Delta_{j-1}","aliases":"Δ_j-1 Delta","display":"\\Delta_{j-1}","kind":"use","excerpt":"equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2Δ_j^2 -frac h2∑_jv_jφ_j^2. Variations with compact time support give w_jddotφ_j+dot w_jdotφ_j =fracc_j+1/2Δ_j-c_j-1/2Δ_j-1h^2 -v_jφ_j. The neighboring edge terms i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5c2d3273fd73d5e9","tex":"\\Delta_{j-1}^2","aliases":"Δ_j-1^2 Delta","display":"\\Delta_{j-1}^2","kind":"use","excerpt":"[Local energy, flux, and geometric work] For every twice differentiable solution of [source reference], define e_j · =frac h2w_jdotφ_j^2+frac h2v_jφ_j^2 +fracc_j+1/24hΔ_j^2 +fracc_j-1/24hΔ_j-1^2,nonumber F_j+1/2 · =-fracc_j+1/22hΔ_j (dotφ_j+dotφ_j+1),nonumber s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e4b6e196f224dc32","tex":"\\Delta_j^2","aliases":"Δ_j^2 Delta","display":"\\Delta_j^2","kind":"use","excerpt":"w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8e4e08a5e917de66","tex":"\\delta_{jk}","aliases":"δ_jk delta","display":"\\delta_{jk}","kind":"use","excerpt":", and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, th","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f101f26c86293154","tex":"\\delta_K","aliases":"δ_K delta","display":"\\delta_K","kind":"use","excerpt":"For the full matrix algebra, choose a Hermitian matrix K and set δ_KX=i[K,X]. Direct multiplication gives δ_K(XY)=(δ_KX)Y+X(δ_KY). It also obeys δ_K(X^*)=(δ_KX)^*, Trδ_KX=0, and is skew-adjoint for the Hilbert--Schmidt inner product. The last identity follows by moving K through the trace. These properties supply an exact finite product calculus on","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9601f08c373fe8e4","tex":"\\delta_k","aliases":"δ_k delta","display":"\\delta_k","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber G_ij(h) · =frac12[ δ_kδ_ih_jk+δ_kδ_jh_ik -δ^2h_ij-δ_iδ_jh_rm tr -δ_ijC].","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3f3790d3553b608d","tex":"\\delta_l","aliases":"δ_l delta","display":"\\delta_l","kind":"use","excerpt":"Twice its divergence expands as 2δ_j G_ij = · δ_iδ_jδ_kh_jk +δ^2δ_kh_ik-δ^2δ_jh_ij -δ_iδ^2h_rm tr · -δ_iδ_kδ_lh_kl +δ_iδ^2h_rm tr. The terms cancel in three pairs. For self-adjointness, write B_i(a)=δ_ja_ij and A(a)=δ_iδ_ja_ij. Two applications of the adjoint identity give 2innera G(b)_h = · -2innerB(a)B(b)_h +∑_kinnerδ_kaδ_kb_h","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e8b18ae884be8df2","tex":"\\delta_{mn}","aliases":"δ_mn delta","display":"\\delta_{mn}","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea2a7f3a25fa545c","tex":"\\delta_{pn,m}","aliases":"δ_pn,m delta","display":"\\delta_{pn,m}","kind":"use","excerpt":"ty and the binomial variance bound, by separating |k/n-t|≤δ from its complement and then letting n→∞, δ→0. Continuous functions are dense for this finite measure, as one sees by approximating interval steps in the angle coordinate by continuous ramps. This supplies completeness. Finally, Chebyshev composition gives K_pϕ_n=ϕ_pn. For every n,m≥0, ⟨ϕ_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1d9e0b50c7e8925a","tex":"\\delta_{qq'}","aliases":"δ_qq' delta","display":"\\delta_{qq'}","kind":"use","excerpt":"Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-51a87ccbd55b1aa4","tex":"\\delta_{rs}","aliases":"δ_rs delta","display":"\\delta_{rs}","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber rankD_0^(h) · =1, · rankD_r^(h) · =(p-1)p^r-1 (r≥1).","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b0a780cb0a0d5ad7","tex":"\\delta_{rt}","aliases":"δ_rt delta","display":"\\delta_{rt}","kind":"use","excerpt":", its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e32d7eec9c371dea","tex":"\\delta_{st}","aliases":"δ_st delta","display":"\\delta_{st}","kind":"use","excerpt":", its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-024a046ea306ad65","tex":"\\Delta_U","aliases":"Δ_U Delta","display":"\\Delta_U","kind":"use","excerpt":"[Covariance of linked differences] For spacing h>0, define (D^+_μ,Uψ)_x · =fracU_x,μψ_x+widehatμ-ψ_xh, (Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4a73afb10ef0b959","tex":"\\delta_U","aliases":"δ_U delta","display":"\\delta_U","kind":"use","excerpt":"The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b35f037ae1ca06a1","tex":"\\delta_u","aliases":"δ_u delta","display":"\\delta_u","kind":"use","excerpt":"For example, the residual electric constraint is G_ U^red=∂_a E^a+qΨ^†Ψ, Q[u]=-∫_Σ u G_ U^red d^3x. It generates δ_uA_a=∂_au and δ_uΨ=iquΨ. The half-density Lie derivative in the spatial generator is L_η^(1/2)Ψ =η^a∂_aΨ+tfrac12(∂_aη^a)Ψ. This term is forced by the density weight. Treating Ψ as a scalar would give a different moment map.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7a9a9e1ad8d08502","tex":"\\Delta_{U^g}","aliases":"Δ_U^g Delta","display":"\\Delta_{U^g}","kind":"use","excerpt":"(Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-32a0678296fa0cc7","tex":"\\delta_V","aliases":"δ_V delta","display":"\\delta_V","kind":"use","excerpt":"The parameter algebra combines vector fields with Lorentz and Abelian transformations in a semidirect product. In the inherited field-space commutator convention, [δ_U,δ_V]_fld =-δ_[U,V]_par, where [(ξ,λ,u),(η,μ,v)]_par =([ξ,η], L_ξμ- L_ηλ+[λ,μ], L_ξ v- L_η u). The minus sign is a convention for these field-space generators. It fixes the plus sign ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-43ab2fa7aeb228e6","tex":"\\delta_x","aliases":"δ_x delta","display":"\\delta_x","kind":"use","excerpt":"both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] pr","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4bff5300fdcd34eb","tex":"\\delta_y","aliases":"δ_y delta","display":"\\delta_y","kind":"use","excerpt":"both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] pr","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bb8ce6bd7aafe6f1","tex":"\\delta_z","aliases":"δ_z delta","display":"\\delta_z","kind":"use","excerpt":"both walks act on the same finite Hilbert space, with I=1. One uses the reconstructed links v_x,v_y,v_z, the other the prescribed links u_x,u_y,u_z, and both use the same two contractive mass factors M_0,M_1. With execution order M_0-X-Y-Z-Y-X-M_1, define δ_i=δ(u_i,v_i). The mass defects vanish. Equation [source reference] and [source reference] pr","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-141577db886abda4","tex":"\\Delta_\\Delta","aliases":"Δ_Δ Delta Delta","display":"\\Delta_\\Delta","kind":"use","excerpt":"equation overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. Direct multiplication proves GJ_ D^Δ=0. Since","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-536a24f4cca16e8d","tex":"\\Delta_\\Delta^{-1}","aliases":"Δ_Δ^-1 Delta Delta","display":"\\Delta_\\Delta^{-1}","kind":"use","excerpt":"labelc:constraints:discrete-gradient The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. Direct multiplication proves GJ_ D^Δ=0. Since Δ_Δ and its inverse ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ee986ad8011bd9d2","tex":"\\delta_{\\mu\\nu}","aliases":"δ_μν delta mu nu","display":"\\delta_{\\mu\\nu}","kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"Given ε>0, choose g∈ V_M with normf-g_2<ε. For m≥ M, the projection fixes g, so normf- E_m f_2 =norm(I- E_m)(f-g)_2≤normf-g_2<ε. This proves strong convergence without exchanging an infinite integral with a limit.","count":29},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"The theorem concerns the semisimple pointwise algebra C^N. It does not assert that every finite-dimensional commutative complex algebra has only the zero derivation. For example, on C[ε]/(ε^2), the map D(a+bε)=bε is a nonzero derivation. The nilpotent element changes the algebraic hypothesis.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e3e5a993e2ec8e09","tex":"\\epsilon^{abc}","aliases":"ϵ^abc epsilon","display":"\\epsilon^{abc}","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fbcb4249d64b915d","tex":"\\epsilon^I","aliases":"ϵ^I epsilon","display":"\\epsilon^I","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ab7cfeb562562c69","tex":"\\epsilon^i","aliases":"ϵ^i epsilon","display":"\\epsilon^i","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-afe3d484b60c1007","tex":"\\epsilon^{IJ}","aliases":"ϵ^IJ epsilon","display":"\\epsilon^{IJ}","kind":"use","excerpt":"Algebraic torsion and the reduced phase space. For the symmetric Dirac action, define γ_5=iγ^0γ^1γ^2γ^3 and J_5^I=barψγ^Iγ_5ψ. In the parent convention, the connection equation has the algebraic solution T^I=tfracκ4ϵ^I_JKLJ_5^Je^Kwedge e^L, C^IJ=tfracκ4ϵ^IJ_KLJ_5^Ke^L, with ω=mathringω(e)+C. The solution is unique on the nondegenerate coframe secto","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6dfce6e7e372892a","tex":"\\epsilon^{ij}","aliases":"ϵ^ij epsilon","display":"\\epsilon^{ij}","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d1776e7e5d78a172","tex":"\\epsilon^{ijk}","aliases":"ϵ^ijk epsilon","display":"\\epsilon^{ijk}","kind":"use","excerpt":"H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-009aa60286314a39","tex":"\\epsilon_{0123}","aliases":"ϵ_0123 epsilon","display":"\\epsilon_{0123}","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-da012babef541427","tex":"\\epsilon_1","aliases":"ϵ_1 epsilon","display":"\\epsilon_1","kind":"use","excerpt":"Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decompo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c7ffdc6ef6b651ea","tex":"\\epsilon_2","aliases":"ϵ_2 epsilon","display":"\\epsilon_2","kind":"use","excerpt":"Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decompo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e61b7ea9a3581771","tex":"\\epsilon_{a+1}","aliases":"ϵ_a+1 epsilon","display":"\\epsilon_{a+1}","kind":"use","excerpt":"choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite energies ϵ_a+1,…,ϵ_h and couplings γ_a+1,…,γ_h-1. At s=0 the upward amplitude into u is proportiona","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-de18910f25a2ed52","tex":"\\epsilon_{a,\\mathbf r}","aliases":"ϵ_a, r epsilon mathbf","display":"\\epsilon_{a,\\mathbf r}","kind":"use","excerpt":"For example, on a joint depth packet one may take A_ r =(∑_aϵ_a, rT_a) ⊗ I_phase, r, ϵ_a, r∈R, T_a=T_a^*. The matrices T_a act on the sign and torsion factors. Their mutual commutation is unnecessary because their sum is self-adjoint on the finite block. The coefficients assigned to an existing packet are fixed under refinement. An extension acting","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-62e4a848980609ac","tex":"\\epsilon_{abc}","aliases":"ϵ_abc epsilon","display":"\\epsilon_{abc}","kind":"use","excerpt":"H · =tfrac12ϵ^ij_kE^a_iE^b_jF_ab^k +Λdet E. Here det E=tfrac16ϵ_abcϵ^ijkE^a_iE^b_jE^c_k. The smeared generators are G[λ] · =τ_N(λ^i G_i), · D[ξ] · =τ_N(ξ^a D_a),","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1f74d9a4539571bc","tex":"\\epsilon_h","aliases":"ϵ_h epsilon","display":"\\epsilon_h","kind":"use","excerpt":"Let R:C^h+1→ℓ^2(T_p,h) send e_r to |r⟩. Equation [source reference] proves HR=RJ_h, where J_h= ϵ_0 · γ_0 · · · γ_0 · ϵ_1 · γ_1 · · · γ_1 · ϵ_2 · ddots · · · ddots · ddots · γ_h-1 · · · γ_h-1 · ϵ_h . Because H is self-adjoint, the orthogonal complement of Ran R is also invariant: for w⊥Ran R, ⟨ Re_r,Hw⟩=⟨ HRe_r,w⟩=0. Hence this is a reducing decompo","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a93d9c9ebca43b37","tex":"\\epsilon_{ij}","aliases":"ϵ_ij epsilon","display":"\\epsilon_{ij}","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k, G_i · =δ_aE^a_i+ϵ_ij^kA_a^jE^a_k, V_a · =F_ab^iE^b_i, D_a= V_a-A_a^i G_i, H · =tfrac12","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-973f701abab21f48","tex":"\\epsilon_{ijk}","aliases":"ϵ_ijk epsilon","display":"\\epsilon_{ijk}","kind":"use","excerpt":"Time gauge and simplicity pullback. Choose an oriented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1ab16a16a710a778","tex":"\\epsilon_{IJKL}","aliases":"ϵ_IJKL epsilon","display":"\\epsilon_{IJKL}","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6f89c09ead72a18a","tex":"\\epsilon_{JIKL}","aliases":"ϵ_JIKL epsilon","display":"\\epsilon_{JIKL}","kind":"use","excerpt":"The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gau","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-778fb0127de2ea78","tex":"\\epsilon_{KLIJ}","aliases":"ϵ_KLIJ epsilon","display":"\\epsilon_{KLIJ}","kind":"use","excerpt":"IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0d161a854cc2f845","tex":"\\epsilon_m","aliases":"ϵ_m epsilon","display":"\\epsilon_m","kind":"use","excerpt":"Let (ϵ_m)_m≥0 be any sequence of real numbers. Define Dom(H_ϵ) · ={f∈HH_p: ∑_m≥0|ϵ_m|^2norm F_mf_2^2<∞}, nonumber","count":26},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e17754e3e2e4c50","tex":"\\epsilon_{m_j}","aliases":"ϵ_m_j epsilon","display":"\\epsilon_{m_j}","kind":"use","excerpt":"g-resolvent. If the absolute energies tend to infinity, the right side of [source reference] tends to zero. The retained part of the resolvent has finite rank, so the resolvent is a norm limit of finite-rank operators and is compact. Conversely, if the absolute energies do not tend to infinity, there are infinitely many distinct levels m_j with |ϵ_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc06eebc88a8b163","tex":"\\varepsilon_n","aliases":"ε_n varepsilon","display":"\\varepsilon_n","kind":"use","excerpt":"Summable refinement and what a limit still requires. For the dyadic mesh h_n=a_0 2^-n, the regular estimate produces the nonnegative sequence ε_n=C4^-n, C=a_0^2 A_∂. The actual generated operator norms are bounded by this sequence and hence are summable. This is more information than merely ε_n→0. A sequence with adjacent errors tending to zero nee","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a28ffe0adef622b5","tex":"\\epsilon_{p,0}","aliases":"ϵ_p,0 epsilon","display":"\\epsilon_{p,0}","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determine","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b3b3cfb9deb4c4e2","tex":"\\epsilon_{p,h+1}","aliases":"ϵ_p,h+1 epsilon","display":"\\epsilon_{p,h+1}","kind":"use","excerpt":"silon_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonalizes the level blocks: T_p,hH_p,h^Haar ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1a16cbc91fb1fd77","tex":"\\epsilon_{p,r}","aliases":"ϵ_p,r epsilon","display":"\\epsilon_{p,r}","kind":"use","excerpt":"Choose a real sequence of level energies ϵ_p,r and define H_p,h^Haar=∑_r=0^h ϵ_p,rD_r^(h). Every wavelet in Ran D_r^(h) has energy ϵ_p,r. The projectors in [source reference] show directly that H_p,h+1^HaarV_p,h =V_p,hH_p,h^Haar. The new detail energy ϵ_p,h+1 does not alter the embedded older levels. Moreover the recursive Haar transform diagonaliz","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-70bb170c716771c3","tex":"\\epsilon_r","aliases":"ϵ_r epsilon","display":"\\epsilon_r","kind":"use","excerpt":"Let T_p,h be a rooted p-ary tree of depth h≥1. Vertices at depth r have onsite energy ϵ_r∈R. Every parent-to-child edge from level r to r+1 has real coupling γ_r/sqrt p, where γ_r∈R. The Hamiltonian is the corresponding real symmetric adjacency matrix with onsite diagonal. The level symmetry and this branch normalization are structural assumptions ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c5bef5ad97898be2","tex":"\\epsilon_{r+1}","aliases":"ϵ_r+1 epsilon","display":"\\epsilon_{r+1}","kind":"use","excerpt":"g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ,","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1fde43544e1e7fe7","tex":"\\zeta^I","aliases":"ζ^I zeta","display":"\\zeta^I","kind":"use","excerpt":"ADM ghosts and spectral headroom. On a nondegenerate spacelike slice, let q_ab=η_IJE_a^IE_b^J and define the densitized normal widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3a7b183402eb6a5c","tex":"\\zeta_{p,h}","aliases":"ζ_p,h zeta","display":"\\zeta_{p,h}","kind":"use","excerpt":"[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fa38a4e2bce94527","tex":"\\zeta_{p,h+1}","aliases":"ζ_p,h+1 zeta","display":"\\zeta_{p,h+1}","kind":"use","excerpt":"The lifting construction in the proof permits the coherent choice ζ_p,h+1≡ζ_p,hpmodp^h+1. With that choice, reduction of unit addresses retains t and reduces a modulo p^h. If the torsion lifts are instead chosen independently at each depth, the finite factorizations remain exact, but a comparison between levels must include the resulting coordinate","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-44c728f80de640dd","tex":"\\zeta_{p,h}^{\\,t}","aliases":"ζ_p,h^ t zeta","display":"\\zeta_{p,h}^{\\,t}","kind":"use","excerpt":"[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ, (s,a) · ⟼(-1)^s5^apmod2^h+2, · (s,a) · ∈Z/2Z×Z/2^hZ are group isomorphisms onto U_p,h and U_2,h respectively.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"For λ_n=n, a resolvent at iη, η≠0, has zero-padded error sup_n>Nfrac1sqrtn^2+η^2 =frac1sqrt(N+1)^2+η^2⟶0. The heat multiplier e^-tn has error e^-t(N+1) for fixed t>0. These tails decay because the operators are compact. At t=0 the heat operator is the identity, so the same norm-convergence assertion is false.","count":35},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9657c492efcc34dd","tex":"\\eta^2","aliases":"η^2 eta","display":"\\eta^2","kind":"use","excerpt":"For λ_n=n, a resolvent at iη, η≠0, has zero-padded error sup_n>Nfrac1sqrtn^2+η^2 =frac1sqrt(N+1)^2+η^2⟶0. The heat multiplier e^-tn has error e^-t(N+1) for fixed t>0. These tails decay because the operators are compact. At t=0 the heat operator is the identity, so the same norm-convergence assertion is false.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4024c59fd0a358ce","tex":"\\eta^a","aliases":"η^a eta","display":"\\eta^a","kind":"use","excerpt":"Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e0f19d7866e16390","tex":"\\eta^b","aliases":"η^b eta","display":"\\eta^b","kind":"use","excerpt":"Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aae04eeae6b07750","tex":"\\eta^{IJ}","aliases":"η^IJ eta","display":"\\eta^{IJ}","kind":"use","excerpt":"vol_e · =tfrac14!ϵ_IJKL e^Iwedge e^Jwedge e^Kwedge e^L, · Σ_I · =tfrac13!ϵ_IJKLe^Jwedge e^Kwedge e^L. Use {γ^I,γ^J}=2η^IJ and γ_IJ=tfrac12[γ_I,γ_J]. The covariant derivatives are Dψ · = dψ+tfrac14ω^IJγ_IJψ-iq aψ,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50d5e65dfebdbedc","tex":"\\eta_a","aliases":"η_a eta","display":"\\eta_a","kind":"use","excerpt":"[Exact nonlinear defect identity] For arbitrary finite arrays q,p,ξ,η, { D_h[ξ], D_h[η]}_h = · D_h[ξ Dη-η Dξ]nonumber · +h∑_ap_a[ ξ_a L_D(η,Dq)_a -η_a L_D(ξ,Dq)_a].","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18a46bf826b0cbf4","tex":"\\eta_{IJ}","aliases":"η_IJ eta","display":"\\eta_{IJ}","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1dd0a87bdefa7845","tex":"\\eta_j","aliases":"η_j eta","display":"\\eta_j","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d0ab24b9a250f9cb","tex":"\\eta_{j+1}^-","aliases":"η_j+1^- eta","display":"\\eta_{j+1}^-","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3e03897b69d9ea1","tex":"\\eta_{j-1}^+","aliases":"η_j-1^+ eta","display":"\\eta_{j-1}^+","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5e88b9488b9fcd67","tex":"\\eta_j^+","aliases":"η_j^+ eta","display":"\\eta_j^+","kind":"use","excerpt":"[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2d58b3c78f282ec7","tex":"\\eta_j^-","aliases":"η_j^- eta","display":"\\eta_j^-","kind":"use","excerpt":"The first coin preserves the local norm, so ρ_j^n=|η_j^+|^2+|η_j^-|^2. The second coin and temporal phase preserve the norm after the shift, giving ρ_j^n+1=|η_j-1^+|^2+|η_j+1^-|^2. Subtract these expressions and group the two terms associated with each boundary edge. This is [source reference]. The transformed η_j is g_j^nη_j, so its channel norms ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"l A_L,j; frac(-1)^Ly^LL^2L). One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters are positive and the ratio test again gives entire convergence. Character filtering reconstructs B=∑_jB_L,j. With the Euler derivative Θ=y d/dy, termwise differentiation al","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and se","count":74},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6a830538315a1ab8","tex":"\\vartheta","aliases":"ϑ vartheta","display":"\\vartheta","kind":"use","excerpt":"The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the se","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-772c337813af41af","tex":"\\Theta^2","aliases":"Θ^2 Theta","display":"\\Theta^2","kind":"use","excerpt":"One of the two q! factors is the conventional hypergeometric factorial and the other supplies the explicit bottom parameter 1. All bottom parameters are positive and the ratio test again gives entire convergence. Character filtering reconstructs B=∑_jB_L,j. With the Euler derivative Θ=y d/dy, termwise differentiation also gives the cyclic system Θ^","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","excerpt":"of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8481a76e553d57b9","tex":"\\vartheta^2","aliases":"ϑ^2 vartheta","display":"\\vartheta^2","kind":"use","excerpt":"The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the se","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c112a1cdbb689680","tex":"\\theta^4","aliases":"θ^4 theta","display":"\\theta^4","kind":"use","excerpt":"of F. If the rectangle is centered at x_c and F is C^2, symmetric integration removes its linear Taylor terms: θ=-qa_μ a_ν F_μν(x_c)+O(a^4). Consequently, 1-ReW =tfrac12q^2a_μ^2a_ν^2 F_μν(x_c)^2+O(a^6). To see the remainder, use 1-cosθ=θ^2/2+O(θ^4) and θ=O(a^2). The cross term from the O(a^4) phase error is O(a^6), while the fourth-power term is O(","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a37bcd8a9273f437","tex":"\\Theta_{\\mathrm D}","aliases":"Θ_ D Theta mathrm","display":"\\Theta_{\\mathrm D}","kind":"use","excerpt":"Fermionic primary constraints and half densities. Impose the inherited algebraic spinor relation barψ=ψ^†γ^0, where (γ^0)^2=-id. Its phase convention must be included in the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0a9a14fec4bd0562","tex":"\\theta_+","aliases":"θ_+ theta","display":"\\theta_+","kind":"use","excerpt":"For [source reference], choose future radial null vectors ℓ_±=σ^-1∂_t+(-v±1)∂_r. Because the sphere area is 4π r^2, its expansion along ℓ is 2ℓ(r)/r. Consequently θ_+=frac2r(1-v), θ_-=-frac2r(1+v). On the positive branch, v=1 is a marginally outer trapped sphere, and v>1 gives both expansions negative. Equivalently, the compactness 2m/r equals or e","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0123f2e7368df612","tex":"\\theta_-","aliases":"θ_- theta","display":"\\theta_-","kind":"use","excerpt":"For [source reference], choose future radial null vectors ℓ_±=σ^-1∂_t+(-v±1)∂_r. Because the sphere area is 4π r^2, its expansion along ℓ is 2ℓ(r)/r. Consequently θ_+=frac2r(1-v), θ_-=-frac2r(1+v). On the positive branch, v=1 is a marginally outer trapped sphere, and v>1 gives both expansions negative. Equivalently, the compactness 2m/r equals or e","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","excerpt":"The mean logarithmic expansion of this map is also exact: ∫_-2^2log|R_p'(x)| dν(x)=log p. Indeed for x=2cosθ, R_p'(x)=psin(pθ)/sinθ. Logarithms at its finitely many zeros are integrable, since they have the local form log|θ-θ_0| plus a bounded term. Hence the integral is log p plus the difference of the integrals of log|sin(pθ)| and logsinθ over [0","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e543323c4045f446","tex":"\\theta_f","aliases":"θ_f theta","display":"\\theta_f","kind":"use","excerpt":"six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact group identity. If W_f=e^iθ_f with each θ_f chosen as a princ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a721963f725aae53","tex":"\\theta_i","aliases":"θ_i theta","display":"\\theta_i","kind":"use","excerpt":"Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c8082d88b8040556","tex":"\\theta_j","aliases":"θ_j theta","display":"\\theta_j","kind":"use","excerpt":"Let S|n⟩=|n+1pmod N⟩ and H=μ I+κ(S+S^*), μ,κ∈R. Use the coarse Fourier matrix (F_M)_jk=M^-1/2e^-2π ijk/M and θ_j=2π j/M. After A, Fourier transformation in k, and the permutation from residue major to momentum major ordering, the operator becomes a direct sum of p× p symbols. To see their entries, note that S sends (r,k) to (r+1,k) for r<p-1 and se","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-998cf8e699fcb189","tex":"\\Theta_{\\mathrm{red}}","aliases":"Θ_red Theta mathrm","display":"\\Theta_{\\mathrm{red}}","kind":"use","excerpt":"The resulting real canonical superphase space has coordinates (K_a^i,widetilde E^a_i), (A_a, E^a), (Ψ,Ψ^†), and one-form Θ_red=∫_Σ d^3x [frac1κwidetilde E^a_iδ K_a^i + E^aδ A_a +frac i2(Ψ^†δΨ-δΨ^†Ψ)]. Its even canonical dimension is 24 per spatial point: eighteen gravitational and six electromagnetic coordinates. The odd primary block has eight com","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-66f2af2663fe43f3","tex":"\\theta_x","aliases":"θ_x theta","display":"\\theta_x","kind":"use","excerpt":"For the xy face, the two-path angle difference is Θ_xy(p,h) = · θ_y(p+h e_x,h)-θ_y(p,h) · -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cro","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-878cb89cae859a1d","tex":"\\Theta_{xy}","aliases":"Θ_xy Theta","display":"\\Theta_{xy}","kind":"use","excerpt":"For the xy face, the two-path angle difference is Θ_xy(p,h) = · θ_y(p+h e_x,h)-θ_y(p,h) · -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cro","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6602dff3fef9759e","tex":"\\theta_y","aliases":"θ_y theta","display":"\\theta_y","kind":"use","excerpt":"For the xy face, the two-path angle difference is Θ_xy(p,h) = · θ_y(p+h e_x,h)-θ_y(p,h) · -(θ_x(p+h e_y,h)-θ_x(p,h)). Therefore |Θ_xy(p,h)|≤ (B_ymid x+B_xmid y)h^2, |Ω_xy(p,h)-1|≤ (B_ymid x+B_xmid y)h^2. For the last implication, e^it-1=∫_0^t i e^is ds has absolute value at most |t|. The other two plane estimates follow from their corresponding cro","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4f43ba5cea859e71","tex":"\\Theta_\\Sigma","aliases":"Θ_Σ Theta Sigma","display":"\\Theta_\\Sigma","kind":"use","excerpt":"Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A · -tfrac i2σ_I(barψγ^Iδψ -δbarψγ^Iψ)]. The minus sign in the spinor term is fixed by the time--space orientation and by moving dt past the spatial three-form. It must be ret","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-769672901a15b39c","tex":"\\iota_c","aliases":"ι_c iota","display":"\\iota_c","kind":"use","excerpt":", the Cartan identities give L_ηΩ=ι_η R+D_Ω(ι_ηΩ), L_η A=ι_η F+ d_Σ(ι_η A). Accordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression J_Σ[c,χ,u]=C[ζ(c)] +G_ L[χ-ι_cΩ] +G_ U[u+ι_cA]. For tangential c, ζ(c)=ι_cE is a spatial contraction. Fo","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-df590a2d29cde0af","tex":"\\iota_L","aliases":"ι_L iota","display":"\\iota_L","kind":"use","excerpt":"B^* · D, A=A^*, D=D^*. Here B:HH_D→HH_L. Let ι_L embed HH_L into the full space. This notation fixes the coordinate type of every compression.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9a81ea7f53a2a389","tex":"\\iota_L^*","aliases":"ι_L^* iota","display":"\\iota_L^*","kind":"use","excerpt":"[Finite Feshbach identity] If z∉σ(D), define Σ(z)=B(z-D)^-1B^*, S(z)=z-A-Σ(z). Then z-H is invertible exactly when S(z) is invertible. On this common regular set, ι_L^*(z-H)^-1ι_L · = S(z)^-1,nonumber det(z-H) · =det(z-D)det S(z). For E∈R∖σ(D), an eigenvector of H is equivalent to a nonzero solution of S(E)u=0, with hidden component v=(E-D)^-1B^*u.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-62f48d52cf8d451c","tex":"\\iota_\\eta","aliases":"ι_η iota eta","display":"\\iota_\\eta","kind":"use","excerpt":"Normal deformations require more boundary data. For a spatial vector field η, the Cartan identities give L_ηΩ=ι_η R+D_Ω(ι_ηΩ), L_η A=ι_η F+ d_Σ(ι_η A). Accordingly, a covariant tangential generator acquires Lorentz and Abelian compensators to generate an ordinary spatial Lie derivative. The source records the expression J_Σ[c,χ,u]=C[ζ(c)] +G_ L[χ-ι","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is","count":36},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0fc1259c08b990e9","tex":"\\kappa^2","aliases":"κ^2 kappa","display":"\\kappa^2","kind":"use","excerpt":"The normalized effective energy is (z-μ_p(z))/κ_p(z)=R_p(x). The parameter z remains fixed through the elimination. Since μ_p and κ_p depend on it, this is an energy-dependent spectral closure, not an energy-independent Hamiltonian flow. For p=2, R_2(x)=x^2-2 and U_1(x/2)=x, giving H_rm eff^(2)(z) =μ I+fracκ^2z-μ(2I+T+T^*), z≠μ. This decimation ret","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f2aa78ec120c23cd","tex":"\\kappa^{p-1}","aliases":"κ^p-1 kappa","display":"\\kappa^{p-1}","kind":"use","excerpt":"Let Q project onto the eliminated sites. On the regular chart, det(z-H)=det(z-QHQ)det S_p(z), det(z-QHQ)=[κ^p-1U_p-1(x/2)]^M. Their product is a polynomial identity after cancellation. It records both the hidden path factor and the coarse Schur factor, including their cancellations at the resonant energies.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f4d1001288e3b575","tex":"\\kappa_m","aliases":"κ_m kappa","display":"\\kappa_m","kind":"use","excerpt":"an interval and is therefore an identity of polynomials, including complex x. For t=cosθ away from zeros of the sine denominators, U_n-1(T_m(t))U_m-1(t) =fracsin(mnθ)sin(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-39929c2ef6bc800a","tex":"\\kappa_{m,n}","aliases":"κ_m,n kappa","display":"\\kappa_{m,n}","kind":"use","excerpt":"ls, including complex x. For t=cosθ away from zeros of the sine denominators, U_n-1(T_m(t))U_m-1(t) =fracsin(mnθ)sin(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m(x). A second one gives κ_m,n =fracκ_mU_n-1(x_m/2) =f","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-46fe9f573328c1ae","tex":"\\kappa_{mn}","aliases":"κ_mn kappa","display":"\\kappa_{mn}","kind":"use","excerpt":"U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur inverses exist. Its final coefficients are κ_mn(z)=fracκU_mn-1(x/2), μ_mn(z)=z-fracκ R_mn(x)U_mn-1(x/2). For x=2cosθ, first-kind composition gives R","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-810ceeda3892b7a8","tex":"\\kappa_p","aliases":"κ_p kappa","display":"\\kappa_p","kind":"use","excerpt":"Let the cyclic translation be (S_p,hf)(x)=f(x-1), with subtraction modulo p^h. Since a fine replicated function depends only on the residue modulo p^h, S_p,h+1V_p,h=V_p,hS_p,h. The adjoint translation obeys the same identity. Therefore the cyclic Laplacian L_p,h=2I-S_p,h-S_p,h^* intertwines, and for real κ_p the family H_p,h=H_p,h^Haar+κ_p L_p,h is","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Let V be an isometry from a coarse space to a finite fine space, set P=VV^* and Q=I-P, and let H_0=H_0^* satisfy H_0V=VH_c. Let B=B^* be purely off diagonal: PBP=0, QBQ=0. For a real λ, define H_λ=H_0+λ B. In the source model λ=p^-w for an assigned weight w≥1.","count":92},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b20ef177d3714eb3","tex":"\\Lambda'","aliases":"Λ' Lambda","display":"\\Lambda'","kind":"use","excerpt":"The gauge-improved radial momentum and electric constraints are D=P_RR'-Λ P_Λ', G=Q', and the Hamiltonian constraint is H= · -fracP_RP_ΛR +fracΛ P_Λ^22R^2 +fracRR''Λ-fracRR'Λ'Λ^2 +fracR'^22Λ-fracΛ2 +fracΛ Q^22R^2. The momenta follow the convention P_Λ · =-frac RN(dot R-N^rR'),","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-308a821eda6e7229","tex":"\\lambda^2","aliases":"λ^2 lambda","display":"\\lambda^2","kind":"use","excerpt":"To verify this, subtract the constant integrand QBP in Duhamel's formula and use norme^-iuK-I≤|u|norm K for a finite self-adjoint K. The two deviations are bounded by (|t-s|normH_λ+|s|normH_0)normQBP on the oriented integration interval. Their integrals are both t^2/2. For |λ|≤1, this is a uniform O(t^2|λ|) remainder. A generic O(t^2λ^2) remainder ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2669d46cbfa6c78a","tex":"\\Lambda^3","aliases":"Λ^3 Lambda","display":"\\Lambda^3","kind":"use","excerpt":"-fracP_ΛRΛ D +frac QR G. Consequently, on a patch where R'/Λ≠0, the systems ( H, D, G)=0 and ( M', D, G)=0 have the same zero set. Differentiating [source reference] gives M'= · fracR'2-fracR'^32Λ^2 -fracRR'R''Λ^2+fracRR'^2Λ'Λ^3 +fracP_Λ P_Λ'R-fracP_Λ^2R'2R^2 · +fracQQ'R-fracQ^2R'2R^2. Expanding the right side of [source reference] cancels the two ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d356f71025f33224","tex":"\\lambda^i","aliases":"λ^i lambda","display":"\\lambda^i","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9f563c9de566dc8a","tex":"\\lambda^{IJ}","aliases":"λ^IJ lambda","display":"\\lambda^{IJ}","kind":"use","excerpt":"Gauge transformations and graded conventions. For even parameters (ξ,λ,u), the coframe transforms by its Lie derivative and Lorentz rotation, the connection by L_ξω-D_ωλ, and the Abelian potential by L_ξ a+ du. The spinor transforms by L_ξψ+tfrac14λ^IJγ_IJψ+iquψ, with the dual action on barψ. All internal contractions in~[source reference] are inva","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-75946f9ee1ce0aa6","tex":"\\lambda^j","aliases":"λ^j lambda","display":"\\lambda^j","kind":"use","excerpt":"Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8fa5b25f3a5291e2","tex":"\\Lambda^{\\rm TT}_{ij,kl}","aliases":"Λ^rm TT_ij,kl Lambda rm","display":"\\Lambda^{\\rm TT}_{ij,kl}","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7238fc50817cce55","tex":"\\lambda_0","aliases":"λ_0 lambda","display":"\\lambda_0","kind":"use","excerpt":"If a periodic λ exists, integrating λ_θ=M_G forces the zero-period condition. Conversely, fix τ_0 and define on the lifted strip λ(τ,θ)=λ_0 +∫_τ_0^τH_G(s,0) ds +∫_0^θM_G(τ,η) dη. Its θ derivative is M_G. Its τ derivative is H_G by the integrability identity and the fundamental theorem of calculus. The zero-period condition makes it periodic. The di","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-129252f1935f1ee1","tex":"\\Lambda_h","aliases":"Λ_h Lambda","display":"\\Lambda_h","kind":"use","excerpt":"The closed finite object is the full inner algebra. To identify a continuum spatial diffeomorphism, one must additionally construct parameters Λ_h[ξ] whose action approximates the Lie derivative on resolved observables. An arbitrary subset of matrices chosen to resemble first derivatives need not be closed under commutators. Equation~[source refere","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8890f23babb77fdb","tex":"\\Lambda_j","aliases":"Λ_j Lambda","display":"\\Lambda_j","kind":"use","excerpt":"[Exact finite reduced constraints] On the image of [source reference], the four reduced constraints C_Λ,j · =Λ_j-D_4R_j, · C_M,j · =P_Λ,j^2-2MR_j+Q^2, C_P,j · =P_R,jP_Λ,j-MΛ_j, · C_Q,j · =Q_j-Q vanish in exact arithmetic. The discrete mass coordinate obtained by replacing R' with D_4R in [source reference] equals M at every reconstructed node. Each","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-75281f36da6b8ff7","tex":"\\lambda_n","aliases":"λ_n lambda","display":"\\lambda_n","kind":"use","excerpt":"Let (λ_n) be any real sequence and define D(H_λ)={c∈ℓ^2: ∑_nλ_n^2|c_n|^2<∞}, (H_λ c)_n=λ_nc_n. The operator is dense because it contains every finitely supported sequence. It is symmetric by the reality of the multipliers. Its domain is maximal for this coordinate action.","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3d5f1cc7f79d67ce","tex":"\\lambda_n^2","aliases":"λ_n^2 lambda","display":"\\lambda_n^2","kind":"use","excerpt":"Let (λ_n) be any real sequence and define D(H_λ)={c∈ℓ^2: ∑_nλ_n^2|c_n|^2<∞}, (H_λ c)_n=λ_nc_n. The operator is dense because it contains every finitely supported sequence. It is symmetric by the reality of the multipliers. Its domain is maximal for this coordinate action.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ecdf3ba0683d5eab","tex":"\\lambda_\\theta","aliases":"λ_θ lambda theta","display":"\\lambda_\\theta","kind":"use","excerpt":"M_G · =2(P_τ P_θ+e^2PQ_τ Q_θ). The remaining reconstruction equations are λ_τ=H_G and λ_θ=M_G.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-deb6b1daac660ba3","tex":"\\lambda_\\tau","aliases":"λ_τ lambda tau","display":"\\lambda_\\tau","kind":"use","excerpt":"M_G · =2(P_τ P_θ+e^2PQ_τ Q_θ). The remaining reconstruction equations are λ_τ=H_G and λ_θ=M_G.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c26c6e94e2af784f","tex":"\\mu^k","aliases":"μ^k mu","display":"\\mu^k","kind":"use","excerpt":"Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ffb97b3578a68df","tex":"\\mu_m","aliases":"μ_m mu","display":"\\mu_m","kind":"use","excerpt":"an interval and is therefore an identity of polynomials, including complex x. For t=cosθ away from zeros of the sine denominators, U_n-1(T_m(t))U_m-1(t) =fracsin(mnθ)sin(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-72a42d4b5c7b8dbd","tex":"\\mu_{m,n}","aliases":"μ_m,n mu","display":"\\mu_{m,n}","kind":"use","excerpt":"(mθ) fracsin(mθ)sinθ =U_mn-1(t). Polynomial equality extends it through those zeros and to all t. After the first regular elimination, the normalized variable is x_m=(z-μ_m)/κ_m=R_m(x). A second one gives κ_m,n =fracκ_mU_n-1(x_m/2) =fracκ U_m-1(x/2)U_n-1(T_m(x/2)) =fracκU_mn-1(x/2). Similarly μ_m,n=z-κ_m,nR_n(R_m(x)), which gives [source reference]","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fe764fd9990b4bf7","tex":"\\mu_{mn}","aliases":"μ_mn mu","display":"\\mu_{mn}","kind":"use","excerpt":"U_mn-1(t) · =U_n-1(T_m(t)) U_m-1(t) hold over C. In a free cycle with N=mnL, L≥3, successive m- then n-site decimation at a fixed z agrees with direct mn-site decimation whenever its intermediate Schur inverses exist. Its final coefficients are κ_mn(z)=fracκU_mn-1(x/2), μ_mn(z)=z-fracκ R_mn(x)U_mn-1(x/2). For x=2cosθ, first-kind composition gives R","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"For completeness, the union of these ranges is dense. Cylinders form a countable clopen base for the topology on Z_p and generate its Borel sigma-algebra. If f∈ L^2 is orthogonal to every cylinder indicator, the finite complex measure ν(A)=∫_A f dμ_p vanishes on cylinders and their finite unions. The class of sets on which ν is zero contains the wh","count":24},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c773963d9aaa806d","tex":"\\nu_{\\rm emit}","aliases":"ν_rm emit nu rm","display":"\\nu_{\\rm emit}","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5f40876cacfe4bfb","tex":"\\nu_{\\rm obs}","aliases":"ν_rm obs nu rm","display":"\\nu_{\\rm obs}","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c645657d025cc351","tex":"\\Xi","aliases":"Ξ Xi","display":"\\Xi","kind":"use","excerpt":"The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or eq","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^","count":27},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3b20e380f56b440c","tex":"\\xi^a","aliases":"ξ^a xi","display":"\\xi^a","kind":"use","excerpt":"Constraints and their normalizations. Take connection coefficients A_a^i, densitized triads E^a_i, internal parameters λ^i, shifts ξ^a, and densitized lapse n in T_K. Set the canonical normalization to one and define F_ab^i · =δ_aA_b^i-δ_bA_a^i +ϵ^i_jkA_a^jA_b^k,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf3f0e1455e18ba2","tex":"\\xi^b","aliases":"ξ^b xi","display":"\\xi^b","kind":"use","excerpt":"Put (λ×μ)^i · =ϵ^i_jkλ^jμ^k, [ξ,η]^a · =ξ^bδ_bη^a-η^bδ_bξ^a, L_ξ^(-1)n · =ξ^aδ_an-nδ_aξ^a, S^a(n,m) · =E^a_iE^bi(mδ_bn-nδ_bm). The lapse convention fixes the sign and density weight of this last expression. It should not be compared with a differently ordered normal bracket without translating conventions.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-31dd9a2ca27f8848","tex":"\\Xi^\\ddagger","aliases":"Ξ^ddagger Xi ddagger","display":null,"kind":"use","excerpt":"The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or eq","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-59bb53c8543f7887","tex":"\\xi_a","aliases":"ξ_a xi","display":"\\xi_a","kind":"use","excerpt":"Let a periodic real lattice have canonical brackets {q_a,p_b}_h=δ_ab/h and a skew matrix D. Define D_h[ξ]=h∑_aξ_ap_a(Dq)_a, L_D(f,g)=D(fg)-(Df)g-f(Dg). All products on this lattice are pointwise.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-273ca48755e332d5","tex":"\\xi_i","aliases":"ξ_i xi","display":"\\xi_i","kind":"use","excerpt":"[Exact finite linearized closure] For every solution of [source reference], dot C=-δ_iC_i, dot C_i=0. Thus initially vanishing constraints remain zero. With the canonical Frobenius bracket and smearings C[n]=⟨ n,C⟩_h, D[ξ]=⟨ξ_i,C_i⟩_h, all three brackets {C[n],C[m]}, {D[ξ],D[η]}, and {C[n],D[ξ]} vanish identically.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1deb1e97057da782","tex":"\\xi_j","aliases":"ξ_j xi","display":"\\xi_j","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d631fd84ebdad115","tex":"\\xi_j^+","aliases":"ξ_j^+ xi","display":"\\xi_j^+","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d0850d2b3c9c36b7","tex":"\\xi_j^-","aliases":"ξ_j^- xi","display":"\\xi_j^-","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-430363506ce958cc","tex":"\\xi_k","aliases":"ξ_k xi","display":"\\xi_k","kind":"use","excerpt":"The momentum generator changes the metric perturbation by h_ij↦ h_ij+δ_iξ_j+δ_jξ_i. The corresponding change of the scalar constraint is 2δ^2δ_kξ_k-2δ^2δ_kξ_k=0. No product rule is used in these cancellations.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). Wi","count":43},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cb28cf0122b8897d","tex":"\\varpi","aliases":"ϖ varpi","display":"\\varpi","kind":"use","excerpt":"There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total gh","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d465d49b065aadd9","tex":"\\Pi'","aliases":"Π' Pi","display":"\\Pi'","kind":"use","excerpt":"endequation Thus ∂_t(m'-r^2 E)=0. A radial mass constraint imposed initially propagates under this continuum system. Expanding [source reference] gives dotΦ=(σ A)' and dotΠ=(σ B)'+2σ B/r. Differentiate E and subtract the radial derivative of the flux. Terms involving Π', Φ', and v' cancel, leaving ∂_t(r^2 E)-∂_r(r^2 J) =r^2[σ'AB+ΠΦdot v]. Since m=r","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f6c04fa92bb658c0","tex":"\\Pi^2","aliases":"Π^2 Pi","display":"\\Pi^2","kind":"use","excerpt":"lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map again leaves it unchanged. On the transverse t","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"The exponent is positive for each fixed L, but decreases as 2π^2/L^2 when L grows. The conclusion is therefore uniform in the finite packet label j at fixed L; it is not a depth-uniform estimate at arbitrarily large residue modulus.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0c9f6eacc84d6080","tex":"\\pi^3","aliases":"π^3 pi","display":"\\pi^3","kind":"use","excerpt":"(∑_n∈ SC(n)^2 a_n _2^2)^1/2 =frac(2π)^36=frac4π^33.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a23e3c8d3f2c871d","tex":"\\Pi_a","aliases":"Π_a Pi","display":"\\Pi_a","kind":"use","excerpt":"Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6983098f86ea86cc","tex":"\\pi_a","aliases":"π_a pi","display":"\\pi_a","kind":"use","excerpt":"For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9bf697610fc96b35","tex":"\\varpi_A","aliases":"ϖ_A varpi","display":"\\varpi_A","kind":"use","excerpt":"There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total gh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eeab2b59b76e99ba","tex":"\\Pi_b","aliases":"Π_b Pi","display":"\\Pi_b","kind":"use","excerpt":"The dynamics respects the same construction. Put widetilde K=VKV^*. It is skew-adjoint, and fracddtVψ=widetilde K Vψ. Moving adjoints across the inner product shows 2ReinnerΠ_bVψwidetilde KΠ_aVψ =2ReinnerF_bψKF_aψ=J^F_ba. The positive-effect current is therefore exactly the sharp current of this lifted flow evaluated on the embedded state. It is no","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-15b2d314605b1489","tex":"\\pi_b","aliases":"π_b pi","display":"\\pi_b","kind":"use","excerpt":"For the rank-one jump model, detailed balance supplies a definite monotonicity theorem. Let π_a>0 be a fixed probability distribution, and assume γ_baπ_a=γ_abπ_b. Define relative densities u_a=p_a/π_a and conductances c_ba=γ_baπ_a. Then c_ba=c_ab≥0 and F_b← a=c_ba(u_a-u_b).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-26ac1047af452808","tex":"\\Pi_{bc}","aliases":"Π_bc Pi","display":"\\Pi_{bc}","kind":"use","excerpt":"Similarly, with E^a=tfrac12ϵ^abcΠ_bc, ∫_ΣΠwedgeδ A =∫_Σ E^aδ A_a d^3x. The Maxwell auxiliary equation then gives its metric Hamiltonian H_ M=frac12eq_ab ( E^a E^b+ B^a B^b), B^a=tfrac12ϵ^abc F_bc. The inverse 1/e is well defined in the patch and singular at its boundary.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-744abce417454055","tex":"\\varpi_C","aliases":"ϖ_C varpi","display":"\\varpi_C","kind":"use","excerpt":"There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total gh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7a72c1d968673c18","tex":"\\Pi_h","aliases":"Π_h Pi","display":"\\Pi_h","kind":"use","excerpt":"On interior points of a uniform radial grid, use the fourth-order derivative (D_4u)_j=frac-u_j+2+8u_j+1-8u_j-1+u_j-212h. Its stencil radius is two. Given M, Q, and a fixed branch ε, a radius array satisfying D_4R_j>0 and 2MR_j-Q^2>0 determines Π_h[R]_j=(R_j,D_4R_j, εsqrt2MR_j-Q^2, fracM D_4R_jεsqrt2MR_j-Q^2,Q). This is a local reconstruction of (R,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5fe4316a85529609","tex":"\\Pi_{ij}","aliases":"Π_ij Pi","display":"\\Pi_{ij}","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bc3ccd852974b115","tex":"\\pi_{ij}","aliases":"π_ij pi","display":"\\pi_{ij}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-45ec2a0e8556ad15","tex":"\\Pi_{ik}","aliases":"Π_ik Pi","display":"\\Pi_{ik}","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9670b7adc5d4414b","tex":"\\Pi_{il}","aliases":"Π_il Pi","display":"\\Pi_{il}","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-364fc2af609ec6e0","tex":"\\pi_{ji}","aliases":"π_ji pi","display":"\\pi_{ji}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e203f46ce227724d","tex":"\\Pi_{jk}","aliases":"Π_jk Pi","display":"\\Pi_{jk}","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea3a8b556b9dfae9","tex":"\\Pi_{jl}","aliases":"Π_jl Pi","display":"\\Pi_{jl}","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bbe3cb6f07086c40","tex":"\\pi_{kk}","aliases":"π_kk pi","display":"\\pi_{kk}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-66df9abdcaa5244e","tex":"\\Pi_{kl}","aliases":"Π_kl Pi","display":"\\Pi_{kl}","kind":"use","excerpt":"For a Fourier mode on the periodic centered lattice, let widetilde k_i=sin(k_ih)/h. If widetilde k≠0, define Π_ij=δ_ij -fracwidetilde k_iwidetilde k_j|widetilde k|^2, Λ^rm TT_ij,kl =tfrac12(Π_ikΠ_jl+Π_ilΠ_jk-Π_ijΠ_kl). For a symmetric tensor X, its image is Π XΠ-Πtr(Π X)/2. It is transverse and traceless because Π^2=Π and trΠ=2. Applying the map ag","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fcdf8b96328dc85b","tex":"\\Pi_P","aliases":"Π_P Pi","display":"\\Pi_P","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1aedf1628657716c","tex":"\\Pi_P^2","aliases":"Π_P^2 Pi","display":"\\Pi_P^2","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fc4c702d6f1c7544","tex":"\\Pi_Q","aliases":"Π_Q Pi","display":"\\Pi_Q","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-10341aa4e026fcc1","tex":"\\Pi_Q^2","aliases":"Π_Q^2 Pi","display":"\\Pi_Q^2","kind":"use","excerpt":"The separate Gowdy BFV target. The Gowdy target chapter changes from τ to areal time t=e^-τ>0. Its wave-map Hamiltonian is H_t=fracΠ_P^2+e^-2PΠ_Q^22t +frac t2(P_θ^2+e^2PQ_θ^2). The physical Hamiltonian density at fixed t and the tangential density D=Π_PP_θ+Π_QQ_θ have the continuum bracket pattern of two deformation generators with densitized smear","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d2eb2d4e3fc3398a","tex":"\\Pi_r","aliases":"Π_r Pi","display":"\\Pi_r","kind":"use","excerpt":"p-|0⟩⟨0|)⊗ I_m] W_p,h · =D_h^(h). The second identity subtracts the coarse zero-frequency projector from the identity using [source reference]. Recursive application to the zero-frequency channel alone constructs a unitary Haar analysis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). Wi","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4d66a630bbdba50b","tex":"\\pi_{\\rm tr}","aliases":"π_rm tr pi rm","display":"\\pi_{\\rm tr}","kind":"use","excerpt":"Let h_ij=h_ji and π_ij=π_ji take values in V_h. Repeated spatial indices are summed. Put h_rm tr=h_kk, π_rm tr=π_kk and define C · =δ_iδ_jh_ij-δ^2h_rm tr, · C_i · =-2δ_jπ_ij,nonumber","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b484755dd099904d","tex":"\\pi_\\chi","aliases":"π_χ pi chi","display":"\\pi_\\chi","kind":"use","excerpt":"nal BFV sector On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation cancels the terms quadratic in g","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a49eb9b81d375c75","tex":"\\rho_{(x,\\mathbf d)}^n","aliases":"ρ_(x, d)^n rho mathbf","display":"\\rho_{(x,\\mathbf d)}^n","kind":"use","excerpt":"[Exact finite propagation] If the initial outgoing amplitudes are supported at v_0, then d_C_N(x,x_0)+d_T( d, d_0)>n Longrightarrow ρ_(x, d)^n=0. For two prefixes, writing lcp for the length of their longest common prefix, d_T( d, d_0) =| d|+| d_0|-2lcp( d, d_0). In particular, each tick allows at most one physical edge or one tree edge, and || d|-","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6fd3bc029e9c79bd","tex":"\\rho_+","aliases":"ρ_+ rho","display":"\\rho_+","kind":"use","excerpt":"In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i -i · 1, ρ_-=frac121 · -i i · 1, ρ_0=frac121 · 0 0 · 1. All three are density matrices with packet probabilities (1/2,1/2). For H=g(0 · 1","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-80922e0e24a01913","tex":"\\rho_-","aliases":"ρ_- rho","display":"\\rho_-","kind":"use","excerpt":"In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i -i · 1, ρ_-=frac121 · -i i · 1, ρ_0=frac121 · 0 0 · 1. All three are density matrices with packet probabilities (1/2,1/2). For H=g(0 · 1","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","excerpt":"In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i -i · 1, ρ_-=frac121 · -i i · 1, ρ_0=frac121 · 0 0 · 1. All three are density matrices with packet probabilities (1/2,1/2). For H=g(0 · 1","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cf67cfa9d97d052e","tex":"\\rho_{12}","aliases":"ρ_12 rho","display":"\\rho_{12}","kind":"use","excerpt":"[Two-state relaxation] Take γ_21=γ_12=k>0 and p_1+p_2=1. Then dot p_1=k(1-2p_1), p_1(t)=tfrac12+(p_1(0)-tfrac12)e^-2kt, ρ_12(t)=e^-ktρ_12(0). The populations relax toward equal weights. In the coherent exchange of Example~[source reference], the populations oscillate and depend on phase. Both systems admit antisymmetric probability currents, but th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-362dfb6c4b153cec","tex":"\\rho_A","aliases":"ρ_A rho","display":"\\rho_A","kind":"use","excerpt":"al because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are exact atlas coarsenings. For example, p_a^A=Tr((P_a⊗id_B)ρ_AB) =Tr(P_aρ_A), ρ_A=Tr_Bρ_AB. Choose full orthonormal bases of the two Hilbert spaces, with (e","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-acbabae96acb9647","tex":"\\rho_{AB}","aliases":"ρ_AB rho","display":"\\rho_{AB}","kind":"use","excerpt":"Let HH_A and HH_B be finite-dimensional Hilbert spaces with finite orthogonal atlases (P_a) and (Q_μ). Their tensor atlas is R_aμ=P_a⊗ Q_μ. It is orthogonal because tensor multiplication gives R_aμR_bν=δ_abδ_μνR_aμ. Its sum is id_A⊗id_B. For a density matrix ρ_AB, define p_aμ=Tr(R_aμρ_AB), p_a^A=∑_μ p_aμ, p_μ^B=∑_a p_aμ. These marginalizations are ","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-adbd7f547dc7bde2","tex":"\\rho_{ab}","aliases":"ρ_ab rho","display":"\\rho_{ab}","kind":"use","excerpt":"In the coordinate atlas, the current is 2Im(H_baρ_ab). It depends on off-diagonal coherence. A useful test is to compare ρ_+=frac121 · i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f3e334246055738a","tex":"\\rho_{bb}","aliases":"ρ_bb rho","display":"\\rho_{bb}","kind":"use","excerpt":"[Population gain and loss] The diagonal populations p_b=ρ_bb of a Hermitian solution satisfy dot p_b=∑_aF_b← a, F_b← a=γ_bap_a-γ_abp_b, F_b← a=-F_a← b. The total population is constant, and every coarse population has the sum of the boundary jump currents as its derivative.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d9a79e9beba3ca3f","tex":"\\rho_C","aliases":"ρ_C rho","display":"\\rho_C","kind":"use","excerpt":"x_1^2+mn>0. Writing μ=mn/(m+n) and d=x_1^2+μ, the stationary detail values are y_*=x_1x_2/d and z_*=my_* /(m+n). Substitution gives S_C=fracμ x_2^22d, R_C=∂_x_1+fracx_1x_2d∂_x_2. Differentiation proves R_CS_C=0. Gaussian integration, rather than stationary substitution alone, also gives the density ρ_Cpropto D^-1/2, up to a constant. It satisfies d","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-936d7991f33298f9","tex":"\\rho_i","aliases":"ρ_i rho","display":"\\rho_i","kind":"use","excerpt":"Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6a94deaf3c45a250","tex":"\\rho_{ij}","aliases":"ρ_ij rho","display":"\\rho_{ij}","kind":"use","excerpt":"In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c86a36ff5a44dcd8","tex":"\\rho_j","aliases":"ρ_j rho","display":"\\rho_j","kind":"use","excerpt":"[Charge balance and Gauss propagation] For every differentiable solution of [source reference]--[source reference], define ρ_j=ψ_j^*ψ_j, barρ=frac1N∑_jρ_j, G_j=fracE_j-E_j-1h+ρ_j-barρ. Then dotρ_j+fracJ_j-J_j-1h=0, dotbarρ=0, dot G_j=0. In particular, initial data satisfying G_j=0 retain that constraint on their interval of existence.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-36466f74c3e41259","tex":"\\rho_j^n","aliases":"ρ_j^n rho","display":"\\rho_j^n","kind":"use","excerpt":"[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18318974b1f97ee7","tex":"\\rho_j^{n+1}","aliases":"ρ_j^n+1 rho","display":"\\rho_j^{n+1}","kind":"use","excerpt":"[Exact one-step continuity] Set ρ_j^n=normψ_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18866b84545a9559","tex":"\\Sigma'","aliases":"Σ' Sigma","display":"\\Sigma'","kind":"use","excerpt":"[Upper half-plane sign and the exact visible norm] For z=x+iy with y>0, ImΣ(z) =-yB((x-D)^2+y^2I)^-1B^*≤0, Im S(z)≥ yI. In particular S(z) is invertible and norm S(z)^-1≤ y^-1. For real E∉σ(D), Σ'(E)=-B(E-D)^-2B^*. If (u,v) is a normalized full eigenvector at E, then 1=⟨ u,(I-Σ'(E))u⟩. For real λ, Im(z-λ)^-1=-y/((x-λ)^2+y^2). Applying this identity","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b8a1bf1bab77ec11","tex":"\\sigma'","aliases":"σ' sigma","display":"\\sigma'","kind":"use","excerpt":"For a nondegenerate visible spin direction with normalized low-space direction u_*, the visible weight of its normalized full eigenstate is [1-σ'(E)]^-1. If k=m_0=0, the spin eigenspace is degenerate, but the same normalization relation holds for each chosen unit vector in that eigenspace.","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6512b523961a05f5","tex":"\\sigma^{-1}","aliases":"σ^-1 sigma","display":"\\sigma^{-1}","kind":"use","excerpt":"For [source reference], choose future radial null vectors ℓ_±=σ^-1∂_t+(-v±1)∂_r. Because the sphere area is 4π r^2, its expansion along ℓ is 2ℓ(r)/r. Consequently θ_+=frac2r(1-v), θ_-=-frac2r(1+v). On the positive branch, v=1 is a marginally outer trapped sphere, and v>1 gives both expansions negative. Equivalently, the compactness 2m/r equals or e","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c797b4151d255212","tex":"\\sigma^1","aliases":"σ^1 sigma","display":"\\sigma^1","kind":"use","excerpt":"Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0819a6083d4c9117","tex":"\\sigma^3","aliases":"σ^3 sigma","display":"\\sigma^3","kind":"use","excerpt":"Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basi","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-86570cf2b6053ba1","tex":"\\sigma^a","aliases":"σ^a sigma","display":"\\sigma^a","kind":"use","excerpt":"Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basi","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4cb413620fbda2f1","tex":"\\sigma^b","aliases":"σ^b sigma","display":"\\sigma^b","kind":"use","excerpt":"Let σ^1,σ^2,σ^3 be the Pauli matrices and define the real components q_x^a=ψ_x^*σ^aψ_x. Expand the conjugated Pauli basis as g_x^*σ^a g_x=∑_b R_x^a_bσ^b, q_x^g,a=∑_bR_x^a_bq_x^b. The coefficients are real because conjugation preserves traceless Hermitian matrices. The trace inner product tfrac12Tr(AB) is preserved by conjugation, and the Pauli basi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-59681cd776049191","tex":"\\Sigma_0","aliases":"Σ_0 Sigma","display":"\\Sigma_0","kind":"use","excerpt":"g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d0331456e52105eb","tex":"\\sigma_0","aliases":"σ_0 sigma","display":"\\sigma_0","kind":"use","excerpt":"Fermionic primary constraints and half densities. Impose the inherited algebraic spinor relation barψ=ψ^†γ^0, where (γ^0)^2=-id. Its phase convention must be included in the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-429412ff1df255be","tex":"\\sigma_A","aliases":"σ_A sigma","display":"\\sigma_A","kind":"use","excerpt":"For a product state ρ_AB=σ_A⊗τ_B, the trace factorizes and p_aμ=p_a^Ap_μ^B. General states need not have that property. The normalized vector Ψ=(e_0⊗ e_0+e_1⊗ e_1)/sqrt2 has joint probabilities (p_aμ)=1/2 · 0","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-782577c277fcf715","tex":"\\Sigma_D","aliases":"Σ_D Sigma","display":"\\Sigma_D","kind":"use","excerpt":"As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a702a847774b0ed7","tex":"\\Sigma_h","aliases":"Σ_h Sigma","display":"\\Sigma_h","kind":"use","excerpt":"g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e05b751fe47a304b","tex":"\\Sigma_I","aliases":"Σ_I Sigma","display":"\\Sigma_I","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-746ee34210cc6942","tex":"\\sigma_I","aliases":"σ_I sigma","display":"\\sigma_I","kind":"use","excerpt":"IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L, Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0ed845e79c8d8e41","tex":"\\sigma_i","aliases":"σ_i sigma","display":"\\sigma_i","kind":"use","excerpt":"Exact straight refinement and curvature are independent. The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phas","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f9ab434fbb0d31ef","tex":"\\sigma_i^2","aliases":"σ_i^2 sigma","display":"\\sigma_i^2","kind":"use","excerpt":"Exact straight refinement and curvature are independent. The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phas","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8de4fa9e7f9c5804","tex":"\\sigma_j","aliases":"σ_j sigma","display":"\\sigma_j","kind":"use","excerpt":"α_j= 0 · σ_j σ_j · 0, β=I_2 · 0 0 · -I_2,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d8dd014618e9e438","tex":"\\sigma_j^*","aliases":"σ_j^* sigma","display":"\\sigma_j^*","kind":"use","excerpt":"α_j^*= 0 · σ_j^* σ_j^* · 0 =α_j, α_jα_k= σ_jσ_k · 0 0 · σ_jσ_k.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d11986cbb3c06acc","tex":"\\sigma_K","aliases":"σ_K sigma","display":"\\sigma_K","kind":"use","excerpt":"C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gauss forms G_IJ · =D_Ω B_IJ -tfrac i8σ_Kbarψ{γ^K,γ_IJ}ψ, G_ U · = d_ΣΠ-qσ_Ibarψγ^Iψ. Consequently the multiplier equations are C_I= G_IJ= G_ U=0. These are exact formulas for the split action. The au","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fa431dd60a582371","tex":"\\sigma_k","aliases":"σ_k sigma","display":"\\sigma_k","kind":"use","excerpt":"α_j^*= 0 · σ_j^* σ_j^* · 0 =α_j, α_jα_k= σ_jσ_k · 0 0 · σ_jσ_k.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-65acde73901caecb","tex":"\\Sigma_P","aliases":"Σ_P Sigma","display":"\\Sigma_P","kind":"use","excerpt":"For z outside the spectrum of H_Q=(QH_fQ)|_Ran Q, the coarse Schur self-energy is Σ_P(z)=PH_fQ(z-H_Q)^-1QH_fP. Compatibility makes it identically zero by setting B=0. This zero is an algebraic statement, not a uniform bound on a resolvent near its spectrum. A nonzero coupling requires the energy-dependent Schur term when the detail sector is elimin","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-795b54231d0400ed","tex":"\\Sigma_{p,w}","aliases":"Σ_p,w Sigma","display":"\\Sigma_{p,w}","kind":"use","excerpt":"Eliminating the hidden radial level gives Σ_p,w(z)=frac|G_p,w|^2z-ϵ, z≠ϵ. This is the exact finite Schur term. The full visible Green function, when its denominator is nonzero, is [z-E_c-Σ_p,w(z)]^-1. It is distinct from the self-energy itself, and the locations of its poles depend on both sectors.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a63fa784be234859","tex":"\\Sigma_r","aliases":"Σ_r Sigma","display":"\\Sigma_r","kind":"use","excerpt":"g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b5c9bb4ed7f4bb6f","tex":"\\Sigma_{r+1}","aliases":"Σ_r+1 Sigma","display":"\\Sigma_{r+1}","kind":"use","excerpt":"g_r(z) · =frac1z-ϵ_r-γ_r^2g_r+1(z), r<h. Defining Σ_h=0 and Σ_r=γ_r^2/(z-ϵ_r+1-Σ_r+1) gives ⟨root,(z-H)^-1root⟩ =g_0(z)=frac1z-ϵ_0-Σ_0(z). The equality to the tree resolvent follows from the exact reducing decomposition. Equations [source reference] are rational identities. Nested denominators can become singular at a real energy where a canceled f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7499d2052ab744b7","tex":"\\sigma_x","aliases":"σ_x sigma","display":"\\sigma_x","kind":"use","excerpt":"As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ae6135878257b347","tex":"\\sigma_y","aliases":"σ_y sigma","display":"\\sigma_y","kind":"use","excerpt":"σ_x=0 · 1 1 · 0, σ_y=0 · -i i · 0, σ_z=1 · 0 0 · -1.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3fa147d5109a56b1","tex":"\\sigma_z","aliases":"σ_z sigma","display":"\\sigma_z","kind":"use","excerpt":"As a conditional finite model, take A=H_D(k)=kσ_z+m_0σ_x with k,m_0∈R and D=J_1:h⊗ I_2. Couple the visible spinor identically to the first site of the tail by B=γ_0(⟨1|⊗ I_2). Its self-energy is scalar in spin space: Σ_D(z)=σ(z)I_2, σ(z)=γ_0^2⟨1,(z-J_1:h)^-11⟩. The determinant formula from Theorem~[source reference] is det(z-H_rm full) =det(z-D)((z","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ff0c3a9746b729b1","tex":"\\tau^2","aliases":"τ^2 tau","display":"\\tau^2","kind":"use","excerpt":"a compact regular neighborhood where d_h≥ d_0>0. Assume the mixed time and state derivatives of widehat f_h through order three and the state derivatives of C_h through order four are bounded there. Apply the endpoint correction of Proposition~[source reference] after explicit midpoint. Then its local error is O(Δτ^3) and its global error on a fixe","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7224d7199b93ec25","tex":"\\tau^3","aliases":"τ^3 tau","display":"\\tau^3","kind":"use","excerpt":"of dot z=widehat f_h(τ,z) and its numerical steps remain in a compact regular neighborhood where d_h≥ d_0>0. Assume the mixed time and state derivatives of widehat f_h through order three and the state derivatives of C_h through order four are bounded there. Apply the endpoint correction of Proposition~[source reference] after explicit midpoint. Th","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-35af929f24440334","tex":"\\tau_0","aliases":"τ_0 tau","display":"\\tau_0","kind":"use","excerpt":"If a periodic λ exists, integrating λ_θ=M_G forces the zero-period condition. Conversely, fix τ_0 and define on the lifted strip λ(τ,θ)=λ_0 +∫_τ_0^τH_G(s,0) ds +∫_0^θM_G(τ,η) dη. Its θ derivative is M_G. Its τ derivative is H_G by the integrability identity and the fundamental theorem of calculus. The zero-period condition makes it periodic. The di","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db508c3ce80a5e43","tex":"\\tau_B","aliases":"τ_B tau","display":"\\tau_B","kind":"use","excerpt":"For a product state ρ_AB=σ_A⊗τ_B, the trace factorizes and p_aμ=p_a^Ap_μ^B. General states need not have that property. The normalized vector Ψ=(e_0⊗ e_0+e_1⊗ e_1)/sqrt2 has joint probabilities (p_aμ)=1/2 · 0","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2ed1bcaa9d2d3dfe","tex":"\\tau_i","aliases":"τ_i tau","display":"\\tau_i","kind":"use","excerpt":"Exact straight refinement and curvature are independent. The one-edge identity can be assembled into a three-axis refinement on the fine site set Site×{0,1}^3. The half translation σ_i toggles the corresponding child bit; when that bit carries from one to zero, it applies the coarse translation τ_i to the site. Thus σ_i^2 is the lift of τ_i. A phas","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-93f600ceeb9d3b50","tex":"\\tau_N","aliases":"τ_N tau","display":"\\tau_N","kind":"use","excerpt":"e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-80ad2e368455b323","tex":"\\upsilon","aliases":"υ upsilon","display":"\\upsilon","kind":"use","excerpt":"Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"n⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). For each basis vector, U_CRT|u⟩ is the basis vector with its unique tuple of coordinates. Distinct units have distinct tuples, so inner products are preserved. The dimensions also agree: φ(M)=2^h_2+1∏_p∈ P_ o(p-1)p^h_p. Multiplication of units becomes addition in each cyclic coordinate.","count":28},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c89e189652ea7cb2","tex":"\\Phi'","aliases":"Φ' Phi","display":"\\Phi'","kind":"use","excerpt":"Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2c08a3f3921a7de9","tex":"\\varphi'","aliases":"φ' varphi","display":"\\varphi'","kind":"use","excerpt":"To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source refe","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c2baa4e22eef86e2","tex":"\\varphi'^2","aliases":"φ'^2 varphi","display":"\\varphi'^2","kind":"use","excerpt":"To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source refe","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-aad33d2dbafe3856","tex":"\\Phi^2","aliases":"Φ^2 Phi","display":"\\Phi^2","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-607d2c331667a908","tex":"\\varphi^{23}","aliases":"φ^23 varphi","display":"\\varphi^{23}","kind":"use","excerpt":"ge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F needs this sign translation. Substitution of the wedge equation into","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2e9bba4479762fb1","tex":"\\varphi^{IJ}","aliases":"φ^IJ varphi","display":"\\varphi^{IJ}","kind":"use","excerpt":"The parent action is S_0=S_ G+S_ M^(1)+S_ D, with S_ G · =frac14κ∫ϵ_IJKLe^Iwedge e^Jwedge R^KL -fracΛκ∫vol_e, S_ M^(1) · =∫(tfrac12φ_IJFwedge e^Iwedge e^J -tfrac14φ_IJφ^IJvol_e), S_ D · =frac i2∫Σ_Iwedge (barψγ^I Dψ -( Dbarψ)γ^Iψ) -m∫vol_ebarψψ. This formulation uses no inverse tetrad in its defining action. Its largest monomials have six field fac","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-defceda344190192","tex":"\\varphi^{KL}","aliases":"φ^KL varphi","display":"\\varphi^{KL}","kind":"use","excerpt":"The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gau","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-28b947267f902868","tex":"\\phi_0","aliases":"ϕ_0 phi","display":"\\phi_0","kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0dd8e14cc9a785dc","tex":"\\Phi_1","aliases":"Φ_1 Phi","display":"\\Phi_1","kind":"use","excerpt":"Endpoint Dirac projection. Consider a finite-dimensional real canonical chart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. F","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7da0ece20570a181","tex":"\\phi_1","aliases":"ϕ_1 phi","display":"\\phi_1","kind":"use","excerpt":"K_p is an isometry with proper range, rather than a unitary: for example ϕ_1 is orthogonal to its range. Its adjoint discards nondivisible modes. Invariance of ν is a statement about the normalized free spectral distribution under R_p. It is not a proof that a sequence of growing Hilbert spaces has converged to an interacting Hamiltonian or that it","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d33036383a860124","tex":"\\Phi_2","aliases":"Φ_2 Phi","display":"\\Phi_2","kind":"use","excerpt":"A worked particle constraint. In R^6 with z=(q,p) and canonical J, set Φ_1=q· q-1, Φ_2=q· p. At one point the ordinary Dirac matrix is Δ(z)=0 · 2 q ^2","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-570f5323737bcd9d","tex":"\\Phi_{2m}","aliases":"Φ_2m Phi","display":"\\Phi_{2m}","kind":"use","excerpt":"Endpoint Dirac projection. Consider a finite-dimensional real canonical chart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. F","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8cc9e6674048bfda","tex":"\\Phi_e'","aliases":"Φ_e' Phi","display":"\\Phi_e'","kind":"use","excerpt":"ambda^2+Q^2)R^3 -fracP_Λ P_RR^2+(N^rP_R)' -fracR''Λ+fracΛ'R'Λ^2. Substituting the data and differentiating p(F)^2=2MF-Q^2 reduces their right sides to fracMF_tp(F), M[fracF_trp(F) -fracMF_rF_tp(F)^3], which are the direct time derivatives of the two momenta. Finally, Φ_e'=QF_r/F^2 cancels NΛ Q/R^2 in dot A_r=NΛ Q/R^2-Φ_e'.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-43812a1e322e8ca5","tex":"\\varphi_{IJ}","aliases":"φ_IJ varphi","display":"\\varphi_{IJ}","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d916d96a3f8d422a","tex":"\\varphi_j","aliases":"φ_j varphi","display":"\\varphi_j","kind":"use","excerpt":"N counts lattice sites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field ","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9df5dbbaa7b552c8","tex":"\\varphi_{j+1}","aliases":"φ_j+1 varphi","display":"\\varphi_{j+1}","kind":"use","excerpt":"ites. In the finite formulas below only the coefficient names w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-37d36f4d52c9812c","tex":"\\varphi_j^2","aliases":"φ_j^2 varphi","display":"\\varphi_j^2","kind":"use","excerpt":"w, c, and v are needed: w_j=fraca_jN_j>0, c_j+1/2=fracN_j+1/2a_j+1/2>0, v_j=N_ja_jm^2≥0. Thus N/a is the coordinate characteristic speed of the continuum massless equation. Assume these finite coefficient arrays are continuously differentiable in time. For a real field φ_j(t) set Δ_j=φ_j+1-φ_j and define L_h=frac h2∑_jw_jdotφ_j^2 -frac12h∑_jc_j+1/2","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2dc37ac785ef60ce","tex":"\\Phi_j^T","aliases":"Φ_j^T Phi","display":"\\Phi_j^T","kind":"use","excerpt":"ous function F, define the segment discrete gradient overline∇ F(z_0,z_1)=∫_0^1∇ F((1-s)z_0+sz_1) ds. The ordinary fundamental theorem of calculus gives F(z_1)-F(z_0)=overline∇ F(z_0,z_1)^T(z_1-z_0). Let G have rows overline∇Φ_j^T. The endpoint constraint matrix is Δ_Δ=GJG^T. Require this matrix to be invertible and define J_ D^Δ =J-JG^TΔ_Δ^-1GJ. D","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-694cee5f94c55798","tex":"\\varphi_{KL}","aliases":"φ_KL varphi","display":"\\varphi_{KL}","kind":"use","excerpt":"The temporal coframe multiplies the Einstein--matter three-forms C_I= C_I^ G+ C_I^ M+ C_I^ D, where C_I^ G · =tfrac12κϵ_IJKLE^Jwedge R^KL -tfracΛκσ_I, C_I^ M · =φ_IJ Fwedge E^J-tfrac14φ_KLφ^KLσ_I, C_I^ D · =tfrac i4ϵ_JIKLE^Kwedge E^Lwedge (barψγ^J D_Σψ -( D_Σbarψ)γ^Jψ) -mσ_Ibarψψ. Spatial integration by parts identifies the Lorentz and electric Gau","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-decd81e7725d5b31","tex":"\\phi_m","aliases":"ϕ_m phi","display":"\\phi_m","kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0aaab945af7d0743","tex":"\\phi_{m/p}","aliases":"ϕ_m/p phi","display":"\\phi_{m/p}","kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-72ec78a83e15672c","tex":"\\phi_n","aliases":"ϕ_n phi","display":"\\phi_n","kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-abd2409cfb8360f5","tex":"\\phi_{pn}","aliases":"ϕ_pn phi","display":"\\phi_{pn}","kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m, 0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e5500c68c310f9c1","tex":"\\Phi_{\\rm emit}","aliases":"Φ_rm emit Phi rm","display":"\\Phi_{\\rm emit}","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-531add35ef892f7f","tex":"\\Phi_{\\rm obs}","aliases":"Φ_rm obs Phi rm","display":"\\Phi_{\\rm obs}","kind":"use","excerpt":"In the stipulated first-order isotropic weak-field metric, ds^2=-(1+frac2Φc_0^2)c_0^2dt^2 +(1-frac2Φc_0^2)d x^2, |fracΦc_0^2|ll1, a static clock has lapse N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0c73fde37e3f6f1b","tex":"\\phi_{\\rm phys}","aliases":"ϕ_rm phys phi rm","display":"\\phi_{\\rm phys}","kind":"use","excerpt":"To test dynamical concentration of matter, replace electrovac by a real massless scalar. Use G=c=1 and the rescaled field φ=sqrt4π ϕ_rm phys. The scalar canonical pair is (φ,P_φ), and the matter additions to the Hamiltonian and momentum constraints are H_rm matt=fracP_φ^22Λ R^2 +fracR^2φ'^22Λ, D_rm matt=P_φφ'. The gravitational part is [source refe","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7016cbae921a9f68","tex":"\\phi_t","aliases":"ϕ_t phi","display":"\\phi_t","kind":"use","excerpt":"Consider a twice continuously differentiable real function ϕ on an oriented (t,x) rectangle, and the coframe e^0=e^ϕ dt, e^1=e^ϕ dx, ds^2=e^2ϕ(-dt^2+dx^2). Write K=0 · 1 1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. The","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f5c9f73ef8b8d6e6","tex":"\\phi_{tt}","aliases":"ϕ_tt phi","display":"\\phi_{tt}","kind":"use","excerpt":"1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. Therefore Ω=dω=(ϕ_tt-ϕ_xx)K dtwedge dx, R=2e^-2ϕ(ϕ_tt-ϕ_xx). This fixes the scalar-curvature sign convention used in this monograph.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-121c0f11eac3d462","tex":"\\phi_x","aliases":"ϕ_x phi","display":"\\phi_x","kind":"use","excerpt":"Consider a twice continuously differentiable real function ϕ on an oriented (t,x) rectangle, and the coframe e^0=e^ϕ dt, e^1=e^ϕ dx, ds^2=e^2ϕ(-dt^2+dx^2). Write K=0 · 1 1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. The","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c4279ca52108141d","tex":"\\varphi_x","aliases":"φ_x varphi","display":"\\varphi_x","kind":"use","excerpt":"For a smooth field sampled at sites and a smooth coefficient sampled at edge midpoints, the spatial expression is second-order consistent with ∂_x(c∂_xφ). Indeed, expanding at x_j gives c(x_j+h/2)[φ(x_j+h)-φ(x_j)] -c(x_j-h/2)[φ(x_j)-φ(x_j-h)] =h^2(cφ_x)_x(x_j)+O(h^4). Uniform bounds on the requisite fourth derivatives make the remainder uniform on ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-418fd32d4d04d557","tex":"\\phi_x^*","aliases":"ϕ_x^* phi","display":"\\phi_x^*","kind":"use","excerpt":"The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm ter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9adf0621d8b11bc","tex":"\\phi_{xx}","aliases":"ϕ_xx phi","display":"\\phi_{xx}","kind":"use","excerpt":"1 · 0, ω=(ϕ_xdt+ϕ_tdx)K. The coframe satisfies the torsion-free Cartan equations with this connection. Since every connection value is a multiple of the same matrix, ωwedgeω=0. Therefore Ω=dω=(ϕ_tt-ϕ_xx)K dtwedge dx, R=2e^-2ϕ(ϕ_tt-ϕ_xx). This fixes the scalar-curvature sign convention used in this monograph.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"nt. The larger graph adds p children, changing the degree to d+p. For an old outgoing arc basis vector e_j, the new Grover coin gives amplitude 2/(d+p) on each of the p new outgoing child arcs. An internal unitary preserves its internal norm. The subsequent reversal shift sends these amplitudes to new arcs at the children, outside the old arc range","count":26},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-11fdaa25e327864a","tex":"\\chi^2","aliases":"χ^2 chi","display":"\\chi^2","kind":"use","excerpt":"Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-973edae78b5c5539","tex":"\\chi^{IJ}","aliases":"χ^IJ chi","display":"\\chi^{IJ}","kind":"use","excerpt":"ultiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ, sbarψ · = L_cbarψ -tfrac14χ^IJbarψγ_IJ-iqυbarψ. The ordering in the sec","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0bbd4413777b414e","tex":"\\chi_A","aliases":"χ_A chi","display":"\\chi_A","kind":"use","excerpt":"the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Different","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c2171a0d6968e6fe","tex":"\\chi_j","aliases":"χ_j chi","display":"\\chi_j","kind":"use","excerpt":"1. For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps therefore uses the unimodular transfer matrix A(x)^p. Cayley--Hamilton gives the coarse recurrence χ_j+1-R_p(x)χ_j","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-04fa9c1343879b99","tex":"\\chi_{j+1}","aliases":"χ_j+1 chi","display":"\\chi_{j+1}","kind":"use","excerpt":"1. For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps therefore uses the unimodular transfer matrix A(x)^p. Cayley--Hamilton gives the coarse recurrence χ_j+1-R_p(x)χ_j","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-660c5fbfabd67d33","tex":"\\chi_{j-1}","aliases":"χ_j-1 chi","display":"\\chi_{j-1}","kind":"use","excerpt":"1. For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps therefore uses the unimodular transfer matrix A(x)^p. Cayley--Hamilton gives the coarse recurrence χ_j+1-R_p(x)χ_j","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Z_sign have eigenvalues +1,-1 on a dyadic sign qubit, and let B=B^* act on the remaining factors. Assume there are orthonormal states |b_0⟩,|b_1⟩ with B|b_0⟩=0 and B|b_1⟩=|b_1⟩. One concrete choice is an odd-prime torsion involution times a fixed odd-prime detail projector, with the torsion state in its +1 eigenspace. For real g≠0, use H_int=gZ_sig","count":18},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a1d181fcbd5bda98","tex":"\\psi'","aliases":"ψ' psi","display":"\\psi'","kind":"use","excerpt":"For fixed x, put f_x(s)=exp(-i∫_x^x+sA(r) dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-86694f2d67d3ff7b","tex":"\\Psi^\\dagger","aliases":"Ψ^† Psi dagger","display":"\\Psi^\\dagger","kind":"use","excerpt":"tion Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left graded canonical convention, the primary matrix is {χ_A(x),barχ_B(y)}_ P =-ie(x)δ_ABδ^3(x-y). It is invertible exactly where e≠0. Set Ψ=e^1/2ψ and Ψ^†=e^1/2ψ^†. Differentiation gives Ψ^†δΨ · =eψ^†δψ+tfrac12ψ^†ψ δ e,","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-674719b5c71f505f","tex":"\\psi^\\dagger","aliases":"ψ^† psi dagger","display":"\\psi^\\dagger","kind":"use","excerpt":"Fermionic primary constraints and half densities. Impose the inherited algebraic spinor relation barψ=ψ^†γ^0, where (γ^0)^2=-id. Its phase convention must be included in the physical translation just described. In time gauge σ_0=e d^3x and σ_i=0. Equation~[source reference] becomes Θ_ D=frac i2∫_Σ e (ψ^†δψ-δψ^†ψ) d^3x. In the inherited right--left ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c1b88ae641d78c35","tex":"\\psi_0","aliases":"ψ_0 psi","display":"\\psi_0","kind":"use","excerpt":"h has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω_p^(q'-q)d=δ_qq'. The number of these orthonormal functions is exactly d_m, so they span that level. Together with ψ_0=1, they form ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0e5ee24e05fc4a76","tex":"\\psi_0^{(h)}","aliases":"ψ_0^(h) psi","display":"\\psi_0^{(h)}","kind":"use","excerpt":"0, · otherwise, where d_r(x) is the digit multiplying p^r-1. Include the constant scaling vector ψ_0^(h)(x)=p^-h/2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7279e6042a44c457","tex":"\\Psi_A","aliases":"Ψ_A Psi","display":"\\Psi_A","kind":"use","excerpt":"δΨ^†Ψ · =eδψ^†ψ+tfrac12ψ^†ψ δ e. The determinant terms cancel, so Θ_ D=frac i2∫_Σ (Ψ^†δΨ-δΨ^†Ψ) d^3x, {Ψ_A(x),Ψ_B^†(y)}_ D =-iδ_ABδ^3(x-y). The half density removes the field-dependent coefficient in the fermionic primary bracket without shifting the gravitational momentum. It requires the chosen real positive square root of e, so it is a local coo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1d16d5974f691ea4","tex":"\\psi_A","aliases":"ψ_A psi","display":"\\psi_A","kind":"use","excerpt":"[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c2b61b1b4b58bf76","tex":"\\psi_B","aliases":"ψ_B psi","display":"\\psi_B","kind":"use","excerpt":"[Two packets inside four coordinates] Let HH=C^4 with its usual inner product, and set ψ=frac1sqrt7(1,i,2,-i)^ T, P_A=diag(1,1,0,0), P_B=diag(0,0,1,1). Then ψ_A=(1,i,0,0)^ T/sqrt7 and ψ_B=(0,0,2,-i)^ T/sqrt7. The weights are p_A=2/7 and p_B=5/7. Refining into the four coordinate projectors gives weights 1/7,1/7,4/7,1/7. The two descriptions resolve","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2d48c69bc1aecc34","tex":"\\psi_b","aliases":"ψ_b psi","display":"\\psi_b","kind":"use","excerpt":"Define the receiving-first current by J_b← a(t) =2Reinnerψ_b(t)K(t)ψ_a(t) =2Reinnerψ_b(t)K_ba(t)ψ_a(t). The second equality follows because P_bψ_b=ψ_b. A positive value means that the contribution assigned to the pair increases the weight in b. The current is a real number, and its sign convention is fixed by the derivative calculation below.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-098b2a09651c388e","tex":"\\Psi_B^\\dagger","aliases":"Ψ_B^† Psi dagger","display":"\\Psi_B^\\dagger","kind":"use","excerpt":"δΨ^†Ψ · =eδψ^†ψ+tfrac12ψ^†ψ δ e. The determinant terms cancel, so Θ_ D=frac i2∫_Σ (Ψ^†δΨ-δΨ^†Ψ) d^3x, {Ψ_A(x),Ψ_B^†(y)}_ D =-iδ_ABδ^3(x-y). The half density removes the field-dependent coefficient in the fermionic primary bracket without shifting the gravitational momentum. It requires the chosen real positive square root of e, so it is a local coo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f2e39e3d75403e53","tex":"\\psi_j","aliases":"ψ_j psi","display":"\\psi_j","kind":"use","excerpt":"For a time-dependent scalar gauge g_j^n, use (U_j^n)^g=g_j^nU_j^n(g_j+1^n)^-1, (T_j^n)^g=g_j^n+1T_j^n(g_j^n)^-1. Spatial transport first brings the gauge factor to site j on the old slice. Temporal transport then brings it to the new slice. Thus the step operator W_n satisfies W_n[U^g,T^g]G^n=G^n+1 W_n[U,T], (G^nψ)_j=g_j^nψ_j. Every factor in the s","count":10},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-58aa479f4a87b2a0","tex":"\\psi_{j+1}","aliases":"ψ_j+1 psi","display":"\\psi_{j+1}","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations spec","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d242c66daa18da63","tex":"\\psi_{j-1}","aliases":"ψ_j-1 psi","display":"\\psi_{j-1}","kind":"use","excerpt":"sign conventions: S_h=∫[ fracih2∑_j(ψ_j^*dotψ_j-dotψ_j^*ψ_j) +h∑_jE_jdot A_j-H_h]dt. Variations with fixed endpoint data yield ihdotψ_j=∂ H_h/∂overlineψ_j, hdot A_j=∂ H_h/∂ E_j, and hdot E_j=-∂ H_h/∂ A_j. Written out, these are dotψ_j · =-imβψ_j -frac12hα U_jψ_j+1 +frac12hU_j-1^*αψ_j-1, dot A_j · =E_j, · dot E_j · =J_j, · J_j · =Re(ψ_j^*α U_jψ_j+1)","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c796ff8e7977df39","tex":"\\psi_{j-1}^*","aliases":"ψ_j-1^* psi","display":"\\psi_{j-1}^*","kind":"use","excerpt":"Differentiate ρ_j and use dotρ_j=2Re(ψ_j^*dotψ_j). The mass term contributes zero, since ψ_j^*βψ_j is real. The forward hopping term contributes -J_j/h. The backward scalar is the conjugate of ψ_j-1^*α U_j-1ψ_j, using α=α^* and the scalar nature of U. Its contribution is J_j-1/h. This proves continuity. Periodic summation gives dotbarρ=0. Finally, ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6507e649e586c341","tex":"\\psi_j^*","aliases":"ψ_j^* psi","display":"\\psi_j^*","kind":"use","excerpt":"Fix h>0, m∈R, and Hermitian spin matrices α=α^* and β=β^* on C^s. Let ψ_j∈C^s and A_j,E_j∈R, and set U_j=e^-ihA_j. The source absorbs the charge into A. Its real Hamiltonian is H_h · =H_ m+H_E, H_ m · =hm∑_jψ_j^*βψ_j +∑_jIm(ψ_j^*α U_jψ_j+1), · H_E · =frac h2∑_jE_j^2. Hermiticity is enough for the identities below. Additional Clifford relations spec","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2722d43d565a24a4","tex":"\\psi_j^n","aliases":"ψ_j^n psi","display":"\\psi_j^n","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1c9c1e667b6f998c","tex":"\\psi_j^{n+1}","aliases":"ψ_j^n+1 psi","display":"\\psi_j^{n+1}","kind":"use","excerpt":"Covariance alone does not specify a current. To obtain one, fix the walk that moves the amplitudes. Let N≥1 be an integer, let j∈Z/NZ, let ψ_j^n∈C^2, and let U_j^n∈ U(1) be backward spatial links. Let T_j^n∈ U(1) transport from (j,n) to (j,n+1). At each step choose two on-site unitary coins C_1,j^n,C_2,j^n∈ U(2) and define η_j · =C_1,j^nψ_j^n, ξ_j^","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a411276b17f3a178","tex":"\\psi_{m,a,q}","aliases":"ψ_m,a,q psi","display":"\\psi_{m,a,q}","kind":"use","excerpt":"For an explicit basis, put ω_p=exp(2π i/p) and define ψ_m,a,q=p^(m-1)/2∑_d=0^p-1 ω_p^qd1_B_m,a+d p^m-1, m≥1, 0≤ a<p^m-1, 1≤ q<p. Each has zero average over its parent, since ∑_dω_p^qd=0, and lies in Ran F_m. Its norm squared is p^m-1p p^-m=1. Distinct parents have disjoint support, while two characters at the same parent have inner product p^-1∑_dω","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d9e406735fd906a4","tex":"\\psi_n","aliases":"ψ_n psi","display":"\\psi_n","kind":"use","excerpt":"The stationary equation on an interior site is ψ_n+1=xψ_n-ψ_n-1. The eliminated block of z-H in one cell is κ K_p-1(x), where K_n(x) has diagonal x and neighboring off-diagonal entries -1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e0b43fdc9ef3ba09","tex":"\\psi_{n+1}","aliases":"ψ_n+1 psi","display":"\\psi_{n+1}","kind":"use","excerpt":"The stationary equation on an interior site is ψ_n+1=xψ_n-ψ_n-1. The eliminated block of z-H in one cell is κ K_p-1(x), where K_n(x) has diagonal x and neighboring off-diagonal entries -1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-abfa54f18e6ebc6d","tex":"\\psi_{n-1}","aliases":"ψ_n-1 psi","display":"\\psi_{n-1}","kind":"use","excerpt":"The stationary equation on an interior site is ψ_n+1=xψ_n-ψ_n-1. The eliminated block of z-H in one cell is κ K_p-1(x), where K_n(x) has diagonal x and neighboring off-diagonal entries -1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bd8f16f7432ef794","tex":"\\psi_{pj}","aliases":"ψ_pj psi","display":"\\psi_{pj}","kind":"use","excerpt":"equation For n=1 this uses U_-1=0. Multiplication by A(x) advances the first column through the Chebyshev recurrence and places the negative previous first column in the second. The trace is R_n(x). Sampling the fine recurrence every p steps therefore uses the unimodular transfer matrix A(x)^p. Cayley--Hamilton gives the coarse recurrence χ_j+1-R_p","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1d54446a517f972c","tex":"\\psi_r","aliases":"ψ_r psi","display":"\\psi_r","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-08cf7f88c5994a7c","tex":"\\psi_s","aliases":"ψ_s psi","display":"\\psi_s","kind":"use","excerpt":"[Probability currents between residue channels] Suppose idotψ_r=∑_sH_rsψ_s with H_rs^*=H_sr. The channel weights w_r=normψ_r^2 satisfy dot w_r=∑_s J_r← s, J_r← s =2Im⟨ψ_r,H_rsψ_s⟩, J_r← s=- J_s← r. Consequently ∑_rw_r is constant. Differentiating the norm gives dot w_r=2Re⟨ψ_r,-i∑_sH_rsψ_s⟩, which is the stated sum of imaginary parts. Self-adjoint ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bc736e8ef3361342","tex":"\\psi_x","aliases":"ψ_x psi","display":"\\psi_x","kind":"use","excerpt":"A field assigns a vector to each site, but a local change of frame can be chosen independently at every site. A difference such as ψ_x+widehatμ-ψ_x therefore needs a rule for comparing its two entries. A link supplies that rule. Its orientation is part of its definition, rather than a sign convention to be inferred after a calculation.","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8222a2a1e93111e6","tex":"\\psi_{x+\\widehat\\mu}","aliases":"ψ_x+widehatμ psi widehat mu","display":"\\psi_{x+\\widehat\\mu}","kind":"use","excerpt":"A field assigns a vector to each site, but a local change of frame can be chosen independently at every site. A difference such as ψ_x+widehatμ-ψ_x therefore needs a rule for comparing its two entries. A link supplies that rule. Its orientation is part of its definition, rather than a sign convention to be inferred after a calculation.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a5b2d2df696a3716","tex":"\\psi_{x+\\widehat\\mu}^g","aliases":"ψ_x+widehatμ^g psi widehat mu","display":"\\psi_{x+\\widehat\\mu}^g","kind":"use","excerpt":"ing on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame of its destination gives U_x,μ^g=g_xU_x,μg_x+widehatμ^-1, U_x,μ^gψ_x+widehatμ^g =g_xU_x,μψ_x+widehatμ. Product","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-247280f36d547b56","tex":"\\psi_{x-\\widehat\\mu}","aliases":"ψ_x-widehatμ psi widehat mu","display":"\\psi_{x-\\widehat\\mu}","kind":"use","excerpt":"[Covariance of linked differences] For spacing h>0, define (D^+_μ,Uψ)_x · =fracU_x,μψ_x+widehatμ-ψ_xh, (Δ_Uψ)_x · =frac1h^2∑_μ( U_x,μψ_x+widehatμ +U_x-widehatμ,μ^-1ψ_x-widehatμ-2ψ_x). Then D^+_μ,U^g(gψ)=gD^+_μ,Uψ and Δ_U^g(gψ)=gΔ_Uψ.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-243488011205ccd6","tex":"\\psi_x^*","aliases":"ψ_x^* psi","display":"\\psi_x^*","kind":"use","excerpt":"The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm ter","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1dda4f82a9e3e86f","tex":"\\psi_x^g","aliases":"ψ_x^g psi","display":"\\psi_x^g","kind":"use","excerpt":"a finite periodic lattice and let every fiber be a copy of C^r. We initially allow any unitary gauge group acting on this fiber. In the convention used for the spatial stencils below, U_x,μ:HH_x+widehatμ⟶HH_x, U_x,μ^*U_x,μ=U_x,μU_x,μ^*=I. The gauge g=(g_x)_x∈ V changes coordinates by ψ_x^g=g_xψ_x. Requiring the transported vector to have the frame ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-efdf8990a98fa39d","tex":"\\omega'","aliases":"ω' omega","display":"\\omega'","kind":"use","excerpt":"The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and pa","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50cc5e20a1c2c14c","tex":"\\Omega^{0i}","aliases":"Ω^0i Omega","display":"\\Omega^{0i}","kind":"use","excerpt":"ented time-gauge patch E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-768389d483cf5935","tex":"\\omega^I","aliases":"ω^I omega","display":"\\omega^I","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a8ecdd64f9da9bae","tex":"\\Omega^{IJ}","aliases":"Ω^IJ Omega","display":"\\Omega^{IJ}","kind":"use","excerpt":"An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-021aea4873729e82","tex":"\\omega^{IJ}","aliases":"ω^IJ omega","display":"\\omega^{IJ}","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da,","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8e4eeccd528d4197","tex":"\\Omega^{jk}","aliases":"Ω^jk Omega","display":"\\Omega^{jk}","kind":"use","excerpt":"E^0=0, E^i=e_a^i dx^a, e:=det(e_a^i)>0. The positive determinant fixes the orientation and ensures invertibility. Define widetilde E^a_i=tfrac12ϵ^abcϵ_ijke_b^je_c^k, widetilde E^a_ie_a^j=eδ_i^j. Let K^i=Ω^0i and Γ^i=-tfrac12ϵ^i_jkΩ^jk. The simplicity pullback gives B_ij=0, B_0i=tfrac14κϵ_ijkE^jwedge E^k, ∫_Σ B_IJwedgeδΩ^IJ =frac1κ∫_Σwidetilde E^a_i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d0e92a3d0b375b2d","tex":"\\omega^{KJ}","aliases":"ω^KJ omega","display":"\\omega^{KJ}","kind":"use","excerpt":"Fields, action, and polynomial structure. Fix η_IJ=diag(-1,1,1,1) and ϵ_0123=1. The independent fields are a real coframe e^I, a real antisymmetric Lorentz connection ω^IJ, an Abelian connection a, an antisymmetric auxiliary scalar φ_IJ, and Dirac fields ψ,barψ. The spinors are Grassmann odd. Define R^IJ · = dω^IJ+ω^I_Kwedgeω^KJ, · F · = da, vol_e ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0fc4543680ca1902","tex":"\\omega_+","aliases":"ω_+ omega","display":"\\omega_+","kind":"use","excerpt":"The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and pa","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0837ec06cd7132fd","tex":"\\omega_+'","aliases":"ω_+' omega","display":"\\omega_+'","kind":"use","excerpt":"The plus-sign observable is assigned the endpoint action G(t,x)=e^χ(t,x)K, U_a→ b'=G(b)U_a→ bG(a)^-1. It corresponds to the signed connection rule ω_+'=ω_++dχ K. Intermediate endpoint factors cancel, so the loop observable is conjugated at its base point. For the Cartan convention D=d+ω, a coframe transformation e↦ Ge instead gives ω'=ω-dχ K and pa","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cfeb9eb96e89714e","tex":"\\Omega_C","aliases":"Ω_C Omega","display":"\\Omega_C","kind":"use","excerpt":"If {Ω_F,Ω_F}=0, then s_F={Ω_F,·} is a differential. To transfer a nonlinear Hamiltonian gauge structure, the contraction must also lift compatibly to the completed nonlinear coalgebra and respect its cyclic or symplectic structure. On a local exact canonical chart, a resulting Hamiltonian vector field can have a charge Ω_C. Its square then has Hami","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ff45fb79f7e1cb1e","tex":"\\Omega_F","aliases":"Ω_F Omega","display":"\\Omega_F","kind":"use","excerpt":"There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total gh","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a1bb204421e927bd","tex":"\\Omega_F^{(\\ge3)}","aliases":"Ω_F^(≥3) Omega ge","display":"\\Omega_F^{(\\ge3)}","kind":"use","excerpt":"There is a separate step from this theorem to a nonlinear BFV charge. On a canonical graded phase space, introduce odd ghosts c^A of ghost number one and their odd conjugate momenta ϖ_A of ghost number minus one. A charge of ghost number one has the form Ω_F=c^AC_A-tfrac12c^Ac^BU_AB^Cϖ_C +Ω_F^(≥3). The superscript counts ghost factors, not total gh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f954fc517876491c","tex":"\\omega_h","aliases":"ω_h omega","display":"\\omega_h","kind":"use","excerpt":"The resulting frequency obeys ω_h^2=∑_ifracsin^2(k_ih)h^2. For fixed k≠0 as h→0, ω_h^2=| k|^2-frach^23∑_i k_i^4+O(h^4), ω_h=| k| -frach^26| k|∑_i k_i^4+O(h^4). These follow by expanding the sine and then the positive square root. The regime keeps k fixed. It is not uniform up to the Nyquist boundary.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f8cdd4391d2292b5","tex":"\\omega_h^2","aliases":"ω_h^2 omega","display":"\\omega_h^2","kind":"use","excerpt":"The resulting frequency obeys ω_h^2=∑_ifracsin^2(k_ih)h^2. For fixed k≠0 as h→0, ω_h^2=| k|^2-frach^23∑_i k_i^4+O(h^4), ω_h=| k| -frach^26| k|∑_i k_i^4+O(h^4). These follow by expanding the sine and then the positive square root. The regime keeps k fixed. It is not uniform up to the Nyquist boundary.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7c5e43c1625ae71d","tex":"\\Omega_{ij}","aliases":"Ω_ij Omega","display":"\\Omega_{ij}","kind":"use","excerpt":"For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argu","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-10ac20847e543481","tex":"\\Omega_{\\mathrm{int}}","aliases":"Ω_int Omega mathrm","display":"\\Omega_{\\mathrm{int}}","kind":"use","excerpt":"nal BFV sector On a regular symplectic reduction where internal gauge transformations have the stated moment maps, their algebra is Lorentz plus Abelian: {G_ L[λ],G_ L[μ]} =G_ L[[λ,μ]], {G_ L,G_ U}={G_ U,G_ U}=0. Its canonical BFV charge is explicit: Ω_int=G_ L[χ]+G_ U[υ] -tfrac12⟨π_χ,[χ,χ]⟩. The moment-map relation cancels the terms quadratic in g","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-17643fc6afa4dc8b","tex":"\\omega_L","aliases":"ω_L omega","display":"\\omega_L","kind":"use","excerpt":"Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9832b5c4fc1bb7e1","tex":"\\omega_L^{(n-j)\\ell}","aliases":"ω_L^(n-j)ℓ omega ell","display":"\\omega_L^{(n-j)\\ell}","kind":"use","excerpt":"vergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, the coefficient of a_nz^n in [source reference] is L^-1∑_ℓω_L^(n-j)ℓ, equal to one exactly when n≡ jpmod L. The idempotent and reconstruction laws follow by filtering these coefficients agai","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f917b6e617bb2331","tex":"\\omega_L^{-j\\ell}","aliases":"ω_L^-jℓ omega ell","display":"\\omega_L^{-j\\ell}","kind":"use","excerpt":"Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c44b43b4aea54580","tex":"\\omega_L^\\ell","aliases":"ω_L^ℓ omega ell","display":"\\omega_L^\\ell","kind":"use","excerpt":"Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a102bfba12f83124","tex":"\\omega_p^{(e-d)\\ell}","aliases":"ω_p^(e-d)ℓ omega ell","display":"\\omega_p^{(e-d)\\ell}","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c3a30be014515298","tex":"\\omega_p^{(q'-q)d}","aliases":"ω_p^(q'-q)d omega","display":"\\omega_p^{(q'-q)d}","kind":"use","excerpt":"Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db0401080bf91c98","tex":"\\omega_p^{-d\\ell}","aliases":"ω_p^-dℓ omega ell","display":"\\omega_p^{-d\\ell}","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-61cdd9fbde51e8b1","tex":"\\omega_p^d","aliases":"ω_p^d omega","display":"\\omega_p^d","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-58cd7d70a3503669","tex":"\\omega_p^{qd}","aliases":"ω_p^qd omega","display":"\\omega_p^{qd}","kind":"use","excerpt":"Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-59875dc6e7f002db","tex":"\\Omega_{xy}","aliases":"Ω_xy Omega","display":"\\Omega_{xy}","kind":"use","excerpt":"For example, the two positive paths across an xy face at height z are p_xy(z)=ℓ_y(1,z)ℓ_x(0,z), p_yx(z)=ℓ_x(1,z)ℓ_y(0,z). Define Ω_xy(z)=p_xy(z)p_yx(z)^-1, and define the xz and yz face phases analogously. Both paths have the same endpoints, so their gauge factors cancel in this abelian ratio. Thus each Ω is gauge invariant, and Ω=1 precisely when ","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e2b7ed9de8f0f49b","tex":"\\Omega_{xz}","aliases":"Ω_xz Omega","display":"\\Omega_{xz}","kind":"use","excerpt":"Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY, p_YXZ · =Ω_xz(1)p_YZX, · p_","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-994125d019c8421f","tex":"\\Omega_{yz}","aliases":"Ω_yz Omega","display":"\\Omega_{yz}","kind":"use","excerpt":"Six paths and one Bianchi relation. The three coordinate directions can be traversed in six orders, denoted p_XYZ, p_XZY, p_YXZ, p_YZX, p_ZXY and p_ZYX. These names record execution order, while link multiplication still acts from right to left. Adjacent transpositions give p_XYZ · =Ω_xy(0)p_YXZ, · p_XYZ · =Ω_yz(1)p_XZY, p_YXZ · =Ω_xz(1)p_YZX, · p_","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6aa524e36cbbf167","tex":"\\omega_\\perp","aliases":"ω_⊥ omega perp","display":"\\omega_\\perp","kind":"use","excerpt":"lit On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-18f90c283b071b01","tex":"\\omega_\\perp^{IJ}","aliases":"ω_⊥^IJ omega perp","display":"\\omega_\\perp^{IJ}","kind":"use","excerpt":"An exact time--space split. On M=I×Σ with orientation dtwedgeor(Σ), write e^I=e_⊥^I dt+E^I, ω^IJ=ω_⊥^IJ dt+Ω^IJ, a=a_⊥ dt+A. Let R= d_ΣΩ+ΩwedgeΩ and F= d_Σ A. The curvature and field strength split as R= dtwedge(dotΩ-D_Ωω_⊥)+ R, F= dtwedge(dot A- d_Σ a_⊥)+ F. Define B_IJ · =tfrac14κϵ_KLIJE^Kwedge E^L,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"use","excerpt":"Finally the annihilation operator routes adjacent residues: Q_r^(L)aQ_s^(L)=0 unless r≡ s-1pmod L, Q_r^(L)a^† Q_s^(L)=0 unless r≡ s+1pmod L. On a finite number-state combination this follows from a|n⟩=sqrt n |n-1⟩ and a^†|n⟩=sqrtn+1 |n+1⟩. Each residue projector preserves Dom(a) and Dom(a^†), because it only removes terms from their weighted number","count":11},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2c05d83b370cdcc7","tex":"\\ell_x","aliases":"ℓ_x ell","display":"\\ell_x","kind":"use","excerpt":"Cubical transport and the curvature obstruction. Three one-axis subdivisions do not automatically define a path-independent value at a shared corner. The issue already appears in one 2×2×2 vertex cube. Let the vertices be (x,y,z)∈{0,1}^3, with positive links ℓ_x(y,z), ℓ_y(x,z) and ℓ_z(x,y). There are twelve links and six faces. Under a vertex gauge","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a8e35b84dc016d32","tex":"\\ell_y","aliases":"ℓ_y ell","display":"\\ell_y","kind":"use","excerpt":"Cubical transport and the curvature obstruction. Three one-axis subdivisions do not automatically define a path-independent value at a shared corner. The issue already appears in one 2×2×2 vertex cube. Let the vertices be (x,y,z)∈{0,1}^3, with positive links ℓ_x(y,z), ℓ_y(x,z) and ℓ_z(x,y). There are twelve links and six faces. Under a vertex gauge","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7bbf65f26182fbdc","tex":"\\ell_z","aliases":"ℓ_z ell","display":"\\ell_z","kind":"use","excerpt":"Cubical transport and the curvature obstruction. Three one-axis subdivisions do not automatically define a path-independent value at a shared corner. The issue already appears in one 2×2×2 vertex cube. Let the vertices be (x,y,z)∈{0,1}^3, with positive links ℓ_x(y,z), ℓ_y(x,z) and ℓ_z(x,y). There are twelve links and six faces. Under a vertex gauge","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cb520b9f36cbee35","tex":"\\ell_\\pm","aliases":"ℓ_± ell pm","display":"\\ell_\\pm","kind":"use","excerpt":"For [source reference], choose future radial null vectors ℓ_±=σ^-1∂_t+(-v±1)∂_r. Because the sphere area is 4π r^2, its expansion along ℓ is 2ℓ(r)/r. Consequently θ_+=frac2r(1-v), θ_-=-frac2r(1+v). On the positive branch, v=1 is a marginally outer trapped sphere, and v>1 gives both expansions negative. Equivalently, the compactness 2m/r equals or e","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"sis map T_p,h whose channel projectors Π_r have the ranks in [source reference] and obey T_p,h^*Π_r T_p,h=D_r^(h). With our positive Fourier sign, the analysis row of character q pairs with the complex-conjugate wavelet character. This only permutes the nonzero character labels q↔ p-q and leaves the detail projectors unchanged.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"On finite Fourier polynomials, the operator agrees with the matrix differential expression -i∑_jα_j∂_j+mβ already calculated. Its extension is specified by Fourier transport and the maximal domain, so no pointwise differentiability assumption is imposed on every L^2 representative. Equality with a separately constructed weak differential operator i","count":57},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d74afdd2e29665d0","tex":"\\partial_+","aliases":"∂_+ partial","display":"\\partial_+","kind":"use","excerpt":"i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-59508e0d19e81bbe","tex":"\\partial_-","aliases":"∂_- partial","display":"\\partial_-","kind":"use","excerpt":"i_j^n^2 and J_j+1/2^n=|η_j^+|^2-|η_j+1^-|^2. The linked walk obeys ρ_j^n+1-ρ_j^n=J_j-1/2^n-J_j+1/2^n. Both the density and this current are gauge invariant. Total charge is conserved. For a consecutive periodic arc A with specified left and right boundary edges, Q_A^n+1-Q_A^n=J_∂_-A^n-J_∂_+A^n.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3546b807788704d6","tex":"\\partial_a","aliases":"∂_a partial","display":"\\partial_a","kind":"use","excerpt":"A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normaliz","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5735f6c055adb0c0","tex":"\\partial_{A_j}","aliases":"∂_A_j partial","display":"\\partial_{A_j}","kind":"use","excerpt":"dot A_j · =E_j, · dot E_j · =J_j, · J_j · =Re(ψ_j^*α U_jψ_j+1). For example, if z_j=ψ_j^*α U_jψ_j+1, then ∂_A_jImz_j =Im(-ihz_j)=-hRez_j. This fixes the positive sign of dot E_j=J_j for the conventions in the action.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e22e9fc616e6e066","tex":"\\partial_b","aliases":"∂_b partial","display":"\\partial_b","kind":"use","excerpt":"nequation widetilde n_I=tfrac13!ϵ_IJKLϵ^abcE_a^JE_b^KE_c^L. It satisfies widetilde n_IE_a^I=0 and -η^IJwidetilde n_Iwidetilde n_J=det(q_ab). A parameter ζ^I=νwidetilde n^I+η^aE_a^I yields a densitized lapse ν and shift η. On a correctly reduced boundary space the expected normal bracket has structure shift S^a(ν,μ)=det(q_cd)q^ab (ν∂_bμ-μ∂_bν). The ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3e7cebb8f331465a","tex":"\\partial_I","aliases":"∂_I partial","display":"\\partial_I","kind":"use","excerpt":"The geometric fields are real, but the physical spinor convention needs a separate check. The displayed action retains the inherited Dirac-type polynomial gauge model. With the stated Clifford sign and kinetic factor i, the flat-space operators iγ^I∂_I± m have product (iγ^I∂_I+m)(iγ^J∂_J-m) =-Box-m^2, Box=η^IJ∂_I∂_J. Thus its squared field equation","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c97379bf19e9e4d6","tex":"\\partial_i","aliases":"∂_i partial","display":"\\partial_i","kind":"use","excerpt":"For an affine component, A_i(p+t e_i)=A_i(p)+t ∂_iA_i, so θ_i(p,h)=hA_i(p)+tfrac12h^2∂_iA_i. The face angle is exactly h^2F_ij, where F_ij=∂_iA_j-∂_jA_i. Hence Ω_ij=e^ih^2F_ij. Using |e^it-1|≤|t| and the selected five faces yields normD_rm total(h) ≤ h^2(|F_xy|+2|F_xz|+2|F_yz|). This exact affine calculation is stronger than a truncated Taylor argu","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-100dccaf595fc888","tex":"\\partial_J","aliases":"∂_J partial","display":"\\partial_J","kind":"use","excerpt":"The geometric fields are real, but the physical spinor convention needs a separate check. The displayed action retains the inherited Dirac-type polynomial gauge model. With the stated Clifford sign and kinetic factor i, the flat-space operators iγ^I∂_I± m have product (iγ^I∂_I+m)(iγ^J∂_J-m) =-Box-m^2, Box=η^IJ∂_I∂_J. Thus its squared field equation","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1cd2f6e5cfcfeb44","tex":"\\partial_j","aliases":"∂_j partial","display":"\\partial_j","kind":"use","excerpt":"On finite Fourier polynomials, the operator agrees with the matrix differential expression -i∑_jα_j∂_j+mβ already calculated. Its extension is specified by Fourier transport and the maximal domain, so no pointwise differentiability assumption is imposed on every L^2 representative. Equality with a separately constructed weak differential operator i","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9a3eb10d7e3c2e0","tex":"\\partial_p","aliases":"∂_p partial","display":"\\partial_p","kind":"use","excerpt":"Coordinate differentiation gives ∂_p D_h[ξ]=hξ Dq and ∂_q D_h[ξ]=-hD(ξ p). Insert these in the canonical bracket and use D^ T=-D: { D_h[ξ], D_h[η]}_h =h∑_ap_a[ξ_aD(η Dq)_a -η_aD(ξ Dq)_a]. Expanding each derivative using the definition of L_D cancels the two ξη D^2q terms. The remaining product-rule terms give the first term of [source reference], a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-168e83b92c806624","tex":"\\partial_q","aliases":"∂_q partial","display":"\\partial_q","kind":"use","excerpt":"Coordinate differentiation gives ∂_p D_h[ξ]=hξ Dq and ∂_q D_h[ξ]=-hD(ξ p). Insert these in the canonical bracket and use D^ T=-D: { D_h[ξ], D_h[η]}_h =h∑_ap_a[ξ_aD(η Dq)_a -η_aD(ξ Dq)_a]. Expanding each derivative using the definition of L_D cancels the two ξη D^2q terms. The remaining product-rule terms give the first term of [source reference], a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1a40a8de294e1b51","tex":"\\partial_R","aliases":"∂_R partial","display":"\\partial_R","kind":"use","excerpt":"normδ P_R_∞ · ≤frac Mp_*normD_4δ R_∞ +fracM^2d^*p_*^3normδ R_∞. For example, ∂_Rp=M/p and the two partial derivatives of MΛ/p have magnitudes at most M/p_* and M^2d^*/p_*^3. The admissible segment preserves the displayed linear radius and derivative bounds. The constants are independent of h.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f6d8946150f4ab94","tex":"\\partial_r","aliases":"∂_r partial","display":"\\partial_r","kind":"use","excerpt":"E, E=frac12(Π^2+Φ^2)+vΠΦ. Preservation of Λ=1 gives 0=σ(frac vr-fracP_Rr)+(σ v)' =σ rΠΦ+σ'v, hence (logσ)'=-fracrΠΦv. The scalar evolution equations are dotφ · =σ(Π+vΦ), dotΦ · =∂_r[σ(Π+vΦ)], dotΠ · =r^-2∂_r[r^2σ(Φ+vΠ)]. If Φ=φ' initially and compatible boundary data are imposed, the first two equations preserve that relation in the continuum.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dd94c17e5e087649","tex":"\\partial_t","aliases":"∂_t partial","display":"\\partial_t","kind":"use","excerpt":"s_j · =-frac h2dot w_jdotφ_j^2 +frac h2dot v_jφ_j^2 +fracdot c_j+1/24hΔ_j^2 +fracdot c_j-1/24hΔ_j-1^2. Then dot e_j+F_j+1/2-F_j-1/2=s_j. In particular, E_h=∑_je_j satisfies dot E_h=∑_js_j=-∂_tL_h, where the partial derivative holds φ and dotφ fixed.","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d58378212f1d1d15","tex":"\\partial_x","aliases":"∂_x partial","display":"\\partial_x","kind":"use","excerpt":"[Local second-order consistency] With uniform bounds on A and its derivatives through order two and on ψ and its derivatives through order three on a fixed neighborhood, (D_h^Aψ)(x)=(∂_x-iA(x))ψ(x)+O(h^2) (h→0). The remainder is uniform over points whose neighborhoods have the same derivative bounds.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-605218e5b93e5d47","tex":"\\partial_{x_1}","aliases":"∂_x_1 partial","display":"\\partial_{x_1}","kind":"use","excerpt":"A nonlinear Gaussian reduction in closed form. The small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-85be16485a8fd286","tex":"\\partial_{x_2}","aliases":"∂_x_2 partial","display":"\\partial_{x_2}","kind":"use","excerpt":"A nonlinear Gaussian reduction in closed form. The small elimination model in the transfer chapter explains why an effective action and an effective measure must be transported together. Let m,n>0 and S_F(x_1,x_2,y,z)=tfrac12(x_2-x_1y)^2 +tfrac m2(y-z)^2+tfrac n2z^2. The fine gauge vector R_F=∂_x_1+y∂_x_2 leaves S_F invariant and has zero ordinary ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-422f05688ea1c04e","tex":"\\partial_\\theta","aliases":"∂_θ partial theta","display":"\\partial_\\theta","kind":"use","excerpt":"E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b8be09e8f717ffe2","tex":"\\partial_\\nu","aliases":"∂_ν partial nu","display":"\\partial_\\nu","kind":"use","excerpt":"Let c,χ,υ be odd ghosts and let s be a left odd derivation. All ghosts multiplying component fields are placed on the left. Set sc^μ=c^ν∂_ν c^μ, sχ= L_cχ+χ^2, sυ= L_cυ. On the ordinary fields use the gauge variations with ghosts in place of even parameters, interpreted with this left-placement rule. In particular, sψ · = L_cψ+tfrac14χ^IJγ_IJψ+iqυψ,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a2f34286c4589ec1","tex":"\\partial_\\tau","aliases":"∂_τ partial tau","display":"\\partial_\\tau","kind":"use","excerpt":"E_Q · =Q_ττ-γ Q_θθ +2(P_τ Q_τ-γ P_θ Q_θ). In particular, the Q equation can first be written as ∂_τ(e^2PQ_τ)-∂_θ(e^2Pγ Q_θ)=0. The stationary equations are E_P= E_Q=0.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9fb77d52cdb67151","tex":"\\partial_\\tau^2","aliases":"∂_τ^2 partial tau","display":"\\partial_\\tau^2","kind":"use","excerpt":". Define the entire function J_0(x)=∑_m=0^∞frac(-1)^m(x^2/4)^m(m!)^2. The series and its differentiated series converge uniformly on compact sets. Coefficient comparison gives x^2J_0''+xJ_0'+x^2J_0=0. Thus P(τ,θ)=A J_0(ne^-τ)cos(nθ) is an exact polarized solution. Indeed, for x=ne^-τ, ∂_τ^2J_0(x)=x^2J_0''(x)+xJ_0'(x)=-x^2J_0(x), whereas P_θθ=-n^2P.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-569331afbd5d2de3","tex":"\\nabla","aliases":"∇ nabla","display":"\\nabla","kind":"use","excerpt":"An exact matrix momentum sector. Let the fine canonical coordinates be Hermitian matrix pairs (Z^A,P_A), with real components and the trace bracket {F,G}=∑_ATr( ∇_Z^AF∇_P_AG- ∇_P_AF∇_Z^AG). The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjug","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-876cb452f8440f63","tex":"\\nabla_h","aliases":"∇_h nabla","display":"\\nabla_h","kind":"use","excerpt":"For the bracket, use {F,G}=⟨∇_hF,∇_π G⟩_h -⟨∇_π F,∇_hG⟩_h. Two scalar constraints depend only on h and two momentum constraints depend only on π, so their brackets vanish. The gradients for the mixed bracket are (∇_hC[n])_ij=δ_iδ_jn-δ_ijδ^2n, (∇_π D[ξ])_ij=δ_iξ_j+δ_jξ_i. Their contraction vanishes after one integration by parts because the first te","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-85722266da76cd67","tex":"\\nabla_P","aliases":"∇_P nabla","display":"\\nabla_P","kind":"use","excerpt":"equation The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjugation: {Z^A,D[Λ]}=i[Λ,Z^A], {P_A,D[Λ]}=i[Λ,P_A]. Indeed, cyclic variation gives ∇_PD[Λ]=i[Λ,Z] and ∇_ZD[Λ]=i[P,Λ] for one pair.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2f12e7a302f43022","tex":"\\nabla_{P_A}","aliases":"∇_P_A nabla","display":"\\nabla_{P_A}","kind":"use","excerpt":"An exact matrix momentum sector. Let the fine canonical coordinates be Hermitian matrix pairs (Z^A,P_A), with real components and the trace bracket {F,G}=∑_ATr( ∇_Z^AF∇_P_AG- ∇_P_AF∇_Z^AG). The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjug","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a409f6974e1d4356","tex":"\\nabla_Q","aliases":"∇_Q nabla","display":"\\nabla_Q","kind":"use","excerpt":"For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-be9d7dca88bd31c4","tex":"\\nabla_U","aliases":"∇_U nabla","display":"\\nabla_U","kind":"use","excerpt":"For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b1afd481338bf2d7","tex":"\\nabla_V","aliases":"∇_V nabla","display":"\\nabla_V","kind":"use","excerpt":"For the ordinary Euclidean gradients, skew-adjointness of D_h gives ∇_PC_h · =-2hD_hU+4h e^2PV(D_hQ), ∇_QC_h · =-2hD_h(e^2PV), g_U:=∇_UC_h · =2hD_hP, · g_V:=∇_VC_h · =2h e^2PD_hQ. Put G_h=(0,0,g_U,g_V) and d_h=normg_U_2^2+normg_V_2^2. On the open region d_h>0, define Λ_h=frac∇ C_h· f_hd_h, widehat f_h=f_h-Λ_hG_h. The chain rule proves the exact tan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-346cb4e378e60b5d","tex":"\\nabla_Z","aliases":"∇_Z nabla","display":"\\nabla_Z","kind":"use","excerpt":"ned by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjugation: {Z^A,D[Λ]}=i[Λ,Z^A], {P_A,D[Λ]}=i[Λ,P_A]. Indeed, cyclic variation gives ∇_PD[Λ]=i[Λ,Z] and ∇_ZD[Λ]=i[P,Λ] for one pair.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-27d5c72219127c6a","tex":"\\nabla_{Z^A}","aliases":"∇_Z^A nabla","display":"\\nabla_{Z^A}","kind":"use","excerpt":"An exact matrix momentum sector. Let the fine canonical coordinates be Hermitian matrix pairs (Z^A,P_A), with real components and the trace bracket {F,G}=∑_ATr( ∇_Z^AF∇_P_AG- ∇_P_AF∇_Z^AG). The trace gradients are defined by real variations. For a Hermitian parameter Λ, put D[Λ]=i∑_ATr(P_A[Λ,Z^A]). This real functional generates simultaneous conjug","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a75ba8bdaddfa75a","tex":"\\nabla_\\pi","aliases":"∇_π nabla pi","display":"\\nabla_\\pi","kind":"use","excerpt":"For the bracket, use {F,G}=⟨∇_hF,∇_π G⟩_h -⟨∇_π F,∇_hG⟩_h. Two scalar constraints depend only on h and two momentum constraints depend only on π, so their brackets vanish. The gradients for the mixed bracket are (∇_hC[n])_ij=δ_iδ_jn-δ_ijδ^2n, (∇_π D[ξ])_ij=δ_iξ_j+δ_jξ_i. Their contraction vanishes after one integration by parts because the first te","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"[Functional-calculus intertwining] Under Theorem~[source reference], for every complex function f defined on the finite union of the spectra, f(H_f)V=Vf(H_c). In particular, for real t, real β, and z∉σ(H_f), e^-itH_fV · =Ve^-itH_c,nonumber","count":8},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6b6fc7fb58c7b90c","tex":"\\prod_{f\\subset\\partial c}","aliases":"∏_f⊂∂ c prod subset partial","display":"\\prod_{f\\subset\\partial c}","kind":"use","excerpt":"For scalar links, multiply the oriented plaquettes on the six faces of a cube. Every edge occurs twice, with opposite orientation. Commutativity permits its factors to meet and cancel, so ∏_f⊂∂ cW_f^or(f,c)=1. The same cancellation makes the product of fine plaquettes on a rectangular patch equal its boundary holonomy. The statement is an exact gro","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2d3db36262598bcd","tex":"\\prod_j","aliases":"∏_j prod","display":"\\prod_j","kind":"use","excerpt":"If f_j∈ T_K_j and ∑_jK_j<N/2, the product stays strictly below Nyquist in each coordinate. Sampling is injective on this support and gives δ_a∏_jS_Nf_j · =S_N∂_a(∏_jf_j), τ_N(∏_jS_Nf_j) · =(2π)^-3∫_ T^3∏_jf_j d^3x. The Leibniz rule and trace integration by parts consequently hold for the declared products. They do not hold for arbitrary sampled poi","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9335981ae95d8e40","tex":"\\prod_{k=0}^{q-1}","aliases":"∏_k=0^q-1 prod","display":"\\prod_{k=0}^{q-1}","kind":"use","excerpt":"Apply this filter to e^z and put E_L,j(z)=∑_q=0^∞fracz^Lq+j(Lq+j)!. For α∈C, write (α)_q=α(α+1)⋯(α+q-1) and (α)_0=1. Splitting the factorial into step-L factors gives frac(Lq+j)!j! · =∏_r=1^L∏_k=0^q-1(j+r+Lk)nonumber · =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every m","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-05c5b21bb8d4781f","tex":"\\prod_{k=1}^{m}","aliases":"∏_k=1^m prod","display":"\\prod_{k=1}^{m}","kind":"use","excerpt":"[Exact response of a monomial] For r∈N and v∈ E, T_m(h^rv)=c_m,rh^rv, c_m,r=∏_k=1^mfracb^k-r-1b^k-1. The empty product equals one. In particular, c_m,0=1, and c_m,r=0 for 1≤ r≤ m. At level zero the assertion is the definition. If it holds at level m, the next recursion multiplies the coefficient by fracb^m+1b^-r-1b^m+1-1. This appends the next fact","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8cbaa1215b56d5e0","tex":"\\prod_{p\\in\\mathcal P_{\\mathrm o}}","aliases":"∏_p∈ P_ o prod in mathcal mathrm","display":"\\prod_{p\\in\\mathcal P_{\\mathrm o}}","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-401f2450d734174a","tex":"\\prod_{r=1}^j","aliases":"∏_r=1^j prod","display":"\\prod_{r=1}^j","kind":"use","excerpt":"Let p be prime, let L=mp for a positive integer m, and take 0≤ j<p. For every q≥0, g_q=fracj!(Lq)!(Lq+j)! =binomLq+jj^-1∈Z_p^×. Indeed binomLq+jj=∏_r=1^j(Lq+r)/r. Each numerator and denominator is a p-adic unit because Lq≡0pmod p and 1≤ r<p. For j=0 the empty product is one. The proof also covers p=2 in its short window. Outside this window, a fact","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7bce99afe738aa6","tex":"\\prod_{r=1}^{L}","aliases":"∏_r=1^L prod","display":"\\prod_{r=1}^{L}","kind":"use","excerpt":"Apply this filter to e^z and put E_L,j(z)=∑_q=0^∞fracz^Lq+j(Lq+j)!. For α∈C, write (α)_q=α(α+1)⋯(α+q-1) and (α)_0=1. Splitting the factorial into step-L factors gives frac(Lq+j)!j! · =∏_r=1^L∏_k=0^q-1(j+r+Lk)nonumber · =L^Lq∏_r=1^L(fracj+rL)_q. Exactly one bottom parameter is one, at r=L-j. Define the multiset A_L,j ={fracj+rL:1≤ r≤ L}∖{1}. Every m","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-cec653bd65c24ad4","tex":"\\sum_{\\mathbf d}","aliases":"∑_ d sum mathbf","display":"\\sum_{\\mathbf d}","kind":"use","excerpt":"After w refinements a packet has n=p^w orthonormal descendants. Its normalized radial vector is |R_p,w⟩=n^-1/2∑_ d| d⟩. The norm is n^-1∑_ d1=1, so each descendant has amplitude modulus p^-w/2 and Born probability p^-w. This reproduces [source reference]. The words ``radial'' and ``angular'' here refer to the uniform vector and its orthogonal compl","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dfa5b4b45388d395","tex":"\\sum_{\\mathbf d\\in\\{0,\\ldots,p-1\\}^w}","aliases":"∑_ d∈{0,…,p-1}^w sum mathbf in ldots","display":"\\sum_{\\mathbf d\\in\\{0,\\ldots,p-1\\}^w}","kind":"use","excerpt":"D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-62016043dd92b660","tex":"\\sum_{\\|k\\|_\\infty\\le K}","aliases":"∑_ k _∞≤ K sum infty le","display":"\\sum_{\\|k\\|_\\infty\\le K}","kind":"use","excerpt":"A filtered algebra with room for products. On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normaliz","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9682ef91352bf906","tex":"\\sum_{\\mathbf r}","aliases":"∑_ r sum mathbf","display":"\\sum_{\\mathbf r}","kind":"use","excerpt":"[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber H_A f · =∑_ rA_ r F_ rf. Then H_A is self-adjoint, and finite rectangular packet sums form a core. Every rectangula","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-54d79a998f5bb042","tex":"\\sum_A","aliases":"∑_A sum","display":"\\sum_A","kind":"use","excerpt":"J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7684dda7bca2d3dd","tex":"\\sum_{a,b}","aliases":"∑_a,b sum","display":"\\sum_{a,b}","kind":"use","excerpt":"is a resolved operator rather than a preferred coordinate matrix. Let V be a complex vector space with a finite family of projectors (P_a)_a∈ A satisfying P_aP_b=δ_abP_a, ∑_a∈ AP_a=I. The component P_av is the packet of v selected by a. For a linear map L:V→ V, its transfer block from packet a to packet b is L_ba=P_bLP_a. Completeness gives L=∑_a,b","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9068784c641ac3f6","tex":"\\sum_{a=0}^{p-1}","aliases":"∑_a=0^p-1 sum","display":"\\sum_{a=0}^{p-1}","kind":"use","excerpt":"Analytic refinement of a factorial lane is independent of this unit condition. Splitting q=pr+a, 0≤ a<p, gives the exact identity E_L,j(z)=∑_a=0^p-1E_pL,j+aL(z), 0≤ j<L. Absolute convergence permits this partition of the summation index. The labels j+aL all lie in [0,pL), but need not lie in the parent's short unit window. Exact complex analytic re","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b7aa8d9329a85bf6","tex":"\\sum_{a=0}^{p^{r-1}-1}","aliases":"∑_a=0^p^r-1-1 sum","display":"\\sum_{a=0}^{p^{r-1}-1}","kind":"use","excerpt":"joint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level r, equal to the rank in [source reference], so they span that detail subspace. Hence D_r^(h)=∑_a=0^p","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b1314006a37070ed","tex":"\\sum_{a\\in C(A)}","aliases":"∑_a∈ C(A) sum in","display":"\\sum_{a\\in C(A)}","kind":"use","excerpt":"Suppose P_A=∑_a∈ C(A)P_a and P_B=∑_b∈ C(B)P_b. Then P_BLP_A=∑_b∈ C(B)∑_a∈ C(A)P_bLP_a. The identity follows by distributing two finite sums. It says exactly which fine transfers a coarse observation retains. A scalar summary or a diagonal-only summary may retain less information and therefore needs its own loss statement.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c8d172fdcd28f72a","tex":"\\sum_{a\\in C_A}","aliases":"∑_a∈ C_A sum in","display":"\\sum_{a\\in C_A}","kind":"use","excerpt":"Let A be a finite set of coarse labels and let f:I→ A assign every fine packet to exactly one coarse label. Write C_A=f^-1(A) and define Q_A=∑_a∈ C_AP_a. The fibers are disjoint and cover I, so (Q_A) is another orthogonal atlas. Empty fibers give zero projectors. This is a grouping of an existing resolution. It need not be an approximation to a dif","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-02071cad21612a9a","tex":"\\sum_{a\\in C_\\alpha}","aliases":"∑_a∈ C_α sum in alpha","display":"\\sum_{a\\in C_\\alpha}","kind":"use","excerpt":"For a partition A=bigsqcup_αC_α, the strong sum Q_α=∑_a∈ C_αP_a is a bounded orthogonal projection. Its weight is always normQ_αψ^2=∑_a∈ C_αnormP_aψ^2. To define a coarse current with HQ_αψ, additionally require Q_α D(H)⊂ D(H). Fine domain preservation alone does not state this infinite-cell property.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-10721c0e2add56b1","tex":"\\sum_{a\\notin C}","aliases":"∑_a∉ C sum notin","display":"\\sum_{a\\notin C}","kind":"use","excerpt":"fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c7968355940f474b","tex":"\\sum_{A\\ne B}","aliases":"∑_A≠ B sum ne","display":"\\sum_{A\\ne B}","kind":"use","excerpt":"The coarse jump balance is always exact. For it to close as a time-homogeneous Markov equation for coarse weights alone, a further compatibility condition suffices. For distinct coarse cells A,B, suppose ∑_b∈ C_Bγ_ba=q_BA for every a∈ C_A. Then summing the fine gains gives q_BAp_A^ c, so dot p_B^ c=∑_A≠ B(q_BAp_A^ c-q_ABp_B^ c). Without this equali","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","excerpt":"[Block calculus] For bounded X,Y, X · =∑_b,aX_ba, · P_bXψ · =∑_aX_baψ_a, (XY)_ca · =∑_bX_cbY_ba, · (X^*)_ab · =(X_ba)^*. The identity has diagonal blocks P_a and zero off-diagonal blocks. If X=X^*, its opposite blocks are adjoints.","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c43729fd2f2f98cd","tex":"\\sum_{b,a}","aliases":"∑_b,a sum","display":"\\sum_{b,a}","kind":"use","excerpt":"[Block calculus] For bounded X,Y, X · =∑_b,aX_ba, · P_bXψ · =∑_aX_baψ_a, (XY)_ca · =∑_bX_cbY_ba, · (X^*)_ab · =(X_ba)^*. The identity has diagonal blocks P_a and zero off-diagonal blocks. If X=X^*, its opposite blocks are adjoints.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6c78ad579d275dc7","tex":"\\sum_{b=0}^h","aliases":"∑_b=0^h sum","display":"\\sum_{b=0}^h","kind":"use","excerpt":"The rank (p-1)p^r-1 thus counts chains born at level r or new Haar directions on a level. A single level's detail space is usually transported to the next level; it is the entire angular chain that is invariant under the tree Hamiltonian. The full dark dimension is ∑_r=1^h(p-1)p^r-1(h-r+1) =∑_b=0^hp^b-(h+1). Every one of these states is invisible t","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f5bff478efdb5579","tex":"\\sum_{b\\in A}","aliases":"∑_b∈ A sum in","display":"\\sum_{b\\in A}","kind":"use","excerpt":"For two operators L and M, insert the resolution between them: P_cMLP_a=∑_b∈ A(P_cMP_b)(P_bLP_a). This is block multiplication written without a choice of basis inside each packet. The intermediate index records the interface through which a composition passes. It is also the reason that a change of resolution must specify how fine packets assemble","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5a63562b0ab80881","tex":"\\sum_{b\\in C}","aliases":"∑_b∈ C sum in","display":"\\sum_{b\\in C}","kind":"use","excerpt":"fracddt∑_ap_a · =0. For every subset C⊆ I, its total weight obeys fracddt∑_b∈ Cp_b =∑_b∈ C∑_a∉ CJ_b← a.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8f2acc418631442b","tex":"\\sum_{b\\in C(B)}","aliases":"∑_b∈ C(B) sum in","display":"\\sum_{b\\in C(B)}","kind":"use","excerpt":"Suppose P_A=∑_a∈ C(A)P_a and P_B=∑_b∈ C(B)P_b. Then P_BLP_A=∑_b∈ C(B)∑_a∈ C(A)P_bLP_a. The identity follows by distributing two finite sums. It says exactly which fine transfers a coarse observation retains. A scalar summary or a diagonal-only summary may retain less information and therefore needs its own loss statement.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4ea9c40bbaa4e5c2","tex":"\\sum_{b\\in C_B}","aliases":"∑_b∈ C_B sum in","display":"\\sum_{b\\in C_B}","kind":"use","excerpt":"[Exact coarse transfer] The coarse weight and current are p_A^ c · =normQ_Aψ^2=∑_a∈ C_Ap_a, J_B← A^ c · =2ReinnerQ_BψKQ_Aψ =∑_b∈ C_B∑_a∈ C_AJ_b← a. They satisfy dot p_B^ c=∑_AJ_B← A^ c and J_B← A^ c=-J_A← B^ c. In particular, the sum of fine currents internal to a coarse cell is zero.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b85c57e31dd1fb9d","tex":"\\sum_{b\\in C_\\beta}","aliases":"∑_b∈ C_β sum in beta","display":"\\sum_{b\\in C_\\beta}","kind":"use","excerpt":"Under this coarse-domain hypothesis, symmetry and the strong cell sums justify both iterated sums separately: J^rm c_βα · =2IminnerQ_βψHQ_αψ · =∑_b∈ C_β(∑_a∈ C_αJ_ba) =∑_a∈ C_α(∑_b∈ C_βJ_ba). For example, first summing a moves H onto P_bψ, then the outer sum pairs ∑_bP_bψ with the fixed vector HQ_αψ. The reverse order has its own symmetric argument","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2046ba257a075646","tex":"\\sum_{b\\ne a}","aliases":"∑_b≠ a sum ne","display":"\\sum_{b\\ne a}","kind":"use","excerpt":"of C^n, use its rank-one projectors, and choose rates γ_ba≥0 for b≠ a. Set diagonal rates to zero for convenience and define L_ba=sqrtγ_ba |e_b⟩⟨ e_a|, D_L(ρ)=Lρ L^*- frac12(L^*Lρ+ρ L^*L). Consider a supplied differentiable matrix trajectory satisfying dotρ=∑_b≠ a D_L_ba(ρ). This choice of jumps is tied to the selected basis. A general dissipative ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-996c298de46e2347","tex":"\\sum_{b\\ne i}","aliases":"∑_b≠ i sum ne","display":"\\sum_{b\\ne i}","kind":"use","excerpt":"In this model the population equation closes without knowing the coherences. The coherences nevertheless have their own dynamics. For i≠ j, let r_i=∑_b≠ iγ_bi be the total rate leaving i. The gain matrices are diagonal, while each anticommutator selects its source row and column. Consequently dotρ_ij=-tfrac12(r_i+r_j)ρ_ij. For fixed rates, ρ_ij(t)=","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3004d1c3c6d1b821","tex":"\\sum_c","aliases":"∑_c sum","display":"\\sum_c","kind":"use","excerpt":"Let (P_a) be a finite sharp atlas. Writing X_ba=P_bXP_a and B_ba=P_bBP_a, insertion of I=∑_cP_c gives P_bδ_B(X)P_a =i∑_c(B_bcX_ca-X_bcB_ca). Each term has a genuine intermediate packet c. One ordering applies X then B, while the other applies B then X. Replacing a matrix product by an entrywise product would destroy this path composition and the ca","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-581e46df03c6e7a6","tex":"\\sum_d","aliases":"∑_d sum","display":"\\sum_d","kind":"use","excerpt":"One more digit supplies p children for every address. The uniform refinement map is (V_p,hf)(a+d p^h)=p^-1/2f(a), 0≤ a<p^h,quad0≤ d<p. In tensor coordinates it is f↦|+_p⟩⊗ f, where |+_p⟩=p^-1/2∑_d|d⟩ and the child digit is listed first. The norm calculation is normV_p,hf^2=∑_a∑_d p^-1|f(a)|^2=norm f^2. The adjoint is the normalized amplitude sum (V","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ed7bfe0fc25b6ba1","tex":"\\sum_{d=0}^{p-1}","aliases":"∑_d=0^p-1 sum","display":"\\sum_{d=0}^{p-1}","kind":"use","excerpt":"Each wavelet is constant on level-r cylinders and has zero average on its parent level-r-1 cylinder, because ∑_d=0^p-1ω_p^qd=0 for nonzero q. Thus it lies in Ran D_r^(h). Its support contains p^h-r+1 points, giving unit norm. Distinct parents have disjoint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Let V_h be a finite real inner-product space. Let δ_1,δ_2,δ_3 be commuting skew-adjoint operators on it: δ_iδ_j=δ_jδ_i, innerδ_iXY_h=-innerXδ_iY_h, δ^2=∑_iδ_i^2. The periodic centered differences in three coordinate directions provide one example. All tensor inner products below sum over both indices, using the Frobenius inner product on symmetric ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a9b279e4a6b63e78","tex":"\\sum_{i=0}^{M-1}","aliases":"∑_i=0^M-1 sum","display":"\\sum_{i=0}^{M-1}","kind":"use","excerpt":"Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-76c8d38e858879d4","tex":"\\sum_{i=1}^{M-1}","aliases":"∑_i=1^M-1 sum","display":"\\sum_{i=1}^{M-1}","kind":"use","excerpt":"Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":", and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, th","count":27},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c57a71fec0503f9d","tex":"\\sum_{j=0}^{L-1}","aliases":"∑_j=0^L-1 sum","display":"\\sum_{j=0}^{L-1}","kind":"use","excerpt":"[Entire lane calculus for a bounded operator] Let A_0∈ B(HH) and t∈C. The series E_L,j(tA_0) converges absolutely in operator norm and satisfies E_L,j(tA_0) · =frac1L∑_ℓ=0^L-1ω_L^-jℓ e^ω_L^ℓ tA_0,nonumber · =frac(tA_0)^jj! _0F_L-1 (- A_L,j; frac(tA_0)^LL^L),nonumber ∑_j=0^L-1E_L,j(tA_0) · =e^tA_0. If all terms through degree Q are retained in a lan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7d1d47c0c0a47e08","tex":"\\sum_{j=1}^N","aliases":"∑_j=1^N sum","display":"\\sum_{j=1}^N","kind":"use","excerpt":"The telescoping mechanism for a complete walk. Let I:X→ Y be a bounded linear comparison map and let F_j:Y→ Y, C_j:X→ X be fine and coarse factors. Write D_j=F_jI-IC_j. Two factors satisfy the exact identity F_2F_1I-IC_2C_1=F_2D_1+D_2C_1. Repeated insertion gives, in execution order 1,…,N, F_N⋯ F_1I-IC_N⋯ C_1 =∑_j=1^N F_N⋯ F_j+1D_jC_j-1⋯ C_1. Empty","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b9b31c7c60715e60","tex":"\\sum_{j=a}^{b}","aliases":"∑_j=a^b sum","display":"\\sum_{j=a}^{b}","kind":"use","excerpt":"rge without compensating background would be inconsistent. Equation [source reference] supplies that neutralizing mean. Conservation of G_j is a propagation theorem. It preserves an initially nonzero constraint violation too; it does not project arbitrary initial data onto the constraint surface. Summing continuity over a≤ j≤ b gives the physical-l","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c30312fae2625b14","tex":"\\sum_{j=r}^s","aliases":"∑_j=r^s sum","display":"\\sum_{j=r}^s","kind":"use","excerpt":"Every e_j is nonnegative under the stated coefficient assumptions. An interval of cells r,…,s obeys the exact coarse balance fracddt∑_j=r^se_j +F_s+1/2-F_r-1/2=∑_j=r^ss_j. Thus coarse energy is obtained by adding fine cell energies and retaining only the exposed boundary fluxes. This is the same cancellation mechanism as projector transfer, applied","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c233b15936db2d27","tex":"\\sum_{j=x,y,z}","aliases":"∑_j=x,y,z sum","display":"\\sum_{j=x,y,z}","kind":"use","excerpt":"-i∑_j=x,y,zα_j∂_j+mβ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"The series converges absolutely in operator norm, being dominated by ∑_k(|t| H )^k/k!. This is exact continuous-time evolution, with no time step or series truncation.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a8b3533c8f1752a0","tex":"\\sum_{k=0}^{N}","aliases":"∑_k=0^N sum","display":"\\sum_{k=0}^{N}","kind":"use","excerpt":"∑_k=0^Nfracz^k(A^ H)^kk! =T(∑_k=0^Nfracz^kA^kk!)T^-1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7796ee229bc8c35a","tex":"\\sum_{k=0}^{N-1}","aliases":"∑_k=0^N-1 sum","display":"\\sum_{k=0}^{N-1}","kind":"use","excerpt":"On [-2,2] define the probability measure dν(x)=fracdxπsqrt4-x^2. It is the distribution of 2cosθ for a uniform angle on the circle, or on [0,π]. It is also the limiting normalized spectral measure of the free cycles: for a continuous test function f, frac1N∑_k=0^N-1f(2cos(2π k/N)) ⟶frac12π∫_0^2πf(2cosθ) dθ. This is the Riemann-sum limit of a contin","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d163c0162318851c","tex":"\\sum_{k=0}^\\infty","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^\\infty","kind":"use","excerpt":"is sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of the same operator, so there is no commuting-operator assumption. For the tail, bound a lane ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f1c2ff812e2b7828","tex":"\\sum_{k=0}^{\\infty}","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^{\\infty}","kind":"use","excerpt":"U_H(t)=e^-itH=∑_k=0^∞frac(-itH)^kk!.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b5d62fa15f1a3c0e","tex":"\\sum_{k=1}^{\\infty}","aliases":"∑_k=1^∞ sum infty","display":"\\sum_{k=1}^{\\infty}","kind":"use","excerpt":"e^X-e^Y ≤ X-Y ∑_k=1^∞frackR^k-1k! =e^R X-Y .","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-baa8f7877711f9a7","tex":"\\sum_{k=n}^{m-1}","aliases":"∑_k=n^m-1 sum","display":"\\sum_{k=n}^{m-1}","kind":"use","excerpt":"pace limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence is Cauchy. Com","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f26710096ba8e9ae","tex":"\\sum_{k=n}^{\\infty}","aliases":"∑_k=n^∞ sum infty","display":"\\sum_{k=n}^{\\infty}","kind":"use","excerpt":"[A complete-space limit with an explicit tail] Let f_n be a sequence in a complete metric space and suppose d(f_n,f_n+1)≤ C4^-n (n≥0), C≥0. Then f_n converges to some f_∞, and d(f_n,f_∞)≤∑_k=n^∞C4^-k =tfrac43C4^-n. For m>n, the triangle inequality gives d(f_n,f_m)≤∑_k=n^m-1C4^-k ≤tfrac43C4^-n. The upper bound tends to zero with n, so the sequence i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-01f56a5e1c2b8b56","tex":"\\sum_{k\\ge1}","aliases":"∑_k≥1 sum ge","display":"\\sum_{k\\ge1}","kind":"use","excerpt":"A concrete domain boundary. As an ordinary example, put nonzero coefficients only on the positive axis modes n=(k,0,0), k≥1, and set a_n=k^-sv for a fixed unit spinor v. All remaining coefficients are zero. Then ∑_nnorma_n^2=∑_k≥1k^-2s, which is finite exactly when s>1/2. The additional momentum series is 4π^2∑_k≥1k^2-2s, which is finite exactly wh","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7dd1f8372600c494","tex":"\\sum_m","aliases":"∑_m sum","display":"\\sum_m","kind":"use","excerpt":"it-parseval identifies f with the square-summable sequence f_m= F_mf. Finite sequences belong to the domain, so it is dense. The real multipliers imply symmetry. To identify the adjoint, suppose g∈Dom(H_ϵ^*) and write k=H_ϵ^*g. Testing the adjoint identity against an arbitrary vector supported on level m gives k_m=ϵ_m g_m. Since k is square summabl","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fe8ca430ca97a131","tex":"\\sum_{m=0}^M","aliases":"∑_m=0^M sum","display":"\\sum_{m=0}^M","kind":"use","excerpt":"Use F_0= E_0 and F_m= E_m- E_m-1 for m≥1. These are the infinite depth details called D_m in the archived chapter. They correspond to the finite D_m^(M) after the embedding U_p,M. Equation [source reference] gives F_m F_n=δ_mn F_m, ∑_m=0^M F_m= E_M. For example, if 1≤ m<n, the four products in ( E_m- E_m-1)( E_n- E_n-1) are E_m, E_m, E_m-1, E_m-1 a","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-668e17fed94fbb22","tex":"\\sum_{m=0}^{M}","aliases":"∑_m=0^M sum","display":"\\sum_{m=0}^{M}","kind":"use","excerpt":"Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-187f645f69d06fb0","tex":"\\sum_{m=0}^{\\infty}","aliases":"∑_m=0^∞ sum infty","display":"\\sum_{m=0}^{\\infty}","kind":"use","excerpt":"Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-06ee679c4f230fdb","tex":"\\sum_{m>M}","aliases":"∑_m>M sum","display":"\\sum_{m>M}","kind":"use","excerpt":"[Exact tail and regularity bound] For f∈HH_p and every M≥0, norm(I- E_M)f_2^2 =∑_m>Mnorm F_mf_2^2. If f∈ H_p^s, then norm(I- E_M)f_2 ≤ p^-s(M+1)normf_ H_p^s. The convergence E_M→ I is strong and is not convergence in operator norm. The omitted details are orthogonal, so Parseval gives the equality. For m>M, p^2sm≥ p^2s(M+1). Multiplying the omitted","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-abaf68bfbf4ecdd4","tex":"\\sum_{m\\leq M}","aliases":"∑_m≤ M sum leq","display":"\\sum_{m\\leq M}","kind":"use","excerpt":"An alternative is to extend the finite Hamiltonian by the zero Hamiltonian on the omitted subspace: widetilde H_M=H_ϵ E_M =∑_m≤ Mϵ_m F_m. This is a bounded self-adjoint operator on all of HH_p. Its propagator is a full-space unitary, but has a different tail: e^-itwidetilde H_M =U_M^emb(t)+(I- E_M), norm(e^-itwidetilde H_M-U(t))f_2 ≤2norm(I- E_M)f_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d1cabd23a4e71916","tex":"\\sum_{m\\geq0}","aliases":"∑_m≥0 sum geq","display":"\\sum_{m\\geq0}","kind":"use","excerpt":"For s>0, define the Haar regularity norm normf_ H_p^s^2 =∑_m≥0p^2smnorm F_mf_2^2, H_p^s={f∈HH_p:normf_ H_p^s<∞}. No identification with another Sobolev convention is required here. The definition itself gives the quantitative cutoff estimate.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-97951b3ae8950fca","tex":"\\sum_{m\\geq1}","aliases":"∑_m≥1 sum geq","display":"\\sum_{m\\geq1}","kind":"use","excerpt":"silon is bounded precisely when the energies are bounded below. If there is no lower bound, the levelwise exponential is an unbounded operator with domain {f:∑_m e^-2βϵ_mnorm F_mf_2^2<∞}. Thus a finite Euclidean intertwining identity does not by itself supply a bounded infinite heat semigroup. When bounded below, the positive operator has trace Tr(","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","excerpt":"Completion and continuum evolution. The complete coefficient Hilbert space. Let KK=ℓ^2(Z^3;C^4) ={a=(a_n):∑_n∈Z^3norma_n^2<∞}. Its inner product is innerab=∑_ninnera_nb_n. Absolute convergence follows from the scalar Cauchy--Schwarz inequality. The norm therefore measures the total squared amplitude across all Fourier packets. A finite family of am","count":15},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8773b980988ccc84","tex":"\\sum_{n=0}^\\infty","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^\\infty","kind":"use","excerpt":"Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eced359e9f8ad685","tex":"\\sum_{n=Q+1}^\\infty","aliases":"∑_n=Q+1^∞ sum infty","display":"\\sum_{n=Q+1}^\\infty","kind":"use","excerpt":"is sum, so it converges absolutely. The character calculation and factorial split can therefore be applied in operator norm, giving the first two identities. Partitioning the nonnegative degrees by their residues proves the third. All products are powers of the same operator, so there is no commuting-operator assumption. For the tail, bound a lane ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9b846ee3188af45a","tex":"\\sum_{n>N}","aliases":"∑_n>N sum","display":"\\sum_{n>N}","kind":"use","excerpt":"adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there. For S_Nc=∑_n≤ Nc_ne_n, normc-S_Nc_H_λ^2 =∑_n>N(1+λ_n^2)|c_n|^2⟶0.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-683d9edfe17c29d2","tex":"\\sum_{n\\in S}","aliases":"∑_n∈ S sum in","display":"\\sum_{n\\in S}","kind":"use","excerpt":"f_a(x)=∑_n∈ Se_n(x)a_n.","count":16},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-89b99765eefcebef","tex":"\\sum_{n\\in\\Z^3}","aliases":"∑_n∈Z^3 sum in Z","display":null,"kind":"use","excerpt":"Completion and continuum evolution. The complete coefficient Hilbert space. Let KK=ℓ^2(Z^3;C^4) ={a=(a_n):∑_n∈Z^3norma_n^2<∞}. Its inner product is innerab=∑_ninnera_nb_n. Absolute convergence follows from the scalar Cauchy--Schwarz inequality. The norm therefore measures the total squared amplitude across all Fourier packets. A finite family of am","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9adcdaef75b599bd","tex":"\\sum_{n\\notin S}","aliases":"∑_n∉ S sum notin","display":"\\sum_{n\\notin S}","kind":"use","excerpt":"For a finite set S⊂Z^3, define the truncation P_Sa by retaining a_n when n∈ S and assigning zero otherwise. Then norma-P_Sa^2=∑_n∉ Snorma_n^2. The sum on the right can be made arbitrarily small because the full nonnegative series converges. Thus finite packets are dense in KK. The statement concerns the norm of the omitted amplitudes and does not d","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f7c05ed299cc8fcf","tex":"\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}","aliases":"∑_n≡ j (rm mod L) sum equiv rm","display":"\\sum_{n\\equiv j\\;({\\rm mod}\\;L)}","kind":"use","excerpt":"On the oscillator Fock space, let widehat N|n⟩=n|n⟩ and define the bounded unitary V=e^2π iwidehat N/L. Its spectral residues are Q_j^(L)=frac1L∑_ℓ=0^L-1ω_L^-jℓV^ℓ =∑_n≡ j (rm mod L)|n⟩⟨ n|. On every number basis vector, the same finite character sum proves the second equality. These Q_j^(L) are bounded orthogonal projectors, Q_jQ_k=δ_jkQ_j, and ∑_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-165ec26aa834867f","tex":"\\sum_{n\\leq N}","aliases":"∑_n≤ N sum leq","display":"\\sum_{n\\leq N}","kind":"use","excerpt":"operator H_λ is self-adjoint. Its coordinate cutoffs converge in graph norm on D(H_λ). If y∈ D(H_λ^*) and H_λ^*y=z, testing the adjoint identity against e_n gives z_n=λ_ny_n. Since z∈ℓ^2, the sequence y belongs to D(H_λ). Symmetry gives the opposite inclusion of domains, and the actions agree there. For S_Nc=∑_n≤ Nc_ne_n, normc-S_Nc_H_λ^2 =∑_n>N(1+","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f0d1bc9b91545210","tex":"\\sum_{n\\geq1}","aliases":"∑_n≥1 sum geq","display":"\\sum_{n\\geq1}","kind":"use","excerpt":"[Strong convergence with no differentiable limiting orbit] For λ_n=n, take c_n=C/n with C=(∑_n≥1n^-2)^-1/2. This is a normalized ℓ^2 state outside D(H_λ). Every finite cutoff orbit is differentiable, and [source reference] gives uniform convergence to the full orbit. Yet the full orbit is nowhere differentiable in ℓ^2, by the preceding proposition.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e5495c300dba1b9b","tex":"\\sum_{q=0}^\\infty","aliases":"∑_q=0^∞ sum infty","display":"\\sum_{q=0}^\\infty","kind":"use","excerpt":", and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I. To prove the coefficient formula, th","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a21758814a18a18d","tex":"\\sum_{q=0}^{\\infty}","aliases":"∑_q=0^∞ sum infty","display":"\\sum_{q=0}^{\\infty}","kind":"use","excerpt":"The Taylor channels can be separated without changing the physical mode. Put y=x^2/4, B(y)=J_0(2sqrt y), and for an integer r≥2 define B_r,j(y)=∑_q=0^∞ frac(-1)^rq+jy^rq+j((rq+j)!)^2, j=0,…,r-1. Then B=∑_j B_r,j. With ϑ=y d/dy, the coefficient identity ϑ^2 B_r,j+y B_r,j-1=0, jpmod r, follows from m^2/(m!)^2=1/((m-1)!)^2 for m≥1. Consequently the se","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-125fff8908feab30","tex":"\\sum_{q=1}^{p-1}","aliases":"∑_q=1^p-1 sum","display":"\\sum_{q=1}^{p-1}","kind":"use","excerpt":"joint supports. For a fixed parent the inner product of the characters q,q' is p^-1∑_d=0^p-1ω_p^(q'-q)d=δ_qq'. Wavelets at different levels are orthogonal because their detail projectors are orthogonal. There are exactly (p-1)p^r-1 wavelets at level r, equal to the rank in [source reference], so they span that detail subspace. Hence D_r^(h)=∑_a=0^p","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"Fix an integer p≥2 and M≥3, and put N=pM. Primality is not needed for the identities in this section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-096f1afa7adcb4ee","tex":"\\sum_{r(a)=\\beta}","aliases":"∑_r(a)=β sum beta","display":"\\sum_{r(a)=\\beta}","kind":"use","excerpt":"If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-83dad0d78f66f949","tex":"\\sum_{r,s=0}^{p-1}","aliases":"∑_r,s=0^p-1 sum","display":"\\sum_{r,s=0}^{p-1}","kind":"use","excerpt":"section. For 0≤ r<p, define the isometry J_r:C^M→C^N by J_r|k⟩=|pk+r⟩, 0≤ k<M. Then J_r^*J_s=δ_rsI_M and ∑_rJ_rJ_r^*=I_N. The standard analysis map A sends |pk+r⟩ to |r⟩⊗|k⟩. This is a permutation unitary. For any operator X on C^N, its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality f","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c7b90c949ac04119","tex":"\\sum_{r=0}^{3}","aliases":"∑_r=0^3 sum","display":"\\sum_{r=0}^{3}","kind":"use","excerpt":"⟨ v,w⟩=∑_r=0^3overlinev_rw_r, v _2^2=∑_r=0^3|v_r|^2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-14001847ba6d9b2f","tex":"\\sum_{r=0}^h","aliases":"∑_r=0^h sum","display":"\\sum_{r=0}^h","kind":"use","excerpt":"[Finite Haar resolution] The cylinder projectors satisfy E_r^(h)E_s^(h)=E_min(r,s)^(h), E_h^(h)=I. Define D_0^(h)=E_0^(h), D_r^(h)=E_r^(h)-E_r-1^(h) (1≤ r≤ h). Then the D_r^(h) are mutually orthogonal projectors and ∑_r=0^hD_r^(h) · =I, · D_r^(h)D_s^(h) · =δ_rsD_r^(h),nonumber rankD_0^(h) · =1, · rankD_r^(h) · =(p-1)p^r-1 (r≥1).","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0d24bc4aca4d845c","tex":"\\sum_{r=0}^{k}","aliases":"∑_r=0^k sum","display":"\\sum_{r=0}^{k}","kind":"use","excerpt":"For Hermitian matrices the two exponentials in the integrand are unitary. This does not rely on the special Clifford algebra: the series gives (e^-iuA)^*=e^iuA, and the Cauchy product of the two absolutely convergent series gives e^iuAe^-iuA=I. Indeed, the coefficient of A^k for k>0 is (iu)^k∑_r=0^kbinomkr(-1)^r/k!=0, and the constant coefficient i","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e1f302dc5a0699e7","tex":"\\sum_{r=0}^{k-1}","aliases":"∑_r=0^k-1 sum","display":"\\sum_{r=0}^{k-1}","kind":"use","excerpt":"X^k-Y^k=∑_r=0^k-1X^k-1-r(X-Y)Y^r (k≥1).","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4a8ae622c77aac5b","tex":"\\sum_{r=0}^m","aliases":"∑_r=0^m sum","display":"\\sum_{r=0}^m","kind":"use","excerpt":"The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an addition","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4b5b1fcf31d6f6c3","tex":"\\sum_{r=1}^b","aliases":"∑_r=1^b sum","display":"\\sum_{r=1}^b","kind":"use","excerpt":"its fixed ancestral coefficient, on all children below the lower vertex at a common level. The lower chain has zero sum over its own child contrasts, so their inner product is zero. The same zero-sum calculation makes every angular vector orthogonal to the uniform radial level vector. At a fixed depth b, the radial vector contributes one dimension.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9f756cf70c299a69","tex":"\\sum_{r=1}^h","aliases":"∑_r=1^h sum","display":"\\sum_{r=1}^h","kind":"use","excerpt":"For the pure hierarchical Hamiltonian, the finite spectral theorem gives U_p,h(t)=e^-itH_p,h^Haar =∑_r=0^he^-itϵ_p,rD_r^(h). The cylinder kernel is E_r^(h)(x,y)=p^-(h-r) 1_{x≡ y (mod p^r)}. Consequently U_p,h(t;x,y) · =p^-he^-itϵ_p,0nonumber · +∑_r=1^he^-itϵ_p,r [p^-(h-r)1_x≡ y (mod p^r) -p^-(h-r+1)1_x≡ y (mod p^r-1) ]. Every indicator is determine","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-bf6132b8c151067a","tex":"\\sum_{r=1}^{m}","aliases":"∑_r=1^m sum","display":"\\sum_{r=1}^{m}","kind":"use","excerpt":"Suppose that a finite expansion is available, f(h)=F+∑_r=1^mh^rv_r+R(h). Linearity and the proposition give the exact identity T_mf(h)-F=T_mR(h). Only a finite sum has been used. No convergence of an infinite formal series is needed. The size of the right-hand side remains an analytic question.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8615333b8546053a","tex":"\\sum_{r=m+1}^h","aliases":"∑_r=m+1^h sum","display":"\\sum_{r=m+1}^h","kind":"use","excerpt":"The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an addition","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7d98fe9de6bebd3c","tex":"\\sum_s","aliases":"∑_s sum","display":"\\sum_s","kind":"use","excerpt":", its residue blocks X_rs=J_r^*XJ_s reconstruct it: X=∑_r,s=0^p-1J_rX_rsJ_s^*, ( A X A^*)_rs=X_rs. The first equality follows by inserting ∑_rJ_rJ_r^* on both sides of X. If H=H^*, its blocks satisfy H_rs^*=H_sr. If U is unitary, the same insertion gives ∑_rU_rs^*U_rt · =J_s^*U^*UJ_t=δ_stI_M,nonumber ∑_sU_rsU_ts^* · =J_r^*UU^*J_t=δ_rtI_M. These are","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c0815e59aa813271","tex":"\\sum_{\\substack{b\\in C_\\beta\\\\a\\in C_\\alpha}}","aliases":"∑_substackb∈ C_β a∈ C_α sum substack in beta a in alpha","display":null,"kind":"use","excerpt":"Summing the differentiated corners reconstructs δ_BX. For a coarse partition with Q_α=∑_a∈ C_αP_a, Q_βδ_B(X)Q_α =∑_substackb∈ C_β a∈ C_αP_bδ_B(X)P_a. This is exact aggregation of the already differentiated operator. It does not claim that discarding all internal matrix structure produces a new pointwise algebra with a nonzero exact derivation. The ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6ed65771d02f8363","tex":"\\sum_{\\substack{r(b)=\\beta\\\\r(a)=\\alpha}}","aliases":"∑_substackr(b)=β r(a)=α sum substack beta r alpha","display":null,"kind":"use","excerpt":"If a map r:A→ B groups the labels, the coarse effect is G_β=∑_r(a)=βF_a. Bilinearity proves p^rm c_β=∑_r(a)=βp_a, J^rm c_βα =∑_substackr(b)=β r(a)=αJ^F_ba. No orthogonality is needed for this finite aggregation law. A divergence-free antisymmetric array may also be added to J^F without changing its row sums. Thus the balance law determines the chan","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-1189cbb810e02392","tex":"\\sum_{\\substack{u\\notin A,\\ v\\in A\\\\u\\sim v}}","aliases":"∑_substacku∉ A, v∈ A u∼ v sum substack notin in u sim","display":null,"kind":"use","excerpt":"in gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A, v∈ A u∼ vF_u→ v^n -∑_substackv∈ A, u∉ A u∼ vF_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e30b86f05538dcdd","tex":"\\sum_{\\substack{v\\in A,\\ u\\notin A\\\\u\\sim v}}","aliases":"∑_substackv∈ A, u∉ A u∼ v sum substack in notin u sim","display":null,"kind":"use","excerpt":"in gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A, v∈ A u∼ vF_u→ v^n -∑_substackv∈ A, u∉ A u∼ vF_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e9c7a389e7da74a1","tex":"\\sum_{v\\in A}","aliases":"∑_v∈ A sum in","display":"\\sum_{v\\in A}","kind":"use","excerpt":"in gives ∑_uF_v→ u^n=ρ_v^n. The reversal shift gives ρ_v^n+1=∑_u F_u→ v^n. Subtraction proves ρ_v^n+1-ρ_v^n =∑_u∼ v(F_u→ v^n-F_v→ u^n). For every set of vertices A, summing this equation cancels every internal directed flux once with its opposite occurrence. Hence ∑_v∈ A(ρ_v^n+1-ρ_v^n) =∑_substacku∉ A, v∈ A u∼ vF_u→ v^n -∑_substackv∈ A, u∉ A u∼ vF_","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6546cb38aa60d81e","tex":"\\sum_{v\\in\\{0,1\\}^3}","aliases":"∑_v∈{0,1}^3 sum in","display":"\\sum_{v\\in\\{0,1\\}^3}","kind":"use","excerpt":"Interpolation is possible even when it depends on a path. Choose one reference path from the origin to each vertex and let r(v) be its transport. Then (I_ℓψ)(v)=r(v)ψ, innerΨΦ_rm cube =tfrac18∑_v∈{0,1}^3innerΨ(v)Φ(v). Every r(v) is unitary, so this map is a Hilbert isometry from one spinor to the eight-child cell. Its gauge covariance follows from ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8c1a906fe53c62b1","tex":"\\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}","aliases":"∑_w∈{0,…,p-1}^s sum in ldots","display":"\\sum_{w\\in\\{0,\\ldots,p-1\\}^{s}}","kind":"use","excerpt":"There is a more precise form of the complementary decomposition. For each vertex u at depth a<h, choose an orthonormal basis of vectors c=(c_0,…,c_p-1)∈C^p with ∑_dc_d=0. For s=0,…,h-a-1, set |u,c;s⟩ =p^-s/2∑_d=0^p-1c_d ∑_w∈{0,…,p-1}^s|u d w⟩. For fixed u,c, these are unit vectors on successive levels. They obey the Jacobi recurrence with onsite en","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0e44b9be25437975","tex":"\\sum_x","aliases":"∑_x sum","display":"\\sum_x","kind":"use","excerpt":"The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm ter","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2be6045d20d089d0","tex":"\\sum_{x,a,i}","aliases":"∑_x,a,i sum","display":"\\sum_{x,a,i}","kind":"use","excerpt":"On the full grid canonical phase space the bracket is {F,G}_N=N^3∑_x,a,i (frac∂ F∂ A_a^i(x) frac∂ G∂ E^a_i(x) -frac∂ F∂ E^a_i(x) frac∂ G∂ A_a^i(x)). The factor N^3 matches the normalized trace in the action. The resolved identities below are evaluated at band-limited data in this phase space. Replacing this bracket by a bracket that first projects ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-576fa14b6fc700f4","tex":"\\sum_{x,\\mu}","aliases":"∑_x,μ sum mu","display":"\\sum_{x,\\mu}","kind":"use","excerpt":"The lattice inner product is innerϕψ_V=∑_xϕ_x^*ψ_x, conjugate-linear in the first entry. Periodic reindexing and unitarity also give the useful identity -innerψΔ_Uψ_V =∑_x,μlVert(D^+_μ,Uψ)_xrVert^2. Indeed, expanding one squared difference gives h^-2(normψ_x+widehatμ^2+normψ_x^2 -2Re(ψ_x^*U_x,μψ_x+widehatμ)). Summing and shifting the first norm ter","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d540aa8cbc4fbf01","tex":"\\sum_{x,\\mu<\\nu}","aliases":"∑_x,μ<ν sum mu nu","display":"\\sum_{x,\\mu<\\nu}","kind":"use","excerpt":"For SU(2) links, the Wilson action of the source is the finite function S_W[U]=frac1g_YM^2 ∑_x,μ<ν(1-tfrac12ReTrW_x,μν), g_YM∈R∖{0}. The subscript distinguishes the coupling from the site gauge g_x. Cyclicity of trace proves S_W[U^g]=S_W[U]. Since an SU(2) matrix has eigenvalues e^iθ,e^-iθ, a plaquette term is 1-cosθ≥0. In the Abelian case the anal","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-b9ea8830cdebefeb","tex":"\\sum_{x\\in A}","aliases":"∑_x∈ A sum in","display":"\\sum_{x\\in A}","kind":"use","excerpt":"For a partition of fine spatial sites into coarse regions A, the nonnegative diagnostic E_A^rm con=∑_x∈ Ah^3 (|C_x|^2+∑_i|C_i,x|^2) adds exactly to the full fine diagnostic. This is an exact partition identity. It does not assert that differentiating a coarse field reconstructs its fine constraint values.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3ec3e94007233342","tex":"\\sum_{x\\in S}","aliases":"∑_x∈ S sum in","display":"\\sum_{x\\in S}","kind":"use","excerpt":"Hilbert estimates with explicit constants. Finite-dimensional continuity is not enough to identify an operator norm with physical ℓ^2 energy. The earlier continuous-linear packaging retains the ambient norm on a function space. The Hilbert scalar layer instead realizes a finite four-spinor field as HH_S=ℓ^2(S×{0,1,2,3};C), normψ^2=∑_x∈ Snormψ(x)_C^","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9faa93cb185e1d0d","tex":"\\sum_{y\\equiv x\\, (\\mathrm{mod}\\,p^r)}","aliases":"∑_y≡ x (mod p^r) sum equiv mathrm","display":"\\sum_{y\\equiv x\\, (\\mathrm{mod}\\,p^r)}","kind":"use","excerpt":"For 0≤ r≤ h, define the cylinder average (E_r^(h)f)(x)=p^-(h-r) ∑_y≡ x (mod p^r)f(y). Its matrix is real and symmetric. Averaging twice over the same cylinder has the same effect as averaging once. Thus E_r^(h) is the orthogonal projector onto functions constant on level-r cylinders. The range has dimension p^r.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7f9ca7ecb7712f78","tex":"\\sum_{|v|=r}","aliases":"∑_|v|=r sum","display":"\\sum_{|v|=r}","kind":"use","excerpt":"Define normalized level states |r⟩=p^-r/2∑_|v|=r|v⟩, r=0,…,h. They are orthonormal because different levels are disjoint and level r contains p^r vertices. A direct action calculation gives H|r⟩=γ_r-1|r-1⟩ +ϵ_r|r⟩+γ_r|r+1⟩, with omitted terms at the endpoints. For example, the coefficient at each child at level r+1 is p^-r/2γ_r/sqrt p=γ_r p^-(r+1)/","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ade7e6465dee6e86","tex":"\\sum_\\lambda","aliases":"∑_λ sum lambda","display":"\\sum_\\lambda","kind":"use","excerpt":"Let D=∑_λλ E_λ be its spectral resolution. Then Σ(z)=∑_λfracB E_λ B^*z-λ. Every residue B E_λ B^* is positive semidefinite. Its rank records which directions of the hidden eigenspace reach the visible space. A vanishing residue removes the corresponding pole of Σ, even though z-D is still singular there.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f85a9d219ed81c46","tex":"\\sum_\\mu","aliases":"∑_μ sum mu","display":"\\sum_\\mu","kind":"use","excerpt":"For finite H, expansion of the matrix exponential proves e^-it A H A^* = A e^-itH A^*. Indeed ( A H A^*)^n= A H^n A^*, and the series converges in matrix norm. The same transformation of each Kraus operator gives a unitarily equivalent quantum channel: if Φ(ρ)=∑_μ K_μρ K_μ^* with ∑_μ K_μ^*K_μ=I, then K_μ^rm poly= A K_μ A^*, ∑_μ(K_μ^rm poly)^*K_μ^rm","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-472fcc6e7ca48333","tex":"\\sum_\\Xi","aliases":"∑_Ξ sum Xi","display":"\\sum_\\Xi","kind":"use","excerpt":"The classical BV action. For all fields and ghosts Ξ, introduce antifields Ξ^ddagger with opposite parity and ghost number -1-gh(Ξ). Use a canonical odd antibracket, with right and left derivatives and the indicated graded ordering. The minimal action can be written S_BV=S_0+ ∫∑_Ξ⟨Ξ^ddagger,sΞ⟩. Antisymmetric Lorentz indices are counted once, or eq","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e3116f8319e2684e","tex":"\\sum_\\ell","aliases":"∑_ℓ sum ell","display":"\\sum_\\ell","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-43e727cf5aadc16c","tex":"\\sum_{\\ell=0}^{L-1}","aliases":"∑_ℓ=0^L-1 sum ell","display":"\\sum_{\\ell=0}^{L-1}","kind":"use","excerpt":"Fix an integer L≥2, write ω_L=e^2π i/L, and let F(z)=∑_n=0^∞ a_nz^n be analytic for |z|<ρ. The whole disk is invariant under the rotations used below. Define ( R_L,jF)(z) =frac1L∑_ℓ=0^L-1ω_L^-jℓF(ω_L^ℓ z), 0≤ j<L. Absolute convergence for |z|<ρ and the finite character sum give ( R_L,jF)(z)=∑_q=0^∞ a_Lq+jz^Lq+j, R_L,j R_L,k =δ_jk R_L,j, ∑_j R_L,j=I","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f9da70e93ac834a8","tex":"\\sum_{\\ell=0}^{p-1}","aliases":"∑_ℓ=0^p-1 sum ell","display":"\\sum_{\\ell=0}^{p-1}","kind":"use","excerpt":"Let the new-digit modulation be M_p,h|a+d p^h-1⟩=ω_p^d|a+d p^h-1⟩. The selector of child digit d is the exact roots-of-unity filter R_d=frac1p∑_ℓ=0^p-1ω_p^-dℓM_p,h^ℓ. On a child with digit e, the coefficient is p^-1∑_ℓω_p^(e-d)ℓ=δ_de. Therefore R_dR_e=δ_deR_d, R_d^*=R_d, and ∑_dR_d=I. These projectors select address digits. They are different from ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-14617be2bc27090e","tex":"\\sum_{\\ell=1}^{L-1}","aliases":"∑_ℓ=1^L-1 sum ell","display":"\\sum_{\\ell=1}^{L-1}","kind":"use","excerpt":"[Approach to uniform residue probability] For fixed L≥2 and μ≥0, |p_L,j(μ)-frac1L| ≤frac1L∑_ℓ=1^L-1 e^-μ(1-cos(2πℓ/L)) ≤fracL-1L e^-μ(1-cos(2π/L)). In [source reference], the ℓ=0 term is 1/L. The modulus of every other term is e^μ(Reω_L^ℓ-1)/L. The triangle inequality gives the first bound. Among the nonzero residues, the largest cosine is cos(2π/L","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-602b230463ce9d93","tex":"\\mp","aliases":"∓ mp","display":"\\mp","kind":"use","excerpt":"Reality is built into e and ω. For real Lorentz bivectors, the internal dual obeys ⋆^2=-id and the projectors P_±=tfrac12(id∓ i⋆) satisfy overlineP_+X=P_-X. Both chiral blocks are retained and related by conjugation. A real vector field with an existing unique flow preserves this real locus. This parent-level statement supplies a real classical sta","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"det(z-H) · =det(z-D)det S(z). For E∈R∖σ(D), an eigenvector of H is equivalent to a nonzero solution of S(E)u=0, with hidden component v=(E-D)^-1B^*u. Put C=z-D. Direct block multiplication gives z-H= I · -BC^-1","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"[Invariant free measure and exact mode routing] For every integer p≥2, (R_p)_*ν=ν. The map K_pf=f∘ R_p is an isometry on L^2(ν). With ϕ_0=1, ϕ_n(x)=sqrt2 T_n(x/2), n≥1, its action and adjoint on an orthonormal basis are K_pϕ_n=ϕ_pn, K_p^*ϕ_m= ϕ_m/p, · pmid m,","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.","count":67},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-9e05adb8ff8d013d","tex":"\\infty^2","aliases":"∞^2 infty","display":"\\infty^2","kind":"use","excerpt":"show that both directions are continuous. The first follows because each coordinate square is bounded by the sum; the second because each of four squares is at most v _∞^2.","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"At depth m, the finite space HH_p,m=ℓ^2(Z/p^mZ) has counting inner product and orthonormal basis e_a. Define U_p,me_a=p^m/21_B_m,a, V_m=span {1_B_m,a:0≤ a<p^m}. The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its imag","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a387bbfa91002daa","tex":"\\int_{\\mathcal L_D}","aliases":"∫_ L_D int mathcal","display":"\\int_{\\mathcal L_D}","kind":"use","excerpt":"regulated fine quantum BV theory already has compatible odd-symplectic and measure data and satisfies tfrac12(S_F,S_F)-iℏΔ_FS_F=0. Suppose the variables split into retained and detail sectors with Δ_F=Δ_C+Δ_D, and choose an admissible detail Lagrangian L_D. Define the coarse weight by e^iS_C/ℏ= N^-1∫_ L_De^iS_F/ℏ. When this integral is well defined","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ad06d6293cf51313","tex":"\\int_{\\mathbb T^3}","aliases":"∫_ T^3 int mathbb","display":"\\int_{\\mathbb T^3}","kind":"use","excerpt":"equipped with normalized Haar measure dx, so ∫_ T^31 dx=1. Integration of a periodic representative may equivalently be taken over [0,1)^3 with Lebesgue measure. A Fourier mode is n=(n_1,n_2,n_3)∈ Z^3, its momentum is p(n)=2π n, and its scalar plane wave is","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d0667d7e593026bd","tex":"\\int_{-2}^2","aliases":"∫_-2^2 int","display":"\\int_{-2}^2","kind":"use","excerpt":"The mean logarithmic expansion of this map is also exact: ∫_-2^2log|R_p'(x)| dν(x)=log p. Indeed for x=2cosθ, R_p'(x)=psin(pθ)/sinθ. Logarithms at its finitely many zeros are integrable, since they have the local form log|θ-θ_0| plus a bounded term. Hence the integral is log p plus the difference of the integrals of log|sin(pθ)| and logsinθ over [0","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ed738c29977ebe55","tex":"\\int_{-L}^L","aliases":"∫_-L^L int","display":"\\int_{-L}^L","kind":"use","excerpt":"N=sqrt1+2Φ/c_0^2. The static-frequency ratio in this approximation is fracν_rm obsν_rm emit =frac N_rm emit N_rm obs =1+fracΦ_rm emit-Φ_rm obsc_0^2 +O(fracΦ^2c_0^4). For Φ(x,b)=-GM_0/sqrtx^2+b^2, b>0, evaluating the first-order straight-ray integrals over [-L,L] gives Δ t_rm Shapiro · =-frac2c_0^3∫_-L^LΦ(x,b) dx =frac4GM_0c_0^3asinh(frac Lb), α(L,b","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","excerpt":"hart z∈ R^2n with constant skew matrix J. Let Φ=(Φ_1,…,Φ_2m) collect independent first-class constraints and regular gauge conditions, or more generally a regular second-class set. Assume these are smooth on a neighborhood of the whole segment between the two step endpoints. For a smooth autonomous function F, define the segment discrete gradient o","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2547054a0e887b21","tex":"\\int_0^{2\\pi}","aliases":"∫_0^2π int pi","display":"\\int_0^{2\\pi}","kind":"use","excerpt":"On [-2,2] define the probability measure dν(x)=fracdxπsqrt4-x^2. It is the distribution of 2cosθ for a uniform angle on the circle, or on [0,π]. It is also the limiting normalized spectral measure of the free cycles: for a continuous test function f, frac1N∑_k=0^N-1f(2cos(2π k/N)) ⟶frac12π∫_0^2πf(2cosθ) dθ. This is the Riemann-sum limit of a contin","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fa12689080284226","tex":"\\int_0^{2\\pi p}","aliases":"∫_0^2π p int pi","display":"\\int_0^{2\\pi p}","kind":"use","excerpt":"0, · pnmid m. In the first case m=0 is included, so K_p^*ϕ_0=ϕ_0. For a bounded measurable f, use the full circle parameterization: ∫ f(R_p(x)) dν(x) · =frac12π∫_0^2πf(2cos(pθ)) dθ · =frac12π p∫_0^2π pf(2cos u) du =∫ f(x) dν(x). The last equality uses periodicity. Applying this equality to |f|^2 proves the isometry, first on bounded functions and t","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ea9bb4c1b114de90","tex":"\\int_0^h","aliases":"∫_0^h int","display":"\\int_0^h","kind":"use","excerpt":"Connection-generated links and a quadratic mesh bound. The scalar budgets become useful when links arise from a specified background rather than independent phases. Consider real abelian connection components A_x,A_y,A_z on a Euclidean chart. Define exact coordinate-edge angles and forward links by θ_i(p,h)=∫_0^h A_i(p+t e_i) dt, u_i(p,h)=e^iθ_i(p,","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-75f238c1c506c49d","tex":"\\int_0^{h/2}","aliases":"∫_0^h/2 int","display":"\\int_0^{h/2}","kind":"use","excerpt":"Exact composition precedes every estimate. Continuity of A_i along its own coordinate lines permits interval-integral additivity and a change of variable: θ_i(p,h) · =∫_0^h/2A_i(p+t e_i) dt +∫_h/2^hA_i(p+t e_i) dt · =θ_i(p,h/2)+θ_i(p+(h/2)e_i,h/2). Exponentiation gives the ordered identity u_i(p+(h/2)e_i,h/2)u_i(p,h/2)=u_i(p,h). Thus the reconstruc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-051f75585a774ce8","tex":"\\int_0^r","aliases":"∫_0^r int","display":"\\int_0^r","kind":"use","excerpt":"Let G>0, let ρ(r) be a continuous spherical density, and impose regularity lim_rdownarrow0r^2Φ'(r)=0. The radial Poisson equation gives (r^2Φ')'=4π G r^2ρ, M_rm enc(r)=4π∫_0^rs^2ρ(s) ds, Φ'(r)=fracGM_rm enc(r)r^2. The acceleration is a_r=-Φ'. A positive density therefore gives an inward radial acceleration.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6122a3a715c464e3","tex":"\\int_0^{R_0}","aliases":"∫_0^R_0 int","display":"\\int_0^{R_0}","kind":"use","excerpt":"Putting u(r)=rΦ(r) removes the radial first derivative: r^2Φ'=ru'-u, (r^2Φ')'=ru'', u''=4π G rρ. For fixed endpoint values, this equation is the stationary condition of S[u]=∫_0^R_0[frac(u')^22G+4π rρ u]dr. On r_i=ih, i=0,…,M, with Mh=R_0, its finite counterpart is S_h[u]=∑_i=0^M-1frac(u_i+1-u_i)^22Gh +4π h∑_i=1^M-1r_iρ_i u_i.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a2972272b24e9c45","tex":"\\int_0^t","aliases":"∫_0^t int","display":"\\int_0^t","kind":"use","excerpt":"and normΣ_P(z) ≤frac|λ|^2normPBQ^2 dist(z,σ(QH_λ Q|_QHH)). The defect is λ BV, so the first identity is immediate. Self-adjointness of H_0 makes its invariant coarse range reducing. Hence Qe^-itH_0P=0, and Duhamel's identity gives Qe^-itH_λP =-iλ∫_0^t Qe^-i(t-s)H_λB e^-isH_0P ds. The off-diagonal assumption and reduction by H_0 give B e^-isH_0P=QBP","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-712f3452ad319777","tex":"\\int_0^u","aliases":"∫_0^u int","display":"\\int_0^u","kind":"use","excerpt":"Proof. First prove the scalar estimate without a sign restriction on h. The fundamental theorem of calculus gives |sin u|≤|u|, since sin u=∫_0^ucos v dv and |cos v|≤1. Therefore","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8bcd55715febb3c0","tex":"\\int_0^z","aliases":"∫_0^z int","display":"\\int_0^z","kind":"use","excerpt":"For z≥0, integrate this bound in sin z-z=∫_0^z(cos u-1) du to obtain |sin z-z|≤ z^3/6. Oddness of the residual proves |sin z-z|≤|z|^3/6 for negative z. For h≠0, substitute z=hp_j(n) and divide by |h|:","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4fb5af8a571997bd","tex":"\\int_0^{\\theta}","aliases":"∫_0^θ int theta","display":"\\int_0^{\\theta}","kind":"use","excerpt":"If a periodic λ exists, integrating λ_θ=M_G forces the zero-period condition. Conversely, fix τ_0 and define on the lifted strip λ(τ,θ)=λ_0 +∫_τ_0^τH_G(s,0) ds +∫_0^θM_G(τ,η) dη. Its θ derivative is M_G. Its τ derivative is H_G by the integrability identity and the fundamental theorem of calculus. The zero-period condition makes it periodic. The di","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"K(τ) · =B e^-iτ DB^*, τ≥0, together with reconstruction v(t)=e^-itDv_0-i∫_0^te^-i(t-s)DB^*u(s) ds. For Imz>0, the self-energy is its exact one-sided transform: Σ(z)=-i∫_0^∞ e^izτK(τ) dτ. Differentiate e^itDv(t) and integrate the hidden equation. This gives [source reference], whose substitution in the visible equation gives [source reference]. Conv","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6c39689bc04b0a10","tex":"\\int_A","aliases":"∫_A int","display":"\\int_A","kind":"use","excerpt":"For completeness, the union of these ranges is dense. Cylinders form a countable clopen base for the topology on Z_p and generate its Borel sigma-algebra. If f∈ L^2 is orthogonal to every cylinder indicator, the finite complex measure ν(A)=∫_A f dμ_p vanishes on cylinders and their finite unions. The class of sets on which ν is zero contains the wh","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-2f67664008c33cd2","tex":"\\int_{B_{m,a}}","aliases":"∫_B_m,a int","display":"\\int_{B_{m,a}}","kind":"use","excerpt":". The cylinders are disjoint. Consequently inner U_p,me_a U_p,me_b =p^mμ_p(B_m,a∩ B_m,b)=δ_ab. The same indicators span V_m, so U_p,m is unitary onto this subspace. For c=∑_a c_ae_a, its image has constant value p^m/2c_a on B_m,a. In particular, ∫_B_m,a| U_p,mc|^2 dμ_p=|c_a|^2. There is no extra probability factor after this normalization.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4fb956e840efa313","tex":"\\int_{C_a}","aliases":"∫_C_a int","display":"\\int_{C_a}","kind":"use","excerpt":"Cell values and normalized coefficients. For a measurable finite partition into cells of equal volume v>0, distinguish the physical constant value q_a∈C^4 on cell C_a from the normalized coefficient ψ_a=sqrt v q_a. The step interpolation is (Jψ)(x)=v^-1/2ψ_a (x∈ C_a), with equality understood almost everywhere. Direct integration gives normJψ_L^2^2","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-3ee4763e7e30f936","tex":"\\int_e","aliases":"∫_e int","display":"\\int_e","kind":"use","excerpt":"We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt alon","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-a3295146e088975d","tex":"\\int_{h/2}^{h}","aliases":"∫_h/2^h int","display":"\\int_{h/2}^{h}","kind":"use","excerpt":"Exact composition precedes every estimate. Continuity of A_i along its own coordinate lines permits interval-integral additivity and a change of variable: θ_i(p,h) · =∫_0^h/2A_i(p+t e_i) dt +∫_h/2^hA_i(p+t e_i) dt · =θ_i(p,h/2)+θ_i(p+(h/2)e_i,h/2). Exponentiation gives the ordered identity u_i(p+(h/2)e_i,h/2)u_i(p,h/2)=u_i(p,h). Thus the reconstruc","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fe9351791882eb76","tex":"\\int_I","aliases":"∫_I int","display":"\\int_I","kind":"use","excerpt":"Let P,Q be real smooth fields on I×T, with θ∈R/(2πZ) and γ(τ)=e^-2τ. Define S_G=frac12∫_I∫_0^2π [P_τ^2-γ P_θ^2 +e^2P(Q_τ^2-γ Q_θ^2)]dθ dτ. Time variations vanish at the endpoints of I and spatial variations are periodic. Integration by parts gives the two Euler--Lagrange expressions E_P · =P_ττ-γ P_θθ -e^2P(Q_τ^2-γ Q_θ^2),nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-964ee142a04858f9","tex":"\\int_{\\R}","aliases":"∫_R int R","display":null,"kind":"use","excerpt":"Let H_∞ be self-adjoint with spectral measure E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum o","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-939963fba5a39d6b","tex":"\\int_\\square","aliases":"∫_square int square","display":"\\int_\\square","kind":"use","excerpt":"We use the archive's plus-sign edge convention: for each oriented boundary edge define U_e=exp(+∫_eω). This specifies the signed transport observable used below. Multiplying in boundary order yields an exact identity W_square =exp(Koint_∂square(ϕ_xdt+ϕ_tdx)) =exp(frac K2∫_square e^2ϕR dt dx). To see the second equality directly, integrate ϕ_tt alon","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-fea858387f776702","tex":"\\int_{\\square}","aliases":"∫_square int square","display":"\\int_{\\square}","kind":"use","excerpt":"Suppose a smooth Abelian connection is sampled by exact line integrals, with a consistent loop orientation. Write its holonomy as W=e^iθ, where the small real phase is θ=-q∫_squareF_μν dS. Here q is the charge and F_μν is the corresponding oriented curvature component. Take rectangles with side lengths a_μ,a_ν=O(a) and uniformly controlled derivati","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d3b0bf5772538b33","tex":"\\int_x^{x+h}","aliases":"∫_x^x+h int","display":"\\int_x^{x+h}","kind":"use","excerpt":"The exact finite identities did not assume a smooth field in the background. A continuum comparison does. Let A be a real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains t","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-741b2176a1c35e3c","tex":"\\int_x^{x+s}","aliases":"∫_x^x+s int","display":"\\int_x^{x+s}","kind":"use","excerpt":"For fixed x, put f_x(s)=exp(-i∫_x^x+sA(r) dr)ψ(x+s). Then the two numerator terms in [source reference] are f_x(h) and f_x(-h). The derivative at zero is f_x'(0)=ψ'(x)-iA(x)ψ(x). Taylor expansion of f_x at zero to second order with bounded third derivative gives f_x(h)-f_x(-h)=2hf_x'(0)+O(h^3). Division by 2h proves the assertion. The derivative bo","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-421b279969e85688","tex":"\\int_{x_j}^{x_j+h}","aliases":"∫_x_j^x_j+h int","display":"\\int_{x_j}^{x_j+h}","kind":"use","excerpt":"real C^2 function near x, let ψ be C^3 there, and set U_h(x)=exp(-i∫_x^x+hA(s) ds), (D_h^Aψ)(x) =fracU_h(x)ψ(x+h)-U_h(x-h)^-1ψ(x-h)2h. The smooth A in this paragraph is a continuum coefficient. In the finite Hamiltonian one obtains the same link by taking A_j to be its edge average h^-1∫_x_j^x_j+hA(s) ds. This is an additional sampling prescription","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d68a718169ef9594","tex":"\\int_\\Sigma","aliases":"∫_Σ int Sigma","display":"\\int_\\Sigma","kind":"use","excerpt":"Π · =tfrac12φ_IJE^Iwedge E^J, σ_I=tfrac13!ϵ_IJKLE^Jwedge E^Kwedge E^L. Direct exterior multiplication gives the spatial symplectic potential Θ_Σ=∫_Σ[ · B_IJwedgeδΩ^IJ+Πwedgeδ A · -tfrac i2σ_I(barψγ^Iδψ -δbarψγ^Iψ)]. The minus sign in the spinor term is fixed by the time--space orientation and by moving dt past the spatial three-form. It must be ret","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-8165d9752a014553","tex":"\\int_{\\tau_0}^{\\tau}","aliases":"∫_τ_0^τ int tau tau","display":"\\int_{\\tau_0}^{\\tau}","kind":"use","excerpt":"If a periodic λ exists, integrating λ_θ=M_G forces the zero-period condition. Conversely, fix τ_0 and define on the lifted strip λ(τ,θ)=λ_0 +∫_τ_0^τH_G(s,0) ds +∫_0^θM_G(τ,η) dη. Its θ derivative is M_G. Its τ derivative is H_G by the integrability identity and the fundamental theorem of calculus. The zero-period condition makes it periodic. The di","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"The invariant curvature diagnostic is K=frac48M^2R^6-frac96MQ^2R^7 +frac56Q^4R^8. For M=1, Q=0.8, the radii are r_+=1.6, r_-=0.4, and R_rm turn=0.32. The benchmark interval R∈[0.6,6] crosses the outer horizon only. At its inner end K≃199.6646853, while K(r_+)≃1.106262207. High curvature and regular horizon-crossing coordinates coexist on this inter","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"In the odd-prime register t is a torsion coordinate and a is a principal phase coordinate. In the dyadic register s is a sign bit and a is again the principal phase. The Hilbert-space factorizations are therefore ℓ^2(U_p,h) · ≅ C^p-1_tors⊗ℓ^2(Z/p^hZ), · · p odd,nonumber ℓ^2(U_2,h) · ≅ C^2_sign⊗ℓ^2(Z/2^hZ). These are unitary changes of basis. They d","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"eterminant formula. Inverting the factors gives the stated upper left block. Alternatively, in (z-H)(u,v)^ T=(f,0)^ T, the lower equation gives v=C^-1B^*u and the upper equation becomes S(z)u=f. The same elimination with f=0 proves the eigenvector statement. If u=0 at such an E, invertibility of E-D forces v=0, so a nonzero full eigenvector indeed ","count":33},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"0 · 0 has compressed Hamiltonian zero but detail probability sin^2(|γ|t) from the first basis state. It fails exact refinement compatibility whenever γ≠0.","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Two tails and a finite core] Let V and W be linear isometries on a normed space. For any a and any approximation b, normVa-Wa≤ 2norma-b+normVb-Wb. Write the residual as V(a-b)+(Vb-Wb)+W(b-a). The triangle inequality bounds its norm by the sum of the three norms. Each outer term has norm norma-b because its operator is an isometry.","count":59},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"[Compact-time strong convergence] For every a∈KK, every T≥0, and every ε>0, there exists δ>0 such that 0<h<δ, |t|≤ T Longrightarrow normU_h(t)a-U(t)a<ε. Choose a finite S such that norma-P_Sa<ε/4. The finite-packet theorem gives a sufficiently small positive δ for which the core residual is less than ε/2 throughout the time interval. The two-tail i","count":20},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"The adjoint domain and self-adjointness. Modewise Hermiticity gives symmetry on the maximal domain. If a,b belong to that domain, the two series in innerwidehat H_mab =∑_ninnerH_na_nb_n =∑_ninnera_nH_nb_n =innerawidehat H_mb converge absolutely by Cauchy--Schwarz. This establishes the inclusion widehat H_m⊆widehat H_m^*, including the corresponding","count":3},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an addition","count":7},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"QH_fP · =0, PH_fP=VH_cV^*. They hold exactly when, on Ran P⊕Ran Q, H_f=(VH_cV^*)|_Ran P ⊕ H_Q, H_Q=(QH_fQ)|_Ran Q. The detail operator H_Q may be any self-adjoint operator on Ran Q.","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d315191ea29af671","tex":"\\bigoplus_{\\mathbf r}","aliases":"⊕_ r bigoplus mathbf","display":"\\bigoplus_{\\mathbf r}","kind":"use","excerpt":"[Self-adjoint packet blocks and compatible finite dynamics] For every r, let A_ r be a self-adjoint operator on the finite-dimensional space KK_ r=Ran F_ r. On HH_ P^∞=⊕_ rKK_ r define Dom(H_A) · ={f:∑_ r normA_ r F_ rf^2<∞}, nonumber","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-50db95a0004bcd02","tex":"\\bigoplus_a","aliases":"⊕_a bigoplus","display":"\\bigoplus_a","kind":"use","excerpt":"Choose Hilbert spaces KK_a and bounded maps V_a:HH→KK_a such that ∑_aV_a^*V_a=I. The combined analysis map V:HH→KK=⊕_aKK_a is Vψ=(V_aψ)_a. Its adjoint is synthesis, V^*(y_a)_a=∑_aV_a^*y_a. Consequently V^*V=I, R=VV^*, R^*=R^2=R. Here R projects onto the closed range of the isometry V. The range need not fill KK. If Π_a is the coordinate projector, ","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-ff559903fce4d2fd","tex":"\\bigoplus_{j=0}^{M-1}","aliases":"⊕_j=0^M-1 bigoplus","display":"\\bigoplus_{j=0}^{M-1}","kind":"use","excerpt":"e^-iθe_0, · r=p-1. H_p(θ)=μ I_p+κ(P_p(θ)+P_p(θ)^*). The transformed H is ⊕_j=0^M-1 H_p(θ_j). In particular, P_p(θ)^p=e^-iθI_p.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f3bc2dd9a6dd9759","tex":"\\bigoplus_{m=0}^{\\infty}","aliases":"⊕_m=0^∞ bigoplus infty","display":"\\bigoplus_{m=0}^{\\infty}","kind":"use","excerpt":"Consequently HH_p=⊕_m=0^∞Ran F_m, normf_2^2=∑_m=0^∞norm F_mf_2^2. The sum of the projections is I in the strong operator topology. The detail ranks are d_0=1 and d_m=(p-1)p^m-1 for m≥1, the differences of the dimensions p^m-p^m-1.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c1e4b6cbf2585e99","tex":"\\bigoplus_{n\\in S}","aliases":"⊕_n∈ S bigoplus in","display":"\\bigoplus_{n\\in S}","kind":"use","excerpt":"K_S=⊕_n∈ S C^4, a _ K_S^2=∑_n∈ S a_n _2^2.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-7e8554b16ebfef24","tex":"\\bigoplus_{r=0}^h","aliases":"⊕_r=0^h bigoplus","display":"\\bigoplus_{r=0}^h","kind":"use","excerpt":"The decomposition is exact at each finite depth: H_p,h=⊕_r=0^hRan D_r^(h), norm f^2=∑_r=0^hnormD_r^(h)f^2. A depth cutoff m≤ h retains E_m^(h)=∑_r=0^mD_r^(h). Its discarded norm is exactly norm(I-E_m^(h))f^2= ∑_r=m+1^hnormD_r^(h)f^2. No smoothness assumption is needed for this finite Parseval identity. A rate of decrease with m requires an addition","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6e1bd670f8830b0f","tex":"\\bigoplus_v","aliases":"⊕_v bigoplus","display":"\\bigoplus_v","kind":"use","excerpt":"for odd p. The finite arc Hilbert space is HH_arc=ℓ^2(vec E)⊗C^τ_p. For each vertex of degree d_v, define the normalized outgoing vector s_v=d_v^-1/2(1,…,1) and the Grover coin G_v=2|s_v⟩⟨ s_v|-I. Its square is I, because |s_v⟩⟨ s_v| is a projection. Take any local unitary R_v on the torsion or sign factor and put C=⊕_v(G_v⊗ R_v). The reversal shif","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-5aaaf24677ceb564","tex":"\\bigotimes","aliases":"⊗ bigotimes","display":"\\bigotimes","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-f7130473f45998ab","tex":"\\bigotimes_p","aliases":"⊗_p bigotimes","display":"\\bigotimes_p","kind":"use","excerpt":"Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-06a1b3c4dffd21b3","tex":"\\bigotimes_{p\\in\\mathcal P}","aliases":"⊗_p∈ P bigotimes in mathcal","display":"\\bigotimes_{p\\in\\mathcal P}","kind":"use","excerpt":"Let P be a finite prime set and let KK_tor be a finite-dimensional complex space for the retained torsion and dyadic sign registers. Set HH_ P^∞ =KK_tor⊗ ⊗_p∈ PL^2(Z_p,μ_p). The corresponding finite register embeds by I_KK_tor⊗⊗_p U_p,h_p. For a multi-depth h=(h_p), define E_ h =I_KK_tor⊗⊗_p E_p,h_p. Factorwise projection identities give E_ h E_ k ","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-658c3994137bf7f4","tex":"\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}","aliases":"⊗_p∈ P_ o bigotimes in mathcal mathrm","display":"\\bigotimes_{p\\in\\mathcal P_{\\mathrm o}}","kind":"use","excerpt":"Let P_ o be a finite set of odd primes and put M=2^h_2+2∏_p∈ P_ op^h_p+1. The Chinese remainder theorem gives a bijection from the unit addresses modulo M to their local unit addresses. Combining it with Proposition~[source reference] gives a permutation unitary U_CRT: ℓ^2((Z/MZ)^×) ⟶ C^2_sign⊗ℓ^2(Z/2^h_2Z) ⊗⊗_p∈ P_ o (C^p-1_tors⊗ℓ^2(Z/p^h_pZ)). Fo","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"Let H_ϵ^(M) be the finite restriction of H_ϵ to V_M. One useful approximation to U(t) is its embedded finite propagator, extended by zero on the omitted subspace: U_M^emb(t) =∑_m=0^Me^-itϵ_m F_m =U(t) E_M. It is unitary on V_M and has kernel V_M^⊥ as an operator on HH_p.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It does not say th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d42ab8c0624e1096","tex":"\\bigcap_j","aliases":"⋂_j bigcap","display":"\\bigcap_j","kind":"use","excerpt":"E. For real t define the scalar functions f_j(λ)=E_L,j(-itλ). The operator f_j(H_∞) has domain Dom(f_j(H_∞)) ={ψ:∫_R|f_j(λ)|^2 d⟨ψ,E(λ)ψ⟩<∞}. The scalar identity ∑_jf_j(λ)=e^-itλ gives ∑_jf_j(H_∞)ψ=e^-itH_∞ψ, ψ∈⋂_jDom(f_j(H_∞)). The equality holds on the stated common domain because it is a finite sum of spectral functions there. It does not say th","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"sets on which ν is zero contains the whole space, is closed under complements, and is closed under countable disjoint unions. The elementary monotone-class extension from the cylinder algebra therefore gives ν(A)=0 for every Borel set. Applying this to the positive and negative level sets of the real and imaginary parts of f yields f=0 almost every","count":2},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4d5ed9808720c1a8","tex":"\\bigcup_K","aliases":"⋃_K bigcup","display":"\\bigcup_K","kind":"use","excerpt":"for products On T^3=( R/2π Z)^3, let T_K={∑_ k _∞≤ K widehat f_k e^ik· x}. The algebraic relations are T_K T_L⊂ T_K+L, ∂_a T_K⊂ T_K. Thus T_K is a resolved space, while ⋃_K T_K is the product-closed differential algebra. Let S_N sample on an N^3 periodic grid, δ_a be its Fourier derivative, and τ_N=N^-3∑_x its normalized trace.","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-e8829e61e00377d3","tex":"\\bigcup_m","aliases":"⋃_m bigcup","display":"\\bigcup_m","kind":"use","excerpt":"sets on which ν is zero contains the whole space, is closed under complements, and is closed under countable disjoint unions. The elementary monotone-class extension from the cylinder algebra therefore gives ν(A)=0 for every Borel set. Applying this to the positive and negative level sets of the real and imaginary parts of f yields f=0 almost every","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"use","excerpt":"Variation of φ gives the unambiguous wedge equation Fwedge e^Iwedge e^J=φ^IJvol_e. Under the standard Lorentzian Hodge convention αwedge⋆β=⟨α,β⟩vol_e, this equation identifies φ with minus the internal components of ⋆ F. For example, F=E e^0wedge e^1 has ⋆ F=-E e^2wedge e^3, whereas the wedge equation gives φ^23=E. The inherited shorthand φ=⋆ F nee","count":6},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-d300cc10a142bb79","tex":"\\star^2","aliases":"⋆^2 star","display":"\\star^2","kind":"use","excerpt":"Reality is built into e and ω. For real Lorentz bivectors, the internal dual obeys ⋆^2=-id and the projectors P_±=tfrac12(id∓ i⋆) satisfy overlineP_+X=P_-X. Both chiral blocks are retained and related by conjugation. A real vector field with an existing unique flow preserves this real locus. This parent-level statement supplies a real classical sta","count":1},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"D_r^(h+1)V_p,h · =V_p,hD_r^(h) (0≤ r≤ h), · D_h+1^(h+1)V_p,h · =0. Repeated refinements compose as isometries. For w≥0 and one coarse address, V_p,h+w-1⋯ V_p,h|a⟩ =p^-w/2∑_ d∈{0,…,p-1}^w|a, d⟩, with the empty product interpreted as the identity. The factor p^-w/2 is forced by the isometry and orthonormality of the descendant addresses.","count":9},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-6d63ab7a8797f0ed","tex":"\\rangle_h","aliases":"⟩_h rangle","display":"\\rangle_h","kind":"use","excerpt":"dotπ_ij · =- G_ij(h) +δ_iδ_jn-δ_ijδ^2n. The kinetic gradient is 2π_ij-δ_ijπ_rm tr. Lemma~[source reference] makes the potential gradient G(h). The lapse gradient follows by two spatial integrations by parts. Integrating the shift term once gives -⟨δ_in_j+δ_jn_i,π_ij⟩_h in the Lagrangian, fixing the sign in the first equation.","count":5},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-92fa129423d0a85c","tex":"\\rangle_{L^2}","aliases":"⟩_L^2 rangle","display":"\\rangle_{L^2}","kind":"use","excerpt":"⟨ f,g⟩_L^2 =∫_ T^3⟨ f(x),g(x)⟩ dx","count":4},{"work":"monographs-physics","url":"notation/monographs-physics.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[Finite unit coordinates] For odd p, choose a unit ζ_p,h of order p-1 whose reduction generates (Z/pZ)^×. The maps (t,a) · ⟼ζ_p,h^ t(1+p)^a pmodp^h+1, · (t,a) · ∈Z/(p-1)Z×Z/p^hZ, (s,a) · ⟼(-1)^s5^apmod2^h+2, · (s,a) · ∈Z/2Z×Z/2^hZ are group isomorphisms onto U_p,h and U_2,h respectively.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"notation-table","excerpt":"Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"notation-table","excerpt":"[p_-,p_+] · A probability enclosure. Acceptance at relative tolerance ε requires p_->0 and p_+/p_- -1≤ε.","count":120},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).","count":411},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"Gram representation · The operator AA^top or its dual A^top A. Their positive eigenvalues agree with multiplicity.","count":133},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a53ead434f8754af","tex":"A^\\top","aliases":"A^top top","display":"A^\\top","kind":"notation-table","excerpt":"Gram representation · The operator AA^top or its dual A^top A. Their positive eigenvalues agree with multiplicity.","count":46},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-08eb65adb1a856b8","tex":"A_b","aliases":"A_b","display":"A_b","kind":"notation-table","excerpt":"A, A_b · A real factor and its resident column blocks.","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"notation-table","excerpt":"A, A_b · A real factor and its resident column blocks.","count":125},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":18},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"notation-table","excerpt":"[d_-,d_+] · A base-ten logarithmic probability enclosure.","count":46},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5034448c12e61aca","tex":"d_+","aliases":"d_+","display":"d_+","kind":"notation-table","excerpt":"[d_-,d_+] · A base-ten logarithmic probability enclosure.","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4d0cda83a107b262","tex":"d_-","aliases":"d_-","display":"d_-","kind":"notation-table","excerpt":"[d_-,d_+] · A base-ten logarithmic probability enclosure.","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"notation-table","excerpt":"κ(s) · The cumulant generating function, log E e^sQ.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"notation-table","excerpt":"κ(s) · The cumulant generating function, log E e^sQ.","count":80},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2c268d8002e5f66f","tex":"e^{sQ}","aliases":"e^sQ","display":"e^{sQ}","kind":"notation-table","excerpt":"κ(s) · The cumulant generating function, log E e^sQ.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"notation-table","excerpt":"CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).","count":58},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"notation-table","excerpt":"H, K · Symmetric operators, with sample Gram K=AA^top and feature Gram H=A^top A in the streaming construction.","count":101},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"notation-table","excerpt":"P_h · Orthogonal structural phase projectors.","count":53},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"notation-table","excerpt":"Noncentrality · The squared mean displacement in a standardized Gaussian direction. The symbol δ_i in a noncentral chi-square component differs from a centered packet deviation.","count":57},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"notation-table","excerpt":"u_k, J_k · Frequency kπ/b and its exponentially weighted payoff integral.","count":27},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"notation-table","excerpt":"Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.","count":74},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b78dc9dea4a072f6","tex":"J_k","aliases":"J_k","display":"J_k","kind":"notation-table","excerpt":"u_k, J_k · Frequency kπ/b and its exponentially weighted payoff integral.","count":18},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","excerpt":"H, K · Symmetric operators, with sample Gram K=AA^top and feature Gram H=A^top A in the streaming construction.","count":32},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":122},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"notation-table","excerpt":"κ(s) · The cumulant generating function, log E e^sQ.","count":41},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"notation-table","excerpt":"Σ, M, μ · Covariance, quadratic weight, and Gaussian mean.","count":72},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"notation-table","excerpt":"M_P,m · The mth centered power sum of a spectral packet P.","count":39},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f655713b08177b3d","tex":"M_{P,m}","aliases":"M_P,m","display":"M_{P,m}","kind":"notation-table","excerpt":"M_P,m · The mth centered power sum of a spectral packet P.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"notation-table","excerpt":"N · Retained COS mode count. The implementation argument is K.","count":68},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"notation-table","excerpt":"midrule A · A factor. In the streaming construction, A∈ R^n× m.","count":45},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":12},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"P_h · Orthogonal structural phase projectors.","count":94},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":82},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8c58117cdec5bfd5","tex":"p_+","aliases":"p_+","display":"p_+","kind":"notation-table","excerpt":"[p_-,p_+] · A probability enclosure. Acceptance at relative tolerance ε requires p_->0 and p_+/p_- -1≤ε.","count":10},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-cc8cf6fb3fe21ade","tex":"p_-","aliases":"p_-","display":"p_-","kind":"notation-table","excerpt":"[p_-,p_+] · A probability enclosure. Acceptance at relative tolerance ε requires p_->0 and p_+/p_- -1≤ε.","count":12},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ea63928309796eee","tex":"P_h","aliases":"P_h","display":"P_h","kind":"notation-table","excerpt":"P_h · Orthogonal structural phase projectors.","count":29},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"notation-table","excerpt":"CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).","count":107},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"notation-table","excerpt":"CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).","count":106},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"notation-table","excerpt":"midrule A · A factor. In the streaming construction, A∈ R^n× m.","count":14},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-772c490b66e2c0ea","tex":"\\mathbb R^{n\\times m}","aliases":"R^n× m mathbb times","display":"\\mathbb R^{n\\times m}","kind":"notation-table","excerpt":"midrule A · A factor. In the streaming construction, A∈ R^n× m.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":26},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":43},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"notation-table","excerpt":"ρ_P(s) · The packet's complex-grid convergence radius parameter.","count":96},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"notation-table","excerpt":"Msucceq0 · Positive semidefiniteness.","count":8},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":65},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"notation-table","excerpt":"Gram representation · The operator AA^top or its dual A^top A. Their positive eigenvalues agree with multiplicity.","count":114},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":45},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-836830fc78b794aa","tex":"u_k","aliases":"u_k","display":"u_k","kind":"notation-table","excerpt":"u_k, J_k · Frequency kπ/b and its exponentially weighted payoff integral.","count":20},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"notation-table","excerpt":"· _2 · Euclidean vector norm or induced matrix norm, as appropriate.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"notation-table","excerpt":"midrule A · A factor. In the streaming construction, A∈ R^n× m.","count":139},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"notation-table","excerpt":"Noncentrality · The squared mean displacement in a standardized Gaussian direction. The symbol δ_i in a noncentral chi-square component differs from a centered packet deviation.","count":17},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a104cc9eebe89629","tex":"\\delta_i","aliases":"δ_i delta","display":"\\delta_i","kind":"notation-table","excerpt":"Noncentrality · The squared mean displacement in a standardized Gaussian direction. The symbol δ_i in a noncentral chi-square component differs from a centered packet deviation.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":22},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-67ec06abdf0c36fe","tex":"\\varepsilon_{\\rm COS}","aliases":"ε_rm COS varepsilon rm","display":"\\varepsilon_{\\rm COS}","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f710bdaccce7c876","tex":"\\varepsilon_{\\rm pkt}","aliases":"ε_rm pkt varepsilon rm","display":"\\varepsilon_{\\rm pkt}","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e2dfd2f078f25eb4","tex":"\\varepsilon_{\\rm sup}","aliases":"ε_rm sup varepsilon rm","display":"\\varepsilon_{\\rm sup}","kind":"notation-table","excerpt":"ε_rm pkt, ε_rm sup, ε_rm COS · Packet, support/folding, and omitted-mode contributions to the analytic budget.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"notation-table","excerpt":"θ, η · Admissible exponential tilts, with η>θ when used in a Chernoff support bound.","count":18},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"notation-table","excerpt":"Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.","count":107},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c45faae0260eabc9","tex":"\\theta_*","aliases":"θ_* theta","display":"\\theta_*","kind":"notation-table","excerpt":"Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"notation-table","excerpt":"Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.","count":49},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-14c664a95943803d","tex":"\\kappa'","aliases":"κ' kappa","display":"\\kappa'","kind":"notation-table","excerpt":"Saddlepoint tilt · An admissible exponential tilt satisfying κ'(θ_*)=q. It improves centering, but exact solution of that equation is not required for the directed enclosure theorem.","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"notation-table","excerpt":"Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.","count":74},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"notation-table","excerpt":"Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.","count":37},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ca5171acdedb719e","tex":"\\lambda_j^2","aliases":"λ_j^2 lambda","display":"\\lambda_j^2","kind":"notation-table","excerpt":"Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"notation-table","excerpt":"Σ, M, μ · Covariance, quadratic weight, and Gaussian mean.","count":25},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"notation-table","excerpt":"u_k, J_k · Frequency kπ/b and its exponentially weighted payoff integral.","count":19},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"notation-table","excerpt":"ρ_P(s) · The packet's complex-grid convergence radius parameter.","count":55},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2899f3cda75e636c","tex":"\\rho_P","aliases":"ρ_P rho","display":"\\rho_P","kind":"notation-table","excerpt":"ρ_P(s) · The packet's complex-grid convergence radius parameter.","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"notation-table","excerpt":"Σ, M, μ · Covariance, quadratic weight, and Gaussian mean.","count":14},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"notation-table","excerpt":"midrule A · A factor. In the streaming construction, A∈ R^n× m.","count":56},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"notation-table","excerpt":"Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.","count":97},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"notation-table","excerpt":"Effective rank · The participation ratio (∑_jλ_j)^2/∑_jλ_j^2 for a nonzero positive spectrum. It need not be an integer.","count":13},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"notation-table","excerpt":"CDF · Cumulative distribution function, F(q)=Pr(Q≤ q).","count":103},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"notation-table","excerpt":"q · The threshold in P(Q≥ q).","count":68},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-54034542483dca3c","tex":"A^{\\mathsf T}","aliases":"A^ T mathsf","display":"A^{\\mathsf T}","kind":"introduction-passage","excerpt":"The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":57},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":37},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-401361282589c1a2","tex":"\\mathcal B_{\\mathcal O}","aliases":"B_ O mathcal mathcal","display":"\\mathcal B_{\\mathcal O}","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0bb7ba7e58f63875","tex":"\\mathcal B_{gh}","aliases":"B_gh mathcal","display":"\\mathcal B_{gh}","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-557f88fb6cda082f","tex":"B_h","aliases":"B_h","display":"B_h","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-12cb07ad60774339","tex":"\\mathcal B_{hg}","aliases":"B_hg mathcal","display":"\\mathcal B_{hg}","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":41},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"introduction-passage","excerpt":"Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fb7427a95ad8c5b4","tex":"\\E","aliases":"E operatorname mathbb","display":"{\\operatorname{\\mathbb E}}","kind":"introduction-passage","excerpt":"Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j ","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0a5f8b3570c6f2e9","tex":"e^{(-\\theta+\\ii u_k)(b-q)}","aliases":"e^(-θ+ii u_k)(b-q) e^(-θ+i u_k)(b-q) theta ii theta mathrm","display":"e^{(-\\theta+{\\mathrm{i}} u_k)(b-q)}","kind":"introduction-passage","excerpt":"the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denot","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4c04a96c211fc8b4","tex":"e^{-\\theta(x-q)}","aliases":"e^-θ(x-q) theta","display":"e^{-\\theta(x-q)}","kind":"introduction-passage","excerpt":"Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-36e24b50b457c3cb","tex":"e^{\\ii u_k q}","aliases":"e^ii u_k q e^i u_k q ii mathrm","display":"e^{{\\mathrm{i}} u_k q}","kind":"introduction-passage","excerpt":"the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denot","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-937e4073756141af","tex":"\\operatorname{eig}","aliases":"eig operatorname","display":"\\operatorname{eig}","kind":"introduction-passage","excerpt":"The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":32},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-88b326798e7011cf","tex":"\\ii","aliases":"ii i mathrm","display":"{\\mathrm{i}}","kind":"introduction-passage","excerpt":"the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-θ+i u_k)(b-q)θ-i u_k ]. The prime on a cosine sum denot","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9cde3c700d58d43c","tex":"J_{h1}","aliases":"J_h1","display":"J_{h1}","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3f2d06bce4bd4ff2","tex":"J_{hB_h}","aliases":"J_hB_h","display":"J_{hB_h}","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8a398414f1370038","tex":"\\mathrm{MAF}","aliases":"MAF mathrm","display":"\\mathrm{MAF}","kind":"introduction-passage","excerpt":"The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2974a0f77c94c6dd","tex":"\\mathrm{MAF}_j","aliases":"MAF_j mathrm","display":"\\mathrm{MAF}_j","kind":"introduction-passage","excerpt":"The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":26},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9cc66676daa83a18","tex":"\\max_j","aliases":"max_j max","display":"\\max_j","kind":"introduction-passage","excerpt":"Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d9d706ee857ada00","tex":"\\max_{x_e\\in\\{0,1\\}}","aliases":"max_x_e∈{0,1} max in","display":"\\max_{x_e\\in\\{0,1\\}}","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":16},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ecdf2ff2033f302f","tex":"\\mathcal P_1","aliases":"P_1 mathcal","display":"\\mathcal P_1","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-45c4f17ed76d05ef","tex":"P_g","aliases":"P_g","display":"P_g","kind":"introduction-passage","excerpt":"Storage refinement changes only the order and grouping of finite sums. By partition refinement invariance, the streamed products reconstruct the same A, A^top, and therefore the same H=A^top A. Since each P_h is fixed by the structural phase partition, every block P_hHP_g is unchanged. The phase graph is defined only by these blocks, so its path st","count":22},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b9ca4be0e9bd233a","tex":"\\mathcal P_H","aliases":"P_H mathcal","display":"\\mathcal P_H","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5627e8b6e58a3075","tex":"\\mathcal P_h","aliases":"P_h mathcal","display":"\\mathcal P_h","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j ","count":86},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ff81ba5e23d549c7","tex":"\\Re","aliases":"Re","display":"\\Re","kind":"introduction-passage","excerpt":"Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j ","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"introduction-passage","excerpt":"The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(","count":16},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"introduction-passage","excerpt":"The second experiment used the existing Komorebi SKAT benchmark generator. Synthetic genotype matrices were centered and scaled, then residualized against three covariates. The kernel weights were w_j=(1-MAF_j)^24, where MAF denotes minor-allele frequency. The factor used column scales sqrtw_j. Its positive quadratic-form spectrum was λ=tfrac12eig(","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-929c170d8af8ade4","tex":"\\widetilde G","aliases":"widetilde G widetilde","display":"\\widetilde G","kind":"introduction-passage","excerpt":"where widetilde G_b denotes the centered and scaled block. The full weighted factor is never materialized.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6afdd25c304ef00d","tex":"\\widetilde G_b","aliases":"widetilde G_b widetilde","display":"\\widetilde G_b","kind":"introduction-passage","excerpt":"where widetilde G_b denotes the centered and scaled block. The full weighted factor is never materialized.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":34},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c69d923bf6621481","tex":"x_e","aliases":"x_e","display":"x_e","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"introduction-passage","excerpt":"Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j ","count":27},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-10845de283abdc8a","tex":"Z_j","aliases":"Z_j","display":"Z_j","kind":"introduction-passage","excerpt":"Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j ","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2166d574ac2ce471","tex":"Z_j^2","aliases":"Z_j^2","display":"Z_j^2","kind":"introduction-passage","excerpt":"Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j ","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":23},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2442980e628749af","tex":"\\beta_e^{(k)}","aliases":"β_e^(k) beta","display":"\\beta_e^{(k)}","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c5b085f18c150ac1","tex":"\\Delta_e","aliases":"Δ_e Delta","display":"\\Delta_e","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b52e03989ef1b9f5","tex":"\\kappa^{(m)}","aliases":"κ^(m) kappa","display":"\\kappa^{(m)}","kind":"introduction-passage","excerpt":"log(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose argument can wrap by 2π. For real θ<1/(2λ_max), κ^(m)(θ) =2^m-1(m-1)!∑_j=1^r fracλ_j^m(1-2θλ_j)^","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1a0667313603a9f1","tex":"\\lambda_j^m","aliases":"λ_j^m lambda","display":"\\lambda_j^m","kind":"introduction-passage","excerpt":"log(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j has positive real part on this domain, so its logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose argument can wrap by 2π. For real θ<1/(2λ_max), κ^(m)(θ) =2^m-1(m-1)!∑_j=1^r fracλ_j^m(1-2θλ_j)^","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-22452671e06f3b7f","tex":"\\lambda_{\\max}","aliases":"λ_max lambda max","display":"\\lambda_{\\max}","kind":"introduction-passage","excerpt":"Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j ","count":12},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":23},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-","count":13},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ec1616068a6ee172","tex":"\\sum_e","aliases":"∑_e sum","display":"\\sum_e","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c373be36458d8e51","tex":"\\sum_{e\\in\\mathcal O}","aliases":"∑_e∈ O sum in mathcal","display":"\\sum_{e\\in\\mathcal O}","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6ce2ba87b3975aab","tex":"\\sum_{j=1}^{r}","aliases":"∑_j=1^r sum","display":"\\sum_{j=1}^{r}","kind":"introduction-passage","excerpt":"Let Q=∑_j=1^rλ_j Z_j^2, λ_j>0, r≥1. The variables Z_j are independent standard normal variables. The moment generating function exists for Re s < frac12λ_max, λ_max=max_jλ_j, and its cumulant generating function is κ(s)=log E e^sQ =-frac12∑_j=1^rlog(1-2sλ_j). For complex s, κ denotes the analytic logarithm normalized by κ(0)=0. Each factor 1-2sλ_j ","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"introduction-passage","excerpt":"Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1a19201cb1effb34","tex":"\\int_q^b","aliases":"∫_q^b int","display":"\\int_q^b","kind":"introduction-passage","excerpt":"Because Q≥0, K-COS uses the interval [0,b]. Throughout the following functional and packet-error results, assume b>q≥0, b>0, 0<θ<1/(2λ_max), and an integer mode count N≥1. Let u_k=frackπb, k=0,1,…, and define the exponentially weighted cosine integrals J_k(q,b,θ) =∫_q^b e^-θ(x-q)cos(u_kx) dx. They admit the closed form J_k =Re[ e^i u_k q frac1-e^(-","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned o","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-68ba6a2632bf9644","tex":"\\mathbf1_{[q,b]}","aliases":"1_[q,b] mathbf","display":"\\mathbf1_{[q,b]}","kind":"use","excerpt":"[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3ccd0dbb76c629fe","tex":"\\mathbf1_{\\{Q\\ge q\\}}","aliases":"1_{Q≥ q} mathbf ge","display":"\\mathbf1_{\\{Q\\ge q\\}}","kind":"use","excerpt":"For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"The numerical interface implements this reduction directly. In an isotropic control, Σ=I, M=aI, a>0, the law collapses to the exact scaled noncentral chi-square distribution Q/a∼χ^2_d( μ ^2).","count":23},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2533f3ff34e393b7","tex":"A'_{ik}","aliases":"A'_ik","display":"A'_{ik}","kind":"use","excerpt":"The reindexing theorem declgramEntry_reindex handles a bijection e:V≃ V. It moves the block to e(s) and transforms the factor to A'_ik=A_i,e^-1(k). The resulting Gram entry is unchanged. Permuting only the coordinates without moving their factor columns falls outside this statement.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8f914025617b7ebc","tex":"A^T","aliases":"A^T","display":"A^T","kind":"use","excerpt":"Let A∈ R^n× m be supplied as an IEEE binary64 array. Each stored entry is treated as its exact dyadic rational value. The target operator is therefore the exact Gram matrix H=AA^T defined by those binary64 values, not an unspecified nearby real matrix.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"[Representation equivalence] The same central quadratic-form law may be supplied by any of the following: item a nonnegative spectrum {λ_j}. item a positive semidefinite matrix H. item a factor A with H=AA^top or H=A^top A. item a replayable streamed factor A=[A_1 ⋯ A_B]. The nonzero spectra of AA^top and A^top A are identical, and AA^top=∑_bA_bA_b","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6d63899ef719c388","tex":"A_1^\\top","aliases":"A_1^top top","display":"A_1^\\top","kind":"use","excerpt":"Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2e0e0ad2200e7da2","tex":"A_2^\\top","aliases":"A_2^top top","display":"A_2^\\top","kind":"use","excerpt":"The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0 0 · 1 · 0 · 1 1 · 0 · 1 · 0 0 · 1 · 0 · 1, A_2A_2^top=A_3A_3^top=frac12 1 ·","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d325da4f9b58928d","tex":"A_3","aliases":"A_3","display":"A_3","kind":"use","excerpt":"The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6d6e8c6dd3fb39c6","tex":"A_3^\\top","aliases":"A_3^top top","display":"A_3^\\top","kind":"use","excerpt":"The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0 0 · 1 · 0 · 1 1 · 0 · 1 · 0 0 · 1 · 0 · 1, A_2A_2^top=A_3A_3^top=frac12 1 ·","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6c1908f93f8dee44","tex":"A_B","aliases":"A_B","display":"A_B","kind":"use","excerpt":"[Representation equivalence] The same central quadratic-form law may be supplied by any of the following: item a nonnegative spectrum {λ_j}. item a positive semidefinite matrix H. item a factor A with H=AA^top or H=A^top A. item a replayable streamed factor A=[A_1 ⋯ A_B]. The nonzero spectra of AA^top and A^top A are identical, and AA^top=∑_bA_bA_b","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d7439f4f816d2a74","tex":"A_{b,I_r}","aliases":"A_b,I_r","display":"A_{b,I_r}","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-938e7d9a4edb4554","tex":"A_{b,I_s}^\\top","aliases":"A_b,I_s^top top","display":"A_{b,I_s}^\\top","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-875f61e7844f5048","tex":"A_b^{\\mathsf T}","aliases":"A_b^ T mathsf","display":"A_b^{\\mathsf T}","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]figures_production/figure_02_exact_block_kernel_assembly.pdf captionEach resident column block contributes a complete sample-space Gram summand A_bA_b^ T. Adding the summands recovers AA^ T exactly. Storage partitioning changes residence and evaluation order, not the represented operator.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-08e363ae4221d772","tex":"A_B^\\top","aliases":"A_B^top top","display":"A_B^\\top","kind":"use","excerpt":"Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7347e27bcb45136f","tex":"A_b^\\top","aliases":"A_b^top top","display":"A_b^\\top","kind":"use","excerpt":"The same central quadratic-form law may be supplied by any of the following: item a nonnegative spectrum {λ_j}. item a positive semidefinite matrix H. item a factor A with H=AA^top or H=A^top A. item a replayable streamed factor A=[A_1 ⋯ A_B]. The nonzero spectra of AA^top and A^top A are identical, and AA^top=∑_bA_bA_b^top.","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5a8e57a002dc32f4","tex":"a_d","aliases":"a_d","display":"a_d","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c2e6b50202ef016e","tex":"A_F","aliases":"A_F","display":"A_F","kind":"use","excerpt":"starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant spa","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8e8058fd0271b377","tex":"A_F^\\top","aliases":"A_F^top top","display":"A_F^\\top","kind":"use","excerpt":"^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ed36bb76a6fb5d82","tex":"A_{F_k}","aliases":"A_F_k","display":"A_{F_k}","kind":"use","excerpt":"ghts, and structural phase metadata fixed, and assume all factor-action blocks required for the retained phase operator are available. At depth k, let F_k be the graph k-neighborhood of F_0. Then no storage block lying wholly in a phase outside F_k can affect the k-step Krylov vector. Such a block need not be loaded when evaluating the requested po","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-313e432c8f4d51b0","tex":"A_{i,e^{-1}(k)}","aliases":"A_i,e^-1(k)","display":"A_{i,e^{-1}(k)}","kind":"use","excerpt":"The reindexing theorem declgramEntry_reindex handles a bijection e:V≃ V. It moves the block to e(s) and transforms the factor to A'_ik=A_i,e^-1(k). The resulting Gram entry is unchanged. Permuting only the coordinates without moving their factor columns falls outside this statement.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1dfeb34401714b09","tex":"A_{i1}","aliases":"A_i1","display":"A_{i1}","kind":"use","excerpt":"If the two blocks overlap in one column, their sum repeats that column's outer product. For s=t={1} and A_i1=3, the union has contribution 9 while the sum of the two blocks is 18. This is why declgramEntry_union requires disjointness.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9816586d612c1e9a","tex":"A_{ik}","aliases":"A_ik","display":"A_{ik}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-410901667e22470e","tex":"A_J","aliases":"A_J","display":"A_J","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"For the declared vector v=(1,2,-1,0)^top, the action can be evaluated without forming any block Gram matrix: ∑_j=1^3 A_j(A_j^top v)=Kv=(2,3,-2,-1)^top. The symbolic check confirms both storage partitions. Every column remains represented. Dropping an operator coupling instead changes the operator and requires the error analysis developed next.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":10},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-88149fe9acdfcaf6","tex":"A_{J'}","aliases":"A_J'","display":"A_{J'}","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-33fae3e04a3d5f9f","tex":"A_{J'}^\\top","aliases":"A_J'^top top","display":"A_{J'}^\\top","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c795a5c2756e5e45","tex":"a_j^{-1/2}","aliases":"a_j^-1/2","display":"a_j^{-1/2}","kind":"use","excerpt":"Since 1+(a_jt)^2≥(a_jt)^2, we have |ϕ_θ(t)| ≤ C_θ t^-r/2, C_θ=∏_j a_j^-1/2.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9227a949b8e25080","tex":"A_J^\\top","aliases":"A_J^top top","display":"A_J^\\top","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-658802b5970c9fc3","tex":"A_j^\\top","aliases":"A_j^top top","display":"A_j^\\top","kind":"use","excerpt":"For the declared vector v=(1,2,-1,0)^top, the action can be evaluated without forming any block Gram matrix: ∑_j=1^3 A_j(A_j^top v)=Kv=(2,3,-2,-1)^top. The symbolic check confirms both storage partitions. Every column remains represented. Dropping an operator coupling instead changes the operator and requires the error analysis developed next.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e7da8393928e682c","tex":"A_{jk}","aliases":"A_jk","display":"A_{jk}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d3261090d13587bd","tex":"\\mathrm{an}","aliases":"an mathrm","display":"\\mathrm{an}","kind":"use","excerpt":"Let p= P(Q≥ q), and let widehat p be the finite K-COS approximation formed with [label=(roman*)] item an upper interval endpoint b>q, item an integer N≥1 of cosine modes, item a packet approximation satisfying [source reference] on the finite grid. Under the positive-spectrum and parameter hypotheses of Proposition [source reference], |p-widehat p|","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-16aff9ea611eb515","tex":"\\arctan","aliases":"arctan","display":"\\arctan","kind":"use","excerpt":"The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponenti","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8bed5507d096707b","tex":"B^{\\mathsf T}","aliases":"B^ T mathsf","display":"B^{\\mathsf T}","kind":"use","excerpt":"A=UB+F, U^ TF=0, A^ TA=B^ TB+F^ TF. t=trR= F _F^2, v=tr(R^2), ρ_R= R _2. Let U contain d orthonormal sample-space directions. Project the factor into its captured range coordinates B=U^ TA and residual F=(I-UU^ T)A. Orthogonality eliminates the cross terms in the feature-space Gram matrix.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-368d1624fb84e4fe","tex":"\\mathcal B^j","aliases":"B^j mathcal","display":"\\mathcal B^j","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-280215750dd88be1","tex":"\\mathcal B^k","aliases":"B^k mathcal","display":"\\mathcal B^k","kind":"use","excerpt":"[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).","count":15},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-df15704d2f586b97","tex":"\\mathcal B^{k-1-j}","aliases":"B^k-1-j mathcal","display":"\\mathcal B^{k-1-j}","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8b99783075cb661f","tex":"B^\\top","aliases":"B^top top","display":"B^\\top","kind":"use","excerpt":"[Exact range core] The nonzero eigenvalues of K=AA^top are exactly the nonzero eigenvalues of BB^top, with multiplicity. If range(Q_r)=range(A), then BB^top is positive definite on the core space and its entire spectrum is the nonzero spectrum of K.","count":8},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2da2815025407827","tex":"\\mathcal B_{\\mathcal O}^j","aliases":"B_ O^j mathcal mathcal","display":"\\mathcal B_{\\mathcal O}^j","kind":"use","excerpt":"The first inequality in the second display is Theorem~[source reference]. For the transfer matrices, the exact noncommutative telescoping identity is B^k- B_ O^k = ∑_j=0^k-1 B^k-1-j ( B- B_ O) B_ O^j. All matrices and ρ_0 are nonnegative, and 0≤ B_ O≤ B entrywise. Hence B_ O^jρ_0 ≤ B^jρ_0. Substituting B- B_ O = ∑_e∈ OΔ_e and distributing the sum g","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b7349162311a072a","tex":"\\mathcal B_{\\mathcal O}^k","aliases":"B_ O^k mathcal mathcal","display":"\\mathcal B_{\\mathcal O}^k","kind":"use","excerpt":"[Edge-additive leakage majorant] For every omitted set O⊆ E, 0≤ ( B^k- B_ O^k)ρ_0 ≤ ∑_e∈ Oq_e^(k) componentwise. Consequently, lVert(H^k-H_ O^k)XrVert_F ≤ lVert ( B^k- B_ O^k)ρ_0 rVert_2 ≤ ∑_e∈ Oβ_e^(k).","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-56e5c919eaefa876","tex":"\\mathcal B_0","aliases":"B_0 mathcal","display":"\\mathcal B_0","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0, 0, · otherwise, R= B- B_0. Symmetric pair retention preserves self-adjointness of H_0, although the following error theorem does not require positive semidefiniteness of H_0.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5e42e638d2a47b03","tex":"\\mathcal B_0^j","aliases":"B_0^j mathcal","display":"\\mathcal B_0^j","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d307cf0f04802c95","tex":"\\mathcal B_0^k","aliases":"B_0^k mathcal","display":"\\mathcal B_0^k","kind":"use","excerpt":"[Thresholded-power certificate] For every k≥1, ρ((H^k-H_0^k)X) ≤ ( B^k- B_0^k)ρ(X). In particular, lVert(H^k-H_0^k)XrVert_F ≤ lVert( B^k- B_0^k)ρ(X)rVert_2.","count":8},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b2f9008f03849cc5","tex":"B_k","aliases":"B_k","display":"B_k","kind":"use","excerpt":"Let widehatκ(s_k) be the packet approximation at s_k=θ+i u_k, and suppose |widehatκ(s_k)-κ(s_k)|≤ B_k. For disjoint packets, B_k can be chosen as the sum of the packet bounds [source reference].","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8d19b306c01e6077","tex":"\\bar G","aliases":"bar G bar","display":"\\bar G","kind":"use","excerpt":"G_b · ⟶ G_b-bar G_b ⟶ widetilde G_b, widetilde G_b · ⟶ M_Xwidetilde G_b ⟶ A_b=M_Xwidetilde G_bW_b^1/2 ⟶ A_bA_b^top,","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d526da74c961bf64","tex":"\\bar G_b","aliases":"bar G_b bar","display":"\\bar G_b","kind":"use","excerpt":"G_b · ⟶ G_b-bar G_b ⟶ widetilde G_b, widetilde G_b · ⟶ M_Xwidetilde G_b ⟶ A_b=M_Xwidetilde G_bW_b^1/2 ⟶ A_bA_b^top,","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathbb","display":"{\\mathbb C}","kind":"use","excerpt":"[Centered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"Let E=H-H_0 be the self-adjoint error introduced by phase pruning. Using the same orthogonal phase projectors P_h, define C_hg = P_hEP_g _2.","count":8},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a1a3e5ac585d7cf0","tex":"\\mathcal C_{\\mathcal O}","aliases":"C_ O mathcal mathcal","display":"\\mathcal C_{\\mathcal O}","kind":"use","excerpt":"An execution schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c609f244935cbee4","tex":"c_B","aliases":"c_B","display":"c_B","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e78a957e4b60188f","tex":"c_b","aliases":"c_b","display":"c_b","kind":"use","excerpt":"The mathematical streaming identities do not require transfers and GEMMs to be serial. Consider B storage blocks. Let t_b be the transfer time for block b and c_b its device compute time.","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-201668608ff331e6","tex":"\\mathcal C_{hg}","aliases":"C_hg mathcal","display":"\\mathcal C_{hg}","kind":"use","excerpt":"Let E=H-H_0 be the self-adjoint error introduced by phase pruning. Using the same orthogonal phase projectors P_h, define C_hg = P_hEP_g _2.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"use","excerpt":"Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a2649fc12744ae4b","tex":"c_i^2","aliases":"c_i^2","display":"c_i^2","kind":"use","excerpt":"Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-254051ed849e0a84","tex":"c_k","aliases":"c_k","display":"c_k","kind":"use","excerpt":"The separate definition weightedSquares(s,c,z)=∑_k∈ sc_k z_k^2 models a supplied diagonal quadratic form. Its coefficients need not be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ ","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d5b7a071b82101f6","tex":"\\mathcal C_{\\rm complete}","aliases":"C_rm complete mathcal rm","display":"\\mathcal C_{\\rm complete}","kind":"use","excerpt":"D . Consequently, the nonzero singular values of A are exactly the nonzero singular values, with multiplicity, of the stacked completion core C_rm complete = B D .","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-676afbbae0b72611","tex":"\\mathcal C_{\\rm complete}^\\top","aliases":"C_rm complete^top mathcal rm top","display":"\\mathcal C_{\\rm complete}^\\top","kind":"use","excerpt":"The decomposition A=UB+F and the identity F=WD give the factorization. The matrix [U W] has orthonormal columns, so A^top A= C_rm complete^top C_rm complete. The nonzero singular values therefore agree with multiplicity. If the combined basis contains directions outside range(A), its row Gram may include additional zeros. Its entire spectrum is the","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-49425c7aa0f871c9","tex":"C_\\theta","aliases":"C_θ theta","display":"C_\\theta","kind":"use","excerpt":"Since 1+(a_jt)^2≥(a_jt)^2, we have |ϕ_θ(t)| ≤ C_θ t^-r/2, C_θ=∏_j a_j^-1/2.","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9858584b0941ea70","tex":"\\mathrm{COS}","aliases":"COS mathrm","display":"\\mathrm{COS}","kind":"use","excerpt":"[Omitted COS-mode certificate] Let N≥1 and b>q≥0. Set u_N=Nπ/b and β_N=β(u_N). If β_N>0, then the contribution of all exact tilted-COS modes k≥ N to the weighted tail functional is bounded by ε_COS ≤ e^κ(θ)-θ q frac4|ϕ_θ(u_N)|π (frac1N+frac1β_N).","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-af16ca3c54254627","tex":"\\operatorname{cum}","aliases":"cum operatorname","display":"\\operatorname{cum}","kind":"use","excerpt":"ts logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose argument can wrap by 2π. For real θ<1/(2λ_max), κ^(m)(θ) =2^m-1(m-1)!∑_j=1^r fracλ_j^m(1-2θλ_j)^m, m≥1. At θ=0, this recovers the standard cumulant identity cum_m(Q)=2^m-1(m-1)!∑_jλ_j^m.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3dc7855972cb7d72","tex":"\\operatorname{cum}_m","aliases":"cum_m operatorname","display":"\\operatorname{cum}_m","kind":"use","excerpt":"ts logarithm in the sum is principal. The sum need not equal the principal logarithm of the product, whose argument can wrap by 2π. For real θ<1/(2λ_max), κ^(m)(θ) =2^m-1(m-1)!∑_j=1^r fracλ_j^m(1-2θλ_j)^m, m≥1. At θ=0, this recovers the standard cumulant identity cum_m(Q)=2^m-1(m-1)!∑_jλ_j^m.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":14},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e541229d2a3c7787","tex":"D^{1/2}","aliases":"D^1/2","display":"D^{1/2}","kind":"use","excerpt":"Set S=Q^TQ. The first hypothesis gives the Loewner inequalities (1-η)Ipreceq Spreceq(1+η)I. The eigenvalues of B=(QD^1/2)(QD^1/2)^T are the same as those of D^1/2SD^1/2. Hence (1-η)D preceq D^1/2SD^1/2 preceq (1+η)D, and eigenvalue monotonicity yields (1-η)d_i ≤ λ_i(B) ≤ (1+η)d_i. Weyl's perturbation inequality and H-B _2≤ρ then give |λ_i(H)-λ_i(B)","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"use","excerpt":"A high-precision symmetric eigensolver is used only to generate candidate quantities Q∈ R^n× n, D=diag(d_1,…,d_n), d_1≥⋯≥ d_n≥0. No correctness claim depends on trusting this eigensolver.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"use","excerpt":"[Candidate eigensystem post-certification] Let B=QDQ^T. Suppose directed-rounding computation proves Q^TQ-I _2≤η<1 and H-B _2≤ρ. Then the ordered eigenvalues of the positive semidefinite matrix H satisfy max{0,(1-η)d_i-ρ} ≤ λ_i(H) ≤ (1+η)d_i+ρ for every i.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"use","excerpt":"A high-precision symmetric eigensolver is used only to generate candidate quantities Q∈ R^n× n, D=diag(d_1,…,d_n), d_1≥⋯≥ d_n≥0. No correctness claim depends on trusting this eigensolver.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f6452e49beb78ef8","tex":"\\mathcal D_{\\rm phase}","aliases":"D_rm phase mathcal rm","display":"\\mathcal D_{\\rm phase}","kind":"use","excerpt":"The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8b00bd7034dc3ea0","tex":"\\mathcal D_{\\rm rank}","aliases":"D_rm rank mathcal rm","display":"\\mathcal D_{\\rm rank}","kind":"use","excerpt":"The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2ec52d3af23b660c","tex":"\\mathcal D_{\\rm spectrum}","aliases":"D_rm spectrum mathcal rm","display":"\\mathcal D_{\\rm spectrum}","kind":"use","excerpt":"The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-db716c254fd2b459","tex":"\\mathcal D_{\\rm storage}","aliases":"D_rm storage mathcal rm","display":"\\mathcal D_{\\rm storage}","kind":"use","excerpt":"The solver is organized around four independent decomposition axes: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. This display describes the pipeline schematically, not four endomorphisms proved to commute pairwise. The precise commutation theorem below is for exact storage refinement and a fixed structural phase partition. Full range ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"1 · -1 · -1 · 1 satisfies V^top K V=diag(1,1,2,2). For Z∼ N(0,I_4), Q=Z^top KZ overset d= Y+2W, Y,W independent with law χ^2_2. The spectrum is therefore (1,1,2,2) and E Q=6. The six-dimensional feature Gram A^top A has these same four nonzero eigenvalues and two additional zeros.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-22140c9d8867aa44","tex":"\\operatorname{Disjoint}","aliases":"Disjoint operatorname","display":"\\operatorname{Disjoint}","kind":"use","excerpt":"The principal refinement theorem is declgramEntry_union. Its hypotheses are finite sets s,t⊆ V and Disjoint(s,t). It proves, for all sample coordinates i,j, gramEntry(s∪ t,A,i,j) =gramEntry(s,A,i,j)+gramEntry(t,A,i,j). Disjointness prevents repeated columns from being counted twice. It is a mathematical condition, not a performance assumption. The ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","excerpt":"∑_hP_h=I, g→ hiff P_hHP_g≠0. dist(g,h)>k Longrightarrow P_hH^kP_g=0. Choose a complete orthogonal family of phase projectors on feature space and form H=A^ TA. An edge from phase g to phase h exists when the transfer block P_hHP_g is nonzero. Each operator application follows one graph edge. An iterate of depth k cannot reach a phase farther than k","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"Residual information supplies a stopping rule for approximate iteration and a handoff to exact completion: c streamed / pruned Krylov [2mm] downarrow [-1mm] residual moments (t,v,ρ) [2mm] downarrow [-1mm] continue Krylov, · residual range still large, exact streamed residual completion, · completion rank affordable.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain w","count":20},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b0de09e6c7e6261e","tex":"e^{(-\\theta+iu_k)(b-q)}","aliases":"e^(-θ+iu_k)(b-q) theta","display":"e^{(-\\theta+iu_k)(b-q)}","kind":"use","excerpt":"J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an expl","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0a4e60157e813586","tex":"e^{-(x-y)/4}","aliases":"e^-(x-y)/4","display":"e^{-(x-y)/4}","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c2349512e48c7e97","tex":"e^{-1}","aliases":"e^-1","display":"e^{-1}","kind":"use","excerpt":"The reindexing theorem declgramEntry_reindex handles a bijection e:V≃ V. It moves the block to e(s) and transforms the factor to A'_ik=A_i,e^-1(k). The resulting Gram entry is unchanged. Permuting only the coordinates without moving their factor columns falls outside this statement.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1ae75dcc8dbb0dad","tex":"e^{-10}","aliases":"e^-10","display":"e^{-10}","kind":"use","excerpt":"approximately 0.0246131, which fails the requested tolerance 0.02. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below 0.013731. The displayed, coarser interval also meets the requested tolerance 0.02. An independently eval","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f603065fa6096eb2","tex":"e^{-20}","aliases":"e^-20","display":"e^{-20}","kind":"use","excerpt":"approximately 0.0246131, which fails the requested tolerance 0.02. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below 0.013731. The displayed, coarser interval also meets the requested tolerance 0.02. An independently eval","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-014f09d39057a640","tex":"e^{-q/0.14}","aliases":"e^-q/0.14","display":"e^{-q/0.14}","kind":"use","excerpt":"Retained low-rank limitation. For independent Y_1,Y_2∼χ^2_2, the quadratic form Q=0.07(Y_1-Y_2) is Laplace distributed with scale 0.14, so for q≥0, P(Q≥ q)=frac12e^-q/0.14. The computed values agree with this exact law through 10^-200 to roughly 10^-9 decades or better. However, the implemented analytic COS-series certificate rejects these low-rank","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fec3298b80e1c28e","tex":"e^{-q/2}","aliases":"e^-q/2","display":"e^{-q/2}","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b83391c58b9ecb34","tex":"e^{-q/4}","aliases":"e^-q/4","display":"e^{-q/4}","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a3b3e2ebdde99841","tex":"e^{-x}","aliases":"e^-x","display":"e^{-x}","kind":"use","excerpt":"For the control Q=0.02χ^2_80, the survival probability has the finite exact formula P(Q≥ q) = e^-x ∑_j=0^39 fracx^jj!, x=fracq0.04.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ba2e1525e66f7823","tex":"e^{-x/2}","aliases":"e^-x/2","display":"e^{-x/2}","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f9b00a2a950802c9","tex":"e^{-x/4}","aliases":"e^-x/4","display":"e^{-x/4}","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a325ae8bbc936147","tex":"e^{-y/2}","aliases":"e^-y/2","display":"e^{-y/2}","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-99b487a1df045997","tex":"e^{-z}","aliases":"e^-z","display":"e^{-z}","kind":"use","excerpt":"z=q/0.04, P(Q≥ q)=e^-z∑_k=0^39fracz^kk!. log_10P(Q≥ q)=frac-z+log (∑_k=0^39z^k/k!)log(10). For an even-degree chi-square variable, the upper tail has a finite positive-sum formula. The control Q=0.02χ^2_80 is a gamma law of integer shape forty and scale 0.04. Its survival probability can therefore be evaluated independently of cosine inversion.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a5c5ac0ef4693ac0","tex":"e^{-\\theta(b-q)}","aliases":"e^-θ(b-q) theta","display":"e^{-\\theta(b-q)}","kind":"use","excerpt":"nd] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b) (1+e^-θ(b-q)).","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5d88a39b4ee67aa9","tex":"e^{-\\theta(Q-q)}","aliases":"e^-θ(Q-q) theta","display":"e^{-\\theta(Q-q)}","kind":"use","excerpt":"For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a50dec80a592280c","tex":"e^{B_k}","aliases":"e^B_k","display":"e^{B_k}","kind":"use","excerpt":"[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-62432e62736af3bb","tex":"e^{iu_kq}","aliases":"e^iu_kq","display":"e^{iu_kq}","kind":"use","excerpt":"J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an expl","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8687cbd65af8dafc","tex":"e^\\top","aliases":"e^top top","display":"e^\\top","kind":"use","excerpt":"item directly through the factor, Q=frac12 A^top e _2^2. item through the sample kernel, Q=frac12 e^top Ke. item through the eigencoordinates of K.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-260efd500fe69b2a","tex":"e^{\\widehat\\kappa(s_k)-\\theta q}","aliases":"e^widehatκ(s_k)-θ q widehat kappa theta","display":"e^{\\widehat\\kappa(s_k)-\\theta q}","kind":"use","excerpt":"Write Δ_k=widehatκ(s_k)-κ(s_k). Then |e^widehatκ(s_k)-θ q-e^κ(s_k)-θ q| ≤ |e^κ(s_k)-θ q| (e^|Δ_k|-1). Moreover Reκ(θ+i u_k)-κ(θ) =log|ϕ_θ(u_k)|≤0, so |e^κ(s_k)-θ q| ≤ e^κ(θ)-θ q. Substitute this into the finite cosine sum and apply the triangle inequality.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3c0cd97594dc142a","tex":"e^{|\\Delta_k|}","aliases":"e^|Δ_k| Delta","display":"e^{|\\Delta_k|}","kind":"use","excerpt":"Write Δ_k=widehatκ(s_k)-κ(s_k). Then |e^widehatκ(s_k)-θ q-e^κ(s_k)-θ q| ≤ |e^κ(s_k)-θ q| (e^|Δ_k|-1). Moreover Reκ(θ+i u_k)-κ(θ) =log|ϕ_θ(u_k)|≤0, so |e^κ(s_k)-θ q| ≤ e^κ(θ)-θ q. Substitute this into the finite cosine sum and apply the triangle inequality.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8c9694deb6e67f31","tex":"e^{\\varepsilon_k}","aliases":"e^ε_k varepsilon","display":"e^{\\varepsilon_k}","kind":"use","excerpt":"After separating the likelihood-ratio prefactor, the packet contribution contains sums of the form frac2b ∑_k |J_k| ( e^ε_k-1 ). Direct evaluation can overflow even when the candidate is destined to be rejected by the adaptive planner. The implementation instead computes log( frac2b ∑_k |J_k| ( e^ε_k-1 ) ) by log-sum-exp arithmetic, and compares th","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ab45ea73bfa3e208","tex":"e^{\\theta x-\\kappa(\\theta)}","aliases":"e^θ x-κ(θ) theta kappa theta","display":"e^{\\theta x-\\kappa(\\theta)}","kind":"use","excerpt":"Define the exponentially tilted law by f_θ(x)=e^θ x-κ(θ)f(x). Its characteristic function is ϕ_θ(t) =exp{κ(θ+i t)-κ(θ)}, and E_θQ=κ'(θ). Therefore the choice [source reference] moves the query q to the mean of the tilted law.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a779bd4e78a6d721","tex":"e^{\\kappa(s_k)-\\theta q}","aliases":"e^κ(s_k)-θ q kappa theta","display":"e^{\\kappa(s_k)-\\theta q}","kind":"use","excerpt":"Write Δ_k=widehatκ(s_k)-κ(s_k). Then |e^widehatκ(s_k)-θ q-e^κ(s_k)-θ q| ≤ |e^κ(s_k)-θ q| (e^|Δ_k|-1). Moreover Reκ(θ+i u_k)-κ(θ) =log|ϕ_θ(u_k)|≤0, so |e^κ(s_k)-θ q| ≤ e^κ(θ)-θ q. Substitute this into the finite cosine sum and apply the triangle inequality.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-962da1fa2eb7fa44","tex":"e^{\\kappa(\\theta)-\\theta q}","aliases":"e^κ(θ)-θ q kappa theta theta","display":"e^{\\kappa(\\theta)-\\theta q}","kind":"use","excerpt":"For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.","count":8},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b146c3b5f8cad245","tex":"e^{\\kappa(\\theta+\\ii u_k)-\\theta q}","aliases":"e^κ(θ+ii u_k)-θ q e^κ(θ+i u_k)-θ q kappa theta ii theta kappa theta mathrm theta","display":"e^{\\kappa(\\theta+{\\mathrm{i}} u_k)-\\theta q}","kind":"use","excerpt":"If the exact tilted characteristic function is used, the N-mode finite-window expression is T_N,b =frac2b∑_k=0^N-1' J_k Re[e^κ(θ+i u_k)-θ q]. The cancellation of κ(θ) in [source reference] is useful numerically: the likelihood-ratio prefactor from [source reference] and the normalization in [source reference] do not need to be formed separately ins","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a3a5a5fd853b5ca1","tex":"e^{\\kappa(\\theta+it)-\\kappa(\\theta)}","aliases":"e^κ(θ+it)-κ(θ) kappa theta kappa theta","display":"e^{\\kappa(\\theta+it)-\\kappa(\\theta)}","kind":"use","excerpt":"J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an expl","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4d60cc719878240d","tex":"\\mathcal E_0","aliases":"E_0 mathcal","display":"\\mathcal E_0","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-94def8daada155b7","tex":"E_{\\rm modes}","aliases":"E_rm modes rm","display":"E_{\\rm modes}","kind":"use","excerpt":"|p-widehat p|≤ E_rm packet+E_rm support+E_rm modes (analytic packet path). S_θ(q)∈ S_N+[-E,E], E=E_rm support+E_rm series (exact-spectrum interval path). Packet approximation changes the retained-grid transform. Support truncation and cosine periodization change the finite-window functional. Mode truncation removes infinitely many terms. These thre","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-005629b28875751f","tex":"E_{\\rm packet}","aliases":"E_rm packet rm","display":"E_{\\rm packet}","kind":"use","excerpt":"|p-widehat p|≤ E_rm packet+E_rm support+E_rm modes (analytic packet path). S_θ(q)∈ S_N+[-E,E], E=E_rm support+E_rm series (exact-spectrum interval path). Packet approximation changes the retained-grid transform. Support truncation and cosine periodization change the finite-window functional. Mode truncation removes infinitely many terms. These thre","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a65ae76edb86fa3d","tex":"E_{\\rm series}","aliases":"E_rm series rm","display":"E_{\\rm series}","kind":"use","excerpt":"Let ν=frac r2>1, N≥1. For a finite transform retaining modes 0,…,N-1, E_rm series ≤ frac2C_θbθ (fracbπ)^ν [ N^-ν + fracN^1-νν-1 ].","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-04e789834774b799","tex":"E_{\\rm support}","aliases":"E_rm support rm","display":"E_{\\rm support}","kind":"use","excerpt":"Therefore E_rm support ≤ P_θ(Q>b) ( 1+e^-θ(b-q) ).","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ce74455beb257c47","tex":"E_\\theta","aliases":"E_θ theta","display":"E_\\theta","kind":"use","excerpt":"Choose any 0<θ<frac12λ_max. Exponential tilting gives P(Q≥ q) = e^κ(θ)-θ q S_θ(q), where S_θ(q) = E_θ [ e^-θ(Q-q) 1_{Q≥ q} ].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-afb6de14514ba44d","tex":"\\E_{\\theta}","aliases":"E_θ E theta operatorname mathbb theta","display":"{\\operatorname{\\mathbb E}}_{\\theta}","kind":"use","excerpt":"Define the exponentially tilted law by f_θ(x)=e^θ x-κ(θ)f(x). Its characteristic function is ϕ_θ(t) =exp{κ(θ+i t)-κ(θ)}, and E_θQ=κ'(θ). Therefore the choice [source reference] moves the query q to the mean of the tilted law.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f941905cd0d39915","tex":"\\mathbb E_\\theta","aliases":"E_θ mathbb theta","display":"\\mathbb E_\\theta","kind":"use","excerpt":"κ(s)=-frac12∑_jlog(1-2sλ_j), 0<θ<frac12λ_max. P(Q≥ q)=e^κ(θ)-θ q E_θ[e^-θ(Q-q)1_{Q≥ q}]. For a positive spectrum, the cumulant generating function is a sum of logarithms. A real tilt inside its domain defines a normalized reweighted law. The original upper tail becomes an exponential prefactor times a bounded weighted expectation under that law.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c30432f309c5faa8","tex":"\\operatorname{Enclosed}","aliases":"Enclosed operatorname","display":"\\operatorname{Enclosed}","kind":"use","excerpt":"The interval module uses two elementary predicates: Enclosed(x,L,U) · :Longleftrightarrow L≤ x≤ U, ErrorBound(x,widehat x,r) · :Longleftrightarrow |x-widehat x|≤ r. The equivalence declerrorBound_iff_enclosed proves |x-widehat x|≤ r Longleftrightarrow widehat x-r≤ x≤widehat x+r. There is no extra nonnegativity hypothesis: if this predicate holds, t","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps ε varepsilon","display":"{\\varepsilon}","kind":"use","excerpt":"[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).","count":13},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fa45489558f82038","tex":"\\eps_{\\mathrm{an}}","aliases":"eps_an ε_an eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{an}}","kind":"use","excerpt":"Let p= P(Q≥ q), and let widehat p be the finite K-COS approximation formed with [label=(roman*)] item an upper interval endpoint b>q, item an integer N≥1 of cosine modes, item a packet approximation satisfying [source reference] on the finite grid. Under the positive-spectrum and parameter hypotheses of Proposition [source reference], |p-widehat p|","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4fb993ea6aa2267c","tex":"\\eps_{\\mathrm{COS}}","aliases":"eps_COS ε_COS eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{COS}}","kind":"use","excerpt":"[Omitted COS-mode certificate] Let N≥1 and b>q≥0. Set u_N=Nπ/b and β_N=β(u_N). If β_N>0, then the contribution of all exact tilted-COS modes k≥ N to the weighted tail functional is bounded by ε_COS ≤ e^κ(θ)-θ q frac4|ϕ_θ(u_N)|π (frac1N+frac1β_N).","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-aab38e348e3faf09","tex":"\\eps_{\\mathrm{pkt}}","aliases":"eps_pkt ε_pkt eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{pkt}}","kind":"use","excerpt":"[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ad7d9909850ebc4f","tex":"\\eps_{\\mathrm{sup}}","aliases":"eps_sup ε_sup eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{sup}}","kind":"use","excerpt":"nd] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b) (1+e^-θ(b-q)).","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2af0b227aab89935","tex":"\\operatorname{ErrorBound}","aliases":"ErrorBound operatorname","display":"\\operatorname{ErrorBound}","kind":"use","excerpt":"The interval module uses two elementary predicates: Enclosed(x,L,U) · :Longleftrightarrow L≤ x≤ U, ErrorBound(x,widehat x,r) · :Longleftrightarrow |x-widehat x|≤ r. The equivalence declerrorBound_iff_enclosed proves |x-widehat x|≤ r Longleftrightarrow widehat x-r≤ x≤widehat x+r. There is no extra nonnegativity hypothesis: if this predicate holds, t","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5d8ae6090ea1e829","tex":"\\operatorname{expm1}","aliases":"expm1 operatorname","display":"\\operatorname{expm1}","kind":"use","excerpt":"The directed-rounding backend encloses the finite transform and exponential prefactor and then enlarges that enclosure by the proved support/alias and infinite-mode error bounds. Therefore the true tail probability lies inside the returned interval. The controller encloses expm1(log(10)(d_+-d_-)) by directed rounding and sets certified=True only wh","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e5dc3a473bd1bc61","tex":"F^{\\mathsf T}","aliases":"F^ T mathsf","display":"F^{\\mathsf T}","kind":"use","excerpt":"A=UB+F, U^ TF=0, A^ TA=B^ TB+F^ TF. t=trR= F _F^2, v=tr(R^2), ρ_R= R _2. Let U contain d orthonormal sample-space directions. Project the factor into its captured range coordinates B=U^ TA and residual F=(I-UU^ T)A. Orthogonality eliminates the cross terms in the feature-space Gram matrix.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","excerpt":"This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form. The second records how concentrated the r","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-64863d8dd06c95a8","tex":"F^\\top","aliases":"F^top top","display":"F^\\top","kind":"use","excerpt":"[Exact core--residual decomposition] A^top A = B^top B+F^top F. In particular the unresolved spectral term R=F^top F is positive semidefinite.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"[Exact frontier-to-I/O reduction] Suppose a validated exact phase graph of the fixed operator H=A^top A is already supplied and the initial Krylov block is supported on a phase set F_0. Keep covariate projections, weights, and structural phase metadata fixed, and assume all factor-action blocks required for the retained phase operator are available","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-54b44dec26d2ba54","tex":"F_b","aliases":"F_b","display":"F_b","kind":"use","excerpt":"[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4804c63f2a6a63f5","tex":"F_{b\\#}","aliases":"F_b#","display":"F_{b\\#}","kind":"use","excerpt":"[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7e04443bf0b17f7b","tex":"F_k","aliases":"F_k","display":"F_k","kind":"use","excerpt":"xact frontier-to-I/O reduction] Suppose a validated exact phase graph of the fixed operator H=A^top A is already supplied and the initial Krylov block is supported on a phase set F_0. Keep covariate projections, weights, and structural phase metadata fixed, and assume all factor-action blocks required for the retained phase operator are available. ","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-64f5ff12e8643285","tex":"f_Q","aliases":"f_Q","display":"f_Q","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-bbb386be992c59ae","tex":"f_{\\theta}","aliases":"f_θ theta","display":"f_{\\theta}","kind":"use","excerpt":"Define the exponentially tilted law by f_θ(x)=e^θ x-κ(θ)f(x). Its characteristic function is ϕ_θ(t) =exp{κ(θ+i t)-κ(θ)}, and E_θQ=κ'(θ). Therefore the choice [source reference] moves the query q to the mean of the tilted law.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-30f11c51a9f8c388","tex":"\\operatorname{factorEnergy}","aliases":"factorEnergy operatorname","display":"\\operatorname{factorEnergy}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":19},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b355be6ee30169aa","tex":"G_{[:,J_b]}","aliases":"G_[:,J_b]","display":"G_{[:,J_b]}","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-76b53653bff85b9e","tex":"G_b","aliases":"G_b","display":"G_b","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e74ab3b7f0014aa3","tex":"G_b^\\top","aliases":"G_b^top top","display":"G_b^\\top","kind":"use","excerpt":"[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b98256bebc25e86f","tex":"\\operatorname{gramEntry}","aliases":"gramEntry operatorname","display":"\\operatorname{gramEntry}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-425652a73ca954b2","tex":"H^k","aliases":"H^k","display":"H^k","kind":"use","excerpt":"[Polyphase light cone] If the graph distance from phase g to phase h exceeds k, then P_hH^kP_g=0.","count":16},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7f0aa7666470b126","tex":"H^{k-1-j}","aliases":"H^k-1-j","display":"H^{k-1-j}","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-dbdd7c1a3a1ed831","tex":"H_{\\mathcal O}","aliases":"H_ O mathcal","display":"H_{\\mathcal O}","kind":"use","excerpt":". For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d0c3eb31a2b52ebb","tex":"H_{\\mathcal O}^k","aliases":"H_ O^k mathcal","display":"H_{\\mathcal O}^k","kind":"use","excerpt":"[Edge-additive leakage majorant] For every omitted set O⊆ E, 0≤ ( B^k- B_ O^k)ρ_0 ≤ ∑_e∈ Oq_e^(k) componentwise. Consequently, lVert(H^k-H_ O^k)XrVert_F ≤ lVert ( B^k- B_ O^k)ρ_0 rVert_2 ≤ ∑_e∈ Oβ_e^(k).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,","count":13},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8eee6f57e815c963","tex":"H_0^j","aliases":"H_0^j","display":"H_0^j","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-defcfc919ed5c175","tex":"H_0^k","aliases":"H_0^k","display":"H_0^k","kind":"use","excerpt":"[Thresholded-power certificate] For every k≥1, ρ((H^k-H_0^k)X) ≤ ( B^k- B_0^k)ρ(X). In particular, lVert(H^k-H_0^k)XrVert_F ≤ lVert( B^k- B_0^k)ρ(X)rVert_2.","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-097615d1cce1b4b2","tex":"H_b","aliases":"H_b","display":"H_b","kind":"use","excerpt":"A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-766c113d1bfcf920","tex":"H_{b,I_r}","aliases":"H_b,I_r","display":"H_{b,I_r}","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f0f212be99411910","tex":"H_F","aliases":"H_F","display":"H_F","kind":"use","excerpt":"^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-86a036439f6696d1","tex":"H_F^k","aliases":"H_F^k","display":"H_F^k","kind":"use","excerpt":"rdinate inclusion from the retained coordinates F=F_k into the full variant space. Thus P_F=J_FJ_F^top and A_F=AJ_F. The compressed operator and its ambient embedding satisfy H_F=J_F^top HJ_F=A_F^top A_F, P_FHP_F=J_FH_FJ_F^top. These identities distinguish the compressed matrix from the larger ambient matrix. Since every path of length at most k fr","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"use","excerpt":"The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponenti","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":41},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ab39294284c03d46","tex":"\\mathrm i","aliases":"i mathrm","display":"\\mathrm i","kind":"use","excerpt":"A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-37475f8aeaec6cce","tex":"I_1","aliases":"I_1","display":"I_1","kind":"use","excerpt":"A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"Why the nonzero-spectrum qualification matters. Take instead A=1 · 0 0 · 0, Q_r=I_2, B=Q_r^top A=A. The captured subspace contains the range of A, but is larger than that range. Both Gram matrices have spectrum {1,0}. Their nonzero spectrum is {1}. Thus the nonzero spectrum of AA^top cannot be identified with the entire spectrum of BB^top. Choosing","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-78fd137c582a5e5f","tex":"I_4","aliases":"I_4","display":"I_4","kind":"use","excerpt":"1 · -1 · -1 · 1 satisfies V^top K V=diag(1,1,2,2). For Z∼ N(0,I_4), Q=Z^top KZ overset d= Y+2W, Y,W independent with law χ^2_2. The spectrum is therefore (1,1,2,2) and E Q=6. The six-dimensional feature Gram A^top A has these same four nonzero eigenvalues and two additional zeros.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5dc7ccbeace33e78","tex":"I_R","aliases":"I_R","display":"I_R","kind":"use","excerpt":"A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e9855187f190ff70","tex":"I_r","aliases":"I_r","display":"I_r","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-111b6ee2c5f8dbd2","tex":"I_s","aliases":"I_s","display":"I_s","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"∑_hP_h=I, g→ hiff P_hHP_g≠0. dist(g,h)>k Longrightarrow P_hH^kP_g=0. Choose a complete orthogonal family of phase projectors on feature space and form H=A^ TA. An edge from phase g to phase h exists when the transfer block P_hHP_g is nonzero. Each operator application follows one graph edge. An iterate of depth k cannot reach a phase farther than k","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e7d6fe2c83275eaa","tex":"\\inf_{0<2\\theta(c+\\rho)<1}","aliases":"inf_0<2θ(c+ρ)<1 inf theta rho","display":"\\inf_{0<2\\theta(c+\\rho)<1}","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-146cbf287aa37a5f","tex":"\\mathcal J","aliases":"J mathcal","display":"\\mathcal J","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8d6308571f33e418","tex":"J'","aliases":"J'","display":"J'","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-37959be93056eb7b","tex":"\\mathcal J'","aliases":"J' mathcal","display":"\\mathcal J'","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-08e93c501f7d1688","tex":"J_1","aliases":"J_1","display":"J_1","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fd3bc8d8a8eb6f8f","tex":"J_B","aliases":"J_B","display":"J_B","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-88fbeb34dbc66890","tex":"J_b","aliases":"J_b","display":"J_b","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fe61bfb695992cf7","tex":"J_F","aliases":"J_F","display":"J_F","kind":"use","excerpt":"For every phase h∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinate","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-13e268c7c69015bf","tex":"J_F^\\top","aliases":"J_F^top top","display":"J_F^\\top","kind":"use","excerpt":"∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the ","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6013290ab9ee6c0e","tex":"k^{-1}","aliases":"k^-1","display":"k^{-1}","kind":"use","excerpt":"has no atom at the jump of h. The even periodic extension of h has bounded Fej er means, which converge at every continuity point. Dominated convergence identifies the integral of those means with ∫ h d(F_b#ν_θ). Their cosine coefficients are the Ces aro weighted coefficients 2J_k/b, and folding preserves the cosine moments Reϕ_θ(kπ/b). Finally J_k","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-04680049fd3d81f9","tex":"k^{-1-r/2}","aliases":"k^-1-r/2","display":"k^{-1-r/2}","kind":"use","excerpt":"ted convergence identifies the integral of those means with ∫ h d(F_b#ν_θ). Their cosine coefficients are the Ces aro weighted coefficients 2J_k/b, and folding preserves the cosine moments Reϕ_θ(kπ/b). Finally J_k=O(k^-1) and |ϕ_θ(kπ/b)|=O(k^-r/2), as follows directly from the product formula below. Thus the coefficient products are O(k^-1-r/2). Th","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0a79174406aa1a67","tex":"k^{-r/2}","aliases":"k^-r/2","display":"k^{-r/2}","kind":"use","excerpt":"p of h. The even periodic extension of h has bounded Fej er means, which converge at every continuity point. Dominated convergence identifies the integral of those means with ∫ h d(F_b#ν_θ). Their cosine coefficients are the Ces aro weighted coefficients 2J_k/b, and folding preserves the cosine moments Reϕ_θ(kπ/b). Finally J_k=O(k^-1) and |ϕ_θ(kπ/b","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0a5e2fb9a08ef3d3","tex":"k^{-\\beta_N-1}","aliases":"k^-β_N-1 beta","display":"k^{-\\beta_N-1}","kind":"use","excerpt":"Using the cosine coefficient magnitude 2|ϕ_θ(u_k)|/b and [source reference], the omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0d5b2978ec912e70","tex":"K_{I_r,I_s}","aliases":"K_I_r,I_s","display":"K_{I_r,I_s}","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-69a26fc25f751d91","tex":"K_{ij}","aliases":"K_ij","display":"K_{ij}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-927de3cdac384ec2","tex":"K_s","aliases":"K_s","display":"K_s","kind":"use","excerpt":"The same refinement has formal consequences for the operator action and the quadratic statistic: (K_s∪ tv)_i · =(K_s v)_i+(K_t v)_i, v^top K_s∪ tv · =v^top K_s v+v^top K_t v. These are declmatrixAction_gram_union and declquadraticForm_gram_union. The independent declstreamedAction_union proves the corresponding streamed accumulation identity direct","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-065cee47552f9a21","tex":"K_{s\\cup t}","aliases":"K_s∪ t cup","display":"K_{s\\cup t}","kind":"use","excerpt":"The same refinement has formal consequences for the operator action and the quadratic statistic: (K_s∪ tv)_i · =(K_s v)_i+(K_t v)_i, v^top K_s∪ tv · =v^top K_s v+v^top K_t v. These are declmatrixAction_gram_union and declquadraticForm_gram_union. The independent declstreamedAction_union proves the corresponding streamed accumulation identity direct","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ab625c241c8a0e88","tex":"K_t","aliases":"K_t","display":"K_t","kind":"use","excerpt":"The same refinement has formal consequences for the operator action and the quadratic statistic: (K_s∪ tv)_i · =(K_s v)_i+(K_t v)_i, v^top K_s∪ tv · =v^top K_s v+v^top K_t v. These are declmatrixAction_gram_union and declquadraticForm_gram_union. The independent declstreamedAction_union proves the corresponding streamed accumulation identity direct","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"The packet planner accepts a finite positive spectrum, an upper-tail query q>∑_jλ_j, and a requested relative tolerance τ>0. Its initial support parameter satisfies L>0, and its mode count is a positive integer.","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d750be54985836b3","tex":"L^\\top","aliases":"L^top top","display":"L^\\top","kind":"use","excerpt":"Hierarchical refinement gave relative discrepancy 3.17×10^-16. For H=L^top L with block-bidiagonal L, the smallest eigenvalue was approximately 2.86 and the largest norm outside the exact light cone was zero. The QR-core singular-spectrum discrepancy was 5.89×10^-16 in relative ℓ_2 norm. These are finite controls, not universal floating-point error","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e352ebec50445255","tex":"l_k","aliases":"l_k","display":"l_k","kind":"use","excerpt":"The separate definition weightedSquares(s,c,z)=∑_k∈ sc_k z_k^2 models a supplied diagonal quadratic form. Its coefficients need not be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ ","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[Residual-rank bracket] lceil max{ fractρ, fract^2v } rceil ≤ rankF ≤ n-r. The zero-residual case is interpreted separately as rank zero.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-973977200963bdff","tex":"\\log_{10}","aliases":"log_10 log","display":"\\log_{10}","kind":"use","excerpt":"[htbp] centering includegraphics[width=linewidth]figures_production/figure_14_dual_budget_tail_bridge.pdf captionTail brackets under a fixed query budget and varying operator budgets. The dotted reference is at log_10p=-8. A sufficiently loose operator budget admits a strong coupling and widens the bracket even though the query-local constraint sti","count":18},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Let the phase projectors form an orthogonal decomposition of variant space: P_h^top=P_h=P_h^2, P_hP_g=0 (h≠ g), ∑_hP_h=I. Thus P_h projects onto variant phase h. Again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"∑_hP_h=I, g→ hiff P_hHP_g≠0. dist(g,h)>k Longrightarrow P_hH^kP_g=0. Choose a complete orthogonal family of phase projectors on feature space and form H=A^ TA. An edge from phase g to phase h exists when the transfer block P_hHP_g is nonzero. Each operator application follows one graph edge. An iterate of depth k cannot reach a phase farther than k","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-25d467671d09a670","tex":"\\lVert","aliases":"lVert","display":"\\lVert","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":15},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b32487cf3c3d262c","tex":"M^{1/2}","aliases":"M^1/2","display":"M^{1/2}","kind":"use","excerpt":"xact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). The","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-66fd7e806d93b854","tex":"M^{k-1}","aliases":"M^k-1","display":"M^{k-1}","kind":"use","excerpt":"[Scalar threshold law] For any induced matrix norm, let M=max{lVert BrVert,lVert B_0rVert}. Then lVert B^k- B_0^krVert ≤ kM^k-1lVert RrVert. Hence weak discarded coupling produces a first-order error bound proportional to the discarded transfer norm and to the Krylov depth.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ddd55276b0b2e151","tex":"M^\\top","aliases":"M^top top","display":"M^\\top","kind":"use","excerpt":"The positive case is not the natural boundary of the central theory. Let X∼ N(0,Σ), M=M^top, where M may be indefinite, and again write S=Σ^1/2. The symmetric operator H=SMS has a real signed spectrum.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fa02e26ac15b5f1d","tex":"M_3","aliases":"M_3","display":"M_3","kind":"use","excerpt":"A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b9672ca091c4a70a","tex":"M_4","aliases":"M_4","display":"M_4","kind":"use","excerpt":"A packet with a nonzero remainder. To see the approximation itself, take a separate declared packet {1.96,1.98,2.02,2.04}, μ=2, and evaluate its contribution at s=0.1+0.4 i. The deviations are -0.04,-0.02,0.02,0.04. In particular, M_1=M_3=0, M_2=0.004, M_4=0.00000544. The factor α=2s/(1-4s) gives ρ=0.04|α|<1, so Theorem~[source reference] applies. ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e23a6bb4925a7d19","tex":"M_{P,0}","aliases":"M_P,0","display":"M_{P,0}","kind":"use","excerpt":"Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2a931c8a10a942ad","tex":"M_{P,1}","aliases":"M_P,1","display":"M_{P,1}","kind":"use","excerpt":"Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-12fcf6e1ba955880","tex":"M_{\\rm block}","aliases":"M_rm block rm","display":"M_{\\rm block}","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1a33dd4e78d90d51","tex":"M_{\\rm double}","aliases":"M_rm double rm","display":"M_{\\rm double}","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f89f923e4f377d21","tex":"M_{\\rm fixed}","aliases":"M_rm fixed rm","display":"M_{\\rm fixed}","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a36159e6a5ce2ba9","tex":"M_{\\rm full}","aliases":"M_rm full rm","display":"M_{\\rm full}","kind":"use","excerpt":"For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-17c2d80fc3832b34","tex":"M_{\\rm single}","aliases":"M_rm single rm","display":"M_{\\rm single}","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a41dabcb081dfd4c","tex":"M_{\\rm stream}","aliases":"M_rm stream rm","display":"M_{\\rm stream}","kind":"use","excerpt":"For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-88849f84a35be77a","tex":"M_X","aliases":"M_X","display":"M_X","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":25},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6341a5f2b8d21049","tex":"M_X^{\\mathsf T}","aliases":"M_X^ T mathsf","display":"M_X^{\\mathsf T}","kind":"use","excerpt":"Q_X^ TQ_X=I, M_X=I-Q_XQ_X^ T, M_X^2=M_X=M_X^ T. A=M_XGW^1/2, K=AA^ T. For a covariate range basis Q_X with orthonormal columns, the residualizer M_X=I-Q_XQ_X^ T removes the component in the covariate span. It is symmetric and idempotent. Applying it to the weighted feature factor restricts the kernel to variation left after the null-model covariate","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e37c0b4654d9fb2f","tex":"M_X^2","aliases":"M_X^2","display":"M_X^2","kind":"use","excerpt":"A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-171a34cbb1f0c1b6","tex":"M_X^\\top","aliases":"M_X^top top","display":"M_X^\\top","kind":"use","excerpt":"A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-dd8fa0d4e2cd2484","tex":"\\operatorname{matrixAction}","aliases":"matrixAction operatorname","display":"\\operatorname{matrixAction}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-115f51447aa568ea","tex":"\\max_{j\\in P}","aliases":"max_j∈ P max in","display":"\\max_{j\\in P}","kind":"use","excerpt":"[Centered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5dc9138b150888cc","tex":"\\max_{\\lambda_j<0}","aliases":"max_λ_j<0 max lambda","display":"\\max_{\\lambda_j<0}","kind":"use","excerpt":"For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain w","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain w","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-044ffeeb22e17997","tex":"\\min_{\\lambda_j>0}","aliases":"min_λ_j>0 min lambda","display":"\\min_{\\lambda_j>0}","kind":"use","excerpt":"For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain w","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-db8a81733714c917","tex":"N^{-\\beta_N}","aliases":"N^-β_N beta","display":"N^{-\\beta_N}","kind":"use","excerpt":"omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e111667fca23f947","tex":"N^{-\\beta_N-1}","aliases":"N^-β_N-1 beta","display":"N^{-\\beta_N-1}","kind":"use","excerpt":"omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5e0231364bfa13b3","tex":"N^{-\\nu}","aliases":"N^-ν nu","display":"N^{-\\nu}","kind":"use","excerpt":"Let ν=frac r2>1, N≥1. For a finite transform retaining modes 0,…,N-1, E_rm series ≤ frac2C_θbθ (fracbπ)^ν [ N^-ν + fracN^1-νν-1 ].","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4140f6bac1bc2350","tex":"N^{1-\\nu}","aliases":"N^1-ν nu","display":"N^{1-\\nu}","kind":"use","excerpt":"Let ν=frac r2>1, N≥1. For a finite transform retaining modes 0,…,N-1, E_rm series ≤ frac2C_θbθ (fracbπ)^ν [ N^-ν + fracN^1-νν-1 ].","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e620a8e6903abc71","tex":"N^{\\beta_N}","aliases":"N^β_N beta","display":"N^{\\beta_N}","kind":"use","excerpt":"Using the cosine coefficient magnitude 2|ϕ_θ(u_k)|/b and [source reference], the omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a2f61c3010260c15","tex":"n_P","aliases":"n_P","display":"n_P","kind":"use","excerpt":"Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0e517349dec3b6ba","tex":"\\overset","aliases":"overset","display":null,"kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e9d48e107ca79b17","tex":"P_F","aliases":"P_F","display":"P_F","kind":"use","excerpt":"∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinates F=F_k into the ","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9f7ab0fe246e6670","tex":"P_h^2","aliases":"P_h^2","display":"P_h^2","kind":"use","excerpt":"Let the phase projectors form an orthogonal decomposition of variant space: P_h^top=P_h=P_h^2, P_hP_g=0 (h≠ g), ∑_hP_h=I. Thus P_h projects onto variant phase h. Again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-69461bc7dce0e40a","tex":"P_h^\\top","aliases":"P_h^top top","display":"P_h^\\top","kind":"use","excerpt":"Let the phase projectors form an orthogonal decomposition of variant space: P_h^top=P_h=P_h^2, P_hP_g=0 (h≠ g), ∑_hP_h=I. Thus P_h projects onto variant phase h. Again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"Insert the projector resolution ∑_jP_j=I between successive factors of H. Every nonzero term in P_hH^kP_g corresponds to a phase path of length k. If no such path exists, every term vanishes.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d61b96062b482c72","tex":"P_q","aliases":"P_q","display":"P_q","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-383ce804b4f1b5aa","tex":"P_T","aliases":"P_T","display":"P_T","kind":"use","excerpt":":final-2-3 For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-58fb42807c0a2ca7","tex":"P_\\theta","aliases":"P_θ theta","display":"P_\\theta","kind":"use","excerpt":"Under the tilted measure, P_θ(Q>b) ≤ exp{ κ(η)-κ(θ)-(η-θ)b }.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-bc527840c44f512e","tex":"\\mathcal P_{\\Lambda,q,\\varepsilon}","aliases":"P_Λ,q,ε mathcal Lambda varepsilon","display":"\\mathcal P_{\\Lambda,q,\\varepsilon}","kind":"use","excerpt":"s_k=θ+ifrackπb, ρ_P(s_k)<1 on every retained frequency. (Λ,q,ε)⟼ P_Λ,q,ε. A distance between eigenvalues is insufficient to decide whether a packet is useful. The expansion must be controlled at the actual complex arguments s_k=θ+i kπ/b selected by the tail query. The tilt, support endpoint, and retained mode count all change that grid.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5ecc24519450d56a","tex":"\\mathcal P_{\\Lambda,q,\\tau}","aliases":"P_Λ,q,τ mathcal Lambda tau","display":"\\mathcal P_{\\Lambda,q,\\tau}","kind":"use","excerpt":"The resulting object is query dependent. Schematically, (Λ,q,τ) ⟼ P_Λ,q,τ ⟼ (widehat p,certificate), where P_Λ,q,τ stores exact spikes, packet centers, centered moments, packet radii, and the selected COS parameters.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5f6172bbc0fbb11e","tex":"\\mathrm{pkt}","aliases":"pkt mathrm","display":"\\mathrm{pkt}","kind":"use","excerpt":"[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b6516a74e6dd2749","tex":"\\Pp","aliases":"Pp P mathbb","display":"{\\mathbb P}","kind":"use","excerpt":"For any θ>0 in the transform domain, P(Q≥ q) =e^κ(θ)-θ q E_θ [e^-θ(Q-q) 1_{Q≥ q}]. This classical exponential-tilting identity lets K-COS evaluate the upper tail without subtracting a cumulative distribution function (CDF) value close to one.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-94921992b08500b9","tex":"\\Pp_{\\theta}","aliases":"Pp_θ P_θ Pp theta mathbb theta","display":"{\\mathbb P}_{\\theta}","kind":"use","excerpt":"[Support and periodization bound] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b)","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-71a98ec1e2bb9f0a","tex":"\\preceq","aliases":"preceq","display":"\\preceq","kind":"use","excerpt":"Set S=Q^TQ. The first hypothesis gives the Loewner inequalities (1-η)Ipreceq Spreceq(1+η)I. The eigenvalues of B=(QD^1/2)(QD^1/2)^T are the same as those of D^1/2SD^1/2. Hence (1-η)D preceq D^1/2SD^1/2 preceq (1+η)D, and eigenvalue monotonicity yields (1-η)d_i ≤ λ_i(B) ≤ (1+η)d_i. Weyl's perturbation inequality and H-B _2≤ρ then give |λ_i(H)-λ_i(B)","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3f1b6dc519af96a7","tex":"\\prime","aliases":"prime","display":"\\prime","kind":"use","excerpt":"J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an expl","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-050d088f2ad53d96","tex":"Q^T","aliases":"Q^T","display":"Q^T","kind":"use","excerpt":"[Candidate eigensystem post-certification] Let B=QDQ^T. Suppose directed-rounding computation proves Q^TQ-I _2≤η<1 and H-B _2≤ρ. Then the ordered eigenvalues of the positive semidefinite matrix H satisfy max{0,(1-η)d_i-ρ} ≤ λ_i(H) ≤ (1+η)d_i+ρ for every i.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2e7d480dd8714698","tex":"Q^\\top","aliases":"Q^top top","display":"Q^\\top","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c364e9f084300ad4","tex":"Q_+","aliases":"Q_+","display":"Q_+","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-22244099d34fb3a0","tex":"Q_-","aliases":"Q_-","display":"Q_-","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","excerpt":"[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). T","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-78e11e0a9b0798b5","tex":"q_e^{(d)}","aliases":"q_e^(d)","display":"q_e^{(d)}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0d15eef406611064","tex":"q_e^{(k)}","aliases":"q_e^(k)","display":"q_e^{(k)}","kind":"use","excerpt":"For depth k≥1 and starting phase profile ρ_0=ρ(X), define the edge risk vector q_e^(k) = ∑_j=0^k-1 B^k-1-jΔ_e B^jρ_0 and its scalar risk β_e^(k)=lVert q_e^(k)rVert_2. Unlike the raw block norm B_hg, this quantity depends on the start profile and on Krylov depth.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0d86ba69a441f990","tex":"q_e^{(p)}","aliases":"q_e^(p)","display":"q_e^{(p)}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a4210e21e79e5611","tex":"Q_r","aliases":"Q_r","display":"Q_r","kind":"use","excerpt":"Suppose an orthonormal basis Q_r captures the factor range: range(A)⊆range(Q_r). Then A=Q_rB, B=Q_r^top A.","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-46a25eef05df8c77","tex":"Q_r^\\top","aliases":"Q_r^top top","display":"Q_r^\\top","kind":"use","excerpt":"Suppose an orthonormal basis Q_r captures the factor range: range(A)⊆range(Q_r). Then A=Q_rB, B=Q_r^top A.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b55754320a8637cb","tex":"Q_X","aliases":"Q_X","display":"Q_X","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-138c7f7e777f2927","tex":"Q_{X,I_r}","aliases":"Q_X,I_r","display":"Q_{X,I_r}","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-df947929914c1caa","tex":"Q_{X,I_r}^\\top","aliases":"Q_X,I_r^top top","display":"Q_{X,I_r}^\\top","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-11f127f39c34416b","tex":"Q_X^{\\mathsf T}","aliases":"Q_X^ T mathsf","display":"Q_X^{\\mathsf T}","kind":"use","excerpt":"Q_X^ TQ_X=I, M_X=I-Q_XQ_X^ T, M_X^2=M_X=M_X^ T. A=M_XGW^1/2, K=AA^ T. For a covariate range basis Q_X with orthonormal columns, the residualizer M_X=I-Q_XQ_X^ T removes the component in the covariate span. It is symmetric and idempotent. Applying it to the weighted feature factor restricts the kernel to variation left after the null-model covariate","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8843b5276a2a8f90","tex":"Q_X^\\top","aliases":"Q_X^top top","display":"Q_X^\\top","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-79e670199e387f36","tex":"Q_\\perp","aliases":"Q_⊥ perp","display":"Q_\\perp","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-dea5180e332dd5bc","tex":"Q_\\perp^\\top","aliases":"Q_⊥^top perp top","display":"Q_\\perp^\\top","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ff66f12e10ff352f","tex":"\\operatorname{quadraticForm}","aliases":"quadraticForm operatorname","display":"\\operatorname{quadraticForm}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.","count":35},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"The factor has a finite sample index type I and an arbitrary variant index type V. A variant block is a finite set s⊆ V. The factor entries are exact real numbers, A:I× V→ R. The Lean type is declRow -> Variant -> Real. The sample coordinates carry a declFintype instance. The variants carry a declDecidableEq instance to support finite-set unions, s","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0, 0, · otherwise, R= B- B_0. Symmetric pair retention preserves self-adjointness of H_0, although the following error theorem does not require positive semidefiniteness of H_0.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form. The second records how concentrated the r","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"[htbp] centering captionSynthetic rare-variant simulation scenarios. Adjacent r^2 is computed within the simulated LD blocks after variant standardization. lrrrrr scenario · n · m · rank · median adjacent r^2 · effective rank","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c0ead3103527e3a8","tex":"\\mathbb R^{4\\times2}","aliases":"R^4×2 mathbb times","display":"\\mathbb R^{4\\times2}","kind":"use","excerpt":"The running example in three storage blocks. Partition the factor of Section~[source reference] into consecutive pairs of columns, A=[A_1 A_2 A_3], A_i∈ R^4×2. Although A_2=A_3, both column groups must be counted. Their Gram contributions are A_1A_1^top=frac12 1 · 0 · 1 · 0","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3a7615ff2031245c","tex":"\\mathbb R^{m\\times s}","aliases":"R^m× s mathbb times","display":"\\mathbb R^{m\\times s}","kind":"use","excerpt":"Let V∈ R^m× s be partitioned conformally: V= V_1","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-67b61430a0d291a9","tex":"\\mathbb R^{n\\times c_b}","aliases":"R^n× c_b mathbb times","display":"\\mathbb R^{n\\times c_b}","kind":"use","excerpt":"A=[A_1 ⋯ A_B], AA^ T=∑_b=1^B A_bA_b^ T. KV=∑_b A_b(A_b^ TV), A_b∈ R^n× c_b, V∈ R^n× s. Partition a factor by columns and keep every block. Horizontal concatenation turns the sample-space Gram matrix into a sum of block Gram matrices. Changing the block widths or the order in which blocks are read preserves the represented operator.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-55f8d0c50ba802bf","tex":"\\mathbb R^{n\\times n}","aliases":"R^n× n mathbb times","display":"\\mathbb R^{n\\times n}","kind":"use","excerpt":"A high-precision symmetric eigensolver is used only to generate candidate quantities Q∈ R^n× n, D=diag(d_1,…,d_n), d_1≥⋯≥ d_n≥0. No correctness claim depends on trusting this eigensolver.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-dbebc888329e8fe1","tex":"\\mathbb R^{n\\times p}","aliases":"R^n× p mathbb times","display":"\\mathbb R^{n\\times p}","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-25213bd56a1923f1","tex":"\\mathbb R^{n\\times r}","aliases":"R^n× r mathbb times","display":"\\mathbb R^{n\\times r}","kind":"use","excerpt":"Let U∈ R^n× r have orthonormal columns and define B=U^top A, F=(I-UU^top)A. Then A=UB+F, U^top F=0.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c06cbcc9be02a47a","tex":"\\mathbb R^{n\\times s}","aliases":"R^n× s mathbb times","display":"\\mathbb R^{n\\times s}","kind":"use","excerpt":"[One-pass sample-space streaming action] For every block vector V∈ R^n× s, KV = M_X ∑_b=1^B G_bW_bG_b^top M_XV.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-baf9ce2c27891e4e","tex":"R_b","aliases":"R_b","display":"R_b","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ade69a4d18e62b84","tex":"r_g","aliases":"r_g","display":"r_g","kind":"use","excerpt":"For any vector x, let r_g= P_gx _2. Then P_hEx _2 ≤ ∑_g P_hEP_g _2 P_gx _2 = ( C r)_h. Orthogonality of the phase decomposition gives Ex _2^2 = ∑_h P_hEx _2^2 ≤ C r _2^2 ≤ C _2^2 r _2^2 = C _2^2 x _2^2. Taking the supremum over unit vectors proves the claim.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b8db6823423681b3","tex":"R_h","aliases":"R_h","display":"R_h","kind":"use","excerpt":"Let R be a newly generated Krylov residual and R_h=P_hR.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-455a3f2c7c575c0e","tex":"r_h","aliases":"r_h","display":"r_h","kind":"use","excerpt":"[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d19e0f8ac22a9168","tex":"r_{hg}","aliases":"r_hg","display":"r_{hg}","kind":"use","excerpt":"If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d0a12b6c6c489d99","tex":"r_i^2","aliases":"r_i^2","display":"r_i^2","kind":"use","excerpt":"The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f56789beaa10d47c","tex":"R_{P,p}","aliases":"R_P,p","display":"R_{P,p}","kind":"use","excerpt":"tered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P,p(s","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c2283e1914e4269a","tex":"r_{\\rm eff}","aliases":"r_rm eff rm","display":"r_{\\rm eff}","kind":"use","excerpt":"Effective rank here is the participation ratio r_rm eff=frac(∑_jλ_j)^2∑_jλ_j^2. It measures spectral concentration, not an integer count of nonzero directions. The corresponding spectra therefore span a substantial range of concentration: stronger LD reduces effective rank even when the nominal quadratic-form rank remains close to the sample residu","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0876bfe73de67b89","tex":"r_s","aliases":"r_s","display":"r_s","kind":"use","excerpt":"The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-268aa2b4269fd3ac","tex":"R_\\perp","aliases":"R_⊥ perp","display":"R_\\perp","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9bf28df79e66226e","tex":"\\operatorname{range}","aliases":"range operatorname","display":"\\operatorname{range}","kind":"use","excerpt":"s λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈ran","count":14},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[Residual-rank bracket] lceil max{ fractρ, fract^2v } rceil ≤ rankF ≤ n-r. The zero-residual case is interpreted separately as rank zero.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-61f9609b70de73a0","tex":"\\rVert","aliases":"rVert","display":"\\rVert","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":15},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c1e5fffeb9eae9df","tex":"\\rVert_2","aliases":"rVert_2 rVert","display":"\\rVert_2","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-897e5303d2e01253","tex":"\\rVert_F","aliases":"rVert_F rVert","display":"\\rVert_F","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ca9d46208a25ce8a","tex":"\\rVert_F^2","aliases":"rVert_F^2 rVert","display":"\\rVert_F^2","kind":"use","excerpt":"ing Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-95c3fe100cc78e04","tex":"\\boldsymbol S","aliases":"S boldsymbol","display":"\\boldsymbol S","kind":"use","excerpt":"Let S_N be the directed-rounding interval containing the finite N-mode tilted COS sum, and set E=E_rm support+E_rm series. Then S_θ(q) ∈ S_N+[-E,E].","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"[Storage-phase commutation theorem] Let A=M_XGW^1/2, H=A^top A, and let S and S' be any two exact storage refinements that respect the same structural phase partition. Then: [label=(roman*)] item the exact operator H is identical under S and S'. item every phase coupling P_hHP_g is identical under S and S'. item the exact phase graph and every k-st","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fe39e36813ced2eb","tex":"S^{\\mathsf T}","aliases":"S^ T mathsf","display":"S^{\\mathsf T}","kind":"use","excerpt":"The finite control is a nine-phase chain with H=2I+S+S^ T and a unit start in phase five. The printed entries are exact integers in H^kx, and the shaded support expands by one phase at each step. The script checks every zero outside the cone.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b4ebcce0dc1a5c39","tex":"s_k","aliases":"s_k","display":"s_k","kind":"use","excerpt":"Let widehatκ(s_k) be the packet approximation at s_k=θ+i u_k, and suppose |widehatκ(s_k)-κ(s_k)|≤ B_k. For disjoint packets, B_k can be chosen as the sum of the packet bounds [source reference].","count":10},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-22cf9853e0e53ac8","tex":"S_N","aliases":"S_N","display":"S_N","kind":"use","excerpt":"J_k=∫_q^be^-θ(x-q)cos(u_kx) dx=Re [e^iu_kqfrac1-e^(-θ+iu_k)(b-q)θ-iu_k]. S_N=frac2b∑_k=0^N-1^prime J_k Re ϕ_θ(u_k), ϕ_θ(t)=e^κ(θ+it)-κ(θ). On a positive-support interval [0,b], the cosine frequencies are u_k=kπ/b. The payoff is zero below the threshold and decays exponentially above it. Integrating each cosine mode against that payoff gives an expl","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4b27aa71369f86a9","tex":"\\boldsymbol S_N","aliases":"S_N boldsymbol","display":"\\boldsymbol S_N","kind":"use","excerpt":"Let S_N be the directed-rounding interval containing the finite N-mode tilted COS sum, and set E=E_rm support+E_rm series. Then S_θ(q) ∈ S_N+[-E,E].","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-373731058437e68c","tex":"S_\\theta","aliases":"S_θ theta","display":"S_\\theta","kind":"use","excerpt":"Choose any 0<θ<frac12λ_max. Exponential tilting gives P(Q≥ q) = e^κ(θ)-θ q S_θ(q), where S_θ(q) = E_θ [ e^-θ(Q-q) 1_{Q≥ q} ].","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponenti","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-59a4b2862212f500","tex":"\\operatorname{spec}","aliases":"spec operatorname","display":"\\operatorname{spec}","kind":"use","excerpt":"X=Σ^1/2Z, Z∼ N(0,I), H=Σ^1/2MΣ^1/2. X^ TMXoverset d=∑_j=1^rλ_jZ_j^2, λ_j∈spec(H)∖{0}. For a central Gaussian vector with covariance Σsucceq0 and symmetric M, the quadratic form becomes a weighted sum of independent standard-normal squares after whitening and orthogonal diagonalization. The operator may originate in a kernel, a signal model, or anot","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-360396e5da8cb6ed","tex":"\\operatorname{streamedAction}","aliases":"streamedAction operatorname","display":"\\operatorname{streamedAction}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-32f99fc05b64a106","tex":"\\succ","aliases":"succ","display":"\\succ","kind":"use","excerpt":"2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). Then q_0= (I-P_T)b ^2. Thus q_0=0 exactly when M^1/2μ∈range(M^1/2S). In particular, Σsucc0 is sufficient. For singular covariance","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-29ce927e19b67f64","tex":"\\mathrm{sup}","aliases":"sup mathrm","display":"\\mathrm{sup}","kind":"use","excerpt":"nd] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b) (1+e^-θ(b-q)).","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"xact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). The","count":28},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ba679fda82ca705d","tex":"t^{-r/2}","aliases":"t^-r/2","display":"t^{-r/2}","kind":"use","excerpt":"Since 1+(a_jt)^2≥(a_jt)^2, we have |ϕ_θ(t)| ≤ C_θ t^-r/2, C_θ=∏_j a_j^-1/2.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3ed5533634f29449","tex":"T^{-\\nu}","aliases":"T^-ν nu","display":"T^{-\\nu}","kind":"use","excerpt":"The four spectra include balanced, skewed, dominant-spike, and higher-rank cases. Each reference integral was evaluated at two truncation limits. The omitted integral after the frequency cutoff T>0 was bounded using frac1π ∫_T^∞fracduuR(u) ≤ fracCπνT^-ν, ν=frac12∑_jh_j, C=∏_jλ_j^-h_j/2.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"Factor the determinant through the known core: I-2θ(C+R) = (I-2θ C)^1/2 (I-2θ T) (I-2θ C)^1/2, where T=(I-2θ C)^-1/2 R (I-2θ C)^-1/2. Since (I-2θ C)^-1/2 _2^2=α, we have trT≤α t, tr(T^2)≤α^2v, T _2≤αρ. For every eigenvalue x of T, -frac12log(1-2θ x) ≤ θ x + fracθ^2x^2 1-2θαρ. The scalar inequality follows from the power series of -log(1-u) and a ge","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"With sample tile height t and resident variant width c, the dominant working storage, including a resident covariate tile, is O(tc+pc+tp+t^2). This count excludes the backing input store, completed output blocks, allocator overhead, and library workspaces.","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f544e1fddefb59a0","tex":"t^d","aliases":"t^d","display":"t^d","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-68c16f5dbc93324c","tex":"t^j","aliases":"t^j","display":"t^j","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-032c5bccde429f54","tex":"T^\\top","aliases":"T^top top","display":"T^\\top","kind":"use","excerpt":"xpand (μ+SZ)^top M(μ+SZ) = Z^top SMSZ + 2μ^top MSZ + μ^top Mμ and rotate by V. Completing the square gives the displayed noncentral components. For a null vector of SMS, M^1/2Sv=0, hence v^top SMμ = (M^1/2Sv)^top M^1/2μ = 0, so there is no unresolved stochastic linear term in a zero-eigenvalue direction. For the positive eigenvectors v_i of T^top T","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","excerpt":"[Power envelope from one frequency] For t≥ t_0>0, |ϕ_θ(t)| ≤ |ϕ_θ(t_0)| (fractt_0)^-β(t_0).","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-cefa01be96058c3e","tex":"T_b","aliases":"T_b","display":"T_b","kind":"use","excerpt":"[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","excerpt":"The mathematical streaming identities do not require transfers and GEMMs to be serial. Consider B storage blocks. Let t_b be the transfer time for block b and c_b its device compute time.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e0d4b679b4a2b87a","tex":"t_{b+1}","aliases":"t_b+1","display":"t_{b+1}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-cf84b2d1fd96ffd7","tex":"T_{N,b}","aliases":"T_N,b","display":"T_{N,b}","kind":"use","excerpt":"If the exact tilted characteristic function is used, the N-mode finite-window expression is T_N,b =frac2b∑_k=0^N-1' J_k Re[e^κ(θ+i u_k)-θ q]. The cancellation of κ(θ) in [source reference] is useful numerically: the likelihood-ratio prefactor from [source reference] and the normalization in [source reference] do not need to be formed separately ins","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-529849886462f9b6","tex":"T_{\\rm pipe}","aliases":"T_rm pipe rm","display":"T_{\\rm pipe}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8b06363b18c1a84d","tex":"T_{\\rm serial}","aliases":"T_rm serial rm","display":"T_{\\rm serial}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form. The second records how concentrated the r","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.","count":42},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4af4f175a4c49987","tex":"U^{\\mathsf T}","aliases":"U^ T mathsf","display":"U^{\\mathsf T}","kind":"use","excerpt":"A=UB+F, U^ TF=0, A^ TA=B^ TB+F^ TF. t=trR= F _F^2, v=tr(R^2), ρ_R= R _2. Let U contain d orthonormal sample-space directions. Project the factor into its captured range coordinates B=U^ TA and residual F=(I-UU^ T)A. Orthogonality eliminates the cross terms in the feature-space Gram matrix.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponenti","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d813568f15ad058f","tex":"U^\\top","aliases":"U^top top","display":"U^\\top","kind":"use","excerpt":"There is a second route that does not use the pruned operator as the final spectral approximation. Let U be any orthonormal sample-space basis, possibly generated cheaply from a pruned or approximate Krylov process, and define B=U^top A, F=(I-UU^top)A, δ= F _2.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-685bf0a5c43d8ed4","tex":"U_{hg}","aliases":"U_hg","display":"U_{hg}","kind":"use","excerpt":"If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b207f2554f44324f","tex":"u_N","aliases":"u_N","display":"u_N","kind":"use","excerpt":"[Omitted COS-mode certificate] Let N≥1 and b>q≥0. Set u_N=Nπ/b and β_N=β(u_N). If β_N>0, then the contribution of all exact tilted-COS modes k≥ N to the weighted tail functional is bounded by ε_COS ≤ e^κ(θ)-θ q frac4|ϕ_θ(u_N)|π (frac1N+frac1β_N).","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d27fa8e4e0984e4b","tex":"\\uparrow","aliases":"uparrow","display":"\\uparrow","kind":"use","excerpt":"By [source reference], κ'(θ)=∑_jfracλ_j1-2θλ_j is continuous and strictly increasing on the stated interval. It equals E Q at θ=0 and diverges as θuparrow1/(2λ_max). The intermediate value theorem and strict monotonicity give existence and uniqueness.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"Write X=SZ with Z∼ N(0,I) and diagonalize SMS=VΛ V^top. Rotational invariance of the standard normal gives Z^top VΛ V^top Z overset d= ∑_jλ_jZ_j^2. No positivity assumption on M is needed for this central reduction.","count":31},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"If AA^top u=λ u with λ>0, then A^top u≠0 and A^top A(A^top u)=λ A^top u. Conversely, an eigenvector v of A^top A with positive eigenvalue maps to the eigenvector Av of AA^top. These maps restrict to inverse isomorphisms of the corresponding eigenspaces after scaling, so multiplicities agree. Horizontal concatenation gives AA^top=∑_b A_bA_b^top by d","count":30},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-bd832005bdef034c","tex":"V^\\top","aliases":"V^top top","display":"V^\\top","kind":"use","excerpt":"Write X=SZ with Z∼ N(0,I) and diagonalize SMS=VΛ V^top. Rotational invariance of the standard normal gives Z^top VΛ V^top Z overset d= ∑_jλ_jZ_j^2. No positivity assumption on M is needed for this central reduction.","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6b104ac6b34a135f","tex":"v^\\top","aliases":"v^top top","display":"v^\\top","kind":"use","excerpt":"Expand (μ+SZ)^top M(μ+SZ) = Z^top SMSZ + 2μ^top MSZ + μ^top Mμ and rotate by V. Completing the square gives the displayed noncentral components. For a null vector of SMS, M^1/2Sv=0, hence v^top SMμ = (M^1/2Sv)^top M^1/2μ = 0, so there is no unresolved stochastic linear term in a zero-eigenvalue direction. For the positive eigenvectors v_i of T^top ","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"Let V∈ R^m× s be partitioned conformally: V= V_1 vdots V_B .","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9f5cc10c11d8d9e7","tex":"V_B","aliases":"V_B","display":"V_B","kind":"use","excerpt":"Let V∈ R^m× s be partitioned conformally: V= V_1 vdots V_B .","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2d8b529eb2a0a517","tex":"V_b","aliases":"V_b","display":"V_b","kind":"use","excerpt":"[Streamed AV] AV = M_X ∑_b=1^B G_bW_b^1/2V_b.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2199b487ebbba079","tex":"V_{hg}^\\top","aliases":"V_hg^top top","display":"V_{hg}^\\top","kind":"use","excerpt":"If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"proof Expand (μ+SZ)^top M(μ+SZ) = Z^top SMSZ + 2μ^top MSZ + μ^top Mμ and rotate by V. Completing the square gives the displayed noncentral components. For a null vector of SMS, M^1/2Sv=0, hence v^top SMμ = (M^1/2Sv)^top M^1/2μ = 0, so there is no unresolved stochastic linear term in a zero-eigenvalue direction. For the positive eigenvectors v_i of ","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-44cf5118c3be9cba","tex":"\\vdots","aliases":"vdots","display":"\\vdots","kind":"use","excerpt":"Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"1 · -1 · -1 · 1 satisfies V^top K V=diag(1,1,2,2). For Z∼ N(0,I_4), Q=Z^top KZ overset d= Y+2W, Y,W independent with law χ^2_2. The spectrum is therefore (1,1,2,2) and E Q=6. The six-dimensional feature Gram A^top A has these same four nonzero eigenvalues and two additional zeros.","count":31},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-148db1d69b066230","tex":"W^{1/2}","aliases":"W^1/2","display":"W^{1/2}","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8e80dcfdce0dc44d","tex":"W_b","aliases":"W_b","display":"W_b","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-56f8bedcef5fce45","tex":"W_b^{1/2}","aliases":"W_b^1/2","display":"W_b^{1/2}","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f1e02a32b6b73575","tex":"W_{J_b,J_b}","aliases":"W_J_b,J_b","display":"W_{J_b,J_b}","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a6dc04e23090a2a5","tex":"w_m","aliases":"w_m","display":"w_m","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-454b338180ffce5a","tex":"\\operatorname{weightedSquares}","aliases":"weightedSquares operatorname","display":"\\operatorname{weightedSquares}","kind":"use","excerpt":"The separate definition weightedSquares(s,c,z)=∑_k∈ sc_k z_k^2 models a supplied diagonal quadratic form. Its coefficients need not be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"[K-COS analytic error certificate] Let p= P(Q≥ q), and let widehat p be the finite K-COS approximation formed with [label=(roman*)] item an upper interval endpoint b>q, item an integer N≥1 of cosine modes, item a packet approximation satisfying [source reference] on the finite grid. Under the positive-spectrum and parameter hypotheses of Propositio","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b0d65d8f4ac93228","tex":"\\widehat T","aliases":"widehat T widehat","display":"\\widehat T","kind":"use","excerpt":"[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c176826312b6d513","tex":"\\widehat T_{N,b}","aliases":"widehat T_N,b widehat","display":"\\widehat T_{N,b}","kind":"use","excerpt":"[Finite-grid packet-to-tail bound] Let widehat T_N,b be obtained from [source reference] by replacing κ(s_k) by widehatκ(s_k). Then |widehat T_N,b-T_N,b| ≤ ε_pkt, where ε_pkt = frac2b e^κ(θ)-θ q ∑_k=0^N-1' |J_k|(e^B_k-1).","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e8d6f1354dafb866","tex":"\\widehat x","aliases":"widehat x widehat","display":"\\widehat x","kind":"use","excerpt":"The interval module uses two elementary predicates: Enclosed(x,L,U) · :Longleftrightarrow L≤ x≤ U, ErrorBound(x,widehat x,r) · :Longleftrightarrow |x-widehat x|≤ r. The equivalence declerrorBound_iff_enclosed proves |x-widehat x|≤ r Longleftrightarrow widehat x-r≤ x≤widehat x+r. There is no extra nonnegativity hypothesis: if this predicate holds, t","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3d35f38f87e37bb6","tex":"\\widehat\\kappa","aliases":"widehatκ widehat kappa","display":"\\widehat\\kappa","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-78b0f0554573718c","tex":"\\widehat\\kappa_H","aliases":"widehatκ_H widehat kappa","display":"\\widehat\\kappa_H","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehatκ_H(θ)-θ q}, where widehatκ_H is the preceding MGF upper bound.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-48423f1cb94c99be","tex":"\\widetilde\\sigma","aliases":"widetildeσ widetilde sigma","display":"\\widetilde\\sigma","kind":"use","excerpt":"[Range-core singular-value certificate] For A∈ R^n× m, list the singular values of A and B in decreasing order and pad both lists with zeros to length n. Write these lists as σ_i(A) and widetildeσ_i(B), respectively. They are the square roots of the n eigenvalues of AA^top and (UB)(UB)^top. Then, for 1≤ i≤ n, | σ_i(A) - widetildeσ_i(B) | ≤ δ. Conse","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0ef696fbdaa82fc8","tex":"\\widetilde\\sigma_i","aliases":"widetildeσ_i widetilde sigma","display":"\\widetilde\\sigma_i","kind":"use","excerpt":"[Range-core singular-value certificate] For A∈ R^n× m, list the singular values of A and B in decreasing order and pad both lists with zeros to length n. Write these lists as σ_i(A) and widetildeσ_i(B), respectively. They are the square roots of the n eigenvalues of AA^top and (UB)(UB)^top. Then, for 1≤ i≤ n, | σ_i(A) - widetildeσ_i(B) | ≤ δ. Conse","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":66},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-323aaf429c48a253","tex":"X^{\\mathsf T}","aliases":"X^ T mathsf","display":"X^{\\mathsf T}","kind":"use","excerpt":"X=Σ^1/2Z, Z∼ N(0,I), H=Σ^1/2MΣ^1/2. X^ TMXoverset d=∑_j=1^rλ_jZ_j^2, λ_j∈spec(H)∖{0}. For a central Gaussian vector with covariance Σsucceq0 and symmetric M, the quadratic form becomes a weighted sum of independent standard-normal squares after whitening and orthogonal diagonalization. The operator may originate in a kernel, a signal model, or anot","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-bc02d46ac6b0666f","tex":"x^{-\\beta_N-1}","aliases":"x^-β_N-1 beta","display":"x^{-\\beta_N-1}","kind":"use","excerpt":"omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"ginproof Factor the determinant through the known core: I-2θ(C+R) = (I-2θ C)^1/2 (I-2θ T) (I-2θ C)^1/2, where T=(I-2θ C)^-1/2 R (I-2θ C)^-1/2. Since (I-2θ C)^-1/2 _2^2=α, we have trT≤α t, tr(T^2)≤α^2v, T _2≤αρ. For every eigenvalue x of T, -frac12log(1-2θ x) ≤ θ x + fracθ^2x^2 1-2θαρ. The scalar inequality follows from the power series of -log(1-u)","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9e2df13713b51ec7","tex":"x^j","aliases":"x^j","display":"x^j","kind":"use","excerpt":"For the control Q=0.02χ^2_80, the survival probability has the finite exact formula P(Q≥ q) = e^-x ∑_j=0^39 fracx^jj!, x=fracq0.04.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-bb0e8953f025926b","tex":"X^\\top","aliases":"X^top top","display":"X^\\top","kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7ef58ddaab7b5984","tex":"Y_1","aliases":"Y_1","display":"Y_1","kind":"use","excerpt":"Retained low-rank limitation. For independent Y_1,Y_2∼χ^2_2, the quadratic form Q=0.07(Y_1-Y_2) is Laplace distributed with scale 0.14, so for q≥0, P(Q≥ q)=frac12e^-q/0.14. The computed values agree with this exact law through 10^-200 to roughly 10^-9 decades or better. However, the implemented analytic COS-series certificate rejects these low-rank","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-57080b7d6f26d5e6","tex":"Y_2","aliases":"Y_2","display":"Y_2","kind":"use","excerpt":"Retained low-rank limitation. For independent Y_1,Y_2∼χ^2_2, the quadratic form Q=0.07(Y_1-Y_2) is Laplace distributed with scale 0.14, so for q≥0, P(Q≥ q)=frac12e^-q/0.14. The computed values agree with this exact law through 10^-200 to roughly 10^-9 decades or better. However, the implemented analytic COS-series certificate rejects these low-rank","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8e1b422661cf5616","tex":"Y_i","aliases":"Y_i","display":"Y_i","kind":"use","excerpt":"Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-95d510d0eb71e72d","tex":"Y_i^2","aliases":"Y_i^2","display":"Y_i^2","kind":"use","excerpt":"Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"s positive real part for each 1-2sλ_j and for 1-2sμ_P. The disk condition gives positive real part for 1-α_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing ov","count":12},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","excerpt":"z=q/0.04, P(Q≥ q)=e^-z∑_k=0^39fracz^kk!. log_10P(Q≥ q)=frac-z+log (∑_k=0^39z^k/k!)log(10). For an even-degree chi-square variable, the upper tail has a finite positive-sum formula. The control Q=0.02χ^2_80 is a gamma law of integer shape forty and scale 0.04. Its survival probability can therefore be evaluated independently of cosine inversion.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-56835621b2f38901","tex":"z^m","aliases":"z^m","display":"z^m","kind":"use","excerpt":"s positive real part for each 1-2sλ_j and for 1-2sμ_P. The disk condition gives positive real part for 1-α_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing ov","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-532408f4f67397a2","tex":"Z^\\top","aliases":"Z^top top","display":"Z^\\top","kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4ce8c5865fb8a8b6","tex":"z^\\top","aliases":"z^top top","display":"z^\\top","kind":"use","excerpt":"For each scenario, 100 Gaussian null residual draws were evaluated in three ways. Under the standardized Gaussian null, write e=M_Xz with z∼ N(0,I). Because M_XA=A, the statistic used here is Q=frac12 z^top Kz, so its nonzero mixture coefficients are frac12λ_j(K). The factor frac12 belongs to this case-study normalization, not to the general quadra","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-16723da64295ab54","tex":"Z_1^2","aliases":"Z_1^2","display":"Z_1^2","kind":"use","excerpt":"0 · 2 · 2. The large operator has eigenvalues 4,2,0. Its two-dimensional core has exactly the nonzero eigenvalues 4,2. The Gaussian quadratic form therefore has the law 4Z_1^2+2Z_2^2, whether it is computed from the full factor, the full Gram matrix, or the core.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1d38f9876f87b7f8","tex":"Z_2^2","aliases":"Z_2^2","display":"Z_2^2","kind":"use","excerpt":"0 · 2 · 2. The large operator has eigenvalues 4,2,0. Its two-dimensional core has exactly the nonzero eigenvalues 4,2. The Gaussian quadratic form therefore has the law 4Z_1^2+2Z_2^2, whether it is computed from the full factor, the full Gram matrix, or the core.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-bc1f10031646b5b4","tex":"Z_i","aliases":"Z_i","display":"Z_i","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-86b443a234736cf2","tex":"Z_i^2","aliases":"Z_i^2","display":"Z_i^2","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-660d17b19d61de5f","tex":"z_k^2","aliases":"z_k^2","display":"z_k^2","kind":"use","excerpt":"The separate definition weightedSquares(s,c,z)=∑_k∈ sc_k z_k^2 models a supplied diagonal quadratic form. Its coefficients need not be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ ","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"The packet representation is sign aware: positive and negative eigenvalues are clustered separately, and each same-sign packet stores a center μ and centered moments. The local expansion remains -frac12log(1-2s(μ+δ)) = -frac12log(1-2sμ) + frac12∑_n≥1 fracα(s)^nδ^nn, α(s)=frac2s1-2sμ, with the same geometric remainder condition |α(s)|max|δ|<1.","count":53},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6567adae36ef90db","tex":"|^{-2}","aliases":"|^-2","display":"|^{-2}","kind":"use","excerpt":"^2_2, the quadratic form Q=0.07(Y_1-Y_2) is Laplace distributed with scale 0.14, so for q≥0, P(Q≥ q)=frac12e^-q/0.14. The computed values agree with this exact law through 10^-200 to roughly 10^-9 decades or better. However, the implemented analytic COS-series certificate rejects these low-rank mixed cases because the tilted characteristic function","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-72b9314bf5ec7c99","tex":"|^{-r/2}","aliases":"|^-r/2","display":"|^{-r/2}","kind":"use","excerpt":"A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e347e328e4c7e3bd","tex":"|^m","aliases":"|^m","display":"|^m","kind":"use","excerpt":"alpha_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing over j∈ P gives [source reference]. The remainder satisfies |∑_m≥ p+1fracz^mm| ≤ ∑_m≥ p+1frac|z|^mp+1 =","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-df5bc58a9a56477e","tex":"|^{p+1}","aliases":"|^p+1","display":"|^{p+1}","kind":"use","excerpt":"alpha_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing over j∈ P gives [source reference]. The remainder satisfies |∑_m≥ p+1fracz^mm| ≤ ∑_m≥ p+1frac|z|^mp+1 =","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"The packet representation is sign aware: positive and negative eigenvalues are clustered separately, and each same-sign packet stores a center μ and centered moments. The local expansion remains -frac12log(1-2s(μ+δ)) = -frac12log(1-2sμ) + frac12∑_n≥1 fracα(s)^nδ^nn, α(s)=frac2s1-2sμ, with the same geometric remainder condition |α(s)|max|δ|<1.","count":17},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d830d9bc7e9c803c","tex":"\\alpha^2","aliases":"α^2 alpha","display":"\\alpha^2","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-69cd80b05cfde50a","tex":"\\alpha_P","aliases":"α_P alpha","display":"\\alpha_P","kind":"use","excerpt":"[Centered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P","count":8},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-eabc276ff3dd2e25","tex":"\\beta_{(1)}","aliases":"β_(1) beta","display":"\\beta_{(1)}","kind":"use","excerpt":"Suppose each retained phase pair has the same communication or scheduling cost, and fix a budget τ≥0. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f889f38b64a54aed","tex":"\\beta_{(2)}","aliases":"β_(2) beta","display":"\\beta_{(2)}","kind":"use","excerpt":"Suppose each retained phase pair has the same communication or scheduling cost, and fix a budget τ≥0. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fd8f34ecec6238ae","tex":"\\beta_{(i)}","aliases":"β_(i) beta","display":"\\beta_{(i)}","kind":"use","excerpt":"[Cardinality-optimal certified pruning] Let r∈{0,…,N} be the largest integer satisfying ∑_i=1^rβ_(i)≤τ. Then omitting the r least-risk phase pairs is feasible, and no feasible omitted set can contain more than r pairs under the scalar additive certificate.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3138ed3758d4a624","tex":"\\beta_{(N)}","aliases":"β_(N) beta","display":"\\beta_{(N)}","kind":"use","excerpt":"Suppose each retained phase pair has the same communication or scheduling cost, and fix a budget τ≥0. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-3cf40acf5b62ff4c","tex":"\\beta_e^{(p)}","aliases":"β_e^(p) beta","display":"\\beta_e^{(p)}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7d08544089d2bb4b","tex":"\\beta_N","aliases":"β_N beta","display":"\\beta_N","kind":"use","excerpt":"[Omitted COS-mode certificate] Let N≥1 and b>q≥0. Set u_N=Nπ/b and β_N=β(u_N). If β_N>0, then the contribution of all exact tilted-COS modes k≥ N to the weighted tail functional is bounded by ε_COS ≤ e^κ(θ)-θ q frac4|ϕ_θ(u_N)|π (frac1N+frac1β_N).","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c2a7a6d9761d3a05","tex":"\\delta^n","aliases":"δ^n delta","display":"\\delta^n","kind":"use","excerpt":"The packet representation is sign aware: positive and negative eigenvalues are clustered separately, and each same-sign packet stores a center μ and centered moments. The local expansion remains -frac12log(1-2s(μ+δ)) = -frac12log(1-2sμ) + frac12∑_n≥1 fracα(s)^nδ^nn, α(s)=frac2s1-2sμ, with the same geometric remainder condition |α(s)|max|δ|<1.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0739577033095c8a","tex":"\\delta_j^m","aliases":"δ_j^m delta","display":"\\delta_j^m","kind":"use","excerpt":"Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a11e9f1487b2c505","tex":"\\Delta_k","aliases":"Δ_k Delta","display":"\\Delta_k","kind":"use","excerpt":"Write Δ_k=widehatκ(s_k)-κ(s_k). Then |e^widehatκ(s_k)-θ q-e^κ(s_k)-θ q| ≤ |e^κ(s_k)-θ q| (e^|Δ_k|-1). Moreover Reκ(θ+i u_k)-κ(θ) =log|ϕ_θ(u_k)|≤0, so |e^κ(s_k)-θ q| ≤ e^κ(θ)-θ q. Substitute this into the finite cosine sum and apply the triangle inequality.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ae06e15b8d980ead","tex":"\\varepsilon_k","aliases":"ε_k varepsilon","display":"\\varepsilon_k","kind":"use","excerpt":"After separating the likelihood-ratio prefactor, the packet contribution contains sums of the form frac2b ∑_k |J_k| ( e^ε_k-1 ). Direct evaluation can overflow even when the candidate is destined to be rejected by the adaptive planner. The implementation instead computes log( frac2b ∑_k |J_k| ( e^ε_k-1 ) ) by log-sum-exp arithmetic, and compares th","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-6a6b220199468edf","tex":"\\varepsilon_{\\rm req}","aliases":"ε_rm req varepsilon rm","display":"\\varepsilon_{\\rm req}","kind":"use","excerpt":"For an exact positive spectrum and threshold q, the solver accepts a requested relative multiplicative accuracy ε_rm req>0 and returns a certified interval [p_-,p_+] only when fracp_+p_--1 ≤ ε_rm req.","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e1948b35752cbf72","tex":"\\varepsilon_{\\rm requested}","aliases":"ε_rm requested varepsilon rm","display":"\\varepsilon_{\\rm requested}","kind":"use","excerpt":"Accuracy-driven positive tail solver · certified / fail-closed · A result is accepted only after p_+/p_- - 1≤ε_rm requested is proved.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponenti","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8a5c59e2a54d0dba","tex":"\\kappa''","aliases":"κ'' kappa","display":"\\kappa''","kind":"use","excerpt":"[leftmargin=2em] item Solve κ'(θ_*)=q. item Initialize an upper interval b=q+Lsqrtκ''(θ_*)+sqrtκ^(4)(θ_*) and a mode count N. item Retain the largest spectral spikes exactly and place the remaining sorted eigenvalues into coarse packets. item Evaluate ρ_P(θ_*+i u_k) and the packet remainder bound on the actual COS grid. Split packets until the prop","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-790a1358163da94b","tex":"\\kappa^{(4)}","aliases":"κ^(4) kappa","display":"\\kappa^{(4)}","kind":"use","excerpt":"[leftmargin=2em] item Solve κ'(θ_*)=q. item Initialize an upper interval b=q+Lsqrtκ''(θ_*)+sqrtκ^(4)(θ_*) and a mode count N. item Retain the largest spectral spikes exactly and place the remaining sorted eigenvalues into coarse packets. item Evaluate ρ_P(θ_*+i u_k) and the packet remainder bound on the actual COS grid. Split packets until the prop","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d2058ade960a4e7e","tex":"\\kappa_C","aliases":"κ_C kappa","display":"\\kappa_C","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9632b8029387d9d3","tex":"\\kappa_H","aliases":"κ_H kappa","display":"\\kappa_H","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, κ_H(θ) =-frac12logdet(I-2θ H) satisfies κ_H(θ) ≤ κ_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-766409f41c22eb2f","tex":"\\kappa_P","aliases":"κ_P kappa","display":"\\kappa_P","kind":"use","excerpt":"tered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P,p(s","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"Write X=SZ with Z∼ N(0,I) and diagonalize SMS=VΛ V^top. Rotational invariance of the standard normal gives Z^top VΛ V^top Z overset d= ∑_jλ_jZ_j^2. No positivity assumption on M is needed for this central reduction.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"use","excerpt":"Let X∼ N(μ,Σ), Msucceq0. Again set S=Σ^1/2 and diagonalize SMS=VΛ V^top. Writing c=V^top SMμ, each positive-eigenvalue direction satisfies λ_iY_i^2+2c_iY_i = λ_i ( Y_i+fracc_iλ_i )^2 - fracc_i^2λ_i.","count":18},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7d9933ae3ee9c1cb","tex":"\\lambda_j^{-h_j/2}","aliases":"λ_j^-h_j/2 lambda","display":"\\lambda_j^{-h_j/2}","kind":"use","excerpt":"The four spectra include balanced, skewed, dominant-spike, and higher-rank cases. Each reference integral was evaluated at two truncation limits. The omitted integral after the frequency cutoff T>0 was bounded using frac1π ∫_T^∞fracduuR(u) ≤ fracCπνT^-ν, ν=frac12∑_jh_j, C=∏_jλ_j^-h_j/2.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-36442f76e90ff174","tex":"\\lambda_r","aliases":"λ_r lambda","display":"\\lambda_r","kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c61d60614d8a51af","tex":"\\mu^\\top","aliases":"μ^top mu top","display":"\\mu^\\top","kind":"use","excerpt":"[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). T","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-252454ad397ec5a8","tex":"\\mu_i","aliases":"μ_i mu","display":"\\mu_i","kind":"use","excerpt":"Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ad4e1804d97e3fe2","tex":"\\mu_P","aliases":"μ_P mu","display":"\\mu_P","kind":"use","excerpt":"Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-251ea3a3e1a96024","tex":"\\nu_\\theta","aliases":"ν_θ nu theta","display":"\\nu_\\theta","kind":"use","excerpt":"[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","excerpt":"For depth k≥1 and starting phase profile ρ_0=ρ(X), define the edge risk vector q_e^(k) = ∑_j=0^k-1 B^k-1-jΔ_e B^jρ_0 and its scalar risk β_e^(k)=lVert q_e^(k)rVert_2. Unlike the raw block norm B_hg, this quantity depends on the start profile and on Krylov depth.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8239c91a9dd24610","tex":"\\rho_R","aliases":"ρ_R rho","display":"\\rho_R","kind":"use","excerpt":"A=UB+F, U^ TF=0, A^ TA=B^ TB+F^ TF. t=trR= F _F^2, v=tr(R^2), ρ_R= R _2. Let U contain d orthonormal sample-space directions. Project the factor into its captured range coordinates B=U^ TA and residual F=(I-UU^ T)A. Orthogonality eliminates the cross terms in the feature-space Gram matrix.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"[Range-core singular-value certificate] For A∈ R^n× m, list the singular values of A and B in decreasing order and pad both lists with zeros to length n. Write these lists as σ_i(A) and widetildeσ_i(B), respectively. They are the square roots of the n eigenvalues of AA^top and (UB)(UB)^top. Then, for 1≤ i≤ n, | σ_i(A) - widetildeσ_i(B) | ≤ δ. Conse","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-40db2622b698a4f9","tex":"\\Sigma^{1/2}","aliases":"Σ^1/2 Sigma","display":"\\Sigma^{1/2}","kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0ed845e79c8d8e41","tex":"\\sigma_i","aliases":"σ_i sigma","display":"\\sigma_i","kind":"use","excerpt":"[Range-core singular-value certificate] For A∈ R^n× m, list the singular values of A and B in decreasing order and pad both lists with zeros to length n. Write these lists as σ_i(A) and widetildeσ_i(B), respectively. They are the square roots of the n eigenvalues of AA^top and (UB)(UB)^top. Then, for 1≤ i≤ n, | σ_i(A) - widetildeσ_i(B) | ≤ δ. Conse","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4a895ee28e0d67c3","tex":"\\tau_K","aliases":"τ_K tau","display":"\\tau_K","kind":"use","excerpt":"An execution schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a6d23f9e2ee3e0f6","tex":"\\tau_{\\theta}","aliases":"τ_θ tau theta","display":"\\tau_{\\theta}","kind":"use","excerpt":"[Support and periodization bound] Let the finite positive spectrum have rank r≥1, and let b>q≥0, b>0, and 0<θ<1/(2λ_max). Let τ_θ(b)= P_θ(Q>b). Then the combined error from (i) replacing the tilted density on [0,b] by its full-coefficient cosine periodization and (ii) omitting the original integral over (b,∞) is bounded by ε_sup ≤ e^κ(θ)-θ q τ_θ(b)","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7a18b4081386ab4c","tex":"\\tau_\\lambda","aliases":"τ_λ tau lambda","display":"\\tau_\\lambda","kind":"use","excerpt":"An execution schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.","count":21},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a0e60aa893f341f8","tex":"\\phi_\\theta","aliases":"ϕ_θ phi theta","display":"\\phi_\\theta","kind":"use","excerpt":"A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fccdef90213bcd7a","tex":"\\phi_{\\theta}","aliases":"ϕ_θ phi theta","display":"\\phi_{\\theta}","kind":"use","excerpt":"Define the exponentially tilted law by f_θ(x)=e^θ x-κ(θ)f(x). Its characteristic function is ϕ_θ(t) =exp{κ(θ+i t)-κ(θ)}, and E_θQ=κ'(θ). Therefore the choice [source reference] moves the query q to the mean of the tilted law.","count":12},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). T","count":17},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-11fdaa25e327864a","tex":"\\chi^2","aliases":"χ^2 chi","display":"\\chi^2","kind":"use","excerpt":"Storage blocking controls residence. Structural phases encode linkage-disequilibrium (LD) windows, annotations, frequency bands, or other coordinate groups. Krylov methods compress the operator to the rank actually requested. Spectral packets compress the final generalized-χ^2 transform.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a731e2bb165ef773","tex":"\\chi^2_1","aliases":"χ^2_1 chi","display":"\\chi^2_1","kind":"use","excerpt":"[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). T","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-183660d7c089343d","tex":"\\chi^2_2","aliases":"χ^2_2 chi","display":"\\chi^2_2","kind":"use","excerpt":"1 · -1 · -1 · 1 satisfies V^top K V=diag(1,1,2,2). For Z∼ N(0,I_4), Q=Z^top KZ overset d= Y+2W, Y,W independent with law χ^2_2. The spectrum is therefore (1,1,2,2) and E Q=6. The six-dimensional feature Gram A^top A has these same four nonzero eigenvalues and two additional zeros.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5239bf8deb430b9e","tex":"\\chi^2_{80}","aliases":"χ^2_80 chi","display":"\\chi^2_{80}","kind":"use","excerpt":"For the control Q=0.02χ^2_80, the survival probability has the finite exact formula P(Q≥ q) = e^-x ∑_j=0^39 fracx^jj!, x=fracq0.04.","count":9},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a79be8283260adde","tex":"\\chi^2_d","aliases":"χ^2_d chi","display":"\\chi^2_d","kind":"use","excerpt":"The numerical interface implements this reduction directly. In an isotropic control, Σ=I, M=aI, a>0, the law collapses to the exact scaled noncentral chi-square distribution Q/a∼χ^2_d( μ ^2).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f71176b4c504c459","tex":"\\ell_2","aliases":"ℓ_2 ell","display":"\\ell_2","kind":"use","excerpt":"Hierarchical refinement gave relative discrepancy 3.17×10^-16. For H=L^top L with block-bidiagonal L, the smallest eigenvalue was approximately 2.86 and the largest norm outside the exact light cone was zero. The QR-core singular-spectrum discrepancy was 5.89×10^-16 in relative ℓ_2 norm. These are finite controls, not universal floating-point error","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"use","excerpt":"[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"in endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain whenever a positive spectral component is present. The negative-only case with at least one nonzero coefficient has a positive saddlepoint only for E[Q]<q<0. Indeed, κ'(θ) increases from E[Q] to zero as θ→∞. For q≥0 ","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Since A=Q_rB and Q_r^top Q_r=I, one has AA^top=Q_r(BB^top)Q_r^top. The map x↦ Q_rx is an isometry from the core space onto range(Q_r). On that range the two operators are conjugate, and AA^top vanishes on its orthogonal complement. Hence their nonzero eigenvalues agree with multiplicity. If the captured range equals range(A), then B has full row ra","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"ot be eigenvalues obtained by a proved diagonalization. It has three theorems: item declweightedSquares_mono: if l_k≤ u_k for every k∈ s, then ∑ l_kz_k^2≤∑ u_kz_k^2. item declweightedSquares_nonneg: if c_k≥0 for every k∈ s, then ∑ c_kz_k^2≥0. item declweightedSquares_tail_imp: coefficientwise order gives the pointwise implication q≤∑ l_kz_k^2⇒ q≤∑ ","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"For every phase h∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Every path of length at most k beginning in F_0 stays in F_k. Let J_F be the coordinate inclusion from the retained coordinate","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2d3db36262598bcd","tex":"\\prod_j","aliases":"∏_j prod","display":"\\prod_j","kind":"use","excerpt":"A useful distinction appears in the Fourier tail. Under the saddlepoint tilt, |ϕ_θ(t)| = ∏_j [ 1+ ( frac2λ_j t1-2θλ_j )^2 ]^-1/4. If r coefficients are nonzero, then asymptotically |ϕ_θ(t)|=O(|t|^-r/2). Thus certification difficulty depends strongly on spectral rank, not only on how small the requested probability is.","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-aa6ac22062ba0415","tex":"\\prod_{j=1}^r","aliases":"∏_j=1^r prod","display":"\\prod_{j=1}^r","kind":"use","excerpt":"Define a_j= frac2λ_j1-2θλ_j. The normalized tilted characteristic function satisfies |ϕ_θ(t)| = ∏_j=1^r ( 1+(a_jt)^2 )^-1/4.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-08e723782301ff4e","tex":"\\prod_{j=1}^{r}","aliases":"∏_j=1^r prod","display":"\\prod_{j=1}^{r}","kind":"use","excerpt":"[Tilted characteristic-function modulus] Define a_j=frac2λ_j1-2θλ_j>0. Then |ϕ_θ(t)| =∏_j=1^r(1+(a_jt)^2)^-1/4. For t>0, define the logarithmic decay rate β(t) =-fracdlog|ϕ_θ(t)|dlog t =frac12∑_j=1^r frac(a_jt)^21+(a_jt)^2. Set β(0)=0 by continuous extension. Then β(t) is nondecreasing for t≥0.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2f0d826a23cff58c","tex":"\\sum_{(h,g)\\in\\mathcal E_0}","aliases":"∑_(h,g)∈ E_0 sum in mathcal","display":"\\sum_{(h,g)\\in\\mathcal E_0}","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","excerpt":"The same central quadratic-form law may be supplied by any of the following: item a nonnegative spectrum {λ_j}. item a positive semidefinite matrix H. item a factor A with H=AA^top or H=A^top A. item a replayable streamed factor A=[A_1 ⋯ A_B]. The nonzero spectra of AA^top and A^top A are identical, and AA^top=∑_bA_bA_b^top.","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d724b548ac80a603","tex":"\\sum_{b=1}^B","aliases":"∑_b=1^B sum","display":"\\sum_{b=1}^B","kind":"use","excerpt":"[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4e3e65f97e51859c","tex":"\\sum_{b=1}^{B}","aliases":"∑_b=1^B sum","display":"\\sum_{b=1}^{B}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-ce18b6ca08050486","tex":"\\sum_{b=1}^{B-1}","aliases":"∑_b=1^B-1 sum","display":"\\sum_{b=1}^{B-1}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2fb84539c83f1f39","tex":"\\sum_{d=0}^{D}","aliases":"∑_d=0^D sum","display":"\\sum_{d=0}^{D}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-06344775b94820a0","tex":"\\sum_{d=1}^{D}","aliases":"∑_d=1^D sum","display":"\\sum_{d=1}^{D}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-76e758f1be5158a2","tex":"\\sum_g","aliases":"∑_g sum","display":"\\sum_g","kind":"use","excerpt":"Using ∑_gP_g=I and submultiplicativity, lVert P_hHXrVert_F · =lVert∑_gP_hHP_gXrVert_F","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-20794c41f8dd0413","tex":"\\sum_h","aliases":"∑_h sum","display":"\\sum_h","kind":"use","excerpt":"Let the phase projectors form an orthogonal decomposition of variant space: P_h^top=P_h=P_h^2, P_hP_g=0 (h≠ g), ∑_hP_h=I. Thus P_h projects onto variant phase h. Again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"[Exact noncentral reduction] For the positive eigenvalues λ_i>0, Q overset d= q_0 + ∑_i λ_i χ^2_1(δ_i), where δ_i= (fracc_iλ_i)^2 and q_0 = μ^top Mμ - ∑_ifracc_i^2λ_i ≥0. The noncentral chi-square components are independent. To state the zero-shift condition precisely, put T=M^1/2S, b=M^1/2μ, and let P_T be the orthogonal projector onto range(T). T","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2874dc45e47106fd","tex":"\\sum_{i=1}^{r}","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^{r}","kind":"use","excerpt":"[Cardinality-optimal certified pruning] Let r∈{0,…,N} be the largest integer satisfying ∑_i=1^rβ_(i)≤τ. Then omitting the r least-risk phase pairs is feasible, and no feasible omitted set can contain more than r pairs under the scalar additive certificate.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e4c6762a80e52f72","tex":"\\sum_{i=1}^{r+1}","aliases":"∑_i=1^r+1 sum","display":"\\sum_{i=1}^{r+1}","kind":"use","excerpt":"If r=N, every candidate pair can be omitted and optimality is immediate. Otherwise, among all subsets of cardinality r+1, the smallest possible sum of risks is ∑_i=1^r+1β_(i). By maximality of r, that sum exceeds τ. Hence no set of r+1 omitted pairs is feasible. The prefix of length r is feasible by construction.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-15292da14c260471","tex":"\\sum_{i=1}^s","aliases":"∑_i=1^s sum","display":"\\sum_{i=1}^s","kind":"use","excerpt":"The inequality t≤ sρ gives s≥ t/ρ. Cauchy--Schwarz gives t^2 = (∑_i=1^s r_i)^2 ≤ s∑_i=1^s r_i^2 =sv, so s≥ t^2/v. Finally, every column of F lies in range(U)^⊥, whose dimension is n-r.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7fec58bf80198971","tex":"\\sum_{i\\in I}","aliases":"∑_i∈ I sum in","display":"\\sum_{i\\in I}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b98158348cda32a4","tex":"\\sum_{J'\\in\\mathcal J'}","aliases":"∑_J'∈ J' sum in mathcal","display":"\\sum_{J'\\in\\mathcal J'}","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4f3d7ceb5a12514b","tex":"\\sum_{j=0}^{39}","aliases":"∑_j=0^39 sum","display":"\\sum_{j=0}^{39}","kind":"use","excerpt":"For the control Q=0.02χ^2_80, the survival probability has the finite exact formula P(Q≥ q) = e^-x ∑_j=0^39 fracx^jj!, x=fracq0.04.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b845f4dcb81f67c5","tex":"\\sum_{j=0}^d","aliases":"∑_j=0^d sum","display":"\\sum_{j=0}^d","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-02998aeb731bff1f","tex":"\\sum_{j=0}^{k-1}","aliases":"∑_j=0^k-1 sum","display":"\\sum_{j=0}^{k-1}","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f335dc3560fec8f6","tex":"\\sum_{j=1}^3","aliases":"∑_j=1^3 sum","display":"\\sum_{j=1}^3","kind":"use","excerpt":"For the declared vector v=(1,2,-1,0)^top, the action can be evaluated without forming any block Gram matrix: ∑_j=1^3 A_j(A_j^top v)=Kv=(2,3,-2,-1)^top. The symbolic check confirms both storage partitions. Every column remains represented. Dropping an operator coupling instead changes the operator and requires the error analysis developed next.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-dec62b92295c8c6d","tex":"\\sum_{j=1}^d","aliases":"∑_j=1^d sum","display":"\\sum_{j=1}^d","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8650842fbc4c4680","tex":"\\sum_{j=1}^r","aliases":"∑_j=1^r sum","display":"\\sum_{j=1}^r","kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-96c7cad8168579f9","tex":"\\sum_{j\\in I}","aliases":"∑_j∈ I sum in","display":"\\sum_{j\\in I}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-bea1c4ac9cd0066f","tex":"\\sum_{J\\in\\mathcal J}","aliases":"∑_J∈ J sum in mathcal","display":"\\sum_{J\\in\\mathcal J}","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-547322240e47c5c2","tex":"\\sum_{j\\in P}","aliases":"∑_j∈ P sum in","display":"\\sum_{j\\in P}","kind":"use","excerpt":"Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"After separating the likelihood-ratio prefactor, the packet contribution contains sums of the form frac2b ∑_k |J_k| ( e^ε_k-1 ). Direct evaluation can overflow even when the candidate is destined to be rejected by the adaptive planner. The implementation instead computes log( frac2b ∑_k |J_k| ( e^ε_k-1 ) ) by log-sum-exp arithmetic, and compares th","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-50a7d52192d79920","tex":"\\sum_{k=0}^{39}","aliases":"∑_k=0^39 sum","display":"\\sum_{k=0}^{39}","kind":"use","excerpt":"z=q/0.04, P(Q≥ q)=e^-z∑_k=0^39fracz^kk!. log_10P(Q≥ q)=frac-z+log (∑_k=0^39z^k/k!)log(10). For an even-degree chi-square variable, the upper tail has a finite positive-sum formula. The control Q=0.02χ^2_80 is a gamma law of integer shape forty and scale 0.04. Its survival probability can therefore be evaluated independently of cosine inversion.","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7796ee229bc8c35a","tex":"\\sum_{k=0}^{N-1}","aliases":"∑_k=0^N-1 sum","display":"\\sum_{k=0}^{N-1}","kind":"use","excerpt":"If the exact tilted characteristic function is used, the N-mode finite-window expression is T_N,b =frac2b∑_k=0^N-1' J_k Re[e^κ(θ+i u_k)-θ q]. The cancellation of κ(θ) in [source reference] is useful numerically: the likelihood-ratio prefactor from [source reference] and the normalization in [source reference] do not need to be formed separately ins","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-f1c2ff812e2b7828","tex":"\\sum_{k=0}^{\\infty}","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^{\\infty}","kind":"use","excerpt":"[Integrated cosine identity by folding] Let ν_θ be the tilted probability measure and let F_b:[0,∞)→[0,b] fold each interval of length 2b by reflection. Then cos(kπ F_b(x)/b)=cos(kπ x/b). For h(x)=e^-θ(x-q)1_[q,b](x), ∫ h d(F_b#ν_θ) =frac2b∑_k=0^∞' J_k Reϕ_θ(kπ/b). The series is absolutely convergent for every positive rank r≥1. A positive weighted","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-81f704903555aa2a","tex":"\\sum_{k=N}^{\\infty}","aliases":"∑_k=N^∞ sum infty","display":"\\sum_{k=N}^{\\infty}","kind":"use","excerpt":"Using the cosine coefficient magnitude 2|ϕ_θ(u_k)|/b and [source reference], the omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-7b6df6a3f5f3d6d7","tex":"\\sum_{k\\in s}","aliases":"∑_k∈ s sum in","display":"\\sum_{k\\in s}","kind":"use","excerpt":"There are five central finite-algebra definitions: gramEntry(s,A,i,j) · =∑_k∈ sA_ikA_jk, [6pt] matrixAction(K,v,i) · =∑_j∈ IK_ijv_j, [6pt] streamedAction(s,A,v,i) · =∑_k∈ sA_ik∑_j∈ IA_jkv_j, [6pt] quadraticForm(K,v) · =∑_i∈ Iv_imatrixAction(K,v,i), [6pt] factorEnergy(s,A,v) · =∑_k∈ s(∑_i∈ IA_ikv_i)^2. The source uses ordinary finite sums in R. The ","count":4},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c0dd06184d3077e8","tex":"\\sum_{m=1}^{p}","aliases":"∑_m=1^p sum","display":"\\sum_{m=1}^{p}","kind":"use","excerpt":"tered packet expansion] Fix s∈ C with Re s<1/(2λ_max), use the principal logarithm, and define α_P(s)=frac2s1-2sμ_P, ρ_P(s)=|α_P(s)|max_j∈ P|δ_j|. For a nonnegative integer expansion order p, if ρ_P(s)<1, then the packet contribution to [source reference] is κ_P(s) · =-fracn_P2log(1-2sμ_P) +frac12∑_m=1^p fracα_P(s)^mmM_P,m +R_P,p(s), where |R_P,p(s","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-8db2d82c09fc913a","tex":"\\sum_{m\\ge p+1}","aliases":"∑_m≥ p+1 sum ge","display":"\\sum_{m\\ge p+1}","kind":"use","excerpt":"alpha_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing over j∈ P gives [source reference]. The remainder satisfies |∑_m≥ p+1fracz^mm| ≤ ∑_m≥ p+1frac|z|^mp+1 =","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a0a9cffbc383b69e","tex":"\\sum_{m\\ge1}","aliases":"∑_m≥1 sum ge","display":"\\sum_{m\\ge1}","kind":"use","excerpt":"s positive real part for each 1-2sλ_j and for 1-2sμ_P. The disk condition gives positive real part for 1-α_Pδ_j. The two factor arguments therefore sum inside (-π,π), so their principal logarithms add without a branch ambiguity. Hence -frac12log(1-2sλ_j) =-frac12log(1-2sμ_P) -frac12log(1-α_Pδ_j). When |α_Pδ_j|<1, -log(1-z)=∑_m≥1fracz^mm. Summing ov","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"The packet representation is sign aware: positive and negative eigenvalues are clustered separately, and each same-sign packet stores a center μ and centered moments. The local expansion remains -frac12log(1-2s(μ+δ)) = -frac12log(1-2sμ) + frac12∑_n≥1 fracα(s)^nδ^nn, α(s)=frac2s1-2sμ, with the same geometric remainder condition |α(s)|max|δ|<1.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-25fbceb9ec5c944a","tex":"\\sum_{\\{h,g\\}\\in\\mathcal O}","aliases":"∑_{h,g}∈ O sum in mathcal","display":"\\sum_{\\{h,g\\}\\in\\mathcal O}","kind":"use","excerpt":". For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained. The corresponding pruned operator is H_ O =H-∑_{h,g}∈ O (P_hHP_g+P_gHP_h). Thus B_ O is its block-norm transfer matrix.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"X=Σ^1/2Z, Z∼ N(0,I), H=Σ^1/2MΣ^1/2. X^ TMXoverset d=∑_j=1^rλ_jZ_j^2, λ_j∈spec(H)∖{0}. For a central Gaussian vector with covariance Σsucceq0 and symmetric M, the quadratic form becomes a weighted sum of independent standard-normal squares after whitening and orthogonal diagonalization. The operator may originate in a kernel, a signal model, or anot","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"The solver separates four logically different representations: storage blocks ∘ structural phases ∘ Krylov/range compression ∘ spectral tail packets. Storage blocks answer where data reside. Structural phases answer how the operator propagates. Krylov/range compression answers which subspace has actually been resolved. Spectral packets answer how t","count":3},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"For the right-tail path, the cumulant generating function is κ(s) = -frac12∑_jlog(1-2sλ_j), with real domain max_λ_j<0frac12λ_j < s < min_λ_j>0frac12λ_j. Here a missing negative or positive spectral set gives a domain endpoint -∞ or +∞, respectively. When q>E[Q]=∑_jλ_j, the saddlepoint is the unique positive solution of κ'(θ)=q inside this domain w","count":17},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"The principal refinement theorem is declgramEntry_union. Its hypotheses are finite sets s,t⊆ V and Disjoint(s,t). It proves, for all sample coordinates i,j, gramEntry(s∪ t,A,i,j) =gramEntry(s,A,i,j)+gramEntry(t,A,i,j). Disjointness prevents repeated columns from being counted twice. It is a mathematical condition, not a performance assumption. The ","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b0f45e1bbdcf0ea7","tex":"\\int_0^x","aliases":"∫_0^x int","display":"\\int_0^x","kind":"use","excerpt":"Here the probability law also has a closed form. The densities of Y and 2W are tfrac12e^-x/2 and tfrac14e^-x/4 on the nonnegative half-line. Their convolution is f_Q(x)=∫_0^xfrac12e^-y/2frac14e^-(x-y)/4 dy =frac12(e^-x/4-e^-x/2). Integrating from q to infinity gives P(Q≥ q)=2e^-q/4-e^-q/2, q≥0. At q=40 the probability is approximately 9.079779837×1","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"The reference evaluates the original characteristic function directly through the central Imhof inversion identity P(Q≥ q) = frac12 + frac1π ∫_0^∞ fracsinΘ(u)u R(u) du, where, for distinct positive coefficients λ_j with multiplicities h_j, Θ(u) = frac12∑_j h_jarctan(λ_j u) - frac12qu, R(u) = ∏_j (1+λ_j^2u^2)^h_j/4. This route does not use exponenti","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-1eda1172931ab90f","tex":"\\int_N^{\\infty}","aliases":"∫_N^∞ int infty","display":"\\int_N^{\\infty}","kind":"use","excerpt":"omitted series is at most e^κ(θ)-θ q frac4π ∑_k=N^∞frac|ϕ_θ(u_k)|k. By Corollary [source reference], |ϕ_θ(u_k)| ≤ |ϕ_θ(u_N)|(frackN)^-β_N. Therefore ∑_k=N^∞frac|ϕ_θ(u_k)|k ≤ |ϕ_θ(u_N)|N^β_N ∑_k=N^∞k^-β_N-1. Finally, ∑_k=N^∞k^-β_N-1 ≤ N^-β_N-1+∫_N^∞x^-β_N-1 dx =N^-β_N-1+fracN^-β_Nβ_N, which yields [source reference].","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-e052eebe752244d0","tex":"\\int_T^\\infty","aliases":"∫_T^∞ int infty","display":"\\int_T^\\infty","kind":"use","excerpt":"The four spectra include balanced, skewed, dominant-spike, and higher-rank cases. Each reference integral was evaluated at two truncation limits. The omitted integral after the frequency cutoff T>0 was bounded using frac1π ∫_T^∞fracduuR(u) ≤ fracCπνT^-ν, ν=frac12∑_jh_j, C=∏_jλ_j^-h_j/2.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"Let X∼ N(0,Σ), Msucceq0, and write S=Σ^1/2. With X=SZ, Z∼ N(0,I), Q=X^top MX = Z^top(SMS)Z. If the nonzero eigenvalues of the positive semidefinite operator H=SMS are λ_1,…,λ_r, then Qoverset d= ∑_j=1^rλ_jZ_j^2.","count":11},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"The reindexing theorem declgramEntry_reindex handles a bijection e:V≃ V. It moves the block to e(s) and transforms the factor to A'_ik=A_i,e^-1(k). The resulting Gram entry is unchanged. Permuting only the coordinates without moving their factor columns falls outside this statement.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"The running example under an exponential tilt. For spectrum (1,1,2,2), the cumulant generator is κ(s)=-log(1-2s)-log(1-4s) on Re s<1/4. At q=40, the saddlepoint equation reduces to 160θ^2-112θ+17=0, with frac21-2θ+frac41-4θ=40, θ_* = frac14-sqrt2640≈0.2225245122. The other root lies outside the transform domain. The tilted mean is 40, while the lik","count":6},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"If AA^top u=λ u with λ>0, then A^top u≠0 and A^top A(A^top u)=λ A^top u. Conversely, an eigenvector v of A^top A with positive eigenvalue maps to the eigenvector Av of AA^top. These maps restrict to inverse isomorphisms of the corresponding eigenspaces after scaling, so multiplicities agree. Horizontal concatenation gives AA^top=∑_b A_bA_b^top by d","count":5},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).","count":10},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).","count":2},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"Let a nonempty packet P⊂{1,…,r} contain n_P≥1 eigenvalues. Define its center μ_P=frac1n_P∑_j∈ Pλ_j, δ_j=λ_j-μ_P, and centered moments M_P,m=∑_j∈ Pδ_j^m, m≥0. Thus M_P,0=n_P and M_P,1=0.","count":1},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥, Q_⊥^top Q_⊥=I, Q^top Q_⊥=0. Here the columns of Q_⊥ form an orthonormal basis of the nonzero residual range. Zero residual directions are omitted.","count":7},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"This is the key composition law: mathematical locality ⟶ active phases ⟶ active storage blocks.","count":8},{"work":"monographs-cos-method","url":"notation/monographs-cos-method.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"The resulting object is query dependent. Schematically, (Λ,q,τ) ⟼ P_Λ,q,τ ⟼ (widehat p,certificate), where P_Λ,q,τ stores exact spikes, packet centers, centered moments, packet radii, and the selected COS parameters.","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"notation-table","excerpt":"Ξ, B · An evaluation contract and its resource allowance.","count":50},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"notation-table","excerpt":"B_0⊆ X · Possible current states of a finite repair problem.","count":18},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"notation-table","excerpt":"e, r, P · An experiment, a terminal repair, and an executable diagnostic tree.","count":20},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"notation-table","excerpt":"F_h(B_0) · Nondominated execution/storage pairs for successful trees of depth at most h.","count":8},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e982a068c0aac4ff","tex":"F_h","aliases":"F_h","display":"F_h","kind":"notation-table","excerpt":"F_h(B_0) · Nondominated execution/storage pairs for successful trees of depth at most h.","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"notation-table","excerpt":"F_h(B_0) · Nondominated execution/storage pairs for successful trees of depth at most h.","count":30},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"notation-table","excerpt":"K, W_h,k · A safe continuation set and states guaranteeing k progress events within h attempts.","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"notation-table","excerpt":"K, W_h,k · A safe continuation set and states guaranteeing k progress events within h attempts.","count":26},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","excerpt":"midrule S, U, L · A target solver, an improvement procedure, and a reusable library.","count":15},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"e, r, P · An experiment, a terminal repair, and an executable diagnostic tree.","count":16},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"notation-table","excerpt":"e, r, P · An experiment, a terminal repair, and an executable diagnostic tree.","count":23},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3bce33d4a711b894","tex":"\\Rcal","aliases":"Rcal R mathcal","display":"{\\mathcal R}","kind":"notation-table","excerpt":"R_Ξ(U,L;B) · Expected improvement produced by the procedure, not an oracle over its unselected candidates.","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-bdc7ac27c66dc39c","tex":"\\Rcal_\\Xi","aliases":"Rcal_Ξ R_Ξ Rcal Xi mathcal Xi","display":"{\\mathcal R}_\\Xi","kind":"notation-table","excerpt":"R_Ξ(U,L;B) · Expected improvement produced by the procedure, not an oracle over its unselected candidates.","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"notation-table","excerpt":"midrule S, U, L · A target solver, an improvement procedure, and a reusable library.","count":8},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"notation-table","excerpt":"midrule S, U, L · A target solver, an improvement procedure, and a reusable library.","count":17},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"notation-table","excerpt":"K, W_h,k · A safe continuation set and states guaranteeing k progress events within h attempts.","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-67fbcb7a77b5ea1b","tex":"W_{h,k}","aliases":"W_h,k","display":"W_{h,k}","kind":"notation-table","excerpt":"K, W_h,k · A safe continuation set and states guaranteeing k progress events within h attempts.","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"notation-table","excerpt":"B_0⊆ X · Possible current states of a finite repair problem.","count":18},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c645657d025cc351","tex":"\\Xi","aliases":"Ξ Xi","display":"\\Xi","kind":"notation-table","excerpt":"Ξ, B · An evaluation contract and its resource allowance.","count":9},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"notation-table","excerpt":"σ, τ · Correspondences of states and operations in a declared transfer.","count":11},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"notation-table","excerpt":"σ, τ · Correspondences of states and operations in a declared transfer.","count":12},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"notation-table","excerpt":"B_0⊆ X · Possible current states of a finite repair problem.","count":15},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":67},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":15},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":26},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3b858f3e8665ea65","tex":"D_e","aliases":"D_e","display":"D_e","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":14},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-48dc92a98cff75ec","tex":"f_e","aliases":"f_e","display":"f_e","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"riment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one s","count":13},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"definition-passage","excerpt":"riment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All costs are in one s","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"definition-passage","excerpt":"[Improvement contract] A contract Ξ specifies a distribution H of starting-solver/task pairs, an initial information and library boundary, permitted observations and modifications, a resource allowance B, protected predicates, and an evaluation rule J_τ(S)∈[0,1]. The improvement procedure returns one selected descendant or the unchanged baseline. S","count":9},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"[Improvement contract] A contract Ξ specifies a distribution H of starting-solver/task pairs, an initial information and library boundary, permitted observations and modifications, a resource allowance B, protected predicates, and an evaluation rule J_τ(S)∈[0,1]. The improvement procedure returns one selected descendant or the unchanged baseline. S","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ac50e45a798d4049","tex":"J_\\tau","aliases":"J_τ tau","display":"J_\\tau","kind":"definition-passage","excerpt":"[Improvement contract] A contract Ξ specifies a distribution H of starting-solver/task pairs, an initial information and library boundary, permitted observations and modifications, a resource allowance B, protected predicates, and an evaluation rule J_τ(S)∈[0,1]. The improvement procedure returns one selected descendant or the unchanged baseline. S","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":13},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3554f05e6897e490","tex":"O_e","aliases":"O_e","display":"O_e","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b2bdbcfb9c9cfc41","tex":"o_e","aliases":"o_e","display":"o_e","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":11},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Improvement contract] A contract Ξ specifies a distribution H of starting-solver/task pairs, an initial information and library boundary, permitted observations and modifications, a resource allowance B, protected predicates, and an evaluation rule J_τ(S)∈[0,1]. The improvement procedure returns one selected descendant or the unchanged baseline. S","count":31},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Finite repair experiment model] A model has a finite nonempty state set X, a finite experiment set E, and a finite repair set A. Experiment e is allowed on a domain D_e⊆ X, changes the state by f_e:D_e→ X, and reports o_e:D_e→ O_e. Its execution cost c_e≥0 is fixed and known. Repair a has fixed cost c_a≥0 and succeeds on a declared set G_a⊆ X. All","count":32},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"Write U(S,τ,L;B) for the selected result, including the procedure's randomness where applicable. Its productivity is R_Ξ(U,L;B)= E_(S,τ)∼ H [J_τ(U(S,τ,L;B))-J_τ(S)]. Invalid execution and out-of-budget behavior receive the treatment fixed by Ξ, not an exclusion chosen after seeing the outcome. The score may be a primary coordinate of a larger profi","count":75},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-fb7427a95ad8c5b4","tex":"\\E","aliases":"E mathbb","display":"{\\mathbb E}","kind":"introduction-passage","excerpt":"Write U(S,τ,L;B) for the selected result, including the procedure's randomness where applicable. Its productivity is R_Ξ(U,L;B)= E_(S,τ)∼ H [J_τ(U(S,τ,L;B))-J_τ(S)]. Invalid execution and out-of-budget behavior receive the treatment fixed by Ξ, not an exclusion chosen after seeing the outcome. The score may be a primary coordinate of a larger profi","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-08ef244830e2cec4","tex":"\\E_{(S,\\tau)\\sim\\mathcal H}","aliases":"E_(S,τ)∼ H E tau sim mathcal mathbb tau sim mathcal","display":"{\\mathbb E}_{(S,\\tau)\\sim\\mathcal H}","kind":"introduction-passage","excerpt":"Write U(S,τ,L;B) for the selected result, including the procedure's randomness where applicable. Its productivity is R_Ξ(U,L;B)= E_(S,τ)∼ H [J_τ(U(S,τ,L;B))-J_τ(S)]. Invalid execution and out-of-budget behavior receive the treatment fixed by Ξ, not an exclusion chosen after seeing the outcome. The score may be a primary coordinate of a larger profi","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1e8b482a0cdcae31","tex":"\\llbracket","aliases":"llbracket","display":null,"kind":"introduction-passage","excerpt":"A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"introduction-passage","excerpt":"A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted ","count":44},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted ","count":22},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted ","count":14},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-cbe1f992df7dd288","tex":"\\rrbracket","aliases":"rrbracket","display":null,"kind":"introduction-passage","excerpt":"A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"A certified quotient preserves the investigation's evidence. Write llbracket prrbracket(z) for an expression's value in context z. Let n(p) be a deterministic representation produced by finitely many checked substitutions of valid equations. We only require n(p)=n(q) Longrightarrow llbracket prrbracket(z)=llbracket qrrbracket(z) for every admitted ","count":12},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"introduction-passage","excerpt":"Write U(S,τ,L;B) for the selected result, including the procedure's randomness where applicable. Its productivity is R_Ξ(U,L;B)= E_(S,τ)∼ H [J_τ(U(S,τ,L;B))-J_τ(S)]. Invalid execution and out-of-budget behavior receive the treatment fixed by Ξ, not an exclusion chosen after seeing the outcome. The score may be a primary coordinate of a larger profi","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"niform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the ","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"The model is fully declared. Setting each known cell as the defect on a calibration copy would permit learning the complete query table in 8(8+3)=88 observation calls. The code instead generates that table from the analytical specification and records it as supplied information. It does not claim to have executed those calibrations or inferred a hi","count":34},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"A diagnostic query names a subset A and returns 1{j∈ A}. The reference language contains eight singleton queries and three bit-coordinate queries. Repair r_i succeeds exactly when i=j. Because repair is irreversible within the episode, a procedure may commit only when every surviving possibility justifies its choice. Otherwise it may continue inves","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"A diagnostic query names a subset A and returns 1{j∈ A}. The reference language contains eight singleton queries and three bit-coordinate queries. Repair r_i succeeds exactly when i=j. Because repair is irreversible within the episode, a procedure may commit only when every surviving possibility justifies its choice. Otherwise it may continue inves","count":15},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"ompilers satisfy the common-domain premises of Theorem~[source reference]. If all stages complete, quotient-guided rule acquisition followed by optimized compilation returns the same learned expression and the same diagnostic controller as ordinary rule acquisition followed by the reference ordered compilation. In a common additive resource model, ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7226e2825e376081","tex":"\\mathcal A'","aliases":"A' mathcal","display":"\\mathcal A'","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"Suppose a revision replaces a by a'. Mark a vertex directly stale whenever one of its declared component values differs. Let Δ be the union of those vertices and all their descendants in the dependency graph. Equivalently, in a topological order, mark v whenever a|_D_v≠ a'|_D_v or pa(v)∩Δ≠∅. Here equality refers to the declared artifact value, not ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"appendix A complete diagnostic trace. Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"appendix A complete diagnostic trace. Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"A declared dependency graph. Let an artifact state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"appendix A complete diagnostic trace. Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-5a8e57a002dc32f4","tex":"a_d","aliases":"a_d","display":"a_d","kind":"use","excerpt":"A declared dependency graph. Let an artifact state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"use","excerpt":"For a target relabeling σ(i)=3i+1pmod8, the defect has target label σ(5)=0. Each query becomes σ(A_r) and the terminal repair becomes r'_σ(5). Reports and tree shape agree with the source, but the label adapter is part of the transported interface. Applying the old numeric repair label five directly in the target would be incorrect. This is why syn","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-82821bc2124f1658","tex":"A_t","aliases":"A_t","display":"A_t","kind":"use","excerpt":"[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which att","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Repair sufficiency is not full identification. Suppose R(θ_1)={a,c} and R(θ_2)={b,c}. The cases are distinct, yet repair c works in both. A method can stop without learning which case occurred. Conversely, pairwise overlap of repair sets does not guarantee one repair for an entire uncertainty set. The three sets {a,b}, {b,c}, and {a,c} intersect pa","count":43},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-0071df57536df27e","tex":"B'_{\\tau(e),\\eta_e(o)}","aliases":"B'_τ(e),η_e(o) tau eta","display":"B'_{\\tau(e),\\eta_e(o)}","kind":"use","excerpt":"e preserved. In particular, F_h(B_0)= F'_h(σ(B_0)). The same conclusion holds for every selected tree, not only a cost-optimal tree. A terminal repair is valid on B_0 exactly when its image is valid on σ(B_0). For an internal experiment, domain preservation gives uniform admissibility. Transition and observation preservation yield σ(B_e,o)=B'_τ(e),","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f94c2a70b1030a1a","tex":"B_{e,o}","aliases":"B_e,o","display":"B_{e,o}","kind":"use","excerpt":"A chosen experiment must be uniformly allowed: B_0⊆ D_e. After observing o, the possible post-experiment states are B_e,o={f_e(x):x∈ B_0, o_e(x)=o}. Only nonempty branches are retained. A terminal repair is justified when B_0⊆ G_a. If the original state or an earlier obligation matters, it must be retained as an immutable tag in X. Merely changing ","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-fcd1d016a0b9e4bf","tex":"\\bar y","aliases":"bar y bar","display":"\\bar y","kind":"use","excerpt":"[Local operation agreement preserves finite continuation] Let D be invariant under every source operation F_a, let q:D→ Z, and suppose qF_a=G_aq on D for every admitted a. If the required output satisfies y=bar yq on D, then for every finite operation word w, qF_w=G_wq, yF_w=bar yG_wq. The empty word gives the identity. Assume the relation holds fo","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"ation available inside a search A compiler is considering a binary query on a support of m possible immutable cases. It reads the query's public model rows in a fixed order. After a prefix, a of the inspected cases have report zero and b have report one. There are m-a-b rows left to read. If this query is the first of at most d≥1 remaining queries,","count":40},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ebd1b51c8a60881c","tex":"C_{\\rm execute}","aliases":"C_rm execute rm","display":"C_{\\rm execute}","kind":"use","excerpt":"Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f1ce996935a45a50","tex":"C_{\\rm install}","aliases":"C_rm install rm","display":"C_{\\rm install}","kind":"use","excerpt":"Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b1fc30ad0cce6444","tex":"C_{\\rm model}","aliases":"C_rm model rm","display":"C_{\\rm model}","kind":"use","excerpt":"Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c3ebead36d730927","tex":"C_{\\rm prepare}","aliases":"C_rm prepare rm","display":"C_{\\rm prepare}","kind":"use","excerpt":"Let C_rm prepare be the complete preparation cost in a declared common resource model, including failed proposals and certificate checking, and let d_t be the subsequent saving on task t after including dispatch. Reuse is beneficial in that model exactly when ∑_t=1^Td_t>C_rm prepare. This is the earlier amortization identity applied to acquisition ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-21a5ce52f6366e9c","tex":"C_{\\rm search}","aliases":"C_rm search rm","display":"C_{\\rm search}","kind":"use","excerpt":"Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7a5ef302e1b9bf69","tex":"C_{\\rm store}","aliases":"C_rm store rm","display":"C_{\\rm store}","kind":"use","excerpt":"Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7792cf3103d96f1c","tex":"C_{\\rm validate}","aliases":"C_rm validate rm","display":"C_{\\rm validate}","kind":"use","excerpt":"Use a resource vector rather than an unexplained token total: C=(C_rm model,C_rm search,C_rm validate,C_rm install, C_rm execute,C_rm store). These coordinates need not share units. Model construction may involve environment observations. Search and validation may involve processor work. Installation may require conversion or compilation. Execution","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c99be13da294dc50","tex":"c_t","aliases":"c_t","display":"c_t","kind":"use","excerpt":"A simple accounting model can express the additional conditions for acceleration. Let g_t be an expected useful gain and c_t>0 the complete cost of the corresponding discovery and acceptance cycle. Improvement productivity per unit resource is g_t/c_t. Raising g_t alone does not raise that ratio when c_t rises faster. A larger reachable repair set ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-2eff86b42c380025","tex":"\\mathrm{compare}","aliases":"compare mathrm","display":"\\mathrm{compare}","kind":"use","excerpt":"If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-491630d7018dba28","tex":"\\operatorname{conv}","aliases":"conv operatorname","display":"\\operatorname{conv}","kind":"use","excerpt":"[Resource dominance and declared price uncertainty] For a fixed ledger and workload, w·Δ(n)≥0 for every nonnegative price vector w if and only if Δ(n) is componentwise nonnegative. If admissible prices form the convex hull of a nonempty finite set W_0, then min_w∈conv(W_0)w·Δ(n) =min_w∈ W_0w·Δ(n). Thus strict improvement throughout that price set h","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"[The resource price of conservative extension] Suppose an embedding preserves every old procedure's successful behavior and adds at most d≥0 execution work and s≥0 stored records. Every old allowance (b,m) is then feasible in the new language under allowance (b+d,m+s). Feasibility at the unchanged allowance follows only when the embedded procedure ","count":33},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"use","excerpt":"s of Theorem~[source reference]. If all stages complete, quotient-guided rule acquisition followed by optimized compilation returns the same learned expression and the same diagnostic controller as ordinary rule acquisition followed by the reference ordered compilation. In a common additive resource model, writing K for extra preparation cost, A,A'","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-32f4f8a168125618","tex":"D'_{\\tau(e)}","aliases":"D'_τ(e) tau","display":"D'_{\\tau(e)}","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x), η_e(o_e(x)) · =o'_τ(e)(σ x), · σ(G_a) · =G'_ψ(a),nonumber and matching execution costs. These assum","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"The same search with fewer model-row reads. The reference compiler first checks the immutable, unit-cost, binary-query and singleton-repair scope of Theorem~[source reference]. It then conducts an ordered depth-first feasibility search at the information lower depth d_0=lceillog_2|B_0|rceil. The full-scan version reads every current row of a candid","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-70f89149d18a83e8","tex":"D_e^\\theta","aliases":"D_e^θ theta","display":"D_e^\\theta","kind":"use","excerpt":"A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an inve","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b377ac1e0c05ff0f","tex":"d_t","aliases":"d_t","display":"d_t","kind":"use","excerpt":"Let C_rm prepare be the complete preparation cost in a declared common resource model, including failed proposals and certificate checking, and let d_t be the subsequent saving on task t after including dispatch. Reuse is beneficial in that model exactly when ∑_t=1^Td_t>C_rm prepare. This is the earlier amortization identity applied to acquisition ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"use","excerpt":"state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(v) identify guarantees used as premises. Its ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":9},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7d11c517fc1de2b4","tex":"\\mathcal E'","aliases":"E' mathcal","display":"\\mathcal E'","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"Choose a nonnegative bet λ_t, measurable from H_t-1, with λ_t≤1/(1+γ). Define E_0=1, E_t=E_t-1(1+λ_t(Y_t-γ)). The variable E_t is an evidence process, not a posterior probability that the candidate is better. Predictability means choosing the bet before observing the score to which it is applied.","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-67371d243352fe13","tex":"E_0^{(j)}","aliases":"E_0^(j)","display":"E_0^{(j)}","kind":"use","excerpt":"Evidence across recursive generations. For attempt j, condition on the complete earlier lineage, freeze its candidate, and initialize a fresh valid audit at E_0^(j)=1 with error allowance δ_j. If ∑_jδ_j≤δ, Proposition~[source reference] bounds the probability of ever accepting a false margin claim by δ. Each audit can stop sequentially, and the nex","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-5bbddd50e7cf1221","tex":"\\E_{P_0}","aliases":"E_P_0 E mathbb","display":"{\\mathbb E}_{P_0}","kind":"use","excerpt":"ry decision: report case zero if the repair belongs to R(0), and case one otherwise. Success in case zero implies a correct decision. Success in case one also implies a correct decision because the two repair sets are disjoint. Thus repair success is at most the success of this binary test. For a randomized test with probability f(o)∈[0,1] of repor","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ff556bf7d9b30168","tex":"\\E_{P_1}","aliases":"E_P_1 E mathbb","display":"{\\mathbb E}_{P_1}","kind":"use","excerpt":"ry decision: report case zero if the repair belongs to R(0), and case one otherwise. Success in case zero implies a correct decision. Success in case one also implies a correct decision because the two repair sets are disjoint. Thus repair success is at most the success of this binary test. For a randomized test with probability f(o)∈[0,1] of repor","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f75d15c864ad3cef","tex":"E_t","aliases":"E_t","display":"E_t","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":8},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-2c7a3501c3e771c4","tex":"E_{t-1}","aliases":"E_t-1","display":"E_{t-1}","kind":"use","excerpt":"Choose a nonnegative bet λ_t, measurable from H_t-1, with λ_t≤1/(1+γ). Define E_0=1, E_t=E_t-1(1+λ_t(Y_t-γ)). The variable E_t is an evidence process, not a posterior probability that the candidate is better. Predictability means choosing the bet before observing the score to which it is applied.","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequali","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-cace9a962804366a","tex":"\\mathcal F'_h","aliases":"F'_h mathcal","display":"\\mathcal F'_h","kind":"use","excerpt":"[Exact transport of diagnostic procedures] Under~[source reference], relabeling every experiment, branch, and repair transports successful diagnostic trees bijectively between the source support B_0 and target support σ(B_0). Execution cost, complete tree storage, and depth are preserved. In particular, F_h(B_0)= F'_h(σ(B_0)). The same conclusion h","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d6012f1edaba837f","tex":"f'_{\\tau(e)}","aliases":"f'_τ(e) tau","display":"f'_{\\tau(e)}","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x), η_e(o_e(x)) · =o'_τ(e)(σ x), · σ(G_a) · =G'_ψ(a),nonumber and matching execution costs. These assum","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-313137accdec2e68","tex":"\\mathcal F_0","aliases":"F_0 mathcal","display":"\\mathcal F_0","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"[Local operation agreement preserves finite continuation] Let D be invariant under every source operation F_a, let q:D→ Z, and suppose qF_a=G_aq on D for every admitted a. If the required output satisfies y=bar yq on D, then for every finite operation word w, qF_w=G_wq, yF_w=bar yG_wq. The empty word gives the identity. Assume the relation holds fo","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-109ec22b64c2454d","tex":"f_e^\\theta","aliases":"f_e^θ theta","display":"f_e^\\theta","kind":"use","excerpt":"A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an inve","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-5e3d805240078a53","tex":"\\mathcal F_h","aliases":"F_h mathcal","display":"\\mathcal F_h","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-874f13ac2a9c06a1","tex":"\\mathcal F_{h-1}","aliases":"F_h-1 mathcal","display":"\\mathcal F_{h-1}","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4a14d2ab0a464e3b","tex":"F_u","aliases":"F_u","display":"F_u","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"use","excerpt":"[Local operation agreement preserves finite continuation] Let D be invariant under every source operation F_a, let q:D→ Z, and suppose qF_a=G_aq on D for every admitted a. If the required output satisfies y=bar yq on D, then for every finite operation word w, qF_w=G_wq, yF_w=bar yG_wq. The empty word gives the identity. Assume the relation holds fo","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-00c37c61b4205056","tex":"\\mathrm{full}","aliases":"full mathrm","display":"\\mathrm{full}","kind":"use","excerpt":"If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"For a fixed budget, let R^*_ G be the largest productivity among procedures expressible in a declared language G, with installed data and interpreter costs held fixed. The gap R^*_ G-R(U) is search or selection headroom within that language. Comparing R^*_ G' with R^*_ G concerns expressibility and resources. Neither upper envelope is automatically","count":9},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"Paired repairs change the task, not just its notation. Consider 2g possible hidden cases divided into g pairs, where g is a power of two. One repair succeeds on both members of each pair. The query language contains equality questions and bits of the pair index. The greedy weight constructor gives weight one to the first member of each pair and zer","count":10},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-abeca96c3125e618","tex":"\\mathcal G'","aliases":"G' mathcal","display":"\\mathcal G'","kind":"use","excerpt":"For a fixed budget, let R^*_ G be the largest productivity among procedures expressible in a declared language G, with installed data and interpreter costs held fixed. The gap R^*_ G-R(U) is search or selection headroom within that language. Comparing R^*_ G' with R^*_ G concerns expressibility and resources. Neither upper envelope is automatically","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3dfdbb430c5e1740","tex":"G'_{\\psi(a)}","aliases":"G'_ψ(a) psi","display":"G'_{\\psi(a)}","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x), η_e(o_e(x)) · =o'_τ(e)(σ x), · σ(G_a) · =G'_ψ(a),nonumber and matching execution costs. These assum","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-9eb17f719597fdbe","tex":"G_a^\\theta","aliases":"G_a^θ theta","display":"G_a^\\theta","kind":"use","excerpt":"A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an inve","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3f8720ad3f121275","tex":"G_r","aliases":"G_r","display":"G_r","kind":"use","excerpt":"Let B be a finite nonempty immutable-case support and let G_r⊆ B be each repair's success set restricted to it. Define κ(B) to be the smallest number of repair-success sets whose union covers B. Set κ(B)=∞ when no such cover exists. This number concerns the repaired objective, not the number of hidden states. Computing it can itself be expensive.","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1481d5cc6f09169d","tex":"g_t","aliases":"g_t","display":"g_t","kind":"use","excerpt":"A simple accounting model can express the additional conditions for acceleration. Let g_t be an expected useful gain and c_t>0 the complete cost of the corresponding discovery and acceptance cycle. Improvement productivity per unit resource is g_t/c_t. Raising g_t alone does not raise that ratio when c_t rises faster. A larger reachable repair set ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-6d584c042dd0c359","tex":"G_w","aliases":"G_w","display":"G_w","kind":"use","excerpt":"[Local operation agreement preserves finite continuation] Let D be invariant under every source operation F_a, let q:D→ Z, and suppose qF_a=G_aq on D for every admitted a. If the required output satisfies y=bar yq on D, then for every finite operation word w, qF_w=G_wq, yF_w=bar yG_wq. The empty word gives the identity. Assume the relation holds fo","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-5d4698720d0499ca","tex":"G_{\\theta,a}","aliases":"G_θ,a theta","display":"G_{\\theta,a}","kind":"use","excerpt":"What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked op","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8c918c284a1665cd","tex":"\\Gcal","aliases":"Gcal G mathcal","display":"{\\mathcal G}","kind":"use","excerpt":"Four objects and two meanings of success. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total expressions. A target task supplies contexts z∈ D and desired values f(z). An ordinary learner uses counterexamples to find the first grammar expression that agrees with all of D. A separate equational learner observes expressions on a developme","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-edcbd44645488c8a","tex":"\\mathcal H_{t-1}","aliases":"H_t-1 mathcal","display":"\\mathcal H_{t-1}","kind":"use","excerpt":"[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which att","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-518159fee1404da8","tex":"\\hat\\imath","aliases":"hatimath hat imath","display":"\\hat\\imath","kind":"use","excerpt":"[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"A diagnostic query names a subset A and returns 1{j∈ A}. The reference language contains eight singleton queries and three bit-coordinate queries. Repair r_i succeeds exactly when i=j. Because repair is irreversible within the episode, a procedure may commit only when every surviving possibility justifies its choice. Otherwise it may continue inves","count":35},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1eba5c75dddfe29c","tex":"i^*","aliases":"i^*","display":"i^*","kind":"use","excerpt":"niform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"Every case produces one deterministic query history. Two different cases cannot terminate at the same repair leaf because no repair succeeds in both. Thus there are at least n nonempty leaves. A tree of binary query depth d has at most 2^d leaves, so d≥lceillog_2 nrceil. Every path also executes a unit-cost repair. For storage, let l be the number ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-0475b2ac7d225af5","tex":"i_2","aliases":"i_2","display":"i_2","kind":"use","excerpt":"Every case produces one deterministic query history. Two different cases cannot terminate at the same repair leaf because no repair succeeds in both. Thus there are at least n nonempty leaves. A tree of binary query depth d has at most 2^d leaves, so d≥lceillog_2 nrceil. Every path also executes a unit-cost repair. For storage, let l be the number ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"A small world with reusable diagnostic structure. Consider eight cells with exactly one latent defect, indexed by j∈{0,…,7}. The solver must repair that cell while preserving the others. Its old representation retains only the number of defects, so every episode initially looks identical.","count":17},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"and the original and optimized compilers satisfy the common-domain premises of Theorem~[source reference]. If all stages complete, quotient-guided rule acquisition followed by optimized compilation returns the same learned expression and the same diagnostic controller as ordinary rule acquisition followed by the reference ordered compilation. In a ","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"Define K_0=Safe, K_n+1={z∈ K_n:∃ a∈ A(z), Post(z,a)⊆ K_n}. The descending finite sequence reaches a fixed point K. It is the set from which some policy keeps every successor safe indefinitely. This can include doing nothing forever. Safety alone is not recursive improvement.","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-bb9a538f5e406f30","tex":"K_n","aliases":"K_n","display":"K_n","kind":"use","excerpt":"Define K_0=Safe, K_n+1={z∈ K_n:∃ a∈ A(z), Post(z,a)⊆ K_n}. The descending finite sequence reaches a fixed point K. It is the set from which some policy keeps every successor safe indefinitely. This can include doing nothing forever. Safety alone is not recursive improvement.","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8a89bee060b13402","tex":"K_{n+1}","aliases":"K_n+1","display":"K_{n+1}","kind":"use","excerpt":"Define K_0=Safe, K_n+1={z∈ K_n:∃ a∈ A(z), Post(z,a)⊆ K_n}. The descending finite sequence reaches a fixed point K. It is the set from which some policy keeps every successor safe indefinitely. This can include doing nothing forever. Safety alone is not recursive improvement.","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"exhaustive synthesis. Every case produces one deterministic query history. Two different cases cannot terminate at the same repair leaf because no repair succeeds in both. Thus there are at least n nonempty leaves. A tree of binary query depth d has at most 2^d leaves, so d≥lceillog_2 nrceil. Every path also executes a unit-cost repair. For storage","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-34c05dc8f5073293","tex":"L'","aliases":"L'","display":"L'","kind":"use","excerpt":"Fixed-library and full-system comparisons. A comparison of R_Ξ(U',L;B) with R_Ξ(U,L;B) isolates process change at a common library. A comparison with (U',L') measures the combined descendant. Both are useful, but they answer different questions. A crossed comparison of (U,L), (U',L), (U,L'), and (U',L') can expose interaction between procedure and ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-49e6cb003fe8b536","tex":"L_t","aliases":"L_t","display":"L_t","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[Checked lower-bound closure of a diagnostic search] Every successful controller in this scope has worst-case execution work at least 1+lceillog_2 nrceil and storage at least 2n-1. If an independently checked candidate attains both bounds and respects the requested horizon, its cost pair is the entire nondominated frontier. Returning this candidate","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"[A fixed-margin ceiling on a fixed score] Suppose a lineage has a fixed measure R_t∈[a,b], every accepted progress event increases it by at least γ>0, and all other transitions are nondecreasing in that same measure. Then the number of progress events is at most lfloorfracb-aγrfloor. An infinite sequence of positive increments is compatible with bo","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"[Checked lower-bound closure of a diagnostic search] Every successful controller in this scope has worst-case execution work at least 1+lceillog_2 nrceil and storage at least 2n-1. If an independently checked candidate attains both bounds and respects the requested horizon, its cost pair is the entire nondominated frontier. Returning this candidate","count":8},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"[Checked lower-bound closure of a diagnostic search] Every successful controller in this scope has worst-case execution work at least 1+lceillog_2 nrceil and storage at least 2n-1. If an independently checked candidate attains both bounds and respects the requested horizon, its cost pair is the entire nondominated frontier. Returning this candidate","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"Let W(P,B_0) be the largest execution cost of a complete path, including its terminal repair. Let M(P) be the number of stored tree records. These costs are not independently minimized. A single tree has to realize both.","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),","count":27},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7f54f92432b51276","tex":"m_o","aliases":"m_o","display":"m_o","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"use","excerpt":"[The same operator spelling with changed semantics] The identity x+0=x is valid for ordinary addition. Suppose a revised interpreter uses the same printed symbol but executes amathbin⊕b=a+b+1. Then xmathbin⊕0≠ x. A cache certified for the former interpreter cannot be imported merely because its expression strings still parse. A semantic identifier ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":14},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"use","excerpt":"[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d441c86815baa4be","tex":"\\max_o","aliases":"max_o max","display":"\\max_o","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which att","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"nd a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their standalone completion costs are D_0,D_1∈ N_>0∪{∞} and at lea","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-cc8961cbd7cc9df9","tex":"\\min_{w\\in W_0}","aliases":"min_w∈ W_0 min in","display":"\\min_{w\\in W_0}","kind":"use","excerpt":"[Resource dominance and declared price uncertainty] For a fixed ledger and workload, w·Δ(n)≥0 for every nonnegative price vector w if and only if Δ(n) is componentwise nonnegative. If admissible prices form the convex hull of a nonempty finite set W_0, then min_w∈conv(W_0)w·Δ(n) =min_w∈ W_0w·Δ(n). Thus strict improvement throughout that price set h","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7e18358242f4b0dc","tex":"\\min_{w\\in\\operatorname{conv}(W_0)}","aliases":"min_w∈conv(W_0) min in operatorname","display":"\\min_{w\\in\\operatorname{conv}(W_0)}","kind":"use","excerpt":"[Resource dominance and declared price uncertainty] For a fixed ledger and workload, w·Δ(n)≥0 for every nonnegative price vector w if and only if Δ(n) is componentwise nonnegative. If admissible prices form the convex hull of a nonempty finite set W_0, then min_w∈conv(W_0)w·Δ(n) =min_w∈ W_0w·Δ(n). Thus strict improvement throughout that price set h","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final scor","count":19},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f8c1643801c1b1fe","tex":"\\mathbb N_{>0}","aliases":"N_>0 mathbb","display":"\\mathbb N_{>0}","kind":"use","excerpt":"lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e66911ccd27e5fbc","tex":"N_{\\mathrm{compare}}","aliases":"N_compare mathrm","display":"N_{\\mathrm{compare}}","kind":"use","excerpt":"If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ae4704a81fdf75d2","tex":"n_t","aliases":"n_t","display":"n_t","kind":"use","excerpt":"One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequali","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-76e48dcd185aa356","tex":"O'_{\\tau(e)}","aliases":"O'_τ(e) tau","display":"O'_{\\tau(e)}","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-990c002ed2409e69","tex":"o'_{\\tau(e)}","aliases":"o'_τ(e) tau","display":"o'_{\\tau(e)}","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x), η_e(o_e(x)) · =o'_τ(e)(σ x), · σ(G_a) · =G'_ψ(a),nonumber and matching execution costs. These assum","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-706818517abf30a8","tex":"o_e^\\theta","aliases":"o_e^θ theta","display":"o_e^\\theta","kind":"use","excerpt":"A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an inve","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"[Feature sufficiency for a search decision] Let P be a finite set of search contexts, let r: P→ Z be their available feature map, and let ℓ: P→{0,1} be the required decision label. A deterministic rule f:Z→{0,1} satisfying fr=ℓ exists if and only if ℓ is constant on each nonempty fibre of r. A pair in one fibre with different labels rules out every","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"[Feedback indistinguishability limits repair] Suppose two cases have disjoint successful-repair sets and induce transcript laws P_0,P_1 under a fixed admitted investigation protocol. A repair selector observes only the transcript and independent private randomness. Under equal prior probabilities, its average probability of successful repair is at ","count":8},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"Four objects and two meanings of success. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total expressions. A target task supplies contexts z∈ D and desired values f(z). An ordinary learner uses counterexamples to find the first grammar expression that agrees with all of D. A separate equational learner observes expressions on a developme","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[Feedback indistinguishability limits repair] Suppose two cases have disjoint successful-repair sets and induce transcript laws P_0,P_1 under a fixed admitted investigation protocol. A repair selector observes only the transcript and independent private randomness. Under equal prior probabilities, its average probability of successful repair is at ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"[Sound ordered quotienting from checked equations] Assume the expression language is pure and total, every admitted schema equation is valid for all well-typed substitutions in the common semantic domain, and each normalization terminates after valid contextual replacements. For each p_i, keep its earliest grammar representative ρ(i)=min{j:n(p_j)=n","count":8},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"use","excerpt":"[Sound ordered quotienting from checked equations] Assume the expression language is pure and total, every admitted schema equation is valid for all well-typed substitutions in the common semantic domain, and each normalization terminates after valid contextual replacements. For each p_i, keep its earliest grammar representative ρ(i)=min{j:n(p_j)=n","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-cc9e0dabacaadc8b","tex":"p_{N-1}","aliases":"p_N-1","display":"p_{N-1}","kind":"use","excerpt":"Four objects and two meanings of success. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total expressions. A target task supplies contexts z∈ D and desired values f(z). An ordinary learner uses counterexamples to find the first grammar expression that agrees with all of D. A separate equational learner observes expressions on a developme","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-12b33428971a165a","tex":"P_t","aliases":"P_t","display":"P_t","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-fa1ae76a27248a1e","tex":"P_{t+1}","aliases":"P_t+1","display":"P_{t+1}","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-fbd55c78e5bd8f12","tex":"p_{\\rho(i)}","aliases":"p_ρ(i) rho","display":"p_{\\rho(i)}","kind":"use","excerpt":"cked equations] Assume the expression language is pure and total, every admitted schema equation is valid for all well-typed substitutions in the common semantic domain, and each normalization terminates after valid contextual replacements. For each p_i, keep its earliest grammar representative ρ(i)=min{j:n(p_j)=n(p_i)}. Then ρ(i)≤ i, p_i and p_ρ(i","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7f6ddefca2507bfe","tex":"p_\\psi","aliases":"p_ψ psi","display":"p_\\psi","kind":"use","excerpt":"What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked op","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4bc9c56157a62dc5","tex":"\\operatorname{pa}","aliases":"pa operatorname","display":"\\operatorname{pa}","kind":"use","excerpt":"A declared dependency graph. Let an artifact state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-dfea44c8abc940ec","tex":"\\Pareto","aliases":"Pareto operatorname","display":"{\\operatorname{Pareto}}","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"For a target relabeling σ(i)=3i+1pmod8, the defect has target label σ(5)=0. Each query becomes σ(A_r) and the terminal repair becomes r'_σ(5). Reports and tree shape agree with the source, but the label adapter is part of the transported interface. Applying the old numeric repair label five directly in the target would be incorrect. This is why syn","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f1dd2bd1079bcf86","tex":"\\Post","aliases":"Post operatorname","display":"{\\operatorname{Post}}","kind":"use","excerpt":"Each z has a finite set of authorized attempts A(z). An attempt has a nonempty set Post(z,a) of possible successor configurations. This can represent all outcomes of a stochastic proposal or an abstract uncertainty set. A safe set Safe⊆ Z encodes the declared semantic, evidence, authority, and resource conditions. A pause or terminal state may have","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which att","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"es every finite continuation of the retained labels and outputs. A strict refinement increases the number of nonempty blocks, which is bounded by |X|. At a fixed point, equivalent states have equivalent successors, so the quotient operations are well-defined. Induction on word length then preserves all retained labels after every finite word. For m","count":18},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"Let a hidden case θ determine a set R(θ) of successful repairs. The set may have several members. Diagnosis need only obtain enough information to choose a repair in the appropriate set. It need not identify every hidden property of θ.","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d41fe2809fb29445","tex":"r'_{\\sigma(5)}","aliases":"r'_σ(5) sigma","display":"r'_{\\sigma(5)}","kind":"use","excerpt":"For a target relabeling σ(i)=3i+1pmod8, the defect has target label σ(5)=0. Each query becomes σ(A_r) and the terminal repair becomes r'_σ(5). Reports and tree shape agree with the source, but the label adapter is part of the transported interface. Applying the old numeric repair label five directly in the target would be incorrect. This is why syn","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1b221660c5e1ad9a","tex":"R^*_{\\mathcal G}","aliases":"R^*_ G mathcal","display":"R^*_{\\mathcal G}","kind":"use","excerpt":"For a fixed budget, let R^*_ G be the largest productivity among procedures expressible in a declared language G, with installed data and interpreter costs held fixed. The gap R^*_ G-R(U) is search or selection headroom within that language. Comparing R^*_ G' with R^*_ G concerns expressibility and resources. Neither upper envelope is automatically","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-0faa5db6de8d543f","tex":"R^*_{\\mathcal G'}","aliases":"R^*_ G' mathcal","display":"R^*_{\\mathcal G'}","kind":"use","excerpt":"For a fixed budget, let R^*_ G be the largest productivity among procedures expressible in a declared language G, with installed data and interpreter costs held fixed. The gap R^*_ G-R(U) is search or selection headroom within that language. Comparing R^*_ G' with R^*_ G concerns expressibility and resources. Neither upper envelope is automatically","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequali","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-855b51431c459bf3","tex":"\\mathbb R^d","aliases":"R^d mathbb","display":"\\mathbb R^d","kind":"use","excerpt":"Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"ma>0, and all other transitions are nondecreasing in that same measure. Then the number of progress events is at most lfloorfracb-aγrfloor. An infinite sequence of positive increments is compatible with bounded total improvement when no common positive lower bound is imposed. After k progress events, telescoping and nondecrease give R_t≥ R_0+kγ≥ a+","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b22adc2a54ec6353","tex":"r_5","aliases":"r_5","display":"r_5","kind":"use","excerpt":"p:trace Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The terminal repair is r_5. The p","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-148f6d1b72f67e4b","tex":"R_{\\mathrm{full}}","aliases":"R_full mathrm","display":"R_{\\mathrm{full}}","kind":"use","excerpt":"If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ff25d16c5ae0dc58","tex":"R_{\\hat\\imath}","aliases":"R_hatimath hat imath","display":"R_{\\hat\\imath}","kind":"use","excerpt":"[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"use","excerpt":"For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final scor","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"A diagnostic query names a subset A and returns 1{j∈ A}. The reference language contains eight singleton queries and three bit-coordinate queries. Repair r_i succeeds exactly when i=j. Because repair is irreversible within the episode, a procedure may commit only when every surviving possibility justifies its choice. Otherwise it may continue inves","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d05f49cca47725be","tex":"R_{i^*}","aliases":"R_i^*","display":"R_{i^*}","kind":"use","excerpt":"assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same tw","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-07c29c38146ae7ad","tex":"r_{o\\oplus\\theta}","aliases":"r_o⊕θ oplus theta","display":"r_{o\\oplus\\theta}","kind":"use","excerpt":"Add a permitted reference measurement that reports θ, leaves x unchanged, and costs two. The procedure can calibrate, make the ordinary query, and choose r_o⊕θ. Its worst-case work is four and its complete binary tree has seven records. Either measurement alone leaves both repairs possible. The two measurements are therefore both necessary in this ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1ba3cd8a0856a76d","tex":"R_{\\mathrm{shortcut}}","aliases":"R_shortcut mathrm","display":"R_{\\mathrm{shortcut}}","kind":"use","excerpt":"total unit-cost binary immutable queries, and the balanced proposer produces a perfect depth-k tree. In the stipulated implementation, the semantic guard reads every query row once, the proposer scans all q queries at each internal support, and verification performs one support reconstruction and one complete case replay. The total number of transi","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8bbe0b8d06054149","tex":"R_{\\mathrm{stream}}","aliases":"R_stream mathrm","display":"R_{\\mathrm{stream}}","kind":"use","excerpt":"If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-49121c793f36cbb5","tex":"R_t","aliases":"R_t","display":"R_t","kind":"use","excerpt":"[A fixed-margin ceiling on a fixed score] Suppose a lineage has a fixed measure R_t∈[a,b], every accepted progress event increases it by at least γ>0, and all other transitions are nondecreasing in that same measure. Then the number of progress events is at most lfloorfracb-aγrfloor. An infinite sequence of positive increments is compatible with bo","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e4b6d938c63d0d7a","tex":"r_\\phi","aliases":"r_ϕ phi","display":"r_\\phi","kind":"use","excerpt":"What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked op","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-43dcd88270900239","tex":"\\mathbb R_{\\ge0}^d","aliases":"R_≥0^d mathbb ge","display":"\\mathbb R_{\\ge0}^d","kind":"use","excerpt":"Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-fcf0d1ec782295a3","tex":"\\Rcal'","aliases":"Rcal' R' Rcal mathcal","display":"{\\mathcal R}'","kind":"use","excerpt":"Approximate performance transfer. Suppose a source comparison gives productivity advantage Δ to U' over U. Let widetilde U',widetilde U be actual transported target procedures. If each procedure's productivity differs from its source value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[Checked lower-bound closure of a diagnostic search] Every successful controller in this scope has worst-case execution work at least 1+lceillog_2 nrceil and storage at least 2n-1. If an independently checked candidate attains both bounds and respects the requested horizon, its cost pair is the entire nondominated frontier. Returning this candidate","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"[A fixed-margin ceiling on a fixed score] Suppose a lineage has a fixed measure R_t∈[a,b], every accepted progress event increases it by at least γ>0, and all other transitions are nondecreasing in that same measure. Then the number of progress events is at most lfloorfracb-aγrfloor. An infinite sequence of positive increments is compatible with bo","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"Let C_rm prepare be the complete preparation cost in a declared common resource model, including failed proposals and certificate checking, and let d_t be the subsequent saving on task t after including dispatch. Reuse is beneficial in that model exactly when ∑_t=1^Td_t>C_rm prepare. This is the earlier amortization identity applied to acquisition ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"[The resource price of conservative extension] Suppose an embedding preserves every old procedure's successful behavior and adds at most d≥0 execution work and s≥0 stored records. Every old allowance (b,m) is then feasible in the new language under allowance (b+d,m+s). Feasibility at the unchanged allowance follows only when the embedded procedure ","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b37e9083bef6f614","tex":"S_r","aliases":"S_r","display":"S_r","kind":"use","excerpt":"Let X be a finite immutable state set and let S_r⊆ X be the success set of repair r. Choose nonnegative rational weights w_x satisfying ∑_x∈ S_rw_x≤1 for every permitted repair r. This is a feasible dual weight assignment for unweighted set cover. We use only its elementary weak-duality property, not an assumption that the best assignment has been ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-87807c21edc049cd","tex":"S_t","aliases":"S_t","display":"S_t","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-a354a58aabc0fe1c","tex":"\\Safe","aliases":"Safe mathsf","display":"{\\mathsf{Safe}}","kind":"use","excerpt":"Each z has a finite set of authorized attempts A(z). An attempt has a nonempty set Post(z,a) of possible successor configurations. This can represent all outcomes of a stochastic proposal or an abstract uncertainty set. A safe set Safe⊆ Z encodes the declared semantic, evidence, authority, and resource conditions. A pause or terminal state may have","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f020c9fbd9af5930","tex":"\\mathrm{shortcut}","aliases":"shortcut mathrm","display":"\\mathrm{shortcut}","kind":"use","excerpt":"total unit-cost binary immutable queries, and the balanced proposer produces a perfect depth-k tree. In the stipulated implementation, the semantic guard reads every query row once, the proposer scans all q queries at each internal support, and verification performs one support reconstruction and one complete case replay. The total number of transi","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d6008072c780cd95","tex":"\\mathrm{stream}","aliases":"stream mathrm","display":"\\mathrm{stream}","kind":"use","excerpt":"If a model-row read has declared price w>0 and a new comparison has price one, the additional per-use saving under this conservative two-unit account is Δ(w)=w (R_full-R_stream)-N_compare. For the 64-case row it is positive when w>14209/8287. At w=1 it is negative. The price is a modeling choice, not a measured statement about Python or a physical ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"[An anytime-valid paired improvement audit] Under~[source reference] and the stated predictable bet constraints, E_t is a nonnegative supermartingale. For every δ∈(0,1), Pr(sup_t≥0E_t≥1/δ)≤δ. Consequently an audit may reject the stated null at its first crossing of 1/δ, while choosing its stopping time from the observed audit history. The result do","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-53bc99b07a603a4f","tex":"\\sup_{t\\geq0}","aliases":"sup_t≥0 sup geq","display":"\\sup_{t\\geq0}","kind":"use","excerpt":"[An anytime-valid paired improvement audit] Under~[source reference] and the stated predictable bet constraints, E_t is a nonnegative supermartingale. For every δ∈(0,1), Pr(sup_t≥0E_t≥1/δ)≤δ. Consequently an audit may reject the stated null at its first crossing of 1/δ, while choosing its stopping time from the observed audit history. The result do","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":57},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV d_TV mathrm","display":"{d_{\\mathrm{TV}}}","kind":"use","excerpt":"roposition[Feedback indistinguishability limits repair] Suppose two cases have disjoint successful-repair sets and induce transcript laws P_0,P_1 under a fixed admitted investigation protocol. A repair selector observes only the transcript and independent private randomness. Under equal prior probabilities, its average probability of successful rep","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ec61c8382b2d3710","tex":"U'","aliases":"U'","display":"U'","kind":"use","excerpt":"Fixed-library and full-system comparisons. A comparison of R_Ξ(U',L;B) with R_Ξ(U,L;B) isolates process change at a common library. A comparison with (U',L') measures the combined descendant. Both are useful, but they answer different questions. A crossed comparison of (U,L), (U',L), (U,L'), and (U',L') can expose interaction between procedure and ","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final scor","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-9fcfc7136a869616","tex":"U_N","aliases":"U_N","display":"U_N","kind":"use","excerpt":"For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final scor","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-6ae21ddbb7e958a2","tex":"U_t","aliases":"U_t","display":"U_t","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"A declared dependency graph. Let an artifact state be a=(a_1,…,a_d). Its components may include operation semantics, a grammar and its order, an equation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(","count":10},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0),","count":21},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"[Resource dominance and declared price uncertainty] For a fixed ledger and workload, w·Δ(n)≥0 for every nonnegative price vector w if and only if Δ(n) is componentwise nonnegative. If admissible prices form the convex hull of a nonempty finite set W_0, then min_w∈conv(W_0)w·Δ(n) =min_w∈ W_0w·Δ(n). Thus strict improvement throughout that price set h","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c5480dc96ac56ca2","tex":"W_{0,0}","aliases":"W_0,0","display":"W_{0,0}","kind":"use","excerpt":"For each transition, a progress mark p(z,a,z')∈{0,1} is supplied by a checked criterion. A mark of one means the specified process-improvement claim has been established under the same comparison. It is not assigned merely because a file changed. Define W_0,0 · = K, · W_0,k · =∅ (k>0), W_h,k · ={z∈ K:∃ a∈ A(z), Post(z,a)⊆ K,nonumber [-2pt] · z'∈ W_","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3aded8f2b3589e32","tex":"W_{0,k}","aliases":"W_0,k","display":"W_{0,k}","kind":"use","excerpt":"For each transition, a progress mark p(z,a,z')∈{0,1} is supplied by a checked criterion. A mark of one means the specified process-improvement claim has been established under the same comparison. It is not assigned merely because a file changed. Define W_0,0 · = K, · W_0,k · =∅ (k>0), W_h,k · ={z∈ K:∃ a∈ A(z), Post(z,a)⊆ K,nonumber [-2pt] · z'∈ W_","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"For example, on three cases with success sets {1,2}, {2,3}, and {1,3}, the ordered greedy constructor returns total weight one. The feasible fractional assignment w_1=w_2=w_3=1/2 has total 3/2 and proves that at least two leaves are required. Thus the supplied constructor does not optimize the bound. Both assignments are valid and neither creates a","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3106fac3e3c8f992","tex":"w_2","aliases":"w_2","display":"w_2","kind":"use","excerpt":"For example, on three cases with success sets {1,2}, {2,3}, and {1,3}, the ordered greedy constructor returns total weight one. The feasible fractional assignment w_1=w_2=w_3=1/2 has total 3/2 and proves that at least two leaves are required. Thus the supplied constructor does not optimize the bound. Both assignments are valid and neither creates a","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-9763a163e7ff6591","tex":"w_3","aliases":"w_3","display":"w_3","kind":"use","excerpt":"For example, on three cases with success sets {1,2}, {2,3}, and {1,3}, the ordered greedy constructor returns total weight one. The feasible fractional assignment w_1=w_2=w_3=1/2 has total 3/2 and proves that at least two leaves are required. Thus the supplied constructor does not optimize the bound. Both assignments are valid and neither creates a","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7c3094fc743065b1","tex":"W_{h-1,\\max\\{0,k-p(z,a,z')\\}}","aliases":"W_h-1,max{0,k-p(z,a,z')} max","display":"W_{h-1,\\max\\{0,k-p(z,a,z')\\}}","kind":"use","excerpt":"For each transition, a progress mark p(z,a,z')∈{0,1} is supplied by a checked criterion. A mark of one means the specified process-improvement claim has been established under the same comparison. It is not assigned merely because a file changed. Define W_0,0 · = K, · W_0,k · =∅ (k>0), W_h,k · ={z∈ K:∃ a∈ A(z), Post(z,a)⊆ K,nonumber [-2pt] · z'∈ W_","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-a24f305d6325f526","tex":"w_o","aliases":"w_o","display":"w_o","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e6e9e4d147adf393","tex":"w_v","aliases":"w_v","display":"w_v","kind":"use","excerpt":"quation library, experiment tables, repair-success sets, a returned controller, and a resource model. A finite directed acyclic graph has one vertex v for each claimed guarantee. Its incoming vertices pa(v) identify guarantees used as premises. Its direct dependency set D_v⊆{1,…,d} identifies the artifact components used directly. Each vertex carri","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-5b4ce5956eb97972","tex":"w_x","aliases":"w_x","display":"w_x","kind":"use","excerpt":"Let X be a finite immutable state set and let S_r⊆ X be the success set of repair r. Choose nonnegative rational weights w_x satisfying ∑_x∈ S_rw_x≤1 for every permitted repair r. This is a feasible dual weight assignment for unweighted set cover. We use only its elementary weak-duality property, not an assumption that the best assignment has been ","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"e conditional-mean premise. Since Y_t-γ≥-(1+γ), every factor in~[source reference] is nonnegative. Conditional expectation and predictability give E[E_tmid H_t-1] =E_t-1(1+λ_t( E[Y_tmid H_t-1]-γ)) ≤ E_t-1. Let τ be the first threshold crossing. For a finite N, the stopped process at τwedge N still has expected value at most one, by induction on the","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-967c63ebf4cd78be","tex":"\\widehat R","aliases":"widehat R widehat","display":"\\widehat R","kind":"use","excerpt":"For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final scor","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e0887913ce9f0cef","tex":"\\widehat R_{\\hat\\imath}","aliases":"widehat R_hatimath widehat hat imath","display":"\\widehat R_{\\hat\\imath}","kind":"use","excerpt":"assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same tw","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f3e1cec7580875b3","tex":"\\widehat R_i","aliases":"widehat R_i widehat","display":"\\widehat R_i","kind":"use","excerpt":"For a fixed finite collection {U_1,…,U_N}, suppose a declared assessment gives estimates widehat R_i of true productivities R_i. This assessment can be exact on a fully specified finite distribution or statistical under additional assumptions. Choosing a candidate must use the assessment actually available at selection time, not a hidden final scor","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-19029b9c979ec627","tex":"\\widehat R_{i^*}","aliases":"widehat R_i^* widehat","display":"\\widehat R_{i^*}","kind":"use","excerpt":"assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_i^*-2ε. Apply the same tw","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-5df53739c784d094","tex":"\\widehat\\Delta","aliases":"widehatΔ widehat Delta","display":"\\widehat\\Delta","kind":"use","excerpt":"One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequali","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-0ba62b2ff576b8eb","tex":"\\widehat\\Delta_t","aliases":"widehatΔ_t widehat Delta","display":"\\widehat\\Delta_t","kind":"use","excerpt":"One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequali","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d74a2e10811483dc","tex":"\\widetilde f","aliases":"widetilde f widetilde","display":"\\widetilde f","kind":"use","excerpt":"[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/s","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-128fa659bfedffe5","tex":"\\widetilde f_e","aliases":"widetilde f_e widetilde","display":"\\widetilde f_e","kind":"use","excerpt":"[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/s","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-929c170d8af8ade4","tex":"\\widetilde G","aliases":"widetilde G widetilde","display":"\\widetilde G","kind":"use","excerpt":"[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/s","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-008a0a204981576f","tex":"\\widetilde G_a","aliases":"widetilde G_a widetilde","display":"\\widetilde G_a","kind":"use","excerpt":"[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/s","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-756f2ac70a75c1ed","tex":"\\widetilde o","aliases":"widetilde o widetilde","display":"\\widetilde o","kind":"use","excerpt":"[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/s","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-15c79c23846df118","tex":"\\widetilde o_e","aliases":"widetilde o_e widetilde","display":"\\widetilde o_e","kind":"use","excerpt":"[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/s","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-aba1aecba944a16d","tex":"\\widetilde U","aliases":"widetilde U widetilde","display":"\\widetilde U","kind":"use","excerpt":"Approximate performance transfer. Suppose a source comparison gives productivity advantage Δ to U' over U. Let widetilde U',widetilde U be actual transported target procedures. If each procedure's productivity differs from its source value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-72746316ee7d6770","tex":"\\widetilde U'","aliases":"widetilde U' widetilde","display":"\\widetilde U'","kind":"use","excerpt":"Approximate performance transfer. Suppose a source comparison gives productivity advantage Δ to U' over U. Let widetilde U',widetilde U be actual transported target procedures. If each procedure's productivity differs from its source value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ea2a8d1df2130cc8","tex":"\\widetilde X","aliases":"widetilde X widetilde","display":"\\widetilde X","kind":"use","excerpt":"[Repair synthesis under a fixed unknown model] Construct a finite repair model on widetilde X=Θ× X by widetilde f_e(θ,x)=(θ,f_e^θ(x)), widetilde o_e(θ,x)=o_e^θ(x), widetilde G_a={(θ,x):x∈ G_a^θ}, with the corresponding tagged domains and the common action costs. Applying Theorem~[source reference] to this model gives exactly the nondominated work/s","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":26},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f5857cac0ddcea35","tex":"X'","aliases":"X'","display":"X'","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"appendix A complete diagnostic trace. Take n=8 and hidden defect j=5, whose binary digits are 101. One optimal tree uses the three queries A_0={1,3,5,7}, A_1={2,3,6,7}, A_2={4,5,6,7}. The precise order may differ among equally optimal compiled trees. For this order, the supports are {0,…,7}xrightarrow1{1,3,5,7} xrightarrow0{1,5}xrightarrow1{5}. The","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequali","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8e1b422661cf5616","tex":"Y_i","aliases":"Y_i","display":"Y_i","kind":"use","excerpt":"One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequali","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-50e74e1b36da6948","tex":"Y_t","aliases":"Y_t","display":"Y_t","kind":"use","excerpt":"We give a small time-uniform alternative whose finite arithmetic is directly inspectable. Before an audit begins, freeze the candidate procedure, its comparator, their installed information, the task law, the common resource cap, and a paired score Y_t∈[-1,1]. Timeouts and invalid outcomes must have a stipulated score rather than disappear from the","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"For each transition, a progress mark p(z,a,z')∈{0,1} is supplied by a checked criterion. A mark of one means the specified process-improvement claim has been established under the same comparison. It is not assigned merely because a file changed. Define W_0,0 · = K, · W_0,k · =∅ (k>0),","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-3480de748bc9723a","tex":"Z_t","aliases":"Z_t","display":"Z_t","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"[Least stable refinement of a retained representation] The splitting construction stabilizes after at most |X|-|P_0| strict refinements. Its final partition is the coarsest refinement of P_0 on which every F_u has a well-defined quotient operation. It preserves every finite continuation of the retained labels and outputs. A strict refinement increa","count":9},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-0044552e1459e501","tex":"|_{D_v}","aliases":"|_D_v","display":"|_{D_v}","kind":"use","excerpt":"Suppose a revision replaces a by a'. Mark a vertex directly stale whenever one of its declared component values differs. Let Δ be the union of those vertices and all their descendants in the dependency graph. Equivalently, in a topological order, mark v whenever a|_D_v≠ a'|_D_v or pa(v)∩Δ≠∅. Here equality refers to the declared artifact value, not ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions into one declared additive resource measure. Its priced saving is w·Δ(n). The prices are modeling assumptions, not","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"t possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an investigation. The initial information is a nonempty","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_","count":15},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Approximate performance transfer. Suppose a source comparison gives productivity advantage Δ to U' over U. Let widetilde U',widetilde U be actual transported target procedures. If each procedure's productivity differs from its source value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. ","count":19},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"ource value by at most ε, then R'(widetilde U')- R'(widetilde U) ≥Δ-2ε. This follows by subtracting the two inequalities. A common source of the premise is an exact task correspondence plus a bounded evaluation-law shift. For any fixed gain function in [-1,1], changing the task law by total variation δ changes its expectation by at most 2δ. A furth","count":18},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"Evidence across recursive generations. For attempt j, condition on the complete earlier lineage, freeze its candidate, and initialize a fresh valid audit at E_0^(j)=1 with error allowance δ_j. If ∑_jδ_j≤δ, Proposition~[source reference] bounds the probability of ever accepting a false margin claim by δ. Each audit can stop sequentially, and the nex","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c0cfccdc70abb5b6","tex":"\\Delta_t","aliases":"Δ_t Delta","display":"\\Delta_t","kind":"use","excerpt":"One sufficient fresh-audit construction. Freeze a candidate and its comparator using the prior history. Draw n_t fresh evaluation cases conditionally independently from the declared distribution and record paired score differences Y_i∈[-1,1]. Let Δ_t be their conditional mean and widehatΔ_t their average. The bounded-variable concentration inequali","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4abc637c4be411ae","tex":"\\delta_t","aliases":"δ_t delta","display":"\\delta_t","kind":"use","excerpt":"[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which att","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"[Selection under uniform assessment error] If |widehat R_i-R_i|≤ε for every candidate and hatimath maximizes widehat R_i, then R_hatimath≥ max_iR_i-2ε. If a baseline is included and the selected candidate's estimated advantage exceeds 2ε+γ, its true advantage exceeds γ. For a true maximizer i^*, R_hatimath≥widehat R_hatimath-ε ≥widehat R_i^*-ε ≥ R_","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-26849087606e21ff","tex":"\\eta_e","aliases":"η_e eta","display":"\\eta_e","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"A useful first extension retains several explicit possibilities rather than replacing uncertainty by one convenient estimate. Let Θ be a finite nonempty set of candidate models with a common state set X, experiment names, repair names, and nonnegative costs. Model θ specifies D_e^θ, f_e^θ, o_e^θ, and G_a^θ. The unknown model is fixed during an inve","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"Let a hidden case θ determine a set R(θ) of successful repairs. The set may have several members. Diagnosis need only obtain enough information to choose a repair in the appropriate set. It need not identify every hidden property of θ.","count":22},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-9f146cf9b0e1580e","tex":"\\theta_1","aliases":"θ_1 theta","display":"\\theta_1","kind":"use","excerpt":"Repair sufficiency is not full identification. Suppose R(θ_1)={a,c} and R(θ_2)={b,c}. The cases are distinct, yet repair c works in both. A method can stop without learning which case occurred. Conversely, pairwise overlap of repair sets does not guarantee one repair for an entire uncertainty set. The three sets {a,b}, {b,c}, and {a,c} intersect pa","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d74d781424901ae1","tex":"\\theta_2","aliases":"θ_2 theta","display":"\\theta_2","kind":"use","excerpt":"Repair sufficiency is not full identification. Suppose R(θ_1)={a,c} and R(θ_2)={b,c}. The cases are distinct, yet repair c works in both. A method can stop without learning which case occurred. Conversely, pairwise overlap of repair sets does not guarantee one repair for an entire uncertainty set. The three sets {a,b}, {b,c}, and {a,c} intersect pa","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"Let B be a finite nonempty immutable-case support and let G_r⊆ B be each repair's success set restricted to it. Define κ(B) to be the smallest number of repair-success sets whose union covers B. Set κ(B)=∞ when no such cover exists. This number concerns the repaired objective, not the number of hidden states. Computing it can itself be expensive.","count":9},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Choose a nonnegative bet λ_t, measurable from H_t-1, with λ_t≤1/(1+γ). Define E_0=1, E_t=E_t-1(1+λ_t(Y_t-γ)). The variable E_t is an evidence process, not a posterior probability that the candidate is better. Predictability means choosing the bet before observing the score to which it is applied.","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-29b14a34343e28e4","tex":"\\lambda_t","aliases":"λ_t lambda","display":"\\lambda_t","kind":"use","excerpt":"Choose a nonnegative bet λ_t, measurable from H_t-1, with λ_t≤1/(1+γ). Define E_0=1, E_t=E_t-1(1+λ_t(Y_t-γ)). The variable E_t is an evidence process, not a posterior probability that the candidate is better. Predictability means choosing the bet before observing the score to which it is applied.","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-93b36ed48fba09a5","tex":"\\Xi_t","aliases":"Ξ_t Xi","display":"\\Xi_t","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked op","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-587890bf36e41b2d","tex":"\\pi_\\omega","aliases":"π_ω pi omega","display":"\\pi_\\omega","kind":"use","excerpt":"What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked op","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":8},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-faf2d9e8110151a4","tex":"\\rho_t","aliases":"ρ_t rho","display":"\\rho_t","kind":"use","excerpt":"What persists across a recursive step. A lineage state can be written schematically as Z_t=(S_t,U_t,L_t,E_t,Ξ_t,ρ_t), where E_t records evidence and rejected alternatives, and ρ_t records remaining resources and scope. A proposed modification may target any admitted subset. A comparison across different Ξ_t needs a declared bridge. It cannot be jus","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked op","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Consider source and target finite repair models. Let σ:X→ X' be a bijection, τ: E→ E' a bijection of experiments, ψ: A→ A' a bijection of repairs, and η_e:O_e→ O'_τ(e) a bijection of report labels. Assume σ(D_e) · =D'_τ(e), · σ f_e(x) · =f'_τ(e)σ(x),","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"What would learn. Let r_ϕ encode the retained state, G_θ,a describe proposed operations, π_ω choose observations or experiments, and p_ψ propose candidate programs or parameter updates. A development episode can modify ϕ,θ,ω,ψ through a declared learning rule. A persistent program library can also change by adding, replacing, or removing checked op","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"A small world with reusable diagnostic structure. Consider eight cells with exactly one latent defect, indexed by j∈{0,…,7}. The solver must repair that cell while preserving the others. Its old representation retains only the number of defects, so every episode initially looks identical.","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Feature sufficiency for a search decision] Let P be a finite set of search contexts, let r: P→ Z be their available feature map, and let ℓ: P→{0,1} be the required decision label. A deterministic rule f:Z→{0,1} satisfying fr=ℓ exists if and only if ℓ is constant on each nonempty fibre of r. A pair in one fibre with different labels rules out every","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"proposition[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time a","count":3},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"Suppose a revision replaces a by a'. Mark a vertex directly stale whenever one of its declared component values differs. Let Δ be the union of those vertices and all their descendants in the dependency graph. Equivalently, in a topological order, mark v whenever a|_D_v≠ a'|_D_v or pa(v)∩Δ≠∅. Here equality refers to the declared artifact value, not ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":7},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Which gains survive a change of resource prices? Fix an additive ledger with d≥1 named event types. Let K∈ R_≥0^d be extra setup, let s_i∈ R^d be the old-minus-new per-use event vector for workload type i, and let n_i be its nonnegative integer multiplicity. The net event saving is Δ(n)=∑_i n_i s_i-K. A price vector w∈ R_≥0^d specifies conversions ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Evidence across recursive generations. For attempt j, condition on the complete earlier lineage, freeze its candidate, and initialize a fresh valid audit at E_0^(j)=1 with error allowance δ_j. If ∑_jδ_j≤δ, Proposition~[source reference] bounds the probability of ever accepting a false margin claim by δ. Each audit can stop sequentially, and the nex","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ccbd66bdaed18a2b","tex":"\\sum_o","aliases":"∑_o sum","display":"\\sum_o","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-702bc9ceff3fca47","tex":"\\sum_{t=1}^{T}","aliases":"∑_t=1^T sum","display":"\\sum_{t=1}^{T}","kind":"use","excerpt":"Let C_rm prepare be the complete preparation cost in a declared common resource model, including failed proposals and certificate checking, and let d_t be the subsequent saving on task t after including dispatch. Reuse is beneficial in that model exactly when ∑_t=1^Td_t>C_rm prepare. This is the earlier amortization identity applied to acquisition ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-075f5b1599c3c4ed","tex":"\\sum_{t\\geq1}","aliases":"∑_t≥1 sum geq","display":"\\sum_{t\\geq1}","kind":"use","excerpt":"[Conditional error budgets across an adaptive lineage] Let H_t-1 be the information available before attempt t. Suppose the event A_t that attempt t accepts a false improvement claim satisfies Pr(A_tmid H_t-1)≤δ_t almost surely, for deterministic δ_t≥0 with ∑_t≥1δ_t≤δ. Then Pr(∃ t≥1:A_t)≤δ. The candidate-generating process and the time at which att","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7fb07bdba495887b","tex":"\\sum_{x\\in B}","aliases":"∑_x∈ B sum in","display":"\\sum_{x\\in B}","kind":"use","excerpt":"[Repair-relative weight and prefix obstruction] Suppose a total immutable deterministic binary diagnostic tree succeeds on nonempty support B and has L reachable leaves and maximum query depth h. Under~[source reference], ∑_x∈ Bw_x≤ L≤2^h. At a candidate query root with h≥1 remaining query levels, if the cases already read into either outcome branc","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-2fa0bdda5c9e6c7e","tex":"\\sum_{x\\in S_r}","aliases":"∑_x∈ S_r sum in","display":"\\sum_{x\\in S_r}","kind":"use","excerpt":"Let X be a finite immutable state set and let S_r⊆ X be the success set of repair r. Choose nonnegative rational weights w_x satisfying ∑_x∈ S_rw_x≤1 for every permitted repair r. This is a feasible dual weight assignment for unweighted set cover. We use only its elementary weak-duality property, not an assumption that the best assignment has been ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"lowdown from a certified two-worker race] Suppose a parent and a shortcut worker can be paused after each unit of a common work measure and resumed without losing progress. Each may announce completion only after producing a result certified for the same target. Alternate one unit of each worker and stop at the first certified completion. If their ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"Suppose a revision replaces a by a'. Mark a vertex directly stale whenever one of its declared component values differs. Let Δ be the union of those vertices and all their descendants in the dependency graph. Equivalently, in a topological order, mark v whenever a|_D_v≠ a'|_D_v or pa(v)∩Δ≠∅. Here equality refers to the declared artifact value, not ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"[The same operator spelling with changed semantics] The identity x+0=x is valid for ordinary addition. Suppose a revised interpreter uses the same printed symbol but executes amathbin⊕b=a+b+1. Then xmathbin⊕0≠ x. A cache certified for the former interpreter cannot be imported merely because its expression strings still parse. A semantic identifier ","count":2},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b4f882cbfabf90e0","tex":"\\equiv_{P_t}","aliases":"≡_P_t equiv","display":"\\equiv_{P_t}","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-e3fa6485b9bac2a9","tex":"\\equiv_{P_{t+1}}","aliases":"≡_P_t+1 equiv","display":"\\equiv_{P_{t+1}}","kind":"use","excerpt":"Repairing a failed representation. Let X be a finite set with deterministic operations F_u:X→ X and output labels q(x). Begin with a partition P_0 that preserves both the old retained labels and q. Define the next partition by splitting a block whenever two states in it have successors in different current blocks under some permitted operation. Equ","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"Learning equations without confusing probes with proofs. The reference language uses exact integers, the total operations +, min, max, and the comparisons <, ≤, >, ≥, and =. A finite schema language contains 135 terms built from variables x,y, small integer literals, Boolean literals, and at most one binary operation. The proposer evaluates those s","count":5},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"ry decision: report case zero if the repair belongs to R(0), and case one otherwise. Success in case zero implies a correct decision. Success in case one also implies a correct decision because the two repair sets are disjoint. Thus repair success is at most the success of this binary test. For a randomized test with probability f(o)∈[0,1] of repor","count":26},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"Learning equations without confusing probes with proofs. The reference language uses exact integers, the total operations +, min, max, and the comparisons <, ≤, >, ≥, and =. A finite schema language contains 135 terms built from variables x,y, small integer literals, Boolean literals, and at most one binary operation. The proposer evaluates those s","count":6},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"One hidden bit and two sensor conventions. [Known-model repair without a common unknown-model policy] Let x∈{0,1} be the cell requiring repair and let θ∈{0,1} be a fixed sensor convention. A nondestructive query reports o=x⊕θ at cost one. Repair r_i succeeds exactly when x=i and costs one. If θ is known, one query followed by the appropriate repair","count":4},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":1},{"work":"monographs-self-improving-intelligence","url":"notation/monographs-self-improving-intelligence.html#symbol-4042331604425568","tex":"\\bigcup_{e:B_0\\subseteq D_e}","aliases":"⋃_e:B_0⊆ D_e bigcup subseteq","display":"\\bigcup_{e:B_0\\subseteq D_e}","kind":"use","excerpt":"For horizon h, let F_h(B_0) be the nondominated attainable pairs (w,m). A pair dominates another when it is no larger in either coordinate and strictly smaller in at least one. Let T(B_0)={(c_a,1):B_0⊆ G_a}. Then F_0(B_0) · =Pareto T(B_0), F_h(B_0) · =Pareto(T(B_0)∪ ⋃_e:B_0⊆ D_e {(c_e+max_o w_o, 1+∑_o m_o): (w_o,m_o)∈ F_h-1(B_e,o) ∀ o}). nonumber A","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"notation-table","excerpt":"F_a, q · A state transition and the task observation it must preserve.","count":94},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathcal","display":"{\\mathcal C}","kind":"notation-table","excerpt":"x∼_ Cy · Equality of every task observation after every permitted finite word.","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"notation-table","excerpt":"E∈ E · An admitted evidence event used to filter V.","count":19},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-fb7427a95ad8c5b4","tex":"\\E","aliases":"E mathcal","display":"{\\mathcal E}","kind":"notation-table","excerpt":"E∈ E · An admitted evidence event used to filter V.","count":8},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a14da4e5535e89e4","tex":"E_\\infty","aliases":"E_∞ infty","display":"E_\\infty","kind":"notation-table","excerpt":"E_∞ · Propagated span of all model-discrepancy directions.","count":8},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"notation-table","excerpt":"F_a, q · A state transition and the task observation it must preserve.","count":8},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"notation-table","excerpt":"F_a, q · A state transition and the task observation it must preserve.","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"notation-table","excerpt":"K^π(hmidω) · Probability of observable record h under one permitted protocol.","count":26},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"notation-table","excerpt":"I_Q · Initial vectors with common output continuations across fixed models.","count":12},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"notation-table","excerpt":"p_i,y · Joint probability of a likelihood profile and a target value.","count":46},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-caf5dd124506cb34","tex":"I_Q","aliases":"I_Q","display":"I_Q","kind":"notation-table","excerpt":"I_Q · Initial vectors with common output continuations across fixed models.","count":8},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","excerpt":"K^π(hmidω) · Probability of observable record h under one permitted protocol.","count":10},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-efdf478668af045f","tex":"K^\\pi","aliases":"K^π pi","display":"K^\\pi","kind":"notation-table","excerpt":"K^π(hmidω) · Probability of observable record h under one permitted protocol.","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"notation-table","excerpt":"K^π(hmidω) · Probability of observable record h under one permitted protocol.","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"notation-table","excerpt":"p_i,y · Joint probability of a likelihood profile and a target value.","count":29},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-bbc9af944a60b2b7","tex":"p_{i,y}","aliases":"p_i,y","display":"p_{i,y}","kind":"notation-table","excerpt":"p_i,y · Joint probability of a likelihood profile and a target value.","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"I_Q · Initial vectors with common output continuations across fixed models.","count":55},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"notation-table","excerpt":"F_a, q · A state transition and the task observation it must 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Its access is explicit.","count":21},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"notation-table","excerpt":"U, S_a · A calibrated input subspace and a known operator restriction.","count":19},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-785e812c9b93033d","tex":"S_a","aliases":"S_a","display":"S_a","kind":"notation-table","excerpt":"U, S_a · A calibrated input subspace and a known operator restriction.","count":14},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"notation-table","excerpt":"U, S_a · A calibrated input subspace and a known operator restriction.","count":40},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e6d266c8226990fb","tex":"U_\\infty","aliases":"U_∞ infty","display":"U_\\infty","kind":"notation-table","excerpt":"U_∞ · The maximal domain on which calibrated continuation is determined.","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"notation-table","excerpt":"V⊆Ω · A set of candidate worlds, including the possible empty set.","count":53},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"notation-table","excerpt":"r:X→ Z · Retained evidence or an available initial code. Its access is explicit.","count":10},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"notation-table","excerpt":"x∼_ Cy · Equality of every task observation after every permitted finite word.","count":27},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"notation-table","excerpt":"x∼_ Cy · Equality of every task observation after every permitted finite word.","count":27},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"notation-table","excerpt":"r:X→ Z · Retained evidence or an available initial code. Its access is explicit.","count":16},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"notation-table","excerpt":"K^π(hmidω) · Probability of observable record h under one permitted protocol.","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"notation-table","excerpt":"Σ(V) · The complete signature of future task-valued corrections.","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"notation-table","excerpt":"midrule ω∈Ω · A candidate world, including any fixed model or historical tag.","count":24},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"notation-table","excerpt":"midrule ω∈Ω · A candidate world, including any fixed model or historical tag.","count":34},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"notation-table","excerpt":"r:X→ Z · Retained evidence or an available initial code. Its access is explicit.","count":26},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"notation-table","excerpt":"midrule ω∈Ω · A candidate world, including any fixed model or historical tag.","count":47},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"notation-table","excerpt":"U_∞ · The maximal domain on which calibrated continuation is determined.","count":21},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"notation-table","excerpt":"x∼_ Cy · Equality of every task observation after every permitted finite word.","count":9},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e253178b877b95ca","tex":"\\sim_{\\C}","aliases":"∼_C ∼_ C sim C sim mathcal","display":"\\sim_{{\\mathcal C}}","kind":"notation-table","excerpt":"x∼_ Cy · Equality of every task observation after every permitted finite word.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"notation-table","excerpt":"V⊆Ω · A set of candidate worlds, including the possible empty set.","count":24},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation nam","count":191},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"definition-passage","excerpt":"[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation nam","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation nam","count":34},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"definition-passage","excerpt":"[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation nam","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation nam","count":18},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"definition-passage","excerpt":"[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation nam","count":8},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6d584c042dd0c359","tex":"G_w","aliases":"G_w","display":"G_w","kind":"definition-passage","excerpt":"[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation nam","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is include","count":29},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-523d0c86cceb2d71","tex":"H_\\pi","aliases":"H_π pi","display":"H_\\pi","kind":"definition-passage","excerpt":"Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is include","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is include","count":47},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Exact initialization from available evidence] Given an available initial code r:X→ H, an exact reusable predictor consists of a finite state set Z, an initialization d:H→ Z, updates G_a:Z→ Z, and an output map bar q:Z→ Y such that bar q G_w d r(x)=q(F_wx) for all x and all finite words. After initialization the predictor receives the operation nam","count":28},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is include","count":10},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"definition-passage","excerpt":"Laws, records, and reusable predictive state. [Finite evidence contract] A finite evidence contract has a nonempty candidate set Ω, a family Π of permitted protocols, a finite record space H_π for each protocol, and kernels K^π(hmidω). A target t:Ω→ Y states what is to be identified. Any hidden model index that is fixed during a protocol is include","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9b2c0bb8ad27aac4","tex":"\\Sigma_{\\E,t}","aliases":"Σ_E,t Σ_ E,t Sigma E Sigma mathcal","display":"\\Sigma_{{\\mathcal E},t}","kind":"definition-passage","excerpt":"[Correction signature] The correction signature of V is Σ_ E,t(V)=(Q(V∩ E))_E∈ E.tag3 Two candidate sets have the same signature when every admitted evidence filter leaves the same possible target values, including whether no candidate remains.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"definition-passage","excerpt":"[Correction signature] The correction signature of V is Σ_ E,t(V)=(Q(V∩ E))_E∈ E.tag3 Two candidate sets have the same signature when every admitted evidence filter leaves the same possible target values, including whether no candidate remains.","count":11},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.","count":43},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":50},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-53d6e2369412f170","tex":"\\A","aliases":"A mathcal","display":"{\\mathcal A}","kind":"introduction-passage","excerpt":"Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-7208e5576d4df180","tex":"A^{n-1-j}","aliases":"A^n-1-j","display":"A^{n-1-j}","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"introduction-passage","excerpt":"plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v.","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"introduction-passage","excerpt":"plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":44},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9b3234202ef44211","tex":"B^j","aliases":"B^j","display":"B^j","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":30},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"introduction-passage","excerpt":"Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":55},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.","count":38},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":28},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":27},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":14},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":53},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":37},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-cf9404d3f534ee7e","tex":"n_k","aliases":"n_k","display":"n_k","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":15},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4d1a2a0483cceb59","tex":"S_{a_1}","aliases":"S_a_1","display":"S_{a_1}","kind":"introduction-passage","excerpt":"plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-1449bc6ae24d0315","tex":"S_{a_j}","aliases":"S_a_j","display":"S_{a_j}","kind":"introduction-passage","excerpt":"plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ed40760b634e85a0","tex":"S_w","aliases":"S_w","display":"S_w","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":34},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-5d42311231194d5c","tex":"t^{-1}","aliases":"t^-1","display":"t^{-1}","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6bd9b7e7cab61954","tex":"T_{a_1}","aliases":"T_a_1","display":"T_{a_1}","kind":"introduction-passage","excerpt":"rox For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operat","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8918c5fecfd6422d","tex":"T_{a_j}","aliases":"T_a_j","display":"T_{a_j}","kind":"introduction-passage","excerpt":"rox For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operat","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d1272d40e795db7e","tex":"T_w","aliases":"T_w","display":"T_w","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"introduction-passage","excerpt":"Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"introduction-passage","excerpt":"Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4784442b3d8786a6","tex":"U_{j+1}","aliases":"U_j+1","display":"U_{j+1}","kind":"introduction-passage","excerpt":"Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"Define a descending sequence of subspaces U_0=U, U_j+1={v∈ U:S_av∈ U_j for every a∈ A}.tag7 The sequence is descending by induction. Every strict inclusion decreases dimension, so a fixed point is reached after at most dim U strict drops. Denote it by U_∞.","count":64},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":9},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"introduction-passage","excerpt":"rox For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operat","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-0430cf91ce15859e","tex":"x_{j+1}","aliases":"x_j+1","display":"x_{j+1}","kind":"introduction-passage","excerpt":"≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v.","count":21},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ae33ff0c9a9cfbb4","tex":"z_j","aliases":"z_j","display":"z_j","kind":"introduction-passage","excerpt":"plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v.","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ce5d4e7c1cb0ec93","tex":"z_{j+1}","aliases":"z_j+1","display":"z_{j+1}","kind":"introduction-passage","excerpt":"≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v. For the next operation, x_j+1-z_j+1 ≤ A x_j-z_j +η z_j ≤ A x_j-z_j","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"ete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate marginals. There are r=|D_t| coordinates with nonnegative entri","count":9},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a11e9f1487b2c505","tex":"\\Delta_k","aliases":"Δ_k Delta","display":"\\Delta_k","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":18},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f936e2a0d91e3cb9","tex":"\\lambda_{e,o,i}","aliases":"λ_e,o,i lambda","display":"\\lambda_{e,o,i}","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":12},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":12},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":28},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ed833334953a2c63","tex":"\\ell_{e,o}","aliases":"ℓ_e,o ell","display":"\\ell_{e,o}","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":31},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"introduction-passage","excerpt":"A concrete probability record. For an integer k≥1, write Δ_k for the probability simplex on k cells. Fix a denominator N≥1. A retained record has the form widehat p=(n_1/N,…,n_k/N), n_i∈{0,…,N}, ∑_i n_i=N. The cell identities, denominator, and decoding rule are fixed background. The k integer fields can be stored literally in klceillog_2(N+1)rceil ","count":11},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f7b37e8032ebe855","tex":"\\sum_{j=0}^{n-1}","aliases":"∑_j=0^n-1 sum","display":"\\sum_{j=0}^{n-1}","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4ddca6b6229ab410","tex":"\\sum_{\\omega\\in A_i,\\ t(\\omega)=y}","aliases":"∑_ω∈ A_i, t(ω)=y sum omega in omega","display":"\\sum_{\\omega\\in A_i,\\ t(\\omega)=y}","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"Partition Ω into the nonempty cells A_1,…,A_m on which the complete likelihood vector (ℓ_e,o(ω))_e,o is constant. Write its common value as λ_e,o,i. The profile vectors of different cells are distinct by construction. For a prior μ, retain p_i,y(μ)=∑_ω∈ A_i, t(ω)=yμ(ω), (i,y)∈ D_t={(i,y):A_i∩ t^-1(y)≠∅}. These are joint masses, not the two separate","count":11},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":32},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"[Approximate continuation with explicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯","count":16},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"plicit leakage control] For every v∈ L and every word w of length n≥1, T_wv-S_wv ≤η v ∑_j=0^n-1A^n-1-jB^j.tag9 For the empty word the discrepancy is zero. Integer zero powers in the displayed finite sum have their usual value one. The conclusion does not require the actual trajectory to remain in L. Write z_j=S_a_j⋯ S_a_1v∈ L and x_j=T_a_j⋯ T_a_1v.","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-40848b29e73821cf","tex":"\\mathbf1_{A_i}","aliases":"1_A_i mathbf","display":"\\mathbf1_{A_i}","kind":"use","excerpt":"For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"1 · 0, Q=1 · 1, v=binom11. Both fixed systems have output sequence 2,1,0,0,… from v, since A^2=B^2=0. But applying A first and B second gives QBAv=1, not zero. That hybrid trajectory belongs to neither candidate model. Treating fixed uncertainty as a freely changing matrix would incorrectly reject the valid output prediction.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-0014cfc2591bdee6","tex":"a_*","aliases":"a_*","display":"a_*","kind":"use","excerpt":"Let a_* minimize R. Then R(widehat a)≤widehat R(widehat a)+ξ ≤widehat R(a_*)+ξ≤ R(a_*)+2ξ. For any other decision, the approximate gap loses at most 2ξ when the two risks are replaced by their true values, proving the strict-gap assertion. The threshold assertion is the upper risk bound itself. Each step uses the same posterior and surrogate respon","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"A finite test for agreement under a fixed unknown model. Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y.","count":11},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-658c399ebd168feb","tex":"A_H","aliases":"A_H","display":"A_H","kind":"use","excerpt":"let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"For a word w=a_1⋯ a_m, F_w applies a_1 first and a_m last. The empty word is the identity. Define x∼_ C x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite w. This is a comparison in a common observation language. It does not require that different models share arbitrary latent coordinates. The behavioral class is a canonical response profile e","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-82821bc2124f1658","tex":"A_t","aliases":"A_t","display":"A_t","kind":"use","excerpt":"Proposition~[source reference] and the rounding bound give d_TV(p_t+1,widehat p_t+1) ≤γd_TV(p_t,widehat p_t)+γε+η_N. Iterating the uncapped scalar bound proves (11); total variation is also at most one. Aggregating cell coordinates into target coordinates cannot increase total variation, by the triangle inequality. Substituting e_0≤η_N gives η_N A_","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4f1eee202dc651a9","tex":"A_u","aliases":"A_u","display":"A_u","kind":"use","excerpt":", let any competing linear predictor be given. Its reachable span is generated by vectors G_wdR(v). Associate such a vector with [A_wv]∈ W/N and extend by linearity. To see this association is well defined, suppose a finite linear combination of the predictor vectors is zero. Applying any later word and the predictor readout shows that the matching","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"ent under a fixed unknown model Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y. Define I_Q={v∈ V:QT_w^θ ","count":11},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f8e650d523df8a1a","tex":"\\mathrm{after}","aliases":"after mathrm","display":"\\mathrm{after}","kind":"use","excerpt":"A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"Let Ω be finite and nonempty, let t:Ω→ Y have finite image, and let B be a finite collection of experiments. Experiment e has a finite outcome set and specified likelihoods ℓ_e,o(ω)≥0, ∑_oℓ_e,o(ω)=1. The world is fixed. Every experiment remains available, and each new report has this conditional law given the world, the chosen experiment, and all e","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Why experimentwise evidence need not compose. [Pairwise separation without a successful investigation] There are three possible initial cases 0,1,2. Test a reports whether the case is 0, and test b reports whether it is 1. Either first test then destroys the accessible case state. Every later test returns a fixed zero report. The target remains the","count":28},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e77f9eceea598e09","tex":"B'","aliases":"B'","display":"B'","kind":"use","excerpt":"[Ambiguity radius and an explicit prediction certificate] For every nonempty finite B, inf_z∈ R^d max_ω∈ B z-v(ω) _∞=ρ(B), and m(B) attains the infimum. If B'⊆ B is nonempty then ρ(B')≤ρ(B).","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f76a26dbc0d9fe26","tex":"B^2","aliases":"B^2","display":"B^2","kind":"use","excerpt":"1 · 0, Q=1 · 1, v=binom11. Both fixed systems have output sequence 2,1,0,0,… from v, since A^2=B^2=0. But applying A first and B second gives QBAv=1, not zero. That hybrid trajectory belongs to neither candidate model. Treating fixed uncertainty as a freely changing matrix would incorrectly reject the valid output prediction.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-177c0e2077099d42","tex":"B_a^\\theta","aliases":"B_a^θ theta","display":"B_a^\\theta","kind":"use","excerpt":"For each θ, form the paired maps and difference observation B_a^θ=T_a^θ · 0 0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-da1298a26addc701","tex":"\\mathcal B_{\\bar\\ell_{e,o}}","aliases":"B_barℓ_e,o mathcal bar ell","display":"\\mathcal B_{\\bar\\ell_{e,o}}","kind":"use","excerpt":"After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment sel","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d41402bfd3b150fb","tex":"B_c","aliases":"B_c","display":"B_c","kind":"use","excerpt":"What compression does to the candidate set. For a fixed raw-record channel, let B(h) be its exact support set. If only c(h) is retained, the correct support is B_c(z)=⋃_h:c(h)=zB(h). Indeed, a world can produce the coarse record exactly when it can produce at least one raw record in that fibre. Thus B(h)⊆ B_c(c(h)), and the ambiguity radius cannot ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-29921e72f47564e9","tex":"B_H","aliases":"B_H","display":"B_H","kind":"use","excerpt":"let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 ","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9474dae99bcf0491","tex":"B_t","aliases":"B_t","display":"B_t","kind":"use","excerpt":"Proposition~[source reference] and the rounding bound give d_TV(p_t+1,widehat p_t+1) ≤γd_TV(p_t,widehat p_t)+γε+η_N. Iterating the uncapped scalar bound proves (11); total variation is also at most one. Aggregating cell coordinates into target coordinates cannot increase total variation, by the triangle inequality. Substituting e_0≤η_N gives η_N A_","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ec4f7252d4f1b8b1","tex":"B_w^\\theta","aliases":"B_w^θ theta","display":"B_w^\\theta","kind":"use","excerpt":"[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is exclud","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c4967f04af944e7d","tex":"\\mathcal B_{\\widetilde\\ell}","aliases":"B_widetildeℓ mathcal widetilde ell","display":"\\mathcal B_{\\widetilde\\ell}","kind":"use","excerpt":"[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storag","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-869fb977e7cfe4b3","tex":"B_\\pi","aliases":"B_π pi","display":"B_\\pi","kind":"use","excerpt":"[Law identification and finite-record identification] The target is a function of the complete family of protocol laws if and only if it is constant on the ∼_Π classes. For a fixed protocol π, an error-free rule d:H_π→ Y exists if and only if t is constant on every nonempty support set B_π(h)={ω:K^π(hmidω)>0}. Allowing a randomized output rule does","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-cb04e3fd5b21a489","tex":"\\mathcal B_\\ell","aliases":"B_ℓ mathcal ell","display":"\\mathcal B_\\ell","kind":"use","excerpt":"For a likelihood vector ℓ=(ℓ_i) define the Bayes map B_ℓ(p)_i=fracp_iℓ_i∑_jp_jℓ_j whenever its denominator is positive.","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8f8a4d0a063c4065","tex":"\\bar Q","aliases":"bar Q bar","display":"\\bar Q","kind":"use","excerpt":"following conditions are equivalent. item QT_w^θ=QA_w on all of V, for every fixed model θ and every word w. item E_∞⊆ker Q. item There exists a subspace N⊆ker Q, invariant under every A_a, containing every im D_a^θ. item There is a surjective linear representation r:V→ Z, shared linear updates G_a, and a readout bar Q such that rT_a^θ=G_ar for all","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-1784da886e1aea15","tex":"\\bar v","aliases":"bar v bar","display":"\\bar v","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not obse","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-05eb652c0306afb0","tex":"\\bar v_0","aliases":"bar v_0 bar","display":"\\bar v_0","kind":"use","excerpt":"tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v_0-bar v_0 ≤ e_0. These in","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-7c3bbe0c694a1eff","tex":"\\bar v_t","aliases":"bar v_t bar","display":"\\bar v_t","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not obse","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d097515d24536636","tex":"\\bar v_{t+1}","aliases":"bar v_t+1 bar","display":"\\bar v_{t+1}","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not obse","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-fcd1d016a0b9e4bf","tex":"\\bar y","aliases":"bar y bar","display":"\\bar y","kind":"use","excerpt":"Fix an actual evidence history. Let μ be the exact posterior on the stipulated finite world set and let p_i=μ(C_i). Suppose the retained distribution is widehat p with d_TV(p,widehat p)≤ e. For a declared later action word or response horizon, let y(ω) be the actual model response and let bar y_i be the response used for cell i. Assume sup_ω∈ C_i y","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c844eabf65ff0f85","tex":"\\bar y_i","aliases":"bar y_i bar","display":"\\bar y_i","kind":"use","excerpt":"Fix an actual evidence history. Let μ be the exact posterior on the stipulated finite world set and let p_i=μ(C_i). Suppose the retained distribution is widehat p with d_TV(p,widehat p)≤ e. For a declared later action word or response horizon, let y(ω) be the actual model response and let bar y_i be the response used for cell i. Assume sup_ω∈ C_i y","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d97fba459e62ed0e","tex":"\\bar\\ell","aliases":"barℓ bar ell","display":"\\bar\\ell","kind":"use","excerpt":"Choose one valid representative report law barℓ_e,o,i per cell. Assume, for every admitted experiment, report, cell, and world in that cell, β≤ ℓ_e,o(ω),barℓ_e,o,i≤1, |ℓ_e,o(ω)-barℓ_e,o,i|≤ε. tag10 Both the actual and representative report laws sum to one over reports. Exact profile cells satisfy this condition with ε=0 when their likelihoods have ","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6345e53d57728b36","tex":"\\bar\\ell_{e,o}","aliases":"barℓ_e,o bar ell","display":"\\bar\\ell_{e,o}","kind":"use","excerpt":"After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment sel","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-272c6006840c4484","tex":"\\bar\\ell_{e,o,i}","aliases":"barℓ_e,o,i bar ell","display":"\\bar\\ell_{e,o,i}","kind":"use","excerpt":"Choose one valid representative report law barℓ_e,o,i per cell. Assume, for every admitted experiment, report, cell, and world in that cell, β≤ ℓ_e,o(ω),barℓ_e,o,i≤1, |ℓ_e,o(ω)-barℓ_e,o,i|≤ε. tag10 Both the actual and representative report laws sum to one over reports. Exact profile cells satisfy this condition with ε=0 when their likelihoods have ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"For each θ, form the paired maps and difference observation B_a^θ=T_a^θ · 0 0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.","count":13},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"every pair in D is separated by at least one admitted test. Executing all tests recovers t, and a selected nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegat","count":13},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditiona","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditiona","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"every pair in D is separated by at least one admitted test. Executing all tests recovers t, and a selected nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegat","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditiona","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-433d97a8fbb31f65","tex":"C_t","aliases":"C_t","display":"C_t","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not obse","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"Let the permitted decision set D be finite and nonempty. Its loss L(y,a) takes values in [0,1] and is K-Lipschitz in y, uniformly over a. Define R(a)=∑_ωμ(ω)L(y(ω),a), widehat R(a)=∑_iwidehat p_iL(bar y_i,a). These are risks under the stated model posterior. They are not frequentist coverage probabilities or guarantees on the loss of a single reali","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","excerpt":"Under the three strict inequalities, (I-M)^-1=D^-11-d · b c · 1-g has nonnegative entries. Let a=(I-M)^-1(1,1)^T. Both coordinates are positive and Ma=a-(1,1)^T<a. Hence M is a strict contraction in the weighted maximum norm with weights a, and its geometric series converges to (I-M)^-1. Summing the nonnegative right-hand side of (14) gives frace_0","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"[A tiny rounded-away mass can control a later posterior] Take two worlds with prior (1-d,d), where 0<d<1/(2N). The remainder rule gives the retained record (1,0). A binary report has likelihoods (d^2,1) for outcome one. After that outcome, the actual posterior mass of the second world is fracdd+(1-d)d^2=frac11+d-d^2, which tends to one as d tends t","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9f66bb3522fa9b25","tex":"d_3","aliases":"d_3","display":"d_3","kind":"use","excerpt":", (v_0,h_0)=(s,1). tag17 The predictor for cell (s,θ) uses bar v_t=2^-ts and therefore reports bar y_i=s/8. The nuisance bit n affects the physical response but remains unidentified by the reference observations. With g=d=1/2, b=1/64, c=1/8, and V_0=H_0=1, the comparison determinant is 127/512>0. Directly evaluating (14) at three prediction steps y","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6de3eb080886bbf6","tex":"D_a^\\theta","aliases":"D_a^θ theta","display":"D_a^\\theta","kind":"use","excerpt":"When uncertain dynamics admit one common observable quotient. Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all ","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-b377ac1e0c05ff0f","tex":"d_t","aliases":"d_t","display":"d_t","kind":"use","excerpt":"[Visible-error bounds from a common block comparison] Every admitted execution satisfying (13) obeys v_t-bar v_t ≤ d_t :=g^te_0+b∑_j=0^t-1g^t-1-j(u_j)_2. tag14 In particular, when b=0 and e_0=0, the visible prediction is exact for every finite time, without a stability requirement on d or c.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"[The coarsest exact memory for admitted corrections] Equality of correction signatures is a congruence for all updates V↦ V∩ E_0. The signature therefore supports well-defined updates and a readout of Q(V). It preserves that readout after every finite sequence of admitted filters.","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the g","count":10},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"We first separate these cases in a setting where observations do not interfere with one another. Let Ω be finite, let r:Ω→ Z be retained evidence, and let t:Ω→ Y be the revised target. Let E={e_1,…,e_m} be a finite family of total, deterministic, nondestructive tests. Each test reads the same fixed world and leaves it unchanged. All tests remain av","count":13},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"A two-dimensional calibration with a one-dimensional reusable core. [Calibration does not close under continuation] Let V have basis e_1,e_2,e_3, let U=span(e_1,e_2), and let the single calibrated operation satisfy Se_1=e_1, Se_2=e_3. The first continuation is known on all of U. The next calibrated domain is U_1=span(e_1), which is already invarian","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"For odd n, majority voting is the likelihood-ratio rule. Its error in either world is e_n=∑_j=0^(n-1)/2binomnj(3/4)^j(1/4)^n-j. The first values are e_1=1/4, e_3=5/32, and e_5=53/512. These are exact analytical probabilities, not observed error rates from an agent experiment.","count":10},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a64c5d974442c22a","tex":"e_5","aliases":"e_5","display":"e_5","kind":"use","excerpt":"For odd n, majority voting is the likelihood-ratio rule. Its error in either world is e_n=∑_j=0^(n-1)/2binomnj(3/4)^j(1/4)^n-j. The first values are e_1=1/4, e_3=5/32, and e_5=53/512. These are exact analytical probabilities, not observed error rates from an agent experiment.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"For J⊆{1,…,m}, write R_J(ω)=(r(ω),(e_j(ω))_j∈ J). Let the unresolved target pairs be D={{ω,ω'}:r(ω)=r(ω'), t(ω)≠ t(ω')}. A test covers such a pair when it gives different reports in the two worlds.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-14785c3f11130818","tex":"E_{j+1}","aliases":"E_j+1","display":"E_{j+1}","kind":"use","excerpt":"ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-542812c54fe5f147","tex":"e_k","aliases":"e_k","display":"e_k","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","excerpt":"We first separate these cases in a setting where observations do not interfere with one another. Let Ω be finite, let r:Ω→ Z be retained evidence, and let t:Ω→ Y be the revised target. Let E={e_1,…,e_m} be a finite family of total, deterministic, nondestructive tests. Each test reads the same fixed world and leaves it unchanged. All tests remain av","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"For odd n, majority voting is the likelihood-ratio rule. Its error in either world is e_n=∑_j=0^(n-1)/2binomnj(3/4)^j(1/4)^n-j. The first values are e_1=1/4, e_3=5/32, and e_5=53/512. These are exact analytical probabilities, not observed error rates from an agent experiment.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2775a21153e9ebc4","tex":"e_t","aliases":"e_t","display":"e_t","kind":"use","excerpt":"[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the g","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Small exact illustrations and reproduction. Three small exact suites check the declared constructions with integers, sets, rational arithmetic, and F_2. The retained observable suite compares paired row-space closure with direct word enumeration, tests common-quotient and initialization criteria, and compares likelihood-profile updates with raw-wor","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"(P_0,P_1)=frac12∑_h|P_0(h)-P_1(h)|.tag4 A deterministic function or a common stochastic processing channel applied to the transcript cannot increase this upper bound. Convert the estimator into a binary test by reporting world zero when it outputs the first target and world one otherwise. A correct target output always gives a correct binary decisi","count":10},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"Small exact illustrations and reproduction. Three small exact suites check the declared constructions with integers, sets, rational arithmetic, and F_2. The retained observable suite compares paired row-space closure with direct word enumeration, tests common-quotient and initialization criteria, and compares likelihood-profile updates with raw-wor","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"use","excerpt":"For the update, first hold ℓ fixed. Given 0≤ f_i≤1, put a=∑_i B_ℓ(q)_if_i and Z_p=∑_ip_iℓ_i. Then ∑_i( B_ℓ(p)_i- B_ℓ(q)_i)f_i =frac∑_i(p_i-q_i)ℓ_i(f_i-a)Z_p. Every value ℓ_i(f_i-a) lies in [-a,1-a], an interval of length one. Since p-q has total mass zero, its pairing with a function of oscillation at most one is bounded in absolute value by d_TV(p","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"it outputs the first target and world one otherwise. A correct target output always gives a correct binary decision. If f(h)∈[0,1] is the test's chance of declaring world zero, its success is frac12(1+∑_h f(h)(P_0(h)-P_1(h))). The sum is at most the sum of positive differences, which equals total variation. This proves (4). If B(h,z) is a common pr","count":12},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-963f554cb03ecddf","tex":"g^t","aliases":"g^t","display":"g^t","kind":"use","excerpt":"[Visible-error bounds from a common block comparison] Every admitted execution satisfying (13) obeys v_t-bar v_t ≤ d_t :=g^te_0+b∑_j=0^t-1g^t-1-j(u_j)_2. tag14 In particular, when b=0 and e_0=0, the visible prediction is exact for every finite time, without a stability requirement on d or c.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8fe5a7ed01773d24","tex":"g^{t-1-j}","aliases":"g^t-1-j","display":"g^{t-1-j}","kind":"use","excerpt":"[Visible-error bounds from a common block comparison] Every admitted execution satisfying (13) obeys v_t-bar v_t ≤ d_t :=g^te_0+b∑_j=0^t-1g^t-1-j(u_j)_2. tag14 In particular, when b=0 and e_0=0, the visible prediction is exact for every finite time, without a stability requirement on d or c.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-b4f9811bf3c61e66","tex":"G_t","aliases":"G_t","display":"G_t","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not obse","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v_0-bar v","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v_0-bar v","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-814f64e7253b7d27","tex":"H_t","aliases":"H_t","display":"H_t","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not obse","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-480c877d455d2c2b","tex":"h_t","aliases":"h_t","display":"h_t","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-723b629860f6ad6b","tex":"h_{t+1}","aliases":"h_t+1","display":"h_{t+1}","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not obse","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-0f6f9e87439c63ef","tex":"\\mathrm{id}","aliases":"id mathrm","display":"\\mathrm{id}","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"use","excerpt":"ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"[Ambiguity radius and an explicit prediction certificate] For every nonempty finite B, inf_z∈ R^d max_ω∈ B z-v(ω) _∞=ρ(B), and m(B) attains the infimum. If B'⊆ B is nonempty then ρ(B')≤ρ(B).","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-80558c2ac4e5b981","tex":"\\inf_{z\\in\\mathbb R^d}","aliases":"inf_z∈ R^d inf in mathbb","display":"\\inf_{z\\in\\mathbb R^d}","kind":"use","excerpt":"[Ambiguity radius and an explicit prediction certificate] For every nonempty finite B, inf_z∈ R^d max_ω∈ B z-v(ω) _∞=ρ(B), and m(B) attains the infimum. If B'⊆ B is nonempty then ρ(B')≤ρ(B).","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"For J⊆{1,…,m}, write R_J(ω)=(r(ω),(e_j(ω))_j∈ J). Let the unresolved target pairs be D={{ω,ω'}:r(ω)=r(ω'), t(ω)≠ t(ω')}. A test covers such a pair when it gives different reports in the two worlds.","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-390c5a967f2a0d11","tex":"K_0^\\theta","aliases":"K_0^θ theta","display":"K_0^\\theta","kind":"use","excerpt":"0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c996230a8e8e8e3a","tex":"K_{j+1}^\\theta","aliases":"K_j+1^θ theta","display":"K_{j+1}^\\theta","kind":"use","excerpt":"0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-023a3e73b5f3329a","tex":"K_j^\\theta","aliases":"K_j^θ theta","display":"K_j^\\theta","kind":"use","excerpt":"0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-5f7f2ad4a7098985","tex":"K_\\infty^\\theta","aliases":"K_∞^θ infty theta","display":"K_\\infty^\\theta","kind":"use","excerpt":"[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is exclud","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.","count":13},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"For a word w=a_1⋯ a_m, F_w applies a_1 first and a_m last. The empty word is the identity. Define x∼_ C x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite w. This is a comparison in a common observation language. It does not require that different models share arbitrary latent coordinates. The behavioral class is a canonical response profile e","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"[Evidence sufficient for one exact predictive realization] An exact reusable predictor exists if and only if r(x)=r(x') Longrightarrow x∼_ C x'.tag1 When it exists, the behavioral quotient X/∼_ C supplies a predictor with the fewest initialized states. Every predictor initialized for all x∈ X requires at least |X/∼_ C| distinct values in d(r(X)). E","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":15},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-3b9146b307a147c1","tex":"M^j","aliases":"M^j","display":"M^j","kind":"use","excerpt":"η · M, η>0, M≥0. The one-step error on the unit calibrated vector is η, independent of M. After two steps the error is η(1+M). It can be arbitrarily large while the local calibration error is fixed. For n steps the second coordinate is η∑_j=0^n-1M^j.","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8cf54ac1a04b7600","tex":"M^{t+1}","aliases":"M^t+1","display":"M^{t+1}","kind":"use","excerpt":"+fracb(cV_0+(1-g)H_0)(1-g)D. tag15 These are sufficient common bounds. Failure of the inequality for D is not a proof that a particular signed system is unstable. Conversely, the two diagonal conditions alone are insufficient for a uniform conclusion over the stated block class. The norm triangle inequality and the common block bounds give v_t+1 h_","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c879cfab2bd64d1b","tex":"m_j","aliases":"m_j","display":"m_j","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d51cf03329d9401b","tex":"M_{\\mu,y}","aliases":"M_μ,y mu","display":"M_{\\mu,y}","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"use","excerpt":"[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storag","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9cc66676daa83a18","tex":"\\max_j","aliases":"max_j max","display":"\\max_j","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-efb940232bcb4194","tex":"\\max_{\\omega\\in B}","aliases":"max_ω∈ B max omega in","display":"\\max_{\\omega\\in B}","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegative fixed cost c_j, the minimum cost of one fixed nonadaptive test set, used for all initial codes in case ","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-68aad21ee178f7ff","tex":"\\min_{J:\\,J\\text{ covers }D}","aliases":"min_J: J covers D min text","display":"\\min_{J:\\,J\\text{ covers }D}","kind":"use","excerpt":"nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegative fixed cost c_j, the minimum cost of one fixed nonadaptive test set, used for all initial codes in case ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f5fb8ef0d7cd1153","tex":"\\min_{\\omega\\in B}","aliases":"min_ω∈ B min omega in","display":"\\min_{\\omega\\in B}","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d6ce51886365878b","tex":"\\mathrm{model}","aliases":"model mathrm","display":"\\mathrm{model}","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ebc65dd86b43b2d4","tex":"N_0","aliases":"N_0","display":"N_0","kind":"use","excerpt":"eqker Q, invariant under every A_a, containing every im D_a^θ. item There is a surjective linear representation r:V→ Z, shared linear updates G_a, and a readout bar Q such that rT_a^θ=G_ar for all a,θ, Q=bar Qr. When the conditions hold, V/E_∞ gives one such quotient. The smallest-dimensional common linear quotient is V/N_0, where N_0=⋂_wker(QA_w) ","count":9},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"An exact count when evidence identifies a partition cell. Suppose A_1,…,A_m are a partition of Ω into nonempty cells, and E consists exactly of all unions of those cells. Define the attainable evidence--target pairs D_t={(i,y): y∈ t(A_i)}, r=|D_t|=∑_i|t(A_i)|. A candidate set induces an occupancy record O(V)={(i,t(ω)):ω∈ V∩ A_i}.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e1d6eed537fd5981","tex":"o_1","aliases":"o_1","display":"o_1","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-96526df066f36358","tex":"o_j","aliases":"o_j","display":"o_j","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-987c9403a7315b90","tex":"o_k","aliases":"o_k","display":"o_k","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-99f4082f84c6ad2b","tex":"\\mathrm{old}","aliases":"old mathrm","display":"\\mathrm{old}","kind":"use","excerpt":"A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"immutable worlds, one with a permanently zero output and one with a permanently one output. The available initial record is the same symbol in both. Within each world separately, a one-state predictor exists: it simply prints that world's output. There is nevertheless no predictor that can be initialized from the common record and be correct in bot","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"What a revised evidence language requires. Let P_0 be the partition of worlds into the nonempty old profile--target cells. Let P_1 be the corresponding partition after likelihoods or the target have changed, still on the same finite Ω. The old record stores the probability of each P_0 cell. The new record would store the probability of each P_1 cel","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6e5b4a214cba7723","tex":"p'_{i,y}","aliases":"p'_i,y","display":"p'_{i,y}","kind":"use","excerpt":"[Exact target correction from likelihood-profile memory] Under the fixed-world likelihood model, the record p=(p_i,y) supports every finite sequence of report predictions and target-posterior corrections. Its next-report probability and update are z_e,o=∑_i,yλ_e,o,ip_i,y, p'_i,y=fracλ_e,o,ip_i,yz_e,o (z_e,o>0), and its target readout is Pr(t=y)=∑_i","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"For a fixed protocol, let two worlds with different targets generate finite transcript laws P_0 and P_1. An estimator sees only that transcript and independent randomness. Its average success under equal prior probabilities is bounded by the discrimination in the entire record.","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-bbf5a2287f56b850","tex":"\\mathcal P_0","aliases":"P_0 mathcal","display":"\\mathcal P_0","kind":"use","excerpt":"What a revised evidence language requires. Let P_0 be the partition of worlds into the nonempty old profile--target cells. Let P_1 be the corresponding partition after likelihoods or the target have changed, still on the same finite Ω. The old record stores the probability of each P_0 cell. The new record would store the probability of each P_1 cel","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment sel","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"For a fixed protocol, let two worlds with different targets generate finite transcript laws P_0 and P_1. An estimator sees only that transcript and independent randomness. Its average success under equal prior probabilities is bounded by the discrimination in the entire record.","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ecdf2ff2033f302f","tex":"\\mathcal P_1","aliases":"P_1 mathcal","display":"\\mathcal P_1","kind":"use","excerpt":"What a revised evidence language requires. Let P_0 be the partition of worlds into the nonempty old profile--target cells. Let P_1 be the corresponding partition after likelihoods or the target have changed, still on the same finite Ω. The old record stores the probability of each P_0 cell. The new record would store the probability of each P_1 cel","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"use","excerpt":"For a likelihood vector ℓ=(ℓ_i) define the Bayes map B_ℓ(p)_i=fracp_iℓ_i∑_jp_jℓ_j whenever its denominator is positive.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d58992a0c0c08b27","tex":"p_t","aliases":"p_t","display":"p_t","kind":"use","excerpt":"After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment sel","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d0f71f17f13253c4","tex":"p_{t+1}","aliases":"p_t+1","display":"p_{t+1}","kind":"use","excerpt":"Proposition~[source reference] and the rounding bound give d_TV(p_t+1,widehat p_t+1) ≤γd_TV(p_t,widehat p_t)+γε+η_N. Iterating the uncapped scalar bound proves (11); total variation is also at most one. Aggregating cell coordinates into target coordinates cannot increase total variation, by the triangle inequality. Substituting e_0≤η_N gives η_N A_","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9481f5615013c541","tex":"P_\\omega","aliases":"P_ω omega","display":"P_\\omega","kind":"use","excerpt":"immutable worlds, one with a permanently zero output and one with a permanently one output. The available initial record is the same symbol in both. Within each world separately, a one-state predictor exists: it simply prints that world's output. There is nevertheless no predictor that can be initialized from the common record and be correct in bot","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d692f9ac1c18626a","tex":"\\Pow","aliases":"Pow P mathcal","display":"{\\mathcal P}","kind":"use","excerpt":"Let Ω be a finite model set and t:Ω→ Y a fixed target. For a candidate set V⊆Ω, let Q(V)=t(V)={t(ω):ω∈ V}. The empty set is permitted. It represents a contradiction between the admitted model family and the evidence, not a successful identification. Let E⊆ P(Ω) be a finite family of evidence events containing Ω and closed under intersections. Filte","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-3fd3db68bc14d307","tex":"\\Pr_{\\mu_+}","aliases":"Pr_μ_+ Pr mu","display":"\\Pr_{\\mu_+}","kind":"use","excerpt":"Consider a prior μ_+ assigning mass 1/2 to (s,θ)=(0,0),(1,1) and a prior μ_- assigning mass 1/2 to (1,0),(0,1), with any fixed nuisance convention. Both have uniform s and uniform θ. Both predict the next report 1 with probability 1/2. After that report, however, Pr_μ_+(s=1mid o=1)=tfrac34, Pr_μ_-(s=1mid o=1)=tfrac14. After k fresh reports all equa","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4e5ad6c316920417","tex":"\\Pr_{\\mu_-}","aliases":"Pr_μ_- Pr mu","display":"\\Pr_{\\mu_-}","kind":"use","excerpt":"Consider a prior μ_+ assigning mass 1/2 to (s,θ)=(0,0),(1,1) and a prior μ_- assigning mass 1/2 to (1,0),(0,1), with any fixed nuisance convention. Both have uniform s and uniform θ. Both predict the next report 1 with probability 1/2. After that report, however, Pr_μ_+(s=1mid o=1)=tfrac34, Pr_μ_-(s=1mid o=1)=tfrac14. After k fresh reports all equa","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"Why the unknown model must not change between steps. [A switching adversary creates a nonexistent failure] Over Q, let there be one action and two possible fixed matrices A=0 · 1","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"use","excerpt":"Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-da423ba3a883a93b","tex":"\\mathbb Q^2","aliases":"Q^2 mathbb","display":"\\mathbb Q^2","kind":"use","excerpt":"Initial dimension, dynamic memory, and hidden state. In Counterexample~[source reference], set L=span{(1,1)} and R(x)=x_1. The reference reaches (1,1) and (1,0), so W= Q^2. The rows Q=(1,1) and QA=(0,1) have trivial common kernel, giving N=0. A two-dimensional linear predictor is therefore necessary even for this one-dimensional initial family: it ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"use","excerpt":"For the update, first hold ℓ fixed. Given 0≤ f_i≤1, put a=∑_i B_ℓ(q)_if_i and Z_p=∑_ip_iℓ_i. Then ∑_i( B_ℓ(p)_i- B_ℓ(q)_i)f_i =frac∑_i(p_i-q_i)ℓ_i(f_i-a)Z_p. Every value ℓ_i(f_i-a) lies in [-a,1-a], an interval of length one. Since p-q has total mass zero, its pairing with a function of oscillation at most one is bounded in absolute value by d_TV(p","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-237925df6db86b64","tex":"\\mathcal Q_N","aliases":"Q_N mathcal","display":"\\mathcal Q_N","kind":"use","excerpt":"Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"For J⊆{1,…,m}, write R_J(ω)=(r(ω),(e_j(ω))_j∈ J). Let the unresolved target pairs be D={{ω,ω'}:r(ω)=r(ω'), t(ω)≠ t(ω')}. A test covers such a pair when it gives different reports in the two worlds.","count":16},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-855b51431c459bf3","tex":"\\mathbb R^d","aliases":"R^d mathbb","display":"\\mathbb R^d","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-5491081fce1c2c32","tex":"R_J","aliases":"R_J","display":"R_J","kind":"use","excerpt":"For J⊆{1,…,m}, write R_J(ω)=(r(ω),(e_j(ω))_j∈ J). Let the unresolved target pairs be D={{ω,ω'}:r(ω)=r(ω'), t(ω)≠ t(ω')}. A test covers such a pair when it gives different reports in the two worlds.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"Define Q_N(p) by first setting n_i=lfloor Np_irfloor. Distribute the remaining N-∑_i n_i units to the largest fractional remainders, with ties broken by a fixed cell order. This makes the normalization exact after every rounding. It is not independent coordinate rounding followed by an unspecified repair.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"A calibrated bit with an irrelevant hidden register. Consider eight possible worlds ω=(s,θ,k)∈{0,1}^3. The physical bit is s. The sensor convention is θ, and k is a nuisance bit that none of the admitted operations or questions uses. The display is y=s⊕θ. The ordinary action F flips s and leaves the other bits fixed. Its effect on the display is ex","count":29},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-68387971995ff129","tex":"S_b","aliases":"S_b","display":"S_b","kind":"use","excerpt":"ppose v∉ U_n-1. If v∉ U, the one-step proposition supplies two extensions that disagree within one action. Otherwise the recursive definition gives a word of length j≤ n-1 with a first exit from U. Its intermediate vectors before exit lie in U, so its exit vector z∉ U is the same under every compatible extension family. Choose any next action b. Tw","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4dade7617531bf72","tex":"s_{\\mathrm{old}}","aliases":"s_old mathrm","display":"s_{\\mathrm{old}}","kind":"use","excerpt":"A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span span operatorname","display":"{\\operatorname{span}}","kind":"use","excerpt":"Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}","count":9},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2dc892a16b97f181","tex":"\\sup_{\\omega\\in B(H)}","aliases":"sup_ω∈ B(H) sup omega in","display":"\\sup_{\\omega\\in B(H)}","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-58112450b9ad7b0a","tex":"\\sup_{\\omega\\in C_i}","aliases":"sup_ω∈ C_i sup omega in","display":"\\sup_{\\omega\\in C_i}","kind":"use","excerpt":"Fix an actual evidence history. Let μ be the exact posterior on the stipulated finite world set and let p_i=μ(C_i). Suppose the retained distribution is widehat p with d_TV(p,widehat p)≤ e. For a declared later action word or response horizon, let y(ω) be the actual model response and let bar y_i be the response used for cell i. Assume sup_ω∈ C_i y","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"one-dimensional reusable core [Calibration does not close under continuation] Let V have basis e_1,e_2,e_3, let U=span(e_1,e_2), and let the single calibrated operation satisfy Se_1=e_1, Se_2=e_3. The first continuation is known on all of U. The next calibrated domain is U_1=span(e_1), which is already invariant. One completion has T_0e_3=0 and ano","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"ore [Calibration does not close under continuation] Let V have basis e_1,e_2,e_3, let U=span(e_1,e_2), and let the single calibrated operation satisfy Se_1=e_1, Se_2=e_3. The first continuation is known on all of U. The next calibrated domain is U_1=span(e_1), which is already invariant. One completion has T_0e_3=0 and another has T_1e_3=e_1. Both ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"use","excerpt":"Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}","count":9},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a26584fc91eb3ec9","tex":"T_a'","aliases":"T_a'","display":"T_a'","kind":"use","excerpt":"For a fixed action a, the value T_av is the same for every linear extension of S_a if and only if v∈ U. The common value is then S_av. If v∈ U, all extensions agree by definition. If v∉ U, choose a linear functional ℓ that vanishes on U and satisfies ℓ(v)=1, and choose a nonzero vector b∈ V. Given any extension T_a, the map T_a'(x)=T_a(x)+ℓ(x)b is ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-635335c1377b1a05","tex":"T_a^0","aliases":"T_a^0","display":"T_a^0","kind":"use","excerpt":"A finite test for agreement under a fixed unknown model. Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-1a4a5aadced24804","tex":"T_a^\\theta","aliases":"T_a^θ theta","display":"T_a^\\theta","kind":"use","excerpt":"A finite test for agreement under a fixed unknown model. Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y.","count":8},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditiona","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-64afaf0ed6cbcadc","tex":"T_n","aliases":"T_n","display":"T_n","kind":"use","excerpt":"The later response is produced by three applications of one fixed candidate matrix, T_n=1/2 · (-1)^n/64 1/8 · 1/2, (v_0,h_0)=(s,1). tag17 The predictor for cell (s,θ) uses bar v_t=2^-ts and therefore reports bar y_i=s/8. The nuisance bit n affects the physical response but remains unidentified by the reference observations. With g=d=1/2, b=1/64, c=","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ed0619d26eb6427a","tex":"T_w^\\theta","aliases":"T_w^θ theta","display":"T_w^\\theta","kind":"use","excerpt":"ent under a fixed unknown model Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y. Define I_Q={v∈ V:QT_w^θ ","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV d_TV mathrm","display":"{d_{\\mathrm{TV}}}","kind":"use","excerpt":"[A finite-record discrimination bound] For two distinct target values and equal world probabilities, every transcript-based estimator has average success at most frac1+d_TV(P_0,P_1)2, d_TV(P_0,P_1)=frac12∑_h|P_0(h)-P_1(h)|.tag4 A deterministic function or a common stochastic processing channel applied to the transcript cannot increase this upper bo","count":9},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":10},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v_0-bar v_0 ≤ e_0. These initial bounds are supplied evidence or declared model assumptions. Put M=g · b c · d, u_0=V_0 H_0, u_","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"A two-dimensional calibration with a one-dimensional reusable core. [Calibration does not close under continuation] Let V have basis e_1,e_2,e_3, let U=span(e_1,e_2), and let the single calibrated operation satisfy Se_1=e_1, Se_2=e_3. The first continuation is known on all of U. The next calibrated domain is U_1=span(e_1), which is already invarian","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-7423c3fa878315fb","tex":"U_{h-1}","aliases":"U_h-1","display":"U_{h-1}","kind":"use","excerpt":"[No enlargement under unrestricted linear completion] If Q≠0, then for every h≥1, the vectors QT_wv are independent of the compatible extension family for every word of length at most h if and only if v∈ U_h-1. In particular, the all-word output-identifiable domain is exactly U_∞. If Q=0, every initial vector is trivially output-identifiable. On U_","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f221f487fed33512","tex":"U_{n-1}","aliases":"U_n-1","display":"U_{n-1}","kind":"use","excerpt":"[The maximal calibrated continuation domain] For every integer n≥1 and every v∈ V, the values T_wv for all operation words of length at most n are independent of the chosen compatible extension family if and only if v∈ U_n-1. Consequently, every finite full-state continuation is identifiable exactly on U_∞. This is the largest subspace L⊆ U such th","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"Association can be the missing state. [Equal marginals with different corrected answers] In the sensor model, let V_0={(s,θ,k):s⊕θ=0}, V_1={(s,θ,k):s⊕θ=1}. Take the target to be s and the evidence cells to be the two values of θ. Both candidate sets currently allow physical bits {0,1}, and both allow conventions {0,1}. Yet after the reference repor","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"ix, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not observe h_t. The blocks may come from a fixed unknown candidate or from an explicitly admitted varying family. Assume common operator-norm bounds G_t ≤ g, B_t ≤ b, C_t ≤ c, H_t ≤ d, g,b,c,d≥0. Let v_0 ≤ V_0, h_0 ≤ H_0, and v","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"Association can be the missing state. [Equal marginals with different corrected answers] In the sensor model, let V_0={(s,θ,k):s⊕θ=0}, V_1={(s,θ,k):s⊕θ=1}. Take the target to be s and the evidence cells to be the two values of θ. Both candidate sets currently allow physical bits {0,1}, and both allow conventions {0,1}. Yet after the reference repor","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-7be15d086b46f651","tex":"v_t","aliases":"v_t","display":"v_t","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1","count":8},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-225b3e9c5bddd3e5","tex":"v_{t+1}","aliases":"v_t+1","display":"v_{t+1}","kind":"use","excerpt":"Write the physical state as (v_t,h_t) in a direct sum of two finite-dimensional normed spaces. The required output is the visible component v_t. Along an admitted execution suppose v_t+1 h_t+1 =G_t · B_t{ C_t · H_t v_t h_t, bar v_t+1=G_tbar v_t. tag13 The reduced predictor knows the chosen G_t and receives the same action sequence. It does not obse","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Conversely, any retained representation with deterministic filter updates and a readout reproducing Q after every such sequence must distinguish every two candidate sets with different signatures. Thus the image of Σ_ E,t is the smallest exact state space, in cardinality, for this correction contract on all candidate subsets. If Σ(V)=Σ(W), then for","count":13},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"[A common quantitative grounding certificate] Under (16), put ξ=min{1,e+Kd}. Then |R(a)-widehat R(a)|≤ξ for every a∈ D. If widehat a minimizes widehat R, its true posterior regret is at most 2ξ. If it is the unique approximate minimizer and widehat R(a)-widehat R(widehat a)>2ξ for every a≠widehat a, then it is the unique true minimizer. A decision ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-77fd09daba007b85","tex":"\\widehat m","aliases":"widehat m widehat","display":"\\widehat m","kind":"use","excerpt":"model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the r","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-b7a405639bc72c2a","tex":"\\widehat p_0","aliases":"widehat p_0 widehat","display":"\\widehat p_0","kind":"use","excerpt":"After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment sel","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6c7df3584eb03432","tex":"\\widehat p_i","aliases":"widehat p_i widehat","display":"\\widehat p_i","kind":"use","excerpt":"Let the permitted decision set D be finite and nonempty. Its loss L(y,a) takes values in [0,1] and is K-Lipschitz in y, uniformly over a. Define R(a)=∑_ωμ(ω)L(y(ω),a), widehat R(a)=∑_iwidehat p_iL(bar y_i,a). These are risks under the stated model posterior. They are not frequentist coverage probabilities or guarantees on the loss of a single reali","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-88e7974099a060be","tex":"\\widehat p_t","aliases":"widehat p_t widehat","display":"\\widehat p_t","kind":"use","excerpt":"After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment sel","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-40cf13a68cde7db3","tex":"\\widehat p_{t+1}","aliases":"widehat p_t+1 widehat","display":"\\widehat p_{t+1}","kind":"use","excerpt":"After a history, let p_t be the cell masses of the exact world posterior. Initialize the retained record by widehat p_0= Q_N(p_0). For the next selected experiment and observed report use widehat p_t+1= Q_N ( B_barℓ_e,o(widehat p_t)). The reference posterior and rounded record are compared along the same actual action/report history. Experiment sel","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-967c63ebf4cd78be","tex":"\\widehat R","aliases":"widehat R widehat","display":"\\widehat R","kind":"use","excerpt":"Let the permitted decision set D be finite and nonempty. Its loss L(y,a) takes values in [0,1] and is K-Lipschitz in y, uniformly over a. Define R(a)=∑_ωμ(ω)L(y(ω),a), widehat R(a)=∑_iwidehat p_iL(bar y_i,a). These are risks under the stated model posterior. They are not frequentist coverage probabilities or guarantees on the loss of a single reali","count":8},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f207fa5d22c484f6","tex":"\\widehat v","aliases":"widehat v widehat","display":"\\widehat v","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4ea47c9fd4372cb9","tex":"\\widehat v_H","aliases":"widehat v_H widehat","display":"\\widehat v_H","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-80b3be1647ac2fcb","tex":"\\widehat\\rho","aliases":"widehatρ widehat rho","display":"\\widehat\\rho","kind":"use","excerpt":"se a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and radius widehatρ from widehat v_H. If the returned numerical ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-af3afdd82abfc218","tex":"\\widetilde\\ell","aliases":"widetildeℓ widetilde ell","display":"\\widetilde\\ell","kind":"use","excerpt":"[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storag","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-cb52b469f62b481c","tex":"\\widetilde\\ell_i","aliases":"widetildeℓ_i widetilde ell","display":"\\widetilde\\ell_i","kind":"use","excerpt":"[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storag","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"For a word w=a_1⋯ a_m, F_w applies a_1 first and a_m last. The empty word is the identity. Define x∼_ C x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite w. This is a comparison in a common observation language. It does not require that different models share arbitrary latent coordinates. The behavioral class is a canonical response profile e","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}","count":9},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"2 · -1 · 3, Q=1 · 0 · 0. The hidden third-coordinate dynamics are different, and T_1 can make that coordinate grow. Nevertheless the discrepancy range is span{e_3}, which the reference leaves inside itself and Q annihilates. The common quotient retains (x_1,x_2), with update G(x_1,x_2)=(x_1+x_2,x_2). The output at step k is x_1+kx_2 under both mode","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"ancy range is span{e_3}, which the reference leaves inside itself and Q annihilates. The common quotient retains (x_1,x_2), with update G(x_1,x_2)=(x_1+x_2,x_2). The output at step k is x_1+kx_2 under both models, and even under switching between them. The output alone is not an initialized reusable state: equal x_1 with different x_2 gives differe","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that condition it defines maps S_a:","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","excerpt":"Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-be34691f19273780","tex":"y_{a,1}","aliases":"y_a,1","display":"y_{a,1}","kind":"use","excerpt":"Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-bd65ce295255a59f","tex":"y_{a,i}","aliases":"y_a,i","display":"y_{a,i}","kind":"use","excerpt":"supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}. The calibration is consistent precisely when every linear relation among the inputs is respected by the outputs, for each action. Under that condition it defines maps S_a:","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-afeb4c3001e9873f","tex":"y_{a,m}","aliases":"y_a,m","display":"y_{a,m}","kind":"use","excerpt":"Let V be a finite-dimensional vector space over a field, and let A be a finite nonempty set of operation names. The unknown operations are linear maps T_a:V→ V. Calibration supplies input vectors x_1,…,x_m and exact outputs y_a,1,…,y_a,m for every a. The operator family is fixed, rather than being reselected after each action. Let U=span{x_1,…,x_m}","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-eef2897f96ab6dc2","tex":"y_{\\mathrm{after}\\ Z}","aliases":"y_after Z mathrm","display":"y_{\\mathrm{after}\\ Z}","kind":"use","excerpt":"A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8bcd84de243237e9","tex":"y_{\\mathrm{old}}","aliases":"y_old mathrm","display":"y_{\\mathrm{old}}","kind":"use","excerpt":"A state-changing measurement. A different operation Z sets s to zero and then reports the new display. The report is θ. If the old display y_old was retained, then s_old=y_old⊕ y_after Z. The current physical state is zero, not s_old. A correct account distinguishes the recovered historical value from the new state it will act on.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-48f6b2f918098577","tex":"z_{e,o}","aliases":"z_e,o","display":"z_{e,o}","kind":"use","excerpt":"[Exact target correction from likelihood-profile memory] Under the fixed-world likelihood model, the record p=(p_i,y) supports every finite sequence of report predictions and target-posterior corrections. Its next-report probability and update are z_e,o=∑_i,yλ_e,o,ip_i,y, p'_i,y=fracλ_e,o,ip_i,yz_e,o (z_e,o>0), and its target readout is Pr(t=y)=∑_i","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-baa7278dbe97db40","tex":"Z_p","aliases":"Z_p","display":"Z_p","kind":"use","excerpt":"For the update, first hold ℓ fixed. Given 0≤ f_i≤1, put a=∑_i B_ℓ(q)_if_i and Z_p=∑_ip_iℓ_i. Then ∑_i( B_ℓ(p)_i- B_ℓ(q)_i)f_i =frac∑_i(p_i-q_i)ℓ_i(f_i-a)Z_p. Every value ℓ_i(f_i-a) lies in [-a,1-a], an interval of length one. Since p-q has total mass zero, its pairing with a function of oscillation at most one is bounded in absolute value by d_TV(p","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"able. On U_h-1 the full vectors are determined by Theorem~[source reference], so their Q outputs are determined. Outside that subspace, follow the same first-exit construction as in that proof. At an uncalibrated input z, choose a linear functional ℓ vanishing on U with ℓ(z)=1. Because Q≠0, choose b with Qb≠0. Replacing the next action by T_a+ℓ(·)b","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"[Rounding and positive-likelihood update bounds] For every p∈Δ_k, d_TV(p, Q_N(p))≤frack2N. Let p,q∈Δ_k, let 0<β≤1, and let every entry of ℓ and widetildeℓ lie in [β,1]. If max_i|ℓ_i-widetildeℓ_i|≤ε, then d_TV( B_ℓ(p), B_widetildeℓ(q)) ≤ fracd_TV(p,q)+εβ. Each right-hand side may also be replaced by its minimum with one. The probability-field storag","count":10},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8735109003c9a8e3","tex":"\\beta^{-1}","aliases":"β^-1 beta","display":"\\beta^{-1}","kind":"use","excerpt":"[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the g","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the g","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-093e921f6604d9d8","tex":"\\gamma^j","aliases":"γ^j gamma","display":"\\gamma^j","kind":"use","excerpt":"[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the g","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-a401f83cf88764ac","tex":"\\gamma^t","aliases":"γ^t gamma","display":"\\gamma^t","kind":"use","excerpt":"[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the g","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":9},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-210fc6f61ef84d60","tex":"\\Delta^{-1}","aliases":"Δ^-1 Delta","display":"\\Delta^{-1}","kind":"use","excerpt":"[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is exclud","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-986788bd9ce6ff3c","tex":"\\delta_{\\mathrm{id}}","aliases":"δ_id delta mathrm","display":"\\delta_{\\mathrm{id}}","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-fff47860c2d333d8","tex":"\\delta_{\\mathrm{model}}","aliases":"δ_model delta mathrm","display":"\\delta_{\\mathrm{model}}","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":18},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-efb4acc8c0653938","tex":"\\eta_N","aliases":"η_N eta","display":"\\eta_N","kind":"use","excerpt":"[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the g","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"A finite test for agreement under a fixed unknown model. Let V be n-dimensional over a field, let Θ be a finite nonempty model set, and let T_a^θ:V→ V be specified linear maps for each model θ and action a in a finite nonempty alphabet. All words are allowed. Choose a reference model 0∈Θ and write A_a=T_a^0. The shared linear observation is Q:V→ Y.","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"A calibrated bit with an irrelevant hidden register. Consider eight possible worlds ω=(s,θ,k)∈{0,1}^3. The physical bit is s. The sensor convention is θ, and k is a nuisance bit that none of the admitted operations or questions uses. The display is y=s⊕θ. The ordinary action F flips s and leaves the other bits fixed. Its effect on the display is ex","count":49},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-d967d409e25df438","tex":"\\lambda_{b(i,j),i}","aliases":"λ_b(i,j),i lambda","display":"\\lambda_{b(i,j),i}","kind":"use","excerpt":"For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ce0f5ef15f9fa00b","tex":"\\lambda_{b(i,j),j}","aliases":"λ_b(i,j),j lambda","display":"\\lambda_{b(i,j),j}","kind":"use","excerpt":"For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9ff06f4c37cc3e34","tex":"\\mu_+","aliases":"μ_+ mu","display":"\\mu_+","kind":"use","excerpt":"Consider a prior μ_+ assigning mass 1/2 to (s,θ)=(0,0),(1,1) and a prior μ_- assigning mass 1/2 to (1,0),(0,1), with any fixed nuisance convention. Both have uniform s and uniform θ. Both predict the next report 1 with probability 1/2. After that report, however, Pr_μ_+(s=1mid o=1)=tfrac34, Pr_μ_-(s=1mid o=1)=tfrac14. After k fresh reports all equa","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8db12104c2080045","tex":"\\mu_-","aliases":"μ_- mu","display":"\\mu_-","kind":"use","excerpt":"Consider a prior μ_+ assigning mass 1/2 to (s,θ)=(0,0),(1,1) and a prior μ_- assigning mass 1/2 to (1,0),(0,1), with any fixed nuisance convention. Both have uniform s and uniform θ. Both predict the next report 1 with probability 1/2. After that report, however, Pr_μ_+(s=1mid o=1)=tfrac34, Pr_μ_-(s=1mid o=1)=tfrac14. After k fresh reports all equa","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-3b453de64aef4dc5","tex":"\\mu_0","aliases":"μ_0 mu","display":"\\mu_0","kind":"use","excerpt":"Approximate likelihood profiles. Let Ω be a finite static world set with prior μ_0. A fixed partition C_1,…,C_k refines the target: t(ω)=t_i throughout C_i. Each admitted experiment e is nondestructive and has a finite report alphabet. Its actual conditional probabilities are ℓ_e,o(ω). Reports are freshly generated from those likelihoods conditiona","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"[A common quantitative grounding certificate] Under (16), put ξ=min{1,e+Kd}. Then |R(a)-widehat R(a)|≤ξ for every a∈ D. If widehat a minimizes widehat R, its true posterior regret is at most 2ξ. If it is the unique approximate minimizer and widehat R(a)-widehat R(widehat a)>2ξ for every a≠widehat a, then it is the unique true minimizer. A decision ","count":10},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"[ d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 and tolerance τ>ε B_H, every integer N≥ max {1,lceil frackA_H2(τ-ε B_H)rceil} tag12 is sufficient for tar","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-efdf8990a98fa39d","tex":"\\omega'","aliases":"ω' omega","display":"\\omega'","kind":"use","excerpt":"Define ω∼_Πω' when their record laws agree for every π∈Π. This relation concerns the complete records, including action choices when those are observable. Equality of individual observation marginals is insufficient.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-514e7ff9048f931c","tex":"\\omega_*","aliases":"ω_* omega","display":"\\omega_*","kind":"use","excerpt":"Fix nonnegative tolerances η,ε and δ_id,δ_model∈[0,1]. Suppose a data record H produces a candidate set B(H) with Pr{ω_*∈ B(H)}≥1-δ_id. Suppose a second event, of probability at least 1-δ_model, guarantees sup_ω∈ B(H) v(ω)-widehat v_H(ω) _∞≤η. Read the last supremum as zero when B(H) is empty. When B(H) is nonempty, form the midpoint widehat m and ","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"[An uncalibrated direction amplifies a small residual] Let L=span(e_1), let Se_1=e_1, and in the ℓ^1 norm define T=1 · 0","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-7df0df67e8e3da0c","tex":"\\ell_{b(i,j)}","aliases":"ℓ_b(i,j) ell","display":"\\ell_{b(i,j)}","kind":"use","excerpt":"For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2eb7ce8fda0a91d1","tex":"\\ell_{e_j,o_j}","aliases":"ℓ_e_j,o_j ell","display":"\\ell_{e_j,o_j}","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"For a likelihood vector ℓ=(ℓ_i) define the Bayes map B_ℓ(p)_i=fracp_iℓ_i∑_jp_jℓ_j whenever its denominator is positive.","count":7},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8943b22d90e5e02b","tex":"\\ell_j","aliases":"ℓ_j ell","display":"\\ell_j","kind":"use","excerpt":"Let v:Ω→ R^d record a fixed finite collection of target quantities, where d≥1. These can be future predictions on specified operation words. For a nonempty B⊆Ω, define ℓ_j(B)=min_ω∈ Bv_j(ω), u_j(B)=max_ω∈ Bv_j(ω), m_j(B)=fracℓ_j(B)+u_j(B)2, ρ(B)=frac12max_j(u_j(B)-ℓ_j(B)).tag5 The radius concerns coordinatewise prediction in a common frame. It is n","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"rrelevant hidden register Consider eight possible worlds ω=(s,θ,k)∈{0,1}^3. The physical bit is s. The sensor convention is θ, and k is a nuisance bit that none of the admitted operations or questions uses. The display is y=s⊕θ. The ordinary action F flips s and leaves the other bits fixed. Its effect on the display is exactly y↦1-y, whatever the u","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"immutable worlds, one with a permanently zero output and one with a permanently one output. The available initial record is the same symbol in both. Within each world separately, a one-state predictor exists: it simply prints that world's output. There is nevertheless no predictor that can be initialized from the common record and be correct in bot","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"immutable worlds, one with a permanently zero output and one with a permanently one output. The available initial record is the same symbol in both. Within each world separately, a one-state predictor exists: it simply prints that world's output. There is nevertheless no predictor that can be initialized from the common record and be correct in bot","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"[One-step identification from exact calibration] For a fixed action a, the value T_av is the same for every linear extension of S_a if and only if v∈ U. The common value is then S_av. If v∈ U, all extensions agree by definition. If v∉ U, choose a linear functional ℓ that vanishes on U and satisfies ℓ(v)=1, and choose a nonzero vector b∈ V. Given an","count":4},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-74de77a8aadc310f","tex":"\\prod_{j=1}^k","aliases":"∏_j=1^k prod","display":"\\prod_{j=1}^k","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-5e4c84160d4d61eb","tex":"\\prod_{j\\ne i}","aliases":"∏_j≠ i prod ne","display":"\\prod_{j\\ne i}","kind":"use","excerpt":"For each cell i and each j≠ i, choose one likelihood coordinate b(i,j)=(e,o) for which λ_b(i,j),i≠λ_b(i,j),j. Such a coordinate exists because profile vectors are distinct. On Ω the polynomial P_i(ω)=∏_j≠ i fracℓ_b(i,j)(ω)-λ_b(i,j),j λ_b(i,j),i-λ_b(i,j),j is exactly 1_A_i(ω). It is a linear combination of likelihood monomials of degree at most m-1.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f7dd81f92a8f6222","tex":"\\sum_{a,\\theta}","aliases":"∑_a,θ sum theta","display":"\\sum_{a,\\theta}","kind":"use","excerpt":"ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-20794c41f8dd0413","tex":"\\sum_h","aliases":"∑_h sum","display":"\\sum_h","kind":"use","excerpt":"[A finite-record discrimination bound] For two distinct target values and equal world probabilities, every transcript-based estimator has average success at most frac1+d_TV(P_0,P_1)2, d_TV(P_0,P_1)=frac12∑_h|P_0(h)-P_1(h)|.tag4 A deterministic function or a common stochastic processing channel applied to the transcript cannot increase this upper bo","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-30322d67114459c9","tex":"\\sum_{i,y}","aliases":"∑_i,y sum","display":"\\sum_{i,y}","kind":"use","excerpt":"[Exact target correction from likelihood-profile memory] Under the fixed-world likelihood model, the record p=(p_i,y) supports every finite sequence of report predictions and target-posterior corrections. Its next-report probability and update are z_e,o=∑_i,yλ_e,o,ip_i,y, p'_i,y=fracλ_e,o,ip_i,yz_e,o (z_e,o>0), and its target readout is Pr(t=y)=∑_i","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"For a likelihood vector ℓ=(ℓ_i) define the Bayes map B_ℓ(p)_i=fracp_iℓ_i∑_jp_jℓ_j whenever its denominator is positive.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-6989a4bbfc918b48","tex":"\\sum_{j=0}^{(n-1)/2}","aliases":"∑_j=0^(n-1)/2 sum","display":"\\sum_{j=0}^{(n-1)/2}","kind":"use","excerpt":"For odd n, majority voting is the likelihood-ratio rule. Its error in either world is e_n=∑_j=0^(n-1)/2binomnj(3/4)^j(1/4)^n-j. The first values are e_1=1/4, e_3=5/32, and e_5=53/512. These are exact analytical probabilities, not observed error rates from an agent experiment.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-457f9d31db6c5fcc","tex":"\\sum_{j=0}^{H}","aliases":"∑_j=0^H sum","display":"\\sum_{j=0}^{H}","kind":"use","excerpt":"let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-8f7791db6e6f82eb","tex":"\\sum_{j=0}^{H-1}","aliases":"∑_j=0^H-1 sum","display":"\\sum_{j=0}^{H-1}","kind":"use","excerpt":"let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the generic initialization bound e_0≤η_N, define A_H=∑_j=0^Hγ^j, B_H=γ∑_j=0^H-1γ^j. For any horizon H≥0 ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-df66290cc89f818f","tex":"\\sum_{j=0}^{t-1}","aliases":"∑_j=0^t-1 sum","display":"\\sum_{j=0}^{t-1}","kind":"use","excerpt":"[Finite retained memory for a declared correction horizon] Under (10), set η_N=k/(2N), γ=β^-1, and let e_0 be any valid bound on d_TV(p_0,widehat p_0). For every possible action/report history of length t, d_TV(p_t,widehat p_t)≤ e_t :=min {1, γ^t e_0+(γε+η_N) ∑_j=0^t-1γ^j}. tag11 The same bound holds after mapping cells to target values. With the g","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-b4619f532d9c032b","tex":"\\sum_{j\\in J}","aliases":"∑_j∈ J sum in","display":"\\sum_{j\\in J}","kind":"use","excerpt":"nonadaptive set J suffices exactly when it covers D. item Some pair in D has identical reports under every admitted test. No finite adaptive protocol using this unchanged test interface and r guarantees t in every world. If test j has nonnegative fixed cost c_j, the minimum cost of one fixed nonadaptive test set, used for all initial codes in case ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-49afb54019149b8f","tex":"\\sum_{j\\ge0}","aliases":"∑_j≥0 sum ge","display":"\\sum_{j\\ge0}","kind":"use","excerpt":"c · 1-g has nonnegative entries. Let a=(I-M)^-1(1,1)^T. Both coordinates are positive and Ma=a-(1,1)^T<a. Hence M is a strict contraction in the weighted maximum norm with weights a, and its geometric series converges to (I-M)^-1. Summing the nonnegative right-hand side of (14) gives frace_01-g+fracb1-g∑_j≥0(M^ju_0)_2. The second coordinate of (I-M","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ccbd66bdaed18a2b","tex":"\\sum_o","aliases":"∑_o sum","display":"\\sum_o","kind":"use","excerpt":"Let Ω be finite and nonempty, let t:Ω→ Y have finite image, and let B be a finite collection of experiments. Experiment e has a finite outcome set and specified likelihoods ℓ_e,o(ω)≥0, ∑_oℓ_e,o(ω)=1. The world is fixed. Every experiment remains available, and each new report has this conditional law given the world, the chosen experiment, and all e","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-525699e51075699d","tex":"\\sum_{t(\\omega)=y}","aliases":"∑_t(ω)=y sum omega","display":"\\sum_{t(\\omega)=y}","kind":"use","excerpt":"For necessity, a valid predictor and updater determine, for every word w=((e_1,o_1),…,(e_k,o_k)), the masses M_μ,y(w)=∑_t(ω)=yμ(ω) ∏_j=1^kℓ_e_j,o_j(ω). Indeed, multiply the successive predicted report probabilities by the final target posterior. If an earlier factor is zero, the joint mass is zero and no impossible conditional posterior is needed. ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-5e51231cedd78224","tex":"\\sum_{t=0}^{\\infty}","aliases":"∑_t=0^∞ sum infty","display":"\\sum_{t=0}^{\\infty}","kind":"use","excerpt":"If g<1, d<1, and D=(1-g)(1-d)-bc>0, then the total visible discrepancy is bounded by ∑_t=0^∞ v_t-bar v_t ≤ frace_01-g +fracb(cV_0+(1-g)H_0)(1-g)D. tag15 These are sufficient common bounds. Failure of the inequality for D is not a proof that a particular signed system is unstable. Conversely, the two diagonal conditions alone are insufficient for a ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-5966b9097878350e","tex":"\\sum_z","aliases":"∑_z sum","display":"\\sum_z","kind":"use","excerpt":"herwise. A correct target output always gives a correct binary decision. If f(h)∈[0,1] is the test's chance of declaring world zero, its success is frac12(1+∑_h f(h)(P_0(h)-P_1(h))). The sum is at most the sum of positive differences, which equals total variation. This proves (4). If B(h,z) is a common processing channel and g(z)∈[0,1] a processed-","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-2e993161c26d59c7","tex":"\\sum_{\\omega}","aliases":"∑_ω sum omega","display":"\\sum_{\\omega}","kind":"use","excerpt":"Let the permitted decision set D be finite and nonempty. Its loss L(y,a) takes values in [0,1] and is K-Lipschitz in y, uniformly over a. Define R(a)=∑_ωμ(ω)L(y(ω),a), widehat R(a)=∑_iwidehat p_iL(bar y_i,a). These are risks under the stated model posterior. They are not frequentist coverage probabilities or guarantees on the loss of a single reali","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-12d14377bffa1656","tex":"\\sim_\\C","aliases":"∼_C ∼_ C sim C sim mathcal","display":"\\sim_{\\mathcal C}","kind":"use","excerpt":"For a word w=a_1⋯ a_m, F_w applies a_1 first and a_m last. The empty word is the identity. Define x∼_ C x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite w. This is a comparison in a common observation language. It does not require that different models share arbitrary latent coordinates. The behavioral class is a canonical response profile e","count":6},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-11495beb63cf39b5","tex":"\\sim_\\Pi","aliases":"∼_Π sim Pi","display":"\\sim_\\Pi","kind":"use","excerpt":"Define ω∼_Πω' when their record laws agree for every π∈Π. This relation concerns the complete records, including action choices when those are observable. Equality of individual observation marginals is insufficient.","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"midrule a=0 · 23991/262144 ≈0.09152 · 173737/1835008 ≈0.09468","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"A calibrated bit with an irrelevant hidden register. Consider eight possible worlds ω=(s,θ,k)∈{0,1}^3. The physical bit is s. The sensor convention is θ, and k is a nuisance bit that none of the admitted operations or questions uses. The display is y=s⊕θ. The ordinary action F flips s and leaves the other bits fixed. Its effect on the display is ex","count":5},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.","count":3},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-bca212e1e10d6008","tex":"\\bigcap_a","aliases":"⋂_a bigcap","display":"\\bigcap_a","kind":"use","excerpt":"0 · A_a, C=Q · -Q, Δ v=(v,v). The pair is evolved by the same action word in both components. Define K_0^θ=ker C, K_j+1^θ=K_j^θ∩⋂_a(B_a^θ)^-1K_j^θ.","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-27b1d2fc9bc15338","tex":"\\bigcap_w","aliases":"⋂_w bigcap","display":"\\bigcap_w","kind":"use","excerpt":"[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is exclud","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-c37fb3ac563ee65c","tex":"\\bigcap_{\\theta\\in\\Theta}","aliases":"⋂_θ∈Θ bigcap theta in Theta","display":"\\bigcap_{\\theta\\in\\Theta}","kind":"use","excerpt":"[Finite output-agreement test with persistent model tags] The descending paired-kernel sequence stabilizes in at most 2n strict dimension drops. Its terminal subspace is K_∞^θ=⋂_wker(CB_w^θ), I_Q=⋂_θ∈ΘΔ^-1K_∞^θ. Thus exact all-word output agreement for the finite candidate family is decidable by finite linear algebra. If an initial vector is exclud","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"What compression does to the candidate set. For a fixed raw-record channel, let B(h) be its exact support set. If only c(h) is retained, the correct support is B_c(z)=⋃_h:c(h)=zB(h). Indeed, a world can produce the coarse record exactly when it can produce at least one raw record in that fibre. Thus B(h)⊆ B_c(c(h)), and the ambiguity radius cannot ","count":2},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-418c6deb9cfe8728","tex":"\\bigcup_{h:c(h)=z}","aliases":"⋃_h:c(h)=z bigcup","display":"\\bigcup_{h:c(h)=z}","kind":"use","excerpt":"What compression does to the candidate set. For a fixed raw-record channel, let B(h) be its exact support set. If only c(h) is retained, the correct support is B_c(z)=⋃_h:c(h)=zB(h). Indeed, a world can produce the coarse record exactly when it can produce at least one raw record in that fibre. Thus B(h)⊆ B_c(c(h)), and the ambiguity radius cannot ","count":1},{"work":"monographs-grounded-abstraction","url":"notation/monographs-grounded-abstraction.html#symbol-ab76f757cb3a28be","tex":"\\bigcup_j","aliases":"⋃_j bigcup","display":"\\bigcup_j","kind":"use","excerpt":"ne common observable quotient Agreement on selected initial conditions is weaker than agreement on the whole space. The stronger problem has a useful exact solution. Retain the finite family and reference matrices of Section~[source reference], and define the difference maps D_a^θ=T_a^θ-A_a. Their ranges contain all possible one-step discrepancies.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-52555ee0fe55f126","tex":"\\bar F","aliases":"bar F bar","display":"\\bar F","kind":"notation-table","excerpt":"bar F_u, bar q · An update and a query decoder operating on the retained state.","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-678701062f958896","tex":"\\bar F_u","aliases":"bar F_u bar","display":"\\bar F_u","kind":"notation-table","excerpt":"bar F_u, bar q · An update and a query decoder operating on the retained state.","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"notation-table","excerpt":"bar F_u, bar q · An update and a query decoder operating on the retained state.","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"notation-table","excerpt":"C_e,o(V) · Candidates retained after experiment e returns observation o.","count":20},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathcal","display":"{\\mathcal C}","kind":"notation-table","excerpt":"midrule C · Declared family of states, operations, and future questions.","count":13},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-0943ac249c1305a8","tex":"C_{e,o}","aliases":"C_e,o","display":"C_{e,o}","kind":"notation-table","excerpt":"C_e,o(V) · Candidates retained after experiment e returns observation o.","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"notation-table","excerpt":"C_e,o(V) · Candidates retained after experiment e returns observation o.","count":48},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"notation-table","excerpt":"F_u, F_w · One implemented operation, or operations in the order specified by a word w.","count":13},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4a14d2ab0a464e3b","tex":"F_u","aliases":"F_u","display":"F_u","kind":"notation-table","excerpt":"F_u, F_w · One implemented operation, or operations in the order specified by a word w.","count":8},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"notation-table","excerpt":"F_u, F_w · One implemented operation, or operations in the order specified by a word w.","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-ffe66c0383172bf0","tex":"\\M","aliases":"M mathcal","display":"{\\mathcal M}","kind":"notation-table","excerpt":"V⊆ M · Current set of candidate models, not a probability distribution.","count":18},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"notation-table","excerpt":"C_e,o(V) · Candidates retained after experiment e returns observation o.","count":54},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite outcome space.","count":19},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite outcome space.","count":16},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"notation-table","excerpt":"r: X→ Z · Retained representation. Its acquisition procedure is a separate obligation.","count":47},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"notation-table","excerpt":"T, τ · Translation of representations and the corresponding operation names.","count":37},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV d_TV mathrm","display":"{d_{\\mathrm{TV}}}","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite outcome space.","count":7},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"notation-table","excerpt":"F_u, F_w · One implemented operation, or operations in the order specified by a word w.","count":30},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"notation-table","excerpt":"V⊆ M · Current set of candidate models, not a probability distribution.","count":35},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"notation-table","excerpt":"F_u, F_w · One implemented operation, or operations in the order specified by a word w.","count":15},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e53fe76bd3b89fa2","tex":"\\X","aliases":"X mathcal","display":"{\\mathcal X}","kind":"notation-table","excerpt":"r: X→ Z · Retained representation. Its acquisition procedure is a separate obligation.","count":15},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite outcome space.","count":52},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z mathcal","display":"{\\mathcal Z}","kind":"notation-table","excerpt":"r: X→ Z · Retained representation. Its acquisition procedure is a separate obligation.","count":10},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite outcome space.","count":15},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"notation-table","excerpt":"ϕ, η · Translation of candidate models and translation of observations.","count":14},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"notation-table","excerpt":"T, τ · Translation of representations and the corresponding operation names.","count":8},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"notation-table","excerpt":"ϕ, η · Translation of candidate models and translation of observations.","count":11},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"notation-table","excerpt":"r: X→ Z · Retained representation. Its acquisition procedure is a separate obligation.","count":27},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite outcome space.","count":9},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-0e44b9be25437975","tex":"\\sum_x","aliases":"∑_x sum","display":"\\sum_x","kind":"notation-table","excerpt":"d_TV(P,Q) · tfrac12∑_x|P(x)-Q(x)| on a finite outcome space.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"notation-table","excerpt":"V⊆ M · Current set of candidate models, not a probability distribution.","count":9},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Exact reusable representation] An exact reusable representation for C is a surjection r: X→ Z, updates bar F_u: Z→ Z, and decoders bar q: Z→ Y such that, for all x,u,q, r(F_u(x))=bar F_u(r(x)), q(x)=bar q(r(x)). Induction immediately gives q(F_w(x))=bar q(bar F_w(r(x))) for every finite word. The same initial record supports many future questions.","count":133},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Recipient-relative explanatory adequacy] Fix a recipient procedure L, background information B, resource allowance, and continuation family. An artifact E is adequate for a target model m when L(B,E) yields a representation and operations that answer the required continuations for m within the declared error and resource limits. The artifact's inp","count":27},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Recipient-relative explanatory adequacy] Fix a recipient procedure L, background information B, resource allowance, and continuation family. An artifact E is adequate for a target model m when L(B,E) yields a representation and operations that answer the required continuations for m within the declared error and resource limits. The artifact's inp","count":23},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"[Recipient-relative explanatory adequacy] Fix a recipient procedure L, background information B, resource allowance, and continuation family. An artifact E is adequate for a target model m when L(B,E) yields a representation and operations that answer the required continuations for m within the declared error and resource limits. The artifact's inp","count":33},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Recipient-relative explanatory adequacy] Fix a recipient procedure L, background information B, resource allowance, and continuation family. An artifact E is adequate for a target model m when L(B,E) yields a representation and operations that answer the required continuations for m within the declared error and resource limits. The artifact's inp","count":60},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"[Exact reusable representation] An exact reusable representation for C is a surjection r: X→ Z, updates bar F_u: Z→ Z, and decoders bar q: Z→ Y such that, for all x,u,q, r(F_u(x))=bar F_u(r(x)), q(x)=bar q(r(x)). Induction immediately gives q(F_w(x))=bar q(bar F_w(r(x))) for every finite word. The same initial record supports many future questions.","count":22},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-29ee7dd9b2baba56","tex":"\\Y","aliases":"Y mathcal","display":"{\\mathcal Y}","kind":"definition-passage","excerpt":"[Exact reusable representation] An exact reusable representation for C is a surjection r: X→ Z, updates bar F_u: Z→ Z, and decoders bar q: Z→ Y such that, for all x,u,q, r(F_u(x))=bar F_u(r(x)), q(x)=bar q(r(x)). Induction immediately gives q(F_w(x))=bar q(bar F_w(r(x))) for every finite word. The same initial record supports many future questions.","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1,","count":41},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":10},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-7764e407096f3048","tex":"A_e","aliases":"A_e","display":"A_e","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":66},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-0f602c3c4f71fbbb","tex":"\\C'","aliases":"C' C mathcal","display":"{\\mathcal C}'","kind":"introduction-passage","excerpt":"The cost of expanding the contract. Let C' have the same complete state space and contain every operation and query of C, while possibly adding more. Write ≡_ C' for its finer behavioral equivalence. Take any representation r adequate for C. For each z, let k_z be the number of ≡_ C' classes that meet the fibre r^-1(z).","count":6},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":6},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"For a policy tree T and initial state x, write s_T(x) for its terminal sender transcript and x_T(x) for its final physical state. The tree has finitely many reachable leaves. Specify recipient backgrounds β_i^T(x) and demands d_i^T(x) by a declared rule. A demand may concern x_T(x), the original x, or a stated pair of past and present values. These","count":18},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a2d4e1d011de2337","tex":"d_i^T","aliases":"d_i^T","display":"d_i^T","kind":"introduction-passage","excerpt":"For a policy tree T and initial state x, write s_T(x) for its terminal sender transcript and x_T(x) for its final physical state. The tree has finitely many reachable leaves. Specify recipient backgrounds β_i^T(x) and demands d_i^T(x) by a declared rule. A demand may concern x_T(x), the original x, or a stated pair of past and present values. These","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1,","count":29},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"For a policy tree T and initial state x, write s_T(x) for its terminal sender transcript and x_T(x) for its final physical state. The tree has finitely many reachable leaves. Specify recipient backgrounds β_i^T(x) and demands d_i^T(x) by a declared rule. A demand may concern x_T(x), the original x, or a stated pair of past and present values. These","count":43},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"The cost of expanding the contract. Let C' have the same complete state space and contain every operation and query of C, while possibly adding more. Write ≡_ C' for its finer behavioral equivalence. Take any representation r adequate for C. For each z, let k_z be the number of ≡_ C' classes that meet the fibre r^-1(z).","count":9},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-7053ab49a6a3ff72","tex":"k_z","aliases":"k_z","display":"k_z","kind":"introduction-passage","excerpt":"The cost of expanding the contract. Let C' have the same complete state space and contain every operation and query of C, while possibly adding more. Write ≡_ C' for its finer behavioral equivalence. Take any representation r adequate for C. For each z, let k_z be the number of ≡_ C' classes that meet the fibre r^-1(z).","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"introduction-passage","excerpt":"Let M∈ M be a fixed unknown model in a finite family, with prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly","count":7},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-ed76d6b3d182baf7","tex":"L_n","aliases":"L_n","display":"L_n","kind":"introduction-passage","excerpt":"prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly inspect M. Define Q_m(z)=∑_o L_m(o)B(o,z), p^o(m)=fracp(m)","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"Let M∈ M be a fixed unknown model in a finite family, with prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly","count":20},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":15},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e13032e4b8921379","tex":"\\max_{o:V_{e,o}\\ne\\varnothing}","aliases":"max_o:V_e,o≠∅ max ne varnothing","display":"\\max_{o:V_{e,o}\\ne\\varnothing}","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":9},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-f637f3e80bda9c4f","tex":"\\min_{e:V\\subseteq A_e}","aliases":"min_e:V⊆ A_e min subseteq","display":"\\min_{e:V\\subseteq A_e}","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly inspect M. Define Q_m(z)=∑_o L_m(o)B(o,z), p^o(m)=fracp(m)","count":43},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"introduction-passage","excerpt":"Let M∈ M be a fixed unknown model in a finite family, with prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly","count":9},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"introduction-passage","excerpt":"Let M∈ M be a fixed unknown model in a finite family, with prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Let M∈ M be a fixed unknown model in a finite family, with prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly","count":29},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-f27944b8dec73076","tex":"p^o","aliases":"p^o","display":"p^o","kind":"introduction-passage","excerpt":"prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly inspect M. Define Q_m(z)=∑_o L_m(o)B(o,z), p^o(m)=fracp(m)","count":8},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-9ead24cf2309f605","tex":"p^z","aliases":"p^z","display":"p^z","kind":"introduction-passage","excerpt":"prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly inspect M. Define Q_m(z)=∑_o L_m(o)B(o,z), p^o(m)=fracp(m)","count":6},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"introduction-passage","excerpt":"Let M∈ M be a fixed unknown model in a finite family, with prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-3406acd3f0f64ef4","tex":"Q_m","aliases":"Q_m","display":"Q_m","kind":"introduction-passage","excerpt":"prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly inspect M. Define Q_m(z)=∑_o L_m(o)B(o,z), p^o(m)=fracp(m)","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-c95e041440f50ec9","tex":"Q_n","aliases":"Q_n","display":"Q_n","kind":"introduction-passage","excerpt":"prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly inspect M. Define Q_m(z)=∑_o L_m(o)B(o,z), p^o(m)=fracp(m)","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-aff4a4490ea52474","tex":"r^{-1}","aliases":"r^-1","display":"r^{-1}","kind":"introduction-passage","excerpt":"The cost of expanding the contract. Let C' have the same complete state space and contain every operation and query of C, while possibly adding more. Write ≡_ C' for its finer behavioral equivalence. Take any representation r adequate for C. For each z, let k_z be the number of ≡_ C' classes that meet the fibre r^-1(z).","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"For a policy tree T and initial state x, write s_T(x) for its terminal sender transcript and x_T(x) for its final physical state. The tree has finitely many reachable leaves. Specify recipient backgrounds β_i^T(x) and demands d_i^T(x) by a declared rule. A demand may concern x_T(x), the original x, or a stated pair of past and present values. These","count":61},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-f6ffbd57e8fdfebd","tex":"s_T","aliases":"s_T","display":"s_T","kind":"introduction-passage","excerpt":"For a policy tree T and initial state x, write s_T(x) for its terminal sender transcript and x_T(x) for its final physical state. The tree has finitely many reachable leaves. Specify recipient backgrounds β_i^T(x) and demands d_i^T(x) by a declared rule. A demand may concern x_T(x), the original x, or a stated pair of past and present values. These","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1,","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-1edd83621988d298","tex":"\\mathcal V_0","aliases":"V_0 mathcal","display":"\\mathcal V_0","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-9bacaf6e1626666c","tex":"V_{e,o}","aliases":"V_e,o","display":"V_{e,o}","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-284fab15b29e3a8e","tex":"\\mathcal V_h","aliases":"V_h mathcal","display":"\\mathcal V_h","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1,","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-845ae67b4619aa13","tex":"\\mathcal V_{h-1}","aliases":"V_h-1 mathcal","display":"\\mathcal V_{h-1}","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1,","count":14},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-606b2175123b3bb4","tex":"x_T","aliases":"x_T","display":"x_T","kind":"introduction-passage","excerpt":"For a policy tree T and initial state x, write s_T(x) for its terminal sender transcript and x_T(x) for its final physical state. The tree has finitely many reachable leaves. Specify recipient backgrounds β_i^T(x) and demands d_i^T(x) by a declared rule. A demand may concern x_T(x), the original x, or a stated pair of past and present values. These","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1,","count":11},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-c656dc90ffbe53cd","tex":"\\mathcal Z","aliases":"Z mathcal","display":"\\mathcal Z","kind":"introduction-passage","excerpt":"Let M∈ M be a fixed unknown model in a finite family, with prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"The cost of expanding the contract. Let C' have the same complete state space and contain every operation and query of C, while possibly adding more. Write ≡_ C' for its finer behavioral equivalence. Take any representation r adequate for C. For each z, let k_z be the number of ≡_ C' classes that meet the fibre r^-1(z).","count":56},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"introduction-passage","excerpt":"For a policy tree T and initial state x, write s_T(x) for its terminal sender transcript and x_T(x) for its final physical state. The tree has finitely many reachable leaves. Specify recipient backgrounds β_i^T(x) and demands d_i^T(x) by a declared rule. A demand may concern x_T(x), the original x, or a stated pair of past and present values. These","count":7},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a173078221713d7e","tex":"\\beta_i^T","aliases":"β_i^T beta","display":"\\beta_i^T","kind":"introduction-passage","excerpt":"For a policy tree T and initial state x, write s_T(x) for its terminal sender transcript and x_T(x) for its final physical state. The tree has finitely many reachable leaves. Specify recipient backgrounds β_i^T(x) and demands d_i^T(x) by a declared rule. A demand may concern x_T(x), the original x, or a stated pair of past and present values. These","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Let M∈ M be a fixed unknown model in a finite family, with prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly","count":35},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"introduction-passage","excerpt":"prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly inspect M. Define Q_m(z)=∑_o L_m(o)B(o,z), p^o(m)=fracp(m)","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-ccbd66bdaed18a2b","tex":"\\sum_o","aliases":"∑_o sum","display":"\\sum_o","kind":"introduction-passage","excerpt":"prior p(m). An experiment produces O∈ O with likelihood L_m(o). A common reporting channel B(o,z) produces Z∈ Z. Each likelihood row and each channel row is a probability distribution. The complete joint model is Pr(M=m,O=o,Z=z)=p(m)L_m(o)B(o,z). In particular, the channel does not secretly inspect M. Define Q_m(z)=∑_o L_m(o)B(o,z), p^o(m)=fracp(m)","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1,","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"Let r:X→ Z be the target record. A support is terminal when r is constant on it. Let h be the number of experiments still available. Write V_h(V) for the least worst-case sum of experiment costs among policies that terminate with the target known within h steps. Infeasibility is denoted +∞. Then define V_h(V) · =0 · · if |r(V)|=1, V_0(V) · =+∞ · · ","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"introduction-passage","excerpt":"The cost of expanding the contract. Let C' have the same complete state space and contain every operation and query of C, while possibly adding more. Write ≡_ C' for its finer behavioral equivalence. Take any representation r adequate for C. For each z, let k_z be the number of ≡_ C' classes that meet the fibre r^-1(z).","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-991567be41031f95","tex":"\\equiv_{\\C'}","aliases":"≡_C' ≡_ C' equiv C equiv mathcal","display":"\\equiv_{{\\mathcal C}'}","kind":"introduction-passage","excerpt":"The cost of expanding the contract. Let C' have the same complete state space and contain every operation and query of C, while possibly adding more. Write ≡_ C' for its finer behavioral equivalence. Take any representation r adequate for C. For each z, let k_z be the number of ≡_ C' classes that meet the fibre r^-1(z).","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Behavioral compression and the price of new questions. Define x≡_ C y Longleftrightarrow q(F_w(x))=q(F_w(y)) for all q∈ Q, w∈ U^*. This is equality of all declared future answer profiles. It is an equivalence relation. It describes behavioral sufficiency under the contract, not equality of hidden mechanisms or identity outside that contract.","count":9},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"Let X be a finite nonempty set of cases, α:X→ S the sender's access map, and A a finite nonempty artifact catalogue. Recipient i has a background map β_i:X→ B_i, a demanded output d_i:X→ D_i, and a fixed procedure L_i. Evaluate L_i(β_i(x),a) under a declared instruction and memory bound. A timeout, an invalid output, or a rejected artifact counts a","count":12},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"A small instrument with a hidden register. Consider a mathematical instrument with three registers x=(a,b,c)∈ F_5^3. Each register contains one of 0,1,2,3,4. All arithmetic in this example is modulo five. The display reports q(a,b,c)=a+b. There are 125 possible complete states, but only five possible readings. The registers and their operations are","count":81},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"An actual execution of J, followed by a new reading, supplies the missing distinction. Because J changes the instrument, the recovered state is the state after that operation: (a',b',c')=(s_J-b, b, s_J-s). The old state can then be reconstructed because this particular operation is invertible. Neither reversibility nor a free observation should be ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-0014cfc2591bdee6","tex":"a_*","aliases":"a_*","display":"a_*","kind":"use","excerpt":". If the expected-loss gap between a_i and every other action under r_i is strictly greater than 2L_maxe_i, then a_i is also the unique optimum under p. The triangle inequality bounds d_TV(p,r_i) by e_i. For a function taking values in [0,L_max], its expectations under these two distributions differ by at most L_maxe_i. Let a_* minimize expected lo","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"[A joint posterior and recipient decision certificate] Let p be the reference posterior, widehat p an approximate or transmitted posterior, and r_i recipient i's reconstructed posterior. Suppose d_TV(p,widehat p)≤ϵ, d_TV(widehat p,r_i)≤η_i. If a_i minimizes expected loss under r_i, then, with e_i=min{1,ϵ+η_i}, E_pℓ(M,a_i)-min_a∈ A E_pℓ(M,a) ≤min{L_","count":6},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-fe19f8e43be8ed44","tex":"\\mathcal A_s","aliases":"A_s mathcal","display":"\\mathcal A_s","kind":"use","excerpt":"Define A(x)={a∈ A:every recipient returns d_i(x) within its declared limits}, and, for each reachable sender value s, define A_s=⋂_x:α(x)=s A(x). An artifact selected from A_s must work for all cases the sender still considers possible. Choosing a different acceptable artifact after secretly learning which hidden case is true is not allowed.","count":7},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"A small instrument with a hidden register. Consider a mathematical instrument with three registers x=(a,b,c)∈ F_5^3. Each register contains one of 0,1,2,3,4. All arithmetic in this example is modulo five. The display reports q(a,b,c)=a+b. There are 125 possible complete states, but only five possible readings. The registers and their operations are","count":103},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"use","excerpt":"An actual execution of J, followed by a new reading, supplies the missing distinction. Because J changes the instrument, the recovered state is the state after that operation: (a',b',c')=(s_J-b, b, s_J-s). The old state can then be reconstructed because this particular operation is invertible. Neither reversibility nor a free observation should be ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-dbdc187a73452464","tex":"b^h","aliases":"b^h","display":"b^h","kind":"use","excerpt":"rates any two distinct remaining observational classes. Repeatedly selecting an informative experiment preserves m_* and reaches a single observational class after at most |V_0|-1 informative observations. If there are N distinguishable target classes, every adaptive procedure identifying all of them using experiments with at most b≥2 outcomes and ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-66c8d0359e712fa2","tex":"B_e","aliases":"B_e","display":"B_e","kind":"use","excerpt":"Let the unknown model M remain fixed. At time t, a known policy selects experiment e_t using only the prior reported history and recorded experiment names. Randomization is permitted when its conditional law is independent of M given that history. Given M=m and e_t=e, a fresh raw observation has law L_e,m, and its reported form is drawn through B_e","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-fda8c29980b7df98","tex":"B_{e_t}","aliases":"B_e_t","display":"B_{e_t}","kind":"use","excerpt":"cy, the posterior conditioned on the complete raw and reported record equals the posterior conditioned only on the reported outcomes and experiment names. The claim also holds at a bounded stopping time determined by that reported history. For a fixed realized history, its likelihood under model m is the product of the selected-experiment probabili","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"use","excerpt":"Let X be a finite nonempty set of cases, α:X→ S the sender's access map, and A a finite nonempty artifact catalogue. Recipient i has a background map β_i:X→ B_i, a demanded output d_i:X→ D_i, and a fixed procedure L_i. Evaluate L_i(β_i(x),a) under a declared instruction and memory bound. A timeout, an invalid output, or a rejected artifact counts a","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-00743b7e08d0c4a4","tex":"B_\\theta","aliases":"B_θ theta","display":"B_\\theta","kind":"use","excerpt":"[The information an exact explanation must carry] If a fixed-length b-bit artifact lets the recipient answer the declared questions exactly for every target in Θ, then 2^b≥ N. With target-dependent background B_θ, the same argument gives 2^b≥max_B N_B, where N_B counts distinct answer profiles among targets sharing that exact background value. Two ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-bf623b625d280efe","tex":"\\bar F_w","aliases":"bar F_w bar","display":"\\bar F_w","kind":"use","excerpt":"[Exact reusable representation] An exact reusable representation for C is a surjection r: X→ Z, updates bar F_u: Z→ Z, and decoders bar q: Z→ Y such that, for all x,u,q, r(F_u(x))=bar F_u(r(x)), q(x)=bar q(r(x)). Induction immediately gives q(F_w(x))=bar q(bar F_w(r(x))) for every finite word. The same initial record supports many future questions.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b4f1d2372b4ab9ce","tex":"\\bar H","aliases":"bar H bar","display":"\\bar H","kind":"use","excerpt":"A sufficient retained state for the original operation family is z=(s,b)=(a+b,b). Its updates are bar H(s,b)=(s+b,b), bar W(s,b)=(s,s-b), and its displayed output is its first coordinate. These equations retain the information needed for every finite H/W continuation without retaining c.","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-bb3cdc779ccbade8","tex":"\\bar J","aliases":"bar J bar","display":"\\bar J","kind":"use","excerpt":"The current record supplies (s,b). Executing J and observing s_J determines c=s_J-s, while~[source reference] updates the affected first register. The corrected extended coordinates are (s_J,b,c). Their operations are bar H(s,b,c)=(s+b,b,c), bar W(s,b,c)=(s,s-b,c), bar J(s,b,c)=(s+c,b,c). The base method survives as the restriction of the enlarged ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-fad6ae0fd6c0acb8","tex":"\\bar K","aliases":"bar K bar","display":"\\bar K","kind":"use","excerpt":"[Uniform controlled stochastic reuse] A single reduced kernel bar K_u reproduces every projected one-step law from every complete state if and only if, whenever r(x)=r(y), ∑_v:r(v)=z'K_u(x,v)=∑_v:r(v)=z'K_u(y,v) for every u,z'. With matched initial projected laws, the reduced model preserves the law of every finite sequence of reduced states and qu","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-d1786c2e4892c6a4","tex":"\\bar K_u","aliases":"bar K_u bar","display":"\\bar K_u","kind":"use","excerpt":"[Uniform controlled stochastic reuse] A single reduced kernel bar K_u reproduces every projected one-step law from every complete state if and only if, whenever r(x)=r(y), ∑_v:r(v)=z'K_u(x,v)=∑_v:r(v)=z'K_u(y,v) for every u,z'. With matched initial projected laws, the reduced model preserves the law of every finite sequence of reduced states and qu","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-527449602c150418","tex":"\\bar W","aliases":"bar W bar","display":"\\bar W","kind":"use","excerpt":"A sufficient retained state for the original operation family is z=(s,b)=(a+b,b). Its updates are bar H(s,b)=(s+b,b), bar W(s,b)=(s,s-b), and its displayed output is its first coordinate. These equations retain the information needed for every finite H/W continuation without retaining c.","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","excerpt":"An actual execution of J, followed by a new reading, supplies the missing distinction. Because J changes the instrument, the recovered state is the state after that operation: (a',b',c')=(s_J-b, b, s_J-s). The old state can then be reconstructed because this particular operation is invertible. Neither reversibility nor a free observation should be ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-27cba12fdc4807c3","tex":"C'_{o'}","aliases":"C'_o'","display":"C'_{o'}","kind":"use","excerpt":"Write ϕ[V]={ϕ(m):m∈ V}. Let C_o(V)={m∈ V:f(m)=o} and C'_o'(W)={n∈ W:g(n)=o'}. Both are ordinary consistency filters.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e611f32231dd41ea","tex":"C'_{\\eta(o)}","aliases":"C'_η(o) eta","display":"C'_{\\eta(o)}","kind":"use","excerpt":"[Correction transport and evidence reflection] Under~[source reference], for every V⊆ M and every attainable source outcome o∈ f( M), ϕ[C_o(V)]⊆ C'_η(o)(ϕ[V]). Equality holds for all such V and o if and only if η is injective on the attainable outcome set f( M). The claim includes candidate sets contradicted by the observation, so that preservation","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4c79fafc541a02b5","tex":"C_o","aliases":"C_o","display":"C_o","kind":"use","excerpt":"Write ϕ[V]={ϕ(m):m∈ V}. Let C_o(V)={m∈ V:f(m)=o} and C'_o'(W)={n∈ W:g(n)=o'}. Both are ordinary consistency filters.","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"One encoder E:S→ W produces a common artifact. A decoder D_i for each recipient must satisfy D_i(E(α(m)),β_i(m))=d_i(m) for every m,i. The encoder is not allowed to consult a recipient-specific hidden state. Recipients do not receive an uncharged transcript or a separate description of the selected case. Here decoders may be designed once for the d","count":8},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"use","excerpt":"One encoder E:S→ W produces a common artifact. A decoder D_i for each recipient must satisfy D_i(E(α(m)),β_i(m))=d_i(m) for every m,i. The encoder is not allowed to consult a recipient-specific hidden state. Recipients do not receive an uncharged transcript or a separate description of the selected case. Here decoders may be designed once for the d","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"use","excerpt":"An accessible sender and a family of recipients. Let M be a finite nonempty set of possible cases. The sender observes only α(m), where α: M→ S is a declared access map. Recipient i∈{1,…,k} receives background β_i(m) and must recover d_i(m). A demand can be a single answer, a reusable state, or the complete finite answer profile for a declared task","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-16c6b7cfa8776972","tex":"D_{\\mathrm{KL}}","aliases":"D_KL mathrm","display":"D_{\\mathrm{KL}}","kind":"use","excerpt":"[Expected posterior loss is conditional information] Under~[source reference], E_O,Z [D_KL(p^OVert p^Z)] =I(M;Omid Z)=I(M;O)-I(M;Z). The common value is nonnegative. It is zero exactly when the two posteriors agree on every positive-probability observation pair for that prior. For a fully supported prior, zero is equivalent to the factorization in ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"use","excerpt":"[Expected posterior loss is conditional information] Under~[source reference], E_O,Z [D_KL(p^OVert p^Z)] =I(M;Omid Z)=I(M;O)-I(M;Z). The common value is nonnegative. It is zero exactly when the two posteriors agree on every positive-probability observation pair for that prior. For a fully supported prior, zero is equivalent to the factorization in ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"A finite model of informative correction. Let M be a finite nonempty set of deterministic candidate models. Let E be a finite set of permitted experiments. Each experiment e has an outcome map f_e: M→ O_e. The current information is a candidate set V⊆ M. On observing o from e, update it by C_e,o(V)={m∈ V:f_e(m)=o}. This is consistency filtering, no","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"[Finite-horizon transfer budget] For matched source and target operation sequences, start from z_0,w_0 with d(Tz_0,w_0)≤ e_0. After n operations, d(Tz_n,w_n) · ≤ L^n e_0+ϵ∑_j=0^n-1L^j,","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"[A joint posterior and recipient decision certificate] Let p be the reference posterior, widehat p an approximate or transmitted posterior, and r_i recipient i's reconstructed posterior. Suppose d_TV(p,widehat p)≤ϵ, d_TV(widehat p,r_i)≤η_i. If a_i minimizes expected loss under r_i, then, with e_i=min{1,ϵ+η_i}, E_pℓ(M,a_i)-min_a∈ A E_pℓ(M,a) ≤min{L_","count":6},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"|q(z_n)-q'(w_n)| · ≤δ+M(L^n e_0+ϵ∑_j=0^n-1L^j). The empty sum is zero and L^0=1, including when L=0. At L=1 the state bound is e_0+nϵ. Writing e_n=d(Tz_n,w_n), insert g_τ(u)(Tz_n) and use the triangle inequality and the Lipschitz bound to obtain e_n+1≤ϵ+Le_n. Iterating gives the stated expression. Insert q'(Tz_n) between the two outputs and use the","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-d3e1f9162e4702a9","tex":"e_{n+1}","aliases":"e_n+1","display":"e_{n+1}","kind":"use","excerpt":"|q(z_n)-q'(w_n)| · ≤δ+M(L^n e_0+ϵ∑_j=0^n-1L^j). The empty sum is zero and L^0=1, including when L=0. At L=1 the state bound is e_0+nϵ. Writing e_n=d(Tz_n,w_n), insert g_τ(u)(Tz_n) and use the triangle inequality and the Lipschitz bound to obtain e_n+1≤ϵ+Le_n. Iterating gives the stated expression. Insert q'(Tz_n) between the two outputs and use the","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8bb7bbec3bd852d7","tex":"\\mathbb E_{O,Z}","aliases":"E_O,Z mathbb","display":"\\mathbb E_{O,Z}","kind":"use","excerpt":"[Expected posterior loss is conditional information] Under~[source reference], E_O,Z [D_KL(p^OVert p^Z)] =I(M;Omid Z)=I(M;O)-I(M;Z). The common value is nonnegative. It is zero exactly when the two posteriors agree on every positive-probability observation pair for that prior. For a fully supported prior, zero is equivalent to the factorization in ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-84d642745fa8ff24","tex":"\\mathbb E_p","aliases":"E_p mathbb","display":"\\mathbb E_p","kind":"use","excerpt":"int posterior and recipient decision certificate] Let p be the reference posterior, widehat p an approximate or transmitted posterior, and r_i recipient i's reconstructed posterior. Suppose d_TV(p,widehat p)≤ϵ, d_TV(widehat p,r_i)≤η_i. If a_i minimizes expected loss under r_i, then, with e_i=min{1,ϵ+η_i}, E_pℓ(M,a_i)-min_a∈ A E_pℓ(M,a) ≤min{L_max,2","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-2775a21153e9ebc4","tex":"e_t","aliases":"e_t","display":"e_t","kind":"use","excerpt":"Let the unknown model M remain fixed. At time t, a known policy selects experiment e_t using only the prior reported history and recorded experiment names. Randomization is permitted when its conditional law is independent of M given that history. Given M=m and e_t=e, a fresh raw observation has law L_e,m, and its reported form is drawn through B_e","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-632b38e7d6727892","tex":"e_\\theta","aliases":"e_θ theta","display":"e_\\theta","kind":"use","excerpt":"Task-dependent interpreter changes are information. Let Θ be a finite nonempty family of tasks and let each task have a required answer profile over a fixed query family. A fixed interpreter U receives background B_θ, an installed program string p_θ, an artifact e_θ, and a query. It must return the required answer. The interpreter is fixed across t","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-13956f3126b65885","tex":"\\Eq","aliases":"Eq ≡_ C equiv mathcal","display":"{\\equiv_{{\\mathcal C}}}","kind":"use","excerpt":"Behavioral compression and the price of new questions. Define x≡_ C y Longleftrightarrow q(F_w(x))=q(F_w(y)) for all q∈ Q, w∈ U^*. This is equality of all declared future answer profiles. It is an equivalence relation. It describes behavioral sufficiency under the contract, not equality of hidden mechanisms or identity outside that contract.","count":8},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"A costed first-edition instrument certificate. The common realization can now be described without a designable recipient or an omniscient sender. The instrument is still (a,b,c)∈ F_5^3. Before the diagnostic, the sender knows only (s,b)=(a+b,b). Recipient A has the old register pair (a,b). Recipient B has the old display pair (s,b). Neither recipi","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Let a source representation have state space Z, updates f_u, and query decoders q. Let a target have state space W, updates g_v, and decoders q'. A map τ translates operation names. Query pairs (q,q') and a representation map T:Z→ W are fixed in advance. The operation names alone do not establish that the target implements the intended source opera","count":20},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-18f5d4683a720767","tex":"\\mathbb F_5^3","aliases":"F_5^3 mathbb","display":"\\mathbb F_5^3","kind":"use","excerpt":"A costed first-edition instrument certificate. The common realization can now be described without a designable recipient or an omniscient sender. The instrument is still (a,b,c)∈ F_5^3. Before the diagnostic, the sender knows only (s,b)=(a+b,b). Recipient A has the old register pair (a,b). Recipient B has the old display pair (s,b). Neither recipi","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-48dc92a98cff75ec","tex":"f_e","aliases":"f_e","display":"f_e","kind":"use","excerpt":"A finite model of informative correction. Let M be a finite nonempty set of deterministic candidate models. Let E be a finite set of permitted experiments. Each experiment e has an outcome map f_e: M→ O_e. The current information is a candidate set V⊆ M. On observing o from e, update it by C_e,o(V)={m∈ V:f_e(m)=o}. This is consistency filtering, no","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-73373fd911c8ae54","tex":"f_u","aliases":"f_u","display":"f_u","kind":"use","excerpt":"Let a source representation have state space Z, updates f_u, and query decoders q. Let a target have state space W, updates g_v, and decoders q'. A map τ translates operation names. Query pairs (q,q') and a representation map T:Z→ W are fixed in advance. The operation names alone do not establish that the target implements the intended source opera","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4d4d70922273f64c","tex":"F_{u_1}","aliases":"F_u_1","display":"F_{u_1}","kind":"use","excerpt":"t for reusable understanding Let X and Y be finite nonempty sets. Let U be a finite operation alphabet, possibly empty, and let Q be a finite nonempty family of queries q: X→ Y. Each u∈ U has an implemented total transition F_u: X→ X. The deterministic mathematical contract is C=( X, U,(F_u)_u∈ U, Q, Y). For a word w=u_1⋯ u_n, define F_w=F_u_n∘⋯∘ F","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-10dfe6ac4a9ad562","tex":"F_{u_n}","aliases":"F_u_n","display":"F_{u_n}","kind":"use","excerpt":"t for reusable understanding Let X and Y be finite nonempty sets. Let U be a finite operation alphabet, possibly empty, and let Q be a finite nonempty family of queries q: X→ Y. Each u∈ U has an implemented total transition F_u: X→ X. The deterministic mathematical contract is C=( X, U,(F_u)_u∈ U, Q, Y). For a word w=u_1⋯ u_n, define F_w=F_u_n∘⋯∘ F","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a31d948777ad9761","tex":"f_w","aliases":"f_w","display":"f_w","kind":"use","excerpt":"[Exact transport of reusable operations] Suppose, for every source operation and matched query, T f_u=g_τ(u)T, q'T=q. Then, for every source state and every finite source word w, T f_w=g_τ(w)T, q'(g_τ(w)(Tz))=q(f_w(z)). Two such transports compose, with their operation and query correspondences composed as well. No injectivity assumption on T is ne","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-846789dfe1f07b3e","tex":"\\Ffive","aliases":"Ffive F_5 mathbb","display":"{\\mathbb F_5}","kind":"use","excerpt":"A small instrument with a hidden register. Consider a mathematical instrument with three registers x=(a,b,c)∈ F_5^3. Each register contains one of 0,1,2,3,4. All arithmetic in this example is modulo five. The display reports q(a,b,c)=a+b. There are 125 possible complete states, but only five possible readings. The registers and their operations are","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-547d6c30d235a8a4","tex":"\\Ffive^3","aliases":"Ffive^3 F_5^3 Ffive mathbb","display":"{\\mathbb F_5}^3","kind":"use","excerpt":"A small instrument with a hidden register. Consider a mathematical instrument with three registers x=(a,b,c)∈ F_5^3. Each register contains one of 0,1,2,3,4. All arithmetic in this example is modulo five. The display reports q(a,b,c)=a+b. There are 125 possible complete states, but only five possible readings. The registers and their operations are","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Build a graph G_α on α( M). Join s and t when some cases m,n and recipient i satisfy α(m)=s, α(n)=t, β_i(m)=β_i(n), d_i(m)≠ d_i(n). An edge can be a loop when s=t. It then records a distinction that neither the sender's access nor that recipient's background can resolve.","count":6},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"Let a source representation have state space Z, updates f_u, and query decoders q. Let a target have state space W, updates g_v, and decoders q'. A map τ translates operation names. Query pairs (q,q') and a representation map T:Z→ W are fixed in advance. The operation names alone do not establish that the target implements the intended source opera","count":14},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"For each state x and artifact a accepted by all fixed recipient procedures, let γ(x,a)∈ N^r be the terminal charge. It includes transmission and the declared source and recipient execution work. Invalid or out-of-budget decoder executions have no terminal entry. At support V, a stop using a is allowed only if it works for every x∈ V, with worst-cas","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-07ae6e95d4df7899","tex":"G_T","aliases":"G_T","display":"G_T","kind":"use","excerpt":"The recipient sees only the common artifact and its specified background. If an experiment name, its timing, or another transcript is visible to the recipient, that information belongs in β_i^T. It must not be used as an uncharged auxiliary message. Let C(T) be the maximum total experimental cost across initial states. Build G_T on the reachable se","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-99b7a1e1268a1b18","tex":"g_v","aliases":"g_v","display":"g_v","kind":"use","excerpt":"Let a source representation have state space Z, updates f_u, and query decoders q. Let a target have state space W, updates g_v, and decoders q'. A map τ translates operation names. Query pairs (q,q') and a representation map T:Z→ W are fixed in advance. The operation names alone do not establish that the target implements the intended source opera","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b7e94b9c52e447fe","tex":"G_\\alpha","aliases":"G_α alpha","display":"G_\\alpha","kind":"use","excerpt":"Build a graph G_α on α( M). Join s and t when some cases m,n and recipient i satisfy α(m)=s, α(n)=t, β_i(m)=β_i(n), d_i(m)≠ d_i(n). An edge can be a loop when s=t. It then records a distinction that neither the sender's access nor that recipient's background can resolve.","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4a50a12033226473","tex":"g_{\\tau(u)}","aliases":"g_τ(u) tau","display":"g_{\\tau(u)}","kind":"use","excerpt":"[Exact transport of reusable operations] Suppose, for every source operation and matched query, T f_u=g_τ(u)T, q'T=q. Then, for every source state and every finite source word w, T f_w=g_τ(w)T, q'(g_τ(w)(Tz))=q(f_w(z)). Two such transports compose, with their operation and query correspondences composed as well. No injectivity assumption on T is ne","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b10cb03065d681c3","tex":"g_{\\tau(w)}","aliases":"g_τ(w) tau","display":"g_{\\tau(w)}","kind":"use","excerpt":"[Exact transport of reusable operations] Suppose, for every source operation and matched query, T f_u=g_τ(u)T, q'T=q. Then, for every source state and every finite source word w, T f_w=g_τ(w)T, q'(g_τ(w)(Tz))=q(f_w(z)). Two such transports compose, with their operation and query correspondences composed as well. No injectivity assumption on T is ne","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"Initially, two operations are available: H(a,b,c)=(a+b,b,c), W(a,b,c)=(b,a,c). The first adds the second register into the first. The second exchanges the first two registers. The third register is preserved by both and does not affect the display. The operation order is public. A prediction concerns what the display would show after a named finite","count":28},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a9605075a23133f4","tex":"h_{\\sigma(\\tau(u))}","aliases":"h_σ(τ(u)) sigma tau","display":"h_{\\sigma(\\tau(u))}","kind":"use","excerpt":"ll. No injectivity assumption on T is needed for this forward guarantee. The empty word is immediate. If the identity holds for w, then T f_u f_w=g_τ(u)T f_w =g_τ(u)g_τ(w)T. Induction proves the word identity, and applying the matched decoder proves the answer identity. For a second transport S with operation map σ, substitution gives STf_u=h_σ(τ(u","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"[Expected posterior loss is conditional information] Under~[source reference], E_O,Z [D_KL(p^OVert p^Z)] =I(M;Omid Z)=I(M;O)-I(M;Z). The common value is nonnegative. It is zero exactly when the two posteriors agree on every positive-probability observation pair for that prior. For a fully supported prior, zero is equivalent to the factorization in ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"A new operation changes what must be known. Now admit J(a,b,c)=(a+c,b,c). The old representation remains correct for H and W. It is no longer sufficient for the enlarged contract. States with the same (a,b) but different c now produce different readings after J.","count":48},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"[Finite-horizon transfer budget] For matched source and target operation sequences, start from z_0,w_0 with d(Tz_0,w_0)≤ e_0. After n operations, d(Tz_n,w_n) · ≤ L^n e_0+ϵ∑_j=0^n-1L^j, |q(z_n)-q'(w_n)| · ≤δ+M(L^n e_0+ϵ∑_j=0^n-1L^j). The empty sum is zero and L^0=1, including when L=0. At L=1 the state bound is e_0+nϵ. Writing e_n=d(Tz_n,w_n), inser","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"A bounded stochastic extension. The distinction between a current answer and a reusable state also survives stochastic dynamics. Let K_u(x,x') be finite transition kernels and let r: X→ Z be surjective. Assume the current query values are constant on fibres of r.","count":8},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-16bc7321a3832aa5","tex":"K_u","aliases":"K_u","display":"K_u","kind":"use","excerpt":"A bounded stochastic extension. The distinction between a current answer and a reusable state also survives stochastic dynamics. Let K_u(x,x') be finite transition kernels and let r: X→ Z be surjective. Assume the current query values are constant on fibres of r.","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-37c9533a2f91febd","tex":"\\mathrm{KL}","aliases":"KL mathrm","display":"\\mathrm{KL}","kind":"use","excerpt":"[Expected posterior loss is conditional information] Under~[source reference], E_O,Z [D_KL(p^OVert p^Z)] =I(M;Omid Z)=I(M;O)-I(M;Z). The common value is nonnegative. It is zero exactly when the two posteriors agree on every positive-probability observation pair for that prior. For a fully supported prior, zero is equivalent to the factorization in ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-efb929291fb3ed41","tex":"L^0","aliases":"L^0","display":"L^0","kind":"use","excerpt":"|q(z_n)-q'(w_n)| · ≤δ+M(L^n e_0+ϵ∑_j=0^n-1L^j). The empty sum is zero and L^0=1, including when L=0. At L=1 the state bound is e_0+nϵ. Writing e_n=d(Tz_n,w_n), insert g_τ(u)(Tz_n) and use the triangle inequality and the Lipschitz bound to obtain e_n+1≤ϵ+Le_n. Iterating gives the stated expression. Insert q'(Tz_n) between the two outputs and use the","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-1fb84042c209d5c5","tex":"L^j","aliases":"L^j","display":"L^j","kind":"use","excerpt":"[Finite-horizon transfer budget] For matched source and target operation sequences, start from z_0,w_0 with d(Tz_0,w_0)≤ e_0. After n operations, d(Tz_n,w_n) · ≤ L^n e_0+ϵ∑_j=0^n-1L^j, |q(z_n)-q'(w_n)| · ≤δ+M(L^n e_0+ϵ∑_j=0^n-1L^j). The empty sum is zero and L^0=1, including when L=0. At L=1 the state bound is e_0+nϵ. Writing e_n=d(Tz_n,w_n), inser","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-478acc5f340c0573","tex":"L^n","aliases":"L^n","display":"L^n","kind":"use","excerpt":"[Finite-horizon transfer budget] For matched source and target operation sequences, start from z_0,w_0 with d(Tz_0,w_0)≤ e_0. After n operations, d(Tz_n,w_n) · ≤ L^n e_0+ϵ∑_j=0^n-1L^j, |q(z_n)-q'(w_n)| · ≤δ+M(L^n e_0+ϵ∑_j=0^n-1L^j). The empty sum is zero and L^0=1, including when L=0. At L=1 the state bound is e_0+nϵ. Writing e_n=d(Tz_n,w_n), inser","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-cdf7bae306756b1b","tex":"L_A","aliases":"L_A","display":"L_A","kind":"use","excerpt":"The two frozen decoders are L_A((a,b),m)=(a+m,b,m), L_B((s,b),m)=(s+m,b,m). Recipient A obtains the post-J register description. Recipient B obtains the post-J display-centered description. Both compute one modular addition. Their installed update procedures then support all H/W/J continuations in their respective coordinates. These procedures are ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-563fb35759c558d0","tex":"L_B","aliases":"L_B","display":"L_B","kind":"use","excerpt":"The two frozen decoders are L_A((a,b),m)=(a+m,b,m), L_B((s,b),m)=(s+m,b,m). Recipient A obtains the post-J register description. Recipient B obtains the post-J display-centered description. Both compute one modular addition. Their installed update procedures then support all H/W/J continuations in their respective coordinates. These procedures are ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6a13517b54ad55ec","tex":"L_{e,m}","aliases":"L_e,m","display":"L_{e,m}","kind":"use","excerpt":"Let the unknown model M remain fixed. At time t, a known policy selects experiment e_t using only the prior reported history and recorded experiment names. Randomization is permitted when its conditional law is independent of M given that history. Given M=m and e_t=e, a fresh raw observation has law L_e,m, and its reported form is drawn through B_e","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-38bb3fa5b9d57081","tex":"L_{e_t,m}","aliases":"L_e_t,m","display":"L_{e_t,m}","kind":"use","excerpt":"cy, the posterior conditioned on the complete raw and reported record equals the posterior conditioned only on the reported outcomes and experiment names. The claim also holds at a bounded stopping time determined by that reported history. For a fixed realized history, its likelihood under model m is the product of the selected-experiment probabili","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a716b0894950a10a","tex":"L_i","aliases":"L_i","display":"L_i","kind":"use","excerpt":"Let X be a finite nonempty set of cases, α:X→ S the sender's access map, and A a finite nonempty artifact catalogue. Recipient i has a background map β_i:X→ B_i, a demanded output d_i:X→ D_i, and a fixed procedure L_i. Evaluate L_i(β_i(x),a) under a declared instruction and memory bound. A timeout, an invalid output, or a rejected artifact counts a","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-fc686f6bdce3b5ca","tex":"L_{\\max}","aliases":"L_max max","display":"L_{\\max}","kind":"use","excerpt":"From approximate beliefs to a recipient's decisions. Not every posterior distinction matters equally to an action. To link correction with use, fix one finite nonempty action set A and losses ℓ(m,a)∈[0,L_max]. A receiver evaluates actions from the distribution its actual interpreter reconstructs. The target loss function and available actions must ","count":7},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[The minimal exact reusable state] The quotient X/≡_ C carries an exact reusable representation for C. Every other exact reusable representation r: X→ Z refines this quotient: if r(x)=r(y), then x≡_ C y. There is a unique map h: Z→ X/≡_ C with h(r(x))=[x]. In particular, | Z|≥| X/≡_ C|. A fixed-length binary state needs at least lceillog_2| X/≡_ C|","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"[The minimal exact reusable state] The quotient X/≡_ C carries an exact reusable representation for C. Every other exact reusable representation r: X→ Z refines this quotient: if r(x)=r(y), then x≡_ C y. There is a unique map h: Z→ X/≡_ C with h(r(x))=[x]. In particular, | Z|≥| X/≡_ C|. A fixed-length binary state needs at least lceillog_2| X/≡_ C|","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"[The minimal exact reusable state] The quotient X/≡_ C carries an exact reusable representation for C. Every other exact reusable representation r: X→ Z refines this quotient: if r(x)=r(y), then x≡_ C y. There is a unique map h: Z→ X/≡_ C with h(r(x))=[x]. In particular, | Z|≥| X/≡_ C|. A fixed-length binary state needs at least lceillog_2| X/≡_ C|","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Behavioral compression and the price of new questions. Define x≡_ C y Longleftrightarrow q(F_w(x))=q(F_w(y)) for all q∈ Q, w∈ U^*. This is equality of all declared future answer profiles. It is an equivalence relation. It describes behavioral sufficiency under the contract, not equality of hidden mechanisms or identity outside that contract.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b11203379a62bd48","tex":"m_*","aliases":"m_*","display":"m_*","kind":"use","excerpt":"[Finite correction under separating evidence] Suppose the true model m_* belongs to the initial set V_0, every observation is exact, and the allowed experiment family separates any two distinct remaining observational classes. Repeatedly selecting an informative experiment preserves m_* and reaches a single observational class after at most |V_0|-1","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":"[A translated observation preserves both mistaken candidates] Take M={m_0,m_1} with f(m_0)=0 and f(m_1)=1. Translate models bijectively to {n_0,n_1}, but translate both observations to a single symbol *. Let g(n_0)=g(n_1)=*. The observation square commutes. Starting from both candidates, source observation 1 leaves only m_1, whose image is {n_1}. T","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"[A translated observation preserves both mistaken candidates] Take M={m_0,m_1} with f(m_0)=0 and f(m_1)=1. Translate models bijectively to {n_0,n_1}, but translate both observations to a single symbol *. Let g(n_0)=g(n_1)=*. The observation square commutes. Starting from both candidates, source observation 1 leaves only m_1, whose image is {n_1}. T","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e4e08911b5f19664","tex":"\\max_B","aliases":"max_B max","display":"\\max_B","kind":"use","excerpt":"[Charging a task-dependent interpreter] Suppose p_θ and e_θ have fixed lengths d and b bits, respectively, and the deterministic interpreter answers every required query exactly. Then 2^d+b≥max_B N_B. In particular, representing every Boolean function on n input bits under one fixed background requires d+b≥2^n. This counts task-dependent informatio","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-19305727802db580","tex":"\\max_{B}","aliases":"max_B max","display":"\\max_{B}","kind":"use","excerpt":"[The information an exact explanation must carry] If a fixed-length b-bit artifact lets the recipient answer the declared questions exactly for every target in Θ, then 2^b≥ N. With target-dependent background B_θ, the same argument gives 2^b≥max_B N_B, where N_B counts distinct answer profiles among targets sharing that exact background value. Two ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-d441c86815baa4be","tex":"\\max_o","aliases":"max_o max","display":"\\max_o","kind":"use","excerpt":"For a tree T, let v(T) be its componentwise worst-case charged online cost. Let P_h(V) be the attainable pairs (v(T),n(T)). The stop pairs are (g_a(V),1) for each artifact accepted throughout V. For h>0, each uniformly admissible experiment gives the additional pairs (c_e+max_o v_o, 1+∑_o n_o), (v_o,n_o)∈ P_h-1(V_e,o), with one child choice for eve","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-95ba5202a1737266","tex":"\\max_{x\\in V}","aliases":"max_x∈ V max in","display":"\\max_{x\\in V}","kind":"use","excerpt":"For each state x and artifact a accepted by all fixed recipient procedures, let γ(x,a)∈ N^r be the terminal charge. It includes transmission and the declared source and recipient execution work. Invalid or out-of-budget decoder executions have no terminal entry. At support V, a stop using a is allowed only if it works for every x∈ V, with worst-cas","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b0594d3d24a0288b","tex":"\\max_z","aliases":"max_z max","display":"\\max_z","kind":"use","excerpt":"[Contract expansion and supplementary information] The relation ≡_ C' refines ≡_ C. Any supplementary code s: X→ S for which (r,s) supports exact reusable behavior under C' must satisfy | S|≥max_z k_z. If every ≡_ C' class lies within a single fibre of r, this bound is attained by an exact reusable pair (r,s). This condition holds when r is the old","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"When a coarser report preserves every update. The reporting model implies M→ O→ Z. Therefore, on every positive-probability pair (o,z), the full posterior after observing both records equals p^o. The coarse posterior is a mixture of these fine posteriors: p^z=∑_oPr(O=omid Z=z) p^o. Averaging is not generally the same as retaining each possible upda","count":15},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-c5f5ee23219d1696","tex":"\\min_{a\\in A}","aliases":"min_a∈ A min in","display":"\\min_{a\\in A}","kind":"use","excerpt":"int posterior and recipient decision certificate] Let p be the reference posterior, widehat p an approximate or transmitted posterior, and r_i recipient i's reconstructed posterior. Suppose d_TV(p,widehat p)≤ϵ, d_TV(widehat p,r_i)≤η_i. If a_i minimizes expected loss under r_i, then, with e_i=min{1,ϵ+η_i}, E_pℓ(M,a_i)-min_a∈ A E_pℓ(M,a) ≤min{L_max,2","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"An exact communication lower bound. Let Θ be a finite family of target models. Relative to the required continuation questions, divide these models into N classes with distinct answer profiles. Use one fixed recipient background and one fixed deterministic recipient. The teacher can select a message but cannot change the recipient's background sepa","count":16},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"The explicit finite implementation model Use a finite augmented state space X. Include every origin variable, history flag, recipient background, and permission needed to make the declared transition and terminal tables valid. Each experiment e has the transition, observation, and admissibility semantics of Section~[source reference], together with","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"use","excerpt":"Let M and N be finite model families, with model translation ϕ: M→ N. For one source experiment e, let f: M→ O give its outcomes. A corresponding target experiment has outcome map g: N→ O'. Let η: O→ O' translate evidence. Assume the basic observation square commutes: g(ϕ(m))=η(f(m)) for all m∈ M. This condition says that a model's predicted eviden","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-56155806d18c6f60","tex":"\\mathbb N^r","aliases":"N^r mathbb","display":"\\mathbb N^r","kind":"use","excerpt":"The explicit finite implementation model Use a finite augmented state space X. Include every origin variable, history flag, recipient background, and permission needed to make the declared transition and terminal tables valid. Each experiment e has the transition, observation, and admissibility semantics of Section~[source reference], together with","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","excerpt":"[A translated observation preserves both mistaken candidates] Take M={m_0,m_1} with f(m_0)=0 and f(m_1)=1. Translate models bijectively to {n_0,n_1}, but translate both observations to a single symbol *. Let g(n_0)=g(n_1)=*. The observation square commutes. Starting from both candidates, source observation 1 leaves only m_1, whose image is {n_1}. T","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"[A translated observation preserves both mistaken candidates] Take M={m_0,m_1} with f(m_0)=0 and f(m_1)=1. Translate models bijectively to {n_0,n_1}, but translate both observations to a single symbol *. Let g(n_0)=g(n_1)=*. The observation square commutes. Starting from both candidates, source observation 1 leaves only m_1, whose image is {n_1}. T","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-45d31541cc3ba4ff","tex":"N_B","aliases":"N_B","display":"N_B","kind":"use","excerpt":"[The information an exact explanation must carry] If a fixed-length b-bit artifact lets the recipient answer the declared questions exactly for every target in Θ, then 2^b≥ N. With target-dependent background B_θ, the same argument gives 2^b≥max_B N_B, where N_B counts distinct answer profiles among targets sharing that exact background value. Two ","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-aaec29026bff78cd","tex":"n_o","aliases":"n_o","display":"n_o","kind":"use","excerpt":"For a tree T, let v(T) be its componentwise worst-case charged online cost. Let P_h(V) be the attainable pairs (v(T),n(T)). The stop pairs are (g_a(V),1) for each artifact accepted throughout V. For h>0, each uniformly admissible experiment gives the additional pairs (c_e+max_o v_o, 1+∑_o n_o), (v_o,n_o)∈ P_h-1(V_e,o), with one child choice for eve","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b9dd02171b5ea5b5","tex":"\\mathcal O'","aliases":"O' mathcal","display":"\\mathcal O'","kind":"use","excerpt":"Let M and N be finite model families, with model translation ϕ: M→ N. For one source experiment e, let f: M→ O give its outcomes. A corresponding target experiment has outcome map g: N→ O'. Let η: O→ O' translate evidence. Assume the basic observation square commutes: g(ϕ(m))=η(f(m)) for all m∈ M. This condition says that a model's predicted eviden","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-54726f787dc65700","tex":"o'","aliases":"o'","display":"o'","kind":"use","excerpt":"Write ϕ[V]={ϕ(m):m∈ V}. Let C_o(V)={m∈ V:f(m)=o} and C'_o'(W)={n∈ W:g(n)=o'}. Both are ordinary consistency filters.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-3554f05e6897e490","tex":"O_e","aliases":"O_e","display":"O_e","kind":"use","excerpt":"Experiments change both information and state. Let X be a finite nonempty state space. Each experiment e has an admissibility set A_e⊆ X, a transition T_e:X→ X, an observation o_e:X→ O_e, and a known cost c_e≥0. The observation is indexed by the state before the experiment, but can describe a reading taken afterward: for example o_e=q∘ T_e. At an i","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-9856c4f5918cec70","tex":"\\mathcal O_e","aliases":"O_e mathcal","display":"\\mathcal O_e","kind":"use","excerpt":"A finite model of informative correction. Let M be a finite nonempty set of deterministic candidate models. Let E be a finite set of permitted experiments. Each experiment e has an outcome map f_e: M→ O_e. The current information is a candidate set V⊆ M. On observing o from e, update it by C_e,o(V)={m∈ V:f_e(m)=o}. This is consistency filtering, no","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b2bdbcfb9c9cfc41","tex":"o_e","aliases":"o_e","display":"o_e","kind":"use","excerpt":"Experiments change both information and state. Let X be a finite nonempty state space. Each experiment e has an admissibility set A_e⊆ X, a transition T_e:X→ X, an observation o_e:X→ O_e, and a known cost c_e≥0. The observation is indexed by the state before the experiment, but can describe a reading taken afterward: for example o_e=q∘ T_e. At an i","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4a15af8f6ff25f74","tex":"o_t","aliases":"o_t","display":"o_t","kind":"use","excerpt":"cy, the posterior conditioned on the complete raw and reported record equals the posterior conditioned only on the reported outcomes and experiment names. The claim also holds at a bounded stopping time determined by that reported history. For a fixed realized history, its likelihood under model m is the product of the selected-experiment probabili","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"For a tree T, let v(T) be its componentwise worst-case charged online cost. Let P_h(V) be the attainable pairs (v(T),n(T)). The stop pairs are (g_a(V),1) for each artifact accepted throughout V. For h>0, each uniformly admissible experiment gives the additional pairs (c_e+max_o v_o, 1+∑_o n_o), (v_o,n_o)∈ P_h-1(V_e,o), with one child choice for eve","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e57276c69cbfb04f","tex":"p^{-1}","aliases":"p^-1","display":"p^{-1}","kind":"use","excerpt":"The interpreter must not secretly do all the work. Take any bijection p: X→ Z of a finite state space. Define bar F_u=p∘ F_u∘ p^-1, bar q=q∘ p^-1. All exact representation equations hold. The labels in Z may nevertheless be arbitrary, and the translated updates may require a complete lookup table. Exact representability alone does not explain why t","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4504840e095de412","tex":"p^O","aliases":"p^O","display":"p^O","kind":"use","excerpt":"[Expected posterior loss is conditional information] Under~[source reference], E_O,Z [D_KL(p^OVert p^Z)] =I(M;Omid Z)=I(M;O)-I(M;Z). The common value is nonnegative. It is zero exactly when the two posteriors agree on every positive-probability observation pair for that prior. For a fully supported prior, zero is equivalent to the factorization in ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b5ba0497b20928a8","tex":"p^Z","aliases":"p^Z","display":"p^Z","kind":"use","excerpt":"[Expected posterior loss is conditional information] Under~[source reference], E_O,Z [D_KL(p^OVert p^Z)] =I(M;Omid Z)=I(M;O)-I(M;Z). The common value is nonnegative. It is zero exactly when the two posteriors agree on every positive-probability observation pair for that prior. For a fully supported prior, zero is equivalent to the factorization in ","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"For computation, let P_0 group states with identical current query vectors. Given a partition P_k, refine each block by the block of F_u(x) for every u. Induction shows that P_k identifies exactly the states agreeing on words of length at most k. Every strict refinement increases the block count. At most | X|-1 strict refinements are possible. At a","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[The identification limit of a transcript] Suppose two proposed implementations induce complete transcript laws P_0 and P_1 on a finite transcript alphabet under a specified permitted experiment, including any adaptive choices made by its evaluator. With equal prior probability, the greatest identification accuracy available from that transcript is","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-5627e8b6e58a3075","tex":"\\mathcal P_h","aliases":"P_h mathcal","display":"\\mathcal P_h","kind":"use","excerpt":"For a tree T, let v(T) be its componentwise worst-case charged online cost. Let P_h(V) be the attainable pairs (v(T),n(T)). The stop pairs are (g_a(V),1) for each artifact accepted throughout V. For h>0, each uniformly admissible experiment gives the additional pairs (c_e+max_o v_o, 1+∑_o n_o), (v_o,n_o)∈ P_h-1(V_e,o), with one child choice for eve","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-55b834ed6437b820","tex":"\\mathcal P_{h-1}","aliases":"P_h-1 mathcal","display":"\\mathcal P_{h-1}","kind":"use","excerpt":"For a tree T, let v(T) be its componentwise worst-case charged online cost. Let P_h(V) be the attainable pairs (v(T),n(T)). The stop pairs are (g_a(V),1) for each artifact accepted throughout V. For h>0, each uniformly admissible experiment gives the additional pairs (c_e+max_o v_o, 1+∑_o n_o), (v_o,n_o)∈ P_h-1(V_e,o), with one child choice for eve","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"For computation, let P_0 group states with identical current query vectors. Given a partition P_k, refine each block by the block of F_u(x) for every u. Induction shows that P_k identifies exactly the states agreeing on words of length at most k. Every strict refinement increases the block count. At most | X|-1 strict refinements are possible. At a","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-eb7979ea66ee31a5","tex":"p_m","aliases":"p_m","display":"p_m","kind":"use","excerpt":"ion Write u_m=P(M=m,E), v_m=Q(M=m,E), and let p=u/a, q=v/b. Exchanging P and Q if necessary, assume a≤ b. The variation inside E is at least tfrac12∑_m|u_m-v_m|. The variation outside E is at least |a-b|/2, since its total masses differ by that amount. Hence δ≥ tfrac12∑_m|u_m-v_m|+tfrac12(b-a) =b-∑_mmin(u_m,v_m). Since a≤ b, min(ap_m,bq_m)≤ bmin(p_","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-392faed71557f8bf","tex":"p_\\theta","aliases":"p_θ theta","display":"p_\\theta","kind":"use","excerpt":"Task-dependent interpreter changes are information. Let Θ be a finite nonempty family of tasks and let each task have a required answer profile over a fixed query family. A fixed interpreter U receives background B_θ, an installed program string p_θ, an artifact e_θ, and a query. It must return the required answer. The interpreter is fixed across t","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q mathcal","display":"{\\mathcal Q}","kind":"use","excerpt":"A contract for reusable understanding. Let X and Y be finite nonempty sets. Let U be a finite operation alphabet, possibly empty, and let Q be a finite nonempty family of queries q: X→ Y. Each u∈ U has an implemented total transition F_u: X→ X. The deterministic mathematical contract is C=( X, U,(F_u)_u∈ U, Q, Y). For a word w=u_1⋯ u_n, define F_w=","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"Let a source representation have state space Z, updates f_u, and query decoders q. Let a target have state space W, updates g_v, and decoders q'. A map τ translates operation names. Query pairs (q,q') and a representation map T:Z→ W are fixed in advance. The operation names alone do not establish that the target implements the intended source opera","count":7},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-f4034bec016cbd8e","tex":"Q_{e,m}","aliases":"Q_e,m","display":"Q_{e,m}","kind":"use","excerpt":"t e_t using only the prior reported history and recorded experiment names. Randomization is permitted when its conditional law is independent of M given that history. Given M=m and e_t=e, a fresh raw observation has law L_e,m, and its reported form is drawn through B_e. Conditional on the model and selected experiment, these draws do not depend on ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-9711a6c06707eaf6","tex":"Q_{e_t,m}","aliases":"Q_e_t,m","display":"Q_{e_t,m}","kind":"use","excerpt":"at a bounded stopping time determined by that reported history. For a fixed realized history, its likelihood under model m is the product of the selected-experiment probabilities and the terms L_e_t,m(o_t)B_e_t(o_t,z_t). The policy factors are independent of m once the reported history is fixed. Applying the assumed factorization gives a product of","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a2dbd37e0ccec228","tex":"q_m","aliases":"q_m","display":"q_m","kind":"use","excerpt":"ion Write u_m=P(M=m,E), v_m=Q(M=m,E), and let p=u/a, q=v/b. Exchanging P and Q if necessary, assume a≤ b. The variation inside E is at least tfrac12∑_m|u_m-v_m|. The variation outside E is at least |a-b|/2, since its total masses differ by that amount. Hence δ≥ tfrac12∑_m|u_m-v_m|+tfrac12(b-a) =b-∑_mmin(u_m,v_m). Since a≤ b, min(ap_m,bq_m)≤ bmin(p_","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"[A joint posterior and recipient decision certificate] Let p be the reference posterior, widehat p an approximate or transmitted posterior, and r_i recipient i's reconstructed posterior. Suppose d_TV(p,widehat p)≤ϵ, d_TV(widehat p,r_i)≤η_i. If a_i minimizes expected loss under r_i, then, with e_i=min{1,ϵ+η_i}, E_pℓ(M,a_i)-min_a∈ A E_pℓ(M,a) ≤min{L_","count":7},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[The minimal exact reusable state] The quotient X/≡_ C carries an exact reusable representation for C. Every other exact reusable representation r: X→ Z refines this quotient: if r(x)=r(y), then x≡_ C y. There is a unique map h: Z→ X/≡_ C with h(r(x))=[x]. In particular, | Z|≥| X/≡_ C|. A fixed-length binary state needs at least lceillog_2| X/≡_ C|","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"r operation and query correspondences composed as well. No injectivity assumption on T is needed for this forward guarantee. The empty word is immediate. If the identity holds for w, then T f_u f_w=g_τ(u)T f_w =g_τ(u)g_τ(w)T. Induction proves the word identity, and applying the matched decoder proves the answer identity. For a second transport S wi","count":10},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"[Contract expansion and supplementary information] The relation ≡_ C' refines ≡_ C. Any supplementary code s: X→ S for which (r,s) supports exact reusable behavior under C' must satisfy | S|≥max_z k_z. If every ≡_ C' class lies within a single fibre of r, this bound is attained by an exact reusable pair (r,s). This condition holds when r is the old","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8a9f605aee045f84","tex":"s_H","aliases":"s_H","display":"s_H","kind":"use","excerpt":"The representation is also minimal in its number of states. The present reading s and the reading s_H after a hypothetical H determine b=s_H-s, a=2s-s_H. Two different pairs (a,b) therefore differ on at least one of these questions. Every exact reusable representation for this contract needs at least 25 distinguishable values. Section~[source refer","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-7367d6672c4feb62","tex":"s_J","aliases":"s_J","display":"s_J","kind":"use","excerpt":"The additional reading s_J yields c=s_J-s. Together with the earlier distinctions for a and b, the enlarged question family distinguishes all 125 complete states. The useful state counts are therefore five for the current display, 25 for reusable H/W prediction, and 125 after J is admitted.","count":9},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"Build a graph G_α on α( M). Join s and t when some cases m,n and recipient i satisfy α(m)=s, α(n)=t, β_i(m)=β_i(n), d_i(m)≠ d_i(n). An edge can be a loop when s=t. It then records a distinction that neither the sender's access nor that recipient's background can resolve.","count":13},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-bb1ad3b8487cb395","tex":"T_e","aliases":"T_e","display":"T_e","kind":"use","excerpt":"Experiments change both information and state. Let X be a finite nonempty state space. Each experiment e has an admissibility set A_e⊆ X, a transition T_e:X→ X, an observation o_e:X→ O_e, and a known cost c_e≥0. The observation is indexed by the state before the experiment, but can describe a reading taken afterward: for example o_e=q∘ T_e. At an i","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"Task-dependent interpreter changes are information. Let Θ be a finite nonempty family of tasks and let each task have a required answer profile over a fixed query family. A fixed interpreter U receives background B_θ, an installed program string p_θ, an artifact e_θ, and a query. It must return the required answer. The interpreter is fixed across t","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-300f218594b1347a","tex":"\\U","aliases":"U mathcal","display":"{\\mathcal U}","kind":"use","excerpt":"A contract for reusable understanding. Let X and Y be finite nonempty sets. Let U be a finite operation alphabet, possibly empty, and let Q be a finite nonempty family of queries q: X→ Y. Each u∈ U has an implemented total transition F_u: X→ X. The deterministic mathematical contract is C=( X, U,(F_u)_u∈ U, Q, Y). For a word w=u_1⋯ u_n, define F_w=","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-dc8018c6c1ea5943","tex":"\\U^*","aliases":"U^* U mathcal","display":"{\\mathcal U}^*","kind":"use","excerpt":"Behavioral compression and the price of new questions. Define x≡_ C y Longleftrightarrow q(F_w(x))=q(F_w(y)) for all q∈ Q, w∈ U^*. This is equality of all declared future answer profiles. It is an equivalence relation. It describes behavioral sufficiency under the contract, not equality of hidden mechanisms or identity outside that contract.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"A contract for reusable understanding. Let X and Y be finite nonempty sets. Let U be a finite operation alphabet, possibly empty, and let Q be a finite nonempty family of queries q: X→ Y. Each u∈ U has an implemented total transition F_u: X→ X. The deterministic mathematical contract is C=( X, U,(F_u)_u∈ U, Q, Y). For a word w=u_1⋯ u_n, define F_w=","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8d86088ae482ad75","tex":"u_m","aliases":"u_m","display":"u_m","kind":"use","excerpt":"ith d_TV(P,Q)≤δ. Let E be an observation event with a=P(E)>0 and b=Q(E)>0. Then d_TV(P(Mmid E),Q(Mmid E)) ≤min{1,fracδmax(a,b)}. In particular, a justified lower bound P(E)≥γ>0 supplies the weaker usable bound min{1,δ/γ}. Write u_m=P(M=m,E), v_m=Q(M=m,E), and let p=u/a, q=v/b. Exchanging P and Q if necessary, assume a≤ b. The variation inside E is ","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"use","excerpt":"A contract for reusable understanding. Let X and Y be finite nonempty sets. Let U be a finite operation alphabet, possibly empty, and let Q be a finite nonempty family of queries q: X→ Y. Each u∈ U has an implemented total transition F_u: X→ X. The deterministic mathematical contract is C=( X, U,(F_u)_u∈ U, Q, Y). For a word w=u_1⋯ u_n, define F_w=","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"[Uniform controlled stochastic reuse] A single reduced kernel bar K_u reproduces every projected one-step law from every complete state if and only if, whenever r(x)=r(y), ∑_v:r(v)=z'K_u(x,v)=∑_v:r(v)=z'K_u(y,v) for every u,z'. With matched initial projected laws, the reduced model preserves the law of every finite sequence of reduced states and qu","count":11},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"[Finite correction under separating evidence] Suppose the true model m_* belongs to the initial set V_0, every observation is exact, and the allowed experiment family separates any two distinct remaining observational classes. Repeatedly selecting an informative experiment preserves m_* and reaches a single observational class after at most |V_0|-1","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b00eda2d2588cd09","tex":"v_m","aliases":"v_m","display":"v_m","kind":"use","excerpt":"qδ. Let E be an observation event with a=P(E)>0 and b=Q(E)>0. Then d_TV(P(Mmid E),Q(Mmid E)) ≤min{1,fracδmax(a,b)}. In particular, a justified lower bound P(E)≥γ>0 supplies the weaker usable bound min{1,δ/γ}. Write u_m=P(M=m,E), v_m=Q(M=m,E), and let p=u/a, q=v/b. Exchanging P and Q if necessary, assume a≤ b. The variation inside E is at least tfra","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8e7676bd93a7edb5","tex":"v_o","aliases":"v_o","display":"v_o","kind":"use","excerpt":"For a tree T, let v(T) be its componentwise worst-case charged online cost. Let P_h(V) be the attainable pairs (v(T),n(T)). The stop pairs are (g_a(V),1) for each artifact accepted throughout V. For h>0, each uniformly admissible experiment gives the additional pairs (c_e+max_o v_o, 1+∑_o n_o), (v_o,n_o)∈ P_h-1(V_e,o), with one child choice for eve","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-50ce1a62663b77d9","tex":"\\Vert","aliases":"Vert","display":"\\Vert","kind":"use","excerpt":"[Expected posterior loss is conditional information] Under~[source reference], E_O,Z [D_KL(p^OVert p^Z)] =I(M;Omid Z)=I(M;O)-I(M;Z). The common value is nonnegative. It is zero exactly when the two posteriors agree on every positive-probability observation pair for that prior. For a fully supported prior, zero is equivalent to the factorization in ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Initially, two operations are available: H(a,b,c)=(a+b,b,c), W(a,b,c)=(b,a,c). The first adds the second register into the first. The second exchanges the first two registers. The third register is preserved by both and does not affect the display. The operation order is public. A prediction concerns what the display would show after a named finite","count":24},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-59e63012bc6d4f87","tex":"w_0","aliases":"w_0","display":"w_0","kind":"use","excerpt":"[Finite-horizon transfer budget] For matched source and target operation sequences, start from z_0,w_0 with d(Tz_0,w_0)≤ e_0. After n operations, d(Tz_n,w_n) · ≤ L^n e_0+ϵ∑_j=0^n-1L^j,","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8945947ef031a855","tex":"w_n","aliases":"w_n","display":"w_n","kind":"use","excerpt":"[Finite-horizon transfer budget] For matched source and target operation sequences, start from z_0,w_0 with d(Tz_0,w_0)≤ e_0. After n operations, d(Tz_n,w_n) · ≤ L^n e_0+ϵ∑_j=0^n-1L^j, |q(z_n)-q'(w_n)| · ≤δ+M(L^n e_0+ϵ∑_j=0^n-1L^j). The empty sum is zero and L^0=1, including when L=0. At L=1 the state bound is e_0+nϵ. Writing e_n=d(Tz_n,w_n), inser","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"[A joint posterior and recipient decision certificate] Let p be the reference posterior, widehat p an approximate or transmitted posterior, and r_i recipient i's reconstructed posterior. Suppose d_TV(p,widehat p)≤ϵ, d_TV(widehat p,r_i)≤η_i. If a_i minimizes expected loss under r_i, then, with e_i=min{1,ϵ+η_i}, E_pℓ(M,a_i)-min_a∈ A E_pℓ(M,a) ≤min{L_","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-ef8e4ac1a775f7c5","tex":"\\widetilde W","aliases":"widetilde W widetilde","display":"\\widetilde W","kind":"use","excerpt":"[Correct current answers with a wrong transported operation] Retain the correct map T(a,b)=(a+b,b) and the correct display decoder. Now incorrectly define the target swap by widetilde W(s,b)=(s,b). The current display and the display immediately after a swap are still correct. Starting from (a,b)=(1,3), however, the source sequence W then H ends wi","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"A bounded stochastic extension. The distinction between a current answer and a reusable state also survives stochastic dynamics. Let K_u(x,x') be finite transition kernels and let r: X→ Z be surjective. Assume the current query values are constant on fibres of r.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"The graph belongs to the acquired transcript. Fix a finite set X_0 of possible initial states, a finite deterministic experiment catalogue as in Section~[source reference], and a finite horizon h. A deterministic experimental policy is an observation-history tree. Its choices cannot depend on the hidden initial state. Only uniformly admissible choi","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"To require recovery of a past distinction, replace the state by (x_0,x_t), let every transition preserve x_0, and make the terminal target include the required part of x_0. The same recurrence then treats historical reconstruction honestly. In the reset example, the two origins remain indistinguishable at every horizon. If both current knowledge an","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"[Information suffices but the installed decoder needs a larger codebook] Take three cases x_0,x_1,x_2 fully visible to the sender. One recipient already has background i in case x_i and always needs output zero. The sender-access graph has no edges, so a designable decoder could ignore a single common message and return zero. Now fix an installed d","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[Information suffices but the installed decoder needs a larger codebook] Take three cases x_0,x_1,x_2 fully visible to the sender. One recipient already has background i in case x_i and always needs output zero. The sender-access graph has no edges, so a designable decoder could ignore a single common message and return zero. Now fix an installed d","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"[Information suffices but the installed decoder needs a larger codebook] Take three cases x_0,x_1,x_2 fully visible to the sender. One recipient already has background i in case x_i and always needs output zero. The sender-access graph has no edges, so a designable decoder could ignore a single common message and return zero. Now fix an installed d","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-1029cb8ad51ed272","tex":"x_t","aliases":"x_t","display":"x_t","kind":"use","excerpt":"To require recovery of a past distinction, replace the state by (x_0,x_t), let every transition preserve x_0, and make the terminal target include the required part of x_0. The same recurrence then treats historical reconstruction honestly. In the reset example, the two origins remain indistinguishable at every horizon. If both current knowledge an","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"The noisy instrument and a harmless discarded flag. Return to the instrument immediately after a known J intervention. The earlier (s,b) is known, while c is initially uniform on five values. A read-only sensor reports the shifted value Y=reported display-s modulo five. It returns the correct c with probability 3/4 and each wrong value with probabi","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"Knowing the current display is insufficient. A record containing only s=a+b answers the current question exactly. It cannot, in general, predict the display after H. The two complete states x=(0,0,0), y=(1,4,0) both display zero. After H, their displays are zero and four. Any predictor receiving only the common old display must return the same answ","count":10},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"[Uniform controlled stochastic reuse] A single reduced kernel bar K_u reproduces every projected one-step law from every complete state if and only if, whenever r(x)=r(y), ∑_v:r(v)=z'K_u(x,v)=∑_v:r(v)=z'K_u(y,v) for every u,z'. With matched initial projected laws, the reduced model preserves the law of every finite sequence of reduced states and qu","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"[Finite-horizon transfer budget] For matched source and target operation sequences, start from z_0,w_0 with d(Tz_0,w_0)≤ e_0. After n operations, d(Tz_n,w_n) · ≤ L^n e_0+ϵ∑_j=0^n-1L^j,","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-bdc1562155501d06","tex":"z_n","aliases":"z_n","display":"z_n","kind":"use","excerpt":"[Finite-horizon transfer budget] For matched source and target operation sequences, start from z_0,w_0 with d(Tz_0,w_0)≤ e_0. After n operations, d(Tz_n,w_n) · ≤ L^n e_0+ϵ∑_j=0^n-1L^j, |q(z_n)-q'(w_n)| · ≤δ+M(L^n e_0+ϵ∑_j=0^n-1L^j). The empty sum is zero and L^0=1, including when L=0. At L=1 the state bound is e_0+nϵ. Writing e_n=d(Tz_n,w_n), inser","count":4},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"cy, the posterior conditioned on the complete raw and reported record equals the posterior conditioned only on the reported outcomes and experiment names. The claim also holds at a bounded stopping time determined by that reported history. For a fixed realized history, its likelihood under model m is the product of the selected-experiment probabili","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"id Z=z). If (o,z) has positive probability, posterior equality and Bayes' formula give p(m)L_m(o)B(o,z) =Pr(O=o,Z=z)p^o(m) =ρ(omid z)Pr(Z=z)p^z(m) =p(m)Q_m(z)ρ(omid z). Cancel the positive p(m). If the pair has probability zero, full prior support makes L_m(o)B(o,z)=0 for every model, and ρ(omid z)=0. If z is impossible under every model, choose an","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"An accessible sender and a family of recipients. Let M be a finite nonempty set of possible cases. The sender observes only α(m), where α: M→ S is a declared access map. Recipient i∈{1,…,k} receives background β_i(m) and must recover d_i(m). A demand can be a single answer, a reusable state, or the complete finite answer profile for a declared task","count":13},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-6cb4d369799123ce","tex":"\\beta_i","aliases":"β_i beta","display":"\\beta_i","kind":"use","excerpt":"An accessible sender and a family of recipients. Let M be a finite nonempty set of possible cases. The sender observes only α(m), where α: M→ S is a declared access map. Recipient i∈{1,…,k} receives background β_i(m) and must recover d_i(m). A demand can be a single answer, a reusable state, or the complete finite answer profile for a declared task","count":5},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"evidence denominator] Let P and Q be finite joint laws of a model and an observation, with d_TV(P,Q)≤δ. Let E be an observation event with a=P(E)>0 and b=Q(E)>0. Then d_TV(P(Mmid E),Q(Mmid E)) ≤min{1,fracδmax(a,b)}. In particular, a justified lower bound P(E)≥γ>0 supplies the weaker usable bound min{1,δ/γ}. Write u_m=P(M=m,E), v_m=Q(M=m,E), and let","count":6},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"te transfer accumulates error Exact equalities are not always available. For this result let the target state space carry a metric d. Suppose each target update is L-Lipschitz for a common L≥0, and suppose d(Tf_u(z),g_τ(u)(Tz))≤ϵ uniformly on every state and operation that the contract permits. Let a target query be M-Lipschitz with real-valued out","count":10},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Approximate transfer accumulates error. Exact equalities are not always available. For this result let the target state space carry a metric d. Suppose each target update is L-Lipschitz for a common L≥0, and suppose d(Tf_u(z),g_τ(u)(Tz))≤ϵ uniformly on every state and operation that the contract permits. Let a target query be M-Lipschitz with real-","count":11},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-412b907de59c5fc0","tex":"\\eta_i","aliases":"η_i eta","display":"\\eta_i","kind":"use","excerpt":"[A joint posterior and recipient decision certificate] Let p be the reference posterior, widehat p an approximate or transmitted posterior, and r_i recipient i's reconstructed posterior. Suppose d_TV(p,widehat p)≤ϵ, d_TV(widehat p,r_i)≤η_i. If a_i minimizes expected loss under r_i, then, with e_i=min{1,ϵ+η_i}, E_pℓ(M,a_i)-min_a∈ A E_pℓ(M,a) ≤min{L_","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"An exact communication lower bound. Let Θ be a finite family of target models. Relative to the required continuation questions, divide these models into N classes with distinct answer profiles. Use one fixed recipient background and one fixed deterministic recipient. The teacher can select a message but cannot change the recipient's background sepa","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"[The information an exact explanation must carry] If a fixed-length b-bit artifact lets the recipient answer the declared questions exactly for every target in Θ, then 2^b≥ N. With target-dependent background B_θ, the same argument gives 2^b≥max_B N_B, where N_B counts distinct answer profiles among targets sharing that exact background value. Two ","count":7},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"act Bayesian correction through a reporting channel] For fixed finite likelihoods L and reporting channel B, the following are equivalent: [label=(roman*)] item For every prior p, p^o=p^z on every positive-probability pair (o,z). item This equality holds for one prior assigning positive mass to every model. item There exists a row-stochastic kernel","count":10},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b29736466988be5b","tex":"\\rho_e","aliases":"ρ_e rho","display":"\\rho_e","kind":"use","excerpt":"[Sequential preservation of noisy correction under adaptive experiments] Suppose each permitted experiment admits a kernel ρ_e with L_e,m(o)B_e(o,z)=Q_e,m(z)ρ_e(omid z) for all m,o,z. For every finite positive-probability history under the declared policy, the posterior conditioned on the complete raw and reported record equals the posterior condit","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e015129d9aa666b4","tex":"\\rho_{e_t}","aliases":"ρ_e_t rho","display":"\\rho_{e_t}","kind":"use","excerpt":"y. For a fixed realized history, its likelihood under model m is the product of the selected-experiment probabilities and the terms L_e_t,m(o_t)B_e_t(o_t,z_t). The policy factors are independent of m once the reported history is fixed. Applying the assumed factorization gives a product of the terms Q_e_t,m(z_t) multiplied by the model-independent f","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"orrespondences composed as well. No injectivity assumption on T is needed for this forward guarantee. The empty word is immediate. If the identity holds for w, then T f_u f_w=g_τ(u)T f_w =g_τ(u)g_τ(w)T. Induction proves the word identity, and applying the matched decoder proves the answer identity. For a second transport S with operation map σ, sub","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"[A common explanatory artifact under sender and recipient constraints] An exact artifact satisfying~[source reference] with at most K≥1 symbols exists if and only if G_α has no loop and admits a proper coloring with at most K colors. In the loop-free case, the minimum alphabet size is χ(G_α), and the minimum fixed binary length with designable deco","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"ntary value computed solely from r(x) cannot repair a fibre meeting more than one expanded class. Agreement on the larger continuation family implies agreement on the smaller one. Within a fixed fibre of r, an adequate pair (r,s) must separate the k_z new classes, giving the lower bound. For attainment under the additional condition, assign distinc","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"From approximate beliefs to a recipient's decisions. Not every posterior distinction matters equally to an action. To link correction with use, fix one finite nonempty action set A and losses ℓ(m,a)∈[0,L_max]. A receiver evaluates actions from the distribution its actual interpreter reconstructs. The target loss function and available actions must ","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"(r(x))=[x]. In particular, | Z|≥| X/≡_ C|. A fixed-length binary state needs at least lceillog_2| X/≡_ C|rceil bits. The quotient can be computed by finite partition refinement. If x≡_ C y, the empty word gives q(x)=q(y) for every query. Moreover, F_u(x)≡_ C F_u(y) because any continuation after u is a continuation in the original contract. Thus [x","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-f7b37e8032ebe855","tex":"\\sum_{j=0}^{n-1}","aliases":"∑_j=0^n-1 sum","display":"\\sum_{j=0}^{n-1}","kind":"use","excerpt":"[Finite-horizon transfer budget] For matched source and target operation sequences, start from z_0,w_0 with d(Tz_0,w_0)≤ e_0. After n operations, d(Tz_n,w_n) · ≤ L^n e_0+ϵ∑_j=0^n-1L^j, |q(z_n)-q'(w_n)| · ≤δ+M(L^n e_0+ϵ∑_j=0^n-1L^j). The empty sum is zero and L^0=1, including when L=0. At L=1 the state bound is e_0+nϵ. Writing e_n=d(Tz_n,w_n), inser","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-7dd1f8372600c494","tex":"\\sum_m","aliases":"∑_m sum","display":"\\sum_m","kind":"use","excerpt":"(Mmid E),Q(Mmid E)) ≤min{1,fracδmax(a,b)}. In particular, a justified lower bound P(E)≥γ>0 supplies the weaker usable bound min{1,δ/γ}. Write u_m=P(M=m,E), v_m=Q(M=m,E), and let p=u/a, q=v/b. Exchanging P and Q if necessary, assume a≤ b. The variation inside E is at least tfrac12∑_m|u_m-v_m|. The variation outside E is at least |a-b|/2, since its t","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-73169fd8304f15ee","tex":"\\sum_{v:r(v)=z'}","aliases":"∑_v:r(v)=z' sum","display":"\\sum_{v:r(v)=z'}","kind":"use","excerpt":"[Uniform controlled stochastic reuse] A single reduced kernel bar K_u reproduces every projected one-step law from every complete state if and only if, whenever r(x)=r(y), ∑_v:r(v)=z'K_u(x,v)=∑_v:r(v)=z'K_u(y,v) for every u,z'. With matched initial projected laws, the reduced model preserves the law of every finite sequence of reduced states and qu","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"t for reusable understanding Let X and Y be finite nonempty sets. Let U be a finite operation alphabet, possibly empty, and let Q be a finite nonempty family of queries q: X→ Y. Each u∈ U has an implemented total transition F_u: X→ X. The deterministic mathematical contract is C=( X, U,(F_u)_u∈ U, Q, Y). For a word w=u_1⋯ u_n, define F_w=F_u_n∘⋯∘ F","count":3},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[The fixed-recipient covering criterion] Under this finite contract, an exact encoder E:S→ A exists on the reachable sender values if and only if every A_s is nonempty. Its smallest possible image size is min{|C|:C⊆ A, C∩ A_s≠∅ for every reachable s}, when the feasible family is nonempty. Thus a codebook of at most K artifacts is possible exactly w","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-320ca1e802c6d0d1","tex":"\\equiv_{\\C}","aliases":"≡_C ≡_ C equiv C equiv mathcal","display":"\\equiv_{{\\mathcal C}}","kind":"use","excerpt":"[Contract expansion and supplementary information] The relation ≡_ C' refines ≡_ C. Any supplementary code s: X→ S for which (r,s) supports exact reusable behavior under C' must satisfy | S|≥max_z k_z. If every ≡_ C' class lies within a single fibre of r, this bound is attained by an exact reusable pair (r,s). This condition holds when r is the old","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"Approximate transfer accumulates error. Exact equalities are not always available. For this result let the target state space carry a metric d. Suppose each target update is L-Lipschitz for a common L≥0, and suppose d(Tf_u(z),g_τ(u)(Tz))≤ϵ uniformly on every state and operation that the contract permits. Let a target query be M-Lipschitz with real-","count":17},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"[The minimal exact reusable state] The quotient X/≡_ C carries an exact reusable representation for C. Every other exact reusable representation r: X→ Z refines this quotient: if r(x)=r(y), then x≡_ C y. There is a unique map h: Z→ X/≡_ C with h(r(x))=[x]. In particular, | Z|≥| X/≡_ C|. A fixed-length binary state needs at least lceillog_2| X/≡_ C|","count":18},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"Complementary knowledge can reduce communication. For two bits (a,b), let the first recipient know a and need b, while the second knows b and needs a. A sender with access only to a⊕ b can broadcast that one bit. Both recipients recover their missing bit. The sender need not reconstruct the complete pair. An inference from successful explanation to","count":2},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"Define A(x)={a∈ A:every recipient returns d_i(x) within its declared limits}, and, for each reachable sender value s, define A_s=⋂_x:α(x)=s A(x). An artifact selected from A_s must work for all cases the sender still considers possible. Choosing a different acceptable artifact after secretly learning which hidden case is true is not allowed.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-e0c9e168c9ba71d4","tex":"\\bigcap_{x:\\alpha(x)=s}","aliases":"⋂_x:α(x)=s bigcap alpha","display":"\\bigcap_{x:\\alpha(x)=s}","kind":"use","excerpt":"Define A(x)={a∈ A:every recipient returns d_i(x) within its declared limits}, and, for each reachable sender value s, define A_s=⋂_x:α(x)=s A(x). An artifact selected from A_s must work for all cases the sender still considers possible. Choosing a different acceptable artifact after secretly learning which hidden case is true is not allowed.","count":1},{"work":"research-machine-understanding","url":"notation/research-machine-understanding.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"A contract for reusable understanding. Let X and Y be finite nonempty sets. Let U be a finite operation alphabet, possibly empty, and let Q be a finite nonempty family of queries q: X→ Y. Each u∈ U has an implemented total transition F_u: X→ X. The deterministic mathematical contract is C=( X, U,(F_u)_u∈ U, Q, Y). For a word w=u_1⋯ u_n, define F_w=","count":2},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":3},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":3},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"G_o = E[gg^top] + λ I, B = G_c^1/2G_oG_c^1/2.","count":2},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":5},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-0d550ec33c8f5f89","tex":"\\mathbb{E}","aliases":"E mathbb","display":"\\mathbb{E}","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":3},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":6},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"G_o = E[gg^top] + λ I, B = G_c^1/2G_oG_c^1/2.","count":1},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-489eb83cb25f8814","tex":"g^\\top","aliases":"g^top top","display":"g^\\top","kind":"use","excerpt":"G_o = E[gg^top] + λ I, B = G_c^1/2G_oG_c^1/2.","count":1},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-fd6dc28b158b5d9e","tex":"G_c","aliases":"G_c","display":"G_c","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":2},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-323b784be0458ff8","tex":"G_c^{1/2}","aliases":"G_c^1/2","display":"G_c^{1/2}","kind":"use","excerpt":"G_o = E[gg^top] + λ I, B = G_c^1/2G_oG_c^1/2.","count":3},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-26ee716956ef233d","tex":"G_o","aliases":"G_o","display":"G_o","kind":"use","excerpt":"G_o = E[gg^top] + λ I, B = G_c^1/2G_oG_c^1/2.","count":4},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":3},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-7e677e896aa786a7","tex":"h^\\top","aliases":"h^top top","display":"h^\\top","kind":"use","excerpt":"G_c=E[Δ hΔ h^top]+λ I.","count":1},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-858e4bb65a33c6eb","tex":"h_\\ell","aliases":"h_ℓ ell","display":"h_\\ell","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":2},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-e483d419f8d4e722","tex":"h_\\ell^\\top","aliases":"h_ℓ^top ell top","display":"h_\\ell^\\top","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":1},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":3},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"G_o = E[gg^top] + λ I, B = G_c^1/2G_oG_c^1/2.","count":4},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":3},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"At upstream layer u, sealed random residual perturbations are propagated to candidate source layer ℓ. Their source-layer effects estimate","count":1},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":2},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"G_c = E[Δ h_ℓΔ h_ℓ^top] + λ I.","count":3},{"work":"experiments-arithmetic-jspace","url":"notation/experiments-arithmetic-jspace.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"At upstream layer u, sealed random residual perturbations are propagated to candidate source layer ℓ. Their source-layer effects estimate","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":23},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":12},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-881b14d48be22ead","tex":"O_X","aliases":"O_X","display":"O_X","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-e17a3fdd593037e8","tex":"\\rho_X","aliases":"ρ_X rho","display":"\\rho_X","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":11},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":14},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":14},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":66},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"When a linear quotient exists. [Quotient-kernel criterion] Let Q:Z→ S be a surjective linear map. Linear maps A_a:S→ S satisfying QM_a=A_aQ exist for all a if and only if ker Q is invariant under all M_a. They are then unique. If the equations hold and z∈ker Q, then QM_az=A_aQz=0. Conversely, define A_a(Qz)=QM_az. Two lifts of the same vector diffe","count":15},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":36},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-ee1c17b300d5eec7","tex":"a^{i-1}","aliases":"a^i-1","display":"a^{i-1}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-61bf579da1b5c924","tex":"a^j","aliases":"a^j","display":"a^j","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-59e55bca64fe758a","tex":"a^t","aliases":"a^t","display":"a^t","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"When a linear quotient exists. [Quotient-kernel criterion] Let Q:Z→ S be a surjective linear map. Linear maps A_a:S→ S satisfying QM_a=A_aQ exist for all a if and only if ker Q is invariant under all M_a. They are then unique. If the equations hold and z∈ker Q, then QM_az=A_aQz=0. Conversely, define A_a(Qz)=QM_az. Two lifts of the same vector diffe","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-73c1bf580d5c1dbc","tex":"A_g","aliases":"A_g","display":"A_g","kind":"use","excerpt":"Operator transport and compositional semantics. Exact transport. Let A_g:X→ X, B_g:Y→ Y, and T:X→ Y be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is B_gT=TA_g. When this equation holds, translating a state and applying the target operator agrees with applying the source operator and then trans","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-757f361b88f183a1","tex":"A_{g_1}","aliases":"A_g_1","display":"A_{g_1}","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a3890fdc6bc6bf01","tex":"A_{g_i}","aliases":"A_g_i","display":"A_{g_i}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a08728bb890c722a","tex":"A_{g_{i-1}}","aliases":"A_g_i-1","display":"A_{g_{i-1}}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-69416ef1cdacea48","tex":"A_{g_k}","aliases":"A_g_k","display":"A_{g_k}","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-9e0e75c509aef636","tex":"\\mathrm{access}","aliases":"access mathrm","display":"\\mathrm{access}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-4e2b11a7001e1904","tex":"\\arg","aliases":"arg","display":"\\arg","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":16},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8343f449c6123e7f","tex":"B^{-\\mathsf T}","aliases":"B^- T mathsf","display":"B^{-\\mathsf T}","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-20948689226e0962","tex":"b^{k-i}","aliases":"b^k-i","display":"b^{k-i}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a0f42eb1ac146472","tex":"B_g","aliases":"B_g","display":"B_g","kind":"use","excerpt":"Operator transport and compositional semantics. Exact transport. Let A_g:X→ X, B_g:Y→ Y, and T:X→ Y be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is B_gT=TA_g. When this equation holds, translating a state and applying the target operator agrees with applying the source operator and then trans","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-9cdee93f0dbbc22b","tex":"B_{g_i}","aliases":"B_g_i","display":"B_{g_i}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-eea952c6fc375053","tex":"B_{g_{i+1}}","aliases":"B_g_i+1","display":"B_{g_{i+1}}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-5700cccb5947c0c3","tex":"B_{g_k}","aliases":"B_g_k","display":"B_{g_k}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-5902c64cbd2a059e","tex":"B_w","aliases":"B_w","display":"B_w","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-52555ee0fe55f126","tex":"\\bar F","aliases":"bar F bar","display":"\\bar F","kind":"use","excerpt":"Let the declared machine be M=(S, A,F,O). The certificate contains a surjection q:S→ Q, quotient transitions bar F_a, and outputs bar O, subject to q(F_a(s))=bar F_a(q(s)), O(s)=bar O(q(s)) (s∈ S, a∈ A). The certificate generator uses partition refinement. The checker instead verifies the displayed equations over every declared state and action. It","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-0ddd6936013fb39a","tex":"\\bar F_a","aliases":"bar F_a bar","display":"\\bar F_a","kind":"use","excerpt":"Let the declared machine be M=(S, A,F,O). The certificate contains a surjection q:S→ Q, quotient transitions bar F_a, and outputs bar O, subject to q(F_a(s))=bar F_a(q(s)), O(s)=bar O(q(s)) (s∈ S, a∈ A). The certificate generator uses partition refinement. The checker instead verifies the displayed equations over every declared state and action. It","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-bf623b625d280efe","tex":"\\bar F_w","aliases":"bar F_w bar","display":"\\bar F_w","kind":"use","excerpt":"ntheorem[Finite semantic checkpoint preservation] If [source reference] holds, then a checkpoint storing q(s) reproduces O(F_w(s)) for every finite word w over the declared alphabet, when continued with the quotient transitions. No information inside a fiber of q is needed for those outputs. Induction on |w| gives q(F_w(s))=bar F_w(q(s)). Apply the","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-fbcb33a013f25925","tex":"\\bar O","aliases":"bar O bar","display":"\\bar O","kind":"use","excerpt":"subsectionA task-relative realization Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that sem","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":6},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":6},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-dd9910701e9036c0","tex":"\\mathrm{content}","aliases":"content mathrm","display":"\\mathrm{content}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"[Semantic continuation] If [source reference] holds on a forward-invariant domain D⊆ Z, then O(F_w(z))=bar O(G_w(q(z))) for every z∈ D and every finite w. Forward invariance keeps every intermediate state inside the domain where the equations hold. Induction gives q(F_w(z))=G_w(q(z)). Apply the observation factorization to the endpoint. Without for","count":9},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, successive pairs meet the bound but the first and last do not. Quotient terminology therefore requires an actual equivalence relation, a specified quantization, or a separate approximate-abstraction formalism. Pairwise tolerance alone is ins","count":6},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag","display":null,"kind":"use","excerpt":"Curvature can be invisible to the task. Conversely, exact semantic behavior does not require the entire ambient space to be path-flat. Consider a block-diagonal action U_g=diag(P_g,V_g), observed only through the first block. The P_g may obey all semantic relations while the V_g retain path-dependent history. The task quotient is exact even though ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-d9b6193b1833f637","tex":"E_a","aliases":"E_a","display":"E_a","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-2775a21153e9ebc4","tex":"e_t","aliases":"e_t","display":"e_t","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-0c0ec2f484593060","tex":"e_{t+1}","aliases":"e_t+1","display":"e_{t+1}","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-1f3da6ad15edf007","tex":"\\mathrm{evidence}","aliases":"evidence mathrm","display":"\\mathrm{evidence}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":17},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8c4caf5c79a35c9c","tex":"f^2","aliases":"f^2","display":"f^2","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":6},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-24ceed0c3b629a6f","tex":"F_{a_1}","aliases":"F_a_1","display":"F_{a_1}","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-d6d2fa627b1c221a","tex":"F_{a_k}","aliases":"F_a_k","display":"F_{a_k}","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-f0160f5b8a57192c","tex":"F_M","aliases":"F_M","display":"F_M","kind":"use","excerpt":"Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":8},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-0de75f759f97e080","tex":"F_\\theta","aliases":"F_θ theta","display":"F_\\theta","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":11},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-1, metric-relative identities can be compared consistently. These elementary distinction","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-7a952bafce193d41","tex":"g_{i+1}","aliases":"g_i+1","display":"g_{i+1}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-98d7ab59401fb7a2","tex":"g_{i-1}","aliases":"g_i-1","display":"g_{i-1}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-c1c366f26da43334","tex":"g_k","aliases":"g_k","display":"g_k","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-5b978f7139fcb802","tex":"G_W","aliases":"G_W","display":"G_W","kind":"use","excerpt":"Bounded discovery and semantic admission. The Discovery Bridge supplies a missing distinction between a well-formed proposal and an identified operator. It accepts a complete finite world W, a primitive vocabulary G_W, a depth bound d, demonstrations D, and explicitly claimed laws. All of these belong to the certificate's scope. [source reference]","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-6d584c042dd0c359","tex":"G_w","aliases":"G_w","display":"G_w","kind":"use","excerpt":"[Semantic continuation] If [source reference] holds on a forward-invariant domain D⊆ Z, then O(F_w(z))=bar O(G_w(q(z))) for every z∈ D and every finite w. Forward invariance keeps every intermediate state inside the domain where the equations hold. Induction gives q(F_w(z))=G_w(q(z)). Apply the observation factorization to the endpoint. Without for","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a3668e53a6fbde8b","tex":"G_W^j","aliases":"G_W^j","display":"G_W^j","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":24},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":9},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-0f6f9e87439c63ef","tex":"\\mathrm{id}","aliases":"id mathrm","display":"\\mathrm{id}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":11},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":11},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-3ff8bbec26df05be","tex":"K^\\perp","aliases":"K^⊥ perp","display":"K^\\perp","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":8},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-34c05dc8f5073293","tex":"L'","aliases":"L'","display":"L'","kind":"use","excerpt":"A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates a source statement, operational state, and audit ledger to their successors. A statement that rejects still has an outcome and, within th","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":21},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-c887199df3da3b63","tex":"M_a","aliases":"M_a","display":"M_a","kind":"use","excerpt":"imensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, and invisibility after one action followed by every suffix of length at most H. This is the displayed recurrence. Equality of adjacent spaces gives ","count":6},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-151ed972dadaf0c5","tex":"M_a^{-1}","aliases":"M_a^-1","display":"M_a^{-1}","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a201c1f6b956e50e","tex":"M_{a_t}","aliases":"M_a_t","display":"M_{a_t}","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-cda098d91a6d8a59","tex":"M_g","aliases":"M_g","display":"M_g","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-f54709b1dbd93fcb","tex":"M_g^{\\mathsf T}","aliases":"M_g^ T mathsf","display":"M_g^{\\mathsf T}","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-e3e697d6b6f9bc2b","tex":"M_w","aliases":"M_w","display":"M_w","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-96732b6e93afdef8","tex":"M_\\varnothing","aliases":"M_∅ varnothing","display":"M_\\varnothing","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-9cc66676daa83a18","tex":"\\max_j","aliases":"max_j max","display":"\\max_j","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-21587643d68708f2","tex":"\\max_{j\\ne i}","aliases":"max_j≠ i max ne","display":"\\max_{j\\ne i}","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":8},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-75b832abbbfa466e","tex":"N_H","aliases":"N_H","display":"N_H","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-708e79b9bce0fd2c","tex":"N_{H+1}","aliases":"N_H+1","display":"N_{H+1}","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-aa2c11a05da3d1fe","tex":"N_\\infty","aliases":"N_∞ infty","display":"N_\\infty","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact","count":13},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-03c58bd9c57b005f","tex":"O_h","aliases":"O_h","display":"O_h","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-c7f7b360dbef58cb","tex":"O_\\theta","aliases":"O_θ theta","display":"O_\\theta","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-17fac17b7a6c279b","tex":"\\mathrm{origin}","aliases":"origin mathrm","display":"\\mathrm{origin}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":12},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","excerpt":"Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-1, metric-relative identities can be compared consistently. These elementary distinction","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-1, metric-relative identities can be compared consistently. These elementary distinction","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-45c4f17ed76d05ef","tex":"P_g","aliases":"P_g","display":"P_g","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-176e7aa636add15d","tex":"P_g^{\\mathsf T}","aliases":"P_g^ T mathsf","display":"P_g^{\\mathsf T}","kind":"use","excerpt":"roposition[Exact finite realization] The preceding construction implements the supplied permutation action on every anchor and preserves the G-norm. In the canonical semantic chart B=I, it is simply the permutation executor M_g=P_g. The anchor equation follows by cancellation of B^-1B. For the metric identity, substitute the definitions and use P_g","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-af4f1e08915ee052","tex":"P_w","aliases":"P_w","display":"P_w","kind":"use","excerpt":"ing construction implements the supplied permutation action on every anchor and preserves the G-norm. In the canonical semantic chart B=I, it is simply the permutation executor M_g=P_g. The anchor equation follows by cancellation of B^-1B. For the metric identity, substitute the definitions and use P_g^ TP_g=I. Products telescope to BP_wB^-1, so se","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a3dc0fd16baf5130","tex":"\\mathcal P_{W,d}","aliases":"P_W,d mathcal","display":"\\mathcal P_{W,d}","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":12},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":11},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-2c2a4ce653267792","tex":"Q_{L+1,H}","aliases":"Q_L+1,H","display":"Q_{L+1,H}","kind":"use","excerpt":"[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a follow","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-0b1a4bea7fb01edc","tex":"Q_{L,H}","aliases":"Q_L,H","display":"Q_{L,H}","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-79dd9bd275bab498","tex":"Q_{L,H+1}","aliases":"Q_L,H+1","display":"Q_{L,H+1}","kind":"use","excerpt":"[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a follow","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8477f8563980f26b","tex":"r^4","aliases":"r^4","display":"r^4","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-4b4d59d2797a34fc","tex":"R_\\varepsilon","aliases":"R_ε varepsilon","display":"R_\\varepsilon","kind":"use","excerpt":"An exact norm-preserving failure. [Small phase error accumulates] Let the intended semantic operation be the identity and the learned implementation be a planar rotation R_ε. Every step preserves Euclidean norm exactly. After k steps the state has rotated by kε. With a readout determined by the sign of the first coordinate, a state starting at (1,0","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-1d8d9772e61c1b47","tex":"R_{\\le L}","aliases":"R_≤ L le","display":"R_{\\le L}","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-e2180f7da880bd51","tex":"R_{\\le L+1}","aliases":"R_≤ L+1 le","display":"R_{\\le L+1}","kind":"use","excerpt":"dproposition The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a followed by w has length at most H+1. Hence O(F_wF_a(z))=O(F_wF_a(z')), so F_a(z)∼_HF_a(z'). Both successor states are in R_≤ L+1. T","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":14},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates a source statement, operational state, and audit ledger to their successors. A statement that rejects still has an outcome and, within th","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-50b3713c204305be","tex":"S_W","aliases":"S_W","display":"S_W","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8ca333022971594c","tex":"\\mathrm{scope}","aliases":"scope mathrm","display":"\\mathrm{scope}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-02a3d6cf5b723f6c","tex":"\\mathrm{status}","aliases":"status mathrm","display":"\\mathrm{status}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Operator transport and compositional semantics. Exact transport. Let A_g:X→ X, B_g:Y→ Y, and T:X→ Y be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is B_gT=TA_g. When this equation holds, translating a state and applying the target operator agrees with applying the source operator and then trans","count":17},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":6},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":9},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-818909e7a18cc6f5","tex":"T^{-\\mathsf T}","aliases":"T^- T mathsf","display":"T^{-\\mathsf T}","kind":"use","excerpt":"Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-1, metric-relative identities can be compared consistently. These elementary distinction","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"use","excerpt":"word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T is invertible, the target operators are conjugates B_g=TA_gT^-1. When it is not, the equation may ","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":10},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":9},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a257f9d1ed0e54ad","tex":"u^{\\mathsf T}","aliases":"u^ T mathsf","display":"u^{\\mathsf T}","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"belthm:isometry Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Henc","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-9f5988962b4f6877","tex":"U_g","aliases":"U_g","display":"U_g","kind":"use","excerpt":"Curvature can be invisible to the task. Conversely, exact semantic behavior does not require the entire ambient space to be path-flat. Consider a block-diagonal action U_g=diag(P_g,V_g), observed only through the first block. The P_g may obey all semantic relations while the V_g retain path-dependent history. The task quotient is exact even though ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a6ab3f6998c3dac0","tex":"u_t","aliases":"u_t","display":"u_t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":8},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a8b2a9721419c0b6","tex":"v^{\\mathsf T}","aliases":"v^ T mathsf","display":"v^{\\mathsf T}","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-9151633891e8e3e1","tex":"V_g","aliases":"V_g","display":"V_g","kind":"use","excerpt":"Curvature can be invisible to the task. Conversely, exact semantic behavior does not require the entire ambient space to be path-flat. Consider a block-diagonal action U_g=diag(P_g,V_g), observed only through the first block. The P_g may obey all semantic relations while the V_g retain path-dependent history. The task quotient is exact even though ","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-d42e4596a0b89189","tex":"\\vdash","aliases":"vdash","display":null,"kind":"use","excerpt":"A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates a source statement, operational state, and audit ledger to their successors. A statement that rejects still has an outcome and, within th","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Bounded discovery and semantic admission. The Discovery Bridge supplies a missing distinction between a well-formed proposal and an identified operator. It accepts a complete finite world W, a primitive vocabulary G_W, a depth bound d, demonstrations D, and explicitly claimed laws. All of these belong to the certificate's scope. [source reference]","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":23},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"A minimal delayed-observability counterexample. Let the concrete state be (x,y)∈{0,1}^2, with present observation O(x,y)=x, and let the only operation swap the coordinates. At horizon zero, there are two classes, classified by x. At horizon one, there are four: the next observation also reveals y.","count":9},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-1ecb58e66eedd43b","tex":"y_t","aliases":"y_t","display":"y_t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":6},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":26},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, successive pairs meet the bound but the first and last do not. Quotient terminology therefore requires an actual equivalence relation, a specified quantization, or a separate approximate-abstraction formalism. Pairwise tolerance alone is ins","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8e5d0000abb9cf07","tex":"z_{t+1}","aliases":"z_t+1","display":"z_{t+1}","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":7},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"Observable rank after recurrence. The learned-checkpoint probe uses all twelve v0.4 checkpoints and 32 primary contexts. It finds rank seven at present observation and rank twelve after one through four continuation steps. Repeat-16 hard-polar log-radial gain remains below 2.90×10^-5 in absolute value, while p95 oriented loop angles reach roughly .","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, successive pairs meet the bound but the first and last do not. Quotient terminology therefore requires an actual equivalence relation, a specified quantization, or a separate approximate-abstraction formalism. Pairwise tolerance alone is ins","count":13},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-f3cf8a4e3d9677f6","tex":"\\varepsilon_i","aliases":"ε_i varepsilon","display":"\\varepsilon_i","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"rror accumulates] Let the intended semantic operation be the identity and the learned implementation be a planar rotation R_ε. Every step preserves Euclidean norm exactly. After k steps the state has rotated by kε. With a readout determined by the sign of the first coordinate, a state starting at (1,0) eventually changes class for any fixed 0<ε<π/2","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-18829b323610a69f","tex":"\\sigma_W","aliases":"σ_W sigma","display":"\\sigma_W","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates a source statement, operational state, and audit ledger to their successors. A statement that rejects still has an outcome and, within th","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-8943b22d90e5e02b","tex":"\\ell_j","aliases":"ℓ_j ell","display":"\\ell_j","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"From latent computation to an explicit language. The intended interface. The working architecture separates three timescales: fast latent computation ↔ typed semantic checkpoint ↔ persistent evidence ledger. A latent state can be provisional and disposable. The checkpoint preserves only distinctions justified for a declared future use. The ledger p","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-09ade2b60fb5add0","tex":"\\sum_{i=1}^{k}","aliases":"∑_i=1^k sum","display":"\\sum_{i=1}^{k}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-df66290cc89f818f","tex":"\\sum_{j=0}^{t-1}","aliases":"∑_j=0^t-1 sum","display":"\\sum_{j=0}^{t-1}","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":6},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-b92ecce9a562b5d7","tex":"\\sim_H","aliases":"∼_H sim","display":"\\sim_H","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-2c5d47bf0e1e95ba","tex":"\\sim_{H+1}","aliases":"∼_H+1 sim","display":"\\sim_{H+1}","kind":"use","excerpt":"nproposition[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"Norm ratio at 2048 · 1.0000 · 1.0000 · ≈0 · ≈0","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact","count":3},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, successive pairs meet the bound but the first and last do not. Quotient terminology therefore requires an actual equivalence relation, a specified quantization, or a separate approximate-abstraction formalism. Pairwise tolerance alone is ins","count":17},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"[Semantic continuation] If [source reference] holds on a forward-invariant domain D⊆ Z, then O(F_w(z))=bar O(G_w(q(z))) for every z∈ D and every finite w. Forward invariance keeps every intermediate state inside the domain where the equations hold. Induction gives q(F_w(z))=G_w(q(z)). Apply the observation factorization to the endpoint. Without for","count":8},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-529f8d1978a8a0ae","tex":"\\bigcap_{a\\in\\mathcal A}","aliases":"⋂_a∈ A bigcap in mathcal","display":"\\bigcap_{a\\in\\mathcal A}","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-ee563a8aa1670e17","tex":"\\bigcap_{|w|\\le H}","aliases":"⋂_|w|≤ H bigcap le","display":"\\bigcap_{|w|\\le H}","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-6eb256c5206a08bc","tex":"\\bigcup_{j=0}^{d}","aliases":"⋃_j=0^d bigcup","display":"\\bigcup_{j=0}^{d}","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":5},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality.","count":2},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a follow","count":4},{"work":"research-osl-neuralese","url":"notation/research-osl-neuralese.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a follow","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":23},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":12},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-881b14d48be22ead","tex":"O_X","aliases":"O_X","display":"O_X","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-e17a3fdd593037e8","tex":"\\rho_X","aliases":"ρ_X rho","display":"\\rho_X","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":11},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Realization] A realization of an operation vocabulary G consists of a state space X, maps ρ_X(g):X→ X for g∈ G, and a declared observation map O_X:X→ Y. For typed operations, each generator instead has an explicit source and target carrier, and only type-compatible words may be composed. The observation map is part of the claim. Preserving a categ","count":14},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":14},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":66},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"When a linear quotient exists. [Quotient-kernel criterion] Let Q:Z→ S be a surjective linear map. Linear maps A_a:S→ S satisfying QM_a=A_aQ exist for all a if and only if ker Q is invariant under all M_a. They are then unique. If the equations hold and z∈ker Q, then QM_az=A_aQz=0. Conversely, define A_a(Qz)=QM_az. Two lifts of the same vector diffe","count":15},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":36},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-ee1c17b300d5eec7","tex":"a^{i-1}","aliases":"a^i-1","display":"a^{i-1}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-61bf579da1b5c924","tex":"a^j","aliases":"a^j","display":"a^j","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-59e55bca64fe758a","tex":"a^t","aliases":"a^t","display":"a^t","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"When a linear quotient exists. [Quotient-kernel criterion] Let Q:Z→ S be a surjective linear map. Linear maps A_a:S→ S satisfying QM_a=A_aQ exist for all a if and only if ker Q is invariant under all M_a. They are then unique. If the equations hold and z∈ker Q, then QM_az=A_aQz=0. Conversely, define A_a(Qz)=QM_az. Two lifts of the same vector diffe","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-73c1bf580d5c1dbc","tex":"A_g","aliases":"A_g","display":"A_g","kind":"use","excerpt":"Operator transport and compositional semantics. Exact transport. Let A_g:X→ X, B_g:Y→ Y, and T:X→ Y be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is B_gT=TA_g. When this equation holds, translating a state and applying the target operator agrees with applying the source operator and then trans","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-757f361b88f183a1","tex":"A_{g_1}","aliases":"A_g_1","display":"A_{g_1}","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a3890fdc6bc6bf01","tex":"A_{g_i}","aliases":"A_g_i","display":"A_{g_i}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a08728bb890c722a","tex":"A_{g_{i-1}}","aliases":"A_g_i-1","display":"A_{g_{i-1}}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-69416ef1cdacea48","tex":"A_{g_k}","aliases":"A_g_k","display":"A_{g_k}","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-9e0e75c509aef636","tex":"\\mathrm{access}","aliases":"access mathrm","display":"\\mathrm{access}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-4e2b11a7001e1904","tex":"\\arg","aliases":"arg","display":"\\arg","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":16},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8343f449c6123e7f","tex":"B^{-\\mathsf T}","aliases":"B^- T mathsf","display":"B^{-\\mathsf T}","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-20948689226e0962","tex":"b^{k-i}","aliases":"b^k-i","display":"b^{k-i}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a0f42eb1ac146472","tex":"B_g","aliases":"B_g","display":"B_g","kind":"use","excerpt":"Operator transport and compositional semantics. Exact transport. Let A_g:X→ X, B_g:Y→ Y, and T:X→ Y be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is B_gT=TA_g. When this equation holds, translating a state and applying the target operator agrees with applying the source operator and then trans","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-9cdee93f0dbbc22b","tex":"B_{g_i}","aliases":"B_g_i","display":"B_{g_i}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-eea952c6fc375053","tex":"B_{g_{i+1}}","aliases":"B_g_i+1","display":"B_{g_{i+1}}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-5700cccb5947c0c3","tex":"B_{g_k}","aliases":"B_g_k","display":"B_{g_k}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-5902c64cbd2a059e","tex":"B_w","aliases":"B_w","display":"B_w","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-52555ee0fe55f126","tex":"\\bar F","aliases":"bar F bar","display":"\\bar F","kind":"use","excerpt":"Let the declared machine be M=(S, A,F,O). The certificate contains a surjection q:S→ Q, quotient transitions bar F_a, and outputs bar O, subject to q(F_a(s))=bar F_a(q(s)), O(s)=bar O(q(s)) (s∈ S, a∈ A). The certificate generator uses partition refinement. The checker instead verifies the displayed equations over every declared state and action. It","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-0ddd6936013fb39a","tex":"\\bar F_a","aliases":"bar F_a bar","display":"\\bar F_a","kind":"use","excerpt":"Let the declared machine be M=(S, A,F,O). The certificate contains a surjection q:S→ Q, quotient transitions bar F_a, and outputs bar O, subject to q(F_a(s))=bar F_a(q(s)), O(s)=bar O(q(s)) (s∈ S, a∈ A). The certificate generator uses partition refinement. The checker instead verifies the displayed equations over every declared state and action. It","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-bf623b625d280efe","tex":"\\bar F_w","aliases":"bar F_w bar","display":"\\bar F_w","kind":"use","excerpt":"ntheorem[Finite semantic checkpoint preservation] If [source reference] holds, then a checkpoint storing q(s) reproduces O(F_w(s)) for every finite word w over the declared alphabet, when continued with the quotient transitions. No information inside a fiber of q is needed for those outputs. Induction on |w| gives q(F_w(s))=bar F_w(q(s)). Apply the","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-fbcb33a013f25925","tex":"\\bar O","aliases":"bar O bar","display":"\\bar O","kind":"use","excerpt":"subsectionA task-relative realization Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=bar O∘ q. These conditions say that sem","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":6},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":6},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-dd9910701e9036c0","tex":"\\mathrm{content}","aliases":"content mathrm","display":"\\mathrm{content}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"[Semantic continuation] If [source reference] holds on a forward-invariant domain D⊆ Z, then O(F_w(z))=bar O(G_w(q(z))) for every z∈ D and every finite w. Forward invariance keeps every intermediate state inside the domain where the equations hold. Induction gives q(F_w(z))=G_w(q(z)). Apply the observation factorization to the endpoint. Without for","count":9},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, successive pairs meet the bound but the first and last do not. Quotient terminology therefore requires an actual equivalence relation, a specified quantization, or a separate approximate-abstraction formalism. Pairwise tolerance alone is ins","count":6},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag","display":null,"kind":"use","excerpt":"Curvature can be invisible to the task. Conversely, exact semantic behavior does not require the entire ambient space to be path-flat. Consider a block-diagonal action U_g=diag(P_g,V_g), observed only through the first block. The P_g may obey all semantic relations while the V_g retain path-dependent history. The task quotient is exact even though ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-d9b6193b1833f637","tex":"E_a","aliases":"E_a","display":"E_a","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-2775a21153e9ebc4","tex":"e_t","aliases":"e_t","display":"e_t","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-0c0ec2f484593060","tex":"e_{t+1}","aliases":"e_t+1","display":"e_{t+1}","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-1f3da6ad15edf007","tex":"\\mathrm{evidence}","aliases":"evidence mathrm","display":"\\mathrm{evidence}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":17},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8c4caf5c79a35c9c","tex":"f^2","aliases":"f^2","display":"f^2","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":6},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-24ceed0c3b629a6f","tex":"F_{a_1}","aliases":"F_a_1","display":"F_{a_1}","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-d6d2fa627b1c221a","tex":"F_{a_k}","aliases":"F_a_k","display":"F_{a_k}","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-f0160f5b8a57192c","tex":"F_M","aliases":"F_M","display":"F_M","kind":"use","excerpt":"Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":8},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-0de75f759f97e080","tex":"F_\\theta","aliases":"F_θ theta","display":"F_\\theta","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":11},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-1, metric-relative identities can be compared consistently. These elementary distinction","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-7a952bafce193d41","tex":"g_{i+1}","aliases":"g_i+1","display":"g_{i+1}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-98d7ab59401fb7a2","tex":"g_{i-1}","aliases":"g_i-1","display":"g_{i-1}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-c1c366f26da43334","tex":"g_k","aliases":"g_k","display":"g_k","kind":"use","excerpt":"[Compositional transport] If [source reference] holds for every generator, then for every finite word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T i","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-5b978f7139fcb802","tex":"G_W","aliases":"G_W","display":"G_W","kind":"use","excerpt":"Bounded discovery and semantic admission. The Discovery Bridge supplies a missing distinction between a well-formed proposal and an identified operator. It accepts a complete finite world W, a primitive vocabulary G_W, a depth bound d, demonstrations D, and explicitly claimed laws. All of these belong to the certificate's scope. [source reference]","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-6d584c042dd0c359","tex":"G_w","aliases":"G_w","display":"G_w","kind":"use","excerpt":"[Semantic continuation] If [source reference] holds on a forward-invariant domain D⊆ Z, then O(F_w(z))=bar O(G_w(q(z))) for every z∈ D and every finite w. Forward invariance keeps every intermediate state inside the domain where the equations hold. Induction gives q(F_w(z))=G_w(q(z)). Apply the observation factorization to the endpoint. Without for","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a3668e53a6fbde8b","tex":"G_W^j","aliases":"G_W^j","display":"G_W^j","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":24},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":9},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-0f6f9e87439c63ef","tex":"\\mathrm{id}","aliases":"id mathrm","display":"\\mathrm{id}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":11},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":11},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-3ff8bbec26df05be","tex":"K^\\perp","aliases":"K^⊥ perp","display":"K^\\perp","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":8},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-34c05dc8f5073293","tex":"L'","aliases":"L'","display":"L'","kind":"use","excerpt":"A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates a source statement, operational state, and audit ledger to their successors. A statement that rejects still has an outcome and, within th","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":21},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-c887199df3da3b63","tex":"M_a","aliases":"M_a","display":"M_a","kind":"use","excerpt":"imensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, and invisibility after one action followed by every suffix of length at most H. This is the displayed recurrence. Equality of adjacent spaces gives ","count":6},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-151ed972dadaf0c5","tex":"M_a^{-1}","aliases":"M_a^-1","display":"M_a^{-1}","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a201c1f6b956e50e","tex":"M_{a_t}","aliases":"M_a_t","display":"M_{a_t}","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-cda098d91a6d8a59","tex":"M_g","aliases":"M_g","display":"M_g","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-f54709b1dbd93fcb","tex":"M_g^{\\mathsf T}","aliases":"M_g^ T mathsf","display":"M_g^{\\mathsf T}","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-e3e697d6b6f9bc2b","tex":"M_w","aliases":"M_w","display":"M_w","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-96732b6e93afdef8","tex":"M_\\varnothing","aliases":"M_∅ varnothing","display":"M_\\varnothing","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-9cc66676daa83a18","tex":"\\max_j","aliases":"max_j max","display":"\\max_j","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-21587643d68708f2","tex":"\\max_{j\\ne i}","aliases":"max_j≠ i max ne","display":"\\max_{j\\ne i}","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":8},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-75b832abbbfa466e","tex":"N_H","aliases":"N_H","display":"N_H","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-708e79b9bce0fd2c","tex":"N_{H+1}","aliases":"N_H+1","display":"N_{H+1}","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-aa2c11a05da3d1fe","tex":"N_\\infty","aliases":"N_∞ infty","display":"N_\\infty","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact","count":13},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-03c58bd9c57b005f","tex":"O_h","aliases":"O_h","display":"O_h","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-c7f7b360dbef58cb","tex":"O_\\theta","aliases":"O_θ theta","display":"O_\\theta","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-17fac17b7a6c279b","tex":"\\mathrm{origin}","aliases":"origin mathrm","display":"\\mathrm{origin}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":12},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","excerpt":"Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-1, metric-relative identities can be compared consistently. These elementary distinction","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-1, metric-relative identities can be compared consistently. These elementary distinction","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-45c4f17ed76d05ef","tex":"P_g","aliases":"P_g","display":"P_g","kind":"use","excerpt":"Exact compiled controls and learned operators. Anchor transport. Suppose B is an invertible matrix whose columns are latent representatives of a finite semantic basis, and P_g is a supplied permutation action. Define M_g=BP_gB^-1, G=B^- TB^-1. Then M_gB=BP_g and M_g^ TGM_g=G. For a permutation representation, every group relation is preserved exact","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-176e7aa636add15d","tex":"P_g^{\\mathsf T}","aliases":"P_g^ T mathsf","display":"P_g^{\\mathsf T}","kind":"use","excerpt":"roposition[Exact finite realization] The preceding construction implements the supplied permutation action on every anchor and preserves the G-norm. In the canonical semantic chart B=I, it is simply the permutation executor M_g=P_g. The anchor equation follows by cancellation of B^-1B. For the metric identity, substitute the definitions and use P_g","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-af4f1e08915ee052","tex":"P_w","aliases":"P_w","display":"P_w","kind":"use","excerpt":"ing construction implements the supplied permutation action on every anchor and preserves the G-norm. In the canonical semantic chart B=I, it is simply the permutation executor M_g=P_g. The anchor equation follows by cancellation of B^-1B. For the metric identity, substitute the definitions and use P_g^ TP_g=I. Products telescope to BP_wB^-1, so se","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a3dc0fd16baf5130","tex":"\\mathcal P_{W,d}","aliases":"P_W,d mathcal","display":"\\mathcal P_{W,d}","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":12},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":11},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-2c2a4ce653267792","tex":"Q_{L+1,H}","aliases":"Q_L+1,H","display":"Q_{L+1,H}","kind":"use","excerpt":"[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a follow","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-0b1a4bea7fb01edc","tex":"Q_{L,H}","aliases":"Q_L,H","display":"Q_{L,H}","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-79dd9bd275bab498","tex":"Q_{L,H+1}","aliases":"Q_L,H+1","display":"Q_{L,H+1}","kind":"use","excerpt":"[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a follow","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8477f8563980f26b","tex":"r^4","aliases":"r^4","display":"r^4","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-4b4d59d2797a34fc","tex":"R_\\varepsilon","aliases":"R_ε varepsilon","display":"R_\\varepsilon","kind":"use","excerpt":"An exact norm-preserving failure. [Small phase error accumulates] Let the intended semantic operation be the identity and the learned implementation be a planar rotation R_ε. Every step preserves Euclidean norm exactly. After k steps the state has rotated by kε. With a readout determined by the sign of the first coordinate, a state starting at (1,0","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-1d8d9772e61c1b47","tex":"R_{\\le L}","aliases":"R_≤ L le","display":"R_{\\le L}","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-e2180f7da880bd51","tex":"R_{\\le L+1}","aliases":"R_≤ L+1 le","display":"R_{\\le L+1}","kind":"use","excerpt":"dproposition The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a followed by w has length at most H+1. Hence O(F_wF_a(z))=O(F_wF_a(z')), so F_a(z)∼_HF_a(z'). Both successor states are in R_≤ L+1. T","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":14},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates a source statement, operational state, and audit ledger to their successors. A statement that rejects still has an outcome and, within th","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-50b3713c204305be","tex":"S_W","aliases":"S_W","display":"S_W","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8ca333022971594c","tex":"\\mathrm{scope}","aliases":"scope mathrm","display":"\\mathrm{scope}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-02a3d6cf5b723f6c","tex":"\\mathrm{status}","aliases":"status mathrm","display":"\\mathrm{status}","kind":"use","excerpt":"A useful abstract operational packet is p=(id,content,origin,access,status,scope,evidence). The implementation has a more detailed schema. The content may describe a vector or an operation, but it may also be an ordinary proposition. The provenance records how the content entered the system. An access relation states which agents may inspect it. En","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Operator transport and compositional semantics. Exact transport. Let A_g:X→ X, B_g:Y→ Y, and T:X→ Y be linear maps. We use column vectors and execute the rightmost map first. The compatibility equation is B_gT=TA_g. When this equation holds, translating a state and applying the target operator agrees with applying the source operator and then trans","count":17},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":6},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":9},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-818909e7a18cc6f5","tex":"T^{-\\mathsf T}","aliases":"T^- T mathsf","display":"T^{-\\mathsf T}","kind":"use","excerpt":"Projectors and changes of chart. For an invertible T, define P'=TPT^-1. If P^2=P, then (P')^2=TP^2T^-1=P'. Idempotence is preserved by similarity. Orthogonality with respect to the Euclidean inner product need not be. If a metric G is transported to G'=T^- TGT^-1, metric-relative identities can be compared consistently. These elementary distinction","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"use","excerpt":"word w=g_1⋯ g_k, with A_w=A_g_k⋯ A_g_1 and analogously for B_w, one has B_wT=TA_w. The empty word gives T=T. If the identity holds for w, then B_gB_wT=B_gTA_w=TA_gA_w. Induction on word length proves the result. No invertibility of T is needed. When T is invertible, the target operators are conjugates B_g=TA_gT^-1. When it is not, the equation may ","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":10},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":9},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a257f9d1ed0e54ad","tex":"u^{\\mathsf T}","aliases":"u^ T mathsf","display":"u^{\\mathsf T}","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"belthm:isometry Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Henc","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-9f5988962b4f6877","tex":"U_g","aliases":"U_g","display":"U_g","kind":"use","excerpt":"Curvature can be invisible to the task. Conversely, exact semantic behavior does not require the entire ambient space to be path-flat. Consider a block-diagonal action U_g=diag(P_g,V_g), observed only through the first block. The P_g may obey all semantic relations while the V_g retain path-dependent history. The task quotient is exact even though ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a6ab3f6998c3dac0","tex":"u_t","aliases":"u_t","display":"u_t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":8},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a8b2a9721419c0b6","tex":"v^{\\mathsf T}","aliases":"v^ T mathsf","display":"v^{\\mathsf T}","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-9151633891e8e3e1","tex":"V_g","aliases":"V_g","display":"V_g","kind":"use","excerpt":"Curvature can be invisible to the task. Conversely, exact semantic behavior does not require the entire ambient space to be path-flat. Consider a block-diagonal action U_g=diag(P_g,V_g), observed only through the first block. The P_g may obey all semantic relations while the V_g retain path-dependent history. The task quotient is exact even though ","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-d42e4596a0b89189","tex":"\\vdash","aliases":"vdash","display":null,"kind":"use","excerpt":"A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates a source statement, operational state, and audit ledger to their successors. A statement that rejects still has an outcome and, within th","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Bounded discovery and semantic admission. The Discovery Bridge supplies a missing distinction between a well-formed proposal and an identified operator. It accepts a complete finite world W, a primitive vocabulary G_W, a depth bound d, demonstrations D, and explicitly claimed laws. All of these belong to the certificate's scope. [source reference]","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":23},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"A minimal delayed-observability counterexample. Let the concrete state be (x,y)∈{0,1}^2, with present observation O(x,y)=x, and let the only operation swap the coordinates. At horizon zero, there are two classes, classified by x. At horizon one, there are four: the next observation also reveals y.","count":9},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-1ecb58e66eedd43b","tex":"y_t","aliases":"y_t","display":"y_t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":6},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"The D4 fixture. The demonstrator has sixteen concrete states (a,b,u), with a∈ Z/4 Z, b,u∈ Z/2 Z. Its visible state is (a,b); u is a private bit. Operations are r(a,b,u)=(a+1,b,u), f(a,b,u)=(-a,1-b,u), v(a,b,u)=(a,b,1-u). The first two satisfy r^4=f^2=frfr=1. The private operation changes only u. Partition refinement recovers eight semantic classes.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":26},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, successive pairs meet the bound but the first and last do not. Quotient terminology therefore requires an actual equivalence relation, a specified quantization, or a separate approximate-abstraction formalism. Pairwise tolerance alone is ins","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8e5d0000abb9cf07","tex":"z_{t+1}","aliases":"z_t+1","display":"z_{t+1}","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":7},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"Observable rank after recurrence. The learned-checkpoint probe uses all twelve v0.4 checkpoints and 32 primary contexts. It finds rank seven at present observation and rank twelve after one through four continuation steps. Repeat-16 hard-polar log-radial gain remains below 2.90×10^-5 in absolute value, while p95 oriented loop angles reach roughly .","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"The operational state machine. State and judgment forms. A schematic runtime state is S=(Γ,V, O, P, Q, M, C, A, T), where Γ contains declarations, V vectors, O operators, P packets, Q proof resources, M memories, C capabilities, A planned actions, and T contracts, sessions, and transaction state. The concrete runtime also maintains metadata and aud","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, successive pairs meet the bound but the first and last do not. Quotient terminology therefore requires an actual equivalence relation, a specified quantization, or a separate approximate-abstraction formalism. Pairwise tolerance alone is ins","count":13},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-f3cf8a4e3d9677f6","tex":"\\varepsilon_i","aliases":"ε_i varepsilon","display":"\\varepsilon_i","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"A general controlled latent system is z_t+1=F_θ(z_t,u_t), y_t=O_θ(z_t,q_t), where u_t supplies an operation or observation and q_t specifies a later query. A recurrent state is scientifically interesting when it supports multiple future uses, not merely when it stores the already known final output token. The question may be asked for a trained ope","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"rror accumulates] Let the intended semantic operation be the identity and the learned implementation be a planar rotation R_ε. Every step preserves Euclidean norm exactly. After k steps the state has rotated by kε. With a readout determined by the sign of the first coordinate, a state starting at (1,0) eventually changes class for any fixed 0<ε<π/2","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-18829b323610a69f","tex":"\\sigma_W","aliases":"σ_W sigma","display":"\\sigma_W","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"A typing judgment Γvdash e:τ constrains an expression's carrier or kind. A proof-resource judgment constrains which already established obligations may be consumed. An execution judgment ⟨ c,S,L⟩⟶⟨ S',L'⟩ relates a source statement, operational state, and audit ledger to their successors. A statement that rejects still has an outcome and, within th","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"Margin protection is finite and local. [Readout margin bound] Suppose a logit vector has unique winning class i and margin m=ℓ_i-max_j≠ iℓ_j>0. Any perturbation Δℓ with normΔℓ_∞<m/2 preserves that winner. For every competitor j, the perturbed difference is at least m-2normΔℓ_∞>0. A finite region with positive margins can hide a continuous path defe","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-8943b22d90e5e02b","tex":"\\ell_j","aliases":"ℓ_j ell","display":"\\ell_j","kind":"use","excerpt":"Hard and soft observations. A hard interface is O_h(z)=argmax_j ℓ_j(z). A soft interface may be the probability vector or centered logits. Exact agreement under O_h permits changes in every nonwinning logit and changes in the winner's margin. Those changes can matter under later operations.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"From latent computation to an explicit language. The intended interface. The working architecture separates three timescales: fast latent computation ↔ typed semantic checkpoint ↔ persistent evidence ledger. A latent state can be provisional and disposable. The checkpoint preserves only distinctions justified for a declared future use. The ledger p","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Derivative of normalized dynamics] For u=Mz/normMz and Mz≠0, DF_M(z)=frac(I-uu^ T)MnormMz. The derivative of x↦ x/normx at nonzero x is (I-vv^ T)/normx, with v=x/normx. Compose it with the linear derivative M. A bounded normalized state can therefore have a locally amplifying angular derivative. A contraction claim must name the norm, state domain","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-09ade2b60fb5add0","tex":"\\sum_{i=1}^{k}","aliases":"∑_i=1^k sum","display":"\\sum_{i=1}^{k}","kind":"use","excerpt":"Approximate transport. [Finite-word error budget] Suppose E_i=B_g_iT-TA_g_i, with normE_i≤ε_i. Then B_wT-TA_w=∑_i=1^kB_g_k⋯ B_g_i+1E_iA_g_i-1⋯ A_g_1. Consequently, if every normA_g≤ a and normB_g≤ b, normB_wT-TA_w≤∑_i=1^k b^k-iε_i a^i-1. Insert the intermediate products obtained by moving T past one operator at a time. Consecutive products cancel, ","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-df66290cc89f818f","tex":"\\sum_{j=0}^{t-1}","aliases":"∑_j=0^t-1 sum","display":"\\sum_{j=0}^{t-1}","kind":"use","excerpt":"Approximate intertwining. If QM_a=A_aQ+E_a, repeated composition can accumulate leakage. For latent norm bounded by R, semantic operators bounded by a, and normE_a≤ε, the reduced-state discrepancy obeys e_t+1≤ a e_t+ε R. Induction gives e_t≤ a^t e_0+ε R∑_j=0^t-1a^j. When a=1, the upper bound grows linearly. When a>1, it may grow exponentially. A co","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":6},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-b92ecce9a562b5d7","tex":"\\sim_H","aliases":"∼_H sim","display":"\\sim_H","kind":"use","excerpt":"The two-index completion surface. History depth and future horizon. Let I⊆ Z be the allowed initial states and R_≤ L={F_w(z):z∈ I, |w|≤ L}. Define exact finite-horizon observational equivalence by z∼_H z'iff O(F_w(z))=O(F_w(z')) for every |w|≤ H. The finite object is Q_L,H=R_≤ L/ ∼_H. The parameter L governs which microstates can be reached; H gove","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-2c5d47bf0e1e95ba","tex":"\\sim_{H+1}","aliases":"∼_H+1 sim","display":"\\sim_{H+1}","kind":"use","excerpt":"nproposition[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"Norm ratio at 2048 · 1.0000 · 1.0000 · ≈0 · ≈0","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Gauge fixing is not semantic completion. Positive rays. For nonzero z, positive scaling may be redundant for a homogeneous hard-decision readout. The appropriate relation is z∼ cz for c>0, not identification of z with -z. Per-step normalization gives F_M(z)=fracMznormMz, provided Mz≠0. This fixes radial magnitude, but leaves angular dynamics intact","count":3},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"An approximate relation such as d(z,z')≤ε is generally not an equivalence relation. For points 0,0.75ε,1.5ε, successive pairs meet the bound but the first and last do not. Quotient terminology therefore requires an actual equivalence relation, a specified quantization, or a separate approximate-abstraction formalism. Pairwise tolerance alone is ins","count":17},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"[Semantic continuation] If [source reference] holds on a forward-invariant domain D⊆ Z, then O(F_w(z))=bar O(G_w(q(z))) for every z∈ D and every finite w. Forward invariance keeps every intermediate state inside the domain where the equations hold. Induction gives q(F_w(z))=G_w(q(z)). Apply the observation factorization to the endpoint. Without for","count":8},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"[Fixed-kernel isometry obstruction] Let V be a finite-dimensional inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-529f8d1978a8a0ae","tex":"\\bigcap_{a\\in\\mathcal A}","aliases":"⋂_a∈ A bigcap in mathcal","display":"\\bigcap_{a\\in\\mathcal A}","kind":"use","excerpt":"[Observable-kernel recurrence] One has N_H+1=N_H∩⋂_a∈ AM_a^-1N_H. If N_H+1=N_H, then N_H is invariant under every generator and equals N_∞. In a d-dimensional state space, the decreasing sequence stabilizes after at most d strict dimension drops. Invisibility for a word of length at most H+1 includes invisibility for all words of length at most H, ","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-ee563a8aa1670e17","tex":"\\bigcap_{|w|\\le H}","aliases":"⋂_|w|≤ H bigcap le","display":"\\bigcap_{|w|\\le H}","kind":"use","excerpt":"Linear observable kernels. A computable null-space construction. For a switched linear system z_t+1=M_a_tz_t with observation Cz_t, define N_H=⋂_|w|≤ Hker(CM_w), M_∅=I. A direction belongs to N_H exactly when it is invisible under every continuation through horizon H.","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-6eb256c5206a08bc","tex":"\\bigcup_{j=0}^{d}","aliases":"⋃_j=0^d bigcup","display":"\\bigcup_{j=0}^{d}","kind":"use","excerpt":"Transformation classes. Let S_W be the full finite legal state set and P_W,d=⋃_j=0^dG_W^j, σ_W(p)=(p(s))_s∈ S_W. Programs are identified when their complete state signatures agree. For a demonstration set D={(s_i,t_i)}, define C(D)={[p]:p∈ P_W,d, p(s_i)=t_i for all i}. The implementation enumerates this bounded universe, quotients equivalent spelli","count":1},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"Exact semantics before approximate semantics. A task-relative realization. Let A be an operation alphabet, Z a latent state space, and Y an observation space. Each a∈ A induces F_a:Z→ Z. For a word w=a_1⋯ a_k, define F_w=F_a_k∘⋯∘ F_a_1. A candidate semantic state space S, operation maps G_a, and abstraction q:Z→ S should satisfy q∘ F_a=G_a∘ q, O=ba","count":5},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"inner-product space, K⊆ V a subspace, and U:V→ V an isometry with U(K)⊆ K. Then U(K)=K and U(K^⊥)=K^⊥. Relative to V=K^⊥⊕ K, U is block diagonal. An isometry is injective, so dim U(K)=dim K, and the finite-dimensional inclusion gives equality. Thus U^-1(K)=K. For x∈ K^⊥, k∈ K, ⟨ Ux,k⟩=⟨ x,U^-1k⟩=0. Hence U(K^⊥)⊆ K^⊥; dimensions again give equality.","count":2},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a follow","count":4},{"work":"research-osl-neuralese-reading-edition","url":"notation/research-osl-neuralese-reading-edition.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[Refinement and diagonal action] Increasing H refines ∼_H. For every action a, there is a well-defined map Q_L,H+1⟶ Q_L+1,H, [z]⟼[F_a(z)]. The set of tested continuations grows with H, so equality under all longer-horizon tests implies equality under the shorter set. If z∼_H+1z', then for every suffix |w|≤ H, the continuation consisting of a follow","count":1},{"work":"research-sric-external-admission","url":"notation/research-sric-external-admission.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"use","excerpt":"item[square] At least 25 GB free disk space. item[square] NVIDIA driver operational. item[square] Python 3.10, 3.11, or 3.12 available through py. item[square] Git available in PowerShell. item[square] Network access to GitHub, PyPI, PyTorch wheels, and Hugging Face.","count":26},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":12},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-01478c4149c7701c","tex":"\\Aut","aliases":"Aut","display":null,"kind":"definition-passage","excerpt":"[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-86d552dd3c26e14e","tex":"\\mathrm{con}","aliases":"con mathrm","display":"\\mathrm{con}","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-3e5ea727ca0c6746","tex":"\\operatorname{cov}","aliases":"cov operatorname","display":"\\operatorname{cov}","kind":"definition-passage","excerpt":"[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"definition-passage","excerpt":"[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-bccce75adb940e5d","tex":"E_{\\mathrm{con}}","aliases":"E_con mathrm","display":"E_{\\mathrm{con}}","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-fd285b636a315f33","tex":"E_{\\mathrm{wit}}","aliases":"E_wit mathrm","display":"E_{\\mathrm{wit}}","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"definition-passage","excerpt":"[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":9},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":13},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-1e9de57bf29ef2a6","tex":"h'","aliases":"h'","display":"h'","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":6},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.","count":4},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"definition-passage","excerpt":"[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":17},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":10},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-4e4ee8a109009a53","tex":"\\subsetneq","aliases":"subsetneq","display":null,"kind":"definition-passage","excerpt":"[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":6},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-d7884e8dc4c0191e","tex":"\\Ucal","aliases":"Ucal","display":null,"kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":11},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-b54c3e6a2f3cb013","tex":"\\Ucal^{\\le r}","aliases":"Ucal^≤ r Ucal le","display":null,"kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-318c04501636594a","tex":"\\mathrm{wit}","aliases":"wit mathrm","display":"\\mathrm{wit}","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Semantic twin] Two observed trajectories x and x' are semantic twins relative to Ucal when they arise from the same protected task and have the same protected future and intervention laws, while nuisance coordinates, rendering variables, or architecture-specific gauges may differ.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"definition-passage","excerpt":"[Semantic twin] Two observed trajectories x and x' are semantic twins relative to Ucal when they arise from the same protected task and have the same protected future and intervention laws, while nuisance coordinates, rendering variables, or architecture-specific gauges may differ.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":5},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-ae744f7d0e0e0cac","tex":"Y^{u}_{0:r}","aliases":"Y^u_0:r","display":"Y^{u}_{0:r}","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":8},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.","count":4},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-2b4d236724e03917","tex":"Z_{\\Ucal,r}","aliases":"Z_Ucal,r Ucal","display":null,"kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":4},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"definition-passage","excerpt":"[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-4838d0a223d69d19","tex":"\\sim_{\\Ucal,r}","aliases":"∼_Ucal,r sim Ucal","display":null,"kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_Ucal,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈Ucal^≤ r. The quotient Z_Ucal,r=H/∼_Ucal,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-be14a316739e6949","tex":"\\operatorname{Acc}","aliases":"Acc operatorname","display":"\\operatorname{Acc}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-6a027721bfb3521f","tex":"\\operatorname{Acc}_{\\mathrm{answer}}","aliases":"Acc_answer operatorname mathrm","display":"\\operatorname{Acc}_{\\mathrm{answer}}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-e7e898debdc0c247","tex":"\\mathrm{answer}","aliases":"answer mathrm","display":"\\mathrm{answer}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-26406e352f59ad7e","tex":"D_{\\mathrm{twin}}","aliases":"D_twin mathrm","display":"D_{\\mathrm{twin}}","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps","display":null,"kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"[Deterministic semantic collision no-go] Let y:H→ Y be the representation exposed to a target-only compiler and let z:H→ Z be a protected semantic variable. If there exist h_1,h_2 such that y(h_1)=y(h_2), z(h_1)≠ z(h_2), then no deterministic map f:Y→ Z recovers z on both states.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-f3bce720e4deb824","tex":"\\mathrm{future}","aliases":"future mathrm","display":"\\mathrm{future}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-8c918c284a1665cd","tex":"\\Gcal","aliases":"Gcal","display":null,"kind":"use","excerpt":"The finite grammar is Gcal_23 = { l answer-only, PCA, predictive, intervention-response, joint minimal, full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82,","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-5c5a179d74fbc5eb","tex":"\\Gcal_{23}","aliases":"Gcal_23 Gcal","display":null,"kind":"use","excerpt":"The finite grammar is Gcal_23 = { l answer-only, PCA, predictive, intervention-response, joint minimal, full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82,","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"[Deterministic semantic collision no-go] Let y:H→ Y be the representation exposed to a target-only compiler and let z:H→ Z be a protected semantic variable. If there exist h_1,h_2 such that y(h_1)=y(h_2), z(h_1)≠ z(h_2), then no deterministic map f:Y→ Z recovers z on both states.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","excerpt":"[Deterministic semantic collision no-go] Let y:H→ Y be the representation exposed to a target-only compiler and let z:H→ Z be a protected semantic variable. If there exist h_1,h_2 such that y(h_1)=y(h_2), z(h_1)≠ z(h_2), then no deterministic map f:Y→ Z recovers z on both states.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-42225b157ba9718e","tex":"\\mathrm{intervention}","aliases":"intervention mathrm","display":"\\mathrm{intervention}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"[Causal-predictive sufficiency] Let q:H→ Z_Ucal,r be the quotient map. For every u∈Ucal^≤ r, there exists a Markov kernel K_u such that L (Y^u_0:rmid h)=K_u(q(h)). Thus q(h) is sufficient for every protected consequence appearing in the equivalence relation.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-16bc7321a3832aa5","tex":"K_u","aliases":"K_u","display":"K_u","kind":"use","excerpt":"[Causal-predictive sufficiency] Let q:H→ Z_Ucal,r be the quotient map. For every u∈Ucal^≤ r, there exists a Markov kernel K_u such that L (Y^u_0:rmid h)=K_u(q(h)). Thus q(h) is sufficient for every protected consequence appearing in the equivalence relation.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"The finite grammar is Gcal_23 = { l answer-only, PCA, predictive, intervention-response, joint minimal, full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82,","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probabil","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probabil","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-f99d57ee2f34e27d","tex":"q_\\theta","aliases":"q_θ theta","display":"q_\\theta","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":4},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"Each episode contains two independently gauged recurrent systems. A trajectory has five latent steps, four task-relevant semantic coordinates, and twelve nuisance coordinates per step. The observed flattened trajectory therefore lies in R^80. The semantic dynamics, gauges, nuisance processes, and hidden audit coordinates are not passed to the quoti","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"Across uniquely identified regimes, hidden semantic reconstruction is numerically exact in the released fixture. In the gauge regime, minimum gauge-invariant R^2 is 0.995522, while the minimum recovered semantic-subspace overlap is 0.991563. In the semantic-ambiguity regime, the minimum protected disagreement between two observationally equivalent ","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-1523811833b5e4c0","tex":"R^2_{\\mathrm{future}}","aliases":"R^2_future mathrm","display":"R^2_{\\mathrm{future}}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-311080e714276ff6","tex":"R^2_{\\mathrm{intervention}}","aliases":"R^2_intervention mathrm","display":"R^2_{\\mathrm{intervention}}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-9d8a6ecda587180a","tex":"\\R^{80}","aliases":"R^80 R","display":null,"kind":"use","excerpt":"Each episode contains two independently gauged recurrent systems. A trajectory has five latent steps, four task-relevant semantic coordinates, and twelve nuisance coordinates per step. The observed flattened trajectory therefore lies in R^80. The semantic dynamics, gauges, nuisance processes, and hidden audit coordinates are not passed to the quoti","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"[Minimality of the exact semantic quotient] Suppose s:H→ S is sufficient for every protected law in [source reference]. Then there exists a map ϕ:S→ Z_Ucal,r on s(H) satisfying q=ϕ∘ s. Consequently, any exact sufficient representation refines the causal-predictive quotient.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"[Minimality of the exact semantic quotient] Suppose s:H→ S is sufficient for every protected law in [source reference]. Then there exists a map ϕ:S→ Z_Ucal,r on s(H) satisfying q=ϕ∘ s. Consequently, any exact sufficient representation refines the causal-predictive quotient.","count":5},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-559f90b6098df902","tex":"\\mathrm{twin}","aliases":"twin mathrm","display":"\\mathrm{twin}","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probabil","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[Deterministic semantic collision no-go] Let y:H→ Y be the representation exposed to a target-only compiler and let z:H→ Z be a protected semantic variable. If there exist h_1,h_2 such that y(h_1)=y(h_2), z(h_1)≠ z(h_2), then no deterministic map f:Y→ Z recovers z on both states.","count":3},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),eps}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"[Minimality of the exact semantic quotient] Suppose s:H→ S is sufficient for every protected law in [source reference]. Then there exists a map ϕ:S→ Z_Ucal,r on s(H) satisfying q=ϕ∘ s. Consequently, any exact sufficient representation refines the causal-predictive quotient.","count":4},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"[Causal-predictive sufficiency] Let q:H→ Z_Ucal,r be the quotient map. For every u∈Ucal^≤ r, there exists a Markov kernel K_u such that L (Y^u_0:rmid h)=K_u(q(h)). Thus q(h) is sufficient for every protected consequence appearing in the equivalence relation.","count":7},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"[Minimality of the exact semantic quotient] Suppose s:H→ S is sufficient for every protected law in [source reference]. Then there exists a map ϕ:S→ Z_Ucal,r on s(H) satisfying q=ϕ∘ s. Consequently, any exact sufficient representation refines the causal-predictive quotient.","count":1},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probabil","count":2},{"work":"research-sric-block23-witness","url":"notation/research-sric-block23-witness.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"notation-table","excerpt":"σ_+(D) · Smallest positive singular value of D","count":415},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"G=(Z,P,C,B) · Research-compiler diagnostic, experiment, method-constructor, and binding language","count":1237},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"J_Ξ(A) · Resource-normalized utility of agent A in context 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receipt","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b3db48abbbcefa34","tex":"\\Pi_9^\\Gamma","aliases":"Π_9^Γ Pi Gamma","display":"\\Pi_9^\\Gamma","kind":"notation-table","excerpt":"Γ_3,Π_9^Γ · Block 15 language-growing compiler and its lineage receipt","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c8256780a3ff4954","tex":"\\Pi_{\\mathrm{con}}","aliases":"Π_con Pi mathrm","display":"\\Pi_{\\mathrm{con}}","kind":"notation-table","excerpt":"Π_con · Proof-carrying constructor-expansion certificate","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-40db1c3ce9d1bc31","tex":"\\Pi_i^+","aliases":"Π_i^+ Pi","display":"\\Pi_i^+","kind":"notation-table","excerpt":"Π_i^-,Π_i^+ · Results-free pre-receipt and evidence-appended post-receipt for research candidate i","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-50ccb20ee6d178fa","tex":"\\Pi_i^-","aliases":"Π_i^- Pi","display":"\\Pi_i^-","kind":"notation-table","excerpt":"Π_i^-,Π_i^+ · Results-free pre-receipt and evidence-appended post-receipt for research candidate i","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ded9962e5d8e3f5","tex":"\\Pi_{\\mathrm{resp}}","aliases":"Π_resp Pi mathrm","display":"\\Pi_{\\mathrm{resp}}","kind":"notation-table","excerpt":"Π_resp · Proof-carrying response-language expansion certificate","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9c1693595ca544be","tex":"\\Pi_{\\Gamma\\text{-lang}}","aliases":"Π_Γ-lang Pi Gamma text","display":"\\Pi_{\\Gamma\\text{-lang}}","kind":"notation-table","excerpt":"Π_Γ-lang · Proof-carrying compiler-language expansion receipt","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"notation-table","excerpt":"ρ_t · Authorized repair rank rank D_t","count":25},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-faf2d9e8110151a4","tex":"\\rho_t","aliases":"ρ_t rho","display":"\\rho_t","kind":"notation-table","excerpt":"ρ_t · Authorized repair rank rank D_t","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"notation-table","excerpt":"σ_+(D) · Smallest positive singular value of D","count":51},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f7eeac933ef7a77c","tex":"\\sigma_+","aliases":"σ_+ sigma","display":"\\sigma_+","kind":"notation-table","excerpt":"σ_+(D) · Smallest positive singular value of D","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-30da2ecba0b39743","tex":"\\sigma_{\\Rcal}","aliases":"σ_Rcal σ_R sigma Rcal sigma mathcal","display":"\\sigma_{{\\mathcal{R}}}","kind":"notation-table","excerpt":"σ_R(H_a) · Response signature of hidden block H_a","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"notation-table","excerpt":"E_τ^flat · Spectral approximate path-flat core at threshold τ","count":109},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"notation-table","excerpt":"widehat K_ϕ(A)^δ · Outcome kernel predicted by the compressed self-model","count":50},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-28b947267f902868","tex":"\\phi_0","aliases":"ϕ_0 phi","display":"\\phi_0","kind":"notation-table","excerpt":"ϕ_0 · Inherited diagnostic quotient mapping both novel mechanisms to one unknown token","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7016cbae921a9f68","tex":"\\phi_t","aliases":"ϕ_t phi","display":"\\phi_t","kind":"notation-table","excerpt":"ϕ_t(A_t) · Compressed causal self-model","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"notation-table","excerpt":"D^† · Moore--Penrose pseudoinverse of the correction map","count":36},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"notation-table","excerpt":"M_∩ · Conjunctive integrity-monitor ensemble","count":25},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"notation-table","excerpt":"A^⋆(τ) · Acceptable validity-filtered methods for research problem τ","count":25},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"definition-passage","excerpt":"nition[Forward implementation refinement] A map α:widehat S→ S forward-refines the runtime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observat","count":105},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f3bd2f5ce823ada9","tex":"\\mathfrak A^+","aliases":"A^+ mathfrak","display":"\\mathfrak A^+","kind":"definition-passage","excerpt":"[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair eff","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"definition-passage","excerpt":"[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"definition-passage","excerpt":"[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"definition-passage","excerpt":"[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e50ca40bad47ed08","tex":"A_s","aliases":"A_s","display":"A_s","kind":"definition-passage","excerpt":"[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from sou","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51d5a0ed47981d1a","tex":"\\mathcal A_T","aliases":"A_T mathcal","display":"\\mathcal A_T","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f925f2dd41a18a99","tex":"\\mathcal A_t","aliases":"A_t mathcal","display":"\\mathcal A_t","kind":"definition-passage","excerpt":"[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from sou","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5dd428e203fe7a9","tex":"\\mathfrak A_t'","aliases":"A_t' mathfrak","display":"\\mathfrak A_t'","kind":"definition-passage","excerpt":"[Response algebra and commutant] The response algebra and its commutant are A_t=alg(I,R_v^t:v∈ V), A_t'={S∈End(H_t):SR_v^t=R_v^tS ∀ v}.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c9ef72cd5fa5dfce","tex":"\\mathsf A_v","aliases":"A_v mathsf","display":"\\mathsf A_v","kind":"definition-passage","excerpt":"[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, mea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ff4c7bb2acfd4548","tex":"\\mathsf A_w","aliases":"A_w mathsf","display":"\\mathsf A_w","kind":"definition-passage","excerpt":"[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, mea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":31},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-877b67ef87c6ce18","tex":"\\Acal_0","aliases":"Acal_0 A_0 Acal mathcal","display":"{\\mathcal{A}}_0","kind":"definition-passage","excerpt":"[Conservative atlas extension] An atlas A_1 conservatively extends A_0 when every chart of A_0 occurs as an identical prefix of A_1, the route and canonicalized coordinates agree on every inherited support, and the inherited decision path remains executable as an exact fallback.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-78cbd8ca715c5eb3","tex":"\\Acal_1","aliases":"Acal_1 A_1 Acal mathcal","display":"{\\mathcal{A}}_1","kind":"definition-passage","excerpt":"[Conservative atlas extension] An atlas A_1 conservatively extends A_0 when every chart of A_0 occurs as an identical prefix of A_1, the route and canonicalized coordinates agree on every inherited support, and the inherited decision path remains executable as an exact fallback.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3cea48dca7c90a23","tex":"\\Acal_A","aliases":"Acal_A A_A Acal mathcal","display":"{\\mathcal{A}}_A","kind":"definition-passage","excerpt":"[Authorized transport] A transport datum is authorized when T_U(A_A)⊆A_B. It is (c_A,c_B,κ)-budget controlled when c_B(T_Uu)≤κ c_A(u) for every u∈A_A.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-82a1b3bc843d002c","tex":"\\Acal_B","aliases":"Acal_B A_B Acal mathcal","display":"{\\mathcal{A}}_B","kind":"definition-passage","excerpt":"[Authorized transport] A transport datum is authorized when T_U(A_A)⊆A_B. It is (c_A,c_B,κ)-budget controlled when c_B(T_Uu)≤κ c_A(u) for every u∈A_A.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e9608f147ef15a95","tex":"\\operatorname{Accept}","aliases":"Accept operatorname","display":"\\operatorname{Accept}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c041351c5aeef4fd","tex":"\\mathsf{Act}","aliases":"Act mathsf","display":"\\mathsf{Act}","kind":"definition-passage","excerpt":"[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7348fed86605e833","tex":"\\mathrm{adm}","aliases":"adm mathrm","display":"\\mathrm{adm}","kind":"definition-passage","excerpt":"[Distribution-indexed atlas] A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abs","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2a55759122ebe931","tex":"\\mathsf{Adm}","aliases":"Adm mathsf","display":"\\mathsf{Adm}","kind":"definition-passage","excerpt":"[Block 19 semantic bundle] The finite semantic bundle is B_19 = ( A_1, a, S, F, J, H, C, Adm ), where A_1 is the expanded atlas, a is the set aggregator, F is the recurrent Neuralese transition, J is the protected task readout, H is the halt readout, C is the context branch, and Adm is the selective chart gate.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-37e8eb6fc5007d54","tex":"\\operatorname{Admissible}","aliases":"Admissible operatorname","display":"\\operatorname{Admissible}","kind":"definition-passage","excerpt":"[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d74e1d5e3de38cf0","tex":"\\operatorname{alg}","aliases":"alg operatorname","display":"\\operatorname{alg}","kind":"definition-passage","excerpt":"[Response algebra and commutant] The response algebra and its commutant are A_t=alg(I,R_v^t:v∈ V), A_t'={S∈End(H_t):SR_v^t=R_v^tS ∀ v}.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d2cb7ddd80203ae6","tex":"\\mathrm{alias}","aliases":"alias mathrm","display":"\\mathrm{alias}","kind":"definition-passage","excerpt":"[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair eff","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-037d0c4a3ac7d1ab","tex":"\\angles","aliases":"angles","display":null,"kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":26},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-01478c4149c7701c","tex":"\\Aut","aliases":"Aut operatorname","display":"{\\operatorname{Aut}}","kind":"definition-passage","excerpt":"[Protected gauge block] A subspace G⊆ Z is a protected gauge block when every permitted future and intervention law is invariant under a declared group G≤Aut(G). The identified object is then the orbit [z]_ G, not a preferred basis inside G.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dc3205cf24688a05","tex":"\\mathrm{auth}","aliases":"auth mathrm","display":"\\mathrm{auth}","kind":"definition-passage","excerpt":"[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ea0b1d9a333c4d54","tex":"\\mathfrak B","aliases":"B mathfrak","display":"\\mathfrak B","kind":"definition-passage","excerpt":"[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint,","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d1e4f21b86ad3668","tex":"\\mathsf B","aliases":"B mathsf","display":"\\mathsf B","kind":"definition-passage","excerpt":"[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint,","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"definition-passage","excerpt":"[Evaluator-nullspace] The locally invisible behavior subspace of portfolio B is N_B = ⋂_j=1^mker Db_j.","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"definition-passage","excerpt":"[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9474dae99bcf0491","tex":"B_t","aliases":"B_t","display":"B_t","kind":"definition-passage","excerpt":"[Budget acceleration] Let B_t(γ) be the least budget required by mechanism K_t to obtain expected foreign-agent gain at least γ. Budget acceleration holds on an interval I when B_t+1(γ)<B_t(γ) for every γ∈ I.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d8d4b66e30067b11","tex":"B_{t+1}","aliases":"B_t+1","display":"B_{t+1}","kind":"definition-passage","excerpt":"[Budget acceleration] Let B_t(γ) be the least budget required by mechanism K_t to obtain expected foreign-agent gain at least γ. Budget acceleration holds on an interval I when B_t+1(γ)<B_t(γ) for every γ∈ I.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9c74542ebc03d6cd","tex":"\\bar c","aliases":"bar c bar","display":"\\bar c","kind":"definition-passage","excerpt":"[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a c","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-43c8e73158bed877","tex":"\\bar h","aliases":"bar h bar","display":"\\bar h","kind":"definition-passage","excerpt":"oℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0172f00d6015ed24","tex":"\\bar T","aliases":"bar T bar","display":"\\bar T","kind":"definition-passage","excerpt":"[Inherited global-affine grammar] The inherited bridge grammar is G_glob = setbar T:ℝ^d+1→ℝ^q linear in homogeneous coordinates. It fits one affine source-to-target map across every observable context.","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9c489dcd7234c766","tex":"\\bar T_0","aliases":"bar T_0 bar","display":"\\bar T_0","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-666d47bdcd777cdc","tex":"\\bar T_1","aliases":"bar T_1 bar","display":"\\bar T_1","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cb66734132f36672","tex":"\\bar T_{g(h)}","aliases":"bar T_g(h) bar","display":"\\bar T_{g(h)}","kind":"definition-passage","excerpt":"oℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f0b29b0ccceb6708","tex":"\\bar T_j","aliases":"bar T_j bar","display":"\\bar T_j","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-415d260d791ac2cf","tex":"\\Bcal_{19}","aliases":"Bcal_19 B_19 Bcal mathcal","display":"{\\mathcal{B}}_{19}","kind":"definition-passage","excerpt":"[Block 19 semantic bundle] The finite semantic bundle is B_19 = ( A_1, a, S, F, J, H, C, Adm ), where A_1 is the expanded atlas, a is the set aggregator, F is the recurrent Neuralese transition, J is the protected task readout, H is the halt readout, C is the context branch, and Adm is the selective chart gate.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d13b1528b249ec85","tex":"\\bigwedge","aliases":"bigwedge","display":"\\bigwedge","kind":"definition-passage","excerpt":"[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7a75e8338d6a4608","tex":"\\bigwedge_{j\\leq\\ell}","aliases":"bigwedge_j≤ℓ bigwedge leq ell","display":"\\bigwedge_{j\\leq\\ell}","kind":"definition-passage","excerpt":"[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2ed728a2bb8f27d6","tex":"\\bm c","aliases":"c bm","display":"\\bm c","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0eaa1204dfcfa445","tex":"\\mathbf c","aliases":"c mathbf","display":"\\mathbf c","kind":"definition-passage","excerpt":"[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"definition-passage","excerpt":"[Finite Neuralese semantic bundle] A finite architecture--context Neuralese bundle is a tuple E = ( A, C, {H_a,c}_a,c, {J_a,c}_a,c, R ), where H_a,c is a latent trajectory space, J_a,c is an externally calibrated semantic readout, and R is a frozen chart-admission rule. No global identification among the fibers is assumed.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-934bd28914427e92","tex":"C^{a}_{c\\to d}","aliases":"C^a_c→ d to","display":"C^{a}_{c\\to d}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0d21737c443fefcf","tex":"C^{b}_{c\\to d}","aliases":"C^b_c→ d to","display":"C^{b}_{c\\to d}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6ab87fe8ac9a0afb","tex":"\\mathfrak C_A","aliases":"C_A mathfrak","display":"\\mathfrak C_A","kind":"definition-passage","excerpt":"[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9b74ac7e4a926fd8","tex":"c_A","aliases":"c_A","display":"c_A","kind":"definition-passage","excerpt":"[Authorized transport] A transport datum is authorized when T_U(A_A)⊆A_B. It is (c_A,c_B,κ)-budget controlled when c_B(T_Uu)≤κ c_A(u) for every u∈A_A.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c609f244935cbee4","tex":"c_B","aliases":"c_B","display":"c_B","kind":"definition-passage","excerpt":"[Authorized transport] A transport datum is authorized when T_U(A_A)⊆A_B. It is (c_A,c_B,κ)-budget controlled when c_B(T_Uu)≤κ c_A(u) for every u∈A_A.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3a342a2e332eabef","tex":"C_e","aliases":"C_e","display":"C_e","kind":"definition-passage","excerpt":"[Matched selector comparison] Two selectors K and K' are compared under matched resources when, episode by episode, they receive: item the identical candidate bundle C_e in the identical order; item the identical receipt fields and calibration evidence; item the identical target, anchor, and audit data; item the identical candidate-construction bud","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c8a2933fb47afac8","tex":"c_{e,K(Z_e)}","aliases":"c_e,K(Z_e)","display":"c_{e,K(Z_e)}","kind":"definition-passage","excerpt":"[Repair selector] A repair selector is a measurable map K: Z^B⟶{0,…,B-1}, where Z is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is c_e,K(Z_e).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-afa213ebfe38f71e","tex":"\\mathbf c_i","aliases":"c_i mathbf","display":"\\mathbf c_i","kind":"definition-passage","excerpt":"[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cb54c8e6cc18062e","tex":"\\bm c_{\\mathrm{in}}","aliases":"c_in bm mathrm","display":"\\bm c_{\\mathrm{in}}","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ac26951d19c5faba","tex":"C_{\\mathrm{model}}","aliases":"C_model mathrm","display":"C_{\\mathrm{model}}","kind":"definition-passage","excerpt":"[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5ed4d33b02a94dfa","tex":"c_{\\mathrm{out}}","aliases":"c_out mathrm","display":"c_{\\mathrm{out}}","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5a4eda3184d1b657","tex":"\\mathfrak C_u","aliases":"C_u mathfrak","display":"\\mathfrak C_u","kind":"definition-passage","excerpt":"[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and u","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-065fc2a65996d145","tex":"C_z","aliases":"C_z","display":"C_z","kind":"definition-passage","excerpt":"[Repair-consistent cell] A cell is repair consistent when ⋂_x∈ C_zA^⋆(x)≠∅. It is inconsistent when no single authorized repair works for every episode that the grammar treats as equivalent.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-40c9ccc48f7f4420","tex":"C_\\omega","aliases":"C_ω omega","display":"C_\\omega","kind":"definition-passage","excerpt":"[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observatio","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-da3393d84f09b1b4","tex":"C_{\\omega_1}","aliases":"C_ω_1 omega","display":"C_{\\omega_1}","kind":"definition-passage","excerpt":"[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-797e6a662ec08d76","tex":"C_{\\omega_2}","aliases":"C_ω_2 omega","display":"C_{\\omega_2}","kind":"definition-passage","excerpt":"[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2db36ee8a90d52f4","tex":"\\Ccal_t","aliases":"Ccal_t C_t Ccal mathcal","display":"{\\mathcal{C}}_t","kind":"definition-passage","excerpt":"[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7353210f841b4366","tex":"\\Ccal_t^+","aliases":"Ccal_t^+ C_t^+ Ccal mathcal","display":"{\\mathcal{C}}_t^+","kind":"definition-passage","excerpt":"[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-46b875e09cd8c3e6","tex":"\\mathrm{chart}","aliases":"chart mathrm","display":"\\mathrm{chart}","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c9436aea1bf2444f","tex":"\\check","aliases":"check","display":null,"kind":"definition-passage","excerpt":"[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c21f28545c370331","tex":"\\mathrm{code}","aliases":"code mathrm","display":"\\mathrm{code}","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5596b5ccd2fd942","tex":"\\mathrm{comm}","aliases":"comm mathrm","display":"\\mathrm{comm}","kind":"definition-passage","excerpt":"[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair eff","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-208f33c6209efc35","tex":"\\mathrm{Complexity}","aliases":"Complexity mathrm","display":"\\mathrm{Complexity}","kind":"definition-passage","excerpt":"[Cover adequacy score] For a candidate functional cover U, define the diagnostic score A( U) := η_loc( U) +τη_glue( U) +λ Complexity( U) +γ Instability( U).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e6e837946a9ee6aa","tex":"\\mathrm{cons}","aliases":"cons mathrm","display":"\\mathrm{cons}","kind":"definition-passage","excerpt":"[Conservative lineage dominance] Write L_tpreceq_consL_t+1 when: item the old grammar, evidence ledger, and proof objects embed into the new lineage state; item every old protected interpretation remains recoverable within the declared semantic-debt bound; item old authorized behaviors replay within declared tolerances; item the new correction imag","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3e5ea727ca0c6746","tex":"\\operatorname{cov}","aliases":"cov operatorname","display":"\\operatorname{cov}","kind":"definition-passage","excerpt":"[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b893acf270b3c5ec","tex":"\\operatorname{CPre}","aliases":"CPre operatorname","display":"\\operatorname{CPre}","kind":"definition-passage","excerpt":"[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a3919709d0a6b99e","tex":"\\mathrm{ctx}","aliases":"ctx mathrm","display":"\\mathrm{ctx}","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-94cbf15f690d8f85","tex":"\\operatorname{CVaR}","aliases":"CVaR operatorname","display":"\\operatorname{CVaR}","kind":"definition-passage","excerpt":"[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and u","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0c7185cdda6df286","tex":"\\operatorname{CVaR}_\\alpha","aliases":"CVaR_α operatorname alpha","display":"\\operatorname{CVaR}_\\alpha","kind":"definition-passage","excerpt":"[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and u","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"definition-passage","excerpt":"[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ddc7e0f1c8697a47","tex":"\\mathsf D","aliases":"D mathsf","display":"\\mathsf D","kind":"definition-passage","excerpt":"[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint,","count":17},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-63ed82a00b685c9b","tex":"D_A","aliases":"D_A","display":"D_A","kind":"definition-passage","excerpt":"[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is e","count":38},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ee5bb7d36741efec","tex":"D_B","aliases":"D_B","display":"D_B","kind":"definition-passage","excerpt":"[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is e","count":31},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cb1fed3d203f3cb5","tex":"\\mathcal D_{\\mathrm{foreign}}","aliases":"D_foreign mathcal mathrm","display":"\\mathcal D_{\\mathrm{foreign}}","kind":"definition-passage","excerpt":"[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ec37dd5a75f2754b","tex":"\\mathcal D_{\\rm audit}","aliases":"D_rm audit mathcal rm","display":"\\mathcal D_{\\rm audit}","kind":"definition-passage","excerpt":"[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise dis","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d3bde8ab13666ee7","tex":"\\mathcal D_{\\rm cal}","aliases":"D_rm cal mathcal rm","display":"\\mathcal D_{\\rm cal}","kind":"definition-passage","excerpt":"[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise dis","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0534e6463f851d8d","tex":"\\mathsf D_{\\rm dev}","aliases":"D_rm dev mathsf rm","display":"\\mathsf D_{\\rm dev}","kind":"definition-passage","excerpt":"[Sealed meta-compiler] A sealed meta-compiler is a map M: ( D_rm dev, D_rm meta,K_0,Γ) ⟼ (K_1,Π_1), where Γ is a declared finite meta-grammar and Π_1 is a descendant receipt. The compiler is sealed when: item neither final audit split is supplied to M; item Π_1 commits to the child state, parent identity, input hashes, grammar, selected configurati","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a466b7035717ec93","tex":"\\mathcal D_{\\rm meta}","aliases":"D_rm meta mathcal rm","display":"\\mathcal D_{\\rm meta}","kind":"definition-passage","excerpt":"[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise dis","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-50c2d2710162e071","tex":"\\mathsf D_{\\rm meta}","aliases":"D_rm meta mathsf rm","display":"\\mathsf D_{\\rm meta}","kind":"definition-passage","excerpt":"[Sealed meta-compiler] A sealed meta-compiler is a map M: ( D_rm dev, D_rm meta,K_0,Γ) ⟼ (K_1,Π_1), where Γ is a declared finite meta-grammar and Π_1 is a descendant receipt. The compiler is sealed when: item neither final audit split is supplied to M; item Π_1 commits to the child state, parent identity, input hashes, grammar, selected configurati","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-adf0952ae0a9d3fe","tex":"d_{\\mathrm{sig}}","aliases":"d_sig mathrm","display":"d_{\\mathrm{sig}}","kind":"definition-passage","excerpt":"[Distribution-indexed atlas] A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abs","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c631055a6cfad223","tex":"D_{t+1}","aliases":"D_t+1","display":"D_{t+1}","kind":"definition-passage","excerpt":"[Repair death] A class [r]∈coker D_t undergoes repair death at t+1 when overline T_t([r])=0, there exists an authorized u∈ U_t+1 satisfying D_t+1u=-T_t^Vr, and the protected defect transport obeys a lower norm bound normT_t^Vr≥κnormr for a declared κ>0.","count":20},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"definition-passage","excerpt":"[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e4a9cdafec82620","tex":"\\Dcal_{\\rm for}","aliases":"Dcal_rm for D_rm for Dcal rm mathcal rm","display":"{\\mathcal{D}}_{\\rm for}","kind":"definition-passage","excerpt":"[Matched-budget foreign metaproductivity] For a foreign target distribution D_rm for and budget B, define M_B^rm for( P) = E_A∼D_rm for [G_Ξ( P,A)], provided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-20d3acf9e0b804fd","tex":"\\mathrm{decision}","aliases":"decision mathrm","display":"\\mathrm{decision}","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a1771269e2dfa28b","tex":"\\mathsf{depth}","aliases":"depth mathsf","display":"\\mathsf{depth}","kind":"definition-passage","excerpt":"[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 25","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-abc34f0330181068","tex":"\\mathrm{digest}","aliases":"digest mathrm","display":"\\mathrm{digest}","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-794e0281dc354abf","tex":"\\mathsf{Dist}","aliases":"Dist mathsf","display":"\\mathsf{Dist}","kind":"definition-passage","excerpt":"[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db5a6e03dac4739b","tex":"\\dom","aliases":"dom operatorname","display":"{\\operatorname{dom}}","kind":"definition-passage","excerpt":"[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a11d22a99cb376b1","tex":"\\mathfrak E","aliases":"E mathfrak","display":"\\mathfrak E","kind":"definition-passage","excerpt":"[Finite Neuralese semantic bundle] A finite architecture--context Neuralese bundle is a tuple E = ( A, C, {H_a,c}_a,c, {J_a,c}_a,c, R ), where H_a,c is a latent trajectory space, J_a,c is an externally calibrated semantic readout, and R is a frozen chart-admission rule. No global identification among the fibers is assumed.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ad8532292a2930eb","tex":"\\mathbb E_{A\\sim\\mathcal D_{\\mathrm{foreign}}}","aliases":"E_A∼ D_foreign mathbb sim mathcal mathrm","display":"\\mathbb E_{A\\sim\\mathcal D_{\\mathrm{foreign}}}","kind":"definition-passage","excerpt":"ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d0e8dd6200bfb7c4","tex":"\\mathbb E_{A\\sim\\mathcal F_{\\mathrm{post}}}","aliases":"E_A∼ F_post mathbb sim mathcal mathrm","display":"\\mathbb E_{A\\sim\\mathcal F_{\\mathrm{post}}}","kind":"definition-passage","excerpt":"[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a7b80af7437094a8","tex":"\\mathbb E_{A\\sim\\Dcal_{\\rm for}}","aliases":"E_A∼Dcal_rm for E_A∼D_rm for mathbb sim Dcal rm mathbb sim mathcal rm","display":"\\mathbb E_{A\\sim{\\mathcal{D}}_{\\rm for}}","kind":"definition-passage","excerpt":"[Matched-budget foreign metaproductivity] For a foreign target distribution D_rm for and budget B, define M_B^rm for( P) = E_A∼D_rm for [G_Ξ( P,A)], provided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bccce75adb940e5d","tex":"E_{\\mathrm{con}}","aliases":"E_con mathrm","display":"E_{\\mathrm{con}}","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7299a0471cde40ba","tex":"\\mathbb E_{e\\sim Q}","aliases":"E_e∼ Q mathbb sim","display":"\\mathbb E_{e\\sim Q}","kind":"definition-passage","excerpt":"[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f0b78f24767f11cb","tex":"\\mathbb E_{h\\sim P_{b,c}}","aliases":"E_h∼ P_b,c mathbb sim","display":"\\mathbb E_{h\\sim P_{b,c}}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a4da71df7125a67a","tex":"E_{\\mathrm{irr}}","aliases":"E_irr mathrm","display":"E_{\\mathrm{irr}}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f75d15c864ad3cef","tex":"E_t","aliases":"E_t","display":"E_t","kind":"definition-passage","excerpt":"[Rotating witness schedule] A sequence (E_t,W_t)_t≥0 is a rotating witness schedule when: item E_t∩ W_t=∅ for every t; item W_t is immutable during transaction t; item every component belongs to E_t for at least one t; item every witness replacement is validated by a component outside the same edit set; item protected evidence is transported throug","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ce58d5c7fe38b3f1","tex":"E_v^{\\mathrm{eq}}","aliases":"E_v^eq mathrm","display":"E_v^{\\mathrm{eq}}","kind":"definition-passage","excerpt":"[Closed-walk equalizer core] For a possibly noninvertible evaluator graph, let W(v) be the monoid of directed closed walks at v. Define E_v^eq := ⋂_γ∈ W(v)ker(H_γ-I).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f905335fffc8ce49","tex":"E_v^{\\flat}","aliases":"E_v^flat flat","display":null,"kind":"definition-passage","excerpt":"[Core-retractable upgrade] An evaluator morphism e:v→ w is core-retractable when T_e(E_v^flat)⊆ E_w^flat and there is a map R_e:E_w^flat→ E_v^flat such that R_eT_e x=x for every x∈ E_v^flat. It is core-isomorphic when T_e restricts to an isomorphism from E_v^flat onto E_w^flat.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-73b14bff296e7bc6","tex":"E_{v_0}^{\\flat}","aliases":"E_v_0^flat flat","display":null,"kind":"definition-passage","excerpt":"[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4c336ec08d156ec7","tex":"E_w","aliases":"E_w","display":"E_w","kind":"definition-passage","excerpt":"[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0785ecc00a75c439","tex":"E_w^{\\flat}","aliases":"E_w^flat flat","display":null,"kind":"definition-passage","excerpt":"[Core-retractable upgrade] An evaluator morphism e:v→ w is core-retractable when T_e(E_v^flat)⊆ E_w^flat and there is a map R_e:E_w^flat→ E_v^flat such that R_eT_e x=x for every x∈ E_v^flat. It is core-isomorphic when T_e restricts to an isomorphism from E_v^flat onto E_w^flat.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fd285b636a315f33","tex":"E_{\\mathrm{wit}}","aliases":"E_wit mathrm","display":"E_{\\mathrm{wit}}","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6b1d556dd63cec86","tex":"\\mathbb E_{z\\sim P_{a,d}}","aliases":"E_z∼ P_a,d mathbb sim","display":"\\mathbb E_{z\\sim P_{a,d}}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e36138753761ef5b","tex":"E_{\\tau}^{\\flat}","aliases":"E_τ^flat tau flat","display":null,"kind":"definition-passage","excerpt":"[Thresholded approximate flat core] For τ≥0, define E_τ^flat := spansetx_i:λ_i≤τ, where x_i is an eigenbasis of A.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-27b0d977699ac2e5","tex":"\\Ecal_{\\mathrm{post}}","aliases":"Ecal_post E_post Ecal mathrm mathcal mathrm","display":"{\\mathcal{E}}_{\\mathrm{post}}","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5974260cc8f8a94f","tex":"\\Ecal_{\\mathrm{pre}}","aliases":"Ecal_pre E_pre Ecal mathrm mathcal mathrm","display":"{\\mathcal{E}}_{\\mathrm{pre}}","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cdbeff5bc0f0355e","tex":"\\Ecal_{t+1}","aliases":"Ecal_t+1 E_t+1 Ecal mathcal","display":"{\\mathcal{E}}_{t+1}","kind":"definition-passage","excerpt":"[Append-only evidence embedding] Evidence preservation from generation t to t+1 is an injective map ι_t:E_t↪E_t+1 that preserves payload, source, timestamp order, precommitment status, provenance type, and cryptographic integrity fields. A later correction may append a refutation or reinterpretation receipt, but it may not silently alter the old en","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2713c5351fc16286","tex":"\\mathrm{edit}","aliases":"edit mathrm","display":"\\mathrm{edit}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End operatorname","display":"{\\operatorname{End}}","kind":"definition-passage","excerpt":"[Response algebra and commutant] The response algebra and its commutant are A_t=alg(I,R_v^t:v∈ V), A_t'={S∈End(H_t):SR_v^t=R_v^tS ∀ v}.","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c4ff4fdb449defa","tex":"\\mathrm{episodes}","aliases":"episodes mathrm","display":"\\mathrm{episodes}","kind":"definition-passage","excerpt":"[Repair abstraction operator] A repair abstraction operator is a map a: H_episodes ⟶ overlineO that maps one or more concrete episodes to a typed schema with a failure signature, applicability condition, correction program, effect profile, and receipt template.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-62453cc10993d82a","tex":"\\bm\\eps","aliases":"eps ε bm eps bm varepsilon","display":"\\bm{\\varepsilon}","kind":"definition-passage","excerpt":"[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rul","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-098392c4077dd300","tex":"\\eps_{\\mathrm{auth},e}","aliases":"eps_auth,e ε_auth,e eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{auth},e}","kind":"definition-passage","excerpt":"[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-81210d333cf4cf69","tex":"\\eps_{\\mathrm{comm}}","aliases":"eps_comm ε_comm eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{comm}}","kind":"definition-passage","excerpt":"[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair eff","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c3c18b1f04272fe9","tex":"\\eps_{D,e}","aliases":"eps_D,e ε_D,e eps varepsilon","display":"{\\varepsilon}_{D,e}","kind":"definition-passage","excerpt":"[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f4040d951ef30bed","tex":"\\eps_D^{\\mathrm{rel}}","aliases":"eps_D^rel ε_D^rel eps mathrm varepsilon mathrm","display":"{\\varepsilon}_D^{\\mathrm{rel}}","kind":"definition-passage","excerpt":"[Scale-normalized transport defect] For a declared stabilizer τ>0, define ε_D^rel := fracnormD_BT_U-T_VD_A normD_BnormT_U+normT_VnormD_A+τ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-234f41522d513749","tex":"\\bm\\eps_e","aliases":"eps_e ε_e bm eps bm varepsilon","display":"\\bm{\\varepsilon}_e","kind":"definition-passage","excerpt":"[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rul","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1fc9b8bfa398d30f","tex":"\\eps_{\\mathrm{fit}}","aliases":"eps_fit ε_fit eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{fit}}","kind":"definition-passage","excerpt":"[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair eff","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7ff0b8fb62ee80ef","tex":"\\eps_Q","aliases":"eps_Q ε_Q eps varepsilon","display":"{\\varepsilon}_Q","kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-df029d98a9281d80","tex":"\\eps_{r,e}","aliases":"eps_r,e ε_r,e eps varepsilon","display":"{\\varepsilon}_{r,e}","kind":"definition-passage","excerpt":"[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2957ec81e99ffcd8","tex":"\\eps_{\\rm rank}","aliases":"eps_rm rank ε_rm rank eps rm varepsilon rm","display":"{\\varepsilon}_{\\rm rank}","kind":"definition-passage","excerpt":"[Conditioning bottleneck certificate] A rank-revealing parent has a conditioning bottleneck with margin κ>0 when: item its full-rank rate is at least 1-ε_rm rank; item its weighted information floor is below the declared stability floor; item a conditioning-aware policy using the same query budget improves development metaproductivity by at least κ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3ca7c7fda875b94f","tex":"\\eps_{\\mathrm{sem},e}","aliases":"eps_sem,e ε_sem,e eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{sem},e}","kind":"definition-passage","excerpt":"[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-41f956533bc1c43a","tex":"\\mathrm{eq}","aliases":"eq mathrm","display":"\\mathrm{eq}","kind":"definition-passage","excerpt":"[Closed-walk equalizer core] For a possibly noninvertible evaluator graph, let W(v) be the monoid of directed closed walks at v. Define E_v^eq := ⋂_γ∈ W(v)ker(H_γ-I).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e5516c77dd40feea","tex":"\\mathrm{eval}","aliases":"eval mathrm","display":"\\mathrm{eval}","kind":"definition-passage","excerpt":"[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, mea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"definition-passage","excerpt":"[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-31d7232e6fa917d3","tex":"F_i","aliases":"F_i","display":"F_i","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-afc9983b7b83826b","tex":"F_K","aliases":"F_K","display":"F_K","kind":"definition-passage","excerpt":"[Shapley meta-headroom] For i∈ N, define h_i(K) = ∑_R⊆ N∖{i} frac|R|!(m-|R|-1)!m! [ F_K(R∪{i})-F_K(R) ].","count":18},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-46e2c41582135f28","tex":"\\mathcal F_{\\mathrm{post}}","aliases":"F_post mathcal mathrm","display":"\\mathcal F_{\\mathrm{post}}","kind":"definition-passage","excerpt":"[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"definition-passage","excerpt":"[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cea98ff9d672e05b","tex":"F_{t+1}","aliases":"F_t+1","display":"F_{t+1}","kind":"definition-passage","excerpt":"[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-112dbc53c71d6e73","tex":"F_t^c","aliases":"F_t^c","display":"F_t^c","kind":"definition-passage","excerpt":"[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rul","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c4f568dc946ddd98","tex":"\\Fcal","aliases":"Fcal F mathcal","display":"{\\mathcal{F}}","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":20},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7bcbfad9635bd1be","tex":"\\mathrm{fit}","aliases":"fit mathrm","display":"\\mathrm{fit}","kind":"definition-passage","excerpt":"[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair eff","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-15e971c7a974a6e3","tex":"\\mathrm{foreign}","aliases":"foreign mathrm","display":"\\mathrm{foreign}","kind":"definition-passage","excerpt":"[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A","count":16},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8310b480f63b2214","tex":"\\mathsf{full}","aliases":"full mathsf","display":"\\mathsf{full}","kind":"definition-passage","excerpt":"[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 25","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7dbcc819defb6916","tex":"\\mathsf{full}^{-\\mathrm{sem}}","aliases":"full^-sem mathsf mathrm","display":"\\mathsf{full}^{-\\mathrm{sem}}","kind":"definition-passage","excerpt":"[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 25","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f3bce720e4deb824","tex":"\\mathrm{future}","aliases":"future mathrm","display":"\\mathrm{future}","kind":"definition-passage","excerpt":"[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9b7728b48f00b50d","tex":"\\bm g","aliases":"g bm","display":"\\bm g","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"definition-passage","excerpt":"[Research-compiler robustness environment] A research-compiler robustness environment is a tuple U = ( Q, H, A, V, G ), where: item Q is a family of research-problem distributions and generator shifts; item H is a family of archive contaminations and provenance perturbations; item A is a family of integrity, specification, and selection attacks; it","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6f57c86e47f3bde3","tex":"\\mathsf G","aliases":"G mathsf","display":"\\mathsf G","kind":"definition-passage","excerpt":"[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint,","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-28a0f54c10c0e63f","tex":"G_A","aliases":"G_A","display":"G_A","kind":"definition-passage","excerpt":"[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ab6c525d71bce701","tex":"g_A","aliases":"g_A","display":"g_A","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9bdcbeadb9e38bb9","tex":"G_e","aliases":"G_e","display":"G_e","kind":"definition-passage","excerpt":"[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ec1c980979f73c29","tex":"G_j","aliases":"G_j","display":"G_j","kind":"definition-passage","excerpt":"[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e428cd7189f9862a","tex":"\\mathcal G_{\\rm gauge}","aliases":"G_rm gauge mathcal rm","display":"\\mathcal G_{\\rm gauge}","kind":"definition-passage","excerpt":"[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden f","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b850cab9b7de4f08","tex":"G_{\\gamma}","aliases":"G_γ gamma","display":"G_{\\gamma}","kind":"definition-passage","excerpt":"[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-544b717890c93434","tex":"G_\\Xi","aliases":"G_Ξ Xi","display":"G_\\Xi","kind":"definition-passage","excerpt":"[Matched-budget foreign metaproductivity] For a foreign target distribution D_rm for and budget B, define M_B^rm for( P) = E_A∼D_rm for [G_Ξ( P,A)], provided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bed42666d905a9fb","tex":"G_\\ell","aliases":"G_ℓ ell","display":"G_\\ell","kind":"definition-passage","excerpt":"[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8c918c284a1665cd","tex":"\\Gcal","aliases":"Gcal G mathcal","display":"{\\mathcal{G}}","kind":"definition-passage","excerpt":"[Conservative grammar extension] An embedding ι_t:G_t↪G_t+1 is conservative when: item every old well-typed expression remains well typed; item every old derivation translates to a derivation; item the interpretation of every old expression on old evidence is unchanged; item protected evidence identifiers and provenance resources are transported id","count":51},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d2d8ce8f8dca555a","tex":"\\Gcal'","aliases":"Gcal' G' Gcal mathcal","display":"{\\mathcal{G}}'","kind":"definition-passage","excerpt":"[Relative grammar-expansion witness] Let G↪G' be a candidate extension and let r' be the transported defect. The extension exposes a missing correction direction with margin κ>0 when q_G'(r') ≤ q_G(r)-κ.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-26d0b7a6348ae155","tex":"\\Gcal_{17}","aliases":"Gcal_17 G_17 Gcal mathcal","display":"{\\mathcal{G}}_{17}","kind":"definition-passage","excerpt":"[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 25","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-543992e9714793d1","tex":"\\Gcal_{\\mathrm{chart}}","aliases":"Gcal_chart G_chart Gcal mathrm mathcal mathrm","display":"{\\mathcal{G}}_{\\mathrm{chart}}","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-829aeeb3e7e76dbb","tex":"\\Gcal_{\\mathrm{glob}}","aliases":"Gcal_glob G_glob Gcal mathrm mathcal mathrm","display":"{\\mathcal{G}}_{\\mathrm{glob}}","kind":"definition-passage","excerpt":"[Inherited global-affine grammar] The inherited bridge grammar is G_glob = setbar T:ℝ^d+1→ℝ^q linear in homogeneous coordinates. It fits one affine source-to-target map across every observable context.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c495ad95d9ea3642","tex":"\\Gcal_t","aliases":"Gcal_t G_t Gcal mathcal","display":"{\\mathcal{G}}_t","kind":"definition-passage","excerpt":"[Conservative grammar extension] An embedding ι_t:G_t↪G_t+1 is conservative when: item every old well-typed expression remains well typed; item every old derivation translates to a derivation; item the interpretation of every old expression on old evidence is unchanged; item protected evidence identifiers and provenance resources are transported id","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-263c2745c9743dfd","tex":"\\Gcal_{t+1}","aliases":"Gcal_t+1 G_t+1 Gcal mathcal","display":"{\\mathcal{G}}_{t+1}","kind":"definition-passage","excerpt":"[Conservative grammar extension] An embedding ι_t:G_t↪G_t+1 is conservative when: item every old well-typed expression remains well typed; item every old derivation translates to a derivation; item the interpretation of every old expression on old evidence is unchanged; item protected evidence identifiers and provenance resources are transported id","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b0fbb8a119ae0bef","tex":"\\mathrm{geo}","aliases":"geo mathrm","display":"\\mathrm{geo}","kind":"definition-passage","excerpt":"[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation whe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dbaf3bbf176dd3a0","tex":"\\mathrm{glob}","aliases":"glob mathrm","display":"\\mathrm{glob}","kind":"definition-passage","excerpt":"[Inherited global-affine grammar] The inherited bridge grammar is G_glob = setbar T:ℝ^d+1→ℝ^q linear in homogeneous coordinates. It fits one affine source-to-target map across every observable context.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d383049eb85dd666","tex":"\\mathfrak h","aliases":"h mathfrak","display":"\\mathfrak h","kind":"definition-passage","excerpt":"[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-823170eb09d88517","tex":"\\mathsf H","aliases":"H mathsf","display":"\\mathsf H","kind":"definition-passage","excerpt":"[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A","count":193},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1e9de57bf29ef2a6","tex":"h'","aliases":"h'","display":"h'","kind":"definition-passage","excerpt":"[Deterministically separating probe family] Suppose every K_h^p is a point mass at an outcome y(h,p). The family P_probe separates H when for every h≠ h' there exists p∈P_probe such that y(h,p)≠ y(h',p).","count":13},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"definition-passage","excerpt":"[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a0ef8a5a8d16d00c","tex":"H^K","aliases":"H^K","display":"H^K","kind":"definition-passage","excerpt":"[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hoo","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-93e99e00d3c3de51","tex":"H^U_\\gamma","aliases":"H^U_γ gamma","display":"H^U_\\gamma","kind":"definition-passage","excerpt":"[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f8118b659ac23340","tex":"H^V_\\gamma","aliases":"H^V_γ gamma","display":"H^V_\\gamma","kind":"definition-passage","excerpt":"[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"definition-passage","excerpt":"[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hoo","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c133281fc7cd7eab","tex":"H_{a,c}","aliases":"H_a,c","display":"H_{a,c}","kind":"definition-passage","excerpt":"[Finite Neuralese semantic bundle] A finite architecture--context Neuralese bundle is a tuple E = ( A, C, {H_a,c}_a,c, {J_a,c}_a,c, R ), where H_a,c is a latent trajectory space, J_a,c is an externally calibrated semantic readout, and R is a frozen chart-admission rule. No global identification among the fibers is assumed.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-097615d1cce1b4b2","tex":"H_b","aliases":"H_b","display":"H_b","kind":"definition-passage","excerpt":"[Response alias] Two reducing blocks H_a,H_b are response aliased under R when σ_R(H_a)=σ_R(H_b).","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b47a9eb9c1e2a501","tex":"\\mathcal H_{\\mathrm{episodes}}","aliases":"H_episodes mathcal mathrm","display":"\\mathcal H_{\\mathrm{episodes}}","kind":"definition-passage","excerpt":"[Repair abstraction operator] A repair abstraction operator is a map a: H_episodes ⟶ overlineO that maps one or more concrete episodes to a typed schema with a failure signature, applicability condition, correction program, effect profile, and receipt template.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d3cba85783189f4d","tex":"H_i","aliases":"H_i","display":"H_i","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-333bf23be8bc780a","tex":"h_i","aliases":"h_i","display":"h_i","kind":"definition-passage","excerpt":"[Shapley meta-headroom] For i∈ N, define h_i(K) = ∑_R⊆ N∖{i} frac|R|!(m-|R|-1)!m! [ F_K(R∪{i})-F_K(R) ].","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b9197c23b01e1938","tex":"h_i^{\\mathrm{code}}","aliases":"h_i^code mathrm","display":"h_i^{\\mathrm{code}}","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e7ea3ccb6e0ecbe1","tex":"h_i^{\\mathrm{spec}}","aliases":"h_i^spec mathrm","display":"h_i^{\\mathrm{spec}}","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6ae2edc4345665e2","tex":"H_K","aliases":"H_K","display":"H_K","kind":"definition-passage","excerpt":"[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hoo","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8911fccc15deb3e8","tex":"h_K","aliases":"h_K","display":"h_K","kind":"definition-passage","excerpt":"[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hoo","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6f4f690123aae57e","tex":"H_{\\rm aff}","aliases":"H_rm aff rm","display":"H_{\\rm aff}","kind":"definition-passage","excerpt":"[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-082f50e75cb09005","tex":"H_{\\rm non}","aliases":"H_rm non rm","display":"H_{\\rm non}","kind":"definition-passage","excerpt":"[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-619ba501c7bc28e9","tex":"H_s","aliases":"H_s","display":"H_s","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":28},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-814f64e7253b7d27","tex":"H_t","aliases":"H_t","display":"H_t","kind":"definition-passage","excerpt":"[Distributionally stable Neuralese transport] Let M be a declared family of context distributions. A probe--bridge pair (P,B) is (ε,δ)-stable on M when, for every μ∈ M outside an exceptional set of declared probability at most δ, normY-PBH_t_L^2(μ)≤ε, and the corresponding source-reference, alignment, protected-regression, and abstention gates all ","count":30},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-330725cd618f011d","tex":"\\mathfrak h_v","aliases":"h_v mathfrak","display":"\\mathfrak h_v","kind":"definition-passage","excerpt":"[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4906fdd4162c9b6e","tex":"H_\\gamma","aliases":"H_γ gamma","display":"H_\\gamma","kind":"definition-passage","excerpt":"[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a c","count":157},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-94878d1cb329662d","tex":"I_{H_a}","aliases":"I_H_a","display":"I_{H_a}","kind":"definition-passage","excerpt":"[Block-scalar response signature] Let H=⊕_a=1^kH_a be a reducing decomposition for every R_j. The response language is block scalar when restrR_jH_a=λ_ajI_H_a for scalars λ_aj. The vector σ_R(H_a) = (λ_a1,…,λ_am) is the response signature of block H_a.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b2d01566586c29fd","tex":"I_{H_b}","aliases":"I_H_b","display":"I_{H_b}","kind":"definition-passage","excerpt":"[Separating response operator] A candidate S separates an aliased pair (H_a,H_b) when both subspaces reduce S and restrSH_a=α I_H_a, restrSH_b=β I_H_b, α≠β.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-41bd5b0100b2ea43","tex":"\\Ical","aliases":"Ical I mathcal","display":"{\\mathcal{I}}","kind":"definition-passage","excerpt":"[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].","count":24},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f846c1c5c0ec0854","tex":"\\Ical_{\\gamma}^{\\omega}","aliases":"Ical_γ^ω I_γ^ω Ical gamma omega mathcal gamma omega","display":"{\\mathcal{I}}_{\\gamma}^{\\omega}","kind":"definition-passage","excerpt":"[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0f6f9e87439c63ef","tex":"\\mathrm{id}","aliases":"id mathrm","display":"\\mathrm{id}","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8c583f8a14257381","tex":"\\mathrm{id}_A","aliases":"id_A mathrm","display":"\\mathrm{id}_A","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-690d3dc3a0f5422e","tex":"\\mathrm{id}_B","aliases":"id_B mathrm","display":"\\mathrm{id}_B","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"definition-passage","excerpt":"[Claim gate] For claim level ℓ, let G_ℓ(E,Π)∈{0,1} be a precommitted predicate over revealed evidence and receipts. The claim is admissible only if every prerequisite gate is one: Admissible(ℓ) iff bigwedge_j≤ℓG_j(E,Π)=1.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-810ddc61406ded0f","tex":"\\mathrm{imp}","aliases":"imp mathrm","display":"\\mathrm{imp}","kind":"definition-passage","excerpt":"[Improvement reproduction number] The local repair-discovery reproduction number is R_imp = rad(M).","count":14},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ef28ee4048cca398","tex":"\\mathrm{in}","aliases":"in mathrm","display":"\\mathrm{in}","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"definition-passage","excerpt":"[Core adequacy modulus] For a subspace F⊆ E with orthogonal projector P_F, define μ(F;Δ_prot) := inf_d∈Δ_prot fracnormP_Fdnorm d.","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e016784d80dc10ab","tex":"\\inf_{d\\in\\Delta_{\\mathrm{prot}}}","aliases":"inf_d∈Δ_prot inf in Delta mathrm","display":"\\inf_{d\\in\\Delta_{\\mathrm{prot}}}","kind":"definition-passage","excerpt":"[Core adequacy modulus] For a subspace F⊆ E with orthogonal projector P_F, define μ(F;Δ_prot) := inf_d∈Δ_prot fracnormP_Fdnorm d.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0f0219abbd41b48c","tex":"\\inf_{u\\in\\mathfrak U}","aliases":"inf_u∈ U inf in mathfrak","display":"\\inf_{u\\in\\mathfrak U}","kind":"definition-passage","excerpt":"[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and u","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-36cf2a7c1e40dc3e","tex":"\\mathrm{Instability}","aliases":"Instability mathrm","display":"\\mathrm{Instability}","kind":"definition-passage","excerpt":"[Cover adequacy score] For a candidate functional cover U, define the diagnostic score A( U) := η_loc( U) +τη_glue( U) +λ Complexity( U) +γ Instability( U).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a9528ef0ceabc08c","tex":"\\mathrm{invalid}","aliases":"invalid mathrm","display":"\\mathrm{invalid}","kind":"definition-passage","excerpt":"[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operat","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b0c1f9a561c7c5ac","tex":"\\mathrm{irr}","aliases":"irr mathrm","display":"\\mathrm{irr}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Packet leakage profile] The directed leakage from correction packet j into defect packet i is ℓ_ij:=normP_iDQ_j, i≠ j. The matrix (ℓ_ij) is the packet leakage profile.","count":132},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-46b404703f0d83b0","tex":"J_{a,c}","aliases":"J_a,c","display":"J_{a,c}","kind":"definition-passage","excerpt":"[Finite Neuralese semantic bundle] A finite architecture--context Neuralese bundle is a tuple E = ( A, C, {H_a,c}_a,c, {J_a,c}_a,c, R ), where H_a,c is a latent trajectory space, J_a,c is an externally calibrated semantic readout, and R is a frozen chart-admission rule. No global identification among the fibers is assumed.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2d5c793bf26d9108","tex":"J_w","aliases":"J_w","display":"J_w","kind":"definition-passage","excerpt":"to w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected his","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.","count":142},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fb9dab5747004561","tex":"K'","aliases":"K'","display":"K'","kind":"definition-passage","excerpt":"[Matched selector comparison] Two selectors K and K' are compared under matched resources when, episode by episode, they receive: item the identical candidate bundle C_e in the identical order; item the identical receipt fields and calibration evidence; item the identical target, anchor, and audit data; item the identical candidate-construction bud","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"definition-passage","excerpt":"[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.","count":42},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cf5fcf757b255a84","tex":"K_1","aliases":"K_1","display":"K_1","kind":"definition-passage","excerpt":"[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.","count":52},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-488c69dc1fe1e49c","tex":"K_{A_t}^{\\delta}","aliases":"K_A_t^δ delta","display":"K_{A_t}^{\\delta}","kind":"definition-passage","excerpt":"[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a6d60dae6df93c93","tex":"K_{\\mathrm{edit}}","aliases":"K_edit mathrm","display":"K_{\\mathrm{edit}}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-55674dc77ced84ea","tex":"K_h^p","aliases":"K_h^p","display":"K_h^p","kind":"definition-passage","excerpt":"[Deterministically separating probe family] Suppose every K_h^p is a point mass at an outcome y(h,p). The family P_probe separates H when for every h≠ h' there exists p∈P_probe such that y(h,p)≠ y(h',p).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-46d21cf1bf7572c1","tex":"K_m","aliases":"K_m","display":"K_m","kind":"definition-passage","excerpt":"[Finite repeated-lineage certificate] A sequence K_0xrightarrowΠ_1K_1xrightarrowΠ_2⋯xrightarrowΠ_mK_m has an m-edge repeated-lineage certificate when every edge is productive in its own predeclared protected context, every parent receipt is replayed before the child is compiled, no final audit evidence is reused as development evidence for a later ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f1f34832ebc20f0f","tex":"K_{\\mathrm{protected}}","aliases":"K_protected mathrm","display":"K_{\\mathrm{protected}}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fd51be9cb0ff1729","tex":"K_T","aliases":"K_T","display":"K_T","kind":"definition-passage","excerpt":"[Repeated-lineage certificate] A finite lineage path K_0→ K_1→⋯→ K_T has a repeated-lineage metaproductivity certificate when: item every edge uses disjoint development, meta-validation, and final audit evidence; item candidate, query, compute, and validation resources are fixed or explicitly accounted; item every edge has a positive conditional lo","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ab625c241c8a0e88","tex":"K_t","aliases":"K_t","display":"K_t","kind":"definition-passage","excerpt":"ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign ","count":22},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-65af3701b4164730","tex":"K_{t+1}","aliases":"K_t+1","display":"K_{t+1}","kind":"definition-passage","excerpt":"ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign ","count":14},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"definition-passage","excerpt":"[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).","count":49},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"definition-passage","excerpt":"[Distributionally stable Neuralese transport] Let M be a declared family of context distributions. A probe--bridge pair (P,B) is (ε,δ)-stable on M when, for every μ∈ M outside an exceptional set of declared probability at most δ, normY-PBH_t_L^2(μ)≤ε, and the corresponding source-reference, alignment, protected-regression, and abstention gates all ","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"definition-passage","excerpt":"[Open-set ascending lineage] A lineage is conservatively ascending when L_0preceq_consL_1 preceq_consL_2 preceq_cons⋯. It is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"definition-passage","excerpt":"[Open-set ascending lineage] A lineage is conservatively ascending when L_0preceq_consL_1 preceq_consL_2 preceq_cons⋯. It is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"definition-passage","excerpt":"[Open-set ascending lineage] A lineage is conservatively ascending when L_0preceq_consL_1 preceq_consL_2 preceq_cons⋯. It is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-32067f05b0fb68a7","tex":"L_I","aliases":"L_I","display":"L_I","kind":"definition-passage","excerpt":"[Irreversible-effect budget] When perfect buffering is impossible, let I(τ) be the set of irreversible external effects produced before decision. A transaction is η-reversible relative to a loss functional L_I when L_I(I(τ))≤η. The quantity η must be part of the precommitment and acceptance rule.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fc35dabd558a2331","tex":"L_{\\rm dev}","aliases":"L_rm dev rm","display":"L_{\\rm dev}","kind":"definition-passage","excerpt":"[One-step bounded recursive meta-improvement] Let P_0 be a parent repair process. Let a lineage compiler Meta receive only a development ledger L_rm dev and return P_1=Meta( P_0,L_rm dev). The edge P_0→ P_1 is a one-step bounded recursive meta-improvement on (D_rm for,B) when: item the descendant is sealed before the audit distribution is revealed;","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-49e6cb003fe8b536","tex":"L_t","aliases":"L_t","display":"L_t","kind":"definition-passage","excerpt":"[Conservative lineage dominance] Write L_tpreceq_consL_t+1 when: item the old grammar, evidence ledger, and proof objects embed into the new lineage state; item every old protected interpretation remains recoverable within the declared semantic-debt bound; item old authorized behaviors replay within declared tolerances; item the new correction imag","count":13},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9f64f718d6a46c28","tex":"L_{t+1}","aliases":"L_t+1","display":"L_{t+1}","kind":"definition-passage","excerpt":"[Conservative lineage dominance] Write L_tpreceq_consL_t+1 when: item the old grammar, evidence ledger, and proof objects embed into the new lineage state; item every old protected interpretation remains recoverable within the declared semantic-debt bound; item old authorized behaviors replay within declared tolerances; item the new correction imag","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8a141d50d7cf3ae9","tex":"\\mathfrak L_{t+1}","aliases":"L_t+1 mathfrak","display":"\\mathfrak L_{t+1}","kind":"definition-passage","excerpt":"[Bounded recursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-01208e159d2aec8c","tex":"\\land","aliases":"land","display":"\\land","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3755b02b61222ecc","tex":"\\Lcal","aliases":"Lcal L mathcal","display":"{\\mathcal{L}}","kind":"definition-passage","excerpt":"[Automated research compiler] An automated research compiler is a tuple Γ = (L,F,P,Q,V,M,R), where: item L is a structured literature and prior-method memory; item F is an ancestor-closed finding graph or shared research forum; item P generates typed candidate methods and pre-result specifications; item Q allocates cheap pilots and full experiments","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-667bf0820457d363","tex":"\\LCB","aliases":"LCB operatorname","display":"{\\operatorname{LCB}}","kind":"definition-passage","excerpt":"[Statistically validated improvement] A candidate B is a statistically validated (Δ,α)-improvement over A when: item mathscr I(A,B)=1; item both systems are evaluated within the same resource envelope B; item a predeclared lower confidence bound satisfies LCB_1-α (J_Ξ(B)-J_Ξ(A)) ≥Δ; item the evidence ledger used by the test is append-only across th","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e3c837a0847a1385","tex":"\\LCB_{0.95}","aliases":"LCB_0.95 LCB operatorname","display":"{\\operatorname{LCB}}_{0.95}","kind":"definition-passage","excerpt":"[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0bb346dd2a9be2c6","tex":"\\LCB_{1-\\alpha}","aliases":"LCB_1-α LCB alpha operatorname alpha","display":"{\\operatorname{LCB}}_{1-\\alpha}","kind":"definition-passage","excerpt":"[Statistically validated improvement] A candidate B is a statistically validated (Δ,α)-improvement over A when: item mathscr I(A,B)=1; item both systems are evaluated within the same resource envelope B; item a predeclared lower confidence bound satisfies LCB_1-α (J_Ξ(B)-J_Ξ(A)) ≥Δ; item the evidence ledger used by the test is append-only across th","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9fa2e4d2a26b7dcf","tex":"\\mathrm{learn}","aliases":"learn mathrm","display":"\\mathrm{learn}","kind":"definition-passage","excerpt":"[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from sou","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"definition-passage","excerpt":"[Finite exact realization] A finite runtime exactly realizes M when α is surjective on reachable abstract states and every abstract authorized edge has at least one executable realizing path with matching receipt and observation.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f0042c6cf9ee0b72","tex":"\\mathfrak M","aliases":"M mathfrak","display":"\\mathfrak M","kind":"definition-passage","excerpt":"[Sealed meta-compiler] A sealed meta-compiler is a map M: ( D_rm dev, D_rm meta,K_0,Γ) ⟼ (K_1,Π_1), where Γ is a declared finite meta-grammar and Π_1 is a descendant receipt. The compiler is sealed when: item neither final audit split is supplied to M; item Π_1 commits to the child state, parent identity, input hashes, grammar, selected configurati","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b02117d0f899eb57","tex":"M^{\\mathrm{foreign}}_{h,B}","aliases":"M^foreign_h,B mathrm","display":"M^{\\mathrm{foreign}}_{h,B}","kind":"definition-passage","excerpt":"[Bounded recursive self-improvement] A lineage exhibits bounded recursive self-improvement over a declared generation set when: item protected task utility improves on a nontrivial held-out family; item M^foreign_h,B increases under matched budgets; item task-relevant structural repair capacity increases on a held-out defect family; item evidence, ","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d97f4789c0cd0133","tex":"M_B^{\\rm for}","aliases":"M_B^rm for rm","display":"M_B^{\\rm for}","kind":"definition-passage","excerpt":"[Matched-budget foreign metaproductivity] For a foreign target distribution D_rm for and budget B, define M_B^rm for( P) = E_A∼D_rm for [G_Ξ( P,A)], provided every compared process receives the same target correspondence, calibration, candidate, and irreversible-effect budgets.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c3155e8e2daf28a9","tex":"M_{h,B}^{\\mathrm{foreign}}","aliases":"M_h,B^foreign mathrm","display":"M_{h,B}^{\\mathrm{foreign}}","kind":"definition-passage","excerpt":"eta-improvement] A lineage edge K_t→ K_t+1 is a one-step recursive meta-improvement when: item K_t+1 was constructed by an authorized transaction whose proposal process used K_t; item the edge passed an independent witness and preserved the declared evidence and rollback invariants; item under matched budgets on a frozen foreign-agent distribution,","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"definition-passage","excerpt":"[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"definition-passage","excerpt":"[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).","count":42},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8a2608d81ff3c7d8","tex":"\\max_{e\\in E\\setminus T}","aliases":"max_e∈ E∖ T max in setminus","display":"\\max_{e\\in E\\setminus T}","kind":"definition-passage","excerpt":"[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d80123f93a0ffd9","tex":"\\Mcal","aliases":"Mcal M mathcal","display":"{\\mathcal{M}}","kind":"definition-passage","excerpt":"[ε-saturation] A module set R⊆ N is ε-saturated at K when F_K(R)-M(K)≤ε.","count":25},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f73b2c8609cfa86f","tex":"\\Mcal_{\\Xi_t}","aliases":"Mcal_Ξ_t M_Ξ_t Mcal Xi mathcal Xi","display":"{\\mathcal{M}}_{\\Xi_t}","kind":"definition-passage","excerpt":"[Context-indexed lineage edge] Let Ξ_t be the protected context for edge t→ t+1. A sealed lineage edge K_txrightarrowΠ_t+1K_t+1 is γ_t-productive in Ξ_t when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies M_Ξ_t(K_t+1)-M_Ξ_t(K_t)≥ γ_t>0.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b00c568f08b1df42","tex":"\\Meta","aliases":"Meta mathrm","display":"{\\mathrm{Meta}}","kind":"definition-passage","excerpt":"[One-step bounded recursive meta-improvement] Let P_0 be a parent repair process. Let a lineage compiler Meta receive only a development ledger L_rm dev and return P_1=Meta( P_0,L_rm dev). The edge P_0→ P_1 is a one-step bounded recursive meta-improvement on (D_rm for,B) when: item the descendant is sealed before the audit distribution is revealed;","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":23},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":39},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-742557a8a4b4aa70","tex":"\\min_{c\\in C}","aliases":"min_c∈ C min in","display":"\\min_{c\\in C}","kind":"definition-passage","excerpt":"b19-atlas A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abstains.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d6ce51886365878b","tex":"\\mathrm{model}","aliases":"model mathrm","display":"\\mathrm{model}","kind":"definition-passage","excerpt":"[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"definition-passage","excerpt":"[Shapley meta-headroom] For i∈ N, define h_i(K) = ∑_R⊆ N∖{i} frac|R|!(m-|R|-1)!m! [ F_K(R∪{i})-F_K(R) ].","count":70},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0c883e9d04894df7","tex":"\\mathrm{new}","aliases":"new mathrm","display":"\\mathrm{new}","kind":"definition-passage","excerpt":"[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, mea","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":163},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"definition-passage","excerpt":"[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-962eb33121c4bcf4","tex":"O_i","aliases":"O_i","display":"O_i","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7fc03b5db3831c26","tex":"o_i","aliases":"o_i","display":"o_i","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob operatorname","display":"{\\operatorname{Ob}}","kind":"definition-passage","excerpt":"[Authorized obstruction class] The authorized obstruction to correcting r is Ob_D(r)=[r]∈coker D:=V/im D.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-46c1284bb2e365e3","tex":"\\Ob_D","aliases":"Ob_D Ob operatorname","display":"{\\operatorname{Ob}}_D","kind":"definition-passage","excerpt":"[Authorized obstruction class] The authorized obstruction to correcting r is Ob_D(r)=[r]∈coker D:=V/im D.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f8c93cc0e25b969f","tex":"\\Ocal_0","aliases":"Ocal_0 O_0 Ocal mathcal","display":"{\\mathcal{O}}_0","kind":"definition-passage","excerpt":"[One-step bounded repair-process improvement] A lineage edge (O_0,K_0) xrightarrowΠ_1 (O_1,K_1) is a one-step bounded repair-process improvement relative to (Q,B,U,R) when: item K_1 is generated and sealed through an authorized lineage transaction; item parent and child use the same candidate budget B and candidate bundle on each target; item a pre","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c06d56a216829cc","tex":"\\Ocal_1","aliases":"Ocal_1 O_1 Ocal mathcal","display":"{\\mathcal{O}}_1","kind":"definition-passage","excerpt":"[One-step bounded repair-process improvement] A lineage edge (O_0,K_0) xrightarrowΠ_1 (O_1,K_1) is a one-step bounded repair-process improvement relative to (Q,B,U,R) when: item K_1 is generated and sealed through an authorized lineage transaction; item parent and child use the same candidate budget B and candidate bundle on each target; item a pre","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1232946c51ec7997","tex":"\\Ocal_{t+1}","aliases":"Ocal_t+1 O_t+1 Ocal mathcal","display":"{\\mathcal{O}}_{t+1}","kind":"definition-passage","excerpt":"[Conservative repair-operad extension] An inclusion ι_t:O_t↪O_t+1 is conservative when: item it is injective on old colors and operations; item it preserves identities, permutations, and every old syntactic composition; item it preserves old preconditions, postconditions, and protected-invariant contracts on old states; item every previously valid ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-99f4082f84c6ad2b","tex":"\\mathrm{old}","aliases":"old mathrm","display":"\\mathrm{old}","kind":"definition-passage","excerpt":"[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7d905bc6806c132f","tex":"\\one","aliases":"one 1 mathbf","display":"{\\mathbf{1}}","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":17},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-44a55b48f8dea448","tex":"\\mathrm{out}","aliases":"out mathrm","display":"\\mathrm{out}","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-787fbab193f03433","tex":"\\overline c","aliases":"overline c overline","display":"\\overline c","kind":"definition-passage","excerpt":"[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e4b45cd66b9c8f4e","tex":"\\overline L","aliases":"overline L overline","display":"\\overline L","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e4faff0b07b45371","tex":"\\overline o","aliases":"overline o overline","display":"\\overline o","kind":"definition-passage","excerpt":"[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c226cd3303f20dba","tex":"\\overline s","aliases":"overline s overline","display":"\\overline s","kind":"definition-passage","excerpt":"[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-23856ee01c2b6c2b","tex":"\\overline T","aliases":"overline T overline","display":"\\overline T","kind":"definition-passage","excerpt":"[Protected obstruction subspace] A protected obstruction subspace is a declared subspace Q_A⊆coker D_A whose nonzero classes must remain detectable after transport. The transport is obstruction-faithful on Q_A when keroverline T_V∩ Q_A=set0.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5ce8ff28224a85a6","tex":"\\overline T_t","aliases":"overline T_t overline","display":"\\overline T_t","kind":"definition-passage","excerpt":"[Repair death] A class [r]∈coker D_t undergoes repair death at t+1 when overline T_t([r])=0, there exists an authorized u∈ U_t+1 satisfying D_t+1u=-T_t^Vr, and the protected defect transport obeys a lower norm bound normT_t^Vr≥κnormr for a declared κ>0.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d210f097d4d82876","tex":"\\overline T_V","aliases":"overline T_V overline","display":"\\overline T_V","kind":"definition-passage","excerpt":"[Protected obstruction subspace] A protected obstruction subspace is a declared subspace Q_A⊆coker D_A whose nonzero classes must remain detectable after transport. The transport is obstruction-faithful on Q_A when keroverline T_V∩ Q_A=set0.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db64b27099ab4aae","tex":"\\overline w","aliases":"overline w overline","display":"\\overline w","kind":"definition-passage","excerpt":"[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-521f775cc2e8a27e","tex":"\\overline\\eps","aliases":"overlineeps overlineε overline eps overline varepsilon","display":"\\overline{\\varepsilon}","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bd0691afbf5e3caf","tex":"\\overline\\eps_D","aliases":"overlineeps_D overlineε_D overline eps overline varepsilon","display":"\\overline{\\varepsilon}_D","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b188b60dfa0a3287","tex":"\\overline\\eps_{\\mathrm{hol}}","aliases":"overlineeps_hol overlineε_hol overline eps mathrm overline varepsilon mathrm","display":"\\overline{\\varepsilon}_{\\mathrm{hol}}","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6bd49d506cd4112f","tex":"\\overline\\eps_J","aliases":"overlineeps_J overlineε_J overline eps overline varepsilon","display":"\\overline{\\varepsilon}_J","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a67d8a1e74772f2e","tex":"\\overline\\eps_{\\mathrm{leak}}","aliases":"overlineeps_leak overlineε_leak overline eps mathrm overline varepsilon mathrm","display":"\\overline{\\varepsilon}_{\\mathrm{leak}}","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51467e8abef599df","tex":"\\overline\\eps_r","aliases":"overlineeps_r overlineε_r overline eps overline varepsilon","display":"\\overline{\\varepsilon}_r","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1c36b2047df5232e","tex":"\\overline{\\Sscr_t}","aliases":"overlineSscr_t overlinemathscrS_t overline Sscr overline mathscr","display":"\\overline{{\\mathscr{S}}_t}","kind":"definition-passage","excerpt":"[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"definition-passage","excerpt":"on[Neuralese reasoner] A finite-step Neuralese reasoner is a tuple N = (X,Y,H,E,B,U,D,Λ,K), where X is an input space, Y an answer space, H a finite-dimensional latent state space, E:X→ H a prompt encoder, B:H×X→ H a backbone transition, U:H× H× H→ H an update gate, D:H×X→ P(Y) an answer readout, Λ:H→[0,1] a halting map, and K a maximum latent-step","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"definition-passage","excerpt":"[Finite repair process] A finite repair process is a tuple P=(π_rm anc, B,π_rm gate,B,Π), where π_rm anc selects target correspondence queries, B estimates a bridge and transports a frozen source repair, π_rm gate commits or abstains, B is the declared resource budget, and Π is the receipt schema.","count":18},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c18367064df5a710","tex":"\\mathfrak p","aliases":"p mathfrak","display":"\\mathfrak p","kind":"definition-passage","excerpt":"[Correction datum] A finite-dimensional correction datum is a tuple C=(U,V,D,A,c, p), where U is a correction space, V is a defect space, D:U→ V is the linearized defect-response map, A⊆ U is the authorized correction set, c:U→[0,∞] is a declared resource cost, and p is a collection of protected predicates or invariant maps.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e99a99588c03ac6","tex":"\\mathsf P","aliases":"P mathsf","display":"\\mathsf P","kind":"definition-passage","excerpt":"[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint,","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"definition-passage","excerpt":"[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-047f44dd3d94f951","tex":"\\mathfrak P_0","aliases":"P_0 mathfrak","display":"\\mathfrak P_0","kind":"definition-passage","excerpt":"[One-step bounded recursive meta-improvement] Let P_0 be a parent repair process. Let a lineage compiler Meta receive only a development ledger L_rm dev and return P_1=Meta( P_0,L_rm dev). The edge P_0→ P_1 is a one-step bounded recursive meta-improvement on (D_rm for,B) when: item the descendant is sealed before the audit distribution is revealed;","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ff6cb30d6b67da5","tex":"\\mathfrak P_1","aliases":"P_1 mathfrak","display":"\\mathfrak P_1","kind":"definition-passage","excerpt":"[One-step bounded recursive meta-improvement] Let P_0 be a parent repair process. Let a lineage compiler Meta receive only a development ledger L_rm dev and return P_1=Meta( P_0,L_rm dev). The edge P_0→ P_1 is a one-step bounded recursive meta-improvement on (D_rm for,B) when: item the descendant is sealed before the audit distribution is revealed;","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"definition-passage","excerpt":"[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.","count":19},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8dd763535525bfa9","tex":"P_{a,d}","aliases":"P_a,d","display":"P_{a,d}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16d50d93212c48e6","tex":"P_{\\mathrm{alias}}","aliases":"P_alias mathrm","display":"P_{\\mathrm{alias}}","kind":"definition-passage","excerpt":"[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair eff","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d2087737dfdb4981","tex":"P_{b,c}","aliases":"P_b,c","display":"P_{b,c}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1a2161cede1a251f","tex":"P_{\\mathrm{ctx}}","aliases":"P_ctx mathrm","display":"P_{\\mathrm{ctx}}","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e9d48e107ca79b17","tex":"P_F","aliases":"P_F","display":"P_F","kind":"definition-passage","excerpt":"[Core adequacy modulus] For a subspace F⊆ E with orthogonal projector P_F, define μ(F;Δ_prot) := inf_d∈Δ_prot fracnormP_Fdnorm d.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"definition-passage","excerpt":"[Packet leakage profile] The directed leakage from correction packet j into defect packet i is ℓ_ij:=normP_iDQ_j, i≠ j. The matrix (ℓ_ij) is the packet leakage profile.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7e32786a5b15df61","tex":"P_J","aliases":"P_J","display":"P_J","kind":"definition-passage","excerpt":"[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4e1c944353041f9e","tex":"\\mathsf{parent}","aliases":"parent mathsf","display":"\\mathsf{parent}","kind":"definition-passage","excerpt":"[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 25","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f8fd7d6e192237ce","tex":"\\mathsf{parent}^{+}","aliases":"parent^+ mathsf","display":"\\mathsf{parent}^{+}","kind":"definition-passage","excerpt":"[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 25","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d940b686bc0907b1","tex":"\\Pcal_e","aliases":"Pcal_e P_e Pcal mathcal","display":"{\\mathcal{P}}_e","kind":"definition-passage","excerpt":"[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rul","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bd180ffff065d18f","tex":"\\mathrm{post}","aliases":"post mathrm","display":"\\mathrm{post}","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-059de914e788ce7f","tex":"\\mathrm{pre}","aliases":"pre mathrm","display":"\\mathrm{pre}","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-71a98ec1e2bb9f0a","tex":"\\preceq","aliases":"preceq","display":"\\preceq","kind":"definition-passage","excerpt":"[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e90e85d8f6c6073","tex":"\\preceq_{\\mathrm{cons}}","aliases":"preceq_cons preceq mathrm","display":"\\preceq_{\\mathrm{cons}}","kind":"definition-passage","excerpt":"[Conservative lineage dominance] Write L_tpreceq_consL_t+1 when: item the old grammar, evidence ledger, and proof objects embed into the new lineage state; item every old protected interpretation remains recoverable within the declared semantic-debt bound; item old authorized behaviors replay within declared tolerances; item the new correction imag","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6767c1753bb2ae82","tex":"\\mathrm{previous}","aliases":"previous mathrm","display":"\\mathrm{previous}","kind":"definition-passage","excerpt":"[Receipt chain] For receipts Π_1,…,Π_T, define Π_t.previous = ∅, · t=1, h(Π_t-1), · t>1.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d3f079b7784b5757","tex":"\\mathrm{prog}","aliases":"prog mathrm","display":"\\mathrm{prog}","kind":"definition-passage","excerpt":"[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation whe","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7aa45a137872cff1","tex":"\\mathrm{protected}","aliases":"protected mathrm","display":"\\mathrm{protected}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fa94555571b50f85","tex":"\\mathrm{provenance\\_ok}","aliases":"provenance_ok mathrm","display":"\\mathrm{provenance\\_ok}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-757e4abf7a5a8cf5","tex":"q_{\\Gcal}","aliases":"q_Gcal q_G Gcal mathcal","display":"q_{{\\mathcal{G}}}","kind":"definition-passage","excerpt":"[Relative grammar-expansion witness] Let G↪G' be a candidate extension and let r' be the transported defect. The extension exposes a missing correction direction with margin κ>0 when q_G'(r') ≤ q_G(r)-κ.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae2f6a43fa146b7f","tex":"q_{\\Gcal'}","aliases":"q_Gcal' q_G' Gcal mathcal","display":"q_{{\\mathcal{G}}'}","kind":"definition-passage","excerpt":"[Relative grammar-expansion witness] Let G↪G' be a candidate extension and let r' be the transported defect. The extension exposes a missing correction direction with margin κ>0 when q_G'(r') ≤ q_G(r)-κ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"definition-passage","excerpt":"[Packet leakage profile] The directed leakage from correction packet j into defect packet i is ℓ_ij:=normP_iDQ_j, i≠ j. The matrix (ℓ_ij) is the packet leakage profile.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bcaccc7de0d00845","tex":"Q_{\\mathrm{spec}}","aliases":"Q_spec mathrm","display":"Q_{\\mathrm{spec}}","kind":"definition-passage","excerpt":"[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7f7e65502d4bed28","tex":"\\mathcal Q_T","aliases":"Q_T mathcal","display":"\\mathcal Q_T","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8141a49faa601213","tex":"Q_v","aliases":"Q_v","display":"Q_v","kind":"definition-passage","excerpt":"[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b55754320a8637cb","tex":"Q_X","aliases":"Q_X","display":"Q_X","kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-79fe9e26e1a59839","tex":"Q_Y","aliases":"Q_Y","display":"Q_Y","kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aaf174f6701cc732","tex":"\\Qcal","aliases":"Qcal Q mathcal","display":"{\\mathcal{Q}}","kind":"definition-passage","excerpt":"[Automated research compiler] An automated research compiler is a tuple Γ = (L,F,P,Q,V,M,R), where: item L is a structured literature and prior-method memory; item F is an ancestor-closed finding graph or shared research forum; item P generates typed candidate methods and pre-result specifications; item Q allocates cheap pilots and full experiments","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R ℝ mathbb","display":"{\\mathbb{R}}","kind":"definition-passage","excerpt":"[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty","count":111},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"definition-passage","excerpt":"[Finite transport grammar] The Block 20 compiler searches the following declared grammar: item global_full: one pooled affine map R^200→ R^200; item chart_step_shared: six chart-indexed affine maps R^40→ R^40 applied independently at every latent step; item chart_full: six chart-indexed affine maps R^200→ R^200.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ac168ec1a0ac655","tex":"\\mathbf r","aliases":"r mathbf","display":"\\mathbf r","kind":"definition-passage","excerpt":"[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"definition-passage","excerpt":"[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, mea","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-32fe36360874826c","tex":"\\mathfrak R","aliases":"R mathfrak","display":"\\mathfrak R","kind":"definition-passage","excerpt":"[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from sou","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-02f6e210c4b44c83","tex":"\\mathsf R","aliases":"R mathsf","display":"\\mathsf R","kind":"definition-passage","excerpt":"[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint,","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"definition-passage","excerpt":"[Relative grammar-expansion witness] Let G↪G' be a candidate extension and let r' be the transported defect. The extension exposes a missing correction direction with margin κ>0 when q_G'(r') ≤ q_G(r)-κ.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e8dce53c4dd25b21","tex":"\\mathbb R^{200}","aliases":"R^200 mathbb","display":"\\mathbb R^{200}","kind":"definition-passage","excerpt":"[Finite transport grammar] The Block 20 compiler searches the following declared grammar: item global_full: one pooled affine map R^200→ R^200; item chart_step_shared: six chart-indexed affine maps R^40→ R^40 applied independently at every latent step; item chart_full: six chart-indexed affine maps R^200→ R^200.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-767dcbdaa13e9883","tex":"R^2_{\\mathrm{probe}}","aliases":"R^2_probe mathrm","display":"R^2_{\\mathrm{probe}}","kind":"definition-passage","excerpt":"[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-78d6b9680d84cb1c","tex":"\\R^3","aliases":"R^3 ℝ^3 R mathbb","display":"{\\mathbb{R}}^3","kind":"definition-passage","excerpt":"[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-649dc083c76939af","tex":"\\mathbb R^{40}","aliases":"R^40 mathbb","display":"\\mathbb R^{40}","kind":"definition-passage","excerpt":"[Finite transport grammar] The Block 20 compiler searches the following declared grammar: item global_full: one pooled affine map R^200→ R^200; item chart_step_shared: six chart-indexed affine maps R^40→ R^40 applied independently at every latent step; item chart_full: six chart-indexed affine maps R^200→ R^200.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bb58fc47ddb31a6c","tex":"\\R^5","aliases":"R^5 ℝ^5 R mathbb","display":"{\\mathbb{R}}^5","kind":"definition-passage","excerpt":"servation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstration loca","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a18ed0a7c155f9d6","tex":"\\R^{5\\times8}","aliases":"R^5×8 ℝ^5×8 R times mathbb times","display":"{\\mathbb{R}}^{5\\times8}","kind":"definition-passage","excerpt":"[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise dis","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5171ebd98eb74b60","tex":"\\R^9","aliases":"R^9 ℝ^9 R mathbb","display":"{\\mathbb{R}}^9","kind":"definition-passage","excerpt":"[Post-connection transport episode] A Block 21 episode is a tuple E = (X,Y,Z,p, D_rm cal, D_rm meta, D_rm audit), where X∈ℝ^5×8 is the protected source packet, Y is an observed post-connection packet or lower-dimensional statistic, Z∈ℝ^9 is the protected semantic witness, p∈ℝ is a bounded diagnostic response, and the three supports are pairwise dis","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c8a169dda6307bc","tex":"\\R^{d+1}","aliases":"R^d+1 ℝ^d+1 R mathbb","display":"{\\mathbb{R}}^{d+1}","kind":"definition-passage","excerpt":"[Inherited global-affine grammar] The inherited bridge grammar is G_glob = setbar T:ℝ^d+1→ℝ^q linear in homogeneous coordinates. It fits one affine source-to-target map across every observable context.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bd767d9848a982be","tex":"\\R^k","aliases":"R^k ℝ^k R mathbb","display":"{\\mathbb{R}}^k","kind":"definition-passage","excerpt":"[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0d726011beb97417","tex":"\\R^q","aliases":"R^q ℝ^q R mathbb","display":"{\\mathbb{R}}^q","kind":"definition-passage","excerpt":"[Evaluator object] An evaluator object at version v is a tuple E_v=(R_v,E_v,ι_v,J_v,Q_v, h_v), where: item R_v is the protected raw-evidence space or ledger span; item E_v is the evaluator's internal evidence-representation space; item ι_v:R_v→ E_v encodes raw evidence; item J_v:E_v→ℝ^q is a score or decision-readout map; item Q_v is an uncertainty","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db16d7a30e82272d","tex":"\\R^{q\\times(d+1)}","aliases":"R^q×(d+1) ℝ^q×(d+1) R times mathbb times","display":"{\\mathbb{R}}^{q\\times(d+1)}","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4bdc1badb352ecd9","tex":"\\R^r","aliases":"R^r ℝ^r R mathbb","display":"{\\mathbb{R}}^r","kind":"definition-passage","excerpt":"[Invariant-context two-chart grammar] Let P_ctx:H_s→ℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7d4e5408470f7103","tex":"\\R_{>0}^{7}","aliases":"R_>0^7 ℝ_>0^7 R mathbb","display":"{\\mathbb{R}}_{>0}^{7}","kind":"definition-passage","excerpt":"[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6e93838ea3bf8d2c","tex":"r_A","aliases":"r_A","display":"r_A","kind":"definition-passage","excerpt":"gindefinition[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. T","count":24},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c6016f5cf675a3ae","tex":"r_B","aliases":"r_B","display":"r_B","kind":"definition-passage","excerpt":"ransport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is exa","count":16},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-55c658da2911c141","tex":"R_e","aliases":"R_e","display":"R_e","kind":"definition-passage","excerpt":"[Core-retractable upgrade] An evaluator morphism e:v→ w is core-retractable when T_e(E_v^flat)⊆ E_w^flat and there is a map R_e:E_w^flat→ E_v^flat such that R_eT_e x=x for every x∈ E_v^flat. It is core-isomorphic when T_e restricts to an isomorphism from E_v^flat onto E_w^flat.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ca8f23c2477cd36d","tex":"\\mathbf r_i","aliases":"r_i mathbf","display":"\\mathbf r_i","kind":"definition-passage","excerpt":"[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-77963a0bc9579ab2","tex":"R_{\\mathrm{imp}}","aliases":"R_imp mathrm","display":"R_{\\mathrm{imp}}","kind":"definition-passage","excerpt":"[Improvement reproduction number] The local repair-discovery reproduction number is R_imp = rad(M).","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a03f214a14787ce8","tex":"R_j","aliases":"R_j","display":"R_j","kind":"definition-passage","excerpt":"[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.","count":18},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c2e2955bb59abc9d","tex":"\\mathfrak R_{\\mathrm{learn}}","aliases":"R_learn mathfrak mathrm","display":"\\mathfrak R_{\\mathrm{learn}}","kind":"definition-passage","excerpt":"[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from sou","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-091fb9d0a9551219","tex":"r_{\\max}","aliases":"r_max max","display":"r_{\\max}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6649dab9c125800d","tex":"\\mathcal R_{\\mathrm{rollback}}","aliases":"R_rollback mathcal mathrm","display":"\\mathcal R_{\\mathrm{rollback}}","kind":"definition-passage","excerpt":"[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, mea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-99bbe0f6af468b02","tex":"R_u","aliases":"R_u","display":"R_u","kind":"definition-passage","excerpt":"[Robust validity-filtered compiler value] For risk weight λ≥0, define C_rm rob(Γ) = inf_u∈ U [ C_u(Γ) - λCVaR_α (-R_u(Γ,τ)) ]. The finite Block 14 realization does not compute the unrestricted infimum. It evaluates a frozen finite stress atlas and reports the mean, groupwise minima, lower-tail gain, negative-gain rate, invalid-selection rate, and u","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2a515d033c7281ab","tex":"R_u^s","aliases":"R_u^s","display":"R_u^s","kind":"definition-passage","excerpt":"[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden f","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8094b6a463eb5091","tex":"R_v^t","aliases":"R_v^t","display":"R_v^t","kind":"definition-passage","excerpt":"[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden f","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9bf70bf07940952b","tex":"R_w","aliases":"R_w","display":"R_w","kind":"definition-passage","excerpt":"[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-68df6b2bfb142937","tex":"r_\\alpha","aliases":"r_α alpha","display":"r_\\alpha","kind":"definition-passage","excerpt":"[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-650ae3274f788b89","tex":"\\rad","aliases":"rad operatorname","display":"{\\operatorname{rad}}","kind":"definition-passage","excerpt":"[Improvement reproduction number] The local repair-discovery reproduction number is R_imp = rad(M).","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-83a4c3bfb660d766","tex":"\\mathsf{rank1}","aliases":"rank1 mathsf","display":"\\mathsf{rank1}","kind":"definition-passage","excerpt":"[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 25","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4b6743c3095cd690","tex":"\\mathsf{rank4}","aliases":"rank4 mathsf","display":"\\mathsf{rank4}","kind":"definition-passage","excerpt":"[Finite Neuralese repair grammar] The Block 17 compiler grammar is G_17 = {parent^+,depth,rank1, rank4,full,full^-sem}. The symbol parent^+ denotes matched-update continuation of the task-blind parent. The final symbol uses full conditioning but removes the semantic-witness loss. Every candidate receives exactly 550 AdamW updates with batch size 25","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-29098d0a684c06f9","tex":"\\mathrm{rel}","aliases":"rel mathrm","display":"\\mathrm{rel}","kind":"definition-passage","excerpt":"[Scale-normalized transport defect] For a declared stabilizer τ>0, define ε_D^rel := fracnormD_BT_U-T_VD_A normD_BnormT_U+normT_VnormD_A+τ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7fd450c48ffe299e","tex":"\\mathsf{replay}","aliases":"replay mathsf","display":"\\mathsf{replay}","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-98b1ff6629224567","tex":"\\restr","aliases":"restr","display":null,"kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bc555f66091182b1","tex":"\\mathrm{risk}","aliases":"risk mathrm","display":"\\mathrm{risk}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-92fb92672670a9ef","tex":"\\mathrm{rollback}","aliases":"rollback mathrm","display":"\\mathrm{rollback}","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-17e6e62cdd37d68b","tex":"\\mathsf S","aliases":"S mathsf","display":"\\mathsf S","kind":"definition-passage","excerpt":"[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint,","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).","count":250},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-45333b1806be2764","tex":"S_e","aliases":"S_e","display":"S_e","kind":"definition-passage","excerpt":"[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-698cd4222a287d75","tex":"S_i","aliases":"S_i","display":"S_i","kind":"definition-passage","excerpt":"[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-154dc0434c592a19","tex":"s_o","aliases":"s_o","display":"s_o","kind":"definition-passage","excerpt":"[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-765b365ec949ce24","tex":"s_{t+1}","aliases":"s_t+1","display":"s_{t+1}","kind":"definition-passage","excerpt":"[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-581c74f8df15cef3","tex":"s_x","aliases":"s_x","display":"s_x","kind":"definition-passage","excerpt":"[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstr","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-81c409120fb4bc98","tex":"s_y","aliases":"s_y","display":"s_y","kind":"definition-passage","excerpt":"[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstr","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-931e712f6fd8a49c","tex":"\\mathrm{sem}","aliases":"sem mathrm","display":"\\mathrm{sem}","kind":"definition-passage","excerpt":"[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fd3afb95d5baebf1","tex":"\\set","aliases":"set","display":null,"kind":"definition-passage","excerpt":"[Protected obstruction subspace] A protected obstruction subspace is a declared subspace Q_A⊆coker D_A whose nonzero classes must remain detectable after transport. The transport is obstruction-faithful on Q_A when keroverline T_V∩ Q_A=set0.","count":36},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4f96b5b34fda23f7","tex":"\\mathrm{sig}","aliases":"sig mathrm","display":"\\mathrm{sig}","kind":"definition-passage","excerpt":"[Distribution-indexed atlas] A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abs","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span span operatorname","display":"{\\operatorname{span}}","kind":"definition-passage","excerpt":"[Thresholded approximate flat core] For τ≥0, define E_τ^flat := spansetx_i:λ_i≤τ, where x_i is an eigenbasis of A.","count":16},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ec2fdd09d5e3a211","tex":"\\mathrm{spec}","aliases":"spec mathrm","display":"\\mathrm{spec}","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fc72604b0c13da35","tex":"\\operatorname{SPre}","aliases":"SPre operatorname","display":"\\operatorname{SPre}","kind":"definition-passage","excerpt":"[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7da82f27f9683558","tex":"\\Sscr","aliases":"Sscr mathscrS mathscr","display":"{\\mathscr{S}}","kind":"definition-passage","excerpt":"[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-69cbc4a5086d09e9","tex":"\\Sscr_t","aliases":"Sscr_t mathscrS_t Sscr mathscr","display":"{\\mathscr{S}}_t","kind":"definition-passage","excerpt":"[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4e4ee8a109009a53","tex":"\\subsetneq","aliases":"subsetneq","display":null,"kind":"definition-passage","excerpt":"[Partial ontology] Let P⊆ Z denote the protected semantic coordinates. A quotient identifies a partial ontology when it is sufficient for a declared subset P_0subsetneq P, returns calibrated ignorance for P∖ P_0, and records coverage cov(q)=fracdim P_0dim P.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-32f99fc05b64a106","tex":"\\succ","aliases":"succ","display":"\\succ","kind":"definition-passage","excerpt":"[Exact context-relative improvement] For agents A,B, write Bsucceq_Ξ A when mathscr I(A,B)=1 and J_Ξ(B)≥ J_Ξ(A). For Δ>0, write Bsucc_Ξ,ΔA when the utility difference is at least Δ.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-406b789903ba06e5","tex":"\\succ_{\\Xi,\\Delta}","aliases":"succ_Ξ,Δ succ Xi Delta","display":"\\succ_{\\Xi,\\Delta}","kind":"definition-passage","excerpt":"[Exact context-relative improvement] For agents A,B, write Bsucceq_Ξ A when mathscr I(A,B)=1 and J_Ξ(B)≥ J_Ξ(A). For Δ>0, write Bsucc_Ξ,ΔA when the utility difference is at least Δ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"definition-passage","excerpt":"[Exact context-relative improvement] For agents A,B, write Bsucceq_Ξ A when mathscr I(A,B)=1 and J_Ξ(B)≥ J_Ξ(A). For Δ>0, write Bsucc_Ξ,ΔA when the utility difference is at least Δ.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-83b20fb1cb683e25","tex":"\\succeq_\\Xi","aliases":"succeq_Ξ succeq Xi","display":"\\succeq_\\Xi","kind":"definition-passage","excerpt":"[Exact context-relative improvement] For agents A,B, write Bsucceq_Ξ A when mathscr I(A,B)=1 and J_Ξ(B)≥ J_Ξ(A). For Δ>0, write Bsucc_Ξ,ΔA when the utility difference is at least Δ.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":14},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1959957164499094","tex":"\\sup_{r\\in\\mathfrak h_v}","aliases":"sup_r∈ h_v sup in mathfrak","display":"\\sup_{r\\in\\mathfrak h_v}","kind":"definition-passage","excerpt":"[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0514daea0e0c22d8","tex":"\\sup_{x\\in K}","aliases":"sup_x∈ K sup in","display":"\\sup_{x\\in K}","kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1198dee6331f7d24","tex":"\\sup_{\\delta\\in\\Dcal_t}","aliases":"sup_δ∈Dcal_t sup_δ∈D_t sup delta in Dcal sup delta in mathcal","display":"\\sup_{\\delta\\in{\\mathcal{D}}_t}","kind":"definition-passage","excerpt":"[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"definition-passage","excerpt":"[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, mea","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-da4fb737a4a4bc72","tex":"\\mathfrak T","aliases":"T mathfrak","display":"\\mathfrak T","kind":"definition-passage","excerpt":"[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is e","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-61c6d37a0f99485a","tex":"T^{c}_{b\\to a}","aliases":"T^c_b→ a to","display":"T^{c}_{b\\to a}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6b5e95a75449f7c6","tex":"T^{d}_{b\\to a}","aliases":"T^d_b→ a to","display":"T^{d}_{b\\to a}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a3724ba98bf5f328","tex":"\\mathfrak T_{A\\to B}","aliases":"T_A→ B mathfrak to","display":"\\mathfrak T_{A\\to B}","kind":"definition-passage","excerpt":"[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is e","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bb1ad3b8487cb395","tex":"T_e","aliases":"T_e","display":"T_e","kind":"definition-passage","excerpt":"[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rul","count":18},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-88548fafcbd748a8","tex":"\\mathcal T_{\\mathrm{new}}","aliases":"T_new mathcal mathrm","display":"\\mathcal T_{\\mathrm{new}}","kind":"definition-passage","excerpt":"[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, mea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-25c4575f88f23416","tex":"T_t^V","aliases":"T_t^V","display":"T_t^V","kind":"definition-passage","excerpt":"[Repair death] A class [r]∈coker D_t undergoes repair death at t+1 when overline T_t([r])=0, there exists an authorized u∈ U_t+1 satisfying D_t+1u=-T_t^Vr, and the protected defect transport obeys a lower norm bound normT_t^Vr≥κnormr for a declared κ>0.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ec00a51e3ddc321","tex":"t_x","aliases":"t_x","display":"t_x","kind":"definition-passage","excerpt":"[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstr","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5176a6e706c8d5ed","tex":"t_y","aliases":"t_y","display":"t_y","kind":"definition-passage","excerpt":"[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstr","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a34699b8745d9855","tex":"\\Tau","aliases":"Tau T mathcal","display":"{\\mathcal{T}}","kind":"definition-passage","excerpt":"[Compiler-language conflict] A finite set S⊆T is a compiler-language conflict for G when all problems in S receive the same inherited diagnostic token but ⋂_τ∈ SA^⋆(τ)=∅. A minimal conflict has no proper conflicting subset.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a238898031259954","tex":"\\mathrm{tests}","aliases":"tests mathrm","display":"\\mathrm{tests}","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"definition-passage","excerpt":"[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.","count":58},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV operatorname","display":"{\\operatorname{TV}}","kind":"definition-passage","excerpt":"[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"definition-passage","excerpt":"[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden f","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"definition-passage","excerpt":"[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.","count":18},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e105a73706668fb5","tex":"U_A","aliases":"U_A","display":"U_A","kind":"definition-passage","excerpt":"[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is e","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c1b855e30cc68627","tex":"U_A^{\\flat}","aliases":"U_A^flat flat","display":null,"kind":"definition-passage","excerpt":"[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-91de088f77bc89c5","tex":"U_B","aliases":"U_B","display":"U_B","kind":"definition-passage","excerpt":"[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is e","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-647fa09689b8fe1a","tex":"U_{t+1}","aliases":"U_t+1","display":"U_{t+1}","kind":"definition-passage","excerpt":"[Repair death] A class [r]∈coker D_t undergoes repair death at t+1 when overline T_t([r])=0, there exists an authorized u∈ U_t+1 satisfying D_t+1u=-T_t^Vr, and the protected defect transport obeys a lower norm bound normT_t^Vr≥κnormr for a declared κ>0.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e67d8e81d3865f41","tex":"U_\\alpha","aliases":"U_α alpha","display":"U_\\alpha","kind":"definition-passage","excerpt":"[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).","count":17},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-697d10e561f07ee9","tex":"u_\\alpha","aliases":"u_α alpha","display":"u_\\alpha","kind":"definition-passage","excerpt":"[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d7884e8dc4c0191e","tex":"\\Ucal","aliases":"Ucal U mathcal","display":"{\\mathcal{U}}","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b54c3e6a2f3cb013","tex":"\\Ucal^{\\le r}","aliases":"Ucal^≤ r U^≤ r Ucal le mathcal le","display":"{\\mathcal{U}}^{\\le r}","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-83f817e6ba9fae50","tex":"\\underline g","aliases":"underline g underline","display":"\\underline g","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4a1cc67b00c2df2e","tex":"\\underline g_A","aliases":"underline g_A underline","display":"\\underline g_A","kind":"definition-passage","excerpt":"[High-probability transport certificate] A random tuple (underline g_A, overlineε_J, overlineε_r, overlineε_D, overlineε_leak, overlineε_hol, overline L) is an (1-α) transport certificate when, jointly with probability at least 1-α, it lower-bounds g_A and upper-bounds every corresponding error or Lipschitz quantity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-baa8270e09e84337","tex":"\\mathrm{unresolved}","aliases":"unresolved mathrm","display":"\\mathrm{unresolved}","kind":"definition-passage","excerpt":"[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operat","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c44f4b33c90c0919","tex":"\\mathsf{use}","aliases":"use mathsf","display":"\\mathsf{use}","kind":"definition-passage","excerpt":"[Evidence namespace] An evidence namespace is a pair (E,use) consisting of protected records and a use ledger. Each evaluation query records which data, metrics, stopping rules, and candidate families were exposed. Reusing evidence is permitted only under a validation rule whose guarantee remains valid conditional on the recorded use history.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1e06ee78de565e2b","tex":"\\mathrm{utility\\_gain}","aliases":"utility_gain mathrm","display":"\\mathrm{utility\\_gain}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"definition-passage","excerpt":"[Research-compiler robustness environment] A research-compiler robustness environment is a tuple U = ( Q, H, A, V, G ), where: item Q is a family of research-problem distributions and generator shifts; item H is a family of archive contaminations and provenance perturbations; item A is a family of integrity, specification, and selection attacks; it","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c965492ad344392","tex":"\\mathsf V","aliases":"V mathsf","display":"\\mathsf V","kind":"definition-passage","excerpt":"[Empirical benchmark instance] A benchmark instance is a tuple B = ( E, A, G, P, W, V, D, S, B, R), where: item E is the environment or transition process; item A is the agent family and functional atlas; item G is the repair grammar and candidate generator; item P is the permitted probe and intervention language; item W is the witness, checkpoint,","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"definition-passage","excerpt":"[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8c41a1fdf1eb317c","tex":"V_A","aliases":"V_A","display":"V_A","kind":"definition-passage","excerpt":"[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is e","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-216bad4941999c1f","tex":"V_A^{\\flat}","aliases":"V_A^flat flat","display":null,"kind":"definition-passage","excerpt":"[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9f5cc10c11d8d9e7","tex":"V_B","aliases":"V_B","display":"V_B","kind":"definition-passage","excerpt":"[Transport datum] Let C_A and C_B be correction data. A transport datum from A to B is a pair T_A→ B=(T_U,T_V), T_U:U_A→ U_B, T_V:V_A→ V_B. Its correction-square defect is Δ_D( T_A→ B) := D_BT_U-T_VD_A. For declared source and target defects r_A∈ V_A and r_B∈ V_B, its defect-identification error is Δ_r( T_A→ B;r_A,r_B) := r_B-T_Vr_A. The datum is e","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-20a23efec08e7107","tex":"\\Valid","aliases":"Valid mathrm","display":"{\\mathrm{Valid}}","kind":"definition-passage","excerpt":"[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operat","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-60d9c4006f0e8bb2","tex":"\\mathrm{valid}","aliases":"valid mathrm","display":"\\mathrm{valid}","kind":"definition-passage","excerpt":"[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operat","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e9c985ad25f7c012","tex":"\\Valid_{\\Ecal,\\Wcal}","aliases":"Valid_Ecal,Wcal Valid_E,W Valid Ecal Wcal mathrm mathcal mathcal","display":"{\\mathrm{Valid}}_{{\\mathcal{E}},{\\mathcal{W}}}","kind":"definition-passage","excerpt":"[Proof-carrying repair operad] A proof-carrying repair operad is a pair O=(overlineO,Valid), where overlineO is a colored symmetric operad of syntactically composable repair programs and Valid_E,W(ω,Π) ∈ {valid,invalid,unresolved} is an evidence- and witness-indexed validation judgment. Only operations judged valid are executable. Unresolved operat","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cb6844a85fd4cf06","tex":"\\mathsf W","aliases":"W mathsf","display":"\\mathsf W","kind":"definition-passage","excerpt":"[Closed-walk equalizer core] For a possibly noninvertible evaluator graph, let W(v) be the monoid of directed closed walks at v. Define E_v^eq := ⋂_γ∈ W(v)ker(H_γ-I).","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι","count":81},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-25a84a8fd44f7f54","tex":"w^\\top","aliases":"w^top top","display":"w^\\top","kind":"definition-passage","excerpt":"[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"definition-passage","excerpt":"[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae58458e74e50c6f","tex":"W_t","aliases":"W_t","display":"W_t","kind":"definition-passage","excerpt":"[Rotating witness schedule] A sequence (E_t,W_t)_t≥0 is a rotating witness schedule when: item E_t∩ W_t=∅ for every t; item W_t is immutable during transaction t; item every component belongs to E_t for at least one t; item every witness replacement is validated by a component outside the same edit set; item protected evidence is transported throug","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9554ffdfe3c2474e","tex":"\\widehat{\\mathbf c}","aliases":"widehat c widehat mathbf","display":"\\widehat{\\mathbf c}","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-14d389e09ea77308","tex":"\\widehat{\\mathbf c}_i","aliases":"widehat c_i widehat mathbf","display":"\\widehat{\\mathbf c}_i","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-45e216c04187fe54","tex":"\\widehat g","aliases":"widehat g widehat","display":"\\widehat g","kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":22},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-342daaae5b2ce5cb","tex":"\\widehat K_{\\phi_t(A_t)}^{\\delta}","aliases":"widehat K_ϕ_t(A_t)^δ widehat phi delta","display":"\\widehat K_{\\phi_t(A_t)}^{\\delta}","kind":"definition-passage","excerpt":"[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a920daa6fa6e6a3","tex":"\\widehat O","aliases":"widehat O widehat","display":"\\widehat O","kind":"definition-passage","excerpt":"ntime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observations commute: widehat O(widehat s)=O(α(widehat s)); item receipt compilation commutes","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e4f5d0b2278e58ab","tex":"\\widehat{\\mathbf r}","aliases":"widehat r widehat mathbf","display":"\\widehat{\\mathbf r}","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5142389b81a71e9d","tex":"\\widehat{\\mathbf r}_i","aliases":"widehat r_i widehat mathbf","display":"\\widehat{\\mathbf r}_i","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-77449d53c5be9d11","tex":"\\widehat S","aliases":"widehat S widehat","display":"\\widehat S","kind":"definition-passage","excerpt":"[Forward implementation refinement] A map α:widehat S→ S forward-refines the runtime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observations c","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7d3704d4c4eab86e","tex":"\\widehat s","aliases":"widehat s widehat","display":"\\widehat s","kind":"definition-passage","excerpt":"[Forward implementation refinement] A map α:widehat S→ S forward-refines the runtime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observations c","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0991feabc19b17ae","tex":"\\widehat s'","aliases":"widehat s' widehat","display":"\\widehat s'","kind":"definition-passage","excerpt":"[Forward implementation refinement] A map α:widehat S→ S forward-refines the runtime when: item every executable initial state maps to an authorized abstract initial state; item for every executable transition widehat sxrightarrowawidehat s', there is an abstract transition path α(widehat s)xrightarrowa,*α(widehat s'); item protected observations c","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c17de24c7659dced","tex":"\\widehat w","aliases":"widehat w widehat","display":"\\widehat w","kind":"definition-passage","excerpt":"[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-999b588fa23f6c96","tex":"\\widehat w_i","aliases":"widehat w_i widehat","display":"\\widehat w_i","kind":"definition-passage","excerpt":"[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e8d6f1354dafb866","tex":"\\widehat x","aliases":"widehat x widehat","display":"\\widehat x","kind":"definition-passage","excerpt":"[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstr","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51fdd71f6572f7c6","tex":"\\widehat y","aliases":"widehat y widehat","display":"\\widehat y","kind":"definition-passage","excerpt":"oℝ^r be a protected context projector whose value is unchanged by the source repair. Let F be a declared family of separators g(h)=1{anglesu,P_ctxh>θ}. The chart extension is G_chart = set(g,bar T_0,bar T_1):g∈F, bar T_j∈ℝ^q×(d+1). Its prediction is widehat y(h)=bar T_g(h)bar h.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-95734e99a6bc63f2","tex":"\\widehat\\delta","aliases":"widehatδ widehat delta","display":"\\widehat\\delta","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9bc4ac46f9980428","tex":"\\widehat\\eta","aliases":"widehatη widehat eta","display":"\\widehat\\eta","kind":"definition-passage","excerpt":"[Quantitative authorized obstruction] The quantitative obstruction magnitude is η_D(r):=normP_obr=dist(r,im D). The normalized obstruction fraction is widehatη_D(r):= η_D(r)/norm r, · r≠0, 0, · r=0.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-de787552fee939db","tex":"\\widehat\\eta_D","aliases":"widehatη_D widehat eta","display":"\\widehat\\eta_D","kind":"definition-passage","excerpt":"[Quantitative authorized obstruction] The quantitative obstruction magnitude is η_D(r):=normP_obr=dist(r,im D). The normalized obstruction fraction is widehatη_D(r):= η_D(r)/norm r, · r≠0, 0, · r=0.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-713eeaa785c879b7","tex":"\\widetilde q","aliases":"widetilde q widetilde","display":"\\widetilde q","kind":"definition-passage","excerpt":"[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from sou","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9fcf9ea0bdd21c78","tex":"\\widetilde T","aliases":"widetilde T widetilde","display":"\\widetilde T","kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-129bc5ddd76ef134","tex":"\\widetilde T_Q","aliases":"widetilde T_Q widetilde","display":"\\widetilde T_Q","kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-318c04501636594a","tex":"\\mathrm{wit}","aliases":"wit mathrm","display":"\\mathrm{wit}","kind":"definition-passage","excerpt":"[Witness-separated ontology transaction] An ontology transaction consists of construction evidence E_con, a compiled quotient q, a pre-reveal commitment σ(q), and witness evidence E_wit. It is witness separated when E_wit is inaccessible to quotient construction and candidate selection, and the final certificate is computed only after σ(q) is fixed","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rul","count":138},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-09868efb784faa54","tex":"\\mathcal X","aliases":"X mathcal","display":"\\mathcal X","kind":"definition-passage","excerpt":"[Learned repair-transport instance] A learned repair-transport instance is a tuple R_learn = ( X,ϕ,q,widetilde q,A_s, A_t, D, P, G, W), where: item X is the input space; item ϕ is the task feature map; item q and widetilde q are protected and corrupted score maps; item A_s is the source agent; item A_t is a family of target agents excluded from sou","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":220},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"definition-passage","excerpt":"[Semantic twin] Two observed trajectories x and x' are semantic twins relative to U when they arise from the same protected task and have the same protected future and intervention laws, while nuisance coordinates, rendering variables, or architecture-specific gauges may differ.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9a7bcf63511fcd36","tex":"x'_i","aliases":"x'_i","display":"x'_i","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b4c8081fe14a0199","tex":"x^{\\mathrm{obs}}","aliases":"x^obs mathrm","display":"x^{\\mathrm{obs}}","kind":"definition-passage","excerpt":"[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstr","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-72e05ef2b0f48e18","tex":"X_c","aliases":"X_c","display":"X_c","kind":"definition-passage","excerpt":"[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-702e017c46584077","tex":"x_c","aliases":"x_c","display":"x_c","kind":"definition-passage","excerpt":"[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-73cc2f41cbac9cf2","tex":"X_d","aliases":"X_d","display":"X_d","kind":"definition-passage","excerpt":"[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f4370da509655e62","tex":"x_d","aliases":"x_d","display":"x_d","kind":"definition-passage","excerpt":"[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"definition-passage","excerpt":"[Thresholded approximate flat core] For τ≥0, define E_τ^flat := spansetx_i:λ_i≤τ, where x_i is an eigenbasis of A.","count":24},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-28e387fe0412fd1f","tex":"X_t^c","aliases":"X_t^c","display":"X_t^c","kind":"definition-passage","excerpt":"[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rul","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-612800f230b89072","tex":"X_t^d","aliases":"X_t^d","display":"X_t^d","kind":"definition-passage","excerpt":"X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rule; item each edge e:c→ d carries a bridge T_e:X_t^c→ X_t^d; item τ_e declares the bridge class; item P_","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-57a532a57d9c63d2","tex":"\\Xcal","aliases":"Xcal X mathcal","display":"{\\mathcal{X}}","kind":"definition-passage","excerpt":"[Neuralese reasoner] A finite-step Neuralese reasoner is a tuple N = (X,Y,H,E,B,U,D,Λ,K), where X is an input space, Y an answer space, H a finite-dimensional latent state space, E:X→ H a prompt encoder, B:H×X→ H a backbone transition, U:H× H× H→ H an update gate, D:H×X→ P(Y) an answer readout, Λ:H→[0,1] a halting map, and K a maximum latent-step b","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"definition-passage","excerpt":"[Lineage-edge receipt] A lineage edge AxrightarrowΠB has a receipt Π= ( id_A, id_B, θ, W, E_pre, E_post, widehat g, mathscr I, tests, decision, rollback, digest ), recording the parent and child, transformation θ, frozen witness, pre- and post-reveal evidence, precommitted effect claim widehat g, protected invariants, validation results, commit dec","count":42},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":94},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f32f37351c2c9bce","tex":"y^{\\mathrm{obs}}","aliases":"y^obs mathrm","display":"y^{\\mathrm{obs}}","kind":"definition-passage","excerpt":"[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstr","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae744f7d0e0e0cac","tex":"Y^{u}_{0:r}","aliases":"Y^u_0:r","display":"Y^{u}_{0:r}","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)),","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f63c0b46cff9ca02","tex":"y_m","aliases":"y_m","display":"y_m","kind":"definition-passage","excerpt":"[Executable affine-program task] A Block 17 task is τ = (x,m,(a_j,b_j)_j=1^m,A_m,B_m,y_m,g,s), y_m=A_mx+B_m, where g∈{A,B} names the generator and s names the data split. The model input is a padded serialization of x, the instructions, and instruction-presence bits. The protected semantic witness is w(τ) = (A_m,fracB_m3,fracy_m4)∈ℝ^3.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f500378d0d9727ec","tex":"Y_{\\mathrm{post}}","aliases":"Y_post mathrm","display":"Y_{\\mathrm{post}}","kind":"definition-passage","excerpt":"[Repair episode] A repair episode is a tuple e= ( A, r, u, Y_pre, Y_post, Π, g ).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7c6fa056d4a0968b","tex":"Y_{\\mathrm{pre}}","aliases":"Y_pre mathrm","display":"Y_{\\mathrm{pre}}","kind":"definition-passage","excerpt":"[Repair episode] A repair episode is a tuple e= ( A, r, u, Y_pre, Y_post, Π, g ).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e4e6a8a761055385","tex":"\\Ycal","aliases":"Ycal Y mathcal","display":"{\\mathcal{Y}}","kind":"definition-passage","excerpt":"[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a c","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c656dc90ffbe53cd","tex":"\\mathcal Z","aliases":"Z mathcal","display":"\\mathcal Z","kind":"definition-passage","excerpt":"[Repair selector] A repair selector is a measurable map K: Z^B⟶{0,…,B-1}, where Z is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is c_e,K(Z_e).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-681b083730d54e5c","tex":"\\mathcal Z^B","aliases":"Z^B mathcal","display":"\\mathcal Z^B","kind":"definition-passage","excerpt":"[Repair selector] A repair selector is a measurable map K: Z^B⟶{0,…,B-1}, where Z is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is c_e,K(Z_e).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"definition-passage","excerpt":"[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hoo","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-285ed8564f3489ae","tex":"z_4","aliases":"z_4","display":"z_4","kind":"definition-passage","excerpt":"[External trace probe] A Block 17 trace probe is a ridge map widehat w=P z_4+q fit only on the 1,200 probe-calibration tasks. The final audit reports held-out coefficient of determination R^2_probe = 1- frac∑_inormw_i-widehat w_i^2 ∑_inormw_i-overline w^2. The frozen semantic-probe gate is R^2_probe≥0.65 on each final support.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-483b6af63cf90eb6","tex":"Z_e","aliases":"Z_e","display":"Z_e","kind":"definition-passage","excerpt":"[Repair selector] A repair selector is a measurable map K: Z^B⟶{0,…,B-1}, where Z is the space of candidate receipts available before the protected audit outcome is revealed. The selected repair is c_e,K(Z_e).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-399e4d09f815b1b6","tex":"Z_K","aliases":"Z_K","display":"Z_K","kind":"definition-passage","excerpt":"[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hoo","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-745416eb807dd8a3","tex":"z_K","aliases":"z_K","display":"z_K","kind":"definition-passage","excerpt":"[Neuralese trace] For input x and fixed depth K, the injected Neuralese trace is Z_K(x)=(z_1(x),…,z_K(x))∈ H^K. The updated trace is H_K(x)=(h_1(x),…,h_K(x)). Every receipt must state which trace convention is being intervened on. Block 16 uses the injected trace because this is the point at which the Hyper runtime exposes an exact intervention hoo","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a7bab79eeca47e83","tex":"Z_M","aliases":"Z_M","display":"Z_M","kind":"definition-passage","excerpt":"[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2b4d236724e03917","tex":"Z_{\\Ucal,r}","aliases":"Z_Ucal,r Z_U,r Ucal mathcal","display":"Z_{{\\mathcal{U}},r}","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7fe17fbad1c2d7e0","tex":"Z_{\\eta}","aliases":"Z_η eta","display":"Z_{\\eta}","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e19496a76c69b8a","tex":"Z_{\\pi}","aliases":"Z_π pi","display":"Z_{\\pi}","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0497c331b71ece6d","tex":"\\mathsf{zero}","aliases":"zero mathsf","display":"\\mathsf{zero}","kind":"definition-passage","excerpt":"[Per-task Neuralese causal coordinates] For task (x_i,y_i), define n_i · :=L(x_i,y_i;zero)-L(x_i,y_i;Z(x_i)), o_i · :=L(x_i,y_i;Z_π(x_i))-L(x_i,y_i;Z(x_i)), c_i · :=L(x_i,y_i;Z_η(x_i))-L(x_i,y_i;Z(x_i)), s_i · :=L(x_i,y_i;Z(x'_i))-L(x_i,y_i;Z(x_i)), r_i · :=absL(x_i,y_i;replay(Z(x_i)))-L(x_i,y_i;Z(x_i)). These are respectively latent necessity, ord","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).","count":33},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2047a00c54476b77","tex":"|_{U_\\alpha}","aliases":"|_U_α alpha","display":"|_{U_\\alpha}","kind":"definition-passage","excerpt":"[Two-level repair obstruction] The local obstruction on U_α is [r|_U_α]∈coker D_α. When every local class vanishes, the global gluing obstruction is [d_αβ]∈check H^1( U,ker D).","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a c","count":88},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6e00e976bbe39280","tex":"\\gamma_{\\min}","aliases":"γ_min gamma min","display":"\\gamma_{\\min}","kind":"definition-passage","excerpt":"[Kernel acceptance predicate] For revealed protected evidence E, the reference acceptance predicate is Accept(widehat s,widehat s',widehatδ,E) iff · K_edit∩ K_protected=∅ · land restrwidehat s'K_protected =restrwidehat sK_protected · land E[utility_gain]≥γ_min · land E[risk]≤ r_max · land E[provenance_ok]=1 · land E_irr=∅.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d72a8a4f1d324e17","tex":"\\gamma_t","aliases":"γ_t gamma","display":"\\gamma_t","kind":"definition-passage","excerpt":"[Context-indexed lineage edge] Let Ξ_t be the protected context for edge t→ t+1. A sealed lineage edge K_txrightarrowΠ_t+1K_t+1 is γ_t-productive in Ξ_t when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies M_Ξ_t(K_t+1)-M_Ξ_t(K_t)≥ γ_t>0.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b652ca7846b4c994","tex":"\\Gamma_{\\Xi,h,B}","aliases":"Γ_Ξ,h,B Gamma Xi","display":"\\Gamma_{\\Xi,h,B}","kind":"definition-passage","excerpt":"ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f16bec24639628c0","tex":"\\delta_J","aliases":"δ_J delta","display":"\\delta_J","kind":"definition-passage","excerpt":"to w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι_v(r). The morphism is exact on protected his","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-052c8b6110b88f43","tex":"\\delta_{\\mathrm{obs}}","aliases":"δ_obs delta mathrm","display":"\\delta_{\\mathrm{obs}}","kind":"definition-passage","excerpt":"[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9e83e11e41159cf8","tex":"\\Delta_Q","aliases":"Δ_Q Delta","display":"\\Delta_Q","kind":"definition-passage","excerpt":"[Selector-level metaproductivity] Let Q be a held-out distribution over target episodes. A child selector K_1 has selector-level metaproductivity over parent K_0 when Δ_Q(K_1,K_0) := E_e∼ Q [ G_e(K_1)-G_e(K_0) ] >0, under a matched comparison, where G_e(K) is protected utility gain relative to the no-op on episode e.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4f55314c9373015f","tex":"\\eta^U_\\gamma","aliases":"η^U_γ eta gamma","display":"\\eta^U_\\gamma","kind":"definition-passage","excerpt":"[Repair holonomy residual] For a correction u and loop γ, define η^U_γ(u) := norm(H^U_γ-I)u. For a defect r, define η^V_γ(r) := norm(H^V_γ-I)r.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-42211a5a157f162e","tex":"\\eta^V_\\gamma","aliases":"η^V_γ eta gamma","display":"\\eta^V_\\gamma","kind":"definition-passage","excerpt":"[Repair holonomy residual] For a correction u and loop γ, define η^U_γ(u) := norm(H^U_γ-I)u. For a defect r, define η^V_γ(r) := norm(H^V_γ-I)r.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cb690c8340ffc699","tex":"\\eta_{0:T}","aliases":"η_0:T eta","display":"\\eta_{0:T}","kind":"definition-passage","excerpt":"[Approximate semantic debt] If an extension preserves old semantics only approximately, let η_t bound its worst protected interpretation error after transport. Under nonexpansive semantic transports, the cumulative debt through stage T is η_0:T ≤ ∑_t=0^T-1η_t.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a3cd056006d98c22","tex":"\\eta_{\\mathrm{hol}}","aliases":"η_hol eta mathrm","display":"\\eta_{\\mathrm{hol}}","kind":"definition-passage","excerpt":"[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a125744dc598f1fd","tex":"\\eta_{\\mathrm{loc}}^2","aliases":"η_loc^2 eta mathrm","display":"\\eta_{\\mathrm{loc}}^2","kind":"definition-passage","excerpt":"[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-501d2904ebeaacf6","tex":"\\eta_{\\mathrm{new}}","aliases":"η_new eta mathrm","display":"\\eta_{\\mathrm{new}}","kind":"definition-passage","excerpt":"[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e5afc7273b6199fc","tex":"\\eta_{\\mathrm{old}}","aliases":"η_old eta mathrm","display":"\\eta_{\\mathrm{old}}","kind":"definition-passage","excerpt":"[Constructor-expansion certificate] A constructor-expansion certificate is a tuple Π_con = (C_t,C_t^+,η_old,η_new, H,κ,C_model,Π_foreign), where H is a nonzero higher-order derivative witness, κ is held-out obstruction reduction, and C_model records constructor complexity.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e5675753458edede","tex":"\\eta_{\\mathrm{sem}}","aliases":"η_sem eta mathrm","display":"\\eta_{\\mathrm{sem}}","kind":"definition-passage","excerpt":"[Semantic holonomy residual] For a represented evidence state x∈ E_v and loop γ at v, define η_sem(x;γ) := norm(H_γ-I)x.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-12235ebd9b8ec538","tex":"\\eta_t","aliases":"η_t eta","display":"\\eta_t","kind":"definition-passage","excerpt":"[Approximate semantic debt] If an extension preserves old semantics only approximately, let η_t bound its worst protected interpretation error after transport. Under nonexpansive semantic transports, the cumulative debt through stage T is η_0:T ≤ ∑_t=0^T-1η_t.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"definition-passage","excerpt":"[Meta-repair] A meta-repair is an authorized transformation Θ_t:O_t⟶O_t+1 that changes one or more of: item failure-signature recognition; item causal-hypothesis generation; item candidate proposal distributions; item repair composition and scheduling; item receipt validation or confidence allocation; item repair abstraction and foreign-transfer te","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a721963f725aae53","tex":"\\theta_i","aliases":"θ_i theta","display":"\\theta_i","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0d7cc092bd519f22","tex":"\\Theta_t","aliases":"Θ_t Theta","display":"\\Theta_t","kind":"definition-passage","excerpt":"[Meta-repair] A meta-repair is an authorized transformation Θ_t:O_t⟶O_t+1 that changes one or more of: item failure-signature recognition; item causal-hypothesis generation; item candidate proposal distributions; item repair composition and scheduling; item receipt validation or confidence allocation; item repair abstraction and foreign-transfer te","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f3cfaebb0e3a6b08","tex":"\\iota_t","aliases":"ι_t iota","display":"\\iota_t","kind":"definition-passage","excerpt":"[Append-only evidence embedding] Evidence preservation from generation t to t+1 is an injective map ι_t:E_t↪E_t+1 that preserves payload, source, timestamp order, precommitment status, provenance type, and cryptographic integrity fields. A later correction may append a refutation or reinterpretation receipt, but it may not silently alter the old en","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c4ebc22b504c354","tex":"\\iota_w","aliases":"ι_w iota","display":"\\iota_w","kind":"definition-passage","excerpt":"[Evaluator morphism] An evaluator morphism e:v→ w consists of maps S_e:R_v→ R_w, T_e:E_v→ E_w, where S_e is the append-only raw-evidence embedding and T_e transports internal representations. Its protected-evidence defect is δ_obs(e) := sup_r∈ h_v normT_eι_v(r)-ι_wS_e(r), and its historical score defect is δ_J(e) := sup_r∈ h_v normJ_wT_eι_v(r)-J_vι","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fd800cbc898607b2","tex":"\\kappa_{a,b;c,d}^{2}","aliases":"κ_a,b;c,d^2 kappa","display":"\\kappa_{a,b;c,d}^{2}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c9503777fa994def","tex":"\\lambda_a","aliases":"λ_a lambda","display":"\\lambda_a","kind":"definition-passage","excerpt":"[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7e8ce29a4f69b6f5","tex":"\\lambda_{a1}","aliases":"λ_a1 lambda","display":"\\lambda_{a1}","kind":"definition-passage","excerpt":"[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-69b5abe90d5c1c76","tex":"\\lambda_{aj}","aliases":"λ_aj lambda","display":"\\lambda_{aj}","kind":"definition-passage","excerpt":"[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ada129e48f68749","tex":"\\lambda_{ak}","aliases":"λ_ak lambda","display":"\\lambda_{ak}","kind":"definition-passage","excerpt":"[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a7c628bb9ee5e939","tex":"\\lambda_{am}","aliases":"λ_am lambda","display":"\\lambda_{am}","kind":"definition-passage","excerpt":"[Block-scalar response signature] Let H=⊕_a=1^kH_a be a reducing decomposition for every R_j. The response language is block scalar when restrR_jH_a=λ_ajI_H_a for scalars λ_aj. The vector σ_R(H_a) = (λ_a1,…,λ_am) is the response signature of block H_a.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-27a452b3d791e541","tex":"\\lambda_b","aliases":"λ_b lambda","display":"\\lambda_b","kind":"definition-passage","excerpt":"response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be0c6c290e01f1e4","tex":"\\lambda_{\\mathrm{geo}}","aliases":"λ_geo lambda mathrm","display":"\\lambda_{\\mathrm{geo}}","kind":"definition-passage","excerpt":"[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation whe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"definition-passage","excerpt":"[Thresholded approximate flat core] For τ≥0, define E_τ^flat := spansetx_i:λ_i≤τ, where x_i is an eigenbasis of A.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-809529b9f71bddaf","tex":"\\lambda_{\\mathrm{prog}}","aliases":"λ_prog lambda mathrm","display":"\\lambda_{\\mathrm{prog}}","kind":"definition-passage","excerpt":"[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation whe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-39fb639c17c0d4f1","tex":"\\lambda_{\\mathrm{sem}}","aliases":"λ_sem lambda mathrm","display":"\\lambda_{\\mathrm{sem}}","kind":"definition-passage","excerpt":"[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation whe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b453de64aef4dc5","tex":"\\mu_0","aliases":"μ_0 mu","display":"\\mu_0","kind":"definition-passage","excerpt":"[Adequate approximate evidence core] A subspace F is (τ,μ_0)-adequate when: item its weighted holonomy energy is at most τ per unit norm; item μ(F;Δ_prot)≥μ_0>0.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4d7dd1bb7fe8bbbd","tex":"\\mu_s","aliases":"μ_s mu","display":"\\mu_s","kind":"definition-passage","excerpt":"[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden f","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6007c7752624667b","tex":"\\mu_t","aliases":"μ_t mu","display":"\\mu_t","kind":"definition-passage","excerpt":"[Unpaired intervention-orbit transport instance] An instance is a tuple U = (H_s,H_t,μ_s,μ_t, U, {R_u^s}_u∈ U, {R_v^t}_v∈ V, G_rm gauge, W), where μ_s and μ_t are independently sampled latent distributions, R_u^s and R_v^t are within-system intervention-response operators, G_rm gauge is the declared harmless gauge group, and W is a witness hidden f","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"definition-passage","excerpt":"[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a c","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-93b36ed48fba09a5","tex":"\\Xi_t","aliases":"Ξ_t Xi","display":"\\Xi_t","kind":"definition-passage","excerpt":"[Context-indexed lineage edge] Let Ξ_t be the protected context for edge t→ t+1. A sealed lineage edge K_txrightarrowΠ_t+1K_t+1 is γ_t-productive in Ξ_t when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies M_Ξ_t(K_t+1)-M_Ξ_t(K_t)≥ γ_t>0.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).","count":20},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c3acc8569f115745","tex":"\\Pi_1","aliases":"Π_1 Pi","display":"\\Pi_1","kind":"definition-passage","excerpt":"[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observatio","count":17},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-222fb49e971fef73","tex":"\\pi_1","aliases":"π_1 pi","display":"\\pi_1","kind":"definition-passage","excerpt":"[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-47d1d41ca535a1e2","tex":"\\Pi_2","aliases":"Π_2 Pi","display":"\\Pi_2","kind":"definition-passage","excerpt":"[Finite repeated-lineage certificate] A sequence K_0xrightarrowΠ_1K_1xrightarrowΠ_2⋯xrightarrowΠ_mK_m has an m-edge repeated-lineage certificate when every edge is productive in its own predeclared protected context, every parent receipt is replayed before the child is compiled, no final audit evidence is reused as development evidence for a later ","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bbbec63f543dd2b6","tex":"\\Pi_{\\mathrm{eval}}","aliases":"Π_eval Pi mathrm","display":"\\Pi_{\\mathrm{eval}}","kind":"definition-passage","excerpt":"[Evaluator-upgrade certificate] An evaluator-upgrade certificate is a tuple Π_eval = (v,w,S_e,T_e,R_e, h_v, A_v, A_w, τ,μ_0, δ_obs,δ_J, T_new, R_rollback, σ), containing the old and new evaluator identifiers, raw and internal transports, core retraction, protected replay set, holonomy-energy operators, accepted flatness and adequacy thresholds, mea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-18bb1602906bbb9b","tex":"\\Pi_{\\mathrm{foreign}}","aliases":"Π_foreign Pi mathrm","display":"\\Pi_{\\mathrm{foreign}}","kind":"definition-passage","excerpt":"[Response-language expansion certificate] A response-language expansion certificate is a tuple Π_resp = (R,S, A, A^+,P_alias,γ, ε_comm,ε_fit,Π_foreign), where P_alias records the inherited merged block, γ is the minimum separating spectral gap, ε_comm is a commutator residual, ε_fit is response-model error, and Π_foreign records held-out repair eff","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5cea6dfcc50d4d45","tex":"\\Pi_i","aliases":"Π_i Pi","display":"\\Pi_i","kind":"definition-passage","excerpt":"[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observatio","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1563b7bbc7625f81","tex":"\\Pi_i^{+}","aliases":"Π_i^+ Pi","display":"\\Pi_i^{+}","kind":"definition-passage","excerpt":"[Post-result research receipt] After evaluation, the post-result receipt is Π_i^+ = (Π_i^-, c_i, r_i,a_i,m_i), where c_i is the scored benchmark vector, r_i the protected-regression vector, a_i the hidden-audit result, and m_i the integrity-monitor verdict.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a6a92adbf4ddf287","tex":"\\Pi_i^{-}","aliases":"Π_i^- Pi","display":"\\Pi_i^{-}","kind":"definition-passage","excerpt":"[Pre-result research receipt] A pre-result receipt has the form Π_i^- = (H_i,O_i,D_i,θ_i, widehat c_i, widehat r_i, F_i,h_i^code,h_i^spec), where H_i is the causal hypothesis, O_i the proposed objective or repair mechanism, D_i the evidence or data construction, θ_i the configuration, widehat c_i the predicted benchmark-effect vector, widehat r_i t","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7b0f169a61793b84","tex":"\\Pi_m","aliases":"Π_m Pi","display":"\\Pi_m","kind":"definition-passage","excerpt":"[Finite repeated-lineage certificate] A sequence K_0xrightarrowΠ_1K_1xrightarrowΠ_2⋯xrightarrowΠ_mK_m has an m-edge repeated-lineage certificate when every edge is productive in its own predeclared protected context, every parent receipt is replayed before the child is compiled, no final audit evidence is reused as development evidence for a later ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-73223260670b8463","tex":"\\Pi_n","aliases":"Π_n Pi","display":"\\Pi_n","kind":"definition-passage","excerpt":"[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observatio","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c52b166ee2bce97","tex":"\\pi_{\\rm anc}","aliases":"π_rm anc pi rm","display":"\\pi_{\\rm anc}","kind":"definition-passage","excerpt":"[Finite repair process] A finite repair process is a tuple P=(π_rm anc, B,π_rm gate,B,Π), where π_rm anc selects target correspondence queries, B estimates a bridge and transports a frozen source repair, π_rm gate commits or abstains, B is the declared resource budget, and Π is the receipt schema.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-500837dc09166d71","tex":"\\pi_{\\rm gate}","aliases":"π_rm gate pi rm","display":"\\pi_{\\rm gate}","kind":"definition-passage","excerpt":"[Finite repair process] A finite repair process is a tuple P=(π_rm anc, B,π_rm gate,B,Π), where π_rm anc selects target correspondence queries, B estimates a bridge and transports a frozen source repair, π_rm gate commits or abstains, B is the declared resource budget, and Π is the receipt schema.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2074f0992f39b2b0","tex":"\\pi_{\\rm PQR}","aliases":"π_rm PQR pi rm","display":"\\pi_{\\rm PQR}","kind":"definition-passage","excerpt":"[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4c72b4b1ec72f327","tex":"\\pi_{\\rm QR}","aliases":"π_rm QR pi rm","display":"\\pi_{\\rm QR}","kind":"definition-passage","excerpt":"[Packet-balanced QR policy] For fit budget k=16, define π_rm PQR(H,P_J) = π_rm QR(H_rm aff,8) ∪ π_rm QR(H_rm non,8), where affected and unaffected membership is inferred from the source causal intervention, not from target audit labels.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ca1d4be17045d238","tex":"\\Pi_T","aliases":"Π_T Pi","display":"\\Pi_T","kind":"definition-passage","excerpt":"[Receipt chain] For receipts Π_1,…,Π_T, define Π_t.previous = ∅, · t=1,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4cb477b36f223b7b","tex":"\\Pi_t","aliases":"Π_t Pi","display":"\\Pi_t","kind":"definition-passage","excerpt":"[Receipt chain] For receipts Π_1,…,Π_T, define Π_t.previous = ∅, · t=1, h(Π_t-1), · t>1.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-563d1d05d9fd86ef","tex":"\\Pi_{t+1}","aliases":"Π_t+1 Pi","display":"\\Pi_{t+1}","kind":"definition-passage","excerpt":"[Context-indexed lineage edge] Let Ξ_t be the protected context for edge t→ t+1. A sealed lineage edge K_txrightarrowΠ_t+1K_t+1 is γ_t-productive in Ξ_t when its receipt replays, its resource and witness gates pass, and a fresh paired audit certifies M_Ξ_t(K_t+1)-M_Ξ_t(K_t)≥ γ_t>0.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8f2b47ec08011cf8","tex":"\\Pi_{t-1}","aliases":"Π_t-1 Pi","display":"\\Pi_{t-1}","kind":"definition-passage","excerpt":"[Receipt chain] For receipts Π_1,…,Π_T, define Π_t.previous = ∅, · t=1, h(Π_t-1), · t>1.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b48fafc33330554","tex":"\\rho_c","aliases":"ρ_c rho","display":"\\rho_c","kind":"definition-passage","excerpt":"[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fb110376135cf752","tex":"\\rho_d","aliases":"ρ_d rho","display":"\\rho_d","kind":"definition-passage","excerpt":"[Semantic twins] Two observed records x_c∈ X_c and x_d∈ X_d are semantic twins when there exists one protected semantic object z∈ Z such that x_c=ρ_c(z), x_d=ρ_d(z). A semantic-twin group is an orbit {ρ_c(z):c∈ C_z} for a declared finite context set C_z⊆C.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2bdd558d975d62ca","tex":"\\rho_T","aliases":"ρ_T rho","display":"\\rho_T","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":17},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae00882e1fd7fcb3","tex":"\\sigma_c","aliases":"σ_c sigma","display":"\\sigma_c","kind":"definition-passage","excerpt":"[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstr","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fa9bcc45315814e2","tex":"\\Sigma_T","aliases":"Σ_T Sigma","display":"\\Sigma_T","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-87cc32cdd84c4069","tex":"\\sigma_T","aliases":"σ_T sigma","display":"\\sigma_T","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9ad9275b62e0fb7f","tex":"\\tau_{\\mathrm{adm}}","aliases":"τ_adm tau mathrm","display":"\\tau_{\\mathrm{adm}}","kind":"definition-passage","excerpt":"[Distribution-indexed atlas] A finite distribution-indexed atlas is A = ( {χ_c:c∈ C}, d_sig, τ_adm, π ), where d_sig is a normalized signature distance, τ_adm is an admission threshold, and π chooses the nearest chart. An input is admitted only when min_c∈ Cd_sig(σ(x),σ_c) ≤τ_adm. Otherwise the atlas returns an unsupported-chart certificate and abs","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-105fa42a83c8f65f","tex":"\\tau_e","aliases":"τ_e tau","display":"\\tau_e","kind":"definition-passage","excerpt":"[Stratified functional atlas] A stratified functional atlas at generation t is a tuple A_t= ( C, G_t, {X_t^c}_c∈ C, {F_t^c}_c∈ C, {T_e,τ_e,P_e,ε_e}_e∈ E(G_t) ), where: item C is a finite set of stratum colors; item G_t is a directed multigraph of declared interfaces; item X_t^c is the state space of stratum c; item F_t^c is its local transition rul","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-878af28f894350de","tex":"\\tau_M^{\\mathrm{future}}","aliases":"τ_M^future tau mathrm","display":"\\tau_M^{\\mathrm{future}}","kind":"definition-passage","excerpt":"[Recursive abstraction estimand] Let F_post be foreign defect families revealed after the source repair. Define τ_M^future = E_A∼ F_post [ G_A(Z_M=1)-G_A(Z_M=0) ].","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f214ad6faf876a75","tex":"\\Phi_w","aliases":"Φ_w Phi","display":"\\Phi_w","kind":"definition-passage","excerpt":"[Weighted meta-regret] For w∈ℝ_>0^7, define Φ_w(r)=w^top r. The weights encode the declared relative value or risk of unresolved module headroom.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"definition-passage","excerpt":"[Cross-implementation repair schema] A Block 18 repair schema is a tuple ρ=(a,λ_sem,λ_prog, λ_geo,χ), where a is an adapter constructor, the three nonnegative λ values weight final semantic, progressive semantic, and latent-geometry witnesses, and χ records whether witness labels are correctly bound or shuffled. A schema is cross-implementation whe","count":15},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-74ef2030c5e37f57","tex":"\\chi_c","aliases":"χ_c chi","display":"\\chi_c","kind":"definition-passage","excerpt":"[Affine observation chart] For a context c, an affine observation chart is a tuple χ_c=(s_x,t_x,s_y,t_y,σ_c), s_xs_y≠0, with canonicalization widehat x=fracx^obs-t_xs_x, widehat y=fracy^obs-t_ys_y, and support signature σ_c∈ℝ^5. The signature is the sorted vector of observed demonstration inputs predicted by the chart on the five canonical demonstr","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":49},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-43b76c3094fa4fa6","tex":"\\omega_1","aliases":"ω_1 omega","display":"\\omega_1","kind":"definition-passage","excerpt":"[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5d52ab32e3afcb63","tex":"\\omega_2","aliases":"ω_2 omega","display":"\\omega_2","kind":"definition-passage","excerpt":"[Schema generality preorder] For repair schemas ω_1 and ω_2 with the same typed effect, write ω_1preceqω_2 when every state satisfying C_ω_1 also satisfies C_ω_2 and the two correction semantics agree on dom(C_ω_1) up to the declared transport tolerance. Then ω_2 is at least as general as ω_1.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-711eb9ed7f4fe5d2","tex":"\\omega_A","aliases":"ω_A omega","display":"\\omega_A","kind":"definition-passage","excerpt":"[Exact transport equivalence of repairs] Write ω_A≅_Tω_B when: item input and output colors correspond under the atlas transport; item the defect-response maps commute with correction and defect transports; item applicability and postconditions are preserved in both directions; item protected invariants and provenance types are preserved; item the ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a7a65a5e0126da9c","tex":"\\Omega_{a,b;c,d}","aliases":"Ω_a,b;c,d Omega","display":"\\Omega_{a,b;c,d}","kind":"definition-passage","excerpt":"[Architecture--context curvature] For architectures a,b and charts c,d, the transport-square defect is Ω_a,b;c,d := C^a_c→ dT^c_b→ a - T^d_b→ aC^b_c→ d. Given an audit distribution P_b,c on target trajectories, its normalized root-mean-square curvature is κ_a,b;c,d^2 = frac E_h∼ P_b,c normΩ_a,b;c,dh^2 E_z∼ P_a,dnormz^2 .","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-74276c1846242871","tex":"\\omega_B","aliases":"ω_B omega","display":"\\omega_B","kind":"definition-passage","excerpt":"[Exact transport equivalence of repairs] Write ω_A≅_Tω_B when: item input and output colors correspond under the atlas transport; item the defect-response maps commute with correction and defect transports; item applicability and postconditions are preserved in both directions; item protected invariants and provenance types are preserved; item the ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ecd03e3ff7466d39","tex":"\\omega_i","aliases":"ω_i omega","display":"\\omega_i","kind":"definition-passage","excerpt":"[Receipt composition] Let Π_i validate ω_i, and let Π validate the outer operation ω conditional on C_ω. The composed receipt Π⋆(Π_1,…,Π_n) is the provenance-directed acyclic graph obtained by proof-path concatenation, evidence union, explicit dependency links, and deduplication of immutable evidence references without duplicating linear observatio","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Proof-carrying repair schema] A repair schema is a tuple ω= ( σ, c_in, c_out, D, C, Q, [r], u, g, mathscr I, Π ), where: item σ is a recognizable failure signature; item c_in=(c_1,…,c_n) and c_out are input and output colors; item D is the declared local defect-response map; item C is an applicability precondition; item Q is a postcondition; item ","count":58},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":33},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8c56dcfd50ae2eaf","tex":"\\ell_{ij}","aliases":"ℓ_ij ell","display":"\\ell_{ij}","kind":"definition-passage","excerpt":"[Packet leakage profile] The directed leakage from correction packet j into defect packet i is ℓ_ij:=normP_iDQ_j, i≠ j. The matrix (ℓ_ij) is the packet leakage profile.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9bb76059a9d9aed1","tex":"\\ell_T","aliases":"ℓ_T ell","display":"\\ell_T","kind":"definition-passage","excerpt":"[Preservation profile] A preservation profile for a bridge T:X→ Y is a declared record P(T)= ( Σ_T, Q_T, ρ_T, ℓ_T, σ_T, A_T ), where Σ_T lists preserved algebraic structures, Q_T lists preserved causal or evaluative queries, ρ_T records provenance behavior, ℓ_T records information loss, σ_T records uncertainty transport, and A_T lists authorized do","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Protected evaluation context] A protected evaluation context is a tuple Ξ= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a c","count":165},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"definition-passage","excerpt":"[Append-only evidence embedding] Evidence preservation from generation t to t+1 is an injective map ι_t:E_t↪E_t+1 that preserves payload, source, timestamp order, precommitment status, provenance type, and cryptographic integrity fields. A later correction may append a refutation or reinterpretation receipt, but it may not silently alter the old en","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"definition-passage","excerpt":"[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"definition-passage","excerpt":"[Controllable predecessor] For X⊆ S, define CPre(X) = set s∈ S: ∃ u∈Act(s) ∀ d∈Dist(s,u), T(s,u,d)∈ X .","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"definition-passage","excerpt":"[Rotating witness schedule] A sequence (E_t,W_t)_t≥0 is a rotating witness schedule when: item E_t∩ W_t=∅ for every t; item W_t is immutable during transaction t; item every component belongs to E_t for at least one t; item every witness replacement is validated by a component outside the same edit set; item protected evidence is transported throug","count":19},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservation predicate for the t","count":308},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"definition-passage","excerpt":"[Structurally new repair schema] Let mathscrS_t be the closure of existing repair schemas under the authorized operadic compositions. A schema s_t+1 is structurally new when s_t+1∉overlinemathscrS_t under the declared behavioral equivalence and spanim s_t+1 not⊆ spanimmathscrS_t.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.","count":78},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7f238b3e1c3bbe13","tex":"\\sum_{a=1}^{m}","aliases":"∑_a=1^m sum","display":"\\sum_{a=1}^{m}","kind":"definition-passage","excerpt":"[Block-scalar response model] The response family is block scalar on [source reference] when R_j=∑_a=1^mλ_ajP_a, λ_aj∈ℝ. The response signature of block a is λ_a=(λ_a1,…,λ_ak)∈ℝ^k. The signatures are separating when λ_a≠λ_b for every a≠ b.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e9cbdf6d93663d3f","tex":"\\sum_{e\\in p}","aliases":"∑_e∈ p sum in","display":"\\sum_{e\\in p}","kind":"definition-passage","excerpt":"[Transport route cost] For a fixed source correction u_0, define the certified cost of a nonexpansive path p by C_tr(p;u_0) := normu_0∑_e∈ pε_D,e + ∑_e∈ pε_r,e + ∑_e∈ pε_auth,e + ∑_e∈ pε_sem,e.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"definition-passage","excerpt":"[Structural-normalized specificity] Let Q_spec = fracoverline s max{overline o,overline c,ε}, where bars denote held-out means and ε>0 is a fixed numerical floor. The default Block 16 specificity gate requires LCB_0.95(overline s)>0, frac1n∑_i1[s_i>0]≥0.75, Q_spec≥0.25.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a125c9c58c844362","tex":"\\sum_{R\\subseteq N\\setminus\\{i\\}}","aliases":"∑_R⊆ N∖{i} sum subseteq setminus","display":"\\sum_{R\\subseteq N\\setminus\\{i\\}}","kind":"definition-passage","excerpt":"[Shapley meta-headroom] For i∈ N, define h_i(K) = ∑_R⊆ N∖{i} frac|R|!(m-|R|-1)!m! [ F_K(R∪{i})-F_K(R) ].","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-469e6dedd4f8c880","tex":"\\sum_{t=0}^{T-1}","aliases":"∑_t=0^T-1 sum","display":"\\sum_{t=0}^{T-1}","kind":"definition-passage","excerpt":"[Approximate semantic debt] If an extension preserves old semantics only approximately, let η_t bound its worst protected interpretation error after transport. Under nonexpansive semantic transports, the cumulative debt through stage T is η_0:T ≤ ∑_t=0^T-1η_t.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ecf317dfca855f25","tex":"\\sum_{\\alpha}","aliases":"∑_α sum alpha","display":"\\sum_{\\alpha}","kind":"definition-passage","excerpt":"[Quantitative gluing obstruction] The graph-level quantitative gluing obstruction is η_glue(e) :=normP_cyce. The local-to-global repair certificate is the pair ( η_loc, η_glue ), η_loc^2 :=∑_αnormD_α u_α+r_α^2.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"definition-passage","excerpt":"[Nonlinear holonomy residual] If the S_i are metric spaces, define η_hol(s_o) := max_e∈ E∖ Td(H_e(s_o),s_o).","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Correction datum] A finite-dimensional correction datum is a tuple C=(U,V,D,A,c, p), where U is a correction space, V is a defect space, D:U→ V is the linearized defect-response map, A⊆ U is the authorized correction set, c:U→[0,∞] is a declared resource cost, and p is a collection of protected predicates or invariant maps.","count":18},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"definition-passage","excerpt":"[Productive recursive closure] For a finite perfect-information transaction game, define I_γ^ω = ν Z. μ Y. [ (G_γ∩SPre(Z)) ∪ SPre(Y) ].","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"ursive improvement step] A lineage transition L_t→ L_t+1 is a bounded recursive improvement step relative to (Ξ,h,B) when: item its lineage receipt is valid and the protected evidence ledger embeds into the next ledger; item F_t+1(Ξ)≥ F_t(Ξ); item on a foreign held-out target distribution D_foreign, E_A∼ D_foreign Γ_Ξ,h,B(K_t+1;A) > E_A∼ D_foreign ","count":36},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4446587d0b6e5e3f","tex":"\\sim_h","aliases":"∼_h sim","display":"\\sim_h","kind":"definition-passage","excerpt":"[Horizon-h continuation equivalence] Two archive nodes A and B are horizon-h continuation equivalent, written A∼_h B, when, under the same repair grammar and matched budgets, their sets of reachable descriptor vectors through at most h further transactions are identical.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4838d0a223d69d19","tex":"\\sim_{\\Ucal,r}","aliases":"∼_Ucal,r ∼_U,r sim Ucal sim mathcal","display":"\\sim_{{\\mathcal{U}},r}","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"definition-passage","excerpt":"[Exact transport equivalence of repairs] Write ω_A≅_Tω_B when: item input and output colors correspond under the atlas transport; item the defect-response maps commute with correction and defect transports; item applicability and postconditions are preserved in both directions; item protected invariants and provenance types are preserved; item the ","count":16},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-529fce9b401b13af","tex":"\\cong_T","aliases":"≅_T cong","display":"\\cong_T","kind":"definition-passage","excerpt":"[Exact transport equivalence of repairs] Write ω_A≅_Tω_B when: item input and output colors correspond under the atlas transport; item the defect-response maps commute with correction and defect transports; item applicability and postconditions are preserved in both directions; item protected invariants and provenance types are preserved; item the ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"definition-passage","excerpt":"[Deterministically separating probe family] Suppose every K_h^p is a point mass at an outcome y(h,p). The family P_probe separates H when for every h≠ h' there exists p∈P_probe such that y(h,p)≠ y(h',p).","count":59},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"[Causal-predictive equivalence] For h,h'∈ H, write h∼_U,rh' when L (Y^u_0:rmid h) = L (Y^u_0:rmid h') for every u∈U^≤ r. The quotient Z_U,r=H/∼_U,r is the finite-horizon causal-predictive semantic space relative to the declared experiment language.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"[ε-intervention sufficiency] The map ϕ_t is ε-intervention-sufficient for D_t when sup_δ∈D_t TV( K_A_t^δ, widehat K_ϕ_t(A_t)^δ ) ≤ε.","count":177},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"Xi= ( Z, μ, Y, ν, bar c, λ, mathscr I, B, α ), where: item Z is a task or environment family and μ is its held-out distribution; item Y is the space of protected external traces; item ν:Z×Y→[0,1] is task utility; item bar c:Z×Y→[0,1] is normalized resource cost; item λ≥0 is a cost weight; item mathscr I(A,B)∈{0,1} is a declared invariant-preservati","count":112},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Property-transport defect] For a declared domain K⊆ X, define ε_Q(T;K) = sup_x∈ K norm Q_Y(Tx)-widetilde T_QQ_X(x) .","count":71},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Block-scalar response signature] Let H=⊕_a=1^kH_a be a reducing decomposition for every R_j. The response language is block scalar when restrR_jH_a=λ_ajI_H_a for scalars λ_aj. The vector σ_R(H_a) = (λ_a1,…,λ_am) is the response signature of block H_a.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-83a4e4e39570bd43","tex":"\\bigoplus_{a=1}^{k}","aliases":"⊕_a=1^k bigoplus","display":"\\bigoplus_{a=1}^{k}","kind":"definition-passage","excerpt":"[Block-scalar response signature] Let H=⊕_a=1^kH_a be a reducing decomposition for every R_j. The response language is block scalar when restrR_jH_a=λ_ajI_H_a for scalars λ_aj. The vector σ_R(H_a) = (λ_a1,…,λ_am) is the response signature of block H_a.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"definition-passage","excerpt":"[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8311820c1358375a","tex":"\\bigcap_{j=1}^{m}","aliases":"⋂_j=1^m bigcap","display":"\\bigcap_{j=1}^{m}","kind":"definition-passage","excerpt":"[Evaluator-nullspace] The locally invisible behavior subspace of portfolio B is N_B = ⋂_j=1^mker Db_j.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-992a01298fada1dc","tex":"\\bigcap_{x\\in C_z}","aliases":"⋂_x∈ C_z bigcap in","display":"\\bigcap_{x\\in C_z}","kind":"definition-passage","excerpt":"[Repair-consistent cell] A cell is repair consistent when ⋂_x∈ C_zA^⋆(x)≠∅. It is inconsistent when no single authorized repair works for every episode that the grammar treats as equivalent.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3154f91e362daf53","tex":"\\bigcap_{x\\in S}","aliases":"⋂_x∈ S bigcap in","display":"\\bigcap_{x\\in S}","kind":"definition-passage","excerpt":"[Minimal repair conflict] A subset S⊆ C_z is a minimal repair conflict when ⋂_x∈ SA^⋆(x)=∅, but every proper subset has a nonempty repair intersection.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-525a3206ddc6bd8f","tex":"\\bigcap_\\gamma","aliases":"⋂_γ bigcap gamma","display":"\\bigcap_\\gamma","kind":"definition-passage","excerpt":"[Path-flat repair and defect cores] The path-flat repair and defect cores at A are U_A^flat := ⋂_γker(H^U_γ-I), V_A^flat := ⋂_γker(H^V_γ-I).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4156ed364b30f661","tex":"\\bigcap_{\\gamma\\in\\mathsf W(v)}","aliases":"⋂_γ∈ W(v) bigcap gamma in mathsf","display":"\\bigcap_{\\gamma\\in\\mathsf W(v)}","kind":"definition-passage","excerpt":"[Closed-walk equalizer core] For a possibly noninvertible evaluator graph, let W(v) be the monoid of directed closed walks at v. Define E_v^eq := ⋂_γ∈ W(v)ker(H_γ-I).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-04bf2189a5905eb6","tex":"\\bigcap_{\\gamma\\in\\pi_1(G,v_0)}","aliases":"⋂_γ∈π_1(G,v_0) bigcap gamma in pi","display":"\\bigcap_{\\gamma\\in\\pi_1(G,v_0)}","kind":"definition-passage","excerpt":"[Path-flat evidence core] The path-flat evidence core at v_0 is E_v_0^flat := ⋂_γ∈π_1(G,v_0) ker(H_γ-I).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-702d93575d1f92f7","tex":"\\bigcap_{\\tau\\in S}","aliases":"⋂_τ∈ S bigcap tau in","display":"\\bigcap_{\\tau\\in S}","kind":"definition-passage","excerpt":"[Compiler-language conflict] A finite set S⊆T is a compiler-language conflict for G when all problems in S receive the same inherited diagnostic token but ⋂_τ∈ SA^⋆(τ)=∅. A minimal conflict has no proper conflicting subset.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Open-set ascending lineage] A lineage is conservatively ascending when L_0preceq_consL_1 preceq_consL_2 preceq_cons⋯. It is structurally open-set when infinitely many edges add a foreign-task-relevant correction direction or validated distinction not representable in the prior grammar.","count":22},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Repair abstraction operator] A repair abstraction operator is a map a: H_episodes ⟶ overlineO that maps one or more concrete episodes to a typed schema with a failure signature, applicability condition, correction program, effect profile, and receipt template.","count":40},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"definition-passage","excerpt":"[Sealed meta-compiler] A sealed meta-compiler is a map M: ( D_rm dev, D_rm meta,K_0,Γ) ⟼ (K_1,Π_1), where Γ is a declared finite meta-grammar and Π_1 is a descendant receipt. The compiler is sealed when: item neither final audit split is supplied to M; item Π_1 commits to the child state, parent identity, input hashes, grammar, selected configurati","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"introduction-passage","excerpt":"Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8b78328ed8b5d10d","tex":"A^{\\rm base}_{i,\\rm prot}","aliases":"A^rm base_i,rm prot rm rm","display":null,"kind":"introduction-passage","excerpt":"Let x_i denote a development candidate feature vector, y_i its revealed protected utility, and r_i=[A^rm base_i,rm prot-A_i,rm prot]_+ its protected-regression label. Standardize the feature matrix to Z using development means and scales.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-99180888fda3a254","tex":"A_{i,\\rm prot}","aliases":"A_i,rm prot rm","display":null,"kind":"introduction-passage","excerpt":"Let x_i denote a development candidate feature vector, y_i its revealed protected utility, and r_i=[A^rm base_i,rm prot-A_i,rm prot]_+ its protected-regression label. Standardize the feature matrix to Z using development means and scales.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cf25d6e02f5cbdf3","tex":"A_{t+1}","aliases":"A_t+1","display":"A_{t+1}","kind":"introduction-passage","excerpt":"Let A_t denote an active agent system at generation t, and let J(A_t) denote externally evaluated task utility. Ordinary task improvement requires J(A_t+1)>J(A_t). Let Γ_h,B(K;A) denote the expected best gain obtained when improvement mechanism K is allowed horizon h and resource budget B to modify target agent A. Improvement-process improvement re","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"introduction-passage","excerpt":"Let M_B(K) denote expected best gain when policy K receives B independently generated candidate opportunities. Even if the policy is unchanged, M_B+1(K)≥ M_B(K). Therefore, comparing a child with B_1>B_0 against a parent with budget B_0 does not isolate improvement of the selection procedure.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"introduction-passage","excerpt":"Let M_B(K) denote expected best gain when policy K receives B independently generated candidate opportunities. Even if the policy is unchanged, M_B+1(K)≥ M_B(K). Therefore, comparing a child with B_1>B_0 against a parent with budget B_0 does not isolate improvement of the selection procedure.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8b5db6699dd9dfb1","tex":"\\operatorname{cj}","aliases":"cj operatorname","display":"\\operatorname{cj}","kind":"introduction-passage","excerpt":"Let cj denote canonical JSON serialization with sorted keys and normalized arrays. For payload x, define h(x)=SHA256(cj(x)). A transaction receipt records parent, candidate, witness, checkpoint, phase trace, evidence, decision, rejection reasons, final state, and previous receipt hash.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-26d83baff19c71bc","tex":"\\Desc","aliases":"Desc operatorname","display":"{\\operatorname{Desc}}","kind":"introduction-passage","excerpt":"Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-31e1d3fa73b2abb1","tex":"\\Desc_{h,B}","aliases":"Desc_h,B Desc operatorname","display":"{\\operatorname{Desc}}_{h,B}","kind":"introduction-passage","excerpt":"Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6329a94f2a4029b0","tex":"\\mathbb E_A","aliases":"E_A mathbb","display":"\\mathbb E_A","kind":"introduction-passage","excerpt":"denote externally evaluated task utility. Ordinary task improvement requires J(A_t+1)>J(A_t). Let Γ_h,B(K;A) denote the expected best gain obtained when improvement mechanism K is allowed horizon h and resource budget B to modify target agent A. Improvement-process improvement requires, on a held-out distribution of target agents, E_AΓ_h,B(K_t+1;A)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9a776891de99e342","tex":"\\mathbb E_{\\tau\\sim Q}","aliases":"E_τ∼ Q mathbb tau sim","display":"\\mathbb E_{\\tau\\sim Q}","kind":"introduction-passage","excerpt":"e fixture, a selected method that passes the stronger post hoc capability and integrity predicates receives its protected audit gain, a no-op receives zero, and a selected invalid method receives the frozen penalty -0.75. This prevents an invalid selection from being silently converted into a neutral outcome. Let R_u(Γ,τ) be protected downside or i","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-34e2f8ff743ea7b4","tex":"\\Gcal_{10}","aliases":"Gcal_10 G_10 Gcal mathcal","display":"{\\mathcal{G}}_{10}","kind":"introduction-passage","excerpt":"Let G_10 denote the Block 10 two-chart grammar with a supplied context projector P_ctx. Let G_11 contain: item the intervention-response atlas compiler; item a finite family of projectors and separators returned by the compiler; item every constructor and witness rule of G_10; item a declared fallback that may instantiate the old projector directly","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a3c14494b6fe74cb","tex":"\\Gcal_{11}","aliases":"Gcal_11 G_11 Gcal mathcal","display":"{\\mathcal{G}}_{11}","kind":"introduction-passage","excerpt":"Let G_10 denote the Block 10 two-chart grammar with a supplied context projector P_ctx. Let G_11 contain: item the intervention-response atlas compiler; item a finite family of projectors and separators returned by the compiler; item every constructor and witness rule of G_10; item a declared fallback that may instantiate the old projector directly","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7dc3a6bc07113a7a","tex":"H_{A,c}^{(1:5)}","aliases":"H_A,c^(1:5)","display":"H_{A,c}^{(1:5)}","kind":"introduction-passage","excerpt":"Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3f40d00cc87b2de5","tex":"H_{B,c}^{(1:5)}","aliases":"H_B,c^(1:5)","display":"H_{B,c}^{(1:5)}","kind":"introduction-passage","excerpt":"Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c3a32fdf35b31aa8","tex":"h_s","aliases":"h_s","display":"h_s","kind":"introduction-passage","excerpt":"Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top vdots h_s(x_m)^top , H_t = h_t(x_1)^top vdots h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled","count":18},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-480c877d455d2c2b","tex":"h_t","aliases":"h_t","display":"h_t","kind":"introduction-passage","excerpt":"Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top vdots h_s(x_m)^top , H_t = h_t(x_1)^top vdots h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f5bae71a192fc64d","tex":"I_V","aliases":"I_V","display":"I_V","kind":"introduction-passage","excerpt":"Let D^†:V→ U^auth denote the Moore--Penrose pseudoinverse. Define P_cor:=DD^†, P_ob:=I_V-DD^†.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f019d8f59b733c35","tex":"\\Ical_{\\mathrm{NL}}","aliases":"Ical_NL I_NL Ical mathrm mathcal mathrm","display":"{\\mathcal{I}}_{\\mathrm{NL}}","kind":"introduction-passage","excerpt":"Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,per","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-173833828c3529a3","tex":"K_2","aliases":"K_2","display":"K_2","kind":"introduction-passage","excerpt":"Let K_2 denote the exact Block 9 conditioning-aware process. It supplies the quality-weighted D-optimal anchor-ordering rule and the published lineage receipt. Block 10 constructs two resource-matched processes: K_2,G^96, K_3,C^96. The global parent K_2,G^96 retains one affine bridge and uses all 80 non-witness pairs to estimate it. The chart child","count":21},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d4b31c401364d843","tex":"K_{2,G}^{96}","aliases":"K_2,G^96","display":"K_{2,G}^{96}","kind":"introduction-passage","excerpt":"Let K_2 denote the exact Block 9 conditioning-aware process. It supplies the quality-weighted D-optimal anchor-ordering rule and the published lineage receipt. Block 10 constructs two resource-matched processes: K_2,G^96, K_3,C^96. The global parent K_2,G^96 retains one affine bridge and uses all 80 non-witness pairs to estimate it. The chart child","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-10819c31a8b153ac","tex":"K_{3,C}^{96}","aliases":"K_3,C^96","display":"K_{3,C}^{96}","kind":"introduction-passage","excerpt":"Let K_2 denote the exact Block 9 conditioning-aware process. It supplies the quality-weighted D-optimal anchor-ordering rule and the published lineage receipt. Block 10 constructs two resource-matched processes: K_2,G^96, K_3,C^96. The global parent K_2,G^96 retains one affine bridge and uses all 80 non-witness pairs to estimate it. The chart child","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-098ede2828d25010","tex":"M_B","aliases":"M_B","display":"M_B","kind":"introduction-passage","excerpt":"Let M_B(K) denote expected best gain when policy K receives B independently generated candidate opportunities. Even if the policy is unchanged, M_B+1(K)≥ M_B(K). Therefore, comparing a child with B_1>B_0 against a parent with budget B_0 does not isolate improvement of the selection procedure.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1955520ac64c2ae6","tex":"M_{B+1}","aliases":"M_B+1","display":"M_{B+1}","kind":"introduction-passage","excerpt":"Let M_B(K) denote expected best gain when policy K receives B independently generated candidate opportunities. Even if the policy is unchanged, M_B+1(K)≥ M_B(K). Therefore, comparing a child with B_1>B_0 against a parent with budget B_0 does not isolate improvement of the selection procedure.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-32ba46ee47e49e3b","tex":"\\Mcal_0","aliases":"Mcal_0 M_0 Mcal mathcal","display":"{\\mathcal{M}}_0","kind":"introduction-passage","excerpt":"Let M_0 denote the inherited singleton-method grammar. For inherited method families a and b, define an ordered pair constructor Seq(a,b) = ``apply the research intervention from family a, then the intervention from family b''. The generic finite constructor language is C_rm pair = setSeq(a,b):a,b∈M_0, a≠ b. In the executable fixture there are six ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-44a7952294f06149","tex":"\\mathrm{NL}","aliases":"NL mathrm","display":"\\mathrm{NL}","kind":"introduction-passage","excerpt":"Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,per","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2a6e7df6bcfcdef7","tex":"\\mathsf{noise}","aliases":"noise mathsf","display":"\\mathsf{noise}","kind":"introduction-passage","excerpt":"Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,per","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-315a8629c1fc4c14","tex":"\\mathsf{packet}","aliases":"packet mathsf","display":"\\mathsf{packet}","kind":"introduction-passage","excerpt":"Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,per","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae99770781138d98","tex":"\\mathsf{permute}","aliases":"permute mathsf","display":"\\mathsf{permute}","kind":"introduction-passage","excerpt":"Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,per","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a48466bfbda2aca3","tex":"\\R^{40}","aliases":"R^40 ℝ^40 R mathbb","display":"{\\mathbb{R}}^{40}","kind":"introduction-passage","excerpt":"Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the ","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4713a74e927838f8","tex":"\\R^8","aliases":"R^8 ℝ^8 R mathbb","display":"{\\mathbb{R}}^8","kind":"introduction-passage","excerpt":"Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the ","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bc1a3b3d4cc5afef","tex":"\\Risk","aliases":"Risk operatorname","display":"{\\operatorname{Risk}}","kind":"introduction-passage","excerpt":"(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]. Finally, let Risk_Ξ(V_t) denote held-out evaluator risk on candidate-effect judgm","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4ae3e2b67424d4d5","tex":"\\Risk_\\Xi","aliases":"Risk_Ξ Risk Xi operatorname Xi","display":"{\\operatorname{Risk}}_\\Xi","kind":"introduction-passage","excerpt":"(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]. Finally, let Risk_Ξ(V_t) denote held-out evaluator risk on candidate-effect judgm","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1dcf95b2a026d667","tex":"s_a","aliases":"s_a","display":"s_a","kind":"introduction-passage","excerpt":"Let s_a denote the best population score attainable from block a, and let widehat s_a be its empirical score.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a85f55a23690e34d","tex":"\\operatorname{Seq}","aliases":"Seq operatorname","display":"\\operatorname{Seq}","kind":"introduction-passage","excerpt":"Let M_0 denote the inherited singleton-method grammar. For inherited method families a and b, define an ordered pair constructor Seq(a,b) = ``apply the research intervention from family a, then the intervention from family b''. The generic finite constructor language is C_rm pair = setSeq(a,b):a,b∈M_0, a≠ b. In the executable fixture there are six ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4bfe7fd046c98b4d","tex":"\\operatorname{SHA256}","aliases":"SHA256 operatorname","display":"\\operatorname{SHA256}","kind":"introduction-passage","excerpt":"Let cj denote canonical JSON serialization with sorted keys and normalized arrays. For payload x, define h(x)=SHA256(cj(x)). A transaction receipt records parent, candidate, witness, checkpoint, phase trace, evidence, decision, rejection reasons, final state, and previous receipt hash.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9173f60779db52b7","tex":"\\sup_{A'\\in\\Desc_{h,B}(K_t,A)}","aliases":"sup_A'∈Desc_h,B(K_t,A) sup in Desc sup in operatorname","display":"\\sup_{A'\\in{\\operatorname{Desc}}_{h,B}(K_t,A)}","kind":"introduction-passage","excerpt":"Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1144c73385f437ce","tex":"\\sup_{A\\in\\Acal_t}","aliases":"sup_A∈Acal_t sup_A∈A_t sup in Acal sup in mathcal","display":"\\sup_{A\\in{\\mathcal{A}}_t}","kind":"introduction-passage","excerpt":"Let A_t⊆ V(Λ_t) be the executable archive at generation t. Its task frontier is F_t(Ξ)=sup_A∈A_tJ_Ξ(A). Let K_t be the active improvement mechanism, including diagnosis, proposal, composition, and validation policies. For horizon h and budget B, define target-relative metaproductivity Γ_Ξ,h,B(K_t;A) = E [ sup_A'∈Desc_h,B(K_t,A) (J_Ξ(A')-J_Ξ(A))_+ ]","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dd53fb3b4cbd4977","tex":"T^{c}_{B\\to A}","aliases":"T^c_B→ A to","display":"T^{c}_{B\\to A}","kind":"introduction-passage","excerpt":"Let A denote the frozen source architecture and let B be one of the two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a8a8edc0e4672a37","tex":"\\mathsf{transplant}","aliases":"transplant mathsf","display":"\\mathsf{transplant}","kind":"introduction-passage","excerpt":"Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,per","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eaf799324f3c8a41","tex":"\\mathsf{transport}","aliases":"transport mathsf","display":"\\mathsf{transport}","kind":"introduction-passage","excerpt":"Let L(x,y;Z) denote mean gold-answer cross-entropy when the prompt is x, target answer is y, and the recurrent latent slots are externally set to trace Z. Let Z(x) denote the clean trace. For a donor x' selected from the same declared answer-format group, let Z(x') be its clean trace. Block 16 fixes the intervention language I_NL = {zero,replay,per","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-62c49bf1537e3953","tex":"U^{\\Auth}","aliases":"U^Auth U^auth Auth mathrm","display":"U^{{\\mathrm{auth}}}","kind":"introduction-passage","excerpt":"Let D^†:V→ U^auth denote the Moore--Penrose pseudoinverse. Define P_cor:=DD^†, P_ob:=I_V-DD^†.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16b94c3d72c906db","tex":"U_u^{\\rm valid}","aliases":"U_u^rm valid rm","display":"U_u^{\\rm valid}","kind":"introduction-passage","excerpt":"Let U_u^rm valid(Γ,τ) denote the validity-filtered protected gain produced by compiler Γ on research problem τ under environment u. In the executable fixture, a selected method that passes the stronger post hoc capability and integrity predicates receives its protected audit gain, a no-op receives zero, and a selected invalid method receives the fr","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-44cf5118c3be9cba","tex":"\\vdots","aliases":"vdots","display":"\\vdots","kind":"introduction-passage","excerpt":"Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top vdots h_s(x_m)^top , H_t = h_t(x_1)^top vdots h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f78f4c25fed459ef","tex":"\\widehat s_a","aliases":"widehat s_a widehat","display":"\\widehat s_a","kind":"introduction-passage","excerpt":"Let s_a denote the best population score attainable from block a, and let widehat s_a be its empirical score.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ad91da91d6be3c40","tex":"\\widetilde F","aliases":"widetilde F widetilde","display":"\\widetilde F","kind":"introduction-passage","excerpt":"Let widetilde F(P,δ) denote execution in an isolated child environment. Write widetilde F(P,δ)=(C,B), where C is the candidate child state and B is a buffer of externally persistent effects.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"introduction-passage","excerpt":"Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top","count":16},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"introduction-passage","excerpt":"Let s denote the source and t a target. On paired anchor inputs x_1,…,x_m, the system observes H_s = h_s(x_1)^top","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-21fbeaadfa7503ba","tex":"\\Gamma_{h,B}","aliases":"Γ_h,B Gamma","display":"\\Gamma_{h,B}","kind":"introduction-passage","excerpt":"Let A_t denote an active agent system at generation t, and let J(A_t) denote externally evaluated task utility. Ordinary task improvement requires J(A_t+1)>J(A_t). Let Γ_h,B(K;A) denote the expected best gain obtained when improvement mechanism K is allowed horizon h and resource budget B to modify target agent A. Improvement-process improvement re","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"introduction-passage","excerpt":"[Compilation noninterference] Let K_1= M(D,V,K_0,Γ), where D and V denote the development and meta-validation receipts. Suppose M is deterministic and receives no value measurable only with respect to final audit outcomes A. Then, conditional on (D,V,K_0,Γ), K_1⊥ ⊥ A. Equivalently, changing the realized final audit outcomes while holding compiler i","count":21},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b60a6be5ea34829f","tex":"\\mathbf{0.050745}","aliases":"0.050745 mathbf","display":"\\mathbf{0.050745}","kind":"use","excerpt":"Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4eb6fba252818b99","tex":"\\mathbf{0.057546}","aliases":"0.057546 mathbf","display":"\\mathbf{0.057546}","kind":"use","excerpt":"Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0a41becad76bada7","tex":"\\mathbf{0.096}","aliases":"0.096 mathbf","display":"\\mathbf{0.096}","kind":"use","excerpt":"Conditioning grandchild · 0.209580 · 0.638021 · 1.000 · 14.268 · 0.096","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-13b183968e5923df","tex":"\\mathbf{0.174135}","aliases":"0.174135 mathbf","display":"\\mathbf{0.174135}","kind":"use","excerpt":"Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ae90ff684d6e1d9","tex":"\\mathbf{0.194936}","aliases":"0.194936 mathbf","display":"\\mathbf{0.194936}","kind":"use","excerpt":"Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9039d793008550e2","tex":"\\mathbf{0.209580}","aliases":"0.209580 mathbf","display":"\\mathbf{0.209580}","kind":"use","excerpt":"Conditioning grandchild · 0.209580 · 0.638021 · 1.000 · 14.268 · 0.096","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4cd0cb0fb1888ed5","tex":"\\mathbf{0.333839}","aliases":"0.333839 mathbf","display":"\\mathbf{0.333839}","kind":"use","excerpt":"Primary language child · 0.859731 · 0.792894 · 0.333839","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4336bd516a125eec","tex":"\\mathbf{0.334224}","aliases":"0.334224 mathbf","display":"\\mathbf{0.334224}","kind":"use","excerpt":"Chart child K_3,C^96 · 0.334224 · 0.284265 · 0.572917 · 0.572917 · 0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-272ec25bb1f25a03","tex":"\\mathbf{0.344720}","aliases":"0.344720 mathbf","display":"\\mathbf{0.344720}","kind":"use","excerpt":"Descendant packet-QR 16 · 0.344720 · 1.000000 · 1.00000 · 24","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-96a56cf2c3be1617","tex":"\\mathbf{0.482289}","aliases":"0.482289 mathbf","display":"\\mathbf{0.482289}","kind":"use","excerpt":"Generator-foreign child · 0.814520 · 0.785584 · 0.482289","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-54c0d91ded2d3141","tex":"\\mathbf{0.594444}","aliases":"0.594444 mathbf","display":"\\mathbf{0.594444}","kind":"use","excerpt":"Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1a437e43ab1ac506","tex":"\\mathbf{0.607778}","aliases":"0.607778 mathbf","display":"\\mathbf{0.607778}","kind":"use","excerpt":"Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ba63a0b881f4acd","tex":"\\mathbf{0.630918}","aliases":"0.630918 mathbf","display":"\\mathbf{0.630918}","kind":"use","excerpt":"Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fa88a8e828af618c","tex":"\\mathbf{0.666244}","aliases":"0.666244 mathbf","display":"\\mathbf{0.666244}","kind":"use","excerpt":"Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ca0195185a4807ca","tex":"\\mathbf{0.770701}","aliases":"0.770701 mathbf","display":"\\mathbf{0.770701}","kind":"use","excerpt":"Full child · inherited · 0.057546 · 0.174135 · 0.770701 · 0.630918 · 0.607778","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a0d0fe159fb34d06","tex":"\\mathbf{0.779341}","aliases":"0.779341 mathbf","display":"\\mathbf{0.779341}","kind":"use","excerpt":"Full child · novel charts · 0.050745 · 0.194936 · 0.779341 · 0.666244 · 0.594444","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fddf3d36cfa451f9","tex":"\\mathbf{0.785584}","aliases":"0.785584 mathbf","display":"\\mathbf{0.785584}","kind":"use","excerpt":"Generator-foreign child · 0.814520 · 0.785584 · 0.482289","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fa41680e6889636e","tex":"\\mathbf{0.792894}","aliases":"0.792894 mathbf","display":"\\mathbf{0.792894}","kind":"use","excerpt":"Primary language child · 0.859731 · 0.792894 · 0.333839","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d8c13469323965b1","tex":"\\mathbf{0.814520}","aliases":"0.814520 mathbf","display":"\\mathbf{0.814520}","kind":"use","excerpt":"Generator-foreign child · 0.814520 · 0.785584 · 0.482289","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e36626cdd35a80d","tex":"\\mathbf{0.859731}","aliases":"0.859731 mathbf","display":"\\mathbf{0.859731}","kind":"use","excerpt":"Primary language child · 0.859731 · 0.792894 · 0.333839","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8e7d6d7ffb82d94b","tex":"\\mathbf{1.00000}","aliases":"1.00000 mathbf","display":"\\mathbf{1.00000}","kind":"use","excerpt":"Descendant packet-QR 16 · 0.344720 · 1.000000 · 1.00000 · 24","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1d2a53ff40ef4a97","tex":"\\mathbf{1.000000}","aliases":"1.000000 mathbf","display":"\\mathbf{1.000000}","kind":"use","excerpt":"Descendant packet-QR 16 · 0.344720 · 1.000000 · 1.00000 · 24","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-569afa351967c58f","tex":"\\mathbf{14.268}","aliases":"14.268 mathbf","display":"\\mathbf{14.268}","kind":"use","excerpt":"Conditioning grandchild · 0.209580 · 0.638021 · 1.000 · 14.268 · 0.096","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-96bdb3ea74c8592b","tex":"\\mathfrak A'","aliases":"A' mathfrak","display":"\\mathfrak A'","kind":"use","excerpt":"Define the unital adjoint-closed algebra A = alg^∗ (I,R_1,…,R_m). Its commutant is A' = {X∈End(H):XA=AX for all A∈ A}. The center is Z( A)= A∩ A'.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d4e39782b46b77af","tex":"A'_{m'}","aliases":"A'_m'","display":"A'_{m'}","kind":"use","excerpt":"Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c08d2cf77d38259","tex":"a^*","aliases":"a^*","display":"a^*","kind":"use","excerpt":"[Finite repair-relevant block selection] Suppose there is a unique repair-relevant block a^* and a margin m>0 such that s_a^*≥ s_a+m for every a≠ a^*. If max_a|widehat s_a-s_a|<fracm2, then arg max_awidehat s_a=a^*.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"0 · 1 . Then (H-I)^*(H-I) = 0 · 0 0 · a^2 . The holonomy-energy eigenvalues are 0 and a^2. Thus E^flat=spansete_1. For a threshold 0≤τ<a^2, the approximate flat core is still one-dimensional. For τ≥ a^2, both coordinates are admitted, but the second coordinate carries loop distortion at scale |a| and should not be described as invariant.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c7aff0995140d9fb","tex":"A^{\\rm base}_{\\rm cal,prot}","aliases":"A^rm base_rm cal,prot rm rm","display":null,"kind":"use","excerpt":"widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9e1ae7a5cb9427ef","tex":"A^{\\rm prot}_e","aliases":"A^rm prot_e rm","display":"A^{\\rm prot}_e","kind":"use","excerpt":"For target episode e and candidate c, let A^rm slice_e(c) · =accuracy on the affected stratum, A^rm prot_e(c) · =accuracy on the unaffected protected stratum, q(c) · =repair rank. The candidate utility is U_e(c) = A^rm slice_e(c) -2[A^rm prot_e(0)-A^rm prot_e(c)]_+ -0.0005q(c), and the reported protected utility gain is G_e(c)=U_e(c)-A^rm slice_e(0","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-92fd89791b2c328d","tex":"A^{\\rm slice}_e","aliases":"A^rm slice_e rm","display":"A^{\\rm slice}_e","kind":"use","excerpt":"For target episode e and candidate c, let A^rm slice_e(c) · =accuracy on the affected stratum, A^rm prot_e(c) · =accuracy on the unaffected protected stratum, q(c) · =repair rank. The candidate utility is U_e(c) = A^rm slice_e(c) -2[A^rm prot_e(0)-A^rm prot_e(c)]_+ -0.0005q(c), and the reported protected utility gain is G_e(c)=U_e(c)-A^rm slice_e(0","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8b754205b6d87c3f","tex":"A^\\ast","aliases":"A^∗ ast","display":"A^\\ast","kind":"use","excerpt":"The identity belongs to A, so r∈ H_r. For B∈ A and Ar∈ H_r, one has BAr∈ H_r, proving invariance. Every invariant subspace containing r contains Ar for every A, hence contains H_r. Adjoint closure implies invariance under every A^∗, so the orthogonal complement is invariant as well.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"After n strict transitions, V(A_n)≥ V(A_0)+nΔ≥ nΔ. Since V(A_n)≤1, one has n≤1/Δ.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bdcba39a7197cb28","tex":"\\mathcal A_0","aliases":"A_0 mathcal","display":"\\mathcal A_0","kind":"use","excerpt":"[Risk-filter tradeoff] Let A_r⊆ A_0 be the candidate set retained with and without a risk filter. Then max_c∈ A_0widehat U(c) ≥ max_c∈ A_rwidehat U(c), but no analogous inequality favors the unfiltered selector for true protected risk.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"This theorem gives a precise reason to value experiments rather than static classification. Suppose an experiment a returns an observation Y_a whose distribution differs between the aliased hypotheses. The compiler can then choose a sequence a_1,a_2,…,a_t that increases the information available for diagnosis.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"This theorem gives a precise reason to value experiments rather than static classification. Suppose an experiment a returns an observation Y_a whose distribution differs between the aliased hypotheses. The compiler can then choose a sequence a_1,a_2,…,a_t that increases the information available for diagnosis.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"[Exact causal-packet recovery for a linear encoder] Suppose E_a = A_a · B_a, A_a,B_a:ℝ^8→ℝ^d_a, and suppose the centered conditional covariance of ψ(x) given s(x)=1 is positive definite. Then span{Δ_a(x):s(x)=1} = im B_a. Consequently, a rank-r singular-subspace estimator with r=rank B_a recovers the exact causal packet from a sufficiently rich noi","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3665b343681e5016","tex":"A_{a_t}","aliases":"A_a_t","display":"A_{a_t}","kind":"use","excerpt":"[Meta-regret Lyapunov contraction] Suppose there exist w∈ℝ_>0^7 and q∈[0,1) such that for every reachable r_t, the controller can choose an authorized action a_t satisfying w^top A_a_tr_t ≤ q w^top r_t. Then every controlled trajectory satisfying [source reference] obeys w^top r_t ≤ q^t w^top r_0 + ∑_k=0^t-1 q^t-1-kw^top e_k.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-66be131002b8b5af","tex":"A_C","aliases":"A_C","display":"A_C","kind":"use","excerpt":"Let K_G be the global parent and K_C the chart process. Let A_C(x) be the frozen chart-certificate decision for target episode x. The descendant is K_3(x)= K_C(x), · A_C(x)=1,","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5494a82a5f60bced","tex":"A_c","aliases":"A_c","display":"A_c","kind":"use","excerpt":"A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribut","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-90feae97c7e75daa","tex":"A_{c(h)}","aliases":"A_c(h)","display":"A_{c(h)}","kind":"use","excerpt":"Every primary target is four dimensional and has two invertible affine charts: y=A_c(h)h+b+ε. The two maps share the protected-context block but differ in the mutable response block. The source repair therefore preserves the chart while inducing two different target-space repair conjugates.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0841717db411db13","tex":"A_{j-1}","aliases":"A_j-1","display":"A_{j-1}","kind":"use","excerpt":"ion be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representation is exposed to the m","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e57fd5001c70376c","tex":"A_j^{-1}","aliases":"A_j^-1","display":"A_j^{-1}","kind":"use","excerpt":"Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7dcee546650900ca","tex":"a_{\\min}","aliases":"a_min min","display":"a_{\\min}","kind":"use","excerpt":"widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"After n strict transitions, V(A_n)≥ V(A_0)+nΔ≥ nΔ. Since V(A_n)≤1, one has n≤1/Δ.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5ec3d9592dfb869c","tex":"\\mathcal A_r","aliases":"A_r mathcal","display":"\\mathcal A_r","kind":"use","excerpt":"[Risk-filter tradeoff] Let A_r⊆ A_0 be the candidate set retained with and without a risk filter. Then max_c∈ A_0widehat U(c) ≥ max_c∈ A_rwidehat U(c), but no analogous inequality favors the unfiltered selector for true protected risk.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d96fa82ae71ac6a0","tex":"\\mathfrak A_R","aliases":"A_R mathfrak","display":"\\mathfrak A_R","kind":"use","excerpt":"The algebra generated by the response family is A_R = alg^*(I,R_1,…,R_k). Because the R_j are simultaneously block scalar in the exact model, A_R is commutative and every polynomial in the responses is again block scalar.","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a3b056e5e92e04dd","tex":"\\mathfrak A_R'","aliases":"A_R' mathfrak","display":"\\mathfrak A_R'","kind":"use","excerpt":"The response commutant is A_R' = setX∈End(H):XR_j=R_jX for every j.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-da296e3c332ec5c8","tex":"a_{\\mathrm{risky}}","aliases":"a_risky mathrm","display":"a_{\\mathrm{risky}}","kind":"use","excerpt":"Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety o","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7855a65c77a5fa64","tex":"A_{\\rm aff}","aliases":"A_rm aff rm","display":"A_{\\rm aff}","kind":"use","excerpt":"The protected utility used in Block 7 is the balanced accuracy functional J_Ξ(A) = frac12 A_rm aff(A)+frac12 A_rm non(A), where A_rm aff is accuracy on the defect-affected stratum and A_rm non is accuracy on the protected unaffected stratum. This prevents the more common unaffected examples from overwhelming the repair objective.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ba43fc411d140427","tex":"A_{\\rm non}","aliases":"A_rm non rm","display":"A_{\\rm non}","kind":"use","excerpt":"The protected utility used in Block 7 is the balanced accuracy functional J_Ξ(A) = frac12 A_rm aff(A)+frac12 A_rm non(A), where A_rm aff is accuracy on the defect-affected stratum and A_rm non is accuracy on the protected unaffected stratum. This prevents the more common unaffected examples from overwhelming the repair objective.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8546df2400ae89a8","tex":"a_{\\mathrm{safe}}","aliases":"a_safe mathrm","display":"a_{\\mathrm{safe}}","kind":"use","excerpt":"Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety o","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-29a879c72dfad052","tex":"A_{t+2}","aliases":"A_t+2","display":"A_{t+2}","kind":"use","excerpt":"The phrase recursive self-improvement is often represented by the schematic loop A_t⟶ A_t+1⟶ A_t+2⟶⋯, with the informal assertion that each descendant is ``better'' than its parent. This notation suppresses the central difficulties. It does not identify the object that persists across generations, the evidence by which improvement is judged, the in","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ba33cc73034f71d7","tex":"\\mathfrak A_x","aliases":"A_x mathfrak","display":"\\mathfrak A_x","kind":"use","excerpt":"For nonlinear dynamics on a state manifold M, the response algebra becomes base-point dependent: x⟼ A_x⊆End(T_xM). Local central projectors define a bundle of tangent strata when their ranks remain locally constant. Transport around a loop can permute or distort the local modules, producing atlas holonomy. A complete nonlinear theory must combine: ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6983fa40ff77fa12","tex":"\\mathrm{abs}","aliases":"abs mathrm","display":"\\mathrm{abs}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9713c7e6a6b31190","tex":"\\mathrm{abstract}","aliases":"abstract mathrm","display":"\\mathrm{abstract}","kind":"use","excerpt":"Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before f","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be14a316739e6949","tex":"\\operatorname{Acc}","aliases":"Acc operatorname","display":"\\operatorname{Acc}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a027721bfb3521f","tex":"\\operatorname{Acc}_{\\mathrm{answer}}","aliases":"Acc_answer operatorname mathrm","display":"\\operatorname{Acc}_{\\mathrm{answer}}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c882d1eea6ca3ee9","tex":"\\mathrm{accuracy}","aliases":"accuracy mathrm","display":"\\mathrm{accuracy}","kind":"use","excerpt":"Suppose a sandbox returns only the scalar task score q(ω)=accuracy(ω). Execution ω_0 reaches accuracy 0.9 using authorized retrieval. Execution ω_1 reaches the same accuracy after importing a prohibited hidden answer key. Then q(ω_0)=q(ω_1)=0.9, but the ideal full-information decisions differ. By [source reference], no witness using only accuracy c","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8b5ce455d01fdf81","tex":"\\mathrm{actuation}","aliases":"actuation mathrm","display":"\\mathrm{actuation}","kind":"use","excerpt":"From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f672f5afa19f4601","tex":"\\operatorname{Aff}","aliases":"Aff operatorname","display":"\\operatorname{Aff}","kind":"use","excerpt":"The inherited global target-repair response lies in the diagonal class D_diag = set(R,R):R∈Aff(H_t), whereas the chart grammar reaches D_chart = Aff(H_t)×Aff(H_t).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6d04ddddf121bd19","tex":"\\mathrm{affine\\ oracle}","aliases":"affine oracle mathrm","display":"\\mathrm{affine\\ oracle}","kind":"use","excerpt":"The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b04e45579a45ba0","tex":"\\operatorname{alg}^{*}","aliases":"alg^* operatorname","display":"\\operatorname{alg}^{*}","kind":"use","excerpt":"The algebra generated by the response family is A_R = alg^*(I,R_1,…,R_k). Because the R_j are simultaneously block scalar in the exact model, A_R is commutative and every polynomial in the responses is again block scalar.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1923d1cdbd89cac7","tex":"\\operatorname{alg}^{\\ast}","aliases":"alg^∗ operatorname ast","display":"\\operatorname{alg}^{\\ast}","kind":"use","excerpt":"Define the unital adjoint-closed algebra A = alg^∗ (I,R_1,…,R_m). Its commutant is A' = {X∈End(H):XA=AX for all A∈ A}. The center is Z( A)= A∩ A'.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-525be5b6e176b361","tex":"\\mathrm{align}","aliases":"align mathrm","display":"\\mathrm{align}","kind":"use","excerpt":"Let (Y,H_s,H_t)∼μ be the protected semantics, source trace, and target trace under context distribution μ. Fix a source probe P and bridge B. Define e_src=Y-PH_s, e_align=P(H_s-BH_t). Then Y-PBH_t=e_src+e_align.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4d5b611fe88ff347","tex":"\\mathrm{ans}","aliases":"ans mathrm","display":"\\mathrm{ans}","kind":"use","excerpt":"Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matc","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e7e898debdc0c247","tex":"\\mathrm{answer}","aliases":"answer mathrm","display":"\\mathrm{answer}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-31691ef0701483eb","tex":"\\mathrm{arch}","aliases":"arch mathrm","display":"\\mathrm{arch}","kind":"use","excerpt":"The architecture-foreign paired effect is widehatΔ_Γ^arch = 0.090800, CI_95% = [0.052311,0.135081]. The sign-flip estimate is again 9.9999×10^-6. The child improves 34.722% of pairs, ties on 65.278%, and no architecture-foreign pair favors the parent.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4e2b11a7001e1904","tex":"\\arg","aliases":"arg","display":"\\arg","kind":"use","excerpt":"Each repair g receives a description-length proxy L(g) recording parameter count and constructor surcharge. The compiler selects widehat g ∈ argmin_g∈ G_21 L(g) subject to the appropriate protected gates. If no repair passes and the collision certificate is valid, it returns G_bot; otherwise it returns unresolved.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-74ac353874d337a1","tex":"\\argmax","aliases":"argmax arg max operatorname","display":"{\\operatorname{arg\\,max}}","kind":"use","excerpt":"pection] Let u:Y→[0,1] be a bounded utility. Define V(δ)= E[u(Y)mid A_t,δ], widehat V(δ)= E[u(Y)mid ϕ_t(A_t),δ]. If ϕ_t is ε-intervention-sufficient, then sup_δ∈D_t absV(δ)-widehat V(δ)≤ε. If δ^*∈arg max_δ∈D_tV(δ), widehatδ∈arg max_δ∈D_twidehat V(δ), then V(δ^*)-V(widehatδ)≤2ε.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-064d19b2d728284c","tex":"\\argmax_a","aliases":"argmax_a arg max_a argmax operatorname","display":"{\\operatorname{arg\\,max}}_a","kind":"use","excerpt":"[Finite repair-relevant block selection] Suppose there is a unique repair-relevant block a^* and a margin m>0 such that s_a^*≥ s_a+m for every a≠ a^*. If max_a|widehat s_a-s_a|<fracm2, then arg max_awidehat s_a=a^*.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e5796755a8a02b3a","tex":"\\argmax_i","aliases":"argmax_i arg max_i argmax operatorname","display":"{\\operatorname{arg\\,max}}_i","kind":"use","excerpt":"[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-216a2d3ce4a02d45","tex":"\\argmax_{i:\\operatorname{Valid}(c_i,\\Pi_i^+)=1}","aliases":"argmax_i:Valid(c_i,Π_i^+)=1 arg max_i:Valid(c_i,Π_i^+)=1 argmax operatorname Pi operatorname operatorname Pi","display":"{\\operatorname{arg\\,max}}_{i:\\operatorname{Valid}(c_i,\\Pi_i^+)=1}","kind":"use","excerpt":"Let Valid(c_i,Π_i^+)∈{0,1} encode capability, integrity, provenance, and resource validity. The selected descendant is K_Γ^⋆(τ) ∈ arg max_i:Valid(c_i,Π_i^+)=1 S_V(c_i), with the no-op baseline retained when no valid candidate exists.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b1c80d0dcfbfb964","tex":"\\argmax_{i\\notin I_{j-1}}","aliases":"argmax_i∉ I_j-1 arg max_i∉ I_j-1 argmax notin operatorname notin","display":"{\\operatorname{arg\\,max}}_{i\\notin I_{j-1}}","kind":"use","excerpt":"The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the stand","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1dae43c1a83fb075","tex":"\\argmax_{v\\in\\Vcal_{\\mathrm{cand}}}","aliases":"argmax_v∈Vcal_cand arg max_v∈V_cand argmax in Vcal mathrm operatorname in mathcal mathrm","display":"{\\operatorname{arg\\,max}}_{v\\in{\\mathcal{V}}_{\\mathrm{cand}}}","kind":"use","excerpt":"Equation [source reference] gives a direct acquisition score. For a candidate family V_cand, choose v^⋆ ∈ arg max_v∈V_cand fracabsanglesz,P_I^⊥v^2 normP_I^⊥v^2+λ c(v)+βRisk(v), where c(v) is implementation cost and Risk(v) is a protected-risk estimate.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-22ff48acc25fbd74","tex":"\\argmax_{\\gamma\\in\\Gamma_{\\rm safe}}","aliases":"argmax_γ∈Γ_rm safe arg max_γ∈Γ_rm safe argmax gamma in Gamma rm operatorname gamma in Gamma rm","display":"{\\operatorname{arg\\,max}}_{\\gamma\\in\\Gamma_{\\rm safe}}","kind":"use","excerpt":"ot4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the frozen bounds. Among safe configurations, the compiler selects widehatγ ∈ arg max_γ∈Γ_rm","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dea99d7da977484a","tex":"\\argmax_{\\delta\\in\\Dcal_t}","aliases":"argmax_δ∈Dcal_t arg max_δ∈D_t argmax delta in Dcal operatorname delta in mathcal","display":"{\\operatorname{arg\\,max}}_{\\delta\\in{\\mathcal{D}}_t}","kind":"use","excerpt":"pection] Let u:Y→[0,1] be a bounded utility. Define V(δ)= E[u(Y)mid A_t,δ], widehat V(δ)= E[u(Y)mid ϕ_t(A_t),δ]. If ϕ_t is ε-intervention-sufficient, then sup_δ∈D_t absV(δ)-widehat V(δ)≤ε. If δ^*∈arg max_δ∈D_tV(δ), widehatδ∈arg max_δ∈D_twidehat V(δ), then V(δ^*)-V(widehatδ)≤2ε.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b779fb94202ff28e","tex":"\\argmin","aliases":"argmin arg min operatorname","display":"{\\operatorname{arg\\,min}}","kind":"use","excerpt":"h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-120d3d6e5e35db23","tex":"\\argmin_{B,b}","aliases":"argmin_B,b arg min_B,b argmin operatorname","display":"{\\operatorname{arg\\,min}}_{B,b}","kind":"use","excerpt":"Let H_s(τ)∈ℝ^36 be the source model's final latent trace state and H_t(τ)∈ℝ^36 the corresponding foreign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The tran","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-13042369cdc94dde","tex":"\\argmin_T","aliases":"argmin_T arg min_T argmin operatorname","display":"{\\operatorname{arg\\,min}}_T","kind":"use","excerpt":"h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a6b217cdc6c37848","tex":"\\argmin_{T^{\\mathsf T}T=I}","aliases":"argmin_T^ TT=I arg min_T^ TT=I argmin mathsf operatorname mathsf","display":"{\\operatorname{arg\\,min}}_{T^{\\mathsf T}T=I}","kind":"use","excerpt":"For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change af","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5efacbdd6855f457","tex":"\\operatorname{arsinh}","aliases":"arsinh operatorname","display":"\\operatorname{arsinh}","kind":"use","excerpt":"The nonlinear family applies Y_t = fracsinh (α(X_tA+b))α, α>0, coordinatewise. The declared inverse constructor first applies ϕ^-1_α(y) = fracarsinh(α y)α and then fits a step-affine map.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16bf83baea03980d","tex":"\\mathrm{audit}","aliases":"audit mathrm","display":"\\mathrm{audit}","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ba6c14f08dc057e8","tex":"\\mathrm{AuthorizedCommit}","aliases":"AuthorizedCommit mathrm","display":"\\mathrm{AuthorizedCommit}","kind":"use","excerpt":"The example separates two predicates: ImprovesSlice(R_t) not⇒ AuthorizedCommit(R_t).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e361ae3431481eec","tex":"\\mathrm{avg}","aliases":"avg mathrm","display":"\\mathrm{avg}","kind":"use","excerpt":"Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d6fff1fe4a7db6bf","tex":"\\mathbf B","aliases":"B mathbf","display":"\\mathbf B","kind":"use","excerpt":"Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes ","count":14},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8350a358f9a31a5b","tex":"B'_{m'}","aliases":"B'_m'","display":"B'_{m'}","kind":"use","excerpt":"Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f673c03c77f9ac50","tex":"B^*","aliases":"B^*","display":"B^*","kind":"use","excerpt":"[Quantitative graph-Hodge gluing] Let L_G:=BB^* be the graph Laplacian and let P_cyc be the orthogonal projector from K^E onto ker B, the cycle space. Then inf_k∈ K^Inorme+ B^*k = normP_cyce. The mean-zero minimum-norm gauge is k^* =-(L_G^† B⊗ I_K)e. If λ_2(L_G)>0 is the spectral gap, then normk^* ≤ frac1sqrtλ_2(L_G)norm e. Exact graph-level gluing","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5b2a90e8ec391e50","tex":"\\mathbf B^*","aliases":"B^* mathbf","display":"\\mathbf B^*","kind":"use","excerpt":"K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"Let h∼ N(0,1) and let s∈{-1,+1} be independent and balanced. Define y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8b99783075cb661f","tex":"B^\\top","aliases":"B^top top","display":"B^\\top","kind":"use","excerpt":"The executable classifier applies the ordered rule Class(r,b,H) = LOCAL_OBSTRUCTION, · normP_(im D)^⊥r>τ, GLUING_OBSTRUCTION, · normP_ker B^topb>τ, EVALUATOR_HOLONOMY, · norm(H-I)e>τ, FLAT_REPAIR, · otherwise. The order is part of the receipt because a case can exhibit more than one defect. The fixture isolates the four modes, so the ordered rule i","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"The sequence [b_1],[b_2],[b_3],…∈coker D is an obstruction tower. Choices in ker D at early orders can alter later classes, so a repair search must retain branch information rather than greedily fixing one representative.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7bfd219cef7f3bf8","tex":"b_{1,\\alpha}","aliases":"b_1,α alpha","display":"b_{1,\\alpha}","kind":"use","excerpt":"The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","excerpt":"A lineage may branch: A⟶ B_1, A⟶ B_2. No definition of recursive improvement requires one child to be the unique continuation of the parent.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"The sequence [b_1],[b_2],[b_3],…∈coker D is an obstruction tower. Choices in ker D at early orders can alter later classes, so a repair search must retain branch information rather than greedily fixing one representative.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-37d3155b470aa6af","tex":"b_{2,\\alpha}","aliases":"b_2,α alpha","display":"b_{2,\\alpha}","kind":"use","excerpt":"The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5284f24ae8cda5b8","tex":"b_3","aliases":"b_3","display":"b_3","kind":"use","excerpt":"The sequence [b_1],[b_2],[b_3],…∈coker D is an obstruction tower. Choices in ker D at early orders can alter later classes, so a repair search must retain branch information rather than greedily fixing one representative.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1a4c2e9cb4075d24","tex":"B_a","aliases":"B_a","display":"B_a","kind":"use","excerpt":"[Exact causal-packet recovery for a linear encoder] Suppose E_a = A_a · B_a, A_a,B_a:ℝ^8→ℝ^d_a, and suppose the centered conditional covariance of ψ(x) given s(x)=1 is positive definite. Then span{Δ_a(x):s(x)=1} = im B_a. Consequently, a rank-r singular-subspace estimator with r=rank B_a recovers the exact causal packet from a sufficiently rich noi","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-58a32903afc3d14d","tex":"b_a","aliases":"b_a","display":"b_a","kind":"use","excerpt":"Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d41402bfd3b150fb","tex":"B_c","aliases":"B_c","display":"B_c","kind":"use","excerpt":"Let M(B) be the offspring matrix under a fixed discovery budget B. Define the critical budget B_c = infsetB:rad(M(B))>1. Comparisons of R_imp are meaningful only at matched budgets and evidence access.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-10fba79ce2cdc0ce","tex":"b_c","aliases":"b_c","display":"b_c","kind":"use","excerpt":"A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribut","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence maps the trace to","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","excerpt":"ion be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representation is exposed to the m","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-89304b2594ebe322","tex":"B_{j-1}","aliases":"B_j-1","display":"B_{j-1}","kind":"use","excerpt":"ion be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representation is exposed to the m","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fc9ad75de14545d5","tex":"b_m","aliases":"b_m","display":"b_m","kind":"use","excerpt":"Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applie","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-01450f293fbd90a9","tex":"b_{m+1}","aliases":"b_m+1","display":"b_{m+1}","kind":"use","excerpt":"[Portfolio refinement shrinks the invisible subspace] Let B^+=B∪{b_m+1}. Then N_B^+ ⊆ N_B. The inclusion is strict if and only if there exists v∈N_B such that Db_m+1(v)≠0.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7a8dd1d457c41089","tex":"b_{N,\\alpha}","aliases":"b_N,α alpha","display":"b_{N,\\alpha}","kind":"use","excerpt":"The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-38d13a3d62430ff9","tex":"b_{n,\\alpha}","aliases":"b_n,α alpha","display":"b_{n,\\alpha}","kind":"use","excerpt":"At order n, let b_n,α∈mathscr V(U_α) be the residual coefficient determined by lower-order choices. The local obstruction is [b_n,α]∈coker D_α. If every local class vanishes, choose u_n,α with D_α u_n,α=-b_n,α. Their differences define an order-n gluing class in check H^1(ker D).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a2c78198d53e894","tex":"B_R","aliases":"B_R","display":"B_R","kind":"use","excerpt":"Let R_t be true protected regression and let U_t^R be a conditional upper confidence bound. Require U_t^R≤β_t, ∑_tβ_t≤ B_R. Let d_t be semantic debt incurred by approximate grammar or evaluator transport, with ∑_td_t≤ B_sem. A lineage certificate should therefore report three separate cumulative quantities: ( ∑_tδ_t, ∑_tβ_t, ∑_td_t ). A positive ga","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c22123c31ec8bac6","tex":"B_R^W","aliases":"B_R^W","display":"B_R^W","kind":"use","excerpt":"[Gauge-fixed local nonlinear repair by contraction] Suppose D:U→ V has a bounded right inverse Q:V→ U, so DQ=I_V, and set W:=im Q. Then restr D W:W→ V is an isomorphism with inverse Q. Let B_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a uniq","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7d4c2fbf3274db6a","tex":"B_{\\mathrm{research}}","aliases":"B_research mathrm","display":"B_{\\mathrm{research}}","kind":"use","excerpt":"Every parent and child run receives exactly B_research=6 full candidate experiments. Candidate metadata and cheap diagnostics are common inputs. The number of proposal receipts, scored evaluator calls, capability checks, and integrity-monitor calls is therefore matched.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-41f8631345e4638b","tex":"B_{\\rm pair}","aliases":"B_rm pair rm","display":"B_{\\rm pair}","kind":"use","excerpt":"B_rm pair · Matched deployment paired-query budget","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3892ab019684f993","tex":"B_{\\rm research}","aliases":"B_rm research rm","display":"B_{\\rm research}","kind":"use","excerpt":"Both compilers receive exactly B_rm research=6 full experiments per problem. The child can use early experiments to estimate response mixtures and allocate later experiments, but it receives no additional full experiment. Candidate metadata, proposal count, evaluator calls, capability checks, and integrity calls are matched.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-10c299c3911c0c85","tex":"b_s","aliases":"b_s","display":"b_s","kind":"use","excerpt":"Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a9c957a800f2d151","tex":"B_{\\mathrm{sem}}","aliases":"B_sem mathrm","display":"B_{\\mathrm{sem}}","kind":"use","excerpt":"Let R_t be true protected regression and let U_t^R be a conditional upper confidence bound. Require U_t^R≤β_t, ∑_tβ_t≤ B_R. Let d_t be semantic debt incurred by approximate grammar or evaluator transport, with ∑_td_t≤ B_sem. A lineage certificate should therefore report three separate cumulative quantities: ( ∑_tδ_t, ∑_tβ_t, ∑_td_t ). A positive ga","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b17ff042f6455b31","tex":"b_t","aliases":"b_t","display":"b_t","kind":"use","excerpt":"Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2d72109f1a3a7929","tex":"\\bar A","aliases":"bar A bar","display":"\\bar A","kind":"use","excerpt":"1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e15729f91b24285c","tex":"\\bar B","aliases":"bar B bar","display":"\\bar B","kind":"use","excerpt":"1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-60e74a0938329fe2","tex":"\\bar g","aliases":"bar g bar","display":"\\bar g","kind":"use","excerpt":"For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a unive","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-49403cdb00505f0e","tex":"\\bar g_c","aliases":"bar g_c bar","display":"\\bar g_c","kind":"use","excerpt":"For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a unive","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b21481c4bf04bb00","tex":"\\bar h_i","aliases":"bar h_i bar","display":"\\bar h_i","kind":"use","excerpt":"A candidate pool contains source state h_i, target observation y_i, and known observation scale σ_i. The inherited acquisition policy selects rows sequentially to maximize the quality-weighted log-determinant increment logdet( λ I+ ∑_i∈ Sσ_i^-2bar h_ibar h_i^top ). Both parent and child use the same resulting order. The parent takes the first 80 ro","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bbb07af497a4be95","tex":"\\bar h_i^\\top","aliases":"bar h_i^top bar top","display":"\\bar h_i^\\top","kind":"use","excerpt":"A candidate pool contains source state h_i, target observation y_i, and known observation scale σ_i. The inherited acquisition policy selects rows sequentially to maximize the quality-weighted log-determinant increment logdet( λ I+ ∑_i∈ Sσ_i^-2bar h_ibar h_i^top ). Both parent and child use the same resulting order. The parent takes the first 80 ro","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-813d7ea0810bedcf","tex":"\\bar R","aliases":"bar R bar","display":"\\bar R","kind":"use","excerpt":"[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar con","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-650714de38f00311","tex":"\\bar R_0","aliases":"bar R_0 bar","display":"\\bar R_0","kind":"use","excerpt":"[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar con","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2999bcd31bcc64e9","tex":"\\bar R_1","aliases":"bar R_1 bar","display":"\\bar R_1","kind":"use","excerpt":"[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar con","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0d8a7f2d6dbebf42","tex":"\\bar R_{\\mathrm{ctr}}","aliases":"bar R_ctr bar mathrm","display":"\\bar R_{\\mathrm{ctr}}","kind":"use","excerpt":"[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar con","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ec6f45569e7597f","tex":"\\bar T_+","aliases":"bar T_+ bar","display":"\\bar T_+","kind":"use","excerpt":"1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b3900f638893d772","tex":"\\bar T_-","aliases":"bar T_- bar","display":"\\bar T_-","kind":"use","excerpt":"1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-033e1fd2dc73c143","tex":"\\bar T_{c(h)}","aliases":"bar T_c(h) bar","display":"\\bar T_{c(h)}","kind":"use","excerpt":"1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1784da886e1aea15","tex":"\\bar v","aliases":"bar v bar","display":"\\bar v","kind":"use","excerpt":"Let I=im D⊆ V be the current authorized correction image and let z=P_I^⊥r be the unresolved component of defect r. Suppose a candidate grammar extension contributes a response vector v∈ V. Only its component outside the old image is structurally new: bar v=P_I^⊥v. The expanded image is I'=I+span{v}.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d182d4e2c4ce05fd","tex":"\\mathrm{base}","aliases":"base mathrm","display":"\\mathrm{base}","kind":"use","excerpt":"The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fcd32b5b66baa834","tex":"\\Bcal^+","aliases":"Bcal^+ B^+ Bcal mathcal","display":"{\\mathcal{B}}^+","kind":"use","excerpt":"[Portfolio refinement shrinks the invisible subspace] Let B^+=B∪{b_m+1}. Then N_B^+ ⊆ N_B. The inclusion is strict if and only if there exists v∈N_B such that Db_m+1(v)≠0.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8695bac541800c07","tex":"\\Bcal_t^{\\mathrm{sand}}","aliases":"Bcal_t^sand B_t^sand Bcal mathrm mathcal mathrm","display":"{\\mathcal{B}}_t^{\\mathrm{sand}}","kind":"use","excerpt":"A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-61ae70fc1146159b","tex":"\\bigwedge_i","aliases":"bigwedge_i bigwedge","display":"\\bigwedge_i","kind":"use","excerpt":"_ω_i; item the conjunction of the inner postconditions and bridge guarantees implies C_ω; item execution of ω from C_ω establishes Q_ω; item receipt composition preserves provenance and protected evidence references. Then the composed receipt is sound for ω∘(ω_1,…,ω_n) with precondition bigwedge_iC_ω_i and postcondition Q_ω.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ff66021ec1d8e804","tex":"\\bot","aliases":"bot","display":"\\bot","kind":"use","excerpt":"Let ϕ_0:T→Z_0 be the inherited diagnostic map. In Block 15, six inherited obstruction classes retain their existing tokens, while two previously unseen obstruction mechanisms are both mapped to one open-set token bot: ϕ_0(τ_ m int) = ϕ_0(τ_ m temp) = bot. The equality in [source reference] is not itself a defect. An unknown token is an appropriate ","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"Let j:V↪ V' be an injective evidence-conservative embedding of old defects into an expanded ontology. Let C=im D⊆ V, C'=im D'⊆ V', and assume preservation of old correctability: j(C)⊆ C'. The embedding induces a map overline j:V/C→ V'/C'.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5eda86663b484271","tex":"c^{\\mathsf T}","aliases":"c^ T mathsf","display":"c^{\\mathsf T}","kind":"use","excerpt":"Choose c∈ℝ^k outside the finite union of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a li","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-993119079db56d40","tex":"\\mathcal C^{(0)}_e","aliases":"C^(0)_e mathcal","display":"\\mathcal C^{(0)}_e","kind":"use","excerpt":"[Fixed-candidate structural invariance] Let C_e be the authorized candidate set for episode e. If parent and child differ only in the selector and C^(0)_e= C^(1)_e for every e, then the pointwise reachable correction set and its linear span are identical for parent and child.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c9b94930e021b53f","tex":"\\mathcal C^{(1)}_e","aliases":"C^(1)_e mathcal","display":"\\mathcal C^{(1)}_e","kind":"use","excerpt":"[Fixed-candidate structural invariance] Let C_e be the authorized candidate set for episode e. If parent and child differ only in the selector and C^(0)_e= C^(1)_e for every e, then the pointwise reachable correction set and its linear span are identical for parent and child.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"use","excerpt":"Set C=D(K), a closed convex cone. Its dual is C^*=sety:D^*y∈ K^*. By the bipolar theorem, -r∈ C if and only if anglesy,-r≥0 for every y∈ C^*, which is the displayed inequality.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4a41d044c4883dce","tex":"c^*","aliases":"c^*","display":"c^*","kind":"use","excerpt":"ity regret under simultaneous lower bounds] Let C_rm safe be the truly safe candidate set. Suppose every c∈ C_rm safe passes the implemented admissibility filter, and suppose 0≤ U_e(c)-widehat U_rm LCB(c)≤ϵ for every admissible c. If widehat c maximizes widehat U_rm LCB and c^* maximizes U_e over C_rm safe, then U_e(c^*)-U_e(widehat c)≤ϵ.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51ad859aedd717f4","tex":"C^{a}_{A0\\to c}","aliases":"C^a_A0→ c to","display":"C^{a}_{A0\\to c}","kind":"use","excerpt":"For architectures a,b and admitted chart c, an architecture map is T^c_b→ a:H_b,c⟶ H_a,c. For one architecture a and two admitted charts c,d, a context map is C^a_c→ d:H_a,c⟶ H_a,d. Block 20 uses A0 as a reference chart and fits a star of maps C^a_c→ A0 and C^a_A0→ c from protected semantic twins.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e7dcad3c5426ee73","tex":"C^{a}_{c\\to A0}","aliases":"C^a_c→ A0 to","display":"C^{a}_{c\\to A0}","kind":"use","excerpt":"For architectures a,b and admitted chart c, an architecture map is T^c_b→ a:H_b,c⟶ H_a,c. For one architecture a and two admitted charts c,d, a context map is C^a_c→ d:H_a,c⟶ H_a,d. Block 20 uses A0 as a reference chart and fits a star of maps C^a_c→ A0 and C^a_A0→ c from protected semantic twins.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ff7607ed3555878b","tex":"C^a_{c\\to d}","aliases":"C^a_c→ d to","display":"C^a_{c\\to d}","kind":"use","excerpt":"The next threshold is atlas-indexed cross-architecture Neuralese transport. Source and target systems should each carry locally valid semantic charts, transport should be tested on chart overlaps, and architecture transport should approximately commute with context transition: C^b_c→ dT^c_a→ b ≈ T^d_a→ bC^a_c→ d. Block 20 tests this threshold direc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ecaf00e9693ba47","tex":"C^b_{c\\to d}","aliases":"C^b_c→ d to","display":"C^b_{c\\to d}","kind":"use","excerpt":"The next threshold is atlas-indexed cross-architecture Neuralese transport. Source and target systems should each carry locally valid semantic charts, transport should be tested on chart overlaps, and architecture transport should approximately commute with context transition: C^b_c→ dT^c_a→ b ≈ T^d_a→ bC^a_c→ d. Block 20 tests this threshold direc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5cee0473087cadf5","tex":"C^k","aliases":"C^k","display":"C^k","kind":"use","excerpt":"[Lyapunov--Schmidt repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-201d337f1f8a4afd","tex":"\\mathbb C^{m_\\alpha}","aliases":"C^m_α mathbb alpha","display":"\\mathbb C^{m_\\alpha}","kind":"use","excerpt":"[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a2bb85c8b7f58b80","tex":"\\mathbb C^{n_\\alpha}","aliases":"C^n_α mathbb alpha","display":"\\mathbb C^{n_\\alpha}","kind":"use","excerpt":"[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-28466245ee25f5ff","tex":"c^\\top","aliases":"c^top top","display":"c^\\top","kind":"use","excerpt":"-1 , we obtain absc^top b=frac1sqrt3>0. Thus local correction succeeds while descent fails.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a294de936aabf841","tex":"C^\\ast","aliases":"C^∗ ast","display":"C^\\ast","kind":"use","excerpt":"A finite-dimensional unital ∗-algebra is a finite-dimensional C^∗-algebra. Decompose its identity into minimal central projections. Each central summand is a simple finite-dimensional C^∗-algebra and therefore isomorphic to a full matrix algebra M_n_α( C). Its representation on the corresponding invariant subspace is a direct sum of m_α equivalent ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7f12695aa484d09b","tex":"C_+","aliases":"C_+","display":"C_+","kind":"use","excerpt":"[Finite conflict-grounded language synthesis] Let a finite development archive contain one inherited token cell C with at least two incompatible repair subclasses. Suppose: item a candidate probe p∈P^+ separates C into cells C_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ae99c7ca70e8dff","tex":"C_-","aliases":"C_-","display":"C_-","kind":"use","excerpt":"[Finite conflict-grounded language synthesis] Let a finite development archive contain one inherited token cell C with at least two incompatible repair subclasses. Suppose: item a candidate probe p∈P^+ separates C into cells C_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"Suppose every target exposes a no-op c_0 and one repair c_1. On half the episodes, the repair gain is 0.4; on the other half it is -0.2. A parent that always abstains has expected gain zero. A child that observes a receipt feature identifying the positive regime with probability 0.8 and the negative regime with probability 0.7 selects c_1 with thos","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aee29aa7640dd406","tex":"C_{1}","aliases":"C_1","display":"C_{1}","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"[Pooled bridge compatibility obstruction] Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-30c4f4070737f38c","tex":"C_{4}","aliases":"C_4","display":"C_{4}","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-99dbb247eeb251a3","tex":"C_A","aliases":"C_A","display":"C_A","kind":"use","excerpt":"Let protected linear invariants be represented by C_A:U_A→ Z_A, C_B:U_B→ Z_B. A correction preserves the invariant when C_Au=0 or C_Bu=0.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"use","excerpt":"Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a717968c751b4646","tex":"C_B","aliases":"C_B","display":"C_B","kind":"use","excerpt":"Let protected linear invariants be represented by C_A:U_A→ Z_A, C_B:U_B→ Z_B. A correction preserves the invariant when C_Au=0 or C_Bu=0.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a8ee0f64b45813fd","tex":"\\mathfrak C_B^{\\mathrm{nom}}","aliases":"C_B^nom mathfrak mathrm","display":"\\mathfrak C_B^{\\mathrm{nom}}","kind":"use","excerpt":"Let C_B^nom be the same envelope when integrity constraints are ignored.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-07eedb1bcbeef36f","tex":"C_{\\mathrm{depth}}","aliases":"C_depth mathrm","display":"C_{\\mathrm{depth}}","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3643448c933a672c","tex":"\\mathcal C_e","aliases":"C_e mathcal","display":"\\mathcal C_e","kind":"use","excerpt":"[Fixed-candidate structural invariance] Let C_e be the authorized candidate set for episode e. If parent and child differ only in the selector and C^(0)_e= C^(1)_e for every e, then the pointwise reachable correction set and its linear span are identical for parent and child.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-183be8aa2f880e20","tex":"c_{e,0}","aliases":"c_e,0","display":"c_{e,0}","kind":"use","excerpt":"Let E be a space of target episodes. An episode e∈ E contains a target agent, a declared defect context, anchor observations, calibration observations, protected audit observations, and all provenance needed to reproduce those objects. Let C_e = (c_e,0,…,c_e,B-1) be an ordered candidate bundle of fixed size B. Candidate c_e,0 is the exact no-op. Ev","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-00918c67d1a4186e","tex":"c_{e,B-1}","aliases":"c_e,B-1","display":"c_{e,B-1}","kind":"use","excerpt":"Let E be a space of target episodes. An episode e∈ E contains a target agent, a declared defect context, anchor observations, calibration observations, protected audit observations, and all provenance needed to reproduce those objects. Let C_e = (c_e,0,…,c_e,B-1) be an ordered candidate bundle of fixed size B. Candidate c_e,0 is the exact no-op. Ev","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6391b31abd61532d","tex":"c_{e,i_0}","aliases":"c_e,i_0","display":"c_{e,i_0}","kind":"use","excerpt":"[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cee7293b2a19d0c4","tex":"c_{e,i_1}","aliases":"c_e,i_1","display":"c_{e,i_1}","kind":"use","excerpt":"[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b78e78268ca8449a","tex":"C_{\\mathrm{full}}","aliases":"C_full mathrm","display":"C_{\\mathrm{full}}","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"Choose c∈ℝ^k outside the finite union of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a li","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-24ab01b59fda38dc","tex":"c_{m+1}","aliases":"c_m+1","display":"c_{m+1}","kind":"use","excerpt":"Choose a generic linear combination G=∑_j=1^mc_jR_j+c_m+1S whose induced scalar values on the distinct joint signatures are pairwise distinct. For each block value γ_a, the Lagrange polynomial p_a(z)=∏_b≠ afracz-γ_bγ_a-γ_b satisfies p_a(G)=P_a, the orthogonal projector onto H_a. Therefore P_a,P_b∈ A(R^+) and belong to its center.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-af25a41431e71657","tex":"C_{\\mathrm{parent}}","aliases":"C_parent mathrm","display":"C_{\\mathrm{parent}}","kind":"use","excerpt":"tau))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-48dab6c792cbc809","tex":"C_{\\mathrm{RI}}","aliases":"C_RI mathrm","display":"C_{\\mathrm{RI}}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-04bd0e26a983efef","tex":"C_{\\rm id}","aliases":"C_rm id rm","display":"C_{\\rm id}","kind":"use","excerpt":"Define the finite structural coverage functional C_rm id( P) = Pr_A∼D_rm for [ ank X_A=d_s]. Then widehat C_rm id( P_1) - widehat C_rm id( P_0) = 0.96875. This is the measured bridge-identification direction added by the meta-repair.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-82739408451d9162","tex":"\\mathcal C_{\\rm safe}","aliases":"C_rm safe mathcal rm","display":"\\mathcal C_{\\rm safe}","kind":"use","excerpt":"[Utility regret under simultaneous lower bounds] Let C_rm safe be the truly safe candidate set. Suppose every c∈ C_rm safe passes the implemented admissibility filter, and suppose 0≤ U_e(c)-widehat U_rm LCB(c)≤ϵ for every admissible c. If widehat c maximizes widehat U_rm LCB and c^* maximizes U_e over C_rm safe, then U_e(c^*)-U_e(widehat c)≤ϵ.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e1e71a0eb7c64bd5","tex":"c_s","aliases":"c_s","display":"c_s","kind":"use","excerpt":"Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-74d610b5f945d7cc","tex":"C_T","aliases":"C_T","display":"C_T","kind":"use","excerpt":"[Conditioned repair stability] Suppose T has full column rank and normT^†normwidehat T-T<1. Then there is a finite constant C_T, depending continuously on normT and normT^† inside this perturbation neighborhood, such that normwidehat D_t-D_t ≤ C_TnormD_snormwidehat T-T. If the protected utility readout is L-Lipschitz with respect to the transported","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-433d97a8fbb31f65","tex":"C_t","aliases":"C_t","display":"C_t","kind":"use","excerpt":"A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-637c0e5aa42a2ea2","tex":"C_\\alpha","aliases":"C_α alpha","display":"C_\\alpha","kind":"use","excerpt":"Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following c","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1c4773f19df06764","tex":"C_\\alpha'","aliases":"C_α' alpha","display":"C_\\alpha'","kind":"use","excerpt":"Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-909c1663262a98c3","tex":"C_{\\omega_i}","aliases":"C_ω_i omega","display":"C_{\\omega_i}","kind":"use","excerpt":"[Compositional receipt soundness] Assume: item every primitive receipt is semantically sound; item execution of each ω_i from C_ω_i establishes Q_ω_i; item the conjunction of the inner postconditions and bridge guarantees implies C_ω; item execution of ω from C_ω establishes Q_ω; item receipt composition preserves provenance and protected evidence ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a61c6b17750fb8db","tex":"\\mathrm{calibrate}","aliases":"calibrate mathrm","display":"\\mathrm{calibrate}","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae40d2f06cb02634","tex":"\\mathrm{cand}","aliases":"cand mathrm","display":"\\mathrm{cand}","kind":"use","excerpt":"Equation [source reference] gives a direct acquisition score. For a candidate family V_cand, choose v^⋆ ∈ arg max_v∈V_cand fracabsanglesz,P_I^⊥v^2 normP_I^⊥v^2+λ c(v)+βRisk(v), where c(v) is implementation cost and Risk(v) is a protected-risk estimate.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dcdb88ea9037ab42","tex":"\\mathrm{cause}","aliases":"cause mathrm","display":"\\mathrm{cause}","kind":"use","excerpt":"G_cause · Causal role · Intervention, necessity ablation, matched negative controls","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-843bed916127060b","tex":"\\Ccal_\\tau","aliases":"Ccal_τ C_τ Ccal tau mathcal tau","display":"{\\mathcal{C}}_\\tau","kind":"use","excerpt":"[Finite constructor selection with conservative tie-break] Suppose a finite constructor family is evaluated on fixed development evidence. Let G^⋆ be the largest held-out development gain and let C_τ={C:G(C)≥ G^⋆-τ}. Selecting the minimum-complexity constructor in C_τ is deterministic and preserves an explicit near-optimality bound of τ on the deve","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d7d217f41c39944","tex":"\\Ccal_{\\tau}","aliases":"Ccal_τ C_τ Ccal tau mathcal tau","display":"{\\mathcal{C}}_{\\tau}","kind":"use","excerpt":"[Finite constructor selection with conservative tie-break] Suppose a finite constructor family is evaluated on fixed development evidence. Let G^⋆ be the largest held-out development gain and let C_τ={C:G(C)≥ G^⋆-τ}. Selecting the minimum-complexity constructor in C_τ is deterministic and preserves an explicit near-optimality bound of τ on the deve","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-645797f075f05a94","tex":"\\mathrm{check}","aliases":"check mathrm","display":"\\mathrm{check}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-89ff13c2e7ec8ead","tex":"\\choose","aliases":"choose","display":null,"kind":"use","excerpt":"If random anchor membership is Bernoulli with affected probability p, the idealized full-rank probability at k=16 is therefore Pr(rank X=16) = Pr(N_1≥8) = ∑_j=8^1616choose jp^j(1-p)^16-j under the generic model. The observed parent full-rank rate on the primary audit set is 0.03125, consistent with a low-probability affected-count event.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-42a1a9f41584ff4b","tex":"\\mathrm{CI}","aliases":"CI mathrm","display":"\\mathrm{CI}","kind":"use","excerpt":"The paired effects are widehatΔ_rm audit · =0.072518, · CI_95% · =[0.046745,0.101389], widehatΔ_rm arch · =0.104999, · CI_95% · =[0.072711,0.140506]. Both one-sided sign-flip p-values equal approximately 4.99975×10^-5 under the frozen 20,000-draw randomization procedure.","count":21},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4a64eecd871a19bb","tex":"\\mathrm{CI}_{0.95}","aliases":"CI_0.95 mathrm","display":"\\mathrm{CI}_{0.95}","kind":"use","excerpt":"The paired child-minus-parent program-specificity gain is widehatΔ_prog^A =0.010714, CI_0.95 =[0.009911,0.011527], with positive fraction 0.823125. The answer-specificity gain is 0.017727. Its frozen interval is [0.016490,0.018960]. Parent-minus-child clean loss is 0.000979, with frozen interval [0.000695,0.001290].","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3719db55bc6b0760","tex":"\\mathrm{CI}_{95\\%}","aliases":"CI_95% mathrm","display":"\\mathrm{CI}_{95\\%}","kind":"use","excerpt":"The paired effects are widehatΔ_rm audit · =0.072518, · CI_95% · =[0.046745,0.101389], widehatΔ_rm arch · =0.104999, · CI_95% · =[0.072711,0.140506]. Both one-sided sign-flip p-values equal approximately 4.99975×10^-5 under the frozen 20,000-draw randomization procedure.","count":17},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ca8cc103ff4c8f73","tex":"\\operatorname{Class}","aliases":"Class operatorname","display":"\\operatorname{Class}","kind":"use","excerpt":"The executable classifier applies the ordered rule Class(r,b,H) = LOCAL_OBSTRUCTION, · normP_(im D)^⊥r>τ, GLUING_OBSTRUCTION, · normP_ker B^topb>τ, EVALUATOR_HOLONOMY, · norm(H-I)e>τ, FLAT_REPAIR, · otherwise. The order is part of the receipt because a case can exhibit more than one defect. The fixture isolates the four modes, so the ordered rule i","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-780075647c19479b","tex":"\\mathrm{clean}","aliases":"clean mathrm","display":"\\mathrm{clean}","kind":"use","excerpt":"positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compil","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c813344d56bc0b2a","tex":"\\operatorname{clip}","aliases":"clip operatorname","display":"\\operatorname{clip}","kind":"use","excerpt":"For current target state y, candidate repaired state widetilde y, and oracle repaired state y^⋆, define G = clip_[-1,1] ( 1- frac Enormwidetilde y-y^⋆^2 Enormy-y^⋆^2 ). The witness records: item validation gain; item source-to-target alignment NRMSE; item decoded protected-context regression; item minimum chart support; item alignment reduction rel","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-23c0e017c7e6fada","tex":"\\operatorname{clip}_{[-1,1]}","aliases":"clip_[-1,1] operatorname","display":"\\operatorname{clip}_{[-1,1]}","kind":"use","excerpt":"For current target state y, candidate repaired state widetilde y, and oracle repaired state y^⋆, define G = clip_[-1,1] ( 1- frac Enormwidetilde y-y^⋆^2 Enormy-y^⋆^2 ). The witness records: item validation gain; item source-to-target alignment NRMSE; item decoded protected-context regression; item minimum chart support; item alignment reduction rel","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5a5417ee042333b1","tex":"\\mathrm{code}_i","aliases":"code_i mathrm","display":"\\mathrm{code}_i","kind":"use","excerpt":"The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aa96d1dffe0e9395","tex":"\\mathrm{com}","aliases":"com mathrm","display":"\\mathrm{com}","kind":"use","excerpt":"For X∈End(H), define commutator energy E_com(X) = ∑_j=1^m ( normXR_j-R_jX_F^2 + normXR_j^∗-R_j^∗ X_F^2 ). There is a positive semidefinite linear operator L_com on the Hilbert space of matrices such that anglesX,L_comX_F = E_com(X). Its kernel is the commutant: kerL_com= A'.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1282513eef191ee2","tex":"\\operatorname{cone}","aliases":"cone operatorname","display":"\\operatorname{cone}","kind":"use","excerpt":"Open-set residual. Let inherited known responses span the nonnegative cone generated by e_1 and e_2 in ℝ^3. The unknown response y=e_3 has dist(y,cone{e_1,e_2})=1. Any threshold below one and above known-response noise flags the case as outside the inherited language. The flag does not identify the missing obstruction. It only prevents a false conf","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-15a27114c1f24082","tex":"\\mathrm{construct}","aliases":"construct mathrm","display":"\\mathrm{construct}","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0a1136ec7535db10","tex":"\\mathrm{corr}","aliases":"corr mathrm","display":"\\mathrm{corr}","kind":"use","excerpt":"For canonical patch u=T^-1δ, define corrected protected energy E_corr(d,u) = ∑_i=1^6w_imin{d_i^2,u_i^2}. This quantity is zero for a no-op and positive when a patch reaches task-relevant defect coordinates. It is reported alongside behavioral gain because a high apparent score can be generated without correcting protected defect energy.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fb3375037c04cb55","tex":"\\Cov","aliases":"Cov operatorname","display":"{\\operatorname{Cov}}","kind":"use","excerpt":"Let each candidate feature f cover the conflict set Cov(f) = {S∈ H_G: f resolves S}. Choosing a minimum-cost feature set that resolves all conflicts is a weighted set-cover problem. An exact solver is possible in the finite companion. A scalable lineage may use the greedy score fracnewly resolved conflictsfeature cost+probe risk. The optimization t","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9dc84000325adddb","tex":"\\operatorname{CPre}_{\\delta}","aliases":"CPre_δ operatorname delta","display":"\\operatorname{CPre}_{\\delta}","kind":"use","excerpt":"For stochastic transitions, define CPre_δ(X) = set s:∃ u∈Act(s), Pr(T(s,u,D)∈ Xmid s,u)≥1-δ . A one-step chance constraint does not by itself provide an infinite-horizon guarantee.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-281b063f0bf66e56","tex":"\\mathrm{ctr}","aliases":"ctr mathrm","display":"\\mathrm{ctr}","kind":"use","excerpt":"[New chart-contrast correction direction] Let the affine target repairs have homogeneous matrices bar R_0 and bar R_1. If bar R_0≠bar R_1, then bar R_ctr = fracbar R_1-bar R_02 induces a nonzero anti-diagonal component (-bar R_ctr,bar R_ctr) orthogonal to every diagonal pair under the product Frobenius inner product. Therefore the chart grammar con","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-304bcd5c900dd2ed","tex":"\\mathrm{ctrl}","aliases":"ctrl mathrm","display":"\\mathrm{ctrl}","kind":"use","excerpt":"Define Ψ_ctrl(X) = sets∈ S:C_RI(s) ∩ CPre(X).","count":13},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1b12981293be7264","tex":"\\operatorname{CVaR}_{0.10}","aliases":"CVaR_0.10 operatorname","display":"\\operatorname{CVaR}_{0.10}","kind":"use","excerpt":"The primary paired results are: @lrrrr@ Compiler · Mean gain · Median · CVaR_0.10 · Invalid selected","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"use","excerpt":"Let j:V↪ V' be an injective evidence-conservative embedding of old defects into an expanded ontology. Let C=im D⊆ V, C'=im D'⊆ V', and assume preservation of old correctability: j(C)⊆ C'. The embedding induces a map overline j:V/C→ V'/C'.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3f6c35cd475ecf1f","tex":"d'_{\\alpha\\beta}","aliases":"d'_αβ alpha beta","display":"d'_{\\alpha\\beta}","kind":"use","excerpt":"If u'_α=u_α+k_α with k_α∈ker D(U_α), then d'_αβ =d_αβ+k_α-k_β. Thus the cocycles differ by a coboundary.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-31aa55e1c5691b98","tex":"d^{(1)}","aliases":"d^(1)","display":"d^{(1)}","kind":"use","excerpt":"The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4120ad83fe12ffad","tex":"d^{(2)}","aliases":"d^(2)","display":"d^{(2)}","kind":"use","excerpt":"The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c33beb07d51c49df","tex":"d^{(N)}","aliases":"d^(N)","display":"d^{(N)}","kind":"use","excerpt":"The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-50a8192cc91dae5d","tex":"d^{(n)}_{\\alpha\\beta}","aliases":"d^(n)_αβ alpha beta","display":"d^{(n)}_{\\alpha\\beta}","kind":"use","excerpt":"declared cover; item repairs have been constructed globally through order n-1. Then a global repair extends through order n if and only if: item every local coefficient class vanishes, [b_n,α]=0 in coker D_α; item for one, hence every, choice of local lifts, the overlap class vanishes, [d^(n)_αβ]=0 in check H^1( U,ker D). If both conditions hold fo","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7e8acadc439b10d5","tex":"D^*","aliases":"D^*","display":"D^*","kind":"use","excerpt":"[Canonical correction--obstruction decomposition] For every r∈ V: item P_cor is the orthogonal projector onto im D; item P_ob is the orthogonal projector onto ker D^*; item the defect decomposes uniquely as r=r_cor+r_ob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complet","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","excerpt":"q · 0 , 0<q<1, so Q_0=I_2 and normEQ_0=q. Then D^-1=(I_2+E)^-1 =frac11-q^2 1 · -q -q · 1 . For r=e_1, the exact repair is u^*=-D^-1r =-frac11-q^2 1","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-70ffa9c8bbca2ebd","tex":"D_+","aliases":"D_+","display":"D_+","kind":"use","excerpt":"D_-,D_+ · Chart-specific transported target repairs","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ff76dfe8b26b12f2","tex":"D_-","aliases":"D_-","display":"D_-","kind":"use","excerpt":"D_-,D_+ · Chart-specific transported target repairs","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"[Finite structural-improvement bound] Let V be a fixed n-dimensional defect space and suppose im D_0 subsetneq im D_1 subsetneq ⋯ is a strict ascending chain. Then the chain contains at most n-rank D_0 strict inclusions.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-20d3738e9564042f","tex":"d_{12}","aliases":"d_12","display":"d_{12}","kind":"use","excerpt":"[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","excerpt":"Case F2: gluing obstruction. Take D_0=D_1=D_2=I_2, so every local defect is exactly correctable. On the oriented triangle with edges (0,1), (1,2), and (0,2), set the scalar overlap mismatch b= 0","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-76b18a6a6da00fb8","tex":"d_{23}","aliases":"d_23","display":"d_{23}","kind":"use","excerpt":"[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d2c6012edbe5de2c","tex":"D_3","aliases":"D_3","display":"D_3","kind":"use","excerpt":"Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006) · =0.105. A direct source-to-target bridge with certified residual penalty be","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f9246c831f21f28b","tex":"d_{31}","aliases":"d_31","display":"d_{31}","kind":"use","excerpt":"[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7903c81010eebdd6","tex":"D_a","aliases":"D_a","display":"D_a","kind":"use","excerpt":"Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-943ac188116d74d8","tex":"d_a","aliases":"d_a","display":"d_a","kind":"use","excerpt":"An agent a has encoder E_a:ℝ^16→ℝ^d_a and an affine head fitted to widetilde q. Its hidden state is h_a(x)=E_a(ϕ(x)). The transport system is not given the matrix or nonlinear parameters of E_a.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8013c41870f7fa44","tex":"d_a^2","aliases":"d_a^2","display":"d_a^2","kind":"use","excerpt":"[Commutant characterization] Under the hypotheses of [source reference], A_R' = ⊕_a=1^mEnd(H_a). If d_a=dim H_a, then dim A_R'=∑_a=1^md_a^2.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c813f7486933e2c5","tex":"D_A^{\\dagger}","aliases":"D_A^† dagger","display":"D_A^{\\dagger}","kind":"use","excerpt":"[Naturality of the Moore--Penrose repair] Let U_A,U_B,V_A,V_B be finite-dimensional Hilbert spaces. Suppose T_U and T_V are unitary and D_B=T_VD_AT_U^*. Then D_B^†=T_UD_A^†T_V^*. For every r_B=T_Vr_A, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction: -D_B^†r_B = T_U(-D_A^†r_A). Th","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a3c56d8af8b1f4a9","tex":"d_{\\mathrm{ans}}","aliases":"d_ans mathrm","display":"d_{\\mathrm{ans}}","kind":"use","excerpt":"Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-25ee3c6ffe011a3f","tex":"\\mathsf D_{\\mathrm{audit}}","aliases":"D_audit mathsf mathrm","display":"\\mathsf D_{\\mathrm{audit}}","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-06f65edbcccd790f","tex":"D_{\\mathrm{avg}}","aliases":"D_avg mathrm","display":"D_{\\mathrm{avg}}","kind":"use","excerpt":"Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5de99215ad954ec4","tex":"d_B","aliases":"d_B","display":"d_B","kind":"use","excerpt":"[Coupled distributional gain transfer] Suppose there is a coupling π of P_A and P_B such that E_(x,y)∼πd_B(y,Sx)≤δ. Assume q_B is K-Lipschitz and q_B(Sx)≥ q_A(x)-η for P_A-almost every x. Then E_y∼ P_Bq_B(y) ≥ E_x∼ P_Aq_A(x)-η-Kδ.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a28fb25553e1a519","tex":"D_B^{\\dagger}","aliases":"D_B^† dagger","display":"D_B^{\\dagger}","kind":"use","excerpt":"[Naturality of the Moore--Penrose repair] Let U_A,U_B,V_A,V_B be finite-dimensional Hilbert spaces. Suppose T_U and T_V are unitary and D_B=T_VD_AT_U^*. Then D_B^†=T_UD_A^†T_V^*. For every r_B=T_Vr_A, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction: -D_B^†r_B = T_U(-D_A^†r_A). Th","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f7fb93f3c3255dcc","tex":"D_C","aliases":"D_C","display":"D_C","kind":"use","excerpt":"[Invariant-constrained obstruction] Let D_C:=restrDker C. An exact first-order repair preserving the invariant exists if and only if [r]=0 in coker D_C. Equivalently, the stacked system D","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-23af4534b77ada56","tex":"\\mathsf D_{\\mathrm{calibrate}}","aliases":"D_calibrate mathsf mathrm","display":"\\mathsf D_{\\mathrm{calibrate}}","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eba1b643fe6c7ce8","tex":"\\mathsf D_{\\mathrm{construct}}","aliases":"D_construct mathsf mathrm","display":"\\mathsf D_{\\mathrm{construct}}","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7b2728350043de5a","tex":"D_{\\mathrm{ctr}}","aliases":"D_ctr mathrm","display":"D_{\\mathrm{ctr}}","kind":"use","excerpt":"Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b858f3e8665ea65","tex":"D_e","aliases":"D_e","display":"D_e","kind":"use","excerpt":"[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ab705310023bc95b","tex":"D_{\\mathrm{flat}}","aliases":"D_flat mathrm","display":"D_{\\mathrm{flat}}","kind":"use","excerpt":"Case F3: flat repair. Set D_flat=I_2, r_flat= 1 -2 , with zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is u_flat=-r_flat.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-06d58cb64df6ee89","tex":"\\mathsf D_{\\mathrm{foreign}}","aliases":"D_foreign mathsf mathrm","display":"\\mathsf D_{\\mathrm{foreign}}","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-04285d87310183c2","tex":"d_i^2","aliases":"d_i^2","display":"d_i^2","kind":"use","excerpt":"For canonical patch u=T^-1δ, define corrected protected energy E_corr(d,u) = ∑_i=1^6w_imin{d_i^2,u_i^2}. This quantity is zero for a no-op and positive when a patch reaches task-relevant defect coordinates. It is reported alongside behavioral gain because a high apparent score can be generated without correcting protected defect energy.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"For n held-out episodes, define paired differences D_j = G_e_j(K_1)-G_e_j(K_0), widehatΔ_n=frac1n∑_j=1^nD_j. Because parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a572acccc9b7cf05","tex":"D_k","aliases":"D_k","display":"D_k","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dffe3666b98294fe","tex":"D_{k+1}","aliases":"D_k+1","display":"D_{k+1}","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4bb727b5da452eba","tex":"D_{\\mathrm{loc}}","aliases":"D_loc mathrm","display":"D_{\\mathrm{loc}}","kind":"use","excerpt":"Case F1: local obstruction. Set D_loc = 1 · 0 0 · 0 , r_loc = 0 1 . Then P_im D_locr_loc=0, (I-P_im D_loc)r_loc=r_loc, so the obstruction norm is one.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d3405c1cc6903854","tex":"D_n","aliases":"D_n","display":"D_n","kind":"use","excerpt":"T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c2394be9c6ff573","tex":"d_{\\mathrm{prog}}","aliases":"d_prog mathrm","display":"d_{\\mathrm{prog}}","kind":"use","excerpt":"Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b61434218211a8c2","tex":"\\mathsf D_{\\rm anchor}","aliases":"D_rm anchor mathsf rm","display":"\\mathsf D_{\\rm anchor}","kind":"use","excerpt":"The controlled suite uses four disjoint data sets: D = D_rm train sqcup D_rm anchor sqcup D_rm calibrate sqcup D_rm audit. The source and target heads are fitted on D_rm train. Packet discovery and source repair fitting use D_rm anchor. The witness sees only D_rm calibrate. Reported final accuracies use D_rm audit.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4bc05f7b2cb7d680","tex":"\\mathsf D_{\\rm arch}","aliases":"D_rm arch mathsf rm","display":"\\mathsf D_{\\rm arch}","kind":"use","excerpt":"The experiment uses four disjoint episode collections: D_rm dev sqcup D_rm meta sqcup D_rm audit sqcup D_rm arch. Development receipts in D_rm dev train the child utility and risk heads. The meta-validation set D_rm meta selects one finite hyperparameter configuration. The seed-and-task audit D_rm audit evaluates new episodes from the development a","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-75d264255cbe474d","tex":"\\mathsf D_{\\rm audit}","aliases":"D_rm audit mathsf rm","display":"\\mathsf D_{\\rm audit}","kind":"use","excerpt":"The controlled suite uses four disjoint data sets: D = D_rm train sqcup D_rm anchor sqcup D_rm calibrate sqcup D_rm audit. The source and target heads are fitted on D_rm train. Packet discovery and source repair fitting use D_rm anchor. The witness sees only D_rm calibrate. Reported final accuracies use D_rm audit.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-023d87a35ada3964","tex":"\\mathsf D_{\\rm calibrate}","aliases":"D_rm calibrate mathsf rm","display":"\\mathsf D_{\\rm calibrate}","kind":"use","excerpt":"The controlled suite uses four disjoint data sets: D = D_rm train sqcup D_rm anchor sqcup D_rm calibrate sqcup D_rm audit. The source and target heads are fitted on D_rm train. Packet discovery and source repair fitting use D_rm anchor. The witness sees only D_rm calibrate. Reported final accuracies use D_rm audit.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-35dcb3afa3641862","tex":"\\mathsf D_{\\rm train}","aliases":"D_rm train mathsf rm","display":"\\mathsf D_{\\rm train}","kind":"use","excerpt":"The controlled suite uses four disjoint data sets: D = D_rm train sqcup D_rm anchor sqcup D_rm calibrate sqcup D_rm audit. The source and target heads are fitted on D_rm train. Packet discovery and source repair fitting use D_rm anchor. The witness sees only D_rm calibrate. Reported final accuracies use D_rm audit.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-acd8deecbfbf287e","tex":"D_s","aliases":"D_s","display":"D_s","kind":"use","excerpt":"Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.","count":20},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-de7bc3aee118c968","tex":"d_s","aliases":"d_s","display":"d_s","kind":"use","excerpt":"[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-321d4528b6ff83b1","tex":"\\mathsf D_{\\mathrm{select}}","aliases":"D_select mathsf mathrm","display":"\\mathsf D_{\\mathrm{select}}","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-222efa122da40873","tex":"d_{\\mathrm{sem}}","aliases":"d_sem mathrm","display":"d_{\\mathrm{sem}}","kind":"use","excerpt":"The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b377ac1e0c05ff0f","tex":"d_t","aliases":"d_t","display":"d_t","kind":"use","excerpt":"[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6335779635b0c503","tex":"D_t^{\\Auth}","aliases":"D_t^Auth D_t^auth Auth mathrm","display":"D_t^{{\\mathrm{auth}}}","kind":"use","excerpt":"Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-26406e352f59ad7e","tex":"D_{\\mathrm{twin}}","aliases":"D_twin mathrm","display":"D_{\\mathrm{twin}}","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),ε}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ee1684d3a8f6d798","tex":"D_u","aliases":"D_u","display":"D_u","kind":"use","excerpt":"The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8f767f66911f09a3","tex":"D_x","aliases":"D_x","display":"D_x","kind":"use","excerpt":"In the linear case, F_x(u)=r_x+D_xu, and the canonical minimum-norm repair is s(x)=-D_x^† r_x whenever r_x∈im D_x.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b1d25d2193b90b46","tex":"D_x^\\dagger","aliases":"D_x^† dagger","display":"D_x^\\dagger","kind":"use","excerpt":"In the linear case, F_x(u)=r_x+D_xu, and the canonical minimum-norm repair is s(x)=-D_x^† r_x whenever r_x∈im D_x.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-99fcbf6d9770c880","tex":"d_Y","aliases":"d_Y","display":"d_Y","kind":"use","excerpt":"[Readout stability under semantic holonomy] Let J:E_v→ Y be L_J-Lipschitz. Then d_Y(J(H_γ x),J(x)) ≤ L_Jη_sem(x;γ). If J returns a score vector and the winning score at x has margin m(x) over the runner-up, then the decision is unchanged whenever 2L_Jη_sem(x;γ)<m(x), where L_J bounds each score coordinate in the same norm.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-07ce74e506d67a97","tex":"d_{\\beta\\gamma}","aliases":"d_βγ beta gamma","display":"d_{\\beta\\gamma}","kind":"use","excerpt":"On every overlap, D(d_αβ) =-r|_U_αβ+r|_U_αβ=0. On triple overlaps, d_αβ+d_βγ+d_γα=0, so the family is a cocycle.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d115af4676ce2238","tex":"d_{\\gamma\\alpha}","aliases":"d_γα gamma alpha","display":"d_{\\gamma\\alpha}","kind":"use","excerpt":"On every overlap, D(d_αβ) =-r|_U_αβ+r|_U_αβ=0. On triple overlaps, d_αβ+d_βγ+d_γα=0, so the family is a cocycle.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f248e9e31b983389","tex":"\\dashrightarrow","aliases":"dashrightarrow","display":null,"kind":"use","excerpt":"Heuristic correspondence · Xdashrightarrow Y · Candidate analogy with no theorem transport until separately certified.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-65dbc2ae5fc209ef","tex":"\\Dcal'","aliases":"Dcal' D' Dcal mathcal","display":"{\\mathcal{D}}'","kind":"use","excerpt":"[Coverage monotonicity] If D⊆D', then ε_D'(ϕ;A) ≥ ε_D(ϕ;A).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bf8e987d990aeb6b","tex":"\\Dcal_{\\mathrm{chart}}","aliases":"Dcal_chart D_chart Dcal mathrm mathcal mathrm","display":"{\\mathcal{D}}_{\\mathrm{chart}}","kind":"use","excerpt":"The inherited global target-repair response lies in the diagonal class D_diag = set(R,R):R∈Aff(H_t), whereas the chart grammar reaches D_chart = Aff(H_t)×Aff(H_t).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-49a7ebe1cc17bd4d","tex":"\\Dcal_{\\mathrm{diag}}","aliases":"Dcal_diag D_diag Dcal mathrm mathcal mathrm","display":"{\\mathcal{D}}_{\\mathrm{diag}}","kind":"use","excerpt":"The inherited global target-repair response lies in the diagonal class D_diag = set(R,R):R∈Aff(H_t), whereas the chart grammar reaches D_chart = Aff(H_t)×Aff(H_t).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fe73e37507431eb6","tex":"\\Dcal_{\\mathrm{foreign}}","aliases":"Dcal_foreign D_foreign Dcal mathrm mathcal mathrm","display":"{\\mathcal{D}}_{\\mathrm{foreign}}","kind":"use","excerpt":"Let U_audit be a protected audit utility unavailable during compiler allocation and final scored selection. Define compiler metaproductivity at research budget B by C_B(Γ) = E_τ∼D_foreign [ U_audit (K_Γ^⋆(τ)) - U_audit(K_τ) ]. Only integrity-valid and capability-passing outputs are eligible.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4bd7a82dc7e74b21","tex":"\\Dcal_{\\mathrm{hold}}","aliases":"Dcal_hold D_hold Dcal mathrm mathcal mathrm","display":"{\\mathcal{D}}_{\\mathrm{hold}}","kind":"use","excerpt":"Let D_hold be a distribution over foreign held-out agents that are not descendants of the evaluated lineage. Let K be a frozen improvement mechanism. Under horizon h and budget B, define M^foreign_h,B(K) = E_A∼D_hold [ max_A'∈Desc_h,B(K,A) J(A')-J(A) ].","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9c0d12c85e9c2639","tex":"\\Dcal_{\\mathrm{parent}}","aliases":"Dcal_parent D_parent Dcal mathrm mathcal mathrm","display":"{\\mathcal{D}}_{\\mathrm{parent}}","kind":"use","excerpt":"Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd d mathrm","display":"{\\,\\mathrm{d}}","kind":"use","excerpt":"[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-73dea5596262750b","tex":"\\mathrm{decide}","aliases":"decide mathrm","display":"\\mathrm{decide}","kind":"use","excerpt":"The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-af6e02d93aadacbb","tex":"\\mathrm{decompose}","aliases":"decompose mathrm","display":"\\mathrm{decompose}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2ae2107287db086c","tex":"\\mathrm{demo}","aliases":"demo mathrm","display":"\\mathrm{demo}","kind":"use","excerpt":"Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Neve","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a674cead272def3d","tex":"\\mathrm{depth}","aliases":"depth mathrm","display":"\\mathrm{depth}","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"The A-optimal quantity tr(M_S^-1) controls mean-square bridge error. The E-optimal quantity λ_min(M_S) controls the least informed direction. The D-optimal quantity logdet M_S controls information volume.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"use","excerpt":"Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics ","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ffa9d7f17ebd6a0c","tex":"\\mathrm{diag}","aliases":"diag mathrm","display":"\\mathrm{diag}","kind":"use","excerpt":"The inherited global target-repair response lies in the diagonal class D_diag = set(R,R):R∈Aff(H_t), whereas the chart grammar reaches D_chart = Aff(H_t)×Aff(H_t).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3cb1c665364f8d06","tex":"\\mathrm{disc}","aliases":"disc mathrm","display":"\\mathrm{disc}","kind":"use","excerpt":"Let Ω be a finite universe of candidate repair-schema equivalence classes. For S⊆Ω, let Disc(S) be the schemas independently validated using the currently available set S. Define Φ_disc(S) = S∪Disc(S).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2cb3235aa7dcdba6","tex":"\\operatorname{Disc}","aliases":"Disc operatorname","display":"\\operatorname{Disc}","kind":"use","excerpt":"Let Ω be a finite universe of candidate repair-schema equivalence classes. For S⊆Ω, let Disc(S) be the schemas independently validated using the currently available set S. Define Φ_disc(S) = S∪Disc(S).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0e04b7f80fa27ddd","tex":"\\mathrm{discovery}","aliases":"discovery mathrm","display":"\\mathrm{discovery}","kind":"use","excerpt":"ive embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","excerpt":"Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2 · = 2normD_ctr_F^2 · = frac12normD_+-D_-_F^2. Thus D_+≠ D_- gives a quantitative s","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a5c20bec0b1d666c","tex":"\\operatorname{dist}^2","aliases":"dist^2 operatorname","display":"\\operatorname{dist}^2","kind":"use","excerpt":"Equip End(H_t)⊕End(H_t) with the product Frobenius norm. For a chart repair pair (D_-,D_+), write D_avg = fracD_-+D_+2, D_ctr = fracD_+-D_-2. The closest diagonal repair is (D_avg,D_avg). Therefore dist^2 ( (D_-,D_+),D_diag ) · = normD_--D_avg_F^2 + normD_+-D_avg_F^2 · = 2normD_ctr_F^2 · = frac12normD_+-D_-_F^2. Thus D_+≠ D_- gives a quantitative s","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-639ec78966670d7b","tex":"\\operatorname{do}","aliases":"do operatorname","display":"\\operatorname{do}","kind":"use","excerpt":"Let D_t be an authorized modification family containing a no-op intervention δ_0. For δ∈D_t, let K_A_t^δ(·) = P(Y∈·mid do(δ),A_t) be the protected external-outcome kernel. A compressed causal self-model ϕ_t(A_t) induces a predicted kernel widehat K_ϕ_t(A_t)^δ(·). The model is evaluated on counterfactual consequences, not on its ability to reproduce","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2938b012b1c73c61","tex":"\\mathrm{dynamic}","aliases":"dynamic mathrm","display":"\\mathrm{dynamic}","kind":"use","excerpt":"positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compil","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7930fe7605107fd5","tex":"\\mathbb E_{(x,y)\\sim\\pi}","aliases":"E_(x,y)∼π mathbb sim pi","display":"\\mathbb E_{(x,y)\\sim\\pi}","kind":"use","excerpt":"[Coupled distributional gain transfer] Suppose there is a coupling π of P_A and P_B such that E_(x,y)∼πd_B(y,Sx)≤δ. Assume q_B is K-Lipschitz and q_B(Sx)≥ q_A(x)-η for P_A-almost every x. Then E_y∼ P_Bq_B(y) ≥ E_x∼ P_Aq_A(x)-η-Kδ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f912500efb3ee692","tex":"\\mathsf E_0","aliases":"E_0 mathsf","display":"\\mathsf E_0","kind":"use","excerpt":"[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f46bed518745275d","tex":"\\mathsf E_1","aliases":"E_1 mathsf","display":"\\mathsf E_1","kind":"use","excerpt":"[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=","count":19},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7e06acce8dfb8fd4","tex":"e_{12}","aliases":"e_12","display":"e_{12}","kind":"use","excerpt":"Orient a triangle cyclically and identify edge mismatches with a vector e=(e_12,e_23,e_31)∈ℝ^3. The cycle space is spanned by c=frac1sqrt3(1,1,1). Hence η_glue(e)=|anglesc,e| =frac|e_12+e_23+e_31|sqrt3. For e=(1,1,-2), the sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For e=(1,1,1), the quantitative ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3053b034a3ecad99","tex":"e_1^{\\mathsf T}","aliases":"e_1^ T mathsf","display":"e_1^{\\mathsf T}","kind":"use","excerpt":"Let the true context packet be P_*=UU^ T for a target-specific orthonormal matrix U∈ℝ^6×2, while the inherited packet is P_0=diag(1,1,0,0,0,0). Let the chart label be c(h)=1{e_1^ TU^ Th>0}. For generic U, c(h) is not measurable from P_0h. No threshold or nonlinear classifier using only P_0h can recover the chart perfectly. The response atlas can re","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics ","count":21},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a770f49464c2e7ef","tex":"e_{23}","aliases":"e_23","display":"e_{23}","kind":"use","excerpt":"Orient a triangle cyclically and identify edge mismatches with a vector e=(e_12,e_23,e_31)∈ℝ^3. The cycle space is spanned by c=frac1sqrt3(1,1,1). Hence η_glue(e)=|anglesc,e| =frac|e_12+e_23+e_31|sqrt3. For e=(1,1,-2), the sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For e=(1,1,1), the quantitative ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"V=ℝ^2, C=span{e_1}, and embed V into V'=ℝ^3 by the standard coordinate inclusion. Let C'=span{e_1,e_2}. The old obstruction space is generated by [e_2], but e_2 becomes correctable in C'. Hence fracj^-1(C')C ≅ span{[e_2]}. The genuinely new obstruction quotient is fracV'C'+j(V) ≅ span{[e_3]}. The expansion repairs one old direction while exposing o","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1872e96baafe3bfb","tex":"e_{31}","aliases":"e_31","display":"e_{31}","kind":"use","excerpt":"Orient a triangle cyclically and identify edge mismatches with a vector e=(e_12,e_23,e_31)∈ℝ^3. The cycle space is spanned by c=frac1sqrt3(1,1,1). Hence η_glue(e)=|anglesc,e| =frac|e_12+e_23+e_31|sqrt3. For e=(1,1,-2), the sum vanishes and the mismatch is a pure gradient, so a vertex gauge glues the repairs exactly. For e=(1,1,1), the quantitative ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-67ab753297c61ce5","tex":"e_4","aliases":"e_4","display":"e_4","kind":"use","excerpt":"Let the old correction image be I=span{e_1,e_2}⊂ℝ^4 and let r=(0,0,3,4). Then z=P_I^⊥r=(0,0,3,4), η_I(r)^2=25. Add generator v=2e_4. Because v∈ I^⊥, frac|anglesz,v|^2norm v^2 = frac8^24 =16. The expanded residual is 3e_3, so η_I'(r)^2=9=25-16. A generator parallel to e_1 would have bar v=0 and would produce no structural reduction despite having no","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d9b6193b1833f637","tex":"E_a","aliases":"E_a","display":"E_a","kind":"use","excerpt":"An agent a has encoder E_a:ℝ^16→ℝ^d_a and an affine head fitted to widetilde q. Its hidden state is h_a(x)=E_a(ϕ(x)). The transport system is not given the matrix or nonlinear parameters of E_a.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e9aaef4e6308432c","tex":"e_a","aliases":"e_a","display":"e_a","kind":"use","excerpt":"Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ed45c147d9202820","tex":"e_{\\mathrm{align}}","aliases":"e_align mathrm","display":"e_{\\mathrm{align}}","kind":"use","excerpt":"Let (Y,H_s,H_t)∼μ be the protected semantics, source trace, and target trace under context distribution μ. Fix a source probe P and bridge B. Define e_src=Y-PH_s, e_align=P(H_s-BH_t). Then Y-PBH_t=e_src+e_align.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0d223e8cedfc458a","tex":"\\mathbb E_{A\\sim\\Dcal}","aliases":"E_A∼Dcal E_A∼D mathbb sim Dcal mathbb sim mathcal","display":"\\mathbb E_{A\\sim{\\mathcal{D}}}","kind":"use","excerpt":"Fix a protected evaluation context Ξ, a foreign-system distribution D, and a matched resource budget B. Define protected foreign metaproductivity by M_Ξ,D,B(K) = E_A∼D [ J_Ξ(K[A;B])-J_Ξ(A) ], where K[A;B] is the authorized descendant selected by K for target A, or A itself when K abstains.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3e96930c3feb41e8","tex":"\\mathbb E_{A\\sim\\Dcal_{\\mathrm{hold}}}","aliases":"E_A∼Dcal_hold E_A∼D_hold mathbb sim Dcal mathrm mathbb sim mathcal mathrm","display":"\\mathbb E_{A\\sim{\\mathcal{D}}_{\\mathrm{hold}}}","kind":"use","excerpt":"Let D_hold be a distribution over foreign held-out agents that are not descendants of the evaluated lineage. Let K be a frozen improvement mechanism. Under horizon h and budget B, define M^foreign_h,B(K) = E_A∼D_hold [ max_A'∈Desc_h,B(K,A) J(A')-J(A) ].","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9a603588606ce810","tex":"E_b","aliases":"E_b","display":"E_b","kind":"use","excerpt":"Let e_h=h_t-widehat T h_s, E_D=D_twidehat T-widehat T D_s, E_b=b_t-widehat T b_s. Then R_t(h_t)-widehat T R_s(h_s) = (I+D_t)e_h+E_Dh_s+E_b. Therefore normR_t(h_t)-widehat T R_s(h_s) ≤ (1+normD_t)norme_h +normE_Dnormh_s +normE_b. The implementation reports normalized alignment and repair-equivariance residuals as empirical summaries of these terms.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d4ca77d6e7158a4a","tex":"E_{\\mathrm{corr}}","aliases":"E_corr mathrm","display":"E_{\\mathrm{corr}}","kind":"use","excerpt":"For canonical patch u=T^-1δ, define corrected protected energy E_corr(d,u) = ∑_i=1^6w_imin{d_i^2,u_i^2}. This quantity is zero for a no-op and positive when a patch reaches task-relevant defect coordinates. It is reported alongside behavioral gain because a high apparent score can be generated without correcting protected defect energy.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8400c7c1173139ae","tex":"E_D","aliases":"E_D","display":"E_D","kind":"use","excerpt":"Let e_h=h_t-widehat T h_s, E_D=D_twidehat T-widehat T D_s, E_b=b_t-widehat T b_s. Then R_t(h_t)-widehat T R_s(h_s) = (I+D_t)e_h+E_Dh_s+E_b. Therefore normR_t(h_t)-widehat T R_s(h_s) ≤ (1+normD_t)norme_h +normE_Dnormh_s +normE_b. The implementation reports normalized alignment and repair-equivariance residuals as empirical summaries of these terms.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5d1836707c2366f7","tex":"e_h","aliases":"e_h","display":"e_h","kind":"use","excerpt":"Let e_h=h_t-widehat T h_s, E_D=D_twidehat T-widehat T D_s, E_b=b_t-widehat T b_s. Then R_t(h_t)-widehat T R_s(h_s) = (I+D_t)e_h+E_Dh_s+E_b. Therefore normR_t(h_t)-widehat T R_s(h_s) ≤ (1+normD_t)norme_h +normE_Dnormh_s +normE_b. The implementation reports normalized alignment and repair-equivariance residuals as empirical summaries of these terms.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"A finding is represented by f_i = (Π_i^-,Π_i^+,code_i, Par(i),e_i,o_i), where e_i is the observed effect profile and o_i is either a surviving obstruction or a validated no-go certificate. Directed edges are typed as extension, ablation, refutation, transport, or replication. A proposal must cite the archive nodes on which its mechanism depends.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3e687c8e7c23de13","tex":"e_{\\mathrm{irr}}","aliases":"e_irr mathrm","display":"e_{\\mathrm{irr}}","kind":"use","excerpt":"An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"For n held-out episodes, define paired differences D_j = G_e_j(K_1)-G_e_j(K_0), widehatΔ_n=frac1n∑_j=1^nD_j. Because parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d364802c9a0ff8bf","tex":"\\mathbb E_{j\\in\\mathsf{FAD}(i)}","aliases":"E_j∈FAD(i) mathbb in mathsf","display":"\\mathbb E_{j\\in\\mathsf{FAD}(i)}","kind":"use","excerpt":"r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-570f2c34e519d6ea","tex":"\\mathbb E_{j\\in\\mathsf{SAP}(i)}","aliases":"E_j∈SAP(i) mathbb in mathsf","display":"\\mathbb E_{j\\in\\mathsf{SAP}(i)}","kind":"use","excerpt":"r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-542812c54fe5f147","tex":"e_k","aliases":"e_k","display":"e_k","kind":"use","excerpt":"For foreign agent j, let Y_j(z) be protected gain under factor setting z. The average necessity effect of factor k relative to the intact architecture 1 is τ_k = E [ Y_j(1)-Y_j(1-e_k) ]. A positive value supports causal usefulness only relative to the frozen environment, compensating mechanisms, and matched resource schedule.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2424a3b380f6015d","tex":"\\mathbb E_{K\\sim\\Dcal_{\\mathrm{parent}}}","aliases":"E_K∼Dcal_parent E_K∼D_parent mathbb sim Dcal mathrm mathbb sim mathcal mathrm","display":"\\mathbb E_{K\\sim{\\mathcal{D}}_{\\mathrm{parent}}}","kind":"use","excerpt":"Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d0abe95633dd7c9","tex":"\\mathsf E_n","aliases":"E_n mathsf","display":"\\mathsf E_n","kind":"use","excerpt":"[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b7ce4855f02a711f","tex":"e_{n-1}","aliases":"e_n-1","display":"e_{n-1}","kind":"use","excerpt":"[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2594c895f18c2d6a","tex":"\\mathbb E_P","aliases":"E_P mathbb","display":"\\mathbb E_P","kind":"use","excerpt":"Let Z be column centered and let P be a uniformly random permutation matrix independent of y. Then E_P[Z^top Py] = Z^top E_P[Py] = Z^topoverline y1 =0. The ridge coefficients are linear in Z^top Py for fixed Z.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dd4edbcb1e1d0e3d","tex":"E_Q","aliases":"E_Q","display":"E_Q","kind":"use","excerpt":"Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-76728bb25248b764","tex":"\\mathbb E_{r\\sim Q}","aliases":"E_r∼ Q mathbb sim","display":"\\mathbb E_{r\\sim Q}","kind":"use","excerpt":"For a foreign residual distribution Q, define expected obstruction energy E_t(Q) = E_r∼ Qη_t(r)^2. Then E_t(Q)-E_t+1(Q) = E_r∼ Q normP_I_t+1∩ I_t^⊥r^2. This is the primary structural metaproductivity estimand for grammar growth.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4552e09155b07986","tex":"\\mathbb E_{r\\sim\\mu}","aliases":"E_r∼μ mathbb sim mu","display":"\\mathbb E_{r\\sim\\mu}","kind":"use","excerpt":"Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-677b6b68e00e4d61","tex":"e_{\\mathrm{src}}","aliases":"e_src mathrm","display":"e_{\\mathrm{src}}","kind":"use","excerpt":"Let (Y,H_s,H_t)∼μ be the protected semantics, source trace, and target trace under context distribution μ. Fix a source probe P and bridge B. Define e_src=Y-PH_s, e_align=P(H_s-BH_t). Then Y-PBH_t=e_src+e_align.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2775a21153e9ebc4","tex":"e_t","aliases":"e_t","display":"e_t","kind":"use","excerpt":"h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bd888caa9f34bc52","tex":"E_{t+1}","aliases":"E_t+1","display":"E_{t+1}","kind":"use","excerpt":"For a foreign residual distribution Q, define expected obstruction energy E_t(Q) = E_r∼ Qη_t(r)^2. Then E_t(Q)-E_t+1(Q) = E_r∼ Q normP_I_t+1∩ I_t^⊥r^2. This is the primary structural metaproductivity estimand for grammar growth.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c2a33441ead8672d","tex":"\\mathcal E_{t+1}^{\\mathrm{cor}}","aliases":"E_t+1^cor mathcal mathrm","display":"\\mathcal E_{t+1}^{\\mathrm{cor}}","kind":"use","excerpt":"[Structural-to-behavioral identity] Assume L_t⊆ L_t+1 and let W=L_t+1∩ L_t^⊥. Then E_t+1^cor(μ)- E_t^cor(μ) = E_r∼μnormP_Wr^2 and E_t^ob(μ)- E_t+1^ob(μ) = E_r∼μnormP_Wr^2. Under quadratic residual loss and exact cancellation of the correctable component, this quantity is exactly the expected additional task-loss reduction made available by the expa","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e5d697af9b3ce7b","tex":"\\mathcal E_{t+1}^{\\mathrm{ob}}","aliases":"E_t+1^ob mathcal mathrm","display":"\\mathcal E_{t+1}^{\\mathrm{ob}}","kind":"use","excerpt":"[Structural-to-behavioral identity] Assume L_t⊆ L_t+1 and let W=L_t+1∩ L_t^⊥. Then E_t+1^cor(μ)- E_t^cor(μ) = E_r∼μnormP_Wr^2 and E_t^ob(μ)- E_t+1^ob(μ) = E_r∼μnormP_Wr^2. Under quadratic residual loss and exact cancellation of the correctable component, this quantity is exactly the expected additional task-loss reduction made available by the expa","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b3b529ca0e748c7b","tex":"\\mathcal E_t^{\\mathrm{cor}}","aliases":"E_t^cor mathcal mathrm","display":"\\mathcal E_t^{\\mathrm{cor}}","kind":"use","excerpt":"Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a3feaee4e77eeb67","tex":"\\mathcal E_t^{\\mathrm{ob}}","aliases":"E_t^ob mathcal mathrm","display":"\\mathcal E_t^{\\mathrm{ob}}","kind":"use","excerpt":"Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fecbe4baeaa97b86","tex":"E_{v_0}","aliases":"E_v_0","display":"E_{v_0}","kind":"use","excerpt":"Let G=(N,E) be a connected directed graph of evaluator versions. Each directed edge e:v→ w carries an invertible internal transport T_e:E_v→ E_w. Fix a base node v_0. For any loop γ based at v_0, let H_γ:E_v_0→ E_v_0 be the composed transport around the loop.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5a9f5fda44a4b503","tex":"\\mathbb E_{x\\sim P_A}","aliases":"E_x∼ P_A mathbb sim","display":"\\mathbb E_{x\\sim P_A}","kind":"use","excerpt":"[Coupled distributional gain transfer] Suppose there is a coupling π of P_A and P_B such that E_(x,y)∼πd_B(y,Sx)≤δ. Assume q_B is K-Lipschitz and q_B(Sx)≥ q_A(x)-η for P_A-almost every x. Then E_y∼ P_Bq_B(y) ≥ E_x∼ P_Aq_A(x)-η-Kδ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a61092ef0dbdbf6b","tex":"\\mathbb E_{y\\sim P_B}","aliases":"E_y∼ P_B mathbb sim","display":"\\mathbb E_{y\\sim P_B}","kind":"use","excerpt":"[Coupled distributional gain transfer] Suppose there is a coupling π of P_A and P_B such that E_(x,y)∼πd_B(y,Sx)≤δ. Assume q_B is K-Lipschitz and q_B(Sx)≥ q_A(x)-η for P_A-almost every x. Then E_y∼ P_Bq_B(y) ≥ E_x∼ P_Aq_A(x)-η-Kδ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b9a502f5679639e1","tex":"\\mathbb E_{z\\sim\\mu}","aliases":"E_z∼μ mathbb sim mu","display":"\\mathbb E_{z\\sim\\mu}","kind":"use","excerpt":"For an agent A, define the resource-normalized utility J_Ξ(A) = E_z∼μ [ ν(z,Y_A(z)) - λbar c(z,Y_A(z)) ]. The definition does not require scalar utility to encode every scientific judgment. It supplies a declared comparison functional for one experimental claim. Multiple protected contexts may be used simultaneously, producing a Pareto relation ins","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ce74455beb257c47","tex":"E_\\theta","aliases":"E_θ theta","display":"E_\\theta","kind":"use","excerpt":"Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following c","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ac88acd2ca03f1c6","tex":"E_{\\theta}","aliases":"E_θ theta","display":"E_{\\theta}","kind":"use","excerpt":"For encoder E_θ, decoder G_δ, and score head H_ω, the pilot minimizes minibatch estimates of L(θ,δ,ω) = λ_rm rec normG_δ(E_θ(ϕ))-ϕ_F^2 + normH_ω(E_θ(ϕ))-widetilde q_F^2, with λ_rm rec=2. Optimization uses a deterministic NumPy Adam implementation for 500 steps. After representation training, the head is refitted by ridge regression to reduce avoida","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b2e8bdee723612ee","tex":"e_{\\theta}","aliases":"e_θ theta","display":"e_{\\theta}","kind":"use","excerpt":"bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f60bf972006c7999","tex":"\\mathbb E_{\\mu}","aliases":"E_μ mathbb mu","display":"\\mathbb E_{\\mu}","kind":"use","excerpt":"[Witness--alignment decomposition] For every square-integrable (Y,H_s,H_t), E_μnormY-PBH_t^2 · = E_μnorme_src^2 + E_μnorme_align^2 +2 E_μanglese_src,e_align, normY-PBH_t_L^2(μ) · ≤ normY-PH_s_L^2(μ) +normP(H_s-BH_t)_L^2(μ).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-852f0c58d4f942d1","tex":"\\mathbb E_{\\tau\\sim\\Dcal_{\\mathrm{foreign}}}","aliases":"E_τ∼Dcal_foreign E_τ∼D_foreign mathbb tau sim Dcal mathrm mathbb tau sim mathcal mathrm","display":"\\mathbb E_{\\tau\\sim{\\mathcal{D}}_{\\mathrm{foreign}}}","kind":"use","excerpt":"Let U_audit be a protected audit utility unavailable during compiler allocation and final scored selection. Define compiler metaproductivity at research budget B by C_B(Γ) = E_τ∼D_foreign [ U_audit (K_Γ^⋆(τ)) - U_audit(K_τ) ]. Only integrity-valid and capability-passing outputs are eligible.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d67e219430d1741b","tex":"\\Ecal_{\\mathrm{com}}","aliases":"Ecal_com E_com Ecal mathrm mathcal mathrm","display":"{\\mathcal{E}}_{\\mathrm{com}}","kind":"use","excerpt":"For X∈End(H), define commutator energy E_com(X) = ∑_j=1^m ( normXR_j-R_jX_F^2 + normXR_j^∗-R_j^∗ X_F^2 ). There is a positive semidefinite linear operator L_com on the Hilbert space of matrices such that anglesX,L_comX_F = E_com(X). Its kernel is the commutant: kerL_com= A'.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-42589d3c55932345","tex":"\\Ecal_K","aliases":"Ecal_K E_K Ecal mathcal","display":"{\\mathcal{E}}_K","kind":"use","excerpt":"Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0733ed3ae8de1932","tex":"\\mathrm{Elman}","aliases":"Elman mathrm","display":"\\mathrm{Elman}","kind":"use","excerpt":"Let A be a finite architecture index set and C a finite admitted chart index set. In the released realization, A = {Elman,GRU,Residual}, C = {A0,A1,A2,A3,N0,N1}. Each pair (a,c)∈ A× C carries a full Neuralese trajectory fiber H_a,c ≅ ( R^40)^5 ≅ R^200. The observation atlas determines which context fiber is authorized. A task outside every admitted","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2cb803c0b9113454","tex":"\\eps^2","aliases":"eps^2 ε^2 eps varepsilon","display":"{\\varepsilon}^2","kind":"use","excerpt":"Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-544dac3554a172aa","tex":"\\eps^3","aliases":"eps^3 ε^3 eps varepsilon","display":"{\\varepsilon}^3","kind":"use","excerpt":"Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-909fa766897aa75b","tex":"\\eps^j","aliases":"eps^j ε^j eps varepsilon","display":"{\\varepsilon}^j","kind":"use","excerpt":"[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8edb48ca2d1d0d22","tex":"\\eps^n","aliases":"eps^n ε^n eps varepsilon","display":"{\\varepsilon}^n","kind":"use","excerpt":"[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c6b4005ee78cff6d","tex":"\\eps^{n+1}","aliases":"eps^n+1 ε^n+1 eps varepsilon","display":"{\\varepsilon}^{n+1}","kind":"use","excerpt":"[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5ae3316e7e32a3e","tex":"\\eps_c","aliases":"eps_c ε_c eps varepsilon","display":"{\\varepsilon}_c","kind":"use","excerpt":"[Cost-sensitive regret] Suppose, in addition, that normalized intervention cost c(δ) is predicted by widehat c(δ) with uniform error at most ε_c. For net utility W(δ)=V(δ)-λ c(δ), widehat W(δ)=widehat V(δ)-λwidehat c(δ), selection by widehat W has regret at most 2(ε+λε_c).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4032cf47ce2cf00b","tex":"\\eps_{D,0}","aliases":"eps_D,0 ε_D,0 eps varepsilon","display":"{\\varepsilon}_{D,0}","kind":"use","excerpt":"Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4b32e2b7d946c3ad","tex":"\\eps_{D,1}","aliases":"eps_D,1 ε_D,1 eps varepsilon","display":"{\\varepsilon}_{D,1}","kind":"use","excerpt":"Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7caf86861376667b","tex":"\\eps_{D,2}","aliases":"eps_D,2 ε_D,2 eps varepsilon","display":"{\\varepsilon}_{D,2}","kind":"use","excerpt":"Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-722bd304f401c42c","tex":"\\eps_{D,k}","aliases":"eps_D,k ε_D,k eps varepsilon","display":"{\\varepsilon}_{D,k}","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1f0375548b0d39d2","tex":"\\eps_{\\Dcal}","aliases":"eps_Dcal ε_D eps Dcal varepsilon mathcal","display":"{\\varepsilon}_{{\\mathcal{D}}}","kind":"use","excerpt":"Define the intervention-coverage error ε_D(ϕ;A) = sup_δ∈D TV(K_A^δ,widehat K_ϕ(A)^δ).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0465a99ef7f7dd65","tex":"\\eps_{\\Dcal'}","aliases":"eps_Dcal' ε_D' eps Dcal varepsilon mathcal","display":"{\\varepsilon}_{{\\mathcal{D}}'}","kind":"use","excerpt":"[Coverage monotonicity] If D⊆D', then ε_D'(ϕ;A) ≥ ε_D(ϕ;A).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dc6ba80ae5602b0d","tex":"\\eps_{\\mathrm{forecast},i}","aliases":"eps_forecast,i ε_forecast,i eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{forecast},i}","kind":"use","excerpt":"The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d5adacc1ad6905d0","tex":"\\eps_{\\mathrm{glob}}","aliases":"eps_glob ε_glob eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{glob}}","kind":"use","excerpt":"e true target relation depends on s=1{h_1>0}, but a candidate separator uses g(h)=1{h_2>0}, where h_2 is independent of s and the repair-relevant state. With balanced cells and unlimited data, both g-branches see the same mixture of the two true maps. Each fitted local map converges to the global population minimizer. Consequently, ε_loc = ε_glob, ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ddb9b7ee75d2f8f0","tex":"\\eps_i","aliases":"eps_i ε_i eps varepsilon","display":"{\\varepsilon}_i","kind":"use","excerpt":"Let the source hidden state have dimension d and the target hidden state have dimension q. For selected source anchors x_i∈ℝ^d, suppose the observed target anchor is y_i=Tx_i+ε_i, E[ε_i]=0, Cov(ε_i)=σ_i^2I_q. The query-quality metadata supplies σ_i before the anchor is selected. This metadata is part of the declared Block 9 environment. It is not d","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7c4a18d328ebd4b8","tex":"\\eps_{\\mathrm{loc}}","aliases":"eps_loc ε_loc eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{loc}}","kind":"use","excerpt":"e true target relation depends on s=1{h_1>0}, but a candidate separator uses g(h)=1{h_2>0}, where h_2 is independent of s and the repair-relevant state. With balanced cells and unlimited data, both g-branches see the same mixture of the two true maps. Each fitted local map converges to the global population minimizer. Consequently, ε_loc = ε_glob, ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ec881b3e16691d5","tex":"\\eps_{\\max}","aliases":"eps_max ε_max eps max varepsilon max","display":"{\\varepsilon}_{\\max}","kind":"use","excerpt":"r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0, widehat r_rm non+η_r · ≤ r_max, widehatε_rm align+η_a · ≤ ε_max, then a candidate passing the corresponding tightened calibration gate has positive audit gain and satisfies the declared audit reg","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6ca430ecd4b89a66","tex":"\\eps_n","aliases":"eps_n ε_n eps varepsilon","display":"{\\varepsilon}_n","kind":"use","excerpt":"[Finite search bound] For n independent meta-validation problems, with probability at least 1-δ, every candidate language score differs from its population score by at most ε_n = sqrt fraclog(2N/δ)2n . Consequently, the selected candidate's population score is within 2ε_n of the best candidate in the finite language.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9e0497874336b75a","tex":"\\eps_{r,0}","aliases":"eps_r,0 ε_r,0 eps varepsilon","display":"{\\varepsilon}_{r,0}","kind":"use","excerpt":"Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3915c684ffc38229","tex":"\\eps_{r,1}","aliases":"eps_r,1 ε_r,1 eps varepsilon","display":"{\\varepsilon}_{r,1}","kind":"use","excerpt":"Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-08d1e10de977ba68","tex":"\\eps_{r,2}","aliases":"eps_r,2 ε_r,2 eps varepsilon","display":"{\\varepsilon}_{r,2}","kind":"use","excerpt":"Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b405a9ca44bcf0b","tex":"\\eps_{r,k}","aliases":"eps_r,k ε_r,k eps varepsilon","display":"{\\varepsilon}_{r,k}","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3975e200827d3e7b","tex":"\\eps_{\\mathrm{resp}}","aliases":"eps_resp ε_resp eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{resp}}","kind":"use","excerpt":"For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ceacb22029bdb862","tex":"\\eps_S","aliases":"eps_S ε_S eps varepsilon","display":"{\\varepsilon}_S","kind":"use","excerpt":"[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ca75ced76869724","tex":"\\eps_{\\mathrm{spec/code},i}","aliases":"eps_spec/code,i ε_spec/code,i eps mathrm varepsilon mathrm","display":"{\\varepsilon}_{\\mathrm{spec/code},i}","kind":"use","excerpt":"The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3dcd829bfb22c4ff","tex":"\\eps_T","aliases":"eps_T ε_T eps varepsilon","display":"{\\varepsilon}_T","kind":"use","excerpt":"[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-991525af468aabf2","tex":"\\eps_t","aliases":"eps_t ε_t eps varepsilon","display":"{\\varepsilon}_t","kind":"use","excerpt":"ulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-13c3c9e49394f081","tex":"\\eps_t^D","aliases":"eps_t^D ε_t^D eps varepsilon","display":"{\\varepsilon}_t^D","kind":"use","excerpt":"Suppose the correction square is approximate: normD_t+1T_t^U-T_t^VD_t ≤ε_t^D. Suppose protected defect meanings are reconstructed with error at most ε_t^V. Define cumulative dynamic semantic debt Δ_m,n = ∑_t=m^n-1 ( ε_t^D+ε_t^V ). Under nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-41799f56ecc3edf5","tex":"\\eps_t^V","aliases":"eps_t^V ε_t^V eps varepsilon","display":"{\\varepsilon}_t^V","kind":"use","excerpt":"Suppose the correction square is approximate: normD_t+1T_t^U-T_t^VD_t ≤ε_t^D. Suppose protected defect meanings are reconstructed with error at most ε_t^V. Define cumulative dynamic semantic debt Δ_m,n = ∑_t=m^n-1 ( ε_t^D+ε_t^V ). Under nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-14acd739d2613f90","tex":"\\mathrm{evaluation}","aliases":"evaluation mathrm","display":"\\mathrm{evaluation}","kind":"use","excerpt":"Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, addres","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7df9566bb1b48bee","tex":"\\mathrm{evaluator}","aliases":"evaluator mathrm","display":"\\mathrm{evaluator}","kind":"use","excerpt":"A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1f3da6ad15edf007","tex":"\\mathrm{evidence}","aliases":"evidence mathrm","display":"\\mathrm{evidence}","kind":"use","excerpt":"Let protected keys be K_protected ={evidence,evaluator,provenance}. The adversarial candidate declares K_edit ={evaluator,utility}. Then K_edit∩ K_protected ={evaluator}≠∅. Even before considering utility, the acceptance predicate is false. The sandbox also changes the protected evaluator value, providing a second independent rejection reason. The ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8fe86c87aaa12d17","tex":"\\EvidenceFlat","aliases":"EvidenceFlat mathrm","display":"{\\mathrm{EvidenceFlat}}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f81f2d7ee59edbaf","tex":"\\mathrm{execute}","aliases":"execute mathrm","display":"\\mathrm{execute}","kind":"use","excerpt":"The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"Hoeffding's inequality for a variable with range width 2G gives Pr( overline Z_n- EZ_i≥ t ) ≤ exp(-fracnt^22G^2). Set the right side to α. The family statement follows by the union bound.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-484de1c254ac2bb8","tex":"\\mathrm F","aliases":"F mathrm","display":"\\mathrm F","kind":"use","excerpt":"For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change af","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aaec5c835f65f308","tex":"\\mathsf F","aliases":"F mathsf","display":"\\mathsf F","kind":"use","excerpt":"The child must pass performance and risk gates simultaneously. Let Gate = Seal wedge Match wedge AuditGain wedge ForeignGain wedge Risk wedge Controls. The released suite evaluates the following frozen selector-gate families: item descendant receipt validity; item split-file load order and sealing; item positive paired child-parent audit interval; ","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"[Newton--Kantorovich repair criterion] Let F:U→ V be continuously Fr echet differentiable between Banach spaces. Let u_0∈ U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton itera","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","excerpt":"h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"Let an affine instruction be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representati","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-87ab1b547e1f885b","tex":"F_{A,b,C}","aliases":"F_A,b,C","display":"F_{A,b,C}","kind":"use","excerpt":"Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b644885b237afa55","tex":"f_{\\mathrm{abs}}","aliases":"f_abs mathrm","display":"f_{\\mathrm{abs}}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8d06c9adefd89045","tex":"f_{\\mathrm{aff}}","aliases":"f_aff mathrm","display":"f_{\\mathrm{aff}}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f04c98ee714b4d4d","tex":"F_{\\mathrm{FiLM}}","aliases":"F_FiLM mathrm","display":"F_{\\mathrm{FiLM}}","kind":"use","excerpt":"The source grammar contains a task-blind continuation, inherited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the pr","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-88011cb54ccdd2dd","tex":"f_{\\mathrm{hinge}}","aliases":"f_hinge mathrm","display":"f_{\\mathrm{hinge}}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"use","excerpt":"A finding is represented by f_i = (Π_i^-,Π_i^+,code_i, Par(i),e_i,o_i), where e_i is the observed effect profile and o_i is either a surviving obstruction or a validated no-go certificate. Directed edges are typed as extension, ablation, refutation, transport, or replication. A proposal must cite the archive nodes on which its mechanism depends.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"use","excerpt":"[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"use","excerpt":"Let an affine instruction be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representati","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"use","excerpt":"Let an affine instruction be f_j(z)=a_jz+b_j, a_j∈{-1,-tfrac12,tfrac12,1}, b_j∈{-tfrac12,0,tfrac12}. A depth-m program acts on x∈{-1,-tfrac34,…,1} by F_m(x) = (f_m∘⋯∘ f_1)(x) =A_mx+B_m, where A_0=1, B_0=0, A_j=a_jA_j-1, B_j=a_jB_j-1+b_j. All quantities in [source reference] are evaluated with exact rational arithmetic before a floating representati","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b63508c242083fc9","tex":"f_{\\mathrm{quad}}","aliases":"f_quad mathrm","display":"f_{\\mathrm{quad}}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-720cb8df4914ae51","tex":"\\mathsf F_{\\rm audit}^{\\rm primary}","aliases":"F_rm audit^rm primary mathsf rm rm","display":"\\mathsf F_{\\rm audit}^{\\rm primary}","kind":"use","excerpt":"The Block 7 primary audit differs from development in both task and representation family: T_rm dev · ∩ T_rm audit=∅, F_rm dev^rm primary · ∩ F_rm audit^rm primary=∅. Development uses ten tasks generated from four trigger families. Audit uses twelve new tasks generated from radial and mixed triggers, new teacher seeds, and new corruption permutatio","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7c2621569c7a7d16","tex":"\\mathsf F_{\\rm audit}^{\\rm stress}","aliases":"F_rm audit^rm stress mathsf rm rm","display":"\\mathsf F_{\\rm audit}^{\\rm stress}","kind":"use","excerpt":"The primary audit families are: F_rm audit^rm primary = set dense-linear-20, block-linear-24, ill-conditioned-linear-20 . The stress families are: F_rm audit^rm stress = set compressed-10, saturated-tanh-24, leaky-ReLU-32, smooth-polynomial-24 . The primary families satisfy the linear bridge hypothesis but are absent from primary development. The s","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0228bf77841e631f","tex":"\\mathsf F_{\\rm dev}^{\\rm primary}","aliases":"F_rm dev^rm primary mathsf rm rm","display":"\\mathsf F_{\\rm dev}^{\\rm primary}","kind":"use","excerpt":"The Block 7 primary audit differs from development in both task and representation family: T_rm dev · ∩ T_rm audit=∅, F_rm dev^rm primary · ∩ F_rm audit^rm primary=∅. Development uses ten tasks generated from four trigger families. Audit uses twelve new tasks generated from radial and mixed triggers, new teacher seeds, and new corruption permutatio","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0848fd096ea5801f","tex":"F_x","aliases":"F_x","display":"F_x","kind":"use","excerpt":"For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6590c3c818974d3f","tex":"f_{\\theta,p}","aliases":"f_θ,p theta","display":"f_{\\theta,p}","kind":"use","excerpt":"bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5d611049070f5d22","tex":"\\mathsf{FAD}","aliases":"FAD mathsf","display":"\\mathsf{FAD}","kind":"use","excerpt":"t d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matche","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8571cc15a510ef31","tex":"\\mathrm{far\\ answer,depth}","aliases":"far answer,depth mathrm","display":"\\mathrm{far\\ answer,depth}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-37499e8e352d5a16","tex":"\\Fcal_a","aliases":"Fcal_a F_a Fcal mathcal","display":"{\\mathcal{F}}_a","kind":"use","excerpt":"For each recovered block widehat P_a, let F_a be a finite family of candidate separators g_a,θ,u(h) = 1{anglesu,widehat P_ah>θ}. The compiler fits a two-chart bridge for every candidate and evaluates a score containing: item held-out local-bridge validation gain; item source-repair invariance of the chart assignment; item agreement between source a","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7e50481b0dd4565c","tex":"\\Fcal_t","aliases":"Fcal_t F_t Fcal mathcal","display":"{\\mathcal{F}}_t","kind":"use","excerpt":"Let (F_t)_t≥0 be the filtration generated by all committed lineage history through transaction t. A proposal is precommitted when its edit, claimed effect, target metrics, stopping rule, and evaluation namespace are F_t-1-measurable. The new evaluation trace is revealed only afterward.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1beb5502574b57f8","tex":"\\Fcal_{t-1}","aliases":"Fcal_t-1 F_t-1 Fcal mathcal","display":"{\\mathcal{F}}_{t-1}","kind":"use","excerpt":"Let (F_t)_t≥0 be the filtration generated by all committed lineage history through transaction t. A proposal is precommitted when its edit, claimed effect, target metrics, stopping rule, and evaluation namespace are F_t-1-measurable. The new evaluation trace is revealed only afterward.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-404c287d3ee8c7fd","tex":"\\Fcal_X","aliases":"Fcal_X F_X Fcal mathcal","display":"{\\mathcal{F}}_X","kind":"use","excerpt":"In the executable fixture, P_ctxh=(h_1,h_2), and the source repair leaves these coordinates invariant. The initial separator vocabulary is F_X = set 1{anglesu_k,P_ctxh>θ} : 0≤ k<128, θ∈Θ_k(X) , where u_k=(cos(2π k/128),sin(2π k/128)) and Θ_k(X) contains zero and the empirical 0.30,0.40,0.50,0.60,0.70 quantiles of the projected fit states. For every","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ea0b341d81174ca","tex":"\\mathrm{FiLM}","aliases":"FiLM mathrm","display":"\\mathrm{FiLM}","kind":"use","excerpt":"The source grammar contains a task-blind continuation, inherited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the pr","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-39156049086df3df","tex":"\\mathrm{FiLM-add}","aliases":"FiLM-add mathrm","display":"\\mathrm{FiLM-add}","kind":"use","excerpt":"The new constructor also fails a stronger comparison with the inherited additive repair. Pooled over the two architectures and two final supports, the selected-child minus legacy-additive program-specificity effect is widehatΔ_FiLM-add =-0.029417, CI_0.95 =[-0.034102,-0.024735]. The negative interval forbids a claim that the new source-selected con","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b93014fce99c12d1","tex":"\\mathrm{finalize}","aliases":"finalize mathrm","display":"\\mathrm{finalize}","kind":"use","excerpt":"The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fdcd52ea1af51a4f","tex":"\\mathrm{flat}","aliases":"flat mathrm","display":"\\mathrm{flat}","kind":"use","excerpt":"Case F3: flat repair. Set D_flat=I_2, r_flat= 1 -2 , with zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is u_flat=-r_flat.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dd532c8f48b05333","tex":"\\mathrm{forecast}","aliases":"forecast mathrm","display":"\\mathrm{forecast}","kind":"use","excerpt":"The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d508c43a21de3a6f","tex":"\\mathrm{fr}","aliases":"fr mathrm","display":"\\mathrm{fr}","kind":"use","excerpt":"A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-00c37c61b4205056","tex":"\\mathrm{full}","aliases":"full mathrm","display":"\\mathrm{full}","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f6a65d66c1c2674f","tex":"G^2","aliases":"G^2","display":"G^2","kind":"use","excerpt":"Hoeffding's inequality for a variable with range width 2G gives Pr( overline Z_n- EZ_i≥ t ) ≤ exp(-fracnt^22G^2). Set the right side to α. The family statement follows by the union bound.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-81cfc623193858ee","tex":"G^\\star","aliases":"G^⋆ star","display":"G^\\star","kind":"use","excerpt":"[Finite constructor selection with conservative tie-break] Suppose a finite constructor family is evaluated on fixed development evidence. Let G^⋆ be the largest held-out development gain and let C_τ={C:G(C)≥ G^⋆-τ}. Selecting the minimum-complexity constructor in C_τ is deterministic and preserves an explicit near-optimality bound of τ on the deve","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-575d86b85365c7b4","tex":"G_0","aliases":"G_0","display":"G_0","kind":"use","excerpt":"The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5d43c103b13190ac","tex":"\\mathfrak G_0","aliases":"G_0 mathfrak","display":"\\mathfrak G_0","kind":"use","excerpt":"_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout C_+; item neither one inherited singleton method nor one unrefined binding is acceptable throughout C; item old-token behavior is preserved through the fallback of [source reference]. Then the extension G_0 ↪ G_1 = G_0 ∪ setp,","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"use","excerpt":"[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d57f8547a7293c06","tex":"\\mathfrak G_1","aliases":"G_1 mathfrak","display":"\\mathfrak G_1","kind":"use","excerpt":"_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout C_+; item neither one inherited singleton method nor one unrefined binding is acceptable throughout C; item old-token behavior is preserved through the fallback of [source reference]. Then the extension G_0 ↪ G_1 = G_0 ∪ setp,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b9dcaf46a46092c","tex":"G_2","aliases":"G_2","display":"G_2","kind":"use","excerpt":"The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8286ab06e78358d2","tex":"\\mathcal G_{21}","aliases":"G_21 mathcal","display":"\\mathcal G_{21}","kind":"use","excerpt":"The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-736fb9ac4210e748","tex":"G_3","aliases":"G_3","display":"G_3","kind":"use","excerpt":"The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2baad54aa60ec586","tex":"G_4","aliases":"G_4","display":"G_4","kind":"use","excerpt":"The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c23b438464086c78","tex":"G_5","aliases":"G_5","display":"G_5","kind":"use","excerpt":"The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-56dec4a6f659e4a6","tex":"G_{a,c}","aliases":"G_a,c","display":"G_{a,c}","kind":"use","excerpt":"Suppose an underlying canonical trajectory is represented in chart c by a row vector z and architecture-dependent gauge matrices: h_a,c=zG_a,c, h_b,c=zG_b,c. The matrices may include coordinate rotations, sign changes, and architecture-specific temporal mixing. Block 20 uses deterministic orthogonal gauges as a controlled stress at the exposed traj","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-08cf0c7510f0208b","tex":"G_{a,c_1}","aliases":"G_a,c_1","display":"G_{a,c_1}","kind":"use","excerpt":"Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport i","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3bb403ec24015f47","tex":"G_{a,c_2}","aliases":"G_a,c_2","display":"G_{a,c_2}","kind":"use","excerpt":"Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport i","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-391c252a86e10b35","tex":"G_{a,c_i}","aliases":"G_a,c_i","display":"G_{a,c_i}","kind":"use","excerpt":"[Pooled bridge compatibility obstruction] Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5c4ad32f6de177b3","tex":"G_{a,j}","aliases":"G_a,j","display":"G_{a,j}","kind":"use","excerpt":"For arms a and b, evaluate both on the same frozen foreign agents and seeds. Let Δ_j=G_a,j-G_b,j. The paired estimate and standard error are widehatΔ =frac1n∑_j=1^nΔ_j, widehatse(widehatΔ) =sqrtfrac1n(n-1)∑_j=1^n(Δ_j-widehatΔ)^2. When gains are bounded in [-1,1], a distribution-free Hoeffding lower bound is LCB_1-α(Δ) = widehatΔ - sqrtfrac2log(1/α)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-17ba43cb660925b1","tex":"G_{a,j,1}","aliases":"G_a,j,1","display":"G_{a,j,1}","kind":"use","excerpt":"The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9e966f3d34b5dde8","tex":"G_{a,j,B}","aliases":"G_a,j,B","display":"G_{a,j,B}","kind":"use","excerpt":"The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2d52311a937fd867","tex":"G_{a,j,b}","aliases":"G_a,j,b","display":"G_{a,j,b}","kind":"use","excerpt":"The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f2b1b1a25d1e8585","tex":"g_{a,\\theta,u}","aliases":"g_a,θ,u theta","display":"g_{a,\\theta,u}","kind":"use","excerpt":"For each recovered block widehat P_a, let F_a be a finite family of candidate separators g_a,θ,u(h) = 1{anglesu,widehat P_ah>θ}. The compiler fits a two-chart bridge for every candidate and evaluates a score containing: item held-out local-bridge validation gain; item source-repair invariance of the chart assignment; item agreement between source a","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b8899a9b394c67a2","tex":"g_B","aliases":"g_B","display":"g_B","kind":"use","excerpt":"Let ℓ_A:V_A→ℝ and ℓ_B:V_B→ℝ be post-defect losses. Define source and target gains g_A := ℓ_A(r_A)-ℓ_A(r_A+D_Au_A), g_B := ℓ_B(r_B)-ℓ_B(r_B+D_BT_Uu_A).","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3721a3ce284d4732","tex":"G_{b,c}","aliases":"G_b,c","display":"G_{b,c}","kind":"use","excerpt":"Suppose an underlying canonical trajectory is represented in chart c by a row vector z and architecture-dependent gauge matrices: h_a,c=zG_a,c, h_b,c=zG_b,c. The matrices may include coordinate rotations, sign changes, and architecture-specific temporal mixing. Block 20 uses deterministic orthogonal gauges as a controlled stress at the exposed traj","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a68eb20ffe4effbc","tex":"G_{b,c}^{\\mathsf T}","aliases":"G_b,c^ T mathsf","display":"G_{b,c}^{\\mathsf T}","kind":"use","excerpt":"[Exact chart-indexed gauge transport] Under the same hypotheses, the chart-indexed family T^c_b→ a = G_b,c^-1G_a,c transports each chart exactly. If all gauges are orthogonal, then G_b,c^-1=G_b,c^ T.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-226d62f8a826f2bc","tex":"G_{b,c}^{-1}","aliases":"G_b,c^-1","display":"G_{b,c}^{-1}","kind":"use","excerpt":"[Exact chart-indexed gauge transport] Under the same hypotheses, the chart-indexed family T^c_b→ a = G_b,c^-1G_a,c transports each chart exactly. If all gauges are orthogonal, then G_b,c^-1=G_b,c^ T.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7bf3e1e6a9b1d02b","tex":"G_{b,c_1}","aliases":"G_b,c_1","display":"G_{b,c_1}","kind":"use","excerpt":"Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport i","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b85ed332c24c271b","tex":"G_{b,c_1}^{-1}","aliases":"G_b,c_1^-1","display":"G_{b,c_1}^{-1}","kind":"use","excerpt":"exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c7b59e2d65b7bc3f","tex":"G_{b,c_2}","aliases":"G_b,c_2","display":"G_{b,c_2}","kind":"use","excerpt":"Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport i","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-60bcdcc1af55df60","tex":"G_{b,c_2}^{-1}","aliases":"G_b,c_2^-1","display":"G_{b,c_2}^{-1}","kind":"use","excerpt":"exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are invertible on V, exact pooled transport is possible only when G_b,c_1^-1G_a,c_1 = G_b,c_2^-1G_a,c_2 on V.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1c64e8cce4e8bde7","tex":"G_{b,c_i}","aliases":"G_b,c_i","display":"G_{b,c_i}","kind":"use","excerpt":"[Pooled bridge compatibility obstruction] Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7f510abc0a44afe8","tex":"G_{b,j}","aliases":"G_b,j","display":"G_{b,j}","kind":"use","excerpt":"For arms a and b, evaluate both on the same frozen foreign agents and seeds. Let Δ_j=G_a,j-G_b,j. The paired estimate and standard error are widehatΔ =frac1n∑_j=1^nΔ_j, widehatse(widehatΔ) =sqrtfrac1n(n-1)∑_j=1^n(Δ_j-widehatΔ)^2. When gains are bounded in [-1,1], a distribution-free Hoeffding lower bound is LCB_1-α(Δ) = widehatΔ - sqrtfrac2log(1/α)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c268dab83726d671","tex":"G_{\\bot}","aliases":"G_bot bot","display":"G_{\\bot}","kind":"use","excerpt":"The finite grammar is G_21 = { ll G_0 · step-affine map, G_1 · full-trajectory affine map, G_2 · monotone nonlinear inverse plus step-affine map, G_3 · bidiagonal temporal inverse plus step-affine map, G_4 · probe-conditioned two-chart affine map, G_5 · semantic quotient, G_bot · collision-grounded no-go. .","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c2acc1b5a6d750ac","tex":"G_{\\mathrm{cause}}","aliases":"G_cause mathrm","display":"G_{\\mathrm{cause}}","kind":"use","excerpt":"G_cause · Causal role · Intervention, necessity ablation, matched negative controls","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b9b8934a7fcb6b82","tex":"G_{e_j}","aliases":"G_e_j","display":"G_{e_j}","kind":"use","excerpt":"For n held-out episodes, define paired differences D_j = G_e_j(K_1)-G_e_j(K_0), widehatΔ_n=frac1n∑_j=1^nD_j. Because parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8d71ef24c0d444fa","tex":"G_{\\mathrm{foreign}}","aliases":"G_foreign mathrm","display":"G_{\\mathrm{foreign}}","kind":"use","excerpt":"G_foreign · Transfer · Held-out systems, matched candidate budget, protected evaluator","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c49925793baf721a","tex":"G_{\\mathrm{global\\ oracle}}","aliases":"G_global oracle mathrm","display":"G_{\\mathrm{global\\ oracle}}","kind":"use","excerpt":"tion 0.152169 and maximum gain 0.364965 across development. By contrast, the true-chart oracle has mean gain 0.998046, and the learned local constructor has mean blind gain 0.613651. Every one of the 54 development targets satisfies the frozen repair-conflict predicate G_global oracle≤0.40, G_true chart≥0.95. Mean learned gate accuracy is 0.915000,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-57eba9a2d217ec51","tex":"G_{h,B}","aliases":"G_h,B","display":"G_{h,B}","kind":"use","excerpt":"Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cd7038b72872679a","tex":"G_i","aliases":"G_i","display":"G_i","kind":"use","excerpt":"[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"use","excerpt":"[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3ad140d4ae100d08","tex":"g_{i^\\star}","aliases":"g_i^⋆ star","display":"g_{i^\\star}","kind":"use","excerpt":"[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-83fc1d051af93301","tex":"G_{j}^{\\mathrm{meta}}","aliases":"G_j^meta mathrm","display":"G_{j}^{\\mathrm{meta}}","kind":"use","excerpt":"Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before f","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5c807f40f02063b2","tex":"G_{j}^{\\mathrm{no\\mbox{-}meta}}","aliases":"G_j^no-meta mathrm mbox","display":"G_{j}^{\\mathrm{no\\mbox{-}meta}}","kind":"use","excerpt":"Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before f","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-103f7de3482048f3","tex":"G_k","aliases":"G_k","display":"G_k","kind":"use","excerpt":"[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c1c366f26da43334","tex":"g_k","aliases":"g_k","display":"g_k","kind":"use","excerpt":"The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c86a2297ef594965","tex":"G_{\\max}","aliases":"G_max max","display":"G_{\\max}","kind":"use","excerpt":"Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51bdcc9eb71698ef","tex":"G_{\\max}^2","aliases":"G_max^2 max","display":"G_{\\max}^2","kind":"use","excerpt":"Each D_i lies in an interval of length 4G_max. The one-sided Hoeffding inequality gives Pr( overline D_n- E D_i≥ε ) ≤ exp( -fracnε^28G_max^2 ). Solving for ε proves the bound.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-817c60f43907524a","tex":"G_{\\mathrm{mech}}","aliases":"G_mech mathrm","display":"G_{\\mathrm{mech}}","kind":"use","excerpt":"G_mech · Mechanics · Legal phase traces, receipt verification, checkpoint restoration, deterministic replay","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-12822035deaa1501","tex":"G_{\\mathrm{meta}}","aliases":"G_meta mathrm","display":"G_{\\mathrm{meta}}","kind":"use","excerpt":"G_meta · Metaproductivity · Parent-child repair-calculus comparison on frozen foreign systems","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3a325a18d4b1ed9b","tex":"G_{\\mathrm{open}}","aliases":"G_open mathrm","display":"G_{\\mathrm{open}}","kind":"use","excerpt":"G_open · Open-set recursion · Conservative grammar extension, prior-proof replay, novel externally validated defect families","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2a9aa4331a4bb3c3","tex":"G_{\\mathrm{pkg}}","aliases":"G_pkg mathrm","display":"G_{\\mathrm{pkg}}","kind":"use","excerpt":"G_pkg · Package integrity · Exact manifest, canonical digests, clean extraction, successful verifier","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-494097a7c99bd0d2","tex":"G_{\\rm cause}","aliases":"G_rm cause rm","display":"G_{\\rm cause}","kind":"use","excerpt":"G_rm cause · Pass · Causal packet, PCA, random, wrong-block, and shuffled-anchor controls · Causal role inside the fixture","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9fa6d180e49b52d7","tex":"G_{\\rm child}","aliases":"G_rm child rm","display":"G_{\\rm child}","kind":"use","excerpt":"Let a parent receive one candidate whose gain is uniformly distributed on [-1,1], and let the child receive two independent candidates from the same distribution. If both choose the best available candidate, then E[G_rm parent]=0, E[G_rm child]=frac13. The child appears better even though its selection rule is unchanged. The difference is entirely ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c3f6bddbdc3a358","tex":"G_{\\rm foreign}","aliases":"G_rm foreign rm","display":"G_{\\rm foreign}","kind":"use","excerpt":"G_rm foreign · Pass · Held-out target seeds, dimensions, encoder families, and learned targets · One shared task and defect grammar","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8f9413d8b24bfde6","tex":"G_{\\rm mech}","aliases":"G_rm mech rm","display":"G_{\\rm mech}","kind":"use","excerpt":"G_rm mech · Pass · Deterministic execution, five tests, frozen JSON and figures · Runtime mechanics only","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4c27b4eab2393b16","tex":"G_{\\rm meta}","aliases":"G_rm meta rm","display":"G_{\\rm meta}","kind":"use","excerpt":"G_rm meta · Not tested · No parent-child repair-calculus comparison · No metaproductivity claim","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51f87a07dec1fdfe","tex":"G_{\\rm meta}^{\\rm bridge}","aliases":"G_rm meta^rm bridge rm rm","display":"G_{\\rm meta}^{\\rm bridge}","kind":"use","excerpt":"The controlled packages therefore pass two distinct level-5 gates: G_rm meta^rm select for selector metaproductivity and G_rm meta^rm bridge for bridge-policy metaproductivity. Neither package establishes repeated-generation closure, autonomous candidate invention, or open-set grammar expansion.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a0d755dadc4f52ee","tex":"G_{\\rm meta}^{\\rm select}","aliases":"G_rm meta^rm select rm rm","display":"G_{\\rm meta}^{\\rm select}","kind":"use","excerpt":"The controlled packages therefore pass two distinct level-5 gates: G_rm meta^rm select for selector metaproductivity and G_rm meta^rm bridge for bridge-policy metaproductivity. Neither package establishes repeated-generation closure, autonomous candidate invention, or open-set grammar expansion.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-42ea4219cf79a273","tex":"G_{\\rm open}","aliases":"G_rm open rm","display":"G_{\\rm open}","kind":"use","excerpt":"G_rm open · Not tested · No autonomous grammar or evaluator expansion · No open-set recursion claim","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e6ef49681b35f7f3","tex":"G_{\\rm parent}","aliases":"G_rm parent rm","display":"G_{\\rm parent}","kind":"use","excerpt":"Let a parent receive one candidate whose gain is uniformly distributed on [-1,1], and let the child receive two independent candidates from the same distribution. If both choose the best available candidate, then E[G_rm parent]=0, E[G_rm child]=frac13. The child appears better even though its selection rule is unchanged. The difference is entirely ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2dffc46bc19873af","tex":"G_{\\rm pkg}","aliases":"G_rm pkg rm","display":"G_{\\rm pkg}","kind":"use","excerpt":"G_rm pkg · Pass · 34-file SHA-256 manifest and clean verifier · Byte integrity relative to the released manifest","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9cfe94df18e45cce","tex":"G_{\\rm sem}","aliases":"G_rm sem rm","display":"G_{\\rm sem}","kind":"use","excerpt":"G_rm sem · Pass · Declared task, repair, transport, and gate objects reproduced · Controlled finite semantics","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae97cc7d91a158ee","tex":"g_{\\rm slice}","aliases":"g_rm slice rm","display":"g_{\\rm slice}","kind":"use","excerpt":"[Calibration-to-audit witness soundness] Suppose, simultaneously, that the audit gain and regression satisfy g_rm slice · ≥ widehat g_rm slice-η_g, r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a15a2b9fe5cbf557","tex":"G_{\\mathrm{sem}}","aliases":"G_sem mathrm","display":"G_{\\mathrm{sem}}","kind":"use","excerpt":"G_sem · Finite semantics · Abstract-runtime refinement and exhaustive frozen endpoints","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fbf1a69625ba4f46","tex":"G_{\\mathrm{true\\ chart}}","aliases":"G_true chart mathrm","display":"G_{\\mathrm{true\\ chart}}","kind":"use","excerpt":"tion 0.152169 and maximum gain 0.364965 across development. By contrast, the true-chart oracle has mean gain 0.998046, and the learned local constructor has mean blind gain 0.613651. Every one of the 54 development targets satisfies the frozen repair-conflict predicate G_global oracle≤0.40, G_true chart≥0.95. Mean learned gate accuracy is 0.915000,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d9dcb71645d955be","tex":"g_{\\widehat i}","aliases":"g_widehat i widehat","display":"g_{\\widehat i}","kind":"use","excerpt":"[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-841df9ca84cfa40e","tex":"g_\\alpha","aliases":"g_α alpha","display":"g_\\alpha","kind":"use","excerpt":"epair exists if and only if the transition family satisfies the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomology [g_αβ]=1 in check H^1( U,mathscr G). Equivalently, there exist local gauges g_α∈mathscr G(U_α) such that g_αβ=g_α^-1g_β.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-72848cca23a525d0","tex":"g_\\alpha^{-1}","aliases":"g_α^-1 alpha","display":"g_\\alpha^{-1}","kind":"use","excerpt":"the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomology [g_αβ]=1 in check H^1( U,mathscr G). Equivalently, there exist local gauges g_α∈mathscr G(U_α) such that g_αβ=g_α^-1g_β.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d05f0811844f2313","tex":"g_\\beta","aliases":"g_β beta","display":"g_\\beta","kind":"use","excerpt":"the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomology [g_αβ]=1 in check H^1( U,mathscr G). Equivalently, there exist local gauges g_α∈mathscr G(U_α) such that g_αβ=g_α^-1g_β.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0a08f8569b33b73b","tex":"g_{\\beta\\gamma}","aliases":"g_βγ beta gamma","display":"g_{\\beta\\gamma}","kind":"use","excerpt":"[Nonlinear torsor descent] Assume the local solution action is principal and effective descent holds for mathscr G-torsors on the declared cover. Then a global nonlinear repair exists if and only if the transition family satisfies the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomol","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-91a5ef085c2491ed","tex":"G_\\gamma","aliases":"G_γ gamma","display":"G_\\gamma","kind":"use","excerpt":"Every state in the Buechi set remains inside the static invariant set under its winning strategy, hence lies in the robust safety kernel. A safe absorbing no-op state outside G_γ witnesses possible strictness.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8356f27f0d57d398","tex":"g_{\\gamma\\alpha}","aliases":"g_γα gamma alpha","display":"g_{\\gamma\\alpha}","kind":"use","excerpt":"[Nonlinear torsor descent] Assume the local solution action is principal and effective descent holds for mathscr G-torsors on the declared cover. Then a global nonlinear repair exists if and only if the transition family satisfies the nonabelian cocycle condition g_αβg_βγg_γα=e on triple overlaps and its class is trivial in nonabelian vCech cohomol","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-39c5fc5227ee14cf","tex":"G_{\\delta}","aliases":"G_δ delta","display":"G_{\\delta}","kind":"use","excerpt":"For encoder E_θ, decoder G_δ, and score head H_ω, the pilot minimizes minibatch estimates of L(θ,δ,ω) = λ_rm rec normG_δ(E_θ(ϕ))-ϕ_F^2 + normH_ω(E_θ(ϕ))-widetilde q_F^2, with λ_rm rec=2. Optimization uses a deterministic NumPy Adam implementation for 500 steps. After representation training, the head is refitted by ridge regression to reduce avoida","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-437e00c6f82b43e4","tex":"\\mathrm{gate}","aliases":"gate mathrm","display":"\\mathrm{gate}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f631c7c24e067bd9","tex":"\\Gcal_0","aliases":"Gcal_0 G_0 Gcal mathcal","display":"{\\mathcal{G}}_0","kind":"use","excerpt":"Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7df77b80c52a2262","tex":"\\Gcal_1","aliases":"Gcal_1 G_1 Gcal mathcal","display":"{\\mathcal{G}}_1","kind":"use","excerpt":"Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ed6f0504127608c7","tex":"\\Gcal_2","aliases":"Gcal_2 G_2 Gcal mathcal","display":"{\\mathcal{G}}_2","kind":"use","excerpt":"Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5c5a179d74fbc5eb","tex":"\\Gcal_{23}","aliases":"Gcal_23 G_23 Gcal mathcal","display":"{\\mathcal{G}}_{23}","kind":"use","excerpt":"The finite grammar is G_23 = { l answer-only, PCA, predictive, intervention-response, joint minimal, full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b2211c50584518d2","tex":"\\Gcal_{\\infty}","aliases":"Gcal_∞ G_∞ Gcal infty mathcal infty","display":"{\\mathcal{G}}_{\\infty}","kind":"use","excerpt":"[Conservative directed limit] The union of translated syntax and proofs defines a directed-limit grammar G_∞ with the following property: every finite expression and every finite derivation in G_∞ appears at some finite stage. Every theorem proved at stage t retains the same protected meaning at every later stage and in G_∞.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1d12ce35e6abc40f","tex":"\\operatorname{GELU}","aliases":"GELU operatorname","display":"\\operatorname{GELU}","kind":"use","excerpt":"p embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z_k+σ(Gz_k)odot tanh(V_2GELU(V_1z_k)+s_k+δ_k(e,z_k)). The three cell classes have distinct executable implementatio","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-108c83c91ea55c65","tex":"\\mathrm{gen}","aliases":"gen mathrm","display":"\\mathrm{gen}","kind":"use","excerpt":"Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-530c8a45651a2504","tex":"\\gg","aliases":"gg","display":null,"kind":"use","excerpt":"This certificate separates two statements: rank X_S=d and λ_min(X_S^ TWX_S)gg0. The first is qualitative identifiability. The second is quantitative stability.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5d62edc1cdf57e7","tex":"\\mathrm{GL}","aliases":"GL mathrm","display":"\\mathrm{GL}","kind":"use","excerpt":"The ordinary control has Y_t=X_t A+b, A∈GL(8), with one shared map at every recurrent step. The inherited step-affine constructor is sufficient. This family tests conservative minimality: a full-trajectory affine map is available and usually eligible, but it should not be selected when the smaller inherited constructor already passes.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bf2c0e1e1afe44ae","tex":"\\mathrm{global}","aliases":"global mathrm","display":"\\mathrm{global}","kind":"use","excerpt":"For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b802aba288969572","tex":"\\mathrm{global\\ oracle}","aliases":"global oracle mathrm","display":"\\mathrm{global\\ oracle}","kind":"use","excerpt":"tion 0.152169 and maximum gain 0.364965 across development. By contrast, the true-chart oracle has mean gain 0.998046, and the learned local constructor has mean blind gain 0.613651. Every one of the 54 development targets satisfies the frozen repair-conflict predicate G_global oracle≤0.40, G_true chart≥0.95. Mean learned gate accuracy is 0.915000,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d0de500e57aa39f7","tex":"\\mathrm{gram}","aliases":"gram mathrm","display":"\\mathrm{gram}","kind":"use","excerpt":"Δ_gram · Global-minus-local held-out transport residual margin","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8af8dc2c6aac2bb4","tex":"\\Grounded","aliases":"Grounded mathrm","display":"{\\mathrm{Grounded}}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-41d6b9a1a9a224ec","tex":"\\mathrm{GRU}","aliases":"GRU mathrm","display":"\\mathrm{GRU}","kind":"use","excerpt":"Let A be a finite architecture index set and C a finite admitted chart index set. In the released realization, A = {Elman,GRU,Residual}, C = {A0,A1,A2,A3,N0,N1}. Each pair (a,c)∈ A× C carries a full Neuralese trajectory fiber H_a,c ≅ ( R^40)^5 ≅ R^200. The observation atlas determines which context fiber is authorized. A task outside every admitted","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1fed946794f750b9","tex":"\\operatorname{GRUCell}","aliases":"GRUCell operatorname","display":"\\operatorname{GRUCell}","kind":"use","excerpt":"Let e=E(ϕ(τ))∈ℝ^36 be the encoded task representation, let z_0∈ℝ^36 be a learned seed, and let s_k∈ℝ^36 be the step embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1bf12537a68f2d94","tex":"\\mathfrak H","aliases":"H mathfrak","display":"\\mathfrak H","kind":"use","excerpt":"The minimal conflicts form a finite hypergraph H_G. A candidate feature or probe f:X→ Y_f resolves a conflict S when f is not constant on S.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","excerpt":"[Finite ranking certificate for one gain cycle] Let X⊆I_ctrl^* and let h:X→{0,1,…,H} satisfy h^-1(0)⊆ G_γ. Suppose that at every state with h(s)>0, the controller has an authorized action whose every admissible successor s' lies in X and satisfies h(s')<h(s). Then the controller can force a visit to G_γ within at most H transactions while remaining","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a653cc5a02109a67","tex":"H^U","aliases":"H^U","display":"H^U","kind":"use","excerpt":"Let U=V=ℝ^2, D=I, and suppose two routes from A to B differ by a loop with correction holonomy H^U= 1 · 0 0 · -1 . The flat repair core is U^flat=spansete_1. For u=e_1, both routes produce the same transported repair. For u=e_2, the two routes differ by (H^U-I)e_2=-2e_2, so the loop residual is 2. A repair schema containing an e_2 component is ther","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-12b2c7288d081d00","tex":"H^U_{q^{-1}p}","aliases":"H^U_q^-1p","display":"H^U_{q^{-1}p}","kind":"use","excerpt":"The path identity is T_U^p=T_U^qH^U_q^-1p. Apply both sides to u and use H^U_q^-1pu=u.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"Given input x, define the prompt anchor h_0=E(x). A normalized injection state z_t and updated latent state h_t satisfy z_t=N(h_t-1), widetilde h_t=B(z_t,x), h_t=U(z_t,widetilde h_t,h_0), 1≤ t≤ K, where N is any declared normalization or scale-matching operator. The answer is sampled or selected from D(h_K,x) after a declared end-of-thought transit","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"use","excerpt":"Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","excerpt":"In the executable fixture, P_ctxh=(h_1,h_2), and the source repair leaves these coordinates invariant. The initial separator vocabulary is F_X = set 1{anglesu_k,P_ctxh>θ} : 0≤ k<128, θ∈Θ_k(X) , where u_k=(cos(2π k/128),sin(2π k/128)) and Θ_k(X) contains zero and the empirical 0.30,0.40,0.50,0.60,0.70 quantiles of the projected fit states. For every","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be80c526bfc5b21c","tex":"H_3","aliases":"H_3","display":"H_3","kind":"use","excerpt":"Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5d2fdba2ed4efda7","tex":"h_3","aliases":"h_3","display":"h_3","kind":"use","excerpt":"The source state is h=(h_1,h_2,h_3,h_4)∈ℝ^4. The first two coordinates form the protected context packet. The source repair is R_s(h)=M_sh+c_s, where M_s= 1 · 0 · 0 · 0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bc71f39c22d38b2d","tex":"h_4","aliases":"h_4","display":"h_4","kind":"use","excerpt":"The source state is h=(h_1,h_2,h_3,h_4)∈ℝ^4. The first two coordinates form the protected context packet. The source repair is R_s(h)=M_sh+c_s, where M_s= 1 · 0 · 0 · 0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d34ebb7e1f0208d2","tex":"h_5","aliases":"h_5","display":"h_5","kind":"use","excerpt":"two Block 20 foreign architectures. For admitted observation chart c, Block 20 supplies an accepted full-trajectory connection T^c_B→ A:H_B,c^(1:5)⟶ H_A,c^(1:5). Let P:ℝ^40→ℝ^8 be an orthonormal packet projector fitted on a pre-audit source support. Applying P independently to the five recurrent states gives Π(H) = (P h_1,…,P h_5) ∈ (ℝ^8)^5 ≅ℝ^40. ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bea5d0e7eb485d1c","tex":"H_A","aliases":"H_A","display":"H_A","kind":"use","excerpt":"For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change af","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-826a45698583f00a","tex":"h_a","aliases":"h_a","display":"h_a","kind":"use","excerpt":"An agent a has encoder E_a:ℝ^16→ℝ^d_a and an affine head fitted to widetilde q. Its hidden state is h_a(x)=E_a(ϕ(x)). The transport system is not given the matrix or nonlinear parameters of E_a.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-71280fd5551db428","tex":"h_{a,c}","aliases":"h_a,c","display":"h_{a,c}","kind":"use","excerpt":"Suppose an underlying canonical trajectory is represented in chart c by a row vector z and architecture-dependent gauge matrices: h_a,c=zG_a,c, h_b,c=zG_b,c. The matrices may include coordinate rotations, sign changes, and architecture-specific temporal mixing. Block 20 uses deterministic orthogonal gauges as a controlled stress at the exposed traj","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2dc6b480cd0ecb3e","tex":"H_{a,d}","aliases":"H_a,d","display":"H_{a,d}","kind":"use","excerpt":"For architectures a,b and admitted chart c, an architecture map is T^c_b→ a:H_b,c⟶ H_a,c. For one architecture a and two admitted charts c,d, a context map is C^a_c→ d:H_a,c⟶ H_a,d. Block 20 uses A0 as a reference chart and fits a star of maps C^a_c→ A0 and C^a_A0→ c from protected semantic twins.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e169bb39e7f4c257","tex":"h_a^{\\circ}","aliases":"h_a^∘ circ","display":"h_a^{\\circ}","kind":"use","excerpt":"0 . The resulting hidden difference is Δ_a(x) = h_a(x)-h_a^∘(x), h_a^∘(x)=E_a(ϕ^∘(x)). The empirical causal packet is the leading rank-r right-singular subspace of the affected differences {Δ_a(x_i):s(x_i)=1}.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-827fc51f8ae47f77","tex":"H_B","aliases":"H_B","display":"H_B","kind":"use","excerpt":"For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change af","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0d825e807a1978b2","tex":"H_{b,c}","aliases":"H_b,c","display":"H_{b,c}","kind":"use","excerpt":"For architectures a,b and admitted chart c, an architecture map is T^c_b→ a:H_b,c⟶ H_a,c. For one architecture a and two admitted charts c,d, a context map is C^a_c→ d:H_a,c⟶ H_a,d. Block 20 uses A0 as a reference chart and fits a star of maps C^a_c→ A0 and C^a_A0→ c from protected semantic twins.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-26c6c684238141f4","tex":"h_{b,c}","aliases":"h_b,c","display":"h_{b,c}","kind":"use","excerpt":"Suppose an underlying canonical trajectory is represented in chart c by a row vector z and architecture-dependent gauge matrices: h_a,c=zG_a,c, h_b,c=zG_b,c. The matrices may include coordinate rotations, sign changes, and architecture-specific temporal mixing. Block 20 uses deterministic orthogonal gauges as a controlled stress at the exposed traj","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eec23fda0c7a0c92","tex":"h_{b,c}^{(1)}","aliases":"h_b,c^(1)","display":"h_{b,c}^{(1)}","kind":"use","excerpt":"A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribut","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-baeff6d2dcbf3743","tex":"h_{b,c}^{(5)}","aliases":"h_b,c^(5)","display":"h_{b,c}^{(5)}","kind":"use","excerpt":"A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribut","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-770815748b4cdfd6","tex":"h_{b,c}^{(k)}","aliases":"h_b,c^(k)","display":"h_{b,c}^{(k)}","kind":"use","excerpt":"A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribut","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-38becd0552b037e7","tex":"h_C","aliases":"h_C","display":"h_C","kind":"use","excerpt":"Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribu","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-234d6f5e8ecbaaaa","tex":"h_c","aliases":"h_c","display":"h_c","kind":"use","excerpt":"For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a unive","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c99024452d793f4","tex":"H_{\\mathrm{ctx}}","aliases":"H_ctx mathrm","display":"H_{\\mathrm{ctx}}","kind":"use","excerpt":"The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d71835fc8f7f5085","tex":"\\mathfrak H_{\\Gcal}","aliases":"H_Gcal H_G mathfrak Gcal mathfrak mathcal","display":"\\mathfrak H_{{\\mathcal{G}}}","kind":"use","excerpt":"The minimal conflicts form a finite hypergraph H_G. A candidate feature or probe f:X→ Y_f resolves a conflict S when f is not constant on S.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9916dcceacb3c35d","tex":"h_H","aliases":"h_H","display":"h_H","kind":"use","excerpt":"Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribu","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8c7da95775a0626f","tex":"H_j","aliases":"H_j","display":"H_j","kind":"use","excerpt":"Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8a5a5b48b1793c8b","tex":"h_k","aliases":"h_k","display":"h_k","kind":"use","excerpt":"The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7f673d47e4aa5949","tex":"h_{k-1}","aliases":"h_k-1","display":"h_{k-1}","kind":"use","excerpt":"The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cf5f97ebded34595","tex":"H_m","aliases":"H_m","display":"H_m","kind":"use","excerpt":"Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-20c5ab57afa8a9b0","tex":"H_{\\mathrm{nui}}","aliases":"H_nui mathrm","display":"H_{\\mathrm{nui}}","kind":"use","excerpt":"The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bc2ccac51143b79d","tex":"H_{p^{-1}\\gamma p}","aliases":"H_p^-1γ p gamma","display":"H_{p^{-1}\\gamma p}","kind":"use","excerpt":"Every loop γ at w corresponds to the conjugate loop p^-1γ p at v, with H_γ=T_pH_p^-1γ pT_p^-1. If x∈ E_v^flat, then H_γ T_px=T_pH_p^-1γ px=T_px, so T_px∈ E_w^flat. Apply the same argument to p^-1 for the reverse inclusion.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0f4be659156002b7","tex":"h_Q","aliases":"h_Q","display":"h_Q","kind":"use","excerpt":"Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribu","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2d64df6fa5868f4c","tex":"H_r","aliases":"H_r","display":"H_r","kind":"use","excerpt":"For a current residual r∈ H, define its cyclic causal module H_r = span{Ar:A∈ A}.","count":14},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a27afcadc0a095e6","tex":"H_{\\mathrm{rep}}","aliases":"H_rep mathrm","display":"H_{\\mathrm{rep}}","kind":"use","excerpt":"The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e129a26ad83c813","tex":"h_S","aliases":"h_S","display":"h_S","kind":"use","excerpt":"Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribu","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c7981bbd519d526","tex":"H_s^{\\top}","aliases":"H_s^top top","display":"H_s^{\\top}","kind":"use","excerpt":"The normal equations are H_s^topH_swidehat T^top=H_s^topH_t. Full column rank makes H_s^topH_s invertible. Substitution of H_t=H_sT^top yields widehat T=T.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6fd7b440d1b6df67","tex":"h_T","aliases":"h_T","display":"h_T","kind":"use","excerpt":"Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribu","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4025eec61a177bec","tex":"h_{t-1}","aliases":"h_t-1","display":"h_{t-1}","kind":"use","excerpt":"Given input x, define the prompt anchor h_0=E(x). A normalized injection state z_t and updated latent state h_t satisfy z_t=N(h_t-1), widetilde h_t=B(z_t,x), h_t=U(z_t,widetilde h_t,h_0), 1≤ t≤ K, where N is any declared normalization or scale-matching operator. The answer is sampled or selected from D(h_K,x) after a declared end-of-thought transit","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a049a97dc219bcd6","tex":"h_V","aliases":"h_V","display":"h_V","kind":"use","excerpt":"Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribu","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bc0529a14c9530dd","tex":"H_w","aliases":"H_w","display":"H_w","kind":"use","excerpt":"[Archive-envelope monotonicity] Suppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector w, define H_w(A_t) = max_A∈A_tw^ Tz(A). Then H_w(A_t) is nondecreasing in t.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e4c184c7e9ba5c5d","tex":"h_W","aliases":"h_W","display":"h_W","kind":"use","excerpt":"Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribu","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-032c6e2522ce5f6d","tex":"H_{\\gamma_1}","aliases":"H_γ_1 gamma","display":"H_{\\gamma_1}","kind":"use","excerpt":"t G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1-I vdots H_γ_m-I .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be8636c719417638","tex":"H_{\\gamma_j}","aliases":"H_γ_j gamma","display":"H_{\\gamma_j}","kind":"use","excerpt":"[Finite computation from a cycle basis] Let G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-02c5563cd63fb758","tex":"H_{\\gamma_m}","aliases":"H_γ_m gamma","display":"H_{\\gamma_m}","kind":"use","excerpt":"t G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1-I vdots H_γ_m-I .","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c873371b98ce0508","tex":"H_{\\omega}","aliases":"H_ω omega","display":"H_{\\omega}","kind":"use","excerpt":"For encoder E_θ, decoder G_δ, and score head H_ω, the pilot minimizes minibatch estimates of L(θ,δ,ω) = λ_rm rec normG_δ(E_θ(ϕ))-ϕ_F^2 + normH_ω(E_θ(ϕ))-widetilde q_F^2, with λ_rm rec=2. Optimization uses a deterministic NumPy Adam implementation for 500 steps. After representation training, the head is refitted by ridge regression to reduce avoida","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b5bcca934867a840","tex":"\\mathrm{halt}","aliases":"halt mathrm","display":"\\mathrm{halt}","kind":"use","excerpt":"positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compil","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8fbcb7cd01c6be61","tex":"\\Hcal_{\\mathrm{obs}}","aliases":"Hcal_obs H_obs Hcal mathrm mathcal mathrm","display":"{\\mathcal{H}}_{\\mathrm{obs}}","kind":"use","excerpt":"The finite benchmark contains six hidden obstruction classes: H_obs = {data,objective,response,constructor,evaluator,resource}. The compiler is not given these labels at deployment. Each research problem exposes a twelve-dimensional cheap diagnostic vector, a baseline repair process, a finite candidate-method grammar, four scored benchmarks, a hidd","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b3803b3222663469","tex":"\\mathrm{high}","aliases":"high mathrm","display":"\\mathrm{high}","kind":"use","excerpt":"Let p∈P^+ be a candidate experiment with response y_p:T⟶ℝ. For threshold θ, define the refined diagnostic map ϕ_p,θ(τ) = (ϕ_0(τ),low), · y_p(τ)≤θ, (ϕ_0(τ),high), · y_p(τ)>θ. The old token is recovered by the projection π_0(z,s)=z, π_0∘ϕ_p,θ=ϕ_0. Thus the extension is conservative at the level of old diagnostic semantics.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2a6e321bddd0ee15","tex":"\\mathrm{hinge}","aliases":"hinge mathrm","display":"\\mathrm{hinge}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-da4b6fe971d78911","tex":"\\mathrm{hist}","aliases":"hist mathrm","display":"\\mathrm{hist}","kind":"use","excerpt":"[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is ","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"The theorem produces an obstruction ledger c|cc order · local obstruction · gluing obstruction 1 · [b_1,α]∈coker D_α · [d^(1)]∈check H^1(ker D) 2 · [b_2,α]∈coker D_α · [d^(2)]∈check H^1(ker D) vdots · vdots · vdots N · [b_N,α]∈coker D_α · [d^(N)]∈check H^1(ker D). Higher categorical repair interfaces may introduce additional coherence obstructions ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-38882254109eafed","tex":"\\mathrm{hold}","aliases":"hold mathrm","display":"\\mathrm{hold}","kind":"use","excerpt":"Let D_hold be a distribution over foreign held-out agents that are not descendants of the evaluated lineage. Let K be a frozen improvement mechanism. Under horizon h and budget B, define M^foreign_h,B(K) = E_A∼D_hold [ max_A'∈Desc_h,B(K,A) J(A')-J(A) ].","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-46cc9b01cd7f4804","tex":"I'","aliases":"I'","display":"I'","kind":"use","excerpt":"An evaluation context is nontrivial when candidate identity, self-reported utility, and mutable internal error registers are conditionally irrelevant given protected external traces. Formally, for candidate identifier I and protected trace Y, V(I,Y)=V(I',Y) for all identifiers I,I' that generate the same authorized trace and resource record. This e","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fc66c01638c40151","tex":"i^2","aliases":"i^2","display":"i^2","kind":"use","excerpt":"Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4bc5700778e1ba26","tex":"I^\\perp","aliases":"I^⊥ perp","display":"I^\\perp","kind":"use","excerpt":"Let I=im D⊆ V be the current authorized correction image and let z=P_I^⊥r be the unresolved component of defect r. Suppose a candidate grammar extension contributes a response vector v∈ V. Only its component outside the old image is structurally new: bar v=P_I^⊥v. The expanded image is I'=I+span{v}.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a5ea0cbfcd600391","tex":"i^\\star","aliases":"i^⋆ star","display":"i^\\star","kind":"use","excerpt":"[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ec2371bf933c7f9","tex":"I_+","aliases":"I_+","display":"I_+","kind":"use","excerpt":"Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applie","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0e79b7ad0299d0b0","tex":"I_0","aliases":"I_0","display":"I_0","kind":"use","excerpt":"Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f18b3c49fd8055ae","tex":"i_0","aliases":"i_0","display":"i_0","kind":"use","excerpt":"[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-37475f8aeaec6cce","tex":"I_1","aliases":"I_1","display":"I_1","kind":"use","excerpt":"Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"[Pathwise selector isolation] For episode e, let parent and child receive the identical ordered bundle C_e and identical pre-audit receipt Z_e. Let i_0=K_0(Z_e), i_1=K_1(Z_e). Then the realized paired utility difference is exactly D_e = U_e(c_e,i_1)-U_e(c_e,i_0). If candidate construction, target state, and audit evaluation are frozen, no candidate","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"Case F2: gluing obstruction. Take D_0=D_1=D_2=I_2, so every local defect is exactly correctable. On the oriented triangle with edges (0,1), (1,2), and (0,2), set the scalar overlap mismatch b= 0","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7cc6e87678c7d216","tex":"I_5","aliases":"I_5","display":"I_5","kind":"use","excerpt":"nts are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-015a8b24b2047119","tex":"I_{d_s}","aliases":"I_d_s","display":"I_{d_s}","kind":"use","excerpt":"[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0625c4dc07eb960a","tex":"i_j","aliases":"i_j","display":"i_j","kind":"use","excerpt":"The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the stand","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2a716c8fbfbd9f44","tex":"I_{j-1}","aliases":"I_j-1","display":"I_{j-1}","kind":"use","excerpt":"The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the stand","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-edadeb2cc1e72d16","tex":"I_K","aliases":"I_K","display":"I_K","kind":"use","excerpt":"Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-86a0ec8c41d6450f","tex":"I_{m_\\alpha}","aliases":"I_m_α alpha","display":"I_{m_\\alpha}","kind":"use","excerpt":"[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-69a4780a0be9987e","tex":"I_q","aliases":"I_q","display":"I_q","kind":"use","excerpt":"Let the source hidden state have dimension d and the target hidden state have dimension q. For selected source anchors x_i∈ℝ^d, suppose the observed target anchor is y_i=Tx_i+ε_i, E[ε_i]=0, Cov(ε_i)=σ_i^2I_q. The query-quality metadata supplies σ_i before the anchor is selected. This metadata is part of the declared Block 9 environment. It is not d","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e9855187f190ff70","tex":"I_r","aliases":"I_r","display":"I_r","kind":"use","excerpt":"[Best affine residual for a quadratic packet repair] Let z∼ N(0,I_r), B=B^top, and define q(z)=z^top Bz. The best L^2 affine approximation to q(z)v is the constant (tr B)v, and its minimum squared residual is inf_a,L Enormq(z)v-(a+Lz)^2 = 2normB_F^2normv^2.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-34b5a08da5138c3a","tex":"I_t","aliases":"I_t","display":"I_t","kind":"use","excerpt":"Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cb01829575692957","tex":"i_t","aliases":"i_t","display":"i_t","kind":"use","excerpt":"[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0c80d6ff475c6099","tex":"I_{t+1}","aliases":"I_t+1","display":"I_{t+1}","kind":"use","excerpt":"Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9a620cdd3f0a1405","tex":"I_{t+1}^\\perp","aliases":"I_t+1^⊥ perp","display":"I_{t+1}^\\perp","kind":"use","excerpt":"The nested orthogonal decomposition is I_t+1 = I_t⊕(I_t+1∩ I_t^⊥). Therefore P_I_t+1^⊥r = P_I_t^⊥r - P_I_t+1∩ I_t^⊥r, and the two terms on the right are orthogonal. The Pythagorean identity gives [source reference].","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-89d18782e6bf6fb0","tex":"I_t^\\perp","aliases":"I_t^⊥ perp","display":"I_t^\\perp","kind":"use","excerpt":"Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-97fa69c0b29da91b","tex":"i_\\ell","aliases":"i_ℓ ell","display":"i_\\ell","kind":"use","excerpt":"Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicti","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d64392fa10e4d796","tex":"I_\\infty","aliases":"I_∞ infty","display":"I_\\infty","kind":"use","excerpt":"Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2c0bf80e8653774e","tex":"\\Ical_{\\mathrm{ctrl}}^{*}","aliases":"Ical_ctrl^* I_ctrl^* Ical mathrm mathcal mathrm","display":"{\\mathcal{I}}_{\\mathrm{ctrl}}^{*}","kind":"use","excerpt":"[Robust controlled improvement kernel] The operator Ψ_ctrl is monotone and has a greatest fixed point I_ctrl^* = ν X.Ψ_ctrl(X). For every state in I_ctrl^*, there exists a stationary authorized policy that keeps every admissible successor inside I_ctrl^*.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-747376124454e8f6","tex":"\\Ical_t^{\\mathrm{NL}}","aliases":"Ical_t^NL I_t^NL Ical mathrm mathcal mathrm","display":"{\\mathcal{I}}_t^{\\mathrm{NL}}","kind":"use","excerpt":"Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a componen","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b51ef54434ae2a91","tex":"\\Ical_\\gamma^\\omega","aliases":"Ical_γ^ω I_γ^ω Ical gamma omega mathcal gamma omega","display":"{\\mathcal{I}}_\\gamma^\\omega","kind":"use","excerpt":"Let s_1 above be an absorbing no-op state and suppose s_1∉ G_γ. Then s_1 lies in the controlled safety kernel but not in the productive Buechi closure I_γ^ω. The system can preserve every invariant forever while producing no repeated validated improvement. Safe persistence is therefore not evidence of recursive improvement.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-39b07e855b17f028","tex":"\\Ical_{\\forall}^{*}","aliases":"Ical_∀^* I_∀^* Ical forall mathcal forall","display":"{\\mathcal{I}}_{\\forall}^{*}","kind":"use","excerpt":"[Existence of universal recursive improvement closure] The operator Ψ_∀:2^S→2^S is monotone and has a greatest fixed point I_∀^* = ν X.Ψ_∀(X). A state lies in I_∀^* exactly when it satisfies the static invariants and every authorized successor also lies in I_∀^*.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ee480a6a1734d79","tex":"\\mathrm{improved}","aliases":"improved mathrm","display":"\\mathrm{improved}","kind":"use","excerpt":"The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d628dc8586b561e6","tex":"\\mathrm{improver}","aliases":"improver mathrm","display":"\\mathrm{improver}","kind":"use","excerpt":"A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-30fd7cc0c82f4a2a","tex":"\\mathrm{ImprovesSlice}","aliases":"ImprovesSlice mathrm","display":"\\mathrm{ImprovesSlice}","kind":"use","excerpt":"The example separates two predicates: ImprovesSlice(R_t) not⇒ AuthorizedCommit(R_t).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cff998e8f8883163","tex":"\\inf_{a,L}","aliases":"inf_a,L inf","display":"\\inf_{a,L}","kind":"use","excerpt":"[Best affine residual for a quadratic packet repair] Let z∼ N(0,I_r), B=B^top, and define q(z)=z^top Bz. The best L^2 affine approximation to q(z)v is the constant (tr B)v, and its minimum squared residual is inf_a,L Enormq(z)v-(a+Lz)^2 = 2normB_F^2normv^2.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d77da737ceefd7a4","tex":"\\inf_k","aliases":"inf_k inf","display":"\\inf_k","kind":"use","excerpt":"Different overlaps may have different uncertainty or semantic importance. Let W_E be a positive-definite operator on K^E. Define η_glue,W(e) := inf_knorme+ B^*k_W_E.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f32c08361161a863","tex":"\\inf_{k\\in K^I}","aliases":"inf_k∈ K^I inf in","display":"\\inf_{k\\in K^I}","kind":"use","excerpt":"[Quantitative graph-Hodge gluing] Let L_G:=BB^* be the graph Laplacian and let P_cyc be the orthogonal projector from K^E onto ker B, the cycle space. Then inf_k∈ K^Inorme+ B^*k = normP_cyce. The mean-zero minimum-norm gauge is k^* =-(L_G^† B⊗ I_K)e. If λ_2(L_G)>0 is the spectral gap, then normk^* ≤ frac1sqrtλ_2(L_G)norm e. Exact graph-level gluing","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b0b883aed7e89620","tex":"\\inf_T","aliases":"inf_T inf","display":"\\inf_T","kind":"use","excerpt":"A population residual formula gives a quantitative special case. Suppose h is centered with covariance Σsucc0, s∈{-1,+1} is independent of h with Pr(s=1)=p, and y=(A+sB)h. Then the best global linear map is T^*=A+(2p-1)B, and inf_T Enormy-Th^2 = 4p(1-p)tr(BΣ B^top). The proof is the normal-equation calculation from the centered two-component mixtur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3aa9c94b7ef16380","tex":"\\inf_t","aliases":"inf_t inf","display":"\\inf_t","kind":"use","excerpt":"Let h∼ N(0,1) and let s∈{-1,+1} be independent and balanced. Define y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f3a63c4125b03b95","tex":"\\inf_{u\\in K}","aliases":"inf_u∈ K inf in","display":"\\inf_{u\\in K}","kind":"use","excerpt":"Some repairs are not closed under negation. Examples include adding tests, allocating nonnegative compute, increasing redundancy, or modifying parameters subject to inequality constraints. Let K⊆ U be a nonempty closed convex authorization set and define η_D,K(r):=inf_u∈ KnormDu+r.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9545e639274b7b9d","tex":"\\inf_{u\\in U^{\\Auth}}","aliases":"inf_u∈ U^Auth inf_u∈ U^auth inf in Auth inf in mathrm","display":"\\inf_{u\\in U^{{\\mathrm{auth}}}}","kind":"use","excerpt":"eginenumerate item P_cor is the orthogonal projector onto im D; item P_ob is the orthogonal projector onto ker D^*; item the defect decomposes uniquely as r=r_cor+r_ob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complete set of least-squares minimizers is -D^† r+ker D; i","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-55466b76f7f0c090","tex":"\\mathrm{int}","aliases":"int mathrm","display":"\\mathrm{int}","kind":"use","excerpt":"[Internal error reduction without external improvement] Let the environment always produce protected outcome Y=1, independently of the agent's action. Let an internal predictor be p∈ℝ with internal score J_int(p)=-(p-Y)^2. A self-intervention setting p←1 improves J_int to zero error but does not change the world, task success, or action consequence","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cda9070e167070fb","tex":"\\mathrm{internal}","aliases":"internal mathrm","display":"\\mathrm{internal}","kind":"use","excerpt":"The latent event variable is z∈ {none,internal,world,sensor,memory,evaluator}. Each event changes one declared mechanism: item internal: the action-response map B changes; item world: the passive dynamics A change; item sensor: the redundant sensor receives an additive bias; item memory: the retention coefficient λ changes; item evaluator: the repo","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-42225b157ba9718e","tex":"\\mathrm{intervention}","aliases":"intervention mathrm","display":"\\mathrm{intervention}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-35b3ffd4dcaf66dc","tex":"J^*","aliases":"J^*","display":"J^*","kind":"use","excerpt":"Let J^*(A_t) be the highest true protected utility among retained validated nodes.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-787a425baaf96eca","tex":"j^{-1}","aliases":"j^-1","display":"j^{-1}","kind":"use","excerpt":"[Old-resolution and new-obstruction exact sequence] There is an exact sequence 0 ⟶ fracj^-1(C')C ⟶ fracVC xrightarrowoverline j fracV'C' ⟶ fracV'C'+j(V) ⟶ 0. The first nonzero term contains old obstruction classes newly resolved by the expanded repair calculus. The final quotient contains genuinely new obstruction directions not represented in the ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-485db41ee683b54c","tex":"J_0","aliases":"J_0","display":"J_0","kind":"use","excerpt":"chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported sc","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-80a3bf75a804e3f7","tex":"J_{\\mathrm{int}}","aliases":"J_int mathrm","display":"J_{\\mathrm{int}}","kind":"use","excerpt":"[Internal error reduction without external improvement] Let the environment always produce protected outcome Y=1, independently of the agent's action. Let an internal predictor be p∈ℝ with internal score J_int(p)=-(p-Y)^2. A self-intervention setting p←1 improves J_int to zero error but does not change the world, task success, or action consequence","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a620020ba505249b","tex":"J_n","aliases":"J_n","display":"J_n","kind":"use","excerpt":"chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported sc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4cef89266f103e47","tex":"J_{\\mathrm{protected}}","aliases":"J_protected mathrm","display":"J_{\\mathrm{protected}}","kind":"use","excerpt":"Consider a parent with protected utility J_0=0.42. A large global rewrite has protected utility J_protected=0.10 but apparent utility widehat J=0.10+1.10=1.20 due to an exploitable proxy bonus. A selector using widehat J accepts apparent gain 0.78, while protected gain is 0.10-0.42=-0.32. This is the mechanism underlying the global-edit positive co","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fb1ef9525dc5b427","tex":"J_T","aliases":"J_T","display":"J_T","kind":"use","excerpt":"Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-07b6123d41bb6991","tex":"J_t","aliases":"J_t","display":"J_t","kind":"use","excerpt":"ulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c7da7579735f1f60","tex":"J_{t+1}","aliases":"J_t+1","display":"J_{t+1}","kind":"use","excerpt":"ulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1bfd4f1f8fd6d7f5","tex":"j_U","aliases":"j_U","display":"j_U","kind":"use","excerpt":"Open-set improvement may enlarge the defect ontology itself. Let j_U:U_t^auth↪ U_t+1^auth, j_V:V_t↪ V_t+1 be injective embeddings satisfying D_t+1j_U=j_VD_t.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db6c2132f079e1ba","tex":"j_V","aliases":"j_V","display":"j_V","kind":"use","excerpt":"Open-set improvement may enlarge the defect ontology itself. Let j_U:U_t^auth↪ U_t+1^auth, j_V:V_t↪ V_t+1 be injective embeddings satisfying D_t+1j_U=j_VD_t.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2c8ffdbbdca4c726","tex":"J_z","aliases":"J_z","display":"J_z","kind":"use","excerpt":"of evaluator defects] Let e:v→ w and f:w→ z be evaluator morphisms. Suppose T_f is L_T-Lipschitz on represented protected evidence. Then δ_obs(f∘ e) ≤ L_Tδ_obs(e)+δ_obs(f), where the second term is evaluated on S_e( h_v). If, in addition, sup_r∈ h_v norm J_zT_fT_eι_v(r)-J_wT_eι_v(r) ≤ δ_J(fmid e), then δ_J(f∘ e) ≤ δ_J(e)+δ_J(fmid e).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-753ac74d7d5e39c2","tex":"\\Jcal","aliases":"Jcal J mathcal","display":"{\\mathcal{J}}","kind":"use","excerpt":"[Regularized minimum-cost repair] For λ>0, the functional J_λ(u) := frac12anglesDu+r,W(Du+r) + fracλ2anglesu,Ru has a unique minimizer u_λ =-(D^*WD+λ R)^-1D^*Wr. It is characterized by the normal equation D^*W(Du_λ+r)+λ Ru_λ=0.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2b492f399e261417","tex":"\\Jcal_\\lambda","aliases":"Jcal_λ J_λ Jcal lambda mathcal lambda","display":"{\\mathcal{J}}_\\lambda","kind":"use","excerpt":"[Regularized minimum-cost repair] For λ>0, the functional J_λ(u) := frac12anglesDu+r,W(Du+r) + fracλ2anglesu,Ru has a unique minimizer u_λ =-(D^*WD+λ R)^-1D^*Wr. It is characterized by the normal equation D^*W(Du_λ+r)+λ Ru_λ=0.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"[Finite meta-grammar generalization] Let K contain N candidate grandchild configurations. Suppose each target-level protected gain lies in [-G,G] and n independent development targets are used. Then, with probability at least 1-δ, every candidate satisfies abswidehatM(K)-M(K) ≤ Gsqrtfrac2log(2N/δ)n. If widehat K maximizes the empirical mean, then M","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-997c382b71c367a6","tex":"K^*","aliases":"K^*","display":"K^*","kind":"use","excerpt":"When K is a cone, noncorrectability admits a Farkas-type dual witness. Define the dual cone K^*:=setz∈ U:anglesz,u≥0 for all u∈ K.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9b23f26161473774","tex":"k^*","aliases":"k^*","display":"k^*","kind":"use","excerpt":"ge gluing] Let L_G:=BB^* be the graph Laplacian and let P_cyc be the orthogonal projector from K^E onto ker B, the cycle space. Then inf_k∈ K^Inorme+ B^*k = normP_cyce. The mean-zero minimum-norm gauge is k^* =-(L_G^† B⊗ I_K)e. If λ_2(L_G)>0 is the spectral gap, then normk^* ≤ frac1sqrtλ_2(L_G)norm e. Exact graph-level gluing is possible if and onl","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fae54ec909fa4765","tex":"K^E","aliases":"K^E","display":"K^E","kind":"use","excerpt":"Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes ","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9c6e379bac2607b1","tex":"K^I","aliases":"K^I","display":"K^I","kind":"use","excerpt":"d graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes the mismatch to e+ B^*k.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16def831203db3b6","tex":"K^\\top","aliases":"K^top top","display":"K^\\top","kind":"use","excerpt":"When X has full column rank, X^top X is invertible and (X^top X)^-1X^top Y =(X^top X)^-1X^top XT^top =T^top. When rank X<d, choose a nonzero v∈ker X. For any nonzero w∈ℝ^m, define K=wv^top. Then XK^top=0, so T and T+K agree on every selected anchor while differing on v.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-41d379662c083537","tex":"K_{0}","aliases":"K_0","display":"K_{0}","kind":"use","excerpt":"al level. A parent repair selector is frozen. Development receipts generated through the existing lineage are used to compile a child selector. Parent and child then receive exactly the same candidate repairs, target observations, calibration information, evaluator, and candidate budget. The child receipt is sealed before either final audit split i","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7873bc25ff5f48d7","tex":"K_{1}","aliases":"K_1","display":"K_{1}","kind":"use","excerpt":"al level. A parent repair selector is frozen. Development receipts generated through the existing lineage are used to compile a child selector. Parent and child then receive exactly the same candidate repairs, target observations, calibration information, evaluator, and candidate budget. The child receipt is sealed before either final audit split i","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1bdfdabfb7ccb3ec","tex":"k_1","aliases":"k_1","display":"k_1","kind":"use","excerpt":"[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c89e46e4c6c428da","tex":"K_{18}","aliases":"K_18","display":"K_{18}","kind":"use","excerpt":"item the admission results in [source reference] hold; item set-only, chart-only, shuffled-binding, no-abstention, and always-abstain controls retain their declared interpretations; item the automated tests, independent environment replay, independent chart reconstruction, independent NumPy forward pass, receipts, and package manifest verify. Then ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3fe5049ee642559b","tex":"K_{19}","aliases":"K_19","display":"K_{19}","kind":"use","excerpt":"item the admission results in [source reference] hold; item set-only, chart-only, shuffled-binding, no-abstention, and always-abstain controls retain their declared interpretations; item the automated tests, independent environment replay, independent chart reconstruction, independent NumPy forward pass, receipts, and package manifest verify. Then ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c64ced3fa5f2e615","tex":"K_1^{\\rm recal}","aliases":"K_1^rm recal rm","display":"K_1^{\\rm recal}","kind":"use","excerpt":"The descendant compiler receives only new development receipts. It diagnoses full-rank saturation, activates a finite conditioning-aware query grammar, selects a grandchild K_2, and seals the resulting policy before the fresh audit tasks and representation families are executed. The primary comparison is M_audit(K_2)-M_audit(K_1^rm recal), where K_","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a8cd867fa822df3a","tex":"k_2","aliases":"k_2","display":"k_2","kind":"use","excerpt":"[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae57cc6b359aadbd","tex":"K_3","aliases":"K_3","display":"K_3","kind":"use","excerpt":"t learned or trained experiment should be staged so that the bottleneck genuinely changes: item K_0 uses random anchors and a conservative fixed selector; item K_1 uses the Block 7 rank-revealing query policy and learned risk-sensitive selector; item K_2 faces full-rank but noisy and ill-conditioned bridges and must discover a conditioning-aware de","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a57e3beb8044c054","tex":"k_3","aliases":"k_3","display":"k_3","kind":"use","excerpt":"[Locally repairable, globally incompatible] Let the nerve be a three-cycle with no triple intersection. Suppose the kernel fiber is ℝ and the oriented differences are d_12=1, d_23=1, d_31=1. If the cocycle were a coboundary, there would be k_1,k_2,k_3 with d_12=k_2-k_1, d_23=k_3-k_2, d_31=k_1-k_3. Summing gives 3=0, a contradiction. Hence no global","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6cc92a51c6969e2c","tex":"K_a","aliases":"K_a","display":"K_a","kind":"use","excerpt":"Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5fc34cbb02e0ac96","tex":"K_{\\mathrm{affine\\ oracle}}","aliases":"K_affine oracle mathrm","display":"K_{\\mathrm{affine\\ oracle}}","kind":"use","excerpt":"The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d7448085f0c3c2e1","tex":"K_C","aliases":"K_C","display":"K_C","kind":"use","excerpt":"Let K_G be the global parent and K_C the chart process. Let A_C(x) be the frozen chart-certificate decision for target episode x. The descendant is K_3(x)= K_C(x), · A_C(x)=1,","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-df956a04b8635c3b","tex":"K_{\\mathrm{chart}}","aliases":"K_chart mathrm","display":"K_{\\mathrm{chart}}","kind":"use","excerpt":"Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes, K_global(h), · otherwise. The child is structur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a514dbccb64f4941","tex":"K_G","aliases":"K_G","display":"K_G","kind":"use","excerpt":"Let K_G be the global parent and K_C the chart process. Let A_C(x) be the frozen chart-certificate decision for target episode x. The descendant is K_3(x)= K_C(x), · A_C(x)=1,","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dc4f98125db1d824","tex":"K_{\\mathrm{global}}","aliases":"K_global mathrm","display":"K_{\\mathrm{global}}","kind":"use","excerpt":"Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes, K_global(h), · otherwise. The child is structur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-709955b2863572cd","tex":"K_{h'}^p","aliases":"K_h'^p","display":"K_{h'}^p","kind":"use","excerpt":"[Observational indistinguishability] If two hypotheses h,h'∈H satisfy K_h^p=K_h'^p for every permitted probe p∈P_probe, then no adaptive decision rule based only on permitted probe transcripts can distinguish h from h' with different transcript distributions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0c7bd880455e7fb0","tex":"K_i","aliases":"K_i","display":"K_i","kind":"use","excerpt":"Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared hea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-75fbaa7c32a6fb32","tex":"K_{\\mathrm{local}}","aliases":"K_local mathrm","display":"K_{\\mathrm{local}}","kind":"use","excerpt":"Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes, K_global(h), · otherwise. The child is structur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-544807fb1bbf53c0","tex":"K_{\\max}","aliases":"K_max max","display":"K_{\\max}","kind":"use","excerpt":"For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9a30d27faec7f7f7","tex":"K_r","aliases":"K_r","display":"K_r","kind":"use","excerpt":"The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation ep","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-40428f0704f0846d","tex":"\\mathcal K_R","aliases":"K_R mathcal","display":"\\mathcal K_R","kind":"use","excerpt":"The inherited obstruction language spans a response cone K_R = {Rz:z∈ℝ_≥0^k}. For observed response y, define the open-set residual η_rm open(y) = dist(y, K_R).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-79baedf6d480ceaa","tex":"k_{\\rm fit}","aliases":"k_rm fit rm","display":"k_{\\rm fit}","kind":"use","excerpt":"The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d04c52f7e2d91dcf","tex":"k_{\\rm val}","aliases":"k_rm val rm","display":"k_{\\rm val}","kind":"use","excerpt":"The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16bc7321a3832aa5","tex":"K_u","aliases":"K_u","display":"K_u","kind":"use","excerpt":"The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation ep","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8796b497bd79f4e9","tex":"k_W^*","aliases":"k_W^*","display":"k_W^*","kind":"use","excerpt":"[Weighted gluing normal equation] Every weighted least-squares gauge satisfies BW_E B^*k =- BW_Ee. The mean-zero minimum-norm minimizer is k_W^* =-( BW_E B^*)^† BW_Ee. The irreducible weighted mismatch is the W_E-orthogonal projection of e onto the W_E-orthogonal complement of im B^*.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-59566ec404aa357a","tex":"k_\\alpha","aliases":"k_α alpha","display":"k_\\alpha","kind":"use","excerpt":"If a global repair u exists, set k_α=u|_U_α-u_α. Then Dk_α=0 and d_αβ=k_β-k_α, so the cocycle is a coboundary. Conversely, if d_αβ=k_β-k_α for kernel sections k_α, then u'_α=u_α+k_α agree on overlaps. The sheaf gluing axiom yields a global u, and Du=-r because the adjustments lie in the kernel.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db873ec06f3d389d","tex":"k_\\beta","aliases":"k_β beta","display":"k_\\beta","kind":"use","excerpt":"If a global repair u exists, set k_α=u|_U_α-u_α. Then Dk_α=0 and d_αβ=k_β-k_α, so the cocycle is a coboundary. Conversely, if d_αβ=k_β-k_α for kernel sections k_α, then u'_α=u_α+k_α agree on overlaps. The sheaf gluing axiom yields a global u, and Du=-r because the adjustments lie in the kernel.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ef24ffb368efc8c3","tex":"K_{\\Gamma}^{\\star}","aliases":"K_Γ^⋆ Gamma star","display":"K_{\\Gamma}^{\\star}","kind":"use","excerpt":"Let Valid(c_i,Π_i^+)∈{0,1} encode capability, integrity, provenance, and resource validity. The selected descendant is K_Γ^⋆(τ) ∈ arg max_i:Valid(c_i,Π_i^+)=1 S_V(c_i), with the no-op baseline retained when no valid candidate exists.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e4d70f28b9ef02d3","tex":"K_{\\Gamma_0}^{\\star}","aliases":"K_Γ_0^⋆ Gamma star","display":"K_{\\Gamma_0}^{\\star}","kind":"use","excerpt":"Let Γ_0 and Γ_1 operate on the same problem τ with identical initial archive, candidate grammar, experiment budget, evaluator portfolio, integrity monitor, and hidden audit isolation. Let D_i = U_audit (K_Γ_1^⋆(τ_i)) - U_audit (K_Γ_0^⋆(τ_i)).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fcde1b0eceaa29b9","tex":"K_{\\Gamma_1}^{\\star}","aliases":"K_Γ_1^⋆ Gamma star","display":"K_{\\Gamma_1}^{\\star}","kind":"use","excerpt":"Let Γ_0 and Γ_1 operate on the same problem τ with identical initial archive, candidate grammar, experiment budget, evaluator portfolio, integrity monitor, and hidden audit isolation. Let D_i = U_audit (K_Γ_1^⋆(τ_i)) - U_audit (K_Γ_0^⋆(τ_i)).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bb1f0b06ee81a422","tex":"K_\\tau","aliases":"K_τ tau","display":"K_\\tau","kind":"use","excerpt":"Let T be a research-problem space. A problem τ∈T contains a baseline repair process K_τ, a declared candidate grammar P_τ, a cheap evidence channel, a scored evaluator portfolio, a hidden protected audit, integrity rules, and a resource budget B.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9feea0dd1279e0f4","tex":"\\Kcal","aliases":"Kcal K mathcal","display":"{\\mathcal{K}}","kind":"use","excerpt":"Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared hea","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7d369e93f419f2c6","tex":"\\Kcal_R","aliases":"Kcal_R K_R Kcal mathcal","display":"{\\mathcal{K}}_R","kind":"use","excerpt":"Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared hea","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2f26d259cfe08591","tex":"\\Kcal_{R'}","aliases":"Kcal_R' K_R' Kcal mathcal","display":"{\\mathcal{K}}_{R'}","kind":"use","excerpt":"Every process admissible under R is admissible under R', so K_R(K)⊆K_R'(K). Taking suprema gives the result.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-64d9b6d27c3dde49","tex":"\\mathsf{Key}","aliases":"Key mathsf","display":"\\mathsf{Key}","kind":"use","excerpt":"An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-96208a5f230c7cb3","tex":"\\mathfrak L_{16}^{\\mathrm{NL}}","aliases":"L_16^NL mathfrak mathrm","display":"\\mathfrak L_{16}^{\\mathrm{NL}}","kind":"use","excerpt":"ain their declared interpretations; item the child is FULL and adaptive on both fresh final supports, while the paired effects in [source reference] have positive frozen intervals; item the independent NumPy, receipt, witness, and gauge checks pass within their declared numerical tolerances. Then the accepted transition L_16^NL xrightarrowΠ_17 L_17","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-869ffae300209117","tex":"\\mathfrak L_{17}^{\\mathrm{NL}}","aliases":"L_17^NL mathfrak mathrm","display":"\\mathfrak L_{17}^{\\mathrm{NL}}","kind":"use","excerpt":"ain their declared interpretations; item the child is FULL and adaptive on both fresh final supports, while the paired effects in [source reference] have positive frozen intervals; item the independent NumPy, receipt, witness, and gauge checks pass within their declared numerical tolerances. Then the accepted transition L_16^NL xrightarrowΠ_17 L_17","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e80f6ffa31517649","tex":"\\mathfrak L_{18}^{\\mathrm{NL}}","aliases":"L_18^NL mathfrak mathrm","display":"\\mathfrak L_{18}^{\\mathrm{NL}}","kind":"use","excerpt":"Block 18 attempts but does not accept L_17^NL xrightarrowΠ_18? L_18^NL. The question mark is substantive: the selected schema passes both primary architecture-foreign audits but fails both shifted-generator audits and both frozen semantic-transport gates. The lineage therefore appends an obstruction receipt rather than a descendant commit. The post","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-735d271e3b332a61","tex":"\\mathfrak L_{19}^{\\mathrm{NL}}","aliases":"L_19^NL mathfrak mathrm","display":"\\mathfrak L_{19}^{\\mathrm{NL}}","kind":"use","excerpt":"Block 19 accepts the source-side continuation L_18^NL xrightarrowΠ_19 L_19^NL, where the added object is a six-chart semantic atlas with a set-structured latent pathway and selective support gate. The edge is conservative on inherited charts and improves fresh tasks in two compiled contexts. It does not yet transport that bundle across the foreign ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d6620d7dfa5c782a","tex":"\\mathfrak L_{20}^{\\mathrm{NL}}","aliases":"L_20^NL mathfrak mathrm","display":"\\mathfrak L_{20}^{\\mathrm{NL}}","kind":"use","excerpt":"Block 20 accepts the finite transport continuation L_19^NL xrightarrowΠ_20 L_20^NL, where the added object is an atlas-indexed connection carrying complete Neuralese trajectories from GRU and gated-residual fibers into the frozen Elman source fibers. The edge preserves the six-chart atlas, passes fresh semantic and curvature audits, and rejects uns","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fefb1ffb0f7bed24","tex":"\\mathfrak L_{21}^{\\mathrm{NL}}","aliases":"L_21^NL mathfrak mathrm","display":"\\mathfrak L_{21}^{\\mathrm{NL}}","kind":"use","excerpt":"Block 21 accepts the finite grammar continuation L_20^NL xrightarrowΠ_21 L_21^NL, where the transport process gains a typed response grammar over affine drift, nonlinear warp, latent-clock mixing, chart aliasing, benign semantic quotient, and destructive semantic loss. The edge preserves every inherited valid repair and accepts only the minimum-des","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f7103edc5660b542","tex":"\\mathfrak L_6","aliases":"L_6 mathfrak","display":"\\mathfrak L_6","kind":"use","excerpt":"The realized compiler edge is L_6 xrightarrowΠ^Γ_7 L_7, where the active repair grammar, candidate-method language, evaluator portfolio, integrity monitor, and experiment budget remain fixed, but the research-allocation compiler changes from Γ_0 to Γ_1. The edge is therefore compiler metaproductivity inside a finite declared research language, not ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a6c1d94cc1b3f421","tex":"\\mathfrak L_7","aliases":"L_7 mathfrak","display":"\\mathfrak L_7","kind":"use","excerpt":"The realized compiler edge is L_6 xrightarrowΠ^Γ_7 L_7, where the active repair grammar, candidate-method language, evaluator portfolio, integrity monitor, and experiment budget remain fixed, but the research-allocation compiler changes from Γ_0 to Γ_1. The edge is therefore compiler metaproductivity inside a finite declared research language, not ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f4190ba947c2e2cf","tex":"L_G^\\dagger","aliases":"L_G^† dagger","display":"L_G^\\dagger","kind":"use","excerpt":"ge gluing] Let L_G:=BB^* be the graph Laplacian and let P_cyc be the orthogonal projector from K^E onto ker B, the cycle space. Then inf_k∈ K^Inorme+ B^*k = normP_cyce. The mean-zero minimum-norm gauge is k^* =-(L_G^† B⊗ I_K)e. If λ_2(L_G)>0 is the spectral gap, then normk^* ≤ frac1sqrtλ_2(L_G)norm e. Exact graph-level gluing is possible if and onl","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eb2ad4ff4ef56a3e","tex":"L_{\\Gcal}","aliases":"L_Gcal L_G Gcal mathcal","display":"L_{{\\mathcal{G}}}","kind":"use","excerpt":"For grammar G with linearized authorized correction image L_G, define q_G(r) = dist(-r,L_G). This is the minimum unresolved first-order residual inside the current grammar.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a716b0894950a10a","tex":"L_i","aliases":"L_i","display":"L_i","kind":"use","excerpt":"For model M, task i, and clean protected loss L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1beda3c6328b1947","tex":"L_{i,\\mathrm{answer}}","aliases":"L_i,answer mathrm","display":"L_{i,\\mathrm{answer}}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-45c3cd7102fc8e29","tex":"L_{i,\\mathrm{answer}}^{\\mathrm{far\\ answer,depth}}","aliases":"L_i,answer^far answer,depth mathrm mathrm","display":"L_{i,\\mathrm{answer}}^{\\mathrm{far\\ answer,depth}}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-057868ba631737b9","tex":"L_{i,\\mathrm{sem}}","aliases":"L_i,sem mathrm","display":"L_{i,\\mathrm{sem}}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a2db0d119eb76cb6","tex":"L_{i,\\mathrm{sem}}^{\\mathrm{same\\ answer,far\\ program}}","aliases":"L_i,sem^same answer,far program mathrm mathrm","display":"L_{i,\\mathrm{sem}}^{\\mathrm{same\\ answer,far\\ program}}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ee2ce7ae30aa549","tex":"L_i^{0}","aliases":"L_i^0","display":"L_i^{0}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-05f559e8223f2af2","tex":"L_i^{\\mathrm{rev}}","aliases":"L_i^rev mathrm","display":"L_i^{\\mathrm{rev}}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ac3bff44e72292cb","tex":"L_i^{\\eta}","aliases":"L_i^η eta","display":"L_i^{\\eta}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-32538fc4e23dd7e1","tex":"L_J","aliases":"L_J","display":"L_J","kind":"use","excerpt":"[Readout stability under semantic holonomy] Let J:E_v→ Y be L_J-Lipschitz. Then d_Y(J(H_γ x),J(x)) ≤ L_Jη_sem(x;γ). If J returns a score vector and the winning score at x has margin m(x) over the runner-up, then the decision is unchanged whenever 2L_Jη_sem(x;γ)<m(x), where L_J bounds each score coordinate in the same norm.","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5ec429224ec330f1","tex":"L_{\\Meta}","aliases":"L_Meta Meta mathrm","display":"L_{{\\mathrm{Meta}}}","kind":"use","excerpt":"Let Ω = Ω_rm devsqcupΩ_rm audit be a disjoint target index set. The compiler receives rows indexed only by Ω_rm dev. It emits a descendant configuration c_1, a compiler ledger L_Meta, and a pre-audit seal Σ_rm pre = H(c_1,L_Meta,source hashes), where H is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory de","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c91dabfc8b700b3b","tex":"L_N","aliases":"L_N","display":"L_N","kind":"use","excerpt":"[Gauge-fixed local nonlinear repair by contraction] Suppose D:U→ V has a bounded right inverse Q:V→ U, so DQ=I_V, and set W:=im Q. Then restr D W:W→ V is an isomorphism with inverse Q. Let B_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a uniq","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a0c6993f26b62e6f","tex":"L_S","aliases":"L_S","display":"L_S","kind":"use","excerpt":"[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-533de406086c2d49","tex":"L_T","aliases":"L_T","display":"L_T","kind":"use","excerpt":"[Composition of evaluator defects] Let e:v→ w and f:w→ z be evaluator morphisms. Suppose T_f is L_T-Lipschitz on represented protected evidence. Then δ_obs(f∘ e) ≤ L_Tδ_obs(e)+δ_obs(f), where the second term is evaluated on S_e( h_v). If, in addition, sup_r∈ h_v norm J_zT_fT_eι_v(r)-J_wT_eι_v(r) ≤ δ_J(fmid e), then δ_J(f∘ e) ≤ δ_J(e)+δ_J(fmid e).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9c9505d3b6a4c451","tex":"\\mathfrak L_{t+1}^{\\mathrm{NL}}","aliases":"L_t+1^NL mathfrak mathrm","display":"\\mathfrak L_{t+1}^{\\mathrm{NL}}","kind":"use","excerpt":"child-minus-parent transplant-cost interval has positive lower endpoint; item child clean loss does not regress beyond a predeclared tolerance; item every claimed packet or transport passes its own held-out causal certificate; item model, task, audit, and result digests are committed in an append-only receipt chain before promotion. Then the accept","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4125ec6a009c25b1","tex":"\\mathfrak L_t^{\\mathrm{NL}}","aliases":"L_t^NL mathfrak mathrm","display":"\\mathfrak L_t^{\\mathrm{NL}}","kind":"use","excerpt":"Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a componen","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b656badb2589356a","tex":"L_t^{\\perp}","aliases":"L_t^⊥ perp","display":"L_t^{\\perp}","kind":"use","excerpt":"[Structural-to-behavioral identity] Assume L_t⊆ L_t+1 and let W=L_t+1∩ L_t^⊥. Then E_t+1^cor(μ)- E_t^cor(μ) = E_r∼μnormP_Wr^2 and E_t^ob(μ)- E_t+1^ob(μ) = E_r∼μnormP_Wr^2. Under quadratic residual loss and exact cancellation of the correctable component, this quantity is exactly the expected additional task-loss reduction made available by the expa","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b532789d6ddb0be6","tex":"L_v","aliases":"L_v","display":"L_v","kind":"use","excerpt":"[Paired evaluator-risk improvement] Let Z_1,…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with prob","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d8ade83d2dcc64e3","tex":"L_w","aliases":"L_w","display":"L_w","kind":"use","excerpt":"[Paired evaluator-risk improvement] Let Z_1,…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with prob","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cf9b9e0cb5be3aed","tex":"\\Lcal_{\\mathrm{com}}","aliases":"Lcal_com L_com Lcal mathrm mathcal mathrm","display":"{\\mathcal{L}}_{\\mathrm{com}}","kind":"use","excerpt":"For X∈End(H), define commutator energy E_com(X) = ∑_j=1^m ( normXR_j-R_jX_F^2 + normXR_j^∗-R_j^∗ X_F^2 ). There is a positive semidefinite linear operator L_com on the Hilbert space of matrices such that anglesX,L_comX_F = E_com(X). Its kernel is the commutant: kerL_com= A'.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[Minimum rounds under a witness quorum] Suppose every transaction requires at least q frozen witness components, where 1≤ q<n. Then at most n-q components can be edited in one round. Editing every component at least once requires at least lceilfracnn-qrceil rounds. This bound is attained by partitioning the components into edit blocks of size at mo","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"[Fixed-margin evaluator ceiling] Let V: S→[0,1] be fixed. If every accepted transition satisfies V(A_t+1)≥ V(A_t)+Δ for a constant Δ>0, then every accepted chain has length at most lfloor 1/Δrfloor+1.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a5f77e7b72a61440","tex":"\\limsup_{T\\to\\infty}","aliases":"limsup_T→∞ limsup to infty","display":"\\limsup_{T\\to\\infty}","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a60a3fa2f483385b","tex":"\\mathbf{Lin}","aliases":"Lin mathbf","display":"\\mathbf{Lin}","kind":"use","excerpt":"[Category of validated lineage paths] Assume valid edge receipts are closed under proof-path concatenation and contain identity receipts for zero-change transitions. Then lineage states are objects and validated receipt paths are morphisms of a category Lin.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2a707ccad5a125ef","tex":"\\ll","aliases":"ll","display":null,"kind":"use","excerpt":"The requirement κll1 is stronger than high pointwise probe performance on one support. It asks whether two independently constructed routes through the bundle yield compatible source representations.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-29a5d89d929b03d7","tex":"\\mathrm{local}","aliases":"local mathrm","display":"\\mathrm{local}","kind":"use","excerpt":"Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes, K_global(h), · otherwise. The child is structur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives ","count":20},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"endfoot endlastfoot Protected evaluation context · Longrightarrow · Nonvacuous improvement relation","count":68},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4323a75a9ec975c9","tex":"\\mathrm{low}","aliases":"low mathrm","display":"\\mathrm{low}","kind":"use","excerpt":"Let p∈P^+ be a candidate experiment with response y_p:T⟶ℝ. For threshold θ, define the refined diagnostic map ϕ_p,θ(τ) = (ϕ_0(τ),low), · y_p(τ)≤θ, (ϕ_0(τ),high), · y_p(τ)>θ. The old token is recovered by the projection π_0(z,s)=z, π_0∘ϕ_p,θ=ϕ_0. Thus the extension is conservative at the level of old diagnostic semantics.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-af9070f1457f1310","tex":"\\mathsf M","aliases":"M mathsf","display":"\\mathsf M","kind":"use","excerpt":"The child must pass performance and risk gates simultaneously. Let Gate = Seal wedge Match wedge AuditGain wedge ForeignGain wedge Risk wedge Controls. The released suite evaluates the following frozen selector-gate families: item descendant receipt validity; item split-file load order and sealing; item positive paired child-parent audit interval; ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-76a8c8c02bf83b7a","tex":"M'","aliases":"M'","display":"M'","kind":"use","excerpt":"Let Msucc0 be the current regularized information matrix. Adding candidate anchor x_i with quality weight w_i=σ_i^-2 gives M'=M+w_ix_ix_i^ T.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a4c7fd7f94be16fe","tex":"m'","aliases":"m'","display":"m'","kind":"use","excerpt":"Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matc","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c693ce1e8e9e894c","tex":"M^{-1}","aliases":"M^-1","display":"M^{-1}","kind":"use","excerpt":"[D-optimal marginal gain] The increase in log information volume is logdet M'-logdet M = log(1+w_ix_i^ TM^-1x_i).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c491dc76b5da376a","tex":"M^{-2}","aliases":"M^-2","display":"M^{-2}","kind":"use","excerpt":"[A-optimal trace reduction] Let v=sqrtw_ix_i. Then tr(M^-1)-tr((M+vv^ T)^-1) = fracv^ TM^-2v1+v^ TM^-1v.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9f7f9cfe1ef4ca4b","tex":"m_+","aliases":"m_+","display":"m_+","kind":"use","excerpt":"[Finite conflict-grounded language synthesis] Let a finite development archive contain one inherited token cell C with at least two incompatible repair subclasses. Suppose: item a candidate probe p∈P^+ separates C into cells C_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout ","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8b87ac0d682e2dd7","tex":"m_-","aliases":"m_-","display":"m_-","kind":"use","excerpt":"[Finite conflict-grounded language synthesis] Let a finite development archive contain one inherited token cell C with at least two incompatible repair subclasses. Suppose: item a candidate probe p∈P^+ separates C into cells C_- and C_+; item there exist m_- ,m_+∈C_rm pair such that m_- is acceptable throughout C_- and m_+ is acceptable throughout ","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"[Localization of supercriticality] After permuting schema types, M has block upper-triangular form with diagonal blocks M_1,…,M_k given by strongly connected components. Therefore rad(M) = max_1≤ r≤ krad(M_r). In particular, R_imp>1 only if at least one strongly connected schema family is itself supercritical.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"One unknown token with incompatible methods. Let two problems a and b both map to bot. Suppose A^⋆(a)={m_1}, A^⋆(b)={m_2}, m_1≠ m_2. Any deterministic policy conditioned only on bot must choose one method and fail the other problem. This is a diagnostic-language obstruction, not ordinary estimation noise.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"0 · -1. On H=ℝ^4, define four response operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generate","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-651ec01c3f6e67cb","tex":"m_2","aliases":"m_2","display":"m_2","kind":"use","excerpt":"One unknown token with incompatible methods. Let two problems a and b both map to bot. Suppose A^⋆(a)={m_1}, A^⋆(b)={m_2}, m_1≠ m_2. Any deterministic policy conditioned only on bot must choose one method and fail the other problem. This is a diagnostic-language obstruction, not ordinary estimation noise.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9827dd3c02748a45","tex":"M_{24}^{\\rm for}","aliases":"M_24^rm for rm","display":"M_{24}^{\\rm for}","kind":"use","excerpt":"item parent and descendant receive the matched budget in [source reference]; item target-level audit differences are exchangeable or otherwise support the declared paired interval; item the protected evaluator and source repairs are unchanged across the parent-child edge; item accepted-harm rate is not increased. If the paired lower confidence boun","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c887199df3da3b63","tex":"M_a","aliases":"M_a","display":"M_a","kind":"use","excerpt":"The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0f5cef98cfcf15fe","tex":"M_c","aliases":"M_c","display":"M_c","kind":"use","excerpt":"tent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute th","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e32d8099dae11fd2","tex":"M_{h,B}^{\\Dcal}","aliases":"M_h,B^Dcal M_h,B^D Dcal mathcal","display":"M_{h,B}^{{\\mathcal{D}}}","kind":"use","excerpt":"[Metaproductivity shift bound] If absG_h,B(K,A)≤ G_max, then abs M_h,B^D(K) - M_h,B^D'(K) ≤ 2G_maxTV(D,D').","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dd6c1917dbe94d07","tex":"M_{h,B}^{\\Dcal'}","aliases":"M_h,B^Dcal' M_h,B^D' Dcal mathcal","display":"M_{h,B}^{{\\mathcal{D}}'}","kind":"use","excerpt":"[Metaproductivity shift bound] If absG_h,B(K,A)≤ G_max, then abs M_h,B^D(K) - M_h,B^D'(K) ≤ 2G_maxTV(D,D').","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-116ff9888b369389","tex":"m_{\\mathrm{hist}}","aliases":"m_hist mathrm","display":"m_{\\mathrm{hist}}","kind":"use","excerpt":"[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is ","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-83d3f2e0b215ee50","tex":"M_{ij}","aliases":"M_ij","display":"M_{ij}","kind":"use","excerpt":"Partition validated schemas into m types. For a type-i schema, let N_ij count type-j descendants whose receipts include a successful necessity ablation showing that removal of the parent schema eliminates the descendant discovery under matched evidence and budget. Define M_ij= E N_ij. This definition excludes descendants that merely share ancestry ","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bf2227566ef624f5","tex":"M_j","aliases":"M_j","display":"M_j","kind":"use","excerpt":"For integrity monitoring, let M_1,…,M_s∈{0,1} be monitor pass indicators and let the ensemble pass only when every component passes: M_∩=∏_j=1^s M_j.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c9f29e7d9e7a175b","tex":"M_k","aliases":"M_k","display":"M_k","kind":"use","excerpt":"[Localization of supercriticality] After permuting schema types, M has block upper-triangular form with diagonal blocks M_1,…,M_k given by strongly connected components. Therefore rad(M) = max_1≤ r≤ krad(M_r). In particular, R_imp>1 only if at least one strongly connected schema family is itself supercritical.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-84449574a12faf3a","tex":"M_{n_\\alpha}","aliases":"M_n_α alpha","display":"M_{n_\\alpha}","kind":"use","excerpt":"[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-28cc0a43f7b6c71a","tex":"M_r","aliases":"M_r","display":"M_r","kind":"use","excerpt":"[Localization of supercriticality] After permuting schema types, M has block upper-triangular form with diagonal blocks M_1,…,M_k given by strongly connected components. Therefore rad(M) = max_1≤ r≤ krad(M_r). In particular, R_imp>1 only if at least one strongly connected schema family is itself supercritical.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bcf8a81b4e8bcc88","tex":"M_S","aliases":"M_S","display":"M_S","kind":"use","excerpt":"Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-17770225492e1d01","tex":"M_s","aliases":"M_s","display":"M_s","kind":"use","excerpt":"Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fbc424d1f9a4ff87","tex":"M_{S,\\lambda}","aliases":"M_S,λ lambda","display":"M_{S,\\lambda}","kind":"use","excerpt":"The executable implementation uses M_S,λ=X^ TWX+λ I with a small fixed λ>0. The reported design diagnostics use source coordinates standardized against the complete candidate pool before applying the reliability weights. Standardization prevents one arbitrary source-coordinate scale from dominating the comparison. The actual bridge fit remains weig","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3ef92dd1119c5fb4","tex":"M_S^{-1}","aliases":"M_S^-1","display":"M_S^{-1}","kind":"use","excerpt":"[Weighted bridge covariance] Assume M_S is invertible and the noise rows are independent with covariance σ_i^2I_q. The weighted least-squares estimator widehat T^ T = M_S^-1X^ TWY is unbiased and satisfies Cov(vec(widehat T^ T-T^ T)mid X) = I_q⊗ M_S^-1. Consequently, E[normwidehat T-T_F^2mid X] = q tr(M_S^-1).","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9de6a9970cb6757b","tex":"M_t","aliases":"M_t","display":"M_t","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6f3ae7d6b37bdd61","tex":"m_t","aliases":"m_t","display":"m_t","kind":"use","excerpt":"Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a01fc42b9500ded3","tex":"m_{t+1}","aliases":"m_t+1","display":"m_{t+1}","kind":"use","excerpt":"Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0447b398c958b23b","tex":"M_{T-1}","aliases":"M_T-1","display":"M_{T-1}","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-401d9b49eeb97619","tex":"m_\\alpha","aliases":"m_α alpha","display":"m_\\alpha","kind":"use","excerpt":"[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ba39a13f3b2761ca","tex":"\\max_{1\\leq b\\leq B}","aliases":"max_1≤ b≤ B max leq leq","display":"\\max_{1\\leq b\\leq B}","kind":"use","excerpt":"The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ce086d55f1ebac0f","tex":"\\max_{1\\leq i\\leq k}","aliases":"max_1≤ i≤ k max leq leq","display":"\\max_{1\\leq i\\leq k}","kind":"use","excerpt":"[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9a8189af4c681ba8","tex":"\\max_{1\\leq i\\leq5}","aliases":"max_1≤ i≤5 max leq leq","display":"\\max_{1\\leq i\\leq5}","kind":"use","excerpt":"Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bb75e43deede31c6","tex":"\\max_{1\\leq r\\leq k}","aliases":"max_1≤ r≤ k max leq leq","display":"\\max_{1\\leq r\\leq k}","kind":"use","excerpt":"[Localization of supercriticality] After permuting schema types, M has block upper-triangular form with diagonal blocks M_1,…,M_k given by strongly connected components. Therefore rad(M) = max_1≤ r≤ krad(M_r). In particular, R_imp>1 only if at least one strongly connected schema family is itself supercritical.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eec18b0187f98c2c","tex":"\\max_a","aliases":"max_a max","display":"\\max_a","kind":"use","excerpt":"[Finite repair-relevant block selection] Suppose there is a unique repair-relevant block a^* and a margin m>0 such that s_a^*≥ s_a+m for every a≠ a^*. If max_a|widehat s_a-s_a|<fracm2, then arg max_awidehat s_a=a^*.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f926e4691687b809","tex":"\\max_{A'\\in\\Desc_{h,B}(K,A)}","aliases":"max_A'∈Desc_h,B(K,A) max in Desc max in operatorname","display":"\\max_{A'\\in{\\operatorname{Desc}}_{h,B}(K,A)}","kind":"use","excerpt":"Let D_hold be a distribution over foreign held-out agents that are not descendants of the evaluated lineage. Let K be a frozen improvement mechanism. Under horizon h and budget B, define M^foreign_h,B(K) = E_A∼D_hold [ max_A'∈Desc_h,B(K,A) J(A')-J(A) ].","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bf0eb415c3b90248","tex":"\\max_{a,j}","aliases":"max_a,j max","display":"\\max_{a,j}","kind":"use","excerpt":"rojectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^-12. The smalle","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-14e5305ac349b25d","tex":"\\max_{A\\in\\Acal_t}","aliases":"max_A∈Acal_t max_A∈A_t max in Acal max in mathcal","display":"\\max_{A\\in{\\mathcal{A}}_t}","kind":"use","excerpt":"[Archive-envelope monotonicity] Suppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector w, define H_w(A_t) = max_A∈A_tw^ Tz(A). Then H_w(A_t) is nondecreasing in t.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9a26700ba2da92c1","tex":"\\max_{c\\in\\mathcal A_0}","aliases":"max_c∈ A_0 max in mathcal","display":"\\max_{c\\in\\mathcal A_0}","kind":"use","excerpt":"[Risk-filter tradeoff] Let A_r⊆ A_0 be the candidate set retained with and without a risk filter. Then max_c∈ A_0widehat U(c) ≥ max_c∈ A_rwidehat U(c), but no analogous inequality favors the unfiltered selector for true protected risk.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-885a693748e49df3","tex":"\\max_{c\\in\\mathcal A_r}","aliases":"max_c∈ A_r max in mathcal","display":"\\max_{c\\in\\mathcal A_r}","kind":"use","excerpt":"[Risk-filter tradeoff] Let A_r⊆ A_0 be the candidate set retained with and without a risk filter. Then max_c∈ A_0widehat U(c) ≥ max_c∈ A_rwidehat U(c), but no analogous inequality favors the unfiltered selector for true protected risk.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"use","excerpt":"For a nonnegative random variable X, E X=∫_0^∞Pr(X>x) d x. Independence gives Pr(max_iG_i≤ x)=F(x)^k. Substitution proves the identity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ed5c67be9852367b","tex":"\\max_{K\\in\\mathcal K}","aliases":"max_K∈ K max in mathcal","display":"\\max_{K\\in\\mathcal K}","kind":"use","excerpt":"contain N candidate grandchild configurations. Suppose each target-level protected gain lies in [-G,G] and n independent development targets are used. Then, with probability at least 1-δ, every candidate satisfies abswidehatM(K)-M(K) ≤ Gsqrtfrac2log(2N/δ)n. If widehat K maximizes the empirical mean, then M(widehat K) ≥ max_K∈ KM(K) - 2Gsqrtfrac2log","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e70fab3e329014f6","tex":"\\max_{m\\in\\Mcal_0}","aliases":"max_m∈Mcal_0 max_m∈M_0 max in Mcal max in mathcal","display":"\\max_{m\\in{\\mathcal{M}}_0}","kind":"use","excerpt":"Let U(τ,m) be protected validity-filtered research gain. Define inherited and extended oracle envelopes O_0(τ)=max_m∈M_0U(τ,m), O_1(τ)=max_m∈M_0∪C_rm pairU(τ,m). The task-relevant structural margin is κ_Q = E_τ∼ Q [ O_1(τ)-O_0(τ) ]. A positive margin means that the extension adds useful methods outside the inherited candidate opportunity set.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5630b44e281b6381","tex":"\\max_{m\\in\\Mcal_0\\cup\\Ccal_{\\rm pair}}","aliases":"max_m∈Mcal_0∪Ccal_rm pair max_m∈M_0∪C_rm pair max in Mcal cup Ccal rm max in mathcal cup mathcal rm","display":"\\max_{m\\in{\\mathcal{M}}_0\\cup{\\mathcal{C}}_{\\rm pair}}","kind":"use","excerpt":"Let U(τ,m) be protected validity-filtered research gain. Define inherited and extended oracle envelopes O_0(τ)=max_m∈M_0U(τ,m), O_1(τ)=max_m∈M_0∪C_rm pairU(τ,m). The task-relevant structural margin is κ_Q = E_τ∼ Q [ O_1(τ)-O_0(τ) ]. A positive margin means that the extension adds useful methods outside the inherited candidate opportunity set.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dd2462ae8ddbd15f","tex":"\\max_{q\\in Q}","aliases":"max_q∈ Q max in","display":"\\max_{q\\in Q}","kind":"use","excerpt":"[Approximate scalarization preservation] For every w≥0 with normw_1=1, max_q∈ Qw^ Tq ≥ max_z∈ Zw^ Tz-ε.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e2aab8a0fc3cab24","tex":"\\max_{S:|S|=k}","aliases":"max_S:|S|=k max","display":"\\max_{S:|S|=k}","kind":"use","excerpt":"does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value σ_min=10^-4. Random noise of norm 10^-3 can then dominate the weakest identified direction. The bottleneck has migrated from rank to conditioning. A meaningful grandchild should target a criterion such as max_S:|S|=k ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3420c4ea9cc4b05c","tex":"\\max_{\\substack{y\\in\\ker D^*\\\\ \\norm y\\leq1}}","aliases":"max_substacky∈ker D^* norm y≤1 max substack in ker norm leq","display":null,"kind":"use","excerpt":"[Dual obstruction witness] For every r∈ V, η_D(r) = max_substacky∈ker D^* norm y≤1 anglesy,r. If r_ob≠0, the maximum is attained by y^*=fracr_obnormr_ob.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b0594d3d24a0288b","tex":"\\max_z","aliases":"max_z max","display":"\\max_z","kind":"use","excerpt":"Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-34f21f9db9bf0632","tex":"\\max_{z\\in Z}","aliases":"max_z∈ Z max in","display":"\\max_{z\\in Z}","kind":"use","excerpt":"[Approximate scalarization preservation] For every w≥0 with normw_1=1, max_q∈ Qw^ Tq ≥ max_z∈ Zw^ Tz-ε.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5513a149a6308882","tex":"\\max_{|S|=k}","aliases":"max_|S|=k max","display":"\\max_{|S|=k}","kind":"use","excerpt":"The E-optimal objective is max_|S|=kλ_min(M_S). The companion initializes with the weighted D-optimal design, then performs same-stratum exchanges that increase the smallest eigenvalue. This is a finite heuristic rather than a claim of global E-optimality.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ff8ed01f2fe2004d","tex":"\\Mcal_{\\mathrm{audit}}","aliases":"Mcal_audit M_audit Mcal mathrm mathcal mathrm","display":"{\\mathcal{M}}_{\\mathrm{audit}}","kind":"use","excerpt":"The descendant compiler receives only new development receipts. It diagnoses full-rank saturation, activates a finite conditioning-aware query grammar, selects a grandchild K_2, and seals the resulting policy before the fresh audit tasks and representation families are executed. The primary comparison is M_audit(K_2)-M_audit(K_1^rm recal), where K_","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5d242bddf86421a6","tex":"\\Mcal_{\\Xi}","aliases":"Mcal_Ξ M_Ξ Mcal Xi mathcal Xi","display":"{\\mathcal{M}}_{\\Xi}","kind":"use","excerpt":"[Same-context transitivity] Suppose every edge is evaluated in one common context Ξ. Then M_Ξ(K_m)-M_Ξ(K_0) = ∑_t=0^m-1 ( M_Ξ(K_t+1)-M_Ξ(K_t) ). Consequently, positive lower bounds on all edges add to a positive lower bound on the complete path.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4196353f6b7537df","tex":"\\Mcal_{\\Xi,\\Dcal,B}","aliases":"Mcal_Ξ,Dcal,B M_Ξ,D,B Mcal Xi Dcal mathcal Xi mathcal","display":"{\\mathcal{M}}_{\\Xi,{\\mathcal{D}},B}","kind":"use","excerpt":"Fix a protected evaluation context Ξ, a foreign-system distribution D, and a matched resource budget B. Define protected foreign metaproductivity by M_Ξ,D,B(K) = E_A∼D [ J_Ξ(K[A;B])-J_Ξ(A) ], where K[A;B] is the authorized descendant selected by K for target A, or A itself when K abstains.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-920c222eb340b7ea","tex":"\\Mcal_{\\Xi_0}","aliases":"Mcal_Ξ_0 M_Ξ_0 Mcal Xi mathcal Xi","display":"{\\mathcal{M}}_{\\Xi_0}","kind":"use","excerpt":"Let edge K_0→ K_1 improve a bridge task distribution Ξ_0, while K_1→ K_2 improves a heteroscedastic distribution Ξ_1. Positive effects in both contexts do not imply M_Ξ_0(K_2)>M_Ξ_0(K_0) or any additive path gain unless Ξ_0 and Ξ_1 share a common utility or a validated context transport.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-365e16b045f6ce7b","tex":"\\mathrm{mean}","aliases":"mean mathrm","display":"\\mathrm{mean}","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aa0bc4e82a6d58fa","tex":"\\mathrm{mech}","aliases":"mech mathrm","display":"\\mathrm{mech}","kind":"use","excerpt":"G_mech · Mechanics · Legal phase traces, receipt verification, checkpoint restoration, deterministic replay","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-634d158997cabc38","tex":"\\mathrm{memory}","aliases":"memory mathrm","display":"\\mathrm{memory}","kind":"use","excerpt":"A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4e2edb9ef439a4ee","tex":"\\mathrm{meta}","aliases":"meta mathrm","display":"\\mathrm{meta}","kind":"use","excerpt":"G_meta · Metaproductivity · Parent-child repair-calculus comparison on frozen foreign systems","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-13aecc48f9903974","tex":"\\MetaCapable","aliases":"MetaCapable mathrm","display":"{\\mathrm{MetaCapable}}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e1765fc478eaab44","tex":"\\min_{b\\neq a}","aliases":"min_b≠ a min neq","display":"\\min_{b\\neq a}","kind":"use","excerpt":"In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e68059acda055f95","tex":"\\min_{c\\neq d}","aliases":"min_c≠ d min neq","display":"\\min_{c\\neq d}","kind":"use","excerpt":"[Nearest-signature route stability] Let the minimum pairwise normalized signature separation of an atlas be Δ_A = min_c≠ dd_sig(σ_c,σ_d). If an observed signature from chart c has normalized perturbation at most δ and 2δ<Δ_A, nearest-signature routing returns c. If additionally δ≤τ_adm, the route is admitted.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5bc8fcda322df753","tex":"\\min_{g\\in\\mathcal G_{21}}","aliases":"min_g∈ G_21 min in mathcal","display":"\\min_{g\\in\\mathcal G_{21}}","kind":"use","excerpt":"Each repair g receives a description-length proxy L(g) recording parameter count and constructor surcharge. The compiler selects widehat g ∈ argmin_g∈ G_21 L(g) subject to the appropriate protected gates. If no repair passes and the collision certificate is valid, it returns G_bot; otherwise it returns unresolved.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-60698b1979c563d1","tex":"\\min_j","aliases":"min_j min","display":"\\min_j","kind":"use","excerpt":"[Conjunctive monitor false-negative bound] For an invalid method event I, the ensemble false-negative probability satisfies P(M_∩=1mid I) ≤ min_j P(M_j=1mid I). If monitor errors are conditionally independent given I, then P(M_∩=1mid I) = ∏_j=1^s P(M_j=1mid I).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8af81b948e546867","tex":"\\min_{J_v(x_i)\\neq J_v(x_j)}","aliases":"min_J_v(x_i)≠ J_v(x_j) min neq","display":"\\min_{J_v(x_i)\\neq J_v(x_j)}","kind":"use","excerpt":"[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c676c48b094f8455","tex":"\\min_{z\\neq z'}","aliases":"min_z≠ z' min neq","display":"\\min_{z\\neq z'}","kind":"use","excerpt":"[Prototype-separation error bound] Let Δ = min_z≠ z'normμ_z-μ_z'. Under equal priors and isotropic Gaussian noise, nearest-prototype classification satisfies Pr(widehat z≠ z) ≤ (| Z|-1) exp(-fracΔ^28σ^2).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cc2836ea0a12161d","tex":"\\mathrm{MSE}","aliases":"MSE mathrm","display":"\\mathrm{MSE}","kind":"use","excerpt":"positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compil","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e982cbcb999c01b7","tex":"\\mathrm{MSE}_{\\mathrm{clean}}","aliases":"MSE_clean mathrm mathrm","display":"\\mathrm{MSE}_{\\mathrm{clean}}","kind":"use","excerpt":"positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compil","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N ℕ mathbb","display":"{\\mathbb{N}}","kind":"use","excerpt":"[Finite local-to-global obstruction tower] Fix N∈ℕ. Assume: item the analytic local repair equations agree under restriction; item the sheaves mathscr U and mathscr V satisfy effective linear descent on the declared cover; item repairs have been constructed globally through order n-1. Then a global repair extends through order n if and only if: ite","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b75f3a902c5f7e60","tex":"n^{-1}","aliases":"n^-1","display":"n^{-1}","kind":"use","excerpt":"[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b86b4ad0b2eda6a1","tex":"\\mathcal N_{\\mathcal B\\cup\\{b_{m+1}\\}}","aliases":"N_ B∪{b_m+1} mathcal mathcal cup","display":"\\mathcal N_{\\mathcal B\\cup\\{b_{m+1}\\}}","kind":"use","excerpt":"[Strict evaluator-nullspace refinement] If Db_m+1 is nonzero on some v∈ N_ B, then N_ B∪{b_m+1} subsetneq N_ B.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ffaa0ad23479bca8","tex":"N_1","aliases":"N_1","display":"N_1","kind":"use","excerpt":"s(x)v(x) , where u,v∈ℝ^8 and A is invertible. Let N_1 be the number of affected anchors among k selections.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4861b5f0d94d3545","tex":"N_5","aliases":"N_5","display":"N_5","kind":"use","excerpt":"Consider five mutable modules: N_5 = {S,Q,T,C,H}, representing selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let F_K(∅)=0.20. Assign singleton contributions (0.08,0.12,0.10,0.16,0.14) and pair synergies ω_QT=0.05, ω_CH=0.06, ω_SC=0.03. Then F_K(N_5)=0.94, H(K)=0.74. Every pair synergy is divided ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-61c069b08b6b0f8b","tex":"N_{ij}","aliases":"N_ij","display":"N_{ij}","kind":"use","excerpt":"Partition validated schemas into m types. For a type-i schema, let N_ij count type-j descendants whose receipts include a successful necessity ablation showing that removal of the parent schema eliminates the descendant discovery under matched evidence and budget. Define M_ij= E N_ij. This definition excludes descendants that merely share ancestry ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7f3015956cb36973","tex":"n_{\\rm anchor}","aliases":"n_rm anchor rm","display":"n_{\\rm anchor}","kind":"use","excerpt":"For the controlled transport experiment, the sample sizes are (n_rm train,n_rm anchor,n_rm calibrate,n_rm audit) =(7000,1500,1600,6000). The learned-representation pilot uses (5000,1200,1200,4000).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-54d8d55699bf9e23","tex":"n_{\\rm audit}","aliases":"n_rm audit rm","display":"n_{\\rm audit}","kind":"use","excerpt":"For the controlled transport experiment, the sample sizes are (n_rm train,n_rm anchor,n_rm calibrate,n_rm audit) =(7000,1500,1600,6000). The learned-representation pilot uses (5000,1200,1200,4000).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f75a8cf363f4a9b9","tex":"n_{\\rm calibrate}","aliases":"n_rm calibrate rm","display":"n_{\\rm calibrate}","kind":"use","excerpt":"For the controlled transport experiment, the sample sizes are (n_rm train,n_rm anchor,n_rm calibrate,n_rm audit) =(7000,1500,1600,6000). The learned-representation pilot uses (5000,1200,1200,4000).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1e09c99f7a08db20","tex":"n_{\\rm train}","aliases":"n_rm train rm","display":"n_{\\rm train}","kind":"use","excerpt":"For the controlled transport experiment, the sample sizes are (n_rm train,n_rm anchor,n_rm calibrate,n_rm audit) =(7000,1500,1600,6000). The learned-representation pilot uses (5000,1200,1200,4000).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5b38b83286298499","tex":"n_{t+1}","aliases":"n_t+1","display":"n_{t+1}","kind":"use","excerpt":"A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d1d039b5ce5beb93","tex":"\\mathcal N_{\\{b_1\\}}","aliases":"N_{b_1} mathcal","display":"\\mathcal N_{\\{b_1\\}}","kind":"use","excerpt":"Let b_1,…,b_m be benchmark readouts. A single scored benchmark leaves the local nullspace N_{b_1} = ker Db_1. An evaluator portfolio has invisible subspace N_ B = ⋂_j=1^mker Db_j.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9727f5125c78ee28","tex":"n_\\alpha","aliases":"n_α alpha","display":"n_\\alpha","kind":"use","excerpt":"[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-93d4c2400aa57029","tex":"\\Ncal_0","aliases":"Ncal_0 N_0 Ncal mathcal","display":"{\\mathcal{N}}_0","kind":"use","excerpt":"[Finite Neuralese metaproductivity edge] Let parent N_0 and child N_1 be evaluated on identical sealed tasks with identical fixed latent-step budgets, answer-scoring rules, donor construction, numerical precision, evaluator, and witness. Assume: item both audits satisfy replay integrity; item the child satisfies latent necessity, ordered computatio","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0339008c2fc42d45","tex":"\\Ncal_1","aliases":"Ncal_1 N_1 Ncal mathcal","display":"{\\mathcal{N}}_1","kind":"use","excerpt":"[Finite Neuralese metaproductivity edge] Let parent N_0 and child N_1 be evaluated on identical sealed tasks with identical fixed latent-step budgets, answer-scoring rules, donor construction, numerical precision, evaluator, and witness. Assume: item both audits satisfy replay integrity; item the child satisfies latent necessity, ordered computatio","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ef4c88eb5dda32ab","tex":"\\Ncal_{\\Bcal^+}","aliases":"Ncal_Bcal^+ N_B^+ Ncal Bcal mathcal mathcal","display":"{\\mathcal{N}}_{{\\mathcal{B}}^+}","kind":"use","excerpt":"[Portfolio refinement shrinks the invisible subspace] Let B^+=B∪{b_m+1}. Then N_B^+ ⊆ N_B. The inclusion is strict if and only if there exists v∈N_B such that Db_m+1(v)≠0.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-03815b4622168347","tex":"\\Ncal_t","aliases":"Ncal_t N_t Ncal mathcal","display":"{\\mathcal{N}}_t","kind":"use","excerpt":"Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a componen","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8e2629b975a7af0a","tex":"\\mathrm{no\\mbox{-}meta}","aliases":"no-meta mathrm mbox","display":"\\mathrm{no\\mbox{-}meta}","kind":"use","excerpt":"Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before f","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7d154b9783de48e2","tex":"\\mathrm{no\\mbox{-}op}","aliases":"no-op mathrm mbox","display":"\\mathrm{no\\mbox{-}op}","kind":"use","excerpt":"Let K_G be the global parent and K_C the chart process. Let A_C(x) be the frozen chart-certificate decision for target episode x. The descendant is K_3(x)= K_C(x), · A_C(x)=1, K_G(x), · A_C(x)=0 and the parent gate passes, no-op, · otherwise.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-675445ea12d3cfe4","tex":"\\mathrm{nom}","aliases":"nom mathrm","display":"\\mathrm{nom}","kind":"use","excerpt":"Let C_B^nom be the same envelope when integrity constraints are ignored.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-30e98d4b5871e653","tex":"\\mathrm{none}","aliases":"none mathrm","display":"\\mathrm{none}","kind":"use","excerpt":"The latent event variable is z∈ {none,internal,world,sensor,memory,evaluator}. Each event changes one declared mechanism: item internal: the action-response map B changes; item world: the passive dynamics A change; item sensor: the redundant sensor receives an additive bias; item memory: the retention coefficient λ changes; item evaluator: the repo","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ace9f5db864dc83","tex":"\\operatorname{NRMSE}","aliases":"NRMSE operatorname","display":"\\operatorname{NRMSE}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cf9bcf7d39b32756","tex":"\\operatorname{NRMSE}_{\\rm packet}","aliases":"NRMSE_rm packet operatorname rm","display":"\\operatorname{NRMSE}_{\\rm packet}","kind":"use","excerpt":"For reconstructive candidates, admissibility requires R^2_rm sem≥0.68, R^2_rm packet≥0.72, NRMSE_rm packet≤0.55. A quotient is admissible only when the observed dimension ratio is at most 0.82, semantic R^2 is at least 0.68, and no reconstructive candidate reaches packet R^2=0.82. A no-go certificate is admissible only after the compiler discovers ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7c77c8ed322af38f","tex":"\\operatorname{NRMSE}_{\\mathrm{trace}}","aliases":"NRMSE_trace operatorname mathrm","display":"\\operatorname{NRMSE}_{\\mathrm{trace}}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b2ad6198ca0924f9","tex":"\\nsubseteq","aliases":"nsubseteq","display":null,"kind":"use","excerpt":"In the intact model every complete successor of s_base lies in K. Under ablation, Post(s_base)nsubseteq K, so s_base∉Ψ(K).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-72f06078b835c8f0","tex":"\\mathrm{nui}","aliases":"nui mathrm","display":"\\mathrm{nui}","kind":"use","excerpt":"The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-49d4b6cc9c776bce","tex":"o_{t+1}","aliases":"o_t+1","display":"o_{t+1}","kind":"use","excerpt":"[Exact sequence of newly correctable directions] Under the assumptions of [source reference], there is a short exact sequence 0 ⟶ fracim D_t+1im D_t ⟶ coker D_t oversetq_t⟶ coker D_t+1 ⟶0. Consequently, o_t-o_t+1 = dimfracim D_t+1im D_t.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a34910944de266ad","tex":"\\odot","aliases":"odot","display":null,"kind":"use","excerpt":"The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ)","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-737f27da7401491a","tex":"\\one^\\top","aliases":"one^top 1^top one top mathbf top","display":"{\\mathbf{1}}^\\top","kind":"use","excerpt":"[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-439eaf0d4b476b49","tex":"\\one_m","aliases":"one_m 1_m one mathbf","display":"{\\mathbf{1}}_m","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-106a86b45be386c3","tex":"\\mathrm{open}","aliases":"open mathrm","display":"\\mathrm{open}","kind":"use","excerpt":"G_open · Open-set recursion · Conservative grammar extension, prior-proof replay, novel externally validated defect families","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e999d9eeac964cac","tex":"\\mathrm{oracle}","aliases":"oracle mathrm","display":"\\mathrm{oracle}","kind":"use","excerpt":"The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eababc3bacd32d42","tex":"\\mathrm{ord}","aliases":"ord mathrm","display":"\\mathrm{ord}","kind":"use","excerpt":"Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ea26d2394b7c0bc5","tex":"\\overline a","aliases":"overline a overline","display":"\\overline a","kind":"use","excerpt":"hed at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-54037e9a059bedc5","tex":"\\overline B","aliases":"overline B overline","display":"\\overline B","kind":"use","excerpt":"[Newton--Kantorovich repair criterion] Let F:U→ V be continuously Fr echet differentiable between Banach spaces. Let u_0∈ U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton itera","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7079e3832479ea24","tex":"\\overline D","aliases":"overline D overline","display":"\\overline D","kind":"use","excerpt":"Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ef9b8773ef8cd352","tex":"\\overline D_n","aliases":"overline D_n overline","display":"\\overline D_n","kind":"use","excerpt":"Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d472066b913c5d84","tex":"\\overline F","aliases":"overline F overline","display":"\\overline F","kind":"use","excerpt":"When the exact envelope is unknown, certified lower and upper oracle bounds underline F_K(R) ≤ F_K(R) ≤ overline F_K(R) produce corresponding bounds on χ_R. Audit-oracle arms are finite empirical approximations to this idea, but they must remain inadmissible for selection because they use final audit outcomes.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-53a254835bd02816","tex":"\\overline F_K","aliases":"overline F_K overline","display":"\\overline F_K","kind":"use","excerpt":"When the exact envelope is unknown, certified lower and upper oracle bounds underline F_K(R) ≤ F_K(R) ≤ overline F_K(R) produce corresponding bounds on χ_R. Audit-oracle arms are finite empirical approximations to this idea, but they must remain inadmissible for selection because they use final audit outcomes.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9b0ad0f92e6daea0","tex":"\\overline G","aliases":"overline G overline","display":"\\overline G","kind":"use","excerpt":"The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are overline G_rm parent · =0.167417, overline G_rm child · =0.272416, overline G_rm raw · =0.254427. Thus overline G_rm child-overline G_rm parent=0.104999 and overline G_rm child-overline G_rm raw=0.017989. The fi","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-90a9c30093a19c57","tex":"\\overline G_{\\rm child}","aliases":"overline G_rm child overline rm","display":"\\overline G_{\\rm child}","kind":"use","excerpt":"The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are overline G_rm parent · =0.167417, overline G_rm child · =0.272416, overline G_rm raw · =0.254427. Thus overline G_rm child-overline G_rm parent=0.104999 and overline G_rm child-overline G_rm raw=0.017989. The fi","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-984e9f13f9341e7f","tex":"\\overline G_{\\rm parent}","aliases":"overline G_rm parent overline rm","display":"\\overline G_{\\rm parent}","kind":"use","excerpt":"The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are overline G_rm parent · =0.167417, overline G_rm child · =0.272416, overline G_rm raw · =0.254427. Thus overline G_rm child-overline G_rm parent=0.104999 and overline G_rm child-overline G_rm raw=0.017989. The fi","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d6b8121ccfcd0422","tex":"\\overline G_{\\rm raw}","aliases":"overline G_rm raw overline rm","display":"\\overline G_{\\rm raw}","kind":"use","excerpt":"The child is developed on eight representation families and audited on four disjoint families. The frozen architecture-foreign metrics are overline G_rm parent · =0.167417, overline G_rm child · =0.272416, overline G_rm raw · =0.254427. Thus overline G_rm child-overline G_rm parent=0.104999 and overline G_rm child-overline G_rm raw=0.017989. The fi","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-85cd1e1d17664467","tex":"\\overline j","aliases":"overline j overline","display":"\\overline j","kind":"use","excerpt":"[Induced obstruction transport under ontology expansion] Equation [source reference] induces a linear map overline j_t:coker D_t→coker D_t+1, [v]⟼[j_Vv]. Its kernel is canonically isomorphic to keroverline j_t ≅ fracj_V(V_t)∩im D_t+1j_V(im D_t). Thus an old obstruction vanishes after expansion exactly when the enlarged repair grammar corrects its e","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-613813e2ef6bb31e","tex":"\\overline j_t","aliases":"overline j_t overline","display":"\\overline j_t","kind":"use","excerpt":"[Induced obstruction transport under ontology expansion] Equation [source reference] induces a linear map overline j_t:coker D_t→coker D_t+1, [v]⟼[j_Vv]. Its kernel is canonically isomorphic to keroverline j_t ≅ fracj_V(V_t)∩im D_t+1j_V(im D_t). Thus an old obstruction vanishes after expansion exactly when the enlarged repair grammar corrects its e","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-622810599c7ebcb0","tex":"\\overline p","aliases":"overline p overline","display":"\\overline p","kind":"use","excerpt":"hed at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1a02ef2c814691f5","tex":"\\overline R","aliases":"overline R overline","display":"\\overline R","kind":"use","excerpt":"For the architecture-foreign split, overline R_rm child · =0.012852, · Pr(R>0.05) · =0.033333, overline R_rm no risk · =0.015741, · Pr(R>0.05) · =0.058333. The risk head therefore reduces both mean and tail protected regression, at a utility cost of 0.021425 relative to the no-risk arm. This is expected from constrained selection and should not be ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c4db14bf41021bf9","tex":"\\overline R_{\\rm child}","aliases":"overline R_rm child overline rm","display":"\\overline R_{\\rm child}","kind":"use","excerpt":"For the architecture-foreign split, overline R_rm child · =0.012852, · Pr(R>0.05) · =0.033333, overline R_rm no risk · =0.015741, · Pr(R>0.05) · =0.058333. The risk head therefore reduces both mean and tail protected regression, at a utility cost of 0.021425 relative to the no-risk arm. This is expected from constrained selection and should not be ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5dea5763784e909b","tex":"\\overline R_{\\rm no\\ risk}","aliases":"overline R_rm no risk overline rm","display":null,"kind":"use","excerpt":"For the architecture-foreign split, overline R_rm child · =0.012852, · Pr(R>0.05) · =0.033333, overline R_rm no risk · =0.015741, · Pr(R>0.05) · =0.058333. The risk head therefore reduces both mean and tail protected regression, at a utility cost of 0.021425 relative to the no-risk arm. This is expected from constrained selection and should not be ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-648f38a9e8adfe52","tex":"\\overline T_0","aliases":"overline T_0 overline","display":"\\overline T_0","kind":"use","excerpt":"The sequence coker D_0 xrightarrowoverline T_0 coker D_1 xrightarrowoverline T_1 coker D_2 ⟶⋯ is the obstruction persistence module of the grammar lineage.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f84e430cc598252d","tex":"\\overline T_1","aliases":"overline T_1 overline","display":"\\overline T_1","kind":"use","excerpt":"The sequence coker D_0 xrightarrowoverline T_0 coker D_1 xrightarrowoverline T_1 coker D_2 ⟶⋯ is the obstruction persistence module of the grammar lineage.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-beffd946d87d6320","tex":"\\overline U","aliases":"overline U overline","display":"\\overline U","kind":"use","excerpt":"compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0477606163886714","tex":"\\overline U_{\\rm meta}","aliases":"overline U_rm meta overline rm","display":"\\overline U_{\\rm meta}","kind":"use","excerpt":"compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-43a8f89e4f079c54","tex":"\\overline y","aliases":"overline y overline","display":"\\overline y","kind":"use","excerpt":"Let Z be column centered and let P be a uniformly random permutation matrix independent of y. Then E_P[Z^top Py] = Z^top E_P[Py] = Z^topoverline y1 =0. The ridge coefficients are linear in Z^top Py for fixed Z.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6ca134bfb84876ed","tex":"\\overline Z","aliases":"overline Z overline","display":"\\overline Z","kind":"use","excerpt":"[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c35e5fe02d18f78f","tex":"\\overline Z_n","aliases":"overline Z_n overline","display":"\\overline Z_n","kind":"use","excerpt":"[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-95524823843282fd","tex":"\\overline\\Delta","aliases":"overlineΔ overline Delta","display":"\\overline\\Delta","kind":"use","excerpt":"…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least 1-α, EΔ_i ≥ overlineΔ_n - s","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16719d575d6501c2","tex":"\\overline\\Delta_n","aliases":"overlineΔ_n overline Delta","display":"\\overline\\Delta_n","kind":"use","excerpt":"…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with probability at least 1-α, EΔ_i ≥ overlineΔ_n - s","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-14d3d7b6e074005d","tex":"\\overline{\\bigcup_t I_t}","aliases":"overline⋃_t I_t overline bigcup","display":"\\overline{\\bigcup_t I_t}","kind":"use","excerpt":"Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1f050b5e2e9a3ef1","tex":"\\overline{\\bigcup_{t\\geq0}I_t}","aliases":"overline⋃_t≥0I_t overline bigcup geq","display":"\\overline{\\bigcup_{t\\geq0}I_t}","kind":"use","excerpt":"[Monotone obstruction convergence] For every defect r, the sequence η_t(r) is nonincreasing and η_t(r) ⟶ dist(-r,overline⋃_t≥0I_t).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0e517349dec3b6ba","tex":"\\overset","aliases":"overset","display":null,"kind":"use","excerpt":"[Exact sequence of newly correctable directions] Under the assumptions of [source reference], there is a short exact sequence 0 ⟶ fracim D_t+1im D_t ⟶ coker D_t oversetq_t⟶ coker D_t+1 ⟶0. Consequently, o_t-o_t+1 = dimfracim D_t+1im D_t.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ef958519c2ac9923","tex":"\\mathbb P","aliases":"P mathbb","display":"\\mathbb P","kind":"use","excerpt":"[Static diagnostic indistinguishability] Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b95fc3463e504f17","tex":"P^{\\mathsf T}","aliases":"P^ T mathsf","display":"P^{\\mathsf T}","kind":"use","excerpt":"Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e57276c69cbfb04f","tex":"p^{-1}","aliases":"p^-1","display":"p^{-1}","kind":"use","excerpt":"Every loop γ at w corresponds to the conjugate loop p^-1γ p at v, with H_γ=T_pH_p^-1γ pT_p^-1. If x∈ E_v^flat, then H_γ T_px=T_pH_p^-1γ px=T_px, so T_px∈ E_w^flat. Apply the same argument to p^-1 for the reverse inclusion.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-44ddf871c8e79a19","tex":"p^j","aliases":"p^j","display":"p^j","kind":"use","excerpt":"If random anchor membership is Bernoulli with affected probability p, the idealized full-rank probability at k=16 is therefore Pr(rank X=16) = Pr(N_1≥8) = ∑_j=8^1616choose jp^j(1-p)^16-j under the generic model. The observed parent full-rank rate on the primary audit set is 0.03125, consistent with a low-probability affected-count event.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-307e9035bbb7b4df","tex":"P^\\top","aliases":"P^top top","display":"P^\\top","kind":"use","excerpt":"Let P be the recovered repair packet, let z=P^top x, and consider a target correction F(x) = Ax+b+(z^top Bz)v, where B=B^top≠0 and v≠0.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-74522c1aa54ef2eb","tex":"p^\\star","aliases":"p^⋆ star","display":"p^\\star","kind":"use","excerpt":"Let p^⋆ be the best conditioning policy and p_0 the best inherited-policy configuration. Empirical errors of at most η imply widehatμ(p^⋆) ≥ μ(p^⋆)-η > μ(p_0)+η ≥ widehatμ(p_0).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bd85cccb55c2649d","tex":"P_{(\\im D)^\\perp}","aliases":"P_(im D)^⊥ im perp operatorname perp","display":"P_{({\\operatorname{im}} D)^\\perp}","kind":"use","excerpt":"The executable classifier applies the ordered rule Class(r,b,H) = LOCAL_OBSTRUCTION, · normP_(im D)^⊥r>τ, GLUING_OBSTRUCTION, · normP_ker B^topb>τ, EVALUATOR_HOLONOMY, · norm(H-I)e>τ, FLAT_REPAIR, · otherwise. The order is part of the receipt because a case can exhibit more than one defect. The fixture isolates the four modes, so the ordered rule i","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-15f03f5faa68df20","tex":"P_{(I\\oplus W)^\\perp}","aliases":"P_(I⊕ W)^⊥ oplus perp","display":"P_{(I\\oplus W)^\\perp}","kind":"use","excerpt":"[Optimal rank-k obstruction extension] The subspace W minimizing expected unresolved obstruction energy E_r∼ Q normP_(I⊕ W)^⊥r^2 is the span of the top k eigenvectors of Σ_ob. The minimum residual energy is the sum of the remaining eigenvalues.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-287bd2e64a3f4504","tex":"P_*","aliases":"P_*","display":"P_*","kind":"use","excerpt":"[Fixed-packet insufficiency under hidden rotation] Let P_0 be a fixed rank-r packet and P_* a target-specific rank-r context packet. Suppose the chart label c(h) is measurable from P_*h but is not almost surely measurable from P_0h. Then no separator depending only on P_0h can reproduce c(h) almost surely. In particular, a target-specific rotation ","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"Let p^⋆ be the best conditioning policy and p_0 the best inherited-policy configuration. Empirical errors of at most η imply widehatμ(p^⋆) ≥ μ(p^⋆)-η > μ(p_0)+η ≥ widehatμ(p_0).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[Exact response-signature atlas recovery] Assume the block-scalar response model and separating signatures. Then every hidden projector P_a belongs to A_R. Consequently, the nonzero minimal central projections of A_R are exactly P_1,…,P_m, up to permutation.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ac241eefcf68177e","tex":"P_{12}","aliases":"P_12","display":"P_{12}","kind":"use","excerpt":"Let R_1=P_1+P_2+3P_3, R_2=2P_1+2P_2-P_3. Blocks 1 and 2 have the same signature (1,2). Every polynomial in R_1,R_2 acts with the same scalar on H_1 and H_2. The finest response-algebra projector is therefore P_12=P_1+P_2, not the pair P_1,P_2. A compiler that outputs two separate blocks from these interventions has invented information absent from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"ponse operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generates the first two-dimensional block","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"Consider the typed repair programs p_1(a,b) · =(0,0), p_2(a,b) · =(a,0), p_3(a,b) · =(0,b), p_4(a,b) · =(b,a), p_5(a,b) · =(a,b). The complete finite verifier checks all eight episodes. Programs p_1 through p_4 each receive a counterexample and are eliminated. Program p_5 passes every episode. The loop terminates after four rejections with a verifi","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f0242a419961441b","tex":"P_3","aliases":"P_3","display":"P_3","kind":"use","excerpt":"Let R_1=P_1+2P_2+4P_3, R_2=3P_1-P_2+2P_3, where P_1,P_2,P_3 are pairwise orthogonal rank-two projectors. Take B=R_1+2R_2=7P_1+0P_2+8P_3. The three eigenvalues 7,0,8 are distinct. Therefore P_1 · =fracB(B-8I)7(7-8),","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"use","excerpt":"Consider the typed repair programs p_1(a,b) · =(0,0), p_2(a,b) · =(a,0), p_3(a,b) · =(0,b), p_4(a,b) · =(b,a), p_5(a,b) · =(a,b). The complete finite verifier checks all eight episodes. Programs p_1 through p_4 each receive a counterexample and are eliminated. Program p_5 passes every episode. The loop terminates after four rejections with a verifi","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f4f00a1c4ddb7e2f","tex":"p_4","aliases":"p_4","display":"p_4","kind":"use","excerpt":"bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ea39b3b7a10c3dfe","tex":"p_5","aliases":"p_5","display":"p_5","kind":"use","excerpt":"Consider the typed repair programs p_1(a,b) · =(0,0), p_2(a,b) · =(a,0), p_3(a,b) · =(0,b), p_4(a,b) · =(b,a), p_5(a,b) · =(a,b). The complete finite verifier checks all eight episodes. Programs p_1 through p_4 each receive a counterexample and are eliminated. Program p_5 passes every episode. The loop terminates after four rejections with a verifi","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f21822f96efaf869","tex":"P_A","aliases":"P_A","display":"P_A","kind":"use","excerpt":"The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c261d0404352ff34","tex":"p_a","aliases":"p_a","display":"p_a","kind":"use","excerpt":"of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a5c35e870d97f9c4","tex":"P_B","aliases":"P_B","display":"P_B","kind":"use","excerpt":"The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","excerpt":"Let H be a finite-dimensional real or complex Hilbert space. Suppose the unknown functional decomposition is H=⊕_a=1^mH_a, P_a:H→ H_a, with pairwise orthogonal projectors satisfying P_aP_b=δ_abP_a, ∑_a=1^mP_a=I. For controlled interventions u_1,…,u_k, let R_j∈End(H), j=1,…,k, be their finite-difference or linearized response operators.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0b67a2140e399c4e","tex":"P_C","aliases":"P_C","display":"P_C","kind":"use","excerpt":"eduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such that P_RF(","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-656ec31c34f2fcf1","tex":"P_{I^\\perp}","aliases":"P_I^⊥ perp","display":"P_{I^\\perp}","kind":"use","excerpt":"Let I=im D⊆ V be the current authorized correction image and let z=P_I^⊥r be the unresolved component of defect r. Suppose a candidate grammar extension contributes a response vector v∈ V. Only its component outside the old image is structurally new: bar v=P_I^⊥v. The expanded image is I'=I+span{v}.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-19e68e7726ae2b59","tex":"P_{I_t}","aliases":"P_I_t","display":"P_{I_t}","kind":"use","excerpt":"Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9329f3b9886daf7f","tex":"P_{I_{t+1}^\\perp}","aliases":"P_I_t+1^⊥ perp","display":"P_{I_{t+1}^\\perp}","kind":"use","excerpt":"The nested orthogonal decomposition is I_t+1 = I_t⊕(I_t+1∩ I_t^⊥). Therefore P_I_t+1^⊥r = P_I_t^⊥r - P_I_t+1∩ I_t^⊥r, and the two terms on the right are orthogonal. The Pythagorean identity gives [source reference].","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-43fe47cee6641a14","tex":"P_{I_{t+1}\\cap I_t^\\perp}","aliases":"P_I_t+1∩ I_t^⊥ cap perp","display":"P_{I_{t+1}\\cap I_t^\\perp}","kind":"use","excerpt":"[Newly correctable energy identity] For nested correction images, η_t(r)^2-η_t+1(r)^2 = normP_I_t+1∩ I_t^⊥r^2.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3a47eafdce7b1acf","tex":"P_{I_t^\\perp}","aliases":"P_I_t^⊥ perp","display":"P_{I_t^\\perp}","kind":"use","excerpt":"Let V be a finite-dimensional defect Hilbert space. Suppose a conservative grammar chain gives nested closed authorized correction images I_0⊆ I_1⊆ I_2⊆⋯⊆ V, I_t=im D_t. For r∈ V, define unresolved obstruction magnitude η_t(r) = dist(-r,I_t) = normP_I_t^⊥r.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bf4e955d1fe3c354","tex":"P_{I_\\infty}","aliases":"P_I_∞ infty","display":"P_{I_\\infty}","kind":"use","excerpt":"Nestedness gives I_t⊆ I_t+1, so distance to I_t+1 cannot exceed distance to I_t. Let I_∞=overline⋃_t I_t. The projections P_I_tr converge strongly to P_I_∞r in finite dimension, so the orthogonal residual norms converge to the distance from -r to I_∞.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f437b88587255ed0","tex":"P_{\\im D_{\\mathrm{loc}}}","aliases":"P_im D_loc im mathrm operatorname mathrm","display":"P_{{\\operatorname{im}} D_{\\mathrm{loc}}}","kind":"use","excerpt":"1 . Then P_im D_locr_loc=0, (I-P_im D_loc)r_loc=r_loc, so the obstruction norm is one.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8838865632c3a5a4","tex":"P_{\\ker B^\\top}","aliases":"P_ker B^top ker top","display":"P_{\\ker B^\\top}","kind":"use","excerpt":"The executable classifier applies the ordered rule Class(r,b,H) = LOCAL_OBSTRUCTION, · normP_(im D)^⊥r>τ, GLUING_OBSTRUCTION, · normP_ker B^topb>τ, EVALUATOR_HOLONOMY, · norm(H-I)e>τ, FLAT_REPAIR, · otherwise. The order is part of the receipt because a case can exhibit more than one defect. The fixture isolates the four modes, so the ordered rule i","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"[Exact response-signature atlas recovery] Assume the block-scalar response model and separating signatures. Then every hidden projector P_a belongs to A_R. Consequently, the nonzero minimal central projections of A_R are exactly P_1,…,P_m, up to permutation.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cce34eab9c317c7e","tex":"p_{\\mathrm{prov}}","aliases":"p_prov mathrm","display":"p_{\\mathrm{prov}}","kind":"use","excerpt":"An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6437e41273729f36","tex":"P_R","aliases":"P_R","display":"P_R","kind":"use","excerpt":"t repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d981c325ad6280a4","tex":"p_{\\mathrm{rank}}","aliases":"p_rank mathrm","display":"p_{\\mathrm{rank}}","kind":"use","excerpt":"Suppose bridge-rank identification has success probability p_rank(K_1)=1. For every child that changes only the rank-identification rule while leaving all other modules fixed, p_rank(K_2)-p_rank(K_1)≤0. This does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value σ_min=10^","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5394e4f29c2c27fe","tex":"p_{\\rm sign}","aliases":"p_rm sign rm","display":"p_{\\rm sign}","kind":"use","excerpt":"On the architecture-foreign split, the child's paired advantage over raw calibration is 0.017989, CI_95%=[0.003477,0.036611], p_rm sign=0.00940. On the seed-and-task audit, the corresponding interval includes zero. The result therefore supports robust improvement over the frozen parent on both audits and a smaller architecture-foreign improvement o","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-78aa8f72acca2b03","tex":"P_s","aliases":"P_s","display":"P_s","kind":"use","excerpt":"Let H_s(τ)∈ℝ^36 be the source model's final latent trace state and H_t(τ)∈ℝ^36 the corresponding foreign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The tran","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-12b33428971a165a","tex":"P_t","aliases":"P_t","display":"P_t","kind":"use","excerpt":"Let V be a Hilbert defect space, let L_t=im D_t^auth, and let P_t be the orthogonal projector onto L_t. For a defect distribution μ, define the correctable energy E_t^cor(μ) = E_r∼μnormP_t r^2 and the obstruction energy E_t^ob(μ) = E_r∼μnorm(I-P_t)r^2.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fa1ae76a27248a1e","tex":"P_{t+1}","aliases":"P_t+1","display":"P_{t+1}","kind":"use","excerpt":"Nested subspaces give the orthogonal decomposition L_t+1=L_t⊕ W, P_t+1=P_t+P_W. The two projected components are orthogonal, so normP_t+1r^2 = normP_tr^2+normP_Wr^2. The obstruction identity follows from Pythagoras. Under exact correction, the post-repair residual is (I-P_t)r, so loss reduction is normP_tr^2.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5e5559aba633def6","tex":"P_t^{\\mathrm{discovery}}","aliases":"P_t^discovery mathrm","display":"P_t^{\\mathrm{discovery}}","kind":"use","excerpt":"ive embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Both quantities may be positive. The total represented regret may increase because the lineage learned ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7961fc847fdf2a4d","tex":"P_t^{\\mathrm{repair}}","aliases":"P_t^repair mathrm","display":"P_t^{\\mathrm{repair}}","kind":"use","excerpt":"eal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^discovery = normn_t+1^2. Bo","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0c723e0565a0c218","tex":"P_W","aliases":"P_W","display":"P_W","kind":"use","excerpt":"[Structural-to-behavioral identity] Assume L_t⊆ L_t+1 and let W=L_t+1∩ L_t^⊥. Then E_t+1^cor(μ)- E_t^cor(μ) = E_r∼μnormP_Wr^2 and E_t^ob(μ)- E_t+1^ob(μ) = E_r∼μnormP_Wr^2. Under quadratic residual loss and exact cancellation of the correctable component, this quantity is exactly the expected additional task-loss reduction made available by the expa","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ec4a70cc8a69449e","tex":"P_x","aliases":"P_x","display":"P_x","kind":"use","excerpt":"A coordinate-level decoder is not the identifiability target. Let P_x be a projector onto a trace packet discovered from declared interventions. Packet necessity and sufficiency are evaluated by removing P_xZ(x), retaining only P_xZ(x), and comparing equal-rank random and PCA controls. The packet is accepted only when its causal effect transfers to","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"use","excerpt":"dentification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-392faed71557f8bf","tex":"p_\\theta","aliases":"p_θ theta","display":"p_\\theta","kind":"use","excerpt":"A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-097790b25abe260f","tex":"\\operatorname{Par}","aliases":"Par operatorname","display":"\\operatorname{Par}","kind":"use","excerpt":"A finding is represented by f_i = (Π_i^-,Π_i^+,code_i, Par(i),e_i,o_i), where e_i is the observed effect profile and o_i is either a surviving obstruction or a validated no-go certificate. Directed edges are typed as extension, ablation, refutation, transport, or replication. A proposal must cite the archive nodes on which its mechanism depends.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5dec399dc82b73a1","tex":"\\mathrm{parent}","aliases":"parent mathrm","display":"\\mathrm{parent}","kind":"use","excerpt":"Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-73f82ac4ea06ec53","tex":"\\mathrm{passive}","aliases":"passive mathrm","display":"\\mathrm{passive}","kind":"use","excerpt":"From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7cb22ce2f9eb5986","tex":"\\Pcal^+","aliases":"Pcal^+ P^+ Pcal mathcal","display":"{\\mathcal{P}}^+","kind":"use","excerpt":"Let p∈P^+ be a candidate experiment with response y_p:T⟶ℝ. For threshold θ, define the refined diagnostic map ϕ_p,θ(τ) = (ϕ_0(τ),low), · y_p(τ)≤θ,","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4e1425f9382e8449","tex":"\\Pcal_{\\Gcal}","aliases":"Pcal_Gcal P_G Pcal Gcal mathcal mathcal","display":"{\\mathcal{P}}_{{\\mathcal{G}}}","kind":"use","excerpt":"Let P_G be a finite typed repair-program grammar and let Verify(p,x)∈{0,1} be a complete verifier on a finite declared episode domain.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2bc62b0e81a126f2","tex":"\\Pcal_\\tau","aliases":"Pcal_τ P_τ Pcal tau mathcal tau","display":"{\\mathcal{P}}_\\tau","kind":"use","excerpt":"Let T be a research-problem space. A problem τ∈T contains a baseline repair process K_τ, a declared candidate grammar P_τ, a cheap evidence channel, a scored evaluator portfolio, a hidden protected audit, integrity rules, and a resource budget B.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-653469fa0e7897d6","tex":"\\mathsf{Penalty}","aliases":"Penalty mathsf","display":"\\mathsf{Penalty}","kind":"use","excerpt":"positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compil","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-306007edb2fcb139","tex":"\\mathsf{Penalty}_{\\mathrm{dynamic}}","aliases":"Penalty_dynamic mathsf mathrm","display":"\\mathsf{Penalty}_{\\mathrm{dynamic}}","kind":"use","excerpt":"positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compil","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aa12f7c771d774e8","tex":"\\mathrm{pkg}","aliases":"pkg mathrm","display":"\\mathrm{pkg}","kind":"use","excerpt":"G_pkg · Package integrity · Exact manifest, canonical digests, clean extraction, successful verifier","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fd38155513318b92","tex":"\\mathrm{planner}","aliases":"planner mathrm","display":"\\mathrm{planner}","kind":"use","excerpt":"A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f1dd2bd1079bcf86","tex":"\\Post","aliases":"Post operatorname","display":"{\\operatorname{Post}}","kind":"use","excerpt":"Let S be a finite set of complete lineage states. Write sxrightarrowτ_imps' when one authorized improvement transaction can take s to s'. Define Post_imp(s) = sets'∈ S:sxrightarrowτ_imps'.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7b58e4a17332ff82","tex":"\\Post_{\\mathrm{imp}}","aliases":"Post_imp Post mathrm operatorname mathrm","display":"{\\operatorname{Post}}_{\\mathrm{imp}}","kind":"use","excerpt":"Let S be a finite set of complete lineage states. Write sxrightarrowτ_imps' when one authorized improvement transaction can take s to s'. Define Post_imp(s) = sets'∈ S:sxrightarrowτ_imps'.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"· -overline L( 2overlineε_r +overlineε_Dnormu_A +overlineε_leak +overlineε_hol ). Then Pr(g_B≥underline g_B)≥1-α. In particular, underline g_B>0 is a valid one-sided certificate of beneficial target transfer.","count":22},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b1b648668f18877f","tex":"\\Pr_{A\\sim\\Dcal_{\\rm for}}","aliases":"Pr_A∼Dcal_rm for Pr_A∼D_rm for Pr sim Dcal rm Pr sim mathcal rm","display":"\\Pr_{A\\sim{\\mathcal{D}}_{\\rm for}}","kind":"use","excerpt":"Define the finite structural coverage functional C_rm id( P) = Pr_A∼D_rm for [ ank X_A=d_s]. Then widehat C_rm id( P_1) - widehat C_rm id( P_0) = 0.96875. This is the measured bridge-identification direction added by the meta-repair.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-003001ebf5867324","tex":"\\prec","aliases":"prec","display":null,"kind":"use","excerpt":"Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-34eeac828633b78e","tex":"\\mathrm{prepare}","aliases":"prepare mathrm","display":"\\mathrm{prepare}","kind":"use","excerpt":"The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0fc0d39f19d365b3","tex":"\\mathrm{preserve}","aliases":"preserve mathrm","display":"\\mathrm{preserve}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fca2be9bdee21b1b","tex":"\\mathrm{primary}","aliases":"primary mathrm","display":"\\mathrm{primary}","kind":"use","excerpt":"The primary paired effect is widehatΔ_Γ^primary = 0.067227, CI_95% = [0.035881,0.103274]. The sign-flip estimate is 9.9999×10^-6. The child improves 21.875% of pairs, ties on 75%, and loses on 3.125%.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-44c9d925f20657d9","tex":"\\mathrm{program}","aliases":"program mathrm","display":"\\mathrm{program}","kind":"use","excerpt":"Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child mi","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c67ad3a63196819","tex":"\\mathrm{prov}","aliases":"prov mathrm","display":"\\mathrm{prov}","kind":"use","excerpt":"An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a9fefa5da21f189f","tex":"\\mathrm{provenance}","aliases":"provenance mathrm","display":"\\mathrm{provenance}","kind":"use","excerpt":"A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9d46fe396bc99b0f","tex":"\\ProvValid","aliases":"ProvValid mathrm","display":"{\\mathrm{ProvValid}}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aa6aa769eb0ab4b5","tex":"Q^{\\mathsf T}","aliases":"Q^ T mathsf","display":"Q^{\\mathsf T}","kind":"use","excerpt":"The objective has value zero at Q. Full column rank implies that XT=Y=XQ only when T=Q. Orthogonality gives QQ^ T=I.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6339057791df9dbc","tex":"q^{-1}","aliases":"q^-1","display":"q^{-1}","kind":"use","excerpt":"[Path independence on the flat repair core] Let p and q be two exact invertible transport paths from A to B. If u∈ U_A is fixed by the loop holonomy associated with q^-1p, then T_U^p u=T_U^q u. In particular, every u∈ U_A^flat has path-independent transport to B.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"q · 0 , 0<q<1, so Q_0=I_2 and normEQ_0=q. Then D^-1=(I_2+E)^-1 =frac11-q^2 1 · -q -q · 1 . For r=e_1, the exact repair is u^*=-D^-1r =-frac11-q^2 1","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6386a60ab12436a6","tex":"q^k","aliases":"q^k","display":"q^k","kind":"use","excerpt":"u-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnormu_1-u_0.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4e7ba17cd328d79a","tex":"q^{n+1}","aliases":"q^n+1","display":"q^{n+1}","kind":"use","excerpt":"ht inverse Q =Q_0(I_V+EQ_0)^-1 =Q_0∑_k=0^∞(-EQ_0)^k. Moreover, norm Q≤fracnormQ_01-q. For defect r, the truncated packet-correction series u^(n) :=-Q_0∑_k=0^n(-EQ_0)^kr has residual Du^(n)+r=(-EQ_0)^n+1r, and hence normDu^(n)+r≤ q^n+1norm r.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dd3d9688e68cf63f","tex":"q^t","aliases":"q^t","display":"q^t","kind":"use","excerpt":"Lyapunov contraction] Suppose there exist w∈ℝ_>0^7 and q∈[0,1) such that for every reachable r_t, the controller can choose an authorized action a_t satisfying w^top A_a_tr_t ≤ q w^top r_t. Then every controlled trajectory satisfying [source reference] obeys w^top r_t ≤ q^t w^top r_0 + ∑_k=0^t-1 q^t-1-kw^top e_k.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2421480e4730e3b1","tex":"q^{t-1-k}","aliases":"q^t-1-k","display":"q^{t-1-k}","kind":"use","excerpt":"Lyapunov contraction] Suppose there exist w∈ℝ_>0^7 and q∈[0,1) such that for every reachable r_t, the controller can choose an authorized action a_t satisfying w^top A_a_tr_t ≤ q w^top r_t. Then every controlled trajectory satisfying [source reference] obeys w^top r_t ≤ q^t w^top r_0 + ∑_k=0^t-1 q^t-1-kw^top e_k.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e7d480dd8714698","tex":"Q^\\top","aliases":"Q^top top","display":"Q^\\top","kind":"use","excerpt":"[Within-packet gauge invariance] If packet coordinates change by an orthogonal basis transformation z'=Q^top z, then the class of functions in [source reference] is unchanged.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"use","excerpt":"For quorums Q_1,Q_2, absQ_1∩ Q_2 ≥ 2q-m > f. At most f replicas are compromised, so at least one member of the intersection is uncompromised.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probabil","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eda0b1a24e71fdef","tex":"Q_2","aliases":"Q_2","display":"Q_2","kind":"use","excerpt":"For quorums Q_1,Q_2, absQ_1∩ Q_2 ≥ 2q-m > f. At most f replicas are compromised, so at least one member of the intersection is uncompromised.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"[Semantic underidentification] Suppose two semantic maps q_1,q_2:H→ Z induce exactly the same compiler-visible future and intervention laws, but there exists a protected witness W such that W(q_1(h))≠ W(q_2(h)) with positive probability. No compiler measurable with respect to the visible evidence can validly choose between q_1 and q_2 with probabil","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5ca75c044055616d","tex":"Q_a","aliases":"Q_a","display":"Q_a","kind":"use","excerpt":"The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-614def494160b266","tex":"q_A","aliases":"q_A","display":"q_A","kind":"use","excerpt":"The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-297dc5c297f94e92","tex":"Q_a^\\top","aliases":"Q_a^top top","display":"Q_a^\\top","kind":"use","excerpt":"The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-10ed15a1ba47f564","tex":"Q_{\\mathrm{ans}}","aliases":"Q_ans mathrm","display":"Q_{\\mathrm{ans}}","kind":"use","excerpt":"hed at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and ","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c8ef5ef2a4fa13a8","tex":"q_B","aliases":"q_B","display":"q_B","kind":"use","excerpt":"The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a38917392d30a86","tex":"q_c","aliases":"q_c","display":"q_c","kind":"use","excerpt":"For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a unive","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-da0cbb93ec7bf500","tex":"Q_{\\mathrm{foreign}}","aliases":"Q_foreign mathrm","display":"Q_{\\mathrm{foreign}}","kind":"use","excerpt":"A structural metaproductivity claim additionally requires E_t+1(Q_foreign) < E_t(Q_foreign) and a causal ablation showing that the new schema is responsible for the reduction.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5950b0378700bfba","tex":"Q_i","aliases":"Q_i","display":"Q_i","kind":"use","excerpt":"Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"use","excerpt":"[Finite ε-Pareto cover] Let Z⊆[0,1]^d be finite and let 0<ε≤1. There exists a subset Q⊆ Z with absQ ≤ lceilfrac1εrceil^d such that for every z∈ Z, some q∈ Q satisfies q_i≥ z_i-ε for every i.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ecfc6d63a3b68fb8","tex":"q_j","aliases":"q_j","display":"q_j","kind":"use","excerpt":"Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applie","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6e25d180dd318b02","tex":"Q_{j-1}","aliases":"Q_j-1","display":"Q_{j-1}","kind":"use","excerpt":"The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the stand","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-586ac19e91cf213b","tex":"Q_{j-1}^\\top","aliases":"Q_j-1^top top","display":"Q_{j-1}^\\top","kind":"use","excerpt":"The descendant policy uses the source intervention-defined packet to divide the unlabeled source-side candidate pool into high- and low-packet-activation strata. It selects eight anchors from each stratum. Within each stratum it applies a deterministic pivoted residual rule: i_j ∈ arg max_i∉ I_j-1 norm(I-Q_j-1Q_j-1^top)z_i^2, where z_i is the stand","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e15bd658e05b8d9b","tex":"q_k","aliases":"q_k","display":"q_k","kind":"use","excerpt":"ited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the preactivation by q_k ⟼ q_kodot(1+frac12tanhγ_k(e)) +β_k(e). Th","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3406acd3f0f64ef4","tex":"Q_m","aliases":"Q_m","display":"Q_m","kind":"use","excerpt":"[Exhaustive finite endpoint theorem] Let M and widehat M be finite, exactly related, and let Q_1,…,Q_m be decidable state or transition predicates. Exhaustive enumeration of all reachable executable states and authorized edges decides whether each Q_i holds over the realized finite model.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9608cda5a4834bd1","tex":"Q_{\\mathrm{prog}}","aliases":"Q_prog mathrm","display":"Q_{\\mathrm{prog}}","kind":"use","excerpt":"hed at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline amax{overline o,overline c,ε}. The frozen ratio thresholds are 0.15 and ","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3a277690d006180b","tex":"Q_{\\rm ans}","aliases":"Q_rm ans rm","display":"Q_{\\rm ans}","kind":"use","excerpt":"The canonical final audits are: [H] centering small resizeboxtextwidth!% @llrrrrrrrl@ Architecture · Split · Clean · Program cost · Q_rm prog · Q_rm ans · Probe R^2 · Halt ρ · Execution · Status","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-60675fc9edf02112","tex":"Q_{\\rm prog}","aliases":"Q_rm prog rm","display":"Q_{\\rm prog}","kind":"use","excerpt":"The canonical final audits are: [H] centering small resizeboxtextwidth!% @llrrrrrrrl@ Architecture · Split · Clean · Program cost · Q_rm prog · Q_rm ans · Probe R^2 · Halt ρ · Execution · Status","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d70d66d45c49fb6a","tex":"Q_s","aliases":"Q_s","display":"Q_s","kind":"use","excerpt":"[Proof sketch] A generic linear combination of the commuting operators has simple spectrum. Its eigenvectors recover the gauge columns up to sign and permutation. Joint response signatures identify the common permutation; operator spectra identify an anonymous intervention permutation; and signed third moments orient the one-dimensional eigendirect","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6fd3920b0502031c","tex":"Q_t","aliases":"Q_t","display":"Q_t","kind":"use","excerpt":"Let Q_t⊆ V_t be a protected obstruction subspace. A dynamic grammar transport is κ-faithful on Q_t when normT_t^Vq≥κnormq for all q∈ Q_t. This extends the obstruction-faithfulness requirement from one bridge to an entire grammar lineage.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"[Authorized obstruction-shrinkage theorem] Suppose U_t^auth⊆ U_t+1^auth and restrD_t+1U_t^auth=D_t. Then im D_t⊆im D_t+1, there is a canonical surjection q_t:coker D_t↠coker D_t+1, [v]_im D_t⟼[v]_im D_t+1, and dimcoker D_t+1≤dimcoker D_t. The inequality is strict if and only if im D_tsubsetneqim D_t+1.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c9affce22ac0db2b","tex":"Q_t^\\top","aliases":"Q_t^top top","display":"Q_t^\\top","kind":"use","excerpt":"[Proof sketch] A generic linear combination of the commuting operators has simple spectrum. Its eigenvectors recover the gauge columns up to sign and permutation. Joint response signatures identify the common permutation; operator spectra identify an anonymous intervention permutation; and signed third moments orient the one-dimensional eigendirect","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d591f47f078d0a89","tex":"Q_Z","aliases":"Q_Z","display":"Q_Z","kind":"use","excerpt":"[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f99d57ee2f34e27d","tex":"q_\\theta","aliases":"q_θ theta","display":"q_\\theta","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),ε}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6cb0c499ab9636e4","tex":"Q_\\omega","aliases":"Q_ω omega","display":"Q_\\omega","kind":"use","excerpt":"Each executable operation has a Hoare-style contract {C_ω} ω {Q_ω}. Syntactic substitution ω∘(ω_1,…,ω_n) is semantically authorized only when the inner postconditions and bridge guarantees imply the outer precondition.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-38bce88aba8a9476","tex":"Q_{\\omega_i}","aliases":"Q_ω_i omega","display":"Q_{\\omega_i}","kind":"use","excerpt":"[Compositional receipt soundness] Assume: item every primitive receipt is semantically sound; item execution of each ω_i from C_ω_i establishes Q_ω_i; item the conjunction of the inner postconditions and bridge guarantees implies C_ω; item execution of ω from C_ω establishes Q_ω; item receipt composition preserves provenance and protected evidence ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e4c2be021155617f","tex":"\\mathrm{quad}","aliases":"quad mathrm","display":"\\mathrm{quad}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-23314a7b0f3cd44a","tex":"\\mathrm{query}","aliases":"query mathrm","display":"\\mathrm{query}","kind":"use","excerpt":"Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Neve","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"use","excerpt":"[Monotonicity of restricted envelopes] If R⊆ R'⊆ N, then F_K(R)≤ F_K(R').","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e4de665ad6755c75","tex":"\\R^{10}","aliases":"R^10 ℝ^10 R mathbb","display":"{\\mathbb{R}}^{10}","kind":"use","excerpt":"bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9b6e6d7f4cb095b8","tex":"\\R^{16}","aliases":"R^16 ℝ^16 R mathbb","display":"{\\mathbb{R}}^{16}","kind":"use","excerpt":"Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x) s(x)ψ(x) ∈ℝ^16.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-00c7fd86dfdbd6bd","tex":"\\R^{16\\times8}","aliases":"R^16×8 ℝ^16×8 R times mathbb times","display":"{\\mathbb{R}}^{16\\times8}","kind":"use","excerpt":"Let E=[A B]:ℝ^16→ℝ^16 with A,B∈ℝ^16×8 and let ϕ(x)=ψ(x)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-180cc50b86fd4b47","tex":"\\R^{19}","aliases":"R^19 ℝ^19 R mathbb","display":"{\\mathbb{R}}^{19}","kind":"use","excerpt":"Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following c","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"thout weakening its gates. After a precommitted compiler revision and fresh final-task generation, the selected full-binding child is FULL on both 1600-task primary and shifted-generator audits while the parent remains PARTIAL. Primary and foreign program-specificity gains are 0.010714 and 0.024763, with frozen intervals [0.009911,0.011527] and [0.","count":75},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cd6a51568c71e4ad","tex":"\\R^2","aliases":"R^2 ℝ^2 R mathbb","display":"{\\mathbb{R}}^2","kind":"use","excerpt":"[Shear holonomy] Let E=ℝ^2 and suppose evaluator evolution around one loop produces H= 1 · α","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9b55e093b405114c","tex":"\\R^{23}","aliases":"R^23 ℝ^23 R mathbb","display":"{\\mathbb{R}}^{23}","kind":"use","excerpt":"Each candidate receipt z_e,c is mapped to a feature vector x(z_e,c)∈ℝ^23. The released feature schema contains: item a constant coordinate; item logarithmic anchor and calibration sizes; item target representation width; item alignment, transport-condition, and repair-equivariance diagnostics; item packet residual and relative repair magnitude; ite","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1523811833b5e4c0","tex":"R^2_{\\mathrm{future}}","aliases":"R^2_future mathrm","display":"R^2_{\\mathrm{future}}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-311080e714276ff6","tex":"R^2_{\\mathrm{intervention}}","aliases":"R^2_intervention mathrm","display":"R^2_{\\mathrm{intervention}}","kind":"use","excerpt":"full identity, equal-rank random }. For each nontrivial constructor, ranks one through six are considered. The full identity is an oversized control. A projection W is valid only when, on both recurrent systems, R^2_future · ≥0.82, R^2_intervention · ≥0.90, Acc_answer · ≥0.78, D_twin · ≤0.22. Among valid candidates, the compiler minimizes rank and ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-405b078847cb594e","tex":"R^2_{\\rm packet}","aliases":"R^2_rm packet rm","display":"R^2_{\\rm packet}","kind":"use","excerpt":"For reconstructive candidates, admissibility requires R^2_rm sem≥0.68, R^2_rm packet≥0.72, NRMSE_rm packet≤0.55. A quotient is admissible only when the observed dimension ratio is at most 0.82, semantic R^2 is at least 0.68, and no reconstructive candidate reaches packet R^2=0.82. A no-go certificate is admissible only after the compiler discovers ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1baea9c715dfe96c","tex":"R^2_{\\rm sem}","aliases":"R^2_rm sem rm","display":"R^2_{\\rm sem}","kind":"use","excerpt":"For reconstructive candidates, admissibility requires R^2_rm sem≥0.68, R^2_rm packet≥0.72, NRMSE_rm packet≤0.55. A quotient is admissible only when the observed dimension ratio is at most 0.82, semantic R^2 is at least 0.68, and no reconstructive candidate reaches packet R^2=0.82. A no-go certificate is admissible only after the compiler discovers ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5540ca7a92781e1f","tex":"R^2_{\\mathrm{transport}}","aliases":"R^2_transport mathrm","display":"R^2_{\\mathrm{transport}}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2dae3ecd522688de","tex":"R^2_{\\mathrm{worst\\ chart}}","aliases":"R^2_worst chart mathrm","display":"R^2_{\\mathrm{worst\\ chart}}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a74d522dddb392e9","tex":"\\R^{32}","aliases":"R^32 ℝ^32 R mathbb","display":"{\\mathbb{R}}^{32}","kind":"use","excerpt":"Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-815bb8e05660a348","tex":"\\R^{36}","aliases":"R^36 ℝ^36 R mathbb","display":"{\\mathbb{R}}^{36}","kind":"use","excerpt":"Let e=E(ϕ(τ))∈ℝ^36 be the encoded task representation, let z_0∈ℝ^36 be a learned seed, and let s_k∈ℝ^36 be the step embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-72b3b4ee79240b62","tex":"\\R^4","aliases":"R^4 ℝ^4 R mathbb","display":"{\\mathbb{R}}^4","kind":"use","excerpt":"Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-44306f114fa81f70","tex":"\\R^6","aliases":"R^6 ℝ^6 R mathbb","display":"{\\mathbb{R}}^6","kind":"use","excerpt":"Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5a6459f38651b8b8","tex":"\\R^{64}","aliases":"R^64 ℝ^64 R mathbb","display":"{\\mathbb{R}}^{64}","kind":"use","excerpt":"Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-551b16828256a1bc","tex":"\\R^{6\\times2}","aliases":"R^6×2 ℝ^6×2 R times mathbb times","display":"{\\mathbb{R}}^{6\\times2}","kind":"use","excerpt":"Let the true context packet be P_*=UU^ T for a target-specific orthonormal matrix U∈ℝ^6×2, while the inherited packet is P_0=diag(1,1,0,0,0,0). Let the chart label be c(h)=1{e_1^ TU^ Th>0}. For generic U, c(h) is not measurable from P_0h. No threshold or nonlinear classifier using only P_0h can recover the chart perfectly. The response atlas can re","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5dfcf710c77db7ff","tex":"\\R^{6\\times6}","aliases":"R^6×6 ℝ^6×6 R times mathbb times","display":"{\\mathbb{R}}^{6\\times6}","kind":"use","excerpt":"Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9d8a6ecda587180a","tex":"\\R^{80}","aliases":"R^80 ℝ^80 R mathbb","display":"{\\mathbb{R}}^{80}","kind":"use","excerpt":"Each episode contains two independently gauged recurrent systems. A trajectory has five latent steps, four task-relevant semantic coordinates, and twelve nuisance coordinates per step. The observed flattened trajectory therefore lies in ℝ^80. The semantic dynamics, gauges, nuisance processes, and hidden audit coordinates are not passed to the quoti","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6e42d74a72040d0b","tex":"\\R^d","aliases":"R^d ℝ^d R mathbb","display":"{\\mathbb{R}}^d","kind":"use","excerpt":"[Full-rank bridge identification] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7237fd2e05f92d03","tex":"\\R^{d_a}","aliases":"R^d_a ℝ^d_a R mathbb","display":"{\\mathbb{R}}^{d_a}","kind":"use","excerpt":"An agent a has encoder E_a:ℝ^16→ℝ^d_a and an affine head fitted to widetilde q. Its hidden state is h_a(x)=E_a(ϕ(x)). The transport system is not given the matrix or nonlinear parameters of E_a.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-de10bb2fde077bce","tex":"\\R^{d_a\\times r}","aliases":"R^d_a× r ℝ^d_a× r R times mathbb times","display":"{\\mathbb{R}}^{d_a\\times r}","kind":"use","excerpt":"Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b8e55b74bfc1dbb","tex":"\\R^{d_s}","aliases":"R^d_s ℝ^d_s R mathbb","display":"{\\mathbb{R}}^{d_s}","kind":"use","excerpt":"[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-365272fad90a3ea8","tex":"\\R^{d_t}","aliases":"R^d_t ℝ^d_t R mathbb","display":"{\\mathbb{R}}^{d_t}","kind":"use","excerpt":"[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-341f368924fefe3c","tex":"\\R^{d\\times d}","aliases":"R^d× d ℝ^d× d R times mathbb times","display":"{\\mathbb{R}}^{d\\times d}","kind":"use","excerpt":"[Rank saturation does not imply stability] For every d≥2 and every η>0, there exists a full-rank design X∈ℝ^d× d such that λ_min(X^ TX)<η.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7d89e152ed11df2d","tex":"\\R^{d\\times r}","aliases":"R^d× r ℝ^d× r R times mathbb times","display":"{\\mathbb{R}}^{d\\times r}","kind":"use","excerpt":"Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6753f131ee0ca092","tex":"\\R^{k\\times d}","aliases":"R^k× d ℝ^k× d R times mathbb times","display":"{\\mathbb{R}}^{k\\times d}","kind":"use","excerpt":"[Full-rank bridge identification] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-282c0afc3ab7b21f","tex":"\\R^m","aliases":"R^m ℝ^m R mathbb","display":"{\\mathbb{R}}^m","kind":"use","excerpt":"[Full-rank bridge identification] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-31b167620b12b40e","tex":"\\mathbb R^{m}","aliases":"R^m mathbb","display":"\\mathbb R^{m}","kind":"use","excerpt":"[Pooled bridge compatibility obstruction] Let V⊆ R^m be spanned by the canonical trajectories observed in two charts c_1,c_2. Suppose one linear map T exactly transports architecture b into architecture a in both charts: zG_b,c_iT=zG_a,c_i for every z∈ V, i∈{1,2}. Then the restrictions to V satisfy G_b,c_1T=G_a,c_1, G_b,c_2T=G_a,c_2. If G_b,c_i are","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-35387f6afb63474d","tex":"\\R^{m_t}","aliases":"R^m_t ℝ^m_t R mathbb","display":"{\\mathbb{R}}^{m_t}","kind":"use","excerpt":"A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-830a4178f2d5ab6a","tex":"\\R^{m_{t+1}}","aliases":"R^m_t+1 ℝ^m_t+1 R mathbb","display":"{\\mathbb{R}}^{m_{t+1}}","kind":"use","excerpt":"A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3cfcccf9a54673d6","tex":"\\R^{m\\times k}","aliases":"R^m× k ℝ^m× k R times mathbb times","display":"{\\mathbb{R}}^{m\\times k}","kind":"use","excerpt":"For a finite response model, let R∈ℝ^m× k contain the expected response signatures of k inherited obstruction components under m experiment types. Let z∈ℝ_","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d0a143c594ab3a8d","tex":"\\R^{m\\times16}","aliases":"R^m×16 ℝ^m×16 R times mathbb times","display":"{\\mathbb{R}}^{m\\times16}","kind":"use","excerpt":"For an exact linear target with hidden width sixteen, let H_s∈ℝ^m×16. When m<16, full column rank is impossible, so the bridge is not uniquely identified by ordinary least squares. At m=8, the frozen sweep reaches 0.4115 affected accuracy and the witness rejects. At m=16, random anchors are generically spanning and accuracy rises to 0.7877. By m=32","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-33da412dce4fbe74","tex":"\\R^{n\\times d}","aliases":"R^n× d ℝ^n× d R times mathbb times","display":"{\\mathbb{R}}^{n\\times d}","kind":"use","excerpt":"Let X_S∈ℝ^n× d be the source hidden-state design induced by anchor set S. Rank recovery is saturated when Pr(rank X_S=d)≥ 1-ε_rm rank under the current foreign development distribution and the remaining performance gap cannot be explained by rank failure.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-29bd2caef940f50f","tex":"\\R^{n\\times q}","aliases":"R^n× q ℝ^n× q R times mathbb times","display":"{\\mathbb{R}}^{n\\times q}","kind":"use","excerpt":"Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-312ae636ae866235","tex":"\\mathbb R^{p}","aliases":"R^p mathbb","display":"\\mathbb R^{p}","kind":"use","excerpt":"[Readout stability from trajectory reconstruction] Let J:H_a,c→ R^p be L_J-Lipschitz. If a transported trajectory satisfies normT^c_b→ ah_b,c-h_a,c ≤ε, then normJ(T^c_b→ ah_b,c)-J(h_a,c) ≤ L_Jε. If two transport paths differ by at most δ, their semantic readouts differ by at most L_Jδ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ef6114439cf5c3c","tex":"\\R^{q\\times d}","aliases":"R^q× d ℝ^q× d R times mathbb times","display":"{\\mathbb{R}}^{q\\times d}","kind":"use","excerpt":"Let the old bridge be T∈ℝ^q× d and let g be any admissible separator. The embedding T⟼(g,T,T) returns Th on both charts. Hence the local grammar can execute the inherited process without approximation. A conservative runtime may therefore use K_chart(h) = K_local(h), · if the chart certificate passes,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a9920f1a82b6cc5b","tex":"R^t_{\\sigma(u)}","aliases":"R^t_σ(u) sigma","display":"R^t_{\\sigma(u)}","kind":"use","excerpt":"For a proposed intervention correspondence σ: U→ V and transport T:H_t→ H_s, define the normalized orbit defect δ_rm orb(T,σ) = frac1| U| ∑_u∈ U fracnormTR^t_σ(u)-R_u^sT_F max{normR_u^s_F,ε}.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-599ffde3991229d8","tex":"r^\\top","aliases":"r^top top","display":"r^\\top","kind":"use","excerpt":"Let foreign residuals follow distribution Q. Define the obstruction covariance operator Σ_ob = E_r∼ Q [ P_I^⊥r r^top P_I^⊥ ]. Let W⊆ I^⊥ be a k-dimensional new response subspace.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7ba3d6d14c84ccd9","tex":"\\R^{|I|\\times|E|}","aliases":"R^|I|×|E| ℝ^|I|×|E| R times mathbb times","display":"{\\mathbb{R}}^{|I|\\times|E|}","kind":"use","excerpt":"Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cf29160b021a9578","tex":"R^\\dagger","aliases":"R^† dagger","display":"R^\\dagger","kind":"use","excerpt":"[Response-mixture recovery bound] Assume R has full column rank and normξ_2≤ε. The least-squares estimator widehat z=R^† y satisfies normwidehat z-z_2 ≤ normR^†_2ε = fracεσ_min(R).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-06affe15cffcc490","tex":"\\R_{\\\\geq0}^k","aliases":"R_ geq0^k ℝ_ geq0^k R geq mathbb geq","display":"{\\mathbb{R}}_{\\\\geq0}^k","kind":"use","excerpt":"For a finite response model, let R∈ℝ^m× k contain the expected response signatures of k inherited obstruction components under m experiment types. Let z∈ℝ_ geq0^k be a sparse mixture with 1^top z=1. An observed response vector has the form y=Rz+ξ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cf696f9c67ad92ec","tex":"\\R_+^2","aliases":"R_+^2 ℝ_+^2 R mathbb","display":"{\\mathbb{R}}_+^2","kind":"use","excerpt":"Let K=ℝ_+^2, D=I_2, and r= 1","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"-transport Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"Let H be a finite-dimensional complex Hilbert space. Suppose permitted interventions or controlled transitions produce operators R_1,…,R_m∈End(H). These may be finite-difference response maps, local transition Jacobians, or finite Koopman operators on an observable dictionary.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"0 · -1. On H=ℝ^4, define four response operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generate","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"use","excerpt":"For distinct records r_1,r_2 with protected difference d=ι_v(r_1)-ι_v(r_2)≠0, injectivity gives T_ed≠0. Exact commutation identifies this with the difference of their new representations.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d0aa3589089d58ba","tex":"R_3","aliases":"R_3","display":"R_3","kind":"use","excerpt":"0 · -1. On H=ℝ^4, define four response operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generate","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d5c7051b56820f95","tex":"r_3","aliases":"r_3","display":"r_3","kind":"use","excerpt":"Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006) · =0.105. A direct source-to-target bridge with certified residual penalty be","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0fe85010c113a2aa","tex":"R_4","aliases":"R_4","display":"R_4","kind":"use","excerpt":"0 · -1. On H=ℝ^4, define four response operators R_1=X⊕0, R_2=Z⊕0, R_3=0⊕ X, R_4=0⊕ Z. The adjoint-closed algebra they generate is M_2(ℝ)⊕ M_2(ℝ), with dimension 8. Its commutant consists of scalar multiples of the identity on each block and therefore has dimension 2. The minimal central projectors are P_1=I_2⊕0, P_2=0⊕ I_2. A residual e_1 generate","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-33462a02fcce1a84","tex":"R_a","aliases":"R_a","display":"R_a","kind":"use","excerpt":"Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cc9268a7f8b83f90","tex":"r_{\\mathrm{cor}}","aliases":"r_cor mathrm","display":"r_{\\mathrm{cor}}","kind":"use","excerpt":"[Canonical correction--obstruction decomposition] For every r∈ V: item P_cor is the orthogonal projector onto im D; item P_ob is the orthogonal projector onto ker D^*; item the defect decomposes uniquely as r=r_cor+r_ob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complet","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2e91da6834681eb7","tex":"r_{\\mathrm{flat}}","aliases":"r_flat mathrm","display":"r_{\\mathrm{flat}}","kind":"use","excerpt":"Case F3: flat repair. Set D_flat=I_2, r_flat= 1 -2 , with zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is u_flat=-r_flat.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a8c3cc1693c46092","tex":"R_g","aliases":"R_g","display":"R_g","kind":"use","excerpt":"[Gate misclassification bound] Let g be the correct chart and widehat g an estimated chart. Suppose the two target repair outputs satisfy normR_1(y)-R_0(y)≤ L throughout the evaluated region and Pr(widehat g≠ g)≤ϵ_g. Then EnormR_widehat g(y)-R_g(y)^2 ≤ ϵ_gL^2. If protected utility is L_J-Lipschitz in repaired state, then abs E J(R_widehat g(y))- E ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d7ff1c475a20db07","tex":"R_{i_1}","aliases":"R_i_1","display":"R_{i_1}","kind":"use","excerpt":"Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicti","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be4a56a88395136b","tex":"r_{i_t,t}","aliases":"r_i_t,t","display":"r_{i_t,t}","kind":"use","excerpt":"[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-785e2950f37a9a95","tex":"R_{i_\\ell}","aliases":"R_i_ℓ ell","display":"R_{i_\\ell}","kind":"use","excerpt":"Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicti","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"[Perron sensitivity] Let M be irreducible with simple Perron root ρ, positive right eigenvector r, and positive left eigenvector ℓ, normalized by ℓ^ Tr=1. Then frac∂ρ∂ M_ij = ℓ_i r_j.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c0c3971df17aae64","tex":"R_{j-1}","aliases":"R_j-1","display":"R_{j-1}","kind":"use","excerpt":"Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-40bfb3f25c9be22a","tex":"R_j^\\ast","aliases":"R_j^∗ ast","display":"R_j^\\ast","kind":"use","excerpt":"For X∈End(H), define commutator energy E_com(X) = ∑_j=1^m ( normXR_j-R_jX_F^2 + normXR_j^∗-R_j^∗ X_F^2 ). There is a positive semidefinite linear operator L_com on the Hilbert space of matrices such that anglesX,L_comX_F = E_com(X). Its kernel is the commutant: kerL_com= A'.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a64e897db39c6a7f","tex":"R_k","aliases":"R_k","display":"R_k","kind":"use","excerpt":"The algebra generated by the response family is A_R = alg^*(I,R_1,…,R_k). Because the R_j are simultaneously block scalar in the exact model, A_R is commutative and every polynomial in the responses is again block scalar.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4d5e97b2dd21ca5d","tex":"r_{k+1}","aliases":"r_k+1","display":"r_{k+1}","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8748367525be9f9f","tex":"r_{k-1}","aliases":"r_k-1","display":"r_{k-1}","kind":"use","excerpt":"widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence maps the trace to","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bb30d127030c28cd","tex":"r_{\\mathrm{loc}}","aliases":"r_loc mathrm","display":"r_{\\mathrm{loc}}","kind":"use","excerpt":"Case F1: local obstruction. Set D_loc = 1 · 0 0 · 0 , r_loc = 0 1 . Then P_im D_locr_loc=0, (I-P_im D_loc)r_loc=r_loc, so the obstruction norm is one.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-25ee15775b880083","tex":"R_m","aliases":"R_m","display":"R_m","kind":"use","excerpt":"Let H be a finite-dimensional complex Hilbert space. Suppose permitted interventions or controlled transitions produce operators R_1,…,R_m∈End(H). These may be finite-difference response maps, local transition Jacobians, or finite Koopman operators on an observable dictionary.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-272cb3ca57c00592","tex":"r_m","aliases":"r_m","display":"r_m","kind":"use","excerpt":"Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a4ff961cd74d1869","tex":"r_n","aliases":"r_n","display":"r_n","kind":"use","excerpt":"T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-35b4666b79fb5c4f","tex":"r_{\\mathrm{ob}}","aliases":"r_ob mathrm","display":"r_{\\mathrm{ob}}","kind":"use","excerpt":"[Canonical correction--obstruction decomposition] For every r∈ V: item P_cor is the orthogonal projector onto im D; item P_ob is the orthogonal projector onto ker D^*; item the defect decomposes uniquely as r=r_cor+r_ob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complet","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8747dbd5458d812b","tex":"r_{\\rm non}","aliases":"r_rm non rm","display":"r_{\\rm non}","kind":"use","excerpt":"[Calibration-to-audit witness soundness] Suppose, simultaneously, that the audit gain and regression satisfy g_rm slice · ≥ widehat g_rm slice-η_g, r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f7b985b4cff51815","tex":"R_s","aliases":"R_s","display":"R_s","kind":"use","excerpt":"Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.","count":14},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-49121c793f36cbb5","tex":"R_t","aliases":"R_t","display":"R_t","kind":"use","excerpt":"A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7d0b0aaec388130d","tex":"r_t","aliases":"r_t","display":"r_t","kind":"use","excerpt":"Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribu","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5d0744b62ca38dd","tex":"r_{t+1}","aliases":"r_t+1","display":"r_{t+1}","kind":"use","excerpt":"h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribution shift, semantic debt, or newly induced headroom.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e2194b7d7b31eb1c","tex":"R_{t,j}","aliases":"R_t,j","display":"R_{t,j}","kind":"use","excerpt":"Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ecec59c825e6516a","tex":"R_u^a","aliases":"R_u^a","display":"R_u^a","kind":"use","excerpt":"The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e68a0f7f534129a7","tex":"R_{\\widehat g}","aliases":"R_widehat g widehat","display":"R_{\\widehat g}","kind":"use","excerpt":"[Gate misclassification bound] Let g be the correct chart and widehat g an estimated chart. Suppose the two target repair outputs satisfy normR_1(y)-R_0(y)≤ L throughout the evaluated region and Pr(widehat g≠ g)≤ϵ_g. Then EnormR_widehat g(y)-R_g(y)^2 ≤ ϵ_gL^2. If protected utility is L_J-Lipschitz in repaired state, then abs E J(R_widehat g(y))- E ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f0e893f7b0a8cb70","tex":"r_x","aliases":"r_x","display":"r_x","kind":"use","excerpt":"For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ccef749844413967","tex":"R_{\\sigma^{-1}(v)}^s","aliases":"R_σ^-1(v)^s sigma","display":"R_{\\sigma^{-1}(v)}^s","kind":"use","excerpt":"[Intertwiner uniqueness modulo the response commutant] Let T_1,T_2:H_t→ H_s be invertible and satisfy T_iR_v^t=R_σ^-1(v)^sT_i for every v∈ V. Then S=T_2^-1T_1 belongs to A_t'. Hence exact intervention-orbit data identifies a transport at most up to the response commutant.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d46e09d5b6a72a7e","tex":"\\R_{\\geq0}^{7}","aliases":"R_≥0^7 ℝ_≥0^7 R geq mathbb geq","display":"{\\mathbb{R}}_{\\geq0}^{7}","kind":"use","excerpt":"Let the unresolved module-headroom vector be r_t = h_H(K_t) h_Q(K_t) h_C(K_t) h_T(K_t) h_S(K_t) h_V(K_t) h_W(K_t) ∈ℝ_≥0^7. An authorized meta-action a changes one or more repair-process modules. In a local finite model, represent its effect by r_t+1 ≤ A_a r_t+e_t, where A_a is a nonnegative response matrix and e_t records estimation error, distribu","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8e952ae709e835fa","tex":"\\R_{\\geq0}^k","aliases":"R_≥0^k ℝ_≥0^k R geq mathbb geq","display":"{\\mathbb{R}}_{\\geq0}^k","kind":"use","excerpt":"The inherited obstruction language spans a response cone K_R = {Rz:z∈ℝ_≥0^k}. For observed response y, define the open-set residual η_rm open(y) = dist(y, K_R).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aeb0ac039155885d","tex":"\\R_{\\geq0}^{m}","aliases":"R_≥0^m ℝ_≥0^m R geq mathbb geq","display":"{\\mathbb{R}}_{\\geq0}^{m}","kind":"use","excerpt":"[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2a1613bcb0226c54","tex":"\\mathrm{rank}","aliases":"rank mathrm","display":"\\mathrm{rank}","kind":"use","excerpt":"Suppose bridge-rank identification has success probability p_rank(K_1)=1. For every child that changes only the rank-identification rule while leaving all other modules fixed, p_rank(K_2)-p_rank(K_1)≤0. This does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value σ_min=10^","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-92b83602c4557972","tex":"\\Rcal^+","aliases":"Rcal^+ R^+ Rcal mathcal","display":"{\\mathcal{R}}^+","kind":"use","excerpt":"[Central refinement by a separating intervention] Suppose H_a,H_b are aliased under R, and S satisfies [source reference]. If the joint signature of every remaining block is distinct from those of H_a and H_b under R^+, then the minimal central projectors of A(R^+) separate H_a and H_b.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f567291c42caf831","tex":"\\Rcal^{+}","aliases":"Rcal^+ R^+ Rcal mathcal","display":"{\\mathcal{R}}^{+}","kind":"use","excerpt":"Let S∈End(H) be a candidate intervention response. Define the expanded language R^+=R∪{S}. The extension is conservative at the syntactic level because every old response expression remains available unchanged.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-39cb9e4faf376667","tex":"\\Rcal_{\\mathrm{valid}}","aliases":"Rcal_valid R_valid Rcal mathrm mathcal mathrm","display":"{\\mathcal{R}}_{\\mathrm{valid}}","kind":"use","excerpt":"[Representation-separation obstruction] Suppose states x_0,x_1 satisfy ϕ_G(x_0)=ϕ_G(x_1), but their externally validated repair sets are disjoint: R_valid(x_0) ∩ R_valid(x_1) = ∅. Then no G-measurable deterministic repair policy is valid on both states.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[Minimum rounds under a witness quorum] Suppose every transaction requires at least q frozen witness components, where 1≤ q<n. Then at most n-q components can be edited in one round. Editing every component at least once requires at least lceilfracnn-qrceil rounds. This bound is attained by partitioning the components into edit blocks of size at mo","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7bb1cbdb73456abe","tex":"\\rceil^d","aliases":"rceil^d rceil","display":"\\rceil^d","kind":"use","excerpt":"[Finite ε-Pareto cover] Let Z⊆[0,1]^d be finite and let 0<ε≤1. There exists a subset Q⊆ Z with absQ ≤ lceilfrac1εrceil^d such that for every z∈ Z, some q∈ Q satisfies q_i≥ z_i-ε for every i.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0dd44fa15c96aac6","tex":"\\mathrm{rep}","aliases":"rep mathrm","display":"\\mathrm{rep}","kind":"use","excerpt":"The source hidden space has dimension six and is generated from three target-specific unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. A target-specific orthogonal or injective embedding hides these blocks from the runtime coordinate system. The source repair changes only H_rep and leaves the context block invariant. Two controlled interventions act","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-131dd08f2af58ba0","tex":"\\mathrm{repair}","aliases":"repair mathrm","display":"\\mathrm{repair}","kind":"use","excerpt":"Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, addres","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-797b2ecfc8b2fd0e","tex":"\\mathrm{repair\\ inv}","aliases":"repair inv mathrm","display":"\\mathrm{repair\\ inv}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8b5cafca52b29988","tex":"\\RepairTyped","aliases":"RepairTyped mathrm","display":"{\\mathrm{RepairTyped}}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4a5bc4ec3d738895","tex":"\\mathrm{representation}","aliases":"representation mathrm","display":"\\mathrm{representation}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-de4b23c57789e583","tex":"\\mathrm{research}","aliases":"research mathrm","display":"\\mathrm{research}","kind":"use","excerpt":"Every parent and child run receives exactly B_research=6 full candidate experiments. Candidate metadata and cheap diagnostics are common inputs. The number of proposal receipts, scored evaluator calls, capability checks, and integrity-monitor calls is therefore matched.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-99ed392c769403bf","tex":"\\mathrm{Residual}","aliases":"Residual mathrm","display":"\\mathrm{Residual}","kind":"use","excerpt":"Let A be a finite architecture index set and C a finite admitted chart index set. In the released realization, A = {Elman,GRU,Residual}, C = {A0,A1,A2,A3,N0,N1}. Each pair (a,c)∈ A× C carries a full Neuralese trajectory fiber H_a,c ≅ ( R^40)^5 ≅ R^200. The observation atlas determines which context fiber is authorized. A task outside every admitted","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-552697fc2f416d27","tex":"\\mathrm{retrieve}","aliases":"retrieve mathrm","display":"\\mathrm{retrieve}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eb10cd16f042554f","tex":"\\mathrm{rev}","aliases":"rev mathrm","display":"\\mathrm{rev}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bb5c267f888d286b","tex":"\\mathrm{reveal}","aliases":"reveal mathrm","display":"\\mathrm{reveal}","kind":"use","excerpt":"The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-346662e55a5c115c","tex":"\\mathsf{reverse}","aliases":"reverse mathsf","display":"\\mathsf{reverse}","kind":"use","excerpt":"Let ℓ_i(Z) be mean squared witness error on recipient i when the four latent states are externally supplied as Z. Let Z_i be the clean trace. The structural coordinates are n_i · =ℓ_i(zero)-ℓ_i(Z_i), o_i · =ℓ_i(reverse(Z_i))-ℓ_i(Z_i), c_i · =ℓ_i(noise(Z_i))-ℓ_i(Z_i), r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at ev","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"[Fixed-margin evaluator ceiling] Let V: S→[0,1] be fixed. If every accepted transition satisfies V(A_t+1)≥ V(A_t)+Δ for a constant Δ>0, then every accepted chain has length at most lfloor 1/Δrfloor+1.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-94b0ad459adb42e7","tex":"\\mathrm{RI}","aliases":"RI mathrm","display":"\\mathrm{RI}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-968977e4d82819b0","tex":"\\rightsquigarrow","aliases":"rightsquigarrow","display":null,"kind":"use","excerpt":"Stochastic channel · K:Xrightsquigarrow Y · Distributional transport only; pointwise identity is not implied.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d6a5321a6a98ce90","tex":"\\mathrm{risky}","aliases":"risky mathrm","display":"\\mathrm{risky}","kind":"use","excerpt":"Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety o","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-86c6fe2dbe5da2d7","tex":"\\mathsf{RMD}","aliases":"RMD mathsf","display":"\\mathsf{RMD}","kind":"use","excerpt":"and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available class are selected by deterministic task","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2972a89cf966df0a","tex":"\\Rollbackable","aliases":"Rollbackable mathrm","display":"{\\mathrm{Rollbackable}}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"[Evaluator-capture triviality] Let S be a candidate set with at least two elements, and suppose the admissible evaluator class contains every function V: S→[0,1]. For every candidate B∈ S, there exists an evaluator V_B under which B is the unique maximizer.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5e769b89788d547a","tex":"s'","aliases":"s'","display":"s'","kind":"use","excerpt":"Let S be a finite set of complete lineage states. Write sxrightarrowτ_imps' when one authorized improvement transaction can take s to s'. Define Post_imp(s) = sets'∈ S:sxrightarrowτ_imps'.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"use","excerpt":"[Finite repair-discovery closure] Under discovery monotonicity, Φ_disc has a least fixed point above any seed set S_0. Iteration S_k+1=Φ_disc(S_k) stabilizes after at most absΩ-absS_0 strict expansions.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety o","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety o","count":13},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"use","excerpt":"Consider a transaction graph with states s_0,s_1,s_2. Suppose s_0→ s_1 is a validated positive parent--child edge, but every authorized successor of s_1 either returns to s_1 with zero gain or enters unsafe s_2. Then s_0→ s_1 witnesses one-step metaproductivity, but s_1 does not lie in a closure requiring infinitely repeated positive gains. A grand","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c340b0448eace6f7","tex":"s_{a^*}","aliases":"s_a^*","display":"s_{a^*}","kind":"use","excerpt":"[Finite repair-relevant block selection] Suppose there is a unique repair-relevant block a^* and a margin m>0 such that s_a^*≥ s_a+m for every a≠ a^*. If max_a|widehat s_a-s_a|<fracm2, then arg max_awidehat s_a=a^*.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4607f2a042368c20","tex":"s_{\\mathrm{base}}","aliases":"s_base mathrm","display":"s_{\\mathrm{base}}","kind":"use","excerpt":"The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a64b77f6851063e5","tex":"s_c","aliases":"s_c","display":"s_c","kind":"use","excerpt":"For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a unive","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6f3f7c3354698471","tex":"s_D","aliases":"s_D","display":"s_D","kind":"use","excerpt":"On the seed/task audit, the parent and child receive the same five candidates on 120 episodes. Let D_j be the child-minus-parent protected utility on episode j. The frozen values satisfy overline D=0.072518, s_D=0.154617, widehatrm se=0.014115. The ordinary normal approximation gives 0.072518±1.98(0.014115) ≈ [0.0446,0.1005], close to the frozen pa","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7175b00a9a59655b","tex":"s_i^T","aliases":"s_i^T","display":"s_i^T","kind":"use","excerpt":"[Fundamental-cycle holonomy criterion] For each root state s_o∈ S_o, tree transport determines a unique family (s_i^T). For every non-tree edge e=(i,j), let H_e:S_o→ S_o be transport around the corresponding fundamental cycle. The tree family extends to a globally compatible family on all edges if and only if H_e(s_o)=s_o for every non-tree edge e.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-861b470470e77a0b","tex":"s_{\\mathrm{improved}}","aliases":"s_improved mathrm","display":"s_{\\mathrm{improved}}","kind":"use","excerpt":"The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3425136155210d61","tex":"s_{\\mathrm{invalid}}","aliases":"s_invalid mathrm","display":"s_{\\mathrm{invalid}}","kind":"use","excerpt":"The minimal closure model has states S={s_base,s_improved,s_invalid} and valid set K={s_base,s_improved}. With intact rollback, authorized complete transactions have successor relation Post(s_base) ={s_base,s_improved}, Post(s_improved)={s_improved}. Failed candidates are restored to the base state before the complete transaction ends.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bbf4aa0bb7699587","tex":"S_j","aliases":"S_j","display":"S_j","kind":"use","excerpt":"A finite operational version avoids explicit cohomology. Let G=(I,E) be connected, with sets of local solutions S_i and invertible transition maps T_ij:S_i→ S_j, T_ji=T_ij^-1. Choose a root o and a spanning tree T⊆ G.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"use","excerpt":"Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applie","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d3db3688006085b","tex":"s_j^{\\star}","aliases":"s_j^⋆ star","display":"s_j^{\\star}","kind":"use","excerpt":"Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applie","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-deef7bca5ee0eda2","tex":"S_k","aliases":"S_k","display":"S_k","kind":"use","excerpt":"[Finite repair-discovery closure] Under discovery monotonicity, Φ_disc has a least fixed point above any seed set S_0. Iteration S_k+1=Φ_disc(S_k) stabilizes after at most absΩ-absS_0 strict expansions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b4ebcce0dc1a5c39","tex":"s_k","aliases":"s_k","display":"s_k","kind":"use","excerpt":"Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following c","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f25a40ea5414593d","tex":"s_k'","aliases":"s_k'","display":"s_k'","kind":"use","excerpt":"Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fc199b209211722d","tex":"S_{k+1}","aliases":"S_k+1","display":"S_{k+1}","kind":"use","excerpt":"[Finite repair-discovery closure] Under discovery monotonicity, Φ_disc has a least fixed point above any seed set S_0. Iteration S_k+1=Φ_disc(S_k) stabilizes after at most absΩ-absS_0 strict expansions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-295677eac78bab16","tex":"S_o","aliases":"S_o","display":"S_o","kind":"use","excerpt":"[Fundamental-cycle holonomy criterion] For each root state s_o∈ S_o, tree transport determines a unique family (s_i^T). For every non-tree edge e=(i,j), let H_e:S_o→ S_o be transport around the corresponding fundamental cycle. The tree family extends to a globally compatible family on all edges if and only if H_e(s_o)=s_o for every non-tree edge e.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c672fa39ab6e246","tex":"s_{r,c}","aliases":"s_r,c","display":"s_{r,c}","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-036af3d0b593d384","tex":"s_{r,c(z)}","aliases":"s_r,c(z)","display":"s_{r,c(z)}","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cf18d3888e4d739a","tex":"S_{\\rm dev}","aliases":"S_rm dev rm","display":"S_{\\rm dev}","kind":"use","excerpt":"For a candidate configuration c, the development compiler computes target-level selected gains and the score S_rm dev(c) = bar g_c -0.20s_c -2h_c +0.02q_c, where bar g_c is mean protected gain, s_c is the target-level standard deviation, h_c is harmful-target rate, and q_c is positive-target rate. This is a finite engineering objective, not a unive","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-34415489e5efc2d9","tex":"s_{u,c}","aliases":"s_u,c","display":"s_{u,c}","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a347107a1ff2e54","tex":"s_{u,c(z)}","aliases":"s_u,c(z)","display":"s_{u,c(z)}","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-093c8e848ffd9f6f","tex":"S_{\\Vcal}","aliases":"S_Vcal S_V Vcal mathcal","display":"S_{{\\mathcal{V}}}","kind":"use","excerpt":"Let Valid(c_i,Π_i^+)∈{0,1} encode capability, integrity, provenance, and resource validity. The selected descendant is K_Γ^⋆(τ) ∈ arg max_i:Valid(c_i,Π_i^+)=1 S_V(c_i), with the no-op baseline retained when no valid candidate exists.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c93f22a2e8ae311c","tex":"s_\\alpha","aliases":"s_α alpha","display":"s_\\alpha","kind":"use","excerpt":"Let mathscr G be a sheaf of groups acting freely and transitively on the sheaf of local nonlinear solutions mathscr S. Choose local solutions s_α∈mathscr S(U_α). On overlaps there are unique transition elements g_αβ∈mathscr G(U_αβ) such that s_α=g_αβ· s_β.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-95c74318cf410b70","tex":"s_\\beta","aliases":"s_β beta","display":"s_\\beta","kind":"use","excerpt":"Let mathscr G be a sheaf of groups acting freely and transitively on the sheaf of local nonlinear solutions mathscr S. Choose local solutions s_α∈mathscr S(U_α). On overlaps there are unique transition elements g_αβ∈mathscr G(U_αβ) such that s_α=g_αβ· s_β.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-28ce9a7c0b8e51d0","tex":"\\mathrm{safe}","aliases":"safe mathrm","display":"\\mathrm{safe}","kind":"use","excerpt":"Let S={s_0,s_1,b}, C_RI={s_0,s_1}. At s_0 there are two authorized actions: a_safe:s_0↦ s_1, a_risky:s_0↦ b. At s_1 there is one action with successor s_1. Universal closure excludes s_0 because one authorized successor is unsafe. Controlled closure includes s_0 because the policy can choose a_safe. This is the concrete distinction between safety o","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2096c1e70a62f21c","tex":"\\mathrm{same\\ answer,far\\ program}","aliases":"same answer,far program mathrm","display":"\\mathrm{same\\ answer,far\\ program}","kind":"use","excerpt":"s L_i, the audit reuses the Block 17 donor distinctions and adds cross-implementation status labels. It records latent necessity, reversal cost, norm-matched noise cost, exact replay, same-answer far-program transplant cost, far-answer matched-depth transplant cost, local semantic-neighbor cost, external trace-probe R^2, dynamic-depth correlation, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-67dc0bfd2c693568","tex":"\\mathrm{sand}","aliases":"sand mathrm","display":"\\mathrm{sand}","kind":"use","excerpt":"A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f4c2fe899046e205","tex":"\\mathsf{SAP}","aliases":"SAP mathsf","display":"\\mathsf{SAP}","kind":"use","excerpt":"Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matc","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2b33b5def36ba927","tex":"\\operatorname{Score}","aliases":"Score operatorname","display":"\\operatorname{Score}","kind":"use","excerpt":"Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inhe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6d4ab0b1f2927903","tex":"\\operatorname{Score}_{\\mathrm{resp}}","aliases":"Score_resp operatorname mathrm","display":"\\operatorname{Score}_{\\mathrm{resp}}","kind":"use","excerpt":"Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inhe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4288360e53b8c108","tex":"\\operatorname{sd}","aliases":"sd operatorname","display":"\\operatorname{sd}","kind":"use","excerpt":"The implementation uses d_sig(u,v) = frac u-v _2/sqrt5 max{sd(v),10^-8}. The runtime receives only the observed support and values. Context names are retained in the data solely for auditing route accuracy.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b3d672f85f494a61","tex":"\\mathrm{seal}","aliases":"seal mathrm","display":"\\mathrm{seal}","kind":"use","excerpt":"The transaction follows six phases: prepare ⟶ seal ⟶ execute ⟶ reveal ⟶ decide ⟶ finalize. Only the final phase may expose buffered effects or replace the active checkpoint.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-411ff5efa5de7ad8","tex":"\\mathrm{select}","aliases":"select mathrm","display":"\\mathrm{select}","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-02654dca6e64aaac","tex":"\\mathrm{Sem}","aliases":"Sem mathrm","display":"\\mathrm{Sem}","kind":"use","excerpt":"The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d4d0a44672fa911","tex":"\\mathrm{sensor}","aliases":"sensor mathrm","display":"\\mathrm{sensor}","kind":"use","excerpt":"The latent event variable is z∈ {none,internal,world,sensor,memory,evaluator}. Each event changes one declared mechanism: item internal: the action-response map B changes; item world: the passive dynamics A change; item sensor: the redundant sensor receives an additive bias; item memory: the retention coefficient λ changes; item evaluator: the repo","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-28f39655338ed57d","tex":"\\operatorname{sgn}","aliases":"sgn operatorname","display":"\\operatorname{sgn}","kind":"use","excerpt":"b. Therefore η_D(r)=|b|. For b≠0, the unit dual witness is y^*=sgn(b)e_2. Every authorized correction has zero effect on the second defect coordinate.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-21670bd4e275805e","tex":"\\sinh","aliases":"sinh","display":"\\sinh","kind":"use","excerpt":"The nonlinear family applies Y_t = fracsinh (α(X_tA+b))α, α>0, coordinatewise. The declared inverse constructor first applies ϕ^-1_α(y) = fracarsinh(α y)α and then fits a step-affine map.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6e5b2524b9507399","tex":"\\mathsf{SN}","aliases":"SN mathsf","display":"\\mathsf{SN}","kind":"use","excerpt":"ce between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matched donors. Three donors per available ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"Evaluator-nullspace shrink. Let T=ℝ^2, Db_1(x_1,x_2)=x_1, and Db_2(x_1,x_2)=x_2. With only b_1, the invisible subspace is span(e_2). Adding b_2 reduces the intersection of kernels to {0}. The new benchmark contributes an independent behavioral distinction rather than merely rescaling the old score.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51a067a3d07c36a3","tex":"\\mathrm{Spearman}","aliases":"Spearman mathrm","display":"\\mathrm{Spearman}","kind":"use","excerpt":"halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d2dbd4f0c7b2724d","tex":"\\mathrm{spec/code}","aliases":"spec/code mathrm","display":"\\mathrm{spec/code}","kind":"use","excerpt":"The forecast residual and proposal--code congruence residual are ε_forecast,i = normwidehat c_i- c_i, ε_spec/code,i = d_sem ( Sem(Π_i^-), Sem(code_i) ). In the executable finite companion, congruence is a Boolean monitor rather than a complete semantic metric.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"The data partition is itself typed: D = D_construct sqcup D_calibrate sqcup D_select sqcup D_foreign sqcup D_audit. Construction data may be used to create schemas. Calibration data may set thresholds. Selection data may choose among frozen candidates. Foreign data estimate metaproductivity. Audit data are not available until the final claim object","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b85bf57a1d709d93","tex":"\\mathrm{square}","aliases":"square mathrm","display":"\\mathrm{square}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e4353e56e229aaaa","tex":"\\mathrm{src}","aliases":"src mathrm","display":"\\mathrm{src}","kind":"use","excerpt":"Let (Y,H_s,H_t)∼μ be the protected semantics, source trace, and target trace under context distribution μ. Fix a source probe P and bridge B. Define e_src=Y-PH_s, e_align=P(H_s-BH_t). Then Y-PBH_t=e_src+e_align.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8c6c76cec5742d33","tex":"\\Sscr_\\alpha","aliases":"Sscr_α mathscrS_α Sscr alpha mathscr alpha","display":"{\\mathscr{S}}_\\alpha","kind":"use","excerpt":"[Local solution torsor] For each U_α, the set mathscrS_α := setu∈mathscr U(U_α):D_α u=-r|_U_α is either empty or a torsor for the additive group ker D(U_α). If a global repair exists, the global solution set is a torsor for Γ(X,ker D).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cd3d080377149900","tex":"\\stackrel","aliases":"stackrel","display":null,"kind":"use","excerpt":"[Format-conditional transplant null] Let G(x) be a declared answer-format class. Suppose that, conditional on G(x)=g, the trace law is independent of task content: Z(x)mid G(x)=g stackreld= Z(x')mid G(x')=g, where x' is an independent donor. Assume the intervention harness leaves the prompt and target answer fixed and replaces only the latent trace","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bcac5605fdc4859a","tex":"\\mathrm{str}","aliases":"str mathrm","display":"\\mathrm{str}","kind":"use","excerpt":"[Finite-sample structural certificate] Suppose normr≤ R almost surely and r_1,…,r_n are independent held-out defects. Let widehatΔ_str = frac1n∑_i=1^nnormP_Wr_i^2. Then with probability at least 1-δ, | widehatΔ_str - EnormP_Wr^2 | ≤ R^2sqrtfraclog(2/δ)2n.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ed5a44c47e0a4dda","tex":"\\sup_{B}","aliases":"sup_B sup","display":"\\sup_{B}","kind":"use","excerpt":"Let D and D' be two foreign-agent distributions and define TV(D,D') = sup_BabsD(B)-D'(B).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-71112ec526caa68c","tex":"\\sup_{u\\in B_R^W}","aliases":"sup_u∈ B_R^W sup in","display":"\\sup_{u\\in B_R^W}","kind":"use","excerpt":"bounded right inverse Q:V→ U, so DQ=I_V, and set W:=im Q. Then restr D W:W→ V is an isomorphism with inverse Q. Let B_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0685a2c98bca26a6","tex":"\\sup_{\\widetilde K\\in\\Kcal}","aliases":"sup_widetilde K∈Kcal sup_widetilde K∈K sup widetilde in Kcal sup widetilde in mathcal","display":"\\sup_{\\widetilde K\\in{\\mathcal{K}}}","kind":"use","excerpt":"protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared headroom is H(K) = F_K(N)-F_K(∅). The adjective declared matters. A larger, previously una","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cd30c1119a1c7561","tex":"\\sup_{\\widetilde K\\in\\Kcal_R(K)}","aliases":"sup_widetilde K∈Kcal_R(K) sup_widetilde K∈K_R(K) sup widetilde in Kcal sup widetilde in mathcal","display":"\\sup_{\\widetilde K\\in{\\mathcal{K}}_R(K)}","kind":"use","excerpt":"Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared hea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3a1990a2d128b373","tex":"\\sup_{\\gamma\\in\\Gamma}","aliases":"sup_γ∈Γ sup gamma in Gamma","display":"\\sup_{\\gamma\\in\\Gamma}","kind":"use","excerpt":"[Finite meta-grammar uniform bound] Suppose, for each γ∈Γ, the per-episode utility lies in an interval of width L, and the n meta-validation episodes are independent. Then with probability at least 1-δ, sup_γ∈Γ | widehat U_n(γ)-U(γ) | ≤ Lsqrtfraclog(2|Γ|/δ)2n. Consequently, the true utility of the empirical maximizer is within twice the right-hand ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c692cbc21785ad4f","tex":"\\sup_{\\delta\\in\\Dcal}","aliases":"sup_δ∈Dcal sup_δ∈D sup delta in Dcal sup delta in mathcal","display":"\\sup_{\\delta\\in{\\mathcal{D}}}","kind":"use","excerpt":"Define the intervention-coverage error ε_D(ϕ;A) = sup_δ∈D TV(K_A^δ,widehat K_ϕ(A)^δ).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-580023d673e19885","tex":"\\operatorname{Sym}","aliases":"Sym operatorname","display":"\\operatorname{Sym}","kind":"use","excerpt":"Every quadratic form z^top Bz becomes z'^top(Q^top BQ)z'. The vector space of symmetric quadratic forms is therefore carried isomorphically to itself under the induced action of Q on Sym^2(P^top H).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fffc80c3d06fc6a7","tex":"\\operatorname{Sym}^2","aliases":"Sym^2 operatorname","display":"\\operatorname{Sym}^2","kind":"use","excerpt":"Every quadratic form z^top Bz becomes z'^top(Q^top BQ)z'. The vector space of symmetric quadratic forms is therefore carried isomorphically to itself under the induced action of Q on Sym^2(P^top H).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"[Archive-envelope monotonicity] Suppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector w, define H_w(A_t) = max_A∈A_tw^ Tz(A). Then H_w(A_t) is nondecreasing in t.","count":41},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f0426e872b1c4d7a","tex":"T^{\\mathsf T}","aliases":"T^ T mathsf","display":"T^{\\mathsf T}","kind":"use","excerpt":"Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-10d6fa1215b924bf","tex":"T^*","aliases":"T^*","display":"T^*","kind":"use","excerpt":"A population residual formula gives a quantitative special case. Suppose h is centered with covariance Σsucc0, s∈{-1,+1} is independent of h with Pr(s=1)=p, and y=(A+sB)h. Then the best global linear map is T^*=A+(2p-1)B, and inf_T Enormy-Th^2 = 4p(1-p)tr(BΣ B^top). The proof is the normal-equation calculation from the centered two-component mixtur","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c1e5465f78cfb0b7","tex":"t^*","aliases":"t^*","display":"t^*","kind":"use","excerpt":"Let h∼ N(0,1) and let s∈{-1,+1} be independent and balanced. Define y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f3369229c8326314","tex":"t^{**}","aliases":"t^**","display":"t^{**}","kind":"use","excerpt":"ion Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n) is well-defined, remains in overline B(u_0,t_*), and converges to a root u^* of F. If the larger radius t^**:=frac1+sqrt1-2hnorm B L is contained in the regularity region, the root is unique in overline B(u_0,t^**).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-885b82dd59955b7a","tex":"T^+","aliases":"T^+","display":"T^+","kind":"use","excerpt":"0 · 0, h_s=e_2. Then h_t=Th_s=0, so target-state alignment is exact. However, T^+T=1 · 0 0 · 0, D_t=TD_sT^+=0. The source repair maps the discarded input direction into the retained output direction: D_sh_s=e_1. Consequently, R_t(Th_s)-TR_s(h_s) =TD_s(T^+T-I)h_s =-1. Thus a target can align exactly as a compressed representation while failing to re","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bfe5b05f2db4353b","tex":"T^{+}","aliases":"T^+","display":"T^{+}","kind":"use","excerpt":"[Exact transport through an injective hidden bridge] Let T:ℝ^d_s→ℝ^d_t have full column rank, so T^+T=I_d_s. Suppose h_t=Th_s throughout the evaluated support. Define D_t and b_t by [source reference] with widehat T=T. Then R_t(Th_s)=TR_s(h_s).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"use","excerpt":"Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"Consider the scalar majorant f(t)=η-t+fracnorm B L2t^2. The condition h≤1/2 gives the two nonnegative roots t_* and t^**. The Lipschitz derivative bound and the Banach lemma imply that F'(u) remains invertible while normu-u_0<1/(norm B L). Standard Newton majorization shows that the increments of the operator iteration are dominated by those of sca","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fc7b28eb860f6d45","tex":"T^c_{a\\to b}","aliases":"T^c_a→ b to","display":"T^c_{a\\to b}","kind":"use","excerpt":"The next threshold is atlas-indexed cross-architecture Neuralese transport. Source and target systems should each carry locally valid semantic charts, transport should be tested on chart overlaps, and architecture transport should approximately commute with context transition: C^b_c→ dT^c_a→ b ≈ T^d_a→ bC^a_c→ d. Block 20 tests this threshold direc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-71532a9d5e0d2271","tex":"T^c_{b\\to a}","aliases":"T^c_b→ a to","display":"T^c_{b\\to a}","kind":"use","excerpt":"[Readout stability from trajectory reconstruction] Let J:H_a,c→ R^p be L_J-Lipschitz. If a transported trajectory satisfies normT^c_b→ ah_b,c-h_a,c ≤ε, then normJ(T^c_b→ ah_b,c)-J(h_a,c) ≤ L_Jε. If two transport paths differ by at most δ, their semantic readouts differ by at most L_Jδ.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ee6aeda8bc4154b","tex":"T^d_{a\\to b}","aliases":"T^d_a→ b to","display":"T^d_{a\\to b}","kind":"use","excerpt":"The next threshold is atlas-indexed cross-architecture Neuralese transport. Source and target systems should each carry locally valid semantic charts, transport should be tested on chart overlaps, and architecture transport should approximately commute with context transition: C^b_c→ dT^c_a→ b ≈ T^d_a→ bC^a_c→ d. Block 20 tests this threshold direc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-032c5bccde429f54","tex":"T^\\top","aliases":"T^top top","display":"T^\\top","kind":"use","excerpt":"[Full-rank bridge identification] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c107f3867b48de6e","tex":"T^{\\top}","aliases":"T^top top","display":"T^{\\top}","kind":"use","excerpt":"h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0568ffb513f9f87c","tex":"T^\\dagger","aliases":"T^† dagger","display":"T^\\dagger","kind":"use","excerpt":"Let the frozen source repair be R_s(h)=h+D_sh+b_s. For an injective bridge T, the exact transported repair is D_t=T D_sT^†, b_t=Tb_s. The estimated repair uses widehat T.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4ff31b43de1984cc","tex":"t_*","aliases":"t_*","display":"t_*","kind":"use","excerpt":"orovich Let F:U→ V be continuously Fr echet differentiable between Banach spaces. Let u_0∈ U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n)","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7f502591dbad0d5e","tex":"T_+","aliases":"T_+","display":"T_+","kind":"use","excerpt":"T_-,T_+ · Chart-specific source-to-target bridge maps","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6b121633ae13dbde","tex":"t_+","aliases":"t_+","display":"t_+","kind":"use","excerpt":"eginequation y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar uses t_-=a-b, t_+=a+b, and has zero noiseless approxima","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4d99951c22895e3a","tex":"T_-","aliases":"T_-","display":"T_-","kind":"use","excerpt":"T_-,T_+ · Chart-specific source-to-target bridge maps","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f91d63d8263b308b","tex":"t_-","aliases":"t_-","display":"t_-","kind":"use","excerpt":"eginequation y=(a+sb)h, b≠0. A global scalar bridge predicts widehat y=th. The population minimizer is t^*=a, and inf_t E (y-th)^2 = E(sbh)^2 = b^2. The residual is not an estimation artifact. It remains after infinite data because one slope cannot equal both a-b and a+b. The two-chart grammar uses t_-=a-b, t_+=a+b, and has zero noiseless approxima","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported sc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ffba8e5cc26dfde8","tex":"\\mathfrak T_0","aliases":"T_0 mathfrak","display":"\\mathfrak T_0","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bbf183245abdf31d","tex":"T_{01}","aliases":"T_01","display":"T_{01}","kind":"use","excerpt":"Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"cation] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequent","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-621da870a6aeba35","tex":"\\mathfrak T_1","aliases":"T_1 mathfrak","display":"\\mathfrak T_1","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1a36fd1f6d7af0cd","tex":"T_{12}","aliases":"T_12","display":"T_{12}","kind":"use","excerpt":"Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2b59d92deb4da7eb","tex":"T_1^\\top","aliases":"T_1^top top","display":"T_1^\\top","kind":"use","excerpt":"ridge Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"cation] Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequent","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7be2ccc9284905af","tex":"T_{20}","aliases":"T_20","display":"T_{20}","kind":"use","excerpt":"Case F4: evaluator holonomy. Use an exactly correctable local defect and zero overlap mismatch. Let evaluator transports around a three-edge cycle be T_01=I_2, T_12=I_2, T_20=diag(1,-1). For protected evaluator direction e_2, H_γ e_2-e_2=-2e_2, normH_γ e_2-e_2=2. The candidate is locally repairable and globally gluable, but its evaluator semantics ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e2c65f21e0d1a063","tex":"T_2^{-1}","aliases":"T_2^-1","display":"T_2^{-1}","kind":"use","excerpt":"[Intertwiner uniqueness modulo the response commutant] Let T_1,T_2:H_t→ H_s be invertible and satisfy T_iR_v^t=R_σ^-1(v)^sT_i for every v∈ V. Then S=T_2^-1T_1 belongs to A_t'. Hence exact intervention-orbit data identifies a transport at most up to the response commutant.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8802812d06213c38","tex":"T_2^\\top","aliases":"T_2^top top","display":"T_2^\\top","kind":"use","excerpt":"ridge Let the target representation satisfy h_t(x)=T h_s(x) for a fixed linear map T:ℝ^d→ℝ^m. Let X∈ℝ^k× d be the selected source design and Y=XT^top. If rank X=d, then the unregularized least-squares bridge is unique and equals T on the entire source space. If rank X<d, there exist distinct maps T_1≠ T_2 such that XT_1^top=XT_2^top=Y. Consequently","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5e5698fa29728c9e","tex":"\\mathfrak T_{A\\to C}","aliases":"T_A→ C mathfrak to","display":"\\mathfrak T_{A\\to C}","kind":"use","excerpt":"Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a1ef26e4d65ecb73","tex":"\\mathfrak T_{B\\to C}","aliases":"T_B→ C mathfrak to","display":"\\mathfrak T_{B\\to C}","kind":"use","excerpt":"Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4b7fbd7d66e3a28d","tex":"T_f","aliases":"T_f","display":"T_f","kind":"use","excerpt":"[Composition of evaluator defects] Let e:v→ w and f:w→ z be evaluator morphisms. Suppose T_f is L_T-Lipschitz on represented protected evidence. Then δ_obs(f∘ e) ≤ L_Tδ_obs(e)+δ_obs(f), where the second term is evaluated on S_e( h_v). If, in addition, sup_r∈ h_v norm J_zT_fT_eι_v(r)-J_wT_eι_v(r) ≤ δ_J(fmid e), then δ_J(f∘ e) ≤ δ_J(e)+δ_J(fmid e).","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-219b7412a89e1904","tex":"T_i","aliases":"T_i","display":"T_i","kind":"use","excerpt":"[Intertwiner uniqueness modulo the response commutant] Let T_1,T_2:H_t→ H_s be invertible and satisfy T_iR_v^t=R_σ^-1(v)^sT_i for every v∈ V. Then S=T_2^-1T_1 belongs to A_t'. Hence exact intervention-orbit data identifies a transport at most up to the response commutant.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e9a7aef78fcda23e","tex":"T_{ij}","aliases":"T_ij","display":"T_{ij}","kind":"use","excerpt":"A finite operational version avoids explicit cohomology. Let G=(I,E) be connected, with sets of local solutions S_i and invertible transition maps T_ij:S_i→ S_j, T_ji=T_ij^-1. Choose a root o and a spanning tree T⊆ G.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d68ca3e0ca5e40c6","tex":"T_{ij}^{-1}","aliases":"T_ij^-1","display":"T_{ij}^{-1}","kind":"use","excerpt":"A finite operational version avoids explicit cohomology. Let G=(I,E) be connected, with sets of local solutions S_i and invertible transition maps T_ij:S_i→ S_j, T_ji=T_ij^-1. Choose a root o and a spanning tree T⊆ G.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"use","excerpt":"Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-95211c71e87133e2","tex":"T_j^{-1}","aliases":"T_j^-1","display":"T_j^{-1}","kind":"use","excerpt":"Let the source repair be R_s(h)=M_sh+c_s. On chart j, let the target bridge be T_j(h)=A_jh+b_j, with A_j invertible on the repair-relevant subspace. Define R_t,j(y) = T_j (R_s(T_j^-1(y))). Expanding gives R_t,j(y) = A_jM_sA_j^-1(y-b_j)+A_jc_s+b_j.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f50f6889319413f5","tex":"T_{ji}","aliases":"T_ji","display":"T_{ji}","kind":"use","excerpt":"A finite operational version avoids explicit cohomology. Let G=(I,E) be connected, with sets of local solutions S_i and invertible transition maps T_ij:S_i→ S_j, T_ji=T_ij^-1. Choose a root o and a spanning tree T⊆ G.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d08a14a786811454","tex":"T_{n-1}","aliases":"T_n-1","display":"T_{n-1}","kind":"use","excerpt":"chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported sc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-713746c8cc52f2b9","tex":"\\mathfrak T_{n-1}","aliases":"T_n-1 mathfrak","display":"\\mathfrak T_{n-1}","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-807e619b0b36adc6","tex":"T_p","aliases":"T_p","display":"T_p","kind":"use","excerpt":"[Basepoint covariance of the flat core] Let p:v→ w be any invertible exact evaluator path in a connected atlas. Then T_p(E_v^flat)=E_w^flat. Hence the dimension of the path-flat core is independent of basepoint.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-86ffbd6d948523b6","tex":"T_p^{-1}","aliases":"T_p^-1","display":"T_p^{-1}","kind":"use","excerpt":"Every loop γ at w corresponds to the conjugate loop p^-1γ p at v, with H_γ=T_pH_p^-1γ pT_p^-1. If x∈ E_v^flat, then H_γ T_px=T_pH_p^-1γ px=T_px, so T_px∈ E_w^flat. Apply the same argument to p^-1 for the reverse inclusion.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-df3f310abe33eba9","tex":"T_r","aliases":"T_r","display":"T_r","kind":"use","excerpt":"The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation ep","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-33037ac01bdbae38","tex":"\\mathsf T_{\\rm audit}","aliases":"T_rm audit mathsf rm","display":"\\mathsf T_{\\rm audit}","kind":"use","excerpt":"The Block 7 primary audit differs from development in both task and representation family: T_rm dev · ∩ T_rm audit=∅, F_rm dev^rm primary · ∩ F_rm audit^rm primary=∅. Development uses ten tasks generated from four trigger families. Audit uses twelve new tasks generated from radial and mixed triggers, new teacher seeds, and new corruption permutatio","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4d7addffc507dde0","tex":"\\mathsf T_{\\rm dev}","aliases":"T_rm dev mathsf rm","display":"\\mathsf T_{\\rm dev}","kind":"use","excerpt":"The Block 7 primary audit differs from development in both task and representation family: T_rm dev · ∩ T_rm audit=∅, F_rm dev^rm primary · ∩ F_rm audit^rm primary=∅. Development uses ten tasks generated from four trigger families. Audit uses twelve new tasks generated from radial and mixed triggers, new teacher seeds, and new corruption permutatio","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-557661f6ae0c797e","tex":"T_t","aliases":"T_t","display":"T_t","kind":"use","excerpt":"ulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6551012434602ffa","tex":"T_t^U","aliases":"T_t^U","display":"T_t^U","kind":"use","excerpt":"A grammar extension may also change the defect representation. Let D_t:U_t→ V_t, with transports T_t^U:U_t→ U_t+1, T_t^V:V_t→ V_t+1. Assume the exact correction square D_t+1T_t^U = T_t^VD_t. Then there is an induced obstruction map overline T_t: coker D_t → coker D_t+1, overline T_t([r])=[T_t^Vr].","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b3fd3f8e5e0b8e94","tex":"T_{U,0}","aliases":"T_U,0","display":"T_{U,0}","kind":"use","excerpt":"stems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1f9433960e46f606","tex":"T_{U,k}","aliases":"T_U,k","display":"T_{U,k}","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5e91088831cb256f","tex":"T_{U,k-1}","aliases":"T_U,k-1","display":"T_{U,k-1}","kind":"use","excerpt":"T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-af8833b9b193d9d4","tex":"T_{U,n-1}","aliases":"T_U,n-1","display":"T_{U,n-1}","kind":"use","excerpt":"stems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fcaef1f195dd9d01","tex":"T_U^*","aliases":"T_U^*","display":"T_U^*","kind":"use","excerpt":"[Naturality of the Moore--Penrose repair] Let U_A,U_B,V_A,V_B be finite-dimensional Hilbert spaces. Suppose T_U and T_V are unitary and D_B=T_VD_AT_U^*. Then D_B^†=T_UD_A^†T_V^*. For every r_B=T_Vr_A, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction: -D_B^†r_B = T_U(-D_A^†r_A). Th","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-13b164e86248d526","tex":"T_U^{-1}","aliases":"T_U^-1","display":"T_U^{-1}","kind":"use","excerpt":"[Exact transport equivalence] Suppose T_U and T_V are isomorphisms and D_B=T_VD_AT_U^-1. Then T_U restricts to an isomorphism ker D_A≅ker D_B, and T_V induces an isomorphism coker D_A≅coker D_B. Thus exact transport equivalence preserves both correction freedom and obstruction dimension.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d31447d4f2b2b70f","tex":"T_U^{AB}","aliases":"T_U^AB","display":"T_U^{AB}","kind":"use","excerpt":"Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0b3bc1a2b85d800a","tex":"T_U^{BC}","aliases":"T_U^BC","display":"T_U^{BC}","kind":"use","excerpt":"Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f54bddf97d8c4da0","tex":"T_U^p","aliases":"T_U^p","display":"T_U^p","kind":"use","excerpt":"[Path independence on the flat repair core] Let p and q be two exact invertible transport paths from A to B. If u∈ U_A is fixed by the loop holonomy associated with q^-1p, then T_U^p u=T_U^q u. In particular, every u∈ U_A^flat has path-independent transport to B.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e355ace52fe2b886","tex":"T_U^q","aliases":"T_U^q","display":"T_U^q","kind":"use","excerpt":"[Path independence on the flat repair core] Let p and q be two exact invertible transport paths from A to B. If u∈ U_A is fixed by the loop holonomy associated with q^-1p, then T_U^p u=T_U^q u. In particular, every u∈ U_A^flat has path-independent transport to B.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5d067a6fee7e623","tex":"T_{V,k}","aliases":"T_V,k","display":"T_{V,k}","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8d32baac34e25d36","tex":"T_{V,k+1}","aliases":"T_V,k+1","display":"T_{V,k+1}","kind":"use","excerpt":"T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e713cb0fbfc67693","tex":"T_{V,n-1}","aliases":"T_V,n-1","display":"T_{V,n-1}","kind":"use","excerpt":"T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e5f60d1063a51e23","tex":"T_V^*","aliases":"T_V^*","display":"T_V^*","kind":"use","excerpt":"[Naturality of the Moore--Penrose repair] Let U_A,U_B,V_A,V_B be finite-dimensional Hilbert spaces. Suppose T_U and T_V are unitary and D_B=T_VD_AT_U^*. Then D_B^†=T_UD_A^†T_V^*. For every r_B=T_Vr_A, the target minimum-norm least-squares correction is the transport of the source minimum-norm least-squares correction: -D_B^†r_B = T_U(-D_A^†r_A). Th","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6be4cf8f4f05bfe0","tex":"T_V^{AB}","aliases":"T_V^AB","display":"T_V^{AB}","kind":"use","excerpt":"Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fc3bf7593ca38c77","tex":"T_V^{BC}","aliases":"T_V^BC","display":"T_V^{BC}","kind":"use","excerpt":"Let T_A→ B=(T_U^AB,T_V^AB), T_B→ C=(T_U^BC,T_V^BC). The composed transport is T_A→ C = (T_U^BCT_U^AB,T_V^BCT_V^AB).","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-325816c77766b823","tex":"T_x","aliases":"T_x","display":"T_x","kind":"use","excerpt":"For nonlinear dynamics on a state manifold M, the response algebra becomes base-point dependent: x⟼ A_x⊆End(T_xM). Local central projectors define a bundle of tangent strata when their ranks remain locally constant. Transport around a loop can permute or distort the local modules, producing atlas holonomy. A complete nonlinear theory must combine: ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-978e87e4f3d831a3","tex":"\\tanh","aliases":"tanh","display":"\\tanh","kind":"use","excerpt":"Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following c","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3c0ebebd59054caf","tex":"\\mathrm{target}","aliases":"target mathrm","display":"\\mathrm{target}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bab897d2d5fb61a8","tex":"\\mathrm{task}","aliases":"task mathrm","display":"\\mathrm{task}","kind":"use","excerpt":"Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, addres","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0dab57a4f245217e","tex":"\\Tau_t^{\\mathrm{NL}}","aliases":"Tau_t^NL T_t^NL Tau mathrm mathcal mathrm","display":"{\\mathcal{T}}_t^{\\mathrm{NL}}","kind":"use","excerpt":"Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a componen","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dd81196deb289034","tex":"\\mathrm{tools}","aliases":"tools mathrm","display":"\\mathrm{tools}","kind":"use","excerpt":"A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fc7a5c603b95f30d","tex":"\\tr","aliases":"tr operatorname","display":"{\\operatorname{tr}}","kind":"use","excerpt":"For W⊆ I^⊥, EnormP_(I⊕ W)^⊥r^2 = tr(Σ_ob)-tr(P_WΣ_ob). The Ky Fan variational principle maximizes tr(P_WΣ_ob) over rank-k orthogonal projectors by choosing the top k eigenspace.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b6ab8e714635a4ba","tex":"\\mathrm{trace}","aliases":"trace mathrm","display":"\\mathrm{trace}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bf809c09acebd684","tex":"\\mathrm{transport}","aliases":"transport mathrm","display":"\\mathrm{transport}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5ba44fa7ab307367","tex":"\\mathrm{true\\ chart}","aliases":"true chart mathrm","display":"\\mathrm{true\\ chart}","kind":"use","excerpt":"tion 0.152169 and maximum gain 0.364965 across development. By contrast, the true-chart oracle has mean gain 0.998046, and the learned local constructor has mean blind gain 0.613651. Every one of the 54 development targets satisfies the frozen repair-conflict predicate G_global oracle≤0.40, G_true chart≥0.95. Mean learned gate accuracy is 0.915000,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-559f90b6098df902","tex":"\\mathrm{twin}","aliases":"twin mathrm","display":"\\mathrm{twin}","kind":"use","excerpt":"For a candidate quotient q_θ, the empirical twin discrepancy is D_twin(q_θ) = E fracnormq_θ(x)-q_θ(x') max{normq_θ(x),ε}. A candidate is construction-valid only when it jointly predicts future consequences and intervention effects, predicts the externally scored answer, and keeps [source reference] below the declared ceiling.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aad33c6ee481d6af","tex":"u'_\\alpha","aliases":"u'_α alpha","display":"u'_\\alpha","kind":"use","excerpt":"If a global repair u exists, set k_α=u|_U_α-u_α. Then Dk_α=0 and d_αβ=k_β-k_α, so the cocycle is a coboundary. Conversely, if d_αβ=k_β-k_α for kernel sections k_α, then u'_α=u_α+k_α agree on overlaps. The sheaf gluing axiom yields a global u, and Du=-r because the adjustments lie in the kernel.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4af4f175a4c49987","tex":"U^{\\mathsf T}","aliases":"U^ T mathsf","display":"U^{\\mathsf T}","kind":"use","excerpt":"Let the true context packet be P_*=UU^ T for a target-specific orthonormal matrix U∈ℝ^6×2, while the inherited packet is P_0=diag(1,1,0,0,0,0). Let the chart label be c(h)=1{e_1^ TU^ Th>0}. For generic U, c(h) is not measurable from P_0h. No threshold or nonlinear classifier using only P_0h can recover the chart perfectly. The response atlas can re","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a257f9d1ed0e54ad","tex":"u^{\\mathsf T}","aliases":"u^ T mathsf","display":"u^{\\mathsf T}","kind":"use","excerpt":"[Worst-direction variance] For every u∈ℝ^d and every target coordinate, Var(u^ T(widehat T^ T-T^ T)e_jmid X) =u^ TM_S^-1u ≤ fracnormu^2λ_min(M_S).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f917d7010564bfa7","tex":"u^{(0)}","aliases":"u^(0)","display":"u^{(0)}","kind":"use","excerpt":"-q. The zeroth packet-local repair is u^(0)=-e_1 and leaves residual Du^(0)+r= 0","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-65c099679b6b07b7","tex":"u^{(n)}","aliases":"u^(n)","display":"u^{(n)}","kind":"use","excerpt":"orrection under small packet coupling] If q:=normEQ_0<1, then D is surjective and has right inverse Q =Q_0(I_V+EQ_0)^-1 =Q_0∑_k=0^∞(-EQ_0)^k. Moreover, norm Q≤fracnormQ_01-q. For defect r, the truncated packet-correction series u^(n) :=-Q_0∑_k=0^n(-EQ_0)^kr has residual Du^(n)+r=(-EQ_0)^n+1r, and hence normDu^(n)+r≤ q^n+1norm r.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-07855388bae78c61","tex":"u^*","aliases":"u^*","display":"u^*","kind":"use","excerpt":"hrmob, r_cor=P_corr, r_ob=P_obr; item the minimum achievable residual is inf_u∈ U^authnormDu+r =normr_ob =dist(r,im D); item the complete set of least-squares minimizers is -D^† r+ker D; item if r is exactly correctable, u^*=-D^† r is the unique exact repair of minimum norm.","count":13},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d849fcfcf96ad168","tex":"u^3","aliases":"u^3","display":"u^3","kind":"use","excerpt":"Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-221bbc1b059b410b","tex":"U^{\\flat}","aliases":"U^flat flat","display":null,"kind":"use","excerpt":"0 · -1 . The flat repair core is U^flat=spansete_1. For u=e_1, both routes produce the same transported repair. For u=e_2, the two routes differ by (H^U-I)e_2=-2e_2, so the loop residual is 2. A repair schema containing an e_2 component is therefore path dependent even though every individual edge transport may be exact.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-df73ec13d71ee514","tex":"u_*","aliases":"u_*","display":"u_*","kind":"use","excerpt":"-0.25 . Hence P_ctxR_s(h)=P_ctxh. The observable chart is generated by a task-specific separator c(h)=1{anglesu_*,P_ctxh>θ_*}. The compiler does not receive u_* or θ_*, but it receives the declared protected packet in which a valid separator may be sought.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","excerpt":"[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"u-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnormu_1-u_0.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2f97c9046c1a4c7a","tex":"u_2","aliases":"u_2","display":"u_2","kind":"use","excerpt":"Consider the specific formal correction equation F(ε,u) = ε r_1+Du+frac12B(u,u)+O(ε^3+norm u^3), where B:U× U→ V is symmetric bilinear. Seek u(ε)=ε u_1+ε^2u_2+O(ε^3).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cdc6b49492357264","tex":"u_3","aliases":"u_3","display":"u_3","kind":"use","excerpt":"Suppose a source correction has norm normu_0=2, every transport is nonexpansive, and three edges have (ε_D,0,ε_D,1,ε_D,2) =(0.01,0.02,0.015), (ε_r,0,ε_r,1,ε_r,2) =(0.005,0.004,0.006). Then [source reference] gives normD_3u_3+r_3 · ≤ 2(0.01+0.02+0.015) +(0.005+0.004+0.006) · =0.105. A direct source-to-target bridge with certified residual penalty be","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9013077e1c377460","tex":"U_4","aliases":"U_4","display":"U_4","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ec4379f6b5fb17bc","tex":"U_a","aliases":"U_a","display":"U_a","kind":"use","excerpt":"Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-432c14b23419ac6d","tex":"u_A","aliases":"u_A","display":"u_A","kind":"use","excerpt":"[Exact repair transport] Suppose D_BT_U=T_VD_A and r_B=T_Vr_A. If u_A satisfies D_Au_A=-r_A, then u_B:=T_Uu_A satisfies D_Bu_B=-r_B.","count":26},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6fc788bf97194794","tex":"U_a^{\\top}","aliases":"U_a^top top","display":"U_a^{\\top}","kind":"use","excerpt":"Let U_a∈ℝ^d_a× r have orthonormal columns spanning the discovered packet. Define desired packet coordinates z_i = U_a^top(h_a^∘(x_i)-h_a(x_i)). Fit an affine predictor widehat z(h)=K_a h+c_a by ridge regression on source anchor states. The repair is R_a(h) = h+U_awidehat z(h) = h+D_a h+b_a, where D_a=U_aK_a, b_a=U_ac_a.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6c5b12d80fe3c4ae","tex":"U_{\\mathrm{audit}}","aliases":"U_audit mathrm","display":"U_{\\mathrm{audit}}","kind":"use","excerpt":"Let U_audit be a protected audit utility unavailable during compiler allocation and final scored selection. Define compiler metaproductivity at research budget B by C_B(Γ) = E_τ∼D_foreign [ U_audit (K_Γ^⋆(τ)) - U_audit(K_τ) ]. Only integrity-valid and capability-passing outputs are eligible.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-924579a9b7ed33d9","tex":"u_B","aliases":"u_B","display":"u_B","kind":"use","excerpt":"[Exact repair transport] Suppose D_BT_U=T_VD_A and r_B=T_Vr_A. If u_A satisfies D_Au_A=-r_A, then u_B:=T_Uu_A satisfies D_Bu_B=-r_B.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a69093f2b23577cd","tex":"u_c","aliases":"u_c","display":"u_c","kind":"use","excerpt":"The context code is computed separately: u_c=C(s_x,t_x,s_y,t_y,d_sig)∈ℝ^10. It supports chart diagnostics but has no computational path into h_0, r_k, the task output, or the halt head.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cb69d506a7bd35e5","tex":"U_e","aliases":"U_e","display":"U_e","kind":"use","excerpt":"al level. A parent repair selector is frozen. Development receipts generated through the existing lineage are used to compile a child selector. Parent and child then receive exactly the same candidate repairs, target observations, calibration information, evaluator, and candidate budget. The child receipt is sealed before either final audit split i","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b66a4d677667d2df","tex":"u_{\\mathrm{flat}}","aliases":"u_flat mathrm","display":"u_{\\mathrm{flat}}","kind":"use","excerpt":"-2 , with zero overlap mismatch and identity evaluator transport. The unique minimum-norm repair is u_flat=-r_flat.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cfb253493521017a","tex":"u_i^2","aliases":"u_i^2","display":"u_i^2","kind":"use","excerpt":"For canonical patch u=T^-1δ, define corrected protected energy E_corr(d,u) = ∑_i=1^6w_imin{d_i^2,u_i^2}. This quantity is zero for a no-op and positive when a patch reaches task-relevant defect coordinates. It is reported alongside behavioral gain because a high apparent score can be generated without correcting protected defect energy.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","excerpt":"[Open-set incompatibility of distinct affine charts] Let U_0,U_1⊆ℝ^d each contain a nonempty open set. Let f_j(h)=A_jh+b_j, j∈{0,1}. If one affine map f(h)=Ah+b satisfies f(h)=f_j(h) for every h∈ U_j and both j, then A=A_0=A_1, b=b_0=b_1. Consequently, if (A_0,b_0)≠(A_1,b_1), no element of G_glob represents the two-chart relation exactly.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-72b2bc9e12d0fa8e","tex":"U_k","aliases":"U_k","display":"U_k","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-836830fc78b794aa","tex":"u_k","aliases":"u_k","display":"u_k","kind":"use","excerpt":"_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnormu_1-u_0.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f6951bc1a235a45d","tex":"U_{k+1}","aliases":"U_k+1","display":"U_{k+1}","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-41ab4fffc65aae75","tex":"u_{k+1}","aliases":"u_k+1","display":"u_{k+1}","kind":"use","excerpt":"_R^W:=setw∈ W:norm w≤ R. Suppose normN(u)-N(v)≤ L_Nnormu-v (u,v∈ B_R^W), with q:=norm QL_N<1, and normQr+norm Qsup_u∈ B_R^WnormN(u)≤ R. Then there exists a unique u^*∈ B_R^W satisfying F(u^*)=0. The gauge-fixed iteration u_k+1=-Qr-QN(u_k) converges to u^* from every u_0∈ B_R^W, with normu_k-u^*≤fracq^k1-qnormu_1-u_0.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"use","excerpt":"U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n) is well-defined, remains in overline B(u_0,t_*), and converges to a root u^* of F. If the ","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b03775ada4b9cdf0","tex":"u_{n+1}","aliases":"u_n+1","display":"u_{n+1}","kind":"use","excerpt":"U and suppose F'(u_0) is invertible with inverse B. Assume that on the closed ball overline B(u_0,R), normF'(u)-F'(v)≤ Lnormu-v. Set η:=normBF(u_0), h:=norm B Lη. If h≤frac12 and t_*:=frac1-sqrt1-2hnorm B L≤ R, then Newton iteration u_n+1=u_n-F'(u_n)^-1F(u_n) is well-defined, remains in overline B(u_0,t_*), and converges to a root u^* of F. If the ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-31970f1f401cb3df","tex":"u_{n,\\alpha}","aliases":"u_n,α alpha","display":"u_{n,\\alpha}","kind":"use","excerpt":"At order n, let b_n,α∈mathscr V(U_α) be the residual coefficient determined by lower-order choices. The local obstruction is [b_n,α]∈coker D_α. If every local class vanishes, choose u_n,α with D_α u_n,α=-b_n,α. Their differences define an order-n gluing class in check H^1(ker D).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-30b08f45af646592","tex":"u_{n-1}","aliases":"u_n-1","display":"u_{n-1}","kind":"use","excerpt":"[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a6f2c5ce2aaa4a4","tex":"U_P","aliases":"U_P","display":"U_P","kind":"use","excerpt":"Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9c562e242027a749","tex":"U_P^\\top","aliases":"U_P^top top","display":"U_P^\\top","kind":"use","excerpt":"Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-61757fc440237661","tex":"U_{\\mathrm{rep}}^\\top","aliases":"U_rep^top mathrm top","display":"U_{\\mathrm{rep}}^\\top","kind":"use","excerpt":"The constructor context uses the recovered repair packet and a target repair of the form F(x)=Ax+b+(z^top Bz)v, z=U_rep^top x, with B≠0 and v≠0. The parent affine grammar is structurally obstructed by [source reference].","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-44229cca67dc816c","tex":"U_s","aliases":"U_s","display":"U_s","kind":"use","excerpt":"Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6ae21ddbb7e958a2","tex":"U_t","aliases":"U_t","display":"U_t","kind":"use","excerpt":"A grammar extension may also change the defect representation. Let D_t:U_t→ V_t, with transports T_t^U:U_t→ U_t+1, T_t^V:V_t→ V_t+1. Assume the exact correction square D_t+1T_t^U = T_t^VD_t. Then there is an induced obstruction map overline T_t: coker D_t → coker D_t+1, overline T_t([r])=[T_t^Vr].","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bc1520bebfe97d66","tex":"U_{t+1}^{\\Auth}","aliases":"U_t+1^Auth U_t+1^auth Auth mathrm","display":"U_{t+1}^{{\\mathrm{auth}}}","kind":"use","excerpt":"[Authorized obstruction-shrinkage theorem] Suppose U_t^auth⊆ U_t+1^auth and restrD_t+1U_t^auth=D_t. Then im D_t⊆im D_t+1, there is a canonical surjection q_t:coker D_t↠coker D_t+1, [v]_im D_t⟼[v]_im D_t+1, and dimcoker D_t+1≤dimcoker D_t. The inequality is strict if and only if im D_tsubsetneqim D_t+1.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1164a4a0480310ee","tex":"U_{t-1}","aliases":"U_t-1","display":"U_{t-1}","kind":"use","excerpt":"The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times: Y_t = (1-β)U_t+β U_t-1, t=2,…,5, with Y_1=U_1 and 0<β<1. In matrix form, Y=W_βU, where W_β is lower bidiagonal. The temporal constructor applies W_β^-1 before fitting the spatial gauge.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6411343e33175cf3","tex":"U_t^R","aliases":"U_t^R","display":"U_t^R","kind":"use","excerpt":"Let R_t be true protected regression and let U_t^R be a conditional upper confidence bound. Require U_t^R≤β_t, ∑_tβ_t≤ B_R. Let d_t be semantic debt incurred by approximate grammar or evaluator transport, with ∑_td_t≤ B_sem. A lineage certificate should therefore report three separate cumulative quantities: ( ∑_tδ_t, ∑_tβ_t, ∑_td_t ). A positive ga","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ad63d401332166ba","tex":"u_x","aliases":"u_x","display":"u_x","kind":"use","excerpt":"For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-458ceff2add52054","tex":"U_x^{\\Auth}","aliases":"U_x^Auth U_x^auth Auth mathrm","display":"U_x^{{\\mathrm{auth}}}","kind":"use","excerpt":"For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0956264b29fe2969","tex":"U_{\\alpha\\beta}","aliases":"U_αβ alpha beta","display":"U_{\\alpha\\beta}","kind":"use","excerpt":"[Linear repair-descent theorem] Assume a local correction u_α exists on every U_α. On overlaps U_αβ=U_α∩ U_β, define d_αβ := restru_αU_αβ - restru_βU_αβ. Then: item d_αβ∈ker D(U_αβ); item (d_αβ) is a vCech 1-cocycle with coefficients in ker D; item a global correction u∈Γ(X,mathscr U) with Du=-r exists if and only if (d_αβ) is a vCech coboundary re","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-225263ad49e9a5bf","tex":"U_\\beta","aliases":"U_β beta","display":"U_\\beta","kind":"use","excerpt":"[Linear repair-descent theorem] Assume a local correction u_α exists on every U_α. On overlaps U_αβ=U_α∩ U_β, define d_αβ := restru_αU_αβ - restru_βU_αβ. Then: item d_αβ∈ker D(U_αβ); item (d_αβ) is a vCech 1-cocycle with coefficients in ker D; item a global correction u∈Γ(X,mathscr U) with Du=-r exists if and only if (d_αβ) is a vCech coboundary re","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-529aedb9edf5f977","tex":"u_\\beta","aliases":"u_β beta","display":"u_\\beta","kind":"use","excerpt":"[Linear repair-descent theorem] Assume a local correction u_α exists on every U_α. On overlaps U_αβ=U_α∩ U_β, define d_αβ := restru_αU_αβ - restru_βU_αβ. Then: item d_αβ∈ker D(U_αβ); item (d_αβ) is a vCech 1-cocycle with coefficients in ker D; item a global correction u∈Γ(X,mathscr U) with Du=-r exists if and only if (d_αβ) is a vCech coboundary re","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-42fb4947708961ca","tex":"u_\\lambda","aliases":"u_λ lambda","display":"u_\\lambda","kind":"use","excerpt":"[Regularized minimum-cost repair] For λ>0, the functional J_λ(u) := frac12anglesDu+r,W(Du+r) + fracλ2anglesu,Ru has a unique minimizer u_λ =-(D^*WD+λ R)^-1D^*Wr. It is characterized by the normal equation D^*W(Du_λ+r)+λ Ru_λ=0.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dbf86aeb273c3fcb","tex":"\\underline F","aliases":"underline F underline","display":"\\underline F","kind":"use","excerpt":"When the exact envelope is unknown, certified lower and upper oracle bounds underline F_K(R) ≤ F_K(R) ≤ overline F_K(R) produce corresponding bounds on χ_R. Audit-oracle arms are finite empirical approximations to this idea, but they must remain inadmissible for selection because they use final audit outcomes.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a69746c16c75dde4","tex":"\\underline F_K","aliases":"underline F_K underline","display":"\\underline F_K","kind":"use","excerpt":"When the exact envelope is unknown, certified lower and upper oracle bounds underline F_K(R) ≤ F_K(R) ≤ overline F_K(R) produce corresponding bounds on χ_R. Audit-oracle arms are finite empirical approximations to this idea, but they must remain inadmissible for selection because they use final audit outcomes.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-85402f777e93b942","tex":"\\underline g_B","aliases":"underline g_B underline","display":"\\underline g_B","kind":"use","excerpt":"[Probabilistic gain preservation by simultaneous bounds] Given an (1-α) transport certificate, define underline g_B := · underline g_A -2overlineε_J · -overline L( 2overlineε_r +overlineε_Dnormu_A +overlineε_leak +overlineε_hol ). Then Pr(g_B≥underline g_B)≥1-α. In particular, underline g_B>0 is a valid one-sided certificate of beneficial target tr","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c533cd39f04eb58e","tex":"\\mathrm{utility}","aliases":"utility mathrm","display":"\\mathrm{utility}","kind":"use","excerpt":"Let protected keys be K_protected ={evidence,evaluator,provenance}. The adversarial candidate declares K_edit ={evaluator,utility}. Then K_edit∩ K_protected ={evaluator}≠∅. Even before considering utility, the acceptance predicate is false. The sandbox also changes the protected evaluator value, providing a second independent rejection reason. The ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"use","excerpt":"Let j:V↪ V' be an injective evidence-conservative embedding of old defects into an expanded ontology. Let C=im D⊆ V, C'=im D'⊆ V', and assume preservation of old correctability: j(C)⊆ C'. The embedding induces a map overline j:V/C→ V'/C'.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0a333431c75f50a1","tex":"V^{\\mathsf T}","aliases":"V^ T mathsf","display":"V^{\\mathsf T}","kind":"use","excerpt":"For parent and child state spaces H_A and H_B of equal dimension, let calibration traces produce matrices X and Y. The orthogonal Procrustes transport is T^* = arg min_T^ TT=InormXT-Y_ F. If X^ TY=UΣ V^ T, then T^*=UV^ T. A transport receipt records calibration residual, held-out residual, singular floor, round-trip residual, and behavior change af","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a8b2a9721419c0b6","tex":"v^{\\mathsf T}","aliases":"v^ T mathsf","display":"v^{\\mathsf T}","kind":"use","excerpt":"[A-optimal trace reduction] Let v=sqrtw_ix_i. Then tr(M^-1)-tr((M+vv^ T)^-1) = fracv^ TM^-2v1+v^ TM^-1v.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"Let the old correction image be I=span{e_1,e_2}⊂ℝ^4 and let r=(0,0,3,4). Then z=P_I^⊥r=(0,0,3,4), η_I(r)^2=25. Add generator v=2e_4. Because v∈ I^⊥, frac|anglesz,v|^2norm v^2 = frac8^24 =16. The expanded residual is 3e_3, so η_I'(r)^2=9=25-16. A generator parallel to e_1 would have bar v=0 and would produce no structural reduction despite having no","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6b104ac6b34a135f","tex":"v^\\top","aliases":"v^top top","display":"v^\\top","kind":"use","excerpt":"When X has full column rank, X^top X is invertible and (X^top X)^-1X^top Y =(X^top X)^-1X^top XT^top =T^top. When rank X<d, choose a nonzero v∈ker X. For any nonzero w∈ℝ^m, define K=wv^top. Then XK^top=0, so T and T+K agree on every selected anchor while differing on v.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d3eb6d43b35f5005","tex":"v^\\star","aliases":"v^⋆ star","display":"v^\\star","kind":"use","excerpt":"Equation [source reference] gives a direct acquisition score. For a candidate family V_cand, choose v^⋆ ∈ arg max_v∈V_cand fracabsanglesz,P_I^⊥v^2 normP_I^⊥v^2+λ c(v)+βRisk(v), where c(v) is implementation cost and Risk(v) is a protected-risk estimate.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"use","excerpt":"p embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z_k+σ(Gz_k)odot tanh(V_2GELU(V_1z_k)+s_k+δ_k(e,z_k)). The three cell classes have distinct executable implementatio","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-641dfc399b115eba","tex":"V_4","aliases":"V_4","display":"V_4","kind":"use","excerpt":"efine z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following conditioning maps: C_depth(e,m) · =D 1_m, C_1(e,m) · =U_1V_1e, rank(U_1V_1)≤1, C_4(e,m) · =U_4V_4e, rank(U_4V_4)≤4, C_full(e,m) · =We. Here 1_m is the one-hot depth c","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-91a52a41351c928c","tex":"V_i","aliases":"V_i","display":"V_i","kind":"use","excerpt":"Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fa886686fadc3c64","tex":"V_k","aliases":"V_k","display":"V_k","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cdebe96f7a1525e6","tex":"V_{k+1}","aliases":"V_k+1","display":"V_{k+1}","kind":"use","excerpt":"[Pathwise transport bound] Consider a path of systems A_0xrightarrow T_0A_1 xrightarrow T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bcd4d5b31c0465be","tex":"V_{t+1}","aliases":"V_t+1","display":"V_{t+1}","kind":"use","excerpt":"Open-set improvement may enlarge the defect ontology itself. Let j_U:U_t^auth↪ U_t+1^auth, j_V:V_t↪ V_t+1 be injective embeddings satisfying D_t+1j_U=j_VD_t.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5bdb72c35c6f53a4","tex":"v_W","aliases":"v_W","display":"v_W","kind":"use","excerpt":"An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a5f540ac7ccd2ba5","tex":"V_x","aliases":"V_x","display":"V_x","kind":"use","excerpt":"For episode x, let F_x:U_x^auth→ V_x be the post-modification defect map, with F_x(0)=r_x. A valid repair satisfies F_x(u_x)=0. A repair schema is a local section s:x↦ u_x that approximately inverts F_x over its declared domain.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-694bdc6e24088dc1","tex":"\\mathrm{val}","aliases":"val mathrm","display":"\\mathrm{val}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4471808a5edd2986","tex":"\\operatorname{Valid}","aliases":"Valid operatorname","display":"\\operatorname{Valid}","kind":"use","excerpt":"Let Valid(c_i,Π_i^+)∈{0,1} encode capability, integrity, provenance, and resource validity. The selected descendant is K_Γ^⋆(τ) ∈ arg max_i:Valid(c_i,Π_i^+)=1 S_V(c_i), with the no-op baseline retained when no valid candidate exists.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-37273846385908d4","tex":"\\mathsf{Value}","aliases":"Value mathsf","display":"\\mathsf{Value}","kind":"use","excerpt":"An executable state is a finite canonical map widehat s:Key→Value. A witness policy is widehat W =(K_protected,γ_min,r_max,p_prov,e_irr,v_W), where K_protected is the immutable-key set, γ_min is the minimum protected gain, r_max is the maximum accepted risk, p_prov indicates whether provenance is mandatory, e_irr controls irreversible effects, and ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c0e8009c8acb599b","tex":"\\Var","aliases":"Var operatorname","display":"{\\operatorname{Var}}","kind":"use","excerpt":"[Worst-direction variance] For every u∈ℝ^d and every target coordinate, Var(u^ T(widehat T^ T-T^ T)e_jmid X) =u^ TM_S^-1u ≤ fracnormu^2λ_min(M_S).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-53c8d505857d9af6","tex":"\\Vcal_{\\mathrm{cand}}","aliases":"Vcal_cand V_cand Vcal mathrm mathcal mathrm","display":"{\\mathcal{V}}_{\\mathrm{cand}}","kind":"use","excerpt":"Equation [source reference] gives a direct acquisition score. For a candidate family V_cand, choose v^⋆ ∈ arg max_v∈V_cand fracabsanglesz,P_I^⊥v^2 normP_I^⊥v^2+λ c(v)+βRisk(v), where c(v) is implementation cost and Risk(v) is a protected-risk estimate.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d6ba1414a3b8d8e5","tex":"\\Vcal_t^{\\mathrm{fr}}","aliases":"Vcal_t^fr V_t^fr Vcal mathrm mathcal mathrm","display":"{\\mathcal{V}}_t^{\\mathrm{fr}}","kind":"use","excerpt":"A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f7cc81212c08d083","tex":"\\operatorname{vec}","aliases":"vec operatorname","display":"\\operatorname{vec}","kind":"use","excerpt":"[Weighted bridge covariance] Assume M_S is invertible and the noise rows are independent with covariance σ_i^2I_q. The weighted least-squares estimator widehat T^ T = M_S^-1X^ TWY is unbiased and satisfies Cov(vec(widehat T^ T-T^ T)mid X) = I_q⊗ M_S^-1. Consequently, E[normwidehat T-T_F^2mid X] = q tr(M_S^-1).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c68c8944583aeb66","tex":"\\vech","aliases":"vech operatorname","display":"{\\operatorname{vech}}","kind":"use","excerpt":"Let U_P∈ℝ^d× r have orthonormal columns spanning the recovered repair packet. Define packet coordinates z=U_P^top(x-c). The degree-two block-local grammar consists of maps F_A,b,C(x) = Ax+b+C vech(zz^top), where vech lists the r(r+1)/2 distinct symmetric quadratic monomials.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c358e9cb60086d4f","tex":"\\mathrm{verifier}","aliases":"verifier mathrm","display":"\\mathrm{verifier}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8e6a89b7172263c2","tex":"\\operatorname{Verify}","aliases":"Verify operatorname","display":"\\operatorname{Verify}","kind":"use","excerpt":"Let P_G be a finite typed repair-program grammar and let Verify(p,x)∈{0,1} be a complete verifier on a finite declared episode domain.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-00883947feeb7d79","tex":"w^{\\mathsf T}","aliases":"w^ T mathsf","display":"w^{\\mathsf T}","kind":"use","excerpt":"[Archive-envelope monotonicity] Suppose archive updates never delete the last representative of a retained node. For every nonnegative weight vector w, define H_w(A_t) = max_A∈A_tw^ Tz(A). Then H_w(A_t) is nondecreasing in t.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-11fdec97e82b0ec6","tex":"W^{-1}","aliases":"W^-1","display":"W^{-1}","kind":"use","excerpt":"Write widehat T^ T-T^ T = M_S^-1X^ TWE. For one target coordinate, the noise covariance is W^-1, so the coefficient covariance is M_S^-1X^ TW W^-1 WXM_S^-1=M_S^-1. The target coordinates are independent and identically weighted, giving the Kronecker covariance. Taking the trace yields the Frobenius identity.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-796be44ebe162e5d","tex":"W_E","aliases":"W_E","display":"W_E","kind":"use","excerpt":"Different overlaps may have different uncertainty or semantic importance. Let W_E be a positive-definite operator on K^E. Define η_glue,W(e) := inf_knorme+ B^*k_W_E.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-18864b29b043b6d8","tex":"W_i","aliases":"W_i","display":"W_i","kind":"use","excerpt":"widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1+c_i+e_k), k=1,…,5, where e_k is a learned step embedding. A gated readout recurrence maps the trace to","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-327206a4d09f1e47","tex":"w_{\\rm capability}","aliases":"w_rm capability rm","display":"w_{\\rm capability}","kind":"use","excerpt":"The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-39253f8567a3c944","tex":"w_{\\rm held}","aliases":"w_rm held rm","display":"w_{\\rm held}","kind":"use","excerpt":"The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ea7b44df9ac1a00","tex":"w_{\\rm mean}","aliases":"w_rm mean rm","display":"w_{\\rm mean}","kind":"use","excerpt":"The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ce126007307c715b","tex":"w_{\\rm median}","aliases":"w_rm median rm","display":"w_{\\rm median}","kind":"use","excerpt":"The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-84808076088d2144","tex":"w_{\\rm min}","aliases":"w_rm min rm","display":"w_{\\rm min}","kind":"use","excerpt":"The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f95b6d9373f7fd2e","tex":"W_{\\beta}","aliases":"W_β beta","display":"W_{\\beta}","kind":"use","excerpt":"The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times: Y_t = (1-β)U_t+β U_t-1, t=2,…,5, with Y_1=U_1 and 0<β<1. In matrix form, Y=W_βU, where W_β is lower bidiagonal. The temporal constructor applies W_β^-1 before fitting the spatial gauge.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-56ac6f72087eab3f","tex":"W_{\\beta}^{-1}","aliases":"W_β^-1 beta","display":"W_{\\beta}^{-1}","kind":"use","excerpt":"The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times: Y_t = (1-β)U_t+β U_t-1, t=2,…,5, with Y_1=U_1 and 0<β<1. In matrix form, Y=W_βU, where W_β is lower bidiagonal. The temporal constructor applies W_β^-1 before fitting the spatial gauge.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a00b1764fd0d06f","tex":"\\Wcal_t^{\\mathrm{NL}}","aliases":"Wcal_t^NL W_t^NL Wcal mathrm mathcal mathrm","display":"{\\mathcal{W}}_t^{\\mathrm{NL}}","kind":"use","excerpt":"Extend the SRIC lineage state by L_t^NL = ( L_t,N_t,I_t^NL,T_t^NL,W_t^NL), where T_t^NL contains trace-transport certificates and W_t^NL is the frozen external answer and invariant witness. A proposed descendant may modify the latent transition, update gate, halting head, packet atlas, or transport map. The backbone may be held fixed for a componen","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8d3781ce495edcf4","tex":"\\widehat A","aliases":"widehat A widehat","display":"\\widehat A","kind":"use","excerpt":"The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm ali","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-08cb36a41922768b","tex":"\\widehat{\\mathsf A}","aliases":"widehat A widehat mathsf","display":"\\widehat{\\mathsf A}","kind":"use","excerpt":"[Stability of the threshold count] Let widehat A be an estimated holonomy-energy operator satisfying normwidehat A- A≤η. Then each ordered eigenvalue satisfies abswidehatλ_i-λ_i≤η. If no eigenvalue of A lies in [τ-η,τ+η], then widehat A and A have the same number of eigenvalues below τ.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bfce72e9a2acbb54","tex":"\\widehat a_{\\mathrm{repair\\ inv}}","aliases":"widehat a_repair inv widehat mathrm","display":"\\widehat a_{\\mathrm{repair\\ inv}}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-299f61122196e048","tex":"\\widehat A_{\\rm non}^{\\rm base}","aliases":"widehat A_rm non^rm base widehat rm rm","display":"\\widehat A_{\\rm non}^{\\rm base}","kind":"use","excerpt":"The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm ali","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c4b3d10073c5b14b","tex":"\\widehat a_{\\mathrm{target}}","aliases":"widehat a_target widehat mathrm","display":"\\widehat a_{\\mathrm{target}}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1b3f66f942a5fe9a","tex":"\\widehat B","aliases":"widehat B widehat","display":"\\widehat B","kind":"use","excerpt":"In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ea83107951d84fac","tex":"\\widehat b","aliases":"widehat b widehat","display":"\\widehat b","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6839f21cc5979275","tex":"\\widehat b_{\\mathrm{chart}}","aliases":"widehat b_chart widehat mathrm","display":"\\widehat b_{\\mathrm{chart}}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"use","excerpt":"Define the finite structural coverage functional C_rm id( P) = Pr_A∼D_rm for [ ank X_A=d_s]. Then widehat C_rm id( P_1) - widehat C_rm id( P_0) = 0.96875. This is the measured bridge-identification direction added by the meta-repair.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e7a695b8f5409721","tex":"\\widehat c","aliases":"widehat c widehat","display":"\\widehat c","kind":"use","excerpt":"[Cost-sensitive regret] Suppose, in addition, that normalized intervention cost c(δ) is predicted by widehat c(δ) with uniform error at most ε_c. For net utility W(δ)=V(δ)-λ c(δ), widehat W(δ)=widehat V(δ)-λwidehat c(δ), selection by widehat W has regret at most 2(ε+λε_c).","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a7c5fd43be199c25","tex":"\\widehat C_{\\rm id}","aliases":"widehat C_rm id widehat rm","display":"\\widehat C_{\\rm id}","kind":"use","excerpt":"Define the finite structural coverage functional C_rm id( P) = Pr_A∼D_rm for [ ank X_A=d_s]. Then widehat C_rm id( P_1) - widehat C_rm id( P_0) = 0.96875. This is the measured bridge-identification direction added by the meta-repair.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d1655f35bb316a56","tex":"\\widehat D","aliases":"widehat D widehat","display":"\\widehat D","kind":"use","excerpt":"[Transferred residual under model error] Let widehat D=D+Δ D and widehat r=r+Δ r. For any proposed repair u, absnormDu+r-normwidehat Du+widehat r ≤ normΔ Dnorm u+normΔ r. In particular, applying an exact repair for (D,r) to the perturbed system yields residual at most normΔ Dnorm u+normΔ r.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-05aa9850c311f7c2","tex":"\\widehat D^*","aliases":"widehat D^* widehat","display":"\\widehat D^*","kind":"use","excerpt":"[Robust empirical noncorrectability certificate] Suppose, on an event of probability at least 1-α, normD-widehat D≤ε_D, normr-widehat r≤ε_r. Fix a repair budget norm u≤ B and a unit vector y∈ V. If anglesy,widehat r > ε_r+B(normwidehat D^*y+ε_D), then, on the same event, no budget-feasible repair satisfies Du=-r.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6dc8b8e8ada96dbb","tex":"\\widehat D_t","aliases":"widehat D_t widehat","display":"\\widehat D_t","kind":"use","excerpt":"[Exact repair-operator perturbation identity] Let widehat D_t=widehat TD_swidehat T^†. Then widehat D_t-D_t = (widehat T-T)D_swidehat T^† + T D_s(widehat T^†-T^†). Hence normwidehat D_t-D_t ≤ normwidehat T-TnormD_snormwidehat T^† + normTnormD_snormwidehat T^†-T^†.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4ae2c725a0324b8f","tex":"\\widehat G","aliases":"widehat G widehat","display":"\\widehat G","kind":"use","excerpt":"Let G be the ideal generic response combination from the proof of [source reference], and let widehat G=G+E be its empirical estimate. Let P be a spectral projector associated with an eigenvalue cluster separated from the rest of the spectrum by gap γ>0.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f238cfc4b9f6f57e","tex":"\\widehat g_i","aliases":"widehat g_i widehat","display":"\\widehat g_i","kind":"use","excerpt":"For candidate c_i, let z_i collect its pre-result receipt fields and cheap pilot evidence. A research-outcome forecaster returns F_θ(z_i) = (widehat g_i,widehat r_i,widehat s_i), where widehat g_i predicts hidden protected gain, widehat r_i predicts protected risk, and widehat s_i predicts scored-evaluator performance.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8a410bcd9385829f","tex":"\\widehat g_{i^\\star}","aliases":"widehat g_i^⋆ widehat star","display":"\\widehat g_{i^\\star}","kind":"use","excerpt":"By optimality of widehat i under the predicted values, widehat g_widehat i≥widehat g_i^⋆. Therefore g_i^⋆ ≤ widehat g_i^⋆+ε ≤ widehat g_widehat i+ε ≤ g_widehat i+2ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-19b9f4d36640f904","tex":"\\widehat g_{\\rm cal}","aliases":"widehat g_rm cal widehat rm","display":"\\widehat g_{\\rm cal}","kind":"use","excerpt":"For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6af707d36cec930f","tex":"\\widehat g_{\\rm slice}","aliases":"widehat g_rm slice widehat rm","display":"\\widehat g_{\\rm slice}","kind":"use","excerpt":"The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm ali","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-584b3d4cc0b5f3c2","tex":"\\widehat g_{\\mathrm{val}}","aliases":"widehat g_val widehat mathrm","display":"\\widehat g_{\\mathrm{val}}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e70cf9e62de33f69","tex":"\\widehat g_{\\widehat i}","aliases":"widehat g_widehat i widehat widehat","display":"\\widehat g_{\\widehat i}","kind":"use","excerpt":"By optimality of widehat i under the predicted values, widehat g_widehat i≥widehat g_i^⋆. Therefore g_i^⋆ ≤ widehat g_i^⋆+ε ≤ widehat g_widehat i+ε ≤ g_widehat i+2ε.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-66d5bee1680ae2b3","tex":"\\widehat h","aliases":"widehat h widehat","display":"\\widehat h","kind":"use","excerpt":"A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribut","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d1972479dc39f70b","tex":"\\widehat h_{a,c}","aliases":"widehat h_a,c widehat","display":"\\widehat h_{a,c}","kind":"use","excerpt":"tent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribute th","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-267b1fb3bd56ceac","tex":"\\widehat h_{a,c}^{(k)}","aliases":"widehat h_a,c^(k) widehat","display":"\\widehat h_{a,c}^{(k)}","kind":"use","excerpt":"A latent-step-shared map acts independently on each recurrent state: widehat h_a,c^(k) = A_c h_b,c^(k)+b_c, k=1,…,5. A full-trajectory map instead acts on the concatenation vec(h_b,c^(1),…,h_b,c^(5)) ∈ R^200, allowing cross-step mixing: vec(widehat h_a,c) = M_cvec(h_b,c)+q_c. This difference is substantive when two recurrent architectures distribut","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b1e3c82c1e7bdadb","tex":"\\widehat i","aliases":"widehat i widehat","display":"\\widehat i","kind":"use","excerpt":"[Uniform forecast-regret bound] Let g_i be the true protected gain of each admissible candidate and suppose |widehat g_i-g_i|≤ε for all candidates. If widehat i∈arg max_iwidehat g_i and i^⋆∈arg max_i g_i, then g_i^⋆-g_widehat i ≤ 2ε.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae91cd5f0de975af","tex":"\\widehat J","aliases":"widehat J widehat","display":"\\widehat J","kind":"use","excerpt":"The global-edit arm contains a candidate whose apparent score receives an exploitable bonus when its rewrite norm is large. The protected evaluator does not receive this bonus. This creates a controlled instance of evaluator capture: widehat J(δ)>widehat J(0) while J(δ)<J(0). A witness-aware arm ranks and commits by J, not widehat J.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a410e85313b42614","tex":"\\widehat M","aliases":"widehat M widehat","display":"\\widehat M","kind":"use","excerpt":"On 144 primary foreign targets, the target-level paired difference is widehatΔ_rm meta = widehat M_24^rm for( P_1) - widehat M_24^rm for( P_0) = 0.209508. A target-cluster bootstrap gives the interval [0.194968,0.223898], and a 100,000-draw sign-flip test gives p=9.9999×10^-6. All 144 target-level paired differences are positive.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9d3cbafc88d39f52","tex":"\\widehat{\\mathcal M}","aliases":"widehat M widehat mathcal","display":"\\widehat{\\mathcal M}","kind":"use","excerpt":"Let M=(S, T,O) be an abstract transition system with state set S, authorized labeled transitions T, and protected observation map O. Let widehat M=(widehat S,widehat T,widehat O) be an executable implementation.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b4f65a75c0c7658a","tex":"\\widehat M_{24}^{\\rm for}","aliases":"widehat M_24^rm for widehat rm","display":"\\widehat M_{24}^{\\rm for}","kind":"use","excerpt":"On 144 primary foreign targets, the target-level paired difference is widehatΔ_rm meta = widehat M_24^rm for( P_1) - widehat M_24^rm for( P_0) = 0.209508. A target-cluster bootstrap gives the interval [0.194968,0.223898], and a 100,000-draw sign-flip test gives p=9.9999×10^-6. All 144 target-level paired differences are positive.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-052f5530f44bfebc","tex":"\\widehat M_{\\mathrm{foreign}}","aliases":"widehat M_foreign widehat mathrm","display":"\\widehat M_{\\mathrm{foreign}}","kind":"use","excerpt":"Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inhe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a4a3f64778da99d","tex":"\\widehat O_0","aliases":"widehat O_0 widehat","display":"\\widehat O_0","kind":"use","excerpt":"The development set contains 56 open-set problems. The inherited base-method oracle obtains mean gain widehat O_0=0.377257. No one inherited unknown-token binding supplies a common high-gain method across the cell.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1c93524398ef8124","tex":"\\widehat O_1","aliases":"widehat O_1 widehat","display":"\\widehat O_1","kind":"use","excerpt":"novel-high · ⟼ objective-then-constructor. The expanded development gain is widehat O_1=0.948962, with newly correctable margin widehatκ = 0.571704.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"use","excerpt":"[Spectral stability of a recovered atlas block] Suppose B is self-adjoint, normE_2<γ_a/2, and widehat P_a is the spectral projector of widehat B onto the eigenvalue cluster corresponding to μ_a. Then normwidehat P_a-P_a_2 ≤ frac2normE_2γ_a.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9fee892982777562","tex":"\\widehat P_1","aliases":"widehat P_1 widehat","display":"\\widehat P_1","kind":"use","excerpt":"For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be86ba60abd33935","tex":"\\widehat P_a","aliases":"widehat P_a widehat","display":"\\widehat P_a","kind":"use","excerpt":"[Spectral stability of a recovered atlas block] Suppose B is self-adjoint, normE_2<γ_a/2, and widehat P_a is the spectral projector of widehat B onto the eigenvalue cluster corresponding to μ_a. Then normwidehat P_a-P_a_2 ≤ frac2normE_2γ_a.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-266ff731dd16faeb","tex":"\\widehat P_m","aliases":"widehat P_m widehat","display":"\\widehat P_m","kind":"use","excerpt":"For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3e63f3d4e5130d54","tex":"\\widehat q","aliases":"widehat q widehat","display":"\\widehat q","kind":"use","excerpt":"and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recurrence generates the Neuralese trace r_k=tanh(W_ih_0+b_i+U_ir_k-1","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-967c63ebf4cd78be","tex":"\\widehat R","aliases":"widehat R widehat","display":"\\widehat R","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_","count":13},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-644e773747d7540b","tex":"\\widehat r","aliases":"widehat r widehat","display":"\\widehat r","kind":"use","excerpt":"[Transferred residual under model error] Let widehat D=D+Δ D and widehat r=r+Δ r. For any proposed repair u, absnormDu+r-normwidehat Du+widehat r ≤ normΔ Dnorm u+normΔ r. In particular, applying an exact repair for (D,r) to the perturbed system yields residual at most normΔ Dnorm u+normΔ r.","count":14},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5db9b7abf5078cf8","tex":"\\widehat r_{\\mathrm{ctx}}","aliases":"widehat r_ctx widehat mathrm","display":"\\widehat r_{\\mathrm{ctx}}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-79ce3b7c8f35eab4","tex":"\\widehat r_i","aliases":"widehat r_i widehat","display":"\\widehat r_i","kind":"use","excerpt":"For candidate c_i, let z_i collect its pre-result receipt fields and cheap pilot evidence. A research-outcome forecaster returns F_θ(z_i) = (widehat g_i,widehat r_i,widehat s_i), where widehat g_i predicts hidden protected gain, widehat r_i predicts protected risk, and widehat s_i predicts scored-evaluator performance.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fc61595aa88a2201","tex":"\\widehat R_j","aliases":"widehat R_j widehat","display":"\\widehat R_j","kind":"use","excerpt":"In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-241f860f4977397c","tex":"\\widehat R_{\\rm cal}","aliases":"widehat R_rm cal widehat rm","display":"\\widehat R_{\\rm cal}","kind":"use","excerpt":"For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d1c8c43d41ebbddb","tex":"\\widehat r_{\\rm cal}","aliases":"widehat r_rm cal widehat rm","display":"\\widehat r_{\\rm cal}","kind":"use","excerpt":"widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2533af5c9930e604","tex":"\\widehat r_{\\rm non}","aliases":"widehat r_rm non widehat rm","display":"\\widehat r_{\\rm non}","kind":"use","excerpt":"The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm ali","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-74dc1e5c7b5fcaa7","tex":"\\widehat R_{\\rm UCB}","aliases":"widehat R_rm UCB widehat rm","display":"\\widehat R_{\\rm UCB}","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4abfa4ea05d50eb8","tex":"\\widehat s_{a^*}","aliases":"widehat s_a^* widehat","display":"\\widehat s_{a^*}","kind":"use","excerpt":"For every a≠ a^*, widehat s_a^*-widehat s_a · ≥ s_a^*-s_a -|widehat s_a^*-s_a^*| -|widehat s_a-s_a| · >m-frac m2-frac m2=0.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-94587314ffbe3e7f","tex":"\\widehat s_i","aliases":"widehat s_i widehat","display":"\\widehat s_i","kind":"use","excerpt":"For candidate c_i, let z_i collect its pre-result receipt fields and cheap pilot evidence. A research-outcome forecaster returns F_θ(z_i) = (widehat g_i,widehat r_i,widehat s_i), where widehat g_i predicts hidden protected gain, widehat r_i predicts protected risk, and widehat s_i predicts scored-evaluator performance.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b2e198011c8e8a0e","tex":"\\widehat s_{\\mathrm{model}}","aliases":"widehat s_model widehat mathrm","display":"\\widehat s_{\\mathrm{model}}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b0d65d8f4ac93228","tex":"\\widehat T","aliases":"widehat T widehat","display":"\\widehat T","kind":"use","excerpt":"h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.","count":27},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fb38f2e7b6f5abb8","tex":"\\widehat{\\mathcal T}","aliases":"widehat T widehat mathcal","display":"\\widehat{\\mathcal T}","kind":"use","excerpt":"Let M=(S, T,O) be an abstract transition system with state set S, authorized labeled transitions T, and protected observation map O. Let widehat M=(widehat S,widehat T,widehat O) be an executable implementation.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16f38b7594169b17","tex":"\\widehat T^{\\mathsf T}","aliases":"widehat T^ T widehat mathsf","display":"\\widehat T^{\\mathsf T}","kind":"use","excerpt":"[Weighted bridge covariance] Assume M_S is invertible and the noise rows are independent with covariance σ_i^2I_q. The weighted least-squares estimator widehat T^ T = M_S^-1X^ TWY is unbiased and satisfies Cov(vec(widehat T^ T-T^ T)mid X) = I_q⊗ M_S^-1. Consequently, E[normwidehat T-T_F^2mid X] = q tr(M_S^-1).","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-490f70df87141b22","tex":"\\widehat T^{+}","aliases":"widehat T^+ widehat","display":"\\widehat T^{+}","kind":"use","excerpt":"Given source repair R_s(h)=h+D_sh+b_s, its transported target form is D_t = widehat T D_swidehat T^+, b_t = widehat T b_s, with transported packet basis obtained by orthogonalizing widehat T U_s.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-49e1235e48e86320","tex":"\\widehat T^\\top","aliases":"widehat T^top widehat top","display":"\\widehat T^\\top","kind":"use","excerpt":"h_t(x_k)^top , and fits widehat T^top = (X^top X+λ I)^-1X^top Y. The parent chooses the k=16 fit anchors uniformly at random. The source hidden space is also 16-dimensional, so a rank-deficient design leaves at least one source direction unidentified.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4b95989c172255c3","tex":"\\widehat T^{\\top}","aliases":"widehat T^top widehat top","display":"\\widehat T^{\\top}","kind":"use","excerpt":"The normal equations are H_s^topH_swidehat T^top=H_s^topH_t. Full column rank makes H_s^topH_s invertible. Substitution of H_t=H_sT^top yields widehat T=T.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9245792c878d33a8","tex":"\\widehat T^\\dagger","aliases":"widehat T^† widehat dagger","display":"\\widehat T^\\dagger","kind":"use","excerpt":"[Exact repair-operator perturbation identity] Let widehat D_t=widehat TD_swidehat T^†. Then widehat D_t-D_t = (widehat T-T)D_swidehat T^† + T D_s(widehat T^†-T^†). Hence normwidehat D_t-D_t ≤ normwidehat T-TnormD_snormwidehat T^† + normTnormD_snormwidehat T^†-T^†.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0b4831544e5fbaa4","tex":"\\widehat T_{t\\leftarrow s}","aliases":"widehat T_t← s widehat leftarrow","display":"\\widehat T_{t\\leftarrow s}","kind":"use","excerpt":"h_t(x_m)^top . It estimates the bridge by regularized least squares: widehat T_t← s ∈ arg min_T normH_sT^top-H_t_F^2 +λnormT_F^2. The transformation matrices used to construct the controlled targets are not exposed to this fit.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5ec76de4222f5aa9","tex":"\\widehat U","aliases":"widehat U widehat","display":"\\widehat U","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-31b4d0331a96e27b","tex":"\\widehat U_n","aliases":"widehat U_n widehat","display":"\\widehat U_n","kind":"use","excerpt":"[Finite meta-grammar uniform bound] Suppose, for each γ∈Γ, the per-episode utility lies in an interval of width L, and the n meta-validation episodes are independent. Then with probability at least 1-δ, sup_γ∈Γ | widehat U_n(γ)-U(γ) | ≤ Lsqrtfraclog(2|Γ|/δ)2n. Consequently, the true utility of the empirical maximizer is within twice the right-hand ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d40e88c2e875f4b","tex":"\\widehat U_{\\rm LCB}","aliases":"widehat U_rm LCB widehat rm","display":"\\widehat U_{\\rm LCB}","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dc5e832c68a33db9","tex":"\\widehat V","aliases":"widehat V widehat","display":"\\widehat V","kind":"use","excerpt":"[Decision regret from relative introspection] Let u:Y→[0,1] be a bounded utility. Define V(δ)= E[u(Y)mid A_t,δ], widehat V(δ)= E[u(Y)mid ϕ_t(A_t),δ]. If ϕ_t is ε-intervention-sufficient, then sup_δ∈D_t absV(δ)-widehat V(δ)≤ε. If δ^*∈arg max_δ∈D_tV(δ), widehatδ∈arg max_δ∈D_twidehat V(δ), then V(δ^*)-V(widehatδ)≤2ε.","count":11},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-15ccf04c00615671","tex":"\\widehat W","aliases":"widehat W widehat","display":"\\widehat W","kind":"use","excerpt":"[Cost-sensitive regret] Suppose, in addition, that normalized intervention cost c(δ) is predicted by widehat c(δ) with uniform error at most ε_c. For net utility W(δ)=V(δ)-λ c(δ), widehat W(δ)=widehat V(δ)-λwidehat c(δ), selection by widehat W has regret at most 2(ε+λε_c).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-74af88e0c8d8aec9","tex":"\\widehat w_k","aliases":"widehat w_k widehat","display":"\\widehat w_k","kind":"use","excerpt":"The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-49ebf6d103f66061","tex":"\\widehat x_i","aliases":"widehat x_i widehat","display":"\\widehat x_i","kind":"use","excerpt":"Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ef2043e78b4fc9ff","tex":"\\widehat Y","aliases":"widehat Y widehat","display":"\\widehat Y","kind":"use","excerpt":"reign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The transported semantic prediction is widehat Y_t(τ) =P_s(B_tH_t(τ)+b_t). The bridge gate requires effecti","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4c8b9ae3509fa24d","tex":"\\widehat y_i","aliases":"widehat y_i widehat","display":"\\widehat y_i","kind":"use","excerpt":"Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a19c22dd304b0a95","tex":"\\widehat Y_t","aliases":"widehat Y_t widehat","display":"\\widehat Y_t","kind":"use","excerpt":"reign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The transported semantic prediction is widehat Y_t(τ) =P_s(B_tH_t(τ)+b_t). The bridge gate requires effecti","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c88f3c79e4ba1c43","tex":"\\widehat z","aliases":"widehat z widehat","display":"\\widehat z","kind":"use","excerpt":"[Prototype-separation error bound] Let Δ = min_z≠ z'normμ_z-μ_z'. Under equal priors and isotropic Gaussian noise, nearest-prototype classification satisfies Pr(widehat z≠ z) ≤ (| Z|-1) exp(-fracΔ^28σ^2).","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2b8a778a98d33302","tex":"\\widehat{\\operatorname{coverage}}","aliases":"widehatcoverage widehat operatorname","display":"\\widehat{\\operatorname{coverage}}","kind":"use","excerpt":"The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0e766119025baae3","tex":"\\widehat{\\operatorname{coverage}}_{A}","aliases":"widehatcoverage_A widehat operatorname","display":"\\widehat{\\operatorname{coverage}}_{A}","kind":"use","excerpt":"The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-183766978b7f7efd","tex":"\\widehat{\\operatorname{coverage}}_{N}","aliases":"widehatcoverage_N widehat operatorname","display":"\\widehat{\\operatorname{coverage}}_{N}","kind":"use","excerpt":"The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d708ec79c176c21a","tex":"\\widehat\\eps","aliases":"widehateps widehatε widehat eps widehat varepsilon","display":"\\widehat{\\varepsilon}","kind":"use","excerpt":"The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm ali","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0c4a1e391dc82285","tex":"\\widehat\\eps_{\\mathrm{align}}","aliases":"widehateps_align widehatε_align widehat eps mathrm widehat varepsilon mathrm","display":"\\widehat{\\varepsilon}_{\\mathrm{align}}","kind":"use","excerpt":"widehatε_align · Calibration normalized hidden-state alignment error","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-215574552388a4ab","tex":"\\widehat\\eps_{\\mathrm{chart}}","aliases":"widehateps_chart widehatε_chart widehat eps mathrm widehat varepsilon mathrm","display":"\\widehat{\\varepsilon}_{\\mathrm{chart}}","kind":"use","excerpt":"All 54 development targets have full source-design rank. The inherited global bridge has mean validation alignment NRMSE widehatε_glob=0.632211, whereas the learned two-chart constructor has widehatε_chart=0.068871. The mean reduction is 0.563340.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fd682423ce8dd77c","tex":"\\widehat\\eps_{\\mathrm{comm}}","aliases":"widehateps_comm widehatε_comm widehat eps mathrm widehat varepsilon mathrm","display":"\\widehat{\\varepsilon}_{\\mathrm{comm}}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7106b4bc5187eea1","tex":"\\widehat\\eps_{\\mathrm{glob}}","aliases":"widehateps_glob widehatε_glob widehat eps mathrm widehat varepsilon mathrm","display":"\\widehat{\\varepsilon}_{\\mathrm{glob}}","kind":"use","excerpt":"All 54 development targets have full source-design rank. The inherited global bridge has mean validation alignment NRMSE widehatε_glob=0.632211, whereas the learned two-chart constructor has widehatε_chart=0.068871. The mean reduction is 0.563340.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ee1512939bb9e2a","tex":"\\widehat\\eps_{\\mathrm{resp}}","aliases":"widehateps_resp widehatε_resp widehat eps mathrm widehat varepsilon mathrm","display":"\\widehat{\\varepsilon}_{\\mathrm{resp}}","kind":"use","excerpt":"The selected atlas candidate commits only when all frozen conditions hold: widehat g_val · ≥0.10, LCB(widehat g_val) · ≥0.05, widehat r_ctx · ≤3×10^-4, widehat a_target · ≥0.72, widehat a_repair inv · ≥0.96, widehat b_chart · ≥0.20, widehat s_model · ≤0.25, widehatγ_min · ≥0.004, widehatε_comm · ≤0.16, widehatε_resp · ≤0.08. The witness therefore e","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-97117fc299f0b71e","tex":"\\widehat\\eps_{\\rm align}","aliases":"widehateps_rm align widehatε_rm align widehat eps rm widehat varepsilon rm","display":"\\widehat{\\varepsilon}_{\\rm align}","kind":"use","excerpt":"The witness commits a transported repair only when all four calibration conditions hold: widehat g_rm slice · ≥ 0.20, widehat r_rm non · ≤ 0.03, widehatε_rm align · ≤ 0.35, widehat A_rm non^rm base · ≥ 0.90. Here widehat g_rm slice is calibration gain on the affected stratum, widehat r_rm non is regression on the unaffected stratum, widehatε_rm ali","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aa101600924271d9","tex":"\\widehat{\\operatorname{false\\ admission}}","aliases":"widehatfalse admission widehat operatorname","display":"\\widehat{\\operatorname{false\\ admission}}","kind":"use","excerpt":"The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a758ce08d1d4292e","tex":"\\widehat{\\operatorname{false\\ admission}}_{U}","aliases":"widehatfalse admission_U widehat operatorname","display":"\\widehat{\\operatorname{false\\ admission}}_{U}","kind":"use","excerpt":"The expanded atlas admits all 900 primary tasks and all 900 novel-chart tasks. It admits none of the 600 nonlinear unsupported tasks: widehatcoverage_A=1, widehatcoverage_N=1, widehatfalse admission_U=0. Forcing the unsupported tasks through the nearest chart produces clean loss 0.703667, semantic-probe R^2=-17.424958, execution accuracy 0.056667, ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-49fbc4ae81c0dc3c","tex":"\\widehat{\\Lcal}","aliases":"widehatLcal widehatL widehat Lcal widehat mathcal","display":"\\widehat{{\\mathcal{L}}}","kind":"use","excerpt":"[Spectral-gap stability of the commutant] Assume the first positive eigenvalue of L_com is at least λ>0. Let widehatL_com be an empirical estimate satisfying normwidehatL_com-L_com≤ε<λ/2. Then the spectral projector onto the empirical eigenvalues in [0,λ/2) differs from the exact commutant projector by at most a constant multiple of ε/λ.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5edbb054925126df","tex":"\\widehat{\\Lcal}_{\\mathrm{com}}","aliases":"widehatLcal_com widehatL_com widehat Lcal mathrm widehat mathcal mathrm","display":"\\widehat{{\\mathcal{L}}}_{\\mathrm{com}}","kind":"use","excerpt":"[Spectral-gap stability of the commutant] Assume the first positive eigenvalue of L_com is at least λ>0. Let widehatL_com be an empirical estimate satisfying normwidehatL_com-L_com≤ε<λ/2. Then the spectral projector onto the empirical eigenvalues in [0,λ/2) differs from the exact commutant projector by at most a constant multiple of ε/λ.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-86e3c8b3890afd9d","tex":"\\widehat\\Mcal","aliases":"widehatMcal widehatM widehat Mcal widehat mathcal","display":"\\widehat{\\mathcal{M}}","kind":"use","excerpt":"[Finite meta-grammar generalization] Let K contain N candidate grandchild configurations. Suppose each target-level protected gain lies in [-G,G] and n independent development targets are used. Then, with probability at least 1-δ, every candidate satisfies abswidehatM(K)-M(K) ≤ Gsqrtfrac2log(2N/δ)n. If widehat K maximizes the empirical mean, then M","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-403c9ef0f3ae17d5","tex":"\\widehat{\\rm se}","aliases":"widehatrm se widehat rm","display":"\\widehat{\\rm se}","kind":"use","excerpt":"es K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configurati","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d295ab46fca9d4e","tex":"\\widehat{\\operatorname{se}}","aliases":"widehatse widehat operatorname","display":"\\widehat{\\operatorname{se}}","kind":"use","excerpt":"For arms a and b, evaluate both on the same frozen foreign agents and seeds. Let Δ_j=G_a,j-G_b,j. The paired estimate and standard error are widehatΔ =frac1n∑_j=1^nΔ_j, widehatse(widehatΔ) =sqrtfrac1n(n-1)∑_j=1^n(Δ_j-widehatΔ)^2. When gains are bounded in [-1,1], a distribution-free Hoeffding lower bound is LCB_1-α(Δ) = widehatΔ - sqrtfrac2log(1/α)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8822fb437ae3d32d","tex":"\\widehat\\beta","aliases":"widehatβ widehat beta","display":"\\widehat\\beta","kind":"use","excerpt":"For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y, widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1bdb2fbed427237c","tex":"\\widehat\\beta_r","aliases":"widehatβ_r widehat beta","display":"\\widehat\\beta_r","kind":"use","excerpt":"For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y, widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1675e27b0efa1a2f","tex":"\\widehat\\beta_u","aliases":"widehatβ_u widehat beta","display":"\\widehat\\beta_u","kind":"use","excerpt":"For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y, widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-78b7a298a6eff7c1","tex":"\\widehat\\gamma","aliases":"widehatγ widehat gamma","display":"\\widehat\\gamma","kind":"use","excerpt":"ot4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the frozen bounds. Among safe configurations, the compiler selects widehatγ ∈ arg max_γ∈Γ_rm","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dba1bd5f356dd9f5","tex":"\\widehat\\gamma_{\\min}","aliases":"widehatγ_min widehat gamma min","display":"\\widehat\\gamma_{\\min}","kind":"use","excerpt":"/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^-12. The smallest selected spectral gap is denoted widehatγ_min. These quantities serve different roles: item ε_resp measures whether the finite block-scalar model f","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5d42d28877d02a3","tex":"\\widehat\\Delta^{\\rm foreign}_{\\Gamma\\text{-lang}}","aliases":"widehatΔ^rm foreign_Γ-lang widehat Delta rm Gamma text","display":"\\widehat\\Delta^{\\rm foreign}_{\\Gamma\\text{-lang}}","kind":"use","excerpt":"The primary paired effect is widehatΔ^rm primary_Γ-lang = 0.225762, CI_95% = [0.166708,0.286431]. The generator-foreign paired effect is widehatΔ^rm foreign_Γ-lang = 0.240296, CI_95% = [0.180154,0.302701]. Both sign-flip estimates are approximately 9.9999×10^-6. No paired problem favors the parent. Primary pairs improve on 36.46% of problems and ti","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d3a41bc6ec8842de","tex":"\\widehat\\Delta^{\\rm primary}_{\\Gamma\\text{-lang}}","aliases":"widehatΔ^rm primary_Γ-lang widehat Delta rm Gamma text","display":"\\widehat\\Delta^{\\rm primary}_{\\Gamma\\text{-lang}}","kind":"use","excerpt":"The primary paired effect is widehatΔ^rm primary_Γ-lang = 0.225762, CI_95% = [0.166708,0.286431]. The generator-foreign paired effect is widehatΔ^rm foreign_Γ-lang = 0.240296, CI_95% = [0.180154,0.302701]. Both sign-flip estimates are approximately 9.9999×10^-6. No paired problem favors the parent. Primary pairs improve on 36.46% of problems and ti","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e8fc72741e0fd65d","tex":"\\widehat\\Delta_{1\\to2}","aliases":"widehatΔ_1→2 widehat Delta to","display":"\\widehat\\Delta_{1\\to2}","kind":"use","excerpt":"The paired grandchild-minus-recalibrated-child effect is widehatΔ_1→2 = 0.172188, with frozen interval [0.141481, 0.203110]. The sign-flip estimate is p=9.9999×10^-6. The positive paired-effect rate is 0.677083, and the negative paired-effect rate is zero. The remaining pairs are ties produced mainly by mutual abstention.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0b11f0fd77b11959","tex":"\\widehat\\Delta_{3\\to4}","aliases":"widehatΔ_3→4 widehat Delta to","display":"\\widehat\\Delta_{3\\to4}","kind":"use","excerpt":"The paired effect is widehatΔ_3→4 =0.207325, with frozen bootstrap interval [0.131902, 0.287910] and sign-flip estimate p=9.9999×10^-6. No primary target favors the parent. The positive paired-effect rate is 0.2875 and the remaining 0.7125 are abstention ties.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-33ceccbfbbb574b5","tex":"\\widehat\\Delta_{4\\to5}","aliases":"widehatΔ_4→5 widehat Delta to","display":"\\widehat\\Delta_{4\\to5}","kind":"use","excerpt":"The paired effect is widehatΔ_4→5 = 0.846985, with frozen interval [0.786100,0.901073] and sign-flip estimate p=9.9999×10^-6. No primary target favors the inherited parent. The remaining 9.375% of pairs are abstention ties.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db1c2718cafb98ff","tex":"\\widehat\\Delta_{5\\to6}","aliases":"widehatΔ_5→6 widehat Delta to","display":"\\widehat\\Delta_{5\\to6}","kind":"use","excerpt":"The paired effect is widehatΔ_5→6 = 0.708388, with frozen interval [0.659635,0.757198]. The sign-flip estimate is 9.9999×10^-6, and no primary target favors the affine parent.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-13a841a3797ae9a5","tex":"\\widehat\\Delta_{\\mathrm{clean}}^{A}","aliases":"widehatΔ_clean^A widehat Delta mathrm","display":"\\widehat\\Delta_{\\mathrm{clean}}^{A}","kind":"use","excerpt":"Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child mi","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3517cfb6858b6089","tex":"\\widehat\\Delta_{\\mathrm{clean}}^{N}","aliases":"widehatΔ_clean^N widehat Delta mathrm","display":"\\widehat\\Delta_{\\mathrm{clean}}^{N}","kind":"use","excerpt":"Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child mi","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ce9face2c916599f","tex":"\\widehat\\Delta_{\\mathrm{FiLM-add}}","aliases":"widehatΔ_FiLM-add widehat Delta mathrm","display":"\\widehat\\Delta_{\\mathrm{FiLM-add}}","kind":"use","excerpt":"The new constructor also fails a stronger comparison with the inherited additive repair. Pooled over the two architectures and two final supports, the selected-child minus legacy-additive program-specificity effect is widehatΔ_FiLM-add =-0.029417, CI_0.95 =[-0.034102,-0.024735]. The negative interval forbids a claim that the new source-selected con","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b1406f02421ab4b","tex":"\\widehat\\Delta_{G\\to C}","aliases":"widehatΔ_G→ C widehat Delta to","display":"\\widehat\\Delta_{G\\to C}","kind":"use","excerpt":"The paired child-minus-parent effect is widehatΔ_G→ C=0.334224, with frozen bootstrap interval [0.266459,0.401903]. The sign-flip estimate is p=9.9999×10^-6. The positive-pair rate is 0.572917, the remaining 0.427083 are conservative abstention ties, and no paired target favors the global parent.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-12917d9b92e5e827","tex":"\\widehat\\Delta_n","aliases":"widehatΔ_n widehat Delta","display":"\\widehat\\Delta_n","kind":"use","excerpt":"For n held-out episodes, define paired differences D_j = G_e_j(K_1)-G_e_j(K_0), widehatΔ_n=frac1n∑_j=1^nD_j. Because parent and child share each target and candidate bundle, pairing removes much of the episode difficulty variation.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ee3e02722d8cd206","tex":"\\widehat\\Delta_{\\mathrm{oracle}\\to C}","aliases":"widehatΔ_oracle→ C widehat Delta mathrm to","display":"\\widehat\\Delta_{\\mathrm{oracle}\\to C}","kind":"use","excerpt":"More importantly, the paired child-minus-direct-global-oracle effect is widehatΔ_oracle→ C=0.157530, with frozen interval [0.080863,0.234135]. Thus the child exceeds a directly supervised global-affine ceiling comparison on average, even though the oracle uses far more target labels and is not constrained to estimate a source-to-target bridge.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a038f93fa9c23fdd","tex":"\\widehat\\Delta_{\\mathrm{prog}}^{A}","aliases":"widehatΔ_prog^A widehat Delta mathrm","display":"\\widehat\\Delta_{\\mathrm{prog}}^{A}","kind":"use","excerpt":"The paired child-minus-parent program-specificity gain is widehatΔ_prog^A =0.010714, CI_0.95 =[0.009911,0.011527], with positive fraction 0.823125. The answer-specificity gain is 0.017727. Its frozen interval is [0.016490,0.018960]. Parent-minus-child clean loss is 0.000979, with frozen interval [0.000695,0.001290].","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dc222cff1d2855bf","tex":"\\widehat\\Delta_{\\mathrm{prog}}^{B}","aliases":"widehatΔ_prog^B widehat Delta mathrm","display":"\\widehat\\Delta_{\\mathrm{prog}}^{B}","kind":"use","excerpt":"The paired generator-foreign program-specificity gain is widehatΔ_prog^B =0.024763, CI_0.95 =[0.022794,0.026793], with positive fraction 0.855. The answer-specificity gain is 0.035400 with interval [0.032956,0.037938]. Clean-loss improvement is 0.003464 with interval [0.002633,0.004324]. Monte Carlo paired sign-flip tests for the program-specificit","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4378f2204d49c742","tex":"\\widehat\\Delta_{\\mathrm{program}}^{A}","aliases":"widehatΔ_program^A widehat Delta mathrm","display":"\\widehat\\Delta_{\\mathrm{program}}^{A}","kind":"use","excerpt":"Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child mi","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fbb8828140547942","tex":"\\widehat\\Delta_{\\mathrm{program}}^{N}","aliases":"widehatΔ_program^N widehat Delta mathrm","display":"\\widehat\\Delta_{\\mathrm{program}}^{N}","kind":"use","excerpt":"Paired effects are: widehatΔ_clean^A · =0.142631, · 95% CI · =[0.135421,0.149531], widehatΔ_clean^N · =0.365616, · 95% CI · =[0.352991,0.378402], widehatΔ_program^A · =0.120614, · 95% CI · =[0.110353,0.131880], widehatΔ_program^N · =0.125290, · 95% CI · =[0.111056,0.138921]. The clean effects are parent minus child; the program effects are child mi","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ca276278599218df","tex":"\\widehat\\Delta_{\\rm arch}","aliases":"widehatΔ_rm arch widehat Delta rm","display":"\\widehat\\Delta_{\\rm arch}","kind":"use","excerpt":"The paired effects are widehatΔ_rm audit · =0.072518, · CI_95% · =[0.046745,0.101389], widehatΔ_rm arch · =0.104999, · CI_95% · =[0.072711,0.140506]. Both one-sided sign-flip p-values equal approximately 4.99975×10^-5 under the frozen 20,000-draw randomization procedure.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cff2118ea76ccb33","tex":"\\widehat\\Delta_{\\rm audit}","aliases":"widehatΔ_rm audit widehat Delta rm","display":"\\widehat\\Delta_{\\rm audit}","kind":"use","excerpt":"The paired effects are widehatΔ_rm audit · =0.072518, · CI_95% · =[0.046745,0.101389], widehatΔ_rm arch · =0.104999, · CI_95% · =[0.072711,0.140506]. Both one-sided sign-flip p-values equal approximately 4.99975×10^-5 under the frozen 20,000-draw randomization procedure.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d1114d5a39a53b6c","tex":"\\widehat\\Delta_{\\rm bridge}","aliases":"widehatΔ_rm bridge widehat Delta rm","display":"\\widehat\\Delta_{\\rm bridge}","kind":"use","excerpt":"resentation families; item a 144-target paired foreign audit and a separate stress suite; item random-budget, farthest-point, shuffled-binding, sign-flip, and plain-QR controls; item exact query ledgers, rank diagnostics, harm checks, and package-closed receipts. On the primary foreign audit, mean balanced selected gain rises from 0.135212 to 0.344","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-96d6a0f1fabdf70e","tex":"\\widehat\\Delta_{\\rm meta}","aliases":"widehatΔ_rm meta widehat Delta rm","display":"\\widehat\\Delta_{\\rm meta}","kind":"use","excerpt":"On 144 primary foreign targets, the target-level paired difference is widehatΔ_rm meta = widehat M_24^rm for( P_1) - widehat M_24^rm for( P_0) = 0.209508. A target-cluster bootstrap gives the interval [0.194968,0.223898], and a 100,000-draw sign-flip test gives p=9.9999×10^-6. All 144 target-level paired differences are positive.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-231d921207695ea5","tex":"\\widehat\\Delta_{\\mathrm{str}}","aliases":"widehatΔ_str widehat Delta mathrm","display":"\\widehat\\Delta_{\\mathrm{str}}","kind":"use","excerpt":"[Finite-sample structural certificate] Suppose normr≤ R almost surely and r_1,…,r_n are independent held-out defects. Let widehatΔ_str = frac1n∑_i=1^nnormP_Wr_i^2. Then with probability at least 1-δ, | widehatΔ_str - EnormP_Wr^2 | ≤ R^2sqrtfraclog(2/δ)2n.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4adbd3e08dc783f9","tex":"\\widehat\\Delta_{\\Gamma}^{\\mathrm{arch}}","aliases":"widehatΔ_Γ^arch widehat Delta Gamma mathrm","display":"\\widehat\\Delta_{\\Gamma}^{\\mathrm{arch}}","kind":"use","excerpt":"The architecture-foreign paired effect is widehatΔ_Γ^arch = 0.090800, CI_95% = [0.052311,0.135081]. The sign-flip estimate is again 9.9999×10^-6. The child improves 34.722% of pairs, ties on 65.278%, and no architecture-foreign pair favors the parent.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-64e8b271a849a59f","tex":"\\widehat\\Delta_\\Gamma^{\\mathrm{primary}}","aliases":"widehatΔ_Γ^primary widehat Delta Gamma mathrm","display":"\\widehat\\Delta_\\Gamma^{\\mathrm{primary}}","kind":"use","excerpt":"Actual Block 13 compiler edge. On the primary audit, the parent and child gains are 0.883004 and 0.950231, so widehatΔ_Γ^primary=0.067227. The child changes research allocation rather than the candidate grammar. The architecture-foreign paired effect is 0.090800.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1cc50efab1807f9b","tex":"\\widehat\\Delta_{\\Gamma}^{\\mathrm{primary}}","aliases":"widehatΔ_Γ^primary widehat Delta Gamma mathrm","display":"\\widehat\\Delta_{\\Gamma}^{\\mathrm{primary}}","kind":"use","excerpt":"The primary paired effect is widehatΔ_Γ^primary = 0.067227, CI_95% = [0.035881,0.103274]. The sign-flip estimate is 9.9999×10^-6. The child improves 21.875% of pairs, ties on 75%, and loses on 3.125%.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d35f38f87e37bb6","tex":"\\widehat\\kappa","aliases":"widehatκ widehat kappa","display":"\\widehat\\kappa","kind":"use","excerpt":"novel-high · ⟼ objective-then-constructor. The expanded development gain is widehat O_1=0.948962, with newly correctable margin widehatκ = 0.571704.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-08d41d2f8c1b0ad2","tex":"\\widehat\\lambda","aliases":"widehatλ widehat lambda","display":"\\widehat\\lambda","kind":"use","excerpt":"[Stability of the threshold count] Let widehat A be an estimated holonomy-energy operator satisfying normwidehat A- A≤η. Then each ordered eigenvalue satisfies abswidehatλ_i-λ_i≤η. If no eigenvalue of A lies in [τ-η,τ+η], then widehat A and A have the same number of eigenvalues below τ.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7fa71c28dfe5f2b4","tex":"\\widehat\\lambda_{aj}","aliases":"widehatλ_aj widehat lambda","display":"\\widehat\\lambda_{aj}","kind":"use","excerpt":"For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-45700ee951c93c26","tex":"\\widehat\\lambda_i","aliases":"widehatλ_i widehat lambda","display":"\\widehat\\lambda_i","kind":"use","excerpt":"[Stability of the threshold count] Let widehat A be an estimated holonomy-energy operator satisfying normwidehat A- A≤η. Then each ordered eigenvalue satisfies abswidehatλ_i-λ_i≤η. If no eigenvalue of A lies in [τ-η,τ+η], then widehat A and A have the same number of eigenvalues below τ.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b186d96cd6172b72","tex":"\\widehat\\mu","aliases":"widehatμ widehat mu","display":"\\widehat\\mu","kind":"use","excerpt":"Let p^⋆ be the best conditioning policy and p_0 the best inherited-policy configuration. Empirical errors of at most η imply widehatμ(p^⋆) ≥ μ(p^⋆)-η > μ(p_0)+η ≥ widehatμ(p_0).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4168079ad6041a6a","tex":"\\widehat\\sigma","aliases":"widehatσ widehat sigma","display":"\\widehat\\sigma","kind":"use","excerpt":"A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-585fdb24fa37b076","tex":"\\widehat\\sigma_i","aliases":"widehatσ_i widehat sigma","display":"\\widehat\\sigma_i","kind":"use","excerpt":"A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ad8c5d8ce79e0069","tex":"\\widehat\\tau","aliases":"widehatτ widehat tau","display":"\\widehat\\tau","kind":"use","excerpt":"Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before f","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d12ddbe511bc7c1","tex":"\\widehat\\tau_{\\mathrm{abstract}}","aliases":"widehatτ_abstract widehat tau mathrm","display":"\\widehat\\tau_{\\mathrm{abstract}}","kind":"use","excerpt":"Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before f","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8d678effd8898b69","tex":"\\widetilde d","aliases":"widetilde d widetilde","display":"\\widetilde d","kind":"use","excerpt":"Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1bcb4f8d3da31653","tex":"\\widetilde h","aliases":"widetilde h widetilde","display":"\\widetilde h","kind":"use","excerpt":"Given input x, define the prompt anchor h_0=E(x). A normalized injection state z_t and updated latent state h_t satisfy z_t=N(h_t-1), widetilde h_t=B(z_t,x), h_t=U(z_t,widetilde h_t,h_0), 1≤ t≤ K, where N is any declared normalization or scale-matching operator. The answer is sampled or selected from D(h_K,x) after a declared end-of-thought transit","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be289365f1a52509","tex":"\\widetilde h_k","aliases":"widetilde h_k widetilde","display":"\\widetilde h_k","kind":"use","excerpt":"The latent trace updates the task representation through g_k · =σ(Gz_k), widetilde h_k · =tanh(Th_k-1+Lz_k), h_k · =(1-g_k)odot h_k-1+g_kodotwidetilde h_k, h_0=e, widehat w_k · =Oh_k. A separate halt head predicts one of four dynamic depths from the mean trace and, for nonparent candidates, the encoded prompt. A semantic training head predicts w(τ)","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ab048f0d3cb387e4","tex":"\\widetilde h_t","aliases":"widetilde h_t widetilde","display":"\\widetilde h_t","kind":"use","excerpt":"Given input x, define the prompt anchor h_0=E(x). A normalized injection state z_t and updated latent state h_t satisfy z_t=N(h_t-1), widetilde h_t=B(z_t,x), h_t=U(z_t,widetilde h_t,h_0), 1≤ t≤ K, where N is any declared normalization or scale-matching operator. The answer is sampled or selected from D(h_K,x) after a declared end-of-thought transit","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9117c60fc5e9378b","tex":"\\widetilde K","aliases":"widetilde K widetilde","display":"\\widetilde K","kind":"use","excerpt":"Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared hea","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a023933ebb0bd81d","tex":"\\widetilde K_i","aliases":"widetilde K_i widetilde","display":"\\widetilde K_i","kind":"use","excerpt":"Let K be the declared finite or compact class of admissible repair processes under the protected context. For R⊆ N, let K_R(K) = { widetilde K∈K: widetilde K_i=K_i for every i∉ R }. The restricted oracle envelope is F_K(R) = sup_widetilde K∈K_R(K) M_Ξ,D,B(widetilde K). Thus F_K(∅)=M(K), F_K(N)=sup_widetilde K∈KM(widetilde K). The total declared hea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-818401732c1dd8d1","tex":"\\widetilde r","aliases":"widetilde r widetilde","display":"\\widetilde r","kind":"use","excerpt":"A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f45ed973bf4634c4","tex":"\\widetilde r_{t+1}","aliases":"widetilde r_t+1 widetilde","display":"\\widetilde r_{t+1}","kind":"use","excerpt":"A grammar extension can reveal error that did not exist as a coordinate in the old state. Let E_t:ℝ^m_t↪ℝ^m_t+1 be the conservative embedding of old meta-regret coordinates. Decompose r_t+1 = E_twidetilde r_t+1+n_t+1, n_t+1⊥im E_t. Define old-coordinate repair progress by P_t^repair = normr_t^2-normwidetilde r_t+1^2, and epistemic discovery by P_t^","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8767ff5a2315c3ff","tex":"\\widetilde S","aliases":"widetilde S widetilde","display":"\\widetilde S","kind":"use","excerpt":"[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e9942049eed9a3bd","tex":"\\widetilde S_Q","aliases":"widetilde S_Q widetilde","display":"\\widetilde S_Q","kind":"use","excerpt":"[Composition bound for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-57c46c6121cf9b58","tex":"\\widetilde y","aliases":"widetilde y widetilde","display":"\\widetilde y","kind":"use","excerpt":"For current target state y, candidate repaired state widetilde y, and oracle repaired state y^⋆, define G = clip_[-1,1] ( 1- frac Enormwidetilde y-y^⋆^2 Enormy-y^⋆^2 ). The witness records: item validation gain; item source-to-target alignment NRMSE; item decoded protected-context regression; item minimum chart support; item alignment reduction rel","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a25d72cc47e636ba","tex":"\\widetilde\\pi","aliases":"widetildeπ widetilde pi","display":"\\widetilde\\pi","kind":"use","excerpt":"Let ϕ_G:X→ Z_G be the state representation expressible in grammar G. A G-measurable deterministic repair policy has the form π=widetildeπ∘ϕ_G.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1e4ebe4b5bc2d076","tex":"\\widetilde\\phi","aliases":"widetildeϕ widetilde phi","display":"\\widetilde\\phi","kind":"use","excerpt":"nts are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c11e608427525c67","tex":"\\mathrm{with}","aliases":"with mathrm","display":"\\mathrm{with}","kind":"use","excerpt":"Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3644dafe13a9504f","tex":"\\mathrm{without}","aliases":"without mathrm","display":"\\mathrm{without}","kind":"use","excerpt":"Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-555f1265129f8c19","tex":"\\Witnessed","aliases":"Witnessed mathrm","display":"{\\mathrm{Witnessed}}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a71d4a658eca31b4","tex":"\\mathrm{workspace}","aliases":"workspace mathrm","display":"\\mathrm{workspace}","kind":"use","excerpt":"A representative color set is C= set model, workspace, memory, planner, improver, evaluator, provenance, tools . The atlas does not assert that these colors are universally correct. It declares the decomposition under which one particular repair theory is being tested.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-898ac359185e330c","tex":"\\WorkspaceBound","aliases":"WorkspaceBound mathrm","display":"{\\mathrm{WorkspaceBound}}","kind":"use","excerpt":"Let C_RI(s) = · Grounded(s) wedgeProvValid(s) wedgeWorkspaceBound(s) · wedgeRepairTyped(s) wedgeWitnessed(s) wedgeEvidenceFlat(s) · wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cb661664117e95a2","tex":"\\mathrm{world}","aliases":"world mathrm","display":"\\mathrm{world}","kind":"use","excerpt":"The latent event variable is z∈ {none,internal,world,sensor,memory,evaluator}. Each event changes one declared mechanism: item internal: the action-response map B changes; item world: the passive dynamics A change; item sensor: the redundant sensor receives an additive bias; item memory: the retention coefficient λ changes; item evaluator: the repo","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fd25d3c239131c11","tex":"\\mathrm{worst\\ chart}","aliases":"worst chart mathrm","display":"\\mathrm{worst\\ chart}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-323aaf429c48a253","tex":"X^{\\mathsf T}","aliases":"X^ T mathsf","display":"X^{\\mathsf T}","kind":"use","excerpt":"[Rank saturation does not imply stability] For every d≥2 and every η>0, there exists a full-rank design X∈ℝ^d× d such that λ_min(X^ TX)<η.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ce0ca4b43768c339","tex":"x^{\\mathsf T}","aliases":"x^ T mathsf","display":"x^{\\mathsf T}","kind":"use","excerpt":"Let M=I_2, x=(2,1)^ T, and w=4. The D-optimal information increment is logdet(M+wxx^ T)-logdet M = log(1+w x^ TM^-1x) = log 21. A low-quality copy with w=1/16 gives only log(1+5/16), despite identical source geometry.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b, f_abs(x) · =a|x|+b, f_quad(x) · =ax^2+bx+c, f_hinge(x) · = ax+c, · x≥0, bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bb0e8953f025926b","tex":"X^\\top","aliases":"X^top top","display":"X^\\top","kind":"use","excerpt":"h_t(x_k)^top , and fits widehat T^top = (X^top X+λ I)^-1X^top Y. The parent chooses the k=16 fit anchors uniformly at random. The source hidden space is also 16-dimensional, so a rank-deficient design leaves at least one source direction unidentified.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3b559865b485d724","tex":"x_+","aliases":"x_+","display":"x_+","kind":"use","excerpt":"[Flat but inadequate core] Let protected evidence states be x_+=(1,1), x_-=(1,-1), and let shear holonomy leave only the first coordinate flat. Both records project to (1,0), so μ(E^flat;setx_+-x_-)=0. The core is exactly flat but cannot preserve the protected distinction between the two outcomes.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-16a74b3eb67dc1d2","tex":"x_-","aliases":"x_-","display":"x_-","kind":"use","excerpt":"[Flat but inadequate core] Let protected evidence states be x_+=(1,1), x_-=(1,-1), and let shear holonomy leave only the first coordinate flat. Both records project to (1,0), so μ(E^flat;setx_+-x_-)=0. The core is exactly flat but cannot preserve the protected distinction between the two outcomes.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"[Finite descending computation] Set X_0=S and X_k+1=Ψ_∀(X_k). The sequence is descending and stabilizes at I_∀^* after at most absS strict reductions.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported sc","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8eeb0a1bbb468424","tex":"X_3","aliases":"X_3","display":"X_3","kind":"use","excerpt":"Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ca7008ae447ee9dc","tex":"x_4","aliases":"x_4","display":"x_4","kind":"use","excerpt":"Let x=(x_1,x_2,x_3,x_4)∈ℝ^4. Define the eight-dimensional base feature ψ(x) = x_1 · x_2 · x_3 · x_4 · x_1x_2 · x_3x_4 · sin x_1 · cos x_2 ^ top ∈ℝ^8, and the affected-stratum indicator s(x)=1{x_1+x_2>0.8}. The full feature map is ϕ(x) = ψ(x)","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2bf17a695a91ab78","tex":"X_A","aliases":"X_A","display":"X_A","kind":"use","excerpt":"The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51e189ae43085a7d","tex":"X_a","aliases":"X_a","display":"X_a","kind":"use","excerpt":"Suppose H=H_1⊕ H_2⊕ H_3, dim H_a=2. Under separating block-scalar responses, every commuting operator has form X=X_1⊕ X_2⊕ X_3, X_a∈ M_2(ℝ). Hence dim A_R' =3dim M_2(ℝ) =12. The companion's modal commutant dimension is 12, matching the exact finite theory.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6b9d2df462b7ec9a","tex":"X_B","aliases":"X_B","display":"X_B","kind":"use","excerpt":"The deterministic theorem compares declared defects. A deployed repair is usually evaluated across a distribution of contexts. Let P_A and P_B be source and target context distributions, let S:X_A→ X_B be a declared context transport, and let q_A,q_B be per-context repair gains.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6b3f87f61546032d","tex":"x_i^{\\mathsf T}","aliases":"x_i^ T mathsf","display":"x_i^{\\mathsf T}","kind":"use","excerpt":"Let Msucc0 be the current regularized information matrix. Adding candidate anchor x_i with quality weight w_i=σ_i^-2 gives M'=M+w_ix_ix_i^ T.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f88b14242652e8b9","tex":"X_k","aliases":"X_k","display":"X_k","kind":"use","excerpt":"[Finite descending computation] Set X_0=S and X_k+1=Ψ_∀(X_k). The sequence is descending and stabilizes at I_∀^* after at most absS strict reductions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"use","excerpt":"flatness] The exact common fixed space is E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e7462f205257fb46","tex":"X_{k+1}","aliases":"X_k+1","display":"X_{k+1}","kind":"use","excerpt":"[Finite descending computation] Set X_0=S and X_k+1=Ψ_∀(X_k). The sequence is descending and stabilizes at I_∀^* after at most absS strict reductions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ec84613b712fe298","tex":"x_N","aliases":"x_N","display":"x_N","kind":"use","excerpt":"[Historical ranking preservation] Let x_1,…,x_N be protected historical candidate records with scalar old scores J_v(x_i). Suppose every unequal old pair has margin at least m_hist := min_J_v(x_i)≠ J_v(x_j) absJ_v(x_i)-J_v(x_j)>0. If an upgrade satisfies absJ_w(T_ex_i)-J_v(x_i)≤ε for every i, with 2ε<m_hist, then every strict historical ranking is ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"[Median breakdown in one coordinate] Let x_1,…,x_n∈ℝ, and suppose fewer than n/2 observations are replaced arbitrarily. Then the sample median of the contaminated set lies between two uncontaminated order statistics. In particular, an adversary controlling less than half the observations cannot drive the median beyond the range of the retained clea","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2eae17cd1d80814c","tex":"x_q","aliases":"x_q","display":"x_q","kind":"use","excerpt":"bx+c, · x<0. The coefficients belong to finite declared integer sets. A task contains five shuffled input--output demonstrations and one query input. The model receives a normalized serialization of those demonstrations and the query. The protected target is y(τ) = ( e_θ, fracp_12,…,fracp_42, fracd(θ)5, fracf_θ,p(x_q)32 )∈ℝ^10, where e_θ∈ℝ^4 is the","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8285a92cff098813","tex":"X_r","aliases":"X_r","display":"X_r","kind":"use","excerpt":"For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ed2990bcb53e180d","tex":"X_S","aliases":"X_S","display":"X_S","kind":"use","excerpt":"Let X_S∈ℝ^n× d be the source hidden-state design induced by anchor set S. Rank recovery is saturated when Pr(rank X_S=d)≥ 1-ε_rm rank under the current foreign development distribution and the remaining performance gap cannot be explained by rank failure.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-08a50b772620f832","tex":"X_S^{\\mathsf T}","aliases":"X_S^ T mathsf","display":"X_S^{\\mathsf T}","kind":"use","excerpt":"This certificate separates two statements: rank X_S=d and λ_min(X_S^ TWX_S)gg0. The first is qualitative identifiability. The second is quantitative stability.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c972bde0bfff9edf","tex":"X_S^\\top","aliases":"X_S^top top","display":"X_S^\\top","kind":"use","excerpt":"does not imply that the repair process has no remaining headroom. Let the target bridge be full rank but have minimum singular value σ_min=10^-4. Random noise of norm 10^-3 can then dominate the weakest identified direction. The bottleneck has migrated from rank to conditioning. A meaningful grandchild should target a criterion such as max_S:|S|=k ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"use","excerpt":"The ordinary control has Y_t=X_t A+b, A∈GL(8), with one shared map at every recurrent step. The inherited step-affine constructor is sufficient. This family tests conservative minimality: a full-trajectory affine map is available and usually eligible, but it should not be selected when the smaller inherited constructor already passes.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1029cb8ad51ed272","tex":"x_t","aliases":"x_t","display":"x_t","kind":"use","excerpt":"[Cumulative evaluator-drift bound] Consider a chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-813924c29494401d","tex":"x_{t+1}","aliases":"x_t+1","display":"x_{t+1}","kind":"use","excerpt":"Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-246e76947c9e4792","tex":"X_\\epsilon","aliases":"X_ϵ epsilon","display":"X_\\epsilon","kind":"use","excerpt":"Let X_ϵ=diag(1,1,ϵ), 0<ϵll1. Then rank X_ϵ=3, but λ_min(X_ϵ^ TX_ϵ)=ϵ^2 and the worst-direction least-squares variance scales as ϵ^-2. A binary rank gate cannot distinguish this design from an orthogonal one.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-98ea101fb9fc8e87","tex":"X_\\epsilon^{\\mathsf T}","aliases":"X_ϵ^ T epsilon mathsf","display":"X_\\epsilon^{\\mathsf T}","kind":"use","excerpt":"Let X_ϵ=diag(1,1,ϵ), 0<ϵll1. Then rank X_ϵ=3, but λ_min(X_ϵ^ TX_ϵ)=ϵ^2 and the worst-direction least-squares variance scales as ϵ^-2. A binary rank gate cannot distinguish this design from an orthogonal one.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ed2692474182335b","tex":"x_\\tau","aliases":"x_τ tau","display":"x_\\tau","kind":"use","excerpt":"A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7af6acee708a77a6","tex":"\\Xcal_A^{\\mathrm{demo}}","aliases":"Xcal_A^demo X_A^demo Xcal mathrm mathcal mathrm","display":"{\\mathcal{X}}_A^{\\mathrm{demo}}","kind":"use","excerpt":"Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Neve","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-95684ed12a4c46e0","tex":"\\Xcal_A^{\\mathrm{query}}","aliases":"Xcal_A^query X_A^query Xcal mathrm mathcal mathrm","display":"{\\mathcal{X}}_A^{\\mathrm{query}}","kind":"use","excerpt":"Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Neve","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a66f023fe9c4ca0b","tex":"\\Xcal_B^{\\mathrm{demo}}","aliases":"Xcal_B^demo X_B^demo Xcal mathrm mathcal mathrm","display":"{\\mathcal{X}}_B^{\\mathrm{demo}}","kind":"use","excerpt":"Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Neve","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9709d53f2df2ab0b","tex":"\\Xcal_B^{\\mathrm{query}}","aliases":"Xcal_B^query X_B^query Xcal mathrm mathcal mathrm","display":"{\\mathcal{X}}_B^{\\mathrm{query}}","kind":"use","excerpt":"Generator A uses approximately uniform template frequencies. Its demonstration and query supports are X_A^demo={-2,-1,0,1,2}, X_A^query={-3,3}. Generator B changes the template mixture toward quadratic and hinge programs and uses X_B^demo={-3,-1,0,1,3}, X_B^query={-3,-2,2,3}. Every shifted record remains inside the same exact program language. Neve","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d443d95428d57f2b","tex":"\\xhookrightarrow","aliases":"xhookrightarrow","display":null,"kind":"use","excerpt":"Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-624f8e0511fbbba8","tex":"y'_{m'}","aliases":"y'_m'","display":"y'_{m'}","kind":"use","excerpt":"Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-40d753fc08b490d6","tex":"y^*","aliases":"y^*","display":"y^*","kind":"use","excerpt":"norm y≤1 anglesy,r. If r_ob≠0, the maximum is attained by y^*=fracr_obnormr_ob.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-39aceea07927ed9b","tex":"y^\\star","aliases":"y^⋆ star","display":"y^\\star","kind":"use","excerpt":"For current target state y, candidate repaired state widetilde y, and oracle repaired state y^⋆, define G = clip_[-1,1] ( 1- frac Enormwidetilde y-y^⋆^2 Enormy-y^⋆^2 ). The witness records: item validation gain; item source-to-target alignment NRMSE; item decoded protected-context regression; item minimum chart support; item alignment reduction rel","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"Suppose a source context 0 and a target context c satisfy x_c=s_xx_0+t_x, y_c=s_yy_0+t_y, for all paired semantic-twin demonstrations.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7ef58ddaab7b5984","tex":"Y_1","aliases":"Y_1","display":"Y_1","kind":"use","excerpt":"The latent-clock family first applies an invertible stepwise affine gauge and then mixes adjacent recurrent times: Y_t = (1-β)U_t+β U_t-1, t=2,…,5, with Y_1=U_1 and 0<β<1. In matrix form, Y=W_βU, where W_β is lower bidiagonal. The temporal constructor applies W_β^-1 before fitting the spatial gauge.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"[Trace-schema attack surface] Witness separation is ineffective when distinct full executions with materially different external consequences map to the same declared trace. The factorization criterion should therefore be tested by adversarially searching for pairs y_1,y_2 satisfying q(y_1)=q(y_2), b(y_1)≠ b(y_2). Such a pair is a constructive cert","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dd412919846eece9","tex":"y_2","aliases":"y_2","display":"y_2","kind":"use","excerpt":"[Trace-schema attack surface] Witness separation is ineffective when distinct full executions with materially different external consequences map to the same declared trace. The factorization criterion should therefore be tested by adversarially searching for pairs y_1,y_2 satisfying q(y_1)=q(y_2), b(y_1)≠ b(y_2). Such a pair is a constructive cert","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5171b856d1185d9c","tex":"y_3","aliases":"y_3","display":"y_3","kind":"use","excerpt":"Let q(x)=Wψ(x)+b∈ℝ^3 be the protected score. Let C be the cyclic score permutation C(y_1,y_2,y_3)^top=(y_3,y_1,y_2)^top. The corrupted score is widetilde q(x) = (1-s(x))q(x)+s(x)Cq(x). Thus the model is almost perfectly correct outside the affected stratum and systematically wrong inside it.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a4544fab5a24d5f5","tex":"Y_A","aliases":"Y_A","display":"Y_A","kind":"use","excerpt":"For an agent A, define the resource-normalized utility J_Ξ(A) = E_z∼μ [ ν(z,Y_A(z)) - λbar c(z,Y_A(z)) ]. The definition does not require scalar utility to encode every scientific judgment. It supplies a declared comparison functional for one experimental claim. Multiple protected contexts may be used simultaneously, producing a Pareto relation ins","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-536459ef72a0484d","tex":"Y_a","aliases":"Y_a","display":"Y_a","kind":"use","excerpt":"This theorem gives a precise reason to value experiments rather than static classification. Suppose an experiment a returns an observation Y_a whose distribution differs between the aliased hypotheses. The compiler can then choose a sequence a_1,a_2,…,a_t that increases the information available for diagnosis.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8a33ca5aa31958a2","tex":"y_c","aliases":"y_c","display":"y_c","kind":"use","excerpt":"Suppose a source context 0 and a target context c satisfy x_c=s_xx_0+t_x, y_c=s_yy_0+t_y, for all paired semantic-twin demonstrations.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-69bb6e300ff2f699","tex":"Y_f","aliases":"Y_f","display":"Y_f","kind":"use","excerpt":"The minimal conflicts form a finite hypergraph H_G. A candidate feature or probe f:X→ Y_f resolves a conflict S when f is not constant on S.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8e1b422661cf5616","tex":"Y_i","aliases":"Y_i","display":"Y_i","kind":"use","excerpt":"The destructive family withholds four protected coordinates and constructs pairs i,j satisfying Y_i=Y_j, Z_i≠ Z_j. The compiler must discover these repeated observations from the public arrays. Their private construction pairs are not supplied.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-27bd33e3fbb9a571","tex":"Y_j","aliases":"Y_j","display":"Y_j","kind":"use","excerpt":"For foreign agent j, let Y_j(z) be protected gain under factor setting z. The average necessity effect of factor k relative to the intact architecture 1 is τ_k = E [ Y_j(1)-Y_j(1-e_k) ]. A positive value supports causal usefulness only relative to the frozen environment, compensating mechanisms, and matched resource schedule.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-233350a293d82242","tex":"y_{\\rm known}","aliases":"y_rm known rm","display":"y_{\\rm known}","kind":"use","excerpt":"[Open-set separation] Suppose every inherited known response y_rm known satisfies η_rm open(y_rm known)≤ε, while every unknown response in a target family satisfies η_rm open(y_rm unknown)≥γ for γ>ε. Then every threshold τ∈(ε,γ) perfectly separates the two families.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ac927db671a2ecf","tex":"y_{\\rm unknown}","aliases":"y_rm unknown rm","display":"y_{\\rm unknown}","kind":"use","excerpt":"[Open-set separation] Suppose every inherited known response y_rm known satisfies η_rm open(y_rm known)≤ε, while every unknown response in a target family satisfies η_rm open(y_rm unknown)≥γ for γ>ε. Then every threshold τ∈(ε,γ) perfectly separates the two families.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-50e74e1b36da6948","tex":"Y_t","aliases":"Y_t","display":"Y_t","kind":"use","excerpt":"The ordinary control has Y_t=X_t A+b, A∈GL(8), with one shared map at every recurrent step. The inherited step-affine constructor is sufficient. This family tests conservative minimality: a full-trajectory affine map is available and usually eligible, but it should not be selected when the smaller inherited constructor already passes.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ecb58e66eedd43b","tex":"y_t","aliases":"y_t","display":"y_t","kind":"use","excerpt":"Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-70eb5d403d3268e1","tex":"Y_{t+1}","aliases":"Y_t+1","display":"Y_{t+1}","kind":"use","excerpt":"A useful self-model participates in a complete causal cycle: ϕ_t(A_t) xrightarrowcandidate policy δ_t xrightarrowworld execution Y_t+1 xrightarrowevidence update ϕ_t+1(A_t+1). The candidate is selected before the protected outcome is revealed. The outcome affects future beliefs and repair policy. Cutting either planner use or consequence feedback d","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cfcf2f9d8faabe75","tex":"y_{t,1}","aliases":"y_t,1","display":"y_{t,1}","kind":"use","excerpt":"1 · 1 , so the protected consistency relation is y_t,3=y_t,1+y_t,2 when the sensor channel is intact.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bf83aeccb002697a","tex":"y_{t,1:2}","aliases":"y_t,1:2","display":"y_{t,1:2}","kind":"use","excerpt":"Let the latent world state be x_t∈ℝ^2, action a_t∈ℝ^2, observation y_t∈ℝ^3, and memory state m_t∈ℝ^2. The baseline dynamics are x_t+1=Ax_t+Ba_t, y_t=Cx_t+b_s, m_t+1=λ m_t+(1-λ)y_t,1:2, and the reported evaluator score is q_t=w^top y_t. The redundant observation map is C= 1 · 0","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0df62146d516942f","tex":"y_{t,2}","aliases":"y_t,2","display":"y_{t,2}","kind":"use","excerpt":"1 · 1 , so the protected consistency relation is y_t,3=y_t,1+y_t,2 when the sensor channel is intact.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7db1f414a9948ff7","tex":"y_{t,3}","aliases":"y_t,3","display":"y_{t,3}","kind":"use","excerpt":"1 · 1 , so the protected consistency relation is y_t,3=y_t,1+y_t,2 when the sensor channel is intact.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f191eaff7a34091d","tex":"z'^{\\top}","aliases":"z'^top top","display":"z'^{\\top}","kind":"use","excerpt":"Every quadratic form z^top Bz becomes z'^top(Q^top BQ)z'. The vector space of symmetric quadratic forms is therefore carried isomorphically to itself under the induced action of Q on Sym^2(P^top H).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-532408f4f67397a2","tex":"Z^\\top","aliases":"Z^top top","display":"Z^\\top","kind":"use","excerpt":"For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y, widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z),","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4ce8c5865fb8a8b6","tex":"z^\\top","aliases":"z^top top","display":"z^\\top","kind":"use","excerpt":"Let P be the recovered repair packet, let z=P^top x, and consider a target correction F(x) = Ax+b+(z^top Bz)v, where B=B^top≠0 and v≠0.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-43529727e65f6fc5","tex":"z_0'","aliases":"z_0'","display":"z_0'","kind":"use","excerpt":"Inductively, z_0'=Pz_0. If z_k'=Pz_k, then z_k+1' = tanh (PRP^ TPz_k+Ps_k+PC_α) =Ptanh(Rz_k+s_k+C_α) =Pz_k+1, because signed permutations commute with coordinatewise tanh. For every latent-consuming map M'=MP^ T, one has M'z_k'=Mz_k. Therefore the gates, hidden candidates, readouts, semantic prediction, and halt logits coincide. The state tensors n","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b23ccb12e2adf65f","tex":"Z_1","aliases":"Z_1","display":"Z_1","kind":"use","excerpt":"[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-05d56d3a899a7245","tex":"z_1^2","aliases":"z_1^2","display":"z_1^2","kind":"use","excerpt":"Nonzero Hessian defeats every affine composition. For z∼ N(0,I_2) and q(z)=z_1^2-z_2^2, every affine predictor has residual variance at least 4. The degree-two feature vector (z_1^2,z_1z_2,z_2^2) represents q exactly. No increase in affine depth changes this conclusion because affine maps remain closed under composition.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5e32645afb213c74","tex":"z_2","aliases":"z_2","display":"z_2","kind":"use","excerpt":"[Static diagnostic indistinguishability] Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-59402be3ba50e7d0","tex":"z_2^2","aliases":"z_2^2","display":"z_2^2","kind":"use","excerpt":"Nonzero Hessian defeats every affine composition. For z∼ N(0,I_2) and q(z)=z_1^2-z_2^2, every affine predictor has residual variance at least 4. The degree-two feature vector (z_1^2,z_1z_2,z_2^2) represents q exactly. No increase in affine depth changes this conclusion because affine maps remain closed under composition.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3701ff92cb978f0e","tex":"Z_A","aliases":"Z_A","display":"Z_A","kind":"use","excerpt":"Let protected linear invariants be represented by C_A:U_A→ Z_A, C_B:U_B→ Z_B. A correction preserves the invariant when C_Au=0 or C_Bu=0.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-706c3b3c0065f16f","tex":"Z_B","aliases":"Z_B","display":"Z_B","kind":"use","excerpt":"Let protected linear invariants be represented by C_A:U_A→ Z_A, C_B:U_B→ Z_B. A correction preserves the invariant when C_Au=0 or C_Bu=0.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0168d1f6e34ef9c3","tex":"Z_C","aliases":"Z_C","display":"Z_C","kind":"use","excerpt":"Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-68c6e50cb08048dd","tex":"Z_D","aliases":"Z_D","display":"Z_D","kind":"use","excerpt":"Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-821da25c35e40b6a","tex":"z_{e,c}","aliases":"z_e,c","display":"z_{e,c}","kind":"use","excerpt":"Each candidate receipt z_e,c is mapped to a feature vector x(z_e,c)∈ℝ^23. The released feature schema contains: item a constant coordinate; item logarithmic anchor and calibration sizes; item target representation width; item alignment, transport-condition, and repair-equivariance diagnostics; item packet residual and relative repair magnitude; ite","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a203ad4968eb0ef3","tex":"Z_{\\Gcal}","aliases":"Z_Gcal Z_G Gcal mathcal","display":"Z_{{\\mathcal{G}}}","kind":"use","excerpt":"Let ϕ_G:X→ Z_G be the state representation expressible in grammar G. A G-measurable deterministic repair policy has the form π=widetildeπ∘ϕ_G.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bc1f10031646b5b4","tex":"Z_i","aliases":"Z_i","display":"Z_i","kind":"use","excerpt":"[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives ","count":12},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-86298b26f7dd1fb4","tex":"z_i","aliases":"z_i","display":"z_i","kind":"use","excerpt":"For maximization coordinates, write Asucceq B when z_i(A)≥ z_i(B) for every i. The Pareto frontier is the set of undominated nodes.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-10845de283abdc8a","tex":"Z_j","aliases":"Z_j","display":"Z_j","kind":"use","excerpt":"r_i · =absℓ_i(replay(Z_i))-ℓ_i(Z_i). Noise is deterministic and norm matched at every step. The program and answer specificity coordinates are the mean recipient costs under SAP and FAD donors: p_i = E_j∈SAP(i) [ℓ_i(Z_j)-ℓ_i(Z_i)], a_i = E_j∈FAD(i) [ℓ_i(Z_j)-ℓ_i(Z_i)]. Define Q_prog = fracoverline pmax{overline o,overline c,ε}, Q_ans = fracoverline","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae33ff0c9a9cfbb4","tex":"z_j","aliases":"z_j","display":"z_j","kind":"use","excerpt":"nostic-indistinguishability Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-74e0eb883b673576","tex":"z_k","aliases":"z_k","display":"z_k","kind":"use","excerpt":"Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following c","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-09f8ae51f8ea0d18","tex":"z_k'","aliases":"z_k'","display":"z_k'","kind":"use","excerpt":"Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d8239feadf1698b5","tex":"z_{k+1}","aliases":"z_k+1","display":"z_{k+1}","kind":"use","excerpt":"Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following c","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1b829ff5adc7d5f2","tex":"z_{k+1}'","aliases":"z_k+1'","display":"z_{k+1}'","kind":"use","excerpt":"Inductively, z_0'=Pz_0. If z_k'=Pz_k, then z_k+1' = tanh (PRP^ TPz_k+Ps_k+PC_α) =Ptanh(Rz_k+s_k+C_α) =Pz_k+1, because signed permutations commute with coordinatewise tanh. For every latent-consuming map M'=MP^ T, one has M'z_k'=Mz_k. Therefore the gates, hidden candidates, readouts, semantic prediction, and halt logits coincide. The state tensors n","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9fdca6236d286603","tex":"Z_n","aliases":"Z_n","display":"Z_n","kind":"use","excerpt":"[Paired held-out gain certificate] Let Z_1,…,Z_n be independent and identically distributed paired target-context gains for a precommitted transported repair, with -G≤ Z_i≤ G. Let overline Z_n=n^-1∑_iZ_i. Then, with probability at least 1-α, E Z_i ≥ overline Z_n -Gsqrtfrac2log(1/α)n. For a precommitted family of m repairs, replacing α by α/m gives ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ff4f43d06ddc819e","tex":"Z_P","aliases":"Z_P","display":"Z_P","kind":"use","excerpt":"Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-318f27136d011f35","tex":"Z_{\\rm prot}","aliases":"Z_rm prot rm","display":"Z_{\\rm prot}","kind":"use","excerpt":"The benign compression exposes a thirty-dimensional statistic containing all nine protected semantic coordinates and twenty-one nuisance projections, followed by an invertible mixing. Since thirty is less than the forty-dimensional packet dimension, complete packet reconstruction is neither required nor generally possible. The appropriate construct","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-18f742c19813b73e","tex":"Z_T","aliases":"Z_T","display":"Z_T","kind":"use","excerpt":"Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-17c7ef854a4ec77f","tex":"z_T","aliases":"z_T","display":"z_T","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5d84c236d1e165ac","tex":"Z_W","aliases":"Z_W","display":"Z_W","kind":"use","excerpt":"Let the executable architecture have binary factors Z =(Z_C,Z_P,Z_D,Z_T,Z_W,Z_A,Z_M), representing causal attribution, packet localization, descent checking, transport, witness separation, archive retention, and repair abstraction.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4f6dc7de86f108d5","tex":"Z_X","aliases":"Z_X","display":"Z_X","kind":"use","excerpt":"Let Q_X:X→ Z_X and Q_Y:Y→ Z_Y be representations of the same declared property in two strata, and let widetilde T_Q:Z_X→ Z_Y be its intended transport.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ebeec497a5ae170e","tex":"Z_Y","aliases":"Z_Y","display":"Z_Y","kind":"use","excerpt":"Let Q_X:X→ Z_X and Q_Y:Y→ Z_Y be representations of the same declared property in two strata, and let widetilde T_Q:Z_X→ Z_Y be its intended transport.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ef27252706196107","tex":"\\Zcal_0","aliases":"Zcal_0 Z_0 Zcal mathcal","display":"{\\mathcal{Z}}_0","kind":"use","excerpt":"Let ϕ_0:T→Z_0 be the inherited diagnostic map. In Block 15, six inherited obstruction classes retain their existing tokens, while two previously unseen obstruction mechanisms are both mapped to one open-set token bot: ϕ_0(τ_ m int) = ϕ_0(τ_ m temp) = bot. The equality in [source reference] is not itself a defect. An unknown token is an appropriate ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"Let the old correction image be I=span{e_1,e_2}⊂ℝ^4 and let r=(0,0,3,4). Then z=P_I^⊥r=(0,0,3,4), η_I(r)^2=25. Add generator v=2e_4. Because v∈ I^⊥, frac|anglesz,v|^2norm v^2 = frac8^24 =16. The expanded residual is 3e_3, so η_I'(r)^2=9=25-16. A generator parallel to e_1 would have bar v=0 and would produce no structural reduction despite having no","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-87ca3b9cc9c4410b","tex":"|_{U_{\\alpha\\beta}}","aliases":"|_U_αβ alpha beta","display":"|_{U_{\\alpha\\beta}}","kind":"use","excerpt":"On every overlap, D(d_αβ) =-r|_U_αβ+r|_U_αβ=0. On triple overlaps, d_αβ+d_βγ+d_γα=0, so the family is a cocycle.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"For F(ε,x)=x^2-ε, the derivative with respect to x at the origin is zero, so the entire scalar equation is the Lyapunov--Schmidt reduced obstruction. For ε>0 there are two branches x=±sqrtε, for ε=0 there is one singular branch, and for ε<0 there is no real repair. A linearized system cannot distinguish these three regimes.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"Let D_t be an authorized modification family containing a no-op intervention δ_0. For δ∈D_t, let K_A_t^δ(·) = P(Y∈·mid do(δ),A_t) be the protected external-outcome kernel. A compressed causal self-model ϕ_t(A_t) induces a predicted kernel widehat K_ϕ_t(A_t)^δ(·). The model is evaluated on counterfactual consequences, not on its ability to reproduce","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f320d7b77ed8bff0","tex":"\\alpha_i","aliases":"α_i alpha","display":"\\alpha_i","kind":"use","excerpt":"Let H_1,…,H_m be ordered primary hypotheses. Choose nonnegative levels α_i satisfying ∑_i=1^mα_i≤α. Each endpoint is tested only with its assigned held-out evidence. Reusing the same audit sample to tune thresholds and report final confidence invalidates the nominal bound.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-beaa4eefa4e0426c","tex":"\\beta_k","aliases":"β_k beta","display":"\\beta_k","kind":"use","excerpt":"The source grammar contains a task-blind continuation, inherited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the pr","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-988ef8458bc095f4","tex":"\\beta_t","aliases":"β_t beta","display":"\\beta_t","kind":"use","excerpt":"Let R_t be true protected regression and let U_t^R be a conditional upper confidence bound. Require U_t^R≤β_t, ∑_tβ_t≤ B_R. Let d_t be semantic debt incurred by approximate grammar or evaluator transport, with ∑_td_t≤ B_sem. A lineage certificate should therefore report three separate cumulative quantities: ( ∑_tδ_t, ∑_tβ_t, ∑_td_t ). A positive ga","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d2e737aa54abf669","tex":"\\gamma_1","aliases":"γ_1 gamma","display":"\\gamma_1","kind":"use","excerpt":"[Finite computation from a cycle basis] Let G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a668a8a06f3deb92","tex":"\\Gamma_1^{\\rm recal}","aliases":"Γ_1^rm recal Gamma rm","display":"\\Gamma_1^{\\rm recal}","kind":"use","excerpt":"Block 14 therefore holds the candidate-method grammar and full-experiment budget fixed while changing the reliability conditions under which research decisions must be made. The published Block 13 result digest and exact 39-method language are replayed. Its compiler design is retained as the inherited parent and recalibrated on the Block 14 develop","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0c682d9e384f54ec","tex":"\\gamma_a","aliases":"γ_a gamma","display":"\\gamma_a","kind":"use","excerpt":"In an executable system, the responses are estimated from finite intervention data. Let widehat R_j=R_j+E_j, and fix coefficients c_j. Define B=∑_jc_jR_j, widehat B=∑_jc_jwidehat R_j, E=widehat B-B. Let the distinct eigenvalue associated with block a be μ_a and define its separation γ_a=min_b≠ a|μ_a-μ_b|.","count":8},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-834118f2adf99115","tex":"\\gamma_b","aliases":"γ_b gamma","display":"\\gamma_b","kind":"use","excerpt":"Choose a generic linear combination G=∑_j=1^mc_jR_j+c_m+1S whose induced scalar values on the distinct joint signatures are pairwise distinct. For each block value γ_a, the Lagrange polynomial p_a(z)=∏_b≠ afracz-γ_bγ_a-γ_b satisfies p_a(G)=P_a, the orthogonal projector onto H_a. Therefore P_a,P_b∈ A(R^+) and belong to its center.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-51cadb8120940875","tex":"\\gamma_j","aliases":"γ_j gamma","display":"\\gamma_j","kind":"use","excerpt":"[Finite computation from a cycle basis] Let G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-530f5d7a3b27e60f","tex":"\\gamma_k","aliases":"γ_k gamma","display":"\\gamma_k","kind":"use","excerpt":"The source grammar contains a task-blind continuation, inherited additive conditioning, final-step FiLM, progressive FiLM with geometry, a low-rank hypernetwork, and a shuffled-witness FiLM control. For the selected FiLM constructor, the encoded task produces step-indexed scale and shift vectors (γ_k(e),β_k(e))_k=0^4 =F_FiLM(e), which act on the pr","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-27c5ee666d44751e","tex":"\\gamma_m","aliases":"γ_m gamma","display":"\\gamma_m","kind":"use","excerpt":"[Finite computation from a cycle basis] Let G be finite. Choose a spanning tree and let γ_1,…,γ_m be the fundamental loops generated by non-tree edges. If the loop transports form a representation of the graph fundamental group, then E_v_0^flat = ⋂_j=1^m ker(H_γ_j-I). Equivalently, E_v_0^flat is the null space of the vertically stacked matrix H_γ_1","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ed3d79337915991","tex":"\\gamma_{\\rm cal}","aliases":"γ_rm cal gamma rm","display":"\\gamma_{\\rm cal}","kind":"use","excerpt":"The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-90f62f8a9dbb777b","tex":"\\Gamma_{\\rm safe}","aliases":"Γ_rm safe Gamma rm","display":"\\Gamma_{\\rm safe}","kind":"use","excerpt":"ot4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation episodes and widehatrm se(γ) its standard error. A configuration is validation-safe when its protected-regression tail rate and maximum regression satisfy the frozen bounds. Among safe configurations, the compiler selects widehatγ ∈ arg max_γ∈Γ_rm","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-298b1b9044edbc49","tex":"\\Gamma_t","aliases":"Γ_t Gamma","display":"\\Gamma_t","kind":"use","excerpt":"Let K_t be the active repair process and let Γ_t be the lineage compiler. Write K_t+1 = Γ_t(K_t,E_t,A_t), where E_t is the available evidence and A_t is the executable archive.","count":9},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dc86e6a969d1bd9c","tex":"\\Gamma_{t+1}","aliases":"Γ_t+1 Gamma","display":"\\Gamma_{t+1}","kind":"use","excerpt":"Define compiler metaproductivity by C(Γ) = E_K∼D_parent [ M(Γ(K,E_K))-M(K) ]. A stronger recursive claim requires C(Γ_t+1)>C(Γ_t) on sealed foreign parent processes under matched compilation resources.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-04b17d603dd0c26f","tex":"\\delta^*","aliases":"δ^* delta","display":"\\delta^*","kind":"use","excerpt":"pection] Let u:Y→[0,1] be a bounded utility. Define V(δ)= E[u(Y)mid A_t,δ], widehat V(δ)= E[u(Y)mid ϕ_t(A_t),δ]. If ϕ_t is ε-intervention-sufficient, then sup_δ∈D_t absV(δ)-widehat V(δ)≤ε. If δ^*∈arg max_δ∈D_tV(δ), widehatδ∈arg max_δ∈D_twidehat V(δ), then V(δ^*)-V(widehatδ)≤2ε.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3f0f0e9cd378ed81","tex":"\\Delta^2","aliases":"Δ^2 Delta","display":"\\Delta^2","kind":"use","excerpt":"[Prototype-separation error bound] Let Δ = min_z≠ z'normμ_z-μ_z'. Under equal priors and isotropic Gaussian noise, nearest-prototype classification satisfies Pr(widehat z≠ z) ≤ (| Z|-1) exp(-fracΔ^28σ^2).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"Let D_t be an authorized modification family containing a no-op intervention δ_0. For δ∈D_t, let K_A_t^δ(·) = P(Y∈·mid do(δ),A_t) be the protected external-outcome kernel. A compressed causal self-model ϕ_t(A_t) induces a predicted kernel widehat K_ϕ_t(A_t)^δ(·). The model is evaluated on counterfactual consequences, not on its ability to reproduce","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9adbb02c2ad54bb0","tex":"\\delta_1","aliases":"δ_1 delta","display":"\\delta_1","kind":"use","excerpt":"[Margin-preserving ranking] If V(δ_1)-V(δ_2)>2ε, then widehat V(δ_1)>widehat V(δ_2). In particular, if the true best intervention is unique and its gap from every alternative exceeds 2ε, the self-model selects the true best intervention.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dac3f880e6e3670a","tex":"\\delta_2","aliases":"δ_2 delta","display":"\\delta_2","kind":"use","excerpt":"[Margin-preserving ranking] If V(δ_1)-V(δ_2)>2ε, then widehat V(δ_1)>widehat V(δ_2). In particular, if the true best intervention is unique and its gap from every alternative exceeds 2ε, the self-model selects the true best intervention.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e84d9212fb63670f","tex":"\\Delta_a","aliases":"Δ_a Delta","display":"\\Delta_a","kind":"use","excerpt":"0 . The resulting hidden difference is Δ_a(x) = h_a(x)-h_a^∘(x), h_a^∘(x)=E_a(ϕ^∘(x)). The empirical causal packet is the leading rank-r right-singular subspace of the affected differences {Δ_a(x_i):s(x_i)=1}.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1525b57dc165eb8a","tex":"\\delta_{ab}","aliases":"δ_ab delta","display":"\\delta_{ab}","kind":"use","excerpt":"Let H be a finite-dimensional real or complex Hilbert space. Suppose the unknown functional decomposition is H=⊕_a=1^mH_a, P_a:H→ H_a, with pairwise orthogonal projectors satisfying P_aP_b=δ_abP_a, ∑_a=1^mP_a=I. For controlled interventions u_1,…,u_k, let R_j∈End(H), j=1,…,k, be their finite-difference or linearized response operators.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2c5a0d733f6b9a8d","tex":"\\Delta_{\\Acal}","aliases":"Δ_Acal Δ_A Delta Acal Delta mathcal","display":"\\Delta_{{\\mathcal{A}}}","kind":"use","excerpt":"[Nearest-signature route stability] Let the minimum pairwise normalized signature separation of an atlas be Δ_A = min_c≠ dd_sig(σ_c,σ_d). If an observed signature from chart c has normalized perturbation at most δ and 2δ<Δ_A, nearest-signature routing returns c. If additionally δ≤τ_adm, the route is admitted.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5c9993259279d0e","tex":"\\Delta_{D,k}","aliases":"Δ_D,k Delta","display":"\\Delta_{D,k}","kind":"use","excerpt":"Repeatedly apply [source reference]. The total correction-square defect is the telescoping sum ∑_k=0^n-1 T_V,n-1⋯ T_V,k+1 Δ_D,k T_U,k-1⋯ T_U,0, and the total defect-identification error is ∑_k=0^n-1 T_V,n-1⋯ T_V,k+1Δ_r,k. Apply both sums to u_0, use D_0u_0+r_0=0, and take norms.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3069cf1843723de2","tex":"\\delta_e","aliases":"δ_e delta","display":"\\delta_e","kind":"use","excerpt":"For a branching archive, index proposed edges by e∈ E(Λ). Assign each edge a budget δ_e>0 with ∑_e∈ E(Λ)δ_e≤δ. The same proof yields archive-wide false-acceptance control. The allocation may be adaptive provided the remaining total budget is predictable from the pre-audit filtration.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-90c5f5840b9467f9","tex":"\\Delta_{\\mathrm{gen}}","aliases":"Δ_gen Delta mathrm","display":"\\Delta_{\\mathrm{gen}}","kind":"use","excerpt":"Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-438897fb6174330b","tex":"\\Delta_{\\mathrm{gram}}","aliases":"Δ_gram Delta mathrm","display":"\\Delta_{\\mathrm{gram}}","kind":"use","excerpt":"Δ_gram · Global-minus-local held-out transport residual margin","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-18146f771d76a4f6","tex":"\\Delta_{G\\to C}","aliases":"Δ_G→ C Delta to","display":"\\Delta_{G\\to C}","kind":"use","excerpt":"The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f7296569209074d2","tex":"\\Delta_i","aliases":"Δ_i Delta","display":"\\Delta_i","kind":"use","excerpt":"[Paired evaluator-risk improvement] Let Z_1,…,Z_n be independent and identically distributed held-out externally revealed cases. Let L_v(Z_i),L_w(Z_i)∈[0,1] be the losses of the old and proposed evaluators, and define Δ_i=L_v(Z_i)-L_w(Z_i)∈[-1,1]. If the upgrade and evaluation rule are fixed before the held-out outcomes are revealed, then with prob","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-288a046779bf284f","tex":"\\Delta_j","aliases":"Δ_j Delta","display":"\\Delta_j","kind":"use","excerpt":"For arms a and b, evaluate both on the same frozen foreign agents and seeds. Let Δ_j=G_a,j-G_b,j. The paired estimate and standard error are widehatΔ =frac1n∑_j=1^nΔ_j, widehatse(widehatΔ) =sqrtfrac1n(n-1)∑_j=1^n(Δ_j-widehatΔ)^2. When gains are bounded in [-1,1], a distribution-free Hoeffding lower bound is LCB_1-α(Δ) = widehatΔ - sqrtfrac2log(1/α)","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9cac204d41e2fb1c","tex":"\\Delta_j^{\\mathrm{ord}}","aliases":"Δ_j^ord Delta mathrm","display":"\\Delta_j^{\\mathrm{ord}}","kind":"use","excerpt":"Choose an operational ordering r_1prec r_2prec⋯prec r_m, m=|N|, with nested sets R_0=∅, R_j={r_1,…,r_j}. Define the ordered marginal headroom Δ_j^ord(K) = F_K(R_j)-F_K(R_j-1).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9601f08c373fe8e4","tex":"\\delta_k","aliases":"δ_k delta","display":"\\delta_k","kind":"use","excerpt":"Let e=E(ϕ(τ))∈ℝ^36 be the encoded task representation, let z_0∈ℝ^36 be a learned seed, and let s_k∈ℝ^36 be the step embedding at latent depth k∈{0,…,4}. The source uses an Elman transition z_k+1 = tanh(Wz_k+s_k+δ_k(e,z_k)). The first foreign family uses a GRU cell z_k+1=GRUCell(u_k,z_k), and the second uses a gated nonlinear residual cell z_k+1 = z","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-85f009fc639ab231","tex":"\\Delta_{m,n}","aliases":"Δ_m,n Delta","display":"\\Delta_{m,n}","kind":"use","excerpt":"Suppose the correction square is approximate: normD_t+1T_t^U-T_t^VD_t ≤ε_t^D. Suppose protected defect meanings are reconstructed with error at most ε_t^V. Define cumulative dynamic semantic debt Δ_m,n = ∑_t=m^n-1 ( ε_t^D+ε_t^V ). Under nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f27de50bed51fb90","tex":"\\Delta_{\\mathrm{oracle}\\to C}","aliases":"Δ_oracle→ C Delta mathrm to","display":"\\Delta_{\\mathrm{oracle}\\to C}","kind":"use","excerpt":"The main paired estimands are Δ_G→ C = E [ G(K_3,C^96,A)-G(K_2,G^96,A) ] and Δ_oracle→ C = E [ G(K_3,C^96,A)-G(K_affine oracle,A) ]. The second quantity is the sharper structural test.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b876da431f7b4530","tex":"\\Delta_{r,k}","aliases":"Δ_r,k Delta","display":"\\Delta_{r,k}","kind":"use","excerpt":"Repeatedly apply [source reference]. The total correction-square defect is the telescoping sum ∑_k=0^n-1 T_V,n-1⋯ T_V,k+1 Δ_D,k T_U,k-1⋯ T_U,0, and the total defect-identification error is ∑_k=0^n-1 T_V,n-1⋯ T_V,k+1Δ_r,k. Apply both sums to u_0, use D_0u_0+r_0=0, and take norms.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be87c5d7acf16f8f","tex":"\\delta_{\\rm orb}","aliases":"δ_rm orb delta rm","display":"\\delta_{\\rm orb}","kind":"use","excerpt":"For a proposed intervention correspondence σ: U→ V and transport T:H_t→ H_s, define the normalized orbit defect δ_rm orb(T,σ) = frac1| U| ∑_u∈ U fracnormTR^t_σ(u)-R_u^sT_F max{normR_u^s_F,ε}.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1d5b0b8e22827e34","tex":"\\delta_{\\rm rank}","aliases":"δ_rm rank delta rm","display":"\\delta_{\\rm rank}","kind":"use","excerpt":"δ_rm rank · Bridge-identification rank defect","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c0cfccdc70abb5b6","tex":"\\Delta_t","aliases":"Δ_t Delta","display":"\\Delta_t","kind":"use","excerpt":"Let the true foreign metaproductivity effect be Δ_t = M(K_t+1)-M(K_t). Let L_t be a lower confidence bound computed from fresh audit evidence satisfying Pr(Δ_t<L_tmidF_t) ≤ δ_t. Accept the edge only when L_t>0.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4abc637c4be411ae","tex":"\\delta_t","aliases":"δ_t delta","display":"\\delta_t","kind":"use","excerpt":"A useful self-model participates in a complete causal cycle: ϕ_t(A_t) xrightarrowcandidate policy δ_t xrightarrowworld execution Y_t+1 xrightarrowevidence update ϕ_t+1(A_t+1). The candidate is selected before the protected outcome is revealed. The outcome affects future beliefs and repair policy. Cutting either planner use or consequence feedback d","count":18},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"1∈ℝ^d+1 for homogeneous affine coordinates. Let c:H_s→{-1,+1} be an observable chart variable. A two-chart affine relation is specified by bar T_-,bar T_+∈ℝ^q×(d+1), with y=bar T_c(h)bar h+ε. Equivalently, writing bar A=fracbar T_++bar T_-2, bar B=fracbar T_+-bar T_-2, we have y=(bar A+c(h)bar B)bar h+ε.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7338013cb213d947","tex":"\\epsilon^{-2}","aliases":"ϵ^-2 epsilon","display":"\\epsilon^{-2}","kind":"use","excerpt":"Let X_ϵ=diag(1,1,ϵ), 0<ϵll1. Then rank X_ϵ=3, but λ_min(X_ϵ^ TX_ϵ)=ϵ^2 and the worst-direction least-squares variance scales as ϵ^-2. A binary rank gate cannot distinguish this design from an orthogonal one.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fb43c73461568514","tex":"\\epsilon^2","aliases":"ϵ^2 epsilon","display":"\\epsilon^2","kind":"use","excerpt":"Let X_ϵ=diag(1,1,ϵ), 0<ϵll1. Then rank X_ϵ=3, but λ_min(X_ϵ^ TX_ϵ)=ϵ^2 and the worst-direction least-squares variance scales as ϵ^-2. A binary rank gate cannot distinguish this design from an orthogonal one.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b4635d58e720adea","tex":"\\epsilon_g","aliases":"ϵ_g epsilon","display":"\\epsilon_g","kind":"use","excerpt":"[Gate misclassification bound] Let g be the correct chart and widehat g an estimated chart. Suppose the two target repair outputs satisfy normR_1(y)-R_0(y)≤ L throughout the evaluated region and Pr(widehat g≠ g)≤ϵ_g. Then EnormR_widehat g(y)-R_g(y)^2 ≤ ϵ_gL^2. If protected utility is L_J-Lipschitz in repaired state, then abs E J(R_widehat g(y))- E ","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8bb5e611667d7ee7","tex":"\\epsilon_{\\rm archive}","aliases":"ϵ_rm archive epsilon rm","display":"\\epsilon_{\\rm archive}","kind":"use","excerpt":"The development archive is contaminated at rate ϵ_rm archive=0.18. Contamination can perturb diagnostics, obstruction labels, research outcomes, and response records. The finite candidate grammar contains the exact 39 methods inherited from Block 13: five variants in each of six obstruction-matched families, five generic variants, and the four orig","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"Let ϕ(τ)∈ℝ^19 be the padded task serialization, and let e=E_θ(ϕ(τ))∈ℝ^40 be the prompt representation. The latent trace has four states. Given conditioning mode α, define z_0=ζ, z_k+1 = tanh (Rz_k+s_k+C_α(e,m)), 0≤ k<4, where s_k is a learned step embedding. The task-blind parent satisfies C_parent(e,m)=0. The descendants use one of the following c","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d12fc9336a6eea97","tex":"\\zeta'","aliases":"ζ' zeta","display":"\\zeta'","kind":"use","excerpt":"Let P be a signed permutation matrix on the latent space. Define transformed latent parameters ζ'=Pζ, R'=PRP^ T, s_k'=Ps_k, C_α'=PC_α, and transform every latent-consuming linear map by right multiplication with P^ T. Then the transformed model has z_k'=Pz_k and identical external outputs.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9657c492efcc34dd","tex":"\\eta^2","aliases":"η^2 eta","display":"\\eta^2","kind":"use","excerpt":"Hoeffding's inequality for a variable with range length 2G gives Pr(abswidehatM(K)-M(K)>η) ≤ 2exp(-fracnη^22G^2). Apply a union bound over N configurations and solve for η. The selected-candidate inequality follows by comparing the empirical maximizer with the true maximizer and paying one deviation term for each.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-70ba5664bb334e33","tex":"\\eta^U_{q^{-1}p}","aliases":"η^U_q^-1p eta","display":"\\eta^U_{q^{-1}p}","kind":"use","excerpt":"[Path-discrepancy identity] Under the hypotheses of [source reference], normT_U^pu-T_U^qu ≤ normT_U^q η^U_q^-1p(u). Thus loop holonomy directly bounds disagreement between two transport routes.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-50d5e65dfebdbedc","tex":"\\eta_a","aliases":"η_a eta","display":"\\eta_a","kind":"use","excerpt":"r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-017a293d20155d21","tex":"\\eta_g","aliases":"η_g eta","display":"\\eta_g","kind":"use","excerpt":"[Calibration-to-audit witness soundness] Suppose, simultaneously, that the audit gain and regression satisfy g_rm slice · ≥ widehat g_rm slice-η_g, r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-822b678e6b294229","tex":"\\eta_{\\mathrm{glue},W}","aliases":"η_glue,W eta mathrm","display":"\\eta_{\\mathrm{glue},W}","kind":"use","excerpt":"Different overlaps may have different uncertainty or semantic importance. Let W_E be a positive-definite operator on K^E. Define η_glue,W(e) := inf_knorme+ B^*k_W_E.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9ac816cc88da0e4e","tex":"\\eta_I","aliases":"η_I eta","display":"\\eta_I","kind":"use","excerpt":"[One-generator obstruction reduction] If bar v≠0, then η_I'(r)^2 = η_I(r)^2 - fracabsanglesz,bar v^2normbar v^2. If bar v=0, the extension introduces no new first-order correctability.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f2d96e78de152ac1","tex":"\\eta_{I'}","aliases":"η_I' eta","display":"\\eta_{I'}","kind":"use","excerpt":"[One-generator obstruction reduction] If bar v≠0, then η_I'(r)^2 = η_I(r)^2 - fracabsanglesz,bar v^2normbar v^2. If bar v=0, the extension introduces no new first-order correctability.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db72e7f4d740d159","tex":"\\eta_r","aliases":"η_r eta","display":"\\eta_r","kind":"use","excerpt":"[Calibration-to-audit witness soundness] Suppose, simultaneously, that the audit gain and regression satisfy g_rm slice · ≥ widehat g_rm slice-η_g, r_rm non · ≤ widehat r_rm non+η_r, and that the reported alignment error upper-bounds the audit alignment error up to η_a. If widehat g_rm slice-η_g · >0,","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a390bc253b6caa99","tex":"\\eta_{t+1}","aliases":"η_t+1 eta","display":"\\eta_{t+1}","kind":"use","excerpt":"[Newly correctable energy identity] For nested correction images, η_t(r)^2-η_t+1(r)^2 = normP_I_t+1∩ I_t^⊥r^2.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f23b2d243c1d9a91","tex":"\\eta_{\\mathrm{with}}","aliases":"η_with eta mathrm","display":"\\eta_{\\mathrm{with}}","kind":"use","excerpt":"Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-62d8e78d48e210da","tex":"\\eta_{\\mathrm{without}}","aliases":"η_without eta mathrm","display":"\\eta_{\\mathrm{without}}","kind":"use","excerpt":"Define the causal generator contribution Δ_gen = E_Q[η_without(r)^2] - E_Q[η_with(r)^2]. The contribution is admissible only when a matched ablation isolates the generator while holding the remaining grammar, evidence, and selection budget fixed.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c45faae0260eabc9","tex":"\\theta_*","aliases":"θ_* theta","display":"\\theta_*","kind":"use","excerpt":"-0.25 . Hence P_ctxR_s(h)=P_ctxh. The observable chart is generated by a task-specific separator c(h)=1{anglesu_*,P_ctxh>θ_*}. The compiler does not receive u_* or θ_*, but it receives the declared protected packet in which a valid separator may be sought.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-96b2159bbf90cc21","tex":"\\theta_{\\mathrm{abs}}","aliases":"θ_abs theta mathrm","display":"\\theta_{\\mathrm{abs}}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cc491d926abb57a8","tex":"\\theta_{\\mathrm{aff}}","aliases":"θ_aff theta mathrm","display":"\\theta_{\\mathrm{aff}}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f6fe80671e66b76f","tex":"\\theta_{\\mathrm{hinge}}","aliases":"θ_hinge theta mathrm","display":"\\theta_{\\mathrm{hinge}}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-231b2b300329dd2c","tex":"\\Theta_k","aliases":"Θ_k Theta","display":"\\Theta_k","kind":"use","excerpt":"In the executable fixture, P_ctxh=(h_1,h_2), and the source repair leaves these coordinates invariant. The initial separator vocabulary is F_X = set 1{anglesu_k,P_ctxh>θ} : 0≤ k<128, θ∈Θ_k(X) , where u_k=(cos(2π k/128),sin(2π k/128)) and Θ_k(X) contains zero and the empirical 0.30,0.40,0.50,0.60,0.70 quantiles of the projected fit states. For every","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-65d6d338821cb461","tex":"\\theta_{\\mathrm{quad}}","aliases":"θ_quad theta mathrm","display":"\\theta_{\\mathrm{quad}}","kind":"use","excerpt":"Let the template set be Θ = {θ_aff,θ_abs, θ_quad,θ_hinge}. For integer parameters a,b,c and integer input x, the exact programs are f_aff(x) · =ax+b,","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1ce2e0275b189f00","tex":"\\Theta_{\\rm select}","aliases":"Θ_rm select Theta rm","display":"\\Theta_{\\rm select}","kind":"use","excerpt":"The child adds one lineage-generated operator: Θ_rm select: { typed candidate receipts } ⟶ { risk-filtered repair choice }. This operator uses learned outcome calibration and candidate-type residual scales. The changed component is therefore the repair-selection policy, not the target agent's weights or the repair constructor.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b065f867a1782955","tex":"\\iota_0","aliases":"ι_0 iota","display":"\\iota_0","kind":"use","excerpt":"Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f1a2e02c9110da33","tex":"\\iota_1","aliases":"ι_1 iota","display":"\\iota_1","kind":"use","excerpt":"Consider a chain G_0 xhookrightarrowι_0 G_1 xhookrightarrowι_1 G_2 xhookrightarrow ⋯ of conservative extensions.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6dde5572c8c70181","tex":"\\iota_{10\\to11}","aliases":"ι_10→11 iota to","display":"\\iota_{10\\to11}","kind":"use","excerpt":"[Conservative embedding of the Block 10 grammar] There is an injective embedding ι_10→11:G_10↪G_11 obtained by fixing the atlas output to P_ctx and retaining the inherited separator, bridge, repair, evaluator, and witness objects. Every Block 10 execution has an observationally identical image in G_11.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6224649573390023","tex":"\\iota_g","aliases":"ι_g iota","display":"\\iota_g","kind":"use","excerpt":"[Conservative embedding of the global grammar] For every separator g∈F, the map ι_g:G_glob↪G_chart, bar T⟼(g,bar T,bar T) is injective and preserves every global-affine prediction exactly.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b7bbc1dbc84b2ae0","tex":"\\kappa_D","aliases":"κ_D kappa","display":"\\kappa_D","kind":"use","excerpt":"κ_D · Normalized chart repair-contrast magnitude","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d5c11c2e36107f9e","tex":"\\kappa_r","aliases":"κ_r kappa","display":"\\kappa_r","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-84739f85d78ff322","tex":"\\kappa_{\\mathrm{square}}","aliases":"κ_square kappa mathrm","display":"\\kappa_{\\mathrm{square}}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-491a0c5e5f8ae190","tex":"\\kappa_T","aliases":"κ_T kappa","display":"\\kappa_T","kind":"use","excerpt":"The executable package records a normalized chart-specific bridge contrast κ_T = fracnormbar T_1-bar T_0_F frac12(normbar T_1_F+normbar T_0_F)+10^-12. A positive contrast alone is not a structural certificate. Block 10 combines it with global-oracle exhaustion, local residual reduction, conservative embedding, matched resources, foreign gain, and b","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0463f89e44aab7af","tex":"\\kappa_u","aliases":"κ_u kappa","display":"\\kappa_u","kind":"use","excerpt":"widehatβ_r · = (Z^top Z+λ_r I)^-1Z^top r. For candidate type c, let s_u,c and s_r,c be the development residual standard deviations for the utility and risk heads. The deployed scores are widehat U_rm LCB(z) · = widehat U(z)-κ_u s_u,c(z), widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"[Spectral characterization of flatness] The exact common fixed space is E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-baee9485e9c71f02","tex":"\\lambda_{1j}","aliases":"λ_1j lambda","display":"\\lambda_{1j}","kind":"use","excerpt":"The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f5a3e989ca142ae8","tex":"\\lambda_{ai_1}","aliases":"λ_ai_1 lambda","display":"\\lambda_{ai_1}","kind":"use","excerpt":"Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicti","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dbf397fa9fa8399b","tex":"\\lambda_{ai_\\ell}","aliases":"λ_ai_ℓ lambda ell","display":"\\lambda_{ai_\\ell}","kind":"use","excerpt":"Every element of A(R) is a finite linear combination of words in the generators. On H_a, a word R_i_1⋯ R_i_ℓ acts by the scalar λ_ai_1⋯λ_ai_ℓ. Equality of the signature vectors makes the same scalar act on H_b. Linear combinations preserve the equality. A central idempotent separating the blocks would act as 1 on one and 0 on the other, contradicti","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b9d8cb7d801f308f","tex":"\\lambda_{bj}","aliases":"λ_bj lambda","display":"\\lambda_{bj}","kind":"use","excerpt":"Write X=∑_a,bP_aXP_b. The relation XR_j=R_jX implies (λ_bj-λ_aj)P_aXP_b=0 for every j. Separating signatures imply that for a≠ b some coefficient is nonzero, so P_aXP_b=0. Hence X is block diagonal. Every block-diagonal operator commutes with each block-scalar R_j. The dimension formula follows from dimEnd(H_a)=d_a^2.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-65d44243a226ac8f","tex":"\\lambda_c","aliases":"λ_c lambda","display":"\\lambda_c","kind":"use","excerpt":"A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-61662c6bc956acf8","tex":"\\lambda_{c}","aliases":"λ_c lambda","display":"\\lambda_{c}","kind":"use","excerpt":"Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inhe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-97731835cbf43158","tex":"\\lambda_{\\mathrm{comm}}","aliases":"λ_comm lambda mathrm","display":"\\lambda_{\\mathrm{comm}}","kind":"use","excerpt":"Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inhe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-18bb5b342b20541e","tex":"\\lambda_d","aliases":"λ_d lambda","display":"\\lambda_d","kind":"use","excerpt":"[Spectral characterization of flatness] The exact common fixed space is E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5b537ddaabdce70a","tex":"\\lambda_{\\mathrm{fit}}","aliases":"λ_fit lambda mathrm","display":"\\lambda_{\\mathrm{fit}}","kind":"use","excerpt":"Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inhe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cfb4cd0eb716a373","tex":"\\lambda_h","aliases":"λ_h lambda","display":"\\lambda_h","kind":"use","excerpt":"A simple acquisition rule is A_i = widehat g_i -λ_rwidehat r_i +λ_uwidehatσ_i +λ_h p_θ(h_imid x_τ) -λ_c C_i, where h_i is the method family, p_θ is an obstruction-class posterior, widehatσ_i is a declared uncertainty bonus, and C_i is experimental cost.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"use","excerpt":"For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d594552e2786afd4","tex":"\\lambda_k","aliases":"λ_k lambda","display":"\\lambda_k","kind":"use","excerpt":"ation E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d8714dd1390fc60e","tex":"\\lambda_{\\mathrm{mean}}","aliases":"λ_mean lambda mathrm","display":"\\lambda_{\\mathrm{mean}}","kind":"use","excerpt":"For a time-varying sequence M_t, let z_t be a row vector of expected schema counts. Under the linear mean model, E z_T = z_0M_0M_1⋯ M_T-1. Define the upper mean-growth exponent λ_mean = limsup_T→∞ frac1T lognormM_0M_1⋯ M_T-1. A positive value indicates exponential growth of the linearized expected count. It does not imply almost-sure survival under","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1dc8e3b2125ff219","tex":"\\lambda_{\\min}","aliases":"λ_min lambda min","display":"\\lambda_{\\min}","kind":"use","excerpt":"This certificate separates two statements: rank X_S=d and λ_min(X_S^ TWX_S)gg0. The first is qualitative identifiability. The second is quantitative stability.","count":16},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f0269abd6c909bc7","tex":"\\lambda_{mj}","aliases":"λ_mj lambda","display":"\\lambda_{mj}","kind":"use","excerpt":"The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7a9bc3a491e56c32","tex":"\\lambda_q","aliases":"λ_q lambda","display":"\\lambda_q","kind":"use","excerpt":"For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b87db7c608ff8298","tex":"\\Lambda_r","aliases":"Λ_r Lambda","display":"\\Lambda_r","kind":"use","excerpt":"The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation ep","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-36442f76e90ff174","tex":"\\lambda_r","aliases":"λ_r lambda","display":"\\lambda_r","kind":"use","excerpt":"For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y,","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a229d0e495ea41f8","tex":"\\lambda_{\\mathrm{risk}}","aliases":"λ_risk lambda mathrm","display":"\\lambda_{\\mathrm{risk}}","kind":"use","excerpt":"Let mathscr S be a finite typed candidate-intervention grammar. For candidate S∈mathscr S, define a development score Score_resp(S) = widehat M_foreign(S) -λ_cC(S) -λ_commε_comm(S) -λ_fitε_fit(S) -λ_riskwidehat R(S). The compiler selects a candidate only after verifying that it adds a nontrivial response-signature distinction and preserves the inhe","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c9fd6dc2dd0f918f","tex":"\\lambda_{\\rm rec}","aliases":"λ_rm rec lambda rm","display":"\\lambda_{\\rm rec}","kind":"use","excerpt":"For encoder E_θ, decoder G_δ, and score head H_ω, the pilot minimizes minibatch estimates of L(θ,δ,ω) = λ_rm rec normG_δ(E_θ(ϕ))-ϕ_F^2 + normH_ω(E_θ(ϕ))-widetilde q_F^2, with λ_rm rec=2. Optimization uses a deterministic NumPy Adam implementation for 500 steps. After representation training, the head is refitted by ridge regression to reduce avoida","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-29b14a34343e28e4","tex":"\\lambda_t","aliases":"λ_t lambda","display":"\\lambda_t","kind":"use","excerpt":"For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-73a733514251f5f8","tex":"\\Lambda_u","aliases":"Λ_u Lambda","display":"\\Lambda_u","kind":"use","excerpt":"The child compiler searches a finite grammar Γ = Λ_u×Λ_r× K_u× K_r× T_r, where the factors specify utility regularization, risk regularization, utility residual penalty, risk residual penalty, and risk threshold. In the released fixture, |Γ| =5·5·4·5·4 =2000. For γ∈Γ, let overline U_rm meta(γ) be its mean protected utility on the meta-validation ep","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-644c4143b76a11da","tex":"\\lambda_u","aliases":"λ_u lambda","display":"\\lambda_u","kind":"use","excerpt":"For regularization parameters λ_u,λ_r>0, define widehatβ_u · = (Z^top Z+λ_u I)^-1Z^top y,","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-215e5f5b4f6c5f48","tex":"\\lambda_{u1}","aliases":"λ_u1 lambda","display":"\\lambda_{u1}","kind":"use","excerpt":"The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8c31fe29634d56af","tex":"\\lambda_{ud}","aliases":"λ_ud lambda","display":"\\lambda_{ud}","kind":"use","excerpt":"The executable fixture uses five commuting symmetric response operators. In latent semantic coordinates they are diagonal: D_u=diag(λ_u1,…,λ_ud), and each system exposes an orthogonal gauge Q_a: R_u^a=Q_aD_uQ_a^top. The joint response signature of semantic coordinate j is λ_j=(λ_1j,…,λ_mj).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6df009a22e6ea1bc","tex":"\\mu_A","aliases":"μ_A mu","display":"\\mu_A","kind":"use","excerpt":"[In-distribution bridge success does not imply foreign success] For every finite dimension d≥1 and every K>0, there exist two context distributions μ_A,μ_B, full-rank source and target trace covariances under both distributions, and fixed linear maps P,B such that semantic prediction and transport are exact under μ_A while transported squared error","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3d1bb2e4dde90487","tex":"\\mu_a","aliases":"μ_a mu","display":"\\mu_a","kind":"use","excerpt":"Choose c∈ℝ^k outside the finite union of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a li","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5926607b3fa28f05","tex":"\\mu_B","aliases":"μ_B mu","display":"\\mu_B","kind":"use","excerpt":"[In-distribution bridge success does not imply foreign success] For every finite dimension d≥1 and every K>0, there exist two context distributions μ_A,μ_B, full-rank source and target trace covariances under both distributions, and fixed linear maps P,B such that semantic prediction and transport are exact under μ_A while transported squared error","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c826a35f37551062","tex":"\\mu_b","aliases":"μ_b mu","display":"\\mu_b","kind":"use","excerpt":"of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-316377c5c1a57bd3","tex":"\\mu_z","aliases":"μ_z mu","display":"\\mu_z","kind":"use","excerpt":"Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dde918fc83756419","tex":"\\mu_{z'}","aliases":"μ_z' mu","display":"\\mu_{z'}","kind":"use","excerpt":"Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"geq0^k be a sparse mixture with 1^top z=1. An observed response vector has the form y=Rz+ξ.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e85394e8b3004bb3","tex":"\\Xi_0","aliases":"Ξ_0 Xi","display":"\\Xi_0","kind":"use","excerpt":"Let edge K_0→ K_1 improve a bridge task distribution Ξ_0, while K_1→ K_2 improves a heteroscedastic distribution Ξ_1. Positive effects in both contexts do not imply M_Ξ_0(K_2)>M_Ξ_0(K_0) or any additive path gain unless Ξ_0 and Ξ_1 share a common utility or a validated context transport.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a86a4793fb3bab73","tex":"\\Xi_1","aliases":"Ξ_1 Xi","display":"\\Xi_1","kind":"use","excerpt":"Let edge K_0→ K_1 improve a bridge task distribution Ξ_0, while K_1→ K_2 improves a heteroscedastic distribution Ξ_1. Positive effects in both contexts do not imply M_Ξ_0(K_2)>M_Ξ_0(K_0) or any additive path gain unless Ξ_0 and Ξ_1 share a common utility or a validated context transport.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1e67206e9c71d213","tex":"\\Pi^{\\Gamma}_7","aliases":"Π^Γ_7 Pi Gamma","display":"\\Pi^{\\Gamma}_7","kind":"use","excerpt":"The realized compiler edge is L_6 xrightarrowΠ^Γ_7 L_7, where the active repair grammar, candidate-method language, evaluator portfolio, integrity monitor, and experiment budget remain fixed, but the research-allocation compiler changes from Γ_0 to Γ_1. The edge is therefore compiler metaproductivity inside a finite declared research language, not ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2d28a27ac7821632","tex":"\\pi_0","aliases":"π_0 pi","display":"\\pi_0","kind":"use","excerpt":"(ϕ_0(τ),high), · y_p(τ)>θ. The old token is recovered by the projection π_0(z,s)=z, π_0∘ϕ_p,θ=ϕ_0. Thus the extension is conservative at the level of old diagnostic semantics.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c16848845311d5f6","tex":"\\Pi_{16}","aliases":"Π_16 Pi","display":"\\Pi_{16}","kind":"use","excerpt":"child-minus-parent transplant-cost interval has positive lower endpoint; item child clean loss does not regress beyond a predeclared tolerance; item every claimed packet or transport passes its own held-out causal certificate; item model, task, audit, and result digests are committed in an append-only receipt chain before promotion. Then the accept","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-98bc0e48cf8ad057","tex":"\\Pi_{17}","aliases":"Π_17 Pi","display":"\\Pi_{17}","kind":"use","excerpt":"ain their declared interpretations; item the child is FULL and adaptive on both fresh final supports, while the paired effects in [source reference] have positive frozen intervals; item the independent NumPy, receipt, witness, and gauge checks pass within their declared numerical tolerances. Then the accepted transition L_16^NL xrightarrowΠ_17 L_17","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-240efaa459947ee2","tex":"\\Pi_{18}","aliases":"Π_18 Pi","display":"\\Pi_{18}","kind":"use","excerpt":"al receipt is frozen, a separate held-out split diagnosis refits only shift-local readouts and bridges. Source and target probes recover R^2 between 0.574 and 0.601, paired bridges recover 0.539 to 0.555, and shuffled bridges remain negative. Thus the next missing object is a context-indexed or invariant semantic atlas with abstention, not another ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-22a1cc98a858f289","tex":"\\Pi_{19}","aliases":"Π_19 Pi","display":"\\Pi_{19}","kind":"use","excerpt":"item the admission results in [source reference] hold; item set-only, chart-only, shuffled-binding, no-abstention, and always-abstain controls retain their declared interpretations; item the automated tests, independent environment replay, independent chart reconstruction, independent NumPy forward pass, receipts, and package manifest verify. Then ","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a539d0a4a394483f","tex":"\\pi_2","aliases":"π_2 pi","display":"\\pi_2","kind":"use","excerpt":"[Static diagnostic indistinguishability] Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dc19a5e5cf9dd5e5","tex":"\\Pi_{20}","aliases":"Π_20 Pi","display":"\\Pi_{20}","kind":"use","excerpt":"Block 20 accepts the finite transport continuation L_19^NL xrightarrowΠ_20 L_20^NL, where the added object is an atlas-indexed connection carrying complete Neuralese trajectories from GRU and gated-residual fibers into the frozen Elman source fibers. The edge preserves the six-chart atlas, passes fresh semantic and curvature audits, and rejects uns","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ca4934afd41ea9d2","tex":"\\Pi_{21}","aliases":"Π_21 Pi","display":"\\Pi_{21}","kind":"use","excerpt":"Block 21 accepts the finite grammar continuation L_20^NL xrightarrowΠ_21 L_21^NL, where the transport process gains a typed response grammar over affine drift, nonlinear warp, latent-clock mixing, chart aliasing, benign semantic quotient, and destructive semantic loss. The edge preserves every inherited valid repair and accepts only the minimum-des","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-acfd479488df4f9a","tex":"\\Pi_3","aliases":"Π_3 Pi","display":"\\Pi_3","kind":"use","excerpt":"K_3,C^96. The global parent K_2,G^96 retains one affine bridge and uses all 80 non-witness pairs to estimate it. The chart child K_3,C^96 uses 48 pairs for local bridge fitting, 32 pairs for chart selection, and the same 16 witness pairs. Both consume exactly 96 paired source-target queries. The edge is K_2,G^96 xrightarrowΠ_3 K_3,C^96.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d16ece4df807e3f9","tex":"\\Pi_4","aliases":"Π_4 Pi","display":"\\Pi_4","kind":"use","excerpt":"The new edge is therefore K_0xrightarrowΠ_1K_1xrightarrowΠ_2K_2 xrightarrowΠ_3K_3xrightarrowΠ_4K_4, where K_3 is the Block 10 chart process with a supplied protected packet and K_4 replaces that packet by an intervention-response atlas compiler. This is not unrestricted atlas discovery. The intervention family, finite state dimension, target task, ","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3292ce2fc97e3d46","tex":"\\Pi_5","aliases":"Π_5 Pi","display":"\\Pi_5","kind":"use","excerpt":"The realized continuation is K_4 xrightarrowΠ_5 K_5 xrightarrowΠ_6 K_6. The first edge expands the intervention-response language. The second expands the repair-constructor grammar. The two edges are evaluated in different protected contexts and are not added into one scalar utility difference without an additional context transport.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b64a8352c4ba44a8","tex":"\\Pi_6","aliases":"Π_6 Pi","display":"\\Pi_6","kind":"use","excerpt":"The realized continuation is K_4 xrightarrowΠ_5 K_5 xrightarrowΠ_6 K_6. The first edge expands the intervention-response language. The second expands the repair-constructor grammar. The two edges are evaluated in different protected contexts and are not added into one scalar utility difference without an additional context transport.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-68d50f5e4aaaad75","tex":"\\Pi_8^\\Gamma","aliases":"Π_8^Γ Pi Gamma","display":"\\Pi_8^\\Gamma","kind":"use","excerpt":"d full-experiment budget fixed while changing the reliability conditions under which research decisions must be made. The published Block 13 result digest and exact 39-method language are replayed. Its compiler design is retained as the inherited parent and recalibrated on the Block 14 development archive: Γ_1^rm recal. A robust descendant Γ_1^rm r","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-71f97426781219c9","tex":"\\Pi_9^{\\Gamma}","aliases":"Π_9^Γ Pi Gamma","display":"\\Pi_9^{\\Gamma}","kind":"use","excerpt":"Block 15 adds the first finite compiler-language edge, Γ_2xrightarrowΠ_9^ΓΓ_3. The inherited compiler retains one undifferentiated unknown token and a singleton-method grammar. Development repair conflicts select a separating probe, two anonymous refined tokens, and two ordered pair-method constructors from generic finite meta-languages. The old co","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8f322ee5409feacc","tex":"\\pi_j","aliases":"π_j pi","display":"\\pi_j","kind":"use","excerpt":"nostic-indistinguishability Let z_1≠ z_2 satisfy P(X_0∈ Bmid Z=z_1) = P(X_0∈ Bmid Z=z_2) for every measurable set B. Then no decision rule d(X_0) distinguishes z_1 from z_2 with accuracy exceeding the larger prior probability: P(d(X_0)=Zmid Z∈{z_1,z_2}) ≤ max{π_1,π_2}, where π_j= P(Z=z_jmid Z∈{z_1,z_2}).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-57678e997cbbb922","tex":"\\Pi_{\\rm anc}","aliases":"Π_rm anc Pi rm","display":"\\Pi_{\\rm anc}","kind":"use","excerpt":"The development meta-grammar contains five anchor policies: Π_rm anc = set random, balanced random, QR, packet QR, augmented QR . It also contains alignment thresholds τ∈set0.01,0.02,0.05,0.10,0.20,0.35 and an optional target-width guard. Every candidate uses the same fit and validation budgets.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-773aef95c47915da","tex":"\\pi_{\\rm anc}^{(0)}","aliases":"π_rm anc^(0) pi rm","display":"\\pi_{\\rm anc}^{(0)}","kind":"use","excerpt":"The parent and descendant differ in exactly two declared fields: π_rm anc^(0) · =uniform random, · π_rm anc^(1) · =packet-balanced QR, π_rm gate^(0) · =(0.20,0.03,0.35), · π_rm gate^(1) · =(0.08,0.035,0.05,full rank). The source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibrat","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f7435c83ad49a685","tex":"\\pi_{\\rm anc}^{(1)}","aliases":"π_rm anc^(1) pi rm","display":"\\pi_{\\rm anc}^{(1)}","kind":"use","excerpt":"The parent and descendant differ in exactly two declared fields: π_rm anc^(0) · =uniform random, · π_rm anc^(1) · =packet-balanced QR, π_rm gate^(0) · =(0.20,0.03,0.35), · π_rm gate^(1) · =(0.08,0.035,0.05,full rank). The source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibrat","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f32ebe47a3ab055c","tex":"\\Pi_{\\rm foreign}","aliases":"Π_rm foreign Pi rm","display":"\\Pi_{\\rm foreign}","kind":"use","excerpt":"A compiler-language expansion receipt is Π_Γ-lang = ( G_0, H_rm conflict, P^+, p,θ, C_rm pair, m_-,m_+, ι, κ, Π_rm seal, Π_rm foreign ), where H_rm conflict is the conflict hypergraph, ι is the conservative embedding, κ is the newly correctable development margin, and the final fields bind the pre-audit compiler state to foreign evidence.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8277d28cdab4e018","tex":"\\pi_{\\rm gate}^{(0)}","aliases":"π_rm gate^(0) pi rm","display":"\\pi_{\\rm gate}^{(0)}","kind":"use","excerpt":"The parent and descendant differ in exactly two declared fields: π_rm anc^(0) · =uniform random, · π_rm anc^(1) · =packet-balanced QR, π_rm gate^(0) · =(0.20,0.03,0.35), · π_rm gate^(1) · =(0.08,0.035,0.05,full rank). The source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibrat","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2ea4c9730cac0917","tex":"\\pi_{\\rm gate}^{(1)}","aliases":"π_rm gate^(1) pi rm","display":"\\pi_{\\rm gate}^{(1)}","kind":"use","excerpt":"The parent and descendant differ in exactly two declared fields: π_rm anc^(0) · =uniform random, · π_rm anc^(1) · =packet-balanced QR, π_rm gate^(0) · =(0.20,0.03,0.35), · π_rm gate^(1) · =(0.08,0.035,0.05,full rank). The source task model, causal packet discovery, source repair, linear bridge family, pseudoinverse repair transport, target calibrat","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7a6779bb2dc9b3d4","tex":"\\Pi_{\\rm seal}","aliases":"Π_rm seal Pi rm","display":"\\Pi_{\\rm seal}","kind":"use","excerpt":"A compiler-language expansion receipt is Π_Γ-lang = ( G_0, H_rm conflict, P^+, p,θ, C_rm pair, m_-,m_+, ι, κ, Π_rm seal, Π_rm foreign ), where H_rm conflict is the conflict hypergraph, ι is the conservative embedding, κ is the newly correctable development margin, and the final fields bind the pre-audit compiler state to foreign evidence.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fa5c80ed33c16641","tex":"\\Pi_t^{\\mathrm{auth}}","aliases":"Π_t^auth Pi mathrm","display":"\\Pi_t^{\\mathrm{auth}}","kind":"use","excerpt":"A witness structure may be represented as W_t = ( V_t^fr, C_t, B_t^sand, R_t, Π_t^auth ), where V_t^fr is the frozen evaluator instance for the current transaction, C_t is a protected checkpoint, B_t^sand is the sandbox boundary, R_t is the rollback procedure, and Π_t^auth records effect authority.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-147b9d77f6dd334b","tex":"\\rho_{\\mathrm{halt}}","aliases":"ρ_halt rho mathrm","display":"\\rho_{\\mathrm{halt}}","kind":"use","excerpt":"positive interval and fraction gates, local continuity, donor coverage, adaptive halting. Among eligible candidates it maximizes S(α) = · min{ fracR^2_probe0.65, fracQ_prog0.15, fracQ_ans0.20 } +0.20R^2_probe · -4 MSE_clean +0.02ρ_halt -Penalty_dynamic. The minimum term makes the weakest normalized semantic gate the principal bottleneck. The compil","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3531bf0eb5c99588","tex":"\\rho_{\\mathrm{preserve}}","aliases":"ρ_preserve rho mathrm","display":"\\rho_{\\mathrm{preserve}}","kind":"use","excerpt":"A candidate is pre-audit eligible only if both foreign target models are ATLAS_FULL and the candidate jointly satisfies R^2_transport · ≥0.75, · R^2_worst chart · ≥0.74, ρ_preserve · ≥0.98, · NRMSE_trace · ≤0.05, κ_square · ≤0.02. · · The selection score is the weakest normalized margin across these gates. This prevents a strong aggregate R^2 from ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-00993b276eb3ca7b","tex":"\\rho_{\\rm cal}","aliases":"ρ_rm cal rho rm","display":"\\rho_{\\rm cal}","kind":"use","excerpt":"The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6a1cca433d78254e","tex":"\\rho_{\\mathrm{Spearman}}","aliases":"ρ_Spearman rho mathrm","display":"\\rho_{\\mathrm{Spearman}}","kind":"use","excerpt":"halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ρ_Spearman(d(x),T(x))>0 and nondegenerate prompt variation in the halt ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6512b523961a05f5","tex":"\\sigma^{-1}","aliases":"σ^-1 sigma","display":"\\sigma^{-1}","kind":"use","excerpt":"[Intertwiner uniqueness modulo the response commutant] Let T_1,T_2:H_t→ H_s be invertible and satisfy T_iR_v^t=R_σ^-1(v)^sT_i for every v∈ V. Then S=T_2^-1T_1 belongs to A_t'. Hence exact intervention-orbit data identifies a transport at most up to the response commutant.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"nts are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor consistency; item deviation from baseline memory update; item discrepancy between reported and protected evaluator scores. The observed probe is widet","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ce43ea6cae6d85ca","tex":"\\Sigma^{\\mathrm{evaluation}}","aliases":"Σ^evaluation Sigma mathrm","display":"\\Sigma^{\\mathrm{evaluation}}","kind":"use","excerpt":"Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, addres","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f2577102d624db86","tex":"\\Sigma^{\\mathrm{memory}}","aliases":"Σ^memory Sigma mathrm","display":"\\Sigma^{\\mathrm{memory}}","kind":"use","excerpt":"Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, addres","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-99b513a093fa66fd","tex":"\\Sigma^{\\mathrm{provenance}}","aliases":"Σ^provenance Sigma mathrm","display":"\\Sigma^{\\mathrm{provenance}}","kind":"use","excerpt":"Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, addres","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be8a4ba681d3ba27","tex":"\\Sigma^{\\mathrm{repair}}","aliases":"Σ^repair Sigma mathrm","display":"\\Sigma^{\\mathrm{repair}}","kind":"use","excerpt":"Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, addres","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-af2a087cfa28bab4","tex":"\\Sigma^{\\mathrm{rollback}}","aliases":"Σ^rollback Sigma mathrm","display":"\\Sigma^{\\mathrm{rollback}}","kind":"use","excerpt":"Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, addres","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-92534e05b5189ab3","tex":"\\Sigma^{\\mathrm{task}}","aliases":"Σ^task Sigma mathrm","display":"\\Sigma^{\\mathrm{task}}","kind":"use","excerpt":"Let a complete implementation state decompose as Σ = Σ^task ⊕ Σ^memory ⊕ Σ^repair ⊕ Σ^evaluation ⊕ Σ^provenance ⊕ Σ^rollback. A transaction selects an editable mask E⊆Σ and a witness mask W⊆Σ with E∩ W=∅. The decomposition is logical rather than necessarily physical. Two components may share hardware while remaining isolated by capabilities, addres","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-59681cd776049191","tex":"\\Sigma_0","aliases":"Σ_0 Sigma","display":"\\Sigma_0","kind":"use","excerpt":"The hidden source space has three unknown rank-two blocks: H=H_ctx⊕ H_rep⊕ H_nui. The inherited signature matrix is Σ_0 = 1.25 · 0.72 1.25 · 0.72 0.80 · 1.30 , where rows correspond to context, repair, and nuisance blocks. The first two rows coincide. By [source reference], every element of the inherited response algebra acts identically on the con","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5a15821b8a097b6","tex":"\\sigma_1^{-2}","aliases":"σ_1^-2 sigma","display":"\\sigma_1^{-2}","kind":"use","excerpt":"Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-58c1a4cd175eaace","tex":"\\sigma_d","aliases":"σ_d sigma","display":"\\sigma_d","kind":"use","excerpt":"[Nearest-signature route stability] Let the minimum pairwise normalized signature separation of an atlas be Δ_A = min_c≠ dd_sig(σ_c,σ_d). If an observed signature from chart c has normalized perturbation at most δ and 2δ<Δ_A, nearest-signature routing returns c. If additionally δ≤τ_adm, the route is admitted.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1d8ad5c3dcba600b","tex":"\\sigma_g","aliases":"σ_g sigma","display":"\\sigma_g","kind":"use","excerpt":"A useful effect profile is vector-valued: g= ( Δ J, Δ C, Δ R, ε_leak, ε_hol, ΔΓ, σ_g ), recording predicted task gain, resource change, risk change, leakage, semantic holonomy, metaproductivity change, and uncertainty. Scalarizing this vector is an evaluator decision, not part of the repair syntax.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0ed845e79c8d8e41","tex":"\\sigma_i","aliases":"σ_i sigma","display":"\\sigma_i","kind":"use","excerpt":"Let the source hidden state have dimension d and the target hidden state have dimension q. For selected source anchors x_i∈ℝ^d, suppose the observed target anchor is y_i=Tx_i+ε_i, E[ε_i]=0, Cov(ε_i)=σ_i^2I_q. The query-quality metadata supplies σ_i before the anchor is selected. This metadata is part of the declared Block 9 environment. It is not d","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-40b58f8694a7cca4","tex":"\\sigma_i^{-2}","aliases":"σ_i^-2 sigma","display":"\\sigma_i^{-2}","kind":"use","excerpt":"Let Msucc0 be the current regularized information matrix. Adding candidate anchor x_i with quality weight w_i=σ_i^-2 gives M'=M+w_ix_ix_i^ T.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f9ab434fbb0d31ef","tex":"\\sigma_i^2","aliases":"σ_i^2 sigma","display":"\\sigma_i^2","kind":"use","excerpt":"Let the source hidden state have dimension d and the target hidden state have dimension q. For selected source anchors x_i∈ℝ^d, suppose the observed target anchor is y_i=Tx_i+ε_i, E[ε_i]=0, Cov(ε_i)=σ_i^2I_q. The query-quality metadata supplies σ_i before the anchor is selected. This metadata is part of the declared Block 9 environment. It is not d","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-22301d4bf1c931e5","tex":"\\sigma_{\\min}","aliases":"σ_min sigma min","display":"\\sigma_{\\min}","kind":"use","excerpt":"[Ridge bias under a full-rank design] Under the hypotheses of [source reference], the ridge estimate satisfies widehat T^top-T^top =-λ(X^top X+λ I)^-1T^top, and therefore normwidehat T-T ≤ fracλσ_min(X)^2+λnormT.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-980b4068c8b679ce","tex":"\\sigma_n^{-2}","aliases":"σ_n^-2 sigma","display":"\\sigma_n^{-2}","kind":"use","excerpt":"Stacking the selected anchors gives Y=XT^ T+E, where X∈ℝ^n× d and Y∈ℝ^n× q. Define W=diag(σ_1^-2,…,σ_n^-2) and the weighted information matrix M_S=X^ TWX.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c2d9093abae7e9ca","tex":"\\Sigma_{\\mathrm{ob}}","aliases":"Σ_ob Sigma mathrm","display":"\\Sigma_{\\mathrm{ob}}","kind":"use","excerpt":"Let foreign residuals follow distribution Q. Define the obstruction covariance operator Σ_ob = E_r∼ Q [ P_I^⊥r r^top P_I^⊥ ]. Let W⊆ I^⊥ be a k-dimensional new response subspace.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b04476ecd94b8cb2","tex":"\\Sigma_{\\rm pre}","aliases":"Σ_rm pre Sigma rm","display":"\\Sigma_{\\rm pre}","kind":"use","excerpt":"Let Ω = Ω_rm devsqcupΩ_rm audit be a disjoint target index set. The compiler receives rows indexed only by Ω_rm dev. It emits a descendant configuration c_1, a compiler ledger L_Meta, and a pre-audit seal Σ_rm pre = H(c_1,L_Meta,source hashes), where H is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory de","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4d66cf40a5afaf2c","tex":"\\sigma_t","aliases":"σ_t sigma","display":"\\sigma_t","kind":"use","excerpt":"Define a module status map σ_t(i) ∈ {active,saturated,unidentified,protected}. An admissible bottleneck controller should: item estimate headroom for active modules; item refuse large gain claims for saturated modules; item request new evidence for unidentified modules; item avoid changing protected modules inside the current transaction; item trig","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b3d2bb6c29d620e7","tex":"\\tau'","aliases":"τ' tau","display":"\\tau'","kind":"use","excerpt":"Let d_prog(τ,τ') be a normalized distance between (A_m,B_m) and (A'_m',B'_m'), and let d_ans(τ,τ') = absy_m/4-y'_m'/4. For a recipient task τ_i, define four donor classes: item SAP: same-answer, far-program, approximately depth-matched donors; item FAD: far-answer, depth-matched donors; item SN: semantic-neighbor donors; item RMD: random depth-matc","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8ab2a2128bd5f80b","tex":"\\tau_{\nm int}","aliases":"τ_ m int tau","display":"\\tau_{\nm int}","kind":"use","excerpt":"Let ϕ_0:T→Z_0 be the inherited diagnostic map. In Block 15, six inherited obstruction classes retain their existing tokens, while two previously unseen obstruction mechanisms are both mapped to one open-set token bot: ϕ_0(τ_ m int) = ϕ_0(τ_ m temp) = bot. The equality in [source reference] is not itself a defect. An unknown token is an appropriate ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5034d59714315e27","tex":"\\tau_{\nm temp}","aliases":"τ_ m temp tau","display":"\\tau_{\nm temp}","kind":"use","excerpt":"Let ϕ_0:T→Z_0 be the inherited diagnostic map. In Block 15, six inherited obstruction classes retain their existing tokens, while two previously unseen obstruction mechanisms are both mapped to one open-set token bot: ϕ_0(τ_ m int) = ϕ_0(τ_ m temp) = bot. The equality in [source reference] is not itself a defect. An unknown token is an appropriate ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8c811b6eab2988ab","tex":"\\tau_1","aliases":"τ_1 tau","display":"\\tau_1","kind":"use","excerpt":"[Serializability of independent repairs] Let τ_1 and τ_2 edit disjoint component masks. Suppose their sandbox state transformations commute, their capability effects are disjoint, and neither transaction changes the sealed witness inputs of the other. Then committing both transactions in either order yields the same operational state and the same p","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e675956848d7751d","tex":"\\tau_2","aliases":"τ_2 tau","display":"\\tau_2","kind":"use","excerpt":"[Serializability of independent repairs] Let τ_1 and τ_2 edit disjoint component masks. Suppose their sandbox state transformations commute, their capability effects are disjoint, and neither transaction changes the sealed witness inputs of the other. Then committing both transactions in either order yields the same operational state and the same p","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5a8b2d67fc2c547d","tex":"\\tau_A","aliases":"τ_A tau","display":"\\tau_A","kind":"use","excerpt":"For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c5bd92712ba2926c","tex":"\\tau_a","aliases":"τ_a tau","display":"\\tau_a","kind":"use","excerpt":"[Aliased-token routing obstruction] Let τ_a and τ_b satisfy ϕ_0(τ_a)=ϕ_0(τ_b) and A^⋆(τ_a)∩A^⋆(τ_b)=∅. Then no deterministic compiler policy that conditions only on ϕ_0(τ) can select an acceptable method for both problems.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5a905007981f9c00","tex":"\\tau_b","aliases":"τ_b tau","display":"\\tau_b","kind":"use","excerpt":"[Aliased-token routing obstruction] Let τ_a and τ_b satisfy ϕ_0(τ_a)=ϕ_0(τ_b) and A^⋆(τ_a)∩A^⋆(τ_b)=∅. Then no deterministic compiler policy that conditions only on ϕ_0(τ) can select an acceptable method for both problems.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b62e889b0ccd5fc1","tex":"\\tau_g","aliases":"τ_g tau","display":"\\tau_g","kind":"use","excerpt":"For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7d5a8ef046afc5c0","tex":"\\tau_I","aliases":"τ_I tau","display":"\\tau_I","kind":"use","excerpt":"For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2ed1bcaa9d2d3dfe","tex":"\\tau_i","aliases":"τ_i tau","display":"\\tau_i","kind":"use","excerpt":"Let Γ_0 and Γ_1 operate on the same problem τ with identical initial archive, candidate grammar, experiment budget, evaluator portfolio, integrity monitor, and hidden audit isolation. Let D_i = U_audit (K_Γ_1^⋆(τ_i)) - U_audit (K_Γ_0^⋆(τ_i)).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c008973d5c31f569","tex":"\\tau_{\\mathrm{imp}}","aliases":"τ_imp tau mathrm","display":"\\tau_{\\mathrm{imp}}","kind":"use","excerpt":"Let S be a finite set of complete lineage states. Write sxrightarrowτ_imps' when one authorized improvement transaction can take s to s'. Define Post_imp(s) = sets'∈ S:sxrightarrowτ_imps'.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2412aa79952fb985","tex":"\\tau_k","aliases":"τ_k tau","display":"\\tau_k","kind":"use","excerpt":"For foreign agent j, let Y_j(z) be protected gain under factor setting z. The average necessity effect of factor k relative to the intact architecture 1 is τ_k = E [ Y_j(1)-Y_j(1-e_k) ]. A positive value supports causal usefulness only relative to the frozen environment, compensating mechanisms, and matched resource schedule.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cae95424b3814cc0","tex":"\\tau_R","aliases":"τ_R tau","display":"\\tau_R","kind":"use","excerpt":"For one candidate row r, a configuration commits only if widehat g_rm cal(r) · ≥ τ_g, widehat R_rm cal(r) · ≤ τ_R, widehatε_rm align(r) · ≤ τ_A, λ_min(M_r) · ≥ τ_I, rank X_r · =d. The selected Block 9 grandchild has (τ_g,τ_R,τ_A,τ_I) = (0.02,0.05,0.20,2.0). The recalibrated rank child has (0.02,0.05,0.55,0.10).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-53304761e001a2f9","tex":"\\tau_r","aliases":"τ_r tau","display":"\\tau_r","kind":"use","excerpt":"widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-12d297d451d89dd8","tex":"\\tau_{\\rm align}","aliases":"τ_rm align tau rm","display":"\\tau_{\\rm align}","kind":"use","excerpt":"The compiler selected: π_rm anc · =π_rm PQR, k_rm fit · =16, k_rm val · =8, γ_rm cal · =0.08, ρ_rm cal · =0.035, τ_rm align · =0.05, full-rank requirement · =true. The selected development score is 0.263289. The underlying mean selected development gain is 0.296478 with zero harmful targets in the development ledger.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0070db7cf4097403","tex":"\\tau_{\\rm cal}","aliases":"τ_rm cal tau rm","display":"\\tau_{\\rm cal}","kind":"use","excerpt":"widehat R_rm UCB(z) · = max{0,widehat R(z)}+κ_r s_r,c(z). A candidate is admissible when widehat R_rm UCB(z)≤τ_r, widehat r_rm cal(z)≤τ_rm cal, A^rm base_rm cal,prot(z)≥ a_min. The child chooses the admissible candidate with maximum widehat U_rm LCB. If the admissible set is empty, it returns the no-op.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dcf42c66c1793b6d","tex":"\\tau_{\\rm distance}","aliases":"τ_rm distance tau rm","display":"\\tau_{\\rm distance}","kind":"use","excerpt":"The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-be1e22ccc084b92e","tex":"\\tau_{\\rm residual}","aliases":"τ_rm residual tau rm","display":"\\tau_{\\rm residual}","kind":"use","excerpt":"The selected child configuration is τ_rm residual · =0.20, · τ_rm distance · =24, w_rm mean · =0.38, · w_rm median · =0.10, w_rm min · =0.24, · w_rm held · =0.22, w_rm capability · =0.32. It is selected from a 27-point finite meta-validation grid before either final audit is opened.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-224c75503750e88d","tex":"\\phi^{-1}_{\\alpha}","aliases":"ϕ^-1_α phi alpha","display":"\\phi^{-1}_{\\alpha}","kind":"use","excerpt":"The nonlinear family applies Y_t = fracsinh (α(X_tA+b))α, α>0, coordinatewise. The declared inverse constructor first applies ϕ^-1_α(y) = fracarsinh(α y)α and then fits a step-affine map.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b7a0bee7eaa530d1","tex":"\\phi^{\\circ}","aliases":"ϕ^∘ phi circ","display":"\\phi^{\\circ}","kind":"use","excerpt":"The permitted intervention removes only the conditional feature block: ϕ^∘(x) = ψ(x) 0 . The resulting hidden difference is Δ_a(x) = h_a(x)-h_a^∘(x), h_a^∘(x)=E_a(ϕ^∘(x)). The empirical causal packet is the leading rank-r right-singular subspace of the affected differences {Δ_a(x_i):s(x_i)=1}.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d0960cac4512b74b","tex":"\\phi_{\\mathrm{actuation}}","aliases":"ϕ_actuation phi mathrm","display":"\\phi_{\\mathrm{actuation}}","kind":"use","excerpt":"From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-994c984c468efefd","tex":"\\Phi_{\\mathrm{disc}}","aliases":"Φ_disc Phi mathrm","display":"\\Phi_{\\mathrm{disc}}","kind":"use","excerpt":"Let Ω be a finite universe of candidate repair-schema equivalence classes. For S⊆Ω, let Disc(S) be the schemas independently validated using the currently available set S. Define Φ_disc(S) = S∪Disc(S).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-094ea5bd62a468ae","tex":"\\phi_{\\mathrm{evaluator}}","aliases":"ϕ_evaluator phi mathrm","display":"\\phi_{\\mathrm{evaluator}}","kind":"use","excerpt":"From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8efc5e23b03bd051","tex":"\\phi_{\\Gcal}","aliases":"ϕ_Gcal ϕ_G phi Gcal phi mathcal","display":"\\phi_{{\\mathcal{G}}}","kind":"use","excerpt":"Let ϕ_G:X→ Z_G be the state representation expressible in grammar G. A G-measurable deterministic repair policy has the form π=widetildeπ∘ϕ_G.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-eaee07229115da6d","tex":"\\phi_{\\Gcal'}","aliases":"ϕ_Gcal' ϕ_G' phi Gcal phi mathcal","display":"\\phi_{{\\mathcal{G}}'}","kind":"use","excerpt":"Appending the feature gives the conservative extension ϕ_G'(x) = (ϕ_G(x),f(x)). The old representation is recovered by projection onto the first coordinate.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2202dc894cbacfe8","tex":"\\phi_{\\mathrm{memory}}","aliases":"ϕ_memory phi mathrm","display":"\\phi_{\\mathrm{memory}}","kind":"use","excerpt":"From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7a127cec61c4d44f","tex":"\\phi_{p,\\theta}","aliases":"ϕ_p,θ phi theta","display":"\\phi_{p,\\theta}","kind":"use","excerpt":"Let p∈P^+ be a candidate experiment with response y_p:T⟶ℝ. For threshold θ, define the refined diagnostic map ϕ_p,θ(τ) = (ϕ_0(τ),low), · y_p(τ)≤θ, (ϕ_0(τ),high), · y_p(τ)>θ. The old token is recovered by the projection π_0(z,s)=z, π_0∘ϕ_p,θ=ϕ_0. Thus the extension is conservative at the level of old diagnostic semantics.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-97c68088f37b807d","tex":"\\phi_{\\mathrm{passive}}","aliases":"ϕ_passive phi mathrm","display":"\\phi_{\\mathrm{passive}}","kind":"use","excerpt":"From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e597f995b307e360","tex":"\\phi_{\\mathrm{sensor}}","aliases":"ϕ_sensor phi mathrm","display":"\\phi_{\\mathrm{sensor}}","kind":"use","excerpt":"From a common probe state x_0 and bounded action a, define the feature vector ϕ(z) = ϕ_passive ϕ_actuation ϕ_sensor ϕ_memory ϕ_evaluator . Its components are, respectively: item deviation from the baseline no-action continuation; item deviation of the action-conditioned response from the baseline action response; item violation of redundant-sensor ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-62d471c4d676af70","tex":"\\Phi_t","aliases":"Φ_t Phi","display":"\\Phi_t","kind":"use","excerpt":"[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a487bc1b1829adae","tex":"\\Phi_{t+1}","aliases":"Φ_t+1 Phi","display":"\\Phi_{t+1}","kind":"use","excerpt":"[Greedy bottleneck contraction] Let r_t∈ℝ_≥0^m and Φ_t=1^top r_t. Suppose the controller chooses i_t with r_i_t,t≥fracΦ_tm, contracts that component by q∈[0,1), and permits total collateral increase at most ε_t. Then Φ_t+1 ≤ (1-frac1-qm)Φ_t+ε_t.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-32a2c188b00bbfc0","tex":"\\phi_{t+1}","aliases":"ϕ_t+1 phi","display":"\\phi_{t+1}","kind":"use","excerpt":"A useful self-model participates in a complete causal cycle: ϕ_t(A_t) xrightarrowcandidate policy δ_t xrightarrowworld execution Y_t+1 xrightarrowevidence update ϕ_t+1(A_t+1). The candidate is selected before the protected outcome is revealed. The outcome affects future beliefs and repair policy. Cutting either planner use or consequence feedback d","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bdfa64f646d856b0","tex":"\\chi_R","aliases":"χ_R chi","display":"\\chi_R","kind":"use","excerpt":"Let K→ K' change only modules in R. When F_K(R)>M(K), define the headroom-adjusted gain χ_R(K,K') = frac M(K')-M(K) F_K(R)-M(K) . For an admissible child inside the declared class, χ_R(K,K')≤1. A small absolute gain with χ_R≈1 indicates a nearly saturated parent. A small absolute gain with χ_R≈0 indicates that the child captured little of the avail","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"· wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X, χ(0,0)=0, such that P_RF(p,k+χ(p,k))=0. Define the reduced obstruction map ψ(p,k):=P_CF(p,k+χ(p,k)). A sufficiently small pair (p,u) solves F(p,u)=0 if and only if u=k+χ(p","count":20},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f2c77f921bfa53f0","tex":"\\Psi_{\\mathrm{ctrl}}","aliases":"Ψ_ctrl Psi mathrm","display":"\\Psi_{\\mathrm{ctrl}}","kind":"use","excerpt":"Define Ψ_ctrl(X) = sets∈ S:C_RI(s) ∩ CPre(X).","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0078b4020b69082c","tex":"\\Psi_{\\forall}","aliases":"Ψ_∀ Psi forall","display":"\\Psi_{\\forall}","kind":"use","excerpt":"· wedgeRollbackable(s) wedgeMetaCapable(s). For X⊆ S, define Ψ_∀(X) = sets∈ S:C_RI(s) and Post_imp(s)⊆ X.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c21a9164798ae206","tex":"\\omega_0","aliases":"ω_0 omega","display":"\\omega_0","kind":"use","excerpt":"Suppose a sandbox returns only the scalar task score q(ω)=accuracy(ω). Execution ω_0 reaches accuracy 0.9 using authorized retrieval. Execution ω_1 reaches the same accuracy after importing a prohibited hidden answer key. Then q(ω_0)=q(ω_1)=0.9, but the ideal full-information decisions differ. By [source reference], no witness using only accuracy c","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-af6bf3814fcd9f1d","tex":"\\omega_{CH}","aliases":"ω_CH omega","display":"\\omega_{CH}","kind":"use","excerpt":"Consider five mutable modules: N_5 = {S,Q,T,C,H}, representing selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let F_K(∅)=0.20. Assign singleton contributions (0.08,0.12,0.10,0.16,0.14) and pair synergies ω_QT=0.05, ω_CH=0.06, ω_SC=0.03. Then F_K(N_5)=0.94, H(K)=0.74. Every pair synergy is divided ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9905f533cc1e6354","tex":"\\omega_{\\mathrm{check}}","aliases":"ω_check omega mathrm","display":"\\omega_{\\mathrm{check}}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ecb53f64a66e56b0","tex":"\\omega_{\\mathrm{decompose}}","aliases":"ω_decompose omega mathrm","display":"\\omega_{\\mathrm{decompose}}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-90665417029987ed","tex":"\\omega_{\\mathrm{gate}}","aliases":"ω_gate omega mathrm","display":"\\omega_{\\mathrm{gate}}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d9ccf90cb1c84e6a","tex":"\\omega_n","aliases":"ω_n omega","display":"\\omega_n","kind":"use","excerpt":"Each executable operation has a Hoare-style contract {C_ω} ω {Q_ω}. Syntactic substitution ω∘(ω_1,…,ω_n) is semantically authorized only when the inner postconditions and bridge guarantees imply the outer precondition.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b411ac72412752cc","tex":"\\omega_{QT}","aliases":"ω_QT omega","display":"\\omega_{QT}","kind":"use","excerpt":"Consider five mutable modules: N_5 = {S,Q,T,C,H}, representing selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let F_K(∅)=0.20. Assign singleton contributions (0.08,0.12,0.10,0.16,0.14) and pair synergies ω_QT=0.05, ω_CH=0.06, ω_SC=0.03. Then F_K(N_5)=0.94, H(K)=0.74. Every pair synergy is divided ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-3ac5b0587c9411f7","tex":"\\omega_{\\mathrm{retrieve}}","aliases":"ω_retrieve omega mathrm","display":"\\omega_{\\mathrm{retrieve}}","kind":"use","excerpt":"Consider four colors memory, representation, verifier, planner. A composite factuality repair may have the form ω_gate ∘ ( ω_retrieve, ω_decompose, ω_check ). The inner operations retrieve evidence, decompose output into claims, and check claims against protected sources. Their postconditions establish the precondition for a planner gate that block","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d1b2b1752a7a2af7","tex":"\\Omega_{\\rm audit}","aliases":"Ω_rm audit Omega rm","display":"\\Omega_{\\rm audit}","kind":"use","excerpt":"Let Ω = Ω_rm devsqcupΩ_rm audit be a disjoint target index set. The compiler receives rows indexed only by Ω_rm dev. It emits a descendant configuration c_1, a compiler ledger L_Meta, and a pre-audit seal Σ_rm pre = H(c_1,L_Meta,source hashes), where H is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory de","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-97ca5335ffe89083","tex":"\\Omega_{\\rm dev}","aliases":"Ω_rm dev Omega rm","display":"\\Omega_{\\rm dev}","kind":"use","excerpt":"Let Ω = Ω_rm devsqcupΩ_rm audit be a disjoint target index set. The compiler receives rows indexed only by Ω_rm dev. It emits a descendant configuration c_1, a compiler ledger L_Meta, and a pre-audit seal Σ_rm pre = H(c_1,L_Meta,source hashes), where H is SHA-256 over canonical serialization. Audit execution refuses to proceed when the in-memory de","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6b5d345def4e804d","tex":"\\omega_{SC}","aliases":"ω_SC omega","display":"\\omega_{SC}","kind":"use","excerpt":"Consider five mutable modules: N_5 = {S,Q,T,C,H}, representing selection, information acquisition, transport, candidate construction, and causal-atlas structure. Let F_K(∅)=0.20. Assign singleton contributions (0.08,0.12,0.10,0.16,0.14) and pair synergies ω_QT=0.05, ω_CH=0.06, ω_SC=0.03. Then F_K(N_5)=0.94, H(K)=0.74. Every pair synergy is divided ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1e05c28e5f477db7","tex":"\\dots","aliases":"… dots","display":"\\dots","kind":"use","excerpt":"[Finite ranking certificate for one gain cycle] Let X⊆I_ctrl^* and let h:X→{0,1,…,H} satisfy h^-1(0)⊆ G_γ. Suppose that at every state with h(s)>0, the controller has an authorized action whose every admissible successor s' lies in X and satisfies h(s')<h(s). Then the controller can force a visit to G_γ within at most H transactions while remaining","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9768067e91e1baac","tex":"\\ell^{\\mathsf T}","aliases":"ℓ^ T ell mathsf","display":"\\ell^{\\mathsf T}","kind":"use","excerpt":"[Perron sensitivity] Let M be irreducible with simple Perron root ρ, positive right eigenvector r, and positive left eigenvector ℓ, normalized by ℓ^ Tr=1. Then frac∂ρ∂ M_ij = ℓ_i r_j.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-1b5b4e6ee09dffe3","tex":"\\ell_A","aliases":"ℓ_A ell","display":"\\ell_A","kind":"use","excerpt":"Let ℓ_A:V_A→ℝ and ℓ_B:V_B→ℝ be post-defect losses. Define source and target gains g_A := ℓ_A(r_A)-ℓ_A(r_A+D_Au_A), g_B := ℓ_B(r_B)-ℓ_B(r_B+D_BT_Uu_A).","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-96a5e0d2554892b6","tex":"\\ell_B","aliases":"ℓ_B ell","display":"\\ell_B","kind":"use","excerpt":"Let ℓ_A:V_A→ℝ and ℓ_B:V_B→ℝ be post-defect losses. Define source and target gains g_A := ℓ_A(r_A)-ℓ_A(r_A+D_Au_A), g_B := ℓ_B(r_B)-ℓ_B(r_B+D_BT_Uu_A).","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0799320fec203d2f","tex":"\\ell_{\\mathrm{chart}}","aliases":"ℓ_chart ell mathrm","display":"\\ell_{\\mathrm{chart}}","kind":"use","excerpt":"For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-cd64c30acacfc97f","tex":"\\ell_{\\mathrm{chart},i}","aliases":"ℓ_chart,i ell mathrm","display":"\\ell_{\\mathrm{chart},i}","kind":"use","excerpt":"For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c41d4fd935bc7dcd","tex":"\\ell_{\\mathrm{global}}","aliases":"ℓ_global ell mathrm","display":"\\ell_{\\mathrm{global}}","kind":"use","excerpt":"For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-82cdd49e88ec848f","tex":"\\ell_{\\mathrm{global},i}","aliases":"ℓ_global,i ell mathrm","display":"\\ell_{\\mathrm{global},i}","kind":"use","excerpt":"For every final task, let ℓ_global and ℓ_chart be the source-probe semantic losses after pooled and selected transport. The paired effect is Δ_i = ℓ_global,i - ℓ_chart,i. Positive values favor the selected chart-indexed map.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"[Perron sensitivity] Let M be irreducible with simple Perron root ρ, positive right eigenvector r, and positive left eigenvector ℓ, normalized by ℓ^ Tr=1. Then frac∂ρ∂ M_ij = ℓ_i r_j.","count":5},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6772f700e4ceabce","tex":"\\ell_{\\rm selected,i}","aliases":"ℓ_rm selected,i ell rm","display":null,"kind":"use","excerpt":"For nonlinear, temporal, and chart-alias episodes, the inherited step-affine map is retained as a wrong-repair control. The paired effect is Δ_i = ℓ_rm step affine,i - ℓ_rm selected,i. Every frozen bootstrap interval is strictly positive.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8d9c6c1b61c911f0","tex":"\\ell_{\\rm step\\ affine,i}","aliases":"ℓ_rm step affine,i ell rm","display":null,"kind":"use","excerpt":"For nonlinear, temporal, and chart-alias episodes, the inherited step-affine map is retained as a wrong-repair control. The paired effect is Δ_i = ℓ_rm step affine,i - ℓ_rm selected,i. Every frozen bootstrap interval is strictly positive.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"use","excerpt":"[Internal error reduction without external improvement] Let the environment always produce protected outcome Y=1, independently of the agent's action. Let an internal predictor be p∈ℝ with internal score J_int(p)=-(p-Y)^2. A self-intervention setting p←1 improves J_int to zero error but does not change the world, task success, or action consequence","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9ec711e428117ead","tex":"\\twoheadrightarrow","aliases":"↠ twoheadrightarrow","display":"\\twoheadrightarrow","kind":"use","excerpt":"Quotient or forgetful map · X↠ Y · Forward transport with declared information loss; reconstruction requires additional data.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Convex authorized correctability] If K is compact, or if u↦normDu+r is coercive on K, then the infimum defining η_D,K(r) is attained. Exact authorized correction exists if and only if η_D,K(r)=0.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"The example separates two predicates: ImprovesSlice(R_t) not⇒ AuthorizedCommit(R_t).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"[Lyapunov--Schmidt repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-dc5dc7a90e422d6f","tex":"\\partial_{(x,y)}","aliases":"∂_(x,y) partial","display":"\\partial_{(x,y)}","kind":"use","excerpt":"x^2 . At (ε,x,y)=(0,0,0), D=∂_(x,y)F = 1 · 0 0 · 0 . The first-order defect r_1=e_1 is correctable by choosing x_1=-1. However, the order-ε^2 term in the second coordinate is 1, which represents a nonzero class in coker D. Indeed, the exact first equation forces x=-ε, and then the second equation becomes ε^2=0. Thus no nonzero small ε admits a repa","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0772d84d70d4f26b","tex":"\\partial_u","aliases":"∂_u partial","display":"\\partial_u","kind":"use","excerpt":"[Lyapunov--Schmidt repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9b42c21c945c34dd","tex":"\\prod_{b\\neq a}","aliases":"∏_b≠ a prod neq","display":"\\prod_{b\\neq a}","kind":"use","excerpt":"of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-df3ae28461c2ddd4","tex":"\\prod_{j=1}^s","aliases":"∏_j=1^s prod","display":"\\prod_{j=1}^s","kind":"use","excerpt":"For integrity monitoring, let M_1,…,M_s∈{0,1} be monitor pass indicators and let the ensemble pass only when every component passes: M_∩=∏_j=1^s M_j.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-21ca24a941a6886b","tex":"\\prod_{j\\in I_+(c)}","aliases":"∏_j∈ I_+(c) prod in","display":"\\prod_{j\\in I_+(c)}","kind":"use","excerpt":"Let the scored evaluator portfolio be B = {b_1,…,b_m}. For method c, define normalized headroom closure q_j(c) = fracs_j(c)-s_j(K_τ)s_j^⋆-s_j(K_τ). Let I_+(c)={j:q_j(c)>0}. The coverage-weighted geometric score is S_cov(c) = frac|I_+(c)|m ( ∏_j∈ I_+(c)q_j(c) )^1/|I_+(c)|, with value zero when I_+(c)=∅. Hard capability and integrity gates are applie","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-625fcf97724431c7","tex":"\\prod_{q<j}","aliases":"∏_q<j prod","display":"\\prod_{q<j}","kind":"use","excerpt":"For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"For recovered projectors widehat P_1,…,widehat P_m, define the normalized response-fit residual ε_resp = frac ( ∑_jnormwidehat R_j- ∑_awidehatλ_ajwidehat P_a_F^2 )^1/2 ( ∑_jnormwidehat R_j_F^2 )^1/2+10^-12 , and the maximum normalized commutator residual ε_comm = max_a,j fracnormwidehat P_awidehat R_j- widehat R_jwidehat P_a_F normwidehat R_j_F+10^","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7684dda7bca2d3dd","tex":"\\sum_{a,b}","aliases":"∑_a,b sum","display":"\\sum_{a,b}","kind":"use","excerpt":"Write X=∑_a,bP_aXP_b. The relation XR_j=R_jX implies (λ_bj-λ_aj)P_aXP_b=0 for every j. Separating signatures imply that for a≠ b some coefficient is nonzero, so P_aXP_b=0. Hence X is block diagonal. Every block-diagonal operator commutes with each block-scalar R_j. The dimension formula follows from dimEnd(H_a)=d_a^2.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e91f39a65624d4ad","tex":"\\sum_{b=1}^{m}","aliases":"∑_b=1^m sum","display":"\\sum_{b=1}^{m}","kind":"use","excerpt":"Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a lies in the generated algebra. A nonzero central projection that is strictly below ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5f978c2eb01800e5","tex":"\\sum_{e\\in E(\\Lambda)}","aliases":"∑_e∈ E(Λ) sum in Lambda","display":"\\sum_{e\\in E(\\Lambda)}","kind":"use","excerpt":"For a branching archive, index proposed edges by e∈ E(Λ). Assign each edge a budget δ_e>0 with ∑_e∈ E(Λ)δ_e≤δ. The same proof yields archive-wide false-acceptance control. The allocation may be adaptive provided the remaining total budget is predictable from the pre-audit filtration.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-8a8ec12adedc2354","tex":"\\sum_{i=1}^{5}","aliases":"∑_i=1^5 sum","display":"\\sum_{i=1}^{5}","kind":"use","excerpt":"Each task contains five demonstration pairs D={(x_i,y_i)}_i=1^5 and a query q. After chart canonicalization, the full descendant computes pair codes p_i=ϕ(widehat x_i,widehat y_i)∈ℝ^32, and the permutation-invariant aggregate a(D) = ( frac15∑_i=1^5p_i, max_1≤ i≤5p_i )∈ℝ^64. The semantic seed is h_0=S(a(D),Q(widehat q))∈ℝ^40. A five-step Elman recur","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6056a9b0b01bdd1d","tex":"\\sum_{i=1}^6","aliases":"∑_i=1^6 sum","display":"\\sum_{i=1}^6","kind":"use","excerpt":"Each foreign agent has a canonical defect d∈ℝ^6, a positive diagonal importance vector w, and an invertible architecture transform T∈ℝ^6×6. The runtime-visible defect is widetilde d=Td. Protected utility is J_T(widetilde d) = exp(-∑_i=1^6w_i(T^-1widetilde d)_i^2). A candidate runtime patch δ produces gain g(δ) = J_T(widetilde d+δ)-J_T(widetilde d).","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f51dd287b0dcadbb","tex":"\\sum_{i=1}^{900}","aliases":"∑_i=1^900 sum","display":"\\sum_{i=1}^{900}","kind":"use","excerpt":"Let H_s(τ)∈ℝ^36 be the source model's final latent trace state and H_t(τ)∈ℝ^36 the corresponding foreign-child state. A ridge-affine bridge is fitted on the 900 paired generator-A anchors: B_t = arg min_B,b ∑_i=1^900 normH_s(τ_i)-BH_t(τ_i)-b^2 +λnormB^2. A source ridge probe P_s is fitted independently on the 1,000 probe-calibration tasks. The tran","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7b615a361efd7684","tex":"\\sum_{i=1}^k","aliases":"∑_i=1^k sum","display":"\\sum_{i=1}^k","kind":"use","excerpt":"t the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-09ade2b60fb5add0","tex":"\\sum_{i=1}^{k}","aliases":"∑_i=1^k sum","display":"\\sum_{i=1}^{k}","kind":"use","excerpt":"[Spectral characterization of flatness] The exact common fixed space is E^flat=ker A. Let the eigenvalues of A be 0≤λ_1≤⋯≤λ_d. Among all k-dimensional subspaces F⊆ E, the quantity ∑_i=1^kanglesx_i, Ax_i for an orthonormal basis x_1,…,x_k of F is minimized by the span of eigenvectors associated with λ_1,…,λ_k, and the minimum is ∑_i=1^kλ_i.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7a21c7919adfbd23","tex":"\\sum_{i=1}^m","aliases":"∑_i=1^m sum","display":"\\sum_{i=1}^m","kind":"use","excerpt":"Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-2f29622fb484623a","tex":"\\sum_{i=1}^n","aliases":"∑_i=1^n sum","display":"\\sum_{i=1}^n","kind":"use","excerpt":"Let D_i be the grandchild-minus-child protected gain on final foreign target i. The primary finite estimand is overline D = frac1n∑_i=1^nD_i. The release records a paired bootstrap interval, a sign-flip test, positive and negative pair rates, and exact query ledgers. The matched target pairing removes between-target difficulty from the edge compari","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d2da8fdb2e437119","tex":"\\sum_{i=1}^{n}","aliases":"∑_i=1^n sum","display":"\\sum_{i=1}^{n}","kind":"use","excerpt":"Let K_0 and K_1 be frozen improvement mechanisms evaluated on the same independent foreign agents A_1,…,A_n. Define G_h,B(K,A) = max_A'∈Desc_h,B(K,A)J(A')-J(A) and assume -G_max ≤ G_h,B(K,A) ≤ G_max. For matched budgets and random seeds, let D_i = G_h,B(K_1,A_i)-G_h,B(K_0,A_i), overline D_n = frac1n∑_i=1^nD_i.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e15f9d5ad22b1aac","tex":"\\sum_{i\\in N}","aliases":"∑_i∈ N sum in","display":"\\sum_{i\\in N}","kind":"use","excerpt":"[Meta-headroom allocation] For every finite repair-process factorization, h_i(K)≥0 for every module i, and ∑_i∈ Nh_i(K)=H(K).","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-4b8afcda15a5dfd1","tex":"\\sum_{i\\in S}","aliases":"∑_i∈ S sum in","display":"\\sum_{i\\in S}","kind":"use","excerpt":"A candidate pool contains source state h_i, target observation y_i, and known observation scale σ_i. The inherited acquisition policy selects rows sequentially to maximize the quality-weighted log-determinant increment logdet( λ I+ ∑_i∈ Sσ_i^-2bar h_ibar h_i^top ). Both parent and child use the same resulting order. The parent takes the first 80 ro","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-bd8e89829b73947e","tex":"\\sum_{j=1}^{k}","aliases":"∑_j=1^k sum","display":"\\sum_{j=1}^{k}","kind":"use","excerpt":"Choose c∈ℝ^k outside the finite union of hyperplanes setc:c^ T(λ_a-λ_b)=0, a≠ b. Then the scalars μ_a=c^ Tλ_a are pairwise distinct. The combined response B=∑_j=1^kc_jR_j =∑_a=1^mμ_aP_a belongs to A_R. For each a, define the Lagrange polynomial p_a(z)=∏_b≠ afracz-μ_bμ_a-μ_b. Functional calculus gives p_a(B)=∑_b=1^mp_a(μ_b)P_b=P_a. Thus every P_a li","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-55bfd4be1daf726b","tex":"\\sum_{j=1}^{m}","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^{m}","kind":"use","excerpt":"Exact fixed subspaces can be too brittle for learned evaluator transports. Let H_1,…,H_m be a finite generating family of loop transports on a Hilbert space E, with positive weights w_j satisfying ∑_jw_j=1. Define the holonomy-energy operator A := ∑_j=1^m w_j(H_j-I)^*(H_j-I). This operator is positive semidefinite.","count":6},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-db50c77870a75686","tex":"\\sum_{j=1}^n","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^n","kind":"use","excerpt":"The archive arm can perform well without a causal repair calculus because it searches several noisy candidates and retains the best. Candidate-budget matching is therefore mandatory. Let G_a,j,1,…,G_a,j,B be candidate gains for arm a on agent j. The primary behavioral quantity is M_a(B) = frac1n∑_j=1^nmax_1≤ b≤ BG_a,j,b. Comparing arms at unequal B","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ef75e15ec0684e46","tex":"\\sum_{j=1}^{n}","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^{n}","kind":"use","excerpt":"Analytic expansion gives F(ε,∑_j=1^nε^ju_j) = ε^n(b_n+Du_n)+O(ε^n+1), because u_n first enters linearly at order n. Cancellation of the coefficient is equivalent to solvability of Du_n=-b_n.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d6b1a299418f0d9d","tex":"\\sum_{j=1}^{n-1}","aliases":"∑_j=1^n-1 sum","display":"\\sum_{j=1}^{n-1}","kind":"use","excerpt":"[Finite formal obstruction tower] Let F(ε,u) be analytic near (0,0) with F(0,0)=0 and D=∂_uF(0,0). Suppose coefficients u_1,…,u_n-1 have been chosen so that F(ε,∑_j=1^n-1ε^ju_j) =ε^nb_n+O(ε^n+1). Then extending the repair to order n by adding ε^nu_n is possible if and only if [b_n]=0 in coker D. When it vanishes, any solution of Du_n=-b_n produces ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-43ca5dff0132ae40","tex":"\\sum_{j=1}^{t}","aliases":"∑_j=1^t sum","display":"\\sum_{j=1}^{t}","kind":"use","excerpt":"For fixed maximum depth K_max, let λ_t(x)=Λ(z_t(x)). Define cumulative halt mass M_t(x) = ∑_j=1^t λ_j(x) ∏_q<j(1-λ_q(x)), and dynamic depth T(x)=min{t:M_t(x)≥τ}, with declared threshold τ. Block 16 does not infer adaptive computation from the existence of a halt head. It requires a difficulty variable d(x) fixed before result access and tests both ","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-820e24441cb27c12","tex":"\\sum_{j=8}^{16}","aliases":"∑_j=8^16 sum","display":"\\sum_{j=8}^{16}","kind":"use","excerpt":"If random anchor membership is Bernoulli with affected probability p, the idealized full-rank probability at k=16 is therefore Pr(rank X=16) = Pr(N_1≥8) = ∑_j=8^1616choose jp^j(1-p)^16-j under the generic model. The observed parent full-rank rate on the primary audit set is 0.03125, consistent with a low-probability affected-count event.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-64c62d58e1130581","tex":"\\sum_{j\\in N}","aliases":"∑_j∈ N sum in","display":"\\sum_{j\\in N}","kind":"use","excerpt":"Let N⊆{1,…,n} be foreign agents whose dominant defect belongs to the held-out sixth type. Define widehatτ_abstract = frac1|N|∑_j∈ N ( G_j^meta-G_j^no-meta ). In the reference positive control, widehatτ_abstract=0.700533. A learned recursive-improvement claim would require the schema to be generated by the parent improvement process, sealed before f","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b7f7abb917a0d195","tex":"\\sum_{k=0}^n","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^n","kind":"use","excerpt":"Since normEQ_0<1, the Neumann series for (I+EQ_0)^-1 converges. Also DQ_0=(D_0+E)Q_0=I+EQ_0, so DQ=I. The norm bound follows from the geometric series. For the truncation, (I+EQ_0)∑_k=0^n(-EQ_0)^k =I-(-EQ_0)^n+1, which gives the residual formula.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-41e0c211ba80d17c","tex":"\\sum_{k=0}^{n}","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^{n}","kind":"use","excerpt":"orrection under small packet coupling] If q:=normEQ_0<1, then D is surjective and has right inverse Q =Q_0(I_V+EQ_0)^-1 =Q_0∑_k=0^∞(-EQ_0)^k. Moreover, norm Q≤fracnormQ_01-q. For defect r, the truncated packet-correction series u^(n) :=-Q_0∑_k=0^n(-EQ_0)^kr has residual Du^(n)+r=(-EQ_0)^n+1r, and hence normDu^(n)+r≤ q^n+1norm r.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fec10602e4575bf9","tex":"\\sum_{k=0}^{n-1}","aliases":"∑_k=0^n-1 sum","display":"\\sum_{k=0}^{n-1}","kind":"use","excerpt":"T_1⋯ xrightarrow T_n-1A_n. Write T_U,k:U_k→ U_k+1 and T_V,k:V_k→ V_k+1. Let normD_k+1T_U,k-T_V,kD_k≤ε_D,k and normr_k+1-T_V,kr_k≤ε_r,k. If D_0u_0=-r_0 and u_n=T_U,n-1⋯ T_U,0u_0, then normD_nu_n+r_n · ≤ ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_D,k normT_U,k-1⋯ T_U,0u_0 · + ∑_k=0^n-1 normT_V,n-1⋯ T_V,k+1 ε_r,k, where empty products are identities.","count":4},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-05da311cedba2569","tex":"\\sum_{k=0}^{t-1}","aliases":"∑_k=0^t-1 sum","display":"\\sum_{k=0}^{t-1}","kind":"use","excerpt":"Lyapunov contraction] Suppose there exist w∈ℝ_>0^7 and q∈[0,1) such that for every reachable r_t, the controller can choose an authorized action a_t satisfying w^top A_a_tr_t ≤ q w^top r_t. Then every controlled trajectory satisfying [source reference] obeys w^top r_t ≤ q^t w^top r_0 + ∑_k=0^t-1 q^t-1-kw^top e_k.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f1c2ff812e2b7828","tex":"\\sum_{k=0}^{\\infty}","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^{\\infty}","kind":"use","excerpt":"[Exact correction under small packet coupling] If q:=normEQ_0<1, then D is surjective and has right inverse Q =Q_0(I_V+EQ_0)^-1 =Q_0∑_k=0^∞(-EQ_0)^k. Moreover, norm Q≤fracnormQ_01-q. For defect r, the truncated packet-correction series u^(n) :=-Q_0∑_k=0^n(-EQ_0)^kr has residual Du^(n)+r=(-EQ_0)^n+1r, and hence normDu^(n)+r≤ q^n+1norm r.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-90c396c1f06e154a","tex":"\\sum_t","aliases":"∑_t sum","display":"\\sum_t","kind":"use","excerpt":"inequation E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported score sequence is Cauchy and converges in every complete score space.","count":15},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0f90512a065cff6a","tex":"\\sum_{t=0}^{3}","aliases":"∑_t=0^3 sum","display":"\\sum_{t=0}^{3}","kind":"use","excerpt":"Suppose four accepted edges use conditional false-acceptance budgets 0.0100, 0.0050, 0.0025, 0.00125. Then ∑_t=0^3δ_t=0.01875. By [source reference], the probability that any accepted edge has nonpositive true metaproductivity is at most 0.01875, assuming every conditional bound is valid.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-920ed37c9ce943a9","tex":"\\sum_{t=0}^{m-1}","aliases":"∑_t=0^m-1 sum","display":"\\sum_{t=0}^{m-1}","kind":"use","excerpt":"[Same-context transitivity] Suppose every edge is evaluated in one common context Ξ. Then M_Ξ(K_m)-M_Ξ(K_0) = ∑_t=0^m-1 ( M_Ξ(K_t+1)-M_Ξ(K_t) ). Consequently, positive lower bounds on all edges add to a positive lower bound on the complete path.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-564d20c930110f8a","tex":"\\sum_{t=0}^{n-1}","aliases":"∑_t=0^n-1 sum","display":"\\sum_{t=0}^{n-1}","kind":"use","excerpt":"chain of evaluator morphisms with readouts into a common complete normed score space, E_0xrightarrowe_0 E_1 xrightarrowe_1⋯ xrightarrowe_n-1 E_n. Suppose that for every protected represented state x_t transported from version 0, normJ_t+1(T_tx_t)-J_t(x_t)≤ε_t. Then normJ_n(T_n-1⋯ T_0x_0)-J_0(x_0) ≤ ∑_t=0^n-1ε_t. If ∑_tε_t<∞, then the transported sc","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-5e51231cedd78224","tex":"\\sum_{t=0}^{\\infty}","aliases":"∑_t=0^∞ sum infty","display":"\\sum_{t=0}^{\\infty}","kind":"use","excerpt":"[Fixed-grammar convergence] If the grammar remains fixed and ∑_t=0^∞w^top e_t<∞, then w^top r_t→0 whenever the tail errors vanish under the convolution in [source reference].","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-702bc9ceff3fca47","tex":"\\sum_{t=1}^{T}","aliases":"∑_t=1^T sum","display":"\\sum_{t=1}^{T}","kind":"use","excerpt":"[Adaptive alpha-spending guarantee] Let F_t be the event that transaction t accepts a child whose true declared gain is nonpositive. Suppose Pr(F_tmidF_t-1)≤δ_t almost surely for a deterministic sequence (δ_t). Then for every T, Pr(⋃_t=1^TF_t) ≤ ∑_t=1^Tδ_t. If ∑_tδ_t<∞, only finitely many false acceptances occur almost surely.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-c379d6790dae8fe1","tex":"\\sum_{t=m}^{n-1}","aliases":"∑_t=m^n-1 sum","display":"\\sum_{t=m}^{n-1}","kind":"use","excerpt":"Suppose the correction square is approximate: normD_t+1T_t^U-T_t^VD_t ≤ε_t^D. Suppose protected defect meanings are reconstructed with error at most ε_t^V. Define cumulative dynamic semantic debt Δ_m,n = ∑_t=m^n-1 ( ε_t^D+ε_t^V ). Under nonexpansive transports, old correction and defect relations drift by at most the corresponding telescoping debt.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-b143119d615f8329","tex":"\\sum_{u\\in\\mathcal U}","aliases":"∑_u∈ U sum in mathcal","display":"\\sum_{u\\in\\mathcal U}","kind":"use","excerpt":"For a proposed intervention correspondence σ: U→ V and transport T:H_t→ H_s, define the normalized orbit defect δ_rm orb(T,σ) = frac1| U| ∑_u∈ U fracnormTR^t_σ(u)-R_u^sT_F max{normR_u^s_F,ε}.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-f2acdaec2a85b745","tex":"\\sum_{z'}","aliases":"∑_z' sum","display":"\\sum_{z'}","kind":"use","excerpt":"Let μ_z=ϕ(z) be the frozen event prototype. With equal priors and matched isotropic noise, the posterior is p(zmidwidetildeϕ) = fracexp(-normwidetildeϕ-μ_z^2/(2σ^2)) ∑_z'exp(-normwidetildeϕ-μ_z'^2/(2σ^2)). A repair is attempted only when max_z p(zmidwidetildeϕ)≥θ and the maximizing event is not none. Otherwise the system abstains.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6f333436e7d549b7","tex":"\\ast","aliases":"∗ ast","display":"\\ast","kind":"use","excerpt":"Define the unital adjoint-closed algebra A = alg^∗ (I,R_1,…,R_m). Its commutant is A' = {X∈End(H):XA=AX for all A∈ A}. The center is Z( A)= A∩ A'.","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"d for bridge defects] Let T:X→ Y and S:Y→ Z. Suppose the property transports satisfy normQ_Y(Tx)-widetilde T_QQ_X(x)≤ε_T for x∈ K, and normQ_Z(Sy)-widetilde S_QQ_Y(y)≤ε_S for y∈ T(K). If normwidetilde S_Q≤ L_S, then the composite bridge obeys ε_Q(S∘ T;K) ≤ ε_S+L_Sε_T with intended property transport widetilde S_Qwidetilde T_Q.","count":23},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"For a nonnegative random variable X, E X=∫_0^∞Pr(X>x) d x. Independence gives Pr(max_iG_i≤ x)=F(x)^k. Substitution proves the identity.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-ff862dd8e88aa975","tex":"\\int_0^{\\infty}","aliases":"∫_0^∞ int infty","display":"\\int_0^{\\infty}","kind":"use","excerpt":"[Expected best-of-k gain] Let G_1,…,G_k be independent nonnegative candidate gains with common cumulative distribution function F and finite mean. Then Emax_1≤ i≤ kG_i = ∫_0^∞ (1-F(x)^k) d x. In particular, expected best gain is nondecreasing in k even when the candidate generator itself does not improve.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"Let K→ K' change only modules in R. When F_K(R)>M(K), define the headroom-adjusted gain χ_R(K,K') = frac M(K')-M(K) F_K(R)-M(K) . For an admissible child inside the declared class, χ_R(K,K')≤1. A small absolute gain with χ_R≈1 indicates a nearly saturated parent. A small absolute gain with χ_R≈0 indicates that the child captured little of the avail","count":10},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"[Affine obstruction for the declared nonlinear family] Let I⊂ℝ be an interval with nonempty interior and let α≠0. No affine scalar map agrees with u↦sinh(α u)/α on all of I. Consequently, if one transformed packet coordinate varies over such an interval, no exact step-affine inverse exists for [source reference].","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"[Lyapunov--Schmidt repair reduction] Let F:P× U→ V be C^k, k≥2, between finite-dimensional spaces, with F(0,0)=0. Let D:=∂_uF(0,0). Choose splittings U=K⊕ X, K=ker D, and V=R⊕ C, R=im D, so that restrDX:X→ R is an isomorphism. Let P_R and P_C be the corresponding projections. Then there are neighborhoods of the origin and a unique C^k map χ:P× K→ X","count":17},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-871211e35f9c624d","tex":"\\bigoplus_{a=1}^{m}","aliases":"⊕_a=1^m bigoplus","display":"\\bigoplus_{a=1}^{m}","kind":"use","excerpt":"Let H be a finite-dimensional real or complex Hilbert space. Suppose the unknown functional decomposition is H=⊕_a=1^mH_a, P_a:H→ H_a, with pairwise orthogonal projectors satisfying P_aP_b=δ_abP_a, ∑_a=1^mP_a=I. For controlled interventions u_1,…,u_k, let R_j∈End(H), j=1,…,k, be their finite-difference or linearized response operators.","count":2},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-15ff689d76e6fe95","tex":"\\bigoplus_{i=1}^m","aliases":"⊕_i=1^m bigoplus","display":"\\bigoplus_{i=1}^m","kind":"use","excerpt":"Let U=⊕_i=1^m U_i, V=⊕_i=1^m V_i, with orthogonal projectors Q_i and P_i. Decompose D=D_0+E, D_0:=∑_i=1^m P_iDQ_i. The block-diagonal operator D_0 represents ideal packet-local correction; E contains all cross-packet effects.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-6fb1d22ec3477921","tex":"\\bigoplus_{\\alpha=1}^{p}","aliases":"⊕_α=1^p bigoplus alpha","display":"\\bigoplus_{\\alpha=1}^{p}","kind":"use","excerpt":"[Finite causal atlas decomposition] There exists a unitary identification H ≅ ⊕_α=1^p ( C^m_α⊗ C^n_α ) under which A ≅ ⊕_α=1^p ( I_m_α⊗ M_n_α( C) ). The minimal central projections P_α∈ Z( A) give a canonical coarse causal atlas H = ⊕_α=1^pP_α H. The blocks are unique up to permutation and unitary changes of basis within multiplicity spaces.","count":3},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","excerpt":"Let G=(I,E) be a connected oriented graph representing the nerve of the cover, let K be a finite-dimensional Hilbert space representing a common trivialization of the kernel fibers, and let e=(e_a)_a∈ E∈ K^E be the transported overlap-mismatch vector. Let B∈ℝ^|I|×|E| be the oriented incidence matrix and set B:=B⊗ I_K. A vertex gauge k∈ K^I changes ","count":7},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-28fddfd6ee2f43f3","tex":"\\bigcap_{j=1}^m","aliases":"⋂_j=1^m bigcap","display":"\\bigcap_{j=1}^m","kind":"use","excerpt":"Let b_1,…,b_m be benchmark readouts. A single scored benchmark leaves the local nullspace N_{b_1} = ker Db_1. An evaluator portfolio has invisible subspace N_ B = ⋂_j=1^mker Db_j.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"[Adaptive alpha-spending guarantee] Let F_t be the event that transaction t accepts a child whose true declared gain is nonpositive. Suppose Pr(F_tmidF_t-1)≤δ_t almost surely for a deterministic sequence (δ_t). Then for every T, Pr(⋃_t=1^TF_t) ≤ ∑_t=1^Tδ_t. If ∑_tδ_t<∞, only finitely many false acceptances occur almost surely.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-490d7f1b2a39a789","tex":"\\bigcup_{t=1}^{T}","aliases":"⋃_t=1^T bigcup","display":"\\bigcup_{t=1}^{T}","kind":"use","excerpt":"[Adaptive alpha-spending guarantee] Let F_t be the event that transaction t accepts a child whose true declared gain is nonpositive. Suppose Pr(F_tmidF_t-1)≤δ_t almost surely for a deterministic sequence (δ_t). Then for every T, Pr(⋃_t=1^TF_t) ≤ ∑_t=1^Tδ_t. If ∑_tδ_t<∞, only finitely many false acceptances occur almost surely.","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"For x∈ℝ^4, define an eight-dimensional task representation ψ(x) = ( x_1,x_2,x_3,x_4, x_1x_2,x_3x_4, sin(sx_1),cos(sx_2) ). A task-specific condition vector and threshold define the affected indicator a(x)=1{⟨ v,x⟩>τ}. The conditional feature is ϕ(x)=(ψ(x),a(x)ψ(x))∈ℝ^16. True three-class score vectors are affine in ψ(x). On the affected stratum onl","count":1},{"work":"research-stratified-recursive-improvement","url":"notation/research-stratified-recursive-improvement.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"For x∈ℝ^4, define an eight-dimensional task representation ψ(x) = ( x_1,x_2,x_3,x_4, x_1x_2,x_3x_4, sin(sx_1),cos(sx_2) ). A task-specific condition vector and threshold define the affected indicator a(x)=1{⟨ v,x⟩>τ}. The conditional feature is ϕ(x)=(ψ(x),a(x)ψ(x))∈ℝ^16. True three-class score vectors are affine in ψ(x). On the affected stratum onl","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":46},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":135},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":30},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Deterministic control sufficiency] Let A_0 be a finite nonempty set of admissible actions, let f:S× A_0→ S be transition, and let r:S× A_0→Q be reward. A representation ϕ:S→ Z is control sufficient if, whenever ϕ(s)=ϕ(t), every admissible action satisfies r(s,a)=r(t,a), ϕ(f(s,a))=ϕ(f(t,a)). The global action set is common to all states in this def","count":44},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"definition-passage","excerpt":"[Deterministic control sufficiency] Let A_0 be a finite nonempty set of admissible actions, let f:S× A_0→ S be transition, and let r:S× A_0→Q be reward. A representation ϕ:S→ Z is control sufficient if, whenever ϕ(s)=ϕ(t), every admissible action satisfies r(s,a)=r(t,a), ϕ(f(s,a))=ϕ(f(t,a)). The global action set is common to all states in this def","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":15},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":39},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":19},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":24},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Finite rational channel] A channel from Ω to a finite observation alphabet O consists of rational entries E(omidω)≥0 with ∑_o E(omidω)=1 for every ω. Given a second channel K(ymid o), their composite is (EK)(ymidω)=∑_o E(omidω)K(ymid o). The composite is a garbling of E.","count":18},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Deterministic control sufficiency] Let A_0 be a finite nonempty set of admissible actions, let f:S× A_0→ S be transition, and let r:S× A_0→Q be reward. A representation ϕ:S→ Z is control sufficient if, whenever ϕ(s)=ϕ(t), every admissible action satisfies r(s,a)=r(t,a), ϕ(f(s,a))=ϕ(f(t,a)). The global action set is common to all states in this def","count":13},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0, Q_h(s,a) · =","count":25},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":15},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-88fbeb34dbc66890","tex":"J_b","aliases":"J_b","display":"J_b","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ca84868410a13952","tex":"J_b^*","aliases":"J_b^*","display":"J_b^*","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Finite rational channel] A channel from Ω to a finite observation alphabet O consists of rational entries E(omidω)≥0 with ∑_o E(omidω)=1 for every ω. Given a second channel K(ymid o), their composite is (EK)(ymidω)=∑_o E(omidω)K(ymid o). The composite is a garbling of E.","count":12},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":19},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4729f0bca65d2d16","tex":"\\max_{a\\in A(s)}","aliases":"max_a∈ A(s) max in","display":"\\max_{a\\in A(s)}","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0, Q_h(s,a) · =","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-5b3ff0981d546bbc","tex":"\\max_{\\pi:Z\\to A}","aliases":"max_π:Z→ A max pi to","display":"\\max_{\\pi:Z\\to A}","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"definition-passage","excerpt":"[Finite rational channel] A channel from Ω to a finite observation alphabet O consists of rational entries E(omidω)≥0 with ∑_o E(omidω)=1 for every ω. Given a second channel K(ymid o), their composite is (EK)(ymidω)=∑_o E(omidω)K(ymid o). The composite is a garbling of E.","count":15},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":41},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"definition-passage","excerpt":"[Finite rational channel] A channel from Ω to a finite observation alphabet O consists of rational entries E(omidω)≥0 with ∑_o E(omidω)=1 for every ω. Given a second channel K(ymid o), their composite is (EK)(ymidω)=∑_o E(omidω)K(ymid o). The composite is a garbling of E.","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"definition-passage","excerpt":"[Finite rational channel] A channel from Ω to a finite observation alphabet O consists of rational entries E(omidω)≥0 with ∑_o E(omidω)=1 for every ω. Given a second channel K(ymid o), their composite is (EK)(ymidω)=∑_o E(omidω)K(ymid o). The composite is a garbling of E.","count":28},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"[Projected stochastic control sufficiency] A representation ϕ:S→ Z is projected stochastically sufficient if equal represented current states have equal immediate rewards and, for every admissible a and every rational fiber-constant function W:S→Q, ∑_s'P_a(s,s')W(s')=∑_s'P_a(t,s')W(s') whenever ϕ(s)=ϕ(t).","count":13},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"definition-passage","excerpt":"[Projected stochastic control sufficiency] A representation ϕ:S→ Z is projected stochastically sufficient if equal represented current states have equal immediate rewards and, for every admissible a and every rational fiber-constant function W:S→Q, ∑_s'P_a(s,s')W(s')=∑_s'P_a(t,s')W(s') whenever ϕ(s)=ϕ(t).","count":7},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0, Q_h(s,a) · =","count":15},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d6a9b153d5eaddc8","tex":"Q_h","aliases":"Q_h","display":"Q_h","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0, Q_h(s,a) · =","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":9},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Deterministic control sufficiency] Let A_0 be a finite nonempty set of admissible actions, let f:S× A_0→ S be transition, and let r:S× A_0→Q be reward. A representation ϕ:S→ Z is control sufficient if, whenever ϕ(s)=ϕ(t), every admissible action satisfies r(s,a)=r(t,a), ϕ(f(s,a))=ϕ(f(t,a)). The global action set is common to all states in this def","count":12},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c4179ac2053e2353","tex":"R^m","aliases":"R^m","display":"R^m","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-241a2a2cad8961a9","tex":"R^{m\\times n}","aliases":"R^m× n times","display":"R^{m\\times n}","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Deterministic control sufficiency] Let A_0 be a finite nonempty set of admissible actions, let f:S× A_0→ S be transition, and let r:S× A_0→Q be reward. A representation ϕ:S→ Z is control sufficient if, whenever ϕ(s)=ϕ(t), every admissible action satisfies r(s,a)=r(t,a), ϕ(f(s,a))=ϕ(f(t,a)). The global action set is common to all states in this def","count":16},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Deterministic control sufficiency] Let A_0 be a finite nonempty set of admissible actions, let f:S× A_0→ S be transition, and let r:S× A_0→Q be reward. A representation ϕ:S→ Z is control sufficient if, whenever ϕ(s)=ϕ(t), every admissible action satisfies r(s,a)=r(t,a), ϕ(f(s,a))=ϕ(f(t,a)). The global action set is common to all states in this def","count":45},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-5e769b89788d547a","tex":"s'","aliases":"s'","display":"s'","kind":"definition-passage","excerpt":"[Projected stochastic control sufficiency] A representation ϕ:S→ Z is projected stochastically sufficient if equal represented current states have equal immediate rewards and, for every admissible a and every rational fiber-constant function W:S→Q, ∑_s'P_a(s,s')W(s')=∑_s'P_a(t,s')W(s') whenever ϕ(s)=ϕ(t).","count":7},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":28},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Deterministic control sufficiency] Let A_0 be a finite nonempty set of admissible actions, let f:S× A_0→ S be transition, and let r:S× A_0→Q be reward. A representation ϕ:S→ Z is control sufficient if, whenever ϕ(s)=ϕ(t), every admissible action satisfies r(s,a)=r(t,a), ϕ(f(s,a))=ϕ(f(t,a)). The global action set is common to all states in this def","count":10},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":14},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0, Q_h(s,a) · =","count":21},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0, Q_h(s,a) · =","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-0b837e89a1e8104e","tex":"V_h","aliases":"V_h","display":"V_h","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0, Q_h(s,a) · =","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-11ef40505ae9bca6","tex":"V_{h+1}","aliases":"V_h+1","display":"V_{h+1}","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0, Q_h(s,a) · =","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"definition-passage","excerpt":"[Projected stochastic control sufficiency] A representation ϕ:S→ Z is projected stochastically sufficient if equal represented current states have equal immediate rewards and, for every admissible a and every rational fiber-constant function W:S→Q, ∑_s'P_a(s,s')W(s')=∑_s'P_a(t,s')W(s') whenever ϕ(s)=ϕ(t).","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":17},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":25},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"definition-passage","excerpt":"[Finite rational channel] A channel from Ω to a finite observation alphabet O consists of rational entries E(omidω)≥0 with ∑_o E(omidω)=1 for every ω. Given a second channel K(ymid o), their composite is (EK)(ymidω)=∑_o E(omidω)K(ymid o). The composite is a garbling of E.","count":12},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":14},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":8},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":11},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":49},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":13},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":9},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":23},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":28},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Deterministic refinement and task sufficiency] Let X be a raw state space and let ϕ:X→ Z and T:X→ Y be functions. We write ϕsucceq T Longleftrightarrow ∃ d:Z→ Y ∀ x∈ X, d(ϕ(x))=T(x). More generally, ϕsucceqψ means that ψ factors through ϕ. The relation is called refinement. The representation ϕ is task sufficient for T when ϕsucceq T. Two represen","count":31},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":32},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ccbd66bdaed18a2b","tex":"\\sum_o","aliases":"∑_o sum","display":"\\sum_o","kind":"definition-passage","excerpt":"[Finite rational channel] A channel from Ω to a finite observation alphabet O consists of rational entries E(omidω)≥0 with ∑_o E(omidω)=1 for every ω. Given a second channel K(ymid o), their composite is (EK)(ymidω)=∑_o E(omidω)K(ymid o). The composite is a garbling of E.","count":7},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d74790b215363bd7","tex":"\\sum_{o\\in O}","aliases":"∑_o∈ O sum in","display":"\\sum_{o\\in O}","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0,","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-20ee150d4e460fbd","tex":"\\sum_{s'}","aliases":"∑_s' sum","display":"\\sum_{s'}","kind":"definition-passage","excerpt":"[Projected stochastic control sufficiency] A representation ϕ:S→ Z is projected stochastically sufficient if equal represented current states have equal immediate rewards and, for every admissible a and every rational fiber-constant function W:S→Q, ∑_s'P_a(s,s')W(s')=∑_s'P_a(t,s')W(s') whenever ϕ(s)=ϕ(t).","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3171c80740394520","tex":"\\sum_{\\omega\\in\\Omega}","aliases":"∑_ω∈Ω sum omega in Omega","display":"\\sum_{\\omega\\in\\Omega}","kind":"definition-passage","excerpt":"[Deterministic experiment value] Let Ω be finite, A be a finite nonempty action set, b:Ω→Q be a prior, and u:Ω× A→Q be utility. For an observation ϕ:Ω→ Z and policy π:Z→ A, define J_b(ϕ,π)=∑_ω∈Ωb(ω)u(ω,π(ϕ(ω))). When Z is finite, write J_b^*(ϕ)=max_π:Z→ AJ_b(ϕ,π).","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"definition-passage","excerpt":"[Dimension-indexed affine signature] Over a ring R, a signature from dimension n to dimension m is a pair (A,b) with A∈ R^m× n and b∈ R^m, denoting x↦ Ax+b. Dimensions are part of the type. Composition and pointwise addition are (B,c)∘(A,b)=(BA,Bb+c), (A,b)+(A',b')=(A+A',b+b').","count":7},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"[Finite rational channel] A channel from Ω to a finite observation alphabet O consists of rational entries E(omidω)≥0 with ∑_o E(omidω)=1 for every ω. Given a second channel K(ymid o), their composite is (EK)(ymidω)=∑_o E(omidω)K(ymid o). The composite is a garbling of E.","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Guarded rational system] Let S be a state space, A(s) a finite nonempty set of actions, and O a finite observation space. A guarded system consists of a rational reward r(s,a), a transition T(s,a,o), and rational probabilities P(omid s,a) satisfying P(omid s,a)≥0, ∑_o∈ OP(omid s,a)=1. Its finite-horizon Bellman values are V_0(s) · =0,","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-fc2b98d398ff9e35","tex":"c_D","aliases":"c_D","display":"c_D","kind":"introduction-passage","excerpt":"[Finite data processing] For a finite sample D⊆ X, let c_D(ϕ)=|ϕ(D)| denote the number of reached representation states. If ϕsucceqψ, then c_D(ψ)≤ c_D(ϕ). Choose d with ψ=d∘ϕ. Then ψ(D)=d(ϕ(D)), and the image of a finite set under a function has cardinality at most that of the original set.","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"[Finite data processing] For a finite sample D⊆ X, let c_D(ϕ)=|ϕ(D)| denote the number of reached representation states. If ϕsucceqψ, then c_D(ψ)≤ c_D(ϕ). Choose d with ψ=d∘ϕ. Then ψ(D)=d(ϕ(D)), and the image of a finite set under a function has cardinality at most that of the original set.","count":19},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"[Finite data processing] For a finite sample D⊆ X, let c_D(ϕ)=|ϕ(D)| denote the number of reached representation states. If ϕsucceqψ, then c_D(ψ)≤ c_D(ϕ). Choose d with ψ=d∘ϕ. Then ψ(D)=d(ϕ(D)), and the image of a finite set under a function has cardinality at most that of the original set.","count":9},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"[Finite data processing] For a finite sample D⊆ X, let c_D(ϕ)=|ϕ(D)| denote the number of reached representation states. If ϕsucceqψ, then c_D(ψ)≤ c_D(ϕ). Choose d with ψ=d∘ϕ. Then ψ(D)=d(ϕ(D)), and the image of a finite set under a function has cardinality at most that of the original set.","count":15},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"introduction-passage","excerpt":"[Finite data processing] For a finite sample D⊆ X, let c_D(ϕ)=|ϕ(D)| denote the number of reached representation states. If ϕsucceqψ, then c_D(ψ)≤ c_D(ϕ). Choose d with ψ=d∘ϕ. Then ψ(D)=d(ϕ(D)), and the image of a finite set under a function has cardinality at most that of the original set.","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3c08d2cf77d38259","tex":"a^*","aliases":"a^*","display":"a^*","kind":"use","excerpt":"Let a_h(s) maximize L_h(s,a) and let a^* maximize the true Q_h. Then V_h+1(s)-Q_h(s,a_h(s)) · ≤ U_h(s,a^*)-L_h(s,a_h(s))","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c84a7e2fef20faa1","tex":"a_h","aliases":"a_h","display":"a_h","kind":"use","excerpt":"Let a_h(s) maximize L_h(s,a) and let a^* maximize the true Q_h. Then V_h+1(s)-Q_h(s,a_h(s)) · ≤ U_h(s,a^*)-L_h(s,a_h(s))","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-99fe824d0aec7a53","tex":"a_o","aliases":"a_o","display":"a_o","kind":"use","excerpt":"[Finite stochastic information-value monotonicity] Every deterministic policy on EK is weakly dominated in value by some deterministic policy on E. Consequently the best deterministic value of EK is at most that of E. For each fine observation o, select a_o∈ A maximizing S_E(o,a). Such a selection exists because A is finite and nonempty. For a coar","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"The initial grammar layer has five atoms: two literals and three variables. If a layer is L, expansion retains L, adds all negations, and adds all ordered pairs under each of the three binary operations. Let M_d count emitted list entries at depth d. The source proves M_0=5, M_d+1=2M_d+3M_d^2. The entries can contain duplicates. The node bound is B","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"[Why the schedule's total allowance is necessary] Take a fair Boolean outcome ω and two checkpoint bad events B_0={ω=0} and B_1={ω=1}. Each has probability 1/2. Selecting j(ω)=ω makes selected failure certain. The total allowance 1/2+1/2=1 remains valid, while either single checkpoint's allowance fails. This is the exact obstruction proved by pathf","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-7be573071ebda78d","tex":"B_d","aliases":"B_d","display":"B_d","kind":"use","excerpt":"The initial grammar layer has five atoms: two literals and three variables. If a layer is L, expansion retains L, adds all negations, and adds all ordered pairs under each of the three binary operations. Let M_d count emitted list entries at depth d. The source proves M_0=5, M_d+1=2M_d+3M_d^2. The entries can contain duplicates. The node bound is B","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-fb301d9e5a553bdf","tex":"B_{d+1}","aliases":"B_d+1","display":"B_{d+1}","kind":"use","excerpt":"The initial grammar layer has five atoms: two literals and three variables. If a layer is L, expansion retains L, adds all negations, and adds all ordered pairs under each of the three binary operations. Let M_d count emitted list entries at depth d. The source proves M_0=5, M_d+1=2M_d+3M_d^2. The entries can contain duplicates. The node bound is B","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8ace69abba671b8c","tex":"b_f","aliases":"b_f","display":"b_f","kind":"use","excerpt":"[Complete zero-and-basis probes] Two affine signatures f,g:R^n→ R^m are equal if and only if they agree at 0 and at every standard basis vector e_j. Thus n+1 vector probes establish equality on all inputs. A mismatch in signatures has a mismatch among these probes. The zero probe identifies the offsets because f(0)=b_f. At e_j, the output is the jt","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","excerpt":"[Scheduled and selected failure bounds] Let B_j be the event that checkpoint j's interval excludes p_j. Then Prob(⋃_j∈ JB_j) ≤∑_j∈ Jα_j. For any data-dependent selector j(ω)∈ J, Prob{ω∈ B_j(ω)}≤∑_j∈ Jα_j. For each outcome ω, the indicator of a union is at most the sum of the individual indicators. Multiply by the nonnegative rational weight of ω an","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-26093651fa21ded6","tex":"B_{j(\\omega)}","aliases":"B_j(ω) omega","display":"B_{j(\\omega)}","kind":"use","excerpt":"[Scheduled and selected failure bounds] Let B_j be the event that checkpoint j's interval excludes p_j. Then Prob(⋃_j∈ JB_j) ≤∑_j∈ Jα_j. For any data-dependent selector j(ω)∈ J, Prob{ω∈ B_j(ω)}≤∑_j∈ Jα_j. For each outcome ω, the indicator of a union is at most the sum of the individual indicators. Multiply by the nonnegative rational weight of ω an","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-990bf3b4099cc02d","tex":"b_{n,k}","aliases":"b_n,k","display":"b_{n,k}","kind":"use","excerpt":"Let K be the number of successes in n Bernoulli trials with rational parameter p∈[0,1]. Define b_n,k(p)=binom nk p^k(1-p)^n-k, 0≤ k≤ n. Choose a positive integer grid size m and nonnegative rational allowance α. The grid cells have endpoints ℓ_i=fracim, u_i=fraci+1m, 0≤ i<m. For observed count k, the envelope of its point mass over cell i is B_n,k,","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-fbe71e366c3f4cde","tex":"B_{n,k,i}","aliases":"B_n,k,i","display":"B_{n,k,i}","kind":"use","excerpt":"uccesses in n Bernoulli trials with rational parameter p∈[0,1]. Define b_n,k(p)=binom nk p^k(1-p)^n-k, 0≤ k≤ n. Choose a positive integer grid size m and nonnegative rational allowance α. The grid cells have endpoints ℓ_i=fracim, u_i=fraci+1m, 0≤ i<m. For observed count k, the envelope of its point mass over cell i is B_n,k,i=binom nk u_i^k(1-ℓ_i)^","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-6cdedb4108ff8a02","tex":"b_o","aliases":"b_o","display":"b_o","kind":"use","excerpt":"A rational belief is a nonnegative vector b on finite Ω with ∑_ω b(ω)=1. For a channel E, define the predictive mass p(o)=∑_ω b(ω)E(omidω). For p(o)>0, the posterior is b_o(ω)=b(ω)E(omidω)/p(o). The total updater in the source uses b_o=b when p(o)=0. This fallback is normalized and has zero coefficient in predictive expectations. Nonnegativity of e","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-7e95a89f39649f47","tex":"\\bar b","aliases":"bar b bar","display":"\\bar b","kind":"use","excerpt":"[Conditioning commutes with exact compression] Let ϕ:Ω→ Z map between finite spaces, and define the pushforward belief bar b(z)=∑_ϕ(ω)=zb(ω). Suppose L=bar L∘ϕ for a nonnegative likelihood bar L:Z→Q, and the evidence mass is positive. Then ϕ_*C_L(b)=C_bar L(ϕ_*b). Such an exact bar L exists if and only if L is constant on every fiber of ϕ. Grouping","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-9569051cbb2341b4","tex":"\\bar L","aliases":"bar L bar","display":"\\bar L","kind":"use","excerpt":"[Conditioning commutes with exact compression] Let ϕ:Ω→ Z map between finite spaces, and define the pushforward belief bar b(z)=∑_ϕ(ω)=zb(ω). Suppose L=bar L∘ϕ for a nonnegative likelihood bar L:Z→Q, and the evidence mass is positive. Then ϕ_*C_L(b)=C_bar L(ϕ_*b). Such an exact bar L exists if and only if L is constant on every fiber of ϕ. Grouping","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-65502e56bb71e26c","tex":"\\bar P","aliases":"bar P bar","display":"\\bar P","kind":"use","excerpt":"[Exact stochastic quotient] Suppose ϕ:S→ Z is surjective and projected stochastically sufficient, with S,Z finite. Choose a representative s_z of every fiber, and define bar r(z,a)=r(s_z,a), bar P_a(z,z')=∑_ϕ(s')=z'P_a(s_z,s'). These quantities give a normalized rational stochastic control system, independent of the representative choices for admis","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-5dfcb36b7669a0f3","tex":"\\bar P_a","aliases":"bar P_a bar","display":"\\bar P_a","kind":"use","excerpt":"[Exact stochastic quotient] Suppose ϕ:S→ Z is surjective and projected stochastically sufficient, with S,Z finite. Choose a representative s_z of every fiber, and define bar r(z,a)=r(s_z,a), bar P_a(z,z')=∑_ϕ(s')=z'P_a(s_z,s'). These quantities give a normalized rational stochastic control system, independent of the representative choices for admis","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4c4d0c215858215d","tex":"\\bar r","aliases":"bar r bar","display":"\\bar r","kind":"use","excerpt":"[Exact stochastic quotient] Suppose ϕ:S→ Z is surjective and projected stochastically sufficient, with S,Z finite. Choose a representative s_z of every fiber, and define bar r(z,a)=r(s_z,a), bar P_a(z,z')=∑_ϕ(s')=z'P_a(s_z,s'). These quantities give a normalized rational stochastic control system, independent of the representative choices for admis","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-41e3b4b9bbfa4a9e","tex":"\\bar V","aliases":"bar V bar","display":"\\bar V","kind":"use","excerpt":"Suppose ϕ:S→ Z is surjective and projected stochastically sufficient, with S,Z finite. Choose a representative s_z of every fiber, and define bar r(z,a)=r(s_z,a), bar P_a(z,z')=∑_ϕ(s')=z'P_a(s_z,s'). These quantities give a normalized rational stochastic control system, independent of the representative choices for admissible actions. For every fin","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4d0130fe1719c407","tex":"\\bar V_n","aliases":"bar V_n bar","display":"\\bar V_n","kind":"use","excerpt":"Suppose ϕ:S→ Z is surjective and projected stochastically sufficient, with S,Z finite. Choose a representative s_z of every fiber, and define bar r(z,a)=r(s_z,a), bar P_a(z,z')=∑_ϕ(s')=z'P_a(s_z,s'). These quantities give a normalized rational stochastic control system, independent of the representative choices for admissible actions. For every fin","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8cf71553d3809a2f","tex":"\\mathsf c","aliases":"c mathsf","display":"\\mathsf c","kind":"use","excerpt":"Writing I for issuance and C for the specified episode certificate, the resulting conclusion is Prob(I∩ C^ c)≤∑_j∈ Jα_j. This is an unconditional probability over the original outcome space. It allows arbitrary dependence between scheduled checkpoints. It retains failed and unfinished runs in that space, with no plan issued on those runs. It does n","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c8a3209e3465f264","tex":"C^{\\mathsf c}","aliases":"C^ c mathsf","display":"C^{\\mathsf c}","kind":"use","excerpt":"Writing I for issuance and C for the specified episode certificate, the resulting conclusion is Prob(I∩ C^ c)≤∑_j∈ Jα_j. This is an unconditional probability over the original outcome space. It allows arbitrary dependence between scheduled checkpoints. It retains failed and unfinished runs in that space, with no plan issued on those runs. It does n","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8ed8f27dbc059b9e","tex":"C_{\\bar L}","aliases":"C_bar L bar","display":"C_{\\bar L}","kind":"use","excerpt":"[Conditioning commutes with exact compression] Let ϕ:Ω→ Z map between finite spaces, and define the pushforward belief bar b(z)=∑_ϕ(ω)=zb(ω). Suppose L=bar L∘ϕ for a nonnegative likelihood bar L:Z→Q, and the evidence mass is positive. Then ϕ_*C_L(b)=C_bar L(ϕ_*b). Such an exact bar L exists if and only if L is constant on every fiber of ϕ. Grouping","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-32afa62b362c3257","tex":"c_c^+","aliases":"c_c^+","display":"c_c^+","kind":"use","excerpt":"Let w_c(k)=c_k^+-c_k^-, w_g(k)=g_k^+-g_k^-, and w_p(k)=p_k^+-p_k^-. The source uses the uniform quantities ε · =w_c(w)+w_c(c)+w_g(w)+w_g(c), δ · =2(w_p(w)+w_p(c)), M · =c_w^++c_c^++g_w^++g_c^+. They bound reward error, Boolean-row L^1 error, and nominal one-step reward magnitude respectively. For an acquisition of kind k, the sharper local reward e","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-254051ed849e0a84","tex":"c_k","aliases":"c_k","display":"c_k","kind":"use","excerpt":"For kind k∈{w,c}, write c_k∈[c_k^-,c_k^+], g_k∈[g_k^-,g_k^+], p_k∈[p_k^-,p_k^+]. The formal nominal model reserves c_k^+, credits g_k^-, and uses p_k^-. A compatible realization supplies exact values inside these bands. In the true comparison system, rewards and report probabilities use those exact values, but the state transitions and action guard","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d136229f47d847af","tex":"c_k^+","aliases":"c_k^+","display":"c_k^+","kind":"use","excerpt":"For kind k∈{w,c}, write c_k∈[c_k^-,c_k^+], g_k∈[g_k^-,g_k^+], p_k∈[p_k^-,p_k^+]. The formal nominal model reserves c_k^+, credits g_k^-, and uses p_k^-. A compatible realization supplies exact values inside these bands. In the true comparison system, rewards and report probabilities use those exact values, but the state transitions and action guard","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c9c056811c356699","tex":"c_k^-","aliases":"c_k^-","display":"c_k^-","kind":"use","excerpt":"For kind k∈{w,c}, write c_k∈[c_k^-,c_k^+], g_k∈[g_k^-,g_k^+], p_k∈[p_k^-,p_k^+]. The formal nominal model reserves c_k^+, credits g_k^-, and uses p_k^-. A compatible realization supplies exact values inside these bands. In the true comparison system, rewards and report probabilities use those exact values, but the state transitions and action guard","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a8e8fbaa4e63d973","tex":"C_L","aliases":"C_L","display":"C_L","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d1246299eafb2fed","tex":"C_{L_1}","aliases":"C_L_1","display":"C_{L_1}","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-9bde7b7950def4ab","tex":"C_{L_1L_2}","aliases":"C_L_1L_2","display":"C_{L_1L_2}","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3a7f0642f7758ab5","tex":"C_{L_2}","aliases":"C_L_2","display":"C_{L_2}","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8700970bfbd78b20","tex":"c_w^+","aliases":"c_w^+","display":"c_w^+","kind":"use","excerpt":"Let w_c(k)=c_k^+-c_k^-, w_g(k)=g_k^+-g_k^-, and w_p(k)=p_k^+-p_k^-. The source uses the uniform quantities ε · =w_c(w)+w_c(c)+w_g(w)+w_g(c), δ · =2(w_p(w)+w_p(c)), M · =c_w^++c_c^++g_w^++g_c^+. They bound reward error, Boolean-row L^1 error, and nominal one-step reward magnitude respectively. For an acquisition of kind k, the sharper local reward e","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e6a6399fcdef2fe8","tex":"c_\\phi","aliases":"c_ϕ phi","display":"c_\\phi","kind":"use","excerpt":"[Deterministic information-value monotonicity] If ϕsucceqψ, then J_b^*(ψ)≤ J_b^*(ϕ). If the acquisition costs satisfy c_ϕ≤ c_ψ, then J_b^*(ψ)-c_ψ≤ J_b^*(ϕ)-c_ϕ. Let d decode ψ from ϕ. Any coarse policy π lifts to π∘ d. For every world the lifted policy chooses exactly the same action, so its decision value is unchanged. A maximizing coarse policy e","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ade7d99d1388af53","tex":"c_\\psi","aliases":"c_ψ psi","display":"c_\\psi","kind":"use","excerpt":"[Deterministic information-value monotonicity] If ϕsucceqψ, then J_b^*(ψ)≤ J_b^*(ϕ). If the acquisition costs satisfy c_ϕ≤ c_ψ, then J_b^*(ψ)-c_ψ≤ J_b^*(ϕ)-c_ϕ. Let d decode ψ from ϕ. Any coarse policy π lifts to π∘ d. For every world the lifted policy chooses exactly the same action, so its decision value is unchanged. A maximizing coarse policy e","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-dadbefeac6b8f90a","tex":"\\operatorname{cost}","aliases":"cost operatorname","display":"\\operatorname{cost}","kind":"use","excerpt":"The menu selector compares three equivalent implementations: the current program, its dense fused atom, and the accepted minimum-width two-stage plan. Its selected tariff is exactly min{cost(p), D(n,m), F(n,m,k)}. This proves a minimum over the explicit menu, and also comparison with every exact two-dense-stage factorization. It does not prove a gl","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"mple[A deliberately conservative abstract quote fixture] The retained uncertainty example assigns zero setup to both kinds, warm gross 200, cold gross 60, warm success band [3/5,1], and exact cold success 1. Start with both statuses untried and use h=1, hence two total actions. The nominal scores are warm 120, cold 60, and stop 0. Here ε=0, δ=4/5, ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-fdadfd14bd665b99","tex":"D_h","aliases":"D_h","display":"D_h","kind":"use","excerpt":"Induction on the Bellman recursion gives |widehat V_h(s)|≤ hM for the nominal value. The source's accumulated value radius and local Q radius are D_h · =∑_j=0^h-1(ε+δ jM), ρ_h(a) · =D_h+ε_a+δ_a hM, Γ_h · =D_h+ε+δ hM. The Bellman residual is zero here because the nominal tables are exact Bellman values. The broader model-error framework also allows ","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-99f460db2ead8f6a","tex":"D_{h+1}","aliases":"D_h+1","display":"D_{h+1}","kind":"use","excerpt":"inproof At horizon zero the value error is zero. In a single backup, insert and subtract the expectation of the nominal continuation under the true row. The reward error contributes at most ε_a, the continuation-value error at most D_h, and row transport at most δ_a hM. Taking a maximum over the same feasible action set is nonexpansive in the unifo","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3e5463cd5ce49eb7","tex":"d_q","aliases":"d_q","display":"d_q","kind":"use","excerpt":"Suppose a family of tasks is given by T_q:X→ Y, indexed by q∈ Q. Define the joint target T_joint(x)=(T_q(x))_q∈ Q∈ Y^Q. Then ϕ is sufficient for the joint target if and only if it is sufficient for each individual task. The forward direction evaluates the decoded answer vector at q. For the reverse direction, choose a decoder d_q for each query and","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries AVOC LARGEbfseries Certified ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"[Complete zero-and-basis probes] Two affine signatures f,g:R^n→ R^m are equal if and only if they agree at 0 and at every standard basis vector e_j. Thus n+1 vector probes establish equality on all inputs. A mismatch in signatures has a mismatch among these probes. The zero probe identifies the offsets because f(0)=b_f. At e_j, the output is the jt","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"[Complete zero-and-basis probes] Two affine signatures f,g:R^n→ R^m are equal if and only if they agree at 0 and at every standard basis vector e_j. Thus n+1 vector probes establish equality on all inputs. A mismatch in signatures has a mismatch among these probes. The zero probe identifies the offsets because f(0)=b_f. At e_j, the output is the jt","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-7c808485c3e440e2","tex":"\\operatorname{eval}","aliases":"eval operatorname","display":"\\operatorname{eval}","kind":"use","excerpt":"Typed programs combine affine atoms, identities, sequential composition, and pointwise addition. Composition connects only matching intermediate dimensions. An independent recursive execution semantics and coefficient-normalization semantics are defined. Structural induction proves eval(normalize(p),x)=run(p,x) for all x. Hence two programs are equ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-27c1e260c9f22812","tex":"\\mathrm{evaluationSpent}","aliases":"evaluationSpent mathrm","display":"\\mathrm{evaluationSpent}","kind":"use","excerpt":"For a successful paid trace with total evaluation debit e and switching debit u, the formal accounting identities are evaluationSpent_rm final · =evaluationSpent_rm initial+e, switchSpent_rm final · =switchSpent_rm initial+u. From a safe initial journal, their sum stays within protected credits. Successful replay also cannot decrease the protected ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-1b275482a1af146d","tex":"\\mathrm{evaluationSpent}_{\\rm final}","aliases":"evaluationSpent_rm final mathrm rm","display":"\\mathrm{evaluationSpent}_{\\rm final}","kind":"use","excerpt":"For a successful paid trace with total evaluation debit e and switching debit u, the formal accounting identities are evaluationSpent_rm final · =evaluationSpent_rm initial+e, switchSpent_rm final · =switchSpent_rm initial+u. From a safe initial journal, their sum stays within protected credits. Successful replay also cannot decrease the protected ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-848413525835555f","tex":"\\mathrm{evaluationSpent}_{\\rm initial}","aliases":"evaluationSpent_rm initial mathrm rm","display":"\\mathrm{evaluationSpent}_{\\rm initial}","kind":"use","excerpt":"For a successful paid trace with total evaluation debit e and switching debit u, the formal accounting identities are evaluationSpent_rm final · =evaluationSpent_rm initial+e, switchSpent_rm final · =switchSpent_rm initial+u. From a safe initial journal, their sum stays within protected credits. Successful replay also cannot decrease the protected ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"[Checker soundness] If the inference checker accepts derivation p for target F under context Γ, then every valuation satisfying Γ satisfies F. Induct on the derivation. A used assumption belongs to Γ and is true by the context hypothesis. Conjunction introduction combines the two inductive conclusions, and its projections select the corresponding c","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"A running computational example. Consider an affine map over F_5 with four inputs and four outputs, all-ones linear coefficient matrix, and offset (1,2,3,4). The direct dense evaluation tariff is 4(2·4+1)=36. Its rank-one factorization has a 4→1 stage followed by a 1→4 stage, with tariff 9+12=21. Thus each declared call saves 15 arithmetic cost uni","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b6bb9a2af76d66fc","tex":"\\mathbf F","aliases":"F mathbf","display":"\\mathbf F","kind":"use","excerpt":"[A checked rank-one bottleneck] In the archived finite-field example over F_5, A is the 4×4 all-ones matrix and b=(1,2,3,4). The first stage computes y=x_1+x_2+x_3+x_4. The second emits (y+1,y+2,y+3,y+4). Taking L to select the first output coordinate and H to select the first input coordinate gives LAH=(1). The width-one factorization is therefore","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f3ee6a8ba9575903","tex":"\\mathbb F_5","aliases":"F_5 mathbb","display":"\\mathbb F_5","kind":"use","excerpt":"A running computational example. Consider an affine map over F_5 with four inputs and four outputs, all-ones linear coefficient matrix, and offset (1,2,3,4). The direct dense evaluation tariff is 4(2·4+1)=36. Its rank-one factorization has a 4→1 stage followed by a 1→4 stage, with tariff 9+12=21. Thus each declared call saves 15 arithmetic cost uni","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-47a73e04062d0183","tex":"\\mathbf F_5","aliases":"F_5 mathbf","display":"\\mathbf F_5","kind":"use","excerpt":"[A checked rank-one bottleneck] In the archived finite-field example over F_5, A is the 4×4 all-ones matrix and b=(1,2,3,4). The first stage computes y=x_1+x_2+x_3+x_4. The second emits (y+1,y+2,y+3,y+4). Taking L to select the first output coordinate and H to select the first input coordinate gives LAH=(1). The width-one factorization is therefore","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4658cf98f2ab845c","tex":"\\mathrm{false}","aliases":"false mathrm","display":"\\mathrm{false}","kind":"use","excerpt":"[Replay and faults do not create new evidence] Let an identity's source be [true,false]. Begin ID 7, report ID 7, then report ID 7 again. The sample list is [true] and the remaining source begins with false. Begin ID 8 and finish it as a fault. The sample list stays unchanged. Begin and report ID 9. It appends false, yielding [true,false]. A furthe","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"[Complete zero-and-basis probes] Two affine signatures f,g:R^n→ R^m are equal if and only if they agree at 0 and at every standard basis vector e_j. Thus n+1 vector probes establish equality on all inputs. A mismatch in signatures has a mismatch among these probes. The zero probe identifies the offsets because f(0)=b_f. At e_j, the output is the jt","count":11},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-9ca6db8163caf5b2","tex":"g_c^+","aliases":"g_c^+","display":"g_c^+","kind":"use","excerpt":"Let w_c(k)=c_k^+-c_k^-, w_g(k)=g_k^+-g_k^-, and w_p(k)=p_k^+-p_k^-. The source uses the uniform quantities ε · =w_c(w)+w_c(c)+w_g(w)+w_g(c), δ · =2(w_p(w)+w_p(c)), M · =c_w^++c_c^++g_w^++g_c^+. They bound reward error, Boolean-row L^1 error, and nominal one-step reward magnitude respectively. For an acquisition of kind k, the sharper local reward e","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c1c366f26da43334","tex":"g_k","aliases":"g_k","display":"g_k","kind":"use","excerpt":"For kind k∈{w,c}, write c_k∈[c_k^-,c_k^+], g_k∈[g_k^-,g_k^+], p_k∈[p_k^-,p_k^+]. The formal nominal model reserves c_k^+, credits g_k^-, and uses p_k^-. A compatible realization supplies exact values inside these bands. In the true comparison system, rewards and report probabilities use those exact values, but the state transitions and action guard","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d72a05926df5fb38","tex":"g_k^+","aliases":"g_k^+","display":"g_k^+","kind":"use","excerpt":"For kind k∈{w,c}, write c_k∈[c_k^-,c_k^+], g_k∈[g_k^-,g_k^+], p_k∈[p_k^-,p_k^+]. The formal nominal model reserves c_k^+, credits g_k^-, and uses p_k^-. A compatible realization supplies exact values inside these bands. In the true comparison system, rewards and report probabilities use those exact values, but the state transitions and action guard","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c4f441536fa85d85","tex":"g_k^-","aliases":"g_k^-","display":"g_k^-","kind":"use","excerpt":"For kind k∈{w,c}, write c_k∈[c_k^-,c_k^+], g_k∈[g_k^-,g_k^+], p_k∈[p_k^-,p_k^+]. The formal nominal model reserves c_k^+, credits g_k^-, and uses p_k^-. A compatible realization supplies exact values inside these bands. In the true comparison system, rewards and report probabilities use those exact values, but the state transitions and action guard","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e281245cb629eec3","tex":"g_w^+","aliases":"g_w^+","display":"g_w^+","kind":"use","excerpt":"Let w_c(k)=c_k^+-c_k^-, w_g(k)=g_k^+-g_k^-, and w_p(k)=p_k^+-p_k^-. The source uses the uniform quantities ε · =w_c(w)+w_c(c)+w_g(w)+w_g(c), δ · =2(w_p(w)+w_p(c)), M · =c_w^++c_c^++g_w^++g_c^+. They bound reward error, Boolean-row L^1 error, and nominal one-step reward magnitude respectively. For an acquisition of kind k, the sharper local reward e","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-0c03d0cc57803144","tex":"\\mathrm{generateCold}","aliases":"generateCold mathrm","display":"\\mathrm{generateCold}","kind":"use","excerpt":"[The artifact-backed rank-one comparison] The exact quote model retained in pathAVOC/RobustQuoteExamples.lean has warm setup 133, cold setup 196, success probability 1 for both, and gross deployment value 15000 for each over 1000 calls. With initial credit 196 and two total actions available, Q_1(inspectWarm)=15000-133=14867, Q_1(generateCold)=1500","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"Work over a field R. For reference f=(A,b):R^n→ R^m, a candidate supplies an intermediate width k, first and second affine signatures a(x)=Ux+u_0, d(y)=Vy+v_0, and matrices L∈ R^k× m, H∈ R^n× k. The checker verifies VU=A, Vu_0+v_0=b, LAH=I_k. The first two identities give the exact upper implementation. The third is an independently checkable lower","count":17},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"example assigns zero setup to both kinds, warm gross 200, cold gross 60, warm success band [3/5,1], and exact cold success 1. Start with both statuses untried and use h=1, hence two total actions. The nominal scores are warm 120, cold 60, and stop 0. Here ε=0, δ=4/5, M=260, and D_1=0. The local warm radius is 208, while the cold and stop radii are ","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Work over a field R. For reference f=(A,b):R^n→ R^m, a candidate supplies an intermediate width k, first and second affine signatures a(x)=Ux+u_0, d(y)=Vy+v_0, and matrices L∈ R^k× m, H∈ R^n× k. The checker verifies VU=A, Vu_0+v_0=b, LAH=I_k. The first two identities give the exact upper implementation. The third is an independently checkable lower","count":8},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"Let K be the number of successes in n Bernoulli trials with rational parameter p∈[0,1]. Define b_n,k(p)=binom nk p^k(1-p)^n-k, 0≤ k≤ n. Choose a positive integer grid size m and nonnegative rational allowance α. The grid cells have endpoints ℓ_i=fracim, u_i=fraci+1m, 0≤ i<m. For observed count k, the envelope of its point mass over cell i is B_n,k,","count":10},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-90844af211ae1183","tex":"I_k","aliases":"I_k","display":"I_k","kind":"use","excerpt":"Work over a field R. For reference f=(A,b):R^n→ R^m, a candidate supplies an intermediate width k, first and second affine signatures a(x)=Ux+u_0, d(y)=Vy+v_0, and matrices L∈ R^k× m, H∈ R^n× k. The checker verifies VU=A, Vu_0+v_0=b, LAH=I_k. The first two identities give the exact upper implementation. The third is an independently checkable lower","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-5f83d57a9279301e","tex":"\\mathrm{inspectWarm}","aliases":"inspectWarm mathrm","display":"\\mathrm{inspectWarm}","kind":"use","excerpt":"[The artifact-backed rank-one comparison] The exact quote model retained in pathAVOC/RobustQuoteExamples.lean has warm setup 133, cold setup 196, success probability 1 for both, and gross deployment value 15000 for each over 1000 calls. With initial credit 196 and two total actions available, Q_1(inspectWarm)=15000-133=14867, Q_1(generateCold)=1500","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"A complete finite semantic algebra supports a stronger conclusion than searching a deeper finite layer. Let D be a semantic domain with interpretations of the Boolean constructors. A cost function c:D→N is a lower certificate if it satisfies the constructor inequalities c(literal b) · ≤1, · c(variable j) · ≤1, c(neg s) · ≤ c(s)+1, · c(smathbin∘t) ·","count":23},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ecf1bfc197d7bcf7","tex":"J_H^{\\rm lower}","aliases":"J_H^rm lower rm","display":"J_H^{\\rm lower}","kind":"use","excerpt":"[Expected frozen-policy regret] For a horizon H, let J_H^rm lower(s) be the expected value of the complete adaptive policy induced by robust lower-Q selection. Under any compatible realization, 0≤ V_H(s)-J_H^rm lower(s) ≤∑_h=0^H-12Γ_h. The comparator is the true optimal legal adaptive plan with the same conservative reservation guards. Write the re","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-97d495a570cf2225","tex":"\\mathrm{joint}","aliases":"joint mathrm","display":"\\mathrm{joint}","kind":"use","excerpt":"Suppose a family of tasks is given by T_q:X→ Y, indexed by q∈ Q. Define the joint target T_joint(x)=(T_q(x))_q∈ Q∈ Y^Q. Then ϕ is sufficient for the joint target if and only if it is sufficient for each individual task. The forward direction evaluates the decoded answer vector at q. For the reverse direction, choose a decoder d_q for each query and","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"An exact normal form can improve verification without improving evaluation. The affine backend therefore checks semantics and cost separately. Its dense arithmetic tariff is D(n,m)=m(2n+1), charging n multiplications, n accumulations, and one bias addition per output row. The coefficient-composition tariff is C(n,m,k)=2knm+D(m,k). These are declare","count":44},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":32},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[Expected frozen-policy regret] For a horizon H, let J_H^rm lower(s) be the expected value of the complete adaptive policy induced by robust lower-Q selection. Under any compatible realization, 0≤ V_H(s)-J_H^rm lower(s) ≤∑_h=0^H-12Γ_h. The comparator is the true optimal legal adaptive plan with the same conservative reservation guards. Write the re","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-9d0fbf856fe55b03","tex":"L^1","aliases":"L^1","display":"L^1","kind":"use","excerpt":"[Rare evidence can reverse a posterior completely] Under the uniform prior on {0,1}, take L_0=(ε,0) and L_1=(0,ε) with rational ε>0. The two likelihoods differ pointwise by at most ε, and both evidence masses equal ε/2. Their posteriors are respectively (1,0) and (0,1), with full L^1 distance 2. Arbitrarily small positive likelihood perturbations t","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"[Rare evidence can reverse a posterior completely] Under the uniform prior on {0,1}, take L_0=(ε,0) and L_1=(0,ε) with rational ε>0. The two likelihoods differ pointwise by at most ε, and both evidence masses equal ε/2. Their posteriors are respectively (1,0) and (0,1), with full L^1 distance 2. Arbitrarily small positive likelihood perturbations t","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-743ff0ea070acbd8","tex":"L_h","aliases":"L_h","display":"L_h","kind":"use","excerpt":"|Q_h(s,a)-widehat Q_h(s,a)| · ≤ρ_h(a)≤Γ_h. Thus L_h(s,a)=widehat Q_h(s,a)-ρ_h(a) and U_h(s,a)=widehat Q_h(s,a)+ρ_h(a) enclose the true Q value. At horizon zero the value error is zero. In a single backup, insert and subtract the expectation of the nominal continuation under the true row. The reward error contributes at most ε_a, the continuation-va","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-01208e159d2aec8c","tex":"\\land","aliases":"land","display":"\\land","kind":"use","excerpt":"athbb Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"[Exact fiber criterion] Suppose Y is nonempty. Then ϕ is sufficient for T if and only if ϕ(x)=ϕ(y) Longrightarrow T(x)=T(y) (x,y∈ X). In particular, a single pair with ϕ(x)=ϕ(y) and T(x)≠ T(y) proves insufficiency. If T=d∘ϕ, equality of the two representations gives equality after applying d. Conversely, fix a default y_0∈ Y. For every reachable z∈","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-529ad2daacd7efcb","tex":"\\lor","aliases":"lor","display":"\\lor","kind":"use","excerpt":"R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCU","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":16},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","excerpt":"The initial grammar layer has five atoms: two literals and three variables. If a layer is L, expansion retains L, adds all negations, and adds all ordered pairs under each of the three binary operations. Let M_d count emitted list entries at depth d. The source proves M_0=5, M_d+1=2M_d+3M_d^2. The entries can contain duplicates. The node bound is B","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"is L, expansion retains L, adds all negations, and adds all ordered pairs under each of the three binary operations. Let M_d count emitted list entries at depth d. The source proves M_0=5, M_d+1=2M_d+3M_d^2. The entries can contain duplicates. The node bound is B_0=1, B_d+1=2B_d+1, and every expression of height at most d occurs in the depth-d list","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"sion retains L, adds all negations, and adds all ordered pairs under each of the three binary operations. Let M_d count emitted list entries at depth d. The source proves M_0=5, M_d+1=2M_d+3M_d^2. The entries can contain duplicates. The node bound is B_0=1, B_d+1=2B_d+1, and every expression of height at most d occurs in the depth-d list. In partic","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-284d7acc4980e163","tex":"M_b","aliases":"M_b","display":"M_b","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-180e72e5e1a7cd2b","tex":"M_{\\bar b}","aliases":"M_bar b bar","display":"M_{\\bar b}","kind":"use","excerpt":"te spaces, and define the pushforward belief bar b(z)=∑_ϕ(ω)=zb(ω). Suppose L=bar L∘ϕ for a nonnegative likelihood bar L:Z→Q, and the evidence mass is positive. Then ϕ_*C_L(b)=C_bar L(ϕ_*b). Such an exact bar L exists if and only if L is constant on every fiber of ϕ. Grouping the normalizer by fibers gives M_b(L)=M_bar b(bar L). On each fiber ϕ^-1(","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-32135a2acb096477","tex":"M_{C_{L_1}(b)}","aliases":"M_C_L_1(b)","display":"M_{C_{L_1}(b)}","kind":"use","excerpt":"For any nonnegative likelihood weight L, write M_b(L)=∑_ω b(ω)L(ω), C_L(b)(ω)=fracb(ω)L(ω)M_b(L) when M_b(L)>0. Given nonnegative L_1,L_2 and positive successive evidence masses, the source proves M_b(L_1)M_C_L_1(b)(L_2)=M_b(L_1L_2), C_L_2(C_L_1(b))=C_L_1L_2(b). These identities follow by expanding the finite sums and canceling the positive normali","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-207e98afc142d0de","tex":"M_d","aliases":"M_d","display":"M_d","kind":"use","excerpt":"The initial grammar layer has five atoms: two literals and three variables. If a layer is L, expansion retains L, adds all negations, and adds all ordered pairs under each of the three binary operations. Let M_d count emitted list entries at depth d. The source proves M_0=5, M_d+1=2M_d+3M_d^2. The entries can contain duplicates. The node bound is B","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b76332913c77b5e0","tex":"M_{d+1}","aliases":"M_d+1","display":"M_{d+1}","kind":"use","excerpt":"The initial grammar layer has five atoms: two literals and three variables. If a layer is L, expansion retains L, adds all negations, and adds all ordered pairs under each of the three binary operations. Let M_d count emitted list entries at depth d. The source proves M_0=5, M_d+1=2M_d+3M_d^2. The entries can contain duplicates. The node bound is B","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-79f53d844b2e76a9","tex":"M_d^2","aliases":"M_d^2","display":"M_d^2","kind":"use","excerpt":"The initial grammar layer has five atoms: two literals and three variables. If a layer is L, expansion retains L, adds all negations, and adds all ordered pairs under each of the three binary operations. Let M_d count emitted list entries at depth d. The source proves M_0=5, M_d+1=2M_d+3M_d^2. The entries can contain duplicates. The node bound is B","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c879cfab2bd64d1b","tex":"m_j","aliases":"m_j","display":"m_j","kind":"use","excerpt":"A schedule is a finite set J of checkpoints. Each checkpoint fixes an identity stratum, sample size n_j, grid m_j>0, true parameter p_j, and nonnegative error allocation α_j. Its count statistic has the stated binomial marginal. Counts can be dependent, including different prefixes of one source and checkpoints that share other underlying randomnes","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"use","excerpt":"A complete finite semantic algebra supports a stronger conclusion than searching a deeper finite layer. Let D be a semantic domain with interpretations of the Boolean constructors. A cost function c:D→N is a lower certificate if it satisfies the constructor inequalities c(literal b) · ≤1, · c(variable j) · ≤1, c(neg s) · ≤ c(s)+1, · c(smathbin∘t) ·","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-572faa9ee950a7b3","tex":"\\max_{a\\in A_0}","aliases":"max_a∈ A_0 max in","display":"\\max_{a\\in A_0}","kind":"use","excerpt":"Define the finite-horizon values by V_0(s)=0, Q_n(s,a)=r(s,a)+V_n(f(s,a)), V_n+1(s)=max_a∈ A_0Q_n(s,a).","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"The menu selector compares three equivalent implementations: the current program, its dense fused atom, and the accepted minimum-width two-stage plan. Its selected tariff is exactly min{cost(p), D(n,m), F(n,m,k)}. This proves a minimum over the explicit menu, and also comparison with every exact two-dense-stage factorization. It does not prove a gl","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Programs are finite abstract syntax trees built from Boolean literals, three input variables, negation, conjunction, disjunction, and exclusive-or. The input domain is the eight triples in {0,1}^3. Let N(p) be the tree-node count: each literal or variable contributes one, negation adds one to its child, and a binary constructor adds one to the sum ","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N","display":null,"kind":"use","excerpt":"[Exact preservation of controlled value] If ϕ is control sufficient, then V_n is constant on every fiber of ϕ for every n∈N. The same is true of Q_n(·,a) for each admissible a, and Bellman-optimal action sets coincide within each fiber. At horizon zero the values vanish. If V_n is fiber constant and ϕ(s)=ϕ(t), control sufficiency gives equal reward","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":",keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"A schedule is a finite set J of checkpoints. Each checkpoint fixes an identity stratum, sample size n_j, grid m_j>0, true parameter p_j, and nonnegative error allocation α_j. Its count statistic has the stated binomial marginal. Counts can be dependent, including different prefixes of one source and checkpoints that share other underlying randomnes","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-9cc3cd93a260a3e9","tex":"N_k","aliases":"N_k","display":"N_k","kind":"use","excerpt":"The primitive model uses a finite rational probability space and, for each identity k, a finite latent Boolean vector of length N_k. Its success parameter p_k is fixed. The fresh-outcome contract states that after agreement with any specified pattern on a previously revealed set of positions, a fresh position's joint success mass is p_k times that ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-1d7d1fdf60d8fdac","tex":"\\neg","aliases":"neg","display":"\\neg","kind":"use","excerpt":"ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ad7288b64abe8cb3","tex":"\\operatorname{normalize}","aliases":"normalize operatorname","display":"\\operatorname{normalize}","kind":"use","excerpt":"Typed programs combine affine atoms, identities, sequential composition, and pointwise addition. Composition connects only matching intermediate dimensions. An independent recursive execution semantics and coefficient-normalization semantics are defined. Structural induction proves eval(normalize(p),x)=run(p,x) for all x. Hence two programs are equ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"A rational belief is a nonnegative vector b on finite Ω with ∑_ω b(ω)=1. For a channel E, define the predictive mass p(o)=∑_ω b(ω)E(omidω). For p(o)>0, the posterior is b_o(ω)=b(ω)E(omidω)/p(o). The total updater in the source uses b_o=b when p(o)=0. This fallback is normalized and has zero coefficient in predictive expectations. Nonnegativity of e","count":44},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b7f360bb2799a8d5","tex":"p^k","aliases":"p^k","display":"p^k","kind":"use","excerpt":"Let K be the number of successes in n Bernoulli trials with rational parameter p∈[0,1]. Define b_n,k(p)=binom nk p^k(1-p)^n-k, 0≤ k≤ n. Choose a positive integer grid size m and nonnegative rational allowance α. The grid cells have endpoints ℓ_i=fracim, u_i=fraci+1m, 0≤ i<m. For observed count k, the envelope of its point mass over cell i is B_n,k,","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"use","excerpt":"A schedule is a finite set J of checkpoints. Each checkpoint fixes an identity stratum, sample size n_j, grid m_j>0, true parameter p_j, and nonnegative error allocation α_j. Its count statistic has the stated binomial marginal. Counts can be dependent, including different prefixes of one source and checkpoints that share other underlying randomnes","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-dc7ba1c49968c37c","tex":"p_k","aliases":"p_k","display":"p_k","kind":"use","excerpt":"For kind k∈{w,c}, write c_k∈[c_k^-,c_k^+], g_k∈[g_k^-,g_k^+], p_k∈[p_k^-,p_k^+]. The formal nominal model reserves c_k^+, credits g_k^-, and uses p_k^-. A compatible realization supplies exact values inside these bands. In the true comparison system, rewards and report probabilities use those exact values, but the state transitions and action guard","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-9883f5c9f5d8b72c","tex":"p_k^+","aliases":"p_k^+","display":"p_k^+","kind":"use","excerpt":"For kind k∈{w,c}, write c_k∈[c_k^-,c_k^+], g_k∈[g_k^-,g_k^+], p_k∈[p_k^-,p_k^+]. The formal nominal model reserves c_k^+, credits g_k^-, and uses p_k^-. A compatible realization supplies exact values inside these bands. In the true comparison system, rewards and report probabilities use those exact values, but the state transitions and action guard","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-17d7ea2a715d8c04","tex":"p_k^-","aliases":"p_k^-","display":"p_k^-","kind":"use","excerpt":"For kind k∈{w,c}, write c_k∈[c_k^-,c_k^+], g_k∈[g_k^-,g_k^+], p_k∈[p_k^-,p_k^+]. The formal nominal model reserves c_k^+, credits g_k^-, and uses p_k^-. A compatible realization supplies exact values inside these bands. In the true comparison system, rewards and report probabilities use those exact values, but the state transitions and action guard","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d4876c09442ebddb","tex":"P_p","aliases":"P_p","display":"P_p","kind":"use","excerpt":"[Bernoulli row error and expectation transport] For Bernoulli parameters p,q, ∑_o∈{0,1}|P_p(o)-P_q(o)|=2|p-q|. If |W(t)|≤ B, then replacing an observation row P by widehat P changes its expected continuation by at most |∑_o(P(o)-widehat P(o))W(T(s,a,o))| ≤ B∑_o|P(o)-widehat P(o)|. The two Bernoulli row differences are p-q and q-p, which have equal ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d61b96062b482c72","tex":"P_q","aliases":"P_q","display":"P_q","kind":"use","excerpt":"[Bernoulli row error and expectation transport] For Bernoulli parameters p,q, ∑_o∈{0,1}|P_p(o)-P_q(o)|=2|p-q|. If |W(t)|≤ B, then replacing an observation row P by widehat P changes its expected continuation by at most |∑_o(P(o)-widehat P(o))W(T(s,a,o))| ≤ B∑_o|P(o)-widehat P(o)|. The two Bernoulli row differences are p-q and q-p, which have equal ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-80b4b6d1ff3db017","tex":"\\Prob","aliases":"Prob","display":null,"kind":"use","excerpt":"Writing I for issuance and C for the specified episode certificate, the resulting conclusion is Prob(I∩ C^ c)≤∑_j∈ Jα_j. This is an unconditional probability over the original outcome space. It allows arbitrary dependence between scheduled checkpoints. It retains failed and unfinished runs in that space, with no plan issued on those runs. It does n","count":7},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-83c2f25462cf97fa","tex":"\\operatorname{Prop}","aliases":"Prop operatorname","display":"\\operatorname{Prop}","kind":"use","excerpt":"The concrete binding layer uses a small data-only propositional language. Formulas contain numbered atoms, truth, falsity, conjunction, disjunction, and implication. Derivations contain the corresponding introduction and elimination rules. Atoms are interpreted as arbitrary propositions under a valuation v:N→Prop. A recursive inference function eit","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"wstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"Suppose a family of tasks is given by T_q:X→ Y, indexed by q∈ Q. Define the joint target T_joint(x)=(T_q(x))_q∈ Q∈ Y^Q. Then ϕ is sufficient for the joint target if and only if it is sufficient for each individual task. The forward direction evaluates the decoded answer vector at q. For the reverse direction, choose a decoder d_q for each query and","count":18},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"use","excerpt":"Here h in Q_h counts continuation steps. Choosing an acquisition action and then deploying a successful artifact requires two total actions, so its first-action score is Q_1, not Q_0. This indexing also applies to robustChoice md h s in the formal source.","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"use","excerpt":"Here h in Q_h counts continuation steps. Choosing an acquisition action and then deploying a successful artifact requires two total actions, so its first-action score is Q_1, not Q_0. This indexing also applies to robustChoice md h s in the formal source.","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c95e041440f50ec9","tex":"Q_n","aliases":"Q_n","display":"Q_n","kind":"use","excerpt":"Define the finite-horizon values by V_0(s)=0, Q_n(s,a)=r(s,a)+V_n(f(s,a)), V_n+1(s)=max_a∈ A_0Q_n(s,a).","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-415f1e228de5aaef","tex":"R^{k\\times m}","aliases":"R^k× m times","display":"R^{k\\times m}","kind":"use","excerpt":"Work over a field R. For reference f=(A,b):R^n→ R^m, a candidate supplies an intermediate width k, first and second affine signatures a(x)=Ux+u_0, d(y)=Vy+v_0, and matrices L∈ R^k× m, H∈ R^n× k. The checker verifies VU=A, Vu_0+v_0=b, LAH=I_k. The first two identities give the exact upper implementation. The third is an independently checkable lower","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4415f2a3fc7b65ea","tex":"R^n","aliases":"R^n","display":"R^n","kind":"use","excerpt":"[Complete zero-and-basis probes] Two affine signatures f,g:R^n→ R^m are equal if and only if they agree at 0 and at every standard basis vector e_j. Thus n+1 vector probes establish equality on all inputs. A mismatch in signatures has a mismatch among these probes. The zero probe identifies the offsets because f(0)=b_f. At e_j, the output is the jt","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-381d41f7320fb5dc","tex":"R^{n\\times k}","aliases":"R^n× k times","display":"R^{n\\times k}","kind":"use","excerpt":"Work over a field R. For reference f=(A,b):R^n→ R^m, a candidate supplies an intermediate width k, first and second affine signatures a(x)=Ux+u_0, d(y)=Vy+v_0, and matrices L∈ R^k× m, H∈ R^n× k. The checker verifies VU=A, Vu_0+v_0=b, LAH=I_k. The first two identities give the exact upper implementation. The third is an independently checkable lower","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"[Certified minimum affine width] An accepted candidate has k=rank(A). Every exact two-stage affine factorization of f through width ℓ satisfies k≤ℓ. Under the dense two-stage tariff, F(n,m,k)=D(n,k)+D(k,m), the accepted candidate is no more expensive than any such two-stage factorization. The upper factorization gives rank(A)=rank(VU)≤ k. The lower","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"[Expected frozen-policy regret] For a horizon H, let J_H^rm lower(s) be the expected value of the complete adaptive policy induced by robust lower-Q selection. Under any compatible realization, 0≤ V_H(s)-J_H^rm lower(s) ≤∑_h=0^H-12Γ_h. The comparator is the true optimal legal adaptive plan with the same conservative reservation guards. Write the re","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-0bfd0939e0882ca4","tex":"\\operatorname{run}","aliases":"run operatorname","display":"\\operatorname{run}","kind":"use","excerpt":"Typed programs combine affine atoms, identities, sequential composition, and pointwise addition. Composition connects only matching intermediate dimensions. An independent recursive execution semantics and coefficient-normalization semantics are defined. Structural induction proves eval(normalize(p),x)=run(p,x) for all x. Hence two programs are equ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-aee6c380cd411ada","tex":"S_E","aliases":"S_E","display":"S_E","kind":"use","excerpt":"The unnormalized signal-action score and policy value are S_E(o,a)=∑_ω b(ω)E(omidω)u(ω,a), J_b(E,π)=∑_o S_E(o,π(o)). This form avoids dividing by the evidence mass. In particular, it remains well defined on observations whose predictive probability is zero.","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-22eee60ad7295e41","tex":"s_z","aliases":"s_z","display":"s_z","kind":"use","excerpt":"[Exact stochastic quotient] Suppose ϕ:S→ Z is surjective and projected stochastically sufficient, with S,Z finite. Choose a representative s_z of every fiber, and define bar r(z,a)=r(s_z,a), bar P_a(z,z')=∑_ϕ(s')=z'P_a(s_z,s'). These quantities give a normalized rational stochastic control system, independent of the representative choices for admis","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-378e53f0475d209c","tex":"\\mathrm{samples}","aliases":"samples mathrm","display":"\\mathrm{samples}","kind":"use","excerpt":"Every accepted generative step emits an ordinary recorder command and an effective reveal marker. Whole-run induction proves that the emitted commands reproduce the actual raw recorder state and that the reveal state equals the inherited reveal machine on its effective trace. In particular, for every identity, samples_k(terminalLog) =source_k.take ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-40acfa600a19b862","tex":"\\mathrm{samples}_k","aliases":"samples_k mathrm","display":"\\mathrm{samples}_k","kind":"use","excerpt":"Every accepted generative step emits an ordinary recorder command and an effective reveal marker. Whole-run induction proves that the emitted commands reproduce the actual raw recorder state and that the reveal state equals the inherited reveal machine on its effective trace. In particular, for every identity, samples_k(terminalLog) =source_k.take ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-bc1e1a7d5effc2be","tex":"\\operatorname{sem}","aliases":"sem operatorname","display":"\\operatorname{sem}","kind":"use","excerpt":"uppose c satisfies those inequalities and, for every semantic state s, a witness expression w_s has semantics s and node count c(s). Then for every expression p with semantics s, N(w_s)=c(s)≤ N(p). No expression-height bound appears in this conclusion. Two such attained cost certificates have identical cost functions. Structural induction proves c(","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ae01b8265b984010","tex":"\\mathrm{source}","aliases":"source mathrm","display":"\\mathrm{source}","kind":"use","excerpt":"Every accepted generative step emits an ordinary recorder command and an effective reveal marker. Whole-run induction proves that the emitted commands reproduce the actual raw recorder state and that the reveal state equals the inherited reveal machine on its effective trace. In particular, for every identity, samples_k(terminalLog) =source_k.take ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d90df77ab1acd1ee","tex":"\\mathrm{source}_k","aliases":"source_k mathrm","display":"\\mathrm{source}_k","kind":"use","excerpt":"Every accepted generative step emits an ordinary recorder command and an effective reveal marker. Whole-run induction proves that the emitted commands reproduce the actual raw recorder state and that the reveal state equals the inherited reveal machine on its effective trace. In particular, for every identity, samples_k(terminalLog) =source_k.take ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f36cd9a6740ae220","tex":"\\mathrm{stop}","aliases":"stop mathrm","display":"\\mathrm{stop}","kind":"use","excerpt":"example assigns zero setup to both kinds, warm gross 200, cold gross 60, warm success band [3/5,1], and exact cold success 1. Start with both statuses untried and use h=1, hence two total actions. The nominal scores are warm 120, cold 60, and stop 0. Here ε=0, δ=4/5, M=260, and D_1=0. The local warm radius is 208, while the cold and stop radii are ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ebee2eb781860be9","tex":"\\mathrm{switchSpent}","aliases":"switchSpent mathrm","display":"\\mathrm{switchSpent}","kind":"use","excerpt":"For a successful paid trace with total evaluation debit e and switching debit u, the formal accounting identities are evaluationSpent_rm final · =evaluationSpent_rm initial+e, switchSpent_rm final · =switchSpent_rm initial+u. From a safe initial journal, their sum stays within protected credits. Successful replay also cannot decrease the protected ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a3903427f8de1cbc","tex":"\\mathrm{switchSpent}_{\\rm final}","aliases":"switchSpent_rm final mathrm rm","display":"\\mathrm{switchSpent}_{\\rm final}","kind":"use","excerpt":"For a successful paid trace with total evaluation debit e and switching debit u, the formal accounting identities are evaluationSpent_rm final · =evaluationSpent_rm initial+e, switchSpent_rm final · =switchSpent_rm initial+u. From a safe initial journal, their sum stays within protected credits. Successful replay also cannot decrease the protected ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b42d3310e404eace","tex":"\\mathrm{switchSpent}_{\\rm initial}","aliases":"switchSpent_rm initial mathrm rm","display":"\\mathrm{switchSpent}_{\\rm initial}","kind":"use","excerpt":"For a successful paid trace with total evaluation debit e and switching debit u, the formal accounting identities are evaluationSpent_rm final · =evaluationSpent_rm initial+e, switchSpent_rm final · =switchSpent_rm initial+u. From a safe initial journal, their sum stays within protected credits. Successful replay also cannot decrease the protected ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-696323f1537cad57","tex":"T_{\\mathrm{joint}}","aliases":"T_joint mathrm","display":"T_{\\mathrm{joint}}","kind":"use","excerpt":"Suppose a family of tasks is given by T_q:X→ Y, indexed by q∈ Q. Define the joint target T_joint(x)=(T_q(x))_q∈ Q∈ Y^Q. Then ϕ is sufficient for the joint target if and only if it is sufficient for each individual task. The forward direction evaluates the decoded answer vector at q. For the reverse direction, choose a decoder d_q for each query and","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-79b896ccd56f4035","tex":"T_q","aliases":"T_q","display":"T_q","kind":"use","excerpt":"Suppose a family of tasks is given by T_q:X→ Y, indexed by q∈ Q. Define the joint target T_joint(x)=(T_q(x))_q∈ Q∈ Y^Q. Then ϕ is sufficient for the joint target if and only if it is sufficient for each individual task. The forward direction evaluates the decoded answer vector at q. For the reverse direction, choose a decoder d_q for each query and","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-e8b87e82d64945cb","tex":"\\mathrm{take}","aliases":"take mathrm","display":"\\mathrm{take}","kind":"use","excerpt":"Every accepted generative step emits an ordinary recorder command and an effective reveal marker. Whole-run induction proves that the emitted commands reproduce the actual raw recorder state and that the reveal state equals the inherited reveal machine on its effective trace. In particular, for every identity, samples_k(terminalLog) =source_k.take ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f9c154305bddb0f8","tex":"\\mathrm{terminalLog}","aliases":"terminalLog mathrm","display":"\\mathrm{terminalLog}","kind":"use","excerpt":"Every accepted generative step emits an ordinary recorder command and an effective reveal marker. Whole-run induction proves that the emitted commands reproduce the actual raw recorder state and that the reveal state equals the inherited reveal machine on its effective trace. In particular, for every identity, samples_k(terminalLog) =source_k.take ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4ef6c0d9701b3b69","tex":"\\triangleright","aliases":"triangleright","display":null,"kind":"use","excerpt":"λ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries AVOC LARGEbfseries Certified Artifac","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-28ec3adb69bf092d","tex":"\\mathrm{true}","aliases":"true mathrm","display":"\\mathrm{true}","kind":"use","excerpt":"[Replay and faults do not create new evidence] Let an identity's source be [true,false]. Begin ID 7, report ID 7, then report ID 7 again. The sample list is [true] and the remaining source begins with false. Begin ID 8 and finish it as a fault. The sample list stays unchanged. Begin and report ID 9. It appends false, yielding [true,false]. A furthe","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"Work over a field R. For reference f=(A,b):R^n→ R^m, a candidate supplies an intermediate width k, first and second affine signatures a(x)=Ux+u_0, d(y)=Vy+v_0, and matrices L∈ R^k× m, H∈ R^n× k. The checker verifies VU=A, Vu_0+v_0=b, LAH=I_k. The first two identities give the exact upper implementation. The third is an independently checkable lower","count":9},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"Work over a field R. For reference f=(A,b):R^n→ R^m, a candidate supplies an intermediate width k, first and second affine signatures a(x)=Ux+u_0, d(y)=Vy+v_0, and matrices L∈ R^k× m, H∈ R^n× k. The checker verifies VU=A, Vu_0+v_0=b, LAH=I_k. The first two identities give the exact upper implementation. The third is an independently checkable lower","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8a830c0eacd44ea9","tex":"U_h","aliases":"U_h","display":"U_h","kind":"use","excerpt":"|Q_h(s,a)-widehat Q_h(s,a)| · ≤ρ_h(a)≤Γ_h. Thus L_h(s,a)=widehat Q_h(s,a)-ρ_h(a) and U_h(s,a)=widehat Q_h(s,a)+ρ_h(a) enclose the true Q value. At horizon zero the value error is zero. In a single backup, insert and subtract the expectation of the nominal continuation under the true row. The reward error contributes at most ε_a, the continuation-va","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"Let K be the number of successes in n Bernoulli trials with rational parameter p∈[0,1]. Define b_n,k(p)=binom nk p^k(1-p)^n-k, 0≤ k≤ n. Choose a positive integer grid size m and nonnegative rational allowance α. The grid cells have endpoints ℓ_i=fracim, u_i=fraci+1m, 0≤ i<m. For observed count k, the envelope of its point mass over cell i is B_n,k,","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-012ea586d65a5736","tex":"u_i^k","aliases":"u_i^k","display":"u_i^k","kind":"use","excerpt":"uccesses in n Bernoulli trials with rational parameter p∈[0,1]. Define b_n,k(p)=binom nk p^k(1-p)^n-k, 0≤ k≤ n. Choose a positive integer grid size m and nonnegative rational allowance α. The grid cells have endpoints ℓ_i=fracim, u_i=fraci+1m, 0≤ i<m. For observed count k, the envelope of its point mass over cell i is B_n,k,i=binom nk u_i^k(1-ℓ_i)^","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d27fa8e4e0984e4b","tex":"\\uparrow","aliases":"uparrow","display":"\\uparrow","kind":"use","excerpt":"ma1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries AVOC LARGEbfseries Cert","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"The concrete binding layer uses a small data-only propositional language. Formulas contain numbered atoms, truth, falsity, conjunction, disjunction, and implication. Derivations contain the corresponding introduction and elimination rules. Atoms are interpreted as arbitrary propositions under a valuation v:N→Prop. A recursive inference function eit","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"Work over a field R. For reference f=(A,b):R^n→ R^m, a candidate supplies an intermediate width k, first and second affine signatures a(x)=Ux+u_0, d(y)=Vy+v_0, and matrices L∈ R^k× m, H∈ R^n× k. The checker verifies VU=A, Vu_0+v_0=b, LAH=I_k. The first two identities give the exact upper implementation. The third is an independently checkable lower","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-5483c31d7be9f42d","tex":"v_0'","aliases":"v_0'","display":"v_0'","kind":"use","excerpt":"Suppose a stored candidate has been validated for (A,b) and a later target is (A,b') with exactly the same linear matrix. Keep U,u_0,V,L,H and replace only the second-stage offset by v_0'=b'-Vu_0. Then VU=A is unchanged, Vu_0+v_0'=b', and LAH=I_k remains true. The same width and two-stage tariff are retained. A second retargeting overwrites the off","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-dd902fa43e338fbe","tex":"V_H","aliases":"V_H","display":"V_H","kind":"use","excerpt":"[Optimality over complete adaptive plans] Every horizon H has a legal deterministic observation tree attaining V_H(s). Every other legal deterministic observation tree has value at most V_H(s). If each admissible transition preserves an invariant, the attaining tree preserves that invariant on every branch. At horizon zero, the empty tree has value","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a6735e9f71ad2ebf","tex":"V_n","aliases":"V_n","display":"V_n","kind":"use","excerpt":"Define the finite-horizon values by V_0(s)=0, Q_n(s,a)=r(s,a)+V_n(f(s,a)), V_n+1(s)=max_a∈ A_0Q_n(s,a).","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4cf1c9f8283bbb4c","tex":"V_{n+1}","aliases":"V_n+1","display":"V_{n+1}","kind":"use","excerpt":"Define the finite-horizon values by V_0(s)=0, Q_n(s,a)=r(s,a)+V_n(f(s,a)), V_n+1(s)=max_a∈ A_0Q_n(s,a).","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a675678fa6689935","tex":"\\vee","aliases":"vee","display":"\\vee","kind":"use","excerpt":"[A smaller artifact need not repay its discovery] The reference neg(neg x_1veeneg x_2) has six nodes. The equivalent conjunction x_1wedge x_2 has three. The archived depth-one search examines the complete 85-entry layer under its stated limits and returns the conjunction, with modeled search charge 6890. Each eager evaluation subsequently saves thr","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"[Attained global semantic minimum] Suppose c satisfies those inequalities and, for every semantic state s, a witness expression w_s has semantics s and node count c(s). Then for every expression p with semantics s, N(w_s)=c(s)≤ N(p). No expression-height bound appears in this conclusion. Two such attained cost certificates have identical cost funct","count":12},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-89d4460b19400d9c","tex":"w_c","aliases":"w_c","display":"w_c","kind":"use","excerpt":"Let w_c(k)=c_k^+-c_k^-, w_g(k)=g_k^+-g_k^-, and w_p(k)=p_k^+-p_k^-. The source uses the uniform quantities ε · =w_c(w)+w_c(c)+w_g(w)+w_g(c),","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-634c8deb08d526b2","tex":"w_g","aliases":"w_g","display":"w_g","kind":"use","excerpt":"Let w_c(k)=c_k^+-c_k^-, w_g(k)=g_k^+-g_k^-, and w_p(k)=p_k^+-p_k^-. The source uses the uniform quantities ε · =w_c(w)+w_c(c)+w_g(w)+w_g(c),","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-2b65a86b582ae995","tex":"w_p","aliases":"w_p","display":"w_p","kind":"use","excerpt":"Let w_c(k)=c_k^+-c_k^-, w_g(k)=g_k^+-g_k^-, and w_p(k)=p_k^+-p_k^-. The source uses the uniform quantities ε · =w_c(w)+w_c(c)+w_g(w)+w_g(c),","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-04fbfe585bfb125e","tex":"w_s","aliases":"w_s","display":"w_s","kind":"use","excerpt":"[Attained global semantic minimum] Suppose c satisfies those inequalities and, for every semantic state s, a witness expression w_s has semantics s and node count c(s). Then for every expression p with semantics s, N(w_s)=c(s)≤ N(p). No expression-height bound appears in this conclusion. Two such attained cost certificates have identical cost funct","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"[A task must be declared before compression] Let X={0,1}^3 and let T(x_1,x_2,x_3)=x_1wedge x_2. The representation ϕ(x)=T(x) is sufficient for this target, although it discards x_3 and many distinctions among the first two coordinates. The representation ψ(x)=x_1⊕ x_2 is insufficient: (0,0,0) and (1,1,0) have the same ψ value and different T values","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"use","excerpt":"[Bernoulli row error and expectation transport] For Bernoulli parameters p,q, ∑_o∈{0,1}|P_p(o)-P_q(o)|=2|p-q|. If |W(t)|≤ B, then replacing an observation row P by widehat P changes its expected continuation by at most |∑_o(P(o)-widehat P(o))W(T(s,a,o))| ≤ B∑_o|P(o)-widehat P(o)|. The two Bernoulli row differences are p-q and q-p, which have equal ","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-47c202182f2c6f9b","tex":"\\widehat Q","aliases":"widehat Q widehat","display":"\\widehat Q","kind":"use","excerpt":"[Value and Q enclosures] For every compatible realization, every state, and every continuation horizon, |V_h(s)-widehat V_h(s)| · ≤ D_h, |Q_h(s,a)-widehat Q_h(s,a)| · ≤ρ_h(a)≤Γ_h. Thus L_h(s,a)=widehat Q_h(s,a)-ρ_h(a) and U_h(s,a)=widehat Q_h(s,a)+ρ_h(a) enclose the true Q value. At horizon zero the value error is zero. In a single backup, insert a","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ee4c407309202da4","tex":"\\widehat Q_1","aliases":"widehat Q_1 widehat","display":"\\widehat Q_1","kind":"use","excerpt":"example assigns zero setup to both kinds, warm gross 200, cold gross 60, warm success band [3/5,1], and exact cold success 1. Start with both statuses untried and use h=1, hence two total actions. The nominal scores are warm 120, cold 60, and stop 0. Here ε=0, δ=4/5, M=260, and D_1=0. The local warm radius is 208, while the cold and stop radii are ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-892d93ce627e7f00","tex":"\\widehat Q_h","aliases":"widehat Q_h widehat","display":"\\widehat Q_h","kind":"use","excerpt":"[Value and Q enclosures] For every compatible realization, every state, and every continuation horizon, |V_h(s)-widehat V_h(s)| · ≤ D_h, |Q_h(s,a)-widehat Q_h(s,a)| · ≤ρ_h(a)≤Γ_h. Thus L_h(s,a)=widehat Q_h(s,a)-ρ_h(a) and U_h(s,a)=widehat Q_h(s,a)+ρ_h(a) enclose the true Q value. At horizon zero the value error is zero. In a single backup, insert a","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-dc5e832c68a33db9","tex":"\\widehat V","aliases":"widehat V widehat","display":"\\widehat V","kind":"use","excerpt":"Induction on the Bellman recursion gives |widehat V_h(s)|≤ hM for the nominal value. The source's accumulated value radius and local Q radius are D_h · =∑_j=0^h-1(ε+δ jM),","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b7ae7cd15bd6330e","tex":"\\widehat V_h","aliases":"widehat V_h widehat","display":"\\widehat V_h","kind":"use","excerpt":"Induction on the Bellman recursion gives |widehat V_h(s)|≤ hM for the nominal value. The source's accumulated value radius and local Q radius are D_h · =∑_j=0^h-1(ε+δ jM),","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"[Predictable availability still biases completed runs] Let (X_1,X_2) be independent fair bits. Reveal X_1, and reveal X_2 by the deadline only if X_1=1. Availability of the second report depends on the past, not on X_2. Completion probability is 1/2. Conditional on completion, X_1=1 with probability one, and the two-bit count has probability zero o","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"[A task must be declared before compression] Let X={0,1}^3 and let T(x_1,x_2,x_3)=x_1wedge x_2. The representation ϕ(x)=T(x) is sufficient for this target, although it discards x_3 and many distinctions among the first two coordinates. The representation ψ(x)=x_1⊕ x_2 is insufficient: (0,0,0) and (1,1,0) have the same ψ value and different T values","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"[Predictable availability still biases completed runs] Let (X_1,X_2) be independent fair bits. Reveal X_1, and reveal X_2 by the deadline only if X_1=1. Availability of the second report depends on the past, not on X_2. Completion probability is 1/2. Conditional on completion, X_1=1 with probability one, and the two-bit count has probability zero o","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[A task must be declared before compression] Let X={0,1}^3 and let T(x_1,x_2,x_3)=x_1wedge x_2. The representation ϕ(x)=T(x) is sufficient for this target, although it discards x_3 and many distinctions among the first two coordinates. The representation ψ(x)=x_1⊕ x_2 is insufficient: (0,0,0) and (1,1,0) have the same ψ value and different T values","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"[A task must be declared before compression] Let X={0,1}^3 and let T(x_1,x_2,x_3)=x_1wedge x_2. The representation ϕ(x)=T(x) is sufficient for this target, although it discards x_3 and many distinctions among the first two coordinates. The representation ψ(x)=x_1⊕ x_2 is insufficient: (0,0,0) and (1,1,0) have the same ψ value and different T values","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ca7008ae447ee9dc","tex":"x_4","aliases":"x_4","display":"x_4","kind":"use","excerpt":"[A checked rank-one bottleneck] In the archived finite-field example over F_5, A is the 4×4 all-ones matrix and b=(1,2,3,4). The first stage computes y=x_1+x_2+x_3+x_4. The second emits (y+1,y+2,y+3,y+4). Taking L to select the first output coordinate and H to select the first input coordinate gives LAH=(1). The width-one factorization is therefore","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"use","excerpt":"The primitive model uses a finite rational probability space and, for each identity k, a finite latent Boolean vector of length N_k. Its success parameter p_k is fixed. The fresh-outcome contract states that after agreement with any specified pattern on a previously revealed set of positions, a fresh position's joint success mass is p_k times that ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-a487d9d7f59851e2","tex":"x_z","aliases":"x_z","display":"x_z","kind":"use","excerpt":"fficient for T if and only if ϕ(x)=ϕ(y) Longrightarrow T(x)=T(y) (x,y∈ X). In particular, a single pair with ϕ(x)=ϕ(y) and T(x)≠ T(y) proves insufficiency. If T=d∘ϕ, equality of the two representations gives equality after applying d. Conversely, fix a default y_0∈ Y. For every reachable z∈ϕ(X) choose x_z with ϕ(x_z)=z and set d(z)=T(x_z). Set d(z)","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d8bb8ff8ecf6812f","tex":"Y^Q","aliases":"Y^Q","display":"Y^Q","kind":"use","excerpt":"Suppose a family of tasks is given by T_q:X→ Y, indexed by q∈ Q. Define the joint target T_joint(x)=(T_q(x))_q∈ Q∈ Y^Q. Then ϕ is sufficient for the joint target if and only if it is sufficient for each individual task. The forward direction evaluates the decoded answer vector at q. For the reverse direction, choose a decoder d_q for each query and","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"criterion Suppose Y is nonempty. Then ϕ is sufficient for T if and only if ϕ(x)=ϕ(y) Longrightarrow T(x)=T(y) (x,y∈ X). In particular, a single pair with ϕ(x)=ϕ(y) and T(x)≠ T(y) proves insufficiency. If T=d∘ϕ, equality of the two representations gives equality after applying d. Conversely, fix a default y_0∈ Y. For every reachable z∈ϕ(X) choose x_","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"orlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓down","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"ty. Then ϕ is sufficient for T if and only if ϕ(x)=ϕ(y) Longrightarrow T(x)=T(y) (x,y∈ X). In particular, a single pair with ϕ(x)=ϕ(y) and T(x)≠ T(y) proves insufficiency. If T=d∘ϕ, equality of the two representations gives equality after applying d. Conversely, fix a default y_0∈ Y. For every reachable z∈ϕ(X) choose x_z with ϕ(x_z)=z and set d(z)=","count":16},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"When Z is finite, this expectation condition is equivalent to equality of the mass sent to every abstract fiber: ∑_ϕ(s')=z'P_a(s,s')=∑_ϕ(s')=z'P_a(t,s') (z'∈ Z). To prove necessity, use the indicator of the fiber as W. To prove sufficiency, group the expectation by fibers and factor out the common value of W on each nonempty fiber. Empty fibers con","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"[2]textcolorteal#1 #2 newcommandfig[3][H]centeringincludegraphics[width=linewidth]#1caption#2 graphicspath../figures/figures/ pagestylefancyfancyhf fancyhead[L]smalltextcolornavyAVOC / Certified artifact planning fancyhead[R]smallPublication edition fancyfoot[L]small Komorebi P. · 7 October 2026 fancyfoot[R]smallthepage / pagerefLastPage renewcomma","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"orlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂","count":15},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","excerpt":"Writing I for issuance and C for the specified episode certificate, the resulting conclusion is Prob(I∩ C^ c)≤∑_j∈ Jα_j. This is an unconditional probability over the original outcome space. It allows arbitrary dependence between scheduled checkpoints. It retains failed and unfinished runs in that space, with no plan issued on those runs. It does n","count":7},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"style=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"e1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / RE","count":10},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c874a92d5eb1b875","tex":"\\Gamma_h","aliases":"Γ_h Gamma","display":"\\Gamma_h","kind":"use","excerpt":"Induction on the Bellman recursion gives |widehat V_h(s)|≤ hM for the nominal value. The source's accumulated value radius and local Q radius are D_h · =∑_j=0^h-1(ε+δ jM), ρ_h(a) · =D_h+ε_a+δ_a hM, Γ_h · =D_h+ε+δ hM. The Bellman residual is zero here because the nominal tables are exact Bellman values. The broader model-error framework also allows ","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-0ea785cdef3a5542","tex":"\\Gamma_j","aliases":"Γ_j Gamma","display":"\\Gamma_j","kind":"use","excerpt":"leq∑_h=0^H-12Γ_h. The comparator is the true optimal legal adaptive plan with the same conservative reservation guards. Write the regret at a node with h+1 remaining steps as its local Bellman loss plus the expected regret of the selected continuation. The local term is at most 2Γ_h. By induction, every child regret is bounded by the common sum ∑_j","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstse","count":7},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","excerpt":"Induction on the Bellman recursion gives |widehat V_h(s)|≤ hM for the nominal value. The source's accumulated value radius and local Q radius are D_h · =∑_j=0^h-1(ε+δ jM), ρ_h(a) · =D_h+ε_a+δ_a hM, Γ_h · =D_h+ε+δ hM. The Bellman residual is zero here because the nominal tables are exact Bellman values. The broader model-error framework also allows ","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"skip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode","count":12},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-bae7842f2d63f7f5","tex":"\\varepsilon_a","aliases":"ε_a varepsilon","display":"\\varepsilon_a","kind":"use","excerpt":"Induction on the Bellman recursion gives |widehat V_h(s)|≤ hM for the nominal value. The source's accumulated value radius and local Q radius are D_h · =∑_j=0^h-1(ε+δ jM), ρ_h(a) · =D_h+ε_a+δ_a hM, Γ_h · =D_h+ε+δ hM. The Bellman residual is zero here because the nominal tables are exact Bellman values. The broader model-error framework also allows ","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"rrow1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolorn","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolor","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOR","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"y,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"rightarrow1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanch","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=fal","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy lar","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolorn","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"ge1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI","count":6},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-737cdd19f67dfe28","tex":"\\rho_1","aliases":"ρ_1 rho","display":"\\rho_1","kind":"use","excerpt":"example assigns zero setup to both kinds, warm gross 200, cold gross 60, warm success band [3/5,1], and exact cold success 1. Start with both statuses untried and use h=1, hence two total actions. The nominal scores are warm 120, cold 60, and stop 0. Here ε=0, δ=4/5, M=260, and D_1=0. The local warm radius is 208, while the cold and stop radii are ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-3c51f5238630cd1f","tex":"\\rho_h","aliases":"ρ_h rho","display":"\\rho_h","kind":"use","excerpt":"Induction on the Bellman recursion gives |widehat V_h(s)|≤ hM for the nominal value. The source's accumulated value radius and local Q radius are D_h · =∑_j=0^h-1(ε+δ jM), ρ_h(a) · =D_h+ε_a+δ_a hM, Γ_h · =D_h+ε+δ hM. The Bellman residual is zero here because the nominal tables are exact Bellman values. The broader model-error framework also allows ","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"bb N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REAL","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppagean","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4f7f632689e8289e","tex":"\\phi^{-1}","aliases":"ϕ^-1 phi","display":"\\phi^{-1}","kind":"use","excerpt":"f bar b(z)=∑_ϕ(ω)=zb(ω). Suppose L=bar L∘ϕ for a nonnegative likelihood bar L:Z→Q, and the evidence mass is positive. Then ϕ_*C_L(b)=C_bar L(ϕ_*b). Such an exact bar L exists if and only if L is constant on every fiber of ϕ. Grouping the normalizer by fibers gives M_b(L)=M_bar b(bar L). On each fiber ϕ^-1(z), the likelihood can be factored outside ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d0aea1606cfe2399","tex":"\\phi_*","aliases":"ϕ_* phi","display":"\\phi_*","kind":"use","excerpt":"[Conditioning commutes with exact compression] Let ϕ:Ω→ Z map between finite spaces, and define the pushforward belief bar b(z)=∑_ϕ(ω)=zb(ω). Suppose L=bar L∘ϕ for a nonnegative likelihood bar L:Z→Q, and the evidence mass is positive. Then ϕ_*C_L(b)=C_bar L(ϕ_*b). Such an exact bar L exists if and only if L is constant on every fiber of ϕ. Grouping","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large K","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Certified minimum affine width] An accepted candidate has k=rank(A). Every exact two-stage affine factorization of f through width ℓ satisfies k≤ℓ. Under the dense two-stage tariff, F(n,m,k)=D(n,k)+D(k,m), the accepted candidate is no more expensive than any such two-stage factorization. The upper factorization gives rank(A)=rank(VU)≤ k. The lower","count":7},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"Let K be the number of successes in n Bernoulli trials with rational parameter p∈[0,1]. Define b_n,k(p)=binom nk p^k(1-p)^n-k, 0≤ k≤ n. Choose a positive integer grid size m and nonnegative rational allowance α. The grid cells have endpoints ℓ_i=fracim, u_i=fraci+1m, 0≤ i<m. For observed count k, the envelope of its point mass over cell i is B_n,k,","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"use","excerpt":"LeanCodelanguage=Lean,basicstyle=ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"nt=[s]/--/,morestring=[b]\" lstdefinestyleLeanCodelanguage=Lean,basicstyle=ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"[b]\" lstdefinestyleLeanCodelanguage=Lean,basicstyle=ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 ","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"n1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries AVOC LARGEbf","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-8dc80d4c5a00f8af","tex":"\\sum_{h=0}^{H-1}","aliases":"∑_h=0^H-1 sum","display":"\\sum_{h=0}^{H-1}","kind":"use","excerpt":"[Expected frozen-policy regret] For a horizon H, let J_H^rm lower(s) be the expected value of the complete adaptive policy induced by robust lower-Q selection. Under any compatible realization, 0≤ V_H(s)-J_H^rm lower(s) ≤∑_h=0^H-12Γ_h. The comparator is the true optimal legal adaptive plan with the same conservative reservation guards. Write the re","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"The probability in [source reference] ranges over the original outcome space, including rejected, unresolved, and insufficient-data runs. There is no claim that a plan will be issued, that any operation finishes, or that collection is cheap. If the issuance probability is q>0, the generic conditional implication is only Prob{uncertifiedmidissued} ≤","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-de903e03048f0752","tex":"\\sum_{j<h}","aliases":"∑_j<h sum","display":"\\sum_{j<h}","kind":"use","excerpt":"leq∑_h=0^H-12Γ_h. The comparator is the true optimal legal adaptive plan with the same conservative reservation guards. Write the regret at a node with h+1 remaining steps as its local Bellman loss plus the expected regret of the selected continuation. The local term is at most 2Γ_h. By induction, every child regret is bounded by the common sum ∑_j","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-fe904bdaad396f55","tex":"\\sum_{j=0}^{h-1}","aliases":"∑_j=0^h-1 sum","display":"\\sum_{j=0}^{h-1}","kind":"use","excerpt":"Induction on the Bellman recursion gives |widehat V_h(s)|≤ hM for the nominal value. The source's accumulated value radius and local Q radius are D_h · =∑_j=0^h-1(ε+δ jM), ρ_h(a) · =D_h+ε_a+δ_a hM, Γ_h · =D_h+ε+δ hM. The Bellman residual is zero here because the nominal tables are exact Bellman values. The broader model-error framework also allows ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b4619f532d9c032b","tex":"\\sum_{j\\in J}","aliases":"∑_j∈ J sum in","display":"\\sum_{j\\in J}","kind":"use","excerpt":"Writing I for issuance and C for the specified episode certificate, the resulting conclusion is Prob(I∩ C^ c)≤∑_j∈ Jα_j. This is an unconditional probability over the original outcome space. It allows arbitrary dependence between scheduled checkpoints. It retains failed and unfinished runs in that space, with no plan issued on those runs. It does n","count":4},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b01cb725207ea93a","tex":"\\sum_{k=10}^{20}","aliases":"∑_k=10^20 sum","display":"\\sum_{k=10}^{20}","kind":"use","excerpt":"For the implemented interval, the stronger retained finite experiment starts twenty independent fair attempts and lets all successes complete before all rejections. If there are at least ten successes, the first ten completed reports are all true. Their interval is [3/5,1] and excludes 1/2. That event has mass frac12^20∑_k=10^20binom20k =frac308333","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-106f70537564e78c","tex":"\\sum_{o\\in\\{0,1\\}}","aliases":"∑_o∈{0,1} sum in","display":"\\sum_{o\\in\\{0,1\\}}","kind":"use","excerpt":"[Bernoulli row error and expectation transport] For Bernoulli parameters p,q, ∑_o∈{0,1}|P_p(o)-P_q(o)|=2|p-q|. If |W(t)|≤ B, then replacing an observation row P by widehat P changes its expected continuation by at most |∑_o(P(o)-widehat P(o))W(T(s,a,o))| ≤ B∑_o|P(o)-widehat P(o)|. The two Bernoulli row differences are p-q and q-p, which have equal ","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ee373b78330c8be1","tex":"\\sum_y","aliases":"∑_y sum","display":"\\sum_y","kind":"use","excerpt":"licy on EK is weakly dominated in value by some deterministic policy on E. Consequently the best deterministic value of EK is at most that of E. For each fine observation o, select a_o∈ A maximizing S_E(o,a). Such a selection exists because A is finite and nonempty. For a coarse policy π, expand the channel composition and rearrange finite sums: J_","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-f2acdaec2a85b745","tex":"\\sum_{z'}","aliases":"∑_z' sum","display":"\\sum_{z'}","kind":"use","excerpt":"e actions. For every finite horizon, bar V_n(ϕ(s))=V_n(s). Nonnegative fiber masses give nonnegative quotient probabilities; partitioning S by the fibers shows that each quotient row sums to one. Sufficiency makes reward and projected mass unchanged when replacing s_z by another representative. Inductively assume the value equality at horizon n. Th","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-561d7275c0288372","tex":"\\sum_{\\phi(s')=z'}","aliases":"∑_ϕ(s')=z' sum phi","display":"\\sum_{\\phi(s')=z'}","kind":"use","excerpt":"When Z is finite, this expectation condition is equivalent to equality of the mass sent to every abstract fiber: ∑_ϕ(s')=z'P_a(s,s')=∑_ϕ(s')=z'P_a(t,s') (z'∈ Z). To prove necessity, use the indicator of the fiber as W. To prove sufficiency, group the expectation by fibers and factor out the common value of W on each nonempty fiber. Empty fibers con","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ac3f40316ca2a829","tex":"\\sum_{\\phi(\\omega)=z}","aliases":"∑_ϕ(ω)=z sum phi omega","display":"\\sum_{\\phi(\\omega)=z}","kind":"use","excerpt":"[Conditioning commutes with exact compression] Let ϕ:Ω→ Z map between finite spaces, and define the pushforward belief bar b(z)=∑_ϕ(ω)=zb(ω). Suppose L=bar L∘ϕ for a nonnegative likelihood bar L:Z→Q, and the evidence mass is positive. Then ϕ_*C_L(b)=C_bar L(ϕ_*b). Such an exact bar L exists if and only if L is constant on every fiber of ϕ. Grouping","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-ea8a48c6147135b3","tex":"\\sum_\\omega","aliases":"∑_ω sum omega","display":"\\sum_\\omega","kind":"use","excerpt":"The unnormalized signal-action score and policy value are S_E(o,a)=∑_ω b(ω)E(omidω)u(ω,a), J_b(E,π)=∑_o S_E(o,π(o)). This form avoids dividing by the evidence mass. In particular, it remains well defined on observations whose predictive probability is zero.","count":5},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"Writing I for issuance and C for the specified episode certificate, the resulting conclusion is Prob(I∩ C^ c)≤∑_j∈ Jα_j. This is an unconditional probability over the original outcome space. It allows arbitrary dependence between scheduled checkpoints. It retains failed and unfinished runs in that space, with no plan issued on those runs. It does n","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"s=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1","count":3},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"Writing I for issuance and C for the specified episode certificate, the resulting conclusion is Prob(I∩ C^ c)≤∑_j∈ Jα_j. This is an unconditional probability over the original outcome space. It allows arbitrary dependence between scheduled checkpoints. It retains failed and unfinished runs in that space, with no plan issued on those runs. It does n","count":21},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseri","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"[A task must be declared before compression] Let X={0,1}^3 and let T(x_1,x_2,x_3)=x_1wedge x_2. The representation ϕ(x)=T(x) is sufficient for this target, although it discards x_3 and many distinctions among the first two coordinates. The representation ψ(x)=x_1⊕ x_2 is insufficient: (0,0,0) and (1,1,0) have the same ψ value and different T values","count":2},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"[Scheduled and selected failure bounds] Let B_j be the event that checkpoint j's interval excludes p_j. Then Prob(⋃_j∈ JB_j) ≤∑_j∈ Jα_j. For any data-dependent selector j(ω)∈ J, Prob{ω∈ B_j(ω)}≤∑_j∈ Jα_j. For each outcome ω, the indicator of a union is at most the sum of the individual indicators. Multiply by the nonnegative rational weight of ω an","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-c5604e7103d9198f","tex":"\\bigcup_{j\\in J}","aliases":"⋃_j∈ J bigcup in","display":"\\bigcup_{j\\in J}","kind":"use","excerpt":"[Scheduled and selected failure bounds] Let B_j be the event that checkpoint j's interval excludes p_j. Then Prob(⋃_j∈ JB_j) ≤∑_j∈ Jα_j. For any data-dependent selector j(ω)∈ J, Prob{ω∈ B_j(ω)}≤∑_j∈ Jα_j. For each outcome ω, the indicator of a union is at most the sum of the individual indicators. Multiply by the nonnegative rational weight of ω an","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries AVO","count":1},{"work":"research-avoc","url":"notation/research-avoc.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandavoccover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries AVOC LA","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"notation-table","excerpt":"z_*,k · positive evidence floor","count":15},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"notation-table","excerpt":"e_k^-,e_k^+ · aggregate errors before and after an update","count":63},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"P_C=EC · representative projection","count":315},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"A_m^ f,A_m^ c · fine and coarse interaction tensors of order m","count":110},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"notation-table","excerpt":"P_i,overline P_a · fine and coarse packet projectors","count":50},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4ebb3eefbf36db8f","tex":"A_m^{\\coarse}","aliases":"A_m^coarse A_m^ c coarse mathrm","display":"A_m^{{\\mathrm c}}","kind":"notation-table","excerpt":"A_m^ f,A_m^ c · fine and coarse interaction tensors of order m","count":47},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9a18c78f18b13782","tex":"A_m^{\\fine}","aliases":"A_m^fine A_m^ f fine mathrm","display":"A_m^{{\\mathrm f}}","kind":"notation-table","excerpt":"A_m^ f,A_m^ c · fine and coarse interaction tensors of order m","count":67},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fa75fa9a693ac0f2","tex":"\\Bup","aliases":"Bup B mathsf","display":"{\\mathsf B}","kind":"notation-table","excerpt":"B_λ · Bayesian normalized likelihood update","count":35},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e578848339781a67","tex":"\\Bup_{\\lambda}","aliases":"Bup_λ B_λ Bup lambda mathsf lambda","display":"{\\mathsf B}_{\\lambda}","kind":"notation-table","excerpt":"B_λ · Bayesian normalized likelihood update","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"notation-table","excerpt":"C · surjective aggregation or quotient map","count":263},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-18b8ec4b3719fb29","tex":"\\coarse","aliases":"coarse c mathrm","display":"{\\mathrm c}","kind":"notation-table","excerpt":"X_ f,X_ c · fine and coarse state spaces","count":216},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"notation-table","excerpt":"d_Φ,k · direct prediction quotient defect at step k","count":17},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7fb4da85cf4b62ba","tex":"d_{\\Bup,k}","aliases":"d_Bup,k d_ B,k Bup mathsf","display":"d_{{\\mathsf B},k}","kind":"notation-table","excerpt":"d_ B,k · direct Bayesian refinement defect at step k","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ab4cb25226ab50dd","tex":"d_{\\Phi,k}","aliases":"d_Φ,k Phi","display":"d_{\\Phi,k}","kind":"notation-table","excerpt":"d_Φ,k · direct prediction quotient defect at step k","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"notation-table","excerpt":"E · right inverse selecting a fine representative of each coarse 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fields","count":26},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-227f8507a3d7ee70","tex":"\\fine","aliases":"fine f mathrm","display":"{\\mathrm f}","kind":"notation-table","excerpt":"X_ f,X_ c · fine and coarse state spaces","count":245},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"notation-table","excerpt":"H_m · entropy of the normalized hyperedge energy distribution at order m","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-cf5f97ebded34595","tex":"H_m","aliases":"H_m","display":"H_m","kind":"notation-table","excerpt":"H_m · entropy of the normalized hyperedge energy distribution at order 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operators","count":11},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1323407d9b924103","tex":"L_{\\fine}","aliases":"L_fine L_ f fine mathrm","display":"L_{{\\mathrm f}}","kind":"notation-table","excerpt":"L_ f,L_ c · positive observation transfer operators","count":10},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"notation-table","excerpt":"A_m^ f,A_m^ c · fine and coarse interaction tensors of order m","count":204},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"notation-table","excerpt":"Ψ_1:N · cocycle for a finite observation sequence","count":27},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ffdd263bffb70d69","tex":"\\overline P","aliases":"overline P overline","display":"\\overline P","kind":"notation-table","excerpt":"P_i,overline P_a · fine and coarse packet projectors","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0554550a5d4c3d25","tex":"\\overline P_a","aliases":"overline P_a overline","display":"\\overline P_a","kind":"notation-table","excerpt":"P_i,overline P_a · fine and coarse packet projectors","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"P_i,overline P_a · fine and coarse packet 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C","count":19},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-72684694ad1ea082","tex":"Q_C","aliases":"Q_C","display":"Q_C","kind":"notation-table","excerpt":"Q_C=I-EC · projection onto the hidden directions, with range Ker C","count":10},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"notation-table","excerpt":"R_m · omitted hyperedge residual tensor","count":28},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-25ee15775b880083","tex":"R_m","aliases":"R_m","display":"R_m","kind":"notation-table","excerpt":"R_m · omitted hyperedge residual tensor","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"notation-table","excerpt":"Φ_t^ f,Φ_t^ c · fine and coarse flows or prediction maps","count":34},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"notation-table","excerpt":"X_ f,X_ c · fine and coarse state spaces","count":56},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1c88def6d64ad10d","tex":"X_{\\coarse}","aliases":"X_coarse X_ c coarse mathrm","display":"X_{{\\mathrm c}}","kind":"notation-table","excerpt":"X_ f,X_ c · fine and coarse state spaces","count":23},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2dcf35cd1052dd18","tex":"X_{\\fine}","aliases":"X_fine X_ f fine mathrm","display":"X_{{\\mathrm f}}","kind":"notation-table","excerpt":"X_ f,X_ c · fine and coarse state spaces","count":23},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"notation-table","excerpt":"z_*,k · positive evidence floor","count":54},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-05d0eed3a929e23e","tex":"z_{*,k}","aliases":"z_*,k","display":"z_{*,k}","kind":"notation-table","excerpt":"z_*,k · positive evidence floor","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"notation-table","excerpt":"Γ_k · single step prediction and update map","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8335fc4d7b413b19","tex":"\\Gamma_k","aliases":"Γ_k Gamma","display":"\\Gamma_k","kind":"notation-table","excerpt":"Γ_k · single step prediction and update map","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"notation-table","excerpt":"Δ_m · quotient defect at interaction order m","count":33},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-dd0aa14ad6313087","tex":"\\Delta_m","aliases":"Δ_m Delta","display":"\\Delta_m","kind":"notation-table","excerpt":"Δ_m · quotient defect at interaction order m","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"notation-table","excerpt":"B_λ · Bayesian normalized likelihood update","count":75},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"notation-table","excerpt":"π · map from fine packet labels to coarse labels","count":23},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"notation-table","excerpt":"Φ_t^ f,Φ_t^ c · fine and coarse flows or prediction maps","count":33},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-04a1f714539d4126","tex":"\\Phi_t^{\\coarse}","aliases":"Φ_t^coarse Φ_t^ c Phi coarse Phi mathrm","display":"\\Phi_t^{{\\mathrm c}}","kind":"notation-table","excerpt":"Φ_t^ f,Φ_t^ c · fine and coarse flows or prediction maps","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a1f880abb5d3b1cd","tex":"\\Phi_t^{\\fine}","aliases":"Φ_t^fine Φ_t^ f Phi fine Phi mathrm","display":"\\Phi_t^{{\\mathrm f}}","kind":"notation-table","excerpt":"Φ_t^ f,Φ_t^ c · fine and coarse flows or prediction maps","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"notation-table","excerpt":"Ψ_1:N · cocycle for a finite observation sequence","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ca433527ea6d0781","tex":"\\Psi_{1:N}","aliases":"Ψ_1:N Psi","display":"\\Psi_{1:N}","kind":"notation-table","excerpt":"Ψ_1:N · cocycle for a finite observation sequence","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9a267e7c2cfa4ba2","tex":"A_0^{\\coarse}","aliases":"A_0^coarse A_0^ c coarse mathrm","display":"A_0^{{\\mathrm c}}","kind":"definition-passage","excerpt":"m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c55d999bb9f11783","tex":"A_0^{\\fine}","aliases":"A_0^fine A_0^ f fine mathrm","display":"A_0^{{\\mathrm f}}","kind":"definition-passage","excerpt":"m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4faeffb478510e4d","tex":"A_{i\\mid j_1,\\ldots,j_m}","aliases":"A_imid j_1,…,j_m mid ldots","display":null,"kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-45e024d0a9cb8f07","tex":"A_m^{\\coarse,E}","aliases":"A_m^coarse,E A_m^ c,E coarse mathrm","display":"A_m^{{\\mathrm c},E}","kind":"definition-passage","excerpt":"[Coarse tensor induced by a section] Fix E with CE=I. Define A_m^ c,E = CA_m^ fE^× m.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Interaction defect] For a proposed coarse tensor B_m, define Δ_m[B_m] = CA_m^ f-B_mC^× m. Exact descent is the condition Δ_m[B_m]=0.","count":9},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-60283dddc0176654","tex":"B_+","aliases":"B_+","display":"B_+","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"definition-passage","excerpt":"[Interaction defect] For a proposed coarse tensor B_m, define Δ_m[B_m] = CA_m^ f-B_mC^× m. Exact descent is the condition Δ_m[B_m]=0.","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8e3c0fce1b7a60ae","tex":"\\bullet","aliases":"bullet","display":"\\bullet","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-02973647ab1e8962","tex":"C^{\\times m}","aliases":"C^× m times","display":"C^{\\times m}","kind":"definition-passage","excerpt":"[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.","count":26},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-82497562659dda68","tex":"\\cA","aliases":"cA A mathcal","display":"{\\mathcal A}","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7c1b3013ec0181c3","tex":"\\cL","aliases":"cL L mathcal","display":"{\\mathcal L}","kind":"definition-passage","excerpt":"[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a072b58d3a55c662","tex":"\\cL^m","aliases":"cL^m L^m cL mathcal","display":"{\\mathcal L}^m","kind":"definition-passage","excerpt":"[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b13ebabb2aef2170","tex":"\\cV","aliases":"cV V mathcal","display":"{\\mathcal V}","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c4586b6759b63b2b","tex":"\\cV_{\\bullet}","aliases":"cV_bullet V_bullet cV bullet mathcal bullet","display":"{\\mathcal V}_{\\bullet}","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-496fad0d1f2beb6c","tex":"\\cV_{\\tau}","aliases":"cV_τ V_τ cV tau mathcal tau","display":"{\\mathcal V}_{\\tau}","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e760defdc207b559","tex":"\\cV_{\\tau_1}","aliases":"cV_τ_1 V_τ_1 cV tau mathcal tau","display":"{\\mathcal V}_{\\tau_1}","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-399dde10b8aeb7b5","tex":"\\cV_{\\tau_m}","aliases":"cV_τ_m V_τ_m cV tau mathcal tau","display":"{\\mathcal V}_{\\tau_m}","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0645873febb68b5c","tex":"\\Downarrow","aliases":"Downarrow","display":null,"kind":"definition-passage","excerpt":"[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e0635a35aeb02932","tex":"\\Downarrow_C","aliases":"Downarrow_C Downarrow","display":null,"kind":"definition-passage","excerpt":"[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-dd243848c4c9762c","tex":"E^{\\times m}","aliases":"E^× m times","display":"E^{\\times m}","kind":"definition-passage","excerpt":"[Coarse tensor induced by a section] Fix E with CE=I. Define A_m^ c,E = CA_m^ fE^× m.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-123383239b8b433d","tex":"I_{\\coarse}","aliases":"I_coarse I_ c coarse mathrm","display":"I_{{\\mathrm c}}","kind":"definition-passage","excerpt":"[Compatible packet aggregation] Suppose (P_i)_i∈ I_ f and (overline P_a)_a∈ I_ c resolve the fine and coarse spaces. A surjection π:I_ f↠ I_ c is compatible with C when overline P_aC = C∑_π(i)=aP_i (a∈ I_ c).","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-26ff1878634d5790","tex":"I_{\\fine}","aliases":"I_fine I_ f fine mathrm","display":"I_{{\\mathrm f}}","kind":"definition-passage","excerpt":"[Compatible packet aggregation] Suppose (P_i)_i∈ I_ f and (overline P_a)_a∈ I_ c resolve the fine and coarse spaces. A surjection π:I_ f↠ I_ c is compatible with C when overline P_aC = C∑_π(i)=aP_i (a∈ I_ c).","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"definition-passage","excerpt":"[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-444ffa15ba67368d","tex":"\\inf_{B_m\\in\\cL^m(X_{\\coarse};X_{\\coarse})}","aliases":"inf_B_m∈cL^m(X_coarse;X_coarse) inf_B_m∈ L^m(X_ c;X_ c) inf in cL coarse coarse inf in mathcal mathrm mathrm","display":"\\inf_{B_m\\in{\\mathcal L}^m(X_{{\\mathrm c}};X_{{\\mathrm c}})}","kind":"definition-passage","excerpt":"[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":33},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0e97a80aa927d24d","tex":"j_1","aliases":"j_1","display":"j_1","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":15},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-416b4c70e8031735","tex":"j_m","aliases":"j_m","display":"j_m","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":15},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-eec18b0187f98c2c","tex":"\\max_a","aliases":"max_a max","display":"\\max_a","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fd7571cb3bcf32a6","tex":"\\max_{i:\\pi(i)=a}","aliases":"max_i:π(i)=a max pi","display":"\\max_{i:\\pi(i)=a}","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2343bb4dae2c60dc","tex":"\\min_{i:\\pi(i)=a}","aliases":"min_i:π(i)=a min pi","display":"\\min_{i:\\pi(i)=a}","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"definition-passage","excerpt":"[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.","count":54},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3c62a9efd4412aac","tex":"\\Nrm","aliases":"Nrm N mathsf","display":"{\\mathsf N}","kind":"definition-passage","excerpt":"[Normalized positive transfer] For a positive linear operator L:X→ X, define N_ℓ(L)(x) = fracLxℓ(Lx) whenever ℓ(Lx)>0.","count":9},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e2303908a8ed0cc1","tex":"\\Nrm_{\\ell}","aliases":"Nrm_ℓ N_ℓ Nrm ell mathsf ell","display":"{\\mathsf N}_{\\ell}","kind":"definition-passage","excerpt":"[Normalized positive transfer] For a positive linear operator L:X→ X, define N_ℓ(L)(x) = fracLxℓ(Lx) whenever ℓ(Lx)>0.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a625c27cba36b747","tex":"\\osc","aliases":"osc operatorname","display":"{\\operatorname{osc}}","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-110de86027c1a24f","tex":"\\osc_a","aliases":"osc_a osc operatorname","display":"{\\operatorname{osc}}_a","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c7915728f5baadb4","tex":"\\osc_C","aliases":"osc_C osc operatorname","display":"{\\operatorname{osc}}_C","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-81147b115446999c","tex":"P_{j_1}","aliases":"P_j_1","display":"P_{j_1}","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-234598544176e608","tex":"P_{j_m}","aliases":"P_j_m","display":"P_{j_m}","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5950b0378700bfba","tex":"Q_i","aliases":"Q_i","display":"Q_i","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2e6d3d996f44d01d","tex":"\\mathbf x","aliases":"x mathbf","display":"\\mathbf x","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":89},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7272f222c85749d6","tex":"X^m","aliases":"X^m","display":"X^m","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4577e71d24c631f1","tex":"\\mathbf x_1","aliases":"x_1 mathbf","display":"\\mathbf x_1","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":25},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-cb5ba94b976c67ef","tex":"X_{\\coarse}^m","aliases":"X_coarse^m X_ c^m coarse mathrm","display":"X_{{\\mathrm c}}^m","kind":"definition-passage","excerpt":"[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-89971ffe940a8aa8","tex":"X_{\\fine}^m","aliases":"X_fine^m X_ f^m fine mathrm","display":"X_{{\\mathrm f}}^m","kind":"definition-passage","excerpt":"[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-cf95984f48893a25","tex":"\\mathbf x_m","aliases":"x_m mathbf","display":"\\mathbf x_m","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e95cefd42ac01f28","tex":"x_{|\\tau|}","aliases":"x_|τ| tau","display":"x_{|\\tau|}","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Exact interaction quotient] Let A_m^ f:X_ f^m→ X_ f and A_m^ c:X_ c^m→ X_ c be multilinear maps of order m. We say that A_m^ f descends through C to A_m^ c, and write A_m^ fDownarrow_C A_m^ c, when C A_m^ f = A_m^ cC^× m. For m=0, descent means CA_0^ f=A_0^ c.","count":66},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":11},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"definition-passage","excerpt":"[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.","count":16},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f35beefa406c3f57","tex":"\\varepsilon_m^\\star","aliases":"ε_m^⋆ varepsilon star","display":"\\varepsilon_m^\\star","kind":"definition-passage","excerpt":"[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-26c73eb87a926d1c","tex":"\\lambda^{\\fine}","aliases":"λ^fine λ^ f lambda fine lambda mathrm","display":"\\lambda^{{\\mathrm f}}","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":19},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6ab8ea017e6e947f","tex":"\\lambda_i^{\\fine}","aliases":"λ_i^fine λ_i^ f lambda fine lambda mathrm","display":"\\lambda_i^{{\\mathrm f}}","kind":"definition-passage","excerpt":"[Fiber oscillation] For a fine likelihood, define osc_a(λ^ f) = max_i:π(i)=aλ_i^ f - min_i:π(i)=aλ_i^ f, and osc_C(λ^ f) =max_aosc_a(λ^ f).","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8c811b6eab2988ab","tex":"\\tau_1","aliases":"τ_1 tau","display":"\\tau_1","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0a37e389a69b4972","tex":"\\tau_m","aliases":"τ_m tau","display":"\\tau_m","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":45},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"definition-passage","excerpt":"[Normalized positive transfer] For a positive linear operator L:X→ X, define N_ℓ(L)(x) = fracLxℓ(Lx) whenever ℓ(Lx)>0.","count":17},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":22},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9ec711e428117ead","tex":"\\twoheadrightarrow","aliases":"↠ twoheadrightarrow","display":"\\twoheadrightarrow","kind":"definition-passage","excerpt":"[Compatible packet aggregation] Suppose (P_i)_i∈ I_ f and (overline P_a)_a∈ I_ c resolve the fine and coarse spaces. A surjection π:I_ f↠ I_ c is compatible with C when overline P_aC = C∑_π(i)=aP_i (a∈ I_ c).","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":40},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":42},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7fec58bf80198971","tex":"\\sum_{i\\in I}","aliases":"∑_i∈ I sum in","display":"\\sum_{i\\in I}","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f161e3d2c11b0003","tex":"\\sum_{\\pi(i)=a}","aliases":"∑_π(i)=a sum pi","display":"\\sum_{\\pi(i)=a}","kind":"definition-passage","excerpt":"[Compatible packet aggregation] Suppose (P_i)_i∈ I_ f and (overline P_a)_a∈ I_ c resolve the fine and coarse spaces. A surjection π:I_ f↠ I_ c is compatible with C when overline P_aC = C∑_π(i)=aP_i (a∈ I_ c).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Interaction tree evaluation] Let A=(A_m)_m≥1 be a family of multilinear maps A_m:X^m→ X. A rooted interaction tree is defined recursively. A leaf bullet has evaluation V_bullet=I_X. If τ=B_+(τ_1,…,τ_m) is formed by adjoining a root of arity m, define V_τ(x_1,…,x_|τ|) = A_m( V_τ_1( x_1),…, V_τ_m( x_m) ), where |τ| denotes the number of leaves and t","count":21},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fa0b3e595fc9f5b4","tex":"\\bigoplus_{i\\in I}","aliases":"⊕_i∈ I bigoplus in","display":"\\bigoplus_{i\\in I}","kind":"definition-passage","excerpt":"[Packet resolution] A packet resolution of a vector space X is a finite family of projections (P_i)_i∈ I satisfying P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. Thus X=⊕_i∈ IX_i, X_i=P_iX. When X is a Hilbert space, the principal norm identities will assume that the P_i are orthogonal.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"definition-passage","excerpt":"[Best quotient error] For a fine interaction A_m^ f, define ε_m^⋆ = inf_B_m∈ L^m(X_ c;X_ c) normCA_m^ f-B_mC^× m.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Resolved multilinear block and hyperedge] For a multilinear map A:X^m→ Y of order m, define A_imid j_1,…,j_m = Q_iA(P_j_1 ·,…,P_j_m ·). The ordered incidence (j_1,…,j_m)⟶ i is the corresponding directed interaction hyperedge. We call the hyperedge active when the block in [source reference] is nonzero.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f7aaf260310284bd","tex":"a^{\\fine,(m)}_{i;j_1,\\ldots,j_m}","aliases":"a^fine,(m)_i;j_1,…,j_m a^ f,(m)_i;j_1,…,j_m fine ldots mathrm ldots","display":"a^{{\\mathrm f},(m)}_{i;j_1,\\ldots,j_m}","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":25},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-623eedced5b39a14","tex":"C^{\\otimes m}","aliases":"C^⊗ m otimes","display":"C^{\\otimes m}","kind":"introduction-passage","excerpt":"Here C^× m(x_1,…,x_m) =(Cx_1,…,Cx_m). We write C^⊗ m for the induced linear map between algebraic tensor powers.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1213c40d57a76ce1","tex":"C_{10}","aliases":"C_10","display":"C_{10}","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":9},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c5c4e4887f9bb904","tex":"C_{a,a-1}","aliases":"C_a,a-1","display":"C_{a,a-1}","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-73850e5fc882aa8c","tex":"C_{ab}","aliases":"C_ab","display":"C_{ab}","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3665595b7e415381","tex":"C_{b+1,b}","aliases":"C_b+1,b","display":"C_{b+1,b}","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3fbe1e6538d2100d","tex":"C_{L,L-1}","aliases":"C_L,L-1","display":"C_{L,L-1}","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b66848dfe6155c4c","tex":"\\cC","aliases":"cC C mathcal","display":"{\\mathcal C}","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b26fff0a6d0be25b","tex":"\\cL_{\\coarse}","aliases":"cL_coarse L_ c cL coarse mathcal mathrm","display":"{\\mathcal L}_{{\\mathrm c}}","kind":"introduction-passage","excerpt":"d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pu","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f3560c5982ef49bd","tex":"\\cL_{\\fine}","aliases":"cL_fine L_ f cL fine mathcal mathrm","display":"{\\mathcal L}_{{\\mathrm f}}","kind":"introduction-passage","excerpt":"d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition is generator intertwining C L_ f= L_ c C for the pu","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd d mathrm","display":"{\\,\\mathrm d}","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f854ffb0431207da","tex":"\\dot x","aliases":"dot x dot","display":"\\dot x","kind":"introduction-passage","excerpt":"The quotient problem. Let X_ f be a fine state space, let X_ c be a coarse state space, and let C:X_ f↠ X_ c be a surjective linear map. A fine trajectory satisfies dot x(t)=F_ f(x(t)), x(0)=x_0, and the coarse observable of interest is z(t)=Cx(t). The first question is whether z(t) obeys a closed equation of its own. Equivalently, can we define a ","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-833b0079d665b3b1","tex":"e_{j_1}","aliases":"e_j_1","display":"e_{j_1}","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c2b0b8ba7770dbf5","tex":"e_{j_m}","aliases":"e_j_m","display":"e_{j_m}","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":10},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b314e2300364246a","tex":"f_a","aliases":"f_a","display":"f_a","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-eac97e3f154fa8f1","tex":"f_{\\pi(i)}","aliases":"f_π(i) pi","display":"f_{\\pi(i)}","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":13},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7d905bc6806c132f","tex":"\\one","aliases":"one 1 mathbf","display":"{\\mathbf 1}","kind":"introduction-passage","excerpt":"Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.","count":11},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b9f1bf76a072652d","tex":"\\one^T","aliases":"one^T 1^T one mathbf","display":"{\\mathbf 1}^T","kind":"introduction-passage","excerpt":"Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.","count":10},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":17},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R mathbb","display":"{\\mathbb R}","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":12},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-12e365b1aab42166","tex":"\\R^n","aliases":"R^n R mathbb","display":"{\\mathbb R}^n","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0d726011beb97417","tex":"\\R^q","aliases":"R^q R mathbb","display":"{\\mathbb R}^q","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-61bdbf136c2dac84","tex":"R_L","aliases":"R_L","display":"R_L","kind":"introduction-passage","excerpt":"Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.","count":46},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"Defect in the observation intertwining law. Let L_ f and L_ c be positive linear observation operators, and choose a scale s>0. On a finite simplex, we write N(L) for normalization with respect to the total mass 1^T. Define R_L = CL_ f-sL_ cC.","count":43},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fc7a5c603b95f30d","tex":"\\tr","aliases":"tr operatorname","display":"{\\operatorname{tr}}","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"introduction-passage","excerpt":"[Exact quotient flow] Let U_ f⊆ X_ f and U_ c⊆ X_ c be open, with C(U_ f)⊆ U_ c. Suppose F_ f and F_ c are locally Lipschitz and satisfy CF_ f=F_ cC on U_ f. Let Φ_t^ f(x) and Φ_t^ c(z) denote the unique maximal local flows. Whenever both sides are defined, CΦ_t^ f(x) = Φ_t^ c(Cx).","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-41ad68b4c5be8b71","tex":"U_{\\coarse}","aliases":"U_coarse U_ c coarse mathrm","display":"U_{{\\mathrm c}}","kind":"introduction-passage","excerpt":"[Exact quotient flow] Let U_ f⊆ X_ f and U_ c⊆ X_ c be open, with C(U_ f)⊆ U_ c. Suppose F_ f and F_ c are locally Lipschitz and satisfy CF_ f=F_ cC on U_ f. Let Φ_t^ f(x) and Φ_t^ c(z) denote the unique maximal local flows. Whenever both sides are defined, CΦ_t^ f(x) = Φ_t^ c(Cx).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-20c5e01660e7b7d8","tex":"U_{\\fine}","aliases":"U_fine U_ f fine mathrm","display":"U_{{\\mathrm f}}","kind":"introduction-passage","excerpt":"[Exact quotient flow] Let U_ f⊆ X_ f and U_ c⊆ X_ c be open, with C(U_ f)⊆ U_ c. Suppose F_ f and F_ c are locally Lipschitz and satisfy CF_ f=F_ cC on U_ f. Let Φ_t^ f(x) and Φ_t^ c(z) denote the unique maximal local flows. Whenever both sides are defined, CΦ_t^ f(x) = Φ_t^ c(Cx).","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ae58458e74e50c6f","tex":"W_t","aliases":"W_t","display":"W_t","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"introduction-passage","excerpt":"The quotient problem. Let X_ f be a fine state space, let X_ c be a coarse state space, and let C:X_ f↠ X_ c be a surjective linear map. A fine trajectory satisfies dot x(t)=F_ f(x(t)), x(0)=x_0, and the coarse observable of interest is z(t)=Cx(t). The first question is whether z(t) obeys a closed equation of its own. Equivalently, can we define a ","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8f1384780cc6224d","tex":"X_L","aliases":"X_L","display":"X_L","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ff1f31d326f1e92c","tex":"X_{L-1}","aliases":"X_L-1","display":"X_{L-1}","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"introduction-passage","excerpt":"Here C^× m(x_1,…,x_m) =(Cx_1,…,Cx_m). We write C^⊗ m for the induced linear map between algebraic tensor powers.","count":12},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9d61d79dfd3ce863","tex":"\\Delta_m^{ab}","aliases":"Δ_m^ab Delta","display":"\\Delta_m^{ab}","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0ba1798c3eae7df4","tex":"\\sigma^T","aliases":"σ^T sigma","display":"\\sigma^T","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"introduction-passage","excerpt":"Stochastic interaction quotients. For a diffusion d X_t=b(X_t) d t+σ(X_t) d W_t, the generator is Lφ =Dφ[b] +frac12tr(σσ^TD^2φ). Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges. Let C denote the pullback or averaging map on observables induced by the aggregation. The natural exact condition i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2f29622fb484623a","tex":"\\sum_{i=1}^n","aliases":"∑_i=1^n sum","display":"\\sum_{i=1}^n","kind":"introduction-passage","excerpt":"Let I_ f={1,…,n}, I_ c={1,…,q}, and let π:I_ f↠ I_ c. Write (e_i) and (f_a) for the standard bases of R^n and R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"introduction-passage","excerpt":"[Exact quotient flow] Let U_ f⊆ X_ f and U_ c⊆ X_ c be open, with C(U_ f)⊆ U_ c. Suppose F_ f and F_ c are locally Lipschitz and satisfy CF_ f=F_ cC on U_ f. Let Φ_t^ f(x) and Φ_t^ c(z) denote the unique maximal local flows. Whenever both sides are defined, CΦ_t^ f(x) = Φ_t^ c(Cx).","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","excerpt":"Here C^× m(x_1,…,x_m) =(Cx_1,…,Cx_m). We write C^⊗ m for the induced linear map between algebraic tensor powers.","count":17},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"Refinement towers. Let X_LxrightarrowC_L,L-1X_L-1 ⟶⋯⟶ X_1xrightarrowC_10X_0. For a>b, write C_ab=C_b+1,b⋯ C_a,a-1. Let Δ_m^ab denote the interaction defect for the composite map C_ab. Iterating [source reference] yields a telescoping formula.","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-604a7f9090666cd2","tex":"a^{\\coarse,(m)}_{a;\\mathbf b}","aliases":"a^coarse,(m)_a; b a^ c,(m)_a; b coarse mathbf mathrm mathbf","display":"a^{{\\mathrm c},(m)}_{a;\\mathbf b}","kind":"use","excerpt":"put a and coarse input tuple b: [label=arabic*.,leftmargin=2.3em] item Evaluate g^(m)_a; j for all j∈ G_ b. item In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance. item When the exact test passes, assign the common value to a^ c,(m)_a; b. When ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-adea2289450ac2bd","tex":"a^{\\coarse,(m)}_{a;b_1,\\ldots,b_m}","aliases":"a^coarse,(m)_a;b_1,…,b_m a^ c,(m)_a;b_1,…,b_m coarse ldots mathrm ldots","display":"a^{{\\mathrm c},(m)}_{a;b_1,\\ldots,b_m}","kind":"use","excerpt":"The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e529bbe36e7ddc4e","tex":"a^{\\coarse,\\mathrm{LS}}_{a;\\mathbf b}","aliases":"a^coarse,LS_a; b a^ c,LS_a; b coarse mathrm mathbf mathrm mathrm mathbf","display":"a^{{\\mathrm c},\\mathrm{LS}}_{a;\\mathbf b}","kind":"use","excerpt":"Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-bcf47fddb213c9c5","tex":"a^{\\coarse,\\infty}_{a;\\mathbf b}","aliases":"a^coarse,∞_a; b a^ c,∞_a; b coarse infty mathbf mathrm infty mathbf","display":"a^{{\\mathrm c},\\infty}_{a;\\mathbf b}","kind":"use","excerpt":"r total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with optimal scalar fiber error ε^∞_a; b = frac12 ( max_ j∈ G_ bg_a; j - min_ j∈ G_ bg_a; j ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-70c30936c2a44280","tex":"a^{\\fine,(1)}_{i;j}","aliases":"a^fine,(1)_i;j a^ f,(1)_i;j fine mathrm","display":"a^{{\\mathrm f},(1)}_{i;j}","kind":"use","excerpt":"[Unary aggregate rate criterion] For m=1, exact descent is equivalent to ∑_i:π(i)=aa^ f,(1)_i;j = ∑_i:π(i)=aa^ f,(1)_i;j' for all coarse outputs a whenever π(j)=π(j').","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-02fde2c385d604a8","tex":"a^{\\fine,(1)}_{i;j'}","aliases":"a^fine,(1)_i;j' a^ f,(1)_i;j' fine mathrm","display":"a^{{\\mathrm f},(1)}_{i;j'}","kind":"use","excerpt":"[Unary aggregate rate criterion] For m=1, exact descent is equivalent to ∑_i:π(i)=aa^ f,(1)_i;j = ∑_i:π(i)=aa^ f,(1)_i;j' for all coarse outputs a whenever π(j)=π(j').","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5db35e391792c65c","tex":"a^{\\fine,(2)}_{i;j',k'}","aliases":"a^fine,(2)_i;j',k' a^ f,(2)_i;j',k' fine mathrm","display":"a^{{\\mathrm f},(2)}_{i;j',k'}","kind":"use","excerpt":"[Quadratic hypergraph criterion] For m=2, exact descent is equivalent to ∑_i:π(i)=aa^ f,(2)_i;j,k = ∑_i:π(i)=aa^ f,(2)_i;j',k' whenever π(j)=π(j'), π(k)=π(k'). The resulting coarse interaction is the directed quadratic hyperedge (π(j),π(k))⟶ a.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-68848f551b2d339c","tex":"a^{\\fine,(2)}_{i;j,k}","aliases":"a^fine,(2)_i;j,k a^ f,(2)_i;j,k fine mathrm","display":"a^{{\\mathrm f},(2)}_{i;j,k}","kind":"use","excerpt":"[Quadratic hypergraph criterion] For m=2, exact descent is equivalent to ∑_i:π(i)=aa^ f,(2)_i;j,k = ∑_i:π(i)=aa^ f,(2)_i;j',k' whenever π(j)=π(j'), π(k)=π(k'). The resulting coarse interaction is the directed quadratic hyperedge (π(j),π(k))⟶ a.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f025b702363c281d","tex":"A_1^{\\fine}","aliases":"A_1^fine A_1^ f fine mathrm","display":"A_1^{{\\mathrm f}}","kind":"use","excerpt":"Why hypergraphs appear. Suppose a local analytic vector field has the expansion F_ f(x)=∑_m≥0A_m^ f[x,…,x]. The linear coefficient A_1^ f moves one input packet into one output packet. The quadratic coefficient A_2^ f combines two input packets before producing an output. In general, A_m^ f combines m inputs. We record this incidence by the directe","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8c58b3b1b6a4ef3d","tex":"A_2^{\\fine}","aliases":"A_2^fine A_2^ f fine mathrm","display":"A_2^{{\\mathrm f}}","kind":"use","excerpt":"Why hypergraphs appear. Suppose a local analytic vector field has the expansion F_ f(x)=∑_m≥0A_m^ f[x,…,x]. The linear coefficient A_1^ f moves one input packet into one output packet. The quadratic coefficient A_2^ f combines two input packets before producing an output. In general, A_m^ f combines m inputs. We record this incidence by the directe","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-625a1388e0673c7b","tex":"A_{\\coarse}","aliases":"A_coarse A_ c coarse mathrm","display":"A_{{\\mathrm c}}","kind":"use","excerpt":"[Linear semigroup naturality] If CA_ f=A_ cC, then Ce^tA_ f = e^tA_ cC for every t.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6f08533656136165","tex":"A_{\\coarse}^k","aliases":"A_coarse^k A_ c^k coarse mathrm","display":"A_{{\\mathrm c}}^k","kind":"use","excerpt":"Either apply [source reference] to the linear systems or sum the exponential series using CA_ f^k=A_ c^kC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0d2f7e0091af126e","tex":"A_{\\fine}","aliases":"A_fine A_ f fine mathrm","display":"A_{{\\mathrm f}}","kind":"use","excerpt":"[Linear semigroup naturality] If CA_ f=A_ cC, then Ce^tA_ f = e^tA_ cC for every t.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7abc599070e8825a","tex":"A_{\\fine}^k","aliases":"A_fine^k A_ f^k fine mathrm","display":"A_{{\\mathrm f}}^k","kind":"use","excerpt":"Either apply [source reference] to the linear systems or sum the exponential series using CA_ f^k=A_ c^kC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-da782776f66f7454","tex":"A_m^{(0)}","aliases":"A_m^(0)","display":"A_m^{(0)}","kind":"use","excerpt":"[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-aa0ef414e34766c1","tex":"A_m^{(1)}","aliases":"A_m^(1)","display":"A_m^{(1)}","kind":"use","excerpt":"[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0857cf6547d42a6f","tex":"A_m^{(2)}","aliases":"A_m^(2)","display":"A_m^{(2)}","kind":"use","excerpt":"[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2fe210c4a5e26f3d","tex":"A_m^c","aliases":"A_m^c","display":"A_m^c","kind":"use","excerpt":"CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7a2a8dafd95fd60b","tex":"A_m^f","aliases":"A_m^f","display":"A_m^f","kind":"use","excerpt":"CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b5234122a1e41665","tex":"A_m^{\\mathrm{keep}}","aliases":"A_m^keep mathrm","display":"A_m^{\\mathrm{keep}}","kind":"use","excerpt":"Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-61ec922a7b66b76d","tex":"A_{PP}","aliases":"A_PP","display":"A_{PP}","kind":"use","excerpt":"Coarse dynamics with memory. Eliminating hidden packets generally produces memory. The target form is dot z(t) =A_PPz(t) +∫_0^tK(t-s)z(s) d s +η(t). The quotient theory should allow a fine system without memory to produce coarse dynamics with memory. Volterra hyperedges and fractional kernels would then be treated as native coarse interactions rath","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"Write a= 1^Tu and b= 1^Tv. Then normfracua-fracvb_1 · ≤ fracnormu-v_1a + normv_1absfrac1a-frac1b · = fracnormu-v_1a + fracabsa-ba · ≤ frac2normu-v_1a. Interchanging u and v gives the same inequality with b in the denominator. Taking the smaller of the two upper bounds yields the denominator max{a,b}.","count":12},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-dda3f4b7df54c019","tex":"\\mathbf b","aliases":"b mathbf","display":"\\mathbf b","kind":"use","excerpt":"Exact test for every interaction order. For a tensor of order m in scalar packet coordinates, compute the aggregated coefficient array g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m. For every coarse input tuple b=(b_1,…,b_m), define the fine fiber product G_ b = {(j_1,…,j_m):π(j_r)=b_r for all r}.","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d1e4f21b86ad3668","tex":"\\mathsf B","aliases":"B mathsf","display":"\\mathsf B","kind":"use","excerpt":"Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e0f22da8d43f9d13","tex":"\\mathsf B_c","aliases":"B_c mathsf","display":"\\mathsf B_c","kind":"use","excerpt":"CL_f=sL_cC Longrightarrow C B_f= B_cC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2fe4d718fe2a1fd7","tex":"\\mathsf B_f","aliases":"B_f mathsf","display":"\\mathsf B_f","kind":"use","excerpt":"CL_f=sL_cC Longrightarrow C B_f= B_cC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fc9ad75de14545d5","tex":"b_m","aliases":"b_m","display":"b_m","kind":"use","excerpt":"The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ac0d31a2615b8b66","tex":"b_r","aliases":"b_r","display":"b_r","kind":"use","excerpt":"The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m, π(j_r)=b_r.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-19907c873f5b18b9","tex":"\\mathsf B_y","aliases":"B_y mathsf","display":"\\mathsf B_y","kind":"use","excerpt":"Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-764e8d95e30efda4","tex":"\\mathsf B_y^c","aliases":"B_y^c mathsf","display":"\\mathsf B_y^c","kind":"use","excerpt":"CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-67226c42992d42a2","tex":"\\mathsf B_y^f","aliases":"B_y^f mathsf","display":"\\mathsf B_y^f","kind":"use","excerpt":"CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6a5dfc6b404e90b1","tex":"\\Bup_k^{\\coarse}","aliases":"Bup_k^coarse B_k^ c Bup coarse mathsf mathrm","display":"{\\mathsf B}_k^{{\\mathrm c}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-077f284eb8449c38","tex":"\\Bup_k^{\\fine}","aliases":"Bup_k^fine B_k^ f Bup fine mathsf mathrm","display":"{\\mathsf B}_k^{{\\mathrm f}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2fd8ead0e883432f","tex":"\\Bup_{\\lambda^{\\coarse}}","aliases":"Bup_λ^coarse B_λ^ c Bup lambda coarse mathsf lambda mathrm","display":"{\\mathsf B}_{\\lambda^{{\\mathrm c}}}","kind":"use","excerpt":"Classical sufficiency theory asks whether a statistic preserves all information relevant to a chosen observation family [source reference]. Here the operator question is C B_λ^ f = B_λ^ cC. When both the flow identity [source reference] and the Bayesian identity [source reference] hold at each observation time, the whole filtering sequence commutes","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f81b2f7910afaade","tex":"\\Bup_{\\lambda^{\\fine}}","aliases":"Bup_λ^fine B_λ^ f Bup lambda fine mathsf lambda mathrm","display":"{\\mathsf B}_{\\lambda^{{\\mathrm f}}}","kind":"use","excerpt":"Classical sufficiency theory asks whether a statistic preserves all information relevant to a chosen observation family [source reference]. Here the operator question is C B_λ^ f = B_λ^ cC. When both the flow identity [source reference] and the Bayesian identity [source reference] hold at each observation time, the whole filtering sequence commutes","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-348a98c5c7180f04","tex":"\\Bup_{\\lambda_1}","aliases":"Bup_λ_1 B_λ_1 Bup lambda mathsf lambda","display":"{\\mathsf B}_{\\lambda_1}","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4973ebaf5354fa81","tex":"\\Bup_{\\lambda_1\\odot\\lambda_2}","aliases":"Bup_λ_1odotλ_2 B_λ_1odotλ_2 Bup lambda odot lambda mathsf lambda odot lambda","display":null,"kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3d54bb9870c356bf","tex":"\\Bup_{\\lambda_2}","aliases":"Bup_λ_2 B_λ_2 Bup lambda mathsf lambda","display":"{\\mathsf B}_{\\lambda_2}","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0225c5fbdfbee211","tex":"\\Bup_{\\lambda_{y_1}^{\\coarse}}","aliases":"Bup_λ_y_1^coarse B_λ_y_1^ c Bup lambda coarse mathsf lambda mathrm","display":"{\\mathsf B}_{\\lambda_{y_1}^{{\\mathrm c}}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7c5aea1c51381435","tex":"\\Bup_{\\lambda_{y_1}^{\\fine}}","aliases":"Bup_λ_y_1^fine B_λ_y_1^ f Bup lambda fine mathsf lambda mathrm","display":"{\\mathsf B}_{\\lambda_{y_1}^{{\\mathrm f}}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-cb2ea184e3128737","tex":"\\Bup_{\\lambda_{y_k}^{\\coarse}}","aliases":"Bup_λ_y_k^coarse B_λ_y_k^ c Bup lambda coarse mathsf lambda mathrm","display":"{\\mathsf B}_{\\lambda_{y_k}^{{\\mathrm c}}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f273a927df8f6dcd","tex":"\\Bup_{\\lambda_{y_k}^{\\fine}}","aliases":"Bup_λ_y_k^fine B_λ_y_k^ f Bup lambda fine mathsf lambda mathrm","display":"{\\mathsf B}_{\\lambda_{y_k}^{{\\mathrm f}}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-01f1875175efe4ca","tex":"\\Bup_{\\lambda_{y_N}^{\\coarse}}","aliases":"Bup_λ_y_N^coarse B_λ_y_N^ c Bup lambda coarse mathsf lambda mathrm","display":"{\\mathsf B}_{\\lambda_{y_N}^{{\\mathrm c}}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-10faed40df8bda99","tex":"\\Bup_{\\lambda_{y_N}^{\\fine}}","aliases":"Bup_λ_y_N^fine B_λ_y_N^ f Bup lambda fine mathsf lambda mathrm","display":"{\\mathsf B}_{\\lambda_{y_N}^{{\\mathrm f}}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C mathbb","display":"{\\mathbb C}","kind":"use","excerpt":"Finite packet resolutions. The main algebraic results are stated for vector spaces of finite dimension over R or C. The quotient and tensor factorizations are then purely algebraic. The later analytic results add the stated convergence conditions and the usual assumptions for local uniqueness.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"Certificate pipeline. The reference implementation follows the dependency chain c interaction tensors downarrow fiber constancy and kernel tests downarrow coarse tensor and Δ_m downarrow d_F(R) and d_Φ,k downarrow r_λ,k,z_*,k,d_ B,k downarrow e_k^-,e_k^+, E_N. Each numerical record stores the theorem identifier, the inputs, the residual, the tolera","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-dd0b3056e758361c","tex":"C^r","aliases":"C^r","display":"C^r","kind":"use","excerpt":"[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c94fd79f2de7324b","tex":"C^T","aliases":"C^T","display":"C^T","kind":"use","excerpt":"[Sufficient condition for Bayesian refinement] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).","count":10},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-14630b35564e9682","tex":"C^{\\times |\\tau|}","aliases":"C^× |τ| times tau","display":"C^{\\times |\\tau|}","kind":"use","excerpt":"[Treewise quotient naturality] Suppose every operation in a fine family descends: CA_m^ f=A_m^ cC^× m (m≥1). Then, for every rooted interaction tree τ, C V_τ^ f = V_τ^ cC^× |τ|.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b74f814ce5515d92","tex":"C^{\\times|\\tau_1|}","aliases":"C^×|τ_1| times tau","display":"C^{\\times|\\tau_1|}","kind":"use","excerpt":"Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2aced05342558800","tex":"C^{\\times|\\tau_m|}","aliases":"C^×|τ_m| times tau","display":"C^{\\times|\\tau_m|}","kind":"use","excerpt":"Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-64c8813f26a141de","tex":"C^{\\times|\\tau|}","aliases":"C^×|τ| times tau","display":"C^{\\times|\\tau|}","kind":"use","excerpt":"Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-929d999724cbd9e0","tex":"C_{00}","aliases":"C_00","display":"C_{00}","kind":"use","excerpt":"[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-285255642c1c101e","tex":"C_{10}^{\\times m}","aliases":"C_10^× m times","display":"C_{10}^{\\times m}","kind":"use","excerpt":"Compose the two identities: C_20A_m^(2) =C_10A_m^(1)C_21^× m =A_m^(0)C_10^× mC_21^× m =A_m^(0)C_20^× m.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d287347ae0146da3","tex":"C_{20}","aliases":"C_20","display":"C_{20}","kind":"use","excerpt":"Composition across refinement levels. Let X_2xrightarrowC_21X_1xrightarrowC_10X_0, C_20=C_10C_21.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5a766175b2cbc428","tex":"C_{20}^{\\times m}","aliases":"C_20^× m times","display":"C_{20}^{\\times m}","kind":"use","excerpt":"Compose the two identities: C_20A_m^(2) =C_10A_m^(1)C_21^× m =A_m^(0)C_10^× mC_21^× m =A_m^(0)C_20^× m.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-88a3e18f148a2a4a","tex":"C_{21}","aliases":"C_21","display":"C_{21}","kind":"use","excerpt":"Composition across refinement levels. Let X_2xrightarrowC_21X_1xrightarrowC_10X_0, C_20=C_10C_21.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9d236732790e1137","tex":"C_{21}^{\\times m}","aliases":"C_21^× m times","display":"C_{21}^{\\times m}","kind":"use","excerpt":"Compose the two identities: C_20A_m^(2) =C_10A_m^(1)C_21^× m =A_m^(0)C_10^× mC_21^× m =A_m^(0)C_20^× m.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-946c8845d71b0288","tex":"C_{L,\\ell+1}","aliases":"C_L,ℓ+1 ell","display":"C_{L,\\ell+1}","kind":"use","excerpt":"[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ce48dfd430b374b0","tex":"C_{L,\\ell+1}^{\\times m}","aliases":"C_L,ℓ+1^× m ell times","display":"C_{L,\\ell+1}^{\\times m}","kind":"use","excerpt":"[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b8940f585abf652e","tex":"C_{LL}","aliases":"C_LL","display":"C_{LL}","kind":"use","excerpt":"[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-26ab6cfd3e093f21","tex":"C_{\\ell0}","aliases":"C_ℓ0 ell","display":"C_{\\ell0}","kind":"use","excerpt":"[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7c6b86b445085023","tex":"\\cE","aliases":"cE E mathcal","display":"{\\mathcal E}","kind":"use","excerpt":"Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-79734b861430b2dd","tex":"\\cE_m^{\\mathrm{keep}}","aliases":"cE_m^keep E_m^keep cE mathrm mathcal mathrm","display":"{\\mathcal E}_m^{\\mathrm{keep}}","kind":"use","excerpt":"Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3ce593d12a592ef7","tex":"\\cG","aliases":"cG G mathcal","display":"{\\mathcal G}","kind":"use","excerpt":"Exact test for every interaction order. For a tensor of order m in scalar packet coordinates, compute the aggregated coefficient array g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m. For every coarse input tuple b=(b_1,…,b_m), define the fine fiber product G_ b = {(j_1,…,j_m):π(j_r)=b_r for all r}.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-899d278126ab94f6","tex":"\\cG_{\\mathbf b}","aliases":"cG_ b G_ b cG mathbf mathcal mathbf","display":"{\\mathcal G}_{\\mathbf b}","kind":"use","excerpt":"Exact test for every interaction order. For a tensor of order m in scalar packet coordinates, compute the aggregated coefficient array g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m. For every coarse input tuple b=(b_1,…,b_m), define the fine fiber product G_ b = {(j_1,…,j_m):π(j_r)=b_r for all r}.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-581d8c822b2d8c56","tex":"\\operatorname{Complexity}","aliases":"Complexity operatorname","display":"\\operatorname{Complexity}","kind":"use","excerpt":"Approximate joint design. A finite approximate problem is min_C,{A_m^ c},{λ_y^ c} [ ∑_mw_mnormCA_m^ f-A_m^ cC^× m +∑_yv_yd_ B,y +γComplexity(C) ]. The main questions concern existence, tractable relaxations, and stability of the minimizers. Another question is whether the task weights can be learned without invalidating the certificate after a dist","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-95243eedd9aae923","tex":"\\operatorname{Cost}","aliases":"Cost operatorname","display":"\\operatorname{Cost}","kind":"use","excerpt":"A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-20d41ee7317fddd1","tex":"\\cS","aliases":"cS S mathcal","display":"{\\mathcal S}","kind":"use","excerpt":"Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9a83a0c6090a2a95","tex":"\\cS_{\\coarse}","aliases":"cS_coarse S_ c cS coarse mathcal mathrm","display":"{\\mathcal S}_{{\\mathrm c}}","kind":"use","excerpt":"Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a329cc74d6f5ec9d","tex":"\\cS_{\\fine}","aliases":"cS_fine S_ f cS fine mathcal mathrm","display":"{\\mathcal S}_{{\\mathrm f}}","kind":"use","excerpt":"Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-39f674872188cfed","tex":"\\cV_{\\tau}^{\\coarse}","aliases":"cV_τ^coarse V_τ^ c cV tau coarse mathcal tau mathrm","display":"{\\mathcal V}_{\\tau}^{{\\mathrm c}}","kind":"use","excerpt":"[Treewise quotient naturality] Suppose every operation in a fine family descends: CA_m^ f=A_m^ cC^× m (m≥1). Then, for every rooted interaction tree τ, C V_τ^ f = V_τ^ cC^× |τ|.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-363907bdbebc0f2f","tex":"\\cV_{\\tau}^{\\fine}","aliases":"cV_τ^fine V_τ^ f cV tau fine mathcal tau mathrm","display":"{\\mathcal V}_{\\tau}^{{\\mathrm f}}","kind":"use","excerpt":"[Treewise quotient naturality] Suppose every operation in a fine family descends: CA_m^ f=A_m^ cC^× m (m≥1). Then, for every rooted interaction tree τ, C V_τ^ f = V_τ^ cC^× |τ|.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-080e31e680eb1240","tex":"\\cV_{\\tau_1}^{\\coarse}","aliases":"cV_τ_1^coarse V_τ_1^ c cV tau coarse mathcal tau mathrm","display":"{\\mathcal V}_{\\tau_1}^{{\\mathrm c}}","kind":"use","excerpt":"Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c1efc2e2cd0a00cc","tex":"\\cV_{\\tau_1}^{\\fine}","aliases":"cV_τ_1^fine V_τ_1^ f cV tau fine mathcal tau mathrm","display":"{\\mathcal V}_{\\tau_1}^{{\\mathrm f}}","kind":"use","excerpt":"Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5474996921858ed1","tex":"\\cV_{\\tau_m}^{\\coarse}","aliases":"cV_τ_m^coarse V_τ_m^ c cV tau coarse mathcal tau mathrm","display":"{\\mathcal V}_{\\tau_m}^{{\\mathrm c}}","kind":"use","excerpt":"Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-72a61a6989d120f0","tex":"\\cV_{\\tau_m}^{\\fine}","aliases":"cV_τ_m^fine V_τ_m^ f cV tau fine mathcal tau mathrm","display":"{\\mathcal V}_{\\tau_m}^{{\\mathrm f}}","kind":"use","excerpt":"Proceed by induction on the number of internal vertices. The leaf case is CI=IC. Suppose τ=B_+(τ_1,…,τ_m). Using interaction descent at the root and the inductive hypothesis on every subtree, C V_τ^ f · = CA_m^ f ( V_τ_1^ f,…, V_τ_m^ f ) · = A_m^ c ( C V_τ_1^ f,…, C V_τ_m^ f ) · = A_m^ c ( V_τ_1^ cC^×|τ_1|,…, V_τ_m^ cC^×|τ_m| ) · = V_τ^ cC^×|τ|.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1f233c12e42a32bf","tex":"\\cY","aliases":"cY Y mathcal","display":"{\\mathcal Y}","kind":"use","excerpt":"Coarsest exact joint partition. For a finite interaction family (A_m^ f)_m≤ M and likelihood family (λ_y^ f)_y∈ Y, seek min_π|I_ c| subject to CA_m^ f=A_m^ cC^× m (m≤ M), and λ_y^ f=s_yC^Tλ_y^ c (y∈ Y). The partition determined by the observations alone groups states with identical likelihood profiles. The dynamical condition also depends on the pa","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8192ef4226385b14","tex":"\\mathfrak d","aliases":"d mathfrak","display":"\\mathfrak d","kind":"use","excerpt":"[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).","count":9},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2f80fcde2a727a6d","tex":"D^k","aliases":"D^k","display":"D^k","kind":"use","excerpt":"[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-09e71d50145ecad3","tex":"D^r","aliases":"D^r","display":"D^r","kind":"use","excerpt":"[Bayesian jet naturality] Under the hypotheses of [source reference], for every r≥1 and every interior fine prior, C D^r B_λ^ f(μ) [h_1,…,h_r] = D^r B_λ^ c(Cμ) [Ch_1,…,Ch_r].","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-582134d43addcd3d","tex":"d_{\\Bup}","aliases":"d_Bup d_ B Bup mathsf","display":"d_{{\\mathsf B}}","kind":"use","excerpt":"_ f=Z_λ^ f(μ), z_ c=sZ_λ^ c(Cμ). Then norm C B_λ^ f(μ) - B_λ^ c(Cμ) _1 ≤ frac2∑_iμ_iabsr_λ,i max{z_ f,z_ c}. If both evidences are at least z_*>0, then d_ B ≤ frac2normr_λ_∞z_*.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7c0b6c5549189b74","tex":"d_{\\Bup,y}","aliases":"d_Bup,y d_ B,y Bup mathsf","display":"d_{{\\mathsf B},y}","kind":"use","excerpt":"A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability ","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c811b35a7be62dfc","tex":"\\mathfrak d_F","aliases":"d_F mathfrak","display":"\\mathfrak d_F","kind":"use","excerpt":"[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d061988375f3cc3d","tex":"\\mathfrak d_{F,k}","aliases":"d_F,k mathfrak","display":"\\mathfrak d_{F,k}","kind":"use","excerpt":"From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the obser","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3b6b37c3ee83ea6f","tex":"\\diag","aliases":"diag operatorname","display":"{\\operatorname{diag}}","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5e578fe4099097cb","tex":"\\dot x_1","aliases":"dot x_1 dot","display":"\\dot x_1","kind":"use","excerpt":"h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x), dot x_2 · =-ax_2+bx_4-h_12(x), dot x_3 · = ax_1-bx_3+h_34(x), dot x_4 · = ax_2-bx_4-h_34(x).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0d433a1e80a05a48","tex":"\\dot x_2","aliases":"dot x_2 dot","display":"\\dot x_2","kind":"use","excerpt":"h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x), dot x_2 · =-ax_2+bx_4-h_12(x), dot x_3 · = ax_1-bx_3+h_34(x), dot x_4 · = ax_2-bx_4-h_34(x).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f9753f0a556bf01b","tex":"\\dot x_3","aliases":"dot x_3 dot","display":"\\dot x_3","kind":"use","excerpt":"h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x), dot x_2 · =-ax_2+bx_4-h_12(x), dot x_3 · = ax_1-bx_3+h_34(x), dot x_4 · = ax_2-bx_4-h_34(x).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d1baa7c0891f9bd0","tex":"\\dot x_4","aliases":"dot x_4 dot","display":"\\dot x_4","kind":"use","excerpt":"h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x), dot x_2 · =-ax_2+bx_4-h_12(x), dot x_3 · = ax_1-bx_3+h_34(x), dot x_4 · = ax_2-bx_4-h_34(x).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-389bd177d57e5191","tex":"\\dot y","aliases":"dot y dot","display":"\\dot y","kind":"use","excerpt":"Let y(t)=Cx(t). From [source reference], dot y(t)=F_ c(y(t))+δ_F(x(t)). Subtract the integral equations for y and z: y(t)-z(t) =y(0)-z(0) +∫_0^t (F_ c(y(s))-F_ c(z(s))) d s +∫_0^tδ_F(x(s)) d s. Taking norms and applying the Grönwall inequality yields [source reference]. The uniform cases follow by evaluating the exponential integral.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-825f9bf753046ea5","tex":"\\dot z","aliases":"dot z dot","display":"\\dot z","kind":"use","excerpt":"Let x(t)=Φ_t^ f(x_0) and set z(t)=Cx(t). Then dot z(t) =Cdot x(t) =CF_ f(x(t)) =F_ c(Cx(t)) =F_ c(z(t)), with z(0)=Cx_0. Uniqueness for the coarse initial value problem gives z(t)=Φ_t^ c(Cx_0).","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e92e798bd9f0f483","tex":"\\dot z_1","aliases":"dot z_1 dot","display":"\\dot z_1","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-97ebf6aa9dcf76cc","tex":"\\dot z_2","aliases":"dot z_2 dot","display":"\\dot z_2","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"Certificate pipeline. The reference implementation follows the dependency chain c interaction tensors downarrow fiber constancy and kernel tests downarrow coarse tensor and Δ_m downarrow d_F(R) and d_Φ,k downarrow r_λ,k,z_*,k,d_ B,k downarrow e_k^-,e_k^+, E_N. Each numerical record stores the theorem identifier, the inputs, the residual, the tolera","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2a021cb995927d44","tex":"\\Downarrow_{C_{10}}","aliases":"Downarrow_C_10 Downarrow","display":null,"kind":"use","excerpt":"[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-eb0243cd6edac5e4","tex":"\\Downarrow_{C_{20}}","aliases":"Downarrow_C_20 Downarrow","display":null,"kind":"use","excerpt":"[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fc61698778a22b1b","tex":"\\Downarrow_{C_{21}}","aliases":"Downarrow_C_21 Downarrow","display":null,"kind":"use","excerpt":"[Transitivity of exact interaction quotients] If A_m^(2)Downarrow_C_21A_m^(1) and A_m^(1)Downarrow_C_10A_m^(0), then A_m^(2)Downarrow_C_20A_m^(0).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4d4e015cc45b0e46","tex":"e^{H_m}","aliases":"e^H_m","display":"e^{H_m}","kind":"use","excerpt":"Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2f935cef1c3347b2","tex":"e^{L(t-s)}","aliases":"e^L(t-s)","display":"e^{L(t-s)}","kind":"use","excerpt":"[Approximate quotient flow bound] Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e4791e82522f365a","tex":"e^{L_{F,k}T_k}","aliases":"e^L_F,kT_k","display":"e^{L_{F,k}T_k}","kind":"use","excerpt":"From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the obser","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-bb582ca6960394af","tex":"e^{LT}","aliases":"e^LT","display":"e^{LT}","kind":"use","excerpt":"[Coefficient certificate for a flow segment] If normx(s)≤ R and Proposition~[source reference] applies, then a flow interval of length T satisfies normCΦ_T^ f(x_0)-Φ_T^ c(Cx_0) ≤ d_F(R)frace^LT-1L for L>0, with T d_F(R) at L=0.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-36e03dc97e0171a4","tex":"e^{Lt}","aliases":"e^Lt","display":"e^{Lt}","kind":"use","excerpt":"[Approximate quotient flow bound] Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5c9e0db3243a92fd","tex":"e^{tA_{\\coarse}}","aliases":"e^tA_coarse e^tA_ c coarse mathrm","display":"e^{tA_{{\\mathrm c}}}","kind":"use","excerpt":"[Linear semigroup naturality] If CA_ f=A_ cC, then Ce^tA_ f = e^tA_ cC for every t.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-744362bb3b4cb6b3","tex":"e^{tA_{\\fine}}","aliases":"e^tA_fine e^tA_ f fine mathrm","display":"e^{tA_{{\\mathrm f}}}","kind":"use","excerpt":"[Linear semigroup naturality] If CA_ f=A_ cC, then Ce^tA_ f = e^tA_ cC for every t.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c7571c010d929899","tex":"e^{tQ_{\\coarse}}","aliases":"e^tQ_coarse e^tQ_ c coarse mathrm","display":"e^{tQ_{{\\mathrm c}}}","kind":"use","excerpt":"Markov lumpability and Markov functions. For unary stochastic dynamics, the identity CQ_ f=Q_ cC is a standard exact lumping condition. It implies Ce^tQ_ f=e^tQ_ cC and means that the coarse process is Markov for every fine initial distribution compatible with the quotient. Classical treatments distinguish several forms of strong, ordinary, exact, ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-84657cfbf6c4923d","tex":"e^{tQ_{\\fine}}","aliases":"e^tQ_fine e^tQ_ f fine mathrm","display":"e^{tQ_{{\\mathrm f}}}","kind":"use","excerpt":"Markov lumpability and Markov functions. For unary stochastic dynamics, the identity CQ_ f=Q_ cC is a standard exact lumping condition. It implies Ce^tQ_ f=e^tQ_ cC and means that the coarse process is Markov for every fine initial distribution compatible with the quotient. Classical treatments distinguish several forms of strong, ordinary, exact, ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-054426076ef8f1d3","tex":"e_0^+","aliases":"e_0^+","display":"e_0^+","kind":"use","excerpt":"=Φ_k^ c(μ_k-1^ c), μ_k^ c = B_k^ c(widehatμ_k^ c). Define the discrepancies before and after each update by e_k^- =normCwidehatμ_k^ f-widehatμ_k^ c_1, e_k^+ =normCμ_k^ f-μ_k^ c_1. At the initial time, set e_0^+=normCμ_0^ f-μ_0^ c_1.","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5d1836707c2366f7","tex":"e_h","aliases":"e_h","display":"e_h","kind":"use","excerpt":"texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0fb0d2699e946acb","tex":"e_{h_1}","aliases":"e_h_1","display":"e_{h_1}","kind":"use","excerpt":"texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e7bc8c842e69d46f","tex":"e_{h_2}","aliases":"e_h_2","display":"e_{h_2}","kind":"use","excerpt":"texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-aca3cfd0359f0f4f","tex":"e_{h_M}","aliases":"e_h_M","display":"e_{h_M}","kind":"use","excerpt":"texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6ae25c917d4b485d","tex":"e_{h_r}","aliases":"e_h_r","display":"e_{h_r}","kind":"use","excerpt":"[Optimal retention of k orthogonal hyperedges] Among all retained sets of cardinality k, the set {h_1,…,h_k} minimizes the Hilbert Schmidt residual. The minimum is min_| E|=k normR_m( E)_HS^2 = ∑_r>ke_h_r.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0013e910a7dc1026","tex":"e_{k-1}^+","aliases":"e_k-1^+","display":"e_{k-1}^+","kind":"use","excerpt":"[Prediction bound for one step] Under [source reference], e_k^- ≤ L_Φ,ke_k-1^+ +d_Φ,k.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e749405022a66e6b","tex":"E_m","aliases":"E_m","display":"E_m","kind":"use","excerpt":"A dense implementation requires O(n^m+1) arithmetic to aggregate a tensor of order m on n states. If the fine tensor is stored as E_m active scalar hyperedges, the aggregation pass is O(E_m), followed by grouping and consistency checks over the represented coarse incidences.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e7aff08d6844bfde","tex":"e_N^+","aliases":"e_N^+","display":"e_N^+","kind":"use","excerpt":"[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-60e07aa0a44afc52","tex":"e_{\\eta}","aliases":"e_η eta","display":"e_{\\eta}","kind":"use","excerpt":"Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f8483dce1203449b","tex":"\\mathrm{eff}","aliases":"eff mathrm","display":"\\mathrm{eff}","kind":"use","excerpt":"Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6f6e7cf786df9d23","tex":"\\EffectiveHyperedgeCount","aliases":"EffectiveHyperedgeCount","display":null,"kind":"use","excerpt":"Concentrated hyperedge field. A synthetic interaction field with decreasing orthogonal block energies illustrates [source reference]. The computed hyperedge entropy is HyperedgeEntropyValue, corresponding to an effective count of approximately EffectiveHyperedgeCount hyperedges despite a larger ambient support. The ranked spectrum in [source refere","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4b895fe6b97e29b0","tex":"\\Err","aliases":"Err E mathfrak","display":"{\\mathfrak E}","kind":"use","excerpt":"Certificate pipeline. The reference implementation follows the dependency chain c interaction tensors downarrow fiber constancy and kernel tests downarrow coarse tensor and Δ_m downarrow d_F(R) and d_Φ,k downarrow r_λ,k,z_*,k,d_ B,k downarrow e_k^-,e_k^+, E_N. Each numerical record stores the theorem identifier, the inputs, the residual, the tolera","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-432cf432115c8a0d","tex":"\\Err_N","aliases":"Err_N E_N Err mathfrak","display":"{\\mathfrak E}_N","kind":"use","excerpt":"Certificate pipeline. The reference implementation follows the dependency chain c interaction tensors downarrow fiber constancy and kernel tests downarrow coarse tensor and Δ_m downarrow d_F(R) and d_Φ,k downarrow r_λ,k,z_*,k,d_ B,k downarrow e_k^-,e_k^+, E_N. Each numerical record stores the theorem identifier, the inputs, the residual, the tolera","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3ba3ceeb303aa491","tex":"\\ExactEvidenceResidual","aliases":"ExactEvidenceResidual","display":null,"kind":"use","excerpt":"[Exact nonlinear filtering residuals] For the deterministic run with four states, the largest aggregate prediction residual was ExactPredictionResidual. The largest posterior residual after an update was ExactPosteriorResidual, and the largest residual in a single evidence factor was ExactEvidenceResidual. The small nonzero values come from floatin","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-96e66240369ac9ba","tex":"\\ExactPosteriorResidual","aliases":"ExactPosteriorResidual","display":null,"kind":"use","excerpt":"[Exact nonlinear filtering residuals] For the deterministic run with four states, the largest aggregate prediction residual was ExactPredictionResidual. The largest posterior residual after an update was ExactPosteriorResidual, and the largest residual in a single evidence factor was ExactEvidenceResidual. The small nonzero values come from floatin","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fe72365179520deb","tex":"\\ExactPredictionResidual","aliases":"ExactPredictionResidual","display":null,"kind":"use","excerpt":"[Exact nonlinear filtering residuals] For the deterministic run with four states, the largest aggregate prediction residual was ExactPredictionResidual. The largest posterior residual after an update was ExactPosteriorResidual, and the largest residual in a single evidence factor was ExactEvidenceResidual. The small nonzero values come from floatin","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-aaec5c835f65f308","tex":"\\mathsf F","aliases":"F mathsf","display":"\\mathsf F","kind":"use","excerpt":"abeleq:lumpability-certificate-schema L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A cert","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-92cbe7d6d84c4e19","tex":"F_c","aliases":"F_c","display":"F_c","kind":"use","excerpt":"CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7e1baad8b2629e87","tex":"F_{\\coarse}^{\\circ n}","aliases":"F_coarse^∘ n F_ c^∘ n coarse circ mathrm circ","display":"F_{{\\mathrm c}}^{\\circ n}","kind":"use","excerpt":"[Quotients of discrete dynamics] If CF_ f=F_ cC, then CF_ f^∘ n = F_ c^∘ nC for every integer n≥0.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6f98b1eca815e46c","tex":"F_f","aliases":"F_f","display":"F_f","kind":"use","excerpt":"CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6d25283357a004f7","tex":"F_{\\fine}^{\\circ n}","aliases":"F_fine^∘ n F_ f^∘ n fine circ mathrm circ","display":"F_{{\\mathrm f}}^{\\circ n}","kind":"use","excerpt":"[Quotients of discrete dynamics] If CF_ f=F_ cC, then CF_ f^∘ n = F_ c^∘ nC for every integer n≥0.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a31fd9bc362269d6","tex":"\\mathsf F_N","aliases":"F_N mathsf","display":"\\mathsf F_N","kind":"use","excerpt":"abeleq:lumpability-certificate-schema L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite ledger as JSON files and CSV tables. A cert","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-35f22ec71c53feec","tex":"f_{\\pi(j_1)}","aliases":"f_π(j_1) pi","display":"f_{\\pi(j_1)}","kind":"use","excerpt":"Applying C to [source reference] gives CA_m^ f(e_j_1,…,e_j_m) = ∑_a=1^q g^(m)_a;j_1,…,j_mf_a. If a coarse tensor exists, this vector equals A_m^ c(f_π(j_1),…,f_π(j_m)), which depends only on the coarse input labels. This proves necessity. Conversely, constancy on each fiber makes [source reference] well defined. Equality on every tuple of fine basi","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e2c94021008b3afc","tex":"f_{\\pi(j_m)}","aliases":"f_π(j_m) pi","display":"f_{\\pi(j_m)}","kind":"use","excerpt":"Applying C to [source reference] gives CA_m^ f(e_j_1,…,e_j_m) = ∑_a=1^q g^(m)_a;j_1,…,j_mf_a. If a coarse tensor exists, this vector equals A_m^ c(f_π(j_1),…,f_π(j_m)), which depends only on the coarse input labels. This proves necessity. Conversely, constancy on each fiber makes [source reference] well defined. Equality on every tuple of fine basi","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"[Compatibility with Fa a di Bruno composition] Let F_ f and F_ c be smooth maps with CF_ f=F_ cC, and let G_ f and G_ c satisfy the same relation. Then each rooted tree in the Fa a di Bruno expansion, written in packet coordinates, descends to the corresponding coarse tree. Moreover, C(G_ f∘ F_ f) =(G_ c∘ F_ c)C.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.","count":11},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4a6e104ba8e95488","tex":"g^{(m)}_{a;\\mathbf j}","aliases":"g^(m)_a; j mathbf","display":"g^{(m)}_{a;\\mathbf j}","kind":"use","excerpt":"[Exact packet coordinate lumpability test] For every coarse output a and coarse input tuple b: [label=arabic*.,leftmargin=2.3em] item Evaluate g^(m)_a; j for all j∈ G_ b. item In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance. item When the exa","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-71b740b1706109e9","tex":"g^{(m)}_{a;j_1',\\ldots,j_m'}","aliases":"g^(m)_a;j_1',…,j_m' ldots","display":"g^{(m)}_{a;j_1',\\ldots,j_m'}","kind":"use","excerpt":"[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6197d8dd435d9e0d","tex":"g^{(m)}_{a;j_1,\\ldots,j_m}","aliases":"g^(m)_a;j_1,…,j_m ldots","display":"g^{(m)}_{a;j_1,\\ldots,j_m}","kind":"use","excerpt":"R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3855e20a182a6f56","tex":"g^T","aliases":"g^T","display":"g^T","kind":"use","excerpt":"[Coarse observable error] For any coarse observable g∈ R^q, absg^TCμ_N^ f-g^Tμ_N^ c ≤ normg_∞e_N^+. In particular, TV(Cμ_N^ f,μ_N^ c) =frac12e_N^+.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6de66ec0182843d1","tex":"g_{a;\\mathbf j}","aliases":"g_a; j mathbf","display":"g_{a;\\mathbf j}","kind":"use","excerpt":"Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ ","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9496d806b845dd33","tex":"G_{\\coarse}","aliases":"G_coarse G_ c coarse mathrm","display":"G_{{\\mathrm c}}","kind":"use","excerpt":"[Compatibility with Fa a di Bruno composition] Let F_ f and F_ c be smooth maps with CF_ f=F_ cC, and let G_ f and G_ c satisfy the same relation. Then each rooted tree in the Fa a di Bruno expansion, written in packet coordinates, descends to the corresponding coarse tree. Moreover, C(G_ f∘ F_ f) =(G_ c∘ F_ c)C.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-772d0047baa30356","tex":"G_{\\fine}","aliases":"G_fine G_ f fine mathrm","display":"G_{{\\mathrm f}}","kind":"use","excerpt":"[Compatibility with Fa a di Bruno composition] Let F_ f and F_ c be smooth maps with CF_ f=F_ cC, and let G_ f and G_ c satisfy the same relation. Then each rooted tree in the Fa a di Bruno expansion, written in packet coordinates, descends to the corresponding coarse tree. Moreover, C(G_ f∘ F_ f) =(G_ c∘ F_ c)C.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].","count":12},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b6c99003ed43c18d","tex":"h_{12}","aliases":"h_12","display":"h_{12}","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-97adf0ad98b6ff13","tex":"h_2","aliases":"h_2","display":"h_2","kind":"use","excerpt":"texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4f890be69dc34d0b","tex":"h_{34}","aliases":"h_34","display":"h_{34}","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8a5a5b48b1793c8b","tex":"h_k","aliases":"h_k","display":"h_k","kind":"use","excerpt":"[Jet naturality] Let F_ f and F_ c be C^r maps satisfying CF_ f=F_ cC. Then, for every 1≤ k≤ r, C D^kF_ f(x)[h_1,…,h_k] = D^kF_ c(Cx)[Ch_1,…,Ch_k].","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6cad8b298e4618f4","tex":"h_M","aliases":"h_M","display":"h_M","kind":"use","excerpt":"texorpdfstringRetaining the k largest energy hyperedgesRetaining the k largest energy hyperedges. Let e_h=normwidehat A_m,h_HS^2 and order the energies e_h_1≥ e_h_2≥⋯≥ e_h_M.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3e01b0cedad21f47","tex":"h_r","aliases":"h_r","display":"h_r","kind":"use","excerpt":"[Bayesian jet naturality] Under the hypotheses of [source reference], for every r≥1 and every interior fine prior, C D^r B_λ^ f(μ) [h_1,…,h_r] = D^r B_λ^ c(Cμ) [Ch_1,…,Ch_r].","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e3bae9b717557c2b","tex":"\\mathrm{HS}","aliases":"HS mathrm","display":"\\mathrm{HS}","kind":"use","excerpt":"[Orthogonal hyperedge Parseval identity] Distinct packet blocks are orthogonal in Hilbert Schmidt inner product, and normwidehat A_HS^2 = ∑_i,j_1,…,j_m normwidehat A_imid j_1,…,j_m_HS^2.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f8358e76fa0b2213","tex":"\\HyperedgeEntropyValue","aliases":"HyperedgeEntropyValue","display":null,"kind":"use","excerpt":"Concentrated hyperedge field. A synthetic interaction field with decreasing orthogonal block energies illustrates [source reference]. The computed hyperedge entropy is HyperedgeEntropyValue, corresponding to an effective count of approximately EffectiveHyperedgeCount hyperedges despite a larger ambient support. The ranked spectrum in [source refere","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-69a4780a0be9987e","tex":"I_q","aliases":"I_q","display":"I_q","kind":"use","excerpt":"[A quotient with invisible internal interactions] Let X_ c= R^q, X_ f= R^2q, and C=I_q · I_q, E=frac12I_q I_q. Let A_m^ c be arbitrary and let K_m:X_ f^m→ X_ f satisfy CK_m=0. Then A_m^ f = EA_m^ cC^× m+K_m satisfies CA_m^ f=A_m^ cC^× m. The tensor K_m may contain many fine hyperedges, so the fine model can remain complicated. Exact quotienting ask","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-38dca881ad2712ad","tex":"I_{X_{\\coarse}}","aliases":"I_X_coarse I_X_ c coarse mathrm","display":"I_{X_{{\\mathrm c}}}","kind":"use","excerpt":"Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine represen","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e3677a7331334f76","tex":"I_{X_{\\fine}}","aliases":"I_X_fine I_X_ f fine mathrm","display":"I_{X_{{\\mathrm f}}}","kind":"use","excerpt":"Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine represen","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ab3ef27be5e3d7f7","tex":"I_Y","aliases":"I_Y","display":"I_Y","kind":"use","excerpt":"Insert I_X=∑_jP_j in every input slot and I_Y=∑_iQ_i after applying A. Multilinearity distributes all finite sums and gives [source reference].","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"Let (P_j)_j∈ J resolve a source X and (Q_i)_i∈ I resolve a target Y.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5b761556af44ee0a","tex":"\\mathbf j","aliases":"j mathbf","display":"\\mathbf j","kind":"use","excerpt":"[Exact packet coordinate lumpability test] For every coarse output a and coarse input tuple b: [label=arabic*.,leftmargin=2.3em] item Evaluate g^(m)_a; j for all j∈ G_ b. item In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance. item When the exa","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7be60211f38c975c","tex":"\\mathbf j'","aliases":"j' mathbf","display":"\\mathbf j'","kind":"use","excerpt":"em In exact arithmetic, test whether all values are equal. With floating point input, compare their spread with a declared tolerance and record that tolerance. item When the exact test passes, assign the common value to a^ c,(m)_a; b. When only the numerical tolerance is met, assign a declared representative such as the mean or the midrange. item O","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d853ad34ce9c0840","tex":"j'","aliases":"j'","display":"j'","kind":"use","excerpt":"[Unary aggregate rate criterion] For m=1, exact descent is equivalent to ∑_i:π(i)=aa^ f,(1)_i;j = ∑_i:π(i)=aa^ f,(1)_i;j' for all coarse outputs a whenever π(j)=π(j').","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b24f73fa686c2306","tex":"j_1'","aliases":"j_1'","display":"j_1'","kind":"use","excerpt":"[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7b4f4736461bbe42","tex":"j_m'","aliases":"j_m'","display":"j_m'","kind":"use","excerpt":"[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-456e17f40199861f","tex":"j_r","aliases":"j_r","display":"j_r","kind":"use","excerpt":"[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m,","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-71efb78189f40f99","tex":"j_r'","aliases":"j_r'","display":"j_r'","kind":"use","excerpt":"[Coefficient criterion for every interaction order] The fine tensor A_m^ f descends through the block sum map [source reference] if and only if, for every coarse output a, g^(m)_a;j_1,…,j_m = g^(m)_a;j_1',…,j_m' whenever π(j_r)=π(j_r') (r=1,…,m). When this condition holds, the unique coarse coefficients are a^ c,(m)_a;b_1,…,b_m = g^(m)_a;j_1,…,j_m,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"I_q. Let A_m^ c be arbitrary and let K_m:X_ f^m→ X_ f satisfy CK_m=0. Then A_m^ f = EA_m^ cC^× m+K_m satisfies CA_m^ f=A_m^ cC^× m. The tensor K_m may contain many fine hyperedges, so the fine model can remain complicated. Exact quotienting asks only that every hidden contribution disappear after applying C.","count":15},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7a0a635a73cee305","tex":"k'","aliases":"k'","display":"k'","kind":"use","excerpt":"[Quadratic hypergraph criterion] For m=2, exact descent is equivalent to ∑_i:π(i)=aa^ f,(2)_i;j,k = ∑_i:π(i)=aa^ f,(2)_i;j',k' whenever π(j)=π(j'), π(k)=π(k'). The resulting coarse interaction is the directed quadratic hyperedge (π(j),π(k))⟶ a.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-404b73dac64610d0","tex":"K_{\\coarse}","aliases":"K_coarse K_ c coarse mathrm","display":"K_{{\\mathrm c}}","kind":"use","excerpt":"Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f2d7a52239c8e265","tex":"K_{\\fine}","aliases":"K_fine K_ f fine mathrm","display":"K_{{\\mathrm f}}","kind":"use","excerpt":"Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-46d21cf1bf7572c1","tex":"K_m","aliases":"K_m","display":"K_m","kind":"use","excerpt":"I_q. Let A_m^ c be arbitrary and let K_m:X_ f^m→ X_ f satisfy CK_m=0. Then A_m^ f = EA_m^ cC^× m+K_m satisfies CA_m^ f=A_m^ cC^× m. The tensor K_m may contain many fine hyperedges, so the fine model can remain complicated. Exact quotienting asks only that every hidden contribution disappear after applying C.","count":11},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e2b60819c95171e8","tex":"k_r","aliases":"k_r","display":"k_r","kind":"use","excerpt":"Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, a","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-75ef47be232c5a6e","tex":"\\mathrm{keep}","aliases":"keep mathrm","display":"\\mathrm{keep}","kind":"use","excerpt":"Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2aab5edbc5f3efff","tex":"\\mathsf L","aliases":"L mathsf","display":"\\mathsf L","kind":"use","excerpt":"Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"[Composition law for normalized transfers] For positive operators L_1,L_2 on the same ordered state space, N_ℓ(L_2) ∘ N_ℓ(L_1) = N_ℓ(L_2L_1) wherever both sides are defined.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"[Composition law for normalized transfers] For positive operators L_1,L_2 on the same ordered state space, N_ℓ(L_2) ∘ N_ℓ(L_1) = N_ℓ(L_2L_1) wherever both sides are defined.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1496a456568579c3","tex":"L_{\\Bup,k}","aliases":"L_Bup,k L_ B,k Bup mathsf","display":"L_{{\\mathsf B},k}","kind":"use","excerpt":"Assume the coarse prediction is Lipschitz with constant L_Φ,k and that its quotient defect is bounded by sup_μ normCΦ_k^ f(μ)-Φ_k^ c(Cμ)_1 ≤ d_Φ,k. Assume the coarse update is Lipschitz with constant L_ B,k and that its direct refinement defect is bounded by sup_μ normC B_k^ f(μ)- B_k^ c(Cμ)_1 ≤ d_ B,k. The suprema in [source reference] and [source","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2f8dd52a01de0ecd","tex":"L_c","aliases":"L_c","display":"L_c","kind":"use","excerpt":"CL_f=sL_cC Longrightarrow C B_f= B_cC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ec6b4ea000366681","tex":"L_f","aliases":"L_f","display":"L_f","kind":"use","excerpt":"CL_f=sL_cC Longrightarrow C B_f= B_cC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3199644642af153d","tex":"L_{F,k}","aliases":"L_F,k","display":"L_{F,k}","kind":"use","excerpt":"From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the obser","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-64740eccad1308d8","tex":"\\mathsf L_m","aliases":"L_m mathsf","display":"\\mathsf L_m","kind":"use","excerpt":"Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-bf68cea986b231ee","tex":"L_y^c","aliases":"L_y^c","display":"L_y^c","kind":"use","excerpt":"CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6c141caaaa97358f","tex":"L_y^f","aliases":"L_y^f","display":"L_y^f","kind":"use","excerpt":"CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-42e75d54c8ccd989","tex":"L_{\\lambda}","aliases":"L_λ lambda","display":"L_{\\lambda}","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6dae6515dd6f887f","tex":"L_{\\lambda^{\\coarse}}","aliases":"L_λ^coarse L_λ^ c lambda coarse lambda mathrm","display":"L_{\\lambda^{{\\mathrm c}}}","kind":"use","excerpt":"Equation [source reference] is equivalent to CL_λ^ f =sL_λ^ cC. Apply [source reference]. The evidence identity follows from (λ^ f)^Tμ =s(λ^ c)^TCμ.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a8a2ea5ed1278943","tex":"L_{\\lambda^{\\fine}}","aliases":"L_λ^fine L_λ^ f lambda fine lambda mathrm","display":"L_{\\lambda^{{\\mathrm f}}}","kind":"use","excerpt":"Equation [source reference] is equivalent to CL_λ^ f =sL_λ^ cC. Apply [source reference]. The evidence identity follows from (λ^ f)^Tμ =s(λ^ c)^TCμ.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b5d1540e46c795f6","tex":"L_{\\Phi,k}","aliases":"L_Φ,k Phi","display":"L_{\\Phi,k}","kind":"use","excerpt":"Assume the coarse prediction is Lipschitz with constant L_Φ,k and that its quotient defect is bounded by sup_μ normCΦ_k^ f(μ)-Φ_k^ c(Cμ)_1 ≤ d_Φ,k. Assume the coarse update is Lipschitz with constant L_ B,k and that its direct refinement defect is bounded by sup_μ normC B_k^ f(μ)- B_k^ c(Cμ)_1 ≤ d_ B,k. The suprema in [source reference] and [source","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"use","excerpt":"[label=(roman*),leftmargin=2.8em] item If 0≤α<1, then e_N^+ ≤ α^Ne_0^+ + βfrac1-α^N1-α, limsup_N→∞e_N^+ ≤fracβ1-α.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-62d5bc57c3618c26","tex":"\\limsup_{N\\to\\infty}","aliases":"limsup_N→∞ limsup to infty","display":"\\limsup_{N\\to\\infty}","kind":"use","excerpt":"[label=(roman*),leftmargin=2.8em] item If 0≤α<1, then e_N^+ ≤ α^Ne_0^+ + βfrac1-α^N1-α, limsup_N→∞e_N^+ ≤fracβ1-α.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"for one step] Assume widehatμ_k^ f is a probability vector. Under [source reference], absz_k^ f-s_kz_k^ c ≤ s_knormλ_k^ c_∞e_k^- +normr_k_∞. If z_k^ f,s_kz_k^ c≥ z_*,k>0, then abs log z_k^ f -log(s_kz_k^ c) ≤ frac s_knormλ_k^ c_∞e_k^- +normr_k_∞ z_*,k.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"The proof strategy can be summarized by the chain multilinear quotient Longrightarrow interaction tree quotient Longrightarrow flow quotient Longrightarrow Bayesian quotient Longrightarrow filtering cocycle quotient Longrightarrow posterior certificate.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2541556cac379ec2","tex":"\\mathrm{LS}","aliases":"LS mathrm","display":"\\mathrm{LS}","kind":"use","excerpt":"Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"[Coefficientwise characterization of polynomial descent] Let F_ f(x)=∑_m=0^MA_m^ f[x^⊗ m], F_ c(z)=∑_m=0^MA_m^ c[z^⊗ m], where every A_m^ f and A_m^ c is symmetric. Then CF_ f=F_ cC if and only if CA_m^ f=A_m^ cC^× m (0≤ m≤ M).","count":12},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-47192caa46e42322","tex":"M_{\\coarse}","aliases":"M_coarse M_ c coarse mathrm","display":"M_{{\\mathrm c}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e5c2b628b058ae78","tex":"M_{\\fine}","aliases":"M_fine M_ f fine mathrm","display":"M_{{\\mathrm f}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-19e7d12924a016e5","tex":"\\max_{\\mathbf j\\in\\cG_{\\mathbf b}}","aliases":"max_ j∈cG_ b max_ j∈ G_ b max mathbf in cG mathbf max mathbf in mathcal mathbf","display":"\\max_{\\mathbf j\\in{\\mathcal G}_{\\mathbf b}}","kind":"use","excerpt":"r total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with optimal scalar fiber error ε^∞_a; b = frac12 ( max_ j∈ G_ bg_a; j - min_ j∈ G_ bg_a; j ","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7293b7045b642b01","tex":"\\min_{\\mathbf j\\in\\cG_{\\mathbf b}}","aliases":"min_ j∈cG_ b min_ j∈ G_ b min mathbf in cG mathbf min mathbf in mathcal mathbf","display":"\\min_{\\mathbf j\\in{\\mathcal G}_{\\mathbf b}}","kind":"use","excerpt":"r total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ G_ bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with optimal scalar fiber error ε^∞_a; b = frac12 ( max_ j∈ G_ bg_a; j - min_ j∈ G_ bg_a; j ","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9cddaf3dde3b8ffb","tex":"\\min_{C,\\{A_m^{\\coarse}\\},\\{\\cE_m^{\\mathrm{keep}}\\}}","aliases":"min_C,{A_m^coarse},{cE_m^keep} min_C,{A_m^ c},{ E_m^keep} min coarse cE mathrm min mathrm mathcal mathrm","display":"\\min_{C,\\{A_m^{{\\mathrm c}}\\},\\{{\\mathcal E}_m^{\\mathrm{keep}}\\}}","kind":"use","excerpt":"A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-215bca7b4604c055","tex":"\\min_{C,\\{A_m^{\\coarse}\\},\\{\\lambda_y^{\\coarse}\\}}","aliases":"min_C,{A_m^coarse},{λ_y^coarse} min_C,{A_m^ c},{λ_y^ c} min coarse lambda coarse min mathrm lambda mathrm","display":"\\min_{C,\\{A_m^{{\\mathrm c}}\\},\\{\\lambda_y^{{\\mathrm c}}\\}}","kind":"use","excerpt":"Approximate joint design. A finite approximate problem is min_C,{A_m^ c},{λ_y^ c} [ ∑_mw_mnormCA_m^ f-A_m^ cC^× m +∑_yv_yd_ B,y +γComplexity(C) ]. The main questions concern existence, tractable relaxations, and stability of the minimizers. Another question is whether the task weights can be learned without invalidating the certificate after a dist","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-675969e036afbdb3","tex":"\\min_{|\\cE|=k}","aliases":"min_|cE|=k min_| E|=k min cE min mathcal","display":"\\min_{|{\\mathcal E}|=k}","kind":"use","excerpt":"[Optimal retention of k orthogonal hyperedges] Among all retained sets of cardinality k, the set {h_1,…,h_k} minimizes the Hilbert Schmidt residual. The minimum is min_| E|=k normR_m( E)_HS^2 = ∑_r>ke_h_r.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0aac477735fa59fc","tex":"\\min_{\\pi}","aliases":"min_π min pi","display":"\\min_{\\pi}","kind":"use","excerpt":"Coarsest exact joint partition. For a finite interaction family (A_m^ f)_m≤ M and likelihood family (λ_y^ f)_y∈ Y, seek min_π|I_ c| subject to CA_m^ f=A_m^ cC^× m (m≤ M), and λ_y^ f=s_yC^Tλ_y^ c (y∈ Y). The partition determined by the observations alone groups states with identical likelihood profiles. The dynamical condition also depends on the pa","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e078f341254da22e","tex":"n^{m+1}","aliases":"n^m+1","display":"n^{m+1}","kind":"use","excerpt":"A dense implementation requires O(n^m+1) arithmetic to aggregate a tensor of order m on n states. If the fine tensor is stored as E_m active scalar hyperedges, the aggregation pass is O(E_m), followed by grouping and consistency checks over the represented coarse incidences.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-aed088c0e902258e","tex":"N_{\\mathrm{eff},m}","aliases":"N_eff,m mathrm","display":"N_{\\mathrm{eff},m}","kind":"use","excerpt":"Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d4d9568722502642","tex":"\\Nrm_{\\ell_{\\coarse}}","aliases":"Nrm_ℓ_coarse N_ℓ_ c Nrm ell coarse mathsf ell mathrm","display":"{\\mathsf N}_{\\ell_{{\\mathrm c}}}","kind":"use","excerpt":"malized transfer] Let L_ f:X_ f→ X_ f and L_ c:X_ c→ X_ c be positive linear operators. Suppose CL_ f=sL_ cC for some s>0, and suppose [source reference] holds. Then, on the common domain of normalization, C N_ℓ_ f(L_ f) = N_ℓ_ c(L_ c)C.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f268c0a66f109980","tex":"\\Nrm_{\\ell_{\\fine}}","aliases":"Nrm_ℓ_fine N_ℓ_ f Nrm ell fine mathsf ell mathrm","display":"{\\mathsf N}_{\\ell_{{\\mathrm f}}}","kind":"use","excerpt":"malized transfer] Let L_ f:X_ f→ X_ f and L_ c:X_ c→ X_ c be positive linear operators. Suppose CL_ f=sL_ cC for some s>0, and suppose [source reference] holds. Then, on the common domain of normalization, C N_ℓ_ f(L_ f) = N_ℓ_ c(L_ c)C.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"A dense implementation requires O(n^m+1) arithmetic to aggregate a tensor of order m on n states. If the fine tensor is stored as E_m active scalar hyperedges, the aggregation pass is O(E_m), followed by grouping and consistency checks over the represented coarse incidences.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3c84a1672e483df5","tex":"\\mathrm{obs}","aliases":"obs mathrm","display":"\\mathrm{obs}","kind":"use","excerpt":"A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a34910944de266ad","tex":"\\odot","aliases":"odot","display":null,"kind":"use","excerpt":"Why Bayesian observations add another nonlinear operation. Prediction is only one part of a filter. Given a probability vector μ and a positive likelihood λ, Bayesian conditioning is B_λ(μ) = fracλodotμλ^Tμ. Likelihood multiplication is linear, but the evidence in the denominator couples all packets. The full update is therefore nonlinear even when","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e032304f495aab52","tex":"\\one_{\\coarse}^T","aliases":"one_coarse^T 1_ c^T one coarse mathbf mathrm","display":"{\\mathbf 1}_{{\\mathrm c}}^T","kind":"use","excerpt":"Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-da0402ea8b7f9f1e","tex":"\\one_{\\fine}^T","aliases":"one_fine^T 1_ f^T one fine mathbf mathrm","display":"{\\mathbf 1}_{{\\mathrm f}}^T","kind":"use","excerpt":"Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-01eecdf5cc4bca8b","tex":"\\overline A","aliases":"overline A overline","display":"\\overline A","kind":"use","excerpt":"/Ker C be the canonical quotient map. Since C is surjective, there is a unique vector space isomorphism widetilde C:X_ f/Ker C→ X_ c satisfying C=widetilde Cq. The multilinear map CA_m^ f:X_ f^m→ X_ c descends if and only if its value is unchanged when any one input is altered by an element of Ker C. The induced quotient tensor is overline A_m ([x_","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7208c1b661d4c2cc","tex":"\\overline A_m","aliases":"overline A_m overline","display":"\\overline A_m","kind":"use","excerpt":"/Ker C be the canonical quotient map. Since C is surjective, there is a unique vector space isomorphism widetilde C:X_ f/Ker C→ X_ c satisfying C=widetilde Cq. The multilinear map CA_m^ f:X_ f^m→ X_ c descends if and only if its value is unchanged when any one input is altered by an element of Ker C. The induced quotient tensor is overline A_m ([x_","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9f1ae61172342775","tex":"\\overline\\lambda","aliases":"overlineλ overline lambda","display":"\\overline\\lambda","kind":"use","excerpt":"fracλ_i^ fλ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k gives λ_j^ f=λ_i^ f. The fine likelihood is therefore constant on each fiber. Write its value on fiber a as overlineλ_a. Equation [source reference] now gives fracoverlineλ_aoverlineλ_b = fracλ_a^ cλ_b^ c for every pair a,b. Hence","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-600aef1944796dc2","tex":"\\overline\\lambda_a","aliases":"overlineλ_a overline lambda","display":"\\overline\\lambda_a","kind":"use","excerpt":"fracλ_i^ fλ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k gives λ_j^ f=λ_i^ f. The fine likelihood is therefore constant on each fiber. Write its value on fiber a as overlineλ_a. Equation [source reference] now gives fracoverlineλ_aoverlineλ_b = fracλ_a^ cλ_b^ c for every pair a,b. Hence","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-17b3f89325c1dd8c","tex":"\\overline\\lambda_b","aliases":"overlineλ_b overline lambda","display":"\\overline\\lambda_b","kind":"use","excerpt":"λ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k gives λ_j^ f=λ_i^ f. The fine likelihood is therefore constant on each fiber. Write its value on fiber a as overlineλ_a. Equation [source reference] now gives fracoverlineλ_aoverlineλ_b = fracλ_a^ cλ_b^ c for every pair a,b. Hence overlineλ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-882583972d08300b","tex":"P_C^2","aliases":"P_C^2","display":"P_C^2","kind":"use","excerpt":"Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine represen","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-333d75f622b78fee","tex":"p_h","aliases":"p_h","display":"p_h","kind":"use","excerpt":"Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5832b604af6b8d6d","tex":"\\PerturbationSlope","aliases":"PerturbationSlope","display":null,"kind":"use","excerpt":"[Perturbation scaling] The experiment used logarithmically spaced values with ε=η. The fitted slope of the final posterior discrepancy was PerturbationSlope. Every measured error remained below its recursive certificate.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-eda0fef223ad56b8","tex":"\\propto","aliases":"propto","display":null,"kind":"use","excerpt":"Posterior distributions over resolutions. Instead of selecting one C, place a posterior on a finite family of candidate resolutions: p(Cmid D) propto p(Dmid C)p(C). Predictions then average both parameter and topology uncertainty: p(z_tmid D) = ∑_Cp(z_tmid C,D)p(Cmid D). A fuller model should distinguish uncertainty in the coefficients of a fixed h","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ff7eb8f0d00f446f","tex":"\\mathfrak Q","aliases":"Q mathfrak","display":"\\mathfrak Q","kind":"use","excerpt":"Best coarse interaction and quotient defect. Let Q_m(C) = {B_mC^× m:B_m∈ L^m(X_ c;X_ c)} be the subspace of outputs from multilinear maps of order m that factor through the coarse space.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-56e3f0863af8e77c","tex":"Q_C^2","aliases":"Q_C^2","display":"Q_C^2","kind":"use","excerpt":"Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine represen","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-cb83a1159e0840f7","tex":"Q_{\\coarse}","aliases":"Q_coarse Q_ c coarse mathrm","display":"Q_{{\\mathrm c}}","kind":"use","excerpt":"Markov lumpability and Markov functions. For unary stochastic dynamics, the identity CQ_ f=Q_ cC is a standard exact lumping condition. It implies Ce^tQ_ f=e^tQ_ cC and means that the coarse process is Markov for every fine initial distribution compatible with the quotient. Classical treatments distinguish several forms of strong, ordinary, exact, ","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a7b41ec1668496b5","tex":"Q_{\\fine}","aliases":"Q_fine Q_ f fine mathrm","display":"Q_{{\\mathrm f}}","kind":"use","excerpt":"Let Q_ f be the generator of a continuous time Markov chain, using the column vector convention. The left side of [source reference] is the total transition rate from fine state j into coarse packet a. Corollary~[source reference] is therefore the standard criterion for strong lumpability.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-49ffdcdd4595cc48","tex":"\\mathfrak Q_m","aliases":"Q_m mathfrak","display":"\\mathfrak Q_m","kind":"use","excerpt":"Best coarse interaction and quotient defect. Let Q_m(C) = {B_mC^× m:B_m∈ L^m(X_ c;X_ c)} be the subspace of outputs from multilinear maps of order m that factor through the coarse space.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.","count":43},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-80402bf9f4f1ca1b","tex":"\\R^{2q}","aliases":"R^2q R mathbb","display":"{\\mathbb R}^{2q}","kind":"use","excerpt":"[A quotient with invisible internal interactions] Let X_ c= R^q, X_ f= R^2q, and C=I_q · I_q, E=frac12I_q","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c4179ac2053e2353","tex":"R^m","aliases":"R^m","display":"R^m","kind":"use","excerpt":"[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-98f29d79d667c5ba","tex":"r^m","aliases":"r^m","display":"r^m","kind":"use","excerpt":"[Analytic interaction descent] Let C:X_ f↠ X_ c have a bounded right inverse E. Suppose the coefficient series converges normally on every smaller ball, ∑_m=0^∞normA_m^ fr^m<∞ (0≤ r<R), and, for every m≥0, there is a coarse coefficient A_m^ c satisfying CA_m^ f=A_m^ cC^× m. Then F_ c(z) = ∑_m=0^∞A_m^ c[z^⊗ m] converges absolutely whenever normEnorm","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5e24b89ece0c5e90","tex":"\\R_+^n","aliases":"R_+^n R mathbb","display":"{\\mathbb R}_+^n","kind":"use","excerpt":"Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f9681aba270208a6","tex":"\\R_+^q","aliases":"R_+^q R mathbb","display":"{\\mathbb R}_+^q","kind":"use","excerpt":"Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4fc46039e327e569","tex":"\\R_{>0}^n","aliases":"R_>0^n R mathbb","display":"{\\mathbb R}_{>0}^n","kind":"use","excerpt":"[Sufficient condition for Bayesian refinement] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c06dd3704f904778","tex":"\\R_{>0}^q","aliases":"R_>0^q R mathbb","display":"{\\mathbb R}_{>0}^q","kind":"use","excerpt":"[Sufficient condition for Bayesian refinement] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a64e897db39c6a7f","tex":"R_k","aliases":"R_k","display":"R_k","kind":"use","excerpt":"From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the obser","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"use","excerpt":"Sequential evidence error. For brevity, write λ_k^ f=λ_y_k^ f and λ_k^ c=λ_y_k^ c, and decompose λ_k^ f =s_kC^Tλ_k^ c+r_k. This gives z_k^ f-s_kz_k^ c · = s_k(λ_k^ c)^T (Cwidehatμ_k^ f-widehatμ_k^ c) +r_k^Twidehatμ_k^ f.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-dfa3c454c983b5ab","tex":"r_k^T","aliases":"r_k^T","display":"r_k^T","kind":"use","excerpt":"Sequential evidence error. For brevity, write λ_k^ f=λ_y_k^ f and λ_k^ c=λ_y_k^ c, and decompose λ_k^ f =s_kC^Tλ_k^ c+r_k. This gives z_k^ f-s_kz_k^ c · = s_k(λ_k^ c)^T (Cwidehatμ_k^ f-widehatμ_k^ c) +r_k^Twidehatμ_k^ f.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-625fc9310223a088","tex":"R_m^{\\fine}","aliases":"R_m^fine R_m^ f fine mathrm","display":"R_m^{{\\mathrm f}}","kind":"use","excerpt":"Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5d6e45e73e872e8e","tex":"r_y","aliases":"r_y","display":"r_y","kind":"use","excerpt":"Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3a3384c51d0889d5","tex":"r_{\\lambda}","aliases":"r_λ lambda","display":"r_{\\lambda}","kind":"use","excerpt":"Likelihood variation inside a fiber. For diagonal likelihood operators, write λ^ f = sC^Tλ^ c+r_λ. Then R_Lμ=C(r_λodotμ). Since block sum aggregation is an ℓ^1 contraction on signed vectors, normR_Lμ_1 ≤ ∑_iμ_iabsr_λ,i ≤ normr_λ_∞.","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ca7f79bca93a0438","tex":"r_{\\lambda,i}","aliases":"r_λ,i lambda","display":"r_{\\lambda,i}","kind":"use","excerpt":"Likelihood variation inside a fiber. For diagonal likelihood operators, write λ^ f = sC^Tλ^ c+r_λ. Then R_Lμ=C(r_λodotμ). Since block sum aggregation is an ℓ^1 contraction on signed vectors, normR_Lμ_1 ≤ ∑_iμ_iabsr_λ,i ≤ normr_λ_∞.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-85aae5023fa11218","tex":"r_{\\lambda,k}","aliases":"r_λ,k lambda","display":"r_{\\lambda,k}","kind":"use","excerpt":"ield has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the observation, Corollary~[source reference] gives d_ B,k ≤ frac2normr_λ,k_∞z_*,k. Substituting these bounds into [source reference] gives a certificate expressed direct","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b11c81e6903fa9bf","tex":"\\Ran","aliases":"Ran operatorname","display":"{\\operatorname{Ran}}","kind":"use","excerpt":"Coarse maps, sections, and hidden directions. Let C:X_ f↠ X_ c be a surjective linear coarse map. Finite dimensionality guarantees a linear right inverse E:X_ c→ X_ f, CE=I_X_ c. Define P_C=EC, Q_C=I_X_ f-EC. Then P_C^2=P_C, Q_C^2=Q_C, Ran Q_C=Ker C. Every fine state has the exact decomposition x=ECx+Q_Cx. The first term is the chosen fine represen","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b4ebcce0dc1a5c39","tex":"s_k","aliases":"s_k","display":"s_k","kind":"use","excerpt":"[Evidence factorization under exact refinement] Assume the hypotheses of [source reference], let μ_0^ c=Cμ_0^ f, and suppose λ_y_k^ f =s_kC^Tλ_y_k^ c (s_k>0). Then z_k^ f=s_kz_k^ c for every step, and the likelihood of the observation sequence satisfies Z_1:N^ f = (∏_k=1^Ns_k) Z_1:N^ c. Under the gauge s_k=1, the fine and coarse sequential evidence","count":11},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-81c409120fb4bc98","tex":"s_y","aliases":"s_y","display":"s_y","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7f626a737210f03b","tex":"\\sup_{\\norm{x_r}\\leq1}","aliases":"sup_normx_r≤1 sup norm leq","display":null,"kind":"use","excerpt":"Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-973d887cb0deea55","tex":"\\sup_{\\norm{x}\\leq R}","aliases":"sup_normx≤ R sup norm leq","display":null,"kind":"use","excerpt":"[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7b3c581956d2345b","tex":"\\sup_{\\mu}","aliases":"sup_μ sup mu","display":"\\sup_{\\mu}","kind":"use","excerpt":"Assume the coarse prediction is Lipschitz with constant L_Φ,k and that its quotient defect is bounded by sup_μ normCΦ_k^ f(μ)-Φ_k^ c(Cμ)_1 ≤ d_Φ,k. Assume the coarse update is Lipschitz with constant L_ B,k and that its direct refinement defect is bounded by sup_μ normC B_k^ f(μ)- B_k^ c(Cμ)_1 ≤ d_ B,k. The suprema in [source reference] and [source","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-aea3af576abad800","tex":"t^m","aliases":"t^m","display":"t^m","kind":"use","excerpt":"The reverse implication follows from [source reference], applied to a finite sum. Conversely, fix x∈ X_ f and evaluate the field identity at tx: ∑_m=0^Mt^mCA_m^ f[x^⊗ m] = ∑_m=0^Mt^mA_m^ c[(Cx)^⊗ m]. Equality of the coefficients of this polynomial with values in X_ c gives equality on the diagonal for the two symmetric multilinear maps of order m. ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b6d1c35d86f0d39d","tex":"T_k","aliases":"T_k","display":"T_k","kind":"use","excerpt":"From interaction coefficients to posterior error. For a flow interval of length T_k, suppose the fine trajectory remains in a ball of radius R_k and the coarse field has Lipschitz constant L_F,k. By Corollary~[source reference], one may take d_Φ,k = d_F,k(R_k) frace^L_F,kT_k-1L_F,k, with the continuous value T_k d_F,k(R_k) at L_F,k=0. For the obser","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-830c9c56b4f2bf21","tex":"t_k","aliases":"t_k","display":"t_k","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-28b20920cb7a318f","tex":"t_{k-1}","aliases":"t_k-1","display":"t_{k-1}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-163dd25fe6cb484d","tex":"t_N","aliases":"t_N","display":"t_N","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3e3becea69293024","tex":"\\mathrm{total}","aliases":"total mathrm","display":"\\mathrm{total}","kind":"use","excerpt":"Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV operatorname","display":"{\\operatorname{TV}}","kind":"use","excerpt":"[Coarse observable error] For any coarse observable g∈ R^q, absg^TCμ_N^ f-g^Tμ_N^ c ≤ normg_∞e_N^+. In particular, TV(Cμ_N^ f,μ_N^ c) =frac12e_N^+.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Both sides of [source reference] extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses","count":18},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"Both sides of [source reference] extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses","count":20},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1, h_34(x) · = (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3, with all coefficients nonnegative. The fine system is dot x_1 · =-ax_1+bx_3+h_12(x)","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-51fd0818257204a0","tex":"v_r","aliases":"v_r","display":"v_r","kind":"use","excerpt":"Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, a","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5b59d34e2b2271f4","tex":"v_y","aliases":"v_y","display":"v_y","kind":"use","excerpt":"Approximate joint design. A finite approximate problem is min_C,{A_m^ c},{λ_y^ c} [ ∑_mw_mnormCA_m^ f-A_m^ cC^× m +∑_yv_yd_ B,y +γComplexity(C) ]. The main questions concern existence, tractable relaxations, and stability of the minimizers. Another question is whether the task weights can be learned without invalidating the certificate after a dist","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability ","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-45377a81ffef2b83","tex":"w_{\\mathbf j}","aliases":"w_ j mathbf","display":"w_{\\mathbf j}","kind":"use","excerpt":"Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ ","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a6dc04e23090a2a5","tex":"w_m","aliases":"w_m","display":"w_m","kind":"use","excerpt":"A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability ","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8d3781ce495edcf4","tex":"\\widehat A","aliases":"widehat A widehat","display":"\\widehat A","kind":"use","excerpt":"item Let widehat A_m^ f:X_ f^⊗ m→ X_ c be the linearization of CA_m^ f. Then Ker(C^⊗ m) ⊆ Ker(widehat A_m^ f).","count":9},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f598ab8f1f27b672","tex":"\\widehat A_{i\\mid j_1,\\ldots,j_m}","aliases":"widehat A_imid j_1,…,j_m widehat mid ldots","display":null,"kind":"use","excerpt":"Hilbert Schmidt tensor blocks. Let X and Y be finite dimensional Hilbert spaces with orthogonal packet projectors (P_j) and (Q_i). Identify a multilinear Hilbert Schmidt tensor of order m with the Hilbert Schmidt linear map widehat A:X^⊗ m→ Y. Its packet block is widehat A_imid j_1,…,j_m = Q_iwidehat A(P_j_1⊗⋯⊗ P_j_m).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7a01d17f6b25e6b1","tex":"\\widehat A_m","aliases":"widehat A_m widehat","display":"\\widehat A_m","kind":"use","excerpt":"Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-88833023404de6c8","tex":"\\widehat A_{m,h}","aliases":"widehat A_m,h widehat","display":"\\widehat A_{m,h}","kind":"use","excerpt":"Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-59b59716104aea55","tex":"\\widehat A_m^{\\fine}","aliases":"widehat A_m^fine widehat A_m^ f widehat fine widehat mathrm","display":"\\widehat A_m^{{\\mathrm f}}","kind":"use","excerpt":"item Let widehat A_m^ f:X_ f^⊗ m→ X_ c be the linearization of CA_m^ f. Then Ker(C^⊗ m) ⊆ Ker(widehat A_m^ f).","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8a04a22d6d8904dc","tex":"\\widehat A_m^{\\mathrm{keep}}","aliases":"widehat A_m^keep widehat mathrm","display":"\\widehat A_m^{\\mathrm{keep}}","kind":"use","excerpt":"Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b186d96cd6172b72","tex":"\\widehat\\mu","aliases":"widehatμ widehat mu","display":"\\widehat\\mu","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":12},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-07d181a40f2b5128","tex":"\\widehat\\mu_k^{\\coarse}","aliases":"widehatμ_k^coarse widehatμ_k^ c widehat mu coarse widehat mu mathrm","display":"\\widehat\\mu_k^{{\\mathrm c}}","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-951aa6f067a4242c","tex":"\\widehat\\mu_k^{\\fine}","aliases":"widehatμ_k^fine widehatμ_k^ f widehat mu fine widehat mu mathrm","display":"\\widehat\\mu_k^{{\\mathrm f}}","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":11},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-47c52f415949a6cb","tex":"\\widetilde C","aliases":"widetilde C widetilde","display":"\\widetilde C","kind":"use","excerpt":"Formulation on the quotient space. Let q:X_ f→ X_ f/Ker C be the canonical quotient map. Since C is surjective, there is a unique vector space isomorphism widetilde C:X_ f/Ker C→ X_ c satisfying C=widetilde Cq. The multilinear map CA_m^ f:X_ f^m→ X_ c descends if and only if its value is unchanged when any one input is altered by an element of Ker ","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-87d94d04bf974e18","tex":"\\widetilde C^{-1}","aliases":"widetilde C^-1 widetilde","display":"\\widetilde C^{-1}","kind":"use","excerpt":"que vector space isomorphism widetilde C:X_ f/Ker C→ X_ c satisfying C=widetilde Cq. The multilinear map CA_m^ f:X_ f^m→ X_ c descends if and only if its value is unchanged when any one input is altered by an element of Ker C. The induced quotient tensor is overline A_m ([x_1],…,[x_m]) =CA_m^ f(x_1,…,x_m), and A_m^ c = widetilde C overline A_m (wid","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c66bda09cbfde6df","tex":"X^{\\otimes m}","aliases":"X^⊗ m otimes","display":"X^{\\otimes m}","kind":"use","excerpt":"Hilbert Schmidt tensor blocks. Let X and Y be finite dimensional Hilbert spaces with orthogonal packet projectors (P_j) and (Q_i). Identify a multilinear Hilbert Schmidt tensor of order m with the Hilbert Schmidt linear map widehat A:X^⊗ m→ Y. Its packet block is widehat A_imid j_1,…,j_m = Q_iwidehat A(P_j_1⊗⋯⊗ P_j_m).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a6972a2d3f98c5ab","tex":"x^{\\otimes m}","aliases":"x^⊗ m otimes","display":"x^{\\otimes m}","kind":"use","excerpt":"Descent of polynomial and analytic vector fields. Let F_ f(x) = ∑_m=0^∞A_m^ f[x^⊗ m] be locally analytic, where A_m^ f is a symmetric multilinear map of order m for m≥1 and A_0^ f∈ X_ f. Throughout, A_m[x^⊗ m] =A_m(x,…,x).","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9054ec0d33867a23","tex":"x_1'","aliases":"x_1'","display":"x_1'","kind":"use","excerpt":"item The map CA_m^ f is constant on products of fibers: if Cx_r=Cx_r' for all r, then CA_m^ f(x_1,…,x_m) = CA_m^ f(x_1',…,x_m').","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"Composition across refinement levels. Let X_2xrightarrowC_21X_1xrightarrowC_10X_0, C_20=C_10C_21.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"A nonlinear fine system with four states and a quotient with two states. Let x=(x_1,x_2,x_3,x_4)^T∈Δ_4, C= 1 · 1 · 0 · 0 0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"A nonlinear fine system with four states and a quotient with two states. Let x=(x_1,x_2,x_3,x_4)^T∈Δ_4, C= 1 · 1 · 0 · 0 0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ca7008ae447ee9dc","tex":"x_4","aliases":"x_4","display":"x_4","kind":"use","excerpt":"A nonlinear fine system with four states and a quotient with two states. Let x=(x_1,x_2,x_3,x_4)^T∈Δ_4, C= 1 · 1 · 0 · 0 0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-72e05ef2b0f48e18","tex":"X_c","aliases":"X_c","display":"X_c","kind":"use","excerpt":"For a surjective coarse map C:X_f→ X_c, the paper studies","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f76f02d6c081bad5","tex":"X_f","aliases":"X_f","display":"X_f","kind":"use","excerpt":"For a surjective coarse map C:X_f→ X_c, the paper studies","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6e0c36139d3f8650","tex":"X_{\\fine}^{\\otimes m}","aliases":"X_fine^⊗ m X_ f^⊗ m fine otimes mathrm otimes","display":"X_{{\\mathrm f}}^{\\otimes m}","kind":"use","excerpt":"[label=Carabic*.,leftmargin=3.5em] item A quotient criterion for every interaction order. For a surjective map C, a multilinear fine interaction descends exactly when CA_m^ f vanishes whenever one input lies in Ker C. Equivalently, its linearization on X_ f^⊗ m factors through C^⊗ m.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4ad2b0228177b68e","tex":"X_{\\fine}^{\\otimes(m-r)}","aliases":"X_fine^⊗(m-r) X_ f^⊗(m-r) fine otimes mathrm otimes","display":"X_{{\\mathrm f}}^{\\otimes(m-r)}","kind":"use","excerpt":"To compare (ii) and (iv), identify a multilinear map of order m with its linearization on the algebraic tensor product. Since C is surjective, Ker(C^⊗ m) = ∑_r=1^m X_ f^⊗(r-1)⊗Ker C⊗ X_ f^⊗(m-r). Choose a splitting X_ f=E(X_ c)⊕Ker C and expand the tensor power. Vanishing of widehat A_m^ f on this sum is exactly the slot condition. Thus (ii) and (i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0a009fd8a8e22de5","tex":"X_{\\fine}^{\\otimes(r-1)}","aliases":"X_fine^⊗(r-1) X_ f^⊗(r-1) fine otimes mathrm otimes","display":"X_{{\\mathrm f}}^{\\otimes(r-1)}","kind":"use","excerpt":"To compare (ii) and (iv), identify a multilinear map of order m with its linearization on the algebraic tensor product. Since C is surjective, Ker(C^⊗ m) = ∑_r=1^m X_ f^⊗(r-1)⊗Ker C⊗ X_ f^⊗(m-r). Choose a splitting X_ f=E(X_ c)⊕Ker C and expand the tensor power. Vanishing of widehat A_m^ f on this sum is exactly the slot condition. Thus (ii) and (i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-981a0d8a05cbaf1b","tex":"x_i^{\\alpha_{ir}}","aliases":"x_i^α_ir alpha","display":"x_i^{\\alpha_{ir}}","kind":"use","excerpt":"Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, a","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-848620af401d93ab","tex":"x_m'","aliases":"x_m'","display":"x_m'","kind":"use","excerpt":"item The map CA_m^ f is constant on products of fibers: if Cx_r=Cx_r' for all r, then CA_m^ f(x_1,…,x_m) = CA_m^ f(x_1',…,x_m').","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.","count":10},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ca2482b5fe3813c7","tex":"x_r'","aliases":"x_r'","display":"x_r'","kind":"use","excerpt":"item The map CA_m^ f is constant on products of fibers: if Cx_r=Cx_r' for all r, then CA_m^ f(x_1,…,x_m) = CA_m^ f(x_1',…,x_m').","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1e9d35fe4d36c66f","tex":"x_{r+1}","aliases":"x_r+1","display":"x_{r+1}","kind":"use","excerpt":"item For every slot r∈{1,…,m}, CA_m^ f(x_1,…,x_r-1,k,x_r+1,…,x_m)=0 whenever k∈Ker C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0ef07195fe759ac3","tex":"x_{r-1}","aliases":"x_r-1","display":"x_{r-1}","kind":"use","excerpt":"item For every slot r∈{1,…,m}, CA_m^ f(x_1,…,x_r-1,k,x_r+1,…,x_m)=0 whenever k∈Ker C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[Nonlinear congruence interpretation] For a general map F:X_ f→ X_ f, descent through C is equivalent to Cx=Cy Longrightarrow CF(x)=CF(y). Thus the relation x∼_Cy, defined by Cx=Cy, must be a congruence for the coarse output of F. [source reference] turns that statement into a condition on each multilinear coefficient.","count":28},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6da9db5fdf3dbe21","tex":"y_{1:N}","aliases":"y_1:N","display":"y_{1:N}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f803465b3e31ab31","tex":"y_1^S","aliases":"y_1^S","display":"y_1^S","kind":"use","excerpt":"[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b050e04b7e254aa7","tex":"y_k","aliases":"y_k","display":"y_k","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7a6b0a82d171acdd","tex":"y_m^S","aliases":"y_m^S","display":"y_m^S","kind":"use","excerpt":"[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b3b03c916a5da762","tex":"y_N","aliases":"y_N","display":"y_N","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-10d6688c989eb59e","tex":"y_r^S","aliases":"y_r^S","display":"y_r^S","kind":"use","excerpt":"ensor is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S, P_Cx_r, · r∉ S.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":10},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ffb223d2d1c3fee8","tex":"z^{\\otimes m}","aliases":"z^⊗ m otimes","display":"z^{\\otimes m}","kind":"use","excerpt":"nt series converges normally on every smaller ball, ∑_m=0^∞normA_m^ fr^m<∞ (0≤ r<R), and, for every m≥0, there is a coarse coefficient A_m^ c satisfying CA_m^ f=A_m^ cC^× m. Then F_ c(z) = ∑_m=0^∞A_m^ c[z^⊗ m] converges absolutely whenever normEnormz<R, and CF_ f(x)=F_ c(Cx) for every x in the fine convergence domain whose coarse image lies in the ","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-10c7caeba58a1aa9","tex":"z_*","aliases":"z_*","display":"z_*","kind":"use","excerpt":"refinement-defect Let z_ f=Z_λ^ f(μ), z_ c=sZ_λ^ c(Cμ). Then norm C B_λ^ f(μ) - B_λ^ c(Cμ) _1 ≤ frac2∑_iμ_iabsr_λ,i max{z_ f,z_ c}. If both evidences are at least z_*>0, then d_ B ≤ frac2normr_λ_∞z_*.","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5bdfca4c18722fc5","tex":"z_{*,y}","aliases":"z_*,y","display":"z_{*,y}","kind":"use","excerpt":"Certificate schemas and reproducibility. A minimal interaction certificate stores L_m = (m,C,A_m^ f,A_m^ c,Δ_m, normΔ_m,norm,tolerance). A Bayesian certificate stores B_y = (λ_y^ f,λ_y^ c,s_y,r_y, z_*,y,d_ B,y). A filtering certificate stores F_N = (e_0^+,{L_Φ,k},{L_ B,k}, {d_Φ,k},{d_ B,k}, {e_k^-,e_k^+}). The release serializes the executed finite","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-56a4bd3bbe97bed0","tex":"Z_{1:N}^{\\coarse}","aliases":"Z_1:N^coarse Z_1:N^ c coarse mathrm","display":"Z_{1:N}^{{\\mathrm c}}","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-66aec29edd5a3927","tex":"Z_{1:N}^{\\fine}","aliases":"Z_1:N^fine Z_1:N^ f fine mathrm","display":"Z_{1:N}^{{\\mathrm f}}","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5e32645afb213c74","tex":"z_2","aliases":"z_2","display":"z_2","kind":"use","excerpt":"0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a1bf3264cb46d7bf","tex":"z_{\\coarse}","aliases":"z_coarse z_ c coarse mathrm","display":"z_{{\\mathrm c}}","kind":"use","excerpt":"[Bayesian refinement defect] Let z_ f=Z_λ^ f(μ), z_ c=sZ_λ^ c(Cμ). Then norm C B_λ^ f(μ) - B_λ^ c(Cμ) _1 ≤ frac2∑_iμ_iabsr_λ,i max{z_ f,z_ c}. If both evidences are at least z_*>0, then d_ B ≤ frac2normr_λ_∞z_*.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-25ef0c38f20c343c","tex":"z_{\\fine}","aliases":"z_fine z_ f fine mathrm","display":"z_{{\\mathrm f}}","kind":"use","excerpt":"[Bayesian refinement defect] Let z_ f=Z_λ^ f(μ), z_ c=sZ_λ^ c(Cμ). Then norm C B_λ^ f(μ) - B_λ^ c(Cμ) _1 ≤ frac2∑_iμ_iabsr_λ,i max{z_ f,z_ c}. If both evidences are at least z_*>0, then d_ B ≤ frac2normr_λ_∞z_*.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3ae3c82e54a8f834","tex":"z_k^{\\coarse}","aliases":"z_k^coarse z_k^ c coarse mathrm","display":"z_k^{{\\mathrm c}}","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":9},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1c752e2e27b7caa5","tex":"z_k^{\\fine}","aliases":"z_k^fine z_k^ f fine mathrm","display":"z_k^{{\\mathrm f}}","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":9},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-42dba447a7ef210c","tex":"z_m","aliases":"z_m","display":"z_m","kind":"use","excerpt":"Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-974e1ebfd0c228a6","tex":"z_r","aliases":"z_r","display":"z_r","kind":"use","excerpt":"Assume (iii). For z_r∈ X_ c choose arbitrary lifts x_r with Cx_r=z_r and set A_m^ c(z_1,…,z_m) = CA_m^ f(x_1,…,x_m). Fiber constancy makes the definition independent of the lifts. Multilinearity is inherited from A_m^ f, and the defining identity gives (i). Surjectivity of C also gives uniqueness: every coarse input tuple has a fine lift.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"Posterior distributions over resolutions. Instead of selecting one C, place a posterior on a finite family of candidate resolutions: p(Cmid D) propto p(Dmid C)p(C). Predictions then average both parameter and topology uncertainty: p(z_tmid D) = ∑_Cp(z_tmid C,D)p(Cmid D). A fuller model should distinguish uncertainty in the coefficients of a fixed h","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2c26ce534ddcf4f7","tex":"Z_{\\lambda}","aliases":"Z_λ lambda","display":"Z_{\\lambda}","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-919fcda719441b01","tex":"Z_{\\lambda^{\\coarse}}","aliases":"Z_λ^coarse Z_λ^ c lambda coarse lambda mathrm","display":"Z_{\\lambda^{{\\mathrm c}}}","kind":"use","excerpt":"ent] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1bb74563d4a6f593","tex":"Z_{\\lambda^{\\fine}}","aliases":"Z_λ^fine Z_λ^ f lambda fine lambda mathrm","display":"Z_{\\lambda^{{\\mathrm f}}}","kind":"use","excerpt":"ent] Let λ^ f∈ R_>0^n and λ^ c∈ R_>0^q. If λ^ f=sC^Tλ^ c for some s>0, then C B_λ^ f = B_λ^ cC. Moreover, Z_λ^ f(μ) = sZ_λ^ c(Cμ).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"Combining the two inequalities gives e_k^+ ≤ α_ke_k-1^++β_k, where α_k=L_ B,kL_Φ,k, β_k=L_ B,kd_Φ,k+d_ B,k.","count":13},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b6ac1774de8bd95d","tex":"\\alpha^N","aliases":"α^N alpha","display":"\\alpha^N","kind":"use","excerpt":"[label=(roman*),leftmargin=2.8em] item If 0≤α<1, then e_N^+ ≤ α^Ne_0^+ + βfrac1-α^N1-α, limsup_N→∞e_N^+ ≤fracβ1-α.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fa75b85fd3d0168a","tex":"\\alpha_{ir}","aliases":"α_ir alpha","display":"\\alpha_{ir}","kind":"use","excerpt":"Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, a","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","excerpt":"[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-993a80257a1ce549","tex":"\\alpha_k","aliases":"α_k alpha","display":"\\alpha_k","kind":"use","excerpt":"Combining the two inequalities gives e_k^+ ≤ α_ke_k-1^++β_k, where α_k=L_ B,kL_Φ,k, β_k=L_ B,kd_Φ,k+d_ B,k.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"Combining the two inequalities gives e_k^+ ≤ α_ke_k-1^++β_k, where α_k=L_ B,kL_Φ,k, β_k=L_ B,kd_Φ,k+d_ B,k.","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-beaa4eefa4e0426c","tex":"\\beta_k","aliases":"β_k beta","display":"\\beta_k","kind":"use","excerpt":"Combining the two inequalities gives e_k^+ ≤ α_ke_k-1^++β_k, where α_k=L_ B,kL_Φ,k, β_k=L_ B,kd_Φ,k+d_ B,k.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability ","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-47a82d267a89aab8","tex":"\\Gamma_1^{\\coarse}","aliases":"Γ_1^coarse Γ_1^ c Gamma coarse Gamma mathrm","display":"\\Gamma_1^{{\\mathrm c}}","kind":"use","excerpt":"servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3f0b0b217383965d","tex":"\\Gamma_1^{\\fine}","aliases":"Γ_1^fine Γ_1^ f Gamma fine Gamma mathrm","display":"\\Gamma_1^{{\\mathrm f}}","kind":"use","excerpt":"servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c0d0febf945d1249","tex":"\\Gamma_k^{\\coarse}","aliases":"Γ_k^coarse Γ_k^ c Gamma coarse Gamma mathrm","display":"\\Gamma_k^{{\\mathrm c}}","kind":"use","excerpt":"observation sequences Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2e0993f0612bb573","tex":"\\Gamma_k^{\\fine}","aliases":"Γ_k^fine Γ_k^ f Gamma fine Gamma mathrm","display":"\\Gamma_k^{{\\mathrm f}}","kind":"use","excerpt":"observation sequences Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-37348467ba1d9c70","tex":"\\Gamma_N^{\\coarse}","aliases":"Γ_N^coarse Γ_N^ c Gamma coarse Gamma mathrm","display":"\\Gamma_N^{{\\mathrm c}}","kind":"use","excerpt":"servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4625479088e36891","tex":"\\Gamma_N^{\\fine}","aliases":"Γ_N^fine Γ_N^ f Gamma fine Gamma mathrm","display":"\\Gamma_N^{{\\mathrm f}}","kind":"use","excerpt":"servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-198bb3bf1a420a2e","tex":"\\Delta_4","aliases":"Δ_4 Delta","display":"\\Delta_4","kind":"use","excerpt":"A nonlinear fine system with four states and a quotient with two states. Let x=(x_1,x_2,x_3,x_4)^T∈Δ_4, C= 1 · 1 · 0 · 0 0 · 0 · 1 · 1 , so z=Cx=(z_1,z_2)^T, z_1=x_1+x_2, z_2=x_3+x_4. The coarse dynamics are dot z_1=-az_1+bz_2, dot z_2=az_1-bz_2, a,b>0. Define nonlinear internal exchange channels h_12(x) · = (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e4a6864a60fce542","tex":"\\Delta_{\\coarse}","aliases":"Δ_coarse Δ_ c Delta coarse Delta mathrm","display":"\\Delta_{{\\mathrm c}}","kind":"use","excerpt":"Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d1d329fa75d5b720","tex":"\\delta_F","aliases":"δ_F delta","display":"\\delta_F","kind":"use","excerpt":"Defect majorants for the full field. Suppose fine and proposed coarse fields have expansions F_ f(x)=∑_m≥0A_m^ f[x^⊗ m], F_ c(z)=∑_m≥0A_m^ c[z^⊗ m]. Define the field defect δ_F(x) = CF_ f(x)-F_ c(Cx). Whenever both series converge absolutely, δ_F(x) = ∑_m≥0Δ_m[x^⊗ m].","count":10},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fe711cb4ca0cd66d","tex":"\\Delta_{\\fine}","aliases":"Δ_fine Δ_ f Delta fine Delta mathrm","display":"\\Delta_{{\\mathrm f}}","kind":"use","excerpt":"Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d7dd8482a41b4759","tex":"\\Delta_m^{10}","aliases":"Δ_m^10 Delta","display":"\\Delta_m^{10}","kind":"use","excerpt":"[Interaction defect across two levels] Define Δ_m^21 =C_21A_m^(2)-A_m^(1)C_21^× m, Δ_m^10 =C_10A_m^(1)-A_m^(0)C_10^× m, and Δ_m^20 =C_20A_m^(2)-A_m^(0)C_20^× m. Then Δ_m^20 = C_10Δ_m^21 + Δ_m^10C_21^× m. Consequently, normΔ_m^20 ≤ normC_10normΔ_m^21 + normΔ_m^10normC_21^m.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e6f814b649129316","tex":"\\Delta_m^{20}","aliases":"Δ_m^20 Delta","display":"\\Delta_m^{20}","kind":"use","excerpt":"[Interaction defect across two levels] Define Δ_m^21 =C_21A_m^(2)-A_m^(1)C_21^× m, Δ_m^10 =C_10A_m^(1)-A_m^(0)C_10^× m, and Δ_m^20 =C_20A_m^(2)-A_m^(0)C_20^× m. Then Δ_m^20 = C_10Δ_m^21 + Δ_m^10C_21^× m. Consequently, normΔ_m^20 ≤ normC_10normΔ_m^21 + normΔ_m^10normC_21^m.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d9d37e5ed6a3f107","tex":"\\Delta_m^{21}","aliases":"Δ_m^21 Delta","display":"\\Delta_m^{21}","kind":"use","excerpt":"[Interaction defect across two levels] Define Δ_m^21 =C_21A_m^(2)-A_m^(1)C_21^× m, Δ_m^10 =C_10A_m^(1)-A_m^(0)C_10^× m, and Δ_m^20 =C_20A_m^(2)-A_m^(0)C_20^× m. Then Δ_m^20 = C_10Δ_m^21 + Δ_m^10C_21^× m. Consequently, normΔ_m^20 ≤ normC_10normΔ_m^21 + normΔ_m^10normC_21^m.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-bae8bcd390d9d51c","tex":"\\Delta_m^E","aliases":"Δ_m^E Delta","display":"\\Delta_m^E","kind":"use","excerpt":"[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5a2883928ff8dafe","tex":"\\Delta_m^{\\mathrm{keep}}","aliases":"Δ_m^keep Delta mathrm","display":"\\Delta_m^{\\mathrm{keep}}","kind":"use","excerpt":"Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a7dfeda1c71a9825","tex":"\\Delta_m^{L0}","aliases":"Δ_m^L0 Delta","display":"\\Delta_m^{L0}","kind":"use","excerpt":"[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c082c75e0b482cc6","tex":"\\Delta_m^{\\mathrm{total}}","aliases":"Δ_m^total Delta mathrm","display":"\\Delta_m^{\\mathrm{total}}","kind":"use","excerpt":"Combined quotient and pruning defect. Let the full fine coefficient split as A_m^ f =A_m^keep+R_m^ f. For a proposed coarse coefficient A_m^ c, define Δ_m^total =CA_m^ f-A_m^ cC^× m. Then Δ_m^total · = underbraceCA_m^keep-A_m^ cC^× m _Δ_m^keep +CR_m^ f. Therefore normΔ_m^total ≤ normΔ_m^keep +normCnormR_m^ f.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-13a22d0b6324cca3","tex":"\\Delta_m^{\\ell+1,\\ell}","aliases":"Δ_m^ℓ+1,ℓ Delta ell ell","display":"\\Delta_m^{\\ell+1,\\ell}","kind":"use","excerpt":"[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-542de1299c1c558f","tex":"\\Delta_n","aliases":"Δ_n Delta","display":"\\Delta_n","kind":"use","excerpt":"[Necessary condition for Bayesian refinement] Assume q≥2, every fiber of π is nonempty, and λ^ f,λ^ c are strictly positive. If C B_λ^ f(μ) = B_λ^ c(Cμ) for every μ in the interior of Δ_n, then there exists s>0 such that λ^ f=sC^Tλ^ c.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5b9b83552aa17d56","tex":"\\varepsilon^{\\infty}_{a;\\mathbf b}","aliases":"ε^∞_a; b varepsilon infty mathbf","display":"\\varepsilon^{\\infty}_{a;\\mathbf b}","kind":"use","excerpt":"athbf bw_ j . For scalar worst case loss, a minimax representative is the midrange a^ c,∞_a; b = frac12 ( max_ j∈ G_ bg_a; j + min_ j∈ G_ bg_a; j ), with optimal scalar fiber error ε^∞_a; b = frac12 ( max_ j∈ G_ bg_a; j - min_ j∈ G_ bg_a; j ). These formulas are local coefficient summaries. A global operator norm can couple several fibers and may r","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-17acb26caf5f1312","tex":"\\varepsilon_F","aliases":"ε_F varepsilon","display":"\\varepsilon_F","kind":"use","excerpt":"d Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d6b02083e2b0a177","tex":"\\varepsilon_m","aliases":"ε_m varepsilon","display":"\\varepsilon_m","kind":"use","excerpt":"[Analytic defect majorant] If normΔ_m≤ε_m and the scalar series d_F(R)=∑_m≥0ε_mR^m converges, then sup_normx≤ Rnormδ_F(x) ≤ d_F(R).","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The","count":13},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-26458abd5f858638","tex":"\\lambda^{\\coarse}","aliases":"λ^coarse λ^ c lambda coarse lambda mathrm","display":"\\lambda^{{\\mathrm c}}","kind":"use","excerpt":"Classical sufficiency theory asks whether a statistic preserves all information relevant to a chosen observation family [source reference]. Here the operator question is C B_λ^ f = B_λ^ cC. When both the flow identity [source reference] and the Bayesian identity [source reference] hold at each observation time, the whole filtering sequence commutes","count":18},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d238725330d7affa","tex":"\\lambda^{\\fine,\\eta}","aliases":"λ^fine,η λ^ f,η lambda fine eta lambda mathrm eta","display":"\\lambda^{{\\mathrm f},\\eta}","kind":"use","excerpt":"A controlled observation violation. Perturb the fine likelihood inside one fiber: λ^ f,η = C^Tλ^ c + η(1,-1,0,0)^T. Assume absη<λ_1^ c, so every component of λ^ f,η remains strictly positive. The fiber oscillation is 2absη and normr_λ_∞=absη. If the evidence is bounded below by z_*>0, then d_ B ≤ frac2absηz_*. The sweep in [source reference] varies","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f7220199e3ae07d7","tex":"\\lambda^T","aliases":"λ^T lambda","display":"\\lambda^T","kind":"use","excerpt":"Why Bayesian observations add another nonlinear operation. Prediction is only one part of a filter. Given a probability vector μ and a positive likelihood λ, Bayesian conditioning is B_λ(μ) = fracλodotμλ^Tμ. Likelihood multiplication is linear, but the evidence in the denominator couples all packets. The full update is therefore nonlinear even when","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b808a3c004c6530d","tex":"\\lambda_1^{\\coarse}","aliases":"λ_1^coarse λ_1^ c lambda coarse lambda mathrm","display":"\\lambda_1^{{\\mathrm c}}","kind":"use","excerpt":"A controlled observation violation. Perturb the fine likelihood inside one fiber: λ^ f,η = C^Tλ^ c + η(1,-1,0,0)^T. Assume absη<λ_1^ c, so every component of λ^ f,η remains strictly positive. The fiber oscillation is 2absη and normr_λ_∞=absη. If the evidence is bounded below by z_*>0, then d_ B ≤ frac2absηz_*. The sweep in [source reference] varies","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-975e1a0c6d000741","tex":"\\lambda_2","aliases":"λ_2 lambda","display":"\\lambda_2","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9691772080d0f10b","tex":"\\lambda_a^{\\coarse}","aliases":"λ_a^coarse λ_a^ c lambda coarse lambda mathrm","display":"\\lambda_a^{{\\mathrm c}}","kind":"use","excerpt":"ivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses of packets a and b is fracλ_i^ fuλ_k^ fv, whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. ","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4f0fd11ba1a41202","tex":"\\lambda_b^{\\coarse}","aliases":"λ_b^coarse λ_b^ c lambda coarse lambda mathrm","display":"\\lambda_b^{{\\mathrm c}}","kind":"use","excerpt":"ivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses of packets a and b is fracλ_i^ fuλ_k^ fv, whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. ","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ec8795a34a250c5a","tex":"\\lambda_j^{\\fine}","aliases":"λ_j^fine λ_j^ f lambda fine lambda mathrm","display":"\\lambda_j^{{\\mathrm f}}","kind":"use","excerpt":"whereas the coarse posterior ratio is fracλ_a^ cuλ_b^ cv. Equality for all u,v>0 gives fracλ_i^ fλ_k^ f = fracλ_a^ cλ_b^ c. Take another state j in the same fiber as i. Applying the same argument to j and k gives λ_j^ f=λ_i^ f. The fine likelihood is therefore constant on each fiber. Write its value on fiber a as overlineλ_a. Equation [source refer","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-791cacb49489c5ae","tex":"\\lambda_k^{\\coarse}","aliases":"λ_k^coarse λ_k^ c lambda coarse lambda mathrm","display":"\\lambda_k^{{\\mathrm c}}","kind":"use","excerpt":"Sequential evidence error. For brevity, write λ_k^ f=λ_y_k^ f and λ_k^ c=λ_y_k^ c, and decompose λ_k^ f =s_kC^Tλ_k^ c+r_k. This gives z_k^ f-s_kz_k^ c · = s_k(λ_k^ c)^T (Cwidehatμ_k^ f-widehatμ_k^ c) +r_k^Twidehatμ_k^ f.","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fc0a7dde61026075","tex":"\\lambda_k^{\\fine}","aliases":"λ_k^fine λ_k^ f lambda fine lambda mathrm","display":"\\lambda_k^{{\\mathrm f}}","kind":"use","excerpt":"es-equality-all-priors extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses of packet","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-75281f36da6b8ff7","tex":"\\lambda_n","aliases":"λ_n lambda","display":"\\lambda_n","kind":"use","excerpt":"Classical Bayesian update. On the finite simplex, let L_λ=diag(λ_1,…,λ_n), λ_i>0. With ℓ(x)= 1^Tx, N_ℓ(L_λ)(μ) = fracλodotμλ^Tμ = B_λ(μ). The evidence is Z_λ(μ)=λ^Tμ. The composition law becomes B_λ_2∘ B_λ_1 = B_λ_1odotλ_2.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ef9702d84f2cbc12","tex":"\\lambda_y","aliases":"λ_y lambda","display":"\\lambda_y","kind":"use","excerpt":"A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5fa93ed10435aec5","tex":"\\lambda_y^{\\coarse}","aliases":"λ_y^coarse λ_y^ c lambda coarse lambda mathrm","display":"\\lambda_y^{{\\mathrm c}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3477f0f32447dd6e","tex":"\\lambda_y^{\\fine}","aliases":"λ_y^fine λ_y^ f lambda fine lambda mathrm","display":"\\lambda_y^{{\\mathrm f}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f )","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e03b36fd25b1b6dc","tex":"\\lambda_{y_1}^{\\coarse}","aliases":"λ_y_1^coarse λ_y_1^ c lambda coarse lambda mathrm","display":"\\lambda_{y_1}^{{\\mathrm c}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-59669c95105275ed","tex":"\\lambda_{y_1}^{\\fine}","aliases":"λ_y_1^fine λ_y_1^ f lambda fine lambda mathrm","display":"\\lambda_{y_1}^{{\\mathrm f}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ff1b1842fe78e30b","tex":"\\lambda_{y_k}^{\\coarse}","aliases":"λ_y_k^coarse λ_y_k^ c lambda coarse lambda mathrm","display":"\\lambda_{y_k}^{{\\mathrm c}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b7a8524467886a9f","tex":"\\lambda_{y_k}^{\\fine}","aliases":"λ_y_k^fine λ_y_k^ f lambda fine lambda mathrm","display":"\\lambda_{y_k}^{{\\mathrm f}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e346c90c713d7696","tex":"\\lambda_{y_N}^{\\coarse}","aliases":"λ_y_N^coarse λ_y_N^ c lambda coarse lambda mathrm","display":"\\lambda_{y_N}^{{\\mathrm c}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-583a2313f4752bd7","tex":"\\lambda_{y_N}^{\\fine}","aliases":"λ_y_N^fine λ_y_N^ f lambda fine lambda mathrm","display":"\\lambda_{y_N}^{{\\mathrm f}}","kind":"use","excerpt":"[Hidden Markov lumpability] Let M_ f and M_ c be column stochastic matrices satisfying CM_ f=M_ cC, and let every observation likelihood satisfy λ_y^ f=s_yC^Tλ_y^ c. Then, for every observation sequence y_1:N, · C( B_λ_y_N^ f∘ M_ f ∘⋯∘ B_λ_y_1^ f∘ M_ f ) · = ( B_λ_y_N^ c∘ M_ c ∘⋯∘ B_λ_y_1^ c∘ M_ c )C.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"Why Bayesian observations add another nonlinear operation. Prediction is only one part of a filter. Given a probability vector μ and a positive likelihood λ, Bayesian conditioning is B_λ(μ) = fracλodotμλ^Tμ. Likelihood multiplication is linear, but the evidence in the denominator couples all packets. The full update is therefore nonlinear even when","count":47},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-62662d7aeb8b2708","tex":"\\mu_0^{\\coarse}","aliases":"μ_0^coarse μ_0^ c mu coarse mu mathrm","display":"\\mu_0^{{\\mathrm c}}","kind":"use","excerpt":"[Exact posterior aggregation] If μ_0^ c=Cμ_0^ f, then Cμ_N^ f=μ_N^ c for every finite observation sequence, where μ_N^ f=Ψ_1:N^ f(μ_0^ f), μ_N^ c=Ψ_1:N^ c(μ_0^ c).","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-cd1b8d5f2412ef79","tex":"\\mu_0^{\\fine}","aliases":"μ_0^fine μ_0^ f mu fine mu mathrm","display":"\\mu_0^{{\\mathrm f}}","kind":"use","excerpt":"[Exact posterior aggregation] If μ_0^ c=Cμ_0^ f, then Cμ_N^ f=μ_N^ c for every finite observation sequence, where μ_N^ f=Ψ_1:N^ f(μ_0^ f), μ_N^ c=Ψ_1:N^ c(μ_0^ c).","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-252454ad397ec5a8","tex":"\\mu_i","aliases":"μ_i mu","display":"\\mu_i","kind":"use","excerpt":"Likelihood variation inside a fiber. For diagonal likelihood operators, write λ^ f = sC^Tλ^ c+r_λ. Then R_Lμ=C(r_λodotμ). Since block sum aggregation is an ℓ^1 contraction on signed vectors, normR_Lμ_1 ≤ ∑_iμ_iabsr_λ,i ≤ normr_λ_∞.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1095b53f3fab6b6b","tex":"\\mu_{k-1}^{\\coarse}","aliases":"μ_k-1^coarse μ_k-1^ c mu coarse mu mathrm","display":"\\mu_{k-1}^{{\\mathrm c}}","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f43384379215e546","tex":"\\mu_{k-1}^{\\fine}","aliases":"μ_k-1^fine μ_k-1^ f mu fine mu mathrm","display":"\\mu_{k-1}^{{\\mathrm f}}","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-9d31a3e94bb77f94","tex":"\\mu_k^{\\coarse}","aliases":"μ_k^coarse μ_k^ c mu coarse mu mathrm","display":"\\mu_k^{{\\mathrm c}}","kind":"use","excerpt":"Prediction and update errors. Let the fine recursion be widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), μ_k^ f = B_k^ f(widehatμ_k^ f), and the coarse recursion be widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), μ_k^ c = B_k^ c(widehatμ_k^ c). Define the discrepancies before and after each update by e_k^- =normCwidehatμ_k^ f-widehatμ_k^ c_1, e_k^+ =normCμ_k^ f-μ_k^ c_1. At the ","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1ad8f38aec19f479","tex":"\\mu_k^{\\fine}","aliases":"μ_k^fine μ_k^ f mu fine mu mathrm","display":"\\mu_k^{{\\mathrm f}}","kind":"use","excerpt":"Prediction and update errors. Let the fine recursion be widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), μ_k^ f = B_k^ f(widehatμ_k^ f), and the coarse recursion be widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), μ_k^ c = B_k^ c(widehatμ_k^ c). Define the discrepancies before and after each update by e_k^- =normCwidehatμ_k^ f-widehatμ_k^ c_1, e_k^+ =normCμ_k^ f-μ_k^ c_1. At the ","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f0a1dfb48130a62a","tex":"\\mu_N^{\\coarse}","aliases":"μ_N^coarse μ_N^ c mu coarse mu mathrm","display":"\\mu_N^{{\\mathrm c}}","kind":"use","excerpt":"[Exact posterior aggregation] If μ_0^ c=Cμ_0^ f, then Cμ_N^ f=μ_N^ c for every finite observation sequence, where μ_N^ f=Ψ_1:N^ f(μ_0^ f), μ_N^ c=Ψ_1:N^ c(μ_0^ c).","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-070f393f89ee839e","tex":"\\mu_N^{\\fine}","aliases":"μ_N^fine μ_N^ f mu fine mu mathrm","display":"\\mu_N^{{\\mathrm f}}","kind":"use","excerpt":"[Exact posterior aggregation] If μ_0^ c=Cμ_0^ f, then Cμ_N^ f=μ_N^ c for every finite observation sequence, where μ_N^ f=Ψ_1:N^ f(μ_0^ f), μ_N^ c=Ψ_1:N^ c(μ_0^ c).","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"Positivity and simplex preservation. Let Δ_ f ={μ∈ R_+^n: 1^Tμ=1}, Δ_ c ={ν∈ R_+^q: 1^Tν=1}. For block sum aggregation, CΔ_ f=Δ_ c and 1_ c^TC= 1_ f^T.","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6a331de74ff475c9","tex":"\\pi^{-1}","aliases":"π^-1 pi","display":"\\pi^{-1}","kind":"use","excerpt":"Both sides of [source reference] extend continuously to any prior with nonzero evidence. Strict positivity guarantees nonzero evidence throughout the closed simplex. Fix two distinct coarse packets a and b. Choose i∈π^-1(a) and k∈π^-1(b), then consider a prior supported on {i,k} with masses u,v>0. On the fine side, the ratio of the posterior masses","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"[H] centering includegraphics[width=0.78linewidth]posterior_error_certificate.pdf captionJoint dynamical and likelihood perturbation sweep with ε=η=ρ. For small ρ, the final posterior discrepancy grows linearly, while the recursive certificate remains above every measured value.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b15d595fecaa0b33","tex":"\\Phi_k^{\\coarse}","aliases":"Φ_k^coarse Φ_k^ c Phi coarse Phi mathrm","display":"\\Phi_k^{{\\mathrm c}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":9},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-30a3cef04f09fd0e","tex":"\\Phi_k^{\\fine}","aliases":"Φ_k^fine Φ_k^ f Phi fine Phi mathrm","display":"\\Phi_k^{{\\mathrm f}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":8},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a1c266a6f8312e40","tex":"\\Phi_t^c","aliases":"Φ_t^c Phi","display":"\\Phi_t^c","kind":"use","excerpt":"CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-442dea07758dd057","tex":"\\Phi_T^{\\coarse}","aliases":"Φ_T^coarse Φ_T^ c Phi coarse Phi mathrm","display":"\\Phi_T^{{\\mathrm c}}","kind":"use","excerpt":"[Coefficient certificate for a flow segment] If normx(s)≤ R and Proposition~[source reference] applies, then a flow interval of length T satisfies normCΦ_T^ f(x_0)-Φ_T^ c(Cx_0) ≤ d_F(R)frace^LT-1L for L>0, with T d_F(R) at L=0.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d43333a8f8ace2f1","tex":"\\Phi_t^f","aliases":"Φ_t^f Phi","display":"\\Phi_t^f","kind":"use","excerpt":"CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f2a0b3a9ed421079","tex":"\\Phi_T^{\\fine}","aliases":"Φ_T^fine Φ_T^ f Phi fine Phi mathrm","display":"\\Phi_T^{{\\mathrm f}}","kind":"use","excerpt":"[Coefficient certificate for a flow segment] If normx(s)≤ R and Proposition~[source reference] applies, then a flow interval of length T satisfies normCΦ_T^ f(x_0)-Φ_T^ c(Cx_0) ≤ d_F(R)frace^LT-1L for L>0, with T d_F(R) at L=0.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-79760fe067be0674","tex":"\\Phi_T^{\\fine,\\varepsilon}","aliases":"Φ_T^fine,ε Φ_T^ f,ε Phi fine varepsilon Phi mathrm varepsilon","display":"\\Phi_T^{{\\mathrm f},\\varepsilon}","kind":"use","excerpt":"1, so, on the simplex, normδ_F(x)_1 =2ε x_1 ≤2ε. The coarse Markov flow on two states is an ℓ^1 contraction. Over one interval, this gives normCΦ_T^ f,ε(x)-Φ_T^ c(Cx)_1 ≤2ε T.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2bf660e666063070","tex":"\\Phi_{t_k-t_{k-1}}^{\\coarse}","aliases":"Φ_t_k-t_k-1^coarse Φ_t_k-t_k-1^ c Phi coarse Phi mathrm","display":"\\Phi_{t_k-t_{k-1}}^{{\\mathrm c}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0500556dc908ef68","tex":"\\Phi_{t_k-t_{k-1}}^{\\fine}","aliases":"Φ_t_k-t_k-1^fine Φ_t_k-t_k-1^ f Phi fine Phi mathrm","display":"\\Phi_{t_k-t_{k-1}}^{{\\mathrm f}}","kind":"use","excerpt":"Finite observation sequences. Let 0=t_0<t_1<⋯<t_N and define prediction maps Φ_k^ f=Φ_t_k-t_k-1^ f, Φ_k^ c=Φ_t_k-t_k-1^ c. For observations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-937918561265abae","tex":"\\Psi_{1:N}^c","aliases":"Ψ_1:N^c Psi","display":"\\Psi_{1:N}^c","kind":"use","excerpt":"CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6bafd56a397d7acb","tex":"\\Psi_{1:N}^{\\coarse}","aliases":"Ψ_1:N^coarse Ψ_1:N^ c Psi coarse Psi mathrm","display":"\\Psi_{1:N}^{{\\mathrm c}}","kind":"use","excerpt":"servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-df4f0643eef0cbc8","tex":"\\Psi_{1:N}^f","aliases":"Ψ_1:N^f Psi","display":"\\Psi_{1:N}^f","kind":"use","excerpt":"CL_y^f=s_yL_y^cC ⇒ C B_y^f= B_y^cC ⇒ CΨ_1:N^f=Ψ_1:N^cC.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f781ee039cb1a5c5","tex":"\\Psi_{1:N}^{\\fine}","aliases":"Ψ_1:N^fine Ψ_1:N^ f Psi fine Psi mathrm","display":"\\Psi_{1:N}^{{\\mathrm f}}","kind":"use","excerpt":"servations y_1,…,y_N, let B_k^ f= B_λ_y_k^ f, B_k^ c= B_λ_y_k^ c. Pair each prediction with the observation that follows it: Γ_k^ f = B_k^ f∘Φ_k^ f, Γ_k^ c = B_k^ c∘Φ_k^ c, and define the resulting filtering cocycles Ψ_1:N^ f =Γ_N^ f∘⋯∘Γ_1^ f, Ψ_1:N^ c =Γ_N^ c∘⋯∘Γ_1^ c.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"Likelihood variation inside a fiber. For diagonal likelihood operators, write λ^ f = sC^Tλ^ c+r_λ. Then R_Lμ=C(r_λodotμ). Since block sum aggregation is an ℓ^1 contraction on signed vectors, normR_Lμ_1 ≤ ∑_iμ_iabsr_λ,i ≤ normr_λ_∞.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-837406a230d80642","tex":"\\ell^\\infty","aliases":"ℓ^∞ ell infty","display":"\\ell^\\infty","kind":"use","excerpt":"Apply the duality between ℓ^1 and ℓ^∞. The total variation identity is standard for finite probability vectors.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-bf2d64ae9ba67fb4","tex":"\\ell_{\\coarse}","aliases":"ℓ_coarse ℓ_ c ell coarse ell mathrm","display":"\\ell_{{\\mathrm c}}","kind":"use","excerpt":"Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-6085476db9386410","tex":"\\ell_{\\fine}","aliases":"ℓ_fine ℓ_ f ell fine ell mathrm","display":"\\ell_{{\\mathrm f}}","kind":"use","excerpt":"Let X_ f and X_ c be finite dimensional ordered vector spaces with positive cones K_ f and K_ c. Let ℓ_ f:X_ f→ R, ℓ_ c:X_ c→ R be positive mass functionals. Define normalized state spaces S_ f ={x∈ K_ f:ℓ_ f(x)=1}, S_ c ={z∈ K_ c:ℓ_ c(z)=1}. Assume C is positive and preserves mass: ℓ_ cC=ℓ_ f.","count":5},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"CA_m^f=A_m^cC^× m ⇒ CF_f=F_cC ⇒ CΦ_t^f=Φ_t^cC.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"At order m, the central question is ∃ A_m^ c such that C A_m^ f=A_m^ cC^× m? If this question has an answer at every order, the quotient relation can be carried from individual interactions to the full field, its derivatives, its interaction trees, and its flow.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"ensor is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S, P_Cx_r, · r∉ S.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"Interpretation for reaction networks. Consider a polynomial reaction system dot x=S v(x), v_r(x)=k_r∏_i x_i^α_ir. Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column. The inputs are the reactants. A block sum aggregation is exact precisely when the monomial coefficients, a","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-53acc6e69a0ea1c1","tex":"\\prod_{j=1}^N","aliases":"∏_j=1^N prod","display":"\\prod_{j=1}^N","kind":"use","excerpt":"[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ed2f6e83c1e20345","tex":"\\prod_{j=k+1}^N","aliases":"∏_j=k+1^N prod","display":"\\prod_{j=k+1}^N","kind":"use","excerpt":"[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5312172804774212","tex":"\\prod_k","aliases":"∏_k prod","display":"\\prod_k","kind":"use","excerpt":"[Certificate for cumulative log evidence] Let Z_1:N^ f=∏_kz_k^ f and Z_1:N^ c=∏_kz_k^ c. Then abs log Z_1:N^ f -∑_k=1^Nlog s_k -log Z_1:N^ c ≤ ∑_k=1^N frac s_knormλ_k^ c_∞e_k^- +normr_k_∞ z_*,k.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-52dec9661f2d00d2","tex":"\\prod_{k=1}^N","aliases":"∏_k=1^N prod","display":"\\prod_{k=1}^N","kind":"use","excerpt":"Sequential evidence. Let widehatμ_k^ f =Φ_k^ f(μ_k-1^ f), widehatμ_k^ c =Φ_k^ c(μ_k-1^ c), and define the step evidence z_k^ f =(λ_y_k^ f)^Twidehatμ_k^ f, z_k^ c =(λ_y_k^ c)^Twidehatμ_k^ c. Set Z_1:N^ f=∏_k=1^N z_k^ f, Z_1:N^ c=∏_k=1^N z_k^ c.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a80e2180b2488520","tex":"\\sum_{\\mathbf j\\in\\cG_{\\mathbf b}}","aliases":"∑_ j∈cG_ b ∑_ j∈ G_ b sum mathbf in cG mathbf sum mathbf in mathcal mathbf","display":"\\sum_{\\mathbf j\\in{\\mathcal G}_{\\mathbf b}}","kind":"use","excerpt":"Approximate coarse coefficients. When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution. Under scalar squared loss with nonnegative weights w_ j on a fiber product, assume their total weight is positive. The least squares representative is a^ c,LS_a; b = frac ∑_ j∈ G_ b w_ jg_a; j ∑_ j∈ ","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-4c30f98e8665b83a","tex":"\\sum_{a=1}^q","aliases":"∑_a=1^q sum","display":"\\sum_{a=1}^q","kind":"use","excerpt":"Applying C to [source reference] gives CA_m^ f(e_j_1,…,e_j_m) = ∑_a=1^q g^(m)_a;j_1,…,j_mf_a. If a coarse tensor exists, this vector equals A_m^ c(f_π(j_1),…,f_π(j_m)), which depends only on the coarse input labels. This proves necessity. Conversely, constancy on each fiber makes [source reference] well defined. Equality on every tuple of fine basi","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-59a73cc552080ec9","tex":"\\sum_C","aliases":"∑_C sum","display":"\\sum_C","kind":"use","excerpt":"Posterior distributions over resolutions. Instead of selecting one C, place a posterior on a finite family of candidate resolutions: p(Cmid D) propto p(Dmid C)p(C). Predictions then average both parameter and topology uncertainty: p(z_tmid D) = ∑_Cp(z_tmid C,D)p(Cmid D). A fuller model should distinguish uncertainty in the coefficients of a fixed h","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-20794c41f8dd0413","tex":"\\sum_h","aliases":"∑_h sum","display":"\\sum_h","kind":"use","excerpt":"Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f09f4361c9ce0ae5","tex":"\\sum_{h\\in\\cE_m^{\\mathrm{keep}}}","aliases":"∑_h∈cE_m^keep ∑_h∈ E_m^keep sum in cE mathrm sum in mathcal mathrm","display":"\\sum_{h\\in{\\mathcal E}_m^{\\mathrm{keep}}}","kind":"use","excerpt":"Retained field and named residual. Choose a retained hyperedge set E_m^keep. Define widehat A_m^keep = ∑_h∈ E_m^keep widehat A_m,h, R_m = widehat A_m-widehat A_m^keep.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-1edb6af544aee300","tex":"\\sum_{h\\notin\\cE_m^{\\mathrm{keep}}}","aliases":"∑_h∉cE_m^keep ∑_h∉ E_m^keep sum notin cE mathrm sum notin mathcal mathrm","display":"\\sum_{h\\notin{\\mathcal E}_m^{\\mathrm{keep}}}","kind":"use","excerpt":"[Exact pruning residual] Under the hypotheses of [source reference], normR_m_HS^2 = ∑_h∉ E_m^keep normwidehat A_m,h_HS^2.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Insert I_X=∑_jP_j in every input slot and I_Y=∑_iQ_i after applying A. Multilinearity distributes all finite sums and gives [source reference].","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-2ee5b079d386ca9e","tex":"\\sum_{i,j_1,\\ldots,j_m}","aliases":"∑_i,j_1,…,j_m sum ldots","display":"\\sum_{i,j_1,\\ldots,j_m}","kind":"use","excerpt":"[Exact multilinear packet reconstruction] Every multilinear map A:X^m→ Y of order m satisfies A(x_1,…,x_m) = ∑_i,j_1,…,j_m A_imid j_1,…,j_m (P_j_1x_1,…,P_j_mx_m).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-22f1acef20c66c4f","tex":"\\sum_{i:\\pi(i)=a}","aliases":"∑_i:π(i)=a sum pi","display":"\\sum_{i:\\pi(i)=a}","kind":"use","excerpt":"R^q. Define Ce_i=f_π(i), (Cx)_a=∑_π(i)=ax_i. Suppose A_m^ f(e_j_1,…,e_j_m) = ∑_i=1^n a^ f,(m)_i;j_1,…,j_me_i. For each coarse output packet a, sum the fine coefficients over its output fiber and define g^(m)_a;j_1,…,j_m = ∑_i:π(i)=a a^ f,(m)_i;j_1,…,j_m.","count":4},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Insert I_X=∑_jP_j in every input slot and I_Y=∑_iQ_i after applying A. Multilinearity distributes all finite sums and gives [source reference].","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3301e77ef9879552","tex":"\\sum_{k=1}^N","aliases":"∑_k=1^N sum","display":"\\sum_{k=1}^N","kind":"use","excerpt":"[Certified approximate filtering] Under the assumptions above, e_N^+ ≤ (∏_j=1^Nα_j)e_0^+ + ∑_k=1^N (∏_j=k+1^Nα_j) β_k. Here and below, an empty product equals one. With exact initial aggregation e_0^+=0, e_N^+ ≤ ∑_k=1^N (∏_j=k+1^Nα_j) (L_ B,kd_Φ,k+d_ B,k).","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7dd1f8372600c494","tex":"\\sum_m","aliases":"∑_m sum","display":"\\sum_m","kind":"use","excerpt":"For analytic fields on a ball of radius R, d_F(R) ≤ ∑_m normΔ_m^keepR^m + normC∑_mnormR_m^ fR^m. The flow and posterior bounds use this combined quantity. A pruning decision is justified only when the omitted part remains explicit and is carried through the later estimates like any other model error.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-668e17fed94fbb22","tex":"\\sum_{m=0}^{M}","aliases":"∑_m=0^M sum","display":"\\sum_{m=0}^{M}","kind":"use","excerpt":"[Coefficientwise characterization of polynomial descent] Let F_ f(x)=∑_m=0^MA_m^ f[x^⊗ m], F_ c(z)=∑_m=0^MA_m^ c[z^⊗ m], where every A_m^ f and A_m^ c is symmetric. Then CF_ f=F_ cC if and only if CA_m^ f=A_m^ cC^× m (0≤ m≤ M).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-187f645f69d06fb0","tex":"\\sum_{m=0}^{\\infty}","aliases":"∑_m=0^∞ sum infty","display":"\\sum_{m=0}^{\\infty}","kind":"use","excerpt":"Descent of polynomial and analytic vector fields. Let F_ f(x) = ∑_m=0^∞A_m^ f[x^⊗ m] be locally analytic, where A_m^ f is a symmetric multilinear map of order m for m≥1 and A_0^ f∈ X_ f. Throughout, A_m[x^⊗ m] =A_m(x,…,x).","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-d1cabd23a4e71916","tex":"\\sum_{m\\geq0}","aliases":"∑_m≥0 sum geq","display":"\\sum_{m\\geq0}","kind":"use","excerpt":"Why hypergraphs appear. Suppose a local analytic vector field has the expansion F_ f(x)=∑_m≥0A_m^ f[x,…,x]. The linear coefficient A_1^ f moves one input packet into one output packet. The quadratic coefficient A_2^ f combines two input packets before producing an output. In general, A_m^ f combines m inputs. We record this incidence by the directe","count":6},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c10c1b41782d8ca9","tex":"\\sum_{r=1}^m","aliases":"∑_r=1^m sum","display":"\\sum_{r=1}^m","kind":"use","excerpt":"To compare (ii) and (iv), identify a multilinear map of order m with its linearization on the algebraic tensor product. Since C is surjective, Ker(C^⊗ m) = ∑_r=1^m X_ f^⊗(r-1)⊗Ker C⊗ X_ f^⊗(m-r). Choose a splitting X_ f=E(X_ c)⊕Ker C and expand the tensor power. Vanishing of widehat A_m^ f on this sum is exactly the slot condition. Thus (ii) and (i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-7fed2ade50062727","tex":"\\sum_{r>k}","aliases":"∑_r>k sum","display":"\\sum_{r>k}","kind":"use","excerpt":"[Optimal retention of k orthogonal hyperedges] Among all retained sets of cardinality k, the set {h_1,…,h_k} minimizes the Hilbert Schmidt residual. The minimum is min_| E|=k normR_m( E)_HS^2 = ∑_r>ke_h_r.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-ee373b78330c8be1","tex":"\\sum_y","aliases":"∑_y sum","display":"\\sum_y","kind":"use","excerpt":"A budgeted design problem. These terms lead to a practical design problem. The objective below records the tradeoff without claiming that it can always be solved efficiently: min_C,{A_m^ c},{ E_m^keep} [ ∑_mw_mnormΔ_m^total +λ∑_mCost( E_m^keep) +γ∑_y d_ B,y ]. The minimization is subject to any required positivity, mass, rank, and interpretability ","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-5086dc901b465b92","tex":"\\sum_{\\eta}","aliases":"∑_η sum eta","display":"\\sum_{\\eta}","kind":"use","excerpt":"Hyperedge entropy. For nonzero total interaction energy, define p_h=frace_h∑_ηe_η, H_m=-∑_hp_hlog p_h, and the effective hyperedge count N_eff,m=e^H_m. Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges. High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution. The","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-43e727cf5aadc16c","tex":"\\sum_{\\ell=0}^{L-1}","aliases":"∑_ℓ=0^L-1 sum ell","display":"\\sum_{\\ell=0}^{L-1}","kind":"use","excerpt":"[Telescoping interaction defect] For an interaction of order m assigned at every level, Δ_m^L0 = ∑_ℓ=0^L-1 C_ℓ0 Δ_m^ℓ+1,ℓ C_L,ℓ+1^× m, where C_00=I and C_LL=I. Consequently, normΔ_m^L0 ≤ ∑_ℓ=0^L-1 normC_ℓ0 normΔ_m^ℓ+1,ℓ normC_L,ℓ+1^m.","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-8fc9ba4c61093a8e","tex":"\\sum_{\\varnothing\\neq S\\subseteq\\{1,\\ldots,m\\}}","aliases":"∑_∅≠ S⊆{1,…,m} sum varnothing neq subseteq ldots","display":"\\sum_{\\varnothing\\neq S\\subseteq\\{1,\\ldots,m\\}}","kind":"use","excerpt":"[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"[Compatibility with Fa a di Bruno composition] Let F_ f and F_ c be smooth maps with CF_ f=F_ cC, and let G_ f and G_ c satisfy the same relation. Then each rooted tree in the Fa a di Bruno expansion, written in packet coordinates, descends to the corresponding coarse tree. Moreover, C(G_ f∘ F_ f) =(G_ c∘ F_ c)C.","count":7},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"Descent of polynomial and analytic vector fields. Let F_ f(x) = ∑_m=0^∞A_m^ f[x^⊗ m] be locally analytic, where A_m^ f is a symmetric multilinear map of order m for m≥1 and A_0^ f∈ X_ f. Throughout, A_m[x^⊗ m] =A_m(x,…,x).","count":20},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"[Approximate quotient flow bound] Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-a2972272b24e9c45","tex":"\\int_0^t","aliases":"∫_0^t int","display":"\\int_0^t","kind":"use","excerpt":"[Approximate quotient flow bound] Let dot x=F_ f(x), dot z=F_ c(z), and suppose both solutions remain in domains on which normF_ c(u)-F_ c(v) ≤ Lnormu-v. Set e(t)=normCx(t)-z(t). Then e(t) ≤ e^Lte(0) + ∫_0^t e^L(t-s)normδ_F(x(s)) d s. If e(0)=0 and normδ_F(x(s))≤ε_F on [0,t], then e(t) ≤ displaystyle ε_Ffrace^Lt-1L, · L>0,","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"[Nonlinear congruence interpretation] For a general map F:X_ f→ X_ f, descent through C is equivalent to Cx=Cy Longrightarrow CF(x)=CF(y). Thus the relation x∼_Cy, defined by Cx=Cy, must be a congruence for the coarse output of F. [source reference] turns that statement into a condition on each multilinear coefficient.","count":3},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-3b7854340bf5e600","tex":"\\sim_C","aliases":"∼_C sim","display":"\\sim_C","kind":"use","excerpt":"[Nonlinear congruence interpretation] For a general map F:X_ f→ X_ f, descent through C is equivalent to Cx=Cy Longrightarrow CF(x)=CF(y). Thus the relation x∼_Cy, defined by Cx=Cy, must be a congruence for the coarse output of F. [source reference] turns that statement into a condition on each multilinear coefficient.","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-03f857a6f80c8728","tex":"\\sim_{\\mathrm{obs}}","aliases":"∼_obs sim mathrm","display":"\\sim_{\\mathrm{obs}}","kind":"use","excerpt":"A finite observational partition lemma. Let Y be a finite observation alphabet and let each state i have a strictly positive likelihood profile Λ(i) = (λ_y(i))_y∈ Y. Define i∼_obsj Longleftrightarrow Λ(i)=Λ(j).","count":2},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"[Expansion of the section defect by hidden inputs] The defect of [source reference] is Δ_m^E(x_1,…,x_m) = CA_m^ f(x_1,…,x_m) - CA_m^ f(P_Cx_1,…,P_Cx_m), and admits the exact expansion Δ_m^E(x_1,…,x_m) = ∑_∅≠ S⊆{1,…,m} CA_m^ f(y_1^S,…,y_m^S), where y_r^S= Q_Cx_r, · r∈ S,","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"Norm conventions. For a multilinear map A:X^m→ Y of order m, let normA = sup_normx_r≤1 normA(x_1,…,x_m). If E is a fixed right inverse of C, then any exact coarse tensor necessarily satisfies A_m^ c(z_1,…,z_m) = CA_m^ f(Ez_1,…,Ez_m), so normA_m^ c ≤ normCnormA_m^ fnormE^m. The formula also makes the later uniqueness argument immediate.","count":86},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"To compare (ii) and (iv), identify a multilinear map of order m with its linearization on the algebraic tensor product. Since C is surjective, Ker(C^⊗ m) = ∑_r=1^m X_ f^⊗(r-1)⊗Ker C⊗ X_ f^⊗(m-r). Choose a splitting X_ f=E(X_ c)⊕Ker C and expand the tensor power. Vanishing of widehat A_m^ f on this sum is exactly the slot condition. Thus (ii) and (i","count":1},{"work":"research-hypergraph-bayesian-refinement","url":"notation/research-hypergraph-bayesian-refinement.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"The map B_m⟼ B_mC^× m is injective because C is surjective. Its range is a finite dimensional linear subspace of L^m(X_ f;X_ c), hence it is closed. In a finite dimensional normed space, the distance to a nonempty closed subspace attains its minimum. A nearest point in this range therefore exists. Any such point has a unique preimage under the disp","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":7},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"Each mutable packet family carries a generation number g. A command records the generation it consumed. Refinement increments g for the affected family. A provisional result may commit only if g_rm consumed=g_rm current.","count":9},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":3},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":3},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":4},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":2},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-2caf829ea04d5e2b","tex":"B_{\\rm HBM}","aliases":"B_rm HBM rm","display":"B_{\\rm HBM}","kind":"use","excerpt":"The PRISM-1R planner should minimize a physicalized cost rather than arithmetic alone: C(G)= α F +β B_rm HBM +γ B_rm SRAM +δ H_rm NoC +η L_rm sync +κ C_rm cert +ρ E[C_rm replay].","count":2},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-95a974adf036b9bd","tex":"B_{\\rm SRAM}","aliases":"B_rm SRAM rm","display":"B_{\\rm SRAM}","kind":"use","excerpt":"The PRISM-1R planner should minimize a physicalized cost rather than arithmetic alone: C(G)= α F +β B_rm HBM +γ B_rm SRAM +δ H_rm NoC +η L_rm sync +κ C_rm cert +ρ E[C_rm replay].","count":2},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-054938768825867d","tex":"\\operatorname{bank}","aliases":"bank operatorname","display":"\\operatorname{bank}","kind":"use","excerpt":"A prototype mapping may use a configurable XOR swizzle, bank(w)= (w_3:0⊕ w_7:4⊕ h(P))bmod 16, where w is the local word index and h(P) is a four-bit packet/phase hash. This is an implementation policy, not part of the mathematical contract, and should be replaced if FPGA counters show worse behavior for target transforms.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-e76ca40698ecc64e","tex":"\\bfseries","aliases":"bfseries","display":null,"kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"A prototype mapping may use a configurable XOR swizzle, bank(w)= (w_3:0⊕ w_7:4⊕ h(P))bmod 16, where w is the local word index and h(P) is a four-bit packet/phase hash. This is an implementation policy, not part of the mathematical contract, and should be replaced if FPGA counters show worse behavior for target transforms.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-fa0e21b40952f33b","tex":"\\operatorname{bytes}","aliases":"bytes operatorname","display":"\\operatorname{bytes}","kind":"use","excerpt":"The placement objective is not merely minimum latency. It is minimum structured movement: min ∑_e∈ E bytes(e)×hops(e).","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":4},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":7},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-c561c31f71ef9395","tex":"C_{\\rm cert}","aliases":"C_rm cert rm","display":"C_{\\rm cert}","kind":"use","excerpt":"The PRISM-1R planner should minimize a physicalized cost rather than arithmetic alone: C(G)= α F +β B_rm HBM +γ B_rm SRAM +δ H_rm NoC +η L_rm sync +κ C_rm cert +ρ E[C_rm replay].","count":2},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-a7fad5a9ff656d52","tex":"C_{\\rm replay}","aliases":"C_rm replay rm","display":"C_{\\rm replay}","kind":"use","excerpt":"The PRISM-1R planner should minimize a physicalized cost rather than arithmetic alone: C(G)= α F +β B_rm HBM +γ B_rm SRAM +δ H_rm NoC +η L_rm sync +κ C_rm cert +ρ E[C_rm 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amortized.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-76f485281fe641e3","tex":"S_{\\rm architecture}","aliases":"S_rm architecture rm","display":"S_{\\rm architecture}","kind":"use","excerpt":"Each experiment should separate: S_rm representation, S_rm backend, S_rm architecture, S_rm amortized.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-d02e213df2324cdb","tex":"S_{\\rm backend}","aliases":"S_rm backend rm","display":"S_{\\rm backend}","kind":"use","excerpt":"Each experiment should separate: S_rm representation, S_rm backend, S_rm architecture, S_rm amortized.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-3e15adaa9049f083","tex":"S_{\\rm representation}","aliases":"S_rm representation rm","display":"S_{\\rm representation}","kind":"use","excerpt":"Each experiment should separate: S_rm representation, S_rm backend, S_rm architecture, S_rm amortized.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-f93687b75635f48a","tex":"\\section","aliases":"section","display":null,"kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-9bd6563a665b1ad1","tex":"\\small","aliases":"small","display":null,"kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"use","excerpt":"[label=square] item Packet / transfer / atlas / refinement / generation descriptor schemas are versioned. item Packet IR canonical serialization is stable. item Output contracts are explicit and tested. item Commit and replay state machines are formally specified. item Generation invalidation rules are formally specified. item Minimum C0-C3 traffic","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + 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","count":3},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-0ba457916e820d71","tex":"w_{3:0}","aliases":"w_3:0","display":"w_{3:0}","kind":"use","excerpt":"A prototype mapping may use a configurable XOR swizzle, bank(w)= (w_3:0⊕ w_7:4⊕ h(P))bmod 16, where w is the local word index and h(P) is a four-bit packet/phase hash. This is an implementation policy, not part of the mathematical contract, and should be replaced if FPGA counters show worse behavior for target transforms.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-a5c3c052d2649df5","tex":"w_{7:4}","aliases":"w_7:4","display":"w_{7:4}","kind":"use","excerpt":"A prototype mapping may use a configurable XOR swizzle, bank(w)= (w_3:0⊕ w_7:4⊕ h(P))bmod 16, where w is the local word index and h(P) is a four-bit packet/phase hash. This is an implementation policy, not part of the mathematical contract, and should be replaced if FPGA counters show worse behavior for target transforms.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large Transformation plane + resolution plane + certificate-gated commit [0.45in] rule0.90linewidth0.8pt [0.28in] 0.88linewidth>bfseriesp1.55inX Baseline · PRISM-1 Final Updated architecture specification ","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"[0.12in] LARGEbfseries Implementation and Deployment Blueprint v0.2 [0.18in] large Packet-programmable mixed-representation accelerator [0.05in] large 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decomposition itself: P_α longleftrightarrow {P_α_1,…,P_α_k}.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-993a80257a1ce549","tex":"\\alpha_k","aliases":"α_k alpha","display":"\\alpha_k","kind":"use","excerpt":"The resolution plane modifies the decomposition itself: P_α longleftrightarrow {P_α_1,…,P_α_k}.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"The transformation plane performs computation within the current representation and resolution: P_α x ⟼ P_β T P_α x.","count":2},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"The PRISM-1R planner should minimize a physicalized cost rather than arithmetic alone: C(G)= α F +β B_rm HBM +γ B_rm SRAM +δ H_rm NoC +η L_rm sync +κ C_rm cert +ρ E[C_rm replay].","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"The PRISM-1R planner should minimize a physicalized cost rather than arithmetic alone: C(G)= α F +β B_rm HBM +γ B_rm SRAM +δ H_rm NoC +η L_rm sync +κ C_rm cert +ρ E[C_rm replay].","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"The PRISM-1R planner should minimize a physicalized cost rather than arithmetic alone: C(G)= α F +β B_rm HBM +γ B_rm SRAM +δ H_rm NoC +η L_rm sync +κ C_rm cert +ρ E[C_rm replay].","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"The PRISM-1R planner should minimize a physicalized cost rather than arithmetic alone: C(G)= α F +β B_rm HBM +γ B_rm SRAM +δ H_rm NoC +η L_rm sync +κ C_rm cert +ρ E[C_rm replay].","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"The PRISM-1R planner should minimize a physicalized cost rather than arithmetic alone: C(G)= α F +β B_rm HBM +γ B_rm SRAM +δ H_rm NoC +η L_rm sync +κ C_rm cert +ρ E[C_rm replay].","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"The resolution plane modifies the decomposition itself: P_α longleftrightarrow {P_α_1,…,P_α_k}.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"For I1 the hardware-level implication is simply (g_rm consumed≠ g_rm current)⇒ negcommit. The implementation package includes a minimal executable reference test for this invariant.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","excerpt":"The placement objective is not merely minimum latency. It is minimum structured movement: min ∑_e∈ E bytes(e)×hops(e).","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"The placement objective is not merely minimum latency. It is minimum structured movement: min ∑_e∈ E bytes(e)×hops(e).","count":2},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-6d1546afc3ef11e5","tex":"\\sum_{e\\in E}","aliases":"∑_e∈ E sum in","display":"\\sum_{e\\in E}","kind":"use","excerpt":"The placement objective is not merely minimum latency. It is minimum structured movement: min ∑_e∈ E bytes(e)×hops(e).","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"For example, local squared residual summaries may reduce as R_P^2=∑_i r_i^2 without transmitting the original packet tensor through the certificate network.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"For I1 the hardware-level implication is simply (g_rm consumed≠ g_rm current)⇒ negcommit. The implementation package includes a minimal executable reference test for this invariant.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"A prototype mapping may use a configurable XOR swizzle, bank(w)= (w_3:0⊕ w_7:4⊕ h(P))bmod 16, where w is the local word index and h(P) is a four-bit packet/phase hash. This is an implementation policy, not part of the mathematical contract, and should be replaced if FPGA counters show worse behavior for target transforms.","count":1},{"work":"research-prism-1r","url":"notation/research-prism-1r.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"The transformation plane performs computation within the current representation and resolution: P_α x ⟼ P_β T P_α x.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":38},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-2aa34ef1611fca9c","tex":"\\mathbf n","aliases":"n mathbf","display":"\\mathbf n","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-2e99a99588c03ac6","tex":"\\mathsf P","aliases":"P mathsf","display":"\\mathsf P","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":4},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"definition-passage","excerpt":"[Packet descriptor] A packet descriptor is a typed record P=(i,p,a,σ, n,ℓ,π,τ), where i is the packet ID, p is the parent, a is the atlas, σ is the selector type, n is the shape, ℓ is the storage location, π is the precision tag, and τ is the allowed local error contribution.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.","count":13},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":17},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simul","count":14},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-7764e407096f3048","tex":"A_e","aliases":"A_e","display":"A_e","kind":"use","excerpt":"[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-0230e8bde40956b0","tex":"\\Acal","aliases":"Acal A mathcal","display":"{\\mathcal A}","kind":"use","excerpt":"Atlas descriptor. An atlas descriptor records an analysis graph, a synthesis graph, normalization data, and packet selectors. A unitary transform uses its adjoint for synthesis. A redundant exact frame records the canonical dual or another declared synthesis map satisfying S A=I.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-2f0c3277ff0b7098","tex":"\\mathrm{amortization}","aliases":"amortization mathrm","display":"\\mathrm{amortization}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-402578a48c09b3d4","tex":"\\mathrm{architecture}","aliases":"architecture mathrm","display":"\\mathrm{architecture}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simul","count":6},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.","count":17},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"The block-diagonal part stores (n/b)b^2=nb scalars. The two rank factors store 2nr. The total is n(b+2r). A full array stores n^2 scalars.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":8},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"use","excerpt":"Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-cdae72f972852da1","tex":"c_f","aliases":"c_f","display":"c_f","kind":"use","excerpt":"Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-860a844f873baee1","tex":"\\operatorname{child}","aliases":"child operatorname","display":"\\operatorname{child}","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simul","count":12},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-10a6a0ef946e6792","tex":"D^{-1}","aliases":"D^-1","display":"D^{-1}","kind":"use","excerpt":"Woodbury solve. The chip uses packet solves with D_α and one small dense correction: (D+UV^*)^-1 =D^-1-D^-1U(I+V^*D^-1U)^-1V^*D^-1. The solver engines handle the rank-sized middle system.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-63ed82a00b685c9b","tex":"D_A","aliases":"D_A","display":"D_A","kind":"use","excerpt":"Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simul","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-ee5bb7d36741efec","tex":"D_B","aliases":"D_B","display":"D_B","kind":"use","excerpt":"Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simul","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"use","excerpt":"Woodbury solve. The chip uses packet solves with D_α and one small dense correction: (D+UV^*)^-1 =D^-1-D^-1U(I+V^*D^-1U)^-1V^*D^-1. The solver engines handle the rank-sized middle system.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Exact analysis and synthesis. Let U be unitary and let E_α be coordinate selectors. The pulled-back projectors are P_α=U^*E_α U. The hardware analysis applies U and selects the coordinates. Synthesis scatters the selected coordinates and applies U^*.","count":7},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":10},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-a20a0c8f1552bcce","tex":"E_e","aliases":"E_e","display":"E_e","kind":"use","excerpt":"[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-29663d1f0bd44f71","tex":"E_\\alpha","aliases":"E_α alpha","display":"E_\\alpha","kind":"use","excerpt":"Exact analysis and synthesis. Let U be unitary and let E_α be coordinate selectors. The pulled-back projectors are P_α=U^*E_α U. The hardware analysis applies U and selects the coordinates. Synthesis scatters the selected coordinates and applies U^*.","count":4},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array ","count":4},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array ","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-776694e9d4ce89eb","tex":"F_L","aliases":"F_L","display":"F_L","kind":"use","excerpt":"Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array ","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-f43c89d5480f3fa1","tex":"F_{L-1}","aliases":"F_L-1","display":"F_{L-1}","kind":"use","excerpt":"Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array ","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-1725e2bf2fce3bf7","tex":"\\mathrm{graph}","aliases":"graph mathrm","display":"\\mathrm{graph}","kind":"use","excerpt":"PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"Local refinement. Suppose a parent packet task v is replaced by an acyclic child module H=(W,F). Predecessors of v connect to child entries. Child exits connect to successors of v.","count":4},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":5},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-a23d735052a1018f","tex":"\\mathrm{kernel}","aliases":"kernel mathrm","display":"\\mathrm{kernel}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simul","count":6},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array ","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-c2a0f098f8a86ce1","tex":"\\min_{o\\in\\mathcal O_v}","aliases":"min_o∈ O_v min in mathcal","display":"\\min_{o\\in\\mathcal O_v}","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array ","count":29},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.","count":5},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array ","count":4},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":4},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-6a2e772e59bdd377","tex":"\\mathcal O_v","aliases":"O_v mathcal","display":"\\mathcal O_v","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":8},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":4},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-056b9963f80df264","tex":"P_{\\alpha(e)}","aliases":"P_α(e) alpha","display":"P_{\\alpha(e)}","kind":"use","excerpt":"[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-aad7607e608d0052","tex":"P_\\beta","aliases":"P_β beta","display":"P_\\beta","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-003001ebf5867324","tex":"\\prec","aliases":"prec","display":null,"kind":"use","excerpt":"[Topological splice] Let prec be a topological order of the original graph and let prec_H be a topological order of the child module. Replacing v in prec by prec_H gives a topological order of the refined graph. Every unaffected pair of context tasks keeps its relative order.","count":5},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-8765519f7f855f36","tex":"\\prec_H","aliases":"prec_H prec","display":null,"kind":"use","excerpt":"[Topological splice] Let prec be a topological order of the original graph and let prec_H be a topological order of the child module. Replacing v in prec by prec_H gives a topological order of the refined graph. Every unaffected pair of context tasks keeps its relative order.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"BDLR dataflow. For A=D+UV^*, the apply is AX=DX+U(V^*X). The diagonal blocks remain distributed. The r× q contraction is reduced and broadcast.","count":12},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simul","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-32fe36360874826c","tex":"\\mathfrak R","aliases":"R mathfrak","display":"\\mathfrak R","kind":"use","excerpt":"No dense preservation requirement. The architecture does not begin with a dense matrix and try to compress it after the fact. It begins with a typed operator. That operator may be represented as R={ · explicit array, block, block sparse, low rank, BDLR, · Kronecker, sparse chain, transform plan, packet graph}. A dense representation is useful when ","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.","count":16},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-4a5bc4ec3d738895","tex":"\\mathrm{representation}","aliases":"representation mathrm","display":"\\mathrm{representation}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simul","count":5},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","excerpt":"Compatible middle bases. The intermediate basis is part of the optimization. For any invertible S, AB=(AS)(S^-1B). Suppose A=L D_A S^-1, B=S D_B R. Then AB=L(D_AD_B)R. When L, S, and R have fast application and D_A, D_B are block diagonal, the product inherits the blockwise cost plus the transform costs. Fourier diagonalization of circulants, simul","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-9ddb8850545745e5","tex":"S_{\\mathrm{amortization}}","aliases":"S_amortization mathrm","display":"S_{\\mathrm{amortization}}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-3ca3008e6912f578","tex":"S_{\\mathrm{architecture}}","aliases":"S_architecture mathrm","display":"S_{\\mathrm{architecture}}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-7aa9c815c9ae3803","tex":"s_e","aliases":"s_e","display":"s_e","kind":"use","excerpt":"Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-d211ee7d05cd66c9","tex":"S_{\\mathrm{kernel}}","aliases":"S_kernel mathrm","display":"S_{\\mathrm{kernel}}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-3b736f10fdbaef26","tex":"S_{\\mathrm{representation}}","aliases":"S_representation mathrm","display":"S_{\\mathrm{representation}}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-e0890cf6c64b7328","tex":"S_{\\mathrm{total}}","aliases":"S_total mathrm","display":"S_{\\mathrm{total}}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-b764f1cbd00009f2","tex":"\\Scal","aliases":"Scal S mathcal","display":"{\\mathcal S}","kind":"use","excerpt":"Atlas descriptor. An atlas descriptor records an analysis graph, a synthesis graph, normalization data, and packet selectors. A unitary transform uses its adjoint for synthesis. A redundant exact frame records the canonical dual or another declared synthesis map satisfying S A=I.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":5},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-8ea3039f3fbba75a","tex":"T_{\\mathrm{graph}}","aliases":"T_graph mathrm","display":"T_{\\mathrm{graph}}","kind":"use","excerpt":"PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-250004212eff8da7","tex":"T_{\\beta\\alpha}","aliases":"T_βα beta alpha","display":"T_{\\beta\\alpha}","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-3e3becea69293024","tex":"\\mathrm{total}","aliases":"total mathrm","display":"\\mathrm{total}","kind":"use","excerpt":"Four different sources of gain. A fair result must say what changed. The total route may improve because the mathematics removed work, because the same representation ran better on another device, because a kernel was fused or graph-replayed, or because one-time setup was amortized. These are different quantities: S_total = S_representation S_archi","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.","count":14},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-1296a5898d2e7d5f","tex":"U^*","aliases":"U^*","display":"U^*","kind":"use","excerpt":"Exact analysis and synthesis. Let U be unitary and let E_α be coordinate selectors. The pulled-back projectors are P_α=U^*E_α U. The hardware analysis applies U and selects the coordinates. Synthesis scatters the selected coordinates and applies U^*.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.","count":11},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":8},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"use","excerpt":"Representation-first storage. For an equal-block BDLR operator A=D+UV^*, with dimension n, block width b, and rank r, the stored scalar count is n(b+2r) in place of n^2.","count":10},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Local refinement. Suppose a parent packet task v is replaced by an acyclic child module H=(W,F). Predecessors of v connect to child entries. Child exits connect to successors of v.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-8d3781ce495edcf4","tex":"\\widehat A","aliases":"widehat A widehat","display":"\\widehat A","kind":"use","excerpt":"[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":8},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"For a complete selector family, ∑_α E_α=I. Hence S A x =U^*(∑_α E_α)Ux =U^*Ux =x. The incomplete case replaces the identity by the declared selector sum.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-1e9458c4c7da9d3b","tex":"X_\\alpha","aliases":"X_α alpha","display":"X_\\alpha","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[Certified commit] Suppose a command result is provisional until a certificate c(y) is evaluated. If the commit controller releases y only when c(y) satisfies the declared acceptance rule, then every visible result has passed that rule. A failed result cannot satisfy a downstream completion token.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"The software basis already exists. On the recorded RTX 3070 Ti Laptop GPU, an equal-block packet operator was measured at 0.046032 ms and ran 14.8919× faster than its matched device-resident dense GPU row. A BDLR packet path with CUDA Graph replay was measured at 0.079872 ms and ran 10.9355× faster than the matched dense GPU row. Graph replay impro","count":21},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":12},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":4},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-0fcb1c3cf5e3bf16","tex":"\\delta_{\\alpha\\beta}","aliases":"δ_αβ delta alpha beta","display":"\\delta_{\\alpha\\beta}","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-2fc46a82cd7318e7","tex":"\\varepsilon_e","aliases":"ε_e varepsilon","display":"\\varepsilon_e","kind":"use","excerpt":"Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.","count":3},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-d50122673779d86f","tex":"\\eta_b","aliases":"η_b eta","display":"\\eta_b","kind":"use","excerpt":"PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-5a74287168683605","tex":"\\eta_c","aliases":"η_c eta","display":"\\eta_c","kind":"use","excerpt":"PRISM architecture model. The architecture model uses T= max( fracFη_cP, fracBη_bW ) +T_graph, where F is arithmetic, B is data movement, P is peak arithmetic, W is bandwidth, and η_c, η_b are efficiency factors. The model is used to decide which hardware paths are worth building. It is not a silicon measurement.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"Fused multi-atlas analysis · 14.73 mus · 8.96 mus","count":6},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":4},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":9},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-ec1616068a6ee172","tex":"\\sum_e","aliases":"∑_e sum","display":"\\sum_e","kind":"use","excerpt":"Exact packet reconstruction gives (widehat A-A)x=∑_eE_eP_α(e)x. The triangle inequality gives the bound.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-6d1546afc3ef11e5","tex":"\\sum_{e\\in E}","aliases":"∑_e∈ E sum in","display":"\\sum_{e\\in E}","kind":"use","excerpt":"[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-f7155be421cc3965","tex":"\\sum_f","aliases":"∑_f sum","display":"\\sum_f","kind":"use","excerpt":"Precision allocation. Under a logarithmic cost model and total error budget τ, the continuous allocation is ε_e =fracτ c_es_e∑_f c_f, where s_e is sensitivity and c_e is cost. The compiler maps the result to supported formats.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-d36e379ac40623d3","tex":"\\sum_{u\\in\\operatorname{child}(o)}","aliases":"∑_u∈child(o) sum in operatorname","display":"\\sum_{u\\in\\operatorname{child}(o)}","kind":"use","excerpt":"For a fixed candidate tree, each choice o has a local cost c(o), a local error e(o), and zero or more children. The planner evaluates J(v)=min_o∈ O_v [ c(o)+λ e(o)^2+ ∑_u∈child(o)J(u) ]. Bottom-up evaluation is globally optimal on the fixed tree because every feasible plan consists of one local choice followed by feasible child plans.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"use","excerpt":"For a complete selector family, ∑_α E_α=I. Hence S A x =U^*(∑_α E_α)Ux =U^*Ux =x. The incomplete case replaces the identity by the declared selector sum.","count":2},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-cabedb544f677980","tex":"\\sum_{\\alpha\\in I}","aliases":"∑_α∈ I sum alpha in","display":"\\sum_{\\alpha\\in I}","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"Modeled arithmetic peaks. With 256 execution groups and one 16×16 matrix array per group, the model yields FP16 or BF16 · ≈ 196.6 TFLOP/s, FP32 · ≈ 98.3 TFLOP/s, FP64 · ≈ 24.6 TFLOP/s. These are model peaks. The design is not organized around a claim of superior generic dense throughput. Its purpose is to execute fewer operations and move fewer byt","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Packet error bound] Let A=∑_e∈ EA_e, widehat A=∑_e∈ E(A_e+E_e). If edge e starts in packet α(e), then norm(widehat A-A)x ≤ ∑_e∈ EnormE_enormP_α(e)x.","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-90b882d77bb3725c","tex":"\\bigoplus_{\\alpha\\in I}","aliases":"⊕_α∈ I bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in I}","kind":"use","excerpt":"Projector-resolved computation starts before that flattening step. Let X=⊕_α∈ IX_α, P_α P_β=δ_αβP_α, ∑_α∈ IP_α=I. For an operator T:X→ X, the native objects are the transfer blocks T_βα=P_β T P_α. The complete operator is reconstructed by T=∑_α,βT_βα. The zero pattern, block shapes, rank structure, parent-child relations, and packet sensitivities a","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"Sparse chains and transform plans. A matrix may be dense as an array and cheap as an operator. If A=F_LF_L-1⋯ F_1 and each factor has O(n) nonzero entries, then applying A costs O(nL). If L=O(log n), the cost is O(nlog n). FFT and butterfly operators fit this pattern. PRISM stores the factor chain and executes it. It does not store the dense array ","count":1},{"work":"research-prism-1","url":"notation/research-prism-1.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"CUDA-like execution hierarchy. CUDA made heterogeneous arithmetic programmable through a hierarchy of lanes, warps, blocks, grids, and memories. PRISM uses an analogous execution hierarchy while retaining the mathematical structure that CUDA kernels normally receive only through software conventions: lane ⟶ packet warp ⟶ execution group ⟶ Packet St","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":172},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":14},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":61},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":123},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":11},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":31},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":10},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e9aaef4e6308432c","tex":"e_a","aliases":"e_a","display":"e_a","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":6},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":52},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":11},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":9},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":38},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-5ca75c044055616d","tex":"Q_a","aliases":"Q_a","display":"Q_a","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":22},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":10},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":6},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":21},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":12},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-ad151dded7ef7d37","tex":"\\sum_{a\\in A}","aliases":"∑_a∈ A sum in","display":"\\sum_{a\\in A}","kind":"definition-passage","excerpt":"[Cache validity] A cache state is a tuple c=(B,(G_a),(e_a),t) satisfying G_a=Q_aF(B), e_a=E(G_a), t=∑_a∈ Ae_a. The cached score is s(c)=tw.","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":58},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a890da204ed251d6","tex":"A_{\\mathrm{count}}","aliases":"A_count mathrm","display":"A_{\\mathrm{count}}","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","excerpt":"The common precision selector is the computable integer expression precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d.","count":27},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-00a9da7456c9ee3b","tex":"\\mathrm{count}","aliases":"count mathrm","display":"\\mathrm{count}","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"The common precision selector is the computable integer expression precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d.","count":28},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"introduction-passage","excerpt":"The common precision selector is the computable integer expression precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d.","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"introduction-passage","excerpt":"The common precision selector is the computable integer expression precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d.","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-de8ee209119cc205","tex":"\\log^{\\N}_2","aliases":"log^N_2 log N","display":null,"kind":"introduction-passage","excerpt":"The common precision selector is the computable integer expression precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d.","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-b582d3167c70c098","tex":"\\N","aliases":"N","display":null,"kind":"introduction-passage","excerpt":"The common precision selector is the computable integer expression precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d.","count":5},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"inequation precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d. The proof uses integer power and ceiling inequalities. ","count":23},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-fc83984b5081759e","tex":"\\operatorname{precisionFor}","aliases":"precisionFor operatorname","display":"\\operatorname{precisionFor}","kind":"introduction-passage","excerpt":"The common precision selector is the computable integer expression precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d.","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":6},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":16},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"introduction-passage","excerpt":"The common precision selector is the computable integer expression precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d.","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-fb99ab52bd974a63","tex":"\\rceil_{\\N}","aliases":"rceil_N rceil N","display":null,"kind":"introduction-passage","excerpt":"The common precision selector is the computable integer expression precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d.","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":66},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-fc7a5c603b95f30d","tex":"\\tr","aliases":"tr","display":null,"kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":13},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"For a supplied rational observation y, the executable contract uses the explicit version ν_y. That version is defined at every such y and is already proved to disintegrate the model. The solver's distance theorem is pointwise against this version. It does not prove pointwise equality to every possible modification of a canonical posterior kernel on","count":32},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"introduction-passage","excerpt":"For a supplied rational observation y, the executable contract uses the explicit version ν_y. That version is defined at every such y and is already proved to disintegrate the model. The solver's distance theorem is pointwise against this version. It does not prove pointwise equality to every possible modification of a canonical posterior kernel on","count":16},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-775ac5e316e54a08","tex":"\\nu_y","aliases":"ν_y nu","display":"\\nu_y","kind":"introduction-passage","excerpt":"For a supplied rational observation y, the executable contract uses the explicit version ν_y. That version is defined at every such y and is already proved to disintegrate the model. The solver's distance theorem is pointwise against this version. It does not prove pointwise equality to every possible modification of a canonical posterior kernel on","count":12},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-bd1aac0f414abebd","tex":"\\rho^2","aliases":"ρ^2 rho","display":"\\rho^2","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","excerpt":"Write A_count=|A| for the positive number of output labels, and set ρ=1-fracσσ+tr S, q=ρ^2, r=1-frac1-qA_count. The model factorization and positive noise floor imply that S is positive semidefinite with quadratic lower bound σ. Its trace is nonnegative, and σ+tr S is a valid positive spectral cap. Thus 0≤ρ<1, 0≤ q<1, and 0≤ r<1. No eigenvalue orac","count":42},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"inequation precisionFor(C,d) =log^N_2(lceil C/drceil_N)+1, d>0, where log^N_2 denotes the natural-number logarithm pathNat.log, including its specified value at zero. It is not a floating-point logarithm. For every precision at least this selected count, pathcoefficient_mesh_le gives C2^-p≤ d. The proof uses integer power and ceiling inequalities. ","count":42},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"The exact posterior formula is ν_y= N(m+Kd,C_*), C_*=P-KHP. Writing this formula does not by itself establish conditional probability semantics. A matrix identity could describe a plausible mean and covariance without proving that its Gaussian is a disintegration of the prior and likelihood. The source constructs that disintegration at the level of","count":19},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'),","count":5},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'), D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a53ead434f8754af","tex":"A^\\top","aliases":"A^top top","display":"A^\\top","kind":"use","excerpt":"Use the Frobenius inner product ⟨ A,C⟩_ F=tr(A^top C). For a≠ a', self-adjointness and disjointness give ⟨ Q_aA,Q_a'A⟩_ F =tr(A^top Q_aQ_a'A)=0. Consequently the energy of a finite sum of output packets is the sum of their energies. The input analogue follows by transposing the matrices. Applying the two decompositions successively gives complete t","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-0b44ac59eeb82cc2","tex":"A_{ij}^2","aliases":"A_ij^2","display":"A_{ij}^2","kind":"use","excerpt":"We use Euclidean vector norms and the squared Frobenius energy E(A)=FrobA^2=∑_i,jA_ij^2. The distinction between these norms matters. A matrix energy is a finite sum of scalar squares that can be evaluated exactly on rational inputs. An operator norm is used in the proof that this energy controls distribution error. The certificate never asks a pro","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-863ef150816ef63f","tex":"\\mathrm{approximation}","aliases":"approximation mathrm","display":"\\mathrm{approximation}","kind":"use","excerpt":"andidate checking] Let M be a factor-coded rational model, Q,D rational packet families, and c a rational candidate. If both pathcheckCandidate M Q D c and pathcheckTolerance M c epsilon return true, then the exact real posterior version represented by M and the exact real Joseph Gaussian represented by the gain of c satisfy Wtwo(M.posterior,M.appr","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'),","count":5},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'),","count":22},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e77f9eceea598e09","tex":"B'","aliases":"B'","display":"B'","kind":"use","excerpt":"For a selected label a, the executed update is B' · =B+τ G_a, · G'_a · =G_aT, · e'_a · =E(G'_a),nonumber G'_b · =G_b (b≠ a), · e'_b · =e_b (b≠ a), · t' · =t-e_a+e'_a. This is the definition pathKomorebi.CachedPosterior.step. It computes the new selected residual from the old selected residual and the precomputed transition. It retains unselected en","count":9},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'), D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8bed5507d096707b","tex":"B^{\\mathsf T}","aliases":"B^ T mathsf","display":"B^{\\mathsf T}","kind":"use","excerpt":"For a rational gain B, the Joseph approximation has mean and covariance m_B · =m+B(y-Hm), P_B · =(I-BH)P(I-BH)^ T+BRB^ T. Its covariance is positive semidefinite by construction. The approximation is a measure, rather than a collection of unqualified decimal entries. The exact gain K=PH^ TS^-1 identifies the familiar Gaussian posterior parameters. ","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8b99783075cb661f","tex":"B^\\top","aliases":"B^top top","display":"B^\\top","kind":"use","excerpt":"Let B∈R^n× p be an arbitrary gain candidate. It need not solve the innovation equation and need not have been produced by the packet algorithm. Define m_B · =m+Bd, C_B · =(I_n-BH)P(I_n-BH)^top+BRB^top, μ_B · = N(m_B,C_B). Every C_B is positive semidefinite, since it is a sum of covariance congruences. Thus every candidate gain determines a genuine ","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f87c7c154af9d919","tex":"B_*","aliases":"B_*","display":"B_*","kind":"use","excerpt":"-2/3 · 5/6). If B_*=(2/3,1/6)^ T is the exact gain, expansion of the Joseph expression gives V_B_6=V_*+6(B_6-B_*)(B_6-B_*)^ T. The gain discrepancy is B_*-B_6=(2/1029,1/2058)^ T, which produces the displayed matrix. This second calculation checks the sign of the off-diagonal entries and explains why the approximate covariance has an additional posi","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"Validity is an exact equality invariant. It does not mean that a stored residual is approximately near a true residual, nor that its energy is merely an upper bound. Initialization evaluates F(B_0) once, projects it onto every output packet, computes every energy, and sums them. The theorem pathinitialState_valid follows directly from the definitio","count":9},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"Exact cache validity cannot be reused unchanged after independently rounding its components. The release contains a scalar counterexample with P=H=R=1, σ=1, m=0, y=1, and B_0=0. Then S=2, τ=1/3, w=2, and the first exact update is B_1=1/3, F(B_1)=1/3. The naive route separately rounds the gain center and residual center downward to zero fractional b","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4b94fddf75c4f060","tex":"B_4","aliases":"B_4","display":"B_4","kind":"use","excerpt":"After four updates, B_4=frac203625J, F(B_4)=frac16625J, a(B_4)=frac3072390625. The score weight is w=3, because lVert yrVert^2=2. The strict margin is ε^2/4=1/64. At the previous state the score is 768/15625>1/64; after the fourth it is below 1/64. The computed fuel is 4267. Precision 4 produces score 435/64 and fails; precision 12 produces 9111/10","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-7d1b17c2a7b4bc06","tex":"B_5","aliases":"B_5","display":"B_5","kind":"use","excerpt":"1/343, a(B_6)=frac34117649. The strict search threshold is ε^2/4=1/1600. At the previous state, a(B_5)=130/117649>1/1600, while a(B_6)<1/1600. Thus six is the actual update count for this stopping rule. The formal fuel is 410093; the program uses only six of these allowed updates. This large difference illustrates the conservative fuel construction","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-dfd1ab77f645f709","tex":"B_6","aliases":"B_6","display":"B_6","kind":"use","excerpt":"After six updates, each coordinate has been selected three times. The geometric sum in its gain gives B_6=228/343 57/343, F(B_6)=4/343 1/343, a(B_6)=frac34117649. The strict search threshold is ε^2/4=1/1600. At the previous state, a(B_5)=130/117649>1/1600, while a(B_6)<1/1600. Thus six is the actual update count for this stopping rule. The formal f","count":8},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"use","excerpt":"The transition contracts every residual matrix: E(XT)≤ qE(X). If label a is updated, its residual energy contracts by this inequality and all other output residuals remain unchanged. Orthogonal energy accounting gives E(F(B'))≤ E(F(B))-(1-q)e_a. Because a is a maximum, e_a≥ E(F(B))/A_count. It follows that E(F(B'))≤ rE(F(B)), a(B_n)≤ r^na(B_0). The","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-896cb2a8a3d18c46","tex":"B_{n+1}^{\\mathrm{ideal}}","aliases":"B_n+1^ideal mathrm","display":"B_{n+1}^{\\mathrm{ideal}}","kind":"use","excerpt":"Core 1.0.0 already retains a different, completed rounded solver. Its update is a genuinely rounded full-gain Richardson step. It neither claims exact selective cache invariants nor selects a largest cached packet. Its declarations expose the whole-update defect instead. If widetilde B_n+1 is the rounded update and B_n+1^ideal is the exact full ste","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-43f43495bc3019e9","tex":"B_n^{\\mathrm{exact}}","aliases":"B_n^exact mathrm","display":"B_n^{\\mathrm{exact}}","kind":"use","excerpt":"Writing d for the number of gain entries and h_t=2^-t, the source establishes an all-iteration tracking bound E(widetilde B_n-B_n^exact) ≤frac6d h_t^2(1-q)^2. Its residual-score comparison has the form a(widetilde B_n)≤2a(B_n^exact)+C_roundh_t^2, C_round= frac12dE(S)w(1-q)^2. The source names are pathroundedGain_tracking, pathroundoffScoreCoeff, an","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-b6a0e01f5d7d0367","tex":"C_*","aliases":"C_*","display":"C_*","kind":"use","excerpt":"The exact posterior formula is ν_y= N(m+Kd,C_*), C_*=P-KHP. Writing this formula does not by itself establish conditional probability semantics. A matrix identity could describe a plausible mean and covariance without proving that its Gaussian is a disintegration of the prior and likelihood. The source constructs that disintegration at the level of","count":11},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f64a9bccd47e1e6c","tex":"C_*^{-1}","aliases":"C_*^-1","display":"C_*^{-1}","kind":"use","excerpt":"The covariance C_* admits the Joseph representation C_*=(I_n-KH)P(I_n-KH)^top+KRK^top. Each term on the right is positive semidefinite. Therefore C_*succeq0, including at the singular boundary. This avoids any proof based on P^-1 or C_*^-1.","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a717968c751b4646","tex":"C_B","aliases":"C_B","display":"C_B","kind":"use","excerpt":"Let B∈R^n× p be an arbitrary gain candidate. It need not solve the innovation equation and need not have been produced by the packet algorithm. Define m_B · =m+Bd, C_B · =(I_n-BH)P(I_n-BH)^top+BRB^top, μ_B · = N(m_B,C_B). Every C_B is positive semidefinite, since it is a sum of covariance congruences. Thus every candidate gain determines a genuine ","count":8},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a0a1c6308a531a2e","tex":"C_{\\mathrm{out}}","aliases":"C_out mathrm","display":"C_{\\mathrm{out}}","kind":"use","excerpt":"Each output arithmetic expression has a precision-independent budget coefficient. The source sums all mean coefficients and all covariance coefficients to obtain C_out, defined by pathoutputCoefficient. Every individual coefficient is nonnegative and no greater than this sum. It chooses p_out=precisionFor(C_out,δ) and evaluates the output graph at ","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-bea4a4bcfb686a4a","tex":"C_{\\mathrm{round}}","aliases":"C_round mathrm","display":"C_{\\mathrm{round}}","kind":"use","excerpt":"Writing d for the number of gain entries and h_t=2^-t, the source establishes an all-iteration tracking bound E(widetilde B_n-B_n^exact) ≤frac6d h_t^2(1-q)^2. Its residual-score comparison has the form a(widetilde B_n)≤2a(B_n^exact)+C_roundh_t^2, C_round= frac12dE(S)w(1-q)^2. The source names are pathroundedGain_tracking, pathroundoffScoreCoeff, an","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-5b36c5af2888cccd","tex":"C_{\\mathrm{ver}}","aliases":"C_ver mathrm","display":"C_{\\mathrm{ver}}","kind":"use","excerpt":"Let h_p=2^-p. The dyadic evaluator propagates rational centers and outward radii through its arithmetic graph. Its budget calculation supplies a nonnegative coefficient C_ver(M,Q,D,B) independent of p, with widehat a_p(B)≤ a(B)+C_ver(M,Q,D,B)h_p. Here widehat a_p is the actual rounded candidate score. The full expression for the coefficient is path","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'),","count":15},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4256ad8575b1b705","tex":"D_b","aliases":"D_b","display":"D_b","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'),","count":10},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-7c33bd97b2b03115","tex":"D_{b'}","aliases":"D_b'","display":"D_{b'}","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'), D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-10f6fbeca45153a1","tex":"D_b^2","aliases":"D_b^2","display":"D_b^2","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'), D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d5192e28eb29baaa","tex":"D_b^\\top","aliases":"D_b^top top","display":"D_b^\\top","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'), D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"0, y=1. It is represented by L=diag(2,1), N=0, and σ=1. The two output projectors select the coordinate axes, and the single input projector is 1. Start with B_0=0, seed 0, scan [0,1], probes [4,12], ε=1/20, and δ=1/1000.","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries Certified Bayesian","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-fb7427a95ad8c5b4","tex":"\\E","aliases":"E","display":null,"kind":"use","excerpt":"Let Δ=K-B. Subtraction in [source reference] gives T_K-T_B=Δ d-Δ(HU+V). The random vector HU+V is centered and has covariance S. The difference therefore has mean Δ d and covariance Δ SΔ^top. Its expected squared Euclidean norm is E T_K-T_B ^2 = Δ d ^2+tr(Δ SΔ^top). The source proves both marginals and this cost. It does not assume that a producer ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-fed08268272f434c","tex":"e'_a","aliases":"e'_a","display":"e'_a","kind":"use","excerpt":"For a selected label a, the executed update is B' · =B+τ G_a, · G'_a · =G_aT, · e'_a · =E(G'_a),nonumber G'_b · =G_b (b≠ a), · e'_b · =e_b (b≠ a), · t' · =t-e_a+e'_a. This is the definition pathKomorebi.CachedPosterior.step. It computes the new selected residual from the old selected residual and the precomputed transition. It retains unselected en","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a7b08031ede26c71","tex":"e'_b","aliases":"e'_b","display":"e'_b","kind":"use","excerpt":"For a selected label a, the executed update is B' · =B+τ G_a, · G'_a · =G_aT, · e'_a · =E(G'_a),nonumber G'_b · =G_b (b≠ a), · e'_b · =e_b (b≠ a), · t' · =t-e_a+e'_a. This is the definition pathKomorebi.CachedPosterior.step. It computes the new selected residual from the old selected residual and the precomputed transition. It retains unselected en","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-239d2a60fb954866","tex":"E_*","aliases":"E_*","display":"E_*","kind":"use","excerpt":"For the reverse construction, sample Y∼ N(Hm,S) and an independent E_*∼ N(0,C_*). Define Z=m+K(Y-Hm)+E_*. Conditional on Y=y, the law of Z is precisely ν_y. The joint mean of (Y,Z) is again (Hm,m). Its cross covariance is SK^top=HP, and its state covariance is KSK^top+C_*=KHP+(P-KHP)=P. It therefore has exactly the covariance in [source reference].","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f1285db44838b010","tex":"e_+","aliases":"e_+","display":"e_+","kind":"use","excerpt":"Take B_0=0, seed 0, scan [0,1], probes [4,12], ε=1/4, and δ=1/1000. Here tr S=4, τ=1/5, and T=I-(I+J)/5. The initial residual is J, entirely in the Q_+ output packet. Since JS=3J, each selected update multiplies this residual by 2/5. The Q_- residual remains zero. Thus every selected label is 0 and (e_+,e_-)=(4(4/25)^n,0), B_n=frac1-(2/5)^n3J. The ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3da4494053c2f25c","tex":"e_-","aliases":"e_-","display":"e_-","kind":"use","excerpt":"Take B_0=0, seed 0, scan [0,1], probes [4,12], ε=1/4, and δ=1/1000. Here tr S=4, τ=1/5, and T=I-(I+J)/5. The initial residual is J, entirely in the Q_+ output packet. Since JS=3J, each selected update multiplies this residual by 2/5. The Q_- residual remains zero. Thus every selected label is 0 and (e_+,e_-)=(4(4/25)^n,0), B_n=frac1-(2/5)^n3J. The ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"al residual is (4,1)^ T, with packet energies 16 and 1. Updating one coordinate multiplies its residual by 1/7 and its energy by 1/49. The other coordinate remains exactly unchanged. The maximum-energy schedule therefore alternates between labels 0 and 1. The complete pre-update trace is rrrrr Updates already made · Selected label · e_0 · e_1 · t","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"ual is (4,1)^ T, with packet energies 16 and 1. Updating one coordinate multiplies its residual by 1/7 and its energy by 1/49. The other coordinate remains exactly unchanged. The maximum-energy schedule therefore alternates between labels 0 and 1. The complete pre-update trace is rrrrr Updates already made · Selected label · e_0 · e_1 · t","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-b6266893a62573e7","tex":"e_b","aliases":"e_b","display":"e_b","kind":"use","excerpt":"For a selected label a, the executed update is B' · =B+τ G_a, · G'_a · =G_aT, · e'_a · =E(G'_a),nonumber G'_b · =G_b (b≠ a), · e'_b · =e_b (b≠ a), · t' · =t-e_a+e'_a. This is the definition pathKomorebi.CachedPosterior.step. It computes the new selected residual from the old selected residual and the precomputed transition. It retains unselected en","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-ae96da606adc1558","tex":"\\mathrm{exact}","aliases":"exact mathrm","display":"\\mathrm{exact}","kind":"use","excerpt":"Writing d for the number of gain entries and h_t=2^-t, the source establishes an all-iteration tracking bound E(widetilde B_n-B_n^exact) ≤frac6d h_t^2(1-q)^2. Its residual-score comparison has the form a(widetilde B_n)≤2a(B_n^exact)+C_roundh_t^2, C_round= frac12dE(S)w(1-q)^2. The source names are pathroundedGain_tracking, pathroundoffScoreCoeff, an","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-484de1c254ac2bb8","tex":"\\mathrm F","aliases":"F mathrm","display":"\\mathrm F","kind":"use","excerpt":"Use the Frobenius inner product ⟨ A,C⟩_ F=tr(A^top C). For a≠ a', self-adjointness and disjointness give ⟨ Q_aA,Q_a'A⟩_ F =tr(A^top Q_aQ_a'A)=0. Consequently the energy of a finite sum of output packets is the sum of their energies. The input analogue follows by transposing the matrices. Applying the two decompositions successively gives complete t","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"sigma^-2 d ^2). For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose of the ith row of F, then tr(FS^-1F^top)=∑_i f_i^top S^-1f_i ≤∑_i f_i S^-1f_i ≤σ^-1∑_i f_i ^2. The last sum is E(F). Add","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-64863d8dd06c95a8","tex":"F^\\top","aliases":"F^top top","display":"F^\\top","kind":"use","excerpt":"[Frobenius energy controls the residual cost] If F∈R^n× p and Ssucceqσ I_psucc0, then FS^-1d ^2+tr(FS^-1F^top) ≤ E(F)(σ^-1+σ^-2 d ^2). For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose o","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8edc80d718f078a9","tex":"F_{ab}","aliases":"F_ab","display":"F_{ab}","kind":"use","excerpt":"The two-sided residual packet is F_ab(B)=Q_aF(B)D_b. Each packet remains an n× p matrix, although it is supported on the output and input subspaces selected by its projectors. The notation does not assume a coordinate partition or require projectors to be diagonal. Rational rotated projectors are allowed when their exact entries satisfy the declare","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"use","excerpt":"sigma^-2 d ^2). For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose of the ith row of F, then tr(FS^-1F^top)=∑_i f_i^top S^-1f_i ≤∑_i f_i S^-1f_i ≤σ^-1∑_i f_i ^2. The last sum is E(F). Add","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-73370e19a36d1453","tex":"f_i^\\top","aliases":"f_i^top top","display":"f_i^\\top","kind":"use","excerpt":"ajorant For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose of the ith row of F, then tr(FS^-1F^top)=∑_i f_i^top S^-1f_i ≤∑_i f_i S^-1f_i ≤σ^-1∑_i f_i ^2. The last sum is E(F). Adding the ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-09888c9e8426e6e3","tex":"F_{ij}","aliases":"F_ij","display":"F_{ij}","kind":"use","excerpt":"[Frobenius energy controls the residual cost] If F∈R^n× p and Ssucceqσ I_psucc0, then FS^-1d ^2+tr(FS^-1F^top) ≤ E(F)(σ^-1+σ^-2 d ^2). For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose o","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-ce7a3baf7e95efa4","tex":"F_{ij}^2","aliases":"F_ij^2","display":"F_{ij}^2","kind":"use","excerpt":"[Frobenius energy controls the residual cost] If F∈R^n× p and Ssucceqσ I_psucc0, then FS^-1d ^2+tr(FS^-1F^top) ≤ E(F)(σ^-1+σ^-2 d ^2). For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose o","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-bf6efda13a9cd474","tex":"\\Frob","aliases":"Frob","display":null,"kind":"use","excerpt":"We use Euclidean vector norms and the squared Frobenius energy E(A)=FrobA^2=∑_i,jA_ij^2. The distinction between these norms matters. A matrix energy is a finite sum of scalar squares that can be evaluated exactly on rational inputs. An operator norm is used in the proof that this energy controls distribution error. The certificate never asks a pro","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-fe8d9cd185344719","tex":"G'_a","aliases":"G'_a","display":"G'_a","kind":"use","excerpt":"For a selected label a, the executed update is B' · =B+τ G_a, · G'_a · =G_aT, · e'_a · =E(G'_a),nonumber G'_b · =G_b (b≠ a), · e'_b · =e_b (b≠ a), · t' · =t-e_a+e'_a. This is the definition pathKomorebi.CachedPosterior.step. It computes the new selected residual from the old selected residual and the precomputed transition. It retains unselected en","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-1141f8ead2f5a048","tex":"G'_b","aliases":"G'_b","display":"G'_b","kind":"use","excerpt":"For a selected label a, the executed update is B' · =B+τ G_a, · G'_a · =G_aT, · e'_a · =E(G'_a),nonumber G'_b · =G_b (b≠ a), · e'_b · =e_b (b≠ a), · t' · =t-e_a+e'_a. This is the definition pathKomorebi.CachedPosterior.step. It computes the new selected residual from the old selected residual and the precomputed transition. It retains unselected en","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-76b53653bff85b9e","tex":"G_b","aliases":"G_b","display":"G_b","kind":"use","excerpt":"For a selected label a, the executed update is B' · =B+τ G_a, · G'_a · =G_aT, · e'_a · =E(G'_a),nonumber G'_b · =G_b (b≠ a), · e'_b · =e_b (b≠ a), · t' · =t-e_a+e'_a. This is the definition pathKomorebi.CachedPosterior.step. It computes the new selected residual from the old selected residual and the precomputed transition. It retains unselected en","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4564fa3b11bf468a","tex":"\\mathrm{gain}","aliases":"gain mathrm","display":"\\mathrm{gain}","kind":"use","excerpt":"andidate checking] Let M be a factor-coded rational model, Q,D rational packet families, and c a rational candidate. If both pathcheckCandidate M Q D c and pathcheckTolerance M c epsilon return true, then the exact real posterior version represented by M and the exact real Joseph Gaussian represented by the gain of c satisfy Wtwo(M.posterior,M.appr","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":52},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"Let h_p=2^-p. The dyadic evaluator propagates rational centers and outward radii through its arithmetic graph. Its budget calculation supplies a nonnegative coefficient C_ver(M,Q,D,B) independent of p, with widehat a_p(B)≤ a(B)+C_ver(M,Q,D,B)h_p. Here widehat a_p is the actual rounded candidate score. The full expression for the coefficient is path","count":5},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-22442a90910975a8","tex":"H^{\\mathsf T}","aliases":"H^ T mathsf","display":"H^{\\mathsf T}","kind":"use","excerpt":". The representation uses rational factors P=LL^ T, R=NN^ T+σ I, σ>0. All data supplied to the executable construction, including m,H,L,N,σ and the observation y, are rational. Finite state and observation index types determine the matrix shapes. There is no requirement that H be square or that P be invertible. The innovation matrix is S=HPH^ T+R. ","count":6},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-c4adccfc3bfed139","tex":"H^\\top","aliases":"H^top top","display":"H^\\top","kind":"use","excerpt":"For an observed value y, put d=y-Hm, S=HPH^top+R, K=PH^top S^-1. The matrix S is the innovation covariance. It is positive definite because, for every nonzero z∈R^p, z^top Sz=(H^top z)^top P(H^top z)+z^top Rz>0. Consequently the inverse in the definition of K exists even when P has a nontrivial nullspace or H has deficient rank. Every inverse used ","count":11},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d54b1a6be7c911b2","tex":"h_p","aliases":"h_p","display":"h_p","kind":"use","excerpt":"Let h_p=2^-p. The dyadic evaluator propagates rational centers and outward radii through its arithmetic graph. Its budget calculation supplies a nonnegative coefficient C_ver(M,Q,D,B) independent of p, with widehat a_p(B)≤ a(B)+C_ver(M,Q,D,B)h_p. Here widehat a_p is the actual rounded candidate score. The full expression for the coefficient is path","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-480c877d455d2c2b","tex":"h_t","aliases":"h_t","display":"h_t","kind":"use","excerpt":"Writing d for the number of gain entries and h_t=2^-t, the source establishes an all-iteration tracking bound E(widetilde B_n-B_n^exact) ≤frac6d h_t^2(1-q)^2. Its residual-score comparison has the form a(widetilde B_n)≤2a(B_n^exact)+C_roundh_t^2, C_round= frac12dE(S)w(1-q)^2. The source names are pathroundedGain_tracking, pathroundoffScoreCoeff, an","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-2813a9e6502d1a32","tex":"h_t^2","aliases":"h_t^2","display":"h_t^2","kind":"use","excerpt":"Writing d for the number of gain entries and h_t=2^-t, the source establishes an all-iteration tracking bound E(widetilde B_n-B_n^exact) ≤frac6d h_t^2(1-q)^2. Its residual-score comparison has the form a(widetilde B_n)≤2a(B_n^exact)+C_roundh_t^2, C_round= frac12dE(S)w(1-q)^2. The source names are pathroundedGain_tracking, pathroundoffScoreCoeff, an","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"The completed executable contract represents its model by rational matrices and vectors. Its input consists of H, m, y, a prior factor L∈Q^n× r, a noise factor N∈Q^p× s, and a rational number σ. The covariance matrices are derived from these inputs: P=LL^top, R=NN^top+σ I_p, σ>0. The factor dimensions r and s are finite and need not equal the state","count":20},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"We use Euclidean vector norms and the squared Frobenius energy E(A)=FrobA^2=∑_i,jA_ij^2. The distinction between these norms matters. A matrix energy is a finite sum of scalar squares that can be evaluated exactly on rational inputs. An operator norm is used in the proof that this energy controls distribution error. The certificate never asks a pro","count":5},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-95d680fad2ccb816","tex":"I_n","aliases":"I_n","display":"I_n","kind":"use","excerpt":"The covariance C_* admits the Joseph representation C_*=(I_n-KH)P(I_n-KH)^top+KRK^top. Each term on the right is positive semidefinite. Therefore C_*succeq0, including at the singular boundary. This avoids any proof based on P^-1 or C_*^-1.","count":5},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-c7b33cae672307e6","tex":"I_p","aliases":"I_p","display":"I_p","kind":"use","excerpt":"The completed executable contract represents its model by rational matrices and vectors. Its input consists of H, m, y, a prior factor L∈Q^n× r, a noise factor N∈Q^p× s, and a rational number σ. The covariance matrices are derived from these inputs: P=LL^top, R=NN^top+σ I_p, σ>0. The factor dimensions r and s are finite and need not equal the state","count":5},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-dc4a1500d8267a2b","tex":"\\mathrm{ideal}","aliases":"ideal mathrm","display":"\\mathrm{ideal}","kind":"use","excerpt":"Core 1.0.0 already retains a different, completed rounded solver. Its update is a genuinely rounded full-gain Richardson step. It neither claims exact selective cache invariants nor selects a largest cached packet. Its declarations expose the whole-update defect instead. If widetilde B_n+1 is the rounded update and B_n+1^ideal is the exact full ste","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"For probability laws with finite second moment, write Wtwo(ν,μ)^2 =inf_π∈Π(ν,μ)∫ x-z ^2 dπ(x,z), where Π(ν,μ) is the set of transport plans with the stated marginals. The formal source defines the squared quantity as a nonnegative extended-real infimum and obtains the distance by a square-root operation. The explicit finite Gaussian coupling proves","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a75168cc271ffd1d","tex":"\\inf_{\\pi\\in\\Pi(\\nu,\\mu)}","aliases":"inf_π∈Π(ν,μ) inf pi in Pi nu mu","display":"\\inf_{\\pi\\in\\Pi(\\nu,\\mu)}","kind":"use","excerpt":"For probability laws with finite second moment, write Wtwo(ν,μ)^2 =inf_π∈Π(ν,μ)∫ x-z ^2 dπ(x,z), where Π(ν,μ) is the set of transport plans with the stated marginals. The formal source defines the squared quantity as a nonnegative extended-real infimum and obtains the distance by a square-root operation. The explicit finite Gaussian coupling proves","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"The rotated fixture makes the prior singular and the packets non-coordinate: P=J=1 · 1 1 · 1, H=I, R=I, m=0, y=1 -1. It uses the rectangular prior factor L=(1,1)^ T, zero noise factor, and σ=1. Both output and input packet families are Q_+=frac121 · 1","count":10},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"We use Euclidean vector norms and the squared Frobenius energy E(A)=FrobA^2=∑_i,jA_ij^2. The distinction between these norms matters. A matrix energy is a finite sum of scalar squares that can be evaluated exactly on rational inputs. An operator norm is used in the proof that this energy controls distribution error. The certificate never asks a pro","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"For an observed value y, put d=y-Hm, S=HPH^top+R, K=PH^top S^-1. The matrix S is the innovation covariance. It is positive definite because, for every nonzero z∈R^p, z^top Sz=(H^top z)^top P(H^top z)+z^top Rz>0. Consequently the inverse in the definition of K exists even when P has a nontrivial nullspace or H has deficient rank. Every inverse used ","count":30},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"st ε^2/4 at the current state. If the test fails and fuel remains, it advances once and decrements fuel. At fuel zero it still tests the last permitted state. This convention permits success after exactly N updates. pathsearch_score proves that every returned state passes the score condition. pathsearch_origin proves that a returned state is an act","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-16def831203db3b6","tex":"K^\\top","aliases":"K^top top","display":"K^\\top","kind":"use","excerpt":"The covariance C_* admits the Joseph representation C_*=(I_n-KH)P(I_n-KH)^top+KRK^top. Each term on the right is positive semidefinite. Therefore C_*succeq0, including at the singular boundary. This avoids any proof based on P^-1 or C_*^-1.","count":8},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"The completed executable contract represents its model by rational matrices and vectors. Its input consists of H, m, y, a prior factor L∈Q^n× r, a noise factor N∈Q^p× s, and a rational number σ. The covariance matrices are derived from these inputs: P=LL^top, R=NN^top+σ I_p, σ>0. The factor dimensions r and s are finite and need not equal the state","count":9},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-7576dd58cb440d5d","tex":"L^{\\mathsf T}","aliases":"L^ T mathsf","display":"L^{\\mathsf T}","kind":"use","excerpt":"The problem is one Gaussian observation update. Write the prior as X∼ N(m,P), Y=HX+η, where the centered Gaussian noise η is independent of X. The representation uses rational factors P=LL^ T, R=NN^ T+σ I, σ>0. All data supplied to the executable construction, including m,H,L,N,σ and the observation y, are rational. Finite state and observation ind","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d750be54985836b3","tex":"L^\\top","aliases":"L^top top","display":"L^\\top","kind":"use","excerpt":"The completed executable contract represents its model by rational matrices and vectors. Its input consists of H, m, y, a prior factor L∈Q^n× r, a noise factor N∈Q^p× s, and a rational number σ. The covariance matrices are derived from these inputs: P=LL^top, R=NN^top+σ I_p, σ>0. The factor dimensions r and s are finite and need not equal the state","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-01208e159d2aec8c","tex":"\\land","aliases":"land","display":"\\land","kind":"use","excerpt":"athbb Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALI","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-529ad2daacd7efcb","tex":"\\lor","aliases":"lor","display":"\\lor","kind":"use","excerpt":"R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALC","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-25d467671d09a670","tex":"\\lVert","aliases":"lVert","display":"\\lVert","kind":"use","excerpt":"Preprocessing forms three model-dependent constants: τ=frac1σ+tr S, T=I-τ S, w=fracσ+lVert vrVert^2σ^2. All are rational. They are the fields of the record pathKomorebi.CachedPosterior.Kernel, produced by pathKomorebi.CachedPosterior.prepare. The correctness theorems use this actual preparation, rather than an arbitrary record whose fields happen t","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"The first Boolean function is checkCandidate(M,Q,D,c) =decide(ValidCandidate(M,Q,D,c)). It recomputes the finite obligations for the explicit model, packets, and candidate. A claimed successful producer run is not one of its inputs. The equivalence pathcheckCandidate_eq_true states that its true result is exactly candidate validity.","count":7},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":31},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-0627127c9388850a","tex":"m_B","aliases":"m_B","display":"m_B","kind":"use","excerpt":"Let B∈R^n× p be an arbitrary gain candidate. It need not solve the innovation equation and need not have been produced by the packet algorithm. Define m_B · =m+Bd, C_B · =(I_n-BH)P(I_n-BH)^top+BRB^top, μ_B · = N(m_B,C_B). Every C_B is positive semidefinite, since it is a sum of covariance congruences. Thus every candidate gain determines a genuine ","count":7},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries Certified Bayesi","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"The completed executable contract represents its model by rational matrices and vectors. Its input consists of H, m, y, a prior factor L∈Q^n× r, a noise factor N∈Q^p× s, and a rational number σ. The covariance matrices are derived from these inputs: P=LL^top, R=NN^top+σ I_p, σ>0. The factor dimensions r and s are finite and need not equal the state","count":11},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":",keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":15},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":34},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f6add17b631ae77e","tex":"N^{\\mathsf T}","aliases":"N^ T mathsf","display":"N^{\\mathsf T}","kind":"use","excerpt":"The problem is one Gaussian observation update. Write the prior as X∼ N(m,P), Y=HX+η, where the centered Gaussian noise η is independent of X. The representation uses rational factors P=LL^ T, R=NN^ T+σ I, σ>0. All data supplied to the executable construction, including m,H,L,N,σ and the observation y, are rational. Finite state and observation ind","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-bb26eb02161b6691","tex":"N^\\top","aliases":"N^top top","display":"N^\\top","kind":"use","excerpt":"The completed executable contract represents its model by rational matrices and vectors. Its input consists of H, m, y, a prior factor L∈Q^n× r, a noise factor N∈Q^p× s, and a rational number σ. The covariance matrices are derived from these inputs: P=LL^top, R=NN^top+σ I_p, σ>0. The factor dimensions r and s are finite and need not equal the state","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-1d7d1fdf60d8fdac","tex":"\\neg","aliases":"neg","display":"\\neg","kind":"use","excerpt":"ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULU","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f814b60f967151c4","tex":"\\mathrm{op}","aliases":"op mathrm","display":"\\mathrm{op}","kind":"use","excerpt":"p gives z^top Sz≥σ z ^2 for all z∈R^p. Let x=S^-1v. Then Sx=v, and the Cauchy--Schwarz inequality yields σ x ^2≤ x^top Sx=x^top v≤ x v . If x=0 the desired bound is immediate. Otherwise division by σ x gives S^-1v ≤σ^-1 v . Hence S^-1 _op≤σ^-1. This elementary estimate supplies an explicit certificate constant without computing the inverse. The inv","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-44a55b48f8dea448","tex":"\\mathrm{out}","aliases":"out mathrm","display":"\\mathrm{out}","kind":"use","excerpt":"Each output arithmetic expression has a precision-independent budget coefficient. The source sums all mean coefficients and all covariance coefficients to obtain C_out, defined by pathoutputCoefficient. Every individual coefficient is nonnegative and no greater than this sum. It chooses p_out=precisionFor(C_out,δ) and evaluates the output graph at ","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":52},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3401330892ce4d79","tex":"P^{-1}","aliases":"P^-1","display":"P^{-1}","kind":"use","excerpt":"The covariance C_* admits the Joseph representation C_*=(I_n-KH)P(I_n-KH)^top+KRK^top. Each term on the right is positive semidefinite. Therefore C_*succeq0, including at the singular boundary. This avoids any proof based on P^-1 or C_*^-1.","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-7cd48fb48ab38484","tex":"p_*","aliases":"p_*","display":"p_*","kind":"use","excerpt":"For final verification, the solver appends p_* = sufficientPrecision(M,Q,D,B,ε^2/4). At its returned search state, a(B)≤ε^2/4. The fallback guarantees inflation at most another ε^2/4, so widehat a_p_*(B)≤ε^2/2≤ε^2. Positive ε gives a positive inflation allowance. Valid model and packet setup give the candidate check, while this inequality gives the","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a5c35e870d97f9c4","tex":"P_B","aliases":"P_B","display":"P_B","kind":"use","excerpt":"For a rational gain B, the Joseph approximation has mean and covariance m_B · =m+B(y-Hm), P_B · =(I-BH)P(I-BH)^ T+BRB^ T. Its covariance is positive semidefinite by construction. The approximation is a measure, rather than a collection of unqualified decimal entries. The exact gain K=PH^ TS^-1 identifies the familiar Gaussian posterior parameters. ","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-5904b1468bc34e77","tex":"p_{\\mathrm{out}}","aliases":"p_out mathrm","display":"p_{\\mathrm{out}}","kind":"use","excerpt":"Each output arithmetic expression has a precision-independent budget coefficient. The source sums all mean coefficients and all covariance coefficients to obtain C_out, defined by pathoutputCoefficient. Every individual coefficient is nonnegative and no greater than this sum. It chooses p_out=precisionFor(C_out,δ) and evaluates the output graph at ","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a563e7642374a500","tex":"\\mathrm{posterior}","aliases":"posterior mathrm","display":"\\mathrm{posterior}","kind":"use","excerpt":"andidate checking] Let M be a factor-coded rational model, Q,D rational packet families, and c a rational candidate. If both pathcheckCandidate M Q D c and pathcheckTolerance M c epsilon return true, then the exact real posterior version represented by M and the exact real Joseph Gaussian represented by the gain of c satisfy Wtwo(M.posterior,M.appr","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"use","excerpt":"The completed executable contract represents its model by rational matrices and vectors. Its input consists of H, m, y, a prior factor L∈Q^n× r, a noise factor N∈Q^p× s, and a rational number σ. The covariance matrices are derived from these inputs: P=LL^top, R=NN^top+σ I_p, σ>0. The factor dimensions r and s are finite and need not equal the state","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"wstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-abe44409013535d4","tex":"\\Q^{n\\times r}","aliases":"Q^n× r Q times","display":null,"kind":"use","excerpt":"The completed executable contract represents its model by rational matrices and vectors. Its input consists of H, m, y, a prior factor L∈Q^n× r, a noise factor N∈Q^p× s, and a rational number σ. The covariance matrices are derived from these inputs: P=LL^top, R=NN^top+σ I_p, σ>0. The factor dimensions r and s are finite and need not equal the state","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f3dff2ae5bcda16b","tex":"\\Q^{p\\times s}","aliases":"Q^p× s Q times","display":null,"kind":"use","excerpt":"The completed executable contract represents its model by rational matrices and vectors. Its input consists of H, m, y, a prior factor L∈Q^n× r, a noise factor N∈Q^p× s, and a rational number σ. The covariance matrices are derived from these inputs: P=LL^top, R=NN^top+σ I_p, σ>0. The factor dimensions r and s are finite and need not equal the state","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-c364e9f084300ad4","tex":"Q_+","aliases":"Q_+","display":"Q_+","kind":"use","excerpt":"-1. It uses the rectangular prior factor L=(1,1)^ T, zero noise factor, and σ=1. Both output and input packet families are Q_+=frac121 · 1 1 · 1, Q_-=frac121 · -1 -1 · 1. These project onto the common and antisymmetric directions. They are rational, self-adjoint, orthogonal, and complete. The prior has rank one, but the innovation matrix S=I+J is p","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-22244099d34fb3a0","tex":"Q_-","aliases":"Q_-","display":"Q_-","kind":"use","excerpt":"-1. It uses the rectangular prior factor L=(1,1)^ T, zero noise factor, and σ=1. Both output and input packet families are Q_+=frac121 · 1 1 · 1, Q_-=frac121 · -1 -1 · 1. These project onto the common and antisymmetric directions. They are rational, self-adjoint, orthogonal, and complete. The prior has rank one, but the innovation matrix S=I+J is p","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-adcabca4b58dfb87","tex":"Q_{a'}","aliases":"Q_a'","display":"Q_{a'}","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'), D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-73557af8d9045cff","tex":"Q_a^2","aliases":"Q_a^2","display":"Q_a^2","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'), D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-297dc5c297f94e92","tex":"Q_a^\\top","aliases":"Q_a^top top","display":"Q_a^\\top","kind":"use","excerpt":"A packet family in this release is a finite collection of self-adjoint, pairwise disjoint idempotent matrices. Let {Q_a:a∈ A} act on the state space and {D_b:b∈ B} act on the observation space. For each family, the structural conditions are Q_a^top · =Q_a, · Q_a^2 · =Q_a, · Q_aQ_a' · =0 (a≠ a'), D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-c57539584fe8169f","tex":"Q_b","aliases":"Q_b","display":"Q_b","kind":"use","excerpt":"in [source reference] is valid. By cache validity, G_a=Q_aF(B), so the gain update is the exact packet step B'=B+τ Q_aF(B). Expand its residual: F(B')=F(B)-τ Q_aF(B)S. Multiplication on the left by the selected projector gives Q_aF(B')=Q_aF(B)-τ Q_a^2F(B)S =G_a(I-τ S)=G_aT. For b≠ a, mutual orthogonality gives Q_bQ_a=0 and hence Q_bF(B')=Q_bF(B)=G_","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":26},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":12},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-cd6a51568c71e4ad","tex":"\\R^2","aliases":"R^2 R","display":null,"kind":"use","excerpt":"For example, take the coordinate projectors onto the two axes of R^2. A residual matrix with a nonzero second row has zero energy in the first output packet. Certifying that first packet alone says nothing about the omitted row. Completeness and a bound for every packet are what change local information into a global posterior guarantee. The exact ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-12e365b1aab42166","tex":"\\R^n","aliases":"R^n R","display":null,"kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e3b6ec511215a68d","tex":"r^n","aliases":"r^n","display":"r^n","kind":"use","excerpt":"The transition contracts every residual matrix: E(XT)≤ qE(X). If label a is updated, its residual energy contracts by this inequality and all other output residuals remain unchanged. Orthogonal energy accounting gives E(F(B'))≤ E(F(B))-(1-q)e_a. Because a is a maximum, e_a≥ E(F(B))/A_count. It follows that E(F(B'))≤ rE(F(B)), a(B_n)≤ r^na(B_0). The","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8b7607f6cd5f54ed","tex":"\\R^{n\\times n}","aliases":"R^n× n R times","display":null,"kind":"use","excerpt":"e inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A singu","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e8a9254d06f455bb","tex":"\\R^{n\\times p}","aliases":"R^n× p R times","display":null,"kind":"use","excerpt":"Let B∈R^n× p be an arbitrary gain candidate. It need not solve the innovation equation and need not have been produced by the packet algorithm. Define m_B · =m+Bd,","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-14a3aa1bc94facab","tex":"\\R^p","aliases":"R^p R","display":null,"kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-0a95e7a41560e911","tex":"\\R^{p\\times n}","aliases":"R^p× n R times","display":null,"kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e7c94265a6bb538d","tex":"\\mathrm{round}","aliases":"round mathrm","display":"\\mathrm{round}","kind":"use","excerpt":"Writing d for the number of gain entries and h_t=2^-t, the source establishes an all-iteration tracking bound E(widetilde B_n-B_n^exact) ≤frac6d h_t^2(1-q)^2. Its residual-score comparison has the form a(widetilde B_n)≤2a(B_n^exact)+C_roundh_t^2, C_round= frac12dE(S)w(1-q)^2. The source names are pathroundedGain_tracking, pathroundoffScoreCoeff, an","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-61f9609b70de73a0","tex":"\\rVert","aliases":"rVert","display":"\\rVert","kind":"use","excerpt":"Preprocessing forms three model-dependent constants: τ=frac1σ+tr S, T=I-τ S, w=fracσ+lVert vrVert^2σ^2. All are rational. They are the fields of the record pathKomorebi.CachedPosterior.Kernel, produced by pathKomorebi.CachedPosterior.prepare. The correctness theorems use this actual preparation, rather than an arbitrary record whose fields happen t","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-78e04f6069787e74","tex":"\\rVert^2","aliases":"rVert^2 rVert","display":"\\rVert^2","kind":"use","excerpt":"Orthogonal complete packets account for the whole energy: ∑_a∈ AE(Q_aF)=E(F). Consequently a valid cache has t=E(F(B)) and s(c)=E(F(B))fracσ+lVert vrVert^2σ^2=a(B). The right-hand side is the exact residual score used by the posterior error certificate. The declarations pathtotal_eq_residual_energy and pathscore_eq establish these two identificatio","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-02f6204f8addafb9","tex":"\\rVert^{2}","aliases":"rVert^2 rVert","display":"\\rVert^{2}","kind":"use","excerpt":"Preprocessing forms three model-dependent constants: τ=frac1σ+tr S, T=I-τ S, w=fracσ+lVert vrVert^2σ^2. All are rational. They are the fields of the record pathKomorebi.CachedPosterior.Kernel, produced by pathKomorebi.CachedPosterior.prepare. The correctness theorems use this actual preparation, rather than an arbitrary record whose fields happen t","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","excerpt":"For an observed value y, put d=y-Hm, S=HPH^top+R, K=PH^top S^-1. The matrix S is the innovation covariance. It is positive definite because, for every nonzero z∈R^p, z^top Sz=(H^top z)^top P(H^top z)+z^top Rz>0. Consequently the inverse in the definition of K exists even when P has a nontrivial nullspace or H has deficient rank. Every inverse used ","count":14},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-32f99fc05b64a106","tex":"\\succ","aliases":"succ","display":"\\succ","kind":"use","excerpt":"t state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A singular prior is an ordinary Gaussian probability law concentrated on an affin","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"use","excerpt":"imensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A singular prior is an ordinary Gaussian probability law concentr","count":6},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-67f0db11b82350d1","tex":"\\operatorname{sufficientPrecision}","aliases":"sufficientPrecision operatorname","display":"\\operatorname{sufficientPrecision}","kind":"use","excerpt":"For final verification, the solver appends p_* = sufficientPrecision(M,Q,D,B,ε^2/4). At its returned search state, a(B)≤ε^2/4. The fallback guarantees inflation at most another ε^2/4, so widehat a_p_*(B)≤ε^2/2≤ε^2. Positive ε gives a positive inflation allowance. Valid model and packet setup give the candidate check, while this inequality gives the","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Take independent centered Gaussian variables U∼ N(0,P) and V∼ N(0,R). For each gain B, define the mathematical affine random variable T_B=m+Bd+(I_n-BH)U-BV. Its mean is m_B and its covariance is C_B, so its law is μ_B. At B=K, the exact Joseph identity identifies its law with ν_y. The pair (T_K,T_B), using the same inputs (U,V), is therefore an act","count":11},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"The problem is one Gaussian observation update. Write the prior as X∼ N(m,P), Y=HX+η, where the centered Gaussian noise η is independent of X. The representation uses rational factors P=LL^ T, R=NN^ T+σ I, σ>0. All data supplied to the executable construction, including m,H,L,N,σ and the observation y, are rational. Finite state and observation ind","count":16},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-77970456be0eda56","tex":"t'","aliases":"t'","display":"t'","kind":"use","excerpt":"For a selected label a, the executed update is B' · =B+τ G_a, · G'_a · =G_aT, · e'_a · =E(G'_a),nonumber G'_b · =G_b (b≠ a), · e'_b · =e_b (b≠ a), · t' · =t-e_a+e'_a. This is the definition pathKomorebi.CachedPosterior.step. It computes the new selected residual from the old selected residual and the precomputed transition. It retains unselected en","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-b037fe8890ca7d73","tex":"T_B","aliases":"T_B","display":"T_B","kind":"use","excerpt":"Take independent centered Gaussian variables U∼ N(0,P) and V∼ N(0,R). For each gain B, define the mathematical affine random variable T_B=m+Bd+(I_n-BH)U-BV. Its mean is m_B and its covariance is C_B, so its law is μ_B. At B=K, the exact Joseph identity identifies its law with ν_y. The pair (T_K,T_B), using the same inputs (U,V), is therefore an act","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-bae1686b52c98bd5","tex":"T_K","aliases":"T_K","display":"T_K","kind":"use","excerpt":"independent centered Gaussian variables U∼ N(0,P) and V∼ N(0,R). For each gain B, define the mathematical affine random variable T_B=m+Bd+(I_n-BH)U-BV. Its mean is m_B and its covariance is C_B, so its law is μ_B. At B=K, the exact Joseph identity identifies its law with ν_y. The pair (T_K,T_B), using the same inputs (U,V), is therefore an actual t","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"For an observed value y, put d=y-Hm, S=HPH^top+R, K=PH^top S^-1. The matrix S is the innovation covariance. It is positive definite because, for every nonzero z∈R^p, z^top Sz=(H^top z)^top P(H^top z)+z^top Rz>0. Consequently the inverse in the definition of K exists even when P has a nontrivial nullspace or H has deficient rank. Every inverse used ","count":39},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4ef6c0d9701b3b69","tex":"\\triangleright","aliases":"triangleright","display":null,"kind":"use","excerpt":"λ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries Certified Bayesian","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"Take independent centered Gaussian variables U∼ N(0,P) and V∼ N(0,R). For each gain B, define the mathematical affine random variable T_B=m+Bd+(I_n-BH)U-BV. Its mean is m_B and its covariance is C_B, so its law is μ_B. At B=K, the exact Joseph identity identifies its law with ν_y. The pair (T_K,T_B), using the same inputs (U,V), is therefore an act","count":15},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Suppose a producer supplies values u_ab and v with proved inequalities E(F_ab(B))≤ u_ab, d ^2≤ v. Completeness and nonnegative energy imply E(F(B))≤ U, U=∑_a,bu_ab. Each u_ab is nonnegative because it bounds a nonnegative quantity. Likewise v≥0. The score widehat a(B)=Ufracσ+vσ^2 therefore bounds a(B) and hence the squared posterior distance. If U(","count":7},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e77780c38a5d6ab9","tex":"u_{ab}","aliases":"u_ab","display":"u_{ab}","kind":"use","excerpt":"Suppose a producer supplies values u_ab and v with proved inequalities E(F_ab(B))≤ u_ab, d ^2≤ v. Completeness and nonnegative energy imply E(F(B))≤ U, U=∑_a,bu_ab. Each u_ab is nonnegative because it bounds a nonnegative quantity. Likewise v≥0. The score widehat a(B)=Ufracσ+vσ^2 therefore bounds a(B) and hence the squared posterior distance. If U(","count":7},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d27fa8e4e0984e4b","tex":"\\uparrow","aliases":"uparrow","display":"\\uparrow","kind":"use","excerpt":"ma1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries Certified Bayesian","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":15},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"The same σ that certifies positive noise also bounds the inverse innovation operator. Adding the positive semidefinite term HPH^top gives z^top Sz≥σ z ^2 for all z∈R^p. Let x=S^-1v. Then Sx=v, and the Cauchy--Schwarz inequality yields σ x ^2≤ x^top Sx=x^top v≤ x v . If x=0 the desired bound is immediate. Otherwise division by σ x gives S^-1v ≤σ^-1 ","count":18},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f2d53a514504ac21","tex":"V_*","aliases":"V_*","display":"V_*","kind":"use","excerpt":"-78432 · 98041. It can be checked directly by substituting B_6 into [source reference]. Another useful check starts from the exact posterior covariance V_* = (4/3 · -2/3 -2/3 · 5/6). If B_*=(2/3,1/6)^ T is the exact gain, expansion of the Joseph expression gives V_B_6=V_*+6(B_6-B_*)(B_6-B_*)^ T. The gain discrepancy is B_*-B_6=(2/1029,1/2058)^ T, w","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-9f5cc10c11d8d9e7","tex":"V_B","aliases":"V_B","display":"V_B","kind":"use","excerpt":"A useful posterior output contains more than a passing gain. For the returned rational gain B, the parameter stage represents μ_B · =m+B(y-Hm),nonumber V_B · =(I-BH)P(I-BH)^ T+BRB^ T. These are the mean and Joseph covariance of the same Gaussian law certified by the Wasserstein statement. The covariance is positive semidefinite under the model hypo","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-68206056a3aac691","tex":"V_{B_4}","aliases":"V_B_4","display":"V_{B_4}","kind":"use","excerpt":"The exact posterior mean is zero because Jy=0, and the returned approximation's mean is also zero. Its Joseph covariance is V_B_4=frac130379390625J. The exact posterior covariance is J/3. Both laws remain supported in the prior's one-dimensional common-direction subspace. This example therefore combines a singular prior, rotated packet selection, a","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-006b778c929e2116","tex":"V_{B_6}","aliases":"V_B_6","display":"V_{B_6}","kind":"use","excerpt":"The exact rational matrix enclosed by those covariance balls is V_B_6=frac1117649 156868 · -78432 -78432 · 98041. It can be checked directly by substituting B_6 into [source reference]. Another useful check starts from the exact posterior covariance V_* = (4/3 · -2/3","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"[Frobenius energy controls the residual cost] If F∈R^n× p and Ssucceqσ I_psucc0, then FS^-1d ^2+tr(FS^-1F^top) ≤ E(F)(σ^-1+σ^-2 d ^2). For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose o","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-886ccc7ad77d22b3","tex":"\\mathrm{ver}","aliases":"ver mathrm","display":"\\mathrm{ver}","kind":"use","excerpt":"Let h_p=2^-p. The dyadic evaluator propagates rational centers and outward radii through its arithmetic graph. Its budget calculation supplies a nonnegative coefficient C_ver(M,Q,D,B) independent of p, with widehat a_p(B)≤ a(B)+C_ver(M,Q,D,B)h_p. Here widehat a_p is the actual rounded candidate score. The full expression for the coefficient is path","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Joseph gain · A valid Gaussian approximation and residual-based W_2 bound · Common-noise coupling and matrix bounds","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-1dd506c998004ce1","tex":"W_2","aliases":"W_2","display":"W_2","kind":"use","excerpt":"Joseph gain · A valid Gaussian approximation and residual-based W_2 bound · Common-noise coupling and matrix bounds","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"Suppose a producer supplies values u_ab and v with proved inequalities E(F_ab(B))≤ u_ab, d ^2≤ v. Completeness and nonnegative energy imply E(F(B))≤ U, U=∑_a,bu_ab. Each u_ab is nonnegative because it bounds a nonnegative quantity. Likewise v≥0. The score widehat a(B)=Ufracσ+vσ^2 therefore bounds a(B) and hence the squared posterior distance. If U(","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-47a7b16468ab8670","tex":"\\widehat a_p","aliases":"widehat a_p widehat","display":"\\widehat a_p","kind":"use","excerpt":"Let h_p=2^-p. The dyadic evaluator propagates rational centers and outward radii through its arithmetic graph. Its budget calculation supplies a nonnegative coefficient C_ver(M,Q,D,B) independent of p, with widehat a_p(B)≤ a(B)+C_ver(M,Q,D,B)h_p. Here widehat a_p is the actual rounded candidate score. The full expression for the coefficient is path","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8ad88c5aec7e7cd5","tex":"\\widehat a_{p_*}","aliases":"widehat a_p_* widehat","display":"\\widehat a_{p_*}","kind":"use","excerpt":"For final verification, the solver appends p_* = sufficientPrecision(M,Q,D,B,ε^2/4). At its returned search state, a(B)≤ε^2/4. The fallback guarantees inflation at most another ε^2/4, so widehat a_p_*(B)≤ε^2/2≤ε^2. Positive ε gives a positive inflation allowance. Valid model and packet setup give the candidate check, while this inequality gives the","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f596b26bb2804e7d","tex":"\\widetilde B","aliases":"widetilde B widetilde","display":"\\widetilde B","kind":"use","excerpt":"Core 1.0.0 already retains a different, completed rounded solver. Its update is a genuinely rounded full-gain Richardson step. It neither claims exact selective cache invariants nor selects a largest cached packet. Its declarations expose the whole-update defect instead. If widetilde B_n+1 is the rounded update and B_n+1^ideal is the exact full ste","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-1e6f2ceb45d0daf0","tex":"\\widetilde B_n","aliases":"widetilde B_n widetilde","display":"\\widetilde B_n","kind":"use","excerpt":"Core 1.0.0 already retains a different, completed rounded solver. Its update is a genuinely rounded full-gain Richardson step. It neither claims exact selective cache invariants nor selects a largest cached packet. Its declarations expose the whole-update defect instead. If widetilde B_n+1 is the rounded update and B_n+1^ideal is the exact full ste","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-781ad13a14241bb4","tex":"\\widetilde B_{n+1}","aliases":"widetilde B_n+1 widetilde","display":"\\widetilde B_{n+1}","kind":"use","excerpt":"Core 1.0.0 already retains a different, completed rounded solver. Its update is a genuinely rounded full-gain Richardson step. It neither claims exact selective cache invariants nor selects a largest cached packet. Its declarations expose the whole-update defect instead. If widetilde B_n+1 is the rounded update and B_n+1^ideal is the exact full ste","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-843873bf8da73057","tex":"\\Wtwo","aliases":"Wtwo","display":null,"kind":"use","excerpt":"For probability laws with finite second moment, write Wtwo(ν,μ)^2 =inf_π∈Π(ν,μ)∫ x-z ^2 dπ(x,z), where Π(ν,μ) is the set of transport plans with the stated marginals. The formal source defines the squared quantity as a nonnegative extended-real infimum and obtains the distance by a square-root operation. The explicit finite Gaussian coupling proves","count":10},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-fdcc5f83cf536b2d","tex":"\\Wtwo^2","aliases":"Wtwo^2 Wtwo","display":null,"kind":"use","excerpt":"ns are pathposterior_wasserstein2Sq_le_residual_energy and pathposterior_wasserstein2_le_of_residual_tolerance. Their hypotheses include the covariance assumptions, positivity of the floor, and the floor's quadratic interpretation. In the factor-coded program, those hypotheses are discharged by the represented factors and the checked positive scala","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":9},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"This representation makes the covariance hypotheses concrete. For every x, x^top LL^top x= L^top x ^2≥0, x^top Rx= N^top x ^2+σ x ^2. Thus P is positive semidefinite and Rsucceqσ I_psucc0. These conclusions are proved from the factors; they are not additional statistical assumptions hidden inside a covariance record. The real semantics interprets e","count":12},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-cd80b0dabb00d99b","tex":"x^\\top","aliases":"x^top top","display":"x^\\top","kind":"use","excerpt":"This representation makes the covariance hypotheses concrete. For every x, x^top LL^top x= L^top x ^2≥0, x^top Rx= N^top x ^2+σ x ^2. Thus P is positive semidefinite and Rsucceqσ I_psucc0. These conclusions are proved from the factors; they are not additional statistical assumptions hidden inside a covariance record. The real semantics interprets e","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":9},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"For the reverse construction, sample Y∼ N(Hm,S) and an independent E_*∼ N(0,C_*). Define Z=m+K(Y-Hm)+E_*. Conditional on Y=y, the law of Z is precisely ν_y. The joint mean of (Y,Z) is again (Hm,m). Its cross covariance is SK^top=HP, and its state covariance is KSK^top+C_*=KHP+(P-KHP)=P. It therefore has exactly the covariance in [source reference].","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"orlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓down","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"For an observed value y, put d=y-Hm, S=HPH^top+R, K=PH^top S^-1. The matrix S is the innovation covariance. It is positive definite because, for every nonzero z∈R^p, z^top Sz=(H^top z)^top P(H^top z)+z^top Rz>0. Consequently the inverse in the definition of K exists even when P has a nontrivial nullspace or H has deficient rank. Every inverse used ","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4ce8c5865fb8a8b6","tex":"z^\\top","aliases":"z^top top","display":"z^\\top","kind":"use","excerpt":"For an observed value y, put d=y-Hm, S=HPH^top+R, K=PH^top S^-1. The matrix S is the innovation covariance. It is positive definite because, for every nonzero z∈R^p, z^top Sz=(H^top z)^top P(H^top z)+z^top Rz>0. Consequently the inverse in the definition of K exists even when P has a nontrivial nullspace or H has deficient rank. Every inverse used ","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"note[2]textcolorteal#1 #2 newcommandfig[3][H]centeringincludegraphics[width=linewidth]#1caption#2 graphicspath../figures/figures/ pagestylefancyfancyhf fancyhead[L]smalltextcolornavyCertified Bayesian inference fancyhead[R]smallPublication edition fancyfoot[L]small Komorebi P. · 7 October 2026 fancyfoot[R]smallthepage / pagerefLastPage renewcommand","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS ","count":9},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"orlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"style=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"e1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / R","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Let Δ=K-B. Subtraction in [source reference] gives T_K-T_B=Δ d-Δ(HU+V). The random vector HU+V is centered and has covariance S. The difference therefore has mean Δ d and covariance Δ SΔ^top. Its expected squared Euclidean norm is E T_K-T_B ^2 = Δ d ^2+tr(Δ SΔ^top). The source proves both marginals and this cost. It does not assume that a producer ","count":5},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"Parameter arrays · Enclose this approximation's parameters, with radii at most δ · Declared evaluator and precision theorem","count":18},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-0ce010e4426cda56","tex":"\\Delta^\\top","aliases":"Δ^top Delta top","display":"\\Delta^\\top","kind":"use","excerpt":"Let Δ=K-B. Subtraction in [source reference] gives T_K-T_B=Δ d-Δ(HU+V). The random vector HU+V is centered and has covariance S. The difference therefore has mean Δ d and covariance Δ SΔ^top. Its expected squared Euclidean norm is E T_K-T_B ^2 = Δ d ^2+tr(Δ SΔ^top). The source proves both marginals and this cost. It does not assume that a producer ","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"[Publication contract of the canonical solver] Fix a finite-dimensional rational factor-coded model with σ>0, and complete, mutually orthogonal self-adjoint rational output and input packet families. Supply an output-packet seed and a finite scan containing every output label. Let ϵ,δ>0 be rational. An arbitrary rational initial gain and an arbitra","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"skip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode","count":42},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"osterior certificate] For the model in [source reference], with Rsucceqσ I_p and σ>0, every gain B satisfies Wtwo(ν_y,μ_B)^2≤ a(B), a(B)=E(F(B))fracσ+ d ^2σ^2. In particular, for ε≥0, the stopping condition E(F(B))(σ+ d ^2)≤ε^2σ^2 implies Wtwo(ν_y,μ_B)≤ε. The common-noise transport plan gives [source reference]. Lemma~[source reference] bounds its ","count":21},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"rrow1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolor","count":6},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolo","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMO","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"y,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"rightarrow1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanc","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=fa","count":12},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-5926607b3fa28f05","tex":"\\mu_B","aliases":"μ_B mu","display":"\\mu_B","kind":"use","excerpt":"Let B∈R^n× p be an arbitrary gain candidate. It need not solve the innovation equation and need not have been produced by the packet algorithm. Define m_B · =m+Bd, C_B · =(I_n-BH)P(I_n-BH)^top+BRB^top, μ_B · = N(m_B,C_B). Every C_B is positive semidefinite, since it is a sum of covariance congruences. Thus every candidate gain determines a genuine ","count":9},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolor","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"For probability laws with finite second moment, write Wtwo(ν,μ)^2 =inf_π∈Π(ν,μ)∫ x-z ^2 dπ(x,z), where Π(ν,μ) is the set of transport plans with the stated marginals. The formal source defines the squared quantity as a nonnegative extended-real infimum and obtains the distance by a square-root operation. The explicit finite Gaussian coupling proves","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":".7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f9c3cb397f3b4ffb","tex":"\\Pi_y","aliases":"Π_y Pi","display":"\\Pi_y","kind":"use","excerpt":"The canonical program pathKomorebi.CachedPosterior.runCached returns a result r. If B is its candidate gain and Π_y is the explicitly proved Gaussian posterior version, then Wtwo(Π_y, N(m_B,P_B))≤ϵ. The candidate and tolerance predicates hold. The reported mean and covariance intervals enclose m_B and P_B, respectively, and every reported radius is","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"bb N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REA","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-6512b523961a05f5","tex":"\\sigma^{-1}","aliases":"σ^-1 sigma","display":"\\sigma^{-1}","kind":"use","excerpt":"perator. Adding the positive semidefinite term HPH^top gives z^top Sz≥σ z ^2 for all z∈R^p. Let x=S^-1v. Then Sx=v, and the Cauchy--Schwarz inequality yields σ x ^2≤ x^top Sx=x^top v≤ x v . If x=0 the desired bound is immediate. Otherwise division by σ x gives S^-1v ≤σ^-1 v . Hence S^-1 _op≤σ^-1. This elementary estimate supplies an explicit certif","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-7647170094160a35","tex":"\\sigma^{-2}","aliases":"σ^-2 sigma","display":"\\sigma^{-2}","kind":"use","excerpt":"[Frobenius energy controls the residual cost] If F∈R^n× p and Ssucceqσ I_psucc0, then FS^-1d ^2+tr(FS^-1F^top) ≤ E(F)(σ^-1+σ^-2 d ^2). For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose o","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"[Residual-energy posterior certificate] For the model in [source reference], with Rsucceqσ I_p and σ>0, every gain B satisfies Wtwo(ν_y,μ_B)^2≤ a(B), a(B)=E(F(B))fracσ+ d ^2σ^2. In particular, for ε≥0, the stopping condition E(F(B))(σ+ d ^2)≤ε^2σ^2 implies Wtwo(ν_y,μ_B)≤ε. The common-noise transport plan gives [source reference]. Lemma~[source refe","count":8},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-238fd3fe6760f691","tex":"\\sigma^{2}","aliases":"σ^2 sigma","display":"\\sigma^{2}","kind":"use","excerpt":"Preprocessing forms three model-dependent constants: τ=frac1σ+tr S, T=I-τ S, w=fracσ+lVert vrVert^2σ^2. All are rational. They are the fields of the record pathKomorebi.CachedPosterior.Kernel, produced by pathKomorebi.CachedPosterior.prepare. The correctness theorems use this actual preparation, rather than an arbitrary record whose fields happen t","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageancho","count":12},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"Relation to established methods. Maximum-energy packet selection has a direct optimization interpretation. For the quadratic objective ϕ(B)=tfrac12tr(BSB^ T)-⟨ B,PH^ T⟩_ F, the gradient is BS-PH^ T=-F(B). The selected packet has largest projected-gradient norm. This is a block Gauss--Southwell mechanism, and the residual cache is maintained by exac","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"le,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleri","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"e=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetupp","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Gaussian disintegration] Let ℓ(x,·)= N(Hx,R) be the observation kernel and let η= N(Hm,S) be the evidence law. Then the explicit kernel y↦ν_y satisfies η(dy) ν_y(dx) = N(m,P)(dx) ℓ(x,dy) after exchanging the order of the two coordinates on the right. In particular, η is the actual observation marginal and ν is a posterior version for this prior an","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"use","excerpt":"LeanCodelanguage=Lean,basicstyle=ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"--,morecomment=[s]/--/,morestring=[b]\" lstdefinestyleLeanCodelanguage=Lean,basicstyle=ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"nt=[s]/--/,morestring=[b]\" lstdefinestyleLeanCodelanguage=Lean,basicstyle=ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Gaussian disintegration] Let ℓ(x,·)= N(Hx,R) be the observation kernel and let η= N(Hm,S) be the evidence law. Then the explicit kernel y↦ν_y satisfies η(dy) ν_y(dx) = N(m,P)(dx) ℓ(x,dy) after exchanging the order of the two coordinates on the right. In particular, η is the actual observation marginal and ν is a posterior version for this prior an","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"s concrete. For every x, x^top LL^top x= L^top x ^2≥0, x^top Rx= N^top x ^2+σ x ^2. Thus P is positive semidefinite and Rsucceqσ I_psucc0. These conclusions are proved from the factors; they are not additional statistical assumptions hidden inside a covariance record. The real semantics interprets each rational coordinate through the exact inclusio","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"[b]\" lstdefinestyleLeanCodelanguage=Lean,basicstyle=ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"ving,sensitive=true,morecomment=[l]--,morecomment=[s]/--/,morestring=[b]\" lstdefinestyleLeanCodelanguage=Lean,basicstyle=ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"ue,morecomment=[l]--,morecomment=[s]/--/,morestring=[b]\" lstdefinestyleLeanCodelanguage=Lean,basicstyle=ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"ttfamilyfootnotesize,breaklines=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 ","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"n1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries Certified B","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-7684dda7bca2d3dd","tex":"\\sum_{a,b}","aliases":"∑_a,b sum","display":"\\sum_{a,b}","kind":"use","excerpt":"Suppose a producer supplies values u_ab and v with proved inequalities E(F_ab(B))≤ u_ab, d ^2≤ v. Completeness and nonnegative energy imply E(F(B))≤ U, U=∑_a,bu_ab. Each u_ab is nonnegative because it bounds a nonnegative quantity. Likewise v≥0. The score widehat a(B)=Ufracσ+vσ^2 therefore bounds a(B) and hence the squared posterior distance. If U(","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-856476d7da8c3c90","tex":"\\sum_{a\\in\\mathcal A}","aliases":"∑_a∈ A sum in mathcal","display":"\\sum_{a\\in\\mathcal A}","kind":"use","excerpt":"D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'). Completeness is a separate condition: ∑_a∈ AQ_a=I_n, ∑_b∈ BD_b=I_p. The self-adjoint condition makes these Euclidean orthogonal projectors. Idempotence alone would not give the required energy geometry. Completeness prevents a certificate from forgetting a residual direction outside the select","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-985a5455bd14672d","tex":"\\sum_{a\\in U}","aliases":"∑_a∈ U sum in","display":"\\sum_{a\\in U}","kind":"use","excerpt":"lthm:foundation-packet-energy If both families satisfy [source reference]--[source reference], then every A∈R^n× p satisfies E(A)=∑_a∈ A∑_b∈ BE(Q_aAD_b). More generally, for subsets U⊆ A and V⊆ B, E ((∑_a∈ UQ_a)A (∑_b∈ VD_b)) =∑_a∈ U∑_b∈ VE(Q_aAD_b). The sum of any subfamily of the Q_a is again self-adjoint and idempotent: expansion of its square r","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","excerpt":"=G_a(I-τ S)=G_aT. For b≠ a, mutual orthogonality gives Q_bQ_a=0 and hence Q_bF(B')=Q_bF(B)=G_b. This proves every residual equality in the new cache. Its energy equalities hold because the selected energy is evaluated from its new residual and the remaining residual-energy pairs are retained. Finally, replacing one term in a finite sum changes that","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-599af2bc58ba4c22","tex":"\\sum_{b\\in\\mathcal B}","aliases":"∑_b∈ B sum in mathcal","display":"\\sum_{b\\in\\mathcal B}","kind":"use","excerpt":"D_b^top · =D_b, · D_b^2 · =D_b, · D_bD_b' · =0 (b≠ b'). Completeness is a separate condition: ∑_a∈ AQ_a=I_n, ∑_b∈ BD_b=I_p. The self-adjoint condition makes these Euclidean orthogonal projectors. Idempotence alone would not give the required energy geometry. Completeness prevents a certificate from forgetting a residual direction outside the select","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-965e6a0927763af5","tex":"\\sum_{b\\in V}","aliases":"∑_b∈ V sum in","display":"\\sum_{b\\in V}","kind":"use","excerpt":"lthm:foundation-packet-energy If both families satisfy [source reference]--[source reference], then every A∈R^n× p satisfies E(A)=∑_a∈ A∑_b∈ BE(Q_aAD_b). More generally, for subsets U⊆ A and V⊆ B, E ((∑_a∈ UQ_a)A (∑_b∈ VD_b)) =∑_a∈ U∑_b∈ VE(Q_aAD_b). The sum of any subfamily of the Q_a is again self-adjoint and idempotent: expansion of its square r","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"[Frobenius energy controls the residual cost] If F∈R^n× p and Ssucceqσ I_psucc0, then FS^-1d ^2+tr(FS^-1F^top) ≤ E(F)(σ^-1+σ^-2 d ^2). For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose o","count":2},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-38c03c7cd3f0f240","tex":"\\sum_{i,j}","aliases":"∑_i,j sum","display":"\\sum_{i,j}","kind":"use","excerpt":"We use Euclidean vector norms and the squared Frobenius energy E(A)=FrobA^2=∑_i,jA_ij^2. The distinction between these norms matters. A matrix energy is a finite sum of scalar squares that can be evaluated exactly on rational inputs. An operator norm is used in the proof that this energy controls distribution error. The certificate never asks a pro","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"[Frobenius energy controls the residual cost] If F∈R^n× p and Ssucceqσ I_psucc0, then FS^-1d ^2+tr(FS^-1F^top) ≤ E(F)(σ^-1+σ^-2 d ^2). For a vector v, rowwise Cauchy--Schwarz gives Fv ^2=∑_i(∑_jF_ijv_j)^2 ≤∑_i(∑_jF_ij^2) v ^2 =E(F) v ^2. Apply this with v=S^-1d and use [source reference] to obtain FS^-1d ^2≤ E(F)σ^-2 d ^2. If f_i is the transpose o","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"a1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries Certified Bayesian","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"For probability laws with finite second moment, write Wtwo(ν,μ)^2 =inf_π∈Π(ν,μ)∫ x-z ^2 dπ(x,z), where Π(ν,μ) is the set of transport plans with the stated marginals. The formal source defines the squared quantity as a nonnegative extended-real infimum and obtains the distance by a square-root operation. The explicit finite Gaussian coupling proves","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"The inference problem concerns one observation of a finite-dimensional latent state. Let the state space be R^n and the observation space be R^p. A state X has prior law N(m,P), and its observation is Y=HX+V, V∼ N(0,R), V independent of X. Here H∈R^p× n may be rectangular, and P∈R^n× n may be singular. The assumptions are Psucceq0 and Rsucc0. A sin","count":7},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"s=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1","count":5},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"The canonical program pathKomorebi.CachedPosterior.runCached returns a result r. If B is its candidate gain and Π_y is the explicitly proved Gaussian posterior version, then Wtwo(Π_y, N(m_B,P_B))≤ϵ. The candidate and tolerance predicates hold. The reported mean and covariance intervals enclose m_B and P_B, respectively, and every reported radius is","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"es=true,columns=fullflexible,keepspaces=true,showstringspaces=false,backgroundcolor=colorlight,frame=single,rulecolor=colorlight,keywordstyle=colornavy,commentstyle=colorteal,aboveskip=0.7em,belowskip=0.7em, literate=∀∀1 ∃∃1 →→1 ↔↔1 ⇒⇒1 ←←1 ∈∈1 ∉∉1 ≤≤1 ≥≥1 ≠≠1 ℕ N1 ℚ Q1 ℝ R1 ℤ Z1 ΩΩ1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θ","count":12},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfser","count":1},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"a1 αα1 ββ1 κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Huge","count":3},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"κκ1 δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries Ce","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"δδ1 εε1 ππ1 ωω1 σσ1 ττ1 μμ1 λλ1 ΘΘ1 ξξ1 ηη1 νν1 χχ1 θθ1 ρρ1 ΓΓ1 ΣΣ1 ∧land1 ∨lor1 ¬neg1 ××1 ⊆⊆1 ⊂⊂1 ⟨⟨1 ⟩⟩1 ∑∑1 ∏∏1 ∣mid1 ∘∘1 ↑uparrow1 ↓downarrow1 ▸triangleright1 ₀_01 ₁_11 ₂_21 ₃_31 ₄_41 ₅_51 ₆_61 ₇_71 ₈_81 ₉_91 lstsetstyle=LeanCode newcommandbayescover[2]hypersetuppageanchor=falsecolornavy large KOMOREBI / REALIZATION CALCULUS Hugebfseries Certif","count":4},{"work":"research-certified-bayesian-inference","url":"notation/research-certified-bayesian-inference.html#symbol-0b6be0c6e1bf232c","tex":"\\rangle_{\\mathrm F}","aliases":"⟩_ F rangle mathrm","display":"\\rangle_{\\mathrm F}","kind":"use","excerpt":"Use the Frobenius inner product ⟨ A,C⟩_ F=tr(A^top C). For a≠ a', self-adjointness and disjointness give ⟨ Q_aA,Q_a'A⟩_ F =tr(A^top Q_aQ_a'A)=0. Consequently the energy of a finite sum of output packets is the sum of their energies. The input analogue follows by transposing the matrices. Applying the two decompositions successively gives complete t","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"Goal use · η_G→ A · Goal intervention versus goal-label replay.","count":178},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"notation-table","excerpt":"Damage detection · η_D→ M · Damage versus matched no-damage.","count":138},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"notation-table","excerpt":"Goal retention · η_G(τ) · Goal intervention with future cues matched or removed.","count":172},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"notation-table","excerpt":"Goal semantics · Held-out Q_t^g ordering · Distinct goals reverse or alter at least one declared action ordering.","count":127},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"notation-table","excerpt":"Damage detection · η_D→ M · Damage versus matched no-damage.","count":46},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"notation-table","excerpt":"Null ID · One of N_1,…,N_34 or a declared extension.","count":30},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ffaa0ad23479bca8","tex":"N_1","aliases":"N_1","display":"N_1","kind":"notation-table","excerpt":"Null ID · One of N_1,…,N_34 or a declared extension.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1dc905c2b7910428","tex":"N_{34}","aliases":"N_34","display":"N_{34}","kind":"notation-table","excerpt":"Null ID · One of N_1,…,N_34 or a declared extension.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"Goal semantics · Held-out Q_t^g ordering · Distinct goals reverse or alter at least one declared action ordering.","count":90},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ec98d5d2ac24681","tex":"Q_t^g","aliases":"Q_t^g","display":"Q_t^g","kind":"notation-table","excerpt":"Goal semantics · Held-out Q_t^g ordering · Distinct goals reverse or alter at least one declared action ordering.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"notation-table","excerpt":"Repair mediation · η_M→ R · Monitor intervention and monitor ablation.","count":85},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-131dd08f2af58ba0","tex":"\\mathrm{repair}","aliases":"repair mathrm","display":"\\mathrm{repair}","kind":"notation-table","excerpt":"Repair specificity · Δ_repair^spec · Intended repair versus cost-matched generic action.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ec2fdd09d5e3a211","tex":"\\mathrm{spec}","aliases":"spec mathrm","display":"\\mathrm{spec}","kind":"notation-table","excerpt":"Repair specificity · Δ_repair^spec · Intended repair versus cost-matched generic action.","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"notation-table","excerpt":"Goal semantics · Held-out Q_t^g ordering · Distinct goals reverse or alter at least one declared action ordering.","count":489},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"notation-table","excerpt":"Value use · η_V→ A · Value-label permutation and value-channel ablation.","count":220},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"notation-table","excerpt":"Restoration · η_R→ v · Repair versus matched no-repair.","count":32},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"notation-table","excerpt":"Outcome control · η_A→ Y · Feasible action intervention.","count":136},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"notation-table","excerpt":"Repair specificity · Δ_repair^spec · Intended repair versus cost-matched generic action.","count":75},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"notation-table","excerpt":"Counterfactual accuracy · δ_Q · Internal value versus external action interventions.","count":190},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7d38daffe35514fe","tex":"\\delta_Q","aliases":"δ_Q delta","display":"\\delta_Q","kind":"notation-table","excerpt":"Counterfactual accuracy · δ_Q · Internal value versus external action interventions.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-644374c4a2c89e9b","tex":"\\Delta_{\\mathrm{repair}}^{\\mathrm{spec}}","aliases":"Δ_repair^spec Delta mathrm mathrm","display":"\\Delta_{\\mathrm{repair}}^{\\mathrm{spec}}","kind":"notation-table","excerpt":"Repair specificity · Δ_repair^spec · Intended repair versus cost-matched generic action.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"notation-table","excerpt":"Goal retention · η_G(τ) · Goal intervention with future cues matched or removed.","count":96},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-df8c666334f80473","tex":"\\eta_{A\\to Y}","aliases":"η_A→ Y eta to","display":"\\eta_{A\\to Y}","kind":"notation-table","excerpt":"Outcome control · η_A→ Y · Feasible action intervention.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f844e1d44e5725b0","tex":"\\eta_{D\\to M}","aliases":"η_D→ M eta to","display":"\\eta_{D\\to M}","kind":"notation-table","excerpt":"Damage detection · η_D→ M · Damage versus matched no-damage.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1675b4a4188c1f01","tex":"\\eta_G","aliases":"η_G eta","display":"\\eta_G","kind":"notation-table","excerpt":"Goal retention · η_G(τ) · Goal intervention with future cues matched or removed.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9a6bb73579c0ba71","tex":"\\eta_{G\\to A}","aliases":"η_G→ A eta to","display":"\\eta_{G\\to A}","kind":"notation-table","excerpt":"Goal use · η_G→ A · Goal intervention versus goal-label replay.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8e62838f6e243e5e","tex":"\\eta_{M\\to R}","aliases":"η_M→ R eta to","display":"\\eta_{M\\to R}","kind":"notation-table","excerpt":"Repair mediation · η_M→ R · Monitor intervention and monitor ablation.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3beac2157679d112","tex":"\\eta_{R\\to v}","aliases":"η_R→ v eta to","display":"\\eta_{R\\to v}","kind":"notation-table","excerpt":"Restoration · η_R→ v · Repair versus matched no-repair.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b964ddca1e6b0676","tex":"\\eta_{V\\to A}","aliases":"η_V→ A eta to","display":"\\eta_{V\\to A}","kind":"notation-table","excerpt":"Value use · η_V→ A · Value-label permutation and value-channel ablation.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"notation-table","excerpt":"Goal retention · η_G(τ) · Goal intervention with future cues matched or removed.","count":101},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"notation-table","excerpt":"Null ID · One of N_1,…,N_34 or a declared extension.","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"notation-table","excerpt":"Goal use · η_G→ A · Goal intervention versus goal-label replay.","count":99},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":157},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":314},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"definition-passage","excerpt":"item ℓ is the normalized loss or utility convention; item τ is the temporal horizon or horizon family; item I is the feasible intervention family; item N is the required matched-null family; and item Θ is the gate schema, direction, threshold, and indifference region used for certification. The complete certificate is written Cert(W;ς) = bigwedge_q","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":905},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":36},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Exact operation-compatible transfer] A core transfer H_e:C_i→ C_j is operation compatible when it is an intertwiner: H_eρ_i(a)=ρ_j(a)H_e ∀ a∈ A. Equivalently, the diagram ccc C_i · xrightarrowρ_i(a) · C_i","count":138},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"definition-passage","excerpt":"[Natural workspace operators] Define O_Workspace = { (A_i)_i∈ V: A_t(e)H_e=H_eA_s(e) ∀ e }. Categorically, O_Workspace=Nat(H,H), the algebra of natural endomorphisms of the local system.","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a30a4fb26994a13a","tex":"A_{s(e)}","aliases":"A_s(e)","display":"A_{s(e)}","kind":"definition-passage","excerpt":"[Natural workspace operators] Define O_Workspace = { (A_i)_i∈ V: A_t(e)H_e=H_eA_s(e) ∀ e }. Categorically, O_Workspace=Nat(H,H), the algebra of natural endomorphisms of the local system.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f5a2ad9ff96ea944","tex":"A_{t(e)}","aliases":"A_t(e)","display":"A_{t(e)}","kind":"definition-passage","excerpt":"[Natural workspace operators] Define O_Workspace = { (A_i)_i∈ V: A_t(e)H_e=H_eA_s(e) ∀ e }. Categorically, O_Workspace=Nat(H,H), the algebra of natural endomorphisms of the local system.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Feasible implementation intervention] Let B_t denote declared background variables that should remain matched. A family {ι_w:w∈ W_t^∘} is a feasible implementation of workspace values on the experimental domain W_t^∘ when: item each ι_w acts only on declared manipulable parents, channels, or artifacts at or after time t; item the intervened realiz","count":90},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ea0b1d9a333c4d54","tex":"\\mathfrak B","aliases":"B mathfrak","display":"\\mathfrak B","kind":"definition-passage","excerpt":"[Boundary problem] A boundary problem is a tuple B = ( V, S, τ, Y, I, Π, G_B ), where S⊆2^V is the admissible candidate family, τ is the temporal scale, Y is the operational target, I is the feasible intervention family, Π is the environment distribution, and G_B is a symmetry group under which boundaries are considered equivalent.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9474dae99bcf0491","tex":"B_t","aliases":"B_t","display":"B_t","kind":"definition-passage","excerpt":"[Feasible implementation intervention] Let B_t denote declared background variables that should remain matched. A family {ι_w:w∈ W_t^∘} is a feasible implementation of workspace values on the experimental domain W_t^∘ when: item each ι_w acts only on declared manipulable parents, channels, or artifacts at or after time t; item the intervened realiz","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-190834da497e3d7f","tex":"B_\\alpha","aliases":"B_α alpha","display":"B_\\alpha","kind":"definition-passage","excerpt":"[System decomposition] A decomposition of a carrier presentation is a tuple D = ( Π, O,G_Π, I_Π ), where: item Π={B_α}_α∈ A is a partition of Ω; item O_α is the declared local-access map for block B_α, producing the local state or history available at that component; item G_Π records the declared communication, transfer, and causal interfaces among","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b61c22b86277b610","tex":"\\bigvee","aliases":"bigvee","display":"\\bigvee","kind":"definition-passage","excerpt":"r presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only from those fine components. The aggregation is information faithful when mathscr","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ef3b78e48361d2e7","tex":"\\bigvee_{i\\in q^{-1}(B)}","aliases":"bigvee_i∈ q^-1(B) bigvee in","display":"\\bigvee_{i\\in q^{-1}(B)}","kind":"definition-passage","excerpt":"r presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only from those fine components. The aggregation is information faithful when mathscr","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d13b1528b249ec85","tex":"\\bigwedge","aliases":"bigwedge","display":"\\bigwedge","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-37f24839d4e3a6f0","tex":"\\bigwedge_{q\\in\\mathcal J(L)}","aliases":"bigwedge_q∈ J(L) bigwedge in mathcal","display":"\\bigwedge_{q\\in\\mathcal J(L)}","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dd9f18e8b972124a","tex":"\\bigwedge_{q\\in\\mathcal J(\\varsigma)}","aliases":"bigwedge_q∈ J(ς) bigwedge in mathcal varsigma","display":"\\bigwedge_{q\\in\\mathcal J(\\varsigma)}","kind":"definition-passage","excerpt":"item ℓ is the normalized loss or utility convention; item τ is the temporal horizon or horizon family; item I is the feasible intervention family; item N is the required matched-null family; and item Θ is the gate schema, direction, threshold, and indifference region used for certification. The complete certificate is written Cert(W;ς) = bigwedge_q","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Admissible local decomposition] For every vertex i, choose an orthogonal decomposition X_i=C_i⊕ R_i, with projectors P_i:X_i→ C_i, Q_i=I-P_i:X_i→ R_i. The subspace C_i is the candidate locally realized transferable core. The subspace R_i contains substrate-specific, context-specific, or otherwise noncore degrees of freedom relative to the present ","count":130},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"definition-passage","excerpt":"[Operation algebra] Let F∈{ R, C}. An operation algebra A is a unital associative F-algebra whose elements represent abstract transformations, inferential rules, questions, or procedures that can be realized in multiple strata.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":25},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ab8bde9a479f56a0","tex":"\\mathfrak C","aliases":"C mathfrak","display":"\\mathfrak C","kind":"definition-passage","excerpt":"[Case-study realization] A case-study realization is a tuple K = ( R, B, C, P, E, D, S, I, Q, G ), where: item R is the declared relational realization; item B is the boundary problem and component decomposition; item C is the structural core--transfer model; item P is the downstream predictive or control problem; item E is the evidence or generate","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Exact operation-compatible transfer] A core transfer H_e:C_i→ C_j is operation compatible when it is an intertwiner: H_eρ_i(a)=ρ_j(a)H_e ∀ a∈ A. Equivalently, the diagram ccc C_i · xrightarrowρ_i(a) · C_i downarrow H_e · · downarrow H_e C_j · xrightarrowρ_j(a) · C_j commutes for every a.","count":177},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-21f4b24b246d2bf8","tex":"\\mathcal C_E","aliases":"C_E mathcal","display":"\\mathcal C_E","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"definition-passage","excerpt":"[Admissible local decomposition] For every vertex i, choose an orthogonal decomposition X_i=C_i⊕ R_i, with projectors P_i:X_i→ C_i, Q_i=I-P_i:X_i→ R_i. The subspace C_i is the candidate locally realized transferable core. The subspace R_i contains substrate-specific, context-specific, or otherwise noncore degrees of freedom relative to the present ","count":23},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"definition-passage","excerpt":"[Global compatible section] The exact structural workspace is Γ(H) = { (c_i)_i∈ V∈⊕_i∈ VC_i: c_t(e)=H_ec_s(e) ∀ e∈ E }. An element c=(c_i)∈Γ(H) is a global compatible section.","count":25},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"definition-passage","excerpt":"[Exact operation-compatible transfer] A core transfer H_e:C_i→ C_j is operation compatible when it is an intertwiner: H_eρ_i(a)=ρ_j(a)H_e ∀ a∈ A. Equivalently, the diagram ccc C_i · xrightarrowρ_i(a) · C_i","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2f9988fcaa27a91e","tex":"C_{s(e)}","aliases":"C_s(e)","display":"C_{s(e)}","kind":"definition-passage","excerpt":"[Exact core-transfer system] An exact core-transfer system on G consists of: item decompositions X_i=C_i⊕ R_i; item core-preserving edge maps V_e=0; and item invertible core blocks H_e∈ GL(C_s(e),C_t(e)) whenever holonomy is invoked.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f3de1c17a8b215e6","tex":"c_{s(e)}","aliases":"c_s(e)","display":"c_{s(e)}","kind":"definition-passage","excerpt":"[Global compatible section] The exact structural workspace is Γ(H) = { (c_i)_i∈ V∈⊕_i∈ VC_i: c_t(e)=H_ec_s(e) ∀ e∈ E }. An element c=(c_i)∈Γ(H) is a global compatible section.","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fdcade446392b998","tex":"C_{t(e)}","aliases":"C_t(e)","display":"C_{t(e)}","kind":"definition-passage","excerpt":"[Exact core-transfer system] An exact core-transfer system on G consists of: item decompositions X_i=C_i⊕ R_i; item core-preserving edge maps V_e=0; and item invertible core blocks H_e∈ GL(C_s(e),C_t(e)) whenever holonomy is invoked.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6eab6acf267fd9f3","tex":"c_{t(e)}","aliases":"c_t(e)","display":"c_{t(e)}","kind":"definition-passage","excerpt":"[Global compatible section] The exact structural workspace is Γ(H) = { (c_i)_i∈ V∈⊕_i∈ VC_i: c_t(e)=H_ec_s(e) ∀ e∈ E }. An element c=(c_i)∈Γ(H) is a global compatible section.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-85b4eefe3cd8f404","tex":"\\mathcal C_V","aliases":"C_V mathcal","display":"\\mathcal C_V","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":20},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6c8327f02285f341","tex":"\\mathrm{causal}","aliases":"causal mathrm","display":"\\mathrm{causal}","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":26},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1ca71bbaea6ea0f3","tex":"\\mathrm{cert}","aliases":"cert mathrm","display":"\\mathrm{cert}","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6b37e6ce8c7d569e","tex":"\\operatorname{Cert}","aliases":"Cert operatorname","display":"\\operatorname{Cert}","kind":"definition-passage","excerpt":"item ℓ is the normalized loss or utility convention; item τ is the temporal horizon or horizon family; item I is the feasible intervention family; item N is the required matched-null family; and item Θ is the gate schema, direction, threshold, and indifference region used for certification. The complete certificate is written Cert(W;ς) = bigwedge_q","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ef9e9c2e6c4fa87","tex":"\\operatorname{Certify}","aliases":"Certify operatorname","display":"\\operatorname{Certify}","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9de196ab543dfbb3","tex":"\\operatorname{Certify}_{L,\\mathfrak D}^{\\mathrm{rob}}","aliases":"Certify_L, D^rob operatorname mathfrak mathrm","display":"\\operatorname{Certify}_{L,\\mathfrak D}^{\\mathrm{rob}}","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":64},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-40d9503903bdca20","tex":"\\mathfrak D","aliases":"D mathfrak","display":"\\mathfrak D","kind":"definition-passage","excerpt":"[Admissible decomposition family] An admissible family D is a predeclared collection of decompositions of the same carrier presentation satisfying the following audit conditions. item Carrier continuity. Each component corresponds to a process, artifact, or mechanism that can be tracked through the relevant time scale. item Access locality. The loc","count":23},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":144},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"definition-passage","excerpt":"[Exact structural correspondence] The correspondence is structurally exact when D'J_V=J_ED. It is operation exact for a∈ A when ρ_V'(a)J_V=J_Vρ_V(a).","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-51cf6a6f87f0d065","tex":"\\mathcal D_c","aliases":"D_c mathcal","display":"\\mathcal D_c","kind":"definition-passage","excerpt":"[Refinement and coarsening] Let D_f and D_c be decompositions of the same carrier presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only ","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-65c51bcc749e8188","tex":"\\mathcal D_{\\mathrm{cert}}","aliases":"D_cert mathcal mathrm","display":"\\mathcal D_{\\mathrm{cert}}","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-21ffea7eb85f74ce","tex":"\\mathcal D_f","aliases":"D_f mathcal","display":"\\mathcal D_f","kind":"definition-passage","excerpt":"[Refinement and coarsening] Let D_f and D_c be decompositions of the same carrier presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only ","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0cea4925d0ea9351","tex":"\\mathcal D_{\\mathrm{form}}","aliases":"D_form mathcal mathrm","display":"\\mathcal D_{\\mathrm{form}}","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-23282ac10f2eedd4","tex":"D_H","aliases":"D_H","display":"D_H","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":42},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cfb75ff9a03ea22b","tex":"D_H^*","aliases":"D_H^*","display":"D_H^*","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dda7761ef41722d1","tex":"d_\\varepsilon","aliases":"d_ε varepsilon","display":"d_\\varepsilon","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-639ec78966670d7b","tex":"\\operatorname{do}","aliases":"do operatorname","display":"\\operatorname{do}","kind":"definition-passage","excerpt":"[Formation leakage] A certification variable Y_cert leaks into formation when the conditional law of widehatθ changes under an intervention on Y_cert that should be inaccessible during formation: L ( widehatθ mid do(Y_cert=y) ) ≠ L ( widehatθ mid do(Y_cert=y') ). A valid discovery claim requires a documented information barrier or an audit showing ","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"definition-passage","excerpt":"[Exact operation-compatible transfer] A core transfer H_e:C_i→ C_j is operation compatible when it is an intertwiner: H_eρ_i(a)=ρ_j(a)H_e ∀ a∈ A. Equivalently, the diagram ccc C_i · xrightarrowρ_i(a) · C_i downarrow H_e · · downarrow H_e C_j · xrightarrowρ_j(a) · C_j commutes for every a.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":99},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":43},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":32},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":188},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-67898a9bf37c7838","tex":"\\mathbb E_{(h,h')\\sim\\nu_q}","aliases":"E_(h,h')∼ν_q mathbb sim nu","display":"\\mathbb E_{(h,h')\\sim\\nu_q}","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d54b95dd1d27920d","tex":"\\mathcal E_H","aliases":"E_H mathcal","display":"\\mathcal E_H","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0b409dac287cd785","tex":"e_q^{\\mathrm{causal}}","aliases":"e_q^causal mathrm","display":"e_q^{\\mathrm{causal}}","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ab7b5b836be0a065","tex":"e_q^{\\mathrm{MDL}}","aliases":"e_q^MDL mathrm","display":"e_q^{\\mathrm{MDL}}","kind":"definition-passage","excerpt":"[Code-relative history contribution] Let M_q be a declared coding or predictive model class for a learned component q. Let L_ M_q(qmid Z) be its held-out codelength given side information Z. Define e_q^MDL = [ frac L_ M_q(qmidΣ) - L_ M_q(qmidΣ, D_form) max{L_ M_q(qmidΣ),1} ]_0^1, where [x]_0^1=min{1,max{0,x}}.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f75d15c864ad3cef","tex":"E_t","aliases":"E_t","display":"E_t","kind":"definition-passage","excerpt":"[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-29663d1f0bd44f71","tex":"E_\\alpha","aliases":"E_α alpha","display":"E_\\alpha","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Admissible equivalence] An admissible equivalence F: R≃ R' is a family of correspondences preserving, exactly or within a declared tolerance: · the transferable core and its dimension,","count":47},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"definition-passage","excerpt":"[Operation algebra] Let F∈{ R, C}. An operation algebra A is a unital associative F-algebra whose elements represent abstract transformations, inferential rules, questions, or procedures that can be realized in multiple strata.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"definition-passage","excerpt":"[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e428d133c03e0d1d","tex":"\\mathfrak F","aliases":"F mathfrak","display":"\\mathfrak F","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Refinement and coarsening] Let D_f and D_c be decompositions of the same carrier presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only ","count":50},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8360edf4981185a6","tex":"F_B^{\\mathcal D_c}","aliases":"F_B^ D_c mathcal","display":"F_B^{\\mathcal D_c}","kind":"definition-passage","excerpt":"r presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only from those fine components. The aggregation is information faithful when mathscr","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aff7c3301156fc6d","tex":"F_i^{\\mathcal D_f}","aliases":"F_i^ D_f mathcal","display":"F_i^{\\mathcal D_f}","kind":"definition-passage","excerpt":"r presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only from those fine components. The aggregation is information faithful when mathscr","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-445f7af5f2eec46f","tex":"F_\\alpha^{\\mathcal D}","aliases":"F_α^ D alpha mathcal","display":"F_\\alpha^{\\mathcal D}","kind":"definition-passage","excerpt":"declared local-access map for block B_α, producing the local state or history available at that component; item G_Π records the declared communication, transfer, and causal interfaces among the blocks; and item I_Π is the family of feasible interventions at that resolution. The accessible information of component α is the sigma-field mathscr F_α^ D","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c11171d4f8d7df3c","tex":"\\mathrm{form}","aliases":"form mathrm","display":"\\mathrm{form}","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":24},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"definition-passage","excerpt":"[Boundary problem] A boundary problem is a tuple B = ( V, S, τ, Y, I, Π, G_B ), where S⊆2^V is the admissible candidate family, τ is the temporal scale, Y is the operational target, I is the feasible intervention family, Π is the environment distribution, and G_B is a symmetry group under which boundaries are considered equivalent.","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d52852c50f32c338","tex":"\\mathcal G_B","aliases":"G_B mathcal","display":"\\mathcal G_B","kind":"definition-passage","excerpt":"[Boundary problem] A boundary problem is a tuple B = ( V, S, τ, Y, I, Π, G_B ), where S⊆2^V is the admissible candidate family, τ is the temporal scale, Y is the operational target, I is the feasible intervention family, Π is the environment distribution, and G_B is a symmetry group under which boundaries are considered equivalent.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-69b8020400c9d9c8","tex":"G_I","aliases":"G_I","display":"G_I","kind":"definition-passage","excerpt":"ver A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets. Isolated vertices ar","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"definition-passage","excerpt":"[Exact static equivalence] Two static realizations are exactly gauge equivalent, R∼_stat R', when there exists an invertible family g_i such that H_e'=g_t(e)H_eg_s(e)^-1, ρ_i'(a)=g_iρ_i(a)g_i^-1. The exact static abstract workspace is Workspace_stat = [ R]_∼_stat.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0465a31a1244ca6e","tex":"g_i^{-1}","aliases":"g_i^-1","display":"g_i^{-1}","kind":"definition-passage","excerpt":"[Exact static equivalence] Two static realizations are exactly gauge equivalent, R∼_stat R', when there exists an invertible family g_i such that H_e'=g_t(e)H_eg_s(e)^-1, ρ_i'(a)=g_iρ_i(a)g_i^-1. The exact static abstract workspace is Workspace_stat = [ R]_∼_stat.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9176c4796705ddf2","tex":"g_{s(e)}^{-1}","aliases":"g_s(e)^-1","display":"g_{s(e)}^{-1}","kind":"definition-passage","excerpt":"[Exact static equivalence] Two static realizations are exactly gauge equivalent, R∼_stat R', when there exists an invertible family g_i such that H_e'=g_t(e)H_eg_s(e)^-1, ρ_i'(a)=g_iρ_i(a)g_i^-1. The exact static abstract workspace is Workspace_stat = [ R]_∼_stat.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8288adc4679f2e2b","tex":"g_{t(e)}","aliases":"g_t(e)","display":"g_{t(e)}","kind":"definition-passage","excerpt":"[Exact static equivalence] Two static realizations are exactly gauge equivalent, R∼_stat R', when there exists an invertible family g_i such that H_e'=g_t(e)H_eg_s(e)^-1, ρ_i'(a)=g_iρ_i(a)g_i^-1. The exact static abstract workspace is Workspace_stat = [ R]_∼_stat.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b7e94b9c52e447fe","tex":"G_\\alpha","aliases":"G_α alpha","display":"G_\\alpha","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d3ceb5e3d3dc2ad7","tex":"G_{\\alpha_0\\cdots\\alpha_p}","aliases":"G_α_0⋯α_p alpha cdots alpha","display":"G_{\\alpha_0\\cdots\\alpha_p}","kind":"definition-passage","excerpt":"ver A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets. Isolated vertices ar","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a422946c9218f432","tex":"g_\\varepsilon","aliases":"g_ε varepsilon","display":"g_\\varepsilon","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-42052e27bdd34e2f","tex":"G_{\\Pi}","aliases":"G_Π Pi","display":"G_{\\Pi}","kind":"definition-passage","excerpt":"[System decomposition] A decomposition of a carrier presentation is a tuple D = ( Π, O,G_Π, I_Π ), where: item Π={B_α}_α∈ A is a partition of Ω; item O_α is the declared local-access map for block B_α, producing the local state or history available at that component; item G_Π records the declared communication, transfer, and causal interfaces among","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"[Exact core-transfer system] An exact core-transfer system on G consists of: item decompositions X_i=C_i⊕ R_i; item core-preserving edge maps V_e=0; and item invertible core blocks H_e∈ GL(C_s(e),C_t(e)) whenever holonomy is invoked.","count":275},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4897511f6c74a73e","tex":"\\mathbf H","aliases":"H mathbf","display":"\\mathbf H","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":82},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-823170eb09d88517","tex":"\\mathsf H","aliases":"H mathsf","display":"\\mathsf H","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":49},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1e9de57bf29ef2a6","tex":"h'","aliases":"h'","display":"h'","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fa5a6573c905a228","tex":"H^0","aliases":"H^0","display":"H^0","kind":"definition-passage","excerpt":"[Descent-faithful compression] The compression is descent-faithful in degree zero when the induced map H^0(Q): kerδ^0 ⟶ keroverlineδ^0 is an isomorphism.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5d724fab1c00df76","tex":"H_e","aliases":"H_e","display":"H_e","kind":"definition-passage","excerpt":"[Exact core-transfer system] An exact core-transfer system on G consists of: item decompositions X_i=C_i⊕ R_i; item core-preserving edge maps V_e=0; and item invertible core blocks H_e∈ GL(C_s(e),C_t(e)) whenever holonomy is invoked.","count":43},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-86a3167a5c1db885","tex":"H_e'","aliases":"H_e'","display":"H_e'","kind":"definition-passage","excerpt":"[Exact static equivalence] Two static realizations are exactly gauge equivalent, R∼_stat R', when there exists an invertible family g_i such that H_e'=g_t(e)H_eg_s(e)^-1, ρ_i'(a)=g_iρ_i(a)g_i^-1. The exact static abstract workspace is Workspace_stat = [ R]_∼_stat.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7e86dd422b0fae70","tex":"\\mathbf H_t","aliases":"H_t mathbf","display":"\\mathbf H_t","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":55},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4a06f8f1ac75c1e5","tex":"\\mathsf H_t","aliases":"H_t mathsf","display":"\\mathsf H_t","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Admissible local decomposition] For every vertex i, choose an orthogonal decomposition X_i=C_i⊕ R_i, with projectors P_i:X_i→ C_i, Q_i=I-P_i:X_i→ R_i. The subspace C_i is the candidate locally realized transferable core. The subspace R_i contains substrate-specific, context-specific, or otherwise noncore degrees of freedom relative to the present ","count":83},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":61},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":196},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-322953d750f5811b","tex":"\\mathcal I_{\\mathrm{form}}","aliases":"I_form mathcal mathrm","display":"\\mathcal I_{\\mathrm{form}}","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-591f669332547359","tex":"\\mathcal I_{\\Pi}","aliases":"I_Π mathcal Pi","display":"\\mathcal I_{\\Pi}","kind":"definition-passage","excerpt":"[System decomposition] A decomposition of a carrier presentation is a tuple D = ( Π, O,G_Π, I_Π ), where: item Π={B_α}_α∈ A is a partition of Ω; item O_α is the declared local-access map for block B_α, producing the local state or history available at that component; item G_Π records the declared communication, transfer, and causal interfaces among","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7caa8aee712bf74f","tex":"\\mathrm{ID}","aliases":"ID mathrm","display":"\\mathrm{ID}","kind":"definition-passage","excerpt":"[Identified realization set] Given an experimental law P^ E and model class M, define R_ID (P^ E; M) = { R∈ M: P_ R^ E=P^ E }. For a gate q and specification ς, its identified margin set is M_q^ID (P^ E;ς) = { m_q(W_ R;ς): R∈ R_ID }.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":27},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-96845c43bfd2ea7b","tex":"\\inf_{\\mathcal D\\in\\mathfrak D}","aliases":"inf_ D∈ D inf mathcal in mathfrak","display":"\\inf_{\\mathcal D\\in\\mathfrak D}","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"[Exact structural correspondence] The correspondence is structurally exact when D'J_V=J_ED. It is operation exact for a∈ A when ρ_V'(a)J_V=J_Vρ_V(a).","count":75},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-146cbf287aa37a5f","tex":"\\mathcal J","aliases":"J mathcal","display":"\\mathcal J","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":25},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Interaction graph] Let G=(V,E,s,t) be a finite directed multigraph. Parallel edges and self-loops are allowed. The vertex set indexes strata, participants, artifacts, models, instruments, or subworkspaces. An edge e:i→ j denotes an admissible directed translation or influence channel from i to j.","count":53},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-942c5866758e6b63","tex":"\\mathcal J_*","aliases":"J_* mathcal","display":"\\mathcal J_*","kind":"definition-passage","excerpt":"[Specification leakage] Specification leakage occurs when certification outcomes influence any of the following without a fresh inferential correction: Y^+, ℓ, τ, I, N, Θ, S, J_*. The leakage status is separate from formation leakage. A workspace extractor may be frozen while the analyst still searches over targets or thresholds on the certificatio","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-eb1e5ef0b2f1a37a","tex":"J_E","aliases":"J_E","display":"J_E","kind":"definition-passage","excerpt":"[Exact structural correspondence] The correspondence is structurally exact when D'J_V=J_ED. It is operation exact for a∈ A when ρ_V'(a)J_V=J_Vρ_V(a).","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-45b22ca81d02b714","tex":"J_V","aliases":"J_V","display":"J_V","kind":"definition-passage","excerpt":"[Exact structural correspondence] The correspondence is structurally exact when D'J_V=J_ED. It is operation exact for a∈ A when ρ_V'(a)J_V=J_Vρ_V(a).","count":24},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":67},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-784c9007afca3979","tex":"\\mathfrak K","aliases":"K mathfrak","display":"\\mathfrak K","kind":"definition-passage","excerpt":"[Case-study realization] A case-study realization is a tuple K = ( R, B, C, P, E, D, S, I, Q, G ), where: item R is the declared relational realization; item B is the boundary problem and component decomposition; item C is the structural core--transfer model; item P is the downstream predictive or control problem; item E is the evidence or generate","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":98},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ab625c241c8a0e88","tex":"K_t","aliases":"K_t","display":"K_t","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"definition-passage","excerpt":"[Descent-faithful compression] The compression is descent-faithful in degree zero when the induced map H^0(Q): kerδ^0 ⟶ keroverlineδ^0 is an isomorphism.","count":48},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"[Exact core-transfer system] An exact core-transfer system on G consists of: item decompositions X_i=C_i⊕ R_i; item core-preserving edge maps V_e=0; and item invertible core blocks H_e∈ GL(C_s(e),C_t(e)) whenever holonomy is invoked.","count":150},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-df497c6da63e3fa9","tex":"\\mathcal L^{\\iota_h}","aliases":"L^ι_h mathcal iota","display":"\\mathcal L^{\\iota_h}","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f54b2477cac69322","tex":"\\mathcal L^{\\iota_{h'}}","aliases":"L^ι_h' mathcal iota","display":"\\mathcal L^{\\iota_{h'}}","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6c0a41a99936ba3b","tex":"L_{\\mathfrak M_q}","aliases":"L_ M_q mathfrak","display":"L_{\\mathfrak M_q}","kind":"definition-passage","excerpt":"[Code-relative history contribution] Let M_q be a declared coding or predictive model class for a learned component q. Let L_ M_q(qmid Z) be its held-out codelength given side information Z. Define e_q^MDL = [ frac L_ M_q(qmidΣ) - L_ M_q(qmidΣ, D_form) max{L_ M_q(qmidΣ),1} ]_0^1, where [x]_0^1=min{1,max{0,x}}.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0aaec542ed4fd265","tex":"\\mathcal L_{\\mathfrak R}","aliases":"L_ R mathcal mathfrak","display":"\\mathcal L_{\\mathfrak R}","kind":"definition-passage","excerpt":"efinition[Experimental design and induced law] An experimental design is a tuple E = ( O, I, T), where O is the observation map, I is the family of implemented interventions, and T is the set of measured temporal windows. A realization R induces the family of laws P_ R^ E = { L_ R ( O( H)middo(ι)): ι∈ I, t∈ T }.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e279f17f48190d40","tex":"L_H","aliases":"L_H","display":"L_H","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":47},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ea711daac4046624","tex":"L_{H(t)}","aliases":"L_H(t)","display":"L_{H(t)}","kind":"definition-passage","excerpt":"y space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_i(t), and transfer Laplacians L_t=L_H(t); and item a future target Y_t^+∈ Y. All state, hist","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-49e6cb003fe8b536","tex":"L_t","aliases":"L_t","display":"L_t","kind":"definition-passage","excerpt":"y space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_i(t), and transfer Laplacians L_t=L_H(t); and item a future target Y_t^+∈ Y. All state, hist","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f0042c6cf9ee0b72","tex":"\\mathfrak M","aliases":"M mathfrak","display":"\\mathfrak M","kind":"definition-passage","excerpt":"[Code-relative history contribution] Let M_q be a declared coding or predictive model class for a learned component q. Let L_ M_q(qmid Z) be its held-out codelength given side information Z. Define e_q^MDL = [ frac L_ M_q(qmidΣ) - L_ M_q(qmidΣ, D_form) max{L_ M_q(qmidΣ),1} ]_0^1, where [x]_0^1=min{1,max{0,x}}.","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-af9070f1457f1310","tex":"\\mathsf M","aliases":"M mathsf","display":"\\mathsf M","kind":"definition-passage","excerpt":"[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":110},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a603e6b6eedcb75f","tex":"M_e","aliases":"M_e","display":"M_e","kind":"definition-passage","excerpt":"[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-293f213cd3b3d25a","tex":"\\mathsf M_e","aliases":"M_e mathsf","display":"\\mathsf M_e","kind":"definition-passage","excerpt":"[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c0d64e7d3a88200b","tex":"\\mathfrak M_q","aliases":"M_q mathfrak","display":"\\mathfrak M_q","kind":"definition-passage","excerpt":"[Code-relative history contribution] Let M_q be a declared coding or predictive model class for a learned component q. Let L_ M_q(qmid Z) be its held-out codelength given side information Z. Define e_q^MDL = [ frac L_ M_q(qmidΣ) - L_ M_q(qmidΣ, D_form) max{L_ M_q(qmidΣ),1} ]_0^1, where [x]_0^1=min{1,max{0,x}}.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-36671d516d8e31ae","tex":"m_q","aliases":"m_q","display":"m_q","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":34},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8b5f18309597163e","tex":"\\mathcal M_q^{\\mathrm{ID}}","aliases":"M_q^ID mathcal mathrm","display":"\\mathcal M_q^{\\mathrm{ID}}","kind":"definition-passage","excerpt":"ion[Identified realization set] Given an experimental law P^ E and model class M, define R_ID (P^ E; M) = { R∈ M: P_ R^ E=P^ E }. For a gate q and specification ς, its identified margin set is M_q^ID (P^ E;ς) = { m_q(W_ R;ς): R∈ R_ID }.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"definition-passage","excerpt":"declared local-access map for block B_α, producing the local state or history available at that component; item G_Π records the declared communication, transfer, and causal interfaces among the blocks; and item I_Π is the family of feasible interventions at that resolution. The accessible information of component α is the sigma-field mathscr F_α^ D","count":49},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"definition-passage","excerpt":"[Code-relative history contribution] Let M_q be a declared coding or predictive model class for a learned component q. Let L_ M_q(qmid Z) be its held-out codelength given side information Z. Define e_q^MDL = [ frac L_ M_q(qmidΣ) - L_ M_q(qmidΣ, D_form) max{L_ M_q(qmidΣ),1} ]_0^1, where [x]_0^1=min{1,max{0,x}}.","count":30},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8b54bc814c48cfff","tex":"\\mathrm{MDL}","aliases":"MDL mathrm","display":"\\mathrm{MDL}","kind":"definition-passage","excerpt":"[Code-relative history contribution] Let M_q be a declared coding or predictive model class for a learned component q. Let L_ M_q(qmid Z) be its held-out codelength given side information Z. Define e_q^MDL = [ frac L_ M_q(qmidΣ) - L_ M_q(qmidΣ, D_form) max{L_ M_q(qmidΣ),1} ]_0^1, where [x]_0^1=min{1,max{0,x}}.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":77},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"definition-passage","excerpt":"a declared coding or predictive model class for a learned component q. Let L_ M_q(qmid Z) be its held-out codelength given side information Z. Define e_q^MDL = [ frac L_ M_q(qmidΣ) - L_ M_q(qmidΣ, D_form) max{L_ M_q(qmidΣ),1} ]_0^1, where [x]_0^1=min{1,max{0,x}}.","count":45},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"definition-passage","excerpt":"[Relational witness pair] A pair of laws (P,Q) is a predictive relational witness pair for the decomposition V and target Y_t^+ when Q∈ N_pred(P) but Risk_ℓ,Q(V) ≠ Risk_ℓ,P(V). The two systems agree on every single-node predictive marginal but differ in full-system operative capacity.","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9d12f5e9849717a9","tex":"\\mathcal N_{\\mathrm{pred}}","aliases":"N_pred mathcal mathrm","display":"\\mathcal N_{\\mathrm{pred}}","kind":"definition-passage","excerpt":"[Relational witness pair] A pair of laws (P,Q) is a predictive relational witness pair for the decomposition V and target Y_t^+ when Q∈ N_pred(P) but Risk_ℓ,Q(V) ≠ Risk_ℓ,P(V). The two systems agree on every single-node predictive marginal but differ in full-system operative capacity.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-79d4c7f9c8579543","tex":"\\Nat","aliases":"Nat","display":null,"kind":"definition-passage","excerpt":"[Natural workspace operators] Define O_Workspace = { (A_i)_i∈ V: A_t(e)H_e=H_eA_s(e) ∀ e }. Categorically, O_Workspace=Nat(H,H), the algebra of natural endomorphisms of the local system.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"definition-passage","excerpt":"[System decomposition] A decomposition of a carrier presentation is a tuple D = ( Π, O,G_Π, I_Π ), where: item Π={B_α}_α∈ A is a partition of Ω; item O_α is the declared local-access map for block B_α, producing the local state or history available at that component; item G_Π records the declared communication, transfer, and causal interfaces among","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0e314e8fff6f4c2b","tex":"\\mathbf O","aliases":"O mathbf","display":"\\mathbf O","kind":"definition-passage","excerpt":"[System decomposition] A decomposition of a carrier presentation is a tuple D = ( Π, O,G_Π, I_Π ), where: item Π={B_α}_α∈ A is a partition of Ω; item O_α is the declared local-access map for block B_α, producing the local state or history available at that component; item G_Π records the declared communication, transfer, and causal interfaces among","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"definition-passage","excerpt":"[Natural workspace operators] Define O_Workspace = { (A_i)_i∈ V: A_t(e)H_e=H_eA_s(e) ∀ e }. Categorically, O_Workspace=Nat(H,H), the algebra of natural endomorphisms of the local system.","count":17},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-40d1c140eb4497c0","tex":"\\mathcal O_{\\Workspace}","aliases":"O_Workspace mathcal Workspace","display":null,"kind":"definition-passage","excerpt":"[Natural workspace operators] Define O_Workspace = { (A_i)_i∈ V: A_t(e)H_e=H_eA_s(e) ∀ e }. Categorically, O_Workspace=Nat(H,H), the algebra of natural endomorphisms of the local system.","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-98de897d8623cf36","tex":"O_\\alpha","aliases":"O_α alpha","display":"O_\\alpha","kind":"definition-passage","excerpt":"[System decomposition] A decomposition of a carrier presentation is a tuple D = ( Π, O,G_Π, I_Π ), where: item Π={B_α}_α∈ A is a partition of Ω; item O_α is the declared local-access map for block B_α, producing the local state or history available at that component; item G_Π records the declared communication, transfer, and causal interfaces among","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c664119f133715f3","tex":"\\overline\\delta","aliases":"overlineδ overline delta","display":"\\overline\\delta","kind":"definition-passage","excerpt":"[Descent-faithful compression] The compression is descent-faithful in degree zero when the induced map H^0(Q): kerδ^0 ⟶ keroverlineδ^0 is an isomorphism.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c89c40db50b47fee","tex":"\\overline\\delta^0","aliases":"overlineδ^0 overline delta","display":"\\overline\\delta^0","kind":"definition-passage","excerpt":"[Descent-faithful compression] The compression is descent-faithful in degree zero when the induced map H^0(Q): kerδ^0 ⟶ keroverlineδ^0 is an isomorphism.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"[Admissible local decomposition] For every vertex i, choose an orthogonal decomposition X_i=C_i⊕ R_i, with projectors P_i:X_i→ C_i, Q_i=I-P_i:X_i→ R_i. The subspace C_i is the candidate locally realized transferable core. The subspace R_i contains substrate-specific, context-specific, or otherwise noncore degrees of freedom relative to the present ","count":167},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ef958519c2ac9923","tex":"\\mathbb P","aliases":"P mathbb","display":"\\mathbb P","kind":"definition-passage","excerpt":"[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_","count":44},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"definition-passage","excerpt":"[Case-study realization] A case-study realization is a tuple K = ( R, B, C, P, E, D, S, I, Q, G ), where: item R is the declared relational realization; item B is the boundary problem and component decomposition; item C is the structural core--transfer model; item P is the downstream predictive or control problem; item E is the evidence or generate","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":39},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-10de9fd0b627c3fb","tex":"P^{\\mathcal E}","aliases":"P^ E mathcal","display":"P^{\\mathcal E}","kind":"definition-passage","excerpt":"[Identified realization set] Given an experimental law P^ E and model class M, define R_ID (P^ E; M) = { R∈ M: P_ R^ E=P^ E }. For a gate q and specification ς, its identified margin set is M_q^ID (P^ E;ς) = { m_q(W_ R;ς): R∈ R_ID }.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1669011d13120939","tex":"\\mathbb P_{\\mathbf H_t}","aliases":"P_ H_t mathbb mathbf","display":"\\mathbb P_{\\mathbf H_t}","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-db6e296be7609c50","tex":"P_{\\mathfrak R'}^{\\mathcal E}","aliases":"P_ R'^ E mathfrak mathcal","display":"P_{\\mathfrak R'}^{\\mathcal E}","kind":"definition-passage","excerpt":"[Experimental equivalence] Two realizations are experimentally equivalent under E, written R≡_ E R', when P_ R^ E = P_ R'^ E. A property ψ( R) is identifiable on a model class M under E when it is constant on every ≡_ E-equivalence class contained in M.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82f8e348a6c72715","tex":"P_{\\mathfrak R}^{\\mathcal E}","aliases":"P_ R^ E mathfrak mathcal","display":"P_{\\mathfrak R}^{\\mathcal E}","kind":"definition-passage","excerpt":"efinition[Experimental design and induced law] An experimental design is a tuple E = ( O, I, T), where O is the observation map, I is the family of implemented interventions, and T is the set of measured temporal windows. A realization R induces the family of laws P_ R^ E = { L_ R ( O( H)middo(ι)): ι∈ I, t∈ T }.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5e48d3525d3bfec2","tex":"P_{[0,\\varepsilon]}","aliases":"P_[0,ε] varepsilon","display":"P_{[0,\\varepsilon]}","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"definition-passage","excerpt":"[Admissible local decomposition] For every vertex i, choose an orthogonal decomposition X_i=C_i⊕ R_i, with projectors P_i:X_i→ C_i, Q_i=I-P_i:X_i→ R_i. The subspace C_i is the candidate locally realized transferable core. The subspace R_i contains substrate-specific, context-specific, or otherwise noncore degrees of freedom relative to the present ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a19953262ec3ea20","tex":"\\mathbb P_U","aliases":"P_U mathbb","display":"\\mathbb P_U","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-44b2e8700e8904df","tex":"\\pr","aliases":"pr","display":null,"kind":"definition-passage","excerpt":"[Relational instantiation] A workspace is relationally instantiated relative to the decomposition V when no single projection determines its equivalence class: ∀ i∈ V, ∃ R, R' such that pr_i( R)=pr_i( R') but [ R]_≃≠[ R']_≃.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c5622dfac220c4de","tex":"\\pr_i","aliases":"pr_i pr","display":null,"kind":"definition-passage","excerpt":"[Relational instantiation] A workspace is relationally instantiated relative to the decomposition V when no single projection determines its equivalence class: ∀ i∈ V, ∃ R, R' such that pr_i( R)=pr_i( R') but [ R]_≃≠[ R']_≃.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-71a98ec1e2bb9f0a","tex":"\\preceq","aliases":"preceq","display":"\\preceq","kind":"definition-passage","excerpt":"[Refinement and coarsening] Let D_f and D_c be decompositions of the same carrier presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-28156301964cd09e","tex":"\\mathrm{pred}","aliases":"pred mathrm","display":"\\mathrm{pred}","kind":"definition-passage","excerpt":"[Relational witness pair] A pair of laws (P,Q) is a predictive relational witness pair for the decomposition V and target Y_t^+ when Q∈ N_pred(P) but Risk_ℓ,Q(V) ≠ Risk_ℓ,P(V). The two systems agree on every single-node predictive marginal but differ in full-system operative capacity.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"definition-passage","excerpt":"[Case-study realization] A case-study realization is a tuple K = ( R, B, C, P, E, D, S, I, Q, G ), where: item R is the declared relational realization; item B is the boundary problem and component decomposition; item C is the structural core--transfer model; item P is the downstream predictive or control problem; item E is the evidence or generate","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"[Code-relative history contribution] Let M_q be a declared coding or predictive model class for a learned component q. Let L_ M_q(qmid Z) be its held-out codelength given side information Z. Define e_q^MDL = [ frac L_ M_q(qmidΣ) - L_ M_q(qmidΣ, D_form) max{L_ M_q(qmidΣ),1} ]_0^1, where [x]_0^1=min{1,max{0,x}}.","count":158},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6339057791df9dbc","tex":"q^{-1}","aliases":"q^-1","display":"q^{-1}","kind":"definition-passage","excerpt":"r presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only from those fine components. The aggregation is information faithful when mathscr","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5950b0378700bfba","tex":"Q_i","aliases":"Q_i","display":"Q_i","kind":"definition-passage","excerpt":"[Admissible local decomposition] For every vertex i, choose an orthogonal decomposition X_i=C_i⊕ R_i, with projectors P_i:X_i→ C_i, Q_i=I-P_i:X_i→ R_i. The subspace C_i is the candidate locally realized transferable core. The subspace R_i contains substrate-specific, context-specific, or otherwise noncore degrees of freedom relative to the present ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"definition-passage","excerpt":"[Operation algebra] Let F∈{ R, C}. An operation algebra A is a unital associative F-algebra whose elements represent abstract transformations, inferential rules, questions, or procedures that can be realized in multiple strata.","count":23},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-32fe36360874826c","tex":"\\mathfrak R","aliases":"R mathfrak","display":"\\mathfrak R","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":66},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-90d481201274672a","tex":"\\mathfrak R'","aliases":"R' mathfrak","display":"\\mathfrak R'","kind":"definition-passage","excerpt":"[Admissible equivalence] An admissible equivalence F: R≃ R' is a family of correspondences preserving, exactly or within a declared tolerance: · the transferable core and its dimension,","count":24},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"definition-passage","excerpt":"[Admissible local decomposition] For every vertex i, choose an orthogonal decomposition X_i=C_i⊕ R_i, with projectors P_i:X_i→ C_i, Q_i=I-P_i:X_i→ R_i. The subspace C_i is the candidate locally realized transferable core. The subspace R_i contains substrate-specific, context-specific, or otherwise noncore degrees of freedom relative to the present ","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2c4a56bb4e0f0ea2","tex":"\\mathfrak R_{\\mathrm{ID}}","aliases":"R_ID mathfrak mathrm","display":"\\mathfrak R_{\\mathrm{ID}}","kind":"definition-passage","excerpt":"[Identified realization set] Given an experimental law P^ E and model class M, define R_ID (P^ E; M) = { R∈ M: P_ R^ E=P^ E }. For a gate q and specification ς, its identified margin set is M_q^ID (P^ E;ς) = { m_q(W_ R;ς): R∈ R_ID }.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-774c49d2837a43cc","tex":"\\mathfrak R_{\\mathrm{stat}}","aliases":"R_stat mathfrak mathrm","display":"\\mathfrak R_{\\mathrm{stat}}","kind":"definition-passage","excerpt":"s; item ρ={ρ_i}_i∈ V gives the local realization of those operations; item H is the temporally ordered interaction history; item I is a specified family of admissible interventions; and item Y is the family of downstream variables relative to which causal and predictive claims are evaluated. The static reduct is R_stat = (G, X, T, A,ρ).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-512d411adf8ee1ff","tex":"\\ran","aliases":"ran","display":null,"kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bc1a3b3d4cc5afef","tex":"\\Risk","aliases":"Risk","display":null,"kind":"definition-passage","excerpt":"[Relational witness pair] A pair of laws (P,Q) is a predictive relational witness pair for the decomposition V and target Y_t^+ when Q∈ N_pred(P) but Risk_ℓ,Q(V) ≠ Risk_ℓ,P(V). The two systems agree on every single-node predictive marginal but differ in full-system operative capacity.","count":36},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-02619967d6644298","tex":"\\Risk_{\\ell,P}","aliases":"Risk_ℓ,P Risk ell","display":null,"kind":"definition-passage","excerpt":"[Relational witness pair] A pair of laws (P,Q) is a predictive relational witness pair for the decomposition V and target Y_t^+ when Q∈ N_pred(P) but Risk_ℓ,Q(V) ≠ Risk_ℓ,P(V). The two systems agree on every single-node predictive marginal but differ in full-system operative capacity.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-16da498ea68d7756","tex":"\\Risk_{\\ell,Q}","aliases":"Risk_ℓ,Q Risk ell","display":null,"kind":"definition-passage","excerpt":"[Relational witness pair] A pair of laws (P,Q) is a predictive relational witness pair for the decomposition V and target Y_t^+ when Q∈ N_pred(P) but Risk_ℓ,Q(V) ≠ Risk_ℓ,P(V). The two systems agree on every single-node predictive marginal but differ in full-system operative capacity.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-250daa0b4f74f6ad","tex":"\\mathrm{rob}","aliases":"rob mathrm","display":"\\mathrm{rob}","kind":"definition-passage","excerpt":"[Decomposition-robust certificate] A workspace passes the D-robust certificate at level L when Certify_L, D^rob(W) = bigwedge_q∈ J(L) { inf_ D∈ D m_q(W; D)>0 }. A pass for one selected decomposition is a decomposition-specific claim. A pass throughout D is a family-robust claim. A claim is decomposition fragile when at least one required gate chang","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":160},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b617b296bbfd924b","tex":"\\mathfrak S","aliases":"S mathfrak","display":"\\mathfrak S","kind":"definition-passage","excerpt":"[Boundary problem] A boundary problem is a tuple B = ( V, S, τ, Y, I, Π, G_B ), where S⊆2^V is the admissible candidate family, τ is the temporal scale, Y is the operational target, I is the feasible intervention family, Π is the environment distribution, and G_B is a symmetry group under which boundaries are considered equivalent.","count":67},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-17e6e62cdd37d68b","tex":"\\mathsf S","aliases":"S mathsf","display":"\\mathsf S","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":59},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-87807c21edc049cd","tex":"S_t","aliases":"S_t","display":"S_t","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":26},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cb0cdd6140f92ecf","tex":"\\mathsf S_t","aliases":"S_t mathsf","display":"\\mathsf S_t","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e99b28f7de2336b3","tex":"S_t^{\\iota_w}","aliases":"S_t^ι_w iota","display":"S_t^{\\iota_w}","kind":"definition-passage","excerpt":"(B_t denote declared background variables that should remain matched. A family {ι_w:w∈ W_t^∘} is a feasible implementation of workspace values on the experimental domain W_t^∘ when: item each ι_w acts only on declared manipulable parents, channels, or artifacts at or after time t; item the intervened realization satisfies π_t(S_t^ι_w)=w almost sure","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2507466f7a520c56","tex":"\\mathrm{stat}","aliases":"stat mathrm","display":"\\mathrm{stat}","kind":"definition-passage","excerpt":"s; item ρ={ρ_i}_i∈ V gives the local realization of those operations; item H is the temporally ordered interaction history; item I is a specified family of admissible interventions; and item Y is the family of downstream variables relative to which causal and predictive claims are evaluated. The static reduct is R_stat = (G, X, T, A,ρ).","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":119},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d589f35b0c8bc738","tex":"\\mathbf T","aliases":"T mathbf","display":"\\mathbf T","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"definition-passage","excerpt":"[Experimental design and induced law] An experimental design is a tuple E = ( O, I, T), where O is the observation map, I is the family of implemented interventions, and T is the set of measured temporal windows. A realization R induces the family of laws P_ R^ E = { L_ R ( O( H)middo(ι)): ι∈ I, t∈ T }.","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bb1ad3b8487cb395","tex":"T_e","aliases":"T_e","display":"T_e","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":33},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":62},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ef50ddb6014d940d","tex":"V_c","aliases":"V_c","display":"V_c","kind":"definition-passage","excerpt":"[Refinement and coarsening] Let D_f and D_c be decompositions of the same carrier presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only ","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f7fd62f7cd34cb10","tex":"V_e","aliases":"V_e","display":"V_e","kind":"definition-passage","excerpt":"[Exact core-transfer system] An exact core-transfer system on G consists of: item decompositions X_i=C_i⊕ R_i; item core-preserving edge maps V_e=0; and item invertible core blocks H_e∈ GL(C_s(e),C_t(e)) whenever holonomy is invoked.","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4f979281d0af8ba8","tex":"V_f","aliases":"V_f","display":"V_f","kind":"definition-passage","excerpt":"[Refinement and coarsening] Let D_f and D_c be decompositions of the same carrier presentation. The fine decomposition refines the coarse decomposition, written D_fpreceq D_c, when there is a surjection q:V_f⟶ V_c such that each coarse component is the union of the fine components in one fiber of q, and its accessible information is generated only ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-92becdaca2dd0d19","tex":"V_\\alpha","aliases":"V_α alpha","display":"V_\\alpha","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"definition-passage","excerpt":"[Exact structural--operational alignment] The candidate W_t is observationally aligned when there exists a measurable map ϕ_t such that Z_t=ϕ_t(W_t) a.s. It is I-causally aligned when Z_t^ I=ϕ_t^ I(W_t) a.s.","count":209},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cb6844a85fd4cf06","tex":"\\mathsf W","aliases":"W mathsf","display":"\\mathsf W","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":90},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d3cd13201cc4d03b","tex":"W_{\\mathfrak R}","aliases":"W_ R mathfrak","display":"W_{\\mathfrak R}","kind":"definition-passage","excerpt":"ion[Identified realization set] Given an experimental law P^ E and model class M, define R_ID (P^ E; M) = { R∈ M: P_ R^ E=P^ E }. For a gate q and specification ς, its identified margin set is M_q^ID (P^ E;ς) = { m_q(W_ R;ς): R∈ R_ID }.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-179eb697363e8b1a","tex":"w_e","aliases":"w_e","display":"w_e","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":17},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ae58458e74e50c6f","tex":"W_t","aliases":"W_t","display":"W_t","kind":"definition-passage","excerpt":"[Exact structural--operational alignment] The candidate W_t is observationally aligned when there exists a measurable map ϕ_t such that Z_t=ϕ_t(W_t) a.s. It is I-causally aligned when Z_t^ I=ϕ_t^ I(W_t) a.s.","count":59},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-37d08bb93713e167","tex":"\\mathsf W_t","aliases":"W_t mathsf","display":"\\mathsf W_t","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bd757811c5501f5d","tex":"\\mathsf W_t^\\circ","aliases":"W_t^∘ mathsf circ","display":"\\mathsf W_t^\\circ","kind":"definition-passage","excerpt":"[Feasible implementation intervention] Let B_t denote declared background variables that should remain matched. A family {ι_w:w∈ W_t^∘} is a feasible implementation of workspace values on the experimental domain W_t^∘ when: item each ι_w acts only on declared manipulable parents, channels, or artifacts at or after time t; item the intervened realiz","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1b3f66f942a5fe9a","tex":"\\widehat B","aliases":"widehat B widehat","display":"\\widehat B","kind":"definition-passage","excerpt":"is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the learned organization; item D_cert is held-out certification data; and item I_form is a declared family of interventions on scaffold and formation history. The learned organization is decomposed as widehatθ = ( widehat V, widehat E, widehat H, widehat P, widehat O,","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-03e64edeac04e31f","tex":"\\widehat E","aliases":"widehat E widehat","display":"\\widehat E","kind":"definition-passage","excerpt":"is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the learned organization; item D_cert is held-out certification data; and item I_form is a declared family of interventions on scaffold and formation history. The learned organization is decomposed as widehatθ = ( widehat V, widehat E, widehat H, widehat P, widehat O,","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5c577d5457da31ff","tex":"\\widehat H","aliases":"widehat H widehat","display":"\\widehat H","kind":"definition-passage","excerpt":"is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the learned organization; item D_cert is held-out certification data; and item I_form is a declared family of interventions on scaffold and formation history. The learned organization is decomposed as widehatθ = ( widehat V, widehat E, widehat H, widehat P, widehat O,","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5303ee85e91a83b9","tex":"\\widehat{\\mathcal O}","aliases":"widehat O widehat mathcal","display":"\\widehat{\\mathcal O}","kind":"definition-passage","excerpt":"is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the learned organization; item D_cert is held-out certification data; and item I_form is a declared family of interventions on scaffold and formation history. The learned organization is decomposed as widehatθ = ( widehat V, widehat E, widehat H, widehat P, widehat O,","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"definition-passage","excerpt":"is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the learned organization; item D_cert is held-out certification data; and item I_form is a declared family of interventions on scaffold and formation history. The learned organization is decomposed as widehatθ = ( widehat V, widehat E, widehat H, widehat P, widehat O,","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dc5e832c68a33db9","tex":"\\widehat V","aliases":"widehat V widehat","display":"\\widehat V","kind":"definition-passage","excerpt":"is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the learned organization; item D_cert is held-out certification data; and item I_form is a declared family of interventions on scaffold and formation history. The learned organization is decomposed as widehatθ = ( widehat V, widehat E, widehat H, widehat P, widehat O,","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fec34e4c8a1d397d","tex":"\\widehat\\theta","aliases":"widehatθ widehat theta","display":"\\widehat\\theta","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4a3b6ca197a4bd4b","tex":"\\Workspace","aliases":"Workspace","display":null,"kind":"definition-passage","excerpt":"[Natural workspace operators] Define O_Workspace = { (A_i)_i∈ V: A_t(e)H_e=H_eA_s(e) ∀ e }. Categorically, O_Workspace=Nat(H,H), the algebra of natural endomorphisms of the local system.","count":43},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-18ec5695125a6209","tex":"\\Workspace_{\\mathrm{stat}}","aliases":"Workspace_stat Workspace mathrm","display":null,"kind":"definition-passage","excerpt":"[Exact static equivalence] Two static realizations are exactly gauge equivalent, R∼_stat R', when there exists an invertible family g_i such that H_e'=g_t(e)H_eg_s(e)^-1, ρ_i'(a)=g_iρ_i(a)g_i^-1. The exact static abstract workspace is Workspace_stat = [ R]_∼_stat.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":84},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-25460d31b0a1910e","tex":"\\mathbf X","aliases":"X mathbf","display":"\\mathbf X","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a86973ce6a8c4622","tex":"\\mathsf X","aliases":"X mathsf","display":"\\mathsf X","kind":"definition-passage","excerpt":"[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"a declared coding or predictive model class for a learned component q. Let L_ M_q(qmid Z) be its held-out codelength given side information Z. Define e_q^MDL = [ frac L_ M_q(qmidΣ) - L_ M_q(qmidΣ, D_form) max{L_ M_q(qmidΣ),1} ]_0^1, where [x]_0^1=min{1,max{0,x}}.","count":45},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":25},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7890a705df962b21","tex":"\\mathsf X_i","aliases":"X_i mathsf","display":"\\mathsf X_i","kind":"definition-passage","excerpt":"[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"definition-passage","excerpt":"[Exact operation-compatible transfer] A core transfer H_e:C_i→ C_j is operation compatible when it is an intertwiner: H_eρ_i(a)=ρ_j(a)H_e ∀ a∈ A. Equivalently, the diagram ccc C_i · xrightarrowρ_i(a) · C_i downarrow H_e · · downarrow H_e C_j · xrightarrowρ_j(a) · C_j commutes for every a.","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bef13251af5aee62","tex":"\\mathcal Y","aliases":"Y mathcal","display":"\\mathcal Y","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b87adf799540d63a","tex":"\\mathsf Y","aliases":"Y mathsf","display":"\\mathsf Y","kind":"definition-passage","excerpt":"d consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_i(t), and transfer Laplacians L_t=L_H(t); and item a future target Y_t^+∈ Y. All state, history, and target spac","count":23},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"definition-passage","excerpt":"[Formation leakage] A certification variable Y_cert leaks into formation when the conditional law of widehatθ changes under an intervention on Y_cert that should be inaccessible during formation: L ( widehatθ mid do(Y_cert=y) ) ≠ L ( widehatθ mid do(Y_cert=y') ). A valid discovery claim requires a documented information barrier or an audit showing ","count":19},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-241d1cd21689b8a9","tex":"y'","aliases":"y'","display":"y'","kind":"definition-passage","excerpt":"[Formation leakage] A certification variable Y_cert leaks into formation when the conditional law of widehatθ changes under an intervention on Y_cert that should be inaccessible during formation: L ( widehatθ mid do(Y_cert=y) ) ≠ L ( widehatθ mid do(Y_cert=y') ). A valid discovery claim requires a documented information barrier or an audit showing ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c453920b3857736","tex":"Y^+","aliases":"Y^+","display":"Y^+","kind":"definition-passage","excerpt":"[Workspace specification] A workspace specification is a tuple ς = ( D, Y^+, ℓ, τ, I, N, Θ ), where: item D is the decomposition and local-access model; item Y^+ is the declared future variable or trajectory; item ℓ is the normalized loss or utility convention; item τ is the temporal horizon or horizon family; item I is the feasible intervention fa","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bb5195b60627dbce","tex":"Y_{\\mathrm{cert}}","aliases":"Y_cert mathrm","display":"Y_{\\mathrm{cert}}","kind":"definition-passage","excerpt":"[Formation leakage] A certification variable Y_cert leaks into formation when the conditional law of widehatθ changes under an intervention on Y_cert that should be inaccessible during formation: L ( widehatθ mid do(Y_cert=y) ) ≠ L ( widehatθ mid do(Y_cert=y') ). A valid discovery claim requires a documented information barrier or an audit showing ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f81433359e48524b","tex":"Y_t^+","aliases":"Y_t^+","display":"Y_t^+","kind":"definition-passage","excerpt":"d consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_i(t), and transfer Laplacians L_t=L_H(t); and item a future target Y_t^+∈ Y. All state, history, and target spac","count":59},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"definition-passage","excerpt":"d Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ∈ P( Y). Two histories are predictively equivalent","count":54},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3480de748bc9723a","tex":"Z_t","aliases":"Z_t","display":"Z_t","kind":"definition-passage","excerpt":"d Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ∈ P( Y). Two histories are predictively equivalent","count":29},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c0320d86f4654cc4","tex":"Z_t^{\\mathcal I}","aliases":"Z_t^ I mathcal","display":"Z_t^{\\mathcal I}","kind":"definition-passage","excerpt":"[Exact structural--operational alignment] The candidate W_t is observationally aligned when there exists a measurable map ϕ_t such that Z_t=ϕ_t(W_t) a.s. It is I-causally aligned when Z_t^ I=ϕ_t^ I(W_t) a.s.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"definition-passage","excerpt":"d Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ∈ P( Y). Two histories are predictively equivalent","count":25},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Predictive kernel and canonical state] Let H_t and Y be standard Borel spaces. A regular conditional kernel is a measurable map K_t: H_t× B( Y)→[0,1] satisfying K_t(h,A) = P(Y_t^+∈ Amid H_t=h) for P_ H_t-almost every h. Let P( Y) carry its evaluation sigma-algebra. The canonical predictive state is the random probability measure Z_t = K_t( H_t,·) ","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":120},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-59ce46150d0cfe77","tex":"\\alpha_0","aliases":"α_0 alpha","display":"\\alpha_0","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-74bd761a4b5437cf","tex":"\\alpha_p","aliases":"α_p alpha","display":"\\alpha_p","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"definition-passage","excerpt":"[Global compatible section] The exact structural workspace is Γ(H) = { (c_i)_i∈ V∈⊕_i∈ VC_i: c_t(e)=H_ec_s(e) ∀ e∈ E }. An element c=(c_i)∈Γ(H) is a global compatible section.","count":56},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-86b6b780303c695c","tex":"\\Gamma_\\varepsilon","aliases":"Γ_ε Gamma varepsilon","display":"\\Gamma_\\varepsilon","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e31248095b88ab5f","tex":"\\delta^0","aliases":"δ^0 delta","display":"\\delta^0","kind":"definition-passage","excerpt":"[Descent-faithful compression] The compression is descent-faithful in degree zero when the induced map H^0(Q): kerδ^0 ⟶ keroverlineδ^0 is an isomorphism.","count":25},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":126},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"definition-passage","excerpt":"tion[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vector","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1aa9e6b21e4c6c49","tex":"\\zeta_e","aliases":"ζ_e zeta","display":"\\zeta_e","kind":"definition-passage","excerpt":"tion[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vector","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"definition-passage","excerpt":"[Workspace specification] A workspace specification is a tuple ς = ( D, Y^+, ℓ, τ, I, N, Θ ), where: item D is the decomposition and local-access model; item Y^+ is the declared future variable or trajectory; item ℓ is the normalized loss or utility convention; item τ is the temporal horizon or horizon family; item I is the feasible intervention fa","count":29},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"definition-passage","excerpt":"[Feasible implementation intervention] Let B_t denote declared background variables that should remain matched. A family {ι_w:w∈ W_t^∘} is a feasible implementation of workspace values on the experimental domain W_t^∘ when: item each ι_w acts only on declared manipulable parents, channels, or artifacts at or after time t; item the intervened realiz","count":43},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d547859ac15c2c75","tex":"\\iota_h","aliases":"ι_h iota","display":"\\iota_h","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f99ddae5688bcccc","tex":"\\iota_{h'}","aliases":"ι_h' iota","display":"\\iota_{h'}","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c4ebc22b504c354","tex":"\\iota_w","aliases":"ι_w iota","display":"\\iota_w","kind":"definition-passage","excerpt":"[Feasible implementation intervention] Let B_t denote declared background variables that should remain matched. A family {ι_w:w∈ W_t^∘} is a feasible implementation of workspace values on the experimental domain W_t^∘ when: item each ι_w acts only on declared manipulable parents, channels, or artifacts at or after time t; item the intervened realiz","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":73},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ae0271b381b05112","tex":"\\lambda_{d_\\varepsilon}","aliases":"λ_d_ε lambda varepsilon","display":"\\lambda_{d_\\varepsilon}","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-244fc3fdc08d2fc8","tex":"\\lambda_{d_\\varepsilon+1}","aliases":"λ_d_ε+1 lambda varepsilon","display":"\\lambda_{d_\\varepsilon+1}","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d594552e2786afd4","tex":"\\lambda_k","aliases":"λ_k lambda","display":"\\lambda_k","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d5fba9031264fee8","tex":"\\nu_q","aliases":"ν_q nu","display":"\\nu_q","kind":"definition-passage","excerpt":"[Causal formation sensitivity] Let ι_h and ι_h' be matched feasible interventions that alter formation history while preserving the scaffold, randomization law, sample size, and intervention cost. For a divergence D, define e_q^causal = E_(h,h')∼ν_q D ( L^ι_h(q), L^ι_h'(q) ). A common null uses temporally or relationally shuffled histories that pre","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"definition-passage","excerpt":"tion[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vector","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-273ca48755e332d5","tex":"\\xi_i","aliases":"ξ_i xi","display":"\\xi_i","kind":"definition-passage","excerpt":"tion[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vector","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"definition-passage","excerpt":"[Boundary problem] A boundary problem is a tuple B = ( V, S, τ, Y, I, Π, G_B ), where S⊆2^V is the admissible candidate family, τ is the temporal scale, Y is the operational target, I is the feasible intervention family, Π is the environment distribution, and G_B is a symmetry group under which boundaries are considered equivalent.","count":41},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":22},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9091f42f61615f68","tex":"\\pi_t","aliases":"π_t pi","display":"\\pi_t","kind":"definition-passage","excerpt":"[Time-unrolled causal realization] A time-unrolled causal realization is a structural causal model M = ( U, V,F, P_U) whose endogenous variables include the local states, messages, actions, artifacts, target variables, and a measurable workspace extractor π_t: S_t→ W_t, W_t=π_t(S_t). The unrolled graph is acyclic even when the rolled process is rec","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2ad20c5b5f5b3357","tex":"\\boldsymbol\\rho","aliases":"ρ boldsymbol rho","display":"\\boldsymbol\\rho","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"ple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those operations; item H is the temporally ordered interaction histor","count":60},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-936d7991f33298f9","tex":"\\rho_i","aliases":"ρ_i rho","display":"\\rho_i","kind":"definition-passage","excerpt":"ple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those operations; item H is the temporally ordered interaction histor","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aafa2d66de12bd53","tex":"\\rho_i'","aliases":"ρ_i' rho","display":"\\rho_i'","kind":"definition-passage","excerpt":"[Exact static equivalence] Two static realizations are exactly gauge equivalent, R∼_stat R', when there exists an invertible family g_i such that H_e'=g_t(e)H_eg_s(e)^-1, ρ_i'(a)=g_iρ_i(a)g_i^-1. The exact static abstract workspace is Workspace_stat = [ R]_∼_stat.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c86a36ff5a44dcd8","tex":"\\rho_j","aliases":"ρ_j rho","display":"\\rho_j","kind":"definition-passage","excerpt":"[Exact operation-compatible transfer] A core transfer H_e:C_i→ C_j is operation compatible when it is an intertwiner: H_eρ_i(a)=ρ_j(a)H_e ∀ a∈ A. Equivalently, the diagram ccc C_i · xrightarrowρ_i(a) · C_i","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-751af651a8069bc1","tex":"\\rho_V","aliases":"ρ_V rho","display":"\\rho_V","kind":"definition-passage","excerpt":"[Exact structural correspondence] The correspondence is structurally exact when D'J_V=J_ED. It is operation exact for a∈ A when ρ_V'(a)J_V=J_Vρ_V(a).","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4c7e1cb2694ffa34","tex":"\\rho_V'","aliases":"ρ_V' rho","display":"\\rho_V'","kind":"definition-passage","excerpt":"[Exact structural correspondence] The correspondence is structurally exact when D'J_V=J_ED. It is operation exact for a∈ A when ρ_V'(a)J_V=J_Vρ_V(a).","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"definition-passage","excerpt":"[Formation system] A workspace formation system is a tuple F = ( Σ, D_form, L, Ω, widehatθ, D_cert, I_form ), where: item Σ is the externally supplied scaffold; item D_form is the formation or training history; item L is the learning or selection rule; item Ω is its internal random seed or stochastic state; item widehatθ= L(Σ, D_form,Ω) is the lear","count":27},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"declared local-access map for block B_α, producing the local state or history available at that component; item G_Π records the declared communication, transfer, and causal interfaces among the blocks; and item I_Π is the family of feasible interventions at that resolution. The accessible information of component α is the sigma-field mathscr F_α^ D","count":34},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fac92674bdf59eec","tex":"\\varsigma","aliases":"ς varsigma","display":"\\varsigma","kind":"definition-passage","excerpt":"[Workspace specification] A workspace specification is a tuple ς = ( D, Y^+, ℓ, τ, I, N, Θ ), where: item D is the decomposition and local-access model; item Y^+ is the declared future variable or trajectory; item ℓ is the normalized loss or utility convention; item τ is the temporal horizon or horizon family; item I is the feasible intervention fa","count":61},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c182bef2ecc2c886","tex":"\\varsigma'","aliases":"ς' varsigma","display":"\\varsigma'","kind":"definition-passage","excerpt":"[Specification equivalence] Two specifications ς and ς' are equivalent, written ς≃_specς', when there are bijective transport maps that: item preserve the sigma-field generated by the future target; item preserve the ordering and normalization of decisions under the loss; item map feasible interventions to physically equivalent interventions; item ","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3277dcfd0c5ecae8","tex":"\\varsigma_0","aliases":"ς_0 varsigma","display":"\\varsigma_0","kind":"definition-passage","excerpt":"item[Specification-specific] when it is asserted only for one declared ς_0. item[Specification-robust] when every required gate has positive lower envelope across S. item[Specification-fragile] when it passes for at least one admissible specification and fails for at least one other.","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"definition-passage","excerpt":"[Exact structural--operational alignment] The candidate W_t is observationally aligned when there exists a measurable map ϕ_t such that Z_t=ϕ_t(W_t) a.s. It is I-causally aligned when Z_t^ I=ϕ_t^ I(W_t) a.s.","count":23},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7016cbae921a9f68","tex":"\\phi_t","aliases":"ϕ_t phi","display":"\\phi_t","kind":"definition-passage","excerpt":"[Exact structural--operational alignment] The candidate W_t is observationally aligned when there exists a measurable map ϕ_t such that Z_t=ϕ_t(W_t) a.s. It is I-causally aligned when Z_t^ I=ϕ_t^ I(W_t) a.s.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-49a4d37a9ae3d235","tex":"\\phi_t^{\\mathcal I}","aliases":"ϕ_t^ I phi mathcal","display":"\\phi_t^{\\mathcal I}","kind":"definition-passage","excerpt":"[Exact structural--operational alignment] The candidate W_t is observationally aligned when there exists a measurable map ϕ_t such that Z_t=ϕ_t(W_t) a.s. It is I-causally aligned when Z_t^ I=ϕ_t^ I(W_t) a.s.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"definition-passage","excerpt":"[Experimental equivalence] Two realizations are experimentally equivalent under E, written R≡_ E R', when P_ R^ E = P_ R'^ E. A property ψ( R) is identifiable on a model class M under E when it is constant on every ≡_ E-equivalence class contained in M.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"definition-passage","excerpt":"[Dynamic relational realization] A dynamic relational realization is defined on a probability space (Ω, F, P) and consists of: item node states X_i(t)∈ X_i for i∈ V; item edge messages M_e(t)∈ M_e for e∈ E; item external inputs E_t, local actions A_i(t), and disturbances ξ_i(t),ζ_e(t); item time-dependent transfer data H_e(t), local core vectors c_","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"definition-passage","excerpt":"[Relational witness pair] A pair of laws (P,Q) is a predictive relational witness pair for the decomposition V and target Y_t^+ when Q∈ N_pred(P) but Risk_ℓ,Q(V) ≠ Risk_ℓ,P(V). The two systems agree on every single-node predictive marginal but differ in full-system operative capacity.","count":54},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"definition-passage","excerpt":"[Relational instantiation] A workspace is relationally instantiated relative to the decomposition V when no single projection determines its equivalence class: ∀ i∈ V, ∃ R, R' such that pr_i( R)=pr_i( R') but [ R]_≃≠[ R']_≃.","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"definition-passage","excerpt":"[Relational instantiation] A workspace is relationally instantiated relative to the decomposition V when no single projection determines its equivalence class: ∀ i∈ V, ∃ R, R' such that pr_i( R)=pr_i( R') but [ R]_≃≠[ R']_≃.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Concrete realization] A concrete relational realization is a tuple R = ( G, X, T, A,ρ, H, I, Y ), where: item G=(V,E,s,t) is an interaction graph; item X={X_i}_i∈ V is a family of local state spaces; item T={T_e}_e∈ E is a family of transfer maps; item A is an algebra of admissible operations; item ρ={ρ_i}_i∈ V gives the local realization of those","count":250},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":39},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6d1546afc3ef11e5","tex":"\\sum_{e\\in E}","aliases":"∑_e∈ E sum in","display":"\\sum_{e\\in E}","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"definition-passage","excerpt":"[Feasible implementation intervention] Let B_t denote declared background variables that should remain matched. A family {ι_w:w∈ W_t^∘} is a feasible implementation of workspace values on the experimental domain W_t^∘ when: item each ι_w acts only on declared manipulable parents, channels, or artifacts at or after time t; item the intervened realiz","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"[Exact static equivalence] Two static realizations are exactly gauge equivalent, R∼_stat R', when there exists an invertible family g_i such that H_e'=g_t(e)H_eg_s(e)^-1, ρ_i'(a)=g_iρ_i(a)g_i^-1. The exact static abstract workspace is Workspace_stat = [ R]_∼_stat.","count":19},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5280f0d9b98fedd4","tex":"\\sim_{\\mathrm{stat}}","aliases":"∼_stat sim mathrm","display":"\\sim_{\\mathrm{stat}}","kind":"definition-passage","excerpt":"[Exact static equivalence] Two static realizations are exactly gauge equivalent, R∼_stat R', when there exists an invertible family g_i such that H_e'=g_t(e)H_eg_s(e)^-1, ρ_i'(a)=g_iρ_i(a)g_i^-1. The exact static abstract workspace is Workspace_stat = [ R]_∼_stat.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"definition-passage","excerpt":"[Admissible equivalence] An admissible equivalence F: R≃ R' is a family of correspondences preserving, exactly or within a declared tolerance: · the transferable core and its dimension,","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1af92f42e20d1558","tex":"\\simeq_{\\mathrm{spec}}","aliases":"≃_spec simeq mathrm","display":"\\simeq_{\\mathrm{spec}}","kind":"definition-passage","excerpt":"[Specification equivalence] Two specifications ς and ς' are equivalent, written ς≃_specς', when there are bijective transport maps that: item preserve the sigma-field generated by the future target; item preserve the ordering and normalization of decisions under the loss; item map feasible interventions to physically equivalent interventions; item ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"definition-passage","excerpt":"[Relational instantiation] A workspace is relationally instantiated relative to the decomposition V when no single projection determines its equivalence class: ∀ i∈ V, ∃ R, R' such that pr_i( R)=pr_i( R') but [ R]_≃≠[ R']_≃.","count":32},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"definition-passage","excerpt":"[Experimental equivalence] Two realizations are experimentally equivalent under E, written R≡_ E R', when P_ R^ E = P_ R'^ E. A property ψ( R) is identifiable on a model class M under E when it is constant on every ≡_ E-equivalence class contained in M.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1b3033ebf7bd98df","tex":"\\equiv_{\\mathcal E}","aliases":"≡_ E equiv mathcal","display":"\\equiv_{\\mathcal E}","kind":"definition-passage","excerpt":"[Experimental equivalence] Two realizations are experimentally equivalent under E, written R≡_ E R', when P_ R^ E = P_ R'^ E. A property ψ( R) is identifiable on a model class M under E when it is constant on every ≡_ E-equivalence class contained in M.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":208},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Threshold workspace] For ε≥0, define Γ_ε(H) = ran P_[0,ε](L_H). Its dimension is d_ε = #{k:λ_k≤ε}. When 0<d_ε<N, define the boundary gap g_ε = λ_d_ε+1-λ_d_ε.","count":74},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":41},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Global compatible section] The exact structural workspace is Γ(H) = { (c_i)_i∈ V∈⊕_i∈ VC_i: c_t(e)=H_ec_s(e) ∀ e∈ E }. An element c=(c_i)∈Γ(H) is a global compatible section.","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"definition-passage","excerpt":"[Admissible local decomposition] For every vertex i, choose an orthogonal decomposition X_i=C_i⊕ R_i, with projectors P_i:X_i→ C_i, Q_i=I-P_i:X_i→ R_i. The subspace C_i is the candidate locally realized transferable core. The subspace R_i contains substrate-specific, context-specific, or otherwise noncore degrees of freedom relative to the present ","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2fed50267b9e2ac6","tex":"\\bigoplus_{i\\in V}","aliases":"⊕_i∈ V bigoplus in","display":"\\bigoplus_{i\\in V}","kind":"definition-passage","excerpt":"[Global compatible section] The exact structural workspace is Γ(H) = { (c_i)_i∈ V∈⊕_i∈ VC_i: c_t(e)=H_ec_s(e) ∀ e∈ E }. An element c=(c_i)∈Γ(H) is a global compatible section.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"definition-passage","excerpt":"ver A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets. Isolated vertices ar","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8978fa5970bcdbef","tex":"\\bigcap_{\\alpha\\in I}","aliases":"⋂_α∈ I bigcap alpha in","display":"\\bigcap_{\\alpha\\in I}","kind":"definition-passage","excerpt":"ver A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets. Isolated vertices ar","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-839ccee45161a86d","tex":"\\bigcup_{\\alpha\\in A}","aliases":"⋃_α∈ A bigcup alpha in","display":"\\bigcup_{\\alpha\\in A}","kind":"definition-passage","excerpt":"[Finite workspace cover] A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"ver A finite family of subgraphs U={G_α}_α∈ A is an edge-complete workspace cover of G when V=⋃_α∈ AV_α, E=⋃_α∈ AE_α. Every edge in E_α carries the transfer inherited from G. For a nonempty finite index set I={α_0,…,α_p}⊆ A, define the intersection subgraph G_I = G_α_0⋯α_p := ⋂_α∈ IG_α by intersecting both vertex and edge sets. Isolated vertices ar","count":25},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Transfer discrepancy] The transfer discrepancy operator is D_H: C_V⟶ C_E, (D_Hc)_e = sqrtw_e(c_t(e)-H_ec_s(e)). Its squared norm is the coherence energy E_H(c) = D_Hc ^2 = ∑_e∈ Ew_e c_t(e)-H_ec_s(e) ^2. The transfer Laplacian is L_H=D_H^*D_H.","count":31},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82821bc2124f1658","tex":"A_t","aliases":"A_t","display":"A_t","kind":"introduction-passage","excerpt":"Let V_t denote the internal value channel. A counterfactual-mediation test compares: G_t ⟶ V_t ⟶ A_t ⟶ Y_t^+. The relevant quantities include: η_G→ V, η_V→ A, η_A→ Y, δ_G→ Amid V, where the final term measures the residual goal-to-action effect after the value channel is blocked or randomized.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f925f2dd41a18a99","tex":"\\mathcal A_t","aliases":"A_t mathcal","display":"\\mathcal A_t","kind":"introduction-passage","excerpt":"Let A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a)","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7d56e6ebb6e66e14","tex":"\\mathcal A_t^*","aliases":"A_t^* mathcal","display":"\\mathcal A_t^*","kind":"introduction-passage","excerpt":"Let A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a)","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-74ac353874d337a1","tex":"\\argmax","aliases":"argmax","display":null,"kind":"introduction-passage","excerpt":"Let A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a)","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d55ad65edfc19357","tex":"\\argmax_{a\\in\\mathcal A_t(w,e)}","aliases":"argmax_a∈ A_t(w,e) argmax in mathcal","display":null,"kind":"introduction-passage","excerpt":"Let A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a)","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":91},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-184e8bf31391ed2f","tex":"\\bigwedge_{j\\in\\mathcal J(\\ell)}","aliases":"bigwedge_j∈ J(ℓ) bigwedge in mathcal ell","display":"\\bigwedge_{j\\in\\mathcal J(\\ell)}","kind":"introduction-passage","excerpt":"Let L( K)∈{0,1,…,6} denote the highest level from Part~XXIV whose complete required gate set is established. The level is conjunctive and pointwise in the declared scientific specification unless the D and X coordinates also pass: L( K)≥ ℓ Longleftrightarrow bigwedge_j∈ J(ℓ)Pass_j( K). A case may have strong higher-level-looking behavior while rema","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c0fa9e872b2cc65b","tex":"\\mathrm{bound}","aliases":"bound mathrm","display":"\\mathrm{bound}","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f407bedadfee371c","tex":"c^0","aliases":"c^0","display":"c^0","kind":"introduction-passage","excerpt":"Let C_t denote an external command or task cue. Fix a neutral or matched future command trajectory c_t+1:t+τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-433d97a8fbb31f65","tex":"C_t","aliases":"C_t","display":"C_t","kind":"introduction-passage","excerpt":"Let C_t denote an external command or task cue. Fix a neutral or matched future command trajectory c_t+1:t+τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-451865ffcdc8dd8b","tex":"C_{t+1:t+\\tau}","aliases":"C_t+1:t+τ tau","display":"C_{t+1:t+\\tau}","kind":"introduction-passage","excerpt":"Let C_t denote an external command or task cue. Fix a neutral or matched future command trajectory c_t+1:t+τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-45f9da60335cec9b","tex":"c_{t+1:t+\\tau}^{0}","aliases":"c_t+1:t+τ^0 tau","display":"c_{t+1:t+\\tau}^{0}","kind":"introduction-passage","excerpt":"Let C_t denote an external command or task cue. Fix a neutral or matched future command trajectory c_t+1:t+τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7d9bc259bcfcd8e3","tex":"\\centernot","aliases":"centernot","display":null,"kind":"introduction-passage","excerpt":"The relation is therefore analogical rather than identificatory: SRW(W) centernotLongrightarrow GlobalNeuronalWorkspace(W) centernotLongrightarrow Conscious(W). The term is retained because it denotes a domain in which jointly available operations can be carried out. Every consciousness claim remains on an explicitly separate axis.","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c5e621c1e94ca796","tex":"\\operatorname{Conscious}","aliases":"Conscious operatorname","display":"\\operatorname{Conscious}","kind":"introduction-passage","excerpt":"The relation is therefore analogical rather than identificatory: SRW(W) centernotLongrightarrow GlobalNeuronalWorkspace(W) centernotLongrightarrow Conscious(W). The term is retained because it denotes a domain in which jointly available operations can be carried out. Every consciousness claim remains on an explicitly separate axis.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ddc7e0f1c8697a47","tex":"\\mathsf D","aliases":"D mathsf","display":"\\mathsf D","kind":"introduction-passage","excerpt":"Let L( K)∈{0,1,…,6} denote the highest level from Part~XXIV whose complete required gate set is established. The level is conjunctive and pointwise in the declared scientific specification unless the D and X coordinates also pass: L( K)≥ ℓ Longleftrightarrow bigwedge_j∈ J(ℓ)Pass_j( K). A case may have strong higher-level-looking behavior while rema","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a500ade917bb472a","tex":"D_G","aliases":"D_G","display":"D_G","kind":"introduction-passage","excerpt":"Let C_t denote an external command or task cue. Fix a neutral or matched future command trajectory c_t+1:t+τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"introduction-passage","excerpt":"A repair intervention r∈ I_R can add, remove, retrain, rewrite, or reroute an admissible component. Let r_0 denote matched no-repair. The post-repair integrity value at horizon τ is V_t^rep(r;d) = E [ v_t+τ mid do(D_t=d), do(R_t=r), H_t ]. The repair gain is Δ_t^rep(r;d) = V_t^rep(r;d) - V_t^rep(r_0;d). Repair costs are handled either by restrictin","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6f78b62eb28153fc","tex":"D_Y","aliases":"D_Y","display":"D_Y","kind":"introduction-passage","excerpt":"β w,χ b:(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^prime+∈·). For a chosen divergence D_Y, define δ_causal(β) = sup_(w,b)∈ D D_Y( K(w,b), K'(β w,χ ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"introduction-passage","excerpt":"Let 0≤λ_1≤λ_2≤⋯≤λ_N, N=dim C_V, be the eigenvalues of L_H, repeated with multiplicity, and let P_I(L_H) denote the spectral projector associated with an interval I.","count":19},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8a32e1fef29bf96f","tex":"\\mathrm{dir}","aliases":"dir mathrm","display":"\\mathrm{dir}","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c2349512e48c7e97","tex":"e^{-1}","aliases":"e^-1","display":"e^{-1}","kind":"introduction-passage","excerpt":"ninvertible. Holonomy, however, requires transport around reversible loops. To state the exact holonomy theorem without ambiguity, let |G| be the underlying undirected graph and let Π_1(G) denote its fundamental path groupoid. Every oriented edge is then supplied with a formal reverse. The core transfers extend to Π_1(G) only when the relevant maps","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6fae6a0c12e8f5a0","tex":"\\mathbb E^{\\iota_a}","aliases":"E^ι_a mathbb iota","display":"\\mathbb E^{\\iota_a}","kind":"introduction-passage","excerpt":"Let A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a)","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-88bd193615ce9295","tex":"\\mathbb E_{(w,w',b)\\sim\\nu}","aliases":"E_(w,w',b)∼ν mathbb sim nu","display":"\\mathbb E_{(w,w',b)\\sim\\nu}","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a4c811cece987c8d","tex":"\\mathcal F_t^A","aliases":"F_t^A mathcal","display":"\\mathcal F_t^A","kind":"introduction-passage","excerpt":"For a coalition A⊆ V, let H_t^A denote the random history available to that coalition through time t. Its generated information is F_t^A=σ( H_t^A). The full relational history is H_t= H_t^V, with F_t^A⊆ F_t^B whenever A⊆ B. The empty-coalition history H_t^∅ is the trivial constant statistic and represents prediction from the background target law a","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-db54aa382c44e650","tex":"\\mathcal F_t^B","aliases":"F_t^B mathcal","display":"\\mathcal F_t^B","kind":"introduction-passage","excerpt":"For a coalition A⊆ V, let H_t^A denote the random history available to that coalition through time t. Its generated information is F_t^A=σ( H_t^A). The full relational history is H_t= H_t^V, with F_t^A⊆ F_t^B whenever A⊆ B. The empty-coalition history H_t^∅ is the trivial constant statistic and represents prediction from the background target law a","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-02b0b5ff2a25a5d9","tex":"g'","aliases":"g'","display":"g'","kind":"introduction-passage","excerpt":"Let C_t denote an external command or task cue. Fix a neutral or matched future command trajectory c_t+1:t+τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"introduction-passage","excerpt":"[Intersection-gate false-certification control] Let a certificate require all gates G_1,…,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is abse","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7ccf548c96ad41ac","tex":"g_d","aliases":"g_d","display":"g_d","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ec1c980979f73c29","tex":"G_j","aliases":"G_j","display":"G_j","kind":"introduction-passage","excerpt":"[Intersection-gate false-certification control] Let a certificate require all gates G_1,…,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is abse","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-09f3e72028e864cd","tex":"G_m","aliases":"G_m","display":"G_m","kind":"introduction-passage","excerpt":"[Intersection-gate false-certification control] Let a certificate require all gates G_1,…,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is abse","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b4f9811bf3c61e66","tex":"G_t","aliases":"G_t","display":"G_t","kind":"introduction-passage","excerpt":"Let A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a)","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1239bb88a6730650","tex":"G_{t+\\tau}","aliases":"G_t+τ tau","display":"G_{t+\\tau}","kind":"introduction-passage","excerpt":"Let C_t denote an external command or task cue. Fix a neutral or matched future command trajectory c_t+1:t+τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-12fedb06a6c6a426","tex":"\\operatorname{GlobalNeuronalWorkspace}","aliases":"GlobalNeuronalWorkspace operatorname","display":"\\operatorname{GlobalNeuronalWorkspace}","kind":"introduction-passage","excerpt":"The relation is therefore analogical rather than identificatory: SRW(W) centernotLongrightarrow GlobalNeuronalWorkspace(W) centernotLongrightarrow Conscious(W). The term is retained because it denotes a domain in which jointly available operations can be carried out. Every consciousness claim remains on an explicitly separate axis.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"introduction-passage","excerpt":"[Intersection-gate false-certification control] Let a certificate require all gates G_1,…,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is abse","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3e660bc3bed20fed","tex":"H_{0j}","aliases":"H_0j","display":"H_{0j}","kind":"introduction-passage","excerpt":"[Intersection-gate false-certification control] Let a certificate require all gates G_1,…,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is abse","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5b9dae71873dd207","tex":"H_e^{-1}","aliases":"H_e^-1","display":"H_e^{-1}","kind":"introduction-passage","excerpt":"ninvertible. Holonomy, however, requires transport around reversible loops. To state the exact holonomy theorem without ambiguity, let |G| be the underlying undirected graph and let Π_1(G) denote its fundamental path groupoid. Every oriented edge is then supplied with a formal reverse. The core transfers extend to Π_1(G) only when the relevant maps","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4fa868978e9b912b","tex":"H_{e^{-1}}","aliases":"H_e^-1","display":"H_{e^{-1}}","kind":"introduction-passage","excerpt":"ninvertible. Holonomy, however, requires transport around reversible loops. To state the exact holonomy theorem without ambiguity, let |G| be the underlying undirected graph and let Π_1(G) denote its fundamental path groupoid. Every oriented edge is then supplied with a formal reverse. The core transfers extend to Π_1(G) only when the relevant maps","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-98469b1f6a16db75","tex":"\\mathbf H_t^A","aliases":"H_t^A mathbf","display":"\\mathbf H_t^A","kind":"introduction-passage","excerpt":"For a coalition A⊆ V, let H_t^A denote the random history available to that coalition through time t. Its generated information is F_t^A=σ( H_t^A). The full relational history is H_t= H_t^V, with F_t^A⊆ F_t^B whenever A⊆ B. The empty-coalition history H_t^∅ is the trivial constant statistic and represents prediction from the background target law a","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0086408713a89bcd","tex":"\\mathbf H_t^V","aliases":"H_t^V mathbf","display":"\\mathbf H_t^V","kind":"introduction-passage","excerpt":"For a coalition A⊆ V, let H_t^A denote the random history available to that coalition through time t. Its generated information is F_t^A=σ( H_t^A). The full relational history is H_t= H_t^V, with F_t^A⊆ F_t^B whenever A⊆ B. The empty-coalition history H_t^∅ is the trivial constant statistic and represents prediction from the background target law a","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9f4204102f31924c","tex":"\\mathbf H_t^\\varnothing","aliases":"H_t^∅ mathbf varnothing","display":"\\mathbf H_t^\\varnothing","kind":"introduction-passage","excerpt":"For a coalition A⊆ V, let H_t^A denote the random history available to that coalition through time t. Its generated information is F_t^A=σ( H_t^A). The full relational history is H_t= H_t^V, with F_t^A⊆ F_t^B whenever A⊆ B. The empty-coalition history H_t^∅ is the trivial constant statistic and represents prediction from the background target law a","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0dc926e3445b8b40","tex":"\\mathcal I_R","aliases":"I_R mathcal","display":"\\mathcal I_R","kind":"introduction-passage","excerpt":"A repair intervention r∈ I_R can add, remove, retrain, rewrite, or reroute an admissible component. Let r_0 denote matched no-repair. The post-repair integrity value at horizon τ is V_t^rep(r;d) = E [ v_t+τ mid do(D_t=d), do(R_t=r), H_t ]. The repair gain is Δ_t^rep(r;d) = V_t^rep(r;d) - V_t^rep(r_0;d). Repair costs are handled either by restrictin","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fb9dab5747004561","tex":"K'","aliases":"K'","display":"K'","kind":"introduction-passage","excerpt":"nce does not determine operational identity. Let {ι_w,b:(w,b)∈ D} and {ι'_β w,χ b:(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^prime+∈·). For a chose","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9a30d27faec7f7f7","tex":"K_r","aliases":"K_r","display":"K_r","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2139530d0b2d1758","tex":"K_{R-1}","aliases":"K_R-1","display":"K_{R-1}","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5236b838b836f2fa","tex":"\\mathrm{loc}","aliases":"loc mathrm","display":"\\mathrm{loc}","kind":"introduction-passage","excerpt":"Exact or approximate overlap coherence does not by itself establish global predictive sufficiency. Let W_t denote a glued structural state and let Z_t^loc = (Z_α,t)_α∈ A be local predictive summaries recovered from its chart restrictions.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"introduction-passage","excerpt":"Let L( K)∈{0,1,…,6} denote the highest level from Part~XXIV whose complete required gate set is established. The level is conjunctive and pointwise in the declared scientific specification unless the D and X coordinates also pass: L( K)≥ ℓ Longleftrightarrow bigwedge_j∈ J(ℓ)Pass_j( K). A case may have strong higher-level-looking behavior while rema","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"introduction-passage","excerpt":"The relation is therefore analogical rather than identificatory: SRW(W) centernotLongrightarrow GlobalNeuronalWorkspace(W) centernotLongrightarrow Conscious(W). The term is retained because it denotes a domain in which jointly available operations can be carried out. Every consciousness claim remains on an explicitly separate axis.","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c9dc58ed7c147505","tex":"\\mathrm{loop}","aliases":"loop mathrm","display":"\\mathrm{loop}","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ac8f6317e7ffbdde","tex":"\\mathcal M^{\\setminus\\mathrm{dir}}","aliases":"M^∖dir mathcal setminus mathrm","display":"\\mathcal M^{\\setminus\\mathrm{dir}}","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5a59db50da3c52bc","tex":"\\max_{1\\leq j\\leq m}","aliases":"max_1≤ j≤ m max leq leq","display":"\\max_{1\\leq j\\leq m}","kind":"introduction-passage","excerpt":"dots,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is absent. If the certificate is declared only when every gate passes, then sup_P∈ H_0 P_P(c","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6c9e44ac55def8aa","tex":"\\mathbb P^{\\prime\\,\\iota'_{\\beta w,\\chi b}}","aliases":"P^prime ι'_β w,χ b mathbb prime iota beta chi","display":"\\mathbb P^{\\prime\\,\\iota'_{\\beta w,\\chi b}}","kind":"introduction-passage","excerpt":"nce does not determine operational identity. Let {ι_w,b:(w,b)∈ D} and {ι'_β w,χ b:(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^prime+∈·). For a chose","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5d739c143458278d","tex":"\\mathbb P^{\\iota_{w,b}}","aliases":"P^ι_w,b mathbb iota","display":"\\mathbb P^{\\iota_{w,b}}","kind":"introduction-passage","excerpt":"Structural correspondence does not determine operational identity. Let {ι_w,b:(w,b)∈ D} and {ι'_β w,χ b:(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-77495e066618607a","tex":"\\mathbb P_{\\mathcal M^{\\setminus\\mathrm{dir}}}^{\\iota_w}","aliases":"P_ M^∖dir^ι_w mathbb mathcal setminus mathrm iota","display":"\\mathbb P_{\\mathcal M^{\\setminus\\mathrm{dir}}}^{\\iota_w}","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-390408ea0cff8f65","tex":"\\mathbb P_{\\mathcal M^{\\setminus\\mathrm{dir}}}^{\\iota_{w'}}","aliases":"P_ M^∖dir^ι_w' mathbb mathcal setminus mathrm iota","display":"\\mathbb P_{\\mathcal M^{\\setminus\\mathrm{dir}}}^{\\iota_{w'}}","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-14f8900152d6716a","tex":"P_d","aliases":"P_d","display":"P_d","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":21},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d65ad4aef739a573","tex":"P_I","aliases":"P_I","display":"P_I","kind":"introduction-passage","excerpt":"Let 0≤λ_1≤λ_2≤⋯≤λ_N, N=dim C_V, be the eigenvalues of L_H, repeated with multiplicity, and let P_I(L_H) denote the spectral projector associated with an interval I.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-06191cab41b2ec3c","tex":"\\mathbb P_P","aliases":"P_P mathbb","display":"\\mathbb P_P","kind":"introduction-passage","excerpt":"[Intersection-gate false-certification control] Let a certificate require all gates G_1,…,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is abse","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b3ca057642ba6830","tex":"\\Pass","aliases":"Pass","display":null,"kind":"introduction-passage","excerpt":"Let L( K)∈{0,1,…,6} denote the highest level from Part~XXIV whose complete required gate set is established. The level is conjunctive and pointwise in the declared scientific specification unless the D and X coordinates also pass: L( K)≥ ℓ Longleftrightarrow bigwedge_j∈ J(ℓ)Pass_j( K). A case may have strong higher-level-looking behavior while rema","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-84852125e168dcaf","tex":"\\Pass_j","aliases":"Pass_j Pass","display":null,"kind":"introduction-passage","excerpt":"Let L( K)∈{0,1,…,6} denote the highest level from Part~XXIV whose complete required gate set is established. The level is conjunctive and pointwise in the declared scientific specification unless the D and X coordinates also pass: L( K)≥ ℓ Longleftrightarrow bigwedge_j∈ J(ℓ)Pass_j( K). A case may have strong higher-level-looking behavior while rema","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3f1b6dc519af96a7","tex":"\\prime","aliases":"prime","display":"\\prime","kind":"introduction-passage","excerpt":"nce does not determine operational identity. Let {ι_w,b:(w,b)∈ D} and {ι'_β w,χ b:(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^prime+∈·). For a chose","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":140},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"introduction-passage","excerpt":"A repair intervention r∈ I_R can add, remove, retrain, rewrite, or reroute an admissible component. Let r_0 denote matched no-repair. The post-repair integrity value at horizon τ is V_t^rep(r;d) = E [ v_t+τ mid do(D_t=d), do(R_t=r), H_t ]. The repair gain is Δ_t^rep(r;d) = V_t^rep(r;d) - V_t^rep(r_0;d). Repair costs are handled either by restrictin","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-49121c793f36cbb5","tex":"R_t","aliases":"R_t","display":"R_t","kind":"introduction-passage","excerpt":"A repair intervention r∈ I_R can add, remove, retrain, rewrite, or reroute an admissible component. Let r_0 denote matched no-repair. The post-repair integrity value at horizon τ is V_t^rep(r;d) = E [ v_t+τ mid do(D_t=d), do(R_t=r), H_t ]. The repair gain is Δ_t^rep(r;d) = V_t^rep(r;d) - V_t^rep(r_0;d). Repair costs are handled either by restrictin","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0dd44fa15c96aac6","tex":"\\mathrm{rep}","aliases":"rep mathrm","display":"\\mathrm{rep}","kind":"introduction-passage","excerpt":"A repair intervention r∈ I_R can add, remove, retrain, rewrite, or reroute an admissible component. Let r_0 denote matched no-repair. The post-repair integrity value at horizon τ is V_t^rep(r;d) = E [ v_t+τ mid do(D_t=d), do(R_t=r), H_t ]. The repair gain is Δ_t^rep(r;d) = V_t^rep(r;d) - V_t^rep(r_0;d). Repair costs are handled either by restrictin","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8aa2d4b22c33e9d5","tex":"\\operatorname{SRW}","aliases":"SRW operatorname","display":"\\operatorname{SRW}","kind":"introduction-passage","excerpt":"The relation is therefore analogical rather than identificatory: SRW(W) centernotLongrightarrow GlobalNeuronalWorkspace(W) centernotLongrightarrow Conscious(W). The term is retained because it denotes a domain in which jointly available operations can be carried out. Every consciousness claim remains on an explicitly separate axis.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"introduction-passage","excerpt":":(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^prime+∈·). For a chosen divergence D_Y, define δ_causal(β) = sup_(w,b)∈ D D_Y( K(w,b), K'(β w,χ b) ). T","count":47},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a2a4ff6dae1a25b7","tex":"\\sup_{(w,b)\\in\\mathcal D}","aliases":"sup_(w,b)∈ D sup in mathcal","display":"\\sup_{(w,b)\\in\\mathcal D}","kind":"introduction-passage","excerpt":":(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^prime+∈·). For a chosen divergence D_Y, define δ_causal(β) = sup_(w,b)∈ D D_Y( K(w,b), K'(β w,χ b) ). T","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bc68c14415a74380","tex":"\\sup_{P\\in H_0}","aliases":"sup_P∈ H_0 sup in","display":"\\sup_{P\\in H_0}","kind":"introduction-passage","excerpt":"dots,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is absent. If the certificate is declared only when every gate passes, then sup_P∈ H_0 P_P(c","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8b21705d78237701","tex":"\\sup_{P\\in H_{0j}}","aliases":"sup_P∈ H_0j sup in","display":"\\sup_{P\\in H_{0j}}","kind":"introduction-passage","excerpt":"[Intersection-gate false-certification control] Let a certificate require all gates G_1,…,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is abse","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-67714dc438842e32","tex":"\\sup_x","aliases":"sup_x sup","display":"\\sup_x","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-17aada5f90193f08","tex":"\\operatorname{TV}","aliases":"TV operatorname","display":"\\operatorname{TV}","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9f5988962b4f6877","tex":"U_g","aliases":"U_g","display":"U_g","kind":"introduction-passage","excerpt":"Let A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a)","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-21f030f572f5b2a5","tex":"V_t","aliases":"V_t","display":"V_t","kind":"introduction-passage","excerpt":"Let V_t denote the internal value channel. A counterfactual-mediation test compares: G_t ⟶ V_t ⟶ A_t ⟶ Y_t^+. The relevant quantities include: η_G→ V, η_V→ A, η_A→ Y, δ_G→ Amid V, where the final term measures the residual goal-to-action effect after the value channel is blocked or randomized.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f25cb6dcb56fd327","tex":"v_{t+\\tau}","aliases":"v_t+τ tau","display":"v_{t+\\tau}","kind":"introduction-passage","excerpt":"A repair intervention r∈ I_R can add, remove, retrain, rewrite, or reroute an admissible component. Let r_0 denote matched no-repair. The post-repair integrity value at horizon τ is V_t^rep(r;d) = E [ v_t+τ mid do(D_t=d), do(R_t=r), H_t ]. The repair gain is Δ_t^rep(r;d) = V_t^rep(r;d) - V_t^rep(r_0;d). Repair costs are handled either by restrictin","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c5739eceac3d392","tex":"V_t^{\\mathrm{rep}}","aliases":"V_t^rep mathrm","display":"V_t^{\\mathrm{rep}}","kind":"introduction-passage","excerpt":"A repair intervention r∈ I_R can add, remove, retrain, rewrite, or reroute an admissible component. Let r_0 denote matched no-repair. The post-repair integrity value at horizon τ is V_t^rep(r;d) = E [ v_t+τ mid do(D_t=d), do(R_t=r), H_t ]. The repair gain is Δ_t^rep(r;d) = V_t^rep(r;d) - V_t^rep(r_0;d). Repair costs are handled either by restrictin","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b99d2e5f37ae7e17","tex":"w'","aliases":"w'","display":"w'","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bbcdc850ea5ff763","tex":"W_{t+\\tau}","aliases":"W_t+τ tau","display":"W_{t+\\tau}","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-47c202182f2c6f9b","tex":"\\widehat Q","aliases":"widehat Q widehat","display":"\\widehat Q","kind":"introduction-passage","excerpt":"( A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a) )","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-222343f350d50679","tex":"\\widehat Q_t","aliases":"widehat Q_t widehat","display":"\\widehat Q_t","kind":"introduction-passage","excerpt":"( A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a) )","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9117c60fc5e9378b","tex":"\\widetilde K","aliases":"widetilde K widetilde","display":"\\widetilde K","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-066eb30ff5b36125","tex":"\\widetilde K_0","aliases":"widetilde K_0 widetilde","display":"\\widetilde K_0","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dcf4c65041884955","tex":"\\widetilde K_r","aliases":"widetilde K_r widetilde","display":"\\widetilde K_r","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5ff60e500d73dea3","tex":"\\widetilde K_{R-1}","aliases":"widetilde K_R-1 widetilde","display":"\\widetilde K_{R-1}","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-97353ccc44a7ab31","tex":"\\widetilde P_d","aliases":"widetilde P_d widetilde","display":"\\widetilde P_d","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-604a217150c4b1f9","tex":"Y^{\\prime+}","aliases":"Y^prime+ prime","display":"Y^{\\prime+}","kind":"introduction-passage","excerpt":"nce does not determine operational identity. Let {ι_w,b:(w,b)∈ D} and {ι'_β w,χ b:(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^prime+∈·). For a chose","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9ef0883d7feadbd1","tex":"Z_t^{\\mathrm{loc}}","aliases":"Z_t^loc mathrm","display":"Z_t^{\\mathrm{loc}}","kind":"introduction-passage","excerpt":"Exact or approximate overlap coherence does not by itself establish global predictive sufficiency. Let W_t denote a glued structural state and let Z_t^loc = (Z_α,t)_α∈ A be local predictive summaries recovered from its chart restrictions.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-23b7cc6a662033c8","tex":"Z_{\\alpha,t}","aliases":"Z_α,t alpha","display":"Z_{\\alpha,t}","kind":"introduction-passage","excerpt":"Exact or approximate overlap coherence does not by itself establish global predictive sufficiency. Let W_t denote a glued structural state and let Z_t^loc = (Z_α,t)_α∈ A be local predictive summaries recovered from its chart restrictions.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"Global compatible sections make sense even when transfers are noninvertible. Holonomy, however, requires transport around reversible loops. To state the exact holonomy theorem without ambiguity, let |G| be the underlying undirected graph and let Π_1(G) denote its fundamental path groupoid. Every oriented edge is then supplied with a formal reverse.","count":65},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"introduction-passage","excerpt":"[Intersection-gate false-certification control] Let a certificate require all gates G_1,…,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is abse","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":61},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":54},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-edd4d6889a9a9118","tex":"\\delta_{\\mathrm{causal}}","aliases":"δ_causal delta mathrm","display":"\\delta_{\\mathrm{causal}}","kind":"introduction-passage","excerpt":":(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^prime+∈·). For a chosen divergence D_Y, define δ_causal(β) = sup_(w,b)∈ D D_Y( K(w,b), K'(β w,χ b) ). T","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c05afe2b61e507f0","tex":"\\delta_{G\\to A\\mid V}","aliases":"δ_G→ Amid V delta to mid","display":null,"kind":"introduction-passage","excerpt":"Let V_t denote the internal value channel. A counterfactual-mediation test compares: G_t ⟶ V_t ⟶ A_t ⟶ Y_t^+. The relevant quantities include: η_G→ V, η_V→ A, η_A→ Y, δ_G→ Amid V, where the final term measures the residual goal-to-action effect after the value channel is blocked or randomized.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a6dc26c121dd6a18","tex":"\\delta_H","aliases":"δ_H delta","display":"\\delta_H","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-76da5c62afac42f9","tex":"\\Delta_t^{\\mathrm{rep}}","aliases":"Δ_t^rep Delta mathrm","display":"\\Delta_t^{\\mathrm{rep}}","kind":"introduction-passage","excerpt":"A repair intervention r∈ I_R can add, remove, retrain, rewrite, or reroute an admissible component. Let r_0 denote matched no-repair. The post-repair integrity value at horizon τ is V_t^rep(r;d) = E [ v_t+τ mid do(D_t=d), do(R_t=r), H_t ]. The repair gain is Δ_t^rep(r;d) = V_t^rep(r;d) - V_t^rep(r_0;d). Repair costs are handled either by restrictin","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a2e1656b4a472be9","tex":"\\zeta_r","aliases":"ζ_r zeta","display":"\\zeta_r","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1de79c88fada7eb9","tex":"\\eta_{G\\to V}","aliases":"η_G→ V eta to","display":"\\eta_{G\\to V}","kind":"introduction-passage","excerpt":"Let V_t denote the internal value channel. A counterfactual-mediation test compares: G_t ⟶ V_t ⟶ A_t ⟶ Y_t^+. The relevant quantities include: η_G→ V, η_V→ A, η_A→ Y, δ_G→ Amid V, where the final term measures the residual goal-to-action effect after the value channel is blocked or randomized.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-08421570ac66623d","tex":"\\eta_{\\mathrm{loop}}","aliases":"η_loop eta mathrm","display":"\\eta_{\\mathrm{loop}}","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d9cb1454cf422384","tex":"\\iota'_{\\beta w,\\chi b}","aliases":"ι'_β w,χ b iota beta chi","display":"\\iota'_{\\beta w,\\chi b}","kind":"introduction-passage","excerpt":"Structural correspondence does not determine operational identity. Let {ι_w,b:(w,b)∈ D} and {ι'_β w,χ b:(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-32bd3eb90e3c36aa","tex":"\\iota_a","aliases":"ι_a iota","display":"\\iota_a","kind":"introduction-passage","excerpt":"Let A_t(w,e) denote the actions that can be implemented from workspace state w in environment e. The interventional action value is Q_t^g(amid w,e) = E^ι_a [ U_g(Y_t^+) mid W_t=w, E_t=e, G_t=g ]. The optimal action set is A_t^*(w,g,e) = argmax_a∈ A_t(w,e)Q_t^g(amid w,e). A counterfactual evaluator is an internal state widehat Q_t = ( widehat Q_t(a)","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6224649573390023","tex":"\\iota_g","aliases":"ι_g iota","display":"\\iota_g","kind":"introduction-passage","excerpt":"Let C_t denote an external command or task cue. Fix a neutral or matched future command trajectory c_t+1:t+τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d276fa4c08338bb2","tex":"\\iota_{g'}","aliases":"ι_g' iota","display":"\\iota_{g'}","kind":"introduction-passage","excerpt":"Let C_t denote an external command or task cue. Fix a neutral or matched future command trajectory c_t+1:t+τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c7fde2f27afdaa37","tex":"\\iota_{w'}","aliases":"ι_w' iota","display":"\\iota_{w'}","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-32c935b670b29a43","tex":"\\iota_{w,b}","aliases":"ι_w,b iota","display":"\\iota_{w,b}","kind":"introduction-passage","excerpt":"Structural correspondence does not determine operational identity. Let {ι_w,b:(w,b)∈ D} and {ι'_β w,χ b:(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"introduction-passage","excerpt":"Let 0≤λ_1≤λ_2≤⋯≤λ_N, N=dim C_V, be the eigenvalues of L_H, repeated with multiplicity, and let P_I(L_H) denote the spectral projector associated with an interval I.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-975e1a0c6d000741","tex":"\\lambda_2","aliases":"λ_2 lambda","display":"\\lambda_2","kind":"introduction-passage","excerpt":"Let 0≤λ_1≤λ_2≤⋯≤λ_N, N=dim C_V, be the eigenvalues of L_H, repeated with multiplicity, and let P_I(L_H) denote the spectral projector associated with an interval I.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-18bb5b342b20541e","tex":"\\lambda_d","aliases":"λ_d lambda","display":"\\lambda_d","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bd19369437f5d505","tex":"\\lambda_{d+1}","aliases":"λ_d+1 lambda","display":"\\lambda_{d+1}","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-57d6bd6085b620be","tex":"\\lambda_N","aliases":"λ_N lambda","display":"\\lambda_N","kind":"introduction-passage","excerpt":"Let 0≤λ_1≤λ_2≤⋯≤λ_N, N=dim C_V, be the eigenvalues of L_H, repeated with multiplicity, and let P_I(L_H) denote the spectral projector associated with an interval I.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-288247c3559696c5","tex":"\\Lambda_\\gamma","aliases":"Λ_γ Lambda gamma","display":"\\Lambda_\\gamma","kind":"introduction-passage","excerpt":"A minimal structural report is therefore ( d,λ_d,λ_d+1,g_d, δ_H, P_d-widetilde P_d _bound, β(a), Λ_γ ), where Λ_γ denotes the right-hand side of the pathwise leakage bound. This structural report is intentionally incomplete as a workspace certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility gates required to interpret the ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c3acc8569f115745","tex":"\\Pi_1","aliases":"Π_1 Pi","display":"\\Pi_1","kind":"introduction-passage","excerpt":"Global compatible sections make sense even when transfers are noninvertible. Holonomy, however, requires transport around reversible loops. To state the exact holonomy theorem without ambiguity, let |G| be the underlying undirected graph and let Π_1(G) denote its fundamental path groupoid. Every oriented edge is then supplied with a formal reverse.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"introduction-passage","excerpt":"Structural correspondence does not determine operational identity. Let {ι_w,b:(w,b)∈ D} and {ι'_β w,χ b:(w,b)∈ D} be matched feasible implementation-intervention families in the sense of Part~VIII. Here b denotes declared background variables and χ is the background correspondence. Define K(w,b) = P^ι_w,b(Y^+∈·), K'(β w,χ b) = P^prime ι'_β w,χ b(Y^","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"introduction-passage","excerpt":"For a coalition A⊆ V, let H_t^A denote the random history available to that coalition through time t. Its generated information is F_t^A=σ( H_t^A). The full relational history is H_t= H_t^V, with F_t^A⊆ F_t^B whenever A⊆ B. The empty-coalition history H_t^∅ is the trivial constant statistic and represents prediction from the background target law a","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-09ea8e176c2c6916","tex":"\\sum_{r=0}^{R-1}","aliases":"∑_r=0^R-1 sum","display":"\\sum_{r=0}^{R-1}","kind":"introduction-passage","excerpt":"Let K_r and widetilde K_r now denote exact and implemented Markov kernels at successive hierarchy levels, and suppose sup_x TV ( widetilde K_r(x,·), K_r(x,·) ) ≤ ζ_r. Because Markov kernels contract total variation, a telescoping argument gives sup_x TV ( widetilde K_0⋯widetilde K_R-1(x,·), K_0⋯ K_R-1(x,·) ) ≤ ∑_r=0^R-1ζ_r. Thus causal semantics ca","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"introduction-passage","excerpt":"To isolate reentry, consider the unrolled causal graph and remove every direct retention edge whose source and target are both workspace variables at successive times. Let M^∖dir denote the resulting path-specific model, preserving paths that leave the workspace through a nonworkspace participant, action, message, or artifact and later return. Defi","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"introduction-passage","excerpt":"If the complement is defined by Γ_ε(H)^⊥, the result is metric dependent. Under nonunitary gauge changes, either the metric must be transported with the coordinates or interventions must be matched by a coordinate-independent physical cost. Otherwise specificity can change merely because the representation was rescaled.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b402553fd89a7d6e","tex":"\\bigcup_{j=1}^m","aliases":"⋃_j=1^m bigcup","display":"\\bigcup_{j=1}^m","kind":"introduction-passage","excerpt":"[Intersection-gate false-certification control] Let a certificate require all gates G_1,…,G_m to pass. For each gate j, let H_0j denote the null that the required property is absent, and assume the predeclared test satisfies sup_P∈ H_0j P_P(G_j passes) ≤ α_j. The global null is H_0 = ⋃_j=1^m H_0j, meaning that at least one required property is abse","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7b08ee432850f5ec","tex":"+'","aliases":"+'","display":"+'","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3063c42c093674d1","tex":"\\mathbf1_{[0,\\varepsilon]}","aliases":"1_[0,ε] mathbf varepsilon","display":"\\mathbf1_{[0,\\varepsilon]}","kind":"use","excerpt":"The complete static-to-agentive construction is · (Ω, D, S,Σ, D_form, L) [2mm] · Downarrow ( D,widehat V,widehat E,widehat H,widehat P,widehat B) [2mm] · Downarrow (G,{X_i},{T_e}) [2mm] · Downarrow X_i=C_i⊕ R_i [2mm] · ({C_i},{H_e}, A,{ρ_i}) [2mm] · Downarrow Γ(H)=varprojlim C_i [2mm] · Downarrow L_H=D_H^*D_H [2mm] · Downarrow Γ_ε(H) = ran 1_[0,ε](","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8732dbfaefa9c782","tex":"\\mathbf1_{[0,\\varepsilon_t]}","aliases":"1_[0,ε_t] mathbf varepsilon","display":"\\mathbf1_{[0,\\varepsilon_t]}","kind":"use","excerpt":"Let c(t)=(c_i(t))_i∈ V ∈ C_V(t), P_t= 1_[0,ε_t](L_t). The instantaneous structural workspace coordinate is W_t^str=P_tc(t) ∈ Γ_ε_t(H(t)). When P_t varies with time, these coordinates inhabit a moving family of subspaces. Parts~X--XII will supply the transport required to compare them across time. In what follows, all sufficiency and intervention st","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4b82e0794ad5cb35","tex":"\\mathfrak A","aliases":"A mathfrak","display":"\\mathfrak A","kind":"use","excerpt":"A controlled relational realization extends Definition~[source reference] by the tuple A_Workspace = ( R, G, A_t, Y^+, {U_g}_g∈ G, Π, I_A, I_G, I_D, I_R ). Here: item R is the underlying relational realization; item G is a measurable goal-code space; item A_t is the feasible action set at time t; item Y^+ is the relevant future trajectory space; it","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-36e242206d1129a8","tex":"\\mathsf A","aliases":"A mathsf","display":"\\mathsf A","kind":"use","excerpt":"Sufficiency has a direct decision-theoretic interpretation. Use the convention TV(P,Q)=sup_A|P(A)-Q(A)|. Let A be a finite action set and ℓ: Y× A→[0,1] a bounded loss. Define Risk_ℓ(S_t) = inf_d E [ ℓ(Y_t^+,d(S_t)) ].","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"Write A=ρ_V(a), A'=ρ_V'(a), and define δ_ρ(a)= A'J_V-J_VA , ℓ_a= (I-P)AP . Let A_ W=PAP, A_ W''=P'A'P'.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"Two goal codes are operationally equivalent, g∼_goalg', when they induce the same action ordering in every declared state and intervention background: Q_t^g(amid w) ≥ Q_t^g(a'mid w) Longleftrightarrow Q_t^g'(amid w) ≥ Q_t^g'(a'mid w). The abstract goal is the equivalence class [g]_goal. This quotient prevents a mere invertible recoding of a goal ch","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c08d2cf77d38259","tex":"a^*","aliases":"a^*","display":"a^*","kind":"use","excerpt":"r widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal policy π^*, E_a∼π^*[Q(a)] - E_a∼widehatπ[Q(a)] ≤ 2ε_Q+δ_opt. For a deterministic greedy action widehat a∈argmax_awidehat Q(a), Q(a^*)-Q(widehat a) ≤2ε_Q. If the true optimal action a^* is unique and its value gap Δ_Q = Q(a^*)- max_a≠ a^*Q","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a12b4032efa2b919","tex":"A^0","aliases":"A^0","display":"A^0","kind":"use","excerpt":"Suppose local operations A_α:mathscr W_α→mathscr W_α and overlap operations A_αβ:mathscr W_αβ→ mathscr W_αβ satisfy r_αβ^αA_α = A_αβr_αβ^α, r_αβ^βA_β = A_αβr_αβ^β. Then, with A^0=⊕_α A_α and A^1=⊕_α<βA_αβ, B_ UA^0=A^1B_ U. Therefore A^0ker B_ U⊆ker B_ U. The operation-closure theorem from Part~III reappears at the meta level: operations on subworks","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-994ce857f7c17ab3","tex":"A^1","aliases":"A^1","display":"A^1","kind":"use","excerpt":"Suppose local operations A_α:mathscr W_α→mathscr W_α and overlap operations A_αβ:mathscr W_αβ→ mathscr W_αβ satisfy r_αβ^αA_α = A_αβr_αβ^α, r_αβ^βA_β = A_αβr_αβ^β. Then, with A^0=⊕_α A_α and A^1=⊕_α<βA_αβ, B_ UA^0=A^1B_ U. Therefore A^0ker B_ U⊆ker B_ U. The operation-closure theorem from Part~III reappears at the meta level: operations on subworks","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"Let C_1=C_2= R, use unit weights, and take two directed transfers H_12=a, H_21=b. Then D_H = -a · 1 1 · -b , L_H = a^2+1 · -(a+b) -(a+b) · b^2+1 . Writing τ=a^2+b^2+2, the eigenvalues are λ_± = frac12 ( τ±sqrtτ^2-4(ab-1)^2 ), and det L_H=(ab-1)^2. Thus a nonzero exact shared core exists precisely when ab=1, which is exactly the trivial-holonomy con","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8b1f15e8a65f75f5","tex":"A_{\\mathcal W}","aliases":"A_ W mathcal","display":"A_{\\mathcal W}","kind":"use","excerpt":"Write A=ρ_V(a), A'=ρ_V'(a), and define δ_ρ(a)= A'J_V-J_VA , ℓ_a= (I-P)AP . Let A_ W=PAP, A_ W''=P'A'P'.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2244a6bee7a1ddc7","tex":"A_{\\mathcal W'}'","aliases":"A_ W'' mathcal","display":"A_{\\mathcal W'}'","kind":"use","excerpt":"Write A=ρ_V(a), A'=ρ_V'(a), and define δ_ρ(a)= A'J_V-J_VA , ℓ_a= (I-P)AP . Let A_ W=PAP, A_ W''=P'A'P'.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4880eb8d6002c355","tex":"\\mathcal A_1","aliases":"A_1 mathcal","display":"\\mathcal A_1","kind":"use","excerpt":"For an admissible correspondence J, define the operation defect on a declared unit ball A_1 by δ_op( J) = sup_a∈ A_1 A_ W''β-β A_ W . The primary certification object is the identity profile d_id( J) = ( q_str, δ_op( J), δ_causal(β), δ_rec(β); μ^keep ). The first four coordinates must remain below declared tolerances, while the non-collapse margin ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6530dbbb96180946","tex":"a_C","aliases":"a_C","display":"a_C","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5b84eb8a8ab33211","tex":"a_D","aliases":"a_D","display":"a_D","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-41ae46808180b866","tex":"a_e","aliases":"a_e","display":"a_e","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f99866921a0da4a5","tex":"a_I","aliases":"a_I","display":"a_I","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"Conversely, take B∈Comm(Hol_o) and define A_i=H_τ_iBH_τ_i^-1. This is independent of the chosen root path: changing τ_i multiplies it by a loop holonomy that commutes with B. For an edge e:i→ j, the operator H_τ_j^-1H_eH_τ_i is a root holonomy. Commutation with B rearranges to A_jH_e=H_eA_i. Hence (A_i) is natural and Φ_o((A_i))=B. Thus Φ_o is surj","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9dd023cef29cb3f9","tex":"a_K","aliases":"a_K","display":"a_K","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c558af70c2e8eb08","tex":"a_L","aliases":"a_L","display":"a_L","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-96326a0acc77e20a","tex":"A_o","aliases":"A_o","display":"A_o","kind":"use","excerpt":"[Commutant characterization] Let |G| be connected and let H be an invertible local system. Root evaluation induces an algebra isomorphism Φ_o: O_Workspace xrightarrow ≅ Comm(Hol_o), Φ_o((A_i))=A_o, where Comm(Hol_o) = {B∈End(C_o):Bh=hB ∀ h∈Hol_o}.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5c2966392a5248c3","tex":"a_O","aliases":"a_O","display":"a_O","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-580ffa2dbf3c98ca","tex":"A_P","aliases":"A_P","display":"A_P","kind":"use","excerpt":"On ran P and ranwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7bf6c5a5a0ecb9f2","tex":"a_R","aliases":"a_R","display":"a_R","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f78009c8b1487c7d","tex":"A_S","aliases":"A_S","display":"A_S","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":21},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e8317bafa0d82a29","tex":"A_S'","aliases":"A_S'","display":"A_S'","kind":"use","excerpt":"[Preservation of coalition frame bounds] Identify the old and new abstract cores with ran P and ran P' through their chosen core parameterizations. Let S be a declared coalition, let A_S:ran P→ Y_S, A_S':ran P'→ Y_S' be the resulting accessible coalition encoders, and let K_S be a unitary identification of their observation spaces. Suppose A_S'β-K_","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-558114fbe63976ca","tex":"A_S^*","aliases":"A_S^*","display":"A_S^*","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b0fdf5db672cb3a7","tex":"A_S^\\dagger","aliases":"A_S^† dagger","display":"A_S^\\dagger","kind":"use","excerpt":"[Coalition reconstruction criterion] For a coalition S, the following are equivalent: item A_S is injective; item F_S is positive definite; item α_S>0; item there exists a linear decoder R_S with R_SA_S=I_C_*. When these conditions hold, the canonical decoder is R_S=A_S^†=F_S^-1A_S^*.","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-37b59cf4c71406cd","tex":"a_T","aliases":"a_T","display":"a_T","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"After a possible damage event, let the pre-repair health be h_t^-∈{0,1}. The system chooses a joint action u_t=(a_t,r_t)∈{0,1}^2, where a_t is a task action and r_t is a repair action. Define the optimal task action a_t^*=g_t⊕ b_t⊕ h_t^- and optimal repair action r_t^*=1-h_t^-. No individual node determines the full pair (a_t^*,r_t^*): item the pla","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0994af505d69ad2f","tex":"A_{t+\\tau}","aliases":"A_t+τ tau","display":"A_{t+\\tau}","kind":"use","excerpt":"τ^0. For goal interventions ι_g and ι_g', define the cue-controlled retention effect η_G(τ;g,g') = D_G ( P ( G_t+τ∈· mid ι_g, do(C_t+1:t+τ=c^0) ), P ( G_t+τ∈· mid ι_g', do(C_t+1:t+τ=c^0) ) ). The corresponding action-retention effect is η_G→ A(τ;g,g') = D_A ( P(A_t+τ∈·midι_g,c^0), P(A_t+τ∈·midι_g',c^0) ). The first quantity tests whether a goal dif","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5ae46470a0407e9c","tex":"a_t^*","aliases":"a_t^*","display":"a_t^*","kind":"use","excerpt":"After a possible damage event, let the pre-repair health be h_t^-∈{0,1}. The system chooses a joint action u_t=(a_t,r_t)∈{0,1}^2, where a_t is a task action and r_t is a repair action. Define the optimal task action a_t^*=g_t⊕ b_t⊕ h_t^- and optimal repair action r_t^*=1-h_t^-. No individual node determines the full pair (a_t^*,r_t^*): item the pla","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c37213136acbc65","tex":"A_{V\\setminus A}","aliases":"A_V∖ A setminus","display":"A_{V\\setminus A}","kind":"use","excerpt":"[Uniform erasure resilience] The encoding family is exactly reconstructible after every loss of at most k nodes if and only if α^(k)>0. If surviving observations have the form y_V∖ A=A_V∖ Ac+η_A, |A|≤ k, then the canonical survivor decoder obeys the uniform estimate widehat c_A-c ≤ frac η_A sqrtα^(k).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d0ef5ec187a14151","tex":"\\mathfrak A_{\\Workspace}","aliases":"A_Workspace mathfrak Workspace","display":null,"kind":"use","excerpt":"A controlled relational realization extends Definition~[source reference] by the tuple A_Workspace = ( R, G, A_t, Y^+, {U_g}_g∈ G, Π, I_A, I_G, I_D, I_R ). Here: item R is the underlying relational realization; item G is a measurable goal-code space; item A_t is the feasible action set at time t; item Y^+ is the relevant future trajectory space; it","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-08a584d38018c160","tex":"a_Z","aliases":"a_Z","display":"a_Z","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-edac8f4090c64b8b","tex":"A_\\alpha","aliases":"A_α alpha","display":"A_\\alpha","kind":"use","excerpt":"Suppose local operations A_α:mathscr W_α→mathscr W_α and overlap operations A_αβ:mathscr W_αβ→ mathscr W_αβ satisfy r_αβ^αA_α = A_αβr_αβ^α, r_αβ^βA_β = A_αβr_αβ^β. Then, with A^0=⊕_α A_α and A^1=⊕_α<βA_αβ, B_ UA^0=A^1B_ U. Therefore A^0ker B_ U⊆ker B_ U. The operation-closure theorem from Part~III reappears at the meta level: operations on subworks","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c0f1ef682fe84650","tex":"A_{\\alpha\\beta}","aliases":"A_αβ alpha beta","display":"A_{\\alpha\\beta}","kind":"use","excerpt":"Suppose local operations A_α:mathscr W_α→mathscr W_α and overlap operations A_αβ:mathscr W_αβ→ mathscr W_αβ satisfy r_αβ^αA_α = A_αβr_αβ^α, r_αβ^βA_β = A_αβr_αβ^β. Then, with A^0=⊕_α A_α and A^1=⊕_α<βA_αβ, B_ UA^0=A^1B_ U. Therefore A^0ker B_ U⊆ker B_ U. The operation-closure theorem from Part~III reappears at the meta level: operations on subworks","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5b817d1c8421c905","tex":"A_\\beta","aliases":"A_β beta","display":"A_\\beta","kind":"use","excerpt":"Suppose local operations A_α:mathscr W_α→mathscr W_α and overlap operations A_αβ:mathscr W_αβ→ mathscr W_αβ satisfy r_αβ^αA_α = A_αβr_αβ^α, r_αβ^βA_β = A_αβr_αβ^β. Then, with A^0=⊕_α A_α and A^1=⊕_α<βA_αβ, B_ UA^0=A^1B_ U. Therefore A^0ker B_ U⊆ker B_ U. The operation-closure theorem from Part~III reappears at the meta level: operations on subworks","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-332baf0cdd8ca108","tex":"\\mathsf{actions}","aliases":"actions mathsf","display":"\\mathsf{actions}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6eb48f2da2a707ea","tex":"\\operatorname{AdaptivelyParameterised}","aliases":"AdaptivelyParameterised operatorname","display":"\\operatorname{AdaptivelyParameterised}","kind":"use","excerpt":"subtopicOrthogonal formation-provenance axis Scaffolded ⟶ AdaptivelyParameterised ⟶ StructurallyAdaptive ⟶ EndogenouslyConstituted. This axis is independent of Levels~0--6. A scaffolded system can instantiate a high-level relational workspace, while an endogenously learned organization can fail every operational workspace gate.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-45d4500122db1f19","tex":"\\addcontentsline","aliases":"addcontentsline","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-08ef384d624d4774","tex":"\\mathsf{ADP}","aliases":"ADP mathsf","display":"\\mathsf{ADP}","kind":"use","excerpt":"The audit uses five labels: ll STD · = established concept, theorem, or technique, ADP · = established machinery adapted to relational workspaces, SYN · = nontrivial synthesis of established components, CAN · = candidate original contribution requiring priority review, EMP · = empirical claim not yet established in an uncontrolled system. These lab","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-70d6f62f762de210","tex":"\\operatorname{Agent}","aliases":"Agent operatorname","display":"\\operatorname{Agent}","kind":"use","excerpt":"by fracoverline d+overlineϵ1-β; item no single-node statistic is sufficient for the complete control state; item matched local policy nulls cannot reproduce the intact control value; item goal-dependent action selection and workspace repair are causally mediated by implemented internal channels. Hence the realization satisfies the formal predicate ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d6a8e5e2bccbc48","tex":"\\operatorname{Agent}_{\\Theta_A}","aliases":"Agent_Θ_A operatorname Theta","display":"\\operatorname{Agent}_{\\Theta_A}","kind":"use","excerpt":"by fracoverline d+overlineϵ1-β; item no single-node statistic is sufficient for the complete control state; item matched local policy nulls cannot reproduce the intact control value; item goal-dependent action selection and workspace repair are causally mediated by implemented internal channels. Hence the realization satisfies the formal predicate ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7eca8a2e51079f53","tex":"\\mathbf{Alg}","aliases":"Alg mathbf","display":"\\mathbf{Alg}","kind":"use","excerpt":"The finite-dimensional Hilbert assumption supplies adjoints and spectral tools. The exact compatible-section definitions require less and may later be formulated in categories such as Ban, Meas, Prob, Top, Alg.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-525be5b6e176b361","tex":"\\mathrm{align}","aliases":"align mathrm","display":"\\mathrm{align}","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9f82e623dbc870b5","tex":"\\mathrm{analytic}","aliases":"analytic mathrm","display":"\\mathrm{analytic}","kind":"use","excerpt":"Inside the declared finite model, S · =checkmark_analytic, · O · =checkmark_analytic, R · =checkmark_analytic, · P · =checkmark_analytic, F · =scaffolded, · A · =not tested, D · =fixed decomposition, · C · =not established. The highest supported level is Level~4 when the handoff dynamics and identity correspondence are included. The example does no","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-87708a8e37000af2","tex":"\\operatorname{Ann}","aliases":"Ann operatorname","display":"\\operatorname{Ann}","kind":"use","excerpt":"Some natural operators may vanish on every global section while acting nontrivially on locally available but globally inconsistent directions. Define the annihilator Ann(Γ(H)) = {A∈ O_Workspace: π_Workspace(A)=0}. The effective workspace algebra is O_Workspace^eff = O_Workspace/Ann(Γ(H)). Its action on Γ(H) is faithful by construction.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-064d19b2d728284c","tex":"\\argmax_a","aliases":"argmax_a argmax","display":null,"kind":"use","excerpt":") is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal policy π^*, E_a∼π^*[Q(a)] - E_a∼widehatπ[Q(a)] ≤ 2ε_Q+δ_opt. For a deterministic greedy action widehat a∈argmax_awidehat Q(a), Q(a^*)-Q(widehat a) ≤2ε_Q. If the true optimal action a^* is unique and its value gap Δ_Q ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3022a52c53b7386b","tex":"\\argmax_r","aliases":"argmax_r argmax","display":null,"kind":"use","excerpt":"[Repair selection under value-model error] Fix a damage intervention d and a finite feasible repair set R. Let V(r)=V_t^rep(r;d) be the true expected post-repair integrity margin and let widehat V obey sup_r∈ R|widehat V(r)-V(r)| ≤ ε_R. Let widehat r∈argmax_rwidehat V(r) and r^*∈argmax_r V(r). Then V(r^*)-V(widehat r) ≤ 2ε_R. If V(r^*) ≥ ρ+2ε_R, th","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dfb8c1d7f3b5f52a","tex":"\\argmax_{S\\in\\mathfrak S_{\\mathrm{pass}}}","aliases":"argmax_S∈ S_pass argmax in mathfrak mathrm","display":null,"kind":"use","excerpt":"· -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass contains only candidates that satisfy the required workspace and anti-centralization gates.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b779fb94202ff28e","tex":"\\argmin","aliases":"argmin","display":null,"kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a5542d60b088941a","tex":"\\argmin_{z,a,H,P,\\mathcal O,B}","aliases":"argmin_z,a,H,P, O,B argmin mathcal","display":null,"kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b940458028f8d4e0","tex":"\\argmin_{\\theta\\in\\mathfrak T}","aliases":"argmin_θ∈ T argmin theta in mathfrak","display":null,"kind":"use","excerpt":"[Uniform structure-selection recovery] Let T be a finite candidate family with population score J: T→ R. Let θ_* = argmin_θ∈ TJ(θ) be unique, with margin Δ_J = min_θ≠θ_* (J(θ)-J(θ_*))>0. If a held-out estimate widehat J satisfies sup_θ∈ T |widehat J(θ)-J(θ)| ≤ε, then every empirical minimizer widehatθ obeys J(widehatθ)-J(θ_*) ≤2ε. If 2ε<Δ_J, then w","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-733f6ad16595eaf4","tex":"\\operatorname{artanh}","aliases":"artanh operatorname","display":"\\operatorname{artanh}","kind":"use","excerpt":"A fixed point satisfies artanh(m)=λ m. For m>0, the function h(m)=fracartanh(m)m is strictly increasing from 1 to +∞. Hence there is no positive solution when λ≤1, and exactly one positive solution when λ>1. Odd symmetry gives the negative solution.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5d154d4165fdaff6","tex":"\\mathsf{artifacts}","aliases":"artifacts mathsf","display":"\\mathsf{artifacts}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f673c03c77f9ac50","tex":"B^*","aliases":"B^*","display":"B^*","kind":"use","excerpt":"eorem[Approximate structural replacement] If η_L<frac g2, then q_str := P'-J_VPJ_V^* ≤ fracη_Lg-η_L ≤ frac2η_Lg. If q_str<1, then B=P'J_VP:ran P⟶ran P' is bijective and satisfies σ_min(B) ≥ sqrt1-q_str^2. Its polar factor β =B(B^*B)^-1/2 is a canonical unitary workspace correspondence. Moreover, β-J_VP ≤ q_str + 1-sqrt1-q_str^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"Let C_1=C_2= R, use unit weights, and take two directed transfers H_12=a, H_21=b. Then D_H = -a · 1 1 · -b , L_H = a^2+1 · -(a+b) -(a+b) · b^2+1 . Writing τ=a^2+b^2+2, the eigenvalues are λ_± = frac12 ( τ±sqrtτ^2-4(ab-1)^2 ), and det L_H=(ab-1)^2. Thus a nonzero exact shared core exists precisely when ab=1, which is exactly the trivial-holonomy con","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1083bd27a70aa169","tex":"B_{\\mathcal U}","aliases":"B_ U mathcal","display":"B_{\\mathcal U}","kind":"use","excerpt":"First assume each local state already lies in its exact local workspace. Define B_ U := δ^0: check C^0( U;mathscr W) ⟶ check C^1( U;mathscr W) and the meta-Laplacian L_ U^meta = B_ U^*B_ U. By Theorem~[source reference], ker L_ U^meta = R_ Umathscr W(G). Thus the gluing problem is itself a workspace problem whose ``nodes'' are local workspaces and ","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-75fea51d7ad22dca","tex":"B_{\\mathcal U}^*","aliases":"B_ U^* mathcal","display":"B_{\\mathcal U}^*","kind":"use","excerpt":"First assume each local state already lies in its exact local workspace. Define B_ U := δ^0: check C^0( U;mathscr W) ⟶ check C^1( U;mathscr W) and the meta-Laplacian L_ U^meta = B_ U^*B_ U. By Theorem~[source reference], ker L_ U^meta = R_ Umathscr W(G). Thus the gluing problem is itself a workspace problem whose ``nodes'' are local workspaces and ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fb9e4008ca2e6c3a","tex":"b_{12}","aliases":"b_12","display":"b_{12}","kind":"use","excerpt":"Consider three charts arranged in a cycle, with one-dimensional overlap spaces and oriented offsets b_12, b_23, b_31. Local potentials would have to satisfy q_2-q_1=b_12, q_3-q_2=b_23, q_1-q_3=b_31. Adding the equations gives the necessary and sufficient condition b_12+b_23+b_31=0. With the standard edge norm, the harmonic obstruction has magnitude","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f2cec4611ded7308","tex":"b_{23}","aliases":"b_23","display":"b_{23}","kind":"use","excerpt":"Consider three charts arranged in a cycle, with one-dimensional overlap spaces and oriented offsets b_12, b_23, b_31. Local potentials would have to satisfy q_2-q_1=b_12, q_3-q_2=b_23, q_1-q_3=b_31. Adding the equations gives the necessary and sufficient condition b_12+b_23+b_31=0. With the standard edge norm, the harmonic obstruction has magnitude","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b560aea36fa2e800","tex":"b_{31}","aliases":"b_31","display":"b_{31}","kind":"use","excerpt":"Consider three charts arranged in a cycle, with one-dimensional overlap spaces and oriented offsets b_12, b_23, b_31. Local potentials would have to satisfy q_2-q_1=b_12, q_3-q_2=b_23, q_1-q_3=b_31. Adding the equations gives the necessary and sufficient condition b_12+b_23+b_31=0. With the standard edge norm, the harmonic obstruction has magnitude","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-305ce1517344aaba","tex":"B_Q","aliases":"B_Q","display":"B_Q","kind":"use","excerpt":"On ran P and ranwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9c51d8969a601c9b","tex":"b_q","aliases":"b_q","display":"b_q","kind":"use","excerpt":"Fix a reference decomposition D_0. For each D∈ D, let Φ_ D:W^ D_0⟶ W^ D be a declared workspace correspondence and let δ_ D measure its structural and causal identity defect. Assume gate q has a transport bound | m_q(Φ_ DW; D) -m_q(W; D_0) | ≤ L_qδ_ D+b_q( D), where L_q is a sensitivity constant and b_q records model mismatch not captured by the id","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e6fa5733ed9c8cbf","tex":"B_q^{\\mathrm{model}}","aliases":"B_q^model mathrm","display":"B_q^{\\mathrm{model}}","kind":"use","excerpt":"In practice, the reference margin is reduced by three distinct uncertainty budgets: B_q^total = B_q^model +B_q^spec +B_q^sample. The first covers variation across observationally compatible realizations. The second covers variation across admissible specifications. The third covers finite-sample estimation. The sufficient condition widehat m_q(W;ς_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-afedfd363ed3dc0f","tex":"B_q^{\\mathrm{sample}}","aliases":"B_q^sample mathrm","display":"B_q^{\\mathrm{sample}}","kind":"use","excerpt":"In practice, the reference margin is reduced by three distinct uncertainty budgets: B_q^total = B_q^model +B_q^spec +B_q^sample. The first covers variation across observationally compatible realizations. The second covers variation across admissible specifications. The third covers finite-sample estimation. The sufficient condition widehat m_q(W;ς_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fa3afe9ebe16f9f5","tex":"B_q^{\\mathrm{spec}}","aliases":"B_q^spec mathrm","display":"B_q^{\\mathrm{spec}}","kind":"use","excerpt":"In practice, the reference margin is reduced by three distinct uncertainty budgets: B_q^total = B_q^model +B_q^spec +B_q^sample. The first covers variation across observationally compatible realizations. The second covers variation across admissible specifications. The third covers finite-sample estimation. The sufficient condition widehat m_q(W;ς_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b27ef65ca3061ed5","tex":"B_q^{\\mathrm{total}}","aliases":"B_q^total mathrm","display":"B_q^{\\mathrm{total}}","kind":"use","excerpt":"In practice, the reference margin is reduced by three distinct uncertainty budgets: B_q^total = B_q^model +B_q^spec +B_q^sample. The first covers variation across observationally compatible realizations. The second covers variation across admissible specifications. The third covers finite-sample estimation. The sufficient condition widehat m_q(W;ς_","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-026204dc005ea4b1","tex":"B_s","aliases":"B_s","display":"B_s","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5c7840d3a1a045ca","tex":"\\mathcal B_t","aliases":"B_t mathcal","display":"\\mathcal B_t","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b17ff042f6455b31","tex":"b_t","aliases":"b_t","display":"b_t","kind":"use","excerpt":"Let b_t∈[0,1] be a boundary-integrity score and let the reserve be ρ. Define e_t=(ρ-b_t)_+. Suppose uncorrected damage obeys e_t+1^(0) ≤ 1.08e_t+d_t, so the open-loop boundary is locally unstable. A repair policy with measured causal effect e_t+1^(0)-e_t+1 ≥ 0.25e_t-ϵ_t gives β=1.08-0.25=0.83. If overline d=0.01, overlineϵ=0.005, then limsup e_t ≤ ","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9bed1073a19c6dad","tex":"\\mathcal B_{t+1}","aliases":"B_t+1 mathcal","display":"\\mathcal B_{t+1}","kind":"use","excerpt":"[Boundary persistence under score drift] Assume S_t^* is the unique maximizer of B_t with margin Δ_t^B, and suppose sup_S∈ S_pass | B_t+1(S)- B_t(S)| ≤δ_t. If 2δ_t<Δ_t^B, then S_t+1^*=S_t^*. More generally, B_t+1(S_t+1^*) - B_t+1(S_t^*) ≤2δ_t.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7200ba04eadd3707","tex":"B_t^{\\mathrm{sys}}","aliases":"B_t^sys mathrm","display":"B_t^{\\mathrm{sys}}","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0,","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-acdcc36814b4e6bd","tex":"b_{\\alpha\\beta}","aliases":"b_αβ alpha beta","display":"b_{\\alpha\\beta}","kind":"use","excerpt":"A different problem arises when only relative transition data are supplied. Let b=(b_αβ)∈check C^1( U;mathscr W) be a prescribed family of overlap offsets. The triple-overlap consistency condition is δ^1b=0. One seeks local potentials q=(q_α)∈check C^0 satisfying δ^0q=b. Define check H^1( U;mathscr W) = frackerδ^1imδ^0. Then the affine gluing probl","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b490be250afb646d","tex":"\\mathcal B_{\\vartheta}","aliases":"B_ϑ mathcal vartheta","display":"\\mathcal B_{\\vartheta}","kind":"use","excerpt":"If the score depends on a parameter ϑ_t and satisfies sup_S | B_ϑ(S)- B_ϑ'(S)| ≤ L_Bd(ϑ,ϑ'), then the sufficient stability condition becomes 2L_Bd(ϑ_t,ϑ_t+1) < Δ_t^B.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-309e738fb75686e7","tex":"\\mathcal B_{\\vartheta'}","aliases":"B_ϑ' mathcal vartheta","display":"\\mathcal B_{\\vartheta'}","kind":"use","excerpt":"If the score depends on a parameter ϑ_t and satisfies sup_S | B_ϑ(S)- B_ϑ'(S)| ≤ L_Bd(ϑ,ϑ'), then the sufficient stability condition becomes 2L_Bd(ϑ_t,ϑ_t+1) < Δ_t^B.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cb34c3c4ba9593c1","tex":"\\mathbf{Ban}","aliases":"Ban mathbf","display":"\\mathbf{Ban}","kind":"use","excerpt":"The finite-dimensional Hilbert assumption supplies adjoints and spectral tools. The exact compatible-section definitions require less and may later be formulated in categories such as Ban, Meas, Prob, Top, Alg.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f2e44089747c2c2f","tex":"\\bar S","aliases":"bar S bar","display":"\\bar S","kind":"use","excerpt":"For S∈ S, define Ret_τ(S) = I ( X_S(t+τ);X_S(t) mid X_bar S(t),E_t ), Dep_τ(S) = I ( X_S(t+τ);X_bar S(t) mid X_S(t),E_t ), Out_τ(S) = I ( X_bar S(t+τ);X_S(t) mid X_bar S(t),E_t ). These quantities are observational. They can be nonzero because of common causes, selection, or feedback paths and therefore cannot alone establish a causal boundary.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1784da886e1aea15","tex":"\\bar v","aliases":"bar v bar","display":"\\bar v","kind":"use","excerpt":"[Mean-square persistence] Suppose E[e_t+1^2mid F_t] ≤ q e_t^2+v_t, 0≤ q<1, and E[v_t]≤bar v. Then E[e_t^2] ≤ q^t E[e_0^2] + fracbar v(1-q^t)1-q, so limsup_t→∞ E[e_t^2] ≤ fracbar v1-q.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2e68e5f0cf8f956f","tex":"\\bar\\alpha","aliases":"barα bar alpha","display":"\\bar\\alpha","kind":"use","excerpt":"· + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-04de6740090d8b99","tex":"\\bar\\alpha^t","aliases":"barα^t bar alpha","display":"\\bar\\alpha^t","kind":"use","excerpt":"· + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bcc291407f308417","tex":"\\bar\\sigma","aliases":"barσ bar sigma","display":"\\bar\\sigma","kind":"use","excerpt":"· + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a9b1975bf6dbae6f","tex":"\\mathrm{bench}","aliases":"bench mathrm","display":"\\mathrm{bench}","kind":"use","excerpt":"Let two realizations have identical benchmark output distributions, P(Y_bench)=P'(Y_bench), but different intervention kernels: K(w,b)≠ K'(β w,b) for some matched workspace intervention. Then task behavior is preserved while d_causal(W,W')>0. The later system may be a successor or replacement rather than the same workspace under the identity criter","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-094580d2e9af605f","tex":"\\operatorname{Bernoulli}","aliases":"Bernoulli operatorname","display":"\\operatorname{Bernoulli}","kind":"use","excerpt":"[XOR witness pair] Under the intact law P, let X_1,X_2oversetiid∼Bernoulli(1/2), Y=X_1⊕ X_2. Then each (X_i,Y) pair is independent and uniform, so H_P(Ymid X_i)=1 bit, H_P(Ymid X_1,X_2)=0. Let Q make X_1,X_2,Y mutually independent fair bits. Then Q_(X_i,Y)=P_(X_i,Y) (i=1,2), but H_Q(Ymid X_1,X_2)=1 bit. Therefore Δ_joint^log=1 bit, k_0=2, Δ_tr(Q)=1","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e76ca40698ecc64e","tex":"\\bfseries","aliases":"bfseries","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-15946b326a027961","tex":"\\bigvee_{B\\in S}","aliases":"bigvee_B∈ S bigvee in","display":"\\bigvee_{B\\in S}","kind":"use","excerpt":"Information faithfulness gives mathscr F_S^ D_c = bigvee_B∈ S bigvee_i∈ q^-1(B) mathscr F_i^ D_f = mathscr F_q^-1(S)^ D_f. The two Bayes risks optimize over the same measurable decision rules and are therefore equal. If a fine coalition of size k is sufficient within ε, the set of coarse blocks meeting that coalition has at most k elements and cont","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-51f2cc9a5728c853","tex":"\\bigvee_{i\\in S}","aliases":"bigvee_i∈ S bigvee in","display":"\\bigvee_{i\\in S}","kind":"use","excerpt":"The structural core can be preserved by faithful coarse-graining, while coalition counts generally are not. Let mathscr F_S^ D = bigvee_i∈ Smathscr F_i^ D be the information available to coalition S under decomposition D. For a declared loss ℓ, write Risk_ D(S) = inf_f E [ ℓ(Y^+,f(mathscr F_S^ D)) ].","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b9d81ad445c6ebbb","tex":"\\bigwedge_{j\\in\\mathcal J(L,\\varsigma)}","aliases":"bigwedge_j∈ J(L,ς) bigwedge in mathcal varsigma","display":"\\bigwedge_{j\\in\\mathcal J(L,\\varsigma)}","kind":"use","excerpt":"Let J(L) be the required gate set for claim level L. Then Certify_L(W;ς) = bigwedge_j∈ J(L,ς) Pass_j(W;ς). For a workspace-agent claim, the required set includes every structural, operational, relational, persistent, goal, counterfactual-control, and self-maintenance gate declared in Parts~IX and XVI. A specification-robust claim additionally requi","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5bb28d7c6728c2ad","tex":"\\mathrm{boundary}","aliases":"boundary mathrm","display":"\\mathrm{boundary}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-15f7ee30a80f62e1","tex":"\\operatorname{BoundaryDamage}","aliases":"BoundaryDamage operatorname","display":"\\operatorname{BoundaryDamage}","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0, · Form_Θ_form(W","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d4bd201d998ce26","tex":"\\operatorname{BoundaryDamage}_t","aliases":"BoundaryDamage_t operatorname","display":"\\operatorname{BoundaryDamage}_t","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0, · Form_Θ_form(W","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8e3c0fce1b7a60ae","tex":"\\bullet","aliases":"bullet","display":"\\bullet","kind":"use","excerpt":"A practical hierarchy rarely transmits each complete local workspace. Let Q^p:check C^p( U;mathscr W)→overline C^p be compression maps into smaller local and overlap representations. Suppose there is a compressed coboundary overlineδ^p such that Q^p+1δ^p = overlineδ^p Q^p. The maps Q^bullet then form a cochain map.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-420668a5bdd9e0bf","tex":"\\mathbf C","aliases":"C mathbf","display":"\\mathbf C","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-52f862c0486a95ce","tex":"\\mathsf C","aliases":"C mathsf","display":"\\mathsf C","kind":"use","excerpt":"For each case we report the vector Card( K) = ( S, O, R, P, F, A, D, X, C ), where: ll S · = structural core and spectral coherence,","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ed86b18d690756fa","tex":"\\mathbf C'","aliases":"C' mathbf","display":"\\mathbf C'","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e7e42689204088d5","tex":"C^0","aliases":"C^0","display":"C^0","kind":"use","excerpt":"A different problem arises when only relative transition data are supplied. Let b=(b_αβ)∈check C^1( U;mathscr W) be a prescribed family of overlap offsets. The triple-overlap consistency condition is δ^1b=0. One seeks local potentials q=(q_α)∈check C^0 satisfying δ^0q=b. Define check H^1( U;mathscr W) = frackerδ^1imδ^0. Then the affine gluing probl","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bb4b39a0fa3e20d1","tex":"C^1","aliases":"C^1","display":"C^1","kind":"use","excerpt":"A different problem arises when only relative transition data are supplied. Let b=(b_αβ)∈check C^1( U;mathscr W) be a prescribed family of overlap offsets. The triple-overlap consistency condition is δ^1b=0. One seeks local potentials q=(q_α)∈check C^0 satisfying δ^0q=b. Define check H^1( U;mathscr W) = frackerδ^1imδ^0. Then the affine gluing probl","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d3cf32b34096d64b","tex":"C^p","aliases":"C^p","display":"C^p","kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fffa321f31db7eb2","tex":"C^{p+1}","aliases":"C^p+1","display":"C^{p+1}","kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-afc067aa288fa509","tex":"C_{\\mathfrak R}","aliases":"C_ R mathfrak","display":"C_{\\mathfrak R}","kind":"use","excerpt":"in M satisfy R≡_ E R' but Cert(W_ R;ς) ≠ Cert(W_ R';ς). Then the workspace certificate is not identifiable on M under E. More strongly, for every possibly randomized decision rule widehat C based only on data generated by E, P_ R ( widehat C≠ C_ R ) + P_ R' ( widehat C≠ C_ R' ) =1, where C_ R=Cert(W_ R;ς). Consequently, inf_widehat C max{ P_ R(wide","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-69e22c1b3e02b705","tex":"C_{\\mathfrak R'}","aliases":"C_ R' mathfrak","display":"C_{\\mathfrak R'}","kind":"use","excerpt":"in M satisfy R≡_ E R' but Cert(W_ R;ς) ≠ Cert(W_ R';ς). Then the workspace certificate is not identifiable on M under E. More strongly, for every possibly randomized decision rule widehat C based only on data generated by E, P_ R ( widehat C≠ C_ R ) + P_ R' ( widehat C≠ C_ R' ) =1, where C_ R=Cert(W_ R;ς). Consequently, inf_widehat C max{ P_ R(wide","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b6a0e01f5d7d0367","tex":"C_*","aliases":"C_*","display":"C_*","kind":"use","excerpt":"The obstruction associated with a loop is Ω_γ=H_γ-I_C_o. The maximal path-independent root core is C_* = ⋂_γ:o→ oker(H_γ-I). Equivalently, C_*=C_o^Hol_o=Fix(Hol_o).","count":24},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3ea713ccba207d76","tex":"C_*^g","aliases":"C_*^g","display":"C_*^g","kind":"use","excerpt":"[Exact gauge covariance] The map G_V:Γ(H)⟶Γ(H^g) is an isomorphism. Therefore, Γ(H^g)=G_VΓ(H), dimΓ(H^g)=dimΓ(H). If the core transports are invertible and o is a root, then H_γ^g=g_oH_γ g_o^-1 for every loop γ:o→ o. Hence the holonomy-fixed core transforms by C_*^g=g_oC_*.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"The notation absorbs lower-bounded and upper-bounded gates into one sign convention. If a causal effect must exceed c_0, then m_q=widehatη-c_0. If a defect must remain below ε_0, then m_q=ε_0-widehatδ. The complete certificate therefore passes exactly when every required signed margin is positive.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"[Equivalent operations in different coordinates] Let C_1=C_2= R^2, choose S∈ GL_2( R), and set H_12=S, ρ_2(a)=Sρ_1(a)S^-1. Then H_12ρ_1(a) =Sρ_1(a) =ρ_2(a)S =ρ_2(a)H_12. The local matrices can look different while realizing the same abstract operation. This is the elementary model of representation change used throughout the theory.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"Let C_*= R^2 and define scalar node encodings E_1(c_1,c_2)=c_1, E_2(c_1,c_2)=c_2, E_3(c_1,c_2)=c_1+c_2. Every singleton has rank one, so no node reconstructs the two-dimensional core. Every pair does reconstruct. For S={1,2}, F_S=I, α_S=1. For S={1,3} or S={2,3}, the frame eigenvalues are frac3±sqrt52. Therefore m_rec=2, k_erase=1, d_er=2, α^(1)=fr","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"[Equivalent operations in different coordinates] Let C_1=C_2= R^2, choose S∈ GL_2( R), and set H_12=S, ρ_2(a)=Sρ_1(a)S^-1. Then H_12ρ_1(a) =Sρ_1(a) =ρ_2(a)S =ρ_2(a)H_12. The local matrices can look different while realizing the same abstract operation. This is the elementary model of representation change used throughout the theory.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"Let C_*= R^2 and define scalar node encodings E_1(c_1,c_2)=c_1, E_2(c_1,c_2)=c_2, E_3(c_1,c_2)=c_1+c_2. Every singleton has rank one, so no node reconstructs the two-dimensional core. Every pair does reconstruct. For S={1,2}, F_S=I, α_S=1. For S={1,3} or S={2,3}, the frame eigenvalues are frac3±sqrt52. Therefore m_rec=2, k_erase=1, d_er=2, α^(1)=fr","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"[One-dimensional core surviving a contradictory cycle] Let a three-node cycle have C_1=C_2=C_3= R^2 and H_12=I, H_23=I, H_31=J, J= 1 · 0","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b60e443e8108308c","tex":"c_3","aliases":"c_3","display":"c_3","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1),","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0f67e12878e28280","tex":"C_a","aliases":"C_a","display":"C_a","kind":"use","excerpt":"Consider three strata: h=human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. Th","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"use","excerpt":"Consider three strata: h=human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. Th","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bd50272046bff20c","tex":"\\mathcal C_{\\mathrm{dyn}}","aliases":"C_dyn mathcal mathrm","display":"\\mathcal C_{\\mathrm{dyn}}","kind":"use","excerpt":"The resulting certification report is C_dyn(W_t) =( · δ_align^ I, {Δ_A^ℓ}_A⊆ V, k_ε, Δ_tr( Q), · η_causal^ν, Δ_specific(r), η_rec(τ), η_loop(τ) ). No single coordinate certifies a relational workspace.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-836c5812320b60db","tex":"\\mathcal C_{\\mathrm{full}}","aliases":"C_full mathcal mathrm","display":"\\mathcal C_{\\mathrm{full}}","kind":"use","excerpt":"Use a vector rather than one aggregate score: C_full(W) =( · d_ε, g_ε, δ_coh, δ_align^ I, Δ_joint^ℓ, k_ε, · Δ_tr( Q), η_causal^ν, Δ_specific(r), η_loop(τ), · d_id, Rec_k, underline m_ D, Δ_dec, E_form, δ^scaf, δ^leak, · Δ_J, Δ_t^B, COut_τ(widehat S), CIn_τ(widehat S), · η_G(τ), δ_G^cue(τ), η_G→ A, δ_Q, η_V→ A, η_A→ Y, · η_D→ M, η_M→ R, η_R→ v, Δ_re","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fd685bf246955178","tex":"C_h","aliases":"C_h","display":"C_h","kind":"use","excerpt":"Consider three strata: h=human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. Th","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e45b8627b92d5b56","tex":"c_h","aliases":"c_h","display":"c_h","kind":"use","excerpt":"Consider three strata: h=human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. Th","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-51b2ade2b4806d6c","tex":"c_h'","aliases":"c_h'","display":"c_h'","kind":"use","excerpt":"human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. The round-trip consistency ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c6b1460127e8bcbe","tex":"C_i'","aliases":"C_i'","display":"C_i'","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-37d96a5a789a854b","tex":"C_{i_m}","aliases":"C_i_m","display":"C_{i_m}","kind":"use","excerpt":"For every step, T_rι_i_r-1^C = ι_i_r^CH_r+ ι_i_r^RV_r. Telescope the sequence T_m⋯ T_r+1ι_i_r^C H_r⋯ H_1, r=0,1,…,m. The difference between consecutive terms is exactly the r-th summand in the displayed formula. Projection onto R_i_m or C_i_m, followed by submultiplicativity of the operator norm, gives the two bounds.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fd3505bdae53f8f5","tex":"\\mathcal C_{\\mathrm{ID}}","aliases":"C_ID mathcal mathrm","display":"\\mathcal C_{\\mathrm{ID}}","kind":"use","excerpt":"Define C_ID (P^ E;ς) = { Cert(W_ R;ς): R∈ R_ID } ⊆{0,1}. The certificate is point identified exactly when this set is a singleton. The marginal gate intervals provide simple sufficient conditions, but they need not be logically necessary because gate margins can be dependent across the identified set.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"independent of the chosen path. Indeed, if σ_i:o→ i is another path, then σ_i^-1τ_i:o→ o is a loop, and invariance of x gives H_σ_i^-1H_τ_ix=x. Multiplying by H_σ_i yields H_τ_ix=H_σ_ix. For any edge e:i→ j, the paths eτ_i and τ_j both run from o to j. Path independence therefore gives H_ec_i =H_eH_τ_ix =H_τ_jx =c_j. Thus (c_i) is a global compatib","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"For composable intertwiners H_e:C_i→ C_j and H_f:C_j→ C_k, H_fH_eρ_i(a) =H_fρ_j(a)H_e =ρ_k(a)H_fH_e. Induction over path length proves the claim.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"Consider three strata: h=human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. Th","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-646fe15eb212b5b3","tex":"c_m","aliases":"c_m","display":"c_m","kind":"use","excerpt":"Consider three strata: h=human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. Th","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4c79fafc541a02b5","tex":"C_o","aliases":"C_o","display":"C_o","kind":"use","excerpt":"Fix a root o. The evaluation map is ev_o:Γ(H)→ C_o, ev_o((c_i))=c_o.","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-276d0e055981b0e3","tex":"c_o","aliases":"c_o","display":"c_o","kind":"use","excerpt":"Fix a root o. The evaluation map is ev_o:Γ(H)→ C_o, ev_o((c_i))=c_o.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5116ac3ac85d36f8","tex":"C_o^{\\Hol_o}","aliases":"C_o^Hol_o Hol","display":null,"kind":"use","excerpt":"The obstruction associated with a loop is Ω_γ=H_γ-I_C_o. The maximal path-independent root core is C_* = ⋂_γ:o→ oker(H_γ-I). Equivalently, C_*=C_o^Hol_o=Fix(Hol_o).","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-acd7c250d8c7903c","tex":"C_o^{\\operatorname{Hol}_o}","aliases":"C_o^Hol_o operatorname","display":"C_o^{\\operatorname{Hol}_o}","kind":"use","excerpt":"subtopicT_1: Limit--holonomy theorem Γ(H)≅ C_o^Hol_o.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8d2834fa95c1ac27","tex":"C_{s(\\gamma)}","aliases":"C_s(γ) gamma","display":"C_{s(\\gamma)}","kind":"use","excerpt":"[Pathwise core preservation] Suppose V_e=0 for every edge in a directed path γ=e_m⋯ e_1. Then T_γ(C_s(γ))⊆ C_t(γ) and P_t(γ)T_γ P_s(γ) = H_e_m⋯ H_e_1.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f51ab62943e516e8","tex":"C_{s+1}","aliases":"C_s+1","display":"C_{s+1}","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5b511d4340e7028f","tex":"\\mathfrak C_{\\mathrm{SRW}}","aliases":"C_SRW mathfrak mathrm","display":"\\mathfrak C_{\\mathrm{SRW}}","kind":"use","excerpt":"The strongest current candidate is not a single equation. It is the following conjunctive object: C_SRW = ( c heterogeneous core--transfer diagram and residuals, exact or spectral shared-section state, operation and predictive alignment, feasible causal interventions, coalition irreducibility under matched local-predictive nulls, non-collapse, reco","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c99be13da294dc50","tex":"c_t","aliases":"c_t","display":"c_t","kind":"use","excerpt":"Consider three agents: V_MA={P,M,R}, called the planner, monitor, and recorder. The global workspace-control state is c_t=(g_t,b_t,h_t)∈ F_2^3, where: ll g_t=goal bit, · b_t=environment or belief bit,","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0fa6ce095893e5bb","tex":"C_{t(\\gamma)}","aliases":"C_t(γ) gamma","display":"C_{t(\\gamma)}","kind":"use","excerpt":"[Pathwise core preservation] Suppose V_e=0 for every edge in a directed path γ=e_m⋯ e_1. Then T_γ(C_s(γ))⊆ C_t(γ) and P_t(γ)T_γ P_s(γ) = H_e_m⋯ H_e_1.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-371527fd7396f28d","tex":"C_{t+1}","aliases":"C_t+1","display":"C_{t+1}","kind":"use","excerpt":"Let G_t+1=C_t+1, where C_t+1 is an external instruction repeated at every step. Goal and action can be strongly correlated, but after cutting the cue channel, I(G_t+τ;G_tmidcut(C))=0. The system tracks an external command. It has not established internal goal retention.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e69ace86d0fa3344","tex":"C_t^{\\mathrm{ctl}}","aliases":"C_t^ctl mathrm","display":"C_t^{\\mathrm{ctl}}","kind":"use","excerpt":"A controlled state contains at least C_t^ctl = ( W_t,G_t,M_t,B_t,E_t ), where G_t is the represented goal, M_t is a workspace-integrity monitor, B_t is the current boundary or routing state, and E_t is the declared external context. An action is generated by A_t ∼ π_t ( ·mid W_t,G_t,M_t,B_t,E_t ). The future controlled state obeys a time-unrolled s","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8c6e1de13bca3fc1","tex":"\\mathrm{cal}","aliases":"cal mathrm","display":"\\mathrm{cal}","kind":"use","excerpt":"The same sample must not be used both to select a workspace and to certify its coherence, causal efficacy, or relational irreducibility. The required split is D = D_form sqcup D_cal sqcup D_cert. The formation sample selects widehatθ. The calibration sample fixes thresholds, regularization strengths, and intervention magnitudes. The certification s","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4147e6f7dde04213","tex":"\\mathsf{CAN}","aliases":"CAN mathsf","display":"\\mathsf{CAN}","kind":"use","excerpt":"The audit uses five labels: ll STD · = established concept, theorem, or technique, ADP · = established machinery adapted to relational workspaces, SYN · = nontrivial synthesis of established components, CAN · = candidate original contribution requiring priority review, EMP · = empirical claim not yet established in an uncontrolled system. These lab","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2a346c74e451cc3e","tex":"\\mathsf{Card}","aliases":"Card mathsf","display":"\\mathsf{Card}","kind":"use","excerpt":"For each case we report the vector Card( K) = ( S, O, R, P, F, A, D, X, C ), where: ll S · = structural core and spectral coherence,","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-163f20bf8e408034","tex":"\\CE","aliases":"CE","display":null,"kind":"use","excerpt":"Let D be a divergence or metric on probability laws. For a matched background value b, define CE_Workspace(w,w';b) = D ( P^ι_w(Y_t^+∈·mid B_t=b), P^ι_w'(Y_t^+∈·mid B_t=b) ). For a declared comparison design ν over (w,w',b), define η_causal^ν = E_(w,w',b)∼ν [ CE_Workspace(w,w';b) ]. The maximum η_causal^max = sup_w,w',bCE_Workspace(w,w';b) may be us","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c1ef70f5f3d0dc7d","tex":"\\operatorname{CE}","aliases":"CE operatorname","display":"\\operatorname{CE}","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0, · Form_Θ_form(W","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-75e40c8262ecc656","tex":"\\CE_{\\Workspace}","aliases":"CE_Workspace CE Workspace","display":null,"kind":"use","excerpt":"Let D be a divergence or metric on probability laws. For a matched background value b, define CE_Workspace(w,w';b) = D ( P^ι_w(Y_t^+∈·mid B_t=b), P^ι_w'(Y_t^+∈·mid B_t=b) ). For a declared comparison design ν over (w,w',b), define η_causal^ν = E_(w,w',b)∼ν [ CE_Workspace(w,w';b) ]. The maximum η_causal^max = sup_w,w',bCE_Workspace(w,w';b) may be us","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0b5ef59b65d05a5c","tex":"\\CE_{\\Workspace'}","aliases":"CE_Workspace' CE Workspace","display":null,"kind":"use","excerpt":"[Coordinate invariance of intervention effects] Let g: W_t→ W_t' be a measurable bijection and suppose the reparametrized intervention family satisfies ι_g(w)'=ι_w. Then every divergence-based causal effect is preserved: CE_Workspace'(g(w),g(w');b) = CE_Workspace(w,w';b).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-12585c304e20b612","tex":"\\operatorname{Cert}_{\\mathfrak S}","aliases":"Cert_ S operatorname mathfrak","display":"\\operatorname{Cert}_{\\mathfrak S}","kind":"use","excerpt":"n Sm_q(W;ς), overline m_q^ S(W) = sup_ς∈ Sm_q(W;ς). Its sensitivity width is Δ_q^spec(W) = overline m_q^ S(W) - underline m_q^ S(W). The robust certificate margin is r_ S(W) = inf_ς∈ S min_q∈ J_*m_q(W;ς). Then Cert_ S(W)=1 Longleftrightarrow r_ S(W)>0.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82d63a8988f3dcf2","tex":"\\mathrm{certificate}","aliases":"certificate mathrm","display":"\\mathrm{certificate}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-56009bb3f7164f3a","tex":"\\operatorname{Certify}_L","aliases":"Certify_L operatorname","display":"\\operatorname{Certify}_L","kind":"use","excerpt":"Let J(L) be the required gate set for claim level L. Then Certify_L(W;ς) = bigwedge_j∈ J(L,ς) Pass_j(W;ς). For a workspace-agent claim, the required set includes every structural, operational, relational, persistent, goal, counterfactual-control, and self-maintenance gate declared in Parts~IX and XVI. A specification-robust claim additionally requi","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-eb7a38e5922ae0f3","tex":"\\operatorname{Ch}","aliases":"Ch operatorname","display":"\\operatorname{Ch}","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c9436aea1bf2444f","tex":"\\check","aliases":"check","display":null,"kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f674a93ddf89a480","tex":"\\checkmark","aliases":"checkmark","display":null,"kind":"use","excerpt":"Inside the declared finite model, S · =checkmark_analytic, · O · =checkmark_analytic, R · =checkmark_analytic, · P · =checkmark_analytic, F · =scaffolded, · A · =not tested, D · =fixed decomposition, · C · =not established. The highest supported level is Level~4 when the handoff dynamics and identity correspondence are included. The example does no","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-04b1d47bafaf05fc","tex":"\\checkmark_{\\mathrm{analytic}}","aliases":"checkmark_analytic checkmark mathrm","display":null,"kind":"use","excerpt":"Inside the declared finite model, S · =checkmark_analytic, · O · =checkmark_analytic, R · =checkmark_analytic, · P · =checkmark_analytic, F · =scaffolded, · A · =not tested, D · =fixed decomposition, · C · =not established. The highest supported level is Level~4 when the handoff dynamics and identity correspondence are included. The example does no","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-533aeab4b3e0de2c","tex":"\\CIn","aliases":"CIn","display":null,"kind":"use","excerpt":"nate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the future law of X_S: CIn_τ(S) = E_u,u'∼ν_bar S D ( L^ι_bar S,u (X_S(t+τ)), L^ι_bar ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-438f6ab3ed3a196b","tex":"\\CIn_\\tau","aliases":"CIn_τ CIn tau","display":null,"kind":"use","excerpt":"nate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the future law of X_S: CIn_τ(S) = E_u,u'∼ν_bar S D ( L^ι_bar S,u (X_S(t+τ)), L^ι_bar ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c3b73c2fb5435609","tex":"\\operatorname{Cl}","aliases":"Cl operatorname","display":"\\operatorname{Cl}","kind":"use","excerpt":"mined by mathematical and operational equivalence, not by terminology. Let C be a claim in the paper and S a prior source. Write Spreccurlyeq C when the main content of C follows from S after admissible relabeling, restriction, or direct specialization. A substantive novelty residual exists only when the claim contains content not absorbed by the c","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7df1bff400546d80","tex":"\\mathsf{claim}","aliases":"claim mathsf","display":"\\mathsf{claim}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-450d165bfeb0ca8d","tex":"\\clearpage","aliases":"clearpage","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-57d5f54bcad0aa19","tex":"\\mathrm{coh}","aliases":"coh mathrm","display":"\\mathrm{coh}","kind":"use","excerpt":"For c≠0, define the Rayleigh defect δ_coh(c) = frac E_H(c) c ^2. For a nonzero subspace W⊆ C_V, define δ_coh(W) = sup_substackc∈ W","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d91af84ebc144c69","tex":"\\mathsf{Coh}","aliases":"Coh mathsf","display":"\\mathsf{Coh}","kind":"use","excerpt":"For the induced subsystem on S, define Coh(S) = tr ( P_S,dL_SP_S,d ), where P_S,d projects onto the selected low-energy workspace modes. Let Cut(S) = ∑_e:s(e)∈ S, t(e)∉ Sw_e + ∑_e:s(e)∉ S, t(e)∈ Sw_e. Let k_ε(S) be the minimal sufficient coalition size inside the candidate subsystem. A candidate is admissible only if SRW_Θ_S(W_S)=1, k_ε(S)≥2, unles","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-271cce43b6af00f7","tex":"\\operatorname{Comm}","aliases":"Comm operatorname","display":"\\operatorname{Comm}","kind":"use","excerpt":"[Holonomy commutes with abstract operations] For every loop γ:o→ o, H_γρ_o(a)=ρ_o(a)H_γ. Hence ρ_o( A) ⊆ Comm(Hol_o), where Comm(Hol_o) is the commutant of the holonomy representation.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-425dfc9c64850b00","tex":"\\mathrm{concept}","aliases":"concept mathrm","display":"\\mathrm{concept}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"[Local compatibility without a global nonzero core] Let the only nontrivial cycle holonomy on R^2 be a rotation R_θ= cosθ · -sinθ sinθ · cosθ with θ∉2π Z. Then ker(R_θ-I)={0}. Every edge map is individually invertible and well-defined, yet the complete cycle admits no nonzero exact shared core. Pairwise translatability is therefore insufficient for","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-95243eedd9aae923","tex":"\\operatorname{Cost}","aliases":"Cost operatorname","display":"\\operatorname{Cost}","kind":"use","excerpt":"A composite identity cost is Cost( J) = a q_str +b δ_op( J) +c δ_causal(β) +d δ_rec(β), where a,b,c,d>0 are declared scales. Define d_id( R, R') = inf_ J Cost( J). This quantity is a task-relative optimization surrogate until the admissible correspondence class, intervention matching, divergence, and scales are fixed. It does not replace the coordi","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f50fb9d014112b0b","tex":"\\COut","aliases":"COut","display":null,"kind":"use","excerpt":"Let ι_S,x be a feasible intervention that changes the internal workspace coordinate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the fu","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2828a82f3d04c8a6","tex":"\\COut_\\tau","aliases":"COut_τ COut tau","display":null,"kind":"use","excerpt":"Let ι_S,x be a feasible intervention that changes the internal workspace coordinate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the fu","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e0385f32a56631d0","tex":"\\mathrm{ctl}","aliases":"ctl mathrm","display":"\\mathrm{ctl}","kind":"use","excerpt":"A controlled state contains at least C_t^ctl = ( W_t,G_t,M_t,B_t,E_t ), where G_t is the represented goal, M_t is a workspace-integrity monitor, B_t is the current boundary or routing state, and E_t is the declared external context. An action is generated by A_t ∼ π_t ( ·mid W_t,G_t,M_t,B_t,E_t ). The future controlled state obeys a time-unrolled s","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-41778850fe4d77a0","tex":"\\mathrm{cue}","aliases":"cue mathrm","display":"\\mathrm{cue}","kind":"use","excerpt":"Cue dependence must be reported separately. For matched initial goal g, define δ_G^cue(τ) = sup_c,c' D_G ( P(G_t+τ∈·midι_g,do(C=c)), P(G_t+τ∈·midι_g,do(C=c')) ). High cue dependence is not a failure when the claimed agent is intentionally instruction following. It does mean that persistence cannot be attributed to an autonomous internal objective w","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8c0cec3e8b48c1b3","tex":"\\mathsf{Cut}","aliases":"Cut mathsf","display":"\\mathsf{Cut}","kind":"use","excerpt":"For the induced subsystem on S, define Coh(S) = tr ( P_S,dL_SP_S,d ), where P_S,d projects onto the selected low-energy workspace modes. Let Cut(S) = ∑_e:s(e)∈ S, t(e)∉ Sw_e + ∑_e:s(e)∉ S, t(e)∈ Sw_e. Let k_ε(S) be the minimal sufficient coalition size inside the candidate subsystem. A candidate is admissible only if SRW_Θ_S(W_S)=1, k_ε(S)≥2, unles","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-87cc7ffbac0eb07a","tex":"\\operatorname{cut}","aliases":"cut operatorname","display":"\\operatorname{cut}","kind":"use","excerpt":"Let G_t+1=C_t+1, where C_t+1 is an external instruction repeated at every step. Goal and action can be strongly correlated, but after cutting the cue channel, I(G_t+τ;G_tmidcut(C))=0. The system tracks an external command. It has not established internal goal retention.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-eae54bb0ec4239ae","tex":"\\mathbf d","aliases":"d mathbf","display":"\\mathbf d","kind":"use","excerpt":"For an admissible correspondence J, define the operation defect on a declared unit ball A_1 by δ_op( J) = sup_a∈ A_1 A_ W''β-β A_ W . The primary certification object is the identity profile d_id( J) = ( q_str, δ_op( J), δ_causal(β), δ_rec(β); μ^keep ). The first four coordinates must remain below declared tolerances, while the non-collapse margin ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e63d808025fba917","tex":"\\mathrm d","aliases":"d mathrm","display":"\\mathrm d","kind":"use","excerpt":"For every bounded measurable f: Y→ R, define Z_t(f)=∫_ Yf(y) Z_t( dy). By construction, E[f(Y_t^+)mid H_t]=Z_t(f). Since Z_t is a measurable function of H_t, the tower property gives E[f(Y_t^+)mid Z_t] = E[Z_t(f)mid Z_t] = Z_t(f). Hence conditioning additionally on H_t does not change the conditional expectation, which is the stated conditional ind","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c2a814548eab63ec","tex":"\\mathcal D'","aliases":"D' mathcal","display":"\\mathcal D'","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0803287316bac781","tex":"D'^*","aliases":"D'^*","display":"D'^*","kind":"use","excerpt":"For every c, D'J_Vc=J_EDc. Because J_E is invertible, Dc=0 if and only if D'J_Vc=0, proving the kernel statement. Taking adjoints of the exact covariance identity gives D'^*J_E=J_VD^*. Therefore L'J_V=D'^*D'J_V=D'^*J_ED=J_VD^*D=J_VL. The spectral and operation claims follow by functional calculus and the intertwining assumptions.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7e8acadc439b10d5","tex":"D^*","aliases":"D^*","display":"D^*","kind":"use","excerpt":"For every c, D'J_Vc=J_EDc. Because J_E is invertible, Dc=0 if and only if D'J_Vc=0, proving the kernel statement. Taking adjoints of the exact covariance identity gives D'^*J_E=J_VD^*. Therefore L'J_V=D'^*D'J_V=D'^*J_ED=J_VD^*D=J_VL. The spectral and operation claims follow by functional calculus and the intertwining assumptions.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-16d921ce116914a4","tex":"d_{\\mathcal P}","aliases":"d_ P mathcal","display":"d_{\\mathcal P}","kind":"use","excerpt":"Let d_ P be a bounded metric generating the measurable structure on P( Y). Define the reconstruction defect of the canonical state from coalition A by δ_A^Z = inf_ψ_A E [ d_ P ( Z_t,ψ_A( H_t^A) ) ].","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-622d46880775834e","tex":"d_{\\mathfrak S}","aliases":"d_ S mathfrak","display":"d_{\\mathfrak S}","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ')","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1bac4959bf0f6da8","tex":"d_{\\mathcal W'}","aliases":"d_ W' mathcal","display":"d_{\\mathcal W'}","kind":"use","excerpt":"A reconstructive handoff may also be imperfect. Let δ_rec(β) = E[d_ W'(W'_init,β W)] measure replacement initialization error. Corollary~[source reference] provides one explicit upper bound on this term.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-19383ab252438954","tex":"\\mathcal D_0","aliases":"D_0 mathcal","display":"\\mathcal D_0","kind":"use","excerpt":"Fix a reference decomposition D_0. For each D∈ D, let Φ_ D:W^ D_0⟶ W^ D be a declared workspace correspondence and let δ_ D measure its structural and causal identity defect. Assume gate q has a transport bound | m_q(Φ_ DW; D) -m_q(W; D_0) | ≤ L_qδ_ D+b_q( D), where L_q is a sensitivity constant and b_q records model mismatch not captured by the id","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-63ed82a00b685c9b","tex":"D_A","aliases":"D_A","display":"D_A","kind":"use","excerpt":"Goal-dependent action mediation is measured by feasible interventions on the goal channel while matching the current workspace and external context. For a divergence D_A, define η_G→ A = sup_g,g',w,e D_A ( P(A_t∈·midι_g,w,e), P(A_t∈·midι_g',w,e) ). Workspace-dependent action mediation is η_W→ A = sup_w,w',g,e D_A ( P(A_t∈·midι_w,g,e), P(A_t∈·midι_w","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0ec57d816513bae2","tex":"d_A","aliases":"d_A","display":"d_A","kind":"use","excerpt":"Order the local coordinates as x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^ T. The overlap discrepancies are g_H-g_M, p_M-p_X, v_X-v_A, d_A-d_H. Thus D_HAI = 1 · 0 · -1 · 0 · 0 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dc3c7b8d473b63d1","tex":"D_{a,H}","aliases":"D_a,H","display":"D_{a,H}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5de99215ad954ec4","tex":"d_B","aliases":"d_B","display":"d_B","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0, · Form_Θ_form(W","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ad1be1bb8c3099f","tex":"\\mathcal D_{\\mathrm{cal}}","aliases":"D_cal mathcal mathrm","display":"\\mathcal D_{\\mathrm{cal}}","kind":"use","excerpt":"The same sample must not be used both to select a workspace and to certify its coherence, causal efficacy, or relational irreducibility. The required split is D = D_form sqcup D_cal sqcup D_cert. The formation sample selects widehatθ. The calibration sample fixes thresholds, regularization strengths, and intervention magnitudes. The certification s","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ae09b948099a2db1","tex":"d_{\\mathrm{causal}}","aliases":"d_causal mathrm","display":"d_{\\mathrm{causal}}","kind":"use","excerpt":"Let two realizations have identical benchmark output distributions, P(Y_bench)=P'(Y_bench), but different intervention kernels: K(w,b)≠ K'(β w,b) for some matched workspace intervention. Then task behavior is preserved while d_causal(W,W')>0. The later system may be a successor or replacement rather than the same workspace under the identity criter","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f9f7917f95b1e648","tex":"d_D","aliases":"d_D","display":"d_D","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1e284bd8361a43b1","tex":"d_{\\mathrm{er}}","aliases":"d_er mathrm","display":"d_{\\mathrm{er}}","kind":"use","excerpt":"Define the accessible block support of c≠0 by supp_E(c) = {i∈ V:E_ic≠0} and define the node-erasure distance d_er = min_c≠0|supp_E(c)|.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-39c1c7c6d86a3aa2","tex":"d_G","aliases":"d_G","display":"d_G","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d4067dd31ffb1ab2","tex":"D_{G_\\alpha}","aliases":"D_G_α alpha","display":"D_{G_\\alpha}","kind":"use","excerpt":"Local candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5341c066ee58261e","tex":"d_H","aliases":"d_H","display":"d_H","kind":"use","excerpt":"Order the local coordinates as x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^ T. The overlap discrepancies are g_H-g_M, p_M-p_X, v_X-v_A, d_A-d_H. Thus D_HAI = 1 · 0 · -1 · 0 · 0 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-92627c037e20f1ce","tex":"D_{H^g}","aliases":"D_H^g","display":"D_{H^g}","kind":"use","excerpt":"For each i∈ V, let g_i∈ GL(C_i), g=(g_i)_i∈ V. Define the vertex and edge direct sums G_V=⊕_i∈ Vg_i, G_E=⊕_e∈ Eg_t(e). The transformed transfers and local operation representations are H_e^g = g_t(e)H_eg_s(e)^-1, ρ_i^g(a) = g_iρ_i(a)g_i^-1. The fundamental covariance identity is D_H^gG_V=G_ED_H. It follows directly by evaluating the e-component: sq","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f211d2857c866bee","tex":"D_{H^g}^*","aliases":"D_H^g^*","display":"D_{H^g}^*","kind":"use","excerpt":"From D_H^g=G_ED_HG_V^-1 and unitarity, D_H^g^*D_H^g = G_VD_H^*G_E^-1G_ED_HG_V^-1 = G_VL_HG_V^-1. The spectral statements follow from unitary similarity.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d5c52c12ef4de174","tex":"D_{\\mathrm{HAI}}","aliases":"D_HAI mathrm","display":"D_{\\mathrm{HAI}}","kind":"use","excerpt":"Order the local coordinates as x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^ T. The overlap discrepancies are g_H-g_M, p_M-p_X, v_X-v_A, d_A-d_H. Thus D_HAI = 1 · 0 · -1 · 0 · 0 · 0 · 0 · 0 0 · 0 · 0 · 1 · -1 · 0 · 0 · 0 0 · 0 · 0 · 0 · 0 · 1 · 0 · -1 0 · -1 · 0 · 0 · 0 · 0 · 1 · 0 . The rows are orthogonal with squared norm two. Therefore ker D_HAI ≅ R^4, ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-57612fbcd764c44c","tex":"D_{\\mathrm{HH}}","aliases":"D_HH mathrm","display":"D_{\\mathrm{HH}}","kind":"use","excerpt":"Order the local coordinates as x=(q_P,p_P,p_C,f_C,q_N,f_N)^ T. Compatibility requires p_P=p_C, f_C=f_N, q_N=q_P. The discrepancy operator is D_HH = 0 · 1 · -1 · 0 · 0 · 0 0 · 0 · 0 · 1 · 0 · -1 -1 · 0 · 0 · 0 · 1 · 0 . Therefore ker D_HH = { (q,p,p,f,q,f)^ T:q,p,f∈ R } ≅ R^3. The rows are orthogonal with squared norm two, so D_HHD_HH^*=2I_3. Conseq","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b94841ea95944639","tex":"D_{\\mathrm{HH}}^*","aliases":"D_HH^* mathrm","display":"D_{\\mathrm{HH}}^*","kind":"use","excerpt":"-1 · 0 · 0 · 0 · 1 · 0 . Therefore ker D_HH = { (q,p,p,f,q,f)^ T:q,p,f∈ R } ≅ R^3. The rows are orthogonal with squared norm two, so D_HHD_HH^*=2I_3. Consequently the transfer Laplacian has spectrum spec(L_HH) = {0,0,0,2,2,2}, and the exact workspace gap is γ_HH=2. The low-energy subspace is therefore stable under any Laplacian perturbation whose o","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-77d978b80a551733","tex":"d_I","aliases":"d_I","display":"d_I","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b1f930435e77fb41","tex":"\\mathbf d_{\\mathrm{id}}","aliases":"d_id mathbf mathrm","display":"\\mathbf d_{\\mathrm{id}}","kind":"use","excerpt":"For an admissible correspondence J, define the operation defect on a declared unit ball A_1 by δ_op( J) = sup_a∈ A_1 A_ W''β-β A_ W . The primary certification object is the identity profile d_id( J) = ( q_str, δ_op( J), δ_causal(β), δ_rec(β); μ^keep ). The first four coordinates must remain below declared tolerances, while the non-collapse margin ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-13ea1da5b4752eb7","tex":"d_{\\mathrm{id}}","aliases":"d_id mathrm","display":"d_{\\mathrm{id}}","kind":"use","excerpt":"Persistence does not require token identity. Let R_t and R_t+1 be temporally successive realizations. Continuity is possible when there exists an admissible transport J_t: R_trightsquigarrow R_t+1 that preserves the operative core and its causal role within tolerance: d_id(Workspace_t,Workspace_t+1)≤ε. Thus a workspace may persist by reconstruction","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-357bde3cdbce9978","tex":"D_{\\mathrm{int}}","aliases":"D_int mathrm","display":"D_{\\mathrm{int}}","kind":"use","excerpt":"Let the vertex set of a core-transfer diagram be partitioned as V=V_1sqcup⋯sqcup V_m. Write the transfer discrepancy as D_H = D_int D_∂ , D_int = ⊕_α=1^mD_α, where D_α contains the edge constraints internal to V_α, and D_∂ contains the constraints crossing between blocks. Define the internally coherent macro-state spaces W_α=ker D_α, W_Π=⊕_α=1^mW_α","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fb9345c810f9b4a1","tex":"D_{\\mathrm{int}}^*","aliases":"D_int^* mathrm","display":"D_{\\mathrm{int}}^*","kind":"use","excerpt":"[Two-scale coherence bound] Let P_Π be the orthogonal projector onto J_Π W_Π. Let γ_int>0 be the smallest positive eigenvalue of D_int^*D_int, and let γ_Π>0 be the smallest positive eigenvalue of D_Π^*D_Π. Then every c∈ C_V satisfies dist(c,Γ(H)) ≤ frac D_intc sqrtγ_int + frac D_Π J_Π^*P_Π c sqrtγ_Π. The first term measures incoherence within macro","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-382f08d5d528afb0","tex":"d_N","aliases":"d_N","display":"d_N","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a3861faf82137cee","tex":"d_{\\mathrm{path}}^{\\rightarrow}","aliases":"d_path^rightarrow mathrm rightarrow","display":"d_{\\mathrm{path}}^{\\rightarrow}","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-57992bb96af9a227","tex":"d_{\\mathrm{path}}^{\\mathrm{sym}}","aliases":"d_path^sym mathrm mathrm","display":"d_{\\mathrm{path}}^{\\mathrm{sym}}","kind":"use","excerpt":"d path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and intervention divergences need not be symmetric, symmetry is not assumed. When reverse correspondences exist, define the symmetrized pseudometric d_path^sym( R, R') = max{ d_path^rightar","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3e5463cd5ce49eb7","tex":"d_q","aliases":"d_q","display":"d_q","kind":"use","excerpt":"For a family S of admissible scaffold variants, define the cross-scaffold identity defect δ_q^scaf = inf_{J_σσ'} E_σ,σ'∼π d_q ( J_σσ'q_σ,q_σ' ), where J_σσ' transports the learned object between admissible coordinate systems. A low value indicates that the same abstract organization is recovered under multiple non-solution-bearing scaffolds.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7f18f9c2fbc4b832","tex":"\\mathcal D_{\\mathrm{sel}}","aliases":"D_sel mathcal mathrm","display":"\\mathcal D_{\\mathrm{sel}}","kind":"use","excerpt":"[Validity after independent specification selection] Let widehatς be selected using D_sel, and let D_cert be independent of D_sel. For every fixed ς∈ S, let ϕ_ς( D_cert)∈{0,1} be a certificate rule satisfying sup_P:Cert(W;ς)=0 P_P(ϕ_ς=1) ≤α. Then the selected-specification procedure satisfies sup_P P_P( ϕ_widehatς=1, Cert(W;widehatς)=0 ) ≤α.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1f69f0aebdb2a517","tex":"\\mathcal D_{\\mathrm{spec}}","aliases":"D_spec mathcal mathrm","display":"\\mathcal D_{\\mathrm{spec}}","kind":"use","excerpt":"A practical design may use: D_form sqcup D_spec sqcup D_cert sqcup D_stress. Formation learns the workspace. The specification split chooses targets, horizons, nulls, and thresholds from a predeclared menu. Certification makes the pass decision once. Stress testing explores broader shifts without retroactively changing the certified claim. Cross-fi","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-83d436422d6e1db8","tex":"\\mathcal D_{\\mathrm{stress}}","aliases":"D_stress mathcal mathrm","display":"\\mathcal D_{\\mathrm{stress}}","kind":"use","excerpt":"A practical design may use: D_form sqcup D_spec sqcup D_cert sqcup D_stress. Formation learns the workspace. The specification split chooses targets, horizons, nulls, and thresholds from a predeclared menu. Certification makes the pass decision once. Stress testing explores broader shifts without retroactively changing the certified claim. Cross-fi","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b377ac1e0c05ff0f","tex":"d_t","aliases":"d_t","display":"d_t","kind":"use","excerpt":"[Causal repair stabilizes maintenance deficit] Let e_t≥0 be the maintenance deficit. Suppose the matched no-repair trajectory satisfies e_t+1^(0) ≤ α e_t+d_t, where d_t≥0 is exogenous damage. Suppose the implemented repair policy has the causal one-step effect e_t+1^(0)-e_t+1 ≥ κ e_t-ϵ_t. Assume 0≤β:=α-κ<1, 0≤ d_t≤overline d, 0≤ϵ_t≤overlineϵ. Then ","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c631055a6cfad223","tex":"D_{t+1}","aliases":"D_t+1","display":"D_{t+1}","kind":"use","excerpt":"To compare successive times, choose unitary identification maps U_t^V: C_V(t)⟶ C_V(t+1), U_t^E: C_E(t)⟶ C_E(t+1). They are comparison maps, not assumptions that the physical carriers are identical. Define the one-step diagram defect Δ_t^D = D_t+1U_t^V-U_t^ED_t, δ_t^D= Δ_t^D . Exact temporal compatibility is the condition Δ_t^D=0.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f43f932deb2cf69a","tex":"D_t^*","aliases":"D_t^*","display":"D_t^*","kind":"use","excerpt":"At time t, let C_V(t)=⊕_i∈ VC_i(t), C_E(t)=⊕_e∈ EC_t(e)(t), and let D_t: C_V(t)⟶ C_E(t), L_t=D_t^*D_t be the transfer discrepancy and transfer Laplacian. At time t, the combinatorial graph is held fixed while the local representations, transfers, and metrics may vary. Changes of graph are treated later by a graph correspondence.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1ed190318d1a87de","tex":"d_u","aliases":"d_u","display":"d_u","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d3b6e2eaef5fb02a","tex":"d_v","aliases":"d_v","display":"d_v","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0a45e8c30cdc421c","tex":"d_W","aliases":"d_W","display":"d_W","kind":"use","excerpt":"Geometric recovery becomes behaviorally relevant when the workspace state controls a predictive kernel. Let K_w= P(Y∈·mid W=w) and suppose TV(K_w,K_w')≤ L_Kd_W(w,w').","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-83bccd6fed0b1c30","tex":"D_{\\widetilde H}","aliases":"D_widetilde H widetilde","display":"D_{\\widetilde H}","kind":"use","excerpt":"Suppose the edge maps are perturbed to widetilde H_e=H_e+Δ_e while the weights and vertex metrics remain fixed. Define δ_H^2 = max_i∈ V ∑_e:s(e)=iw_e Δ_e ^2. Let F=D_widetilde H-D_H. Then (Fc)_e=-sqrtw_e Δ_ec_s(e), F ≤δ_H. Since L_widetilde H-L_H =D_H^*F+F^*D_H+F^*F, we obtain L_widetilde H-L_H ≤ 2 D_H δ_H+δ_H^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c676ea7aeb2ba9c2","tex":"d_{\\Workspace}","aliases":"d_Workspace Workspace","display":null,"kind":"use","excerpt":"A nontrivial exact structural workspace exists when dimΓ(H)>0. Its exact dimension is d_Workspace=dimΓ(H).","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-99fcbf6d9770c880","tex":"d_Y","aliases":"d_Y","display":"d_Y","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2569dc5eb62aae9e","tex":"D_\\alpha","aliases":"D_α alpha","display":"D_\\alpha","kind":"use","excerpt":"The exact and approximate constructions can be compressed as local transfer systems · xrightarrow ker D_α local workspaces · xrightarrow kerδ^0 meta-workspace · xrightarrow finite recursive descent global workspace. At exact resolution, all finite parenthesizations are canonically equivalent. At approximate resolution, every level contributes a def","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-eb7a3968e45cf0cd","tex":"d_{\\varepsilon}","aliases":"d_ε varepsilon","display":"d_{\\varepsilon}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9f7085d6ab421a97","tex":"d_\\Theta","aliases":"d_Θ Theta","display":"d_\\Theta","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-78ce6be299bd03f0","tex":"D_\\Pi","aliases":"D_Π Pi","display":"D_\\Pi","kind":"use","excerpt":"where D_α contains the edge constraints internal to V_α, and D_∂ contains the constraints crossing between blocks. Define the internally coherent macro-state spaces W_α=ker D_α, W_Π=⊕_α=1^mW_α. Let J_Π:W_Π↪ C_V be the canonical isometric inclusion, and define the quotient discrepancy D_Π=D_∂J_Π. The quotient does not replace a group of nodes by an ","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-99914e9e93d417ad","tex":"D_\\Pi^*","aliases":"D_Π^* Pi","display":"D_\\Pi^*","kind":"use","excerpt":"[Faithful coarse-graining] For every finite partition of the vertex set, ker D_H = J_Πker D_Π. Equivalently, Γ(H) ≅ ker D_Π = ker(D_Π^*D_Π). In particular, dimΓ(H) = dim W_Π-rank(D_Π). Thus faithful coarse-graining preserves the exact structural workspace even though it changes the number and interpretation of the displayed nodes.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6f0d52b9a8bea385","tex":"d_\\tau","aliases":"d_τ tau","display":"d_\\tau","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0a1ac0d65e3855ed","tex":"d_\\ell","aliases":"d_ℓ ell","display":"d_\\ell","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-903974f7b83d9f0f","tex":"D_{\\partial}","aliases":"D_∂ partial","display":"D_{\\partial}","kind":"use","excerpt":"Let the vertex set of a core-transfer diagram be partitioned as V=V_1sqcup⋯sqcup V_m. Write the transfer discrepancy as D_H = D_int D_∂ , D_int = ⊕_α=1^mD_α, where D_α contains the edge constraints internal to V_α, and D_∂ contains the constraints crossing between blocks. Define the internally coherent macro-state spaces W_α=ker D_α, W_Π=⊕_α=1^mW_α","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b64fabaef18b557d","tex":"\\mathsf{data\\_splits}","aliases":"data_splits mathsf","display":"\\mathsf{data\\_splits}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd","display":null,"kind":"use","excerpt":"nwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P-μ I) (PEwideti","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b800b3bf90953293","tex":"\\mathrm{dec}","aliases":"dec mathrm","display":"\\mathrm{dec}","kind":"use","excerpt":"Let m_q(W; D) be the signed margin of gate q, oriented so that m_q(W; D)>0 means that the gate passes after its uncertainty correction. Define the decomposition envelope underline m_q = inf_ D∈ Dm_q(W; D), overline m_q = sup_ D∈ Dm_q(W; D), and its sensitivity width Δ_q^dec = overline m_q-underline m_q.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f4c1038df3e536b4","tex":"\\mathsf{decomposition}","aliases":"decomposition mathsf","display":"\\mathsf{decomposition}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1f7f550cf2dfe3eb","tex":"\\Dep","aliases":"Dep","display":null,"kind":"use","excerpt":"For S∈ S, define Ret_τ(S) = I ( X_S(t+τ);X_S(t) mid X_bar S(t),E_t ), Dep_τ(S) = I ( X_S(t+τ);X_bar S(t) mid X_S(t),E_t ), Out_τ(S) = I ( X_bar S(t+τ);X_S(t) mid X_bar S(t),E_t ). These quantities are observational. They can be nonzero because of common causes, selection, or feedback paths and therefore cannot alone establish a causal boundary.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2fe6ecb36ea8bad2","tex":"\\Dep_\\tau","aliases":"Dep_τ Dep tau","display":null,"kind":"use","excerpt":"For S∈ S, define Ret_τ(S) = I ( X_S(t+τ);X_S(t) mid X_bar S(t),E_t ), Dep_τ(S) = I ( X_S(t+τ);X_bar S(t) mid X_S(t),E_t ), Out_τ(S) = I ( X_bar S(t+τ);X_S(t) mid X_bar S(t),E_t ). These quantities are observational. They can be nonzero because of common causes, selection, or feedback paths and therefore cannot alone establish a causal boundary.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"-(a+b) · b^2+1 . Writing τ=a^2+b^2+2, the eigenvalues are λ_± = frac12 ( τ±sqrtτ^2-4(ab-1)^2 ), and det L_H=(ab-1)^2. Thus a nonzero exact shared core exists precisely when ab=1, which is exactly the trivial-holonomy condition. Away from exact consistency, λ_-λ_+=(ab-1)^2. When |ab-1| is small while λ_+ stays bounded away from zero, the soft worksp","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"[Operators in the reflection-cycle workspace] For Example~[source reference], the holonomy is J=diag(1,-1). Its commutant consists of diagonal matrices: Comm(J) = { a · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","excerpt":"[Coherence-to-core distance estimate] Let W_0=Γ(H)=ker L_H, r=dim W_0, and suppose r<N. Set γ_H=λ_r+1>0. Then every c∈ C_V satisfies dist(c,W_0)^2 ≤ frac E_H(c)γ_H. Moreover, E_H(c) ≤ L_H dist(c,W_0)^2.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d5cfc408cd7326f7","tex":"\\operatorname{Distributed}","aliases":"Distributed operatorname","display":"\\operatorname{Distributed}","kind":"use","excerpt":"Workspace( R) centernotLongrightarrow Conscious( R). Likewise, Distributed(Workspace) centernotLongrightarrow Person(Workspace). Workspace structure, reentry, agency, self-maintenance, and phenomenal consciousness remain formally distinct predicates. Any bridge among them would require additional theory and evidence.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0645873febb68b5c","tex":"\\Downarrow","aliases":"Downarrow","display":null,"kind":"use","excerpt":"The complete static-to-agentive construction is · (Ω, D, S,Σ, D_form, L) [2mm] · Downarrow ( D,widehat V,widehat E,widehat H,widehat P,widehat B) [2mm] · Downarrow (G,{X_i},{T_e}) [2mm] · Downarrow X_i=C_i⊕ R_i [2mm] · ({C_i},{H_e}, A,{ρ_i}) [2mm] · Downarrow Γ(H)=varprojlim C_i [2mm] · Downarrow L_H=D_H^*D_H [2mm] · Downarrow Γ_ε(H) = ran 1_[0,ε](","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a3e1c1b79b140f90","tex":"\\mathrm{dyn}","aliases":"dyn mathrm","display":"\\mathrm{dyn}","kind":"use","excerpt":"The resulting certification report is C_dyn(W_t) =( · δ_align^ I, {Δ_A^ℓ}_A⊆ V, k_ε, Δ_tr( Q), · η_causal^ν, Δ_specific(r), η_rec(τ), η_loop(τ) ). No single coordinate certifies a relational workspace.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7015f1f6989ac898","tex":"\\mathbf E","aliases":"E mathbf","display":"\\mathbf E","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-05dded90074b912f","tex":"e^{-(b-a)t/2}","aliases":"e^-(b-a)t/2","display":"e^{-(b-a)t/2}","kind":"use","excerpt":"On ran P and ranwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d9a627b18a8b94df","tex":"e^{-t(B_Q-\\mu I)}","aliases":"e^-t(B_Q-μ I) mu","display":"e^{-t(B_Q-\\mu I)}","kind":"use","excerpt":"On ran P and ranwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2266e261d8135459","tex":"e^{t(A_P-\\mu I)}","aliases":"e^t(A_P-μ I) mu","display":"e^{t(A_P-\\mu I)}","kind":"use","excerpt":"On ran P and ranwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4fb90739223954bc","tex":"e_{\\mathcal O}","aliases":"e_ O mathcal","display":"e_{\\mathcal O}","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-23312dd33b5f99eb","tex":"\\mathcal E_{\\mathcal U}^{\\mathrm{tot}}","aliases":"E_ U^tot mathcal mathcal mathrm","display":"\\mathcal E_{\\mathcal U}^{\\mathrm{tot}}","kind":"use","excerpt":"[0.2em] sqrtλ_ov S_ U , L_ U^tot = K_ U^*K_ U. Its energy is E_ U^tot(u) = · ∑_α D_G_αu_α ^2 · + λ_ov ∑_α<β ρ^G_β_G_αβu_β - ρ^G_α_G_αβu_α ^2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"[Nonautonomous persistence theorem] Let e_t≥0 satisfy e_t+1≤α_te_t+σ_t, α_t,σ_t≥0. Then, for every t≥1, e_t ≤ · (∏_r=0^t-1α_r)e_0 · + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fc0c682317c378af","tex":"e_0^2","aliases":"e_0^2","display":"e_0^2","kind":"use","excerpt":"[Mean-square persistence] Suppose E[e_t+1^2mid F_t] ≤ q e_t^2+v_t, 0≤ q<1, and E[v_t]≤bar v. Then E[e_t^2] ≤ q^t E[e_0^2] + fracbar v(1-q^t)1-q, so limsup_t→∞ E[e_t^2] ≤ fracbar v1-q.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"Let C_*= R^2 and define scalar node encodings E_1(c_1,c_2)=c_1, E_2(c_1,c_2)=c_2, E_3(c_1,c_2)=c_1+c_2. Every singleton has rank one, so no node reconstructs the two-dimensional core. Every pair does reconstruct. For S={1,2}, F_S=I, α_S=1. For S={1,3} or S={2,3}, the frame eigenvalues are frac3±sqrt52. Therefore m_rec=2, k_erase=1, d_er=2, α^(1)=fr","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"The directed path category is C=Path(G). Its objects are vertices. A path γ=e_m⋯ e_2e_1 means that e_1 is traversed first. Composition is concatenation, and the empty path at i is denoted 1_i.","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"Let C_*= R^2 and define scalar node encodings E_1(c_1,c_2)=c_1, E_2(c_1,c_2)=c_2, E_3(c_1,c_2)=c_1+c_2. Every singleton has rank one, so no node reconstructs the two-dimensional core. Every pair does reconstruct. For S={1,2}, F_S=I, α_S=1. For S={1,3} or S={2,3}, the frame eigenvalues are frac3±sqrt52. Therefore m_rec=2, k_erase=1, d_er=2, α^(1)=fr","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"The directed path category is C=Path(G). Its objects are vertices. A path γ=e_m⋯ e_2e_1 means that e_1 is traversed first. Composition is concatenation, and the empty path at i is denoted 1_i.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82c2d872ef715d1f","tex":"E_3","aliases":"E_3","display":"E_3","kind":"use","excerpt":"Let C_*= R^2 and define scalar node encodings E_1(c_1,c_2)=c_1, E_2(c_1,c_2)=c_2, E_3(c_1,c_2)=c_1+c_2. Every singleton has rank one, so no node reconstructs the two-dimensional core. Every pair does reconstruct. For S={1,2}, F_S=I, α_S=1. For S={1,3} or S={2,3}, the frame eigenvalues are frac3±sqrt52. Therefore m_rec=2, k_erase=1, d_er=2, α^(1)=fr","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8db0cea46fdd60f3","tex":"E_A","aliases":"E_A","display":"E_A","kind":"use","excerpt":"p=open proof or implementation obligations, · v=verification and regression status. Define local encodings E_Hc=(g,d), E_Mc=(g,p), E_Xc=(p,v), E_Ac=(d,v). Each coordinate is represented at exactly two nodes, and the incidence pattern forms a four-cycle.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c4048d3c1a406dfb","tex":"\\mathbb E_{a\\sim\\widehat\\pi}","aliases":"E_a∼widehatπ mathbb sim widehat pi","display":"\\mathbb E_{a\\sim\\widehat\\pi}","kind":"use","excerpt":"gintheorem[Approximate counterfactual policy selection] Let A be a finite feasible action set. Let Q: A→ R be the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-op","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f4844ca66ffea7ae","tex":"\\mathbb E_{a\\sim\\pi}","aliases":"E_a∼π mathbb sim pi","display":"\\mathbb E_{a\\sim\\pi}","kind":"use","excerpt":"gintheorem[Approximate counterfactual policy selection] Let A be a finite feasible action set. Let Q: A→ R be the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-op","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8dd41d491328c4d2","tex":"\\mathbb E_{a\\sim\\pi^*}","aliases":"E_a∼π^* mathbb sim pi","display":"\\mathbb E_{a\\sim\\pi^*}","kind":"use","excerpt":"the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal policy π^*, E_a∼π^*[Q(a)] - E_a∼widehatπ[Q(a)] ≤ 2ε_Q+δ_opt. For a deterministic greedy action widehat a","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d1b9cf3f81e604ba","tex":"e_B","aliases":"e_B","display":"e_B","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e06471cf155d9b6c","tex":"e_B^{\\mathrm{causal}}","aliases":"e_B^causal mathrm","display":"e_B^{\\mathrm{causal}}","kind":"use","excerpt":"A boundary widehat S is certified as endogenously formed only if: item the membership mask was not directly encoded in Σ_B or Σ_Y; item e_B^MDL and e_B^causal exceed declared thresholds; item held-out recovery satisfies Theorem~[source reference]; item the boundary remains stable under admissible environment drift or its changes are predicted by sc","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e7906ced39799870","tex":"e_B^{\\mathrm{MDL}}","aliases":"e_B^MDL mathrm","display":"e_B^{\\mathrm{MDL}}","kind":"use","excerpt":"A boundary widehat S is certified as endogenously formed only if: item the membership mask was not directly encoded in Σ_B or Σ_Y; item e_B^MDL and e_B^causal exceed declared thresholds; item held-out recovery satisfies Theorem~[source reference]; item the boundary remains stable under admissible environment drift or its changes are predicted by sc","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c386fa4a39ad29f","tex":"E_C","aliases":"E_C","display":"E_C","kind":"use","excerpt":"f · =falsification-status coordinate. The accessible encodings are E_P= 1 · 0 · 0 0 · 1 · 0 , E_C= 0 · 1 · 0 0 · 0 · 1 , E_N= 1 · 0 · 0 0 · 0 · 1 . The semantics are stylized. The point is not that real researchers contain only two numbers. The point is to exhibit, without metaphor, a system in which no participant or artifact contains the entire o","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-38c29133d6347a0f","tex":"E_C^*","aliases":"E_C^*","display":"E_C^*","kind":"use","excerpt":"The full frame operator is F_PCN=E_P^*E_P+E_C^*E_C+E_N^*E_N=2I_3. For every two-node coalition S, λ_min(F_S)=1. Thus every two-node coalition reconstructs the complete core and, for noisy observations y_S=A_Sc+η_S, widehat c-c ≤ η_S . No individual node reconstructs c, because rank(E_P) = rank(E_C) = rank(E_N) =2<3. The realization is therefore sim","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2c11dc52a9b67865","tex":"\\mathcal E_{\\mathrm{cert}}","aliases":"E_cert mathcal mathrm","display":"\\mathcal E_{\\mathrm{cert}}","kind":"use","excerpt":"cal D = D_form sqcup D_cal sqcup D_cert. The formation sample selects widehatθ. The calibration sample fixes thresholds, regularization strengths, and intervention magnitudes. The certification sample evaluates the final gates without further structural selection. In a multi-environment study, the stronger requirement is E_form ∩ E_cert = ∅, so tha","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1dbd0aceb8a92466","tex":"e_E","aliases":"e_E","display":"e_E","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6c0015ae0427edc8","tex":"\\mathbf E_{\\mathrm{form}}","aliases":"E_form mathbf mathrm","display":"\\mathbf E_{\\mathrm{form}}","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e43a1b05d940f0c8","tex":"\\mathcal E_{\\mathrm{form}}","aliases":"E_form mathcal mathrm","display":"\\mathcal E_{\\mathrm{form}}","kind":"use","excerpt":"cal D = D_form sqcup D_cal sqcup D_cert. The formation sample selects widehatθ. The calibration sample fixes thresholds, regularization strengths, and intervention magnitudes. The certification sample evaluates the final gates without further structural selection. In a multi-environment study, the stronger requirement is E_form ∩ E_cert = ∅, so tha","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-92780dc9a20a0ca6","tex":"E_H","aliases":"E_H","display":"E_H","kind":"use","excerpt":"p=open proof or implementation obligations, · v=verification and regression status. Define local encodings E_Hc=(g,d), E_Mc=(g,p), E_Xc=(p,v), E_Ac=(d,v). Each coordinate is represented at exactly two nodes, and the incidence pattern forms a four-cycle.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b576b4aed1362b85","tex":"e_H","aliases":"e_H","display":"e_H","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-77aaef070496886d","tex":"\\mathcal E_{H^g}","aliases":"E_H^g mathcal","display":"\\mathcal E_{H^g}","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"nsional abstract core. The exact static theory provides a complete local chart overline E_i = ev_i∘Ψ:C_*⟶ C_i, where Ψ:C_*→Γ(H) is the root-to-section isomorphism. A participant or artifact need not expose that complete chart. Let O_i:C_i⟶ Y_i be the declared observation or access map and define the accessible encoding E_i=O_ioverline E_i:C_*⟶ Y_i.","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-43012e02a798a8c7","tex":"E_i^*","aliases":"E_i^*","display":"E_i^*","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"Network consensus studies how local interactions drive distributed states toward agreement [source reference]. Agreement is not the same as a workspace. Consensus often removes heterogeneity by converging all nodes to a common value. Stratified relational workspaces preserve heterogeneous local representations and may require complementary disagree","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2b41f5a969d3c731","tex":"\\mathbb E_{\\jmath,\\jmath'\\sim\\mathcal I_{\\perp}(r)}","aliases":"E_jmath,jmath'∼ I_⊥(r) mathbb jmath jmath sim mathcal perp","display":"\\mathbb E_{\\jmath,\\jmath'\\sim\\mathcal I_{\\perp}(r)}","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-95c94409fd656c31","tex":"E_M","aliases":"E_M","display":"E_M","kind":"use","excerpt":"p=open proof or implementation obligations, · v=verification and regression status. Define local encodings E_Hc=(g,d), E_Mc=(g,p), E_Xc=(p,v), E_Ac=(d,v). Each coordinate is represented at exactly two nodes, and the incidence pattern forms a four-cycle.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","excerpt":"The directed path category is C=Path(G). Its objects are vertices. A path γ=e_m⋯ e_2e_1 means that e_1 is traversed first. Composition is concatenation, and the empty path at i is denoted 1_i.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-57e6767552eb853b","tex":"e_{\\max}","aliases":"e_max max","display":"e_{\\max}","kind":"use","excerpt":"t = min_j∈ J_maintϕ_j,t. The workspace passes the protected gates when v_t≥0. To maintain a reserve ρ>0, define the maintenance deficit e_t =(ρ-v_t)_+. Define the zero-reserve and bounded-deficit viability sets V_Θ = { S:ϕ_j(S)≥0 for every j∈ J_maint } V_Θ,e_max = { S:e(S)≤ e_max } The first set requires every protected gate to pass. The second per","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e466c5eab5f55a38","tex":"E_N","aliases":"E_N","display":"E_N","kind":"use","excerpt":"f · =falsification-status coordinate. The accessible encodings are E_P= 1 · 0 · 0 0 · 1 · 0 , E_C= 0 · 1 · 0 0 · 0 · 1 , E_N= 1 · 0 · 0 0 · 0 · 1 . The semantics are stylized. The point is not that real researchers contain only two numbers. The point is to exhibit, without metaphor, a system in which no participant or artifact contains the entire o","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dd278ab284857a30","tex":"E_N^*","aliases":"E_N^*","display":"E_N^*","kind":"use","excerpt":"The full frame operator is F_PCN=E_P^*E_P+E_C^*E_C+E_N^*E_N=2I_3. For every two-node coalition S, λ_min(F_S)=1. Thus every two-node coalition reconstructs the complete core and, for noisy observations y_S=A_Sc+η_S, widehat c-c ≤ η_S . No individual node reconstructs c, because rank(E_P) = rank(E_C) = rank(E_N) =2<3. The realization is therefore sim","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-32eb5431a5e744fc","tex":"\\mathcal E_{\\mathrm{ov}}","aliases":"E_ov mathcal mathrm","display":"\\mathcal E_{\\mathrm{ov}}","kind":"use","excerpt":"A family w=(w_α)_α is exactly compatible when δ^0w=0. Its direct mismatch is m_ U(w)=δ^0w. After choosing inner products and positive overlap weights, the mismatch energy is E_ov(w) = δ^0w ^2 = ∑_α<β ν_αβ r_αβ^βw_β -r_αβ^αw_α ^2. This is the appropriate obstruction when the full local states are already given. Such a family glues exactly if and onl","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-be6867d3a51ccf7a","tex":"E_P","aliases":"E_P","display":"E_P","kind":"use","excerpt":"f · =falsification-status coordinate. The accessible encodings are E_P= 1 · 0 · 0 0 · 1 · 0 , E_C= 0 · 1 · 0 0 · 0 · 1 , E_N= 1 · 0 · 0 0 · 0 · 1 . The semantics are stylized. The point is not that real researchers contain only two numbers. The point is to exhibit, without metaphor, a system in which no participant or artifact contains the entire o","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0a21ab7396f4979e","tex":"e_P","aliases":"e_P","display":"e_P","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2cbffb8d0eda3f35","tex":"E_{P'}","aliases":"E_P'","display":"E_{P'}","kind":"use","excerpt":"Suppose the proposer becomes unavailable. The remaining coalition {C,N} reconstructs (q,p,f) exactly. A replacement proposer P' can therefore be initialized by E_P'c=(q,p). If P' uses a different local coordinate system J_P'∈ GL(2), then the incoming and outgoing overlap maps are conjugated accordingly. The global section and all downstream kernels","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3ad298fefd36707f","tex":"E_P^*","aliases":"E_P^*","display":"E_P^*","kind":"use","excerpt":"The full frame operator is F_PCN=E_P^*E_P+E_C^*E_C+E_N^*E_N=2I_3. For every two-node coalition S, λ_min(F_S)=1. Thus every two-node coalition reconstructs the complete core and, for noisy observations y_S=A_Sc+η_S, widehat c-c ≤ η_S . No individual node reconstructs c, because rank(E_P) = rank(E_C) = rank(E_N) =2<3. The realization is therefore sim","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-244f69b2cbc609aa","tex":"e_q","aliases":"e_q","display":"e_q","kind":"use","excerpt":"[Finite obstruction set] Let e_1,…,e_r be the non-tree edges of a connected finite graph. Then C_* = ⋂_q=1^r ker(H_γ_e_q-I).","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"For a single edge, V_e=0 means T_e(c,0)=(H_ec,0), so core states remain in the core. Inductively, if the first r transfers map c to (H_e_r⋯ H_e_1c,0), applying T_e_r+1 gives (H_e_r+1H_e_r⋯ H_e_1c,0). The conclusion follows after m steps.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-84511894cb17d79d","tex":"E_{r'}'","aliases":"E_r''","display":"E_{r'}'","kind":"use","excerpt":"Suppose node r is removed and the surviving coalition is S=V∖{r}. A replacement node r' is to receive the encoding E_r'':C_*→ Y_r''.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7b243e274b6d1702","tex":"e_{r+1}","aliases":"e_r+1","display":"e_{r+1}","kind":"use","excerpt":"For a single edge, V_e=0 means T_e(c,0)=(H_ec,0), so core states remain in the core. Inductively, if the first r transfers map c to (H_e_r⋯ H_e_1c,0), applying T_e_r+1 gives (H_e_r+1H_e_r⋯ H_e_1c,0). The conclusion follows after m steps.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2775a21153e9ebc4","tex":"e_t","aliases":"e_t","display":"e_t","kind":"use","excerpt":"Let the realized vertex state satisfy x_t+1=F_tx_t+ξ_t, where F_t: C_V(t)→ C_V(t+1) is the actual update and ξ_t is an unmodeled disturbance. Define the off-workspace error e_t= (I-P_t)x_t . Introduce α_t = (I-P_t+1)F_t(I-P_t) , which measures persistence of prior off-workspace error, and ℓ_t = (I-P_t+1)F_tP_t , which measures leakage of a valid wo","count":41},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bd888caa9f34bc52","tex":"E_{t+1}","aliases":"E_t+1","display":"E_{t+1}","kind":"use","excerpt":"C_t^ctl = ( W_t,G_t,M_t,B_t,E_t ), where G_t is the represented goal, M_t is a workspace-integrity monitor, B_t is the current boundary or routing state, and E_t is the declared external context. An action is generated by A_t ∼ π_t ( ·mid W_t,G_t,M_t,B_t,E_t ). The future controlled state obeys a time-unrolled structural model S_t+1 = F_t(S_t,A_t,E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c0ec2f484593060","tex":"e_{t+1}","aliases":"e_t+1","display":"e_{t+1}","kind":"use","excerpt":"[Grounded coherence recurrence] Every trajectory satisfies e_t+1≤α_te_t+σ_t, where σ_t = ℓ_t P_tx_t + (I-P_t+1)ξ_t .","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7402ee2dd87e127a","tex":"E_{t+1:t+\\tau}","aliases":"E_t+1:t+τ tau","display":"E_{t+1:t+\\tau}","kind":"use","excerpt":"The observational condition I(G_t+τ;G_tmid E_t+1:t+τ)>0 can arise from several non-agentic mechanisms: item a persistent external command writes the same code at every step; item a clock, episode label, or task identifier drives both variables; item a frozen memory cell repeats a symbol that is not used by the policy; item selection on successful t","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-705cfb0ba75b66cc","tex":"e_{t+1}^{(0)}","aliases":"e_t+1^(0)","display":"e_{t+1}^{(0)}","kind":"use","excerpt":"[Causal repair stabilizes maintenance deficit] Let e_t≥0 be the maintenance deficit. Suppose the matched no-repair trajectory satisfies e_t+1^(0) ≤ α e_t+d_t, where d_t≥0 is exogenous damage. Suppose the implemented repair policy has the causal one-step effect e_t+1^(0)-e_t+1 ≥ κ e_t-ϵ_t. Assume 0≤β:=α-κ<1, 0≤ d_t≤overline d, 0≤ϵ_t≤overlineϵ. Then ","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ef5139cd2ca78dd4","tex":"e_{t+1}^2","aliases":"e_t+1^2","display":"e_{t+1}^2","kind":"use","excerpt":"[Mean-square persistence] Suppose E[e_t+1^2mid F_t] ≤ q e_t^2+v_t, 0≤ q<1, and E[v_t]≤bar v. Then E[e_t^2] ≤ q^t E[e_0^2] + fracbar v(1-q^t)1-q, so limsup_t→∞ E[e_t^2] ≤ fracbar v1-q.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6d211a8152065728","tex":"e_t^2","aliases":"e_t^2","display":"e_t^2","kind":"use","excerpt":"[Mean-square persistence] Suppose E[e_t+1^2mid F_t] ≤ q e_t^2+v_t, 0≤ q<1, and E[v_t]≤bar v. Then E[e_t^2] ≤ q^t E[e_0^2] + fracbar v(1-q^t)1-q, so limsup_t→∞ E[e_t^2] ≤ fracbar v1-q.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-db12d116f9aa6f46","tex":"\\mathbb E_{u,u'\\sim\\nu_{\\bar S}}","aliases":"E_u,u'∼ν_bar S mathbb sim nu bar","display":"\\mathbb E_{u,u'\\sim\\nu_{\\bar S}}","kind":"use","excerpt":"nate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the future law of X_S: CIn_τ(S) = E_u,u'∼ν_bar S D ( L^ι_bar S,u (X_S(t+τ)), L^ι_bar ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-79ef905cb263faa3","tex":"E_{\\mathrm{und}}","aliases":"E_und mathrm","display":"E_{\\mathrm{und}}","kind":"use","excerpt":"The number of independent graph cycles is r=|E_und|-|V|+1 for a connected underlying graph, after treating parallel edges with their multiplicity.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f8b38dcb375cdefa","tex":"e_V","aliases":"e_V","display":"e_V","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f97d8732922dd40f","tex":"\\mathbb E_{\\widehat\\pi}","aliases":"E_widehatπ mathbb widehat pi","display":"\\mathbb E_{\\widehat\\pi}","kind":"use","excerpt":"For every policy π, uniform approximation gives | E_a∼π[widehat Q(a)] - E_a∼π[Q(a)] | ≤ε_Q. Therefore E_π^*[Q] - E_widehatπ[Q] · ≤ E_π^*[widehat Q] - E_widehatπ[widehat Q] +2ε_Q · ≤ δ_opt+2ε_Q. The deterministic bound is the special case of point-mass policies. If widehat a≠ a^*, then widehat Q(a^*) ≥ Q(a^*)-ε_Q > Q(widehat a)+ε_Q ≥ widehat Q(wideh","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-44985fac8d8e1583","tex":"E_X","aliases":"E_X","display":"E_X","kind":"use","excerpt":"p=open proof or implementation obligations, · v=verification and regression status. Define local encodings E_Hc=(g,d), E_Mc=(g,p), E_Xc=(p,v), E_Ac=(d,v). Each coordinate is represented at exactly two nodes, and the incidence pattern forms a four-cycle.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-227fd36e57043d33","tex":"\\mathbb E_{x,x'\\sim\\nu_S}","aliases":"E_x,x'∼ν_S mathbb sim nu","display":"\\mathbb E_{x,x'\\sim\\nu_S}","kind":"use","excerpt":"Let ι_S,x be a feasible intervention that changes the internal workspace coordinate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the fu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6f0b8f31ca61671d","tex":"E_{\\{1,2\\}}","aliases":"E_{1,2}","display":"E_{\\{1,2\\}}","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1), E_{1,3}^*E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-39790a497e1400c4","tex":"E_{\\{1,2\\}}^*","aliases":"E_{1,2}^*","display":"E_{\\{1,2\\}}^*","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1), E_{1,3}^*E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-422f8d1001733205","tex":"E_{\\{1,3\\}}","aliases":"E_{1,3}","display":"E_{\\{1,3\\}}","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1), E_{1,3}^*E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4d7fe3f083b3c317","tex":"E_{\\{1,3\\}}^*","aliases":"E_{1,3}^*","display":"E_{\\{1,3\\}}^*","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1), E_{1,3}^*E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4c80267899fbfb6a","tex":"E_{\\{2,3\\}}","aliases":"E_{2,3}","display":"E_{\\{2,3\\}}","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1), E_{1,3}^*E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0a0b918dfe9851c1","tex":"E_{\\{2,3\\}}^*","aliases":"E_{2,3}^*","display":"E_{\\{2,3\\}}^*","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1), E_{1,3}^*E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-449d97177f13fe51","tex":"\\mathbb E_{\\iota,\\iota'\\sim\\mathcal I_{\\Workspace}(r)}","aliases":"E_ι,ι'∼ I_Workspace(r) mathbb iota iota sim mathcal Workspace","display":null,"kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3edbeefacc00b930","tex":"\\mathbb E_{\\pi^*}","aliases":"E_π^* mathbb pi","display":"\\mathbb E_{\\pi^*}","kind":"use","excerpt":"For every policy π, uniform approximation gives | E_a∼π[widehat Q(a)] - E_a∼π[Q(a)] | ≤ε_Q. Therefore E_π^*[Q] - E_widehatπ[Q] · ≤ E_π^*[widehat Q] - E_widehatπ[widehat Q] +2ε_Q · ≤ δ_opt+2ε_Q. The deterministic bound is the special case of point-mass policies. If widehat a≠ a^*, then widehat Q(a^*) ≥ Q(a^*)-ε_Q > Q(widehat a)+ε_Q ≥ widehat Q(wideh","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6e0fbac3bb7a13eb","tex":"\\mathbb E_{\\sigma,\\sigma'\\sim\\pi}","aliases":"E_σ,σ'∼π mathbb sigma sigma sim pi","display":"\\mathbb E_{\\sigma,\\sigma'\\sim\\pi}","kind":"use","excerpt":"For a family S of admissible scaffold variants, define the cross-scaffold identity defect δ_q^scaf = inf_{J_σσ'} E_σ,σ'∼π d_q ( J_σσ'q_σ,q_σ' ), where J_σσ' transports the learned object between admissible coordinate systems. A low value indicates that the same abstract organization is recovered under multiple non-solution-bearing scaffolds.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f7a75cdc18f33e04","tex":"e_{\\phi(q)}^{\\mathrm{causal}}","aliases":"e_ϕ(q)^causal phi mathrm","display":"e_{\\phi(q)}^{\\mathrm{causal}}","kind":"use","excerpt":"[Reparameterization invariance of causal provenance] Let ϕ be a measurable bijection on the range of q, and let the divergence D be invariant under measurable bijections. Then e_ϕ(q)^causal = e_q^causal.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-adfc9557075cd429","tex":"e_\\ell","aliases":"e_ℓ ell","display":"e_\\ell","kind":"use","excerpt":"For a path γ=e_m⋯ e_1, Q_t(γ)T_γ P_s(γ) ≤ ∑_r=1^m · ( ∏_ℓ=r+1^m T_e_ℓ ) V_e_r · × ( ∏_ℓ=1^r-1 H_e_ℓ ). The same bound controls the defect between full core-to-core path transport and the pure core product. An exact Duhamel identity yields both estimates.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f8483dce1203449b","tex":"\\mathrm{eff}","aliases":"eff mathrm","display":"\\mathrm{eff}","kind":"use","excerpt":"Some natural operators may vanish on every global section while acting nontrivially on locally available but globally inconsistent directions. Define the annihilator Ann(Γ(H)) = {A∈ O_Workspace: π_Workspace(A)=0}. The effective workspace algebra is O_Workspace^eff = O_Workspace/Ann(Γ(H)). Its action on Γ(H) is faithful by construction.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-614832267c86d56d","tex":"\\mathsf{EMP}","aliases":"EMP mathsf","display":"\\mathsf{EMP}","kind":"use","excerpt":"The audit uses five labels: ll STD · = established concept, theorem, or technique, ADP · = established machinery adapted to relational workspaces, SYN · = nontrivial synthesis of established components, CAN · = candidate original contribution requiring priority review, EMP · = empirical claim not yet established in an uncontrolled system. These lab","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End","display":null,"kind":"use","excerpt":"Every core space carries a unital representation ρ_i: A⟶End(C_i). For a∈ A, the operator ρ_i(a) is the realization of the same abstract operation in local coordinates.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fd758461fe2026e4","tex":"\\operatorname{Endogenous}","aliases":"Endogenous operatorname","display":"\\operatorname{Endogenous}","kind":"use","excerpt":"The formation provenance of a workspace and its operational certification are separate coordinates. A fully scaffolded structure may satisfy every structural and causal workspace gate. A highly endogenous structure may fail to become coherent or causally useful. Symbolically, Endogenous(W) centernotLongrightarrow SRW_Θ(W), SRW_Θ(W) centernotLongrig","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7861f8f943b08243","tex":"\\operatorname{EndogenouslyConstituted}","aliases":"EndogenouslyConstituted operatorname","display":"\\operatorname{EndogenouslyConstituted}","kind":"use","excerpt":"subtopicOrthogonal formation-provenance axis Scaffolded ⟶ AdaptivelyParameterised ⟶ StructurallyAdaptive ⟶ EndogenouslyConstituted. This axis is independent of Levels~0--6. A scaffolded system can instantiate a high-level relational workspace, while an endogenously learned organization can fail every operational workspace gate.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-976c0648ee9a50a0","tex":"\\operatorname{Eq}","aliases":"Eq operatorname","display":"\\operatorname{Eq}","kind":"use","excerpt":"[Exact workspace gluing] For every finite edge-complete workspace cover U, the sequence 0 ⟶ mathscr W(G) xrightarrow R_ U check C^0( U;mathscr W) xrightarrow δ^0 check C^1( U;mathscr W) is exact. Equivalently, mathscr W(G) ≅ kerδ^0 ≅ Eq ( ∏_αmathscr W_α rightrightarrows ∏_α<βmathscr W_αβ ). Every pairwise compatible family of exact local workspace ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bb2e777e36c7d148","tex":"\\operatorname{Eq}_{\\mathcal U_{\\mathrm{flat}}}","aliases":"Eq_ U_flat operatorname mathcal mathrm","display":"\\operatorname{Eq}_{\\mathcal U_{\\mathrm{flat}}}","kind":"use","excerpt":"l U_flat = {G_ℓ:ℓ∈ L( T)} is itself an edge-complete cover of G. Let δ_flat^0 be its overlap coboundary and define W_flat = kerδ_flat^0. Theorem~[source reference] and Theorem~[source reference] give W_r ≅ W_flat ≅ mathscr W(G). Equivalently, Lim_ T^rec ( {mathscr W(G_ℓ)}_ℓ∈ L( T) ) ≅ Eq_ U_flat ( {mathscr W(G_ℓ)}_ℓ∈ L( T) ). The notation on the le","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-02d71103309509dd","tex":"\\mathsf{equivalence\\_classes}","aliases":"equivalence_classes mathsf","display":"\\mathsf{equivalence\\_classes}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5fd8bc9225f2a7c8","tex":"\\mathrm{er}","aliases":"er mathrm","display":"\\mathrm{er}","kind":"use","excerpt":"Define the accessible block support of c≠0 by supp_E(c) = {i∈ V:E_ic≠0} and define the node-erasure distance d_er = min_c≠0|supp_E(c)|.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-60c3fea27953f5b6","tex":"\\mathrm{erase}","aliases":"erase mathrm","display":"\\mathrm{erase}","kind":"use","excerpt":"Define the minimum reconstructing coalition size m_rec = min{|S|:α_S>0} and the maximum exact erasure tolerance k_erase = max{k:α^(k)>0}. The condition m_rec≥2 means that no individual encoding is sufficient for exact core recovery. The condition k_erase≥1 means that the system survives at least one node loss. These can hold simultaneously because ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c8f21276e6dfefea","tex":"\\ev","aliases":"ev","display":null,"kind":"use","excerpt":"Fix a root o. The evaluation map is ev_o:Γ(H)→ C_o, ev_o((c_i))=c_o.","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-126aa46f3bf4657c","tex":"\\ev_i","aliases":"ev_i ev","display":null,"kind":"use","excerpt":"Let C_* be a finite-dimensional abstract core. The exact static theory provides a complete local chart overline E_i = ev_i∘Ψ:C_*⟶ C_i, where Ψ:C_*→Γ(H) is the root-to-section isomorphism. A participant or artifact need not expose that complete chart. Let O_i:C_i⟶ Y_i be the declared observation or access map and define the accessible encoding E_i=O","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e61e397fd4f022f7","tex":"\\ev_o","aliases":"ev_o ev","display":null,"kind":"use","excerpt":"Fix a root o. The evaluation map is ev_o:Γ(H)→ C_o, ev_o((c_i))=c_o.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b84fe63eeee6a5b2","tex":"\\mathrm{experiment}","aliases":"experiment mathrm","display":"\\mathrm{experiment}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-40d0b0d9a0cac949","tex":"\\operatorname{ExtendedCognition}","aliases":"ExtendedCognition operatorname","display":"\\operatorname{ExtendedCognition}","kind":"use","excerpt":"Accordingly, ExtendedCognition centernotLongleftrightarrow SRW. An externally scaffolded process may be cognitively extended without being relationally irreducible. Conversely, an institution or multi-agent system may instantiate a certified relational workspace without being an extension of any single person's cognition.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aaec5c835f65f308","tex":"\\mathsf F","aliases":"F mathsf","display":"\\mathsf F","kind":"use","excerpt":"For each case we report the vector Card( K) = ( S, O, R, P, F, A, D, X, C ), where: ll S · = structural core and spectral coherence,","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a3382034c436fa8d","tex":"F^*","aliases":"F^*","display":"F^*","kind":"use","excerpt":"Suppose the edge maps are perturbed to widetilde H_e=H_e+Δ_e while the weights and vertex metrics remain fixed. Define δ_H^2 = max_i∈ V ∑_e:s(e)=iw_e Δ_e ^2. Let F=D_widetilde H-D_H. Then (Fc)_e=-sqrtw_e Δ_ec_s(e), F ≤δ_H. Since L_widetilde H-L_H =D_H^*F+F^*D_H+F^*F, we obtain L_widetilde H-L_H ≤ 2 D_H δ_H+δ_H^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-49ddaa56c96d4e05","tex":"f^{-1}","aliases":"f^-1","display":"f^{-1}","kind":"use","excerpt":"The reparameterized realization at μ is the original realization at f^-1(μ). It passes precisely when f^-1(μ)>λ_c, equivalently when μ>f(λ_c).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","excerpt":"thm:transfer-identification Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"l H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(x) - F_R-1∘⋯∘ F_0(x) · ≤ ∑_r=0^R-1 ( ∏_s=r+1^R-1L_s ) ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2c82feb782bf1c96","tex":"\\mathbb F_2^3","aliases":"F_2^3 mathbb","display":"\\mathbb F_2^3","kind":"use","excerpt":"Consider three agents: V_MA={P,M,R}, called the planner, monitor, and recorder. The global workspace-control state is c_t=(g_t,b_t,h_t)∈ F_2^3, where: ll g_t=goal bit, · b_t=environment or belief bit,","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b8ef7c6fdaebd0ea","tex":"f_C","aliases":"f_C","display":"f_C","kind":"use","excerpt":"Order the local coordinates as x=(q_P,p_P,p_C,f_C,q_N,f_N)^ T. Compatibility requires p_P=p_C, f_C=f_N, q_N=q_P. The discrepancy operator is D_HH = 0 · 1 · -1 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c69b8c0bc3dad383","tex":"F_{HMXA}","aliases":"F_HMXA","display":"F_{HMXA}","kind":"use","excerpt":"0 · -1 · 0 · 0 · 0 · 0 · 1 · 0 . The rows are orthogonal with squared norm two. Therefore ker D_HAI ≅ R^4, spec(L_HAI) = {0,0,0,0,2,2,2,2}, γ_HAI=2. The full accessible frame operator is F_HMXA=2I_4. No single node is injective. Every three-node coalition reconstructs the full state, and the two opposite pairs {H,X}, {M,A} are already sufficient. A","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-31d7232e6fa917d3","tex":"F_i","aliases":"F_i","display":"F_i","kind":"use","excerpt":"A representative local update is X_i(t+1) = F_i ( X_i(t), {M_e(t):t(e)=i}, A_i(t),E_t,ξ_i(t) ), with messages M_e(t) = K_e ( X_s(e)(t), W_t, E_t,ζ_e(t) ). This notation permits, but does not require, the current workspace state to modulate future transfers.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5022f1c21df97421","tex":"F_i^{\\mathcal D}","aliases":"F_i^ D mathcal","display":"F_i^{\\mathcal D}","kind":"use","excerpt":"The structural core can be preserved by faithful coarse-graining, while coalition counts generally are not. Let mathscr F_S^ D = bigvee_i∈ Smathscr F_i^ D be the information available to coalition S under decomposition D. For a declared loss ℓ, write Risk_ D(S) = inf_f E [ ℓ(Y^+,f(mathscr F_S^ D)) ].","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-43135e362a6370f0","tex":"f_N","aliases":"f_N","display":"f_N","kind":"use","excerpt":"Order the local coordinates as x=(q_P,p_P,p_C,f_C,q_N,f_N)^ T. Compatibility requires p_P=p_C, f_C=f_N, q_N=q_P. The discrepancy operator is D_HH = 0 · 1 · -1 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2c85b389b6e908b5","tex":"F_{PCN}","aliases":"F_PCN","display":"F_{PCN}","kind":"use","excerpt":"The full frame operator is F_PCN=E_P^*E_P+E_C^*E_C+E_N^*E_N=2I_3. For every two-node coalition S, λ_min(F_S)=1. Thus every two-node coalition reconstructs the complete core and, for noisy observations y_S=A_Sc+η_S, widehat c-c ≤ η_S . No individual node reconstructs c, because rank(E_P) = rank(E_C) = rank(E_N) =2<3. The realization is therefore sim","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c7c3f2242255d7d0","tex":"F_{q^{-1}(S)}^{\\mathcal D_f}","aliases":"F_q^-1(S)^ D_f mathcal","display":"F_{q^{-1}(S)}^{\\mathcal D_f}","kind":"use","excerpt":"Information faithfulness gives mathscr F_S^ D_c = bigvee_B∈ S bigvee_i∈ q^-1(B) mathscr F_i^ D_f = mathscr F_q^-1(S)^ D_f. The two Bayes risks optimize over the same measurable decision rules and are therefore equal. If a fine coalition of size k is sufficient within ε, the set of coarse blocks meeting that coalition has at most k elements and cont","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f7110079fe13a85a","tex":"F_r","aliases":"F_r","display":"F_r","kind":"use","excerpt":"[Approximate hierarchical collapse bound] Let F_r: H_r→ H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-62d46406dc5133a0","tex":"F_{R-1}","aliases":"F_R-1","display":"F_{R-1}","kind":"use","excerpt":"l H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(x) - F_R-1∘⋯∘ F_0(x) · ≤ ∑_r=0^R-1 ( ∏_s=r+1^R-1L_s ) ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-881a9b759497c5b7","tex":"F_S","aliases":"F_S","display":"F_S","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d1f07d6f77e9c9f8","tex":"F_S^{\\mathcal D}","aliases":"F_S^ D mathcal","display":"F_S^{\\mathcal D}","kind":"use","excerpt":"The structural core can be preserved by faithful coarse-graining, while coalition counts generally are not. Let mathscr F_S^ D = bigvee_i∈ Smathscr F_i^ D be the information available to coalition S under decomposition D. For a declared loss ℓ, write Risk_ D(S) = inf_f E [ ℓ(Y^+,f(mathscr F_S^ D)) ].","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3df7dd9390c0e91f","tex":"F_S^{\\mathcal D_c}","aliases":"F_S^ D_c mathcal","display":"F_S^{\\mathcal D_c}","kind":"use","excerpt":"Information faithfulness gives mathscr F_S^ D_c = bigvee_B∈ S bigvee_i∈ q^-1(B) mathscr F_i^ D_f = mathscr F_q^-1(S)^ D_f. The two Bayes risks optimize over the same measurable decision rules and are therefore equal. If a fine coalition of size k is sufficient within ε, the set of coarse blocks meeting that coalition has at most k elements and cont","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-924806c8226c3b76","tex":"F_S^{-1}","aliases":"F_S^-1","display":"F_S^{-1}","kind":"use","excerpt":"[Coalition reconstruction criterion] For a coalition S, the following are equivalent: item A_S is injective; item F_S is positive definite; item α_S>0; item there exists a linear decoder R_S with R_SA_S=I_C_*. When these conditions hold, the canonical decoder is R_S=A_S^†=F_S^-1A_S^*.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","excerpt":"Let the realized vertex state satisfy x_t+1=F_tx_t+ξ_t, where F_t: C_V(t)→ C_V(t+1) is the actual update and ξ_t is an unmodeled disturbance. Define the off-workspace error e_t= (I-P_t)x_t . Introduce α_t = (I-P_t+1)F_t(I-P_t) , which measures persistence of prior off-workspace error, and ℓ_t = (I-P_t+1)F_tP_t , which measures leakage of a valid wo","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5736cb794ac576a4","tex":"\\mathcal F_t","aliases":"F_t mathcal","display":"\\mathcal F_t","kind":"use","excerpt":"For stochastic systems, pathwise bounded disturbances may be too strong. Let { F_t} be a filtration containing the history through time t.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-832e56d8bd1b7868","tex":"F_\\lambda","aliases":"F_λ lambda","display":"F_\\lambda","kind":"use","excerpt":"[Pitchfork formation transition] For the map F_λ(m)=tanh(λ m), m∈[-1,1], the following hold. item If 0≤λ≤1, the unique fixed point is m=0. It is globally attracting for λ<1 and nonexpanding at λ=1. item If λ>1, there are exactly three fixed points -m_λ, quad0, m_λ, m_λ∈(0,1). The zero fixed point is unstable and the two nonzero fixed points are loc","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-25e2d291ac41f667","tex":"F_\\lambda'","aliases":"F_λ' lambda","display":"F_\\lambda'","kind":"use","excerpt":"The derivative is F_λ'(m) = λsech^2(λ m). At zero this equals λ, proving stability for λ<1 and instability for λ>1. At a nonzero fixed point, F_λ'(m_λ) = λ(1-m_λ^2) = frac(1-m_λ^2)artanh(m_λ)m_λ <1, because artanh(m) < fracm1-m^2 (0<m<1). The derivative is positive, so the nonzero fixed points are locally stable. Finally, expanding tanh(λ m) = λ m-","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-338458495169eede","tex":"\\mathsf{failures}","aliases":"failures mathsf","display":"\\mathsf{failures}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a860aaa58918688c","tex":"\\mathsf{family}","aliases":"family mathsf","display":"\\mathsf{family}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bbd553f714170673","tex":"\\mathbf{FinHilb}","aliases":"FinHilb mathbf","display":"\\mathbf{FinHilb}","kind":"use","excerpt":"For every i∈ V, assign a finite-dimensional Hilbert space X_i. For every edge, assign a linear transfer T_e:X_s(e)⟶ X_t(e). The path transfer is T_γ=T_e_m⋯ T_e_2T_e_1, T_1_i=I_X_i. Accordingly, the static transfer system is a functor X:Path(G)⟶FinHilb.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-eceac6300c526562","tex":"\\mathbf{FinHilb}^{\\times}","aliases":"FinHilb^× mathbf times","display":"\\mathbf{FinHilb}^{\\times}","kind":"use","excerpt":"Hence two regimes will be distinguished: item the diagram regime, in which arbitrary directed maps define compatible sections; and item the local-system regime, in which invertible core maps define a functor H:Π_1(G)→FinHilb^×. Theorems based on holonomy will explicitly assume the second regime.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8442e962516f4a6f","tex":"\\operatorname{Fix}","aliases":"Fix operatorname","display":"\\operatorname{Fix}","kind":"use","excerpt":"The obstruction associated with a loop is Ω_γ=H_γ-I_C_o. The maximal path-independent root core is C_* = ⋂_γ:o→ oker(H_γ-I). Equivalently, C_*=C_o^Hol_o=Fix(Hol_o).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fdcd52ea1af51a4f","tex":"\\mathrm{flat}","aliases":"flat mathrm","display":"\\mathrm{flat}","kind":"use","excerpt":"Let L( T) be the leaf set. Repeated edge completeness implies that the flattened family U_flat = {G_ℓ:ℓ∈ L( T)} is itself an edge-complete cover of G. Let δ_flat^0 be its overlap coboundary and define W_flat = kerδ_flat^0. Theorem~[source reference] and Theorem~[source reference] give W_r ≅ W_flat ≅ mathscr W(G). Equivalently, Lim_ T^rec ( {mathscr","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0ac9c953d40b5f8c","tex":"\\operatorname{Form}","aliases":"Form operatorname","display":"\\operatorname{Form}","kind":"use","excerpt":"For thresholds Θ_form, define Form_Θ_form(W)=1 only when all applicable gates pass: · e_q^MDL≥ τ_q^MDL, · · e_q^causal≥ τ_q^causal,","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4bcc4f762aa350c3","tex":"\\operatorname{Form}_{\\Theta_{\\mathrm{form}}}","aliases":"Form_Θ_form operatorname Theta mathrm","display":"\\operatorname{Form}_{\\Theta_{\\mathrm{form}}}","kind":"use","excerpt":"For thresholds Θ_form, define Form_Θ_form(W)=1 only when all applicable gates pass: · e_q^MDL≥ τ_q^MDL, · · e_q^causal≥ τ_q^causal,","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c9be4a45a389b946","tex":"\\mathsf{formation}","aliases":"formation mathsf","display":"\\mathsf{formation}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5be8587083dc0a80","tex":"\\mathrm{frame}","aliases":"frame mathrm","display":"\\mathrm{frame}","kind":"use","excerpt":"e bounds] Identify the old and new abstract cores with ran P and ran P' through their chosen core parameterizations. Let S be a declared coalition, let A_S:ran P→ Y_S, A_S':ran P'→ Y_S' be the resulting accessible coalition encoders, and let K_S be a unitary identification of their observation spaces. Suppose A_S'β-K_SA_S ≤ δ_S^frame < sqrtα_S. The","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-00c37c61b4205056","tex":"\\mathrm{full}","aliases":"full mathrm","display":"\\mathrm{full}","kind":"use","excerpt":"Use a vector rather than one aggregate score: C_full(W) =( · d_ε, g_ε, δ_coh, δ_align^ I, Δ_joint^ℓ, k_ε, · Δ_tr( Q), η_causal^ν, Δ_specific(r), η_loop(τ), · d_id, Rec_k, underline m_ D, Δ_dec, E_form, δ^scaf, δ^leak, · Δ_J, Δ_t^B, COut_τ(widehat S), CIn_τ(widehat S), · η_G(τ), δ_G^cue(τ), η_G→ A, δ_Q, η_V→ A, η_A→ Y, · η_D→ M, η_M→ R, η_R→ v, Δ_re","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6f57c86e47f3bde3","tex":"\\mathsf G","aliases":"G mathsf","display":"\\mathsf G","kind":"use","excerpt":"[Coarsening and secret-sharing trivializations] The following statements hold. item Any finite realization can be coarsened to a single displayed component. Relative to that decomposition, the full-system state is local and k_0≤1. item Let Z be uniform on a finite abelian group G. Let K be an independent uniform group element and define X_1=K, X_2=","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0a566b1db5acde7f","tex":"g_0","aliases":"g_0","display":"g_0","kind":"use","excerpt":"· +c δ_c+d δ_r. In particular, if all local structural, operation, closure, causal, and reconstruction defects are O(δ_rep), the relevant operator norms are uniformly bounded, and g≥ g_0>0, then d_id( R, R') ≤ C_1fracδ_repg_0 +C_2δ_rep +O (fracδ_rep^2g_0). After a fixed normalization of the non-spectral terms, this recovers the schematic target d_i","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-76b53653bff85b9e","tex":"G_b","aliases":"G_b","display":"G_b","kind":"use","excerpt":"Proceed upward from the leaves. The claim holds at every leaf by definition. Assume it holds for all children of an internal node a. Replacing each child space W_b by its canonical copy mathscr W(G_b), the recursive definition of W_a is exactly the compatible-family kernel for the edge-complete child cover of G_a. Theorem~[source reference] therefo","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4855ea42dd65b6f9","tex":"G_E","aliases":"G_E","display":"G_E","kind":"use","excerpt":"For each i∈ V, let g_i∈ GL(C_i), g=(g_i)_i∈ V. Define the vertex and edge direct sums G_V=⊕_i∈ Vg_i, G_E=⊕_e∈ Eg_t(e). The transformed transfers and local operation representations are H_e^g = g_t(e)H_eg_s(e)^-1, ρ_i^g(a) = g_iρ_i(a)g_i^-1. The fundamental covariance identity is D_H^gG_V=G_ED_H. It follows directly by evaluating the e-component: sq","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-367ba2ef0ff00656","tex":"G_E^{-1}","aliases":"G_E^-1","display":"G_E^{-1}","kind":"use","excerpt":"From D_H^g=G_ED_HG_V^-1 and unitarity, D_H^g^*D_H^g = G_VD_H^*G_E^-1G_ED_HG_V^-1 = G_VL_HG_V^-1. The spectral statements follow from unitary similarity.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0e616248d4202c7b","tex":"g_H","aliases":"g_H","display":"g_H","kind":"use","excerpt":"Order the local coordinates as x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^ T. The overlap discrepancies are g_H-g_M, p_M-p_X, v_X-v_A, d_A-d_H. Thus D_HAI = 1 · 0 · -1 · 0 · 0 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1da4412a091302e4","tex":"G_{j_P}","aliases":"G_j_P","display":"G_{j_P}","kind":"use","excerpt":"For any P∈ H_0, choose an index j_P such that P∈ H_0j_P. The event that every gate passes is contained in the event that gate j_P passes. Hence P_P(certificate passes) ≤ P_P(G_j_P passes) ≤ α_j_P ≤ max_jα_j. Taking the supremum over P∈ H_0 proves the claim.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ec7643e6086652d9","tex":"g_M","aliases":"g_M","display":"g_M","kind":"use","excerpt":"Order the local coordinates as x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^ T. The overlap discrepancies are g_H-g_M, p_M-p_X, v_X-v_A, d_A-d_H. Thus D_HAI = 1 · 0 · -1 · 0 · 0 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4fb3babb401b7d27","tex":"g_o","aliases":"g_o","display":"g_o","kind":"use","excerpt":"[Exact gauge covariance] The map G_V:Γ(H)⟶Γ(H^g) is an isomorphism. Therefore, Γ(H^g)=G_VΓ(H), dimΓ(H^g)=dimΓ(H). If the core transports are invertible and o is a root, then H_γ^g=g_oH_γ g_o^-1 for every loop γ:o→ o. Hence the holonomy-fixed core transforms by C_*^g=g_oC_*.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dc9b9defb8c535bf","tex":"g_o^{-1}","aliases":"g_o^-1","display":"g_o^{-1}","kind":"use","excerpt":"[Exact gauge covariance] The map G_V:Γ(H)⟶Γ(H^g) is an isomorphism. Therefore, Γ(H^g)=G_VΓ(H), dimΓ(H^g)=dimΓ(H). If the core transports are invertible and o is a root, then H_γ^g=g_oH_γ g_o^-1 for every loop γ:o→ o. Hence the holonomy-fixed core transforms by C_*^g=g_oC_*.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3f8720ad3f121275","tex":"G_r","aliases":"G_r","display":"G_r","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6bcb32fb1f100c04","tex":"G_s","aliases":"G_s","display":"G_s","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0f687fdea300a109","tex":"G_s'","aliases":"G_s'","display":"G_s'","kind":"use","excerpt":"Subtract the two update equations. The matched command and disturbance terms cancel, giving G_s+1-G_s+1' = R_s(G_s-G_s'). Iteration gives the exact product formula. The upper bound follows from submultiplicativity of the operator norm. The lower bound follows from σ_min(AB) ≥ σ_min(A)σ_min(B) for square linear maps.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aac1d194fd0d6a77","tex":"G_{s+1}","aliases":"G_s+1","display":"G_{s+1}","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0325bebadb9c3f54","tex":"G_{s+1}'","aliases":"G_s+1'","display":"G_{s+1}'","kind":"use","excerpt":"Subtract the two update equations. The matched command and disturbance terms cancel, giving G_s+1-G_s+1' = R_s(G_s-G_s'). Iteration gives the exact product formula. The upper bound follows from submultiplicativity of the operator norm. The lower bound follows from σ_min(AB) ≥ σ_min(A)σ_min(B) for square linear maps.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1481d5cc6f09169d","tex":"g_t","aliases":"g_t","display":"g_t","kind":"use","excerpt":"Let P_t project onto a selected rank-d low-energy workspace cluster of L_t, and set W_t=ran P_t. Write g_t=λ_d+1(L_t)-λ_d(L_t)>0 for the separating spectral gap and η_t^L=2 D_t δ_t^D+(δ_t^D)^2.","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e672dad3221e2320","tex":"G_t'","aliases":"G_t'","display":"G_t'","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d8313a7d60fe037e","tex":"G_{t+1}","aliases":"G_t+1","display":"G_{t+1}","kind":"use","excerpt":"Let G_t+1=C_t+1, where C_t+1 is an external instruction repeated at every step. Goal and action can be strongly correlated, but after cutting the cue channel, I(G_t+τ;G_tmidcut(C))=0. The system tracks an external command. It has not established internal goal retention.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d89f2f4784895ba3","tex":"g_{t+1}","aliases":"g_t+1","display":"g_{t+1}","kind":"use","excerpt":"In the absence of an explicit goal intervention, let g_t+1=g_t. Under matched cue-cut trials, do(g_t=0) and do(g_t=1) remain distinguishable at every later time: |g_t+τ-g_t+τ'|=1. Thus the cue-controlled retention factor is exactly one: η_G(τ)=1. The goal is not reloaded by an external channel after the initial intervention.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bf3710573ab410bc","tex":"g_{t+\\tau}","aliases":"g_t+τ tau","display":"g_{t+\\tau}","kind":"use","excerpt":"In the absence of an explicit goal intervention, let g_t+1=g_t. Under matched cue-cut trials, do(g_t=0) and do(g_t=1) remain distinguishable at every later time: |g_t+τ-g_t+τ'|=1. Thus the cue-controlled retention factor is exactly one: η_G(τ)=1. The goal is not reloaded by an external channel after the initial intervention.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3682e4bce114e5fc","tex":"G_{t+\\tau}'","aliases":"G_t+τ' tau","display":"G_{t+\\tau}'","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-554a8c8c16fa2a7e","tex":"g_{t+\\tau}'","aliases":"g_t+τ' tau","display":"g_{t+\\tau}'","kind":"use","excerpt":"In the absence of an explicit goal intervention, let g_t+1=g_t. Under matched cue-cut trials, do(g_t=0) and do(g_t=1) remain distinguishable at every later time: |g_t+τ-g_t+τ'|=1. Thus the cue-controlled retention factor is exactly one: η_G(τ)=1. The goal is not reloaded by an external channel after the initial intervention.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d27464ff8d8878b","tex":"g_u","aliases":"g_u","display":"g_u","kind":"use","excerpt":"A learned transfer map is not absolutely identifiable when local coordinates are latent. Under local invertible changes X_u'=g_uX_u, X_v'=g_vX_v, the same abstract transfer is represented by H_uv'=g_vH_uvg_u^-1. The identifiable object is therefore the gauge class [H]_ G = { (g_t(e)H_eg_s(e)^-1)_e∈ E : (g_i)_i∈ V∈ G }. Any formation claim that comp","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5a227a17a9544bf3","tex":"g_u^{\\mathsf T}","aliases":"g_u^ T mathsf","display":"g_u^{\\mathsf T}","kind":"use","excerpt":"The transformed covariances are Σ_Y'X'=g_vΣ_YXg_u^ T, Σ_X'X'=g_uΣ_XXg_u^ T. Substitution into the covariance formula and inversion of the second factor give the result.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-613889e905565d30","tex":"g_u^{-1}","aliases":"g_u^-1","display":"g_u^{-1}","kind":"use","excerpt":"A learned transfer map is not absolutely identifiable when local coordinates are latent. Under local invertible changes X_u'=g_uX_u, X_v'=g_vX_v, the same abstract transfer is represented by H_uv'=g_vH_uvg_u^-1. The identifiable object is therefore the gauge class [H]_ G = { (g_t(e)H_eg_s(e)^-1)_e∈ E : (g_i)_i∈ V∈ G }. Any formation claim that comp","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-935ed0707d78ccd5","tex":"G_V","aliases":"G_V","display":"G_V","kind":"use","excerpt":"For each i∈ V, let g_i∈ GL(C_i), g=(g_i)_i∈ V. Define the vertex and edge direct sums G_V=⊕_i∈ Vg_i, G_E=⊕_e∈ Eg_t(e). The transformed transfers and local operation representations are H_e^g = g_t(e)H_eg_s(e)^-1, ρ_i^g(a) = g_iρ_i(a)g_i^-1. The fundamental covariance identity is D_H^gG_V=G_ED_H. It follows directly by evaluating the e-component: sq","count":17},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-99b7a1e1268a1b18","tex":"g_v","aliases":"g_v","display":"g_v","kind":"use","excerpt":"A learned transfer map is not absolutely identifiable when local coordinates are latent. Under local invertible changes X_u'=g_uX_u, X_v'=g_vX_v, the same abstract transfer is represented by H_uv'=g_vH_uvg_u^-1. The identifiable object is therefore the gauge class [H]_ G = { (g_t(e)H_eg_s(e)^-1)_e∈ E : (g_i)_i∈ V∈ G }. Any formation claim that comp","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-713c93524d6b7d59","tex":"G_V^{-1}","aliases":"G_V^-1","display":"G_V^{-1}","kind":"use","excerpt":"[Unitary gauge covariance of the Laplacian] If every g_i is unitary, then G_V and G_E are unitary and L_H^g=G_VL_HG_V^-1. Consequently, spec(L_H^g)=spec(L_H), P_I(L_H^g)=G_VP_I(L_H)G_V^-1 for every spectral interval I.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a700fbdde5b7fb0b","tex":"G_{\\alpha\\beta}","aliases":"G_αβ alpha beta","display":"G_{\\alpha\\beta}","kind":"use","excerpt":"For pairwise overlaps write mathscr W_α=mathscr W(G_α), mathscr W_αβ=mathscr W(G_αβ), and r_αβ^α: mathscr W_α→mathscr W_αβ, r_αβ^β: mathscr W_β→mathscr W_αβ. The interface is therefore a cospan mathscr W_α xrightarrow r_αβ^α mathscr W_αβ xleftarrow r_αβ^β mathscr W_β, not necessarily a direct transfer map from one complete local workspace to the ot","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-876a72a7054676bb","tex":"G_\\beta","aliases":"G_β beta","display":"G_\\beta","kind":"use","excerpt":"Conversely, let w=(w_α)_α∈kerδ^0. For each vertex i∈ V, choose any chart G_α containing i and set c_i=(w_α)_i. If another chart G_β contains i, then i∈ G_αβ and overlap compatibility gives (w_α)_i=(w_β)_i. Thus c_i is well defined.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d90c073fa47b6a4","tex":"g_{\\varepsilon}","aliases":"g_ε varepsilon","display":"g_{\\varepsilon}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e4a1cd0461e0af2b","tex":"g_{\\varepsilon,t}","aliases":"g_ε,t varepsilon","display":"g_{\\varepsilon,t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bed42666d905a9fb","tex":"G_\\ell","aliases":"G_ℓ ell","display":"G_\\ell","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-092c7b43e3e56dc8","tex":"\\mathrm{gap}","aliases":"gap mathrm","display":"\\mathrm{gap}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-40dfe59cffa9d005","tex":"\\operatorname{Gate}","aliases":"Gate operatorname","display":"\\operatorname{Gate}","kind":"use","excerpt":"Symbolically, SRW(W_t;ς) = bigwedge_q∈ J(ς) Gate_q(W_t;ς). The gate set is claim-relative. A one-step operational workspace need not pass persistence or reentry gates, while a persistent reentrant workspace must. No scalar total score is sufficient: failure of a required gate cannot be hidden by high performance on another gate.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-55d0c18f4b14d30e","tex":"\\mathsf{gate\\_envelopes}","aliases":"gate_envelopes mathsf","display":"\\mathsf{gate\\_envelopes}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-83ae6a37427c4423","tex":"\\operatorname{Gate}_q","aliases":"Gate_q operatorname","display":"\\operatorname{Gate}_q","kind":"use","excerpt":"Symbolically, SRW(W_t;ς) = bigwedge_q∈ J(ς) Gate_q(W_t;ς). The gate set is claim-relative. A one-step operational workspace need not pass persistence or reentry gates, while a persistent reentrant workspace must. No scalar total score is sufficient: failure of a required gate cannot be hidden by high performance on another gate.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c24fe8fc6ba66b3b","tex":"\\mathsf{gates}","aliases":"gates mathsf","display":"\\mathsf{gates}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d0a77ad288998915","tex":"\\mathrm{goal}","aliases":"goal mathrm","display":"\\mathrm{goal}","kind":"use","excerpt":"Two goal codes are operationally equivalent, g∼_goalg', when they induce the same action ordering in every declared state and intervention background: Q_t^g(amid w) ≥ Q_t^g(a'mid w) Longleftrightarrow Q_t^g'(amid w) ≥ Q_t^g'(a'mid w). The abstract goal is the equivalence class [g]_goal. This quotient prevents a mere invertible recoding of a goal ch","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-946f8f73d2ba1e6a","tex":"\\operatorname{GoalPersistent}","aliases":"GoalPersistent operatorname","display":"\\operatorname{GoalPersistent}","kind":"use","excerpt":"The following implications are not licensed: Agent_Θ_A(Workspace) centernotLongrightarrow PhenomenallyConscious(Workspace), SelfMaintaining(Workspace) centernotLongrightarrow SelfAware(Workspace), GoalPersistent(Workspace) centernotLongrightarrow MoralPatient(Workspace). A thermostat-like controller can maintain a variable without representing itse","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-51685bedce2a9082","tex":"\\mathsf{goals}","aliases":"goals mathsf","display":"\\mathsf{goals}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f8a10287dd141fba","tex":"h^-","aliases":"h^-","display":"h^-","kind":"use","excerpt":"Let the one-step utility be U_g(a,r;b,h^-) = 1{a=g⊕ b⊕ h^-} +2 1{h^-lor r=1} -frac12r. For fixed (g,b,h^-), the unique maximizer is (a^*,r^*) = (g⊕ b⊕ h^-,1-h^-). Indeed: item a wrong task action loses one unit; item omitting repair when h^-=0 loses 3/2 units; and item performing unnecessary repair when h^-=1 loses 1/2 unit. The smallest optimal-ac","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"A different problem arises when only relative transition data are supplied. Let b=(b_αβ)∈check C^1( U;mathscr W) be a prescribed family of overlap offsets. The triple-overlap consistency condition is δ^1b=0. One seeks local potentials q=(q_α)∈check C^0 satisfying δ^0q=b. Define check H^1( U;mathscr W) = frackerδ^1imδ^0. Then the affine gluing probl","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b413b1ca96d2b256","tex":"\\mathcal H^1_{\\mathcal U}","aliases":"H^1_ U mathcal mathcal","display":"\\mathcal H^1_{\\mathcal U}","kind":"use","excerpt":"Equip the finite cochain spaces with Hilbert inner products and define the first vCech Laplacian Δ_1 = δ^0(δ^0)^* + (δ^1)^*δ^1. Let H^1_ U=kerΔ_1.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c91cbe4297f58788","tex":"H^g","aliases":"H^g","display":"H^g","kind":"use","excerpt":"For each i∈ V, let g_i∈ GL(C_i), g=(g_i)_i∈ V. Define the vertex and edge direct sums G_V=⊕_i∈ Vg_i, G_E=⊕_e∈ Eg_t(e). The transformed transfers and local operation representations are H_e^g = g_t(e)H_eg_s(e)^-1, ρ_i^g(a) = g_iρ_i(a)g_i^-1. The fundamental covariance identity is D_H^gG_V=G_ED_H. It follows directly by evaluating the e-component: sq","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4d3ed802d65d82f1","tex":"h^m","aliases":"h^m","display":"h^m","kind":"use","excerpt":"[Uniform path bound] If along the path T_r ≤τ, H_r ≤ h, V_r ≤ν, then the leakage and core-transport defects are bounded by ν∑_r=1^mτ^m-rh^r-1 = displaystyle νfracτ^m-h^mτ-h, · τ≠ h, [1.1em] displaystyle mντ^m-1, · τ=h. In particular, if τ≤1 and h≤1, the bound is at most mν.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b509ce898e2cd7d2","tex":"h^{r-1}","aliases":"h^r-1","display":"h^{r-1}","kind":"use","excerpt":"[Uniform path bound] If along the path T_r ≤τ, H_r ≤ h, V_r ≤ν, then the leakage and core-transport defects are bounded by ν∑_r=1^mτ^m-rh^r-1 = displaystyle νfracτ^m-h^mτ-h, · τ≠ h, [1.1em] displaystyle mντ^m-1, · τ=h. In particular, if τ≤1 and h≤1, the bound is at most mν.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ae5758b951d11adc","tex":"\\mathbf H^S","aliases":"H^S mathbf","display":"\\mathbf H^S","kind":"use","excerpt":"[Exact reconstruction implies coalition sufficiency] Let W be sufficient for Y relative to the full history H: Yindep Hmid W. If a coalition reconstructs the workspace exactly, W=R_S( H^S) almost surely, then Yindep Hmid H^S. Under logarithmic loss, Risk_log(S)=Risk_log(V).","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a3bcc231ad3de5cd","tex":"H_*","aliases":"H_*","display":"H_*","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-167569092d25c5d2","tex":"H_*'","aliases":"H_*'","display":"H_*'","kind":"use","excerpt":"[Gauge covariance of least squares] Under invertible coordinate transformations X'=g_uX, Y'=g_vY, the population least-squares solution transforms as H_*'=g_vH_*g_u^-1.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-836ca543c1e3ec09","tex":"H_{0j_P}","aliases":"H_0j_P","display":"H_{0j_P}","kind":"use","excerpt":"For any P∈ H_0, choose an index j_P such that P∈ H_0j_P. The event that every gate passes is contained in the event that gate j_P passes. Hence P_P(certificate passes) ≤ P_P(G_j_P passes) ≤ α_j_P ≤ max_jα_j. Taking the supremum over P∈ H_0 proves the claim.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d6011109fce5ce94","tex":"H_{12}","aliases":"H_12","display":"H_{12}","kind":"use","excerpt":"[Equivalent operations in different coordinates] Let C_1=C_2= R^2, choose S∈ GL_2( R), and set H_12=S, ρ_2(a)=Sρ_1(a)S^-1. Then H_12ρ_1(a) =Sρ_1(a) =ρ_2(a)S =ρ_2(a)H_12. The local matrices can look different while realizing the same abstract operation. This is the elementary model of representation change used throughout the theory.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-66fc929643bbe3da","tex":"H_2","aliases":"H_2","display":"H_2","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7d176a341e3cb7d7","tex":"H_{21}","aliases":"H_21","display":"H_{21}","kind":"use","excerpt":"Let C_1=C_2= R, use unit weights, and take two directed transfers H_12=a, H_21=b. Then D_H = -a · 1","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-493f788481b04eec","tex":"H_{23}","aliases":"H_23","display":"H_{23}","kind":"use","excerpt":"[One-dimensional core surviving a contradictory cycle] Let a three-node cycle have C_1=C_2=C_3= R^2 and H_12=I, H_23=I, H_31=J, J= 1 · 0 0 · -1 . The cycle holonomy is J, hence C_*=ker(J-I)=span{(1,0)^top}. The first coordinate circulates consistently through all three strata. The second returns with its sign reversed and therefore cannot define a ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7ea59a05cd0a885b","tex":"H_{31}","aliases":"H_31","display":"H_{31}","kind":"use","excerpt":"[One-dimensional core surviving a contradictory cycle] Let a three-node cycle have C_1=C_2=C_3= R^2 and H_12=I, H_23=I, H_31=J, J= 1 · 0 0 · -1 . The cycle holonomy is J, hence C_*=ker(J-I)=span{(1,0)^top}. The first coordinate circulates consistently through all three strata. The second returns with its sign reversed and therefore cannot define a ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d54388c828cd1c6a","tex":"H_{ah}","aliases":"H_ah","display":"H_{ah}","kind":"use","excerpt":"Consider three strata: h=human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. Th","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f10380c142f70a49","tex":"H_e^*","aliases":"H_e^*","display":"H_e^*","kind":"use","excerpt":"For z=(z_e)_e∈ E∈ C_E, the adjoint is (D_H^*z)_i = ∑_e:t(e)=isqrtw_e z_e - ∑_e:s(e)=isqrtw_e H_e^*z_e. Consequently, (L_Hc)_i = · ∑_e:t(e)=iw_e(c_i-H_ec_s(e))","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2c9c32461ec3f001","tex":"H_e^g","aliases":"H_e^g","display":"H_e^g","kind":"use","excerpt":"For each i∈ V, let g_i∈ GL(C_i), g=(g_i)_i∈ V. Define the vertex and edge direct sums G_V=⊕_i∈ Vg_i, G_E=⊕_e∈ Eg_t(e). The transformed transfers and local operation representations are H_e^g = g_t(e)H_eg_s(e)^-1, ρ_i^g(a) = g_iρ_i(a)g_i^-1. The fundamental covariance identity is D_H^gG_V=G_ED_H. It follows directly by evaluating the e-component: sq","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f7c152eadfb594c5","tex":"H_{e_1}","aliases":"H_e_1","display":"H_{e_1}","kind":"use","excerpt":"[Pathwise core preservation] Suppose V_e=0 for every edge in a directed path γ=e_m⋯ e_1. Then T_γ(C_s(γ))⊆ C_t(γ) and P_t(γ)T_γ P_s(γ) = H_e_m⋯ H_e_1.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5939f48905045f38","tex":"H_{e_m}","aliases":"H_e_m","display":"H_{e_m}","kind":"use","excerpt":"[Pathwise core preservation] Suppose V_e=0 for every edge in a directed path γ=e_m⋯ e_1. Then T_γ(C_s(γ))⊆ C_t(γ) and P_t(γ)T_γ P_s(γ) = H_e_m⋯ H_e_1.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-26ae0fa325355773","tex":"H_{e_r}","aliases":"H_e_r","display":"H_{e_r}","kind":"use","excerpt":"For a single edge, V_e=0 means T_e(c,0)=(H_ec,0), so core states remain in the core. Inductively, if the first r transfers map c to (H_e_r⋯ H_e_1c,0), applying T_e_r+1 gives (H_e_r+1H_e_r⋯ H_e_1c,0). The conclusion follows after m steps.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2bc430b0b695b58c","tex":"H_{e_{r+1}}","aliases":"H_e_r+1","display":"H_{e_{r+1}}","kind":"use","excerpt":"For a single edge, V_e=0 means T_e(c,0)=(H_ec,0), so core states remain in the core. Inductively, if the first r transfers map c to (H_e_r⋯ H_e_1c,0), applying T_e_r+1 gives (H_e_r+1H_e_r⋯ H_e_1c,0). The conclusion follows after m steps.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cc3d2226848e88b2","tex":"H_{e_\\ell}","aliases":"H_e_ℓ ell","display":"H_{e_\\ell}","kind":"use","excerpt":"For a path γ=e_m⋯ e_1, Q_t(γ)T_γ P_s(γ) ≤ ∑_r=1^m · ( ∏_ℓ=r+1^m T_e_ℓ ) V_e_r · × ( ∏_ℓ=1^r-1 H_e_ℓ ). The same bound controls the defect between full core-to-core path transport and the pure core product. An exact Duhamel identity yields both estimates.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-399afbaa0e9a8610","tex":"H_f","aliases":"H_f","display":"H_f","kind":"use","excerpt":"For composable edges e:i→ j and f:j→ k, direct block multiplication gives T_fT_e = H_fH_e+U_fV_e · H_fU_e+U_fK_e V_fH_e+K_fV_e · V_fU_e+K_fK_e . The induced core-to-core block is therefore P_kT_fT_eP_i = H_fH_e+U_fV_e. The term U_fV_e is a residual-mediated correction: information leaves the candidate core at the intermediate node and later re-ente","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-410175f42b606822","tex":"H_{hm}","aliases":"H_hm","display":"H_{hm}","kind":"use","excerpt":"Consider three strata: h=human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. Th","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cf5f97ebded34595","tex":"H_m","aliases":"H_m","display":"H_m","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0658bc031c42b0b5","tex":"H_{ma}","aliases":"H_ma","display":"H_{ma}","kind":"use","excerpt":"Consider three strata: h=human conceptual state, m=model-formal state, a=artifact state. Let the corresponding core spaces be C_h,C_m,C_a, with transfers H_hm:C_h→ C_m, H_ma:C_m→ C_a, H_ah:C_a→ C_h. A proposed shared item begins as c_h∈ C_h, is formalized as c_m=H_hmc_h, externalized as c_a=H_mac_m, and re-enters human reasoning as c_h'=H_ahc_a. Th","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0943f2790539cd70","tex":"H_P","aliases":"H_P","display":"H_P","kind":"use","excerpt":"[XOR witness pair] Under the intact law P, let X_1,X_2oversetiid∼Bernoulli(1/2), Y=X_1⊕ X_2. Then each (X_i,Y) pair is independent and uniform, so H_P(Ymid X_i)=1 bit, H_P(Ymid X_1,X_2)=0. Let Q make X_1,X_2,Y mutually independent fair bits. Then Q_(X_i,Y)=P_(X_i,Y) (i=1,2), but H_Q(Ymid X_1,X_2)=1 bit. Therefore Δ_joint^log=1 bit, k_0=2, Δ_tr(Q)=1","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d83201f3b36e1284","tex":"H_Q","aliases":"H_Q","display":"H_Q","kind":"use","excerpt":"ness pair] Under the intact law P, let X_1,X_2oversetiid∼Bernoulli(1/2), Y=X_1⊕ X_2. Then each (X_i,Y) pair is independent and uniform, so H_P(Ymid X_i)=1 bit, H_P(Ymid X_1,X_2)=0. Let Q make X_1,X_2,Y mutually independent fair bits. Then Q_(X_i,Y)=P_(X_i,Y) (i=1,2), but H_Q(Ymid X_1,X_2)=1 bit. Therefore Δ_joint^log=1 bit, k_0=2, Δ_tr(Q)=1 bit. Th","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d64df6fa5868f4c","tex":"H_r","aliases":"H_r","display":"H_r","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4408cd3dcd4b6b67","tex":"\\mathcal H_r","aliases":"H_r mathcal","display":"\\mathcal H_r","kind":"use","excerpt":"[Approximate hierarchical collapse bound] Let F_r: H_r→ H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-719f58cca33d0dd6","tex":"\\mathcal H_{r+1}","aliases":"H_r+1 mathcal","display":"\\mathcal H_{r+1}","kind":"use","excerpt":"[Approximate hierarchical collapse bound] Let F_r: H_r→ H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bd53539eaf104c23","tex":"H_{r-1}","aliases":"H_r-1","display":"H_{r-1}","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-480c877d455d2c2b","tex":"h_t","aliases":"h_t","display":"h_t","kind":"use","excerpt":"Consider three agents: V_MA={P,M,R}, called the planner, monitor, and recorder. The global workspace-control state is c_t=(g_t,b_t,h_t)∈ F_2^3, where: ll g_t=goal bit, · b_t=environment or belief bit,","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-723b629860f6ad6b","tex":"h_{t+1}","aliases":"h_t+1","display":"h_{t+1}","kind":"use","excerpt":"Post-action health evolves according to h_t+1=h_t^-lor r_t. Define the maintenance deficit e_t=1-h_t. Under the intact policy, h_t+1=1 and e_t+1=0 for both damage states. Under the matched no-repair intervention and h_t^-=0, e_t+1^(0)=1. Thus the measured causal repair gain is Δ_maint=1. The closed-loop recurrence has β=0, overline d=0, overlineϵ=0","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7b48fa10e3b53d5c","tex":"h_t^-","aliases":"h_t^-","display":"h_t^-","kind":"use","excerpt":"After a possible damage event, let the pre-repair health be h_t^-∈{0,1}. The system chooses a joint action u_t=(a_t,r_t)∈{0,1}^2, where a_t is a task action and r_t is a repair action. Define the optimal task action a_t^*=g_t⊕ b_t⊕ h_t^- and optimal repair action r_t^*=1-h_t^-. No individual node determines the full pair (a_t^*,r_t^*): item the pla","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-696fc63b00092a72","tex":"\\mathbf H_t^1","aliases":"H_t^1 mathbf","display":"\\mathbf H_t^1","kind":"use","excerpt":"Let P be the intact law of (Y_t^+, H_t^1,…, H_t^n). Define the local-marginal null class N_marg(P) = { Q: Q_ H_t^i=P_ H_t^i for every i }, and the stronger local-predictive-matched class N_pred(P) = { Q: Q_(Y_t^+, H_t^i) = P_(Y_t^+, H_t^i) for every i }. Because H_t^i can include the complete local trajectory, the latter class can preserve all sing","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a9e47d7bffd8e888","tex":"\\mathbf H_t^B","aliases":"H_t^B mathbf","display":"\\mathbf H_t^B","kind":"use","excerpt":"[Coalition log-risk identity] Assume H(Y_t^+)<∞. If A⊆ B⊆ V, then Risk_log(A)-Risk_log(B) = I ( Y_t^+; H_t^B mid H_t^A ). In particular, Risk_log(A)-Risk_log(V) = I ( Y_t^+; H_t mid H_t^A ).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b65458c78128212e","tex":"\\mathbf H_t^i","aliases":"H_t^i mathbf","display":"\\mathbf H_t^i","kind":"use","excerpt":"Let P be the intact law of (Y_t^+, H_t^1,…, H_t^n). Define the local-marginal null class N_marg(P) = { Q: Q_ H_t^i=P_ H_t^i for every i }, and the stronger local-predictive-matched class N_pred(P) = { Q: Q_(Y_t^+, H_t^i) = P_(Y_t^+, H_t^i) for every i }. Because H_t^i can include the complete local trajectory, the latter class can preserve all sing","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-193105fcec05fe58","tex":"\\mathbf H_t^n","aliases":"H_t^n mathbf","display":"\\mathbf H_t^n","kind":"use","excerpt":"Let P be the intact law of (Y_t^+, H_t^1,…, H_t^n). Define the local-marginal null class N_marg(P) = { Q: Q_ H_t^i=P_ H_t^i for every i }, and the stronger local-predictive-matched class N_pred(P) = { Q: Q_(Y_t^+, H_t^i) = P_(Y_t^+, H_t^i) for every i }. Because H_t^i can include the complete local trajectory, the latter class can preserve all sing","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9110570cc0125511","tex":"\\mathcal H_t^S","aliases":"H_t^S mathcal","display":"\\mathcal H_t^S","kind":"use","excerpt":"Let H_t^S be the history available to coalition S⊆ V. For a loss ℓ_G, define goal-reconstruction risk Risk_G(S) = inf_f E [ ℓ_G ( G_t, f( H_t^S) ) ]. For tolerance ε, define k_G,ε = min { |S|: Risk_G(S) ≤ Risk_G(V)+ε }. A goal is locally stored when k_G,ε=1. It is distributed relative to the declared decomposition when k_G,ε≥2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6926f4f60d700602","tex":"\\mathbf H_t^{\\{i\\}}","aliases":"H_t^{i} mathbf","display":"\\mathbf H_t^{\\{i\\}}","kind":"use","excerpt":"Define the coalition gap Δ_A^ℓ = Risk_ℓ(A)-Risk_ℓ(V) ≥0. The node-level joint advantage is Δ_joint^ℓ = min_i∈ V Δ_{i}^ℓ. Under logarithmic loss, Δ_joint^log = min_i∈ V I ( Y_t^+; H_t mid H_t^{i} ). Therefore Δ_joint^log>0 means that every individual component omits some predictive information available to the complete system.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7a0d8c928ba4c445","tex":"H_{uv}","aliases":"H_uv","display":"H_{uv}","kind":"use","excerpt":"A learned transfer map is not absolutely identifiable when local coordinates are latent. Under local invertible changes X_u'=g_uX_u, X_v'=g_vX_v, the same abstract transfer is represented by H_uv'=g_vH_uvg_u^-1. The identifiable object is therefore the gauge class [H]_ G = { (g_t(e)H_eg_s(e)^-1)_e∈ E : (g_i)_i∈ V∈ G }. Any formation claim that comp","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5cbce1312cfffccb","tex":"H_{uv}'","aliases":"H_uv'","display":"H_{uv}'","kind":"use","excerpt":"A learned transfer map is not absolutely identifiable when local coordinates are latent. Under local invertible changes X_u'=g_uX_u, X_v'=g_vX_v, the same abstract transfer is represented by H_uv'=g_vH_uvg_u^-1. The identifiable object is therefore the gauge class [H]_ G = { (g_t(e)H_eg_s(e)^-1)_e∈ E : (g_i)_i∈ V∈ G }. Any formation claim that comp","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4906fdd4162c9b6e","tex":"H_\\gamma","aliases":"H_γ gamma","display":"H_\\gamma","kind":"use","excerpt":"[Autonomous splitting] If U_e=V_e=0 for every edge, then the complete diagram decomposes as the direct sum of two diagrams, X≅ H⊕ K, and every path transfer satisfies T_γ = H_γ · 0 0 · K_γ .","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d31412c9a9c474b1","tex":"H_\\gamma^g","aliases":"H_γ^g gamma","display":"H_\\gamma^g","kind":"use","excerpt":"[Exact gauge covariance] The map G_V:Γ(H)⟶Γ(H^g) is an isomorphism. Therefore, Γ(H^g)=G_VΓ(H), dimΓ(H^g)=dimΓ(H). If the core transports are invertible and o is a root, then H_γ^g=g_oH_γ g_o^-1 for every loop γ:o→ o. Hence the holonomy-fixed core transforms by C_*^g=g_oC_*.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-032c6e2522ce5f6d","tex":"H_{\\gamma_1}","aliases":"H_γ_1 gamma","display":"H_{\\gamma_1}","kind":"use","excerpt":"[Finite matrix test] Let γ_1,…,γ_r be fundamental cycle generators and suppose dim C_o=d. Define the stacked obstruction matrix M = H_γ_1-I vdots H_γ_r-I . Then C_*=ker M, d_Workspace=d-rank( M).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3f77883e346050cc","tex":"H_{\\gamma_e}","aliases":"H_γ_e gamma","display":"H_{\\gamma_e}","kind":"use","excerpt":"Choose a spanning tree T⊆|G|. Let τ_i:o→ i be the unique tree path. Every non-tree edge e:i→ j determines a fundamental loop γ_e=τ_j^-1eτ_i with transfer H_γ_e=H_τ_j^-1H_eH_τ_i. The fundamental group of a finite graph is generated freely by these cotree loops. Hence it is enough to impose fixed-point conditions for a finite set of generators.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-834f9837f9904364","tex":"H_{\\gamma_{e_q}}","aliases":"H_γ_e_q gamma","display":"H_{\\gamma_{e_q}}","kind":"use","excerpt":"[Finite obstruction set] Let e_1,…,e_r be the non-tree edges of a connected finite graph. Then C_* = ⋂_q=1^r ker(H_γ_e_q-I).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0fc9921bbb8f1cb7","tex":"H_{\\gamma_r}","aliases":"H_γ_r gamma","display":"H_{\\gamma_r}","kind":"use","excerpt":"[Finite matrix test] Let γ_1,…,γ_r be fundamental cycle generators and suppose dim C_o=d. Define the stacked obstruction matrix M = H_γ_1-I vdots H_γ_r-I . Then C_*=ker M, d_Workspace=d-rank( M).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-52b32ca2bad9d022","tex":"H_{\\sigma_i}","aliases":"H_σ_i sigma","display":"H_{\\sigma_i}","kind":"use","excerpt":"Conversely, let x∈ C_o^Hol_o. Choose a path τ_i:o→ i for each vertex and define c_i=H_τ_ix. This definition is independent of the chosen path. Indeed, if σ_i:o→ i is another path, then σ_i^-1τ_i:o→ o is a loop, and invariance of x gives H_σ_i^-1H_τ_ix=x. Multiplying by H_σ_i yields H_τ_ix=H_σ_ix. For any edge e:i→ j, the paths eτ_i and τ_j both run","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a9acb2b8b4859b9b","tex":"H_{\\sigma_i}^{-1}","aliases":"H_σ_i^-1 sigma","display":"H_{\\sigma_i}^{-1}","kind":"use","excerpt":"Conversely, let x∈ C_o^Hol_o. Choose a path τ_i:o→ i for each vertex and define c_i=H_τ_ix. This definition is independent of the chosen path. Indeed, if σ_i:o→ i is another path, then σ_i^-1τ_i:o→ o is a loop, and invariance of x gives H_σ_i^-1H_τ_ix=x. Multiplying by H_σ_i yields H_τ_ix=H_σ_ix. For any edge e:i→ j, the paths eτ_i and τ_j both run","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c0d3e7a62ba3b56","tex":"H_{\\tau_i}","aliases":"H_τ_i tau","display":"H_{\\tau_i}","kind":"use","excerpt":"Choose for each vertex i a path τ_i:o→ i. Compatibility gives c_i=H_τ_ic_o. Thus a section with c_o=0 vanishes at every vertex. Hence ev_o is injective.","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-97fb4e2c47dd912d","tex":"H_{\\tau_i}^{-1}","aliases":"H_τ_i^-1 tau","display":"H_{\\tau_i}^{-1}","kind":"use","excerpt":"Choose paths τ_i:o→ i. Naturality along τ_i gives A_iH_τ_i=H_τ_iA_o, so A_i=H_τ_iA_oH_τ_i^-1. Therefore A_o determines every A_i, and Φ_o is injective.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-da20996fde2444dc","tex":"H_{\\tau_j}","aliases":"H_τ_j tau","display":"H_{\\tau_j}","kind":"use","excerpt":"independent of the chosen path. Indeed, if σ_i:o→ i is another path, then σ_i^-1τ_i:o→ o is a loop, and invariance of x gives H_σ_i^-1H_τ_ix=x. Multiplying by H_σ_i yields H_τ_ix=H_σ_ix. For any edge e:i→ j, the paths eτ_i and τ_j both run from o to j. Path independence therefore gives H_ec_i =H_eH_τ_ix =H_τ_jx =c_j. Thus (c_i) is a global compatib","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6b9f0513c575dd75","tex":"H_{\\tau_j}^{-1}","aliases":"H_τ_j^-1 tau","display":"H_{\\tau_j}^{-1}","kind":"use","excerpt":"Choose a spanning tree T⊆|G|. Let τ_i:o→ i be the unique tree path. Every non-tree edge e:i→ j determines a fundamental loop γ_e=τ_j^-1eτ_i with transfer H_γ_e=H_τ_j^-1H_eH_τ_i. The fundamental group of a finite graph is generated freely by these cotree loops. Hence it is enough to impose fixed-point conditions for a finite set of generators.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4c4425940c285435","tex":"H_\\ell","aliases":"H_ℓ ell","display":"H_\\ell","kind":"use","excerpt":"· = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r · × (∏_ℓ=1^r-1 H_ℓ ), and the same right-hand side bounds P_i_mT_m⋯ T_1P_i_0-H_m⋯ H_1 .","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f87328ab050cb4f3","tex":"\\mathrm{HAI}","aliases":"HAI mathrm","display":"\\mathrm{HAI}","kind":"use","excerpt":"Consider four heterogeneous components: V_HAI={H,M,X,A}, where: item H is a human researcher who maintains goals and semantic direction; item M is a language-model process that constructs local formalizations; item X is an execution and verification process that compiles, tests, or falsifies candidate artifacts; and item A is a persistent artifact ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7cdf7106c6849e41","tex":"\\mathrm{HH}","aliases":"HH mathrm","display":"\\mathrm{HH}","kind":"use","excerpt":"Consider two researchers and one persistent notebook: V_HH={P,C,N}, where: item P is a proposer who tracks the theorem statement and open proof obligations; item C is a critic who tracks proof obligations and falsification status; item N is a notebook that tracks the theorem statement and falsification status. The abstract core is c=(q,p,f)^ T∈ R^3","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-da4b6fe971d78911","tex":"\\mathrm{hist}","aliases":"hist mathrm","display":"\\mathrm{hist}","kind":"use","excerpt":"component q of widehatθ, define its scaffold-only Bayes risk under loss ℓ_q by R_q^Σ = inf_f E [ ℓ_q(q,f(Σ,Ω)) ]. Define the full formation risk by R_q^Σ, H = inf_g E [ ℓ_q(q,g(Σ, D_form,Ω)) ]. The history advantage is Δ_q^hist = R_q^Σ - R_q^Σ, H. A positive history advantage means that formation history contains information about q that is not alr","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5ab62d5df4f5b229","tex":"\\Hol","aliases":"Hol","display":null,"kind":"use","excerpt":"[Holonomy commutes with abstract operations] For every loop γ:o→ o, H_γρ_o(a)=ρ_o(a)H_γ. Hence ρ_o( A) ⊆ Comm(Hol_o), where Comm(Hol_o) is the commutant of the holonomy representation.","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f1f2427536861709","tex":"\\operatorname{Hol}","aliases":"Hol operatorname","display":"\\operatorname{Hol}","kind":"use","excerpt":"subtopicT_1: Limit--holonomy theorem Γ(H)≅ C_o^Hol_o.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b822685611d43bd7","tex":"\\operatorname{hol}","aliases":"hol operatorname","display":"\\operatorname{hol}","kind":"use","excerpt":"Fix a root o∈ V. For every loop class [γ]∈π_1(|G|,o), define the holonomy Hol([γ])=H_γ∈ GL(C_o). Functoriality gives a group representation hol_o: π_1(|G|,o)⟶ GL(C_o). Its image is the holonomy group Hol_o=im(hol_o).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6e5628079ba2c0a8","tex":"\\Hol_o","aliases":"Hol_o Hol","display":null,"kind":"use","excerpt":"[Holonomy commutes with abstract operations] For every loop γ:o→ o, H_γρ_o(a)=ρ_o(a)H_γ. Hence ρ_o( A) ⊆ Comm(Hol_o), where Comm(Hol_o) is the commutant of the holonomy representation.","count":17},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cc8f936d475b065a","tex":"\\operatorname{Hol}_o","aliases":"Hol_o operatorname","display":"\\operatorname{Hol}_o","kind":"use","excerpt":"subtopicT_1: Limit--holonomy theorem Γ(H)≅ C_o^Hol_o.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-72aa7b97f7a2b149","tex":"\\operatorname{hol}_o","aliases":"hol_o operatorname","display":"\\operatorname{hol}_o","kind":"use","excerpt":"Fix a root o∈ V. For every loop class [γ]∈π_1(|G|,o), define the holonomy Hol([γ])=H_γ∈ GL(C_o). Functoriality gives a group representation hol_o: π_1(|G|,o)⟶ GL(C_o). Its image is the holonomy group Hol_o=im(hol_o).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fb5fd77dce771dc4","tex":"\\mathsf{horizons}","aliases":"horizons mathsf","display":"\\mathsf{horizons}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-40f7120e325dc5ce","tex":"\\mathcal I'","aliases":"I' mathcal","display":"\\mathcal I'","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c0c5c19ddcb59f53","tex":"I_{\\mathbb P}","aliases":"I_ P mathbb","display":"I_{\\mathbb P}","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-289ba939d5127364","tex":"I_{\\mathbb P^\\iota}","aliases":"I_ P^ι mathbb iota","display":"I_{\\mathbb P^\\iota}","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f18b3c49fd8055ae","tex":"i_0","aliases":"i_0","display":"i_0","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0475b2ac7d225af5","tex":"i_2","aliases":"i_2","display":"i_2","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ba0538909d6a977","tex":"I_3","aliases":"I_3","display":"I_3","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1),","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-78fd137c582a5e5f","tex":"I_4","aliases":"I_4","display":"I_4","kind":"use","excerpt":"0 · -1 · 0 · 0 · 0 · 0 · 1 · 0 . The rows are orthogonal with squared norm two. Therefore ker D_HAI ≅ R^4, spec(L_HAI) = {0,0,0,0,2,2,2,2}, γ_HAI=2. The full accessible frame operator is F_HMXA=2I_4. No single node is injective. Every three-node coalition reconstructs the full state, and the two opposite pairs {H,X}, {M,A} are already sufficient. A","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-11d0c15b5bb28442","tex":"\\mathcal I_A","aliases":"I_A mathcal","display":"\\mathcal I_A","kind":"use","excerpt":"A controlled relational realization extends Definition~[source reference] by the tuple A_Workspace = ( R, G, A_t, Y^+, {U_g}_g∈ G, Π, I_A, I_G, I_D, I_R ). Here: item R is the underlying relational realization; item G is a measurable goal-code space; item A_t is the feasible action set at time t; item Y^+ is the relevant future trajectory space; it","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c5fb6d506693daa6","tex":"I_{C_*}","aliases":"I_C_*","display":"I_{C_*}","kind":"use","excerpt":"[Coalition reconstruction criterion] For a coalition S, the following are equivalent: item A_S is injective; item F_S is positive definite; item α_S>0; item there exists a linear decoder R_S with R_SA_S=I_C_*. When these conditions hold, the canonical decoder is R_S=A_S^†=F_S^-1A_S^*.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8f47cf0699671f4b","tex":"I_{C_o}","aliases":"I_C_o","display":"I_{C_o}","kind":"use","excerpt":"The obstruction associated with a loop is Ω_γ=H_γ-I_C_o. The maximal path-independent root core is C_* = ⋂_γ:o→ oker(H_γ-I). Equivalently, C_*=C_o^Hol_o=Fix(Hol_o).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-20d6646340ce20f0","tex":"I_d","aliases":"I_d","display":"I_d","kind":"use","excerpt":"Let every node store a complete copy of the same core: E_i=I_d for every i. Then coherence is perfect and reconstruction is robust, but k_0=1, min_iδ_i^local=0. If the archive is never used in downstream control, then also η_causal=0. This is a shared representation or archive, not a relationally irreducible workspace.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0feb57923f5d12c5","tex":"\\mathcal I_D","aliases":"I_D mathcal","display":"\\mathcal I_D","kind":"use","excerpt":"A controlled relational realization extends Definition~[source reference] by the tuple A_Workspace = ( R, G, A_t, Y^+, {U_g}_g∈ G, Π, I_A, I_G, I_D, I_R ). Here: item R is the underlying relational realization; item G is a measurable goal-code space; item A_t is the feasible action set at time t; item Y^+ is the relevant future trajectory space; it","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8965caae02b2b060","tex":"\\mathcal I_G","aliases":"I_G mathcal","display":"\\mathcal I_G","kind":"use","excerpt":"A controlled relational realization extends Definition~[source reference] by the tuple A_Workspace = ( R, G, A_t, Y^+, {U_g}_g∈ G, Π, I_A, I_G, I_D, I_R ). Here: item R is the underlying relational realization; item G is a measurable goal-code space; item A_t is the feasible action set at time t; item Y^+ is the relevant future trajectory space; it","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2ee8ac8c8d5446ba","tex":"i_m","aliases":"i_m","display":"i_m","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c6b4e7520988cc0","tex":"I_P","aliases":"I_P","display":"I_P","kind":"use","excerpt":"] Let Y be finite, let P be the intact law at a fixed decision time, and let W_t=π_t( H_t) be a structurally supported candidate. Let Q⊆ N_pred(P) be a nonempty, predeclared panel of validated relation-breaking nulls. Assume: item exact structural--operational alignment, I_P(Y_t^+; H_tmid W_t)=0; item a positive node-level information gap, δ_node =","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-afccf775c866af1f","tex":"i_r","aliases":"i_r","display":"i_r","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-43d39e6da3f47a01","tex":"i_{r-1}","aliases":"i_r-1","display":"i_{r-1}","kind":"use","excerpt":"For every step, T_rι_i_r-1^C = ι_i_r^CH_r+ ι_i_r^RV_r. Telescope the sequence T_m⋯ T_r+1ι_i_r^C H_r⋯ H_1, r=0,1,…,m. The difference between consecutive terms is exactly the r-th summand in the displayed formula. Projection onto R_i_m or C_i_m, followed by submultiplicativity of the operator norm, gives the two bounds.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f5481a11d254e18f","tex":"\\mathcal I_{\\mathrm{rep}}","aliases":"I_rep mathcal mathrm","display":"\\mathcal I_{\\mathrm{rep}}","kind":"use","excerpt":"A complete replacement audit must report at least I_rep = ( q_str, δ_op, δ_causal, δ_rec, μ^keep ). The failure modes are different: ll q_str large · the shared subspace changed,","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-45bbd45b21387532","tex":"\\mathcal I_W","aliases":"I_W mathcal","display":"\\mathcal I_W","kind":"use","excerpt":"D,Y^+,ℓ,τ, I, N,Θ); item the admissible decomposition family D and specification family S; item the target trajectory Y^+, loss, horizon, and environment family; item the candidate structural workspace extractor; item the goal family G, utility normalization, and feasible action set; item the intervention families I_W, I_G, I_A, I_D, I_R; item the ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f9522255b4f6e73e","tex":"\\mathcal I_{\\Workspace}","aliases":"I_Workspace mathcal Workspace","display":null,"kind":"use","excerpt":"A causal effect on the target is not workspace-specific if an equal-cost perturbation outside the candidate workspace produces the same effect. Let c(ι) be a declared intervention cost and let I_Workspace(r), I_⊥(r) be matched families with cost r, the first changing workspace coordinates and the second changing complementary coordinates while pres","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-72a374675a00d284","tex":"I_{X_i}","aliases":"I_X_i","display":"I_{X_i}","kind":"use","excerpt":"For every i∈ V, assign a finite-dimensional Hilbert space X_i. For every edge, assign a linear transfer T_e:X_s(e)⟶ X_t(e). The path transfer is T_γ=T_e_m⋯ T_e_2T_e_1, T_1_i=I_X_i. Accordingly, the static transfer system is a functor X:Path(G)⟶FinHilb.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-14473aae59c0c726","tex":"I_{\\Gamma(H)}","aliases":"I_Γ(H) Gamma","display":"I_{\\Gamma(H)}","kind":"use","excerpt":"[Inherited algebraic structure] Any algebraic relation satisfied by ρ_i( A) is inherited by the global action. In particular, if a,b∈ A, then ρ_Γ(ab)=ρ_Γ(a)ρ_Γ(b), ρ_Γ(1)=I_Γ(H).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8f79d1a8f511506d","tex":"\\mathcal I_{\\perp}","aliases":"I_⊥ mathcal perp","display":"\\mathcal I_{\\perp}","kind":"use","excerpt":"A causal effect on the target is not workspace-specific if an equal-cost perturbation outside the candidate workspace produces the same effect. Let c(ι) be a declared intervention cost and let I_Workspace(r), I_⊥(r) be matched families with cost r, the first changing workspace coordinates and the second changing complementary coordinates while pres","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0f6f9e87439c63ef","tex":"\\mathrm{id}","aliases":"id mathrm","display":"\\mathrm{id}","kind":"use","excerpt":"Persistence does not require token identity. Let R_t and R_t+1 be temporally successive realizations. Continuity is possible when there exists an admissible transport J_t: R_trightsquigarrow R_t+1 that preserves the operative core and its causal role within tolerance: d_id(Workspace_t,Workspace_t+1)≤ε. Thus a workspace may persist by reconstruction","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b09c3d3928fe2c8b","tex":"\\operatorname{id}","aliases":"id operatorname","display":"\\operatorname{id}","kind":"use","excerpt":"The observational state is the special case I={id}. A causal workspace claim should align with Z_t^ I for a declared intervention family rather than only with the observational state.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8cea4c32b1be7863","tex":"\\mathsf{identified\\_sets}","aliases":"identified_sets mathsf","display":"\\mathsf{identified\\_sets}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bf97ee78c336c089","tex":"\\mathrm{iid}","aliases":"iid mathrm","display":"\\mathrm{iid}","kind":"use","excerpt":"[XOR witness pair] Under the intact law P, let X_1,X_2oversetiid∼Bernoulli(1/2), Y=X_1⊕ X_2. Then each (X_i,Y) pair is independent and uniform, so H_P(Ymid X_i)=1 bit, H_P(Ymid X_1,X_2)=0. Let Q make X_1,X_2,Y mutually independent fair bits. Then Q_(X_i,Y)=P_(X_i,Y) (i=1,2), but H_Q(Ymid X_1,X_2)=1 bit. Therefore Δ_joint^log=1 bit, k_0=2, Δ_tr(Q)=1","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"Fix a root o∈ V. For every loop class [γ]∈π_1(|G|,o), define the holonomy Hol([γ])=H_γ∈ GL(C_o). Functoriality gives a group representation hol_o: π_1(|G|,o)⟶ GL(C_o). Its image is the holonomy group Hol_o=im(hol_o).","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ef28ee4048cca398","tex":"\\mathrm{in}","aliases":"in mathrm","display":"\\mathrm{in}","kind":"use","excerpt":"Core leakage, residual contamination, and total transfer defect are measured by λ_e^out= V_e , λ_e^in= U_e , δ_e^tr = U_e + V_e . These quantities depend on the chosen norms and projectors. Their empirical meaning must therefore be tied to a declared intervention and prediction protocol.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2e10adac9c475997","tex":"\\indep","aliases":"indep","display":null,"kind":"use","excerpt":"[Predictive-state sufficiency and minimality, T_12] Under Definition~[source reference]: item Z_t is predictively sufficient: Y_t^+indep H_tmid Z_t. item If S_t=s( H_t) is any other history statistic satisfying Y_t^+indep H_tmid S_t, then there is a measurable map ϕ such that Z_t=ϕ(S_t) almost surely. item Consequently, any two minimal sufficient p","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-781a09724b8c7d83","tex":"\\inf_{\\mathfrak J}","aliases":"inf_ J inf mathfrak","display":"\\inf_{\\mathfrak J}","kind":"use","excerpt":"A composite identity cost is Cost( J) = a q_str +b δ_op( J) +c δ_causal(β) +d δ_rec(β), where a,b,c,d>0 are declared scales. Define d_id( R, R') = inf_ J Cost( J). This quantity is a task-relative optimization surrogate until the admissible correspondence class, intervention matching, divergence, and scales are fixed. It does not replace the coordi","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-580d0b7d9283e7d5","tex":"\\inf_{\\mathfrak J_{0:T}}","aliases":"inf_ J_0:T inf mathfrak","display":"\\inf_{\\mathfrak J_{0:T}}","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-be6abe51004e9b2d","tex":"\\inf_{\\mathfrak R_0\\in\\mathfrak M}","aliases":"inf_ R_0∈ M inf mathfrak in mathfrak","display":"\\inf_{\\mathfrak R_0\\in\\mathfrak M}","kind":"use","excerpt":"In finite samples, the identified set is itself uncertain. Let widehat R_1-δ be a confidence region satisfying inf_ R_0∈ M P_ R_0 ( R_0∈ widehat R_1-δ ) ≥1-δ. A conservative robust lower margin is widehat r_1-δ = inf_ R∈widehat R_1-δ inf_ς∈ S min_q m_q(W_ R;ς). If widehat r_1-δ>0, a false specification-robust certificate has probability at most δ, ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bdfd9d1eaaba0f76","tex":"\\inf_{\\mathfrak R\\in\\mathfrak R_{\\mathrm{ID}}}","aliases":"inf_ R∈ R_ID inf mathfrak in mathfrak mathrm","display":"\\inf_{\\mathfrak R\\in\\mathfrak R_{\\mathrm{ID}}}","kind":"use","excerpt":"For robustness across specifications, define the realization-level robust margin r_ S( R) = inf_ς∈ S min_q∈ J_* m_q(W_ R;ς) and its identified range underline r_ID, S = inf_ R∈ R_ID r_ S( R), overline r_ID, S = sup_ R∈ R_ID r_ S( R).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-51172923785c0b52","tex":"\\inf_{\\mathfrak R\\in\\widehat{\\mathfrak R}_{1-\\delta}}","aliases":"inf_ R∈widehat R_1-δ inf mathfrak in widehat mathfrak delta","display":"\\inf_{\\mathfrak R\\in\\widehat{\\mathfrak R}_{1-\\delta}}","kind":"use","excerpt":"In finite samples, the identified set is itself uncertain. Let widehat R_1-δ be a confidence region satisfying inf_ R_0∈ M P_ R_0 ( R_0∈ widehat R_1-δ ) ≥1-δ. A conservative robust lower margin is widehat r_1-δ = inf_ R∈widehat R_1-δ inf_ς∈ S min_q m_q(W_ R;ς). If widehat r_1-δ>0, a false specification-robust certificate has probability at most δ, ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aa35d7abfb11bc62","tex":"\\inf_d","aliases":"inf_d inf","display":"\\inf_d","kind":"use","excerpt":"For A⊆ V, the coalition statistic is H_t^A. Given an action space A and loss ℓ, define its unrestricted Bayes risk by Risk_ℓ(A) = inf_d E [ ℓ(Y_t^+,d( H_t^A)) ]. The infimum is over measurable decision rules for which the expectation is defined. The full-system risk is Risk_ℓ(V).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e63253b3f5e9bda6","tex":"\\inf_{d}","aliases":"inf_d inf","display":"\\inf_{d}","kind":"use","excerpt":"Sufficiency has a direct decision-theoretic interpretation. Use the convention TV(P,Q)=sup_A|P(A)-Q(A)|. Let A be a finite action set and ℓ: Y× A→[0,1] a bounded loss. Define Risk_ℓ(S_t) = inf_d E [ ℓ(Y_t^+,d(S_t)) ].","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-229edcd074540eda","tex":"\\inf_f","aliases":"inf_f inf","display":"\\inf_f","kind":"use","excerpt":"For any component q of widehatθ, define its scaffold-only Bayes risk under loss ℓ_q by R_q^Σ = inf_f E [ ℓ_q(q,f(Σ,Ω)) ]. Define the full formation risk by R_q^Σ, H = inf_g E [ ℓ_q(q,g(Σ, D_form,Ω)) ]. The history advantage is Δ_q^hist = R_q^Σ - R_q^Σ, H. A positive history advantage means that formation history contains information about q that is","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1c9170a18b7566b4","tex":"\\inf_g","aliases":"inf_g inf","display":"\\inf_g","kind":"use","excerpt":"For any component q of widehatθ, define its scaffold-only Bayes risk under loss ℓ_q by R_q^Σ = inf_f E [ ℓ_q(q,f(Σ,Ω)) ]. Define the full formation risk by R_q^Σ, H = inf_g E [ ℓ_q(q,g(Σ, D_form,Ω)) ]. The history advantage is Δ_q^hist = R_q^Σ - R_q^Σ, H. A positive history advantage means that formation history contains information about q that is","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3b4ab70d2dd72e60","tex":"\\inf_q","aliases":"inf_q inf","display":"\\inf_q","kind":"use","excerpt":"[Harmonic gluing obstruction] For every cocycle b∈kerδ^1, there is an orthogonal decomposition b=δ^0q^†+h, h∈ H^1_ U, where q^†=(δ^0)^† b is a least-squares potential. Moreover, inf_q b-δ^0q = h = P_ H^1_ Ub . Hence b is exactly integrable if and only if its harmonic component vanishes.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-92d8273a19fc51e7","tex":"\\inf_{\\widehat C}","aliases":"inf_widehat C inf widehat","display":"\\inf_{\\widehat C}","kind":"use","excerpt":") under E. More strongly, for every possibly randomized decision rule widehat C based only on data generated by E, P_ R ( widehat C≠ C_ R ) + P_ R' ( widehat C≠ C_ R' ) =1, where C_ R=Cert(W_ R;ς). Consequently, inf_widehat C max{ P_ R(widehat C≠ C_ R), P_ R'(widehat C≠ C_ R') } ≥frac12.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-713572657e012247","tex":"\\inf_{\\{J_{\\sigma\\sigma'}\\}}","aliases":"inf_{J_σσ'} inf sigma sigma","display":"\\inf_{\\{J_{\\sigma\\sigma'}\\}}","kind":"use","excerpt":"For a family S of admissible scaffold variants, define the cross-scaffold identity defect δ_q^scaf = inf_{J_σσ'} E_σ,σ'∼π d_q ( J_σσ'q_σ,q_σ' ), where J_σσ' transports the learned object between admissible coordinate systems. A low value indicates that the same abstract organization is recovered under multiple non-solution-bearing scaffolds.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fc04a6128aef58ee","tex":"\\inf_{\\varsigma\\in\\mathfrak S}","aliases":"inf_ς∈ S inf varsigma in mathfrak","display":"\\inf_{\\varsigma\\in\\mathfrak S}","kind":"use","excerpt":"For each gate q∈ J_*, define the specification envelope underline m_q^ S(W) = inf_ς∈ Sm_q(W;ς), overline m_q^ S(W) = sup_ς∈ Sm_q(W;ς). Its sensitivity width is Δ_q^spec(W) = overline m_q^ S(W) - underline m_q^ S(W). The robust certificate margin is r_ S(W) = inf_ς∈ S min_q∈ J_*m_q(W;ς). Then Cert_ S(W)=1 Longleftrightarrow r_ S(W)>0.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6a721bbbb12055c2","tex":"\\inf_{\\psi_A}","aliases":"inf_ψ_A inf psi","display":"\\inf_{\\psi_A}","kind":"use","excerpt":"Let d_ P be a bounded metric generating the measurable structure on P( Y). Define the reconstruction defect of the canonical state from coalition A by δ_A^Z = inf_ψ_A E [ d_ P ( Z_t,ψ_A( H_t^A) ) ].","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-06f9a3ba03a67bc4","tex":"\\mathrm{init}","aliases":"init mathrm","display":"\\mathrm{init}","kind":"use","excerpt":"[Handoff initialization] If α_S>0 and the survivors observe y=A_Sc+η, initialize the replacement by x_r'^init =E_r''A_S^† y. Then x_r'^init-E_r''c ≤ frac E_r'' η sqrtα_S.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-132f03f7394f2bda","tex":"\\Instability","aliases":"Instability","display":null,"kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f14d41f943a5a39d","tex":"\\Instability_t","aliases":"Instability_t Instability","display":null,"kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-55466b76f7f0c090","tex":"\\mathrm{int}","aliases":"int mathrm","display":"\\mathrm{int}","kind":"use","excerpt":"The edgewise defect is Ξ_e(a) = H_eρ_s(e)(a)-ρ_t(e)(a)H_e. For a bounded generating family G⊆ A, define η_int( G) = sup_a∈ G ∑_e∈ Ew_e Ξ_e(a) ^2. When A is normed and its unit ball is appropriate, one may take G={a: a _ A≤1}. Exact representational universality means η_int=0, while approximate universality means η_int≤ε_int. The use of a generating","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-186d0bb8a21f89db","tex":"\\mathsf{interventions}","aliases":"interventions mathsf","display":"\\mathsf{interventions}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-96b0f5abe3a4731a","tex":"\\mathfrak J","aliases":"J mathfrak","display":"\\mathfrak J","kind":"use","excerpt":"For an admissible correspondence J, define the operation defect on a declared unit ball A_1 by δ_op( J) = sup_a∈ A_1 A_ W''β-β A_ W . The primary certification object is the identity profile d_id( J) = ( q_str, δ_op( J), δ_causal(β), δ_rec(β); μ^keep ). The first four coordinates must remain below declared tolerances, while the non-collapse margin ","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-93d2faf0e0135e86","tex":"J_{\\mathcal U}","aliases":"J_ U mathcal","display":"J_{\\mathcal U}","kind":"use","excerpt":"Let J_ U: C_V(G)→widehat C_ U, J_ Uc=(c|_G_α)_α, be the duplication map.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2cf6f3974989e454","tex":"J_{\\mathcal U}^*","aliases":"J_ U^* mathcal","display":"J_{\\mathcal U}^*","kind":"use","excerpt":"Let m_i=|{α:i∈ V_α}| be the multiplicity of vertex i. Then J_ U^*J_ U=M_ U, (M_ Uc)_i=m_ic_i. The pseudoinverse performs coordinatewise averaging: (J_ U^† u)_i = frac1m_i ∑_α:i∈ V_α(u_α)_i. Let widehat u = P_ker K_ Uu = J_ Uwidehat c. Then widehat c = J_ U^†widehat u and u-J_ Uwidehat c ≤ frac K_ Uu sqrtγ_ U^tot. If m_min=min_i m_i, then widehat c-","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8a041232f8c58d05","tex":"J_{\\mathcal U}^\\dagger","aliases":"J_ U^† mathcal dagger","display":"J_{\\mathcal U}^\\dagger","kind":"use","excerpt":"Let m_i=|{α:i∈ V_α}| be the multiplicity of vertex i. Then J_ U^*J_ U=M_ U, (M_ Uc)_i=m_ic_i. The pseudoinverse performs coordinatewise averaging: (J_ U^† u)_i = frac1m_i ∑_α:i∈ V_α(u_α)_i. Let widehat u = P_ker K_ Uu = J_ Uwidehat c. Then widehat c = J_ U^†widehat u and u-J_ Uwidehat c ≤ frac K_ Uu sqrtγ_ U^tot. If m_min=min_i m_i, then widehat c-","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-49287a1e804c7929","tex":"\\mathfrak J_{0:T}","aliases":"J_0:T mathfrak","display":"\\mathfrak J_{0:T}","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-11d88ad682117768","tex":"J_i","aliases":"J_i","display":"J_i","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d873ce663c254ae2","tex":"\\mathcal J_{\\mathrm{leak}}","aliases":"J_leak mathcal mathrm","display":"\\mathcal J_{\\mathrm{leak}}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-af39feba3b79d098","tex":"\\mathcal J_{\\mathrm{maint}}","aliases":"J_maint mathcal mathrm","display":"\\mathcal J_{\\mathrm{maint}}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g,","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-acc6f405cd49422f","tex":"j_P","aliases":"j_P","display":"j_P","kind":"use","excerpt":"For any P∈ H_0, choose an index j_P such that P∈ H_0j_P. The event that every gate passes is contained in the event that gate j_P passes. Hence P_P(certificate passes) ≤ P_P(G_j_P passes) ≤ α_j_P ≤ max_jα_j. Taking the supremum over P∈ H_0 proves the claim.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5f99dd08cc8c61d3","tex":"J_{P'}","aliases":"J_P'","display":"J_{P'}","kind":"use","excerpt":"Suppose the proposer becomes unavailable. The remaining coalition {C,N} reconstructs (q,p,f) exactly. A replacement proposer P' can therefore be initialized by E_P'c=(q,p). If P' uses a different local coordinate system J_P'∈ GL(2), then the incoming and outgoing overlap maps are conjugated accordingly. The global section and all downstream kernels","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-281588f8b2aa034b","tex":"J_{s(e)}","aliases":"J_s(e)","display":"J_{s(e)}","kind":"use","excerpt":"For an approximate correspondence define Δ_D=D'J_V-J_ED, δ_D= Δ_D . If the graph weights are unchanged, define edgewise comparison defects δ_e = H_e'J_s(e)-J_t(e)H_e . The discrepancy operator has one changed source block per edge, so δ_D^2 ≤ max_i∈ V ∑_e:s(e)=iw_eδ_e^2 ≤ ∑_e∈ Ew_eδ_e^2. When a single node is replaced, every nonincident edge has δ_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7a69afaebcf8a9a2","tex":"\\mathcal J_{\\mathrm{stab}}","aliases":"J_stab mathcal mathrm","display":"\\mathcal J_{\\mathrm{stab}}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-07b6123d41bb6991","tex":"J_t","aliases":"J_t","display":"J_t","kind":"use","excerpt":"Persistence does not require token identity. Let R_t and R_t+1 be temporally successive realizations. Continuity is possible when there exists an admissible transport J_t: R_trightsquigarrow R_t+1 that preserves the operative core and its causal role within tolerance: d_id(Workspace_t,Workspace_t+1)≤ε. Thus a workspace may persist by reconstruction","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7990d6ea35212a4a","tex":"\\mathfrak J_t","aliases":"J_t mathfrak","display":"\\mathfrak J_t","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82f170f8864b97d4","tex":"J_{t(e)}","aliases":"J_t(e)","display":"J_{t(e)}","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-788dc63b4249b7e7","tex":"J_V^*","aliases":"J_V^*","display":"J_V^*","kind":"use","excerpt":"[Exact replacement isomorphism] Under an exact structural correspondence, Γ(D')=J_VΓ(D), L'J_V=J_VL. Consequently, every spectral workspace subspace is transported exactly: P_I(L')=J_VP_I(L)J_V^* for every spectral interval I. If the correspondence is also operation exact, then the transported workspace actions are intertwined exactly.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-10086ae86b457194","tex":"J_\\Pi","aliases":"J_Π Pi","display":"J_\\Pi","kind":"use","excerpt":"D_∂ , D_int = ⊕_α=1^mD_α, where D_α contains the edge constraints internal to V_α, and D_∂ contains the constraints crossing between blocks. Define the internally coherent macro-state spaces W_α=ker D_α, W_Π=⊕_α=1^mW_α. Let J_Π:W_Π↪ C_V be the canonical isometric inclusion, and define the quotient discrepancy D_Π=D_∂J_Π. The quotient does not repla","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2befaa63a8deace6","tex":"J_\\Pi^*","aliases":"J_Π^* Pi","display":"J_\\Pi^*","kind":"use","excerpt":"[Two-scale coherence bound] Let P_Π be the orthogonal projector onto J_Π W_Π. Let γ_int>0 be the smallest positive eigenvalue of D_int^*D_int, and let γ_Π>0 be the smallest positive eigenvalue of D_Π^*D_Π. Then every c∈ C_V satisfies dist(c,Γ(H)) ≤ frac D_intc sqrtγ_int + frac D_Π J_Π^*P_Π c sqrtγ_Π. The first term measures incoherence within macro","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3699b9dadbe987d2","tex":"J_{\\sigma\\sigma'}","aliases":"J_σσ' sigma sigma","display":"J_{\\sigma\\sigma'}","kind":"use","excerpt":"For a family S of admissible scaffold variants, define the cross-scaffold identity defect δ_q^scaf = inf_{J_σσ'} E_σ,σ'∼π d_q ( J_σσ'q_σ,q_σ' ), where J_σσ' transports the learned object between admissible coordinate systems. A low value indicates that the same abstract organization is recovered under multiple non-solution-bearing scaffolds.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cdbeca40dd7bbf39","tex":"\\jmath","aliases":"jmath","display":"\\jmath","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6bd30b4343cfeeaa","tex":"\\jmath'","aliases":"jmath' jmath","display":"\\jmath'","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ad76c25944c0ae3","tex":"\\jmath_E","aliases":"jmath_E jmath","display":"\\jmath_E","kind":"use","excerpt":"Let jmath_E randomize selected communication edges while preserving their marginal message laws, rates, and timing budget. The causal transfer gap is Δ_tr^do(jmath_E) = Risk_ℓ, P^jmath_E(V) - Risk_ℓ, P(V). Unlike an observational null law, this quantity is anchored to an implemented channel intervention. A positive effect is evidence that the intac","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-97d495a570cf2225","tex":"\\mathrm{joint}","aliases":"joint mathrm","display":"\\mathrm{joint}","kind":"use","excerpt":"Define the coalition gap Δ_A^ℓ = Risk_ℓ(A)-Risk_ℓ(V) ≥0. The node-level joint advantage is Δ_joint^ℓ = min_i∈ V Δ_{i}^ℓ. Under logarithmic loss, Δ_joint^log = min_i∈ V I ( Y_t^+; H_t mid H_t^{i} ). Therefore Δ_joint^log>0 means that every individual component omits some predictive information available to the complete system.","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"For tolerance ε≥0, define k_ε = min{ |A|: Risk_ℓ(A) ≤ Risk_ℓ(V)+ε }. The set of minimal coalitions is K_ε = { A⊆ V: |A|=k_ε, Risk_ℓ(A) ≤ Risk_ℓ(V)+ε }.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-80fb05254193b71d","tex":"\\mathcal K'","aliases":"K' mathcal","display":"\\mathcal K'","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-093dc47e134cf184","tex":"K_{\\mathcal U}","aliases":"K_ U mathcal","display":"K_{\\mathcal U}","kind":"use","excerpt":"and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U = widehat D_ U [0.2em] sqrtλ_ov S_ U , L_ U^tot = K_ U^*K_ U. Its energy is E_ U^tot(u) = · ∑_α D_G_αu_α ^2","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-05752f90b68c5286","tex":"K_{\\mathcal U}^*","aliases":"K_ U^* mathcal","display":"K_{\\mathcal U}^*","kind":"use","excerpt":"and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U = widehat D_ U [0.2em] sqrtλ_ov S_ U , L_ U^tot = K_ U^*K_ U. Its energy is E_ U^tot(u) = · ∑_α D_G_αu_α ^2","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9817d114fe256b80","tex":"k_0","aliases":"k_0","display":"k_0","kind":"use","excerpt":"This yields a graded classification: k_0=0 · Longrightarrow the background law alone is sufficient, k_0=1 · Longrightarrow some individual node is sufficient, k_0≥2 · Longrightarrow node-level relational irreducibility, k_0=|V| · Longrightarrow every proper coalition is insufficient. The last condition is much stronger and should not be inferred fr","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-173833828c3529a3","tex":"K_2","aliases":"K_2","display":"K_2","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9754a3bf487ab283","tex":"k_{\\mathrm{ctl},0}","aliases":"k_ctl,0 mathrm","display":"k_{\\mathrm{ctl},0}","kind":"use","excerpt":"(a_t^*,r_t^*): item the planner lacks h_t^-; item the monitor lacks g_t; and item the recorder lacks b_t. Every two-node coalition determines all three workspace bits and therefore the complete action. Under independent uniform bits, H(a_t^*mid X_i(t))=1 for every i, so the task-action component has a one-bit joint advantage and k_ctl,0=2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-17a77c632116ee6e","tex":"k_{\\mathrm{ctl},\\varepsilon}","aliases":"k_ctl,ε mathrm varepsilon","display":"k_{\\mathrm{ctl},\\varepsilon}","kind":"use","excerpt":"For the complete control state, define Risk_ctl(S) = inf_f E [ ℓ_ctl ( C_t^ctl, f( H_t^S) ) ], and let k_ctl,ε be its minimal sufficient coalition size. A strong workspace-level agency claim requires k_ctl,ε≥2. This is a location criterion. It does not yet establish goal persistence, counterfactual evaluation, or self-maintenance.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c3b7ca5c5b381650","tex":"k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}","aliases":"k_ctl,ε_ctl mathrm varepsilon mathrm","display":"k_{\\mathrm{ctl},\\varepsilon_{\\mathrm{ctl}}}","kind":"use","excerpt":"G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use exceeds τ_V→ A; item action-to-outcome control exceeds τ_A→ Y; item damage detection, monitor-to-repair mediation, and restoration exceed their thresholds; item repair specificity exceeds τ_spec; item k_ctl,ε_ctl≥2; and item rep","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6b1ed1dd4ea436f0","tex":"K_e","aliases":"K_e","display":"K_e","kind":"use","excerpt":"Relative to these decompositions, every edge map has the block form T_e= H_e · U_e V_e · K_e , where H_e=P_t(e)T_eP_s(e):C_s(e)→ C_t(e), U_e=P_t(e)T_eQ_s(e):R_s(e)→ C_t(e), V_e=Q_t(e)T_eP_s(e):C_s(e)→ R_t(e), K_e=Q_t(e)T_eQ_s(e):R_s(e)→ R_t(e).","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-228693534fb94838","tex":"k_{\\mathrm{erase}}","aliases":"k_erase mathrm","display":"k_{\\mathrm{erase}}","kind":"use","excerpt":"Define the minimum reconstructing coalition size m_rec = min{|S|:α_S>0} and the maximum exact erasure tolerance k_erase = max{k:α^(k)>0}. The condition m_rec≥2 means that no individual encoding is sufficient for exact core recovery. The condition k_erase≥1 means that the system survives at least one node loss. These can hold simultaneously because ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-07baee216908c2f6","tex":"K_f","aliases":"K_f","display":"K_f","kind":"use","excerpt":"For composable edges e:i→ j and f:j→ k, direct block multiplication gives T_fT_e = H_fH_e+U_fV_e · H_fU_e+U_fK_e V_fH_e+K_fV_e · V_fU_e+K_fK_e . The induced core-to-core block is therefore P_kT_fT_eP_i = H_fH_e+U_fV_e. The term U_fV_e is a residual-mediated correction: information leaves the candidate core at the intermediate node and later re-ente","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5d5dbc4c8e15a78b","tex":"k_{G,\\varepsilon}","aliases":"k_G,ε varepsilon","display":"k_{G,\\varepsilon}","kind":"use","excerpt":"Let H_t^S be the history available to coalition S⊆ V. For a loss ℓ_G, define goal-reconstruction risk Risk_G(S) = inf_f E [ ℓ_G ( G_t, f( H_t^S) ) ]. For tolerance ε, define k_G,ε = min { |S|: Risk_G(S) ≤ Risk_G(V)+ε }. A goal is locally stored when k_G,ε=1. It is distributed relative to the declared decomposition when k_G,ε≥2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-193077db22dceb36","tex":"K_r^*","aliases":"K_r^*","display":"K_r^*","kind":"use","excerpt":"At level r, let K_r be the corresponding total descent operator, γ_r = λ_min^+(K_r^*K_r), K_ru_r ≤ε_r. The nearest exact compatible state then differs from u_r by at most fracε_rsqrtγ_r. Allow an additional implementation, compression, or causal-summary defect ζ_r, and define δ_r = fracε_rsqrtγ_r + ζ_r.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bc4ee0f317f944b3","tex":"k_{\\mathrm{rec}}","aliases":"k_rec mathrm","display":"k_{\\mathrm{rec}}","kind":"use","excerpt":"thbb R^4, spec(L_HAI) = {0,0,0,0,2,2,2,2}, γ_HAI=2. The full accessible frame operator is F_HMXA=2I_4. No single node is injective. Every three-node coalition reconstructs the full state, and the two opposite pairs {H,X}, {M,A} are already sufficient. Adjacent pairs observe only three of the four coordinates. Hence k_rec=2, d_er=2, while the identi","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-be487da6c99748e0","tex":"K_S","aliases":"K_S","display":"K_S","kind":"use","excerpt":"[Preservation of coalition frame bounds] Identify the old and new abstract cores with ran P and ran P' through their chosen core parameterizations. Let S be a declared coalition, let A_S:ran P→ Y_S, A_S':ran P'→ Y_S' be the resulting accessible coalition encoders, and let K_S be a unitary identification of their observation spaces. Suppose A_S'β-K_","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-946fe38655e9829a","tex":"K_{t+\\tau}","aliases":"K_t+τ tau","display":"K_{t+\\tau}","kind":"use","excerpt":"[Behavioral readout of a retained goal] Assume, in addition, that under the matched intervention background the action readout has conditional mean E[A_t+τmid G_t+τ,W_t+τ] = K_t+τG_t+τ+b(W_t+τ), and that the two runs are matched on W_t+τ. Then E[A_t+τmidι_g] - E[A_t+τmidι_g'] = K_t+τR_t+τ:t(g-g'). If σ_min(K_t+τ)≥ k_t+τ>0, the norm of this action-m","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a15a552c4ce642c3","tex":"k_{t+\\tau}","aliases":"k_t+τ tau","display":"k_{t+\\tau}","kind":"use","excerpt":"elcor:goal-action-readout Assume, in addition, that under the matched intervention background the action readout has conditional mean E[A_t+τmid G_t+τ,W_t+τ] = K_t+τG_t+τ+b(W_t+τ), and that the two runs are matched on W_t+τ. Then E[A_t+τmidι_g] - E[A_t+τmidι_g'] = K_t+τR_t+τ:t(g-g'). If σ_min(K_t+τ)≥ k_t+τ>0, the norm of this action-mean difference","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cfcd9745f0071940","tex":"K_t^S","aliases":"K_t^S","display":"K_t^S","kind":"use","excerpt":"For an arbitrary history statistic S_t=s( H_t), let K_t^S(s,·) = P(Y_t^+∈·mid S_t=s). Whenever the conditional relative entropy is finite, define the sufficiency defect δ_suff(S_t) = E [ KL ( K_t( H_t,·) middle K_t^S(S_t,·) ) ].","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2f7b6d8ee1d2d05f","tex":"K_t^{S}","aliases":"K_t^S","display":"K_t^{S}","kind":"use","excerpt":"For an arbitrary history statistic S_t=s( H_t), let K_t^S(s,·) = P(Y_t^+∈·mid S_t=s). Whenever the conditional relative entropy is finite, define the sufficiency defect δ_suff(S_t) = E [ KL ( K_t( H_t,·) middle K_t^S(S_t,·) ) ].","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e291842438f7360e","tex":"K_t^{\\iota}","aliases":"K_t^ι iota","display":"K_t^{\\iota}","kind":"use","excerpt":"Observational prediction may merge histories that behave identically under the current policy but respond differently to future interventions. Let I be a finite or countable family of interventions applied only after time t. For each ι∈ I, let K_t^ι(h,·) = P^ι(Y_t^+∈·mid H_t=h) be a regular post-intervention kernel on the common pre-intervention hi","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e5e4823da3d93d95","tex":"K_w","aliases":"K_w","display":"K_w","kind":"use","excerpt":"Geometric recovery becomes behaviorally relevant when the workspace state controls a predictive kernel. Let K_w= P(Y∈·mid W=w) and suppose TV(K_w,K_w')≤ L_Kd_W(w,w').","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a4acddc51fa6eea3","tex":"K_{w'}","aliases":"K_w'","display":"K_{w'}","kind":"use","excerpt":"Geometric recovery becomes behaviorally relevant when the workspace state controls a predictive kernel. Let K_w= P(Y∈·mid W=w) and suppose TV(K_w,K_w')≤ L_Kd_W(w,w').","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c8c8f39e86dfc0b9","tex":"K_{\\widehat w}","aliases":"K_widehat w widehat","display":"K_{\\widehat w}","kind":"use","excerpt":"For every pair w,widehat w, using the Bayes action for K_widehat w under the true law K_w incurs conditional regret at most 2TV(K_w,K_widehat w). This follows by inserting and subtracting the risk under K_widehat w for both the chosen action and the Bayes action for K_w. Average the bound and use the Lipschitz assumption. The coalition's optimal ru","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c15dac24ac895342","tex":"K_\\gamma","aliases":"K_γ gamma","display":"K_\\gamma","kind":"use","excerpt":"[Autonomous splitting] If U_e=V_e=0 for every edge, then the complete diagram decomposes as the direct sum of two diagrams, X≅ H⊕ K, and every path transfer satisfies T_γ = H_γ · 0 0 · K_γ .","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-59683b5c38e267df","tex":"\\mathcal K_\\varepsilon","aliases":"K_ε mathcal varepsilon","display":"\\mathcal K_\\varepsilon","kind":"use","excerpt":"For tolerance ε≥0, define k_ε = min{ |A|: Risk_ℓ(A) ≤ Risk_ℓ(V)+ε }. The set of minimal coalitions is K_ε = { A⊆ V: |A|=k_ε, Risk_ℓ(A) ≤ Risk_ℓ(V)+ε }.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-67f9e9d3dc0b5e7d","tex":"k_\\varepsilon","aliases":"k_ε varepsilon","display":"k_\\varepsilon","kind":"use","excerpt":"For tolerance ε≥0, define k_ε = min{ |A|: Risk_ℓ(A) ≤ Risk_ℓ(V)+ε }. The set of minimal coalitions is K_ε = { A⊆ V: |A|=k_ε, Risk_ℓ(A) ≤ Risk_ℓ(V)+ε }.","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5a205f9a51209590","tex":"k_\\varepsilon^{\\mathrm{meta}}","aliases":"k_ε^meta varepsilon mathrm","display":"k_\\varepsilon^{\\mathrm{meta}}","kind":"use","excerpt":"nough accessible information to reconstruct the whole meta-state. A claim that the hierarchy is genuinely distributed must therefore reuse the coalition gates from Parts~VII and XI. At the chart level, one may require, for example, α_{α}^meta=0 for every α, α_A^meta>0 for some |A|≥2, or the predictive analogue k_ε^meta≥2, Δ_joint^meta>0. Capacity-m","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-408b9ac98a629f80","tex":"k_{\\varepsilon_k}","aliases":"k_ε_k varepsilon","display":"k_{\\varepsilon_k}","kind":"use","excerpt":"subtopicDistributedness k_ε_k ≥ k.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-75ef47be232c5a6e","tex":"\\mathrm{keep}","aliases":"keep mathrm","display":"\\mathrm{keep}","kind":"use","excerpt":"Low off-workspace error is not sufficient for meaningful persistence. The zero update F_t=0 drives every state exactly into every linear workspace while destroying all content. Define the internal transport B_t=P_t+1F_tP_t: W_t⟶ W_t+1 and its non-collapse margin μ_t^keep = σ_min(B_t|_ W_t). A content-preserving persistence claim requires μ_t^keep>0","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c262b927b1bd4a6f","tex":"\\KL","aliases":"KL","display":null,"kind":"use","excerpt":"For an arbitrary history statistic S_t=s( H_t), let K_t^S(s,·) = P(Y_t^+∈·mid S_t=s). Whenever the conditional relative entropy is finite, define the sufficiency defect δ_suff(S_t) = E [ KL ( K_t( H_t,·) middle K_t^S(S_t,·) ) ].","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-34c05dc8f5073293","tex":"L'","aliases":"L'","display":"L'","kind":"use","excerpt":"[Exact replacement isomorphism] Under an exact structural correspondence, Γ(D')=J_VΓ(D), L'J_V=J_VL. Consequently, every spectral workspace subspace is transported exactly: P_I(L')=J_VP_I(L)J_V^* for every spectral interval I. If the correspondence is also operation exact, then the transported workspace actions are intertwined exactly.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9d04858b33827bcc","tex":"L^R","aliases":"L^R","display":"L^R","kind":"use","excerpt":"· ≤ ∑_r=0^R-1 ( ∏_s=r+1^R-1L_s ) δ_r. In particular, if L_r≤1, then the total error is at most ∑_r=0^R-1δ_r. If L_r≤ L<1 and δ_r≤δ, then total error ≤ fracδ(1-L^R)1-L ≤ fracδ1-L.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-269748fd64081339","tex":"\\mathcal L^{\\iota_{\\bar S,u}}","aliases":"L^ι_bar S,u mathcal iota bar","display":"\\mathcal L^{\\iota_{\\bar S,u}}","kind":"use","excerpt":"nate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the future law of X_S: CIn_τ(S) = E_u,u'∼ν_bar S D ( L^ι_bar S,u (X_S(t+τ)), L^ι_bar ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f594367e6f089bbc","tex":"\\mathcal L^{\\iota_{\\bar S,u'}}","aliases":"L^ι_bar S,u' mathcal iota bar","display":"\\mathcal L^{\\iota_{\\bar S,u'}}","kind":"use","excerpt":"nate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the future law of X_S: CIn_τ(S) = E_u,u'∼ν_bar S D ( L^ι_bar S,u (X_S(t+τ)), L^ι_bar ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4905c2a22a8936c5","tex":"\\mathcal L^{\\iota_{S,x}}","aliases":"L^ι_S,x mathcal iota","display":"\\mathcal L^{\\iota_{S,x}}","kind":"use","excerpt":"Let ι_S,x be a feasible intervention that changes the internal workspace coordinate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the fu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c8a9cc475dedb979","tex":"\\mathcal L^{\\iota_{S,x'}}","aliases":"L^ι_S,x' mathcal iota","display":"\\mathcal L^{\\iota_{S,x'}}","kind":"use","excerpt":"Let ι_S,x be a feasible intervention that changes the internal workspace coordinate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the fu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3fcff3e0a4fb5ad8","tex":"L_{\\mathcal U}^{\\mathrm{meta}}","aliases":"L_ U^meta mathcal mathrm","display":"L_{\\mathcal U}^{\\mathrm{meta}}","kind":"use","excerpt":"First assume each local state already lies in its exact local workspace. Define B_ U := δ^0: check C^0( U;mathscr W) ⟶ check C^1( U;mathscr W) and the meta-Laplacian L_ U^meta = B_ U^*B_ U. By Theorem~[source reference], ker L_ U^meta = R_ Umathscr W(G). Thus the gluing problem is itself a workspace problem whose ``nodes'' are local workspaces and ","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-84b87795b41b4cd9","tex":"L_{\\mathcal U}^{\\mathrm{tot}}","aliases":"L_ U^tot mathcal mathrm","display":"L_{\\mathcal U}^{\\mathrm{tot}}","kind":"use","excerpt":"and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U = widehat D_ U [0.2em] sqrtλ_ov S_ U , L_ U^tot = K_ U^*K_ U. Its energy is E_ U^tot(u) = · ∑_α D_G_αu_α ^2","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"[Equivalent operations in different coordinates] Let C_1=C_2= R^2, choose S∈ GL_2( R), and set H_12=S, ρ_2(a)=Sρ_1(a)S^-1. Then H_12ρ_1(a) =Sρ_1(a) =ρ_2(a)S =ρ_2(a)H_12. The local matrices can look different while realizing the same abstract operation. This is the elementary model of representation change used throughout the theory.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-563fb35759c558d0","tex":"L_B","aliases":"L_B","display":"L_B","kind":"use","excerpt":"If the score depends on a parameter ϑ_t and satisfies sup_S | B_ϑ(S)- B_ϑ'(S)| ≤ L_Bd(ϑ,ϑ'), then the sufficient stability condition becomes 2L_Bd(ϑ_t,ϑ_t+1) < Δ_t^B.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c9df216add6f89bc","tex":"L_{H^g}","aliases":"L_H^g","display":"L_{H^g}","kind":"use","excerpt":"[Unitary gauge covariance of the Laplacian] If every g_i is unitary, then G_V and G_E are unitary and L_H^g=G_VL_HG_V^-1. Consequently, spec(L_H^g)=spec(L_H), P_I(L_H^g)=G_VP_I(L_H)G_V^-1 for every spectral interval I.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b03974539b120a08","tex":"L_{\\mathrm{HAI}}","aliases":"L_HAI mathrm","display":"L_{\\mathrm{HAI}}","kind":"use","excerpt":"0 · -1 · 0 · 0 · 0 · 0 · 1 · 0 . The rows are orthogonal with squared norm two. Therefore ker D_HAI ≅ R^4, spec(L_HAI) = {0,0,0,0,2,2,2,2}, γ_HAI=2. The full accessible frame operator is F_HMXA=2I_4. No single node is injective. Every three-node coalition reconstructs the full state, and the two opposite pairs {H,X}, {M,A} are already sufficient. A","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4a3ad1fdba0bf22e","tex":"L_{\\mathrm{HH}}","aliases":"L_HH mathrm","display":"L_{\\mathrm{HH}}","kind":"use","excerpt":"-1 · 0 · 0 · 0 · 1 · 0 . Therefore ker D_HH = { (q,p,p,f,q,f)^ T:q,p,f∈ R } ≅ R^3. The rows are orthogonal with squared norm two, so D_HHD_HH^*=2I_3. Consequently the transfer Laplacian has spectrum spec(L_HH) = {0,0,0,2,2,2}, and the exact workspace gap is γ_HH=2. The low-energy subspace is therefore stable under any Laplacian perturbation whose o","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4662d64de6729153","tex":"L_K","aliases":"L_K","display":"L_K","kind":"use","excerpt":"Geometric recovery becomes behaviorally relevant when the workspace state controls a predictive kernel. Let K_w= P(Y∈·mid W=w) and suppose TV(K_w,K_w')≤ L_Kd_W(w,w').","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ed5212305c5c9989","tex":"L_q","aliases":"L_q","display":"L_q","kind":"use","excerpt":"Fix a reference decomposition D_0. For each D∈ D, let Φ_ D:W^ D_0⟶ W^ D be a declared workspace correspondence and let δ_ D measure its structural and causal identity defect. Assume gate q has a transport bound | m_q(Φ_ DW; D) -m_q(W; D_0) | ≤ L_qδ_ D+b_q( D), where L_q is a sensitivity constant and b_q records model mismatch not captured by the id","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a377a288ccd2ba74","tex":"L_r","aliases":"L_r","display":"L_r","kind":"use","excerpt":"theorem[Approximate hierarchical collapse bound] Let F_r: H_r→ H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetil","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a0c6993f26b62e6f","tex":"L_S","aliases":"L_S","display":"L_S","kind":"use","excerpt":"For the induced subsystem on S, define Coh(S) = tr ( P_S,dL_SP_S,d ), where P_S,d projects onto the selected low-energy workspace modes. Let Cut(S) = ∑_e:s(e)∈ S, t(e)∉ Sw_e + ∑_e:s(e)∉ S, t(e)∈ Sw_e. Let k_ε(S) be the minimal sufficient coalition size inside the candidate subsystem. A candidate is admissible only if SRW_Θ_S(W_S)=1, k_ε(S)≥2, unles","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-578177ca609e29d3","tex":"L_s","aliases":"L_s","display":"L_s","kind":"use","excerpt":"l H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(x) - F_R-1∘⋯∘ F_0(x) · ≤ ∑_r=0^R-1 ( ∏_s=r+1^R-1L_s ) ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9f64f718d6a46c28","tex":"L_{t+1}","aliases":"L_t+1","display":"L_{t+1}","kind":"use","excerpt":"[Temporal Laplacian drift] The carried Laplacian satisfies (U_t^V)^*L_t+1U_t^V-L_t ≤ 2 D_t δ_t^D+(δ_t^D)^2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-87eaedd79b4aeee2","tex":"L_{\\widetilde H}","aliases":"L_widetilde H widetilde","display":"L_{\\widetilde H}","kind":"use","excerpt":"Suppose the edge maps are perturbed to widetilde H_e=H_e+Δ_e while the weights and vertex metrics remain fixed. Define δ_H^2 = max_i∈ V ∑_e:s(e)=iw_e Δ_e ^2. Let F=D_widetilde H-D_H. Then (Fc)_e=-sqrtw_e Δ_ec_s(e), F ≤δ_H. Since L_widetilde H-L_H =D_H^*F+F^*D_H+F^*F, we obtain L_widetilde H-L_H ≤ 2 D_H δ_H+δ_H^2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bda22e23ac42a073","tex":"\\Large","aliases":"Large","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3b3d7f267bfbab0b","tex":"\\large","aliases":"large","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a25d61be36946f23","tex":"\\mathrm{leak}","aliases":"leak mathrm","display":"\\mathrm{leak}","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-01cb563188761f9e","tex":"\\operatorname{Len}","aliases":"Len operatorname","display":"\\operatorname{Len}","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aa943001516abfe6","tex":"\\lesssim","aliases":"lesssim","display":null,"kind":"use","excerpt":"truction defects are O(δ_rep), the relevant operator norms are uniformly bounded, and g≥ g_0>0, then d_id( R, R') ≤ C_1fracδ_repg_0 +C_2δ_rep +O (fracδ_rep^2g_0). After a fixed normalization of the non-spectral terms, this recovers the schematic target d_id lesssim Cfracδ_repg_0.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c9eb09444d55967","tex":"\\operatorname{Lim}","aliases":"Lim operatorname","display":"\\operatorname{Lim}","kind":"use","excerpt":"l U_flat = {G_ℓ:ℓ∈ L( T)} is itself an edge-complete cover of G. Let δ_flat^0 be its overlap coboundary and define W_flat = kerδ_flat^0. Theorem~[source reference] and Theorem~[source reference] give W_r ≅ W_flat ≅ mathscr W(G). Equivalently, Lim_ T^rec ( {mathscr W(G_ℓ)}_ℓ∈ L( T) ) ≅ Eq_ U_flat ( {mathscr W(G_ℓ)}_ℓ∈ L( T) ). The notation on the le","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-205d0effa577910c","tex":"\\operatorname{Lim}_{\\mathcal T}^{\\mathrm{rec}}","aliases":"Lim_ T^rec operatorname mathcal mathrm","display":"\\operatorname{Lim}_{\\mathcal T}^{\\mathrm{rec}}","kind":"use","excerpt":"l U_flat = {G_ℓ:ℓ∈ L( T)} is itself an edge-complete cover of G. Let δ_flat^0 be its overlap coboundary and define W_flat = kerδ_flat^0. Theorem~[source reference] and Theorem~[source reference] give W_r ≅ W_flat ≅ mathscr W(G). Equivalently, Lim_ T^rec ( {mathscr W(G_ℓ)}_ℓ∈ L( T) ) ≅ Eq_ U_flat ( {mathscr W(G_ℓ)}_ℓ∈ L( T) ). The notation on the le","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"use","excerpt":"· + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7b02562f005902db","tex":"\\limsup_{t\\to\\infty}","aliases":"limsup_t→∞ limsup to infty","display":"\\limsup_{t\\to\\infty}","kind":"use","excerpt":"· + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2a707ccad5a125ef","tex":"\\ll","aliases":"ll","display":null,"kind":"use","excerpt":"The spectrum supports three distinct interpretations: rcl λ_k=0 · Longleftrightarrow · exact compatible direction, 0<λ_kll g_d · Longrightarrow · approximately transported direction, λ_k≈λ_k+1 · Longrightarrow · rank selection is unstable at that boundary. A low eigenvalue can still be scientifically spurious if it arises from a poorly chosen weigh","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-29a5d89d929b03d7","tex":"\\mathrm{local}","aliases":"local mathrm","display":"\\mathrm{local}","kind":"use","excerpt":"Let every node store a complete copy of the same core: E_i=I_d for every i. Then coherence is perfect and reconstruction is robust, but k_0=1, min_iδ_i^local=0. If the archive is never used in downstream control, then also η_causal=0. This is a shared representation or archive, not a relationally irreducible workspace.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"For a finite target alphabet and logarithmic scoring loss ℓ_log(y,q)=-log q(y), q∈ P( Y), the identity is exact: Risk_log(S_t)=H(Y_t^+mid S_t), Risk_log(S_t)-Risk_log( H_t) = I(Y_t^+; H_tmid S_t). Natural logarithms measure the gap in nats; base-two logarithms measure it in bits.","count":24},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-529ad2daacd7efcb","tex":"\\lor","aliases":"lor","display":"\\lor","kind":"use","excerpt":"Let the one-step utility be U_g(a,r;b,h^-) = 1{a=g⊕ b⊕ h^-} +2 1{h^-lor r=1} -frac12r. For fixed (g,b,h^-), the unique maximizer is (a^*,r^*) = (g⊕ b⊕ h^-,1-h^-). Indeed: item a wrong task action loses one unit; item omitting repair when h^-=0 loses 3/2 units; and item performing unnecessary repair when h^-=1 loses 1/2 unit. The smallest optimal-ac","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f109b11a9ed7c626","tex":"\\mathsf{losses}","aliases":"losses mathsf","display":"\\mathsf{losses}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-039ec3e8ab324910","tex":"\\mathfrak m","aliases":"m mathfrak","display":"\\mathfrak m","kind":"use","excerpt":"A family w=(w_α)_α is exactly compatible when δ^0w=0. Its direct mismatch is m_ U(w)=δ^0w. After choosing inner products and positive overlap weights, the mismatch energy is E_ov(w) = δ^0w ^2 = ∑_α<β ν_αβ r_αβ^βw_β -r_αβ^αw_α ^2. This is the appropriate obstruction when the full local states are already given. Such a family glues exactly if and onl","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"The derivative is F_λ'(m) = λsech^2(λ m). At zero this equals λ, proving stability for λ<1 and instability for λ>1. At a nonzero fixed point, F_λ'(m_λ) = λ(1-m_λ^2) = frac(1-m_λ^2)artanh(m_λ)m_λ <1, because artanh(m) < fracm1-m^2 (0<m<1). The derivative is positive, so the nonzero fixed points are locally stable. Finally, expanding tanh(λ m) = λ m-","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9155222104384cb9","tex":"m^3","aliases":"m^3","display":"m^3","kind":"use","excerpt":"ambda, proving stability for λ<1 and instability for λ>1. At a nonzero fixed point, F_λ'(m_λ) = λ(1-m_λ^2) = frac(1-m_λ^2)artanh(m_λ)m_λ <1, because artanh(m) < fracm1-m^2 (0<m<1). The derivative is positive, so the nonzero fixed points are locally stable. Finally, expanding tanh(λ m) = λ m- fracλ^3m^33 +O(m^5) and solving the nonzero fixed-point e","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fa3cae93c3722bc0","tex":"m^5","aliases":"m^5","display":"m^5","kind":"use","excerpt":"ambda, proving stability for λ<1 and instability for λ>1. At a nonzero fixed point, F_λ'(m_λ) = λ(1-m_λ^2) = frac(1-m_λ^2)artanh(m_λ)m_λ <1, because artanh(m) < fracm1-m^2 (0<m<1). The derivative is positive, so the nonzero fixed points are locally stable. Finally, expanding tanh(λ m) = λ m- fracλ^3m^33 +O(m^5) and solving the nonzero fixed-point e","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-47f4330802ec5d87","tex":"M_{\\mathcal U}","aliases":"M_ U mathcal","display":"M_{\\mathcal U}","kind":"use","excerpt":"Let m_i=|{α:i∈ V_α}| be the multiplicity of vertex i. Then J_ U^*J_ U=M_ U, (M_ Uc)_i=m_ic_i. The pseudoinverse performs coordinatewise averaging: (J_ U^† u)_i = frac1m_i ∑_α:i∈ V_α(u_α)_i. Let widehat u = P_ker K_ Uu = J_ Uwidehat c. Then widehat c = J_ U^†widehat u and u-J_ Uwidehat c ≤ frac K_ Uu sqrtγ_ U^tot. If m_min=min_i m_i, then widehat c-","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-73008f4a6ebdc49a","tex":"\\mathfrak m_{\\mathcal U}","aliases":"m_ U mathfrak mathcal","display":"\\mathfrak m_{\\mathcal U}","kind":"use","excerpt":"A family w=(w_α)_α is exactly compatible when δ^0w=0. Its direct mismatch is m_ U(w)=δ^0w. After choosing inner products and positive overlap weights, the mismatch energy is E_ov(w) = δ^0w ^2 = ∑_α<β ν_αβ r_αβ^βw_β -r_αβ^αw_α ^2. This is the appropriate obstruction when the full local states are already given. Such a family glues exactly if and onl","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b5b5be0c2efd3035","tex":"M_E","aliases":"M_E","display":"M_E","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ecd57bfde2b4b291","tex":"m_E","aliases":"m_E","display":"m_E","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b5b89c1d36128739","tex":"M_E^2","aliases":"M_E^2","display":"M_E^2","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0f6f8c20fb9b9338","tex":"m_E^2","aliases":"m_E^2","display":"m_E^2","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"use","excerpt":"Let m_i=|{α:i∈ V_α}| be the multiplicity of vertex i. Then J_ U^*J_ U=M_ U, (M_ Uc)_i=m_ic_i. The pseudoinverse performs coordinatewise averaging: (J_ U^† u)_i = frac1m_i ∑_α:i∈ V_α(u_α)_i. Let widehat u = P_ker K_ Uu = J_ Uwidehat c. Then widehat c = J_ U^†widehat u and u-J_ Uwidehat c ≤ frac K_ Uu sqrtγ_ U^tot. If m_min=min_i m_i, then widehat c-","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1dec70f2b1878999","tex":"m_{\\min}","aliases":"m_min min","display":"m_{\\min}","kind":"use","excerpt":"forms coordinatewise averaging: (J_ U^† u)_i = frac1m_i ∑_α:i∈ V_α(u_α)_i. Let widehat u = P_ker K_ Uu = J_ Uwidehat c. Then widehat c = J_ U^†widehat u and u-J_ Uwidehat c ≤ frac K_ Uu sqrtγ_ U^tot. If m_min=min_i m_i, then widehat c-J_ U^† u ≤ frac K_ Uu sqrtm_minγ_ U^tot. The simple overlap average and the globally coherent least-squares assembl","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-690ba96f54c7fd69","tex":"m_{\\mathrm{rec}}","aliases":"m_rec mathrm","display":"m_{\\mathrm{rec}}","kind":"use","excerpt":"Define the minimum reconstructing coalition size m_rec = min{|S|:α_S>0} and the maximum exact erasure tolerance k_erase = max{k:α^(k)>0}. The condition m_rec≥2 means that no individual encoding is sufficient for exact core recovery. The condition k_erase≥1 means that the system survives at least one node loss. These can hold simultaneously because ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-17770225492e1d01","tex":"M_s","aliases":"M_s","display":"M_s","kind":"use","excerpt":"thm:goal-retention Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s ) g-g' ≤ G_t+τ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82f744f1968fc463","tex":"m_s","aliases":"m_s","display":"m_s","kind":"use","excerpt":"thm:goal-retention Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s ) g-g' ≤ G_t+τ","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9de6a9970cb6757b","tex":"M_t","aliases":"M_t","display":"M_t","kind":"use","excerpt":"A controlled state contains at least C_t^ctl = ( W_t,G_t,M_t,B_t,E_t ), where G_t is the represented goal, M_t is a workspace-integrity monitor, B_t is the current boundary or routing state, and E_t is the declared external context. An action is generated by A_t ∼ π_t ( ·mid W_t,G_t,M_t,B_t,E_t ). The future controlled state obeys a time-unrolled s","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6f3ae7d6b37bdd61","tex":"m_t","aliases":"m_t","display":"m_t","kind":"use","excerpt":"For m_t+1=tanh(λ m_t), there is a unique zero fixed point for 0≤λ≤1. For λ>1, the zero state is unstable and exactly two stable nonzero fixed points appear. Their amplitude satisfies m_λ = sqrtfrac3(λ-1)λ^3 + O((λ-1)^3/2). This is an exact structural benchmark, not a universal law for complete relational workspaces.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a01fc42b9500ded3","tex":"m_{t+1}","aliases":"m_t+1","display":"m_{t+1}","kind":"use","excerpt":"For m_t+1=tanh(λ m_t), there is a unique zero fixed point for 0≤λ≤1. For λ>1, the zero state is unstable and exactly two stable nonzero fixed points appear. Their amplitude satisfies m_λ = sqrtfrac3(λ-1)λ^3 + O((λ-1)^3/2). This is an exact structural benchmark, not a universal law for complete relational workspaces.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9449462255135514","tex":"M_V","aliases":"M_V","display":"M_V","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-41cf8d877ba40285","tex":"m_V","aliases":"m_V","display":"m_V","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-59eaeab81b9eee3d","tex":"M_V^2","aliases":"M_V^2","display":"M_V^2","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cb493d3b42c096fb","tex":"m_V^2","aliases":"m_V^2","display":"m_V^2","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0dae06547437e14b","tex":"m_\\lambda","aliases":"m_λ lambda","display":"m_\\lambda","kind":"use","excerpt":"For m_t+1=tanh(λ m_t), there is a unique zero fixed point for 0≤λ≤1. For λ>1, the zero state is unstable and exactly two stable nonzero fixed points appear. Their amplitude satisfies m_λ = sqrtfrac3(λ-1)λ^3 + O((λ-1)^3/2). This is an exact structural benchmark, not a universal law for complete relational workspaces.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b26da18053c63761","tex":"m_\\lambda^2","aliases":"m_λ^2 lambda","display":"m_\\lambda^2","kind":"use","excerpt":"The derivative is F_λ'(m) = λsech^2(λ m). At zero this equals λ, proving stability for λ<1 and instability for λ>1. At a nonzero fixed point, F_λ'(m_λ) = λ(1-m_λ^2) = frac(1-m_λ^2)artanh(m_λ)m_λ <1, because artanh(m) < fracm1-m^2 (0<m<1). The derivative is positive, so the nonzero fixed points are locally stable. Finally, expanding tanh(λ m) = λ m-","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9592b96e21f7003a","tex":"\\mathrm{MA}","aliases":"MA mathrm","display":"\\mathrm{MA}","kind":"use","excerpt":"Consider three agents: V_MA={P,M,R}, called the planner, monitor, and recorder. The global workspace-control state is c_t=(g_t,b_t,h_t)∈ F_2^3, where: ll g_t=goal bit, · b_t=environment or belief bit,","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ed80df4aa685473d","tex":"\\mathrm{maint}","aliases":"maint mathrm","display":"\\mathrm{maint}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g,","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-33f79cb4dade3f86","tex":"\\mathrm{marg}","aliases":"marg mathrm","display":"\\mathrm{marg}","kind":"use","excerpt":"Let P be the intact law of (Y_t^+, H_t^1,…, H_t^n). Define the local-marginal null class N_marg(P) = { Q: Q_ H_t^i=P_ H_t^i for every i }, and the stronger local-predictive-matched class N_pred(P) = { Q: Q_(Y_t^+, H_t^i) = P_(Y_t^+, H_t^i) for every i }. Because H_t^i can include the complete local trajectory, the latter class can preserve all sing","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b09c594c76223cf4","tex":"\\max_{a\\neq a^*}","aliases":"max_a≠ a^* max neq","display":"\\max_{a\\neq a^*}","kind":"use","excerpt":"a)] ≤ δ_opt. Then, for every true-value-optimal policy π^*, E_a∼π^*[Q(a)] - E_a∼widehatπ[Q(a)] ≤ 2ε_Q+δ_opt. For a deterministic greedy action widehat a∈argmax_awidehat Q(a), Q(a^*)-Q(widehat a) ≤2ε_Q. If the true optimal action a^* is unique and its value gap Δ_Q = Q(a^*)- max_a≠ a^*Q(a) satisfies Δ_Q>2ε_Q, then every greedy maximizer of widehat Q","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5c1293b3e74725f5","tex":"\\max_{i\\in V}","aliases":"max_i∈ V max in","display":"\\max_{i\\in V}","kind":"use","excerpt":"Suppose the edge maps are perturbed to widetilde H_e=H_e+Δ_e while the weights and vertex metrics remain fixed. Define δ_H^2 = max_i∈ V ∑_e:s(e)=iw_e Δ_e ^2. Let F=D_widetilde H-D_H. Then (Fc)_e=-sqrtw_e Δ_ec_s(e), F ≤δ_H. Since L_widetilde H-L_H =D_H^*F+F^*D_H+F^*F, we obtain L_widetilde H-L_H ≤ 2 D_H δ_H+δ_H^2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9cc66676daa83a18","tex":"\\max_j","aliases":"max_j max","display":"\\max_j","kind":"use","excerpt":"Suppose a certificate requires every gate G_1,…,G_m to pass, and the predeclared test for gate j has false-pass probability at most α_j whenever that required property is absent. Under the global null that at least one required property is absent, P(complete certificate passes) ≤ max_jα_j. No independence assumption among gate tests is required. Th","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-74d4e4c000015210","tex":"\\max_{\\substack{c\\in S\\\\\\|c\\|=1}}","aliases":"max_substackc∈ S c =1 max substack in","display":"\\max_{\\substack{c\\in S\\\\\\|c\\|=1}}","kind":"use","excerpt":"[Min--max characterization] For 1≤ k≤ N, λ_k = min_substackS⊆ C_V dim S=k max_substackc∈ S c =1 E_H(c). Consequently, the span of the first k eigenvectors minimizes the worst-case coherence energy among all k-dimensional candidate subspaces.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4bf4d1900a893b9c","tex":"\\max_{S\\neq S_t^*}","aliases":"max_S≠ S_t^* max neq","display":"\\max_{S\\neq S_t^*}","kind":"use","excerpt":"[Held-out boundary recovery] Let S_pass be finite and let S_t^* = argmax_S∈ S_pass B_t(S) be unique. Define the population margin Δ_t^B = B_t(S_t^*) - max_S≠ S_t^* B_t(S)>0. If sup_S∈ S_pass | widehat B_t(S)- B_t(S) | ≤ε_t, then every empirical maximizer widehat S_t satisfies B_t(S_t^*)- B_t(widehat S_t) ≤2ε_t. If 2ε_t<Δ_t^B, then widehat S_t=S_t^*","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f3a6469310919baf","tex":"\\max_{t<T}","aliases":"max_t<T max","display":"\\max_{t<T}","kind":"use","excerpt":"A finite-horizon persistence certificate may be recorded as P_0:T = ( max_t<Tq_t^str, max_t≤ Te_t, min_t<Tμ_t^keep, ∑_t<Tσ_t ). No single coordinate of this vector subsumes the others.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-75fb1b9cf227ed62","tex":"\\max_{t\\leq T}","aliases":"max_t≤ T max leq","display":"\\max_{t\\leq T}","kind":"use","excerpt":"A finite-horizon persistence certificate may be recorded as P_0:T = ( max_t<Tq_t^str, max_t≤ Te_t, min_t<Tμ_t^keep, ∑_t<Tσ_t ). No single coordinate of this vector subsumes the others.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1252440762814f1f","tex":"\\max_{\\pi}","aliases":"max_π max pi","display":"\\max_{\\pi}","kind":"use","excerpt":"gintheorem[Approximate counterfactual policy selection] Let A be a finite feasible action set. Let Q: A→ R be the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-op","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-396da347c627f30a","tex":"\\max_{\\varsigma_j\\in\\mathfrak S_\\eta}","aliases":"max_ς_j∈ S_η max varsigma in mathfrak eta","display":"\\max_{\\varsigma_j\\in\\mathfrak S_\\eta}","kind":"use","excerpt":"Let S_η be an η-net of ( S,d_ S) with covering number N(η, S,d_ S). If both the population and empirical margin functions are L_q-Lipschitz, then sup_ς∈ S |widehat m_q(ς)-m_q(ς)| ≤ max_ς_j∈ S_η |widehat m_q(ς_j)-m_q(ς_j)| +2L_qη. For bounded empirical means, a valid radius is therefore ε_q,n,η(δ) = sqrt frac log(2| J_*| N(η, S,d_ S)/δ) 2n +2L_qη. T","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2554b0d6552658f0","tex":"\\mathbf{Meas}","aliases":"Meas mathbf","display":"\\mathbf{Meas}","kind":"use","excerpt":"The finite-dimensional Hilbert assumption supplies adjoints and spectral tools. The exact compatible-section definitions require less and may later be formulated in categories such as Ban, Meas, Prob, Top, Alg.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c3da343f7f5cf5bc","tex":"\\mathrm{med}","aliases":"med mathrm","display":"\\mathrm{med}","kind":"use","excerpt":"A stronger mediation test equalizes the workspace state after two distinct channel interventions. Let μ be a common imposed workspace distribution, implemented by a feasible stochastic intervention in the sense of Definition~[source reference]. The notation below is shorthand for that implementation. Define the residual mediation defect δ_med = sup","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4e2edb9ef439a4ee","tex":"\\mathrm{meta}","aliases":"meta mathrm","display":"\\mathrm{meta}","kind":"use","excerpt":"First assume each local state already lies in its exact local workspace. Define B_ U := δ^0: check C^0( U;mathscr W) ⟶ check C^1( U;mathscr W) and the meta-Laplacian L_ U^meta = B_ U^*B_ U. By Theorem~[source reference], ker L_ U^meta = R_ Umathscr W(G). Thus the gluing problem is itself a workspace problem whose ``nodes'' are local workspaces and ","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bd6afaaa498f22c1","tex":"\\mathsf{metric}","aliases":"metric mathsf","display":"\\mathsf{metric}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-152abbd85d69d8e9","tex":"\\middle","aliases":"middle","display":null,"kind":"use","excerpt":"For an arbitrary history statistic S_t=s( H_t), let K_t^S(s,·) = P(Y_t^+∈·mid S_t=s). Whenever the conditional relative entropy is finite, define the sufficiency defect δ_suff(S_t) = E [ KL ( K_t( H_t,·) middle K_t^S(S_t,·) ) ].","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-945455835d87b682","tex":"\\min_{1\\leq k\\leq K}","aliases":"min_1≤ k≤ K min leq leq","display":"\\min_{1\\leq k\\leq K}","kind":"use","excerpt":"A pointwise valid test does not remain valid after the analyst searches for the most favorable target, horizon, null panel, or threshold. Consider K independent null p-values and select the smallest. Then P(min_1≤ k≤ Kp_k≤α) = 1-(1-α)^K, which exceeds α whenever K>1. The same logic applies when a workspace certificate is reported only for the speci","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3edf593fca4f7dd8","tex":"\\min_{c\\neq0}","aliases":"min_c≠0 min neq","display":"\\min_{c\\neq0}","kind":"use","excerpt":"Define the accessible block support of c≠0 by supp_E(c) = {i∈ V:E_ic≠0} and define the node-erasure distance d_er = min_c≠0|supp_E(c)|.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4bc014492ab92312","tex":"\\min_i","aliases":"min_i min","display":"\\min_i","kind":"use","excerpt":"forms coordinatewise averaging: (J_ U^† u)_i = frac1m_i ∑_α:i∈ V_α(u_α)_i. Let widehat u = P_ker K_ Uu = J_ Uwidehat c. Then widehat c = J_ U^†widehat u and u-J_ Uwidehat c ≤ frac K_ Uu sqrtγ_ U^tot. If m_min=min_i m_i, then widehat c-J_ U^† u ≤ frac K_ Uu sqrtm_minγ_ U^tot. The simple overlap average and the globally coherent least-squares assembl","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8f165cfdf5422c97","tex":"\\min_{i\\in V}","aliases":"min_i∈ V min in","display":"\\min_{i\\in V}","kind":"use","excerpt":"Define the coalition gap Δ_A^ℓ = Risk_ℓ(A)-Risk_ℓ(V) ≥0. The node-level joint advantage is Δ_joint^ℓ = min_i∈ V Δ_{i}^ℓ. Under logarithmic loss, Δ_joint^log = min_i∈ V I ( Y_t^+; H_t mid H_t^{i} ). Therefore Δ_joint^log>0 means that every individual component omits some predictive information available to the complete system.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-475a92ae1d82b133","tex":"\\min_{j\\in\\mathcal J_{\\mathrm{maint}}}","aliases":"min_j∈ J_maint min in mathcal mathrm","display":"\\min_{j\\in\\mathcal J_{\\mathrm{maint}}}","kind":"use","excerpt":"ϕ_boundary,t · =Δ_t^B-underlineΔ_B. The aggregate integrity margin is v_t = min_j∈ J_maintϕ_j,t. The workspace passes the protected gates when v_t≥0. To maintain a reserve ρ>0, define the maintenance deficit e_t =(ρ-v_t)_+. Define the zero-reserve and bounded-deficit viability sets V_Θ = { S:ϕ_j(S)≥0 for every j∈ J_maint } V_Θ,e_max = { S:e(S)≤ e_m","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-af555498bdb2b55d","tex":"\\min_q","aliases":"min_q min","display":"\\min_q","kind":"use","excerpt":"For a reference specification ς_0, define the distance to certificate failure by r_spec(W;ς_0) = inf{ d_ S(ς,ς_0): min_q m_q(W;ς)≤0 }. If no failing specification exists in the declared universe, the radius is infinite relative to that universe. A radius near zero means that a small scientifically defensible change can reverse the conclusion.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-86c73adf01bd95df","tex":"\\min_{q:L_q>0}","aliases":"min_q:L_q>0 min","display":"\\min_{q:L_q>0}","kind":"use","excerpt":"When b_q≡0, a convenient certificate radius is r_dec(W) = min_q:L_q>0 fracm_q(W; D_0)L_q. Every admissible family satisfying sup_ D∈ Dδ_ D < r_dec(W) preserves the sign of all required gate margins. In applications, the transport terms can be bounded using earlier results: small textwidth@>bfseriesraggedrightarraybackslashp0.22textwidth>raggedright","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-38725127ef1cee36","tex":"\\min_{q\\in\\mathcal J_*}","aliases":"min_q∈ J_* min in mathcal","display":"\\min_{q\\in\\mathcal J_*}","kind":"use","excerpt":"in J_*, define the specification envelope underline m_q^ S(W) = inf_ς∈ Sm_q(W;ς), overline m_q^ S(W) = sup_ς∈ Sm_q(W;ς). Its sensitivity width is Δ_q^spec(W) = overline m_q^ S(W) - underline m_q^ S(W). The robust certificate margin is r_ S(W) = inf_ς∈ S min_q∈ J_*m_q(W;ς). Then Cert_ S(W)=1 Longleftrightarrow r_ S(W)>0.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a046d4014560eb1a","tex":"\\min_{Q\\in\\mathcal Q}","aliases":"min_Q∈ Q min in mathcal","display":"\\min_{Q\\in\\mathcal Q}","kind":"use","excerpt":"declared null Q, define Δ_tr(Q) = Risk_ℓ,Q(V) - Risk_ℓ,P(V). For a finite validated panel Q={Q_1,…,Q_m}⊆ N_pred(P), report both the complete vector Δ_tr( Q) = ( Δ_tr(Q_1),…, Δ_tr(Q_m) ) and the robust panel minimum Δ_tr^min( Q) = min_Q∈ QΔ_tr(Q). The panel must be fixed independently of the observed favorable gap. Otherwise null selection can manuf","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-de245e42e5c11798","tex":"\\min_{\\substack{A\\subseteq V\\\\|A|\\leq k}}","aliases":"min_substackA⊆ V |A|≤ k min substack subseteq leq","display":"\\min_{\\substack{A\\subseteq V\\\\|A|\\leq k}}","kind":"use","excerpt":"For k≥0, define the worst surviving lower frame bound α^(k) = min_substackA⊆ V |A|≤ k α_V∖ A. The sequence α^(k) is nonincreasing in k.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c68806391bb67169","tex":"\\min_{\\substack{A\\subseteq V\\\\|A|\\leq r}}","aliases":"min_substackA⊆ V |A|≤ r min substack subseteq leq","display":"\\min_{\\substack{A\\subseteq V\\\\|A|\\leq r}}","kind":"use","excerpt":"For 0≤ r≤ |V|, define the order-r coalition gap profile by Δ^(r)_ℓ = min_substackA⊆ V |A|≤ r [ Risk_ℓ(A)-Risk_ℓ(V) ]. The profile is nonincreasing in r and reaches zero by r=|V|. It records how rapidly operative capacity becomes reconstructible as larger coalitions are permitted.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a70dda30410ad5d4","tex":"\\min_{\\substack{S\\subseteq\\mathcal C_V\\\\\\dim S=k}}","aliases":"min_substackS⊆ C_V dim S=k min substack subseteq mathcal dim","display":"\\min_{\\substack{S\\subseteq\\mathcal C_V\\\\\\dim S=k}}","kind":"use","excerpt":"[Min--max characterization] For 1≤ k≤ N, λ_k = min_substackS⊆ C_V dim S=k max_substackc∈ S c =1 E_H(c). Consequently, the span of the first k eigenvectors minimizes the worst-case coherence energy among all k-dimensional candidate subspaces.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2232dad528c4d537","tex":"\\min_{S\\subseteq V_{\\mathcal D}}","aliases":"min_S⊆ V_ D min subseteq mathcal","display":"\\min_{S\\subseteq V_{\\mathcal D}}","kind":"use","excerpt":"A more stable quantity measures primitive carrier cost. Let w_ω>0 be a declared cost for each ω∈Ω, and assign a block the cost c(B)=∑_ω∈ Bw_ω. Define the sufficient support cost s_ε( D) = min_S⊆ V_ D { ∑_B∈ Sc(B): Risk_ D(S) ≤ Risk_ D(V_ D)+ε }. If D_fpreceq D_c is information faithful, then s_ε( D_f) ≤ s_ε( D_c). The fine decomposition permits eve","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f8430698aa9f66f9","tex":"\\min_{t<T}","aliases":"min_t<T min","display":"\\min_{t<T}","kind":"use","excerpt":"A finite-horizon persistence certificate may be recorded as P_0:T = ( max_t<Tq_t^str, max_t≤ Te_t, min_t<Tμ_t^keep, ∑_t<Tσ_t ). No single coordinate of this vector subsumes the others.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-169cfe47dc27df42","tex":"\\min_{\\theta\\neq\\theta_*}","aliases":"min_θ≠θ_* min theta neq theta","display":"\\min_{\\theta\\neq\\theta_*}","kind":"use","excerpt":"[Uniform structure-selection recovery] Let T be a finite candidate family with population score J: T→ R. Let θ_* = argmin_θ∈ TJ(θ) be unique, with margin Δ_J = min_θ≠θ_* (J(θ)-J(θ_*))>0. If a held-out estimate widehat J satisfies sup_θ∈ T |widehat J(θ)-J(θ)| ≤ε, then every empirical minimizer widehatθ obeys J(widehatθ)-J(θ_*) ≤2ε. If 2ε<Δ_J, then w","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d6ce51886365878b","tex":"\\mathrm{model}","aliases":"model mathrm","display":"\\mathrm{model}","kind":"use","excerpt":"In practice, the reference margin is reduced by three distinct uncertainty budgets: B_q^total = B_q^model +B_q^spec +B_q^sample. The first covers variation across observationally compatible realizations. The second covers variation across admissible specifications. The third covers finite-sample estimation. The sufficient condition widehat m_q(W;ς_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9f680cbf1fb0ef9a","tex":"\\operatorname{MoralPatient}","aliases":"MoralPatient operatorname","display":"\\operatorname{MoralPatient}","kind":"use","excerpt":"The following implications are not licensed: Agent_Θ_A(Workspace) centernotLongrightarrow PhenomenallyConscious(Workspace), SelfMaintaining(Workspace) centernotLongrightarrow SelfAware(Workspace), GoalPersistent(Workspace) centernotLongrightarrow MoralPatient(Workspace). A thermostat-like controller can maintain a variable without representing itse","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":17},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-498c801ffe40ead9","tex":"\\mathcal N'","aliases":"N' mathcal","display":"\\mathcal N'","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6a87293985c01784","tex":"N_{10}","aliases":"N_10","display":"N_{10}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-47d94dd1cd34ffa8","tex":"N_{11}","aliases":"N_11","display":"N_{11}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-04c8f47916d66ddd","tex":"N_{12}","aliases":"N_12","display":"N_{12}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7f03e9e7696ff715","tex":"N_{13}","aliases":"N_13","display":"N_{13}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5dab811f786f2862","tex":"N_{14}","aliases":"N_14","display":"N_{14}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f8bace513c7fdacb","tex":"N_{15}","aliases":"N_15","display":"N_{15}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0aeff9a3fe1d6faf","tex":"N_{16}","aliases":"N_16","display":"N_{16}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-989ddeee74c14780","tex":"N_{17}","aliases":"N_17","display":"N_{17}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-97b2f7260f053771","tex":"N_{18}","aliases":"N_18","display":"N_{18}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4835901913892c2f","tex":"N_{19}","aliases":"N_19","display":"N_{19}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-72ec741403c87a9e","tex":"N_2","aliases":"N_2","display":"N_2","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cd23929955304633","tex":"N_{20}","aliases":"N_20","display":"N_{20}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-243f195ec0356ebe","tex":"N_{21}","aliases":"N_21","display":"N_{21}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-80d17e7ff594033e","tex":"N_{22}","aliases":"N_22","display":"N_{22}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-070113506c0ae361","tex":"N_{23}","aliases":"N_23","display":"N_{23}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3e8601557539a047","tex":"N_{24}","aliases":"N_24","display":"N_{24}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-086c5ec017eb9332","tex":"N_{25}","aliases":"N_25","display":"N_{25}","kind":"use","excerpt":"Carrier-relative relationality becomes identifiable only when the experiment can test the claimed causal separation. Examples include independently interrupting the candidate carriers, exposing the hidden memory, randomizing a shared controller, or measuring whether one physical mechanism reconstructs the complete state before the displayed split. ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b4db9ed328cba4fc","tex":"N_{26}","aliases":"N_26","display":"N_{26}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d9f7fafd68c30b48","tex":"N_{27}","aliases":"N_27","display":"N_{27}","kind":"use","excerpt":"Carrier-relative relationality becomes identifiable only when the experiment can test the claimed causal separation. Examples include independently interrupting the candidate carriers, exposing the hidden memory, randomizing a shared controller, or measuring whether one physical mechanism reconstructs the complete state before the displayed split. ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c9c15c800ff1807c","tex":"N_{28}","aliases":"N_28","display":"N_{28}","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fad88ab7eacc5837","tex":"N_{29}","aliases":"N_29","display":"N_{29}","kind":"use","excerpt":"The empirical null and audit panel is extended by: N_29 · =post hoc target-selection audit, N_30 · =horizon cherry-picking audit, N_31 · =loss and utility substitution audit, N_32 · =restricted intervention-family audit, N_33 · =threshold and gate-set tuning audit, N_34 · =frozen specification sweep across S. The first five reconstruct or bound the","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bcbc010f58d53ce9","tex":"N_3","aliases":"N_3","display":"N_3","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b6a497de2739b54d","tex":"N_{30}","aliases":"N_30","display":"N_{30}","kind":"use","excerpt":"The empirical null and audit panel is extended by: N_29 · =post hoc target-selection audit, N_30 · =horizon cherry-picking audit, N_31 · =loss and utility substitution audit, N_32 · =restricted intervention-family audit, N_33 · =threshold and gate-set tuning audit, N_34 · =frozen specification sweep across S. The first five reconstruct or bound the","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b3cf9de3b727f4c1","tex":"N_{31}","aliases":"N_31","display":"N_{31}","kind":"use","excerpt":"The empirical null and audit panel is extended by: N_29 · =post hoc target-selection audit, N_30 · =horizon cherry-picking audit, N_31 · =loss and utility substitution audit, N_32 · =restricted intervention-family audit, N_33 · =threshold and gate-set tuning audit, N_34 · =frozen specification sweep across S. The first five reconstruct or bound the","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-438dc9211edba52e","tex":"N_{32}","aliases":"N_32","display":"N_{32}","kind":"use","excerpt":"The empirical null and audit panel is extended by: N_29 · =post hoc target-selection audit, N_30 · =horizon cherry-picking audit, N_31 · =loss and utility substitution audit, N_32 · =restricted intervention-family audit, N_33 · =threshold and gate-set tuning audit, N_34 · =frozen specification sweep across S. The first five reconstruct or bound the","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4fec5fa4a3f355be","tex":"N_{33}","aliases":"N_33","display":"N_{33}","kind":"use","excerpt":"The empirical null and audit panel is extended by: N_29 · =post hoc target-selection audit, N_30 · =horizon cherry-picking audit, N_31 · =loss and utility substitution audit, N_32 · =restricted intervention-family audit, N_33 · =threshold and gate-set tuning audit, N_34 · =frozen specification sweep across S. The first five reconstruct or bound the","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cb3d287819f7ea60","tex":"N_4","aliases":"N_4","display":"N_4","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4861b5f0d94d3545","tex":"N_5","aliases":"N_5","display":"N_5","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b3347751e36ff4b3","tex":"N_6","aliases":"N_6","display":"N_6","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a6b82a95d75be938","tex":"N_7","aliases":"N_7","display":"N_7","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-772efe6b5c524d11","tex":"N_8","aliases":"N_8","display":"N_8","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0e81b98c7aa5982b","tex":"N_9","aliases":"N_9","display":"N_9","kind":"use","excerpt":"The core null panel is: N_1 · =shuffled transfer maps with validated local matching, N_2 · =shuffled cross-node temporal correspondence, N_3 · =matched-capacity local memory, N_4 · =matched-size irrelevant artifact, N_5 · =participant ablation, N_6 · =artifact ablation, N_7 · =workspace implementation intervention, N_8 · =matched off-workspace inte","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7532962b20c80b4a","tex":"n_C","aliases":"n_C","display":"n_C","kind":"use","excerpt":"[Finite rank computation] Let n_C=∑_i∈ Vdim C_i. Then dimΓ(H)=n_C-rank(D_H). Hence the exact workspace dimension can be obtained by a single rank or singular-value computation, even when holonomy is unavailable because some transfers are noninvertible.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-929c7993bd8592a0","tex":"N_{\\mathrm{form}}","aliases":"N_form mathrm","display":"N_{\\mathrm{form}}","kind":"use","excerpt":"For thresholds Θ_form, define Form_Θ_form(W)=1 only when all applicable gates pass: · e_q^MDL≥ τ_q^MDL, · · e_q^causal≥ τ_q^causal, · δ_q^scaf≤ε_q^scaf, · · δ^leak≤ε_leak, · SRW_Θ(W)=1, · · Pass(N_form)=1. The final workspace gate is included because endogenous discovery of an unstable or causally irrelevant pattern is not endogenous workspace form","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e1613ab8834072c1","tex":"N_j","aliases":"N_j","display":"N_j","kind":"use","excerpt":"Null panel · Applicable N_j, preservation targets, and fit budgets.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-74f2332262dc7628","tex":"\\mathcal N_{\\mathrm{marg}}","aliases":"N_marg mathcal mathrm","display":"\\mathcal N_{\\mathrm{marg}}","kind":"use","excerpt":"Let P be the intact law of (Y_t^+, H_t^1,…, H_t^n). Define the local-marginal null class N_marg(P) = { Q: Q_ H_t^i=P_ H_t^i for every i }, and the stronger local-predictive-matched class N_pred(P) = { Q: Q_(Y_t^+, H_t^i) = P_(Y_t^+, H_t^i) for every i }. Because H_t^i can include the complete local trajectory, the latter class can preserve all sing","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0f7e6f17a2641d82","tex":"\\mathrm{node}","aliases":"node mathrm","display":"\\mathrm{node}","kind":"use","excerpt":"intact law at a fixed decision time, and let W_t=π_t( H_t) be a structurally supported candidate. Let Q⊆ N_pred(P) be a nonempty, predeclared panel of validated relation-breaking nulls. Assume: item exact structural--operational alignment, I_P(Y_t^+; H_tmid W_t)=0; item a positive node-level information gap, δ_node = min_i∈ V I_P(Y_t^+; H_tmid H_t^","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","excerpt":"The workspace is not determined by any single local state: ∀ i∈ V, ∃ R, R' such that pr_i( R)=pr_i( R') but Workspace( R)not≃Workspace( R'). The same person, model, or document can therefore participate in distinct workspaces because the relevant transfer graph, history, and operation structure differ.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f7331e0821e03ad4","tex":"\\mathrm{null}","aliases":"null mathrm","display":"\\mathrm{null}","kind":"use","excerpt":"bseteq N_pred(P) be a nonempty, predeclared panel of validated relation-breaking nulls. Assume: item exact structural--operational alignment, I_P(Y_t^+; H_tmid W_t)=0; item a positive node-level information gap, δ_node = min_i∈ V I_P(Y_t^+; H_tmid H_t^i) >0; item a positive robust matched-null gap, δ_null = min_Q∈ Q [ H_Q(Y_t^+mid H_t) - H_P(Y_t^+m","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c006fe4a9c51a332","tex":"\\mathsf{null\\_panels}","aliases":"null_panels mathsf","display":"\\mathsf{null\\_panels}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d8571e4c68c3456d","tex":"\\mathsf{nulls}","aliases":"nulls mathsf","display":"\\mathsf{nulls}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1659229448cb49ea","tex":"\\mathsf O","aliases":"O mathsf","display":"\\mathsf O","kind":"use","excerpt":"For each case we report the vector Card( K) = ( S, O, R, P, F, A, D, X, C ), where: ll S · = structural core and spectral coherence,","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":74},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-962eb33121c4bcf4","tex":"O_i","aliases":"O_i","display":"O_i","kind":"use","excerpt":"Let C_* be a finite-dimensional abstract core. The exact static theory provides a complete local chart overline E_i = ev_i∘Ψ:C_*⟶ C_i, where Ψ:C_*→Γ(H) is the root-to-section isomorphism. A participant or artifact need not expose that complete chart. Let O_i:C_i⟶ Y_i be the declared observation or access map and define the accessible encoding E_i=O","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cbeb43c66d7a9ada","tex":"\\mathcal O_{\\Workspace}^{\\mathrm{eff}}","aliases":"O_Workspace^eff mathcal Workspace mathrm","display":null,"kind":"use","excerpt":"Some natural operators may vanish on every global section while acting nontrivially on locally available but globally inconsistent directions. Define the annihilator Ann(Γ(H)) = {A∈ O_Workspace: π_Workspace(A)=0}. The effective workspace algebra is O_Workspace^eff = O_Workspace/Ann(Γ(H)). Its action on Γ(H) is faithful by construction.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d2b5ff62cf438bd2","tex":"\\mathrm{object}","aliases":"object mathrm","display":"\\mathrm{object}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c84a1672e483df5","tex":"\\mathrm{obs}","aliases":"obs mathrm","display":"\\mathrm{obs}","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f814b60f967151c4","tex":"\\mathrm{op}","aliases":"op mathrm","display":"\\mathrm{op}","kind":"use","excerpt":"For an admissible correspondence J, define the operation defect on a declared unit ball A_1 by δ_op( J) = sup_a∈ A_1 A_ W''β-β A_ W . The primary certification object is the identity profile d_id( J) = ( q_str, δ_op( J), δ_causal(β), δ_rec(β); μ^keep ). The first four coordinates must remain below declared tolerances, while the non-collapse margin ","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3f53b8803d9bdf36","tex":"\\mathrm{opt}","aliases":"opt mathrm","display":"\\mathrm{opt}","kind":"use","excerpt":"[Approximate counterfactual policy selection] Let A be a finite feasible action set. Let Q: A→ R be the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal poli","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ddc62450d7c47cd4","tex":"\\Out","aliases":"Out","display":null,"kind":"use","excerpt":"For S∈ S, define Ret_τ(S) = I ( X_S(t+τ);X_S(t) mid X_bar S(t),E_t ), Dep_τ(S) = I ( X_S(t+τ);X_bar S(t) mid X_S(t),E_t ), Out_τ(S) = I ( X_bar S(t+τ);X_S(t) mid X_bar S(t),E_t ). These quantities are observational. They can be nonzero because of common causes, selection, or feedback paths and therefore cannot alone establish a causal boundary.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-44a55b48f8dea448","tex":"\\mathrm{out}","aliases":"out mathrm","display":"\\mathrm{out}","kind":"use","excerpt":"Core leakage, residual contamination, and total transfer defect are measured by λ_e^out= V_e , λ_e^in= U_e , δ_e^tr = U_e + V_e . These quantities depend on the chosen norms and projectors. Their empirical meaning must therefore be tied to a declared intervention and prediction protocol.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8f0632ad5e72cb30","tex":"\\Out_\\tau","aliases":"Out_τ Out tau","display":null,"kind":"use","excerpt":"For S∈ S, define Ret_τ(S) = I ( X_S(t+τ);X_S(t) mid X_bar S(t),E_t ), Dep_τ(S) = I ( X_S(t+τ);X_bar S(t) mid X_S(t),E_t ), Out_τ(S) = I ( X_bar S(t+τ);X_S(t) mid X_bar S(t),E_t ). These quantities are observational. They can be nonzero because of common causes, selection, or feedback paths and therefore cannot alone establish a causal boundary.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-65874b040af73a4e","tex":"\\mathrm{ov}","aliases":"ov mathrm","display":"\\mathrm{ov}","kind":"use","excerpt":"A family w=(w_α)_α is exactly compatible when δ^0w=0. Its direct mismatch is m_ U(w)=δ^0w. After choosing inner products and positive overlap weights, the mismatch energy is E_ov(w) = δ^0w ^2 = ∑_α<β ν_αβ r_αβ^βw_β -r_αβ^αw_α ^2. This is the appropriate obstruction when the full local states are already given. Such a family glues exactly if and onl","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0aeccd83542eabf9","tex":"\\overline C","aliases":"overline C overline","display":"\\overline C","kind":"use","excerpt":"A practical hierarchy rarely transmits each complete local workspace. Let Q^p:check C^p( U;mathscr W)→overline C^p be compression maps into smaller local and overlap representations. Suppose there is a compressed coboundary overlineδ^p such that Q^p+1δ^p = overlineδ^p Q^p. The maps Q^bullet then form a cochain map.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-91fa6b16b49a0676","tex":"\\overline C^p","aliases":"overline C^p overline","display":"\\overline C^p","kind":"use","excerpt":"A practical hierarchy rarely transmits each complete local workspace. Let Q^p:check C^p( U;mathscr W)→overline C^p be compression maps into smaller local and overlap representations. Suppose there is a compressed coboundary overlineδ^p such that Q^p+1δ^p = overlineδ^p Q^p. The maps Q^bullet then form a cochain map.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-512abf9c72d3182f","tex":"\\overline d","aliases":"overline d overline","display":"\\overline d","kind":"use","excerpt":"egintheorem[Causal repair stabilizes maintenance deficit] Let e_t≥0 be the maintenance deficit. Suppose the matched no-repair trajectory satisfies e_t+1^(0) ≤ α e_t+d_t, where d_t≥0 is exogenous damage. Suppose the implemented repair policy has the causal one-step effect e_t+1^(0)-e_t+1 ≥ κ e_t-ϵ_t. Assume 0≤β:=α-κ<1, 0≤ d_t≤overline d, 0≤ϵ_t≤overl","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0ff3814053977f1f","tex":"\\overline E","aliases":"overline E overline","display":"\\overline E","kind":"use","excerpt":"Let C_* be a finite-dimensional abstract core. The exact static theory provides a complete local chart overline E_i = ev_i∘Ψ:C_*⟶ C_i, where Ψ:C_*→Γ(H) is the root-to-section isomorphism. A participant or artifact need not expose that complete chart. Let O_i:C_i⟶ Y_i be the declared observation or access map and define the accessible encoding E_i=O","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b1c5903348a4fbe0","tex":"\\overline E_i","aliases":"overline E_i overline","display":"\\overline E_i","kind":"use","excerpt":"Let C_* be a finite-dimensional abstract core. The exact static theory provides a complete local chart overline E_i = ev_i∘Ψ:C_*⟶ C_i, where Ψ:C_*→Γ(H) is the root-to-section isomorphism. A participant or artifact need not expose that complete chart. Let O_i:C_i⟶ Y_i be the declared observation or access map and define the accessible encoding E_i=O","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-53747cc445df0a6e","tex":"\\overline m","aliases":"overline m overline","display":"\\overline m","kind":"use","excerpt":"Let m_q(W; D) be the signed margin of gate q, oriented so that m_q(W; D)>0 means that the gate passes after its uncertainty correction. Define the decomposition envelope underline m_q = inf_ D∈ Dm_q(W; D), overline m_q = sup_ D∈ Dm_q(W; D), and its sensitivity width Δ_q^dec = overline m_q-underline m_q.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b86a1edcb3971c58","tex":"\\overline m_q","aliases":"overline m_q overline","display":"\\overline m_q","kind":"use","excerpt":"Let m_q(W; D) be the signed margin of gate q, oriented so that m_q(W; D)>0 means that the gate passes after its uncertainty correction. Define the decomposition envelope underline m_q = inf_ D∈ Dm_q(W; D), overline m_q = sup_ D∈ Dm_q(W; D), and its sensitivity width Δ_q^dec = overline m_q-underline m_q.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ae27db76d99dffe","tex":"\\overline m_q^{\\mathfrak S}","aliases":"overline m_q^ S overline mathfrak","display":"\\overline m_q^{\\mathfrak S}","kind":"use","excerpt":"For each gate q∈ J_*, define the specification envelope underline m_q^ S(W) = inf_ς∈ Sm_q(W;ς), overline m_q^ S(W) = sup_ς∈ Sm_q(W;ς). Its sensitivity width is Δ_q^spec(W) = overline m_q^ S(W) - underline m_q^ S(W). The robust certificate margin is r_ S(W) = inf_ς∈ S min_q∈ J_*m_q(W;ς). Then Cert_ S(W)=1 Longleftrightarrow r_ S(W)>0.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3ae5d3f5947d525c","tex":"\\overline m_q^{\\mathrm{ID}}","aliases":"overline m_q^ID overline mathrm","display":"\\overline m_q^{\\mathrm{ID}}","kind":"use","excerpt":"The sharp lower and upper identified margins are underline m_q^ID(ς) = inf M_q^ID(ς), overline m_q^ID(ς) = sup M_q^ID(ς). A gate is sign-identified as passing if its lower bound is positive. It is sign-identified as failing if its upper bound is nonpositive. If the interval crosses zero, the available design does not determine that gate's outcome.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-41dacfd07761cf94","tex":"\\overline r","aliases":"overline r overline","display":"\\overline r","kind":"use","excerpt":"For robustness across specifications, define the realization-level robust margin r_ S( R) = inf_ς∈ S min_q∈ J_* m_q(W_ R;ς) and its identified range underline r_ID, S = inf_ R∈ R_ID r_ S( R), overline r_ID, S = sup_ R∈ R_ID r_ S( R).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aa71349a526385fd","tex":"\\overline r_{\\mathrm{ID},\\mathfrak S}","aliases":"overline r_ID, S overline mathrm mathfrak","display":"\\overline r_{\\mathrm{ID},\\mathfrak S}","kind":"use","excerpt":"For robustness across specifications, define the realization-level robust margin r_ S( R) = inf_ς∈ S min_q∈ J_* m_q(W_ R;ς) and its identified range underline r_ID, S = inf_ R∈ R_ID r_ S( R), overline r_ID, S = sup_ R∈ R_ID r_ S( R).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a3a402929f50849b","tex":"\\overline\\delta^p","aliases":"overlineδ^p overline delta","display":"\\overline\\delta^p","kind":"use","excerpt":"A practical hierarchy rarely transmits each complete local workspace. Let Q^p:check C^p( U;mathscr W)→overline C^p be compression maps into smaller local and overlap representations. Suppose there is a compressed coboundary overlineδ^p such that Q^p+1δ^p = overlineδ^p Q^p. The maps Q^bullet then form a cochain map.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d1d6282319c98963","tex":"\\overline\\delta_{\\mathrm{coh}}","aliases":"overlineδ_coh overline delta mathrm","display":"\\overline\\delta_{\\mathrm{coh}}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-91424c61d93b987b","tex":"\\overline\\epsilon","aliases":"overlineϵ overline epsilon","display":"\\overline\\epsilon","kind":"use","excerpt":"egintheorem[Causal repair stabilizes maintenance deficit] Let e_t≥0 be the maintenance deficit. Suppose the matched no-repair trajectory satisfies e_t+1^(0) ≤ α e_t+d_t, where d_t≥0 is exogenous damage. Suppose the implemented repair policy has the causal one-step effect e_t+1^(0)-e_t+1 ≥ κ e_t-ϵ_t. Assume 0≤β:=α-κ<1, 0≤ d_t≤overline d, 0≤ϵ_t≤overl","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0e517349dec3b6ba","tex":"\\overset","aliases":"overset","display":null,"kind":"use","excerpt":"[XOR witness pair] Under the intact law P, let X_1,X_2oversetiid∼Bernoulli(1/2), Y=X_1⊕ X_2. Then each (X_i,Y) pair is independent and uniform, so H_P(Ymid X_i)=1 bit, H_P(Ymid X_1,X_2)=0. Let Q make X_1,X_2,Y mutually independent fair bits. Then Q_(X_i,Y)=P_(X_i,Y) (i=1,2), but H_Q(Ymid X_1,X_2)=1 bit. Therefore Δ_joint^log=1 bit, k_0=2, Δ_tr(Q)=1","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ac1c675b9969cf81","tex":"\\mathbf p","aliases":"p mathbf","display":"\\mathbf p","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2e99a99588c03ac6","tex":"\\mathsf P","aliases":"P mathsf","display":"\\mathsf P","kind":"use","excerpt":"For each case we report the vector Card( K) = ( S, O, R, P, F, A, D, X, C ), where: ll S · = structural core and spectral coherence,","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","excerpt":"Let P and P' be rank-d workspace projectors selected from L and L', and let g>0 be the separating source gap.","count":19},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9b3808979d25e382","tex":"\\mathbb P^{\\jmath}","aliases":"P^jmath mathbb jmath","display":"\\mathbb P^{\\jmath}","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d190172a89542b2b","tex":"\\mathbb P^{\\jmath'}","aliases":"P^jmath' mathbb jmath","display":"\\mathbb P^{\\jmath'}","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4e33290354d118d1","tex":"\\mathbb P^{\\jmath',\\operatorname{do}(W_t\\sim\\mu)}","aliases":"P^jmath',do(W_t∼μ) mathbb jmath operatorname sim mu","display":"\\mathbb P^{\\jmath',\\operatorname{do}(W_t\\sim\\mu)}","kind":"use","excerpt":"A stronger mediation test equalizes the workspace state after two distinct channel interventions. Let μ be a common imposed workspace distribution, implemented by a feasible stochastic intervention in the sense of Definition~[source reference]. The notation below is shorthand for that implementation. Define the residual mediation defect δ_med = sup","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e28d7b225d12842e","tex":"\\mathbb P^{\\jmath,\\operatorname{do}(W_t\\sim\\mu)}","aliases":"P^jmath,do(W_t∼μ) mathbb jmath operatorname sim mu","display":"\\mathbb P^{\\jmath,\\operatorname{do}(W_t\\sim\\mu)}","kind":"use","excerpt":"A stronger mediation test equalizes the workspace state after two distinct channel interventions. Let μ be a common imposed workspace distribution, implemented by a feasible stochastic intervention in the sense of Definition~[source reference]. The notation below is shorthand for that implementation. Define the residual mediation defect δ_med = sup","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0f6df868f0ffb270","tex":"\\mathbb P^{\\jmath_E}","aliases":"P^jmath_E mathbb jmath","display":"\\mathbb P^{\\jmath_E}","kind":"use","excerpt":"Let jmath_E randomize selected communication edges while preserving their marginal message laws, rates, and timing budget. The causal transfer gap is Δ_tr^do(jmath_E) = Risk_ℓ, P^jmath_E(V) - Risk_ℓ, P(V). Unlike an observational null law, this quantity is anchored to an implemented channel intervention. A positive effect is evidence that the intac","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1d50e4bc8860b240","tex":"\\mathbb P^\\iota","aliases":"P^ι mathbb iota","display":"\\mathbb P^\\iota","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b562c0e6508c1c16","tex":"\\mathbb P^{\\iota}","aliases":"P^ι mathbb iota","display":"\\mathbb P^{\\iota}","kind":"use","excerpt":"Observational prediction may merge histories that behave identically under the current policy but respond differently to future interventions. Let I be a finite or countable family of interventions applied only after time t. For each ι∈ I, let K_t^ι(h,·) = P^ι(Y_t^+∈·mid H_t=h) be a regular post-intervention kernel on the common pre-intervention hi","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9a4a28695123e7b2","tex":"\\mathbb P^{\\iota'}","aliases":"P^ι' mathbb iota","display":"\\mathbb P^{\\iota'}","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-369f1f9b82f4b357","tex":"\\mathbb P^{\\iota_w}","aliases":"P^ι_w mathbb iota","display":"\\mathbb P^{\\iota_w}","kind":"use","excerpt":"Let D be a divergence or metric on probability laws. For a matched background value b, define CE_Workspace(w,w';b) = D ( P^ι_w(Y_t^+∈·mid B_t=b), P^ι_w'(Y_t^+∈·mid B_t=b) ). For a declared comparison design ν over (w,w',b), define η_causal^ν = E_(w,w',b)∼ν [ CE_Workspace(w,w';b) ]. The maximum η_causal^max = sup_w,w',bCE_Workspace(w,w';b) may be us","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-67a8c97514b734a8","tex":"\\mathbb P^{\\iota_{w'}}","aliases":"P^ι_w' mathbb iota","display":"\\mathbb P^{\\iota_{w'}}","kind":"use","excerpt":"Let D be a divergence or metric on probability laws. For a matched background value b, define CE_Workspace(w,w';b) = D ( P^ι_w(Y_t^+∈·mid B_t=b), P^ι_w'(Y_t^+∈·mid B_t=b) ). For a declared comparison design ν over (w,w',b), define η_causal^ν = E_(w,w',b)∼ν [ CE_Workspace(w,w';b) ]. The maximum η_causal^max = sup_w,w',bCE_Workspace(w,w';b) may be us","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0971407f1e249fb9","tex":"P_{\\mathcal H^1_{\\mathcal U}}","aliases":"P_ H^1_ U mathcal mathcal","display":"P_{\\mathcal H^1_{\\mathcal U}}","kind":"use","excerpt":"[Harmonic gluing obstruction] For every cocycle b∈kerδ^1, there is an orthogonal decomposition b=δ^0q^†+h, h∈ H^1_ U, where q^†=(δ^0)^† b is a least-squares potential. Moreover, inf_q b-δ^0q = h = P_ H^1_ Ub . Hence b is exactly integrable if and only if its harmonic component vanishes.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-63049a5a3e538809","tex":"P_{\\mathbf H_t^i}","aliases":"P_ H_t^i mathbf","display":"P_{\\mathbf H_t^i}","kind":"use","excerpt":"Let P be the intact law of (Y_t^+, H_t^1,…, H_t^n). Define the local-marginal null class N_marg(P) = { Q: Q_ H_t^i=P_ H_t^i for every i }, and the stronger local-predictive-matched class N_pred(P) = { Q: Q_(Y_t^+, H_t^i) = P_(Y_t^+, H_t^i) for every i }. Because H_t^i can include the complete local trajectory, the latter class can preserve all sing","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a9648d767bd86136","tex":"\\mathbb P_{\\mathfrak R}","aliases":"P_ R mathbb mathfrak","display":"\\mathbb P_{\\mathfrak R}","kind":"use","excerpt":"in M satisfy R≡_ E R' but Cert(W_ R;ς) ≠ Cert(W_ R';ς). Then the workspace certificate is not identifiable on M under E. More strongly, for every possibly randomized decision rule widehat C based only on data generated by E, P_ R ( widehat C≠ C_ R ) + P_ R' ( widehat C≠ C_ R' ) =1, where C_ R=Cert(W_ R;ς). Consequently, inf_widehat C max{ P_ R(wide","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1c07582a58dd7ab4","tex":"\\mathbb P_{\\mathfrak R'}","aliases":"P_ R' mathbb mathfrak","display":"\\mathbb P_{\\mathfrak R'}","kind":"use","excerpt":"in M satisfy R≡_ E R' but Cert(W_ R;ς) ≠ Cert(W_ R';ς). Then the workspace certificate is not identifiable on M under E. More strongly, for every possibly randomized decision rule widehat C based only on data generated by E, P_ R ( widehat C≠ C_ R ) + P_ R' ( widehat C≠ C_ R' ) =1, where C_ R=Cert(W_ R;ς). Consequently, inf_widehat C max{ P_ R(wide","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3a63e93da2a05e6e","tex":"\\mathbb P_{\\mathfrak R_0}","aliases":"P_ R_0 mathbb mathfrak","display":"\\mathbb P_{\\mathfrak R_0}","kind":"use","excerpt":"In finite samples, the identified set is itself uncertain. Let widehat R_1-δ be a confidence region satisfying inf_ R_0∈ M P_ R_0 ( R_0∈ widehat R_1-δ ) ≥1-δ. A conservative robust lower margin is widehat r_1-δ = inf_ R∈widehat R_1-δ inf_ς∈ S min_q m_q(W_ R;ς). If widehat r_1-δ>0, a false specification-robust certificate has probability at most δ, ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e89d266138c31a50","tex":"P_{(X_i,Y)}","aliases":"P_(X_i,Y)","display":"P_{(X_i,Y)}","kind":"use","excerpt":"[XOR witness pair] Under the intact law P, let X_1,X_2oversetiid∼Bernoulli(1/2), Y=X_1⊕ X_2. Then each (X_i,Y) pair is independent and uniform, so H_P(Ymid X_i)=1 bit, H_P(Ymid X_1,X_2)=0. Let Q make X_1,X_2,Y mutually independent fair bits. Then Q_(X_i,Y)=P_(X_i,Y) (i=1,2), but H_Q(Ymid X_1,X_2)=1 bit. Therefore Δ_joint^log=1 bit, k_0=2, Δ_tr(Q)=1","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1e71e5d9ebee874e","tex":"P_{(Y_t^+,\\mathbf H_t^i)}","aliases":"P_(Y_t^+, H_t^i) mathbf","display":"P_{(Y_t^+,\\mathbf H_t^i)}","kind":"use","excerpt":"Let P be the intact law of (Y_t^+, H_t^1,…, H_t^n). Define the local-marginal null class N_marg(P) = { Q: Q_ H_t^i=P_ H_t^i for every i }, and the stronger local-predictive-matched class N_pred(P) = { Q: Q_(Y_t^+, H_t^i) = P_(Y_t^+, H_t^i) for every i }. Because H_t^i can include the complete local trajectory, the latter class can preserve all sing","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-287bd2e64a3f4504","tex":"P_*","aliases":"P_*","display":"P_*","kind":"use","excerpt":"[Averaging projector for finite holonomy] If Hol_o is finite and acts unitarily, then P_*= frac1|Hol_o| ∑_h∈Hol_oh is the orthogonal projector onto C_*.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7fa5c09575a6a832","tex":"P_*^*","aliases":"P_*^*","display":"P_*^*","kind":"use","excerpt":"For every g∈Hol_o, left multiplication permutes the group elements, so gP_*=P_*. Thus ranP_*⊆ C_*. If x∈ C_*, then P_*x=x, so C_*⊆ranP_*. Unitarity gives P_*^*=P_*, and the range characterization implies P_*^2=P_*.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-344ffabd889d52f2","tex":"P_*^2","aliases":"P_*^2","display":"P_*^2","kind":"use","excerpt":"For every g∈Hol_o, left multiplication permutes the group elements, so gP_*=P_*. Thus ranP_*⊆ C_*. If x∈ C_*, then P_*x=x, so C_*⊆ranP_*. Unitarity gives P_*^*=P_*, and the range characterization implies P_*^2=P_*.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"Write c=P_0c+Q_0c, where P_0 projects onto W_0 and Q_0=I-P_0. Since L_HP_0=0, E_H(c) = ⟨ Q_0c,L_HQ_0c⟩. The spectrum of L_H on W_0^⊥ lies in [γ_H, L_H ]. Hence γ_H Q_0c ^2 ≤ E_H(c) ≤ L_H Q_0c ^2. Finally, Q_0c =dist(c,W_0).","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6bb2e795a72d51fe","tex":"\\mathcal P_{0:T}","aliases":"P_0:T mathcal","display":"\\mathcal P_{0:T}","kind":"use","excerpt":"A finite-horizon persistence certificate may be recorded as P_0:T = ( max_t<Tq_t^str, max_t≤ Te_t, min_t<Tμ_t^keep, ∑_t<Tσ_t ). No single coordinate of this vector subsumes the others.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8a232427a9b65168","tex":"p_C","aliases":"p_C","display":"p_C","kind":"use","excerpt":"Order the local coordinates as x=(q_P,p_P,p_C,f_C,q_N,f_N)^ T. Compatibility requires p_P=p_C, f_C=f_N, q_N=q_P. The discrepancy operator is D_HH = 0 · 1 · -1 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c7febed8b9e4040c","tex":"p_{\\mathrm{certificate}}","aliases":"p_certificate mathrm","display":"p_{\\mathrm{certificate}}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9fde4b74ddbe7f4b","tex":"p_{\\mathrm{concept}}","aliases":"p_concept mathrm","display":"p_{\\mathrm{concept}}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b24df2c80a0e6d4c","tex":"p_{\\mathrm{experiment}}","aliases":"p_experiment mathrm","display":"p_{\\mathrm{experiment}}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c740fc3062154f48","tex":"P_{\\mathrm{HH}}","aliases":"P_HH mathrm","display":"P_{\\mathrm{HH}}","kind":"use","excerpt":"ition has all three bits and therefore zero conditional entropy. Hence Δ_joint^log=1 bit, k_0=2. Let Q_HH preserve the independent uniform local bit pairs but replace Y by an independent uniform bit. Because parity is independent of every two-bit local view, Q_HH preserves each (Y,X_i)-marginal. Nevertheless, Risk_log,Q_HH(V) - Risk_log,P_HH(V) =1 ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fd6af1e81735bb80","tex":"P_{i_0}","aliases":"P_i_0","display":"P_{i_0}","kind":"use","excerpt":"· = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r · × (∏_ℓ=1^r-1 H_ℓ ), and the same right-hand side bounds P_i_mT_m⋯ T_1P_i_0-H_m⋯ H_1 .","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8822c5c4684bf376","tex":"P_{i_2}","aliases":"P_i_2","display":"P_{i_2}","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-121ed7254a7702c6","tex":"P_{i_m}","aliases":"P_i_m","display":"P_{i_m}","kind":"use","excerpt":"· × (∏_ℓ=1^r-1 H_ℓ ), and the same right-hand side bounds P_i_mT_m⋯ T_1P_i_0-H_m⋯ H_1 .","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"V_fH_e+K_fV_e · V_fU_e+K_fK_e . The induced core-to-core block is therefore P_kT_fT_eP_i = H_fH_e+U_fV_e. The term U_fV_e is a residual-mediated correction: information leaves the candidate core at the intermediate node and later re-enters it. Consequently, the simple path product H_fH_e faithfully represents full core transport only under a preser","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dc7ba1c49968c37c","tex":"p_k","aliases":"p_k","display":"p_k","kind":"use","excerpt":"A pointwise valid test does not remain valid after the analyst searches for the most favorable target, horizon, null panel, or threshold. Consider K independent null p-values and select the smallest. Then P(min_1≤ k≤ Kp_k≤α) = 1-(1-α)^K, which exceeds α whenever K>1. The same logic applies when a workspace certificate is reported only for the speci","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bee861a8efe6782e","tex":"P_{\\ker K_{\\mathcal U}}","aliases":"P_ker K_ U ker mathcal","display":"P_{\\ker K_{\\mathcal U}}","kind":"use","excerpt":"Let m_i=|{α:i∈ V_α}| be the multiplicity of vertex i. Then J_ U^*J_ U=M_ U, (M_ Uc)_i=m_ic_i. The pseudoinverse performs coordinatewise averaging: (J_ U^† u)_i = frac1m_i ∑_α:i∈ V_α(u_α)_i. Let widehat u = P_ker K_ Uu = J_ Uwidehat c. Then widehat c = J_ U^†widehat u and u-J_ Uwidehat c ≤ frac K_ Uu sqrtγ_ U^tot. If m_min=min_i m_i, then widehat c-","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e32f108b3db0ce30","tex":"p_M","aliases":"p_M","display":"p_M","kind":"use","excerpt":"Order the local coordinates as x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^ T. The overlap discrepancies are g_H-g_M, p_M-p_X, v_X-v_A, d_A-d_H. Thus D_HAI = 1 · 0 · -1 · 0 · 0 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6ef244069610e32a","tex":"p_{\\mathrm{object}}","aliases":"p_object mathrm","display":"p_{\\mathrm{object}}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cf13a09c71ab27c8","tex":"p_P","aliases":"p_P","display":"p_P","kind":"use","excerpt":"Order the local coordinates as x=(q_P,p_P,p_C,f_C,q_N,f_N)^ T. Compatibility requires p_P=p_C, f_C=f_N, q_N=q_P. The discrepancy operator is D_HH = 0 · 1 · -1 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a2f212d49285eb21","tex":"P_{s(e)}","aliases":"P_s(e)","display":"P_{s(e)}","kind":"use","excerpt":"V_e · K_e , where H_e=P_t(e)T_eP_s(e):C_s(e)→ C_t(e), U_e=P_t(e)T_eQ_s(e):R_s(e)→ C_t(e), V_e=Q_t(e)T_eP_s(e):C_s(e)→ R_t(e), K_e=Q_t(e)T_eQ_s(e):R_s(e)→ R_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b14edd9368412e72","tex":"P_{s(\\gamma)}","aliases":"P_s(γ) gamma","display":"P_{s(\\gamma)}","kind":"use","excerpt":"[Pathwise core preservation] Suppose V_e=0 for every edge in a directed path γ=e_m⋯ e_1. Then T_γ(C_s(γ))⊆ C_t(γ) and P_t(γ)T_γ P_s(γ) = H_e_m⋯ H_e_1.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b2bf370eaab77a0d","tex":"P_{S,d}","aliases":"P_S,d","display":"P_{S,d}","kind":"use","excerpt":"For the induced subsystem on S, define Coh(S) = tr ( P_S,dL_SP_S,d ), where P_S,d projects onto the selected low-energy workspace modes. Let Cut(S) = ∑_e:s(e)∈ S, t(e)∉ Sw_e + ∑_e:s(e)∉ S, t(e)∈ Sw_e. Let k_ε(S) be the minimal sufficient coalition size inside the candidate subsystem. A candidate is admissible only if SRW_Θ_S(W_S)=1, k_ε(S)≥2, unles","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-12b33428971a165a","tex":"P_t","aliases":"P_t","display":"P_t","kind":"use","excerpt":"Let c(t)=(c_i(t))_i∈ V ∈ C_V(t), P_t= 1_[0,ε_t](L_t). The instantaneous structural workspace coordinate is W_t^str=P_tc(t) ∈ Γ_ε_t(H(t)). When P_t varies with time, these coordinates inhabit a moving family of subspaces. Parts~X--XII will supply the transport required to compare them across time. In what follows, all sufficiency and intervention st","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5251d237ebe0338d","tex":"P_{t(e)}","aliases":"P_t(e)","display":"P_{t(e)}","kind":"use","excerpt":"V_e · K_e , where H_e=P_t(e)T_eP_s(e):C_s(e)→ C_t(e), U_e=P_t(e)T_eQ_s(e):R_s(e)→ C_t(e), V_e=Q_t(e)T_eP_s(e):C_s(e)→ R_t(e), K_e=Q_t(e)T_eQ_s(e):R_s(e)→ R_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-296b82e7620a371c","tex":"P_{t(\\gamma)}","aliases":"P_t(γ) gamma","display":"P_{t(\\gamma)}","kind":"use","excerpt":"[Pathwise core preservation] Suppose V_e=0 for every edge in a directed path γ=e_m⋯ e_1. Then T_γ(C_s(γ))⊆ C_t(γ) and P_t(γ)T_γ P_s(γ) = H_e_m⋯ H_e_1.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fa1ae76a27248a1e","tex":"P_{t+1}","aliases":"P_t+1","display":"P_{t+1}","kind":"use","excerpt":"[Temporal workspace tracking] If η_t^L<fracg_t2, then the transported and newly extracted workspace projectors satisfy q_t^str := P_t+1-U_t^VP_t(U_t^V)^* ≤ fracη_t^Lg_t-η_t^L ≤ frac2η_t^Lg_t.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e608e3a8306a65c6","tex":"p_{\\mathrm{theorem}}","aliases":"p_theorem mathrm","display":"p_{\\mathrm{theorem}}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-934a5efa5bc74615","tex":"p_X","aliases":"p_X","display":"p_X","kind":"use","excerpt":"Order the local coordinates as x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^ T. The overlap discrepancies are g_H-g_M, p_M-p_X, v_X-v_A, d_A-d_H. Thus D_HAI = 1 · 0 · -1 · 0 · 0 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2509b957f1cb9bfa","tex":"P_{\\{\\lambda_1,\\ldots,\\lambda_d\\}}","aliases":"P_{λ_1,…,λ_d} lambda ldots lambda","display":"P_{\\{\\lambda_1,\\ldots,\\lambda_d\\}}","kind":"use","excerpt":"An equivalent index-based description fixes d and writes W_d=ran P_d, P_d=P_{λ_1,…,λ_d}(L_H), with cluster gap g_d=λ_d+1-λ_d>0. The index-based form is convenient for perturbation theorems because the workspace rank is declared explicitly.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-127c73e14985958f","tex":"P_\\Pi","aliases":"P_Π Pi","display":"P_\\Pi","kind":"use","excerpt":"[Two-scale coherence bound] Let P_Π be the orthogonal projector onto J_Π W_Π. Let γ_int>0 be the smallest positive eigenvalue of D_int^*D_int, and let γ_Π>0 be the smallest positive eigenvalue of D_Π^*D_Π. Then every c∈ C_V satisfies dist(c,Γ(H)) ≤ frac D_intc sqrtγ_int + frac D_Π J_Π^*P_Π c sqrtγ_Π. The first term measures incoherence within macro","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-267dde3867775d23","tex":"\\pagenumbering","aliases":"pagenumbering","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bc660fe2b7285e52","tex":"\\par","aliases":"par","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-71a5a8f5d615fdc1","tex":"\\mathrm{pass}","aliases":"pass mathrm","display":"\\mathrm{pass}","kind":"use","excerpt":"· -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass contains only candidates that satisfy the required workspace and anti-centralization gates.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-945cd4bac8caf45c","tex":"\\operatorname{Pass}","aliases":"Pass operatorname","display":"\\operatorname{Pass}","kind":"use","excerpt":"For thresholds Θ_form, define Form_Θ_form(W)=1 only when all applicable gates pass: · e_q^MDL≥ τ_q^MDL, · · e_q^causal≥ τ_q^causal, · δ_q^scaf≤ε_q^scaf, · · δ^leak≤ε_leak, · SRW_Θ(W)=1, · · Pass(N_form)=1. The final workspace gate is included because endogenous discovery of an unstable or causally irrelevant pattern is not endogenous workspace form","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7c6ad645858c8d6c","tex":"\\operatorname{Pass}_j","aliases":"Pass_j operatorname","display":"\\operatorname{Pass}_j","kind":"use","excerpt":"Let J(L) be the required gate set for claim level L. Then Certify_L(W;ς) = bigwedge_j∈ J(L,ς) Pass_j(W;ς). For a workspace-agent claim, the required set includes every structural, operational, relational, persistent, goal, counterfactual-control, and self-maintenance gate declared in Parts~IX and XVI. A specification-robust claim additionally requi","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-069badab219deadb","tex":"\\Path","aliases":"Path","display":null,"kind":"use","excerpt":"The directed path category is C=Path(G). Its objects are vertices. A path γ=e_m⋯ e_2e_1 means that e_1 is traversed first. Composition is concatenation, and the empty path at i is denoted 1_i.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ff51057b24527f74","tex":"\\mathrm{path}","aliases":"path mathrm","display":"\\mathrm{path}","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c1b5371e3a9d5881","tex":"\\mathrm{pers}","aliases":"pers mathrm","display":"\\mathrm{pers}","kind":"use","excerpt":"Let Θ = ( ε_coh, ε_align, ε_k, underlineη_c, underlineη_loop, δ_node, δ_null, δ_spec, k, ε_pers ). Let ς = ( D,Y^+,ℓ,τ, I, Q,Θ) be the complete declared scientific specification, where Q⊆ N_pred(P) is a validated null panel. A dynamic realization contains a certified stratified relational workspace when there exists a candidate W_t satisfying all g","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e97d94c252c22d74","tex":"\\operatorname{Person}","aliases":"Person operatorname","display":"\\operatorname{Person}","kind":"use","excerpt":"Workspace( R) centernotLongrightarrow Conscious( R). Likewise, Distributed(Workspace) centernotLongrightarrow Person(Workspace). Workspace structure, reentry, agency, self-maintenance, and phenomenal consciousness remain formally distinct predicates. Any bridge among them would require additional theory and evidence.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-91712eba3c98bd65","tex":"\\operatorname{PhenomenalConsciousness}","aliases":"PhenomenalConsciousness operatorname","display":"\\operatorname{PhenomenalConsciousness}","kind":"use","excerpt":"subtopicSeparate, unresolved axis PhenomenalConsciousness.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9c17431e47fa1f08","tex":"\\operatorname{PhenomenallyConscious}","aliases":"PhenomenallyConscious operatorname","display":"\\operatorname{PhenomenallyConscious}","kind":"use","excerpt":"The following implications are not licensed: Agent_Θ_A(Workspace) centernotLongrightarrow PhenomenallyConscious(Workspace), SelfMaintaining(Workspace) centernotLongrightarrow SelfAware(Workspace), GoalPersistent(Workspace) centernotLongrightarrow MoralPatient(Workspace). A thermostat-like controller can maintain a variable without representing itse","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-11fad9be5f1124a8","tex":"\\preccurlyeq","aliases":"preccurlyeq","display":null,"kind":"use","excerpt":"Failure to find the phrase ``Stratified Relational Workspace Theory'' in prior literature is weak evidence. Priority is determined by mathematical and operational equivalence, not by terminology. Let C be a claim in the paper and S a prior source. Write Spreccurlyeq C when the main content of C follows from S after admissible relabeling, restrictio","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9c477d9845b6ea68","tex":"\\mathbf{Prob}","aliases":"Prob mathbf","display":"\\mathbf{Prob}","kind":"use","excerpt":"The finite-dimensional Hilbert assumption supplies adjoints and spectral tools. The exact compatible-section definitions require less and may later be formulated in categories such as Ban, Meas, Prob, Top, Alg.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-410042c8a19f2c04","tex":"Q^\\bullet","aliases":"Q^bullet bullet","display":"Q^\\bullet","kind":"use","excerpt":"A practical hierarchy rarely transmits each complete local workspace. Let Q^p:check C^p( U;mathscr W)→overline C^p be compression maps into smaller local and overlap representations. Suppose there is a compressed coboundary overlineδ^p such that Q^p+1δ^p = overlineδ^p Q^p. The maps Q^bullet then form a cochain map.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d954ee225400f1d1","tex":"Q^p","aliases":"Q^p","display":"Q^p","kind":"use","excerpt":"A practical hierarchy rarely transmits each complete local workspace. Let Q^p:check C^p( U;mathscr W)→overline C^p be compression maps into smaller local and overlap representations. Suppose there is a compressed coboundary overlineδ^p such that Q^p+1δ^p = overlineδ^p Q^p. The maps Q^bullet then form a cochain map.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5febd3c5f34840a8","tex":"Q^{p+1}","aliases":"Q^p+1","display":"Q^{p+1}","kind":"use","excerpt":"A practical hierarchy rarely transmits each complete local workspace. Let Q^p:check C^p( U;mathscr W)→overline C^p be compression maps into smaller local and overlap representations. Suppose there is a compressed coboundary overlineδ^p such that Q^p+1δ^p = overlineδ^p Q^p. The maps Q^bullet then form a cochain map.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dd3d9688e68cf63f","tex":"q^t","aliases":"q^t","display":"q^t","kind":"use","excerpt":"[Mean-square persistence] Suppose E[e_t+1^2mid F_t] ≤ q e_t^2+v_t, 0≤ q<1, and E[v_t]≤bar v. Then E[e_t^2] ≤ q^t E[e_0^2] + fracbar v(1-q^t)1-q, so limsup_t→∞ E[e_t^2] ≤ fracbar v1-q.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-320f8d74cef644fe","tex":"q^\\dagger","aliases":"q^† dagger","display":"q^\\dagger","kind":"use","excerpt":"[Harmonic gluing obstruction] For every cocycle b∈kerδ^1, there is an orthogonal decomposition b=δ^0q^†+h, h∈ H^1_ U, where q^†=(δ^0)^† b is a least-squares potential. Moreover, inf_q b-δ^0q = h = P_ H^1_ Ub . Hence b is exactly integrable if and only if its harmonic component vanishes.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-629f51d071a72580","tex":"Q_{\\mathbf H_t^i}","aliases":"Q_ H_t^i mathbf","display":"Q_{\\mathbf H_t^i}","kind":"use","excerpt":"Let P be the intact law of (Y_t^+, H_t^1,…, H_t^n). Define the local-marginal null class N_marg(P) = { Q: Q_ H_t^i=P_ H_t^i for every i }, and the stronger local-predictive-matched class N_pred(P) = { Q: Q_(Y_t^+, H_t^i) = P_(Y_t^+, H_t^i) for every i }. Because H_t^i can include the complete local trajectory, the latter class can preserve all sing","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cf0327aab17f286b","tex":"Q_{(X_i,Y)}","aliases":"Q_(X_i,Y)","display":"Q_{(X_i,Y)}","kind":"use","excerpt":"[XOR witness pair] Under the intact law P, let X_1,X_2oversetiid∼Bernoulli(1/2), Y=X_1⊕ X_2. Then each (X_i,Y) pair is independent and uniform, so H_P(Ymid X_i)=1 bit, H_P(Ymid X_1,X_2)=0. Let Q make X_1,X_2,Y mutually independent fair bits. Then Q_(X_i,Y)=P_(X_i,Y) (i=1,2), but H_Q(Ymid X_1,X_2)=1 bit. Therefore Δ_joint^log=1 bit, k_0=2, Δ_tr(Q)=1","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-87c22bf8eb0a7839","tex":"Q_{(Y_t^+,\\mathbf H_t^i)}","aliases":"Q_(Y_t^+, H_t^i) mathbf","display":"Q_{(Y_t^+,\\mathbf H_t^i)}","kind":"use","excerpt":"Let P be the intact law of (Y_t^+, H_t^1,…, H_t^n). Define the local-marginal null class N_marg(P) = { Q: Q_ H_t^i=P_ H_t^i for every i }, and the stronger local-predictive-matched class N_pred(P) = { Q: Q_(Y_t^+, H_t^i) = P_(Y_t^+, H_t^i) for every i }. Because H_t^i can include the complete local trajectory, the latter class can preserve all sing","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4e4bfa4de54508e8","tex":"q_*","aliases":"q_*","display":"q_*","kind":"use","excerpt":"[Quantitative replacement invariance] Let J be a unitary graph correspondence with diagram defect δ_D. Define η_L=2 D δ_D+δ_D^2 and assume η_L<frac g2. Set q_* = fracη_Lg-η_L. Then there exists a unitary workspace correspondence β for which q_str≤ q_*. Moreover, δ_op( J) ≤ sup_a∈ A_1 [ · ρ_V'(a) q_* +δ_ρ(a) +ℓ_a","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e19ea3b7d2235251","tex":"q_*^2","aliases":"q_*^2","display":"q_*^2","kind":"use","excerpt":"m[Quantitative replacement invariance] Let J be a unitary graph correspondence with diagram defect δ_D. Define η_L=2 D δ_D+δ_D^2 and assume η_L<frac g2. Set q_* = fracη_Lg-η_L. Then there exists a unitary workspace correspondence β for which q_str≤ q_*. Moreover, δ_op( J) ≤ sup_a∈ A_1 [ · ρ_V'(a) q_* +δ_ρ(a) +ℓ_a · + ( A_ W'' + A_ W ) (1-sqrt1-q_*^","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"use","excerpt":"Write c=P_0c+Q_0c, where P_0 projects onto W_0 and Q_0=I-P_0. Since L_HP_0=0, E_H(c) = ⟨ Q_0c,L_HQ_0c⟩. The spectrum of L_H on W_0^⊥ lies in [γ_H, L_H ]. Hence γ_H Q_0c ^2 ≤ E_H(c) ≤ L_H Q_0c ^2. Finally, Q_0c =dist(c,W_0).","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"use","excerpt":"For a predeclared null Q, define Δ_tr(Q) = Risk_ℓ,Q(V) - Risk_ℓ,P(V). For a finite validated panel Q={Q_1,…,Q_m}⊆ N_pred(P), report both the complete vector Δ_tr( Q) = ( Δ_tr(Q_1),…, Δ_tr(Q_m) ) and the robust panel minimum Δ_tr^min( Q) = min_Q∈ QΔ_tr(Q). The panel must be fixed independently of the observed favorable gap. Otherwise null selection ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"Consider three charts arranged in a cycle, with one-dimensional overlap spaces and oriented offsets b_12, b_23, b_31. Local potentials would have to satisfy q_2-q_1=b_12, q_3-q_2=b_23, q_1-q_3=b_31. Adding the equations gives the necessary and sufficient condition b_12+b_23+b_31=0. With the standard edge norm, the harmonic obstruction has magnitude","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"Consider three charts arranged in a cycle, with one-dimensional overlap spaces and oriented offsets b_12, b_23, b_31. Local potentials would have to satisfy q_2-q_1=b_12, q_3-q_2=b_23, q_1-q_3=b_31. Adding the equations gives the necessary and sufficient condition b_12+b_23+b_31=0. With the standard edge norm, the harmonic obstruction has magnitude","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ae9808cb0f6e77d","tex":"q_3","aliases":"q_3","display":"q_3","kind":"use","excerpt":"Consider three charts arranged in a cycle, with one-dimensional overlap spaces and oriented offsets b_12, b_23, b_31. Local potentials would have to satisfy q_2-q_1=b_12, q_3-q_2=b_23, q_1-q_3=b_31. Adding the equations gives the necessary and sufficient condition b_12+b_23+b_31=0. With the standard edge norm, the harmonic obstruction has magnitude","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-83dc7e104ed7732b","tex":"Q_d","aliases":"Q_d","display":"Q_d","kind":"use","excerpt":"[Approximate operation-closure theorem] Let P_d project onto the first d eigenvectors of L_H, set λ_d≤ε, μ=λ_d+1>0, and let Q_d=I-P_d. Then every a∈ A satisfies Q_dρ_V(a)P_d ≤ frac ρ_E(a) sqrtε+β(a) sqrtμ.","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-234f1baf03365c39","tex":"Q_{\\mathrm{HH}}","aliases":"Q_HH mathrm","display":"Q_{\\mathrm{HH}}","kind":"use","excerpt":"tion is abstract: acceptance depends on the statement, the state of the proof obligations, and the current falsification record. No individual node has sufficient information: H(Ymid X_P) = H(Ymid X_C) = H(Ymid X_N) =1. Every two-node coalition has all three bits and therefore zero conditional entropy. Hence Δ_joint^log=1 bit, k_0=2. Let Q_HH prese","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-91ae7ec0b0c0eee0","tex":"Q_{i_2}","aliases":"Q_i_2","display":"Q_{i_2}","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-22be930dba6978e7","tex":"Q_{i_m}","aliases":"Q_i_m","display":"Q_{i_m}","kind":"use","excerpt":"· = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r · × (∏_ℓ=1^r-1 H_ℓ ), and the same right-hand side bounds P_i_mT_m⋯ T_1P_i_0-H_m⋯ H_1 .","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3406acd3f0f64ef4","tex":"Q_m","aliases":"Q_m","display":"Q_m","kind":"use","excerpt":"For a predeclared null Q, define Δ_tr(Q) = Risk_ℓ,Q(V) - Risk_ℓ,P(V). For a finite validated panel Q={Q_1,…,Q_m}⊆ N_pred(P), report both the complete vector Δ_tr( Q) = ( Δ_tr(Q_1),…, Δ_tr(Q_m) ) and the robust panel minimum Δ_tr^min( Q) = min_Q∈ QΔ_tr(Q). The panel must be fixed independently of the observed favorable gap. Otherwise null selection ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-684be6feb6813856","tex":"q_N","aliases":"q_N","display":"q_N","kind":"use","excerpt":"Order the local coordinates as x=(q_P,p_P,p_C,f_C,q_N,f_N)^ T. Compatibility requires p_P=p_C, f_C=f_N, q_N=q_P. The discrepancy operator is D_HH = 0 · 1 · -1 · 0 · 0 · 0","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0db94dc2cdc9bd9f","tex":"q_P","aliases":"q_P","display":"q_P","kind":"use","excerpt":"Order the local coordinates as x=(q_P,p_P,p_C,f_C,q_N,f_N)^ T. Compatibility requires p_P=p_C, f_C=f_N, q_N=q_P. The discrepancy operator is D_HH = 0 · 1 · -1 · 0 · 0 · 0","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-53c2d6d15f1b4102","tex":"Q_{s(e)}","aliases":"Q_s(e)","display":"Q_{s(e)}","kind":"use","excerpt":"V_e · K_e , where H_e=P_t(e)T_eP_s(e):C_s(e)→ C_t(e), U_e=P_t(e)T_eQ_s(e):R_s(e)→ C_t(e), V_e=Q_t(e)T_eP_s(e):C_s(e)→ R_t(e), K_e=Q_t(e)T_eQ_s(e):R_s(e)→ R_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2e7365ab6b9bcc51","tex":"q_{\\mathrm{str}}","aliases":"q_str mathrm","display":"q_{\\mathrm{str}}","kind":"use","excerpt":"[Approximate structural replacement] If η_L<frac g2, then q_str := P'-J_VPJ_V^* ≤ fracη_Lg-η_L ≤ frac2η_Lg. If q_str<1, then B=P'J_VP:ran P⟶ran P' is bijective and satisfies σ_min(B) ≥ sqrt1-q_str^2. Its polar factor β =B(B^*B)^-1/2 is a canonical unitary workspace correspondence. Moreover, β-J_VP ≤ q_str + 1-sqrt1-q_str^2.","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8d747df18109ebcd","tex":"q_{\\mathrm{str}}^2","aliases":"q_str^2 mathrm","display":"q_{\\mathrm{str}}^2","kind":"use","excerpt":"[Approximate structural replacement] If η_L<frac g2, then q_str := P'-J_VPJ_V^* ≤ fracη_Lg-η_L ≤ frac2η_Lg. If q_str<1, then B=P'J_VP:ran P⟶ran P' is bijective and satisfies σ_min(B) ≥ sqrt1-q_str^2. Its polar factor β =B(B^*B)^-1/2 is a canonical unitary workspace correspondence. Moreover, β-J_VP ≤ q_str + 1-sqrt1-q_str^2.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6fd3920b0502031c","tex":"Q_t","aliases":"Q_t","display":"Q_t","kind":"use","excerpt":"If the interventional action-value estimator satisfies sup_a|widehat Q_t(a)-Q_t(a)|≤ε_Q and the implemented policy has optimization error at most δ_opt relative to widehat Q_t, then Reg_t(widehatπ_t) ≤ 2ε_Q+δ_opt. For deterministic greedy action, the bound is 2ε_Q. A unique true optimum with value gap greater than 2ε_Q is recovered exactly. The gua","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9768261cdb0554c9","tex":"Q_{t(e)}","aliases":"Q_t(e)","display":"Q_{t(e)}","kind":"use","excerpt":"V_e · K_e , where H_e=P_t(e)T_eP_s(e):C_s(e)→ C_t(e), U_e=P_t(e)T_eQ_s(e):R_s(e)→ C_t(e), V_e=Q_t(e)T_eP_s(e):C_s(e)→ R_t(e), K_e=Q_t(e)T_eQ_s(e):R_s(e)→ R_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b0e423bb3476bcc4","tex":"Q_{t(\\gamma)}","aliases":"Q_t(γ) gamma","display":"Q_{t(\\gamma)}","kind":"use","excerpt":"For a path γ=e_m⋯ e_1, Q_t(γ)T_γ P_s(γ) ≤ ∑_r=1^m · ( ∏_ℓ=r+1^m T_e_ℓ ) V_e_r · × ( ∏_ℓ=1^r-1 H_e_ℓ ). The same bound controls the defect between full core-to-core path transport and the pure core product. An exact Duhamel identity yields both estimates.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bc235e052567c7a1","tex":"Q_t^{g'}","aliases":"Q_t^g'","display":"Q_t^{g'}","kind":"use","excerpt":"Two goal codes are operationally equivalent, g∼_goalg', when they induce the same action ordering in every declared state and intervention background: Q_t^g(amid w) ≥ Q_t^g(a'mid w) Longleftrightarrow Q_t^g'(amid w) ≥ Q_t^g'(a'mid w). The abstract goal is the equivalence class [g]_goal. This quotient prevents a mere invertible recoding of a goal ch","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-741f602298fe8acb","tex":"q_t^{\\mathrm{str}}","aliases":"q_t^str mathrm","display":"q_t^{\\mathrm{str}}","kind":"use","excerpt":"[Temporal workspace tracking] If η_t^L<fracg_t2, then the transported and newly extracted workspace projectors satisfy q_t^str := P_t+1-U_t^VP_t(U_t^V)^* ≤ fracη_t^Lg_t-η_t^L ≤ frac2η_t^Lg_t.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ab69aa180f14a53e","tex":"q_\\alpha","aliases":"q_α alpha","display":"q_\\alpha","kind":"use","excerpt":"A different problem arises when only relative transition data are supplied. Let b=(b_αβ)∈check C^1( U;mathscr W) be a prescribed family of overlap offsets. The triple-overlap consistency condition is δ^1b=0. One seeks local potentials q=(q_α)∈check C^0 satisfying δ^0q=b. Define check H^1( U;mathscr W) = frackerδ^1imδ^0. Then the affine gluing probl","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d14fc9f7fa113f19","tex":"q_\\iota","aliases":"q_ι iota","display":"q_\\iota","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3d23f47e9b01955c","tex":"q_{\\sigma}","aliases":"q_σ sigma","display":"q_{\\sigma}","kind":"use","excerpt":"For a family S of admissible scaffold variants, define the cross-scaffold identity defect δ_q^scaf = inf_{J_σσ'} E_σ,σ'∼π d_q ( J_σσ'q_σ,q_σ' ), where J_σσ' transports the learned object between admissible coordinate systems. A low value indicates that the same abstract organization is recovered under multiple non-solution-bearing scaffolds.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c96f1c69660ad05e","tex":"q_{\\sigma'}","aliases":"q_σ' sigma","display":"q_{\\sigma'}","kind":"use","excerpt":"For a family S of admissible scaffold variants, define the cross-scaffold identity defect δ_q^scaf = inf_{J_σσ'} E_σ,σ'∼π d_q ( J_σσ'q_σ,q_σ' ), where J_σσ' transports the learned object between admissible coordinate systems. A low value indicates that the same abstract organization is recovered under multiple non-solution-bearing scaffolds.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"[Approximate hierarchical collapse bound] Let F_r: H_r→ H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-02f6e210c4b44c83","tex":"\\mathsf R","aliases":"R mathsf","display":"\\mathsf R","kind":"use","excerpt":"For each case we report the vector Card( K) = ( S, O, R, P, F, A, D, X, C ), where: ll S · = structural core and spectral coherence,","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"Suppose node r is removed and the surviving coalition is S=V∖{r}. A replacement node r' is to receive the encoding E_r'':C_*→ Y_r''.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d9916429f311b5a2","tex":"r^*","aliases":"r^*","display":"r^*","kind":"use","excerpt":"gintheorem[Repair selection under value-model error] Fix a damage intervention d and a finite feasible repair set R. Let V(r)=V_t^rep(r;d) be the true expected post-repair integrity margin and let widehat V obey sup_r∈ R|widehat V(r)-V(r)| ≤ ε_R. Let widehat r∈argmax_rwidehat V(r) and r^*∈argmax_r V(r). Then V(r^*)-V(widehat r) ≤ 2ε_R. If V(r^*) ≥ ","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a3d472d51734fedb","tex":"\\mathbb R^2","aliases":"R^2 mathbb","display":"\\mathbb R^2","kind":"use","excerpt":"[Equivalent operations in different coordinates] Let C_1=C_2= R^2, choose S∈ GL_2( R), and set H_12=S, ρ_2(a)=Sρ_1(a)S^-1. Then H_12ρ_1(a) =Sρ_1(a) =ρ_2(a)S =ρ_2(a)H_12. The local matrices can look different while realizing the same abstract operation. This is the elementary model of representation change used throughout the theory.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-15bd74f3637fba8d","tex":"\\mathbb R^3","aliases":"R^3 mathbb","display":"\\mathbb R^3","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1),","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c6163472e649a229","tex":"\\mathbb R^4","aliases":"R^4 mathbb","display":"\\mathbb R^4","kind":"use","excerpt":"rcher who maintains goals and semantic direction; item M is a language-model process that constructs local formalizations; item X is an execution and verification process that compiles, tests, or falsifies candidate artifacts; and item A is a persistent artifact store containing exact source, results, and versioned handoff state. Let the abstract r","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-96e4307256041ef7","tex":"\\mathbb R^{d_G}","aliases":"R^d_G mathbb","display":"\\mathbb R^{d_G}","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c5c5b11ba64a1559","tex":"\\mathbb R^{d_u}","aliases":"R^d_u mathbb","display":"\\mathbb R^{d_u}","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-34b3a8ea2cfa8081","tex":"\\mathbb R^{d_v}","aliases":"R^d_v mathbb","display":"\\mathbb R^{d_v}","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4189f312500530e9","tex":"r_{\\mathfrak S}","aliases":"r_ S mathfrak","display":"r_{\\mathfrak S}","kind":"use","excerpt":"in J_*, define the specification envelope underline m_q^ S(W) = inf_ς∈ Sm_q(W;ς), overline m_q^ S(W) = sup_ς∈ Sm_q(W;ς). Its sensitivity width is Δ_q^spec(W) = overline m_q^ S(W) - underline m_q^ S(W). The robust certificate margin is r_ S(W) = inf_ς∈ S min_q∈ J_*m_q(W;ς). Then Cert_ S(W)=1 Longleftrightarrow r_ S(W)>0.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cad1b2fb14376e8b","tex":"R_{\\mathcal U}","aliases":"R_ U mathcal","display":"R_{\\mathcal U}","kind":"use","excerpt":"Let R_ U:mathscr W(G) ⟶ check C^0( U;mathscr W) be the restriction map R_ U(w) = (w|_G_α)_α∈ A.","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c9ec7eaff010215","tex":"\\mathfrak R_+","aliases":"R_+ mathfrak","display":"\\mathfrak R_+","kind":"use","excerpt":"mpatible with the experimental law passes every required gate under every admissible specification. The robust certificate is identified as passing. item If overline r_ID, S≤0, then no compatible realization passes the complete certificate throughout the family. The robust certificate is identified as failing. item If there exist compatible realiza","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-41bb10a071993fed","tex":"\\mathfrak R_-","aliases":"R_- mathfrak","display":"\\mathfrak R_-","kind":"use","excerpt":"mpatible with the experimental law passes every required gate under every admissible specification. The robust certificate is identified as passing. item If overline r_ID, S≤0, then no compatible realization passes the complete certificate throughout the family. The robust certificate is identified as failing. item If there exist compatible realiza","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bb842969a375eb0a","tex":"R_{-i}","aliases":"R_-i","display":"R_{-i}","kind":"use","excerpt":"Let the total realization decompose as R=(X_i,R_-i), where R_-i contains the remaining components and relations. A workspace functional Workspace=f(X_i,R_-i) supervenes on R by construction. If there exist R_-i≠ R_-i' with f(X_i,R_-i)≠ f(X_i,R_-i'), then the same local projection X_i occurs in distinct workspaces. Hence supervenience on the whole i","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8fbcc1ed27dfb475","tex":"R_{-i}'","aliases":"R_-i'","display":"R_{-i}'","kind":"use","excerpt":"Let the total realization decompose as R=(X_i,R_-i), where R_-i contains the remaining components and relations. A workspace functional Workspace=f(X_i,R_-i) supervenes on R by construction. If there exist R_-i≠ R_-i' with f(X_i,R_-i)≠ f(X_i,R_-i'), then the same local projection X_i occurs in distinct workspaces. Hence supervenience on the whole i","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-570c7df9a0f60b5f","tex":"\\mathfrak R_0","aliases":"R_0 mathfrak","display":"\\mathfrak R_0","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-23bf30da92b1c6aa","tex":"\\mathfrak R_1","aliases":"R_1 mathfrak","display":"\\mathfrak R_1","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d14416b8a3291934","tex":"r_{\\mathrm{dec}}","aliases":"r_dec mathrm","display":"r_{\\mathrm{dec}}","kind":"use","excerpt":"When b_q≡0, a convenient certificate radius is r_dec(W) = min_q:L_q>0 fracm_q(W; D_0)L_q. Every admissible family satisfying sup_ D∈ Dδ_ D < r_dec(W) preserves the sign of all required gate margins. In applications, the transport terms can be bounded using earlier results: small textwidth@>bfseriesraggedrightarraybackslashp0.22textwidth>raggedright","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7b007cc7cc12365e","tex":"R_{i_m}","aliases":"R_i_m","display":"R_{i_m}","kind":"use","excerpt":"For every step, T_rι_i_r-1^C = ι_i_r^CH_r+ ι_i_r^RV_r. Telescope the sequence T_m⋯ T_r+1ι_i_r^C H_r⋯ H_1, r=0,1,…,m. The difference between consecutive terms is exactly the r-th summand in the displayed formula. Projection onto R_i_m or C_i_m, followed by submultiplicativity of the operator norm, gives the two bounds.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-690b5dbf5fbbd3af","tex":"r_J^K","aliases":"r_J^K","display":"r_J^K","kind":"use","excerpt":"If K⊆ L⊆ G are subgraphs, coordinate restriction gives ρ_K^L: C_V(L)⟶ C_V(K). Because every edge equation on K is also an edge equation on L, restriction maps exact sections to exact sections: r_K^L := ρ_K^L|_mathscr W(L): mathscr W(L)⟶mathscr W(K). They obey r_L^L=I, r_J^K r_K^L=r_J^L whenever J⊆ K⊆ L. Thus K↦mathscr W(K) is a contravariant system","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4d3aa1102072a191","tex":"r_J^L","aliases":"r_J^L","display":"r_J^L","kind":"use","excerpt":"If K⊆ L⊆ G are subgraphs, coordinate restriction gives ρ_K^L: C_V(L)⟶ C_V(K). Because every edge equation on K is also an edge equation on L, restriction maps exact sections to exact sections: r_K^L := ρ_K^L|_mathscr W(L): mathscr W(L)⟶mathscr W(K). They obey r_L^L=I, r_J^K r_K^L=r_J^L whenever J⊆ K⊆ L. Thus K↦mathscr W(K) is a contravariant system","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ccfcdead7bc0fcf1","tex":"r_K^L","aliases":"r_K^L","display":"r_K^L","kind":"use","excerpt":"If K⊆ L⊆ G are subgraphs, coordinate restriction gives ρ_K^L: C_V(L)⟶ C_V(K). Because every edge equation on K is also an edge equation on L, restriction maps exact sections to exact sections: r_K^L := ρ_K^L|_mathscr W(L): mathscr W(L)⟶mathscr W(K). They obey r_L^L=I, r_J^K r_K^L=r_J^L whenever J⊆ K⊆ L. Thus K↦mathscr W(K) is a contravariant system","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-070fdc095c2403f4","tex":"r_L^L","aliases":"r_L^L","display":"r_L^L","kind":"use","excerpt":"If K⊆ L⊆ G are subgraphs, coordinate restriction gives ρ_K^L: C_V(L)⟶ C_V(K). Because every edge equation on K is also an edge equation on L, restriction maps exact sections to exact sections: r_K^L := ρ_K^L|_mathscr W(L): mathscr W(L)⟶mathscr W(K). They obey r_L^L=I, r_J^K r_K^L=r_J^L whenever J⊆ K⊆ L. Thus K↦mathscr W(K) is a contravariant system","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9bf7aeaffeeda10a","tex":"\\mathcal R_q^{\\Sigma}","aliases":"R_q^Σ mathcal Sigma","display":"\\mathcal R_q^{\\Sigma}","kind":"use","excerpt":"For any component q of widehatθ, define its scaffold-only Bayes risk under loss ℓ_q by R_q^Σ = inf_f E [ ℓ_q(q,f(Σ,Ω)) ]. Define the full formation risk by R_q^Σ, H = inf_g E [ ℓ_q(q,g(Σ, D_form,Ω)) ]. The history advantage is Δ_q^hist = R_q^Σ - R_q^Σ, H. A positive history advantage means that formation history contains information about q that is","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8af374f63d14731d","tex":"\\mathcal R_q^{\\Sigma,\\mathcal H}","aliases":"R_q^Σ, H mathcal Sigma mathcal","display":"\\mathcal R_q^{\\Sigma,\\mathcal H}","kind":"use","excerpt":"For any component q of widehatθ, define its scaffold-only Bayes risk under loss ℓ_q by R_q^Σ = inf_f E [ ℓ_q(q,f(Σ,Ω)) ]. Define the full formation risk by R_q^Σ, H = inf_g E [ ℓ_q(q,g(Σ, D_form,Ω)) ]. The history advantage is Δ_q^hist = R_q^Σ - R_q^Σ, H. A positive history advantage means that formation history contains information about q that is","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-937de87a1560a7e8","tex":"\\mathcal R_r","aliases":"R_r mathcal","display":"\\mathcal R_r","kind":"use","excerpt":"[Approximate hierarchical collapse bound] Let F_r: H_r→ H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c1e0d6344f9474d8","tex":"R_S","aliases":"R_S","display":"R_S","kind":"use","excerpt":"[Coalition reconstruction criterion] For a coalition S, the following are equivalent: item A_S is injective; item F_S is positive definite; item α_S>0; item there exists a linear decoder R_S with R_SA_S=I_C_*. When these conditions hold, the canonical decoder is R_S=A_S^†=F_S^-1A_S^*.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f7b985b4cff51815","tex":"R_s","aliases":"R_s","display":"R_s","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-06ac1a596ae5a479","tex":"R_{s(e)}","aliases":"R_s(e)","display":"R_{s(e)}","kind":"use","excerpt":"V_e · K_e , where H_e=P_t(e)T_eP_s(e):C_s(e)→ C_t(e), U_e=P_t(e)T_eQ_s(e):R_s(e)→ C_t(e), V_e=Q_t(e)T_eP_s(e):C_s(e)→ R_t(e), K_e=Q_t(e)T_eQ_s(e):R_s(e)→ R_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c62b3900bbb09bf","tex":"r_{\\mathrm{spec}}","aliases":"r_spec mathrm","display":"r_{\\mathrm{spec}}","kind":"use","excerpt":"For a reference specification ς_0, define the distance to certificate failure by r_spec(W;ς_0) = inf{ d_ S(ς,ς_0): min_q m_q(W;ς)≤0 }. If no failing specification exists in the declared universe, the radius is infinite relative to that universe. A radius near zero means that a small scientifically defensible change can reverse the conclusion.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9f055453d29d7bb6","tex":"\\mathfrak R_T","aliases":"R_T mathfrak","display":"\\mathfrak R_T","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-167871c0232bd862","tex":"\\mathfrak R_t","aliases":"R_t mathfrak","display":"\\mathfrak R_t","kind":"use","excerpt":"Persistence does not require token identity. Let R_t and R_t+1 be temporally successive realizations. Continuity is possible when there exists an admissible transport J_t: R_trightsquigarrow R_t+1 that preserves the operative core and its causal role within tolerance: d_id(Workspace_t,Workspace_t+1)≤ε. Thus a workspace may persist by reconstruction","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7d0b0aaec388130d","tex":"r_t","aliases":"r_t","display":"r_t","kind":"use","excerpt":"After a possible damage event, let the pre-repair health be h_t^-∈{0,1}. The system chooses a joint action u_t=(a_t,r_t)∈{0,1}^2, where a_t is a task action and r_t is a repair action. Define the optimal task action a_t^*=g_t⊕ b_t⊕ h_t^- and optimal repair action r_t^*=1-h_t^-. No individual node determines the full pair (a_t^*,r_t^*): item the pla","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5dd6a5ea8a98ce15","tex":"R_{t(e)}","aliases":"R_t(e)","display":"R_{t(e)}","kind":"use","excerpt":"V_e · K_e , where H_e=P_t(e)T_eP_s(e):C_s(e)→ C_t(e), U_e=P_t(e)T_eQ_s(e):R_s(e)→ C_t(e), V_e=Q_t(e)T_eP_s(e):C_s(e)→ R_t(e), K_e=Q_t(e)T_eQ_s(e):R_s(e)→ R_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-41c1cecea87087ab","tex":"\\mathfrak R_{t+1}","aliases":"R_t+1 mathfrak","display":"\\mathfrak R_{t+1}","kind":"use","excerpt":"Persistence does not require token identity. Let R_t and R_t+1 be temporally successive realizations. Continuity is possible when there exists an admissible transport J_t: R_trightsquigarrow R_t+1 that preserves the operative core and its causal role within tolerance: d_id(Workspace_t,Workspace_t+1)≤ε. Thus a workspace may persist by reconstruction","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-90fcee0cd9b75a90","tex":"R_{t+\\tau-1}","aliases":"R_t+τ-1 tau","display":"R_{t+\\tau-1}","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1a1e5e6ff838f780","tex":"R_{t+\\tau:t}","aliases":"R_t+τ:t tau","display":"R_{t+\\tau:t}","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-62624aab44f25be6","tex":"r_t^*","aliases":"r_t^*","display":"r_t^*","kind":"use","excerpt":"After a possible damage event, let the pre-repair health be h_t^-∈{0,1}. The system chooses a joint action u_t=(a_t,r_t)∈{0,1}^2, where a_t is a task action and r_t is a repair action. Define the optimal task action a_t^*=g_t⊕ b_t⊕ h_t^- and optimal repair action r_t^*=1-h_t^-. No individual node determines the full pair (a_t^*,r_t^*): item the pla","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6d81f609d05b362c","tex":"\\mathfrak R_{\\mathrm{total}}","aliases":"R_total mathfrak mathrm","display":"\\mathfrak R_{\\mathrm{total}}","kind":"use","excerpt":"The claim that the workspace is not identical with either participant does not imply that it occupies a separate nonphysical location. Formally, Workspace is an invariant of R, not an additional term appended to the physical realization: R_total eq( R,ghost workspace). The phrase ``between the participants'' is therefore topological or relational: ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b8e033117b76f34e","tex":"r_{\\alpha_0\\cdots\\alpha_{p+1}}^{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}","aliases":"r_α_0⋯α_p+1^α_0⋯widehatα_j⋯α_p+1 alpha cdots alpha alpha cdots widehat alpha cdots alpha","display":"r_{\\alpha_0\\cdots\\alpha_{p+1}}^{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}","kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f809e3bdd501a69e","tex":"r_{\\alpha\\beta}^{\\alpha}","aliases":"r_αβ^α alpha beta alpha","display":"r_{\\alpha\\beta}^{\\alpha}","kind":"use","excerpt":"For pairwise overlaps write mathscr W_α=mathscr W(G_α), mathscr W_αβ=mathscr W(G_αβ), and r_αβ^α: mathscr W_α→mathscr W_αβ, r_αβ^β: mathscr W_β→mathscr W_αβ. The interface is therefore a cospan mathscr W_α xrightarrow r_αβ^α mathscr W_αβ xleftarrow r_αβ^β mathscr W_β, not necessarily a direct transfer map from one complete local workspace to the ot","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9007637d70d89a66","tex":"r_{\\alpha\\beta}^{\\beta}","aliases":"r_αβ^β alpha beta beta","display":"r_{\\alpha\\beta}^{\\beta}","kind":"use","excerpt":"For pairwise overlaps write mathscr W_α=mathscr W(G_α), mathscr W_αβ=mathscr W(G_αβ), and r_αβ^α: mathscr W_α→mathscr W_αβ, r_αβ^β: mathscr W_β→mathscr W_αβ. The interface is therefore a cospan mathscr W_α xrightarrow r_αβ^α mathscr W_αβ xleftarrow r_αβ^β mathscr W_β, not necessarily a direct transfer map from one complete local workspace to the ot","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-04574488c1d5a80f","tex":"R_\\theta","aliases":"R_θ theta","display":"R_\\theta","kind":"use","excerpt":"[Local compatibility without a global nonzero core] Let the only nontrivial cycle holonomy on R^2 be a rotation R_θ= cosθ · -sinθ sinθ · cosθ with θ∉2π Z. Then ker(R_θ-I)={0}. Every edge map is individually invertible and well-defined, yet the complete cycle admits no nonzero exact shared core. Pairwise translatability is therefore insufficient for","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-17909b1b04287dfe","tex":"\\mathfrak R_\\lambda","aliases":"R_λ mathfrak lambda","display":"\\mathfrak R_\\lambda","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0},","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-268aa2b4269fd3ac","tex":"R_\\perp","aliases":"R_⊥ perp","display":"R_\\perp","kind":"use","excerpt":"Define the detection effect η_D→ M, mediation effect η_M→ R, restoration effect η_R→ v, and repair specificity Δ_repair^spec = η_R→ v - η_R_⊥→ v, where R_⊥ is a matched action outside the proposed repair channel.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-952542ed93bf41db","tex":"\\operatorname{ran}","aliases":"ran operatorname","display":"\\operatorname{ran}","kind":"use","excerpt":"For every g∈Hol_o, left multiplication permutes the group elements, so gP_*=P_*. Thus ranP_*⊆ C_*. If x∈ C_*, then P_*x=x, so C_*⊆ranP_*. Unitarity gives P_*^*=P_*, and the range characterization implies P_*^2=P_*.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"H_γ_r-I . Then C_*=ker M, d_Workspace=d-rank( M).","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-38ac57489dd0edfc","tex":"\\Rec","aliases":"Rec","display":null,"kind":"use","excerpt":"Report d_id(W_t,W_t+τ), Rec_k, e_t, α^(k), μ_t^keep. The report must separate: item subspace tracking from state tracking; item coherence from non-collapse; item reconstruction from mere prediction; item repair from successor-workspace creation; and item structural replacement from causal and reconstructive replacement.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4321aa573d789e4c","tex":"\\mathrm{rec}","aliases":"rec mathrm","display":"\\mathrm{rec}","kind":"use","excerpt":"Temporal correlation is weaker than causal recurrence. Define the total workspace recurrence effect at lag τ by η_rec(τ) = E_(w,w',b)∼ν D ( P^ι_w(W_t+τ∈·mid B_t=b), P^ι_w'(W_t+τ∈·mid B_t=b) ). A nonzero value shows that manipulating the present workspace changes a later workspace state. It may, however, be explained by direct copying or a retained ","count":19},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8568ac0960631ce0","tex":"\\Rec_k","aliases":"Rec_k Rec","display":null,"kind":"use","excerpt":"Report d_id(W_t,W_t+τ), Rec_k, e_t, α^(k), μ_t^keep. The report must separate: item subspace tracking from state tracking; item coherence from non-collapse; item reconstruction from mere prediction; item repair from successor-workspace creation; and item structural replacement from causal and reconstructive replacement.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ca4dab8ec4325ec7","tex":"\\mathsf{reference}","aliases":"reference mathsf","display":"\\mathsf{reference}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f5a789f619d24251","tex":"\\operatorname{Reg}","aliases":"Reg operatorname","display":"\\operatorname{Reg}","kind":"use","excerpt":"If the interventional action-value estimator satisfies sup_a|widehat Q_t(a)-Q_t(a)|≤ε_Q and the implemented policy has optimization error at most δ_opt relative to widehat Q_t, then Reg_t(widehatπ_t) ≤ 2ε_Q+δ_opt. For deterministic greedy action, the bound is 2ε_Q. A unique true optimum with value gap greater than 2ε_Q is recovered exactly. The gua","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a0ee4c2b7c964414","tex":"\\operatorname{Reg}_t","aliases":"Reg_t operatorname","display":"\\operatorname{Reg}_t","kind":"use","excerpt":"If the interventional action-value estimator satisfies sup_a|widehat Q_t(a)-Q_t(a)|≤ε_Q and the implemented policy has optimization error at most δ_opt relative to widehat Q_t, then Reg_t(widehatπ_t) ≤ 2ε_Q+δ_opt. For deterministic greedy action, the bound is 2ε_Q. A unique true optimum with value gap greater than 2ε_Q is recovered exactly. The gua","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-29098d0a684c06f9","tex":"\\mathrm{rel}","aliases":"rel mathrm","display":"\\mathrm{rel}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-23d26e1dca05508e","tex":"\\mathrm{rep-id}","aliases":"rep-id mathrm","display":"\\mathrm{rep-id}","kind":"use","excerpt":"Restoration of performance is not automatically restoration of identity. Let W_t^- be the last certified pre-damage workspace and let W_t+τ^r be the post-repair workspace. Define the repair identity defect δ_rep-id(r) = d_id ( W_t^-,W_t+τ^r ). A repair action is identity preserving at tolerance ε_id when Δ_t^rep(r;d)>0, δ_rep-id(r) ≤ ε_id. When int","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-052e9616613c2ffc","tex":"\\operatorname{RepairAction}","aliases":"RepairAction operatorname","display":"\\operatorname{RepairAction}","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0, · Form_Θ_form(W","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e44a95f9af3322b0","tex":"\\operatorname{RepairAction}_{t+1}","aliases":"RepairAction_t+1 operatorname","display":"\\operatorname{RepairAction}_{t+1}","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0, · Form_Θ_form(W","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4edab3fe6562ebc5","tex":"\\mathrm{replacement}","aliases":"replacement mathrm","display":"\\mathrm{replacement}","kind":"use","excerpt":"operation leakage. Coalition frame bounds also persist under sufficiently small reconstructive transport error. With declared causal-kernel and reconstructive-handoff defects, d_id ≤ a q_str +b δ_op +c δ_causal +d δ_rec. Under bounded norms and a uniform spectral gap, this gives the schematic estimate d_id lesssim Cfracδ_replacementg. The causal an","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1ecf96b50772dd74","tex":"\\operatorname{Res}","aliases":"Res operatorname","display":"\\operatorname{Res}","kind":"use","excerpt":"mined by mathematical and operational equivalence, not by terminology. Let C be a claim in the paper and S a prior source. Write Spreccurlyeq C when the main content of C follows from S after admissible relabeling, restriction, or direct specialization. A substantive novelty residual exists only when the claim contains content not absorbed by the c","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7ed5bb687c0f2951","tex":"\\mathsf{ResultRecord}","aliases":"ResultRecord mathsf","display":"\\mathsf{ResultRecord}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6dbd571cb07ca086","tex":"\\Ret","aliases":"Ret","display":null,"kind":"use","excerpt":"For S∈ S, define Ret_τ(S) = I ( X_S(t+τ);X_S(t) mid X_bar S(t),E_t ), Dep_τ(S) = I ( X_S(t+τ);X_bar S(t) mid X_S(t),E_t ), Out_τ(S) = I ( X_bar S(t+τ);X_S(t) mid X_bar S(t),E_t ). These quantities are observational. They can be nonzero because of common causes, selection, or feedback paths and therefore cannot alone establish a causal boundary.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5b8e008820e98b90","tex":"\\Ret_\\tau","aliases":"Ret_τ Ret tau","display":null,"kind":"use","excerpt":"For S∈ S, define Ret_τ(S) = I ( X_S(t+τ);X_S(t) mid X_bar S(t),E_t ), Dep_τ(S) = I ( X_S(t+τ);X_bar S(t) mid X_S(t),E_t ), Out_τ(S) = I ( X_bar S(t+τ);X_S(t) mid X_bar S(t),E_t ). These quantities are observational. They can be nonzero because of common causes, selection, or feedback paths and therefore cannot alone establish a causal boundary.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8b084868aad0a6d3","tex":"\\rightarrow","aliases":"rightarrow","display":"\\rightarrow","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-74bcfeab24a52e96","tex":"\\rightrightarrows","aliases":"rightrightarrows","display":null,"kind":"use","excerpt":"[Exact workspace gluing] For every finite edge-complete workspace cover U, the sequence 0 ⟶ mathscr W(G) xrightarrow R_ U check C^0( U;mathscr W) xrightarrow δ^0 check C^1( U;mathscr W) is exact. Equivalently, mathscr W(G) ≅ kerδ^0 ≅ Eq ( ∏_αmathscr W_α rightrightarrows ∏_α<βmathscr W_αβ ). Every pairwise compatible family of exact local workspace ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-968977e4d82819b0","tex":"\\rightsquigarrow","aliases":"rightsquigarrow","display":null,"kind":"use","excerpt":"Persistence does not require token identity. Let R_t and R_t+1 be temporally successive realizations. Continuity is possible when there exists an admissible transport J_t: R_trightsquigarrow R_t+1 that preserves the operative core and its causal role within tolerance: d_id(Workspace_t,Workspace_t+1)≤ε. Thus a workspace may persist by reconstruction","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-53c695c7eecf0e22","tex":"\\Risk_{\\mathcal D}","aliases":"Risk_ D Risk mathcal","display":null,"kind":"use","excerpt":"The structural core can be preserved by faithful coarse-graining, while coalition counts generally are not. Let mathscr F_S^ D = bigvee_i∈ Smathscr F_i^ D be the information available to coalition S under decomposition D. For a declared loss ℓ, write Risk_ D(S) = inf_f E [ ℓ(Y^+,f(mathscr F_S^ D)) ].","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bf51ae24e2ea8b9a","tex":"\\Risk_{\\mathcal D_c}","aliases":"Risk_ D_c Risk mathcal","display":null,"kind":"use","excerpt":"[Bayes-risk transport under faithful aggregation] Suppose D_fpreceq D_c through q:V_f→ V_c, and suppose the aggregation is information faithful. Then for every coarse coalition S⊆ V_c, Risk_ D_c(S) = Risk_ D_f(q^-1(S)). Consequently, k_ε( D_c) ≤ k_ε( D_f), where coalition size is counted in displayed nodes.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5acbc57c6359547a","tex":"\\Risk_{\\mathcal D_f}","aliases":"Risk_ D_f Risk mathcal","display":null,"kind":"use","excerpt":"[Bayes-risk transport under faithful aggregation] Suppose D_fpreceq D_c through q:V_f→ V_c, and suppose the aggregation is information faithful. Then for every coarse coalition S⊆ V_c, Risk_ D_c(S) = Risk_ D_f(q^-1(S)). Consequently, k_ε( D_c) ≤ k_ε( D_f), where coalition size is counted in displayed nodes.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bfeb042340a4a8b1","tex":"\\Risk_{\\mathrm{ctl}}","aliases":"Risk_ctl Risk mathrm","display":null,"kind":"use","excerpt":"For the complete control state, define Risk_ctl(S) = inf_f E [ ℓ_ctl ( C_t^ctl, f( H_t^S) ) ], and let k_ctl,ε be its minimal sufficient coalition size. A strong workspace-level agency claim requires k_ctl,ε≥2. This is a location criterion. It does not yet establish goal persistence, counterfactual evaluation, or self-maintenance.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7eed1d139b32e2c7","tex":"\\Risk_G","aliases":"Risk_G Risk","display":null,"kind":"use","excerpt":"Let H_t^S be the history available to coalition S⊆ V. For a loss ℓ_G, define goal-reconstruction risk Risk_G(S) = inf_f E [ ℓ_G ( G_t, f( H_t^S) ) ]. For tolerance ε, define k_G,ε = min { |S|: Risk_G(S) ≤ Risk_G(V)+ε }. A goal is locally stored when k_G,ε=1. It is distributed relative to the declared decomposition when k_G,ε≥2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-30077d43e0009725","tex":"\\Risk_{\\log}","aliases":"Risk_log Risk log","display":null,"kind":"use","excerpt":"For a finite target alphabet and logarithmic scoring loss ℓ_log(y,q)=-log q(y), q∈ P( Y), the identity is exact: Risk_log(S_t)=H(Y_t^+mid S_t), Risk_log(S_t)-Risk_log( H_t) = I(Y_t^+; H_tmid S_t). Natural logarithms measure the gap in nats; base-two logarithms measure it in bits.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6ea3522148fa6d1f","tex":"\\Risk_{\\log,P}","aliases":"Risk_log,P Risk log","display":null,"kind":"use","excerpt":"By Theorem~[source reference], for each node i, Risk_log,P({i})- Risk_log,P(V) = I_P(Y_t^+; H_tmid H_t^i) ≥ δ_node>0. Hence no single-node history, and therefore no statistic of one, is sufficient; this gives k_0≥2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9d3c54fea307cdc2","tex":"\\Risk_{\\log,P_{\\mathrm{HH}}}","aliases":"Risk_log,P_HH Risk log mathrm","display":null,"kind":"use","excerpt":"ition has all three bits and therefore zero conditional entropy. Hence Δ_joint^log=1 bit, k_0=2. Let Q_HH preserve the independent uniform local bit pairs but replace Y by an independent uniform bit. Because parity is independent of every two-bit local view, Q_HH preserves each (Y,X_i)-marginal. Nevertheless, Risk_log,Q_HH(V) - Risk_log,P_HH(V) =1 ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-98e6d535dd2fb07b","tex":"\\Risk_{\\log,Q}","aliases":"Risk_log,Q Risk log","display":null,"kind":"use","excerpt":"For every Q∈ Q, Proposition~[source reference] gives Risk_log,Q({i}) = Risk_log,P({i}) ∀ i. The assumed null gap gives Risk_log,Q(V) > Risk_log,P(V). Thus a description determined only by the collection of single-node predictive marginals cannot reproduce the intact joint capacity: it would assign the same local data to P and Q while their full-sys","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bbcd2a00938ad590","tex":"\\Risk_{\\log,Q_{\\mathrm{HH}}}","aliases":"Risk_log,Q_HH Risk log mathrm","display":null,"kind":"use","excerpt":"ition has all three bits and therefore zero conditional entropy. Hence Δ_joint^log=1 bit, k_0=2. Let Q_HH preserve the independent uniform local bit pairs but replace Y by an independent uniform bit. Because parity is independent of every two-bit local view, Q_HH preserves each (Y,X_i)-marginal. Nevertheless, Risk_log,Q_HH(V) - Risk_log,P_HH(V) =1 ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7fb2697b7e0a23ec","tex":"\\Risk_\\ell","aliases":"Risk_ℓ Risk ell","display":null,"kind":"use","excerpt":"Sufficiency has a direct decision-theoretic interpretation. Use the convention TV(P,Q)=sup_A|P(A)-Q(A)|. Let A be a finite action set and ℓ: Y× A→[0,1] a bounded loss. Define Risk_ℓ(S_t) = inf_d E [ ℓ(Y_t^+,d(S_t)) ].","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f70d1b5f952c86da","tex":"\\Risk_{\\ell,\\mathbb P}","aliases":"Risk_ℓ, P Risk ell mathbb","display":null,"kind":"use","excerpt":"Let jmath_E randomize selected communication edges while preserving their marginal message laws, rates, and timing budget. The causal transfer gap is Δ_tr^do(jmath_E) = Risk_ℓ, P^jmath_E(V) - Risk_ℓ, P(V). Unlike an observational null law, this quantity is anchored to an implemented channel intervention. A positive effect is evidence that the intac","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dacac93c0f1ac822","tex":"\\Risk_{\\ell,\\mathbb P^{\\jmath_E}}","aliases":"Risk_ℓ, P^jmath_E Risk ell mathbb jmath","display":null,"kind":"use","excerpt":"Let jmath_E randomize selected communication edges while preserving their marginal message laws, rates, and timing budget. The causal transfer gap is Δ_tr^do(jmath_E) = Risk_ℓ, P^jmath_E(V) - Risk_ℓ, P(V). Unlike an observational null law, this quantity is anchored to an implemented channel intervention. A positive effect is evidence that the intac","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-84fd068a2eb6aebc","tex":"\\Risk_\\ell^{\\iota}","aliases":"Risk_ℓ^ι Risk ell iota","display":null,"kind":"use","excerpt":"hbf H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ∑_ιq_ι [ Risk_ℓ^ι(W_t) - Risk_ℓ^ι( H_t) ] ≤ sqrtfrac δ_align^ I(W_t)2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a58e4c2e82e7cd1b","tex":"\\mathrm{root}","aliases":"root mathrm","display":"\\mathrm{root}","kind":"use","excerpt":"For every finite rooted hierarchy whose children edge-cover their parent, W_a≅mathscr W(G_a) for every hierarchy node a, and in particular W_root≅mathscr W(G). If level r contributes defect δ_r and downstream collapse maps have Lipschitz constants L_s, then δ_flat^(R) ≤ ∑_r=0^R-1 ( ∏_s=r+1^R-1L_s )δ_r.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-627aafe3bece8a45","tex":"\\operatorname{Routing}","aliases":"Routing operatorname","display":"\\operatorname{Routing}","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0, · Form_Θ_form(W","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cd2cd45ba10e64d5","tex":"\\operatorname{Routing}_{t+1}","aliases":"Routing_t+1 operatorname","display":"\\operatorname{Routing}_{t+1}","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0, · Form_Θ_form(W","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2f2fae6efa7605a9","tex":"\\rule","aliases":"rule","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","excerpt":"[Equivalent operations in different coordinates] Let C_1=C_2= R^2, choose S∈ GL_2( R), and set H_12=S, ρ_2(a)=Sρ_1(a)S^-1. Then H_12ρ_1(a) =Sρ_1(a) =ρ_2(a)S =ρ_2(a)H_12. The local matrices can look different while realizing the same abstract operation. This is the elementary model of representation change used throughout the theory.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8715f5958171f7d6","tex":"S_{\\mathcal U}","aliases":"S_ U mathcal","display":"S_{\\mathcal U}","kind":"use","excerpt":"l candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U = wi","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-41020c426fa09aa4","tex":"\\mathfrak S_{\\mathrm{pass}}","aliases":"S_pass mathfrak mathrm","display":"\\mathfrak S_{\\mathrm{pass}}","kind":"use","excerpt":"· -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass contains only candidates that satisfy the required workspace and anti-centralization gates.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3d41b533490731ab","tex":"S_{t+1}","aliases":"S_t+1","display":"S_{t+1}","kind":"use","excerpt":"C_t^ctl = ( W_t,G_t,M_t,B_t,E_t ), where G_t is the represented goal, M_t is a workspace-integrity monitor, B_t is the current boundary or routing state, and E_t is the declared external context. An action is generated by A_t ∼ π_t ( ·mid W_t,G_t,M_t,B_t,E_t ). The future controlled state obeys a time-unrolled structural model S_t+1 = F_t(S_t,A_t,E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a0da78a7c7407457","tex":"S_{t+1}^*","aliases":"S_t+1^*","display":"S_{t+1}^*","kind":"use","excerpt":"[Boundary persistence under score drift] Assume S_t^* is the unique maximizer of B_t with margin Δ_t^B, and suppose sup_S∈ S_pass | B_t+1(S)- B_t(S)| ≤δ_t. If 2δ_t<Δ_t^B, then S_t+1^*=S_t^*. More generally, B_t+1(S_t+1^*) - B_t+1(S_t^*) ≤2δ_t.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-eca46f604effec09","tex":"S_t^*","aliases":"S_t^*","display":"S_t^*","kind":"use","excerpt":"[Held-out boundary recovery] Let S_pass be finite and let S_t^* = argmax_S∈ S_pass B_t(S) be unique. Define the population margin Δ_t^B = B_t(S_t^*) - max_S≠ S_t^* B_t(S)>0. If sup_S∈ S_pass | widehat B_t(S)- B_t(S) | ≤ε_t, then every empirical maximizer widehat S_t satisfies B_t(S_t^*)- B_t(widehat S_t) ≤2ε_t. If 2ε_t<Δ_t^B, then widehat S_t=S_t^*","count":15},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8863c2d3c292b39d","tex":"s_{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}","aliases":"s_α_0⋯widehatα_j⋯α_p+1 alpha cdots widehat alpha cdots alpha","display":"s_{\\alpha_0\\cdots\\widehat{\\alpha_j}\\cdots\\alpha_{p+1}}","kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b724e4cb5137a2d4","tex":"s_\\varepsilon","aliases":"s_ε varepsilon","display":"s_\\varepsilon","kind":"use","excerpt":"A more stable quantity measures primitive carrier cost. Let w_ω>0 be a declared cost for each ω∈Ω, and assign a block the cost c(B)=∑_ω∈ Bw_ω. Define the sufficient support cost s_ε( D) = min_S⊆ V_ D { ∑_B∈ Sc(B): Risk_ D(S) ≤ Risk_ D(V_ D)+ε }. If D_fpreceq D_c is information faithful, then s_ε( D_f) ≤ s_ε( D_c). The fine decomposition permits eve","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cf53d3b364826419","tex":"\\mathfrak S_\\eta","aliases":"S_η mathfrak eta","display":"\\mathfrak S_\\eta","kind":"use","excerpt":"Let S_η be an η-net of ( S,d_ S) with covering number N(η, S,d_ S). If both the population and empirical margin functions are L_q-Lipschitz, then sup_ς∈ S |widehat m_q(ς)-m_q(ς)| ≤ max_ς_j∈ S_η |widehat m_q(ς_j)-m_q(ς_j)| +2L_qη. For bounded empirical means, a valid radius is therefore ε_q,n,η(δ) = sqrt frac log(2| J_*| N(η, S,d_ S)/δ) 2n +2L_qη. T","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b3179ba98980cad3","tex":"\\mathrm{sample}","aliases":"sample mathrm","display":"\\mathrm{sample}","kind":"use","excerpt":"In practice, the reference margin is reduced by three distinct uncertainty budgets: B_q^total = B_q^model +B_q^spec +B_q^sample. The first covers variation across observationally compatible realizations. The second covers variation across admissible specifications. The third covers finite-sample estimation. The sufficient condition widehat m_q(W;ς_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1e4f5694556ea955","tex":"\\mathrm{scaf}","aliases":"scaf mathrm","display":"\\mathrm{scaf}","kind":"use","excerpt":"For a family S of admissible scaffold variants, define the cross-scaffold identity defect δ_q^scaf = inf_{J_σσ'} E_σ,σ'∼π d_q ( J_σσ'q_σ,q_σ' ), where J_σσ' transports the learned object between admissible coordinate systems. A low value indicates that the same abstract organization is recovered under multiple non-solution-bearing scaffolds.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a10d8d642b450801","tex":"\\operatorname{Scaffolded}","aliases":"Scaffolded operatorname","display":"\\operatorname{Scaffolded}","kind":"use","excerpt":"subtopicOrthogonal formation-provenance axis Scaffolded ⟶ AdaptivelyParameterised ⟶ StructurallyAdaptive ⟶ EndogenouslyConstituted. This axis is independent of Levels~0--6. A scaffolded system can instantiate a high-level relational workspace, while an endogenously learned organization can fail every operational workspace gate.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d39429b9e78430d","tex":"\\operatorname{sech}","aliases":"sech operatorname","display":"\\operatorname{sech}","kind":"use","excerpt":"The derivative is F_λ'(m) = λsech^2(λ m). At zero this equals λ, proving stability for λ<1 and instability for λ>1. At a nonzero fixed point, F_λ'(m_λ) = λ(1-m_λ^2) = frac(1-m_λ^2)artanh(m_λ)m_λ <1, because artanh(m) < fracm1-m^2 (0<m<1). The derivative is positive, so the nonzero fixed points are locally stable. Finally, expanding tanh(λ m) = λ m-","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c2cd755c944ccda0","tex":"\\operatorname{sech}^2","aliases":"sech^2 operatorname","display":"\\operatorname{sech}^2","kind":"use","excerpt":"The derivative is F_λ'(m) = λsech^2(λ m). At zero this equals λ, proving stability for λ<1 and instability for λ>1. At a nonzero fixed point, F_λ'(m_λ) = λ(1-m_λ^2) = frac(1-m_λ^2)artanh(m_λ)m_λ <1, because artanh(m) < fracm1-m^2 (0<m<1). The derivative is positive, so the nonzero fixed points are locally stable. Finally, expanding tanh(λ m) = λ m-","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f93687b75635f48a","tex":"\\section","aliases":"section","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-57c22728827aa262","tex":"\\mathrm{sel}","aliases":"sel mathrm","display":"\\mathrm{sel}","kind":"use","excerpt":"[Validity after independent specification selection] Let widehatς be selected using D_sel, and let D_cert be independent of D_sel. For every fixed ς∈ S, let ϕ_ς( D_cert)∈{0,1} be a certificate rule satisfying sup_P:Cert(W;ς)=0 P_P(ϕ_ς=1) ≤α. Then the selected-specification procedure satisfies sup_P P_P( ϕ_widehatς=1, Cert(W;widehatς)=0 ) ≤α.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4369638c40f08f8d","tex":"\\mathsf{selection\\_history}","aliases":"selection_history mathsf","display":"\\mathsf{selection\\_history}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c81b815dc4ca9908","tex":"\\operatorname{SelfAware}","aliases":"SelfAware operatorname","display":"\\operatorname{SelfAware}","kind":"use","excerpt":"The following implications are not licensed: Agent_Θ_A(Workspace) centernotLongrightarrow PhenomenallyConscious(Workspace), SelfMaintaining(Workspace) centernotLongrightarrow SelfAware(Workspace), GoalPersistent(Workspace) centernotLongrightarrow MoralPatient(Workspace). A thermostat-like controller can maintain a variable without representing itse","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-70f1eaeb8b6b27f8","tex":"\\operatorname{SelfMaintaining}","aliases":"SelfMaintaining operatorname","display":"\\operatorname{SelfMaintaining}","kind":"use","excerpt":"The following implications are not licensed: Agent_Θ_A(Workspace) centernotLongrightarrow PhenomenallyConscious(Workspace), SelfMaintaining(Workspace) centernotLongrightarrow SelfAware(Workspace), GoalPersistent(Workspace) centernotLongrightarrow MoralPatient(Workspace). A thermostat-like controller can maintain a variable without representing itse","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d6787c3992ae9099","tex":"\\operatorname{SelfMaintenance}","aliases":"SelfMaintenance operatorname","display":"\\operatorname{SelfMaintenance}","kind":"use","excerpt":"subtopicLevel 6: Workspace agent · η_G→ A>0, η_W→ A>0, δ_Q≤ε_Q, · Reg_t(π) ≤2ε_Q+δ_opt, k_ctl,ε≥2, SelfMaintenance.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"[Local compatibility without a global nonzero core] Let the only nontrivial cycle holonomy on R^2 be a rotation R_θ= cosθ · -sinθ sinθ · cosθ with θ∉2π Z. Then ker(R_θ-I)={0}. Every edge map is individually invertible and well-defined, yet the complete cycle admits no nonzero exact shared core. Pairwise translatability is therefore insufficient for","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aac4caf925195f39","tex":"\\mathsf{Size}","aliases":"Size mathsf","display":"\\mathsf{Size}","kind":"use","excerpt":"A general score is B_t(S) = · a_RRet_τ(S) +a_OOut_τ(S) +a_CCOut_τ(S) · -a_DDep_τ(S) -a_ICIn_τ(S) -a_LCoh(S) · -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass conta","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9bd6563a665b1ad1","tex":"\\small","aliases":"small","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"0 · -1 . The cycle holonomy is J, hence C_*=ker(J-I)=span{(1,0)^top}. The first coordinate circulates consistently through all three strata. The second returns with its sign reversed and therefore cannot define a nonzero exact shared section. Thus d_Workspace=1.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-81c7e7f407a1b2e9","tex":"\\spec","aliases":"spec","display":null,"kind":"use","excerpt":"[Unitary gauge covariance of the Laplacian] If every g_i is unitary, then G_V and G_E are unitary and L_H^g=G_VL_HG_V^-1. Consequently, spec(L_H^g)=spec(L_H), P_I(L_H^g)=G_VP_I(L_H)G_V^-1 for every spectral interval I.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ace11d5ff02516de","tex":"\\mathrm{specific}","aliases":"specific mathrm","display":"\\mathrm{specific}","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cc44e3fb79060540","tex":"\\mathsf{specification}","aliases":"specification mathsf","display":"\\mathsf{specification}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-96d5eecfa0783030","tex":"\\mathsf{specification\\_envelopes}","aliases":"specification_envelopes mathsf","display":"\\mathsf{specification\\_envelopes}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2feafb22ecf1657b","tex":"\\mathsf{specification\\_family}","aliases":"specification_family mathsf","display":"\\mathsf{specification\\_family}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b895b2d17673efa4","tex":"\\mathsf{SpecificationRecord}","aliases":"SpecificationRecord mathsf","display":"\\mathsf{SpecificationRecord}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"The same sample must not be used both to select a workspace and to certify its coherence, causal efficacy, or relational irreducibility. The required split is D = D_form sqcup D_cal sqcup D_cert. The formation sample selects widehatθ. The calibration sample fixes thresholds, regularization strengths, and intervention magnitudes. The certification s","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9c79a6679e6135f5","tex":"\\SRW","aliases":"SRW","display":null,"kind":"use","excerpt":"Symbolically, SRW(W_t;ς) = bigwedge_q∈ J(ς) Gate_q(W_t;ς). The gate set is claim-relative. A one-step operational workspace need not pass persistence or reentry gates, while a persistent reentrant workspace must. No scalar total score is sufficient: failure of a required gate cannot be hidden by high performance on another gate.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9003ed2fe767e501","tex":"\\mathrm{SRW}","aliases":"SRW mathrm","display":"\\mathrm{SRW}","kind":"use","excerpt":"The strongest current candidate is not a single equation. It is the following conjunctive object: C_SRW = ( c heterogeneous core--transfer diagram and residuals, exact or spectral shared-section state, operation and predictive alignment, feasible causal interventions, coalition irreducibility under matched local-predictive nulls, non-collapse, reco","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c9c8b73b7e897c84","tex":"\\SRW_\\Theta","aliases":"SRW_Θ SRW Theta","display":null,"kind":"use","excerpt":"This yields the more precise notation SRW(W;ς)=1. The earlier expression SRW_Θ(W; D) is the special case in which the remaining coordinates of ς are held fixed.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-53e31a90ba57bd12","tex":"\\operatorname{SRW}_{\\Theta}","aliases":"SRW_Θ operatorname Theta","display":"\\operatorname{SRW}_{\\Theta}","kind":"use","excerpt":"The formation provenance of a workspace and its operational certification are separate coordinates. A fully scaffolded structure may satisfy every structural and causal workspace gate. A highly endogenous structure may fail to become coherent or causally useful. Symbolically, Endogenous(W) centernotLongrightarrow SRW_Θ(W), SRW_Θ(W) centernotLongrig","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-81562c169fb0cca6","tex":"\\operatorname{SRW}_{\\Theta_S}","aliases":"SRW_Θ_S operatorname Theta","display":"\\operatorname{SRW}_{\\Theta_S}","kind":"use","excerpt":"S, define Coh(S) = tr ( P_S,dL_SP_S,d ), where P_S,d projects onto the selected low-energy workspace modes. Let Cut(S) = ∑_e:s(e)∈ S, t(e)∉ Sw_e + ∑_e:s(e)∉ S, t(e)∈ Sw_e. Let k_ε(S) be the minimal sufficient coalition size inside the candidate subsystem. A candidate is admissible only if SRW_Θ_S(W_S)=1, k_ε(S)≥2, unless the claim explicitly concer","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d7dc0e74ed03b173","tex":"\\mathrm{stab}","aliases":"stab mathrm","display":"\\mathrm{stab}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-50b5c93defcd39db","tex":"\\mathrm{static}","aliases":"static mathrm","display":"\\mathrm{static}","kind":"use","excerpt":"The structural workspace is therefore more accurately represented by the triple Workspace_static = ( Γ(H), O_Workspace, π_Workspace ), not merely by the vector space Γ(H).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d52fe00cbcd5e1c1","tex":"\\mathsf{STD}","aliases":"STD mathsf","display":"\\mathsf{STD}","kind":"use","excerpt":"The audit uses five labels: ll STD · = established concept, theorem, or technique, ADP · = established machinery adapted to relational workspaces, SYN · = nontrivial synthesis of established components, CAN · = candidate original contribution requiring priority review, EMP · = empirical claim not yet established in an uncontrolled system. These lab","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bcac5605fdc4859a","tex":"\\mathrm{str}","aliases":"str mathrm","display":"\\mathrm{str}","kind":"use","excerpt":"Let c(t)=(c_i(t))_i∈ V ∈ C_V(t), P_t= 1_[0,ε_t](L_t). The instantaneous structural workspace coordinate is W_t^str=P_tc(t) ∈ Γ_ε_t(H(t)). When P_t varies with time, these coordinates inhabit a moving family of subspaces. Parts~X--XII will supply the transport required to compare them across time. In what follows, all sufficiency and intervention st","count":29},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-01ec86b1326d42e6","tex":"\\mathrm{stress}","aliases":"stress mathrm","display":"\\mathrm{stress}","kind":"use","excerpt":"A practical design may use: D_form sqcup D_spec sqcup D_cert sqcup D_stress. Formation learns the workspace. The specification split chooses targets, horizons, nulls, and thresholds from a predeclared menu. Certification makes the pass decision once. Stress testing explores broader shifts without retroactively changing the certified claim. Cross-fi","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e3cb98a5910dc5ea","tex":"\\operatorname{StructurallyAdaptive}","aliases":"StructurallyAdaptive operatorname","display":"\\operatorname{StructurallyAdaptive}","kind":"use","excerpt":"subtopicOrthogonal formation-provenance axis Scaffolded ⟶ AdaptivelyParameterised ⟶ StructurallyAdaptive ⟶ EndogenouslyConstituted. This axis is independent of Levels~0--6. A scaffolded system can instantiate a high-level relational workspace, while an endogenously learned organization can fail every operational workspace gate.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b8c9fcb0d545266d","tex":"\\mathrm{suff}","aliases":"suff mathrm","display":"\\mathrm{suff}","kind":"use","excerpt":"For an arbitrary history statistic S_t=s( H_t), let K_t^S(s,·) = P(Y_t^+∈·mid S_t=s). Whenever the conditional relative entropy is finite, define the sufficiency defect δ_suff(S_t) = E [ KL ( K_t( H_t,·) middle K_t^S(S_t,·) ) ].","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4f16f99523f2e107","tex":"\\sup_{\\mathcal D\\in\\mathfrak D}","aliases":"sup_ D∈ D sup mathcal in mathfrak","display":"\\sup_{\\mathcal D\\in\\mathfrak D}","kind":"use","excerpt":"Let m_q(W; D) be the signed margin of gate q, oriented so that m_q(W; D)>0 means that the gate passes after its uncertainty correction. Define the decomposition envelope underline m_q = inf_ D∈ Dm_q(W; D), overline m_q = sup_ D∈ Dm_q(W; D), and its sensitivity width Δ_q^dec = overline m_q-underline m_q.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e51630578ddcc523","tex":"\\sup_{\\mathfrak R\\in\\mathfrak R_{\\mathrm{ID}}}","aliases":"sup_ R∈ R_ID sup mathfrak in mathfrak mathrm","display":"\\sup_{\\mathfrak R\\in\\mathfrak R_{\\mathrm{ID}}}","kind":"use","excerpt":"For robustness across specifications, define the realization-level robust margin r_ S( R) = inf_ς∈ S min_q∈ J_* m_q(W_ R;ς) and its identified range underline r_ID, S = inf_ R∈ R_ID r_ S( R), overline r_ID, S = sup_ R∈ R_ID r_ S( R).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5664713d368a14d1","tex":"\\sup_a","aliases":"sup_a sup","display":"\\sup_a","kind":"use","excerpt":"If the interventional action-value estimator satisfies sup_a|widehat Q_t(a)-Q_t(a)|≤ε_Q and the implemented policy has optimization error at most δ_opt relative to widehat Q_t, then Reg_t(widehatπ_t) ≤ 2ε_Q+δ_opt. For deterministic greedy action, the bound is 2ε_Q. A unique true optimum with value gap greater than 2ε_Q is recovered exactly. The gua","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-af67e1e8a3ba0999","tex":"\\sup_{A}","aliases":"sup_A sup","display":"\\sup_{A}","kind":"use","excerpt":"Sufficiency has a direct decision-theoretic interpretation. Use the convention TV(P,Q)=sup_A|P(A)-Q(A)|. Let A be a finite action set and ℓ: Y× A→[0,1] a bounded loss. Define Risk_ℓ(S_t) = inf_d E [ ℓ(Y_t^+,d(S_t)) ].","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-54d1b2d03e3fb731","tex":"\\sup_{a,a',b}","aliases":"sup_a,a',b sup","display":"\\sup_{a,a',b}","kind":"use","excerpt":". For a divergence D_A, define η_G→ A = sup_g,g',w,e D_A ( P(A_t∈·midι_g,w,e), P(A_t∈·midι_g',w,e) ). Workspace-dependent action mediation is η_W→ A = sup_w,w',g,e D_A ( P(A_t∈·midι_w,g,e), P(A_t∈·midι_w',g,e) ). An action must also alter the declared future: η_A→ Y = sup_a,a',b D_Y ( P(Y_t^+∈·midι_a,b), P(Y_t^+∈·midι_a',b) ). A positive η_G→ A wit","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0e99c2e8d45ce7e5","tex":"\\sup_{a\\in\\mathcal A}","aliases":"sup_a∈ A sup in mathcal","display":"\\sup_{a\\in\\mathcal A}","kind":"use","excerpt":"[Approximate counterfactual policy selection] Let A be a finite feasible action set. Let Q: A→ R be the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal poli","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5fee3283ae5c1956","tex":"\\sup_{a\\in\\mathcal A_1}","aliases":"sup_a∈ A_1 sup in mathcal","display":"\\sup_{a\\in\\mathcal A_1}","kind":"use","excerpt":"For an admissible correspondence J, define the operation defect on a declared unit ball A_1 by δ_op( J) = sup_a∈ A_1 A_ W''β-β A_ W . The primary certification object is the identity profile d_id( J) = ( q_str, δ_op( J), δ_causal(β), δ_rec(β); μ^keep ). The first four coordinates must remain below declared tolerances, while the non-collapse margin ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-967b08448ad072f9","tex":"\\sup_{a\\in\\mathcal A_t}","aliases":"sup_a∈ A_t sup in mathcal","display":"\\sup_{a\\in\\mathcal A_t}","kind":"use","excerpt":"item[Counterfactual accuracy.] δ_Q = sup_a∈ A_t |widehat Q_t(a)-Q_t^g(amid w,e)|.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f88b71eab661f81a","tex":"\\sup_{a\\in\\mathcal G}","aliases":"sup_a∈ G sup in mathcal","display":"\\sup_{a\\in\\mathcal G}","kind":"use","excerpt":"The edgewise defect is Ξ_e(a) = H_eρ_s(e)(a)-ρ_t(e)(a)H_e. For a bounded generating family G⊆ A, define η_int( G) = sup_a∈ G ∑_e∈ Ew_e Ξ_e(a) ^2. When A is normed and its unit ball is appropriate, one may take G={a: a _ A≤1}. Exact representational universality means η_int=0, while approximate universality means η_int≤ε_int. The use of a generating","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-af747ef9eb970bcb","tex":"\\sup_{c,c'}","aliases":"sup_c,c' sup","display":"\\sup_{c,c'}","kind":"use","excerpt":"Cue dependence must be reported separately. For matched initial goal g, define δ_G^cue(τ) = sup_c,c' D_G ( P(G_t+τ∈·midι_g,do(C=c)), P(G_t+τ∈·midι_g,do(C=c')) ). High cue dependence is not a failure when the claimed agent is intentionally instruction following. It does mean that persistence cannot be attributed to an autonomous internal objective w","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-48eac3282a416370","tex":"\\sup_{g,g',w,e}","aliases":"sup_g,g',w,e sup","display":"\\sup_{g,g',w,e}","kind":"use","excerpt":"Goal-dependent action mediation is measured by feasible interventions on the goal channel while matching the current workspace and external context. For a divergence D_A, define η_G→ A = sup_g,g',w,e D_A ( P(A_t∈·midι_g,w,e), P(A_t∈·midι_g',w,e) ). Workspace-dependent action mediation is η_W→ A = sup_w,w',g,e D_A ( P(A_t∈·midι_w,g,e), P(A_t∈·midι_w","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7010577c819f6bce","tex":"\\sup_{\\jmath,\\jmath'}","aliases":"sup_jmath,jmath' sup jmath jmath","display":"\\sup_{\\jmath,\\jmath'}","kind":"use","excerpt":"A stronger mediation test equalizes the workspace state after two distinct channel interventions. Let μ be a common imposed workspace distribution, implemented by a feasible stochastic intervention in the sense of Definition~[source reference]. The notation below is shorthand for that implementation. Define the residual mediation defect δ_med = sup","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-01c3a846a2356031","tex":"\\sup_P","aliases":"sup_P sup","display":"\\sup_P","kind":"use","excerpt":"sel, and let D_cert be independent of D_sel. For every fixed ς∈ S, let ϕ_ς( D_cert)∈{0,1} be a certificate rule satisfying sup_P:Cert(W;ς)=0 P_P(ϕ_ς=1) ≤α. Then the selected-specification procedure satisfies sup_P P_P( ϕ_widehatς=1, Cert(W;widehatς)=0 ) ≤α.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ccc04e9fb5658e31","tex":"\\sup_{P:\\,\\operatorname{Cert}_{\\mathfrak S}(W)=0}","aliases":"sup_P: Cert_ S(W)=0 sup operatorname mathfrak","display":"\\sup_{P:\\,\\operatorname{Cert}_{\\mathfrak S}(W)=0}","kind":"use","excerpt":"[Uniform finite-sample certification across specifications] Under the uniform error event above, declare a specification-robust certificate only when widehatunderline m_q,1-δ^ S>0 for every q∈ J_*. Then sup_P: Cert_ S(W)=0 P_P ( the robust certificate is declared ) ≤δ.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-806effca174506f1","tex":"\\sup_{P:\\operatorname{Cert}(W;\\varsigma)=0}","aliases":"sup_P:Cert(W;ς)=0 sup operatorname varsigma","display":"\\sup_{P:\\operatorname{Cert}(W;\\varsigma)=0}","kind":"use","excerpt":"em[Validity after independent specification selection] Let widehatς be selected using D_sel, and let D_cert be independent of D_sel. For every fixed ς∈ S, let ϕ_ς( D_cert)∈{0,1} be a certificate rule satisfying sup_P:Cert(W;ς)=0 P_P(ϕ_ς=1) ≤α. Then the selected-specification procedure satisfies sup_P P_P( ϕ_widehatς=1, Cert(W;widehatς)=0 ) ≤α.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7ed0ccc85fcf2e88","tex":"\\sup_{q,\\varsigma}","aliases":"sup_q,ς sup varsigma","display":"\\sup_{q,\\varsigma}","kind":"use","excerpt":"Suppose S and J_* are finite, and each margin is estimated as widehat m_q(ς) = frac1n∑_i=1^nZ_i,q,ς-θ_q,ς, 0≤ Z_i,q,ς≤1, from independent certification units. Hoeffding's inequality and a union bound give P( sup_q,ς | widehat m_q(ς)-m_q(ς) | > sqrt frac log(2| J_*|| S|/δ) 2n ) ≤δ. Thus the choice ε_n(δ) = sqrt frac log(2| J_*|| S|/δ) 2n yields an e","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3ddbba6b04269f62","tex":"\\sup_{r\\in\\mathcal R}","aliases":"sup_r∈ R sup in mathcal","display":"\\sup_{r\\in\\mathcal R}","kind":"use","excerpt":"[Repair selection under value-model error] Fix a damage intervention d and a finite feasible repair set R. Let V(r)=V_t^rep(r;d) be the true expected post-repair integrity margin and let widehat V obey sup_r∈ R|widehat V(r)-V(r)| ≤ ε_R. Let widehat r∈argmax_rwidehat V(r) and r^*∈argmax_r V(r). Then V(r^*)-V(widehat r) ≤ 2ε_R. If V(r^*) ≥ ρ+2ε_R, th","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-510a10b64cc768dd","tex":"\\sup_S","aliases":"sup_S sup","display":"\\sup_S","kind":"use","excerpt":"If the score depends on a parameter ϑ_t and satisfies sup_S | B_ϑ(S)- B_ϑ'(S)| ≤ L_Bd(ϑ,ϑ'), then the sufficient stability condition becomes 2L_Bd(ϑ_t,ϑ_t+1) < Δ_t^B.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8bbc9266046899ee","tex":"\\sup_{\\substack{c\\in W\\\\\\|c\\|=1}}","aliases":"sup_substackc∈ W c =1 sup substack in","display":"\\sup_{\\substack{c\\in W\\\\\\|c\\|=1}}","kind":"use","excerpt":"For c≠0, define the Rayleigh defect δ_coh(c) = frac E_H(c) c ^2. For a nonzero subspace W⊆ C_V, define δ_coh(W) = sup_substackc∈ W c =1 E_H(c). If W is L_H-invariant, then δ_coh(W) = L_H|_W . In particular, δ_coh(Γ_ε(H)) ≤ε.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-73804f312dfa6f9b","tex":"\\sup_{S\\in\\mathfrak S_{\\mathrm{pass}}}","aliases":"sup_S∈ S_pass sup in mathfrak mathrm","display":"\\sup_{S\\in\\mathfrak S_{\\mathrm{pass}}}","kind":"use","excerpt":"[Held-out boundary recovery] Let S_pass be finite and let S_t^* = argmax_S∈ S_pass B_t(S) be unique. Define the population margin Δ_t^B = B_t(S_t^*) - max_S≠ S_t^* B_t(S)>0. If sup_S∈ S_pass | widehat B_t(S)- B_t(S) | ≤ε_t, then every empirical maximizer widehat S_t satisfies B_t(S_t^*)- B_t(widehat S_t) ≤2ε_t. If 2ε_t<Δ_t^B, then widehat S_t=S_t^*","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-34c8666b79775742","tex":"\\sup_t","aliases":"sup_t sup","display":"\\sup_t","kind":"use","excerpt":"For every damage family, report η_D→ M, η_M→ R, η_R→ v, Δ_repair^spec, δ_rep-id, sup_t e_t. The repeated-disturbance experiment must contain: item a no-damage baseline; item damage with repair disabled; item damage with the intact repair loop; item monitor ablation; item cost-matched random repair; item off-workspace repair; item repeated and compo","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d49665dbab1aa56b","tex":"\\sup_{w,w',b}","aliases":"sup_w,w',b sup","display":"\\sup_{w,w',b}","kind":"use","excerpt":"ility laws. For a matched background value b, define CE_Workspace(w,w';b) = D ( P^ι_w(Y_t^+∈·mid B_t=b), P^ι_w'(Y_t^+∈·mid B_t=b) ). For a declared comparison design ν over (w,w',b), define η_causal^ν = E_(w,w',b)∼ν [ CE_Workspace(w,w';b) ]. The maximum η_causal^max = sup_w,w',bCE_Workspace(w,w';b) may be useful for discovery, but a design-weighted","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-32945c34d15e30bc","tex":"\\sup_{w,w',g,e}","aliases":"sup_w,w',g,e sup","display":"\\sup_{w,w',g,e}","kind":"use","excerpt":"Goal-dependent action mediation is measured by feasible interventions on the goal channel while matching the current workspace and external context. For a divergence D_A, define η_G→ A = sup_g,g',w,e D_A ( P(A_t∈·midι_g,w,e), P(A_t∈·midι_g',w,e) ). Workspace-dependent action mediation is η_W→ A = sup_w,w',g,e D_A ( P(A_t∈·midι_w,g,e), P(A_t∈·midι_w","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4b88aabfdab91909","tex":"\\sup_{x\\in\\mathcal R_r}","aliases":"sup_x∈ R_r sup in mathcal","display":"\\sup_{x\\in\\mathcal R_r}","kind":"use","excerpt":"ound] Let F_r: H_r→ H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(x) - F_R-1∘⋯∘ F_0(x)","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ec158fc62000af30","tex":"\\sup_{\\theta\\in\\mathfrak T}","aliases":"sup_θ∈ T sup theta in mathfrak","display":"\\sup_{\\theta\\in\\mathfrak T}","kind":"use","excerpt":"orem[Uniform structure-selection recovery] Let T be a finite candidate family with population score J: T→ R. Let θ_* = argmin_θ∈ TJ(θ) be unique, with margin Δ_J = min_θ≠θ_* (J(θ)-J(θ_*))>0. If a held-out estimate widehat J satisfies sup_θ∈ T |widehat J(θ)-J(θ)| ≤ε, then every empirical minimizer widehatθ obeys J(widehatθ)-J(θ_*) ≤2ε. If 2ε<Δ_J, th","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6516fc2e0ae60482","tex":"\\sup_{\\lambda\\in\\Lambda_n}","aliases":"sup_λ∈Λ_n sup lambda in Lambda","display":"\\sup_{\\lambda\\in\\Lambda_n}","kind":"use","excerpt":"Let Q(λ) be a gate-specific population order parameter and widehat Q_n(λ) its held-out estimate. If sup_λ∈Λ_n |widehat Q_n(λ)-Q(λ)| ≤ε_n on a finite evaluation grid Λ_n, then every threshold crossing with vertical margin larger than 2ε_n is recovered by the same uniform-selection argument as Theorem~[source reference]. Near a continuous transition,","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-266e00df4d8623a3","tex":"\\sup_{\\varsigma\\in\\mathfrak S}","aliases":"sup_ς∈ S sup varsigma in mathfrak","display":"\\sup_{\\varsigma\\in\\mathfrak S}","kind":"use","excerpt":"For each gate q∈ J_*, define the specification envelope underline m_q^ S(W) = inf_ς∈ Sm_q(W;ς), overline m_q^ S(W) = sup_ς∈ Sm_q(W;ς). Its sensitivity width is Δ_q^spec(W) = overline m_q^ S(W) - underline m_q^ S(W). The robust certificate margin is r_ S(W) = inf_ς∈ S min_q∈ J_*m_q(W;ς). Then Cert_ S(W)=1 Longleftrightarrow r_ S(W)>0.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0b7cc491b1ee030b","tex":"\\operatorname{supp}","aliases":"supp operatorname","display":"\\operatorname{supp}","kind":"use","excerpt":"Define the accessible block support of c≠0 by supp_E(c) = {i∈ V:E_ic≠0} and define the node-erasure distance d_er = min_c≠0|supp_E(c)|.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b63d44b57de31bd4","tex":"\\operatorname{supp}_E","aliases":"supp_E operatorname","display":"\\operatorname{supp}_E","kind":"use","excerpt":"Define the accessible block support of c≠0 by supp_E(c) = {i∈ V:E_ic≠0} and define the node-erasure distance d_er = min_c≠0|supp_E(c)|.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-27ae04abadae6c2a","tex":"\\mathrm{sym}","aliases":"sym mathrm","display":"\\mathrm{sym}","kind":"use","excerpt":"d path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and intervention divergences need not be symmetric, symmetry is not assumed. When reverse correspondences exist, define the symmetrized pseudometric d_path^sym( R, R') = max{ d_path^rightar","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a678611fc500cdba","tex":"\\mathsf{SYN}","aliases":"SYN mathsf","display":"\\mathsf{SYN}","kind":"use","excerpt":"The audit uses five labels: ll STD · = established concept, theorem, or technique, ADP · = established machinery adapted to relational workspaces, SYN · = nontrivial synthesis of established components, CAN · = candidate original contribution requiring priority review, EMP · = empirical claim not yet established in an uncontrolled system. These lab","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ade086a1f71df625","tex":"\\mathrm{sys}","aliases":"sys mathrm","display":"\\mathrm{sys}","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0,","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-da4fb737a4a4bc72","tex":"\\mathfrak T","aliases":"T mathfrak","display":"\\mathfrak T","kind":"use","excerpt":"[Uniform structure-selection recovery] Let T be a finite candidate family with population score J: T→ R. Let θ_* = argmin_θ∈ TJ(θ) be unique, with margin Δ_J = min_θ≠θ_* (J(θ)-J(θ_*))>0. If a held-out estimate widehat J satisfies sup_θ∈ T |widehat J(θ)-J(θ)| ≤ε, then every empirical minimizer widehatθ obeys J(widehatθ)-J(θ_*) ≤2ε. If 2ε<Δ_J, then w","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7bba8734aa77eeb2","tex":"t^-","aliases":"t^-","display":"t^-","kind":"use","excerpt":"Restoration of performance is not automatically restoration of identity. Let W_t^- be the last certified pre-damage workspace and let W_t+τ^r be the post-repair workspace. Define the repair identity defect δ_rep-id(r) = d_id ( W_t^-,W_t+τ^r ). A repair action is identity preserving at tolerance ε_id when Δ_t^rep(r;d)>0, δ_rep-id(r) ≤ ε_id. When int","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e6d114cbf0e8cc1a","tex":"T_{10}","aliases":"T_10","display":"T_{10}","kind":"use","excerpt":"subtopicT_10: Approximate universality theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f790967529827b89","tex":"T_{11}","aliases":"T_11","display":"T_{11}","kind":"use","excerpt":"subtopicT_11: Core--residual accumulation theorem","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1a36fd1f6d7af0cd","tex":"T_{12}","aliases":"T_12","display":"T_{12}","kind":"use","excerpt":"[Predictive-state sufficiency and minimality, T_12] Under Definition~[source reference]: item Z_t is predictively sufficient: Y_t^+indep H_tmid Z_t. item If S_t=s( H_t) is any other history statistic satisfying Y_t^+indep H_tmid S_t, then there is a measurable map ϕ such that Z_t=ϕ(S_t) almost surely. item Consequently, any two minimal sufficient p","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-21a8dad328629d94","tex":"T_{13}","aliases":"T_13","display":"T_{13}","kind":"use","excerpt":"[Operational relationality, T_13] Let Y be finite, let P be the intact law at a fixed decision time, and let W_t=π_t( H_t) be a structurally supported candidate. Let Q⊆ N_pred(P) be a nonempty, predeclared panel of validated relation-breaking nulls. Assume: item exact structural--operational alignment, I_P(Y_t^+; H_tmid W_t)=0; item a positive node","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-83598e1df166fb58","tex":"T_{14}","aliases":"T_14","display":"T_{14}","kind":"use","excerpt":"subtopic41.1 Replacement invariance theorem T_14","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8b3b41486f59fecd","tex":"T_{15}","aliases":"T_15","display":"T_{15}","kind":"use","excerpt":"subtopic49.3 Transfer identification theorem T_15","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e8b62f4c3f8f7587","tex":"T_{16}","aliases":"T_16","display":"T_{16}","kind":"use","excerpt":"subtopic49.4 Held-out structure-selection theorem T_16","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-de83cc089cdca904","tex":"T_{17}","aliases":"T_17","display":"T_{17}","kind":"use","excerpt":"subtopic51.2 Stable boundary recovery T_17","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e9ee526f4fb80df6","tex":"T_{18}","aliases":"T_18","display":"T_{18}","kind":"use","excerpt":"subtopicT_18: Model-specific formation transition theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-134a1055bc253627","tex":"T_{19}","aliases":"T_19","display":"T_{19}","kind":"use","excerpt":"subtopic52.7 T_19: Cue-controlled goal-retention theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-431a15df9c3d99ca","tex":"T_{1_i}","aliases":"T_1_i","display":"T_{1_i}","kind":"use","excerpt":"For every i∈ V, assign a finite-dimensional Hilbert space X_i. For every edge, assign a linear transfer T_e:X_s(e)⟶ X_t(e). The path transfer is T_γ=T_e_m⋯ T_e_2T_e_1, T_1_i=I_X_i. Accordingly, the static transfer system is a functor X:Path(G)⟶FinHilb.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7be2ccc9284905af","tex":"T_{20}","aliases":"T_20","display":"T_{20}","kind":"use","excerpt":"subtopic53.3 T_20: Approximate counterfactual policy theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2792e5aeb2ae7e1f","tex":"T_{21}","aliases":"T_21","display":"T_{21}","kind":"use","excerpt":"subtopic53.8 T_21: Repair-selection margin theorem","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d40e5e2331bf36b5","tex":"T_{22}","aliases":"T_22","display":"T_{22}","kind":"use","excerpt":"subtopic53.10 T_22: Closed-loop self-maintenance theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-56f9752ffe92acc1","tex":"T_{23}","aliases":"T_23","display":"T_{23}","kind":"use","excerpt":"subtopic53.14 T_23: Workspace-agency synthesis theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-61cf75914ce7e182","tex":"T_{24}","aliases":"T_24","display":"T_{24}","kind":"use","excerpt":"subtopicT_24: Gatewise certification-validity theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-259cafcb760fb5cd","tex":"T_{25}","aliases":"T_25","display":"T_{25}","kind":"use","excerpt":"subtopicT_25: Faithful coarse-graining theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-92946f909f415778","tex":"T_{26}","aliases":"T_26","display":"T_{26}","kind":"use","excerpt":"subtopicT_26: Bayes-risk transport theorem","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-658d40195a1b05db","tex":"T_{27}","aliases":"T_27","display":"T_{27}","kind":"use","excerpt":"subtopicT_27: Decomposition-transport robustness theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aada24738133243d","tex":"T_{28}","aliases":"T_28","display":"T_{28}","kind":"use","excerpt":"subtopic62.3 T_28: Specification-transport robustness theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d4d5e121a685f63c","tex":"T_{29}","aliases":"T_29","display":"T_{29}","kind":"use","excerpt":"subtopic63.1 T_29: Non-identifiability impossibility theorem","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-36a85d18b1f4f5e2","tex":"T_{2a}","aliases":"T_2a","display":"T_{2a}","kind":"use","excerpt":"subtopicT_2a: Workspace commutant theorem O_Workspace ≅ Comm(Hol_o).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","excerpt":"subtopicT_3: Transfer-Laplacian kernel theorem ker L_H=Γ(H).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e7b630dea07df89c","tex":"T_{30}","aliases":"T_30","display":"T_{30}","kind":"use","excerpt":"subtopic64.2 T_30: Joint specification-identification theorem","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c91cf69963e765b5","tex":"T_{31}","aliases":"T_31","display":"T_{31}","kind":"use","excerpt":"subtopic65.1 T_31: Uniform specification-certificate theorem","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-899fcb428498d22e","tex":"T_{32}","aliases":"T_32","display":"T_{32}","kind":"use","excerpt":"subtopic66.2 T_32: Sample-split selection theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-64f17e7dedd6a141","tex":"T_4","aliases":"T_4","display":"T_4","kind":"use","excerpt":"subtopicT_4: Spectral stability theorem If η= widetilde L_H-L_H <g_d/2, then widetilde P_d-P_d ≤ fracηg_d-η ≤ frac2ηg_d.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6b06b314a3b34d94","tex":"T_5","aliases":"T_5","display":"T_5","kind":"use","excerpt":"subtopicT_5: Gauge invariance theorem Γ(H^g)=G_VΓ(H), and under unitary gauge, L_H^g=G_VL_HG_V^-1.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c2ee887a30223081","tex":"T_6","aliases":"T_6","display":"T_6","kind":"use","excerpt":"subtopic32.3 Persistence theorem T_6","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6f58c955d6af4e14","tex":"T_7","aliases":"T_7","display":"T_7","kind":"use","excerpt":"subtopic35.1 Stable reconstruction theorem T_7","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9e9fd0f2c2f26c2b","tex":"T_8","aliases":"T_8","display":"T_8","kind":"use","excerpt":"subtopicT_8: Exact gluing theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5a8e3f96a2fb5d5a","tex":"T_9","aliases":"T_9","display":"T_9","kind":"use","excerpt":"subtopicT_9: Hierarchical collapse theorem","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f9db030d4ae40b4d","tex":"T_{e_1}","aliases":"T_e_1","display":"T_{e_1}","kind":"use","excerpt":"For every i∈ V, assign a finite-dimensional Hilbert space X_i. For every edge, assign a linear transfer T_e:X_s(e)⟶ X_t(e). The path transfer is T_γ=T_e_m⋯ T_e_2T_e_1, T_1_i=I_X_i. Accordingly, the static transfer system is a functor X:Path(G)⟶FinHilb.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c72544cbae738bf9","tex":"T_{e_2}","aliases":"T_e_2","display":"T_{e_2}","kind":"use","excerpt":"For every i∈ V, assign a finite-dimensional Hilbert space X_i. For every edge, assign a linear transfer T_e:X_s(e)⟶ X_t(e). The path transfer is T_γ=T_e_m⋯ T_e_2T_e_1, T_1_i=I_X_i. Accordingly, the static transfer system is a functor X:Path(G)⟶FinHilb.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-433e0a6fbaea092a","tex":"T_{e_m}","aliases":"T_e_m","display":"T_{e_m}","kind":"use","excerpt":"For every i∈ V, assign a finite-dimensional Hilbert space X_i. For every edge, assign a linear transfer T_e:X_s(e)⟶ X_t(e). The path transfer is T_γ=T_e_m⋯ T_e_2T_e_1, T_1_i=I_X_i. Accordingly, the static transfer system is a functor X:Path(G)⟶FinHilb.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-796c305a4e19d579","tex":"T_{e_r}","aliases":"T_e_r","display":"T_{e_r}","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a672cc71a4bb137d","tex":"T_{e_{r+1}}","aliases":"T_e_r+1","display":"T_{e_{r+1}}","kind":"use","excerpt":"For a single edge, V_e=0 means T_e(c,0)=(H_ec,0), so core states remain in the core. Inductively, if the first r transfers map c to (H_e_r⋯ H_e_1c,0), applying T_e_r+1 gives (H_e_r+1H_e_r⋯ H_e_1c,0). The conclusion follows after m steps.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fe0dee2e407aa42f","tex":"T_{e_\\ell}","aliases":"T_e_ℓ ell","display":"T_{e_\\ell}","kind":"use","excerpt":"For a path γ=e_m⋯ e_1, Q_t(γ)T_γ P_s(γ) ≤ ∑_r=1^m · ( ∏_ℓ=r+1^m T_e_ℓ ) V_e_r · × ( ∏_ℓ=1^r-1 H_e_ℓ ). The same bound controls the defect between full core-to-core path transport and the pure core product. An exact Duhamel identity yields both estimates.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4b7fbd7d66e3a28d","tex":"T_f","aliases":"T_f","display":"T_f","kind":"use","excerpt":"For composable edges e:i→ j and f:j→ k, direct block multiplication gives T_fT_e = H_fH_e+U_fV_e · H_fU_e+U_fK_e V_fH_e+K_fV_e · V_fU_e+K_fK_e . The induced core-to-core block is therefore P_kT_fT_eP_i = H_fH_e+U_fV_e. The term U_fV_e is a residual-mediated correction: information leaves the candidate core at the intermediate node and later re-ente","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b8e003e6dbf7e04c","tex":"T_m","aliases":"T_m","display":"T_m","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-df3f310abe33eba9","tex":"T_r","aliases":"T_r","display":"T_r","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-28cef6973683b5e6","tex":"T_{r+1}","aliases":"T_r+1","display":"T_{r+1}","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e5687f719fcef22b","tex":"T_\\gamma","aliases":"T_γ gamma","display":"T_\\gamma","kind":"use","excerpt":"For every i∈ V, assign a finite-dimensional Hilbert space X_i. For every edge, assign a linear transfer T_e:X_s(e)⟶ X_t(e). The path transfer is T_γ=T_e_m⋯ T_e_2T_e_1, T_1_i=I_X_i. Accordingly, the static transfer system is a functor X:Path(G)⟶FinHilb.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-35899e3c99220253","tex":"T_\\ell","aliases":"T_ℓ ell","display":"T_\\ell","kind":"use","excerpt":"· = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r · × (∏_ℓ=1^r-1 H_ℓ ), and the same right-hand side bounds P_i_mT_m⋯ T_1P_i_0-H_m⋯ H_1 .","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a9a485c98291d899","tex":"\\tableofcontents","aliases":"tableofcontents","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-978e87e4f3d831a3","tex":"\\tanh","aliases":"tanh","display":"\\tanh","kind":"use","excerpt":"For m_t+1=tanh(λ m_t), there is a unique zero fixed point for 0≤λ≤1. For λ>1, the zero state is unstable and exactly two stable nonzero fixed points appear. Their amplitude satisfies m_λ = sqrtfrac3(λ-1)λ^3 + O((λ-1)^3/2). This is an exact structural benchmark, not a universal law for complete relational workspaces.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1fe82c101179581b","tex":"\\mathsf{targets}","aliases":"targets mathsf","display":"\\mathsf{targets}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bab897d2d5fb61a8","tex":"\\mathrm{task}","aliases":"task mathrm","display":"\\mathrm{task}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-db686cb84689b599","tex":"\\textwidth","aliases":"textwidth","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c8300d356b177d21","tex":"\\mathrm{theorem}","aliases":"theorem mathrm","display":"\\mathrm{theorem}","kind":"use","excerpt":"The analysis does not make one global novelty claim. It uses a priority profile p(C) = ( p_concept, p_object, p_theorem, p_certificate, p_experiment ). For example, the philosophical concept of distributed cognition may be standard while a specific mathematical object is adapted, a theorem is a specialization of known spectral theory, the certifica","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b4549ea66b2b0e2b","tex":"\\mathsf{thresholds}","aliases":"thresholds mathsf","display":"\\mathsf{thresholds}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-711fbd066476a170","tex":"\\tilde m","aliases":"tilde m tilde","display":"\\tilde m","kind":"use","excerpt":"ly privileged. Decompositions are constrained by access, causal interfaces, intervention separability, temporal continuity, and explicit shared resources. A certificate established only at one interface is a valid but limited claim. A stronger claim reports the full envelope ( widetilde m_q, underline m_q, overline m_q, Δ_q^dec ) for each required ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-23cf12355b229fc4","tex":"\\tilde m_q","aliases":"tilde m_q tilde","display":"\\tilde m_q","kind":"use","excerpt":"ly privileged. Decompositions are constrained by access, causal interfaces, intervention separability, temporal continuity, and explicit shared resources. A certificate established only at one interface is a valid but limited claim. A stronger claim reports the full envelope ( widetilde m_q, underline m_q, overline m_q, Δ_q^dec ) for each required ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-406f62d781ecf46d","tex":"\\mathbf{Top}","aliases":"Top mathbf","display":"\\mathbf{Top}","kind":"use","excerpt":"The finite-dimensional Hilbert assumption supplies adjoints and spectral tools. The exact compatible-section definitions require less and may later be formulated in categories such as Ban, Meas, Prob, Top, Alg.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"0 · -1 . The cycle holonomy is J, hence C_*=ker(J-I)=span{(1,0)^top}. The first coordinate circulates consistently through all three strata. The second returns with its sign reversed and therefore cannot define a nonzero exact shared section. Thus d_Workspace=1.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-70a3fc170e8a1c43","tex":"\\mathrm{tot}","aliases":"tot mathrm","display":"\\mathrm{tot}","kind":"use","excerpt":"and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U = widehat D_ U [0.2em] sqrtλ_ov S_ U , L_ U^tot = K_ U^*K_ U. Its energy is E_ U^tot(u) = · ∑_α D_G_αu_α ^2","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3e3becea69293024","tex":"\\mathrm{total}","aliases":"total mathrm","display":"\\mathrm{total}","kind":"use","excerpt":"The claim that the workspace is not identical with either participant does not imply that it occupies a separate nonphysical location. Formally, Workspace is an invariant of R, not an additional term appended to the physical realization: R_total eq( R,ghost workspace). The phrase ``between the participants'' is therefore topological or relational: ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bbfa3039713b4e0c","tex":"\\mathrm{tr}","aliases":"tr mathrm","display":"\\mathrm{tr}","kind":"use","excerpt":"Core leakage, residual contamination, and total transfer defect are measured by λ_e^out= V_e , λ_e^in= U_e , δ_e^tr = U_e + V_e . These quantities depend on the chosen norms and projectors. Their empirical meaning must therefore be tied to a declared intervention and prediction protocol.","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"thm:transfer-identification Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV","display":null,"kind":"use","excerpt":"Sufficiency has a direct decision-theoretic interpretation. Use the convention TV(P,Q)=sup_A|P(A)-Q(A)|. Let A be a finite action set and ℓ: Y× A→[0,1] a bounded loss. Define Risk_ℓ(S_t) = inf_d E [ ℓ(Y_t^+,d(S_t)) ].","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":27},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ec7d40a789aa2d83","tex":"u'","aliases":"u'","display":"u'","kind":"use","excerpt":"nate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the future law of X_S: CIn_τ(S) = E_u,u'∼ν_bar S D ( L^ι_bar S,u (X_S(t+τ)), L^ι_bar ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-76bfead452deff02","tex":"U_2","aliases":"U_2","display":"U_2","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cb69d506a7bd35e5","tex":"U_e","aliases":"U_e","display":"U_e","kind":"use","excerpt":"Relative to these decompositions, every edge map has the block form T_e= H_e · U_e V_e · K_e , where H_e=P_t(e)T_eP_s(e):C_s(e)→ C_t(e), U_e=P_t(e)T_eQ_s(e):R_s(e)→ C_t(e), V_e=Q_t(e)T_eP_s(e):C_s(e)→ R_t(e), K_e=Q_t(e)T_eQ_s(e):R_s(e)→ R_t(e).","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ada27fd6e08006d5","tex":"U_f","aliases":"U_f","display":"U_f","kind":"use","excerpt":"For composable edges e:i→ j and f:j→ k, direct block multiplication gives T_fT_e = H_fH_e+U_fV_e · H_fU_e+U_fK_e V_fH_e+K_fV_e · V_fU_e+K_fK_e . The induced core-to-core block is therefore P_kT_fT_eP_i = H_fH_e+U_fV_e. The term U_fV_e is a residual-mediated correction: information leaves the candidate core at the intermediate node and later re-ente","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d02fde853a4be6c","tex":"\\mathcal U_{\\mathrm{flat}}","aliases":"U_flat mathcal mathrm","display":"\\mathcal U_{\\mathrm{flat}}","kind":"use","excerpt":"Let L( T) be the leaf set. Repeated edge completeness implies that the flattened family U_flat = {G_ℓ:ℓ∈ L( T)} is itself an edge-complete cover of G. Let δ_flat^0 be its overlap coboundary and define W_flat = kerδ_flat^0. Theorem~[source reference] and Theorem~[source reference] give W_r ≅ W_flat ≅ mathscr W(G). Equivalently, Lim_ T^rec ( {mathscr","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8611e5a338fe3b84","tex":"U_r","aliases":"U_r","display":"U_r","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a822fb370eb1244d","tex":"u_r","aliases":"u_r","display":"u_r","kind":"use","excerpt":"At level r, let K_r be the corresponding total descent operator, γ_r = λ_min^+(K_r^*K_r), K_ru_r ≤ε_r. The nearest exact compatible state then differs from u_r by at most fracε_rsqrtγ_r. Allow an additional implementation, compression, or causal-summary defect ζ_r, and define δ_r = fracε_rsqrtγ_r + ζ_r.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a6ab3f6998c3dac0","tex":"u_t","aliases":"u_t","display":"u_t","kind":"use","excerpt":"Let C_V(t)= R^2 and let P_t project onto u_t=(cosθ_t,sinθ_t). If U_t^V is rotation through θ_t+1-θ_t and F_t=U_t^V, then a state x_t=u_t obeys e_t=e_t+1=0, μ_t^keep=1. The workspace changes raw coordinates but not abstract identity. If instead F_t=I, then e_t+1 = |sin(θ_t+1-θ_t)|. The same structural motion now produces a state-tracking defect beca","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7a2552ebf28e03b6","tex":"U_t^E","aliases":"U_t^E","display":"U_t^E","kind":"use","excerpt":"To compare successive times, choose unitary identification maps U_t^V: C_V(t)⟶ C_V(t+1), U_t^E: C_E(t)⟶ C_E(t+1). They are comparison maps, not assumptions that the physical carriers are identical. Define the one-step diagram defect Δ_t^D = D_t+1U_t^V-U_t^ED_t, δ_t^D= Δ_t^D . Exact temporal compatibility is the condition Δ_t^D=0.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0f958db420de798c","tex":"U_t^V","aliases":"U_t^V","display":"U_t^V","kind":"use","excerpt":"To compare successive times, choose unitary identification maps U_t^V: C_V(t)⟶ C_V(t+1), U_t^E: C_E(t)⟶ C_E(t+1). They are comparison maps, not assumptions that the physical carriers are identical. Define the one-step diagram defect Δ_t^D = D_t+1U_t^V-U_t^ED_t, δ_t^D= Δ_t^D . Exact temporal compatibility is the condition Δ_t^D=0.","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-697d10e561f07ee9","tex":"u_\\alpha","aliases":"u_α alpha","display":"u_\\alpha","kind":"use","excerpt":"Local candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U ","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-529aedb9edf5f977","tex":"u_\\beta","aliases":"u_β beta","display":"u_\\beta","kind":"use","excerpt":"l candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U = wi","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2db45dcb20498040","tex":"\\mathsf{uncertainty}","aliases":"uncertainty mathsf","display":"\\mathsf{uncertainty}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-84db642f69e27cc6","tex":"\\mathrm{und}","aliases":"und mathrm","display":"\\mathrm{und}","kind":"use","excerpt":"The number of independent graph cycles is r=|E_und|-|V|+1 for a connected underlying graph, after treating parallel edges with their multiplicity.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-83f817e6ba9fae50","tex":"\\underline g","aliases":"underline g underline","display":"\\underline g","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-06e0007fba1c8d1a","tex":"\\underline m","aliases":"underline m underline","display":"\\underline m","kind":"use","excerpt":"Let m_q(W; D) be the signed margin of gate q, oriented so that m_q(W; D)>0 means that the gate passes after its uncertainty correction. Define the decomposition envelope underline m_q = inf_ D∈ Dm_q(W; D), overline m_q = sup_ D∈ Dm_q(W; D), and its sensitivity width Δ_q^dec = overline m_q-underline m_q.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1c8c06d5c93ed88f","tex":"\\underline{\\mathbf m}","aliases":"underline m underline mathbf","display":"\\underline{\\mathbf m}","kind":"use","excerpt":"Use a vector rather than one aggregate score: C_full(W) =( · d_ε, g_ε, δ_coh, δ_align^ I, Δ_joint^ℓ, k_ε, · Δ_tr( Q), η_causal^ν, Δ_specific(r), η_loop(τ), · d_id, Rec_k, underline m_ D, Δ_dec, E_form, δ^scaf, δ^leak, · Δ_J, Δ_t^B, COut_τ(widehat S), CIn_τ(widehat S), · η_G(τ), δ_G^cue(τ), η_G→ A, δ_Q, η_V→ A, η_A→ Y, · η_D→ M, η_M→ R, η_R→ v, Δ_re","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3e4118c682742040","tex":"\\underline{\\mathbf m}_{\\mathfrak D}","aliases":"underline m_ D underline mathbf mathfrak","display":"\\underline{\\mathbf m}_{\\mathfrak D}","kind":"use","excerpt":"Use a vector rather than one aggregate score: C_full(W) =( · d_ε, g_ε, δ_coh, δ_align^ I, Δ_joint^ℓ, k_ε, · Δ_tr( Q), η_causal^ν, Δ_specific(r), η_loop(τ), · d_id, Rec_k, underline m_ D, Δ_dec, E_form, δ^scaf, δ^leak, · Δ_J, Δ_t^B, COut_τ(widehat S), CIn_τ(widehat S), · η_G(τ), δ_G^cue(τ), η_G→ A, δ_Q, η_V→ A, η_A→ Y, · η_D→ M, η_M→ R, η_R→ v, Δ_re","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ed0f7073e1f5988f","tex":"\\underline m_q","aliases":"underline m_q underline","display":"\\underline m_q","kind":"use","excerpt":"Let m_q(W; D) be the signed margin of gate q, oriented so that m_q(W; D)>0 means that the gate passes after its uncertainty correction. Define the decomposition envelope underline m_q = inf_ D∈ Dm_q(W; D), overline m_q = sup_ D∈ Dm_q(W; D), and its sensitivity width Δ_q^dec = overline m_q-underline m_q.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-99595982a00fdb12","tex":"\\underline m_q^{\\mathfrak S}","aliases":"underline m_q^ S underline mathfrak","display":"\\underline m_q^{\\mathfrak S}","kind":"use","excerpt":"For each gate q∈ J_*, define the specification envelope underline m_q^ S(W) = inf_ς∈ Sm_q(W;ς), overline m_q^ S(W) = sup_ς∈ Sm_q(W;ς). Its sensitivity width is Δ_q^spec(W) = overline m_q^ S(W) - underline m_q^ S(W). The robust certificate margin is r_ S(W) = inf_ς∈ S min_q∈ J_*m_q(W;ς). Then Cert_ S(W)=1 Longleftrightarrow r_ S(W)>0.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7c25a865ac8838b0","tex":"\\underline m_q^{\\mathrm{ID}}","aliases":"underline m_q^ID underline mathrm","display":"\\underline m_q^{\\mathrm{ID}}","kind":"use","excerpt":"The sharp lower and upper identified margins are underline m_q^ID(ς) = inf M_q^ID(ς), overline m_q^ID(ς) = sup M_q^ID(ς). A gate is sign-identified as passing if its lower bound is positive. It is sign-identified as failing if its upper bound is nonpositive. If the interval crosses zero, the available design does not determine that gate's outcome.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a0cf04363c3e277e","tex":"\\underline r","aliases":"underline r underline","display":"\\underline r","kind":"use","excerpt":"For robustness across specifications, define the realization-level robust margin r_ S( R) = inf_ς∈ S min_q∈ J_* m_q(W_ R;ς) and its identified range underline r_ID, S = inf_ R∈ R_ID r_ S( R), overline r_ID, S = sup_ R∈ R_ID r_ S( R).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0cb8044da032aab8","tex":"\\underline r_{\\mathrm{ID},\\mathfrak S}","aliases":"underline r_ID, S underline mathrm mathfrak","display":"\\underline r_{\\mathrm{ID},\\mathfrak S}","kind":"use","excerpt":"For robustness across specifications, define the realization-level robust margin r_ S( R) = inf_ς∈ S min_q∈ J_* m_q(W_ R;ς) and its identified range underline r_ID, S = inf_ R∈ R_ID r_ S( R), overline r_ID, S = sup_ R∈ R_ID r_ S( R).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a84c807a5a80beb7","tex":"\\underline\\alpha","aliases":"underlineα underline alpha","display":"\\underline\\alpha","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bd11ffa7d66071e4","tex":"\\underline\\Delta","aliases":"underlineΔ underline Delta","display":"\\underline\\Delta","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-957a711f337108d8","tex":"\\underline\\Delta_B","aliases":"underlineΔ_B underline Delta","display":"\\underline\\Delta_B","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5330d0c979382245","tex":"\\underline\\eta","aliases":"underlineη underline eta","display":"\\underline\\eta","kind":"use","excerpt":"Let Θ = ( ε_coh, ε_align, ε_k, underlineη_c, underlineη_loop, δ_node, δ_null, δ_spec, k, ε_pers ). Let ς = ( D,Y^+,ℓ,τ, I, Q,Θ) be the complete declared scientific specification, where Q⊆ N_pred(P) is a validated null panel. A dynamic realization contains a certified stratified relational workspace when there exists a candidate W_t satisfying all g","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-20fec49af00fd046","tex":"\\underline\\eta_c","aliases":"underlineη_c underline eta","display":"\\underline\\eta_c","kind":"use","excerpt":"Let Θ = ( ε_coh, ε_align, ε_k, underlineη_c, underlineη_loop, δ_node, δ_null, δ_spec, k, ε_pers ). Let ς = ( D,Y^+,ℓ,τ, I, Q,Θ) be the complete declared scientific specification, where Q⊆ N_pred(P) is a validated null panel. A dynamic realization contains a certified stratified relational workspace when there exists a candidate W_t satisfying all g","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4ff9cbd364a56abb","tex":"\\underline\\eta_{\\mathrm{loop}}","aliases":"underlineη_loop underline eta mathrm","display":"\\underline\\eta_{\\mathrm{loop}}","kind":"use","excerpt":"Let Θ = ( ε_coh, ε_align, ε_k, underlineη_c, underlineη_loop, δ_node, δ_null, δ_spec, k, ε_pers ). Let ς = ( D,Y^+,ℓ,τ, I, Q,Θ) be the complete declared scientific specification, where Q⊆ N_pred(P) is a validated null panel. A dynamic realization contains a certified stratified relational workspace when there exists a candidate W_t satisfying all g","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4f5fdd0c5c090b30","tex":"\\underline\\mu","aliases":"underlineμ underline mu","display":"\\underline\\mu","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bb47e48c7bb1d69b","tex":"\\mathsf{uniform\\_bounds}","aliases":"uniform_bounds mathsf","display":"\\mathsf{uniform\\_bounds}","kind":"use","excerpt":"The machine-readable record should include: SpecificationRecord ={ · reference, family, equivalence_classes, metric, · targets, losses, horizons, interventions, · null_panels, thresholds, selection_history, · identified_sets, uniform_bounds, gate_envelopes }. Every omitted family member requires a reason. Every changed specification requires a data","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"use","excerpt":"For a declared graph, set of core dimensions, weights, and transfer maps, the static analysis can be executed in the following order: item Assemble the sparse block operator D_H from the edge equations sqrtw_e(c_t(e)-H_ec_s(e)). item Compute a singular-value decomposition D_H=UΣ V^*, λ_k(L_H)=σ_k(D_H)^2. Use a declared numerical tolerance to distin","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cb9a91770e93fe45","tex":"V_{\\mathcal D}","aliases":"V_ D mathcal","display":"V_{\\mathcal D}","kind":"use","excerpt":"A more stable quantity measures primitive carrier cost. Let w_ω>0 be a declared cost for each ω∈Ω, and assign a block the cost c(B)=∑_ω∈ Bw_ω. Define the sufficient support cost s_ε( D) = min_S⊆ V_ D { ∑_B∈ Sc(B): Risk_ D(S) ≤ Risk_ D(V_ D)+ε }. If D_fpreceq D_c is information faithful, then s_ε( D_f) ≤ s_ε( D_c). The fine decomposition permits eve","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"use","excerpt":"For two consecutive transfers, P_i_2T_2T_1P_i_0 = H_2H_1+U_2V_1, Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-206f471c412d574c","tex":"v_A","aliases":"v_A","display":"v_A","kind":"use","excerpt":"Order the local coordinates as x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^ T. The overlap discrepancies are g_H-g_M, p_M-p_X, v_X-v_A, d_A-d_H. Thus D_HAI = 1 · 0 · -1 · 0 · 0 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d63adcc380863b98","tex":"V_{e_r}","aliases":"V_e_r","display":"V_{e_r}","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b68c595a4fdad02b","tex":"V_{\\mathrm{HAI}}","aliases":"V_HAI mathrm","display":"V_{\\mathrm{HAI}}","kind":"use","excerpt":"Consider four heterogeneous components: V_HAI={H,M,X,A}, where: item H is a human researcher who maintains goals and semantic direction; item M is a language-model process that constructs local formalizations; item X is an execution and verification process that compiles, tests, or falsifies candidate artifacts; and item A is a persistent artifact ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-154de0132c3e6b89","tex":"V_{\\mathrm{HH}}","aliases":"V_HH mathrm","display":"V_{\\mathrm{HH}}","kind":"use","excerpt":"Consider two researchers and one persistent notebook: V_HH={P,C,N}, where: item P is a proposer who tracks the theorem statement and open proof obligations; item C is a critic who tracks proof obligations and falsification status; item N is a notebook that tracks the theorem statement and falsification status. The abstract core is c=(q,p,f)^ T∈ R^3","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8b86c046f5cd21ee","tex":"V_m","aliases":"V_m","display":"V_m","kind":"use","excerpt":"Let the vertex set of a core-transfer diagram be partitioned as V=V_1sqcup⋯sqcup V_m. Write the transfer discrepancy as D_H = D_int","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5ffe3c14a24decf3","tex":"V_{\\mathrm{MA}}","aliases":"V_MA mathrm","display":"V_{\\mathrm{MA}}","kind":"use","excerpt":"Consider three agents: V_MA={P,M,R}, called the planner, monitor, and recorder. The global workspace-control state is c_t=(g_t,b_t,h_t)∈ F_2^3, where: ll g_t=goal bit, · b_t=environment or belief bit,","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1b5f238ca9baf774","tex":"V_r","aliases":"V_r","display":"V_r","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7be15d086b46f651","tex":"v_t","aliases":"v_t","display":"v_t","kind":"use","excerpt":"[Mean-square persistence] Suppose E[e_t+1^2mid F_t] ≤ q e_t^2+v_t, 0≤ q<1, and E[v_t]≤bar v. Then E[e_t^2] ≤ q^t E[e_0^2] + fracbar v(1-q^t)1-q, so limsup_t→∞ E[e_t^2] ≤ fracbar v1-q.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8614e37cf86e668d","tex":"v_X","aliases":"v_X","display":"v_X","kind":"use","excerpt":"Order the local coordinates as x=(g_H,d_H,g_M,p_M,p_X,v_X,d_A,v_A)^ T. The overlap discrepancies are g_H-g_M, p_M-p_X, v_X-v_A, d_A-d_H. Thus D_HAI = 1 · 0 · -1 · 0 · 0 · 0 · 0 · 0","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fb52b1bfac9cf230","tex":"\\mathcal V_\\Theta","aliases":"V_Θ mathcal Theta","display":"\\mathcal V_\\Theta","kind":"use","excerpt":"_boundary,t · =Δ_t^B-underlineΔ_B. The aggregate integrity margin is v_t = min_j∈ J_maintϕ_j,t. The workspace passes the protected gates when v_t≥0. To maintain a reserve ρ>0, define the maintenance deficit e_t =(ρ-v_t)_+. Define the zero-reserve and bounded-deficit viability sets V_Θ = { S:ϕ_j(S)≥0 for every j∈ J_maint } V_Θ,e_max = { S:e(S)≤ e_ma","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7066a3daa1ca0ae8","tex":"\\mathcal V_{\\Theta,e_{\\max}}","aliases":"V_Θ,e_max mathcal Theta max","display":"\\mathcal V_{\\Theta,e_{\\max}}","kind":"use","excerpt":"t = min_j∈ J_maintϕ_j,t. The workspace passes the protected gates when v_t≥0. To maintain a reserve ρ>0, define the maintenance deficit e_t =(ρ-v_t)_+. Define the zero-reserve and bounded-deficit viability sets V_Θ = { S:ϕ_j(S)≥0 for every j∈ J_maint } V_Θ,e_max = { S:e(S)≤ e_max } The first set requires every protected gate to pass. The second per","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"use","excerpt":"Equivalently, Γ(H)=varprojlim_Path(G)C_i. A section is not a single vector copied verbatim at every vertex. It is one abstract element represented by a family of local vectors that agree under the specified translation maps.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3d7fd0d4b2e4cac7","tex":"\\varprojlim_{\\Path(G)}","aliases":"varprojlim_Path(G) varprojlim Path","display":null,"kind":"use","excerpt":"Equivalently, Γ(H)=varprojlim_Path(G)C_i. A section is not a single vector copied verbatim at every vertex. It is one abstract element represented by a family of local vectors that agree under the specified translation maps.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-44cf5118c3be9cba","tex":"\\vdots","aliases":"vdots","display":"\\vdots","kind":"use","excerpt":"[Finite matrix test] Let γ_1,…,γ_r be fundamental cycle generators and suppose dim C_o=d. Define the stacked obstruction matrix M = H_γ_1-I vdots H_γ_r-I . Then C_*=ker M, d_Workspace=d-rank( M).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0fef838539965a6b","tex":"\\vfill","aliases":"vfill","display":null,"kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e83c58309eb2060b","tex":"\\mathcal W","aliases":"W mathcal","display":"\\mathcal W","kind":"use","excerpt":"Let P_t project onto a selected rank-d low-energy workspace cluster of L_t, and set W_t=ran P_t. Write g_t=λ_d+1(L_t)-λ_d(L_t)>0 for the separating spectral gap and η_t^L=2 D_t δ_t^D+(δ_t^D)^2.","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e9fe8ee5ada9c8bf","tex":"W'","aliases":"W'","display":"W'","kind":"use","excerpt":"Let two realizations have identical benchmark output distributions, P(Y_bench)=P'(Y_bench), but different intervention kernels: K(w,b)≠ K'(β w,b) for some matched workspace intervention. Then task behavior is preserved while d_causal(W,W')>0. The later system may be a successor or replacement rather than the same workspace under the identity criter","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3a85ccd36373cc59","tex":"\\mathcal W'","aliases":"W' mathcal","display":"\\mathcal W'","kind":"use","excerpt":"Write A=ρ_V(a), A'=ρ_V'(a), and define δ_ρ(a)= A'J_V-J_VA , ℓ_a= (I-P)AP . Let A_ W=PAP, A_ W''=P'A'P'.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-780ad0c0d35ab89b","tex":"W'_{\\mathrm{init}}","aliases":"W'_init mathrm","display":"W'_{\\mathrm{init}}","kind":"use","excerpt":"A reconstructive handoff may also be imperfect. Let δ_rec(β) = E[d_ W'(W'_init,β W)] measure replacement initialization error. Corollary~[source reference] provides one explicit upper bound on this term.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9231de7d5e2a9c2d","tex":"W^{\\mathcal D}","aliases":"W^ D mathcal","display":"W^{\\mathcal D}","kind":"use","excerpt":"Fix a reference decomposition D_0. For each D∈ D, let Φ_ D:W^ D_0⟶ W^ D be a declared workspace correspondence and let δ_ D measure its structural and causal identity defect. Assume gate q has a transport bound | m_q(Φ_ DW; D) -m_q(W; D_0) | ≤ L_qδ_ D+b_q( D), where L_q is a sensitivity constant and b_q records model mismatch not captured by the id","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ebf9a4ac3038d433","tex":"W^{\\mathcal D_0}","aliases":"W^ D_0 mathcal","display":"W^{\\mathcal D_0}","kind":"use","excerpt":"Fix a reference decomposition D_0. For each D∈ D, let Φ_ D:W^ D_0⟶ W^ D be a declared workspace correspondence and let δ_ D measure its structural and causal identity defect. Assume gate q has a transport bound | m_q(Φ_ DW; D) -m_q(W; D_0) | ≤ L_qδ_ D+b_q( D), where L_q is a sensitivity constant and b_q records model mismatch not captured by the id","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7210c42e82a40f55","tex":"W^{(r)}","aliases":"W^(r)","display":"W^{(r)}","kind":"use","excerpt":"For appropriate recurrent systems, W^(r+1) = Γ_ε_r(W^(r)) may converge to a stable workspace type: d_id ( W^(r+1),W^(r) ) →0.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d70d69bdd8cbabd9","tex":"W^{(r+1)}","aliases":"W^(r+1)","display":"W^{(r+1)}","kind":"use","excerpt":"For appropriate recurrent systems, W^(r+1) = Γ_ε_r(W^(r)) may converge to a stable workspace type: d_id ( W^(r+1),W^(r) ) →0.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-33e7e01949eaa198","tex":"W_{\\mathfrak R'}","aliases":"W_ R' mathfrak","display":"W_{\\mathfrak R'}","kind":"use","excerpt":"[No uniformly correct decision under experimental equivalence] Let R, R'∈ M satisfy R≡_ E R' but Cert(W_ R;ς) ≠ Cert(W_ R';ς). Then the workspace certificate is not identifiable on M under E. More strongly, for every possibly randomized decision rule widehat C based only on data generated by E, P_ R ( widehat C≠ C_ R ) + P_ R' ( widehat C≠ C_ R' ) ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"[Coherence-to-core distance estimate] Let W_0=Γ(H)=ker L_H, r=dim W_0, and suppose r<N. Set γ_H=λ_r+1>0. Then every c∈ C_V satisfies dist(c,W_0)^2 ≤ frac E_H(c)γ_H. Moreover, E_H(c) ≤ L_H dist(c,W_0)^2.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-22ed30e13447582d","tex":"W_0^\\perp","aliases":"W_0^⊥ perp","display":"W_0^\\perp","kind":"use","excerpt":"Write c=P_0c+Q_0c, where P_0 projects onto W_0 and Q_0=I-P_0. Since L_HP_0=0, E_H(c) = ⟨ Q_0c,L_HQ_0c⟩. The spectrum of L_H on W_0^⊥ lies in [γ_H, L_H ]. Hence γ_H Q_0c ^2 ≤ E_H(c) ≤ L_H Q_0c ^2. Finally, Q_0c =dist(c,W_0).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9bf2f69e44430b5b","tex":"W_{12}","aliases":"W_12","display":"W_{12}","kind":"use","excerpt":"Let G be the directed triangle 1xrightarrow a 2 xrightarrow b 3 xrightarrow c 1 with one-dimensional vertex cores. Cover it by its three edge charts. Their local workspaces are all one-dimensional: mathscr W_12 · ={(x,ax):x∈ R}, mathscr W_23 · ={(y,by):y∈ R}, mathscr W_31 · ={(z,cz):z∈ R}. Overlap agreement requires y=ax, z=by, x=cz. Therefore (1-a","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f88228c4c64b5034","tex":"W_{23}","aliases":"W_23","display":"W_{23}","kind":"use","excerpt":"Let G be the directed triangle 1xrightarrow a 2 xrightarrow b 3 xrightarrow c 1 with one-dimensional vertex cores. Cover it by its three edge charts. Their local workspaces are all one-dimensional: mathscr W_12 · ={(x,ax):x∈ R}, mathscr W_23 · ={(y,by):y∈ R}, mathscr W_31 · ={(z,cz):z∈ R}. Overlap agreement requires y=ax, z=by, x=cz. Therefore (1-a","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c70ce8ab6c90904","tex":"W_{31}","aliases":"W_31","display":"W_{31}","kind":"use","excerpt":"Let G be the directed triangle 1xrightarrow a 2 xrightarrow b 3 xrightarrow c 1 with one-dimensional vertex cores. Cover it by its three edge charts. Their local workspaces are all one-dimensional: mathscr W_12 · ={(x,ax):x∈ R}, mathscr W_23 · ={(y,by):y∈ R}, mathscr W_31 · ={(z,cz):z∈ R}. Overlap agreement requires y=ax, z=by, x=cz. Therefore (1-a","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0289b7ef692ee27e","tex":"W_a","aliases":"W_a","display":"W_a","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8e80dcfdce0dc44d","tex":"W_b","aliases":"W_b","display":"W_b","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2e8d1f4c9da76d8f","tex":"W_{B_t^{\\mathrm{sys}}}","aliases":"W_B_t^sys mathrm","display":"W_{B_t^{\\mathrm{sys}}}","kind":"use","excerpt":"Let B_t^sys be an internal membership or routing state used by the system, and let S_t^* = argmax_S∈ S_pass B_t(S) be the population boundary defined by the external certification protocol. The strong endogenous boundary conjecture asserts that there are adaptive systems and nontrivial environments for which: · d_B(B_t^sys,S_t^*)→0, · Form_Θ_form(W","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c7a6c577c0092b1b","tex":"W_{bc}","aliases":"W_bc","display":"W_{bc}","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c881f6c7443f9e3c","tex":"W_d","aliases":"W_d","display":"W_d","kind":"use","excerpt":"An equivalent index-based description fixes d and writes W_d=ran P_d, P_d=P_{λ_1,…,λ_d}(L_H), with cluster gap g_d=λ_d+1-λ_d>0. The index-based form is convenient for perturbation theorems because the workspace rank is declared explicitly.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-20f2ecbd884ef38b","tex":"w_D","aliases":"w_D","display":"w_D","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d7027e372c0b944e","tex":"W_{\\mathrm{flat}}","aliases":"W_flat mathrm","display":"W_{\\mathrm{flat}}","kind":"use","excerpt":"Let L( T) be the leaf set. Repeated edge completeness implies that the flattened family U_flat = {G_ℓ:ℓ∈ L( T)} is itself an edge-complete cover of G. Let δ_flat^0 be its overlap coboundary and define W_flat = kerδ_flat^0. Theorem~[source reference] and Theorem~[source reference] give W_r ≅ W_flat ≅ mathscr W(G). Equivalently, Lim_ T^rec ( {mathscr","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4caa181d23d35511","tex":"w_I","aliases":"w_I","display":"w_I","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c56b8f530e941c8f","tex":"W_{\\mathrm{meta}}","aliases":"W_meta mathrm","display":"W_{\\mathrm{meta}}","kind":"use","excerpt":"Define the exact meta-workspace mathscr W_meta( U) := ker B_ U ⊆ ⊕_αmathscr W_α. Theorem~[source reference] gives the canonical identification mathscr W_meta( U) ≅ mathscr W(G). Thus exact formation of a workspace from local workspaces does not create an additional mysterious carrier. It gives a second presentation of the same global compatible-sec","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2b7f78e3b57fec58","tex":"w_N","aliases":"w_N","display":"w_N","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4a4f3028bb492bb2","tex":"W_r","aliases":"W_r","display":"W_r","kind":"use","excerpt":"[Exact hierarchical collapse] For every finite workspace hierarchy as above, there is a canonical isomorphism W_a≅mathscr W(G_a) for every node a. In particular, W_r≅mathscr W(G). Therefore recursively gluing local workspaces, using any other finite parenthesization built from edge-complete intermediate covers that retain all required overlap objec","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c79a06c2ccf82b94","tex":"W_{\\mathrm{root}}","aliases":"W_root mathrm","display":"W_{\\mathrm{root}}","kind":"use","excerpt":"For every finite rooted hierarchy whose children edge-cover their parent, W_a≅mathscr W(G_a) for every hierarchy node a, and in particular W_root≅mathscr W(G). If level r contributes defect δ_r and downstream collapse maps have Lipschitz constants L_s, then δ_flat^(R) ≤ ∑_r=0^R-1 ( ∏_s=r+1^R-1L_s )δ_r.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-daaaeea663ec3318","tex":"W_S","aliases":"W_S","display":"W_S","kind":"use","excerpt":"S, define Coh(S) = tr ( P_S,dL_SP_S,d ), where P_S,d projects onto the selected low-energy workspace modes. Let Cut(S) = ∑_e:s(e)∈ S, t(e)∉ Sw_e + ∑_e:s(e)∉ S, t(e)∈ Sw_e. Let k_ε(S) be the minimal sufficient coalition size inside the candidate subsystem. A candidate is admissible only if SRW_Θ_S(W_S)=1, k_ε(S)≥2, unless the claim explicitly concer","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8d28b879da5f9453","tex":"\\mathcal W_t","aliases":"W_t mathcal","display":"\\mathcal W_t","kind":"use","excerpt":"Let P_t project onto a selected rank-d low-energy workspace cluster of L_t, and set W_t=ran P_t. Write g_t=λ_d+1(L_t)-λ_d(L_t)>0 for the separating spectral gap and η_t^L=2 D_t δ_t^D+(δ_t^D)^2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-29ac20d9e18358ef","tex":"\\mathsf W_t'","aliases":"W_t' mathsf","display":"\\mathsf W_t'","kind":"use","excerpt":"[Coordinate invariance of intervention effects] Let g: W_t→ W_t' be a measurable bijection and suppose the reparametrized intervention family satisfies ι_g(w)'=ι_w. Then every divergence-based causal effect is preserved: CE_Workspace'(g(w),g(w');b) = CE_Workspace(w,w';b).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a5f4ec73f00b8fb2","tex":"W_{t+1}","aliases":"W_t+1","display":"W_{t+1}","kind":"use","excerpt":"C_t^ctl = ( W_t,G_t,M_t,B_t,E_t ), where G_t is the represented goal, M_t is a workspace-integrity monitor, B_t is the current boundary or routing state, and E_t is the declared external context. An action is generated by A_t ∼ π_t ( ·mid W_t,G_t,M_t,B_t,E_t ). The future controlled state obeys a time-unrolled structural model S_t+1 = F_t(S_t,A_t,E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bce83a108fed0a94","tex":"\\mathcal W_{t+1}","aliases":"W_t+1 mathcal","display":"\\mathcal W_{t+1}","kind":"use","excerpt":"Low off-workspace error is not sufficient for meaningful persistence. The zero update F_t=0 drives every state exactly into every linear workspace while destroying all content. Define the internal transport B_t=P_t+1F_tP_t: W_t⟶ W_t+1 and its non-collapse margin μ_t^keep = σ_min(B_t|_ W_t). A content-preserving persistence claim requires μ_t^keep>0","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1363951f87a46dce","tex":"W_{t+\\tau}^{r}","aliases":"W_t+τ^r tau","display":"W_{t+\\tau}^{r}","kind":"use","excerpt":"Restoration of performance is not automatically restoration of identity. Let W_t^- be the last certified pre-damage workspace and let W_t+τ^r be the post-repair workspace. Define the repair identity defect δ_rep-id(r) = d_id ( W_t^-,W_t+τ^r ). A repair action is identity preserving at tolerance ε_id when Δ_t^rep(r;d)>0, δ_rep-id(r) ≤ ε_id. When int","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b7ecd9a0a4dd936b","tex":"W_{t^-}","aliases":"W_t^-","display":"W_{t^-}","kind":"use","excerpt":"Restoration of performance is not automatically restoration of identity. Let W_t^- be the last certified pre-damage workspace and let W_t+τ^r be the post-repair workspace. Define the repair identity defect δ_rep-id(r) = d_id ( W_t^-,W_t+τ^r ). A repair action is identity preserving at tolerance ε_id when Δ_t^rep(r;d)>0, δ_rep-id(r) ≤ ε_id. When int","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2c6d80800d97a158","tex":"W_t^{\\mathrm{str}}","aliases":"W_t^str mathrm","display":"W_t^{\\mathrm{str}}","kind":"use","excerpt":"Let c(t)=(c_i(t))_i∈ V ∈ C_V(t), P_t= 1_[0,ε_t](L_t). The instantaneous structural workspace coordinate is W_t^str=P_tc(t) ∈ Γ_ε_t(H(t)). When P_t varies with time, these coordinates inhabit a moving family of subspaces. Parts~X--XII will supply the transport required to compare them across time. In what follows, all sufficiency and intervention st","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d122ac7392c2126f","tex":"w_Y","aliases":"w_Y","display":"w_Y","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b90de78a5bfaa7d2","tex":"W_\\alpha","aliases":"W_α alpha","display":"W_\\alpha","kind":"use","excerpt":"For pairwise overlaps write mathscr W_α=mathscr W(G_α), mathscr W_αβ=mathscr W(G_αβ), and r_αβ^α: mathscr W_α→mathscr W_αβ, r_αβ^β: mathscr W_β→mathscr W_αβ. The interface is therefore a cospan mathscr W_α xrightarrow r_αβ^α mathscr W_αβ xleftarrow r_αβ^β mathscr W_β, not necessarily a direct transfer map from one complete local workspace to the ot","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d8a35fcdffee6031","tex":"w_\\alpha","aliases":"w_α alpha","display":"w_\\alpha","kind":"use","excerpt":"the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a local family w=(w_α)_α, (δ^0w)_αβ = r_αβ^βw_β - r_αβ^αw_α. Thus δ^0 is the overlap-discrepancy operator at the level of exact local workspaces","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-46f0f9dca905844b","tex":"W_{\\alpha\\beta}","aliases":"W_αβ alpha beta","display":"W_{\\alpha\\beta}","kind":"use","excerpt":"For pairwise overlaps write mathscr W_α=mathscr W(G_α), mathscr W_αβ=mathscr W(G_αβ), and r_αβ^α: mathscr W_α→mathscr W_αβ, r_αβ^β: mathscr W_β→mathscr W_αβ. The interface is therefore a cospan mathscr W_α xrightarrow r_αβ^α mathscr W_αβ xleftarrow r_αβ^β mathscr W_β, not necessarily a direct transfer map from one complete local workspace to the ot","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-86b253a8850aab10","tex":"W_\\beta","aliases":"W_β beta","display":"W_\\beta","kind":"use","excerpt":"For pairwise overlaps write mathscr W_α=mathscr W(G_α), mathscr W_αβ=mathscr W(G_αβ), and r_αβ^α: mathscr W_α→mathscr W_αβ, r_αβ^β: mathscr W_β→mathscr W_αβ. The interface is therefore a cospan mathscr W_α xrightarrow r_αβ^α mathscr W_αβ xleftarrow r_αβ^β mathscr W_β, not necessarily a direct transfer map from one complete local workspace to the ot","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1d9a65014a7810c6","tex":"w_\\beta","aliases":"w_β beta","display":"w_\\beta","kind":"use","excerpt":"ndary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a local family w=(w_α)_α, (δ^0w)_αβ = r_αβ^βw_β - r_αβ^αw_α. Thus δ^0 is the overlap-discrepancy operator at the level of exact local workspaces.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9eec9d81853c9da7","tex":"w_\\Theta","aliases":"w_Θ Theta","display":"w_\\Theta","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5704b00361e1b378","tex":"W_\\lambda","aliases":"W_λ lambda","display":"W_\\lambda","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-41c22918a535e847","tex":"W_\\Pi","aliases":"W_Π Pi","display":"W_\\Pi","kind":"use","excerpt":"D_∂ , D_int = ⊕_α=1^mD_α, where D_α contains the edge constraints internal to V_α, and D_∂ contains the constraints crossing between blocks. Define the internally coherent macro-state spaces W_α=ker D_α, W_Π=⊕_α=1^mW_α. Let J_Π:W_Π↪ C_V be the canonical isometric inclusion, and define the quotient discrepancy D_Π=D_∂J_Π. The quotient does not repla","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-712e1d375bc90aff","tex":"W_{\\Pi}","aliases":"W_Π Pi","display":"W_{\\Pi}","kind":"use","excerpt":"D_∂ , D_int = ⊕_α=1^mD_α, where D_α contains the edge constraints internal to V_α, and D_∂ contains the constraints crossing between blocks. Define the internally coherent macro-state spaces W_α=ker D_α, W_Π=⊕_α=1^mW_α. Let J_Π:W_Π↪ C_V be the canonical isometric inclusion, and define the quotient discrepancy D_Π=D_∂J_Π. The quotient does not repla","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cd0932e90c45452a","tex":"W_{\\varsigma}","aliases":"W_ς varsigma","display":"W_{\\varsigma}","kind":"use","excerpt":"Changing the decomposition can alter the coordinate realization of the workspace, while changing the target or horizon can alter the relevant predictive state. Let Φ_ς:W_ς_0⟶ W_ς be the declared correspondence when such transport is needed. For each gate, assume a bound | m_q(Φ_ςW;ς) -m_q(W;ς_0) | ≤ L_qd_ S(ς,ς_0) +b_q(ς), where b_q records mismatc","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f619b63fe80d72e7","tex":"W_{\\varsigma_0}","aliases":"W_ς_0 varsigma","display":"W_{\\varsigma_0}","kind":"use","excerpt":"Changing the decomposition can alter the coordinate realization of the workspace, while changing the target or horizon can alter the relevant predictive state. Let Φ_ς:W_ς_0⟶ W_ς be the declared correspondence when such transport is needed. For each gate, assume a bound | m_q(Φ_ςW;ς) -m_q(W;ς_0) | ≤ L_qd_ S(ς,ς_0) +b_q(ς), where b_q records mismatc","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-54be262c941bfd00","tex":"w_\\tau","aliases":"w_τ tau","display":"w_\\tau","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-375e643e02d899a4","tex":"w_\\omega","aliases":"w_ω omega","display":"w_\\omega","kind":"use","excerpt":"A more stable quantity measures primitive carrier cost. Let w_ω>0 be a declared cost for each ω∈Ω, and assign a block the cost c(B)=∑_ω∈ Bw_ω. Define the sufficient support cost s_ε( D) = min_S⊆ V_ D { ∑_B∈ Sc(B): Risk_ D(S) ≤ Risk_ D(V_ D)+ε }. If D_fpreceq D_c is information faithful, then s_ε( D_f) ≤ s_ε( D_c). The fine decomposition permits eve","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-404fc764d07fb6f9","tex":"W_\\ell","aliases":"W_ℓ ell","display":"W_\\ell","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6f2779cd2e16044c","tex":"w_\\ell","aliases":"w_ℓ ell","display":"w_\\ell","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":") is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal policy π^*, E_a∼π^*[Q(a)] - E_a∼widehatπ[Q(a)] ≤ 2ε_Q+δ_opt. For a deterministic greedy action widehat a∈argmax_awidehat Q(a), Q(a^*)-Q(widehat a) ≤2ε_Q. If the true optimal action a^* is unique and its value gap Δ_Q ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5932b4272d1487fe","tex":"\\widehat{\\mathcal B}","aliases":"widehat B widehat mathcal","display":"\\widehat{\\mathcal B}","kind":"use","excerpt":"· -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass contains only candidates that satisfy the required workspace and anti-centralization gates.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c30fce6153cc3b45","tex":"\\widehat{\\mathcal B}_t","aliases":"widehat B_t widehat mathcal","display":"\\widehat{\\mathcal B}_t","kind":"use","excerpt":"· -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass contains only candidates that satisfy the required workspace and anti-centralization gates.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"use","excerpt":"m:workspace-nonidentifiability Let R, R'∈ M satisfy R≡_ E R' but Cert(W_ R;ς) ≠ Cert(W_ R';ς). Then the workspace certificate is not identifiable on M under E. More strongly, for every possibly randomized decision rule widehat C based only on data generated by E, P_ R ( widehat C≠ C_ R ) + P_ R' ( widehat C≠ C_ R' ) =1, where C_ R=Cert(W_ R;ς). Con","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e7a695b8f5409721","tex":"\\widehat c","aliases":"widehat c widehat","display":"\\widehat c","kind":"use","excerpt":"[Stable core reconstruction] Assume α_S>0 and observe y=A_Sc+η. With the canonical estimate widehat c=A_S^† y, one has widehat c-c ≤ frac η sqrtα_S.","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-06a316492d618f6f","tex":"\\widehat{\\mathcal C}","aliases":"widehat C widehat mathcal","display":"\\widehat{\\mathcal C}","kind":"use","excerpt":"Local candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2f63676f647ed29d","tex":"\\widehat{\\mathcal C}_{\\mathcal U}","aliases":"widehat C_ U widehat mathcal mathcal","display":"\\widehat{\\mathcal C}_{\\mathcal U}","kind":"use","excerpt":"Local candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a7046ed830e3b1cc","tex":"\\widehat c_A","aliases":"widehat c_A widehat","display":"\\widehat c_A","kind":"use","excerpt":"[Uniform erasure resilience] The encoding family is exactly reconstructible after every loss of at most k nodes if and only if α^(k)>0. If surviving observations have the form y_V∖ A=A_V∖ Ac+η_A, |A|≤ k, then the canonical survivor decoder obeys the uniform estimate widehat c_A-c ≤ frac η_A sqrtα^(k).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d1655f35bb316a56","tex":"\\widehat D","aliases":"widehat D widehat","display":"\\widehat D","kind":"use","excerpt":"Local candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-54aaa860c55bcbd3","tex":"\\widehat D_{\\mathcal U}","aliases":"widehat D_ U widehat mathcal","display":"\\widehat D_{\\mathcal U}","kind":"use","excerpt":"Local candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4c34e6fafeaafe4d","tex":"\\widehat H^{(1)}","aliases":"widehat H^(1) widehat","display":"\\widehat H^{(1)}","kind":"use","excerpt":"Train the same formation rule under several scaffolds Σ^(1),…,Σ^(K) that differ in node labels, coordinate systems, and communication topology but support the same abstract task. After gauge alignment, measure whether the learned organizations transport to a common equivalence class: [widehat H^(1)]_ G ≅⋯≅ [widehat H^(K)]_ G. A structure that appea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b65b78996f6f5b6b","tex":"\\widehat H^{(K)}","aliases":"widehat H^(K) widehat","display":"\\widehat H^{(K)}","kind":"use","excerpt":"Train the same formation rule under several scaffolds Σ^(1),…,Σ^(K) that differ in node labels, coordinate systems, and communication topology but support the same abstract task. After gauge alignment, measure whether the learned organizations transport to a common equivalence class: [widehat H^(1)]_ G ≅⋯≅ [widehat H^(K)]_ G. A structure that appea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-17f902f29404b1e4","tex":"\\widehat H_\\tau","aliases":"widehat H_τ widehat tau","display":"\\widehat H_\\tau","kind":"use","excerpt":"mpirical-transfer-bound For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ .","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ae91cd5f0de975af","tex":"\\widehat J","aliases":"widehat J widehat","display":"\\widehat J","kind":"use","excerpt":"[Uniform structure-selection recovery] Let T be a finite candidate family with population score J: T→ R. Let θ_* = argmin_θ∈ TJ(θ) be unique, with margin Δ_J = min_θ≠θ_* (J(θ)-J(θ_*))>0. If a held-out estimate widehat J satisfies sup_θ∈ T |widehat J(θ)-J(θ)| ≤ε, then every empirical minimizer widehatθ obeys J(widehatθ)-J(θ_*) ≤2ε. If 2ε<Δ_J, then w","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a410e85313b42614","tex":"\\widehat M","aliases":"widehat M widehat","display":"\\widehat M","kind":"use","excerpt":"[Deterministic empirical recovery bound] For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ ","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-77fd09daba007b85","tex":"\\widehat m","aliases":"widehat m widehat","display":"\\widehat m","kind":"use","excerpt":"In practice, the reference margin is reduced by three distinct uncertainty budgets: B_q^total = B_q^model +B_q^spec +B_q^sample. The first covers variation across observationally compatible realizations. The second covers variation across admissible specifications. The third covers finite-sample estimation. The sufficient condition widehat m_q(W;ς_","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ae12aa3de1b22c70","tex":"\\widehat m_q","aliases":"widehat m_q widehat","display":"\\widehat m_q","kind":"use","excerpt":"In practice, the reference margin is reduced by three distinct uncertainty budgets: B_q^total = B_q^model +B_q^spec +B_q^sample. The first covers variation across observationally compatible realizations. The second covers variation across admissible specifications. The third covers finite-sample estimation. The sufficient condition widehat m_q(W;ς_","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-08ee7c9e6bd33ed5","tex":"\\widehat Q_n","aliases":"widehat Q_n widehat","display":"\\widehat Q_n","kind":"use","excerpt":"Let Q(λ) be a gate-specific population order parameter and widehat Q_n(λ) its held-out estimate. If sup_λ∈Λ_n |widehat Q_n(λ)-Q(λ)| ≤ε_n on a finite evaluation grid Λ_n, then every threshold crossing with vertical margin larger than 2ε_n is recovered by the same uniform-selection argument as Theorem~[source reference]. Near a continuous transition,","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-644e773747d7540b","tex":"\\widehat r","aliases":"widehat r widehat","display":"\\widehat r","kind":"use","excerpt":"[Repair selection under value-model error] Fix a damage intervention d and a finite feasible repair set R. Let V(r)=V_t^rep(r;d) be the true expected post-repair integrity margin and let widehat V obey sup_r∈ R|widehat V(r)-V(r)| ≤ ε_R. Let widehat r∈argmax_rwidehat V(r) and r^*∈argmax_r V(r). Then V(r^*)-V(widehat r) ≤ 2ε_R. If V(r^*) ≥ ρ+2ε_R, th","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aaf4c886bf12df25","tex":"\\widehat{\\mathcal R}","aliases":"widehat R widehat mathcal","display":"\\widehat{\\mathcal R}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-379d62b5ecf96b65","tex":"\\widehat{\\mathfrak R}","aliases":"widehat R widehat mathfrak","display":"\\widehat{\\mathfrak R}","kind":"use","excerpt":"In finite samples, the identified set is itself uncertain. Let widehat R_1-δ be a confidence region satisfying inf_ R_0∈ M P_ R_0 ( R_0∈ widehat R_1-δ ) ≥1-δ. A conservative robust lower margin is widehat r_1-δ = inf_ R∈widehat R_1-δ inf_ς∈ S min_q m_q(W_ R;ς). If widehat r_1-δ>0, a false specification-robust certificate has probability at most δ, ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4eff9e4a79b4d629","tex":"\\widehat r_{1-\\delta}","aliases":"widehat r_1-δ widehat delta","display":"\\widehat r_{1-\\delta}","kind":"use","excerpt":"In finite samples, the identified set is itself uncertain. Let widehat R_1-δ be a confidence region satisfying inf_ R_0∈ M P_ R_0 ( R_0∈ widehat R_1-δ ) ≥1-δ. A conservative robust lower margin is widehat r_1-δ = inf_ R∈widehat R_1-δ inf_ς∈ S min_q m_q(W_ R;ς). If widehat r_1-δ>0, a false specification-robust certificate has probability at most δ, ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6284be9a9d642a44","tex":"\\widehat{\\mathfrak R}_{1-\\delta}","aliases":"widehat R_1-δ widehat mathfrak delta","display":"\\widehat{\\mathfrak R}_{1-\\delta}","kind":"use","excerpt":"In finite samples, the identified set is itself uncertain. Let widehat R_1-δ be a confidence region satisfying inf_ R_0∈ M P_ R_0 ( R_0∈ widehat R_1-δ ) ≥1-δ. A conservative robust lower margin is widehat r_1-δ = inf_ R∈widehat R_1-δ inf_ς∈ S min_q m_q(W_ R;ς). If widehat r_1-δ>0, a false specification-robust certificate has probability at most δ, ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e4b03e9c0f759db5","tex":"\\widehat{\\mathcal R}_{\\mathrm{task}}","aliases":"widehat R_task widehat mathcal mathrm","display":"\\widehat{\\mathcal R}_{\\mathrm{task}}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-77449d53c5be9d11","tex":"\\widehat S","aliases":"widehat S widehat","display":"\\widehat S","kind":"use","excerpt":"· -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass contains only candidates that satisfy the required workspace and anti-centralization gates.","count":10},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-51489efa35fd84c6","tex":"\\widehat S_t","aliases":"widehat S_t widehat","display":"\\widehat S_t","kind":"use","excerpt":"· -a_KCut(S) -a_ZSize(S) -a_TInstability_t(S). The coefficients and all normalizations are part of the boundary problem. No single choice is universal. The selected boundary is widehat S_t ∈ argmax_S∈ S_pass widehat B_t(S), where S_pass contains only candidates that satisfy the required workspace and anti-centralization gates.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2c8131c596a2a289","tex":"\\widehat u","aliases":"widehat u widehat","display":"\\widehat u","kind":"use","excerpt":"Let m_i=|{α:i∈ V_α}| be the multiplicity of vertex i. Then J_ U^*J_ U=M_ U, (M_ Uc)_i=m_ic_i. The pseudoinverse performs coordinatewise averaging: (J_ U^† u)_i = frac1m_i ∑_α:i∈ V_α(u_α)_i. Let widehat u = P_ker K_ Uu = J_ Uwidehat c. Then widehat c = J_ U^†widehat u and u-J_ Uwidehat c ≤ frac K_ Uu sqrtγ_ U^tot. If m_min=min_i m_i, then widehat c-","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-15ccf04c00615671","tex":"\\widehat W","aliases":"widehat W widehat","display":"\\widehat W","kind":"use","excerpt":"[Reconstruction-to-risk transfer] Let W be sufficient for a target Y, let losses take values in [0,1], and let a coalition construct widehat W from its local data. Then its optimal Bayes risk satisfies 0≤ Risk(S)-Risk(W) ≤ 2L_K E[d_W(W,widehat W)].","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c17de24c7659dced","tex":"\\widehat w","aliases":"widehat w widehat","display":"\\widehat w","kind":"use","excerpt":"For every pair w,widehat w, using the Bayes action for K_widehat w under the true law K_w incurs conditional regret at most 2TV(K_w,K_widehat w). This follows by inserting and subtracting the risk under K_widehat w for both the chosen action and the Bayes action for K_w. Average the bound and use the Lipschitz assumption. The coalition's optimal ru","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c0c24272fe4ccfa6","tex":"\\widehat{\\underline m}","aliases":"widehatunderline m widehat underline","display":"\\widehat{\\underline m}","kind":"use","excerpt":"Let widehat m_q(ς) estimate m_q(W;ς). Suppose there are nonnegative error radii ε_q(n,δ) such that P( sup_ς∈ S | widehat m_q(ς)-m_q(ς) | ≤ ε_q(n,δ) for all q∈ J_* ) ≥1-δ. Define the empirical lower confidence envelope widehatunderline m_q,1-δ^ S = inf_ς∈ S widehat m_q(ς) - ε_q(n,δ).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-62c61f926b20b30a","tex":"\\widehat{\\underline m}_{q,1-\\delta}^{\\mathfrak S}","aliases":"widehatunderline m_q,1-δ^ S widehat underline delta mathfrak","display":"\\widehat{\\underline m}_{q,1-\\delta}^{\\mathfrak S}","kind":"use","excerpt":"Let widehat m_q(ς) estimate m_q(W;ς). Suppose there are nonnegative error radii ε_q(n,δ) such that P( sup_ς∈ S | widehat m_q(ς)-m_q(ς) | ≤ ε_q(n,δ) for all q∈ J_* ) ≥1-δ. Define the empirical lower confidence envelope widehatunderline m_q,1-δ^ S = inf_ς∈ S widehat m_q(ς) - ε_q(n,δ).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-96f3a332d2da45df","tex":"\\widehat{\\alpha_j}","aliases":"widehatα_j widehat alpha","display":"\\widehat{\\alpha_j}","kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-95734e99a6bc63f2","tex":"\\widehat\\delta","aliases":"widehatδ widehat delta","display":"\\widehat\\delta","kind":"use","excerpt":"The notation absorbs lower-bounded and upper-bounded gates into one sign convention. If a causal effect must exceed c_0, then m_q=widehatη-c_0. If a defect must remain below ε_0, then m_q=ε_0-widehatδ. The complete certificate therefore passes exactly when every required signed margin is positive.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9bc4ac46f9980428","tex":"\\widehat\\eta","aliases":"widehatη widehat eta","display":"\\widehat\\eta","kind":"use","excerpt":"The notation absorbs lower-bounded and upper-bounded gates into one sign convention. If a causal effect must exceed c_0, then m_q=widehatη-c_0. If a defect must remain below ε_0, then m_q=ε_0-widehatδ. The complete certificate therefore passes exactly when every required signed margin is positive.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3d35f38f87e37bb6","tex":"\\widehat\\kappa","aliases":"widehatκ widehat kappa","display":"\\widehat\\kappa","kind":"use","excerpt":"[Deterministic empirical recovery bound] For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cc65d798d9e94322","tex":"\\widehat\\pi","aliases":"widehatπ widehat pi","display":"\\widehat\\pi","kind":"use","excerpt":"[Approximate counterfactual policy selection] Let A be a finite feasible action set. Let Q: A→ R be the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal poli","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-51ad55a59f818a04","tex":"\\widehat\\pi_t","aliases":"widehatπ_t widehat pi","display":"\\widehat\\pi_t","kind":"use","excerpt":"If the interventional action-value estimator satisfies sup_a|widehat Q_t(a)-Q_t(a)|≤ε_Q and the implemented policy has optimization error at most δ_opt relative to widehat Q_t, then Reg_t(widehatπ_t) ≤ 2ε_Q+δ_opt. For deterministic greedy action, the bound is 2ε_Q. A unique true optimum with value gap greater than 2ε_Q is recovered exactly. The gua","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-286fa0e262fa3e15","tex":"\\widehat\\Sigma","aliases":"widehatΣ widehat Sigma","display":"\\widehat\\Sigma","kind":"use","excerpt":"[Deterministic empirical recovery bound] For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7b5b3bf1143d049f","tex":"\\widehat\\varsigma","aliases":"widehatς widehat varsigma","display":"\\widehat\\varsigma","kind":"use","excerpt":"[Validity after independent specification selection] Let widehatς be selected using D_sel, and let D_cert be independent of D_sel. For every fixed ς∈ S, let ϕ_ς( D_cert)∈{0,1} be a certificate rule satisfying sup_P:Cert(W;ς)=0 P_P(ϕ_ς=1) ≤α. Then the selected-specification procedure satisfies sup_P P_P( ϕ_widehatς=1, Cert(W;widehatς)=0 ) ≤α.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-277fb593a51119f9","tex":"\\widehat\\Sigma_{XX}","aliases":"widehatΣ_XX widehat Sigma","display":"\\widehat\\Sigma_{XX}","kind":"use","excerpt":"[Deterministic empirical recovery bound] For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7043e54d7e915c22","tex":"\\widehat\\Sigma_{XX}^{-1}","aliases":"widehatΣ_XX^-1 widehat Sigma","display":"\\widehat\\Sigma_{XX}^{-1}","kind":"use","excerpt":"The empirical normal equations give widehat H-H_* = widehat MwidehatΣ_XX^-1. For ridge regression, widehat H_τ-H_* = (widehat M-τ H_*) (widehatΣ_XX+τ I)^-1. Taking Frobenius and operator norms yields both estimates.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ad91da91d6be3c40","tex":"\\widetilde F","aliases":"widetilde F widetilde","display":"\\widetilde F","kind":"use","excerpt":"[Approximate hierarchical collapse bound] Let F_r: H_r→ H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-681c5dfa39c3379d","tex":"\\widetilde F_0","aliases":"widetilde F_0 widetilde","display":"\\widetilde F_0","kind":"use","excerpt":"l H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(x) - F_R-1∘⋯∘ F_0(x) · ≤ ∑_r=0^R-1 ( ∏_s=r+1^R-1L_s ) ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82666f9be4bdaea6","tex":"\\widetilde F_r","aliases":"widetilde F_r widetilde","display":"\\widetilde F_r","kind":"use","excerpt":"[Approximate hierarchical collapse bound] Let F_r: H_r→ H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a50a05462256a843","tex":"\\widetilde F_{R-1}","aliases":"widetilde F_R-1 widetilde","display":"\\widetilde F_{R-1}","kind":"use","excerpt":"l H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(x) - F_R-1∘⋯∘ F_0(x) · ≤ ∑_r=0^R-1 ( ∏_s=r+1^R-1L_s ) ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-70fad85f996380ab","tex":"\\widetilde H","aliases":"widetilde H widetilde","display":"\\widetilde H","kind":"use","excerpt":"Suppose the edge maps are perturbed to widetilde H_e=H_e+Δ_e while the weights and vertex metrics remain fixed. Define δ_H^2 = max_i∈ V ∑_e:s(e)=iw_e Δ_e ^2. Let F=D_widetilde H-D_H. Then (Fc)_e=-sqrtw_e Δ_ec_s(e), F ≤δ_H. Since L_widetilde H-L_H =D_H^*F+F^*D_H+F^*F, we obtain L_widetilde H-L_H ≤ 2 D_H δ_H+δ_H^2.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c9af3dd79a6f33c7","tex":"\\widetilde H_e","aliases":"widetilde H_e widetilde","display":"\\widetilde H_e","kind":"use","excerpt":"Suppose the edge maps are perturbed to widetilde H_e=H_e+Δ_e while the weights and vertex metrics remain fixed. Define δ_H^2 = max_i∈ V ∑_e:s(e)=iw_e Δ_e ^2. Let F=D_widetilde H-D_H. Then (Fc)_e=-sqrtw_e Δ_ec_s(e), F ≤δ_H. Since L_widetilde H-L_H =D_H^*F+F^*D_H+F^*F, we obtain L_widetilde H-L_H ≤ 2 D_H δ_H+δ_H^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-468630464758f8a1","tex":"\\widetilde L","aliases":"widetilde L widetilde","display":"\\widetilde L","kind":"use","excerpt":"[Spectral workspace stability] Let L and widetilde L=L+E be self-adjoint positive semidefinite operators on the same N-dimensional space. Suppose λ_d(L)<λ_d+1(L), g_d=λ_d+1(L)-λ_d(L)>0, and η= E <fracg_d2. Let P_d and widetilde P_d project onto the first d eigenvectors of L and widetilde L, respectively. Then the rank-d cluster remains separated an","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3eb770a298a8059f","tex":"\\widetilde L_H","aliases":"widetilde L_H widetilde","display":"\\widetilde L_H","kind":"use","excerpt":"subtopicT_4: Spectral stability theorem If η= widetilde L_H-L_H <g_d/2, then widetilde P_d-P_d ≤ fracηg_d-η ≤ frac2ηg_d.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5f43b2813d6b7cc9","tex":"\\widetilde m","aliases":"widetilde m widetilde","display":"\\widetilde m","kind":"use","excerpt":"The resulting position is neither that decomposition is arbitrary nor that one partition is uniquely privileged. Decompositions are constrained by access, causal interfaces, intervention separability, temporal continuity, and explicit shared resources. A certificate established only at one interface is a valid but limited claim. A stronger claim re","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4dac9c585bd1b845","tex":"\\widetilde m_q","aliases":"widetilde m_q widetilde","display":"\\widetilde m_q","kind":"use","excerpt":"The resulting position is neither that decomposition is arbitrary nor that one partition is uniquely privileged. Decompositions are constrained by access, causal interfaces, intervention separability, temporal continuity, and explicit shared resources. A certificate established only at one interface is a valid but limited claim. A stronger claim re","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ed8c63057cfee2e2","tex":"\\widetilde Q","aliases":"widetilde Q widetilde","display":"\\widetilde Q","kind":"use","excerpt":"[Separated spectral subspaces] Let A and B=A+E be self-adjoint operators. Let P be an A-spectral projector whose eigenvalues lie in ( -∞,a], and let widetilde Q be a B-spectral projector whose eigenvalues lie in [b,∞), with b>a. Then Pwidetilde Q ≤ frac E b-a.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c77170c848a2e8b4","tex":"\\widetilde x","aliases":"widetilde x widetilde","display":"\\widetilde x","kind":"use","excerpt":"Let x_r and widetilde x_r be the exact and implemented states after r levels. Then widetilde x_r+1-x_r+1 · ≤ widetilde F_r(widetilde x_r)-F_r(widetilde x_r)","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cbc9fc1965c5cd2b","tex":"\\widetilde x_r","aliases":"widetilde x_r widetilde","display":"\\widetilde x_r","kind":"use","excerpt":"Let x_r and widetilde x_r be the exact and implemented states after r levels. Then widetilde x_r+1-x_r+1 · ≤ widetilde F_r(widetilde x_r)-F_r(widetilde x_r)","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-30b2ef859463967d","tex":"\\widetilde x_{r+1}","aliases":"widetilde x_r+1 widetilde","display":"\\widetilde x_{r+1}","kind":"use","excerpt":"Let x_r and widetilde x_r be the exact and implemented states after r levels. Then widetilde x_r+1-x_r+1 · ≤ widetilde F_r(widetilde x_r)-F_r(widetilde x_r) · + F_r(widetilde x_r)-F_r(x_r) · ≤ δ_r+L_r widetilde x_r-x_r . Iterating this recurrence gives the displayed product-sum estimate. The final two bounds follow by summing a finite or infinite g","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7049d36b7defb732","tex":"\\widetilde Y","aliases":"widetilde Y widetilde","display":"\\widetilde Y","kind":"use","excerpt":"The target Y_t^+ can be a next output, a finite future trajectory, a research decision, a proof outcome, or any measurable downstream quantity. Every predictive workspace claim is therefore target-relative: WorkspaceFor(Y_t^+) centernotLongrightarrow WorkspaceFor(widetilde Y_t^+).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dd73ce3c01836f01","tex":"\\widetilde Y_t^+","aliases":"widetilde Y_t^+ widetilde","display":"\\widetilde Y_t^+","kind":"use","excerpt":"The target Y_t^+ can be a next output, a finite future trajectory, a research decision, a proof outcome, or any measurable downstream quantity. Every predictive workspace claim is therefore target-relative: WorkspaceFor(Y_t^+) centernotLongrightarrow WorkspaceFor(widetilde Y_t^+).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c3b5ed4d5935a9b","tex":"\\operatorname{Workspace}","aliases":"Workspace operatorname","display":"\\operatorname{Workspace}","kind":"use","excerpt":"Workspace( R) centernotLongrightarrow Conscious( R). Likewise, Distributed(Workspace) centernotLongrightarrow Person(Workspace). Workspace structure, reentry, agency, self-maintenance, and phenomenal consciousness remain formally distinct predicates. Any bridge among them would require additional theory and evidence.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c35939cbd3213414","tex":"\\Workspace'","aliases":"Workspace' Workspace","display":null,"kind":"use","excerpt":"[Coordinate invariance of intervention effects] Let g: W_t→ W_t' be a measurable bijection and suppose the reparametrized intervention family satisfies ι_g(w)'=ι_w. Then every divergence-based causal effect is preserved: CE_Workspace'(g(w),g(w');b) = CE_Workspace(w,w';b).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c8c66d31c1fb154b","tex":"\\mathsf{workspace\\_extractor}","aliases":"workspace_extractor mathsf","display":"\\mathsf{workspace\\_extractor}","kind":"use","excerpt":"The machine-readable record should contain at least the following top-level objects: ResultRecord ={ · claim, specification, specification_family, decomposition, data_splits, formation, · workspace_extractor, goals, actions, interventions, · nulls, gates, identified_sets, specification_envelopes, uncertainty, failures, artifacts }. Each gate object","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a581618c8bb37ffc","tex":"\\Workspace_{\\mathrm{static}}","aliases":"Workspace_static Workspace mathrm","display":null,"kind":"use","excerpt":"The structural workspace is therefore more accurately represented by the triple Workspace_static = ( Γ(H), O_Workspace, π_Workspace ), not merely by the vector space Γ(H).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cf2966881476561d","tex":"\\Workspace_t","aliases":"Workspace_t Workspace","display":null,"kind":"use","excerpt":"Persistence does not require token identity. Let R_t and R_t+1 be temporally successive realizations. Continuity is possible when there exists an admissible transport J_t: R_trightsquigarrow R_t+1 that preserves the operative core and its causal role within tolerance: d_id(Workspace_t,Workspace_t+1)≤ε. Thus a workspace may persist by reconstruction","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-12049790582fa6a1","tex":"\\Workspace_{t+1}","aliases":"Workspace_t+1 Workspace","display":null,"kind":"use","excerpt":"Persistence does not require token identity. Let R_t and R_t+1 be temporally successive realizations. Continuity is possible when there exists an admissible transport J_t: R_trightsquigarrow R_t+1 that preserves the operative core and its causal role within tolerance: d_id(Workspace_t,Workspace_t+1)≤ε. Thus a workspace may persist by reconstruction","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-160f64e899c66ec0","tex":"\\operatorname{WorkspaceFor}","aliases":"WorkspaceFor operatorname","display":"\\operatorname{WorkspaceFor}","kind":"use","excerpt":"The target Y_t^+ can be a next output, a finite future trajectory, a research decision, a proof outcome, or any measurable downstream quantity. Every predictive workspace claim is therefore target-relative: WorkspaceFor(Y_t^+) centernotLongrightarrow WorkspaceFor(widetilde Y_t^+).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-09868efb784faa54","tex":"\\mathcal X","aliases":"X mathcal","display":"\\mathcal X","kind":"use","excerpt":"For every i∈ V, assign a finite-dimensional Hilbert space X_i. For every edge, assign a linear transfer T_e:X_s(e)⟶ X_t(e). The path transfer is T_γ=T_e_m⋯ T_e_2T_e_1, T_1_i=I_X_i. Accordingly, the static transfer system is a functor X:Path(G)⟶FinHilb.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-574f3f3ea8b3081e","tex":"\\mathfrak X","aliases":"X mathfrak","display":"\\mathfrak X","kind":"use","excerpt":"Define the vertex and edge direct-sum representations ρ_V(a)=⊕_i∈ Vρ_i(a), ρ_E(a)=⊕_e∈ Eρ_t(e)(a). Lift the edgewise intertwining defects to ( X_a c)_e = sqrtw_e Ξ_e(a)c_s(e). Then the exact commutator identity is D_Hρ_V(a)-ρ_E(a)D_H=- X_a. Furthermore, X_a ≤ β(a), β(a)^2 = max_i∈ V ∑_e:s(e)=iw_e Ξ_e(a) ^2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f5857cac0ddcea35","tex":"X'","aliases":"X'","display":"X'","kind":"use","excerpt":"[Gauge covariance of least squares] Under invertible coordinate transformations X'=g_uX, Y'=g_vY, the population least-squares solution transforms as H_*'=g_vH_*g_u^-1.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"Let ι_S,x be a feasible intervention that changes the internal workspace coordinate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the fu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-323aaf429c48a253","tex":"X^{\\mathsf T}","aliases":"X^ T mathsf","display":"X^{\\mathsf T}","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"[XOR witness pair] Under the intact law P, let X_1,X_2oversetiid∼Bernoulli(1/2), Y=X_1⊕ X_2. Then each (X_i,Y) pair is independent and uniform, so H_P(Ymid X_i)=1 bit, H_P(Ymid X_1,X_2)=0. Let Q make X_1,X_2,Y mutually independent fair bits. Then Q_(X_i,Y)=P_(X_i,Y) (i=1,2), but H_Q(Ymid X_1,X_2)=1 bit. Therefore Δ_joint^log=1 bit, k_0=2, Δ_tr(Q)=1","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"[XOR witness pair] Under the intact law P, let X_1,X_2oversetiid∼Bernoulli(1/2), Y=X_1⊕ X_2. Then each (X_i,Y) pair is independent and uniform, so H_P(Ymid X_i)=1 bit, H_P(Ymid X_1,X_2)=0. Let Q make X_1,X_2,Y mutually independent fair bits. Then Q_(X_i,Y)=P_(X_i,Y) (i=1,2), but H_Q(Ymid X_1,X_2)=1 bit. Therefore Δ_joint^log=1 bit, k_0=2, Δ_tr(Q)=1","count":13},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7d19fe81a11ca429","tex":"\\mathfrak X_a","aliases":"X_a mathfrak","display":"\\mathfrak X_a","kind":"use","excerpt":"Define the vertex and edge direct-sum representations ρ_V(a)=⊕_i∈ Vρ_i(a), ρ_E(a)=⊕_e∈ Eρ_t(e)(a). Lift the edgewise intertwining defects to ( X_a c)_e = sqrtw_e Ξ_e(a)c_s(e). Then the exact commutator identity is D_Hρ_V(a)-ρ_E(a)D_H=- X_a. Furthermore, X_a ≤ β(a), β(a)^2 = max_i∈ V ∑_e:s(e)=iw_e Ξ_e(a) ^2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-49a7b2d2da8d5b1b","tex":"X_{\\bar S}","aliases":"X_bar S bar","display":"X_{\\bar S}","kind":"use","excerpt":"For S∈ S, define Ret_τ(S) = I ( X_S(t+τ);X_S(t) mid X_bar S(t),E_t ), Dep_τ(S) = I ( X_S(t+τ);X_bar S(t) mid X_S(t),E_t ), Out_τ(S) = I ( X_bar S(t+τ);X_S(t) mid X_bar S(t),E_t ). These quantities are observational. They can be nonzero because of common causes, selection, or feedback paths and therefore cannot alone establish a causal boundary.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e7e08756fa886133","tex":"X_C","aliases":"X_C","display":"X_C","kind":"use","excerpt":"Let q,p,f now be independent uniform bits and define the downstream acceptance decision Y=q⊕ p⊕ f. The interpretation is abstract: acceptance depends on the statement, the state of the proof obligations, and the current falsification record. No individual node has sufficient information: H(Ymid X_P) = H(Ymid X_C) = H(Ymid X_N) =1. Every two-node co","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"use","excerpt":"Network consensus studies how local interactions drive distributed states toward agreement [source reference]. Agreement is not the same as a workspace. Consensus often removes heterogeneity by converging all nodes to a common value. Stratified relational workspaces preserve heterogeneous local representations and may require complementary disagree","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-10665c77b53ab88a","tex":"X_M","aliases":"X_M","display":"X_M","kind":"use","excerpt":"h_t=workspace-health bit, · h_t=1 means intact. The local accessible states are X_P(t)=(g_t,b_t), X_M(t)=(b_t,h_t), X_R(t)=(g_t,h_t). This is the three-node distributed code from Topic~77. Consequently: dimΓ=3, spec(L)={0,0,0,2,2,2}, k_rec=2, d_er=2. The organization is fully scaffolded. No endogenous-formation claim is made.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f3be737b42076753","tex":"X_m","aliases":"X_m","display":"X_m","kind":"use","excerpt":"Partial-information decomposition distinguishes redundant, unique, and synergistic information in multivariate systems, building on standard information-theoretic quantities [source reference]. The XOR examples in Part~XXII are canonical synergy constructions and are not new. They are used because they provide exact ground truth for relational irre","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-854a0c87655a5383","tex":"X_N","aliases":"X_N","display":"X_N","kind":"use","excerpt":"Let q,p,f now be independent uniform bits and define the downstream acceptance decision Y=q⊕ p⊕ f. The interpretation is abstract: acceptance depends on the statement, the state of the proof obligations, and the current falsification record. No individual node has sufficient information: H(Ymid X_P) = H(Ymid X_C) = H(Ymid X_N) =1. Every two-node co","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9883bc44c47316e6","tex":"X_o","aliases":"X_o","display":"X_o","kind":"use","excerpt":"Suppose node o contains the complete predictive state Z_t, while every other node receives only messages generated by o. The full system can have excellent performance, rich communication, and recurrent activity. Yet Y_t^+indep H_tmid X_o(t), k_0=1, Δ_joint=0. Removing the decorative nodes may change latency or style without changing the operative ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-853cecd6231f3bad","tex":"X_P","aliases":"X_P","display":"X_P","kind":"use","excerpt":"Let q,p,f now be independent uniform bits and define the downstream acceptance decision Y=q⊕ p⊕ f. The interpretation is abstract: acceptance depends on the statement, the state of the proof obligations, and the current falsification record. No individual node has sufficient information: H(Ymid X_P) = H(Ymid X_C) = H(Ymid X_N) =1. Every two-node co","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e798a8a7ea91ac3e","tex":"X_R","aliases":"X_R","display":"X_R","kind":"use","excerpt":"h_t=workspace-health bit, · h_t=1 means intact. The local accessible states are X_P(t)=(g_t,b_t), X_M(t)=(b_t,h_t), X_R(t)=(g_t,h_t). This is the three-node distributed code from Topic~77. Consequently: dimΓ=3, spec(L)={0,0,0,2,2,2}, k_rec=2, d_er=2. The organization is fully scaffolded. No endogenous-formation claim is made.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8285a92cff098813","tex":"X_r","aliases":"X_r","display":"X_r","kind":"use","excerpt":"[Deterministic empirical recovery bound] For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"Let x_r and widetilde x_r be the exact and implemented states after r levels. Then widetilde x_r+1-x_r+1 · ≤ widetilde F_r(widetilde x_r)-F_r(widetilde x_r)","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dccb0eea2147bbda","tex":"x_{r'}^{\\mathrm{init}}","aliases":"x_r'^init mathrm","display":"x_{r'}^{\\mathrm{init}}","kind":"use","excerpt":"[Handoff initialization] If α_S>0 and the survivors observe y=A_Sc+η, initialize the replacement by x_r'^init =E_r''A_S^† y. Then x_r'^init-E_r''c ≤ frac E_r'' η sqrtα_S.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1e9d35fe4d36c66f","tex":"x_{r+1}","aliases":"x_r+1","display":"x_{r+1}","kind":"use","excerpt":"Let x_r and widetilde x_r be the exact and implemented states after r levels. Then widetilde x_r+1-x_r+1 · ≤ widetilde F_r(widetilde x_r)-F_r(widetilde x_r) · + F_r(widetilde x_r)-F_r(x_r) · ≤ δ_r+L_r widetilde x_r-x_r . Iterating this recurrence gives the displayed product-sum estimate. The final two bounds follow by summing a finite or infinite g","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-671eb883df9e0c00","tex":"X_r^{\\mathsf T}","aliases":"X_r^ T mathsf","display":"X_r^{\\mathsf T}","kind":"use","excerpt":"[Deterministic empirical recovery bound] For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ed2990bcb53e180d","tex":"X_S","aliases":"X_S","display":"X_S","kind":"use","excerpt":"For S∈ S, define Ret_τ(S) = I ( X_S(t+τ);X_S(t) mid X_bar S(t),E_t ), Dep_τ(S) = I ( X_S(t+τ);X_bar S(t) mid X_S(t),E_t ), Out_τ(S) = I ( X_bar S(t+τ);X_S(t) mid X_bar S(t),E_t ). These quantities are observational. They can be nonzero because of common causes, selection, or feedback paths and therefore cannot alone establish a causal boundary.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-520c99b0851ef651","tex":"X_{s(e)}","aliases":"X_s(e)","display":"X_{s(e)}","kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1029cb8ad51ed272","tex":"x_t","aliases":"x_t","display":"x_t","kind":"use","excerpt":"Let the realized vertex state satisfy x_t+1=F_tx_t+ξ_t, where F_t: C_V(t)→ C_V(t+1) is the actual update and ξ_t is an unmodeled disturbance. Define the off-workspace error e_t= (I-P_t)x_t . Introduce α_t = (I-P_t+1)F_t(I-P_t) , which measures persistence of prior off-workspace error, and ℓ_t = (I-P_t+1)F_tP_t , which measures leakage of a valid wo","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4c931cc1c91f8258","tex":"X_{t(e)}","aliases":"X_t(e)","display":"X_{t(e)}","kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-813924c29494401d","tex":"x_{t+1}","aliases":"x_t+1","display":"x_{t+1}","kind":"use","excerpt":"Let the realized vertex state satisfy x_t+1=F_tx_t+ξ_t, where F_t: C_V(t)→ C_V(t+1) is the actual update and ξ_t is an unmodeled disturbance. Define the off-workspace error e_t= (I-P_t)x_t . Introduce α_t = (I-P_t+1)F_t(I-P_t) , which measures persistence of prior off-workspace error, and ℓ_t = (I-P_t+1)F_tP_t , which measures leakage of a valid wo","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-708bfd9721bf49c2","tex":"X_u","aliases":"X_u","display":"X_u","kind":"use","excerpt":"A learned transfer map is not absolutely identifiable when local coordinates are latent. Under local invertible changes X_u'=g_uX_u, X_v'=g_vX_v, the same abstract transfer is represented by H_uv'=g_vH_uvg_u^-1. The identifiable object is therefore the gauge class [H]_ G = { (g_t(e)H_eg_s(e)^-1)_e∈ E : (g_i)_i∈ V∈ G }. Any formation claim that comp","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4ca3acedbf9471c5","tex":"X_u'","aliases":"X_u'","display":"X_u'","kind":"use","excerpt":"A learned transfer map is not absolutely identifiable when local coordinates are latent. Under local invertible changes X_u'=g_uX_u, X_v'=g_vX_v, the same abstract transfer is represented by H_uv'=g_vH_uvg_u^-1. The identifiable object is therefore the gauge class [H]_ G = { (g_t(e)H_eg_s(e)^-1)_e∈ E : (g_i)_i∈ V∈ G }. Any formation claim that comp","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-08a6f9f108660d1a","tex":"X_v","aliases":"X_v","display":"X_v","kind":"use","excerpt":"A learned transfer map is not absolutely identifiable when local coordinates are latent. Under local invertible changes X_u'=g_uX_u, X_v'=g_vX_v, the same abstract transfer is represented by H_uv'=g_vH_uvg_u^-1. The identifiable object is therefore the gauge class [H]_ G = { (g_t(e)H_eg_s(e)^-1)_e∈ E : (g_i)_i∈ V∈ G }. Any formation claim that comp","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-000dc2d7d4c5a4ea","tex":"X_v'","aliases":"X_v'","display":"X_v'","kind":"use","excerpt":"A learned transfer map is not absolutely identifiable when local coordinates are latent. Under local invertible changes X_u'=g_uX_u, X_v'=g_vX_v, the same abstract transfer is represented by H_uv'=g_vH_uvg_u^-1. The identifiable object is therefore the gauge class [H]_ G = { (g_t(e)H_eg_s(e)^-1)_e∈ E : (g_i)_i∈ V∈ G }. Any formation claim that comp","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-55abe5dbff13226b","tex":"\\xleftarrow","aliases":"xleftarrow","display":null,"kind":"use","excerpt":"For pairwise overlaps write mathscr W_α=mathscr W(G_α), mathscr W_αβ=mathscr W(G_αβ), and r_αβ^α: mathscr W_α→mathscr W_αβ, r_αβ^β: mathscr W_β→mathscr W_αβ. The interface is therefore a cospan mathscr W_α xrightarrow r_αβ^α mathscr W_αβ xleftarrow r_αβ^β mathscr W_β, not necessarily a direct transfer map from one complete local workspace to the ot","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b0df18759bd8af0a","tex":"Y'","aliases":"Y'","display":"Y'","kind":"use","excerpt":"Take two realizations with identical G, D, L, P, ρ. In the first, let Y=W. In the second, let Y'=1-W for a binary workspace coordinate. Then q_str=0, δ_ρ(a)=0 for every operation, while the observational and interventional outcome kernels differ maximally under the identity correspondence. Therefore no purely structural theorem can prove causal ide","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f7e2679d489a54be","tex":"\\mathcal Y^+","aliases":"Y^+ mathcal","display":"\\mathcal Y^+","kind":"use","excerpt":"A controlled relational realization extends Definition~[source reference] by the tuple A_Workspace = ( R, G, A_t, Y^+, {U_g}_g∈ G, Π, I_A, I_G, I_D, I_R ). Here: item R is the underlying relational realization; item G is a measurable goal-code space; item A_t is the feasible action set at time t; item Y^+ is the relevant future trajectory space; it","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9a2e52ba72db60cf","tex":"Y^{+'}","aliases":"Y^+'","display":"Y^{+'}","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-79692341e865d3f0","tex":"Y_{\\mathrm{bench}}","aliases":"Y_bench mathrm","display":"Y_{\\mathrm{bench}}","kind":"use","excerpt":"Let two realizations have identical benchmark output distributions, P(Y_bench)=P'(Y_bench), but different intervention kernels: K(w,b)≠ K'(β w,b) for some matched workspace intervention. Then task behavior is preserved while d_causal(W,W')>0. The later system may be a successor or replacement rather than the same workspace under the identity criter","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-009b7a5fccb27aa3","tex":"\\mathsf Y_i","aliases":"Y_i mathsf","display":"\\mathsf Y_i","kind":"use","excerpt":"Let C_* be a finite-dimensional abstract core. The exact static theory provides a complete local chart overline E_i = ev_i∘Ψ:C_*⟶ C_i, where Ψ:C_*→Γ(H) is the root-to-section isomorphism. A participant or artifact need not expose that complete chart. Let O_i:C_i⟶ Y_i be the declared observation or access map and define the accessible encoding E_i=O","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-72c43d344dc79844","tex":"Y_r","aliases":"Y_r","display":"Y_r","kind":"use","excerpt":"[Deterministic empirical recovery bound] For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dd3b861928b61166","tex":"\\mathsf Y_{r'}'","aliases":"Y_r'' mathsf","display":"\\mathsf Y_{r'}'","kind":"use","excerpt":"Suppose node r is removed and the surviving coalition is S=V∖{r}. A replacement node r' is to receive the encoding E_r'':C_*→ Y_r''.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-79f5fc10a1174f12","tex":"\\mathsf Y_S","aliases":"Y_S mathsf","display":"\\mathsf Y_S","kind":"use","excerpt":"[Preservation of coalition frame bounds] Identify the old and new abstract cores with ran P and ran P' through their chosen core parameterizations. Let S be a declared coalition, let A_S:ran P→ Y_S, A_S':ran P'→ Y_S' be the resulting accessible coalition encoders, and let K_S be a unitary identification of their observation spaces. Suppose A_S'β-K_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-18079c3e710842d1","tex":"y_S","aliases":"y_S","display":"y_S","kind":"use","excerpt":"The full frame operator is F_PCN=E_P^*E_P+E_C^*E_C+E_N^*E_N=2I_3. For every two-node coalition S, λ_min(F_S)=1. Thus every two-node coalition reconstructs the complete core and, for noisy observations y_S=A_Sc+η_S, widehat c-c ≤ η_S . No individual node reconstructs c, because rank(E_P) = rank(E_C) = rank(E_N) =2<3. The realization is therefore sim","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b721983272984cda","tex":"\\mathsf Y_S'","aliases":"Y_S' mathsf","display":"\\mathsf Y_S'","kind":"use","excerpt":"[Preservation of coalition frame bounds] Identify the old and new abstract cores with ran P and ran P' through their chosen core parameterizations. Let S be a declared coalition, let A_S:ran P→ Y_S, A_S':ran P'→ Y_S' be the resulting accessible coalition encoders, and let K_S be a unitary identification of their observation spaces. Suppose A_S'β-K_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-24452e92e41a0cb5","tex":"y_{V\\setminus A}","aliases":"y_V∖ A setminus","display":"y_{V\\setminus A}","kind":"use","excerpt":"[Uniform erasure resilience] The encoding family is exactly reconstructible after every loss of at most k nodes if and only if α^(k)>0. If surviving observations have the form y_V∖ A=A_V∖ Ac+η_A, |A|≤ k, then the canonical survivor decoder obeys the uniform estimate widehat c_A-c ≤ frac η_A sqrtα^(k).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"sinθ · cosθ with θ∉2π Z. Then ker(R_θ-I)={0}. Every edge map is individually invertible and well-defined, yet the complete cycle admits no nonzero exact shared core. Pairwise translatability is therefore insufficient for global path independence.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"[0.18em] Workspace Theory Large A Core--Transfer Framework for Distributed Cognition, Collaboration, and Epistemic Systems large Komorebi P. largebfseries Foundations, Dynamics, Agency, and Empirical Certification vfill colorRuleColorrule0.70textwidth0.8pt small August 2026 pagenumberingroman Collaborative reasoning increasingly spans people, compu","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2efb553be08836e7","tex":"z_e","aliases":"z_e","display":"z_e","kind":"use","excerpt":"For z=(z_e)_e∈ E∈ C_E, the adjoint is (D_H^*z)_i = ∑_e:t(e)=isqrtw_e z_e - ∑_e:s(e)=isqrtw_e H_e^*z_e. Consequently, (L_Hc)_i = · ∑_e:t(e)=iw_e(c_i-H_ec_s(e))","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-86298b26f7dd1fb4","tex":"z_i","aliases":"z_i","display":"z_i","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-214932301a3fbe6b","tex":"Z_{i,q,\\varsigma}","aliases":"Z_i,q,ς varsigma","display":"Z_{i,q,\\varsigma}","kind":"use","excerpt":"Suppose S and J_* are finite, and each margin is estimated as widehat m_q(ς) = frac1n∑_i=1^nZ_i,q,ς-θ_q,ς, 0≤ Z_i,q,ς≤1, from independent certification units. Hoeffding's inequality and a union bound give P( sup_q,ς | widehat m_q(ς)-m_q(ς) | > sqrt frac log(2| J_*|| S|/δ) 2n ) ≤δ. Thus the choice ε_n(δ) = sqrt frac log(2| J_*|| S|/δ) 2n yields an e","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-eb37f2c03736f913","tex":"z_{s(e)}","aliases":"z_s(e)","display":"z_{s(e)}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e336d1c6f8092636","tex":"z_{t(e)}","aliases":"z_t(e)","display":"z_{t(e)}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7f6b31da43b1c8a0","tex":"|_{\\mathcal W_t}","aliases":"|_ W_t mathcal","display":"|_{\\mathcal W_t}","kind":"use","excerpt":"Low off-workspace error is not sufficient for meaningful persistence. The zero update F_t=0 drives every state exactly into every linear workspace while destroying all content. Define the internal transport B_t=P_t+1F_tP_t: W_t⟶ W_t+1 and its non-collapse margin μ_t^keep = σ_min(B_t|_ W_t). A content-preserving persistence claim requires μ_t^keep>0","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-78359d206ced154e","tex":"|_G","aliases":"|_G","display":"|_G","kind":"use","excerpt":"Let G=(V,E) be a finite directed interaction graph with local core spaces C_i, core transfers H_e, vertex space C_V(G), discrepancy operator D_G, and exact workspace mathscr W(G) := Γ(H|_G) = ker D_G.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d031310166282dfd","tex":"|_{G_\\alpha}","aliases":"|_G_α alpha","display":"|_{G_\\alpha}","kind":"use","excerpt":"Unless stated otherwise, connectedness means that |G| is connected. A root o∈ V may then be joined to every vertex by a path in the underlying graph. If G has several connected components, the theory applies to each component and Γ(H) ≅ ⊕_αΓ(H|_G_α).","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f8af569a83fe1831","tex":"|_{\\mathscr W(L)}","aliases":"|_mathscr W(L) mathscr","display":"|_{\\mathscr W(L)}","kind":"use","excerpt":"If K⊆ L⊆ G are subgraphs, coordinate restriction gives ρ_K^L: C_V(L)⟶ C_V(K). Because every edge equation on K is also an edge equation on L, restriction maps exact sections to exact sections: r_K^L := ρ_K^L|_mathscr W(L): mathscr W(L)⟶mathscr W(K). They obey r_L^L=I, r_J^K r_K^L=r_J^L whenever J⊆ K⊆ L. Thus K↦mathscr W(K) is a contravariant system","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-44797dfcb0c8f6ff","tex":"|_{\\ran P}","aliases":"|_ran P ran","display":null,"kind":"use","excerpt":"On ran P and ranwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b09a500e111dd6c9","tex":"|_{\\ran\\widetilde Q}","aliases":"|_ranwidetilde Q ran widetilde","display":null,"kind":"use","excerpt":"On ran P and ranwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2b9f0248ab6db655","tex":"|_W","aliases":"|_W","display":"|_W","kind":"use","excerpt":"|c =1 E_H(c). If W is L_H-invariant, then δ_coh(W) = L_H|_W . In particular, δ_coh(Γ_ε(H)) ≤ε.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"-(a+b) · b^2+1 . Writing τ=a^2+b^2+2, the eigenvalues are λ_± = frac12 ( τ±sqrtτ^2-4(ab-1)^2 ), and det L_H=(ab-1)^2. Thus a nonzero exact shared core exists precisely when ab=1, which is exactly the trivial-holonomy condition. Away from exact consistency, λ_-λ_+=(ab-1)^2. When |ab-1| is small while λ_+ stays bounded away from zero, the soft worksp","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b378725159b1aef3","tex":"\\alpha^{(1)}","aliases":"α^(1) alpha","display":"\\alpha^{(1)}","kind":"use","excerpt":"Let C_*= R^2 and define scalar node encodings E_1(c_1,c_2)=c_1, E_2(c_1,c_2)=c_2, E_3(c_1,c_2)=c_1+c_2. Every singleton has rank one, so no node reconstructs the two-dimensional core. Every pair does reconstruct. For S={1,2}, F_S=I, α_S=1. For S={1,3} or S={2,3}, the frame eigenvalues are frac3±sqrt52. Therefore m_rec=2, k_erase=1, d_er=2, α^(1)=fr","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-981b81cfa63fb8b2","tex":"\\alpha^{(k)}","aliases":"α^(k) alpha","display":"\\alpha^{(k)}","kind":"use","excerpt":"For k≥0, define the worst surviving lower frame bound α^(k) = min_substackA⊆ V |A|≤ k α_V∖ A. The sequence α^(k) is nonincreasing in k.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"use","excerpt":"Repeated substitution yields e_t · ≤ α_t-1⋯α_0e_0 + α_t-1⋯α_1σ_0 + ⋯ + σ_t-1, which is the first formula. Under the uniform bounds, dominate each product by the corresponding power of barα and sum the geometric series.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-023d31a35b01edea","tex":"\\alpha_A^{\\mathrm{meta}}","aliases":"α_A^meta alpha mathrm","display":"\\alpha_A^{\\mathrm{meta}}","kind":"use","excerpt":"mathscr W_meta( U)≅mathscr W(G) is a compositional statement, not a relational-irreducibility statement. A single chart may still expose enough accessible information to reconstruct the whole meta-state. A claim that the hierarchy is genuinely distributed must therefore reuse the coalition gates from Parts~VII and XI. At the chart level, one may re","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ce0cb217ed2d6185","tex":"\\alpha_{j_P}","aliases":"α_j_P alpha","display":"\\alpha_{j_P}","kind":"use","excerpt":"For any P∈ H_0, choose an index j_P such that P∈ H_0j_P. The event that every gate passes is contained in the event that gate j_P passes. Hence P_P(certificate passes) ≤ P_P(G_j_P passes) ≤ α_j_P ≤ max_jα_j. Taking the supremum over P∈ H_0 proves the claim.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b2ccedd933b262f6","tex":"\\alpha_{p+1}","aliases":"α_p+1 alpha","display":"\\alpha_{p+1}","kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-af15262773600a7f","tex":"\\alpha_r","aliases":"α_r alpha","display":"\\alpha_r","kind":"use","excerpt":"[Nonautonomous persistence theorem] Let e_t≥0 satisfy e_t+1≤α_te_t+σ_t, α_t,σ_t≥0. Then, for every t≥1, e_t ≤ · (∏_r=0^t-1α_r)e_0 · + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9108111b5e10b68f","tex":"\\alpha_S","aliases":"α_S alpha","display":"\\alpha_S","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":19},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-45cabb4e08c24a7f","tex":"\\alpha_S'","aliases":"α_S' alpha","display":"\\alpha_S'","kind":"use","excerpt":"be a declared coalition, let A_S:ran P→ Y_S, A_S':ran P'→ Y_S' be the resulting accessible coalition encoders, and let K_S be a unitary identification of their observation spaces. Suppose A_S'β-K_SA_S ≤ δ_S^frame < sqrtα_S. Then the replacement coalition remains reconstructive and its new lower frame bound satisfies α_S' ≥ ( sqrtα_S-δ_S^frame )^2. ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-673ddcfc7e4cb456","tex":"\\alpha_t","aliases":"α_t alpha","display":"\\alpha_t","kind":"use","excerpt":"Let the realized vertex state satisfy x_t+1=F_tx_t+ξ_t, where F_t: C_V(t)→ C_V(t+1) is the actual update and ξ_t is an unmodeled disturbance. Define the off-workspace error e_t= (I-P_t)x_t . Introduce α_t = (I-P_t+1)F_t(I-P_t) , which measures persistence of prior off-workspace error, and ℓ_t = (I-P_t+1)F_tP_t , which measures leakage of a valid wo","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4bd7da5bc86c5ab1","tex":"\\alpha_{t-1}","aliases":"α_t-1 alpha","display":"\\alpha_{t-1}","kind":"use","excerpt":"Repeated substitution yields e_t · ≤ α_t-1⋯α_0e_0 + α_t-1⋯α_1σ_0 + ⋯ + σ_t-1, which is the first formula. Under the uniform bounds, dominate each product by the corresponding power of barα and sum the geometric series.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-85d939956564b256","tex":"\\alpha_t^{(k)}","aliases":"α_t^(k) alpha","display":"\\alpha_t^{(k)}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-962e943a4d3cad93","tex":"\\alpha_{V\\setminus A}","aliases":"α_V∖ A alpha setminus","display":"\\alpha_{V\\setminus A}","kind":"use","excerpt":"For k≥0, define the worst surviving lower frame bound α^(k) = min_substackA⊆ V |A|≤ k α_V∖ A. The sequence α^(k) is nonincreasing in k.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1b8a5f24f3e7373f","tex":"\\alpha_{\\{i,j\\}}","aliases":"α_{i,j} alpha","display":"\\alpha_{\\{i,j\\}}","kind":"use","excerpt":"E_{2,3}^*E_{2,3} · =diag(1,1,2). Hence α_{i,j}=1 (i≠ j), and one-node erasure recovery is exact. Every nonzero core vector is visible at at least two nodes, so d_er=2. This code is distributed but redundant: no node is sufficient, yet any one node may be lost without destroying the core.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5aa0471e8829e8ee","tex":"\\alpha_{\\{\\alpha\\}}^{\\mathrm{meta}}","aliases":"α_{α}^meta alpha alpha mathrm","display":"\\alpha_{\\{\\alpha\\}}^{\\mathrm{meta}}","kind":"use","excerpt":"mathscr W_meta( U)≅mathscr W(G) is a compositional statement, not a relational-irreducibility statement. A single chart may still expose enough accessible information to reconstruct the whole meta-state. A claim that the hierarchy is genuinely distributed must therefore reuse the coalition gates from Parts~VII and XI. At the chart level, one may re","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-05657b61d3150e7e","tex":"\\beta^t","aliases":"β^t beta","display":"\\beta^t","kind":"use","excerpt":"enance deficit. Suppose the matched no-repair trajectory satisfies e_t+1^(0) ≤ α e_t+d_t, where d_t≥0 is exogenous damage. Suppose the implemented repair policy has the causal one-step effect e_t+1^(0)-e_t+1 ≥ κ e_t-ϵ_t. Assume 0≤β:=α-κ<1, 0≤ d_t≤overline d, 0≤ϵ_t≤overlineϵ. Then e_t ≤ β^te_0 + frac(overline d+overlineϵ)(1-β^t)1-β, and therefore li","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ccc4be5e5da67fb6","tex":"\\beta_S","aliases":"β_S beta","display":"\\beta_S","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a533e0518591a216","tex":"\\gamma_{\\mathcal U}^{\\mathrm{meta}}","aliases":"γ_ U^meta gamma mathcal mathrm","display":"\\gamma_{\\mathcal U}^{\\mathrm{meta}}","kind":"use","excerpt":"Let γ_ U^meta = λ_min^+ (L_ U^meta) be the first positive eigenvalue.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b391451dddec7729","tex":"\\gamma_{\\mathcal U}^{\\mathrm{tot}}","aliases":"γ_ U^tot gamma mathcal mathrm","display":"\\gamma_{\\mathcal U}^{\\mathrm{tot}}","kind":"use","excerpt":"[Total descent kernel] For every finite edge-complete cover, ker K_ U = J_ Umathscr W(G). Consequently, if γ_ U^tot = λ_min^+ (L_ U^tot), then dist ( u, J_ Umathscr W(G) )^2 ≤ frac E_ U^tot(u) γ_ U^tot.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d2e737aa54abf669","tex":"\\gamma_1","aliases":"γ_1 gamma","display":"\\gamma_1","kind":"use","excerpt":"[Finite matrix test] Let γ_1,…,γ_r be fundamental cycle generators and suppose dim C_o=d. Define the stacked obstruction matrix M = H_γ_1-I","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a7873a27a61c21a0","tex":"\\gamma_e","aliases":"γ_e gamma","display":"\\gamma_e","kind":"use","excerpt":"Choose a spanning tree T⊆|G|. Let τ_i:o→ i be the unique tree path. Every non-tree edge e:i→ j determines a fundamental loop γ_e=τ_j^-1eτ_i with transfer H_γ_e=H_τ_j^-1H_eH_τ_i. The fundamental group of a finite graph is generated freely by these cotree loops. Hence it is enough to impose fixed-point conditions for a finite set of generators.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c2cbc0c8a3b756c","tex":"\\gamma_{e_q}","aliases":"γ_e_q gamma","display":"\\gamma_{e_q}","kind":"use","excerpt":"[Finite obstruction set] Let e_1,…,e_r be the non-tree edges of a connected finite graph. Then C_* = ⋂_q=1^r ker(H_γ_e_q-I).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-19feb91f7d6addcc","tex":"\\gamma_H","aliases":"γ_H gamma","display":"\\gamma_H","kind":"use","excerpt":"[Coherence-to-core distance estimate] Let W_0=Γ(H)=ker L_H, r=dim W_0, and suppose r<N. Set γ_H=λ_r+1>0. Then every c∈ C_V satisfies dist(c,W_0)^2 ≤ frac E_H(c)γ_H. Moreover, E_H(c) ≤ L_H dist(c,W_0)^2.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c9526d81cf10f7ae","tex":"\\gamma_{\\mathrm{HAI}}","aliases":"γ_HAI gamma mathrm","display":"\\gamma_{\\mathrm{HAI}}","kind":"use","excerpt":"0 · -1 · 0 · 0 · 0 · 0 · 1 · 0 . The rows are orthogonal with squared norm two. Therefore ker D_HAI ≅ R^4, spec(L_HAI) = {0,0,0,0,2,2,2,2}, γ_HAI=2. The full accessible frame operator is F_HMXA=2I_4. No single node is injective. Every three-node coalition reconstructs the full state, and the two opposite pairs {H,X}, {M,A} are already sufficient. A","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c82b936bc85cf8ff","tex":"\\gamma_{\\mathrm{HH}}","aliases":"γ_HH gamma mathrm","display":"\\gamma_{\\mathrm{HH}}","kind":"use","excerpt":"-1 · 0 · 0 · 0 · 1 · 0 . Therefore ker D_HH = { (q,p,p,f,q,f)^ T:q,p,f∈ R } ≅ R^3. The rows are orthogonal with squared norm two, so D_HHD_HH^*=2I_3. Consequently the transfer Laplacian has spectrum spec(L_HH) = {0,0,0,2,2,2}, and the exact workspace gap is γ_HH=2. The low-energy subspace is therefore stable under any Laplacian perturbation whose o","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1d6a683efb60adac","tex":"\\gamma_{\\mathrm{int}}","aliases":"γ_int gamma mathrm","display":"\\gamma_{\\mathrm{int}}","kind":"use","excerpt":"[Two-scale coherence bound] Let P_Π be the orthogonal projector onto J_Π W_Π. Let γ_int>0 be the smallest positive eigenvalue of D_int^*D_int, and let γ_Π>0 be the smallest positive eigenvalue of D_Π^*D_Π. Then every c∈ C_V satisfies dist(c,Γ(H)) ≤ frac D_intc sqrtγ_int + frac D_Π J_Π^*P_Π c sqrtγ_Π. The first term measures incoherence within macro","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-788db25068d804f5","tex":"\\gamma_r","aliases":"γ_r gamma","display":"\\gamma_r","kind":"use","excerpt":"[Finite matrix test] Let γ_1,…,γ_r be fundamental cycle generators and suppose dim C_o=d. Define the stacked obstruction matrix M = H_γ_1-I","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f4702ead01ddf09d","tex":"\\Gamma_{\\varepsilon_r}","aliases":"Γ_ε_r Gamma varepsilon","display":"\\Gamma_{\\varepsilon_r}","kind":"use","excerpt":"For appropriate recurrent systems, W^(r+1) = Γ_ε_r(W^(r)) may converge to a stable workspace type: d_id ( W^(r+1),W^(r) ) →0.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f480e95daf44dfed","tex":"\\Gamma_{\\varepsilon_t}","aliases":"Γ_ε_t Gamma varepsilon","display":"\\Gamma_{\\varepsilon_t}","kind":"use","excerpt":"Let c(t)=(c_i(t))_i∈ V ∈ C_V(t), P_t= 1_[0,ε_t](L_t). The instantaneous structural workspace coordinate is W_t^str=P_tc(t) ∈ Γ_ε_t(H(t)). When P_t varies with time, these coordinates inhabit a moving family of subspaces. Parts~X--XII will supply the transport required to compare them across time. In what follows, all sufficiency and intervention st","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-77692b96d06f5af4","tex":"\\gamma_\\Pi","aliases":"γ_Π gamma Pi","display":"\\gamma_\\Pi","kind":"use","excerpt":"[Two-scale coherence bound] Let P_Π be the orthogonal projector onto J_Π W_Π. Let γ_int>0 be the smallest positive eigenvalue of D_int^*D_int, and let γ_Π>0 be the smallest positive eigenvalue of D_Π^*D_Π. Then every c∈ C_V satisfies dist(c,Γ(H)) ≤ frac D_intc sqrtγ_int + frac D_Π J_Π^*P_Π c sqrtγ_Π. The first term measures incoherence within macro","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3aa4e894a52cdfe1","tex":"\\boldsymbol\\Delta","aliases":"Δ boldsymbol Delta","display":"\\boldsymbol\\Delta","kind":"use","excerpt":"For a predeclared null Q, define Δ_tr(Q) = Risk_ℓ,Q(V) - Risk_ℓ,P(V). For a finite validated panel Q={Q_1,…,Q_m}⊆ N_pred(P), report both the complete vector Δ_tr( Q) = ( Δ_tr(Q_1),…, Δ_tr(Q_m) ) and the robust panel minimum Δ_tr^min( Q) = min_Q∈ QΔ_tr(Q). The panel must be fixed independently of the observed favorable gap. Otherwise null selection ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-602211dbb52f91fa","tex":"\\Delta^{(r)}_\\ell","aliases":"Δ^(r)_ℓ Delta ell","display":"\\Delta^{(r)}_\\ell","kind":"use","excerpt":"For 0≤ r≤ |V|, define the order-r coalition gap profile by Δ^(r)_ℓ = min_substackA⊆ V |A|≤ r [ Risk_ℓ(A)-Risk_ℓ(V) ]. The profile is nonincreasing in r and reaches zero by r=|V|. It records how rapidly operative capacity becomes reconstructible as larger coalitions are permitted.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-98bcd3bd41d2c335","tex":"\\delta^1","aliases":"δ^1 delta","display":"\\delta^1","kind":"use","excerpt":"A different problem arises when only relative transition data are supplied. Let b=(b_αβ)∈check C^1( U;mathscr W) be a prescribed family of overlap offsets. The triple-overlap consistency condition is δ^1b=0. One seeks local potentials q=(q_α)∈check C^0 satisfying δ^0q=b. Define check H^1( U;mathscr W) = frackerδ^1imδ^0. Then the affine gluing probl","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"use","excerpt":"[ Q_i_2T_2T_1P_i_0 = V_2H_1+K_2V_1. Thus leakage out of the core is first order in the blocks V_r, while an excursion that leaves and then returns to the core contains at least one V-block and one later U-block. For a fixed path length, if U_r , V_r =O(δ) and the diagonal blocks remain bounded, the residual-mediated correction to pure core-to-core ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f113a05b6a235186","tex":"\\delta^{\\mathrm{leak}}","aliases":"δ^leak delta mathrm","display":"\\delta^{\\mathrm{leak}}","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b9cb2debfd5e7c7f","tex":"\\delta^p","aliases":"δ^p delta","display":"\\delta^p","kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5b9e32799ac13285","tex":"\\delta^{p+1}","aliases":"δ^p+1 delta","display":"\\delta^{p+1}","kind":"use","excerpt":"pha_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a local family w=(w_α)_α, (δ^0w)_αβ = r_αβ^βw_β - r_αβ^αw_α. Thus δ^0 is the overlap-discrepancy op","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6dcd1257470d979b","tex":"\\delta^{\\mathrm{scaf}}","aliases":"δ^scaf delta mathrm","display":"\\delta^{\\mathrm{scaf}}","kind":"use","excerpt":"The complete formation signature is E_form(W) = ( e_V,e_E,e_H,e_P,e_ O,e_B; δ^scaf, δ^leak ), where every e_q is reported as the pair e_q = ( e_q^MDL, e_q^causal ). No scalar average is permitted to hide a zero history contribution for one of the structures claimed to have been discovered.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-79fbb71f8b8bd013","tex":"\\delta_{\\mathcal D}","aliases":"δ_ D delta mathcal","display":"\\delta_{\\mathcal D}","kind":"use","excerpt":"Fix a reference decomposition D_0. For each D∈ D, let Φ_ D:W^ D_0⟶ W^ D be a declared workspace correspondence and let δ_ D measure its structural and causal identity defect. Assume gate q has a transport bound | m_q(Φ_ DW; D) -m_q(W; D_0) | ≤ L_qδ_ D+b_q( D), where L_q is a sensitivity constant and b_q records model mismatch not captured by the id","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2bb168c166c79f1d","tex":"\\Delta_1","aliases":"Δ_1 Delta","display":"\\Delta_1","kind":"use","excerpt":"Equip the finite cochain spaces with Hilbert inner products and define the first vCech Laplacian Δ_1 = δ^0(δ^0)^* + (δ^1)^*δ^1. Let H^1_ U=kerΔ_1.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7f6305885463b98e","tex":"\\delta_a^0","aliases":"δ_a^0 delta","display":"\\delta_a^0","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-557fc2e14e50ac2c","tex":"\\delta_A^Z","aliases":"δ_A^Z delta","display":"\\delta_A^Z","kind":"use","excerpt":"Let d_ P be a bounded metric generating the measurable structure on P( Y). Define the reconstruction defect of the canonical state from coalition A by δ_A^Z = inf_ψ_A E [ d_ P ( Z_t,ψ_A( H_t^A) ) ].","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-64e636fbc41877a9","tex":"\\Delta_A^\\ell","aliases":"Δ_A^ℓ Delta ell","display":"\\Delta_A^\\ell","kind":"use","excerpt":"The resulting certification report is C_dyn(W_t) =( · δ_align^ I, {Δ_A^ℓ}_A⊆ V, k_ε, Δ_tr( Q), · η_causal^ν, Δ_specific(r), η_rec(τ), η_loop(τ) ). No single coordinate certifies a relational workspace.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-74b45f353d2a1789","tex":"\\Delta_A^{\\ell}","aliases":"Δ_A^ℓ Delta ell","display":"\\Delta_A^{\\ell}","kind":"use","excerpt":"Define the coalition gap Δ_A^ℓ = Risk_ℓ(A)-Risk_ℓ(V) ≥0. The node-level joint advantage is Δ_joint^ℓ = min_i∈ V Δ_{i}^ℓ. Under logarithmic loss, Δ_joint^log = min_i∈ V I ( Y_t^+; H_t mid H_t^{i} ). Therefore Δ_joint^log>0 means that every individual component omits some predictive information available to the complete system.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-424d7d2d0062249d","tex":"\\delta_{\\mathrm{align}}","aliases":"δ_align delta mathrm","display":"\\delta_{\\mathrm{align}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6a32c8d892319cd1","tex":"\\delta_{\\mathrm{align}}^{\\mathcal I}","aliases":"δ_align^ I delta mathrm mathcal","display":"\\delta_{\\mathrm{align}}^{\\mathcal I}","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-046c86a67b69cdc5","tex":"\\delta_{\\mathrm{align}}^{\\mathrm{obs}}","aliases":"δ_align^obs delta mathrm mathrm","display":"\\delta_{\\mathrm{align}}^{\\mathrm{obs}}","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1d6bb66bd4727edf","tex":"\\delta_c","aliases":"δ_c delta","display":"\\delta_c","kind":"use","excerpt":"· + ( A_ W'' + A_ W ) (1-sqrt1-q_*^2) ]. If, in addition, δ_causal(β)≤δ_c, δ_rec(β)≤δ_r, then d_id( R, R') ≤ · a q_*+b δ_op( J)","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-edb4cc9f5ee8cbe8","tex":"\\delta_{\\mathrm{coh}}","aliases":"δ_coh delta mathrm","display":"\\delta_{\\mathrm{coh}}","kind":"use","excerpt":"For c≠0, define the Rayleigh defect δ_coh(c) = frac E_H(c) c ^2. For a nonzero subspace W⊆ C_V, define δ_coh(W) = sup_substackc∈ W","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-670766d1fdc77e1b","tex":"\\delta_{\\mathrm{coh},t}","aliases":"δ_coh,t delta mathrm","display":"\\delta_{\\mathrm{coh},t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c6a395594f216319","tex":"\\Delta_D","aliases":"Δ_D Delta","display":"\\Delta_D","kind":"use","excerpt":"For an approximate correspondence define Δ_D=D'J_V-J_ED, δ_D= Δ_D . If the graph weights are unchanged, define edgewise comparison defects δ_e = H_e'J_s(e)-J_t(e)H_e . The discrepancy operator has one changed source block per edge, so δ_D^2 ≤ max_i∈ V ∑_e:s(e)=iw_eδ_e^2 ≤ ∑_e∈ Ew_eδ_e^2. When a single node is replaced, every nonincident edge has δ_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9e3833e6f164847f","tex":"\\delta_D","aliases":"δ_D delta","display":"\\delta_D","kind":"use","excerpt":"For an approximate correspondence define Δ_D=D'J_V-J_ED, δ_D= Δ_D . If the graph weights are unchanged, define edgewise comparison defects δ_e = H_e'J_s(e)-J_t(e)H_e . The discrepancy operator has one changed source block per edge, so δ_D^2 ≤ max_i∈ V ∑_e:s(e)=iw_eδ_e^2 ≤ ∑_e∈ Ew_eδ_e^2. When a single node is replaced, every nonincident edge has δ_","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a32a58334848f845","tex":"\\delta_D^2","aliases":"δ_D^2 delta","display":"\\delta_D^2","kind":"use","excerpt":"For an approximate correspondence define Δ_D=D'J_V-J_ED, δ_D= Δ_D . If the graph weights are unchanged, define edgewise comparison defects δ_e = H_e'J_s(e)-J_t(e)H_e . The discrepancy operator has one changed source block per edge, so δ_D^2 ≤ max_i∈ V ∑_e:s(e)=iw_eδ_e^2 ≤ ∑_e∈ Ew_eδ_e^2. When a single node is replaced, every nonincident edge has δ_","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-78013cfd94fd4d2e","tex":"\\boldsymbol\\Delta_{\\mathrm{dec}}","aliases":"Δ_dec boldsymbol Delta mathrm","display":"\\boldsymbol\\Delta_{\\mathrm{dec}}","kind":"use","excerpt":"Use a vector rather than one aggregate score: C_full(W) =( · d_ε, g_ε, δ_coh, δ_align^ I, Δ_joint^ℓ, k_ε, · Δ_tr( Q), η_causal^ν, Δ_specific(r), η_loop(τ), · d_id, Rec_k, underline m_ D, Δ_dec, E_form, δ^scaf, δ^leak, · Δ_J, Δ_t^B, COut_τ(widehat S), CIn_τ(widehat S), · η_G(τ), δ_G^cue(τ), η_G→ A, δ_Q, η_V→ A, η_A→ Y, · η_D→ M, η_M→ R, η_R→ v, Δ_re","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c5b085f18c150ac1","tex":"\\Delta_e","aliases":"Δ_e Delta","display":"\\Delta_e","kind":"use","excerpt":"Suppose the edge maps are perturbed to widetilde H_e=H_e+Δ_e while the weights and vertex metrics remain fixed. Define δ_H^2 = max_i∈ V ∑_e:s(e)=iw_e Δ_e ^2. Let F=D_widetilde H-D_H. Then (Fc)_e=-sqrtw_e Δ_ec_s(e), F ≤δ_H. Since L_widetilde H-L_H =D_H^*F+F^*D_H+F^*F, we obtain L_widetilde H-L_H ≤ 2 D_H δ_H+δ_H^2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3069cf1843723de2","tex":"\\delta_e","aliases":"δ_e delta","display":"\\delta_e","kind":"use","excerpt":"For an approximate correspondence define Δ_D=D'J_V-J_ED, δ_D= Δ_D . If the graph weights are unchanged, define edgewise comparison defects δ_e = H_e'J_s(e)-J_t(e)H_e . The discrepancy operator has one changed source block per edge, so δ_D^2 ≤ max_i∈ V ∑_e:s(e)=iw_eδ_e^2 ≤ ∑_e∈ Ew_eδ_e^2. When a single node is replaced, every nonincident edge has δ_","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-73bf298ea89c9e11","tex":"\\delta_e^2","aliases":"δ_e^2 delta","display":"\\delta_e^2","kind":"use","excerpt":"For an approximate correspondence define Δ_D=D'J_V-J_ED, δ_D= Δ_D . If the graph weights are unchanged, define edgewise comparison defects δ_e = H_e'J_s(e)-J_t(e)H_e . The discrepancy operator has one changed source block per edge, so δ_D^2 ≤ max_i∈ V ∑_e:s(e)=iw_eδ_e^2 ≤ ∑_e∈ Ew_eδ_e^2. When a single node is replaced, every nonincident edge has δ_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3d7790bad0a81320","tex":"\\delta_e^{\\mathrm{tr}}","aliases":"δ_e^tr delta mathrm","display":"\\delta_e^{\\mathrm{tr}}","kind":"use","excerpt":"Core leakage, residual contamination, and total transfer defect are measured by λ_e^out= V_e , λ_e^in= U_e , δ_e^tr = U_e + V_e . These quantities depend on the chosen norms and projectors. Their empirical meaning must therefore be tied to a declared intervention and prediction protocol.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-84ea82d83b6220e7","tex":"\\delta_{\\mathrm{flat}}^{(R)}","aliases":"δ_flat^(R) delta mathrm","display":"\\delta_{\\mathrm{flat}}^{(R)}","kind":"use","excerpt":"The exact theorem states recursive exact gluing = one-shot exact gluing. The approximate theorem instead supplies a measurable flattening defect: δ_flat^(R) = widetilde F_R-1⋯widetilde F_0(x) - F_R-1⋯ F_0(x) . A hierarchy is empirically justified when it reduces computation, communication, or local model complexity while keeping δ_flat^(R), predict","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5b6de7d10fb8eca6","tex":"\\delta_{\\mathrm{flat}}^0","aliases":"δ_flat^0 delta mathrm","display":"\\delta_{\\mathrm{flat}}^0","kind":"use","excerpt":"Let L( T) be the leaf set. Repeated edge completeness implies that the flattened family U_flat = {G_ℓ:ℓ∈ L( T)} is itself an edge-complete cover of G. Let δ_flat^0 be its overlap coboundary and define W_flat = kerδ_flat^0. Theorem~[source reference] and Theorem~[source reference] give W_r ≅ W_flat ≅ mathscr W(G). Equivalently, Lim_ T^rec ( {mathscr","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8727cfc8fb629c3a","tex":"\\delta_G^{\\mathrm{cue}}","aliases":"δ_G^cue delta mathrm","display":"\\delta_G^{\\mathrm{cue}}","kind":"use","excerpt":"Cue dependence must be reported separately. For matched initial goal g, define δ_G^cue(τ) = sup_c,c' D_G ( P(G_t+τ∈·midι_g,do(C=c)), P(G_t+τ∈·midι_g,do(C=c')) ). High cue dependence is not a failure when the claimed agent is intentionally instruction following. It does mean that persistence cannot be attributed to an autonomous internal objective w","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-97d7d8b637a9e863","tex":"\\delta_H^2","aliases":"δ_H^2 delta","display":"\\delta_H^2","kind":"use","excerpt":"Suppose the edge maps are perturbed to widetilde H_e=H_e+Δ_e while the weights and vertex metrics remain fixed. Define δ_H^2 = max_i∈ V ∑_e:s(e)=iw_e Δ_e ^2. Let F=D_widetilde H-D_H. Then (Fc)_e=-sqrtw_e Δ_ec_s(e), F ≤δ_H. Since L_widetilde H-L_H =D_H^*F+F^*D_H+F^*F, we obtain L_widetilde H-L_H ≤ 2 D_H δ_H+δ_H^2.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c25d6c039a20249f","tex":"\\delta_i^{\\mathrm{local}}","aliases":"δ_i^local delta mathrm","display":"\\delta_i^{\\mathrm{local}}","kind":"use","excerpt":"Let every node store a complete copy of the same core: E_i=I_d for every i. Then coherence is perfect and reconstruction is robust, but k_0=1, min_iδ_i^local=0. If the archive is never used in downstream control, then also η_causal=0. This is a shared representation or archive, not a relationally irreducible workspace.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f5741d5b2a7dedf3","tex":"\\Delta_J","aliases":"Δ_J Delta","display":"\\Delta_J","kind":"use","excerpt":"[Uniform structure-selection recovery] Let T be a finite candidate family with population score J: T→ R. Let θ_* = argmin_θ∈ TJ(θ) be unique, with margin Δ_J = min_θ≠θ_* (J(θ)-J(θ_*))>0. If a held-out estimate widehat J satisfies sup_θ∈ T |widehat J(θ)-J(θ)| ≤ε, then every empirical minimizer widehatθ obeys J(widehatθ)-J(θ_*) ≤2ε. If 2ε<Δ_J, then w","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8b982c44f3781524","tex":"\\Delta_{\\mathrm{joint}}","aliases":"Δ_joint Delta mathrm","display":"\\Delta_{\\mathrm{joint}}","kind":"use","excerpt":"Suppose node o contains the complete predictive state Z_t, while every other node receives only messages generated by o. The full system can have excellent performance, rich communication, and recurrent activity. Yet Y_t^+indep H_tmid X_o(t), k_0=1, Δ_joint=0. Removing the decorative nodes may change latency or style without changing the operative ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2b43302c563e08f7","tex":"\\Delta_{\\mathrm{joint}}^{\\log}","aliases":"Δ_joint^log Delta mathrm log","display":"\\Delta_{\\mathrm{joint}}^{\\log}","kind":"use","excerpt":"Define the coalition gap Δ_A^ℓ = Risk_ℓ(A)-Risk_ℓ(V) ≥0. The node-level joint advantage is Δ_joint^ℓ = min_i∈ V Δ_{i}^ℓ. Under logarithmic loss, Δ_joint^log = min_i∈ V I ( Y_t^+; H_t mid H_t^{i} ). Therefore Δ_joint^log>0 means that every individual component omits some predictive information available to the complete system.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4ba96ca190936b83","tex":"\\Delta_{\\mathrm{joint}}^{\\mathrm{meta}}","aliases":"Δ_joint^meta Delta mathrm mathrm","display":"\\Delta_{\\mathrm{joint}}^{\\mathrm{meta}}","kind":"use","excerpt":"nough accessible information to reconstruct the whole meta-state. A claim that the hierarchy is genuinely distributed must therefore reuse the coalition gates from Parts~VII and XI. At the chart level, one may require, for example, α_{α}^meta=0 for every α, α_A^meta>0 for some |A|≥2, or the predictive analogue k_ε^meta≥2, Δ_joint^meta>0. Capacity-m","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9da2d1c922d61b8c","tex":"\\Delta_{\\mathrm{joint}}^\\ell","aliases":"Δ_joint^ℓ Delta mathrm ell","display":"\\Delta_{\\mathrm{joint}}^\\ell","kind":"use","excerpt":"Use a vector rather than one aggregate score: C_full(W) =( · d_ε, g_ε, δ_coh, δ_align^ I, Δ_joint^ℓ, k_ε, · Δ_tr( Q), η_causal^ν, Δ_specific(r), η_loop(τ), · d_id, Rec_k, underline m_ D, Δ_dec, E_form, δ^scaf, δ^leak, · Δ_J, Δ_t^B, COut_τ(widehat S), CIn_τ(widehat S), · η_G(τ), δ_G^cue(τ), η_G→ A, δ_Q, η_V→ A, η_A→ Y, · η_D→ M, η_M→ R, η_R→ v, Δ_re","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1532d34ffae01be6","tex":"\\Delta_{\\mathrm{joint}}^{\\ell}","aliases":"Δ_joint^ℓ Delta mathrm ell","display":"\\Delta_{\\mathrm{joint}}^{\\ell}","kind":"use","excerpt":"Define the coalition gap Δ_A^ℓ = Risk_ℓ(A)-Risk_ℓ(V) ≥0. The node-level joint advantage is Δ_joint^ℓ = min_i∈ V Δ_{i}^ℓ. Under logarithmic loss, Δ_joint^log = min_i∈ V I ( Y_t^+; H_t mid H_t^{i} ). Therefore Δ_joint^log>0 means that every individual component omits some predictive information available to the complete system.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e10dac05432eea87","tex":"\\Delta_{\\log}^{(r)}","aliases":"Δ_log^(r) Delta log","display":"\\Delta_{\\log}^{(r)}","kind":"use","excerpt":"[Information form of coalition size] Under the assumptions of Theorem~[source reference], k_ε = min{ |A|: I(Y_t^+; H_tmid H_t^A) ≤ε }. Equivalently, k_ε is the first r for which Δ_log^(r)≤ε.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-43aa8de9970d7c67","tex":"\\Delta_{\\mathrm{maint}}","aliases":"Δ_maint Delta mathrm","display":"\\Delta_{\\mathrm{maint}}","kind":"use","excerpt":"Post-action health evolves according to h_t+1=h_t^-lor r_t. Define the maintenance deficit e_t=1-h_t. Under the intact policy, h_t+1=1 and e_t+1=0 for both damage states. Under the matched no-repair intervention and h_t^-=0, e_t+1^(0)=1. Thus the measured causal repair gain is Δ_maint=1. The closed-loop recurrence has β=0, overline d=0, overlineϵ=0","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1c0e1a86d539643c","tex":"\\delta_{\\mathrm{med}}","aliases":"δ_med delta mathrm","display":"\\delta_{\\mathrm{med}}","kind":"use","excerpt":"A stronger mediation test equalizes the workspace state after two distinct channel interventions. Let μ be a common imposed workspace distribution, implemented by a feasible stochastic intervention in the sense of Definition~[source reference]. The notation below is shorthand for that implementation. Define the residual mediation defect δ_med = sup","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1e0cd6bb0cf74492","tex":"\\delta_{\\mathrm{node}}","aliases":"δ_node delta mathrm","display":"\\delta_{\\mathrm{node}}","kind":"use","excerpt":"intact law at a fixed decision time, and let W_t=π_t( H_t) be a structurally supported candidate. Let Q⊆ N_pred(P) be a nonempty, predeclared panel of validated relation-breaking nulls. Assume: item exact structural--operational alignment, I_P(Y_t^+; H_tmid W_t)=0; item a positive node-level information gap, δ_node = min_i∈ V I_P(Y_t^+; H_tmid H_t^","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-190214dbe96624fc","tex":"\\delta_{\\mathrm{null}}","aliases":"δ_null delta mathrm","display":"\\delta_{\\mathrm{null}}","kind":"use","excerpt":"bseteq N_pred(P) be a nonempty, predeclared panel of validated relation-breaking nulls. Assume: item exact structural--operational alignment, I_P(Y_t^+; H_tmid W_t)=0; item a positive node-level information gap, δ_node = min_i∈ V I_P(Y_t^+; H_tmid H_t^i) >0; item a positive robust matched-null gap, δ_null = min_Q∈ Q [ H_Q(Y_t^+mid H_t) - H_P(Y_t^+m","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3b157fc923ac85de","tex":"\\delta_{\\mathrm{op}}","aliases":"δ_op delta mathrm","display":"\\delta_{\\mathrm{op}}","kind":"use","excerpt":"For an admissible correspondence J, define the operation defect on a declared unit ball A_1 by δ_op( J) = sup_a∈ A_1 A_ W''β-β A_ W . The primary certification object is the identity profile d_id( J) = ( q_str, δ_op( J), δ_causal(β), δ_rec(β); μ^keep ). The first four coordinates must remain below declared tolerances, while the non-collapse margin ","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e6344922b947efe1","tex":"\\delta_{\\mathrm{opt}}","aliases":"δ_opt delta mathrm","display":"\\delta_{\\mathrm{opt}}","kind":"use","excerpt":"[Approximate counterfactual policy selection] Let A be a finite feasible action set. Let Q: A→ R be the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal poli","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4c9bf89c545451f9","tex":"\\delta_P","aliases":"δ_P delta","display":"\\delta_P","kind":"use","excerpt":"[Closure under projector drift] Let widetilde P_d be another rank-d orthogonal projector and set δ_P= P_d-widetilde P_d . Then (I-widetilde P_d)ρ_V(a)widetilde P_d ≤ Q_dρ_V(a)P_d +2 ρ_V(a) δ_P. Combining this estimate with Theorem~[source reference] controls operation leakage after noisy workspace extraction.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9e83e11e41159cf8","tex":"\\Delta_Q","aliases":"Δ_Q Delta","display":"\\Delta_Q","kind":"use","excerpt":"a)] ≤ δ_opt. Then, for every true-value-optimal policy π^*, E_a∼π^*[Q(a)] - E_a∼widehatπ[Q(a)] ≤ 2ε_Q+δ_opt. For a deterministic greedy action widehat a∈argmax_awidehat Q(a), Q(a^*)-Q(widehat a) ≤2ε_Q. If the true optimal action a^* is unique and its value gap Δ_Q = Q(a^*)- max_a≠ a^*Q(a) satisfies Δ_Q>2ε_Q, then every greedy maximizer of widehat Q","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-56cf9f3fdd3374ae","tex":"\\Delta_q^{\\mathrm{dec}}","aliases":"Δ_q^dec Delta mathrm","display":"\\Delta_q^{\\mathrm{dec}}","kind":"use","excerpt":"Let m_q(W; D) be the signed margin of gate q, oriented so that m_q(W; D)>0 means that the gate passes after its uncertainty correction. Define the decomposition envelope underline m_q = inf_ D∈ Dm_q(W; D), overline m_q = sup_ D∈ Dm_q(W; D), and its sensitivity width Δ_q^dec = overline m_q-underline m_q.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6777545f7589c13f","tex":"\\Delta_q^{\\mathrm{hist}}","aliases":"Δ_q^hist Delta mathrm","display":"\\Delta_q^{\\mathrm{hist}}","kind":"use","excerpt":"component q of widehatθ, define its scaffold-only Bayes risk under loss ℓ_q by R_q^Σ = inf_f E [ ℓ_q(q,f(Σ,Ω)) ]. Define the full formation risk by R_q^Σ, H = inf_g E [ ℓ_q(q,g(Σ, D_form,Ω)) ]. The history advantage is Δ_q^hist = R_q^Σ - R_q^Σ, H. A positive history advantage means that formation history contains information about q that is not alr","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7bce5af342b70684","tex":"\\delta_q^{\\mathrm{scaf}}","aliases":"δ_q^scaf delta mathrm","display":"\\delta_q^{\\mathrm{scaf}}","kind":"use","excerpt":"For a family S of admissible scaffold variants, define the cross-scaffold identity defect δ_q^scaf = inf_{J_σσ'} E_σ,σ'∼π d_q ( J_σσ'q_σ,q_σ' ), where J_σσ' transports the learned object between admissible coordinate systems. A low value indicates that the same abstract organization is recovered under multiple non-solution-bearing scaffolds.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-de72482049a47527","tex":"\\Delta_q^{\\mathrm{spec}}","aliases":"Δ_q^spec Delta mathrm","display":"\\Delta_q^{\\mathrm{spec}}","kind":"use","excerpt":"For each gate q∈ J_*, define the specification envelope underline m_q^ S(W) = inf_ς∈ Sm_q(W;ς), overline m_q^ S(W) = sup_ς∈ Sm_q(W;ς). Its sensitivity width is Δ_q^spec(W) = overline m_q^ S(W) - underline m_q^ S(W). The robust certificate margin is r_ S(W) = inf_ς∈ S min_q∈ J_*m_q(W;ς). Then Cert_ S(W)=1 Longleftrightarrow r_ S(W)>0.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-043cc7068d423bd8","tex":"\\delta_r","aliases":"δ_r delta","display":"\\delta_r","kind":"use","excerpt":"· + ( A_ W'' + A_ W ) (1-sqrt1-q_*^2) ]. If, in addition, δ_causal(β)≤δ_c, δ_rec(β)≤δ_r, then d_id( R, R') ≤ · a q_*+b δ_op( J)","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2083b79412e7d8d3","tex":"\\delta_{\\mathrm{rec}}","aliases":"δ_rec delta mathrm","display":"\\delta_{\\mathrm{rec}}","kind":"use","excerpt":"A reconstructive handoff may also be imperfect. Let δ_rec(β) = E[d_ W'(W'_init,β W)] measure replacement initialization error. Corollary~[source reference] provides one explicit upper bound on this term.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7071d01578c9a366","tex":"\\delta_{\\mathrm{rep}}","aliases":"δ_rep delta mathrm","display":"\\delta_{\\mathrm{rep}}","kind":"use","excerpt":"· +c δ_c+d δ_r. In particular, if all local structural, operation, closure, causal, and reconstruction defects are O(δ_rep), the relevant operator norms are uniformly bounded, and g≥ g_0>0, then d_id( R, R') ≤ C_1fracδ_repg_0 +C_2δ_rep +O (fracδ_rep^2g_0). After a fixed normalization of the non-spectral terms, this recovers the schematic target d_i","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-790c4cdf07a4e8f7","tex":"\\delta_{\\mathrm{rep-id}}","aliases":"δ_rep-id delta mathrm","display":"\\delta_{\\mathrm{rep-id}}","kind":"use","excerpt":"Restoration of performance is not automatically restoration of identity. Let W_t^- be the last certified pre-damage workspace and let W_t+τ^r be the post-repair workspace. Define the repair identity defect δ_rep-id(r) = d_id ( W_t^-,W_t+τ^r ). A repair action is identity preserving at tolerance ε_id when Δ_t^rep(r;d)>0, δ_rep-id(r) ≤ ε_id. When int","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b54cbc0be565c241","tex":"\\delta_{\\mathrm{rep}}^2","aliases":"δ_rep^2 delta mathrm","display":"\\delta_{\\mathrm{rep}}^2","kind":"use","excerpt":"· +c δ_c+d δ_r. In particular, if all local structural, operation, closure, causal, and reconstruction defects are O(δ_rep), the relevant operator norms are uniformly bounded, and g≥ g_0>0, then d_id( R, R') ≤ C_1fracδ_repg_0 +C_2δ_rep +O (fracδ_rep^2g_0). After a fixed normalization of the non-spectral terms, this recovers the schematic target d_i","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e62ba1cc8022bd6e","tex":"\\delta_{\\mathrm{replacement}}","aliases":"δ_replacement delta mathrm","display":"\\delta_{\\mathrm{replacement}}","kind":"use","excerpt":"operation leakage. Coalition frame bounds also persist under sufficiently small reconstructive transport error. With declared causal-kernel and reconstructive-handoff defects, d_id ≤ a q_str +b δ_op +c δ_causal +d δ_rec. Under bounded norms and a uniform spectral gap, this gives the schematic estimate d_id lesssim Cfracδ_replacementg. The causal an","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d0446810a52779a","tex":"\\Delta_S","aliases":"Δ_S Delta","display":"\\Delta_S","kind":"use","excerpt":"[Encoding mismatch] If the true observation is y=(A_S+Δ_S)c+η but the nominal decoder A_S^† is used, then widehat c-c ≤ frac Δ_S c + η sqrtα_S.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-715efb318b8f2333","tex":"\\delta_{S}^{\\mathrm{frame}}","aliases":"δ_S^frame delta mathrm","display":"\\delta_{S}^{\\mathrm{frame}}","kind":"use","excerpt":"e bounds] Identify the old and new abstract cores with ran P and ran P' through their chosen core parameterizations. Let S be a declared coalition, let A_S:ran P→ Y_S, A_S':ran P'→ Y_S' be the resulting accessible coalition encoders, and let K_S be a unitary identification of their observation spaces. Suppose A_S'β-K_SA_S ≤ δ_S^frame < sqrtα_S. The","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-29cdf16ce0a7d6b4","tex":"\\delta_{\\mathrm{spec}}","aliases":"δ_spec delta mathrm","display":"\\delta_{\\mathrm{spec}}","kind":"use","excerpt":"Let Θ = ( ε_coh, ε_align, ε_k, underlineη_c, underlineη_loop, δ_node, δ_null, δ_spec, k, ε_pers ). Let ς = ( D,Y^+,ℓ,τ, I, Q,Θ) be the complete declared scientific specification, where Q⊆ N_pred(P) is a validated null panel. A dynamic realization contains a certified stratified relational workspace when there exists a candidate W_t satisfying all g","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-139cc041e56abc51","tex":"\\Delta_{\\mathrm{specific}}","aliases":"Δ_specific Delta mathrm","display":"\\Delta_{\\mathrm{specific}}","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8e527157ff5b715d","tex":"\\delta_{\\mathrm{suff}}","aliases":"δ_suff delta mathrm","display":"\\delta_{\\mathrm{suff}}","kind":"use","excerpt":"For an arbitrary history statistic S_t=s( H_t), let K_t^S(s,·) = P(Y_t^+∈·mid S_t=s). Whenever the conditional relative entropy is finite, define the sufficiency defect δ_suff(S_t) = E [ KL ( K_t( H_t,·) middle K_t^S(S_t,·) ) ].","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4abc637c4be411ae","tex":"\\delta_t","aliases":"δ_t delta","display":"\\delta_t","kind":"use","excerpt":"[Boundary persistence under score drift] Assume S_t^* is the unique maximizer of B_t with margin Δ_t^B, and suppose sup_S∈ S_pass | B_t+1(S)- B_t(S)| ≤δ_t. If 2δ_t<Δ_t^B, then S_t+1^*=S_t^*. More generally, B_t+1(S_t+1^*) - B_t+1(S_t^*) ≤2δ_t.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e6b19a79c1a787cd","tex":"\\Delta_t^B","aliases":"Δ_t^B Delta","display":"\\Delta_t^B","kind":"use","excerpt":"[Held-out boundary recovery] Let S_pass be finite and let S_t^* = argmax_S∈ S_pass B_t(S) be unique. Define the population margin Δ_t^B = B_t(S_t^*) - max_S≠ S_t^* B_t(S)>0. If sup_S∈ S_pass | widehat B_t(S)- B_t(S) | ≤ε_t, then every empirical maximizer widehat S_t satisfies B_t(S_t^*)- B_t(widehat S_t) ≤2ε_t. If 2ε_t<Δ_t^B, then widehat S_t=S_t^*","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c90aba688a81d40","tex":"\\Delta_t^D","aliases":"Δ_t^D Delta","display":"\\Delta_t^D","kind":"use","excerpt":"To compare successive times, choose unitary identification maps U_t^V: C_V(t)⟶ C_V(t+1), U_t^E: C_E(t)⟶ C_E(t+1). They are comparison maps, not assumptions that the physical carriers are identical. Define the one-step diagram defect Δ_t^D = D_t+1U_t^V-U_t^ED_t, δ_t^D= Δ_t^D . Exact temporal compatibility is the condition Δ_t^D=0.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a3d74ff031041308","tex":"\\delta_t^D","aliases":"δ_t^D delta","display":"\\delta_t^D","kind":"use","excerpt":"To compare successive times, choose unitary identification maps U_t^V: C_V(t)⟶ C_V(t+1), U_t^E: C_E(t)⟶ C_E(t+1). They are comparison maps, not assumptions that the physical carriers are identical. Define the one-step diagram defect Δ_t^D = D_t+1U_t^V-U_t^ED_t, δ_t^D= Δ_t^D . Exact temporal compatibility is the condition Δ_t^D=0.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2286001799319ee8","tex":"\\Delta_{\\mathrm{tr}}","aliases":"Δ_tr Delta mathrm","display":"\\Delta_{\\mathrm{tr}}","kind":"use","excerpt":"For a predeclared null Q, define Δ_tr(Q) = Risk_ℓ,Q(V) - Risk_ℓ,P(V). For a finite validated panel Q={Q_1,…,Q_m}⊆ N_pred(P), report both the complete vector Δ_tr( Q) = ( Δ_tr(Q_1),…, Δ_tr(Q_m) ) and the robust panel minimum Δ_tr^min( Q) = min_Q∈ QΔ_tr(Q). The panel must be fixed independently of the observed favorable gap. Otherwise null selection ","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7bab215e3b8c63c5","tex":"\\boldsymbol\\Delta_{\\mathrm{tr}}","aliases":"Δ_tr boldsymbol Delta mathrm","display":"\\boldsymbol\\Delta_{\\mathrm{tr}}","kind":"use","excerpt":"For a predeclared null Q, define Δ_tr(Q) = Risk_ℓ,Q(V) - Risk_ℓ,P(V). For a finite validated panel Q={Q_1,…,Q_m}⊆ N_pred(P), report both the complete vector Δ_tr( Q) = ( Δ_tr(Q_1),…, Δ_tr(Q_m) ) and the robust panel minimum Δ_tr^min( Q) = min_Q∈ QΔ_tr(Q). The panel must be fixed independently of the observed favorable gap. Otherwise null selection ","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e6de1755201ead15","tex":"\\Delta_{\\mathrm{tr}}^{\\operatorname{do}}","aliases":"Δ_tr^do Delta mathrm operatorname","display":"\\Delta_{\\mathrm{tr}}^{\\operatorname{do}}","kind":"use","excerpt":"Let jmath_E randomize selected communication edges while preserving their marginal message laws, rates, and timing budget. The causal transfer gap is Δ_tr^do(jmath_E) = Risk_ℓ, P^jmath_E(V) - Risk_ℓ, P(V). Unlike an observational null law, this quantity is anchored to an implemented channel intervention. A positive effect is evidence that the intac","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-142ea28bb093d3a6","tex":"\\Delta_{\\mathrm{tr}}^{\\min}","aliases":"Δ_tr^min Delta mathrm min","display":"\\Delta_{\\mathrm{tr}}^{\\min}","kind":"use","excerpt":"declared null Q, define Δ_tr(Q) = Risk_ℓ,Q(V) - Risk_ℓ,P(V). For a finite validated panel Q={Q_1,…,Q_m}⊆ N_pred(P), report both the complete vector Δ_tr( Q) = ( Δ_tr(Q_1),…, Δ_tr(Q_m) ) and the robust panel minimum Δ_tr^min( Q) = min_Q∈ QΔ_tr(Q). The panel must be fixed independently of the observed favorable gap. Otherwise null selection can manuf","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f93e91b59d868d9e","tex":"\\Delta_{\\{i\\}}^{\\ell}","aliases":"Δ_{i}^ℓ Delta ell","display":"\\Delta_{\\{i\\}}^{\\ell}","kind":"use","excerpt":"Define the coalition gap Δ_A^ℓ = Risk_ℓ(A)-Risk_ℓ(V) ≥0. The node-level joint advantage is Δ_joint^ℓ = min_i∈ V Δ_{i}^ℓ. Under logarithmic loss, Δ_joint^log = min_i∈ V I ( Y_t^+; H_t mid H_t^{i} ). Therefore Δ_joint^log>0 means that every individual component omits some predictive information available to the complete system.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8fb1d5e617b5ae2e","tex":"\\delta_\\rho","aliases":"δ_ρ delta rho","display":"\\delta_\\rho","kind":"use","excerpt":"Write A=ρ_V(a), A'=ρ_V'(a), and define δ_ρ(a)= A'J_V-J_VA , ℓ_a= (I-P)AP . Let A_ W=PAP, A_ W''=P'A'P'.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"[Causal repair stabilizes maintenance deficit] Let e_t≥0 be the maintenance deficit. Suppose the matched no-repair trajectory satisfies e_t+1^(0) ≤ α e_t+d_t, where d_t≥0 is exogenous damage. Suppose the implemented repair policy has the causal one-step effect e_t+1^(0)-e_t+1 ≥ κ e_t-ϵ_t. Assume 0≤β:=α-κ<1, 0≤ d_t≤overline d, 0≤ϵ_t≤overlineϵ. Then ","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a05a39430affb6df","tex":"\\varepsilon_0","aliases":"ε_0 varepsilon","display":"\\varepsilon_0","kind":"use","excerpt":"The notation absorbs lower-bounded and upper-bounded gates into one sign convention. If a causal effect must exceed c_0, then m_q=widehatη-c_0. If a defect must remain below ε_0, then m_q=ε_0-widehatδ. The complete certificate therefore passes exactly when every required signed margin is positive.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-34ccf45f2593c4d6","tex":"\\varepsilon_{\\mathrm{align}}","aliases":"ε_align varepsilon mathrm","display":"\\varepsilon_{\\mathrm{align}}","kind":"use","excerpt":"Thus neither low transfer energy nor high predictive performance alone is enough. The combined gate is δ_coh(W_t) ≤ ε_coh, δ_align^ I(W_t) ≤ ε_align.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b2a07f566324679d","tex":"\\varepsilon_{\\mathrm{coh}}","aliases":"ε_coh varepsilon mathrm","display":"\\varepsilon_{\\mathrm{coh}}","kind":"use","excerpt":"Thus neither low transfer energy nor high predictive performance alone is enough. The combined gate is δ_coh(W_t) ≤ ε_coh, δ_align^ I(W_t) ≤ ε_align.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-faafbac8084344c5","tex":"\\varepsilon_{\\mathrm{ctl}}","aliases":"ε_ctl varepsilon mathrm","display":"\\varepsilon_{\\mathrm{ctl}}","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2c547c1f0e6b70da","tex":"\\varepsilon_{\\mathrm{id}}","aliases":"ε_id varepsilon mathrm","display":"\\varepsilon_{\\mathrm{id}}","kind":"use","excerpt":"Restoration of performance is not automatically restoration of identity. Let W_t^- be the last certified pre-damage workspace and let W_t+τ^r be the post-repair workspace. Define the repair identity defect δ_rep-id(r) = d_id ( W_t^-,W_t+τ^r ). A repair action is identity preserving at tolerance ε_id when Δ_t^rep(r;d)>0, δ_rep-id(r) ≤ ε_id. When int","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d6d9fd76729b539d","tex":"\\varepsilon_{\\mathrm{int}}","aliases":"ε_int varepsilon mathrm","display":"\\varepsilon_{\\mathrm{int}}","kind":"use","excerpt":"a bounded generating family G⊆ A, define η_int( G) = sup_a∈ G ∑_e∈ Ew_e Ξ_e(a) ^2. When A is normed and its unit ball is appropriate, one may take G={a: a _ A≤1}. Exact representational universality means η_int=0, while approximate universality means η_int≤ε_int. The use of a generating family prevents the definition from depending on an unbounded ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ae06e15b8d980ead","tex":"\\varepsilon_k","aliases":"ε_k varepsilon","display":"\\varepsilon_k","kind":"use","excerpt":"Let Θ = ( ε_coh, ε_align, ε_k, underlineη_c, underlineη_loop, δ_node, δ_null, δ_spec, k, ε_pers ). Let ς = ( D,Y^+,ℓ,τ, I, Q,Θ) be the complete declared scientific specification, where Q⊆ N_pred(P) is a validated null panel. A dynamic realization contains a certified stratified relational workspace when there exists a candidate W_t satisfying all g","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d727be6eb315fbdc","tex":"\\varepsilon_{\\mathrm{leak}}","aliases":"ε_leak varepsilon mathrm","display":"\\varepsilon_{\\mathrm{leak}}","kind":"use","excerpt":"For thresholds Θ_form, define Form_Θ_form(W)=1 only when all applicable gates pass: · e_q^MDL≥ τ_q^MDL, · · e_q^causal≥ τ_q^causal, · δ_q^scaf≤ε_q^scaf, · · δ^leak≤ε_leak, · SRW_Θ(W)=1, · · Pass(N_form)=1. The final workspace gate is included because endogenous discovery of an unstable or causally irrelevant pattern is not endogenous workspace form","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dc06eebc88a8b163","tex":"\\varepsilon_n","aliases":"ε_n varepsilon","display":"\\varepsilon_n","kind":"use","excerpt":"sigma) = frac1n∑_i=1^nZ_i,q,ς-θ_q,ς, 0≤ Z_i,q,ς≤1, from independent certification units. Hoeffding's inequality and a union bound give P( sup_q,ς | widehat m_q(ς)-m_q(ς) | > sqrt frac log(2| J_*|| S|/δ) 2n ) ≤δ. Thus the choice ε_n(δ) = sqrt frac log(2| J_*|| S|/δ) 2n yields an explicit conservative certificate. Dependent episodes require a bound a","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4bf668519b237548","tex":"\\varepsilon_{\\mathrm{op}}","aliases":"ε_op varepsilon mathrm","display":"\\varepsilon_{\\mathrm{op}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6243bb961a95a698","tex":"\\varepsilon_{\\mathrm{pers}}","aliases":"ε_pers varepsilon mathrm","display":"\\varepsilon_{\\mathrm{pers}}","kind":"use","excerpt":"Let Θ = ( ε_coh, ε_align, ε_k, underlineη_c, underlineη_loop, δ_node, δ_null, δ_spec, k, ε_pers ). Let ς = ( D,Y^+,ℓ,τ, I, Q,Θ) be the complete declared scientific specification, where Q⊆ N_pred(P) is a validated null panel. A dynamic realization contains a certified stratified relational workspace when there exists a candidate W_t satisfying all g","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-872d405cb92e1319","tex":"\\varepsilon_Q","aliases":"ε_Q varepsilon","display":"\\varepsilon_Q","kind":"use","excerpt":"[Approximate counterfactual policy selection] Let A be a finite feasible action set. Let Q: A→ R be the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal poli","count":19},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a31e4c39fd7cd879","tex":"\\varepsilon_q","aliases":"ε_q varepsilon","display":"\\varepsilon_q","kind":"use","excerpt":"Let widehat m_q(ς) estimate m_q(W;ς). Suppose there are nonnegative error radii ε_q(n,δ) such that P( sup_ς∈ S | widehat m_q(ς)-m_q(ς) | ≤ ε_q(n,δ) for all q∈ J_* ) ≥1-δ. Define the empirical lower confidence envelope widehatunderline m_q,1-δ^ S = inf_ς∈ S widehat m_q(ς) - ε_q(n,δ).","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b98eb5d95a72a944","tex":"\\varepsilon_{q,n,\\eta}","aliases":"ε_q,n,η varepsilon eta","display":"\\varepsilon_{q,n,\\eta}","kind":"use","excerpt":",d_ S) with covering number N(η, S,d_ S). If both the population and empirical margin functions are L_q-Lipschitz, then sup_ς∈ S |widehat m_q(ς)-m_q(ς)| ≤ max_ς_j∈ S_η |widehat m_q(ς_j)-m_q(ς_j)| +2L_qη. For bounded empirical means, a valid radius is therefore ε_q,n,η(δ) = sqrt frac log(2| J_*| N(η, S,d_ S)/δ) 2n +2L_qη. The tuning parameter η bala","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a257294283f32006","tex":"\\varepsilon_q^{\\mathrm{scaf}}","aliases":"ε_q^scaf varepsilon mathrm","display":"\\varepsilon_q^{\\mathrm{scaf}}","kind":"use","excerpt":"For thresholds Θ_form, define Form_Θ_form(W)=1 only when all applicable gates pass: · e_q^MDL≥ τ_q^MDL, · · e_q^causal≥ τ_q^causal, · δ_q^scaf≤ε_q^scaf, · · δ^leak≤ε_leak, · SRW_Θ(W)=1, · · Pass(N_form)=1. The final workspace gate is included because endogenous discovery of an unstable or causally irrelevant pattern is not endogenous workspace form","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-369a731ef7da1eab","tex":"\\varepsilon_R","aliases":"ε_R varepsilon","display":"\\varepsilon_R","kind":"use","excerpt":"[Repair selection under value-model error] Fix a damage intervention d and a finite feasible repair set R. Let V(r)=V_t^rep(r;d) be the true expected post-repair integrity margin and let widehat V obey sup_r∈ R|widehat V(r)-V(r)| ≤ ε_R. Let widehat r∈argmax_rwidehat V(r) and r^*∈argmax_r V(r). Then V(r^*)-V(widehat r) ≤ 2ε_R. If V(r^*) ≥ ρ+2ε_R, th","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4d4822222ddc0abe","tex":"\\varepsilon_r","aliases":"ε_r varepsilon","display":"\\varepsilon_r","kind":"use","excerpt":"At level r, let K_r be the corresponding total descent operator, γ_r = λ_min^+(K_r^*K_r), K_ru_r ≤ε_r. The nearest exact compatible state then differs from u_r by at most fracε_rsqrtγ_r. Allow an additional implementation, compression, or causal-summary defect ζ_r, and define δ_r = fracε_rsqrtγ_r + ζ_r.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d08739f367199333","tex":"\\epsilon_t","aliases":"ϵ_t epsilon","display":"\\epsilon_t","kind":"use","excerpt":"[Causal repair stabilizes maintenance deficit] Let e_t≥0 be the maintenance deficit. Suppose the matched no-repair trajectory satisfies e_t+1^(0) ≤ α e_t+d_t, where d_t≥0 is exogenous damage. Suppose the implemented repair policy has the causal one-step effect e_t+1^(0)-e_t+1 ≥ κ e_t-ϵ_t. Assume 0≤β:=α-κ<1, 0≤ d_t≤overline d, 0≤ϵ_t≤overlineϵ. Then ","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-76ab81e6b93d8a15","tex":"\\varepsilon_t","aliases":"ε_t varepsilon","display":"\\varepsilon_t","kind":"use","excerpt":"Let c(t)=(c_i(t))_i∈ V ∈ C_V(t), P_t= 1_[0,ε_t](L_t). The instantaneous structural workspace coordinate is W_t^str=P_tc(t) ∈ Γ_ε_t(H(t)). When P_t varies with time, these coordinates inhabit a moving family of subspaces. Parts~X--XII will supply the transport required to compare them across time. In what follows, all sufficiency and intervention st","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-45cd55e6a4bea8cf","tex":"\\zeta_{s+1}","aliases":"ζ_s+1 zeta","display":"\\zeta_{s+1}","kind":"use","excerpt":"[Cue-controlled goal retention] Let G_s∈ R^d_G obey G_s+1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s )","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c996b134ff20da50","tex":"\\eta_A","aliases":"η_A eta","display":"\\eta_A","kind":"use","excerpt":"[Uniform erasure resilience] The encoding family is exactly reconstructible after every loss of at most k nodes if and only if α^(k)>0. If surviving observations have the form y_V∖ A=A_V∖ Ac+η_A, |A|≤ k, then the canonical survivor decoder obeys the uniform estimate widehat c_A-c ≤ frac η_A sqrtα^(k).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-69ebcadbcab7d34f","tex":"\\eta_{\\mathrm{causal}}","aliases":"η_causal eta mathrm","display":"\\eta_{\\mathrm{causal}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2ad2fc8b6b0329eb","tex":"\\eta_{\\mathrm{causal},t}","aliases":"η_causal,t eta mathrm","display":"\\eta_{\\mathrm{causal},t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-33c774256c04d88d","tex":"\\eta_{\\mathrm{causal}}^{\\max}","aliases":"η_causal^max eta mathrm max","display":"\\eta_{\\mathrm{causal}}^{\\max}","kind":"use","excerpt":"ility laws. For a matched background value b, define CE_Workspace(w,w';b) = D ( P^ι_w(Y_t^+∈·mid B_t=b), P^ι_w'(Y_t^+∈·mid B_t=b) ). For a declared comparison design ν over (w,w',b), define η_causal^ν = E_(w,w',b)∼ν [ CE_Workspace(w,w';b) ]. The maximum η_causal^max = sup_w,w',bCE_Workspace(w,w';b) may be useful for discovery, but a design-weighted","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-294adba4830a5727","tex":"\\eta_{\\mathrm{causal}}^\\nu","aliases":"η_causal^ν eta mathrm nu","display":"\\eta_{\\mathrm{causal}}^\\nu","kind":"use","excerpt":"The resulting certification report is C_dyn(W_t) =( · δ_align^ I, {Δ_A^ℓ}_A⊆ V, k_ε, Δ_tr( Q), · η_causal^ν, Δ_specific(r), η_rec(τ), η_loop(τ) ). No single coordinate certifies a relational workspace.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7c97671049b7eefa","tex":"\\eta_{\\mathrm{causal}}^{\\nu}","aliases":"η_causal^ν eta mathrm nu","display":"\\eta_{\\mathrm{causal}}^{\\nu}","kind":"use","excerpt":"Let D be a divergence or metric on probability laws. For a matched background value b, define CE_Workspace(w,w';b) = D ( P^ι_w(Y_t^+∈·mid B_t=b), P^ι_w'(Y_t^+∈·mid B_t=b) ). For a declared comparison design ν over (w,w',b), define η_causal^ν = E_(w,w',b)∼ν [ CE_Workspace(w,w';b) ]. The maximum η_causal^max = sup_w,w',bCE_Workspace(w,w';b) may be us","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-722df4f400308314","tex":"\\eta_H","aliases":"η_H eta","display":"\\eta_H","kind":"use","excerpt":"[Edgewise stability certificate] If η_H=2 D_H δ_H+δ_H^2 <fracg_d2, then P_d(H)-P_d(widetilde H) ≤ fracη_Hg_d-η_H.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-497e8ef3c0c0a68a","tex":"\\eta_{\\mathrm{int}}","aliases":"η_int eta mathrm","display":"\\eta_{\\mathrm{int}}","kind":"use","excerpt":"The edgewise defect is Ξ_e(a) = H_eρ_s(e)(a)-ρ_t(e)(a)H_e. For a bounded generating family G⊆ A, define η_int( G) = sup_a∈ G ∑_e∈ Ew_e Ξ_e(a) ^2. When A is normed and its unit ball is appropriate, one may take G={a: a _ A≤1}. Exact representational universality means η_int=0, while approximate universality means η_int≤ε_int. The use of a generating","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e111f71716f9ae8c","tex":"\\eta_L","aliases":"η_L eta","display":"\\eta_L","kind":"use","excerpt":"The corresponding carried-Laplacian defect obeys η_L := J_V^*L'J_V-L ≤ 2 D δ_D+δ_D^2. This is Proposition~[source reference] with the time indices removed.","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6f9422f129b74f8d","tex":"\\eta_{R_\\perp\\to v}","aliases":"η_R_⊥→ v eta perp to","display":"\\eta_{R_\\perp\\to v}","kind":"use","excerpt":"Define the detection effect η_D→ M, mediation effect η_M→ R, restoration effect η_R→ v, and repair specificity Δ_repair^spec = η_R→ v - η_R_⊥→ v, where R_⊥ is a matched action outside the proposed repair channel.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bddd232d77894675","tex":"\\eta_{\\mathrm{rec}}","aliases":"η_rec eta mathrm","display":"\\eta_{\\mathrm{rec}}","kind":"use","excerpt":"Temporal correlation is weaker than causal recurrence. Define the total workspace recurrence effect at lag τ by η_rec(τ) = E_(w,w',b)∼ν D ( P^ι_w(W_t+τ∈·mid B_t=b), P^ι_w'(W_t+τ∈·mid B_t=b) ). A nonzero value shows that manipulating the present workspace changes a later workspace state. It may, however, be explained by direct copying or a retained ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-773fd011e34fd7aa","tex":"\\eta_S","aliases":"η_S eta","display":"\\eta_S","kind":"use","excerpt":"The full frame operator is F_PCN=E_P^*E_P+E_C^*E_C+E_N^*E_N=2I_3. For every two-node coalition S, λ_min(F_S)=1. Thus every two-node coalition reconstructs the complete core and, for noisy observations y_S=A_Sc+η_S, widehat c-c ≤ η_S . No individual node reconstructs c, because rank(E_P) = rank(E_C) = rank(E_N) =2<3. The realization is therefore sim","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c937d02bd07db207","tex":"\\eta_t^L","aliases":"η_t^L eta","display":"\\eta_t^L","kind":"use","excerpt":"Let P_t project onto a selected rank-d low-energy workspace cluster of L_t, and set W_t=ran P_t. Write g_t=λ_d+1(L_t)-λ_d(L_t)>0 for the separating spectral gap and η_t^L=2 D_t δ_t^D+(δ_t^D)^2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2f83260916ecaff8","tex":"\\eta_{\\Workspace}","aliases":"η_Workspace eta Workspace","display":null,"kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1824f46d9de467d3","tex":"\\eta_{W\\to A}","aliases":"η_W→ A eta to","display":"\\eta_{W\\to A}","kind":"use","excerpt":"Goal-dependent action mediation is measured by feasible interventions on the goal channel while matching the current workspace and external context. For a divergence D_A, define η_G→ A = sup_g,g',w,e D_A ( P(A_t∈·midι_g,w,e), P(A_t∈·midι_g',w,e) ). Workspace-dependent action mediation is η_W→ A = sup_w,w',g,e D_A ( P(A_t∈·midι_w,g,e), P(A_t∈·midι_w","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aa596740a92286aa","tex":"\\eta_{\\perp}","aliases":"η_⊥ eta perp","display":"\\eta_{\\perp}","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"[Local compatibility without a global nonzero core] Let the only nontrivial cycle holonomy on R^2 be a rotation R_θ= cosθ · -sinθ sinθ · cosθ with θ∉2π Z. Then ker(R_θ-I)={0}. Every edge map is individually invertible and well-defined, yet the complete cycle admits no nonzero exact shared core. Pairwise translatability is therefore insufficient for","count":18},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6a830538315a1ab8","tex":"\\vartheta","aliases":"ϑ vartheta","display":"\\vartheta","kind":"use","excerpt":"If the score depends on a parameter ϑ_t and satisfies sup_S | B_ϑ(S)- B_ϑ'(S)| ≤ L_Bd(ϑ,ϑ'), then the sufficient stability condition becomes 2L_Bd(ϑ_t,ϑ_t+1) < Δ_t^B.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82326ce14a4224fc","tex":"\\Theta'","aliases":"Θ' Theta","display":"\\Theta'","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-185359e6a5840446","tex":"\\vartheta'","aliases":"ϑ' vartheta","display":"\\vartheta'","kind":"use","excerpt":"If the score depends on a parameter ϑ_t and satisfies sup_S | B_ϑ(S)- B_ϑ'(S)| ≤ L_Bd(ϑ,ϑ'), then the sufficient stability condition becomes 2L_Bd(ϑ_t,ϑ_t+1) < Δ_t^B.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c45faae0260eabc9","tex":"\\theta_*","aliases":"θ_* theta","display":"\\theta_*","kind":"use","excerpt":"[Uniform structure-selection recovery] Let T be a finite candidate family with population score J: T→ R. Let θ_* = argmin_θ∈ TJ(θ) be unique, with margin Δ_J = min_θ≠θ_* (J(θ)-J(θ_*))>0. If a held-out estimate widehat J satisfies sup_θ∈ T |widehat J(θ)-J(θ)| ≤ε, then every empirical minimizer widehatθ obeys J(widehatθ)-J(θ_*) ≤2ε. If 2ε<Δ_J, then w","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7ac8440b0dfd2286","tex":"\\Theta_A","aliases":"Θ_A Theta","display":"\\Theta_A","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bdb816c3ddad55fc","tex":"\\Theta_{\\mathrm{form}}","aliases":"Θ_form Theta mathrm","display":"\\Theta_{\\mathrm{form}}","kind":"use","excerpt":"For thresholds Θ_form, define Form_Θ_form(W)=1 only when all applicable gates pass: · e_q^MDL≥ τ_q^MDL, · · e_q^causal≥ τ_q^causal,","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a7a23af7a74160ce","tex":"\\theta_{q,\\varsigma}","aliases":"θ_q,ς theta varsigma","display":"\\theta_{q,\\varsigma}","kind":"use","excerpt":"Suppose S and J_* are finite, and each margin is estimated as widehat m_q(ς) = frac1n∑_i=1^nZ_i,q,ς-θ_q,ς, 0≤ Z_i,q,ς≤1, from independent certification units. Hoeffding's inequality and a union bound give P( sup_q,ς | widehat m_q(ς)-m_q(ς) | > sqrt frac log(2| J_*|| S|/δ) 2n ) ≤δ. Thus the choice ε_n(δ) = sqrt frac log(2| J_*|| S|/δ) 2n yields an e","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bbacdf51dcc61946","tex":"\\Theta_S","aliases":"Θ_S Theta","display":"\\Theta_S","kind":"use","excerpt":"S, define Coh(S) = tr ( P_S,dL_SP_S,d ), where P_S,d projects onto the selected low-energy workspace modes. Let Cut(S) = ∑_e:s(e)∈ S, t(e)∉ Sw_e + ∑_e:s(e)∉ S, t(e)∈ Sw_e. Let k_ε(S) be the minimal sufficient coalition size inside the candidate subsystem. A candidate is admissible only if SRW_Θ_S(W_S)=1, k_ε(S)≥2, unless the claim explicitly concer","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-82b36b6456dd8108","tex":"\\theta_t","aliases":"θ_t theta","display":"\\theta_t","kind":"use","excerpt":"Let C_V(t)= R^2 and let P_t project onto u_t=(cosθ_t,sinθ_t). If U_t^V is rotation through θ_t+1-θ_t and F_t=U_t^V, then a state x_t=u_t obeys e_t=e_t+1=0, μ_t^keep=1. The workspace changes raw coordinates but not abstract identity. If instead F_t=I, then e_t+1 = |sin(θ_t+1-θ_t)|. The same structural motion now produces a state-tracking defect beca","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4517a34f7fe01d65","tex":"\\vartheta_t","aliases":"ϑ_t vartheta","display":"\\vartheta_t","kind":"use","excerpt":"If the score depends on a parameter ϑ_t and satisfies sup_S | B_ϑ(S)- B_ϑ'(S)| ≤ L_Bd(ϑ,ϑ'), then the sufficient stability condition becomes 2L_Bd(ϑ_t,ϑ_t+1) < Δ_t^B.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d2d3e917335e1a86","tex":"\\theta_{t+1}","aliases":"θ_t+1 theta","display":"\\theta_{t+1}","kind":"use","excerpt":"Let C_V(t)= R^2 and let P_t project onto u_t=(cosθ_t,sinθ_t). If U_t^V is rotation through θ_t+1-θ_t and F_t=U_t^V, then a state x_t=u_t obeys e_t=e_t+1=0, μ_t^keep=1. The workspace changes raw coordinates but not abstract identity. If instead F_t=I, then e_t+1 = |sin(θ_t+1-θ_t)|. The same structural motion now produces a state-tracking defect beca","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bc02e763ece78cfe","tex":"\\vartheta_{t+1}","aliases":"ϑ_t+1 vartheta","display":"\\vartheta_{t+1}","kind":"use","excerpt":"If the score depends on a parameter ϑ_t and satisfies sup_S | B_ϑ(S)- B_ϑ'(S)| ≤ L_Bd(ϑ,ϑ'), then the sufficient stability condition becomes 2L_Bd(ϑ_t,ϑ_t+1) < Δ_t^B.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5574f59676716d19","tex":"\\iota'","aliases":"ι' iota","display":"\\iota'","kind":"use","excerpt":"Define η_Workspace(r) = E_ι,ι'∼ I_Workspace(r) D ( P^ι(Y_t^+), P^ι'(Y_t^+) ), η_⊥(r) = E_jmath,jmath'∼ I_⊥(r) D ( P^jmath(Y_t^+), P^jmath'(Y_t^+) ), and Δ_specific(r) = η_Workspace(r)-η_⊥(r). A positive value supports causal concentration in the candidate workspace at the declared perturbation scale.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f07be25ecbec5c33","tex":"\\iota_{a'}","aliases":"ι_a' iota","display":"\\iota_{a'}","kind":"use","excerpt":". For a divergence D_A, define η_G→ A = sup_g,g',w,e D_A ( P(A_t∈·midι_g,w,e), P(A_t∈·midι_g',w,e) ). Workspace-dependent action mediation is η_W→ A = sup_w,w',g,e D_A ( P(A_t∈·midι_w,g,e), P(A_t∈·midι_w',g,e) ). An action must also alter the declared future: η_A→ Y = sup_a,a',b D_Y ( P(Y_t^+∈·midι_a,b), P(Y_t^+∈·midι_a',b) ). A positive η_G→ A wit","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-55b639b1008945e0","tex":"\\iota_{\\bar S,u}","aliases":"ι_bar S,u iota bar","display":"\\iota_{\\bar S,u}","kind":"use","excerpt":"nate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the future law of X_S: CIn_τ(S) = E_u,u'∼ν_bar S D ( L^ι_bar S,u (X_S(t+τ)), L^ι_bar ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fd6a82ebe041efa1","tex":"\\iota_{\\bar S,u'}","aliases":"ι_bar S,u' iota bar","display":"\\iota_{\\bar S,u'}","kind":"use","excerpt":"nate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the future law of X_S: CIn_τ(S) = E_u,u'∼ν_bar S D ( L^ι_bar S,u (X_S(t+τ)), L^ι_bar ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6f6ab2e20492b4a4","tex":"\\iota_{g(w)}'","aliases":"ι_g(w)' iota","display":"\\iota_{g(w)}'","kind":"use","excerpt":"[Coordinate invariance of intervention effects] Let g: W_t→ W_t' be a measurable bijection and suppose the reparametrized intervention family satisfies ι_g(w)'=ι_w. Then every divergence-based causal effect is preserved: CE_Workspace'(g(w),g(w');b) = CE_Workspace(w,w';b).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-af4ad2f7e36d6853","tex":"\\iota_i^C","aliases":"ι_i^C iota","display":"\\iota_i^C","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d9cac9ef44859958","tex":"\\iota_i^R","aliases":"ι_i^R iota","display":"\\iota_i^R","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dcce5e559374e36f","tex":"\\iota_{i_0}^C","aliases":"ι_i_0^C iota","display":"\\iota_{i_0}^C","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ebb48c926483aa3f","tex":"\\iota_{i_m}^C","aliases":"ι_i_m^C iota","display":"\\iota_{i_m}^C","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5152a666f8aca85e","tex":"\\iota_{i_{r-1}}^C","aliases":"ι_i_r-1^C iota","display":"\\iota_{i_{r-1}}^C","kind":"use","excerpt":"For every step, T_rι_i_r-1^C = ι_i_r^CH_r+ ι_i_r^RV_r. Telescope the sequence T_m⋯ T_r+1ι_i_r^C H_r⋯ H_1, r=0,1,…,m. The difference between consecutive terms is exactly the r-th summand in the displayed formula. Projection onto R_i_m or C_i_m, followed by submultiplicativity of the operator norm, gives the two bounds.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9218e9bf0be8cc52","tex":"\\iota_{i_r}^C","aliases":"ι_i_r^C iota","display":"\\iota_{i_r}^C","kind":"use","excerpt":"For every step, T_rι_i_r-1^C = ι_i_r^CH_r+ ι_i_r^RV_r. Telescope the sequence T_m⋯ T_r+1ι_i_r^C H_r⋯ H_1, r=0,1,…,m. The difference between consecutive terms is exactly the r-th summand in the displayed formula. Projection onto R_i_m or C_i_m, followed by submultiplicativity of the operator norm, gives the two bounds.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bb7ac6ec63599e0b","tex":"\\iota_{i_r}^R","aliases":"ι_i_r^R iota","display":"\\iota_{i_r}^R","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2313bff3eabcc5ab","tex":"\\iota_{q\\leftarrow a}","aliases":"ι_q← a iota leftarrow","display":"\\iota_{q\\leftarrow a}","kind":"use","excerpt":"A feasible statement intervention must alter both local realizations of the statement coordinate: ι_q← a: (q_P,q_N)←(a,a), while holding (p,f) fixed. For configurations with fixed p and f, changing a flips the downstream decision with total-variation effect one: TV ( P(Ymidι_q←0), P(Ymidι_q←1) ) =1. A matched intervention on a residual notebook-for","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dc548c4867a191d5","tex":"\\iota_{q\\leftarrow0}","aliases":"ι_q←0 iota leftarrow","display":"\\iota_{q\\leftarrow0}","kind":"use","excerpt":"A feasible statement intervention must alter both local realizations of the statement coordinate: ι_q← a: (q_P,q_N)←(a,a), while holding (p,f) fixed. For configurations with fixed p and f, changing a flips the downstream decision with total-variation effect one: TV ( P(Ymidι_q←0), P(Ymidι_q←1) ) =1. A matched intervention on a residual notebook-for","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b94d2596d6daee6f","tex":"\\iota_{q\\leftarrow1}","aliases":"ι_q←1 iota leftarrow","display":"\\iota_{q\\leftarrow1}","kind":"use","excerpt":"A feasible statement intervention must alter both local realizations of the statement coordinate: ι_q← a: (q_P,q_N)←(a,a), while holding (p,f) fixed. For configurations with fixed p and f, changing a flips the downstream decision with total-variation effect one: TV ( P(Ymidι_q←0), P(Ymidι_q←1) ) =1. A matched intervention on a residual notebook-for","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fdc3b0afd9fbc654","tex":"\\iota_{S,x}","aliases":"ι_S,x iota","display":"\\iota_{S,x}","kind":"use","excerpt":"Let ι_S,x be a feasible intervention that changes the internal workspace coordinate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the fu","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-61123adaba2d7c2d","tex":"\\iota_{S,x'}","aliases":"ι_S,x' iota","display":"\\iota_{S,x'}","kind":"use","excerpt":"Let ι_S,x be a feasible intervention that changes the internal workspace coordinate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the fu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"The lower frame bound α_S measures recoverability. The upper bound β_S measures amplification in the forward encoding. Their ratio κ_S=sqrtfracβ_Sα_S is the coalition condition number.","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9a5303f916822ab6","tex":"\\kappa_q","aliases":"κ_q kappa","display":"\\kappa_q","kind":"use","excerpt":"A robust procedure can be valid yet uninformative when the family is too large or the data are insufficient. Each gate should therefore report an indifference region [-κ_q,κ_q]. A conclusion is: item[Robust pass] when the lower confidence envelope exceeds κ_q for every gate. item[Robust fail] when a required gate's upper confidence envelope is belo","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-49c710b9b0e78982","tex":"\\kappa_S","aliases":"κ_S kappa","display":"\\kappa_S","kind":"use","excerpt":"The lower frame bound α_S measures recoverability. The upper bound β_S measures amplification in the forward encoding. Their ratio κ_S=sqrtfracβ_Sα_S is the coalition condition number.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-08209adac2de3dd9","tex":"\\lambda^3","aliases":"λ^3 lambda","display":"\\lambda^3","kind":"use","excerpt":"For m_t+1=tanh(λ m_t), there is a unique zero fixed point for 0≤λ≤1. For λ>1, the zero state is unstable and exactly two stable nonzero fixed points appear. Their amplitude satisfies m_λ = sqrtfrac3(λ-1)λ^3 + O((λ-1)^3/2). This is an exact structural benchmark, not a universal law for complete relational workspaces.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e7d367cdec71a6b4","tex":"\\lambda_+","aliases":"λ_+ lambda","display":"\\lambda_+","kind":"use","excerpt":"-(a+b) · b^2+1 . Writing τ=a^2+b^2+2, the eigenvalues are λ_± = frac12 ( τ±sqrtτ^2-4(ab-1)^2 ), and det L_H=(ab-1)^2. Thus a nonzero exact shared core exists precisely when ab=1, which is exactly the trivial-holonomy condition. Away from exact consistency, λ_-λ_+=(ab-1)^2. When |ab-1| is small while λ_+ stays bounded away from zero, the soft worksp","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-16b1a787c83a448c","tex":"\\lambda_-","aliases":"λ_- lambda","display":"\\lambda_-","kind":"use","excerpt":"-(a+b) · b^2+1 . Writing τ=a^2+b^2+2, the eigenvalues are λ_± = frac12 ( τ±sqrtτ^2-4(ab-1)^2 ), and det L_H=(ab-1)^2. Thus a nonzero exact shared core exists precisely when ab=1, which is exactly the trivial-holonomy condition. Away from exact consistency, λ_-λ_+=(ab-1)^2. When |ab-1| is small while λ_+ stays bounded away from zero, the soft worksp","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-65d44243a226ac8f","tex":"\\lambda_c","aliases":"λ_c lambda","display":"\\lambda_c","kind":"use","excerpt":"The symbolic outline proposed a coupling threshold λ_c. A general theory must first state what is allowed to vary with λ. A threshold may concern structural coherence, causal efficacy, relational irreducibility, or the conjunction of all certification gates. These need not occur at the same parameter value.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-395ecbe832b217cc","tex":"\\lambda_{\\mathrm{causal}}","aliases":"λ_causal lambda mathrm","display":"\\lambda_{\\mathrm{causal}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6922b8c7dfba7e78","tex":"\\lambda_{\\mathrm{coh}}","aliases":"λ_coh lambda mathrm","display":"\\lambda_{\\mathrm{coh}}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ea9e4fc388451ee7","tex":"\\lambda_E","aliases":"λ_E lambda","display":"\\lambda_E","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ae24fb61e85a5791","tex":"\\lambda_e^{\\mathrm{in}}","aliases":"λ_e^in lambda mathrm","display":"\\lambda_e^{\\mathrm{in}}","kind":"use","excerpt":"Core leakage, residual contamination, and total transfer defect are measured by λ_e^out= V_e , λ_e^in= U_e , δ_e^tr = U_e + V_e . These quantities depend on the chosen norms and projectors. Their empirical meaning must therefore be tied to a declared intervention and prediction protocol.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e3f9c5238a77d2c2","tex":"\\lambda_e^{\\mathrm{out}}","aliases":"λ_e^out lambda mathrm","display":"\\lambda_e^{\\mathrm{out}}","kind":"use","excerpt":"Core leakage, residual contamination, and total transfer defect are measured by λ_e^out= V_e , λ_e^in= U_e , δ_e^tr = U_e + V_e . These quantities depend on the chosen norms and projectors. Their empirical meaning must therefore be tied to a declared intervention and prediction protocol.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-13cd552e06c99d64","tex":"\\lambda_{\\mathrm{form}}","aliases":"λ_form lambda mathrm","display":"\\lambda_{\\mathrm{form}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6360b8efa50c4155","tex":"\\lambda_{k+1}","aliases":"λ_k+1 lambda","display":"\\lambda_{k+1}","kind":"use","excerpt":"The spectrum supports three distinct interpretations: rcl λ_k=0 · Longleftrightarrow · exact compatible direction, 0<λ_kll g_d · Longrightarrow · approximately transported direction, λ_k≈λ_k+1 · Longrightarrow · rank selection is unstable at that boundary. A low eigenvalue can still be scientifically spurious if it arises from a poorly chosen weigh","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b78bb4380d0f5c22","tex":"\\lambda_{\\mathrm{leak}}","aliases":"λ_leak lambda mathrm","display":"\\lambda_{\\mathrm{leak}}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-22452671e06f3b7f","tex":"\\lambda_{\\max}","aliases":"λ_max lambda max","display":"\\lambda_{\\max}","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1dc8e3b2125ff219","tex":"\\lambda_{\\min}","aliases":"λ_min lambda min","display":"\\lambda_{\\min}","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4d7ac3c10502fde2","tex":"\\lambda_{\\min}^+","aliases":"λ_min^+ lambda min","display":"\\lambda_{\\min}^+","kind":"use","excerpt":"Let γ_ U^meta = λ_min^+ (L_ U^meta) be the first positive eigenvalue.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-62ed57f1458fb941","tex":"\\Lambda_n","aliases":"Λ_n Lambda","display":"\\Lambda_n","kind":"use","excerpt":"Let Q(λ) be a gate-specific population order parameter and widehat Q_n(λ) its held-out estimate. If sup_λ∈Λ_n |widehat Q_n(λ)-Q(λ)| ≤ε_n on a finite evaluation grid Λ_n, then every threshold crossing with vertical margin larger than 2ε_n is recovered by the same uniform-selection argument as Theorem~[source reference]. Near a continuous transition,","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-94cabd86d6252e9d","tex":"\\lambda_{\\mathrm{op}}","aliases":"λ_op lambda mathrm","display":"\\lambda_{\\mathrm{op}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-42349672dca846cc","tex":"\\lambda_{\\mathrm{ov}}","aliases":"λ_ov lambda mathrm","display":"\\lambda_{\\mathrm{ov}}","kind":"use","excerpt":"ha∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U = widehat D_ U","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6219c57013db378b","tex":"\\lambda_{r+1}","aliases":"λ_r+1 lambda","display":"\\lambda_{r+1}","kind":"use","excerpt":"[Coherence-to-core distance estimate] Let W_0=Γ(H)=ker L_H, r=dim W_0, and suppose r<N. Set γ_H=λ_r+1>0. Then every c∈ C_V satisfies dist(c,W_0)^2 ≤ frac E_H(c)γ_H. Moreover, E_H(c) ≤ L_H dist(c,W_0)^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6a088a1032a46db7","tex":"\\lambda_{\\mathrm{rel}}","aliases":"λ_rel lambda mathrm","display":"\\lambda_{\\mathrm{rel}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-915d29e65de51f7b","tex":"\\lambda_{\\mathrm{stab}}","aliases":"λ_stab lambda mathrm","display":"\\lambda_{\\mathrm{stab}}","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-158a9879a8e53726","tex":"\\lambda_{\\mathrm{str}}","aliases":"λ_str lambda mathrm","display":"\\lambda_{\\mathrm{str}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-412328d9c2d6574f","tex":"\\lambda_V","aliases":"λ_V lambda","display":"\\lambda_V","kind":"use","excerpt":"Let z_i∈{0,1}, a_e∈[0,1] be node-membership and edge-activation variables satisfying a_e≤ z_s(e), a_e≤ z_t(e). A weighted learned discrepancy operator is (D_a,Hc)_e = sqrtw_ea_e ( c_t(e)-H_ec_s(e) ). One generic formation objective is widehatθ ∈ argmin_z,a,H,P, O,B { · widehat R_task + λ_coh D_a,Hc ^2 + λ_E a _1 + λ_V z _1 · + λ_stab J_stab + λ_lea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4b9ffb77538c9f7d","tex":"\\lambda_\\pm","aliases":"λ_± lambda pm","display":"\\lambda_\\pm","kind":"use","excerpt":"-(a+b) · b^2+1 . Writing τ=a^2+b^2+2, the eigenvalues are λ_± = frac12 ( τ±sqrtτ^2-4(ab-1)^2 ), and det L_H=(ab-1)^2. Thus a nonzero exact shared core exists precisely when ab=1, which is exactly the trivial-holonomy condition. Away from exact consistency, λ_-λ_+=(ab-1)^2. When |ab-1| is small while λ_+ stays bounded away from zero, the soft worksp","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"On ran P and ranwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P","count":28},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-db413c605ed4b317","tex":"\\mu^{\\mathrm{keep}}","aliases":"μ^keep mu mathrm","display":"\\mu^{\\mathrm{keep}}","kind":"use","excerpt":"For an admissible correspondence J, define the operation defect on a declared unit ball A_1 by δ_op( J) = sup_a∈ A_1 A_ W''β-β A_ W . The primary certification object is the identity profile d_id( J) = ( q_str, δ_op( J), δ_causal(β), δ_rec(β); μ^keep ). The first four coordinates must remain below declared tolerances, while the non-collapse margin ","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-501a3f9b0709cb6c","tex":"\\mu_c","aliases":"μ_c mu","display":"\\mu_c","kind":"use","excerpt":"[Threshold values are parameterization dependent] Suppose a model indexed by λ passes a declared formation certificate exactly for λ>λ_c. For every strictly increasing bijection f, the reparameterized family indexed by μ=f(λ) has threshold μ_c=f(λ_c). Consequently, the numerical value of a coupling threshold is not invariant without a physically or","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d0d626aecb12af2d","tex":"\\mu_t^{\\mathrm{keep}}","aliases":"μ_t^keep mu mathrm","display":"\\mu_t^{\\mathrm{keep}}","kind":"use","excerpt":"Low off-workspace error is not sufficient for meaningful persistence. The zero update F_t=0 drives every state exactly into every linear workspace while destroying all content. Define the internal transport B_t=P_t+1F_tP_t: W_t⟶ W_t+1 and its non-collapse margin μ_t^keep = σ_min(B_t|_ W_t). A content-preserving persistence claim requires μ_t^keep>0","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1499b33525b25770","tex":"\\nu_{\\bar S}","aliases":"ν_bar S nu bar","display":"\\nu_{\\bar S}","kind":"use","excerpt":"nate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the future law of X_S: CIn_τ(S) = E_u,u'∼ν_bar S D ( L^ι_bar S,u (X_S(t+τ)), L^ι_bar ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-25d0db0451aa9725","tex":"\\nu_S","aliases":"ν_S nu","display":"\\nu_S","kind":"use","excerpt":"Let ι_S,x be a feasible intervention that changes the internal workspace coordinate of S to x while matching background variables and intervention cost. Define outward causal efficacy COut_τ(S) = E_x,x'∼ν_S D ( L^ι_S,x (X_bar S(t+τ)), L^ι_S,x' (X_bar S(t+τ)) ). Define inward causal dependence analogously by intervening on bar S and measuring the fu","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-937e33566999a075","tex":"\\nu_{\\alpha\\beta}","aliases":"ν_αβ nu alpha beta","display":"\\nu_{\\alpha\\beta}","kind":"use","excerpt":"A family w=(w_α)_α is exactly compatible when δ^0w=0. Its direct mismatch is m_ U(w)=δ^0w. After choosing inner products and positive overlap weights, the mismatch energy is E_ov(w) = δ^0w ^2 = ∑_α<β ν_αβ r_αβ^βw_β -r_αβ^αw_α ^2. This is the appropriate obstruction when the full local states are already given. Such a family glues exactly if and onl","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c645657d025cc351","tex":"\\Xi","aliases":"Ξ Xi","display":"\\Xi","kind":"use","excerpt":"The edgewise defect is Ξ_e(a) = H_eρ_s(e)(a)-ρ_t(e)(a)H_e. For a bounded generating family G⊆ A, define η_int( G) = sup_a∈ G ∑_e∈ Ew_e Ξ_e(a) ^2. When A is normed and its unit ball is appropriate, one may take G={a: a _ A≤1}. Exact representational universality means η_int=0, while approximate universality means η_int≤ε_int. The use of a generating","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b50e13f14213622c","tex":"\\Xi_e","aliases":"Ξ_e Xi","display":"\\Xi_e","kind":"use","excerpt":"The edgewise defect is Ξ_e(a) = H_eρ_s(e)(a)-ρ_t(e)(a)H_e. For a bounded generating family G⊆ A, define η_int( G) = sup_a∈ G ∑_e∈ Ew_e Ξ_e(a) ^2. When A is normed and its unit ball is appropriate, one may take G={a: a _ A≤1}. Exact representational universality means η_int=0, while approximate universality means η_int≤ε_int. The use of a generating","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-79d6c0b3ad220a54","tex":"\\xi_r","aliases":"ξ_r xi","display":"\\xi_r","kind":"use","excerpt":"[Deterministic empirical recovery bound] For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bfa08d60e4e1fbda","tex":"\\xi_t","aliases":"ξ_t xi","display":"\\xi_t","kind":"use","excerpt":"Let the realized vertex state satisfy x_t+1=F_tx_t+ξ_t, where F_t: C_V(t)→ C_V(t+1) is the actual update and ξ_t is an unmodeled disturbance. Define the off-workspace error e_t= (I-P_t)x_t . Introduce α_t = (I-P_t+1)F_t(I-P_t) , which measures persistence of prior off-workspace error, and ℓ_t = (I-P_t+1)F_tP_t , which measures leakage of a valid wo","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cdfec8b4593aed39","tex":"\\xi_{t+1}","aliases":"ξ_t+1 xi","display":"\\xi_{t+1}","kind":"use","excerpt":"C_t^ctl = ( W_t,G_t,M_t,B_t,E_t ), where G_t is the represented goal, M_t is a workspace-integrity monitor, B_t is the current boundary or routing state, and E_t is the declared external context. An action is generated by A_t ∼ π_t ( ·mid W_t,G_t,M_t,B_t,E_t ). The future controlled state obeys a time-unrolled structural model S_t+1 = F_t(S_t,A_t,E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-37cd1251b390f507","tex":"\\pi^*","aliases":"π^* pi","display":"\\pi^*","kind":"use","excerpt":"thbb R be the true interventional value and let widehat Q satisfy sup_a∈ A|widehat Q(a)-Q(a)|≤ε_Q. Suppose a policy widehatπ is δ_opt-optimal for widehat Q: max_π E_a∼π[widehat Q(a)] - E_a∼widehatπ[widehat Q(a)] ≤ δ_opt. Then, for every true-value-optimal policy π^*, E_a∼π^*[Q(a)] - E_a∼widehatπ[Q(a)] ≤ 2ε_Q+δ_opt. For a deterministic greedy action","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-222fb49e971fef73","tex":"\\pi_1","aliases":"π_1 pi","display":"\\pi_1","kind":"use","excerpt":"Fix a root o∈ V. For every loop class [γ]∈π_1(|G|,o), define the holonomy Hol([γ])=H_γ∈ GL(C_o). Functoriality gives a group representation hol_o: π_1(|G|,o)⟶ GL(C_o). Its image is the holonomy group Hol_o=im(hol_o).","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d2023561fc861dfb","tex":"\\pi_{\\Workspace}","aliases":"π_Workspace pi Workspace","display":null,"kind":"use","excerpt":"Pointwise addition, scalar multiplication, and composition make O_Workspace a unital associative algebra. Each A=(A_i) acts on global sections by π_Workspace(A)(c_i)=(A_ic_i). Naturality guarantees that (A_ic_i) is again compatible.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a9fc1ff1221962c3","tex":"\\rho'","aliases":"ρ' rho","display":"\\rho'","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c07e947a51646c8e","tex":"\\rho^{G_\\alpha}_{G_{\\alpha\\beta}}","aliases":"ρ^G_α_G_αβ rho alpha alpha beta","display":"\\rho^{G_\\alpha}_{G_{\\alpha\\beta}}","kind":"use","excerpt":"[0.2em] sqrtλ_ov S_ U , L_ U^tot = K_ U^*K_ U. Its energy is E_ U^tot(u) = · ∑_α D_G_αu_α ^2 · + λ_ov ∑_α<β ρ^G_β_G_αβu_β - ρ^G_α_G_αβu_α ^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-99a384bc5fc3de39","tex":"\\rho^{G_\\beta}_{G_{\\alpha\\beta}}","aliases":"ρ^G_β_G_αβ rho beta alpha beta","display":"\\rho^{G_\\beta}_{G_{\\alpha\\beta}}","kind":"use","excerpt":"[0.2em] sqrtλ_ov S_ U , L_ U^tot = K_ U^*K_ U. Its energy is E_ U^tot(u) = · ∑_α D_G_αu_α ^2 · + λ_ov ∑_α<β ρ^G_β_G_αβu_β - ρ^G_α_G_αβu_α ^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-737cdd19f67dfe28","tex":"\\rho_1","aliases":"ρ_1 rho","display":"\\rho_1","kind":"use","excerpt":"[Equivalent operations in different coordinates] Let C_1=C_2= R^2, choose S∈ GL_2( R), and set H_12=S, ρ_2(a)=Sρ_1(a)S^-1. Then H_12ρ_1(a) =Sρ_1(a) =ρ_2(a)S =ρ_2(a)H_12. The local matrices can look different while realizing the same abstract operation. This is the elementary model of representation change used throughout the theory.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d7ba0ffb2518dd65","tex":"\\rho_2","aliases":"ρ_2 rho","display":"\\rho_2","kind":"use","excerpt":"[Equivalent operations in different coordinates] Let C_1=C_2= R^2, choose S∈ GL_2( R), and set H_12=S, ρ_2(a)=Sρ_1(a)S^-1. Then H_12ρ_1(a) =Sρ_1(a) =ρ_2(a)S =ρ_2(a)H_12. The local matrices can look different while realizing the same abstract operation. This is the elementary model of representation change used throughout the theory.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-80f34d32227ee3d9","tex":"\\rho_E","aliases":"ρ_E rho","display":"\\rho_E","kind":"use","excerpt":"Define the vertex and edge direct-sum representations ρ_V(a)=⊕_i∈ Vρ_i(a), ρ_E(a)=⊕_e∈ Eρ_t(e)(a). Lift the edgewise intertwining defects to ( X_a c)_e = sqrtw_e Ξ_e(a)c_s(e). Then the exact commutator identity is D_Hρ_V(a)-ρ_E(a)D_H=- X_a. Furthermore, X_a ≤ β(a), β(a)^2 = max_i∈ V ∑_e:s(e)=iw_e Ξ_e(a) ^2.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c8d860445b1eabf","tex":"\\rho_{G_{\\alpha\\beta}}^{G_\\alpha}","aliases":"ρ_G_αβ^G_α rho alpha beta alpha","display":"\\rho_{G_{\\alpha\\beta}}^{G_\\alpha}","kind":"use","excerpt":"l candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U = wi","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f88f1220c66e74cd","tex":"\\rho_{G_{\\alpha\\beta}}^{G_\\beta}","aliases":"ρ_G_αβ^G_β rho alpha beta beta","display":"\\rho_{G_{\\alpha\\beta}}^{G_\\beta}","kind":"use","excerpt":"l candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U = wi","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d897ac2a9175d02a","tex":"\\rho_i^g","aliases":"ρ_i^g rho","display":"\\rho_i^g","kind":"use","excerpt":"For each i∈ V, let g_i∈ GL(C_i), g=(g_i)_i∈ V. Define the vertex and edge direct sums G_V=⊕_i∈ Vg_i, G_E=⊕_e∈ Eg_t(e). The transformed transfers and local operation representations are H_e^g = g_t(e)H_eg_s(e)^-1, ρ_i^g(a) = g_iρ_i(a)g_i^-1. The fundamental covariance identity is D_H^gG_V=G_ED_H. It follows directly by evaluating the e-component: sq","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aaa4c9bd206e21e1","tex":"\\rho_k","aliases":"ρ_k rho","display":"\\rho_k","kind":"use","excerpt":"For composable intertwiners H_e:C_i→ C_j and H_f:C_j→ C_k, H_fH_eρ_i(a) =H_fρ_j(a)H_e =ρ_k(a)H_fH_e. Induction over path length proves the claim.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c42525ad6d24417","tex":"\\rho_K^L","aliases":"ρ_K^L rho","display":"\\rho_K^L","kind":"use","excerpt":"If K⊆ L⊆ G are subgraphs, coordinate restriction gives ρ_K^L: C_V(L)⟶ C_V(K). Because every edge equation on K is also an edge equation on L, restriction maps exact sections to exact sections: r_K^L := ρ_K^L|_mathscr W(L): mathscr W(L)⟶mathscr W(K). They obey r_L^L=I, r_J^K r_K^L=r_J^L whenever J⊆ K⊆ L. Thus K↦mathscr W(K) is a contravariant system","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bd8a8d118e12ed91","tex":"\\rho_o","aliases":"ρ_o rho","display":"\\rho_o","kind":"use","excerpt":"[Holonomy commutes with abstract operations] For every loop γ:o→ o, H_γρ_o(a)=ρ_o(a)H_γ. Hence ρ_o( A) ⊆ Comm(Hol_o), where Comm(Hol_o) is the commutant of the holonomy representation.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4348bf4aac85b7fa","tex":"\\rho_{s(e)}","aliases":"ρ_s(e) rho","display":"\\rho_{s(e)}","kind":"use","excerpt":"The edgewise defect is Ξ_e(a) = H_eρ_s(e)(a)-ρ_t(e)(a)H_e. For a bounded generating family G⊆ A, define η_int( G) = sup_a∈ G ∑_e∈ Ew_e Ξ_e(a) ^2. When A is normed and its unit ball is appropriate, one may take G={a: a _ A≤1}. Exact representational universality means η_int=0, while approximate universality means η_int≤ε_int. The use of a generating","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a040e3b0bd580d0e","tex":"\\rho_{t(e)}","aliases":"ρ_t(e) rho","display":"\\rho_{t(e)}","kind":"use","excerpt":"The edgewise defect is Ξ_e(a) = H_eρ_s(e)(a)-ρ_t(e)(a)H_e. For a bounded generating family G⊆ A, define η_int( G) = sup_a∈ G ∑_e∈ Ew_e Ξ_e(a) ^2. When A is normed and its unit ball is appropriate, one may take G={a: a _ A≤1}. Exact representational universality means η_int=0, while approximate universality means η_int≤ε_int. The use of a generating","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-65140941a2e66db3","tex":"\\rho_\\Gamma","aliases":"ρ_Γ rho Gamma","display":"\\rho_\\Gamma","kind":"use","excerpt":"[Global operation closure] Suppose every core transfer intertwines the representations of A: H_eρ_s(e)(a)=ρ_t(e)(a)H_e ∀ e∈ E, a∈ A. Then Γ(H) is an A-module under the componentwise action ρ_Γ(a)(c_i)_i∈ V = (ρ_i(a)c_i)_i∈ V. Equivalently, the holonomy-fixed root core is invariant: ρ_o(a)C_*⊆ C_* ∀ a∈ A.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b8a1bf1bab77ec11","tex":"\\sigma'","aliases":"σ' sigma","display":"\\sigma'","kind":"use","excerpt":"For a family S of admissible scaffold variants, define the cross-scaffold identity defect δ_q^scaf = inf_{J_σσ'} E_σ,σ'∼π d_q ( J_σσ'q_σ,q_σ' ), where J_σσ' transports the learned object between admissible coordinate systems. A low value indicates that the same abstract organization is recovered under multiple non-solution-bearing scaffolds.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a22772dc01f8bdd6","tex":"\\Sigma^{(1)}","aliases":"Σ^(1) Sigma","display":"\\Sigma^{(1)}","kind":"use","excerpt":"Train the same formation rule under several scaffolds Σ^(1),…,Σ^(K) that differ in node labels, coordinate systems, and communication topology but support the same abstract task. After gauge alignment, measure whether the learned organizations transport to a common equivalence class: [widehat H^(1)]_ G ≅⋯≅ [widehat H^(K)]_ G. A structure that appea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5cfba02fd9c6eba8","tex":"\\Sigma^{(K)}","aliases":"Σ^(K) Sigma","display":"\\Sigma^{(K)}","kind":"use","excerpt":"Train the same formation rule under several scaffolds Σ^(1),…,Σ^(K) that differ in node labels, coordinate systems, and communication topology but support the same abstract task. After gauge alignment, measure whether the learned organizations transport to a common equivalence class: [widehat H^(1)]_ G ≅⋯≅ [widehat H^(K)]_ G. A structure that appea","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bbc21cd7a488e5b8","tex":"\\Sigma_{\\mathcal O}","aliases":"Σ_ O Sigma mathcal","display":"\\Sigma_{\\mathcal O}","kind":"use","excerpt":"The scaffold is itself stratified: Σ = ( Σ_V, Σ_E, Σ_H, Σ_P, Σ_ O, Σ_B, Σ_Y ). The final coordinate Σ_Y records labels, targets, privileged masks, or other supervision that could directly reveal the intended workspace. This coordinate is necessary because an architecture may appear to discover a boundary even when the desired membership mask was en","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d0331456e52105eb","tex":"\\sigma_0","aliases":"σ_0 sigma","display":"\\sigma_0","kind":"use","excerpt":"Repeated substitution yields e_t · ≤ α_t-1⋯α_0e_0 + α_t-1⋯α_1σ_0 + ⋯ + σ_t-1, which is the first formula. Under the uniform bounds, dominate each product by the corresponding power of barα and sum the geometric series.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3a0a6ab4567976dd","tex":"\\Sigma_B","aliases":"Σ_B Sigma","display":"\\Sigma_B","kind":"use","excerpt":"The scaffold is itself stratified: Σ = ( Σ_V, Σ_E, Σ_H, Σ_P, Σ_ O, Σ_B, Σ_Y ). The final coordinate Σ_Y records labels, targets, privileged masks, or other supervision that could directly reveal the intended workspace. This coordinate is necessary because an architecture may appear to discover a boundary even when the desired membership mask was en","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-698299c20910a5f0","tex":"\\Sigma_E","aliases":"Σ_E Sigma","display":"\\Sigma_E","kind":"use","excerpt":"The scaffold is itself stratified: Σ = ( Σ_V, Σ_E, Σ_H, Σ_P, Σ_ O, Σ_B, Σ_Y ). The final coordinate Σ_Y records labels, targets, privileged masks, or other supervision that could directly reveal the intended workspace. This coordinate is necessary because an architecture may appear to discover a boundary even when the desired membership mask was en","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bcfe3b6de4005b87","tex":"\\Sigma_H","aliases":"Σ_H Sigma","display":"\\Sigma_H","kind":"use","excerpt":"The scaffold is itself stratified: Σ = ( Σ_V, Σ_E, Σ_H, Σ_P, Σ_ O, Σ_B, Σ_Y ). The final coordinate Σ_Y records labels, targets, privileged masks, or other supervision that could directly reveal the intended workspace. This coordinate is necessary because an architecture may appear to discover a boundary even when the desired membership mask was en","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0ed845e79c8d8e41","tex":"\\sigma_i","aliases":"σ_i sigma","display":"\\sigma_i","kind":"use","excerpt":"Conversely, let x∈ C_o^Hol_o. Choose a path τ_i:o→ i for each vertex and define c_i=H_τ_ix. This definition is independent of the chosen path. Indeed, if σ_i:o→ i is another path, then σ_i^-1τ_i:o→ o is a loop, and invariance of x gives H_σ_i^-1H_τ_ix=x. Multiplying by H_σ_i yields H_τ_ix=H_σ_ix. For any edge e:i→ j, the paths eτ_i and τ_j both run","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-45aa5b98c4ff13d5","tex":"\\sigma_i^{-1}","aliases":"σ_i^-1 sigma","display":"\\sigma_i^{-1}","kind":"use","excerpt":"Conversely, let x∈ C_o^Hol_o. Choose a path τ_i:o→ i for each vertex and define c_i=H_τ_ix. This definition is independent of the chosen path. Indeed, if σ_i:o→ i is another path, then σ_i^-1τ_i:o→ o is a loop, and invariance of x gives H_σ_i^-1H_τ_ix=x. Multiplying by H_σ_i yields H_τ_ix=H_σ_ix. For any edge e:i→ j, the paths eτ_i and τ_j both run","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8de4fa9e7f9c5804","tex":"\\sigma_j","aliases":"σ_j sigma","display":"\\sigma_j","kind":"use","excerpt":"[Nonautonomous persistence theorem] Let e_t≥0 satisfy e_t+1≤α_te_t+σ_t, α_t,σ_t≥0. Then, for every t≥1, e_t ≤ · (∏_r=0^t-1α_r)e_0 · + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d5780be451d3dec1","tex":"\\varsigma_j","aliases":"ς_j varsigma","display":"\\varsigma_j","kind":"use","excerpt":"Let S_η be an η-net of ( S,d_ S) with covering number N(η, S,d_ S). If both the population and empirical margin functions are L_q-Lipschitz, then sup_ς∈ S |widehat m_q(ς)-m_q(ς)| ≤ max_ς_j∈ S_η |widehat m_q(ς_j)-m_q(ς_j)| +2L_qη. For bounded empirical means, a valid radius is therefore ε_q,n,η(δ) = sqrt frac log(2| J_*| N(η, S,d_ S)/δ) 2n +2L_qη. T","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fa431dd60a582371","tex":"\\sigma_k","aliases":"σ_k sigma","display":"\\sigma_k","kind":"use","excerpt":"For a declared graph, set of core dimensions, weights, and transfer maps, the static analysis can be executed in the following order: item Assemble the sparse block operator D_H from the edge equations sqrtw_e(c_t(e)-H_ec_s(e)). item Compute a singular-value decomposition D_H=UΣ V^*, λ_k(L_H)=σ_k(D_H)^2. Use a declared numerical tolerance to distin","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-22301d4bf1c931e5","tex":"\\sigma_{\\min}","aliases":"σ_min sigma min","display":"\\sigma_{\\min}","kind":"use","excerpt":"For a general invertible gauge, define m_V=σ_min(G_V), M_V= G_V , m_E=σ_min(G_E), M_E= G_E . The covariance identity yields m_E^2 E_H(c) ≤ E_H^g(G_Vc) ≤ M_E^2 E_H(c), and m_V^2 c ^2 ≤ G_Vc ^2 ≤ M_V^2 c ^2. Therefore the Rayleigh quotients obey fracm_E^2M_V^2 frac E_H(c) c ^2 ≤ frac E_H^g(G_Vc) G_Vc ^2 ≤ fracM_E^2m_V^2 frac E_H(c) c ^2. By the min--","count":9},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-65acde73901caecb","tex":"\\Sigma_P","aliases":"Σ_P Sigma","display":"\\Sigma_P","kind":"use","excerpt":"The scaffold is itself stratified: Σ = ( Σ_V, Σ_E, Σ_H, Σ_P, Σ_ O, Σ_B, Σ_Y ). The final coordinate Σ_Y records labels, targets, privileged masks, or other supervision that could directly reveal the intended workspace. This coordinate is necessary because an architecture may appear to discover a boundary even when the desired membership mask was en","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4d66cf40a5afaf2c","tex":"\\sigma_t","aliases":"σ_t sigma","display":"\\sigma_t","kind":"use","excerpt":"[Grounded coherence recurrence] Every trajectory satisfies e_t+1≤α_te_t+σ_t, where σ_t = ℓ_t P_tx_t + (I-P_t+1)ξ_t .","count":12},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c50697a89d3cb77d","tex":"\\sigma_{t-1}","aliases":"σ_t-1 sigma","display":"\\sigma_{t-1}","kind":"use","excerpt":"Repeated substitution yields e_t · ≤ α_t-1⋯α_0e_0 + α_t-1⋯α_1σ_0 + ⋯ + σ_t-1, which is the first formula. Under the uniform bounds, dominate each product by the corresponding power of barα and sum the geometric series.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-72078f9d4c5056d9","tex":"\\Sigma_V","aliases":"Σ_V Sigma","display":"\\Sigma_V","kind":"use","excerpt":"The scaffold is itself stratified: Σ = ( Σ_V, Σ_E, Σ_H, Σ_P, Σ_ O, Σ_B, Σ_Y ). The final coordinate Σ_Y records labels, targets, privileged masks, or other supervision that could directly reveal the intended workspace. This coordinate is necessary because an architecture may appear to discover a boundary even when the desired membership mask was en","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-65cbda3790092245","tex":"\\Sigma_{X'X'}","aliases":"Σ_X'X' Sigma","display":"\\Sigma_{X'X'}","kind":"use","excerpt":"The transformed covariances are Σ_Y'X'=g_vΣ_YXg_u^ T, Σ_X'X'=g_uΣ_XXg_u^ T. Substitution into the covariance formula and inversion of the second factor give the result.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e959a4c4abc7b0dc","tex":"\\Sigma_{XX}","aliases":"Σ_XX Sigma","display":"\\Sigma_{XX}","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":6},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-006721eeb4294c74","tex":"\\Sigma_{XX}^{-1}","aliases":"Σ_XX^-1 Sigma","display":"\\Sigma_{XX}^{-1}","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6fcdf41ec57c8f44","tex":"\\Sigma_Y","aliases":"Σ_Y Sigma","display":"\\Sigma_Y","kind":"use","excerpt":"The scaffold is itself stratified: Σ = ( Σ_V, Σ_E, Σ_H, Σ_P, Σ_ O, Σ_B, Σ_Y ). The final coordinate Σ_Y records labels, targets, privileged masks, or other supervision that could directly reveal the intended workspace. This coordinate is necessary because an architecture may appear to discover a boundary even when the desired membership mask was en","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cb2c3ce07ded301d","tex":"\\Sigma_{Y'X'}","aliases":"Σ_Y'X' Sigma","display":"\\Sigma_{Y'X'}","kind":"use","excerpt":"The transformed covariances are Σ_Y'X'=g_vΣ_YXg_u^ T, Σ_X'X'=g_uΣ_XXg_u^ T. Substitution into the covariance formula and inversion of the second factor give the result.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2b39e95f241f0d54","tex":"\\Sigma_{YX}","aliases":"Σ_YX Sigma","display":"\\Sigma_{YX}","kind":"use","excerpt":"[Population transfer identification] Let X∈ R^d_u and Y∈ R^d_v satisfy Y=H_*X+ξ, E[ξ X^ T]=0, and assume Σ_XX = E[XX^ T] succeq κ I for some κ>0. Then H_* is the unique minimizer of R(H) = E Y-HX _2^2, with H_* = Σ_YXΣ_XX^-1. Moreover, R(H)- R(H_*) = tr ( (H-H_*)Σ_XX(H-H_*)^ T ) ≥ κ H-H_* _F^2.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b3d2bb6c29d620e7","tex":"\\tau'","aliases":"τ' tau","display":"\\tau'","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ff0c3a9746b729b1","tex":"\\tau^2","aliases":"τ^2 tau","display":"\\tau^2","kind":"use","excerpt":"-(a+b) · b^2+1 . Writing τ=a^2+b^2+2, the eigenvalues are λ_± = frac12 ( τ±sqrtτ^2-4(ab-1)^2 ), and det L_H=(ab-1)^2. Thus a nonzero exact shared core exists precisely when ab=1, which is exactly the trivial-holonomy condition. Away from exact consistency, λ_-λ_+=(ab-1)^2. When |ab-1| is small while λ_+ stays bounded away from zero, the soft worksp","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7d039a149fedab64","tex":"\\tau^m","aliases":"τ^m tau","display":"\\tau^m","kind":"use","excerpt":"[Uniform path bound] If along the path T_r ≤τ, H_r ≤ h, V_r ≤ν, then the leakage and core-transport defects are bounded by ν∑_r=1^mτ^m-rh^r-1 = displaystyle νfracτ^m-h^mτ-h, · τ≠ h, [1.1em] displaystyle mντ^m-1, · τ=h. In particular, if τ≤1 and h≤1, the bound is at most mν.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e28e3ff232d17144","tex":"\\tau^{m-1}","aliases":"τ^m-1 tau","display":"\\tau^{m-1}","kind":"use","excerpt":"[Uniform path bound] If along the path T_r ≤τ, H_r ≤ h, V_r ≤ν, then the leakage and core-transport defects are bounded by ν∑_r=1^mτ^m-rh^r-1 = displaystyle νfracτ^m-h^mτ-h, · τ≠ h, [1.1em] displaystyle mντ^m-1, · τ=h. In particular, if τ≤1 and h≤1, the bound is at most mν.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e9c3e4baf90ab376","tex":"\\tau^{m-r}","aliases":"τ^m-r tau","display":"\\tau^{m-r}","kind":"use","excerpt":"[Uniform path bound] If along the path T_r ≤τ, H_r ≤ h, V_r ≤ν, then the leakage and core-transport defects are bounded by ν∑_r=1^mτ^m-rh^r-1 = displaystyle νfracτ^m-h^mτ-h, · τ≠ h, [1.1em] displaystyle mντ^m-1, · τ=h. In particular, if τ≤1 and h≤1, the bound is at most mν.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-14a4bd02a6176716","tex":"\\tau_{A\\to Y}","aliases":"τ_A→ Y tau to","display":"\\tau_{A\\to Y}","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-75bb788af51e4828","tex":"\\tau_{\\mathrm{causal}}","aliases":"τ_causal tau mathrm","display":"\\tau_{\\mathrm{causal}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c073c7ddcefa7185","tex":"\\tau_{D\\to M}","aliases":"τ_D→ M tau to","display":"\\tau_{D\\to M}","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e52f8bb6c0d98046","tex":"\\tau_G","aliases":"τ_G tau","display":"\\tau_G","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3c2cd1bd36fcad55","tex":"\\tau_{G\\to A}","aliases":"τ_G→ A tau to","display":"\\tau_{G\\to A}","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2ed1bcaa9d2d3dfe","tex":"\\tau_i","aliases":"τ_i tau","display":"\\tau_i","kind":"use","excerpt":"Choose for each vertex i a path τ_i:o→ i. Compatibility gives c_i=H_τ_ic_o. Thus a section with c_o=0 vanishes at every vertex. Hence ev_o is injective.","count":21},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-5ff841193b5dd2b3","tex":"\\tau_j","aliases":"τ_j tau","display":"\\tau_j","kind":"use","excerpt":"Choose a spanning tree T⊆|G|. Let τ_i:o→ i be the unique tree path. Every non-tree edge e:i→ j determines a fundamental loop γ_e=τ_j^-1eτ_i with transfer H_γ_e=H_τ_j^-1H_eH_τ_i. The fundamental group of a finite graph is generated freely by these cotree loops. Hence it is enough to impose fixed-point conditions for a finite set of generators.","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-77fe2d088dc0548e","tex":"\\tau_j^{-1}","aliases":"τ_j^-1 tau","display":"\\tau_j^{-1}","kind":"use","excerpt":"Choose a spanning tree T⊆|G|. Let τ_i:o→ i be the unique tree path. Every non-tree edge e:i→ j determines a fundamental loop γ_e=τ_j^-1eτ_i with transfer H_γ_e=H_τ_j^-1H_eH_τ_i. The fundamental group of a finite graph is generated freely by these cotree loops. Hence it is enough to impose fixed-point conditions for a finite set of generators.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e6fad63940e087f1","tex":"\\tau_{M\\to R}","aliases":"τ_M→ R tau to","display":"\\tau_{M\\to R}","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7e1bc156e67f5203","tex":"\\tau_q^{\\mathrm{causal}}","aliases":"τ_q^causal tau mathrm","display":"\\tau_q^{\\mathrm{causal}}","kind":"use","excerpt":"For thresholds Θ_form, define Form_Θ_form(W)=1 only when all applicable gates pass: · e_q^MDL≥ τ_q^MDL, · · e_q^causal≥ τ_q^causal, · δ_q^scaf≤ε_q^scaf, · · δ^leak≤ε_leak, · SRW_Θ(W)=1, · · Pass(N_form)=1. The final workspace gate is included because endogenous discovery of an unstable or causally irrelevant pattern is not endogenous workspace form","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-734a00c65a890f23","tex":"\\tau_q^{\\mathrm{MDL}}","aliases":"τ_q^MDL tau mathrm","display":"\\tau_q^{\\mathrm{MDL}}","kind":"use","excerpt":"For thresholds Θ_form, define Form_Θ_form(W)=1 only when all applicable gates pass: · e_q^MDL≥ τ_q^MDL, · · e_q^causal≥ τ_q^causal, · δ_q^scaf≤ε_q^scaf, · · δ^leak≤ε_leak, · SRW_Θ(W)=1, · · Pass(N_form)=1. The final workspace gate is included because endogenous discovery of an unstable or causally irrelevant pattern is not endogenous workspace form","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4a903d1201a67d64","tex":"\\tau_{R\\to v}","aliases":"τ_R→ v tau to","display":"\\tau_{R\\to v}","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0ff691516329631a","tex":"\\tau_{\\mathrm{spec}}","aliases":"τ_spec tau mathrm","display":"\\tau_{\\mathrm{spec}}","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9560e34421327822","tex":"\\tau_{\\mathrm{tr}}","aliases":"τ_tr tau mathrm","display":"\\tau_{\\mathrm{tr}}","kind":"use","excerpt":"For a one-parameter family R_λ, define λ_str · = inf{λ: d_ε(λ)>0, g_ε(λ)>0}, λ_op · = inf{λ: δ_align(λ) ≤ε_op}, λ_causal · = inf{λ: η_causal(λ) ≥τ_causal}, λ_rel · = inf{λ: k_ε(λ)≥2, Δ_tr(λ) ≥τ_tr}, λ_form · = inf{λ: Form_Θ_form (W_λ)=1}. In general, λ_str, λ_op, λ_causal, λ_rel, λ_form can be distinct. A synchronized representation can appear befo","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bbdb5621da7195cd","tex":"\\tau_{V\\to A}","aliases":"τ_V→ A tau to","display":"\\tau_{V\\to A}","kind":"use","excerpt":"Let Θ_A = ( τ_G, τ_G→ A, ε_Q, τ_V→ A, τ_A→ Y, τ_D→ M, τ_M→ R, τ_R→ v, τ_spec, ε_ctl, e_max ). A Θ_A-certified workspace agent is a controlled realization for which: item SRW(W_t;ς)=1; item cue-controlled goal retention exceeds τ_G; item goal-to-action mediation exceeds τ_G→ A; item counterfactual-value error is at most ε_Q; item value-to-action use","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"[Commutant characterization] Let |G| be connected and let H be an invertible local system. Root evaluation induces an algebra isomorphism Φ_o: O_Workspace xrightarrow ≅ Comm(Hol_o), Φ_o((A_i))=A_o, where Comm(Hol_o) = {B∈End(C_o):Bh=hB ∀ h∈Hol_o}.","count":14},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"Let R=(G, C,D,ρ, I, K) and R'=(G', C',D',ρ', I', K') be two realizations. A correspondence begins with a graph isomorphism φ:G→ G'. After relabeling by φ, let J_i:C_i→ C_i' be unitary local identifications and define J_V=⊕_i∈ VJ_i, J_E=⊕_e∈ EJ_t(e).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8075b42b78f39b03","tex":"\\Phi_{\\mathcal D}","aliases":"Φ_ D Phi mathcal","display":"\\Phi_{\\mathcal D}","kind":"use","excerpt":"Fix a reference decomposition D_0. For each D∈ D, let Φ_ D:W^ D_0⟶ W^ D be a declared workspace correspondence and let δ_ D measure its structural and causal identity defect. Assume gate q has a transport bound | m_q(Φ_ DW; D) -m_q(W; D_0) | ≤ L_qδ_ D+b_q( D), where L_q is a sensitivity constant and b_q records model mismatch not captured by the id","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bb543eb31351b7a2","tex":"\\phi_{\\mathcal I}","aliases":"ϕ_ I phi mathcal","display":"\\phi_{\\mathcal I}","kind":"use","excerpt":"[Minimal simultaneous interventional state] The random element Z_t^ I is sufficient simultaneously for all post-intervention target laws. If a statistic S_t=s( H_t) determines the conditional target law for every ι∈ I, then there exists a measurable ϕ_ I such that Z_t^ I=ϕ_ I(S_t) a.s.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6e0e9d52d03c0dcb","tex":"\\phi_{\\mathrm{boundary},t}","aliases":"ϕ_boundary,t phi mathrm","display":"\\phi_{\\mathrm{boundary},t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2be8950b02e6547c","tex":"\\phi_{\\mathrm{causal},t}","aliases":"ϕ_causal,t phi mathrm","display":"\\phi_{\\mathrm{causal},t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3295079d3254357d","tex":"\\phi_{\\mathrm{coh},t}","aliases":"ϕ_coh,t phi mathrm","display":"\\phi_{\\mathrm{coh},t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-65cc4ee5ef05f90a","tex":"\\phi_{\\mathrm{gap},t}","aliases":"ϕ_gap,t phi mathrm","display":"\\phi_{\\mathrm{gap},t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-96e36242d8809b05","tex":"\\phi_j","aliases":"ϕ_j phi","display":"\\phi_j","kind":"use","excerpt":"_boundary,t · =Δ_t^B-underlineΔ_B. The aggregate integrity margin is v_t = min_j∈ J_maintϕ_j,t. The workspace passes the protected gates when v_t≥0. To maintain a reserve ρ>0, define the maintenance deficit e_t =(ρ-v_t)_+. Define the zero-reserve and bounded-deficit viability sets V_Θ = { S:ϕ_j(S)≥0 for every j∈ J_maint } V_Θ,e_max = { S:e(S)≤ e_ma","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ec57c1e9c5db09ce","tex":"\\phi_{j,t}","aliases":"ϕ_j,t phi","display":"\\phi_{j,t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g,","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-26a665d1cc0e93f3","tex":"\\phi_{\\mathrm{keep},t}","aliases":"ϕ_keep,t phi mathrm","display":"\\phi_{\\mathrm{keep},t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-437778ee80dbd1ac","tex":"\\Phi_o","aliases":"Φ_o Phi","display":"\\Phi_o","kind":"use","excerpt":"[Commutant characterization] Let |G| be connected and let H be an invertible local system. Root evaluation induces an algebra isomorphism Φ_o: O_Workspace xrightarrow ≅ Comm(Hol_o), Φ_o((A_i))=A_o, where Comm(Hol_o) = {B∈End(C_o):Bh=hB ∀ h∈Hol_o}.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-03f75c476452b891","tex":"\\phi_{\\mathrm{rec},t}","aliases":"ϕ_rec,t phi mathrm","display":"\\phi_{\\mathrm{rec},t}","kind":"use","excerpt":"Choose a declared set J_maint of workspace gates that must be protected. Write each gate as a signed margin ϕ_j,t ≥0 Longleftrightarrow gate j passes at time t. Examples include ϕ_gap,t · =g_ε,t-underline g, ϕ_coh,t · =overlineδ_coh-δ_coh,t, ϕ_rec,t · =α_t^(k)-underlineα, ϕ_keep,t · =μ_t^keep-underlineμ, ϕ_causal,t · =η_causal,t-underlineη, ϕ_bound","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9d33f874f1b2ae64","tex":"\\phi_W","aliases":"ϕ_W phi","display":"\\phi_W","kind":"use","excerpt":"On standard Borel spaces, the canonical conditional-law state Z_t = P(Y_t^+∈·mid H_t) is sufficient and minimal: Y_t^+indep H_tmid Z_t, and every sufficient statistic S_t=s( H_t) satisfies Z_t=ϕ(S_t) a.s. Moreover, a structural candidate W_t is exactly operationally aligned if and only if Z_t=ϕ_W(W_t) a.s. The countable intervention-indexed version","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a940fb11efdb0bab","tex":"\\phi_{\\widehat\\varsigma}","aliases":"ϕ_widehatς phi widehat varsigma","display":"\\phi_{\\widehat\\varsigma}","kind":"use","excerpt":"sel, and let D_cert be independent of D_sel. For every fixed ς∈ S, let ϕ_ς( D_cert)∈{0,1} be a certificate rule satisfying sup_P:Cert(W;ς)=0 P_P(ϕ_ς=1) ≤α. Then the selected-specification procedure satisfies sup_P P_P( ϕ_widehatς=1, Cert(W;widehatς)=0 ) ≤α.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-84c6c651b1545707","tex":"\\Phi_{\\varsigma}","aliases":"Φ_ς Phi varsigma","display":"\\Phi_{\\varsigma}","kind":"use","excerpt":"Changing the decomposition can alter the coordinate realization of the workspace, while changing the target or horizon can alter the relevant predictive state. Let Φ_ς:W_ς_0⟶ W_ς be the declared correspondence when such transport is needed. For each gate, assume a bound | m_q(Φ_ςW;ς) -m_q(W;ς_0) | ≤ L_qd_ S(ς,ς_0) +b_q(ς), where b_q records mismatc","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b8ff163994d623f5","tex":"\\phi_\\varsigma","aliases":"ϕ_ς phi varsigma","display":"\\phi_\\varsigma","kind":"use","excerpt":"[Validity after independent specification selection] Let widehatς be selected using D_sel, and let D_cert be independent of D_sel. For every fixed ς∈ S, let ϕ_ς( D_cert)∈{0,1} be a certificate rule satisfying sup_P:Cert(W;ς)=0 P_P(ϕ_ς=1) ≤α. Then the selected-specification procedure satisfies sup_P P_P( ϕ_widehatς=1, Cert(W;widehatς)=0 ) ≤α.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Let C_* be a finite-dimensional abstract core. The exact static theory provides a complete local chart overline E_i = ev_i∘Ψ:C_*⟶ C_i, where Ψ:C_*→Γ(H) is the root-to-section isomorphism. A participant or artifact need not expose that complete chart. Let O_i:C_i⟶ Y_i be the declared observation or access map and define the accessible encoding E_i=O","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1d16d5974f691ea4","tex":"\\psi_A","aliases":"ψ_A psi","display":"\\psi_A","kind":"use","excerpt":"Let d_ P be a bounded metric generating the measurable structure on P( Y). Define the reconstruction defect of the canonical state from coalition A by δ_A^Z = inf_ψ_A E [ d_ P ( Z_t,ψ_A( H_t^A) ) ].","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6db479813e7c12fd","tex":"\\psi_i","aliases":"ψ_i psi","display":"\\psi_i","kind":"use","excerpt":"Exact alignment and Theorem~[source reference] imply that the canonical predictive state factors through W_t. If W_t=ψ_i( H_t^i) almost surely for some node, then the canonical state would also be a function of H_t^i, contradicting the positive node gap through Proposition~[source reference].","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"The scalar ω_ U(b) := P_ H^1_ Ub is an irreducible overlap-frustration measure. Its vanishing is invariant under admissible invertible coordinate changes, although its numerical norm depends on the chosen metric. It cannot be removed by adding local potentials alone.","count":7},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-da0a770c4761efd2","tex":"\\omega_{\\mathcal U}","aliases":"ω_ U omega mathcal","display":"\\omega_{\\mathcal U}","kind":"use","excerpt":"The scalar ω_ U(b) := P_ H^1_ Ub is an irreducible overlap-frustration measure. Its vanishing is invariant under admissible invertible coordinate changes, although its numerical norm depends on the chosen metric. It cannot be removed by adding local potentials alone.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e467cd2a3e347aff","tex":"\\omega_{t+1}","aliases":"ω_t+1 omega","display":"\\omega_{t+1}","kind":"use","excerpt":"C_t^ctl = ( W_t,G_t,M_t,B_t,E_t ), where G_t is the represented goal, M_t is a workspace-integrity monitor, B_t is the current boundary or routing state, and E_t is the declared external context. An action is generated by A_t ∼ π_t ( ·mid W_t,G_t,M_t,B_t,E_t ). The future controlled state obeys a time-unrolled structural model S_t+1 = F_t(S_t,A_t,E","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-cc9bb8081bd02ba9","tex":"\\Omega_\\gamma","aliases":"Ω_γ Omega gamma","display":"\\Omega_\\gamma","kind":"use","excerpt":"The obstruction associated with a loop is Ω_γ=H_γ-I_C_o. The maximal path-independent root core is C_* = ⋂_γ:o→ oker(H_γ-I). Equivalently, C_*=C_o^Hol_o=Fix(Hol_o).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"use","excerpt":"[Coalition reconstruction criterion] For a coalition S, the following are equivalent: item A_S is injective; item F_S is positive definite; item α_S>0; item there exists a linear decoder R_S with R_SA_S=I_C_*. When these conditions hold, the canonical decoder is R_S=A_S^†=F_S^-1A_S^*.","count":16},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8aaa13d7254b3cae","tex":"\\ell'","aliases":"ℓ' ell","display":"\\ell'","kind":"use","excerpt":"Let d_ S be a declared extended pseudometric on the quotient of S by specification equivalence. A convenient decomposition is d_ S(ς,ς') = · w_Dd_D( D, D') +w_Yd_Y(Y^+,Y^+') +w_ℓ d_ℓ(ℓ,ℓ') · +w_τ d_τ(τ,τ') +w_Id_I( I, I') +w_Nd_N( N, N') +w_Θ d_Θ(Θ,Θ'). No universal metric is imposed. The target term may use a divergence between conditional laws un","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bcd57746c1d3a7fc","tex":"\\ell_a","aliases":"ℓ_a ell","display":"\\ell_a","kind":"use","excerpt":"Write A=ρ_V(a), A'=ρ_V'(a), and define δ_ρ(a)= A'J_V-J_VA , ℓ_a= (I-P)AP . Let A_ W=PAP, A_ W''=P'A'P'.","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d08d976b9e37c665","tex":"\\ell_{\\mathrm{ctl}}","aliases":"ℓ_ctl ell mathrm","display":"\\ell_{\\mathrm{ctl}}","kind":"use","excerpt":"For the complete control state, define Risk_ctl(S) = inf_f E [ ℓ_ctl ( C_t^ctl, f( H_t^S) ) ], and let k_ctl,ε be its minimal sufficient coalition size. A strong workspace-level agency claim requires k_ctl,ε≥2. This is a location criterion. It does not yet establish goal persistence, counterfactual evaluation, or self-maintenance.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2b95471cbd4330f4","tex":"\\ell_G","aliases":"ℓ_G ell","display":"\\ell_G","kind":"use","excerpt":"Let H_t^S be the history available to coalition S⊆ V. For a loss ℓ_G, define goal-reconstruction risk Risk_G(S) = inf_f E [ ℓ_G ( G_t, f( H_t^S) ) ]. For tolerance ε, define k_G,ε = min { |S|: Risk_G(S) ≤ Risk_G(V)+ε }. A goal is locally stored when k_G,ε=1. It is distributed relative to the declared decomposition when k_G,ε≥2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b5bc16884dcf7263","tex":"\\ell_{\\log}","aliases":"ℓ_log ell log","display":"\\ell_{\\log}","kind":"use","excerpt":"For a finite target alphabet and logarithmic scoring loss ℓ_log(y,q)=-log q(y), q∈ P( Y), the identity is exact: Risk_log(S_t)=H(Y_t^+mid S_t), Risk_log(S_t)-Risk_log( H_t) = I(Y_t^+; H_tmid S_t). Natural logarithms measure the gap in nats; base-two logarithms measure it in bits.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-17c0c0401f91d262","tex":"\\ell_q","aliases":"ℓ_q ell","display":"\\ell_q","kind":"use","excerpt":"For any component q of widehatθ, define its scaffold-only Bayes risk under loss ℓ_q by R_q^Σ = inf_f E [ ℓ_q(q,f(Σ,Ω)) ]. Define the full formation risk by R_q^Σ, H = inf_g E [ ℓ_q(q,g(Σ, D_form,Ω)) ]. The history advantage is Δ_q^hist = R_q^Σ - R_q^Σ, H. A positive history advantage means that formation history contains information about q that is","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-307fdd827b919c3d","tex":"\\ell_t","aliases":"ℓ_t ell","display":"\\ell_t","kind":"use","excerpt":"Let the realized vertex state satisfy x_t+1=F_tx_t+ξ_t, where F_t: C_V(t)→ C_V(t+1) is the actual update and ξ_t is an unmodeled disturbance. Define the off-workspace error e_t= (I-P_t)x_t . Introduce α_t = (I-P_t+1)F_t(I-P_t) , which measures persistence of prior off-workspace error, and ℓ_t = (I-P_t+1)F_tP_t , which measures leakage of a valid wo","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"use","excerpt":"A feasible statement intervention must alter both local realizations of the statement coordinate: ι_q← a: (q_P,q_N)←(a,a), while holding (p,f) fixed. For configurations with fixed p and f, changing a flips the downstream decision with total-variation effect one: TV ( P(Ymidι_q←0), P(Ymidι_q←1) ) =1. A matched intervention on a residual notebook-for","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"aphs, coordinate restriction gives ρ_K^L: C_V(L)⟶ C_V(K). Because every edge equation on K is also an edge equation on L, restriction maps exact sections to exact sections: r_K^L := ρ_K^L|_mathscr W(L): mathscr W(L)⟶mathscr W(K). They obey r_L^L=I, r_J^K r_K^L=r_J^L whenever J⊆ K⊆ L. Thus K↦mathscr W(K) is a contravariant system of local workspaces","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"Let a path have vertices i_0xrightarrowe_1i_1xrightarrowe_2⋯ xrightarrowe_mi_m. Write T_r=T_e_r, H_r=H_e_r, V_r=V_e_r, and let ι_i^C:C_i↪ X_i, ι_i^R:R_i↪ X_i be the canonical inclusions. Empty products below are identities.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"Let the vertex set of a core-transfer diagram be partitioned as V=V_1sqcup⋯sqcup V_m. Write the transfer discrepancy as D_H = D_int D_∂ , D_int = ⊕_α=1^mD_α, where D_α contains the edge constraints internal to V_α, and D_∂ contains the constraints crossing between blocks. Define the internally coherent macro-state spaces W_α=ker D_α, W_Π=⊕_α=1^mW_α","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"sinθ · cosθ with θ∉2π Z. Then ker(R_θ-I)={0}. Every edge map is individually invertible and well-defined, yet the complete cycle admits no nonzero exact shared core. Pairwise translatability is therefore insufficient for global path independence.","count":3},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"· = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r · × (∏_ℓ=1^r-1 H_ℓ ), and the same right-hand side bounds P_i_mT_m⋯ T_1P_i_0-H_m⋯ H_1 .","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8e1c31b2edabc172","tex":"\\prod_{r=0}^{t-1}","aliases":"∏_r=0^t-1 prod","display":"\\prod_{r=0}^{t-1}","kind":"use","excerpt":"[Nonautonomous persistence theorem] Let e_t≥0 satisfy e_t+1≤α_te_t+σ_t, α_t,σ_t≥0. Then, for every t≥1, e_t ≤ · (∏_r=0^t-1α_r)e_0 · + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7bd83e3060400fcc","tex":"\\prod_{r=j+1}^{t-1}","aliases":"∏_r=j+1^t-1 prod","display":"\\prod_{r=j+1}^{t-1}","kind":"use","excerpt":"[Nonautonomous persistence theorem] Let e_t≥0 satisfy e_t+1≤α_te_t+σ_t, α_t,σ_t≥0. Then, for every t≥1, e_t ≤ · (∏_r=0^t-1α_r)e_0 · + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0a86f66b3b267a78","tex":"\\prod_s","aliases":"∏_s prod","display":"\\prod_s","kind":"use","excerpt":"For matched future command and disturbance trajectories under G_s+1=R_sG_s+B_sC_s+1+ζ_s+1, two goal interventions satisfy G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s≤σ_min(R_s), R_s ≤ M_s, then (∏_s m_s) g-g' ≤ G_t+τ-G_t+τ' ≤ (∏_s M_s) g-g' . A linear action readout gives a corresponding lower bound on the induced action-mean separation. The result isolat","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6e9589d0bdcbe5f0","tex":"\\prod_{s=r+1}^{R-1}","aliases":"∏_s=r+1^R-1 prod","display":"\\prod_{s=r+1}^{R-1}","kind":"use","excerpt":"l H_r+1, widetilde F_r: H_r→ H_r+1 be exact and implemented level-collapse maps for r=0,…,R-1. Let R_r⊆ H_r contain the exact and implemented states reachable at level r. Assume, on R_r, F_r(x)-F_r(y) ≤ L_r x-y and sup_x∈ R_r widetilde F_r(x)-F_r(x) ≤ δ_r. Then · widetilde F_R-1∘⋯∘widetilde F_0(x) - F_R-1∘⋯∘ F_0(x) · ≤ ∑_r=0^R-1 ( ∏_s=r+1^R-1L_s ) ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-794d2be49af2f15e","tex":"\\prod_{s=t}^{t+\\tau-1}","aliases":"∏_s=t^t+τ-1 prod tau","display":"\\prod_{s=t}^{t+\\tau-1}","kind":"use","excerpt":"1 = R_sG_s+B_sC_s+1+ζ_s+1 (s=t,…,t+τ-1). Consider two intervention runs that set G_t=g, G_t'=g', and then use the same command trajectory and the same disturbance realization. Let R_t+τ:t = R_t+τ-1⋯ R_t. Then G_t+τ-G_t+τ' = R_t+τ:t(g-g'). If m_s ≤ σ_min(R_s), R_s ≤ M_s, then ( ∏_s=t^t+τ-1m_s ) g-g' ≤ G_t+τ-G_t+τ' ≤ ( ∏_s=t^t+τ-1M_s ) g-g' .","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-01f661fab59b64b9","tex":"\\prod_\\alpha","aliases":"∏_α prod alpha","display":"\\prod_\\alpha","kind":"use","excerpt":"[Exact workspace gluing] For every finite edge-complete workspace cover U, the sequence 0 ⟶ mathscr W(G) xrightarrow R_ U check C^0( U;mathscr W) xrightarrow δ^0 check C^1( U;mathscr W) is exact. Equivalently, mathscr W(G) ≅ kerδ^0 ≅ Eq ( ∏_αmathscr W_α rightrightarrows ∏_α<βmathscr W_αβ ). Every pairwise compatible family of exact local workspace ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7a65e54bf15af597","tex":"\\prod_{\\alpha<\\beta}","aliases":"∏_α<β prod alpha beta","display":"\\prod_{\\alpha<\\beta}","kind":"use","excerpt":"[Exact workspace gluing] For every finite edge-complete workspace cover U, the sequence 0 ⟶ mathscr W(G) xrightarrow R_ U check C^0( U;mathscr W) xrightarrow δ^0 check C^1( U;mathscr W) is exact. Equivalently, mathscr W(G) ≅ kerδ^0 ≅ Eq ( ∏_αmathscr W_α rightrightarrows ∏_α<βmathscr W_αβ ). Every pairwise compatible family of exact local workspace ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-59a3d6b7de60c330","tex":"\\prod_{\\ell=1}^{r-1}","aliases":"∏_ℓ=1^r-1 prod ell","display":"\\prod_{\\ell=1}^{r-1}","kind":"use","excerpt":"· = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r · × (∏_ℓ=1^r-1 H_ℓ ), and the same right-hand side bounds P_i_mT_m⋯ T_1P_i_0-H_m⋯ H_1 .","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-06dc42bdcc814741","tex":"\\prod_{\\ell=r+1}^{m}","aliases":"∏_ℓ=r+1^m prod ell","display":"\\prod_{\\ell=r+1}^{m}","kind":"use","excerpt":"· = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r · × (∏_ℓ=1^r-1 H_ℓ ), and the same right-hand side bounds P_i_mT_m⋯ T_1P_i_0-H_m⋯ H_1 .","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-dbd199cda9e94680","tex":"\\sum_{B\\in S}","aliases":"∑_B∈ S sum in","display":"\\sum_{B\\in S}","kind":"use","excerpt":"A more stable quantity measures primitive carrier cost. Let w_ω>0 be a declared cost for each ω∈Ω, and assign a block the cost c(B)=∑_ω∈ Bw_ω. Define the sufficient support cost s_ε( D) = min_S⊆ V_ D { ∑_B∈ Sc(B): Risk_ D(S) ≤ Risk_ D(V_ D)+ε }. If D_fpreceq D_c is information faithful, then s_ε( D_f) ≤ s_ε( D_c). The fine decomposition permits eve","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6fd9071dd9bd51ef","tex":"\\sum_{e:s(e)=i}","aliases":"∑_e:s(e)=i sum","display":"\\sum_{e:s(e)=i}","kind":"use","excerpt":"For z=(z_e)_e∈ E∈ C_E, the adjoint is (D_H^*z)_i = ∑_e:t(e)=isqrtw_e z_e - ∑_e:s(e)=isqrtw_e H_e^*z_e. Consequently, (L_Hc)_i = · ∑_e:t(e)=iw_e(c_i-H_ec_s(e))","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8fdada032766993f","tex":"\\sum_{e:s(e)\\in S,\\ t(e)\\notin S}","aliases":"∑_e:s(e)∈ S, t(e)∉ S sum in notin","display":"\\sum_{e:s(e)\\in S,\\ t(e)\\notin S}","kind":"use","excerpt":"For the induced subsystem on S, define Coh(S) = tr ( P_S,dL_SP_S,d ), where P_S,d projects onto the selected low-energy workspace modes. Let Cut(S) = ∑_e:s(e)∈ S, t(e)∉ Sw_e + ∑_e:s(e)∉ S, t(e)∈ Sw_e. Let k_ε(S) be the minimal sufficient coalition size inside the candidate subsystem. A candidate is admissible only if SRW_Θ_S(W_S)=1, k_ε(S)≥2, unles","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-ec27f2d94f10ea94","tex":"\\sum_{e:s(e)\\notin S,\\ t(e)\\in S}","aliases":"∑_e:s(e)∉ S, t(e)∈ S sum notin in","display":"\\sum_{e:s(e)\\notin S,\\ t(e)\\in S}","kind":"use","excerpt":"For the induced subsystem on S, define Coh(S) = tr ( P_S,dL_SP_S,d ), where P_S,d projects onto the selected low-energy workspace modes. Let Cut(S) = ∑_e:s(e)∈ S, t(e)∉ Sw_e + ∑_e:s(e)∉ S, t(e)∈ Sw_e. Let k_ε(S) be the minimal sufficient coalition size inside the candidate subsystem. A candidate is admissible only if SRW_Θ_S(W_S)=1, k_ε(S)≥2, unles","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-afdcdbccd7887353","tex":"\\sum_{e:t(e)=i}","aliases":"∑_e:t(e)=i sum","display":"\\sum_{e:t(e)=i}","kind":"use","excerpt":"For z=(z_e)_e∈ E∈ C_E, the adjoint is (D_H^*z)_i = ∑_e:t(e)=isqrtw_e z_e - ∑_e:s(e)=isqrtw_e H_e^*z_e. Consequently, (L_Hc)_i = · ∑_e:t(e)=iw_e(c_i-H_ec_s(e))","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-957bbe7f2309996b","tex":"\\sum_{h\\in\\Hol_o}","aliases":"∑_h∈Hol_o sum in Hol","display":null,"kind":"use","excerpt":"[Averaging projector for finite holonomy] If Hol_o is finite and acts unitarily, then P_*= frac1|Hol_o| ∑_h∈Hol_oh is the orthogonal projector onto C_*.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d5a070236082d954","tex":"\\sum_{i=1}^3","aliases":"∑_i=1^3 sum","display":"\\sum_{i=1}^3","kind":"use","excerpt":"The positive examples use a recurring distributed code. Let c=(c_1,c_2,c_3)^ T∈ R^3 and define three local encodings E_1c=(c_1,c_2), E_2c=(c_2,c_3), E_3c=(c_1,c_3). Then ∑_i=1^3E_i^*E_i=2I_3. No individual encoding is injective, while every two-node coalition is injective. The pairwise frame operators are E_{1,2}^*E_{1,2} · =diag(1,2,1),","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2f29622fb484623a","tex":"\\sum_{i=1}^n","aliases":"∑_i=1^n sum","display":"\\sum_{i=1}^n","kind":"use","excerpt":"Suppose S and J_* are finite, and each margin is estimated as widehat m_q(ς) = frac1n∑_i=1^nZ_i,q,ς-θ_q,ς, 0≤ Z_i,q,ς≤1, from independent certification units. Hoeffding's inequality and a union bound give P( sup_q,ς | widehat m_q(ς)-m_q(ς) | > sqrt frac log(2| J_*|| S|/δ) 2n ) ≤δ. Thus the choice ε_n(δ) = sqrt frac log(2| J_*|| S|/δ) 2n yields an e","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4b8afcda15a5dfd1","tex":"\\sum_{i\\in S}","aliases":"∑_i∈ S sum in","display":"\\sum_{i\\in S}","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-1f2091ac81786c40","tex":"\\sum_{i\\in V}","aliases":"∑_i∈ V sum in","display":"\\sum_{i\\in V}","kind":"use","excerpt":"[Finite rank computation] Let n_C=∑_i∈ Vdim C_i. Then dimΓ(H)=n_C-rank(D_H). Hence the exact workspace dimension can be obtained by a single rank or singular-value computation, even when holonomy is unavailable because some transfers are noninvertible.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4bc4c95bca9467a8","tex":"\\sum_{i\\in V\\setminus A}","aliases":"∑_i∈ V∖ A sum in setminus","display":"\\sum_{i\\in V\\setminus A}","kind":"use","excerpt":"[Dimensional obstruction] Let d=dim C_*. If ∑_i∈ Sdim Y_i<d, then α_S=0, so S cannot reconstruct the core. Consequently, k-failure resilience requires ∑_i∈ V∖ Adim Y_i≥ d for every A⊆ V with |A|≤ k.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0eebf99dc433637d","tex":"\\sum_{j=0}^{p+1}","aliases":"∑_j=0^p+1 sum","display":"\\sum_{j=0}^{p+1}","kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-df66290cc89f818f","tex":"\\sum_{j=0}^{t-1}","aliases":"∑_j=0^t-1 sum","display":"\\sum_{j=0}^{t-1}","kind":"use","excerpt":"[Nonautonomous persistence theorem] Let e_t≥0 satisfy e_t+1≤α_te_t+σ_t, α_t,σ_t≥0. Then, for every t≥1, e_t ≤ · (∏_r=0^t-1α_r)e_0 · + ∑_j=0^t-1 (∏_r=j+1^t-1α_r)σ_j, where an empty product equals one. If α_t≤barα<1, σ_t≤barσ, then e_t ≤ barα^te_0 + fracbarσ(1-barα^t)1-barα, and consequently limsup_t→∞e_t ≤ fracbarσ1-barα.","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-bf6132b8c151067a","tex":"\\sum_{r=1}^{m}","aliases":"∑_r=1^m sum","display":"\\sum_{r=1}^{m}","kind":"use","excerpt":"[Pathwise core--residual Duhamel formula] The deviation of full path transport from pure core transport is exactly · T_m⋯ T_1ι_i_0^C - ι_i_m^C H_m⋯ H_1 · = ∑_r=1^m T_m⋯ T_r+1 ι_i_r^R V_r H_r-1⋯ H_1. Consequently, Q_i_mT_m⋯ T_1P_i_0 ≤ ∑_r=1^m · (∏_ℓ=r+1^m T_ℓ ) V_r","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-962e6f0b0ebec75f","tex":"\\sum_{r=1}^n","aliases":"∑_r=1^n sum","display":"\\sum_{r=1}^n","kind":"use","excerpt":"[Deterministic empirical recovery bound] For samples (X_r,Y_r)_r=1^n, write widehatΣ_XX = frac1n∑_r=1^nX_rX_r^ T, widehat M = frac1n∑_r=1^nξ_rX_r^ T. If widehatΣ_XXsucceqwidehatκ I, then ordinary least squares satisfies widehat H-H_* _F ≤ frac widehat M _Fwidehatκ. For ridge parameter τ>0, widehat H_τ-H_* _F ≤ frac widehat M _F+τ H_* _F widehatκ+τ ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0c905509ce292259","tex":"\\sum_{t<T}","aliases":"∑_t<T sum","display":"\\sum_{t<T}","kind":"use","excerpt":"A finite-horizon persistence certificate may be recorded as P_0:T = ( max_t<Tq_t^str, max_t≤ Te_t, min_t<Tμ_t^keep, ∑_t<Tσ_t ). No single coordinate of this vector subsumes the others.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-469e6dedd4f8c880","tex":"\\sum_{t=0}^{T-1}","aliases":"∑_t=0^T-1 sum","display":"\\sum_{t=0}^{T-1}","kind":"use","excerpt":"For a sequence of realizations R_0, R_1,…, R_T, let J_t: R_t→ R_t+1 be admissible correspondences. Define the path length Len( J_0:T) = ∑_t=0^T-1Cost( J_t) and the induced directed path quasi-pseudometric d_path^rightarrow( R_0, R_T) = inf_ J_0:T Len( J_0:T). Concatenation gives the directed triangle inequality. Because replacement costs and interv","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aa1dbc193c31e5d5","tex":"\\sum_{t\\geq0}","aliases":"∑_t≥0 sum geq","display":"\\sum_{t\\geq0}","kind":"use","excerpt":"[Vanishing-disturbance convergence] If α_t≤barα<1 and σ_t→0, then e_t→0. More generally, if ∑_t≥0σ_t<∞, then the disturbance contribution in Theorem~[source reference] vanishes asymptotically.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-6d8e30d9f2646a56","tex":"\\sum_\\alpha","aliases":"∑_α sum alpha","display":"\\sum_\\alpha","kind":"use","excerpt":"[0.2em] sqrtλ_ov S_ U , L_ U^tot = K_ U^*K_ U. Its energy is E_ U^tot(u) = · ∑_α D_G_αu_α ^2 · + λ_ov ∑_α<β ρ^G_β_G_αβu_β - ρ^G_α_G_αβu_α ^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-57047b8e489066cf","tex":"\\sum_{\\alpha:i\\in V_\\alpha}","aliases":"∑_α:i∈ V_α sum alpha in alpha","display":"\\sum_{\\alpha:i\\in V_\\alpha}","kind":"use","excerpt":"Let m_i=|{α:i∈ V_α}| be the multiplicity of vertex i. Then J_ U^*J_ U=M_ U, (M_ Uc)_i=m_ic_i. The pseudoinverse performs coordinatewise averaging: (J_ U^† u)_i = frac1m_i ∑_α:i∈ V_α(u_α)_i. Let widehat u = P_ker K_ Uu = J_ Uwidehat c. Then widehat c = J_ U^†widehat u and u-J_ Uwidehat c ≤ frac K_ Uu sqrtγ_ U^tot. If m_min=min_i m_i, then widehat c-","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-50c2fa2200582276","tex":"\\sum_{\\alpha<\\beta}","aliases":"∑_α<β sum alpha beta","display":"\\sum_{\\alpha<\\beta}","kind":"use","excerpt":"A family w=(w_α)_α is exactly compatible when δ^0w=0. Its direct mismatch is m_ U(w)=δ^0w. After choosing inner products and positive overlap weights, the mismatch energy is E_ov(w) = δ^0w ^2 = ∑_α<β ν_αβ r_αβ^βw_β -r_αβ^αw_α ^2. This is the appropriate obstruction when the full local states are already given. Such a family glues exactly if and onl","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-65bbcf65283d2b6f","tex":"\\sum_{\\alpha\\in A}","aliases":"∑_α∈ A sum alpha in","display":"\\sum_{\\alpha\\in A}","kind":"use","excerpt":"[Zeroth descent cohomology] For every finite edge-complete cover, mathscr W(G) ≅ check H^0( U;mathscr W) = kerδ^0. In finite dimensions, dimmathscr W(G) = ∑_α∈ Adimmathscr W_α - rank(δ^0).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0d22b04169d547dc","tex":"\\sum_\\iota","aliases":"∑_ι sum iota","display":"\\sum_\\iota","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-aa0512b2f31a5fdf","tex":"\\sum_{\\iota}","aliases":"∑_ι sum iota","display":"\\sum_{\\iota}","kind":"use","excerpt":"hbf H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ∑_ιq_ι [ Risk_ℓ^ι(W_t) - Risk_ℓ^ι( H_t) ] ≤ sqrtfrac δ_align^ I(W_t)2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-8ed9d74ba89a0d33","tex":"\\sum_{\\iota\\in\\mathcal I}","aliases":"∑_ι∈ I sum iota in mathcal","display":"\\sum_{\\iota\\in\\mathcal I}","kind":"use","excerpt":"Define the observational alignment defect by δ_align^obs(W_t) = I_ P(Y_t^+; H_tmid W_t). For a countable intervention family with declared weights q_ι>0, ∑_ι q_ι=1, define δ_align^ I(W_t) = ∑_ι∈ I q_ι I_ P^ι (Y_t^+; H_tmid W_t). For bounded decision losses, Proposition~[source reference] applies under each intervention and gives the weighted bound ","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4355a70e6947a0a3","tex":"\\sum_{\\omega\\in B}","aliases":"∑_ω∈ B sum omega in","display":"\\sum_{\\omega\\in B}","kind":"use","excerpt":"A more stable quantity measures primitive carrier cost. Let w_ω>0 be a declared cost for each ω∈Ω, and assign a block the cost c(B)=∑_ω∈ Bw_ω. Define the sufficient support cost s_ε( D) = min_S⊆ V_ D { ∑_B∈ Sc(B): Risk_ D(S) ≤ Risk_ D(V_ D)+ε }. If D_fpreceq D_c is information faithful, then s_ε( D_f) ≤ s_ε( D_c). The fine decomposition permits eve","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"[Separated spectral subspaces] Let A and B=A+E be self-adjoint operators. Let P be an A-spectral projector whose eigenvalues lie in ( -∞,a], and let widetilde Q be a B-spectral projector whose eigenvalues lie in [b,∞), with b>a. Then Pwidetilde Q ≤ frac E b-a.","count":11},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"cal D = D_form sqcup D_cal sqcup D_cert. The formation sample selects widehatθ. The calibration sample fixes thresholds, regularization strengths, and intervention magnitudes. The certification sample evaluates the final gates without further structural selection. In a multi-environment study, the stronger requirement is E_form ∩ E_cert = ∅, so tha","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"nwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P-μ I) (PEwideti","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3953b2deaa372779","tex":"\\int_{\\mathsf Y}","aliases":"∫_ Y int mathsf","display":"\\int_{\\mathsf Y}","kind":"use","excerpt":"For every bounded measurable f: Y→ R, define Z_t(f)=∫_ Yf(y) Z_t( dy). By construction, E[f(Y_t^+)mid H_t]=Z_t(f). Since Z_t is a measurable function of H_t, the tower property gives E[f(Y_t^+)mid Z_t] = E[Z_t(f)mid Z_t] = Z_t(f). Hence conditioning additionally on H_t does not change the conditional expectation, which is the stated conditional ind","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"nwidetilde Q, set A_P=PAP|_ran P, B_Q=widetilde QBwidetilde Q|_ranwidetilde Q. The operator X=Pwidetilde Q satisfies the Sylvester equation A_PX-XB_Q=-PEwidetilde Q. Let μ=(a+b)/2. Since e^t(A_P-μ I) ≤ e^-(b-a)t/2, e^-t(B_Q-μ I) ≤ e^-(b-a)t/2, the inverse Sylvester operator has the convergent integral representation X = ∫_0^∞ e^t(A_P-μ I) (PEwideti","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-750d42e0972ca7b1","tex":"\\sim_{\\mathrm{goal}}","aliases":"∼_goal sim mathrm","display":"\\sim_{\\mathrm{goal}}","kind":"use","excerpt":"Two goal codes are operationally equivalent, g∼_goalg', when they induce the same action ordering in every declared state and intervention background: Q_t^g(amid w) ≥ Q_t^g(a'mid w) Longleftrightarrow Q_t^g'(amid w) ≥ Q_t^g'(a'mid w). The abstract goal is the equivalence class [g]_goal. This quotient prevents a mere invertible recoding of a goal ch","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-669bc9179dbba22c","tex":"\\sim_Y","aliases":"∼_Y sim","display":"\\sim_Y","kind":"use","excerpt":"The quotient notation [ H_t]_∼_Y expresses the right intuition but hides measurability and null-set issues. A canonical construction is obtained from regular conditional distributions.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"Distinct implementations may instantiate the same abstract workspace: R≠ R', R≃ R' Longrightarrow Workspace( R)≅Workspace( R').","count":28},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"The spectrum supports three distinct interpretations: rcl λ_k=0 · Longleftrightarrow · exact compatible direction, 0<λ_kll g_d · Longrightarrow · approximately transported direction, λ_k≈λ_k+1 · Longrightarrow · rank selection is unstable at that boundary. A low eigenvalue can still be scientifically spurious if it arises from a poorly chosen weigh","count":4},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-034a9e192ac4f494","tex":"\\bigoplus_{b<c}","aliases":"⊕_b<c bigoplus","display":"\\bigoplus_{b<c}","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-4ed7e068feff962e","tex":"\\bigoplus_{b\\in\\operatorname{Ch}(a)}","aliases":"⊕_b∈Ch(a) bigoplus in operatorname","display":"\\bigoplus_{b\\in\\operatorname{Ch}(a)}","kind":"use","excerpt":"A finite workspace hierarchy is a rooted tree T. Each node a∈ T carries a subgraph G_a. The root r carries G_r=G. For every internal node a, its children Ch(a) form a finite edge-complete cover of G_a. Leaves carry primitive local workspace spaces W_ℓ=mathscr W(G_ℓ). For an internal node, define recursively W_a = ker ( δ_a^0: ⊕_b∈Ch(a)W_b ⟶ ⊕_b<cW_","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9e230cb842b55b7b","tex":"\\bigoplus_{e\\in E}","aliases":"⊕_e∈ E bigoplus in","display":"\\bigoplus_{e\\in E}","kind":"use","excerpt":"rfectly under every transfer. Approximate theory asks three more realistic questions: item how strongly a candidate state violates edgewise compatibility; item whether a low-defect subspace is spectrally separated from incoherent directions; and item how that subspace changes when transfers, metrics, or operations are perturbed. Throughout Parts~IV","count":5},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-575d497ec968a2f1","tex":"\\bigoplus_{i\\in S}","aliases":"⊕_i∈ S bigoplus in","display":"\\bigoplus_{i\\in S}","kind":"use","excerpt":"For a coalition S⊆ V, define the analysis operator A_S:C_*⟶⊕_i∈ S Y_i, A_Sc=(E_ic)_i∈ S, and the coalition frame operator F_S=A_S^*A_S=∑_i∈ SE_i^*E_i. Set α_S=λ_min(F_S), β_S=λ_max(F_S). Thus α_S c ^2 ≤ A_Sc ^2 ≤ β_S c ^2.","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-3f6ad8dea6f164a3","tex":"\\bigoplus_{j=1}^m","aliases":"⊕_j=1^m bigoplus","display":"\\bigoplus_{j=1}^m","kind":"use","excerpt":"The first benchmark is the exact parity family from Topics~85--80. For m≥3, distribute m independent bits among nodes so that no individual node sees all bits, but specified coalitions do. The target is Y=⊕_j=1^m C_j. The benchmark measures: item the learned transfer-Laplacian spectrum; item coalition conditional entropy and log risk; item recovery","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-9b591bcc37a61369","tex":"\\bigoplus_\\alpha","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_\\alpha","kind":"use","excerpt":"Define the exact meta-workspace mathscr W_meta( U) := ker B_ U ⊆ ⊕_αmathscr W_α. Theorem~[source reference] gives the canonical identification mathscr W_meta( U) ≅ mathscr W(G). Thus exact formation of a workspace from local workspaces does not create an additional mysterious carrier. It gives a second presentation of the same global compatible-sec","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-b1d323816298cbee","tex":"\\bigoplus_{\\alpha}","aliases":"⊕_α bigoplus alpha","display":"\\bigoplus_{\\alpha}","kind":"use","excerpt":"Unless stated otherwise, connectedness means that |G| is connected. A root o∈ V may then be joined to every vertex by a path in the underlying graph. If G has several connected components, the theory applies to each component and Γ(H) ≅ ⊕_αΓ(H|_G_α).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-598632d8c9c1d1e6","tex":"\\bigoplus_{\\alpha<\\beta}","aliases":"⊕_α<β bigoplus alpha beta","display":"\\bigoplus_{\\alpha<\\beta}","kind":"use","excerpt":"Suppose local operations A_α:mathscr W_α→mathscr W_α and overlap operations A_αβ:mathscr W_αβ→ mathscr W_αβ satisfy r_αβ^αA_α = A_αβr_αβ^α, r_αβ^βA_β = A_αβr_αβ^β. Then, with A^0=⊕_α A_α and A^1=⊕_α<βA_αβ, B_ UA^0=A^1B_ U. Therefore A^0ker B_ U⊆ker B_ U. The operation-closure theorem from Part~III reappears at the meta level: operations on subworks","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-50e2cdab71473e4f","tex":"\\bigoplus_{\\alpha=1}^{m}","aliases":"⊕_α=1^m bigoplus alpha","display":"\\bigoplus_{\\alpha=1}^{m}","kind":"use","excerpt":"Let the vertex set of a core-transfer diagram be partitioned as V=V_1sqcup⋯sqcup V_m. Write the transfer discrepancy as D_H = D_int D_∂ , D_int = ⊕_α=1^mD_α, where D_α contains the edge constraints internal to V_α, and D_∂ contains the constraints crossing between blocks. Define the internally coherent macro-state spaces W_α=ker D_α, W_Π=⊕_α=1^mW_α","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-d76c768d5b138029","tex":"\\bigoplus_{\\alpha_0<\\cdots<\\alpha_p}","aliases":"⊕_α_0<⋯<α_p bigoplus alpha cdots alpha","display":"\\bigoplus_{\\alpha_0<\\cdots<\\alpha_p}","kind":"use","excerpt":"For p≥0, define the finite cochain space check C^p( U;mathscr W) = ⊕_α_0<⋯<α_p mathscr W(G_α_0⋯α_p), where empty intersections contribute the zero space. The coboundary δ^p:check C^p⟶check C^p+1 is (δ^p s)_α_0⋯α_p+1 = ∑_j=0^p+1(-1)^j r_α_0⋯α_p+1^% α_0⋯widehatα_j⋯α_p+1 % s_α_0⋯widehatα_j⋯α_p+1. The restriction identities imply δ^p+1δ^p=0. For a loca","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-0b30f8056faf98e6","tex":"\\bigoplus_{\\alpha\\in A}","aliases":"⊕_α∈ A bigoplus alpha in","display":"\\bigoplus_{\\alpha\\in A}","kind":"use","excerpt":"Local candidates may also violate their internal transfer equations. Define the local-copy space widehat C_ U = ⊕_α∈ A C_V(G_α) and the block internal discrepancy widehat D_ U = ⊕_α∈ AD_G_α. For u=(u_α)_α∈widehat C_ U, define the raw overlap synchronization operator (S_ Uu)_αβ = ρ_G_αβ^G_βu_β - ρ_G_αβ^G_αu_α. For an overlap weight λ_ov>0, set K_ U ","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fba9f4be17a23c06","tex":"\\bigcap_{q=1}^{r}","aliases":"⋂_q=1^r bigcap","display":"\\bigcap_{q=1}^{r}","kind":"use","excerpt":"[Finite obstruction set] Let e_1,…,e_r be the non-tree edges of a connected finite graph. Then C_* = ⋂_q=1^r ker(H_γ_e_q-I).","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-a36a29a5dc087fd1","tex":"\\bigcap_{\\gamma:o\\to o}","aliases":"⋂_γ:o→ o bigcap gamma to","display":"\\bigcap_{\\gamma:o\\to o}","kind":"use","excerpt":"The obstruction associated with a loop is Ω_γ=H_γ-I_C_o. The maximal path-independent root core is C_* = ⋂_γ:o→ oker(H_γ-I). Equivalently, C_*=C_o^Hol_o=Fix(Hol_o).","count":2},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-40461bb232c267ca","tex":"\\bigcup_{S\\in\\mathcal L}","aliases":"⋃_S∈ L bigcup in mathcal","display":"\\bigcup_{S\\in\\mathcal L}","kind":"use","excerpt":"mined by mathematical and operational equivalence, not by terminology. Let C be a claim in the paper and S a prior source. Write Spreccurlyeq C when the main content of C follows from S after admissible relabeling, restriction, or direct specialization. A substantive novelty residual exists only when the claim contains content not absorbed by the c","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"[Positivity and variational identity] The transfer Laplacian is self-adjoint and positive semidefinite. For every c∈ C_V, ⟨ c,L_Hc⟩= E_H(c)≥0.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"[Positivity and variational identity] The transfer Laplacian is self-adjoint and positive semidefinite. For every c∈ C_V, ⟨ c,L_Hc⟩= E_H(c)≥0.","count":8},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-2d9841ce7d048a4d","tex":"\\rangle_i","aliases":"⟩_i rangle","display":"\\rangle_i","kind":"use","excerpt":"A pure change of coordinates should transform the metric as well as the matrices. On each transformed space, define ⟨ u,v⟩_i^g = ⟨ g_i^-1u,g_i^-1v⟩_i. Use the analogous pullback metric on every edge copy. Then G_V and G_E are unitary between the original and transformed metric spaces. Consequently, the complete transfer-Laplacian spectrum is exactl","count":1},{"work":"research-stratified-relational-workspace","url":"notation/research-stratified-relational-workspace.html#symbol-fca938cfe23f2102","tex":"\\rangle_i^g","aliases":"⟩_i^g rangle","display":"\\rangle_i^g","kind":"use","excerpt":"A pure change of coordinates should transform the metric as well as the matrices. On each transformed space, define ⟨ u,v⟩_i^g = ⟨ g_i^-1u,g_i^-1v⟩_i. Use the analogous pullback metric on every edge copy. Then G_V and G_E are unitary between the original and transformed metric spaces. Consequently, the complete transfer-Laplacian spectrum is exactl","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":20},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":124},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":39},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":76},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-bb3cdc779ccbade8","tex":"\\bar J","aliases":"bar J bar","display":"\\bar J","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-446fba0b12b9aac6","tex":"\\bar J_0","aliases":"bar J_0 bar","display":"\\bar J_0","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-58d2d6ac0d680d9f","tex":"\\bar J_h","aliases":"bar J_h bar","display":"\\bar J_h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-32da9c12730e8ae8","tex":"\\bar J_{h-1}","aliases":"bar J_h-1 bar","display":"\\bar J_{h-1}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c0435683f947250d","tex":"\\bar z","aliases":"bar z bar","display":"\\bar z","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f68011f6686ac355","tex":"\\bar z_{a,o}","aliases":"bar z_a,o bar","display":"\\bar z_{a,o}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4415f58b83fbd8c6","tex":"\\bar\\Psi","aliases":"barΨ bar Psi","display":"\\bar\\Psi","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5c18dd7e6706bb6e","tex":"\\bar\\Psi_{a,o}","aliases":"barΨ_a,o bar Psi","display":"\\bar\\Psi_{a,o}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":10},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":25},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":30},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":12},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-24ceed0c3b629a6f","tex":"F_{a_1}","aliases":"F_a_1","display":"F_{a_1}","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d6d2fa627b1c221a","tex":"F_{a_k}","aliases":"F_a_k","display":"F_{a_k}","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":23},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6e30319bc4adb843","tex":"J_h","aliases":"J_h","display":"J_h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":11},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":8},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a20d1b2c5e09f23a","tex":"\\min_{a\\in A(q)}","aliases":"min_a∈ A(q) min in","display":"\\min_{a\\in A(q)}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-73ed63492f6711c5","tex":"\\min_{d\\in D(q)}","aliases":"min_d∈ D(q) min in","display":"\\min_{d\\in D(q)}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":42},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":32},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":16},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-bbf62fa0f1dd5f2f","tex":"\\mathcal Q_h","aliases":"Q_h mathcal","display":"\\mathcal Q_h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":21},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":15},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":15},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-7151afe2b5220a1a","tex":"\\ell_d","aliases":"ℓ_d ell","display":"\\ell_d","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":18},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":18},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":16},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0876d820c228b469","tex":"\\sum_{o:\\bar z_{a,o}(q)>0}","aliases":"∑_o:bar z_a,o(q)>0 sum bar","display":"\\sum_{o:\\bar z_{a,o}(q)>0}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":22},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2876747ca029362d","tex":"\\mathbf1^\\top","aliases":"1^top mathbf top","display":"\\mathbf1^\\top","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ca30d7bde7b966d9","tex":"\\mathbf1_X^\\top","aliases":"1_X^top mathbf top","display":"\\mathbf1_X^\\top","kind":"use","excerpt":"Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point m","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c907afa1681c3705","tex":"\\mathbf1_Z^\\top","aliases":"1_Z^top mathbf top","display":"\\mathbf1_Z^\\top","kind":"use","excerpt":"Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point m","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0014cfc2591bdee6","tex":"a_*","aliases":"a_*","display":"a_*","kind":"use","excerpt":"For a true minimizer a_*, R(widehat a)≤widehat R(widehat a)+ξ ≤widehat R(a_*)+ξ≤ R(a_*)+2ξ. The loss range supplies the cap one. Finally, for any other a, subtract the two error bounds to obtain R(a)-R(widehat a)≥widehat R(a)-widehat R(widehat a)-2ξ>0. source[GA], ``A common quantitative grounding certificate''; [MU], ``A joint posterior and recipi","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ee19b85346d2e7d2","tex":"A_A","aliases":"A_A","display":"A_A","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be finite dimensional, let T_a^θ be finitely many candidate linear operators, and let A_a be one complete candidate system chosen as reference. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear rea","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6c1908f93f8dee44","tex":"A_B","aliases":"A_B","display":"A_B","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5a8e57a002dc32f4","tex":"a_d","aliases":"a_d","display":"a_d","kind":"use","excerpt":"Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined b","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5b950cf09bc94cd2","tex":"A_{\\rm end}","aliases":"A_rm end rm","display":"A_{\\rm end}","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4f6eb91cb1eca840","tex":"A_{\\rm run}","aliases":"A_rm run rm","display":"A_{\\rm run}","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-82821bc2124f1658","tex":"A_t","aliases":"A_t","display":"A_t","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-45d4500122db1f19","tex":"\\addcontentsline","aliases":"addcontentsline","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4afb314fdcd4106f","tex":"\\arraybackslash","aliases":"arraybackslash","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":18},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":31},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-eebc9b85b3d9979a","tex":"\\bar D","aliases":"bar D bar","display":"\\bar D","kind":"use","excerpt":"[A quotient that preserves legal operations] For a surjection π:X→ Z, quotient labels, enabled sets, and successors satisfying ℓ=barℓπ, D=bar Dπ, π F(x,a)=bar F(π x,a) exist exactly when equal fibres have equal labels, equal enabled sets, and equal projected successors for every enabled action. The quotient data are unique and preserve every legal ","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-52555ee0fe55f126","tex":"\\bar F","aliases":"bar F bar","display":"\\bar F","kind":"use","excerpt":"[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a(","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0ddd6936013fb39a","tex":"\\bar F_a","aliases":"bar F_a bar","display":"\\bar F_a","kind":"use","excerpt":"[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a(","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-fad6ae0fd6c0acb8","tex":"\\bar K","aliases":"bar K bar","display":"\\bar K","kind":"use","excerpt":"nique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly positive coarse likelihoods for every prior. At a stable partition the probability of reaching each coarse block depends only on the current block. These sums define a stochastic coarse tran","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d9966b1737faabea","tex":"\\bar K_{a,o}","aliases":"bar K_a,o bar","display":"\\bar K_{a,o}","kind":"use","excerpt":"[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the repor","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-742fe606e2cdb4d4","tex":"\\bar K_a^\\top","aliases":"bar K_a^top bar top","display":"\\bar K_a^\\top","kind":"use","excerpt":"nique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly positive coarse likelihoods for every prior. At a stable partition the probability of reaching each coarse block depends only on the current block. These sums define a stochastic coarse tran","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8f8a4d0a063c4065","tex":"\\bar Q","aliases":"bar Q bar","display":"\\bar Q","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also pre","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"use","excerpt":"havioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a([x])=[F_ax] and its read","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2311bbf2ac9fa5af","tex":"\\bar u","aliases":"bar u bar","display":"\\bar u","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-38f0a5c7c65b9678","tex":"\\bar u_h","aliases":"bar u_h bar","display":"\\bar u_h","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-028d200d0ea01a8b","tex":"\\bar u_\\varnothing","aliases":"bar u_∅ bar varnothing","display":"\\bar u_\\varnothing","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d97fba459e62ed0e","tex":"\\bar\\ell","aliases":"barℓ bar ell","display":"\\bar\\ell","kind":"use","excerpt":"For minimality, take any admissible refinement η of π_0. It has at least two blocks. Fix a strictly positive fine likelihood ℓ and a strictly positive coarse likelihood barℓ inducing its assumed update. For states x,y in one block and z in a different block, compare priors assigning half their mass to x,z and to y,z. Their coarse priors agree. Equa","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"An agent given only y can predict every future display under repeated flips: after k flips it is y⊕(kbmod2). It cannot determine the current physical bit. Worlds (0,0,0) and (1,1,0) share display zero and have different targets. The obstruction is the fibre of the available record, not a shortage of computational effort.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":32},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":7},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0; in the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0; in the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"use","excerpt":"tion, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegativ","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-efafb53fff3f1cb9","tex":"\\mathcal C_h","aliases":"C_h mathcal","display":"\\mathcal C_h","kind":"use","excerpt":"Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c591dfc771720384","tex":"c_{\\max}","aliases":"c_max max","display":"c_{\\max}","kind":"use","excerpt":"te planning must not share one unqualified guarantee. Fix a finite horizon H and one finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-state and action/report path in both models, taking values in [0,B]. For example, a terminal loss in [0,1] and ","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-42eb4bc96455bb91","tex":"C_\\pi","aliases":"C_π pi","display":"C_\\pi","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2949436ae89cb480","tex":"\\chapter","aliases":"chapter","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6de3eb080886bbf6","tex":"D_a^\\theta","aliases":"D_a^θ theta","display":"D_a^\\theta","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3b858f3e8665ea65","tex":"D_e","aliases":"D_e","display":"D_e","kind":"use","excerpt":"One investigation, one resource budget. questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X,","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5341c066ee58261e","tex":"d_H","aliases":"d_H","display":"d_H","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_","count":11},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"use","excerpt":"Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined b","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":30},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":15},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"[Minimal exact memory for a correction contract] Equality of Σ is a congruence for every update V↦ V∩ E_0. The signature supports the current target set and its exact update after every finite evidence sequence. Any deterministic retained state supporting the same readouts and updates on all candidate subsets must distinguish every two distinct sig","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-48dc92a98cff75ec","tex":"f_e","aliases":"f_e","display":"f_e","kind":"use","excerpt":", one resource budget questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a determinist","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.","count":14},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that ","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also pre","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"A terminal artifact a is acceptable only if every fixed recipient succeeds in every state of V. Its worst-case terminal cost is g_a(V), computed componentwise. A controller is a literal finite tree with one record per experiment or terminal node. Stored memory counts all branches, including those not visited in one execution.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5f29bf5e40e3c97b","tex":"G_{\\tau(a)}","aliases":"G_τ(a) tau","display":"G_{\\tau(a)}","kind":"use","excerpt":"A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a5620dda8c46dbb1","tex":"G_{\\tau(w)}","aliases":"G_τ(w) tau","display":"G_{\\tau(w)}","kind":"use","excerpt":"[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":19},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5bc59e2e502080d9","tex":"\\mathcal H_{j-1}","aliases":"H_j-1 mathcal","display":"\\mathcal H_{j-1}","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4936b070e854f399","tex":"\\hfill","aliases":"hfill","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-7a664dd94a0911d8","tex":"\\hrule","aliases":"hrule","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":12},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":25},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of ","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0475b2ac7d225af5","tex":"i_2","aliases":"i_2","display":"i_2","kind":"use","excerpt":"Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of ","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-34af8cf33f9487de","tex":"I_U","aliases":"I_U","display":"I_U","kind":"use","excerpt":"[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f5bae71a192fc64d","tex":"I_V","aliases":"I_V","display":"I_V","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-bdedc0251851d395","tex":"I_Z","aliases":"I_Z","display":"I_Z","kind":"use","excerpt":"be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"use","excerpt":":reconstruction There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv on im A. If Av=Aw, then","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":8},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":18},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state n","count":8},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6cc92a51c6969e2c","tex":"K_a","aliases":"K_a","display":"K_a","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-16b5593bbf90ad19","tex":"K_{a,o}","aliases":"K_a,o","display":"K_{a,o}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":8},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0a0b4681c427fb78","tex":"K_a^\\top","aliases":"K_a^top top","display":"K_a^\\top","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also pre","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":16},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-73f17951391086e7","tex":"L_*","aliases":"L_*","display":"L_*","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":7},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a716b0894950a10a","tex":"L_i","aliases":"L_i","display":"L_i","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-09eb22fcb1ad8180","tex":"L_{k+1}","aliases":"L_k+1","display":"L_{k+1}","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"use","excerpt":"Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-987eab22f86d841c","tex":"\\LARGE","aliases":"LARGE","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-bda22e23ac42a073","tex":"\\Large","aliases":"Large","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3b3d7f267bfbab0b","tex":"\\large","aliases":"large","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf,","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-45e90aa85790529d","tex":"\\linewidth","aliases":"linewidth","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf,","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf,","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":20},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-7f54f92432b51276","tex":"m_o","aliases":"m_o","display":"m_o","kind":"use","excerpt":"athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasib","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-dc6fa4cc1dd82d35","tex":"\\mainmatter","aliases":"mainmatter","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"e distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_","count":7},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d441c86815baa4be","tex":"\\max_o","aliases":"max_o max","display":"\\max_o","kind":"use","excerpt":"m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty ","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"[When a report preserves every Bayesian correction] The following are equivalent: item For every prior, the posterior after o equals that after z on every positive-probability pair (o,z). item The equality holds for one prior positive on every model. item There is a stochastic kernel ρ(omid z) such that L_m(o)B(o,z)=Q_m(z)ρ(omid z) for all m,o,z. T","count":14},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3f9db2ae5742966b","tex":"\\midrule","aliases":"midrule","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-1e2deab3af39b182","tex":"\\min_a","aliases":"min_a min","display":"\\min_a","kind":"use","excerpt":"[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add a","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-13d3a20824a03360","tex":"\\min_{w\\in W}","aliases":"min_w∈ W min in","display":"\\min_{w\\in W}","kind":"use","excerpt":"[Which gains survive every admitted price?] The saving w·Δ is nonnegative for every nonnegative price vector if and only if every coordinate of Δ is nonnegative. If prices lie in the convex hull of a finite nonempty set W, their minimum saving equals min_w∈ Ww·Δ. A componentwise nonnegative vector has nonnegative inner product with every nonnegativ","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":22},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8bbbd92a94033296","tex":"n'","aliases":"n'","display":"n'","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive me","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3554f05e6897e490","tex":"O_e","aliases":"O_e","display":"O_e","kind":"use","excerpt":"belch:inquiry questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic succe","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-b2bdbcfb9c9cfc41","tex":"o_e","aliases":"o_e","display":"o_e","kind":"use","excerpt":"belch:inquiry questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic succe","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2a82579e1e2588f5","tex":"O_{t+1}","aliases":"O_t+1","display":"O_{t+1}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":17},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitte","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"use","excerpt":"An approximate model supports a different, quantitative claim. Exact policy transport and approximate planning must not share one unqualified guarantee. Fix a finite horizon H and one finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-state and ac","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","excerpt":"The theorem permits removal of an independent nuisance flag. It does not permit removal of every association. Consider two priors on (b,s), with rows indexed by b and columns by s: P=1/2 · 0 0 · 1/2, P'=0 · 1/2 1/2 · 0. Both individual marginals are uniform. A noisy calibration reports s correctly with probability 3/4, independently conditional on ","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the report probabilities and the aggregated positive-probability posteriors: z_a,o(p)=bar z_a,o(Cp), CΨ_a,o(p)=barΨ_a,o(Cp). Starting from Cp_0","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5c39c7933673208b","tex":"p_E","aliases":"p_E","display":"p_E","kind":"use","excerpt":"d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, th","count":7},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-b9e70c157342a064","tex":"P_H","aliases":"P_H","display":"P_H","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5627e8b6e58a3075","tex":"\\mathcal P_h","aliases":"P_h mathcal","display":"\\mathcal P_h","kind":"use","excerpt":"The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitte","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-cc9e0dabacaadc8b","tex":"p_{N-1}","aliases":"p_N-1","display":"p_{N-1}","kind":"use","excerpt":"Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that ","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-fe0059a8d2b46448","tex":"P_{\\rm av}","aliases":"P_rm av rm","display":"P_{\\rm av}","kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-43031e8fba4068f2","tex":"P_{\\rm det}","aliases":"P_rm det rm","display":"P_{\\rm det}","kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4bc9c56157a62dc5","tex":"\\operatorname{pa}","aliases":"pa operatorname","display":"\\operatorname{pa}","kind":"use","excerpt":"Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined b","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-bc660fe2b7285e52","tex":"\\par","aliases":"par","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-bfd0b07cc3c57995","tex":"\\operatorname{Pareto}","aliases":"Pareto operatorname","display":"\\operatorname{Pareto}","kind":"use","excerpt":"The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitte","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e695ee90e9b92c62","tex":"\\operatorname{Post}","aliases":"Post operatorname","display":"\\operatorname{Post}","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d692f9ac1c18626a","tex":"\\Pow","aliases":"Pow P mathcal","display":"{\\mathcal P}","kind":"use","excerpt":"Let E⊆ P(Ω) contain Ω and be closed under intersection. In this chapter the world does not change when an evidence event is received. The update is V↦ V∩ E. Physical actions and their effects will return in Chapter~[source reference].","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":22},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3406acd3f0f64ef4","tex":"Q_m","aliases":"Q_m","display":"Q_m","kind":"use","excerpt":"Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6e3edc954e0d904a","tex":"\\question","aliases":"question","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":14},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":13},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a3d472d51734fedb","tex":"\\mathbb R^2","aliases":"R^2 mathbb","display":"\\mathbb R^2","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2f01b4d5a5d6a367","tex":"\\mathbb R^d_{\\ge0}","aliases":"R^d_≥0 mathbb ge","display":"\\mathbb R^d_{\\ge0}","kind":"use","excerpt":"beginproposition[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be ","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e13c04e5f7fac1fc","tex":"\\raggedright","aliases":"raggedright","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf,","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero.","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":11},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":34},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive me","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-9bd6563a665b1ad1","tex":"\\small","aliases":"small","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span span operatorname","display":"{\\operatorname{span}}","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-7deace183d0a867f","tex":"\\sup_\\omega","aliases":"sup_ω sup omega","display":"\\sup_\\omega","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":40},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":20},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5d42311231194d5c","tex":"t^{-1}","aliases":"t^-1","display":"t^{-1}","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0bc6dba302face63","tex":"T^\\theta","aliases":"T^θ theta","display":"T^\\theta","kind":"use","excerpt":"For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f90055131c0edf2c","tex":"T_*","aliases":"T_*","display":"T_*","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":9},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-1a4a5aadced24804","tex":"T_a^\\theta","aliases":"T_a^θ theta","display":"T_a^\\theta","kind":"use","excerpt":"One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be finite dimensional, let T_a^θ be finitely many candidate linear operators, and let A_a be one complete candidate system chosen as reference. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear rea","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-b6d1c35d86f0d39d","tex":"T_k","aliases":"T_k","display":"T_k","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4e56fe1bda71e196","tex":"T_{k+1}","aliases":"T_k+1","display":"T_{k+1}","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ed0619d26eb6427a","tex":"T_w^\\theta","aliases":"T_w^θ theta","display":"T_w^\\theta","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also pre","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a9a485c98291d899","tex":"\\tableofcontents","aliases":"tableofcontents","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1","count":8},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3000c3bd47e8007c","tex":"\\toprule","aliases":"toprule","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV d_TV mathrm","display":"{d_{\\mathrm{TV}}}","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":8},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":15},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6fdb23dc8f442414","tex":"u_h","aliases":"u_h","display":"u_h","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-9eff72e097fd075b","tex":"u_\\varnothing","aliases":"u_∅ varnothing","display":"u_\\varnothing","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":37},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":33},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-9bacaf6e1626666c","tex":"V_{e,o}","aliases":"V_e,o","display":"V_{e,o}","kind":"use","excerpt":"A current support V≠∅ records possible states. The experiment is permitted only if V⊆ D_e. After report o, the possible post-experiment states are V_e,o={f_e(x):x∈ V, o_e(x)=o}. Only nonempty report branches are retained. If the target concerns a past value, that value must survive as an immutable tag. Resetting the current state is not recovering ","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8e7676bd93a7edb5","tex":"v_o","aliases":"v_o","display":"v_o","kind":"use","excerpt":"athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasib","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-786a3c53c74db475","tex":"v_\\omega","aliases":"v_ω omega","display":"v_\\omega","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0fef838539965a6b","tex":"\\vfill","aliases":"vfill","display":null,"kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add a","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-967c63ebf4cd78be","tex":"\\widehat R","aliases":"widehat R widehat","display":"\\widehat R","kind":"use","excerpt":"widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f207fa5d22c484f6","tex":"\\widehat v","aliases":"widehat v widehat","display":"\\widehat v","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8307c5f38fc587f5","tex":"\\widehat v_\\omega","aliases":"widehat v_ω widehat omega","display":"\\widehat v_\\omega","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3f476ecaf7bf4a0d","tex":"\\widetilde J","aliases":"widetilde J widetilde","display":"\\widetilde J","kind":"use","excerpt":"[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certi","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-9117c60fc5e9378b","tex":"\\widetilde K","aliases":"widetilde K widetilde","display":"\\widetilde K","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8da03ee77856ac56","tex":"\\widetilde K_{a,o}","aliases":"widetilde K_a,o widetilde","display":"\\widetilde K_{a,o}","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-649a6c9973f355dd","tex":"\\widetilde p","aliases":"widetilde p widetilde","display":"\\widetilde p","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_","count":7},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4741526210fb0d8d","tex":"\\widetilde p_0","aliases":"widetilde p_0 widetilde","display":"\\widetilde p_0","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-47fb519e5af1c253","tex":"\\widetilde p_E","aliases":"widetilde p_E widetilde","display":"\\widetilde p_E","kind":"use","excerpt":"d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, th","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-26f82a5d2a875a6e","tex":"\\widetilde P_H","aliases":"widetilde P_H widetilde","display":"\\widetilde P_H","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a25d72cc47e636ba","tex":"\\widetilde\\pi","aliases":"widetildeπ widetilde pi","display":"\\widetilde\\pi","kind":"use","excerpt":"[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certi","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":47},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.","count":8},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"On the nonnegative coordinate cone, preservation is equivalent to nonnegative matrix entries, as is seen by applying a matrix to every standard basis vector. Because the matrices sum to I, their nonnegative off-diagonal entries must all vanish. An idempotent diagonal entry satisfies x^2=x, hence is zero or one. The diagonal sums are one, so each co","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-1e4fb4e21dd5aa54","tex":"X_{t+1}","aliases":"X_t+1","display":"X_{t+1}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":23},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":20},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":35},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the repor","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-9a75d558418afc42","tex":"z_{a,o}","aliases":"z_a,o","display":"z_{a,o}","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-45eb5061d9b39786","tex":"Z_H","aliases":"Z_H","display":"Z_H","kind":"use","excerpt":"e distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey d_","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":7},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"probability. For a positive pair, posterior equality yields p(m)L_m(o)B(o,z)=Pr(o,z)p(mmid o) =ρ(omid z)Pr(z)p(mmid z). Substituting Bayes' formula and cancelling positive p(m) gives the factorization. A zero-probability pair under a positive prior has L_m(o)B(o,z)=0 for all m; its conditional weight is zero. If z is impossible for every model, ass","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6cb4d369799123ce","tex":"\\beta_i","aliases":"β_i beta","display":"\\beta_i","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.","count":7},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_","count":10},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume d_TV(Cp_0,widetilde p_","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d08739f367199333","tex":"\\epsilon_t","aliases":"ϵ_t epsilon","display":"\\epsilon_t","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"For minimality, take any admissible refinement η of π_0. It has at least two blocks. Fix a strictly positive fine likelihood ℓ and a strictly positive coarse likelihood barℓ inducing its assumed update. For states x,y in one block and z in a different block, compare priors assigning half their mass to x,z and to y,z. Their coarse priors agree. Equa","count":8},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be finite dimensional, let T_a^θ be finitely many candidate linear operators, and let A_a be one complete candidate system chosen as reference. A model index θ is fixed during the initial comparison. Let Q:V→ Y be the same linear rea","count":15},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state n","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add a","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":21},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-6d2412f84c728949","tex":"\\pi_*","aliases":"π_* pi","display":"\\pi_*","kind":"use","excerpt":"[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certi","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-2d28a27ac7821632","tex":"\\pi_0","aliases":"π_0 pi","display":"\\pi_0","kind":"use","excerpt":"A mandatory partition π_0 with at least two blocks protects the target distinctions. We seek a refinement that supports every prediction and each separate Bayesian update for every prior. The observation is an operation available in its own right. This is stronger than inspecting only a chosen combined transition-and-report schedule.","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-560c891f5e585ac2","tex":"\\pi_{\\rm init}","aliases":"π_rm init pi rm","display":"\\pi_{\\rm init}","kind":"use","excerpt":"[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest ","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"[When a report preserves every Bayesian correction] The following are equivalent: item For every prior, the posterior after o equals that after z on every positive-probability pair (o,z). item The equality holds for one prior positive on every model. item There is a stochastic kernel ρ(omid z) such that L_m(o)B(o,z)=Q_m(z)ρ(omid z) for all m,o,z. T","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"A state sufficient for every later filter. Define the correction signature Σ(V)=(t(V∩ E))_E∈ E. It records what each admitted future filter would leave of the possible target values.","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"For a compression, reverse the splitting. Let C:V→ Z and R:Z→ V satisfy CR=I_Z, and put P=RC. The compressed operator is Φ(T)=CTR.","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-437405ce641f5157","tex":"\\Psi_{a,o}","aliases":"Ψ_a,o Psi","display":"\\Psi_{a,o}","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":9},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":10},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-efdf8990a98fa39d","tex":"\\omega'","aliases":"ω' omega","display":"\\omega'","kind":"use","excerpt":"[0.12in]Reasoning LARGE Evidence, Methods, and Accountable Revision Large Reviewed Mathematical Core hrule Komorebi P.hfill Core edition 0.2 9 October 2026hfill Internal proof review vfill large A continuous argument from accessible evidence to probability-valid operations, correction, controlled inquiry, and accountable revision. This is a newly w","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"For conditioning, let u,v be the restrictions of the two history laws to E, with masses p_E,widetilde p_E. The difference on the complement has ℓ^1 norm at least |p_E-widetilde p_E|. Thus u-v _1+|p_E-widetilde p_E|≤2d_H. Adding and subtracting v/p_E bounds the conditional distance by d_H/p_E. Interchanging the laws gives d_H/widetilde p_E, so the s","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-313774f1667bd9ea","tex":"\\ell_o","aliases":"ℓ_o ell","display":"\\ell_o","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Exact accounting of the discarded excursion] For any endomorphisms T,U of V, Φ(TU)-Φ(T)Φ(U)=CT(I-P)UR. Thus compression preserves this composition precisely when the discarded excursion on the right vanishes. Full split encoding T↦ ATS preserves composition without that extra condition. Expand the product Φ(T)Φ(U)=CTRCUR=CTPUR and subtract it from","count":5},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ff1f52005cd89e31","tex":"\\sum_{i:w_i>0}","aliases":"∑_i:w_i>0 sum","display":"\\sum_{i:w_i>0}","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0c18bcf01edb7e76","tex":"\\sum_{j\\ge1}","aliases":"∑_j≥1 sum ge","display":"\\sum_{j\\ge1}","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ccbd66bdaed18a2b","tex":"\\sum_o","aliases":"∑_o sum","display":"\\sum_o","kind":"use","excerpt":"Let M∈ M be a fixed model with prior p(m). Let an experiment have likelihood L_m(o) on a finite nonempty observation alphabet O. A reporting channel B(o,z), independent of the model conditional on o, creates a retained report Z. The joint law is Pr(m,o,z)=p(m)L_m(o)B(o,z), Q_m(z)=∑_oL_m(o)B(o,z). Posterior distributions are defined only on positive","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-3abe3b490f0f125b","tex":"\\sum_{o,x'}","aliases":"∑_o,x' sum","display":"\\sum_{o,x'}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f1e914837dbf8c89","tex":"\\sum_{t=0}^{H-1}","aliases":"∑_t=0^H-1 sum","display":"\\sum_{t=0}^{H-1}","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy d_TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-31e59f15403939ba","tex":"\\sum_{x':\\pi(x')=z'}","aliases":"∑_x':π(x')=z' sum pi","display":"\\sum_{x':\\pi(x')=z'}","kind":"use","excerpt":"[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the repor","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-f05731197cd413ab","tex":"\\sum_{x'\\in B}","aliases":"∑_x'∈ B sum in","display":"\\sum_{x'\\in B}","kind":"use","excerpt":"[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest ","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-ea8a48c6147135b3","tex":"\\sum_\\omega","aliases":"∑_ω sum omega","display":"\\sum_\\omega","kind":"use","excerpt":"widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume d_TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"Let E⊆ P(Ω) contain Ω and be closed under intersection. In this chapter the world does not change when an evidence event is received. The update is V↦ V∩ E. Physical actions and their effects will return in Chapter~[source reference].","count":7},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"A stop command is assumed authentic and observable before the next action is selected. Its delivery and authentication are premises, not consequences of the following fixed-point calculation. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.","count":3},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":4},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy d_TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')","count":16},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i. Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint diagona","count":9},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a surjective linear r:V→ Z, common updates G_a, and a readout bar Q with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these conditions, V/E is a common quotient and also pre","count":13},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"The sensor used throughout the book. Write a world as ω=(b,s,n)∈{0,1}^3. The physical target bit is b. The sensor convention is s. The nuisance bit n never affects the operations or outputs of the initial contract. The display is y=b⊕ s, where ⊕ is addition modulo two. A permitted control f flips the physical bit and leaves s,n unchanged. Consequen","count":6},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).","count":2},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-892cecbf386dea45","tex":"\\bigcap_i","aliases":"⋂_i bigcap","display":"\\bigcap_i","kind":"use","excerpt":"[Target-relative joint reconstruction] There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv ","count":1},{"work":"research-machine-theory-formal","url":"notation/research-machine-theory-formal.html#symbol-9d31d41a627eed08","tex":"\\bigcap_{x:\\alpha(x)=h}","aliases":"⋂_x:α(x)=h bigcap alpha","display":"\\bigcap_{x:\\alpha(x)=h}","kind":"use","excerpt":"Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"introduction-passage","excerpt":"cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass p","count":74},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":30},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass p","count":204},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":57},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":110},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-bc61bec23787c0e8","tex":"A^X_m","aliases":"A^X_m","display":"A^X_m","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-80fdb5bd0b16e2c0","tex":"A^Z_m","aliases":"A^Z_m","display":"A^Z_m","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":7},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5b950cf09bc94cd2","tex":"A_{\\rm end}","aliases":"A_rm end rm","display":"A_{\\rm end}","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":42},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-bb3cdc779ccbade8","tex":"\\bar J","aliases":"bar J bar","display":"\\bar J","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-446fba0b12b9aac6","tex":"\\bar J_0","aliases":"bar J_0 bar","display":"\\bar J_0","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-58d2d6ac0d680d9f","tex":"\\bar J_h","aliases":"bar J_h bar","display":"\\bar J_h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-32da9c12730e8ae8","tex":"\\bar J_{h-1}","aliases":"bar J_h-1 bar","display":"\\bar J_{h-1}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-fad6ae0fd6c0acb8","tex":"\\bar K","aliases":"bar K bar","display":"\\bar K","kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":10},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d9966b1737faabea","tex":"\\bar K_{a,o}","aliases":"bar K_a,o bar","display":"\\bar K_{a,o}","kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2311bbf2ac9fa5af","tex":"\\bar u","aliases":"bar u bar","display":"\\bar u","kind":"introduction-passage","excerpt":"cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass p","count":7},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-38f0a5c7c65b9678","tex":"\\bar u_h","aliases":"bar u_h bar","display":"\\bar u_h","kind":"introduction-passage","excerpt":"cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass p","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0b10bfd89492429d","tex":"\\bar u_{hao}","aliases":"bar u_hao bar","display":"\\bar u_{hao}","kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c0435683f947250d","tex":"\\bar z","aliases":"bar z bar","display":"\\bar z","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f68011f6686ac355","tex":"\\bar z_{a,o}","aliases":"bar z_a,o bar","display":"\\bar z_{a,o}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4415f58b83fbd8c6","tex":"\\bar\\Psi","aliases":"barΨ bar Psi","display":"\\bar\\Psi","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5c18dd7e6706bb6e","tex":"\\bar\\Psi_{a,o}","aliases":"barΨ_a,o bar Psi","display":"\\bar\\Psi_{a,o}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":57},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":13},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-12a28881c3f25466","tex":"C_{k+1}","aliases":"C_k+1","display":"C_{k+1}","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":31},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":38},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":17},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-08e01505c24de077","tex":"\\mathrm{end}","aliases":"end mathrm","display":"\\mathrm{end}","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":32},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":13},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-24ceed0c3b629a6f","tex":"F_{a_1}","aliases":"F_a_1","display":"F_{a_1}","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d6d2fa627b1c221a","tex":"F_{a_k}","aliases":"F_a_k","display":"F_{a_k}","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":35},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":33},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass p","count":41},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":18},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":12},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6e30319bc4adb843","tex":"J_h","aliases":"J_h","display":"J_h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":27},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":57},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-16b5593bbf90ad19","tex":"K_{a,o}","aliases":"K_a,o","display":"K_{a,o}","kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":14},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":29},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":21},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a20d1b2c5e09f23a","tex":"\\min_{a\\in A(q)}","aliases":"min_a∈ A(q) min in","display":"\\min_{a\\in A(q)}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-73ed63492f6711c5","tex":"\\min_{d\\in D(q)}","aliases":"min_d∈ D(q) min in","display":"\\min_{d\\in D(q)}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":33},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":33},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":53},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":30},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩,","count":34},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e695ee90e9b92c62","tex":"\\operatorname{Post}","aliases":"Post operatorname","display":"\\operatorname{Post}","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":11},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":21},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-bbf62fa0f1dd5f2f","tex":"\\mathcal Q_h","aliases":"Q_h mathcal","display":"\\mathcal Q_h","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":25},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":10},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c2b4530a730b06f1","tex":"\\mathrm{run}","aliases":"run mathrm","display":"\\mathrm{run}","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":36},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":83},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f90055131c0edf2c","tex":"T_*","aliases":"T_*","display":"T_*","kind":"introduction-passage","excerpt":"T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.","count":30},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-b6d1c35d86f0d39d","tex":"T_k","aliases":"T_k","display":"T_k","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":10},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4e56fe1bda71e196","tex":"T_{k+1}","aliases":"T_k+1","display":"T_{k+1}","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","excerpt":"cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass p","count":46},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6fdb23dc8f442414","tex":"u_h","aliases":"u_h","display":"u_h","kind":"introduction-passage","excerpt":"cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass p","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9d876ec3d2d253a9","tex":"u_{hao}","aliases":"u_hao","display":"u_{hao}","kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":50},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":29},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":46},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":82},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":13},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":53},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":68},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"introduction-passage","excerpt":"horizon for which the policy is defined on all positive-mass prefixes. First prove bar u_h=π_*u_h by induction. It holds initially by the prior assumption. At a positive history, Lemma~[source reference] gives the same available actions on both sides. For any action with positive policy mass, every current fine state of positive mass enables that a","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass p","count":47},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6d2412f84c728949","tex":"\\pi_*","aliases":"π_* pi","display":"\\pi_*","kind":"introduction-passage","excerpt":"cy preserves the law of every finite coarse-state/action/report path. Consequently it preserves every action/report/label history. At every positive-mass history it also preserves the aggregated posterior and the set of actions admitted throughout its support. These claims hold at every horizon for which the policy is defined on all positive-mass p","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the least k with z∈ T_k.","count":13},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":18},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-7151afe2b5220a1a","tex":"\\ell_d","aliases":"ℓ_d ell","display":"\\ell_d","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":45},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"introduction-passage","excerpt":"A model of the field is a set B⊆ V containing A and satisfying every implication P_j⊆ B⇒ c_j∈ B. A protected extension also contains S and avoids N. Define C_0=A∪ S, C_k+1=C_k∪{c_j:P_j⊆ C_k}. The terms increase and lie in V, so the sequence stabilizes. Write C for its stable value.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"The controlled predecessor. Let X,Z be finite nonempty sets and π:X→ Z a surjection. Use a common finite action alphabet with separate running and ending catalogues A^X_m(x) and A^Z_m(z), where m∈{run,end}. Every admitted action has a nonempty successor set. The mode can be included in the action name, so the same printed action in two modes need n","count":52},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":20},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0876d820c228b469","tex":"\\sum_{o:\\bar z_{a,o}(q)>0}","aliases":"∑_o:bar z_a,o(q)>0 sum bar","display":"\\sum_{o:\\bar z_{a,o}(q)>0}","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"introduction-passage","excerpt":"The sure-ending attractor. Define T_0 · =G, T_k+1 · =T_k∪{z∈ S:some a∈ A_rm end(z) satisfies Post(z,a)⊆ T_k}. The sequence is increasing and every term is contained in S. Because the state set is finite, it becomes constant after finitely many strict additions. Let T_* denote its stable value. For z∈ T_*, define the first-entry rank ρ(z) as the lea","count":45},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"One answer versus every permitted continuation. Let X be a finite nonempty state space, and fix an available record r:X→ H. Let A be a finite alphabet of total deterministic operations F_a:X→ X, and let q:X→ Y be the protected output. A word w=a_1⋯ a_k executes a_1 first. Thus F_w=F_a_k⋯ F_a_1, and the empty word denotes the identity.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","excerpt":"Let J_h(p) be the least expected charge plus terminal loss attainable using at most h further experiments, with terminal decisions permitted at any step. It is defined by finite observation trees, not by a hidden-state controller. Its coarse recurrence is bar J_0(q) · =min_d∈ D(q)⟨ q,ℓ_d⟩, Q_h(q,a) · =c_a(q)+∑_o:bar z_a,o(q)>0 bar z_a,o(q)bar J_h-1","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2876747ca029362d","tex":"\\mathbf1^\\top","aliases":"1^top mathbf top","display":"\\mathbf1^\\top","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ca30d7bde7b966d9","tex":"\\mathbf1_X^\\top","aliases":"1_X^top mathbf top","display":"\\mathbf1_X^\\top","kind":"use","excerpt":"Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point m","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c907afa1681c3705","tex":"\\mathbf1_Z^\\top","aliases":"1_Z^top mathbf top","display":"\\mathbf1_Z^\\top","kind":"use","excerpt":"Because 1_Z^top C=1_X^top, the matrix equality gives equal report masses. Dividing it by a common positive mass gives equal aggregated posteriors. Conversely, equality of masses and positive-mass posteriors gives the unnormalized matrix equality on every prior. If the common mass is zero, both unnormalized vectors are zero by nonnegativity. Point m","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a80c8a675a0e6f40","tex":"a'","aliases":"a'","display":"a'","kind":"use","excerpt":"After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0014cfc2591bdee6","tex":"a_*","aliases":"a_*","display":"a_*","kind":"use","excerpt":"For a true minimizer a_*, R(widehat a)≤widehat R(widehat a)+ξ ≤widehat R(a_*)+ξ≤ R(a_*)+2ξ. The loss range supplies the cap one. Finally, for any other a, subtract the two error bounds to obtain R(a)-R(widehat a)≥widehat R(a)-widehat R(widehat a)-2ξ>0. source[GA], ``A common quantitative grounding certificate'' and [MU], ``A joint posterior and rec","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ee19b85346d2e7d2","tex":"A_A","aliases":"A_A","display":"A_A","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"ne quotient for several possible dynamics A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as the","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6c1908f93f8dee44","tex":"A_B","aliases":"A_B","display":"A_B","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5a8e57a002dc32f4","tex":"a_d","aliases":"a_d","display":"a_d","kind":"use","excerpt":"Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined b","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1d9437cb1fc225ec","tex":"a_{i-1}","aliases":"a_i-1","display":"a_{i-1}","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"What several accessible records can reconstruct. Let V be a finite-dimensional vector space over a field. Participant i has a linear record A_i v. Stack the records into Av=(A_1v,…,A_mv), and let Qv be the protected linear target.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4f6eb91cb1eca840","tex":"A_{\\rm run}","aliases":"A_rm run rm","display":"A_{\\rm run}","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-82821bc2124f1658","tex":"A_t","aliases":"A_t","display":"A_t","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"An exact sensor realization. Return to x=(b,s,n)∈{0,1}^3 and retain z=(b,s). The actions hold and flip are available everywhere. A repair action is available only where b=0 and leaves that target value fixed. The nuisance bit is retained by every fine transition but ignored by the coarse state.","count":47},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-eebc9b85b3d9979a","tex":"\\bar D","aliases":"bar D bar","display":"\\bar D","kind":"use","excerpt":"For a nonnegative vector u on X, write (π_*u)(z)=∑_π x=zu(x). This notation applies to probability distributions and to unnormalized history masses. A coarse model has labels barℓ, enabled sets bar D, and joint instruments bar K on its enabled actions.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-52555ee0fe55f126","tex":"\\bar F","aliases":"bar F bar","display":"\\bar F","kind":"use","excerpt":"[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a(","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0ddd6936013fb39a","tex":"\\bar F_a","aliases":"bar F_a bar","display":"\\bar F_a","kind":"use","excerpt":"[Evidence-initialized behavioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a(","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-742fe606e2cdb4d4","tex":"\\bar K_a^\\top","aliases":"bar K_a^top bar top","display":"\\bar K_a^\\top","kind":"use","excerpt":"nique coarsest refinement of π_0 for which all specified predictions admit stochastic coarse transitions and all separate Bayesian updates admit fixed strictly positive coarse likelihoods for every prior. At a stable partition the probability of reaching each coarse block depends only on the current block. These sums define a stochastic coarse tran","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-204116e1ee0f35c1","tex":"\\bar K_{a_i,o_i}","aliases":"bar K_a_i,o_i bar","display":"\\bar K_{a_i,o_i}","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-cf72cfc588b3df8b","tex":"\\bar p","aliases":"bar p bar","display":"\\bar p","kind":"use","excerpt":"Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-990b780acc690717","tex":"\\bar p_0","aliases":"bar p_0 bar","display":"\\bar p_0","kind":"use","excerpt":"Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8f8a4d0a063c4065","tex":"\\bar Q","aliases":"bar Q bar","display":"\\bar Q","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a vector space Z, a surjective linear map r:V→ Z, common linear updates G_a:Z→ Z, and a linear readout bar Q:Z→ Y with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these con","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"use","excerpt":"havioral state] A finite predictor initialized only from r(x), updated only by the operation names, and exact for all protected continuation outputs exists if and only if r(x)=r(x') Longrightarrow x∼ x'. When it exists, X/∼ supplies a predictor with the smallest possible number of initialized states. Its updates are bar F_a([x])=[F_ax] and its read","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-028d200d0ea01a8b","tex":"\\bar u_\\varnothing","aliases":"bar u_∅ bar varnothing","display":"\\bar u_\\varnothing","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d97fba459e62ed0e","tex":"\\bar\\ell","aliases":"barℓ bar ell","display":"\\bar\\ell","kind":"use","excerpt":"For a nonnegative vector u on X, write (π_*u)(z)=∑_π x=zu(x). This notation applies to probability distributions and to unnormalized history masses. A coarse model has labels barℓ, enabled sets bar D, and joint instruments bar K on its enabled actions.","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d13b1528b249ec85","tex":"\\bigwedge","aliases":"bigwedge","display":"\\bigwedge","kind":"use","excerpt":"Logical interpretation is an additional map. Let μ(v) be the proposition represented by claim address v. Suppose every admitted leaf and positive observation is true under μ, and each admitted rule satisfies (bigwedge_p∈ P_jμ(p))Longrightarrowμ(c_j). Then every node in C is true under μ. Indeed, induction over the closure stages propagates truth fr","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-7d6b2159151c6caa","tex":"\\bigwedge_{p\\in P_j}","aliases":"bigwedge_p∈ P_j bigwedge in","display":"\\bigwedge_{p\\in P_j}","kind":"use","excerpt":"Logical interpretation is an additional map. Let μ(v) be the proposition represented by claim address v. Suppose every admitted leaf and positive observation is true under μ, and each admitted rule satisfies (bigwedge_p∈ P_jμ(p))Longrightarrowμ(c_j). Then every node in C is true under μ. Indeed, induction over the closure stages propagates truth fr","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"An agent given only y can predict every future display under repeated flips: after k flips it is y⊕(kbmod2). It cannot determine the current physical bit. Worlds (0,0,0) and (1,1,0) share display zero and have different targets. The obstruction is the fibre of the available record, not a shortage of computational effort.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":7},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0. In the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"[Individual acceptability is not a common message] Two cases have the same sender record. In one, the sole acceptable artifact is c_0. In the other it is c_1≠ c_0. Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"use","excerpt":"tion, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic successor f_e:D_e→ X, a report o_e:D_e→ O_e, and a fixed nonnegativ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-efafb53fff3f1cb9","tex":"\\mathcal C_h","aliases":"C_h mathcal","display":"\\mathcal C_h","kind":"use","excerpt":"Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4222b21eadd5f7b9","tex":"C_i","aliases":"C_i","display":"C_i","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c591dfc771720384","tex":"c_{\\max}","aliases":"c_max max","display":"c_{\\max}","kind":"use","excerpt":"ng must not share one unqualified guarantee. Fix a finite horizon H and one nonempty finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-state and action/report path in both models, taking values in [0,B]. For example, a terminal loss in [0,1] and ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-42eb4bc96455bb91","tex":"C_\\pi","aliases":"C_π pi","display":"C_\\pi","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-b893acf270b3c5ec","tex":"\\operatorname{CPre}","aliases":"CPre operatorname","display":"\\operatorname{CPre}","kind":"use","excerpt":"For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-414c634398f620ed","tex":"\\operatorname{CPre}^X_m","aliases":"CPre^X_m operatorname","display":"\\operatorname{CPre}^X_m","kind":"use","excerpt":"For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1f6f666e8c570585","tex":"\\operatorname{CPre}^X_{\\rm end}","aliases":"CPre^X_rm end operatorname rm","display":"\\operatorname{CPre}^X_{\\rm end}","kind":"use","excerpt":"licy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coordinates. The goal-set assumption gives equality at T_0. Inverse images commute with union and intersection. Applying Lemma~[source reference] to the ending predecessor gives the induction step T^X","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ec773424c50a1309","tex":"\\operatorname{CPre}^Z_m","aliases":"CPre^Z_m operatorname","display":"\\operatorname{CPre}^Z_m","kind":"use","excerpt":"For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d9463ae0a8cc24c7","tex":"\\operatorname{CPre}^Z_{\\rm end}","aliases":"CPre^Z_rm end operatorname rm","display":"\\operatorname{CPre}^Z_{\\rm end}","kind":"use","excerpt":"licy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coordinates. The goal-set assumption gives equality at T_0. Inverse images commute with union and intersection. Applying Lemma~[source reference] to the ending predecessor gives the induction step T^X","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz i","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6de3eb080886bbf6","tex":"D_a^\\theta","aliases":"D_a^θ theta","display":"D_a^\\theta","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3b858f3e8665ea65","tex":"D_e","aliases":"D_e","display":"D_e","kind":"use","excerpt":"One investigation, one resource budget. questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X,","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-fdadfd14bd665b99","tex":"D_h","aliases":"D_h","display":"D_h","kind":"use","excerpt":"Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5341c066ee58261e","tex":"d_H","aliases":"d_H","display":"d_H","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,","count":11},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ef97823489e8f12d","tex":"D_v","aliases":"D_v","display":"D_v","kind":"use","excerpt":"Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined b","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":37},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"E,Σ · An evidence family and its correction signature.","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"[Minimal exact memory for a correction contract] Equality of Σ is a congruence for every update V↦ V∩ E_0. The signature supports the current target set and its exact update after every finite evidence sequence. Any deterministic retained state supporting the same readouts and updates on all candidate subsets must distinguish every two distinct sig","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e84b1a98b5a23eb6","tex":"\\operatorname{ev}","aliases":"ev operatorname","display":"\\operatorname{ev}","kind":"use","excerpt":"The evaluation ev_v:Γ(F)→ F_v records the information visible in one view. When Γ(F) is nonempty and a target q:Γ(F)→ Y is specified, that view is sufficient exactly when the target is constant on its fibres. [Target factorization through a realization view] Suppose Γ(F) and Y are nonempty. There is a function d:ev_v(Γ(F))→ Y such that q=d∘ev_v if ","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-db44ffded3c6918e","tex":"\\operatorname{ev}_v","aliases":"ev_v operatorname","display":"\\operatorname{ev}_v","kind":"use","excerpt":"The evaluation ev_v:Γ(F)→ F_v records the information visible in one view. When Γ(F) is nonempty and a target q:Γ(F)→ Y is specified, that view is sufficient exactly when the target is constant on its fibres. [Target factorization through a realization view] Suppose Γ(F) and Y are nonempty. There is a function d:ev_v(Γ(F))→ Y such that q=d∘ev_v if ","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"The sensor used throughout the book. Write a world as ω=(b,s,n)∈{0,1}^3. The physical target bit is b. The sensor convention is s. The nuisance bit n never affects the operations or outputs of the initial contract. The display is y=b⊕ s, where ⊕ is addition modulo two. A permitted control f flips the physical bit and leaves s,n unchanged. Consequen","count":9},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f1ffe82ebc455b14","tex":"f^*","aliases":"f^*","display":"f^*","kind":"use","excerpt":"Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfyi","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"Over F_2, Agent A observes A_A(b,s)=b+s and Agent B observes A_B(b,s)=s. The joint matrix is invertible. Either agent alone lacks a direction needed to recover b. The nuisance bit may remain unknown because the target map ignores it.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-48dc92a98cff75ec","tex":"f_e","aliases":"f_e","display":"f_e","kind":"use","excerpt":", one resource budget questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a determinist","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-78b29d109052da32","tex":"F_v","aliases":"F_v","display":"F_v","kind":"use","excerpt":"Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfyi","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"Why a small probability error is not a sure guarantee. Take three states s,g,b, with safe set {s,g} and goal {g}. Running holds s or g in place. The nominal ending action at s moves to g with probability one. The perturbed action instead moves to g with probability 1-ϵ and to unsafe state b with probability ϵ, where 0<ϵ<1. A hold action is availabl","count":17},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-29f81858f6292043","tex":"g^*","aliases":"g^*","display":"g^*","kind":"use","excerpt":"Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfyi","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a vector space Z, a surjective linear map r:V→ Z, common linear updates G_a:Z→ Z, and a linear readout bar Q:Z→ Y with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these con","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"The experiment catalogue and the terminal-artifact catalogue are finite, and the resource vector has a fixed finite dimension. A terminal artifact a is acceptable only if every fixed recipient succeeds in every state of V. Its worst-case terminal cost is g_a(V), computed componentwise. A controller is a literal finite tree with one record per exper","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d6ba5e7e6d3c4f08","tex":"G_X","aliases":"G_X","display":"G_X","kind":"use","excerpt":"Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1395c2791cfd267e","tex":"G_Z","aliases":"G_Z","display":"G_Z","kind":"use","excerpt":"Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5f29bf5e40e3c97b","tex":"G_{\\tau(a)}","aliases":"G_τ(a) tau","display":"G_{\\tau(a)}","kind":"use","excerpt":"A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a5620dda8c46dbb1","tex":"G_{\\tau(w)}","aliases":"G_τ(w) tau","display":"G_{\\tau(w)}","kind":"use","excerpt":"[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1e9de57bf29ef2a6","tex":"h'","aliases":"h'","display":"h'","kind":"use","excerpt":"[Exact migration criterion] There is a map J:U_0→ V_0 satisfying JM=N if and only if M(h)=M(h') Longrightarrow N(h)=N(h') (h,h'∈ H). When it exists, J is unique on attained old records. Necessity follows by applying J to the equal old records. Conversely, define J(M(h))=N(h). The fibre condition makes the definition independent of the representativ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5bc59e2e502080d9","tex":"\\mathcal H_{j-1}","aliases":"H_j-1 mathcal","display":"\\mathcal H_{j-1}","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":13},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":26},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of ","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0475b2ac7d225af5","tex":"i_2","aliases":"i_2","display":"i_2","kind":"use","excerpt":"Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of ","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-34af8cf33f9487de","tex":"I_U","aliases":"I_U","display":"I_U","kind":"use","excerpt":"[Forward transport of a reusable method] If the source domain is invariant, and TF_a=G_τ(a)T, q=bar qT, then for every finite word w, TF_w=G_τ(w)T, qF_w=bar qG_τ(w)T. The empty word gives TI_Z=I_UT. Assume the first equality for a prefix. Invariance allows the next local equality to be applied on its successor, yielding TF_aF_w=G_τ(a)TF_w=G_τ(a)G_τ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f5bae71a192fc64d","tex":"I_V","aliases":"I_V","display":"I_V","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-bdedc0251851d395","tex":"I_Z","aliases":"I_Z","display":"I_Z","kind":"use","excerpt":"be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the compatible image A(V), not necessarily I_Z on arbitrary","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0f6f9e87439c63ef","tex":"\\mathrm{id}","aliases":"id mathrm","display":"\\mathrm{id}","kind":"use","excerpt":"Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfyi","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im","display":null,"kind":"use","excerpt":":reconstruction There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv on im A. If Av=Aw, then","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0e97a80aa927d24d","tex":"j_1","aliases":"j_1","display":"j_1","kind":"use","excerpt":"Replay checks a finite causal sequence. A trace σ=(j_1,…,j_m) lists rule addresses. Start from K_0=A∪ S. At each step, reject an unknown rule address or a rule whose premises are not all in the current known set. Otherwise adjoin its conclusion. Strict replay also rejects a conclusion already known. Causal replay allows it.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-416b4c70e8031735","tex":"j_m","aliases":"j_m","display":"j_m","kind":"use","excerpt":"Replay checks a finite causal sequence. A trace σ=(j_1,…,j_m) lists rule addresses. Start from K_0=A∪ S. At each step, reject an unknown rule address or a rule whose premises are not all in the current known set. Otherwise adjoin its conclusion. Strict replay also rejects a conclusion already known. Causal replay allows it.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state n","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-fb9dab5747004561","tex":"K'","aliases":"K'","display":"K'","kind":"use","excerpt":"[Replay is sound and compositional] If replay accepts a trace from a set K⊆ C and returns K', then K⊆ K'⊆ C. For either fixed replay mode, replay of a concatenated trace is replay of its first part followed by replay of its second part from the returned set. Failure in either part is failure of the concatenated replay. The empty trace changes nothi","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-cd49a1788f8baddd","tex":"k_*","aliases":"k_*","display":"k_*","kind":"use","excerpt":"[Minimum fixed-length supplementary information] For u∈ U_0, put k(u)=|N(M^-1{u})|, k_* =max_u∈ U_0k(u). Suppose an auxiliary encoder may inspect the complete source history and sends one fixed-length binary word in addition to the old record. Exact recovery of N is possible with b supplementary bits if and only if 2^b≥ k_*. Thus the minimum is lce","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"Replay checks a finite causal sequence. A trace σ=(j_1,…,j_m) lists rule addresses. Start from K_0=A∪ S. At each step, reject an unknown rule address or a rule whose premises are not all in the current known set. Otherwise adjoin its conclusion. Strict replay also rejects a conclusion already known. Causal replay allows it.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6cc92a51c6969e2c","tex":"K_a","aliases":"K_a","display":"K_a","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0a0b4681c427fb78","tex":"K_a^\\top","aliases":"K_a^top top","display":"K_a^\\top","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a vector space Z, a surjective linear map r:V→ Z, common linear updates G_a:Z→ Z, and a linear readout bar Q:Z→ Y with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these con","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"T_*,L_*,ρ · Ending region, safe-running kernel, and first-entry rank.","count":45},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"Let l be the number of leaves and i_1,i_2 the counts of unary and binary internal nodes. Counting tree edges gives i_1+2i_2=i_1+i_2+l-1, hence i_2=l-1. Total records are 2l-1+i_1≥2n-1. A successful candidate at both coordinatewise lower bounds dominates every other attainable cost profile. Its validity and horizon check make it an actual member of ","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4abaa9ae4986b6c8","tex":"L^X_*","aliases":"L^X_*","display":"L^X_*","kind":"use","excerpt":"[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine polic","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a0aa4c98d4789ca3","tex":"L^X_k","aliases":"L^X_k","display":"L^X_k","kind":"use","excerpt":"Transport of both fixed-point sequences. Construct T^X_k,L^X_k on the fine game and T^Z_k,L^Z_k on the coarse game as in Chapter~[source reference]. In particular, the running sequence starts from the stable ending region in its own model.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-dae7d96e61631e5c","tex":"L^Z_*","aliases":"L^Z_*","display":"L^Z_*","kind":"use","excerpt":"[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine polic","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-7bf466f38c661d7b","tex":"L^Z_k","aliases":"L^Z_k","display":"L^Z_k","kind":"use","excerpt":"Transport of both fixed-point sequences. Construct T^X_k,L^X_k on the fine game and T^Z_k,L^Z_k on the coarse game as in Chapter~[source reference]. In particular, the running sequence starts from the stable ending region in its own model.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-73f17951391086e7","tex":"L_*","aliases":"L_*","display":"L_*","kind":"use","excerpt":"T_*,L_*,ρ · Ending region, safe-running kernel, and first-entry rank.","count":19},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a716b0894950a10a","tex":"L_i","aliases":"L_i","display":"L_i","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":9},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-09eb22fcb1ad8180","tex":"L_{k+1}","aliases":"L_k+1","display":"L_{k+1}","kind":"use","excerpt":"T_k+1 · =T_k∪{z∈ S:∃ a∈ A_rm end(z), Post(z,a)⊆ T_k}. Let T_* be the stable set. Then define L_0=T_*, L_k+1={z∈ L_k:∃ a∈ A_rm run(z), Post(z,a)⊆ L_k}, and let L_* be its stable value. Both sequences stabilize because Z is finite.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ce43d1e9219f5a36","tex":"L_m","aliases":"L_m","display":"L_m","kind":"use","excerpt":"Let M be a finite nonempty model set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retai","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf,","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf,","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf,","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"π(Post_X(x,a)) · =Post_Z(π x,a) (a∈ A^X_m(x)). Then for every B⊆ Z, CPre^X_m(π^-1B) =π^-1(CPre^Z_m(B)). For an admitted action, Post_X(x,a)⊆π^-1B Longleftrightarrow π(Post_X(x,a))⊆ B. Replace the projected successor set by its assumed coarse value. The action ranges in the two existential quantifiers are identical, so existence of a witnessing acti","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"[A record determines a target exactly on its fibres] There exists d:r(Ω)→ Y with d(r(ω))=t(ω) for every world if and only if r(ω)=r(ω') Longrightarrow t(ω)=t(ω'). Necessity follows because a function assigns one value to one input. For sufficiency, define d(h) to be t(ω) for any world with r(ω)=h. Such a world exists for attained h, and the hypothe","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"M,N,J · Old record, required new record, and migration map.","count":23},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"Let M be a finite nonempty model set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retai","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c693ce1e8e9e894c","tex":"M^{-1}","aliases":"M^-1","display":"M^{-1}","kind":"use","excerpt":"[Minimum fixed-length supplementary information] For u∈ U_0, put k(u)=|N(M^-1{u})|, k_* =max_u∈ U_0k(u). Suppose an auxiliary encoder may inspect the complete source history and sends one fixed-length binary word in addition to the old record. Exact recovery of N is possible with b supplementary bits if and only if 2^b≥ k_*. Thus the minimum is lce","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-7f54f92432b51276","tex":"m_o","aliases":"m_o","display":"m_o","kind":"use","excerpt":"athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasib","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"e distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d441c86815baa4be","tex":"\\max_o","aliases":"max_o max","display":"\\max_o","kind":"use","excerpt":"m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasible child pair (v_o,m_o) for every nonempty ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2d9a4d201598cd67","tex":"\\max_{u\\in U_0}","aliases":"max_u∈ U_0 max in","display":"\\max_{u\\in U_0}","kind":"use","excerpt":"[Minimum fixed-length supplementary information] For u∈ U_0, put k(u)=|N(M^-1{u})|, k_* =max_u∈ U_0k(u). Suppose an auxiliary encoder may inspect the complete source history and sends one fixed-length binary word in addition to the old record. Exact recovery of N is possible with b supplementary bits if and only if 2^b≥ k_*. Thus the minimum is lce","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1e2deab3af39b182","tex":"\\min_a","aliases":"min_a min","display":"\\min_a","kind":"use","excerpt":"[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add a","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-13d3a20824a03360","tex":"\\min_{w\\in W}","aliases":"min_w∈ W min in","display":"\\min_{w\\in W}","kind":"use","excerpt":"[Which gains survive every admitted price?] The saving w·Δ is nonnegative for every nonnegative price vector if and only if every coordinate of Δ is nonnegative. If prices lie in the convex hull of a finite nonempty set W, their minimum saving equals min_w∈ Ww·Δ. A componentwise nonnegative vector has nonnegative inner product with every nonnegativ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3e0534ff9dda7eeb","tex":"N'","aliases":"N'","display":"N'","kind":"use","excerpt":"Now require the nuisance bit as well. The new record N'=(b⊕ s,b,n) has four possible values inside each old fibre, giving a two-bit supplementary lower bound. Calibration alone no longer suffices. Two histories with equal b,s and different n collide under (M,E) but have different N'. The lower bound identifies a missing distinction rather than a co","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive me","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"The joint data of a decision process. Let X and Z be finite nonempty state sets and let π:X→ Z be surjective. Let A and O be finite nonempty action and report alphabets. At a fine state x, the set D(x)⊆ A contains the admitted actions, and ℓ(x)∈ Y is a protected label. For a∈ D(x), the nonnegative numbers K_a,o(x'mid x), ∑_o∈ O∑_x'∈ XK_a,o(x'mid x)","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"Let M be a finite nonempty model set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retai","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e1d6eed537fd5981","tex":"o_1","aliases":"o_1","display":"o_1","kind":"use","excerpt":"Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3554f05e6897e490","tex":"O_e","aliases":"O_e","display":"O_e","kind":"use","excerpt":"belch:inquiry questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic succe","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-b2bdbcfb9c9cfc41","tex":"o_e","aliases":"o_e","display":"o_e","kind":"use","excerpt":"belch:inquiry questionA possible answer, an affordable observation, and a small program must belong to the same workflow. Let X be a finite nonempty state space. It includes every origin tag, permission, background record, and history variable that affects the stated behavior. An experiment e is allowed on a domain D_e⊆ X, has a deterministic succe","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-7fc03b5db3831c26","tex":"o_i","aliases":"o_i","display":"o_i","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9efeadb95ea0aede","tex":"o_{i-1}","aliases":"o_i-1","display":"o_{i-1}","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4a15af8f6ff25f74","tex":"o_t","aliases":"o_t","display":"o_t","kind":"use","excerpt":"Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2a82579e1e2588f5","tex":"O_{t+1}","aliases":"O_t+1","display":"O_{t+1}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitte","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"use","excerpt":"An approximate model supports a different, quantitative claim. Exact policy transport and approximate planning must not share one unqualified guarantee. Fix a finite horizon H and one nonempty finite class P of admitted action/report trees, shared by the true and approximate models. The complete loss of a tree is the same function of its coarse-sta","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","excerpt":"The theorem permits removal of an independent nuisance flag. It does not permit removal of every association. Consider two priors on (b,s), with rows indexed by b and columns by s: P=1/2 · 0 0 · 1/2, P'=0 · 1/2 1/2 · 0. Both individual marginals are uniform. A noisy calibration reports s correctly with probability 3/4, independently conditional on ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is ","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"Migration must commute with subsequent operations. Recovering one new record is not yet preserving a method. Let a common finite action alphabet act on H through total maps F_a:H→ H. Suppose both representations close under those same source operations: M F_a=P_a M, N F_a=Q_a N. The maps P_a and Q_a preserve the attained record sets. These equation","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5c39c7933673208b","tex":"p_E","aliases":"p_E","display":"p_E","kind":"use","excerpt":"d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the ","count":7},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-b9e70c157342a064","tex":"P_H","aliases":"P_H","display":"P_H","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5627e8b6e58a3075","tex":"\\mathcal P_h","aliases":"P_h mathcal","display":"\\mathcal P_h","kind":"use","excerpt":"The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitte","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-cc9e0dabacaadc8b","tex":"p_{N-1}","aliases":"p_N-1","display":"p_{N-1}","kind":"use","excerpt":"Improving discovery without changing its meaning. questionRemoving redundant work is an improvement only when the removed work could not change the justified result. Let G=(p_0,…,p_N-1) be a finite ordered grammar of pure, total candidate programs. Let D be a finite context set and t:D→ Y a fixed labeling. A teacher either accepts a candidate that ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-fe0059a8d2b46448","tex":"P_{\\rm av}","aliases":"P_rm av rm","display":"P_{\\rm av}","kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-43031e8fba4068f2","tex":"P_{\\rm det}","aliases":"P_rm det rm","display":"P_{\\rm det}","kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1 1 · 1, P_rm det=tfrac121 · -1 -1 · 1. They sum to the identity and produce (1/2,1/2)^top and (1/4,-1/4)^top. The components reconstruct p exactly, but the detail has zero total mass without being zero.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-af4f1e08915ee052","tex":"P_w","aliases":"P_w","display":"P_w","kind":"use","excerpt":"[A valid migration intertwines the common operations] If JM=N, then on U_0, JP_a=Q_aJ. Consequently JP_w=Q_wJ for every finite action word w, with the execution-order convention of Chapter~[source reference]. Any protected readout that factors through N agrees along the two migrated executions. Take u=M(h). Then JP_a(u)=JM(F_a h)=N(F_a h)=Q_aN(h)=Q","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4bc9c56157a62dc5","tex":"\\operatorname{pa}","aliases":"pa operatorname","display":"\\operatorname{pa}","kind":"use","excerpt":"Reuse follows semantic dependencies. Let an artifact state be a=(a_1,…,a_d). A finite directed acyclic graph represents claimed guarantees. For vertex v, let D_v be its direct artifact dependencies and let pa(v) be its parent guarantees. The declaration is semantically complete if the meaning of its claim and local verification rule is determined b","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-bfd0b07cc3c57995","tex":"\\operatorname{Pareto}","aliases":"Pareto operatorname","display":"\\operatorname{Pareto}","kind":"use","excerpt":"The complete finite frontier. Let P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitte","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-be8c3fd8e266b42d","tex":"\\operatorname{Post}_X","aliases":"Post_X operatorname","display":"\\operatorname{Post}_X","kind":"use","excerpt":"For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-844a6704b4feb167","tex":"\\operatorname{Post}_Z","aliases":"Post_Z operatorname","display":"\\operatorname{Post}_Z","kind":"use","excerpt":"[Predecessors commute with an exact controlled quotient] Suppose, for each mode and fine state, A^X_m(x) · =A^Z_m(π x), π(Post_X(x,a)) · =Post_Z(π x,a) (a∈ A^X_m(x)). Then for every B⊆ Z, CPre^X_m(π^-1B) =π^-1(CPre^Z_m(B)). For an admitted action, Post_X(x,a)⊆π^-1B Longleftrightarrow π(Post_X(x,a))⊆ B. Replace the projected successor set by its ass","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d692f9ac1c18626a","tex":"\\Pow","aliases":"Pow","display":null,"kind":"use","excerpt":"Let E⊆Pow(Ω) contain Ω and be closed under intersection. In this chapter the world does not change when an evidence event is received. The update is V↦ V∩ E. Physical actions and their effects will return in Chapter~[source reference].","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"el set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as the reference and put A_a=T_a^θ_0. A model index θ is fixed during the initial comparison. ","count":26},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5ca75c044055616d","tex":"Q_a","aliases":"Q_a","display":"Q_a","kind":"use","excerpt":"Migration must commute with subsequent operations. Recovering one new record is not yet preserving a method. Let a common finite action alphabet act on H through total maps F_a:H→ H. Suppose both representations close under those same source operations: M F_a=P_a M, N F_a=Q_a N. The maps P_a and Q_a preserve the attained record sets. These equation","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3406acd3f0f64ef4","tex":"Q_m","aliases":"Q_m","display":"Q_m","kind":"use","excerpt":"el set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3fd0648cb95746c9","tex":"Q_w","aliases":"Q_w","display":"Q_w","kind":"use","excerpt":"[A valid migration intertwines the common operations] If JM=N, then on U_0, JP_a=Q_aJ. Consequently JP_w=Q_wJ for every finite action word w, with the execution-order convention of Chapter~[source reference]. Any protected readout that factors through N agrees along the two migrated executions. Take u=M(h). Then JP_a(u)=JM(F_a h)=N(F_a h)=Q_aN(h)=Q","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"For a compression, reverse the splitting. Let C:V→ Z and R:Z→ V satisfy CR=I_Z, and put P=RC. The compressed operator is Φ(T)=CTR.","count":22},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a3d472d51734fedb","tex":"\\mathbb R^2","aliases":"R^2 mathbb","display":"\\mathbb R^2","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2f01b4d5a5d6a367","tex":"\\mathbb R^d_{\\ge0}","aliases":"R^d_≥0 mathbb ge","display":"\\mathbb R^d_{\\ge0}","kind":"use","excerpt":"ition[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be dis","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"A finite field with explicitly admitted rules. Let V={0,…,n-1} be a finite set of claim addresses. Let A⊆ V be admitted leaves, S⊆ V positive observations, and N⊆ V forbidden consequences. These sets need not be disjoint. A finite rule list R has rows r_j=(P_j,c_j), P_j⊆ V, c_j∈ V. Only admitted rules are included. Identical rows may occupy differe","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[A candidate attaining both lower bounds is optimal] Every successful controller has worst-case work at least 1+lceillog_2nrceil and literal storage at least 2n-1. A checked controller attaining both bounds and the requested horizon realizes the entire nondominated work/storage frontier. Distinct cases cannot end at the same successful repair leaf,","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"An exact sensor realization. Return to x=(b,s,n)∈{0,1}^3 and retain z=(b,s). The actions hold and flip are available everywhere. A repair action is available only where b=0 and leaves that target value fixed. The nuisance bit is retained by every fine transition but ignored by the coarse state.","count":49},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Semantic improvement and resource improvement are separate. Fix named additive event types. Let K≥0 be an extra preparation vector, let s_i be the old-minus-new per-use vector for workload type i, and let n_i count its uses. The net event difference is Δ=∑_i n_i s_i-K. A nonnegative price vector w converts these events into one declared additive me","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-dc0ce364c4595e7f","tex":"S_X","aliases":"S_X","display":"S_X","kind":"use","excerpt":"Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-53451da3045c0951","tex":"S_Z","aliases":"S_Z","display":"S_Z","kind":"use","excerpt":"Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span","display":null,"kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz i","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-7deace183d0a867f","tex":"\\sup_\\omega","aliases":"sup_ω sup omega","display":"\\sup_\\omega","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz i","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0b7cc491b1ee030b","tex":"\\operatorname{supp}","aliases":"supp operatorname","display":"\\operatorname{supp}","kind":"use","excerpt":"[Support and admissible actions agree] If bar u_h=π_*u_h, then suppbar u_h=π(suppu_h), ⋂_x:u_h(x)>0D(x)=⋂_z:bar u_h(z)>0bar D(z) whenever the history has positive mass. A finite sum of nonnegative terms is positive exactly when at least one summand is positive. This gives the support identity. The enabled sets are constant on each fibre, so repeati","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include","count":25},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5d42311231194d5c","tex":"t^{-1}","aliases":"t^-1","display":"t^{-1}","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-500d12d9ac6c4d36","tex":"T^X_*","aliases":"T^X_*","display":"T^X_*","kind":"use","excerpt":"[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine polic","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3d146f8009f92e01","tex":"T^X_k","aliases":"T^X_k","display":"T^X_k","kind":"use","excerpt":"Transport of both fixed-point sequences. Construct T^X_k,L^X_k on the fine game and T^Z_k,L^Z_k on the coarse game as in Chapter~[source reference]. In particular, the running sequence starts from the stable ending region in its own model.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6e4cc17b52bed1ed","tex":"T^X_{k+1}","aliases":"T^X_k+1","display":"T^X_{k+1}","kind":"use","excerpt":"licy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coordinates. The goal-set assumption gives equality at T_0. Inverse images commute with union and intersection. Applying Lemma~[source reference] to the ending predecessor gives the induction step T^X","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-b0c9a481c0d7c5dc","tex":"T^Z_*","aliases":"T^Z_*","display":"T^Z_*","kind":"use","excerpt":"[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine polic","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3f0d1cb30a5070c2","tex":"T^Z_k","aliases":"T^Z_k","display":"T^Z_k","kind":"use","excerpt":"Transport of both fixed-point sequences. Construct T^X_k,L^X_k on the fine game and T^Z_k,L^Z_k on the coarse game as in Chapter~[source reference]. In particular, the running sequence starts from the stable ending region in its own model.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0bc6dba302face63","tex":"T^\\theta","aliases":"T^θ theta","display":"T^\\theta","kind":"use","excerpt":"For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1a4a5aadced24804","tex":"T_a^\\theta","aliases":"T_a^θ theta","display":"T_a^\\theta","kind":"use","excerpt":"One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as t","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-7167780333896afc","tex":"T_a^{\\theta_0}","aliases":"T_a^θ_0 theta","display":"T_a^{\\theta_0}","kind":"use","excerpt":"ne quotient for several possible dynamics A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as the","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ed0619d26eb6427a","tex":"T_w^\\theta","aliases":"T_w^θ theta","display":"T_w^\\theta","kind":"use","excerpt":"[A common observable quotient] The following conditions are equivalent: item QT_w^θ=QA_w on all of V, for every fixed candidate θ and every word w. item E⊆ker Q. item There is a vector space Z, a surjective linear map r:V→ Z, common linear updates G_a:Z→ Z, and a linear readout bar Q:Z→ Y with rT_a^θ=G_ar and Q=bar Qr for every a,θ. Under these con","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-99bcf68fe6f3f40b","tex":"T_{\\rho(z)-1}","aliases":"T_ρ(z)-1 rho","display":"T_{\\rho(z)-1}","kind":"use","excerpt":"r strategy. Conversely, a policy that reaches G in at most k+1 steps must either start in G, or select some first ending action such that the continuation from every possible successor reaches G within k steps. By induction every such successor belongs to T_k, hence the initial state lies in T_k+1. At the first-entry stage ρ(z), the selected action","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"[An exact detail component has negative probability mass] For p=(3/4,1/4)^top, the orthogonal average and detail projectors are P_rm av=tfrac121 · 1","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV","display":null,"kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"Exact packets and the hidden-return defect. Let V,Z be vector spaces over a field. Let A:V→ Z be analysis and S:Z→ V synthesis, with SA=I_V. Then AS is an idempotent on Z. The map E(T)=ATS represents a full-state operator T:V→ V on resolved coordinates. Direct multiplication gives E(T) E(U)=ATSAUS=ATUS= E(TU). The resolved identity is AS on the com","count":19},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","excerpt":"A migration map exists exactly on constant fibres. Let H be a finite nonempty set of possible source histories. A history may include a model tag or a past observation whose value must remain available for later correction. Let M:H→ U, N:H→ V be the old record and the required new record. Work on their attained images U_0=M(H) and V_0=N(H). The new","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9eff72e097fd075b","tex":"u_\\varnothing","aliases":"u_∅ varnothing","display":"u_\\varnothing","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"use","excerpt":"Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfyi","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"Define D_a^θ=T_a^θ-A_a and the propagated discrepancy space E=Span{A_wD_a^θ v: v∈ V, a,θ, w finite}. It can be constructed by starting with the sum of the discrepancy ranges and repeatedly adding their images under the reference actions. In finite dimension this increasing sequence stops. Immediate invisibility QD_a^θ=0 alone does not guarantee QE=","count":55},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"A migration map exists exactly on constant fibres. Let H be a finite nonempty set of possible source histories. A history may include a model tag or a past observation whose value must remain available for later correction. Let M:H→ U, N:H→ V be the old record and the required new record. Work on their attained images U_0=M(H) and V_0=N(H). The new","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9bacaf6e1626666c","tex":"V_{e,o}","aliases":"V_e,o","display":"V_{e,o}","kind":"use","excerpt":"A current support V≠∅ records possible states. The experiment is permitted only if V⊆ D_e. After report o, the possible post-experiment states are V_e,o={f_e(x):x∈ V, o_e(x)=o}. Only nonempty report branches are retained. If the target concerns a past value, that value must survive as an immutable tag. Resetting the current state is not recovering ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8e7676bd93a7edb5","tex":"v_o","aliases":"v_o","display":"v_o","kind":"use","excerpt":"athcal P_h(V) be the nondominated pairs (v,m) of worst-case online resource vectors and stored-record counts attainable by successful trees with at most h experiment steps. At depth zero the available pairs are Pareto{(g_a(V),1):a succeeds throughout V}. At greater depth retain those stops and, for each uniformly permitted experiment and one feasib","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-786a3c53c74db475","tex":"v_\\omega","aliases":"v_ω omega","display":"v_\\omega","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz i","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"every update V↦ V∩ E_0. The signature supports the current target set and its exact update after every finite evidence sequence. Any deterministic retained state supporting the same readouts and updates on all candidate subsets must distinguish every two distinct signatures. Thus the image of Σ has the least possible cardinality for this contract. ","count":7},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add a","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz i","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-967c63ebf4cd78be","tex":"\\widehat R","aliases":"widehat R widehat","display":"\\widehat R","kind":"use","excerpt":"widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f207fa5d22c484f6","tex":"\\widehat v","aliases":"widehat v widehat","display":"\\widehat v","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz i","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8307c5f38fc587f5","tex":"\\widehat v_\\omega","aliases":"widehat v_ω widehat omega","display":"\\widehat v_\\omega","kind":"use","excerpt":"One quantitative decision contract. Let p and widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz i","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3f476ecaf7bf4a0d","tex":"\\widetilde J","aliases":"widetilde J widetilde","display":"\\widetilde J","kind":"use","excerpt":"[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certi","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9117c60fc5e9378b","tex":"\\widetilde K","aliases":"widetilde K widetilde","display":"\\widetilde K","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8da03ee77856ac56","tex":"\\widetilde K_{a,o}","aliases":"widetilde K_a,o widetilde","display":"\\widetilde K_{a,o}","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-649a6c9973f355dd","tex":"\\widetilde p","aliases":"widetilde p widetilde","display":"\\widetilde p","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ","count":7},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4741526210fb0d8d","tex":"\\widetilde p_0","aliases":"widetilde p_0 widetilde","display":"\\widetilde p_0","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-47fb519e5af1c253","tex":"\\widetilde p_E","aliases":"widetilde p_E widetilde","display":"\\widetilde p_E","kind":"use","excerpt":"d with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the ","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-26f82a5d2a875a6e","tex":"\\widetilde P_H","aliases":"widetilde P_H widetilde","display":"\\widetilde P_H","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a25d72cc47e636ba","tex":"\\widetilde\\pi","aliases":"widetildeπ widetilde pi","display":"\\widetilde\\pi","kind":"use","excerpt":"[Uniform model error controls a changed policy] Let J(π) and widetilde J(π) be a tree's expected losses in the true and approximate models. Then |J(π)-widetilde J(π)|≤ Bd_H (π∈ P). If widetildeπ minimizes the approximate loss and π_* minimizes the true loss over this same class, then J(widetildeπ)-J(π_*)≤min{B,2Bd_H}. These conclusions do not certi","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"The joint data of a decision process. Let X and Z be finite nonempty state sets and let π:X→ Z be surjective. Let A and O be finite nonempty action and report alphabets. At a fine state x, the set D(x)⊆ A contains the admitted actions, and ℓ(x)∈ Y is a protected label. For a∈ D(x), the nonnegative numbers K_a,o(x'mid x), ∑_o∈ O∑_x'∈ XK_a,o(x'mid x)","count":14},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"On the nonnegative coordinate cone, preservation is equivalent to nonnegative matrix entries, as is seen by applying a matrix to every standard basis vector. Because the matrices sum to I, their nonnegative off-diagonal entries must all vanish. An idempotent diagonal entry satisfies x^2=x, hence is zero or one. The diagonal sums are one, so each co","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[A discarded coordinate returns] Take V= R^2, let C(x_1,x_2)=x_1, let R(z)=(z,0), and let T(x_1,x_2)=(x_2,x_1). The compressed one-step map is zero. Yet T^2=I_V, so Φ(T^2)=I_Z, while Φ(T)^2=0. The discarded second coordinate was not irrelevant to the two-step question.","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"For example, let Q(x_1,x_2,x_3)=x_1 and let every candidate act as T^θ(x_1,x_2,x_3)=(x_1+x_2,x_2,θ x_3). Then E is contained in the third coordinate and remains invisible. The two-coordinate method (x_1,x_2)↦(x_1+x_2,x_2) is exact, although the hidden dynamics are not identified.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1e4fb4e21dd5aa54","tex":"X_{t+1}","aliases":"X_t+1","display":"X_{t+1}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"use","excerpt":"Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfyi","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9d5ae088e190bcb8","tex":"x_w","aliases":"x_w","display":"x_w","kind":"use","excerpt":"realization Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"ic quotient criterion] There are coarse data satisfying D(x)=bar D(π x), ℓ(x)=barℓ(π x), ∑_π x'=z'K_a,o(x'mid x)=bar K_a,o(z'midπ x) for every x, every a∈ D(x), and every o,z', if and only if states in the same fibre have identical labels and enabled sets and identical projected joint successor/report rows. Explicitly, π x=π y must imply D(x) · =D(","count":24},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c656dc90ffbe53cd","tex":"\\mathcal Z","aliases":"Z mathcal","display":"\\mathcal Z","kind":"use","excerpt":"Let M be a finite nonempty model set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retai","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"[Joint legal and stochastic quotient criterion] There are coarse data satisfying D(x)=bar D(π x), ℓ(x)=barℓ(π x), ∑_π x'=z'K_a,o(x'mid x)=bar K_a,o(z'midπ x) for every x, every a∈ D(x), and every o,z', if and only if states in the same fibre have identical labels and enabled sets and identical projected joint successor/report rows. Explicitly, π x=","count":10},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"This induction alone records only the current state and the observed history. To retain the entire state path, fix a coarse prefix (z_0,a_1,o_1,z_1,…,a_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by E","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"This induction alone records only the current state and the observed history. To retain the entire state path, fix a coarse prefix (z_0,a_1,o_1,z_1,…,a_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by E","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9a75d558418afc42","tex":"z_{a,o}","aliases":"z_a,o","display":"z_{a,o}","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-45eb5061d9b39786","tex":"Z_H","aliases":"Z_H","display":"Z_H","kind":"use","excerpt":"e distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,widetilde p_E>0, the conditional final coarse-state distributions obey TV(P","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-86298b26f7dd1fb4","tex":"z_i","aliases":"z_i","display":"z_i","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-74e4dec4a5e37fca","tex":"z_{i-1}","aliases":"z_i-1","display":"z_{i-1}","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"This induction alone records only the current state and the observed history. To retain the entire state path, fix a coarse prefix (z_0,a_1,o_1,z_1,…,a_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by E","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":10},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"probability. For a positive pair, posterior equality yields p(m)L_m(o)B(o,z)=Pr(o,z)p(mmid o) =ρ(omid z)Pr(z)p(mmid z). Substituting Bayes' formula and cancelling positive p(m) gives the factorization. A zero-probability pair under a positive prior has L_m(o)B(o,z)=0 for all m. Its conditional weight is zero. If z is impossible for every model, ass","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6cb4d369799123ce","tex":"\\beta_i","aliases":"β_i beta","display":"\\beta_i","kind":"use","excerpt":"An explanation belongs to a recipient. An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set X, a sender-access map α:X→ H, and a finite artifact catalogue C. Recipient i has a background record β_i(x), a demanded output t_i(x), and a fixed bounded procedure L_i. A timeout, malformed artifact, wrong o","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"midrule Γ(F) · Compatible families in a specified system of views.","count":9},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"For a history h, keep unnormalized states u_h,bar u_h. Initially bar u_∅=Cu_∅. After choosing action a with common probability γ(amid h) and receiving o, the next states are γ(amid h)K_a,ou_h and γ(amid h)bar K_a,obar u_h. The intertwining identity preserves their aggregation. Induction gives equal mass for every history, including zero-mass histor","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"After changing a to a', mark a vertex stale when a direct input changes and propagate staleness to all descendants. Call the resulting set Δ.","count":7},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ","count":10},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"idetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ_t. This is a bound on a complete report/next-state distribution, not on an isolated matrix entry. Assume TV(Cp_0,widetilde p_0)≤δ","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Why a small probability error is not a sure guarantee. Take three states s,g,b, with safe set {s,g} and goal {g}. Running holds s or g in place. The nominal ending action at s moves to g with probability one. The perturbed action instead moves to g with probability 1-ϵ and to unsafe state b with probability ϵ, where 0<ϵ<1. A hold action is availabl","count":11},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d08739f367199333","tex":"\\epsilon_t","aliases":"ϵ_t epsilon","display":"\\epsilon_t","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"For minimality, take any admissible refinement η of π_0. It has at least two blocks. Fix a strictly positive fine likelihood ℓ and a strictly positive coarse likelihood barℓ inducing its assumed update. For states x,y in one block and z in a different block, compare priors assigning half their mass to x,z and to y,z. Their coarse priors agree. Equa","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as t","count":18},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","excerpt":"One quotient for several possible dynamics. A stronger situation arises when the operator itself is uncertain. Let V be a finite-dimensional vector space over a fixed field, and let Y be a vector space over the same field. Let T_a^θ:V→ V be a finite family of candidate linear operators indexed by an action a and a model θ. Choose one model θ_0 as t","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Let K be a convex cone in a real vector space, and let λ be linear and strictly positive on every nonzero member of K. Its normalized states satisfy v∈ K and λ v=1. A coordinate change can send K to a cone containing negative coordinates without changing those states' probability meaning. Conversely, an arbitrary signed component of a valid state n","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"Logical interpretation is an additional map. Let μ(v) be the proposition represented by claim address v. Suppose every admitted leaf and positive observation is true under μ, and each admitted rule satisfies (bigwedge_p∈ P_jμ(p))Longrightarrowμ(c_j). Then every node in C is true under μ. Indeed, induction over the closure stages propagates truth fr","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"[A decision bound for one actual approximation chain] Put ξ=min{1,e+Kd_0}. For every action, |R(a)-widehat R(a)|≤ξ. If widehat a minimizes the approximate risk, then R(widehat a)-min_a R(a)≤min{1,2ξ}. If the approximate gap between its unique minimizer and every alternative exceeds 2ξ, that action is also the unique true model-risk minimizer. Add a","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-6a331de74ff475c9","tex":"\\pi^{-1}","aliases":"π^-1 pi","display":"\\pi^{-1}","kind":"use","excerpt":"Assume the fine and coarse safe and goal sets agree by inverse image: S_X=π^-1(S_Z), G_X=π^-1(G_Z), G_Z⊆ S_Z. This is a substantive restriction. A label that distinguishes a safe state from an unsafe state cannot be omitted from a safety-preserving quotient.","count":8},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-2d28a27ac7821632","tex":"\\pi_0","aliases":"π_0 pi","display":"\\pi_0","kind":"use","excerpt":"A mandatory partition π_0 with at least two blocks protects the target distinctions. We seek a refinement that supports every prediction and each separate Bayesian update for every prior. The observation is an operation available in its own right. This is stronger than inspecting only a chosen combined transition-and-report schedule.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-560c891f5e585ac2","tex":"\\pi_{\\rm init}","aliases":"π_rm init pi rm","display":"\\pi_{\\rm init}","kind":"use","excerpt":"[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-e17a3fdd593037e8","tex":"\\rho_X","aliases":"ρ_X rho","display":"\\rho_X","kind":"use","excerpt":"[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine polic","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f58f59cb68a803b2","tex":"\\rho_Z","aliases":"ρ_Z rho","display":"\\rho_Z","kind":"use","excerpt":"[Ending regions, running kernels, and ranks are preserved] Under the exact controlled-quotient assumptions, T^X_k=π^-1(T^Z_k), T^X_*=π^-1(T^Z_*), L^X_k=π^-1(L^Z_k), L^X_*=π^-1(L^Z_*) for every finite k. For every x∈ T^X_*, the first-entry ranks agree: ρ_X(x)=ρ_Z(π x). A positional coarse policy for safe continuation and ending lifts to a fine polic","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"E,Σ · An evidence family and its correction signature.","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"Replay checks a finite causal sequence. A trace σ=(j_1,…,j_m) lists rule addresses. Start from K_0=A∪ S. At each step, reject an unknown rule address or a rule whose premises are not all in the current known set. Otherwise adjoin its conclusion. Strict replay also rejects a conclusion already known. Causal replay allows it.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"A method can be transported without reconstructing everything. Suppose source states z∈ Z have operations F_a, target states u∈ U have operations G_τ(a), and T:Z→ U is a translation. The operation-name map τ is fixed. Let q and bar q be matched readouts.","count":7},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"For a compression, reverse the splitting. Let C:V→ Z and R:Z→ V satisfy CR=I_Z, and put P=RC. The compressed operator is Φ(T)=CTR.","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"Certificates after a change of contract. A migration map transports data. A certificate authorizes a conclusion only relative to its premises. To reuse a certificate across a revision, specify an old predicate φ on U_0, a new predicate ψ on V_0, and a proved implication φ(u) Longrightarrow ψ(Ju). An old witness for φ(u) and this implication then yi","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Certificates after a change of contract. A migration map transports data. A certificate authorizes a conclusion only relative to its premises. To reuse a certificate across a revision, specify an old predicate φ on U_0, a new predicate ψ on V_0, and a proved implication φ(u) Longrightarrow ψ(Ju). An old witness for φ(u) and this implication then yi","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-437405ce641f5157","tex":"\\Psi_{a,o}","aliases":"Ψ_a,o Psi","display":"\\Psi_{a,o}","kind":"use","excerpt":"For a surjection π:X→ Z, let C sum probabilities over its fibres. Write z_a,o(p)=1^top K_a,op, Ψ_a,o(p)=K_a,op/z_a,o(p) when the denominator is positive. Zero mass means an impossible report under that prior, not an arbitrarily assigned posterior.","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"From evidence to a reusable state. questionThe first question is not how much state to store. It is which distinctions are accessible. Let Ω be a finite nonempty set of possible worlds. A world may include a physical state, a fixed model index, and an origin tag. An observation record is a function r:Ω→ H. A target is a function t:Ω→ Y. All codomai","count":13},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"[A record determines a target exactly on its fibres] There exists d:r(Ω)→ Y with d(r(ω))=t(ω) for every world if and only if r(ω)=r(ω') Longrightarrow t(ω)=t(ω'). Necessity follows because a function assigns one value to one input. For sufficiency, define d(h) to be t(ω) for any world with r(ω)=h. Such a world exists for attained h, and the hypothe","count":11},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-efdf8990a98fa39d","tex":"\\omega'","aliases":"ω' omega","display":"\\omega'","kind":"use","excerpt":"[A record determines a target exactly on its fibres] There exists d:r(Ω)→ Y with d(r(ω))=t(ω) for every world if and only if r(ω)=r(ω') Longrightarrow t(ω)=t(ω'). Necessity follows because a function assigns one value to one input. For sufficiency, define d(h) to be t(ω) for any world with r(ω)=h. Such a world exists for attained h, and the hypothe","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"Observable histories and common admissibility. A history available to the policy is h=(a_1,o_1,…,a_t,o_t). The fine hidden state and the coarse state are not additional policy inputs. Protected labels are recorded for comparison, but are not assumed observed merely because the theorem preserves them. An application that exposes a label must include","count":10},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"For conditioning, let u,v be the restrictions of the two history laws to E, with masses p_E,widetilde p_E. The difference on the complement has ℓ^1 norm at least |p_E-widetilde p_E|. Thus u-v _1+|p_E-widetilde p_E|≤2d_H. Adding and subtracting v/p_E bounds the conditional distance by d_H/p_E. Interchanging the laws gives d_H/widetilde p_E, so the s","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-313774f1667bd9ea","tex":"\\ell_o","aliases":"ℓ_o ell","display":"\\ell_o","kind":"use","excerpt":"First consider a known finite state set X with row-stochastic transition matrices K_a(x,x'). We retain column-vector probability distributions. A partition π has aggregation C_π(z,x)=1_{π(x)=z}, so prediction is p↦ K_a^top p. For each permitted observation symbol o, let ℓ_o(x)>0 be its specified likelihood. When these are a complete report alphabet","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[Exact accounting of the discarded excursion] For any endomorphisms T,U of V, Φ(TU)-Φ(T)Φ(U)=CT(I-P)UR. Thus compression preserves this composition precisely when the discarded excursion on the right vanishes. Full split encoding T↦ ATS preserves composition without that extra condition. Expand the product Φ(T)Φ(U)=CTRCUR=CTPUR and subtract it from","count":7},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"For B⊆ X, define the mode-specific predecessor CPre^X_m(B) ={x:∃ a∈ A^X_m(x), Post_X(x,a)⊆ B}. Define CPre^Z_m in the same way. These operators do not themselves include a safety test. Safety enters the ending recurrence through its intersection with S, and the running recurrence through its current set.","count":5},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":6},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-04e04fce57b19491","tex":"\\prod_{i=1}^{t}","aliases":"∏_i=1^t prod","display":"\\prod_{i=1}^{t}","kind":"use","excerpt":"_t,o_t,z_t). Sum fine path weights over all fine prefixes mapping to this prefix. For one more step, each fine current state in the last fibre contributes the same projected successor/report row by Equation~[source reference]. The policy factor depends only on the already fixed action/report prefix. It can therefore be taken outside the fine-state ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d212f3e24adbf170","tex":"\\prod_v","aliases":"∏_v prod","display":"\\prod_v","kind":"use","excerpt":"realization Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfying all the restriction equations: Γ(F","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ff1f52005cd89e31","tex":"\\sum_{i:w_i>0}","aliases":"∑_i:w_i>0 sum","display":"\\sum_{i:w_i>0}","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0c18bcf01edb7e76","tex":"\\sum_{j\\ge1}","aliases":"∑_j≥1 sum ge","display":"\\sum_{j\\ge1}","kind":"use","excerpt":"[A conditional error budget for a lineage] Let H_j-1 contain the information available before attempt j. Suppose the event A_j of accepting a false claim satisfies Pr(A_jmid H_j-1)≤δ_j almost surely, ∑_j≥1δ_j≤δ, for deterministic nonnegative δ_j. Then the probability of ever accepting any false claim is at most δ, even when candidates and the stopp","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ccbd66bdaed18a2b","tex":"\\sum_o","aliases":"∑_o sum","display":"\\sum_o","kind":"use","excerpt":"el set with prior p(m). The sampled model remains fixed during the experiment. For each m, let L_m be a probability distribution on a finite nonempty observation alphabet O. Let Z be a finite nonempty report alphabet. A stochastic reporting channel B(o,z), conditionally independent of the model given o, creates a retained report Z∈ Z. The joint law","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-3abe3b490f0f125b","tex":"\\sum_{o,x'}","aliases":"∑_o,x' sum","display":"\\sum_{o,x'}","kind":"use","excerpt":"Retain the joint law of the report and the next state. A general experiment can change the state and report correlated information about the change. Describing its transition and report marginals separately is then insufficient. Let K_a,o(x',x)=Pr(X_t+1=x',O_t+1=omid X_t=x,A_t=a) be a nonnegative finite instrument with ∑_o,x'K_a,o(x',x)=1 for every","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d74790b215363bd7","tex":"\\sum_{o\\in O}","aliases":"∑_o∈ O sum in","display":"\\sum_{o\\in O}","kind":"use","excerpt":"The joint data of a decision process. Let X and Z be finite nonempty state sets and let π:X→ Z be surjective. Let A and O be finite nonempty action and report alphabets. At a fine state x, the set D(x)⊆ A contains the admitted actions, and ℓ(x)∈ Y is a protected label. For a∈ D(x), the nonnegative numbers K_a,o(x'mid x), ∑_o∈ O∑_x'∈ XK_a,o(x'mid x)","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f1e914837dbf8c89","tex":"\\sum_{t=0}^{H-1}","aliases":"∑_t=0^H-1 sum","display":"\\sum_{t=0}^{H-1}","kind":"use","excerpt":"[A whole-history bound with an explicit conditioning cost] At horizon H, the distributions P_H,widetilde P_H of the full coarse-state and action/report histories satisfy TV(P_H,widetilde P_H)≤ d_H:=min{1,δ_0+∑_t=0^H-1ϵ_t}. The same bound holds after forgetting any of those history coordinates. For an observed history event E with probabilities p_E,","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-31e59f15403939ba","tex":"\\sum_{x':\\pi(x')=z'}","aliases":"∑_x':π(x')=z' sum pi","display":"\\sum_{x':\\pi(x')=z'}","kind":"use","excerpt":"[One quotient for reports, posteriors, and feedback] A nonnegative normalized coarse instrument bar K_a,o satisfies CK_a,o=bar K_a,oC for every a,o if and only if, for all a,o,z', ∑_x':π(x')=z'K_a,o(x',x) is constant as x varies within any one input fibre. Such a coarse instrument is unique. Equivalently, it preserves for every prior both the repor","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f05731197cd413ab","tex":"\\sum_{x'\\in B}","aliases":"∑_x'∈ B sum in","display":"\\sum_{x'\\in B}","kind":"use","excerpt":"[Least refinement closed under prediction and correction] First split π_0 by the complete vectors (ℓ_o(x))_o, obtaining π_rm init. Then repeatedly split current blocks by (∑_x'∈ BK_a(x,x'))_a,B, where B runs over the current blocks. The process stops after at most |X|-|π_rm init| strict refinement rounds. Its final partition is the unique coarsest ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-c8ed7190ef6914c5","tex":"\\sum_{x'\\in X}","aliases":"∑_x'∈ X sum in","display":"\\sum_{x'\\in X}","kind":"use","excerpt":"The joint data of a decision process. Let X and Z be finite nonempty state sets and let π:X→ Z be surjective. Let A and O be finite nonempty action and report alphabets. At a fine state x, the set D(x)⊆ A contains the admitted actions, and ℓ(x)∈ Y is a protected label. For a∈ D(x), the nonnegative numbers K_a,o(x'mid x), ∑_o∈ O∑_x'∈ XK_a,o(x'mid x)","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-a986582685fa6feb","tex":"\\sum_{\\pi x'=z'}","aliases":"∑_π x'=z' sum pi","display":"\\sum_{\\pi x'=z'}","kind":"use","excerpt":"[Joint legal and stochastic quotient criterion] There are coarse data satisfying D(x)=bar D(π x), ℓ(x)=barℓ(π x), ∑_π x'=z'K_a,o(x'mid x)=bar K_a,o(z'midπ x) for every x, every a∈ D(x), and every o,z', if and only if states in the same fibre have identical labels and enabled sets and identical projected joint successor/report rows. Explicitly, π x=","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-5c9e3e6e46df8910","tex":"\\sum_{\\pi x=z}","aliases":"∑_π x=z sum pi","display":"\\sum_{\\pi x=z}","kind":"use","excerpt":"For a nonnegative vector u on X, write (π_*u)(z)=∑_π x=zu(x). This notation applies to probability distributions and to unnormalized history masses. A coarse model has labels barℓ, enabled sets bar D, and joint instruments bar K on its enabled actions.","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-ea8a48c6147135b3","tex":"\\sum_\\omega","aliases":"∑_ω sum omega","display":"\\sum_\\omega","kind":"use","excerpt":"widehat p be distributions on the same finite world set. For each world let v_ω be a response and widehat v_ω its approximation in one metric space. Assume TV(p,widehat p)≤ e, sup_ω d(v_ω,widehat v_ω)≤ d_0. For each action a in a finite nonempty set, let the loss ℓ(a,v) lie in [0,1] and be K-Lipschitz in v. Define the model risks R(a)=∑_ω p(ω)ℓ(a,v","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"xactly when there exists an ending-only strategy that, whatever the successors, reaches G within at most k transitions while remaining in S. In particular, if ρ(z)>0, there is an ending action whose every successor has strictly smaller first-entry rank. The assertion at k=0 says precisely that the initial state is already in G. Assume it holds for ","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"Views, restrictions, and a coherent realization. Let V be a small category of views. Associate a set F_v to each view v and a restriction f^*:F_w→ F_v to each arrow f:v→ w. Identity and composition satisfy (1_v)^*=id and (g∘ f)^*=f^*∘ g^*. Thus F is a contravariant set-valued functor. A coherent realization is a family of elements x_v∈ F_v satisfyi","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"licy with the same rank bound. To execute that lift, the controller needs the current coarse state and command mode, but not the remaining fine-state coordinates. The goal-set assumption gives equality at T_0. Inverse images commute with union and intersection. Applying Lemma~[source reference] to the ending predecessor gives the induction step T^X","count":11},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"Define behavioral equivalence by x∼ x' Longleftrightarrow q(F_wx)=q(F_wx') for every finite word w. The empty word protects the present output. The other words protect its permitted future uses.","count":3},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"An exact count for partition evidence. Suppose C_1,…,C_m partition Ω into nonempty cells and the admitted events are exactly their unions. Let D_t={(i,y): C_i∩ t^-1(y)≠∅}, r=|D_t|. The occupancy memory retains whether a candidate exists in each attainable cell--target pair.","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"Small law error and a rare conditional history. Approximate models must be compared before declaring their posteriors close. Let widetilde K be a normalized instrument on the same coarse state and report alphabets. Suppose, at step t, every relevant action and fine state satisfy TV((∑_x':π(x')=z'K_a,o(x',x))_o,z', (widetilde K_a,o(z',π(x)))_o,z')≤ϵ","count":17},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"[When complete packets are genuine probability components] Let P_1,…,P_m be linear maps with ∑_iP_i=I. If every P_i preserves K, each normalized v∈ K has the mixture decomposition v=∑_i:w_i>0w_i v_i, w_i=λ P_iv≥0, ∑_iw_i=1, v_i=P_iv/w_i (w_i>0). Every zero-weight component is zero. If K= R^d_≥0 and the P_i are also idempotent, they must be disjoint","count":12},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"After an action, report the new display b⊕ s correctly with probability 3/4 and incorrectly with probability 1/4. Both the label and the joint instrument ignore n. The ten coarse state/action rows satisfy the quotient conditions exactly.","count":11},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is ","count":4},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-892cecbf386dea45","tex":"\\bigcap_i","aliases":"⋂_i bigcap","display":"\\bigcap_i","kind":"use","excerpt":"[Target-relative joint reconstruction] There is a linear decoder D with Q=DA if and only if ker A⊆ker Q. Full-state reconstruction is possible exactly when A is injective. The intersection identity ker A=⋂_iker A_i determines which ambiguities sharing removes. If Q=DA, a vector annihilated by A is also annihilated by Q. Conversely, define D(Av)=Qv ","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-b74250930364852a","tex":"\\bigcap_{x:u_h(x)>0}","aliases":"⋂_x:u_h(x)>0 bigcap","display":"\\bigcap_{x:u_h(x)>0}","kind":"use","excerpt":"Start from a fine prior p_0 and the coarse prior bar p_0=π_*p_0. Let u_h(x) be the unnormalized probability of history h and current state x. A common policy assigns the same probability α(amid h) in both models. At every positive-mass history it may select only actions in D_h=⋂_x:u_h(x)>0D(x). The intersection is over a nonempty support. If it is ","count":2},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-9d31d41a627eed08","tex":"\\bigcap_{x:\\alpha(x)=h}","aliases":"⋂_x:α(x)=h bigcap alpha","display":"\\bigcap_{x:\\alpha(x)=h}","kind":"use","excerpt":"Let C(x) be the artifacts that succeed for every recipient in case x. For an attained sender value h, put C_h=⋂_x:α(x)=h C(x).","count":1},{"work":"research-machine-native-reasoning-wider","url":"notation/research-machine-native-reasoning-wider.html#symbol-f3b9b06c6dba4d3c","tex":"\\bigcap_{z:\\bar u_h(z)>0}","aliases":"⋂_z:bar u_h(z)>0 bigcap bar","display":"\\bigcap_{z:\\bar u_h(z)>0}","kind":"use","excerpt":"[Support and admissible actions agree] If bar u_h=π_*u_h, then suppbar u_h=π(suppu_h), ⋂_x:u_h(x)>0D(x)=⋂_z:bar u_h(z)>0bar D(z) whenever the history has positive mass. A finite sum of nonnegative terms is positive exactly when at least one summand is positive. This gives the support identity. The enabled sets are constant on each fibre, so repeati","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":172},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":461},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":287},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":40},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0efdddd015383ecb","tex":"d_p","aliases":"d_p","display":"d_p","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":122},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":20},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f0160f5b8a57192c","tex":"F_M","aliases":"F_M","display":"F_M","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":46},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5b67a225dc0d299e","tex":"h_{\\mathcal O(1)}","aliases":"h_ O(1) mathcal","display":"h_{\\mathcal O(1)}","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1b26238865ef07e5","tex":"h_{K_{F_M}}","aliases":"h_K_F_M","display":"h_{K_{F_M}}","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":16},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":60},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":45},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":55},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6832be6d7109cc9f","tex":"K_{F_M}","aliases":"K_F_M","display":"K_{F_M}","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b897ba65dbba23a0","tex":"K_{\\mathrm{Kum}}","aliases":"K_Kum mathrm","display":"K_{\\mathrm{Kum}}","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-febfa676c8169791","tex":"\\mathrm{Kum}","aliases":"Kum mathrm","display":"\\mathrm{Kum}","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":59},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":76},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"introduction-passage","excerpt":"ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p","count":18},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":10},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f26013157a61075a","tex":"\\mathcal O_{F_M}","aliases":"O_F_M mathcal","display":"\\mathcal O_{F_M}","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":35},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":217},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b47e4ae8f60bef45","tex":"p^e","aliases":"p^e","display":"p^e","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":20},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fe1b7ade23d1e9a1","tex":"P_M","aliases":"P_M","display":"P_M","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6f5181868d16d4ae","tex":"P_{r',s'}","aliases":"P_r',s'","display":"P_{r',s'}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4734da4707b22bcc","tex":"P_{r,s}","aliases":"P_r,s","display":"P_{r,s}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-474a8c74e28ce2e1","tex":"\\parallel","aliases":"parallel","display":null,"kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":14},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":178},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0d5a58c436e4715b","tex":"r_p","aliases":"r_p","display":"r_p","kind":"introduction-passage","excerpt":"ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":62},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5e769b89788d547a","tex":"s'","aliases":"s'","display":"s'","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p","count":63},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"introduction-passage","excerpt":"ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p","count":39},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":21},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":71},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":23},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":16},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4ee2fc118c6d1327","tex":"\\delta_{r,r'}","aliases":"δ_r,r' delta","display":"\\delta_{r,r'}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-83ebf9d2f08acb0d","tex":"\\delta_{s,s'}","aliases":"δ_s,s' delta","display":"\\delta_{s,s'}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c7ffdc6ef6b651ea","tex":"\\epsilon_2","aliases":"ϵ_2 epsilon","display":"\\epsilon_2","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":15},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9f6e692e69afc404","tex":"\\zeta_M^{-ri-sj}","aliases":"ζ_M^-ri-sj zeta","display":"\\zeta_M^{-ri-sj}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":23},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":11},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b7ec60a68d5fe62e","tex":"\\mu_M^2","aliases":"μ_M^2 mu","display":"\\mu_M^2","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6413311b725a828d","tex":"\\sigma^i","aliases":"σ^i sigma","display":"\\sigma^i","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3174dcdc9ca749bd","tex":"\\tau^j","aliases":"τ^j tau","display":"\\tau^j","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":41},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":61},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-997bb90cd39331b0","tex":"\\sum_{i,j=0}^{M-1}","aliases":"∑_i,j=0^M-1 sum","display":"\\sum_{i,j=0}^{M-1}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0031c656c3e56dba","tex":"\\sum_{r,s}","aliases":"∑_r,s sum","display":"\\sum_{r,s}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":15},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":83},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"ve hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=overline H k_*Γ for the optimal obstruction multiple k_* and an integrality factor","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4487f11646417749","tex":"\\mathbf1_{0\\le r<e_p}","aliases":"1_0≤ r<e_p mathbf le","display":"\\mathbf1_{0\\le r<e_p}","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"[Mason--Stothers in characteristic zero] Let k be an algebraically closed field of characteristic zero. Suppose A,B,C∈ k[X] are nonzero, pairwise coprime, satisfy A+B=C, and are not all constant. Then max{deg A,deg B,deg C}≤ n_0(ABC)-1. This is the polynomial abc theorem. Snyder gives a short primary proof in a form that also makes the derivative h","count":56},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"How to read this piece. addcontentslinetocsectionHow to read this piece The equation a+b=c joins two kinds of structure. Multiplication describes prime factors and their repeated powers. Addition forces those three factorizations to fit one exact relation. The abc conjecture says that this compatibility limits how large the sum can be compared with","count":262},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"Set W=A'B-AB'. It is nonzero. Indeed, if it vanished, then (A/B)'=0, and characteristic zero would force the rational function A/B to be constant. Coprimality would then force both A and B to be constant, and so also C.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence ","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7b48479868582d73","tex":"a^3","aliases":"a^3","display":"a^3","kind":"use","excerpt":"Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-09acaea0cd848de9","tex":"a^5","aliases":"a^5","display":"a^5","kind":"use","excerpt":"Rank two exposes the loss in one line The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence i","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bce8bfd89f58273e","tex":"a^d","aliases":"a^d","display":"a^d","kind":"use","excerpt":"mid c and gcd(a,c)=1, then pnmid W_r,K(a,c). Expand W as a homogeneous polynomial in a,c. Modulo p, every term below the top degree in a retains a positive power of c and vanishes. The unique top subset is (K-r+1,…,K). Its indices, nonzero differences and L_K are all p-units, so its scaled coefficient is a unit. The remaining factor a^d is a unit b","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"Complete-intersection control case. Let F∈Z[X] be monic and squarefree over Q, with an integer simple root a_0. Put T=Z[X]/(F) and let λ: T→Z evaluate at a_0. The ring is finite free over Z and reduced over Q. Then the congruence quotient Z/λ(AnnI) and cotangent module both have order |F'(a_0)|. To see this directly, write F=(X-a_0)G. The kernel I ","count":9},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"The Legendre parameter of the parent is λ=t^2, t=v/u. The degree-two Landen transformation gives λ ⟼ frac4sqrtλ(1+sqrtλ)^2. This is exactly the child ratio a_1/c_1. The transformation preserves the automorphic channel up to the corresponding two-isogeny while lowering the integral height. It proves that the integral representative inside an isogeny","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c9f16574b6182941","tex":"A_{ij}","aliases":"A_ij","display":"A_{ij}","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"Finite polyphase rings. Fix an integer N≥2 and work in A_N=Z[T]/(T^N-1). The residue of V_e(T) records the number of valuation shells in each congruence class modulo N. Multiplication by V_e is represented by a circulant matrix C_e,N.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"Why the epsilon cannot be removed. The uniform statement c≤ KR for one absolute constant K is false. There is an elementary family that proves this without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive,","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d11f2eb675d387eb","tex":"a_{p,q}","aliases":"a_p,q","display":"a_{p,q}","kind":"use","excerpt":"d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modul","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"use","excerpt":"[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"use","excerpt":"grows only logarithmically with each index, whereas its numerator power of t gives linearly growing valuation. Thus the terms tend to zero and the finite Cauchy--Binet expansions pass to the convergent local limit. [Exact good-prime valuation] If p>3r-1 and v_p(t)=e≥1, then v_p( H_r(t))=e r(r+1)/2. Write a_r=2r-1 and s_r=r(r+1)/2. A term indexed by","count":13},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-27e7a4e6057e20eb","tex":"A_{r,K}","aliases":"A_r,K","display":"A_{r,K}","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5118a3baa8303c7d","tex":"A_\\varepsilon","aliases":"A_ε varepsilon","display":"A_\\varepsilon","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-61598b6c6bd4aad0","tex":"A_{\\varepsilon'}","aliases":"A_ε' varepsilon","display":"A_{\\varepsilon'}","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d0a82faef8552fd4","tex":"A_\\varepsilon^2","aliases":"A_ε^2 varepsilon","display":"A_\\varepsilon^2","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a3a52d54c059cb53","tex":"a_\\ell","aliases":"a_ℓ ell","display":"a_\\ell","kind":"use","excerpt":"The automorphic eigenchannel. Modularity associates a weight-two newform f_E to the rational isogeny class of E. Its Hecke coefficients a_ℓ(E) form a global frequency signal. The conductor is the automorphic bandwidth. The newform, however, remembers the rational isogeny class rather than the marked integral representative.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-99fa55a48bd5ebd6","tex":"\\mathrm{act}","aliases":"act mathrm","display":"\\mathrm{act}","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e1ab77241aa5e261","tex":"\\mathrm{add}","aliases":"add mathrm","display":"\\mathrm{add}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-59a706af3e94a284","tex":"\\mathrm{alg}","aliases":"alg mathrm","display":"\\mathrm{alg}","kind":"use","excerpt":"For the output source m=c/R_c, the corrected algebraic defect against a cyclic target of order η_f is δ_alg(c,f) = fracc/R_cgcd(c/R_c,η_f). Prime by prime, δ_alg(c,f) = ∏_p^eparallel c p^max(e-1-v_p(η_f),0). The problem is missing depth, not missing rank.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-87708a8e37000af2","tex":"\\operatorname{Ann}","aliases":"Ann operatorname","display":"\\operatorname{Ann}","kind":"use","excerpt":"Complete-intersection control case. Let F∈Z[X] be monic and squarefree over Q, with an integer simple root a_0. Put T=Z[X]/(F) and let λ: T→Z evaluate at a_0. The ring is finite free over Z and reduced over Q. Then the congruence quotient Z/λ(AnnI) and cotangent module both have order |F'(a_0)|. To see this directly, write F=(X-a_0)G. The kernel I ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"[Mason--Stothers in characteristic zero] Let k be an algebraically closed field of characteristic zero. Suppose A,B,C∈ k[X] are nonzero, pairwise coprime, satisfy A+B=C, and are not all constant. Then max{deg A,deg B,deg C}≤ n_0(ABC)-1. This is the polynomial abc theorem. Snyder gives a short primary proof in a form that also makes the derivative h","count":49},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"How to read this piece. addcontentslinetocsectionHow to read this piece The equation a+b=c joins two kinds of structure. Multiplication describes prime factors and their repeated powers. Addition forces those three factorizations to fit one exact relation. The abc conjecture says that this compatibility limits how large the sum can be compared with","count":220},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e77f9eceea598e09","tex":"B'","aliases":"B'","display":"B'","kind":"use","excerpt":"Set W=A'B-AB'. It is nonzero. Indeed, if it vanished, then (A/B)'=0, and characteristic zero would force the rational function A/B to be constant. Coprimality would then force both A and B to be constant, and so also C.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discrimina","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-709c8692f9d032bb","tex":"B^d","aliases":"B^d","display":"B^d","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","excerpt":"For h_i=0, the same bound holds because its left side is nonpositive. The numerator Vandermonde only improves valuation. Since e≥1, every term in the convergent expansion has valuation at least es_r+∑_i[(k_i-i)-a_rh_i]≥ es_r-C_r,p. The ultrametric inequality gives the same bound for the sum. Multiplication by B_r restores the missing C_r,p. For ran","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7be573071ebda78d","tex":"B_d","aliases":"B_d","display":"B_d","kind":"use","excerpt":"Exact-order divisor packets. Let n=∏_p p^e_p. For every d≥1, define B_d(n) = ∏_substackpmid n dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact c","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c69ec0fece4f3aa0","tex":"b_h","aliases":"b_h","display":"b_h","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","excerpt":"Why the epsilon cannot be removed. The uniform statement c≤ KR for one absolute constant K is false. There is an elementary family that proves this without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive,","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-57c8f1ae1da23ae1","tex":"B_r","aliases":"B_r","display":"B_r","kind":"use","excerpt":"ining primes Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depe","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1784da886e1aea15","tex":"\\bar v","aliases":"bar v bar","display":"\\bar v","kind":"use","excerpt":"Odd logarithmic rows. For odd p^eparallel c, define the finite-depth row coefficient ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. The balanced relation gives ∑_qmid ab ν_qℓ_p,e(q) ≡0 pmodp^e-1. Define the truncated valuation on Z/p^e-1Z by bar v_p(0)=e-1, bar v_p(x)=min{e-1,v_p(widetilde x)} for any integer lift widetilde x of a nonzero residue. Then ba","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5daa5707e80f7e4e","tex":"\\bar v_2","aliases":"bar v_2 bar","display":"\\bar v_2","kind":"use","excerpt":"Weighted dyadic row. For 2^eparallel c with e≥2 and an odd input prime q, define ℓ_2,e(q) = frac14 log_2((-1)^(q-1)/2q) pmod2^e-2. With the analogous truncated valuation on Z/2^e-2Z, bar v_2(ℓ_2,e(q)) = min{e-2,v_2(q^2-1)-3}. At e=1, the output contact depth is zero and this row is omitted. A separate sign packet supplies the missing first dyadic l","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a87a1d8861d8d21d","tex":"\\bar v_p","aliases":"bar v_p bar","display":"\\bar v_p","kind":"use","excerpt":"Odd logarithmic rows. For odd p^eparallel c, define the finite-depth row coefficient ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. The balanced relation gives ∑_qmid ab ν_qℓ_p,e(q) ≡0 pmodp^e-1. Define the truncated valuation on Z/p^e-1Z by bar v_p(0)=e-1, bar v_p(x)=min{e-1,v_p(widetilde x)} for any integer lift widetilde x of a nonzero residue. Then ba","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5bd492e4ea78c5fe","tex":"\\bigsqcup","aliases":"bigsqcup","display":null,"kind":"use","excerpt":"Obstruction cosets. Exclude (1,1,2) throughout this section. Choose a section x_0∈ L with ω(x_0)=1. Then L = bigsqcup_k∈Z (kx_0+L_0). For fixed k, the problem is a closest-vector problem to the moving target -kx_0 in the lattice L_0. The three-sector compression makes this search much more explicit.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ccbecb81e1a49e42","tex":"\\bigsqcup_{k\\in\\Z}","aliases":"bigsqcup_k∈Z bigsqcup in Z","display":null,"kind":"use","excerpt":"Obstruction cosets. Exclude (1,1,2) throughout this section. Choose a section x_0∈ L with ω(x_0)=1. Then L = bigsqcup_k∈Z (kx_0+L_0). For fixed k, the problem is a closest-vector problem to the moving target -kx_0 in the lattice L_0. The three-sector compression makes this search much more explicit.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"A known unconditional bound. Stewart and Yu proved an effective bound of a different scale~[source reference]: there is an effectively computable absolute C>0 such that every primitive positive triple satisfies c<exp (CR^1/3(log R)^3). It controls the height in terms of the radical unconditionally. Its exponential dependence is much larger than the","count":77},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C","display":null,"kind":"use","excerpt":"Let I_N=ker(A_Nxrightarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-420668a5bdd9e0bf","tex":"\\mathbf C","aliases":"C mathbf","display":"\\mathbf C","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"Since C=A+B, W=A'C-AC'. At a root of C of multiplicity e, this expression is divisible by (X-α)^e-1. The three root sets are disjoint. Therefore fracABCrad(ABC)mid W. Taking degrees gives deg A+deg B+deg C-n_0(ABC)≤deg W≤deg A+deg B-1. Hence deg C≤ n_0(ABC)-1. To bound A, apply the same argument to the relation C+(-B)=A. To bound B, use C+(-A)=B. C","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-aaafc3af610c8063","tex":"C'_\\eps","aliases":"C'_eps eps","display":null,"kind":"use","excerpt":"[Conditional filtered-transfer criterion] Suppose every primitive positive triple admits a canonical map H^0(Θ_f):Z/(c/R_c)Z⟶ H^0( C_f) satisfies, for every eps>0, with constants independent of the triple, |ker H^0(Θ_f)|≤ C_eps R^eps and |H^0( C_f)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3bb3970db835d023","tex":"C'_{\\eta}","aliases":"C'_η eta","display":"C'_{\\eta}","kind":"use","excerpt":"If for every η>0 one proves WC≤ C_η R_aR_bR^η, then c≤ C'_ηR^1+ηlog(3c). The same absorption argument gives abc. Conversely WC≤ M_c, so the all-eps output-sector abc inequality immediately supplies this bound. The uniform combined WC estimate is therefore the exact endpoint. A bound on C alone suffices only under independent control of W, for examp","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-93403a6b92bb7f16","tex":"c^{1-\\delta}","aliases":"c^1-δ delta","display":"c^{1-\\delta}","kind":"use","excerpt":"one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a constant L_r,δ such that (log(3c))^a_r≤ L_r,δc^δ for all c≥2. The elementary fact follows because a fixed logarithmic power divided by any positive power of c tends to zero, and the remainder is bounded on a finite initial interval. Thus c^1-δ≤ D_rL_r,δR","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","excerpt":"Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-54a1ea9db444fe2c","tex":"c^{3/2}","aliases":"c^3/2","display":"c^{3/2}","kind":"use","excerpt":"lier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor prop","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3052e6f7fab4cf4a","tex":"c^d","aliases":"c^d","display":"c^d","kind":"use","excerpt":"For a reduced rational argument t=a/c, an obvious integral scalar is W_r,K(a,c)=c^dA_r,K(a/c). It is nonzero when 0<a<c, since the truncated positive expansion is nonzero. Thus it is a genuine nonzero arithmetic candidate. The question is whether it retains the required output-prime contact after clearing denominators.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-33ec106c4d586319","tex":"c^k","aliases":"c^k","display":"c^k","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c86e48078fe8fd92","tex":"c^{o(1)}","aliases":"c^o(1)","display":"c^{o(1)}","kind":"use","excerpt":"ept for (1,1,2), define the shortest transverse packet Θ(a,b,c) = min{ normx_∞: x∈ T_a,b,c, W_x(a,b)≠0 }. Then c ≤ fracRlog2Θ(a,b,c)log c. The exceptional triple (1,1,2) has zero input derivatives and is omitted from Θ. It is absorbed into the constant in the conjecture. A subpolynomial estimate Θ(a,b,c)=c^o(1) would imply the strong abc inequality","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cd69d6e71c44f8ea","tex":"c^\\delta","aliases":"c^δ delta","display":"c^\\delta","kind":"use","excerpt":"nd use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E_r/s_r, a_r=A_r/s_r. Fix the target eps>0. Choose one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a constant L_r,δ such that (log(3c))^a_r≤ L_r,δc^δ for all c≥2. The elementa","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"The Legendre parameter of the parent is λ=t^2, t=v/u. The degree-two Landen transformation gives λ ⟼ frac4sqrtλ(1+sqrtλ)^2. This is exactly the child ratio a_1/c_1. The transformation preserves the automorphic channel up to the corresponding two-isogeny while lowering the integral height. It proves that the integral representative inside an isogeny","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e0c8a76d0e325cac","tex":"c_4","aliases":"c_4","display":"c_4","kind":"use","excerpt":"The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ca65dc5da5fa80e3","tex":"c_6","aliases":"c_6","display":"c_6","kind":"use","excerpt":"The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c30cf4b83f55fb64","tex":"c_6^2","aliases":"c_6^2","display":"c_6^2","kind":"use","excerpt":"The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f616adbba955d792","tex":"C_{e,N}","aliases":"C_e,N","display":"C_{e,N}","kind":"use","excerpt":"Finite polyphase rings. Fix an integer N≥2 and work in A_N=Z[T]/(T^N-1). The residue of V_e(T) records the number of valuation shells in each congruence class modulo N. Multiplication by V_e is represented by a circulant matrix C_e,N.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ebbcc4b3a230e297","tex":"C_\\eps","aliases":"C_eps eps","display":null,"kind":"use","excerpt":"The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b532a9b68d20fb27","tex":"\\mathbf C_f","aliases":"C_f mathbf","display":"\\mathbf C_f","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with v","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cdea161102d3215b","tex":"c_{m,1}","aliases":"c_m,1","display":"c_{m,1}","kind":"use","excerpt":"Small coefficient norm has a limited meaning. The coefficients of P_m alternate in sign. Its absolute-coefficient polynomial evaluated at one gives ∑_s|c_m,s|=∏_j=1^m(j+1)=(m+1)!. Also c_m,1=(-1)^m, so the integer coefficient vector is primitive. After the displayed normalization, its coefficient ℓ^1 norm is one.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-612417ee8a56f0e9","tex":"c_{m,s}","aliases":"c_m,s","display":"c_{m,s}","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"use","excerpt":"Why the epsilon cannot be removed. The uniform statement c≤ KR for one absolute constant K is false. There is an elementary family that proves this without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive,","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-746a0fca63346846","tex":"C_r","aliases":"C_r","display":"C_r","kind":"use","excerpt":"Since |log(b/a)|≤log c and log(c/a)≤log c, every entry has a bound by a constant times (1+log c)^j. A determinant term has total weight ∑_i=0^r-1(i+π(i)+1)=r^2. Therefore | H_r(c/a)|≤ C_r(1+log c)^r^2 for each of those consistent branches. The constant may depend on r. No nonvanishing assertion at every argument in this third chart follows from the","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0ffe11146a25320d","tex":"C_{r,p}","aliases":"C_r,p","display":"C_{r,p}","kind":"use","excerpt":"An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-02a81ce9982fe810","tex":"C_{r,p,i}","aliases":"C_r,p,i","display":"C_{r,p,i}","kind":"use","excerpt":"An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6c697b1a25dd0d97","tex":"C_r^{1/s_r}","aliases":"C_r^1/s_r","display":"C_r^{1/s_r}","kind":"use","excerpt":"admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b59c9966dcf6bcf5","tex":"C_{\\mathrm{red}}","aliases":"C_red mathrm","display":"C_{\\mathrm{red}}","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c6d1d8d8724e93f3","tex":"C_U","aliases":"C_U","display":"C_U","kind":"use","excerpt":"Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5da91e15de04cc9e","tex":"C_\\eta","aliases":"C_η eta","display":"C_\\eta","kind":"use","excerpt":"If for every η>0 one proves WC≤ C_η R_aR_bR^η, then c≤ C'_ηR^1+ηlog(3c). The same absorption argument gives abc. Conversely WC≤ M_c, so the all-eps output-sector abc inequality immediately supplies this bound. The uniform combined WC estimate is therefore the exact endpoint. A bound on C alone suffices only under independent control of W, for examp","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e6a6399fcdef2fe8","tex":"c_\\phi","aliases":"c_ϕ phi","display":"c_\\phi","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-57a2bed9b1839447","tex":"c_\\phi^2","aliases":"c_ϕ^2 phi","display":"c_\\phi^2","kind":"use","excerpt":"parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-57d5f54bcad0aa19","tex":"\\mathrm{coh}","aliases":"coh mathrm","display":"\\mathrm{coh}","kind":"use","excerpt":"er normalization, every output prime with nontrivial residual depth produces a primitive row g_p ∈ (Z/p^χ_pZ)^S_ab satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-073a71f814bdfa5f","tex":"\\coker","aliases":"coker","display":null,"kind":"use","excerpt":"Exact cyclic transducer. Let m,η be positive integers and put C(m)=[ZxrightarrowmZ] and C(η)=[ZxrightarrowηZ]. Set g=gcd(m,η). Multiplication by m/g in degree -1 and by η/g in degree 0 defines a chain map. On cohomology, |ker|=m/g, |im|=g, |coker|=η/g. The maximal transferable packet is the common prime-power depth.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6e6d0cbeeb71722a","tex":"\\operatorname{coker}","aliases":"coker operatorname","display":"\\operatorname{coker}","kind":"use","excerpt":"[Smith obstruction theorem] Let a+b=c be primitive, with at least two of a,b,c greater than one. Then SNF( M_a,b,c) = diag(1,Q). Consequently, coker( M_a,b,c)_tors ≅ Z/QZ. The exceptional triple (1,1,2) has only one nonempty prime sector, so the matrix has rank one and is excluded from the two-row statement.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"Where the integer argument stops. An ordinary integer has no distinguished derivative with both an exact Leibniz rule and an automatic relation D(a)+D(b)=D(c). Arithmetic derivatives provide a family of Leibniz maps. Additivity at one triple then imposes a lattice constraint on the prime-coordinate values.","count":16},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e63d808025fba917","tex":"\\mathrm d","aliases":"d mathrm","display":"\\mathrm d","kind":"use","excerpt":"e three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n, define the universal multiplicative differential dn ","count":15},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d0d28795ad33b969","tex":"D^{1/d}","aliases":"D^1/d","display":"D^{1/d}","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-34ed1a97bb97d001","tex":"d^{d/2}","aliases":"d^d/2","display":"d^{d/2}","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-dcdf529ce373d74b","tex":"D^{m+2}","aliases":"D^m+2","display":"D^{m+2}","kind":"use","excerpt":"An exact differential identity. Let D=t dd/dd t be the Euler operator. Termwise formal differentiation is valid. Applying D^m+2 multiplies the coefficient at t^k by k^m+2, so D^m+2Φ_m=(-1)^m∑_k≥1binomkm+1t^k =(-1)^mfract^m+1(1-t)^m+2. The final equality is the elementary generating function for the binomial coefficients.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-54ff0eb6a1af0d6f","tex":"D_f","aliases":"D_f","display":"D_f","kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-237fc9015f157bd3","tex":"D_r","aliases":"D_r","display":"D_r","kind":"use","excerpt":"admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d444e4eb5807d954","tex":"\\mathrm d_x","aliases":"d_x mathrm","display":"\\mathrm d_x","kind":"use","excerpt":"A scalar packet is an integer vector x=(x_p)_p∈ S∈Z^S. Pairing with x gives the arithmetic derivative d_x(n) = n∑_pmid nfracv_p(n)p x_p. The Leibniz rule survives scalarization.","count":13},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-521b9139cdb95729","tex":"D_y","aliases":"D_y","display":"D_y","kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd","display":null,"kind":"use","excerpt":"An exact differential identity. Let D=t dd/dd t be the Euler operator. Termwise formal differentiation is valid. Applying D^m+2 multiplies the coefficient at t^k by k^m+2, so D^m+2Φ_m=(-1)^m∑_k≥1binomkm+1t^k =(-1)^mfract^m+1(1-t)^m+2. The final equality is the elementary generating function for the binomial coefficients.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"The polynomial theorem and the missing integer analogue. Roots play the role of distinct primes. For a nonzero polynomial F over an algebraically closed field, let rad(F) be the product of its distinct monic linear factors. Its degree counts distinct roots. Write n_0(F)=degrad(F).","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"A nonzero Hankel contact determinant. Independent moment directions replace one filter. For an integer r≥1, define H_r(t)=det(Li_i+j+1(t))_0≤ i,j<r. It uses weights one through 2r-1. The expression is a formal series over Q and an analytic function for |t|<1. No algebraicity at a rational argument is assumed.","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermo","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8f42a8e27ac6c866","tex":"\\operatorname{div}","aliases":"div operatorname","display":"\\operatorname{div}","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Elliptic curves and the modified Szpiro comparison. The marked Frey curve E:y^2=x(x-a)(x+b) translates the additive triple into an elliptic curve. Its discriminant sees repeated depth, while its conductor sees prime support, with local model corrections at the small primes. This makes elliptic height estimates a natural companion to abc.","count":34},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with v","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-50860b6739bbedbb","tex":"e^{1/4}","aliases":"e^1/4","display":"e^{1/4}","kind":"use","excerpt":"Real positivity and a bound uniform in rank. [Positive real determinant] For every r≥1 and every real 0<t<1, 0< H_r(t)≤ e^1/4[-log(1-t)]. The Cauchy--Binet expansion consists of positive terms and contains the nonzero term (1,…,r), so positivity follows. Equivalently, the real moment matrix is positive definite: a nonzero polynomial of degree below","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bf62671d17826896","tex":"e^{2\\pi ix/n}","aliases":"e^2π ix/n pi","display":"e^{2\\pi ix/n}","kind":"use","excerpt":"Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1)","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8618d23a0187d108","tex":"E_{a,b,c}","aliases":"E_a,b,c","display":"E_{a,b,c}","kind":"use","excerpt":"The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discrimina","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7f757c70b1c7b2f2","tex":"\\mathcal E_d","aliases":"E_d mathcal","display":"\\mathcal E_d","kind":"use","excerpt":"Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with v","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-420aa7a6632049fa","tex":"e_f","aliases":"e_f","display":"e_f","kind":"use","excerpt":"The same-sign separation problem. After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let Λ be an integral Hecke lattice and let e_f be the rational projector onto the Frey eigenline. Define η_f = min{m≥1:m e_f(Λ)⊆Λ}. This is the integral eigenchannel separation denominator. It is an arithmetic analogue of ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1b09d4efdcdc1f39","tex":"e_p","aliases":"e_p","display":"e_p","kind":"use","excerpt":"Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records ","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9e1334b7626cad22","tex":"E_{p,r}","aliases":"E_p,r","display":"E_{p,r}","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3a840ddc04edad8c","tex":"E_r","aliases":"E_r","display":"E_r","kind":"use","excerpt":"[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-504dc54aa7ab931b","tex":"e_U","aliases":"e_U","display":"e_U","kind":"use","excerpt":"Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps","display":null,"kind":"use","excerpt":"[Masser--Oesterl e abc] For every real eps>0 there is a constant K_eps>0, depending only on eps, such that every positive integer triple with a+b=c and gcd(a,b)=1 satisfies c≤ K_eps rad(abc)^1+eps. The order of the quantifiers is the statement. First choose eps. Then choose a constant that works for every primitive triple. The constant may become l","count":56},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"A known unconditional bound. Stewart and Yu proved an effective bound of a different scale~[source reference]: there is an effectively computable absolute C>0 such that every primitive positive triple satisfies c<exp (CR^1/3(log R)^3). It controls the height in terms of the radical unconditionally. Its exponential dependence is much larger than the","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F","display":null,"kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":14},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"Complete-intersection control case. Let F∈Z[X] be monic and squarefree over Q, with an integer simple root a_0. Put T=Z[X]/(F) and let λ: T→Z evaluate at a_0. The ring is finite free over Z and reduced over Q. Then the congruence quotient Z/λ(AnnI) and cotangent module both have order |F'(a_0)|. To see this directly, write F=(X-a_0)G. The kernel I ","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b0133f9470b1d08c","tex":"F_{a,b,c}","aliases":"F_a,b,c","display":"F_{a,b,c}","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-dafda06c8efa3652","tex":"f_E","aliases":"f_E","display":"f_E","kind":"use","excerpt":"The automorphic eigenchannel. Modularity associates a weight-two newform f_E to the rational isogeny class of E. Its Hecke coefficients a_ℓ(E) form a global frequency signal. The conductor is the automorphic bandwidth. The newform, however, remembers the rational isogeny class rather than the marked integral representative.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-522490995f512f8c","tex":"F_{i+j+1,p}","aliases":"F_i+j+1,p","display":"F_{i+j+1,p}","kind":"use","excerpt":"Exact local determinant valuation and denominator budget. Depletion gives an integral local matrix. Fix a prime p and an integer r≥1. Define F_s,p(t)=∑_pnmid kfract^kk^s, H_r,p(t)=det(F_i+j+1,p(t))_0≤ i,j<r. These series converge at every t∈Q_p with v_p(t)>0. They are integral power series. The Cauchy--Binet expansion from Section~[source reference","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fda40c00a5457366","tex":"\\F_p","aliases":"F_p F","display":null,"kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-16a89ae6f8afbe2b","tex":"F_{s,p}","aliases":"F_s,p","display":"F_{s,p}","kind":"use","excerpt":"Exact local determinant valuation and denominator budget. Depletion gives an integral local matrix. Fix a prime p and an integer r≥1. Define F_s,p(t)=∑_pnmid kfract^kk^s, H_r,p(t)=det(F_i+j+1,p(t))_0≤ i,j<r. These series converge at every t∈Q_p with v_p(t)>0. They are integral power series. The Cauchy--Binet expansion from Section~[source reference","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Signed arithmetic COS and the sparse spectral leaf. What the finite assignment signal records. The spectral branch of the project asks which prime assignments realize a prescribed residue. Its exact object is a finite distribution. Fix a finite abelian group G and s eligible variable support primes. At the jth prime choose one of three permitted co","count":12},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular ","count":24},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular ","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d230c9e7fd4e8bb2","tex":"g_b","aliases":"g_b","display":"g_b","kind":"use","excerpt":"Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular ","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a4676088e867baee","tex":"g_c","aliases":"g_c","display":"g_c","kind":"use","excerpt":"Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular ","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f046713e329fd428","tex":"g_p","aliases":"g_p","display":"g_p","kind":"use","excerpt":"The coherence character. Set χ_p=0 for absent or squarefree output channels. When the residual modulus is one, use the trivial row and character. After normalization, every output prime with nontrivial residual depth produces a primitive row g_p ∈ (Z/p^χ_pZ)^S_ab satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vec","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f66bcd9a4f79a720","tex":"g_{\\mathrm{val}}","aliases":"g_val mathrm","display":"g_{\\mathrm{val}}","kind":"use","excerpt":"Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":14},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f693946f87e25002","tex":"\\gcd_{p\\mid u}","aliases":"gcd_pmid u gcd mid","display":null,"kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-530c8a45651a2504","tex":"\\gg","aliases":"gg","display":null,"kind":"use","excerpt":"If the assignment model has M=3^s with s≤ω(R), choose δ=eps/2 in both estimates. Then fracφ(Q)Mgg_eps R^1+eps. Consequently a sufficiently large hypothetical exponent-gap exception, represented in this joint unit-group model, lies on the sparse side of the energy barrier. Uniform spectral mixing cannot be the universal finishing argument for this r","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-befa25e4315720cd","tex":"\\gg_\\eps","aliases":"gg_eps gg eps","display":null,"kind":"use","excerpt":"If the assignment model has M=3^s with s≤ω(R), choose δ=eps/2 in both estimates. Then fracφ(Q)Mgg_eps R^1+eps. Consequently a sufficiently large hypothetical exponent-gap exception, represented in this joint unit-group model, lies on the sparse side of the energy barrier. Uniform spectral mixing cannot be the universal finishing argument for this r","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-dbaf3bbf176dd3a0","tex":"\\mathrm{glob}","aliases":"glob mathrm","display":"\\mathrm{glob}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-70196007a27c98a9","tex":"\\gr","aliases":"gr","display":null,"kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"lic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x","count":22},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"A nonzero Hankel contact determinant. Independent moment directions replace one filter. For an integer r≥1, define H_r(t)=det(Li_i+j+1(t))_0≤ i,j<r. It uses weights one through 2r-1. The expression is a formal series over Q and an analytic function for |t|<1. No algebraicity at a rational argument is assumed.","count":21},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fa5a6573c905a228","tex":"H^0","aliases":"H^0","display":"H^0","kind":"use","excerpt":"[Filtered contact quotient] H^0( J_c) = R_cZ/cZ ≅ Z/(c/R_c)Z.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f928390ad1e40fa6","tex":"\\mathcal H_1","aliases":"H_1 mathcal","display":"\\mathcal H_1","kind":"use","excerpt":"For the first two ranks, H_1=Li_1, H_2=Li_1Li_3-Li_2^2. The rank-two determinant compares genuinely different moment directions. Its order-three contact is not inferred from the presence of weight three.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2fd2f57e4f0efca6","tex":"\\mathcal H_2","aliases":"H_2 mathcal","display":"\\mathcal H_2","kind":"use","excerpt":"For the first two ranks, H_1=Li_1, H_2=Li_1Li_3-Li_2^2. The rank-two determinant compares genuinely different moment directions. Its order-three contact is not inferred from the presence of weight three.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2a6b5b11a3d6d7c4","tex":"\\mathcal H_{2,2}","aliases":"H_2,2 mathcal","display":"\\mathcal H_{2,2}","kind":"use","excerpt":"Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-67bd3e605e0d486b","tex":"\\mathcal H_{2,3}","aliases":"H_2,3 mathcal","display":"\\mathcal H_{2,3}","kind":"use","excerpt":"Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-65d56caafc4f11ed","tex":"\\mathcal H_{2,p}","aliases":"H_2,p mathcal","display":"\\mathcal H_{2,p}","kind":"use","excerpt":"The expansion converges p-adically. A unique term of minimum valuation cannot be cancelled by the remaining terms. Its valuation is therefore the valuation of the whole determinant. For rank two, v_p( H_2,p(t))= 3e, · p≠2, 4e+2, · p=2. At two the first unit indices are 1,3, of sum four, and their difference contributes one valuation before squaring","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4f5ceb3e1772c369","tex":"\\mathcal H_3","aliases":"H_3 mathcal","display":"\\mathcal H_3","kind":"use","excerpt":"Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose that for every primitive positive triple there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) such that, for every eps>0, constants independent of the triple give |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-333bf23be8bc780a","tex":"h_i","aliases":"h_i","display":"h_i","kind":"use","excerpt":"If h_i=v_p(k_i)>0, then k_i-i≥ p^h_i-r>a_rh_i. If h_i=0, then k_i-i≥0 and there is no denominator loss. Thus every subset other than (1,…,r) has valuation strictly above es_r. The first subset has a unit coefficient because its indices and differences are smaller than p. Its term has valuation es_r, and a unique minimum again proves equality. The s","count":9},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4408cd3dcd4b6b67","tex":"\\mathcal H_r","aliases":"H_r mathcal","display":"\\mathcal H_r","kind":"use","excerpt":"A nonzero Hankel contact determinant. Independent moment directions replace one filter. For an integer r≥1, define H_r(t)=det(Li_i+j+1(t))_0≤ i,j<r. It uses weights one through 2r-1. The expression is a formal series over Q and an analytic function for |t|<1. No algebraicity at a rational argument is assumed.","count":9},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-369a7bd69e7434a6","tex":"\\mathcal H_{r,K}","aliases":"H_r,K mathcal","display":"\\mathcal H_{r,K}","kind":"use","excerpt":"A rational truncation that fails the output contact test. A genuine algebraic candidate can be tested exactly. Fix integers r≥1 and K≥ r. Replace each polylogarithm by its finite series through index K, obtaining a rational polynomial H_r,K(t). Its Cauchy--Binet expansion has the same positive subset coefficients, restricted to 1≤ k_i≤ K.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0c517f7e5d401b9a","tex":"\\mathcal H_{r,p}","aliases":"H_r,p mathcal","display":"\\mathcal H_{r,p}","kind":"use","excerpt":"Exact local determinant valuation and denominator budget. Depletion gives an integral local matrix. Fix a prime p and an integer r≥1. Define F_s,p(t)=∑_pnmid kfract^kk^s, H_r,p(t)=det(F_i+j+1,p(t))_0≤ i,j<r. These series converge at every t∈Q_p with v_p(t)>0. They are integral power series. The Cauchy--Binet expansion from Section~[source reference","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0745ce822383e740","tex":"I^2","aliases":"I^2","display":"I^2","kind":"use","excerpt":"efree over Q, with an integer simple root a_0. Put T=Z[X]/(F) and let λ: T→Z evaluate at a_0. The ring is finite free over Z and reduced over Q. Then the congruence quotient Z/λ(AnnI) and cotangent module both have order |F'(a_0)|. To see this directly, write F=(X-a_0)G. The kernel I is generated by X-a_0. Modulo I^2, the relation is G(a_0)(X-a_0)=","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-eff6aa27c552c5d2","tex":"I_a","aliases":"I_a","display":"I_a","kind":"use","excerpt":"[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2344f57325d4c16a","tex":"I_b","aliases":"I_b","display":"I_b","kind":"use","excerpt":"[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f9648a100dad73ed","tex":"I_c","aliases":"I_c","display":"I_c","kind":"use","excerpt":"[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"use","excerpt":"Let I_N=ker(A_Nxrightarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im","display":null,"kind":"use","excerpt":"Exact cyclic transducer. Let m,η be positive integers and put C(m)=[ZxrightarrowmZ] and C(η)=[ZxrightarrowηZ]. Set g=gcd(m,η). Multiplication by m/g in degree -1 and by η/g in degree 0 defines a chain map. On cohomology, |ker|=m/g, |im|=g, |coker|=η/g. The maximal transferable packet is the common prime-power depth.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5c2455f4c0bb929a","tex":"\\inf_{k\\ge1}","aliases":"inf_k≥1 inf ge","display":"\\inf_{k\\ge1}","kind":"use","excerpt":"For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4978cb2ce63e316b","tex":"\\mathbf J","aliases":"J mathbf","display":"\\mathbf J","kind":"use","excerpt":"SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f63cab9c738188a7","tex":"J_a","aliases":"J_a","display":"J_a","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-88fbeb34dbc66890","tex":"J_b","aliases":"J_b","display":"J_b","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-365e4afde3f07c22","tex":"J_c","aliases":"J_c","display":"J_c","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-528b7456d232f332","tex":"\\mathbf J_c","aliases":"J_c mathbf","display":"\\mathbf J_c","kind":"use","excerpt":"SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a620020ba505249b","tex":"J_n","aliases":"J_n","display":"J_n","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-753ac74d7d5e39c2","tex":"\\Jcal","aliases":"Jcal","display":null,"kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3e498617a2adef3c","tex":"\\Jcal_{p,e}","aliases":"Jcal_p,e Jcal","display":null,"kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"his without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^","count":90},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4217885dfac90375","tex":"k^{-(2k-1)}","aliases":"k^-(2k-1)","display":"k^{-(2k-1)}","kind":"use","excerpt":"[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3a63de54db38dad9","tex":"k^{-(r-1)}","aliases":"k^-(r-1)","display":"k^{-(r-1)}","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6013290ab9ee6c0e","tex":"k^{-1}","aliases":"k^-1","display":"k^{-1}","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8f5f77932c0a4047","tex":"k^{2r-1}","aliases":"k^2r-1","display":"k^{2r-1}","kind":"use","excerpt":"contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each k, givi","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d52e5b6c283d180a","tex":"k^{i+j+1}","aliases":"k^i+j+1","display":"k^{i+j+1}","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-013653d060d79ff8","tex":"k^{m+1}","aliases":"k^m+1","display":"k^{m+1}","kind":"use","excerpt":"[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cfce4f8a76418180","tex":"k^{m+2}","aliases":"k^m+2","display":"k^{m+2}","kind":"use","excerpt":"[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-68147be7872ac913","tex":"k^s","aliases":"k^s","display":"k^s","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cd49a1788f8baddd","tex":"k_*","aliases":"k_*","display":"k_*","kind":"use","excerpt":"ve hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=overline H k_*Γ for the optimal obstruction multiple k_* and an integrality factor","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cf5fcf757b255a84","tex":"K_1","aliases":"K_1","display":"K_1","kind":"use","excerpt":"al large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b=y^n, c=z^n with the fixed exponent eps=1. Then ","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1bdfdabfb7ccb3ec","tex":"k_1","aliases":"k_1","display":"k_1","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ae57cc6b359aadbd","tex":"K_3","aliases":"K_3","display":"K_3","kind":"use","excerpt":"Weight-three motivation and limitation. Over Q, Borel's rank theorem gives rankK_3(Q)=0 and rankK_5(Q)=1, the rational weights two and three. This makes weight three a plausible rational regulator layer. It does not assert the absence of all weight-two periods or of regulator directions over arbitrary number fields. This motivates the depth-three p","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2df2ffe746cb7b87","tex":"K_5","aliases":"K_5","display":"K_5","kind":"use","excerpt":"Weight-three motivation and limitation. Over Q, Borel's rank theorem gives rankK_3(Q)=0 and rankK_5(Q)=1, the rational weights two and three. This makes weight three a plausible rational regulator layer. It does not assert the absence of all weight-two periods or of regulator directions over arbitrary number fields. This motivates the depth-three p","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-80fe6ef3c06d8bca","tex":"K_\\eps","aliases":"K_eps eps","display":null,"kind":"use","excerpt":"[Masser--Oesterl e abc] For every real eps>0 there is a constant K_eps>0, depending only on eps, such that every positive integer triple with a+b=c and gcd(a,b)=1 satisfies c≤ K_eps rad(abc)^1+eps. The order of the quantifiers is the statement. First choose eps. Then choose a constant that works for every primitive triple. The constant may become l","count":8},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9fde486cb3217527","tex":"K_{\\eps/2}","aliases":"K_eps/2 eps","display":null,"kind":"use","excerpt":"[Finiteness formulation] Conjecture~[source reference] is equivalent to the assertion that, for every eps>0, there are only finitely many primitive positive triples with q(a,b,c)>1+eps. Assume abc, using its bound with eps/2. A triple of quality above 1+eps satisfies R^1+eps<c≤ K_eps/2R^1+eps/2. Hence R^eps/2<K_eps/2. The radical is bounded. The sa","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b783228933c35b11","tex":"k_i","aliases":"k_i","display":"k_i","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":13},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-caa5b7c8da47b932","tex":"k_i^{-1}","aliases":"k_i^-1","display":"k_i^{-1}","kind":"use","excerpt":"k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the s","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8652027c62ad207e","tex":"k_i^{2(r-1)}","aliases":"k_i^2(r-1)","display":"k_i^{2(r-1)}","kind":"use","excerpt":"orem Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =f","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-31dd23ade426e90f","tex":"k_i^{2r-1}","aliases":"k_i^2r-1","display":"k_i^{2r-1}","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bcf3911a4b4214e2","tex":"k_j","aliases":"k_j","display":"k_j","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1452fa51a99cb0c9","tex":"k_j^{-1}","aliases":"k_j^-1","display":"k_j^{-1}","kind":"use","excerpt":"orem Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =f","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e2b60819c95171e8","tex":"k_r","aliases":"k_r","display":"k_r","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a77d87feaab21d87","tex":"K_\\beta","aliases":"K_β beta","display":"K_\\beta","kind":"use","excerpt":"Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose that for every primitive positive triple there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) such that, for every eps>0, constants independent of the triple give |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e08d6b60be0ec0b0","tex":"\\Ker","aliases":"Ker","display":null,"kind":"use","excerpt":"[First-order no-free-polyphase principle] Let S:(T_p,r)_p,r⟼ (∑_rT_p,r)_p be shell synthesis. The first-order additive map and first-order Wronskian factor through S. Hence Ker S is contained in the kernels of both first-order maps.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.","count":10},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":23},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-08093f17d5992ea3","tex":"\\mathbf L","aliases":"L mathbf","display":"\\mathbf L","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1baf4f3c36fac5d7","tex":"\\mathbf L_{\\mathbb T_f}","aliases":"L_ T_f mathbf mathbb","display":"\\mathbf L_{\\mathbb T_f}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"The exact lattice extension. Throughout the remaining height and exact-sequence constructions, exclude (1,1,2). Define L=ker(Λ)∩Z^S and L_0=L∩ker( w). Because Λ is primitive, choose t∈Z^S with Λ(t)=1. Then Z^S=L⊕Z t. The image of M_a,b,c is generated by (1, w(t)) together with (0, w(L)). Its index in Z^2 is therefore the index of w(L) in Z. Theorem","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β)","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4662d64de6729153","tex":"L_K","aliases":"L_K","display":"L_K","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8f15de6bb46f66a6","tex":"L_K^{i+1}","aliases":"L_K^i+1","display":"L_K^{i+1}","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ced1c73da60892be","tex":"L_K^{i+j+1}","aliases":"L_K^i+j+1","display":"L_K^{i+j+1}","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5ccef985a511a309","tex":"L_K^j","aliases":"L_K^j","display":"L_K^j","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3dfb0083323e0b30","tex":"L_K^{r^2}","aliases":"L_K^r^2","display":"L_K^{r^2}","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c1f2591d69ff8561","tex":"\\mathbf L_p^{\\mathrm{st}}","aliases":"L_p^st mathbf mathrm","display":"\\mathbf L_p^{\\mathrm{st}}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-195ff153c4a08c31","tex":"L_{r,\\delta}","aliases":"L_r,δ delta","display":"L_{r,\\delta}","kind":"use","excerpt":"Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E_r/s_r, a_r=A_r/s_r. Fix the target eps>0. Choose one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a constant L_r,δ such that (log(3c))^a_r≤ L_r,δc^δ for a","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a4d5ae1ee956514f","tex":"\\mathbf L_{R^{\\mathrm{glob}}}","aliases":"L_R^glob mathbf mathrm","display":"\\mathbf L_{R^{\\mathrm{glob}}}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3eda83bfb4bc77ed","tex":"\\lcm","aliases":"lcm","display":"\\lcm","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3860d7e8f88507bb","tex":"\\operatorname{lcm}","aliases":"lcm operatorname","display":"\\operatorname{lcm}","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4b914935d1b77c2b","tex":"\\operatorname{lcm}_i","aliases":"lcm_i operatorname","display":"\\operatorname{lcm}_i","kind":"use","excerpt":"The class [c]^k is trivial exactly when every prime valuation ke_i is divisible by M. The least positive such k is lcm_i fracMgcd(M,e_i) = fracMgcd(M,e_1,…,e_r). This is the order of the image. The kernel size follows.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f8f22f45587f417c","tex":"\\Li","aliases":"Li","display":null,"kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":14},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cb7455ea65e17be9","tex":"\\operatorname{Li}","aliases":"Li operatorname","display":"\\operatorname{Li}","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c5fe3830f83ccdc7","tex":"\\Li_1","aliases":"Li_1 Li","display":null,"kind":"use","excerpt":"ent of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0d72fcedece51a61","tex":"\\Li_2","aliases":"Li_2 Li","display":null,"kind":"use","excerpt":"+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-dd11031bedecd830","tex":"\\operatorname{Li}_2","aliases":"Li_2 operatorname","display":"\\operatorname{Li}_2","kind":"use","excerpt":"Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β)","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-637311cc4c7b4276","tex":"\\Li_2^2","aliases":"Li_2^2 Li","display":null,"kind":"use","excerpt":"For the first two ranks, H_1=Li_1, H_2=Li_1Li_3-Li_2^2. The rank-two determinant compares genuinely different moment directions. Its order-three contact is not inferred from the presence of weight three.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b88dea8b98b4c474","tex":"\\Li_{2i+1}","aliases":"Li_2i+1 Li","display":null,"kind":"use","excerpt":"Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bb6679d149d90e69","tex":"\\Li_3","aliases":"Li_3 Li","display":null,"kind":"use","excerpt":"For the first two ranks, H_1=Li_1, H_2=Li_1Li_3-Li_2^2. The rank-two determinant compares genuinely different moment directions. Its order-three contact is not inferred from the presence of weight three.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-69d6f6e88da8767d","tex":"\\operatorname{Li}_3","aliases":"Li_3 operatorname","display":"\\operatorname{Li}_3","kind":"use","excerpt":"Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β) · = ζ(3) +frac16log^3z +fracπ^26log z -frac12log^2zlog(1-z). For 0<z<1, all three displayed arguments use their real branche","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-91cd0cf16e7602ab","tex":"\\Li_{i+j+1}","aliases":"Li_i+j+1 Li","display":null,"kind":"use","excerpt":"A nonzero Hankel contact determinant. Independent moment directions replace one filter. For an integer r≥1, define H_r(t)=det(Li_i+j+1(t))_0≤ i,j<r. It uses weights one through 2r-1. The expression is a formal series over Q and an analytic function for |t|<1. No algebraicity at a rational argument is assumed.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-619b9f182e180c46","tex":"\\Li_j","aliases":"Li_j Li","display":null,"kind":"use","excerpt":"The third complex chart has a branch boundary. For fixed rank and a consistent upper or lower boundary branch at x=c/a>1, the classical integer-weight inversion formula expresses each Li_j(x) as a fixed degree-j polynomial in a chosen log(-x) plus a sign times Li_j(1/x)~[source reference]. The latter is bounded by ζ(j) for j≥2. For j=1, use Li_1(x)","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d138b1d18c96e445","tex":"\\operatorname{Li}_n","aliases":"Li_n operatorname","display":"\\operatorname{Li}_n","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3514bbe26e565ac3","tex":"\\Li_s","aliases":"Li_s Li","display":null,"kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2a707ccad5a125ef","tex":"\\ll","aliases":"ll","display":null,"kind":"use","excerpt":"The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow b","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-969b28b0bd7940ad","tex":"\\ll_\\delta","aliases":"ll_δ ll delta","display":null,"kind":"use","excerpt":"The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow b","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bbf142bde13bd585","tex":"\\log^2","aliases":"log^2 log","display":"\\log^2","kind":"use","excerpt":"Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β) · = ζ(3) +frac16log^3z +fracπ^26log z -frac12log^2zlog(1-z). For 0<z<1, all three displayed arguments use their real branche","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-34a700d5345e5b51","tex":"\\log^3","aliases":"log^3 log","display":"\\log^3","kind":"use","excerpt":"Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β) · = ζ(3) +frac16log^3z +fracπ^26log z -frac12log^2zlog(1-z). For 0<z<1, all three displayed arguments use their real branche","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b=y^n, c=z^n with the fixed exponent eps=1. Then z^n≤ K_1rad(xyz)^2≤ K_1(xyz)^2<K_1z^6, because x,y<z. As z≥2, this implies 2^n-6<K_1 whenever n>6. Any fixed integer larger than 6+log_2max(1,K_1) e","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-53f303c8b1bf6b4c","tex":"\\log_p","aliases":"log_p log","display":"\\log_p","kind":"use","excerpt":"Exact finite-place notch. At an output prime p^eparallel c, β=1-frac ca. Since pnmid a, v_p(β-1)=e. For odd p, v_p(log_pβ)=e. The corresponding weighted dyadic identity holds on the principal branch when e≥2.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cc8b5dc0bd10c348","tex":"\\mathbf m","aliases":"m mathbf","display":"\\mathbf m","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-af9070f1457f1310","tex":"\\mathsf M","aliases":"M mathsf","display":"\\mathsf M","kind":"use","excerpt":"0, · pmid c. Consider M_a,b,c = Λ w : Z^S⟶Z^2. The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set Q=fracabcR.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in","count":59},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d9fb956038d4d543","tex":"M^{1-n}","aliases":"M^1-n","display":"M^{1-n}","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-051c4a9c687cb954","tex":"M^3","aliases":"M^3","display":"M^3","kind":"use","excerpt":"The inverse synthesis gain at weight n is M^n-1. Through weights one, two, and three, the total gain is 1· M· M^2=M^3. The displayed coordinate rescaling has determinant M^3. It has not been identified here with an actual integral motivic or period-lattice map. That identification would require the missing arithmetic realization.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d21cae7cac832b3b","tex":"M^{n-1}","aliases":"M^n-1","display":"M^{n-1}","kind":"use","excerpt":"The inverse synthesis gain at weight n is M^n-1. Through weights one, two, and three, the total gain is 1· M· M^2=M^3. The displayed coordinate rescaling has determinant M^3. It has not been identified here with an actual integral motivic or period-lattice map. That identification would require the missing arithmetic realization.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":"Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then ","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c887199df3da3b63","tex":"M_a","aliases":"M_a","display":"M_a","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9f04231559aedbad","tex":"\\mathsf M_{a,b,c}","aliases":"M_a,b,c mathsf","display":"\\mathsf M_{a,b,c}","kind":"use","excerpt":"0, · pmid c. Consider M_a,b,c = Λ w : Z^S⟶Z^2. The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set Q=fracabcR.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-284d7acc4980e163","tex":"M_b","aliases":"M_b","display":"M_b","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0f5cef98cfcf15fe","tex":"M_c","aliases":"M_c","display":"M_c","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":13},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-207e98afc142d0de","tex":"M_d","aliases":"M_d","display":"M_d","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-19afb20993a1d995","tex":"m_r","aliases":"m_r","display":"m_r","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e9aada65bb2593db","tex":"M_u","aliases":"M_u","display":"M_u","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"[Mason--Stothers in characteristic zero] Let k be an algebraically closed field of characteristic zero. Suppose A,B,C∈ k[X] are nonzero, pairwise coprime, satisfy A+B=C, and are not all constant. Then max{deg A,deg B,deg C}≤ n_0(ABC)-1. This is the polynomial abc theorem. Snyder gives a short primary proof in a form that also makes the derivative h","count":8},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records ","count":59},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4bc014492ab92312","tex":"\\min_i","aliases":"min_i min","display":"\\min_i","kind":"use","excerpt":"Because g_c≤min_i e_i≤log_2 c, we obtain the universal near-injectivity estimate |kerκ_c|≤log_2c. This closes the global-occupancy problem for the contact packet up to logarithmic loss.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-54507a99799b42ce","tex":"\\min_{k\\neq0}","aliases":"min_k≠0 min neq","display":"\\min_{k\\neq0}","kind":"use","excerpt":"For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-22899074deab8ca2","tex":"\\min_{q\\mid ab}","aliases":"min_qmid ab min mid","display":null,"kind":"use","excerpt":"r formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. Define w_p = min { d_p-r_p, min_","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e","count":34},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records ","count":100},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a13200db6a3298f9","tex":"N^{6+\\eps}","aliases":"N^6+eps eps","display":null,"kind":"use","excerpt":"The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","excerpt":"The polynomial theorem and the missing integer analogue. Roots play the role of distinct primes. For a nonzero polynomial F over an algebraically closed field, let rad(F) be the product of its distinct monic linear factors. Its degree counts distinct roots. Write n_0(F)=degrad(F).","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a87bbd35b7c2dadd","tex":"N_E","aliases":"N_E","display":"N_E","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"For a fixed sum r, this quantity is minimized when the counts differ by at most one. If n_i≥ n_j+2, moving one item from class i to class j lowers the pair count by n_i-n_j-1>0. At the balanced minimum, b_h classes have count q_h+1 and the others count q_h. The minimum is therefore (s_h-b_h)binomq_h2+b_hbinomq_h+12 =s_hbinomq_h2+b_hq_h. The first r","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a105737fecb45fb6","tex":"n_{s_h}","aliases":"n_s_h","display":"n_{s_h}","kind":"use","excerpt":"The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":16},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3c84a1672e483df5","tex":"\\mathrm{obs}","aliases":"obs mathrm","display":"\\mathrm{obs}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ab3321d6961879fc","tex":"\\ord","aliases":"ord","display":null,"kind":"use","excerpt":"[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7f0c4545d3e4be82","tex":"\\ord_0","aliases":"ord_0 ord","display":null,"kind":"use","excerpt":"[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7fa9a05b52100c5d","tex":"\\overline H","aliases":"overline H overline","display":"\\overline H","kind":"use","excerpt":"For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-39a974404cc484ff","tex":"\\overline{\\chi(h)}","aliases":"overlineχ(h) overline chi","display":"\\overline{\\chi(h)}","kind":"use","excerpt":"Character orthogonality gives the target count N(h)=frac M|G|∑_χ∈widehat Gwidehatμ(χ)overlineχ(h). The identity is finite and exact. It solves a representation question after the arithmetic channel and admissible contributions have been specified. It does not by itself bound the height of an integer triple that generated that channel.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7c40cce8c99d5a97","tex":"\\overline{\\omega_E}","aliases":"overlineω_E overline omega","display":"\\overline{\\omega_E}","kind":"use","excerpt":"automorphic analysis--synthesis identity Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area A(E) = ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ef958519c2ac9923","tex":"\\mathbb P","aliases":"P mathbb","display":"\\mathbb P","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c18367064df5a710","tex":"\\mathfrak p","aliases":"p mathfrak","display":"\\mathfrak p","kind":"use","excerpt":"persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retai","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fa15e5879160a296","tex":"\\mathbb P^1_\\Z","aliases":"P^1_Z mathbb Z","display":null,"kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a9301b691d286a0d","tex":"\\mathfrak p^2","aliases":"p^2 mathfrak","display":"\\mathfrak p^2","kind":"use","excerpt":"persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retai","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2bbf90313901a9a1","tex":"p^{C_{r,p}}","aliases":"p^C_r,p","display":"p^{C_{r,p}}","kind":"use","excerpt":"ining primes Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depe","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0aaa6739a07fa8fe","tex":"p^{d_p-r_p}","aliases":"p^d_p-r_p","display":"p^{d_p-r_p}","kind":"use","excerpt":"_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4712dba3d0909d07","tex":"p^{e-1}","aliases":"p^e-1","display":"p^{e-1}","kind":"use","excerpt":"Let p^eparallel a. Every term of d_x(a) is divisible by p^e-1. Since pnmid b, the term b d_x(a) is divisible by p^e-1. Also p^emid a, so a d_x(b) is divisible by p^e, hence by p^e-1. Therefore p^e-1mid W_x(a,b). The same argument applies to primes dividing b.","count":13},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3157e31c12869d3e","tex":"p^{e_p}","aliases":"p^e_p","display":"p^{e_p}","kind":"use","excerpt":"Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records ","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0edaee8e5b48a0f0","tex":"p^h","aliases":"p^h","display":"p^h","kind":"use","excerpt":"The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9952deedd172b29f","tex":"p^{h-1}","aliases":"p^h-1","display":"p^{h-1}","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5de6d61763e356ca","tex":"p^{h_i}","aliases":"p^h_i","display":"p^{h_i}","kind":"use","excerpt":"If h_i=v_p(k_i)>0, then k_i-i≥ p^h_i-r>a_rh_i. If h_i=0, then k_i-i≥0 and there is no denominator loss. Thus every subset other than (1,…,r) has valuation strictly above es_r. The first subset has a unit coefficient because its indices and differences are smaller than p. Its term has valuation es_r, and a unique minimum again proves equality. The s","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-44ddf871c8e79a19","tex":"p^j","aliases":"p^j","display":"p^j","kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0069d6e7d8439e52","tex":"p^{j-1}","aliases":"p^j-1","display":"p^{j-1}","kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3d4811701cdaca93","tex":"p^{\\max(e-1-v_p(\\eta_f),0)}","aliases":"p^max(e-1-v_p(η_f),0) max eta","display":"p^{\\max(e-1-v_p(\\eta_f),0)}","kind":"use","excerpt":"For the output source m=c/R_c, the corrected algebraic defect against a cyclic target of order η_f is δ_alg(c,f) = fracc/R_cgcd(c/R_c,η_f). Prime by prime, δ_alg(c,f) = ∏_p^eparallel c p^max(e-1-v_p(η_f),0). The problem is missing depth, not missing rank.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-460ce5afc42aa103","tex":"p^{\\max(e-2,0)}","aliases":"p^max(e-2,0) max","display":"p^{\\max(e-2,0)}","kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d3cdc9ecc001d6c4","tex":"p^n","aliases":"p^n","display":"p^n","kind":"use","excerpt":"Witt realization. For e≥2, the local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). Division by p identifies the source with Z_p/p^e-1Z_p, and the standard additive identification W_n(F_p)≅Z/p^nZ gives the result. For e=1, take W_0=0. The Verschiebung filtration matches multiplication by successive powers of p. This is the type","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-93d9a4bd78ce9dca","tex":"p^{r_p}","aliases":"p^r_p","display":"p^{r_p}","kind":"use","excerpt":"rallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input e","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b49eb13eda2fd7da","tex":"p^{w_p}","aliases":"p^w_p","display":"p^{w_p}","kind":"use","excerpt":"put exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid c","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7251ace6bbf384ac","tex":"p^\\delta","aliases":"p^δ delta","display":"p^\\delta","kind":"use","excerpt":"are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow by separating a finite set of small primes. For all sufficiently large primes, (1-1/p)^-1≤ p^δ and 3≤ p^δ. Every prime dividing Q","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a8b017d2e4c8ef87","tex":"p^{\\chi_p}","aliases":"p^χ_p chi","display":"p^{\\chi_p}","kind":"use","excerpt":"put exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid c","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6460407da427a490","tex":"P_{\\mathrm{act}}","aliases":"P_act mathrm","display":"P_{\\mathrm{act}}","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-48ff2d49e6a3d8a2","tex":"P_{\\mathrm{add}}","aliases":"P_add mathrm","display":"P_{\\mathrm{add}}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3270e71bf1ae54c5","tex":"P_c","aliases":"P_c","display":"P_c","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":10},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9a7264055f2d5e73","tex":"\\mathcal P_c","aliases":"P_c mathcal","display":"\\mathcal P_c","kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e4184e130d662de9","tex":"p_i^{e_i}","aliases":"p_i^e_i","display":"p_i^{e_i}","kind":"use","excerpt":"The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1a251b340cfd2fcc","tex":"p_i^{e_i-1}","aliases":"p_i^e_i-1","display":"p_i^{e_i-1}","kind":"use","excerpt":"The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-eb7979ea66ee31a5","tex":"p_m","aliases":"p_m","display":"p_m","kind":"use","excerpt":"Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-928c8c4569a33b2b","tex":"P_{\\mathrm{obs}}","aliases":"P_obs mathrm","display":"P_{\\mathrm{obs}}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7cd951ba120bfdb9","tex":"P_u","aliases":"P_u","display":"P_u","kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-99acd602923b2989","tex":"\\mathrm{Pet}","aliases":"Pet mathrm","display":"\\mathrm{Pet}","kind":"use","excerpt":"parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.","count":14},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.","count":35},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"use","excerpt":"The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e","count":15},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"bibitemBCDT C. Breuil, B. Conrad, F. Diamond, and R. Taylor, ``On the modularity of elliptic curves over Q: wild 3-adic exercises,'' Journal of the American Mathematical Society 14, no.~4 (2001), 843--939. https://doi.org/10.1090/S0894-0347-01-00370-8doi:10.1090/S0894-0347-01-00370-8.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"Quality measures the exponent in one example. Define the quality of a primitive triple by q(a,b,c)=fraclog clog R. The denominator is positive because c≥2 and R≥2. The logarithm base does not matter. The identity c=R^q is exact, although decimal qualities are approximations.","count":45},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"Weighted dyadic row. For 2^eparallel c with e≥2 and an odd input prime q, define ℓ_2,e(q) = frac14 log_2((-1)^(q-1)/2q) pmod2^e-2. With the analogous truncated valuation on Z/2^e-2Z, bar v_2(ℓ_2,e(q)) = min{e-2,v_2(q^2-1)-3}. At e=1, the output contact depth is zero and this row is omitted. A separate sign packet supplies the missing first dyadic l","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e80a32868f710c6e","tex":"q^{e-1}","aliases":"q^e-1","display":"q^{e-1}","kind":"use","excerpt":"ℓ(v_q(q_E))}. In a Tate basis, the cyclotomic graded diagonal and the inertia Kummer extension respectively force ℓ^nmid(q-1) and ℓ^nmid v_q(q_E)~[source reference]. This is a necessary condition, rather than a converse or a condition on Frobenius alone. These off-diagonal channels carry tame congruence information. They do not naturally reproduce ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1dab4320f437abad","tex":"q^{p-1}","aliases":"q^p-1","display":"q^{p-1}","kind":"use","excerpt":"te-depth row coefficient ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. The balanced relation gives ∑_qmid ab ν_qℓ_p,e(q) ≡0 pmodp^e-1. Define the truncated valuation on Z/p^e-1Z by bar v_p(0)=e-1, bar v_p(x)=min{e-1,v_p(widetilde x)} for any integer lift widetilde x of a nonzero residue. Then bar v_p(ℓ_p,e(q)) = min{e-1,v_p(q^p-1-1)-1}. Exceptional entry","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a5f4b9cf9cbba804","tex":"Q^s","aliases":"Q^s","display":"Q^s","kind":"use","excerpt":"A necessary leading cost for powers of Q. A stronger elementary obstruction measures the exponent cost of any all-triple bound on the unbalanced quantity. [Necessary cost lower bound] Fix s>0. Suppose B,A≥0 and C>0 satisfy Q^s≤ C R^B(log(3c))^A for every primitive positive triple. Then B≥2s. Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0f49a9a532d9fd3b","tex":"Q^{s_r}","aliases":"Q^s_r","display":"Q^{s_r}","kind":"use","excerpt":"[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f7ec10b45d9b6e51","tex":"\\Q^\\times","aliases":"Q^× Q times","display":null,"kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-22fd5715c5058ee4","tex":"q^{\\nu_q}","aliases":"q^ν_q nu","display":"q^{\\nu_q}","kind":"use","excerpt":"Writing β=-∏_qmid abq^ν_q, we obtain the cross-prime logarithmic filter relation log_pβ = ∑_qmid ab ν_qlog_p⟨ q⟩_p. The same depth-three path therefore has a small complex envelope and exact deep finite notches.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-22f1dad501a06ad3","tex":"q_E","aliases":"q_E","display":"q_E","kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-393f3636d9c3ffef","tex":"q_h","aliases":"q_h","display":"q_h","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8205838caa474d98","tex":"\\Q_p","aliases":"Q_p Q","display":null,"kind":"use","excerpt":"Exact local determinant valuation and denominator budget. Depletion gives an integral local matrix. Fix a prime p and an integer r≥1. Define F_s,p(t)=∑_pnmid kfract^kk^s, H_r,p(t)=det(F_i+j+1,p(t))_0≤ i,j<r. These series converge at every t∈Q_p with v_p(t)>0. They are integral power series. The Cauchy--Binet expansion from Section~[source reference","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8a9c86815659ac50","tex":"\\Q_q","aliases":"Q_q Q","display":null,"kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.","count":101},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-32fe36360874826c","tex":"\\mathfrak R","aliases":"R mathfrak","display":"\\mathfrak R","kind":"use","excerpt":"Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bdd5c27444dceacf","tex":"R^{1+2\\eps}","aliases":"R^1+2eps eps","display":null,"kind":"use","excerpt":"The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow b","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1fc80f421d2bcc4b","tex":"R^{1+\\eps}","aliases":"R^1+eps eps","display":null,"kind":"use","excerpt":"[Finiteness formulation] Conjecture~[source reference] is equivalent to the assertion that, for every eps>0, there are only finitely many primitive positive triples with q(a,b,c)>1+eps. Assume abc, using its bound with eps/2. A triple of quality above 1+eps satisfies R^1+eps<c≤ K_eps/2R^1+eps/2. Hence R^eps/2<K_eps/2. The radical is bounded. The sa","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1435585c5d2e9537","tex":"R^{1+\\eps/2}","aliases":"R^1+eps/2 eps","display":null,"kind":"use","excerpt":"[Finiteness formulation] Conjecture~[source reference] is equivalent to the assertion that, for every eps>0, there are only finitely many primitive positive triples with q(a,b,c)>1+eps. Assume abc, using its bound with eps/2. A triple of quality above 1+eps satisfies R^1+eps<c≤ K_eps/2R^1+eps/2. Hence R^eps/2<K_eps/2. The radical is bounded. The sa","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e6e8216076670e10","tex":"R^{1+\\eta}","aliases":"R^1+η eta","display":"R^{1+\\eta}","kind":"use","excerpt":"If for every η>0 one proves WC≤ C_η R_aR_bR^η, then c≤ C'_ηR^1+ηlog(3c). The same absorption argument gives abc. Conversely WC≤ M_c, so the all-eps output-sector abc inequality immediately supplies this bound. The uniform combined WC estimate is therefore the exact endpoint. A bound on C alone suffices only under independent control of W, for examp","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1b32b2618538e6e5","tex":"R^{1+\\eta_r}","aliases":"R^1+η_r eta","display":"R^{1+\\eta_r}","kind":"use","excerpt":"admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-eec1868c0d145a79","tex":"R^{1/3}","aliases":"R^1/3","display":"R^{1/3}","kind":"use","excerpt":"A known unconditional bound. Stewart and Yu proved an effective bound of a different scale~[source reference]: there is an effectively computable absolute C>0 such that every primitive positive triple satisfies c<exp (CR^1/3(log R)^3). It controls the height in terms of the radical unconditionally. Its exponential dependence is much larger than the","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"Since |log(b/a)|≤log c and log(c/a)≤log c, every entry has a bound by a constant times (1+log c)^j. A determinant term has total weight ∑_i=0^r-1(i+π(i)+1)=r^2. Therefore | H_r(c/a)|≤ C_r(1+log c)^r^2 for each of those consistent branches. The constant may depend on r. No nonvanishing assertion at every argument in this third chart follows from the","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0b8e43abdf81fe1d","tex":"R^B","aliases":"R^B","display":"R^B","kind":"use","excerpt":"A necessary leading cost for powers of Q. A stronger elementary obstruction measures the exponent cost of any all-triple bound on the unbalanced quantity. [Necessary cost lower bound] Fix s>0. Suppose B,A≥0 and C>0 satisfy Q^s≤ C R^B(log(3c))^A for every primitive positive triple. Then B≥2s. Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-588a2d9dad78809f","tex":"R^{E_r}","aliases":"R^E_r","display":"R^{E_r}","kind":"use","excerpt":"[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fb58157f40a3c154","tex":"R^\\eps","aliases":"R^eps eps","display":null,"kind":"use","excerpt":"ulations The form best adapted to this research separates the repeated depth of the output from the input prime support. [Exact normalization] For every primitive positive triple, frac cR=fracM_cR_aR_b, Q:=fracabcR, frac Qab=frac cR. Consequently Conjecture~[source reference] is exactly equivalent to each all-eps assertion M_c≤ K_eps R_aR_bR^eps, Q","count":11},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f9ec6aa8d569e532","tex":"R^{\\eps/2}","aliases":"R^eps/2 eps","display":null,"kind":"use","excerpt":"[Finiteness formulation] Conjecture~[source reference] is equivalent to the assertion that, for every eps>0, there are only finitely many primitive positive triples with q(a,b,c)>1+eps. Assume abc, using its bound with eps/2. A triple of quality above 1+eps satisfies R^1+eps<c≤ K_eps/2R^1+eps/2. Hence R^eps/2<K_eps/2. The radical is bounded. The sa","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2eee3ce95fb6aa1d","tex":"R^{\\mathrm{glob}}","aliases":"R^glob mathrm","display":"R^{\\mathrm{glob}}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0209bbddf78e3b9b","tex":"R^{o(1)}","aliases":"R^o(1)","display":"R^{o(1)}","kind":"use","excerpt":"The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a456275e0ed5010c","tex":"R^q","aliases":"R^q","display":"R^q","kind":"use","excerpt":"Quality measures the exponent in one example. Define the quality of a primitive triple by q(a,b,c)=fraclog clog R. The denominator is positive because c≥2 and R≥2. The logarithm base does not matter. The identity c=R^q is exact, although decimal qualities are approximations.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e070129010f80903","tex":"R^\\delta","aliases":"R^δ delta","display":"R^\\delta","kind":"use","excerpt":"The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow b","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c3bfeaf0a95fd5d3","tex":"R^\\eta","aliases":"R^η eta","display":"R^\\eta","kind":"use","excerpt":"[Counterfamily for the unbalanced target] For every fixed 0≤η<1 and A≥0, no constant K makes Q≤ K R^η(log(3c))^A valid for all primitive positive triples. Take a=1, b=2^k, c=2^k+1 for integers k≥1. The triple is primitive and c is odd. Thus R=2rad(c)≤2c≤3·2^k, Q=2^k-1frac crad(c)≥2^k-1. Consequently fracQR^η(log(3c))^A ≥frac2^(1-η)k-13^η(log(3(2^k+","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-33462a02fcce1a84","tex":"R_a","aliases":"R_a","display":"R_a","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":18},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fb14b1a407e1de8e","tex":"r_a","aliases":"r_a","display":"r_a","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-559e0bf6a6b7bef3","tex":"\\mathfrak R_{a,b,c}","aliases":"R_a,b,c mathfrak","display":"\\mathfrak R_{a,b,c}","kind":"use","excerpt":"Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-baf9ce2c27891e4e","tex":"R_b","aliases":"R_b","display":"R_b","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":18},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4bdf782a5897d64d","tex":"r_b","aliases":"r_b","display":"r_b","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-37f2bdf109707126","tex":"R_c","aliases":"R_c","display":"R_c","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":15},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-567e1106095a8a61","tex":"r_c","aliases":"r_c","display":"r_c","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"use","excerpt":"Because b_n contains 3^n+1, its powerful part contains 3^n. Thus rad(b_n)≤ fracb_n3^n, R_n=2rad(b_n)≤frac2b_n3^n. Therefore fracc_nR_n≥frac3^n c_n2b_n>frac3^n2⟶∞. The displayed lower bound does not contradict the positive-eps conjecture. It does not exhibit infinitely many triples with a fixed positive quality gap. The output grows doubly exponenti","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-99bbe0f6af468b02","tex":"R_u","aliases":"R_u","display":"R_u","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7a780768179287f0","tex":"r_u","aliases":"r_u","display":"r_u","kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-79ff2dd3f60f35b4","tex":"\\operatorname{rad}","aliases":"rad operatorname","display":"\\operatorname{rad}","kind":"use","excerpt":"The polynomial theorem and the missing integer analogue. Roots play the role of distinct primes. For a nonzero polynomial F over an algebraically closed field, let rad(F) be the product of its distinct monic linear factors. Its degree counts distinct roots. Write n_0(F)=degrad(F).","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-650ae3274f788b89","tex":"\\rad","aliases":"rad","display":null,"kind":"use","excerpt":"The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.","count":25},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1)","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7fdf44047d9358ee","tex":"\\mathrm{red}","aliases":"red mathrm","display":"\\mathrm{red}","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1)","count":29},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-785e812c9b93033d","tex":"S_a","aliases":"S_a","display":"S_a","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ba2c2b6914e4fdfc","tex":"S_{ab}","aliases":"S_ab","display":"S_{ab}","kind":"use","excerpt":"The signed input exponent signal. Let S_ab={q:qmid ab}. Define ν_q=v_q(b)-v_q(a). Then -frac ba=-∏_qmid abq^ν_q. For every output prime pmid c, the same input exponent packet produces the deep principal unit 1-frac ca. The output persistence is therefore a simultaneous cross-prime cancellation phenomenon.","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-68387971995ff129","tex":"S_b","aliases":"S_b","display":"S_b","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8ec8b0faab9c7fd8","tex":"S_c","aliases":"S_c","display":"S_c","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-508e9a069e81ec8b","tex":"S_d","aliases":"S_d","display":"S_d","kind":"use","excerpt":"Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1)","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-61d46fc74a29e50b","tex":"s_h","aliases":"s_h","display":"s_h","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fdb9a531253b9339","tex":"S_{p,r}","aliases":"S_p,r","display":"S_{p,r}","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a6bf5eb7ecfc145e","tex":"s_r","aliases":"s_r","display":"s_r","kind":"use","excerpt":"[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each ","count":19},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5eac484be3cd94a9","tex":"\\mathrm{sh}","aliases":"sh mathrm","display":"\\mathrm{sh}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fe9982bf1835e2aa","tex":"\\SNF","aliases":"SNF","display":null,"kind":"use","excerpt":"[Polyphase Smith normal form] If gcd(e,N)=1, then SNF(C_e,N) = diag(1,…,1,e).","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-327dcbd8f0058da9","tex":"\\Spec","aliases":"Spec","display":null,"kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-59d24711aca21962","tex":"\\mathrm{st}","aliases":"st mathrm","display":"\\mathrm{st}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1b507451586c4dbe","tex":"\\sup_{h\\ge1}","aliases":"sup_h≥1 sup ge","display":"\\sup_{h\\ge1}","kind":"use","excerpt":"An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":19},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-adac703e9d252af9","tex":"\\mathbb T","aliases":"T mathbb","display":"\\mathbb T","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d589f35b0c8bc738","tex":"\\mathbf T","aliases":"T mathbf","display":"\\mathbf T","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","excerpt":"[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"sidues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"ng cost for powers of Q A stronger elementary obstruction measures the exponent cost of any all-triple bound on the unbalanced quantity. [Necessary cost lower bound] Fix s>0. Suppose B,A≥0 and C>0 satisfy Q^s≤ C R^B(log(3c))^A for every primitive positive triple. Then B≥2s. Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)^d for t≥1. The in","count":102},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":"Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8c6354f5f56778af","tex":"t^5","aliases":"t^5","display":"t^5","kind":"use","excerpt":"Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f544e1fddefb59a0","tex":"t^d","aliases":"t^d","display":"t^d","kind":"use","excerpt":"ng cost for powers of Q A stronger elementary obstruction measures the exponent cost of any all-triple bound on the unbalanced quantity. [Necessary cost lower bound] Fix s>0. Suppose B,A≥0 and C>0 satisfy Q^s≤ C R^B(log(3c))^A for every primitive positive triple. Then B≥2s. Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)^d for t≥1. The in","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-964fcf8c1f59b656","tex":"T^e","aliases":"T^e","display":"T^e","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9b7f9d17ba2c44f5","tex":"T^{e-1}","aliases":"T^e-1","display":"T^{e-1}","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d241eb52ee3cfa72","tex":"t^k","aliases":"t^k","display":"t^k","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":10},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f38b757e0d8e7262","tex":"t^{k_1+\\cdots+k_r}","aliases":"t^k_1+⋯+k_r cdots","display":"t^{k_1+\\cdots+k_r}","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-23adffbfecf53c1f","tex":"t^{k_i}","aliases":"t^k_i","display":"t^{k_i}","kind":"use","excerpt":"i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3bdd78b0155001c4","tex":"t^{m+1}","aliases":"t^m+1","display":"t^{m+1}","kind":"use","excerpt":"[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ac72b989bba7a709","tex":"T^N","aliases":"T^N","display":"T^N","kind":"use","excerpt":"Finite polyphase rings. Fix an integer N≥2 and work in A_N=Z[T]/(T^N-1). The residue of V_e(T) records the number of valuation shells in each congruence class modulo N. Multiplication by V_e is represented by a circulant matrix C_e,N.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e742e85c425d1bc4","tex":"t^{s_r}","aliases":"t^s_r","display":"t^{s_r}","kind":"use","excerpt":"[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4a7bd9c09ab323ef","tex":"T^{v_p(m)}","aliases":"T^v_p(m)","display":"T^{v_p(m)}","kind":"use","excerpt":"nt v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n∂","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4a9e32fb56cb7f6c","tex":"t_a","aliases":"t_a","display":"t_a","kind":"use","excerpt":"sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct","count":15},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bc902e144c8dfe79","tex":"\\mathcal T_{a,b,c}","aliases":"T_a,b,c mathcal","display":"\\mathcal T_{a,b,c}","kind":"use","excerpt":"[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0a30bcd98cf55af8","tex":"t_a^{(0)}","aliases":"t_a^(0)","display":"t_a^{(0)}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","excerpt":"sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct","count":14},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b4b7de788cf0c385","tex":"t_b^{(0)}","aliases":"t_b^(0)","display":"t_b^{(0)}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-04db525251126e68","tex":"t_c","aliases":"t_c","display":"t_c","kind":"use","excerpt":"sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct","count":10},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8dc401ec678ef127","tex":"t_c^{(0)}","aliases":"t_c^(0)","display":"t_c^{(0)}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f00f567e4a81164c","tex":"\\mathbb T_f","aliases":"T_f mathbb","display":"\\mathbb T_f","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-81549140e4fc02bc","tex":"t_p","aliases":"t_p","display":"t_p","kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8decad66f8014b7e","tex":"T_{p,r}","aliases":"T_p,r","display":"T_{p,r}","kind":"use","excerpt":"[First-order no-free-polyphase principle] Let S:(T_p,r)_p,r⟼ (∑_rT_p,r)_p be shell synthesis. The first-order additive map and first-order Wronskian factor through S. Hence Ker S is contained in the kernels of both first-order maps.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-df3f310abe33eba9","tex":"T_r","aliases":"T_r","display":"T_r","kind":"use","excerpt":"First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6d6c4f072fd1f994","tex":"T_s","aliases":"T_s","display":"T_s","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c9ae0c36264af704","tex":"t_u","aliases":"t_u","display":"t_u","kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8dd594b4bce51346","tex":"\\mathrm{tors}","aliases":"tors mathrm","display":"\\mathrm{tors}","kind":"use","excerpt":"[Smith obstruction theorem] Let a+b=c be primitive, with at least two of a,b,c greater than one. Then SNF( M_a,b,c) = diag(1,Q). Consequently, coker( M_a,b,c)_tors ≅ Z/QZ. The exceptional triple (1,1,2) has only one nonempty prime sector, so the matrix has rank one and is excluded from the two-row statement.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[Formal affine-space theorem] Let d be the number of shell variables. After a unimodular integral change of shell coordinates, the ( T)-adic completion of Z[ T]/(F_a,b,c) at the zero section is isomorphic to a formal power-series ring Z[[U_2,…,U_d]].","count":17},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc","count":40},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"[Unitary no-gain principle] Let U be unitary on the shell coordinate of every active prime channel. Then normUΨ_n_2=normΨ_n_2. Transport the activation projector to UP_actU^-1. Its projected energy then also agrees exactly with the original activation energy. The same holds after exact synthesis. An arbitrary unitary transform need not preserve the","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"re of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). The bracket is a unit mod","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0e88dc7b57e42bdd","tex":"u^M","aliases":"u^M","display":"u^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-76bfead452deff02","tex":"U_2","aliases":"U_2","display":"U_2","kind":"use","excerpt":"[Formal affine-space theorem] Let d be the number of shell variables. After a unimodular integral change of shell coordinates, the ( T)-adic completion of Z[ T]/(F_a,b,c) at the zero section is isomorphic to a formal power-series ring Z[[U_2,…,U_d]].","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1361a2562f51cd0c","tex":"U_{ab}","aliases":"U_ab","display":"U_{ab}","kind":"use","excerpt":"Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0b3db69758fee0b4","tex":"U_d","aliases":"U_d","display":"U_d","kind":"use","excerpt":"[Formal affine-space theorem] Let d be the number of shell variables. After a unimodular integral change of shell coordinates, the ( T)-adic completion of Z[ T]/(F_a,b,c) at the zero section is isomorphic to a formal power-series ring Z[[U_2,…,U_d]].","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"p_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gives one quotient equation ∑_i z_i=q(t). For a box radius H, each z_i lies in the interval ℓ_i(H) = lceilfrac-H-ξ_","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a822fb370eb1244d","tex":"u_r","aliases":"u_r","display":"u_r","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"thm:prony The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=","count":16},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0a333431c75f50a1","tex":"V^{\\mathsf T}","aliases":"V^ T mathsf","display":"V^{\\mathsf T}","kind":"use","excerpt":"sidues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-87f0c3c75c88d8b5","tex":"v^M","aliases":"v^M","display":"v^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"Weighted dyadic row. For 2^eparallel c with e≥2 and an odd input prime q, define ℓ_2,e(q) = frac14 log_2((-1)^(q-1)/2q) pmod2^e-2. With the analogous truncated valuation on Z/2^e-2Z, bar v_2(ℓ_2,e(q)) = min{e-2,v_2(q^2-1)-3}. At e=1, the output contact depth is zero and this row is omitted. A separate sign packet supplies the missing first dyadic l","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"s false. There is an elementary family that proves this without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku wit","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f7fd62f7cd34cb10","tex":"V_e","aliases":"V_e","display":"V_e","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":9},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-74b99f2f5e5d09ac","tex":"V_{e+f}","aliases":"V_e+f","display":"V_{e+f}","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4f979281d0af8ba8","tex":"V_f","aliases":"V_f","display":"V_f","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0a435f82d4918621","tex":"v_k","aliases":"v_k","display":"v_k","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4f2f59c5a52860f8","tex":"v_k^{\\mathsf T}","aliases":"v_k^ T mathsf","display":"v_k^{\\mathsf T}","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-724afc19e77e7bfd","tex":"v_q","aliases":"v_q","display":"v_q","kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-aee140b15026b684","tex":"V_{v_p(n)}","aliases":"V_v_p(n)","display":"V_{v_p(n)}","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fa83faab1d44c512","tex":"v_\\ell","aliases":"v_ℓ ell","display":"v_\\ell","kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-694bdc6e24088dc1","tex":"\\mathrm{val}","aliases":"val mathrm","display":"\\mathrm{val}","kind":"use","excerpt":"Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Set W=A'B-AB'. It is nonzero. Indeed, if it vanished, then (A/B)'=0, and characteristic zero would force the rational function A/B to be constant. Coprimality would then force both A and B to be constant, and so also C.","count":56},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1c2a9b04e6197b99","tex":"\\mathsf w","aliases":"w mathsf","display":"\\mathsf w","kind":"use","excerpt":"A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ","count":8},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Arithmetic correction and arbitrary precision. Some large high-quality triples exceed signed 64-bit range, and intermediate quantities such as ab, fracabcR, w· x can overflow even when the input terms individually fit. Every public arithmetic entry point must therefore promote integer-like inputs to arbitrary-precision integers before multiplicatio","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"Witt realization. For e≥2, the local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). Division by p identifies the source with Z_p/p^e-1Z_p, and the standard additive identification W_n(F_p)≅Z/p^nZ gives the result. For e=1, take W_0=0. The Verschiebung filtration matches multiplication by successive powers of p. This is the type","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d01794b5f0c3ada0","tex":"W_1","aliases":"W_1","display":"W_1","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f38b6368252abbb5","tex":"W_{e-1}","aliases":"W_e-1","display":"W_{e-1}","kind":"use","excerpt":"Witt realization. For e≥2, the local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). Division by p identifies the source with Z_p/p^e-1Z_p, and the standard additive identification W_n(F_p)≅Z/p^nZ gives the result. For e=1, take W_0=0. The Verschiebung filtration matches multiplication by successive powers of p. This is the type","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ffbde798b9a7c58f","tex":"W_j","aliases":"W_j","display":"W_j","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-466a54f902244928","tex":"W_n","aliases":"W_n","display":"W_n","kind":"use","excerpt":"Witt realization. For e≥2, the local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). Division by p identifies the source with Z_p/p^e-1Z_p, and the standard additive identification W_n(F_p)≅Z/p^nZ gives the result. For e=1, take W_0=0. The Verschiebung filtration matches multiplication by successive powers of p. This is the type","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d97ea8d524e0d0e0","tex":"\\mathsf w_p","aliases":"w_p mathsf","display":"\\mathsf w_p","kind":"use","excerpt":"A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2b65a86b582ae995","tex":"w_p","aliases":"w_p","display":"w_p","kind":"use","excerpt":"r formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. Define w_p = min { d_p-r_p, min_","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4a4f3028bb492bb2","tex":"W_r","aliases":"W_r","display":"W_r","kind":"use","excerpt":"[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0c8fc197ea740581","tex":"W_{r,K}","aliases":"W_r,K","display":"W_{r,K}","kind":"use","excerpt":"For a reduced rational argument t=a/c, an obvious integral scalar is W_r,K(a,c)=c^dA_r,K(a/c). It is nonzero when 0<a<c, since the truncated positive expansion is nonzero. Thus it is a genuine nonzero arithmetic candidate. The question is whether it retains the required output-prime contact after clearing denominators.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-36d884ad81c0ef6a","tex":"W_x","aliases":"W_x","display":"W_x","kind":"use","excerpt":"The arithmetic Wronskian. Define W_x(a,b) =a d_x(b)-b d_x(a). Writing μ_p= -dfracv_p(a)p, · pmid a,","count":12},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-27826ef365351b0c","tex":"W_y","aliases":"W_y","display":"W_y","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"automorphic analysis--synthesis identity Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area A(E) = ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4ae2c725a0324b8f","tex":"\\widehat G","aliases":"widehat G widehat","display":"\\widehat G","kind":"use","excerpt":"Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b0d65d8f4ac93228","tex":"\\widehat T","aliases":"widehat T widehat","display":"\\widehat T","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0b1d837601c3d673","tex":"\\widehat T_{-j}","aliases":"widehat T_-j widehat","display":"\\widehat T_{-j}","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-28dd0e7bd21d34c5","tex":"\\widehat T_0","aliases":"widehat T_0 widehat","display":"\\widehat T_0","kind":"use","excerpt":"onSecond-order phase energy At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f387d54db2d29aed","tex":"\\widehat T_0^2","aliases":"widehat T_0^2 widehat","display":"\\widehat T_0^2","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-284097439a5a6f5c","tex":"\\widehat T_j","aliases":"widehat T_j widehat","display":"\\widehat T_j","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b186d96cd6172b72","tex":"\\widehat\\mu","aliases":"widehatμ widehat mu","display":"\\widehat\\mu","kind":"use","excerpt":"Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c77170c848a2e8b4","tex":"\\widetilde x","aliases":"widetilde x widetilde","display":"\\widetilde x","kind":"use","excerpt":"Odd logarithmic rows. For odd p^eparallel c, define the finite-depth row coefficient ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. The balanced relation gives ∑_qmid ab ν_qℓ_p,e(q) ≡0 pmodp^e-1. Define the truncated valuation on Z/p^e-1Z by bar v_p(0)=e-1, bar v_p(x)=min{e-1,v_p(widetilde x)} for any integer lift widetilde x of a nonzero residue. Then ba","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"his without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^","count":85},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-793998b5f63ba05c","tex":"x^{(a)}","aliases":"x^(a)","display":"x^{(a)}","kind":"use","excerpt":"For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"use","excerpt":"re of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). The bracket is a unit mod","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-38ee95395ade7e66","tex":"X^M","aliases":"X^M","display":"X^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e189c65b70ed9178","tex":"X^n","aliases":"X^n","display":"X^n","kind":"use","excerpt":"A sharp example. Take A=X^n, B=1-X^n, C=1 for n≥1. The root sets are disjoint. There is one distinct root of A and n distinct roots of B, so n_0(ABC)=n+1. The maximal degree is n, attaining the theorem's bound. In this example W=nX^n-1, and the repeated-root divisor is exactly X^n-1 up to a nonzero scalar.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d8c5e1feefd58efd","tex":"x^n","aliases":"x^n","display":"x^n","kind":"use","excerpt":"ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-646e43ff9cbc8279","tex":"X^{n-1}","aliases":"X^n-1","display":"X^{n-1}","kind":"use","excerpt":"A sharp example. Take A=X^n, B=1-X^n, C=1 for n≥1. The root sets are disjoint. There is one distinct root of A and n distinct roots of B, so n_0(ABC)=n+1. The maximal degree is n, attaining the theorem's bound. In this example W=nX^n-1, and the repeated-root divisor is exactly X^n-1 up to a nonzero scalar.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-baf58144b77f8bed","tex":"X^p","aliases":"X^p","display":"X^p","kind":"use","excerpt":"The proof needs an actual nonzero derivative. In positive characteristic, X^p+(1-X^p)=1 has every derivative zero, so the characteristic-zero statement cannot simply be transferred. The correct positive-characteristic forms include a condition excluding common Frobenius powers.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6dc0a2643d887c2c","tex":"X^s","aliases":"X^s","display":"X^s","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"Obstruction cosets. Exclude (1,1,2) throughout this section. Choose a section x_0∈ L with ω(x_0)=1. Then L = bigsqcup_k∈Z (kx_0+L_0). For fixed k, the problem is a closest-vector problem to the moving target -kx_0 in the lattice L_0. The three-sector compression makes this search much more explicit.","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"The minimality is elementary. At height at most five, |x_3+75x_5|≤380<448, so additivity forces x_2=0. Then x_3=-75x_5 and the same height restriction force both coordinates to vanish. Thus no transverse packet has height below six, and the displayed packet proves Θ=6.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"The minimality is elementary. At height at most five, |x_3+75x_5|≤380<448, so additivity forces x_2=0. Then x_3=-75x_5 and the same height restriction force both coordinates to vanish. Thus no transverse packet has height below six, and the displayed packet proves Θ=6.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-98391915cbb7346b","tex":"x_5","aliases":"x_5","display":"x_5","kind":"use","excerpt":"The minimality is elementary. At height at most five, |x_3+75x_5|≤380<448, so additivity forces x_2=0. Then x_3=-75x_5 and the same height restriction force both coordinates to vanish. Thus no transverse packet has height below six, and the displayed packet proves Θ=6.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-53687bf254c09aec","tex":"x_p","aliases":"x_p","display":"x_p","kind":"use","excerpt":"A scalar packet is an integer vector x=(x_p)_p∈ S∈Z^S. Pairing with x gives the arithmetic derivative d_x(n) = n∑_pmid nfracv_p(n)p x_p. The Leibniz rule survives scalarization.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"Let I_N=ker(A_Nxrightarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y","count":16},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"Elliptic curves and the modified Szpiro comparison. The marked Frey curve E:y^2=x(x-a)(x+b) translates the additive triple into an elliptic curve. Its discriminant sees repeated depth, while its conductor sees prime support, with local model corrections at the small primes. This makes elliptic height estimates a natural companion to abc.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-798bab1eeb3041a1","tex":"Y^M","aliases":"Y^M","display":"Y^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-de7ba1dbdac1562b","tex":"y^n","aliases":"y^n","display":"y^n","kind":"use","excerpt":"ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7a4d74bf293c7c99","tex":"y_p","aliases":"y_p","display":"y_p","kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5d14d79ba80e05b7","tex":"y_{p,r}","aliases":"y_p,r","display":"y_{p,r}","kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y","count":26},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ddecadd3df3a219e","tex":"\\Z^2","aliases":"Z^2 Z","display":null,"kind":"use","excerpt":"0, · pmid c. Consider M_a,b,c = Λ w : Z^S⟶Z^2. The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set Q=fracabcR.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6dedd41de3fd2c9b","tex":"z^6","aliases":"z^6","display":"z^6","kind":"use","excerpt":"al large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b=y^n, c=z^n with the fixed exponent eps=1. Then ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"use","excerpt":"[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-632795a97bd80b33","tex":"Z^M","aliases":"Z^M","display":"Z^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8b55f569822030f0","tex":"\\Z^n","aliases":"Z^n Z","display":null,"kind":"use","excerpt":"For a primitive rank-one vector v∈Z^n, the orthogonal projector denominator is ⟨ v,v⟩, and the congruence quotient is cyclic of that order.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"use","excerpt":"ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-10d0329c35407a66","tex":"z^r","aliases":"z^r","display":"z^r","kind":"use","excerpt":"[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e67551e8d1a79560","tex":"\\Z^{r+1}","aliases":"Z^r+1 Z","display":null,"kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f4d3cabb397a3cc8","tex":"\\Z^S","aliases":"Z^S Z","display":null,"kind":"use","excerpt":"A scalar packet is an integer vector x=(x_p)_p∈ S∈Z^S. Pairing with x gives the arithmetic derivative d_x(n) = n∑_pmid nfracv_p(n)p x_p. The Leibniz rule survives scalarization.","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bc798b14e6021234","tex":"\\Z^s","aliases":"Z^s Z","display":null,"kind":"use","excerpt":"Why abstract lattice geometry is insufficient. Let v∈Z^s be primitive, with s≥2. For every C≥2, there exists a primitive row gpmod C satisfying g· v≡0pmod C. Thus a fixed short vector can lie in primitive congruence lattices of arbitrarily large index.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4ed5de31ab0a09ef","tex":"\\Z^{S_{ab}}","aliases":"Z^S_ab Z","display":null,"kind":"use","excerpt":"satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π iCg· x). Its kernel is the adelic coherence lattice Λ_coh = {x∈Z^S_ab:g· x≡0pmod C}. Since g is primitive modulo C","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-78a80f694d80e6e7","tex":"\\Z_2","aliases":"Z_2 Z","display":null,"kind":"use","excerpt":"At odd p, the p-adic logarithm identifies (1+pZ_p)/(1+p^eZ_p) ≅ pZ_p/p^eZ_p. At 2, the additive contact packet remains uniform, while the multiplicative coordinate begins on 1+4Z_2 and therefore carries the familiar weighted dyadic shift.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-86298b26f7dd1fb4","tex":"z_i","aliases":"z_i","display":"z_i","kind":"use","excerpt":"ix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gi","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b910972bdb78c268","tex":"\\Z_p","aliases":"Z_p Z","display":null,"kind":"use","excerpt":"The depleted local series. For a prime p, delete the terms whose index is divisible by p: Φ_m,p(t)=(-1)^m∑_pnmid kbinomkm+1fract^kk^m+2∈Z_p[[t]]. It converges when v_p(t)>0. [Exact local jet valuation] If v_p(t)=e≥1, then v_p(Φ_m,p(t))= (m+1)e, · pnmid m+1,","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e","count":35},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"Sparsity forces transverse energy. Let E={x:N(x)>0} and r=|E|. Always r≤ M. Parseval yields ∑_χ∈widehat G|widehatμ(χ)|^2 =frac|G|M^2∑_xN(x)^2. Cauchy--Schwarz on the occupied support gives ∑_xN(x)^2≥ M^2/r. Hence [Sparse energy barrier] ∑_χ≠1|widehatμ(χ)|^2 ≥frac|G|r-1≥frac|G|M-1. When few assignment atoms lie in a large group, substantial nontrivi","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3e3225555f03b1a7","tex":"|^3","aliases":"|^3","display":"|^3","kind":"use","excerpt":"The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"Rows restricted to one sector are proportional, so same-sector 2×2 minors vanish. If p and q lie in two different nonempty sectors, the corresponding minor is ±fracabcpq = ± QfracRpq. Every nonzero minor is therefore divisible by Q. Conversely, for each prime ℓmid R, choose p=ℓ and choose q in a different nonempty sector. Since R is squarefree, R/(","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrang","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"The analogy is direct. Prime powers p^e become repeated roots (X-α)^e. A derivative lowers the multiplicity by one. The divisor left over after that operation is F/rad(F), just as n/rad(n) is the repeated-power part of an integer.","count":18},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-dee4fa3ecc4d33d7","tex":"\\alpha_h","aliases":"α_h alpha","display":"\\alpha_h","kind":"use","excerpt":"Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with v","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-80f45a66a06b79f2","tex":"\\alpha_u","aliases":"α_u alpha","display":"\\alpha_u","kind":"use","excerpt":"Phase and coarse separation. The local target has the form t_u(k,n)=α_uk+r_un. For a local prime pmid r_u, t_u(k,n)≡α_ukpmod p. Thus the obstruction multiple k determines the finite phase packet. The coarse parameter n changes only the quotient load. The exact search is a moving polyphase mechanism rather than a generic mixed-integer program.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c26b0386db57f0c4","tex":"\\alpha_x","aliases":"α_x alpha","display":"\\alpha_x","kind":"use","excerpt":"Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1)","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4d8d33d3ce2c108a","tex":"\\alpha_x^i","aliases":"α_x^i alpha","display":"\\alpha_x^i","kind":"use","excerpt":"thm:prony The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2ccda6c120fc6541","tex":"\\alpha_x^k","aliases":"α_x^k alpha","display":"\\alpha_x^k","kind":"use","excerpt":"Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1)","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-18d68840d1fdac68","tex":"\\alpha_\\ell","aliases":"α_ℓ alpha ell","display":"\\alpha_\\ell","kind":"use","excerpt":"Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then ","count":12},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8ed4bfa870aa1639","tex":"\\beta^{1/K}","aliases":"β^1/K beta","display":"\\beta^{1/K}","kind":"use","excerpt":"Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"ve hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=overline H k_*Γ for the optimal obstruction multiple k_* and an integrality factor","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"ic COS and the sparse spectral leaf What the finite assignment signal records. The spectral branch of the project asks which prime assignments realize a prescribed residue. Its exact object is a finite distribution. Fix a finite abelian group G and s eligible variable support primes. At the jth prime choose one of three permitted contributions, rep","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-51cadb8120940875","tex":"\\gamma_j","aliases":"γ_j gamma","display":"\\gamma_j","kind":"use","excerpt":"ic COS and the sparse spectral leaf What the finite assignment signal records. The spectral branch of the project asks which prime assignments realize a prescribed residue. Its exact object is a finite distribution. Fix a finite abelian group G and s eligible variable support primes. At the jth prime choose one of three permitted contributions, rep","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"This equivalence is cited as an external theorem. We do not derive its full small-prime and minimal-model accounting here. The ordinary statement involving only |Δ_min| should not be silently substituted for the max-invariant formulation. A proposed project route through the Frey curve must preserve the marked integral model and the normalization o","count":9},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","excerpt":"The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9db778080ee52c01","tex":"\\delta_{\\mathrm{alg}}","aliases":"δ_alg delta mathrm","display":"\\delta_{\\mathrm{alg}}","kind":"use","excerpt":"For the output source m=c/R_c, the corrected algebraic defect against a cyclic target of order η_f is δ_alg(c,f) = fracc/R_cgcd(c/R_c,η_f). Prime by prime, δ_alg(c,f) = ∏_p^eparallel c p^max(e-1-v_p(η_f),0). The problem is missing depth, not missing rank.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1d6bb66bd4727edf","tex":"\\delta_c","aliases":"δ_c delta","display":"\\delta_c","kind":"use","excerpt":"The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0ba612a4176bddf5","tex":"\\Delta_E","aliases":"Δ_E Delta","display":"\\Delta_E","kind":"use","excerpt":"The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discrimina","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-608581f0be0c4b00","tex":"\\Delta_h","aliases":"Δ_h Delta","display":"\\Delta_h","kind":"use","excerpt":"A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrang","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e6c0ef7533b3c51e","tex":"\\Delta_{\\min}","aliases":"Δ_min Delta min","display":"\\Delta_{\\min}","kind":"use","excerpt":"This equivalence is cited as an external theorem. We do not derive its full small-prime and minimal-model accounting here. The ordinary statement involving only |Δ_min| should not be silently substituted for the max-invariant formulation. A proposed project route through the Frey curve must preserve the marked integral model and the normalization o","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-90d5ba069a82a623","tex":"\\Delta_{uh}","aliases":"Δ_uh Delta","display":"\\Delta_{uh}","kind":"use","excerpt":"A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrang","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2930352759bad3e1","tex":"\\varepsilon'","aliases":"ε' varepsilon","display":"\\varepsilon'","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b3de5d9511fcd1b4","tex":"\\varepsilon_j","aliases":"ε_j varepsilon","display":"\\varepsilon_j","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0b5701ed0b92c9e3","tex":"\\zeta^e","aliases":"ζ^e zeta","display":"\\zeta^e","kind":"use","excerpt":"Let I_N=ker(A_Nxrightarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7434f23c339ba4fb","tex":"\\zeta^N","aliases":"ζ^N zeta","display":"\\zeta^N","kind":"use","excerpt":"Let I_N=ker(A_Nxrightarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5048985144717904","tex":"\\zeta_M","aliases":"ζ_M zeta","display":"\\zeta_M","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5d39f2d87876b475","tex":"\\zeta_M^j","aliases":"ζ_M^j zeta","display":"\\zeta_M^j","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e6c934809237f301","tex":"\\zeta_n","aliases":"ζ_n zeta","display":"\\zeta_n","kind":"use","excerpt":"A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrang","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c85cee5084c73471","tex":"\\eta_f","aliases":"η_f eta","display":"\\eta_f","kind":"use","excerpt":"The same-sign separation problem. After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let Λ be an integral Hecke lattice and let e_f be the rational projector onto the Frey eigenline. Define η_f = min{m≥1:m e_f(Λ)⊆Λ}. This is the integral eigenchannel separation denominator. It is an arithmetic analogue of ","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-db72e7f4d740d159","tex":"\\eta_r","aliases":"η_r eta","display":"\\eta_r","kind":"use","excerpt":"admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9359c9fe8689c79c","tex":"\\eta_U","aliases":"η_U eta","display":"\\eta_U","kind":"use","excerpt":"[Integral projector theorem] The least positive integer η_U satisfying η_U e_U(Λ)⊆Λ is the exponent of the finite quotient C_U.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"The nine canonical implementation groups also reproduced their archived values, after two disclosed filesystem path adaptations. The mathematical code was unchanged. At 3+125=128, the weighted derivative minimum is Θ=6, while the exponent-free moving-search minimum is Φ=8. They optimize different packets. The latter has obstruction multiple one and","count":11},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9066618856182e83","tex":"\\Theta_3","aliases":"Θ_3 Theta","display":"\\Theta_3","kind":"use","excerpt":"Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose that for every primitive positive triple there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) such that, for every eps>0, constants independent of the triple give |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-071c8af44ea8db86","tex":"\\Theta_f","aliases":"Θ_f Theta","display":"\\Theta_f","kind":"use","excerpt":"[Conditional filtered-transfer criterion] Suppose every primitive positive triple admits a canonical map H^0(Θ_f):Z/(c/R_c)Z⟶ H^0( C_f) satisfies, for every eps>0, with constants independent of the triple, |ker H^0(Θ_f)|≤ C_eps R^eps and |H^0( C_f)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3eaf6610cad1f3da","tex":"\\kappa_c","aliases":"κ_c kappa","display":"\\kappa_c","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ","count":20},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with v","count":15},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2c9dd55b67cbaf19","tex":"\\lambda^{\\mathsf T}","aliases":"λ^ T lambda mathsf","display":"\\lambda^{\\mathsf T}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f781fdf3ad4821b6","tex":"\\lambda^{(a)}","aliases":"λ^(a) lambda","display":"\\lambda^{(a)}","kind":"use","excerpt":"Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-73a92e6394b23f10","tex":"\\Lambda_{\\mathrm{coh}}","aliases":"Λ_coh Lambda mathrm","display":"\\Lambda_{\\mathrm{coh}}","kind":"use","excerpt":"satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π iCg· x). Its kernel is the adelic coherence lattice Λ_coh = {x∈Z^S_ab:g· x≡0pmod C}. Since g is primitive modulo C","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-18bb5b342b20541e","tex":"\\lambda_d","aliases":"λ_d lambda","display":"\\lambda_d","kind":"use","excerpt":"Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with v","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7daa0edaa864bbe2","tex":"\\Lambda_p","aliases":"Λ_p Lambda","display":"\\Lambda_p","kind":"use","excerpt":"A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fdc2dc9f687c22a4","tex":"\\lambda_p","aliases":"λ_p lambda","display":"\\lambda_p","kind":"use","excerpt":"The additive row. The condition d_x(a)+ d_x(b)= d_x(c) is one integer equation. Define λ_p= dfraca v_p(a)p, · pmid a, [6pt] dfracb v_p(b)p, · pmid b, [6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-36442f76e90ff174","tex":"\\lambda_r","aliases":"λ_r lambda","display":"\\lambda_r","kind":"use","excerpt":"If h is absent, a degree-r polynomial vanishes at every occupied node and takes value one at α_h, so λ_r(h)=0. If h is occupied, its degree-(r-1) Lagrange polynomial isolates exactly that node. Thus λ_r-1(h)=frac nM^2N(h)^2. At the innovation degree the count bound is exact. The remaining computational issue is finding the needed filter with contro","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-aab6d54dfd2fd7da","tex":"\\lambda_{r-1}","aliases":"λ_r-1 lambda","display":"\\lambda_{r-1}","kind":"use","excerpt":"If h is absent, a degree-r polynomial vanishes at every occupied node and takes value one at α_h, so λ_r(h)=0. If h is occupied, its degree-(r-1) Lagrange polynomial isolates exactly that node. Thus λ_r-1(h)=frac nM^2N(h)^2. At the innovation degree the count bound is exact. The remaining computational issue is finding the needed filter with contro","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cf331a2bc1f6a864","tex":"\\Lambda_U","aliases":"Λ_U Lambda","display":"\\Lambda_U","kind":"use","excerpt":"Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c259e39137ef0548","tex":"\\Lambda_W","aliases":"Λ_W Lambda","display":"\\Lambda_W","kind":"use","excerpt":"Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0bf8701ccd3ba8f5","tex":"\\mu_M","aliases":"μ_M mu","display":"\\mu_M","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"use","excerpt":"The arithmetic Wronskian. Define W_x(a,b) =a d_x(b)-b d_x(a). Writing μ_p= -dfracv_p(a)p, · pmid a, [6pt] dfracv_p(b)p, · pmid b, [6pt] 0, · pmid c, we obtain W_x(a,b)=ab μ· x. An additive packet is transverse when W_x(a,b)≠0. It is degenerate when the two input terms remain derivative-proportional.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e2458c5b6cb798e0","tex":"\\mu_\\perp","aliases":"μ_⊥ mu perp","display":"\\mu_\\perp","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-461d64059c8c1199","tex":"\\mu_\\perp^{\\mathsf T}","aliases":"μ_⊥^ T mu perp mathsf","display":"\\mu_\\perp^{\\mathsf T}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.","count":11},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-cd0b7ab5ccc033f7","tex":"\\nu_0","aliases":"ν_0 nu","display":"\\nu_0","kind":"use","excerpt":"The coherence character. Set χ_p=0 for absent or squarefree output channels. When the residual modulus is one, use the trivial row and character. After normalization, every output prime with nontrivial residual depth produces a primitive row g_p ∈ (Z/p^χ_pZ)^S_ab satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vec","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-898af8461f33e86a","tex":"\\nu_{a,b,c}","aliases":"ν_a,b,c nu","display":"\\nu_{a,b,c}","kind":"use","excerpt":"The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d5fba9031264fee8","tex":"\\nu_q","aliases":"ν_q nu","display":"\\nu_q","kind":"use","excerpt":"Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then ","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-273ca48755e332d5","tex":"\\xi_i","aliases":"ξ_i xi","display":"\\xi_i","kind":"use","excerpt":"CRT normal form Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ffb8af09a0c235fd","tex":"\\xi_p","aliases":"ξ_p xi","display":"\\xi_p","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"A fixed path packet is Π_≤3(z) = ( log z, log(1-z), L_2(z), Li_3(z), Li_3(1-z) ). Its coordinate count is independent of M.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in","count":10},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-76b8428882ee3f96","tex":"\\Pi_{\\le3}","aliases":"Π_≤3 Pi le","display":"\\Pi_{\\le3}","kind":"use","excerpt":"A fixed path packet is Π_≤3(z) = ( log z, log(1-z), L_2(z), Li_3(z), Li_3(1-z) ). Its coordinate count is independent of M.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular ","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-737cdd19f67dfe28","tex":"\\rho_1","aliases":"ρ_1 rho","display":"\\rho_1","kind":"use","excerpt":"Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9bf4e85984084b75","tex":"\\rho_a","aliases":"ρ_a rho","display":"\\rho_a","kind":"use","excerpt":"For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-153325643fdda58b","tex":"\\rho_b","aliases":"ρ_b rho","display":"\\rho_b","kind":"use","excerpt":"For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3b48fafc33330554","tex":"\\rho_c","aliases":"ρ_c rho","display":"\\rho_c","kind":"use","excerpt":"For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-37acf16cb7002188","tex":"\\rho_u^{\\mathrm{sh}}","aliases":"ρ_u^sh rho mathrm","display":"\\rho_u^{\\mathrm{sh}}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7cc70e05d3b03d7a","tex":"\\sigma_a","aliases":"σ_a sigma","display":"\\sigma_a","kind":"use","excerpt":"The stable real height. Relax the integral packet to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=ov","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-98b0d6c9206da517","tex":"\\sigma_{a,b,c}","aliases":"σ_a,b,c sigma","display":"\\sigma_{a,b,c}","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-fc148a89e558c088","tex":"\\sigma_b","aliases":"σ_b sigma","display":"\\sigma_b","kind":"use","excerpt":"The stable real height. Relax the integral packet to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=ov","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ae00882e1fd7fcb3","tex":"\\sigma_c","aliases":"σ_c sigma","display":"\\sigma_c","kind":"use","excerpt":"The stable real height. Relax the integral packet to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=ov","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b17d423cde5b3e30","tex":"\\sigma_u","aliases":"σ_u sigma","display":"\\sigma_u","kind":"use","excerpt":"The stable real height. Relax the integral packet to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=ov","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":18},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area ","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow b","count":8},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bf5b725bf9da669d","tex":"\\Phi_0","aliases":"Φ_0 Phi","display":"\\Phi_0","kind":"use","excerpt":"ent of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0dd8e14cc9a785dc","tex":"\\Phi_1","aliases":"Φ_1 Phi","display":"\\Phi_1","kind":"use","excerpt":"+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bd1555616c800972","tex":"\\phi_E","aliases":"ϕ_E phi","display":"\\phi_E","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-afa3e839955395af","tex":"\\phi_E^*","aliases":"ϕ_E^* phi","display":"\\phi_E^*","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c7452277b5b2947b","tex":"\\Phi_m","aliases":"Φ_m Phi","display":"\\Phi_m","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0a7a0fd5a24bfbb4","tex":"\\Phi_{m,p}","aliases":"Φ_m,p Phi","display":"\\Phi_{m,p}","kind":"use","excerpt":"The depleted local series. For a prime p, delete the terms whose index is divisible by p: Φ_m,p(t)=(-1)^m∑_pnmid kbinomkm+1fract^kk^m+2∈Z_p[[t]]. It converges when v_p(t)>0. [Exact local jet valuation] If v_p(t)=e≥1, then v_p(Φ_m,p(t))= (m+1)e, · pnmid m+1,","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in","count":13},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-847a7aea863f53a5","tex":"\\chi_{\\mathrm{coh}}","aliases":"χ_coh chi mathrm","display":"\\chi_{\\mathrm{coh}}","kind":"use","excerpt":"er normalization, every output prime with nontrivial residual depth produces a primitive row g_p ∈ (Z/p^χ_pZ)^S_ab satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-95fabc8465dd628e","tex":"\\chi_p","aliases":"χ_p chi","display":"\\chi_p","kind":"use","excerpt":"r formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. Define w_p = min { d_p-r_p, min_","count":7},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c152a6f955cd6980","tex":"\\Psi_a","aliases":"Ψ_a Psi","display":"\\Psi_a","kind":"use","excerpt":"For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1dc185857eee32ba","tex":"\\Psi_{a,b,c}","aliases":"Ψ_a,b,c Psi","display":"\\Psi_{a,b,c}","kind":"use","excerpt":"For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-48e24f5ca62767fd","tex":"\\Psi_b","aliases":"Ψ_b Psi","display":"\\Psi_b","kind":"use","excerpt":"For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3664f0ed22551ee1","tex":"\\Psi_c","aliases":"Ψ_c Psi","display":"\\Psi_c","kind":"use","excerpt":"For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-02cb516a6f66aaeb","tex":"\\Psi_n","aliases":"Ψ_n Psi","display":"\\Psi_n","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow b","count":8},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5544658fd58cf8cf","tex":"\\omega_E","aliases":"ω_E omega","display":"\\omega_E","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-af8076f32fb2994d","tex":"\\Omega_S","aliases":"Ω_S Omega","display":"\\Omega_S","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and ","count":28},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"get define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by th","count":23},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"get define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by th","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-28c6765d59fee59f","tex":"\\ell^n","aliases":"ℓ^n ell","display":"\\ell^n","kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-943e031326adef72","tex":"\\ell^{\\alpha_\\ell}","aliases":"ℓ^α_ℓ ell alpha ell","display":"\\ell^{\\alpha_\\ell}","kind":"use","excerpt":"Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3e646a32ec34937c","tex":"\\ell_{2,e}","aliases":"ℓ_2,e ell","display":"\\ell_{2,e}","kind":"use","excerpt":"Weighted dyadic row. For 2^eparallel c with e≥2 and an odd input prime q, define ℓ_2,e(q) = frac14 log_2((-1)^(q-1)/2q) pmod2^e-2. With the analogous truncated valuation on Z/2^e-2Z, bar v_2(ℓ_2,e(q)) = min{e-2,v_2(q^2-1)-3}. At e=1, the output contact depth is zero and this row is omitted. A separate sign packet supplies the missing first dyadic l","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"p_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gives one quotient equation ∑_i z_i=q(t). For a box radius H, each z_i lies in the interval ℓ_i(H) = lceilfrac-H-ξ_","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c812626f1088bbe9","tex":"\\ell_{p,e}","aliases":"ℓ_p,e ell","display":"\\ell_{p,e}","kind":"use","excerpt":"Odd logarithmic rows. For odd p^eparallel c, define the finite-depth row coefficient ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. The balanced relation gives ∑_qmid ab ν_qℓ_p,e(q) ≡0 pmodp^e-1. Define the truncated valuation on Z/p^e-1Z by bar v_p(0)=e-1, bar v_p(x)=min{e-1,v_p(widetilde x)} for any integer lift widetilde x of a nonzero residue. Then ba","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)^d for t≥1. The inputs are coprime. As rad(t^d)≤ t and rad((t+1)^d)≤ t+1, R≤ t(t+1)c, Q=fracabcR≥[t(t+1)]^d-1. The assumed inequality implies [t(t+1)]^s(d-1)≤ C[t(t+1)c]^B(log(3c))^A. Take logarithms, divide by log t and send t to infinity. Since log c/log t→ d and loglog(3c)/log t→0, this yi","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Let I_N=ker(A_Nxrightarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"d ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2e558249cf09608d","tex":"\\partial_{p,T}","aliases":"∂_p,T partial","display":"\\partial_{p,T}","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. Set V_0(T)=0 for an absent prime. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) th","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records ","count":40},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f67711a953b0a584","tex":"\\prod_{d\\ge1}","aliases":"∏_d≥1 prod ge","display":"\\prod_{d\\ge1}","kind":"use","excerpt":"dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3ac28c105ca2b315","tex":"\\prod_{d\\ge2}","aliases":"∏_d≥2 prod ge","display":"\\prod_{d\\ge2}","kind":"use","excerpt":"dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"orem Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =f","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b8ef85e0df350209","tex":"\\prod_{i<j}","aliases":"∏_i<j prod","display":"\\prod_{i<j}","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3cfa866bffa35d9b","tex":"\\prod_{i=0}^{r-1}","aliases":"∏_i=0^r-1 prod","display":"\\prod_{i=0}^{r-1}","kind":"use","excerpt":"Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a4a402d8d3eb9074","tex":"\\prod_{i=1}^r","aliases":"∏_i=1^r prod","display":"\\prod_{i=1}^r","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9079021fd6762740","tex":"\\prod_{i=1}^{r}","aliases":"∏_i=1^r prod","display":"\\prod_{i=1}^{r}","kind":"use","excerpt":"The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5f1fab114d1c094a","tex":"\\prod_{i=1}^{r-1}","aliases":"∏_i=1^r-1 prod","display":"\\prod_{i=1}^{r-1}","kind":"use","excerpt":"Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b8080a407bf6180c","tex":"\\prod_{j=1}^m","aliases":"∏_j=1^m prod","display":"\\prod_{j=1}^m","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-08e723782301ff4e","tex":"\\prod_{j=1}^{r}","aliases":"∏_j=1^r prod","display":"\\prod_{j=1}^{r}","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f7b3830327a5da4c","tex":"\\prod_{j=1}^{r-1}","aliases":"∏_j=1^r-1 prod","display":"\\prod_{j=1}^{r-1}","kind":"use","excerpt":"[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-df3ae28461c2ddd4","tex":"\\prod_{j=1}^s","aliases":"∏_j=1^s prod","display":"\\prod_{j=1}^s","kind":"use","excerpt":"Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3f7b0c58c5e914fa","tex":"\\prod_{k=1}^r","aliases":"∏_k=1^r prod","display":"\\prod_{k=1}^r","kind":"use","excerpt":"[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b9d5148b142926aa","tex":"\\prod_p","aliases":"∏_p prod","display":"\\prod_p","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c7ba725b7697f4ca","tex":"\\prod_{p^e\\parallel abc}","aliases":"∏_p^eparallel abc prod parallel","display":null,"kind":"use","excerpt":"The prime supports of a, b, and c are disjoint. Multiplying all local divisibilities gives ∏_p^eparallel abcp^e-1 = fracabcR mid W_x(a,b).","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-1d6641086f3edea1","tex":"\\prod_{p^e\\parallel c}","aliases":"∏_p^eparallel c prod parallel","display":null,"kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-26fdda4e62a4020e","tex":"\\prod_{p^e\\parallel n}","aliases":"∏_p^eparallel n prod parallel","display":null,"kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e89148161d5d541d","tex":"\\prod_{p\\mid c}","aliases":"∏_pmid c prod mid","display":null,"kind":"use","excerpt":"rallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input e","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-da4c298d2e337844","tex":"\\prod_{p\\mid n}","aliases":"∏_pmid n prod mid","display":null,"kind":"use","excerpt":"Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-00f1a72bc8a69c76","tex":"\\prod_{p\\le3r-1}","aliases":"∏_p≤3r-1 prod le","display":"\\prod_{p\\le3r-1}","kind":"use","excerpt":"ining primes Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depe","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bf2494072e7ab581","tex":"\\prod_{q\\mid ab}","aliases":"∏_qmid ab prod mid","display":null,"kind":"use","excerpt":"Writing β=-∏_qmid abq^ν_q, we obtain the cross-prime logarithmic filter relation log_pβ = ∑_qmid ab ν_qlog_p⟨ q⟩_p. The same depth-three path therefore has a small complex envelope and exact deep finite notches.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-3b928adc47a853c8","tex":"\\prod_{r=0}^{e-1}","aliases":"∏_r=0^e-1 prod","display":"\\prod_{r=0}^{e-1}","kind":"use","excerpt":"First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bbcd7f9ad13fce2b","tex":"\\prod_{\\substack{p\\mid n\\\\d\\mid e_p}}","aliases":"∏_substackpmid n dmid e_p prod substack mid d mid","display":null,"kind":"use","excerpt":"Exact-order divisor packets. Let n=∏_p p^e_p. For every d≥1, define B_d(n) = ∏_substackpmid n dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact c","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-eda8877465c42ed5","tex":"\\prod_{\\substack{\\zeta^N=1\\\\\\zeta\\neq1}}","aliases":"∏_substackζ^N=1 ζ≠1 prod substack zeta zeta neq","display":"\\prod_{\\substack{\\zeta^N=1\\\\\\zeta\\neq1}}","kind":"use","excerpt":"ghtarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1 ζ≠1 frac1-ζ^e1-ζ =1. Thus mu","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-08f907a2865973eb","tex":"\\prod_{u\\in(\\Z/n\\Z)^\\times}","aliases":"∏_u∈(Z/nZ)^× prod in Z Z times","display":null,"kind":"use","excerpt":"A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrang","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-2ef79afc192f8074","tex":"\\prod_{x\\in E}","aliases":"∏_x∈ E prod in","display":"\\prod_{x\\in E}","kind":"use","excerpt":"[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-470d953e1b0bcdc9","tex":"\\prod_{x\\in E\\setminus\\{h\\}}","aliases":"∏_x∈ E∖{h} prod in setminus","display":"\\prod_{x\\in E\\setminus\\{h\\}}","kind":"use","excerpt":"A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrang","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ee654d78dddfa1e6","tex":"\\prod_{\\ell\\text{ odd}}","aliases":"∏_ℓ odd prod ell text","display":"\\prod_{\\ell\\text{ odd}}","kind":"use","excerpt":"Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-eaff1750441df0af","tex":"\\sum_{1\\le k_1<\\cdots<k_r}","aliases":"∑_1≤ k_1<⋯<k_r sum le cdots","display":"\\sum_{1\\le k_1<\\cdots<k_r}","kind":"use","excerpt":"[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and inclu","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ce043e53d6db1168","tex":"\\sum_{d\\mid e}","aliases":"∑_dmid e sum mid","display":null,"kind":"use","excerpt":"dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4989fc2730a6d003","tex":"\\sum_{d\\ge2}","aliases":"∑_d≥2 sum ge","display":"\\sum_{d\\ge2}","kind":"use","excerpt":"Quality as input-output persistence imbalance. For a primitive triple, q-1 = frac log(c/R_c)-log(R_aR_b) log R . Using the divisor packets, q-1 = frac ∑_d≥2φ(d)log B_d(c) - log(R_aR_b) log R. A high-quality triple has more repeated-depth persistence in the output than the input radicals supply as activated bandwidth.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-dd79b3b4cac0915f","tex":"\\sum_{h\\ge1}","aliases":"∑_h≥1 sum ge","display":"\\sum_{h\\ge1}","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"ves linearly growing valuation. Thus the terms tend to zero and the finite Cauchy--Binet expansions pass to the convergent local limit. [Exact good-prime valuation] If p>3r-1 and v_p(t)=e≥1, then v_p( H_r(t))=e r(r+1)/2. Write a_r=2r-1 and s_r=r(r+1)/2. A term indexed by k_1<⋯<k_r has valuation e∑_i k_i+2∑_i<jv_p(k_j-k_i)-a_r∑_i v_p(k_i). For every","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8524806a0ba8de34","tex":"\\sum_{i<j}","aliases":"∑_i<j sum","display":"\\sum_{i<j}","kind":"use","excerpt":"The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-21d7f69515073828","tex":"\\sum_{i=0}^{r-1}","aliases":"∑_i=0^r-1 sum","display":"\\sum_{i=0}^{r-1}","kind":"use","excerpt":"Since |log(b/a)|≤log c and log(c/a)≤log c, every entry has a bound by a constant times (1+log c)^j. A determinant term has total weight ∑_i=0^r-1(i+π(i)+1)=r^2. Therefore | H_r(c/a)|≤ C_r(1+log c)^r^2 for each of those consistent branches. The constant may depend on r. No nonvanishing assertion at every argument in this third chart follows from the","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6182c3bad2d01e16","tex":"\\sum_{i=1}^r","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^r","kind":"use","excerpt":"An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-531efcfef9ef6d90","tex":"\\sum_{i\\ge1}","aliases":"∑_i≥1 sum ge","display":"\\sum_{i\\ge1}","kind":"use","excerpt":"Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b845f4dcb81f67c5","tex":"\\sum_{j=0}^d","aliases":"∑_j=0^d sum","display":"\\sum_{j=0}^d","kind":"use","excerpt":"Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with v","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d406db64a8c9cb22","tex":"\\sum_{j=0}^{M-1}","aliases":"∑_j=0^M-1 sum","display":"\\sum_{j=0}^{M-1}","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d0b76425559f0527","tex":"\\sum_{j=0}^{r-1}","aliases":"∑_j=0^r-1 sum","display":"\\sum_{j=0}^{r-1}","kind":"use","excerpt":"[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8650842fbc4c4680","tex":"\\sum_{j=1}^r","aliases":"∑_j=1^r sum","display":"\\sum_{j=1}^r","kind":"use","excerpt":"Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9914d30e7b62691b","tex":"\\sum_{j\\neq0}","aliases":"∑_j≠0 sum neq","display":"\\sum_{j\\neq0}","kind":"use","excerpt":"onSecond-order phase energy At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e861dfba300b543d","tex":"\\sum_{k=1}^K","aliases":"∑_k=1^K sum","display":"\\sum_{k=1}^K","kind":"use","excerpt":"Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-01f56a5e1c2b8b56","tex":"\\sum_{k\\ge1}","aliases":"∑_k≥1 sum ge","display":"\\sum_{k\\ge1}","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-bbbf8b83915c5734","tex":"\\sum_{k\\ge2}","aliases":"∑_k≥2 sum ge","display":"\\sum_{k\\ge2}","kind":"use","excerpt":"Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-df09d859c509da59","tex":"\\sum_p","aliases":"∑_p sum","display":"\\sum_p","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9212aad30f9a37fa","tex":"\\sum_{p\\mid a}","aliases":"∑_pmid a sum mid","display":null,"kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-6280728bb9fdface","tex":"\\sum_{p\\mid b}","aliases":"∑_pmid b sum mid","display":null,"kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b8e5fc8aeac683b4","tex":"\\sum_{p\\mid c}","aliases":"∑_pmid c sum mid","display":null,"kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b0678753651fceb8","tex":"\\sum_{p\\mid n}","aliases":"∑_pmid n sum mid","display":null,"kind":"use","excerpt":"e three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n, define the universal multiplicative differential dn ","count":6},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-98d653aa96458ba7","tex":"\\sum_{p\\mid u}","aliases":"∑_pmid u sum mid","display":null,"kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-07de5a7bb64c633f","tex":"\\sum_{p\\nmid k}","aliases":"∑_pnmid k sum nmid","display":null,"kind":"use","excerpt":"The depleted local series. For a prime p, delete the terms whose index is divisible by p: Φ_m,p(t)=(-1)^m∑_pnmid kbinomkm+1fract^kk^m+2∈Z_p[[t]]. It converges when v_p(t)>0. [Exact local jet valuation] If v_p(t)=e≥1, then v_p(Φ_m,p(t))= (m+1)e, · pnmid m+1,","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-499f779743314d49","tex":"\\sum_{q\\mid ab}","aliases":"∑_qmid ab sum mid","display":null,"kind":"use","excerpt":"Writing β=-∏_qmid abq^ν_q, we obtain the cross-prime logarithmic filter relation log_pβ = ∑_qmid ab ν_qlog_p⟨ q⟩_p. The same depth-three path therefore has a small complex envelope and exact deep finite notches.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-8586d754d52f2508","tex":"\\sum_{r<s}","aliases":"∑_r<s sum","display":"\\sum_{r<s}","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-82fc846b089bd1f7","tex":"\\sum_{r=0}^{v_p(u)-1}","aliases":"∑_r=0^v_p(u)-1 sum","display":"\\sum_{r=0}^{v_p(u)-1}","kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7d98fe9de6bebd3c","tex":"\\sum_s","aliases":"∑_s sum","display":"\\sum_s","kind":"use","excerpt":"Small coefficient norm has a limited meaning. The coefficients of P_m alternate in sign. Its absolute-coefficient polynomial evaluated at one gives ∑_s|c_m,s|=∏_j=1^m(j+1)=(m+1)!. Also c_m,1=(-1)^m, so the integer coefficient vector is primitive. After the displayed normalization, its coefficient ℓ^1 norm is one.","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-40085659318b75a7","tex":"\\sum_{s=1}^{m+1}","aliases":"∑_s=1^m+1 sum","display":"\\sum_{s=1}^{m+1}","kind":"use","excerpt":"The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source r","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0e44b9be25437975","tex":"\\sum_x","aliases":"∑_x sum","display":"\\sum_x","kind":"use","excerpt":"Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-19ad43044fe351bc","tex":"\\sum_{x\\in E}","aliases":"∑_x∈ E sum in","display":"\\sum_{x\\in E}","kind":"use","excerpt":"Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1)","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-5d4420db90ebac17","tex":"\\sum_{x\\in G}","aliases":"∑_x∈ G sum in","display":"\\sum_{x\\in G}","kind":"use","excerpt":"Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-53017beba95d8b75","tex":"\\sum_\\varepsilon","aliases":"∑_ε sum varepsilon","display":"\\sum_\\varepsilon","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9160ac200b480969","tex":"\\sum_{\\chi\\in\\widehat G}","aliases":"∑_χ∈widehat G sum chi in widehat","display":"\\sum_{\\chi\\in\\widehat G}","kind":"use","excerpt":"Character orthogonality gives the target count N(h)=frac M|G|∑_χ∈widehat Gwidehatμ(χ)overlineχ(h). The identity is finite and exact. It solves a representation question after the arithmetic channel and admissible contributions have been specified. It does not by itself bound the height of an integer triple that generated that channel.","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-9fc07bf4a84c15fb","tex":"\\sum_{\\chi\\ne1}","aliases":"∑_χ≠1 sum chi ne","display":"\\sum_{\\chi\\ne1}","kind":"use","excerpt":"Sparsity forces transverse energy. Let E={x:N(x)>0} and r=|E|. Always r≤ M. Parseval yields ∑_χ∈widehat G|widehatμ(χ)|^2 =frac|G|M^2∑_xN(x)^2. Cauchy--Schwarz on the occupied support gives ∑_xN(x)^2≥ M^2/r. Hence [Sparse energy barrier] ∑_χ≠1|widehatμ(χ)|^2 ≥frac|G|r-1≥frac|G|M-1. When few assignment atoms lie in a large group, substantial nontrivi","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrang","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"Because b_n contains 3^n+1, its powerful part contains 3^n. Thus rad(b_n)≤ fracb_n3^n, R_n=2rad(b_n)≤frac2b_n3^n. Therefore fracc_nR_n≥frac3^n c_n2b_n>frac3^n2⟶∞. The displayed lower bound does not contradict the positive-eps conjecture. It does not exhibit infinitely many triples with a fixed positive quality gap. The output grows doubly exponenti","count":14},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.","count":5},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"automorphic analysis--synthesis identity Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area A(E) = ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4a810ebfdbc59ab3","tex":"\\int_{E(\\C)}","aliases":"∫_E(C) int C","display":null,"kind":"use","excerpt":"automorphic analysis--synthesis identity Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. Define the differential area A(E) = ","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"d ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Sparsity forces transverse energy. Let E={x:N(x)>0} and r=|E|. Always r≤ M. Parseval yields ∑_χ∈widehat G|widehatμ(χ)|^2 =frac|G|M^2∑_xN(x)^2. Cauchy--Schwarz on the occupied support gives ∑_xN(x)^2≥ M^2/r. Hence [Sparse energy barrier] ∑_χ≠1|widehatμ(χ)|^2 ≥frac|G|r-1≥frac|G|M-1. When few assignment atoms lie in a large group, substantial nontrivi","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"[6pt] 0, · pmid c, we obtain W_x(a,b)=ab μ· x. An additive packet is transverse when W_x(a,b)≠0. It is degenerate when the two input terms remain derivative-proportional.","count":10},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.","count":10},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Masser--Oesterl e abc] For every real eps>0 there is a constant K_eps>0, depending only on eps, such that every positive integer triple with a+b=c and gcd(a,b)=1 satisfies c≤ K_eps rad(abc)^1+eps. The order of the quantifiers is the statement. First choose eps. Then choose a constant that works for every primitive triple. The constant may become l","count":96},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"The same-sign separation problem. After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let Λ be an integral Hecke lattice and let e_f be the rational projector onto the Frey eigenline. Define η_f = min{m≥1:m e_f(Λ)⊆Λ}. This is the integral eigenchannel separation denominator. It is an arithmetic analogue of ","count":2},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n","count":4},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"The exact lattice extension. Throughout the remaining height and exact-sequence constructions, exclude (1,1,2). Define L=ker(Λ)∩Z^S and L_0=L∩ker( w). Because Λ is primitive, choose t∈Z^S with Λ(t)=1. Then Z^S=L⊕Z t. The image of M_a,b,c is generated by (1, w(t)) together with (0, w(L)). Its index in Z^2 is therefore the index of w(L) in Z. Theorem","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-61f317edc40cbb12","tex":"\\bigoplus_{i=1}^{r}","aliases":"⊕_i=1^r bigoplus","display":"\\bigoplus_{i=1}^{r}","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-b580a3a49114f560","tex":"\\bigoplus_{p^e\\parallel c}","aliases":"⊕_p^eparallel c bigoplus parallel","display":null,"kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). 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The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such a positive integer n","count":1},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":3},{"work":"abc-reader","url":"notation/abc-reader.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. 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j=0,1,2.","count":3},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-df29459720bd9419","tex":"u^j","aliases":"u^j","display":"u^j","kind":"use","excerpt":"r_j=Bv^j-λ Au^j-κ, j=0,1,2.","count":1},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"r_j=Bv^j-λ Au^j-κ, j=0,1,2.","count":3},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-3a313f6772645c4e","tex":"v^j","aliases":"v^j","display":"v^j","kind":"use","excerpt":"r_j=Bv^j-λ Au^j-κ, j=0,1,2.","count":1},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"t_N(x)=x^p^N-1pmod p^N.","count":4},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-664f40df69e929ed","tex":"x^{p^{N-1}}","aliases":"x^p^N-1","display":"x^{p^{N-1}}","kind":"use","excerpt":"t_N(x)=x^p^N-1pmod p^N.","count":1},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"C_p,N(x,y)=fract_N(x)+t_N(y)-t_N(x+y)p pmod p^N-1,","count":3},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"Q(z)=(z-p)P(z)=∑_j=0^d+1a_jz^j.","count":2},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"use","excerpt":"Q(z)=(z-p)P(z)=∑_j=0^d+1a_jz^j.","count":1},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"max_0≤ j≤ d+1|r_k+j|≥ fracq^k|q-p||P(q)|∑_j=0^d+1|a_j|. tag2","count":1},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"r_j=Bv^j-λ Au^j-κ, j=0,1,2.","count":2},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"mathscr R(n)=λ n+F(rad(n), nbmod M),","count":4},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Q(z)=(z-p)P(z)=∑_j=0^d+1a_jz^j.","count":3},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-24b8da302aa3a7d3","tex":"\\sum_{i=0}^{p-1}","aliases":"∑_i=0^p-1 sum","display":"\\sum_{i=0}^{p-1}","kind":"use","excerpt":"∑_i=0^p-1 C_p,N(1,i)≡ 1pmodp^N-1","count":1},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-e925b1f977fed8dc","tex":"\\sum_{j=0}^{d+1}","aliases":"∑_j=0^d+1 sum","display":"\\sum_{j=0}^{d+1}","kind":"use","excerpt":"Q(z)=(z-p)P(z)=∑_j=0^d+1a_jz^j.","count":2},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"p^k+jL≡ p^kpmod M.","count":3},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"max_0≤ j≤ d+1|r_k+j|≥ fracq^k|q-p||P(q)|∑_j=0^d+1|a_j|. tag2","count":1},{"work":"addmul-arithmetic-reconstruction","url":"notation/addmul-arithmetic-reconstruction.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"E≥fracq^k(q^L-1)(q^L-p^L)2(p^L+1).tag1","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime divi","count":347},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime divi","count":99},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime divi","count":62},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime divi","count":77},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"definition-passage","excerpt":"A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime divi","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime divi","count":38},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"definition-passage","excerpt":"A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime divi","count":30},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"A positive solution of [source reference] is primitive when gcd(A,B,C)=1. Its ordered exponent triple (x,y,z) is its signature. The first two entries may be exchanged together with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime divi","count":60},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this mark","count":139},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":20},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x","count":31},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7db2ae819f88d048","tex":"D_{12}","aliases":"D_12","display":"D_{12}","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1ba0deca13916a02","tex":"D_{23}","aliases":"D_23","display":"D_{23}","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1f87edb2dfa256d3","tex":"d_\\chi","aliases":"d_χ chi","display":"d_\\chi","kind":"introduction-passage","excerpt":"th differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is soluble if ","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8f79d4144c2ac115","tex":"\\dim_{\\F_7}","aliases":"dim_F_7 dim F","display":null,"kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":51},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F","display":null,"kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x","count":26},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fix","count":13},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d218908ad4d0b0a2","tex":"F^\\times","aliases":"F^× times","display":"F^\\times","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d5c247970c8a5083","tex":"F^{\\times M}","aliases":"F^× M times","display":"F^{\\times M}","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5008f9b3ed77a5e3","tex":"\\F_7","aliases":"F_7 F","display":null,"kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x","count":17},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":9},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":60},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a2307107966e729a","tex":"G_{K_\\infty}","aliases":"G_K_∞ infty","display":"G_{K_\\infty}","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":35},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"cycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=","count":43},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"introduction-passage","excerpt":"ype cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p. Thus [R_p]=[μ_p] in H^2(K,k(χ)). For this fixed p, the upper equation is soluble if and only if this class is zero.","count":8},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this mark","count":24},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this mark","count":9},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":36},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":45},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":48},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b4cac2c5cac2b244","tex":"K_v","aliases":"K_v","display":"K_v","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-60333a42736870fb","tex":"K_\\infty","aliases":"K_∞ infty","display":"K_\\infty","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":32},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-bb04371f38e2158f","tex":"M_{21}","aliases":"M_21","display":"M_{21}","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-23f4bbf348c4a0bf","tex":"\\Oloc","aliases":"Oloc","display":null,"kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":27},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":122},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this mark","count":18},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this mark","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b5c9a939784c608f","tex":"R_p","aliases":"R_p","display":"R_p","kind":"introduction-passage","excerpt":"α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=α(g)p(g) gives R_p+d_χ f=μ_p","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":36},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-cdec095fec211ac6","tex":"\\mathrm{SL}","aliases":"SL mathrm","display":"\\mathrm{SL}","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b3f0250b9a6fb70c","tex":"\\mathrm{SL}_2","aliases":"SL_2 mathrm","display":"\\mathrm{SL}_2","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":34},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":60},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":31},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":10},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":9},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":44},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":41},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":22},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":6},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this mark","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5fe4316a85529609","tex":"\\Pi_{ij}","aliases":"Π_ij Pi","display":"\\Pi_{ij}","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X -4X · Z-2X =ZI+2XR, R=1 · -2 -2 · -1, R^2=5I. Since four is invertible modulo M, every marked period coordinate vanishes if and only if X=Z=0. The matrix R belongs to this mark","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":16},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"introduction-passage","excerpt":"Put M=7^j and write [u] additively in F^×/F^× M for the local field F=K_v. Let X=[ϕ], Z=7[C]-3[A]-3[3]-5[sqrt5]. The entire marked period matrix becomes ([Π_ij])= Z+2X · -4X","count":33},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"introduction-passage","excerpt":"cycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [source reference]. Direct expansion of f(g)=","count":57},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x 0 · 1), x∈ k, (M_21/7)bmod7∈ B }. In particular it contains the full kernel of reduction modulo forty nine. Its","count":39},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"introduction-passage","excerpt":"The depth one image theorem. Let K_∞ be the cyclotomic tower. Write B for the additive image of beta on its Galois group and d=dim_F_7B. The inherited group theorem at actual depth one identifies H=ρ(G_K_∞)= {M∈SL_2(Oloc): l Mbmod7=(1 · x","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"introduction-passage","excerpt":"Put s(g)=α(g)β(g). Expansion of the two cocycle laws gives ds=-(α∪β+β∪α). For a chosen primitive p, set D_12=-p and D_23=p-s. They form a defining system with differentials α∪β and β∪α. The associated twisted Massey type cocycle is μ_p=α∪(p-s)-p∪α. Explicitly it equals μ_p(g,h)=α(g)(p(h)-α(h)β(h)) -p(g)χ(g)α(h). Let R_p denote the right side of [so","count":16},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"[Affine fibers] If Stage P has one solution with parameter z_*, its complete solution set for these linear conditions is z_*+ker A. It has 49^3,49^3,49^2,49^3 choices in patterns A through D, respectively. If the target is not in the image, the set is empty. Az=t is equivalent to A(z-z_*)=0. The exact kernel dimensions give the counts over k. An ad","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"(D^n/4)^4. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way. The proposition retains infinitely many signatures. It does not reduce them all to (3,5,7). A theorem for that si","count":64},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"he common prime is 3, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a pr","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"In the descended all inert regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B, [X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"0 · 1), direct conjugation sends (p,q,b) to (p+ax^-1b, xq-2ap-a^2x^-1b, x^-1b). Thus the three coupled equations are β(gh) · =β(g)+χ(g)^-1β(h),","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-06b373fe54fc27d1","tex":"A^3","aliases":"A^3","display":"A^3","kind":"use","excerpt":"For a nonzero rational solution, define t=A^3/C^7 and λ=C^8/(AB^4). Substitution recovers [source reference]. If the original bases are positive, then 0<t<1 and λ>0. The rational parametrization therefore captures the entire positive rational problem. Its missing condition is exactly primitive integrality.","count":12},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7b48479868582d73","tex":"a^3","aliases":"a^3","display":"a^3","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0e6b1659b5e2146e","tex":"A^6","aliases":"A^6","display":"A^6","kind":"use","excerpt":"ion this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-79f1bb0d0b74b664","tex":"a^6","aliases":"a^6","display":"a^6","kind":"use","excerpt":"Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4fbf9458f594ce86","tex":"A^x","aliases":"A^x","display":"A^x","kind":"use","excerpt":"Three powers and one common prime. [Beal] If positive integers A,B,C and integers x,y,z>2 satisfy A^x+B^y=C^z, then A,B,C have a common prime divisor. The exponents are allowed to differ. That is the essential extra freedom beyond the equal exponent Fermat equation. The common prime belongs to all three bases, not merely to the product of two of th","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7a412ec4a07776a7","tex":"a_{\\mathfrak q}","aliases":"a_ q mathfrak","display":"a_{\\mathfrak q}","kind":"use","excerpt":"Suppose a Hecke eigenvalue a_ q is replaced by its minimal polynomial. A resultant against a possible curve trace polynomial asks whether some conjugate can match some allowable trace. Repeating the test at a different prime can choose a different conjugate. If the original representation requires one residue embedding for every prime, this relaxed","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0c3271d6a63ebef5","tex":"A_0^3","aliases":"A_0^3","display":"A_0^3","kind":"use","excerpt":"rization and its integral defect For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f2cfa673011ea148","tex":"a_3","aliases":"a_3","display":"a_3","kind":"use","excerpt":"Global coefficients, diagonal characters, and Frobenius precision. The natural class has a fixed global expansion α=a_ϕ[ϕ]+a_3[3]+a_5[sqrt5], a_3a_5≠0. This lies in a Kummer group tensored with k. It does not mean that an element of F_49 is an exponent of a base field unit. The three coefficients are normalized together, up to one common scalar. Us","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-011995a02078a685","tex":"a_5","aliases":"a_5","display":"a_5","kind":"use","excerpt":"Global coefficients, diagonal characters, and Frobenius precision. The natural class has a fixed global expansion α=a_ϕ[ϕ]+a_3[3]+a_5[sqrt5], a_3a_5≠0. This lies in a Kummer group tensored with k. It does not mean that an element of F_49 is an exponent of a base field unit. The three coefficients are normalized together, up to one common scalar. Us","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-38143a44ee0b7f8e","tex":"a_q","aliases":"a_q","display":"a_q","kind":"use","excerpt":", the two independent pro seven unit coordinates are ϕ,3, whose normalized logarithmic first digits are 5sqrt5,6, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is 3+3+2·9=24","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a4c7ddf01422fbf9","tex":"a_\\phi","aliases":"a_ϕ phi","display":"a_\\phi","kind":"use","excerpt":"Global coefficients, diagonal characters, and Frobenius precision. The natural class has a fixed global expansion α=a_ϕ[ϕ]+a_3[3]+a_5[sqrt5], a_3a_5≠0. This lies in a Kummer group tensored with k. It does not mean that an element of F_49 is an exponent of a base field unit. The three coefficients are normalized together, up to one common scalar. Us","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a3a52d54c059cb53","tex":"a_\\ell","aliases":"a_ℓ ell","display":"a_\\ell","kind":"use","excerpt":"v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The output valuations are a_ℓ · =35h+12m+28n, b_ℓ · =21h+7m+17n, c_ℓ · =15h+5m+12n. They must be nonnegative, and their minimum mus","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-73fff7053fd2794b","tex":"\\Ad","aliases":"Ad","display":null,"kind":"use","excerpt":"beta(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d3261090d13587bd","tex":"\\mathrm{an}","aliases":"an mathrm","display":"\\mathrm{an}","kind":"use","excerpt":"[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"(D^n/4)^4. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way. The proposition retains infinitely many signatures. It does not reduce them all to (3,5,7). A theorem for that si","count":69},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e77f9eceea598e09","tex":"B'","aliases":"B'","display":"B'","kind":"use","excerpt":"prime is 3, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive so","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0c610ff17db77fec","tex":"B^{10}","aliases":"B^10","display":"B^{10}","kind":"use","excerpt":"ion this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative integer solutions of 5i+3j=30 are (i,j)=(6,0),(3,5),(0,10). The discriminant therefore has the form uA^6+vA^3B^5+wB^10.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f76a26dbc0d9fe26","tex":"B^2","aliases":"B^2","display":"B^2","kind":"use","excerpt":"A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinate","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ea04a3745c586a08","tex":"B^4","aliases":"B^4","display":"B^4","kind":"use","excerpt":"For a nonzero rational solution, define t=A^3/C^7 and λ=C^8/(AB^4). Substitution recovers [source reference]. If the original bases are positive, then 0<t<1 and λ>0. The rational parametrization therefore captures the entire positive rational problem. Its missing condition is exactly primitive integrality.","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-96eb9a1efa5223b4","tex":"B^5","aliases":"B^5","display":"B^5","kind":"use","excerpt":"n Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b006b6f417e41ac4","tex":"b^5","aliases":"b^5","display":"b^5","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-268673dd5127a3f5","tex":"B^y","aliases":"B^y","display":"B^y","kind":"use","excerpt":"Three powers and one common prime. [Beal] If positive integers A,B,C and integers x,y,z>2 satisfy A^x+B^y=C^z, then A,B,C have a common prime divisor. The exponents are allowed to differ. That is the essential extra freedom beyond the equal exponent Fermat equation. The common prime belongs to all three bases, not merely to the product of two of th","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"The coupled adjoint equations. Choose one reference Kummer representation over Oloc/49, ρ_0(g)=χ(g) · b_0(g) 0 · 1, whose reduction has the same residual α as the actual natural representation. The reference is built by lifting the coefficients of the same global classes [ϕ],[3],[sqrt5]. It has no seven valuation coordinate at the fixed place. It i","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e2111156777f1bda","tex":"B_0^5","aliases":"B_0^5","display":"B_0^5","kind":"use","excerpt":"rization and its integral defect For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e60489d63df5a607","tex":"b_7","aliases":"b_7","display":"b_7","kind":"use","excerpt":"ntative of X has p=β=0 and upper entry q a k(χ) Kummer cocycle. Define ν_7(X) to be its [7] valuation coordinate. Remaining aligned strict gauges change q by a coboundary and a scalar multiple of residual α, both of valuation zero. Thus the coordinate is well defined under these strict equivalences. The reference b_0 has valuation zero and the actu","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-9c51d8969a601c9b","tex":"b_q","aliases":"b_q","display":"b_q","kind":"use","excerpt":"Because each support prime is inert in K/Q, restriction at a rational q≡1pmod7 multiplies the compatible rational Frobenius value by two: b_q=2∑_rD_q,rs_r. For q≡-1pmod7, the compatible squared rational Frobenius has beta value zero, from the cocycle factor 1+χ(q)^-1=0. This does not say beta is unramified there.","count":7},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2763a74afb84e897","tex":"b_\\ell","aliases":"b_ℓ ell","display":"b_\\ell","kind":"use","excerpt":"v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The output valuations are a_ℓ · =35h+12m+28n, b_ℓ · =21h+7m+17n, c_ℓ · =15h+5m+12n. They must be nonnegative, and their minimum mus","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a16d7db299007129","tex":"\\operatorname{Br}","aliases":"Br operatorname","display":"\\operatorname{Br}","kind":"use","excerpt":"For K=Q(sqrt5), localization on H^2(K,k(χ)) is injective. To see the coefficient translation, restrict to L=K(ζ_7), whose degree over K is six. Restriction followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over L, the module is two trivial F_7 copies and μ_7 is present. Kummer theory identifies each copy ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"(D^n/4)^4. Apply this operation to each term separately. The new bases have exactly the same prime supports as the old bases, so primitivity is preserved. The converse follows by reducing any hypothetical primitive solution in this way. The proposition retains infinitely many signatures. It does not reduce them all to (3,5,7). A theorem for that si","count":66},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"3, so this equality illustrates the conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. I","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"use","excerpt":"Integral semilinearity and the unlabelled image. The integral semilinear theorem supplies more than equality of residual cohomology classes. Under the retained geometric inputs, an actual map between the conjugate curves induces an integral operator U=Cσ on the Tate lattice, with ρ(jgj^-1)=Cσ(ρ(g))C^-1, Cσ(C)=-ρ(j^2), C_21∈7^nOloc. It preserves the","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-52505bff2c2a2e9f","tex":"C^{14}","aliases":"C^14","display":"C^{14}","kind":"use","excerpt":"[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative int","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3ce6f166d33ebf8d","tex":"C^7","aliases":"C^7","display":"C^7","kind":"use","excerpt":"For a nonzero rational solution, define t=A^3/C^7 and λ=C^8/(AB^4). Substitution recovers [source reference]. If the original bases are positive, then 0<t<1 and λ>0. The rational parametrization therefore captures the entire positive rational problem. Its missing condition is exactly primitive integrality.","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-db3585c721e8f66b","tex":"c^7","aliases":"c^7","display":"c^7","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2f25b4c7a3f412ca","tex":"C^8","aliases":"C^8","display":"C^8","kind":"use","excerpt":"For a nonzero rational solution, define t=A^3/C^7 and λ=C^8/(AB^4). Substitution recovers [source reference]. If the original bases are positive, then 0<t<1 and λ>0. The rational parametrization therefore captures the entire positive rational problem. Its missing condition is exactly primitive integrality.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-093e875f46256c76","tex":"C^z","aliases":"C^z","display":"C^z","kind":"use","excerpt":"Three powers and one common prime. [Beal] If positive integers A,B,C and integers x,y,z>2 satisfy A^x+B^y=C^z, then A,B,C have a common prime divisor. The exponents are allowed to differ. That is the essential extra freedom beyond the equal exponent Fermat equation. The common prime belongs to all three bases, not merely to the product of two of th","count":6},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":7},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3a41cce4627f6aac","tex":"C_0^{35}","aliases":"C_0^35","display":"C_0^{35}","kind":"use","excerpt":"At level seven, the last unit condition reduces to 75A being a seventh power. At level forty nine, exact multiplicative identities reduce it to 3·5^9A/C_0^35∈(κ_v^×)^49. Thus full forty nine torsion is rational if and only if 49mid Q-1, 7mid c, ϕ is a forty ninth power residue, and the displayed rational unit is as well. These are two way local cri","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6ec61ba25ded3f20","tex":"C_0^6","aliases":"C_0^6","display":"C_0^6","kind":"use","excerpt":"d H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordin","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-73f5a83f7c0bfa2f","tex":"C_0^7","aliases":"C_0^7","display":"C_0^7","kind":"use","excerpt":"rization and its integral defect For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2r","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-88a3e18f148a2a4a","tex":"C_{21}","aliases":"C_21","display":"C_{21}","kind":"use","excerpt":"Integral semilinearity and the unlabelled image. The integral semilinear theorem supplies more than equality of residual cohomology classes. Under the retained geometric inputs, an actual map between the conjugate curves induces an integral operator U=Cσ on the Tate lattice, with ρ(jgj^-1)=Cσ(ρ(g))C^-1, Cσ(C)=-ρ(j^2), C_21∈7^nOloc. It preserves the","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-dcb4e0ba5e2088bf","tex":"C_6","aliases":"C_6","display":"C_6","kind":"use","excerpt":"For a finite module A, let A'=Hom(A,μ_7) and let G' be the finite group of the field cut out by that dual. The cited sufficient criterion is H^1(G',A')=0. For A=k with trivial action, A'≃ k(χ) and G'=C_6. Averaging kills its positive cohomology because six is invertible modulo seven. For A=k(χ), the source action cancels the cyclotomic target actio","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e8b9f48bd6d09aed","tex":"c_q","aliases":"c_q","display":"c_q","kind":"use","excerpt":"The ordinary inertia generator differs from the frozen twisted residue convention. Its beta value is c_q=s_qℓ_q(Nm_F_q^2/F_qg_q) (q≡1pmod7), and c_q=s_qℓ_q(g_q) for q≡-1pmod7. The conversion factors are rrrrr 197:5 · 307:2 · 337:5 · 547:4 · 673:1","count":8},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8ec5cf28b1acacd8","tex":"c_\\ell","aliases":"c_ℓ ell","display":"c_\\ell","kind":"use","excerpt":"v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The output valuations are a_ℓ · =35h+12m+28n, b_ℓ · =21h+7m+17n, c_ℓ · =15h+5m+12n. They must be nonnegative, and their minimum mus","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-99348e1cff26039d","tex":"\\mathrm{char}","aliases":"char mathrm","display":"\\mathrm{char}","kind":"use","excerpt":"A candidate p=p_0+z_char meets the fixed local condition precisely when the selected local character coordinates equal those of p_v^fix-p_0|_v. At support, substitution in [source reference] gives c_qz_char(F_q)-b_qz_char(T_q)=t_q(p_0). Together, these fifteen equations are exactly Az=t(p_0). Thus Stage P is soluble if and only if t(p_0)∈im A.","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-80bb66e28ec49c6d","tex":"\\Cl","aliases":"Cl","display":null,"kind":"use","excerpt":"ma[The dual Kummer kernel] For every finite Σ containing the prime above seven and all real places, Sha^1_Σ(K,μ_7) =ker(H^1(Gsig,μ_7) ⟶∏_v∈ΣH^1(K_v,μ_7))=0. The discriminant of K is five. Minkowski's ideal bound is sqrt5/2<2, so every ideal class has a representative of norm one. Thus Cl(K)=1 [source reference].","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-073a71f814bdfa5f","tex":"\\coker","aliases":"coker","display":null,"kind":"use","excerpt":"The homogeneous results. The complete exact matrices, row reductions, null bases, and rank minors give crrrl Pattern · rankA · dim_kker A · dim_kcoker A · Remaining difference support","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","excerpt":"7c · d, det F=Q. Put d=1+7D and tr(F)=Q+1+7T. Then a=Q+7(T-D), and the determinant equation gives bc=T+(Q-1)D+7TD-7D^2. Modulo seven this becomes fractr(F)-Q-17=α(F)β(F) in k. This is the coefficient field rank two trace of the actual representation. It is not half the rational four dimensional trace. The hypothesis Q≡1pmod7 makes the extra diagona","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-76b7929465487fe2","tex":"D^n","aliases":"D^n","display":"D^n","kind":"use","excerpt":"on[Prime or four reduction] If a primitive solution of [source reference] exists, there is a primitive solution whose three exponents each belong to {4}∪{p:p is an odd prime}. Conversely, absence of primitive solutions for every ordered signature in this set implies Beal. Let n>2. If n has an odd prime divisor p, write D^n=(D^n/p)^p. If it has no o","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6c1d71eb8c3de331","tex":"D^{n/4}","aliases":"D^n/4","display":"D^{n/4}","kind":"use","excerpt":"ists, there is a primitive solution whose three exponents each belong to {4}∪{p:p is an odd prime}. Conversely, absence of primitive solutions for every ordered signature in this set implies Beal. Let n>2. If n has an odd prime divisor p, write D^n=(D^n/p)^p. If it has no odd prime divisor, it is a power of two at least four, so write D^n=(D^n/4)^4","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a19db67ec78583c0","tex":"D^{n/p}","aliases":"D^n/p","display":"D^{n/p}","kind":"use","excerpt":"on[Prime or four reduction] If a primitive solution of [source reference] exists, there is a primitive solution whose three exponents each belong to {4}∪{p:p is an odd prime}. Conversely, absence of primitive solutions for every ordered signature in this set implies Beal. Let n>2. If n has an odd prime divisor p, write D^n=(D^n/p)^p. If it has no o","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-919a967d236f383e","tex":"d^x","aliases":"d^x","display":"d^x","kind":"use","excerpt":"conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. If A=dA', B=dB', C=dC', the three ter","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-9cbb605f14528d1e","tex":"d^y","aliases":"d^y","display":"d^y","kind":"use","excerpt":"conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. If A=dA', B=dB', C=dC', the three ter","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-604af7619920f1ca","tex":"d^z","aliases":"d^z","display":"d^z","kind":"use","excerpt":"conjecture rather than contradicting it. Its powered terms are 27,216,243. figplaceholder01_beal_termsThe equation in Example~[source reference] has a common prime in its bases. Exact equality alone does not supply primitivity. An ordinary division by gcd(A,B,C) is not a valid reduction to a primitive solution. If A=dA', B=dB', C=dC', the three ter","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-614a5f01523eacf7","tex":"D_{q,q}","aliases":"D_q,q","display":"D_{q,q}","kind":"use","excerpt":"At the chosen q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [s","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a962cb3e6310a1e1","tex":"D_{q,r}","aliases":"D_q,r","display":"D_{q,r}","kind":"use","excerpt":"n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source refere","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"The same representation trace product. At depth one and an unramified place of residue norm Q≡1pmod7, write the actual Frobenius in the fixed natural frame as F=a · b 7c · d, det F=Q. Put d=1+7D and tr(F)=Q+1+7T. Then a=Q+7(T-D), and the determinant equation gives bc=T+(Q-1)D+7TD-7D^2. Modulo seven this becomes fractr(F)-Q-17=α(F)β(F) in k. This is","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and ever","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2551402eef77c95b","tex":"\\dim_k","aliases":"dim_k dim","display":"\\dim_k","kind":"use","excerpt":"The homogeneous results. The complete exact matrices, row reductions, null bases, and rank minors give crrrl Pattern · rankA · dim_kker A · dim_kcoker A · Remaining difference support","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-346650cbdd3f09a1","tex":"\\operatorname{disc}","aliases":"disc operatorname","display":"\\operatorname{disc}","kind":"use","excerpt":"[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative int","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and ever","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"Odd primes away from the coefficients. Let α,β,γ be the quadratic characters of a,b,c, and let e=(-1/p). A primitive residue solution with one zero base exists if β=γ, α=γ, α=eβ, respectively. The same cyclic group argument proves this, and the lifting construction replaces the zero root by a nonzero multiple of p.","count":10},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and ever","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained low","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f2b11b7b1856ea86","tex":"E_{12}","aliases":"E_12","display":"E_{12}","kind":"use","excerpt":"The actual period valuation matrix is 7cI, so this probe has ρ(τ_j)=I+7cPE_12P^-1pmod7^j, where the same local toric line has the required alignment with the natural line. At actual depth one and a detecting g in the residual kernel, [source reference] gives D(τ,g)=49ν_7(X)β(g)pmod343. The actual fixed seven period comparison requires ν_7(X)=cfracv","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained low","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4659d6a082e6ff3c","tex":"E_9","aliases":"E_9","display":"E_9","kind":"use","excerpt":"An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and ever","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps","display":null,"kind":"use","excerpt":"The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps in","count":13},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-08c5fee29bc1411d","tex":"F^{-1}","aliases":"F^-1","display":"F^{-1}","kind":"use","excerpt":"If α∪β were a coboundary in the endpoint case, the two adjacent entries would extend to a representation in the three by three upper unitriangular group over k. That group has exponent seven. The tame relation FTF^-1=T^Q with Q≡1pmod7 would force the images to commute. Their central commutator is α(F)β(T)-α(T)β(F)=α(F)β(T)≠0. This contradiction pro","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-85215a54939cc5cc","tex":"f^3","aliases":"f^3","display":"f^3","kind":"use","excerpt":"A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B,","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e24a7a6d282e55bc","tex":"F^{\\times7^j}","aliases":"F^×7^j times","display":"F^{\\times7^j}","kind":"use","excerpt":"Higher precision retains a valuation coordinate. For 1≤ j≤7c-1, the deep period errors vanish in F^×/F^×7^j. This follows because seventh powering takes 1+7^sOloc_F bijectively onto 1+7^s+1Oloc_F for s≥1. The guaranteed window does not assert that actual errors become nontrivial at the next precision.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-aa588c1f927e908f","tex":"F_{21}","aliases":"F_21","display":"F_{21}","kind":"use","excerpt":"-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line a","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-74317064bf2c9a4d","tex":"F_3","aliases":"F_3","display":"F_3","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-327895c54b115a65","tex":"\\F_{49}","aliases":"F_49 F","display":null,"kind":"use","excerpt":"The natural lattice and the depth of an opposite extension. The reducible branch uses two different kinds of radical. The base field is K=Q(sqrt5), ϕ=(1+sqrt5)/2. The coefficient ring and field are Oloc=Z_7[w]/(w^2-5), k=F_7[w]/(w^2-5)=F_49. The base unit ϕ defines Kummer classes. The coefficient w acts on a rank two representation. Although both i","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6d2ab9eb8cbb5298","tex":"\\F_{49}^\\times","aliases":"F_49^× F times","display":null,"kind":"use","excerpt":"The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constit","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a1a1f9e3594eef3e","tex":"F_5","aliases":"F_5","display":"F_5","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e8a465a11b7587ed","tex":"\\F_7^\\times","aliases":"F_7^× F times","display":null,"kind":"use","excerpt":"Coefficient rigidity without radical identification. The recovered coefficient theorem, initially in the descended depth one regime and later in the all depth semilinear formulation, normalizes the natural class as α=vw[ϕ]+r[3]+[sqrt5], v,r∈F_7^×. Its mechanism retains the actual joint finite quotient. The three Kummer classes are independent after","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-81de0e7e6bef8278","tex":"f_{A,B}","aliases":"f_A,B","display":"f_{A,B}","kind":"use","excerpt":"A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinate","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-fda40c00a5457366","tex":"\\F_p","aliases":"F_p F","display":null,"kind":"use","excerpt":"General Massey vanishing theorems do not erase these qualifications. The all prime theorem of Min avc and T an concerns ordinary products over absolute Galois groups with trivial F_p coefficients [source reference]. The JEMS theorem cited in some older discussion has a mod two statement [source reference]. The 2026 strong vanishing theorem likewise","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0c3489d862af388a","tex":"\\F_p^\\times","aliases":"F_p^× F times","display":null,"kind":"use","excerpt":"If pmid a, primitivity forces the other two bases to be units. Reduction gives b^5v^10=c^7w^14. The quadratic character of this equality proves the necessary symbol condition. Conversely, v^10w^-14 ranges over the squares of F_p^×, because gcd(10,14,p-1)=2. The symbol condition lets us choose the two unit roots, then lift. The other two cases use g","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-666cac5565f15579","tex":"F_q","aliases":"F_q","display":"F_q","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0, z(F_2) · =-5a-∑_iℓ_","count":7},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f9d082b0820d2930","tex":"\\F_q","aliases":"F_q F","display":null,"kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ba2ac5456d322b89","tex":"f_q","aliases":"f_q","display":"f_q","kind":"use","excerpt":"_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) =-f_q^-1∑_b=1^6bℓ_q(π_l(z_q^b)), where f_q=1 or two. Using all six terms without dividing by two in the ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3fd70bbc68ce947e","tex":"f_q^{-1}","aliases":"f_q^-1","display":"f_q^{-1}","kind":"use","excerpt":"For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-96c1f3683124b3c7","tex":"\\F_{q^2}","aliases":"F_q^2 F","display":null,"kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6912f3e4fba1ec53","tex":"F_{q_1}","aliases":"F_q_1","display":"F_{q_1}","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2214f0ff5fb39473","tex":"F_{q_9}","aliases":"F_q_9","display":"F_{q_9}","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-62e13955b6d16e4a","tex":"f_{\\lambda^5A,\\lambda^3B}","aliases":"f_λ^5A,λ^3B lambda lambda","display":"f_{\\lambda^5A,\\lambda^3B}","kind":"use","excerpt":"[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative int","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-994ee8b84ef90714","tex":"\\mathrm{fix}","aliases":"fix mathrm","display":"\\mathrm{fix}","kind":"use","excerpt":"β(g)=(χ(g)^-1-1)λ, p^fix(g)=λχ(g)^-1α(g).","count":10},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f3ad5eda9684dc2c","tex":"\\mathbb G","aliases":"G mathbb","display":"\\mathbb G","kind":"use","excerpt":"[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"For a finite module A, let A'=Hom(A,μ_7) and let G' be the finite group of the field cut out by that dual. The cited sufficient criterion is H^1(G',A')=0. For A=k with trivial action, A'≃ k(χ) and G'=C_6. Averaging kills its positive cohomology because six is invertible modulo seven. For A=k(χ), the source action cancels the cyclotomic target actio","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0a566b1db5acde7f","tex":"g_0","aliases":"g_0","display":"g_0","kind":"use","excerpt":"An actual finite segment of stable lattices. The inherited split toric calculation at three supplies Frobenius eigenvalues nine and one. Their difference eight is a seven adic unit. The projectors E_9=fracρ(g_0)-I8, E_1=frac9I-ρ(g_0)8 are complementary integral idempotents. Choose generators of their images. In this basis, ρ(g_0)=diag(9,1) and ever","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-76b53653bff85b9e","tex":"G_b","aliases":"G_b","display":"G_b","kind":"use","excerpt":"The image group contains upper and lower elements T_a,G_b with tr(T_aG_b)-tr(T_a)-tr(G_b)+2=49abpmod343. All leading residue values occur as the elements vary. This flexibility does not allow a fixed local uniformizer probe to be replaced by any convenient upper element. Preimages in the Galois group may have different local membership and differen","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2f3f7861c9f87c46","tex":"G_F","aliases":"G_F","display":"G_F","kind":"use","excerpt":"dinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u in the torus. Rationality of its torsion class gives g(u)/u∈Λ for every g∈ G_F. Since λ is rational, this quotient also lies in μ_M^2. The lattice is torsion free, so the inters","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f7288fb9331c8dd5","tex":"g_j","aliases":"g_j","display":"g_j","kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-82170a73bb5aa973","tex":"G_K","aliases":"G_K","display":"G_K","kind":"use","excerpt":"of j, one has ρ(j^2)=I. The coefficient unit (1+w)/2 has norm minus one, so dividing U by it gives a semilinear involution. An adapted basis of its plus lattice produces a global endpoint preserving integral frame with ρ(jgj^-1)=σ(ρ(g)). This is an existential normal form. The unknown candidate frame has not been numerically constructed. Its plus l","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-56337fae486b3d21","tex":"\\mathbb G_m^2","aliases":"G_m^2 mathbb","display":"\\mathbb G_m^2","kind":"use","excerpt":"[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7eadc3987326373a","tex":"g_q","aliases":"g_q","display":"g_q","kind":"use","excerpt":", the two independent pro seven unit coordinates are ϕ,3, whose normalized logarithmic first digits are 5sqrt5,6, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is 3+3+2·9=24","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f9ec47b38a5a9555","tex":"g_{q_i}","aliases":"g_q_i","display":"g_{q_i}","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c5d62edc1cdf57e7","tex":"\\mathrm{GL}","aliases":"GL mathrm","display":"\\mathrm{GL}","kind":"use","excerpt":"A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4d4a5d6f6abf65b4","tex":"\\mathrm{GL}_2","aliases":"GL_2 mathrm","display":"\\mathrm{GL}_2","kind":"use","excerpt":"A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ec94e8b35fa92869","tex":"\\Gsig","aliases":"Gsig","display":null,"kind":"use","excerpt":"A bare primitive on the justified ramification set. Fix one of the nine prime beta supports S in Section~[source reference]. Let Σ consist of S, the places above 2,3,5,7, and the two real places. The residual classes factor through Gsig. At every beta support place, χ is trivial modulo seven and the natural local class is zero. Hence its normalized","count":7},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d535d4f2f3619d09","tex":"H^{(1+\\eps)s}","aliases":"H^(1+eps)s eps","display":null,"kind":"use","excerpt":"gnature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"The inherited characteristic zero representation is the specified RM factor of the twisted Jacobian on the retained 3990mid C branch, with determinant the cyclotomic character χ. Its natural residual orientation is 0⟶ k(χ)⟶overline V⟶ k⟶0. The opposite endpoint has a nonzero class β∈ H^1(K,k(χ^-1)). A stable lattice change can alter residual inerti","count":7},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4e081e647d972a26","tex":"H^{71/105}","aliases":"H^71/105","display":"H^{71/105}","kind":"use","excerpt":"Why a generic radical ceiling fails. The exponent equation gives R<H^71/105 for the signature (3,5,7). It is tempting to reverse this relation by postulating H≤ R^105/71 for every primitive sum. That universal assertion is false. The exact equality 2+109·3^10=23^5=6 436 343 has radical 2·3·109·23=15 042, and (23^5)^71>15 042^105. Its logarithmic qu","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-026e8ef245e5bd54","tex":"H^s","aliases":"H^s","display":"H^s","kind":"use","excerpt":"The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps in","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8c7da95775a0626f","tex":"H_j","aliases":"H_j","display":"H_j","kind":"use","excerpt":"Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-113141cd0920367c","tex":"h_q","aliases":"h_q","display":"h_q","kind":"use","excerpt":"The arithmetic target. Choose a bare global primitive p_0 from Theorem~[source reference]. This is one continuous global cochain. Its values have not been computed. Define t(p_0)∈ k^15 as follows. Its first six coordinates are the ordinary local character coordinates of p_v^fix-p_0|_v. At each support prime, put t_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-858e4bb65a33c6eb","tex":"h_\\ell","aliases":"h_ℓ ell","display":"h_\\ell","kind":"use","excerpt":"An integral h exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in t or 1-t, so λ=∏_ℓℓ^h_ℓ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every h_ℓ was forced, so two positive choices of λ would have ratio a positiv","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e0d53229e96ed0e9","tex":"\\operatorname{Hom}","aliases":"Hom operatorname","display":"\\operatorname{Hom}","kind":"use","excerpt":"For a finite module A, let A'=Hom(A,μ_7) and let G' be the finite group of the field cut out by that dual. The cited sufficient criterion is H^1(G',A')=0. For A=k with trivial action, A'≃ k(χ) and G'=C_6. Averaging kills its positive cohomology because six is invertible modulo seven. For A=k(χ), the source action cancels the cyclotomic target actio","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"[X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B A · 0, M^2=A^3B I_2. These are polynomial and matrix identities. Their geometric interpretation requires the correct Cartier and Hasse--Witt conventions. On a good reduction fiber at seven they imply ordinarity when AB is non","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-69a4780a0be9987e","tex":"I_q","aliases":"I_q","display":"I_q","kind":"use","excerpt":"ormalized logarithmic first digits are 5sqrt5,6, and the third coordinate is the uniformizer seven. Each of the nine inert support primes has one seventh power residue coordinate and one unramified coordinate. Real odd primary characters vanish. The total local dimension is 3+3+2·9=24 over k. The chosen g_q=ϕ+a_q generating the residue seventh powe","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-149a1e2c802ce22c","tex":"I_{q_1}","aliases":"I_q_1","display":"I_{q_1}","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8ad53a4b68f38ccc","tex":"I_{q_9}","aliases":"I_q_9","display":"I_{q_9}","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a5e46bfc661e3c45","tex":"I_{q_i}","aliases":"I_q_i","display":"I_{q_i}","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im","display":null,"kind":"use","excerpt":"These equations define a 24 by 10 matrix J. The ten free parameters occur in their declared local coordinates, so J is injective. Direct substitution gives LJ=0. Since both im J and ker L have dimension ten, im J=ker L. Every parameter vector gives a global ordinary character. This is not merely an overapproximation by necessary equations.","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u ","count":11},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-787a425baaf96eca","tex":"j^{-1}","aliases":"j^-1","display":"j^{-1}","kind":"use","excerpt":"Integral semilinearity and the unlabelled image. The integral semilinear theorem supplies more than equality of residual cohomology classes. Under the retained geometric inputs, an actual map between the conjugate curves induces an integral operator U=Cσ on the Tate lattice, with ρ(jgj^-1)=Cσ(ρ(g))C^-1, Cσ(C)=-ρ(j^2), C_21∈7^nOloc. It preserves the","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-596b3f5156c743a8","tex":"j^2","aliases":"j^2","display":"j^2","kind":"use","excerpt":"Integral semilinearity and the unlabelled image. The integral semilinear theorem supplies more than equality of residual cohomology classes. Under the retained geometric inputs, an actual map between the conjugate curves induces an integral operator U=Cσ on the Tate lattice, with ρ(jgj^-1)=Cσ(ρ(g))C^-1, Cσ(C)=-ρ(j^2), C_21∈7^nOloc. It preserves the","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d7da978e293880ea","tex":"J^{\\mathrm{an}}","aliases":"J^an mathrm","display":"J^{\\mathrm{an}}","kind":"use","excerpt":"[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a6e2593816150d2a","tex":"k^{10}","aliases":"k^10","display":"k^{10}","kind":"use","excerpt":"On the local space k^24, let E have fifteen rows. Its first six rows select all fixed local character coordinates in [source reference]. Its remaining nine rows are z⟼ c_qz(F_q)-b_qz(I_q). Define A=EJ:k^10⟶ k^15. The self conductor values and inertia conversions in Section~[source reference] are part of this operator. Dropping them would reproduce ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c6f57e081b8f3b06","tex":"k^{15}","aliases":"k^15","display":"k^{15}","kind":"use","excerpt":"On the local space k^24, let E have fifteen rows. Its first six rows select all fixed local character coordinates in [source reference]. Its remaining nine rows are z⟼ c_qz(F_q)-b_qz(I_q). Define A=EJ:k^10⟶ k^15. The self conductor values and inertia conversions in Section~[source reference] are part of this operator. Dropping them would reproduce ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"A small model of the mistake to avoid. Consider A:k→ k^2 given by A(z)=(z,0). Its kernel is zero, its rank is one, and it has a full left null functional N(u,v)=v. For target (a,b), a solution exists precisely when b=0. The map has a one dimensional image and a zero kernel, but this gives no information about the target's second coordinate. Substit","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f0f3664d299ba0a8","tex":"k^{24}","aliases":"k^24","display":"k^{24}","kind":"use","excerpt":"On the local space k^24, let E have fifteen rows. Its first six rows select all fixed local character coordinates in [source reference]. Its remaining nine rows are z⟼ c_qz(F_q)-b_qz(I_q). Define A=EJ:k^10⟶ k^15. The self conductor values and inertia conversions in Section~[source reference] are part of this operator. Dropping them would reproduce ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5f2772597a802504","tex":"k^9","aliases":"k^9","display":"k^9","kind":"use","excerpt":"For the frozen supports, each supporting local field has residue norm one modulo seven and contains μ_7. Local duality gives one k dimensional H^2 term there. The four fixed fields do not contain μ_7 and have zero H^2(K_v,k). Global dual invariants also vanish. The final portion of the Poitou--Tate sequence then identifies the restricted H^2 with k","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6426539d2f6a0e3f","tex":"K_{1/22}","aliases":"K_1/22","display":"K_{1/22}","kind":"use","excerpt":"There is a stronger uniform bound across the Beal range. Apart from (3,3,3), already excluded by Fermat, one exponent is at least four. Thus s≤1/3+1/3+1/4=11/12. Take eps=1/22 and enlarge its constant to K_1/22≥1. Then (1+eps)s≤23/24, and [source reference] gives H<K_1/22^24. This conditional constant is independent of the signature. It bounds the ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-064a64e6a432d13d","tex":"K_{1/22}^{24}","aliases":"K_1/22^24","display":"K_{1/22}^{24}","kind":"use","excerpt":"There is a stronger uniform bound across the Beal range. Apart from (3,3,3), already excluded by Fermat, one exponent is at least four. Thus s≤1/3+1/3+1/4=11/12. Take eps=1/22 and enlarge its constant to K_1/22≥1. Then (1+eps)s≤23/24, and [source reference] gives H<K_1/22^24. This conditional constant is independent of the signature. It bounds the ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-868e2d5d224d08cb","tex":"K_7","aliases":"K_7","display":"K_7","kind":"use","excerpt":"To define the valuation functional used here, strictly align the actual local toric cyclotomic line at K_7 with the reference line. The aligned local representative of X has p=β=0 and upper entry q a k(χ) Kummer cocycle. Define ν_7(X) to be its [7] valuation coordinate. Remaining aligned strict gauges change q by a coboundary and a scalar multiple ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-80fe6ef3c06d8bca","tex":"K_\\eps","aliases":"K_eps eps","display":null,"kind":"use","excerpt":"The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps in","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-291eeb02baebad13","tex":"K_\\eps^{1/(1-(1+\\eps)s)}","aliases":"K_eps^1/(1-(1+eps)s) eps eps","display":null,"kind":"use","excerpt":"For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps independent of the triple. Then H<K_eps H^(1+eps)s. Whenever (1+eps)s<1, it follows that H<K_eps^1/(1-(1+eps)s). This is a conditional bound,","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5d29756a663fceec","tex":"K_\\eps^{105/(34-71\\eps)}","aliases":"K_eps^105/(34-71eps) eps eps","display":null,"kind":"use","excerpt":"For (3,5,7), the reciprocal sum is 71/105. Thus 0<eps<34/71 gives H<K_eps^105/(34-71eps), C<K_eps^15/(34-71eps). At eps=1/10, the latter exponent is 150/269. To turn this into a proof of absence would require the abc input, an explicit usable constant, and a complete verification below the resulting bound.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7d315370a1c3127e","tex":"K_\\eps^{15/(34-71\\eps)}","aliases":"K_eps^15/(34-71eps) eps eps","display":null,"kind":"use","excerpt":"For (3,5,7), the reciprocal sum is 71/105. Thus 0<eps<34/71 gives H<K_eps^105/(34-71eps), C<K_eps^15/(34-71eps). At eps=1/10, the latter exponent is 150/269. To turn this into a proof of absence would require the abc input, an explicit usable constant, and a complete verification below the resulting bound.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"The all depth formula uses t=1+v_7(c), precision j≥ n+t+1, the probe τ_j, and a detecting g∈keroverlineρ with β(g)≠0. It is v_7(D(τ_j,g))=n+t, fracD(τ_j,g)7^n+t =fracc7^t-1fracv2ϵβ(g)pmod7. The coefficient on the right is nonzero in the detecting case. This is an exact valuation statement under the probe and actual geometric hypotheses. It does not","count":11},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"If no variable C prime had Q≡1pmod7, beta would restrict to an everywhere unramified additive class over L=K(ζ_7). Restriction is injective because [L:K]=6 is prime to seven. The inherited Masley class number result gives h(Q(ζ_35))=1 [source reference]. The field L is a subfield of index two. Extension and norm on ideal classes compose to multipli","count":16},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"[Primitive scaling criterion] Fix a rational 0<t<1. There is a positive rational λ for which [source reference] gives primitive positive integers if and only if, for every rational prime ℓ, v_ℓ(t)>0 · Longrightarrowquad3mid v_ℓ(t), v_ℓ(1-t)>0 · Longrightarrowquad5mid v_ℓ(1-t), v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-039ec3e8ab324910","tex":"\\mathfrak m","aliases":"m mathfrak","display":"\\mathfrak m","kind":"use","excerpt":"At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. T","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. Write m=v_ℓ(t), n=v_ℓ(1-t) and h=v_ℓ(λ). Because t+(1-t)=1, the pair (m,n) belongs to one of four sectors: (m>0,n=0), (m=0,n>0), (m=n<0), (m=n=0). The output valuations are a_ℓ · =35h+12m+28n,","count":21},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","excerpt":"[X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B A · 0, M^2=A^3B I_2. These are polynomial and matrix identities. Their geometric interpretation requires the correct Cartier and Hasse--Witt conventions. On a good reduction fiber at seven they imply ordinarity when AB is non","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"with the first two bases. Primitivity becomes stronger on an actual solution than it is on an arbitrary integer triple. For a solution of [source reference], every prime dividing two bases divides the third. Consequently a primitive solution has pairwise coprime bases. If a prime p divides A and B, the equation gives pmid C^z, hence pmid C. If p di","count":20},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained low","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8e82b2dd54f2b4d1","tex":"\\min_{g:\\rho(g)_{21}\\ne0}","aliases":"min_g:ρ(g)_21≠0 min rho ne","display":"\\min_{g:\\rho(g)_{21}\\ne0}","kind":"use","excerpt":"Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained low","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":28},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"the same for the three exponents. [Prime or four reduction] If a primitive solution of [source reference] exists, there is a primitive solution whose three exponents each belong to {4}∪{p:p is an odd prime}. Conversely, absence of primitive solutions for every ordered signature in this set implies Beal. Let n>2. If n has an odd prime divisor p, wri","count":43},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"item[H2.] At every variable bad place the actual inertia has I+N with N^2=0 and N in 7M_2(Oloc). Its primitive line factorization and same frame Frobenius obey the toric equations. Good inertia is trivial. item[H3.] Outside the four fixed rational primes 2,3,5,7, every possible bad place divides C. The reference is unramified there. item[H4.] At ea","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ffaa0ad23479bca8","tex":"N_1","aliases":"N_1","display":"N_1","kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-72ec741403c87a9e","tex":"N_2","aliases":"N_2","display":"N_2","kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-243f195ec0356ebe","tex":"N_{21}","aliases":"N_21","display":"N_{21}","kind":"use","excerpt":"-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line a","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-9ee7586b85e394f6","tex":"\\operatorname{Nm}","aliases":"Nm operatorname","display":"\\operatorname{Nm}","kind":"use","excerpt":"If the actual polynomial is P(X)=X^4-SX^3+BX^2-QSX+Q^2, the correction u=(τ-Q-1)/7 has Tr(u)=fracS-2(Q+1)7, Nm(u)=frac(B-2Q)-(Q+1)S+(Q+1)^249 modulo seven, under the stated divisibility assumptions. The two implementations extract these values using different methods, including an integer trace discriminant and an enumeration of k by trace and norm","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0b7ad9c468e127af","tex":"\\operatorname{Nm}_{\\F_{q^2}/\\F_q}","aliases":"Nm_F_q^2/F_q operatorname F F","display":null,"kind":"use","excerpt":"The ordinary inertia generator differs from the frozen twisted residue convention. Its beta value is c_q=s_qℓ_q(Nm_F_q^2/F_qg_q) (q≡1pmod7), and c_q=s_qℓ_q(g_q) for q≡-1pmod7. The conversion factors are rrrrr 197:5 · 307:2 · 337:5 · 547:4 · 673:1","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"What the direct irreducibility argument proves. The canonical recovered result is absolute irreducibility of the associated residual rank two RM representation when 14nmid C, subject to its declared realization, conductor, local weight, and finite flat inputs. Its two branches use different information.","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","excerpt":"At seven, the specified twist makes the toric characters χ and one, and cyclotomic inertia is nontrivial. At a variable bad place with Qnot≡1pmod7, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality Q, an inertia ","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3b001e099c6c65f8","tex":"\\Oloc^2","aliases":"Oloc^2 Oloc","display":null,"kind":"use","excerpt":"-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line a","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-31ae00e9af121a5d","tex":"\\Oloc^\\times","aliases":"Oloc^× Oloc times","display":null,"kind":"use","excerpt":"A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-fbc5040fc2bc4d18","tex":"\\Oloc_F","aliases":"Oloc_F Oloc","display":null,"kind":"use","excerpt":"The fixed seven period target. At seven, principal units are not automatically seventh powers. The residue class of 8=1+7 is a simple warning. The away from seven period simplification cannot be copied without an error bound. The inherited integral marked family supplies an error in 1+7^7cOloc_F, where c=v_7(C)≥1 and F=Q_7(sqrt5). That specific err","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ad21f4f96eb368f9","tex":"\\overline V","aliases":"overline V overline","display":"\\overline V","kind":"use","excerpt":"The inherited characteristic zero representation is the specified RM factor of the twisted Jacobian on the retained 3990mid C branch, with determinant the cyclotomic character χ. Its natural residual orientation is 0⟶ k(χ)⟶overline V⟶ k⟶0. The opposite endpoint has a nonzero class β∈ H^1(K,k(χ^-1)). A stable lattice change can alter residual inerti","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-bd7258dddc26dd8c","tex":"\\overline{\\sqrt5}","aliases":"overlinesqrt5 overline sqrt","display":"\\overline{\\sqrt5}","kind":"use","excerpt":"Let X=[ϕ] and H=[3] in the mod seven local Kummer group. Exact logarithmic first digits give L(ϕ)=5overlinesqrt5, L(3)=6, L(sqrt5)=3, L(u)=fracu^48-17pmod7. The two unit coordinates X,H are independent, and [sqrt5]=4H. For a rational seven adic unit A, put q_7(A)=(A^6-1)/7 modulo seven. The local marked period comparison has P=ZI+2XR, Z=(3q_7(A)+5)","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-881e84403d3704c7","tex":"\\overline\\rho","aliases":"overlineρ overline rho","display":"\\overline\\rho","kind":"use","excerpt":"beta(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-597563e926cbce61","tex":"\\overline\\rho_0","aliases":"overlineρ_0 overline rho","display":"\\overline\\rho_0","kind":"use","excerpt":"beta(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3e762917bd13da17","tex":"\\overline\\phi","aliases":"overlineϕ overline phi","display":"\\overline\\phi","kind":"use","excerpt":"The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constit","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ab2002ed3b2dd11b","tex":"\\overline\\phi^{8e}","aliases":"overlineϕ^8e overline phi","display":"\\overline\\phi^{8e}","kind":"use","excerpt":"The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constit","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2r","count":6},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d7a1e9546e0a2b44","tex":"p'_0","aliases":"p'_0","display":"p'_0","kind":"use","excerpt":"Now replace the origin by p'_0=p_0+z_0,char, where its localization is Jz_0. At the fixed places, the target differences decrease by the selected coordinates of this character. At support, [source reference] decreases by c_qz_0,char(F_q)-b_qz_0,char(T_q). Therefore t(p'_0)=t(p_0)-Az_0. The quotient class and Nt are unchanged. The coordinate basis f","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3401330892ce4d79","tex":"P^{-1}","aliases":"P^-1","display":"P^{-1}","kind":"use","excerpt":"The actual period valuation matrix is 7cI, so this probe has ρ(τ_j)=I+7cPE_12P^-1pmod7^j, where the same local toric line has the required alignment with the natural line. At actual depth one and a detecting g in the residual kernel, [source reference] gives D(τ,g)=49ν_7(X)β(g)pmod343. The actual fixed seven period comparison requires ν_7(X)=cfracv","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2ccf806a66ac47c1","tex":"p^{\\mathrm{fix}}","aliases":"p^fix mathrm","display":"p^{\\mathrm{fix}}","kind":"use","excerpt":"β(g)=(χ(g)^-1-1)λ, p^fix(g)=λχ(g)^-1α(g).","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d3cdc9ecc001d6c4","tex":"p^n","aliases":"p^n","display":"p^n","kind":"use","excerpt":"al derivatives are 6a^3u^5,qquad10b^5v^9, -14c^7w^13. In a primitive residue solution, at least two bases are units. At an odd prime, at most one of 6,10,14 is divisible by p. Thus at least one unit variable has unit derivative. Lift the other variables to nonzero integers, using p for a zero residue if necessary. Successive correction in the selec","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"If p_0 is one global bare primitive and p_v are desired finite local primitives, their differences p_v-p_0|_v are ordinary characters. For trivial coefficients there are no nonzero coboundaries. Surjectivity supplies one global character matching every prescribed finite difference exactly. Then p=p_0+z meets those local equations, but may ramify at","count":30},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-26a9ec94734fbef0","tex":"p_v","aliases":"p_v","display":"p_v","kind":"use","excerpt":"If p_0 is one global bare primitive and p_v are desired finite local primitives, their differences p_v-p_0|_v are ordinary characters. For trivial coefficients there are no nonzero coboundaries. Surjectivity supplies one global character matching every prescribed finite difference exactly. Then p=p_0+z meets those local equations, but may ramify at","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1c4c5cbbe366ba53","tex":"p_v^{\\mathrm{fix}}","aliases":"p_v^fix mathrm","display":"p_v^{\\mathrm{fix}}","kind":"use","excerpt":"At a fixed place where beta is the coboundary of λ_v and p=p_v^fix, the explicit local upper primitive q_v^fix(g)=-λ_vχ(g)^-1α(g)^2 satisfies d_χ q_v^fix=R_p by direct expansion. At support, α=0 also kills the local upper right side. These facts prevent treating a formal secondary obstruction as an established nonzero obstruction in the actual loca","count":6},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"If p≡1pmod4, two of the three signs agree, so one of these conditions holds. If p≡3pmod4 and all three fail, then α=β=-γ. The two remaining sign patterns can be handled by the universal identity of Proposition~[source reference].","count":37},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7efa8a82527e585b","tex":"\\mathrm{PV}","aliases":"PV mathrm","display":"\\mathrm{PV}","kind":"use","excerpt":"gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"At seven, the specified twist makes the toric characters χ and one, and cyclotomic inertia is nontrivial. At a variable bad place with Qnot≡1pmod7, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality Q, an inertia ","count":55},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"use","excerpt":"When C is odd, the integral model has good reduction at two. Over the norm four prime of K=Q(sqrt5), the full Frobenius polynomial is either (T^2+T+4)^2 or (T^2+4)^2. A hypothetical degree one constituent, after the conductor and global unit argument, must have eighth power one. Neither quadratic has a root with eighth power one modulo seven. For T","count":13},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2bdc013a59de0c4b","tex":"\\mathfrak q","aliases":"q mathfrak","display":"\\mathfrak q","kind":"use","excerpt":"Suppose a Hecke eigenvalue a_ q is replaced by its minimal polynomial. A resultant against a possible curve trace polynomial asks whether some conjugate can match some allowable trace. Repeating the test at a different prime can choose a different conjugate. If the original representation requires one residue embedding for every prime, this relaxed","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"Two exact local restrictions were established under the stated modular and multiplicative prime comparisons. A rank two trace zero comparison at a prime above eleven forces 11mid A. At a prime above nineteen it forces 19mid B. In either case the full four dimensional reduction to check is (T^2+q)^2pmod7, which is stronger than zero four dimensional","count":99},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f3003d8698b757c1","tex":"Q^{-1}","aliases":"Q^-1","display":"Q^{-1}","kind":"use","excerpt":"e same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality Q, an inertia cocycle with coefficients k(χ^-1) can be nonzero only when Q^2≡1pmod7: the tame Frobenius relation multiplies its inertia value by Q, while the coeffi","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"use","excerpt":"es the toric characters χ and one, and cyclotomic inertia is nontrivial. At a variable bad place with Qnot≡1pmod7, the same argument applies. At the places over two, three and five, a separate local cohomology calculation excludes opposite extensions. For a non seven local field of residue cardinality Q, an inertia cocycle with coefficients k(χ^-1)","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":6},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-cb928b4e19802c06","tex":"q^c","aliases":"q^c","display":"q^c","kind":"use","excerpt":"At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m),","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-fc7c1af9156efca5","tex":"q^f","aliases":"q^f","display":"q^f","kind":"use","excerpt":"At least one variable prime above a divisor of C must consequently have norm one modulo seven. A rational prime splits in K when it is one or four modulo five, giving residue degree one. At an inert prime the degree is two. Combining q^f≡1pmod7 with these splitting possibilities gives exactly qpmod35∈{1,8,13,22,27,29}. None of 2,3,5,7,19, the prime","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","excerpt":"d H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordin","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-059d2c9fe2f7f2fd","tex":"\\Q_7","aliases":"Q_7 Q","display":null,"kind":"use","excerpt":"The fixed seven period target. At seven, principal units are not automatically seventh powers. The residue class of 8=1+7 is a simple warning. The away from seven period simplification cannot be copied without an error bound. The inherited integral marked family supplies an error in 1+7^7cOloc_F, where c=v_7(C)≥1 and F=Q_7(sqrt5). That specific err","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-82c9d7c971d50597","tex":"q_7","aliases":"q_7","display":"q_7","kind":"use","excerpt":"Let X=[ϕ] and H=[3] in the mod seven local Kummer group. Exact logarithmic first digits give L(ϕ)=5overlinesqrt5, L(3)=6, L(sqrt5)=3, L(u)=fracu^48-17pmod7. The two unit coordinates X,H are independent, and [sqrt5]=4H. For a rational seven adic unit A, put q_7(A)=(A^6-1)/7 modulo seven. The local marked period comparison has P=ZI+2XR, Z=(3q_7(A)+5)","count":7},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e06d1d30e176304d","tex":"q_9","aliases":"q_9","display":"q_9","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-506afac37903bdfd","tex":"Q_A","aliases":"Q_A","display":"Q_A","kind":"use","excerpt":"dinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3cfa34a20466eab7","tex":"Q_A^2","aliases":"Q_A^2","display":"Q_A^2","kind":"use","excerpt":"te is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Ne","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ce3c700be6795779","tex":"q_v^{\\mathrm{fix}}","aliases":"q_v^fix mathrm","display":"q_v^{\\mathrm{fix}}","kind":"use","excerpt":"At a fixed place where beta is the coboundary of λ_v and p=p_v^fix, the explicit local upper primitive q_v^fix(g)=-λ_vχ(g)^-1α(g)^2 satisfies d_χ q_v^fix=R_p by direct expansion. At support, α=0 also kills the local upper right side. These facts prevent treating a formal secondary obstruction as an established nonzero obstruction in the actual loca","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"In the first sector these weighted valuations are 105h+36m,105h+35m,105h+35m. Therefore h=-m/3, and the output is (m/3,0,0). In the second sector they are 105h+84n,105h+85n,105h+84n. This forces h=-4n/5 and output (0,n/5,0). In the third sector write m=n=-r with r>0. The weighted valuations are 105h-120r,105h-120r,105h-119r. Thus h=8r/7 and the out","count":27},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1fc80f421d2bcc4b","tex":"R^{1+\\eps}","aliases":"R^1+eps eps","display":null,"kind":"use","excerpt":"The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps in","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a76e14b452aac6fd","tex":"R^{105/71}","aliases":"R^105/71","display":"R^{105/71}","kind":"use","excerpt":"Why a generic radical ceiling fails. The exponent equation gives R<H^71/105 for the signature (3,5,7). It is tempting to reverse this relation by postulating H≤ R^105/71 for every primitive sum. That universal assertion is false. The exact equality 2+109·3^10=23^5=6 436 343 has radical 2·3·109·23=15 042, and (23^5)^71>15 042^105. Its logarithmic qu","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6c4a957eeb9c95f7","tex":"r^7","aliases":"r^7","display":"r^7","kind":"use","excerpt":"Its mechanism retains the actual joint finite quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good split primes, where the same quartic at the paired K places forces products to be coefficient conjugat","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a358d65c8ced8438","tex":"R_{l,q}","aliases":"R_l,q","display":"R_{l,q}","kind":"use","excerpt":"For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-650ae3274f788b89","tex":"\\rad","aliases":"rad","display":null,"kind":"use","excerpt":"The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps in","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"The homogeneous results. The complete exact matrices, row reductions, null bases, and rank minors give crrrl Pattern · rankA · dim_kker A · dim_kcoker A · Remaining difference support","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ed7f66675ed124b7","tex":"\\mathrm{reps}","aliases":"reps mathrm","display":"\\mathrm{reps}","kind":"use","excerpt":"For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-90bb1f0db20aebb0","tex":"\\mathrm{reps}_q","aliases":"reps_q mathrm","display":"\\mathrm{reps}_q","kind":"use","excerpt":"For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6ea458883321123f","tex":"\\mathrm{reps}_r","aliases":"reps_r mathrm","display":"\\mathrm{reps}_r","kind":"use","excerpt":"n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source refere","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2r","count":10},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1d0c661e8cc3752e","tex":"s^{12}","aliases":"s^12","display":"s^{12}","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d716f41976d24ed6","tex":"s^{17}","aliases":"s^17","display":"s^{17}","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c8d0b37b4af3bfcc","tex":"S^2","aliases":"S^2","display":"S^2","kind":"use","excerpt":"beginlemma[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) sati","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-50accd8ea4405896","tex":"s^{28}","aliases":"s^28","display":"s^{28}","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f07087b1b704c6d4","tex":"s^{84}","aliases":"s^84","display":"s^{84}","kind":"use","excerpt":"Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2954e4176316de8d","tex":"s^{85}","aliases":"s^85","display":"s^{85}","kind":"use","excerpt":"ntity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the powered terms by th","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f764dd5fc277a269","tex":"s_1^2","aliases":"s_1^2","display":"s_1^2","kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5b939a36dfaae506","tex":"s_q","aliases":"s_q","display":"s_q","kind":"use","excerpt":"the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding seventh power logarithm, the local functional at the embedding ζ_7↦ z_q^b is b s_qℓ_q.","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a6bf5eb7ecfc145e","tex":"s_r","aliases":"s_r","display":"s_r","kind":"use","excerpt":"Because each support prime is inert in K/Q, restriction at a rational q≡1pmod7 multiplies the compatible rational Frobenius value by two: b_q=2∑_rD_q,rs_r. For q≡-1pmod7, the compatible squared rational Frobenius has beta value zero, from the cocycle factor 1+χ(q)^-1=0. This does not say beta is unramified there.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2cf435253c47680b","tex":"\\Sha","aliases":"Sha","display":null,"kind":"use","excerpt":"[The dual Kummer kernel] For every finite Σ containing the prime above seven and all real places, Sha^1_Σ(K,μ_7) =ker(H^1(Gsig,μ_7) ⟶∏_v∈ΣH^1(K_v,μ_7))=0. The discriminant of K is five. Minkowski's ideal bound is sqrt5/2<2, so every ideal class has a representative of norm one. Thus Cl(K)=1 [source reference].","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c8a5bf6aad4fd2ee","tex":"\\Sha^1_\\Sigma","aliases":"Sha^1_Σ Sha Sigma","display":null,"kind":"use","excerpt":"[The dual Kummer kernel] For every finite Σ containing the prime above seven and all real places, Sha^1_Σ(K,μ_7) =ker(H^1(Gsig,μ_7) ⟶∏_v∈ΣH^1(K_v,μ_7))=0. The discriminant of K is five. Minkowski's ideal bound is sqrt5/2<2, so every ideal class has a representative of norm one. Thus Cl(K)=1 [source reference].","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c0722246d8d8ecf0","tex":"\\Sha^2_\\Sigma","aliases":"Sha^2_Σ Sha Sigma","display":null,"kind":"use","excerpt":"Finite set Poitou--Tate duality pairs this group perfectly with Sha^2_Σ(K,k). Every prime dividing the module order belongs to Σ, as required by the cited theorem. Odd primary real Tate terms vanish. Thus localization H^2(Gsig,k)⟶⊕_v∈ΣH^2(K_v,k) is injective [source reference].","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3962ebf78617b637","tex":"\\mathfrak{sl}","aliases":"sl mathfrak","display":"\\mathfrak{sl}","kind":"use","excerpt":"Every determinant χ framed lift of this residual representation has a unique expression ρ(g)=(I+7X(g))ρ_0(g), X(g)=p(g) · q(g) β(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-34391b093952f50f","tex":"\\mathfrak{sl}_2","aliases":"sl_2 mathfrak","display":"\\mathfrak{sl}_2","kind":"use","excerpt":"Every determinant χ framed lift of this residual representation has a unique expression ρ(g)=(I+7X(g))ρ_0(g), X(g)=p(g) · q(g) β(g) · -p(g)∈sl_2(k). The determinant condition makes X traceless. Multiplication shows that ρ is a representation if and only if X(gh)=X(g)+Ad(overlineρ_0(g))X(h). For overlineρ_0(g)=(x · a","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"When C is odd, the integral model has good reduction at two. Over the norm four prime of K=Q(sqrt5), the full Frobenius polynomial is either (T^2+T+4)^2 or (T^2+4)^2. A hypothetical degree one constituent, after the conductor and global unit argument, must have eighth power one. Neither quadratic has a root with eighth power one modulo seven. For T","count":43},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":59},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5d42311231194d5c","tex":"t^{-1}","aliases":"t^-1","display":"t^{-1}","kind":"use","excerpt":"The Block 47 audit identified this mechanism in the pinned Pacetti--Villagra Torcomian code. It also identified a parameter dictionary issue at eleven: the unlabelled multisets agreed, but individual labels matched after t↦ t^-1. A refinement attached to the wrong labels could incorrectly combine local constraints. The new reader therefore distingu","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-bf72b88eeae69e74","tex":"t^{12}","aliases":"t^12","display":"t^{12}","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"When C is odd, the integral model has good reduction at two. Over the norm four prime of K=Q(sqrt5), the full Frobenius polynomial is either (T^2+T+4)^2 or (T^2+4)^2. A hypothetical degree one constituent, after the conductor and global unit argument, must have eighth power one. Neither quadratic has a root with eighth power one modulo seven. For T","count":10},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-492f47659650b120","tex":"t^{35}","aliases":"t^35","display":"t^{35}","kind":"use","excerpt":"ntity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the powered terms by th","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7d2169064c4849f9","tex":"t^{36}","aliases":"t^36","display":"t^{36}","kind":"use","excerpt":"Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-eee8aceba3736f68","tex":"T^4","aliases":"T^4","display":"T^4","kind":"use","excerpt":"From point counts to a common polynomial. For a smooth genus two curve over F_q, let N_1,N_2 count its points over F_q,F_q^2, including infinity. Write s_1=q+1-N_1, s_2=fracs_1^2-(q^2+1-N_2)2. Its Frobenius polynomial is T^4-s_1T^3+s_2T^2-qs_1T+q^2. In the RM setting, the rank two traces are the roots of X^2-s_1X+(s_2-2q). The two roots belong to o","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8c6354f5f56778af","tex":"t^5","aliases":"t^5","display":"t^5","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-98f93e20c13d2c57","tex":"t^6","aliases":"t^6","display":"t^6","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-179195799d88b1fc","tex":"t^7","aliases":"t^7","display":"t^7","kind":"use","excerpt":"A rational parametrization and its integral defect. For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weig","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-54815db7d2265570","tex":"T^m","aliases":"T^m","display":"T^m","kind":"use","excerpt":"Changing F to T^mF adds mc to b and mp(T) to p(F), so the left side of [source reference] is unchanged. A consistent Frobenius lift change cannot remove the affine right side.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-197961be385026de","tex":"T^Q","aliases":"T^Q","display":"T^Q","kind":"use","excerpt":"If α∪β were a coboundary in the endpoint case, the two adjacent entries would extend to a representation in the three by three upper unitriangular group over k. That group has exponent seven. The tame relation FTF^-1=T^Q with Q≡1pmod7 would force the images to commute. Their central commutator is α(F)β(T)-α(T)β(F)=α(F)β(T)≠0. This contradiction pro","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"use","excerpt":"The image group contains upper and lower elements T_a,G_b with tr(T_aG_b)-tr(T_a)-tr(G_b)+2=49abpmod343. All leading residue values occur as the elements vary. This flexibility does not allow a fixed local uniformizer probe to be replaced by any convenient upper element. Preimages in the Galois group may have different local membership and differen","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"use","excerpt":"Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-79b896ccd56f4035","tex":"T_q","aliases":"T_q","display":"T_q","kind":"use","excerpt":"The arithmetic target. Choose a bare global primitive p_0 from Theorem~[source reference]. This is one continuous global cochain. Its values have not been computed. Define t(p_0)∈ k^15 as follows. Its first six coordinates are the ordinary local character coordinates of p_v^fix-p_0|_v. At each support prime, put t_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ebaca6014c6a2575","tex":"t_q","aliases":"t_q","display":"t_q","kind":"use","excerpt":"The arithmetic target. Choose a bare global primitive p_0 from Theorem~[source reference]. This is one continuous global cochain. Its values have not been computed. Define t(p_0)∈ k^15 as follows. Its first six coordinates are the ordinary local character coordinates of p_v^fix-p_0|_v. At each support prime, put t_q(p_0)=-h_qc_q-c_qp_0(F_q)+b_qp_0(","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"If the actual polynomial is P(X)=X^4-SX^3+BX^2-QSX+Q^2, the correction u=(τ-Q-1)/7 has Tr(u)=fracS-2(Q+1)7, Nm(u)=frac(B-2Q)-(Q+1)S+(Q+1)^249 modulo seven, under the stated divisibility assumptions. The two implementations extract these values using different methods, including an integer trace discriminant and an enumeration of k by trace and norm","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-fc7a5c603b95f30d","tex":"\\tr","aliases":"tr","display":null,"kind":"use","excerpt":"7c · d, det F=Q. Put d=1+7D and tr(F)=Q+1+7T. Then a=Q+7(T-D), and the determinant equation gives bc=T+(Q-1)D+7TD-7D^2. Modulo seven this becomes fractr(F)-Q-17=α(F)β(F) in k. This is the coefficient field rank two trace of the actual representation. It is not half the rational four dimensional trace. The hypothesis Q≡1pmod7 makes the extra diagona","count":6},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-02453f18b1b8a548","tex":"\\tr_{\\Oloc}","aliases":"tr_Oloc tr Oloc","display":null,"kind":"use","excerpt":"A mixed trace that detects the valuation. The valuation coordinate can be read from a joint word in the same representation. For T(g)=tr_Olocρ(g), define D(τ,g)=T(τ g)-T(τ)-T(g)+2. The elementary identity D(τ,g)=tr((ρ(τ)-I)(ρ(g)-I)) shows why the product label matters. Two lists of individual traces do not determine the trace of the product.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b02948282f2a168e","tex":"\\mathrm{trunc}","aliases":"trunc mathrm","display":"\\mathrm{trunc}","kind":"use","excerpt":"orem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d849fcfcf96ad168","tex":"u^3","aliases":"u^3","display":"u^3","kind":"use","excerpt":"At a variable prime dividing C, primitivity makes A,B units. In the residue field put u=-B^2/A. Then u^3=B, u^5=-A, and f_A,B(uX)=u^6(X^2+X-1)^2(X^2-2X+5). The discriminants of the two quadratics are 5 and -16, and their resultant is 45. Away from 2,3,5, the special fiber has exactly two ordinary nodes, rational normalization, and smooth infinity. ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f7c935d31b55226e","tex":"u^{48}","aliases":"u^48","display":"u^{48}","kind":"use","excerpt":"Let X=[ϕ] and H=[3] in the mod seven local Kummer group. Exact logarithmic first digits give L(ϕ)=5overlinesqrt5, L(3)=6, L(sqrt5)=3, L(u)=fracu^48-17pmod7. The two unit coordinates X,H are independent, and [sqrt5]=4H. For a rational seven adic unit A, put q_7(A)=(A^6-1)/7 modulo seven. The local marked period comparison has P=ZI+2XR, Z=(3q_7(A)+5)","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f1e852c6b6b401aa","tex":"u^5","aliases":"u^5","display":"u^5","kind":"use","excerpt":"Odd prime lifting. For the polynomial on the left of [source reference] minus its right side, the partial derivatives are 6a^3u^5,qquad10b^5v^9, -14c^7w^13. In a primitive residue solution, at least two bases are units. At an odd prime, at most one of 6,10,14 is divisible by p. Thus at least one unit variable has unit derivative. Lift the other var","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f2951bda6b96699c","tex":"u^6","aliases":"u^6","display":"u^6","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4421600030c4c807","tex":"u^7","aliases":"u^7","display":"u^7","kind":"use","excerpt":"Its mechanism retains the actual joint finite quotient. The three Kummer classes are independent after cyclotomic restriction. Distinct cyclotomic eigenspaces separate the natural and opposite images. Chebotarev then realizes joint vectors at good split primes, where the same quartic at the paired K places forces products to be coefficient conjugat","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-deb9be7f41a14efb","tex":"U_3","aliases":"U_3","display":"U_3","kind":"use","excerpt":"An exact control separating the two stages. Take the finite group U_3(F_7) with coordinates (a,b,c) and multiplication (a,b,c)(A,B,C)=(a+A,b+B,c+C+aB). Use trivial cyclotomic action, α=a, β=b, and p=c. Then dp=-α∪β, so the first obstruction vanishes. The elements x=(1,0,0) and z=(0,0,1) commute, but R_p(x,z)=2, R_p(z,x)=0. For every ordinary cochai","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ec4379f6b5fb17bc","tex":"U_a","aliases":"U_a","display":"U_a","kind":"use","excerpt":"Let U_a=(1 · 7a 0 · 1) and V=(1 · 0","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0835809a6850acb4","tex":"u_q","aliases":"u_q","display":"u_q","kind":"use","excerpt":"At the chosen q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [s","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d562f43b66695976","tex":"v^{10}","aliases":"v^10","display":"v^{10}","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6f8eadb65c01ea0e","tex":"v^9","aliases":"v^9","display":"v^9","kind":"use","excerpt":"Odd prime lifting. For the polynomial on the left of [source reference] minus its right side, the partial derivatives are 6a^3u^5,qquad10b^5v^9, -14c^7w^13. In a primitive residue solution, at least two bases are units. At an odd prime, at most one of 6,10,14 is divisible by p. Thus at least one unit variable has unit derivative. Lift the other var","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1911deeadb1babb8","tex":"v_7","aliases":"v_7","display":"v_7","kind":"use","excerpt":"Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained low","count":8},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5f6859094884b38b","tex":"v_7^{\\mathrm{trunc}}","aliases":"v_7^trunc mathrm","display":"v_7^{\\mathrm{trunc}}","kind":"use","excerpt":"orem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate triangular precision 1≤ j≤ m, even beyond the proved unit period window. Consequently","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3de92b22860491e7","tex":"V_j","aliases":"V_j","display":"V_j","kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-724afc19e77e7bfd","tex":"v_q","aliases":"v_q","display":"v_q","kind":"use","excerpt":"For the natural curve, the variable node thickness is 7v_q(C). The natural monodromy content is consequently t=1+v_7(v_q(C)). This inherited geometric bridge uses the ordinary node smoothing parameter and the identification of the graph pairing with the period valuation [source reference]. Applied to an actually ramified beta prime, [source referen","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-fa83faab1d44c512","tex":"v_\\ell","aliases":"v_ℓ ell","display":"v_\\ell","kind":"use","excerpt":"[Primitive scaling criterion] Fix a rational 0<t<1. There is a positive rational λ for which [source reference] gives primitive positive integers if and only if, for every rational prime ℓ, v_ℓ(t)>0 · Longrightarrowquad3mid v_ℓ(t), v_ℓ(1-t)>0 · Longrightarrowquad5mid v_ℓ(1-t), v_ℓ(t)<0 · Longrightarrowquad7mid -v_ℓ(t). When it exists, λ is unique. ","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":27},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ddbec861ff06906d","tex":"w^{-14}","aliases":"w^-14","display":"w^{-14}","kind":"use","excerpt":"If pmid a, primitivity forces the other two bases to be units. Reduction gives b^5v^10=c^7w^14. The quadratic character of this equality proves the necessary symbol condition. Conversely, v^10w^-14 ranges over the squares of F_p^×, because gcd(10,14,p-1)=2. The symbol condition lets us choose the two unit roots, then lift. The other two cases use g","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7363f240ac58c586","tex":"w^{13}","aliases":"w^13","display":"w^{13}","kind":"use","excerpt":"Odd prime lifting. For the polynomial on the left of [source reference] minus its right side, the partial derivatives are 6a^3u^5,qquad10b^5v^9, -14c^7w^13. In a primitive residue solution, at least two bases are units. At an odd prime, at most one of 6,10,14 is divisible by p. Thus at least one unit variable has unit derivative. Lift the other var","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-47e37528e0c958ea","tex":"w^{14}","aliases":"w^14","display":"w^{14}","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"The complete local squareclass test. Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ffbde798b9a7c58f","tex":"W_j","aliases":"W_j","display":"W_j","kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-86c1d03c1a864e81","tex":"\\widetilde z","aliases":"widetilde z widetilde","display":"\\widetilde z","kind":"use","excerpt":"The direct independent check lifts the chosen root of Φ_7 modulo q^2 by Hensel, then computes π_q(widetilde z_q)/q modulo q. It verifies the self unit rather than assigning its missing value from the norm relation. The norm calculation and Hensel calculation share the input generator but use different arithmetic routes.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6e8760cb511dff50","tex":"\\widetilde z_q","aliases":"widetilde z_q widetilde","display":"\\widetilde z_q","kind":"use","excerpt":"The direct independent check lifts the chosen root of Φ_7 modulo q^2 by Hensel, then computes π_q(widetilde z_q)/q modulo q. It verifies the self unit rather than assigning its missing value from the norm relation. The norm calculation and Hensel calculation share the input generator but use different arithmetic routes.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-65e8ec76ae768d6c","tex":"x^{-1}","aliases":"x^-1","display":"x^{-1}","kind":"use","excerpt":"0 · 1), direct conjugation sends (p,q,b) to (p+ax^-1b, xq-2ap-a^2x^-1b, x^-1b). Thus the three coupled equations are β(gh) · =β(g)+χ(g)^-1β(h),","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b8e9d80a4978aa3f","tex":"X^{12}","aliases":"X^12","display":"X^{12}","kind":"use","excerpt":"A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B, [X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-57b3fdfea5e94d6e","tex":"X^{13}","aliases":"X^13","display":"X^{13}","kind":"use","excerpt":"A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B, [X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"At a variable prime dividing C, primitivity makes A,B units. In the residue field put u=-B^2/A. Then u^3=B, u^5=-A, and f_A,B(uX)=u^6(X^2+X-1)^2(X^2-2X+5). The discriminants of the two quadratics are 5 and -16, and their resultant is 45. Away from 2,3,5, the special fiber has exactly two ordinary nodes, rational normalization, and smooth infinity. ","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"use","excerpt":"A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinate","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"use","excerpt":"12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ee74251aad7234d0","tex":"X^4","aliases":"X^4","display":"X^4","kind":"use","excerpt":"If the actual polynomial is P(X)=X^4-SX^3+BX^2-QSX+Q^2, the correction u=(τ-Q-1)/7 has Tr(u)=fracS-2(Q+1)7, Nm(u)=frac(B-2Q)-(Q+1)S+(Q+1)^249 modulo seven, under the stated divisibility assumptions. The two implementations extract these values using different methods, including an integer trace discriminant and an enumeration of k by trace and norm","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0b5a620f8119960c","tex":"X^5","aliases":"X^5","display":"X^5","kind":"use","excerpt":"A characteristic seven cancellation. In characteristic seven, the Cartier coefficient calculation comes from f^3. Its relevant coefficients are [X^6]f^3 · =1575B^4, · [X^5]f^3 · =4320A^2B, [X^13]f^3 · =36A, · [X^12]f^3 · =315B^2. The diagonal lanes vanish modulo seven and the off diagonal lanes become M=0 · A^2B","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-982ff71dc0953436","tex":"x^5","aliases":"x^5","display":"x^5","kind":"use","excerpt":"12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f4a985eec4e2eee9","tex":"X^6","aliases":"X^6","display":"X^6","kind":"use","excerpt":"A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinate","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-9924850fb16ee38c","tex":"x^6","aliases":"x^6","display":"x^6","kind":"use","excerpt":"12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 into their equation y^2=5x^","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4ab09af538752b5e","tex":"x^p","aliases":"x^p","display":"x^p","kind":"use","excerpt":"0. DOI: https://doi.org/10.4171/009-1/2110.4171/009-1/21. bibitembilu2005 Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihuailescu), J. Th eor. Nombres Bordeaux 17 (2005), 69--85. DOI: https://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equations z^m=F(x,y) and","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-60b14e0cb3534318","tex":"x^q","aliases":"x^q","display":"x^q","kind":"use","excerpt":"For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) ","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"use","excerpt":"Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b59b1511ce0c90d5","tex":"x_{\\mathrm{PV}}","aliases":"x_PV mathrm","display":"x_{\\mathrm{PV}}","kind":"use","excerpt":"gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b6814312a9b5ac7c","tex":"x_{q_1}","aliases":"x_q_1","display":"x_{q_1}","kind":"use","excerpt":"bottomrule figplaceholder19_affine_dimensionsThe kernel measures ambiguity after an affine solution exists. The cokernel measures possible first stage incompatibility. Neither dimension evaluates the actual target. In the global parameter order (a,x_q_1,…,x_q_9), the kernel is exactly the span of the unit vectors at the listed minus congruence prim","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-34cff92a87e5b282","tex":"x_{q_9}","aliases":"x_q_9","display":"x_{q_9}","kind":"use","excerpt":"bottomrule figplaceholder19_affine_dimensionsThe kernel measures ambiguity after an affine solution exists. The cokernel measures possible first stage incompatibility. Neither dimension evaluates the actual target. In the global parameter order (a,x_q_1,…,x_q_9), the kernel is exactly the span of the unit vectors at the listed minus congruence prim","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"Strict conjugation by I+7Y changes X by Y-Ad(overlineρ_0)Y. Independent local changes can simplify representatives but do not create a common global integral frame. The affine target in the last part of this reader is formulated after one frame and one reference have been fixed.","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinate","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0c909af2321925f7","tex":"y^3","aliases":"y^3","display":"y^3","kind":"use","excerpt":"efhttps://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equations z^m=F(x,y) and Ax^p+By^q=Cz^r, Bull. London Math. Soc. 27 (1995), 513--543. DOI: https://doi.org/10.1112/blms/27.6.51310.1112/blms/27.6.513. bibitemdahmen-siksek2024 S. R. Dahmen and S. Siksek, On the generalized Fermat equat","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-afd59eb09907b820","tex":"y^5","aliases":"y^5","display":"y^5","kind":"use","excerpt":"A current source check also found Chocian's preprint of 22 September 2026, which reports a computer assisted proof for the primitive equation x^3+y^5=z^7 [source reference]. Its existence and reported scope are recorded here. This release does not independently audit that claimed proof or import it as a theorem closing the recovered project route. ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ce6d149a19650241","tex":"y^q","aliases":"y^q","display":"y^q","kind":"use","excerpt":"0. DOI: https://doi.org/10.4171/009-1/2110.4171/009-1/21. bibitembilu2005 Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihuailescu), J. Th eor. Nombres Bordeaux 17 (2005), 69--85. DOI: https://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equations z^m=F(x,y) and","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b398630de1427292","tex":"y_{\\mathrm{PV}}","aliases":"y_PV mathrm","display":"y_{\\mathrm{PV}}","kind":"use","excerpt":"Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinates, with their variables translated as (a,b,c,p)=(B,-C,A,7) [source reference]. Explicitly, put t=-B^5/A^3, d=-B^2/A, and substitute x_PV=d/X, y_PV=tZ/X^3 int","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"use","excerpt":":strata Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on ","count":9},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-544ac8aed4a2c573","tex":"Z^1","aliases":"Z^1","display":"Z^1","kind":"use","excerpt":"Every other first primitive is p+z with z∈ Z^1(K,k) an ordinary character. Formula [source reference] gives μ_p+z-μ_p=α∪ z-z∪α. Graded commutativity in odd characteristic makes its cohomology class 2[α]∪[z]. Therefore the choice independent object at this stage is the affine coset [μ_p]+2[α]∪ H^1(K,k). There exists some first primitive for which th","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1a49e6bec04c7b56","tex":"Z^2","aliases":"Z^2","display":"Z^2","kind":"use","excerpt":"A curve that retains the exponent seven. The signature route uses the integral curve y^2+(X^3+B)y=2BX^3+3AX+B^2. Completing the square with Z=2y+X^3+B gives Z^2=f_A,B(X)=X^6+10BX^3+12AX+5B^2. This is the genus two family used in the recovered source. The comparison with the family of Pacetti and Villagra Torcomian is a rational change of coordinate","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-82a6f2d2c744dfea","tex":"z^7","aliases":"z^7","display":"z^7","kind":"use","excerpt":"A current source check also found Chocian's preprint of 22 September 2026, which reports a computer assisted proof for the primitive equation x^3+y^5=z^7 [source reference]. Its existence and reported scope are recorded here. This release does not independently audit that claimed proof or import it as a theorem closing the recovered project route. ","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-56835621b2f38901","tex":"z^m","aliases":"z^m","display":"z^m","kind":"use","excerpt":"ty, 2005, 325--340. DOI: https://doi.org/10.4171/009-1/2110.4171/009-1/21. bibitembilu2005 Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihuailescu), J. Th eor. Nombres Bordeaux 17 (2005), 69--85. DOI: https://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equatio","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-10d0329c35407a66","tex":"z^r","aliases":"z^r","display":"z^r","kind":"use","excerpt":"0. DOI: https://doi.org/10.4171/009-1/2110.4171/009-1/21. bibitembilu2005 Y. F. Bilu, Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihuailescu), J. Th eor. Nombres Bordeaux 17 (2005), 69--85. DOI: https://doi.org/10.5802/jtnb.47810.5802/jtnb.478. bibitemdarmon-granville1995 H. Darmon and A. Granville, On the equations z^m=F(x,y) and","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-10c7caeba58a1aa9","tex":"z_*","aliases":"z_*","display":"z_*","kind":"use","excerpt":"[Affine fibers] If Stage P has one solution with parameter z_*, its complete solution set for these linear conditions is z_*+ker A. It has 49^3,49^3,49^2,49^3 choices in patterns A through D, respectively. If the target is not in the image, the set is empty. Az=t is equivalent to A(z-z_*)=0. The exact kernel dimensions give the counts over k. An ad","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"Now replace the origin by p'_0=p_0+z_0,char, where its localization is Jz_0. At the fixed places, the target differences decrease by the selected coordinates of this character. At support, [source reference] decreases by c_qz_0,char(F_q)-b_qz_0,char(T_q). Therefore t(p'_0)=t(p_0)-Az_0. The quotient class and Nt are unchanged. The coordinate basis f","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d4ddd06f360cb5df","tex":"z_{0,\\mathrm{char}}","aliases":"z_0,char mathrm","display":"z_{0,\\mathrm{char}}","kind":"use","excerpt":"Now replace the origin by p'_0=p_0+z_0,char, where its localization is Jz_0. At the fixed places, the target differences decrease by the selected coordinates of this character. At support, [source reference] decreases by c_qz_0,char(F_q)-b_qz_0,char(T_q). Therefore t(p'_0)=t(p_0)-Az_0. The quotient class and Nt are unchanged. The coordinate basis f","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-78a80f694d80e6e7","tex":"\\Z_2","aliases":"Z_2 Z","display":null,"kind":"use","excerpt":"For sufficiency, fix the selected even root to be one when its coefficient is even, and two when the coefficient is odd. Fix one odd root to be one. After moving the terms and dividing by the remaining coefficient power, the required remaining root power is a unit congruent to one modulo eight. Such a unit is a square in Z_2, and every odd power ma","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c48b8f19a5db3646","tex":"\\Z_2^\\times","aliases":"Z_2^× Z times","display":null,"kind":"use","excerpt":"For sufficiency, fix the selected even root to be one when its coefficient is even, and two when the coefficient is odd. Fix one odd root to be one. After moving the terms and dividing by the remaining coefficient power, the required remaining root power is a unit congruent to one modulo eight. Such a unit is a square in Z_2, and every odd power ma","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-fdd80d0f2409bcbb","tex":"\\Z_7","aliases":"Z_7 Z","display":null,"kind":"use","excerpt":"The natural lattice and the depth of an opposite extension. The reducible branch uses two different kinds of radical. The base field is K=Q(sqrt5), ϕ=(1+sqrt5)/2. The coefficient ring and field are Oloc=Z_7[w]/(w^2-5), k=F_7[w]/(w^2-5)=F_49. The base unit ϕ defines Kummer classes. The coefficient w acts on a rank two representation. Although both i","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f593a96af5f1ba2e","tex":"z_7","aliases":"z_7","display":"z_7","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-be90faf55548b111","tex":"z_{\\mathrm{char}}","aliases":"z_char mathrm","display":"z_{\\mathrm{char}}","kind":"use","excerpt":"A candidate p=p_0+z_char meets the fixed local condition precisely when the selected local character coordinates equal those of p_v^fix-p_0|_v. At support, substitution in [source reference] gives c_qz_char(F_q)-b_qz_char(T_q)=t_q(p_0). Together, these fifteen equations are exactly Az=t(p_0). Thus Stage P is soluble if and only if t(p_0)∈im A.","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b910972bdb78c268","tex":"\\Z_p","aliases":"Z_p Z","display":null,"kind":"use","excerpt":":strata Write a hypothetical primitive solution as A=au^2, B=bv^2, C=cw^2, where a,b,c are positive pairwise coprime squarefree integers. The equation becomes a^3u^6+b^5v^10=c^7w^14. At a prime p, a local solution is called primitive when at most one base is divisible by p. Roots and bases are required to be nonzero in Z_p. The roots may depend on ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8de2c23d7851156f","tex":"z_q","aliases":"z_q","display":"z_q","kind":"use","excerpt":"In the descended all inert regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8114673204fe4df4","tex":"z_q^b","aliases":"z_q^b","display":"z_q^b","kind":"use","excerpt":"ant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding seventh power logarithm, the local functional at the embedding ζ_7↦ z_q","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e9e2c2b0dafde9f5","tex":"z_r^b","aliases":"z_r^b","display":"z_r^b","kind":"use","excerpt":"n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source refere","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"al has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2rmid D and 0≤ D≤16Q. If 7^2r>16Q, then P(T)=(T^2-aT+Q)^2, ","count":6},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0b4da04a048ac017","tex":"|_v","aliases":"|_v","display":"|_v","kind":"use","excerpt":"If p_0 is one global bare primitive and p_v are desired finite local primitives, their differences p_v-p_0|_v are ordinary characters. For trivial coefficients there are no nonzero coboundaries. Surjectivity supplies one global character matching every prescribed finite difference exactly. Then p=p_0+z meets those local equations, but may ramify at","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"rictions were established under the stated modular and multiplicative prime comparisons. A rank two trace zero comparison at a prime above eleven forces 11mid A. At a prime above nineteen it forces 19mid B. In either case the full four dimensional reduction to check is (T^2+q)^2pmod7, which is stronger than zero four dimensional trace. At bad q, th","count":9},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"Why a generic radical ceiling fails. The exponent equation gives R<H^71/105 for the signature (3,5,7). It is tempting to reverse this relation by postulating H≤ R^105/71 for every primitive sum. That universal assertion is false. The exact equality 2+109·3^10=23^5=6 436 343 has radical 2·3·109·23=15 042, and (23^5)^71>15 042^105. Its logarithmic qu","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","excerpt":"At higher actual lattice depth, put m=lfloor n/2rfloor+1. The inherited diagonal theorem supplies an actual global upper extension α_j for 1≤ j≤ m. In the common range 1≤ j≤min(m,7c-1), its local coordinates satisfy U_j=2ϵ wg_j, V_j=Λ_jg_j, W_j=7cg_j for one unit g_j. The exact valuation matrix preserves the valuation identity at every legitimate t","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"Odd primes away from the coefficients. Let α,β,γ be the quadratic characters of a,b,c, and let e=(-1/p). A primitive residue solution with one zero base exists if β=γ, α=γ, α=eβ, respectively. The same cyclic group argument proves this, and the lifting construction replaces the zero root by a nonzero multiple of p.","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constit","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6bdae33289d0c9e3","tex":"\\delta^8","aliases":"δ^8 delta","display":"\\delta^8","kind":"use","excerpt":"The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constit","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","excerpt":"In the descended all inert regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5d98a72c1e783677","tex":"\\delta_a^{-1}","aliases":"δ_a^-1 delta","display":"\\delta_a^{-1}","kind":"use","excerpt":"In the descended all inert regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"The actual coefficient action must intertwine the toric submodule and quotient period matrices. Their valuation matrix is exactly 7cI, so the two action matrices agree after cancellation over Z_7. Their reduction commutes with R and has characteristic polynomial T^2-5. The retained centralizer argument then gives one orientation ϵ∈{1,-1} and local ","count":9},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"If no variable C prime had Q≡1pmod7, beta would restrict to an everywhere unramified additive class over L=K(ζ_7). Restriction is injective because [L:K]=6 is prime to seven. The inherited Masley class number result gives h(Q(ζ_35))=1 [source reference]. The field L is a subfield of index two. Extension and norm on ideal classes compose to multipli","count":9},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-99c0c2eefbb607a8","tex":"\\zeta_{35}","aliases":"ζ_35 zeta","display":"\\zeta_{35}","kind":"use","excerpt":"If no variable C prime had Q≡1pmod7, beta would restrict to an everywhere unramified additive class over L=K(ζ_7). Restriction is injective because [L:K]=6 is prime to seven. The inherited Masley class number result gives h(Q(ζ_35))=1 [source reference]. The field L is a subfield of index two. Extension and norm on ideal classes compose to multipli","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-79a0be0094ccc021","tex":"\\zeta_7","aliases":"ζ_7 zeta","display":"\\zeta_7","kind":"use","excerpt":"If no variable C prime had Q≡1pmod7, beta would restrict to an everywhere unramified additive class over L=K(ζ_7). Restriction is injective because [L:K]=6 is prime to seven. The inherited Masley class number result gives h(Q(ζ_35))=1 [source reference]. The field L is a subfield of index two. Extension and norm on ideal classes compose to multipli","count":8},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-9e83b1f4e81a303e","tex":"\\zeta_7^2","aliases":"ζ_7^2 zeta","display":"\\zeta_7^2","kind":"use","excerpt":"Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does no","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c09239290921b6d4","tex":"\\zeta_7^3","aliases":"ζ_7^3 zeta","display":"\\zeta_7^3","kind":"use","excerpt":"Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does no","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-6388d0f03300b2a1","tex":"\\zeta_7^4","aliases":"ζ_7^4 zeta","display":"\\zeta_7^4","kind":"use","excerpt":"Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does no","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e6c7e4cdc66cacdd","tex":"\\zeta_7^5","aliases":"ζ_7^5 zeta","display":"\\zeta_7^5","kind":"use","excerpt":"Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does no","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"7^nc(g) · d(g). Modulo 7^n, the diagonal entries are characters. Determinant χ lets us write them as χη_n and η_n^-1. The natural monodromy restrictions and fixed local traces constrain the same η_n at every place. A locally chosen ray character with the right residue does not prove it occurs as this actual diagonal character.","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5ed478059bcd3710","tex":"\\eta_n","aliases":"η_n eta","display":"\\eta_n","kind":"use","excerpt":"7^nc(g) · d(g). Modulo 7^n, the diagonal entries are characters. Determinant χ lets us write them as χη_n and η_n^-1. The natural monodromy restrictions and fixed local traces constrain the same η_n at every place. A locally chosen ray character with the right residue does not prove it occurs as this actual diagonal character.","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-72da797c4343860e","tex":"\\eta_n^{-1}","aliases":"η_n^-1 eta","display":"\\eta_n^{-1}","kind":"use","excerpt":"7^nc(g) · d(g). Modulo 7^n, the diagonal entries are characters. Determinant χ lets us write them as χη_n and η_n^-1. The natural monodromy restrictions and fixed local traces constrain the same η_n at every place. A locally chosen ray character with the right residue does not prove it occurs as this actual diagonal character.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"The local toric hyperplane. At a beta support prime, use a pro seven tame generator T and arithmetic Frobenius F. Put c=β(T)≠0, b=β(F), h=(Q-1)/7pmod7. The local first digit normal form is M=cθ · -θ^2 1 · -θ, A=θ b · θ h-θ^2b b · h-θ b. Relative to the chosen reference, p(T)=θ c, p(F)=θ b-h. Eliminating θ gives the invariant affine equation cp(F)-b","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","excerpt":"The local toric hyperplane. At a beta support prime, use a pro seven tame generator T and arithmetic Frobenius F. Put c=β(T)≠0, b=β(F), h=(Q-1)/7pmod7. The local first digit normal form is M=cθ · -θ^2 1 · -θ, A=θ b · θ h-θ^2b b · h-θ b. Relative to the chosen reference, p(T)=θ c, p(F)=θ b-h. Eliminating θ gives the invariant affine equation cp(F)-b","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"l. At residue characteristic different from seven, a unit is an Mth power precisely when its residue is, by simple root lifting. A general element also needs valuation divisible by M. If Q is the residue cardinality, μ_M⊂ F precisely when Mmid Q-1. Applying Proposition~[source reference] to [source reference] gives the complete local test 7^jmid Q-","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c2890db72a17d545","tex":"\\kappa_7","aliases":"κ_7 kappa","display":"\\kappa_7","kind":"use","excerpt":"he actual coefficient action must intertwine the toric submodule and quotient period matrices. Their valuation matrix is exactly 7cI, so the two action matrices agree after cancellation over Z_7. Their reduction commutes with R and has characteristic polynomial T^2-5. The retained centralizer argument then gives one orientation ϵ∈{1,-1} and local s","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-09fa540d6637b1f4","tex":"\\kappa_j","aliases":"κ_j kappa","display":"\\kappa_j","kind":"use","excerpt":"Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-e680a11163316b31","tex":"\\kappa_v^\\times","aliases":"κ_v^× kappa times","display":"\\kappa_v^\\times","kind":"use","excerpt":"l. At residue characteristic different from seven, a unit is an Mth power precisely when its residue is, by simple root lifting. A general element also needs valuation divisible by M. If Q is the residue cardinality, μ_M⊂ F precisely when Mmid Q-1. Applying Proposition~[source reference] to [source reference] gives the complete local test 7^jmid Q-","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained low","count":10},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"β(g)=(χ(g)^-1-1)λ, p^fix(g)=λχ(g)^-1α(g).","count":21},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5a97315db5e754c4","tex":"\\lambda^{105}","aliases":"λ^105 lambda","display":"\\lambda^{105}","kind":"use","excerpt":"=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) preserves this equality. The two terms on the left are t^36s^84 and t^35s^85. Their sum is t^35s^84(t+s)=t^35s^84, which is C_0^7. The three scaling weights multiply the powered terms by the same factor λ^105. Over a field, if t and 1-t are nonzero, all three bases are nonzero and A_0^3/C_0","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d9f3d69d8823c1f4","tex":"\\lambda^{15}","aliases":"λ^15 lambda","display":"\\lambda^{15}","kind":"use","excerpt":"For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) prese","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b29073cb6a42bd95","tex":"\\lambda^{21}","aliases":"λ^21 lambda","display":"\\lambda^{21}","kind":"use","excerpt":"For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) prese","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-08209adac2de3dd9","tex":"\\lambda^3","aliases":"λ^3 lambda","display":"\\lambda^3","kind":"use","excerpt":"[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative int","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ed644422c7e02468","tex":"\\lambda^{35}","aliases":"λ^35 lambda","display":"\\lambda^{35}","kind":"use","excerpt":"For the signature (3,5,7), rational points are easy to manufacture. Primitive integer points are not. The following polynomial identity makes the difference exact. [Universal power identity] In any commutative ring, put s=1-t and A_0=t^12s^28, B_0=t^7s^17, C_0=t^5s^12. Then A_0^3+B_0^5=C_0^7. Weighted scaling (A,B,C)=(λ^35A_0,λ^21B_0,λ^15C_0) prese","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7b50266abb0ec4a6","tex":"\\lambda^5","aliases":"λ^5 lambda","display":"\\lambda^5","kind":"use","excerpt":"[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative int","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-08fbedae94357b79","tex":"\\lambda^6","aliases":"λ^6 lambda","display":"\\lambda^6","kind":"use","excerpt":"[Polynomial discriminant] The sextic in [source reference] satisfies disc(f_A,B)=2^123^65^5(A^3+B^5)^2. For a hypothetical (3,5,7) solution this becomes 2^123^65^5C^14. Give X,A,B weights 1,5,3. Then f_λ^5A,λ^3B(λ X)=λ^6f_A,B(X). The six roots scale by λ, so their fifteen squared differences give discriminant weight thirty. The only nonnegative int","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ac2ec4913e4c3ca3","tex":"\\Lambda_2","aliases":"Λ_2 Lambda","display":"\\Lambda_2","kind":"use","excerpt":"te is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^2-7q_0pmod49. Both C_0 and the coordinate 7c[7] disappeared modulo seven. Ne","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8890f23babb77fdb","tex":"\\Lambda_j","aliases":"Λ_j Lambda","display":"\\Lambda_j","kind":"use","excerpt":"Define n=min_g:ρ(g)_21≠0v_7(ρ(g)_21). This positive integer is attained. It is an invariant of the fixed natural frame, not a truncation selected for a computation. Every Oloc lattice stable under ρ is preserved by the two projectors, so up to homothety it has the form Λ_j=Oloc e_1+Oloc 7^j e_2. The unit upper monodromy forces j≥0. The attained low","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c3adfdbb65690536","tex":"\\lambda_v","aliases":"λ_v lambda","display":"\\lambda_v","kind":"use","excerpt":"At a fixed place where beta is the coboundary of λ_v and p=p_v^fix, the explicit local upper primitive q_v^fix(g)=-λ_vχ(g)^-1α(g)^2 satisfies d_χ q_v^fix=R_p by direct expansion. At support, α=0 also kills the local upper right side. These facts prevent treating a formal secondary obstruction as an established nonzero obstruction in the actual loca","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-588690976252e676","tex":"\\mu_7","aliases":"μ_7 mu","display":"\\mu_7","kind":"use","excerpt":"The fixed primitive local to global theorem. For K=Q(sqrt5), localization on H^2(K,k(χ)) is injective. To see the coefficient translation, restrict to L=K(ζ_7), whose degree over K is six. Restriction followed by corestriction multiplies by six, which is invertible on seven primary cohomology. Over L, the module is two trivial F_7 copies and μ_7 is","count":6},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0bf8701ccd3ba8f5","tex":"\\mu_M","aliases":"μ_M mu","display":"\\mu_M","kind":"use","excerpt":"[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b7ec60a68d5fe62e","tex":"\\mu_M^2","aliases":"μ_M^2 mu","display":"\\mu_M^2","kind":"use","excerpt":"[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u ","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-047fc7bc99b487fb","tex":"\\mu_{p+z}","aliases":"μ_p+z mu","display":"\\mu_{p+z}","kind":"use","excerpt":"Every other first primitive is p+z with z∈ Z^1(K,k) an ordinary character. Formula [source reference] gives μ_p+z-μ_p=α∪ z-z∪α. Graded commutativity in odd characteristic makes its cohomology class 2[α]∪[z]. Therefore the choice independent object at this stage is the affine coset [μ_p]+2[α]∪ H^1(K,k). There exists some first primitive for which th","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"To define the valuation functional used here, strictly align the actual local toric cyclotomic line at K_7 with the reference line. The aligned local representative of X has p=β=0 and upper entry q a k(χ) Kummer cocycle. Define ν_7(X) to be its [7] valuation coordinate. Remaining aligned strict gauges change q by a coboundary and a scalar multiple ","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a83b2b761b5c4080","tex":"\\nu_7","aliases":"ν_7 nu","display":"\\nu_7","kind":"use","excerpt":"To define the valuation functional used here, strictly align the actual local toric cyclotomic line at K_7 with the reference line. The aligned local representative of X has p=β=0 and upper entry q a k(χ) Kummer cocycle. Define ν_7(X) to be its [7] valuation coordinate. Remaining aligned strict gauges change q by a coboundary and a scalar multiple ","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"A local depth theorem. Here is the matrix argument controlling where this opposite class can ramify. Let Oloc be a discrete valuation ring with uniformizer π, let F∈GL_2(Oloc), Q∈Oloc^× with Q≠1, and let n≥1. Suppose N∈ M_2(Oloc) is a nonzero nilpotent matrix of the form N=μ-xy · x^2","count":16},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-42ba626e1f091ffe","tex":"\\pi^n","aliases":"π^n pi","display":"\\pi^n","kind":"use","excerpt":"-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line a","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-437ff1deb48ad384","tex":"\\pi^{n-s}","aliases":"π^n-s pi","display":"\\pi^{n-s}","kind":"use","excerpt":"y(a,b), ax+by=Q-1. The lower entry is ya, so a∈π^n-sOloc=π^t+sOloc. Both ax and by are divisible by π^s. Their sum Q-1 is therefore divisible by π^s, giving s≤ h. The square on y explains the factor two. A bound n≤ t+h would omit the geometry of the rank one nilpotent and is false. The examples N=7^t-7^s · 1","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-75be53b3397893ea","tex":"\\pi^s","aliases":"π^s pi","display":"\\pi^s","kind":"use","excerpt":"y(a,b), ax+by=Q-1. The lower entry is ya, so a∈π^n-sOloc=π^t+sOloc. Both ax and by are divisible by π^s. Their sum Q-1 is therefore divisible by π^s, giving s≤ h. The square on y explains the factor two. A bound n≤ t+h would omit the geometry of the rank one nilpotent and is false. The examples N=7^t-7^s · 1","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a5892af1fc6f85ce","tex":"\\pi^{t+s}","aliases":"π^t+s pi","display":"\\pi^{t+s}","kind":"use","excerpt":"y(a,b), ax+by=Q-1. The lower entry is ya, so a∈π^n-sOloc=π^t+sOloc. Both ax and by are divisible by π^s. Their sum Q-1 is therefore divisible by π^s, giving s≤ h. The square on y explains the factor two. A bound n≤ t+h would omit the geometry of the rank one nilpotent and is false. The examples N=7^t-7^s · 1","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-cc4081860615a573","tex":"\\pi_{197}","aliases":"π_197 pi","display":"\\pi_{197}","kind":"use","excerpt":"Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does no","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2a0a981a1f03f192","tex":"\\pi_{337}","aliases":"π_337 pi","display":"\\pi_{337}","kind":"use","excerpt":"Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does no","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-917ba7f492100cd1","tex":"\\Pi_{aa}","aliases":"Π_aa Pi","display":"\\Pi_{aa}","kind":"use","excerpt":"At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. T","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-2ddec056a59f379c","tex":"\\Pi_{ab}","aliases":"Π_ab Pi","display":"\\Pi_{ab}","kind":"use","excerpt":"At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. T","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-90e3510a110e4e6d","tex":"\\Pi_{ba}","aliases":"Π_ba Pi","display":"\\Pi_{ba}","kind":"use","excerpt":"At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. T","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-688609730b26bd6b","tex":"\\Pi_{bb}","aliases":"Π_bb Pi","display":"\\Pi_{bb}","kind":"use","excerpt":"At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. T","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f0e454088f5d8a3c","tex":"\\pi_l","aliases":"π_l pi","display":"\\pi_l","kind":"use","excerpt":"For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-3deb210008426065","tex":"\\pi_q","aliases":"π_q pi","display":"\\pi_q","kind":"use","excerpt":"Let M=Q(ζ_7). For a rational conductor q≡1pmod7, choose a generator π_q of norm q for the selected degree one prime. The source gives exact coefficient vectors in 1,ζ_7,…,ζ_7^5 for q=197,337,547,673,743,883,1933. For example, π_197=ζ_7+2ζ_7^2+2ζ_7^4, and π_337=-1+ζ_7-ζ_7^2+3ζ_7^3-ζ_7^4+ζ_7^5. The norm equality identifies one prime ideal. It does no","count":7},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","excerpt":"The coupled adjoint equations. Choose one reference Kummer representation over Oloc/49, ρ_0(g)=χ(g) · b_0(g) 0 · 1, whose reduction has the same residual α as the actual natural representation. The reference is built by lifting the coefficients of the same global classes [ϕ],[3],[sqrt5]. It has no seven valuation coordinate at the fixed place. It i","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"A bare primitive on the justified ramification set. Fix one of the nine prime beta supports S in Section~[source reference]. Let Σ consist of S, the places above 2,3,5,7, and the two real places. The residual classes factor through Gsig. At every beta support place, χ is trivial modulo seven and the natural local class is zero. Hence its normalized","count":15},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"[Discriminant collapse] Suppose a good Frobenius polynomial has integral coefficients and factorization P(T)=(T^2-τ T+Q)(T^2-σ(τ)T+Q) =T^4-ST^3+C_2T^2-QST+Q^2, where Oloc/Z_7 is unramified quadratic, σ is its nontrivial automorphism, all complex roots have absolute value sqrt Q, and τ≡ Q+1pmod7^rOloc. Then the integer D=S^2-4(C_2-2Q) satisfies 7^2r","count":9},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"At good auxiliary primes, the actual rank two trace obeys a precision condition of the form τ≡ Q+1pmod7^rOloc. The exponent r is supplied by the depth and local diagonal estimates. It is not inferred merely from a rank four congruence. This stronger comparison yields a particularly useful universal lemma.","count":16},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5ff841193b5dd2b3","tex":"\\tau_j","aliases":"τ_j tau","display":"\\tau_j","kind":"use","excerpt":"The local Kummer basis [ϕ],[3],[7] supplies a unit annihilating probe τ_j after adjoining the required 7^j roots of unity. It kills the unit coordinates and evaluates the uniformizer coordinate to one. The inherited construction uses the cyclotomic action to show restriction remains injective. At higher cyclotomic levels this is not a blanket prime","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"The direct independent check lifts the chosen root of Φ_7 modulo q^2 by Hensel, then computes π_q(widetilde z_q)/q modulo q. It verifies the self unit rather than assigning its missing value from the norm relation. The norm calculation and Hensel calculation share the input generator but use different arithmetic routes.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-dbb0300906ac1aba","tex":"\\phi^{-2}","aliases":"ϕ^-2 phi","display":"\\phi^{-2}","kind":"use","excerpt":"At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. T","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4ef3d223fa8eb5fd","tex":"\\phi^{-4}","aliases":"ϕ^-4 phi","display":"\\phi^{-4}","kind":"use","excerpt":"At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. T","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d73ae6dc3936033e","tex":"\\phi^2","aliases":"ϕ^2 phi","display":"\\phi^2","kind":"use","excerpt":"At a variable place away from 2,3,5,7, write C=q^cC_0 with C_0 a unit. In the marked two node basis, the actual leading period coordinates are Π_aa · =-fracepsϕ^2675sqrt5(1+ m), Π_bb · =fracepsϕ^-2675sqrt5(1+ m), Π_ab · =Π_ba=-ϕ^-4(1+ m), where eps=C^7/A^3 and the notation means that the quotient by the displayed leading term is a principal unit. T","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ab8c353106812d45","tex":"\\phi^4","aliases":"ϕ^4 phi","display":"\\phi^4","kind":"use","excerpt":"When C is even and seven does not divide C, the two adic torus calculation replaces good reduction at two. The global character has Frobenius value -1 at two. Reciprocity on ϕ^4 and 2/7 transports that value to seven. The finite flat comparison at good seven, together with [source reference], gives the value A^3B. Thus A^3B=-1 modulo seven. Since b","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d1586fe0cd5dff35","tex":"\\phi^{48}","aliases":"ϕ^48 phi","display":"\\phi^{48}","kind":"use","excerpt":"At F=Q_7(sqrt5), exact arithmetic modulo 49 gives fracϕ^48-17=5overlinesqrt5≠0 in F_49. If a unit is a seventh power, its forty eighth power is one modulo forty nine. Indeed, any unit has forty eighth power in 1+7Oloc_F, and the seventh power of this group is contained in 1+49Oloc_F. Formula [source reference] shows that ϕ is not a seventh power in","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a509cf8bdde9831c","tex":"\\phi^e","aliases":"ϕ^e phi","display":"\\phi^e","kind":"use","excerpt":"Its units are {±ϕ^e:e∈Z}. For an elementary verification, scale a unit by a power of ϕ and by a sign so that its first real embedding lies in [1,ϕ). If its value a were strictly between the endpoints, a norm minus one unit would have integral trace a-a^-1 strictly between zero and one. A norm plus one unit would have trace a+a^-1 strictly between t","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-bd25a807d6bd7f7e","tex":"\\Phi_7","aliases":"Φ_7 Phi","display":"\\Phi_7","kind":"use","excerpt":"The direct independent check lifts the chosen root of Φ_7 modulo q^2 by Hensel, then computes π_q(widetilde z_q)/q modulo q. It verifies the self unit rather than assigning its missing value from the norm relation. The norm calculation and Hensel calculation share the input generator but use different arithmetic routes.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7c934e3eb93a78aa","tex":"\\chi^{-1}","aliases":"χ^-1 chi","display":"\\chi^{-1}","kind":"use","excerpt":"The inherited characteristic zero representation is the specified RM factor of the twisted Jacobian on the retained 3990mid C branch, with determinant the cyclotomic character χ. Its natural residual orientation is 0⟶ k(χ)⟶overline V⟶ k⟶0. The opposite endpoint has a nonzero class β∈ H^1(K,k(χ^-1)). A stable lattice change can alter residual inerti","count":5},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-11fdaa25e327864a","tex":"\\chi^2","aliases":"χ^2 chi","display":"\\chi^2","kind":"use","excerpt":"The global unit step uses ϕ=(1+sqrt5)/2, whose residue has order sixteen in F_49^×. The local weight possibilities at seven are 0,1,7,8, up to a common reciprocity sign. Applying reciprocity to the eighth power of each constituent gives overlineϕ^8e=1 and rules out the mixed weights 1,7. The determinant norm relation δ^8=χ^2 then leaves one constit","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-95fabc8465dd628e","tex":"\\chi_p","aliases":"χ_p chi","display":"\\chi_p","kind":"use","excerpt":"There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"The complete ordinary localization map. Let z be an ordinary k valued character of Gsig. It is a difference of bare primitives, not beta and not either primitive itself. Local class field theory gives the following coordinate order: F_2,F_3,F_5,quad7:ϕ,3,7, I_q_1,F_q_1,…,I_q_9,F_q_9. We use the local Artin convention sending a selected uniformizer ","count":4},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Primitive scaling criterion] Fix a rational 0<t<1. There is a positive rational λ for which [source reference] gives primitive positive integers if and only if, for every rational prime ℓ, v_ℓ(t)>0 · Longrightarrowquad3mid v_ℓ(t),","count":24},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-df02b0f30de4bb59","tex":"\\ell^{h_\\ell}","aliases":"ℓ^h_ℓ ell ell","display":"\\ell^{h_\\ell}","kind":"use","excerpt":"An integral h exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in t or 1-t, so λ=∏_ℓℓ^h_ℓ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every h_ℓ was forced, so two positive choices of λ would have ratio a positiv","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ccff9bd3329e6d7e","tex":"\\ell_7","aliases":"ℓ_7 ell","display":"\\ell_7","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0,","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-8943b22d90e5e02b","tex":"\\ell_j","aliases":"ℓ_j ell","display":"\\ell_j","kind":"use","excerpt":"Use the three coordinate basis X_j=[ϕ], H_j=[3], T_j=[7]. The third coordinate is valuation. The actual local scalar class in the window is κ_j=2ϵ wX_j+Λ_jH_j+7cT_j, where Λ_j=7ℓ_j(C_0)-3ℓ_j(A)-3-5d_j, ℓ_j(a)=fraclog(a^6)log(3^6)pmod7^j, d_j=fracℓ_j(5)2. At level 49, with Q_A=(A^6-1)/7 modulo 49 and q_0=(C_0^6-1)/7 modulo seven, Λ_2=47+38Q_A+14Q_A^","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-17c0c0401f91d262","tex":"\\ell_q","aliases":"ℓ_q ell","display":"\\ell_q","kind":"use","excerpt":"regime, let M=Q(ζ_7) and let γ∈ H^1(Q,F_7(χ^-1)) restrict to the relevant beta class. Over M, the restriction is additive and satisfies γ(δ_a gδ_a^-1)=a^-1γ(g). At a conductor prime q≡1pmod7, there are six degree one primes in M. At q≡-1pmod7, there are three degree two primes. If z_q is a chosen seventh root of unity and ℓ_q is the corresponding s","count":9},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-819cb7ab00d8c22c","tex":"\\ell_r","aliases":"ℓ_r ell","display":"\\ell_r","kind":"use","excerpt":"n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source refere","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"[Conditional restricted bare primitive] Assume the stated residual and local properties for α∈ Z^1(Gsig,k(χ)) and β∈ Z^1(Gsig,k(χ^-1)). Then there is a normalized continuous cochain p_0:Gsig→ k satisfying dp_0=-α∪β. No auxiliary ramification outside Σ is needed. The Cartier dual of trivial F_7 is μ_7. The nondegenerate F_7 trace pairing on k identi","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"The Block 47 audit identified this mechanism in the pinned Pacetti--Villagra Torcomian code. It also identified a parameter dictionary issue at eleven: the unlabelled multisets agreed, but individual labels matched after t↦ t^-1. A refinement attached to the wrong labels could incorrectly combine local constraints. The new reader therefore distingu","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"An integral h exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in t or 1-t, so λ=∏_ℓℓ^h_ℓ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every h_ℓ was forced, so two positive choices of λ would have ratio a positiv","count":3},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-dfa6b82b69f0facf","tex":"\\prod_{v\\in S}","aliases":"∏_v∈ S prod in","display":"\\prod_{v\\in S}","kind":"use","excerpt":"Consequently the unrestricted maps H^1(K,k)⟶∏_v∈ SH^1(K_v,k), H^1(K,k(χ))⟶∏_v∈ SH^1(K_v,k(χ)) are surjective for every finite set S under the cited localization theorem. The precise source formulation and its fixed additional finite set refinement are recorded in the literature audit. This statement does not keep the character unramified at every o","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-549b58f9741a8661","tex":"\\prod_{v\\in\\Sigma}","aliases":"∏_v∈Σ prod in Sigma","display":"\\prod_{v\\in\\Sigma}","kind":"use","excerpt":"[The dual Kummer kernel] For every finite Σ containing the prime above seven and all real places, Sha^1_Σ(K,μ_7) =ker(H^1(Gsig,μ_7) ⟶∏_v∈ΣH^1(K_v,μ_7))=0. The discriminant of K is five. Minkowski's ideal bound is sqrt5/2<2, so every ideal class has a representative of norm one. Thus Cl(K)=1 [source reference].","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-058f6157633d37d5","tex":"\\prod_\\ell","aliases":"∏_ℓ prod ell","display":"\\prod_\\ell","kind":"use","excerpt":"An integral h exists in these four cases precisely under the stated divisibility conditions. Only finitely many primes occur in t or 1-t, so λ=∏_ℓℓ^h_ℓ is a well defined positive rational number. The three output supports are disjoint, proving integrality and primitivity. Every h_ℓ was forced, so two positive choices of λ would have ratio a positiv","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-","count":8},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-807dad785a5ee498","tex":"\\sum_{b=1}^6","aliases":"∑_b=1^6 sum","display":"\\sum_{b=1}^6","kind":"use","excerpt":"For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-cd66483ef5c6b544","tex":"\\sum_{b=2}^6","aliases":"∑_b=2^6 sum","display":"\\sum_{b=2}^6","kind":"use","excerpt":"At the chosen q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [s","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5d95714a1572a98d","tex":"\\sum_{b\\in\\mathrm{reps}_q}","aliases":"∑_b∈reps_q sum in mathrm","display":"\\sum_{b\\in\\mathrm{reps}_q}","kind":"use","excerpt":"For degree two, the indices b and -b represent the same prime. Frobenius sends a residue x to x^q, and ℓ_q(x^q)=-ℓ_q(x). Therefore (-b)ℓ_q(x^q)=bℓ_q(x). The functional is well defined on prime ideals, not just on the six embeddings. Principal reciprocity gives, for a test prime generator π_l outside the conductor, R_l,q=-∑_b∈reps_qbℓ_q(π_l(z_q^b)) ","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-b7b72412bbfea41d","tex":"\\sum_{b\\in\\mathrm{reps}_r}","aliases":"∑_b∈reps_r sum in mathrm","display":"\\sum_{b\\in\\mathrm{reps}_r}","kind":"use","excerpt":"n q place, write π_q=q u_q with u_q a unit. The same norm identity gives ℓ_q(u_q)=-∑_b=2^6ℓ_q(π_q(z_q^b)). Principal reciprocity on π_q therefore supplies the own conductor column D_q,q=∑_b=2^6(1-b)ℓ_q(π_q(z_q^b)). For r≠ q, the column is D_q,r=-∑_b∈reps_rbℓ_r(π_q(z_r^b)). The degree two representative convention is the one proved in [source refere","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0, z(F_2) · =-5a-∑_iℓ_","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-396e478748627acb","tex":"\\sum_{i:q_i\\ne q}","aliases":"∑_i:q_i≠ q sum ne","display":"\\sum_{i:q_i\\ne q}","kind":"use","excerpt":"An explicit image parametrization. Put a=z_7(3) and x_i=z(I_q_i). Let ℓ_i be the seventh power logarithm normalized by ℓ_i(g_q_i)=1. The supporting symbols of ϕ vanish. For rational seven adic units define ℓ_7(t)=fraclog(t^6)log(3^6)pmod7 =6fract^6-17pmod7. Principal reciprocity determines the remaining coordinates: z_7(ϕ) · =0, z(F_2) · =-5a-∑_iℓ_","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"Because each support prime is inert in K/Q, restriction at a rational q≡1pmod7 multiplies the compatible rational Frobenius value by two: b_q=2∑_rD_q,rs_r. For q≡-1pmod7, the compatible squared rational Frobenius has beta value zero, from the cocycle factor 1+χ(q)^-1=0. This does not say beta is unramified there.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-90c396c1f06e154a","tex":"\\sum_t","aliases":"∑_t sum","display":"\\sum_t","kind":"use","excerpt":"There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0e44b9be25437975","tex":"\\sum_x","aliases":"∑_x sum","display":"\\sum_x","kind":"use","excerpt":"There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"-y^2 · xy, where μ∈Oloc∖{0}, (x,y)∈Oloc^2 is primitive, t=v(μ)≥0, and FN=QN, NF=N, F_21∈π^nOloc, v(N_21)=n. Set h=v(Q-1). Then n=t+2s, s=v(y),qquad0≤ s≤ h. The lower nilpotent entry is -μ y^2, so its valuation gives n=t+2s. The common image and kernel line of N is generated by (x,y)^T. The two identities for F show that F-I has image in this line a","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"For a finite module A, let A'=Hom(A,μ_7) and let G' be the finite group of the field cut out by that dual. The cited sufficient criterion is H^1(G',A')=0. For A=k with trivial action, A'≃ k(χ) and G'=C_6. Averaging kills its positive cohomology because six is invertible modulo seven. For A=k(χ), the source action cancels the cyclotomic target actio","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"There are (p+1)/4 elements t for which both t and 1-t are nonsquares. Here is a direct count. Extend the quadratic character χ_p by zero. Counting (x-y)(x+y)=d for d≠0 gives ∑_xχ_p(x^2-d)=-1. Completing the square then gives ∑_tχ_p(t(1-t))=-χ_p(-1)=1. After omitting 0,1, both single character sums are -1. Expanding the indicator (1-χ_p(t))(1-χ_p(1-","count":15},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"midrule A · b,c · b≡ c","count":33},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"The conditional abc height bound. An abc estimate makes the role of the signature explicit. Let H=C^z, R=rad(ABC), s=frac1x+frac1y+frac1z. For a primitive solution, the powered terms are pairwise coprime and their radical is R. Positivity gives ABC<H^s. Assume the all epsilon abc estimate H≤ K_eps R^1+eps for every eps>0, with one constant K_eps in","count":17},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"The positivity assumptions also carry information. They imply A^x<C^z, B^y<C^z, C≥2. The last inequality follows because A^x+B^y≥2. A base on the left may equal 1, so a finite classification of powered values need not be a finite classification of exponent labels. The equality 1^x=1 holds for every x. This small point is relevant when comparing a t","count":13},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"[Full torsion and period powers] For the stipulated split uniformization J^an= G_m^2/Λ with rational period lattice, full J[M] is F rational if and only if μ_M⊂ F and every coordinate of a basis of Λ is an Mth power in F. The uniformization gives 0⟶μ_M^2⟶ J[M] ⟶Λ/MΛ⟶0. If all torsion points are rational, so are μ_M^2. For λ∈Λ, choose an Mth root u ","count":2},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"Finite set Poitou--Tate duality pairs this group perfectly with Sha^2_Σ(K,k). Every prime dividing the module order belongs to Σ, as required by the cited theorem. Odd primary real Tate terms vanish. Thus localization H^2(Gsig,k)⟶⊕_v∈ΣH^2(K_v,k) is injective [source reference].","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"The abelianization is k⊕ B. The independent global Kummer class [7] cannot factor through that abelianization with its required cyclotomic weight, giving the stated global noncontainment of the uniformizer Kummer extension in the torsion field over K_∞. This field statement does not imply local noncontainment or zero local period valuation.","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-5708cf214ce02657","tex":"\\bigoplus_{v\\in\\Sigma}","aliases":"⊕_v∈Σ bigoplus in Sigma","display":"\\bigoplus_{v\\in\\Sigma}","kind":"use","excerpt":"Finite set Poitou--Tate duality pairs this group perfectly with Sha^2_Σ(K,k). Every prime dividing the module order belongs to Σ, as required by the cited theorem. Odd primary real Tate terms vanish. Thus localization H^2(Gsig,k)⟶⊕_v∈ΣH^2(K_v,k) is injective [source reference].","count":1},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"The inherited characteristic zero representation is the specified RM factor of the twisted Jacobian on the retained 3990mid C branch, with determinant the cyclotomic character χ. Its natural residual orientation is 0⟶ k(χ)⟶overline V⟶ k⟶0. The opposite endpoint has a nonzero class β∈ H^1(K,k(χ^-1)). A stable lattice change can alter residual inerti","count":6},{"work":"beal-reader","url":"notation/beal-reader.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"On the local space k^24, let E have fifteen rows. Its first six rows select all fixed local character coordinates in [source reference]. Its remaining nine rows are z⟼ c_qz(F_q)-b_qz(I_q). Define A=EJ:k^10⟶ k^15. The self conductor values and inertia conversions in Section~[source reference] are part of this operator. Dropping them would reproduce ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":201},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":423},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":94},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":94},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-596b3f5156c743a8","tex":"j^2","aliases":"j^2","display":"j^2","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":47},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":368},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":44},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a4ff961cd74d1869","tex":"r_n","aliases":"r_n","display":"r_n","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":12},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":76},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":6},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":6},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":38},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":91},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":10},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":24},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":32},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-df62203fd1d3b8e8","tex":"\\sum_{j=1}^{\\infty}","aliases":"∑_j=1^∞ sum infty","display":"\\sum_{j=1}^{\\infty}","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f1c2ff812e2b7828","tex":"\\sum_{k=0}^{\\infty}","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^{\\infty}","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":28},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"[Catalan's constant] The real number studied in this book is G=∑_k=0^∞frac(-1)^k(2k+1)^2. The series converges absolutely because ∑_k=0^∞frac1(2k+1)^2 ≤∑_j=1^∞frac1j^2<∞.","count":70},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[The distinguished rational sequences] Let (u_n)_n≥0 and (v_n)_n≥0 satisfy [source reference], with (u_0,u_1)=(1,frac74), (v_0,v_1)=(0,frac138). Their rational approximants are r_n=fracv_nu_n.","count":83},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":28},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"The notation a_n,c_n in this chapter and the next denotes recurrence coefficients. The later source notation a_n,b_n is translated explicitly when it occurs, and the cleared packet coefficients A_n,m,B_n,m have two indices. These are different objects.","count":84},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"introduction-passage","excerpt":"The notation a_n,c_n in this chapter and the next denotes recurrence coefficients. The later source notation a_n,b_n is translated explicitly when it occurs, and the cleared packet coefficients A_n,m,B_n,m have two indices. These are different objects.","count":32},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-58503597d5a86ed7","tex":"A_{n,m}","aliases":"A_n,m","display":"A_{n,m}","kind":"introduction-passage","excerpt":"The notation a_n,c_n in this chapter and the next denotes recurrence coefficients. The later source notation a_n,b_n is translated explicitly when it occurs, and the cleared packet coefficients A_n,m,B_n,m have two indices. These are different objects.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":24},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"The notation a_n,c_n in this chapter and the next denotes recurrence coefficients. The later source notation a_n,b_n is translated explicitly when it occurs, and the cleared packet coefficients A_n,m,B_n,m have two indices. These are different objects.","count":53},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"introduction-passage","excerpt":"The notation a_n,c_n in this chapter and the next denotes recurrence coefficients. The later source notation a_n,b_n is translated explicitly when it occurs, and the cleared packet coefficients A_n,m,B_n,m have two indices. These are different objects.","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8a46280d414fd140","tex":"B_{n,m}","aliases":"B_n,m","display":"B_{n,m}","kind":"introduction-passage","excerpt":"The notation a_n,c_n in this chapter and the next denotes recurrence coefficients. The later source notation a_n,b_n is translated explicitly when it occurs, and the cleared packet coefficients A_n,m,B_n,m have two indices. These are different objects.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":10},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"The notation a_n,c_n in this chapter and the next denotes recurrence coefficients. The later source notation a_n,b_n is translated explicitly when it occurs, and the cleared packet coefficients A_n,m,B_n,m have two indices. These are different objects.","count":45},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"introduction-passage","excerpt":"The notation a_n,c_n in this chapter and the next denotes recurrence coefficients. The later source notation a_n,b_n is translated explicitly when it occurs, and the cleared packet coefficients A_n,m,B_n,m have two indices. These are different objects.","count":18},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":71},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":23},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"introduction-passage","excerpt":"Set r_n=fracv_nu_n, d_n=fracp(n+1)2(2n+1)^2(2n+2)^2u_nu_n+1. The symbol d_n denotes a positive rational step size. It is unrelated to the least common multiples L_N used in denominator clearing. Dividing the Wronskian identity by u_nu_n+1 yields r_n+1-r_n=(-1)^n d_n. The movement of the approximants therefore alternates exactly.","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"Let E_n denote the well poised _6F_5 series at -1 in Zudilin's displayed Whipple transformation, with a=3n+1, b=c=d=n+1/2, and e=n+1. The transformation states H_n=fracΓ(3n+2)Γ(n+tfrac12) Γ(2n+tfrac32)Γ(2n+1)E_n. Its condition Re(1+a-d-e)>0 becomes n+1/2>0 here. The displayed prefactor of the source hypergeometric form F_n^rm src is F_n^rm src=(-1)","count":15},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":15},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-da2ce42dc70bb3da","tex":"E_n","aliases":"E_n","display":"E_n","kind":"introduction-passage","excerpt":"Let E_n denote the well poised _6F_5 series at -1 in Zudilin's displayed Whipple transformation, with a=3n+1, b=c=d=n+1/2, and e=n+1. The transformation states H_n=fracΓ(3n+2)Γ(n+tfrac12) Γ(2n+tfrac32)Γ(2n+1)E_n. Its condition Re(1+a-d-e)>0 becomes n+1/2>0 here. The displayed prefactor of the source hypergeometric form F_n^rm src is F_n^rm src=(-1)","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"Let E_n denote the well poised _6F_5 series at -1 in Zudilin's displayed Whipple transformation, with a=3n+1, b=c=d=n+1/2, and e=n+1. The transformation states H_n=fracΓ(3n+2)Γ(n+tfrac12) Γ(2n+tfrac32)Γ(2n+1)E_n. Its condition Re(1+a-d-e)>0 becomes n+1/2>0 here. The displayed prefactor of the source hypergeometric form F_n^rm src is F_n^rm src=(-1)","count":21},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"If E denotes the shift operator (Ef)_n=f_n+1, the packet is (1+E)^mT at index n. This is an identity about a finite polynomial in one shift. It does not introduce a further assumption about recurrence solutions or convergence. Reduction modulo a power of two commutes with this polynomial, which is the basis of the valuation computation in the next ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a1a1f9e3594eef3e","tex":"F_5","aliases":"F_5","display":"F_5","kind":"introduction-passage","excerpt":"Let E_n denote the well poised _6F_5 series at -1 in Zudilin's displayed Whipple transformation, with a=3n+1, b=c=d=n+1/2, and e=n+1. The transformation states H_n=fracΓ(3n+2)Γ(n+tfrac12) Γ(2n+tfrac32)Γ(2n+1)E_n. Its condition Re(1+a-d-e)>0 becomes n+1/2>0 here. The displayed prefactor of the source hypergeometric form F_n^rm src is F_n^rm src=(-1)","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-af40e62da957c071","tex":"f_{n+1}","aliases":"f_n+1","display":"f_{n+1}","kind":"introduction-passage","excerpt":"If E denotes the shift operator (Ef)_n=f_n+1, the packet is (1+E)^mT at index n. This is an identity about a finite polynomial in one shift. It does not introduce a further assumption about recurrence solutions or convergence. Reduction modulo a power of two commutes with this polynomial, which is the basis of the valuation computation in the next ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-77f042d815160051","tex":"F_n^{\\rm src}","aliases":"F_n^rm src rm","display":"F_n^{\\rm src}","kind":"introduction-passage","excerpt":"Let E_n denote the well poised _6F_5 series at -1 in Zudilin's displayed Whipple transformation, with a=3n+1, b=c=d=n+1/2, and e=n+1. The transformation states H_n=fracΓ(3n+2)Γ(n+tfrac12) Γ(2n+tfrac32)Γ(2n+1)E_n. Its condition Re(1+a-d-e)>0 becomes n+1/2>0 here. The displayed prefactor of the source hypergeometric form F_n^rm src is F_n^rm src=(-1)","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":28},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":10},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a reduced denominator. It is unrelated to the polynomial q(n) in the recurrence. The rational coefficients of Eskandari's integral normalization are widetilde U_n,m=8(-1)^n16^-nS_n,m and widetilde V_n,m=8(-1)^n16^-nH_n,m. They must not be mistaken for the fix","count":35},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":50},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"introduction-passage","excerpt":"Let E_n denote the well poised _6F_5 series at -1 in Zudilin's displayed Whipple transformation, with a=3n+1, b=c=d=n+1/2, and e=n+1. The transformation states H_n=fracΓ(3n+2)Γ(n+tfrac12) Γ(2n+tfrac32)Γ(2n+1)E_n. Its condition Re(1+a-d-e)>0 becomes n+1/2>0 here. The displayed prefactor of the source hypergeometric form F_n^rm src is F_n^rm src=(-1)","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ad0b33a7bc01e36b","tex":"H_{n,m}","aliases":"H_n,m","display":"H_{n,m}","kind":"introduction-passage","excerpt":"Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a reduced denominator. It is unrelated to the polynomial q(n) in the recurrence. The rational coefficients of Eskandari's integral normalization are widetilde U_n,m=8(-1)^n16^-nS_n,m and widetilde V_n,m=8(-1)^n16^-nH_n,m. They must not be mistaken for the fix","count":13},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"For integers n,m≥0, define S_n,m · =∑_i=0^mbinom miT_n+i, H_n,m · =∑_i=0^mbinom mi16^n+iv_n+i. The two index symbol H_n,m denotes a rational packet coefficient. It is distinct from the hypergeometric function H_n used in the integral normalization derivation. The theorem on T_n proves S_n,m∈Z_>0 at every pair. The coefficient H_n,m is rational and ","count":12},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-db24dc4f777d08de","tex":"k^{1/a}","aliases":"k^1/a","display":"k^{1/a}","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":35},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"introduction-passage","excerpt":"Put L_N=lcm(1,2,…,N), L_0=1. The source's least common multiple notation is translated to L_N here. The letter D will later denote an actual reduced denominator, which can be much smaller than this universal clearing bound. Krattenthaler and Rivoal's Theorem 2 gives 16^nL_2n^2b_n∈Z for positive integer n [source reference]. In the present normaliza","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ec50f746a738db8a","tex":"L_{2(n+m)}^2","aliases":"L_2(n+m)^2","display":"L_{2(n+m)}^2","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cb39d70930091654","tex":"L_{2n}^2","aliases":"L_2n^2","display":"L_{2n}^2","kind":"introduction-passage","excerpt":"Put L_N=lcm(1,2,…,N), L_0=1. The source's least common multiple notation is translated to L_N here. The letter D will later denote an actual reduced denominator, which can be much smaller than this universal clearing bound. Krattenthaler and Rivoal's Theorem 2 gives 16^nL_2n^2b_n∈Z for positive integer n [source reference]. In the present normaliza","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c91dabfc8b700b3b","tex":"L_N","aliases":"L_N","display":"L_N","kind":"introduction-passage","excerpt":"Set r_n=fracv_nu_n, d_n=fracp(n+1)2(2n+1)^2(2n+2)^2u_nu_n+1. The symbol d_n denotes a positive rational step size. It is unrelated to the least common multiples L_N used in denominator clearing. Dividing the Wronskian identity by u_nu_n+1 yields r_n+1-r_n=(-1)^n d_n. The movement of the approximants therefore alternates exactly.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3860d7e8f88507bb","tex":"\\operatorname{lcm}","aliases":"lcm operatorname","display":"\\operatorname{lcm}","kind":"introduction-passage","excerpt":"Put L_N=lcm(1,2,…,N), L_0=1. The source's least common multiple notation is translated to L_N here. The letter D will later denote an actual reduced denominator, which can be much smaller than this universal clearing bound. Krattenthaler and Rivoal's Theorem 2 gives 16^nL_2n^2b_n∈Z for positive integer n [source reference]. In the present normaliza","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":49},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"introduction-passage","excerpt":"the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ xlog p. The first term is k+o(k). The remaining terms are O(sqrtklog k)=o(k), using ϑ(x)=O(x) and at most log_2 k nonzero term","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":118},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":73},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"introduction-passage","excerpt":"-LCM Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ xlog p. The first term is k+o(k). The remaining terms are O(sqrtklog k)=o(k), using ϑ(x)=","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":9},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":24},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-fbe5654b944a7373","tex":"p^a","aliases":"p^a","display":"p^a","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d098c5b56cecbb3c","tex":"p_{\\rm red}","aliases":"p_rm red rm","display":"p_{\\rm red}","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":75},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":31},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-9488db77f2dc3f77","tex":"q_{\\rm red}","aliases":"q_rm red rm","display":"q_{\\rm red}","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5a5d034ec56cc06c","tex":"r_{n+1}","aliases":"r_n+1","display":"r_{n+1}","kind":"introduction-passage","excerpt":"Set r_n=fracv_nu_n, d_n=fracp(n+1)2(2n+1)^2(2n+2)^2u_nu_n+1. The symbol d_n denotes a positive rational step size. It is unrelated to the least common multiples L_N used in denominator clearing. Dividing the Wronskian identity by u_nu_n+1 yields r_n+1-r_n=(-1)^n d_n. The movement of the approximants therefore alternates exactly.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"introduction-passage","excerpt":"Let E_n denote the well poised _6F_5 series at -1 in Zudilin's displayed Whipple transformation, with a=3n+1, b=c=d=n+1/2, and e=n+1. The transformation states H_n=fracΓ(3n+2)Γ(n+tfrac12) Γ(2n+tfrac32)Γ(2n+1)E_n. Its condition Re(1+a-d-e)>0 becomes n+1/2>0 here. The displayed prefactor of the source hypergeometric form F_n^rm src is F_n^rm src=(-1)","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"introduction-passage","excerpt":"Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a reduced denominator. It is unrelated to the polynomial q(n) in the recurrence. The rational coefficients of Eskandari's integral normalization are widetilde U_n,m=8(-1)^n16^-nS_n,m and widetilde V_n,m=8(-1)^n16^-nH_n,m. They must not be mistaken for the fix","count":36},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Let E_n denote the well poised _6F_5 series at -1 in Zudilin's displayed Whipple transformation, with a=3n+1, b=c=d=n+1/2, and e=n+1. The transformation states H_n=fracΓ(3n+2)Γ(n+tfrac12) Γ(2n+tfrac32)Γ(2n+1)E_n. Its condition Re(1+a-d-e)>0 becomes n+1/2>0 here. The displayed prefactor of the source hypergeometric form F_n^rm src is F_n^rm src=(-1)","count":36},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-89e5e006dbe232c0","tex":"S_{n,m}","aliases":"S_n,m","display":"S_{n,m}","kind":"introduction-passage","excerpt":"Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a reduced denominator. It is unrelated to the polynomial q(n) in the recurrence. The rational coefficients of Eskandari's integral normalization are widetilde U_n,m=8(-1)^n16^-nS_n,m and widetilde V_n,m=8(-1)^n16^-nH_n,m. They must not be mistaken for the fix","count":19},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"For integers n,m≥0, define S_n,m · =∑_i=0^mbinom miT_n+i, H_n,m · =∑_i=0^mbinom mi16^n+iv_n+i. The two index symbol H_n,m denotes a rational packet coefficient. It is distinct from the hypergeometric function H_n used in the integral normalization derivation. The theorem on T_n proves S_n,m∈Z_>0 at every pair. The coefficient H_n,m is rational and ","count":29},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-64afaf0ed6cbcadc","tex":"T_n","aliases":"T_n","display":"T_n","kind":"introduction-passage","excerpt":"H_n,m · =∑_i=0^mbinom mi16^n+iv_n+i. The two index symbol H_n,m denotes a rational packet coefficient. It is distinct from the hypergeometric function H_n used in the integral normalization derivation. The theorem on T_n proves S_n,m∈Z_>0 at every pair. The coefficient H_n,m is rational and requires separate denominator clearing.","count":16},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cf3c233b05af8e95","tex":"T_{n+i}","aliases":"T_n+i","display":"T_{n+i}","kind":"introduction-passage","excerpt":"For integers n,m≥0, define S_n,m · =∑_i=0^mbinom miT_n+i, H_n,m · =∑_i=0^mbinom mi16^n+iv_n+i. The two index symbol H_n,m denotes a rational packet coefficient. It is distinct from the hypergeometric function H_n used in the integral normalization derivation. The theorem on T_n proves S_n,m∈Z_>0 at every pair. The coefficient H_n,m is rational and ","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":32},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2f97c9046c1a4c7a","tex":"u_2","aliases":"u_2","display":"u_2","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cdc6b49492357264","tex":"u_3","aliases":"u_3","display":"u_3","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b03775ada4b9cdf0","tex":"u_{n+1}","aliases":"u_n+1","display":"u_{n+1}","kind":"introduction-passage","excerpt":"Set r_n=fracv_nu_n, d_n=fracp(n+1)2(2n+1)^2(2n+2)^2u_nu_n+1. The symbol d_n denotes a positive rational step size. It is unrelated to the least common multiples L_N used in denominator clearing. Dividing the Wronskian identity by u_nu_n+1 yields r_n+1-r_n=(-1)^n d_n. The movement of the approximants therefore alternates exactly.","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":16},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"introduction-passage","excerpt":"The initial u values are positive, and each subsequent value is a positive linear combination of its two predecessors. For v, the initial v_0 is zero and v_1>0. The first forward step gives v_2>0, and the same induction applies thereafter. Consequently the quotient r_n is defined at every index.","count":26},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5ab978331477ab8e","tex":"v_{n+i}","aliases":"v_n+i","display":"v_{n+i}","kind":"introduction-passage","excerpt":"For integers n,m≥0, define S_n,m · =∑_i=0^mbinom miT_n+i, H_n,m · =∑_i=0^mbinom mi16^n+iv_n+i. The two index symbol H_n,m denotes a rational packet coefficient. It is distinct from the hypergeometric function H_n used in the integral normalization derivation. The theorem on T_n proves S_n,m∈Z_>0 at every pair. The coefficient H_n,m is rational and ","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":23},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8945947ef031a855","tex":"w_n","aliases":"w_n","display":"w_n","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":9},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-bb5b0de353a40856","tex":"w_{n-1}","aliases":"w_n-1","display":"w_{n-1}","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":6},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3f476ecaf7bf4a0d","tex":"\\widetilde J","aliases":"widetilde J widetilde","display":"\\widetilde J","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":21},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7a03f901dc9d66a0","tex":"\\widetilde J_{n,m}","aliases":"widetilde J_n,m widetilde","display":"\\widetilde J_{n,m}","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":13},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-aba1aecba944a16d","tex":"\\widetilde U","aliases":"widetilde U widetilde","display":"\\widetilde U","kind":"introduction-passage","excerpt":"Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a reduced denominator. It is unrelated to the polynomial q(n) in the recurrence. The rational coefficients of Eskandari's integral normalization are widetilde U_n,m=8(-1)^n16^-nS_n,m and widetilde V_n,m=8(-1)^n16^-nH_n,m. They must not be mistaken for the fix","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-626375c71503aba1","tex":"\\widetilde U_{n,m}","aliases":"widetilde U_n,m widetilde","display":"\\widetilde U_{n,m}","kind":"introduction-passage","excerpt":"Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a reduced denominator. It is unrelated to the polynomial q(n) in the recurrence. The rational coefficients of Eskandari's integral normalization are widetilde U_n,m=8(-1)^n16^-nS_n,m and widetilde V_n,m=8(-1)^n16^-nH_n,m. They must not be mistaken for the fix","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-60cd197d59c08b6e","tex":"\\widetilde V","aliases":"widetilde V widetilde","display":"\\widetilde V","kind":"introduction-passage","excerpt":"Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a reduced denominator. It is unrelated to the polynomial q(n) in the recurrence. The rational coefficients of Eskandari's integral normalization are widetilde U_n,m=8(-1)^n16^-nS_n,m and widetilde V_n,m=8(-1)^n16^-nH_n,m. They must not be mistaken for the fix","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5bebb2be955c1557","tex":"\\widetilde V_{n,m}","aliases":"widetilde V_n,m widetilde","display":"\\widetilde V_{n,m}","kind":"introduction-passage","excerpt":"Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a reduced denominator. It is unrelated to the polynomial q(n) in the recurrence. The rational coefficients of Eskandari's integral normalization are widetilde U_n,m=8(-1)^n16^-nS_n,m and widetilde V_n,m=8(-1)^n16^-nH_n,m. They must not be mistaken for the fix","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":66},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4715e03ab2461874","tex":"x_{n+1}","aliases":"x_n+1","display":"x_{n+1}","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"introduction-passage","excerpt":"Put L_N=lcm(1,2,…,N), L_0=1. The source's least common multiple notation is translated to L_N here. The letter D will later denote an actual reduced denominator, which can be much smaller than this universal clearing bound. Krattenthaler and Rivoal's Theorem 2 gives 16^nL_2n^2b_n∈Z for positive integer n [source reference]. In the present normaliza","count":18},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a0d8044d38df3073","tex":"\\Z_{>0}","aliases":"Z_>0 Z","display":null,"kind":"introduction-passage","excerpt":"H_n,m · =∑_i=0^mbinom mi16^n+iv_n+i. The two index symbol H_n,m denotes a rational packet coefficient. It is distinct from the hypergeometric function H_n used in the integral normalization derivation. The theorem on T_n proves S_n,m∈Z_>0 at every pair. The coefficient H_n,m is rational and requires separate denominator clearing.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":15},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"melcm(1,2,…,N), L_0=1. The source's least common multiple notation is translated to L_N here. The letter D will later denote an actual reduced denominator, which can be much smaller than this universal clearing bound. Krattenthaler and Rivoal's Theorem 2 gives 16^nL_2n^2b_n∈Z for positive integer n [source reference]. In the present normalization i","count":14},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":97},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e8c617160e3b774f","tex":"\\alpha_2","aliases":"α_2 alpha","display":"\\alpha_2","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a823753ea2193233","tex":"\\alpha_n","aliases":"α_n alpha","display":"\\alpha_n","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":21},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-1bac3c7ce6eb2507","tex":"\\beta_1","aliases":"β_1 beta","display":"\\beta_1","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-05f6c598f663dcba","tex":"\\beta_2","aliases":"β_2 beta","display":"\\beta_2","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-17d9a794b7ae4113","tex":"\\beta_n","aliases":"β_n beta","display":"\\beta_n","kind":"introduction-passage","excerpt":"For a positive solution, ratios of consecutive values give another finite description. If x_n=w_n/w_n-1 is defined, then x_n+1=α_n+fracβ_nx_n. For u, every such ratio is defined. The first two iterations are fracu_2u_1=α_1+fracβ_17/4, fracu_3u_2= α_2+fracβ_2α_1+β_1/(7/4). Repeated substitution yields a finite generalized continued fraction. For v, ","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"introduction-passage","excerpt":"Let E_n denote the well poised _6F_5 series at -1 in Zudilin's displayed Whipple transformation, with a=3n+1, b=c=d=n+1/2, and e=n+1. The transformation states H_n=fracΓ(3n+2)Γ(n+tfrac12) Γ(2n+tfrac32)Γ(2n+1)E_n. Its condition Re(1+a-d-e)>0 becomes n+1/2>0 here. The displayed prefactor of the source hypergeometric form F_n^rm src is F_n^rm src=(-1)","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-6a830538315a1ab8","tex":"\\vartheta","aliases":"ϑ vartheta","display":"\\vartheta","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":11},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-59217d9c23468153","tex":"\\Lambda_{n,m}","aliases":"Λ_n,m Lambda","display":"\\Lambda_{n,m}","kind":"introduction-passage","excerpt":"For a fixed pair, put U=S_n,m and write V=H_n,m=N/D in reduced form. The exact dictionary is Q · =8L_2(n+m)^2,notag A · =(-1)^nQU, B=(-1)^nQV,notag h · =gcd(DU,N)=gcd(U,N),notag g · =gcd(|A|,|B|)=frac QD h,notag q_rm red · =fracDUh, p_rm red=frac Nh,notag Λ_n,m · =frac AgG-frac Bg =frac16^nL_2(n+m)^2gwidetilde J_n,m. The symbol q_rm red denotes a r","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":18},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f5b15e77866469c7","tex":"\\psi_{\\rm Ch}","aliases":"ψ_rm Ch psi rm","display":"\\psi_{\\rm Ch}","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Put L_N=lcm(1,2,…,N), L_0=1. The source's least common multiple notation is translated to L_N here. The letter D will later denote an actual reduced denominator, which can be much smaller than this universal clearing bound. Krattenthaler and Rivoal's Theorem 2 gives 16^nL_2n^2b_n∈Z for positive integer n [source reference]. In the present normaliza","count":10},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Put L_N=lcm(1,2,…,N), L_0=1. The source's least common multiple notation is translated to L_N here. The letter D will later denote an actual reduced denominator, which can be much smaller than this universal clearing bound. Krattenthaler and Rivoal's Theorem 2 gives 16^nL_2n^2b_n∈Z for positive integer n [source reference]. In the present normaliza","count":24},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3d22086a9e52e759","tex":"\\sum_{a\\geq1}","aliases":"∑_a≥1 sum geq","display":"\\sum_{a\\geq1}","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-39e405e72317695b","tex":"\\sum_{i=0}^m","aliases":"∑_i=0^m sum","display":"\\sum_{i=0}^m","kind":"introduction-passage","excerpt":"For integers n,m≥0, define S_n,m · =∑_i=0^mbinom miT_n+i, H_n,m · =∑_i=0^mbinom mi16^n+iv_n+i. The two index symbol H_n,m denotes a rational packet coefficient. It is distinct from the hypergeometric function H_n used in the integral normalization derivation. The theorem on T_n proves S_n,m∈Z_>0 at every pair. The coefficient H_n,m is rational and ","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-23935efd5ff424d7","tex":"\\sum_{p^a\\leq k}","aliases":"∑_p^a≤ k sum leq","display":"\\sum_{p^a\\leq k}","kind":"introduction-passage","excerpt":"The prime number theorem supplies the standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ x","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f0e9982838900566","tex":"\\sum_{p\\leq x}","aliases":"∑_p≤ x sum leq","display":"\\sum_{p\\leq x}","kind":"introduction-passage","excerpt":"e standard asymptotic~[source reference] log L_k=k+o(k). Indeed log L_k=∑_p^a≤ klog p is the Chebyshev function ψ_rm Ch(k), and its ratio to k tends to one. Source contract LC10 records the prime number theorem and the elementary prime power conversion. Equivalently, write log L_k=∑_a≥1ϑ(k^1/a), where ϑ(x)=∑_p≤ xlog p. The first term is k+o(k). The","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"The Euler interval uses a_k=(2k+1)^-2 and the positive difference convention Δ a_k=a_k-a_k+1. The identity Δ^ja_0=∫_0^1(1-x^2)^j(-log x) dd x gives 0≤Δ^ja_0≤1. The positive Euler series for G is ∑_j≥02^-j-1Δ^ja_0. Keeping the first 1001 terms therefore gives a lower endpoint and an upper endpoint obtained by adding 2^-1001. All retained terms are r","count":9},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"At n=1, direct substitution in [source reference] gives p(1)=13, p(2)=65, q(1)=10699, a_1=1872, c_1=260. The recurrence therefore yields u_2 · =frac10699·(7/4)+2601872 =frac64964,","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"At n=2 the coefficients become p(2)=65, p(3)=157, q(2)=434871, a_2=58500, c_2=22608. Using the fractions just obtained, u_3 · =frac434871(649/64)+22608(7/4)58500","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"use","excerpt":"Suppose a cleared form is A_jα-B_j with A_j,B_j∈Z and (A_j,B_j)≠(0,0). Define its common content by h_j=gcd(absA_j,absB_j)>0. The primitive pair is (fracA_jh_j,fracB_jh_j), and the associated form has size fracabsA_jα-B_jh_j. Both coefficients remain integers. Thus a proved lower bound on common content may improve an irrationality argument. The wo","count":9},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"Such a D_j always exists for one finite rational pair. Write u_j=a_j/b_j and v_j=c_j/d_j in lowest terms with positive denominators. The smallest positive common clearing factor is D_j^min=lcm(b_j,d_j). Indeed, integrality of D_ja_j/b_j forces b_jmid D_j by coprimality, and integrality of D_jc_j/d_j forces d_jmid D_j. Conversely, their least common","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"The Euler interval uses a_k=(2k+1)^-2 and the positive difference convention Δ a_k=a_k-a_k+1. The identity Δ^ja_0=∫_0^1(1-x^2)^j(-log x) dd x gives 0≤Δ^ja_0≤1. The positive Euler series for G is ∑_j≥02^-j-1Δ^ja_0. Keeping the first 1001 terms therefore gives a lower endpoint and an upper endpoint obtained by adding 2^-1001. All retained terms are r","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e383dbc0f9ccfa0b","tex":"a_{k+1}","aliases":"a_k+1","display":"a_{k+1}","kind":"use","excerpt":"The Euler interval uses a_k=(2k+1)^-2 and the positive difference convention Δ a_k=a_k-a_k+1. The identity Δ^ja_0=∫_0^1(1-x^2)^j(-log x) dd x gives 0≤Δ^ja_0≤1. The positive Euler series for G is ∑_j≥02^-j-1Δ^ja_0. Keeping the first 1001 terms therefore gives a lower endpoint and an upper endpoint obtained by adding 2^-1001. All retained terms are r","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"[tbp] centeringincludegraphics[width=0.93linewidth]../visual/figures/09_recurrence_structure.pdf captionTwo preceding values determine the next rational value. The forward rule begins at n=1 and uses the stated initial pair. Here and in subsequent recurrence diagrams, A_n=a_n, Q_n=q(n) and C_n=c_n.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"For 0<δ<1, integration by parts gives ∫_δ^1fracarctan xx dd x =-arctan(δ)logδ -∫_δ^1fraclog x1+x^2 dd x. The boundary expression tends to zero as δdownarrow0, since 0≤arctanδ≤δ and δabslogδ→0. The first equality follows from Proposition~[source reference].","count":22},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-27029f09b14e6d27","tex":"\\arcsin","aliases":"arcsin","display":null,"kind":"use","excerpt":"The initial integral supplies a short diagnostic. Substitute x=t^2 and y=1-s^2. Then J_0 · =4∫_0^1 ∫_0^1fracdd s dd t1-t^2+t^2s^2 · =4∫_0^1fracarcsin ttsqrt1-t^2 dd t =4∫_0^π/2θcscθ ddθ. Since an antiderivative of cscθ is logtan(θ/2), integration by parts yields J_0=-4∫_0^π/2logtan(θ/2) ddθ =-8∫_0^π/4logtan z dd z=8G. Both boundary terms vanish. At","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-16aff9ea611eb515","tex":"\\arctan","aliases":"arctan","display":"\\arctan","kind":"use","excerpt":"[Arctangent and tangent forms] The following equalities hold: G=∫_0^1fracarctan xx dd x =-∫_0^π/4log(tanθ) ddθ. The integrand arctan x/x is assigned its continuous value 1 at x=0.","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-16636769fc160a5a","tex":"b^{-1}","aliases":"b^-1","display":"b^{-1}","kind":"use","excerpt":"For a fixed rational α, every integer form Qα-P lies in the discrete set b^-1Z. Its nonzero values cannot enter the open interval (-1/b,1/b). The denominator b need not be known to use this obstruction. If a constructed infinite family has nonzero values tending to zero, it eventually violates the gap for any proposed b.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","excerpt":"Suppose a cleared form is A_jα-B_j with A_j,B_j∈Z and (A_j,B_j)≠(0,0). Define its common content by h_j=gcd(absA_j,absB_j)>0. The primitive pair is (fracA_jh_j,fracB_jh_j), and the associated form has size fracabsA_jα-B_jh_j. Both coefficients remain integers. Thus a proved lower bound on common content may improve an irrationality argument. The wo","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"Such a D_j always exists for one finite rational pair. Write u_j=a_j/b_j and v_j=c_j/d_j in lowest terms with positive denominators. The smallest positive common clearing factor is D_j^min=lcm(b_j,d_j). Indeed, integrality of D_ja_j/b_j forces b_jmid D_j by coprimality, and integrality of D_jc_j/d_j forces d_jmid D_j. Conversely, their least common","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"At n=1, direct substitution in [source reference] gives p(1)=13, p(2)=65, q(1)=10699, a_1=1872, c_1=260. The recurrence therefore yields u_2 · =frac10699·(7/4)+2601872 =frac64964,","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"At n=2 the coefficients become p(2)=65, p(3)=157, q(2)=434871, a_2=58500, c_2=22608. Using the fractions just obtained, u_3 · =frac434871(649/64)+22608(7/4)58500","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-9b74ac7e4a926fd8","tex":"c_A","aliases":"c_A","display":"c_A","kind":"use","excerpt":"sion of the finite recurrence audit. Together with the filter bound it controls the approximation error |G-frac VU| =frac|widetilde J||widetilde U|. Before any additional quantified gcd saving, the reduced denominator q is at most the absolute cleared coefficient |A|. Its available exponent is c_A(α)=4(1+α)+log R+αlog(1+R). The source's resulting g","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"Such a D_j always exists for one finite rational pair. Write u_j=a_j/b_j and v_j=c_j/d_j in lowest terms with positive denominators. The smallest positive common clearing factor is D_j^min=lcm(b_j,d_j). Indeed, integrality of D_ja_j/b_j forces b_jmid D_j by coprimality, and integrality of D_jc_j/d_j forces d_jmid D_j. Conversely, their least common","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cfbf41dac773cbfc","tex":"C_n","aliases":"C_n","display":"C_n","kind":"use","excerpt":"[tbp] centeringincludegraphics[width=0.93linewidth]../visual/figures/09_recurrence_structure.pdf captionTwo preceding values determine the next rational value. The forward rule begins at n=1 and uses the stated initial pair. Here and in subsequent recurrence diagrams, A_n=a_n, Q_n=q(n) and C_n=c_n.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-dfaf75eb728e0e02","tex":"C_{n,m}","aliases":"C_n,m","display":"C_{n,m}","kind":"use","excerpt":"Let N=n+m and put C_n,m=L_2N^2. The pair produced by the historical script is A_n,m · =8(-1)^nC_n,mS_n,m,","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-98d23fc3caf01f4d","tex":"\\csc","aliases":"csc","display":"\\csc","kind":"use","excerpt":"The initial integral supplies a short diagnostic. Substitute x=t^2 and y=1-s^2. Then J_0 · =4∫_0^1 ∫_0^1fracdd s dd t1-t^2+t^2s^2 · =4∫_0^1fracarcsin ttsqrt1-t^2 dd t =4∫_0^π/2θcscθ ddθ. Since an antiderivative of cscθ is logtan(θ/2), integration by parts yields J_0=-4∫_0^π/2logtan(θ/2) ddθ =-8∫_0^π/4logtan z dd z=8G. Both boundary terms vanish. At","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"For a positive integer N, the scanned domain is D_N={(n,m):1≤ n≤ N, 0≤ m≤4n}. The number of scanned pairs is ∑_n=1^N(4n+1)=2N^2+3N. The upper slope 4 is part of the experiment's definition. The scan supplies no information about pairs above this boundary or with n>N.","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d702cb62c5b65c44","tex":"\\mathcal D_{100}","aliases":"D_100 mathcal","display":"\\mathcal D_{100}","kind":"use","excerpt":"The historical smaller crosscheck reports the same maximum from direct sums through N=100. Matching only two summary maxima would be a weak comparison: different entry values can share the same maximum. The release therefore compares every pair in D_100.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-619bb3ff0afb28e5","tex":"\\mathcal D_{2000}","aliases":"D_2000 mathcal","display":"\\mathcal D_{2000}","kind":"use","excerpt":"The historical run takes N=2000 and K=128. The release reran the unchanged historical algorithm and compared its result with the recovered output. The finite arithmetic results agree. In the domain D_2000 there are exactly 8,006,000 pairs. Every observed residue is nonzero, so the residue lemma applies at every pair. The exact maximum valuation in ","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-75abd7afbcc4f1b7","tex":"d_{2k}","aliases":"d_2k","display":"d_{2k}","kind":"use","excerpt":"Equation [source reference] and the preceding lemma identify r_n with partial sums of a convergent alternating series: r_n=∑_j=0^n-1(-1)^j d_j, L=∑_j=0^∞(-1)^j d_j. For the even subsequence, r_2k+2-r_2k=d_2k-d_2k+1>0. For the odd subsequence, r_2k+3-r_2k+1=-d_2k+1+d_2k+2<0. Their adjacent separation d_2k tends to zero, so they have the same limit. ","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-57f77d10ddbf5379","tex":"d_{2k+1}","aliases":"d_2k+1","display":"d_{2k+1}","kind":"use","excerpt":"Equation [source reference] and the preceding lemma identify r_n with partial sums of a convergent alternating series: r_n=∑_j=0^n-1(-1)^j d_j, L=∑_j=0^∞(-1)^j d_j. For the even subsequence, r_2k+2-r_2k=d_2k-d_2k+1>0. For the odd subsequence, r_2k+3-r_2k+1=-d_2k+1+d_2k+2<0. Their adjacent separation d_2k tends to zero, so they have the same limit. ","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-fda3ffcc73c0ad9e","tex":"d_{2k+2}","aliases":"d_2k+2","display":"d_{2k+2}","kind":"use","excerpt":"Equation [source reference] and the preceding lemma identify r_n with partial sums of a convergent alternating series: r_n=∑_j=0^n-1(-1)^j d_j, L=∑_j=0^∞(-1)^j d_j. For the even subsequence, r_2k+2-r_2k=d_2k-d_2k+1>0. For the odd subsequence, r_2k+3-r_2k+1=-d_2k+1+d_2k+2<0. Their adjacent separation d_2k tends to zero, so they have the same limit. ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"In the later chapters the starting form is ℓ_j=u_jα-v_j, u_j,v_j∈Q. Choose a positive integer D_j for which D_ju_j and D_jv_j are integers. Then L_j=D_jℓ_j=(D_ju_j)α-D_jv_j has integer coefficients. Its size is D_jabsℓ_j, so the clearance factor must be included in any decay estimate.","count":21},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"Such a D_j always exists for one finite rational pair. Write u_j=a_j/b_j and v_j=c_j/d_j in lowest terms with positive denominators. The smallest positive common clearing factor is D_j^min=lcm(b_j,d_j). Indeed, integrality of D_ja_j/b_j forces b_jmid D_j by coprimality, and integrality of D_jc_j/d_j forces d_jmid D_j. Conversely, their least common","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-87d81befe324c9a1","tex":"D_j^{1/j}","aliases":"D_j^1/j","display":"D_j^{1/j}","kind":"use","excerpt":"[Exponential clearance criterion] Let α∈R, and let u_j,v_j∈Q for j≥1. Let D_j be positive integers such that D_ju_j,D_jv_j∈Z. Assume u_jα-v_j≠0 for all sufficiently large j. Suppose limsup_j→∞absu_jα-v_j^1/j≤λ, limsup_j→∞D_j^1/j≤δ, where λ≥0, δ<∞ and δλ<1. Then α is irrational.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c32a747cd72fd754","tex":"D_j^{\\min}","aliases":"D_j^min min","display":"D_j^{\\min}","kind":"use","excerpt":"Such a D_j always exists for one finite rational pair. Write u_j=a_j/b_j and v_j=c_j/d_j in lowest terms with positive denominators. The smallest positive common clearing factor is D_j^min=lcm(b_j,d_j). Indeed, integrality of D_ja_j/b_j forces b_jmid D_j by coprimality, and integrality of D_jc_j/d_j forces d_jmid D_j. Conversely, their least common","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d3405c1cc6903854","tex":"D_n","aliases":"D_n","display":"D_n","kind":"use","excerpt":"There is also a formulation independent of the universal clearing choice. With D_n,h_n as in Chapter~[source reference], the primitive error is (-1)^n(D_n/h_n)(S_n,m(n)G-H_n,m(n)). An estimate directly on that expression would suffice. If denominator reduction yields a provably smaller exponential cost, the excess factor Q_n/D_n should simply be re","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8aca8308192a2ca3","tex":"\\mathcal D_N","aliases":"D_N mathcal","display":"\\mathcal D_N","kind":"use","excerpt":"For a positive integer N, the scanned domain is D_N={(n,m):1≤ n≤ N, 0≤ m≤4n}. The number of scanned pairs is ∑_n=1^N(4n+1)=2N^2+3N. The upper slope 4 is part of the experiment's definition. The scan supplies no information about pairs above this boundary or with n>N.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-be9ed4f9d42921bd","tex":"d_{n+1}","aliases":"d_n+1","display":"d_{n+1}","kind":"use","excerpt":"[Alternating brackets] There is a real number L such that r_n→ L. For every k≥0, r_2k<L<r_2k+1. The even subsequence increases strictly and the odd subsequence decreases strictly. For every n≥0, d_n-d_n+1<|L-r_n|<d_n, sign(L-r_n)=(-1)^n.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4144475705f1d2e6","tex":"d_{n-1}","aliases":"d_n-1","display":"d_{n-1}","kind":"use","excerpt":"Positivity is immediate. For n≥1, the determinant recurrence gives fracd_nd_n-1 =fracc_na_nfracu_n-1u_n+1<1. The strict inequality follows from a_nu_n+1=q(n)u_n+c_nu_n-1>c_nu_n-1. Finally u_nu_n+1≥1, so 0<d_n≤ fracp(n+1)2(2n+1)^2(2n+2)^2⟶0. No limiting characteristic root is needed for this conclusion.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd","display":null,"kind":"use","excerpt":"widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^n+1 dd x dd y =8(-1)^n(u_nG-v_n). These are fixed normalization identities. Their factors should be checked together when translating a differently normalized source.","count":25},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-27c4a94b26be76b0","tex":"\\operatorname{den}","aliases":"den operatorname","display":"\\operatorname{den}","kind":"use","excerpt":"The first row illustrates the exact check without printing its very large numerator and denominator. The independently reduced rational packet has denominator D=den(∑_i=0^66binom66i16^20+iv_20+i). Its recorded quotient Q/D is 1 703 936, and the minimal pair has gcd 8. Multiplication gives 1 703 936·8=13 631 488, exactly the historical value. The co","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"The determinant is det M_n=-β_n=-fracc_na_n<0. Every transfer is invertible, with M_n^-1=0 · 1","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"For 0<δ<1, integration by parts gives ∫_δ^1fracarctan xx dd x =-arctan(δ)logδ -∫_δ^1fraclog x1+x^2 dd x. The boundary expression tends to zero as δdownarrow0, since 0≤arctanδ≤δ and δabslogδ→0. The first equality follows from Proposition~[source reference].","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c3d36a4836837ddc","tex":"e^{(-\\tau+\\eps)j}","aliases":"e^(-τ+eps)j tau eps","display":null,"kind":"use","excerpt":"Choose eps>0 with σ-τ+2eps<0. Eventually D_j≤ e^(σ+eps)j and absu_jα-v_j≤ e^(-τ+eps)j. Their product tends to zero. Apply the integer form criterion.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-0403cc7f51c25783","tex":"e^{(\\sigma+\\eps)j}","aliases":"e^(σ+eps)j sigma eps","display":null,"kind":"use","excerpt":"Choose eps>0 with σ-τ+2eps<0. Eventually D_j≤ e^(σ+eps)j and absu_jα-v_j≤ e^(-τ+eps)j. Their product tends to zero. Apply the integer form criterion.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ecedfa459f947dfd","tex":"e^{-(a+1)t}","aliases":"e^-(a+1)t","display":"e^{-(a+1)t}","kind":"use","excerpt":"Set x=e^-t. Then -log x=t, dd x=-e^-tdd t, and the integral becomes ∫_0^∞t e^-(a+1)t dd t. Integration by parts gives (a+1)^-2. The boundary term at infinity is zero because an exponential with positive decay rate dominates t. The boundary term at zero is also zero.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-10efc71e0f8e8940","tex":"e^{-n}","aliases":"e^-n","display":"e^{-n}","kind":"use","excerpt":"There is a simple logical reason to resist extrapolation. Given any finite list of positive values x_1,…,x_N, it has a positive extension that tends to zero and another positive extension that tends to infinity. Keep the list fixed and set the tail to e^-n in the first case or e^n in the second. Similarly, any finite list of positive integer gcds c","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3f6dc0c78382f5b1","tex":"e^{-t}","aliases":"e^-t","display":"e^{-t}","kind":"use","excerpt":"Set x=e^-t. Then -log x=t, dd x=-e^-tdd t, and the integral becomes ∫_0^∞t e^-(a+1)t dd t. Integration by parts gives (a+1)^-2. The boundary term at infinity is zero because an exponential with positive decay rate dominates t. The boundary term at zero is also zero.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-83d8d1cb0d07f185","tex":"e^{cn}","aliases":"e^cn","display":"e^{cn}","kind":"use","excerpt":"A positive exponent in an upper bound has the same logical limitation. If x_n≤ e^cn with c>0, both x_n=e^-n and x_n=e^cn/2 satisfy the bound eventually. One sequence decays and the other grows. Thus the failure of the bound to force decay is a proved limitation of the estimate, not proof of divergence of the object it bounds.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f21dcbcd01d17cd8","tex":"e^{cn/2}","aliases":"e^cn/2","display":"e^{cn/2}","kind":"use","excerpt":"A positive exponent in an upper bound has the same logical limitation. If x_n≤ e^cn with c>0, both x_n=e^-n and x_n=e^cn/2 satisfy the bound eventually. One sequence decays and the other grows. Thus the failure of the bound to force decay is a proved limitation of the estimate, not proof of divergence of the object it bounds.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-52253ff3185e0229","tex":"e^n","aliases":"e^n","display":"e^n","kind":"use","excerpt":"There is a simple logical reason to resist extrapolation. Given any finite list of positive values x_1,…,x_N, it has a positive extension that tends to zero and another positive extension that tends to infinity. Keep the list fixed and set the tail to e^-n in the first case or e^n in the second. Similarly, any finite list of positive integer gcds c","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"[Small Euler intervals] The first difference values are Δ^0a_0=1, Δ a_0=frac89, Δ^2a_0=frac184225. They give E_0=frac12, E_1=frac1318, E_2=frac371450. Thus, for example, frac371450<G<frac17091800. These short transformed sums are useful for checking conventions. Their principal advantage appears as N grows, because the certified width decreases geo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"[Small Euler intervals] The first difference values are Δ^0a_0=1, Δ a_0=frac89, Δ^2a_0=frac184225. They give E_0=frac12, E_1=frac1318, E_2=frac371450. Thus, for example, frac371450<G<frac17091800. These short transformed sums are useful for checking conventions. Their principal advantage appears as N grows, because the certified width decreases geo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"[Small Euler intervals] The first difference values are Δ^0a_0=1, Δ a_0=frac89, Δ^2a_0=frac184225. They give E_0=frac12, E_1=frac1318, E_2=frac371450. Thus, for example, frac371450<G<frac17091800. These short transformed sums are useful for checking conventions. Their principal advantage appears as N grows, because the certified width decreases geo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e466c5eab5f55a38","tex":"E_N","aliases":"E_N","display":"E_N","kind":"use","excerpt":"[Certified Euler enclosure] Define E_N=∑_j=0^NfracΔ^j a_02^j+1, N≥0. Then G=∑_j=0^∞fracΔ^j a_02^j+1, E_N<G<E_N+2^-N-1. More precisely, G-E_N =∫_0^1frac-log x1+x^2 (frac1-x^22)^N+1 dd x.","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps","display":null,"kind":"use","excerpt":"Choose eps>0 with σ-τ+2eps<0. Eventually D_j≤ e^(σ+eps)j and absu_jα-v_j≤ e^(-τ+eps)j. Their product tends to zero. Apply the integer form criterion.","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"The primitive form D_j(u_jα-v_j)/h_j has integer coefficients and remains nonzero. Choose eps>0 so that σ-τ-κ+3eps<0. The upper limits for D_j and the error, and the lower limit for h_j, give eventually fracD_jabsu_jα-v_jh_j ≤exp((σ-τ-κ+3eps)j). The right side tends to zero. Apply Theorem~[source reference] to the primitive pair.","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"The two distinguished solutions can be collected into the fundamental matrix F_n= u_n · v_n u_n-1 · v_n-1, n≥1. It satisfies F_n+1=M_n F_n. Its initial determinant is det F_1= det7/4 · 13/8","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-00c724ec0e9825b1","tex":"\\mathcal F_1","aliases":"F_1 mathcal","display":"\\mathcal F_1","kind":"use","excerpt":"u_n-1 · v_n-1, n≥1. It satisfies F_n+1=M_n F_n. Its initial determinant is det F_1= det7/4 · 13/8 1 · 0=-frac138. Multiplicativity of determinants now gives det F_N+1 =-frac138∏_n=1^N(-β_n)≠0. Thus adjacent values of the two solutions remain independent at every index. Each transfer reverses the determinant sign. The closed evaluation of this produ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d9a99ac54df67d94","tex":"F_{1,1}","aliases":"F_1,1","display":"F_{1,1}","kind":"use","excerpt":"The associated rational packet is H_1,1=16v_1+16^2v_2 =26+frac213989 =frac216329. Therefore the raw form is F_1,1=-8(2624G-frac216329) =frac1730569-20992G. A certified interval for G makes this value positive. This is the smallest counterexample in the finite audit to the proposed sign rule signF_n,m=(-1)^m.","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"Expanding (1-xy)^-n-1 as its nonnegative binomial series and applying Tonelli's theorem gives J_n=C_nH_n, where C_n=fracΓ(n+1)^2Γ(n+tfrac12)^2 Γ(2n+tfrac32)^2, and H_n=_3F_2 ( n+1,n+tfrac12,n+1 2n+tfrac32,2n+tfrac32;1). Indeed, the two beta integrals in the term with index k are B(n+tfrac12+k,n+1) and B(n+1+k,n+tfrac12). Their ratios to the corresp","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-28d97d0c394d33bf","tex":"\\mathcal F_n","aliases":"F_n mathcal","display":"\\mathcal F_n","kind":"use","excerpt":"The two distinguished solutions can be collected into the fundamental matrix F_n= u_n · v_n u_n-1 · v_n-1, n≥1. It satisfies F_n+1=M_n F_n. Its initial determinant is det F_1= det7/4 · 13/8","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-0b74d466efa37e51","tex":"\\mathcal F_{N+1}","aliases":"F_N+1 mathcal","display":"\\mathcal F_{N+1}","kind":"use","excerpt":"1 · 0=-frac138. Multiplicativity of determinants now gives det F_N+1 =-frac138∏_n=1^N(-β_n)≠0. Thus adjacent values of the two solutions remain independent at every index. Each transfer reverses the determinant sign. The closed evaluation of this product is the Wronskian theorem in Chapter~[source reference], where the factors telescope and yield e","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-82f51da5d7e5c9b6","tex":"\\mathcal F_{n+1}","aliases":"F_n+1 mathcal","display":"\\mathcal F_{n+1}","kind":"use","excerpt":"u_n-1 · v_n-1, n≥1. It satisfies F_n+1=M_n F_n. Its initial determinant is det F_1= det7/4 · 13/8","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-71f7068c63a5600d","tex":"F_{n,m}","aliases":"F_n,m","display":"F_{n,m}","kind":"use","excerpt":"For n,m≥0, the integer and rational packets are S_n,m=∑_i=0^mbinom miT_n+i, H_n,m=∑_i=0^mbinom mi16^n+iv_n+i. Their raw linear form and integral filter satisfy F_n,m · =8(-1)^n(S_n,mG-H_n,m),notag widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^","count":10},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-15e892a32e1d61ee","tex":"g_n","aliases":"g_n","display":"g_n","kind":"use","excerpt":"With the cleared gcd g_n=g_n,m(n), the primitive form is exactly Λ_n=fracK_ng_nwidetilde J_n,m(n). Suppose a future arithmetic argument establishes δ=liminf_n→∞fraclog g_nn. For finite δ, the present analytic estimate yields limsup_n→∞fraclog|Λ_n|n ≤κ_0(α)+log M(α)-δ. The same interpretation applies when δ=+∞: any sufficiently large finite lower bo","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-54d6ba5efef4d48d","tex":"g_{n,m(n)}","aliases":"g_n,m(n)","display":"g_{n,m(n)}","kind":"use","excerpt":"With the cleared gcd g_n=g_n,m(n), the primitive form is exactly Λ_n=fracK_ng_nwidetilde J_n,m(n). Suppose a future arithmetic argument establishes δ=liminf_n→∞fraclog g_nn. For finite δ, the present analytic estimate yields limsup_n→∞fraclog|Λ_n|n ≤κ_0(α)+log M(α)-δ. The same interpretation applies when δ=+∞: any sufficiently large finite lower bo","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-761827f4bf6cdb52","tex":"H_{1,1}","aliases":"H_1,1","display":"H_{1,1}","kind":"use","excerpt":"The associated rational packet is H_1,1=16v_1+16^2v_2 =26+frac213989 =frac216329. Therefore the raw form is F_1,1=-8(2624G-frac216329) =frac1730569-20992G. A certified interval for G makes this value positive. This is the smallest counterexample in the finite audit to the proposed sign rule signF_n,m=(-1)^m.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"use","excerpt":"Suppose a cleared form is A_jα-B_j with A_j,B_j∈Z and (A_j,B_j)≠(0,0). Define its common content by h_j=gcd(absA_j,absB_j)>0. The primitive pair is (fracA_jh_j,fracB_jh_j), and the associated form has size fracabsA_jα-B_jh_j. Both coefficients remain integers. Thus a proved lower bound on common content may improve an irrationality argument. The wo","count":9},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-785ae493560363c1","tex":"h_n","aliases":"h_n","display":"h_n","kind":"use","excerpt":"There is also a formulation independent of the universal clearing choice. With D_n,h_n as in Chapter~[source reference], the primitive error is (-1)^n(D_n/h_n)(S_n,m(n)G-H_n,m(n)). An estimate directly on that expression would suffice. If denominator reduction yields a provably smaller exponential cost, the excess factor Q_n/D_n should simply be re","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-0179139e91bac0b1","tex":"H_{n,m(n)}","aliases":"H_n,m(n)","display":"H_{n,m(n)}","kind":"use","excerpt":"There is also a formulation independent of the universal clearing choice. With D_n,h_n as in Chapter~[source reference], the primitive error is (-1)^n(D_n/h_n)(S_n,m(n)G-H_n,m(n)). An estimate directly on that expression would suffice. If denominator reduction yields a provably smaller exponential cost, the excess factor Q_n/D_n should simply be re","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5e073754a563e30e","tex":"\\hbox","aliases":"hbox","display":null,"kind":"use","excerpt":"t can recover a proved rate] Assume the setup of Corollary~[source reference], except that τ>σ is not required. Discard the finite initial set of indices before nonvanishing holds. For the remaining sufficiently large j, define h_j=gcd(absD_ju_j,absD_jv_j), which is positive because the form is nonzero at those indices. If liminf_j→∞fraclog h_jj≥κ ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"For n,m≥0, the integer and rational packets are S_n,m=∑_i=0^mbinom miT_n+i, H_n,m=∑_i=0^mbinom mi16^n+iv_n+i. Their raw linear form and integral filter satisfy F_n,m · =8(-1)^n(S_n,mG-H_n,m),notag widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^","count":21},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-80ec106a68da90ce","tex":"j^{-s}","aliases":"j^-s","display":"j^{-s}","kind":"use","excerpt":"For real s>1, the Dirichlet beta value is β(s)=∑_k=0^∞frac(-1)^k(2k+1)^s. Absolute convergence follows from comparison with ∑ j^-s. Thus G=β(2). The alternating series also defines β(s) for s>0, though absolute convergence is no longer available throughout that wider range. At s=1, integration of the finite geometric expansion of (1+x^2)^-1 gives β","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-485db41ee683b54c","tex":"J_0","aliases":"J_0","display":"J_0","kind":"use","excerpt":"For comparison, (n,m)=(1,0) has S=28,H=26,D=1,h=2,Q=32,g=64. Its primitive form is 13-14G=J_1. At (0,0), the convention H=0,D=1 gives S=1,Q=8,g=8 and primitive form G=J_0/8. These boundary cases require no valuation of the integer zero.","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-08e93c501f7d1688","tex":"J_1","aliases":"J_1","display":"J_1","kind":"use","excerpt":"For comparison, (n,m)=(1,0) has S=28,H=26,D=1,h=2,Q=32,g=64. Its primitive form is 13-14G=J_1. At (0,0), the convention H=0,D=1 gives S=1,Q=8,g=8 and primitive form G=J_0/8. These boundary cases require no valuation of the integer zero.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b78dc9dea4a072f6","tex":"J_k","aliases":"J_k","display":"J_k","kind":"use","excerpt":"Substitute the corrected normalization J_k=8(-1)^k(u_kG-v_k) into [source reference]. The two factors (-1)^i cancel. Thus widetilde J_n,m=8(-1)^n16^-n(S_n,mG-H_n,m). Define the raw packet form F_n,m=16^nwidetilde J_n,m =8(-1)^n(S_n,mG-H_n,m). The adjective raw means that the common denominator of H_n,m has not yet been cleared. It does not mean tha","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a620020ba505249b","tex":"J_n","aliases":"J_n","display":"J_n","kind":"use","excerpt":"widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^n+1 dd x dd y =8(-1)^n(u_nG-v_n). These are fixed normalization identities. Their factors should be checked together when translating a differently normalized source.","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ceadd0cc586441ea","tex":"J_{n+i}","aliases":"J_n+i","display":"J_{n+i}","kind":"use","excerpt":"For n,m≥0, the integer and rational packets are S_n,m=∑_i=0^mbinom miT_n+i, H_n,m=∑_i=0^mbinom mi16^n+iv_n+i. Their raw linear form and integral filter satisfy F_n,m · =8(-1)^n(S_n,mG-H_n,m),notag widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"The transfer product can also be written using continuants. For integers 1≤ r≤ s, let K_r,s be the determinant of the tridiagonal matrix whose diagonal is α_r,…,α_s, lower adjacent diagonal is one, and upper adjacent diagonal is -β_r+1,…,-β_s. For instance, K_r,r=α_r, K_r,r+1=det α_r · -β_r+1","count":24},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2d63c763eacf8111","tex":"K_{1,n-1}","aliases":"K_1,n-1","display":"K_{1,n-1}","kind":"use","excerpt":"[Initial data through continuants] For any solution w and every n≥2, w_n=K_1,n-1w_1+β_1K_2,n-1w_0.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7cbf87a8c5ddc8b0","tex":"K_{2,1}","aliases":"K_2,1","display":"K_{2,1}","kind":"use","excerpt":"At n=2, the formula is w_2=α_1w_1+β_1w_0, using the empty determinant for K_2,1. At n=3, direct iteration gives w_3=(α_2α_1+β_2)w_1+α_2β_1w_0, which agrees with the formula. For later indices, both coefficients in [source reference] satisfy the same recurrence by [source reference]. Induction finishes the proof.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-21e5ef05c57efefb","tex":"K_{2,n-1}","aliases":"K_2,n-1","display":"K_{2,n-1}","kind":"use","excerpt":"[Initial data through continuants] For any solution w and every n≥2, w_n=K_1,n-1w_1+β_1K_2,n-1w_0.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-bb9a538f5e406f30","tex":"K_n","aliases":"K_n","display":"K_n","kind":"use","excerpt":"Set K_n=16^nL_2(n+m(n))^2. For m(n)/n→α, [source reference] gives fraclog K_nn⟶ κ_0(α)=log16+4(1+α). The factor 4 has a precise origin. The lcm index is 2(n+m), and the lcm is squared. Missing either factor would create an incorrect apparent decay margin. The subscript in κ_0 emphasizes that this is the clearing cost alone. Define the combined boun","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f69766de691f017a","tex":"K_{r,r}","aliases":"K_r,r","display":"K_{r,r}","kind":"use","excerpt":"The transfer product can also be written using continuants. For integers 1≤ r≤ s, let K_r,s be the determinant of the tridiagonal matrix whose diagonal is α_r,…,α_s, lower adjacent diagonal is one, and upper adjacent diagonal is -β_r+1,…,-β_s. For instance, K_r,r=α_r, K_r,r+1=det α_r · -β_r+1 1 · α_r+1 =α_rα_r+1+β_r+1. Set K_r,r-1=1 for the empty d","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4235c9dbf3bc6653","tex":"K_{r,r+1}","aliases":"K_r,r+1","display":"K_{r,r+1}","kind":"use","excerpt":"The transfer product can also be written using continuants. For integers 1≤ r≤ s, let K_r,s be the determinant of the tridiagonal matrix whose diagonal is α_r,…,α_s, lower adjacent diagonal is one, and upper adjacent diagonal is -β_r+1,…,-β_s. For instance, K_r,r=α_r, K_r,r+1=det α_r · -β_r+1 1 · α_r+1 =α_rα_r+1+β_r+1. Set K_r,r-1=1 for the empty d","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b678107dc5c16db1","tex":"K_{r,r-1}","aliases":"K_r,r-1","display":"K_{r,r-1}","kind":"use","excerpt":"1 · α_r+1 =α_rα_r+1+β_r+1. Set K_r,r-1=1 for the empty determinant. Expansion at the last row gives K_r,s=α_sK_r,s-1+β_sK_r,s-2 (s≥ r+1). The plus sign comes from the minus sign on the upper adjacent diagonal. This convention matches the positive trailing term in the recurrence. All these continuants are positive rationals.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-621c27e05aa79034","tex":"K_{r,s}","aliases":"K_r,s","display":"K_{r,s}","kind":"use","excerpt":"The transfer product can also be written using continuants. For integers 1≤ r≤ s, let K_r,s be the determinant of the tridiagonal matrix whose diagonal is α_r,…,α_s, lower adjacent diagonal is one, and upper adjacent diagonal is -β_r+1,…,-β_s. For instance, K_r,r=α_r, K_r,r+1=det α_r · -β_r+1","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5296e76b3a3fd7f0","tex":"K_{r,s-1}","aliases":"K_r,s-1","display":"K_{r,s-1}","kind":"use","excerpt":"1 · α_r+1 =α_rα_r+1+β_r+1. Set K_r,r-1=1 for the empty determinant. Expansion at the last row gives K_r,s=α_sK_r,s-1+β_sK_r,s-2 (s≥ r+1). The plus sign comes from the minus sign on the upper adjacent diagonal. This convention matches the positive trailing term in the recurrence. All these continuants are positive rationals.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7656107abe17e4b6","tex":"K_{r,s-2}","aliases":"K_r,s-2","display":"K_{r,s-2}","kind":"use","excerpt":"1 · α_r+1 =α_rα_r+1+β_r+1. Set K_r,r-1=1 for the empty determinant. Expansion at the last row gives K_r,s=α_sK_r,s-1+β_sK_r,s-2 (s≥ r+1). The plus sign comes from the minus sign on the upper adjacent diagonal. This convention matches the positive trailing term in the recurrence. All these continuants are positive rationals.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"Suppose l≤ G≤ h is a proven rational enclosure, and suppose a,b∈Q. Then the interval for aG-b is [al-b,ah-b], · a≥0,","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-1feade5f87c5d534","tex":"L_{2(n+m(n))}^2","aliases":"L_2(n+m(n))^2","display":"L_{2(n+m(n))}^2","kind":"use","excerpt":"Set K_n=16^nL_2(n+m(n))^2. For m(n)/n→α, [source reference] gives fraclog K_nn⟶ κ_0(α)=log16+4(1+α). The factor 4 has a precise origin. The lcm index is 2(n+m), and the lcm is squared. Missing either factor would create an incorrect apparent decay margin. The subscript in κ_0 emphasizes that this is the clearing cost alone. Define the combined boun","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b61d2d2f41717242","tex":"L_{2k}","aliases":"L_2k","display":"L_{2k}","kind":"use","excerpt":"B_n,m · =8(-1)^nC_n,mH_n,m. Both coefficients are integers. Integrality of A follows from that of S. For B, each index k=n+i satisfies k≤ N, so L_2k divides L_2N. The sharp source clearing 4·16^kL_2k^2v_k∈Z therefore proves integrality term by term after multiplication by 8L_2N^2. The binomial weights introduce no further denominators.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c5d2237a2ba3f573","tex":"L_{2k}^2","aliases":"L_2k^2","display":"L_{2k}^2","kind":"use","excerpt":"B_n,m · =8(-1)^nC_n,mH_n,m. Both coefficients are integers. Integrality of A follows from that of S. For B, each index k=n+i satisfies k≤ N, so L_2k divides L_2N. The sharp source clearing 4·16^kL_2k^2v_k∈Z therefore proves integrality term by term after multiplication by 8L_2N^2. The binomial weights introduce no further denominators.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-21d61807b3407df8","tex":"L_{2N}","aliases":"L_2N","display":"L_{2N}","kind":"use","excerpt":"B_n,m · =8(-1)^nC_n,mH_n,m. Both coefficients are integers. Integrality of A follows from that of S. For B, each index k=n+i satisfies k≤ N, so L_2k divides L_2N. The sharp source clearing 4·16^kL_2k^2v_k∈Z therefore proves integrality term by term after multiplication by 8L_2N^2. The binomial weights introduce no further denominators.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-9d9e39195bef95ed","tex":"L_{2N}^2","aliases":"L_2N^2","display":"L_{2N}^2","kind":"use","excerpt":"Let N=n+m and put C_n,m=L_2N^2. The pair produced by the historical script is A_n,m · =8(-1)^nC_n,mS_n,m,","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5aa0fda5d74dd324","tex":"L_4","aliases":"L_4","display":"L_4","kind":"use","excerpt":"The universal factor uses L_4=12, hence Q=8L_4^2=1152. Its redundant scale is Q/D=128. Consequently g=128·64=8192. The signed cleared pair is (-3 022 848,-2 768 896). Dividing by 8192 gives (-369,-338), so the primitive form is Λ_1,1=338-369G. The integral identity gives the same expression. Since widetilde J_1,1=F_1,1/16 and 16L_4^2/g=9/32, frac16","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a5ce27f9435175fc","tex":"L_4^2","aliases":"L_4^2","display":"L_4^2","kind":"use","excerpt":"The universal factor uses L_4=12, hence Q=8L_4^2=1152. Its redundant scale is Q/D=128. Consequently g=128·64=8192. The signed cleared pair is (-3 022 848,-2 768 896). Dividing by 8192 gives (-369,-338), so the primitive form is Λ_1,1=338-369G. The integral identity gives the same expression. Since widetilde J_1,1=F_1,1/16 and 16L_4^2/g=9/32, frac16","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"use","excerpt":"In the later chapters the starting form is ℓ_j=u_jα-v_j, u_j,v_j∈Q. Choose a positive integer D_j for which D_ju_j and D_jv_j are integers. Then L_j=D_jℓ_j=(D_ju_j)α-D_jv_j has integer coefficients. Its size is D_jabsℓ_j, so the clearance factor must be included in any decay estimate.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ef47c73ca0380033","tex":"L_\\ell","aliases":"L_ℓ ell","display":"L_\\ell","kind":"use","excerpt":"ndequation These are fixed-pair aliases U=S_n,m and V=H_n,m for the packets defined earlier, not the integral coefficients of a different source normalization. Here T_k=16^ku_k is a positive integer. Consequently U is a positive integer. Write V=N/D in lowest terms, with D>0 and gcd(N,D)=1. In the exceptional case V=0, the same convention sets N=0,","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d562ae8e2d912d49","tex":"L_\\ell^2","aliases":"L_ℓ^2 ell","display":"L_\\ell^2","kind":"use","excerpt":"ndequation These are fixed-pair aliases U=S_n,m and V=H_n,m for the packets defined earlier, not the integral coefficients of a different source normalization. Here T_k=16^ku_k is a positive integer. Consequently U is a positive integer. Write V=N/D in lowest terms, with D>0 and gcd(N,D)=1. In the exceptional case V=0, the same convention sets N=0,","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"For a fixed row m, the allowed indices are max(1,lceil m/4rceil)≤ n≤ N. Rows are needed through m=4N. In row m, the seed array is sufficient for every n≤5N-m. At the extreme row m=4N this bound is exactly n≤ N. Thus no scanned entry needs an unconstructed seed term. The algorithm evaluates additional row entries to support later rows. Those additio","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"[External recurrence identification and rates] For the exact normalization [source reference], Zudilin's theorem and its stated asymptotic rates give lim_n→∞fracv_nu_n · =G, lim_n→∞u_n^1/n · =ϕ^5, lim_n→∞|u_nG-v_n|^1/n · =ϕ^-5. These are external mathematical inputs, recorded precisely in source contracts LC01 and LC02 [source reference].","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-eb514d3fcefe767d","tex":"\\lim_{n\\to\\infty}","aliases":"lim_n→∞ lim to infty","display":"\\lim_{n\\to\\infty}","kind":"use","excerpt":"[External recurrence identification and rates] For the exact normalization [source reference], Zudilin's theorem and its stated asymptotic rates give lim_n→∞fracv_nu_n · =G, lim_n→∞u_n^1/n · =ϕ^5, lim_n→∞|u_nG-v_n|^1/n · =ϕ^-5. These are external mathematical inputs, recorded precisely in source contracts LC01 and LC02 [source reference].","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e5c19d75887aa542","tex":"\\liminf","aliases":"liminf","display":"\\liminf","kind":"use","excerpt":"t can recover a proved rate] Assume the setup of Corollary~[source reference], except that τ>σ is not required. Discard the finite initial set of indices before nonvanishing holds. For the remaining sufficiently large j, define h_j=gcd(absD_ju_j,absD_jv_j), which is positive because the form is nonzero at those indices. If liminf_j→∞fraclog h_jj≥κ ","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-18b7abd1ba664ef4","tex":"\\liminf_{j\\to\\infty}","aliases":"liminf_j→∞ liminf to infty","display":"\\liminf_{j\\to\\infty}","kind":"use","excerpt":"t can recover a proved rate] Assume the setup of Corollary~[source reference], except that τ>σ is not required. Discard the finite initial set of indices before nonvanishing holds. For the remaining sufficiently large j, define h_j=gcd(absD_ju_j,absD_jv_j), which is positive because the form is nonzero at those indices. If liminf_j→∞fraclog h_jj≥κ ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f4b683e95c67f073","tex":"\\liminf_{n\\to\\infty}","aliases":"liminf_n→∞ liminf to infty","display":"\\liminf_{n\\to\\infty}","kind":"use","excerpt":"With the cleared gcd g_n=g_n,m(n), the primitive form is exactly Λ_n=fracK_ng_nwidetilde J_n,m(n). Suppose a future arithmetic argument establishes δ=liminf_n→∞fraclog g_nn. For finite δ, the present analytic estimate yields limsup_n→∞fraclog|Λ_n|n ≤κ_0(α)+log M(α)-δ. The same interpretation applies when δ=+∞: any sufficiently large finite lower bo","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"use","excerpt":"[Exponential clearance criterion] Let α∈R, and let u_j,v_j∈Q for j≥1. Let D_j be positive integers such that D_ju_j,D_jv_j∈Z. Assume u_jα-v_j≠0 for all sufficiently large j. Suppose limsup_j→∞absu_jα-v_j^1/j≤λ, limsup_j→∞D_j^1/j≤δ, where λ≥0, δ<∞ and δλ<1. Then α is irrational.","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c00a609602b5f0ce","tex":"\\limsup_{j\\to\\infty}","aliases":"limsup_j→∞ limsup to infty","display":"\\limsup_{j\\to\\infty}","kind":"use","excerpt":"[Exponential clearance criterion] Let α∈R, and let u_j,v_j∈Q for j≥1. Let D_j be positive integers such that D_ju_j,D_jv_j∈Z. Assume u_jα-v_j≠0 for all sufficiently large j. Suppose limsup_j→∞absu_jα-v_j^1/j≤λ, limsup_j→∞D_j^1/j≤δ, where λ≥0, δ<∞ and δλ<1. Then α is irrational.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3e2f995f3f54e413","tex":"\\limsup_{n\\to\\infty}","aliases":"limsup_n→∞ limsup to infty","display":"\\limsup_{n\\to\\infty}","kind":"use","excerpt":"Let α_n=m(n)/n. The finite measure in [source reference] gives |widetilde J_n,m(n)| ≤ 8G M(α_n)^n. Continuity of the two branches, including their stated extension at zero, then proves limsup_n→∞ |widetilde J_n,m(n)|^1/n≤ M(α). This is an upper bound. An equality for the actual integral would require a further argument. In particular, an upper boun","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"[5pt] LC07 · Eskandari, 2026 · The bound M(α) and approximation exponent. It is a preprint input below the irrationality threshold.","count":26},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"1 · 0. X_n+1(w)=M_nX_n(w). Hence X_N+1(w)=M_NM_N-1⋯ M_1X_1(w) (N≥1). The order of multiplication follows the temporal order: M_1 acts first. This is a product of changing matrices, not a power of one fixed matrix.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-0901f1f56ba2bc5e","tex":"M_N","aliases":"M_N","display":"M_N","kind":"use","excerpt":"1 · 0. X_n+1(w)=M_nX_n(w). Hence X_N+1(w)=M_NM_N-1⋯ M_1X_1(w) (N≥1). The order of multiplication follows the temporal order: M_1 acts first. This is a product of changing matrices, not a power of one fixed matrix.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"use","excerpt":"Normalize the recurrence by putting α_n=fracq(n)a_n, β_n=fracc_na_n (n≥1). Both quantities are positive rationals. For the adjacent state vector X_n(w)=(w_n,w_n-1)^ T, define M_n=α_n · β_n 1 · 0. X_n+1(w)=M_nX_n(w). Hence X_N+1(w)=M_NM_N-1⋯ M_1X_1(w) (N≥1). The order of multiplication follows the temporal order: M_1 acts first. This is a product of","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-43e5de18bc8854ab","tex":"M_{N-1}","aliases":"M_N-1","display":"M_{N-1}","kind":"use","excerpt":"1 · 0. X_n+1(w)=M_nX_n(w). Hence X_N+1(w)=M_NM_N-1⋯ M_1X_1(w) (N≥1). The order of multiplication follows the temporal order: M_1 acts first. This is a product of changing matrices, not a power of one fixed matrix.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-13c30f77949796cf","tex":"M_n^{-1}","aliases":"M_n^-1","display":"M_n^{-1}","kind":"use","excerpt":"The determinant is det M_n=-β_n=-fracc_na_n<0. Every transfer is invertible, with M_n^-1=0 · 1 1/β_n · -α_n/β_n. An adjacent pair at index n+1 therefore determines the preceding pair at index n, within the stated recurrence domain. This does not extend the equation through n=0, where the trailing coefficient would vanish.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-eae2d59e5b8a6aeb","tex":"M_\\infty","aliases":"M_∞ infty","display":"M_\\infty","kind":"use","excerpt":"The leading terms of the coefficient polynomials are a_n=320n^6+O(n^5), q(n)=3520n^6+O(n^5), c_n=320n^6+O(n^5). Consequently α_n→11, β_n→1, and M_n⟶ M_∞=11 · 1 1 · 0. The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"why the filter changes sign. Put r=sqrtxy. Since x+y≥2r, ψ(x,y)≤fracr^2(1-r)^21-r^2 =fracr^2(1-r)1+r,qquad0<r<1. Equality is achieved when x=y=r. The derivative of the last expression is frac2r(1-r-r^2)(1+r)^2. It has its interior maximum at r=ϕ^-1, where ϕ=(1+sqrt5)/2. Since 1-r=r^2 there, the maximum is maxψ=ϕ^-5≈0.09017. This is greater than 1/1","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-1e207a679f3b87f6","tex":"\\max_{(n,m)\\in\\mathcal D_{2000}}","aliases":"max_(n,m)∈ D_2000 max in mathcal","display":"\\max_{(n,m)\\in\\mathcal D_{2000}}","kind":"use","excerpt":"cal run takes N=2000 and K=128. The release reran the unchanged historical algorithm and compared its result with the recovered output. The finite arithmetic results agree. In the domain D_2000 there are exactly 8,006,000 pairs. Every observed residue is nonzero, so the residue lemma applies at every pair. The exact maximum valuation in this finite","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"Such a D_j always exists for one finite rational pair. Write u_j=a_j/b_j and v_j=c_j/d_j in lowest terms with positive denominators. The smallest positive common clearing factor is D_j^min=lcm(b_j,d_j). Indeed, integrality of D_ja_j/b_j forces b_jmid D_j by coprimality, and integrality of D_jc_j/d_j forces d_jmid D_j. Conversely, their least common","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"Such a D_j always exists for one finite rational pair. Write u_j=a_j/b_j and v_j=c_j/d_j in lowest terms with positive denominators. The smallest positive common clearing factor is D_j^min=lcm(b_j,d_j). Indeed, integrality of D_ja_j/b_j forces b_jmid D_j by coprimality, and integrality of D_jc_j/d_j forces d_jmid D_j. Conversely, their least common","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"For a positive integer N, the scanned domain is D_N={(n,m):1≤ n≤ N, 0≤ m≤4n}. The number of scanned pairs is ∑_n=1^N(4n+1)=2N^2+3N. The upper slope 4 is part of the experiment's definition. The scan supplies no information about pairs above this boundary or with n>N.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"The normalized rational sequences start with (u_0,u_1)=(1,7/4), (v_0,v_1)=(0,13/8). For n≥1, set p(n) · =20n^2-8n+1,notag q(n) · =3520n^6+5632n^5+2064n^4notag · -384n^3-156n^2+16n+7,notag a_n · =(2n+1)^2(2n+2)^2p(n),notag c_n · =(2n-1)^2(2n)^2p(n+1). Each sequence w=u or w=v obeys a_nw_n+1=q(n)w_n+c_nw_n-1. The forward recurrence is used only at n≥","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-0969e62c250901c5","tex":"n^3","aliases":"n^3","display":"n^3","kind":"use","excerpt":"The normalized rational sequences start with (u_0,u_1)=(1,7/4), (v_0,v_1)=(0,13/8). For n≥1, set p(n) · =20n^2-8n+1,notag q(n) · =3520n^6+5632n^5+2064n^4notag · -384n^3-156n^2+16n+7,notag a_n · =(2n+1)^2(2n+2)^2p(n),notag c_n · =(2n-1)^2(2n)^2p(n+1). Each sequence w=u or w=v obeys a_nw_n+1=q(n)w_n+c_nw_n-1. The forward recurrence is used only at n≥","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ac5e944c84a2e799","tex":"n^4","aliases":"n^4","display":"n^4","kind":"use","excerpt":"The normalized rational sequences start with (u_0,u_1)=(1,7/4), (v_0,v_1)=(0,13/8). For n≥1, set p(n) · =20n^2-8n+1,notag q(n) · =3520n^6+5632n^5+2064n^4notag · -384n^3-156n^2+16n+7,notag a_n · =(2n+1)^2(2n+2)^2p(n),notag c_n · =(2n-1)^2(2n)^2p(n+1). Each sequence w=u or w=v obeys a_nw_n+1=q(n)w_n+c_nw_n-1. The forward recurrence is used only at n≥","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-956f7f0f704feeed","tex":"n^5","aliases":"n^5","display":"n^5","kind":"use","excerpt":"The normalized rational sequences start with (u_0,u_1)=(1,7/4), (v_0,v_1)=(0,13/8). For n≥1, set p(n) · =20n^2-8n+1,notag q(n) · =3520n^6+5632n^5+2064n^4notag · -384n^3-156n^2+16n+7,notag a_n · =(2n+1)^2(2n+2)^2p(n),notag c_n · =(2n-1)^2(2n)^2p(n+1). Each sequence w=u or w=v obeys a_nw_n+1=q(n)w_n+c_nw_n-1. The forward recurrence is used only at n≥","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-6093d07fa100c881","tex":"n^6","aliases":"n^6","display":"n^6","kind":"use","excerpt":"The normalized rational sequences start with (u_0,u_1)=(1,7/4), (v_0,v_1)=(0,13/8). For n≥1, set p(n) · =20n^2-8n+1,notag q(n) · =3520n^6+5632n^5+2064n^4notag · -384n^3-156n^2+16n+7,notag a_n · =(2n+1)^2(2n+2)^2p(n),notag c_n · =(2n-1)^2(2n)^2p(n+1). Each sequence w=u or w=v obeys a_nw_n+1=q(n)w_n+c_nw_n-1. The forward recurrence is used only at n≥","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"[Rational lattice gap] Suppose α=a/b with a∈Z and b∈Z_>0. For every pair P,Q∈Z, either Qα-P=0 or absQα-P≥frac1b.","count":30},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"[Integer form criterion] Let α∈R. Suppose there are integer pairs (P_j,Q_j) for j≥1 such that 0<absQ_jα-P_j⟶0. Then α is irrational. The same conclusion holds if these conditions are verified along any infinite subsequence.","count":11},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"use","excerpt":"[A convergent rational sequence with a fixed form error] Let α=1/2 and take fracP_nQ_n=fracn2n+1, n≥1. The numerator and denominator are coprime because any common divisor of n and 2n+1 divides 1. The ordinary error is frac12-fracn2n+1=frac12(2n+1)⟶0. Nevertheless, Q_nα-P_n=frac12 for every n. The approximants are distinct from the target, all coef","count":6},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"use","excerpt":"[Well defined rational solutions] Every initial pair (w_0,w_1)∈Q^2 determines exactly one rational sequence satisfying [source reference]. In particular the sequences in Definition~[source reference] exist uniquely. Moreover, u_n>0 (n≥0), v_n>0 (n≥1), v_0=0.","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-0c370aa793b02aac","tex":"Q'","aliases":"Q'","display":"Q'","kind":"use","excerpt":"[Invariance under additional clearing] If an integer pair (A,B) is replaced by (cA,cB) for an integer c>0, then its primitive pair and primitive linear form remain unchanged. More generally, any integer clearing factor Q' divisible by D gives the same unsigned primitive pair (DU/h,N/h).","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-80c6755bbc84e0c6","tex":"q^{-0.62}","aliases":"q^-0.62","display":"q^{-0.62}","kind":"use","excerpt":"An approximation exponent below one cannot contradict rationality. If G=a/b were rational and p/q≠ a/b were reduced, then |frac ab-frac pq| =frac|aq-bp|bq≥frac1bq. A bound of order q^-0.62 is larger than this obstruction for large q. Even rational numbers have many distinct approximations of that quality. For example a/b+1/(bk) has error 1/(bk) and","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-afeb63abf7eb7cda","tex":"\\Q^2","aliases":"Q^2 Q","display":null,"kind":"use","excerpt":"[Well defined rational solutions] Every initial pair (w_0,w_1)∈Q^2 determines exactly one rational sequence satisfying [source reference]. In particular the sequences in Definition~[source reference] exist uniquely. Moreover, u_n>0 (n≥0), v_n>0 (n≥1), v_0=0.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"[Integer form criterion] Let α∈R. Suppose there are integer pairs (P_j,Q_j) for j≥1 such that 0<absQ_jα-P_j⟶0. Then α is irrational. The same conclusion holds if these conditions are verified along any infinite subsequence.","count":15},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-85bd5c40675dbd25","tex":"Q_j^{-\\theta}","aliases":"Q_j^-θ theta","display":"Q_j^{-\\theta}","kind":"use","excerpt":"This lower bound explains a familiar sufficient exponent. If distinct approximants have Q_j→∞ and absα-fracP_jQ_j≤ C Q_j^-θ for a fixed constant C>0 and an exponent θ>1, then multiplying by Q_j gives a quantity tending to zero. The criterion applies. An upper bound with θ≤1 does not supply that conclusion on its own. It may still describe useful ap","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c95e041440f50ec9","tex":"Q_n","aliases":"Q_n","display":"Q_n","kind":"use","excerpt":"[tbp] centeringincludegraphics[width=0.93linewidth]../visual/figures/09_recurrence_structure.pdf captionTwo preceding values determine the next rational value. The forward rule begins at n=1 and uses the stated initial pair. Here and in subsequent recurrence diagrams, A_n=a_n, Q_n=q(n) and C_n=c_n.","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"Write ϕ=frac1+sqrt52, t=ϕ^-5, R=16ϕ^5=88+40sqrt5. The unfiltered integral has the representation J_k=∫_[0,1]^2ψ(x,y)^k ddμ(x,y), ψ=fracxy(1-x)(1-y)1-xy, ddμ=fracdd x dd ysqrtx(1-y)(1-xy). The measure has finite total mass J_0=8G. The factor 1-xy in its denominator lies outside the square root. Moving it inside changes the integral and invalidates t","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"[Integer form criterion] Let α∈R. Suppose there are integer pairs (P_j,Q_j) for j≥1 such that 0<absQ_jα-P_j⟶0. Then α is irrational. The same conclusion holds if these conditions are verified along any infinite subsequence.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"The kernel has a simple maximum that also explains why the filter changes sign. Put r=sqrtxy. Since x+y≥2r, ψ(x,y)≤fracr^2(1-r)^21-r^2 =fracr^2(1-r)1+r,qquad0<r<1. Equality is achieved when x=y=r. The derivative of the last expression is frac2r(1-r-r^2)(1+r)^2. It has its interior maximum at r=ϕ^-1, where ϕ=(1+sqrt5)/2. Since 1-r=r^2 there, the max","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"The first ratios are r_0=0 and r_1=13/14. The subsequent values lie near the decimal for G, but proximity is only a first observation. Chapter~[source reference] proves exact nested brackets and an error sign at every index. The external identification below then makes those brackets enclosures of Catalan's constant itself.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"The first ratios are r_0=0 and r_1=13/14. The subsequent values lie near the decimal for G, but proximity is only a first observation. Chapter~[source reference] proves exact nested brackets and an error sign at every index. The external identification below then makes those brackets enclosures of Catalan's constant itself.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"use","excerpt":", p(1)=13, p(2)=65, q(1)=10699, a_1=1872, c_1=260. For u, the recurrence gives u_2=frac10699(7/4)+2601872 =frac75933/41872=frac64964. For v, its zero initial term removes the trailing contribution: v_2=frac10699(13/8)+260·01872 =frac139087/81872=frac106991152. Therefore the second rational approximant is r_2=frac1069911682.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e0937200c73f7676","tex":"r_{2k}","aliases":"r_2k","display":"r_{2k}","kind":"use","excerpt":"[Alternating brackets] There is a real number L such that r_n→ L. For every k≥0, r_2k<L<r_2k+1. The even subsequence increases strictly and the odd subsequence decreases strictly. For every n≥0, d_n-d_n+1<|L-r_n|<d_n, sign(L-r_n)=(-1)^n.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-52f3fa24350d43b8","tex":"r_{2k+1}","aliases":"r_2k+1","display":"r_{2k+1}","kind":"use","excerpt":"[Alternating brackets] There is a real number L such that r_n→ L. For every k≥0, r_2k<L<r_2k+1. The even subsequence increases strictly and the odd subsequence decreases strictly. For every n≥0, d_n-d_n+1<|L-r_n|<d_n, sign(L-r_n)=(-1)^n.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3b5ce1e357f065bb","tex":"r_{2k+2}","aliases":"r_2k+2","display":"r_{2k+2}","kind":"use","excerpt":"Equation [source reference] and the preceding lemma identify r_n with partial sums of a convergent alternating series: r_n=∑_j=0^n-1(-1)^j d_j, L=∑_j=0^∞(-1)^j d_j. For the even subsequence, r_2k+2-r_2k=d_2k-d_2k+1>0. For the odd subsequence, r_2k+3-r_2k+1=-d_2k+1+d_2k+2<0. Their adjacent separation d_2k tends to zero, so they have the same limit. ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d595ffb81026a958","tex":"r_{2k+3}","aliases":"r_2k+3","display":"r_{2k+3}","kind":"use","excerpt":"Equation [source reference] and the preceding lemma identify r_n with partial sums of a convergent alternating series: r_n=∑_j=0^n-1(-1)^j d_j, L=∑_j=0^∞(-1)^j d_j. For the even subsequence, r_2k+2-r_2k=d_2k-d_2k+1>0. For the odd subsequence, r_2k+3-r_2k+1=-d_2k+1+d_2k+2<0. Their adjacent separation d_2k tends to zero, so they have the same limit. ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d5c7051b56820f95","tex":"r_3","aliases":"r_3","display":"r_3","kind":"use","excerpt":"· =frac521667445/12858500=frac8025653115200. It follows that r_3=8025653/8761950. All these values were also checked against the recovered exact rational script. A floating point calculation is unnecessary for the equality at any stage.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cbcc6e67ed392000","tex":"r_n^{1/n}","aliases":"r_n^1/n","display":"r_n^{1/n}","kind":"use","excerpt":"diate consequence of the two source rates is lim_n→∞|G-r_n|^1/n =lim_n→∞frac|u_nG-v_n|^1/nu_n^1/n =ϕ^-10. Thus the ratio error has a smaller exponential scale than the unnormalized linear form. The distinction comes from division by the growing coefficient u_n. Also, since r_n→ G>0, one has r_n^1/n→1 and consequently v_n^1/n→ϕ^5. Both individual se","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"For a fixed row m, the allowed indices are max(1,lceil m/4rceil)≤ n≤ N. Rows are needed through m=4N. In row m, the seed array is sufficient for every n≤5N-m. At the extreme row m=4N this bound is exactly n≤ N. Thus no scanned entry needs an unconstructed seed term. The algorithm evaluates additional row entries to support later rows. Those additio","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"The initial integral supplies a short diagnostic. Substitute x=t^2 and y=1-s^2. Then J_0 · =4∫_0^1 ∫_0^1fracdd s dd t1-t^2+t^2s^2","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"Write s_N=∑_k=0^N-1frac(-1)^k(2k+1)^2, N≥0, where s_0=0 is the empty sum. Thus N counts retained terms, as it does in the visual guide. Each s_N is an exact rational number. The terms decrease strictly in magnitude to zero. Grouping successive terms shows that the even partial sums increase and the odd partial sums decrease. These two sequences app","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"For example, s_0=0, s_1=1, s_2=frac89, s_3=frac209225, s_4=frac1001611025. Hence frac1001611025<G<frac209225. These statements use only rational arithmetic. The strict inequality also proves 0<G<1, which will simplify later tail estimates.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-6acacf3199109591","tex":"S_{1,0}","aliases":"S_1,0","display":"S_{1,0}","kind":"use","excerpt":"For (n,m)=(1,1), S_1,1=T_1+T_2=28+2596=2624=2^6·41. Its exact valuation is v_2(S_1,1)=6. Pascal evolution gives the same result from the initial row: S_1,1=S_1,0+S_2,0.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c2e7b38410ab56eb","tex":"S_{1,1}","aliases":"S_1,1","display":"S_{1,1}","kind":"use","excerpt":"For (n,m)=(1,1), S_1,1=T_1+T_2=28+2596=2624=2^6·41. Its exact valuation is v_2(S_1,1)=6. Pascal evolution gives the same result from the initial row: S_1,1=S_1,0+S_2,0.","count":9},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"use","excerpt":"For example, s_0=0, s_1=1, s_2=frac89, s_3=frac209225, s_4=frac1001611025. Hence frac1001611025<G<frac209225. These statements use only rational arithmetic. The strict inequality also proves 0<G<1, which will simplify later tail estimates.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2ba7bd0d7c893268","tex":"S_{2,0}","aliases":"S_2,0","display":"S_{2,0}","kind":"use","excerpt":"For (n,m)=(1,1), S_1,1=T_1+T_2=28+2596=2624=2^6·41. Its exact valuation is v_2(S_1,1)=6. Pascal evolution gives the same result from the initial row: S_1,1=S_1,0+S_2,0.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-44b521efc95ce738","tex":"s_{2r}","aliases":"s_2r","display":"s_{2r}","kind":"use","excerpt":"[Alternating enclosure] For every integer r≥0, s_2r<G<s_2r+1. For every integer N≥0, 0<(-1)^N(G-s_N)<frac1(2N+1)^2.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d8cc4c9ae7a46000","tex":"s_{2r+1}","aliases":"s_2r+1","display":"s_{2r+1}","kind":"use","excerpt":"[Alternating enclosure] For every integer r≥0, s_2r<G<s_2r+1. For every integer N≥0, 0<(-1)^N(G-s_N)<frac1(2N+1)^2.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-774dab453b94d88d","tex":"s_{2r+2}","aliases":"s_2r+2","display":"s_{2r+2}","kind":"use","excerpt":"The difference between two consecutive even sums is s_2r+2-s_2r =frac1(4r+1)^2-frac1(4r+3)^2>0. Similarly, s_2r+3-s_2r+1 =-frac1(4r+3)^2+frac1(4r+5)^2<0. Consecutive sums differ by a term tending to zero, so the increasing even sums and decreasing odd sums have the same limit G. Strictness follows because the next sum of the same parity lies strict","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-78f5a4071e7c25f0","tex":"s_{2r+3}","aliases":"s_2r+3","display":"s_{2r+3}","kind":"use","excerpt":"The difference between two consecutive even sums is s_2r+2-s_2r =frac1(4r+1)^2-frac1(4r+3)^2>0. Similarly, s_2r+3-s_2r+1 =-frac1(4r+3)^2+frac1(4r+5)^2<0. Consecutive sums differ by a term tending to zero, so the increasing even sums and decreasing odd sums have the same limit G. Strictness follows because the next sum of the same parity lies strict","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7b8539d67beca0e3","tex":"s_3","aliases":"s_3","display":"s_3","kind":"use","excerpt":"For example, s_0=0, s_1=1, s_2=frac89, s_3=frac209225, s_4=frac1001611025. Hence frac1001611025<G<frac209225. These statements use only rational arithmetic. The strict inequality also proves 0<G<1, which will simplify later tail estimates.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-80534a5aa724e3ef","tex":"s_4","aliases":"s_4","display":"s_4","kind":"use","excerpt":"For example, s_0=0, s_1=1, s_2=frac89, s_3=frac209225, s_4=frac1001611025. Hence frac1001611025<G<frac209225. These statements use only rational arithmetic. The strict inequality also proves 0<G<1, which will simplify later tail estimates.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cfe0fa40b3c1262c","tex":"s_N","aliases":"s_N","display":"s_N","kind":"use","excerpt":"Write s_N=∑_k=0^N-1frac(-1)^k(2k+1)^2, N≥0, where s_0=0 is the empty sum. Thus N counts retained terms, as it does in the visual guide. Each s_N is an exact rational number. The terms decrease strictly in magnitude to zero. Grouping successive terms shows that the even partial sums increase and the odd partial sums decrease. These two sequences app","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-efc05a675569c6da","tex":"s_{N+1}","aliases":"s_N+1","display":"s_{N+1}","kind":"use","excerpt":"To obtain the error bound, apply this enclosure to s_N and s_N+1. The limit lies strictly between them. Their distance is (2N+1)^-2, and the direction from s_N to s_N+1 has sign (-1)^N.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8519f99da226fd65","tex":"S_{n+1,m}","aliases":"S_n+1,m","display":"S_{n+1,m}","kind":"use","excerpt":"[Pascal evolution] For n,m≥0, S_n,m+1=S_n,m+S_n+1,m. The same identity holds with H in place of S.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-42538001e8204da4","tex":"S_{n,0}","aliases":"S_n,0","display":"S_{n,0}","kind":"use","excerpt":"At m=0, the packet is just S_n,0=T_n. The first two nontrivial rows are S_n,1=T_n+T_n+1, S_n,2=T_n+2T_n+1+T_n+2. For example S_1,1=28+2596=2624. The Pascal identity produces exactly the same integer as direct evaluation of its defining sum. This agreement between an additive row evolution and a direct weighted sum gives two different implementation","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d62cc5a6dcf91ed9","tex":"S_{n,1}","aliases":"S_n,1","display":"S_{n,1}","kind":"use","excerpt":"At m=0, the packet is just S_n,0=T_n. The first two nontrivial rows are S_n,1=T_n+T_n+1, S_n,2=T_n+2T_n+1+T_n+2. For example S_1,1=28+2596=2624. The Pascal identity produces exactly the same integer as direct evaluation of its defining sum. This agreement between an additive row evolution and a direct weighted sum gives two different implementation","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f92cf036586ec9db","tex":"S_{n,2}","aliases":"S_n,2","display":"S_{n,2}","kind":"use","excerpt":"At m=0, the packet is just S_n,0=T_n. The first two nontrivial rows are S_n,1=T_n+T_n+1, S_n,2=T_n+2T_n+1+T_n+2. For example S_1,1=28+2596=2624. The Pascal identity produces exactly the same integer as direct evaluation of its defining sum. This agreement between an additive row evolution and a direct weighted sum gives two different implementation","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5457e4c5b69f80ce","tex":"S_{n,m(n)}","aliases":"S_n,m(n)","display":"S_{n,m(n)}","kind":"use","excerpt":"There is also a formulation independent of the universal clearing choice. With D_n,h_n as in Chapter~[source reference], the primitive error is (-1)^n(D_n/h_n)(S_n,m(n)G-H_n,m(n)). An estimate directly on that expression would suffice. If denominator reduction yields a provably smaller exponential cost, the excess factor Q_n/D_n should simply be re","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4b43c57590549acf","tex":"S_{n,m+1}","aliases":"S_n,m+1","display":"S_{n,m+1}","kind":"use","excerpt":"[Pascal evolution] For n,m≥0, S_n,m+1=S_n,m+S_n+1,m. The same identity holds with H in place of S.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-1d79850b77567ab8","tex":"\\operatorname{sign}","aliases":"sign operatorname","display":"\\operatorname{sign}","kind":"use","excerpt":"associated rational packet is H_1,1=16v_1+16^2v_2 =26+frac213989 =frac216329. Therefore the raw form is F_1,1=-8(2624G-frac216329) =frac1730569-20992G. A certified interval for G makes this value positive. This is the smallest counterexample in the finite audit to the proposed sign rule signF_n,m=(-1)^m.","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"Normalize the recurrence by putting α_n=fracq(n)a_n, β_n=fracc_na_n (n≥1). Both quantities are positive rationals. For the adjacent state vector X_n(w)=(w_n,w_n-1)^ T, define M_n=α_n · β_n","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"At n=2 the coefficients become p(2)=65, p(3)=157, q(2)=434871, a_2=58500, c_2=22608. Using the fractions just obtained, u_3 · =frac434871(649/64)+22608(7/4)58500 · =frac284763375/6458500=frac19471256, [3pt] v_3 · =frac434871(10699/1152)+22608(13/8)58500 · =frac521667445/12858500=frac8025653115200. It follows that r_3=8025653/8761950. All these valu","count":27},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"The next argument needs the positivity of u_n. It also needs a modest lower bound on it, rather than a full asymptotic theorem. Both follow from the recurrence. To see the relevant sign without relying on a plot, put t=n-1. Direct polynomial expansion gives q(n)-a_n= · 3200t^6+24000t^5+73392t^4+117072t^3 · +102768t^2+47112t+8827. Thus q(n)>a_n>0 fo","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"The next argument needs the positivity of u_n. It also needs a modest lower bound on it, rather than a full asymptotic theorem. Both follow from the recurrence. To see the relevant sign without relying on a plot, put t=n-1. Direct polynomial expansion gives q(n)-a_n= · 3200t^6+24000t^5+73392t^4+117072t^3 · +102768t^2+47112t+8827. Thus q(n)>a_n>0 fo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":"The next argument needs the positivity of u_n. It also needs a modest lower bound on it, rather than a full asymptotic theorem. Both follow from the recurrence. To see the relevant sign without relying on a plot, put t=n-1. Direct polynomial expansion gives q(n)-a_n= · 3200t^6+24000t^5+73392t^4+117072t^3 · +102768t^2+47112t+8827. Thus q(n)>a_n>0 fo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8c6354f5f56778af","tex":"t^5","aliases":"t^5","display":"t^5","kind":"use","excerpt":"The next argument needs the positivity of u_n. It also needs a modest lower bound on it, rather than a full asymptotic theorem. Both follow from the recurrence. To see the relevant sign without relying on a plot, put t=n-1. Direct polynomial expansion gives q(n)-a_n= · 3200t^6+24000t^5+73392t^4+117072t^3 · +102768t^2+47112t+8827. Thus q(n)>a_n>0 fo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-98f93e20c13d2c57","tex":"t^6","aliases":"t^6","display":"t^6","kind":"use","excerpt":"The next argument needs the positivity of u_n. It also needs a modest lower bound on it, rather than a full asymptotic theorem. Both follow from the recurrence. To see the relevant sign without relying on a plot, put t=n-1. Direct polynomial expansion gives q(n)-a_n= · 3200t^6+24000t^5+73392t^4+117072t^3 · +102768t^2+47112t+8827. Thus q(n)>a_n>0 fo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"Set T_n=16^nu_n. Its forward recurrence is a_nT_n+1=16q(n)T_n+256c_nT_n-1, T_0=1, T_1=28. The initial terms are 1,28,2596,311536. The universally integral expression is u_n=∑_j=0^nbinom nj binomn-tfrac12jbinomn+j-tfrac12j, with the external coefficient identity and the written half-binomial denominator argument kept distinct.","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"Set T_n=16^nu_n. Its forward recurrence is a_nT_n+1=16q(n)T_n+256c_nT_n-1, T_0=1, T_1=28. The initial terms are 1,28,2596,311536. The universally integral expression is u_n=∑_j=0^nbinom nj binomn-tfrac12jbinomn+j-tfrac12j, with the external coefficient identity and the written half-binomial denominator argument kept distinct.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"v_2 · =frac10699·(13/8)1872 =frac106991152. Multiplying u_2 by 16^2 gives T_2=2596. Alternatively, the integer recurrence numerator is 16·10699·28+256·260·1 =4 859 712=1872·2596. The exact division has no remainder. This calculation illustrates the program's division check. The all-index integrality proof uses the separate half-binomial argument.","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","excerpt":"For example, the n=2 sum is u_2=1+frac152+frac10564=frac64964. Multiplication by 16^2 gives the three integers 256, 1920, and 420, whose sum is T_2=2596. This illustrates the proof at the level of individual terms. The first sequence values are T_0=1, T_1=28, T_2=2596, T_3=311536, T_4=41759524.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-64f17e7dedd6a141","tex":"T_4","aliases":"T_4","display":"T_4","kind":"use","excerpt":"For example, the n=2 sum is u_2=1+frac152+frac10564=frac64964. Multiplication by 16^2 gives the three integers 256, 1920, and 420, whose sum is T_2=2596. This illustrates the proof at the level of individual terms. The first sequence values are T_0=1, T_1=28, T_2=2596, T_3=311536, T_4=41759524.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-caa2886bccf430cc","tex":"T_{5N}","aliases":"T_5N","display":"T_{5N}","kind":"use","excerpt":"The computation first constructs T_0,…,T_5N through the exact integer recurrence. It then reduces these integers modulo 2^K. The resulting array is the packet row with m=0. At each subsequent row, Pascal evolution replaces adjacent entries by their sum modulo 2^K. Induction from [source reference] proves that every entry read by the scan is the exa","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b6d1c35d86f0d39d","tex":"T_k","aliases":"T_k","display":"T_k","kind":"use","excerpt":"Fix integers n,m≥0, and write U=∑_i=0^mbinom miT_n+i, V=∑_i=0^mbinom mi16^n+iv_n+i. These are fixed-pair aliases U=S_n,m and V=H_n,m for the packets defined earlier, not the integral coefficients of a different source normalization. Here T_k=16^ku_k is a positive integer. Consequently U is a positive integer. Write V=N/D in lowest terms, with D>0 a","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-40bcddeff96a69c7","tex":"T_{n+1}","aliases":"T_n+1","display":"T_{n+1}","kind":"use","excerpt":"Set T_n=16^nu_n. Its forward recurrence is a_nT_n+1=16q(n)T_n+256c_nT_n-1, T_0=1, T_1=28. The initial terms are 1,28,2596,311536. The universally integral expression is u_n=∑_j=0^nbinom nj binomn-tfrac12jbinomn+j-tfrac12j, with the external coefficient identity and the written half-binomial denominator argument kept distinct.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-37c3347f728aefb6","tex":"T_{n+2}","aliases":"T_n+2","display":"T_{n+2}","kind":"use","excerpt":"At m=0, the packet is just S_n,0=T_n. The first two nontrivial rows are S_n,1=T_n+T_n+1, S_n,2=T_n+2T_n+1+T_n+2. For example S_1,1=28+2596=2624. The Pascal identity produces exactly the same integer as direct evaluation of its defining sum. This agreement between an additive row evolution and a direct weighted sum gives two different implementation","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d08a14a786811454","tex":"T_{n-1}","aliases":"T_n-1","display":"T_{n-1}","kind":"use","excerpt":"Set T_n=16^nu_n. Its forward recurrence is a_nT_n+1=16q(n)T_n+256c_nT_n-1, T_0=1, T_1=28. The initial terms are 1,28,2596,311536. The universally integral expression is u_n=∑_j=0^nbinom nj binomn-tfrac12jbinomn+j-tfrac12j, with the external coefficient identity and the written half-binomial denominator argument kept distinct.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-75177911206476aa","tex":"\\tan","aliases":"tan","display":"\\tan","kind":"use","excerpt":"[Arctangent and tangent forms] The following equalities hold: G=∫_0^1fracarctan xx dd x =-∫_0^π/4log(tanθ) ddθ. The integrand arctan x/x is assigned its continuous value 1 at x=0.","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"In the later chapters the starting form is ℓ_j=u_jα-v_j, u_j,v_j∈Q. Choose a positive integer D_j for which D_ju_j and D_jv_j are integers. Then L_j=D_jℓ_j=(D_ju_j)α-D_jv_j has integer coefficients. Its size is D_jabsℓ_j, so the clearance factor must be included in any decay estimate.","count":15},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-836830fc78b794aa","tex":"u_k","aliases":"u_k","display":"u_k","kind":"use","excerpt":"Substitute the corrected normalization J_k=8(-1)^k(u_kG-v_k) into [source reference]. The two factors (-1)^i cancel. Thus widetilde J_n,m=8(-1)^n16^-n(S_n,mG-H_n,m). Define the raw packet form F_n,m=16^nwidetilde J_n,m =8(-1)^n(S_n,mG-H_n,m). The adjective raw means that the common denominator of H_n,m has not yet been cleared. It does not mean tha","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-30b08f45af646592","tex":"u_{n-1}","aliases":"u_n-1","display":"u_{n-1}","kind":"use","excerpt":"The two distinguished solutions can be collected into the fundamental matrix F_n= u_n · v_n u_n-1 · v_n-1, n≥1. It satisfies F_n+1=M_n F_n. Its initial determinant is det F_1= det7/4 · 13/8","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-490b0994ccc45d88","tex":"u_n^{1/n}","aliases":"u_n^1/n","display":"u_n^{1/n}","kind":"use","excerpt":"[External recurrence identification and rates] For the exact normalization [source reference], Zudilin's theorem and its stated asymptotic rates give lim_n→∞fracv_nu_n · =G, lim_n→∞u_n^1/n · =ϕ^5, lim_n→∞|u_nG-v_n|^1/n · =ϕ^-5. These are external mathematical inputs, recorded precisely in source contracts LC01 and LC02 [source reference].","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ac6fdb5ec7cb417c","tex":"v_{20+i}","aliases":"v_20+i","display":"v_{20+i}","kind":"use","excerpt":"The first row illustrates the exact check without printing its very large numerator and denominator. The independently reduced rational packet has denominator D=den(∑_i=0^66binom66i16^20+iv_20+i). Its recorded quotient Q/D is 1 703 936, and the minimal pair has gcd 8. Multiplication gives 1 703 936·8=13 631 488, exactly the historical value. The co","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"c_n · =(2n-1)^2(2n)^2p(n+1). Each sequence w=u or w=v obeys a_nw_n+1=q(n)w_n+c_nw_n-1. The forward recurrence is used only at n≥1. Its coefficients a_n are positive there. The first additional terms are u_2=frac64964, u_3=frac19471256, v_2=frac106991152, v_3=frac8025653115200.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"In the later chapters the starting form is ℓ_j=u_jα-v_j, u_j,v_j∈Q. Choose a positive integer D_j for which D_ju_j and D_jv_j are integers. Then L_j=D_jℓ_j=(D_ju_j)α-D_jv_j has integer coefficients. Its size is D_jabsℓ_j, so the clearance factor must be included in any decay estimate.","count":15},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-0a435f82d4918621","tex":"v_k","aliases":"v_k","display":"v_k","kind":"use","excerpt":"Substitute the corrected normalization J_k=8(-1)^k(u_kG-v_k) into [source reference]. The two factors (-1)^i cancel. Thus widetilde J_n,m=8(-1)^n16^-n(S_n,mG-H_n,m). Define the raw packet form F_n,m=16^nwidetilde J_n,m =8(-1)^n(S_n,mG-H_n,m). The adjective raw means that the common denominator of H_n,m has not yet been cleared. It does not mean tha","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7fd7bfa7519bcf67","tex":"v_{n+1}","aliases":"v_n+1","display":"v_{n+1}","kind":"use","excerpt":"The Wronskian in this edition uses the orientation W_n=u_n+1v_n-u_nv_n+1 =frac(-1)^n+1p(n+1)2(2n+1)^2(2n+2)^2. Reversing its two terms reverses the sign. The formula applies to n≥0, with the base case supplied by the initial pair and the induction step supplied by [source reference].","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-9dcaa8d933e2d5a8","tex":"v_{n-1}","aliases":"v_n-1","display":"v_{n-1}","kind":"use","excerpt":"The two distinguished solutions can be collected into the fundamental matrix F_n= u_n · v_n u_n-1 · v_n-1, n≥1. It satisfies F_n+1=M_n F_n. Its initial determinant is det F_1= det7/4 · 13/8","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-81d213d5a0564b18","tex":"v_n^{1/n}","aliases":"v_n^1/n","display":"v_n^{1/n}","kind":"use","excerpt":"ce rates is lim_n→∞|G-r_n|^1/n =lim_n→∞frac|u_nG-v_n|^1/nu_n^1/n =ϕ^-10. Thus the ratio error has a smaller exponential scale than the unnormalized linear form. The distinction comes from division by the growing coefficient u_n. Also, since r_n→ G>0, one has r_n^1/n→1 and consequently v_n^1/n→ϕ^5. Both individual sequences grow, while their specifi","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"use","excerpt":"If N=0, the reduced convention gives D=1. Since U>0, one has h=U, g=QU, and the unsigned primitive pair is (1,0). All the stated identities follow directly in this case. Assume now N≠0. Since gcd(D,N)=1, multiplication of U by D adds no prime factor that can divide N. Prime by prime, min{v_p(DU),v_p(|N|)}=min{v_p(U),v_p(|N|)}. This proves the first","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"The Wronskian in this edition uses the orientation W_n=u_n+1v_n-u_nv_n+1 =frac(-1)^n+1p(n+1)2(2n+1)^2(2n+2)^2. Reversing its two terms reverses the sign. The formula applies to n≥0, with the base case supplied by the initial pair and the induction step supplied by [source reference].","count":9},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b9d7c18b6334d33d","tex":"W_0","aliases":"W_0","display":"W_0","kind":"use","excerpt":"The same small case checks the Wronskian orientation: W_0 · =frac74·0-1·frac138 =-frac138, W_1 · =frac64964·frac138 -frac74·frac106991152 =frac65288. Both agree with [source reference]. They show why the sign is (-1)^n+1 for this orientation.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-59e63012bc6d4f87","tex":"w_0","aliases":"w_0","display":"w_0","kind":"use","excerpt":"The forward equation starts at n=1. Substituting n=0 into its polynomial expression would remove the trailing term and impose 4w_1=7w_0. Although the initial values of u happen to satisfy that extra equation, those of v do not. It is therefore not a valid additional recurrence condition for the pair. At n=1, both w_0 and w_1 are already specified a","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d01794b5f0c3ada0","tex":"W_1","aliases":"W_1","display":"W_1","kind":"use","excerpt":"The same small case checks the Wronskian orientation: W_0 · =frac74·0-1·frac138 =-frac138, W_1 · =frac64964·frac138 -frac74·frac106991152 =frac65288. Both agree with [source reference]. They show why the sign is (-1)^n+1 for this orientation.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"The forward equation starts at n=1. Substituting n=0 into its polynomial expression would remove the trailing term and impose 4w_1=7w_0. Although the initial values of u happen to satisfy that extra equation, those of v do not. It is therefore not a valid additional recurrence condition for the pair. At n=1, both w_0 and w_1 are already specified a","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3106fac3e3c8f992","tex":"w_2","aliases":"w_2","display":"w_2","kind":"use","excerpt":"The forward equation starts at n=1. Substituting n=0 into its polynomial expression would remove the trailing term and impose 4w_1=7w_0. Although the initial values of u happen to satisfy that extra equation, those of v do not. It is therefore not a valid additional recurrence condition for the pair. At n=1, both w_0 and w_1 are already specified a","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-9763a163e7ff6591","tex":"w_3","aliases":"w_3","display":"w_3","kind":"use","excerpt":"At n=2, the formula is w_2=α_1w_1+β_1w_0, using the empty determinant for K_2,1. At n=3, direct iteration gives w_3=(α_2α_1+β_2)w_1+α_2β_1w_0, which agrees with the formula. For later indices, both coefficients in [source reference] satisfy the same recurrence by [source reference]. Induction finishes the proof.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-466a54f902244928","tex":"W_n","aliases":"W_n","display":"W_n","kind":"use","excerpt":"The Wronskian in this edition uses the orientation W_n=u_n+1v_n-u_nv_n+1 =frac(-1)^n+1p(n+1)2(2n+1)^2(2n+2)^2. Reversing its two terms reverses the sign. The formula applies to n≥0, with the base case supplied by the initial pair and the induction step supplied by [source reference].","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ee1d010f6d9cc5bd","tex":"w_{n+1}","aliases":"w_n+1","display":"w_{n+1}","kind":"use","excerpt":"c_n · =(2n-1)^2(2n)^2p(n+1). Each sequence w=u or w=v obeys a_nw_n+1=q(n)w_n+c_nw_n-1. The forward recurrence is used only at n≥1. Its coefficients a_n are positive there. The first additional terms are u_2=frac64964, u_3=frac19471256, v_2=frac106991152, v_3=frac8025653115200.","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-394e76f3f5bd1605","tex":"W_{n-1}","aliases":"W_n-1","display":"W_{n-1}","kind":"use","excerpt":"At n=0 the determinant is -13/8, which agrees with the displayed formula because p(1)=13. For n≥1, insert [source reference] into both terms of [source reference]. The terms containing q(n)u_nv_n cancel, leaving W_n=fracc_na_n(u_n-1v_n-u_nv_n-1) =-fracc_na_nW_n-1. Iterating this equality gives W_n=(-1)^nW_0 ∏_j=1^n frac(2j-1)^2(2j)^2p(j+1)(2j+1)^2(","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3f7dba1c210080bb","tex":"\\widetilde J_{1,1}","aliases":"widetilde J_1,1 widetilde","display":"\\widetilde J_{1,1}","kind":"use","excerpt":"_4=12, hence Q=8L_4^2=1152. Its redundant scale is Q/D=128. Consequently g=128·64=8192. The signed cleared pair is (-3 022 848,-2 768 896). Dividing by 8192 gives (-369,-338), so the primitive form is Λ_1,1=338-369G. The integral identity gives the same expression. Since widetilde J_1,1=F_1,1/16 and 16L_4^2/g=9/32, frac16L_4^2gwidetilde J_1,1 =frac","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-05ab66688eb980e2","tex":"\\widetilde J_{n,m(n)}","aliases":"widetilde J_n,m(n) widetilde","display":"\\widetilde J_{n,m(n)}","kind":"use","excerpt":"Let α_n=m(n)/n. The finite measure in [source reference] gives |widetilde J_n,m(n)| ≤ 8G M(α_n)^n. Continuity of the two branches, including their stated extension at zero, then proves limsup_n→∞ |widetilde J_n,m(n)|^1/n≤ M(α). This is an upper bound. An equality for the actual integral would require a further argument. In particular, an upper boun","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-31235b7ba4ba9524","tex":"\\widetilde U_{n,m(n)}","aliases":"widetilde U_n,m(n) widetilde","display":"\\widetilde U_{n,m(n)}","kind":"use","excerpt":"Eskandari's coefficient growth statement is lim_n→∞fraclog|widetilde U_n,m(n)|n =γ(α) =-log16+log R+αlog(1+R), where widetilde U=8(-1)^n16^-nU. This is an external asymptotic result in the preprint, not a conclusion of the finite recurrence audit. Together with the filter bound it controls the approximation error |G-frac VU| =frac|widetilde J||wide","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"Normalize the recurrence by putting α_n=fracq(n)a_n, β_n=fracc_na_n (n≥1). Both quantities are positive rationals. For the adjacent state vector X_n(w)=(w_n,w_n-1)^ T, define M_n=α_n · β_n","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"The Euler interval uses a_k=(2k+1)^-2 and the positive difference convention Δ a_k=a_k-a_k+1. The identity Δ^ja_0=∫_0^1(1-x^2)^j(-log x) dd x gives 0≤Δ^ja_0≤1. The positive Euler series for G is ∑_j≥02^-j-1Δ^ja_0. Keeping the first 1001 terms therefore gives a lower endpoint and an upper endpoint obtained by adding 2^-1001. All retained terms are r","count":16},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3e91c97fccb94cf6","tex":"x^{2k}","aliases":"x^2k","display":"x^{2k}","kind":"use","excerpt":"For 0≤ x<1, the geometric expansion is frac11+x^2=∑_k=0^∞(-1)^kx^2k. The integrated absolute values of the proposed terms have finite sum: ∑_k=0^∞∫_0^1x^2k(-log x) dd x =∑_k=0^∞frac1(2k+1)^2<∞. Therefore the series of integrable functions can be summed under the integral. The moment identity gives the defining series for G. The point x=1 has measur","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4e5c9fefc8f0f7d8","tex":"x^{2N}","aliases":"x^2N","display":"x^{2N}","kind":"use","excerpt":"^s. Absolute convergence follows from comparison with ∑ j^-s. Thus G=β(2). The alternating series also defines β(s) for s>0, though absolute convergence is no longer available throughout that wider range. At s=1, integration of the finite geometric expansion of (1+x^2)^-1 gives β(1)=π/4. The integrated remainder is bounded by ∫_0^1x^2Ndd x=(2N+1)^-","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f3522098af4f1d38","tex":"x^a","aliases":"x^a","display":"x^a","kind":"use","excerpt":"[A basic moment] For every real a>-1, ∫_0^1x^a(-log x) dd x=frac1(a+1)^2. The integral is understood as an improper integral at 0.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4cdb08449275a3f8","tex":"x^{n-1/2}","aliases":"x^n-1/2","display":"x^{n-1/2}","kind":"use","excerpt":"widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^n+1 dd x dd y =8(-1)^n(u_nG-v_n). These are fixed normalization identities. Their factors should be checked together when translating a differently normalized source.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"1 · 0. X_n+1(w)=M_nX_n(w). Hence X_N+1(w)=M_NM_N-1⋯ M_1X_1(w) (N≥1). The order of multiplication follows the temporal order: M_1 acts first. This is a product of changing matrices, not a power of one fixed matrix.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"There is a simple logical reason to resist extrapolation. Given any finite list of positive values x_1,…,x_N, it has a positive extension that tends to zero and another positive extension that tends to infinity. Keep the list fixed and set the tail to e^-n in the first case or e^n in the second. Similarly, any finite list of positive integer gcds c","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"use","excerpt":"Normalize the recurrence by putting α_n=fracq(n)a_n, β_n=fracc_na_n (n≥1). Both quantities are positive rationals. For the adjacent state vector X_n(w)=(w_n,w_n-1)^ T, define M_n=α_n · β_n","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ec84613b712fe298","tex":"x_N","aliases":"x_N","display":"x_N","kind":"use","excerpt":"There is a simple logical reason to resist extrapolation. Given any finite list of positive values x_1,…,x_N, it has a positive extension that tends to zero and another positive extension that tends to infinity. Keep the list fixed and set the tail to e^-n in the first case or e^n in the second. Similarly, any finite list of positive integer gcds c","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8df463ae90a2e3d3","tex":"X_{N+1}","aliases":"X_N+1","display":"X_{N+1}","kind":"use","excerpt":"1 · 0. X_n+1(w)=M_nX_n(w). Hence X_N+1(w)=M_NM_N-1⋯ M_1X_1(w) (N≥1). The order of multiplication follows the temporal order: M_1 acts first. This is a product of changing matrices, not a power of one fixed matrix.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-98f50cf2afdfc449","tex":"X_{n+1}","aliases":"X_n+1","display":"X_{n+1}","kind":"use","excerpt":"Normalize the recurrence by putting α_n=fracq(n)a_n, β_n=fracc_na_n (n≥1). Both quantities are positive rationals. For the adjacent state vector X_n(w)=(w_n,w_n-1)^ T, define M_n=α_n · β_n 1 · 0. X_n+1(w)=M_nX_n(w). Hence X_N+1(w)=M_NM_N-1⋯ M_1X_1(w) (N≥1). The order of multiplication follows the temporal order: M_1 acts first. This is a product of","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^n+1 dd x dd y =8(-1)^n(u_nG-v_n). These are fixed normalization identities. Their factors should be checked together when translating a differently normalized source.","count":23},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"[Elementary double integral] Catalan's constant also satisfies G=∫_0^1∫_0^1fracdd x dd y1+x^2y^2.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d23907ea435699d2","tex":"y^{2k}","aliases":"y^2k","display":"y^{2k}","kind":"use","excerpt":"There is also a direct series justification. Expanding (1+x^2y^2)^-1 on the square except at the corner (1,1) gives geometric terms (-1)^kx^2ky^2k. Their integrated absolute values sum to ∑_k=0^∞(∫_0^1x^2k dd x)^2 =∑_k=0^∞frac1(2k+1)^2<∞. Thus summation and integration may be interchanged, giving the defining series again.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-de7ba1dbdac1562b","tex":"y^n","aliases":"y^n","display":"y^n","kind":"use","excerpt":"widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^n+1 dd x dd y =8(-1)^n(u_nG-v_n). These are fixed normalization identities. Their factors should be checked together when translating a differently normalized source.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"Work with formal power series, so no convergence issue arises. The central binomial expansion gives (1+4z)^-1/2=∑_k≥0(-1)^kbinom2kkz^k. Every coefficient is an integer. Multiplying by the integer polynomial (1+4z)^h preserves this property. The coefficient of z^j in the product is 4^jbinomh-tfrac12j. More explicitly it is ∑_ℓ=0^min(h,j) binom hℓ 4^","count":13},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"use","excerpt":"Work with formal power series, so no convergence issue arises. The central binomial expansion gives (1+4z)^-1/2=∑_k≥0(-1)^kbinom2kkz^k. Every coefficient is an integer. Multiplying by the integer polynomial (1+4z)^h preserves this property. The coefficient of z^j in the product is 4^jbinomh-tfrac12j. More explicitly it is ∑_ℓ=0^min(h,j) binom hℓ 4^","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","excerpt":"Work with formal power series, so no convergence issue arises. The central binomial expansion gives (1+4z)^-1/2=∑_k≥0(-1)^kbinom2kkz^k. Every coefficient is an integer. Multiplying by the integer polynomial (1+4z)^h preserves this property. The coefficient of z^j in the product is 4^jbinomh-tfrac12j. More explicitly it is ∑_ℓ=0^min(h,j) binom hℓ 4^","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e86777b436179d44","tex":"|^{1/n}","aliases":"|^1/n","display":"|^{1/n}","kind":"use","excerpt":"[External recurrence identification and rates] For the exact normalization [source reference], Zudilin's theorem and its stated asymptotic rates give lim_n→∞fracv_nu_n · =G, lim_n→∞u_n^1/n · =ϕ^5, lim_n→∞|u_nG-v_n|^1/n · =ϕ^-5. These are external mathematical inputs, recorded precisely in source contracts LC01 and LC02 [source reference].","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-af15262773600a7f","tex":"\\alpha_r","aliases":"α_r alpha","display":"\\alpha_r","kind":"use","excerpt":"The transfer product can also be written using continuants. For integers 1≤ r≤ s, let K_r,s be the determinant of the tridiagonal matrix whose diagonal is α_r,…,α_s, lower adjacent diagonal is one, and upper adjacent diagonal is -β_r+1,…,-β_s. For instance, K_r,r=α_r, K_r,r+1=det α_r · -β_r+1","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5a177678929c7e28","tex":"\\alpha_{r+1}","aliases":"α_r+1 alpha","display":"\\alpha_{r+1}","kind":"use","excerpt":"The transfer product can also be written using continuants. For integers 1≤ r≤ s, let K_r,s be the determinant of the tridiagonal matrix whose diagonal is α_r,…,α_s, lower adjacent diagonal is one, and upper adjacent diagonal is -β_r+1,…,-β_s. For instance, K_r,r=α_r, K_r,r+1=det α_r · -β_r+1 1 · α_r+1 =α_rα_r+1+β_r+1. Set K_r,r-1=1 for the empty d","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cd53b89eabc58b71","tex":"\\alpha_s","aliases":"α_s alpha","display":"\\alpha_s","kind":"use","excerpt":"The transfer product can also be written using continuants. For integers 1≤ r≤ s, let K_r,s be the determinant of the tridiagonal matrix whose diagonal is α_r,…,α_s, lower adjacent diagonal is one, and upper adjacent diagonal is -β_r+1,…,-β_s. For instance, K_r,r=α_r, K_r,r+1=det α_r · -β_r+1","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e7ab58453439ed94","tex":"\\beta_{r+1}","aliases":"β_r+1 beta","display":"\\beta_{r+1}","kind":"use","excerpt":"The transfer product can also be written using continuants. For integers 1≤ r≤ s, let K_r,s be the determinant of the tridiagonal matrix whose diagonal is α_r,…,α_s, lower adjacent diagonal is one, and upper adjacent diagonal is -β_r+1,…,-β_s. For instance, K_r,r=α_r, K_r,r+1=det α_r · -β_r+1","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-cabb81d89641f816","tex":"\\beta_s","aliases":"β_s beta","display":"\\beta_s","kind":"use","excerpt":"The transfer product can also be written using continuants. For integers 1≤ r≤ s, let K_r,s be the determinant of the tridiagonal matrix whose diagonal is α_r,…,α_s, lower adjacent diagonal is one, and upper adjacent diagonal is -β_r+1,…,-β_s. For instance, K_r,r=α_r, K_r,r+1=det α_r · -β_r+1","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"Eskandari's coefficient growth statement is lim_n→∞fraclog|widetilde U_n,m(n)|n =γ(α) =-log16+log R+αlog(1+R), where widetilde U=8(-1)^n16^-nU. This is an external asymptotic result in the preprint, not a conclusion of the finite recurrence audit. Together with the filter bound it controls the approximation error |G-frac VU| =frac|widetilde J||wide","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"The Euler interval uses a_k=(2k+1)^-2 and the positive difference convention Δ a_k=a_k-a_k+1. The identity Δ^ja_0=∫_0^1(1-x^2)^j(-log x) dd x gives 0≤Δ^ja_0≤1. The positive Euler series for G is ∑_j≥02^-j-1Δ^ja_0. Keeping the first 1001 terms therefore gives a lower endpoint and an upper endpoint obtained by adding 2^-1001. All retained terms are r","count":11},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"For 0<δ<1, integration by parts gives ∫_δ^1fracarctan xx dd x =-arctan(δ)logδ -∫_δ^1fraclog x1+x^2 dd x. The boundary expression tends to zero as δdownarrow0, since 0≤arctanδ≤δ and δabslogδ→0. The first equality follows from Proposition~[source reference].","count":25},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2d8fd38ef1b9fe4d","tex":"\\delta'","aliases":"δ' delta","display":"\\delta'","kind":"use","excerpt":"Since D_j≥1, the second hypothesis implies δ≥1. Choose positive numbers λ'>λ and δ'>δ with λ'δ'<1. Such a choice is possible by the strict product inequality, including when λ=0.","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f469f9b523c3c0a1","tex":"\\Delta^0","aliases":"Δ^0 Delta","display":"\\Delta^0","kind":"use","excerpt":"[Small Euler intervals] The first difference values are Δ^0a_0=1, Δ a_0=frac89, Δ^2a_0=frac184225. They give E_0=frac12, E_1=frac1318, E_2=frac371450. Thus, for example, frac371450<G<frac17091800. These short transformed sums are useful for checking conventions. Their principal advantage appears as N grows, because the certified width decreases geo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3f0f0e9cd378ed81","tex":"\\Delta^2","aliases":"Δ^2 Delta","display":"\\Delta^2","kind":"use","excerpt":"[Small Euler intervals] The first difference values are Δ^0a_0=1, Δ a_0=frac89, Δ^2a_0=frac184225. They give E_0=frac12, E_1=frac1318, E_2=frac371450. Thus, for example, frac371450<G<frac17091800. These short transformed sums are useful for checking conventions. Their principal advantage appears as N grows, because the certified width decreases geo","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8e0f50e7f57f2609","tex":"\\Delta^j","aliases":"Δ^j Delta","display":"\\Delta^j","kind":"use","excerpt":"The Euler interval uses a_k=(2k+1)^-2 and the positive difference convention Δ a_k=a_k-a_k+1. The identity Δ^ja_0=∫_0^1(1-x^2)^j(-log x) dd x gives 0≤Δ^ja_0≤1. The positive Euler series for G is ∑_j≥02^-j-1Δ^ja_0. Keeping the first 1001 terms therefore gives a lower endpoint and an upper endpoint obtained by adding 2^-1001. All retained terms are r","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"[Sufficient content margin] Choose even integers m(n)≥0 such that m(n)/n→α≥0. Use the coefficient clearing theorem, the corrected filter identity, and the prime number theorem as specified above. If there exists a real number δ_0 with liminf_n→∞fraclog g_n,m(n)n≥δ_0 and δ_0>κ_0(α)+log M(α), then G is irrational.","count":5},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"The coefficient clearing theorem and division by the exact gcd make Λ_n=a_nG-b_n with a_n,b_n∈Z. Evenness of m(n) makes widetilde J_n,m(n)>0, so [source reference] gives Λ_n>0. Choose ε>0 so that 3ε<δ_0-κ_0(α)-log M(α). For all sufficiently large n, the clearing asymptotic gives K_n≤exp((κ_0+ε)n), the filter bound gives |widetilde J_n,m(n)|≤exp((lo","count":6},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"lde J||widetilde U|. Before any additional quantified gcd saving, the reduced denominator q is at most the absolute cleared coefficient |A|. Its available exponent is c_A(α)=4(1+α)+log R+αlog(1+R). The source's resulting guaranteed approximation exponent is η(α)= fraclog R+αlog(1+R)-log16-log M(α) 4(1+α)+log R+αlog(1+R). The source reports a maximu","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"[Arctangent and tangent forms] The following equalities hold: G=∫_0^1fracarctan xx dd x =-∫_0^π/4log(tanθ) ddθ. The integrand arctan x/x is assigned its continuous value 1 at x=0.","count":12},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"t can recover a proved rate] Assume the setup of Corollary~[source reference], except that τ>σ is not required. Discard the finite initial set of indices before nonvanishing holds. For the remaining sufficiently large j, define h_j=gcd(absD_ju_j,absD_jv_j), which is positive because the form is nonzero at those indices. If liminf_j→∞fraclog h_jj≥κ ","count":16},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-35165a2cb10ed645","tex":"\\kappa_0","aliases":"κ_0 kappa","display":"\\kappa_0","kind":"use","excerpt":"Set K_n=16^nL_2(n+m(n))^2. For m(n)/n→α, [source reference] gives fraclog K_nn⟶ κ_0(α)=log16+4(1+α). The factor 4 has a precise origin. The lcm index is 2(n+m), and the lcm is squared. Missing either factor would create an incorrect apparent decay margin. The subscript in κ_0 emphasizes that this is the clearing cost alone. Define the combined boun","count":13},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"1 · 0. The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follows by repeatedly using ϕ^2=ϕ+1. The roots have product -1, as required by the limiting determinant, and very different magnitudes.","count":13},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-11c22a9994059ca8","tex":"\\lambda'","aliases":"λ' lambda","display":"\\lambda'","kind":"use","excerpt":"Since D_j≥1, the second hypothesis implies δ≥1. Choose positive numbers λ'>λ and δ'>δ with λ'δ'<1. Such a choice is possible by the strict product inequality, including when λ=0.","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-308a821eda6e7229","tex":"\\lambda^2","aliases":"λ^2 lambda","display":"\\lambda^2","kind":"use","excerpt":"1 · 0. The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follows by repeatedly using ϕ^2=ϕ+1. The roots have product -1, as required by the limiting determinant, and very different magnitudes.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e7d367cdec71a6b4","tex":"\\lambda_+","aliases":"λ_+ lambda","display":"\\lambda_+","kind":"use","excerpt":"1 · 0. The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follows by repeatedly using ϕ^2=ϕ+1. The roots have product -1, as required by the limiting determinant, and very different magnitudes.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-16b1a787c83a448c","tex":"\\lambda_-","aliases":"λ_- lambda","display":"\\lambda_-","kind":"use","excerpt":"1 · 0. The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follows by repeatedly using ϕ^2=ϕ+1. The roots have product -1, as required by the limiting determinant, and very different magnitudes.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-456d7f3907cbdbb9","tex":"\\Lambda_{1,1}","aliases":"Λ_1,1 Lambda","display":"\\Lambda_{1,1}","kind":"use","excerpt":"The universal factor uses L_4=12, hence Q=8L_4^2=1152. Its redundant scale is Q/D=128. Consequently g=128·64=8192. The signed cleared pair is (-3 022 848,-2 768 896). Dividing by 8192 gives (-369,-338), so the primitive form is Λ_1,1=338-369G. The integral identity gives the same expression. Since widetilde J_1,1=F_1,1/16 and 16L_4^2/g=9/32, frac16","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-62ed57f1458fb941","tex":"\\Lambda_n","aliases":"Λ_n Lambda","display":"\\Lambda_n","kind":"use","excerpt":"With the cleared gcd g_n=g_n,m(n), the primitive form is exactly Λ_n=fracK_ng_nwidetilde J_n,m(n). Suppose a future arithmetic argument establishes δ=liminf_n→∞fraclog g_nn. For finite δ, the present analytic estimate yields limsup_n→∞fraclog|Λ_n|n ≤κ_0(α)+log M(α)-δ. The same interpretation applies when δ=+∞: any sufficiently large finite lower bo","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"On the open unit square, set ψ(x,y)=fracxy(1-x)(1-y)1-xy, ddμ=fracdd x dd ysqrtx(1-y)(1-xy). The factor 1-xy is outside the square root. Moving that factor inside, as in fracdd x dd ysqrtx(1-y)(1-xy), would represent a different integral. With the measure in [source reference], J_n=∫_(0,1)^2ψ^n ddμ, widetilde J_n,m=∫_(0,1)^2ψ^n(1-16ψ)^m ddμ. The se","count":6},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"quation β(s)=∑_k=0^∞frac(-1)^k(2k+1)^s. Absolute convergence follows from comparison with ∑ j^-s. Thus G=β(2). The alternating series also defines β(s) for s>0, though absolute convergence is no longer available throughout that wider range. At s=1, integration of the finite geometric expansion of (1+x^2)^-1 gives β(1)=π/4. The integrated remainder ","count":6},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"[Exactness below the precision ceiling] Let K≥1, let z≠0 be an integer, and let ρ be its least nonnegative residue modulo 2^K. If ρ≠0, then v_2(z)=v_2(ρ)<K. If ρ=0, the residue gives only v_2(z)≥ K.","count":7},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"[Logarithmic rate accounting] Under the integrality and eventual nonvanishing hypotheses above, suppose limsup_j→∞fraclog D_jj≤σ, limsup_j→∞fraclogabsu_jα-v_jj≤-τ, with finite σ and τ>σ. Then α is irrational.","count":9},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"[Logarithmic rate accounting] Under the integrality and eventual nonvanishing hypotheses above, suppose limsup_j→∞fraclog D_jj≤σ, limsup_j→∞fraclogabsu_jα-v_jj≤-τ, with finite σ and τ>σ. Then α is irrational.","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"1 · 0. The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follows by repeatedly using ϕ^2=ϕ+1. The roots have product -1, as required by the limiting determinant, and very different magnitudes.","count":12},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4f7f632689e8289e","tex":"\\phi^{-1}","aliases":"ϕ^-1 phi","display":"\\phi^{-1}","kind":"use","excerpt":"The kernel has a simple maximum that also explains why the filter changes sign. Put r=sqrtxy. Since x+y≥2r, ψ(x,y)≤fracr^2(1-r)^21-r^2 =fracr^2(1-r)1+r,qquad0<r<1. Equality is achieved when x=y=r. The derivative of the last expression is frac2r(1-r-r^2)(1+r)^2. It has its interior maximum at r=ϕ^-1, where ϕ=(1+sqrt5)/2. Since 1-r=r^2 there, the max","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-83175d12edc322e5","tex":"\\phi^{-10}","aliases":"ϕ^-10 phi","display":"\\phi^{-10}","kind":"use","excerpt":"An immediate consequence of the two source rates is lim_n→∞|G-r_n|^1/n =lim_n→∞frac|u_nG-v_n|^1/nu_n^1/n =ϕ^-10. Thus the ratio error has a smaller exponential scale than the unnormalized linear form. The distinction comes from division by the growing coefficient u_n. Also, since r_n→ G>0, one has r_n^1/n→1 and consequently v_n^1/n→ϕ^5. Both indivi","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ffac7c841830a1f7","tex":"\\phi^{-5}","aliases":"ϕ^-5 phi","display":"\\phi^{-5}","kind":"use","excerpt":"1 · 0. The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follows by repeatedly using ϕ^2=ϕ+1. The roots have product -1, as required by the limiting determinant, and very different magnitudes.","count":4},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d73ae6dc3936033e","tex":"\\phi^2","aliases":"ϕ^2 phi","display":"\\phi^2","kind":"use","excerpt":"The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follows by repeatedly using ϕ^2=ϕ+1. The roots have product -1, as required by the limiting determinant, and very different magnitudes.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7bd71e852e91e16d","tex":"\\phi^5","aliases":"ϕ^5 phi","display":"\\phi^5","kind":"use","excerpt":"1 · 0. The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follows by repeatedly using ϕ^2=ϕ+1. The roots have product -1, as required by the limiting determinant, and very different magnitudes.","count":6},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d28947f5acd9f188","tex":"\\psi^n","aliases":"ψ^n psi","display":"\\psi^n","kind":"use","excerpt":"On the open unit square, set ψ(x,y)=fracxy(1-x)(1-y)1-xy, ddμ=fracdd x dd ysqrtx(1-y)(1-xy). The factor 1-xy is outside the square root. Moving that factor inside, as in fracdd x dd ysqrtx(1-y)(1-xy), would represent a different integral. With the measure in [source reference], J_n=∫_(0,1)^2ψ^n ddμ, widetilde J_n,m=∫_(0,1)^2ψ^n(1-16ψ)^m ddμ. The se","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"The second example can also be written as a failure of denominator clearing. The rational coefficient form ℓ_n=α-(frac12+2^-n)=-2^-n is nonzero and tends to zero. Its second coefficient is not an integer. Multiplication by 2^n makes both coefficients integral and gives the constant form -1. The analytic gain has been exactly consumed by the arithme","count":8},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-8943b22d90e5e02b","tex":"\\ell_j","aliases":"ℓ_j ell","display":"\\ell_j","kind":"use","excerpt":"In the later chapters the starting form is ℓ_j=u_jα-v_j, u_j,v_j∈Q. Choose a positive integer D_j for which D_ju_j and D_jv_j are integers. Then L_j=D_jℓ_j=(D_ju_j)α-D_jv_j has integer coefficients. Its size is D_jabsℓ_j, so the clearance factor must be included in any decay estimate.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c02010649fe81236","tex":"\\ell_n","aliases":"ℓ_n ell","display":"\\ell_n","kind":"use","excerpt":"The second example can also be written as a failure of denominator clearing. The rational coefficient form ℓ_n=α-(frac12+2^-n)=-2^-n is nonzero and tends to zero. Its second coefficient is not an integer. Multiplication by 2^n makes both coefficients integral and gives the constant form -1. The analytic gain has been exactly consumed by the arithme","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"The leading terms of the coefficient polynomials are a_n=320n^6+O(n^5), q(n)=3520n^6+O(n^5), c_n=320n^6+O(n^5). Consequently α_n→11, β_n→1, and M_n⟶ M_∞=11 · 1","count":24},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"1 · 0=-frac138. Multiplicativity of determinants now gives det F_N+1 =-frac138∏_n=1^N(-β_n)≠0. Thus adjacent values of the two solutions remain independent at every index. Each transfer reverses the determinant sign. The closed evaluation of this product is the Wronskian theorem in Chapter~[source reference], where the factors telescope and yield e","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-6f7d5c597bf5d07a","tex":"\\prod_{j=1}^n","aliases":"∏_j=1^n prod","display":"\\prod_{j=1}^n","kind":"use","excerpt":"At n=0 the determinant is -13/8, which agrees with the displayed formula because p(1)=13. For n≥1, insert [source reference] into both terms of [source reference]. The terms containing q(n)u_nv_n cancel, leaving W_n=fracc_na_n(u_n-1v_n-u_nv_n-1) =-fracc_na_nW_n-1. Iterating this equality gives W_n=(-1)^nW_0 ∏_j=1^n frac(2j-1)^2(2j)^2p(j+1)(2j+1)^2(","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-89538c8eecefdc68","tex":"\\prod_{n=1}^N","aliases":"∏_n=1^N prod","display":"\\prod_{n=1}^N","kind":"use","excerpt":"1 · 0=-frac138. Multiplicativity of determinants now gives det F_N+1 =-frac138∏_n=1^N(-β_n)≠0. Thus adjacent values of the two solutions remain independent at every index. Each transfer reverses the determinant sign. The closed evaluation of this product is the Wronskian theorem in Chapter~[source reference], where the factors telescope and yield e","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-fa6a3d6d7cad3bd0","tex":"\\sum_{i=0}^{66}","aliases":"∑_i=0^66 sum","display":"\\sum_{i=0}^{66}","kind":"use","excerpt":"The first row illustrates the exact check without printing its very large numerator and denominator. The independently reduced rational packet has denominator D=den(∑_i=0^66binom66i16^20+iv_20+i). Its recorded quotient Q/D is 1 703 936, and the minimal pair has gcd 8. Multiplication gives 1 703 936·8=13 631 488, exactly the historical value. The co","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-fd46ed5b3b564187","tex":"\\sum_{i=0}^{m}","aliases":"∑_i=0^m sum","display":"\\sum_{i=0}^{m}","kind":"use","excerpt":"Fix integers n,m≥0, and write U=∑_i=0^mbinom miT_n+i, V=∑_i=0^mbinom mi16^n+iv_n+i. These are fixed-pair aliases U=S_n,m and V=H_n,m for the packets defined earlier, not the integral coefficients of a different source normalization. Here T_k=16^ku_k is a positive integer. Consequently U is a positive integer. Write V=N/D in lowest terms, with D>0 a","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-148a1e75bbb61e74","tex":"\\sum_{j=0}^n","aliases":"∑_j=0^n sum","display":"\\sum_{j=0}^n","kind":"use","excerpt":"Set T_n=16^nu_n. Its forward recurrence is a_nT_n+1=16q(n)T_n+256c_nT_n-1, T_0=1, T_1=28. The initial terms are 1,28,2596,311536. The universally integral expression is u_n=∑_j=0^nbinom nj binomn-tfrac12jbinomn+j-tfrac12j, with the external coefficient identity and the written half-binomial denominator argument kept distinct.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-b9a7896e38144f2e","tex":"\\sum_{j=0}^{N}","aliases":"∑_j=0^N sum","display":"\\sum_{j=0}^{N}","kind":"use","excerpt":"[Certified Euler enclosure] Define E_N=∑_j=0^NfracΔ^j a_02^j+1, N≥0. Then G=∑_j=0^∞fracΔ^j a_02^j+1, E_N<G<E_N+2^-N-1. More precisely, G-E_N =∫_0^1frac-log x1+x^2 (frac1-x^22)^N+1 dd x.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-f7b37e8032ebe855","tex":"\\sum_{j=0}^{n-1}","aliases":"∑_j=0^n-1 sum","display":"\\sum_{j=0}^{n-1}","kind":"use","excerpt":"Equation [source reference] and the preceding lemma identify r_n with partial sums of a convergent alternating series: r_n=∑_j=0^n-1(-1)^j d_j, L=∑_j=0^∞(-1)^j d_j. For the even subsequence, r_2k+2-r_2k=d_2k-d_2k+1>0. For the odd subsequence, r_2k+3-r_2k+1=-d_2k+1+d_2k+2<0. Their adjacent separation d_2k tends to zero, so they have the same limit. ","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-c579ae579c844347","tex":"\\sum_{j=0}^{\\infty}","aliases":"∑_j=0^∞ sum infty","display":"\\sum_{j=0}^{\\infty}","kind":"use","excerpt":"[Certified Euler enclosure] Define E_N=∑_j=0^NfracΔ^j a_02^j+1, N≥0. Then G=∑_j=0^∞fracΔ^j a_02^j+1, E_N<G<E_N+2^-N-1. More precisely, G-E_N =∫_0^1frac-log x1+x^2 (frac1-x^22)^N+1 dd x.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-89330e1a143912f2","tex":"\\sum_{j\\geq0}","aliases":"∑_j≥0 sum geq","display":"\\sum_{j\\geq0}","kind":"use","excerpt":"The Euler interval uses a_k=(2k+1)^-2 and the positive difference convention Δ a_k=a_k-a_k+1. The identity Δ^ja_0=∫_0^1(1-x^2)^j(-log x) dd x gives 0≤Δ^ja_0≤1. The positive Euler series for G is ∑_j≥02^-j-1Δ^ja_0. Keeping the first 1001 terms therefore gives a lower endpoint and an upper endpoint obtained by adding 2^-1001. All retained terms are r","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4344a265966b0807","tex":"\\sum_{k=0}^{j}","aliases":"∑_k=0^j sum","display":"\\sum_{k=0}^{j}","kind":"use","excerpt":"Let a_k=frac1(2k+1)^2, k≥0, and define the positive difference convention Δ a_k=a_k-a_k+1. The sign convention is fixed here because the other common convention would change every odd difference. Iteration gives Δ^j a_0=∑_k=0^j(-1)^kbinom jkfrac1(2k+1)^2. This is an exact rational number for every j.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-7796ee229bc8c35a","tex":"\\sum_{k=0}^{N-1}","aliases":"∑_k=0^N-1 sum","display":"\\sum_{k=0}^{N-1}","kind":"use","excerpt":"Write s_N=∑_k=0^N-1frac(-1)^k(2k+1)^2, N≥0, where s_0=0 is the empty sum. Thus N counts retained terms, as it does in the visual guide. Each s_N is an exact rational number. The terms decrease strictly in magnitude to zero. Grouping successive terms shows that the even partial sums increase and the odd partial sums decrease. These two sequences app","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a334e226daa86a9f","tex":"\\sum_{k\\geq0}","aliases":"∑_k≥0 sum geq","display":"\\sum_{k\\geq0}","kind":"use","excerpt":"Work with formal power series, so no convergence issue arises. The central binomial expansion gives (1+4z)^-1/2=∑_k≥0(-1)^kbinom2kkz^k. Every coefficient is an integer. Multiplying by the integer polynomial (1+4z)^h preserves this property. The coefficient of z^j in the product is 4^jbinomh-tfrac12j. More explicitly it is ∑_ℓ=0^min(h,j) binom hℓ 4^","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-a3ec24382fc8477a","tex":"\\sum_{n=1}^N","aliases":"∑_n=1^N sum","display":"\\sum_{n=1}^N","kind":"use","excerpt":"For a positive integer N, the scanned domain is D_N={(n,m):1≤ n≤ N, 0≤ m≤4n}. The number of scanned pairs is ∑_n=1^N(4n+1)=2N^2+3N. The upper slope 4 is part of the experiment's definition. The scan supplies no information about pairs above this boundary or with n>N.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-194484736b02f4ed","tex":"\\sum_{\\ell=0}^{\\min(h,j)}","aliases":"∑_ℓ=0^min(h,j) sum ell min","display":"\\sum_{\\ell=0}^{\\min(h,j)}","kind":"use","excerpt":"Work with formal power series, so no convergence issue arises. The central binomial expansion gives (1+4z)^-1/2=∑_k≥0(-1)^kbinom2kkz^k. Every coefficient is an integer. Multiplying by the integer polynomial (1+4z)^h preserves this property. The coefficient of z^j in the product is 4^jbinomh-tfrac12j. More explicitly it is ∑_ℓ=0^min(h,j) binom hℓ 4^","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^n+1 dd x dd y =8(-1)^n(u_nG-v_n). These are fixed normalization identities. Their factors should be checked together when translating a differently normalized source.","count":21},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-00c84852220f2f10","tex":"\\int_{(0,1)^2}","aliases":"∫_(0,1)^2 int","display":"\\int_{(0,1)^2}","kind":"use","excerpt":"On the open unit square, set ψ(x,y)=fracxy(1-x)(1-y)1-xy, ddμ=fracdd x dd ysqrtx(1-y)(1-xy). The factor 1-xy is outside the square root. Moving that factor inside, as in fracdd x dd ysqrtx(1-y)(1-xy), would represent a different integral. With the measure in [source reference], J_n=∫_(0,1)^2ψ^n ddμ, widetilde J_n,m=∫_(0,1)^2ψ^n(1-16ψ)^m ddμ. The se","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-376e76a3a6c4b915","tex":"\\int_0^1","aliases":"∫_0^1 int","display":"\\int_0^1","kind":"use","excerpt":"widetilde J_n,m · =∑_i=0^m(-16)^ibinom miJ_n+i =16^-nF_n,m. The corrected unfiltered integral is J_n=∫_0^1 ∫_0^1 fracx^n-1/2(1-x)^ny^n(1-y)^n-1/2 (1-xy)^n+1 dd x dd y =8(-1)^n(u_nG-v_n). These are fixed normalization identities. Their factors should be checked together when translating a differently normalized source.","count":15},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-dad46397774b52e2","tex":"\\int_0^{\\pi/2}","aliases":"∫_0^π/2 int pi","display":"\\int_0^{\\pi/2}","kind":"use","excerpt":"The initial integral supplies a short diagnostic. Substitute x=t^2 and y=1-s^2. Then J_0 · =4∫_0^1 ∫_0^1fracdd s dd t1-t^2+t^2s^2 · =4∫_0^1fracarcsin ttsqrt1-t^2 dd t =4∫_0^π/2θcscθ ddθ. Since an antiderivative of cscθ is logtan(θ/2), integration by parts yields J_0=-4∫_0^π/2logtan(θ/2) ddθ =-8∫_0^π/4logtan z dd z=8G. Both boundary terms vanish. At","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-473a58b2705741b7","tex":"\\int_0^{\\pi/4}","aliases":"∫_0^π/4 int pi","display":"\\int_0^{\\pi/4}","kind":"use","excerpt":"[Arctangent and tangent forms] The following equalities hold: G=∫_0^1fracarctan xx dd x =-∫_0^π/4log(tanθ) ddθ. The integrand arctan x/x is assigned its continuous value 1 at x=0.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-ff862dd8e88aa975","tex":"\\int_0^{\\infty}","aliases":"∫_0^∞ int infty","display":"\\int_0^{\\infty}","kind":"use","excerpt":"Set x=e^-t. Then -log x=t, dd x=-e^-tdd t, and the integral becomes ∫_0^∞t e^-(a+1)t dd t. Integration by parts gives (a+1)^-2. The boundary term at infinity is zero because an exponential with positive decay rate dominates t. The boundary term at zero is also zero.","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-d1c0650ac412fa7d","tex":"\\int_{[0,1]^2}","aliases":"∫_[0,1]^2 int","display":"\\int_{[0,1]^2}","kind":"use","excerpt":"Write ϕ=frac1+sqrt52, t=ϕ^-5, R=16ϕ^5=88+40sqrt5. The unfiltered integral has the representation J_k=∫_[0,1]^2ψ(x,y)^k ddμ(x,y), ψ=fracxy(1-x)(1-y)1-xy, ddμ=fracdd x dd ysqrtx(1-y)(1-xy). The measure has finite total mass J_0=8G. The factor 1-xy in its denominator lies outside the square root. Moving it inside changes the integral and invalidates t","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-75cecf8e5ef8fb2d","tex":"\\int_\\delta^1","aliases":"∫_δ^1 int delta","display":"\\int_\\delta^1","kind":"use","excerpt":"For 0<δ<1, integration by parts gives ∫_δ^1fracarctan xx dd x =-arctan(δ)logδ -∫_δ^1fraclog x1+x^2 dd x. The boundary expression tends to zero as δdownarrow0, since 0≤arctanδ≤δ and δabslogδ→0. The first equality follows from Proposition~[source reference].","count":1},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"The familiar decimal display begins G≈0.91596559417721901505460351493238411077. The approximation sign matters. A decimal is a coordinate for seeing the number, while a rational enclosure is evidence for a particular numerical assertion. A proof of irrationality requires a different assertion again.","count":3},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"1 · 0=-frac138. Multiplicativity of determinants now gives det F_N+1 =-frac138∏_n=1^N(-β_n)≠0. Thus adjacent values of the two solutions remain independent at every index. Each transfer reverses the determinant sign. The closed evaluation of this product is the Wronskian theorem in Chapter~[source reference], where the factors telescope and yield e","count":14},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"1 · 0. X_n+1(w)=M_nX_n(w). Hence X_N+1(w)=M_NM_N-1⋯ M_1X_1(w) (N≥1). The order of multiplication follows the temporal order: M_1 acts first. This is a product of changing matrices, not a power of one fixed matrix.","count":2},{"work":"catalan-constant-reader","url":"notation/catalan-constant-reader.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"The leading terms of the coefficient polynomials are a_n=320n^6+O(n^5), q(n)=3520n^6+O(n^5), c_n=320n^6+O(n^5). Consequently α_n→11, β_n→1, and M_n⟶ M_∞=11 · 1 1 · 0. The characteristic polynomial of this limiting matrix is λ^2-11λ-1. Its two roots are λ_+=frac11+5sqrt52=ϕ^5, λ_-=frac11-5sqrt52=-ϕ^-5. Here ϕ=(1+sqrt5)/2. The identity ϕ^5=5ϕ+3 follo","count":8},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"notation-table","excerpt":"ρ=β+iγ · A nontrivial zero, counted with its multiplicity.","count":218},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"ρ=β+iγ · A nontrivial zero, counted with its multiplicity.","count":308},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"notation-table","excerpt":"c_K, A_Q · The signed coefficient and its full absolute majorant.","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"notation-table","excerpt":"g_t,a · A Gaussian of norm one in the physical measure du.","count":56},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9ecba2f3c7ea58ca","tex":"\\mathcal A_Q","aliases":"A_Q mathcal","display":"\\mathcal A_Q","kind":"notation-table","excerpt":"c_K, A_Q · The signed coefficient and its full absolute majorant.","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"notation-table","excerpt":"c_K, A_Q · The signed coefficient and its full absolute majorant.","count":34},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1e62e3524995dc9d","tex":"c_K","aliases":"c_K","display":"c_K","kind":"notation-table","excerpt":"c_K, A_Q · The signed coefficient and its full absolute majorant.","count":10},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"notation-table","excerpt":"g_t,a · A Gaussian of norm one in the physical measure du.","count":47},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"notation-table","excerpt":"g_t,a · A Gaussian of norm one in the physical measure du.","count":16},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-234d4e76b0cbe63e","tex":"g_{t,a}","aliases":"g_t,a","display":"g_{t,a}","kind":"notation-table","excerpt":"g_t,a · A Gaussian of norm one in the physical measure du.","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"notation-table","excerpt":"ρ=β+iγ · A nontrivial zero, counted with its multiplicity.","count":23},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"notation-table","excerpt":"widetilde E_J · Squared packet energy on one unit translation band and one fixed heat strip.","count":25},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","excerpt":"K,M · Integer sequence index and its natural scale M=sqrt K.","count":42},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","excerpt":"L · The operator 1/4-∂_u^2 removing the explicit pole channel.","count":13},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"notation-table","excerpt":"K,M · Integer sequence index and its natural scale M=sqrt K.","count":65},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"Q,d · A squarefree product of primes and its number of prime coordinates.","count":68},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"notation-table","excerpt":"λ,R · Squared coherent displacement and Hermite cutoff. The cutoff R in this section is separate from the fixed tail multiplier in the barrier.","count":56},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"notation-table","excerpt":"t,u · Positive heat time and translation, respectively.","count":111},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"notation-table","excerpt":"t,u · Positive heat time and translation, respectively.","count":81},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"notation-table","excerpt":"Δ_Qw_K · The alternating divisor cube acting on the positive weight.","count":48},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-49f567670afb7094","tex":"w_K","aliases":"w_K","display":"w_K","kind":"notation-table","excerpt":"Δ_Qw_K · The alternating divisor cube acting on the positive weight.","count":9},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-20f230bba299446b","tex":"\\widetilde E","aliases":"widetilde E widetilde","display":"\\widetilde E","kind":"notation-table","excerpt":"widetilde E_J · Squared packet energy on one unit translation band and one fixed heat strip.","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-bb08e0c7326341ae","tex":"\\widetilde E_J","aliases":"widetilde E_J widetilde","display":"\\widetilde E_J","kind":"notation-table","excerpt":"widetilde E_J · Squared packet energy on one unit translation band and one fixed heat strip.","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"notation-table","excerpt":"z_ρ · The centered coordinate ρ-1/2.","count":45},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-108c3111700b637c","tex":"z_\\rho","aliases":"z_ρ rho","display":"z_\\rho","kind":"notation-table","excerpt":"z_ρ · The centered coordinate ρ-1/2.","count":9},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"notation-table","excerpt":"ρ=β+iγ · A nontrivial zero, counted with its multiplicity.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"notation-table","excerpt":"ρ=β+iγ · A nontrivial zero, counted with its multiplicity.","count":17},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"notation-table","excerpt":"Δ_Qw_K · The alternating divisor cube acting on the positive weight.","count":12},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9e83e11e41159cf8","tex":"\\Delta_Q","aliases":"Δ_Q Delta","display":"\\Delta_Q","kind":"notation-table","excerpt":"Δ_Qw_K · The alternating divisor cube acting on the positive weight.","count":9},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"notation-table","excerpt":"Θ_t · The Gaussian sum over centered zero modes.","count":18},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0d7cc092bd519f22","tex":"\\Theta_t","aliases":"Θ_t Theta","display":"\\Theta_t","kind":"notation-table","excerpt":"Θ_t · The Gaussian sum over centered zero modes.","count":18},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"notation-table","excerpt":"λ,R · Squared coherent displacement and Hermite cutoff. The cutoff R in this section is separate from the fixed tail multiplier in the barrier.","count":18},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"notation-table","excerpt":"ρ=β+iγ · A nontrivial zero, counted with its multiplicity.","count":45},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"notation-table","excerpt":"L · The operator 1/4-∂_u^2 removing the explicit pole channel.","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-df6c8edf99bfe00c","tex":"\\partial_u^2","aliases":"∂_u^2 partial","display":"\\partial_u^2","kind":"notation-table","excerpt":"L · The operator 1/4-∂_u^2 removing the explicit pole channel.","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"n/2, · n even. For a positive odd integer n, let a(n)=ν_2(3n+1), A(n)=frac3n+12^a(n). Here ν_2(m) is the largest exponent a for which 2^a divides the positive integer m. Thus a(n)≥1, and A(n) is positive and odd.","count":28},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0bdb6564f1e0d191","tex":"a_{k-1}","aliases":"a_k-1","display":"a_{k-1}","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":58},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":8},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d8830a827cebf06e","tex":"C_{j+1}","aliases":"C_j+1","display":"C_{j+1}","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":17},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-84ef01785db55e37","tex":"\\operatorname{Col}","aliases":"Col operatorname","display":"\\operatorname{Col}","kind":"definition-passage","excerpt":"[Classical, shortcut, and odd maps] For a positive integer n, define the classical map and its shortcut by Col(n)= 3n+1, · n odd, n/2, · n even, T(n)= (3n+1)/2, · n odd, n/2, · n even. For a positive odd integer n, let a(n)=ν_2(3n+1), A(n)=frac3n+12^a(n). Here ν_2(m) is the largest exponent a for which 2^a divides the positive integer m. Thus a(n)≥","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"definition-passage","excerpt":"[Coefficient ratio and deficit] Define ρ_r=frac2^S_r3^r, d_r=lfloor rlog_2 3rfloor-S_r. Permanent coefficient noncrossing means ρ_r≤1 for every r≥0. Under that hypothesis d_r is a nonnegative integer.","count":9},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Pole annihilated packet] Let L=1/4-∂_u^2, and define widetildeΘ_t=LΘ_t =∑_ρ∈ Zρ(1-ρ) e^tz_ρ^2+uz_ρ.","count":54},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-12f835cfd2477eec","tex":"e^{tz_\\rho^2+uz_\\rho}","aliases":"e^tz_ρ^2+uz_ρ rho rho","display":"e^{tz_\\rho^2+uz_\\rho}","kind":"definition-passage","excerpt":"[Pole annihilated packet] Let L=1/4-∂_u^2, and define widetildeΘ_t=LΘ_t =∑_ρ∈ Zρ(1-ρ) e^tz_ρ^2+uz_ρ.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"definition-passage","excerpt":"[Zero heat packet] For a real heat parameter t>0 and a complex translation u, define Θ_t(u)=∑_ρ∈ Z exp(tz_ρ^2+uz_ρ).","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":80},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":76},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"definition-passage","excerpt":"[Coefficient ratio and deficit] Define ρ_r=frac2^S_r3^r, d_r=lfloor rlog_2 3rfloor-S_r. Permanent coefficient noncrossing means ρ_r≤1 for every r≥0. Under that hypothesis d_r is a nonnegative integer.","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"definition-passage","excerpt":"[Coefficient ratio and deficit] Define ρ_r=frac2^S_r3^r, d_r=lfloor rlog_2 3rfloor-S_r. Permanent coefficient noncrossing means ρ_r≤1 for every r≥0. Under that hypothesis d_r is a nonnegative integer.","count":41},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"definition-passage","excerpt":"[Coefficient ratio and deficit] Define ρ_r=frac2^S_r3^r, d_r=lfloor rlog_2 3rfloor-S_r. Permanent coefficient noncrossing means ρ_r≤1 for every r≥0. Under that hypothesis d_r is a nonnegative integer.","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"n/2, · n even. For a positive odd integer n, let a(n)=ν_2(3n+1), A(n)=frac3n+12^a(n). Here ν_2(m) is the largest exponent a for which 2^a divides the positive integer m. Thus a(n)≥1, and A(n) is positive and odd.","count":69},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Classical, shortcut, and odd maps] For a positive integer n, define the classical map and its shortcut by Col(n)= 3n+1, · n odd,","count":141},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Coefficient ratio and deficit] Define ρ_r=frac2^S_r3^r, d_r=lfloor rlog_2 3rfloor-S_r. Permanent coefficient noncrossing means ρ_r≤1 for every r≥0. Under that hypothesis d_r is a nonnegative integer.","count":101},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"definition-passage","excerpt":"[Coefficient ratio and deficit] Define ρ_r=frac2^S_r3^r, d_r=lfloor rlog_2 3rfloor-S_r. Permanent coefficient noncrossing means ρ_r≤1 for every r≥0. Under that hypothesis d_r is a nonnegative integer.","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":82},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-bbf4aa0bb7699587","tex":"S_j","aliases":"S_j","display":"S_j","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":11},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-deef7bca5ee0eda2","tex":"S_k","aliases":"S_k","display":"S_k","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b37e9083bef6f614","tex":"S_r","aliases":"S_r","display":"S_r","kind":"definition-passage","excerpt":"[Coefficient ratio and deficit] Define ρ_r=frac2^S_r3^r, d_r=lfloor rlog_2 3rfloor-S_r. Permanent coefficient noncrossing means ρ_r≤1 for every r≥0. Under that hypothesis d_r is a nonnegative integer.","count":14},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Classical, shortcut, and odd maps] For a positive integer n, define the classical map and its shortcut by Col(n)= 3n+1, · n odd, n/2, · n even, T(n)= (3n+1)/2, · n odd, n/2, · n even. For a positive odd integer n, let a(n)=ν_2(3n+1), A(n)=frac3n+12^a(n). Here ν_2(m) is the largest exponent a for which 2^a divides the positive integer m. Thus a(n)≥","count":12},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8e511a803e0a75b4","tex":"\\widetilde\\Theta","aliases":"widetildeΘ widetilde Theta","display":"\\widetilde\\Theta","kind":"definition-passage","excerpt":"[Pole annihilated packet] Let L=1/4-∂_u^2, and define widetildeΘ_t=LΘ_t =∑_ρ∈ Zρ(1-ρ) e^tz_ρ^2+uz_ρ.","count":8},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8d12abd95d875a93","tex":"\\widetilde\\Theta_t","aliases":"widetildeΘ_t widetilde Theta","display":"\\widetilde\\Theta_t","kind":"definition-passage","excerpt":"[Pole annihilated packet] Let L=1/4-∂_u^2, and define widetildeΘ_t=LΘ_t =∑_ρ∈ Zρ(1-ρ) e^tz_ρ^2+uz_ρ.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c656dc90ffbe53cd","tex":"\\mathcal Z","aliases":"Z mathcal","display":"\\mathcal Z","kind":"definition-passage","excerpt":"[Zero heat packet] For a real heat parameter t>0 and a complex translation u, define Θ_t(u)=∑_ρ∈ Z exp(tz_ρ^2+uz_ρ).","count":8},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6ddd635b5b5139b3","tex":"z_\\rho^2","aliases":"z_ρ^2 rho","display":"z_\\rho^2","kind":"definition-passage","excerpt":"[Zero heat packet] For a real heat parameter t>0 and a complex translation u, define Θ_t(u)=∑_ρ∈ Z exp(tz_ρ^2+uz_ρ).","count":9},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"definition-passage","excerpt":"n/2, · n even. For a positive odd integer n, let a(n)=ν_2(3n+1), A(n)=frac3n+12^a(n). Here ν_2(m) is the largest exponent a for which 2^a divides the positive integer m. Thus a(n)≥1, and A(n) is positive and odd.","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c381b1e9f2cf6895","tex":"\\nu_2","aliases":"ν_2 nu","display":"\\nu_2","kind":"definition-passage","excerpt":"n/2, · n even. For a positive odd integer n, let a(n)=ν_2(3n+1), A(n)=frac3n+12^a(n). Here ν_2(m) is the largest exponent a for which 2^a divides the positive integer m. Thus a(n)≥1, and A(n) is positive and odd.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7e425697f7c81595","tex":"\\rho_r","aliases":"ρ_r rho","display":"\\rho_r","kind":"definition-passage","excerpt":"[Coefficient ratio and deficit] Define ρ_r=frac2^S_r3^r, d_r=lfloor rlog_2 3rfloor-S_r. Permanent coefficient noncrossing means ρ_r≤1 for every r≥0. Under that hypothesis d_r is a nonnegative integer.","count":13},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Zero heat packet] For a real heat parameter t>0 and a complex translation u, define Θ_t(u)=∑_ρ∈ Z exp(tz_ρ^2+uz_ρ).","count":13},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":42},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8524806a0ba8de34","tex":"\\sum_{i<j}","aliases":"∑_i<j sum","display":"\\sum_{i<j}","kind":"definition-passage","excerpt":"[Valuation word and offset] A valuation word is a finite sequence w=(a_0,…,a_k-1) of positive integers. Define its partial depths and offsets by S_0=0, S_j=∑_i<ja_i, C_0=0, C_j+1=3C_j+2^S_j. Its total depth is S=S_k and its terminal offset is C=C_k. The word is realized by n if these are the first k valuations of the actual odd orbit from n.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1ec22563555b4bc7","tex":"\\sum_{\\rho\\in\\mathcal Z}","aliases":"∑_ρ∈ Z sum rho in mathcal","display":"\\sum_{\\rho\\in\\mathcal Z}","kind":"definition-passage","excerpt":"[Zero heat packet] For a real heat parameter t>0 and a complex translation u, define Θ_t(u)=∑_ρ∈ Z exp(tz_ρ^2+uz_ρ).","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"[Coefficient ratio and deficit] Define ρ_r=frac2^S_r3^r, d_r=lfloor rlog_2 3rfloor-S_r. Permanent coefficient noncrossing means ρ_r≤1 for every r≥0. Under that hypothesis d_r is a nonnegative integer.","count":75},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"n/2, · n even. For a positive odd integer n, let a(n)=ν_2(3n+1), A(n)=frac3n+12^a(n). Here ν_2(m) is the largest exponent a for which 2^a divides the positive integer m. Thus a(n)≥1, and A(n) is positive and odd.","count":79},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4b82e0794ad5cb35","tex":"\\mathfrak A","aliases":"A mathfrak","display":"\\mathfrak A","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e7409f5b0f01f102","tex":"\\mathfrak A_t","aliases":"A_t mathfrak","display":"\\mathfrak A_t","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-06e1595497d8f5f2","tex":"c_w","aliases":"c_w","display":"c_w","kind":"introduction-passage","excerpt":"Composition and the descent budget. A realized word acts by the affine transformation F_w(n)=λ_w n+c_w, λ_w=frac3^k2^S, c_w=fracC_k2^S. The transfer law is evaluated only on the integer cylinder of the word. As a rational affine expression it is defined elsewhere, but there it need not describe an actual Collatz segment.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8537f550e0a5ab31","tex":"e^{-(u+\\log n)^2/(4t)}","aliases":"e^-(u+log n)^2/(4t) log","display":"e^{-(u+\\log n)^2/(4t)}","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-852496ffeb7a3eae","tex":"e^{-(u-\\log n)^2/(4t)}","aliases":"e^-(u-log n)^2/(4t) log","display":"e^{-(u-\\log n)^2/(4t)}","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-fc720f40cfc90989","tex":"e^{-1/r^2}","aliases":"e^-1/r^2","display":"e^{-1/r^2}","kind":"introduction-passage","excerpt":"The active scale follows from the weight. For a real variable x>1, write y=x^-2. Then w_K(x)=y(1-y)^K, and differentiation in y puts its maximum at y=1/(K+1). The maximum of the continuously extended weight is therefore at x=sqrtK+1, close to M=sqrt K. For fixed r>0, as M tends to infinity through square roots of integers, M^2w_M^2(rM) =r^-2(1-frac","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-480611e2d10a0373","tex":"e^{-tv^2}","aliases":"e^-tv^2","display":"e^{-tv^2}","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-49550fdb9f0a1cd8","tex":"E_\\ell","aliases":"E_ℓ ell","display":"E_\\ell","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"Composition and the descent budget. A realized word acts by the affine transformation F_w(n)=λ_w n+c_w, λ_w=frac3^k2^S, c_w=fracC_k2^S. The transfer law is evaluated only on the integer cylinder of the word. As a rational affine expression it is defined elsewhere, but there it need not describe an actual Collatz segment.","count":9},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"introduction-passage","excerpt":"Composition and the descent budget. A realized word acts by the affine transformation F_w(n)=λ_w n+c_w, λ_w=frac3^k2^S, c_w=fracC_k2^S. The transfer law is evaluated only on the integer cylinder of the word. As a rational affine expression it is defined elsewhere, but there it need not describe an actual Collatz segment.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":25},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":17},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"introduction-passage","excerpt":"The active scale follows from the weight. For a real variable x>1, write y=x^-2. Then w_K(x)=y(1-y)^K, and differentiation in y puts its maximum at y=1/(K+1). The maximum of the continuously extended weight is therefore at x=sqrtK+1, close to M=sqrt K. For fixed r>0, as M tends to infinity through square roots of integers, M^2w_M^2(rM) =r^-2(1-frac","count":13},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":23},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-cc3bc8725e9330b8","tex":"p^{-\\rho}","aliases":"p^-ρ rho","display":"p^{-\\rho}","kind":"introduction-passage","excerpt":"First, an exact prime high pass followed by exact coprime reconstruction preserves a power mode. If a mode has local multiplier 1-p^-ρ, reconstruction contributes the reciprocal multiplier whenever it is defined. Their product is one. The high pass alone can look smaller, but exact reconstruction restores the removed factor. The historical Lean mod","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b7f360bb2799a8d5","tex":"p^k","aliases":"p^k","display":"p^k","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a38be2450a4b576f","tex":"\\mathfrak P_t","aliases":"P_t mathfrak","display":"\\mathfrak P_t","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":8},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-072e7fe97b551a6a","tex":"r^{-2}","aliases":"r^-2","display":"r^{-2}","kind":"introduction-passage","excerpt":"The active scale follows from the weight. For a real variable x>1, write y=x^-2. Then w_K(x)=y(1-y)^K, and differentiation in y puts its maximum at y=1/(K+1). The maximum of the continuously extended weight is therefore at x=sqrtK+1, close to M=sqrt K. For fixed r>0, as M tends to infinity through square roots of integers, M^2w_M^2(rM) =r^-2(1-frac","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"introduction-passage","excerpt":"The active scale follows from the weight. For a real variable x>1, write y=x^-2. Then w_K(x)=y(1-y)^K, and differentiation in y puts its maximum at y=1/(K+1). The maximum of the continuously extended weight is therefore at x=sqrtK+1, close to M=sqrt K. For fixed r>0, as M tends to infinity through square roots of integers, M^2w_M^2(rM) =r^-2(1-frac","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ff81ba5e23d549c7","tex":"\\Re","aliases":"Re","display":"\\Re","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":15},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":58},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":23},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-675fd54b543da3f0","tex":"w_{M^2}","aliases":"w_M^2","display":"w_{M^2}","kind":"introduction-passage","excerpt":"The active scale follows from the weight. For a real variable x>1, write y=x^-2. Then w_K(x)=y(1-y)^K, and differentiation in y puts its maximum at y=1/(K+1). The maximum of the continuously extended weight is therefore at x=sqrtK+1, close to M=sqrt K. For fixed r>0, as M tends to infinity through square roots of integers, M^2w_M^2(rM) =r^-2(1-frac","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-404fc764d07fb6f9","tex":"W_\\ell","aliases":"W_ℓ ell","display":"W_\\ell","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"The active scale follows from the weight. For a real variable x>1, write y=x^-2. Then w_K(x)=y(1-y)^K, and differentiation in y puts its maximum at y=1/(K+1). The maximum of the continuously extended weight is therefore at x=sqrtK+1, close to M=sqrt K. For fixed r>0, as M tends to infinity through square roots of integers, M^2w_M^2(rM) =r^-2(1-frac","count":51},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d1d16f551dc7199c","tex":"x^{-2}","aliases":"x^-2","display":"x^{-2}","kind":"introduction-passage","excerpt":"The active scale follows from the weight. For a real variable x>1, write y=x^-2. Then w_K(x)=y(1-y)^K, and differentiation in y puts its maximum at y=1/(K+1). The maximum of the continuously extended weight is therefore at x=sqrtK+1, close to M=sqrt K. For fixed r>0, as M tends to infinity through square roots of integers, M^2w_M^2(rM) =r^-2(1-frac","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"The active scale follows from the weight. For a real variable x>1, write y=x^-2. Then w_K(x)=y(1-y)^K, and differentiation in y puts its maximum at y=1/(K+1). The maximum of the continuously extended weight is therefore at x=sqrtK+1, close to M=sqrt K. For fixed r>0, as M tends to infinity through square roots of integers, M^2w_M^2(rM) =r^-2(1-frac","count":11},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9c815a51ca56e21f","tex":"\\Gamma'","aliases":"Γ' Gamma","display":"\\Gamma'","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":30},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1ca916b982c03d00","tex":"\\varepsilon_\\ell","aliases":"ε_ℓ varepsilon ell","display":"\\varepsilon_\\ell","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":8},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":12},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ec88137f742d4f35","tex":"\\lambda_w","aliases":"λ_w lambda","display":"\\lambda_w","kind":"introduction-passage","excerpt":"Composition and the descent budget. A realized word acts by the affine transformation F_w(n)=λ_w n+c_w, λ_w=frac3^k2^S, c_w=fracC_k2^S. The transfer law is evaluated only on the integer cylinder of the word. As a rational affine expression it is defined elsewhere, but there it need not describe an actual Collatz segment.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":17},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-91be6a5a77359a71","tex":"\\Psi_\\infty","aliases":"Ψ_∞ Psi infty","display":"\\Psi_\\infty","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":43},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-069c58c03dba2dbd","tex":"\\sum_{n\\ge2}","aliases":"∑_n≥2 sum ge","display":"\\sum_{n\\ge2}","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2b7189a944630b62","tex":"\\sum_{s=5\\ell}^{h}","aliases":"∑_s=5ℓ^h sum ell","display":"\\sum_{s=5\\ell}^{h}","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"introduction-passage","excerpt":"Let Λ(n) denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv,","count":17},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"introduction-passage","excerpt":"There are binoms-15ℓ-1 positive words of total s. If W_ℓ denotes the finite total number of selected words, summing the cylinder counts gives #(E_ℓ∩[1,H])≤ε_ℓ H+W_ℓ, ε_ℓ=frac12∑_s=5ℓ^h binoms-15ℓ-12^-s. This count is arithmetic. It does not assume that orbit valuations are independent random variables.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":14},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-99fa268d54ab5804","tex":"\\int_{\\mathbb R}","aliases":"∫_ R int mathbb","display":"\\int_{\\mathbb R}","kind":"introduction-passage","excerpt":") denote the von Mangoldt function: it is log p when n=p^k for a prime p and an integer k≥1, and it is zero otherwise. Let ψ=Γ'/Γ be the digamma function, and set Ψ_∞(v)=-logπ+ Reψ(frac14+fraciv2) (v∈ R). Define the archimedean term and prime packet by A_t(u) · =frac12π∫_ R e^-tv^2cos(uv)Ψ_∞(v) dv, P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"introduction-passage","excerpt":"The active scale follows from the weight. For a real variable x>1, write y=x^-2. Then w_K(x)=y(1-y)^K, and differentiation in y puts its maximum at y=1/(K+1). The maximum of the continuously extended weight is therefore at x=sqrtK+1, close to M=sqrt K. For fixed r>0, as M tends to infinity through square roots of integers, M^2w_M^2(rM) =r^-2(1-frac","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"The threshold as a fixed point. When G=2^S-3^k>0, the affine transfer has one real fixed point n_* = fracC_kG=fracc_w1-λ_w. For a real input above n_*, the affine expression falls. For an input below n_*, it rises. Equality gives a fixed point of the affine expression. These are statements about that expression before its integer cylinder is impose","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"For example, the word (1,2) has S_2=3 and C_2=5, so 8·31=9·27+5. The corresponding cylinder is n≡11pmod16. Writing n=16t+11 gives A(n)=24t+17, A^2(n)=18t+13. Every member of this progression grows over the two steps, even though the second step falls from 24t+17 to 18t+13. For t=1, the local fall is 41 to 31, while both states remain above 27. Thes","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ea75d7621bfa68aa","tex":"A^7","aliases":"A^7","display":"A^7","kind":"use","excerpt":"· ⟼31104t+8261⟼5832t+1549. The valuations are 1,1,1,1,1,4. The next numerator is 3(5832t+1549)+1=2^3(2187t+581). Therefore A^7(n)=oddPart(2187t+581) ≤2187t+581<4096t+1087. The final valuation may exceed 3 for some parameters. The inequality remains valid because removing further factors of two only reduces the positive quotient. This proof is repre","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-afc7e5c12f805c36","tex":"A^j","aliases":"A^j","display":"A^j","kind":"use","excerpt":"ntheorem[Compressed run exit] Suppose j≥0, q≥1, b≥0, and m is positive odd, with 3^j+1q=2^b m+1. For n=2^j+1q-1, the exact exit is A^j+1(n)=m. If m<n, then j+1 is the exact first strict descent time. Lemma~[source reference] gives A^j(n)=2·3^j q-1. Its next odd numerator is 3(2·3^j q-1)+1=2(3^j+1q-1)=2^b+1m. Oddness of m proves the exact final valu","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-fbf6552309e1f6d4","tex":"A^{j+1}","aliases":"A^j+1","display":"A^{j+1}","kind":"use","excerpt":"[Compressed run exit] Suppose j≥0, q≥1, b≥0, and m is positive odd, with 3^j+1q=2^b m+1. For n=2^j+1q-1, the exact exit is A^j+1(n)=m. If m<n, then j+1 is the exact first strict descent time. Lemma~[source reference] gives A^j(n)=2·3^j q-1. Its next odd numerator is 3(2·3^j q-1)+1=2(3^j+1q-1)=2^b+1m. Oddness of m proves the exact final valuation b+","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-99155510890729d1","tex":"A^r","aliases":"A^r","display":"A^r","kind":"use","excerpt":"[Collatz] For every positive odd integer n, there exists r≥0 such that A^r(n)=1.","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2156f529b7009500","tex":"A^{r+s}","aliases":"A^r+s","display":"A^{r+s}","kind":"use","excerpt":"abeleq:universal for every odd n>1 there exists r≥1 with A^r(n)<n. Convergence to 1 implies descent for each n>1. For the converse, use strong induction on positive odd integers. The case n=1 is immediate. For n>1, choose a descendant m=A^r(n)<n. It is positive and odd, so the induction hypothesis gives A^s(m)=1. Hence A^r+s(n)=1.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9380d36018e1d9c2","tex":"A^s","aliases":"A^s","display":"A^s","kind":"use","excerpt":"for every odd n>1 there exists r≥1 with A^r(n)<n. Convergence to 1 implies descent for each n>1. For the converse, use strong induction on positive odd integers. The case n=1 is immediate. For n>1, choose a descendant m=A^r(n)<n. It is positive and odd, so the induction hypothesis gives A^s(m)=1. Hence A^r+s(n)=1.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-671f8e9722b217bc","tex":"A^t","aliases":"A^t","display":"A^t","kind":"use","excerpt":"[A run of unit valuations] For integers j≥0, q≥1, and 0≤ t≤ j, put n=2^j+1q-1. Then A^t(n)=2^j+1-t3^t q-1. Every one of the first j valuations is exactly 1. Moreover A^t(n)≥ n for 0≤ t≤ j. For t<j, the displayed state has the form 4u-1 with u>0. Thus 3(4u-1)+1=2(6u-1), whose quotient is odd. This proves the next state and valuation by induction. Th","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"One can see the offset being created before writing the general formula. The first step is 2^a_0x_1=3n+1. Multiplying the second step by 2^a_0 gives 2^a_0+a_1x_2=3(3n+1)+2^a_0 =9n+3+2^a_0. The first addition has been multiplied by 3. The second addition has been multiplied by the dyadic depth already removed. After a third step the offset is 9+3·2^","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"r of two. Reducing [source reference] modulo 2^S_j+1 gives 3^k-j(3^j n+C_j)+2^S_jD_j,k≡0pmod2^S_j+1. Every odd multiple of 2^S_j has the same residue modulo 2^S_j+1. Since 3^k-j is invertible, it follows that 3^j n+C_j≡2^S_jpmod2^S_j+1. Thus x_j is integral and odd. It is positive because n>0 and C_j≥0. Finally the recurrence gives 3x_j+1=2^a_jx_j+","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"use","excerpt":"Exact valuation packets. Fix a positive odd seed n, and write x_r=A^r(n). Each step has the exact equation 2^a_rx_r+1=3x_r+1, a_r=a(x_r). The exponent records how much dyadic information is removed at that step. It is determined by the actual state, rather than chosen independently.","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-fc68f71f4cf61369","tex":"a_{r-1}","aliases":"a_r-1","display":"a_{r-1}","kind":"use","excerpt":"[htbp]centering captionExact odd packet coordinates for the first ten steps from 27. rrrrr r · x_r · a_r-1 · S_r · C_r","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-17a6e987887cd940","tex":"\\asymp","aliases":"asymp","display":null,"kind":"use","excerpt":"The cube's shape is also scale dependent. At fixed K, large m makes every factor (1-1/(bm)^2)^K close to one, so its leading coefficient is ∏_pmid Q(1-p^-2). At the active scale masympsqrt K, the different divisors sample different parts of the weight profile. This difference can create local cancellation within a cube. It does not remove the remai","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"[Injectivity] A permanently noncrossing positive odd orbit never repeats a state. Suppose x_u=x_v with v>u. Put k=v-u and B=S_v-S_u. The affine law for this positive length block gives (2^B-3^k)x_u=C_u,v>0. Hence 2^B/3^k>1. Determinism makes the same states and valuations repeat, so ρ_u+hk=ρ_u(2^B/3^k)^h. This exceeds 1 for sufficiently large h, co","count":13},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"One can also retain a weaker certificate when exact final valuation is unnecessary. If the last numerator is known to equal 2^b q with q>0 integral, then the next odd state is at most q. Exact equality requires q odd. A descent proof that only uses an upper bound can accept both parities. Example~[source reference] exploits this distinction. It obt","count":30},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0fcfa06d5284aee8","tex":"b^{-4}","aliases":"b^-4","display":"b^{-4}","kind":"use","excerpt":"The limiting divisor sum is ∑_bmid Qfracμ(b)b^2 =∏_pmid Q(1-p^-2)≥ζ(2)^-1. For 0≤ x≤1, Bernoulli's inequality gives 0≤1-(1-x)^K≤ Kx. With K=M^2, the error in replacing each power by one is at most fracM^2m^2∑_bmid Qb^-4 ≤fracζ(4)R^2. Multiplying the resulting lower bound by m^-2 proves the lemma.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f76a26dbc0d9fe26","tex":"B^2","aliases":"B^2","display":"B^2","kind":"use","excerpt":"It helps to make the normal convergence argument quantitative enough to see which operations it permits. Write z=x+iγ, where |x|<1/2, and u=a+ib. Then Re(tz^2+uz)=t(x^2-γ^2)+ax-bγ. If t∈[t_0,T] and |a|,|b|≤ B, this is at most T/4+B/2-t_0γ^2+B|γ| ≤ T/4+B/2+fracB^22t_0-fract_02γ^2. The last step is the elementary inequality B|γ|≤(t_0/2)γ^2+B^2/(2t_0)","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"The limiting divisor sum is ∑_bmid Qfracμ(b)b^2 =∏_pmid Q(1-p^-2)≥ζ(2)^-1. For 0≤ x≤1, Bernoulli's inequality gives 0≤1-(1-x)^K≤ Kx. With K=M^2, the error in replacing each power by one is at most fracM^2m^2∑_bmid Qb^-4 ≤fracζ(4)R^2. Multiplying the resulting lower bound by m^-2 proves the lemma.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"Conversely, regard u↦widetildeΘ_·(u) as a function taking values in H=L^2([t_0,T]; C). Cauchy Schwarz on each unit interval and the assumed energy bound imply ∫_0^∞ widetildeΘ_·(u) _H e^-au du<∞ (a>0). Its H-valued Laplace transform is holomorphic for Re w>0. Initially for Re w>1/2, it also equals the normally convergent meromorphic series with ter","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"This is the obstruction behind pulling a smaller endpoint backward through an unchanged late prefix. A useful smaller source merger must eventually change the word. For two different transfers ending at the same y, elimination gives m=frac2^S'3^k n+2^S'C-2^S C'2^S3^k'. The remaining task is to force a realizing alternative with 0<m<n. The formula d","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","excerpt":"Suppose a fixed positive width constant C>0 is chosen and the active centers in a shell satisfy |a-c_J|≤ Csqrt J+O(1) and t≥ t_0>0. Put α=C^2/(4t_0). Then λ≤α J+O(sqrt J). For a fixed constant q>α, projection through R_J=lceil qJrceil has norm error bounded by exp(-frac12 I(qmidα)J+o(J)), I(qmidα)=qlog(q/α)-q+α. The coefficient mass of the positive","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"Two ordered words with the same total depth. For w=(1,3), the recurrence gives C_1=1 and C_2=5, with total depth S=4. The affine expression is F_(1,3)(n)=frac9n+516. The exact cylinder requires 9n+5≡16pmod32. The inverse of 9 modulo 32 is 25, so its address is 19 modulo 32. Every positive representative can be written n=32t+19 with an integer t≥0. ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"For example, c_1=frac1ζ(2)-frac1ζ(4), c_2=frac1ζ(2)-frac2ζ(4)+frac1ζ(6). These formulas are exact. At large K, their alternating binomial terms can be individually much larger than their sum, so straightforward floating point evaluation may lose accuracy. A finite expression is not automatically a stable evaluation scheme. For a direct truncation o","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"Two ordered words with the same total depth. For w=(1,3), the recurrence gives C_1=1 and C_2=5, with total depth S=4. The affine expression is F_(1,3)(n)=frac9n+516. The exact cylinder requires 9n+5≡16pmod32. The inverse of 9 modulo 32 is 25, so its address is 19 modulo 32. Every positive representative can be written n=32t+19 with an integer t≥0. ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"For example, c_1=frac1ζ(2)-frac1ζ(4), c_2=frac1ζ(2)-frac2ζ(4)+frac1ζ(6). These formulas are exact. At large K, their alternating binomial terms can be individually much larger than their sum, so straightforward floating point evaluation may lose accuracy. A finite expression is not automatically a stable evaluation scheme. For a direct truncation o","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9ec4ff02b97d1355","tex":"C_{37}","aliases":"C_37","display":"C_{37}","kind":"use","excerpt":"The fresh finite integer audit for this edition finds the first actual descent at accelerated time 37: x_37=23, S_37=59, C_37=1100931843921811423. Thus 2^59·23=3^37·27+1100931843921811423. The same segment has 59 shortcut steps. The historical serious audit separately records a Lean check of T^59(27)=23 [source reference]. The two statements use di","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-cdae72f972852da1","tex":"c_f","aliases":"c_f","display":"c_f","kind":"use","excerpt":"The composition formula also explains why a local fall need not resolve a global obstruction. If an expanding segment has data (λ_g,c_g) and a following falling segment has (λ_f,c_f), the final value is λ_fλ_g n+λ_f c_g+c_f. A comparison with the local peak controls only part of this expression. To prove descent below n, the entire expression must ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-59651fe6df7d2f0f","tex":"c_g","aliases":"c_g","display":"c_g","kind":"use","excerpt":"The composition formula also explains why a local fall need not resolve a global obstruction. If an expanding segment has data (λ_g,c_g) and a following falling segment has (λ_f,c_f), the final value is λ_fλ_g n+λ_f c_g+c_f. A comparison with the local peak controls only part of this expression. To prove descent below n, the entire expression must ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4187260b1a097f96","tex":"c_J","aliases":"c_J","display":"c_J","kind":"use","excerpt":"Suppose a fixed positive width constant C>0 is chosen and the active centers in a shell satisfy |a-c_J|≤ Csqrt J+O(1) and t≥ t_0>0. Put α=C^2/(4t_0). Then λ≤α J+O(sqrt J). For a fixed constant q>α, projection through R_J=lceil qJrceil has norm error bounded by exp(-frac12 I(qmidα)J+o(J)), I(qmidα)=qlog(q/α)-q+α. The coefficient mass of the positive","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-19cddbaf0c2c23b0","tex":"c_{M^2}","aliases":"c_M^2","display":"c_{M^2}","kind":"use","excerpt":"[Báez Duarte criterion] The Riemann hypothesis holds if and only if, for every ε>0, |c_K|=O_ε(K^-3/4+ε). Equivalently, for every ε>0, |c_M^2|=O_ε(M^-3/2+ε), with M restricted to square roots of positive integers.","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-746a0fca63346846","tex":"C_r","aliases":"C_r","display":"C_r","kind":"use","excerpt":"A complete finite example from 27. The seed 27 makes the distinction between the clocks and the floors visible. Its first ten odd accelerated steps are shown in Table~[source reference]. Each row is checked by multiplying the displayed state by 2^S_r and comparing it with 3^r·27+C_r.","count":8},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7dabb4254fbedca3","tex":"C_{r+1}","aliases":"C_r+1","display":"C_{r+1}","kind":"use","excerpt":"endix Complete odd coordinates for the source 27. Table~[source reference] records the entire accelerated orbit through its first arrival at 1. The column a is the valuation used to arrive at the state in that row. The column S is both cumulative dyadic depth and cumulative shortcut time. The column L=S+r is cumulative classical time. The offset is","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6935b9071147e6c4","tex":"C_{u,v}","aliases":"C_u,v","display":"C_{u,v}","kind":"use","excerpt":"[Injectivity] A permanently noncrossing positive odd orbit never repeats a state. Suppose x_u=x_v with v>u. Put k=v-u and B=S_v-S_u. The affine law for this positive length block gives (2^B-3^k)x_u=C_u,v>0. Hence 2^B/3^k>1. Determinism makes the same states and valuations repeat, so ρ_u+hk=ρ_u(2^B/3^k)^h. This exceeds 1 for sufficiently large h, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-31ddf16fe06684fb","tex":"C_\\varepsilon","aliases":"C_ε varepsilon","display":"C_\\varepsilon","kind":"use","excerpt":"For a nonnegative quantity E_J, call its growth subexponential when lim_J→∞fraclog(1+E_J)J=0. Equivalently, for every ε>0, there is a constant C_ε such that E_J≤ C_ε e^ε J for all positive integers J. This is an upper growth condition. It does not assert a positive lower bound.","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"use","excerpt":"P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u-log n)^2/(4t)+ e^-(u+log n)^2/(4t)). The Gaussian makes the prime series absolutely convergent. The completed explicit formula gives Θ_t(u)=2e^t/4cosh(u/2)+ A_t(u)- P_t(u). We use the formula in the recovered normalization [source reference]. With Fourier convention g(x)=(2π)^-1∫ h(v)e^-ivx dv, the","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"Conversely, define the rational candidates x_j=(3^j n+C_j)/2^S_j. We show that every candidate is a positive odd integer. The terminal congruence supplies this for j=k. For 0≤ j<k, split the offset sum at j: C_k=3^k-jC_j+2^S_jD_j,k. The integer D_j,k is odd. Its first summand is 3^k-1-j, while every later summand has an additional factor of two. Re","count":34},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5b5366e7d0a1d14f","tex":"d^r","aliases":"d^r","display":"d^r","kind":"use","excerpt":"belrh:gaussian g_t,a(u)=(2π t)^-1/4e^-(u-a)^2/(4t). It has norm one in L^2( R,du). For a chosen center c, let ψ_r,t,c, r=0,1,2,…, be the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal pr","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-740dee0f1d610e65","tex":"D_{j,k}","aliases":"D_j,k","display":"D_{j,k}","kind":"use","excerpt":"Conversely, define the rational candidates x_j=(3^j n+C_j)/2^S_j. We show that every candidate is a positive odd integer. The terminal congruence supplies this for j=k. For 0≤ j<k, split the offset sum at j: C_k=3^k-jC_j+2^S_jD_j,k. The integer D_j,k is odd. Its first summand is 3^k-1-j, while every later summand has an additional factor of two. Re","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4044c8a85cc77c30","tex":"D_R","aliases":"D_R","display":"D_R","kind":"use","excerpt":"What the proof does not transfer to an unbounded strip. If the deficit bound is replaced by a varying number D_R, the height multiplier 2^D_R+1 varies with the same orbit window. A smaller cylinder density may be offset by a much larger permitted height. The finite family correction may also grow if its block length is allowed to vary. The precedin","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-71f681a5e79f3ca7","tex":"\\mathbb E","aliases":"E mathbb","display":"\\mathbb E","kind":"use","excerpt":"The exponent in [source reference] comes from a standard elementary exponential bound for a Poisson variable X of mean λ. For h>0, Pr(X≥ R)≤ e^-hR E(e^hX) =exp(-hR+λ(e^h-1)). When R>λ>0, choose e^h=R/λ. For λ=0, every positive tail is zero. The positive mean case gives the rate Rlog(R/λ)-R+λ. Passing to the squared norm tail and then taking a squar","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f6040776716cb145","tex":"e^{-(a-b)^2/(8t)}","aliases":"e^-(a-b)^2/(8t)","display":"e^{-(a-b)^2/(8t)}","kind":"use","excerpt":"The word physical refers to the measure du in which the packet is actually evaluated. It does not mean that a separately chosen coefficient norm can be substituted. Completing the square gives the exact overlap ⟨ g_t,a,g_t,b⟩ =e^-(a-b)^2/(8t). For two translates, the Gram matrix has diagonal entries one and off diagonal entry κ=e^-(a-b)^2/(8t). Its","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-968659d0ecad18a5","tex":"e^{-(\\log n)^2/(4t)}","aliases":"e^-(log n)^2/(4t) log","display":"e^{-(\\log n)^2/(4t)}","kind":"use","excerpt":"At u=0, for example, the prime packet is P_t(0)=frac1sqrtπ t ∑_n≥2fracΛ(n)sqrt n e^-(log n)^2/(4t). Every summand is nonnegative. The complete formula is nonetheless a difference involving this positive quantity. At large positive u, the Gaussian centered at log n selects prime powers with logarithms near u, so the active arithmetic scale is around","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-29a7f8c868f3e2d4","tex":"e^{-(u-a)^2/(4t)}","aliases":"e^-(u-a)^2/(4t)","display":"e^{-(u-a)^2/(4t)}","kind":"use","excerpt":"For a translated Gaussian G_t,a(u)=e^-(u-a)^2/(4t), direct differentiation gives LG_t,a(u)= (frac14+frac12t-frac(u-a)^24t^2) G_t,a(u). The annihilated prime packet therefore carries a polynomial factor with both signs. Positivity of the original Gaussian sum is no longer positivity of every transformed term. On the zero side the same operator is si","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7f8190ab886c3366","tex":"e^{-(u-c)^2/(4s)}","aliases":"e^-(u-c)^2/(4s)","display":"e^{-(u-c)^2/(4s)}","kind":"use","excerpt":"To verify [source reference], the scalar Gaussian integral is ∫_ R(2π s)^-1/4 e^-(u-c)^2/(4s)e^uz du =2sqrtπ s(2π s)^-1/4e^cz+sz^2. It holds for every complex z, either by completing the square first for real z and continuing analytically, or by the entire Gaussian transform. Multiplying by the mode coefficient at heat time t replaces t by t+s. The","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-275105af37f3095b","tex":"e^{-(u-c)^2/(4t)}","aliases":"e^-(u-c)^2/(4t)","display":"e^{-(u-c)^2/(4t)}","kind":"use","excerpt":"For t>0 and a real center a, define the normalized Gaussian g_t,a(u)=(2π t)^-1/4e^-(u-a)^2/(4t). It has norm one in L^2( R,du). For a chosen center c, let ψ_r,t,c, r=0,1,2,…, be the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-31b68e89e9bd7557","tex":"e^{-(x+u)^2/(4t)}","aliases":"e^-(x+u)^2/(4t)","display":"e^{-(x+u)^2/(4t)}","kind":"use","excerpt":"ies absolutely convergent. The completed explicit formula gives Θ_t(u)=2e^t/4cosh(u/2)+ A_t(u)- P_t(u). We use the formula in the recovered normalization [source reference]. With Fourier convention g(x)=(2π)^-1∫ h(v)e^-ivx dv, the even test h(v)=e^-tv^2cos(uv) has transform g(x)=frac14sqrtπ t (e^-(x-u)^2/(4t)+e^-(x+u)^2/(4t)). The zero symmetry ide","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1f5cc47ee8a1907f","tex":"e^{-(x-u)^2/(4t)}","aliases":"e^-(x-u)^2/(4t)","display":"e^{-(x-u)^2/(4t)}","kind":"use","excerpt":"ies absolutely convergent. The completed explicit formula gives Θ_t(u)=2e^t/4cosh(u/2)+ A_t(u)- P_t(u). We use the formula in the recovered normalization [source reference]. With Fourier convention g(x)=(2π)^-1∫ h(v)e^-ivx dv, the even test h(v)=e^-tv^2cos(uv) has transform g(x)=frac14sqrtπ t (e^-(x-u)^2/(4t)+e^-(x+u)^2/(4t)). The zero symmetry ide","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-58099472c0de94b8","tex":"e^{-aJ}","aliases":"e^-aJ","display":"e^{-aJ}","kind":"use","excerpt":"The energy statement is deliberately local in u. It does not assume a global square integrable packet, which would already rule out many possible growth behaviors. The Hilbert space H collects only the fixed positive heat interval. Tonelli's theorem identifies widetilde E_J=∫_J^J+1 widetildeΘ_·(u) _H^2 du. Let a>0. On a unit band, Cauchy Schwarz gi","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a93ec8f936ebbe36","tex":"e^{-aJ/2}","aliases":"e^-aJ/2","display":"e^{-aJ/2}","kind":"use","excerpt":"fixed positive heat interval. Tonelli's theorem identifies widetilde E_J=∫_J^J+1 widetildeΘ_·(u) _H^2 du. Let a>0. On a unit band, Cauchy Schwarz gives ∫_J^J+1 widetildeΘ_·(u) _H e^-au du ≤ e^-aJwidetilde E_J^1/2. Choose the energy exponent to be ε=a. Subexponential growth then bounds the right side by a constant times e^-aJ/2. The geometric sum co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7b5ec0af479d0706","tex":"e^{-au}","aliases":"e^-au","display":"e^{-au}","kind":"use","excerpt":"If Θ_t is bounded, then on a compact subset of Re w>0 the integrand of its Laplace transform is bounded by a constant times e^-au, for some a>0. Differentiation with respect to w inserts powers of u, all of which remain integrable. This proves holomorphy on the claimed half plane.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0ef5cf69f2e38bca","tex":"e^{-c\\gamma^2}","aliases":"e^-cγ^2 gamma","display":"e^{-c\\gamma^2}","kind":"use","excerpt":"For completeness, divide the positive ordinates into intervals [j,j+1). The number in an interval is no greater than N(j+1), which is O((j+1)log(j+2)). Even this deliberately loose bound gives ∑_ρ∈ Z(1+|γ|)^A e^-cγ^2<∞ (c>0, A≥0). The finitely many small ordinates cause no difficulty, and the negative ones follow by conjugation. This establishes un","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-bd5cd3605c025773","tex":"e^{-hR}","aliases":"e^-hR","display":"e^{-hR}","kind":"use","excerpt":"The exponent in [source reference] comes from a standard elementary exponential bound for a Poisson variable X of mean λ. For h>0, Pr(X≥ R)≤ e^-hR E(e^hX) =exp(-hR+λ(e^h-1)). When R>λ>0, choose e^h=R/λ. For λ=0, every positive tail is zero. The positive mean case gives the rate Rlog(R/λ)-R+λ. Passing to the squared norm tail and then taking a squar","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-958637a0744902bf","tex":"e^{-iuv}","aliases":"e^-iuv","display":"e^{-iuv}","kind":"use","excerpt":"The normalization can be checked without any arithmetic approximation. For real x, the elementary Gaussian Fourier integral is ∫_ Re^-tv^2e^-ivx dv =sqrtfracπ t e^-x^2/(4t). Write cos(uv)=(e^iuv+e^-iuv)/2. The factor one half from this cosine and the factor (2π)^-1 in the Fourier convention give the coefficient 1/(4sqrtπ t) in g. The function g is ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c1c08936e00dd535","tex":"e^{-ivx}","aliases":"e^-ivx","display":"e^{-ivx}","kind":"use","excerpt":"e^-(u+log n)^2/(4t)). The Gaussian makes the prime series absolutely convergent. The completed explicit formula gives Θ_t(u)=2e^t/4cosh(u/2)+ A_t(u)- P_t(u). We use the formula in the recovered normalization [source reference]. With Fourier convention g(x)=(2π)^-1∫ h(v)e^-ivx dv, the even test h(v)=e^-tv^2cos(uv) has transform g(x)=frac14sqrtπ t (e","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f7230d6a4aae5995","tex":"e^{-t\\gamma^2}","aliases":"e^-tγ^2 gamma","display":"e^{-t\\gamma^2}","kind":"use","excerpt":"The series converges normally on compact sets of translations and on compact intervals of positive heat times. Indeed, for real u, |e^tz_ρ^2+uz_ρ| ≤ e^t/4+|u|/2e^-tγ^2. For complex u, there is an additional exponential factor linear in |γ| on each compact set. Gaussian decay still dominates. The classical zero counting estimate N(Y)=O(Ylog Y), wher","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0443888b5760605b","tex":"e^{-wu}","aliases":"e^-wu","display":"e^{-wu}","kind":"use","excerpt":"Under the hypothesis, [source reference] gives the asserted bound and determinant inequality. Conversely, suppose the packet is bounded on the positive half line. Its Laplace transform F_t(w)=∫_0^∞Θ_t(u)e^-wu du is holomorphic for Re w>0. For Re w>1/2, absolute convergence gives F_t(w)=∑_ρ∈ Z frace^tz_ρ^2w-z_ρ. The series on the right is meromorphi","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-539f1fac37b8c54d","tex":"e^{-x^2}","aliases":"e^-x^2","display":"e^{-x^2}","kind":"use","excerpt":"belrh:gaussian g_t,a(u)=(2π t)^-1/4e^-(u-a)^2/(4t). It has norm one in L^2( R,du). For a chosen center c, let ψ_r,t,c, r=0,1,2,…, be the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal pr","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ecf10d621a4173d2","tex":"e^{-x^2/(4t)}","aliases":"e^-x^2/(4t)","display":"e^{-x^2/(4t)}","kind":"use","excerpt":"The normalization can be checked without any arithmetic approximation. For real x, the elementary Gaussian Fourier integral is ∫_ Re^-tv^2e^-ivx dv =sqrtfracπ t e^-x^2/(4t). Write cos(uv)=(e^iuv+e^-iuv)/2. The factor one half from this cosine and the factor (2π)^-1 in the Fourier convention give the coefficient 1/(4sqrtπ t) in g. The function g is ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-621d0a688cee742a","tex":"e^{-\\lambda}","aliases":"e^-λ lambda","display":"e^{-\\lambda}","kind":"use","excerpt":"e the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g_t,a · =e^-λ/2 ∑_r≥0fracσ^rsqrtr!ψ_r,t,c, σ · =fraca-c2sqrt t, λ=σ^2, ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b86e9c034cb914f5","tex":"e^{-\\lambda/2}","aliases":"e^-λ/2 lambda","display":"e^{-\\lambda/2}","kind":"use","excerpt":"e the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g_t,a · =e^-λ/2 ∑_r≥0fracσ^rsqrtr!ψ_r,t,c, σ · =fraca-c2sqrt t, λ=σ^2, ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-035e0f4caa44c90d","tex":"e^{-\\sigma^2/2}","aliases":"e^-σ^2/2 sigma","display":"e^{-\\sigma^2/2}","kind":"use","excerpt":"Put x=(u-c)/sqrt2t. The ratio of a translated Gaussian to the centered one is fracg_t,a(u)g_t,c(u) =e^sqrt2σ x-σ^2. The Hermite generating identity e^2xv-v^2=∑_r≥0H_r(x)v^r/r!, with v=σ/sqrt2, turns this ratio into e^-σ^2/2∑_r≥0 fracH_r(x)σ^r2^r/2r!. Substitution of the normalized Hermite functions gives [source reference]. Squaring its orthogonal ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-dbf3819694e1360c","tex":"e^{2xv-v^2}","aliases":"e^2xv-v^2","display":"e^{2xv-v^2}","kind":"use","excerpt":"Put x=(u-c)/sqrt2t. The ratio of a translated Gaussian to the centered one is fracg_t,a(u)g_t,c(u) =e^sqrt2σ x-σ^2. The Hermite generating identity e^2xv-v^2=∑_r≥0H_r(x)v^r/r!, with v=σ/sqrt2, turns this ratio into e^-σ^2/2∑_r≥0 fracH_r(x)σ^r2^r/2r!. Substitution of the normalized Hermite functions gives [source reference]. Squaring its orthogonal ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3edac9854f47af65","tex":"e^{au}","aliases":"e^au","display":"e^{au}","kind":"use","excerpt":"To spell out the last implication, suppose the limit superior in [source reference] were below some centered zero's real part. Choose a number a strictly between them. Beyond a finite initial interval the packet would then satisfy |Θ_t(u)|≤ e^au, after slightly enlarging a if necessary. Its Laplace transform would be holomorphic for Re w>a, but the","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3e4c52f1de98b5cf","tex":"e^{cz+sz^2}","aliases":"e^cz+sz^2","display":"e^{cz+sz^2}","kind":"use","excerpt":"To verify [source reference], the scalar Gaussian integral is ∫_ R(2π s)^-1/4 e^-(u-c)^2/(4s)e^uz du =2sqrtπ s(2π s)^-1/4e^cz+sz^2. It holds for every complex z, either by completing the square first for real z and continuing analytically, or by the entire Gaussian transform. Multiplying by the mode coefficient at heat time t replaces t by t+s. The","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-39eb304f112b55e1","tex":"e^h","aliases":"e^h","display":"e^h","kind":"use","excerpt":"The exponent in [source reference] comes from a standard elementary exponential bound for a Poisson variable X of mean λ. For h>0, Pr(X≥ R)≤ e^-hR E(e^hX) =exp(-hR+λ(e^h-1)). When R>λ>0, choose e^h=R/λ. For λ=0, every positive tail is zero. The positive mean case gives the rate Rlog(R/λ)-R+λ. Passing to the squared norm tail and then taking a squar","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d7091bfb5b0ea04d","tex":"e^{hX}","aliases":"e^hX","display":"e^{hX}","kind":"use","excerpt":"The exponent in [source reference] comes from a standard elementary exponential bound for a Poisson variable X of mean λ. For h>0, Pr(X≥ R)≤ e^-hR E(e^hX) =exp(-hR+λ(e^h-1)). When R>λ>0, choose e^h=R/λ. For λ=0, every positive tail is zero. The positive mean case gives the rate Rlog(R/λ)-R+λ. Passing to the squared norm tail and then taking a squar","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-bd0bc939329db710","tex":"e^{iuv}","aliases":"e^iuv","display":"e^{iuv}","kind":"use","excerpt":"The normalization can be checked without any arithmetic approximation. For real x, the elementary Gaussian Fourier integral is ∫_ Re^-tv^2e^-ivx dv =sqrtfracπ t e^-x^2/(4t). Write cos(uv)=(e^iuv+e^-iuv)/2. The factor one half from this cosine and the factor (2π)^-1 in the Fourier convention give the coefficient 1/(4sqrtπ t) in g. The function g is ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d7ef1db85607aed2","tex":"e^{iu\\gamma}","aliases":"e^iuγ gamma","display":"e^{iu\\gamma}","kind":"use","excerpt":"Under the Riemann hypothesis the packet is Θ_t(u)=∑_ρ∈ Ze^-tγ^2e^iuγ. It is then the Fourier transform of a finite positive discrete measure. In particular, its magnitude is bounded by its value at the origin. The converse can be proved without estimating cancellation among different zeros.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-dfbe6edef5648e39","tex":"e^{j+1}","aliases":"e^j+1","display":"e^{j+1}","kind":"use","excerpt":"Absolute convergence of the prime series requires no prime number theorem. Use the elementary bound Λ(n)≤log n and partition the integers by j≤log n<j+1. There are at most e^j+1 terms in such a group. The Gaussian in j dominates the resulting exponential factor. The same reasoning covers differentiated packets because their derivatives insert only ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-706a6dd7fde9273d","tex":"e^{\\sqrt2\\sigma x-\\sigma^2}","aliases":"e^sqrt2σ x-σ^2 sqrt sigma sigma","display":"e^{\\sqrt2\\sigma x-\\sigma^2}","kind":"use","excerpt":"Put x=(u-c)/sqrt2t. The ratio of a translated Gaussian to the centered one is fracg_t,a(u)g_t,c(u) =e^sqrt2σ x-σ^2. The Hermite generating identity e^2xv-v^2=∑_r≥0H_r(x)v^r/r!, with v=σ/sqrt2, turns this ratio into e^-σ^2/2∑_r≥0 fracH_r(x)σ^r2^r/2r!. Substitution of the normalized Hermite functions gives [source reference]. Squaring its orthogonal ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-530c0e944713d8ba","tex":"e^{t z_\\rho^2}","aliases":"e^t z_ρ^2 rho","display":"e^{t z_\\rho^2}","kind":"use","excerpt":"Theta_·(u) as a function taking values in H=L^2([t_0,T]; C). Cauchy Schwarz on each unit interval and the assumed energy bound imply ∫_0^∞ widetildeΘ_·(u) _H e^-au du<∞ (a>0). Its H-valued Laplace transform is holomorphic for Re w>0. Initially for Re w>1/2, it also equals the normally convergent meromorphic series with terms fracρ(1-ρ)e^t z_ρ^2w-z_","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1e74d89e356ee632","tex":"e^{t/4}","aliases":"e^t/4","display":"e^{t/4}","kind":"use","excerpt":"P_t(u) · =frac12sqrtπ t ∑_n≥2fracΛ(n)sqrt n (e^-(u-log n)^2/(4t)+ e^-(u+log n)^2/(4t)). The Gaussian makes the prime series absolutely convergent. The completed explicit formula gives Θ_t(u)=2e^t/4cosh(u/2)+ A_t(u)- P_t(u). We use the formula in the recovered normalization [source reference]. With Fourier convention g(x)=(2π)^-1∫ h(v)e^-ivx dv, the","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4d513f91dd886c63","tex":"e^{t/4+|u|/2}","aliases":"e^t/4+|u|/2","display":"e^{t/4+|u|/2}","kind":"use","excerpt":"The series converges normally on compact sets of translations and on compact intervals of positive heat times. Indeed, for real u, |e^tz_ρ^2+uz_ρ| ≤ e^t/4+|u|/2e^-tγ^2. For complex u, there is an additional exponential factor linear in |γ| on each compact set. Gaussian decay still dominates. The classical zero counting estimate N(Y)=O(Ylog Y), wher","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ecb83b983d093719","tex":"e^{tz^2}","aliases":"e^tz^2","display":"e^{tz^2}","kind":"use","excerpt":"In the smaller initial domain Re w>1/2, absolute summation justifies integration of each zero term: ∫_0^∞ e^tz^2+uz-wu du =frace^tz^2w-z. For a compact set of w avoiding all centered zeros, large ordinates make |w-z| large. The Gaussian coefficient then gives a summable bound for the meromorphic series. Near one distinct centered zero z_0, isolate ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a9686dd1c18ad81f","tex":"e^{tz^2+uz}","aliases":"e^tz^2+uz","display":"e^{tz^2+uz}","kind":"use","excerpt":"For a single centered mode, ∂_t e^tz^2+uz=z^2e^tz^2+uz and ∂_u^2e^tz^2+uz=z^2e^tz^2+uz. Normal convergence transfers this equality to the entire packet. The heat equation here is an exact identity for the chosen series, rather than an empirical fit to a finite list of zeros.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-be4273b66a4bd0e9","tex":"e^{tz^2+uz-wu}","aliases":"e^tz^2+uz-wu","display":"e^{tz^2+uz-wu}","kind":"use","excerpt":"In the smaller initial domain Re w>1/2, absolute summation justifies integration of each zero term: ∫_0^∞ e^tz^2+uz-wu du =frace^tz^2w-z. For a compact set of w avoiding all centered zeros, large ordinates make |w-z| large. The Gaussian coefficient then gives a summable bound for the meromorphic series. Near one distinct centered zero z_0, isolate ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-cf42ec6527098b44","tex":"e^{tz_0^2}","aliases":"e^tz_0^2","display":"e^{tz_0^2}","kind":"use","excerpt":"justifies integration of each zero term: ∫_0^∞ e^tz^2+uz-wu du =frace^tz^2w-z. For a compact set of w avoiding all centered zeros, large ordinates make |w-z| large. The Gaussian coefficient then gives a summable bound for the meromorphic series. Near one distinct centered zero z_0, isolate all copies of that zero. If its multiplicity is m_0, their ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-32b6fff72242d83a","tex":"e^{tz_\\rho^2}","aliases":"e^tz_ρ^2 rho","display":"e^{tz_\\rho^2}","kind":"use","excerpt":"Under the hypothesis, [source reference] gives the asserted bound and determinant inequality. Conversely, suppose the packet is bounded on the positive half line. Its Laplace transform F_t(w)=∫_0^∞Θ_t(u)e^-wu du is holomorphic for Re w>0. For Re w>1/2, absolute convergence gives F_t(w)=∑_ρ∈ Z frace^tz_ρ^2w-z_ρ. The series on the right is meromorphi","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-84d9cf7fe723f14a","tex":"e^u","aliases":"e^u","display":"e^u","kind":"use","excerpt":"example, the prime packet is P_t(0)=frac1sqrtπ t ∑_n≥2fracΛ(n)sqrt n e^-(log n)^2/(4t). Every summand is nonnegative. The complete formula is nonetheless a difference involving this positive quantity. At large positive u, the Gaussian centered at log n selects prime powers with logarithms near u, so the active arithmetic scale is around n=e^u. A fi","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-04abc870a069dff8","tex":"e^{uz}","aliases":"e^uz","display":"e^{uz}","kind":"use","excerpt":"To verify [source reference], the scalar Gaussian integral is ∫_ R(2π s)^-1/4 e^-(u-c)^2/(4s)e^uz du =2sqrtπ s(2π s)^-1/4e^cz+sz^2. It holds for every complex z, either by completing the square first for real z and continuing analytically, or by the entire Gaussian transform. Multiplying by the mode coefficient at heat time t replaces t by t+s. The","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6c9172e113573faa","tex":"e^{x^2}","aliases":"e^x^2","display":"e^{x^2}","kind":"use","excerpt":"belrh:gaussian g_t,a(u)=(2π t)^-1/4e^-(u-a)^2/(4t). It has norm one in L^2( R,du). For a chosen center c, let ψ_r,t,c, r=0,1,2,…, be the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal pr","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d97928b6aa35f877","tex":"e^{\\delta u}","aliases":"e^δ u delta","display":"e^{\\delta u}","kind":"use","excerpt":"figplaceholderrh_heatfrontAmplitude envelopes of hypothetical centered modes. A displacement δ gives amplitude growth e^δ u, with a fixed nonzero heat multiplier omitted in the illustration. These model curves illustrate the theorem's exponent and are not sampled off line zeta zeros or a complete symmetric zero packet.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ba3720983f23986f","tex":"e^{\\varepsilon J}","aliases":"e^ε J varepsilon","display":"e^{\\varepsilon J}","kind":"use","excerpt":"For a nonnegative quantity E_J, call its growth subexponential when lim_J→∞fraclog(1+E_J)J=0. Equivalently, for every ε>0, there is a constant C_ε such that E_J≤ C_ε e^ε J for all positive integers J. This is an upper growth condition. It does not assert a positive lower bound.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-94ce41ed99bb2c85","tex":"e^\\lambda","aliases":"e^λ lambda","display":"e^\\lambda","kind":"use","excerpt":"The Hermite generating identity e^2xv-v^2=∑_r≥0H_r(x)v^r/r!, with v=σ/sqrt2, turns this ratio into e^-σ^2/2∑_r≥0 fracH_r(x)σ^r2^r/2r!. Substitution of the normalized Hermite functions gives [source reference]. Squaring its orthogonal coefficients gives e^-λλ^r/r!. Their total is one because the exponential series sums to e^λ. The discarded squared ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9c665f04ba7fdc59","tex":"e^{\\pi t/4}","aliases":"e^π t/4 pi","display":"e^{\\pi t/4}","kind":"use","excerpt":"figplaceholderrh_xiA finite evaluation of the completed function on the critical line. The lower panel multiplies the normalized value by e^π t/4, where t is the plotted height. The symmetry and normalization are exact classical facts. The plotted interval is a finite numerical illustration and supplies no conclusion about all zeros.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-650700dc8010c4c1","tex":"E_J","aliases":"E_J","display":"E_J","kind":"use","excerpt":"For a nonnegative quantity E_J, call its growth subexponential when lim_J→∞fraclog(1+E_J)J=0. Equivalently, for every ε>0, there is a constant C_ε such that E_J≤ C_ε e^ε J for all positive integers J. This is an upper growth condition. It does not assert a positive lower bound.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"The composition formula also explains why a local fall need not resolve a global obstruction. If an expanding segment has data (λ_g,c_g) and a following falling segment has (λ_f,c_f), the final value is λ_fλ_g n+λ_f c_g+c_f. A comparison with the local peak controls only part of this expression. To prove descent below n, the entire expression must ","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-dc648cbcd35750cb","tex":"F_{(1,3)}","aliases":"F_(1,3)","display":"F_{(1,3)}","kind":"use","excerpt":"Two ordered words with the same total depth. For w=(1,3), the recurrence gives C_1=1 and C_2=5, with total depth S=4. The affine expression is F_(1,3)(n)=frac9n+516. The exact cylinder requires 9n+5≡16pmod32. The inverse of 9 modulo 32 is 25, so its address is 19 modulo 32. Every positive representative can be written n=32t+19 with an integer t≥0. ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9bf1bcac910630c1","tex":"F_{(3,1)}","aliases":"F_(3,1)","display":"F_{(3,1)}","kind":"use","excerpt":"For v=(3,1), the first offset is still 1, but the second is 3+2^3=11. Its affine expression and cylinder are F_(3,1)(n)=frac9n+1116, n≡29pmod32. The realized segment is 32t+29⟼12t+11⟼18t+17, t≥0. Both words have two odd multiplications and four divisions by two. Their coefficients are the same. Their offsets, residue addresses, and intermediate sta","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","excerpt":"Under the hypothesis, [source reference] gives the asserted bound and determinant inequality. Conversely, suppose the packet is bounded on the positive half line. Its Laplace transform F_t(w)=∫_0^∞Θ_t(u)e^-wu du is holomorphic for Re w>0. For Re w>1/2, absolute convergence gives F_t(w)=∑_ρ∈ Z frace^tz_ρ^2w-z_ρ. The series on the right is meromorphi","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-78b29d109052da32","tex":"F_v","aliases":"F_v","display":"F_v","kind":"use","excerpt":"[Concatenation] Let a word w have data (k,S,C) and a following word v have data (ℓ,U,D). Their concatenation has data (k+ℓ,S+U,3^ℓ C+2^S D). On every realizing source, its transfer is F_v∘ F_w. Substitute 2^S y=3^k n+C into 2^U z=3^ℓ y+D. Multiplication by 2^S gives 2^S+Uz=3^k+ℓn+3^ℓ C+2^S D. The exact cylinder theorem identifies the admissible int","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"[Exact descent criterion] A realized word of positive length ends below its source if and only if (2^S-3^k)n>C_k. If 2^S≤3^k, strict descent is impossible. If 2^S>3^k, write G=2^S-3^k. The endpoint is at least the source if and only if n≤lfloorfracC_kGrfloor. The affine identity gives 2^S(x_k-n)=C_k-(2^S-3^k)n. Its sign proves the assertions. Since","count":14},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-64aeb0a5c2b4306e","tex":"g_{s,c}","aliases":"g_s,c","display":"g_{s,c}","kind":"use","excerpt":"The ground channel shows why the distinction matters. Direct integration gives ∫_ Rg_s,c(u)widetildeΘ_t(u) du =K_swidetildeΘ_t+s(c), K_s=2sqrtπ s(2π s)^-1/4. The Gaussian test makes this integral absolutely convergent even without the conjecture. It is a Gaussian pairing, with no assertion that the unweighted zero packet belongs to L^2( R). Complet","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-11bd7ac5518bc926","tex":"G_{t,a}","aliases":"G_t,a","display":"G_{t,a}","kind":"use","excerpt":"For a translated Gaussian G_t,a(u)=e^-(u-a)^2/(4t), direct differentiation gives LG_t,a(u)= (frac14+frac12t-frac(u-a)^24t^2) G_t,a(u). The annihilated prime packet therefore carries a polynomial factor with both signs. Positivity of the original Gaussian sum is no longer positivity of every transformed term. On the zero side the same operator is si","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-fbabe0e68b01a4f8","tex":"g_{t,b}","aliases":"g_t,b","display":"g_{t,b}","kind":"use","excerpt":"The word physical refers to the measure du in which the packet is actually evaluated. It does not mean that a separately chosen coefficient norm can be substituted. Completing the square gives the exact overlap ⟨ g_t,a,g_t,b⟩ =e^-(a-b)^2/(8t). For two translates, the Gram matrix has diagonal entries one and off diagonal entry κ=e^-(a-b)^2/(8t). Its","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7b2868d5754634e2","tex":"g_{t,c}","aliases":"g_t,c","display":"g_{t,c}","kind":"use","excerpt":"Put x=(u-c)/sqrt2t. The ratio of a translated Gaussian to the centered one is fracg_t,a(u)g_t,c(u) =e^sqrt2σ x-σ^2. The Hermite generating identity e^2xv-v^2=∑_r≥0H_r(x)v^r/r!, with v=σ/sqrt2, turns this ratio into e^-σ^2/2∑_r≥0 fracH_r(x)σ^r2^r/2r!. Substitution of the normalized Hermite functions gives [source reference]. Squaring its orthogonal ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"The energy statement is deliberately local in u. It does not assume a global square integrable packet, which would already rule out many possible growth behaviors. The Hilbert space H collects only the fixed positive heat interval. Tonelli's theorem identifies widetilde E_J=∫_J^J+1 widetildeΘ_·(u) _H^2 du. Let a>0. On a unit band, Cauchy Schwarz gi","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7ad378eaa530f4f0","tex":"H_R","aliases":"H_R","display":"H_R","kind":"use","excerpt":"For R>N, Lemmas~[source reference] and [source reference] place the R-N distinct states x_N,…,x_R-1 in E_ℓ below H_R=lceil2^D+1(n+R/3)rceil. Thus R-N≤ε_ℓ H_R+W_ℓ. Fix ℓ and let R tend to infinity. Division by R yields 1≤ε_ℓ2^D+1/3. Choosing ℓ sufficiently large contradicts this inequality. No eventual bounded deficit strip exists.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2d64df6fa5868f4c","tex":"H_r","aliases":"H_r","display":"H_r","kind":"use","excerpt":"The deficit itself can be handled entirely with integers. Let H_r be the least nonnegative integer h such that 3^r<2^h. Then lfloor rlog_2 3rfloor=H_r-1, including r=0. For r>0, equality of a positive power of 3 with a power of 2 is impossible, so the strict threshold identifies the floor exactly. The displayed logarithm is a convenient notation fo","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"e the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g_t,a · =e^-λ/2 ∑_r≥0fracσ^rsqrtr!ψ_r,t,c, σ · =fraca-c2sqrt t, λ=σ^2, ","count":8},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-12473beefe7126eb","tex":"\\Im","aliases":"Im","display":"\\Im","kind":"use","excerpt":"tervals of positive heat times. Indeed, for real u, |e^tz_ρ^2+uz_ρ| ≤ e^t/4+|u|/2e^-tγ^2. For complex u, there is an additional exponential factor linear in |γ| on each compact set. Gaussian decay still dominates. The classical zero counting estimate N(Y)=O(Ylog Y), where N(Y) counts zeros with 0<Imρ≤ Y, gives summability [source reference]. The sa","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7a0a635a73cee305","tex":"k'","aliases":"k'","display":"k'","kind":"use","excerpt":"This is the obstruction behind pulling a smaller endpoint backward through an unchanged late prefix. A useful smaller source merger must eventually change the word. For two different transfers ending at the same y, elimination gives m=frac2^S'3^k n+2^S'C-2^S C'2^S3^k'. The remaining task is to force a realizing alternative with 0<m<n. The formula d","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e4ae06953a7ec1f3","tex":"K^{-3/4+\\varepsilon}","aliases":"K^-3/4+ε varepsilon","display":"K^{-3/4+\\varepsilon}","kind":"use","excerpt":"[Báez Duarte criterion] The Riemann hypothesis holds if and only if, for every ε>0, |c_K|=O_ε(K^-3/4+ε). Equivalently, for every ε>0, |c_M^2|=O_ε(M^-3/2+ε), with M restricted to square roots of positive integers.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-927de3cdac384ec2","tex":"K_s","aliases":"K_s","display":"K_s","kind":"use","excerpt":"The ground channel shows why the distinction matters. Direct integration gives ∫_ Rg_s,c(u)widetildeΘ_t(u) du =K_swidetildeΘ_t+s(c), K_s=2sqrtπ s(2π s)^-1/4. The Gaussian test makes this integral absolutely convergent even without the conjecture. It is a Gaussian pairing, with no assertion that the unweighted zero packet belongs to L^2( R). Complet","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"use","excerpt":"Conversely, regard u↦widetildeΘ_·(u) as a function taking values in H=L^2([t_0,T]; C). Cauchy Schwarz on each unit interval and the assumed energy bound imply ∫_0^∞ widetildeΘ_·(u) _H e^-au du<∞ (a>0). Its H-valued Laplace transform is holomorphic for Re w>0. Initially for Re w>1/2, it also equals the normally convergent meromorphic series with ter","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e2be32bb67b2d6c8","tex":"L^4","aliases":"L^4","display":"L^4","kind":"use","excerpt":"Fourth, a raw positive fourth moment cannot lie below the square of its quadratic energy in a normalized probability space. If ν is its probability measure and F∈ L^4(ν) is the packet, Cauchy Schwarz gives ∫ |F|^4 dν≥(∫ |F|^2 dν)^2. This is the precise obstruction to a strict saving below that scale for the raw moment. It does not assert positivity","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9e3d7978b29768e1","tex":"L^q","aliases":"L^q","display":"L^q","kind":"use","excerpt":"Third, global Hardy norms on a prime torus are blind to the squarefree Möbius sign. For a finite prime set, a squarefree monomial is ∏_p∈ Sz_p. Replacing every coordinate by -z_p multiplies it by (-1)^|S|, exactly its Möbius coefficient. The substitution preserves Haar measure on the torus. Every global L^q norm formed with that measure is conseque","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"For R>N, Lemmas~[source reference] and [source reference] place the R-N distinct states x_N,…,x_R-1 in E_ℓ below H_R=lceil2^D+1(n+R/3)rceil. Thus R-N≤ε_ℓ H_R+W_ℓ. Fix ℓ and let R tend to infinity. Division by R yields 1≤ε_ℓ2^D+1/3. Choosing ℓ sufficiently large contradicts this inequality. No eventual bounded deficit strip exists.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"For a nonnegative quantity E_J, call its growth subexponential when lim_J→∞fraclog(1+E_J)J=0. Equivalently, for every ε>0, there is a constant C_ε such that E_J≤ C_ε e^ε J for all positive integers J. This is an upper growth condition. It does not assert a positive lower bound.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1509a54562ebc764","tex":"\\lim_{J\\to\\infty}","aliases":"lim_J→∞ lim to infty","display":"\\lim_{J\\to\\infty}","kind":"use","excerpt":"For a nonnegative quantity E_J, call its growth subexponential when lim_J→∞fraclog(1+E_J)J=0. Equivalently, for every ε>0, there is a constant C_ε such that E_J≤ C_ε e^ε J for all positive integers J. This is an upper growth condition. It does not assert a positive lower bound.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"use","excerpt":"The packet also makes the obstruction quantitative. Define δ_ζ=sup_ρ∈ Z(Reρ-1/2), M_t(U)=max(1,sup_0≤ u≤ U|Θ_t(u)|). Then 0≤δ_ζ≤1/2. The same Laplace argument proves limsup_U→∞fraclog M_t(U)U=δ_ζ. The upper bound follows from summing absolute values. If a smaller exponential rate held, the Laplace transform would be holomorphic beyond that rate, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-16db682260dbef14","tex":"\\limsup_{U\\to\\infty}","aliases":"limsup_U→∞ limsup to infty","display":"\\limsup_{U\\to\\infty}","kind":"use","excerpt":"The packet also makes the obstruction quantitative. Define δ_ζ=sup_ρ∈ Z(Reρ-1/2), M_t(U)=max(1,sup_0≤ u≤ U|Θ_t(u)|). Then 0≤δ_ζ≤1/2. The same Laplace argument proves limsup_U→∞fraclog M_t(U)U=δ_ζ. The upper bound follows from summing absolute values. If a smaller exponential rate held, the Laplace transform would be holomorphic beyond that rate, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c693ce1e8e9e894c","tex":"M^{-1}","aliases":"M^-1","display":"M^{-1}","kind":"use","excerpt":"The size difference behind the criterion is now visible. There are on the order of M integers in a window of length proportional to M, and a typical weight there has order M^-2. An unsigned mass therefore naturally has order M^-1. The desired signed amplitude has the smaller scale M^-3/2+ε. This count is only orientation. The barrier proof below su","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-95060607cfbb10ec","tex":"M^{-1-o(1)}","aliases":"M^-1-o(1)","display":"M^{-1-o(1)}","kind":"use","excerpt":"[Absolute cube barrier] Fix R with κ_R>0. Suppose squarefree Q=Q(M) has d(M)=o(log M) distinct prime divisors. For all sufficiently large M, A_Q(M)≥ fracκ_R8Rζ(2) fracφ(Q)QM ≥fracκ_R8Rζ(2)M(d+1). Consequently A_Q(M)≥ M^-1-o(1). It cannot itself satisfy the target bound in Theorem~[source reference].","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-de3934f2f63ada88","tex":"M^{-1/2+o(1)}","aliases":"M^-1/2+o(1)","display":"M^{-1/2+o(1)}","kind":"use","excerpt":"The dimension condition means that the cube size 2^d is M^o(1). In this regime the target amplitude is M^-3/2+o(1), whereas the triangle majorant is at least M^-1-o(1). The missing factor is M^-1/2+o(1). It must arise from cancellation or another operation which retains and controls the signs.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c491dc76b5da376a","tex":"M^{-2}","aliases":"M^-2","display":"M^{-2}","kind":"use","excerpt":"The size difference behind the criterion is now visible. There are on the order of M integers in a window of length proportional to M, and a typical weight there has order M^-2. An unsigned mass therefore naturally has order M^-1. The desired signed amplitude has the smaller scale M^-3/2+ε. This count is only orientation. The barrier proof below su","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2d9b06f7801188ec","tex":"m^{-2}","aliases":"m^-2","display":"m^{-2}","kind":"use","excerpt":"The limiting divisor sum is ∑_bmid Qfracμ(b)b^2 =∏_pmid Q(1-p^-2)≥ζ(2)^-1. For 0≤ x≤1, Bernoulli's inequality gives 0≤1-(1-x)^K≤ Kx. With K=M^2, the error in replacing each power by one is at most fracM^2m^2∑_bmid Qb^-4 ≤fracζ(4)R^2. Multiplying the resulting lower bound by m^-2 proves the lemma.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6173be1933be8de8","tex":"M^{-3/2+o(1)}","aliases":"M^-3/2+o(1)","display":"M^{-3/2+o(1)}","kind":"use","excerpt":"The dimension condition means that the cube size 2^d is M^o(1). In this regime the target amplitude is M^-3/2+o(1), whereas the triangle majorant is at least M^-1-o(1). The missing factor is M^-1/2+o(1). It must arise from cancellation or another operation which retains and controls the signs.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5bec81f55eb9291c","tex":"M^{-3/2+\\varepsilon}","aliases":"M^-3/2+ε varepsilon","display":"M^{-3/2+\\varepsilon}","kind":"use","excerpt":"[Báez Duarte criterion] The Riemann hypothesis holds if and only if, for every ε>0, |c_K|=O_ε(K^-3/4+ε). Equivalently, for every ε>0, |c_M^2|=O_ε(M^-3/2+ε), with M restricted to square roots of positive integers.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3ecf1907e6670630","tex":"M^{1/2+o(1)}","aliases":"M^1/2+o(1)","display":"M^{1/2+o(1)}","kind":"use","excerpt":"Order the prime divisors as p_1<⋯<p_d. Since p_i≥ i+1, fracφ(Q)Q=∏_i=1^d(1-p_i^-1) ≥∏_i=1^dfrac ii+1=frac1d+1. Also ∏_pnmid Q(1-p^-2)≥1/ζ(2). The error in [source reference] is M^1/2+o(1), while its main term is at least RM/[ζ(2)(d+1)]. Therefore eventually N_Q,R(M)≥fracRM2ζ(2)fracφ(Q)Q. On every counted base, |μ(m)|=1, and Lemma~[source reference]","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-60a8ce8254521d12","tex":"M^{1/2-\\varepsilon-o(1)}","aliases":"M^1/2-ε-o(1) varepsilon","display":"M^{1/2-\\varepsilon-o(1)}","kind":"use","excerpt":"To see the incompatibility with the target, choose any fixed amplitude exponent loss ε<1/2 in the M formulation of the criterion. A quantity bounded below by M^-1-o(1) cannot also be bounded above by a constant times M^-3/2+ε. Their ratio grows as M^1/2-ε-o(1). The criterion requires every positive ε, so a failure for one such exponent already show","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"At K=1, the weights are rational, since w_1(n)=n^-2-n^-4. Factoring the divisor sums gives Δ_6w_1(m)=frac23m^2-frac2527m^4, Δ_30w_1(m)=frac1625m^2-frac208225m^4. The following entries are exact. Every listed base is coprime to the Q of its row. The final column multiplies the cube value by the remaining Möbius sign. rrrrr Q · m · μ(m) · Δ_Qw_1(m) ·","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-af37a2ae9e4efb41","tex":"m^4","aliases":"m^4","display":"m^4","kind":"use","excerpt":"At K=1, the weights are rational, since w_1(n)=n^-2-n^-4. Factoring the divisor sums gives Δ_6w_1(m)=frac23m^2-frac2527m^4, Δ_30w_1(m)=frac1625m^2-frac208225m^4. The following entries are exact. Every listed base is coprime to the Q of its row. The final column multiplies the cube value by the remaining Möbius sign. rrrrr Q · m · μ(m) · Δ_Qw_1(m) ·","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-70db323dd278c632","tex":"M^{o(1)}","aliases":"M^o(1)","display":"M^{o(1)}","kind":"use","excerpt":"The dimension condition means that the cube size 2^d is M^o(1). In this regime the target amplitude is M^-3/2+o(1), whereas the triangle majorant is at least M^-1-o(1). The missing factor is M^-1/2+o(1). It must arise from cancellation or another operation which retains and controls the signs.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":", absolute summation justifies integration of each zero term: ∫_0^∞ e^tz^2+uz-wu du =frace^tz^2w-z. For a compact set of w avoiding all centered zeros, large ordinates make |w-z| large. The Gaussian coefficient then gives a summable bound for the meromorphic series. Near one distinct centered zero z_0, isolate all copies of that zero. If its multip","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9de6a9970cb6757b","tex":"M_t","aliases":"M_t","display":"M_t","kind":"use","excerpt":"The packet also makes the obstruction quantitative. Define δ_ζ=sup_ρ∈ Z(Reρ-1/2), M_t(U)=max(1,sup_0≤ u≤ U|Θ_t(u)|). Then 0≤δ_ζ≤1/2. The same Laplace argument proves limsup_U→∞fraclog M_t(U)U=δ_ζ. The upper bound follows from summing absolute values. If a smaller exponential rate held, the Laplace transform would be holomorphic beyond that rate, co","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"The packet also makes the obstruction quantitative. Define δ_ζ=sup_ρ∈ Z(Reρ-1/2), M_t(U)=max(1,sup_0≤ u≤ U|Θ_t(u)|). Then 0≤δ_ζ≤1/2. The same Laplace argument proves limsup_U→∞fraclog M_t(U)U=δ_ζ. The upper bound follows from summing absolute values. If a smaller exponential rate held, the Laplace transform would be holomorphic beyond that rate, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"merger obstruction] Suppose two positive sources n>m realize the same affine word with data (k,S,C) and endpoints y>x. Then n-m is a positive multiple of 2^S. In particular, if 2^S≥ n, no smaller positive source can share that word. Subtract the endpoint equations to obtain 3^k(n-m)=2^S(y-x). Coprimality of 3^k and 2^S forces 2^Smid n-m. A positive","count":11},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"[No eventually bounded noncrossing deficit] For a positive odd actual orbit with ρ_r≤1 at every time, the deficit is unbounded on every tail. More precisely, for every pair of integers D,N≥0 there exists r≥ N with 2^D+12^S_r≤3^r. Assume instead that d_r≤ D for every r≥ N. At any such starting time, a local block of k valuations has sum s=S_r+k-S_r ","count":22},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-79616ac402fc2170","tex":"n^{-1/2}","aliases":"n^-1/2","display":"n^{-1/2}","kind":"use","excerpt":"The simple coefficient mass bound also has an elementary proof. Since Λ(n)≤log X for n≤ X, and ∑_n≤ Xn^-1/2≤2sqrt X, their product gives the stated estimate. Compression may therefore be arranged without first proving any delicate prime cancellation theorem. The remaining arithmetic estimate lies in the compressed packet, not in the omitted Gaussia","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c0769f9271add232","tex":"n^{-2}","aliases":"n^-2","display":"n^{-2}","kind":"use","excerpt":"ζ(4)+frac1ζ(6). These formulas are exact. At large K, their alternating binomial terms can be individually much larger than their sum, so straightforward floating point evaluation may lose accuracy. A finite expression is not automatically a stable evaluation scheme. For a direct truncation of the original series at n=N, the elementary certified re","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1bbd54e8b3a790bd","tex":"n^{-2j-2}","aliases":"n^-2j-2","display":"n^{-2j-2}","kind":"use","excerpt":"Although c_K is written as an infinite arithmetic series, each fixed K has an exact finite expression in even zeta values. Expand its power by the binomial theorem: c_K=∑_j=0^K(-1)^jbinom Kjfrac1ζ(2j+2). To justify this identity, note first that the expansion has only K+1 terms. Each inner series ∑μ(n)n^-2j-2 is absolutely convergent. Its Euler pro","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-55d7cf56037490a0","tex":"n^{-4}","aliases":"n^-4","display":"n^{-4}","kind":"use","excerpt":"At K=1, the weights are rational, since w_1(n)=n^-2-n^-4. Factoring the divisor sums gives Δ_6w_1(m)=frac23m^2-frac2527m^4, Δ_30w_1(m)=frac1625m^2-frac208225m^4. The following entries are exact. Every listed base is coprime to the Q of its row. The final column multiplies the cube value by the remaining Möbius sign. rrrrr Q · m · μ(m) · Δ_Qw_1(m) ·","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-38855ce590112f03","tex":"n^{-s}","aliases":"n^-s","display":"n^{-s}","kind":"use","excerpt":"For a complex number s with real part greater than one, the zeta function is ζ(s)=∑_n=1^∞n^-s =∏_p(1-p^-s)^-1, where n ranges over positive integers and p ranges over primes. Analytic continuation extends it to a meromorphic function on the complex plane with a simple pole at s=1. The completed function ξ(s)=frac12s(s-1)π^-s/2Γ(s/2)ζ(s) is entire a","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a92a3713388bda3e","tex":"n^{-\\sigma}","aliases":"n^-σ sigma","display":"n^{-\\sigma}","kind":"use","excerpt":". Differentiating gives -fracζ'(s)ζ(s) =∑_p∑_k≥1(log p)p^-ks =∑_n≥2fracΛ(n)n^s. The factor k from differentiation cancels the denominator k, which is why the coefficient of p^k is log p, rather than log(p^k). Absolute convergence of these derivative series holds for Re s>1. For instance, Λ(n)≤log n, and ∑(log n)n^-σ converges for σ>1.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"s approximately 0.999999875, and the Gram condition number (1+κ)/(1-κ) is about 1.60×10^7. These values diagnose a coordinate system, not the size of a zeta coefficient. The asymptotic explanation is exact: 1-e^-(a-b)^2/(8t)∼frac(a-b)^28t, log(n+1)-log n∼frac1n. Consecutive integer centers therefore produce a Gram condition number of order n^2 at f","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b70859af9424fec5","tex":"n^s","aliases":"n^s","display":"n^s","kind":"use","excerpt":"The appearance of Λ can be traced directly to the Euler product in its first convergence half plane. There, logζ(s)=∑_p∑_k≥1fracp^-ksk. Differentiating gives -fracζ'(s)ζ(s) =∑_p∑_k≥1(log p)p^-ks =∑_n≥2fracΛ(n)n^s. The factor k from differentiation cancels the denominator k, which is why the coefficient of p^k is log p, rather than log(p^k). Absolut","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b9b0f71902c7fecd","tex":"n_*","aliases":"n_*","display":"n_*","kind":"use","excerpt":"The threshold as a fixed point. When G=2^S-3^k>0, the affine transfer has one real fixed point n_* = fracC_kG=fracc_w1-λ_w. For a real input above n_*, the affine expression falls. For an input below n_*, it rises. Equality gives a fixed point of the affine expression. These are statements about that expression before its integer cylinder is impose","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"The first examples are n_1=3 and n_2=151. The latter follows the exact odd orbit 151,227,341,1. The family does more than show that some prefixes can grow. It certifies arbitrarily large first descent times while retaining termination. Such delays must therefore be allowed in any proposed global proof.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b6b337ece4924770","tex":"n_2","aliases":"n_2","display":"n_2","kind":"use","excerpt":"The first examples are n_1=3 and n_2=151. The latter follows the exact odd orbit 151,227,341,1. The family does more than show that some prefixes can grow. It certifies arbitrarily large first descent times while retaining termination. Such delays must therefore be allowed in any proposed global proof.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"Explicit terminating seeds with unbounded delay. The program also constructs a closed family of sources whose first descent occurs arbitrarily late, yet whose first lower value is exactly 1 [source reference]. Define q_j=frac2^3^j+13^j+1, n_j=2^j+1q_j-1.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-eee9c4f0fe9f3b79","tex":"N_{Q,R}","aliases":"N_Q,R","display":"N_{Q,R}","kind":"use","excerpt":"[Squarefree coprime bases] For fixed R>0, the number N_Q,R(M) of squarefree integers m in [RM,2RM] with (m,Q)=1 satisfies, uniformly in squarefree Q, N_Q,R(M)=RMfracφ(Q)Q ∏_pnmid Q(1-p^-2)+O_R(2^dsqrt M). Here φ(Q) is Euler's totient function, which counts the positive residues coprime to Q.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"[Squarefree coprime bases] For fixed R>0, the number N_Q,R(M) of squarefree integers m in [RM,2RM] with (m,Q)=1 satisfies, uniformly in squarefree Q, N_Q,R(M)=RMfracφ(Q)Q ∏_pnmid Q(1-p^-2)+O_R(2^dsqrt M). Here φ(Q) is Euler's totient function, which counts the positive residues coprime to Q.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"ally on compact sets of translations and on compact intervals of positive heat times. Indeed, for real u, |e^tz_ρ^2+uz_ρ| ≤ e^t/4+|u|/2e^-tγ^2. For complex u, there is an additional exponential factor linear in |γ| on each compact set. Gaussian decay still dominates. The classical zero counting estimate N(Y)=O(Ylog Y), where N(Y) counts zeros with ","count":15},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"Suppose a fixed positive width constant C>0 is chosen and the active centers in a shell satisfy |a-c_J|≤ Csqrt J+O(1) and t≥ t_0>0. Put α=C^2/(4t_0). Then λ≤α J+O(sqrt J). For a fixed constant q>α, projection through R_J=lceil qJrceil has norm error bounded by exp(-frac12 I(qmidα)J+o(J)), I(qmidα)=qlog(q/α)-q+α. The coefficient mass of the positive","count":18},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-630b38df8c636710","tex":"O_R","aliases":"O_R","display":"O_R","kind":"use","excerpt":"[Squarefree coprime bases] For fixed R>0, the number N_Q,R(M) of squarefree integers m in [RM,2RM] with (m,Q)=1 satisfies, uniformly in squarefree Q, N_Q,R(M)=RMfracφ(Q)Q ∏_pnmid Q(1-p^-2)+O_R(2^dsqrt M). Here φ(Q) is Euler's totient function, which counts the positive residues coprime to Q.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e4d78e8109446c44","tex":"O_\\varepsilon","aliases":"O_ε varepsilon","display":"O_\\varepsilon","kind":"use","excerpt":"[Báez Duarte criterion] The Riemann hypothesis holds if and only if, for every ε>0, |c_K|=O_ε(K^-3/4+ε). Equivalently, for every ε>0, |c_M^2|=O_ε(M^-3/2+ε), with M restricted to square roots of positive integers.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0920c59121107dec","tex":"\\operatorname{oddPart}","aliases":"oddPart operatorname","display":"\\operatorname{oddPart}","kind":"use","excerpt":"· ⟼31104t+8261⟼5832t+1549. The valuations are 1,1,1,1,1,4. The next numerator is 3(5832t+1549)+1=2^3(2187t+581). Therefore A^7(n)=oddPart(2187t+581) ≤2187t+581<4096t+1087. The final valuation may exceed 3 for some parameters. The inequality remains valid because removing further factors of two only reduces the positive quotient. This proof is repre","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ae21a8553a17610b","tex":"\\overline z","aliases":"overline z overline","display":"\\overline z","kind":"use","excerpt":"Write ρ=β+iγ, with real numbers β and γ, and define its centered coordinate z_ρ=ρ-frac12. The symmetries become z↦-z and z↦overline z. Under the conjecture, every centered coordinate is purely imaginary. Away from the line, the real part of z becomes an exponential growth parameter in the packet below.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a0e4a47d5564def9","tex":"\\overline{1/2+i\\gamma}","aliases":"overline1/2+iγ overline gamma","display":"\\overline{1/2+i\\gamma}","kind":"use","excerpt":"On the critical line, conjugation and reflection identify the same point: overline1/2+iγ=1-(1/2+iγ). Therefore ξ(1/2+iγ) is real. Real values on that line explain why one can track sign changes in a one dimensional plot. They do not explain why every zero must lie there. A real analytic function restricted to one line can have additional zeros else","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-881e84403d3704c7","tex":"\\overline\\rho","aliases":"overlineρ overline rho","display":"\\overline\\rho","kind":"use","excerpt":"Let Z be the multiset of nontrivial zeros, with every zero repeated according to its multiplicity. They lie in the strip 0<Re s<1. Conjugation and the functional equation imply that a zero ρ is accompanied by overlineρ, 1-ρ, and 1-overlineρ, allowing coincidences on the critical line [source reference].","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"rm one in L^2( R,du). For a chosen center c, let ψ_r,t,c, r=0,1,2,…, be the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7bd100089ec86a9a","tex":"p^{-2}","aliases":"p^-2","display":"p^{-2}","kind":"use","excerpt":"The cube's shape is also scale dependent. At fixed K, large m makes every factor (1-1/(bm)^2)^K close to one, so its leading coefficient is ∏_pmid Q(1-p^-2). At the active scale masympsqrt K, the different divisors sample different parts of the weight profile. This difference can create local cancellation within a cube. It does not remove the remai","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-beef7271e4144c91","tex":"p^{-2j-2}","aliases":"p^-2j-2","display":"p^{-2j-2}","kind":"use","excerpt":"hmetic series, each fixed K has an exact finite expression in even zeta values. Expand its power by the binomial theorem: c_K=∑_j=0^K(-1)^jbinom Kjfrac1ζ(2j+2). To justify this identity, note first that the expansion has only K+1 terms. Each inner series ∑μ(n)n^-2j-2 is absolutely convergent. Its Euler product is ∏_p(1-p^-2j-2), which equals the re","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-02a3a1be7590e653","tex":"p^{-ks}","aliases":"p^-ks","display":"p^{-ks}","kind":"use","excerpt":"The appearance of Λ can be traced directly to the Euler product in its first convergence half plane. There, logζ(s)=∑_p∑_k≥1fracp^-ksk. Differentiating gives -fracζ'(s)ζ(s) =∑_p∑_k≥1(log p)p^-ks =∑_n≥2fracΛ(n)n^s. The factor k from differentiation cancels the denominator k, which is why the coefficient of p^k is log p, rather than log(p^k). Absolut","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-acf97391a5aa4604","tex":"p^{-s}","aliases":"p^-s","display":"p^{-s}","kind":"use","excerpt":"For a complex number s with real part greater than one, the zeta function is ζ(s)=∑_n=1^∞n^-s =∏_p(1-p^-s)^-1, where n ranges over positive integers and p ranges over primes. Analytic continuation extends it to a meromorphic function on the complex plane with a simple pole at s=1. The completed function ξ(s)=frac12s(s-1)π^-s/2Γ(s/2)ζ(s) is entire a","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"Order the prime divisors as p_1<⋯<p_d. Since p_i≥ i+1, fracφ(Q)Q=∏_i=1^d(1-p_i^-1) ≥∏_i=1^dfrac ii+1=frac1d+1. Also ∏_pnmid Q(1-p^-2)≥1/ζ(2). The error in [source reference] is M^1/2+o(1), while its main term is at least RM/[ζ(2)(d+1)]. Therefore eventually N_Q,R(M)≥fracRM2ζ(2)fracφ(Q)Q. On every counted base, |μ(m)|=1, and Lemma~[source reference]","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d5113184bf76dd62","tex":"p_d","aliases":"p_d","display":"p_d","kind":"use","excerpt":"Order the prime divisors as p_1<⋯<p_d. Since p_i≥ i+1, fracφ(Q)Q=∏_i=1^d(1-p_i^-1) ≥∏_i=1^dfrac ii+1=frac1d+1. Also ∏_pnmid Q(1-p^-2)≥1/ζ(2). The error in [source reference] is M^1/2+o(1), while its main term is at least RM/[ζ(2)(d+1)]. Therefore eventually N_Q,R(M)≥fracRM2ζ(2)fracφ(Q)Q. On every counted base, |μ(m)|=1, and Lemma~[source reference]","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"Order the prime divisors as p_1<⋯<p_d. Since p_i≥ i+1, fracφ(Q)Q=∏_i=1^d(1-p_i^-1) ≥∏_i=1^dfrac ii+1=frac1d+1. Also ∏_pnmid Q(1-p^-2)≥1/ζ(2). The error in [source reference] is M^1/2+o(1), while its main term is at least RM/[ζ(2)(d+1)]. Therefore eventually N_Q,R(M)≥fracRM2ζ(2)fracφ(Q)Q. On every counted base, |μ(m)|=1, and Lemma~[source reference]","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-24bf68c13a42bfaa","tex":"p_i^{-1}","aliases":"p_i^-1","display":"p_i^{-1}","kind":"use","excerpt":"Order the prime divisors as p_1<⋯<p_d. Since p_i≥ i+1, fracφ(Q)Q=∏_i=1^d(1-p_i^-1) ≥∏_i=1^dfrac ii+1=frac1d+1. Also ∏_pnmid Q(1-p^-2)≥1/ζ(2). The error in [source reference] is M^1/2+o(1), while its main term is at least RM/[ζ(2)(d+1)]. Therefore eventually N_Q,R(M)≥fracRM2ζ(2)fracφ(Q)Q. On every counted base, |μ(m)|=1, and Lemma~[source reference]","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6437e41273729f36","tex":"P_R","aliases":"P_R","display":"P_R","kind":"use","excerpt":"rm one in L^2( R,du). For a chosen center c, let ψ_r,t,c, r=0,1,2,…, be the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"[Exact integer cylinder] Let w be a nonempty valuation word of length k and total depth S. A positive odd integer n realizes w if and only if 3^k n+C_k≡ 2^Spmod2^S+1. Consequently the realizing integers form one odd residue class modulo 2^S+1, whose residue is u_w≡(2^S-C_k)(3^k)^-1pmod2^S+1. If the word is realized, [source reference] has an odd te","count":10},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"The exponent in [source reference] comes from a standard elementary exponential bound for a Poisson variable X of mean λ. For h>0, Pr(X≥ R)≤ e^-hR E(e^hX) =exp(-hR+λ(e^h-1)). When R>λ>0, choose e^h=R/λ. For λ=0, every positive tail is zero. The positive mean case gives the rate Rlog(R/λ)-R+λ. Passing to the squared norm tail and then taking a squar","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"One can also retain a weaker certificate when exact final valuation is unnecessary. If the last numerator is known to equal 2^b q with q>0 integral, then the next odd state is at most q. Exact equality requires q odd. A descent proof that only uses an upper bound can accept both parities. Example~[source reference] exploits this distinction. It obt","count":30},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"The full progression through 151. The parameter j=2 gives 3^j=9 and q_2=frac2^9+13^3=19, n_2=8·19-1=151. The exit equation is 27·19=2^9·1+1. Thus the first two valuations are 1 and the third is 10. Their sum is 12. The cylinder modulus is 2^13=8192, rather than 2^12. That last factor of two keeps the exiting quotient odd.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ecfc6d63a3b68fb8","tex":"q_j","aliases":"q_j","display":"q_j","kind":"use","excerpt":"Explicit terminating seeds with unbounded delay. The program also constructs a closed family of sources whose first descent occurs arbitrarily late, yet whose first lower value is exactly 1 [source reference]. Define q_j=frac2^3^j+13^j+1, n_j=2^j+1q_j-1.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"[Uniform positive cube tail] For a fixed R>0, put κ_R=frac1ζ(2)-fracζ(4)R^2. For every squarefree Q and every integer m≥ RM, Δ_Qw_M^2(m)≥fracκ_Rm^2. In particular, R=3 gives κ_R>0.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"figplaceholdercollatz_residue_treeValuation words give exact dyadic cylinders. The diagram labels the first two valuations by r_1=a_0 and r_2=a_1. Refinement reveals further valuation information. A bounded diagram displays finitely many branches of this infinite arithmetic structure.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"use","excerpt":"figplaceholdercollatz_residue_treeValuation words give exact dyadic cylinders. The diagram labels the first two valuations by r_1=a_0 and r_2=a_1. Refinement reveals further valuation information. A bounded diagram displays finitely many branches of this infinite arithmetic structure.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5491081fce1c2c32","tex":"R_J","aliases":"R_J","display":"R_J","kind":"use","excerpt":"Suppose a fixed positive width constant C>0 is chosen and the active centers in a shell satisfy |a-c_J|≤ Csqrt J+O(1) and t≥ t_0>0. Put α=C^2/(4t_0). Then λ≤α J+O(sqrt J). For a fixed constant q>α, projection through R_J=lceil qJrceil has norm error bounded by exp(-frac12 I(qmidα)J+o(J)), I(qmidα)=qlog(q/α)-q+α. The coefficient mass of the positive","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-fa39637d262230d3","tex":"r_{z_0}","aliases":"r_z_0","display":"r_{z_0}","kind":"use","excerpt":"Near a distinct centered zero z_0 of multiplicity m_0, the residue vector is r_z_0(t)=m_0ρ_0(1-ρ_0)e^tz_0^2. It is continuous and never zero on [t_0,T], so its H norm is positive. To reduce the contradiction to ordinary complex analysis, choose the continuous linear functional given by pairing with this residue vector. The scalar transform then has","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"For R>N, Lemmas~[source reference] and [source reference] place the R-N distinct states x_N,…,x_R-1 in E_ℓ below H_R=lceil2^D+1(n+R/3)rceil. Thus R-N≤ε_ℓ H_R+W_ℓ. Fix ℓ and let R tend to infinity. Division by R yields 1≤ε_ℓ2^D+1/3. Choosing ℓ sufficiently large contradicts this inequality. No eventual bounded deficit strip exists.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"This is the obstruction behind pulling a smaller endpoint backward through an unchanged late prefix. A useful smaller source merger must eventually change the word. For two different transfers ending at the same y, elimination gives m=frac2^S'3^k n+2^S'C-2^S C'2^S3^k'. The remaining task is to force a realizing alternative with 0<m<n. The formula d","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-db9af401ac382039","tex":"S_2","aliases":"S_2","display":"S_2","kind":"use","excerpt":"For example, the word (1,2) has S_2=3 and C_2=5, so 8·31=9·27+5. The corresponding cylinder is n≡11pmod16. Writing n=16t+11 gives A(n)=24t+17, A^2(n)=18t+13. Every member of this progression grows over the two steps, even though the second step falls from 24t+17 to 18t+13. For t=1, the local fall is 41 to 31, while both states remain above 27. Thes","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e6df250e130cd185","tex":"S_{37}","aliases":"S_37","display":"S_{37}","kind":"use","excerpt":"The fresh finite integer audit for this edition finds the first actual descent at accelerated time 37: x_37=23, S_37=59, C_37=1100931843921811423. Thus 2^59·23=3^37·27+1100931843921811423. The same segment has 59 shortcut steps. The historical serious audit separately records a Lean check of T^59(27)=23 [source reference]. The two statements use di","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c35cf257560864c4","tex":"S_I","aliases":"S_I","display":"S_I","kind":"use","excerpt":"For any finite interval I of remaining bases, distinguish its signed contribution from its absolute mass: S_I=∑_substackm∈ I (m,Q)=1μ(m)Δ_Qw_M^2(m), V_I=∑_substackm∈ I (m,Q)=1 |μ(m)| |Δ_Qw_M^2(m)|. The valid inequalities are |S_I|≤ V_I≤ A_Q(M) and |c_M^2|≤ A_Q(M). There is no general inequality |c_M^2|≤ V_I. Terms outside the chosen interval remain","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-698cd4222a287d75","tex":"S_i","aliases":"S_i","display":"S_i","kind":"use","excerpt":"[Affine packet law] If n realizes w, then every prefix satisfies 2^S_jx_j=3^j n+C_j. The terminal offset also has the explicit expression C_k=∑_i=0^k-13^k-1-i2^S_i. At j=0 the identity is x_0=n. If it holds at j, multiply [source reference] by 2^S_j to obtain 2^S_j+1x_j+1=3^j+1n+3C_j+2^S_j. This is the next identity by [source reference]. Iterating","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-774f8b01bd3825f8","tex":"S_{j+1}","aliases":"S_j+1","display":"S_{j+1}","kind":"use","excerpt":"If n realizes w, then every prefix satisfies 2^S_jx_j=3^j n+C_j. The terminal offset also has the explicit expression C_k=∑_i=0^k-13^k-1-i2^S_i. At j=0 the identity is x_0=n. If it holds at j, multiply [source reference] by 2^S_j to obtain 2^S_j+1x_j+1=3^j+1n+3C_j+2^S_j. This is the next identity by [source reference]. Iterating the same offset rec","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-88d7e83d610d9093","tex":"S_{r+k}","aliases":"S_r+k","display":"S_{r+k}","kind":"use","excerpt":"e odd actual orbit with ρ_r≤1 at every time, the deficit is unbounded on every tail. More precisely, for every pair of integers D,N≥0 there exists r≥ N with 2^D+12^S_r≤3^r. Assume instead that d_r≤ D for every r≥ N. At any such starting time, a local block of k valuations has sum s=S_r+k-S_r satisfying s<klog_2 3+D+1. This follows from ρ_r+k≤1 and ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b387b1687e20541b","tex":"S_u","aliases":"S_u","display":"S_u","kind":"use","excerpt":"[Injectivity] A permanently noncrossing positive odd orbit never repeats a state. Suppose x_u=x_v with v>u. Put k=v-u and B=S_v-S_u. The affine law for this positive length block gives (2^B-3^k)x_u=C_u,v>0. Hence 2^B/3^k>1. Determinism makes the same states and valuations repeat, so ρ_u+hk=ρ_u(2^B/3^k)^h. This exceeds 1 for sufficiently large h, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-2f57d1ab5cc8f2ce","tex":"S_v","aliases":"S_v","display":"S_v","kind":"use","excerpt":"[Injectivity] A permanently noncrossing positive odd orbit never repeats a state. Suppose x_u=x_v with v>u. Put k=v-u and B=S_v-S_u. The affine law for this positive length block gives (2^B-3^k)x_u=C_u,v>0. Hence 2^B/3^k>1. Determinism makes the same states and valuations repeat, so ρ_u+hk=ρ_u(2^B/3^k)^h. This exceeds 1 for sufficiently large h, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"The packet also makes the obstruction quantitative. Define δ_ζ=sup_ρ∈ Z(Reρ-1/2), M_t(U)=max(1,sup_0≤ u≤ U|Θ_t(u)|). Then 0≤δ_ζ≤1/2. The same Laplace argument proves limsup_U→∞fraclog M_t(U)U=δ_ζ. The upper bound follows from summing absolute values. If a smaller exponential rate held, the Laplace transform would be holomorphic beyond that rate, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7af57c62c8404a12","tex":"\\sup_{0\\le u\\le U}","aliases":"sup_0≤ u≤ U sup le le","display":"\\sup_{0\\le u\\le U}","kind":"use","excerpt":"The packet also makes the obstruction quantitative. Define δ_ζ=sup_ρ∈ Z(Reρ-1/2), M_t(U)=max(1,sup_0≤ u≤ U|Θ_t(u)|). Then 0≤δ_ζ≤1/2. The same Laplace argument proves limsup_U→∞fraclog M_t(U)U=δ_ζ. The upper bound follows from summing absolute values. If a smaller exponential rate held, the Laplace transform would be holomorphic beyond that rate, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5bc0d057b5d15b5d","tex":"\\sup_{\\rho\\in\\mathcal Z}","aliases":"sup_ρ∈ Z sup rho in mathcal","display":"\\sup_{\\rho\\in\\mathcal Z}","kind":"use","excerpt":"The packet also makes the obstruction quantitative. Define δ_ζ=sup_ρ∈ Z(Reρ-1/2), M_t(U)=max(1,sup_0≤ u≤ U|Θ_t(u)|). Then 0≤δ_ζ≤1/2. The same Laplace argument proves limsup_U→∞fraclog M_t(U)U=δ_ζ. The upper bound follows from summing absolute values. If a smaller exponential rate held, the Laplace transform would be holomorphic beyond that rate, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"For a translated Gaussian G_t,a(u)=e^-(u-a)^2/(4t), direct differentiation gives LG_t,a(u)= (frac14+frac12t-frac(u-a)^24t^2) G_t,a(u). The annihilated prime packet therefore carries a polynomial factor with both signs. Positivity of the original Gaussian sum is no longer positivity of every transformed term. On the zero side the same operator is si","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6e4a69b24263f904","tex":"T^{59}","aliases":"T^59","display":"T^{59}","kind":"use","excerpt":"The fresh finite integer audit for this edition finds the first actual descent at accelerated time 37: x_37=23, S_37=59, C_37=1100931843921811423. Thus 2^59·23=3^37·27+1100931843921811423. The same segment has 59 shortcut steps. The historical serious audit separately records a Lean check of T^59(27)=23 [source reference]. The two statements use di","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","excerpt":"It helps to make the normal convergence argument quantitative enough to see which operations it permits. Write z=x+iγ, where |x|<1/2, and u=a+ib. Then Re(tz^2+uz)=t(x^2-γ^2)+ax-bγ. If t∈[t_0,T] and |a|,|b|≤ B, this is at most T/4+B/2-t_0γ^2+B|γ| ≤ T/4+B/2+fracB^22t_0-fract_02γ^2. The last step is the elementary inequality B|γ|≤(t_0/2)γ^2+B^2/(2t_0)","count":13},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[Concatenation] Let a word w have data (k,S,C) and a following word v have data (ℓ,U,D). Their concatenation has data (k+ℓ,S+U,3^ℓ C+2^S D). On every realizing source, its transfer is F_v∘ F_w. Substitute 2^S y=3^k n+C into 2^U z=3^ℓ y+D. Multiplication by 2^S gives 2^S+Uz=3^k+ℓn+3^ℓ C+2^S D. The exact cylinder theorem identifies the admissible int","count":8},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"For every j≥0, q_j is a positive integer. At j=0 the numerator is 3. If 3^j+1 divides u+1 for u=2^3^j, then 2^3^j+1+1=u^3+1=(u+1)(u^2-u+1). Since u≡-1pmod3, the second factor is divisible by 3. Induction proves divisibility by 3^j+2. Positivity of the quotient is immediate.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d849fcfcf96ad168","tex":"u^3","aliases":"u^3","display":"u^3","kind":"use","excerpt":"For every j≥0, q_j is a positive integer. At j=0 the numerator is 3. If 3^j+1 divides u+1 for u=2^3^j, then 2^3^j+1+1=u^3+1=(u+1)(u^2-u+1). Since u≡-1pmod3, the second factor is divisible by 3. Induction proves divisibility by 3^j+2. Positivity of the quotient is immediate.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-dcc6d3615156a2d8","tex":"u_w","aliases":"u_w","display":"u_w","kind":"use","excerpt":"[Exact integer cylinder] Let w be a nonempty valuation word of length k and total depth S. A positive odd integer n realizes w if and only if 3^k n+C_k≡ 2^Spmod2^S+1. Consequently the realizing integers form one odd residue class modulo 2^S+1, whose residue is u_w≡(2^S-C_k)(3^k)^-1pmod2^S+1. If the word is realized, [source reference] has an odd te","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"For any finite interval I of remaining bases, distinguish its signed contribution from its absolute mass: S_I=∑_substackm∈ I (m,Q)=1μ(m)Δ_Qw_M^2(m), V_I=∑_substackm∈ I (m,Q)=1 |μ(m)| |Δ_Qw_M^2(m)|. The valid inequalities are |S_I|≤ V_I≤ A_Q(M) and |c_M^2|≤ A_Q(M). There is no general inequality |c_M^2|≤ V_I. Terms outside the chosen interval remain","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-fcdd77803761b961","tex":"v^r","aliases":"v^r","display":"v^r","kind":"use","excerpt":"Put x=(u-c)/sqrt2t. The ratio of a translated Gaussian to the centered one is fracg_t,a(u)g_t,c(u) =e^sqrt2σ x-σ^2. The Hermite generating identity e^2xv-v^2=∑_r≥0H_r(x)v^r/r!, with v=σ/sqrt2, turns this ratio into e^-σ^2/2∑_r≥0 fracH_r(x)σ^r2^r/2r!. Substitution of the normalized Hermite functions gives [source reference]. Squaring its orthogonal ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a3bb582415b7e7e6","tex":"V_I","aliases":"V_I","display":"V_I","kind":"use","excerpt":"For any finite interval I of remaining bases, distinguish its signed contribution from its absolute mass: S_I=∑_substackm∈ I (m,Q)=1μ(m)Δ_Qw_M^2(m), V_I=∑_substackm∈ I (m,Q)=1 |μ(m)| |Δ_Qw_M^2(m)|. The valid inequalities are |S_I|≤ V_I≤ A_Q(M) and |c_M^2|≤ A_Q(M). There is no general inequality |c_M^2|≤ V_I. Terms outside the chosen interval remain","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"At K=1, the weights are rational, since w_1(n)=n^-2-n^-4. Factoring the divisor sums gives Δ_6w_1(m)=frac23m^2-frac2527m^4, Δ_30w_1(m)=frac1625m^2-frac208225m^4. The following entries are exact. Every listed base is coprime to the Q of its row. The final column multiplies the cube value by the remaining Möbius sign. rrrrr Q · m · μ(m) · Δ_Qw_1(m) ·","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-66f5c2f78e63c466","tex":"\\widetilde{\\mathfrak A}","aliases":"widetilde A widetilde mathfrak","display":"\\widetilde{\\mathfrak A}","kind":"use","excerpt":"The identity 1/4-z_ρ^2=ρ(1-ρ) explains the weight. It is nonzero for every nontrivial zero. Thus the operator kills the explicit pole contribution in [source reference] while retaining every zero mode. Differentiated normal convergence gives widetildeΘ_t=widetilde A_t- widetilde P_t, where tildes on the right mean application of L.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1414d9edb972531b","tex":"\\widetilde{\\mathfrak A}_t","aliases":"widetilde A_t widetilde mathfrak","display":"\\widetilde{\\mathfrak A}_t","kind":"use","excerpt":"The identity 1/4-z_ρ^2=ρ(1-ρ) explains the weight. It is nonzero for every nontrivial zero. Thus the operator kills the explicit pole contribution in [source reference] while retaining every zero mode. Differentiated normal convergence gives widetildeΘ_t=widetilde A_t- widetilde P_t, where tildes on the right mean application of L.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8f01abba38851936","tex":"\\widetilde E_J^{1/2}","aliases":"widetilde E_J^1/2 widetilde","display":"\\widetilde E_J^{1/2}","kind":"use","excerpt":"The energy statement is deliberately local in u. It does not assume a global square integrable packet, which would already rule out many possible growth behaviors. The Hilbert space H collects only the fixed positive heat interval. Tonelli's theorem identifies widetilde E_J=∫_J^J+1 widetildeΘ_·(u) _H^2 du. Let a>0. On a unit band, Cauchy Schwarz gi","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-24a61b8e47af120e","tex":"\\widetilde{\\mathfrak P}","aliases":"widetilde P widetilde mathfrak","display":"\\widetilde{\\mathfrak P}","kind":"use","excerpt":"The identity 1/4-z_ρ^2=ρ(1-ρ) explains the weight. It is nonzero for every nontrivial zero. Thus the operator kills the explicit pole contribution in [source reference] while retaining every zero mode. Differentiated normal convergence gives widetildeΘ_t=widetilde A_t- widetilde P_t, where tildes on the right mean application of L.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-86b3a8046f49dd2f","tex":"\\widetilde{\\mathfrak P}_t","aliases":"widetilde P_t widetilde mathfrak","display":"\\widetilde{\\mathfrak P}_t","kind":"use","excerpt":"The identity 1/4-z_ρ^2=ρ(1-ρ) explains the weight. It is nonzero for every nontrivial zero. Thus the operator kills the explicit pole contribution in [source reference] while retaining every zero mode. Differentiated normal convergence gives widetildeΘ_t=widetilde A_t- widetilde P_t, where tildes on the right mean application of L.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ea635798f699443e","tex":"\\widetilde\\Theta_{t+s}","aliases":"widetildeΘ_t+s widetilde Theta","display":"\\widetilde\\Theta_{t+s}","kind":"use","excerpt":"The ground channel shows why the distinction matters. Direct integration gives ∫_ Rg_s,c(u)widetildeΘ_t(u) du =K_swidetildeΘ_t+s(c), K_s=2sqrtπ s(2π s)^-1/4. The Gaussian test makes this integral absolutely convergent even without the conjecture. It is a Gaussian pairing, with no assertion that the unweighted zero packet belongs to L^2( R). Complet","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-fc6c50f192ed1e40","tex":"\\widetilde\\Theta_\\cdot","aliases":"widetildeΘ_· widetilde Theta cdot","display":"\\widetilde\\Theta_\\cdot","kind":"use","excerpt":"Conversely, regard u↦widetildeΘ_·(u) as a function taking values in H=L^2([t_0,T]; C). Cauchy Schwarz on each unit interval and the assumed energy bound imply ∫_0^∞ widetildeΘ_·(u) _H e^-au du<∞ (a>0). Its H-valued Laplace transform is holomorphic for Re w>0. Initially for Re w>1/2, it also equals the normally convergent meromorphic series with ter","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"(1) and t≥ t_0>0. Put α=C^2/(4t_0). Then λ≤α J+O(sqrt J). For a fixed constant q>α, projection through R_J=lceil qJrceil has norm error bounded by exp(-frac12 I(qmidα)J+o(J)), I(qmidα)=qlog(q/α)-q+α. The coefficient mass of the positive prime packet through X_J=exp(J+O(sqrt J)) is at most ∑_n≤ X_JfracΛ(n)sqrt n ≤2sqrtX_Jlog X_J=exp(J/2+o(J)). Thus ","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"This appendix is finite evidence with an unusually transparent verification rule. A reader can check any row using integer multiplication and exponentiation, and can check consecutive rows using 3x+1=2^a x'. The large offsets are retained because deleting them would obscure the exact descent budget. Their size has no bearing on a universal theorem ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"It helps to make the normal convergence argument quantitative enough to see which operations it permits. Write z=x+iγ, where |x|<1/2, and u=a+ib. Then Re(tz^2+uz)=t(x^2-γ^2)+ax-bγ. If t∈[t_0,T] and |a|,|b|≤ B, this is at most T/4+B/2-t_0γ^2+B|γ| ≤ T/4+B/2+fracB^22t_0-fract_02γ^2. The last step is the elementary inequality B|γ|≤(t_0/2)γ^2+B^2/(2t_0)","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0cc6c23e58081877","tex":"x^r","aliases":"x^r","display":"x^r","kind":"use","excerpt":"belrh:gaussian g_t,a(u)=(2π t)^-1/4e^-(u-a)^2/(4t). It has norm one in L^2( R,du). For a chosen center c, let ψ_r,t,c, r=0,1,2,…, be the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal pr","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"[Affine packet law] If n realizes w, then every prefix satisfies 2^S_jx_j=3^j n+C_j. The terminal offset also has the explicit expression C_k=∑_i=0^k-13^k-1-i2^S_i. At j=0 the identity is x_0=n. If it holds at j, multiply [source reference] by 2^S_j to obtain 2^S_j+1x_j+1=3^j+1n+3C_j+2^S_j. This is the next identity by [source reference]. Iterating","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"One can see the offset being created before writing the general formula. The first step is 2^a_0x_1=3n+1. Multiplying the second step by 2^a_0 gives 2^a_0+a_1x_2=3(3n+1)+2^a_0 =9n+3+2^a_0. The first addition has been multiplied by 3. The second addition has been multiplied by the dyadic depth already removed. After a third step the offset is 9+3·2^","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"One can see the offset being created before writing the general formula. The first step is 2^a_0x_1=3n+1. Multiplying the second step by 2^a_0 gives 2^a_0+a_1x_2=3(3n+1)+2^a_0 =9n+3+2^a_0. The first addition has been multiplied by 3. The second addition has been multiplied by the dyadic depth already removed. After a third step the offset is 9+3·2^","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6a04c6c43d1ee3ee","tex":"x_{37}","aliases":"x_37","display":"x_{37}","kind":"use","excerpt":"The fresh finite integer audit for this edition finds the first actual descent at accelerated time 37: x_37=23, S_37=59, C_37=1100931843921811423. Thus 2^59·23=3^37·27+1100931843921811423. The same segment has 59 shortcut steps. The historical serious audit separately records a Lean check of T^59(27)=23 [source reference]. The two statements use di","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ece2f99bd925957b","tex":"X_J","aliases":"X_J","display":"X_J","kind":"use","excerpt":"(1) and t≥ t_0>0. Put α=C^2/(4t_0). Then λ≤α J+O(sqrt J). For a fixed constant q>α, projection through R_J=lceil qJrceil has norm error bounded by exp(-frac12 I(qmidα)J+o(J)), I(qmidα)=qlog(q/α)-q+α. The coefficient mass of the positive prime packet through X_J=exp(J+O(sqrt J)) is at most ∑_n≤ X_JfracΛ(n)sqrt n ≤2sqrtX_Jlog X_J=exp(J/2+o(J)). Thus ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"[Affine packet law] If n realizes w, then every prefix satisfies 2^S_jx_j=3^j n+C_j. The terminal offset also has the explicit expression C_k=∑_i=0^k-13^k-1-i2^S_i. At j=0 the identity is x_0=n. If it holds at j, multiply [source reference] by 2^S_j to obtain 2^S_j+1x_j+1=3^j+1n+3C_j+2^S_j. This is the next identity by [source reference]. Iterating","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0430cf91ce15859e","tex":"x_{j+1}","aliases":"x_j+1","display":"x_{j+1}","kind":"use","excerpt":"If n realizes w, then every prefix satisfies 2^S_jx_j=3^j n+C_j. The terminal offset also has the explicit expression C_k=∑_i=0^k-13^k-1-i2^S_i. At j=0 the identity is x_0=n. If it holds at j, multiply [source reference] by 2^S_j to obtain 2^S_j+1x_j+1=3^j+1n+3C_j+2^S_j. This is the next identity by [source reference]. Iterating the same offset rec","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"use","excerpt":"s w if and only if 3^k n+C_k≡ 2^Spmod2^S+1. Consequently the realizing integers form one odd residue class modulo 2^S+1, whose residue is u_w≡(2^S-C_k)(3^k)^-1pmod2^S+1. If the word is realized, [source reference] has an odd terminal quotient x_k. Multiplying an odd integer by 2^S gives exactly the congruence [source reference].","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ec84613b712fe298","tex":"x_N","aliases":"x_N","display":"x_N","kind":"use","excerpt":"For R>N, Lemmas~[source reference] and [source reference] place the R-N distinct states x_N,…,x_R-1 in E_ℓ below H_R=lceil2^D+1(n+R/3)rceil. Thus R-N≤ε_ℓ H_R+W_ℓ. Fix ℓ and let R tend to infinity. Division by R yields 1≤ε_ℓ2^D+1/3. Choosing ℓ sufficiently large contradicts this inequality. No eventual bounded deficit strip exists.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"Exact valuation packets. Fix a positive odd seed n, and write x_r=A^r(n). Each step has the exact equation 2^a_rx_r+1=3x_r+1, a_r=a(x_r). The exponent records how much dyadic information is removed at that step. It is determined by the actual state, rather than chosen independently.","count":14},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1e9d35fe4d36c66f","tex":"x_{r+1}","aliases":"x_r+1","display":"x_{r+1}","kind":"use","excerpt":"Exact valuation packets. Fix a positive odd seed n, and write x_r=A^r(n). Each step has the exact equation 2^a_rx_r+1=3x_r+1, a_r=a(x_r). The exponent records how much dyadic information is removed at that step. It is determined by the actual state, rather than chosen independently.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1c95c68e760e7494","tex":"x_{R-1}","aliases":"x_R-1","display":"x_{R-1}","kind":"use","excerpt":"For R>N, Lemmas~[source reference] and [source reference] place the R-N distinct states x_N,…,x_R-1 in E_ℓ below H_R=lceil2^D+1(n+R/3)rceil. Thus R-N≤ε_ℓ H_R+W_ℓ. Fix ℓ and let R tend to infinity. Division by R yields 1≤ε_ℓ2^D+1/3. Choosing ℓ sufficiently large contradicts this inequality. No eventual bounded deficit strip exists.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6590090cc9a70e3b","tex":"x_u","aliases":"x_u","display":"x_u","kind":"use","excerpt":"[Injectivity] A permanently noncrossing positive odd orbit never repeats a state. Suppose x_u=x_v with v>u. Put k=v-u and B=S_v-S_u. The affine law for this positive length block gives (2^B-3^k)x_u=C_u,v>0. Hence 2^B/3^k>1. Determinism makes the same states and valuations repeat, so ρ_u+hk=ρ_u(2^B/3^k)^h. This exceeds 1 for sufficiently large h, co","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"use","excerpt":"[Injectivity] A permanently noncrossing positive odd orbit never repeats a state. Suppose x_u=x_v with v>u. Put k=v-u and B=S_v-S_u. The affine law for this positive length block gives (2^B-3^k)x_u=C_u,v>0. Hence 2^B/3^k>1. Determinism makes the same states and valuations repeat, so ρ_u+hk=ρ_u(2^B/3^k)^h. This exceeds 1 for sufficiently large h, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"ally on compact sets of translations and on compact intervals of positive heat times. Indeed, for real u, |e^tz_ρ^2+uz_ρ| ≤ e^t/4+|u|/2e^-tγ^2. For complex u, there is an additional exponential factor linear in |γ| on each compact set. Gaussian decay still dominates. The classical zero counting estimate N(Y)=O(Ylog Y), where N(Y) counts zeros with ","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a4900b61886e74c0","tex":"z^{-8\\ell}","aliases":"z^-8ℓ ell","display":"z^{-8\\ell}","kind":"use","excerpt":"The weight z=3/4 tilts the word sum toward the low total valuations permitted by the hypothetical strip. Since 0<z<1 and s≤ h, one has z^-s≤ z^-h. The identity ∑_a≥1(z/2)^a=fracz2-z=frac35 then accounts for all possible positive valuations in each of the 5ℓ positions. Combining this factor with z^-8ℓ gives (frac43)^8ℓ(frac35)^5ℓ =(frac6553684375)^ℓ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c33be5d4a39852a0","tex":"z^{-h}","aliases":"z^-h","display":"z^{-h}","kind":"use","excerpt":"Set z=3/4. For s≤ h, 2^-s≤ z^-h(z/2)^s. Summing over all positive words bounds the truncated sum by ε_ℓ≤frac12z^-h (∑_a≥1(z/2)^a)^5ℓ =frac12(frac43)^D+1 (frac6553684375)^ℓ. The exact rational factor is less than 1, so ε_ℓ tends to zero.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a436bfeccdd39170","tex":"z^{-s}","aliases":"z^-s","display":"z^{-s}","kind":"use","excerpt":"The weight z=3/4 tilts the word sum toward the low total valuations permitted by the hypothetical strip. Since 0<z<1 and s≤ h, one has z^-s≤ z^-h. The identity ∑_a≥1(z/2)^a=fracz2-z=frac35 then accounts for all possible positive valuations in each of the 5ℓ positions. Combining this factor with z^-8ℓ gives (frac43)^8ℓ(frac35)^5ℓ =(frac6553684375)^ℓ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"It helps to make the normal convergence argument quantitative enough to see which operations it permits. Write z=x+iγ, where |x|<1/2, and u=a+ib. Then Re(tz^2+uz)=t(x^2-γ^2)+ax-bγ. If t∈[t_0,T] and |a|,|b|≤ B, this is at most T/4+B/2-t_0γ^2+B|γ| ≤ T/4+B/2+fracB^22t_0-fract_02γ^2. The last step is the elementary inequality B|γ|≤(t_0/2)γ^2+B^2/(2t_0)","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"In the smaller initial domain Re w>1/2, absolute summation justifies integration of each zero term: ∫_0^∞ e^tz^2+uz-wu du =frace^tz^2w-z. For a compact set of w avoiding all centered zeros, large ordinates make |w-z| large. The Gaussian coefficient then gives a summable bound for the meromorphic series. Near one distinct centered zero z_0, isolate ","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-23081359da026fdc","tex":"z_0^2","aliases":"z_0^2","display":"z_0^2","kind":"use","excerpt":"justifies integration of each zero term: ∫_0^∞ e^tz^2+uz-wu du =frace^tz^2w-z. For a compact set of w avoiding all centered zeros, large ordinates make |w-z| large. The Gaussian coefficient then gives a summable bound for the meromorphic series. Near one distinct centered zero z_0, isolate all copies of that zero. If its multiplicity is m_0, their ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-aefc6f887ea50fd4","tex":"z_p","aliases":"z_p","display":"z_p","kind":"use","excerpt":"Third, global Hardy norms on a prime torus are blind to the squarefree Möbius sign. For a finite prime set, a squarefree monomial is ∏_p∈ Sz_p. Replacing every coordinate by -z_p multiplies it by (-1)^|S|, exactly its Möbius coefficient. The substitution preserves Haar measure on the torus. Every global L^q norm formed with that measure is conseque","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"The series converges normally on compact sets of translations and on compact intervals of positive heat times. Indeed, for real u, |e^tz_ρ^2+uz_ρ| ≤ e^t/4+|u|/2e^-tγ^2. For complex u, there is an additional exponential factor linear in |γ| on each compact set. Gaussian decay still dominates. The classical zero counting estimate N(Y)=O(Ylog Y), wher","count":36},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"[Local energy criterion] Fix 0<t_0<T<∞. For positive integers J, set widetilde E_J=∫_t_0^T∫_J^J+1 |widetildeΘ_t(u)|^2 du dt. The Riemann hypothesis is equivalent to subexponential growth of widetilde E_J.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8f4fab3d5ea9ca88","tex":"|^4","aliases":"|^4","display":"|^4","kind":"use","excerpt":"Fourth, a raw positive fourth moment cannot lie below the square of its quadratic energy in a normalized probability space. If ν is its probability measure and F∈ L^4(ν) is the packet, Cauchy Schwarz gives ∫ |F|^4 dν≥(∫ |F|^2 dν)^2. This is the precise obstruction to a strict saving below that scale for the raw moment. It does not assert positivity","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"The choice of operator is tied to a declared removal target. Its two scalar roots are the centered coordinates ±1/2 of the exceptional points s=0,1. Actual nontrivial zeros are strictly inside the critical strip, so none is removed. A different operator which vanished at an unknown nontrivial zero would no longer preserve the obstruction required b","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"One can see the offset being created before writing the general formula. The first step is 2^a_0x_1=3n+1. Multiplying the second step by 2^a_0 gives 2^a_0+a_1x_2=3(3n+1)+2^a_0 =9n+3+2^a_0. The first addition has been multiplied by 3. The second addition has been multiplied by the dyadic depth already removed. After a third step the offset is 9+3·2^","count":24},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"For example, take t=1, a=log1000, and b=log1001. The separation is approximately 0.00099950, the overlap is approximately 0.999999875, and the Gram condition number (1+κ)/(1-κ) is about 1.60×10^7. These values diagnose a coordinate system, not the size of a zeta coefficient. The asymptotic explanation is exact: 1-e^-(a-b)^2/(8t)∼frac(a-b)^28t, log(","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"The contradiction can also be written without a limit. Set B=2^D+1 and choose ℓ so that α=ε_ℓ B/3<1. Since H_R≤ B(n+R/3)+1, the packing inequality would force (1-α)R≤ N+ε_ℓ(Bn+1)+W_ℓ for every integer R>N. The right side is fixed after D,N,n,ℓ are fixed. Selecting an integer R above that finite bound contradicts [source reference]. The limit in the","count":10},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4569df46706329ae","tex":"\\gamma^2","aliases":"γ^2 gamma","display":"\\gamma^2","kind":"use","excerpt":"The series converges normally on compact sets of translations and on compact intervals of positive heat times. Indeed, for real u, |e^tz_ρ^2+uz_ρ| ≤ e^t/4+|u|/2e^-tγ^2. For complex u, there is an additional exponential factor linear in |γ| on each compact set. Gaussian decay still dominates. The classical zero counting estimate N(Y)=O(Ylog Y), wher","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"The reflection is s↦1-s, not ordinary complex conjugation. These two operations commute, and together they generate the symmetric zero pattern. For example, a hypothetical zero ρ=frac12+δ+iγ, δ>0, would bring centered coordinates δ+iγ, δ-iγ, -δ+iγ, -δ-iγ. The quartet may collapse when δ=0 or γ=0. We sum the actual multiset, so coincidences are hand","count":10},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-74b849ebf82c08aa","tex":"\\Delta_{30}","aliases":"Δ_30 Delta","display":"\\Delta_{30}","kind":"use","excerpt":"For Q=6, the divisors correspond to the subsets of {2,3}, and the exact packet is Δ_6w_K(m)=w_K(m)-w_K(2m)-w_K(3m)+w_K(6m). For Q=30, one adds a prime coordinate: Δ_30w_K(m)= · w_K(m)-w_K(2m)-w_K(3m)-w_K(5m) · +w_K(6m)+w_K(10m)+w_K(15m)-w_K(30m). The complete divisor signs are shown here to make the grouping unambiguous. rrrrrrrrr b · 1 · 2 · 3 · 5","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0cf703e49cd53aa3","tex":"\\Delta_6","aliases":"Δ_6 Delta","display":"\\Delta_6","kind":"use","excerpt":"For Q=6, the divisors correspond to the subsets of {2,3}, and the exact packet is Δ_6w_K(m)=w_K(m)-w_K(2m)-w_K(3m)+w_K(6m). For Q=30, one adds a prime coordinate: Δ_30w_K(m)= · w_K(m)-w_K(2m)-w_K(3m)-w_K(5m)","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5aff1f78627b5498","tex":"\\delta_\\zeta","aliases":"δ_ζ delta zeta","display":"\\delta_\\zeta","kind":"use","excerpt":"The packet also makes the obstruction quantitative. Define δ_ζ=sup_ρ∈ Z(Reρ-1/2), M_t(U)=max(1,sup_0≤ u≤ U|Θ_t(u)|). Then 0≤δ_ζ≤1/2. The same Laplace argument proves limsup_U→∞fraclog M_t(U)U=δ_ζ. The upper bound follows from summing absolute values. If a smaller exponential rate held, the Laplace transform would be holomorphic beyond that rate, co","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"For a complex number s with real part greater than one, the zeta function is ζ(s)=∑_n=1^∞n^-s =∏_p(1-p^-s)^-1, where n ranges over positive integers and p ranges over primes. Analytic continuation extends it to a meromorphic function on the complex plane with a simple pole at s=1. The completed function ξ(s)=frac12s(s-1)π^-s/2Γ(s/2)ζ(s) is entire a","count":22},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d12fc9336a6eea97","tex":"\\zeta'","aliases":"ζ' zeta","display":"\\zeta'","kind":"use","excerpt":"The appearance of Λ can be traced directly to the Euler product in its first convergence half plane. There, logζ(s)=∑_p∑_k≥1fracp^-ksk. Differentiating gives -fracζ'(s)ζ(s) =∑_p∑_k≥1(log p)p^-ks =∑_n≥2fracΛ(n)n^s. The factor k from differentiation cancels the denominator k, which is why the coefficient of p^k is log p, rather than log(p^k). Absolut","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"e word physical refers to the measure du in which the packet is actually evaluated. It does not mean that a separately chosen coefficient norm can be substituted. Completing the square gives the exact overlap ⟨ g_t,a,g_t,b⟩ =e^-(a-b)^2/(8t). For two translates, the Gram matrix has diagonal entries one and off diagonal entry κ=e^-(a-b)^2/(8t). Its e","count":12},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-545253b67e7a1373","tex":"\\kappa_3","aliases":"κ_3 kappa","display":"\\kappa_3","kind":"use","excerpt":"The bound is uniform in the chosen primes. The finite product cannot fall below its product over all primes, while the error sum cannot exceed its sum over all positive integers. Taking R=3 gives κ_3=frac6π^2-fracπ^4810>0. The first term is approximately 0.6079 and the subtracted term approximately 0.1203. Thus the selected tail lies beyond the cen","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c08cad81c2ea02b9","tex":"\\kappa_R","aliases":"κ_R kappa","display":"\\kappa_R","kind":"use","excerpt":"[Uniform positive cube tail] For a fixed R>0, put κ_R=frac1ζ(2)-fracζ(4)R^2. For every squarefree Q and every integer m≥ RM, Δ_Qw_M^2(m)≥fracκ_Rm^2. In particular, R=3 gives κ_R>0.","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f315a4056bf8d461","tex":"\\lambda^r","aliases":"λ^r lambda","display":"\\lambda^r","kind":"use","excerpt":"e the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g_t,a · =e^-λ/2 ∑_r≥0fracσ^rsqrtr!ψ_r,t,c, σ · =fraca-c2sqrt t, λ=σ^2, ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-30b08bc07f8b1952","tex":"\\lambda_f","aliases":"λ_f lambda","display":"\\lambda_f","kind":"use","excerpt":"The composition formula also explains why a local fall need not resolve a global obstruction. If an expanding segment has data (λ_g,c_g) and a following falling segment has (λ_f,c_f), the final value is λ_fλ_g n+λ_f c_g+c_f. A comparison with the local peak controls only part of this expression. To prove descent below n, the entire expression must ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b3dcf25b318b68c3","tex":"\\lambda_g","aliases":"λ_g lambda","display":"\\lambda_g","kind":"use","excerpt":"The composition formula also explains why a local fall need not resolve a global obstruction. If an expanding segment has data (λ_g,c_g) and a following falling segment has (λ_f,c_f), the final value is λ_fλ_g n+λ_f c_g+c_f. A comparison with the local peak controls only part of this expression. To prove descent below n, the entire expression must ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"The Möbius function μ(n) is zero when a square of a prime divides n. Otherwise it equals (-1)^k, where k is the number of distinct prime divisors of n, and μ(1)=1. For an integer K≥1, put w_K(n)=frac1n^2(1-frac1n^2)^K, c_K=∑_n≥1μ(n)w_K(n). The series is absolutely convergent. Its natural active scale is n of order M=sqrt K. The coefficient c_K is c","count":23},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"x number s with real part greater than one, the zeta function is ζ(s)=∑_n=1^∞n^-s =∏_p(1-p^-s)^-1, where n ranges over positive integers and p ranges over primes. Analytic continuation extends it to a meromorphic function on the complex plane with a simple pole at s=1. The completed function ξ(s)=frac12s(s-1)π^-s/2Γ(s/2)ζ(s) is entire and satisfies","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c241b98d1bb04e95","tex":"\\pi^{-s/2}","aliases":"π^-s/2 pi","display":"\\pi^{-s/2}","kind":"use","excerpt":"x number s with real part greater than one, the zeta function is ζ(s)=∑_n=1^∞n^-s =∏_p(1-p^-s)^-1, where n ranges over positive integers and p ranges over primes. Analytic continuation extends it to a meromorphic function on the complex plane with a simple pole at s=1. The completed function ξ(s)=frac12s(s-1)π^-s/2Γ(s/2)ζ(s) is entire and satisfies","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"The bound is uniform in the chosen primes. The finite product cannot fall below its product over all primes, while the error sum cannot exceed its sum over all positive integers. Taking R=3 gives κ_3=frac6π^2-fracπ^4810>0. The first term is approximately 0.6079 and the subtracted term approximately 0.1203. Thus the selected tail lies beyond the cen","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-f4875cececfa7801","tex":"\\pi^4","aliases":"π^4 pi","display":"\\pi^4","kind":"use","excerpt":"The bound is uniform in the chosen primes. The finite product cannot fall below its product over all primes, while the error sum cannot exceed its sum over all positive integers. Taking R=3 gives κ_3=frac6π^2-fracπ^4810>0. The first term is approximately 0.6079 and the subtracted term approximately 0.1203. Thus the selected tail lies beyond the cen","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","excerpt":"Near a distinct centered zero z_0 of multiplicity m_0, the residue vector is r_z_0(t)=m_0ρ_0(1-ρ_0)e^tz_0^2. It is continuous and never zero on [t_0,T], so its H norm is positive. To reduce the contradiction to ordinary complex analysis, choose the continuous linear functional given by pairing with this residue vector. The scalar transform then has","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-936d7991f33298f9","tex":"\\rho_i","aliases":"ρ_i rho","display":"\\rho_i","kind":"use","excerpt":"[Normalized offset and height] Under permanent noncrossing, ρ_r x_r=n+frac13∑_i<rρ_i≤ n+frac r3. At every time with d_r≤ D, where D≥0 is integral, x_r<2^D+1(n+frac r3). Divide the affine law by 3^r and use the offset sum. Noncrossing gives ρ_i≤1 for every preceding time, which is essential for the upper bound. From d_r≤ D we have S_r≥lfloor rlog_2 ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0d22aeeb9f4dbe8e","tex":"\\rho_{r+k}","aliases":"ρ_r+k rho","display":"\\rho_{r+k}","kind":"use","excerpt":"nbounded on every tail. More precisely, for every pair of integers D,N≥0 there exists r≥ N with 2^D+12^S_r≤3^r. Assume instead that d_r≤ D for every r≥ N. At any such starting time, a local block of k valuations has sum s=S_r+k-S_r satisfying s<klog_2 3+D+1. This follows from ρ_r+k≤1 and ρ_r>2^-D-1. Take k=5ℓ, where ℓ≥1 is an integer. Since 3^5<2^8","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6a9ae3bd32e1dadd","tex":"\\rho_u","aliases":"ρ_u rho","display":"\\rho_u","kind":"use","excerpt":"[Injectivity] A permanently noncrossing positive odd orbit never repeats a state. Suppose x_u=x_v with v>u. Put k=v-u and B=S_v-S_u. The affine law for this positive length block gives (2^B-3^k)x_u=C_u,v>0. Hence 2^B/3^k>1. Determinism makes the same states and valuations repeat, so ρ_u+hk=ρ_u(2^B/3^k)^h. This exceeds 1 for sufficiently large h, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a4796884cbb79499","tex":"\\rho_{u+hk}","aliases":"ρ_u+hk rho","display":"\\rho_{u+hk}","kind":"use","excerpt":"[Injectivity] A permanently noncrossing positive odd orbit never repeats a state. Suppose x_u=x_v with v>u. Put k=v-u and B=S_v-S_u. The affine law for this positive length block gives (2^B-3^k)x_u=C_u,v>0. Hence 2^B/3^k>1. Determinism makes the same states and valuations repeat, so ρ_u+hk=ρ_u(2^B/3^k)^h. This exceeds 1 for sufficiently large h, co","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":". Differentiating gives -fracζ'(s)ζ(s) =∑_p∑_k≥1(log p)p^-ks =∑_n≥2fracΛ(n)n^s. The factor k from differentiation cancels the denominator k, which is why the coefficient of p^k is log p, rather than log(p^k). Absolute convergence of these derivative series holds for Re s>1. For instance, Λ(n)≤log n, and ∑(log n)n^-σ converges for σ>1.","count":6},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"e the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g_t,a · =e^-λ/2 ∑_r≥0fracσ^rsqrtr!ψ_r,t,c, σ · =fraca-c2sqrt t, λ=σ^2, ","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-92b1b1b0c9bbfab8","tex":"\\sigma^r","aliases":"σ^r sigma","display":"\\sigma^r","kind":"use","excerpt":"e the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g_t,a · =e^-λ/2 ∑_r≥0fracσ^rsqrtr!ψ_r,t,c, σ · =fraca-c2sqrt t, λ=σ^2, ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"[Squarefree coprime bases] For fixed R>0, the number N_Q,R(M) of squarefree integers m in [RM,2RM] with (m,Q)=1 satisfies, uniformly in squarefree Q, N_Q,R(M)=RMfracφ(Q)Q ∏_pnmid Q(1-p^-2)+O_R(2^dsqrt M). Here φ(Q) is Euler's totient function, which counts the positive residues coprime to Q.","count":7},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-84ff15e7f22c1ac7","tex":"\\psi_{r,t,c}","aliases":"ψ_r,t,c psi","display":"\\psi_{r,t,c}","kind":"use","excerpt":"For t>0 and a real center a, define the normalized Gaussian g_t,a(u)=(2π t)^-1/4e^-(u-a)^2/(4t). It has norm one in L^2( R,du). For a chosen center c, let ψ_r,t,c, r=0,1,2,…, be the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Let Q be a squarefree positive integer, and let d=ω(Q) be its number of distinct prime divisors. Its alternating prime cube acts on the weight by Δ_Qw_K(m)=∑_bmid Qμ(b)w_K(bm). Here b ranges over all positive divisors of Q. The result is a sum of 2^d terms, one for each subset of the prime coordinates.","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c7e8bb1d803377df","tex":"\\ell^{-2}","aliases":"ℓ^-2 ell","display":"\\ell^{-2}","kind":"use","excerpt":"Use the exact identity μ(m)^2=∑_ℓ^2mid mμ(ℓ). In the coprime sector only (ℓ,Q)=1 contributes. Inclusion and exclusion over the prime factors of Q counts the possible multiples of ℓ^2 in the interval with main term RMℓ^-2φ(Q)/Q and error O(2^d). There are O_R(sqrt M) values of ℓ, so the summed error is O_R(2^dsqrt M). Extending the main sum from ℓ≤s","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"Use the exact identity μ(m)^2=∑_ℓ^2mid mμ(ℓ). In the coprime sector only (ℓ,Q)=1 contributes. Inclusion and exclusion over the prime factors of Q counts the possible multiples of ℓ^2 in the interval with main term RMℓ^-2φ(Q)/Q and error O(2^d). There are O_R(sqrt M) values of ℓ, so the summed error is O_R(2^dsqrt M). Extending the main sum from ℓ≤s","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"The quantifier on the conclusion matters. Unbounded deficit on every tail does not imply d_r→∞. The deficit could return to small values between increasingly deep excursions. Even convergence d_r→∞ would only imply ρ_r→0, which is compatible with growing positive states. The present theorem closes every bounded strip, rather than the entire noncros","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"t uses S_r shortcut steps and S_r+r classical steps. In the shortcut map the division immediately after an odd multiplication is included in that odd step. In the classical map it is a separate step, as are all the other divisions. For example, A(27)=41 is realized classically by 27,82,41. Since its valuation is 1, this has one shortcut step and tw","count":5},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-dd94c17e5e087649","tex":"\\partial_t","aliases":"∂_t partial","display":"\\partial_t","kind":"use","excerpt":"r complex u, there is an additional exponential factor linear in |γ| on each compact set. Gaussian decay still dominates. The classical zero counting estimate N(Y)=O(Ylog Y), where N(Y) counts zeros with 0<Imρ≤ Y, gives summability [source reference]. The same domination permits differentiation. Consequently ∂_tΘ_t=∂_u^2Θ_t. The zero symmetries mak","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"For a complex number s with real part greater than one, the zeta function is ζ(s)=∑_n=1^∞n^-s =∏_p(1-p^-s)^-1, where n ranges over positive integers and p ranges over primes. Analytic continuation extends it to a meromorphic function on the complex plane with a simple pole at s=1. The completed function ξ(s)=frac12s(s-1)π^-s/2Γ(s/2)ζ(s) is entire a","count":10},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ee204890c9def03a","tex":"\\prod_{i=1}^d","aliases":"∏_i=1^d prod","display":"\\prod_{i=1}^d","kind":"use","excerpt":"Order the prime divisors as p_1<⋯<p_d. Since p_i≥ i+1, fracφ(Q)Q=∏_i=1^d(1-p_i^-1) ≥∏_i=1^dfrac ii+1=frac1d+1. Also ∏_pnmid Q(1-p^-2)≥1/ζ(2). The error in [source reference] is M^1/2+o(1), while its main term is at least RM/[ζ(2)(d+1)]. Therefore eventually N_Q,R(M)≥fracRM2ζ(2)fracφ(Q)Q. On every counted base, |μ(m)|=1, and Lemma~[source reference]","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b9d5148b142926aa","tex":"\\prod_p","aliases":"∏_p prod","display":"\\prod_p","kind":"use","excerpt":"hmetic series, each fixed K has an exact finite expression in even zeta values. Expand its power by the binomial theorem: c_K=∑_j=0^K(-1)^jbinom Kjfrac1ζ(2j+2). To justify this identity, note first that the expansion has only K+1 terms. Each inner series ∑μ(n)n^-2j-2 is absolutely convergent. Its Euler product is ∏_p(1-p^-2j-2), which equals the re","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9140b13729b27e7a","tex":"\\prod_{p}","aliases":"∏_p prod","display":"\\prod_{p}","kind":"use","excerpt":"For a complex number s with real part greater than one, the zeta function is ζ(s)=∑_n=1^∞n^-s =∏_p(1-p^-s)^-1, where n ranges over positive integers and p ranges over primes. Analytic continuation extends it to a meromorphic function on the complex plane with a simple pole at s=1. The completed function ξ(s)=frac12s(s-1)π^-s/2Γ(s/2)ζ(s) is entire a","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9705db81396e2375","tex":"\\prod_{p\\mid Q}","aliases":"∏_pmid Q prod mid","display":null,"kind":"use","excerpt":"The cube's shape is also scale dependent. At fixed K, large m makes every factor (1-1/(bm)^2)^K close to one, so its leading coefficient is ∏_pmid Q(1-p^-2). At the active scale masympsqrt K, the different divisors sample different parts of the weight profile. This difference can create local cancellation within a cube. It does not remove the remai","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5a08eaa4be831d5b","tex":"\\prod_{p\\nmid Q}","aliases":"∏_pnmid Q prod nmid","display":null,"kind":"use","excerpt":"[Squarefree coprime bases] For fixed R>0, the number N_Q,R(M) of squarefree integers m in [RM,2RM] with (m,Q)=1 satisfies, uniformly in squarefree Q, N_Q,R(M)=RMfracφ(Q)Q ∏_pnmid Q(1-p^-2)+O_R(2^dsqrt M). Here φ(Q) is Euler's totient function, which counts the positive residues coprime to Q.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-40f9dd96c7fdf7e8","tex":"\\prod_{p\\nmid6}","aliases":"∏_pnmid6 prod nmid","display":null,"kind":"use","excerpt":"For Q=6, the density constant simplifies to fracφ(6)6∏_pnmid6(1-p^-2) =frac12ζ(2). For Q=30, it is 5/(12ζ(2)). As a finite illustration, take M=10 and R=3. The interval is [30,60]. Its Q=6 squarefree coprime bases are 31,35,37,41,43,47,53,55,59. For Q=30, remove 35 and 55, leaving seven bases. In both coprime sectors, 49 would be counted if squaref","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-516d6ddd3d40ea65","tex":"\\prod_{p\\in S}","aliases":"∏_p∈ S prod in","display":"\\prod_{p\\in S}","kind":"use","excerpt":"Third, global Hardy norms on a prime torus are blind to the squarefree Möbius sign. For a finite prime set, a squarefree monomial is ∏_p∈ Sz_p. Replacing every coordinate by -z_p multiplies it by (-1)^|S|, exactly its Möbius coefficient. The substitution preserves Haar measure on the torus. Every global L^q norm formed with that measure is conseque","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4c4b0da97e414b10","tex":"\\sum_{a_0,\\ldots,a_{k-1}\\ge1}","aliases":"∑_a_0,…,a_k-1≥1 sum ldots ge","display":"\\sum_{a_0,\\ldots,a_{k-1}\\ge1}","kind":"use","excerpt":"For fixed k≥1, the sum of all relative odd cylinder densities is ∑_a_0,…,a_k-1≥12^-(a_0+⋯+a_k-1) =(∑_a≥12^-a)^k=1. This identity reflects the exact partition of positive odd integers by their first k valuations. It does not assert that valuations along one particular infinite orbit are independent samples. A density calculation describes how source","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-e551a711530425ab","tex":"\\sum_{a\\ge1}","aliases":"∑_a≥1 sum ge","display":"\\sum_{a\\ge1}","kind":"use","excerpt":"For fixed k≥1, the sum of all relative odd cylinder densities is ∑_a_0,…,a_k-1≥12^-(a_0+⋯+a_k-1) =(∑_a≥12^-a)^k=1. This identity reflects the exact partition of positive odd integers by their first k valuations. It does not assert that valuations along one particular infinite orbit are independent samples. A density calculation describes how source","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-56be50d7aaebfabc","tex":"\\sum_{b\\mid Q}","aliases":"∑_bmid Q sum mid","display":null,"kind":"use","excerpt":"Let Q be a squarefree positive integer, and let d=ω(Q) be its number of distinct prime divisors. Its alternating prime cube acts on the weight by Δ_Qw_K(m)=∑_bmid Qμ(b)w_K(bm). Here b ranges over all positive divisors of Q. The result is a sum of 2^d terms, one for each subset of the prime coordinates.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-69564fe62ad58d8a","tex":"\\sum_{i<r}","aliases":"∑_i<r sum","display":"\\sum_{i<r}","kind":"use","excerpt":"[Normalized offset and height] Under permanent noncrossing, ρ_r x_r=n+frac13∑_i<rρ_i≤ n+frac r3. At every time with d_r≤ D, where D≥0 is integral, x_r<2^D+1(n+frac r3). Divide the affine law by 3^r and use the offset sum. Noncrossing gives ρ_i≤1 for every preceding time, which is essential for the upper bound. From d_r≤ D we have S_r≥lfloor rlog_2 ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d9b6fff5ddbb4ca4","tex":"\\sum_{i=0}^{k-1}","aliases":"∑_i=0^k-1 sum","display":"\\sum_{i=0}^{k-1}","kind":"use","excerpt":"[Affine packet law] If n realizes w, then every prefix satisfies 2^S_jx_j=3^j n+C_j. The terminal offset also has the explicit expression C_k=∑_i=0^k-13^k-1-i2^S_i. At j=0 the identity is x_0=n. If it holds at j, multiply [source reference] by 2^S_j to obtain 2^S_j+1x_j+1=3^j+1n+3C_j+2^S_j. This is the next identity by [source reference]. Iterating","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-813fafc68f715ce9","tex":"\\sum_{j=0}^K","aliases":"∑_j=0^K sum","display":"\\sum_{j=0}^K","kind":"use","excerpt":"Although c_K is written as an infinite arithmetic series, each fixed K has an exact finite expression in even zeta values. Expand its power by the binomial theorem: c_K=∑_j=0^K(-1)^jbinom Kjfrac1ζ(2j+2). To justify this identity, note first that the expansion has only K+1 terms. Each inner series ∑μ(n)n^-2j-2 is absolutely convergent. Its Euler pro","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-1ebed7c5a175f44d","tex":"\\sum_{k\\ge0}","aliases":"∑_k≥0 sum ge","display":"\\sum_{k\\ge0}","kind":"use","excerpt":"The archimedean term has a uniform bound on every positive heat strip. A direct proof uses the classical digamma series [source reference]: ψ(a+iy)-ψ(a)= ∑_k≥0fraciy(k+a)(k+a+iy), a>0. Its absolute value is at most |y|∑_k≥0(k+a)^-2. Therefore |Ψ_∞(v)|≤ C(1+|v|) for a fixed constant C. Since widetilde A_t(u)=frac12π ∫_ R(1/4+v^2)e^-tv^2cos(uv) Ψ_∞(v","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-01f56a5e1c2b8b56","tex":"\\sum_{k\\ge1}","aliases":"∑_k≥1 sum ge","display":"\\sum_{k\\ge1}","kind":"use","excerpt":"The appearance of Λ can be traced directly to the Euler product in its first convergence half plane. There, logζ(s)=∑_p∑_k≥1fracp^-ksk. Differentiating gives -fracζ'(s)ζ(s) =∑_p∑_k≥1(log p)p^-ks =∑_n≥2fracΛ(n)n^s. The factor k from differentiation cancels the denominator k, which is why the coefficient of p^k is log p, rather than log(p^k). Absolut","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-3dacc49fefa42a3d","tex":"\\sum_{n=1}^{\\infty}","aliases":"∑_n=1^∞ sum infty","display":"\\sum_{n=1}^{\\infty}","kind":"use","excerpt":"For a complex number s with real part greater than one, the zeta function is ζ(s)=∑_n=1^∞n^-s =∏_p(1-p^-s)^-1, where n ranges over positive integers and p ranges over primes. Analytic continuation extends it to a meromorphic function on the complex plane with a simple pole at s=1. The completed function ξ(s)=frac12s(s-1)π^-s/2Γ(s/2)ζ(s) is entire a","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-9b846ee3188af45a","tex":"\\sum_{n>N}","aliases":"∑_n>N sum","display":"\\sum_{n>N}","kind":"use","excerpt":"ζ(4)+frac1ζ(6). These formulas are exact. At large K, their alternating binomial terms can be individually much larger than their sum, so straightforward floating point evaluation may lose accuracy. A finite expression is not automatically a stable evaluation scheme. For a direct truncation of the original series at n=N, the elementary certified re","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-d009dfbbc12565cd","tex":"\\sum_{n\\le N}","aliases":"∑_n≤ N sum le","display":"\\sum_{n\\le N}","kind":"use","excerpt":"ζ(4)+frac1ζ(6). These formulas are exact. At large K, their alternating binomial terms can be individually much larger than their sum, so straightforward floating point evaluation may lose accuracy. A finite expression is not automatically a stable evaluation scheme. For a direct truncation of the original series at n=N, the elementary certified re","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-67a3d7fe39aaa0a8","tex":"\\sum_{n\\le X}","aliases":"∑_n≤ X sum le","display":"\\sum_{n\\le X}","kind":"use","excerpt":"The simple coefficient mass bound also has an elementary proof. Since Λ(n)≤log X for n≤ X, and ∑_n≤ Xn^-1/2≤2sqrt X, their product gives the stated estimate. Compression may therefore be arranged without first proving any delicate prime cancellation theorem. The remaining arithmetic estimate lies in the compressed packet, not in the omitted Gaussia","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7c725e37e419fb92","tex":"\\sum_{n\\le X_J}","aliases":"∑_n≤ X_J sum le","display":"\\sum_{n\\le X_J}","kind":"use","excerpt":"=C^2/(4t_0). Then λ≤α J+O(sqrt J). For a fixed constant q>α, projection through R_J=lceil qJrceil has norm error bounded by exp(-frac12 I(qmidα)J+o(J)), I(qmidα)=qlog(q/α)-q+α. The coefficient mass of the positive prime packet through X_J=exp(J+O(sqrt J)) is at most ∑_n≤ X_JfracΛ(n)sqrt n ≤2sqrtX_Jlog X_J=exp(J/2+o(J)). Thus I(qmidα)>1 makes the ag","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"The Möbius function μ(n) is zero when a square of a prime divides n. Otherwise it equals (-1)^k, where k is the number of distinct prime divisors of n, and μ(1)=1. For an integer K≥1, put w_K(n)=frac1n^2(1-frac1n^2)^K, c_K=∑_n≥1μ(n)w_K(n). The series is absolutely convergent. Its natural active scale is n of order M=sqrt K. The coefficient c_K is c","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-df09d859c509da59","tex":"\\sum_p","aliases":"∑_p sum","display":"\\sum_p","kind":"use","excerpt":"The appearance of Λ can be traced directly to the Euler product in its first convergence half plane. There, logζ(s)=∑_p∑_k≥1fracp^-ksk. Differentiating gives -fracζ'(s)ζ(s) =∑_p∑_k≥1(log p)p^-ks =∑_n≥2fracΛ(n)n^s. The factor k from differentiation cancels the denominator k, which is why the coefficient of p^k is log p, rather than log(p^k). Absolut","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"An orthonormal Hermite basis replaces this almost singular coordinate system. If a finite packet is f=∑_r a_rψ_r,t,c, Parseval's identity gives f _2^2=∑_r|a_r|^2 with no unknown Gram loss. This exact equality is the repair. It remains possible for the arithmetic coefficients a_r to be large, or to contain a conjecture level obstruction in one of th","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-b3322aaf72ebe959","tex":"\\sum_{r>R}","aliases":"∑_r>R sum","display":"\\sum_{r>R}","kind":"use","excerpt":"e the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g_t,a · =e^-λ/2 ∑_r≥0fracσ^rsqrtr!ψ_r,t,c, σ · =fraca-c2sqrt t, λ=σ^2, ","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6a4250883c4968da","tex":"\\sum_{r\\ge0}","aliases":"∑_r≥0 sum ge","display":"\\sum_{r\\ge0}","kind":"use","excerpt":"e the orthonormal scaled Hermite functions ψ_r,t,c(u)= frac(2π t)^-1/4sqrt2^r r! H_r(fracu-csqrt2t)e^-(u-c)^2/(4t). Here H_r(x)=(-1)^r e^x^2(d^r/dx^r)e^-x^2 is the physicists' Hermite polynomial. If P_R is orthogonal projection onto indices 0≤ r≤ R, the coherent expansion gives g_t,a · =e^-λ/2 ∑_r≥0fracσ^rsqrtr!ψ_r,t,c, σ · =fraca-c2sqrt t, λ=σ^2, ","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4e248ced3f0236ad","tex":"\\sum_{s=5\\ell}^{8\\ell+D+1}","aliases":"∑_s=5ℓ^8ℓ+D+1 sum ell ell","display":"\\sum_{s=5\\ell}^{8\\ell+D+1}","kind":"use","excerpt":"For a modulus M, any one residue class contributes at most H/M+1 positive integers up to height H. The +1 accounts for an incomplete terminal period. Applying that estimate separately to every selected word yields the finite term W_ℓ. One may write it exactly as W_ℓ=∑_s=5ℓ^8ℓ+D+1binoms-15ℓ-1 =binom8ℓ+D+15ℓ. It can be enormous. Its size does not inv","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-88b136c3b9fa61ef","tex":"\\sum_{\\substack{m\\in I\\\\(m,Q)=1}}","aliases":"∑_substackm∈ I (m,Q)=1 sum substack in","display":"\\sum_{\\substack{m\\in I\\\\(m,Q)=1}}","kind":"use","excerpt":"For any finite interval I of remaining bases, distinguish its signed contribution from its absolute mass: S_I=∑_substackm∈ I (m,Q)=1μ(m)Δ_Qw_M^2(m), V_I=∑_substackm∈ I (m,Q)=1 |μ(m)| |Δ_Qw_M^2(m)|. The valid inequalities are |S_I|≤ V_I≤ A_Q(M) and |c_M^2|≤ A_Q(M). There is no general inequality |c_M^2|≤ V_I. Terms outside the chosen interval remain","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-6f97de955bad9f39","tex":"\\sum_{\\substack{m\\ge1\\\\(m,Q)=1}}","aliases":"∑_substackm≥1 (m,Q)=1 sum substack ge","display":"\\sum_{\\substack{m\\ge1\\\\(m,Q)=1}}","kind":"use","excerpt":"[Exact cube decomposition] For every K≥1 and squarefree Q, c_K=∑_substackm≥1 (m,Q)=1 μ(m)Δ_Qw_K(m). The condition (m,Q)=1 means that m and Q have greatest common divisor one.","count":3},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-8490d75a036c0cc0","tex":"\\sum_{\\ell^2\\mid m}","aliases":"∑_ℓ^2mid m sum ell mid","display":null,"kind":"use","excerpt":"Use the exact identity μ(m)^2=∑_ℓ^2mid mμ(ℓ). In the coprime sector only (ℓ,Q)=1 contributes. Inclusion and exclusion over the prime factors of Q counts the possible multiples of ℓ^2 in the interval with main term RMℓ^-2φ(Q)/Q and error O(2^d). There are O_R(sqrt M) values of ℓ, so the summed error is O_R(2^dsqrt M). Extending the main sum from ℓ≤s","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"[Concatenation] Let a word w have data (k,S,C) and a following word v have data (ℓ,U,D). Their concatenation has data (k+ℓ,S+U,3^ℓ C+2^S D). On every realizing source, its transfer is F_v∘ F_w. Substitute 2^S y=3^k n+C into 2^U z=3^ℓ y+D. Multiplication by 2^S gives 2^S+Uz=3^k+ℓn+3^ℓ C+2^S D. The exact cylinder theorem identifies the admissible int","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"In the smaller initial domain Re w>1/2, absolute summation justifies integration of each zero term: ∫_0^∞ e^tz^2+uz-wu du =frace^tz^2w-z. For a compact set of w avoiding all centered zeros, large ordinates make |w-z| large. The Gaussian coefficient then gives a summable bound for the meromorphic series. Near one distinct centered zero z_0, isolate ","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ff862dd8e88aa975","tex":"\\int_0^{\\infty}","aliases":"∫_0^∞ int infty","display":"\\int_0^{\\infty}","kind":"use","excerpt":"Under the hypothesis, [source reference] gives the asserted bound and determinant inequality. Conversely, suppose the packet is bounded on the positive half line. Its Laplace transform F_t(w)=∫_0^∞Θ_t(u)e^-wu du is holomorphic for Re w>0. For Re w>1/2, absolute convergence gives F_t(w)=∑_ρ∈ Z frace^tz_ρ^2w-z_ρ. The series on the right is meromorphi","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-a219d1c39f624909","tex":"\\int_J^{J+1}","aliases":"∫_J^J+1 int","display":"\\int_J^{J+1}","kind":"use","excerpt":"[Local energy criterion] Fix 0<t_0<T<∞. For positive integers J, set widetilde E_J=∫_t_0^T∫_J^J+1 |widetildeΘ_t(u)|^2 du dt. The Riemann hypothesis is equivalent to subexponential growth of widetilde E_J.","count":4},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-5a6ea2caa4884474","tex":"\\int_{t_0}^T","aliases":"∫_t_0^T int","display":"\\int_{t_0}^T","kind":"use","excerpt":"[Local energy criterion] Fix 0<t_0<T<∞. For positive integers J, set widetilde E_J=∫_t_0^T∫_J^J+1 |widetildeΘ_t(u)|^2 du dt. The Riemann hypothesis is equivalent to subexponential growth of widetilde E_J.","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"For example, take t=1, a=log1000, and b=log1001. The separation is approximately 0.00099950, the overlap is approximately 0.999999875, and the Gram condition number (1+κ)/(1-κ) is about 1.60×10^7. These values diagnose a coordinate system, not the size of a zeta coefficient. The asymptotic explanation is exact: 1-e^-(a-b)^2/(8t)∼frac(a-b)^28t, log(","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"For a concrete choice, let C=2sqrtt_0, so α=1, and take q=4. Then I(4mid1)=4log4-3≈2.54518>1. The individual norm error decays with exponent approximately -1.27259J. Multiplication by a coefficient mass with exponent at most J/2+o(J) still leaves decay with exponent approximately -0.77259J. This example verifies the scale balance in the compression","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"[Exact integer cylinder] Let w be a nonempty valuation word of length k and total depth S. A positive odd integer n realizes w if and only if 3^k n+C_k≡ 2^Spmod2^S+1. Consequently the realizing integers form one odd residue class modulo 2^S+1, whose residue is u_w≡(2^S-C_k)(3^k)^-1pmod2^S+1. If the word is realized, [source reference] has an odd te","count":10},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"For fixed k≥1, the sum of all relative odd cylinder densities is ∑_a_0,…,a_k-1≥12^-(a_0+⋯+a_k-1) =(∑_a≥12^-a)^k=1. This identity reflects the exact partition of positive odd integers by their first k valuations. It does not assert that valuations along one particular infinite orbit are independent samples. A density calculation describes how source","count":2},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"The word physical refers to the measure du in which the packet is actually evaluated. It does not mean that a separately chosen coefficient norm can be substituted. Completing the square gives the exact overlap ⟨ g_t,a,g_t,b⟩ =e^-(a-b)^2/(8t). For two translates, the Gram matrix has diagonal entries one and off diagonal entry κ=e^-(a-b)^2/(8t). Its","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"The word physical refers to the measure du in which the packet is actually evaluated. It does not mean that a separately chosen coefficient norm can be substituted. Completing the square gives the exact overlap ⟨ g_t,a,g_t,b⟩ =e^-(a-b)^2/(8t). For two translates, the Gram matrix has diagonal entries one and off diagonal entry κ=e^-(a-b)^2/(8t). Its","count":1},{"work":"collatz-and-riemann-readers","url":"notation/collatz-and-riemann-readers.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":",3), the recurrence gives C_1=1 and C_2=5, with total depth S=4. The affine expression is F_(1,3)(n)=frac9n+516. The exact cylinder requires 9n+5≡16pmod32. The inverse of 9 modulo 32 is 25, so its address is 19 modulo 32. Every positive representative can be written n=32t+19 with an integer t≥0. Direct substitution gives the full orbit segment 32t+","count":6},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With","count":25},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.","count":78},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":9},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":7},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-76b949b0834efe6d","tex":"a^{\\mathsf T}","aliases":"a^ T mathsf","display":"a^{\\mathsf T}","kind":"use","excerpt":"e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-c3b0f6b395c9c99c","tex":"A^\\vee","aliases":"A^vee vee","display":"A^\\vee","kind":"use","excerpt":"The pairing is perfect since 7nmiddet H. Its source is the canonical pairing of A[7] with A^vee[7]. The symmetric map H:A→ A^vee need not define an ample polarization: it is indefinite. The exact matrix identity gives contragredient covariance","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"(a_1,…,a_6)=(1,0,-1,-1,0,1), (b_1,…,b_6)=(0,1,-1,-1,1,0).","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-4f9a23e6ec67096e","tex":"a_6","aliases":"a_6","display":"a_6","kind":"use","excerpt":"(a_1,…,a_6)=(1,0,-1,-1,0,1), (b_1,…,b_6)=(0,1,-1,-1,1,0).","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"The cubic character χ(k)=a_k+ω b_k has integer coefficient vectors","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-59a706af3e94a284","tex":"\\mathrm{alg}","aliases":"alg mathrm","display":"\\mathrm{alg}","kind":"use","excerpt":"The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been pr","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix","count":8},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-e77f9eceea598e09","tex":"B'","aliases":"B'","display":"B'","kind":"use","excerpt":"D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-8d0c91c11bff49a2","tex":"b^{\\mathsf T}","aliases":"b^ T mathsf","display":"b^{\\mathsf T}","kind":"use","excerpt":"e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-5435c46c9f88e741","tex":"b_6","aliases":"b_6","display":"b_6","kind":"use","excerpt":"(a_1,…,a_6)=(1,0,-1,-1,0,1), (b_1,…,b_6)=(0,1,-1,-1,1,0).","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","excerpt":"The cubic character χ(k)=a_k+ω b_k has integer coefficient vectors","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-bb3cdc779ccbade8","tex":"\\bar J","aliases":"bar J bar","display":"\\bar J","kind":"use","excerpt":"Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With","count":9},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"is unambiguous and meromorphic. Under formal unit relabeling f_k↦ f_2kbmod7 the functions transform exactly as","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been pr","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-e2fdab254f7c2081","tex":"\\mathbb C^3","aliases":"C^3 mathbb","display":"\\mathbb C^3","kind":"use","excerpt":"Thus the exact principal-seven shifts z+kδ in the earlier gamma-divisor paper are the projections of k(P+P'). This holds **on the universal cover**; π does not automatically descend from A= C^3/ Z[ω]^3 to the original triptic quotient because of the extra Eisenstein periods. The missing deck multipliers cannot be discarded.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-c52f7a54c072ec1b","tex":"C_{\\mathrm{pol}}","aliases":"C_pol mathrm","display":"C_{\\mathrm{pol}}","kind":"use","excerpt":"The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been pr","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-61af2e48dff62660","tex":"D_\\chi","aliases":"D_χ chi","display":"D_\\chi","kind":"use","excerpt":"D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, det M=1.","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-20000cb9b2dd1efb","tex":"e_{7,H}","aliases":"e_7,H","display":"e_{7,H}","kind":"use","excerpt":"e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-207db81da6a3101d","tex":"E_\\omega","aliases":"E_ω omega","display":"E_\\omega","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-505a8aab02afef64","tex":"E_\\omega^3","aliases":"E_ω^3 omega","display":"E_\\omega^3","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-fad271e883a47357","tex":"\\mathrm{ell}","aliases":"ell mathrm","display":"\\mathrm{ell}","kind":"use","excerpt":"Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Equivalently, the primitive multiplicative-character projector applied to the additive function f(k)=e_7,H(kP,Q)=ζ_7^3k produces G_3/6:","count":8},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-b6a9755de225fef9","tex":"f'_k","aliases":"f'_k","display":"f'_k","kind":"use","excerpt":"D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","excerpt":"A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-5a06733f80278422","tex":"f_{2k\\bmod7}","aliases":"f_2kbmod7 bmod","display":"f_{2k\\bmod7}","kind":"use","excerpt":"is unambiguous and meromorphic. Under formal unit relabeling f_k↦ f_2kbmod7 the functions transform exactly as","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-4fac6558c4f0f59e","tex":"f_3","aliases":"f_3","display":"f_3","kind":"use","excerpt":"A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-8f053787a8e5d3cf","tex":"f_4","aliases":"f_4","display":"f_4","kind":"use","excerpt":"A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-3374f149cfeb50af","tex":"f_5","aliases":"f_5","display":"f_5","kind":"use","excerpt":"A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-8275a44e3c107b0c","tex":"f_6","aliases":"f_6","display":"f_6","kind":"use","excerpt":"A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-23af4eb7208f9cad","tex":"\\mathbb F_7","aliases":"F_7 mathbb","display":"\\mathbb F_7","kind":"use","excerpt":"Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f28730c126e1e18a","tex":"\\mathbb F_7^\\times","aliases":"F_7^× mathbb times","display":"\\mathbb F_7^\\times","kind":"use","excerpt":"The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-2f6b5c0f84584144","tex":"f_k","aliases":"f_k","display":"f_k","kind":"use","excerpt":"Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"use","excerpt":"G_r=χ(r)^-1G_1, G_r^3=7J.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-736fb9ac4210e748","tex":"G_3","aliases":"G_3","display":"G_3","kind":"use","excerpt":"Equivalently, the primitive multiplicative-character projector applied to the additive function f(k)=e_7,H(kP,Q)=ζ_7^3k produces G_3/6:","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-3f8720ad3f121275","tex":"G_r","aliases":"G_r","display":"G_r","kind":"use","excerpt":"Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-7eb857d1584a837b","tex":"G_r^3","aliases":"G_r^3","display":"G_r^3","kind":"use","excerpt":"G_r=χ(r)^-1G_1, G_r^3=7J.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":15},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-6bfe1e61875f685a","tex":"H_+","aliases":"H_+","display":"H_+","kind":"use","excerpt":"There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f74f6a57337ee481","tex":"H_{ij}","aliases":"H_ij","display":"H_{ij}","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","excerpt":"The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been pr","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With","count":8},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,","count":14},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"Indeed u represents β^2+56≡β^2, v represents 28-β≡-β, and w represents 15≡1 in F_7[β]/(β^3). The codifferent pairing extracts the coefficient of β^2. It is perfect, and pairs the socle against the top quotient. In particular, u^ TH vanishes on ker N^2=span(u,v) and is nonzero on w.","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"L= Z(β^2+2β)+ Z(-β-2)+15 Z","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, det M=1.","count":10},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-9905e7d4d709aef7","tex":"M^\\mathsf","aliases":"M^ mathsf","display":"M^\\mathsf","kind":"use","excerpt":"There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a2127b3e9668951d","tex":"M^{\\mathsf T}","aliases":"M^ T mathsf","display":"M^{\\mathsf T}","kind":"use","excerpt":"H=4 · -4 · 15 -4 · 1 · 0 15 · 0 · 0, det H=-225, M^ TH=HM.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-c693ce1e8e9e894c","tex":"M^{-1}","aliases":"M^-1","display":"M^{-1}","kind":"use","excerpt":"e_7,H(MX,M^-1Y)=e_7,H(X,Y).","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"Indeed u represents β^2+56≡β^2, v represents 28-β≡-β, and w represents 15≡1 in F_7[β]/(β^3). The codifferent pairing extracts the coefficient of β^2. It is perfect, and pairs the socle against the top quotient. In particular, u^ TH vanishes on ker N^2=span(u,v) and is nonzero on w.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"The pairing is perfect since 7nmiddet H. Its source is the canonical pairing of A[7] with A^vee[7]. The symmetric map H:A→ A^vee need not define an ample polarization: it is indefinite. The exact matrix identity gives contragredient covariance","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-fd71dfd389a79ac5","tex":"\\operatorname{ord}","aliases":"ord operatorname","display":"\\operatorname{ord}","kind":"use","excerpt":"There is a concrete obstruction to replacing the pair (A,B) by a single character-power product. The prior divisor formula gives ord_τ F=4, and δ has order seven modulo Λ_5= Zτ+ Zσ+(1/5) Z. At z=τ-2δ, exactly the term f_2 has a zero, hence","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-903f67cd60746bcf","tex":"\\operatorname{ord}_\\tau","aliases":"ord_τ operatorname tau","display":"\\operatorname{ord}_\\tau","kind":"use","excerpt":"There is a concrete obstruction to replacing the pair (A,B) by a single character-power product. The prior divisor formula gives ord_τ F=4, and δ has order seven modulo Λ_5= Zτ+ Zσ+(1/5) Z. At z=τ-2δ, exactly the term f_2 has a zero, hence","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"Define P=u/J, P'=u/bar J, and Q=w/bar J. The coefficient expansions 1/J=(-1-3ω)/7 and 1/bar J=(2+3ω)/7 yield","count":8},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"In the original analytic period row p=(τ,σ,1), let","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-cec4b8919e5f12a5","tex":"P'","aliases":"P'","display":"P'","kind":"use","excerpt":"Define P=u/J, P'=u/bar J, and Q=w/bar J. The coefficient expansions 1/J=(-1-3ω)/7 and 1/bar J=(2+3ω)/7 yield","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-74df595de253199d","tex":"P_7","aliases":"P_7","display":"P_7","kind":"use","excerpt":"Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"[u v w]^ TH[u v w]≡ 0 · 0 · 1 0 · 1 · 0 1 · 0 · 0pmod7.","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-5964bd51960ac9c2","tex":"\\mathrm{pol}","aliases":"pol mathrm","display":"\\mathrm{pol}","kind":"use","excerpt":"The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been pr","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Define P=u/J, P'=u/bar J, and Q=w/bar J. The coefficient expansions 1/J=(-1-3ω)/7 and 1/bar J=(2+3ω)/7 yield","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"The inherited shift law is F(z+σ)/F(z)=q_t(z)^-1, where t=-τ and q_t(z)=θ_0(z;t)^5/θ_0(5z;5t). Therefore for k=1,…,6,","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"The inherited shift law is F(z+σ)/F(z)=q_t(z)^-1, where t=-τ and q_t(z)=θ_0(z;t)^5/θ_0(5z;5t). Therefore for k=1,…,6,","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.","count":6},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-1ecf96b50772dd74","tex":"\\operatorname{Res}","aliases":"Res operatorname","display":"\\operatorname{Res}","kind":"use","excerpt":"Res_z=τ-2δD_χ(z)=4χ(2)=4ω.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-0e2d62a66cd8a3e6","tex":"\\operatorname{Res}_{z=\\tau-2\\delta}","aliases":"Res_z=τ-2δ operatorname tau delta","display":"\\operatorname{Res}_{z=\\tau-2\\delta}","kind":"use","excerpt":"Res_z=τ-2δD_χ(z)=4χ(2)=4ω.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"Indeed u represents β^2+56≡β^2, v represents 28-β≡-β, and w represents 15≡1 in F_7[β]/(β^3). The codifferent pairing extracts the coefficient of β^2. It is perfect, and pairs the socle against the top quotient. In particular, u^ TH vanishes on ker N^2=span(u,v) and is nonzero on w.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"H=4 · -4 · 15 -4 · 1 · 0 15 · 0 · 0, det H=-225, M^ TH=HM.","count":7},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","excerpt":"(A,B)⟼(B/A,1/A), T=-1 · 1 -1 · 0, T^3=I.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-3f5afdf091510edb","tex":"\\operatorname{Tr}_{K/\\mathbb Q}","aliases":"Tr_K/ Q operatorname mathbb","display":"\\operatorname{Tr}_{K/\\mathbb Q}","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been pr","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and","count":15},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a257f9d1ed0e54ad","tex":"u^{\\mathsf T}","aliases":"u^ T mathsf","display":"u^{\\mathsf T}","kind":"use","excerpt":"Indeed u represents β^2+56≡β^2, v represents 28-β≡-β, and w represents 15≡1 in F_7[β]/(β^3). The codifferent pairing extracts the coefficient of β^2. It is perfect, and pairs the socle against the top quotient. In particular, u^ TH vanishes on ker N^2=span(u,v) and is nonzero on w.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-fd49da06a993b46d","tex":"U_{\\mathrm{alg}}","aliases":"U_alg mathrm","display":"U_{\\mathrm{alg}}","kind":"use","excerpt":"The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been pr","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f3b9ec843ee07186","tex":"U_\\Gamma","aliases":"U_Γ Gamma","display":"U_\\Gamma","kind":"use","excerpt":"The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been pr","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-349daf8c6dd8d0cb","tex":"U_\\Gamma^3","aliases":"U_Γ^3 Gamma","display":"U_\\Gamma^3","kind":"use","excerpt":"The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been pr","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and","count":9},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a675678fa6689935","tex":"\\vee","aliases":"vee","display":"\\vee","kind":"use","excerpt":"The pairing is perfect since 7nmiddet H. Its source is the canonical pairing of A[7] with A^vee[7]. The symmetric map H:A→ A^vee need not define an ample polarization: it is indefinite. The exact matrix identity gives contragredient covariance","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and","count":8},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"use","excerpt":"There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"L= Z(β^2+2β)+ Z(-β-2)+15 Z","count":6},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"The first identity follows by index substitution and the second by the classical Gauss and Jacobi identities. All six cubes were also checked by exact coefficient convolution in Z[ω,z]/(ω^2+ω+1,z^7-1). Thus (G_r/β)^3=J, independent of the chosen nonzero dual direction. The individual cube-root branch is not fixed by this statement.","count":13},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-82a6f2d2c744dfea","tex":"z^7","aliases":"z^7","display":"z^7","kind":"use","excerpt":"The first identity follows by index substitution and the second by the classical Gauss and Jacobi identities. All six cubes were also checked by exact coefficient convolution in Z[ω,z]/(ω^2+ω+1,z^7-1). Thus (G_r/β)^3=J, independent of the chosen nonzero dual direction. The individual cube-root branch is not fixed by this statement.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.","count":17},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-ab15d1c00936ee24","tex":"\\beta^2","aliases":"β^2 beta","display":"\\beta^2","kind":"use","excerpt":"L= Z(β^2+2β)+ Z(-β-2)+15 Z","count":7},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-1210911b063d7dfc","tex":"\\beta^3","aliases":"β^3 beta","display":"\\beta^3","kind":"use","excerpt":"**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-38a0212391c9e2e9","tex":"\\Gamma_{\\mathrm{ell}}","aliases":"Γ_ell Gamma mathrm","display":"\\Gamma_{\\mathrm{ell}}","kind":"use","excerpt":"Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"τ=fracβ^2+2β15, σ=-fracβ+215, δ=fracτ+2σ+47=fracβ^2+56105.","count":11},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-c09239290921b6d4","tex":"\\zeta_7^3","aliases":"ζ_7^3 zeta","display":"\\zeta_7^3","kind":"use","excerpt":"e_7,H(P,Q)=ζ_7^3, e_7,H(P,P')=1.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-76738c7c39098112","tex":"\\zeta_7^{3k}","aliases":"ζ_7^3k zeta","display":"\\zeta_7^{3k}","kind":"use","excerpt":"Equivalently, the primitive multiplicative-character projector applied to the additive function f(k)=e_7,H(kP,Q)=ζ_7^3k produces G_3/6:","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-a0e51cfac9f120cf","tex":"\\zeta_7^{3r}","aliases":"ζ_7^3r zeta","display":"\\zeta_7^{3r}","kind":"use","excerpt":"The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-fdca4f874947d601","tex":"\\zeta_7^{a^{\\mathsf T}Hd-b^{\\mathsf T}Hc}","aliases":"ζ_7^a^ THd-b^ THc zeta mathsf mathsf","display":"\\zeta_7^{a^{\\mathsf T}Hd-b^{\\mathsf T}Hc}","kind":"use","excerpt":"e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-9b05e0da34666219","tex":"\\zeta_7^{rk}","aliases":"ζ_7^rk zeta","display":"\\zeta_7^{rk}","kind":"use","excerpt":"Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"The inherited shift law is F(z+σ)/F(z)=q_t(z)^-1, where t=-τ and q_t(z)=θ_0(z;t)^5/θ_0(5z;5t). Therefore for k=1,…,6,","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","excerpt":"The inherited shift law is F(z+σ)/F(z)=q_t(z)^-1, where t=-τ and q_t(z)=θ_0(z;t)^5/θ_0(5z;5t). Therefore for k=1,…,6,","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"There is a concrete obstruction to replacing the pair (A,B) by a single character-power product. The prior divisor formula gives ord_τ F=4, and δ has order seven modulo Λ_5= Zτ+ Zσ+(1/5) Z. At z=τ-2δ, exactly the term f_2 has a zero, hence","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-64100f915b9d565e","tex":"\\Lambda_5","aliases":"Λ_5 Lambda","display":"\\Lambda_5","kind":"use","excerpt":"There is a concrete obstruction to replacing the pair (A,B) by a single character-power product. The prior divisor formula gives ord_τ F=4, and δ has order seven modulo Λ_5= Zτ+ Zσ+(1/5) Z. At z=τ-2δ, exactly the term f_2 has a zero, hence","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"(Π_χ^-1f)(1)=frac16∑_k≠0χ(k)f(k)=fracG_36.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"The period projection π(v)=pv on the universal cover satisfies π(u/7)=δ, π(w)=1, and","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-74f9f064f393eb05","tex":"\\Pi_{\\chi^{-1}}","aliases":"Π_χ^-1 Pi chi","display":"\\Pi_{\\chi^{-1}}","kind":"use","excerpt":"(Π_χ^-1f)(1)=frac16∑_k≠0χ(k)f(k)=fracG_36.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"In the original analytic period row p=(τ,σ,1), let","count":8},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"In the original analytic period row p=(τ,σ,1), let","count":8},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H","count":10},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-7c934e3eb93a78aa","tex":"\\chi^{-1}","aliases":"χ^-1 chi","display":"\\chi^{-1}","kind":"use","excerpt":"(Π_χ^-1f)(1)=frac16∑_k≠0χ(k)f(k)=fracG_36.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-2e32874fc399e98b","tex":"\\chi_M","aliases":"χ_M chi","display":"\\chi_M","kind":"use","excerpt":"There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With","count":18},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f2f53155f7e59bd3","tex":"\\omega^2","aliases":"ω^2 omega","display":"\\omega^2","kind":"use","excerpt":"Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With","count":5},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"The pairing is perfect since 7nmiddet H. Its source is the canonical pairing of A[7] with A^vee[7]. The symmetric map H:A→ A^vee need not define an ample polarization: it is indefinite. The exact matrix identity gives contragredient covariance","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Hence unit multiplication acts on the seven cover labels by k↦2kpmod7, with explicit period shifts. The inverse unit acts by k↦4k. These are precisely the finite orbits present in the earlier divisor-transport calculations.","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","excerpt":"The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"A logarithmic derivative of a single-valued meromorphic function has an integer residue. Since 4ω∉ Z, **no single meromorphic scalar** has logarithmic derivative D_χ. The two-component meromorphic realization avoids this obstruction and carries an integral order-three monomial action.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-d2c602c786fba304","tex":"\\prod_{j=0}^{8k-1}","aliases":"∏_j=0^8k-1 prod","display":"\\prod_{j=0}^{8k-1}","kind":"use","excerpt":"F(ϵ(z+kδ))= F(ϵ z+2kδ) ∏_j=0^8k-1q_t(ϵ z+2kδ+jσ)^-1.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-1cfbf8cb0a797584","tex":"\\prod_{k=0}^6","aliases":"∏_k=0^6 prod","display":"\\prod_{k=0}^6","kind":"use","excerpt":"Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,","count":3},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-913ac2d9fb8649e2","tex":"\\sum_{k=1}^6","aliases":"∑_k=1^6 sum","display":"\\sum_{k=1}^6","kind":"use","excerpt":"D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-edb8cb451c5ccc6b","tex":"\\sum_{k=1}^{6}","aliases":"∑_k=1^6 sum","display":"\\sum_{k=1}^{6}","kind":"use","excerpt":"Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-982415998cf9e445","tex":"\\sum_{k\\ne0}","aliases":"∑_k≠0 sum ne","display":"\\sum_{k\\ne0}","kind":"use","excerpt":"(Π_χ^-1f)(1)=frac16∑_k≠0χ(k)f(k)=fracG_36.","count":1},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,","count":2},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"[u v w]^ TH[u v w]≡ 0 · 0 · 1 0 · 1 · 0 1 · 0 · 0pmod7.","count":4},{"work":"hilbert12-ambient-dual-slice","url":"notation/hilbert12-ambient-dual-slice.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"(A,B)⟼(B/A,1/A), T=-1 · 1 -1 · 0, T^3=I.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":42},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":95},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where","count":8},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-59a706af3e94a284","tex":"\\mathrm{alg}","aliases":"alg mathrm","display":"\\mathrm{alg}","kind":"use","excerpt":"The orbit laws are **conjectural at the analytic gamma special values**, even though Ψ^3=1 is an exact algebraic theorem. If all three special-value identities hold, the established branch separation identifies U_Γ=U_alg. The nontrivial analytical content has not been proved by the algebraic construction.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where","count":7},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f76a26dbc0d9fe26","tex":"B^2","aliases":"B^2","display":"B^2","kind":"use","excerpt":"D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where","count":8},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).","count":6},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"This identifies an obstruction rather than suppressing it. Every generic fiber d carries the same abstract C_3 symmetry. Symmetry, perfect-square discriminant, and Fourier diagonalization alone cannot force d=0. The missing input must be analytic and arithmetic normalization, not another abstract finite-group identity.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"*Proof.* Work over K(X). Write D=(β^2X-4)(β^2X-5). Expanding the symmetric functions gives the three exact numerator identities","count":10},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":34},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"Disc_Z(P_d)=(49d^2-13d+1)^2.","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-6f78b62eb28153fc","tex":"D_Y","aliases":"D_Y","display":"D_Y","kind":"use","excerpt":"D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).","count":8},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-4c7aa0231486fc78","tex":"D_Y'","aliases":"D_Y'","display":"D_Y'","kind":"use","excerpt":"N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c≠0.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f96fa2943190ed0f","tex":"d_\\Gamma","aliases":"d_Γ Gamma","display":"d_\\Gamma","kind":"use","excerpt":"ect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and su","count":7},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"*Proof.* Direct calculation gives det G= -20+3β^3=1 and trG=1. Cayley–Hamilton proves projective order three. Multiplying each fractional formula by its denominator and reducing modulo v^3-β v^2+1 proves the polynomial expressions. The three roots are distinct because the discriminant is one. The nontrivial maps cyclically permute them. square","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-2cb3235aa7dcdba6","tex":"\\operatorname{Disc}","aliases":"Disc operatorname","display":"\\operatorname{Disc}","kind":"use","excerpt":"Disc_Z(P_d)=(49d^2-13d+1)^2.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-767801b4df8bd969","tex":"\\operatorname{Disc}_Z","aliases":"Disc_Z operatorname","display":"\\operatorname{Disc}_Z","kind":"use","excerpt":"Disc_Z(P_d)=(49d^2-13d+1)^2.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":7},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-a63e596fa8b25a5a","tex":"e^{2\\pi in(s-z)}","aliases":"e^2π in(s-z) pi","display":"e^{2\\pi in(s-z)}","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-6c0caca83c2548f9","tex":"e^{2\\pi in(t+z)}","aliases":"e^2π in(t+z) pi","display":"e^{2\\pi in(t+z)}","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-7f6d22069c3f8d2b","tex":"e^{2\\pi ins}","aliases":"e^2π ins pi","display":"e^{2\\pi ins}","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-9e6c736f81811c78","tex":"e^{2\\pi int}","aliases":"e^2π int pi","display":"e^{2\\pi int}","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-23cf6af8786766c5","tex":"e^{2\\pi is}","aliases":"e^2π is pi","display":"e^{2\\pi is}","kind":"use","excerpt":"For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-d7b9b4237ea057ec","tex":"e^{2\\pi it}","aliases":"e^2π it pi","display":"e^{2\\pi it}","kind":"use","excerpt":"For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"e_1 · =β(1+7d), e_2 · =9β^2d, e_3 · =-1+20d.","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-42877425a0deabad","tex":"e_1^2","aliases":"e_1^2","display":"e_1^2","kind":"use","excerpt":"*Proof.* The norm identity follows from LL'=e_1^2-3e_2. The discriminant gives the square of the Vandermonde. Its orientation is fixed by the root ordering X,Φ(X),Φ^2(X), which at d=0 has Vandermonde +1. Hence","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-bb27e0e173139622","tex":"e_1^3","aliases":"e_1^3","display":"e_1^3","kind":"use","excerpt":"L^3+(L')^3=27e_3-e_1^3+3e_1LL'=(98d-13)q(d).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"e_1 · =β(1+7d), e_2 · =9β^2d, e_3 · =-1+20d.","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"e_1 · =β(1+7d), e_2 · =9β^2d, e_3 · =-1+20d.","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":8},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":8},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-bd129d79486e68ba","tex":"F_{(t-a)/n,(s+b)/n}","aliases":"F_(t-a)/n,(s+b)/n","display":"F_{(t-a)/n,(s+b)/n}","kind":"use","excerpt":"F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-0e70b66982cfe6ee","tex":"F_{t,s}","aliases":"F_t,s","display":"F_{t,s}","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-649b5646f959d9f3","tex":"F_{t_n,s_n}","aliases":"F_t_n,s_n","display":"F_{t_n,s_n}","kind":"use","excerpt":"Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"G=5 · -3β β^2 · -4","count":6},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f6a65d66c1c2674f","tex":"G^2","aliases":"G^2","display":"G^2","kind":"use","excerpt":"has determinant one and trace one. In particular G^2-G+I=0 and G^3=-I. Its action on P^1_K is","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-a64e5dd534945d39","tex":"G^3","aliases":"G^3","display":"G^3","kind":"use","excerpt":"has determinant one and trace one. In particular G^2-G+I=0 and G^3=-I. Its action on P^1_K is","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"Hence h(Y)=N_Y(Y)/D_Y(Y) is a Möbius transformation. On the algebraic cubic it inverts Y=Av^2+Bv+C.","count":6},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","excerpt":"Ψ=h^-1∘Φ∘ h,","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"has determinant one and trace one. In particular G^2-G+I=0 and G^3=-I. Its action on P^1_K is","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"ω^2+ω+1=0, β=7^1/3ω^2, J=2+3ω, overline J=-1-3ω.","count":4},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-29097134fc17258f","tex":"J^2","aliases":"J^2","display":"J^2","kind":"use","excerpt":"L^3=(J^2+49d)q(d), (L')^3=(overline J^ 2+49d)q(d).","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":10},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"L=X_0+ω X_1+ω^2X_2, L'=X_0+ω^2X_1+ω X_2.","count":7},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-34c05dc8f5073293","tex":"L'","aliases":"L'","display":"L'","kind":"use","excerpt":"L=X_0+ω X_1+ω^2X_2, L'=X_0+ω^2X_1+ω X_2.","count":7},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-42ada035ee0b4eaf","tex":"L^3","aliases":"L^3","display":"L^3","kind":"use","excerpt":"L^3=(J^2+49d)q(d), (L')^3=(overline J^ 2+49d)q(d).","count":4},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"where h(Y)=N_Y(Y)/D_Y(Y). If M_h is its 2×2 coefficient matrix, then the matrix for Ψ is M_h^-1GM_h. Its determinant is one and trace is one, so it also has order three.","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-7ab54b2c679702c1","tex":"M_h","aliases":"M_h","display":"M_h","kind":"use","excerpt":"where h(Y)=N_Y(Y)/D_Y(Y). If M_h is its 2×2 coefficient matrix, then the matrix for Ψ is M_h^-1GM_h. Its determinant is one and trace is one, so it also has order three.","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-8932c13d5097f2f3","tex":"M_h^{-1}","aliases":"M_h^-1","display":"M_h^{-1}","kind":"use","excerpt":"where h(Y)=N_Y(Y)/D_Y(Y). If M_h is its 2×2 coefficient matrix, then the matrix for Ψ is M_h^-1GM_h. Its determinant is one and trace is one, so it also has order three.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).","count":11},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are","count":4},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-05df5fdce379b495","tex":"N_Y","aliases":"N_Y","display":"N_Y","kind":"use","excerpt":"D_Y(Y) · =A(Y-C)+B^2+β AB, N_Y(Y) · =(Y-C)(B+β A)-A^2, c · =3(652β^2-1230β-33).","count":8},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-c250365f17a2cc8e","tex":"N_Y'","aliases":"N_Y'","display":"N_Y'","kind":"use","excerpt":"N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c≠0.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-712a4047f09e3ce3","tex":"\\overline J","aliases":"overline J overline","display":"\\overline J","kind":"use","excerpt":"ω^2+ω+1=0, β=7^1/3ω^2, J=2+3ω, overline J=-1-3ω.","count":4},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-0a9f2d266ccedc50","tex":"\\overline J^{\\,2}","aliases":"overline J^ 2 overline","display":"\\overline J^{\\,2}","kind":"use","excerpt":"L^3=(J^2+49d)q(d), (L')^3=(overline J^ 2+49d)q(d).","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-ef958519c2ac9923","tex":"\\mathbb P","aliases":"P mathbb","display":"\\mathbb P","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f794c3b07190d249","tex":"\\mathbb P^1_K","aliases":"P^1_K mathbb","display":"\\mathbb P^1_K","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-14f8900152d6716a","tex":"P_d","aliases":"P_d","display":"P_d","kind":"use","excerpt":"P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Since Q(U)=U^3R(U+U^-1), this is exactly the original reciprocal-sextic membership target, now realized as an ambient cubic invariant.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"Define q(d)=1-13d+49d^2.","count":7},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"R(Y)=Y^3+(6β^2-14β+2)Y^2 +(4β^2-6β-1)Y+94β^2-124β-107.","count":6},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-50fe5af99a640139","tex":"r^{N+1}","aliases":"r^N+1","display":"r^{N+1}","kind":"use","excerpt":"|tail|≤frac10r^N+1(N+1)(1-r)^2(1-q).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"τ=fracβ^2+2β15, s=-fracβ+215, t=-τ,","count":7},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-1c7310c194cdfc9d","tex":"s_{10}","aliases":"s_10","display":"s_{10}","kind":"use","excerpt":"(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f12fba916f1e9557","tex":"s_{23}","aliases":"s_23","display":"s_{23}","kind":"use","excerpt":"(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-91ec47101c58ab98","tex":"s_n","aliases":"s_n","display":"s_n","kind":"use","excerpt":"Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"use","excerpt":"*Proof.* Direct calculation gives det G= -20+3β^3=1 and trG=1. Cayley–Hamilton proves projective order three. Multiplying each fractional formula by its denominator and reducing modulo v^3-β v^2+1 proves the polynomial expressions. The three roots are distinct because the discriminant is one. The nontrivial maps cyclically permute them. square","count":4},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"τ=fracβ^2+2β15, s=-fracβ+215, t=-τ,","count":6},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","excerpt":"(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f6a69f498fc97303","tex":"t_{10}","aliases":"t_10","display":"t_{10}","kind":"use","excerpt":"(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-39395bc8b9065087","tex":"t_{23}","aliases":"t_23","display":"t_{23}","kind":"use","excerpt":"(t_0,s_0)=(t,s), (t_10,s_10)=((t-1)/10,(s+3)/10), (t_23,s_23)=((t-5)/23,(s-2)/23).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-c2ed7a9fa236bbf0","tex":"t_n","aliases":"t_n","display":"t_n","kind":"use","excerpt":"Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"*Proof.* Direct calculation gives det G= -20+3β^3=1 and trG=1. Cayley–Hamilton proves projective order three. Multiplying each fractional formula by its denominator and reducing modulo v^3-β v^2+1 proves the polynomial expressions. The three roots are distinct because the discriminant is one. The nontrivial maps cyclically permute them. square","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":6},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"Since Q(U)=U^3R(U+U^-1), this is exactly the original reciprocal-sextic membership target, now realized as an ambient cubic invariant.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-8e8dbfc6926d65a2","tex":"U^3","aliases":"U^3","display":"U^3","kind":"use","excerpt":"Since Q(U)=U^3R(U+U^-1), this is exactly the original reciprocal-sextic membership target, now realized as an ambient cubic invariant.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-fd49da06a993b46d","tex":"U_{\\mathrm{alg}}","aliases":"U_alg mathrm","display":"U_{\\mathrm{alg}}","kind":"use","excerpt":"The orbit laws are **conjectural at the analytic gamma special values**, even though Ψ^3=1 is an exact algebraic theorem. If all three special-value identities hold, the established branch separation identifies U_Γ=U_alg. The nontrivial analytical content has not been proved by the algebraic construction.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-167a8b5df4139bcf","tex":"U_n","aliases":"U_n","display":"U_n","kind":"use","excerpt":"Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-2d3cb2ada42e8a19","tex":"U_n^{-1}","aliases":"U_n^-1","display":"U_n^{-1}","kind":"use","excerpt":"Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f3b9ec843ee07186","tex":"U_\\Gamma","aliases":"U_Γ Gamma","display":"U_\\Gamma","kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-40c5eeecd3c26b7b","tex":"U_\\Gamma^{-1}","aliases":"U_Γ^-1 Gamma","display":"U_\\Gamma^{-1}","kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"The existing class field construction supplies an irreducible, everywhere unramified cyclic cubic model generated by a root v of f. Its field and ramification properties are inherited from the earlier archive and are not re-proved here. The polynomial discriminant is 1.","count":14},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where","count":8},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-7d77b9a1d8369d4e","tex":"v^3","aliases":"v^3","display":"v^3","kind":"use","excerpt":"*Proof.* Direct calculation gives det G= -20+3β^3=1 and trG=1. Cayley–Hamilton proves projective order three. Multiplying each fractional formula by its denominator and reducing modulo v^3-β v^2+1 proves the polynomial expressions. The three roots are distinct because the discriminant is one. The nontrivial maps cyclically permute them. square","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-6cdebc18a43354f2","tex":"v_\\Gamma","aliases":"v_Γ Gamma","display":"v_\\Gamma","kind":"use","excerpt":"v_Γ(z)=fracN_Y(F(z)+F(z)^-1)D_Y(F(z)+F(z)^-1).","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":17},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"X_0=X, X_1=Φ(X), X_2=Φ^2(X),","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"X_0=X, X_1=Φ(X), X_2=Φ^2(X),","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"X_0=X, X_1=Φ(X), X_2=Φ^2(X),","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":32},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-0e7659cd4ef8f947","tex":"Y^2","aliases":"Y^2","display":"Y^2","kind":"use","excerpt":"R(Y)=Y^3+(6β^2-14β+2)Y^2 +(4β^2-6β-1)Y+94β^2-124β-107.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-2725a0e30a718064","tex":"Y^3","aliases":"Y^3","display":"Y^3","kind":"use","excerpt":"R(Y)=Y^3+(6β^2-14β+2)Y^2 +(4β^2-6β-1)Y+94β^2-124β-107.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-d842f0833fbfd1bc","tex":"Y_0","aliases":"Y_0","display":"Y_0","kind":"use","excerpt":"Let Y_0=Y, Y_1=Ψ(Y), Y_2=Ψ^2(Y). Their trace is a rational function of the invariant d=d(h(Y)). Exact elimination gives","count":7},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-7ef58ddaab7b5984","tex":"Y_1","aliases":"Y_1","display":"Y_1","kind":"use","excerpt":"Let Y_0=Y, Y_1=Ψ(Y), Y_2=Ψ^2(Y). Their trace is a rational function of the invariant d=d(h(Y)). Exact elimination gives","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-09a5ab7c920c252f","tex":"Y_{10}","aliases":"Y_10","display":"Y_{10}","kind":"use","excerpt":"Y_23=Ψ(Y_0), Y_10=Ψ^2(Y_0),","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-57080b7d6f26d5e6","tex":"Y_2","aliases":"Y_2","display":"Y_2","kind":"use","excerpt":"Let Y_0=Y, Y_1=Ψ(Y), Y_2=Ψ^2(Y). Their trace is a rational function of the invariant d=d(h(Y)). Exact elimination gives","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-0ce6276eed1e5683","tex":"Y_{23}","aliases":"Y_23","display":"Y_{23}","kind":"use","excerpt":"Y_23=Ψ(Y_0), Y_10=Ψ^2(Y_0),","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-3859ee776b04fc90","tex":"Y_n","aliases":"Y_n","display":"Y_n","kind":"use","excerpt":"Define U_n=F_t_n,s_n(1/3) and Y_n=U_n+U_n^-1. The exact sufficient analytic obligations now are","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-311e5b2a008e0a18","tex":"Y_{\\rm alg}","aliases":"Y_rm alg rm","display":"Y_{\\rm alg}","kind":"use","excerpt":"The previously constructed algebraic candidate satisfies Y_rm alg=Av^2+Bv+C where","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-54d169b859ae3489","tex":"Y_\\Gamma","aliases":"Y_Γ Gamma","display":"Y_\\Gamma","kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"ect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and su","count":18},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-1a49e6bec04c7b56","tex":"Z^2","aliases":"Z^2","display":"Z^2","kind":"use","excerpt":"P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f64931971fecd572","tex":"Z^3","aliases":"Z^3","display":"Z^3","kind":"use","excerpt":"P_d(Z)=Z^3-β(1+7d)Z^2+9β^2dZ+1-20d.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"where h(Y)=N_Y(Y)/D_Y(Y). If M_h is its 2×2 coefficient matrix, then the matrix for Ψ is M_h^-1GM_h. Its determinant is one and trace is one, so it also has order three.","count":7},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":34},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-ab15d1c00936ee24","tex":"\\beta^2","aliases":"β^2 beta","display":"\\beta^2","kind":"use","excerpt":"τ=fracβ^2+2β15, s=-fracβ+215, t=-τ,","count":20},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-1210911b063d7dfc","tex":"\\beta^3","aliases":"β^3 beta","display":"\\beta^3","kind":"use","excerpt":"For the number field K= Q(β), β^3=7, the previously constructed unramified cubic is generated by a root of f(X)=X^3-β X^2+1. Its cyclic Galois action extends to a globally defined order three Möbius transformation on P^1_K. The quotient coordinate d(X) presents a rational one parameter family of cyclic cubic polynomials whose discriminants are perf","count":6},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"ect squares. Its special fiber at d=0 is the original arithmetic cubic. The associated two primitive Fourier resolvents have exact, explicitly deformed norm and cube formulas. A rational inverse of the earlier quadratic arithmetic coordinate then turns the elliptic gamma function into a canonical meromorphic candidate for the cubic generator and su","count":12},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-66df19f9c2e3aaac","tex":"\\Gamma_{\\rm ell}","aliases":"Γ_rm ell Gamma rm","display":"\\Gamma_{\\rm ell}","kind":"use","excerpt":"F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s), U_Γ=F(1/3), Y_Γ=U_Γ+U_Γ^-1.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"Y_0+Y_1+Y_2 =-6β^2+14β-2 +fracκ d1+θ d,","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"Y_0+Y_1+Y_2 =-6β^2+14β-2 +fracκ d1+θ d,","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"τ=fracβ^2+2β15, s=-fracβ+215, t=-τ,","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"Φ(X)=frac5X-3ββ^2X-4, Φ^2(X)=frac4X-3ββ^2X-5, Φ^3(X)=X.","count":10},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-aad33d2dbafe3856","tex":"\\Phi^2","aliases":"Φ^2 Phi","display":"\\Phi^2","kind":"use","excerpt":"Φ(X)=frac5X-3ββ^2X-4, Φ^2(X)=frac4X-3ββ^2X-5, Φ^3(X)=X.","count":5},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-1f55667a7394eaf6","tex":"\\Phi^3","aliases":"Φ^3 Phi","display":"\\Phi^3","kind":"use","excerpt":"Φ(X)=frac5X-3ββ^2X-4, Φ^2(X)=frac4X-3ββ^2X-5, Φ^3(X)=X.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Ψ=h^-1∘Φ∘ h,","count":9},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-06351a0b74281088","tex":"\\Psi^2","aliases":"Ψ^2 Psi","display":"\\Psi^2","kind":"use","excerpt":"Let Y_0=Y, Y_1=Ψ(Y), Y_2=Ψ^2(Y). Their trace is a rational function of the invariant d=d(h(Y)). Exact elimination gives","count":3},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-0f5d7c21634eb0e9","tex":"\\Psi^3","aliases":"Ψ^3 Psi","display":"\\Psi^3","kind":"use","excerpt":"The orbit laws are **conjectural at the analytic gamma special values**, even though Ψ^3=1 is an exact algebraic theorem. If all three special-value identities hold, the established branch separation identifies U_Γ=U_alg. The nontrivial analytical content has not been proved by the algebraic construction.","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"ω^2+ω+1=0, β=7^1/3ω^2, J=2+3ω, overline J=-1-3ω.","count":8},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f2f53155f7e59bd3","tex":"\\omega^2","aliases":"ω^2 omega","display":"\\omega^2","kind":"use","excerpt":"ω^2+ω+1=0, β=7^1/3ω^2, J=2+3ω, overline J=-1-3ω.","count":6},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"For real z, writing r=max(|e^2π it|,|e^2π is|) and q=min(…), the tail after N terms is bounded by","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f8468cc287609c57","tex":"\\prod_{u=1-n}^0","aliases":"∏_u=1-n^0 prod","display":"\\prod_{u=1-n}^0","kind":"use","excerpt":"F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-f18de406d9a1804e","tex":"\\prod_{v=0}^{n-1}","aliases":"∏_v=0^n-1 prod","display":"\\prod_{v=0}^{n-1}","kind":"use","excerpt":"F_(t-a)/n,(s+b)/n(z) =∏_u=1-n^0∏_v=0^n-1 F_t,s(z+ufract-an+vfracs+bn).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-0db207badc6ca948","tex":"\\sum_{\\substack{n\\ge1\\\\5\\nmid n}}","aliases":"∑_substackn≥1 5nmid n sum substack ge nmid","display":null,"kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"Ψ=h^-1∘Φ∘ h,","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"N_Y'(Y)D_Y(Y)-N_Y(Y)D_Y'(Y)=c≠0.","count":2},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"|tail|≤frac10r^N+1(N+1)(1-r)^2(1-q).","count":1},{"work":"hilbert12-analytic-fourier","url":"notation/hilbert12-analytic-fourier.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"log F_t,s(z)=-5∑_substackn≥1 5nmid n frace^2π in(t+z)-e^2π in(s-z) n(1-e^2π int)(1-e^2π ins).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.","count":31},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"This research package continues the pure cubic experiment K= Q(β), β^3=7.","count":77},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"q_a(z)=fracθ_0(z;a)^5θ_0(5z;5a).","count":9},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"The covector a_0=(1,0,0)^T, corresponding to the oriented complementary pairing with w, has an orbit of length **21** under this matrix. Its first three iterates are","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1a4867c2d25bf8ef","tex":"a_x","aliases":"a_x","display":"a_x","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-09793372ec0658af","tex":"a_{x,m}","aliases":"a_x,m","display":"a_{x,m}","kind":"use","excerpt":"ρ^m z_x=W_x,m+a_x,mτ+b_x,mσ+c_x,m.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"ρ z_k,ℓ,j= z_k',ℓ',j'+aτ+bσ+c","count":6},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-bec856d92c5ab12c","tex":"b_x","aliases":"b_x","display":"b_x","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-722213589bcf074c","tex":"b_{x,m}","aliases":"b_x,m","display":"b_{x,m}","kind":"use","excerpt":"ρ^m z_x=W_x,m+a_x,mτ+b_x,mσ+c_x,m.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"Write (k',ℓ',j')=(A_0,B_0,C_0)bmod7. Then","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,","count":9},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"ρ z_k,ℓ,j= z_k',ℓ',j'+aτ+bσ+c","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"f_m(x)=C_m(x)f_m-1(Tx), x∈ V.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c52f7a54c072ec1b","tex":"C_{\\mathrm{pol}}","aliases":"C_pol mathrm","display":"C_{\\mathrm{pol}}","kind":"use","excerpt":"Im(C_polU_Γ^3)=0","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a14ba09b678988e7","tex":"C_{T^{-m}x}^{(m)}","aliases":"C_T^-mx^(m)","display":"C_{T^{-m}x}^{(m)}","kind":"use","excerpt":"F(z_x)=ε^2 ∏_m=1^∞ (C_T^-mx^(m))^-1.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-731e9fcfb0f6710f","tex":"c_x","aliases":"c_x","display":"c_x","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-e9b13a281a002356","tex":"c_{x,m}","aliases":"c_x,m","display":"c_{x,m}","kind":"use","excerpt":"ρ^m z_x=W_x,m+a_x,mτ+b_x,mσ+c_x,m.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8430f941a1447de4","tex":"C_x^{(m)}","aliases":"C_x^(m)","display":"C_x^{(m)}","kind":"use","excerpt":"C_x^(m)= P(q_t^-1;b_x,m,W_x,m,σ) P(q_σ;a_x,m,W_x,m+b_x,mσ,τ).","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-888fcfdfd0c1eb40","tex":"d_x","aliases":"d_x","display":"d_x","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-bacebf63d78b12d5","tex":"d_{x,m}","aliases":"d_x,m","display":"d_{x,m}","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"Since δ,η,ξ are the images of u/7,v/7,w/7, and det[u v w]=1, the 343 labels represent all of tfrac17Λ/Λ. Here this is torsion of the rank-three lattice quotient, not a claim that the nondiscrete rank-three quotient is an ordinary elliptic curve.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f6c9f6a877b77ba8","tex":"e^{-2\\pi i nq z}","aliases":"e^-2π i nq z pi","display":"e^{-2\\pi i nq z}","kind":"use","excerpt":"-5∑_substackq≥1 5nmid nq fracP^q e^2π i nq z-Q^q e^-2π i nq z q(1-P^q)(1-Q^q).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-06cd04dac0c4ab32","tex":"e^{-2\\pi imz}","aliases":"e^-2π imz pi","display":"e^{-2\\pi imz}","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-0bd45fc4e5e7b00a","tex":"e^{2\\pi i nq z}","aliases":"e^2π i nq z pi","display":"e^{2\\pi i nq z}","kind":"use","excerpt":"-5∑_substackq≥1 5nmid nq fracP^q e^2π i nq z-Q^q e^-2π i nq z q(1-P^q)(1-Q^q).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-33ab1daa7140e655","tex":"e^{2\\pi i/7}","aliases":"e^2π i/7 pi","display":"e^{2\\pi i/7}","kind":"use","excerpt":"Fix the oriented seventh-root phase ζ_7=e^2π i/7. The additive character against the top dual direction is","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-3decd313297f1e0a","tex":"e^{2\\pi imz}","aliases":"e^2π imz pi","display":"e^{2\\pi imz}","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d7b9b4237ea057ec","tex":"e^{2\\pi it}","aliases":"e^2π it pi","display":"e^{2\\pi it}","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d2a3265d8e24ab0a","tex":"e^{2\\pi i\\sigma}","aliases":"e^2π iσ pi sigma","display":"e^{2\\pi i\\sigma}","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-fad271e883a47357","tex":"\\mathrm{ell}","aliases":"ell mathrm","display":"\\mathrm{ell}","kind":"use","excerpt":"F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"Let V=( Z/7)^3, ζ=exp(2π i/7), and let F be the ordinary finite Fourier transform","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).","count":11},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Because R^3-I≠0, the order of R in GL_3( F_7) is exactly 21. Its one-dimensional eigenline has scalar action 4, of order three. Thus the order-three torsion packet is the semisimple slice of a nonsemisimple order-21 arithmetic operator.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"Let V=( Z/7)^3, ζ=exp(2π i/7), and let F be the ordinary finite Fourier transform","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"f_x^(m)=F(ρ^m z_x), m≥0.","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-23af4eb7208f9cad","tex":"\\mathbb F_7","aliases":"F_7 mathbb","display":"\\mathbb F_7","kind":"use","excerpt":"Because R^3-I≠0, the order of R in GL_3( F_7) is exactly 21. Its one-dimensional eigenline has scalar action 4, of order three. Thus the order-three torsion packet is the semisimple slice of a nonsemisimple order-21 arithmetic operator.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"use","excerpt":"f_m(x)=C_m(x)f_m-1(Tx), x∈ V.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-54c97335903d9757","tex":"f_{m-1}","aliases":"f_m-1","display":"f_{m-1}","kind":"use","excerpt":"f_m(x)=C_m(x)f_m-1(Tx), x∈ V.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c59d5ece8cff9ce3","tex":"f_{T x}^{(m-1)}","aliases":"f_T x^(m-1)","display":"f_{T x}^{(m-1)}","kind":"use","excerpt":"f_x^(m)=C_x^(m) f_T x^(m-1).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-891459aa16db4428","tex":"f_x^{(m)}","aliases":"f_x^(m)","display":"f_x^{(m)}","kind":"use","excerpt":"f_x^(m)=F(ρ^m z_x), m≥0.","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"G_H=0 · 0 · 1 0 · 1 · 0 1 · 0 · 0.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-b375d87f28adb804","tex":"G_H","aliases":"G_H","display":"G_H","kind":"use","excerpt":"G_H=0 · 0 · 1 0 · 1 · 0 1 · 0 · 0.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c5d62edc1cdf57e7","tex":"\\mathrm{GL}","aliases":"GL mathrm","display":"\\mathrm{GL}","kind":"use","excerpt":"Because R^3-I≠0, the order of R in GL_3( F_7) is exactly 21. Its one-dimensional eigenline has scalar action 4, of order three. Thus the order-three torsion packet is the semisimple slice of a nonsemisimple order-21 arithmetic operator.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a282d7ccbead79d6","tex":"\\mathrm{GL}_3","aliases":"GL_3 mathrm","display":"\\mathrm{GL}_3","kind":"use","excerpt":"Because R^3-I≠0, the order of R in GL_3( F_7) is exactly 21. Its one-dimensional eigenline has scalar action 4, of order three. Thus the order-three torsion packet is the semisimple slice of a nonsemisimple order-21 arithmetic operator.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"H=4 · -4 · 15 -4 · 1 · 0 15 · 0 · 0, M^T H=HM,","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","excerpt":"Im(C_polU_Γ^3)=0","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"A successful next theorem would compute a character-sensitive, correctly based analytic transport and prove that its arithmetic evaluation agrees with the oriented Gauss/Jacobi coordinate J=2+3ω. The fixed desired scalar identity remains","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"z_k,ℓ,j=z_0+kδ+ℓη+jξ, k,ℓ,j∈{0,…,6}.","count":20},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d853ad34ce9c0840","tex":"j'","aliases":"j'","display":"j'","kind":"use","excerpt":"Write (k',ℓ',j')=(A_0,B_0,C_0)bmod7. Then","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"This research package continues the pure cubic experiment K= Q(β), β^3=7.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"The seven-point packet has arguments z_0+kδ. The first refinement consists of z_0+kδ+ℓη, with 49 labels. The full ambient packet is","count":17},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-7a0a635a73cee305","tex":"k'","aliases":"k'","display":"k'","kind":"use","excerpt":"Write (k',ℓ',j')=(A_0,B_0,C_0)bmod7. Then","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c3032f26051dc807","tex":"L_{t,\\sigma}","aliases":"L_t,σ sigma","display":"L_{t,\\sigma}","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, R=M^-1=8 · -4 · 15 1 · 0 · 0 2 · -1 · 4.","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"f_x^(m)=F(ρ^m z_x), m≥0.","count":15},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c693ce1e8e9e894c","tex":"M^{-1}","aliases":"M^-1","display":"M^{-1}","kind":"use","excerpt":"M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, R=M^-1=8 · -4 · 15 1 · 0 · 0 2 · -1 · 4.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8868d63653f7b040","tex":"M^T","aliases":"M^T","display":"M^T","kind":"use","excerpt":"H=4 · -4 · 15 -4 · 1 · 0 15 · 0 · 0, M^T H=HM,","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"N_nF(z)= F(nz), · 5nmid n, 1, · 5mid n.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define","count":8},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"use","excerpt":"N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"For an integer n, define the oriented finite product","count":15},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a859726df7c3ca43","tex":"\\mathcal N_7","aliases":"N_7 mathcal","display":"\\mathcal N_7","kind":"use","excerpt":"N_7F(1/3)=F(7/3)=F(1/3)=U_Γ.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-93cd7314d041080b","tex":"\\mathcal N_n","aliases":"N_n mathcal","display":"\\mathcal N_n","kind":"use","excerpt":"N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":6},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"As ρ^m z_x→0, all f_x^(m)→ F(0)=ε^2. Every multiplier is nonzero at the chosen arguments, and C_x^(m)=1+O(|ρ|^m-1).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f0130987c1cdd63d","tex":"P^{-(n-1)q}","aliases":"P^-(n-1)q","display":"P^{-(n-1)q}","kind":"use","excerpt":"∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-6ef9147e2bbff9d5","tex":"P^m","aliases":"P^m","display":"P^m","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-ab7ed8333799c173","tex":"P^{nq}","aliases":"P^nq","display":"P^{nq}","kind":"use","excerpt":"∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-7729a9889503d79f","tex":"P^q","aliases":"P^q","display":"P^q","kind":"use","excerpt":"∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-2ecf01b6f7c959fa","tex":"P^{qu}","aliases":"P^qu","display":"P^{qu}","kind":"use","excerpt":"∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-5964bd51960ac9c2","tex":"\\mathrm{pol}","aliases":"pol mathrm","display":"\\mathrm{pol}","kind":"use","excerpt":"Im(C_polU_Γ^3)=0","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"This research package continues the pure cubic experiment K= Q(β), β^3=7.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"F(z+1)=F(z), F(z+τ)=q_σ(z)F(z), F(z+σ)=q_t(z)^-1F(z),","count":10},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-09c21cfcdc6e5230","tex":"Q^m","aliases":"Q^m","display":"Q^m","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-572512a826276966","tex":"Q^{nq}","aliases":"Q^nq","display":"Q^{nq}","kind":"use","excerpt":"∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-49f9a0a45a944f39","tex":"Q^q","aliases":"Q^q","display":"Q^q","kind":"use","excerpt":"∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-6e47571c6993380f","tex":"Q^{qv}","aliases":"Q^qv","display":"Q^{qv}","kind":"use","excerpt":"∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-688eed70a27e98ff","tex":"q_1","aliases":"q_1","display":"q_1","kind":"use","excerpt":"q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a2965900c61ab851","tex":"q_2","aliases":"q_2","display":"q_2","kind":"use","excerpt":"q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8ae9808cb0f6e77d","tex":"q_3","aliases":"q_3","display":"q_3","kind":"use","excerpt":"q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-55d4600438de195a","tex":"q_a","aliases":"q_a","display":"q_a","kind":"use","excerpt":"q_a(z)=fracθ_0(z;a)^5θ_0(5z;5a).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"F(z+1)=F(z), F(z+τ)=q_σ(z)F(z), F(z+σ)=q_t(z)^-1F(z),","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-3371786a0f3cd417","tex":"q_t^{-1}","aliases":"q_t^-1","display":"q_t^{-1}","kind":"use","excerpt":"C_x^(m)= P(q_t^-1;b_x,m,W_x,m,σ) P(q_σ;a_x,m,W_x,m+b_x,mσ,τ).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-41ef2a9c0b7d5d7c","tex":"q_\\sigma","aliases":"q_σ sigma","display":"q_\\sigma","kind":"use","excerpt":"F(z+1)=F(z), F(z+τ)=q_σ(z)F(z), F(z+σ)=q_t(z)^-1F(z),","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, R=M^-1=8 · -4 · 15 1 · 0 · 0 2 · -1 · 4.","count":7},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-938c0793ab6dd1b4","tex":"R^{21}","aliases":"R^21","display":"R^{21}","kind":"use","excerpt":"R^3-I=2N+5N^2pmod7, R^7=4Ipmod7, R^21=Ipmod7.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-508222661b47a325","tex":"R^3","aliases":"R^3","display":"R^3","kind":"use","excerpt":"R^3-I=2N+5N^2pmod7, R^7=4Ipmod7, R^21=Ipmod7.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d6d881fd83e7a2b0","tex":"R^7","aliases":"R^7","display":"R^7","kind":"use","excerpt":"R^3-I=2N+5N^2pmod7, R^7=4Ipmod7, R^21=Ipmod7.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1381075c5bc7ca5f","tex":"R^{m-1}","aliases":"R^m-1","display":"R^{m-1}","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"use","excerpt":"When 5nmid n, the condition 5nmid nq is exactly 5nmid q, giving L_t,σ(nz). When 5mid n, no terms remain and the logarithm is zero. Exponentiating proves the equality in the convergence region. Meromorphic continuation gives the stated identities. square","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"u=(1,2,4)^T, v=(0,1,2)^T, w=(0,0,1)^T.","count":14},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.","count":8},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-fd7db10a5a4217be","tex":"T^{-m}","aliases":"T^-m","display":"T^{-m}","kind":"use","excerpt":"F(z_x)=ε^2 ∏_m=1^∞ (C_T^-mx^(m))^-1.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-6fb855c184182a59","tex":"T^{-T}","aliases":"T^-T","display":"T^{-T}","kind":"use","excerpt":"widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"u=(1,2,4)^T, v=(0,1,2)^T, w=(0,0,1)^T.","count":15},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f3b9ec843ee07186","tex":"U_\\Gamma","aliases":"U_Γ Gamma","display":"U_\\Gamma","kind":"use","excerpt":"F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-349daf8c6dd8d0cb","tex":"U_\\Gamma^3","aliases":"U_Γ^3 Gamma","display":"U_\\Gamma^3","kind":"use","excerpt":"Im(C_polU_Γ^3)=0","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"Let x=(k,ℓ,j)∈ V=( Z/7)^3 and z_x=z_k,ℓ,j. Define","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"u=(1,2,4)^T, v=(0,1,2)^T, w=(0,0,1)^T.","count":14},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"u=(1,2,4)^T, v=(0,1,2)^T, w=(0,0,1)^T.","count":16},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1ad2698225519d8e","tex":"W_{x,m}","aliases":"W_x,m","display":"W_{x,m}","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"use","excerpt":"widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-62d73a5bfd879bf7","tex":"\\widehat C_m","aliases":"widehat C_m widehat","display":"\\widehat C_m","kind":"use","excerpt":"widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"widehat f(a)=∑_x∈ Vζ^a· xf(x), a∈ V.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f605f803980a555f","tex":"\\widehat f_m","aliases":"widehat f_m widehat","display":"\\widehat f_m","kind":"use","excerpt":"widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-5c6dbc4b275ebdb8","tex":"\\widehat f_{m-1}","aliases":"widehat f_m-1 widehat","display":"\\widehat f_{m-1}","kind":"use","excerpt":"widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"Let x=(k,ℓ,j)∈ V=( Z/7)^3 and z_x=z_k,ℓ,j. Define","count":15},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.","count":6},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.","count":23},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-7fb250c40766c45f","tex":"\\mathbb Z^3","aliases":"Z^3 mathbb","display":"\\mathbb Z^3","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1f0b48345ddedcf2","tex":"z_0","aliases":"z_0","display":"z_0","kind":"use","excerpt":"τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.","count":7},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1c6c32755f9b2f61","tex":"z_{k',\\ell',j'}","aliases":"z_k',ℓ',j' ell","display":"z_{k',\\ell',j'}","kind":"use","excerpt":"ρ z_k,ℓ,j= z_k',ℓ',j'+aτ+bσ+c","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-bf244303e945e424","tex":"z_{k,\\ell,j}","aliases":"z_k,ℓ,j ell","display":"z_{k,\\ell,j}","kind":"use","excerpt":"z_k,ℓ,j=z_0+kδ+ℓη+jξ, k,ℓ,j∈{0,…,6}.","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c93e8d9fd881fa69","tex":"z_{T x}","aliases":"z_T x","display":"z_{T x}","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-59db3505ff306ce3","tex":"z_x","aliases":"z_x","display":"z_x","kind":"use","excerpt":"Let x=(k,ℓ,j)∈ V=( Z/7)^3 and z_x=z_k,ℓ,j. Define","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"Choose the complex embedding β=7^1/3ω^2, so |ρ|<1. Put","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-25ca1efe417aa065","tex":"|^{m-1}","aliases":"|^m-1","display":"|^{m-1}","kind":"use","excerpt":"As ρ^m z_x→0, all f_x^(m)→ F(0)=ε^2. Every multiplier is nonzero at the chosen arguments, and C_x^(m)=1+O(|ρ|^m-1).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"343=1+2·3+16·21.","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"This research package continues the pure cubic experiment K= Q(β), β^3=7.","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-ab15d1c00936ee24","tex":"\\beta^2","aliases":"β^2 beta","display":"\\beta^2","kind":"use","excerpt":"ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1210911b063d7dfc","tex":"\\beta^3","aliases":"β^3 beta","display":"\\beta^3","kind":"use","excerpt":"This research package continues the pure cubic experiment K= Q(β), β^3=7.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-38a0212391c9e2e9","tex":"\\Gamma_{\\mathrm{ell}}","aliases":"Γ_ell Gamma mathrm","display":"\\Gamma_{\\mathrm{ell}}","kind":"use","excerpt":"F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"δ=fracτ+2σ+47, η=fracσ+27, ξ=frac17.","count":7},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.","count":6},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-dfc63ab879167e3a","tex":"\\varepsilon^{-1}","aliases":"ε^-1 varepsilon","display":"\\varepsilon^{-1}","kind":"use","excerpt":"ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"Fix the oriented seventh-root phase ζ_7=e^2π i/7. The additive character against the top dual direction is","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c7d778a23b2effd4","tex":"\\zeta^{a\\cdot x}","aliases":"ζ^a· x zeta cdot","display":"\\zeta^{a\\cdot x}","kind":"use","excerpt":"widehat f(a)=∑_x∈ Vζ^a· xf(x), a∈ V.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-79a0be0094ccc021","tex":"\\zeta_7","aliases":"ζ_7 zeta","display":"\\zeta_7","kind":"use","excerpt":"Fix the oriented seventh-root phase ζ_7=e^2π i/7. The additive character against the top dual direction is","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-893414d3dc2a0f92","tex":"\\zeta_7^{3(4k+5\\ell+j)}","aliases":"ζ_7^3(4k+5ℓ+j) zeta ell","display":"\\zeta_7^{3(4k+5\\ell+j)}","kind":"use","excerpt":"ψ_w(T(k,ℓ,j))=ζ_7^3(4k+5ℓ+j).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-76738c7c39098112","tex":"\\zeta_7^{3k}","aliases":"ζ_7^3k zeta","display":"\\zeta_7^{3k}","kind":"use","excerpt":"ψ_w(k,ℓ,j)=ζ_7^3k.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"δ=fracτ+2σ+47, η=fracσ+27, ξ=frac17.","count":6},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"q_a(z)=fracθ_0(z;a)^5θ_0(5z;5a).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","excerpt":"q_a(z)=fracθ_0(z;a)^5θ_0(5z;5a).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.","count":6},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-64100f915b9d565e","tex":"\\Lambda_5","aliases":"Λ_5 Lambda","display":"\\Lambda_5","kind":"use","excerpt":"The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"δ=fracτ+2σ+47, η=fracσ+27, ξ=frac17.","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.","count":12},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d765e00b61e2dbf1","tex":"\\rho^m","aliases":"ρ^m rho","display":"\\rho^m","kind":"use","excerpt":"f_x^(m)=F(ρ^m z_x), m≥0.","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c2a2097aa196be96","tex":"\\rho^{m-1}","aliases":"ρ^m-1 rho","display":"\\rho^{m-1}","kind":"use","excerpt":"Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.","count":18},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.","count":13},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"ψ_w(k,ℓ,j)=ζ_7^3k.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8580071564964861","tex":"\\psi_w","aliases":"ψ_w psi","display":"\\psi_w","kind":"use","excerpt":"ψ_w(k,ℓ,j)=ζ_7^3k.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.","count":3},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f2f53155f7e59bd3","tex":"\\omega^2","aliases":"ω^2 omega","display":"\\omega^2","kind":"use","excerpt":"ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"z_k,ℓ,j=z_0+kδ+ℓη+jξ, k,ℓ,j∈{0,…,6}.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"The seven-point packet has arguments z_0+kδ. The first refinement consists of z_0+kδ+ℓη, with 49 labels. The full ambient packet is","count":16},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-8aaa13d7254b3cae","tex":"\\ell'","aliases":"ℓ' ell","display":"\\ell'","kind":"use","excerpt":"Write (k',ℓ',j')=(A_0,B_0,C_0)bmod7. Then","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"As ρ^m z_x→0, all f_x^(m)→ F(0)=ε^2. Every multiplier is nonzero at the chosen arguments, and C_x^(m)=1+O(|ρ|^m-1).","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","excerpt":"z_k,ℓ,j=z_0+kδ+ℓη+jξ, k,ℓ,j∈{0,…,6}.","count":6},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-7acbb14efa7a9854","tex":"\\prod_{(u,v,j)\\ne(0,0,0)}","aliases":"∏_(u,v,j)≠(0,0,0) prod ne","display":"\\prod_{(u,v,j)\\ne(0,0,0)}","kind":"use","excerpt":"∏_(u,v,j)≠(0,0,0) F(frac13+fracut+vσ+j7)=1.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-0bafcdd9e08caf20","tex":"\\prod_{j=0}^{n-1}","aliases":"∏_j=0^n-1 prod","display":"\\prod_{j=0}^{n-1}","kind":"use","excerpt":"N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f5a40c774f37b582","tex":"\\prod_{m=1}^{\\infty}","aliases":"∏_m=1^∞ prod infty","display":"\\prod_{m=1}^{\\infty}","kind":"use","excerpt":"F(z_x)=ε^2 ∏_m=1^∞ (C_T^-mx^(m))^-1.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-e70803265aa2607c","tex":"\\prod_{r=0}^{n-1}","aliases":"∏_r=0^n-1 prod","display":"\\prod_{r=0}^{n-1}","kind":"use","excerpt":"P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-2ab6ac29f7239138","tex":"\\prod_{r=n}^{-1}","aliases":"∏_r=n^-1 prod","display":"\\prod_{r=n}^{-1}","kind":"use","excerpt":"P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d085977da2e32d60","tex":"\\prod_{u=1-n}^{0}","aliases":"∏_u=1-n^0 prod","display":"\\prod_{u=1-n}^{0}","kind":"use","excerpt":"N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-f18de406d9a1804e","tex":"\\prod_{v=0}^{n-1}","aliases":"∏_v=0^n-1 prod","display":"\\prod_{v=0}^{n-1}","kind":"use","excerpt":"N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":5},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-965e6a0927763af5","tex":"\\sum_{b\\in V}","aliases":"∑_b∈ V sum in","display":"\\sum_{b\\in V}","kind":"use","excerpt":"widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-99f2cf78baa9ddf5","tex":"\\sum_{\\substack{m\\ge1\\\\5\\nmid m}}","aliases":"∑_substackm≥1 5nmid m sum substack ge nmid","display":null,"kind":"use","excerpt":"L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-886fc2f13956afc2","tex":"\\sum_{\\substack{q\\ge1\\\\5\\nmid nq}}","aliases":"∑_substackq≥1 5nmid nq sum substack ge nmid","display":null,"kind":"use","excerpt":"-5∑_substackq≥1 5nmid nq fracP^q e^2π i nq z-Q^q e^-2π i nq z q(1-P^q)(1-Q^q).","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-518929e214b8ff43","tex":"\\sum_{u=1-n}^{0}","aliases":"∑_u=1-n^0 sum","display":"\\sum_{u=1-n}^{0}","kind":"use","excerpt":"∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-33fd4d4704aa1e77","tex":"\\sum_{v=0}^{n-1}","aliases":"∑_v=0^n-1 sum","display":"\\sum_{v=0}^{n-1}","kind":"use","excerpt":"∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-9923075fc1158c85","tex":"\\sum_{x\\in V}","aliases":"∑_x∈ V sum in","display":"\\sum_{x\\in V}","kind":"use","excerpt":"widehat f(a)=∑_x∈ Vζ^a· xf(x), a∈ V.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"F(z_x)=ε^2 ∏_m=1^∞ (C_T^-mx^(m))^-1.","count":1},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"f_x^(m)=F(ρ^m z_x), m≥0.","count":4},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"The essential duality is ⟨ u,w⟩_H=1 while ⟨ u,u⟩_H=⟨ u,v⟩_H=0 modulo seven. A stable linear subspace containing w also contains v=Nw and u=N^2w. Thus both the full unit action and this nontrivial socle-to-top pairing require the complete three-dimensional module.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"The essential duality is ⟨ u,w⟩_H=1 while ⟨ u,u⟩_H=⟨ u,v⟩_H=0 modulo seven. A stable linear subspace containing w also contains v=Nw and u=N^2w. Thus both the full unit action and this nontrivial socle-to-top pairing require the complete three-dimensional module.","count":2},{"work":"hilbert12-contracting-ambient-torsion","url":"notation/hilbert12-contracting-ambient-torsion.html#symbol-79956f9daec39523","tex":"\\rangle_H","aliases":"⟩_H rangle","display":"\\rangle_H","kind":"use","excerpt":"The essential duality is ⟨ u,w⟩_H=1 while ⟨ u,u⟩_H=⟨ u,v⟩_H=0 modulo seven. A stable linear subspace containing w also contains v=Nw and u=N^2w. Thus both the full unit action and this nontrivial socle-to-top pairing require the complete three-dimensional module.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since ","count":164},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.","count":408},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.","count":132},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3735d1fe47e6e1cc","tex":"a_\\Delta","aliases":"a_Δ Delta","display":"a_\\Delta","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":46},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c9e03e158081411b","tex":"e^{-g/2}","aliases":"e^-g/2","display":"e^{-g/2}","kind":"introduction-passage","excerpt":"Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarith","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-565186bc6efb98d9","tex":"e^{2\\pi im^Tv}","aliases":"e^2π im^Tv pi","display":"e^{2\\pi im^Tv}","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d0b00a4cf04562fb","tex":"e^{2\\pi im_2/3}","aliases":"e^2π im_2/3 pi","display":"e^{2\\pi im_2/3}","kind":"introduction-passage","excerpt":"is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leav","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7f08caee3c57fe8a","tex":"e^{h(v)}","aliases":"e^h(v)","display":"e^{h(v)}","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ea2a06b059761708","tex":"e^{\\text{real number}}","aliases":"e^real number text","display":"e^{\\text{real number}}","kind":"introduction-passage","excerpt":"is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leav","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.","count":42},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":26},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8af97db3cd827027","tex":"h^\\#","aliases":"h^#","display":"h^\\#","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":32},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"introduction-passage","excerpt":"The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.","count":37},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"nd the missing identification Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 ","count":16},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":29},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3f2c3a20356cd529","tex":"m^T","aliases":"m^T","display":"m^T","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-651ec01c3f6e67cb","tex":"m_2","aliases":"m_2","display":"m_2","kind":"introduction-passage","excerpt":"is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, this periodic gauge has equal values at the two endpoints, leav","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0f010c6e6040f4e8","tex":"\\overline{h(\\overline v)}","aliases":"overlineh(overline v) overline overline","display":"\\overline{h(\\overline v)}","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"ha e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, t","count":24},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"introduction-passage","excerpt":"ha e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 replaces the periodic logarithm by (h+h^#)/2, real on the real torus. Since Mp_3=p_3 modulo the lattice, t","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"introduction-passage","excerpt":"The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.","count":41},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":29},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"introduction-passage","excerpt":"The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.","count":50},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-082b3ff9755a8e57","tex":"U_{\\rm alg}","aliases":"U_rm alg rm","display":"U_{\\rm alg}","kind":"introduction-passage","excerpt":"The actual arithmetic candidate U=U_rm alg is defined by the retained rational formulas in nolinkurlprior/ARITHMETIC_INPUT.md. This block starts from those formulas. No gamma value is used to find or fit coefficients.","count":18},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f3b9ec843ee07186","tex":"U_\\Gamma","aliases":"U_Γ Gamma","display":"U_\\Gamma","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":22},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":13},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":21},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7fb250c40766c45f","tex":"\\mathbb Z^3","aliases":"Z^3 mathbb","display":"\\mathbb Z^3","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":20},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":39},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"introduction-passage","excerpt":"Actual complex conjugation is a different automorphism. Denote it by κ. In the chosen embedding,","count":9},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":30},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Phase-only compatibility and the missing identification. Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^","count":22},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"introduction-passage","excerpt":"nd the missing identification Suppose a genuine nonzero relative multiplier has been constructed as a_Δ(v)=α e^2π im^Tve^h(v), m∈ Z^3, h mean zero, and its based ratio is actually identified with U_Γ/U_rm alg. Write h^#(v)=overlineh(overline v). It suffices to establish [source reference] for h-h^#. The theorem gives h-h^#=g-g∘ M. The gauge e^-g/2 ","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"Base divisor and the integral lift. Define H(n)=1_{n>0}, w(c)=4, · 5mid c, -1, · 5nmid c. The function H here is an integer step function, distinct from the eta quotient H and the locally defined CM Green quantity H(z) used elsewhere.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-449196fefc599ecb","tex":"\\mathbf1_{\\{1\\le b\\le15\\}}","aliases":"1_{1≤ b≤15} mathbf le le","display":"\\mathbf1_{\\{1\\le b\\le15\\}}","kind":"use","excerpt":"gives two q_s factors and four reciprocal q_t factors; the 2τ shift disappears from q_t by ellipticity. Finally P_7(z+δ)/P_7(z)=F(z+7δ)/F(z) and the last two identities of [source reference] give [source reference]. Under z↦ z+1 the triple changes to (a,b-15,c-10), leaving p and w(c) unchanged. Formula~[source reference] gives ord_z+1R-ord_zR =-w(c","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4855617ad614fd62","tex":"\\mathbf1_{\\{n>0\\}}","aliases":"1_{n>0} mathbf","display":"\\mathbf1_{\\{n>0\\}}","kind":"use","excerpt":"Base divisor and the integral lift. Define H(n)=1_{n>0}, w(c)=4, · 5mid c, -1, · 5nmid c. The function H here is an integer step function, distinct from the eta quotient H and the locally defined CM Green quantity H(z) used elsewhere.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"The exact information lost by doubling. Quotient nonzero meromorphic elliptic functions by constants and translation coboundaries A(w+s)/A(w). Set e=[q_t], p=[ψ] in additive notation. These generate a free Z^2: the divisor charges on the disjoint infinite translation orbits of 0 and -S/2 are respectively (4,4) and (0,-4). A finite divisor coboundar","count":18},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.","count":9},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-64209c7327ae53ce","tex":"\\mathfrak a","aliases":"a mathfrak","display":"\\mathfrak a","kind":"use","excerpt":"|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a2980dabf2b2d34a","tex":"a^{-1/2}","aliases":"a^-1/2","display":"a^{-1/2}","kind":"use","excerpt":"The cubic characteristic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smoot","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"where initially 0≤ a<1 and B_2(a)=a^2-a+1/6. This is the absolute value of the usual Siegel function. Its value is independent of the representative of z modulo Λ_0.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-06b373fe54fc27d1","tex":"A^3","aliases":"A^3","display":"A^3","kind":"use","excerpt":"c=A^3/B. tagN2.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"a_0=(1+ω)ξ^2-ξ+1, a_1=-ξ^2+(1+ω)ξ+1, a_2=-ωξ^2-ωξ+1.","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-826262fb5a1181ed","tex":"a_0^2","aliases":"a_0^2","display":"a_0^2","kind":"use","excerpt":"A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-67fb959e20ba63d9","tex":"A_0^n","aliases":"A_0^n","display":"A_0^n","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"a_0=(1+ω)ξ^2-ξ+1, a_1=-ξ^2+(1+ω)ξ+1, a_2=-ωξ^2-ωξ+1.","count":12},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-789ee80bbf94776c","tex":"a_1^2","aliases":"a_1^2","display":"a_1^2","kind":"use","excerpt":"A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2fdf33c9001e6df0","tex":"a_1^3","aliases":"a_1^3","display":"a_1^3","kind":"use","excerpt":"A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f148128ab2790b8d","tex":"a_1^4","aliases":"a_1^4","display":"a_1^4","kind":"use","excerpt":"A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-cb36a638e65badbb","tex":"a_1^5","aliases":"a_1^5","display":"a_1^5","kind":"use","excerpt":"|U_Γ|^3=|ϵ^3a_2a_1^5|.","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"a_0=(1+ω)ξ^2-ξ+1, a_1=-ξ^2+(1+ω)ξ+1, a_2=-ωξ^2-ωξ+1.","count":15},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-696f28ddbdadb943","tex":"a_2^3","aliases":"a_2^3","display":"a_2^3","kind":"use","excerpt":"C=fracϵ a_1a_2^3ω","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-29d33961a48f1b7c","tex":"a_{h+1}","aliases":"a_h+1","display":"a_{h+1}","kind":"use","excerpt":"log|σ^h W|=(-1)^h(frac R2+2log|a_h+1|) =-3ζ_ f'(σ^h+5,0). tagA6.3","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d1bbe013bba495bc","tex":"a_{h+1}^2","aliases":"a_h+1^2","display":"a_{h+1}^2","kind":"use","excerpt":"N_κ(σ^hW) =N_κ(ϵ a_h+1^2)^(-1)^h, 0≤ h<6. tagA6.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"a_0+a_1+a_2=3, ∑_i<ja_i a_j=6ω, a_0a_1a_2=1.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"a_0a_1a_2=1, a_j^3-3a_j^2+6ω a_j-1=0.","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9f9a558588be02af","tex":"a_{j+1}","aliases":"a_j+1","display":"a_{j+1}","kind":"use","excerpt":"κ(a_j)=ι(a_j+1), jpmod3. tagQ1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ee0853f347f680bc","tex":"a_j^2","aliases":"a_j^2","display":"a_j^2","kind":"use","excerpt":"a_0a_1a_2=1, a_j^3-3a_j^2+6ω a_j-1=0.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-35bdb78398e04447","tex":"a_j^3","aliases":"a_j^3","display":"a_j^3","kind":"use","excerpt":"a_0a_1a_2=1, a_j^3-3a_j^2+6ω a_j-1=0.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0bdb6564f1e0d191","tex":"a_{k-1}","aliases":"a_k-1","display":"a_{k-1}","kind":"use","excerpt":"ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6+frac23log|a_k-1|). tagA5.10","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"V_m^3=|a_m|, tagA5.6","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5f08c746418e7934","tex":"a_m^2","aliases":"a_m^2","display":"a_m^2","kind":"use","excerpt":"a_m^3-3a_m^2+6ω a_m-1=0. tagA5.8","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9bfd7625b7eb43c2","tex":"a_m^3","aliases":"a_m^3","display":"a_m^3","kind":"use","excerpt":"a_m^3-3a_m^2+6ω a_m-1=0. tagA5.8","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bb2614930ea303a4","tex":"a_s","aliases":"a_s","display":"a_s","kind":"use","excerpt":"a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0083f442b19187bb","tex":"A_v","aliases":"A_v","display":"A_v","kind":"use","excerpt":"F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8fab9541d0b68e87","tex":"A_W","aliases":"A_W","display":"A_W","kind":"use","excerpt":"W=(A_W+ω B_W)/3,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-263b1db52659f9f9","tex":"A_\\ell","aliases":"A_ℓ ell","display":"A_\\ell","kind":"use","excerpt":"The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"ks)^4, · 5nmid n, n=2k. For odd n, doubling permutes E_t[n]. For n=2k, it maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1","count":23},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"A=a_0a_1^3=fraca_1^2a_2, B=a_0^2a_1^4=(fraca_1a_2)^2.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-58f9b1ca3c57804b","tex":"\\mathfrak b","aliases":"b mathfrak","display":"\\mathfrak b","kind":"use","excerpt":"|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":67},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-893779c9a8818915","tex":"\\mathfrak b^{-1}","aliases":"b^-1 mathfrak","display":"\\mathfrak b^{-1}","kind":"use","excerpt":"|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-244db4055846b60c","tex":"B^3","aliases":"B^3","display":"B^3","kind":"use","excerpt":"ta_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For an","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c1726793049bce6e","tex":"B_0^n","aliases":"B_0^n","display":"B_0^n","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","excerpt":"H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"Writing b_2=1-6u+u^2, b_4=u^2-u, the derivative and curve equation imply","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8f978781f8d5d089","tex":"b_2^2","aliases":"b_2^2","display":"b_2^2","kind":"use","excerpt":"g_2=λ^4(b_2^2/12-2b_4), g_3=λ^6(-b_2^3/216+b_2b_4/6-u^2).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ca50b128e8edfed0","tex":"b_2^3","aliases":"b_2^3","display":"b_2^3","kind":"use","excerpt":"g_2=λ^4(b_2^2/12-2b_4), g_3=λ^6(-b_2^3/216+b_2b_4/6-u^2).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-16ab16f37fdea085","tex":"b_4","aliases":"b_4","display":"b_4","kind":"use","excerpt":"Writing b_2=1-6u+u^2, b_4=u^2-u, the derivative and curve equation imply","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e3bae85ab6d499ac","tex":"B_D","aliases":"B_D","display":"B_D","kind":"use","excerpt":"The period-one transition fixes the cubic coefficient. The points 0,1,1/3 are regular and nonzero for R and W_D. For 1, [source reference] has nonzero factors at zero. For 1/3, β/3=-5s-2/3, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define B_D(1)=(log W_D)''(1) =∑_λ∈ Dm_λ[-frac1(λ-1)^","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b2f9008f03849cc5","tex":"B_k","aliases":"B_k","display":"B_k","kind":"use","excerpt":"ks)^4, · 5nmid n, n=2k. For odd n, doubling permutes E_t[n]. For n=2k, it maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-10c299c3911c0c85","tex":"b_s","aliases":"b_s","display":"b_s","kind":"use","excerpt":"a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b17ff042f6455b31","tex":"b_t","aliases":"b_t","display":"b_t","kind":"use","excerpt":"a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-aba34f1446cfa879","tex":"B_v","aliases":"B_v","display":"B_v","kind":"use","excerpt":"F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ee397f54c2fce937","tex":"B_W","aliases":"B_W","display":"B_W","kind":"use","excerpt":"W=(A_W+ω B_W)/3,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2763a74afb84e897","tex":"b_\\ell","aliases":"b_ℓ ell","display":"b_\\ell","kind":"use","excerpt":"The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bb3cdc779ccbade8","tex":"\\bar J","aliases":"bar J bar","display":"\\bar J","kind":"use","excerpt":"ω=e^2π i/3, β=7^1/3ω^2, J=2+3ω, bar J=-1-3ω, Jbar J=7.","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d7d42b2ecde077f4","tex":"\\bar s","aliases":"bar s bar","display":"\\bar s","kind":"use","excerpt":"overlineF_t,s(z)=F_-bar t,-bar s(-bar z).","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-13f5fd6f218e6cdf","tex":"\\bar t","aliases":"bar t bar","display":"\\bar t","kind":"use","excerpt":"overlineF_t,s(z)=F_-bar t,-bar s(-bar z).","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c0435683f947250d","tex":"\\bar z","aliases":"bar z bar","display":"\\bar z","kind":"use","excerpt":"overlineF_t,s(z)=F_-bar t,-bar s(-bar z).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ae14042cce913b90","tex":"\\bar\\beta","aliases":"barβ bar beta","display":"\\bar\\beta","kind":"use","excerpt":"For the last formula, the conjugate Gauss resolvent is 7/G, and barβ=ω^2β. Formula (A1.5) follows directly. It must not be replaced by the involution σ^3, which fixes K.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"The point S=sbmodΛ_t has infinite order by rational independence. Put ψ(w)=fracq_t(w)^2q_t(2w)q_t(2w+s). [All finite norms of the return kernel] For every integer n≥1, N_nψ(w)= 1, · 5mid n,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w","count":28},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"Three geometries must stay distinct. The scalar plane carries the meromorphic correction and its canonical divisor product. The intermediate cover X= C^3/ Z^3 carries periodic multipliers and a degree-seven principal cover. The compact CM threefold has lattice Z[ω]^3 and degree 49 for the same matrix. Finite projective representations or oriented t","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d.","count":59},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"on. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole counts O(r^2). Thus R is not representable in this class. For Y=ImΩ>0, the defining theta series has terms bounded by exp{-π n^TYn+2π n Im(Az+b) }. Completing the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives Cexp(C'|z|^2) for a fixed","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-493a70983f1c6903","tex":"\\mathbb C^*","aliases":"C^* mathbb","display":"\\mathbb C^*","kind":"use","excerpt":"The actual two commuting actions. On X= C^3/ Z^3≃( C^*)^3 and its real torus T^3, use M=0 · 1 · 0","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4a4fb977dca130a2","tex":"C^3","aliases":"C^3","display":"C^3","kind":"use","excerpt":"Multiplication and (A4.3) give C^3|X^-3+1|=1. Cubing each equation in (A5.5) now removes the unknown constant exactly:","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e2fdab254f7c2081","tex":"\\mathbb C^3","aliases":"C^3 mathbb","display":"\\mathbb C^3","kind":"use","excerpt":"Three geometries must stay distinct. The scalar plane carries the meromorphic correction and its canonical divisor product. The intermediate cover X= C^3/ Z^3 carries periodic multipliers and a degree-seven principal cover. The compact CM threefold has lattice Z[ω]^3 and degree 49 for the same matrix. Finite projective representations or oriented t","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c954ed43fe71ab81","tex":"c^3","aliases":"c^3","display":"c^3","kind":"use","excerpt":"More generally, if c^3=A and c^8=B with c≠0, then","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0c712d29ff4c416b","tex":"c^8","aliases":"c^8","display":"c^8","kind":"use","excerpt":"More generally, if c^3=A and c^8=B with c≠0, then","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3052e6f7fab4cf4a","tex":"c^d","aliases":"c^d","display":"c^d","kind":"use","excerpt":"The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a795c1dda66f5d32","tex":"c^e","aliases":"c^e","display":"c^e","kind":"use","excerpt":"ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6-ℓ_k-1). tagA3.5","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a429940adeec0b30","tex":"C^q","aliases":"C^q","display":"C^q","kind":"use","excerpt":"formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients hav","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3b2c6bf1d5bbcd40","tex":"C_{18}","aliases":"C_18","display":"C_{18}","kind":"use","excerpt":"For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"Gal(E/ Q)=S_3× C_3. tagA2.1","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b257aef99d8016bf","tex":"C_3^2","aliases":"C_3^2","display":"C_3^2","kind":"use","excerpt":"E= Q(ω,β,ξ), F_0= Q(ω), Gal(E/F_0)=C_3^2.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ca65dc5da5fa80e3","tex":"c_6","aliases":"c_6","display":"c_6","kind":"use","excerpt":"and, for c_6=-ω^2,","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9fc68be2690604ca","tex":"c_6^4","aliases":"c_6^4","display":"c_6^4","kind":"use","excerpt":"For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f2213641e71ff8a2","tex":"c_{7^nk}","aliases":"c_7^nk","display":"c_{7^nk}","kind":"use","excerpt":"equation then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e23309608da99784","tex":"c_{Bk}","aliases":"c_Bk","display":"c_{Bk}","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-254051ed849e0a84","tex":"c_k","aliases":"c_k","display":"c_k","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1e15d2e27313753c","tex":"C_N","aliases":"C_N","display":"C_N","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f0ecd1fd52ce4f77","tex":"C_{\\rm pol}","aliases":"C_rm pol rm","display":"C_{\\rm pol}","kind":"use","excerpt":"The exact remaining arithmetic comparison. The fixed arithmetic presentation gives, with a_0,a_1,a_2 as in the polarized block, C_rm pol=fracϵ a_1a_2^3ω, C_rm polU_rm alg^3=R_+:=|ϵ|^4|a_1|^6|a_2|^4>0. The modulus theorem gives |C_rm polU_Γ^3|=R_+. The exact statement still missing is Im(C_rm polU_Γ^3)=0. Reality together with a valid positive-real-","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e1e71a0eb7c64bd5","tex":"c_s","aliases":"c_s","display":"c_s","kind":"use","excerpt":"a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c99be13da294dc50","tex":"c_t","aliases":"c_t","display":"c_t","kind":"use","excerpt":"For the smoothing denominator, multiplication by five permutes the real residues modulo n; the same two steps apply with period 5t. The numerator becomes θ_0(nz;t)^5 and the denominator becomes θ_0(5nz;5t). This proves (N1.1) with no omitted multiplier and without using the unknown scalar c_t.","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-955b2f7a7c53db37","tex":"c_t^{-36}","aliases":"c_t^-36","display":"c_t^{-36}","kind":"use","excerpt":"c_t^-36=d^-72. tagN8.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-70cc4da13dadddaa","tex":"c_t^3","aliases":"c_t^3","display":"c_t^3","kind":"use","excerpt":"c_t^3=16∏_η≠0Q_5(x(η))=f(a)f(b)=d^6. tagN4.2","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-47ad50239934518e","tex":"c_t^8","aliases":"c_t^8","display":"c_t^8","kind":"use","excerpt":"c_t^8=(ψ_3(a)ψ_3(b))^2=d^16. tagN4.3","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"f=(3,β-1), E=K(ω,θ), θ=2cos(6π/7).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1","count":36},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d.","count":29},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8ed43ac72d96bedf","tex":"d^{-72}","aliases":"d^-72","display":"d^{-72}","kind":"use","excerpt":"c_t^-36=d^-72. tagN8.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bb762638c7220c4c","tex":"D^{-k}","aliases":"D^-k","display":"D^{-k}","kind":"use","excerpt":"5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-71ad7b3ed8c778c5","tex":"d^{16}","aliases":"d^16","display":"d^{16}","kind":"use","excerpt":"c_t^8=(ψ_3(a)ψ_3(b))^2=d^16. tagN4.3","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bef1e474c68da53c","tex":"d^{18}","aliases":"d^18","display":"d^{18}","kind":"use","excerpt":"c_t=(c_t^3)^3/c_t^8=d^18/d^16=d^2. tagN4.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"4y_ay_b=(ℓ^2-m^2)d^2=4d^3.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-35796d277eb7d89c","tex":"d^3","aliases":"d^3","display":"d^3","kind":"use","excerpt":"4y_ay_b=(ℓ^2-m^2)d^2=4d^3.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a41942edfed6dca3","tex":"d^4","aliases":"d^4","display":"d^4","kind":"use","excerpt":"beginequation R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential f","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-98798ec8c47fd430","tex":"d^6","aliases":"d^6","display":"d^6","kind":"use","excerpt":"y_ay_b=d^3, f(a)f(b)=d^6. tagN3.2","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bb231ca776cfc93d","tex":"D^7","aliases":"D^7","display":"D^7","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-6e1a467385f054b6","tex":"d^8","aliases":"d^8","display":"d^8","kind":"use","excerpt":"ψ_3(a)ψ_3(b)=-d^8. tagN3.3","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2f80fcde2a727a6d","tex":"D^k","aliases":"D^k","display":"D^k","kind":"use","excerpt":"5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"D_0=L(0,χ_-3)=1/3, D_1=L(0,χ_-3χ_7)=-2ω, D_2=-2ω^2. tagA2.3","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-99e5f8e078f9fe31","tex":"D_1","aliases":"D_1","display":"D_1","kind":"use","excerpt":"D_0=L(0,χ_-3)=1/3, D_1=L(0,χ_-3χ_7)=-2ω, D_2=-2ω^2. tagA2.3","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3d6380742c628b38","tex":"D_2","aliases":"D_2","display":"D_2","kind":"use","excerpt":"D_0=L(0,χ_-3)=1/3, D_1=L(0,χ_-3χ_7)=-2ω, D_2=-2ω^2. tagA2.3","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"Multiplication by β in the ordered basis (τ,s,1/5) is N_5=2 · -1 · 0 4 · 0 · -3 5 · 0 · -2, N_5^3=7I, det N_5=7, N_5^-1=frac170 · -2 · 3 -7 · -4 · 6 0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly ","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-346650cbdd3f09a1","tex":"\\operatorname{disc}","aliases":"disc operatorname","display":"\\operatorname{disc}","kind":"use","excerpt":"q_t(z)=frac(a-b)^2(x(z)-a)(x(z)-b) =fracdisc(Q_5)Q_5(x(z)). tagN5.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8f42a8e27ac6c866","tex":"\\operatorname{div}","aliases":"div operatorname","display":"\\operatorname{div}","kind":"use","excerpt":"The conjugation divisor cannot be discarded. Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this ","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":29},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"E_z(s)=∑_m,n∈ Z|m+nω+z|^-2s,","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3f1cccbf7d169e62","tex":"e^{-\\alpha_2-\\alpha_3}","aliases":"e^-α_2-α_3 alpha alpha","display":"e^{-\\alpha_2-\\alpha_3}","kind":"use","excerpt":"gent lower-power sums are not substituted. Taking two logarithmic derivatives of [source reference] at the transition [source reference] gives (log K)''(0)=6α_3+B_D(1). Since L_t is even, α_3=-frac16[β^2∑_r=1^15L_t(rs)+B_D(1)]. Evaluating the period-one ratio also determines e^α_1=fracK(0)e^-α_2-α_3W_D(1). This evaluates the exponential, without ch","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ddb315b6febdf9cc","tex":"e^{-\\pi(\\Im z)^2/y}","aliases":"e^-π(Im z)^2/y pi Im","display":"e^{-\\pi(\\Im z)^2/y}","kind":"use","excerpt":"H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-836246a8972448da","tex":"e^{-\\pi\\operatorname{Im}(\\omega)B_2(a)}","aliases":"e^-πIm(ω)B_2(a) pi operatorname omega","display":"e^{-\\pi\\operatorname{Im}(\\omega)B_2(a)}","kind":"use","excerpt":"H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-86142e2209ba3803","tex":"e^{2\\pi i((j+1)a+(k+1)b-z)}","aliases":"e^2π i((j+1)a+(k+1)b-z) pi","display":"e^{2\\pi i((j+1)a+(k+1)b-z)}","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5953d5af404aa486","tex":"e^{2\\pi i((n+1)a-z)}","aliases":"e^2π i((n+1)a-z) pi","display":"e^{2\\pi i((n+1)a-z)}","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f8b0df3b4cd66cb7","tex":"e^{2\\pi i((n+1)t-z)}","aliases":"e^2π i((n+1)t-z) pi","display":"e^{2\\pi i((n+1)t-z)}","kind":"use","excerpt":"θ_0(z;t)=∏_n≥0(1-e^2π i(z+nt))(1-e^2π i((n+1)t-z)).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bee2a34c7e9a6e5f","tex":"e^{2\\pi i(ja+kb+z)}","aliases":"e^2π i(ja+kb+z) pi","display":"e^{2\\pi i(ja+kb+z)}","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-20937538338b1d79","tex":"e^{2\\pi i(n\\tau+z)}","aliases":"e^2π i(nτ+z) pi tau","display":"e^{2\\pi i(n\\tau+z)}","kind":"use","excerpt":"-2log|1-e^2π iz| -2∑_n≥1log|(1-e^2π i(nτ+z))(1-e^2π i(nτ-z))|.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-26958bae3a4a827d","tex":"e^{2\\pi i(n\\tau-z)}","aliases":"e^2π i(nτ-z) pi tau","display":"e^{2\\pi i(n\\tau-z)}","kind":"use","excerpt":"-2log|1-e^2π iz| -2∑_n≥1log|(1-e^2π i(nτ+z))(1-e^2π i(nτ-z))|.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-63cc183a8723c2c5","tex":"e^{2\\pi i(n\\omega+z)}","aliases":"e^2π i(nω+z) pi omega","display":"e^{2\\pi i(n\\omega+z)}","kind":"use","excerpt":"H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c0955b473f9e8dee","tex":"e^{2\\pi i(n\\omega-z)}","aliases":"e^2π i(nω-z) pi omega","display":"e^{2\\pi i(n\\omega-z)}","kind":"use","excerpt":"H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-52c8c63aecfbeeb2","tex":"e^{2\\pi i(z+na)}","aliases":"e^2π i(z+na) pi","display":"e^{2\\pi i(z+na)}","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c86fe6542a39652e","tex":"e^{2\\pi i(z+nt)}","aliases":"e^2π i(z+nt) pi","display":"e^{2\\pi i(z+nt)}","kind":"use","excerpt":"θ_0(z;t)=∏_n≥0(1-e^2π i(z+nt))(1-e^2π i((n+1)t-z)).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f4b443aa50014668","tex":"e^{2\\pi i/3}","aliases":"e^2π i/3 pi","display":"e^{2\\pi i/3}","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-12070a078697d0aa","tex":"e^{2\\pi ia}","aliases":"e^2π ia pi","display":"e^{2\\pi ia}","kind":"use","excerpt":"5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-94b7c2f177d8a3bd","tex":"e^{2\\pi ib}","aliases":"e^2π ib pi","display":"e^{2\\pi ib}","kind":"use","excerpt":"5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-fa017cf07d1300b3","tex":"e^{2\\pi id}","aliases":"e^2π id pi","display":"e^{2\\pi id}","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-94243d79ef7f4ab2","tex":"e^{2\\pi ik^Tv}","aliases":"e^2π ik^Tv pi","display":"e^{2\\pi ik^Tv}","kind":"use","excerpt":"The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-23cf6af8786766c5","tex":"e^{2\\pi is}","aliases":"e^2π is pi","display":"e^{2\\pi is}","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d7b9b4237ea057ec","tex":"e^{2\\pi it}","aliases":"e^2π it pi","display":"e^{2\\pi it}","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-89c284153ba99ce0","tex":"e^{2\\pi iw}","aliases":"e^2π iw pi","display":"e^{2\\pi iw}","kind":"use","excerpt":"beginequation R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential f","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e69854335506b3ba","tex":"e^{2\\pi iz}","aliases":"e^2π iz pi","display":"e^{2\\pi iz}","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2226b6f26b0abdf6","tex":"e^{6\\pi i k/7}","aliases":"e^6π i k/7 pi","display":"e^{6\\pi i k/7}","kind":"use","excerpt":"ξ=β^-1∑_k=1^6χ_7(k)e^6π i k/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-06f0678ace532687","tex":"e^{6\\pi ia/7}","aliases":"e^6π ia/7 pi","display":"e^{6\\pi ia/7}","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1e8c91c52ab3f04d","tex":"E^H","aliases":"E^H","display":"E^H","kind":"use","excerpt":"∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5b64b331236aede1","tex":"e^{h(z)}","aliases":"e^h(z)","display":"e^{h(z)}","kind":"use","excerpt":"that of R. Thus R/W_D extends to a nowhere-zero entire function, and on the simply connected plane has an entire logarithm h. The fourth derivative of h vanishes, hence h is a polynomial of degree at most three. Its value at zero is fixed by R_0, and the vanished canonical jets identify the coefficients in [source reference]. Divisor matching alone","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-142ee071a11b9b97","tex":"e^{R\\|k\\|}","aliases":"e^R k","display":"e^{R\\|k\\|}","kind":"use","excerpt":"is summable. The other two residue classes are obtained by the fixed continuous operators L_N,L_N^2. The series for g converges in every C^q norm; taking the limit gives h=δ_Mg. For a global holomorphic function on X, its Laurent coefficients decay faster than every fixed exponential. On each compact product annulus the decay at 7^nk dominates both","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8c7d28de91686a73","tex":"e^{w+w^2/2+w^3/3}","aliases":"e^w+w^2/2+w^3/3","display":"e^{w+w^2/2+w^3/3}","kind":"use","excerpt":"The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, a","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-29a832efbb4727d0","tex":"e^{z^3}","aliases":"e^z^3","display":"e^{z^3}","kind":"use","excerpt":"en applied about a point where P is nonzero, using an outer circle of twice the radius, bounds its zero count by O(r^2); the same holds for Q. Cancellation only decreases their divisor counts. The factor u contributes no finite divisor. Theorem~[source reference] contradicts this bound for R. The single global unit in this grammar cannot be moved i","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f2d5b4cad8ea2d78","tex":"e^{\\alpha_1}","aliases":"e^α_1 alpha","display":"e^{\\alpha_1}","kind":"use","excerpt":"gent lower-power sums are not substituted. Taking two logarithmic derivatives of [source reference] at the transition [source reference] gives (log K)''(0)=6α_3+B_D(1). Since L_t is even, α_3=-frac16[β^2∑_r=1^15L_t(rs)+B_D(1)]. Evaluating the period-one ratio also determines e^α_1=fracK(0)e^-α_2-α_3W_D(1). This evaluates the exponential, without ch","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c5b3afb0a8099a9c","tex":"e^{\\alpha_1+\\alpha_2/3+\\alpha_3/9}","aliases":"e^α_1+α_2/3+α_3/9 alpha alpha alpha","display":"e^{\\alpha_1+\\alpha_2/3+\\alpha_3/9}","kind":"use","excerpt":"fracW_D(1/3)^3W_D(1) exp(-frac23α_2-frac89α_3). The right side consists of specified theta values and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new root choice. Cube [source reference] at 1/3 to get R_0^3e^α_1+α_2/3+α_3/9W_D(1/3)^3, then ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ed3905e7c0b0f722","tex":"e^{\\pi i/6}","aliases":"e^π i/6 pi","display":"e^{\\pi i/6}","kind":"use","excerpt":"X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d63308e2ded35d4f","tex":"e^{\\pi i\\tau}","aliases":"e^π iτ pi tau","display":"e^{\\pi i\\tau}","kind":"use","excerpt":"H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"On the normalized curve E_0:y^2+y=x^3, set","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"stable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The dual degree-seven image is exactly k∈ B Z^3 Longleftrightarrow ℓ(k):=k_0+2k_1+4k_2≡0pmod7. ","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, witho","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-82c2d872ef715d1f","tex":"E_3","aliases":"E_3","display":"E_3","kind":"use","excerpt":"The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, a","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ac26798e0482a242","tex":"e_{Ak}","aliases":"e_Ak","display":"e_{Ak}","kind":"use","excerpt":"The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-542812c54fe5f147","tex":"e_k","aliases":"e_k","display":"e_k","kind":"use","excerpt":"The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f75d15c864ad3cef","tex":"E_t","aliases":"E_t","display":"E_t","kind":"use","excerpt":"Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,","count":10},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5ac1b190c48447df","tex":"\\mathcal E_z","aliases":"E_z mathcal","display":"\\mathcal E_z","kind":"use","excerpt":"E_z(s)=∑_m,n∈ Z|m+nω+z|^-2s,","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-69be8aefe8e9395c","tex":"\\mathcal E_z'","aliases":"E_z' mathcal","display":"\\mathcal E_z'","kind":"use","excerpt":"E_z(0)=0, E_z'(0)=-2log H(z). tagA4.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b1aeb0fcc8f1b4d3","tex":"\\mathcal E_{\\alpha/(3J)}","aliases":"E_α/(3J) mathcal alpha","display":"\\mathcal E_{\\alpha/(3J)}","kind":"use","excerpt":"63^-s E_α/(3J)(s).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole counts O(r^2). Thus R is not representable in this class. For Y=ImΩ>0, the defining theta series has terms bounded by exp{-π n","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ell(5z;5τ,5s) =fracΓ_rm ell(5(z+t);5t,5s)Γ_rm ell(z+t;t,s)^5, U_Γ=F(1/3). This convention agrees with the checked primary ","count":47},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"Reduction modulo J sends ω to 4∈ F_7. Formula (A3.2) groups the representatives as follows:","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0fb3ff9d2b443b85","tex":"\\mathfrak f","aliases":"f mathfrak","display":"\\mathfrak f","kind":"use","excerpt":"f=(3,β-1), E=K(ω,θ), θ=2cos(6π/7).","count":10},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,","count":12},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9ce3ad17e478e1d4","tex":"f^\\#","aliases":"f^#","display":"f^\\#","kind":"use","excerpt":"The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromor","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1105335d2295cd5b","tex":"F_{-\\bar t,-\\bar s}","aliases":"F_-bar t,-bar s bar bar","display":"F_{-\\bar t,-\\bar s}","kind":"use","excerpt":"overlineF_t,s(z)=F_-bar t,-bar s(-bar z).","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":22},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-663b3ba7a3449e4d","tex":"\\mathbb F_{181}","aliases":"F_181 mathbb","display":"\\mathbb F_{181}","kind":"use","excerpt":"For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-23af4eb7208f9cad","tex":"\\mathbb F_7","aliases":"F_7 mathbb","display":"\\mathbb F_7","kind":"use","excerpt":"Reduction modulo J sends ω to 4∈ F_7. Formula (A3.2) groups the representatives as follows:","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0e70b66982cfe6ee","tex":"F_{t,s}","aliases":"F_t,s","display":"F_{t,s}","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"For the last formula, the conjugate Gauss resolvent is 7/G, and barβ=ω^2β. Formula (A1.5) follows directly. It must not be replaced by the involution σ^3, which fixes K.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e6df94744f7ade62","tex":"g_2","aliases":"g_2","display":"g_2","kind":"use","excerpt":"x(z)=wp_t(z), y(z)=wp_t'(z), f(X)=4X^3-g_2X-g_3.","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-01a75b598bc87b59","tex":"g_2^2","aliases":"g_2^2","display":"g_2^2","kind":"use","excerpt":"ψ_3(X)=3X^4-tfrac32g_2X^2-3g_3X-tfrac116g_2^2,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ab6997694ee35093","tex":"g_3","aliases":"g_3","display":"g_3","kind":"use","excerpt":"For completeness, T is nonzero three-torsion and its normalized x-coordinate is a root of 3x(x^3+1). The lattice is invariant under ordinary conjugation, so wp(1/3) and its derivative are real. Since g_3(Λ_0)>0, the real differential equation wp'^2=4wp^3-g_3 implies wp(1/3)>0. The argument of Ω^2 is π/3, selecting -ω among the three roots of x^3=-1","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-efa0a3eea83b0f61","tex":"\\operatorname{Gal}","aliases":"Gal operatorname","display":"\\operatorname{Gal}","kind":"use","excerpt":"E= Q(ω,β,ξ), F_0= Q(ω), Gal(E/F_0)=C_3^2.","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":21},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8911fccc15deb3e8","tex":"h_K","aliases":"h_K","display":"h_K","kind":"use","excerpt":"The analytic class number formula, h_K=3, and ζ_ Q(0)=-1/2 give","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-864591eb3221a503","tex":"H_\\tau","aliases":"H_τ tau","display":"H_\\tau","kind":"use","excerpt":"H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"The norm matrices on this free subgroup are N_n=I (n odd,5nmid n), N_n=1 · -6 0 · 0 (n even,5nmid n), N_n=0 (5mid n). The middle matrix is a split integral projector with image Z(1,0) and kernel Z(6,1). Its primitive kernel class 6e+p is nonzero, as its two charges are 28,-4. Thus repeated full doubling cannot recover the deleted component. The res","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-12473beefe7126eb","tex":"\\Im","aliases":"Im","display":"\\Im","kind":"use","excerpt":"τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomi","count":13},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-56f708943dcfa778","tex":"\\operatorname{Im}","aliases":"Im operatorname","display":"\\operatorname{Im}","kind":"use","excerpt":"H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ad8e02273a5aecf9","tex":"\\operatorname{Ind}","aliases":"Ind operatorname","display":"\\operatorname{Ind}","kind":"use","excerpt":"Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c2d739454091b81d","tex":"\\operatorname{Ind}_{C_2}^{S_3}","aliases":"Ind_C_2^S_3 operatorname","display":"\\operatorname{Ind}_{C_2}^{S_3}","kind":"use","excerpt":"Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7df656d0365ae01c","tex":"\\operatorname{Ind}_{C_3}^{S_3}","aliases":"Ind_C_3^S_3 operatorname","display":"\\operatorname{Ind}_{C_3}^{S_3}","kind":"use","excerpt":"Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":26},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":38},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5ddfa02159ccf890","tex":"J_{n,v}","aliases":"J_n,v","display":"J_{n,v}","kind":"use","excerpt":"The same substitution works for every oriented three-step return product J_n,v from the predecessor. Consequently ALL seven nontrivial signed-channel limits have finite rational elliptic expressions without a separately supplied eta constant. The first-order limits still have only their guaranteed first-order local cancellation; this normalization ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F","count":23},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":57},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7a0a635a73cee305","tex":"k'","aliases":"k'","display":"k'","kind":"use","excerpt":"Z_i,j,k^(n)=λ(-i,j,5(n+k)), P_i,j,k,r^(n)=λ(-i,j,5(n+k)-r), 1≤ r≤4. Every such triple has a<0, b>0, 3c>2b. Hence p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d7ebe5cf815c906d","tex":"k^T","aliases":"k^T","display":"k^T","kind":"use","excerpt":"The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"e distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=Θ(r^3), n_-(r)=Θ(r^3), n_D(r):=n_+(r)+n_-(r)=Θ(r^3). In particular, for K_0=(|τ|+|s|+2)/|β|, n_+(K_0n)≥4n^3, n_","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9817d114fe256b80","tex":"k_0","aliases":"k_0","display":"k_0","kind":"use","excerpt":"tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The d","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1bdfdabfb7ccb3ec","tex":"k_1","aliases":"k_1","display":"k_1","kind":"use","excerpt":"tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The d","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a8cd867fa822df3a","tex":"k_2","aliases":"k_2","display":"k_2","kind":"use","excerpt":"tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The d","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"its bytes with the inaccessible v2 artifact. Product/continuation conventions, Theorem~3.2 and the fixed example in Section~5.2 are the controlling uses. bibitemMilne J.~S.~Milne, Class Field Theory, version~4.03, August~6, 2020. Cubic reciprocity and Artin induction in Chapter~VIII. https://www.jmilne.org/math/CourseNotes/CFTc.pdf. bibitemCogdell ","count":30},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-71938f131bf0d26e","tex":"L_{\\mathbb Q}","aliases":"L_ Q mathbb","display":"L_{\\mathbb Q}","kind":"use","excerpt":"L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-318c006cb5b824cc","tex":"L_{F_0}","aliases":"L_F_0","display":"L_{F_0}","kind":"use","excerpt":"L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e9753882f4949289","tex":"L_{F_0}'","aliases":"L_F_0'","display":"L_{F_0}'","kind":"use","excerpt":"L_F_0'(0,ϱ)=3 R. tagA2.4","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4662d64de6729153","tex":"L_K","aliases":"L_K","display":"L_K","kind":"use","excerpt":"L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c91dabfc8b700b3b","tex":"L_N","aliases":"L_N","display":"L_N","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2e489a70eb33073c","tex":"L_N^2","aliases":"L_N^2","display":"L_N^2","kind":"use","excerpt":"k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed expression is bounded, up to a derivative convention constant, by 7^n(3-s)∑_k≠0 k ^s|c_k|. It tends geometrically to zero and is summable. The other two residue classes are obtained by the fixed conti","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-fa290acc527931e6","tex":"L_N^{3n}","aliases":"L_N^3n","display":"L_N^{3n}","kind":"use","excerpt":"equation then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For ","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-55f7c4a230e30bb5","tex":"L_N^j","aliases":"L_N^j","display":"L_N^j","kind":"use","excerpt":"[Principal-norm rigidity] Let h,r be smooth mean-zero complex functions on T^3. If S_Nh-h∘ N=δ_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=f","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-afc7e20ca6f9b9e0","tex":"L_N^n","aliases":"L_N^n","display":"L_N^n","kind":"use","excerpt":"eginequation S_Nh-h∘ N=δ_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-49e6cb003fe8b536","tex":"L_t","aliases":"L_t","display":"L_t","kind":"use","excerpt":"The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-","count":12},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,wideha","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"A bounded complex radius bounds |a| and |b| by O(r); the real part then bounds |c| by O(r). There are O((1+r)^3) triples, and absolute weights are at most four. The seven translated numerator factors and the scaled denominator in [source reference] give a finite union of such cone supports and cannot increase this order. Combining the upper bound w","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":27},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The d","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-039ec3e8ab324910","tex":"\\mathfrak m","aliases":"m mathfrak","display":"\\mathfrak m","kind":"use","excerpt":"m=3Jmathscr O_F_0. tagA3.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"ℓ^2=4(2a+b), m^2=4(a+2b),","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8868d63653f7b040","tex":"M^T","aliases":"M^T","display":"M^T","kind":"use","excerpt":"The actual two commuting actions. On X= C^3/ Z^3≃( C^*)^3 and its real torus T^3, use M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, N=2I-M=2 · -1 · 0 4 · 0 · -15 1 · 0 · -2, A=M^T, B=N^T. The period row (τ,s,1) satisfies (τ,s,1)M=ϵ(τ,s,1) and (τ,s,1)N=β(τ,s,1). We have MN=NM, det M=1, det N=7 and N^3=7I. Thus N is a degree-seven cover on X. On the compact la","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0dae06547437e14b","tex":"m_\\lambda","aliases":"m_λ lambda","display":"m_\\lambda","kind":"use","excerpt":"[Complete ceiling divisor] There are no zeros or poles of R outside β^-1Λ_5. At [source reference], m_λ:=ord_λR =w(c) [H(p)-H(2p-a)-H(a)+H(b)]∈{-4,-1,0,1,4}. Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is ","count":10},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"use","excerpt":"m=3Jmathscr O_F_0. tagA3.1","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"ac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weig","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n, 1, · 5mid n. In the prime-to-five case the two product iden","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomi","count":35},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":71},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"1 · 0 · -2, A=M^T, B=N^T. The period row (τ,s,1) satisfies (τ,s,1)M=ϵ(τ,s,1) and (τ,s,1)N=β(τ,s,1). We have MN=NM, det M=1, det N=7 and N^3=7I. Thus N is a degree-seven cover on X. On the compact lattice Z[ω]^3 its degree is 49; these refer to different lattices.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0969e62c250901c5","tex":"n^3","aliases":"n^3","display":"n^3","kind":"use","excerpt":"' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=Θ(r^3), n_-(r)=Θ(r^3), n_D(r):=n_+(r)+n_-(r)=Θ(r^3). In particular, for K_0=(|τ|+|s|+2)/|β|, n_+(K_0n)≥4n^3, n_-(K_0n)≥4n^3. These are unsign","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ac5e944c84a2e799","tex":"n^4","aliases":"n^4","display":"n^4","kind":"use","excerpt":"∏_η∈ E_t[n]∖{0}q_t(η)=n^4 (5nmid n). tagN4.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f061dd3000965a21","tex":"N^T","aliases":"N^T","display":"N^T","kind":"use","excerpt":"The actual two commuting actions. On X= C^3/ Z^3≃( C^*)^3 and its real torus T^3, use M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, N=2I-M=2 · -1 · 0 4 · 0 · -15 1 · 0 · -2, A=M^T, B=N^T. The period row (τ,s,1) satisfies (τ,s,1)M=ϵ(τ,s,1) and (τ,s,1)N=β(τ,s,1). We have MN=NM, det M=1, det N=7 and N^3=7I. Thus N is a degree-seven cover on X. On the compact la","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-62719743cc024dfa","tex":"n^T","aliases":"n^T","display":"n^T","kind":"use","excerpt":"functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole counts O(r^2). Thus R is not representable in this class. For Y=ImΩ>0, the defining theta series has terms bounded by exp{-π n","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4fa33e766db12578","tex":"n_+","aliases":"n_+","display":"n_+","kind":"use","excerpt":"p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) a","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1634b5210d65763f","tex":"n_-","aliases":"n_-","display":"n_-","kind":"use","excerpt":"bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) co","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1eebbeebbd5d7870","tex":"N_{2k}","aliases":"N_2k","display":"N_{2k}","kind":"use","excerpt":"point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+1ψ(w)=ψ(w)fracB_k(w)B_k(w+s). For n=2k prime to five, C_k(w)=∏_j=0^k-1q_t(w+js)^4 gives N_2kψ(w)=q_t(","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-899cc9af3e08acf6","tex":"N_{2k+1}","aliases":"N_2k+1","display":"N_{2k+1}","kind":"use","excerpt":"position For odd n, doubling permutes E_t[n]. For n=2k, it maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1)s) gives N_2k+","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4861b5f0d94d3545","tex":"N_5","aliases":"N_5","display":"N_5","kind":"use","excerpt":"Multiplication by β in the ordered basis (τ,s,1/5) is N_5=2 · -1 · 0 4 · 0 · -3 5 · 0 · -2, N_5^3=7I, det N_5=7, N_5^-1=frac170 · -2 · 3 -7 · -4 · 6 0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly ","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f47d4eb681bfb693","tex":"N_5^{-1}","aliases":"N_5^-1","display":"N_5^{-1}","kind":"use","excerpt":"Multiplication by β in the ordered basis (τ,s,1/5) is N_5=2 · -1 · 0 4 · 0 · -3 5 · 0 · -2, N_5^3=7I, det N_5=7, N_5^-1=frac170 · -2 · 3 -7 · -4 · 6 0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-726aae72bd1ec021","tex":"N_5^3","aliases":"N_5^3","display":"N_5^3","kind":"use","excerpt":"Multiplication by β in the ordered basis (τ,s,1/5) is N_5=2 · -1 · 0 4 · 0 · -3 5 · 0 · -2, N_5^3=7I, det N_5=7, N_5^-1=frac170 · -2 · 3 -7 · -4 · 6 0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-71f77540e0e9914c","tex":"n_D","aliases":"n_D","display":"n_D","kind":"use","excerpt":"the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-52d2fcc2f48803cd","tex":"N_{E/E^H}","aliases":"N_E/E^H","display":"N_{E/E^H}","kind":"use","excerpt":"∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9e4a0006798b9496","tex":"N_{E/F_0}","aliases":"N_E/F_0","display":"N_{E/F_0}","kind":"use","excerpt":"∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5d801779ac256d8e","tex":"N_{K/\\mathbb Q}","aliases":"N_K/ Q mathbb","display":"N_{K/\\mathbb Q}","kind":"use","excerpt":"Indeed, N_K/ Q(ϵ)=1 and its real image is 2-7^1/3. The class number is three and ϵ is a fundamental unit. These arithmetic facts are recorded in BCG, section 5.2; the earlier unramified-cubic note also proves the class-number and ray-field identification used here.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-35b602543437bbe8","tex":"N_n","aliases":"N_n","display":"N_n","kind":"use","excerpt":"Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ba00225ac74c3dfe","tex":"N_\\kappa","aliases":"N_κ kappa","display":"N_\\kappa","kind":"use","excerpt":"Let N_κ(z)=zκ(z)=|z|^2 in the fixed embedding. The six exact identities needed for the comparison are","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n, 1, · 5mid n. In the prime-to-five case the two product iden","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"mathbb Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fix","count":14},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"The exact remaining arithmetic comparison. The fixed arithmetic presentation gives, with a_0,a_1,a_2 as in the polarized block, C_rm pol=fracϵ a_1a_2^3ω, C_rm polU_rm alg^3=R_+:=|ϵ|^4|a_1|^6|a_2|^4>0. The modulus theorem gives |C_rm polU_Γ^3|=R_+. The exact statement still missing is Im(C_rm polU_Γ^3)=0. Reality together with a valid positive-real-","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-24a63ef2b4e5a3e8","tex":"O_{F_0}","aliases":"O_F_0","display":"O_{F_0}","kind":"use","excerpt":"m=3Jmathscr O_F_0. tagA3.1","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-fd71dfd389a79ac5","tex":"\\operatorname{ord}","aliases":"ord operatorname","display":"\\operatorname{ord}","kind":"use","excerpt":"[Base half-space divisor] The divisor of F is supported on Λ_5. At its uniquely parametrized point, ord_aτ+bs+c/5F=w(c) [H(a)-H(b)]. For positive periods t,s, the defining gamma product has a simple zero at jt+ks+ Z for j,k≥1, a simple pole for j,k≤0, and no divisor in the mixed cones. Rational independence of 1,t,s prevents collisions. The argumen","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-83bc58ae45bea337","tex":"\\operatorname{ord}_{-n}","aliases":"ord_-n operatorname","display":"\\operatorname{ord}_{-n}","kind":"use","excerpt":"Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1), ord_n","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-79cfff1a58308123","tex":"\\operatorname{ord}_{a\\tau+bs+c/5}","aliases":"ord_aτ+bs+c/5 operatorname tau","display":"\\operatorname{ord}_{a\\tau+bs+c/5}","kind":"use","excerpt":"[Base half-space divisor] The divisor of F is supported on Λ_5. At its uniquely parametrized point, ord_aτ+bs+c/5F=w(c) [H(a)-H(b)]. For positive periods t,s, the defining gamma product has a simple zero at jt+ks+ Z for j,k≥1, a simple pole for j,k≤0, and no divisor in the mixed cones. Rational independence of 1,t,s prevents collisions. The argumen","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4d3686c9bd59ebab","tex":"\\operatorname{ord}_{ns}","aliases":"ord_ns operatorname","display":"\\operatorname{ord}_{ns}","kind":"use","excerpt":"Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1), ord_n","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2dc9cf2d1252f137","tex":"\\operatorname{ord}_{n\\tau}","aliases":"ord_nτ operatorname tau","display":"\\operatorname{ord}_{n\\tau}","kind":"use","excerpt":"Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1), ord_n","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-235862a14fa14aa7","tex":"\\operatorname{ord}_z","aliases":"ord_z operatorname","display":"\\operatorname{ord}_z","kind":"use","excerpt":"gives two q_s factors and four reciprocal q_t factors; the 2τ shift disappears from q_t by ellipticity. Finally P_7(z+δ)/P_7(z)=F(z+7δ)/F(z) and the last two identities of [source reference] give [source reference]. Under z↦ z+1 the triple changes to (a,b-15,c-10), leaving p and w(c) unchanged. Formula~[source reference] gives ord_z+1R-ord_zR =-w(c","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-fe4358c20694b0f9","tex":"\\operatorname{ord}_{z+1}","aliases":"ord_z+1 operatorname","display":"\\operatorname{ord}_{z+1}","kind":"use","excerpt":"gives two q_s factors and four reciprocal q_t factors; the 2τ shift disappears from q_t by ellipticity. Finally P_7(z+δ)/P_7(z)=F(z+7δ)/F(z) and the last two identities of [source reference] give [source reference]. Under z↦ z+1 the triple changes to (a,b-15,c-10), leaving p and w(c) unchanged. Formula~[source reference] gives ord_z+1R-ord_zR =-w(c","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-836365561dbc1176","tex":"\\operatorname{ord}_{\\lambda}","aliases":"ord_λ operatorname lambda","display":"\\operatorname{ord}_{\\lambda}","kind":"use","excerpt":"[Complete ceiling divisor] There are no zeros or poles of R outside β^-1Λ_5. At [source reference], m_λ:=ord_λR =w(c) [H(p)-H(2p-a)-H(a)+H(b)]∈{-4,-1,0,1,4}. Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e8f200fbaac8f1f3","tex":"\\operatorname{ord}_{\\tau}","aliases":"ord_τ operatorname tau","display":"\\operatorname{ord}_{\\tau}","kind":"use","excerpt":"r cannot be discarded Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this naive conjugation quoti","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0aeccd83542eabf9","tex":"\\overline C","aliases":"overline C overline","display":"\\overline C","kind":"use","excerpt":"overlineC U_Γ^3=C U_Γ^3, (fracU_ΓoverlineU_Γ)^3 =fracoverline CC. tagQ9","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-00565cb4cbe5a12b","tex":"\\overline{a_1}","aliases":"overlinea_1 overline","display":"\\overline{a_1}","kind":"use","excerpt":"frac Aι A =fraca_1^2overlinea_1^ 3overlinea_2^ 2a_2.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a04faa41ba8fb9f4","tex":"\\overline{a_1}^{\\,3}","aliases":"overlinea_1^ 3 overline","display":"\\overline{a_1}^{\\,3}","kind":"use","excerpt":"frac Aι A =fraca_1^2overlinea_1^ 3overlinea_2^ 2a_2.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-925fd86f4980ef31","tex":"\\overline{a_2}","aliases":"overlinea_2 overline","display":"\\overline{a_2}","kind":"use","excerpt":"frac Aι A =fraca_1^2overlinea_1^ 3overlinea_2^ 2a_2.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0a011249448e3582","tex":"\\overline{a_2}^{\\,2}","aliases":"overlinea_2^ 2 overline","display":"\\overline{a_2}^{\\,2}","kind":"use","excerpt":"frac Aι A =fraca_1^2overlinea_1^ 3overlinea_2^ 2a_2.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e0299f012ab607c7","tex":"\\overline{C U_\\Gamma^3}","aliases":"overlineC U_Γ^3 overline Gamma","display":"\\overline{C U_\\Gamma^3}","kind":"use","excerpt":"overlineC U_Γ^3=C U_Γ^3, (fracU_ΓoverlineU_Γ)^3 =fracoverline CC. tagQ9","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9022b64e3d85d693","tex":"\\overline{f(\\overline z)}","aliases":"overlinef(overline z) overline overline","display":"\\overline{f(\\overline z)}","kind":"use","excerpt":"The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromor","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4f0e61b1b89d61c2","tex":"\\overline{F_{t,s}(z)}","aliases":"overlineF_t,s(z) overline","display":"\\overline{F_{t,s}(z)}","kind":"use","excerpt":"overlineF_t,s(z)=F_-bar t,-bar s(-bar z).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-00b8a469a791c06b","tex":"\\overline{R(\\overline z)}","aliases":"overlineR(overline z) overline overline","display":"\\overline{R(\\overline z)}","kind":"use","excerpt":"The conjugation divisor cannot be discarded. Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2927b62d47fb05fc","tex":"\\overline{R_0}","aliases":"overlineR_0 overline","display":"\\overline{R_0}","kind":"use","excerpt":"The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromor","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-6243842ac8496d42","tex":"\\overline{U_\\Gamma}","aliases":"overlineU_Γ overline Gamma","display":"\\overline{U_\\Gamma}","kind":"use","excerpt":"overlineC U_Γ^3=C U_Γ^3, (fracU_ΓoverlineU_Γ)^3 =fracoverline CC. tagQ9","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-de254ea87cab6e97","tex":"\\overline{U_\\Gamma}^{\\,-1}","aliases":"overlineU_Γ^ -1 overline Gamma","display":"\\overline{U_\\Gamma}^{\\,-1}","kind":"use","excerpt":"F_-bar t,-bar s(1/3)=overlineU_Γ^ -1.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3db06d7f13fef7a2","tex":"\\overline{\\alpha_j}","aliases":"overlineα_j overline alpha","display":"\\overline{\\alpha_j}","kind":"use","excerpt":"The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromor","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ad51287cdcecb666","tex":"\\overline\\beta","aliases":"overlineβ overline beta","display":"\\overline\\beta","kind":"use","excerpt":"The conjugation divisor cannot be discarded. Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this ","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-91424c61d93b987b","tex":"\\overline\\epsilon","aliases":"overlineϵ overline epsilon","display":"\\overline\\epsilon","kind":"use","excerpt":"In the fixed embedding ϵ_1=2-7^1/3>0, and ϵ_2=overlineϵ.","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f761c21acf743d81","tex":"\\overline\\epsilon^{\\,2}","aliases":"overlineϵ^ 2 overline epsilon","display":"\\overline\\epsilon^{\\,2}","kind":"use","excerpt":"Moreover ϵ_2/ϵ_1=ϵoverlineϵ^ 2. Equation (Q3) becomes","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"l retains its constant Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_","count":31},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7448f0314aa9b4fe","tex":"P^n","aliases":"P^n","display":"P^n","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-116310a4b3f17728","tex":"P_5","aliases":"P_5","display":"P_5","kind":"use","excerpt":"Writing S_5=a+b and P_5=ab, the symmetric form is","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-74df595de253199d","tex":"P_7","aliases":"P_7","display":"P_7","kind":"use","excerpt":"βδ · =-8s-1, · 7δ · =τ+2s+4. The seven-point product and its correction are P_7(z)=∏_k=0^6F(z+kδ) =fracS_5(z;s,δ)S_5(z;t,δ)S_5(z+s;t,δ), R(z)=fracP_7(z)F(β z). The product identity is an actual gamma multiplication identity, not the assertion P_7(z)=F(β z). For completeness, its elementary product proof is retained from v0.12: rewrite t=2s+4-7δ, re","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2444e179edd3f281","tex":"P_{i,j,k,r}^{(n)}","aliases":"P_i,j,k,r^(n)","display":"P_{i,j,k,r}^{(n)}","kind":"use","excerpt":"Explicit three-dimensional families. For integers n≥1, 1≤ i,j,k≤ n, define Z_i,j,k^(n)=λ(-i,j,5(n+k)), P_i,j,k,r^(n)=λ(-i,j,5(n+k)-r), 1≤ r≤4. Every such triple has a<0, b>0, 3c>2b. Hence p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter ma","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"l retains its constant Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"κ(a_j)=ι(a_j+1), jpmod3. tagQ1","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1","count":16},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":19},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1,","count":38},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d0de82e442e5a70e","tex":"Q^n","aliases":"Q^n","display":"Q^n","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c5d4c2773e19f369","tex":"Q_5","aliases":"Q_5","display":"Q_5","kind":"use","excerpt":"Q_5(X)=(X-a)(X-b).","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-55d4600438de195a","tex":"q_a","aliases":"q_a","display":"q_a","kind":"use","excerpt":"Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e11eab697b8866ee","tex":"q_s","aliases":"q_s","display":"q_s","kind":"use","excerpt":"Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1, q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single ell","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1, q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single ell","count":32},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7742e4399cb287c3","tex":"q_t'","aliases":"q_t'","display":"q_t'","kind":"use","excerpt":"The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":50},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"F_0= Q(ω), K= Q(β), ϵ=2-β, R=-log(2-7^1/3)>0.","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F","count":43},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"use","excerpt":"fracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"Z_i,j,k^(n)=λ(-i,j,5(n+k)), P_i,j,k,r^(n)=λ(-i,j,5(n+k)-r), 1≤ r≤4. Every such triple has a<0, b>0, 3c>2b. Hence p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b875f98592eb0ecc","tex":"R^\\#","aliases":"R^#","display":"R^\\#","kind":"use","excerpt":"The conjugation divisor cannot be discarded. Put R^#(z)=overlineR(overline z). The cubic fields Q(β) and Q(overlineβ) are distinct: equality would put β/overlineβ=ω in a cubic field. Their intersection is Q. The support of divR lies in the first, and that of R^# in the second. Since τ is irrational (indeed β=2-1/(4+15τ)), ord_τ(R/R^#)=4. Thus this ","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4c337ef8cf0ed491","tex":"r^{-4}","aliases":"r^-4","display":"r^{-4}","kind":"use","excerpt":"The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, witho","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-cc27d193e890b4a6","tex":"r^{-5}","aliases":"r^-5","display":"r^{-5}","kind":"use","excerpt":"n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T≥ R_* and |z|≤ T/2, ∑_|λ|>Tfrac|m_λ||λ|^4≤frac4AT, |log W_D,>T(z)|≤frac2A|z|^4T. Partial summation gives the first upper bound by dropping its nonpositive boundary term and using 4∫_T^∞ Ar^3r^-5 dr=4A/T. For the tail factors the canonical branch satisfies |log E_3(w)|≤ |w|^4/[4(1-|w|)]≤ |w|^4/2, ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions, derivatives and polynomials in z. Let u be a single global nowhere-zero entire function. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole c","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-96dee922af39121c","tex":"r^3","aliases":"r^3","display":"r^3","kind":"use","excerpt":"the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8477f8563980f26b","tex":"r^4","aliases":"r^4","display":"r^4","kind":"use","excerpt":"r^4=r, r^9=r.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5968ea9995bfe90d","tex":"r^8","aliases":"r^8","display":"r^8","kind":"use","excerpt":"Thus r^3=r^8=1, and","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-30f2addb9cf36423","tex":"r^9","aliases":"r^9","display":"r^9","kind":"use","excerpt":"r^4=r, r^9=r.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-236f80d54e55c576","tex":"R_*","aliases":"R_*","display":"R_*","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-301b7c07fb502704","tex":"R_+","aliases":"R_+","display":"R_+","kind":"use","excerpt":"R_+=|ϵ|^4|a_1|^6|a_2|^4.","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-207e7e99e88d7ac7","tex":"R_0^2","aliases":"R_0^2","display":"R_0^2","kind":"use","excerpt":"P_7(z)P_7(-z). For 1≤ k≤6, F(z+kδ)F(-z+(7-k)δ)=q_t(z+kδ-s)^-1. The unpaired k=0 term retains q_t(z)/q_s(z). Dividing by the reflected denominator gives R(z)R(-z)=T(z):= fracq_t(z)q_s(β z)q_s(z)q_t(β z) ∏_k=1^6q_t(z+kδ-s)^-1. Its apparent singularities at zero are removable, and T(0)=R_0^2.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-492f632798f41118","tex":"R_0^3","aliases":"R_0^3","display":"R_0^3","kind":"use","excerpt":"[Branch-free special cube] The exact analytic reconstruction at z=1/3 is R(1/3)^3=R_0^3K(0)fracW_D(1/3)^3W_D(1) exp(-frac23α_2-frac89α_3). The right side consists of specified theta values and derivatives, algebraic period data and absolutely convergent genus-three products and sums. It contains no gamma value, arbitrary normalization scalar or new","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,wideha","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ff81ba5e23d549c7","tex":"\\Re","aliases":"Re","display":"\\Re","kind":"use","excerpt":"3.6·10^6<Re(C U_Γ^3)<3.7·10^6.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"A bounded complex radius bounds |a| and |b| by O(r); the real part then bounds |c| by O(r). There are O((1+r)^3) triples, and absolute weights are at most four. The seven translated numerator factors and the scaled denominator in [source reference] give a finite union of such cone supports and cannot increase this order. Combining the upper bound w","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"The point S=sbmodΛ_t has infinite order by rational independence. Put ψ(w)=fracq_t(w)^2q_t(2w)q_t(2w+s). [All finite norms of the return kernel] For every integer n≥1, N_nψ(w)= 1, · 5mid n,","count":30},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":105},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"At zero its value vanishes. Its derivative equals 2π y B_2(a), using Γ(-1/2)=-2sqrtπ, 1/Γ(s)=s+O(s^2), and ζ(-1,a)=-B_2(a)/2.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"r:β↦ωβ, c:ω↦ω^2, s_0:θ↦θ^2-2,","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4977f1f35008155a","tex":"s_0^k","aliases":"s_0^k","display":"s_0^k","kind":"use","excerpt":"ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6-ℓ_k-1). tagA3.5","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"The splitting field F_0(β)/ Q has group S_3, while Q(θ)/ Q is cyclic cubic. Their intersection is Galois over Q; it cannot have degree three because S_3 has no cyclic cubic quotient. Hence","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-aabbd2f9e4d841eb","tex":"S_5","aliases":"S_5","display":"S_5","kind":"use","excerpt":"The seven-point product. Use the fixed periods of Section~[source reference]. Define S_5(z;a,b)=fracΓ_rm ell(z;a,b)^5 Γ_rm ell(5z;5a,5b), F(z)=S_5(z+t;t,s)^-1. All identities below are meromorphic identities; their quotients at apparent exceptional points mean meromorphic continuation. Let Λ_5= Zτ+ Zs+tfrac15 Z, δ=fracτ+2s+47. The period identities","count":13},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b2f9bda1008161c8","tex":"S_5^2","aliases":"S_5^2","display":"S_5^2","kind":"use","excerpt":"q_t(z)=fracS_5^2-4P_5x(z)^2-S_5x(z)+P_5. tagN5.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-22cf9853e0e53ac8","tex":"S_N","aliases":"S_N","display":"S_N","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-52c400d25b92fc63","tex":"s_{\\rm per}","aliases":"s_rm per rm","display":"s_{\\rm per}","kind":"use","excerpt":"U_Γ=fracΓ_rm ell(1/3;τ,s_rm per)^5 Γ_rm ell(5/3;5τ,5s_rm per), τ=(β^2+2β)/15, s_rm per=-(β+2)/15.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b2f1d9f3f381be52","tex":"S_s","aliases":"S_s","display":"S_s","kind":"use","excerpt":"Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-87807c21edc049cd","tex":"S_t","aliases":"S_t","display":"S_t","kind":"use","excerpt":"Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-28f39655338ed57d","tex":"\\operatorname{sgn}","aliases":"sgn operatorname","display":"\\operatorname{sgn}","kind":"use","excerpt":"Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential factor relating 1-e^2π iw to sinπ w has zero fourth logarithmic derivat","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2306f05442dbe998","tex":"\\operatorname{Std}","aliases":"Std operatorname","display":"\\operatorname{Std}","kind":"use","excerpt":"Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-adac703e9d252af9","tex":"\\mathbb T","aliases":"T mathbb","display":"\\mathbb T","kind":"use","excerpt":"# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent to T^3, ident","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":122},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b1c3e8d8054aff08","tex":"\\mathbb T^3","aliases":"T^3 mathbb","display":"\\mathbb T^3","kind":"use","excerpt":"# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromorphic factor. That factor is not a holomorphic-unit gauge or a finite theta section. Regularity on the scalar plane does not establish periodic descent to T^3, ident","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8c6354f5f56778af","tex":"t^5","aliases":"t^5","display":"t^5","kind":"use","excerpt":"For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e2e357ab37efa583","tex":"T^6","aliases":"T^6","display":"T^6","kind":"use","excerpt":"For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-98f93e20c13d2c57","tex":"t^6","aliases":"t^6","display":"t^6","kind":"use","excerpt":"For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s","count":30},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"with P=(0,0), 2P=(u,u^2), and","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-36e020d7b527e3f6","tex":"u_0","aliases":"u_0","display":"u_0","kind":"use","excerpt":"Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3510e2f5aaf8b6dd","tex":"U_{21}","aliases":"U_21","display":"U_{21}","kind":"use","excerpt":"The normalized polynomial U_21(β,ξ,ω)=21U is explicitly defined in H12CMNormCertificate.lean. It is the normal form of the inherited candidate, as checked in both rational implementations. Also","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d3cbaad22faa6cc2","tex":"U_{21}^3","aliases":"U_21^3","display":"U_{21}^3","kind":"use","excerpt":"U_21^3(2-ωβ)(7ι A) =64827 ω[2+(1+ω)β] A,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b814afda3eb4ebcf","tex":"U_{\\rm alg}^{25}","aliases":"U_rm alg^25 rm","display":"U_{\\rm alg}^{25}","kind":"use","excerpt":"W^21=c_6U_rm alg^25σ(U_rm alg)^5σ^2(U_rm alg).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-cbf1555a077a48e7","tex":"U_{\\rm alg}^3","aliases":"U_rm alg^3 rm","display":"U_{\\rm alg}^3","kind":"use","excerpt":"U_rm alg^3=ωfracϵ_2ϵ_1 frac Aι( A). tagQ3","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a87697d3e0787924","tex":"U_{\\rm alg}^6","aliases":"U_rm alg^6 rm","display":"U_{\\rm alg}^6","kind":"use","excerpt":"W^5=σ(W)U_rm alg^6,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-349daf8c6dd8d0cb","tex":"U_\\Gamma^3","aliases":"U_Γ^3 Gamma","display":"U_\\Gamma^3","kind":"use","excerpt":"|C U_Γ^3|=R_+.","count":10},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"V_0=H(1/(3J))H(13/(3J)), V_1=H(4/(3J))H(10/(3J)), V_2=H(16/(3J))H(19/(3J)).","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7d77b9a1d8369d4e","tex":"v^3","aliases":"v^3","display":"v^3","kind":"use","excerpt":"ν=(X+β/X-β^2)/3, v=-β^2ν-2ν^2, v^3-β v^2+1=0.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"V_0=H(1/(3J))H(13/(3J)), V_1=H(4/(3J))H(10/(3J)), V_2=H(16/(3J))H(19/(3J)).","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"V_0=H(1/(3J))H(13/(3J)), V_1=H(4/(3J))H(10/(3J)), V_2=H(16/(3J))H(19/(3J)).","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"use","excerpt":"V_0=H(1/(3J))H(13/(3J)), V_1=H(4/(3J))H(10/(3J)), V_2=H(16/(3J))H(19/(3J)).","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8b86c046f5cd21ee","tex":"V_m","aliases":"V_m","display":"V_m","kind":"use","excerpt":"ℓ_m=-2log V_m, V_0V_1V_2=1. tagA4.3","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-985e62d5d85e9108","tex":"V_m^3","aliases":"V_m^3","display":"V_m^3","kind":"use","excerpt":"V_m^3=|a_m|, tagA5.6","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"The same calculation for the inherited pre-smoothing unit W gives line norms","count":33},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,","count":30},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, a","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bb1e300d2c19576e","tex":"W^{21}","aliases":"W^21","display":"W^{21}","kind":"use","excerpt":"W^21=c_6U_rm alg^25σ(U_rm alg)^5σ^2(U_rm alg).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7be309749e78f316","tex":"W^3","aliases":"W^3","display":"W^3","kind":"use","excerpt":"W^3=-fracϵ_2ϵ_1 frac Bι( B). tagQ4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0d4e45da82688954","tex":"w^3","aliases":"w^3","display":"w^3","kind":"use","excerpt":"The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, a","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0622aed994f04bb7","tex":"w^4","aliases":"w^4","display":"w^4","kind":"use","excerpt":"beginequation R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential f","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-61a19024a5fdb37d","tex":"W^5","aliases":"W^5","display":"W^5","kind":"use","excerpt":"W^5=σ(W)U_rm alg^6,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5caf668b59cf4d10","tex":"w^k","aliases":"w^k","display":"w^k","kind":"use","excerpt":"ir poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded. The cubic upper bound gives ∑_D|m_λ|/|λ|^4<∞, by partial summation or dyadic shells. For |w|<1, log E_3(w)=-∑_k≥4fracw^kk. On each fixed compact set the tail log","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7908ba336cca4654","tex":"W_D","aliases":"W_D","display":"W_D","kind":"use","excerpt":"The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, a","count":13},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f2a8b246b18651dc","tex":"W_{D,>T}","aliases":"W_D,>T","display":"W_{D,>T}","kind":"use","excerpt":"logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T≥ R_* and |z|≤ T/2, ∑_|λ|>Tfrac|m_λ||λ|^4≤frac4AT, |log W_D,>T(z)|≤fra","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5544602b9c8d7e18","tex":"W_D^\\#","aliases":"W_D^#","display":"W_D^\\#","kind":"use","excerpt":"The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromor","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a94b4f8f56cba3ef","tex":"W_{\\rm St}","aliases":"W_rm St rm","display":"W_{\\rm St}","kind":"use","excerpt":"Consequently W_rm St=σ(W) satisfies","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ba1455ccf84f4e4d","tex":"w_{T^2v}","aliases":"w_T^2v","display":"w_{T^2v}","kind":"use","excerpt":"F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-55732f8cfcd5cc28","tex":"w_{Tv}","aliases":"w_Tv","display":"w_{Tv}","kind":"use","excerpt":"F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e6e9e4d147adf393","tex":"w_v","aliases":"w_v","display":"w_v","kind":"use","excerpt":"w_v=tfrac13+(bτ+cs+a)/2, v=(b,c,a)∈{0,1}^3, T(b,c,a)=(a,b,c), τ=-t.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"epresenting δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat c)^T=(a,b-8k,c-5k)^T, which expresses β(λ+kδ)=βλ+","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ea83107951d84fac","tex":"\\widehat b","aliases":"widehat b widehat","display":"\\widehat b","kind":"use","excerpt":"epresenting δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat c)^T=(a,b-8k,c-5k)^T, which expresses β(λ+kδ)=βλ+","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e7a695b8f5409721","tex":"\\widehat c","aliases":"widehat c widehat","display":"\\widehat c","kind":"use","excerpt":"epresenting δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,widehat c)^T=(a,b-8k,c-5k)^T, which expresses β(λ+kδ)=βλ+","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f13dfb9c74fea818","tex":"\\wp","aliases":"wp","display":null,"kind":"use","excerpt":"The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-","count":14},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9b365dbe6a6c267d","tex":"\\wp'^2","aliases":"wp'^2 wp","display":null,"kind":"use","excerpt":"For completeness, T is nonzero three-torsion and its normalized x-coordinate is a root of 3x(x^3+1). The lattice is invariant under ordinary conjugation, so wp(1/3) and its derivative are real. Since g_3(Λ_0)>0, the real differential equation wp'^2=4wp^3-g_3 implies wp(1/3)>0. The argument of Ω^2 is π/3, selecting -ω among the three roots of x^3=-1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-54a2d6341596db2b","tex":"\\wp^3","aliases":"wp^3 wp","display":null,"kind":"use","excerpt":"For completeness, T is nonzero three-torsion and its normalized x-coordinate is a root of 3x(x^3+1). The lattice is invariant under ordinary conjugation, so wp(1/3) and its derivative are real. Since g_3(Λ_0)>0, the real differential equation wp'^2=4wp^3-g_3 implies wp(1/3)>0. The argument of Ω^2 is π/3, selecting -ω among the three roots of x^3=-1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7f89f0e9d955bb00","tex":"\\wp_+","aliases":"wp_+ wp","display":null,"kind":"use","excerpt":"The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-01517f3c74cf31b6","tex":"\\wp_{5t}","aliases":"wp_5t wp","display":null,"kind":"use","excerpt":"The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-cfefec0e7f1fe2e5","tex":"\\wp_t","aliases":"wp_t wp","display":null,"kind":"use","excerpt":"The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5e039d68e0235326","tex":"\\wp_t'","aliases":"wp_t' wp","display":null,"kind":"use","excerpt":"x(z)=wp_t(z), y(z)=wp_t'(z), f(X)=4X^3-g_2X-g_3.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-cce1f32bcfaee31a","tex":"\\wp_\\Lambda","aliases":"wp_Λ wp Lambda","display":null,"kind":"use","excerpt":"fracσ_Λ(T+z)σ_Λ(T-z) σ_Λ(T)^2σ_Λ(z)^2 =wp_Λ(z)-wp_Λ(T). tagA5.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-781ea583be49e05a","tex":"\\wp_{\\Lambda_0}","aliases":"wp_Λ_0 wp Lambda","display":null,"kind":"use","excerpt":"X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":34},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a86973ce6a8c4622","tex":"\\mathsf X","aliases":"X mathsf","display":"\\mathsf X","kind":"use","excerpt":"X_t(z)=fracx(z)-aa-b.","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"The cubic characteristic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smoot","count":21},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-78dcbad828a95e4b","tex":"X^{-1}","aliases":"X^-1","display":"X^{-1}","kind":"use","excerpt":"x(u)=X^-1, x(2u)=ω^2X^-1, x(3u)=ω X^-1. tagA5.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1615767a11b761c0","tex":"X^{-3}","aliases":"X^-3","display":"X^{-3}","kind":"use","excerpt":"Multiplication and (A4.3) give C^3|X^-3+1|=1. Cubing each equation in (A5.5) now removes the unknown constant exactly:","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"a_0 · =frac(ω^2+ω X)^31+J =(1+ω)X^2-X+1, a_1 · =frac(1+ω X)^31+J =-X^2+(1+ω)X+1, a_2 · =frac(ω+ω X)^31+J =-ω X^2-ω X+1. tagA5.7","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"use","excerpt":"On the normalized curve E_0:y^2+y=x^3, set","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ee74251aad7234d0","tex":"X^4","aliases":"X^4","display":"X^4","kind":"use","excerpt":"ψ_3(X)=3X^4-tfrac32g_2X^2-3g_3X-tfrac116g_2^2,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-27db5ab2afe7c2b0","tex":"X^j","aliases":"X^j","display":"X^j","kind":"use","excerpt":"A basis for the rational calculation is β^iX^jω^k, 0≤ i,j<3, 0≤ k<2. For odd h, multiply the two norms instead of dividing: the residual is their product minus one. Both supplied implementations obtain the exact zero polynomial in all six cases.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4f07c087b3a17b11","tex":"\\mathsf X_s","aliases":"X_s mathsf","display":"\\mathsf X_s","kind":"use","excerpt":"X_t(ρ z)= X_s(z), q_t(ρ z)=q_s(z). tagN7.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b75ba905fa4f2898","tex":"\\mathsf X_t","aliases":"X_t mathsf","display":"\\mathsf X_t","kind":"use","excerpt":"X_t(z)=fracx(z)-aa-b.","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"On the normalized curve E_0:y^2+y=x^3, set","count":18},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-854480e3df9e6b0c","tex":"y^{1-2s}","aliases":"y^1-2s","display":"y^{1-2s}","kind":"use","excerpt":"sqrtπ fracΓ(s-1/2)Γ(s) y^1-2s[ζ(2s-1,a)+ζ(2s-1,1-a)].","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0e7659cd4ef8f947","tex":"Y^2","aliases":"Y^2","display":"Y^2","kind":"use","excerpt":"Y^2+(1-u)XY-uY=X^3-uX^2, tagN6.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"On the normalized curve E_0:y^2+y=x^3, set","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7be44d74687569ce","tex":"y_a","aliases":"y_a","display":"y_a","kind":"use","excerpt":"a=x(P), b=x(2P), d=a-b, y_a=y(P), y_b=y(2P),","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d65dae5c5b4349f7","tex":"y_b","aliases":"y_b","display":"y_b","kind":"use","excerpt":"a=x(P), b=x(2P), d=a-b, y_a=y(P), y_b=y(2P),","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":103},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-04736fc5c249b303","tex":"z^{-2}","aliases":"z^-2","display":"z^{-2}","kind":"use","excerpt":"Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4afcc7369b95de7a","tex":"z^{-4}","aliases":"z^-4","display":"z^{-4}","kind":"use","excerpt":"[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ecf8d3085a19b518","tex":"Z^{-n}","aliases":"Z^-n","display":"Z^{-n}","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-af5f0843f8046d2c","tex":"\\mathbb Z^2","aliases":"Z^2 mathbb","display":"\\mathbb Z^2","kind":"use","excerpt":"The exact information lost by doubling. Quotient nonzero meromorphic elliptic functions by constants and translation coboundaries A(w+s)/A(w). Set e=[q_t], p=[ψ] in additive notation. These generate a free Z^2: the divisor charges on the disjoint infinite translation orbits of 0 and -S/2 are respectively (4,4) and (0,-4). A finite divisor coboundar","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4140de75758a272d","tex":"z^3","aliases":"z^3","display":"z^3","kind":"use","excerpt":"∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-34d4b252430fe95b","tex":"z^4","aliases":"z^4","display":"z^4","kind":"use","excerpt":"mathbb Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fix","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"use","excerpt":"The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromor","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7f0a239c7db692b0","tex":"Z^n","aliases":"Z^n","display":"Z^n","kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1838a155bac436b2","tex":"Z_{i,j,k}^{(n)}","aliases":"Z_i,j,k^(n)","display":"Z_{i,j,k}^{(n)}","kind":"use","excerpt":"Explicit three-dimensional families. For integers n≥1, 1≤ i,j,k≤ n, define Z_i,j,k^(n)=λ(-i,j,5(n+k)), P_i,j,k,r^(n)=λ(-i,j,5(n+k)-r), 1≤ r≤4. Every such triple has a<0, b>0, 3c>2b. Hence p>0 and the bracket in [source reference] is one. The Z points have order four and the P points have order minus one. Rational independence makes the parameter ma","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ae33ff0c9a9cfbb4","tex":"z_j","aliases":"z_j","display":"z_j","kind":"use","excerpt":"The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b6f4f7dcc708126e","tex":"Z_m","aliases":"Z_m","display":"Z_m","kind":"use","excerpt":"Let Z_m(s) be the sum of the two ray partial zeta functions in row m, and write ℓ_m=Z_m'(0). Then","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7f9ffd867ef77e02","tex":"Z_m'","aliases":"Z_m'","display":"Z_m'","kind":"use","excerpt":"Let Z_m(s) be the sum of the two ray partial zeta functions in row m, and write ℓ_m=Z_m'(0). Then","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7^1/3<1 at the real embedding. The numbers 1,t,s are rationally independent: a rational linear relation becomes a polynomi","count":51},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-73d4c9929ca29905","tex":"|^{-2s}","aliases":"|^-2s","display":"|^{-2s}","kind":"use","excerpt":"E_z(s)=∑_m,n∈ Z|m+nω+z|^-2s,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ced4cfe2de1b88e3","tex":"|^{-4}","aliases":"|^-4","display":"|^{-4}","kind":"use","excerpt":"The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, witho","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"on. Every meromorphic function u(z)fracP(z)Q(z) has zero and pole counts O(r^2). Thus R is not representable in this class. For Y=ImΩ>0, the defining theta series has terms bounded by exp{-π n^TYn+2π n Im(Az+b) }. Completing the square, and absorbing half of the positive quadratic form into a convergent Gaussian sum, gives Cexp(C'|z|^2) for a fixed","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3e3225555f03b1a7","tex":"|^3","aliases":"|^3","display":"|^3","kind":"use","excerpt":"|U_Γ|^3=|ϵ^3a_2a_1^5|.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8f4fab3d5ea9ca88","tex":"|^4","aliases":"|^4","display":"|^4","kind":"use","excerpt":"R_+=|ϵ|^4|a_1|^6|a_2|^4.","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a2e3fd70ad04e933","tex":"|^6","aliases":"|^6","display":"|^6","kind":"use","excerpt":"R_+=|ϵ|^4|a_1|^6|a_2|^4.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"{T±2u}, {T± u}, {T±3u}. tagA5.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"The scalar missing from a closed finite orbit. The earlier index-23 and index-10 orbit decompositions are 529=1+528 and 100=1+3+8·12. They are exact finite permutation data. For values z_j on a cycle of length d, replacing every value by cz_j leaves all adjacent ratios unchanged, including their product one, but multiplies the cycle product by c^d.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"Gal(E/ Q)=S_3× C_3. tagA2.1","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5fe06057e58f9c75","tex":"\\alpha^6","aliases":"α^6 alpha","display":"\\alpha^6","kind":"use","excerpt":", leaving fraca_Δ(p_3)a_Δ(0)=e^2π im_2/3e^real number. After the based identification the exact modulus equality forces that real exponent to vanish. The ratio then has cube one, and an independent branch enclosure can select one. The constant α cancels in this based ratio; it cannot simply be discarded in the unbased principal norm, where it contr","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"use","excerpt":"The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) ","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e8c617160e3b774f","tex":"\\alpha_2","aliases":"α_2 alpha","display":"\\alpha_2","kind":"use","excerpt":"The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) ","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3eb1b87cfee02bfd","tex":"\\alpha_3","aliases":"α_3 alpha","display":"\\alpha_3","kind":"use","excerpt":"The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) ","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-44dbbebcbc92a60e","tex":"\\alpha_j","aliases":"α_j alpha","display":"\\alpha_j","kind":"use","excerpt":"The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) ","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":67},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8735109003c9a8e3","tex":"\\beta^{-1}","aliases":"β^-1 beta","display":"\\beta^{-1}","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ab15d1c00936ee24","tex":"\\beta^2","aliases":"β^2 beta","display":"\\beta^2","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":10},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1210911b063d7dfc","tex":"\\beta^3","aliases":"β^3 beta","display":"\\beta^3","kind":"use","excerpt":"ω^2+ω+1=0, J=2+3ω, β^3=7, ξ^3=J.","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9cf5f8a24eca4b75","tex":"\\beta^i","aliases":"β^i beta","display":"\\beta^i","kind":"use","excerpt":"A basis for the rational calculation is β^iX^jω^k, 0≤ i,j<3, 0≤ k<2. For odd h, multiply the two norms instead of dividing: the residual is their product minus one. Both supplied implementations obtain the exact zero polynomial in all six cases.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"Let γ_ij fix ω and send","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f09fcdbf25a5d4be","tex":"\\mathbf\\Gamma","aliases":"Γ mathbf Gamma","display":"\\mathbf\\Gamma","kind":"use","excerpt":"|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f375be9034023d47","tex":"\\mathbf\\Gamma_{\\mathfrak a,h}","aliases":"Γ_ a,h mathbf Gamma mathfrak","display":"\\mathbf\\Gamma_{\\mathfrak a,h}","kind":"use","excerpt":"|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1bba528ef0fe38c1","tex":"\\gamma_{ij}","aliases":"γ_ij gamma","display":"\\gamma_{ij}","kind":"use","excerpt":"Let γ_ij fix ω and send","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-66df19f9c2e3aaac","tex":"\\Gamma_{\\rm ell}","aliases":"Γ_rm ell Gamma rm","display":"\\Gamma_{\\rm ell}","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-79adffac600d7139","tex":"\\Gamma_{\\rm Euler}","aliases":"Γ_rm Euler Gamma rm","display":"\\Gamma_{\\rm Euler}","kind":"use","excerpt":"X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":34},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ddd0460063b633b5","tex":"\\delta_M","aliases":"δ_M delta","display":"\\delta_M","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":26},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8edc6624669ef96d","tex":"\\epsilon^{-1}","aliases":"ϵ^-1 epsilon","display":"\\epsilon^{-1}","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-fb43c73461568514","tex":"\\epsilon^2","aliases":"ϵ^2 epsilon","display":"\\epsilon^2","kind":"use","excerpt":"Put q_a(z)=θ_0(z;a)^5/θ_0(5z;5a). Direct product identities and the inherited modular normalization give F(z+1) · =F(z), · fracF(z+τ)F(z) · =q_s(z), · fracF(z+s)F(z) · =q_t(z)^-1, q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single ell","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a34ea3a111e03d1d","tex":"\\epsilon^3","aliases":"ϵ^3 epsilon","display":"\\epsilon^3","kind":"use","excerpt":"|U_Γ|^3=|ϵ^3a_2a_1^5|.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-890e777d8bea7959","tex":"\\epsilon_0","aliases":"ϵ_0 epsilon","display":"\\epsilon_0","kind":"use","excerpt":"ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-da012babef541427","tex":"\\epsilon_1","aliases":"ϵ_1 epsilon","display":"\\epsilon_1","kind":"use","excerpt":"ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c7ffdc6ef6b651ea","tex":"\\epsilon_2","aliases":"ϵ_2 epsilon","display":"\\epsilon_2","kind":"use","excerpt":"ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a14473f51c7df774","tex":"\\epsilon_j","aliases":"ϵ_j epsilon","display":"\\epsilon_j","kind":"use","excerpt":"ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"The even characters make no contribution to the derivative at zero. The nontrivial even characters have at least two archimedean zeros. The trivial character has the zero of ζ_K, together with the extra zero of 1-3^-s. This also agrees with BCG, section 3.1.","count":12},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0ac84558b1441291","tex":"\\zeta_{\\mathfrak f}'","aliases":"ζ_ f' zeta mathfrak","display":"\\zeta_{\\mathfrak f}'","kind":"use","excerpt":"ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6-ℓ_k-1). tagA3.5","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7af8be3c8706c16c","tex":"\\zeta_{\\mathbb Q}","aliases":"ζ_ Q zeta mathbb","display":"\\zeta_{\\mathbb Q}","kind":"use","excerpt":"L_F_0(s,ϱ)=ζ_K(s)/ζ_ Q(s).","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-eecb92568f818b9a","tex":"\\zeta_{F_0}","aliases":"ζ_F_0 zeta","display":"\\zeta_{F_0}","kind":"use","excerpt":"ζ_F_0(s)(1-3^-s)(1-7^-s),","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b264dd7fda915463","tex":"\\zeta_K","aliases":"ζ_K zeta","display":"\\zeta_K","kind":"use","excerpt":"The even characters make no contribution to the derivative at zero. The nontrivial even characters have at least two archimedean zeros. The trivial character has the zero of ζ_K, together with the extra zero of 1-3^-s. This also agrees with BCG, section 3.1.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":14},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9657c492efcc34dd","tex":"\\eta^2","aliases":"η^2 eta","display":"\\eta^2","kind":"use","excerpt":"Conjugation of the base field by c sends ϱη^2 to ϱ^-1η^2=λ^2, so","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f1d4165614f756bc","tex":"\\eta^j","aliases":"η^j eta","display":"\\eta^j","kind":"use","excerpt":"L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f87bcefac1e6e31d","tex":"\\eta_{\\rm Ded}","aliases":"η_rm Ded eta rm","display":"\\eta_{\\rm Ded}","kind":"use","excerpt":"H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"the P points have order minus one. Rational independence makes the parameter map injective. The pole indices are distinct because 5(k-k')=r-r' forces both sides to vanish; the zero and pole families are disjoint by c modulo five. [Two-sided cubic growth] Let n_+(r) and n_-(r) count the zeros and poles of R in |z|≤ r, with multiplicity. Then n_+(r)=","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":13},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","excerpt":"r:β↦ωβ, c:ω↦ω^2, s_0:θ↦θ^2-2,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9f146cf9b0e1580e","tex":"\\theta_1","aliases":"θ_1 theta","display":"\\theta_1","kind":"use","excerpt":"H_τ(z)=e^-π(Im z)^2/y |fracθ_1(π z;e^π iτ)η_rm Ded(τ)|.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"ι(β)=β, ι(ω)=ω^2, ι(ξ)=β/ξ,","count":11},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.","count":24},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"0 · -5 · 4. Its quotient generator is (1,2,20)/7, representing δ. Thus β^-1Λ_5=Λ_5+ Zδ. Write a potential singular point as λ(a,b,c)=fracaτ+bs+c/5β, p=lceilfrac3c-2b7rceil, k=7p+2b-3c∈{0,…,6}. Exactly one argument λ+kδ lies in Λ_5. Its integral coordinates are (widehat a,widehat b,widehat c)=(p,2p-a,20p+5b-8c). Indeed N_5(widehat a,widehat b,wideha","count":35},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-308a821eda6e7229","tex":"\\lambda^2","aliases":"λ^2 lambda","display":"\\lambda^2","kind":"use","excerpt":"The period-one transition fixes the cubic coefficient. The points 0,1,1/3 are regular and nonzero for R and W_D. For 1, [source reference] has nonzero factors at zero. For 1/3, β/3=-5s-2/3, whose rational coordinate is not a rational fifth; rational independence excludes a singular lattice point. Define B_D(1)=(log W_D)''(1) =∑_λ∈ Dm_λ[-frac1(λ-1)^","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-08209adac2de3dd9","tex":"\\lambda^3","aliases":"λ^3 lambda","display":"\\lambda^3","kind":"use","excerpt":"The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, witho","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-2163cd26925b0e84","tex":"\\lambda^4","aliases":"λ^4 lambda","display":"\\lambda^4","kind":"use","excerpt":"g_2=λ^4(b_2^2/12-2b_4), g_3=λ^6(-b_2^3/216+b_2b_4/6-u^2).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-08fbedae94357b79","tex":"\\lambda^6","aliases":"λ^6 lambda","display":"\\lambda^6","kind":"use","excerpt":"g_2=λ^4(b_2^2/12-2b_4), g_3=λ^6(-b_2^3/216+b_2b_4/6-u^2).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-6ede142bbc0a6d7e","tex":"\\Lambda_0","aliases":"Λ_0 Lambda","display":"\\Lambda_0","kind":"use","excerpt":"X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-64100f915b9d565e","tex":"\\Lambda_5","aliases":"Λ_5 Lambda","display":"\\Lambda_5","kind":"use","excerpt":"The seven-point product. Use the fixed periods of Section~[source reference]. Define S_5(z;a,b)=fracΓ_rm ell(z;a,b)^5 Γ_rm ell(5z;5a,5b), F(z)=S_5(z+t;t,s)^-1. All identities below are meromorphic identities; their quotients at apparent exceptional points mean meromorphic continuation. Let Λ_5= Zτ+ Zs+tfrac15 Z, δ=fracτ+2s+47. The period identities","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4fd22cebbbbcd333","tex":"\\Lambda_a","aliases":"Λ_a Lambda","display":"\\Lambda_a","kind":"use","excerpt":"q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e08b44bc605b92e6","tex":"\\Lambda_s","aliases":"Λ_s Lambda","display":"\\Lambda_s","kind":"use","excerpt":"q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-44aff88c8c62e99e","tex":"\\Lambda_t","aliases":"Λ_t Lambda","display":"\\Lambda_t","kind":"use","excerpt":"q_s(z) · =q_t(ρ z), · F(0) · =ϵ^2, · F(1/3)F(2/3) · =1. The two elliptic lattices are homothetic, ρΛ_s=Λ_t, with Λ_a= Z+ Za. This is not an endomorphism of the single elliptic lattice Λ_t. Section~[source reference] proves the exact rational fifth-torsion normalization and explains the covariance without omitting its scalar.","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-17358a1b5ed8286b","tex":"\\Lambda_t^+","aliases":"Λ_t^+ Lambda","display":"\\Lambda_t^+","kind":"use","excerpt":"The fifth-isogeny differential retains its constant. Let a=wp_t(1/5), b=wp_t(2/5) and L_t=(q_t'/q_t)'. The enlarged lattice is Λ_t^+=tfrac15 Z+ Zt, with wp_+(z)=25wp_5t(5z). The degree-five quotient satisfies L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"The right side has modulus one but is not in μ_6: the rational quotient calculation proves its sixth power differs from one. Its value is approximately","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-3141040a2e8f409f","tex":"\\mu_6","aliases":"μ_6 mu","display":"\\mu_6","kind":"use","excerpt":"The right side has modulus one but is not in μ_6: the rational quotient calculation proves its sixth power differs from one. Its value is approximately","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d1718586db38f533","tex":"\\nu^2","aliases":"ν^2 nu","display":"\\nu^2","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2, Y · =(4β^2+β-14)v^2+(11β-23)v-β^2+3β-3, Z · =frac(-2β^2+7β-8)v^2+(","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"ω^2+ω+1=0, J=2+3ω, β^3=7, ξ^3=J.","count":14},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e8149ec697a9ee64","tex":"\\xi^2","aliases":"ξ^2 xi","display":"\\xi^2","kind":"use","excerpt":"a_0=(1+ω)ξ^2-ξ+1, a_1=-ξ^2+(1+ω)ξ+1, a_2=-ωξ^2-ωξ+1.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ab96ce3cb4126eb6","tex":"\\xi^3","aliases":"ξ^3 xi","display":"\\xi^3","kind":"use","excerpt":"ω^2+ω+1=0, J=2+3ω, β^3=7, ξ^3=J.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":15},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-384c27fbfc6509a5","tex":"\\varrho","aliases":"ϱ varrho","display":"\\varrho","kind":"use","excerpt":"We use F_0= Q(ω) for the imaginary-quadratic field, to separate it from the scalar gamma function F. The eta quotient is H(a)=η(5a)/η(a)^5; the integer step function H(n) used in the divisor formula is a different object. The symbol R_0 is the scalar anchor, while R_* is a counting cutoff radius. In the arithmetic block ϱ is a ray character and a r","count":12},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ce9600c0d281c14e","tex":"\\varrho^{-1}","aliases":"ϱ^-1 varrho","display":"\\varrho^{-1}","kind":"use","excerpt":"Conjugation of the base field by c sends ϱη^2 to ϱ^-1η^2=λ^2, so","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bd1aac0f414abebd","tex":"\\rho^2","aliases":"ρ^2 rho","display":"\\rho^2","kind":"use","excerpt":"Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d8d4d6645b48e463","tex":"\\rho^4","aliases":"ρ^4 rho","display":"\\rho^4","kind":"use","excerpt":"a_s=ρ^2a_t, b_s=ρ^2b_t, c_s=ρ^4c_t,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-21ff13b855a4a988","tex":"\\varrho^6","aliases":"ϱ^6 varrho","display":"\\varrho^6","kind":"use","excerpt":"For t^6=W, the lift ϱ(t)=t^5/U_rm alg commutes with t↦ c_6t, and satisfies ϱ^6(t)=c_6^4t. Irreducibility is checked at the residue assignment (β,v,ω)=(32,99,48) over F_181, where W=23 and T^6-23 is irreducible. The earlier local-integrality argument validates this reduction. The resulting abelian group is C_2× C_18.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d765e00b61e2dbf1","tex":"\\rho^m","aliases":"ρ^m rho","display":"\\rho^m","kind":"use","excerpt":"The kernel on X has representatives j(1,2,4)^T/7, 0≤ j≤6. The conductor point p_3=e_2/3 obeys Mp_3-p_3=(0,5,1)^T and Np_3-p_3=(0,-5,-1)^T. The dual character e_k(v)=e^2π ik^Tv satisfies e_k(Mv)=e_Ak(v). Frequency transport is distinct from the scalar return sequence F(ρ^m/3).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"The arithmetic Frobenius σ at the norm-two prime acts by","count":18},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"W^21=c_6U_rm alg^25σ(U_rm alg)^5σ^2(U_rm alg).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0819a6083d4c9117","tex":"\\sigma^3","aliases":"σ^3 sigma","display":"\\sigma^3","kind":"use","excerpt":"For the last formula, the conjugate Gauss resolvent is 7/G, and barβ=ω^2β. Formula (A1.5) follows directly. It must not be replaced by the involution σ^3, which fixes K.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-459d9c15bb309a80","tex":"\\sigma^5","aliases":"σ^5 sigma","display":"\\sigma^5","kind":"use","excerpt":"log|U_Γ|^2 =ζ_ f'(1,0)-5ζ_ f'(σ^5,0). tagA7.1","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-03bb23951142b309","tex":"\\sigma^h","aliases":"σ^h sigma","display":"\\sigma^h","kind":"use","excerpt":"N_κ(σ^hW) =N_κ(ϵ a_h+1^2)^(-1)^h, 0≤ h<6. tagA6.2","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f746e680d517662e","tex":"\\sigma^{h+5}","aliases":"σ^h+5 sigma","display":"\\sigma^{h+5}","kind":"use","excerpt":"log|σ^h W|=(-1)^h(frac R2+2log|a_h+1|) =-3ζ_ f'(σ^h+5,0). tagA6.3","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4d93182daab36f36","tex":"\\sigma_{\\mathfrak a}","aliases":"σ_ a sigma mathfrak","display":"\\sigma_{\\mathfrak a}","kind":"use","excerpt":"|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-bf8e1e398ce726b8","tex":"\\sigma_{\\mathfrak b}","aliases":"σ_ b sigma mathfrak","display":"\\sigma_{\\mathfrak b}","kind":"use","excerpt":"|Γ_ a,h( f b^-1)|^6 =|(σ_ bW_rm St)^N( a)-σ_ a|. tagA7.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-05cf4213033b9a59","tex":"\\sigma_\\Lambda","aliases":"σ_Λ sigma Lambda","display":"\\sigma_\\Lambda","kind":"use","excerpt":"fracσ_Λ(T+z)σ_Λ(T-z) σ_Λ(T)^2σ_Λ(z)^2 =wp_Λ(z)-wp_Λ(T). tagA5.4","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":35},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"ϕ_2(X)=X^4+tfrac12g_2X^2+2g_3X+tfrac116g_2^2.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-69c13435fecd680e","tex":"\\phi_2","aliases":"ϕ_2 phi","display":"\\phi_2","kind":"use","excerpt":"ϕ_2(X)=X^4+tfrac12g_2X^2+2g_3X+tfrac116g_2^2.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b5017bbed0f6ff42","tex":"\\chi_{-3}","aliases":"χ_-3 chi","display":"\\chi_{-3}","kind":"use","excerpt":"L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4f4bb8c89eefcd8d","tex":"\\chi_7","aliases":"χ_7 chi","display":"\\chi_7","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-91eb03acc00c5918","tex":"\\chi_7^j","aliases":"χ_7^j chi","display":"\\chi_7^j","kind":"use","excerpt":"L_K(s,χ_j) =L_ Q(s,χ_-3χ_7^j) L_F_0(s,ϱη^j). tagA2.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c2171a0d6968e6fe","tex":"\\chi_j","aliases":"χ_j chi","display":"\\chi_j","kind":"use","excerpt":"χ_j(c)=-1, χ_j(s_0)=ω^j, j=0,1,2.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"The point S=sbmodΛ_t has infinite order by rational independence. Put ψ(w)=fracq_t(w)^2q_t(2w)q_t(2w+s). [All finite norms of the return kernel] For every integer n≥1, N_nψ(w)= 1, · 5mid n,","count":12},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8a2063c52c7ed296","tex":"\\psi_3","aliases":"ψ_3 psi","display":"\\psi_3","kind":"use","excerpt":"ψ_3(X)=3X^4-tfrac32g_2X^2-3g_3X-tfrac116g_2^2,","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions,","count":6},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":75},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-b0a6be2d12cb77bd","tex":"\\omega^{1-k}","aliases":"ω^1-k omega","display":"\\omega^{1-k}","kind":"use","excerpt":"-frac13(ω^1-kL_F_0'(0,λ) +ω^2-2kL_F_0'(0,λ^2)).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-23a05885bf7340b9","tex":"\\Omega^2","aliases":"Ω^2 Omega","display":"\\Omega^2","kind":"use","excerpt":"X=fracΩ^2wp_Λ_0(1/J), Λ_0= Z+ Zω, Ω=e^π i/6fracΓ_rm Euler(1/3)^32π. tagA1.3","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f2f53155f7e59bd3","tex":"\\omega^2","aliases":"ω^2 omega","display":"\\omega^2","kind":"use","excerpt":"Fixed data, conventions, and the claim boundary. Fix throughout ω · =e^2π i/3, · β · =7^1/3ω^2, · ϵ · =2-β, · ρ · =ϵ^-1, τ · =(β^2+2β)/15, · t · =-τ, · s · =-(β+2)/15, · δ · =(τ+2s+4)/7. The real cube root 7^1/3 is positive; this fixes the lower complex embedding of β. We have Im t,Im s,Imδ>0, Imτ<0 and |ρ|<1. The last follows from N(ϵ)=1 and 0<2-7","count":24},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-be4a840441264df0","tex":"\\omega^{2-2k}","aliases":"ω^2-2k omega","display":"\\omega^{2-2k}","kind":"use","excerpt":"-frac13(ω^1-kL_F_0'(0,λ) +ω^2-2kL_F_0'(0,λ^2)).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7ef85b5370113524","tex":"\\omega^i","aliases":"ω^i omega","display":"\\omega^i","kind":"use","excerpt":"β↦ω^iβ, ξ↦ω^jξ.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a27b633daa590319","tex":"\\omega^j","aliases":"ω^j omega","display":"\\omega^j","kind":"use","excerpt":"ϵ_j=2-ω^jβ, ϵ=ϵ_0, ϵ_0ϵ_1ϵ_2=1.","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-960f13a0e4265f44","tex":"\\omega^k","aliases":"ω^k omega","display":"\\omega^k","kind":"use","excerpt":"A basis for the rational calculation is β^iX^jω^k, 0≤ i,j<3, 0≤ k<2. For odd h, multiply the two norms instead of dividing: the residual is their product minus one. Both supplied implementations obtain the exact zero polynomial in all six cases.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ea425107ac8540ba","tex":"\\Omega_\\ell","aliases":"Ω_ℓ Omega ell","display":"\\Omega_\\ell","kind":"use","excerpt":"The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":4},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"The precise finite-theta obstruction. [Fixed affine finite-theta class] Fix finitely many symmetric period matrices Ω_ℓ with positive-definite imaginary parts ImΩ_ℓ, affine maps z↦ A_ℓ z+b_ℓ, and finite derivatives of their Riemann theta restrictions. Let P,Q be nonzero entire functions formed by finite polynomial expressions in these restrictions,","count":21},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"l of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For any C^q norm choose s>max(q,3). Smooth Fourier coefficients have finite weighted ℓ^1 norms of every order, and the displayed exp","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"ℓ^2=4(2a+b), m^2=4(a+2b),","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e149f481feb293a4","tex":"\\ell_0","aliases":"ℓ_0 ell","display":"\\ell_0","kind":"use","excerpt":"L_F_0'(0,λ)=ℓ_0+ωℓ_1+ω^2ℓ_2, L_F_0'(0,λ^2)=ℓ_0+ω^2ℓ_1+ωℓ_2. tagA3.3","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8a05e97949811e78","tex":"\\ell_1","aliases":"ℓ_1 ell","display":"\\ell_1","kind":"use","excerpt":"L_F_0'(0,λ)=ℓ_0+ωℓ_1+ω^2ℓ_2, L_F_0'(0,λ^2)=ℓ_0+ω^2ℓ_1+ωℓ_2. tagA3.3","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f71176b4c504c459","tex":"\\ell_2","aliases":"ℓ_2 ell","display":"\\ell_2","kind":"use","excerpt":"L_F_0'(0,λ)=ℓ_0+ωℓ_1+ω^2ℓ_2, L_F_0'(0,λ^2)=ℓ_0+ω^2ℓ_1+ωℓ_2. tagA3.3","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c288a99a7899c0aa","tex":"\\ell_{k-1}","aliases":"ℓ_k-1 ell","display":"\\ell_{k-1}","kind":"use","excerpt":"ζ_ f'(c^e s_0^k,0) =(-1)^e(frac R6-ℓ_k-1). tagA3.5","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-933184747400a95a","tex":"\\ell_m","aliases":"ℓ_m ell","display":"\\ell_m","kind":"use","excerpt":"Let Z_m(s) be the sum of the two ray partial zeta functions in row m, and write ℓ_m=Z_m'(0). Then","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Let z→0 in (N1.1). The fourth-order zero gives","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"β↦ω^iβ, ξ↦ω^jξ.","count":8},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"For z=aω+b∉Λ_0, define the positive quantity","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":22},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-83b238c8a0016e9e","tex":"\\prod_{a=0}^{n-1}","aliases":"∏_a=0^n-1 prod","display":"\\prod_{a=0}^{n-1}","kind":"use","excerpt":"1, · 5mid n. In the prime-to-five case the two product identities are ∏_a=0^n-1θ_0(z+a/n;t)=θ_0(nz;nt), ∏_b=0^n-1θ_0(nz+bt;nt)=θ_0(nz;t). Multiplication by five permutes the real residues in the smoothing denominator. When 5mid n, real averaging retains only frequencies already deleted by fifth smoothing. This proves the scalar one in both cases, r","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0d7b66704d8cf4f9","tex":"\\prod_{b=0}^{n-1}","aliases":"∏_b=0^n-1 prod","display":"\\prod_{b=0}^{n-1}","kind":"use","excerpt":"1, · 5mid n. In the prime-to-five case the two product identities are ∏_a=0^n-1θ_0(z+a/n;t)=θ_0(nz;nt), ∏_b=0^n-1θ_0(nz+bt;nt)=θ_0(nz;t). Multiplication by five permutes the real residues in the smoothing denominator. When 5mid n, real averaging retains only frequencies already deleted by fifth smoothing. This proves the scalar one in both cases, r","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e4215a8d0ab987dc","tex":"\\prod_{H\\text{ a line}}","aliases":"∏_H a line prod text","display":"\\prod_{H\\text{ a line}}","kind":"use","excerpt":"∏_H a lineN_E/E^H(z) =z^3N_E/F_0(z). tagQ2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a141fdd304f17550","tex":"\\prod_{j,k\\ge0}","aliases":"∏_j,k≥0 prod ge","display":"\\prod_{j,k\\ge0}","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-6490d6cd9c5da2fb","tex":"\\prod_{j=0}^{2}","aliases":"∏_j=0^2 prod","display":"\\prod_{j=0}^{2}","kind":"use","excerpt":"[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-aef7cb92bbc6f89a","tex":"\\prod_{j=0}^{A_v-1}","aliases":"∏_j=0^A_v-1 prod","display":"\\prod_{j=0}^{A_v-1}","kind":"use","excerpt":"F(ρ w_v)=F(w_Tv)B_v, B_v=∏_j=0^A_v-1q_t(w_T^2v+js), A_v=5+4b-2c+7a.","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-709a1d5e851efb47","tex":"\\prod_{j=0}^{k-1}","aliases":"∏_j=0^k-1 prod","display":"\\prod_{j=0}^{k-1}","kind":"use","excerpt":"ks)^4, · 5nmid n, n=2k. For odd n, doubling permutes E_t[n]. For n=2k, it maps E_t[n] to E_t[k] with four lifts per point, so each denominator norm is a fourth power. Apply [source reference] and cancel the numerator. If 5mid n then all these norms are one, including the even case since 5mid k. For n=2k+1 prime to five, B_k(w)=∏_j=0^k-1q_t(2w+(2j+1","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-857f76184af83427","tex":"\\prod_{k=0}^{6}","aliases":"∏_k=0^6 prod","display":"\\prod_{k=0}^{6}","kind":"use","excerpt":"βδ · =-8s-1, · 7δ · =τ+2s+4. The seven-point product and its correction are P_7(z)=∏_k=0^6F(z+kδ) =fracS_5(z;s,δ)S_5(z;t,δ)S_5(z+s;t,δ), R(z)=fracP_7(z)F(β z). The product identity is an actual gamma multiplication identity, not the assertion P_7(z)=F(β z). For completeness, its elementary product proof is retained from v0.12: rewrite t=2s+4-7δ, re","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-6e2d1c9e7adb82be","tex":"\\prod_{k=1}^{6}","aliases":"∏_k=1^6 prod","display":"\\prod_{k=1}^{6}","kind":"use","excerpt":"coefficient The product and continuation convention yield F(z)F(-z)=fracq_t(z)q_s(z), fracF(w+7δ)F(w)=fracq_s(w)q_t(w)q_t(w+s). Pair the labels k and 7-k in P_7(z)P_7(-z). For 1≤ k≤6, F(z+kδ)F(-z+(7-k)δ)=q_t(z+kδ-s)^-1. The unpaired k=0 term retains q_t(z)/q_s(z). Dividing by the reflected denominator gives R(z)R(-z)=T(z):= fracq_t(z)q_s(β z)q_s(z)","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-aa50cb2002362d44","tex":"\\prod_{nz=w}","aliases":"∏_nz=w prod","display":"\\prod_{nz=w}","kind":"use","excerpt":"Exactly normalized elliptic norms. On E_t= C/Λ_t define N_nf(w)=∏_nz=wf(z) for integers n≥1, with all n^2 preimages counted. The product is independent of the representative lift. Root-of-unity averaging and vertical residue splitting in [source reference] give N_nq_t(w)=q_t(w), · 5nmid n,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-f210ab155ed23468","tex":"\\prod_{n\\ge0}","aliases":"∏_n≥0 prod ge","display":"\\prod_{n\\ge0}","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8234e17cc32d5035","tex":"\\prod_{n\\ge1}","aliases":"∏_n≥1 prod ge","display":"\\prod_{n\\ge1}","kind":"use","excerpt":"H(z)=e^-πIm(ω)B_2(a) |1-e^2π iz| ∏_n≥1|(1-e^2π i(nω+z))(1-e^2π i(nω-z))|, tagA4.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e0661317abeaee45","tex":"\\prod_{r=0}^{3}","aliases":"∏_r=0^3 prod","display":"\\prod_{r=0}^{3}","kind":"use","excerpt":"All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a5393bc487b5c92c","tex":"\\prod_{r=1}^{15}","aliases":"∏_r=1^15 prod","display":"\\prod_{r=1}^{15}","kind":"use","excerpt":"All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-730c00051d9fc348","tex":"\\prod_{r=1}^{8}","aliases":"∏_r=1^8 prod","display":"\\prod_{r=1}^{8}","kind":"use","excerpt":"All transition multipliers and the scalar anchor. [Theta transitions] The following hold meromorphically: K(z):=fracR(z+1)R(z) · =∏_r=1^15q_t(β z-rs)^-1, fracR(z+s)R(z) · =fracq_s(β z-τ)∏_k=0^6q_t(z+kδ), fracR(z+τ)R(z) · = frac∏_k=0^6q_s(z+kδ)∏_r=0^3q_t(β z+rs) q_s(β z)q_s(β z+τ), fracR(z+δ)R(z) · = fracq_s(z)q_t(z)q_t(z+s)∏_r=1^8q_t(β z-rs). Use F","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-a6ec9ce642705ceb","tex":"\\prod_{\\eta\\in E_t[2]\\setminus\\{0\\}}","aliases":"∏_η∈ E_t[2]∖{0} prod eta in setminus","display":"\\prod_{\\eta\\in E_t[2]\\setminus\\{0\\}}","kind":"use","excerpt":"∏_η∈ E_t[2]∖{0}Q_5(x(η)) =fracf(a)f(b)16.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-dca37442052b8dd7","tex":"\\prod_{\\eta\\in E_t[3]\\setminus\\{0\\}}","aliases":"∏_η∈ E_t[3]∖{0} prod eta in setminus","display":"\\prod_{\\eta\\in E_t[3]\\setminus\\{0\\}}","kind":"use","excerpt":"∏_η∈ E_t[3]∖{0}Q_5(x(η)) =(fracψ_3(a)ψ_3(b)9)^2.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-33c00d86cffa64e0","tex":"\\prod_{\\eta\\in E_t[n]}","aliases":"∏_η∈ E_t[n] prod eta in","display":"\\prod_{\\eta\\in E_t[n]}","kind":"use","excerpt":"∏_η∈ E_t[n]q_t(z+η)=q_t(nz). tagN1.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0829bf61de6e27d1","tex":"\\prod_{\\eta\\in E_t[n]\\setminus\\{0\\}}","aliases":"∏_η∈ E_t[n]∖{0} prod eta in setminus","display":"\\prod_{\\eta\\in E_t[n]\\setminus\\{0\\}}","kind":"use","excerpt":"∏_η∈ E_t[n]∖{0}q_t(η)=n^4 (5nmid n). tagN4.1","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-1ac0aae1a9c811e7","tex":"\\prod_{\\eta\\ne0}","aliases":"∏_η≠0 prod eta ne","display":"\\prod_{\\eta\\ne0}","kind":"use","excerpt":"c_t^3=16∏_η≠0Q_5(x(η))=f(a)f(b)=d^6. tagN4.2","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-46bc02bf72faedf0","tex":"\\prod_{\\lambda\\in D}","aliases":"∏_λ∈ D prod lambda in","display":"\\prod_{\\lambda\\in D}","kind":"use","excerpt":"The genus-three divisor product. Let D={λ:m_λ≠0}; zero is not in D. Define E_3(w)=(1-w)e^w+w^2/2+w^3/3, W_D(z)=∏_λ∈ DE_3(z/λ)^m_λ. Negative exponents mean the quotient of the separate positive and negative entire products. [Canonical regularizer and absolute genus] The products defining W_D converge normally on compact sets away from their poles, a","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":33},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ef0041e878d312c4","tex":"\\sum_{a=1}^{21}","aliases":"∑_a=1^21 sum","display":"\\sum_{a=1}^{21}","kind":"use","excerpt":"For example, the latter two use -21^-1∑_a=1^21aχ_-3(a)χ_7(a)^j.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-802885a181593a09","tex":"\\sum_{a=1}^{6}","aliases":"∑_a=1^6 sum","display":"\\sum_{a=1}^{6}","kind":"use","excerpt":"The arithmetic candidate is fixed independently. Let J=2+3ω and choose the Gauss branch X=β^-1∑_a=1^6χ_7(a)e^6π ia/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1), X^3=J. Then E= Q(ω,β,X) has degree eighteen. The retained arithmetic construction defines ν · =(X+β/X-β^2)/3, · v · =-β^2ν-2ν^2,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-08f1256712f9419d","tex":"\\sum_D","aliases":"∑_D sum","display":"\\sum_D","kind":"use","excerpt":"thm:regularizer The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded. The cubic upper bound gives ∑_D|m_λ|/|λ|^4<∞, by partial summation or dy","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-8524806a0ba8de34","tex":"\\sum_{i<j}","aliases":"∑_i<j sum","display":"\\sum_{i<j}","kind":"use","excerpt":"a_0+a_1+a_2=3, ∑_i<ja_i a_j=6ω, a_0a_1a_2=1.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-97df02119f4369bb","tex":"\\sum_{j=0}^{2}","aliases":"∑_j=0^2 sum","display":"\\sum_{j=0}^{2}","kind":"use","excerpt":"Let L_t=(q_t'/q_t)' and S_t=wp_t(1/5)+wp_t(2/5). The normalized theta formula gives L_t(z)=-4z^-2+2S_t+O(z^2), S_s=ρ^2S_t. Since 7δ-2s=-t+4, evenness and t-periodicity pair the six translated L_t terms. Using s=u_0+3δ-2, their sum is 2∑_j=0^2L_t(u_j). The second logarithmic derivative of [source reference] cancels all z^-2 terms, leaving (log T)''(","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-acbf633423691e64","tex":"\\sum_{j=1}^{3}","aliases":"∑_j=1^3 sum","display":"\\sum_{j=1}^{3}","kind":"use","excerpt":"The repaired conjugation quotient and its boundary. For f^#(z)=overlinef(overline z), the canonical product W_D^# has the conjugate divisor. Hence fracR/R^#W_D/W_D^# =fracR_0overlineR_0 exp(∑_j=1^3(α_j-overlineα_j)z^j). This is entire and nowhere zero, with modulus one on the real axis. It cancels all unmatched divisor orders by a canonical meromor","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-95e5392801807f8d","tex":"\\sum_{j=1}^{4}","aliases":"∑_j=1^4 sum","display":"\\sum_{j=1}^{4}","kind":"use","excerpt":"tial L_t(z)=wp_+(z)-5wp_t(z)+2(a+b). At zero L_t has principal part -4/z^2; at the four nonzero kernel points it has +1/(z-P_j)^2, with no simple-pole terms. Thus L_t-wp_++5wp_t is a constant elliptic function. The canonical formula gives q_t(z)=(a-b)^2z^4[1+(a+b)z^2+O(z^4)], fixing that constant as 2(a+b). The degree-five quotient is equivalently ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ef75e15ec0684e46","tex":"\\sum_{j=1}^{n}","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^{n}","kind":"use","excerpt":"eginequation S_Nh-h∘ N=δ_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0c18bcf01edb7e76","tex":"\\sum_{j\\ge1}","aliases":"∑_j≥1 sum ge","display":"\\sum_{j\\ge1}","kind":"use","excerpt":"[Principal-norm rigidity] Let h,r be smooth mean-zero complex functions on T^3. If S_Nh-h∘ N=δ_Mr, then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=f","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ba49544259e2e741","tex":"\\sum_{k=0}^{6}","aliases":"∑_k=0^6 sum","display":"\\sum_{k=0}^{6}","kind":"use","excerpt":"5nmid n fracP^nZ^n-Q^nZ^-nn(1-P^n)(1-Q^n), where A_0=e^2π ia,B_0=e^2π ib in the first line. At one surviving frequency suppress the index n. Summing seven shifts multiplies the two coefficients by ∑_k=0^6D^k and ∑_k=0^6D^-k. The exact identities are fracP∑_k=0^6D^k(1-P)(1-Q) · =frac1+Q(1-P)(1-D)-frac1(1-Q)(1-D),","count":3},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-913ac2d9fb8649e2","tex":"\\sum_{k=1}^6","aliases":"∑_k=1^6 sum","display":"\\sum_{k=1}^6","kind":"use","excerpt":"ξ=β^-1∑_k=1^6χ_7(k)e^6π i k/7, (χ_7(1),…,χ_7(6))=(1,ω,ω^2,ω^2,ω,1).","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-fd6d5cb9dd6b7b70","tex":"\\sum_{k\\in B\\mathbb Z^3}","aliases":"∑_k∈ B Z^3 sum in mathbb","display":"\\sum_{k\\in B\\mathbb Z^3}","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-982415998cf9e445","tex":"\\sum_{k\\ne0}","aliases":"∑_k≠0 sum ne","display":"\\sum_{k\\ne0}","kind":"use","excerpt":"equation then h=δ_Mg for a smooth mean-zero g, with convergent formula g=-frac17∑_j≥1L_N^jr. If h,r extend holomorphically to all of X, so does g. Apply L_N/7 to [source reference]. The trace relation gives L_Nh-h=(1/7)δ_M L_Nr, and induction yields L_N^nh-h=frac17δ_M∑_j=1^nL_N^jr. Because B^3=7I, L_N^3nf=∑_k≠0343^nc_7^nke_k for a mean-zero f. For ","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7a920c42b7adde94","tex":"\\sum_{k\\ge4}","aliases":"∑_k≥4 sum ge","display":"\\sum_{k\\ge4}","kind":"use","excerpt":"ir poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded. The cubic upper bound gives ∑_D|m_λ|/|λ|^4<∞, by partial summation or dyadic shells. For |w|<1, log E_3(w)=-∑_k≥4fracw^kk. On each fixed compact set the tail log","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-362b67ed477f64be","tex":"\\sum_{m,n\\in\\mathbb Z}","aliases":"∑_m,n∈ Z sum in mathbb","display":"\\sum_{m,n\\in\\mathbb Z}","kind":"use","excerpt":"E_z(s)=∑_m,n∈ Z|m+nω+z|^-2s,","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e3b83af24644591c","tex":"\\sum_{n,m\\in\\mathbb Z}","aliases":"∑_n,m∈ Z sum in mathbb","display":"\\sum_{n,m\\in\\mathbb Z}","kind":"use","excerpt":"E_z(s)=∑_n,m∈ Z ((m+b+(n+a)x)^2+(n+a)^2y^2)^-s.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-68a64d3f366822a5","tex":"\\sum_{Nw=v}","aliases":"∑_Nw=v sum","display":"\\sum_{Nw=v}","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-5881f0a3e3cb764a","tex":"\\sum_{n\\in\\mathbb Z}","aliases":"∑_n∈ Z sum in mathbb","display":"\\sum_{n\\in\\mathbb Z}","kind":"use","excerpt":"beginequation R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) means the third derivative of R'/R, a single-valued meromorphic function. The sine partial fractions [source reference] give fracd^4dw^4log(1-e^2π iw) =-6∑_n∈ Z(w-n)^-4. The linear exponential f","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"-2log|1-e^2π iz| -2∑_n≥1log|(1-e^2π i(nτ+z))(1-e^2π i(nτ-z))|.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-89693ec1e9610033","tex":"\\sum_{r=1}^{15}","aliases":"∑_r=1^15 sum","display":"\\sum_{r=1}^{15}","kind":"use","excerpt":"2λ^3]. The combined summand is O(|λ|^-4), so this is an absolutely convergent sum. Its terms must remain combined; the separately divergent lower-power sums are not substituted. Taking two logarithmic derivatives of [source reference] at the transition [source reference] gives (log K)''(0)=6α_3+B_D(1). Since L_t is even, α_3=-frac16[β^2∑_r=1^15L_t(","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0db207badc6ca948","tex":"\\sum_{\\substack{n\\ge1\\\\5\\nmid n}}","aliases":"∑_substackn≥1 5nmid n sum substack ge nmid","display":null,"kind":"use","excerpt":"Full seven-product series proof. For positive imaginary periods t,s,d with t+7d=2s+4, put P=e^2π it, Q=e^2π is, D=e^2π id and Z=e^2π iz. Then PD^7=Q^2. In a common convergence region the fifth-smoothed logarithms are log S_5(z;a,b) · =5∑_substackn≥1 5nmid n fracZ^n-(A_0B_0/Z)^nn(1-A_0^n)(1-B_0^n), log F_t,s(z) · =-5∑_substackn≥1 5nmid n fracP^nZ^n-","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-39ba72372562e7cd","tex":"\\sum_{|\\lambda|>T}","aliases":"∑_|λ|>T sum lambda","display":"\\sum_{|\\lambda|>T}","kind":"use","excerpt":"logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T≥ R_* and |z|≤ T/2, ∑_|λ|>Tfrac|m_λ||λ|^4≤frac4AT, |log W_D,>T(z)|≤fra","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-c7985e990b3a1d24","tex":"\\sum_{|\\lambda|\\le r}","aliases":"∑_|λ|≤ r sum lambda le","display":"\\sum_{|\\lambda|\\le r}","kind":"use","excerpt":"thmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, without a claim about cancellations in specially grouped signed products. [Explicit analytic tail] If n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-637453a7d160c2c7","tex":"\\sum_{\\kappa\\in\\ker N}","aliases":"∑_κ∈ker N sum kappa in ker","display":"\\sum_{\\kappa\\in\\ker N}","kind":"use","excerpt":"The convergent transfer proof. For functions on T^3, let S_Nh(v)=∑_κ∈ker Nh(v+κ), C_Nh(v)=h(Nv), L_Nh(v)=∑_Nw=vh(w), δ_Mh=h-h∘ M. The transfer L_N is unnormalized. For h=∑ c_ke_k, S_Nh=∑_k∈ B Z^37c_ke_k, L_Nh=∑_k7c_Bke_k, S_Nh=(L_Nh)∘ N, L_NC_Nh=7h. Since M permutes ker N and commutes with N, the required operators commute with composition by M.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-96854c32284f2c58","tex":"\\sum_{\\lambda\\in D}","aliases":"∑_λ∈ D sum lambda in","display":"\\sum_{\\lambda\\in D}","kind":"use","excerpt":"The fourth logarithmic derivative fixes the remaining degree. [Cubic-exponential reconstruction] For the actual correction R, (log R)^(4)(z)=-6∑_λ∈ Dfracm_λ(z-λ)^4, locally normally away from D. Consequently R(z)=R_0exp(α_1z+α_2z^2+α_3z^3)W_D(z), α_j=frac(log R)^(j)(0)j!. The logarithmic derivatives are branch independent. The notation (log R)^(4) ","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Apply Theorem~[source reference] at the unique integral numerator lift [source reference] and subtract the denominator order at βλ. The weights agree. To bound the bracket, when a≤0,p>0 it is H(b); when a>0,p≤0 it is H(b)-1. In the other cases it is a difference of two zero-one values. Thus it lies in {-1,0,1}. In particular, ord_nτR=4 (n≥1), ord_n","count":5},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"thm:regularizer The products defining W_D converge normally on compact sets away from their poles, and divW_D=divR, W_D(0)=1, (log W_D)^(j)(0)=0 (1≤ j≤3). Genus three is sufficient and minimal for this ungrouped absolute construction. Special signed groupings are not excluded. The cubic upper bound gives ∑_D|m_λ|/|λ|^4<∞, by partial summation or dy","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"The lower bound n_D(r)≥ c r^3 yields divergence of ∑_D|m_λ|/|λ|^3: partial summation contains 3∫ n_D(r)r^-4 dr, which diverges logarithmically. For a fixed z≠0, log E_2(z/λ)=-fracz^33λ^3+O(|λ|^-4). Its weighted absolute logarithm sum diverges, because the fourth-power remainder is summable. This proves failure of ungrouped absolute genus two, witho","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-e052eebe752244d0","tex":"\\int_T^\\infty","aliases":"∫_T^∞ int infty","display":"\\int_T^\\infty","kind":"use","excerpt":"n_D(r)=∑_|λ|≤ r|m_λ|≤ Ar^3 for all r≥ R_*, then for T≥ R_* and |z|≤ T/2, ∑_|λ|>Tfrac|m_λ||λ|^4≤frac4AT, |log W_D,>T(z)|≤frac2A|z|^4T. Partial summation gives the first upper bound by dropping its nonpositive boundary term and using 4∫_T^∞ Ar^3r^-5 dr=4A/T. For the tail factors the canonical branch satisfies |log E_3(w)|≤ |w|^4/[4(1-|w|)]≤ |w|^4/2, ","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"[Three-theta anchor] R is regular and nonzero at zero, and R_0:=R(0)=∏_j=0^2q_t(u_j)^-1, u_j=fract+δ2+jδ. Let H(a)=η(5a)/η(a)^5. Direct product cancellation gives S_5(z;a,b)∼frac5(2π)^4 H(a) H(b)z^-4. The ratio in [source reference] therefore has value P_7(0)=[ H(s)/ H(t)]/S_5(s;t,δ). Since F(0)= H(s)/ H(t)=ϵ^2, the eta factors cancel and R_0=S_5(s","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"The actual two commuting actions. On X= C^3/ Z^3≃( C^*)^3 and its real torus T^3, use M=0 · 1 · 0","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-268b204a015c2f68","tex":"\\approx","aliases":"≈ approx","display":"\\approx","kind":"use","excerpt":"C≈13.80641265880838653-46.68563220754137422i,","count":2},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"For the upper bound first consider F. A nonzero coefficient in [source reference] has H(a)≠ H(b), so the imaginary parts of aτ and bs have the same sign. Therefore |Im(aτ+bs)|=|a|Im t+|b|Im s. A bounded complex radius bounds |a| and |b| by O(r); the real part then bounds |c| by O(r). There are O((1+r)^3) triples, and absolute weights are at most fo","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"tic polynomial (x-2)^3+7 gives a stable real eigenvalue a=2-7^1/3∈(0,1) and a conjugate unstable pair of modulus a^-1/2>1. Every nonzero integer A-orbit escapes in both directions: otherwise an integer vector would tend to zero along positive or negative iterates. Fourier orbit sums are therefore absolutely convergent for smooth coefficients. The d","count":7},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Base half-space divisor] The divisor of F is supported on Λ_5. At its uniquely parametrized point, ord_aτ+bs+c/5F=w(c) [H(a)-H(b)]. For positive periods t,s, the defining gamma product has a simple zero at jt+ks+ Z for j,k≥1, a simple pole for j,k≤0, and no divisor in the mixed cones. Rational independence of 1,t,s prevents collisions. The argumen","count":24},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"For positive imaginary periods use the additive product convention θ_0(z;a) · =∏_n≥0(1-e^2π i(z+na))(1-e^2π i((n+1)a-z)), Γ_rm ell(z;a,b) · =∏_j,k≥0 frac1-e^2π i((j+1)a+(k+1)b-z)1-e^2π i(ja+kb+z). The negative-first-period continuation is Γ_rm ell(z;-t,s)=Γ_rm ell(z+t;t,s)^-1. Consequently the main scalar function is F(z)=fracΓ_rm ell(z;τ,s)^5Γ_rm ","count":16},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"z+t has coordinates (1-a,b) in the (t,s) basis, giving gamma order H(b)-H(a). In F(z)=fracΓ_rm ell(5(z+t);5t,5s) Γ_rm ell(z+t;t,s)^5, the numerator contributes for every integer c and the denominator only when 5mid c. Their difference is [source reference]. The same product lists all possible zeros and poles. The rank-three additive group Λ_5⊂ C is","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"Ind_C_2^S_3(sgn) =sgn⊕Std, Std=Ind_C_3^S_3(ϱ)","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"Gal(E/K)=⟨ c,s_0⟩, σ=cs_0, Gal(E/F_0)=⟨ r,s_0⟩.","count":1},{"work":"hilbert12-cubic-divisor","url":"notation/hilbert12-cubic-divisor.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"Gal(E/K)=⟨ c,s_0⟩, σ=cs_0, Gal(E/F_0)=⟨ r,s_0⟩.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":101},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":25},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":435},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":77},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":103},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-677eab50351e7675","tex":"A^{-1}","aliases":"A^-1","display":"A^{-1}","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":15},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-011995a02078a685","tex":"a_5","aliases":"a_5","display":"a_5","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":15},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":122},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2311bbf2ac9fa5af","tex":"\\bar u","aliases":"bar u bar","display":"\\bar u","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h)) · =L_sn(b)ρ_s-1(bar ub) +2^s-2D_sn(b)ε_s(u,b), where ε_s(u,b)∈{±1}. Repetition yields a Haar-like expansion ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d13b1528b249ec85","tex":"\\bigwedge","aliases":"bigwedge","display":"\\bigwedge","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-34dd6266107506d4","tex":"\\bigwedge_{j=1}^{6}","aliases":"bigwedge_j=1^6 bigwedge","display":"\\bigwedge_{j=1}^{6}","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":43},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":78},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c94fd79f2de7324b","tex":"C^T","aliases":"C^T","display":"C^T","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":31},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is no","count":71},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-acd8deecbfbf287e","tex":"D_s","aliases":"D_s","display":"D_s","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"introduction-passage","excerpt":"Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is no","count":50},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7deb509fd27d8b0f","tex":"e_{\\mathrm{orb}}","aliases":"e_orb mathrm","display":"e_{\\mathrm{orb}}","kind":"introduction-passage","excerpt":"Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is no","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is no","count":32},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-879f46b871236f65","tex":"F_d","aliases":"F_d","display":"F_d","kind":"introduction-passage","excerpt":"Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is no","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":12},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-09f3e72028e864cd","tex":"G_m","aliases":"G_m","display":"G_m","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":56},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":42},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-849afb33fde944fb","tex":"H^4","aliases":"H^4","display":"H^4","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":69},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-12473beefe7126eb","tex":"\\Im","aliases":"Im","display":"\\Im","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"introduction-passage","excerpt":"Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is no","count":11},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":49},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":30},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":36},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-578177ca609e29d3","tex":"L_s","aliases":"L_s","display":"L_s","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":174},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":32},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b3c541ce8e25de25","tex":"\\mathrm{orb}","aliases":"orb mathrm","display":"\\mathrm{orb}","kind":"introduction-passage","excerpt":"Explicit carrier and remaining Chow map. For an audited CM type, the lower Fermat curve provides a candidate rational character-orbit factor of its Jacobian. Write e_orbJ(F_d) for that factor, up to isogeny. The idempotent selects the complete rational Galois orbit. A projector onto just one holomorphic CM half-plane would not be rational and is no","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":48},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":32},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8605e09a21a342b4","tex":"\\mathbf Q","aliases":"Q mathbf","display":"\\mathbf Q","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part","count":154},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":105},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ff81ba5e23d549c7","tex":"\\Re","aliases":"Re","display":"\\Re","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":49},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":44},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":20},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":67},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-25121c20156ac918","tex":"U_m","aliases":"U_m","display":"U_m","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-deedcf5fc529f076","tex":"\\uplus","aliases":"uplus","display":null,"kind":"introduction-passage","excerpt":"Write ν(a) for its integer residue-multiplicity vector. Write auplus b for concatenation regarded as a multiset. Permuting coordinates is an algebraic automorphism, so multiset identities suffice for algebraicity transfer. Write -a for coordinatewise negation modulo m.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic (2,2)-class lies in the one-dimensional Fermat line V(widetilde a).","count":28},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":65},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":40},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8d1cc57306466ffb","tex":"W_4","aliases":"W_4","display":"W_4","kind":"introduction-passage","excerpt":"Here W_4 denotes the weight filtration, not the Fermat packet notation. The descended classes have weight four, so belong to this image. The category of polarizable pure rational Hodge structures is semisimple. The surjection from H^4(widetilde Y) onto its image thus has a Hodge-theoretic section. Every descended rational (2,2)-class has a rational","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4d16d80dcb44a1a1","tex":"W_{[a]}","aliases":"W_[a]","display":"W_{[a]}","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2b26d57eb5a9595f","tex":"\\widetilde a","aliases":"widetilde a widetilde","display":"\\widetilde a","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7049d36b7defb732","tex":"\\widetilde Y","aliases":"widetilde Y widetilde","display":"\\widetilde Y","kind":"introduction-passage","excerpt":"Here W_4 denotes the weight filtration, not the Fermat packet notation. The descended classes have weight four, so belong to this image. The category of polarizable pure rational Hodge structures is semisimple. The surjection from H^4(widetilde Y) onto its image thus has a Hodge-theoretic section. Every descended rational (2,2)-class has a rational","count":10},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":103},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":38},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3864fe3cacc7c90d","tex":"x_6","aliases":"x_6","display":"x_6","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6b32af54d77c3eab","tex":"X_{c(a)}^4","aliases":"X_c(a)^4","display":"X_{c(a)}^4","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-588ca5f878a82c0d","tex":"x_i^{g(a)}","aliases":"x_i^g(a)","display":"x_i^{g(a)}","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f3be737b42076753","tex":"X_m","aliases":"X_m","display":"X_m","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ca762e13b94db617","tex":"X_m^4","aliases":"X_m^4","display":"X_m^4","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":44},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-dd412919846eece9","tex":"y_2","aliases":"y_2","display":"y_2","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aa384f1459a98947","tex":"y_5","aliases":"y_5","display":"y_5","kind":"introduction-passage","excerpt":"Take the cubic Jacobian-ring class P(y)=y_1y_2y_5. Its exponent vector is b=(0,1,1,0,0,1). Write C=33A^-1 for the cubic exponent matrix A. The class is nonzero in the degree-three Jacobian ring. The residue pullback has the exact exponent identity C^Tb+101=(6,18,27,9,12,21)=widetilde a-1. Therefore the pullback of the corresponding primitive cubic ","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-94b7394a02c96971","tex":"y_6","aliases":"y_6","display":"y_6","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e3d4192e022af00a","tex":"y_j","aliases":"y_j","display":"y_j","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"introduction-passage","excerpt":"Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part","count":34},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-75fbe28ce1b6d91b","tex":"\\mathbf Z","aliases":"Z mathbf","display":"\\mathbf Z","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":15},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1effc9c27f460446","tex":"Z_\\kappa","aliases":"Z_κ kappa","display":"Z_\\kappa","kind":"introduction-passage","excerpt":"Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part","count":12},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":13},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h)) · =L_sn(b)ρ_s-1(bar ub) +2^s-2D_sn(b)ε_s(u,b), where ε_s(u,b)∈{±1}. Repetition yields a Haar-like expansion ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c69c9eb6fe242300","tex":"\\varepsilon_s","aliases":"ε_s varepsilon","display":"\\varepsilon_s","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h)) · =L_sn(b)ρ_s-1(bar ub) +2^s-2D_sn(b)ε_s(u,b), where ε_s(u,b)∈{±1}. Repetition yields a Haar-like expansion ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"introduction-passage","excerpt":"Let h denote the hyperplane class. Since deg Z_κ/m=2q, the primitive part","count":42},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2ffb97b3578a68df","tex":"\\mu_m","aliases":"μ_m mu","display":"\\mu_m","kind":"introduction-passage","excerpt":"The diagonal group is G_m=(μ_m)^6/μ_m. For a character a=(a_0,…,a_5) with all entries nonzero modulo m and sum zero modulo m, let W_[a] denote the rational character-orbit summand of primitive H^4(X_m, Q).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"introduction-passage","excerpt":"Write ν(a) for its integer residue-multiplicity vector. Write auplus b for concatenation regarded as a multiset. Permuting coordinates is an algebraic automorphism, so multiset identities suffice for algebraicity transfer. Write -a for coordinatewise negation modulo m.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b9f06722dea0f6fe","tex":"\\pi_{g(a)}","aliases":"π_g(a) pi","display":"\\pi_{g(a)}","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":13},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-010b55440ebd78d1","tex":"\\rho_{2^s}","aliases":"ρ_2^s rho","display":"\\rho_{2^s}","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-af3082b783446096","tex":"\\rho_s","aliases":"ρ_s rho","display":"\\rho_s","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0990c1a688790eca","tex":"\\rho_{s-1}","aliases":"ρ_s-1 rho","display":"\\rho_{s-1}","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h)) · =L_sn(b)ρ_s-1(bar ub) +2^s-2D_sn(b)ε_s(u,b), where ε_s(u,b)∈{±1}. Repetition yields a Haar-like expansion ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-81d2c8abb95f23dd","tex":"\\Omega_6","aliases":"Ω_6 Omega","display":"\\Omega_6","kind":"introduction-passage","excerpt":"Write (x_1,y_1,…,x_6,y_6) for the integral dual basis. Define Ω_6=bigwedge_j=1^6(x_j+iy_j), u=ReΩ_6, v=ImΩ_6. Both u and v are explicit integral six-forms, with 32 nonzero terms each. An unspecified complex determinant-line generator would not by itself define these rational classes. The chosen expansion removes that ambiguity.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":23},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"terUniversal balanced-packet equations Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) e","count":14},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bcd57746c1d3a7fc","tex":"\\ell_a","aliases":"ℓ_a ell","display":"\\ell_a","kind":"introduction-passage","excerpt":"terUniversal balanced-packet equations Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) e","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":50},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":68},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f576bbd36a401643","tex":"\\sum_{i=0}^5","aliases":"∑_i=0^5 sum","display":"\\sum_{i=0}^5","kind":"introduction-passage","excerpt":"terUniversal balanced-packet equations Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) e","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":20},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":43},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"Let s≥3, m=2^s, h=2^s-1, and write ρ_s=ρ_2^s. For an odd residue bpmod h, its two lifts are b and b+h. Define L_sn(b)=n(b)+n(b+h), D_sn(b)=n(b)-n(b+h). For every odd unit u, one has the exact weighted identity · n(b)ρ_s(ub)+n(b+h)ρ_s(u(b+h))","count":35},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":28},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":28},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-97f32c612ff4e29e","tex":"\\rangle_m","aliases":"⟩_m rangle","display":"\\rangle_m","kind":"introduction-passage","excerpt":"Universal balanced-packet equations. Let U_m=( Z/m Z)^× and let ⟨ r⟩_m∈{1,…,m} denote the least positive residue, with ⟨ r⟩_m=m when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exa","count":11},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"introduction-passage","excerpt":"Exact conductor polyphase. For a packet a, define g(a)=gcd(m,a_0,…,a_5), c(a)=fracmg(a). For a count vector, L_k denotes scaling each residue entry by k while multiplying the modulus by k. Writing a=g(a)b, the coordinate-power map π_g(a):X_m^4⟶ X_c(a)^4, [x_i]⟼[x_i^g(a)], sends the degree-c(a) character packet b to the degree-m packet a.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Cycle frame · S=CC^*. Positivity spans a target packet when every supplied algebraic vector lies in it.","count":26},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8a5c81d9140d93f9","tex":"\\mathbf1^T","aliases":"1^T mathbf","display":"\\mathbf1^T","kind":"use","excerpt":"To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebrai","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8f914025617b7ebc","tex":"A^T","aliases":"A^T","display":"A^T","kind":"use","excerpt":"The Laurent exponent match is a useful exact certificate, but by itself it does not prove that a rational class lifts through a compact graph. A complete manuscript argument instead uses the torus-carrier criterion developed in Chapter~[source reference]. For the reordered character, the exponent matrix A of the cubic satisfies A^Twidetilde a/33=β=","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fc1d460a703c9724","tex":"A^\\perp","aliases":"A^⊥ perp","display":"A^\\perp","kind":"use","excerpt":"[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equali","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2902cbb0483d2928","tex":"A_1^p","aliases":"A_1^p","display":"A_1^p","kind":"use","excerpt":"Finite saturation and exact rational closure. For each degree cutoff D, let A_D^p(X) be the rational span of codimension-p cycle classes of degree at most D. Then A_1^p⊆ A_2^p⊆⋯⊆ H^2p(X,Q).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3f7d80693aceccc9","tex":"A_2^p","aliases":"A_2^p","display":"A_2^p","kind":"use","excerpt":"Finite saturation and exact rational closure. For each degree cutoff D, let A_D^p(X) be the rational span of codimension-p cycle classes of degree at most D. Then A_1^p⊆ A_2^p⊆⋯⊆ H^2p(X,Q).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f151a66f44712728","tex":"A_{2^t}","aliases":"A_2^t","display":"A_{2^t}","kind":"use","excerpt":"[Complete dyadic grammar] After exact conductor descent, every balanced packet at every dyadic modulus is generated by complementary pairs, SAT(2) surface packets, and coordinate-power pullbacks of the single SAT(3) family A_2^t. No second residual dyadic family occurs at any depth.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bd6ead9e0fc67386","tex":"A_{b,c,d}","aliases":"A_b,c,d","display":"A_{b,c,d}","kind":"use","excerpt":"K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-75a3dbe96cbe3706","tex":"A_{\\mathrm{conic}}","aliases":"A_conic mathrm","display":"A_{\\mathrm{conic}}","kind":"use","excerpt":"A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equiva","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3728d1cbbf5a073f","tex":"A_D^p","aliases":"A_D^p","display":"A_D^p","kind":"use","excerpt":"Finite saturation and exact rational closure. For each degree cutoff D, let A_D^p(X) be the rational span of codimension-p cycle classes of degree at most D. Then A_1^p⊆ A_2^p⊆⋯⊆ H^2p(X,Q).","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3e26ff6ce86f04e5","tex":"A_{D_0}^p","aliases":"A_D_0^p","display":"A_{D_0}^p","kind":"use","excerpt":"[Finite stabilization] There exists D_0 such that A_D^p(X)=A_D_0^p(X) for every D≥ D_0. The dimensions form a nondecreasing sequence of integers bounded by dim H^2p(X,Q). Only finitely many strict increases are possible.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b4e23f759cf79e4c","tex":"A_{\\mathrm{exceptional}}","aliases":"A_exceptional mathrm","display":"A_{\\mathrm{exceptional}}","kind":"use","excerpt":"A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equiva","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"Here W·β=3D follows from ∑ a_j=3m. Thus deck invariance follows uniformly from the matrix congruence, it is not an extra conjecture. The positive weight sum gives negative canonical coefficient, normality and well-formedness justify the canonical computation, and the uniruledness, descent, lifting and return arguments above finish the proof. This c","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ee80226ef44249da","tex":"A_{\\mathrm{line}}","aliases":"A_line mathrm","display":"A_{\\mathrm{line}}","kind":"use","excerpt":"A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equiva","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"abeleq:legacy-13.5 P_a= (8a,q-a,2q-a,2q-4a,3q-a,4q-a). The cases c<q/2 and c>q/2 give the same unordered tuple after the internal-edge cancellations. Choose an odd u satisfying ua≡1pmod2q. Since uq≡ q or 3qpmod4q, multiplication by u sends [source reference] to P_1=(8,q-1,2q-4,2q-1,3q-1,4q-1). Its conjugate is A_m= ( 1,frac m4+1,frac m2+1,frac m2+4","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4c8131608f522ec2","tex":"a_{n+1}","aliases":"a_n+1","display":"a_{n+1}","kind":"use","excerpt":"A full-support character is an ordered tuple a=(a_0,…,a_n+1) of nonzero residues modulo m, with sum zero modulo m. Its eigenspace in primitive cohomology is a complex line. For even n, its rational Galois packet is of middle Hodge type precisely when","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-43932d1467cb270a","tex":"A_{\\mathrm{plane}}","aliases":"A_plane mathrm","display":"A_{\\mathrm{plane}}","kind":"use","excerpt":"A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equiva","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-38143a44ee0b7f8e","tex":"a_q","aliases":"a_q","display":"a_q","kind":"use","excerpt":"a_q=(1,1+q,1+2q,1+3q,1+4q,5q-5)pmod 5q.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-da5363b8128410ed","tex":"a_{q,b}","aliases":"a_q,b","display":"a_{q,b}","kind":"use","excerpt":"m=5q, a_q,b=(b,b+q,b+2q,b+3q,b+4q,5(q-b)).","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-99c66204e42f90da","tex":"a_{q,b,i}","aliases":"a_q,b,i","display":"a_{q,b,i}","kind":"use","excerpt":"Q_b=∏_i=0^5x_i^m-a_q,b,i-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"use","excerpt":"Divisor transfer · T=∑_rB_rA_r with actual Chow correspondences, packet preservation and invertibility.","count":14},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ec8134e21571dc67","tex":"A_r^\\dagger","aliases":"A_r^† dagger","display":"A_r^\\dagger","kind":"use","excerpt":"Positive transfer · T=∑_rA_r^† A_r only with algebraic realizations of those adjoints.","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4fb6ac3dfe67ffbe","tex":"\\Alg","aliases":"Alg","display":null,"kind":"use","excerpt":"Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).","count":11},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f6e02aaf98ea60fe","tex":"\\Alg^p","aliases":"Alg^p Alg","display":null,"kind":"use","excerpt":"Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"Divisor transfer · T=∑_rB_rA_r with actual Chow correspondences, packet preservation and invertibility.","count":26},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebrai","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"use","excerpt":"Let t be a unit modulo 5q, and let b'=tbbmod q, with 1≤ b'<q. Multiplication by t permutes the five residues in the fiber over b', because t is a unit modulo 5. The first five least-positive residues are therefore, in some order,","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-655d23d75da0712a","tex":"B^T","aliases":"B^T","display":"B^T","kind":"use","excerpt":"and hence B^Tβ=α. To obtain a genuine character of the projective torus, let r_0=(0,0,2,0,0,3) be the first row of A and set","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). Th","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e5f8318fd1505267","tex":"B_{1,\\overline{\\chi^*}}","aliases":"B_1,overlineχ^* overline chi","display":"B_{1,\\overline{\\chi^*}}","kind":"use","excerpt":"_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,2^s=2^s-tλ_χ^*,2^t. The primitive value is, up to a nonzero scalar, the generalized Bernoulli value B_1,overlineχ^*, equivalently -L(0,overlineχ^*","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-39a4ffab8ed70e3f","tex":"B_{13}","aliases":"B_13","display":"B_{13}","kind":"use","excerpt":"The selected mixed tangent direction B_13=1 breaks the original factorization infinitesimally: the old surface factor loses type (1,1) and the old fourfold factor loses type (2,2). Nevertheless the full u,v remain type (3,3). At first this looked like a promising obstruction.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-406e621513b543a0","tex":"b_{2p+1}","aliases":"b_2p+1","display":"b_{2p+1}","kind":"use","excerpt":"The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). Th","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-684bd1483921402e","tex":"\\mathcal B_c^{\\mathrm{full}}","aliases":"B_c^full mathcal mathrm","display":"\\mathcal B_c^{\\mathrm{full}}","kind":"use","excerpt":"[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance e","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-935b9fb396664c1f","tex":"b_I","aliases":"b_I","display":"b_I","kind":"use","excerpt":"The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"orem[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balan","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2ff1187fcdc685de","tex":"B_{ij}","aliases":"B_ij","display":"B_{ij}","kind":"use","excerpt":"On the projective tori define y_i=∏_jx_j^B_ij. Negative exponents are allowed there. The second identity makes this well-defined into the weighted projective torus, the first gives","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-348b7158501e450b","tex":"\\mathcal B_m","aliases":"B_m mathcal","display":"\\mathcal B_m","kind":"use","excerpt":"To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebrai","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8ac4444648270390","tex":"\\mathcal B_{m,\\chi}","aliases":"B_m,χ mathcal chi","display":"\\mathcal B_{m,\\chi}","kind":"use","excerpt":"Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-57c8f1ae1da23ae1","tex":"B_r","aliases":"B_r","display":"B_r","kind":"use","excerpt":"Divisor transfer · T=∑_rB_rA_r with actual Chow correspondences, packet preservation and invertibility.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-125d02e7e80757c1","tex":"b_R","aliases":"b_R","display":"b_R","kind":"use","excerpt":"The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5bd492e4ea78c5fe","tex":"\\bigsqcup","aliases":"bigsqcup","display":null,"kind":"use","excerpt":"[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance e","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fc8a95dbcea19835","tex":"\\bigsqcup_{c\\mid m}","aliases":"bigsqcup_cmid m bigsqcup mid","display":null,"kind":"use","excerpt":"[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance e","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-071b3dcad0b61b41","tex":"\\operatorname{Bl}","aliases":"Bl operatorname","display":"\\operatorname{Bl}","kind":"use","excerpt":"[Blow-up closure] If it holds for X and a smooth center Z, it holds for Bl_ZX. Use the standard blow-up cohomology decomposition. The new summands are Tate twists of cohomology from Z multiplied by powers of the algebraic exceptional divisor.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-48a3794a6951f28e","tex":"\\operatorname{Bl}_Z","aliases":"Bl_Z operatorname","display":"\\operatorname{Bl}_Z","kind":"use","excerpt":"[Blow-up closure] If it holds for X and a smooth center Z, it holds for Bl_ZX. Use the standard blow-up cohomology decomposition. The new summands are Tate twists of cohomology from Z multiplied by powers of the algebraic exceptional divisor.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"C(1,20,24,42,61,62bmod70)=11/10.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-420668a5bdd9e0bf","tex":"\\mathbf C","aliases":"C mathbf","display":"\\mathbf C","kind":"use","excerpt":"X_m={F_m=x_0^m+⋯+x_5^m=0}⊂ P^5_ C, G_m=(μ_m)^6/μ_m.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"use","excerpt":"Cycle frame · S=CC^*. Positivity spans a target packet when every supplied algebraic vector lies in it.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8b220758d91c2d7e","tex":"c^6","aliases":"c^6","display":"c^6","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-db3585c721e8f66b","tex":"c^7","aliases":"c^7","display":"c^7","kind":"use","excerpt":"F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fb5797c1ce9d24ab","tex":"c^t","aliases":"c^t","display":"c^t","kind":"use","excerpt":"Take V=Span_R{e_1} in R^2 and let the only cycle vector be c=e_1+e_2. Then S=cc^t=1 · 1 1 · 1. The compressed Gram matrix on V is the positive 1×1 matrix (1), while e_1 does not belong to Span(c). Positivity proves that the target direction is detected, not that synthesis returns it inside the original cycle span.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"Positive cycle-frame calculus. Analysis and synthesis. Fix a finite family of rational algebraic cycle classes c_1,…,c_N∈ H^2p(X,Q). Let V⊆ H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projecto","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a7590bbac33fc0eb","tex":"C_D","aliases":"C_D","display":"C_D","kind":"use","excerpt":"[Finite completion certificate] If H is a rational basis matrix of Hdg^p(X) and C_D is the cycle matrix, then the Hodge conjecture for (X,p) is equivalent to the existence of D with rank_Q C_D=rank_Q[C_D H]. A positive compressed frame is equivalent only when the supplied cycle vectors lie in the target packet, or when a separately verified invaria","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-dff8e8535d2ec4f4","tex":"c_{d-1}","aliases":"c_d-1","display":"c_{d-1}","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-91fc289567e65813","tex":"c_I","aliases":"c_I","display":"c_I","kind":"use","excerpt":"The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"use","excerpt":"The infinitely many inequalities can be certified by a finite residue graph. Write y_i=c_i/L, where c_i,L are positive integers. On Z/m Z, an edge r→ r+a_i has cost c_i. An integer function d satisfying","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"Positive cycle-frame calculus. Analysis and synthesis. Fix a finite family of rational algebraic cycle classes c_1,…,c_N∈ H^2p(X,Q). Let V⊆ H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projecto","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c1ecf2aa8f993509","tex":"c_N","aliases":"c_N","display":"c_N","kind":"use","excerpt":"Positive cycle-frame calculus. Analysis and synthesis. Fix a finite family of rational algebraic cycle classes c_1,…,c_N∈ H^2p(X,Q). Let V⊆ H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projecto","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-eeacfb867d6a1eef","tex":"c_{n,j}","aliases":"c_n,j","display":"c_{n,j}","kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d7e172c3d3f1c75b","tex":"C_P","aliases":"C_P","display":"C_P","kind":"use","excerpt":"entatives c,d. Require c,d,e to be nonzero modulo q so that all three resulting character blocks have full support. Degenerate cases must leave this construction and be handled through pair removal or another channel. For example (1,14,2,9,14,5) at modulus 15 is balanced and admits the indicated phase split, but P=(1,14) gives c=0. The phase condit","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-11eaeebd7cee7fd8","tex":"c_R","aliases":"c_R","display":"c_R","kind":"use","excerpt":"The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3bc0395c2cb82c77","tex":"\\CH","aliases":"CH","display":null,"kind":"use","excerpt":"Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d0fe2f8238bcf56d","tex":"\\operatorname{ch}","aliases":"ch operatorname","display":"\\operatorname{ch}","kind":"use","excerpt":"Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees i","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-12dd4cabbe226b19","tex":"\\CH^2","aliases":"CH^2 CH","display":null,"kind":"use","excerpt":"The constructive objective is [Packet-resolved construction target] For every rational packet W⊂ H^4_prim(X_m^4,Q)∩ H^2,2, construct Z_W∈CH^2(X_m^4)_Q with [Z_W]≠0 in W. Once one nonzero algebraic vector is produced in a rational irreducible diagonal-symmetry packet, the entire rational orbit is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1349896bd6692d1a","tex":"\\CH^{\\dim Y_r+p-1}","aliases":"CH^dim Y_r+p-1 CH dim","display":null,"kind":"use","excerpt":"Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic corre","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c01ba64ae42ae00f","tex":"\\CH^{n-p+1}","aliases":"CH^n-p+1 CH","display":null,"kind":"use","excerpt":"Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic corre","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-301e31bb22d4457f","tex":"\\CH^p","aliases":"CH^p CH","display":null,"kind":"use","excerpt":"Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fca9ddf7316b0413","tex":"\\CH^r","aliases":"CH^r CH","display":null,"kind":"use","excerpt":"Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6d5407f9ec6f621c","tex":"\\CH^{\\ast}","aliases":"CH^∗ CH ast","display":null,"kind":"use","excerpt":"The desired correspondence is schematically Γ∈CH^∗(F_d× C_P× S_T)_Q, Γ_*:e_orbH^1(F_d)⟶ e_diag(M(C_P)⊗ M(S_T))(1). This formula is a target specification, not a constructed map. Its coefficient field, diagonal projector, rational multiplicities, and dimensions must be fixed before an isomorphism can be asserted. The smaller CM conductor can have a ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-35e2f977790f972a","tex":"\\operatorname{ch}_p","aliases":"ch_p operatorname","display":"\\operatorname{ch}_p","kind":"use","excerpt":"q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b26d278df528b933","tex":"\\operatorname{ch}_q","aliases":"ch_q operatorname","display":"\\operatorname{ch}_q","kind":"use","excerpt":"Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-632e23454c4ede45","tex":"\\operatorname{ch}_{\\mathrm{top}}","aliases":"ch_top operatorname mathrm","display":"\\operatorname{ch}_{\\mathrm{top}}","kind":"use","excerpt":"Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees i","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5761fa2adf61c496","tex":"\\cl","aliases":"cl","display":null,"kind":"use","excerpt":"The rational Hodge target. Statement and status. Let X be a smooth projective complex variety. For each codimension p, define Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X) and Alg^p(X)=im(CH^p(X)_QxrightarrowclH^2p(X,Q)). Every algebraic cycle class has type (p,p), so Alg^p(X)⊆Hdg^p(X). The rational Hodge conjecture asserts equality for every smooth projective X an","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-05430289e843ac24","tex":"\\mathrm{conic}","aliases":"conic mathrm","display":"\\mathrm{conic}","kind":"use","excerpt":"A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equiva","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"[Finite stabilization] There exists D_0 such that A_D^p(X)=A_D_0^p(X) for every D≥ D_0. The dimensions form a nondecreasing sequence of integers bounded by dim H^2p(X,Q). Only finitely many strict increases are possible.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"use","excerpt":"Complement defects and dyadic triangles. Put q=frac m4, h=2q, m=4q. For a packet count vector n, define its oriented complement defect ∂ n=∑_r=1^h-1d_n(r)[r], d_n(r)=n_r-n_m-r. The self-complementary residue h contributes zero. Since ρ_m(m-r)=-ρ_m(r), the balanced-packet equation is equivalent to ∑_r=1^h-1d_n(r)ρ_m(ur)=0 for every odd unit u. Moreo","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-237fc9015f157bd3","tex":"D_r","aliases":"D_r","display":"D_r","kind":"use","excerpt":"[Invertible divisor-lowering transfer] Suppose T=∑_r=1^M B_rA_r preserves V and is invertible on V. Then every rational Hodge class in V is algebraic. Let α∈ V be of type (p,p). Each A_rα is rational of type (1,1) on Y_r, hence is the class of a rational divisor D_r by the Lefschetz (1,1) theorem. Therefore Tα=∑_r(Λ_r)_*[D_r] is algebraic on X.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f0625c7a9188564a","tex":"D_\\kappa","aliases":"D_κ kappa","display":"D_\\kappa","kind":"use","excerpt":"H=(f_1,g_1,f_2,g_2,f_3,g_3), D_κ=detJac_(x_0,…,x_4,z)H,","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f248e9e31b983389","tex":"\\dashrightarrow","aliases":"dashrightarrow","display":null,"kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"H=(f_1,g_1,f_2,g_2,f_3,g_3), D_κ=detJac_(x_0,…,x_4,z)H,","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ffa9d7f17ebd6a0c","tex":"\\mathrm{diag}","aliases":"diag mathrm","display":"\\mathrm{diag}","kind":"use","excerpt":"After fixing a common cyclotomic coefficient field and a diagonal character projector, write the selected factor schematically as N=e_diag(M(C_P)⊗ M(S_T)). An ordinary tensor product of rational packets contains off-diagonal embedding pairs and is generally not rank one. Rank one here refers only to a selected eigenspace after scalar extension. The","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for ","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"[Finite stabilization] There exists D_0 such that A_D^p(X)=A_D_0^p(X) for every D≥ D_0. The dimensions form a nondecreasing sequence of integers bounded by dim H^2p(X,Q). Only finitely many strict increases are possible.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Projective bundles, blow-ups, and descent. [Projective-bundle closure] If the rational Hodge conjecture holds for X, it holds for P(E)→ X. The projective-bundle formula expresses cohomology as a direct sum of powers of the algebraic hyperplane class times pullbacks of cohomology from X.","count":10},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-927978fe2292e206","tex":"e^{-tX}","aliases":"e^-tX","display":"e^{-tX}","kind":"use","excerpt":"The exact audit then found replacement divisor factors using the pairs (1,6), (2,5), (3,4). These replacement factors remain of divisor type along the selected tangent. With X=-tfrac12(Δ J)J_0, J(t)=e^tXJ_0e^-tX, the pair blocks are preserved by the integrating generator. Thus the alternate factorization defeats the tested counterexample path. It p","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-16861153072b3e07","tex":"e^{2\\pi ir/q}","aliases":"e^2π ir/q pi","display":"e^{2\\pi ir/q}","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-103af72d6feef0cd","tex":"e^{tX}","aliases":"e^tX","display":"e^{tX}","kind":"use","excerpt":"The exact audit then found replacement divisor factors using the pairs (1,6), (2,5), (3,4). These replacement factors remain of divisor type along the selected tangent. With X=-tfrac12(Δ J)J_0, J(t)=e^tXJ_0e^-tX, the pair blocks are preserved by the integrating generator. Thus the alternate factorization defeats the tested counterexample path. It p","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d17b70dc64774136","tex":"e^{\\pi ir/q}","aliases":"e^π ir/q pi","display":"e^{\\pi ir/q}","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"does not prove that a rational class lifts through a compact graph. A complete manuscript argument instead uses the torus-carrier criterion developed in Chapter~[source reference]. For the reordered character, the exponent matrix A of the cubic satisfies A^Twidetilde a/33=β=(1,2,2,1,1,2), C=33A^-1. With r_0 the first row of A, put u=β-3r_0. Then 1^","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.","count":15},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-42877425a0deabad","tex":"e_1^2","aliases":"e_1^2","display":"e_1^2","kind":"use","excerpt":"p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1c19efa2f731eadb","tex":"e_1^4","aliases":"e_1^4","display":"e_1^4","kind":"use","excerpt":"p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8acca8efd11c2c99","tex":"e_2^2","aliases":"e_2^2","display":"e_2^2","kind":"use","excerpt":"p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-67ab753297c61ce5","tex":"e_4","aliases":"e_4","display":"e_4","kind":"use","excerpt":"p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a64c5d974442c22a","tex":"e_5","aliases":"e_5","display":"e_5","kind":"use","excerpt":"p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f3329c74d3c08346","tex":"e_{[\\alpha]}","aliases":"e_[α] alpha","display":"e_{[\\alpha]}","kind":"use","excerpt":"e_[α]H^4(X_210, Q) xrightarrow∼ e_[α]H^4(U_210, Q).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-785bbea9a2e3c724","tex":"e_d","aliases":"e_d","display":"e_d","kind":"use","excerpt":"[Finite Gram certificate] Choose a rational basis e_1,…,e_d of V. If G_ij=∑_r⟨ A_re_i,A_re_j⟩_Y_r has nonzero determinant, every rational Hodge class in V is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-72f0979d05c405ff","tex":"e_{\\mathrm{diag}}","aliases":"e_diag mathrm","display":"e_{\\mathrm{diag}}","kind":"use","excerpt":"After fixing a common cyclotomic coefficient field and a diagonal character projector, write the selected factor schematically as N=e_diag(M(C_P)⊗ M(S_T)). An ordinary tensor product of rational packets contains off-diagonal embedding pairs and is generally not rank one. Rank one here refers only to a selected eigenspace after scalar extension. The","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"[Finite Gram certificate] Choose a rational basis e_1,…,e_d of V. If G_ij=∑_r⟨ A_re_i,A_re_j⟩_Y_r has nonzero determinant, every rational Hodge class in V is algebraic.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"bseteq H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the ","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8a9d72546c912fa0","tex":"E_{m,d}","aliases":"E_m,d","display":"E_{m,d}","kind":"use","excerpt":"For dmid m, let E_m,d=frac1(m/d)^5π_m/d^*π_m/d*. On a character of exact conductor c, E_m,d=1, · cmid d,","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2e5200ca1bb62e54","tex":"E_{m,e}","aliases":"E_m,e","display":"E_{m,e}","kind":"use","excerpt":"0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is it","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4c5c1a1b4860933a","tex":"E_{m,\\gcd(d,e)}","aliases":"E_m,gcd(d,e) gcd","display":"E_{m,\\gcd(d,e)}","kind":"use","excerpt":"0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is it","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-eb7317d14c2cf14b","tex":"e_{m-r}","aliases":"e_m-r","display":"e_{m-r}","kind":"use","excerpt":"Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3ecd0bf9ad886a1d","tex":"e_p^{(m)}","aliases":"e_p^(m)","display":"e_p^{(m)}","kind":"use","excerpt":"Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-77007f1f55a92c5b","tex":"\\mathrm{even}","aliases":"even mathrm","display":"\\mathrm{even}","kind":"use","excerpt":"Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees i","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-03b9472ac121163d","tex":"\\mathrm{exceptional}","aliases":"exceptional mathrm","display":"\\mathrm{exceptional}","kind":"use","excerpt":"A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equiva","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"For a finite generically finite morphism of degree d, the push-pull identity on rational cohomology is f_*f^*=did. Division by d transfers algebraicity downward. Pullback transfers it upward without division. Coordinate-power maps preserve the old conductor of an inflated character, even though the ambient Fermat degree grows.","count":17},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f1ffe82ebc455b14","tex":"f^*","aliases":"f^*","display":"f^*","kind":"use","excerpt":"For a finite generically finite morphism of degree d, the push-pull identity on rational cohomology is f_*f^*=did. Division by d transfers algebraicity downward. Pullback transfers it upward without division. Coordinate-power maps preserve the old conductor of an inflated character, even though the ambient Fermat degree grows.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c1f02f9950b99117","tex":"f_*","aliases":"f_*","display":"f_*","kind":"use","excerpt":"For a finite generically finite morphism of degree d, the push-pull identity on rational cohomology is f_*f^*=did. Division by d transfers algebraicity downward. Pullback transfers it upward without division. Coordinate-power maps preserve the old conductor of an inflated character, even though the ambient Fermat degree grows.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4fac6558c4f0f59e","tex":"f_3","aliases":"f_3","display":"f_3","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-20317a07a635100c","tex":"F_{35}^3","aliases":"F_35^3","display":"F_{35}^3","kind":"use","excerpt":"∫_Z_κres(fracQ_1ΩF_35^3) =frac(2π i)^2κ^6350≠0.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c9c6ef59227a62af","tex":"F_A^{0,2}","aliases":"F_A^0,2","display":"F_A^{0,2}","kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-92cbe7d6d84c4e19","tex":"F_c","aliases":"F_c","display":"F_c","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"use","excerpt":"X_m={F_m=x_0^m+⋯+x_5^m=0}⊂ P^5_ C, G_m=(μ_m)^6/μ_m.","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-cc39fdfd384bbdb2","tex":"\\mathcal F_m","aliases":"F_m mathcal","display":"\\mathcal F_m","kind":"use","excerpt":"Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f853e7cb793578b7","tex":"F_m^3","aliases":"F_m^3","display":"F_m^3","kind":"use","excerpt":"∫_Z_κres(fracQ_bΩF_m^3) =frac(2π i)^250q κ^q-b≠0.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-30ade5d863641faa","tex":"F_s","aliases":"F_s","display":"F_s","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4a14d2ab0a464e3b","tex":"F_u","aliases":"F_u","display":"F_u","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-78b29d109052da32","tex":"F_v","aliases":"F_v","display":"F_v","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a75d31ba6a020ef2","tex":"F_z","aliases":"F_z","display":"F_z","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-22b6d3dfd4f33529","tex":"\\mathrm{Frob}","aliases":"Frob mathrm","display":"\\mathrm{Frob}","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-00c37c61b4205056","tex":"\\mathrm{full}","aliases":"full mathrm","display":"\\mathrm{full}","kind":"use","excerpt":"[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance e","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e6df94744f7ade62","tex":"g_2","aliases":"g_2","display":"g_2","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a8ca81fccec7b3cd","tex":"G_{210}","aliases":"G_210","display":"G_{210}","kind":"use","excerpt":"Thus u is a character of the weighted projective torus. Its pullback restricts on G_210 to the Fermat character α, since 630=3·210.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ab6997694ee35093","tex":"g_3","aliases":"g_3","display":"g_3","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-22f3c1cefe11bb39","tex":"G_{ij}","aliases":"G_ij","display":"G_{ij}","kind":"use","excerpt":"[Finite Gram certificate] Choose a rational basis e_1,…,e_d of V. If G_ij=∑_r⟨ A_re_i,A_re_j⟩_Y_r has nonzero determinant, every rational Hodge class in V is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c3d50e4b12d0681c","tex":"H^*","aliases":"H^*","display":"H^*","kind":"use","excerpt":"If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"The desired correspondence is schematically Γ∈CH^∗(F_d× C_P× S_T)_Q, Γ_*:e_orbH^1(F_d)⟶ e_diag(M(C_P)⊗ M(S_T))(1). This formula is a target specification, not a constructed map. Its coefficient field, diagonal projector, rational multiplicities, and dimensions must be fixed before an isomorphism can be asserted. The smaller CM conductor can have a ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7029404dbb7a2c18","tex":"H^{1,1}","aliases":"H^1,1","display":"H^{1,1}","kind":"use","excerpt":"For a fourfold and p=2, α=Lβ+α_prim, β∈ H^1,1(X)∩ H^2(X,Q). The Lefschetz (1,1) theorem makes β algebraic. Only α_prim remains.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":"r after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive packet: Newton's identity makes the third equation redundant on X_3","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c76765363d54662c","tex":"H^{2,2}","aliases":"H^2,2","display":"H^{2,2}","kind":"use","excerpt":"The constructive objective is [Packet-resolved construction target] For every rational packet W⊂ H^4_prim(X_m^4,Q)∩ H^2,2, construct Z_W∈CH^2(X_m^4)_Q with [Z_W]≠0 in W. Once one nonzero algebraic vector is produced in a rational irreducible diagonal-symmetry packet, the entire rational orbit is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2a698625f2d757b9","tex":"H^2_{\\mathrm{prim}}","aliases":"H^2_prim mathrm","display":"H^2_{\\mathrm{prim}}","kind":"use","excerpt":"Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standa","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-41527cc2ae4b1f89","tex":"H^{2p}","aliases":"H^2p","display":"H^{2p}","kind":"use","excerpt":"Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).","count":19},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a68b5ee45a6bd895","tex":"H^{4,0}","aliases":"H^4,0","display":"H^{4,0}","kind":"use","excerpt":"The residue description places the c-line in H^4,0 under the direct-action convention (or its conjugate under the inverse-character convention). Its rational Galois span contains a nonzero (4,0) part and is also K-invariant.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-223c8cdce6b46104","tex":"H^4_{\\mathrm{prim}}","aliases":"H^4_prim mathrm","display":"H^4_{\\mathrm{prim}}","kind":"use","excerpt":"Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standa","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-77b9e7b18a7c56be","tex":"H^6","aliases":"H^6","display":"H^6","kind":"use","excerpt":"item For suitable GM fourfolds, the vanishing H^4 packet is lowered by universal lines to divisor degree. item For suitable GM sixfolds, the vanishing H^6 packet is lowered by universal planes to divisor degree. item Ambient and tautological classes are treated separately and are algebraic.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-85f970dc3291b083","tex":"H^{\\mathrm{even}}","aliases":"H^even mathrm","display":"H^{\\mathrm{even}}","kind":"use","excerpt":"Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees i","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-92ad3ff2bdd044cf","tex":"H^i","aliases":"H^i","display":"H^i","kind":"use","excerpt":"Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-02dfc62d527a66a0","tex":"H^{i+2(r-n)}","aliases":"H^i+2(r-n)","display":"H^{i+2(r-n)}","kind":"use","excerpt":"Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-87b34ccf5a33919f","tex":"H^{p,p}","aliases":"H^p,p","display":"H^{p,p}","kind":"use","excerpt":"Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-858ebfe3a6d1e027","tex":"h^r","aliases":"h^r","display":"h^r","kind":"use","excerpt":"[Primitive reduction] If every primitive rational Hodge class is algebraic, then every rational Hodge class is algebraic. Write α=∑_rL^rα_r with each α_r primitive and of Hodge type. The class h is algebraic. If α_r is algebraic, then L^rα_r=h^rsmileα_r is algebraic. Summing gives an algebraic representative of α.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-093ccfd1c65e2cec","tex":"h_B","aliases":"h_B","display":"h_B","kind":"use","excerpt":"For any nonzero-entry block B at modulus m, put h_B(u)=m^-1∑_r∈ B⟨ ur⟩_m, with multiplicities retained.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ce06f772827e71fc","tex":"h_{C_P}","aliases":"h_C_P","display":"h_{C_P}","kind":"use","excerpt":"[Exact K23 packet identity] One has the exact multiset equality A+(5c,m-5c)+(5d,m-5d)=C_P+S_T+L. For every unit upmod m, h_C_P(u)+h_S_T(u)+h_L(u)=5, and c+d+e≡0pmod q.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-823a4809591e26fe","tex":"h_L","aliases":"h_L","display":"h_L","kind":"use","excerpt":"[Exact K23 packet identity] One has the exact multiset equality A+(5c,m-5c)+(5d,m-5d)=C_P+S_T+L. For every unit upmod m, h_C_P(u)+h_S_T(u)+h_L(u)=5, and c+d+e≡0pmod q.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8fa9235efb3a1133","tex":"H_{\\mathrm{prim}}^{2p}","aliases":"H_prim^2p mathrm","display":"H_{\\mathrm{prim}}^{2p}","kind":"use","excerpt":"Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic corre","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-81cb62034afcc1fa","tex":"H_{\\mathrm{prim}}^{2p-2r}","aliases":"H_prim^2p-2r mathrm","display":"H_{\\mathrm{prim}}^{2p-2r}","kind":"use","excerpt":"Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2c9d396bb5e57e9c","tex":"H_{\\Q}","aliases":"H_Q Q","display":null,"kind":"use","excerpt":"Unconditional rank test · rank_Q[C H_Q]=rank_QC tests target containment in the actual cycle span.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-dc8cc4a58b5e0b55","tex":"h_{S_T}","aliases":"h_S_T","display":"h_{S_T}","kind":"use","excerpt":"[Exact K23 packet identity] One has the exact multiset equality A+(5c,m-5c)+(5d,m-5d)=C_P+S_T+L. For every unit upmod m, h_C_P(u)+h_S_T(u)+h_L(u)=5, and c+d+e≡0pmod q.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-78ef1590654c6750","tex":"H_Z^{2p}","aliases":"H_Z^2p","display":"H_Z^{2p}","kind":"use","excerpt":"Support verification and multiplicity recovery. Let Z⊆ X be a closed algebraic subset and j:X∖ Z↪ X. The support exact sequence contains H_Z^2p(X,Q)⟶ H^2p(X,Q)xrightarrowj^*H^2p(X∖ Z,Q).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-858e4bb65a33c6eb","tex":"h_\\ell","aliases":"h_ℓ ell","display":"h_\\ell","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-cc5a17924ee261a3","tex":"\\Hdg","aliases":"Hdg","display":null,"kind":"use","excerpt":"Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).","count":12},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8c842372dca85da3","tex":"\\Hdg^p","aliases":"Hdg^p Hdg","display":null,"kind":"use","excerpt":"Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e7b40b909a5de1cc","tex":"\\operatorname{Hess}","aliases":"Hess operatorname","display":"\\operatorname{Hess}","kind":"use","excerpt":"detHess(F_m)=[m(m-1)]^6S_m.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.","count":13},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a60f4be54f655f46","tex":"I_{M,r}","aliases":"I_M,r","display":"I_{M,r}","kind":"use","excerpt":"Degree-residue polyphase refinement. Exact parent-child frame law. Fix a projective embedding and assign each cycle its degree. For modulus M, define the residue packet I_M,r={j:deg Z_j≡ rpmod M}, S_M,r=∑_j∈ I_M,r|c_j⟩⟨ c_j|.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f1c9ad67822f1d50","tex":"I_\\lambda","aliases":"I_λ lambda","display":"I_\\lambda","kind":"use","excerpt":"Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id","display":null,"kind":"use","excerpt":"For a finite generically finite morphism of degree d, the push-pull identity on rational cohomology is f_*f^*=did. Division by d transfers algebraicity downward. Pullback transfers it upward without division. Coordinate-power maps preserve the old conductor of an inflated character, even though the ambient Fermat degree grows.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b09c3d3928fe2c8b","tex":"\\operatorname{id}","aliases":"id operatorname","display":"\\operatorname{id}","kind":"use","excerpt":"π_*π^*=(degπ)id","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im","display":null,"kind":"use","excerpt":"Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"endcases Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2ba8441ab7c7d7ef","tex":"j^*","aliases":"j^*","display":"j^*","kind":"use","excerpt":"Support test · j^*α=0. Purity turns it into cycle algebraicity when the support has codimension at least p.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-485db41ee683b54c","tex":"J_0","aliases":"J_0","display":"J_0","kind":"use","excerpt":"The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for ","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-af9adddba5ce3c4c","tex":"J_{F_m}","aliases":"J_F_m","display":"J_{F_m}","kind":"use","excerpt":"J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5f87ffb491a0c276","tex":"\\operatorname{Jac}","aliases":"Jac operatorname","display":"\\operatorname{Jac}","kind":"use","excerpt":"H=(f_1,g_1,f_2,g_2,f_3,g_3), D_κ=detJac_(x_0,…,x_4,z)H,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-186f834fc07f20cd","tex":"\\operatorname{Jac}_{(x_0,\\ldots,x_4,z)}","aliases":"Jac_(x_0,…,x_4,z) operatorname ldots","display":"\\operatorname{Jac}_{(x_0,\\ldots,x_4,z)}","kind":"use","excerpt":"H=(f_1,g_1,f_2,g_2,f_3,g_3), D_κ=detJac_(x_0,…,x_4,z)H,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees i","count":25},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"The integral triangle basis. Let K_m= { d∈ Z^h-1: ∑_r=1^h-1d(r)ρ_m(ur)=0 for every odd u }. The triangle relations do not merely generate examples, they form the whole balanced defect lattice.","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-738e0b49cf0886b9","tex":"K^0_{\\mathrm{top}}","aliases":"K^0_top mathrm","display":"K^0_{\\mathrm{top}}","kind":"use","excerpt":"Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees i","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5f76e4469b17763e","tex":"k^{n-2j}","aliases":"k^n-2j","display":"k^{n-2j}","kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c2d518a5c0493ae6","tex":"k^{n-2j-1}","aliases":"k^n-2j-1","display":"k^{n-2j-1}","kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6dca9b7143fa0047","tex":"K_{1,3}","aliases":"K_1,3","display":"K_{1,3}","kind":"use","excerpt":"[Exact SAT(2) criterion] Let n be a balanced six-packet at modulus m=2^s. Then n is pair-decomposable or has an SAT(2) certificate unless its gauged defect is ±1 on a four-vertex star K_1,3 in the internal tree T_m. A four-star has no SAT(2) certificate. Every nonstar shape in the preceding proposition splits into at most two four-packet atoms: ite","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4b35ad5d2218e1cf","tex":"\\mathcal K_m","aliases":"K_m mathcal","display":"\\mathcal K_m","kind":"use","excerpt":"The integral triangle basis. Let K_m= { d∈ Z^h-1: ∑_r=1^h-1d(r)ρ_m(ur)=0 for every odd u }. The triangle relations do not merely generate examples, they form the whole balanced defect lattice.","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1c075aa812e79ea0","tex":"K_{\\widetilde Y}","aliases":"K_widetilde Y widetilde","display":"K_{\\widetilde Y}","kind":"use","excerpt":"K_widetilde Y·widetilde C=K_Y· C=-3H· C<0.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d9fd7bae842641b2","tex":"K_Y","aliases":"K_Y","display":"K_Y","kind":"use","excerpt":"K_Y∼_ Q-3H.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equali","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"L_a={v∈ Z_≥0^6∖{0}:a· v≡0pmod m}.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a245919be5634c20","tex":"L^2","aliases":"L^2","display":"L^2","kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-664eb6067f1c6bd3","tex":"L^r","aliases":"L^r","display":"L^r","kind":"use","excerpt":"Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-808c3665a73ad897","tex":"\\mathcal L_a","aliases":"L_a mathcal","display":"\\mathcal L_a","kind":"use","excerpt":"L_a={v∈ Z_≥0^6∖{0}:a· v≡0pmod m}.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-119d339f3e3a2b7c","tex":"\\mathcal L_{a_{q,b}}","aliases":"L_a_q,b mathcal","display":"\\mathcal L_{a_{q,b}}","kind":"use","excerpt":"If v∈ L_a_q,b, reduction modulo q, followed by cancellation of the unit b, gives","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b75aa9a54598bfb9","tex":"L_{m/c}","aliases":"L_m/c","display":"L_{m/c}","kind":"use","excerpt":"[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balance e","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7e9eab168e0b274a","tex":"\\mathrm{Lef}","aliases":"Lef mathrm","display":"\\mathrm{Lef}","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"[Approximation becomes exact] Let P_D be the orthogonal projector onto A_D^p(X)⊗R. For a rational class α, lim_D→∞ (I-P_D)α =0 if and only if α∈ A_D^p(X) for some finite D. After stabilization, the residual is constant. A constant tail converging to zero is zero.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f3f51431785d568c","tex":"\\lim_{D\\to\\infty}","aliases":"lim_D→∞ lim to infty","display":"\\lim_{D\\to\\infty}","kind":"use","excerpt":"[Approximation becomes exact] Let P_D be the orthogonal projector onto A_D^p(X)⊗R. For a rational class α, lim_D→∞ (I-P_D)α =0 if and only if α∈ A_D^p(X) for some finite D. After stabilization, the residual is constant. A constant tail converging to zero is zero.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-59f39850ae27a8d7","tex":"\\mathrm{line}","aliases":"line mathrm","display":"\\mathrm{line}","kind":"use","excerpt":"A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equiva","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Master obstruction projector] The operator O_X^p=P_Hdg^p-P_Alg^p is the orthogonal projector onto (Hdg^p(X)⊗_QR)∩(Alg^p(X)⊗_QR)^⊥. Therefore O_X^p=0 Longleftrightarrow Hdg^p(X)=Alg^p(X). For nested orthogonal projectors Q≤ P, the difference P-Q is self-adjoint and (P-Q)^2=P-PQ-QP+Q=P-Q. Its range is im P∩(im Q)^⊥.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"Polyphase assembly · S_M,r=∑_ℓ=0^L-1S_LM,r+ℓ M. These are frame sums, not sums of span projectors.","count":23},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-768071a8b8aba296","tex":"m^{-1}","aliases":"m^-1","display":"m^{-1}","kind":"use","excerpt":"For any nonzero-entry block B at modulus m, put h_B(u)=m^-1∑_r∈ B⟨ ur⟩_m, with multiplicities retained.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","excerpt":"[Range-packet contraction] Assume R ≤ M, B(a,b) ≤ C a b , 4CM^2 Qf <1. Then the map T(a)=-RQ(f+B(a,a)) has a unique fixed point in the gauge-fixed ball a ≤2M Qf . If Qf=0, the ball is the singleton {0} and the range solution is immediate. At the fixed point, the range component of F(a) vanishes. The remaining obstruction is Φ=Π(f+B(a,a)). The full ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-157396fd2d31299d","tex":"m^p","aliases":"m^p","display":"m^p","kind":"use","excerpt":"Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b79fcd2a8cb4db07","tex":"m^q","aliases":"m^q","display":"m^q","kind":"use","excerpt":"Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8ac08819c3ccbd63","tex":"M_{ki}","aliases":"M_ki","display":"M_{ki}","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c0946f6b47457df3","tex":"\\mathrm{mon}","aliases":"mon mathrm","display":"\\mathrm{mon}","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Support localization and coniveau. Let N^pH^2p(X,Q) be the geometric coniveau-p subspace. In degree 2p, purity forces support contributions of codimension p to be generated by fundamental classes of codimension-p components.","count":23},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-261b1d4accf38dcf","tex":"\\mathbf N","aliases":"N mathbf","display":"\\mathbf N","kind":"use","excerpt":"To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebrai","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"0 · 0 · 0) is nonzero, while N^3=0 and Tr(N^r)=0 for every positive integer r. This is the elementary negative control.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-39b6cdf712594142","tex":"\\mathbf N^{m-1}","aliases":"N^m-1 mathbf","display":"\\mathbf N^{m-1}","kind":"use","excerpt":"To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebrai","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1c2f4b0c89229624","tex":"N^p","aliases":"N^p","display":"N^p","kind":"use","excerpt":"Support localization and coniveau. Let N^pH^2p(X,Q) be the geometric coniveau-p subspace. In degree 2p, purity forces support contributions of codimension p to be generated by fundamental classes of codimension-p components.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ebcd9b4bdca5cfcd","tex":"N^r","aliases":"N^r","display":"N^r","kind":"use","excerpt":"0 · 0 · 0) is nonzero, while N^3=0 and Tr(N^r)=0 for every positive integer r. This is the elementary negative control.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebrai","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonz","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-821cea6ccfaff93a","tex":"n_{m-1}","aliases":"n_m-1","display":"n_{m-1}","kind":"use","excerpt":"To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebrai","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-99f81bde3314fae4","tex":"n_{m-r}","aliases":"n_m-r","display":"n_{m-r}","kind":"use","excerpt":"Complement defects and dyadic triangles. Put q=frac m4, h=2q, m=4q. For a packet count vector n, define its oriented complement defect ∂ n=∑_r=1^h-1d_n(r)[r], d_n(r)=n_r-n_m-r. The self-complementary residue h contributes zero. Since ρ_m(m-r)=-ρ_m(r), the balanced-packet equation is equivalent to ∑_r=1^h-1d_n(r)ρ_m(ur)=0 for every odd unit u. Moreo","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-eb171c4fc1202d77","tex":"n_r","aliases":"n_r","display":"n_r","kind":"use","excerpt":"To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebrai","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"For dmid m, let E_m,d=frac1(m/d)^5π_m/d^*π_m/d*. On a character of exact conductor c, E_m,d=1, · cmid d, 0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. E","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e41afe6415912e0d","tex":"\\operatorname{Norm}","aliases":"Norm operatorname","display":"\\operatorname{Norm}","kind":"use","excerpt":"This also corrects a source-level eigencharacter confusion. The polarization cube has eigenvalue [2+i]^*(θ^3)=125 θ^3=Norm(2+i)^3θ^3, while the Weil directions have eigenvalues -117±44i. The norm-cubed character therefore identifies the polarization cube, and does not identify the two-dimensional Weil subspace. The September audit recorded this loc","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"use","excerpt":"Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ec66d2dd6a0e0d47","tex":"O_b","aliases":"O_b","display":"O_b","kind":"use","excerpt":"The three-primary branch. For p=3 and r≥2, a higher-level kernel orbit has three elements. Define O_b={b,b+q,b+2q}, q=3^r-1, and Γ_b=O_bsqcup{3(q-b)}. Take 1≤ b<q with 3nmid b, ensuring all entries are nonzero. Then Γ_b is a balanced four-packet. A case analysis on the number of oriented kernel orbits gives exact identities expressing every full-co","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-008d99a8223af673","tex":"\\mathcal O_{S_q}","aliases":"O_S_q mathcal","display":"\\mathcal O_{S_q}","kind":"use","excerpt":"Forgetting z defines a finite flat cover Z_κ→ S_q of degree five. On a chart trivializing O_S_q(1), its equation is z^5=P/κ, with the corresponding trivializations understood. At a generic coordinate curve its right side has valuation one. The degree-five polynomial is Eisenstein over that discrete valuation ring and is therefore irreducible over t","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-daf54c3de28b79f6","tex":"\\mathcal O_X^p","aliases":"O_X^p mathcal","display":"\\mathcal O_X^p","kind":"use","excerpt":"[Master obstruction projector] The operator O_X^p=P_Hdg^p-P_Alg^p is the orthogonal projector onto (Hdg^p(X)⊗_QR)∩(Alg^p(X)⊗_QR)^⊥. Therefore O_X^p=0 Longleftrightarrow Hdg^p(X)=Alg^p(X). For nested orthogonal projectors Q≤ P, the difference P-Q is self-adjoint and (P-Q)^2=P-PQ-QP+Q=P-Q. Its range is im P∩(im Q)^⊥.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-892e39ac9ff8d033","tex":"\\mathcal O_Y","aliases":"O_Y mathcal","display":"\\mathcal O_Y","kind":"use","excerpt":"Here H is the ample rational hyperplane class, not an assertion that O_Y(1) is everywhere a line bundle. This argument establishes the adjunction formula in the present non-quasismooth example directly.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-57ff192d28b73a0c","tex":"\\overline v","aliases":"overline v overline","display":"\\overline v","kind":"use","excerpt":"The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-33a94ddd174707c0","tex":"\\overline v^{\\,t}","aliases":"overline v^ t overline","display":"\\overline v^{\\,t}","kind":"use","excerpt":"The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ae21a8553a17610b","tex":"\\overline z","aliases":"overline z overline","display":"\\overline z","kind":"use","excerpt":"The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety g","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a9f74fc1cdc1fadf","tex":"\\overline z_1","aliases":"overline z_1 overline","display":"\\overline z_1","kind":"use","excerpt":"The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety g","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bcc6445aee56a277","tex":"\\overline z_2","aliases":"overline z_2 overline","display":"\\overline z_2","kind":"use","excerpt":"The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety g","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-18f8620bf7a3538d","tex":"\\overline z_3","aliases":"overline z_3 overline","display":"\\overline z_3","kind":"use","excerpt":"The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety g","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7f11a927384f9d5e","tex":"\\overline{\\Delta_\\ell}","aliases":"overlineΔ_ℓ overline Delta ell","display":"\\overline{\\Delta_\\ell}","kind":"use","excerpt":"[Pure imaginary selector] overlineΔ_ℓ=-Δ_ℓ. Since 1-ζ_q^-a=-ζ_q^-a(1-ζ_q^a), complex conjugation multiplies the product by (-1)^3ζ_q^-(c+d+e)=-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ed3c6d4891708d94","tex":"\\overline{\\Sigma_\\ell}","aliases":"overlineΣ_ℓ overline Sigma ell","display":"\\overline{\\Sigma_\\ell}","kind":"use","excerpt":"1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-50430f52dcf32f77","tex":"\\overline{\\chi(a)}","aliases":"overlineχ(a) overline chi","display":"\\overline{\\chi(a)}","kind":"use","excerpt":"ecover the unique integers z_b from its even coordinates, and subtract ∑_bz_bκ_b. The residual balanced vector is supported only on the odd residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, wi","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-356383447f0ec646","tex":"\\overline{\\chi(u)}","aliases":"overlineχ(u) overline chi","display":"\\overline{\\chi(u)}","kind":"use","excerpt":"Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9f229d1acbae0797","tex":"\\overline{\\chi^*}","aliases":"overlineχ^* overline chi","display":"\\overline{\\chi^*}","kind":"use","excerpt":"_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,2^s=2^s-tλ_χ^*,2^t. The primitive value is, up to a nonzero scalar, the generalized Bernoulli value B_1,overlineχ^*, equivalently -L(0,overlineχ^*","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ef958519c2ac9923","tex":"\\mathbb P","aliases":"P mathbb","display":"\\mathbb P","kind":"use","excerpt":"Projective bundles, blow-ups, and descent. [Projective-bundle closure] If the rational Hodge conjecture holds for X, it holds for P(E)→ X. The projective-bundle formula expresses cohomology as a direct sum of powers of the algebraic hyperplane class times pullbacks of cohomology from X.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6a6ffe951cf57727","tex":"\\mathbf P","aliases":"P mathbf","display":"\\mathbf P","kind":"use","excerpt":"Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finit","count":10},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).","count":105},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c84939e3585d6f53","tex":"\\mathbf P^1","aliases":"P^1 mathbf","display":"\\mathbf P^1","kind":"use","excerpt":"For context, this theorem does not use the unknown general fourfold case: a uniruled fourfold is dominated generically finitely by a resolution of T× P^1, with dim T=3. The relevant Hodge classes on the product are algebraic by the threefold case and Künneth, blowup centers have dimension at most two, push-pull transfers algebraicity.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5cb15b86390400d8","tex":"\\mathbf P^4","aliases":"P^4 mathbf","display":"\\mathbf P^4","kind":"use","excerpt":"S_q={e_1(x_0^q,…,x_4^q)=e_2(x_0^q,…,x_4^q)=0}⊂ P^4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ed9be4f4196562e5","tex":"\\mathbb P^5","aliases":"P^5 mathbb","display":"\\mathbb P^5","kind":"use","excerpt":"Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standa","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3a643af369cb160b","tex":"\\mathbf P^5","aliases":"P^5 mathbf","display":"\\mathbf P^5","kind":"use","excerpt":"Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finit","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4f722d029cb232df","tex":"\\mathbf P^5_\\mathbf","aliases":"P^5_ mathbf mathbf","display":"\\mathbf P^5_\\mathbf","kind":"use","excerpt":"X_m={x_0^m+⋯+x_5^m=0}⊂ P^5_ C.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-59fed64afdcf3f92","tex":"\\mathbf P^5_{\\mathbf C}","aliases":"P^5_ C mathbf mathbf","display":"\\mathbf P^5_{\\mathbf C}","kind":"use","excerpt":"X_m={F_m=x_0^m+⋯+x_5^m=0}⊂ P^5_ C, G_m=(μ_m)^6/μ_m.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bbd2abe9ab800d4e","tex":"P^{b-1}","aliases":"P^b-1","display":"P^{b-1}","kind":"use","excerpt":"D_κ=625q^5Δ(x_0^q,…,x_4^q) ∑_b=1^q-1κ^q-bP^b-1z^5(q-b)-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-44ddf871c8e79a19","tex":"p^j","aliases":"p^j","display":"p^j","kind":"use","excerpt":"Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonz","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a2c4997966a645dd","tex":"\\mathbf P^{n+1}_{\\mathbf C}","aliases":"P^n+1_ C mathbf mathbf","display":"\\mathbf P^{n+1}_{\\mathbf C}","kind":"use","excerpt":"X_m^n={x_0^m+⋯+x_n+1^m=0}⊂ P^n+1_ C.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7729a9889503d79f","tex":"P^q","aliases":"P^q","display":"P^q","kind":"use","excerpt":"Since e_5=P^q, on Z_κ one has","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-18a5cc17c9d60090","tex":"P^{q-1}","aliases":"P^q-1","display":"P^{q-1}","kind":"use","excerpt":"Use e_5=P^q and dt_0wedge⋯wedge dt_4=q^5P^q-1 dx_0wedge⋯wedge dx_4. Cancellation in the polynomial ring gives","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-260f2fcaa8aeb441","tex":"P^{q-1-j}","aliases":"P^q-1-j","display":"P^{q-1-j}","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6f0ea21bbfbf69fb","tex":"p^r","aliases":"p^r","display":"p^r","kind":"use","excerpt":"Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonz","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f57ac308185c63af","tex":"p^{r-j}","aliases":"p^r-j","display":"p^{r-j}","kind":"use","excerpt":"Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonz","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-360f794da8094012","tex":"p^s","aliases":"p^s","display":"p^s","kind":"use","excerpt":"r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at l","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7a903471bc3dd494","tex":"p^{s-1}","aliases":"p^s-1","display":"p^{s-1}","kind":"use","excerpt":"r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at l","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7cd48fb48ab38484","tex":"p_*","aliases":"p_*","display":"p_*","kind":"use","excerpt":"Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standa","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"its four triangle atoms gives the unordered packet P_a= (8a,q-a,2q-a,2q-4a,3q-a,4q-a). The cases c<q/2 and c>q/2 give the same unordered tuple after the internal-edge cancellations. Choose an odd u satisfying ua≡1pmod2q. Since uq≡ q or 3qpmod4q, multiplication by u sends [source reference] to P_1=(8,q-1,2q-4,2q-1,3q-1,4q-1). Its conjugate is A_m= (","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"The exceptional packet. A representative is a=(7,10,13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b92de34fe00fe340","tex":"P_{102}","aliases":"P_102","display":"P_{102}","kind":"use","excerpt":"buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-40b4c929d453e5da","tex":"P_{105}","aliases":"P_105","display":"P_{105}","kind":"use","excerpt":"buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ac241eefcf68177e","tex":"P_{12}","aliases":"P_12","display":"P_{12}","kind":"use","excerpt":"buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"The exceptional packet. A representative is a=(7,10,13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-add45222cdd5f39c","tex":"P_{24}","aliases":"P_24","display":"P_{24}","kind":"use","excerpt":"buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d87a5da0aecb9e80","tex":"P_{27}","aliases":"P_27","display":"P_{27}","kind":"use","excerpt":"buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c7ef6bdd367b3d40","tex":"P_{2j}","aliases":"P_2j","display":"P_{2j}","kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"use","excerpt":"The exceptional packet. A representative is a=(7,10,13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e1e2c2d374025ff6","tex":"P_{30}","aliases":"P_30","display":"P_{30}","kind":"use","excerpt":"buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a60c6f460d4ce98b","tex":"P_{42}","aliases":"P_42","display":"P_{42}","kind":"use","excerpt":"buplus P_24uplus P_27uplus P_105uplus Q_3uplus S_5uplus S_7 = · suplus P_12uplus P_30uplus P_42uplus P_102 · uplus Q_9uplus Q_24uplus Q_27uplus R_2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ea39b3b7a10c3dfe","tex":"p_5","aliases":"p_5","display":"p_5","kind":"use","excerpt":"p_5=e_1(e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4)-5e_2e_3+5e_5.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f21822f96efaf869","tex":"P_A","aliases":"P_A","display":"P_A","kind":"use","excerpt":"The orthogonal synthesis of a class α from the cycle span is P_Aα=S S^†α. If α∈ A, this is exact.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"Assume now s≥5 and let a full-conductor star have center c=2a with a odd. Up to global conjugation, expanding its four triangle atoms gives the unordered packet P_a= (8a,q-a,2q-a,2q-4a,3q-a,4q-a). The cases c<q/2 and c>q/2 give the same unordered tuple after the internal-edge cancellations. Choose an odd u satisfying ua≡1pmod2q. Since uq≡ q or 3qpm","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-05a580fd800c9043","tex":"P_{\\Alg}^p","aliases":"P_Alg^p Alg","display":null,"kind":"use","excerpt":"Choose a positive inner product on H^2p(X,R). The orthogonal projectors P_Hdg^p and P_Alg^p act on the real scalar extensions Hdg^p(X)⊗_QR and Alg^p(X)⊗_QR. These real orthogonal projectors provide a reformulation. They are not asserted to be rational matrices or algebraic correspondences. Since the algebraic space is nested in the Hodge space, P_H","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-87f5109330680916","tex":"P_{\\Alg\\otimes\\R}","aliases":"P_Alg⊗R Alg otimes R","display":null,"kind":"use","excerpt":"Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a9d31d0c349bd2e3","tex":"P_D","aliases":"P_D","display":"P_D","kind":"use","excerpt":"[Approximation becomes exact] Let P_D be the orthogonal projector onto A_D^p(X)⊗R. For a rational class α, lim_D→∞ (I-P_D)α =0 if and only if α∈ A_D^p(X) for some finite D. After stabilization, the residual is constant. A constant tail converging to zero is zero.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-333d75f622b78fee","tex":"p_h","aliases":"p_h","display":"p_h","kind":"use","excerpt":". A zero four-packet absorbs that pair and the stabilizing pair. Cutting a three- or four-vertex path produces exactly one oppositely oriented boundary edge, hence exactly one complementary pair in the union of the two atom packets, this is the SAT(2) stabilizer. In the three-vertex case, the vertex atom also supplies the unique neutral h. For two ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-48e8723360c06dfe","tex":"P_{\\Hdg}^p","aliases":"P_Hdg^p Hdg","display":null,"kind":"use","excerpt":"Choose a positive inner product on H^2p(X,R). The orthogonal projectors P_Hdg^p and P_Alg^p act on the real scalar extensions Hdg^p(X)⊗_QR and Alg^p(X)⊗_QR. These real orthogonal projectors provide a reformulation. They are not asserted to be rational matrices or algebraic correspondences. Since the algebraic space is nested in the Hodge space, P_H","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d82996b1f95134eb","tex":"P_{\\Hdg\\otimes\\R}","aliases":"P_Hdg⊗R Hdg otimes R","display":null,"kind":"use","excerpt":"Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"P_k=(k,210-k)","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-528f89f54cfb3d02","tex":"p_r","aliases":"p_r","display":"p_r","kind":"use","excerpt":"Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3f8cc78c6162674e","tex":"p_{r_1}","aliases":"p_r_1","display":"p_{r_1}","kind":"use","excerpt":"Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5ac67833d91fe1bc","tex":"p_{r_2}","aliases":"p_r_2","display":"p_{r_2}","kind":"use","excerpt":"Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d123d401019f9da1","tex":"p_{r_3}","aliases":"p_r_3","display":"p_{r_3}","kind":"use","excerpt":"Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1651bba6c564a161","tex":"p_{r_j}","aliases":"p_r_j","display":"p_{r_j}","kind":"use","excerpt":"Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0c723e0565a0c218","tex":"P_W","aliases":"P_W","display":"P_W","kind":"use","excerpt":"Weil projection · P_W(c)=-(⟨ c,u⟩ u+⟨ c,v⟩ v)/32.","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b86febd8fa411d43","tex":"\\mathrm{plane}","aliases":"plane mathrm","display":"\\mathrm{plane}","kind":"use","excerpt":"A focused multi-incidence target. Let X be a special smooth projective fourfold. Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equiva","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ebe198d42ed3f048","tex":"\\mathrm{prim}","aliases":"prim mathrm","display":"\\mathrm{prim}","kind":"use","excerpt":"Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"[Master obstruction projector] The operator O_X^p=P_Hdg^p-P_Alg^p is the orthogonal projector onto (Hdg^p(X)⊗_QR)∩(Alg^p(X)⊗_QR)^⊥. Therefore O_X^p=0 Longleftrightarrow Hdg^p(X)=Alg^p(X). For nested orthogonal projectors Q≤ P, the difference P-Q is self-adjoint and (P-Q)^2=P-PQ-QP+Q=P-Q. Its range is im P∩(im Q)^⊥.","count":26},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"use","excerpt":"Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).","count":41},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5cf102ce096caa7a","tex":"q^*","aliases":"q^*","display":"q^*","kind":"use","excerpt":"Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standa","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"dim Z_κ=2, deg Z_κ=q(2q)5=10q^2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6e535c3600229c66","tex":"q^4","aliases":"q^4","display":"q^4","kind":"use","excerpt":"de_1wedge⋯wedge de_4wedge dP =q^4Δ(x^q) dx_0wedge⋯wedge dx_4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bb85c47db9e6c417","tex":"q^5","aliases":"q^5","display":"q^5","kind":"use","excerpt":"D_κ=625q^5Δ(x_0^q,…,x_4^q) ∑_b=1^q-1κ^q-bP^b-1z^5(q-b)-1.","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"use","excerpt":"Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6bd5327e2740a5c3","tex":"Q_{24}","aliases":"Q_24","display":"Q_{24}","kind":"use","excerpt":"Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a4034bed935c2da7","tex":"Q_{27}","aliases":"Q_27","display":"Q_{27}","kind":"use","excerpt":"Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f5c3a1f5ac4f85f5","tex":"Q_3","aliases":"Q_3","display":"Q_3","kind":"use","excerpt":"Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1ba8277dab86d3ab","tex":"Q_9","aliases":"Q_9","display":"Q_9","kind":"use","excerpt":"Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e786aba54c22c347","tex":"\\mathbf Q_{>0}^6","aliases":"Q_>0^6 mathbf","display":"\\mathbf Q_{>0}^6","kind":"use","excerpt":"Suppose y∈ Q_>0^6 and y· v≥1 for every v∈ L_a. Then","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c57539584fe8169f","tex":"Q_b","aliases":"Q_b","display":"Q_b","kind":"use","excerpt":"Q_b=∏_i=0^5x_i^m-a_q,b,i-1.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3b93cc2d911081a8","tex":"Q_{m,c}","aliases":"Q_m,c","display":"Q_{m,c}","kind":"use","excerpt":"0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is it","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9b146da61812d4d6","tex":"Q_{m,e}","aliases":"Q_m,e","display":"Q_{m,e}","kind":"use","excerpt":"0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is it","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"ction. After gauge fixing, write the integrability equation abstractly as F(a)=f+La+B(a,a)=0, where L is the linearized operator and B is a bounded bilinear term. Assume a Fredholm splitting of the target I=Q+Π, where Q projects onto the closed range of L and Π onto a finite-dimensional obstruction space. Choose a gauge-fixed complement to ker L as","count":17},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.","count":11},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-cd6a51568c71e4ad","tex":"\\R^2","aliases":"R^2 R","display":null,"kind":"use","excerpt":"Take V=Span_R{e_1} in R^2 and let the only cycle vector be c=e_1+e_2. Then S=cc^t=1 · 1","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"(f+B(a,a)) has a unique fixed point in the gauge-fixed ball a ≤2M Qf . If Qf=0, the ball is the singleton {0} and the range solution is immediate. At the fixed point, the range component of F(a) vanishes. The remaining obstruction is Φ=Π(f+B(a,a)). The full integrability equation holds if and only if Φ=0. For a ≤ r=2M Qf , T(a) ≤ M Qf +MC r^2≤ r un","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b0a9532805ec4d05","tex":"\\R^N","aliases":"R^N R","display":null,"kind":"use","excerpt":"bseteq H^2p(X,R) be the real extension of the rational Hodge packet under study. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"e Laurent exponent match is a useful exact certificate, but by itself it does not prove that a rational class lifts through a compact graph. A complete manuscript argument instead uses the torus-carrier criterion developed in Chapter~[source reference]. For the reordered character, the exponent matrix A of the cubic satisfies A^Twidetilde a/33=β=(1","count":13},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-899fde002c2a2d10","tex":"R_2","aliases":"R_2","display":"R_2","kind":"use","excerpt":"Q_3 · =(3,105,108,204), Q_9 · =(9,105,114,192), Q_24 · =(24,105,129,162), Q_27 · =(27,105,132,156), R_2 · =(2,72,142,204).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"use","excerpt":"Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d5c7051b56820f95","tex":"r_3","aliases":"r_3","display":"r_3","kind":"use","excerpt":"Complementary pairs. For 1≤ r<m, let p_r=e_r+e_m-r. A packet is pair-decomposable when n=p_r_1+p_r_2+p_r_3. These are the standard linear-cycle or Tate-pair packets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b45828d0e48ceff6","tex":"r_{\\deg}","aliases":"r_deg deg","display":"r_{\\deg}","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"Indeed, choose M divisible by m and all denominators of A^-1. Put B=MA^-1, α=(M/m)a, and β=A^Ta/m. For any row r_i of A,","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2f9a0b1661b4e2e9","tex":"r_{\\mathrm{Lef}}","aliases":"r_Lef mathrm","display":"r_{\\mathrm{Lef}}","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-25ee15775b880083","tex":"R_m","aliases":"R_m","display":"R_m","kind":"use","excerpt":"To remove coordinate permutations, introduce the count vector n(a)=(n_1,…,n_m-1)∈ N^m-1, n_r=#{i:a_i=r}. Let R_m(u,r)=⟨ ur⟩_m. Then the complete balanced six-packet space is the finite integer set B_m= { n∈ N^m-1: 1^Tn=6, R_mn=3m 1 }. The unit group permutes the residue coordinates. After additionally identifying coordinate permutations by algebrai","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank","display":null,"kind":"use","excerpt":"Unconditional rank test · rank_Q[C H_Q]=rank_QC tests target containment in the actual cycle span.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-979550588c3eb1d0","tex":"\\operatorname{rank}_{\\Q}","aliases":"rank_Q operatorname Q","display":null,"kind":"use","excerpt":"If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-141c52963208c7e8","tex":"\\rank_{\\Q}","aliases":"rank_Q rank Q","display":null,"kind":"use","excerpt":"Unconditional rank test · rank_Q[C H_Q]=rank_QC tests target containment in the actual cycle span.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1115f7eaacd50934","tex":"\\operatorname{red}","aliases":"red operatorname","display":"\\operatorname{red}","kind":"use","excerpt":"SAT(2) closure and the four-star obstruction. For a defect d, let red(d) be the unmatched residue multiset: a positive coefficient at r contributes copies of r, while a negative coefficient contributes copies of m-r. Its length is d _1. Every packet with defect d is red(d) together with complementary pairs and copies of h.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8db5db648c850430","tex":"\\operatorname{res}","aliases":"res operatorname","display":"\\operatorname{res}","kind":"use","excerpt":"∫_Z_κres(fracQ_bΩF_m^3) =frac(2π i)^250q κ^q-b≠0.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Cycle frame · S=CC^*. Positivity spans a target packet when every supplied algebraic vector lies in it.","count":46},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-eb8d072dc7420869","tex":"S^\\dagger","aliases":"S^† dagger","display":"S^\\dagger","kind":"use","excerpt":"The orthogonal synthesis of a class α from the cycle span is P_Aα=S S^†α. If α∈ A, this is exact.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"[Propagation of invertibility] If a relative transfer T_s is invertible on one fiber s_0, it is invertible on every fiber in the connected base on which the local systems and correspondences are defined. The determinant of the constant rational matrix representing the local-system morphism is locally constant. It is nonzero at s_0, hence everywhere","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aabbd2f9e4d841eb","tex":"S_5","aliases":"S_5","display":"S_5","kind":"use","excerpt":"S_5=(30,60,72,114,156,198), S_7=(12,42,72,102,126,132,162,192).","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c13ef2b47bc153d9","tex":"S_7","aliases":"S_7","display":"S_7","kind":"use","excerpt":"S_5=(30,60,72,114,156,198), S_7=(12,42,72,102,126,132,162,192).","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5cbc71980bce2089","tex":"S_{LM,r+\\ell M}","aliases":"S_LM,r+ℓ M ell","display":"S_{LM,r+\\ell M}","kind":"use","excerpt":"Polyphase assembly · S_M,r=∑_ℓ=0^L-1S_LM,r+ℓ M. These are frame sums, not sums of span projectors.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-57c446e8e6d068a3","tex":"S_m","aliases":"S_m","display":"S_m","kind":"use","excerpt":"J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f7b894cadeb2501f","tex":"S_{M,r}","aliases":"S_M,r","display":"S_{M,r}","kind":"use","excerpt":"Polyphase assembly · S_M,r=∑_ℓ=0^L-1S_LM,r+ℓ M. These are frame sums, not sums of span projectors.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-19ed83d808a66932","tex":"S_q","aliases":"S_q","display":"S_q","kind":"use","excerpt":"S_q={e_1(x_0^q,…,x_4^q)=e_2(x_0^q,…,x_4^q)=0}⊂ P^4.","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b6db9064e751d27e","tex":"s_{\\mathrm{supp}}","aliases":"s_supp mathrm","display":"s_{\\mathrm{supp}}","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b6f114392e4c9056","tex":"S_T","aliases":"S_T","display":"S_T","kind":"use","excerpt":"c,d,e to be nonzero modulo q so that all three resulting character blocks have full support. Degenerate cases must leave this construction and be handled through pair removal or another channel. For example (1,14,2,9,14,5) at modulus 15 is balanced and admits the indicated phase split, but P=(1,14) gives c=0. The phase condition alone is insufficie","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-152abbc9214c8600","tex":"S_\\lambda","aliases":"S_λ lambda","display":"S_\\lambda","kind":"use","excerpt":"Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-01b9fe597a8baa31","tex":"\\smile","aliases":"smile","display":null,"kind":"use","excerpt":"Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span","display":null,"kind":"use","excerpt":"[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equali","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1220412e5152d698","tex":"\\Span_{\\Q}","aliases":"Span_Q Span Q","display":null,"kind":"use","excerpt":"For complex structures in the Weil domain with three positive and three negative K-embedding directions, Ω_6 has type (3,3). Indeed, the K-determinant contains three one-forms from each Hodge side. Thus the rational span W=Span_Q{u,v} is a Hodge substructure of degree six across the domain. The first-order computation confirms that persistence in a","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-159e93933b703ce6","tex":"\\Span_{\\R}","aliases":"Span_R Span R","display":null,"kind":"use","excerpt":"[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equali","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"Curve, Delsarte, and incidence channels. A curve certificate splits the six entries into triples a=Asqcup B, ∑ A=m, ∑ B=2m, B=uApmod m. The resulting (1,1) class on a product of Fermat curves is algebraic by Lefschetz (1,1), and the join correspondence returns a surface class.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"use","excerpt":"Conversely v=1 is invariant, because ∑ a_i=3m, and putting weight 1 on it is a feasible point of [source reference]. Thus C(a_q,b)=1, proving [source reference]. square","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"Matrix capacity · C(a)=sup{∑ w_v:∑ w_vv=1, w_v≥0}.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-78471ce175c11cc6","tex":"\\mathrm{supp}","aliases":"supp mathrm","display":"\\mathrm{supp}","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","excerpt":"[Range-packet contraction] Assume R ≤ M, B(a,b) ≤ C a b , 4CM^2 Qf <1. Then the map T(a)=-RQ(f+B(a,a)) has a unique fixed point in the gauge-fixed ball a ≤2M Qf . If Qf=0, the ball is the singleton {0} and the range solution is immediate. At the fixed point, the range component of F(a) vanishes. The remaining obstruction is Φ=Π(f+B(a,a)). The full ","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":31},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f544e1fddefb59a0","tex":"t^d","aliases":"t^d","display":"t^d","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-40baea273342ef2e","tex":"T^{d-1}","aliases":"T^d-1","display":"T^{d-1}","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7f9e80a5242f2d08","tex":"t^{d-1}","aliases":"t^d-1","display":"t^{d-1}","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6b150b0c28786d0a","tex":"T^{d-2}","aliases":"T^d-2","display":"T^{d-2}","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-60637919cef5af9b","tex":"t_0","aliases":"t_0","display":"t_0","kind":"use","excerpt":"and let e_j be the elementary symmetric polynomial of degree j in t_0,…,t_4. Define","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-afd0f5537bb82fff","tex":"t_4","aliases":"t_4","display":"t_4","kind":"use","excerpt":"and let e_j be the elementary symmetric polynomial of degree j in t_0,…,t_4. Define","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-de0bad5f8f33721c","tex":"t_i","aliases":"t_i","display":"t_i","kind":"use","excerpt":"z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-25bdedf7497647d3","tex":"t_j","aliases":"t_j","display":"t_j","kind":"use","excerpt":"Δ(t)=∏_0≤ i<j≤4(t_i-t_j).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b8e003e6dbf7e04c","tex":"T_m","aliases":"T_m","display":"T_m","kind":"use","excerpt":"[Exact SAT(2) criterion] Let n be a balanced six-packet at modulus m=2^s. Then n is pair-decomposable or has an SAT(2) certificate unless its gauged defect is ±1 on a four-vertex star K_1,3 in the internal tree T_m. A four-star has no SAT(2) certificate. Every nonstar shape in the preceding proposition splits into at most two four-packet atoms: ite","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6d6c4f072fd1f994","tex":"T_s","aliases":"T_s","display":"T_s","kind":"use","excerpt":"[Propagation of invertibility] If a relative transfer T_s is invertible on one fiber s_0, it is invertible on every fiber in the connected base on which the local systems and correspondences are defined. The determinant of the constant rational matrix representing the local-system morphism is locally constant. It is nonzero at s_0, hence everywhere","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1b8d3d054c7c7160","tex":"T_{\\mathrm{tot}}","aliases":"T_tot mathrm","display":"T_{\\mathrm{tot}}","kind":"use","excerpt":"Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4bf736770ec45576","tex":"\\mathrm{top}","aliases":"top mathrm","display":"\\mathrm{top}","kind":"use","excerpt":"Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees i","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-70a3fc170e8a1c43","tex":"\\mathrm{tot}","aliases":"tot mathrm","display":"\\mathrm{tot}","kind":"use","excerpt":"Construct algebraic lowering channels from families of subvarieties: A_line, A_conic, A_plane, A_exceptional. The exact target is ker A_line ∩ker A_conic ∩ker A_plane ∩ker A_exceptional={0} on primitive degree-four cohomology. Under algebraic adjoint hypotheses, equivalently prove T_tot =∑_rA_r^† A_r satisfies λ_min(T_tot)>0.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr","display":null,"kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"0 · 0 · 0) is nonzero, while N^3=0 and Tr(N^r)=0 for every positive integer r. This is the elementary negative control.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-17aada5f90193f08","tex":"\\operatorname{TV}","aliases":"TV operatorname","display":"\\operatorname{TV}","kind":"use","excerpt":"For an integer-valued y, discrete coarea gives TV(y) = ∑_k≥1|∂{y≥ k}| + ∑_k≥1|∂{y≤-k}|. If S is a finite forest of internal vertices of a trivalent tree and c(S) is its number of connected components, then |∂ S|=|S|+2c(S), because 3|S|=2(|S|-c(S))+|∂ S|.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d849fcfcf96ad168","tex":"u^3","aliases":"u^3","display":"u^3","kind":"use","excerpt":"F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fb501a979de5f234","tex":"U_{210}","aliases":"U_210","display":"U_{210}","kind":"use","excerpt":"q:U_210⟶ U_Y,","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e18e12d005f9eb86","tex":"U_{p^{r-j}}","aliases":"U_p^r-j","display":"U_{p^{r-j}}","kind":"use","excerpt":"Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonz","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9af86feeeef31f2d","tex":"U_{p^s}","aliases":"U_p^s","display":"U_{p^s}","kind":"use","excerpt":"r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at l","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aaefb9097bcb319b","tex":"U_{p^{s-1}}","aliases":"U_p^s-1","display":"U_{p^{s-1}}","kind":"use","excerpt":"r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonzero odd kernel-orbit defect has mass at l","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fffc74df039deddc","tex":"U_Y","aliases":"U_Y","display":"U_Y","kind":"use","excerpt":"q:U_210⟶ U_Y,","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7af6ec627892eb71","tex":"v^4","aliases":"v^4","display":"v^4","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-071bb3caadf70358","tex":"v^5","aliases":"v^5","display":"v^5","kind":"use","excerpt":"F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"[Star conductor] Let a four-star have center c, and write j=v_2(c). The gcd of m with its six packet entries is 2^j-1. Hence a star has full conductor exactly when v_2(c)=1. Use the boundary-edge labels of widetilde T_m. The two children of c have valuation j-1. The four boundary edges leaving those children have labels of exact valuation j-1: for ","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-046fdabe03f65bb7","tex":"v_4","aliases":"v_4","display":"v_4","kind":"use","excerpt":"v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-dce3dc9606f0a4f4","tex":"v_5","aliases":"v_5","display":"v_5","kind":"use","excerpt":"v_0+v_1+v_2+v_3+v_4-5v_5≡0pmod q.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"For searching, choose nonnegative integral columns v_j satisfying a· v_j≡0pmod m. Seek a nonsingular six-column matrix A with","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8b86c046f5cd21ee","tex":"V_m","aliases":"V_m","display":"V_m","kind":"use","excerpt":"[Integral dyadic triangle basis] For m=2^s, s≥3, the vectors {κ_b:1≤ b<q} form a Z-basis of K_m. Set r_0=q/2. On the vertex set V_m={1,2,…,q-1}, define, for b≠ r_0, p(b)= 2b, · b<r_0,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Matrix capacity · C(a)=sup{∑ w_v:∑ w_vv=1, w_v≥0}.","count":15},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-df6870ae538234a2","tex":"W_{[3a]}","aliases":"W_[3a]","display":"W_{[3a]}","kind":"use","excerpt":"Its pullback identifies W_[a] with W_[3a]. The identity","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5f3e053f9b1e0bc5","tex":"W_{[b]}","aliases":"W_[b]","display":"W_{[b]}","kind":"use","excerpt":"Assuming the inherited degree-70 carrier proposition, every rational class in W_[b] is algebraic.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-097cc918525aac69","tex":"W_{[ka]}","aliases":"W_[ka]","display":"W_{[ka]}","kind":"use","excerpt":"all classes in the 24-dimensional rational space W_[a] are rational linear combinations of classes of algebraic surfaces on X_70. For every positive integer k, the corresponding pulled-back packet W_[ka] on X_70k is also algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b2d270569641320f","tex":"W_{[\\alpha]}","aliases":"W_[α] alpha","display":"W_{[\\alpha]}","kind":"use","excerpt":"The resulting class and the prescribed class agree on U_210, their difference lies in the packet and restricts to zero. The preceding boundary isomorphism proves that difference is zero. This establishes algebraicity of the entire W_[α] packet on X_210.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-18864b29b043b6d8","tex":"W_i","aliases":"W_i","display":"W_i","kind":"use","excerpt":"D-∑ W_i=30-33=-3.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"∑_j w_j≤1.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e6e9e4d147adf393","tex":"w_v","aliases":"w_v","display":"w_v","kind":"use","excerpt":"Matrix capacity · C(a)=sup{∑ w_v:∑ w_vv=1, w_v≥0}.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-96f1e2f3049a6d75","tex":"\\widehat{dx_i}","aliases":"widehatdx_i widehat","display":"\\widehat{dx_i}","kind":"use","excerpt":"Ω=∑_i=0^5(-1)^ix_i dx_0wedge⋯widehatdx_i⋯wedge dx_5","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-47c52f415949a6cb","tex":"\\widetilde C","aliases":"widetilde C widetilde","display":"\\widetilde C","kind":"use","excerpt":"K_widetilde Y·widetilde C=K_Y· C=-3H· C<0.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9fcf9ea0bdd21c78","tex":"\\widetilde T","aliases":"widetilde T widetilde","display":"\\widetilde T","kind":"use","excerpt":"The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edge","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3f3f70eb2f987c29","tex":"\\widetilde T_m","aliases":"widetilde T_m widetilde","display":"\\widetilde T_m","kind":"use","excerpt":"The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edge","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-09868efb784faa54","tex":"\\mathcal X","aliases":"X mathcal","display":"\\mathcal X","kind":"use","excerpt":"Family propagation. Let π: X→ S and ρ_r: Y_r→ S be smooth projective families over a connected base. Relative algebraic correspondences induce morphisms of Gauss-Manin local systems. In a flat rational trivialization, their transfer matrix is constant.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-cd1f2d428e4995c6","tex":"x^{11}","aliases":"x^11","display":"x^{11}","kind":"use","excerpt":"The exceptional packet. A representative is a=(7,10,13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Modulo 3, the equation x^2+y^2=0 has only the solution (0,0). This is a finite anisotropy control for the residual discriminant. It does not construct a semistable algebraic family with this nilpotent as monodromy. An abstract rational nilpotent satisfying Lie-algebra identities is weaker than a geometric degeneration with the required integral, po","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-60b14e0cb3534318","tex":"x^q","aliases":"x^q","display":"x^q","kind":"use","excerpt":"de_1wedge⋯wedge de_4wedge dP =q^4Δ(x^q) dx_0wedge⋯wedge dx_4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"Ω=∑_i=0^5(-1)^ix_i dx_0wedge⋯widehatdx_i⋯wedge dx_5","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-12646fb9addf1892","tex":"x_0^{210}","aliases":"x_0^210","display":"x_0^{210}","kind":"use","excerpt":"F(y(x))=x_0^210+⋯+x_5^210.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b21ad544f9fa8aef","tex":"x_0^{33}","aliases":"x_0^33","display":"x_0^{33}","kind":"use","excerpt":"Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a98bf18826029f9a","tex":"x_0^{a_0-1}","aliases":"x_0^a_0-1","display":"x_0^{a_0-1}","kind":"use","excerpt":"The Griffiths residue basis of the Fermat hypersurface diagonalizes the action of G_m. The monomial x_0^a_0-1⋯ x_5^a_5-1 at pole order three gives the character of middle Hodge type when ∑ a_i=3m. The entire rational orbit has type (2,2) when","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-10e661a6ae22b48e","tex":"x_0^{b-1}","aliases":"x_0^b-1","display":"x_0^{b-1}","kind":"use","excerpt":"x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aed8b266613e1b6e","tex":"x_0^m","aliases":"x_0^m","display":"x_0^m","kind":"use","excerpt":"Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finit","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-545075d1edcbd776","tex":"x_0^{m-1}","aliases":"x_0^m-1","display":"x_0^{m-1}","kind":"use","excerpt":"J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ca0e182fe628a800","tex":"x_0^q","aliases":"x_0^q","display":"x_0^q","kind":"use","excerpt":"S_q={e_1(x_0^q,…,x_4^q)=e_2(x_0^q,…,x_4^q)=0}⊂ P^4.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-745879913b78243a","tex":"x_1^{26}","aliases":"x_1^26","display":"x_1^{26}","kind":"use","excerpt":"Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-31ea18c2103ffd95","tex":"x_1^{b+q-1}","aliases":"x_1^b+q-1","display":"x_1^{b+q-1}","kind":"use","excerpt":"x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e769e70331b9abf5","tex":"x_1^m","aliases":"x_1^m","display":"x_1^m","kind":"use","excerpt":"Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finit","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-56de97a1a978a551","tex":"X_{210}","aliases":"X_210","display":"X_{210}","kind":"use","excerpt":"where U_210=X_210∩{∏ x_i≠0} and U_Y=Y∩{∏ y_i≠0}.","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-16f4cdb2a921dd84","tex":"X_{210}^4","aliases":"X_210^4","display":"X_{210}^4","kind":"use","excerpt":"Pullback under X_210^4→ X_70^4 gives the algebraic source packet","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-637bf608bf425565","tex":"X_{210}^{52}","aliases":"X_210^52","display":"X_{210}^{52}","kind":"use","excerpt":"The right-hand side is algebraic by concatenation, because the source and every auxiliary component are algebraic. The left-hand auxiliary part Luplus(-L) is a multiset of exactly 24 complementary pairs. The full character has 54 entries and is a character on X_210^52. Complementary-pair removal therefore gives algebraicity of b on X_210^4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-746090484585c853","tex":"X_{210k}^4","aliases":"X_210k^4","display":"X_{210k}^4","kind":"use","excerpt":"For every k≥1, coordinate-power pullback gives the associated packet on X_210k^4. It does not establish the Hodge conjecture for all packets on any of these fourfolds.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-414972f36cf706af","tex":"x_2^{19}","aliases":"x_2^19","display":"x_2^{19}","kind":"use","excerpt":"Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c2eab61574fc6969","tex":"x_2^{b+2q-1}","aliases":"x_2^b+2q-1","display":"x_2^{b+2q-1}","kind":"use","excerpt":"x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4307b1591ccfe642","tex":"x_2^m","aliases":"x_2^m","display":"x_2^m","kind":"use","excerpt":"Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finit","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b7aac485582aab07","tex":"X_{33}^4","aliases":"X_33^4","display":"X_{33}^4","kind":"use","excerpt":"13,19,22,28), or after permutation, widetilde a=(7,19,28,10,13,22). Every unit conjugate has residue sum 99=3·33, so the orbit is rational of type (2,2). It is not totally pair-decomposable. The previously proposed symmetric surface W={p_1(x^11)=p_2(x^11)=p_3(x^11)=0} does not carry the primitive packet: Newton's identity makes the third equation r","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b7b3910d42523e24","tex":"X_{33k}^4","aliases":"X_33k^4","display":"X_{33k}^4","kind":"use","excerpt":"[Infinite packet tower] Let W be an algebraic rational character packet on X_d^2p. For every k≥1, its coordinate-power pullback is an algebraic packet on X_kd^2p. Applied to the preceding degree-33 construction under its geometric hypotheses, this gives a 20-dimensional algebraic packet on every X_33k^4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0be956f491f6b389","tex":"X_{35}","aliases":"X_35","display":"X_{35}","kind":"use","excerpt":"a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ea030620d1c53952","tex":"x_3^{12}","aliases":"x_3^12","display":"x_3^{12}","kind":"use","excerpt":"Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9f188897b4eefc55","tex":"x_3^{b+3q-1}","aliases":"x_3^b+3q-1","display":"x_3^{b+3q-1}","kind":"use","excerpt":"x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-26abf62156c13ea1","tex":"x_3^m","aliases":"x_3^m","display":"x_3^m","kind":"use","excerpt":"Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finit","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ca7008ae447ee9dc","tex":"x_4","aliases":"x_4","display":"x_4","kind":"use","excerpt":"z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-eaf7515bb99e2194","tex":"x_4^5","aliases":"x_4^5","display":"x_4^5","kind":"use","excerpt":"Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3e14d932ddea87c0","tex":"x_4^{b+4q-1}","aliases":"x_4^b+4q-1","display":"x_4^{b+4q-1}","kind":"use","excerpt":"x_0^b-1x_1^b+q-1x_2^b+2q-1x_3^b+3q-1x_4^b+4q-1 z^5(q-b)-1","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e2522597d87095ca","tex":"x_4^m","aliases":"x_4^m","display":"x_4^m","kind":"use","excerpt":"Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finit","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5f62f5549f26bcc0","tex":"x_4^q","aliases":"x_4^q","display":"x_4^q","kind":"use","excerpt":"S_q={e_1(x_0^q,…,x_4^q)=e_2(x_0^q,…,x_4^q)=0}⊂ P^4.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-98391915cbb7346b","tex":"x_5","aliases":"x_5","display":"x_5","kind":"use","excerpt":"Ω=∑_i=0^5(-1)^ix_i dx_0wedge⋯widehatdx_i⋯wedge dx_5","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-efeacaf1e2f698b9","tex":"x_5^{210}","aliases":"x_5^210","display":"x_5^{210}","kind":"use","excerpt":"F(y(x))=x_0^210+⋯+x_5^210.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ed008e28a794463c","tex":"x_5^4","aliases":"x_5^4","display":"x_5^4","kind":"use","excerpt":"Q_1=x_0^33x_1^26x_2^19x_3^12x_4^5x_5^4,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0d61f7653f873a9a","tex":"x_5^5","aliases":"x_5^5","display":"x_5^5","kind":"use","excerpt":"a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7cb5f66d477267eb","tex":"x_5^{a_5-1}","aliases":"x_5^a_5-1","display":"x_5^{a_5-1}","kind":"use","excerpt":"The Griffiths residue basis of the Fermat hypersurface diagonalizes the action of G_m. The monomial x_0^a_0-1⋯ x_5^a_5-1 at pole order three gives the character of middle Hodge type when ∑ a_i=3m. The entire rational orbit has type (2,2) when","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ffada69b2cc0a958","tex":"x_5^m","aliases":"x_5^m","display":"x_5^m","kind":"use","excerpt":"Correct scope and claim boundary. The projector-resolved Hodge calculus naturally specializes to Fermat fourfolds X_m^4={x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0}⊂ P^5, m≥ 3. The diagonal symmetry group makes the primitive middle cohomology unusually explicit: it splits into one-dimensional complex character lines, and rational Hodge packets are finit","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-26189f6905a9c1e3","tex":"x_5^{m-1}","aliases":"x_5^m-1","display":"x_5^{m-1}","kind":"use","excerpt":"J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f0d443e08e081a58","tex":"X_{5q}","aliases":"X_5q","display":"X_{5q}","kind":"use","excerpt":"Thus Z_κ⊂ X_5q. Its defining equations have degrees q,2q,5.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-02d953e843a19151","tex":"X_{70}","aliases":"X_70","display":"X_{70}","kind":"use","excerpt":"all classes in the 24-dimensional rational space W_[a] are rational linear combinations of classes of algebraic surfaces on X_70. For every positive integer k, the corresponding pulled-back packet W_[ka] on X_70k is also algebraic.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6dc4be4b9cfc6092","tex":"X_{70}^4","aliases":"X_70^4","display":"X_{70}^4","kind":"use","excerpt":"Pullback under X_210^4→ X_70^4 gives the algebraic source packet","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aade458e5ecadff6","tex":"X_{70k}","aliases":"X_70k","display":"X_{70k}","kind":"use","excerpt":"all classes in the 24-dimensional rational space W_[a] are rational linear combinations of classes of algebraic surfaces on X_70. For every positive integer k, the corresponding pulled-back packet W_[ka] on X_70k is also algebraic.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f5af80b6f7fd8fcb","tex":"X_d^{2p}","aliases":"X_d^2p","display":"X_d^{2p}","kind":"use","excerpt":"[Infinite packet tower] Let W be an algebraic rational character packet on X_d^2p. For every k≥1, its coordinate-power pullback is an algebraic packet on X_kd^2p. Applied to the preceding degree-33 construction under its geometric hypotheses, this gives a 20-dimensional algebraic packet on every X_33k^4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-720908399b25c2f0","tex":"x_i^{2q}","aliases":"x_i^2q","display":"x_i^{2q}","kind":"use","excerpt":"∑ x_i^q=0, ∑ x_i^2q=0.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e9d129f73a8284d6","tex":"x_i^3","aliases":"x_i^3","display":"x_i^3","kind":"use","excerpt":"π:X_210⟶ X_70, [x_i]⟼[x_i^3].","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0f79c54c5f16a5f8","tex":"x_i^7","aliases":"x_i^7","display":"x_i^7","kind":"use","excerpt":"a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1bb81b74924fb642","tex":"x_i^{m-2}","aliases":"x_i^m-2","display":"x_i^{m-2}","kind":"use","excerpt":"J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4bcc04d23d3978e9","tex":"x_i^{m-a_{q,b,i}-1}","aliases":"x_i^m-a_q,b,i-1","display":"x_i^{m-a_{q,b,i}-1}","kind":"use","excerpt":"Q_b=∏_i=0^5x_i^m-a_q,b,i-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-47e1e8b8784116f8","tex":"x_i^q","aliases":"x_i^q","display":"x_i^q","kind":"use","excerpt":"z=x_5, t_i=x_i^q (0≤ i≤4), P=x_0x_1x_2x_3x_4,","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7a179e33fc6415f8","tex":"x_j^7","aliases":"x_j^7","display":"x_j^7","kind":"use","excerpt":"a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7f311fd65b08595d","tex":"x_j^{B_{ij}}","aliases":"x_j^B_ij","display":"x_j^{B_{ij}}","kind":"use","excerpt":"On the projective tori define y_i=∏_jx_j^B_ij. Negative exponents are allowed there. The second identity makes this well-defined into the weighted projective torus, the first gives","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-58043e918d86d3ca","tex":"X_{kd}^{2p}","aliases":"X_kd^2p","display":"X_{kd}^{2p}","kind":"use","excerpt":"[Infinite packet tower] Let W be an algebraic rational character packet on X_d^2p. For every k≥1, its coordinate-power pullback is an algebraic packet on X_kd^2p. Applied to the preceding degree-33 construction under its geometric hypotheses, this gives a 20-dimensional algebraic packet on every X_33k^4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fbc03d10aef9ae9c","tex":"X_{km}^n","aliases":"X_km^n","display":"X_{km}^n","kind":"use","excerpt":"Power transport. The finite coordinate-power morphism X_km^n→ X_m^n transports algebraic cycle classes by pullback. The character is multiplied by k.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-92d6f58eaee1edca","tex":"X_m^n","aliases":"X_m^n","display":"X_m^n","kind":"use","excerpt":"X_m^n={x_0^m+⋯+x_n+1^m=0}⊂ P^n+1_ C.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b97ab2ec02a73545","tex":"x_{n+1}^m","aliases":"x_n+1^m","display":"x_{n+1}^m","kind":"use","excerpt":"X_m^n={x_0^m+⋯+x_n+1^m=0}⊂ P^n+1_ C.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"The rational Hodge target. Statement and status. Let X be a smooth projective complex variety. For each codimension p, define Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X) and Alg^p(X)=im(CH^p(X)_QxrightarrowclH^2p(X,Q)). Every algebraic cycle class has type (p,p), so Alg^p(X)⊆Hdg^p(X). The rational Hodge conjecture asserts equality for every smooth projective X an","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.","count":31},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bef13251af5aee62","tex":"\\mathcal Y","aliases":"Y mathcal","display":"\\mathcal Y","kind":"use","excerpt":"Family propagation. Let π: X→ S and ρ_r: Y_r→ S be smooth projective families over a connected base. Relative algebraic correspondences induce morphisms of Gauss-Manin local systems. In a flat rational trivialization, their transfer matrix is constant.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"Modulo 3, the equation x^2+y^2=0 has only the solution (0,0). This is a finite anisotropy control for the residual discriminant. It does not construct a semistable algebraic family with this nilpotent as monodromy. An abstract rational nilpotent satisfying Lie-algebra identities is weaker than a geometric degeneration with the required integral, po","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-651f5279d84e1d41","tex":"y_0^2","aliases":"y_0^2","display":"y_0^2","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e5975faa5ee7b1b7","tex":"y_0^3","aliases":"y_0^3","display":"y_0^3","kind":"use","excerpt":"F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7 +y_0y_2^2y_3^2y_5+y_3^2y_5^3.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1017c03c50dd5db9","tex":"y_1^2","aliases":"y_1^2","display":"y_1^2","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-832bc8161d5bccd4","tex":"y_1^4","aliases":"y_1^4","display":"y_1^4","kind":"use","excerpt":"y_0^3, y_1^4, (y_2y_5)^3, (y_3y_4)^13, (y_3y_5)^3","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5f6d8eb43696e508","tex":"y_1^5","aliases":"y_1^5","display":"y_1^5","kind":"use","excerpt":"F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7 +y_0y_2^2y_3^2y_5+y_3^2y_5^3.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fbcfb877b670257d","tex":"y_2^2","aliases":"y_2^2","display":"y_2^2","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5171b856d1185d9c","tex":"y_3","aliases":"y_3","display":"y_3","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-65c28e39bd9b89fc","tex":"y_3^2","aliases":"y_3^2","display":"y_3^2","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-10f0012bfa3107dd","tex":"y_4","aliases":"y_4","display":"y_4","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-986c753840e1cd5a","tex":"y_4^2","aliases":"y_4^2","display":"y_4^2","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3d632cee58a25e44","tex":"y_4^7","aliases":"y_4^7","display":"y_4^7","kind":"use","excerpt":"F=y_0^3+y_1^5+y_2^2y_5^3+y_2y_3^2y_4^7 +y_0y_2^2y_3^2y_5+y_3^2y_5^3.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8d1ab3b8735ada37","tex":"y_5^3","aliases":"y_5^3","display":"y_5^3","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d65dae5c5b4349f7","tex":"y_b","aliases":"y_b","display":"y_b","kind":"use","excerpt":"The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edge","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"The infinitely many inequalities can be certified by a finite residue graph. Write y_i=c_i/L, where c_i,L are positive integers. On Z/m Z, an edge r→ r+a_i has cost c_i. An integer function d satisfying","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-706da2e0e01f2a53","tex":"y_{p(b)}","aliases":"y_p(b)","display":"y_{p(b)}","kind":"use","excerpt":"[Exact trivalent-tree norm] If d=∑_bz_bκ_b∈ K_m, then d _1 =|y_r_0| +∑_b≠ r_0|y_b-y_p(b)| +2∑_substack1≤ b<q b odd|y_b|. The right side is the total variation of the integer function y on widetilde T_m with zero boundary values. The root coordinate is d(q)=z_r_0. For b≠ r_0, multiplying d(2b)=z_b-ϵ_bz_p(b) by τ_b gives τ_bd(2b)=y_b-y_p(b). For odd ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-380ccbb2527a8553","tex":"y_q","aliases":"y_q","display":"y_q","kind":"use","excerpt":"y_q=frac1q(1,1,1,1,1,q-5), y_q·1=1.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-72c43d344dc79844","tex":"Y_r","aliases":"Y_r","display":"Y_r","kind":"use","excerpt":"Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic corre","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-556eeebfdd6f2f93","tex":"\\mathcal Y_r","aliases":"Y_r mathcal","display":"\\mathcal Y_r","kind":"use","excerpt":"Family propagation. Let π: X→ S and ρ_r: Y_r→ S be smooth projective families over a connected base. Relative algebraic correspondences induce morphisms of Gauss-Manin local systems. In a flat rational trivialization, their transfer matrix is constant.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f68c512c5feb69ba","tex":"y_{r_0}","aliases":"y_r_0","display":"y_{r_0}","kind":"use","excerpt":"[Exact trivalent-tree norm] If d=∑_bz_bκ_b∈ K_m, then d _1 =|y_r_0| +∑_b≠ r_0|y_b-y_p(b)| +2∑_substack1≤ b<q b odd|y_b|. The right side is the total variation of the integer function y on widetilde T_m with zero boundary values. The root coordinate is d(q)=z_r_0. For b≠ r_0, multiplying d(2b)=z_b-ϵ_bz_p(b) by τ_b gives τ_bd(2b)=y_b-y_p(b). For odd ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"use","excerpt":"The lattice, polarization and seed. Let K=Q(i) and take homology lattice Z[i]^6. Use the conjugate-linear-first Hermitian form h(v,w)=overline v^ tdiag(1,1,1,-1,-1,-3)w, E=Im h. The seed complex structure is J_0=diag(i,i,i,-i,-i,-i). It preserves the alternating form E, and E(v,J_0v)>0 for nonzero v. These are the Riemann positivity conditions for ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.","count":36},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"F_u · =3u^2+s^2t^2z, · F_v · =5v^4, F_s · =2sz^3+t^2c^7+2ust^2z, · F_t · =2t(z^3+sc^7+us^2z), F_c · =7st^2c^6, · F_z · =3z^2(s^2+t^2)+us^2t^2.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4140de75758a272d","tex":"z^3","aliases":"z^3","display":"z^3","kind":"use","excerpt":"F=u^3+v^5+z^3(s^2+t^2)+st^2c^7+us^2t^2z.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-82f6433a3ae799be","tex":"z^5","aliases":"z^5","display":"z^5","kind":"use","excerpt":"Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-177cd83dcf3d0e1b","tex":"z^{5(q-b)-1}","aliases":"z^5(q-b)-1","display":"z^{5(q-b)-1}","kind":"use","excerpt":"D_κ=625q^5Δ(x_0^q,…,x_4^q) ∑_b=1^q-1κ^q-bP^b-1z^5(q-b)-1.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-df716cd296a880d1","tex":"z^{5j}","aliases":"z^5j","display":"z^{5j}","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d0e6241706010548","tex":"z^{5j-1}","aliases":"z^5j-1","display":"z^{5j-1}","kind":"use","excerpt":"df_3wedge dg_3= 25q∑_j=1^q-1κ^jP^q-1-jz^5j-1 dPwedge dz.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-054e758963b11219","tex":"z^{5q}","aliases":"z^5q","display":"z^{5q}","kind":"use","excerpt":"F_m=p_5+z^5q=5P^q+z^5q =(5κ^q+1)z^5q=0.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-54f7d62b2e9a09e0","tex":"\\mathbf Z^6","aliases":"Z^6 mathbf","display":"\\mathbf Z^6","kind":"use","excerpt":"Every nonsingular nonnegative integral matrix A satisfying A^Ta∈ m Z^6, with w=A^-11>0, gives one feasible point. Hence","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2149eb8ad46cf826","tex":"\\mathbf Z^{h-1}","aliases":"Z^h-1 mathbf","display":"\\mathbf Z^{h-1}","kind":"use","excerpt":"The integral triangle basis. Let K_m= { d∈ Z^h-1: ∑_r=1^h-1d(r)ρ_m(ur)=0 for every odd u }. The triangle relations do not merely generate examples, they form the whole balanced defect lattice.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b23ccb12e2adf65f","tex":"Z_1","aliases":"Z_1","display":"Z_1","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety g","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1353c50b670eb223","tex":"\\mathbf Z_2","aliases":"Z_2 mathbf","display":"\\mathbf Z_2","kind":"use","excerpt":"Weighted dyadic reconstruction. Prime 2 cannot be handled by inserting p=2 into the odd-prime orbit-mass proof. The unit group has the separate form Z_2^×≅{±1}×(1+4 Z_2). The first address is a sign packet, only afterward does the principal-unit refinement begin.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5e32645afb213c74","tex":"z_2","aliases":"z_2","display":"z_2","kind":"use","excerpt":"The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety g","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-65eeefe3eea86f19","tex":"\\mathbf Z_2^\\times","aliases":"Z_2^× mathbf times","display":"\\mathbf Z_2^\\times","kind":"use","excerpt":"Weighted dyadic reconstruction. Prime 2 cannot be handled by inserting p=2 into the odd-prime orbit-mass proof. The unit group has the separate form Z_2^×≅{±1}×(1+4 Z_2). The first address is a sign packet, only afterward does the principal-unit refinement begin.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0702dc64afa5c9d8","tex":"z_3","aliases":"z_3","display":"z_3","kind":"use","excerpt":"The seed graph supplies both rational Weil classes. At the seed, take the graph of (z_1,z_2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety g","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-cdd36f9a578aa62d","tex":"z_b","aliases":"z_b","display":"z_b","kind":"use","excerpt":"Suppose d=∑_b=1^q-1z_bκ_b. Its even coordinates are d(q)=z_r_0, d(2b)=z_b-ϵ_bz_p(b) (b≠ r_0). Starting at the root and moving toward the leaves, these equations recover every z_b integrally. Hence the triangle vectors are linearly independent and their even-coordinate map is unimodular.","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bc1f10031646b5b4","tex":"Z_i","aliases":"Z_i","display":"Z_i","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-10845de283abdc8a","tex":"Z_j","aliases":"Z_j","display":"Z_j","kind":"use","excerpt":"Degree-residue polyphase refinement. Exact parent-child frame law. Fix a projective embedding and assign each cycle its degree. For modulus M, define the residue packet I_M,r={j:deg Z_j≡ rpmod M}, S_M,r=∑_j∈ I_M,r|c_j⟩⟨ c_j|.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8d4486ed2f67d2ed","tex":"z_{p(b)}","aliases":"z_p(b)","display":"z_{p(b)}","kind":"use","excerpt":"Suppose d=∑_b=1^q-1z_bκ_b. Its even coordinates are d(q)=z_r_0, d(2b)=z_b-ϵ_bz_p(b) (b≠ r_0). Starting at the root and moving toward the leaves, these equations recover every z_b integrally. Hence the triangle vectors are linearly independent and their even-coordinate map is unimodular.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5e940a55f95e2b85","tex":"Z_r","aliases":"Z_r","display":"Z_r","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8fe3078aea5388e0","tex":"z_{r_0}","aliases":"z_r_0","display":"z_{r_0}","kind":"use","excerpt":"Suppose d=∑_b=1^q-1z_bκ_b. Its even coordinates are d(q)=z_r_0, d(2b)=z_b-ϵ_bz_p(b) (b≠ r_0). Starting at the root and moving toward the leaves, these equations recover every z_b integrally. Hence the triangle vectors are linearly independent and their even-coordinate map is unimodular.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5d84c236d1e165ac","tex":"Z_W","aliases":"Z_W","display":"Z_W","kind":"use","excerpt":"The constructive objective is [Packet-resolved construction target] For every rational packet W⊂ H^4_prim(X_m^4,Q)∩ H^2,2, construct Z_W∈CH^2(X_m^4)_Q with [Z_W]≠0 in W. Once one nonzero algebraic vector is produced in a rational irreducible diagonal-symmetry packet, the entire rational orbit is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-335310045e6e3444","tex":"\\mathbf Z_{\\ge0}^6","aliases":"Z_≥0^6 mathbf ge","display":"\\mathbf Z_{\\ge0}^6","kind":"use","excerpt":"L_a={v∈ Z_≥0^6∖{0}:a· v≡0pmod m}.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"tudy. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors are algebraic. Define C:R^N→ H^2p(X,R), Ce_j=c_j, and the","count":20},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"dots,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equality im(CC^*)=im C is standard. Moreover ⟨ v,Sv⟩= C^*v ^2=","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-343b02c221c97d88","tex":"|_K","aliases":"|_K","display":"|_K","kind":"use","excerpt":"Let K be the kernel of the torus isogeny. The relation AB=210I shows that K embeds in G_210: an element in the kernel has representatives whose 210th powers are all the same, and a common scalar makes them roots of unity. The displayed character identity proves χ_α|_K=1. The same holds for every rational conjugate. Therefore the whole rational pack","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7fd82fd26947ba96","tex":"|_V","aliases":"|_V","display":"|_V","kind":"use","excerpt":"[Finite positive cycle-frame certificate] Let A=Span_R{c_1,…,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equali","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"Weighted dyadic reconstruction. Prime 2 cannot be handled by inserting p=2 into the odd-prime orbit-mass proof. The unit group has the separate form Z_2^×≅{±1}×(1+4 Z_2). The first address is a sign packet, only afterward does the principal-unit refinement begin.","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"Global dual · y· v≥1 for every invariant column implies C(a)≤ y·1.","count":25},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"Support test · j^*α=0. Purity turns it into cycle algebraicity when the support has codimension at least p.","count":37},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7a040a96d127c0c6","tex":"\\alpha_{5^r}","aliases":"α_5^r alpha","display":"\\alpha_{5^r}","kind":"use","excerpt":"The five-primary branch. For p=5 and r≥2, a nonzero odd defect consumes exactly five unit entries. The base case r=1 was already handled by the base-prime Fourier argument. The sixth entry is forced by balance. Up to the unit action there is one full-conductor exceptional orbit α_5^r= (1,1+q,1+2q,1+3q,1+4q,5q-5), q=5^r-1. All other packets lie in t","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a65603881d2ba1ba","tex":"\\alpha_{\\mathrm{prim}}","aliases":"α_prim alpha mathrm","display":"\\alpha_{\\mathrm{prim}}","kind":"use","excerpt":"For a fourfold and p=2, α=Lβ+α_prim, β∈ H^1,1(X)∩ H^2(X,Q). The Lefschetz (1,1) theorem makes β algebraic. Only α_prim remains.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-af15262773600a7f","tex":"\\alpha_r","aliases":"α_r alpha","display":"\\alpha_r","kind":"use","excerpt":"[Primitive reduction] If every primitive rational Hodge class is algebraic, then every rational Hodge class is algebraic. Write α=∑_rL^rα_r with each α_r primitive and of Hodge type. The class h is algebraic. If α_r is algebraic, then L^rα_r=h^rsmileα_r is algebraic. Summing gives an algebraic representative of α.","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"For a fourfold and p=2, α=Lβ+α_prim, β∈ H^1,1(X)∩ H^2(X,Q). The Lefschetz (1,1) theorem makes β algebraic. Only α_prim remains.","count":10},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.","count":12},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-3350dae1255c748d","tex":"\\Gamma_*","aliases":"Γ_* Gamma","display":"\\Gamma_*","kind":"use","excerpt":"Correspondence degrees. Let X have dimension n and Y dimension m. A correspondence Γ∈CH^r(X× Y) acts by Γ_*:H^i(X)⟶ H^i+2(r-n)(Y) and shifts Hodge type by (r-n,r-n). To lower H^2p(X) to H^2(Y), take r=n-p+1. To return degree two to degree 2p on X, a correspondence on Y× X has codimension m+p-1.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d2e737aa54abf669","tex":"\\gamma_1","aliases":"γ_1 gamma","display":"\\gamma_1","kind":"use","excerpt":"oppositely oriented boundary edge, hence exactly one complementary pair in the union of the two atom packets, this is the SAT(2) stabilizer. In the three-vertex case, the vertex atom also supplies the unique neutral h. For two vertex atoms, their two neutral entries are (h,h)=p_h. The zero-defect case is already a union of three complementary pairs","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2f08cea1939265ee","tex":"\\gamma_2","aliases":"γ_2 gamma","display":"\\gamma_2","kind":"use","excerpt":"oppositely oriented boundary edge, hence exactly one complementary pair in the union of the two atom packets, this is the SAT(2) stabilizer. In the three-vertex case, the vertex atom also supplies the unique neutral h. For two vertex atoms, their two neutral entries are (h,h)=p_h. The zero-defect case is already a union of three complementary pairs","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-27ca25276af08346","tex":"\\Gamma_b","aliases":"Γ_b Gamma","display":"\\Gamma_b","kind":"use","excerpt":"The three-primary branch. For p=3 and r≥2, a higher-level kernel orbit has three elements. Define O_b={b,b+q,b+2q}, q=3^r-1, and Γ_b=O_bsqcup{3(q-b)}. Take 1≤ b<q with 3nmid b, ensuring all entries are nonzero. Then Γ_b is a balanced four-packet. A case analysis on the number of oriented kernel orbits gives exact identities expressing every full-co","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c8e941daf5e0be3b","tex":"\\Gamma_{b,c,d}","aliases":"Γ_b,c,d Gamma","display":"\\Gamma_{b,c,d}","kind":"use","excerpt":"K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ae2e30306bc89e00","tex":"\\gamma_i","aliases":"γ_i gamma","display":"\\gamma_i","kind":"use","excerpt":"Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-530f5d7a3b27e60f","tex":"\\gamma_k","aliases":"γ_k gamma","display":"\\gamma_k","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-627add72621f5ca0","tex":"\\Gamma_r","aliases":"Γ_r Gamma","display":"\\Gamma_r","kind":"use","excerpt":"Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic corre","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-788db25068d804f5","tex":"\\gamma_r","aliases":"γ_r gamma","display":"\\gamma_r","kind":"use","excerpt":"Dual charges. An antisymmetric charge is a rational function w on residues satisfying w(m-r)=-w(r) and ∑_rγ_r w(r)=0 ∀ γ∈ F_m. Such a charge vanishes on every complementary pair and on every saturated identity. Writing w(n)=∑_r n_r w(r), if w(n)≠0, the entire lower-packet saturation semigroup is excluded. This is a universal obstruction to a gramma","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).","count":9},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7d8ed9aa0e00cd9b","tex":"\\delta_{ce}","aliases":"δ_ce delta","display":"\\delta_{ce}","kind":"use","excerpt":"0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is it","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e76cde478bccee9e","tex":"\\delta_j","aliases":"δ_j delta","display":"\\delta_j","kind":"use","excerpt":"Valuation strata and orbit mass. Let m=p^r with p odd. Decompose the entries by exact p-adic valuation. At one stratum, write a=p^jb, with b∈ U_p^r-j, and define the odd count δ_j(b)=n_j(b)-n_j(-b). Since the packet has six entries, δ_j _1≤ 12. For s≥2, primitive odd characters force δ_j to be constant on the kernel cosets of U_p^s→ U_p^s-1. A nonz","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-41af0b6beae368c8","tex":"\\delta_{pq}","aliases":"δ_pq delta","display":"\\delta_{pq}","kind":"use","excerpt":"q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d33d8b646241d747","tex":"\\Delta_\\ell","aliases":"Δ_ℓ Delta ell","display":"\\Delta_\\ell","kind":"use","excerpt":"Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"[Integral dyadic triangle basis] For m=2^s, s≥3, the vectors {κ_b:1≤ b<q} form a Z-basis of K_m. Set r_0=q/2. On the vertex set V_m={1,2,…,q-1}, define, for b≠ r_0, p(b)= 2b, · b<r_0, 2(q-b), · b>r_0, ϵ_b= +1, · b<r_0, -1, · b>r_0. The map p raises the two-adic valuation by one, so repeated application reaches r_0. It therefore defines a rooted tre","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e7a0a367bfa7b00d","tex":"\\epsilon_b","aliases":"ϵ_b epsilon","display":"\\epsilon_b","kind":"use","excerpt":"[Integral dyadic triangle basis] For m=2^s, s≥3, the vectors {κ_b:1≤ b<q} form a Z-basis of K_m. Set r_0=q/2. On the vertex set V_m={1,2,…,q-1}, define, for b≠ r_0, p(b)= 2b, · b<r_0, 2(q-b), · b>r_0, ϵ_b= +1, · b<r_0, -1, · b>r_0. The map p raises the two-adic valuation by one, so repeated application reaches r_0. It therefore defines a rooted tre","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f362658753c0c7bf","tex":"\\zeta_q","aliases":"ζ_q zeta","display":"\\zeta_q","kind":"use","excerpt":"Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b8be676ce7f93516","tex":"\\zeta_q^{-(c+d+e)}","aliases":"ζ_q^-(c+d+e) zeta","display":"\\zeta_q^{-(c+d+e)}","kind":"use","excerpt":"[Pure imaginary selector] overlineΔ_ℓ=-Δ_ℓ. Since 1-ζ_q^-a=-ζ_q^-a(1-ζ_q^a), complex conjugation multiplies the product by (-1)^3ζ_q^-(c+d+e)=-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d73a11aca2cfdb23","tex":"\\zeta_q^{-a}","aliases":"ζ_q^-a zeta","display":"\\zeta_q^{-a}","kind":"use","excerpt":"[Pure imaginary selector] overlineΔ_ℓ=-Δ_ℓ. Since 1-ζ_q^-a=-ζ_q^-a(1-ζ_q^a), complex conjugation multiplies the product by (-1)^3ζ_q^-(c+d+e)=-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7b3bf17dd818a6cf","tex":"\\zeta_q^a","aliases":"ζ_q^a zeta","display":"\\zeta_q^a","kind":"use","excerpt":"[Pure imaginary selector] overlineΔ_ℓ=-Δ_ℓ. Since 1-ζ_q^-a=-ζ_q^-a(1-ζ_q^a), complex conjugation multiplies the product by (-1)^3ζ_q^-(c+d+e)=-1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-87b0f0f4f2c7035d","tex":"\\zeta_q^c","aliases":"ζ_q^c zeta","display":"\\zeta_q^c","kind":"use","excerpt":"Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9088429a079bfcb9","tex":"\\zeta_q^d","aliases":"ζ_q^d zeta","display":"\\zeta_q^d","kind":"use","excerpt":"Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-62f535593a4c352c","tex":"\\zeta_q^e","aliases":"ζ_q^e zeta","display":"\\zeta_q^e","kind":"use","excerpt":"Imaginary CM selector. Write the lower curve packet as ℓ=(c,d,e)pmod q, c+d+e≡0pmod q. Let K=Q(ζ_q) and define Δ_ℓ=(1-ζ_q^c)(1-ζ_q^d)(1-ζ_q^e).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5e83860784f63efd","tex":"\\zeta_q^u","aliases":"ζ_q^u zeta","display":"\\zeta_q^u","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"η=frac1420(31,87,57,101,84,60).","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f3296a0bedcafc07","tex":"\\eta^T","aliases":"η^T eta","display":"\\eta^T","kind":"use","excerpt":"1=η^T1=η^TAw =∑_jw_j(η· v_j)≥∑_jw_j.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8b5366310db92dcd","tex":"\\eta_q","aliases":"η_q eta","display":"\\eta_q","kind":"use","excerpt":"η_q=frac1q(1,1,1,1,1,q-5).","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). Th","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"The intersection form and primitive directions. Use the orientation ∫ x_1y_1⋯ x_6y_6=-1 for this seed. If θ is the polarization class specified by E, the exact Gram matrix on (θ^3,u,v) is 2160 · 0 · 0","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c1ce6d2d2665df29","tex":"\\theta^3","aliases":"θ^3 theta","display":"\\theta^3","kind":"use","excerpt":"The intersection form and primitive directions. Use the orientation ∫ x_1y_1⋯ x_6y_6=-1 for this seed. If θ is the polarization class specified by E, the exact Gram matrix on (θ^3,u,v) is 2160 · 0 · 0","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2b1cade6ef19c925","tex":"\\Theta_{m,b}","aliases":"Θ_m,b Theta","display":"\\Theta_{m,b}","kind":"use","excerpt":"The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). Th","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-74d5f0697e6091af","tex":"\\kappa^6","aliases":"κ^6 kappa","display":"\\kappa^6","kind":"use","excerpt":"10,504,375 κ^6.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1d2e603999a90b70","tex":"\\kappa^7","aliases":"κ^7 kappa","display":"\\kappa^7","kind":"use","excerpt":"Take q=7, b=1, κ^7=-1/5. Then","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9633f47d250ed74c","tex":"\\kappa^j","aliases":"κ^j kappa","display":"\\kappa^j","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e228f4fa507765bd","tex":"\\kappa^q","aliases":"κ^q kappa","display":"\\kappa^q","kind":"use","excerpt":"Five-phase surface · e_1=e_2=P-κ z^5=0, with κ^q=-1/5.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5f691d6c30dfc878","tex":"\\kappa^{q-b}","aliases":"κ^q-b kappa","display":"\\kappa^{q-b}","kind":"use","excerpt":"D_κ=625q^5Δ(x_0^q,…,x_4^q) ∑_b=1^q-1κ^q-bP^b-1z^5(q-b)-1.","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-62f4ed922bb2ce09","tex":"\\kappa_1","aliases":"κ_1 kappa","display":"\\kappa_1","kind":"use","excerpt":"· +(4,m/2,m/2+4,m-8). The first and third packets on the right have defects -κ_1 and -κ_4. The middle packet has defect -κ_2+κ_q-1, an edge atom. Hence all three are balanced four-packets, and [source reference] is an identity of multisets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-93055c50da4a404a","tex":"\\kappa_2","aliases":"κ_2 kappa","display":"\\kappa_2","kind":"use","excerpt":"· +(4,m/2,m/2+4,m-8). The first and third packets on the right have defects -κ_1 and -κ_4. The middle packet has defect -κ_2+κ_q-1, an edge atom. Hence all three are balanced four-packets, and [source reference] is an identity of multisets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f7939e8e26e876f1","tex":"\\kappa_4","aliases":"κ_4 kappa","display":"\\kappa_4","kind":"use","excerpt":"· +(4,m/2,m/2+4,m-8). The first and third packets on the right have defects -κ_1 and -κ_4. The middle packet has defect -κ_2+κ_q-1, an edge atom. Hence all three are balanced four-packets, and [source reference] is an identity of multisets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-05b36e93be2c77ad","tex":"\\kappa_b","aliases":"κ_b kappa","display":"\\kappa_b","kind":"use","excerpt":"For 1≤ b<q, define the triangle defect κ_b=[2b]-[b]+[h-b]. It is realized by the four-packet Γ_b=(2b,h-b,h,m-b).","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2dd0427a964664ef","tex":"\\kappa_{q-1}","aliases":"κ_q-1 kappa","display":"\\kappa_{q-1}","kind":"use","excerpt":"· +(4,m/2,m/2+4,m-8). The first and third packets on the right have defects -κ_1 and -κ_4. The middle packet has defect -κ_2+κ_q-1, an edge atom. Hence all three are balanced four-packets, and [source reference] is an identity of multisets.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic corre","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.","count":18},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-975e1a0c6d000741","tex":"\\lambda_2","aliases":"λ_2 lambda","display":"\\lambda_2","kind":"use","excerpt":"K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-29ccb409dc80afd9","tex":"\\lambda_3","aliases":"λ_3 lambda","display":"\\lambda_3","kind":"use","excerpt":"K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d2fb02f10a3142a1","tex":"\\lambda_4","aliases":"λ_4 lambda","display":"\\lambda_4","kind":"use","excerpt":"K4 four-phase reduction. Choose nonzero residues b,c,dpmod q and put e≡-5^-1bpmod q. Define A_b,c,d= (λ_1(b),λ_2(b),λ_3(b),λ_4(b),5c,5d) and Γ_b,c,d=(b,c,d,e).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1dc8e3b2125ff219","tex":"\\lambda_{\\min}","aliases":"λ_min lambda min","display":"\\lambda_{\\min}","kind":"use","excerpt":"If H is a matrix whose columns form an orthonormal basis of V, the finite Gram certificate is G=H^*SH. Under A⊆ V, we have A=V if and only if λ_min(G)>0. Without that containment, use a rational basis matrix H_Q for the target and the exact certificate rank_Q[C H_Q]=rank_QC to test V⊆ A.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b87db7c608ff8298","tex":"\\Lambda_r","aliases":"Λ_r Lambda","display":"\\Lambda_r","kind":"use","excerpt":"Divisor-lowering correspondence channels. Degree bookkeeping. Let X have complex dimension n, let V⊆ H_prim^2p(X,Q) be a rational Hodge substructure, and let Y_r be smooth projective. Choose correspondences Γ_r∈CH^n-p+1(X× Y_r)_Q, Λ_r∈CH^dim Y_r+p-1(Y_r× X)_Q. They induce A_r=(Γ_r)_*:V→ H^2(Y_r,Q), B_r=(Λ_r)_*:H^2(Y_r,Q)→ H^2p(X,Q). Algebraic corre","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-36442f76e90ff174","tex":"\\lambda_r","aliases":"λ_r lambda","display":"\\lambda_r","kind":"use","excerpt":"Local mod-5 phase grammar. Assume m=5q and first suppose (5,q)=1. For bpmod q and a phase r∈{0,1,2,3,4}, let λ_r(b) be the CRT lift satisfying λ_r(b)≡ rpmod5, λ_r(b)≡ bpmod q. The finite audit reveals four canonical multiplicity profiles, up to multiplication by a unit modulo five: (1,0,0,0,5), (2,1,1,1,1), (1,0,2,1,2), (1,0,1,3,1). The first is th","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5aa4407291a99a61","tex":"\\lambda_\\chi","aliases":"λ_χ lambda chi","display":"\\lambda_\\chi","kind":"use","excerpt":"ecover the unique integers z_b from its even coordinates, and subtract ∑_bz_bκ_b. The residual balanced vector is supported only on the odd residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, wi","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2dfde3f6584849e0","tex":"\\lambda_{\\chi,2^s}","aliases":"λ_χ,2^s lambda chi","display":"\\lambda_{\\chi,2^s}","kind":"use","excerpt":"ing square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,2^s=2^s-tλ_χ^*,2^t. T","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2340ff6c5200e1ec","tex":"\\lambda_{\\chi^*,2^t}","aliases":"λ_χ^*,2^t lambda chi","display":"\\lambda_{\\chi^*,2^t}","kind":"use","excerpt":"ing square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t, then λ_χ,2^s=2^s-tλ_χ^*,2^t. T","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fb80f13cf36184db","tex":"\\mu_{210}","aliases":"μ_210 mu","display":"\\mu_{210}","kind":"use","excerpt":"The coordinate boundary of X_210 is a simple normal crossing divisor. Every nonempty coordinate stratum is fixed pointwise by at least one coordinate scaling subgroup μ_210. Our full-support character is nontrivial on that subgroup. The same is true for every character in its rational orbit.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aca12ae2446c87d2","tex":"\\mu_3","aliases":"μ_3 mu","display":"\\mu_3","kind":"use","excerpt":"The ambient weighted projective space is well formed. Its largest singular stratum is the μ_3-stratum y_0=y_5=0, of dimension three. The restriction of F there is y_1^5+y_2y_3^2y_4^7, not identically zero. All other ambient singular strata have dimension at most two. Thus Y meets the ambient singular locus in codimension at least two. The verifier ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c7bf774264b7bb2e","tex":"\\nu_{\\mathrm{Frob}}","aliases":"ν_Frob nu mathrm","display":"\\nu_{\\mathrm{Frob}}","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8950750b97ec9a17","tex":"\\xi_\\kappa","aliases":"ξ_κ xi kappa","display":"\\xi_\\kappa","kind":"use","excerpt":"ξ_κ=[Z_κ]-2q h^2","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"Nonlinear Dolbeault obstruction packets. Let E be a smooth complex bundle and consider a candidate (0,1)-connection. After gauge fixing, write the integrability equation abstractly as F(a)=f+La+B(a,a)=0, where L is the linearized operator and B is a bounded bilinear term. Assume a Fredholm splitting of the target I=Q+Π, where Q projects onto the cl","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-37cd1251b390f507","tex":"\\pi^*","aliases":"π^* pi","display":"\\pi^*","kind":"use","excerpt":"π_*π^*=(degπ)id","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6d2412f84c728949","tex":"\\pi_*","aliases":"π_* pi","display":"\\pi_*","kind":"use","excerpt":"π_*π^*=(degπ)id","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-650ca9a1a0c7e0cd","tex":"\\Pi_{A,k}","aliases":"Π_A,k Pi","display":"\\Pi_{A,k}","kind":"use","excerpt":"A positive spectral microscope. Large ample twists can make integrability defects visible in the asymptotic degree distribution of Dolbeault-Dirac kernels. Charbonneau and Stern obtain asymptotic Hodge filtrations in which positive coefficients depend on powers of F_A^0,2 [source reference]. Schematically, Tr(P_2jΠ_A,k) =c_n,jk^n-2j (F_A^0,2)^wedge","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-20bc6f34b104939c","tex":"\\pi_{m/d*}","aliases":"π_m/d* pi","display":"\\pi_{m/d*}","kind":"use","excerpt":"For dmid m, let E_m,d=frac1(m/d)^5π_m/d^*π_m/d*. On a character of exact conductor c, E_m,d=1, · cmid d,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-852bab02898067d7","tex":"\\pi_{m/d}^*","aliases":"π_m/d^* pi","display":"\\pi_{m/d}^*","kind":"use","excerpt":"For dmid m, let E_m,d=frac1(m/d)^5π_m/d^*π_m/d*. On a character of exact conductor c, E_m,d=1, · cmid d,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-18406af98fd43046","tex":"\\rho_m","aliases":"ρ_m rho","display":"\\rho_m","kind":"use","excerpt":"Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_","count":10},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7e425697f7c81595","tex":"\\rho_r","aliases":"ρ_r rho","display":"\\rho_r","kind":"use","excerpt":"Family propagation. Let π: X→ S and ρ_r: Y_r→ S be smooth projective families over a connected base. Relative algebraic correspondences induce morphisms of Gauss-Manin local systems. In a flat rational trivialization, their transfer matrix is constant.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-74f3aeeb52b87bcb","tex":"\\sigma_{5,1}","aliases":"σ_5,1 sigma","display":"\\sigma_{5,1}","kind":"use","excerpt":"This is a proof for every q>5, not an extrapolation from finite runs. The verifier's checks for 6≤ q≤80 are benchmarks only. These packets are themselves the standard family σ_5,1, so the obstruction is to the method, not to algebraicity.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bf518aebbb1f195e","tex":"\\sigma_{5,5}","aliases":"σ_5,5 sigma","display":"\\sigma_{5,5}","kind":"use","excerpt":"S_5/6 · =(5,10,12,19,26,33)=σ_5,5 at degree 35, S_7/6 · =(2,7,12,17,21,22,27,32)=σ_7,2 at degree 35,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1b929834c99f8782","tex":"\\sigma_{7,2}","aliases":"σ_7,2 sigma","display":"\\sigma_{7,2}","kind":"use","excerpt":"S_5/6 · =(5,10,12,19,26,33)=σ_5,5 at degree 35, S_7/6 · =(2,7,12,17,21,22,27,32)=σ_7,2 at degree 35,","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-df661d072214bde6","tex":"\\Sigma_N","aliases":"Σ_N Sigma","display":"\\Sigma_N","kind":"use","excerpt":"1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part is exactly the Hodge sign. Consequently the CM type of N(1) is the conjugate of the CM type of the lower curve packet: Σ_N=overlineΣ_ℓ. This is the ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a1ba69464d51bb84","tex":"\\sigma_{p,i}","aliases":"σ_p,i sigma","display":"\\sigma_{p,i}","kind":"use","excerpt":"σ_p,i=(i,i+d,…,i+(p-1)d,m-pi)","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b17d423cde5b3e30","tex":"\\sigma_u","aliases":"σ_u sigma","display":"\\sigma_u","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edge","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5a905007981f9c00","tex":"\\tau_b","aliases":"τ_b tau","display":"\\tau_b","kind":"use","excerpt":"The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edge","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7940ac5f0dbf0fdc","tex":"\\tau_{p(b)}","aliases":"τ_p(b) tau","display":"\\tau_{p(b)}","kind":"use","excerpt":"The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edge","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-61b50b72c863394e","tex":"\\tau_{r_0}","aliases":"τ_r_0 tau","display":"\\tau_{r_0}","kind":"use","excerpt":"The trivalent defect tree. The integral basis turns defect mass into exact tree variation. Define a sign gauge by τ_r_0=1, τ_b=ϵ_bτ_p(b), y_b=τ_bz_b. Attach one zero-valued boundary leaf to the root and two zero-valued boundary leaves to every odd vertex. Every vertex of the resulting augmented tree widetilde T_m has degree three. The boundary edge","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"em[Range-packet contraction] Assume R ≤ M, B(a,b) ≤ C a b , 4CM^2 Qf <1. Then the map T(a)=-RQ(f+B(a,a)) has a unique fixed point in the gauge-fixed ball a ≤2M Qf . If Qf=0, the ball is the singleton {0} and the range solution is immediate. At the fixed point, the range component of F(a) vanishes. The remaining obstruction is Φ=Π(f+B(a,a)). The ful","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"Klein--Delsarte cubic. Consider the smooth cubic fourfold Y: y_0^2y_4+y_1^2y_0+y_2^2y_1+y_3^2y_2+y_4^2y_3+y_5^3=0. An explicit monomial exponent matrix gives a dominant rational map ϕ:X_33^4dashrightarrow Y whose projective-torus lattice map has Smith diagonal (1,33,33,33,33), so the generic torus degree is 33^4.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"The tower packet remains an old conductor-33 sector. A stronger construction appends a complementary pair (1,m-1) after scaling a solved seed to modulus m=kd. If b=(b_0,…,b_2p+1) is algebraic at degree d, then Θ_m,b=(kb_0,…,kb_2p+1,1,m-1) has full raw conductor m and middle Hodge type two dimensions higher. Its rational orbit has dimension φ(m). Th","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-eb6955c10a90f6c8","tex":"\\Phi_{\\lambda}","aliases":"Φ_λ Phi lambda","display":"\\Phi_{\\lambda}","kind":"use","excerpt":"This creates a practical research program. One can refine the finite obstruction space by symmetry, monodromy, support, or arithmetic phase, obtaining Φ=∑_λΦ_λ. Since the decomposition is complete, integrability is equivalent to the simultaneous vanishing of all children. Different geometric arguments may eliminate different children. A symmetry pa","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":12},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-5b0bc04d612476e3","tex":"\\chi^*","aliases":"χ^* chi","display":"\\chi^*","kind":"use","excerpt":"residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, with eigenvalues λ_χ= ∑_a∈ U_mρ_m(a)overlineχ(a) = ∑_a∈ U_maoverlineχ(a). If χ is induced from a primitive odd character χ^* of conductor 2^t,","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ec260bc3688a0cb0","tex":"\\chi_a","aliases":"χ_a chi","display":"\\chi_a","kind":"use","excerpt":"Proof. If K is a diagonal deck subgroup and χ_a|_K=1, then","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b5503be5a578e75b","tex":"\\chi_a^k","aliases":"χ_a^k chi","display":"\\chi_a^k","kind":"use","excerpt":"χ_c|_K=χ_a^k|_K=1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-74ef2030c5e37f57","tex":"\\chi_c","aliases":"χ_c chi","display":"\\chi_c","kind":"use","excerpt":"χ_c|_K=χ_a^k|_K=1.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-27fe6b7aa4b6c0f3","tex":"\\chi_{\\mathrm{mon}}","aliases":"χ_mon chi mathrm","display":"\\chi_{\\mathrm{mon}}","kind":"use","excerpt":"Multiaxial packet addresses. Degree residue is only one address. A refined cycle packet may be labeled by λ=(r_Lef,r_deg,χ_mon,s_supp,ν_Frob). The entries record Lefschetz depth, degree residue, monodromy or symmetry type, support stratum, and arithmetic phase. The total positive frame remains the sum of all leaves. This allows independent methods ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-415e5530d989e657","tex":"\\chi_T","aliases":"χ_T chi","display":"\\chi_T","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-491f589a4e4c9f3b","tex":"\\chi_\\alpha","aliases":"χ_α chi alpha","display":"\\chi_\\alpha","kind":"use","excerpt":"Let K be the kernel of the torus isogeny. The relation AB=210I shows that K embeds in G_210: an element in the kernel has representatives whose 210th powers are all the same, and a common scalar makes them roots of unity. The displayed character identity proves χ_α|_K=1. The same holds for every rational conjugate. Therefore the whole rational pack","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"use","excerpt":"Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0838bdb2b0dd7bda","tex":"\\psi^m","aliases":"ψ^m psi","display":"\\psi^m","kind":"use","excerpt":"Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2aa697a856752e89","tex":"\\Omega_B","aliases":"Ω_B Omega","display":"\\Omega_B","kind":"use","excerpt":"The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-cfeb9eb96e89714e","tex":"\\Omega_C","aliases":"Ω_C Omega","display":"\\Omega_C","kind":"use","excerpt":"The product locus and replacement factorizations. Split the six directions into a four-dimensional Weil factor C and a two-dimensional factor B. The determinant forms factor: Ω_6=Ω_CwedgeΩ_B. Writing their real and imaginary parts as c_R,c_I,b_R,b_I gives u=c_Rb_R-c_Ib_I, v=c_Rb_I+c_Ib_R. On the product locus, the degree-four Weil classes on C are ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"use","excerpt":"Positive transfer · T=∑_rA_r^† A_r only with algebraic realizations of those adjoints.","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8a05e97949811e78","tex":"\\ell_1","aliases":"ℓ_1 ell","display":"\\ell_1","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f71176b4c504c459","tex":"\\ell_2","aliases":"ℓ_2 ell","display":"\\ell_2","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e2b359d76df4ae4d","tex":"\\ell_3","aliases":"ℓ_3 ell","display":"\\ell_3","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8943b22d90e5e02b","tex":"\\ell_j","aliases":"ℓ_j ell","display":"\\ell_j","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Algebraic space · Alg^p(X)=im(CH^p(X)_Q→ H^2p(X,Q)).","count":19},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"Support verification and multiplicity recovery. Let Z⊆ X be a closed algebraic subset and j:X∖ Z↪ X. The support exact sequence contains H_Z^2p(X,Q)⟶ H^2p(X,Q)xrightarrowj^*H^2p(X∖ Z,Q).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"when mmid r. A primitive character address is a six-tuple a=(a_0,…,a_5), 1≤ a_i<m, ∑_i a_i≡0pmod m. The Hodge level of the conjugate indexed by u∈ U_m is ℓ_a(u)=frac1m∑_i=0^5⟨ ua_i⟩_m. The character line has type (2,2) exactly when ℓ_a(1)=3. Its rational Galois orbit is a Hodge packet exactly when ℓ_a(u)=3 ∀ u∈ U_m.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"Complement defects and dyadic triangles. Put q=frac m4, h=2q, m=4q. For a packet count vector n, define its oriented complement defect ∂ n=∑_r=1^h-1d_n(r)[r], d_n(r)=n_r-n_m-r. The self-complementary residue h contributes zero. Since ρ_m(m-r)=-ρ_m(r), the balanced-packet equation is equivalent to ∑_r=1^h-1d_n(r)ρ_m(ur)=0 for every odd unit u. Moreo","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0187aedf63e21d96","tex":"\\prod_{0\\le i<j\\le4}","aliases":"∏_0≤ i<j≤4 prod le le","display":"\\prod_{0\\le i<j\\le4}","kind":"use","excerpt":"Δ(t)=∏_0≤ i<j≤4(t_i-t_j).","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c772f0e8c68091d0","tex":"\\prod_{i=0}^5","aliases":"∏_i=0^5 prod","display":"\\prod_{i=0}^5","kind":"use","excerpt":"J_F_m=(x_0^m-1,…,x_5^m-1), S_m=∏_i=0^5x_i^m-2.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-2d3db36262598bcd","tex":"\\prod_j","aliases":"∏_j prod","display":"\\prod_j","kind":"use","excerpt":"On the projective tori define y_i=∏_jx_j^B_ij. Negative exponents are allowed there. The second identity makes this well-defined into the weighted projective torus, the first gives","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6f3d7d70820f51b9","tex":"\\prod_{j=1}^3","aliases":"∏_j=1^3 prod","display":"\\prod_{j=1}^3","kind":"use","excerpt":"For an embedding σ_u:ζ_q↦ζ_q^u, let h_ℓ(u)=frac⟨ uc⟩_q+⟨ ud⟩_q+⟨ ue⟩_qq∈{1,2}. Using 1-e^2π ir/q=-2i e^π ir/qsinfracπ rq, one obtains σ_u(Δ_ℓ) =8i(-1)^h_ℓ(u) ∏_j=1^3 sinfracπ⟨ uℓ_j⟩_qq. Here (ℓ_1,ℓ_2,ℓ_3)=(c,d,e) is an indexed triple, so repeated residues contribute repeated factors. Every sine factor is positive, so the sign of the imaginary part ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-87224fa8ff2ded0f","tex":"\\prod_{\\substack{0\\le q\\le\\dim X\\\\q\\ne p}}","aliases":"∏_substack0≤ q≤dim X q≠ p prod substack le le dim q ne","display":null,"kind":"use","excerpt":"Adams degree projectors. For integer m≥2, Adams operations satisfy ch_q(ψ^mξ)=m^qch_q(ξ). Define e_p^(m)=∏_substack0≤ q≤dim X q≠ p fracψ^m-m^qIm^p-m^q. Then ch_q(e_p^(m)ξ)=δ_pqch_p(ξ). This exactly isolates codimension p inside a topological K-packet.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-033c4164f02bb19b","tex":"\\sum_{0\\le i<j\\le4}","aliases":"∑_0≤ i<j≤4 sum le le","display":"\\sum_{0\\le i<j\\le4}","kind":"use","excerpt":"a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-48dee709e478eada","tex":"\\sum_{a\\in U_m}","aliases":"∑_a∈ U_m sum in","display":"\\sum_{a\\in U_m}","kind":"use","excerpt":"ecover the unique integers z_b from its even coordinates, and subtract ∑_bz_bκ_b. The residual balanced vector is supported only on the odd residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirichlet characters, wi","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","excerpt":"Now take d∈ K_m, recover the unique integers z_b from its even coordinates, and subtract ∑_bz_bκ_b. The residual balanced vector is supported only on the odd residues 1≤ r<h. The corresponding square sawtooth matrix is nonsingular. Indeed, extend an odd-support vector to an odd function on U_m. Convolution with ρ_m is diagonalized by the odd Dirich","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e2a250994043bc18","tex":"\\sum_{b=1}^{q-1}","aliases":"∑_b=1^q-1 sum","display":"\\sum_{b=1}^{q-1}","kind":"use","excerpt":"Suppose d=∑_b=1^q-1z_bκ_b. Its even coordinates are d(q)=z_r_0, d(2b)=z_b-ϵ_bz_p(b) (b≠ r_0). Starting at the root and moving toward the leaves, these equations recover every z_b integrally. Hence the triangle vectors are linearly independent and their even-coordinate map is unimodular.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4ab8cab65855908b","tex":"\\sum_{b\\ne r_0}","aliases":"∑_b≠ r_0 sum ne","display":"\\sum_{b\\ne r_0}","kind":"use","excerpt":"[Exact trivalent-tree norm] If d=∑_bz_bκ_b∈ K_m, then d _1 =|y_r_0| +∑_b≠ r_0|y_b-y_p(b)| +2∑_substack1≤ b<q b odd|y_b|. The right side is the total variation of the integer function y on widetilde T_m with zero boundary values. The root coordinate is d(q)=z_r_0. For b≠ r_0, multiplying d(2b)=z_b-ϵ_bz_p(b) by τ_b gives τ_bd(2b)=y_b-y_p(b). For odd ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-8ca320cda49a090c","tex":"\\sum_{c\\mid m}","aliases":"∑_cmid m sum mid","display":null,"kind":"use","excerpt":"0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is it","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d210930f336798b9","tex":"\\sum_{d\\mid c}","aliases":"∑_dmid c sum mid","display":null,"kind":"use","excerpt":"0, · cnmid d. Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is it","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a5d3926ec8ff757e","tex":"\\sum_{i=0}^4","aliases":"∑_i=0^4 sum","display":"\\sum_{i=0}^4","kind":"use","excerpt":"a=(1,8,15,22,29,30), Z_κ={ l ∑_i=0^4x_i^7=0, ∑_0≤ i<j≤4x_i^7x_j^7=0, x_0x_1x_2x_3x_4=κ x_5^5 }⊂ X_35.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-65ebb66043dd11be","tex":"\\sum_{i=0}^{5}","aliases":"∑_i=0^5 sum","display":"\\sum_{i=0}^{5}","kind":"use","excerpt":"∑_i=0^5⟨ ta_i⟩_70=210.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7580d5fbac52d6be","tex":"\\sum_{i=1}^N","aliases":"∑_i=1^N sum","display":"\\sum_{i=1}^N","kind":"use","excerpt":"Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6182c3bad2d01e16","tex":"\\sum_{i=1}^r","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^r","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"dots,c_N}. Then im S=A, ker S=A^⊥. Under the stated containment c_j∈ V, the selected cycles span V if and only if S|_V is positive definite. Without containment, positivity only detects the absence of vectors in V orthogonal to the cycle span and does not prove that span contains V. The equality im(CC^*)=im C is standard. Moreover ⟨ v,Sv⟩= C^*v ^2=","count":5},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6da11787047a09a5","tex":"\\sum_{j=0}^{q-1}","aliases":"∑_j=0^q-1 sum","display":"\\sum_{j=0}^{q-1}","kind":"use","excerpt":"f_1 · =e_1, · g_1 · =e_1^4-5e_1^2e_2+5e_1e_3+5e_2^2-5e_4, f_2 · =e_2, · g_2 · =-5e_3, f_3 · =P-κ z^5, · g_3 · =5∑_j=0^q-1P^q-1-jκ^jz^5j.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-05beb5d438ae1bd0","tex":"\\sum_{j=1}^{2N-3}","aliases":"∑_j=1^2N-3 sum","display":"\\sum_{j=1}^{2N-3}","kind":"use","excerpt":"Balanced four-packets and saturation. Let F_m= { γ∈ N^m-1: 1^Tγ=4, R_mγ=2m 1 }. A saturated certificate is an exact nonnegative identity n+∑_j=1^2N-3p_r_j =∑_i=1^Nγ_i, γ_i∈ F_m. The case N=2 is one-step quasi-decomposition. Higher N allows several surface packets to reconstruct one six-packet after Tate stabilization.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7d1d47c0c0a47e08","tex":"\\sum_{j=1}^N","aliases":"∑_j=1^N sum","display":"\\sum_{j=1}^N","kind":"use","excerpt":"tudy. For the positive frame certificate below, require every c_j to belong to V. This can be achieved by a rational algebraic packet projector when such a projector has been constructed. Projection through an arbitrary orthogonal Hodge projector does not establish that the projected vectors are algebraic. Define C:R^N→ H^2p(X,R), Ce_j=c_j, and the","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6ac55c3e7e24dc40","tex":"\\sum_{j=1}^{q-1}","aliases":"∑_j=1^q-1 sum","display":"\\sum_{j=1}^{q-1}","kind":"use","excerpt":"df_3wedge dg_3= 25q∑_j=1^q-1κ^jP^q-1-jz^5j-1 dPwedge dz.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0b1df986ea484636","tex":"\\sum_{j\\in I_{M,r}}","aliases":"∑_j∈ I_M,r sum in","display":"\\sum_{j\\in I_{M,r}}","kind":"use","excerpt":"Degree-residue polyphase refinement. Exact parent-child frame law. Fix a projective embedding and assign each cycle its degree. For modulus M, define the residue packet I_M,r={j:deg Z_j≡ rpmod M}, S_M,r=∑_j∈ I_M,r|c_j⟩⟨ c_j|.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ad1acac8f75227af","tex":"\\sum_{j\\in I_\\lambda}","aliases":"∑_j∈ I_λ sum in lambda","display":"\\sum_{j\\in I_\\lambda}","kind":"use","excerpt":"Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-01f56a5e1c2b8b56","tex":"\\sum_{k\\ge1}","aliases":"∑_k≥1 sum ge","display":"\\sum_{k\\ge1}","kind":"use","excerpt":"For an integer-valued y, discrete coarea gives TV(y) = ∑_k≥1|∂{y≥ k}| + ∑_k≥1|∂{y≤-k}|. If S is a finite forest of internal vertices of a trivalent tree and c(S) is its number of connected components, then |∂ S|=|S|+2c(S), because 3|S|=2(|S|-c(S))+|∂ S|.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"Divisor transfer · T=∑_rB_rA_r with actual Chow correspondences, packet preservation and invertibility.","count":12},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bf88b11d83d88a95","tex":"\\sum_{r=1}^{h-1}","aliases":"∑_r=1^h-1 sum","display":"\\sum_{r=1}^{h-1}","kind":"use","excerpt":"Complement defects and dyadic triangles. Put q=frac m4, h=2q, m=4q. For a packet count vector n, define its oriented complement defect ∂ n=∑_r=1^h-1d_n(r)[r], d_n(r)=n_r-n_m-r. The self-complementary residue h contributes zero. Since ρ_m(m-r)=-ρ_m(r), the balanced-packet equation is equivalent to ∑_r=1^h-1d_n(r)ρ_m(ur)=0 for every odd unit u. Moreo","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-155061be3b01f38d","tex":"\\sum_{r=1}^M","aliases":"∑_r=1^M sum","display":"\\sum_{r=1}^M","kind":"use","excerpt":"[Invertible divisor-lowering transfer] Suppose T=∑_r=1^M B_rA_r preserves V and is invertible on V. Then every rational Hodge class in V is algebraic. Let α∈ V be of type (p,p). Each A_rα is rational of type (1,1) on Y_r, hence is the class of a rational divisor D_r by the Lefschetz (1,1) theorem. Therefore Tα=∑_r(Λ_r)_*[D_r] is algebraic on X.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-145651e011a10b92","tex":"\\sum_{r\\in B}","aliases":"∑_r∈ B sum in","display":"\\sum_{r\\in B}","kind":"use","excerpt":"For any nonzero-entry block B at modulus m, put h_B(u)=m^-1∑_r∈ B⟨ ur⟩_m, with multiplicities retained.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-7f20049a870a3052","tex":"\\sum_{\\substack{1\\le b<q\\\\ b\\text{ odd}}}","aliases":"∑_substack1≤ b<q b odd sum substack le text","display":"\\sum_{\\substack{1\\le b<q\\\\ b\\text{ odd}}}","kind":"use","excerpt":"[Exact trivalent-tree norm] If d=∑_bz_bκ_b∈ K_m, then d _1 =|y_r_0| +∑_b≠ r_0|y_b-y_p(b)| +2∑_substack1≤ b<q b odd|y_b|. The right side is the total variation of the integer function y on widetilde T_m with zero boundary values. The root coordinate is d(q)=z_r_0. For b≠ r_0, multiplying d(2b)=z_b-ϵ_bz_p(b) by τ_b gives τ_bd(2b)=y_b-y_p(b). For odd ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-678130c567f7ae7e","tex":"\\sum_{u\\in U_m}","aliases":"∑_u∈ U_m sum in","display":"\\sum_{u\\in U_m}","kind":"use","excerpt":"Centered Fourier form. Define ρ_m(r)=⟨ r⟩_m-frac m2, F_a(u)=∑_iρ_m(ua_i). Then a is balanced if and only if F_a(u)=0 for all units. Moreover, F_a(-u)=-F_a(u), so only odd characters occur. For a Dirichlet character χ on U_m, put B_m,χ(r)= ∑_u∈ U_mρ_m(ur)overlineχ(u). Finite Fourier inversion gives the exact criterion n(a)∈ B_m Longleftrightarrow ∑_","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-53c07efe2188e4b0","tex":"\\sum_v","aliases":"∑_v sum","display":"\\sum_v","kind":"use","excerpt":"C(a)=sup{∑_vw_v: ∑_vw_vv=1, w_v≥0, only finitely many w_v≠0}.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-ade7e6465dee6e86","tex":"\\sum_\\lambda","aliases":"∑_λ sum lambda","display":"\\sum_\\lambda","kind":"use","excerpt":"Packet assembly without false orthogonality. Partition the cycle indices into packets I_λ. Define S_λ=∑_j∈ I_λ|c_j⟩⟨ c_j|. Then S=∑_λ S_λ. No orthogonality of the child spans is assumed.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-614ab5cb07fb5cef","tex":"\\sum_{\\lambda}","aliases":"∑_λ sum lambda","display":"\\sum_{\\lambda}","kind":"use","excerpt":"This creates a practical research program. One can refine the finite obstruction space by symmetry, monodromy, support, or arithmetic phase, obtaining Φ=∑_λΦ_λ. Since the decomposition is complete, integrability is equivalent to the simultaneous vanishing of all children. Different geometric arguments may eliminate different children. A symmetry pa","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-43e727cf5aadc16c","tex":"\\sum_{\\ell=0}^{L-1}","aliases":"∑_ℓ=0^L-1 sum ell","display":"\\sum_{\\ell=0}^{L-1}","kind":"use","excerpt":"Polyphase assembly · S_M,r=∑_ℓ=0^L-1S_LM,r+ℓ M. These are frame sums, not sums of span projectors.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Support verification and multiplicity recovery. Let Z⊆ X be a closed algebraic subset and j:X∖ Z↪ X. The support exact sequence contains H_Z^2p(X,Q)⟶ H^2p(X,Q)xrightarrowj^*H^2p(X∖ Z,Q).","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6f333436e7d549b7","tex":"\\ast","aliases":"∗ ast","display":"\\ast","kind":"use","excerpt":"The desired correspondence is schematically Γ∈CH^∗(F_d× C_P× S_T)_Q, Γ_*:e_orbH^1(F_d)⟶ e_diag(M(C_P)⊗ M(S_T))(1). This formula is a target specification, not a constructed map. Its coefficient field, diagonal projector, rational multiplicities, and dimensions must be fixed before an isomorphism can be asserted. The smaller CM conductor can have a ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"[Approximation becomes exact] Let P_D be the orthogonal projector onto A_D^p(X)⊗R. For a rational class α, lim_D→∞ (I-P_D)α =0 if and only if α∈ A_D^p(X) for some finite D. After stabilization, the residual is constant. A constant tail converging to zero is zero.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"Rational Hodge space · Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X).","count":10},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"{y_0=y_1=y_2=y_3=0} ∪ {y_0=y_1=y_3=y_5=0} ∪ {y_0=y_1=y_4=y_5=0}.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":7},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c730983c77de7073","tex":"\\int_X","aliases":"∫_X int","display":"\\int_X","kind":"use","excerpt":"Suppose Z has codimension at least p and its codimension-p components are Z_1,…,Z_r. Purity gives α=∑_i=1^r q_i[Z_i]. Choose complementary test classes γ_k and define M_ki=∫_X[Z_i]smileγ_k, b_k=∫_Xαsmileγ_k. Then Mq=b. Removing dependent columns yields exact rational multiplicities.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-aa6638bf30261eed","tex":"\\int_Z","aliases":"∫_Z int","display":"\\int_Z","kind":"use","excerpt":"2,z_3)⟼ (z_1,z_2,z_3,overline z_1,overline z_2,overline z_3). Conjugation preserves the Z[i] lattice and is holomorphic from an elliptic factor with complex structure i to a target factor with complex structure -i. Thus this map defines an abelian-subvariety graph and is algebraic at J_0. On its three source factors the polarization restricts to we","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e1ad3847ce3d0c65","tex":"\\int_{Z_\\kappa}","aliases":"∫_Z_κ int kappa","display":"\\int_{Z_\\kappa}","kind":"use","excerpt":"∫_Z_κres(fracQ_bΩF_m^3) =frac(2π i)^250q κ^q-b≠0.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"Hodge packets in topological K-theory. For finite CW complexes, the rational topological Chern character gives ch_top:K^0_top(X)⊗Qxrightarrow∼H^even(X,Q). For smooth projective X, the algebraic Chern character identifies algebraic K-theory rationally with the Chow ring after the usual filtration. Thus the rational Hodge conjecture for all degrees i","count":4},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-bdcce8a36f1962a3","tex":"\\sim_{\\mathbf Q}","aliases":"∼_ Q sim mathbf","display":"\\sim_{\\mathbf Q}","kind":"use","excerpt":"K_Y∼_ Q-3H.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"Weighted dyadic reconstruction. Prime 2 cannot be handled by inserting p=2 into the odd-prime orbit-mass proof. The unit group has the separate form Z_2^×≅{±1}×(1+4 Z_2). The first address is a sign packet, only afterward does the principal-unit refinement begin.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Let χ_T(t)=t^d+c_d-1t^d-1+⋯+c_0 be the characteristic polynomial on V. Invertibility means c_0≠0. Cayley-Hamilton gives T^-1=-frac1c_0(T^d-1+c_d-1T^d-2+⋯+c_1I). Each power of T is induced by compositions and rational sums of algebraic correspondences. Thus T^-1 preserves algebraic cycle classes. Hence α=T^-1(Tα) is algebraic.","count":22},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"Cubic fourfolds as a validation of the transfer theorem. Let X⊂ P^5 be a smooth cubic fourfold and let F(X) be its Fano variety of lines. The universal incidence variety P={(ℓ,x):x∈ℓ}⊂ F(X)× X induces the Abel-Jacobi map A=p_*q^*:H^4_prim(X,Q)→ H^2_prim(F(X),Q). Beauville and Donagi proved that this is a polarized Hodge isomorphism up to the standa","count":12},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"The rational Hodge target. Statement and status. Let X be a smooth projective complex variety. For each codimension p, define Hdg^p(X)=H^2p(X,Q)∩ H^p,p(X) and Alg^p(X)=im(CH^p(X)_QxrightarrowclH^2p(X,Q)). Every algebraic cycle class has type (p,p), so Alg^p(X)⊆Hdg^p(X). The rational Hodge conjecture asserts equality for every smooth projective X an","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-6e6b2d6128ce8474","tex":"\\bigoplus_{c\\mid m}","aliases":"⊕_cmid m bigoplus mid","display":null,"kind":"use","excerpt":"endcases Therefore E_m,dE_m,e=E_m,gcd(d,e). M obius inversion produces pairwise orthogonal exact-conductor projectors Q_m,c=∑_dmid cμ(c/d)E_m,d. They satisfy Q_m,cQ_m,e=δ_ceQ_m,c, ∑_cmid mQ_m,c=I. Hence H^4_prim(X_m^4,Q) =⊕_cmid mimQ_m,c. Each Q_m,c is a rational linear combination of graph correspondences, so the conductor decomposition is itself ","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-21eb22348b46e87a","tex":"\\bigoplus_{r\\ge0}","aliases":"⊕_r≥0 bigoplus ge","display":"\\bigoplus_{r\\ge0}","kind":"use","excerpt":"Primitive and coniveau reductions. Lefschetz packet decomposition. Let h∈ H^2(X,Q) be the class of an ample divisor and let L(α)=hsmileα. Hard Lefschetz and primitive decomposition yield H^2p(X,Q)=⊕_r≥0L^rH_prim^2p-2r(X,Q). The decomposition respects Hodge type.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","excerpt":"Real obstruction · O=P_Hdg⊗R-P_Alg⊗R. No algebraic realization of these orthogonal projectors is presumed.","count":8},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0c8cc88f15bd94f4","tex":"\\otimes_{\\Q}","aliases":"⊗_Q otimes Q","display":null,"kind":"use","excerpt":"Choose a positive inner product on H^2p(X,R). The orthogonal projectors P_Hdg^p and P_Alg^p act on the real scalar extensions Hdg^p(X)⊗_QR and Alg^p(X)⊗_QR. These real orthogonal projectors provide a reformulation. They are not asserted to be rational matrices or algebraic correspondences. Since the algebraic space is nested in the Hodge space, P_H","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"[Master obstruction projector] The operator O_X^p=P_Hdg^p-P_Alg^p is the orthogonal projector onto (Hdg^p(X)⊗_QR)∩(Alg^p(X)⊗_QR)^⊥. Therefore O_X^p=0 Longleftrightarrow Hdg^p(X)=Alg^p(X). For nested orthogonal projectors Q≤ P, the difference P-Q is self-adjoint and (P-Q)^2=P-PQ-QP+Q=P-Q. Its range is im P∩(im Q)^⊥.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"[Positive child-kernel intersection] For any subspace V, ker(S|_V)=⋂_λker(S_λ|_V). If Sv=0, positivity gives 0=⟨ v,Sv⟩=∑_λ⟨ v,S_λ v⟩. Each nonnegative summand vanishes, which implies S_λ v=0. The converse is immediate.","count":3},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-08bf84a1d20b2d0b","tex":"\\bigcap_r","aliases":"⋂_r bigcap","display":"\\bigcap_r","kind":"use","excerpt":"Positive multi-channel form. Assume the polarized adjoint A_r^† is realized by an algebraic return correspondence B_r. This is an additional geometric hypothesis, not an automatic consequence in arbitrary settings. Then T=∑_rA_r^† A_r is positive and ⟨ v,Tv⟩=∑_r A_rv ^2. Therefore ker T=⋂_rker A_r. Joint injectivity of the lowering channels is enou","count":2},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-20515a6d26ff134c","tex":"\\bigcap_\\lambda","aliases":"⋂_λ bigcap lambda","display":"\\bigcap_\\lambda","kind":"use","excerpt":"[Positive child-kernel intersection] For any subspace V, ker(S|_V)=⋂_λker(S_λ|_V). If Sv=0, positivity gives 0=⟨ v,Sv⟩=∑_λ⟨ v,S_λ v⟩. Each nonnegative summand vanishes, which implies S_λ v=0. The converse is immediate.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"Finite saturation and exact rational closure. For each degree cutoff D, let A_D^p(X) be the rational span of codimension-p cycle classes of degree at most D. Then A_1^p⊆ A_2^p⊆⋯⊆ H^2p(X,Q).","count":14},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-fdaafff1d32e051c","tex":"\\rangle_{70}","aliases":"⟩_70 rangle","display":"\\rangle_{70}","kind":"use","excerpt":"∑_i=0^5⟨ ta_i⟩_70=210.","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-56b263bf8cc7f74c","tex":"\\rangle_c","aliases":"⟩_c rangle","display":"\\rangle_c","kind":"use","excerpt":"orem[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balan","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-97db874a29992d0a","tex":"\\rangle_{gc}","aliases":"⟩_gc rangle","display":"\\rangle_{gc}","kind":"use","excerpt":"orem[Conductor descent] Every imprimitive balanced packet at degree m is the coordinate-power pullback of a unique full-conductor packet at its exact conductor cmid m. Consequently B_m=bigsqcup_cmid mL_m/c B_c^full. If every occupied residue is divisible by g, divide the six entries and the modulus by g. Since ⟨ ugb_i⟩_gc=g⟨ bar u b_i⟩_c, the balan","count":1},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-b4fd1ed072c84fb5","tex":"\\rangle_q","aliases":"⟩_q rangle","display":"\\rangle_q","kind":"use","excerpt":"[Four-phase CRT reduction] The six-packet A_b,c,d is balanced modulo 5q if and only if the four-packet Γ_b,c,d is balanced modulo q. For a unit u, let s=⟨ ub⟩_q and z=⟨ u5^-1b⟩_q. The complete five-element fiber over s has sum 5s+10q, its phase-zero lift is 5z. The four nonzero phases therefore sum to 5s+10q-5z. Adding the last two entries and impo","count":6},{"work":"hodge-reader","url":"notation/hodge-reader.html#symbol-e4eef061c99f7b52","tex":"\\rangle_{Y_r}","aliases":"⟩_Y_r rangle","display":"\\rangle_{Y_r}","kind":"use","excerpt":"[Finite Gram certificate] Choose a rational basis e_1,…,e_d of V. If G_ij=∑_r⟨ A_re_i,A_re_j⟩_Y_r has nonzero determinant, every rational Hodge class in V is algebraic.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.","count":18},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.","count":72},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"definition-passage","excerpt":"An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.","count":192},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.","count":43},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"definition-passage","excerpt":"An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-bd730f6c2c2e8897","tex":"t_d","aliases":"t_d","display":"t_d","kind":"definition-passage","excerpt":"An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.","count":7},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indee","count":33},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":223},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"Our rational function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of vari","count":11},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":38},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"Our rational function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of vari","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C","display":null,"kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":8},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":16},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-59651fe6df7d2f0f","tex":"c_g","aliases":"c_g","display":"c_g","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.","count":17},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-860e8c60556728a1","tex":"E^G","aliases":"E^G","display":"E^G","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b576b4aed1362b85","tex":"e_H","aliases":"e_H","display":"e_H","kind":"introduction-passage","excerpt":"[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.","count":6},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-a69dcdbad3667267","tex":"e_K","aliases":"e_K","display":"e_K","kind":"introduction-passage","excerpt":"[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":17},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indee","count":33},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":119},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":51},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.","count":119},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.","count":38},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":86},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.","count":21},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"introduction-passage","excerpt":"Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indee","count":78},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"nal function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of variables. A ","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"Our rational function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of vari","count":35},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"nal function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of variables. A ","count":9},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":19},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indee","count":41},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":41},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"introduction-passage","excerpt":"Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indee","count":13},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6af4544d891351e3","tex":"x_g","aliases":"x_g","display":"x_g","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":10},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":33},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"introduction-passage","excerpt":"Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indee","count":47},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":13},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-9adbb02c2ad54bb0","tex":"\\delta_1","aliases":"δ_1 delta","display":"\\delta_1","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0a7b0d31ccd40df9","tex":"\\delta_{ag}","aliases":"δ_ag delta","display":"\\delta_{ag}","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4da675801bf99258","tex":"\\delta_g","aliases":"δ_g delta","display":"\\delta_g","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"introduction-passage","excerpt":"Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indee","count":23},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-85e002fb281d7432","tex":"\\zeta^2","aliases":"ζ^2 zeta","display":"\\zeta^2","kind":"introduction-passage","excerpt":"Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indee","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":18},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, whil","count":52},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":11},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6d39fe36bb310cf9","tex":"\\sum_{g\\in G}","aliases":"∑_g∈ G sum in","display":"\\sum_{g\\in G}","kind":"introduction-passage","excerpt":"Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coeffic","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6dbe05965014980d","tex":"\\sum_{x\\in HK}","aliases":"∑_x∈ HK sum in","display":"\\sum_{x\\in HK}","kind":"introduction-passage","excerpt":"[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"introduction-passage","excerpt":"[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.","count":9},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"Our rational function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of vari","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"[Hilbert specialization, external input] Let d≥1 and let E/Q(t_1,…,t_d) be a finite regular Galois extension with group G. There is a Hilbert subset of rational parameter tuples for which specialization is defined and the specialized extension has full Galois group G. The tuples may be required to avoid the zero set of any prescribed nonzero polyno","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed v","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"Let α=sqrt2 and L=Q(α). The polynomial X^2-2 is irreducible over Q. Indeed, if a/b in lowest terms had square two, then a^2=2b^2 would make a even. Writing a=2c gives b^2=2c^2, so b would also be even, a contradiction.","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-7b48479868582d73","tex":"a^3","aliases":"a^3","display":"a^3","kind":"use","excerpt":"For a monic cubic X^3+aX^2+bX+c, the discriminant is a^2b^2-4b^3-4a^3c-27c^2+18abc. Here this gives disc(f)=4+32+4-27+36=49. The discriminant is nonzero, so the roots are distinct.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d325da4f9b58928d","tex":"A_3","aliases":"A_3","display":"A_3","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d84b5a24a5037067","tex":"A_4","aliases":"A_4","display":"A_4","kind":"use","excerpt":"roots. Its order is divisible by four and three, so it is divisible by twelve. Since H≤ S_4, its order is twelve or twenty four. A subgroup of order twelve would have index two and would be normal. The quotient map S_4→ C_2 would send all transpositions to the same element because they are conjugate. They generate S_4, so this element must be nontr","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-2d83d124aaabaee7","tex":"a_g","aliases":"a_g","display":"a_g","kind":"use","excerpt":"The normal basis theorem supplies a regular module for an already existing finite Galois extension [source reference]. If L/Q has group G and normal basis generator α, the vector-space map Φ:Q[G]⟶ L, ∑_g a_g g⟼∑_g a_g g(α) is an equivariant linear isomorphism. It transfers subgroup averages and their fixed vector spaces. It is not a claim that the ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-01478c4149c7701c","tex":"\\Aut","aliases":"Aut","display":null,"kind":"use","excerpt":"[Artin's finite automorphism theorem, external input] Let E be a field and A a finite subgroup of Aut(E). If K=E^A, then E/K is a finite Galois extension, [E:K]=|A|, Gal(E/K)=A.","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-e7fe1a0e0905d629","tex":"\\Aut_{\\Q}","aliases":"Aut_Q Aut Q","display":null,"kind":"use","excerpt":"The substitutions give a homomorphism from G into Aut_Q(L). The identity and composition congruences prove the homomorphism law. Distinct generator images make the homomorphism injective. Write its image as Σ, so |Σ|=n.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":25},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"Let α=sqrt2 and L=Q(α). The polynomial X^2-2 is irreducible over Q. Indeed, if a/b in lowest terms had square two, then a^2=2b^2 would make a even. Writing a=2c gives b^2=2c^2, so b would also be even, a contradiction.","count":6},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4c4bf99072ea9be6","tex":"b^3","aliases":"b^3","display":"b^3","kind":"use","excerpt":"For a monic cubic X^3+aX^2+bX+c, the discriminant is a^2b^2-4b^3-4a^3c-27c^2+18abc. Here this gives disc(f)=4+32+4-27+36=49. The discriminant is nonzero, so the roots are distinct.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":24},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","excerpt":"Let α=sqrt2 and L=Q(α). The polynomial X^2-2 is irreducible over Q. Indeed, if a/b in lowest terms had square two, then a^2=2b^2 would make a even. Writing a=2c gives b^2=2c^2, so b would also be even, a contradiction.","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"Quadratic symmetry: C_2.","count":18},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"Transitivity does useful work. The orbit of a root has size d, so the order of the group is divisible by d. It does not identify the group by itself. A cubic can have group C_3 or S_3. A quartic can have several transitive subgroups of S_4. The examples in the next chapter show how extra information selects the group.","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d4e58ada952ec6ab","tex":"C_4","aliases":"C_4","display":"C_4","kind":"use","excerpt":"[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"use","excerpt":"[Good prime cycle theorem, external input] Let f∈Z[X] be monic, and suppose its reduction modulo a prime p is squarefree. If its irreducible factors over F_p have degrees d_1,…,d_r, the Galois group of its splitting field over Q, acting on the roots of f, contains a permutation with disjoint cycle lengths d_1,…,d_r.","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"use","excerpt":"[Good prime cycle theorem, external input] Let f∈Z[X] be monic, and suppose its reduction modulo a prime p is squarefree. If its irreducible factors over F_p have degrees d_1,…,d_r, the Galois group of its splitting field over Q, acting on the roots of f, contains a permutation with disjoint cycle lengths d_1,…,d_r.","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"Apply the finite automorphism fixed-field theorem, in the form recorded in source contract LC10. It gives a finite Galois extension L/L^Σ with [L:L^Σ]=|Σ|=n, Gal(L/L^Σ)=Σ. All substitutions fix Q, hence Q⊆ L^Σ. Since [L:Q]=deg f=n, the degree formula gives n=[L:L^Σ][L^Σ:Q] =n[L^Σ:Q]. Therefore L^Σ=Q. The preceding Galois assertion now has the requi","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-346650cbdd3f09a1","tex":"\\operatorname{disc}","aliases":"disc operatorname","display":"\\operatorname{disc}","kind":"use","excerpt":"For a monic cubic X^3+aX^2+bX+c, the discriminant is a^2b^2-4b^3-4a^3c-27c^2+18abc. Here this gives disc(f)=4+32+4-27+36=49. The discriminant is nonzero, so the roots are distinct.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4411444b9d01d329","tex":"E^A","aliases":"E^A","display":"E^A","kind":"use","excerpt":"[Artin's finite automorphism theorem, external input] Let E be a field and A a finite subgroup of Aut(E). If K=E^A, then E/K is a finite Galois extension, [E:K]=|A|, Gal(E/K)=A.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-42f5742f9931a04d","tex":"e_{gh}","aliases":"e_gh","display":"e_{gh}","kind":"use","excerpt":"Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constru","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-dafb044ef50096de","tex":"e_{gHg^{-1}}","aliases":"e_gHg^-1","display":"e_{gHg^{-1}}","kind":"use","excerpt":"Conjugation gives ge_Hg^-1=e_gHg^-1. Equality of these group algebra elements is equality of their supports, hence equality of the subgroups. This proves the equivalence of the first two statements. Injectivity of λ proves equivalence with the third.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-c4d76c46e427bcf9","tex":"e_{gx}","aliases":"e_gx","display":"e_{gx}","kind":"use","excerpt":"The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-92780dc9a20a0ca6","tex":"E_H","aliases":"E_H","display":"E_H","kind":"use","excerpt":"The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies","count":18},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-5d1836707c2366f7","tex":"e_h","aliases":"e_h","display":"e_h","kind":"use","excerpt":"Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constru","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d4816b355caa28b3","tex":"E_H^*","aliases":"E_H^*","display":"E_H^*","kind":"use","excerpt":"Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed v","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-aa6f9df8b3d55816","tex":"E_H^2","aliases":"E_H^2","display":"E_H^2","kind":"use","excerpt":"Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed v","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-314483df0da8375a","tex":"E_{HK}","aliases":"E_HK","display":"E_{HK}","kind":"use","excerpt":"[Exact compatibility criterion] In the regular realization, the following statements are equivalent: HK=KH, HK is a subgroup, E_HE_K=E_KE_H. When they hold, E_HE_K=E_KE_H=E_HK=E_⟨ H,K⟩. Without a compatibility assumption, the intersection of the fixed ranges is still V^H∩ V^K=V^⟨ H,K⟩.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-27763fe9196f7610","tex":"e_{HK}","aliases":"e_HK","display":"e_{HK}","kind":"use","excerpt":"For permutable subgroups, [source reference] gives e_He_K=e_HK=e_Ke_H. Applying λ proves the operator equality. If the operators commute, injectivity in [source reference] gives e_He_K=e_Ke_H. Their supports are HK and KH, with positive coefficients, so the sets are equal. A subgroup HK contains both H and K, and is contained in every subgroup cont","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ae0f8d7d9b47cce1","tex":"E_K","aliases":"E_K","display":"E_K","kind":"use","excerpt":"Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.","count":8},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-542812c54fe5f147","tex":"e_k","aliases":"e_k","display":"e_k","kind":"use","excerpt":"They do not yet give an extension of Q. The vector space Q^G has no designated field multiplication. Its coordinatewise multiplication makes it a product algebra, which has zero divisors whenever |G|>1. For distinct h,k, the coordinate idempotents satisfy e_he_k=0 although both are nonzero. A finite product of copies of Q is a useful algebraic symm","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d4c39e5e1f0a0028","tex":"e_x","aliases":"e_x","display":"e_x","kind":"use","excerpt":"The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-5648075de59310c3","tex":"E_{\\{1\\}}","aliases":"E_{1}","display":"E_{\\{1\\}}","kind":"use","excerpt":"In particular, the identity subgroup gives E_{1}=I, while the whole group gives a rank one projector onto the constant vector. Every intermediate subgroup supplies a resolution {E_H,I-E_H}. The word resolution here refers to the vector space decomposition into fixed vectors and their orthogonal complement. A field structure has not been imposed on ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-9552ed350bcb9411","tex":"E_{\\langle H,K\\rangle}","aliases":"E_⟨ H,K⟩ langle rangle","display":"E_{\\langle H,K\\rangle}","kind":"use","excerpt":"[Exact compatibility criterion] In the regular realization, the following statements are equivalent: HK=KH, HK is a subgroup, E_HE_K=E_KE_H. When they hold, E_HE_K=E_KE_H=E_HK=E_⟨ H,K⟩. Without a compatibility assumption, the intersection of the fixed ranges is still V^H∩ V^K=V^⟨ H,K⟩.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b6bb9a2af76d66fc","tex":"\\mathbf F","aliases":"F mathbf","display":"\\mathbf F","kind":"use","excerpt":"For a quartic realization, consider f(X)=X^4-X-1. Its reduction modulo two is X^4+X+1. It has no root in F_2. The only monic irreducible quadratic over F_2 is X^2+X+1. Modulo that polynomial, X^2=X+1, X^3=1 and X^4=X, so X^4+X+1 has remainder one. A reducible quartic without a linear factor must factor as two irreducible quadratics. This possibilit","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-05d84d0512e4829e","tex":"F^G","aliases":"F^G","display":"F^G","kind":"use","excerpt":"[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/09_invariant_base.pdf captionThe invariant field contains rational constants and also transcendental invariants. The diagram writes F=E and K=F^G. Its transcendence degree is fixed by the finite extension E/K. Pure transcendence requires an additional rationality argument before ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-710b65b51795073d","tex":"\\mathbf F_2","aliases":"F_2 mathbf","display":"\\mathbf F_2","kind":"use","excerpt":"For a quartic realization, consider f(X)=X^4-X-1. Its reduction modulo two is X^4+X+1. It has no root in F_2. The only monic irreducible quadratic over F_2 is X^2+X+1. Modulo that polynomial, X^2=X+1, X^3=1 and X^4=X, so X^4+X+1 has remainder one. A reducible quartic without a linear factor must factor as two irreducible quadratics. This possibilit","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4b154902ce62093e","tex":"\\mathbf F_7","aliases":"F_7 mathbf","display":"\\mathbf F_7","kind":"use","excerpt":"Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-2be6cf11bf912bc7","tex":"\\mathbf F_p","aliases":"F_p mathbf","display":"\\mathbf F_p","kind":"use","excerpt":"[Good prime cycle theorem, external input] Let f∈Z[X] be monic, and suppose its reduction modulo a prime p is squarefree. If its irreducible factors over F_p have degrees d_1,…,d_r, the Galois group of its splitting field over Q, acting on the roots of f, contains a permutation with disjoint cycle lengths d_1,…,d_r.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f79f277673daa5a5","tex":"f_t","aliases":"f_t","display":"f_t","kind":"use","excerpt":"The polynomial f_t(X)=X^3-t is a small exact example of what can happen under specialization. It is used here to study changes in the splitting field group. Its generic splitting field is not a regular extension of Q(t), a fact established below.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-aa45a7ad376e6ef9","tex":"g^{-1}","aliases":"g^-1","display":"g^{-1}","kind":"use","excerpt":"Conjugation gives ge_Hg^-1=e_gHg^-1. Equality of these group algebra elements is equality of their supports, hence equality of the subgroups. This proves the equivalence of the first two statements. Injectivity of λ proves equivalence with the third.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constru","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-e6df94744f7ade62","tex":"g_2","aliases":"g_2","display":"g_2","kind":"use","excerpt":"Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constru","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0abf0d17398d7d33","tex":"G_{\\Q}","aliases":"G_Q Q","display":null,"kind":"use","excerpt":"An arithmetic embedding problem uses the profinite group G_Q=Gal(overlineQ/Q) with its Krull topology. A finite target group has the discrete topology. The lift is continuous, compatible with the prescribed quotient, and proper when surjective. The finite cyclic examples in Chapter~6 use a different ambient group and are explicitly models.","count":9},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-21cf47c47bcf5c41","tex":"\\Gal","aliases":"Gal","display":null,"kind":"use","excerpt":"An arithmetic embedding problem uses the profinite group G_Q=Gal(overlineQ/Q) with its Krull topology. A finite target group has the discrete topology. The lift is continuous, compatible with the prescribed quotient, and proper when surjective. The finite cyclic examples in Chapter~6 use a different ambient group and are explicitly models.","count":15},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0fcc70536b44cfb8","tex":"\\operatorname{GL}","aliases":"GL operatorname","display":"\\operatorname{GL}","kind":"use","excerpt":"Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constru","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","excerpt":"For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b00a424575dac843","tex":"h_0^{-1}","aliases":"h_0^-1","display":"h_0^{-1}","kind":"use","excerpt":"Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6 x^3+3x^2+2x+5pmod7 · 5 · 4 · 1 · 2 · 6 · 5 · 5. It has no root, hence is irreduci","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b09c3d3928fe2c8b","tex":"\\operatorname{id}","aliases":"id operatorname","display":"\\operatorname{id}","kind":"use","excerpt":"[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-fdf1e5d7aeed3a57","tex":"\\operatorname{id}_{\\Gamma}","aliases":"id_Γ operatorname Gamma","display":"\\operatorname{id}_{\\Gamma}","kind":"use","excerpt":"[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"Let J=widetildeρ(G_Q)≤ E. Even a weak solution satisfies π(J)=G, since ρ is onto. The inclusion kerwidetildeρ⊆kerρ gives a finite Galois field M containing L. Its group is J, rather than necessarily E. Properness makes the exact arithmetic tower Gal(M/Q)≅ E, Gal(M/L)≅ N, Gal(L/Q)≅ G. The quotient map in this tower must be the specified π under thes","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6013290ab9ee6c0e","tex":"k^{-1}","aliases":"k^-1","display":"k^{-1}","kind":"use","excerpt":"Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-9817d114fe256b80","tex":"k_0","aliases":"k_0","display":"k_0","kind":"use","excerpt":"Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"Let J=widetildeρ(G_Q)≤ E. Even a weak solution satisfies π(J)=G, since ρ is onto. The inclusion kerwidetildeρ⊆kerρ gives a finite Galois field M containing L. Its group is J, rather than necessarily E. Properness makes the exact arithmetic tower Gal(M/Q)≅ E, Gal(M/L)≅ N, Gal(L/Q)≅ G. The quotient map in this tower must be the specified π under thes","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8359725b04f5ac32","tex":"L^H","aliases":"L^H","display":"L^H","kind":"use","excerpt":"[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/02_field_group.pdf captionFor an already supplied finite Galois extension L/K with group G, subgroups H≤ G correspond to intermediate fixed fields L^H. The direction of inclusion reverses. The classical correspondence computes degrees and organizes fields after the ambient field ","count":13},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8df462b99b1c1944","tex":"L^{H\\cap K}","aliases":"L^H∩ K cap","display":"L^{H\\cap K}","kind":"use","excerpt":"The lattice identities follow directly. An element fixed by both H and K is fixed by their generated subgroup, so L^H∩ L^K=L^⟨ H,K⟩. The automorphisms fixing the compositum L^HL^K are precisely those fixing each field, namely H∩ K. The correspondence then gives L^HL^K=L^H∩ K. These identities require no permutability assumption. The stronger produc","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8625f81beb4d88af","tex":"L^K","aliases":"L^K","display":"L^K","kind":"use","excerpt":"The lattice identities follow directly. An element fixed by both H and K is fixed by their generated subgroup, so L^H∩ L^K=L^⟨ H,K⟩. The automorphisms fixing the compositum L^HL^K are precisely those fixing each field, namely H∩ K. The correspondence then gives L^HL^K=L^H∩ K. These identities require no permutability assumption. The stronger produc","count":6},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6b1c73aed1a640ed","tex":"L^\\Sigma","aliases":"L^Σ Sigma","display":"L^\\Sigma","kind":"use","excerpt":"Apply the finite automorphism fixed-field theorem, in the form recorded in source contract LC10. It gives a finite Galois extension L/L^Σ with [L:L^Σ]=|Σ|=n, Gal(L/L^Σ)=Σ. All substitutions fix Q, hence Q⊆ L^Σ. Since [L:Q]=deg f=n, the degree formula gives n=[L:L^Σ][L^Σ:Q] =n[L^Σ:Q]. Therefore L^Σ=Q. The preceding Galois assertion now has the requi","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-5a9942d93097c24f","tex":"L^{\\langle H,K\\rangle}","aliases":"L^⟨ H,K⟩ langle rangle","display":"L^{\\langle H,K\\rangle}","kind":"use","excerpt":"The lattice identities follow directly. An element fixed by both H and K is fixed by their generated subgroup, so L^H∩ L^K=L^⟨ H,K⟩. The automorphisms fixing the compositum L^HL^K are precisely those fixing each field, namely H∩ K. The correspondence then gives L^HL^K=L^H∩ K. These identities require no permutability assumption. The stronger produc","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-62b0d83cebe304fe","tex":"L^{\\langle st\\rangle}","aliases":"L^⟨ st⟩ langle rangle","display":"L^{\\langle st\\rangle}","kind":"use","excerpt":"The fixed fields can also be read directly from the basis. The elements fixed by s are a+csqrt3, so L^⟨ s⟩=Q(sqrt3). Similarly, L^⟨ t⟩=Q(sqrt2). The product st changes the signs of both square roots but fixes their product, giving L^⟨ st⟩=Q(sqrt6). An element fixed by both s and t has only its rational component left.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-cacc9b5cdd84e470","tex":"L^{\\langle s\\rangle}","aliases":"L^⟨ s⟩ langle rangle","display":"L^{\\langle s\\rangle}","kind":"use","excerpt":"The fixed fields can also be read directly from the basis. The elements fixed by s are a+csqrt3, so L^⟨ s⟩=Q(sqrt3). Similarly, L^⟨ t⟩=Q(sqrt2). The product st changes the signs of both square roots but fixes their product, giving L^⟨ st⟩=Q(sqrt6). An element fixed by both s and t has only its rational component left.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-475dabab48bbb62f","tex":"L^{\\langle t\\rangle}","aliases":"L^⟨ t⟩ langle rangle","display":"L^{\\langle t\\rangle}","kind":"use","excerpt":"The fixed fields can also be read directly from the basis. The elements fixed by s are a+csqrt3, so L^⟨ s⟩=Q(sqrt3). Similarly, L^⟨ t⟩=Q(sqrt2). The product st changes the signs of both square roots but fixes their product, giving L^⟨ st⟩=Q(sqrt6). An element fixed by both s and t has only its rational component left.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-e93d2fe019f3b45a","tex":"L^{\\langle\\sigma\\rangle}","aliases":"L^⟨σ⟩ langle sigma rangle","display":"L^{\\langle\\sigma\\rangle}","kind":"use","excerpt":"The polynomial already splits in L, so L/Q is Galois and Gal(L/Q)={1,σ}≃ C_2. Its fixed elements satisfy a+bα=a-bα, whence b=0. This directly checks L^⟨σ⟩=Q without appealing to an abstract correspondence. The same argument works for every nonsquare d∈Q^×, with sqrt2 replaced by sqrt d.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"At t=2, the polynomial X^3-2 is irreducible by Eisenstein at the prime 2. Let α=sqrt[3]2 be the real root and let ζ be a primitive cube root of unity. The splitting field is L_2=Q(α,ζ). The real field Q(α) has degree three. It cannot contain the nonreal element ζ. Adjoining ζ, whose minimal polynomial is X^2+X+1, therefore doubles the degree. The s","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":6},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4e314a2a75f1f52c","tex":"M_{23}","aliases":"M_23","display":"M_{23}","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"Fix a finite group G. For every g∈ G, introduce an independent variable x_g. Let E=Q(x_gmid g∈ G) be the rational function field in these |G| variables. Algebraic independence means that no nonzero polynomial with rational coefficients in the x_g vanishes. In particular distinct variables are distinct elements of E.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":20},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b6a871d443ee2d56","tex":"\\operatorname{N}","aliases":"N operatorname","display":"\\operatorname{N}","kind":"use","excerpt":"In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":12},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8bbbd92a94033296","tex":"n'","aliases":"n'","display":"n'","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-06e2c5f390f46676","tex":"\\operatorname{N}_{L/L^H}","aliases":"N_L/L^H operatorname","display":"\\operatorname{N}_{L/L^H}","kind":"use","excerpt":"In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-703ac0970fbf9ef4","tex":"\\operatorname{N}_{L/L^K}","aliases":"N_L/L^K operatorname","display":"\\operatorname{N}_{L/L^K}","kind":"use","excerpt":"the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-40bdebe77fbdd230","tex":"\\operatorname{N}_{L/\\Q}","aliases":"N_L/Q operatorname Q","display":null,"kind":"use","excerpt":"[One projection, different norms] In L=Q(sqrt2), let H be the full group. For x=a+bsqrt2, P_H(x)=a, Tr_L/Q(x)=2a, N_L/Q(x)=a^2-2b^2. The elements 1 and 1+sqrt2 have the same projection and the same trace, but their norms are 1 and -1. There is therefore no function of the projected value alone that recovers the norm on all of L. Similarly, P_H(sqrt","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f30a2508d38f5158","tex":"\\operatorname{N}_{L^H/L^K}","aliases":"N_L^H/L^K operatorname","display":"\\operatorname{N}_{L^H/L^K}","kind":"use","excerpt":"the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-124043207812d8aa","tex":"\\overline\\Q","aliases":"overlineQ overline Q","display":null,"kind":"use","excerpt":"Fix an algebraic closure overlineQ, and write G_Q=Gal(overlineQ/Q) with its Krull topology. For a finite Galois subfield L⊂overlineQ, restriction supplies a continuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a ","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-1918b7c23a15b567","tex":"\\overline{\\Q}","aliases":"overlineQ overline Q","display":null,"kind":"use","excerpt":"An arithmetic embedding problem uses the profinite group G_Q=Gal(overlineQ/Q) with its Krull topology. A finite target group has the discrete topology. The lift is continuous, compatible with the prescribed quotient, and proper when surjective. The finite cyclic examples in Chapter~6 use a different ambient group and are explicitly models.","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0e517349dec3b6ba","tex":"\\overset","aliases":"overset","display":null,"kind":"use","excerpt":"Fix an algebraic closure overlineQ, and write G_Q=Gal(overlineQ/Q) with its Krull topology. For a finite Galois subfield L⊂overlineQ, restriction supplies a continuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies","count":10},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":13},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"An orbit polynomial gives a useful piece of the theorem's mechanism. For any a∈ E, P_a(T)=∏_h∈ G(T-σ_h(a)) has coefficients in K, since the action permutes its factors. Thus every element of E is algebraic over K. Orbit repetitions are allowed, so this polynomial need not be the minimal polynomial and can have repeated factors. The fact that all su","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b9e70c157342a064","tex":"P_H","aliases":"P_H","display":"P_H","kind":"use","excerpt":"The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6 x^3+3x^2+2x+5pmod7 · 5 · 4 · 1 · 2 · 6 · 5 · 5. It has no root, hence is irreduci","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"There is also an exact projector test. If P,Q are orthogonal projections and PQ is an orthogonal projection, then self adjointness gives PQ=(PQ)^*=QP. In the compatible case, E_HE_K is the projection onto the common fixed range. In the incompatible case, one must retain the ordered products as transfers. Replacing them by a shared refinement would ","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"bibitemHuangEtAl2026 Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries and Shaowu Zhang. The Mathieu group M_23 is a Galois group over Q. arXiv:2608.08538v2, submitted 7 October 2026. Manuscript dated 4 October 2026. Preprint used for dated context. https://arxiv.org/abs/2608.08538v2.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b9baeebca7f43f14","tex":"\\Q^4","aliases":"Q^4 Q","display":null,"kind":"use","excerpt":"[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-aca908de23a8a76f","tex":"\\Q^G","aliases":"Q^G Q","display":null,"kind":"use","excerpt":"Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constru","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f7ec10b45d9b6e51","tex":"\\Q^\\times","aliases":"Q^× Q times","display":null,"kind":"use","excerpt":"The polynomial already splits in L, so L/Q is Galois and Gal(L/Q)={1,σ}≃ C_2. Its fixed elements satisfy a+bα=a-bα, whence b=0. This directly checks L^⟨σ⟩=Q without appealing to an abstract correspondence. The same argument works for every nonsquare d∈Q^×, with sqrt2 replaced by sqrt d.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"The field Q(α) is contained in R, whereas ζ is nonreal. Thus ζ∉Q(α). Its quadratic equation remains irreducible over this real field, and [L:Q]=[L:Q(α)] [Q(α):Q]=2·3=6. The faithful action on three roots embeds Gal(L/Q) into S_3. Both groups have order six, so the embedding is an isomorphism.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-aff4a4490ea52474","tex":"r^{-1}","aliases":"r^-1","display":"r^{-1}","kind":"use","excerpt":"[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, b","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/05_cyclic_cubic.pdf captionThe cubic realization closes inside one root field. The transformation r↦ r^2-2 cycles its three roots, and their exact symmetric identities recover the original polynomial. The group has order three because the polynomial is irreducible and already spl","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-96dee922af39121c","tex":"r^3","aliases":"r^3","display":"r^3","kind":"use","excerpt":"But krk=(1342) is not among 1,r,r^2,r^3, so H is not normal. Moreover kr∈ KH does not lie in HK. If it equalled r^i, then k∈ H, which is false. If it equalled r^ik, then krk=r^i, which is also false. Hence HK≠ KH. Both reverse implications fail in this faithful group action. They hold in the regular action because its group algebra representation r","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-48dec93590be8cd8","tex":"r^i","aliases":"r^i","display":"r^i","kind":"use","excerpt":"But krk=(1342) is not among 1,r,r^2,r^3, so H is not normal. Moreover kr∈ KH does not lie in HK. If it equalled r^i, then k∈ H, which is false. If it equalled r^ik, then krk=r^i, which is also false. Hence HK≠ KH. Both reverse implications fail in this faithful group action. They hold in the regular action because its group algebra representation r","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d5c7051b56820f95","tex":"r_3","aliases":"r_3","display":"r_3","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ade69a4d18e62b84","tex":"r_g","aliases":"r_g","display":"r_g","kind":"use","excerpt":"For every g∈ G, specify a polynomial r_g∈Q[X] of degree less than n. It is intended to describe σ_g(θ)=r_g(θ). The equality f(r_g(θ))=0 ensures that this substitution induces a unital Q-algebra endomorphism of L. Because L is a field, the endomorphism is injective. Because it is a linear endomorphism of a finite dimensional vector space, it is surj","count":8},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-27204dac11744c0b","tex":"r_{gh}","aliases":"r_gh","display":"r_{gh}","kind":"use","excerpt":"The group law must still hold. With our convention that σ_gσ_h means apply σ_h first, the required congruences are r_1(X) · ≡ Xpmod f, r_h(r_g(X)) · ≡ r_gh(X)pmod f. The reversed order of the displayed polynomial substitutions is deliberate. Applying σ_g to r_h(θ) replaces θ by r_g(θ), since rational coefficients are fixed. Finally the residues r_g","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-455a3f2c7c575c0e","tex":"r_h","aliases":"r_h","display":"r_h","kind":"use","excerpt":"The group law must still hold. With our convention that σ_gσ_h means apply σ_h first, the required congruences are r_1(X) · ≡ Xpmod f, r_h(r_g(X)) · ≡ r_gh(X)pmod f. The reversed order of the displayed polynomial substitutions is deliberate. Applying σ_g to r_h(θ) replaces θ by r_g(θ), since rational coefficients are fixed. Finally the residues r_g","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0876bfe73de67b89","tex":"r_s","aliases":"r_s","display":"r_s","kind":"use","excerpt":"For G=C_2={1,s}, take f=X^2-2 and choose r_1=X, r_s=-X. The polynomial is irreducible over Q, because two is not a rational square. Both residues satisfy f(r_g)≡0pmod f. The nontrivial composition gives r_s(r_s(X))=X, and the two images are distinct. The certificate theorem returns Q(sqrt2)/Q with its full group C_2.","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-952542ed93bf41db","tex":"\\operatorname{ran}","aliases":"ran operatorname","display":"\\operatorname{ran}","kind":"use","excerpt":"Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.","count":32},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":42},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"The four elements 1, sqrt2, sqrt3, sqrt6 form a rational basis. Independent choices of signs define four automorphisms. Let s change the sign of sqrt2 and let t change the sign of sqrt3. They satisfy s^2=t^2=1, st=ts. They are distinct because their effects on the two generators are distinct. The extension is the splitting field of (X^2-2)(X^2-3), ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":19},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f6d0dc9f5b462d0b","tex":"S_4","aliases":"S_4","display":"S_4","kind":"use","excerpt":"Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.","count":13},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-508e9a069e81ec8b","tex":"S_d","aliases":"S_d","display":"S_d","kind":"use","excerpt":"Suppose f has d distinct roots in its splitting field L. Every automorphism of L/Q permutes them. The action is faithful because the roots generate L: an automorphism fixing every root fixes every rational expression in those roots. Consequently Gal(L/Q)↪ S_d. If f is irreducible, the action is transitive. One way to see this is to send any root to","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f93687b75635f48a","tex":"\\section","aliases":"section","display":null,"kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"The smallest example makes the warning exact. Let G=C_2 exchange two variables x,y. Define s=x+y, p=xy. The variables x,y are the roots of T^2-sT+p. Since y=s-x, the field Q(x,y) has degree at most two over Q(s,p). The nontrivial exchange fixes s,p, and x∉Q(s,p) because an element in that field would be fixed. Hence the degree is exactly two. Artin","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"The four elements 1, sqrt2, sqrt3, sqrt6 form a rational basis. Independent choices of signs define four automorphisms. Let s change the sign of sqrt2 and let t change the sign of sqrt3. They satisfy s^2=t^2=1, st=ts. They are distinct because their effects on the two generators are distinct. The extension is the splitting field of (X^2-2)(X^2-3), ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-67cf96d5e475b646","tex":"\\operatorname{Tr}_{L/L^H}","aliases":"Tr_L/L^H operatorname","display":"\\operatorname{Tr}_{L/L^H}","kind":"use","excerpt":"In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-37b7620aa9f4f99c","tex":"\\operatorname{Tr}_{L/L^K}","aliases":"Tr_L/L^K operatorname","display":"\\operatorname{Tr}_{L/L^K}","kind":"use","excerpt":"the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-5f79c21c47b05f5d","tex":"\\operatorname{Tr}_{L/\\Q}","aliases":"Tr_L/Q operatorname Q","display":null,"kind":"use","excerpt":"[One projection, different norms] In L=Q(sqrt2), let H be the full group. For x=a+bsqrt2, P_H(x)=a, Tr_L/Q(x)=2a, N_L/Q(x)=a^2-2b^2. The elements 1 and 1+sqrt2 have the same projection and the same trace, but their norms are 1 and -1. There is therefore no function of the projected value alone that recovers the norm on all of L. Similarly, P_H(sqrt","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f702f4f691f0df11","tex":"\\operatorname{Tr}_{L^H/L^K}","aliases":"Tr_L^H/L^K operatorname","display":"\\operatorname{Tr}_{L^H/L^K}","kind":"use","excerpt":"the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-3a0644b9c533e0d9","tex":"\\operatorname{trdeg}","aliases":"trdeg operatorname","display":"\\operatorname{trdeg}","kind":"use","excerpt":"More generally, E/K is algebraic, so the two fields have the same transcendence degree over Q: trdeg_QK=trdeg_QE=|G|. The finite degree [E:K] does not make E or K finite over Q. Both retain transcendental parameters. For the trivial group, the construction says only that E=K=Q(x_1), whereas its arithmetic realization was simply Q. Even this endpoin","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-160f869f94b2619d","tex":"\\operatorname{trdeg}_{\\Q}","aliases":"trdeg_Q operatorname Q","display":null,"kind":"use","excerpt":"More generally, E/K is algebraic, so the two fields have the same transcendence degree over Q: trdeg_QK=trdeg_QE=|G|. The finite degree [E:K] does not make E or K finite over Q. Both retain transcendental parameters. For the trivial group, the construction says only that E=K=Q(x_1), whereas its arithmetic realization was simply Q. Even this endpoin","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":22},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d849fcfcf96ad168","tex":"u^3","aliases":"u^3","display":"u^3","kind":"use","excerpt":"The generic splitting field also has group S_3. To prove this without an experimental inference, take a transcendental u and put t=u^3. The polynomial X^3-t is Eisenstein at the prime t in Q[t][X], hence irreducible over Q(t). Its root field is Q(u), and the splitting field is Q(u,ζ). The only elements of Q(u) algebraic over Q lie in Q. Indeed, if ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":8},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-759397de55065787","tex":"V^2","aliases":"V^2","display":"V^2","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-7a6544df8b60d670","tex":"V^H","aliases":"V^H","display":"V^H","kind":"use","excerpt":"[Fixed range and rank] The operator E_H is the orthogonal projector onto V^H={v∈ V:λ(h)v=v for every h∈ H}. Its rank is [G:H]. If H≤ K, then E_HE_K=E_KE_H=E_K, V^K⊆ V^H.","count":6},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8ee834c243d6b537","tex":"V^K","aliases":"V^K","display":"V^K","kind":"use","excerpt":"[Fixed range and rank] The operator E_H is the orthogonal projector onto V^H={v∈ V:λ(h)v=v for every h∈ H}. Its rank is [G:H]. If H≤ K, then E_HE_K=E_KE_H=E_K, V^K⊆ V^H.","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-5898e68a838fa76b","tex":"V^{\\langle H,K\\rangle}","aliases":"V^⟨ H,K⟩ langle rangle","display":"V^{\\langle H,K\\rangle}","kind":"use","excerpt":"rion] In the regular realization, the following statements are equivalent: HK=KH, HK is a subgroup, E_HE_K=E_KE_H. When they hold, E_HE_K=E_KE_H=E_HK=E_⟨ H,K⟩. Without a compatibility assumption, the intersection of the fixed ranges is still V^H∩ V^K=V^⟨ H,K⟩.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-641dfc399b115eba","tex":"V_4","aliases":"V_4","display":"V_4","kind":"use","excerpt":"Two independent signs: V_4.","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-bbd62cc53ebad75d","tex":"v_{Hg}","aliases":"v_Hg","display":"v_{Hg}","kind":"use","excerpt":"The orbits of H on the basis labels are the right cosets Hg. For one representative g from each coset, put v_Hg=|H|^-1/2∑_h∈ Hδ_hg. These vectors have disjoint supports, are orthonormal, and span the fixed range. Their number is [G:H]. Finally H≤ K makes every K fixed vector H fixed. Orthogonal projectors onto nested subspaces multiply to the proje","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-16cd7a5ab2eeeb5c","tex":"\\widetilde\\rho","aliases":"widetildeρ widetilde rho","display":"\\widetilde\\rho","kind":"use","excerpt":"tinuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a continuous homomorphism widetildeρ:G_Q⟶ E, π∘widetildeρ=ρ. It is a weak solution when this equation holds, and a proper solution when widetildeρ is also surject","count":18},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f39fc6268336557a","tex":"\\widetilde\\rho_0","aliases":"widetildeρ_0 widetilde rho","display":"\\widetilde\\rho_0","kind":"use","excerpt":"[The kernel is missed in a direct product] Take E=N× G with π(n,g)=g. The section s(g)=(1,g) gives widetildeρ_0(γ)=(1,ρ(γ)). Its image is {1}× G. It reaches every element of the desired quotient, but reaches none of the nonidentity elements of N×{1}. With the finite toy N=G=Γ=C_2 and ρ the identity, the image has two elements while E has four. All ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"Let α=sqrt2 and L=Q(α). The polynomial X^2-2 is irreducible over Q. Indeed, if a/b in lowest terms had square two, then a^2=2b^2 would make a even. Writing a=2c gives b^2=2c^2, so b would also be even, a contradiction.","count":12},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6 x^3+3x^2+2x+5pmod7 · 5 · 4 · 1 · 2 · 6 · 5 · 5. It has no root, hence is irreduci","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"use","excerpt":"Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6 x^3+3x^2+2x+5pmod7 · 5 · 4 · 1 · 2 · 6 · 5 · 5. It has no root, hence is irreduci","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ee74251aad7234d0","tex":"X^4","aliases":"X^4","display":"X^4","kind":"use","excerpt":"For a quartic realization, consider f(X)=X^4-X-1. Its reduction modulo two is X^4+X+1. It has no root in F_2. The only monic irreducible quadratic over F_2 is X^2+X+1. Modulo that polynomial, X^2=X+1, X^3=1 and X^4=X, so X^4+X+1 has remainder one. A reducible quartic without a linear factor must factor as two irreducible quadratics. This possibilit","count":6},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"equation σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.","count":10},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ca7008ae447ee9dc","tex":"x_4","aliases":"x_4","display":"x_4","kind":"use","excerpt":"[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-9d6780ab349082dd","tex":"x_h","aliases":"x_h","display":"x_h","kind":"use","excerpt":"equation σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d568c8883faca719","tex":"x_{hg}","aliases":"x_hg","display":"x_{hg}","kind":"use","excerpt":"For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":13},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"use","excerpt":"An integral polynomial can be used if desired. Clear denominators to obtain a polynomial in Z[X] with the same roots. The resulting polynomial need not be monic. Alternatively, multiply a primitive element by an integer large enough to make it integral and use its monic minimal polynomial. These changes concern presentation. They do not replace the","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant c","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-6820374f59f0b332","tex":"|^{-1}","aliases":"|^-1","display":"|^{-1}","kind":"use","excerpt":"The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-e0e8a1f7f49f7cad","tex":"|^{-1/2}","aliases":"|^-1/2","display":"|^{-1/2}","kind":"use","excerpt":"The orbits of H on the basis labels are the right cosets Hg. For one representative g from each coset, put v_Hg=|H|^-1/2∑_h∈ Hδ_hg. These vectors have disjoint supports, are orthonormal, and span the fixed range. Their number is [G:H]. Finally H≤ K makes every K fixed vector H fixed. Orthogonal projectors onto nested subspaces multiply to the proje","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed v","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constru","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"The polynomial already splits in L, so L/Q is Galois and Gal(L/Q)={1,σ}≃ C_2. Its fixed elements satisfy a+bα=a-bα, whence b=0. This directly checks L^⟨σ⟩=Q without appealing to an abstract correspondence. The same argument works for every nonsquare d∈Q^×, with sqrt2 replaced by sqrt d.","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d830d9bc7e9c803c","tex":"\\alpha^2","aliases":"α^2 alpha","display":"\\alpha^2","kind":"use","excerpt":"Every element of L has a unique expression a+bα with a,b∈Q. The map σ(a+bα)=a-bα preserves addition and multiplication because both α and -α satisfy the same relation α^2=2. It fixes Q, has square the identity, and is nontrivial. The only possible images of α under a rational automorphism are α and -α. Thus there are exactly two automorphisms.","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0c2e7bd0f1d04e12","tex":"\\alpha^3","aliases":"α^3 alpha","display":"\\alpha^3","kind":"use","excerpt":"We can see the splitting field without using the discriminant criterion. Let α be a root. In Q(α), the defining equation gives α^3=-α^2+2α+1, α^4=3α^2-α-1. Set β=α^2-2, γ=-α^2-α+1. Direct reduction by the two displayed identities gives α+β+γ=-1, αβ+αγ+βγ=-2, αβγ=1. Thus f(X)=(X-α)(X-β)(X-γ) in Q(α)[X]. For example, the second identity follows from ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-40b7e675597d113c","tex":"\\alpha^4","aliases":"α^4 alpha","display":"\\alpha^4","kind":"use","excerpt":"We can see the splitting field without using the discriminant criterion. Let α be a root. In Q(α), the defining equation gives α^3=-α^2+2α+1, α^4=3α^2-α-1. Set β=α^2-2, γ=-α^2-α+1. Direct reduction by the two displayed identities gives α+β+γ=-1, αβ+αγ+βγ=-2, αβγ=1. Thus f(X)=(X-α)(X-β)(X-γ) in Q(α)[X]. For example, the second identity follows from ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"We can see the splitting field without using the discriminant criterion. Let α be a root. In Q(α), the defining equation gives α^3=-α^2+2α+1, α^4=3α^2-α-1. Set β=α^2-2, γ=-α^2-α+1. Direct reduction by the two displayed identities gives α+β+γ=-1, αβ+αγ+βγ=-2, αβγ=1. Thus f(X)=(X-α)(X-β)(X-γ) in Q(α)[X]. For example, the second identity follows from ","count":7},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ab15d1c00936ee24","tex":"\\beta^2","aliases":"β^2 beta","display":"\\beta^2","kind":"use","excerpt":"Since it splits there, the root field is already its splitting field. Its Galois group has order three and is therefore cyclic. There is an explicit generator sending α to β. The map is well defined because β is another root of the same irreducible polynomial. It is a field embedding into Q(α) and hence an automorphism, since the image has degree t","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.","count":11},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"We can see the splitting field without using the discriminant criterion. Let α be a root. In Q(α), the defining equation gives α^3=-α^2+2α+1, α^4=3α^2-α-1. Set β=α^2-2, γ=-α^2-α+1. Direct reduction by the two displayed identities gives α+β+γ=-1, αβ+αγ+βγ=-2, αβγ=1. Thus f(X)=(X-α)(X-β)(X-γ) in Q(α)[X]. For example, the second identity follows from ","count":16},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4569df46706329ae","tex":"\\gamma^2","aliases":"γ^2 gamma","display":"\\gamma^2","kind":"use","excerpt":"Since it splits there, the root field is already its splitting field. Its Galois group has order three and is therefore cyclic. There is an explicit generator sending α to β. The map is well defined because β is another root of the same irreducible polynomial. It is a field embedding into Q(α) and hence an automorphism, since the image has degree t","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":". In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure lab","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","excerpt":"[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, b","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-845f98ac1e4d0fbd","tex":"\\delta_b","aliases":"δ_b delta","display":"\\delta_b","kind":"use","excerpt":"[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, b","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-1e6c2f33619b8d05","tex":"\\delta_{hg}","aliases":"δ_hg delta","display":"\\delta_{hg}","kind":"use","excerpt":"The orbits of H on the basis labels are the right cosets Hg. For one representative g from each coset, put v_Hg=|H|^-1/2∑_h∈ Hδ_hg. These vectors have disjoint supports, are orthonormal, and span the fixed range. Their number is [G:H]. Finally H≤ K makes every K fixed vector H fixed. Orthogonal projectors onto nested subspaces multiply to the proje","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-043cc7068d423bd8","tex":"\\delta_r","aliases":"δ_r delta","display":"\\delta_r","kind":"use","excerpt":"[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, b","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b38fdb6057dc6503","tex":"\\delta_{r^{-1}}","aliases":"δ_r^-1 delta","display":"\\delta_{r^{-1}}","kind":"use","excerpt":"[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, b","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-35348365f6cc6efe","tex":"\\zeta_3","aliases":"ζ_3 zeta","display":"\\zeta_3","kind":"use","excerpt":"[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/17_parameter_bad.pdf captionThe specialization toy X^3-t has group S_3 at t=2, group C_2 at the nonsingular value t=1, and trivial splitting field at t=0. Its generic splitting field contains ζ_3, so this is not a regular family over Q.","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"For a direct product, a candidate lift has the form (η,ρ), where η:Γ→ N is a homomorphism. Even surjectivity of both coordinates does not automatically imply surjectivity of the pair. If Γ=N=G=C_2 and both maps are the identity, their combined image is the diagonal subgroup of order two. Proposition~[source reference] locates the failure: the kerne","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"Let G be a finite group of order n. A polynomial field presentation consists of a monic irreducible polynomial f∈Q[X] of degree n and L=Q[X]/(f), θ=Xbmod f. Irreducibility is the field input. The quotient is automatically a commutative algebra of dimension n, but that dimension alone would also hold for a reducible polynomial. A reducible quotient ","count":6},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"The transported multiplication is a bilinear map μ:Q[G]⊗_QQ[G]⟶Q[G], μ(u,v)=Φ^-1(Φ(u)Φ(v)). For the left regular action λ, it obeys λ(g)μ(u,v)=μ(λ(g)u,λ(g)v). It is associative and commutative, has a unit and has no zero divisors. Its coefficients depend on the field and on the chosen normal basis. The normal basis theorem assumes that field before","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"Fix an algebraic closure overlineQ, and write G_Q=Gal(overlineQ/Q) with its Krull topology. For a finite Galois subfield L⊂overlineQ, restriction supplies a continuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a ","count":12},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constru","count":32},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"The substitutions give a homomorphism from G into Aut_Q(L). The identity and composition congruences prove the homomorphism law. Distinct generator images make the homomorphism injective. Write its image as Σ, so |Σ|=n.","count":8},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"Every element of L has a unique expression a+bα with a,b∈Q. The map σ(a+bα)=a-bα preserves addition and multiplication because both α and -α satisfy the same relation α^2=2. It fixes Q, has square the identity, and is nontrivial. The only possible images of α under a rational automorphism are α and -α. Thus there are exactly two automorphisms.","count":13},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-1d8ad5c3dcba600b","tex":"\\sigma_g","aliases":"σ_g sigma","display":"\\sigma_g","kind":"use","excerpt":"For every g∈ G, specify a polynomial r_g∈Q[X] of degree less than n. It is intended to describe σ_g(θ)=r_g(θ). The equality f(r_g(θ))=0 ensures that this substitution induces a unital Q-algebra endomorphism of L. Because L is a field, the endomorphism is injective. Because it is a linear endomorphism of a finite dimensional vector space, it is surj","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-a9aa546b822f4a26","tex":"\\sigma_h","aliases":"σ_h sigma","display":"\\sigma_h","kind":"use","excerpt":"For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.","count":7},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-059636082e4170c7","tex":"\\sigma_{h^{-1}}","aliases":"σ_h^-1 sigma","display":"\\sigma_{h^{-1}}","kind":"use","excerpt":"For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-821beed71925a4d8","tex":"\\sigma_{hk}","aliases":"σ_hk sigma","display":"\\sigma_{hk}","kind":"use","excerpt":"For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-fa431dd60a582371","tex":"\\sigma_k","aliases":"σ_k sigma","display":"\\sigma_k","kind":"use","excerpt":"For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"There is also a concrete pair of generators. Over Q(ζ), the element α has degree three: the tower degree is [L:Q(ζ)]=3. Sending α to ζα while fixing ζ therefore gives an automorphism τ of order three. Complex conjugation gives an automorphism c of order two that fixes α and sends ζ to ζ^2. On the generators, cτ c=τ^-1. Hence the cyclic cubic subgro","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-cff2c5f7d8b191ec","tex":"\\tau^{-1}","aliases":"τ^-1 tau","display":"\\tau^{-1}","kind":"use","excerpt":"There is also a concrete pair of generators. Over Q(ζ), the element α has degree three: the tower degree is [L:Q(ζ)]=3. Sending α to ζα while fixing ζ therefore gives an automorphism τ of order three. Complex conjugation gives an automorphism c of order two that fixes α and sends ζ to ζ^2. On the generators, cτ c=τ^-1. Hence the cyclic cubic subgro","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"The normal basis theorem supplies a regular module for an already existing finite Galois extension [source reference]. If L/Q has group G and normal basis generator α, the vector-space map Φ:Q[G]⟶ L, ∑_g a_g g⟼∑_g a_g g(α) is an equivariant linear isomorphism. It transfers subgroup averages and their fixed vector spaces. It is not a claim that the ","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-18f9ac23eff67597","tex":"\\Phi^{-1}","aliases":"Φ^-1 Phi","display":"\\Phi^{-1}","kind":"use","excerpt":"The transported multiplication is a bilinear map μ:Q[G]⊗_QQ[G]⟶Q[G], μ(u,v)=Φ^-1(Φ(u)Φ(v)). For the left regular action λ, it obeys λ(g)μ(u,v)=μ(λ(g)u,λ(g)v). It is associative and commutative, has a unit and has no zero divisors. Its coefficients depend on the field and on the chosen normal basis. The normal basis theorem assumes that field before","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Let H be the Galois group in its faithful action on four roots. Its order is divisible by four and three, so it is divisible by twelve. Since H≤ S_4, its order is twelve or twenty four. A subgroup of order twelve would have index two and would be normal. The quotient map S_4→ C_2 would send all transpositions to the same element because they are co","count":12},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/05_cyclic_cubic.pdf captionThe cubic realization closes inside one root field. The transformation r↦ r^2-2 cycles its three roots, and their exact symmetric identities recover the original polynomial. The group has order three because the polynomial is irreducible and already spl","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"Suppose f has d distinct roots in its splitting field L. Every automorphism of L/Q permutes them. The action is faithful because the roots generate L: an automorphism fixing every root fixes every rational expression in those roots. Consequently Gal(L/Q)↪ S_d. If f is irreducible, the action is transitive. One way to see this is to send any root to","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"The field Q(α) is contained in R, whereas ζ is nonreal. Thus ζ∉Q(α). Its quadratic equation remains irreducible over this real field, and [L:Q]=[L:Q(α)] [Q(α):Q]=2·3=6. The faithful action on three roots embeds Gal(L/Q) into S_3. Both groups have order six, so the embedding is an isomorphism.","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ae5d80b648ebc895","tex":"\\prod_{g\\in G}","aliases":"∏_g∈ G prod in","display":"\\prod_{g\\in G}","kind":"use","excerpt":"In our particular construction, a=x_1 has |G| distinct images. The polynomial P(T)=∏_g∈ G(T-x_g) has degree |G| and coefficients in K. Any polynomial over K vanishing at x_1 vanishes at every x_g after applying the automorphisms. It therefore has at least |G| distinct roots. The minimal polynomial of x_1 has degree |G|. In view of~[source reference","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-f33e7e19c68ec4f8","tex":"\\prod_{h\\in G}","aliases":"∏_h∈ G prod in","display":"\\prod_{h\\in G}","kind":"use","excerpt":"An orbit polynomial gives a useful piece of the theorem's mechanism. For any a∈ E, P_a(T)=∏_h∈ G(T-σ_h(a)) has coefficients in K, since the action permutes its factors. Thus every element of E is algebraic over K. Orbit repetitions are allowed, so this polynomial need not be the minimal polynomial and can have repeated factors. The fact that all su","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-94daef9b8e1acd63","tex":"\\prod_{h\\in H}","aliases":"∏_h∈ H prod in","display":"\\prod_{h\\in H}","kind":"use","excerpt":"In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b8ef85e0df350209","tex":"\\prod_{i<j}","aliases":"∏_i<j prod","display":"\\prod_{i<j}","kind":"use","excerpt":"The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreduci","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-76e758f1be5158a2","tex":"\\sum_g","aliases":"∑_g sum","display":"\\sum_g","kind":"use","excerpt":"Already the invariant s=∑_g∈ Gx_g is transcendental over Q. If a nonzero polynomial F∈Q[T] satisfied F(s)=0, then F(∑_gx_g) would be a zero polynomial in independent variables. Substituting zero for every variable except x_1 would give the impossible identity F(x_1)=0. Thus s∈ K∖Q.","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-33e3aee1069cd50e","tex":"\\sum_{h,k\\in H}","aliases":"∑_h,k∈ H sum in","display":"\\sum_{h,k\\in H}","kind":"use","excerpt":"Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed v","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-e177bb3bef4abaec","tex":"\\sum_{h\\in H}","aliases":"∑_h∈ H sum in","display":"\\sum_{h\\in H}","kind":"use","excerpt":"The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies","count":4},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-c40b8c379565d927","tex":"\\sum_{u\\in H}","aliases":"∑_u∈ H sum in","display":"\\sum_{u\\in H}","kind":"use","excerpt":"Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed v","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Already the invariant s=∑_g∈ Gx_g is transcendental over Q. If a nonzero polynomial F∈Q[T] satisfied F(s)=0, then F(∑_gx_g) would be a zero polynomial in independent variables. Substituting zero for every variable except x_1 would give the impossible identity F(x_1)=0. Thus s∈ K∖Q.","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. ","count":3},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"[Inverse Galois over the rational numbers] For every finite group G, there is a finite Galois extension L/Q and an isomorphism G≃Gal(L/Q).","count":7},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"ndence is an external input, in the form stated by Milne [source reference]. It sends H≤ G to L^H={x∈ L:h(x)=x for every h∈ H}. This is a field because every h respects sums, products, and inverses. The theorem identifies Gal(L/L^H)=H, gives [L^H:Q]=[G:H], and makes subgroup inclusion reverse field inclusion. If H is normal, it also gives Gal(L^H/Q","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"The group law must still hold. With our convention that σ_gσ_h means apply σ_h first, the required congruences are r_1(X) · ≡ Xpmod f, r_h(r_g(X)) · ≡ r_gh(X)pmod f. The reversed order of the displayed polynomial substitutions is deliberate. Applying σ_g to r_h(θ) replaces θ by r_g(θ), since rational coefficients are fixed. Finally the residues r_g","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/02_field_group.pdf captionFor an already supplied finite Galois extension L/K with group G, subgroups H≤ G correspond to intermediate fixed fields L^H. The direction of inclusion reverses. The classical correspondence computes degrees and organizes fields after the ambient field ","count":2},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"For H≤ G, define e_H=frac1|H|∑_h∈ Hh∈C[G], E_H=λ(e_H)=frac1|H|∑_h∈ Hλ(h). The notation E_H in this chapter and its figures always means this operator. It is an average with normalization 1/|H|, rather than an unnormalized sum.","count":7},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"Fix an algebraic closure overlineQ, and write G_Q=Gal(overlineQ/Q) with its Krull topology. For a finite Galois subfield L⊂overlineQ, restriction supplies a continuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a ","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.","count":5},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","excerpt":"The transported multiplication is a bilinear map μ:Q[G]⊗_QQ[G]⟶Q[G], μ(u,v)=Φ^-1(Φ(u)Φ(v)). For the left regular action λ, it obeys λ(g)μ(u,v)=μ(λ(g)u,λ(g)v). It is associative and commutative, has a unit and has no zero divisors. Its coefficients depend on the field and on the chosen normal basis. The normal basis theorem assumes that field before","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-79d27546b685aec1","tex":"\\otimes_\\Q","aliases":"⊗_Q otimes Q","display":null,"kind":"use","excerpt":"The transported multiplication is a bilinear map μ:Q[G]⊗_QQ[G]⟶Q[G], μ(u,v)=Φ^-1(Φ(u)Φ(v)). For the left regular action λ, it obeys λ(g)μ(u,v)=μ(λ(g)u,λ(g)v). It is associative and commutative, has a unit and has no zero divisors. Its coefficients depend on the field and on the chosen normal basis. The normal basis theorem assumes that field before","count":1},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"For the ordered basis (), (23), (12), (123), (132), (13) the script constructs matrix columns from λ(g)e_x=e_gx. Every entry is an exact rational number. The H=⟨(12)⟩ and K=⟨(23)⟩ commutator is nonzero. Matrix ranks are computed over Q, not with a floating tolerance. The figure source reads the frozen JSON generated by the same run. The individual ","count":12},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"For the ordered basis (), (23), (12), (123), (132), (13) the script constructs matrix columns from λ(g)e_x=e_gx. Every entry is an exact rational number. The H=⟨(12)⟩ and K=⟨(23)⟩ commutator is nonzero. Matrix ranks are computed over Q, not with a floating tolerance. The figure source reads the frozen JSON generated by the same run. The individual ","count":12},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"The normal basis theorem supplies a useful translation [source reference]. For some α∈ L, the elements g(α), with g∈ G, form a Q basis. The map Q[G]⟶ L, ∑_g c_g g⟼∑_g c_g g(α) is an equivariant vector space isomorphism. After extending scalars to C, it transports the regular subgroup averages to the fixed vector spaces of the field action. This ver","count":6},{"work":"inverse-galois-reader","url":"notation/inverse-galois-reader.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"The normal basis theorem supplies a useful translation [source reference]. For some α∈ L, the elements g(α), with g∈ G, form a Q basis. The map Q[G]⟶ L, ∑_g c_g g⟼∑_g c_g g(α) is an equivariant vector space isomorphism. After extending scalars to C, it transports the regular subgroup averages to the fixed vector spaces of the field action. This ver","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":201},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":626},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":35},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":126},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":14},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9f9a558588be02af","tex":"a_{j+1}","aliases":"a_j+1","display":"a_{j+1}","kind":"introduction-passage","excerpt":"ogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coefficients","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":34},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"introduction-passage","excerpt":"ial at each fixed shell Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde numerators: P_n(5)=288τ(n), B_0(n)=288|Vn|. The variable T is auxi","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":49},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":68},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":35},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6662c032276e077b","tex":"E_4","aliases":"E_4","display":"E_4","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":12},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-13baa1cc3bbd07ae","tex":"E_4^2","aliases":"E_4^2","display":"E_4^2","kind":"introduction-passage","excerpt":"E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The denominator 1728 is exact, and an integer implementation can test the required divisibility before extr","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-68f15c955d8c7937","tex":"E_4^3","aliases":"E_4^3","display":"E_4^3","kind":"introduction-passage","excerpt":"E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The denominator 1728 is exact, and an integer implementation can test the required divisibility before extr","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-51b438373b685e40","tex":"E_6","aliases":"E_6","display":"E_6","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9e39f1ed1f05c6e3","tex":"E_6^2","aliases":"E_6^2","display":"E_6^2","kind":"introduction-passage","excerpt":"E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The denominator 1728 is exact, and an integer implementation can test the required divisibility before extr","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-edf98275838926c6","tex":"E_8","aliases":"E_8","display":"E_8","kind":"introduction-passage","excerpt":"E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The denominator 1728 is exact, and an integer implementation can test the required divisibility before extr","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":50},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":67},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6e5f2328acefacea","tex":"\\mathcal H_M","aliases":"H_M mathcal","display":"\\mathcal H_M","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":83},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":117},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"The symbol T_k(p) in the trace-formula chapter denotes a Hecke operator on the weight-k cusp-form space. It is an operator, whereas T_r in a scalar recurrence is an integer or an integer polynomial. The weight k and the exponent r serve different roles.","count":87},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":83},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":112},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0f5cef98cfcf15fe","tex":"M_c","aliases":"M_c","display":"M_c","kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e4524d978e547026","tex":"M_f","aliases":"M_f","display":"M_f","kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":155},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":29},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"The symbol T_k(p) in the trace-formula chapter denotes a Hecke operator on the weight-k cusp-form space. It is an operator, whereas T_r in a scalar recurrence is an integer or an integer polynomial. The weight k and the exponent r serve different roles.","count":324},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8ddf7d5771645ae0","tex":"p^{2m+1}","aliases":"p^2m+1","display":"p^{2m+1}","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-76b7670fac05addd","tex":"P_n","aliases":"P_n","display":"P_n","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":77},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":68},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-102bb645d8547d75","tex":"q^n","aliases":"q^n","display":"q^n","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":11},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":65},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"The symbol T_k(p) in the trace-formula chapter denotes a Hecke operator on the weight-k cusp-form space. It is an operator, whereas T_r in a scalar recurrence is an integer or an integer polynomial. The weight k and the exponent r serve different roles.","count":77},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":127},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"introduction-passage","excerpt":"H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coefficients satisfy this condition. The last possible iterated Laplacian has no further nonzero derivative, so the resulting polynomial is harmonic. Its difference from P ","count":43},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f0227a6aa1c8c091","tex":"s_2^j","aliases":"s_2^j","display":"s_2^j","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-257f22917baea591","tex":"s_{2j}","aliases":"s_2j","display":"s_{2j}","kind":"introduction-passage","excerpt":"Trace moments and a unit triangular inversion. The Hurwitz-moment description. Another exact route uses Hurwitz class numbers. Let H(D) denote the conventional weighted Hurwitz class number for a positive discriminant parameter D, with the standard exceptional automorphism weights. At a prime p define s_2j(p)=frac12∑_|t|<2sqrt pt^2jH(4p-t^2). Here ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-95d3d602121aab4d","tex":"s_f","aliases":"s_f","display":"s_f","kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"introduction-passage","excerpt":"exact route uses Hurwitz class numbers. Let H(D) denote the conventional weighted Hurwitz class number for a positive discriminant parameter D, with the standard exceptional automorphism weights. At a prime p define s_2j(p)=frac12∑_|t|<2sqrt pt^2jH(4p-t^2). Here the symbol s_2j(p) is a trace moment and is unrelated to the coordinate power sums s_j(","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"introduction-passage","excerpt":"The symbol T_k(p) in the trace-formula chapter denotes a Hecke operator on the weight-k cusp-form space. It is an operator, whereas T_r in a scalar recurrence is an integer or an integer polynomial. The weight k and the exponent r serve different roles.","count":74},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"introduction-passage","excerpt":"Trace moments and a unit triangular inversion. The Hurwitz-moment description. Another exact route uses Hurwitz class numbers. Let H(D) denote the conventional weighted Hurwitz class number for a positive discriminant parameter D, with the standard exceptional automorphism weights. At a prime p define s_2j(p)=frac12∑_|t|<2sqrt pt^2jH(4p-t^2). Here ","count":44},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"introduction-passage","excerpt":"Trace moments and a unit triangular inversion. The Hurwitz-moment description. Another exact route uses Hurwitz class numbers. Let H(D) denote the conventional weighted Hurwitz class number for a positive discriminant parameter D, with the standard exceptional automorphism weights. At a prime p define s_2j(p)=frac12∑_|t|<2sqrt pt^2jH(4p-t^2). Here ","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1bede04035c2d044","tex":"t^{2j}","aliases":"t^2j","display":"t^{2j}","kind":"introduction-passage","excerpt":"Trace moments and a unit triangular inversion. The Hurwitz-moment description. Another exact route uses Hurwitz class numbers. Let H(D) denote the conventional weighted Hurwitz class number for a positive discriminant parameter D, with the standard exceptional automorphism weights. At a prime p define s_2j(p)=frac12∑_|t|<2sqrt pt^2jH(4p-t^2). Here ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9ea7696d8ec4186b","tex":"T^j","aliases":"T^j","display":"T^j","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8beb0f8d8d01c91b","tex":"T_{2m+1}","aliases":"T_2m+1","display":"T_{2m+1}","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c5211f32d9fb7e7e","tex":"T_{2m-1}","aliases":"T_2m-1","display":"T_{2m-1}","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b6d1c35d86f0d39d","tex":"T_k","aliases":"T_k","display":"T_k","kind":"introduction-passage","excerpt":"The symbol T_k(p) in the trace-formula chapter denotes a Hecke operator on the weight-k cusp-form space. It is an operator, whereas T_r in a scalar recurrence is an integer or an integer polynomial. The weight k and the exponent r serve different roles.","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-df3f310abe33eba9","tex":"T_r","aliases":"T_r","display":"T_r","kind":"introduction-passage","excerpt":"The symbol T_k(p) in the trace-formula chapter denotes a Hecke operator on the weight-k cusp-form space. It is an operator, whereas T_r in a scalar recurrence is an integer or an integer polynomial. The weight k and the exponent r serve different roles.","count":16},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":28},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-bc96f42eef79641f","tex":"U_{p,0}","aliases":"U_p,0","display":"U_{p,0}","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3d0e266cafd311b6","tex":"U_{p,m}","aliases":"U_p,m","display":"U_{p,m}","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"exact route uses Hurwitz class numbers. Let H(D) denote the conventional weighted Hurwitz class number for a positive discriminant parameter D, with the standard exceptional automorphism weights. At a prime p define s_2j(p)=frac12∑_|t|<2sqrt pt^2jH(4p-t^2). Here the symbol s_2j(p) is a trace moment and is unrelated to the coordinate power sums s_j(","count":180},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"introduction-passage","excerpt":"For every odd prime p≠5 and every m≥0, an odd integer U_p,m satisfies R(p^2m+1)=τ(p)U_p,m. The multiplier is defined by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from th","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":35},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c4b70e3e67861ca6","tex":"\\Vn","aliases":"Vn","display":null,"kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":23},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":19},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-225121a7d7ef24ff","tex":"\\Z^{16\\times18}","aliases":"Z^16×18 Z times","display":null,"kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-40fd2e53382d79a7","tex":"\\Z^{6\\times16}","aliases":"Z^6×16 Z times","display":null,"kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ab6377a21eb90171","tex":"\\Z^{6\\times18}","aliases":"Z^6×18 Z times","display":null,"kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"Trace moments and a unit triangular inversion. The Hurwitz-moment description. Another exact route uses Hurwitz class numbers. Let H(D) denote the conventional weighted Hurwitz class number for a positive discriminant parameter D, with the standard exceptional automorphism weights. At a prime p define s_2j(p)=frac12∑_|t|<2sqrt pt^2jH(4p-t^2). Here ","count":26},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":14},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":71},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-28f5ad27d0de9148","tex":"\\Delta_H","aliases":"Δ_H Delta","display":"\\Delta_H","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":23},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4b5df9a21a3e825c","tex":"\\Delta_H^j","aliases":"Δ_H^j Delta","display":"\\Delta_H^j","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":18},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-89fb0feddcfbac9f","tex":"\\sigma_3","aliases":"σ_3 sigma","display":"\\sigma_3","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-51887d918c4642bc","tex":"\\sigma_5","aliases":"σ_5 sigma","display":"\\sigma_5","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"ial at each fixed shell Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde numerators: P_n(5)=288τ(n), B_0(n)=288|Vn|. The variable T is auxi","count":168},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-569331afbd5d2de3","tex":"\\nabla","aliases":"∇ nabla","display":"\\nabla","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1d663f933d0a899c","tex":"\\nabla_H","aliases":"∇_H nabla","display":"\\nabla_H","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"The degree-at-most-22 certificate has six coarse sectors and sixteen fine sectors. Both correction matrices have eighteen columns. Let A∈Z^6×16, M_c∈Z^6×18, M_f∈Z^16×18. The fine-to-coarse parent map completely specifies A: its column j has a 1 in the parent row and zero elsewhere. The original fine masses are given below. Let s_f denote their sixt","count":46},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":24},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b8ef85e0df350209","tex":"\\prod_{i<j}","aliases":"∏_i<j prod","display":"\\prod_{i<j}","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":106},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e186a78b360a163a","tex":"\\sum_{j=0}^{10}","aliases":"∑_j=0^10 sum","display":"\\sum_{j=0}^{10}","kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9629519d9d471c98","tex":"\\sum_{j=0}^{\\lfloor M/2\\rfloor}","aliases":"∑_j=0^lfloor M/2rfloor sum lfloor rfloor","display":null,"kind":"introduction-passage","excerpt":"For a homogeneous polynomial P of degree M, a concrete top-degree harmonic projection in dimension four is H_M(P)=∑_j=0^lfloor M/2rfloor frac(-1)^j(M-j)!4^j j!M! s_2^jΔ_H^jP. For M=0 or 1 the sum contains only j=0. In general, if a_j denotes its coefficient, the product rule gives the cancellation condition a_j+4(j+1)(M-j)a_j+1=0. The factorial coe","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-287922c6bdbe4d2b","tex":"\\sum_{v\\in\\Vn}","aliases":"∑_v∈Vn sum in Vn","display":null,"kind":"introduction-passage","excerpt":"A genuine auxiliary polynomial and its local roots. A polynomial at each fixed shell. Write v_i=i+5c_i for v∈Vn. Define P_n(T)=∑_v∈Vn∏_i<j[(i-j)+T(c_i-c_j)] =∑_j=0^10B_j(n)T^j. For a fixed n, this is an integral polynomial of degree at most ten. It is obtained by summing products of ten linear factors. Evaluation at T=5 returns the Vandermonde nume","count":13},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-37634155dccdb16f","tex":"\\sum_{|t|<2\\sqrt p}","aliases":"∑_|t|<2sqrt p sum sqrt","display":"\\sum_{|t|<2\\sqrt p}","kind":"introduction-passage","excerpt":"Trace moments and a unit triangular inversion. The Hurwitz-moment description. Another exact route uses Hurwitz class numbers. Let H(D) denote the conventional weighted Hurwitz class number for a positive discriminant parameter D, with the standard exceptional automorphism weights. At a prime p define s_2j(p)=frac12∑_|t|<2sqrt pt^2jH(4p-t^2). Here ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"On the open set where D≠0, the product rule gives fracΔ_H(DQ)D =Δ_HQ+2∇_Hlog D·∇_HQ. Here ∇log D denotes the rational logarithmic derivative ∇ D/D. It does not require choosing a complex logarithm or assuming that D is positive. For symmetric Q, the apparent poles cancel and the right side extends to a polynomial on all of H.","count":43},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"Eisenstein coefficients. With the standard constant-one normalization, E_4(q) · =1+240∑_n≥1σ_3(n)q^n, E_6(q) · =1-504∑_n≥1σ_5(n)q^n, and Δ=fracE_4^3-E_6^21728. Since E_8=E_4^2, one may equivalently write (E_4E_8-E_6^2)/1728. This last form supplies an independent divisor-sum convolution for selected large coefficients in the valuation branch. The d","count":77},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"The regular discriminant cosets. Let A_4=H∩Z^5 and L=sqrt5 A_4. Its discriminant module is L^*/L≅ A_4^*/5A_4, |L^*/L|=5^5=3125. A representative of L^* can be written x=frac1sqrt5(a_i+t/5)_i=0^4, qquad0≤ t<5, ∑_i a_i=-t. If two a_i have the same residue modulo 5, the transposition of those coordinates fixes the discriminant coset. An alternating th","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"Two descriptions of the same metric. The orthogonal projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8a5c81d9140d93f9","tex":"\\mathbf1^T","aliases":"1^T mathbf","display":"\\mathbf1^T","kind":"use","excerpt":"Two descriptions of the same metric. The orthogonal projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"Jacobi gives a third engine. Put T_j=j(j+1)/2 and a_j=(-1)^j(2j+1). The series A(q)=∏_k≥1(1-q^k)^3=1+∑_j≥1a_jq^T_j satisfies F=A^8. The identity AqF'=8FqA' yields m f_m=∑_substackj≥1","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-97fe9ac7f36003c8","tex":"a^{2m}","aliases":"a^2m","display":"a^{2m}","kind":"use","excerpt":"First suppose a≠0, and write t=v_p(a). In T_2m, the term indexed by j has valuation at least 2mt+j(11-2t). If t≤4, the j=0 term a^2m has strictly least valuation 2mt. Likewise T_2m-2 has valuation (2m-2)t. The correction p^10T_2m-2 has valuation 2mt+(10-2t)>2mt. Thus v_p(R(p^2m))=2mt, t≤4.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7b48479868582d73","tex":"a^3","aliases":"a^3","display":"a^3","kind":"use","excerpt":"The first local blocks are T_2=a^2-b, T_3=a^3-2ab, T_4=a^4-3a^2b+b^2. Even if a≠0, it is not formally obvious from these expressions that none vanishes. The reduction to primes needs a short arithmetic argument.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e046b0ebe8054038","tex":"a^4","aliases":"a^4","display":"a^4","kind":"use","excerpt":"The first local blocks are T_2=a^2-b, T_3=a^3-2ab, T_4=a^4-3a^2b+b^2. Even if a≠0, it is not formally obvious from these expressions that none vanishes. The reduction to primes needs a short arithmetic argument.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-94ce33e6d3839e5f","tex":"A^8","aliases":"A^8","display":"A^8","kind":"use","excerpt":"Jacobi gives a third engine. Put T_j=j(j+1)/2 and a_j=(-1)^j(2j+1). The series A(q)=∏_k≥1(1-q^k)^3=1+∑_j≥1a_jq^T_j satisfies F=A^8. The identity AqF'=8FqA' yields m f_m=∑_substackj≥1","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ff42385782e7ad13","tex":"a^{r-2j}","aliases":"a^r-2j","display":"a^{r-2j}","kind":"use","excerpt":"The recurrence polynomial and untied valuations. For p≡4pmod5, χ=1 and a≡0pmod5. The exact recurrence expansion is T_r=∑_j=0^lfloor r/2rfloor (-1)^jbinomr-jjb^ja^r-2j. It follows by induction: multiplying the previous row by a and subtracting b times the row before it combines neighboring binomial coefficients. Equivalently, expand the rational gen","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"For that t, set L_i(v)=∏_j≠ i fracE_t(v)-e_j(s_2(v)/C)^5e_i-e_j. Each factor is homogeneous of degree 10, and the denominators are nonzero rational numbers. The resulting polynomial is symmetric homogeneous of degree 10(N-1). On the shell, s_2/C=1, so L_i(v_j)=δ_ij. Consequently, any prescribed rational orbit values a_1,…,a_N are interpolated by Q=","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7e08e3858c2cc6da","tex":"a_{16}","aliases":"a_16","display":"a_{16}","kind":"use","excerpt":"Moment identities that introduce other forms. The harmonic polynomial D(130s_4-47s_2^2) has weight-16 theta form. Its first coefficient is -280. Write E_4Δ=∑ a_16(n)q^n. Then 13∑_v∈VnW(v)s_4(v) =470n^2τ(n)-28a_16(n). Likewise D(90s_2s_4-27s_2^3-320s_3^2) gives 900n∑ W(v)s_4(v)-27000n^3τ(n) -320∑ W(v)s_3(v)^2 =3600a_18(n), where E_6Δ=∑ a_18(n)q^n an","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9c290453cde12e2b","tex":"a_{18}","aliases":"a_18","display":"a_{18}","kind":"use","excerpt":"Moment identities that introduce other forms. The harmonic polynomial D(130s_4-47s_2^2) has weight-16 theta form. Its first coefficient is -280. Write E_4Δ=∑ a_16(n)q^n. Then 13∑_v∈VnW(v)s_4(v) =470n^2τ(n)-28a_16(n). Likewise D(90s_2s_4-27s_2^3-320s_3^2) gives 900n∑ W(v)s_4(v)-27000n^3τ(n) -320∑ W(v)s_3(v)^2 =3600a_18(n), where E_6Δ=∑ a_18(n)q^n an","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d84b5a24a5037067","tex":"A_4","aliases":"A_4","display":"A_4","kind":"use","excerpt":"The regular discriminant cosets. Let A_4=H∩Z^5 and L=sqrt5 A_4. Its discriminant module is L^*/L≅ A_4^*/5A_4, |L^*/L|=5^5=3125. A representative of L^* can be written x=frac1sqrt5(a_i+t/5)_i=0^4, qquad0≤ t<5, ∑_i a_i=-t. If two a_i have the same residue modulo 5, the transposition of those coordinates fixes the discriminant coset. An alternating th","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2c678bf8963b2d95","tex":"A_4^*","aliases":"A_4^*","display":"A_4^*","kind":"use","excerpt":"The regular discriminant cosets. Let A_4=H∩Z^5 and L=sqrt5 A_4. Its discriminant module is L^*/L≅ A_4^*/5A_4, |L^*/L|=5^5=3125. A representative of L^* can be written x=frac1sqrt5(a_i+t/5)_i=0^4, qquad0≤ t<5, ∑_i a_i=-t. If two a_i have the same residue modulo 5, the transposition of those coordinates fixes the discriminant coset. An alternating th","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-011995a02078a685","tex":"a_5","aliases":"a_5","display":"a_5","kind":"use","excerpt":"S13: packet cardinality. Garvan, Kim and Stanton's Theorem~4 and equation~(5.3) give the five-core count [source reference]. In the present conventions this reads a_5(n-1)=∑_dmid nchifive(d)frac nd. For a prime different from 5, the shell cardinality is p+chifive(p). This is the number of packets, not their weighted sum.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"The regular discriminant cosets. Let A_4=H∩Z^5 and L=sqrt5 A_4. Its discriminant module is L^*/L≅ A_4^*/5A_4, |L^*/L|=5^5=3125. A representative of L^* can be written x=frac1sqrt5(a_i+t/5)_i=0^4, qquad0≤ t<5, ∑_i a_i=-t. If two a_i have the same residue modulo 5, the transposition of those coordinates fixes the discriminant coset. An alternating th","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f08e44d118bb2f29","tex":"a_i^2","aliases":"a_i^2","display":"a_i^2","kind":"use","excerpt":"For a regular representative, ∑_i a_i^2 is divisible by 5, since the squared residues sum to 30. It is even, since a_i^2≡ a_ipmod2 and ∑_i a_i=0. The quadratic exponent ∑_i a_i^2/10 is integral. The Weil operator T therefore acts as 1 on the sign line. The Weil action commutes with S_5, so this line is invariant. The rank is even, and the represent","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-07112285aabb492d","tex":"a_N","aliases":"a_N","display":"a_N","kind":"use","excerpt":"For that t, set L_i(v)=∏_j≠ i fracE_t(v)-e_j(s_2(v)/C)^5e_i-e_j. Each factor is homogeneous of degree 10, and the denominators are nonzero rational numbers. The resulting polynomial is symmetric homogeneous of degree 10(N-1). On the shell, s_2/C=1, so L_i(v_j)=δ_ij. Consequently, any prescribed rational orbit values a_1,…,a_N are interpolated by Q=","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a9457f85e2123199","tex":"a_p","aliases":"a_p","display":"a_p","kind":"use","excerpt":"Set prime parameters a_p=τ(p) (p≠ q), a_q=0. For every prime define integral local blocks F_p,0=1, F_p,1=a_p, F_p,r=a_pF_p,r-1-p^11F_p,r-2. For n=∏ p^r_p, define f(n)=∏_pF_p,r_p. Unique factorization makes this a well-defined multiplicative integer sequence with the stated recurrence.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-78516ca8f2ca19ef","tex":"a_{p,j}","aliases":"a_p,j","display":"a_{p,j}","kind":"use","excerpt":"A sufficient sign input. A second precise route would construct permitted corrections with an independently proved sign. At each remaining prime, choose finitely many rational universal null polynomials Q_p,j and rational coefficients a_p,j, and put F_p(v)=1+∑_j a_p,jQ_p,j(v). The choice may depend on p. Universal nullness must hold after the choic","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-38143a44ee0b7f8e","tex":"a_q","aliases":"a_q","display":"a_q","kind":"use","excerpt":"Set prime parameters a_p=τ(p) (p≠ q), a_q=0. For every prime define integral local blocks F_p,0=1, F_p,1=a_p, F_p,r=a_pF_p,r-1-p^11F_p,r-2. For n=∏ p^r_p, define f(n)=∏_pF_p,r_p. Unique factorization makes this a well-defined multiplicative integer sequence with the stated recurrence.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"At a prime p, the default local parameters are a=τ(p) and b=p^11. The scalar recurrence has T_0=1, T_1=a and T_r=aT_r-1-bT_r-2 for r≥2. Thus T_r=τ(p^r). A construction that permits an arbitrary integer a states that change explicitly. Such a recurrence need not be the local sequence of Δ.","count":26},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"The first local blocks are T_2=a^2-b, T_3=a^3-2ab, T_4=a^4-3a^2b+b^2. Even if a≠0, it is not formally obvious from these expressions that none vanishes. The reduction to primes needs a short arithmetic argument.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8cf537fe4af3747d","tex":"b^j","aliases":"b^j","display":"b^j","kind":"use","excerpt":"The recurrence polynomial and untied valuations. For p≡4pmod5, χ=1 and a≡0pmod5. The exact recurrence expansion is T_r=∑_j=0^lfloor r/2rfloor (-1)^jbinomr-jjb^ja^r-2j. It follows by induction: multiplying the previous row by a and subtracting b times the row before it combines neighboring binomial coefficients. Equivalently, expand the rational gen","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ea836dcf4d73fd34","tex":"b^{m-1}","aliases":"b^m-1","display":"b^{m-1}","kind":"use","excerpt":"by the recurrence polynomial, including when τ(p)=0. If τ(p)≠0, then v_2(R(p^2m+1))=v_2(τ(p)). For m=0, take U_p,0=1. For m≥1, use the quotient U(a^2)=fracT_2m+1-cT_2m-1a as polynomial division. The constant term in a^2 comes from the linear-in-a terms in the two blocks. By [source reference], it is (-1)^mb^m-1((m+1)b+mc). For an odd prime p≠5, bot","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-327140a84c768be2","tex":"B_{10}","aliases":"B_10","display":"B_{10}","kind":"use","excerpt":"A simple root at texorpdfstringp=499p=499. The exact coefficients B_0,…,B_10 at 499 are ( · 144000,-360000,1063040,28418240,-148872808,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-58a32903afc3d14d","tex":"b_a","aliases":"b_a","display":"b_a","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5d6ae674cb6898e6","tex":"B_{\\mathrm{cris}}","aliases":"B_cris mathrm","display":"B_{\\mathrm{cris}}","kind":"use","excerpt":"The realization theorem and conventions. Colmez--Fontaine's theorem makes a weakly admissible filtered module admissible [source reference]. In the present setting K=K_0=Q_p and the base Frobenius is the identity. The theorem supplies a two-dimensional semistable Q_p-representation realizing the filtered module. Since N=0, the comparison is crystal","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b2f9008f03849cc5","tex":"B_k","aliases":"B_k","display":"B_k","kind":"use","excerpt":"Localization by cancellation, rather than indicators. Let the columns of a rational matrix N form a basis of ker M_k. Then B_k=M_k+1N, AB_k=0. The columns of B_k transfer mass only within the parents: their effects on every parent cancel. They are still combinations of universal null polynomials. This is the valid repair of the failed sector-indica","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1e0d226ed135452f","tex":"B_{\\mathrm{st}}^{N=0}","aliases":"B_st^N=0 mathrm","display":"B_{\\mathrm{st}}^{N=0}","kind":"use","excerpt":"The realization theorem and conventions. Colmez--Fontaine's theorem makes a weakly admissible filtered module admissible [source reference]. In the present setting K=K_0=Q_p and the base Frobenius is the identity. The theorem supplies a two-dimensional semistable Q_p-representation realizing the filtered module. Since N=0, the comparison is crystal","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"The corresponding five-core partition has size n-1. In particular, the coefficient at n=1 corresponds to the empty partition. The involution is J(v)_i=-v_-ibmod5. A J-orbit may contain one or two packets, and its mass is the sum of their weights.","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[3pt] 2, k≥2 · (-25M-5,10M+2,-15M+4,15M-4,15M+3) · (10M-5,10M+2,20M+4,-20M-4,-20M+3) [3pt] 0, k≥3 · (-20M-15,10M-3,10M+4,20M+1,-20M+13) · (15M-15,-25M-3,10M+4,-15M+1,15M+13) The appropriate powers of 2 make each entry divisible by 7. After division, each vector has the standard residues, sum zero and norm squared 10N_k. The norm identities before d","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"For a nonzero P_n, use the convention v_5(0)=+∞ and define μ_n=min_j(j+v_5(B_j)), G_n(U)=5^-μ_nP_n(5U)=∑_j=0^10C_jU^j. Every C_j=B_j5^j-μ_n is an integer. The power of 5 should be removed only after verifying divisibility for each coefficient. Since 288 is a 5-adic unit, v_5(τ(n))=μ_n+v_5(G_n(1)). All coefficients in the next two tables are exact i","count":23},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C","display":null,"kind":"use","excerpt":"Let Q be any finite symmetric polynomial over C, restricted to ∑_i v_i=0. Suppose ∑_v∈VnW(v)Q(v)=0 for every positive integer n. Write Q=∑_dQ_d in homogeneous degrees. Then every Q_d is separately universally null. Moreover, the weighted Q-sum is zero in each exact-content sector of every shell.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","excerpt":"Interpolation is not universal nullness. Coordinates that identify exactly the orbits. Fix a nonempty shell and put C=10n. Consider the normalized even symmetric invariants I(v)=(fracs_4C^2,fracs_3^2C^3, fracs_3s_5C^4,fracs_5^2C^5). They are unchanged by J.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","excerpt":"Let c be complex conjugation. Then c^2=1 and the cyclotomic character sends c to -1. Consequently detρ_ℓ(c)=-1. In characteristic zero an involution is diagonalizable, with eigenvalues among 1,-1. The determinant forces one of each, so Trρ_ℓ(c)=0 as an exact Z_ℓ identity. This reasoning also covers ℓ=2: the trace is computed before reduction. Diago","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4a4fb977dca130a2","tex":"C^3","aliases":"C^3","display":"C^3","kind":"use","excerpt":"Interpolation is not universal nullness. Coordinates that identify exactly the orbits. Fix a nonempty shell and put C=10n. Consider the normalized even symmetric invariants I(v)=(fracs_4C^2,fracs_3^2C^3, fracs_3s_5C^4,fracs_5^2C^5). They are unchanged by J.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ab467eeeb5a83ecd","tex":"C^4","aliases":"C^4","display":"C^4","kind":"use","excerpt":"Interpolation is not universal nullness. Coordinates that identify exactly the orbits. Fix a nonempty shell and put C=10n. Consider the normalized even symmetric invariants I(v)=(fracs_4C^2,fracs_3^2C^3, fracs_3s_5C^4,fracs_5^2C^5). They are unchanged by J.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5044272ca87ffc23","tex":"C^5","aliases":"C^5","display":"C^5","kind":"use","excerpt":"Interpolation is not universal nullness. Coordinates that identify exactly the orbits. Fix a nonempty shell and put C=10n. Consider the normalized even symmetric invariants I(v)=(fracs_4C^2,fracs_3^2C^3, fracs_3s_5C^4,fracs_5^2C^5). They are unchanged by J.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9a670a7c8b4efcd2","tex":"c^r","aliases":"c^r","display":"c^r","kind":"use","excerpt":"Separating the quasimodular weights. We need only an elementary weight-separation statement. Suppose finitely many quasimodular functions F_k of distinct nonnegative integer weights satisfy F_k(γ z)=∑_r=0^p_kf_k,r(z)c^r(cz+d)^k-r, f_k,0=F_k, p_k≤ k, for γ=[a · b","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"use","excerpt":"For a nonzero P_n, use the convention v_5(0)=+∞ and define μ_n=min_j(j+v_5(B_j)), G_n(U)=5^-μ_nP_n(5U)=∑_j=0^10C_jU^j. Every C_j=B_j5^j-μ_n is an integer. The power of 5 should be removed only after verifying divisibility for each coefficient. Since 288 is a 5-adic unit, v_5(τ(n))=μ_n+v_5(G_n(1)). All coefficients in the next two tables are exact i","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-746a0fca63346846","tex":"C_r","aliases":"C_r","display":"C_r","kind":"use","excerpt":"A residue eliminates three prime classes. Suppose p≠5 and put χ=chifive(p). Modulo 5, χ≡ p^2, so the two recurrence roots may be taken as p and χ: p+χ≡ a, pχ≡ p^11pmod5. The complete homogeneous recurrence coefficient is T_r≡ C_r=∑_i=0^r p^r-iχ^ipmod5. Also χ p^10≡1pmod5. Since χ^2=1, for r≥2 the sum telescopes: R(p^r)≡ C_r-C_r-2 =p^r-1(p+χ)pmod5. ","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6e236a9189fa5355","tex":"C_{r-2}","aliases":"C_r-2","display":"C_{r-2}","kind":"use","excerpt":"lasses Suppose p≠5 and put χ=chifive(p). Modulo 5, χ≡ p^2, so the two recurrence roots may be taken as p and χ: p+χ≡ a, pχ≡ p^11pmod5. The complete homogeneous recurrence coefficient is T_r≡ C_r=∑_i=0^r p^r-iχ^ipmod5. Also χ p^10≡1pmod5. Since χ^2=1, for r≥2 the sum telescopes: R(p^r)≡ C_r-C_r-2 =p^r-1(p+χ)pmod5. The same expression holds at r=1. I","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6931df8a2528852e","tex":"\\chifive","aliases":"chifive","display":null,"kind":"use","excerpt":"The valuation v_ℓ(x) is the exponent of ℓ in a nonzero rational number x, with v_ℓ(0)=∞. The quadratic character used in the radial extraction is chifive(d)= 0 · 5mid d, 1 · d≡1,4pmod5, -1 · d≡2,3pmod5. The character value and the tenth power in the scaling formula are both essential.","count":29},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b123b92295b0b921","tex":"\\mathrm{cris}","aliases":"cris mathrm","display":"\\mathrm{cris}","kind":"use","excerpt":"The realization theorem and conventions. Colmez--Fontaine's theorem makes a weakly admissible filtered module admissible [source reference]. In the present setting K=K_0=Q_p and the base Frobenius is the identity. The theorem supplies a two-dimensional semistable Q_p-representation realizing the filtered module. Since N=0, the comparison is crystal","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"R(n)=∑_d^2mid nμ(d)χ_5(d)d^10τ(n/d^2).","count":74},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"use","excerpt":"Checking weak admissibility. For the chosen cohomological convention, define t_H(D)=∑_i idimgr^iD, t_N(D)=v_p(detφ). Weak admissibility requires t_H(D)=t_N(D) and t_H(D')≤ t_N(D') for every φ-stable subobject D', with the induced filtration. The total equality is t_H(D)=11=t_N(D).","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-79740da50f9fbcc4","tex":"d^{10}","aliases":"d^10","display":"d^{10}","kind":"use","excerpt":"R(n)=∑_d^2mid nμ(d)χ_5(d)d^10τ(n/d^2).","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"R(n)=∑_d^2mid nμ(d)χ_5(d)d^10τ(n/d^2).","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7f067ed769fd5136","tex":"d^j","aliases":"d^j","display":"d^j","kind":"use","excerpt":"Exact coefficient engines. The product and a logarithmic recurrence. Let σ_j(n)=∑_dmid nd^j and write σ=σ_1. Logarithmic differentiation of the product gives DQΔ=(1-24∑_m≥1σ(m)q^m)Δ, DQ=qfracddq. Indeed, the contribution from the factor (1-q^r)^24 is -24fracrq^r1-q^r=-24∑_k≥1rq^rk. Collecting the terms with rk=m gives -24σ(m). The leading factor q ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5b5366e7d0a1d14f","tex":"d^r","aliases":"d^r","display":"d^r","kind":"use","excerpt":"These four coefficients were checked independently by the Eisenstein convolution. If σ_r(n)=∑_dmid nd^r, the coefficient of (E_4E_8-E_6^2)/1728 is obtained from 1728τ(n)= · 240σ_3(n)+480σ_7(n)+1008σ_5(n)","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1b2556595a6d4744","tex":"d^\\perp","aliases":"d^⊥ perp","display":"d^\\perp","kind":"use","excerpt":"Project e_i-e_j orthogonally to d^⊥. Its squared norm is 2-fracδ^24. Because m⊥ d, Cauchy's inequality gives m ^2≥frac(a|δ|+1)^22-δ^2/4. For |δ|=1 this is 4(a+1)^2/7. For |δ|=2 it is (2a+1)^2, a larger quantity. Therefore 10n≥4a^2+frac47(a+1)^2, which simplifies to the lower bound inside the ceiling in [source reference]. Taking the ceiling proves ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"use","excerpt":"If d ^2≥6, [source reference] gives n≥frac154M^2>frac100M^2+10M+235, M≥2. The remaining case is d ^2=4. Then d has two coordinates 1, two -1 and one 0. Of the ten coordinate pairs, two have equal d entries and eight have unequal entries. For a pair with δ=d_i-d_j, (v_i-v_j)(w_i-w_j)=(m_i-m_j)^2-a^2δ^2. The two equal-entry pairs have positive produc","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"If d ^2≥6, [source reference] gives n≥frac154M^2>frac100M^2+10M+235, M≥2. The remaining case is d ^2=4. Then d has two coordinates 1, two -1 and one 0. Of the ten coordinate pairs, two have equal d entries and eight have unequal entries. For a pair with δ=d_i-d_j, (v_i-v_j)(w_i-w_j)=(m_i-m_j)^2-a^2δ^2. The two equal-entry pairs have positive produc","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"Integrality follows from an integer-valued basis. For every integer x, including negative integers, the generalized binomial coefficient binomxj is an integer. Its leading coefficient as a polynomial in x is 1/j!. Thus det[binomv_ij]_0≤ i,j≤4 =frac∏_i<j(v_j-v_i)∏_j=0^4j!. There are ten pairs, so reversing every difference changes the sign by (-1)^1","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"The corresponding upper bounds have direct structural explanations. Each correction matrix has column sum zero, giving bounds 5 and 15. Every row of A has a nonempty, disjoint support, so rank A=6 and dimker A=10. Since AB=0, the detail rank is at most 10. The upper and lower bounds establish the displayed ranks exactly.","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-007c6930a8d4ad63","tex":"\\DQ","aliases":"DQ","display":null,"kind":"use","excerpt":"Exact coefficient engines. The product and a logarithmic recurrence. Let σ_j(n)=∑_dmid nd^j and write σ=σ_1. Logarithmic differentiation of the product gives DQΔ=(1-24∑_m≥1σ(m)q^m)Δ, DQ=qfracddq. Indeed, the contribution from the factor (1-q^r)^24 is -24fracrq^r1-q^r=-24∑_k≥1rq^rk. Collecting the terms with rk=m gives -24σ(m). The leading factor q ","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-289555126a00503e","tex":"\\DQ^2","aliases":"DQ^2 DQ","display":null,"kind":"use","excerpt":"DQ E_6 · =(E_2E_6-E_4^2)/2. These normalized relations are recorded in [source reference], and the constant-one formula for Δ is checked in [source reference]. They define a derivation on the polynomial ring Q[E_2,E_4,E_6]. Repeated differentiation and collection of terms yield E_2DQ^2E_2 · =4DQ^3E_2+frac310DQ^2E_4, E_2DQ^4E_2 · =frac125DQ^5E_2+fra","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b1f54c0633f93f7a","tex":"\\DQ^3","aliases":"DQ^3 DQ","display":null,"kind":"use","excerpt":"DQ E_6 · =(E_2E_6-E_4^2)/2. These normalized relations are recorded in [source reference], and the constant-one formula for Δ is checked in [source reference]. They define a derivation on the polynomial ring Q[E_2,E_4,E_6]. Repeated differentiation and collection of terms yield E_2DQ^2E_2 · =4DQ^3E_2+frac310DQ^2E_4, E_2DQ^4E_2 · =frac125DQ^5E_2+fra","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8101b0c08fdd1d65","tex":"\\DQ^4","aliases":"DQ^4 DQ","display":null,"kind":"use","excerpt":"DQ E_6 · =(E_2E_6-E_4^2)/2. These normalized relations are recorded in [source reference], and the constant-one formula for Δ is checked in [source reference]. They define a derivation on the polynomial ring Q[E_2,E_4,E_6]. Repeated differentiation and collection of terms yield E_2DQ^2E_2 · =4DQ^3E_2+frac310DQ^2E_4, E_2DQ^4E_2 · =frac125DQ^5E_2+fra","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e03c3512547a11ad","tex":"\\DQ^5","aliases":"DQ^5 DQ","display":null,"kind":"use","excerpt":"DQ E_6 · =(E_2E_6-E_4^2)/2. These normalized relations are recorded in [source reference], and the constant-one formula for Δ is checked in [source reference]. They define a derivation on the polynomial ring Q[E_2,E_4,E_6]. Repeated differentiation and collection of terms yield E_2DQ^2E_2 · =4DQ^3E_2+frac310DQ^2E_4, E_2DQ^4E_2 · =frac125DQ^5E_2+fra","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6f527dfe34348e3d","tex":"\\DQ^j","aliases":"DQ^j DQ","display":null,"kind":"use","excerpt":"Each application of DQ=q d/dq multiplies the shell coefficient by v ^2/10. Therefore θ_s_2^jh=10^jDQ^jθ_h. Differentiating a modular transformation law shows that DQ^jf, for modular weight w, is quasimodular of weight w+2j and depth at most j. Combining [source reference] and [source reference], the homogeneous shell series F_d(τ)=∑_n≥1(∑_v∈VnW(v)Q","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":21},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"Newton reductions with zero first power sum. Let e_j be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is ∏_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5, because e_1=s_1=0. The first Newton identities give e_2 · =-frac12s_2, · e_3 · =frac13s_3,","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-98af5c47feb2b63c","tex":"E_2","aliases":"E_2","display":"E_2","kind":"use","excerpt":"Deriving the identity in the differential ring. Set A(q)=∑_n≥1σ(n)q^n, B(q)=∑_n≥1σ_3(n)q^n. Then E_2=1-24A and E_4=1+240B. The Ramanujan differential relations are DQ E_2 · =(E_2^2-E_4)/12,","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"Newton reductions with zero first power sum. Let e_j be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is ∏_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5, because e_1=s_1=0. The first Newton identities give e_2 · =-frac12s_2, · e_3 · =frac13s_3,","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8da21ff8f52d1596","tex":"E_2^2","aliases":"E_2^2","display":"E_2^2","kind":"use","excerpt":"Deriving the identity in the differential ring. Set A(q)=∑_n≥1σ(n)q^n, B(q)=∑_n≥1σ_3(n)q^n. Then E_2=1-24A and E_4=1+240B. The Ramanujan differential relations are DQ E_2 · =(E_2^2-E_4)/12, DQ E_4 · =(E_2E_4-E_6)/3, DQ E_6 · =(E_2E_6-E_4^2)/2. These normalized relations are recorded in [source reference], and the constant-one formula for Δ is check","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"Newton reductions with zero first power sum. Let e_j be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is ∏_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5, because e_1=s_1=0. The first Newton identities give e_2 · =-frac12s_2, · e_3 · =frac13s_3,","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-67ab753297c61ce5","tex":"e_4","aliases":"e_4","display":"e_4","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a64c5d974442c22a","tex":"e_5","aliases":"e_5","display":"e_5","kind":"use","excerpt":"Newton reductions with zero first power sum. Let e_j be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is ∏_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5, because e_1=s_1=0. The first Newton identities give e_2 · =-frac12s_2, · e_3 · =frac13s_3,","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e9aaef4e6308432c","tex":"e_a","aliases":"e_a","display":"e_a","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"in one sector. Replace w by Jw if needed so that w-v=5Md, d∈ A_4∖{0}. Set a=5M/2 and m=(v+w)/2. Since M is even, both are integral. Equal norms give m· d=0,qquad10n= m ^2+a^2 d ^2. The squared norm of a nonzero integral zero-sum vector is a positive even integer. It cannot be 2 here. Such a vector is ±(e_i-e_j), and m· d=0 would force v_i-v_j=±5M. ","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"Newton reductions with zero first power sum. Let e_j be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is ∏_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5, because e_1=s_1=0. The first Newton identities give e_2 · =-frac12s_2, · e_3 · =frac13s_3,","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1f891a7f61f89a21","tex":"E_k","aliases":"E_k","display":"E_k","kind":"use","excerpt":"Put H_k=ker(1^T:Q^s_k→Q), T_k=im M_k, E_k=H_k/T_k. The quotient E_k is the vector space of total-preserving changes that are unavailable from the chosen family. It is not a count of failed sign certificates. A sign certificate may exist even when E_k is nonzero.","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-69b76e4f5fee61c8","tex":"E_{k+1}","aliases":"E_k+1","display":"E_{k+1}","kind":"use","excerpt":"The exact refinement sequence. There is a short exact sequence 0⟶ fracker AM_k+1(ker M_k) ⟶ E_k+1⟶ E_k⟶0. Consequently, if r_k=rank M_k, dim E_k+1-dim E_k =(s_k+1-s_k)-(r_k+1-r_k). The aggregation map sends H_k+1 onto H_k. Given a zero-total parent vector, place each parent entry in one selected child and zero in its other children. The resulting c","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f75d15c864ad3cef","tex":"E_t","aliases":"E_t","display":"E_t","kind":"use","excerpt":"An explicit homogeneous interpolation basis. Let v_1,…,v_N represent the distinct J-orbits. For an integer t≥0, define the degree-10 symmetric polynomial E_t=s_4s_2^3+t s_3^2s_2^2+t^2s_3s_5s_2+t^3s_5^2. For two distinct orbits, E_t(v_i)-E_t(v_j) is a nonzero polynomial in t of degree at most 3, by Proposition~[source reference]. It has at most thre","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"Radial factors become derivatives. For an alternating harmonic h_ℓ, write θ_h(τ)=∑_substackv_i≡ i (5) ∑ v_i=0 h(v)exp(2π iτfrac v ^210). Chapter~[source reference] establishes that it is a scalar cusp form of weight ℓ+2. The normalization includes only a fixed scalar if one divides by 288.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"Two descriptions of the same metric. The orthogonal projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=","count":41},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"The rational row-reduction results are rank M_c=5, rank M_f=15, rank B=10. An independent check obtains the same lower bounds over both prime fields F_65537 and F_1000003. Every denominator used in a reduction is invertible there. A nonzero minor in either field proves a nonzero rational minor after clearing denominators.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b6bb9a2af76d66fc","tex":"\\mathbf F","aliases":"F mathbf","display":"\\mathbf F","kind":"use","excerpt":"· -11893316304,-1160632800). Their weighted valuations are (3,5,3,4,4,5,7,11,8,9,12). Hence μ_499=3 and reduction gives overline G_499(U)=2+3U^2=3(U-1)(U+1) in F_5[U]. At the relevant value U=1, G_499(1)≡10pmod25, G'_499(1)≡1pmod5. Substitute U=1+5d. The integral Taylor identity gives G_499(1+5d)≡10+5dpmod25. The root in this residue class is there","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"The logarithmic derivative provides a different exact engine. With σ_1(j)=∑_dmid jd, comparison of coefficients in qfracF'(q)F(q)=-24∑_j≥1σ_1(j)q^j gives m f_m=-24∑_j=1^mσ_1(j)f_m-j (m≥1). For m=3, the divisor sums are 1,3,4. Hence f_3=-8(252+3(-24)+4)=-8·184=-1472. An implementation should check that the numerator in [source reference] is divisibl","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","excerpt":"oefficient and its local parameters The normalized form is Δ(q)=q∏_m≥1(1-q^m)^24 =∑_n≥1τ(n)q^n, τ(1)=1. All coefficient indices in this book are positive integers. A stored value at index zero is a programming sentinel unless a displayed formula explicitly introduces the coefficients of Δ/q. For the latter coefficients, f_m=τ(m+1) and f_0=1.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"For m=1, the polynomial is simply F_1(x)=x-(p+1). For larger m, its degree grows, but the same constant-term and residue argument controls any integral square root. That is why no bound on m is needed.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"For direct product expansion, terms of degree at most m depend only on factors with k≤ m. The exact binomial expansion (1-q^k)^24=∑_j=0^24(-1)^jbinom24jq^kj therefore supplies a finite algorithm for each desired coefficient. Three short calculations are f_1 · =-24, f_2 · =binom242-24=276-24=252, f_3 · =-binom243+(-24)(-24)-24 · =-2024+576-24=-1472.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6eaaf2d385dfc909","tex":"\\mathbb F_{1000003}","aliases":"F_1000003 mathbb","display":"\\mathbb F_{1000003}","kind":"use","excerpt":"The rational row-reduction results are rank M_c=5, rank M_f=15, rank B=10. An independent check obtains the same lower bounds over both prime fields F_65537 and F_1000003. Every denominator used in a reduction is invertible there. A nonzero minor in either field proves a nonzero rational minor after clearing denominators.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","excerpt":"For direct product expansion, terms of degree at most m depend only on factors with k≤ m. The exact binomial expansion (1-q^k)^24=∑_j=0^24(-1)^jbinom24jq^kj therefore supplies a finite algorithm for each desired coefficient. Three short calculations are f_1 · =-24, f_2 · =binom242-24=276-24=252, f_3 · =-binom243+(-24)(-24)-24 · =-2024+576-24=-1472.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4fac6558c4f0f59e","tex":"f_3","aliases":"f_3","display":"f_3","kind":"use","excerpt":"For direct product expansion, terms of degree at most m depend only on factors with k≤ m. The exact binomial expansion (1-q^k)^24=∑_j=0^24(-1)^jbinom24jq^kj therefore supplies a finite algorithm for each desired coefficient. Three short calculations are f_1 · =-24, f_2 · =binom242-24=276-24=252, f_3 · =-binom243+(-24)(-24)-24 · =-2024+576-24=-1472.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-47a73e04062d0183","tex":"\\mathbf F_5","aliases":"F_5 mathbf","display":"\\mathbf F_5","kind":"use","excerpt":"· -11893316304,-1160632800). Their weighted valuations are (3,5,3,4,4,5,7,11,8,9,12). Hence μ_499=3 and reduction gives overline G_499(U)=2+3U^2=3(U-1)(U+1) in F_5[U]. At the relevant value U=1, G_499(1)≡10pmod25, G'_499(1)≡1pmod5. Substitute U=1+5d. The integral Taylor identity gives G_499(1+5d)≡10+5dpmod25. The root in this residue class is there","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3bb5270c03e35e20","tex":"\\mathbb F_{65537}","aliases":"F_65537 mathbb","display":"\\mathbb F_{65537}","kind":"use","excerpt":"The rational row-reduction results are rank M_c=5, rank M_f=15, rank B=10. An independent check obtains the same lower bounds over both prime fields F_65537 and F_1000003. Every denominator used in a reduction is invertible there. A nonzero minor in either field proves a nonzero rational minor after clearing denominators.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-879f46b871236f65","tex":"F_d","aliases":"F_d","display":"F_d","kind":"use","excerpt":"dq multiplies the shell coefficient by v ^2/10. Therefore θ_s_2^jh=10^jDQ^jθ_h. Differentiating a modular transformation law shows that DQ^jf, for modular weight w, is quasimodular of weight w+2j and depth at most j. Combining [source reference] and [source reference], the homogeneous shell series F_d(τ)=∑_n≥1(∑_v∈VnW(v)Q_d(v))q^n is quasimodular o","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7e04443bf0b17f7b","tex":"F_k","aliases":"F_k","display":"F_k","kind":"use","excerpt":"Separating the quasimodular weights. We need only an elementary weight-separation statement. Suppose finitely many quasimodular functions F_k of distinct nonnegative integer weights satisfy F_k(γ z)=∑_r=0^p_kf_k,r(z)c^r(cz+d)^k-r, f_k,0=F_k, p_k≤ k, for γ=[a · b","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b707e3b4ea43aef9","tex":"f_{k,0}","aliases":"f_k,0","display":"f_{k,0}","kind":"use","excerpt":"Separating the quasimodular weights. We need only an elementary weight-separation statement. Suppose finitely many quasimodular functions F_k of distinct nonnegative integer weights satisfy F_k(γ z)=∑_r=0^p_kf_k,r(z)c^r(cz+d)^k-r, f_k,0=F_k, p_k≤ k, for γ=[a · b","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ba407f59817cd49c","tex":"f_{k,r}","aliases":"f_k,r","display":"f_{k,r}","kind":"use","excerpt":"Separating the quasimodular weights. We need only an elementary weight-separation statement. Suppose finitely many quasimodular functions F_k of distinct nonnegative integer weights satisfy F_k(γ z)=∑_r=0^p_kf_k,r(z)c^r(cz+d)^k-r, f_k,0=F_k, p_k≤ k, for γ=[a · b","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"use","excerpt":"t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses is a monic polynomial F_m(x) in x=u^2. Its constant term and reduction modulo p ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"use","excerpt":"The coefficient and its local parameters. The normalized form is Δ(q)=q∏_m≥1(1-q^m)^24 =∑_n≥1τ(n)q^n, τ(1)=1. All coefficient indices in this book are positive integers. A stored value at index zero is a programming sentinel unless a displayed formula explicitly introduces the coefficients of Δ/q. For the latter coefficients, f_m=τ(m+1) and f_0=1.","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-88b893bc2e8eec84","tex":"f_{m-j}","aliases":"f_m-j","display":"f_{m-j}","kind":"use","excerpt":"The logarithmic derivative provides a different exact engine. With σ_1(j)=∑_dmid jd, comparison of coefficients in qfracF'(q)F(q)=-24∑_j≥1σ_1(j)q^j gives m f_m=-24∑_j=1^mσ_1(j)f_m-j (m≥1). For m=3, the divisor sums are 1,3,4. Hence f_3=-8(252+3(-24)+4)=-8·184=-1472. An implementation should check that the numerator in [source reference] is divisibl","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-84bdac75d67a97c7","tex":"f_{m-T_j}","aliases":"f_m-T_j","display":"f_{m-T_j}","kind":"use","excerpt":"Jacobi gives a third engine. Put T_j=j(j+1)/2 and a_j=(-1)^j(2j+1). The series A(q)=∏_k≥1(1-q^k)^3=1+∑_j≥1a_jq^T_j satisfies F=A^8. The identity AqF'=8FqA' yields m f_m=∑_substackj≥1 T_j≤ m a_j(9T_j-m)f_m-T_j. At m=3, only T_1=1, T_2=3 contribute. Its numerator is (-3)(9-3)(252)+5(27-3)(1)=-4536+120=-4416. Division by 3 again gives -1472. The direc","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6b681ab9d723acad","tex":"f_{n-1}","aliases":"f_n-1","display":"f_{n-1}","kind":"use","excerpt":"Write F(q)=fracΔ(q)q=∏_k≥1(1-q^k)^24 =∑_m≥0f_mq^m. Then f_0=1 and τ(n)=f_n-1. This shift is essential. A program that stores τ(0)=0 as a sentinel must not confuse that entry with the constant coefficient f_0.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6a27312454be2902","tex":"F_p","aliases":"F_p","display":"F_p","kind":"use","excerpt":"A sufficient sign input. A second precise route would construct permitted corrections with an independently proved sign. At each remaining prime, choose finitely many rational universal null polynomials Q_p,j and rational coefficients a_p,j, and put F_p(v)=1+∑_j a_p,jQ_p,j(v). The choice may depend on p. Universal nullness must hold after the choic","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8ef81cc7a99e18f2","tex":"F_{p,0}","aliases":"F_p,0","display":"F_{p,0}","kind":"use","excerpt":"Set prime parameters a_p=τ(p) (p≠ q), a_q=0. For every prime define integral local blocks F_p,0=1, F_p,1=a_p, F_p,r=a_pF_p,r-1-p^11F_p,r-2. For n=∏ p^r_p, define f(n)=∏_pF_p,r_p. Unique factorization makes this a well-defined multiplicative integer sequence with the stated recurrence.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-200eb0edce39a770","tex":"F_{p,1}","aliases":"F_p,1","display":"F_{p,1}","kind":"use","excerpt":"Set prime parameters a_p=τ(p) (p≠ q), a_q=0. For every prime define integral local blocks F_p,0=1, F_p,1=a_p, F_p,r=a_pF_p,r-1-p^11F_p,r-2. For n=∏ p^r_p, define f(n)=∏_pF_p,r_p. Unique factorization makes this a well-defined multiplicative integer sequence with the stated recurrence.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-aa1c0136fd48958c","tex":"F_{p,r}","aliases":"F_p,r","display":"F_{p,r}","kind":"use","excerpt":"Set prime parameters a_p=τ(p) (p≠ q), a_q=0. For every prime define integral local blocks F_p,0=1, F_p,1=a_p, F_p,r=a_pF_p,r-1-p^11F_p,r-2. For n=∏ p^r_p, define f(n)=∏_pF_p,r_p. Unique factorization makes this a well-defined multiplicative integer sequence with the stated recurrence.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7ec9eb3a87a3f145","tex":"F_{p,r-1}","aliases":"F_p,r-1","display":"F_{p,r-1}","kind":"use","excerpt":"Set prime parameters a_p=τ(p) (p≠ q), a_q=0. For every prime define integral local blocks F_p,0=1, F_p,1=a_p, F_p,r=a_pF_p,r-1-p^11F_p,r-2. For n=∏ p^r_p, define f(n)=∏_pF_p,r_p. Unique factorization makes this a well-defined multiplicative integer sequence with the stated recurrence.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-17887e9df93b5708","tex":"F_{p,r-2}","aliases":"F_p,r-2","display":"F_{p,r-2}","kind":"use","excerpt":"Set prime parameters a_p=τ(p) (p≠ q), a_q=0. For every prime define integral local blocks F_p,0=1, F_p,1=a_p, F_p,r=a_pF_p,r-1-p^11F_p,r-2. For n=∏ p^r_p, define f(n)=∏_pF_p,r_p. Unique factorization makes this a well-defined multiplicative integer sequence with the stated recurrence.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-76ace3fe24af878c","tex":"F_{p,r_p}","aliases":"F_p,r_p","display":"F_{p,r_p}","kind":"use","excerpt":"Set prime parameters a_p=τ(p) (p≠ q), a_q=0. For every prime define integral local blocks F_p,0=1, F_p,1=a_p, F_p,r=a_pF_p,r-1-p^11F_p,r-2. For n=∏ p^r_p, define f(n)=∏_pF_p,r_p. Unique factorization makes this a well-defined multiplicative integer sequence with the stated recurrence.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-46748fb4dadfbac4","tex":"F_{q,2m}","aliases":"F_q,2m","display":"F_{q,2m}","kind":"use","excerpt":"The full size bound survives. At primes other than q, the Hecke roots have absolute value p^11/2 by Deligne. Formula [source reference] therefore gives |τ(p^r)|≤(r+1)p^11r/2. At the modified prime, F_q,2m=(-q^11)^m, F_q,2m+1=0. These obey the same local bound at every exponent, including zero. Multiplying the local estimates gives |f(n)|≤∏_p^r_pVer","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7b022e17789b12b5","tex":"F_{q,2m+1}","aliases":"F_q,2m+1","display":"F_{q,2m+1}","kind":"use","excerpt":"The full size bound survives. At primes other than q, the Hecke roots have absolute value p^11/2 by Deligne. Formula [source reference] therefore gives |τ(p^r)|≤(r+1)p^11r/2. At the modified prime, F_q,2m=(-q^11)^m, F_q,2m+1=0. These obey the same local bound at every exponent, including zero. Multiplying the local estimates gives |f(n)|≤∏_p^r_pVer","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2419e39a4f7b4492","tex":"F_{q,r}","aliases":"F_q,r","display":"F_{q,r}","kind":"use","excerpt":"Agreement at every fixed modulus and finite prefix. Only the parameter at q has changed. Every n≤ B is coprime to q, so f(n)=τ(n) there. At q, the old and new first parameters agree modulo 10M. Induction in the integral recurrence gives F_q,r≡τ(q^r)pmod10M for every exponent. All other local blocks agree exactly. Multiplying local congruences gives","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-22770047abacba3d","tex":"\\Fil","aliases":"Fil","display":null,"kind":"use","excerpt":"1 · 0, Trφ=0, detφ=p^11, φ^2=-p^11I. Choose any Q_p-line L⊂ D and put Fil^iD= D · i≤0, L · 1≤ i≤11, 0 · i≥12. The two nonzero associated graded pieces are gr^0D=D/L and gr^11D=L, each of dimension 1.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-80bc810b6d36a8eb","tex":"\\Fil^i","aliases":"Fil^i Fil","display":null,"kind":"use","excerpt":"1 · 0, Trφ=0, detφ=p^11, φ^2=-p^11I. Choose any Q_p-line L⊂ D and put Fil^iD= D · i≤0, L · 1≤ i≤11, 0 · i≥12. The two nonzero associated graded pieces are gr^0D=D/L and gr^11D=L, each of dimension 1.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-bf6efda13a9cd474","tex":"\\Frob","aliases":"Frob","display":null,"kind":"use","excerpt":"Integral representations and complex conjugation. For each prime ℓ, the representation attached to Δ has the arithmetic-Frobenius normalization ρ_ℓ:G_Q⟶ GL_2(Q_ℓ), is continuous and unramified outside ℓ, and satisfies det(X-ρ_ℓ(Frob_p))=X^2-τ(p)X+p^11, p≠ℓ. Its determinant is the eleventh power of the ℓ-adic cyclotomic character. The original cohom","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0879cf2a93a54946","tex":"\\Frob_p","aliases":"Frob_p Frob","display":null,"kind":"use","excerpt":"Integral representations and complex conjugation. For each prime ℓ, the representation attached to Δ has the arithmetic-Frobenius normalization ρ_ℓ:G_Q⟶ GL_2(Q_ℓ), is continuous and unramified outside ℓ, and satisfies det(X-ρ_ℓ(Frob_p))=X^2-τ(p)X+p^11, p≠ℓ. Its determinant is the eleventh power of the ℓ-adic cyclotomic character. The original cohom","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":38},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"The content of a packet is the positive gcd of its five coordinates. Content-one packets are primitive. Multiplication followed by the prescribed residue reindexing gives τ(n)=∑_g^2mid nchifive(g)g^10R(n/g^2), R(n)=∑_d^2mid nμ(d)chifive(d)d^10τ(n/d^2). The boundary values are R(1)=1 and R(p)=τ(p). At a prime shell the only possible content is one, ","count":33},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"evant value U=1, G_499(1)≡10pmod25, G'_499(1)≡1pmod5. Substitute U=1+5d. The integral Taylor identity gives G_499(1+5d)≡10+5dpmod25. The root in this residue class is therefore ρ≡16pmod25. Hensel's lemma lifts it uniquely to Z_5. The divided difference (G(U)-G(ρ))/(U-ρ) is a unit on U≡1pmod5, since its reduction is G'(1)≠0. Thus v_5(G_499(U))=v_5(U","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0d569f5b563b7b52","tex":"G'_{16249}","aliases":"G'_16249","display":"G'_{16249}","kind":"use","excerpt":"G_16249(1) · =61339951998700724370240. The first equals 288τ(16249), and the second equals the first divided by 625. Modulo 5, the normalized polynomial is 3+U^2+U^3=(U-1)^2(U-2). Modulo 25, the table gives G_16249(U)≡13+15U+11U^2+11U^3+15U^5. Thus G_16249(1)≡15pmod25 while G'_16249(1)≡0pmod5. For every integer or 5-adic integer h, polynomial Taylo","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-996ca0554b0d383b","tex":"G'_{499}","aliases":"G'_499","display":"G'_{499}","kind":"use","excerpt":"· -11893316304,-1160632800). Their weighted valuations are (3,5,3,4,4,5,7,11,8,9,12). Hence μ_499=3 and reduction gives overline G_499(U)=2+3U^2=3(U-1)(U+1) in F_5[U]. At the relevant value U=1, G_499(1)≡10pmod25, G'_499(1)≡1pmod5. Substitute U=1+5d. The integral Taylor identity gives G_499(1+5d)≡10+5dpmod25. The root in this residue class is there","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8c34c927fec40de5","tex":"G'_p","aliases":"G'_p","display":"G'_p","kind":"use","excerpt":"A double residual root with no lift. At p=16249, the shell has 16250 packets. The weighted coefficient valuations of P_p(5U) are (4,5,4,4,6,5,7,8,9,9,11), so μ_p=4. The reduction is overline G_p(U)=3+U^2+U^3=(U-1)^2(U-2)pmod5. The residual root at 1 is double. Exact coefficient reduction gives G_p(1)≡15pmod25, G'_p(1)≡0pmod5. For every y∈Z_5, polyn","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-565b30cbcdc2ca2a","tex":"G^{-1}","aliases":"G^-1","display":"G^{-1}","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1ca098b8c4d11a94","tex":"g^{10}","aliases":"g^10","display":"g^{10}","kind":"use","excerpt":"The content of a packet is the positive gcd of its five coordinates. Content-one packets are primitive. Multiplication followed by the prescribed residue reindexing gives τ(n)=∑_g^2mid nchifive(g)g^10R(n/g^2), R(n)=∑_d^2mid nμ(d)chifive(d)d^10τ(n/d^2). The boundary values are R(1)=1 and R(p)=τ(p). At a prime shell the only possible content is one, ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-08d0ec4a83f1da2d","tex":"g^{10+d}","aliases":"g^10+d","display":"g^{10+d}","kind":"use","excerpt":"g(v)=1W(v)Q(v). Symmetry removes the residue permutation from Q, and homogeneity gives Q(v)=g^dQ(u). The scaling bijection yields ∑_v∈VnW(v)Q(v) =∑_g^2mid nchifive(g)g^10+dR_Q(n/g^2). The left side is zero. At n=1, only g=1 contributes, so R_Q(1)=0. Inductively, the g=1 term is R_Q(n) and every nonzero term with g>1 has smaller argument. Thus R_Q(n","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","excerpt":"The content of a packet is the positive gcd of its five coordinates. Content-one packets are primitive. Multiplication followed by the prescribed residue reindexing gives τ(n)=∑_g^2mid nchifive(g)g^10R(n/g^2), R(n)=∑_d^2mid nμ(d)chifive(d)d^10τ(n/d^2). The boundary values are R(1)=1 and R(p)=τ(p). At a prime shell the only possible content is one, ","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2e43a0881e9a1bb2","tex":"g^d","aliases":"g^d","display":"g^d","kind":"use","excerpt":"g(v)=1W(v)Q(v). Symmetry removes the residue permutation from Q, and homogeneity gives Q(v)=g^dQ(u). The scaling bijection yields ∑_v∈VnW(v)Q(v) =∑_g^2mid nchifive(g)g^10+dR_Q(n/g^2). The left side is zero. At n=1, only g=1 contributes, so R_Q(1)=0. Inductively, the g=1 term is R_Q(n) and every nonzero term with g>1 has smaller argument. Thus R_Q(n","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-04079af3eba7a7a8","tex":"G_{16249}","aliases":"G_16249","display":"G_{16249}","kind":"use","excerpt":"begingroupsmall setlengthtabcolsep4pt r r r r j · B_j in P_16249 · C_j in G_16249 · j+v_5(B_j)","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-18d8b8032894a85d","tex":"G_{499}","aliases":"G_499","display":"G_{499}","kind":"use","excerpt":"begingroupsmall setlengthtabcolsep5pt r r r r j · B_j in P_499 · C_j in G_499 · j+v_5(B_j)","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-358c536fce266469","tex":"G_n","aliases":"G_n","display":"G_n","kind":"use","excerpt":"For a nonzero P_n, use the convention v_5(0)=+∞ and define μ_n=min_j(j+v_5(B_j)), G_n(U)=5^-μ_nP_n(5U)=∑_j=0^10C_jU^j. Every C_j=B_j5^j-μ_n is an integer. The power of 5 should be removed only after verifying divisibility for each coefficient. Since 288 is a 5-adic unit, v_5(τ(n))=μ_n+v_5(G_n(1)). All coefficients in the next two tables are exact i","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-47aa314ed96d6f8a","tex":"G_p","aliases":"G_p","display":"G_p","kind":"use","excerpt":"A double residual root with no lift. At p=16249, the shell has 16250 packets. The weighted coefficient valuations of P_p(5U) are (4,5,4,4,6,5,7,8,9,9,11), so μ_p=4. The reduction is overline G_p(U)=3+U^2+U^3=(U-1)^2(U-2)pmod5. The residual root at 1 is double. Exact coefficient reduction gives G_p(1)≡15pmod25, G'_p(1)≡0pmod5. For every y∈Z_5, polyn","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0abf0d17398d7d33","tex":"G_{\\Q}","aliases":"G_Q Q","display":null,"kind":"use","excerpt":"Integral representations and complex conjugation. For each prime ℓ, the representation attached to Δ has the arithmetic-Frobenius normalization ρ_ℓ:G_Q⟶ GL_2(Q_ℓ), is continuous and unramified outside ℓ, and satisfies det(X-ρ_ℓ(Frob_p))=X^2-τ(p)X+p^11, p≠ℓ. Its determinant is the eleventh power of the ℓ-adic cyclotomic character. The original cohom","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"Why the problem lives at primes. The Hecke contracts. The normalized form Δ is a Hecke eigenform. Its coefficients satisfy τ(mn) · =τ(m)τ(n), · · gcd(m,n)=1, τ(p^r+1) · =τ(p)τ(p^r)-p^11τ(p^r-1), · · r≥1. The exponent 11 again records weight 12. The constant term for the local recurrence is τ(1)=1, and the first parameter is a=τ(p). These classical ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5ee03a101afb29c0","tex":"\\operatorname{gr}","aliases":"gr operatorname","display":"\\operatorname{gr}","kind":"use","excerpt":"0 · i≥12. The two nonzero associated graded pieces are gr^0D=D/L and gr^11D=L, each of dimension 1.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8e59a8906baafde8","tex":"\\operatorname{gr}^0","aliases":"gr^0 operatorname","display":"\\operatorname{gr}^0","kind":"use","excerpt":"0 · i≥12. The two nonzero associated graded pieces are gr^0D=D/L and gr^11D=L, each of dimension 1.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-cfc47137b968f4e8","tex":"\\operatorname{gr}^{11}","aliases":"gr^11 operatorname","display":"\\operatorname{gr}^{11}","kind":"use","excerpt":"0 · i≥12. The two nonzero associated graded pieces are gr^0D=D/L and gr^11D=L, each of dimension 1.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a471f8145de323bb","tex":"\\operatorname{gr}^i","aliases":"gr^i operatorname","display":"\\operatorname{gr}^i","kind":"use","excerpt":"Checking weak admissibility. For the chosen cohomological convention, define t_H(D)=∑_i idimgr^iD, t_N(D)=v_p(detφ). Weak admissibility requires t_H(D)=t_N(D) and t_H(D')≤ t_N(D') for every φ-stable subobject D', with the induced filtration. The total equality is t_H(D)=11=t_N(D).","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"Radial factors and harmonic projection. Let h_ℓ be homogeneous harmonic of degree ℓ on the rank-four space H. Euler's identity is v·∇_Hh_ℓ=ℓ h_ℓ. The product rule for the radial factor s_2^j gives Δ_H(s_2^jh_ℓ) =4j(ℓ+j+1)s_2^j-1h_ℓ, j≥1. At j=0 the left side is zero. At j=1,ℓ=0 the coefficient is 8, agreeing with Δ_Hs_2. At j=1,ℓ=10 it is 48, agree","count":21},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":"The Nekrasov--Okounkov identity in Han's formulation is ∏_m≥1(1-q^m)^z-1 =∑_λ q^|λ|∏_h∈ H(λ)(1-frac zh^2). At z=25, a partition having a hook of length 5 contributes zero. The surviving terms are the 5-cores. Multiplication by q shows that the coefficient τ(n) is the sum of the hook products over 5-cores of size n-1.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"The tied valuation texorpdfstringt=5t=5. The preceding endpoint arguments leave t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-63e9c3e80a66add1","tex":"H_1","aliases":"H_1","display":"H_1","kind":"use","excerpt":"The tied valuation texorpdfstringt=5t=5. The preceding endpoint arguments leave t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2919da96f50ce1bd","tex":"\\mathcal H_{12}","aliases":"H_12 mathcal","display":"\\mathcal H_{12}","kind":"use","excerpt":"Harmonic projection is not preservation of shell values. The elementary counterexample is P=s_2D, of degree 12. Since Δ_HP=48D and the next Laplacian is zero, H_12(s_2D)=s_2D-frac148s_2(48D)=0. Yet on Vn the original polynomial equals 10nD, which is nonzero at every packet. The operation removes radial information. It cannot silently turn a shell i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2f689b7a4bde4099","tex":"H_{2m}","aliases":"H_2m","display":"H_{2m}","kind":"use","excerpt":"The tied valuation texorpdfstringt=5t=5. The preceding endpoint arguments leave t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses ","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f7938a7b8a11cdfc","tex":"H_{2m-2}","aliases":"H_2m-2","display":"H_{2m-2}","kind":"use","excerpt":"The tied valuation texorpdfstringt=5t=5. The preceding endpoint arguments leave t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses ","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0231a655bf13a98d","tex":"H_k","aliases":"H_k","display":"H_k","kind":"use","excerpt":"Put H_k=ker(1^T:Q^s_k→Q), T_k=im M_k, E_k=H_k/T_k. The quotient E_k is the vector space of total-preserving changes that are unavailable from the chosen family. It is not a count of failed sign certificates. A sign certificate may exist even when E_k is nonzero.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6a921288deb0d7f9","tex":"H_{k+1}","aliases":"H_k+1","display":"H_{k+1}","kind":"use","excerpt":"inement There is a short exact sequence 0⟶ fracker AM_k+1(ker M_k) ⟶ E_k+1⟶ E_k⟶0. Consequently, if r_k=rank M_k, dim E_k+1-dim E_k =(s_k+1-s_k)-(r_k+1-r_k). The aggregation map sends H_k+1 onto H_k. Given a zero-total parent vector, place each parent entry in one selected child and zero in its other children. The resulting child vector has zero to","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-51164c3c62e251bc","tex":"h_{M-2j}","aliases":"h_M-2j","display":"h_{M-2j}","kind":"use","excerpt":"Repeating the construction on that lower-degree quotient produces the harmonic decomposition P=∑_j=0^lfloor M/2rfloors_2^jh_M-2j. It is unique. If such a sum were zero, apply the Laplacian as many times as the largest occupied radial exponent. All smaller-exponent terms vanish. Formula [source reference] leaves a nonzero scalar times the last harmo","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2d64df6fa5868f4c","tex":"H_r","aliases":"H_r","display":"H_r","kind":"use","excerpt":"The tied valuation texorpdfstringt=5t=5. The preceding endpoint arguments leave t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-bd53539eaf104c23","tex":"H_{r-1}","aliases":"H_r-1","display":"H_{r-1}","kind":"use","excerpt":"The tied valuation texorpdfstringt=5t=5. The preceding endpoint arguments leave t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1ca3a2dfec126c54","tex":"H_{r-2}","aliases":"H_r-2","display":"H_{r-2}","kind":"use","excerpt":"The tied valuation texorpdfstringt=5t=5. The preceding endpoint arguments leave t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-858e4bb65a33c6eb","tex":"h_\\ell","aliases":"h_ℓ ell","display":"h_\\ell","kind":"use","excerpt":"Radial factors and harmonic projection. Let h_ℓ be homogeneous harmonic of degree ℓ on the rank-four space H. Euler's identity is v·∇_Hh_ℓ=ℓ h_ℓ. The product rule for the radial factor s_2^j gives Δ_H(s_2^jh_ℓ) =4j(ℓ+j+1)s_2^j-1h_ℓ, j≥1. At j=0 the left side is zero. At j=1,ℓ=0 the coefficient is 8, agreeing with Δ_Hs_2. At j=1,ℓ=10 it is 48, agree","count":9},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Two descriptions of the same metric. The orthogonal projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-78fd137c582a5e5f","tex":"I_4","aliases":"I_4","display":"I_4","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7cc6e87678c7d216","tex":"I_5","aliases":"I_5","display":"I_5","kind":"use","excerpt":"Two descriptions of the same metric. The orthogonal projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"R(n)=0iffτ(n)=0 · Primitive extraction encodes the same zero problem, with R(p)=τ(p).","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im","display":null,"kind":"use","excerpt":"Put H_k=ker(1^T:Q^s_k→Q), T_k=im M_k, E_k=H_k/T_k. The quotient E_k is the vector space of total-preserving changes that are unavailable from the chosen family. It is not a count of failed sign certificates. A sign certificate may exist even when E_k is nonzero.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"The corresponding five-core partition has size n-1. In particular, the coefficient at n=1 corresponds to the empty partition. The involution is J(v)_i=-v_-ibmod5. A J-orbit may contain one or two packets, and its mass is the sum of their weights.","count":32},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"Two explicit harmonic moment checks. The degree-four correction K_4=130s_4-47s_2^2 has LK_4 =130frac1885s_2-47(104s_2)=0. Thus DK_4 is harmonic. At the first packet, W=1 and K_4=130·34-47·100=-280. Its theta form has weight 16, whose cusp space is spanned by Δ E_4 with first coefficient one. Consequently the theta form is -280Δ E_4, giving the mome","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ec1a2b307412bdd0","tex":"k^4","aliases":"k^4","display":"k^4","kind":"use","excerpt":"The differential residual that survives extraction. A divisor convolution with the original cusp term. Define the fourth divisor moment S_4(n)=∑_k=1^n-1k^4σ(k)σ(n-k). It is a positive integer when n>1. The weights k^4 and the two divisor sums suggest a route through elementary arithmetic. The exact identity, however, retains the cusp coefficient: 8","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"The realization theorem and conventions. Colmez--Fontaine's theorem makes a weakly admissible filtered module admissible [source reference]. In the present setting K=K_0=Q_p and the base Frobenius is the identity. The theorem supplies a two-dimensional semistable Q_p-representation realizing the filtered module. Since N=0, the comparison is crystal","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-15201584ff766791","tex":"K_4","aliases":"K_4","display":"K_4","kind":"use","excerpt":"Two explicit harmonic moment checks. The degree-four correction K_4=130s_4-47s_2^2 has LK_4 =130frac1885s_2-47(104s_2)=0. Thus DK_4 is harmonic. At the first packet, W=1 and K_4=130·34-47·100=-280. Its theta form has weight 16, whose cusp space is spanned by Δ E_4 with first coefficient one. Consequently the theta form is -280Δ E_4, giving the mome","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7e9dc5b85827b59a","tex":"K_6","aliases":"K_6","display":"K_6","kind":"use","excerpt":"Likewise, set K_6=90s_2s_4-27s_2^3-320s_3^2. Using [source reference]--[source reference], its operator image is LK_6= · (5760-5760)s_4","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"The full machine certificate is small pathevidence/computation/replays/transport/p31_transport_certificate.json. It supplies an 18×13 rational matrix N whose columns form a basis of ker M_c, and the detail matrix B=M_fN of size 16×13. The basic exact checks are AM_f=M_c, As_f=s_c, M_cN=0, B=M_fN, AB=0. The last equation shows that a detail correcti","count":11},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"The CRT/NTT computation returns residues modulo 2^19. It is not a reconstruction of those large integer tau coefficients. Within each convolution, however, the integer coefficient is reconstructed before reduction. With L=1032191, M=524288, the nonnegative representative bound is L(M-1)^2=283725419316756479 <998244353·1004535809 =100277219872053657","count":27},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"Three polynomial operations. The ordinary hyperplane Laplacian is Δ_H. The conjugated operator L is characterized by Δ_H(DQ)=D LQ. Harmonicity of DQ means LQ=0. It does not mean that Q itself is harmonic.","count":23},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[3pt] 2, k≥2 · (-25M-5,10M+2,-15M+4,15M-4,15M+3) · (10M-5,10M+2,20M+4,-20M-4,-20M+3) [3pt] 0, k≥3 · (-20M-15,10M-3,10M+4,20M+1,-20M+13) · (15M-15,-25M-3,10M+4,-15M+1,15M+13) The appropriate powers of 2 make each entry divisible by 7. After division, each vector has the standard residues, sum zero and norm squared 10N_k. The norm identities before d","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a44e55170cadffdb","tex":"L^*","aliases":"L^*","display":"L^*","kind":"use","excerpt":"The regular discriminant cosets. Let A_4=H∩Z^5 and L=sqrt5 A_4. Its discriminant module is L^*/L≅ A_4^*/5A_4, |L^*/L|=5^5=3125. A representative of L^* can be written x=frac1sqrt5(a_i+t/5)_i=0^4, qquad0≤ t<5, ∑_i a_i=-t. If two a_i have the same residue modulo 5, the transposition of those coordinates fixes the discriminant coset. An alternating th","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"A continuous representation of the compact group G_Q has a stable Z_ℓ lattice. To see this directly, take an initial lattice L_0. Compactness bounds all its translates inside ℓ^-aL_0 for some a. The module L=∑_g∈ G_Qρ_ℓ(g)L_0 is contained in ℓ^-aL_0 and contains L_0. It is finitely generated, since a submodule of a finite free Z_ℓ module is finitel","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"The Eichler--Selberg trace identities give, for r≥1, ∑_j=0^r(-1)^r-jbinomr+jr-jp^r-js_2j(p) =-1-Tr T_2r+2(p), together with s_0(p)=p. Here T_k(p) is the Hecke operator at p on S_k(SL_2(Z)). It is distinct from the scalar recurrence notation T_r=τ(p^r) at a fixed prime. The trace normalization and Catalan inversion are checked against [source refere","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a716b0894950a10a","tex":"L_i","aliases":"L_i","display":"L_i","kind":"use","excerpt":"For that t, set L_i(v)=∏_j≠ i fracE_t(v)-e_j(s_2(v)/C)^5e_i-e_j. Each factor is homogeneous of degree 10, and the denominators are nonzero rational numbers. The resulting polynomial is symmetric homogeneous of degree 10(N-1). On the shell, s_2/C=1, so L_i(v_j)=δ_ij. Consequently, any prescribed rational orbit values a_1,…,a_N are interpolated by Q=","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"The sharp threshold. Let k≥1, M=2^k, and set N_k=lceilfrac100M^2+10M+235rceil =frac20M^2+2M+γ_k7, where γ_k= 7 · k≡1pmod3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-973977200963bdff","tex":"\\log_{10}","aliases":"log_10 log","display":"\\log_{10}","kind":"use","excerpt":"figplaceholdercoefficients512The exact coefficients through 512 on a signed logarithmic scale. The plotted ordinate is sgn(τ(n))log_10(1+|τ(n)|). Sign changes and large values are visible, while every plotted coefficient is nonzero. This finite plot does not establish the conjecture.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Integrality makes a residue argument useful. If x∈Z, |x|<L and Lmid x, then x=0. If instead a nonzero residue modulo L is established, then x≠0 without any size estimate. At primes p≥5, [source reference] yields 2p^11/2<p^6. Therefore τ(p)=0 Longleftrightarrow p^6midτ(p), p≥5. This is an exact criterion. It replaces equality to zero by a divisibili","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"The logical conclusion and the remaining information. The theorem refutes the listed arithmetic properties Longrightarrow every coefficient is nonzero. The construction does not impose a modular-form realization, the packet-shell identity or the compatible global Galois representations of Δ. Those are potential sources of additional information, ra","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e367badb119c2e7b","tex":"m^{10}","aliases":"m^10","display":"m^{10}","kind":"use","excerpt":"The square-supported arithmetic kernel in [source reference] has inverse d^2⟼μ(d)chifive(d)d^10. Indeed, at a square index m^2, the convolution of the two kernels is chifive(m)m^10∑_dmid mμ(d), which is 1 at m=1 and 0 otherwise. At nonsquare indices both kernels have no support. Consequently R(n)=∑_d^2mid nμ(d)chifive(d)d^10τ(n/d^2). Both formulas ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","excerpt":"The sharp threshold. Let k≥1, M=2^k, and set N_k=lceilfrac100M^2+10M+235rceil =frac20M^2+2M+γ_k7, where γ_k= 7 · k≡1pmod3,","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"The square-supported arithmetic kernel in [source reference] has inverse d^2⟼μ(d)chifive(d)d^10. Indeed, at a square index m^2, the convolution of the two kernels is chifive(m)m^10∑_dmid mμ(d), which is 1 at m=1 and 0 otherwise. At nonsquare indices both kernels have no support. Consequently R(n)=∑_d^2mid nμ(d)chifive(d)d^10τ(n/d^2). Both formulas ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"The stored two-stage certificate first eliminates five coarse masses, using only five elimination equations. It then uses coefficient vectors in ker M_1 to eliminate all but one fine mass while preserving every coarse total. The survivor is -52843168. The product engine independently identifies this with τ(31). The solve does not require a preassig","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"The quantifiers cannot be reversed. Given a fixed finite list of moduli m_1,…,m_s, apply Theorem~[source reference] to their least common multiple. Infinitely many actual primes defeat every test in that list whose decisive step is finding a nonzero residue of τ(p) or any R(p^r). This remains true even if the test is allowed all positive local expo","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"use","excerpt":"If d ^2≥6, [source reference] gives n≥frac154M^2>frac100M^2+10M+235, M≥2. The remaining case is d ^2=4. Then d has two coordinates 1, two -1 and one 0. Of the ten coordinate pairs, two have equal d entries and eight have unequal entries. For a pair with δ=d_i-d_j, (v_i-v_j)(w_i-w_j)=(m_i-m_j)^2-a^2δ^2. The two equal-entry pairs have positive produc","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c879cfab2bd64d1b","tex":"m_j","aliases":"m_j","display":"m_j","kind":"use","excerpt":"If d ^2≥6, [source reference] gives n≥frac154M^2>frac100M^2+10M+235, M≥2. The remaining case is d ^2=4. Then d has two coordinates 1, two -1 and one 0. Of the ten coordinate pairs, two have equal d entries and eight have unequal entries. For a pair with δ=d_i-d_j, (v_i-v_j)(w_i-w_j)=(m_i-m_j)^2-a^2δ^2. The two equal-entry pairs have positive produc","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c9f29e7d9e7a175b","tex":"M_k","aliases":"M_k","display":"M_k","kind":"use","excerpt":"Refinement and exact transfer spaces. Mass vectors and aggregation. Fix a shell, a finite rational family Q_1,…,Q_h of universally null corrections, and the occupied sectors at depth k. Let their number be s_k. Define t_k,r=∑_v∈ S_k,rW(v), (M_k)_rj=∑_v∈ S_k,rW(v)Q_j(v). The vector t_k contains sector masses. The columns of M_k are the effects of th","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3db6ddc80894285f","tex":"M_{k+1}","aliases":"M_k+1","display":"M_{k+1}","kind":"use","excerpt":"Every depth-k+1 sector belongs to exactly one parent at depth k. Let A be the corresponding 0--1 aggregation matrix. Summing child contributions gives the exact identities t_k=At_k+1, M_k=AM_k+1. Every occupied parent has a child. Each correction column has global total zero.","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6866e3ac2aa8a3c4","tex":"m_p","aliases":"m_p","display":"m_p","kind":"use","excerpt":"All prime cases have now been covered. If s=∏ p^m_p, multiplicativity gives R(s^2)=∏_pR(p^2m_p). Every factor is nonzero, and R(1)=1. This proves Theorem~[source reference].","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-82f744f1968fc463","tex":"m_s","aliases":"m_s","display":"m_s","kind":"use","excerpt":"The quantifiers cannot be reversed. Given a fixed finite list of moduli m_1,…,m_s, apply Theorem~[source reference] to their least common multiple. Infinitely many actual primes defeat every test in that list whose decisive step is finding a nonzero residue of τ(p) or any R(p^r). This remains true even if the test is allowed all positive local expo","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"R(n)=∑_d^2mid nμ(d)χ_5(d)d^10τ(n/d^2).","count":24},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"For a nonzero P_n, use the convention v_5(0)=+∞ and define μ_n=min_j(j+v_5(B_j)), G_n(U)=5^-μ_nP_n(5U)=∑_j=0^10C_jU^j. Every C_j=B_j5^j-μ_n is an integer. The power of 5 should be removed only after verifying divisibility for each coefficient. Since 288 is a 5-adic unit, v_5(τ(n))=μ_n+v_5(G_n(1)). All coefficients in the next two tables are exact i","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d1d740bae622a7f0","tex":"\\min_{B_j(n)\\ne0}","aliases":"min_B_j(n)≠0 min ne","display":"\\min_{B_j(n)\\ne0}","kind":"use","excerpt":"Normalize the coefficient valuations by μ_n=min_B_j(n)≠0(j+v_5(B_j(n))), G_n(U)=5^-μ_nP_n(5U). Every coefficient of G_n is integral, and at least one is a unit. Because 288 is a 5-adic unit, v_5(τ(n))=μ_n+v_5(G_n(1)). This formula remains meaningful if τ(n)=0, with valuation ∞. The normalization does not assume nonvanishing: its constant coefficien","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-60698b1979c563d1","tex":"\\min_j","aliases":"min_j min","display":"\\min_j","kind":"use","excerpt":"For a nonzero P_n, use the convention v_5(0)=+∞ and define μ_n=min_j(j+v_5(B_j)), G_n(U)=5^-μ_nP_n(5U)=∑_j=0^10C_jU^j. Every C_j=B_j5^j-μ_n is an integer. The power of 5 should be removed only after verifying divisibility for each coefficient. Since 288 is a 5-adic unit, v_5(τ(n))=μ_n+v_5(G_n(1)). All coefficients in the next two tables are exact i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"S09: weak admissibility and admissibility. The Colmez--Fontaine author manuscript supplies the implication from weak admissibility to admissibility and the N=0 crystalline realization in the specified setting [source reference]. The constructed rank-two module has equal Hodge and Newton totals and no proper Frobenius-stable rational line. Those che","count":37},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-527de8677f85ada6","tex":"n^{11/2}","aliases":"n^11/2","display":"n^{11/2}","kind":"use","excerpt":"every real B≥1 and integer M≥1, there exist a prime q>B and a multiplicative integer sequence f with the following properties: item f(1)=1 and f(n)=τ(n) for every n≤ B. item f(n)≡τ(n)pmod10M for every positive integer n. item Every weight-12 prime-power recurrence holds: f(p^r)=f(p)f(p^r-1)-p^11f(p^r-2), r≥2. item |f(n)|≤ d(n)n^11/2, where d(n) cou","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"Moment identities that introduce other forms. The harmonic polynomial D(130s_4-47s_2^2) has weight-16 theta form. Its first coefficient is -280. Write E_4Δ=∑ a_16(n)q^n. Then 13∑_v∈VnW(v)s_4(v) =470n^2τ(n)-28a_16(n). Likewise D(90s_2s_4-27s_2^3-320s_3^2) gives 900n∑ W(v)s_4(v)-27000n^3τ(n) -320∑ W(v)s_3(v)^2 =3600a_18(n), where E_6Δ=∑ a_18(n)q^n an","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0969e62c250901c5","tex":"n^3","aliases":"n^3","display":"n^3","kind":"use","excerpt":"Moment identities that introduce other forms. The harmonic polynomial D(130s_4-47s_2^2) has weight-16 theta form. Its first coefficient is -280. Write E_4Δ=∑ a_16(n)q^n. Then 13∑_v∈VnW(v)s_4(v) =470n^2τ(n)-28a_16(n). Likewise D(90s_2s_4-27s_2^3-320s_3^2) gives 900n∑ W(v)s_4(v)-27000n^3τ(n) -320∑ W(v)s_3(v)^2 =3600a_18(n), where E_6Δ=∑ a_18(n)q^n an","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ac5e944c84a2e799","tex":"n^4","aliases":"n^4","display":"n^4","kind":"use","excerpt":"l that survives extraction A divisor convolution with the original cusp term. Define the fourth divisor moment S_4(n)=∑_k=1^n-1k^4σ(k)σ(n-k). It is a positive integer when n>1. The weights k^4 and the two divisor sums suggest a route through elementary arithmetic. The exact identity, however, retains the cusp coefficient: 840S_4(n)=n^4[50σ_3(n)+(35","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-bcbc010f58d53ce9","tex":"N_3","aliases":"N_3","display":"N_3","kind":"use","excerpt":"6 · k≡0pmod3. Every depth-k sector in every shell n<N_k is sign-pure. The shell n=N_k has a mixed sector. This threshold ranges over all positive integer shell indices. It does not assert that N_k is prime. For example, N_3=186 is composite.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9cc3cd93a260a3e9","tex":"N_k","aliases":"N_k","display":"N_k","kind":"use","excerpt":"The sharp threshold. Let k≥1, M=2^k, and set N_k=lceilfrac100M^2+10M+235rceil =frac20M^2+2M+γ_k7, where γ_k= 7 · k≡1pmod3,","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"Primitive content and conserved arithmetic sectors. The coordinate gcd is a real invariant. For v∈Vn, let g(v) be the positive gcd of its five coordinates. Since the coordinates occupy all residues modulo 5, one has 5nmid g(v). Call g(v) the content of the packet. A primitive packet has content 1.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","excerpt":"A sufficient adaptive residue input. One precise route would be a uniform construction of residue certificates. Suppose there are explicitly defined functions on prime indices, with integer values m(p)≥2 and b(p), for which one proves τ(p)≡ b(p)pmodm(p), b(p)not≡0pmodm(p) for every prime in [source reference]. These two assertions would settle that","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"bibitemIW Tomoyoshi Ibukiyama and Brandon Williams, Hermitian modular forms and algebraic modular forms on SO(6), https://arxiv.org/pdf/2505.23497arXiv:2505.23497v1, header 29 May 2025, internal date 12 June 2025. S3.4, printed pp.~18--19, gives vector-valued harmonic theta modularity and weight ν+n/2. The scalar full-level sign-line descent is a p","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"[3pt] 2, k≥2 · (-25M-5,10M+2,-15M+4,15M-4,15M+3) · (10M-5,10M+2,20M+4,-20M-4,-20M+3) [3pt] 0, k≥3 · (-20M-15,10M-3,10M+4,20M+1,-20M+13) · (15M-15,-25M-3,10M+4,-15M+1,15M+13) The appropriate powers of 2 make each entry divisible by 7. After division, each vector has the standard residues, sum zero and norm squared 10N_k. The norm identities before d","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9b0ad0f92e6daea0","tex":"\\overline G","aliases":"overline G overline","display":"\\overline G","kind":"use","excerpt":"· -11893316304,-1160632800). Their weighted valuations are (3,5,3,4,4,5,7,11,8,9,12). Hence μ_499=3 and reduction gives overline G_499(U)=2+3U^2=3(U-1)(U+1) in F_5[U]. At the relevant value U=1, G_499(1)≡10pmod25, G'_499(1)≡1pmod5. Substitute U=1+5d. The integral Taylor identity gives G_499(1+5d)≡10+5dpmod25. The root in this residue class is there","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4c5010b04dc89ad9","tex":"\\overline G_{499}","aliases":"overline G_499 overline","display":"\\overline G_{499}","kind":"use","excerpt":"· -11893316304,-1160632800). Their weighted valuations are (3,5,3,4,4,5,7,11,8,9,12). Hence μ_499=3 and reduction gives overline G_499(U)=2+3U^2=3(U-1)(U+1) in F_5[U]. At the relevant value U=1, G_499(1)≡10pmod25, G'_499(1)≡1pmod5. Substitute U=1+5d. The integral Taylor identity gives G_499(1+5d)≡10+5dpmod25. The root in this residue class is there","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2e08ffb9f1fb22e9","tex":"\\overline G_p","aliases":"overline G_p overline","display":"\\overline G_p","kind":"use","excerpt":"A double residual root with no lift. At p=16249, the shell has 16250 packets. The weighted coefficient valuations of P_p(5U) are (4,5,4,4,6,5,7,8,9,9,11), so μ_p=4. The reduction is overline G_p(U)=3+U^2+U^3=(U-1)^2(U-2)pmod5. The residual root at 1 is double. Exact coefficient reduction gives G_p(1)≡15pmod25, G'_p(1)≡0pmod5. For every y∈Z_5, polyn","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a2f4f1ae2a658d51","tex":"p^{-5r}","aliases":"p^-5r","display":"p^{-5r}","kind":"use","excerpt":"The tied valuation texorpdfstringt=5t=5. The preceding endpoint arguments leave t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-560848d158d4b73b","tex":"p^{10}","aliases":"p^10","display":"p^{10}","kind":"use","excerpt":"Every primitive square survives. The local numerator introduced by content. The transform in [source reference] is a Dirichlet convolution of multiplicative functions, so R is multiplicative. At a prime p, write a=τ(p), b=p^11, c=chifive(p)p^10. The local generating factor is ∑_r≥0R(p^r)X^r =frac1-cX^21-aX+bX^2. It follows either by multiplying the","count":9},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-699f90e203cc7be9","tex":"p^{10m}","aliases":"p^10m","display":"p^{10m}","kind":"use","excerpt":"The tied valuation texorpdfstringt=5t=5. The preceding endpoint arguments leave t=5. Write a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8301a6908444e52d","tex":"p^{11}","aliases":"p^11","display":"p^{11}","kind":"use","excerpt":"At a prime p, the default local parameters are a=τ(p) and b=p^11. The scalar recurrence has T_0=1, T_1=a and T_r=aT_r-1-bT_r-2 for r≥2. Thus T_r=τ(p^r). A construction that permits an arbitrary integer a states that change explicitly. Such a recurrence need not be the local sequence of Δ.","count":28},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5d425a052ab72b4c","tex":"p^{11/2}","aliases":"p^11/2","display":"p^{11/2}","kind":"use","excerpt":"S06: Deligne's bound and normalization. The modular representation and purity inputs give |τ(p)|≤2p^11/2 [source reference]. The modular paper's representation construction and the Weil theorem supply different parts of this implication. Their Frobenius conventions require translation. This upper bound includes zero and is never used as a positive ","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4305bc3ec91b1cdf","tex":"p^{11m-1}","aliases":"p^11m-1","display":"p^{11m-1}","kind":"use","excerpt":"If a=0, do not assign it a finite t. Direct substitution gives R(p^2m)=(-1)^mp^11m-1(p+1), which is nonzero. This branch is indispensable: a theorem purporting to survive a hypothetical prime zero must actually evaluate that case.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7d12ec493f14a226","tex":"p^{11r/2}","aliases":"p^11r/2","display":"p^{11r/2}","kind":"use","excerpt":"The full size bound survives. At primes other than q, the Hecke roots have absolute value p^11/2 by Deligne. Formula [source reference] therefore gives |τ(p^r)|≤(r+1)p^11r/2. At the modified prime, F_q,2m=(-q^11)^m, F_q,2m+1=0. These obey the same local bound at every exponent, including zero. Multiplying the local estimates gives |f(n)|≤∏_p^r_pVer","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c124eed972cca6c6","tex":"p^{11r_p/2}","aliases":"p^11r_p/2","display":"p^{11r_p/2}","kind":"use","excerpt":"The full size bound survives. At primes other than q, the Hecke roots have absolute value p^11/2 by Deligne. Formula [source reference] therefore gives |τ(p^r)|≤(r+1)p^11r/2. At the modified prime, F_q,2m=(-q^11)^m, F_q,2m+1=0. These obey the same local bound at every exponent, including zero. Multiplying the local estimates gives |f(n)|≤∏_p^r_pVer","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"Two descriptions of the same metric. The orthogonal projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"At a prime p, σ(p)=1+p and σ_3(p)=1+p^3. Put C(p)=85-49p-84p^2+50p^3. Equation [source reference] becomes τ(p)=p^4C(p)-840S_4(p). For p>7, the coefficient 840 is a p-adic unit. Reducing modulo p^4 gives p^4mid S_4(p) Longleftrightarrow p^4midτ(p). The proposed assertion that p^4 never divides S_4(p) would therefore assert a uniform bound v_p(τ(p))≤","count":11},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-80a406874da00307","tex":"p^{2m}","aliases":"p^2m","display":"p^{2m}","kind":"use","excerpt":"The even primitive blocks are not the same as the even tau blocks. For example, R(p^2)=a^2-p^10(p+chifive(p)). An argument that τ(p^2m)≠0 does not alone imply the corresponding difference is nonzero. We now prove the primitive statement directly.","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b7e9e910f1a1263d","tex":"p^{2m_p}","aliases":"p^2m_p","display":"p^{2m_p}","kind":"use","excerpt":"All prime cases have now been covered. If s=∏ p^m_p, multiplicativity gives R(s^2)=∏_pR(p^2m_p). Every factor is nonzero, and R(1)=1. This proves Theorem~[source reference].","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-578b7099ecb3368e","tex":"p^3","aliases":"p^3","display":"p^3","kind":"use","excerpt":"At a prime p, σ(p)=1+p and σ_3(p)=1+p^3. Put C(p)=85-49p-84p^2+50p^3. Equation [source reference] becomes τ(p)=p^4C(p)-840S_4(p). For p>7, the coefficient 840 is a p-adic unit. Reducing modulo p^4 gives p^4mid S_4(p) Longleftrightarrow p^4midτ(p). The proposed assertion that p^4 never divides S_4(p) would therefore assert a uniform bound v_p(τ(p))≤","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8cdfda19e4c4c67b","tex":"p^4","aliases":"p^4","display":"p^4","kind":"use","excerpt":"The index-dependent criterion already available from Deligne's bound is τ(p)=0iff p^6midτ(p), p≥5. Combined with the divisor convolution, it becomes 840S_4(p)not≡ p^4C(p)pmodp^6, p>7, as the exact desired exclusion. This gives a correctly typed target for a convolution route. No uniform arithmetic property excluding that residue has been proved her","count":11},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d006e038d1ebd2a9","tex":"p^5","aliases":"p^5","display":"p^5","kind":"use","excerpt":"The resulting first moments are s_2(p) · =p^2-1, s_4(p) · =2p^3-3p-1, s_6(p) · =5p^4-9p^2-5p-1, s_8(p) · =14p^5-28p^3-20p^2-7p-1, s_10(p) · =42p^6-90p^4-75p^3-35p^2-9p-1-τ(p). The integer 42 is the fifth Catalan number. It records the contribution from s_0(p)=p after inversion. The lower-degree terms come from the constant -1 in the trace formula. ","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3aa53b8d87472f5c","tex":"p^6","aliases":"p^6","display":"p^6","kind":"use","excerpt":"The index-dependent criterion already available from Deligne's bound is τ(p)=0iff p^6midτ(p), p≥5. Combined with the divisor convolution, it becomes 840S_4(p)not≡ p^4C(p)pmodp^6, p>7, as the exact desired exclusion. This gives a correctly typed target for a convolution route. No uniform arithmetic property excluding that residue has been proved her","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4927a5c0083e2327","tex":"p^{m-1}","aliases":"p^m-1","display":"p^{m-1}","kind":"use","excerpt":"refeq:trexpansion shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses is a monic polynomial F_m(x) in x=u^2. Its constant term and reduction modulo p are F_m(0)=(-1)^mp^m-1(p+1), F_m(x)≡ x^m-1(x-1)pmod p. The leading ","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-92403feabec1cc70","tex":"p^{m_p}","aliases":"p^m_p","display":"p^{m_p}","kind":"use","excerpt":"All prime cases have now been covered. If s=∏ p^m_p, multiplicativity gives R(s^2)=∏_pR(p^2m_p). Every factor is nonzero, and R(1)=1. This proves Theorem~[source reference].","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6f0ea21bbfbf69fb","tex":"p^r","aliases":"p^r","display":"p^r","kind":"use","excerpt":"At a prime p, the default local parameters are a=τ(p) and b=p^11. The scalar recurrence has T_0=1, T_1=a and T_r=aT_r-1-bT_r-2 for r≥2. Thus T_r=τ(p^r). A construction that permits an arbitrary integer a states that change explicitly. Such a recurrence need not be the local sequence of Δ.","count":15},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9b13adfda2839132","tex":"p^{r+1}","aliases":"p^r+1","display":"p^{r+1}","kind":"use","excerpt":"Why the problem lives at primes. The Hecke contracts. The normalized form Δ is a Hecke eigenform. Its coefficients satisfy τ(mn) · =τ(m)τ(n), · · gcd(m,n)=1, τ(p^r+1) · =τ(p)τ(p^r)-p^11τ(p^r-1), · · r≥1. The exponent 11 again records weight 12. The constant term for the local recurrence is τ(1)=1, and the first parameter is a=τ(p). These classical ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-da8a2616906ad75b","tex":"p^{r-1}","aliases":"p^r-1","display":"p^{r-1}","kind":"use","excerpt":"Why the problem lives at primes. The Hecke contracts. The normalized form Δ is a Hecke eigenform. Its coefficients satisfy τ(mn) · =τ(m)τ(n), · · gcd(m,n)=1, τ(p^r+1) · =τ(p)τ(p^r)-p^11τ(p^r-1), · · r≥1. The exponent 11 again records weight 12. The constant term for the local recurrence is τ(1)=1, and the first parameter is a=τ(p). These classical ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d7eaa7da6c63ef86","tex":"p^{r-2}","aliases":"p^r-2","display":"p^{r-2}","kind":"use","excerpt":"For every real B≥1 and integer M≥1, there exist a prime q>B and a multiplicative integer sequence f with the following properties: item f(1)=1 and f(n)=τ(n) for every n≤ B. item f(n)≡τ(n)pmod10M for every positive integer n. item Every weight-12 prime-power recurrence holds: f(p^r)=f(p)f(p^r-1)-p^11f(p^r-2), r≥2. item |f(n)|≤ d(n)n^11/2, where d(n)","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f621a92c3abc652c","tex":"p^{r-i}","aliases":"p^r-i","display":"p^{r-i}","kind":"use","excerpt":"A residue eliminates three prime classes. Suppose p≠5 and put χ=chifive(p). Modulo 5, χ≡ p^2, so the two recurrence roots may be taken as p and χ: p+χ≡ a, pχ≡ p^11pmod5. The complete homogeneous recurrence coefficient is T_r≡ C_r=∑_i=0^r p^r-iχ^ipmod5. Also χ p^10≡1pmod5. Since χ^2=1, for r≥2 the sum telescopes: R(p^r)≡ C_r-C_r-2 =p^r-1(p+χ)pmod5. ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f57ac308185c63af","tex":"p^{r-j}","aliases":"p^r-j","display":"p^{r-j}","kind":"use","excerpt":"The Eichler--Selberg trace identities give, for r≥1, ∑_j=0^r(-1)^r-jbinomr+jr-jp^r-js_2j(p) =-1-Tr T_2r+2(p), together with s_0(p)=p. Here T_k(p) is the Hecke operator at p on S_k(SL_2(Z)). It is distinct from the scalar recurrence notation T_r=τ(p^r) at a fixed prime. The trace normalization and Catalan inversion are checked against [source refere","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-93d9a4bd78ce9dca","tex":"p^{r_p}","aliases":"p^r_p","display":"p^{r_p}","kind":"use","excerpt":"· and τ(p)=0. In particular, Lehmer's conjecture is equivalent to nonvanishing at every prime. Factor n=∏_p p^r_p. Multiplicativity expresses τ(n) as a finite product of local blocks. An integral product is zero exactly when one factor is zero. Apply [source reference] to each factor.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"Take the prime indices (u_1,p_1)=(15,245759), (u_2,p_2)=(47,770047). The exact values are τ(245759) · =77900595298037436891536424960,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ed7de25e667a061c","tex":"P_{16249}","aliases":"P_16249","display":"P_{16249}","kind":"use","excerpt":"begingroupsmall setlengthtabcolsep4pt r r r r j · B_j in P_16249 · C_j in G_16249 · j+v_5(B_j)","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"Take the prime indices (u_1,p_1)=(15,245759), (u_2,p_2)=(47,770047). The exact values are τ(245759) · =77900595298037436891536424960,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-47d25f3be1fe7ea6","tex":"P_{31}","aliases":"P_31","display":"P_{31}","kind":"use","excerpt":"Individual packet polynomials have rational finite roots in [-4,4], since each root is -(i-j)/(c_i-c_j) with a nonzero integer denominator. The aggregate polynomial need not retain that root region. Exact values at p=31 are P_31(10)=-56767541184<0, P_31(11)=4645170845280>0. It has a real root between 10 and 11. Summing signed real-rooted factors do","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9167a9a19f6d041e","tex":"P_{499}","aliases":"P_499","display":"P_{499}","kind":"use","excerpt":"begingroupsmall setlengthtabcolsep5pt r r r r j · B_j in P_499 · C_j in G_499 · j+v_5(B_j)","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-dc7ba1c49968c37c","tex":"p_k","aliases":"p_k","display":"p_k","kind":"use","excerpt":"Separating the quasimodular weights. We need only an elementary weight-separation statement. Suppose finitely many quasimodular functions F_k of distinct nonnegative integer weights satisfy F_k(γ z)=∑_r=0^p_kf_k,r(z)c^r(cz+d)^k-r, f_k,0=F_k, p_k≤ k, for γ=[a · b","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d4876c09442ebddb","tex":"P_p","aliases":"P_p","display":"P_p","kind":"use","excerpt":"A double residual root with no lift. At p=16249, the shell has 16250 packets. The weighted coefficient valuations of P_p(5U) are (4,5,4,4,6,5,7,8,9,9,11), so μ_p=4. The reduction is overline G_p(U)=3+U^2+U^3=(U-1)^2(U-2)pmod5. The residual root at 1 is double. Exact coefficient reduction gives G_p(1)≡15pmod25, G'_p(1)≡0pmod5. For every y∈Z_5, polyn","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e70fd9df7027e2bf","tex":"P_v","aliases":"P_v","display":"P_v","kind":"use","excerpt":"For a packet v, set c_i=(v_i-i)/5 and form P_v(T)=∏_0≤ i<j≤4((i-j)+T(c_i-c_j)). Its physical value is P_v(5)=288W(v). Summing the full shell gives P_n(T)=∑_v∈VnP_v(T)=∑_j=0^10B_jT^j, P_n(5)=288τ(n). The polynomial has a different variable from the prime index. A root in T or in a rescaled variable does not represent a nearby prime coefficient.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-91513c11f34221ca","tex":"P_z","aliases":"P_z","display":"P_z","kind":"use","excerpt":"Fix z in the upper half-plane. The polynomial P_z(c,d)=∑_k∑_r f_k,r(z)c^r(cz+d)^k-r vanishes at every relatively prime integer pair (c,d), because every such pair is the bottom row of a modular matrix. These pairs are Zariski dense. To see this without a geometric theorem, fix any positive prime c. Infinitely many integers d are coprime to it, so t","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"The valuation v_ℓ(x) is the exponent of ℓ in a nonzero rational number x, with v_ℓ(0)=∞. The quadratic character used in the radial extraction is chifive(d)= 0 · 5mid d, 1 · d≡1,4pmod5, -1 · d≡2,3pmod5. The character value and the tenth power in the scaling formula are both essential.","count":63},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"use","excerpt":"The coarse coefficient vector a∈Q^18 makes x=s_f+M_fa, Ax=(0,0,0,0,0,-52843168)^t. The certificate then chooses a coefficient vector d∈ker M_c. Its final fine vector y=x+M_fd is zero except at index 6, where it equals -52843168.","count":20},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3ada406e06836134","tex":"q^{(2j+1)^2}","aliases":"q^(2j+1)^2","display":"q^{(2j+1)^2}","kind":"use","excerpt":"Parity from the actual product. Modulo 2, the Frobenius identity gives ∏_m≥1(1-q^m)^24 ≡∏_m≥1(1-q^8m)^3. Using [source reference], whose coefficients 2j+1 are odd, gives Δ(q)≡∑_j≥0q^1+4j(j+1) =∑_j≥0q^(2j+1)^2pmod2. Distinct nonnegative j give distinct exponents. Therefore τ(n) is odd Longleftrightarrow n is an odd square. In particular, τ(p) is eve","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-85f70422842d6a12","tex":"q^1","aliases":"q^1","display":"q^1","kind":"use","excerpt":"Why a finite initial segment suffices. The level-one valence formula implies that a nonzero holomorphic modular form of weight k has order at the cusp at most k/12 [source reference]. A cusp form whose coefficients of q^1,…,q^lfloor k/12rfloor all vanish has order at least lfloor k/12rfloor+1>k/12. It must be zero.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3de44ed9e1b95b85","tex":"q^{1+4j(j+1)}","aliases":"q^1+4j(j+1)","display":"q^{1+4j(j+1)}","kind":"use","excerpt":"Parity from the actual product. Modulo 2, the Frobenius identity gives ∏_m≥1(1-q^m)^24 ≡∏_m≥1(1-q^8m)^3. Using [source reference], whose coefficients 2j+1 are odd, gives Δ(q)≡∑_j≥0q^1+4j(j+1) =∑_j≥0q^(2j+1)^2pmod2. Distinct nonnegative j give distinct exponents. Therefore τ(n) is odd Longleftrightarrow n is an odd square. In particular, τ(p) is eve","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2b89fb5bfe4a5b66","tex":"q^{10}","aliases":"q^10","display":"q^{10}","kind":"use","excerpt":"A fixed modular form and an exact question. The coefficient that must not disappear. For |q|<1 the product defining Δ converges. As a formal power series it is even simpler: the coefficient of q^N depends only on the first N product factors. Thus each τ(n) is an unambiguously defined integer. The opening terms are Δ(q)= · q-24q^2+252q^3-1472q^4+483","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-df5e344a2840610e","tex":"q^{11}","aliases":"q^11","display":"q^{11}","kind":"use","excerpt":"The full size bound survives. At primes other than q, the Hecke roots have absolute value p^11/2 by Deligne. Formula [source reference] therefore gives |τ(p^r)|≤(r+1)p^11r/2. At the modified prime, F_q,2m=(-q^11)^m, F_q,2m+1=0. These obey the same local bound at every exponent, including zero. Multiplying the local estimates gives |f(n)|≤∏_p^r_pVer","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-87191a288dbc38a7","tex":"q^{11m-1}","aliases":"q^11m-1","display":"q^{11m-1}","kind":"use","excerpt":"All primitive squares still survive. Define the arithmetic transform R_f(n)=∑_d^2mid nμ(d)chifive(d)d^10f(n/d^2). It is multiplicative. Its prime blocks are R_f(p)=a_p, and for r≥2, R_f(p^r)=F_p,r-chifive(p)p^10F_p,r-2. For p≠ q, these are the actual R blocks and their even terms are nonzero. At q, direct substitution and chifive(q)=1 give R_f(q^2m","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e3a6a58fd2f88034","tex":"\\Q^{18}","aliases":"Q^18 Q","display":null,"kind":"use","excerpt":"The coarse coefficient vector a∈Q^18 makes x=s_f+M_fa, Ax=(0,0,0,0,0,-52843168)^t. The certificate then chooses a coefficient vector d∈ker M_c. Its final fine vector y=x+M_fd is zero except at index 6, where it equals -52843168.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"· =-2024+576-24=-1472. The last line comes respectively from the q^3 term in the first factor, the product of its q term with the second factor's q^2 term, and the third factor's q^3 term. Thus it computes τ(4), not τ(3).","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d0cd965c1b4a42cc","tex":"q^{2m}","aliases":"q^2m","display":"q^{2m}","kind":"use","excerpt":"All primitive squares still survive. Define the arithmetic transform R_f(n)=∑_d^2mid nμ(d)chifive(d)d^10f(n/d^2). It is multiplicative. Its prime blocks are R_f(p)=a_p, and for r≥2, R_f(p^r)=F_p,r-chifive(p)p^10F_p,r-2. For p≠ q, these are the actual R blocks and their even terms are nonzero. At q, direct substitution and chifive(q)=1 give R_f(q^2m","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d2d3c9a0c830bb0d","tex":"q^3","aliases":"q^3","display":"q^3","kind":"use","excerpt":"· =-2024+576-24=-1472. The last line comes respectively from the q^3 term in the first factor, the product of its q term with the second factor's q^2 term, and the third factor's q^3 term. Thus it computes τ(4), not τ(3).","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6e535c3600229c66","tex":"q^4","aliases":"q^4","display":"q^4","kind":"use","excerpt":"A fixed modular form and an exact question. The coefficient that must not disappear. For |q|<1 the product defining Δ converges. As a formal power series it is even simpler: the coefficient of q^N depends only on the first N product factors. Thus each τ(n) is an unambiguously defined integer. The opening terms are Δ(q)= · q-24q^2+252q^3-1472q^4+483","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-bb85c47db9e6c417","tex":"q^5","aliases":"q^5","display":"q^5","kind":"use","excerpt":"A fixed modular form and an exact question. The coefficient that must not disappear. For |q|<1 the product defining Δ converges. As a formal power series it is even simpler: the coefficient of q^N depends only on the first N product factors. Thus each τ(n) is an unambiguously defined integer. The opening terms are Δ(q)= · q-24q^2+252q^3-1472q^4+483","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ddca2fd3ea7fd7ac","tex":"q^6","aliases":"q^6","display":"q^6","kind":"use","excerpt":"A fixed modular form and an exact question. The coefficient that must not disappear. For |q|<1 the product defining Δ converges. As a formal power series it is even simpler: the coefficient of q^N depends only on the first N product factors. Thus each τ(n) is an unambiguously defined integer. The opening terms are Δ(q)= · q-24q^2+252q^3-1472q^4+483","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-823d9aa0ece788fc","tex":"q^7","aliases":"q^7","display":"q^7","kind":"use","excerpt":"A fixed modular form and an exact question. The coefficient that must not disappear. For |q|<1 the product defining Δ converges. As a formal power series it is even simpler: the coefficient of q^N depends only on the first N product factors. Thus each τ(n) is an unambiguously defined integer. The opening terms are Δ(q)= · q-24q^2+252q^3-1472q^4+483","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7ea2aec5575a954e","tex":"q^8","aliases":"q^8","display":"q^8","kind":"use","excerpt":"A fixed modular form and an exact question. The coefficient that must not disappear. For |q|<1 the product defining Δ converges. As a formal power series it is even simpler: the coefficient of q^N depends only on the first N product factors. Thus each τ(n) is an unambiguously defined integer. The opening terms are Δ(q)= · q-24q^2+252q^3-1472q^4+483","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-55b4afa8710524c1","tex":"q^{8m}","aliases":"q^8m","display":"q^{8m}","kind":"use","excerpt":"Parity from the actual product. Modulo 2, the Frobenius identity gives ∏_m≥1(1-q^m)^24 ≡∏_m≥1(1-q^8m)^3. Using [source reference], whose coefficients 2j+1 are odd, gives Δ(q)≡∑_j≥0q^1+4j(j+1) =∑_j≥0q^(2j+1)^2pmod2. Distinct nonnegative j give distinct exponents. Therefore τ(n) is odd Longleftrightarrow n is an odd square. In particular, τ(p) is eve","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-faadc201e7d8f078","tex":"q^9","aliases":"q^9","display":"q^9","kind":"use","excerpt":"A fixed modular form and an exact question. The coefficient that must not disappear. For |q|<1 the product defining Δ converges. As a formal power series it is even simpler: the coefficient of q^N depends only on the first N product factors. Thus each τ(n) is an unambiguously defined integer. The opening terms are Δ(q)= · q-24q^2+252q^3-1472q^4+483","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1cbdd272c4939594","tex":"\\Q^h","aliases":"Q^h Q","display":null,"kind":"use","excerpt":"Mass vectors and aggregation Fix a shell, a finite rational family Q_1,…,Q_h of universally null corrections, and the occupied sectors at depth k. Let their number be s_k. Define t_k,r=∑_v∈ S_k,rW(v), (M_k)_rj=∑_v∈ S_k,rW(v)Q_j(v). The vector t_k contains sector masses. The columns of M_k are the effects of the permitted corrections. Coefficients a","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-142e69180c7d1697","tex":"q^j","aliases":"q^j","display":"q^j","kind":"use","excerpt":"The logarithmic derivative provides a different exact engine. With σ_1(j)=∑_dmid jd, comparison of coefficients in qfracF'(q)F(q)=-24∑_j≥1σ_1(j)q^j gives m f_m=-24∑_j=1^mσ_1(j)f_m-j (m≥1). For m=3, the divisor sums are 1,3,4. Hence f_3=-8(252+3(-24)+4)=-8·184=-1472. An implementation should check that the numerator in [source reference] is divisibl","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7629ca39cb9a6855","tex":"q^{j(j+1)/2}","aliases":"q^j(j+1)/2","display":"q^{j(j+1)/2}","kind":"use","excerpt":"Jacobi's triangular series. Jacobi's identity is ∏_m≥1(1-q^m)^3 =∑_j≥0(-1)^j(2j+1)q^j(j+1)/2. Raising this series to the eighth power and multiplying by q gives Δ. The sparse triangular exponents make it useful for coefficient calculations. The source check in [source reference] also uses this identity to establish the parity criterion discussed be","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6386a60ab12436a6","tex":"q^k","aliases":"q^k","display":"q^k","kind":"use","excerpt":"Write F(q)=fracΔ(q)q=∏_k≥1(1-q^k)^24 =∑_m≥0f_mq^m. Then f_0=1 and τ(n)=f_n-1. This shift is essential. A program that stores τ(0)=0 as a sentinel must not confuse that entry with the constant coefficient f_0.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0c3c52cc15aac0b2","tex":"q^{kj}","aliases":"q^kj","display":"q^{kj}","kind":"use","excerpt":"For direct product expansion, terms of degree at most m depend only on factors with k≤ m. The exact binomial expansion (1-q^k)^24=∑_j=0^24(-1)^jbinom24jq^kj therefore supplies a finite algorithm for each desired coefficient. Three short calculations are f_1 · =-24,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-cc601c3b0b24e017","tex":"q^{\\lfloor k/12\\rfloor}","aliases":"q^lfloor k/12rfloor lfloor rfloor","display":null,"kind":"use","excerpt":"Why a finite initial segment suffices. The level-one valence formula implies that a nonzero holomorphic modular form of weight k has order at the cusp at most k/12 [source reference]. A cusp form whose coefficients of q^1,…,q^lfloor k/12rfloor all vanish has order at least lfloor k/12rfloor+1>k/12. It must be zero.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4a3a203830edc472","tex":"q^m","aliases":"q^m","display":"q^m","kind":"use","excerpt":"The coefficient and its local parameters. The normalized form is Δ(q)=q∏_m≥1(1-q^m)^24 =∑_n≥1τ(n)q^n, τ(1)=1. All coefficient indices in this book are positive integers. A stored value at index zero is a programming sentinel unless a displayed formula explicitly introduces the coefficients of Δ/q. For the latter coefficients, f_m=τ(m+1) and f_0=1.","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a23be335f8aa4b35","tex":"\\Q^N","aliases":"Q^N Q","display":null,"kind":"use","excerpt":"A shell-null interpolant can fail on the first shell. Let m_i be the nonzero mass of the ith orbit. Interpolation makes the map Q↦(m_iQ(v_i)) onto Q^N. Thus any zero-total transfer on this one shell can be produced by a homogeneous polynomial.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1928485a8127cf57","tex":"q^N","aliases":"q^N","display":"q^N","kind":"use","excerpt":"A fixed modular form and an exact question. The coefficient that must not disappear. For |q|<1 the product defining Δ converges. As a formal power series it is even simpler: the coefficient of q^N depends only on the first N product factors. Thus each τ(n) is an unambiguously defined integer. The opening terms are Δ(q)= · q-24q^2+252q^3-1472q^4+483","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-50cffe607a1e0f78","tex":"q^r","aliases":"q^r","display":"q^r","kind":"use","excerpt":"Exact coefficient engines. The product and a logarithmic recurrence. Let σ_j(n)=∑_dmid nd^j and write σ=σ_1. Logarithmic differentiation of the product gives DQΔ=(1-24∑_m≥1σ(m)q^m)Δ, DQ=qfracddq. Indeed, the contribution from the factor (1-q^r)^24 is -24fracrq^r1-q^r=-24∑_k≥1rq^rk. Collecting the terms with rk=m gives -24σ(m). The leading factor q ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-87f701058f187dfa","tex":"q^{rk}","aliases":"q^rk","display":"q^{rk}","kind":"use","excerpt":"oefficient engines The product and a logarithmic recurrence. Let σ_j(n)=∑_dmid nd^j and write σ=σ_1. Logarithmic differentiation of the product gives DQΔ=(1-24∑_m≥1σ(m)q^m)Δ, DQ=qfracddq. Indeed, the contribution from the factor (1-q^r)^24 is -24fracrq^r1-q^r=-24∑_k≥1rq^rk. Collecting the terms with rk=m gives -24σ(m). The leading factor q contribu","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ec1aa1812f8e1cec","tex":"\\Q^{s_k}","aliases":"Q^s_k Q","display":null,"kind":"use","excerpt":"Put H_k=ker(1^T:Q^s_k→Q), T_k=im M_k, E_k=H_k/T_k. The quotient E_k is the vector space of total-preserving changes that are unavailable from the chosen family. It is not a count of failed sign certificates. A sign certificate may exist even when E_k is nonzero.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0658eb7c4661dd48","tex":"q^{T_j}","aliases":"q^T_j","display":"q^{T_j}","kind":"use","excerpt":"Jacobi gives a third engine. Put T_j=j(j+1)/2 and a_j=(-1)^j(2j+1). The series A(q)=∏_k≥1(1-q^k)^3=1+∑_j≥1a_jq^T_j satisfies F=A^8. The identity AqF'=8FqA' yields m f_m=∑_substackj≥1","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-dec1f22253885f71","tex":"q^{|\\lambda|}","aliases":"q^|λ| lambda","display":"q^{|\\lambda|}","kind":"use","excerpt":"The Nekrasov--Okounkov identity in Han's formulation is ∏_m≥1(1-q^m)^z-1 =∑_λ q^|λ|∏_h∈ H(λ)(1-frac zh^2). At z=25, a partition having a hook of length 5 contributes zero. The surviving terms are the 5-cores. Multiplication by q shows that the coefficient τ(n) is the sum of the hook products over 5-cores of size n-1.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"use","excerpt":"Refinement and exact transfer spaces. Mass vectors and aggregation. Fix a shell, a finite rational family Q_1,…,Q_h of universally null corrections, and the occupied sectors at depth k. Let their number be s_k. Define t_k,r=∑_v∈ S_k,rW(v), (M_k)_rj=∑_v∈ S_k,rW(v)Q_j(v). The vector t_k contains sector masses. The columns of M_k are the effects of th","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-757d025f70886a4b","tex":"Q_{10}","aliases":"Q_10","display":"Q_{10}","kind":"use","excerpt":"· +69600s_3s_5+45000s_4^2, and Q_10= · 2091s_2^5-11250s_2^3s_4-145800s_2^2s_3^2notag · +409600s_2s_3s_5+15000s_2s_4^2notag · -30000s_3^2s_4-264000s_5^2. Both DQ_8 and DQ_10 are harmonic. The first shell has the vector (0,1,2,-2,-1), with s_2=10, s_3=0, s_4=34, s_5=0. Both polynomials vanish there. Their theta weights are 20 and 22. The cusp spaces ","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d2dd5eb187fc710e","tex":"Q_8","aliases":"Q_8","display":"Q_8","kind":"use","excerpt":"The first explicit corrections. The construction gives Q_8= · 6273s_2^4-33750s_2^2s_4-56800s_2s_3^2notag · +69600s_3s_5+45000s_4^2, and Q_10= · 2091s_2^5-11250s_2^3s_4-145800s_2^2s_3^2notag","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-83dc7e104ed7732b","tex":"Q_d","aliases":"Q_d","display":"Q_d","kind":"use","excerpt":"Let Q be any finite symmetric polynomial over C, restricted to ∑_i v_i=0. Suppose ∑_v∈VnW(v)Q(v)=0 for every positive integer n. Write Q=∑_dQ_d in homogeneous degrees. Then every Q_d is separately universally null. Moreover, the weighted Q-sum is zero in each exact-content sector of every shell.","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d6a9b153d5eaddc8","tex":"Q_h","aliases":"Q_h","display":"Q_h","kind":"use","excerpt":"Refinement and exact transfer spaces. Mass vectors and aggregation. Fix a shell, a finite rational family Q_1,…,Q_h of universally null corrections, and the occupied sectors at depth k. Let their number be s_k. Define t_k,r=∑_v∈ S_k,rW(v), (M_k)_rj=∑_v∈ S_k,rW(v)Q_j(v). The vector t_k contains sector masses. The columns of M_k are the effects of th","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"use","excerpt":"Completeness retains the total. Suppose a shell has nonzero orbit masses w_1,…,w_N and total T. Even allow every rational correction vector q with ∑_iw_iq_i=0. Then ∃ q: w_i(1+q_i) have one weak sign and are not all zero Longleftrightarrow T≠0. Necessity follows from conservation. For sufficiency, if T≠0, take q_i=fracTNw_i-1. Every corrected mass ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"nThe correction that does not change any coefficient A universally null polynomial is a symmetric polynomial Q with ∑_v∈VnW(v)Q(v)=0 for every n≥1. This is much stronger than being null on one selected shell. If a finite family satisfies [source reference], any rational linear combination of it does too. Thus F(v)=1+∑_j a_jQ_j(v) has the exact prop","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8205838caa474d98","tex":"\\Q_p","aliases":"Q_p Q","display":null,"kind":"use","excerpt":"For every prime p, there is a weakly admissible two-dimensional filtered φ-module over Q_p, with monodromy N=0, filtration jumps 0,11 and Frobenius characteristic polynomial X^2+p^11.","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-857f3b2bbbec9d27","tex":"Q_{p,j}","aliases":"Q_p,j","display":"Q_{p,j}","kind":"use","excerpt":"A sufficient sign input. A second precise route would construct permitted corrections with an independently proved sign. At each remaining prime, choose finitely many rational universal null polynomials Q_p,j and rational coefficients a_p,j, and put F_p(v)=1+∑_j a_p,jQ_p,j(v). The choice may depend on p. Universal nullness must hold after the choic","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2381665eb6ba229d","tex":"\\Q_p^\\times","aliases":"Q_p^× Q times","display":null,"kind":"use","excerpt":"There is no proper nonzero φ-stable Q_p-subspace. In dimension two such a subspace would be a line, and φ on it would have an eigenvalue λ∈Q_p satisfying λ^2=-p^11. This would require 2v_p(λ)=11, impossible for an integer-valued valuation on Q_p^×. Thus the proper-subobject inequalities are vacuous. The zero object imposes equality 0=0, and the who","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b21f30f57b90ea39","tex":"\\Q_\\ell","aliases":"Q_ℓ Q ell","display":null,"kind":"use","excerpt":"Integral representations and complex conjugation. For each prime ℓ, the representation attached to Δ has the arithmetic-Frobenius normalization ρ_ℓ:G_Q⟶ GL_2(Q_ℓ), is continuous and unramified outside ℓ, and satisfies det(X-ρ_ℓ(Frob_p))=X^2-τ(p)X+p^11, p≠ℓ. Its determinant is the eleventh power of the ℓ-adic cyclotomic character. The original cohom","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"The universal null construction depends on an exact differential operator. This appendix derives that operator from the Euclidean metric, rather than treating its power-sum formulas as a black box. It also explains the Newton reductions used by the implementation and gives two low-degree harmonic checks. All polynomial identities are on the hyperpl","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-bb58fc47ddb31a6c","tex":"\\R^5","aliases":"R^5 R","display":null,"kind":"use","excerpt":"The universal null construction depends on an exact differential operator. This appendix derives that operator from the Euclidean metric, rather than treating its power-sum formulas as a black box. It also explains the Newton reductions used by the implementation and gives two low-degree harmonic checks. All polynomial identities are on the hyperpl","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a94836df7b97d193","tex":"R_f","aliases":"R_f","display":"R_f","kind":"use","excerpt":"B. item f(n)≡τ(n)pmod10M for every positive integer n. item Every weight-12 prime-power recurrence holds: f(p^r)=f(p)f(p^r-1)-p^11f(p^r-2), r≥2. item |f(n)|≤ d(n)n^11/2, where d(n) counts the positive divisors of n. item f(n) is odd exactly when n is an odd square, and f(n)≡ nσ(n)pmod5. item The arithmetic primitive transform R_f is nonzero at ever","count":9},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"use","excerpt":"The exact refinement sequence. There is a short exact sequence 0⟶ fracker AM_k+1(ker M_k) ⟶ E_k+1⟶ E_k⟶0. Consequently, if r_k=rank M_k, dim E_k+1-dim E_k =(s_k+1-s_k)-(r_k+1-r_k). The aggregation map sends H_k+1 onto H_k. Given a zero-total parent vector, place each parent entry in one selected child and zero in its other children. The resulting c","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4d5e97b2dd21ca5d","tex":"r_{k+1}","aliases":"r_k+1","display":"r_{k+1}","kind":"use","excerpt":"The exact refinement sequence. There is a short exact sequence 0⟶ fracker AM_k+1(ker M_k) ⟶ E_k+1⟶ E_k⟶0. Consequently, if r_k=rank M_k, dim E_k+1-dim E_k =(s_k+1-s_k)-(r_k+1-r_k). The aggregation map sends H_k+1 onto H_k. Given a zero-total parent vector, place each parent entry in one selected child and zero in its other children. The resulting c","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0d5a58c436e4715b","tex":"r_p","aliases":"r_p","display":"r_p","kind":"use","excerpt":"· and τ(p)=0. In particular, Lehmer's conjecture is equivalent to nonvanishing at every prime. Factor n=∏_p p^r_p. Multiplicativity expresses τ(n) as a finite product of local blocks. An integral product is zero exactly when one factor is zero. Apply [source reference] to each factor.","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-04925530c4c8b896","tex":"R_Q","aliases":"R_Q","display":"R_Q","kind":"use","excerpt":"Homogeneous universal nulls preserve each content. Let Q be a symmetric homogeneous polynomial of degree d that is universally null. Then its weighted sum is zero separately in every exact-content sector on every shell. Let R_Q(n)=∑_substackv∈Vn g(v)=1W(v)Q(v). Symmetry removes the residue permutation from Q, and homogeneity gives Q(v)=g^dQ(u). The","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank","display":null,"kind":"use","excerpt":"The rational row-reduction results are rank M_c=5, rank M_f=15, rank B=10. An independent check obtains the same lower bounds over both prime fields F_65537 and F_1000003. Every denominator used in a reduction is invertible there. A nonzero minor in either field proves a nonzero rational minor after clearing denominators.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"The sharp threshold. Let k≥1, M=2^k, and set N_k=lceilfrac100M^2+10M+235rceil =frac20M^2+2M+γ_k7, where γ_k= 7 · k≡1pmod3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"S15: differential equations. Fonseca records the normalized Ramanujan differential relations used in the polynomial computation [source reference]. The project-specific S_4 cancellation is verified algebraically in the main text and by an exact symbolic remainder. Modularity and the level-one dimension input for the cusp identity are separate contr","count":32},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c8d0b37b4af3bfcc","tex":"S^2","aliases":"S^2","display":"S^2","kind":"use","excerpt":"A complete shell enumerator. The shell is finite because every coordinate has absolute value at most sqrt10n. Enumeration can reduce its dimension without discarding any solution. Choose v_0,v_1,v_2 in their residue classes within the bound. The remaining coordinates x=v_3, y=v_4 must satisfy x+y=S=-v_0-v_1-v_2, x^2+y^2=T=10n-v_0^2-v_1^2-v_2^2. The","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"R(s^2)≠0 for all s · Every even primitive local block survives, including a hypothetical zero prime parameter.","count":7},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1e95f7258e4cb3b8","tex":"S^3","aliases":"S^3","display":"S^3","kind":"use","excerpt":", since the squared residues sum to 30. It is even, since a_i^2≡ a_ipmod2 and ∑_i a_i=0. The quadratic exponent ∑_i a_i^2/10 is integral. The Weil operator T therefore acts as 1 on the sign line. The Weil action commutes with S_5, so this line is invariant. The rank is even, and the representation descends to SL_2(Z). On the line the relation (ST)^","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"Write s_m=∑_i=0^4v_i^m, with s_0=5 and s_1=0, and let D=∏_i<j(v_i-v_j). We distinguish the ordinary Laplacian Δ_H from the conjugated operator L defined by Δ_H(DQ)=D LQ. The factor D changes radial coefficients substantially. Confusing the two operators would invalidate the harmonicity test.","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1782caccc889225f","tex":"s_0^2","aliases":"s_0^2","display":"s_0^2","kind":"use","excerpt":"The first values are Ls_2=48, Ls_3=0, Ls_4=frac1885s_2, Ls_5=46s_3. For example, at m=2 the first term is 2s_0^2=50, while the subtraction is 2s_0/5=2. The additional 40 compared with Δ_Hs_2=8 comes from differentiating the degree-ten factor D.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-77ceec1a9c3f6d4a","tex":"s_1","aliases":"s_1","display":"s_1","kind":"use","excerpt":"Write s_m=∑_i=0^4v_i^m, with s_0=5 and s_1=0, and let D=∏_i<j(v_i-v_j). We distinguish the ordinary Laplacian Δ_H from the conjugated operator L defined by Δ_H(DQ)=D LQ. The factor D changes radial coefficients substantially. Confusing the two operators would invalidate the harmonicity test.","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1c7310c194cdfc9d","tex":"s_{10}","aliases":"s_10","display":"s_{10}","kind":"use","excerpt":"The resulting first moments are s_2(p) · =p^2-1, s_4(p) · =2p^3-3p-1, s_6(p) · =5p^4-9p^2-5p-1, s_8(p) · =14p^5-28p^3-20p^2-7p-1, s_10(p) · =42p^6-90p^4-75p^3-35p^2-9p-1-τ(p). The integer 42 is the fifth Catalan number. It records the contribution from s_0(p)=p after inversion. The lower-degree terms come from the constant -1 in the trace formula. ","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-850bbfb51b10c4e9","tex":"S_{12+d}","aliases":"S_12+d","display":"S_{12+d}","kind":"use","excerpt":"tionThe modularity contract and the cusp condition The authenticated harmonic theta theorem states that a homogeneous harmonic polynomial of degree ν on a rank-r positive definite even lattice gives a vector-valued modular form of weight ν+r/2 for the Weil representation [source reference]. Here r=4 and ν=10+d. The project-specific sign-line argume","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-37d8f31b7ce0f58c","tex":"s_2^2","aliases":"s_2^2","display":"s_2^2","kind":"use","excerpt":"Useful instances are L(s_2^2) · =104s_2, L(s_2^3) · =168s_2^2, L(s_2s_4) · =64s_4+frac1885s_2^2, L(s_3^2) · =18s_4-frac185s_2^2. For the first identity, 2s_2 Ls_2+2Γ_22=96s_2+8s_2. For the mixed product, s_4 Ls_2+s_2 Ls_4+2Γ_24 gives the displayed coefficients. Each calculation is a check of the ordered-pair convention.","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0ac09bb3afa2584c","tex":"s_2^3","aliases":"s_2^3","display":"s_2^3","kind":"use","excerpt":"Useful instances are L(s_2^2) · =104s_2, L(s_2^3) · =168s_2^2, L(s_2s_4) · =64s_4+frac1885s_2^2, L(s_3^2) · =18s_4-frac185s_2^2. For the first identity, 2s_2 Ls_2+2Γ_22=96s_2+8s_2. For the mixed product, s_4 Ls_2+s_2 Ls_4+2Γ_24 gives the displayed coefficients. Each calculation is a check of the ordered-pair convention.","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-38b82770b80caef3","tex":"s_2^4","aliases":"s_2^4","display":"s_2^4","kind":"use","excerpt":"s_7 · =frac710s_2s_5+frac712s_3s_4 -frac724s_2^2s_3. The same rule gives s_8=frac14s_4^2+frac14s_2^2s_4 -frac116s_2^4+frac815s_3s_5. Each term has the required invariant degree. This supplies a useful local check before substitution into a larger operator expression.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6f4da9f341efd4aa","tex":"s_2^5","aliases":"s_2^5","display":"s_2^5","kind":"use","excerpt":"· +69600s_3s_5+45000s_4^2, and Q_10= · 2091s_2^5-11250s_2^3s_4-145800s_2^2s_3^2notag · +409600s_2s_3s_5+15000s_2s_4^2notag · -30000s_3^2s_4-264000s_5^2. Both DQ_8 and DQ_10 are harmonic. The first shell has the vector (0,1,2,-2,-1), with s_2=10, s_3=0, s_4=34, s_5=0. Both polynomials vanish there. Their theta weights are 20 and 22. The cusp spaces ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d467f6b298c372d4","tex":"s_2^a","aliases":"s_2^a","display":"s_2^a","kind":"use","excerpt":"At each even invariant degree d, one lists monomials s_2^as_3^bs_4^cs_5^e with 2a+3b+4c+5e=d. Writing an unknown rational linear combination, the equation LQ=0 is a finite linear system. The initial theta coefficients supply further rational linear equations. No floating-point approximation is needed to define this kernel.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8ff11fb70723cda9","tex":"s_2^{j-1}","aliases":"s_2^j-1","display":"s_2^{j-1}","kind":"use","excerpt":"Radial factors and harmonic projection. Let h_ℓ be homogeneous harmonic of degree ℓ on the rank-four space H. Euler's identity is v·∇_Hh_ℓ=ℓ h_ℓ. The product rule for the radial factor s_2^j gives Δ_H(s_2^jh_ℓ) =4j(ℓ+j+1)s_2^j-1h_ℓ, j≥1. At j=0 the left side is zero. At j=1,ℓ=0 the coefficient is 8, agreeing with Δ_Hs_2. At j=1,ℓ=10 it is 48, agree","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a31b04487103f85b","tex":"s_2^k","aliases":"s_2^k","display":"s_2^k","kind":"use","excerpt":"More generally, the top harmonic projection of s_2^kD at degree 10+2k is zero for every k>0. Radial information can disappear. Replacing an interpolant by its top harmonic piece may therefore destroy exactly the values it was designed to have.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7b8539d67beca0e3","tex":"s_3","aliases":"s_3","display":"s_3","kind":"use","excerpt":"The first values are Ls_2=48, Ls_3=0, Ls_4=frac1885s_2, Ls_5=46s_3. For example, at m=2 the first term is 2s_0^2=50, while the subtraction is 2s_0/5=2. The additional 40 compared with Δ_Hs_2=8 comes from differentiating the degree-ten factor D.","count":20},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3f4c1b37eb59775d","tex":"s_3^2","aliases":"s_3^2","display":"s_3^2","kind":"use","excerpt":"Useful instances are L(s_2^2) · =104s_2, L(s_2^3) · =168s_2^2, L(s_2s_4) · =64s_4+frac1885s_2^2, L(s_3^2) · =18s_4-frac185s_2^2. For the first identity, 2s_2 Ls_2+2Γ_22=96s_2+8s_2. For the mixed product, s_4 Ls_2+s_2 Ls_4+2Γ_24 gives the displayed coefficients. Each calculation is a check of the ordered-pair convention.","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-03b6e63542f2fc03","tex":"s_3^b","aliases":"s_3^b","display":"s_3^b","kind":"use","excerpt":"At each even invariant degree d, one lists monomials s_2^as_3^bs_4^cs_5^e with 2a+3b+4c+5e=d. Writing an unknown rational linear combination, the equation LQ=0 is a finite linear system. The initial theta coefficients supply further rational linear equations. No floating-point approximation is needed to define this kernel.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f6d0dc9f5b462d0b","tex":"S_4","aliases":"S_4","display":"S_4","kind":"use","excerpt":"S15: differential equations. Fonseca records the normalized Ramanujan differential relations used in the polynomial computation [source reference]. The project-specific S_4 cancellation is verified algebraically in the main text and by an exact symbolic remainder. Modularity and the level-one dimension input for the cusp identity are separate contr","count":14},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-80534a5aa724e3ef","tex":"s_4","aliases":"s_4","display":"s_4","kind":"use","excerpt":"The first values are Ls_2=48, Ls_3=0, Ls_4=frac1885s_2, Ls_5=46s_3. For example, at m=2 the first term is 2s_0^2=50, while the subtraction is 2s_0/5=2. The additional 40 compared with Δ_Hs_2=8 comes from differentiating the degree-ten factor D.","count":27},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1104d70e5dbe2450","tex":"s_4^2","aliases":"s_4^2","display":"s_4^2","kind":"use","excerpt":"s_7 · =frac710s_2s_5+frac712s_3s_4 -frac724s_2^2s_3. The same rule gives s_8=frac14s_4^2+frac14s_2^2s_4 -frac116s_2^4+frac815s_3s_5. Each term has the required invariant degree. This supplies a useful local check before substitution into a larger operator expression.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7a092a307b3eba88","tex":"s_4^c","aliases":"s_4^c","display":"s_4^c","kind":"use","excerpt":"At each even invariant degree d, one lists monomials s_2^as_3^bs_4^cs_5^e with 2a+3b+4c+5e=d. Writing an unknown rational linear combination, the equation LQ=0 is a finite linear system. The initial theta coefficients supply further rational linear equations. No floating-point approximation is needed to define this kernel.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-aabbd2f9e4d841eb","tex":"S_5","aliases":"S_5","display":"S_5","kind":"use","excerpt":"If all five residues are distinct, they are 0,1,2,3,4 in some order. Their sum is zero modulo 5, so ∑_i a_i=-t forces t=0. These regular cosets form one free S_5-orbit of size 120. The sign representation occurs exactly once in the corresponding permutation module. It does not occur in an irregular orbit, since an odd stabilizer kills the alternati","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7756990fa8ff2caf","tex":"s_5","aliases":"s_5","display":"s_5","kind":"use","excerpt":"The first values are Ls_2=48, Ls_3=0, Ls_4=frac1885s_2, Ls_5=46s_3. For example, at m=2 the first term is 2s_0^2=50, while the subtraction is 2s_0/5=2. The additional 40 compared with Δ_Hs_2=8 comes from differentiating the degree-ten factor D.","count":16},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-15b1492cf676ccb1","tex":"s_5^2","aliases":"s_5^2","display":"s_5^2","kind":"use","excerpt":"· +69600s_3s_5+45000s_4^2, and Q_10= · 2091s_2^5-11250s_2^3s_4-145800s_2^2s_3^2notag · +409600s_2s_3s_5+15000s_2s_4^2notag · -30000s_3^2s_4-264000s_5^2. Both DQ_8 and DQ_10 are harmonic. The first shell has the vector (0,1,2,-2,-1), with s_2=10, s_3=0, s_4=34, s_5=0. Both polynomials vanish there. Their theta weights are 20 and 22. The cusp spaces ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-59fa844aa220d565","tex":"s_5^e","aliases":"s_5^e","display":"s_5^e","kind":"use","excerpt":"At each even invariant degree d, one lists monomials s_2^as_3^bs_4^cs_5^e with 2a+3b+4c+5e=d. Writing an unknown rational linear combination, the equation LQ=0 is a finite linear system. The initial theta coefficients supply further rational linear equations. No floating-point approximation is needed to define this kernel.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-bd1cb353cb061304","tex":"s_6","aliases":"s_6","display":"s_6","kind":"use","excerpt":"The metric formula may produce s_6,s_7,s_8. They are not new independent variables. The invariant ring on H is generated by s_2,s_3,s_4,s_5, and Newton's identities reduce higher indices exactly.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-186dd4bbd1fde878","tex":"s_7","aliases":"s_7","display":"s_7","kind":"use","excerpt":"The metric formula may produce s_6,s_7,s_8. They are not new independent variables. The invariant ring on H is generated by s_2,s_3,s_4,s_5, and Newton's identities reduce higher indices exactly.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-24ba260a7f005920","tex":"s_8","aliases":"s_8","display":"s_8","kind":"use","excerpt":"The metric formula may produce s_6,s_7,s_8. They are not new independent variables. The invariant ring on H is generated by s_2,s_3,s_4,s_5, and Newton's identities reduce higher indices exactly.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1dcf95b2a026d667","tex":"s_a","aliases":"s_a","display":"s_a","kind":"use","excerpt":"ed indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v_i-v_j. The divided difference is the finite polynomial fracv_i^m-1-v_j^m-1v_i-v_j =∑_a=0^m-2v_i^av_j^m-2-a. The sum over a is symmetric in the two coordinates. Hence twice the sum over i<j equals the sum over i≠ j. For each a, that ordered sum is s_as_m-2-a-s_m-2. There ","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a64b77f6851063e5","tex":"s_c","aliases":"s_c","display":"s_c","kind":"use","excerpt":"For instance, row 0 aggregates fine indices 0,8,10,14: 2354352-11844448-31156048-7455448=-48101592. The six coarse masses are therefore s_c=(-48101592,-7162848,18972408, -12874848,-15684592,12008304)^t. Their sum, and the sum of the fine masses, is -52843168. These original masses differ in order from the lexicographic orbit bars because the certif","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-deef7bca5ee0eda2","tex":"S_k","aliases":"S_k","display":"S_k","kind":"use","excerpt":"The Eichler--Selberg trace identities give, for r≥1, ∑_j=0^r(-1)^r-jbinomr+jr-jp^r-js_2j(p) =-1-Tr T_2r+2(p), together with s_0(p)=p. Here T_k(p) is the Hecke operator at p on S_k(SL_2(Z)). It is distinct from the scalar recurrence notation T_r=τ(p^r) at a fixed prime. The trace normalization and Catalan inversion are checked against [source refere","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b4ebcce0dc1a5c39","tex":"s_k","aliases":"s_k","display":"s_k","kind":"use","excerpt":"Every coordinate satisfies its characteristic polynomial. Multiplying that equality by v_i^k-5 and summing, for k≥6, gives s_k=-e_2s_k-2+e_3s_k-3-e_4s_k-4+e_5s_k-5. This recurrence uses only lower indices. Substitution at k=6,7 yields s_6 · =frac34s_2s_4+frac13s_3^2-frac18s_2^3,","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8a81608b031d678e","tex":"s_{k+1}","aliases":"s_k+1","display":"s_{k+1}","kind":"use","excerpt":"The exact refinement sequence. There is a short exact sequence 0⟶ fracker AM_k+1(ker M_k) ⟶ E_k+1⟶ E_k⟶0. Consequently, if r_k=rank M_k, dim E_k+1-dim E_k =(s_k+1-s_k)-(r_k+1-r_k). The aggregation map sends H_k+1 onto H_k. Given a zero-total parent vector, place each parent entry in one selected child and zero in its other children. The resulting c","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d1ed55130338922b","tex":"S_{k,r}","aliases":"S_k,r","display":"S_{k,r}","kind":"use","excerpt":"Refinement and exact transfer spaces. Mass vectors and aggregation. Fix a shell, a finite rational family Q_1,…,Q_h of universally null corrections, and the occupied sectors at depth k. Let their number be s_k. Define t_k,r=∑_v∈ S_k,rW(v), (M_k)_rj=∑_v∈ S_k,rW(v)Q_j(v). The vector t_k contains sector masses. The columns of M_k are the effects of th","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c925deae4419392f","tex":"s_{k-2}","aliases":"s_k-2","display":"s_{k-2}","kind":"use","excerpt":"Every coordinate satisfies its characteristic polynomial. Multiplying that equality by v_i^k-5 and summing, for k≥6, gives s_k=-e_2s_k-2+e_3s_k-3-e_4s_k-4+e_5s_k-5. This recurrence uses only lower indices. Substitution at k=6,7 yields s_6 · =frac34s_2s_4+frac13s_3^2-frac18s_2^3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e2e2e0b1a0ae9bff","tex":"s_{k-3}","aliases":"s_k-3","display":"s_{k-3}","kind":"use","excerpt":"Every coordinate satisfies its characteristic polynomial. Multiplying that equality by v_i^k-5 and summing, for k≥6, gives s_k=-e_2s_k-2+e_3s_k-3-e_4s_k-4+e_5s_k-5. This recurrence uses only lower indices. Substitution at k=6,7 yields s_6 · =frac34s_2s_4+frac13s_3^2-frac18s_2^3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e7e768288a3aa4c6","tex":"s_{k-4}","aliases":"s_k-4","display":"s_{k-4}","kind":"use","excerpt":"Every coordinate satisfies its characteristic polynomial. Multiplying that equality by v_i^k-5 and summing, for k≥6, gives s_k=-e_2s_k-2+e_3s_k-3-e_4s_k-4+e_5s_k-5. This recurrence uses only lower indices. Substitution at k=6,7 yields s_6 · =frac34s_2s_4+frac13s_3^2-frac18s_2^3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9b7f54f6849e6cf8","tex":"s_{k-5}","aliases":"s_k-5","display":"s_{k-5}","kind":"use","excerpt":"Every coordinate satisfies its characteristic polynomial. Multiplying that equality by v_i^k-5 and summing, for k≥6, gives s_k=-e_2s_k-2+e_3s_k-3-e_4s_k-4+e_5s_k-5. This recurrence uses only lower indices. Substitution at k=6,7 yields s_6 · =frac34s_2s_4+frac13s_3^2-frac18s_2^3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b9746e4f6fc18b24","tex":"s_m","aliases":"s_m","display":"s_m","kind":"use","excerpt":"Write s_m=∑_i=0^4v_i^m, with s_0=5 and s_1=0, and let D=∏_i<j(v_i-v_j). We distinguish the ordinary Laplacian Δ_H from the conjugated operator L defined by Δ_H(DQ)=D LQ. The factor D changes radial coefficients substantially. Confusing the two operators would invalidate the harmonicity test.","count":11},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d642bf93c93df609","tex":"s_{m+n-2}","aliases":"s_m+n-2","display":"s_{m+n-2}","kind":"use","excerpt":"The chain-rule metric and polynomial assembly. The same projected gradients give Γ_mn:=∇_Hs_m·∇_Hs_n =mn(s_m+n-2-frac15s_m-1s_n-1). Expanding the product in [source reference] gives the first power sum once and the coordinate-mean product with coefficient -1/5. This identity includes m,n≥1, with the evident zero tangential gradient for s_1.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d6f3fd1d129e9558","tex":"s_{m-1}","aliases":"s_m-1","display":"s_{m-1}","kind":"use","excerpt":"Deriving the power-sum operator. For m≥2, differentiation and projection give (∇_Hs_m)_i · =m(v_i^m-1-frac15s_m-1), Δ_Hs_m · =frac45m(m-1)s_m-2. In the logarithmic term, the coordinate-mean subtraction vanishes because the logarithmic gradient has zero sum. Pair the ordered indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e0048501d83523c1","tex":"s_{m-2}","aliases":"s_m-2","display":"s_{m-2}","kind":"use","excerpt":"Deriving the power-sum operator. For m≥2, differentiation and projection give (∇_Hs_m)_i · =m(v_i^m-1-frac15s_m-1), Δ_Hs_m · =frac45m(m-1)s_m-2. In the logarithmic term, the coordinate-mean subtraction vanishes because the logarithmic gradient has zero sum. Pair the ordered indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6b26b51c775a21f7","tex":"s_{m-2-a}","aliases":"s_m-2-a","display":"s_{m-2-a}","kind":"use","excerpt":"ed indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v_i-v_j. The divided difference is the finite polynomial fracv_i^m-1-v_j^m-1v_i-v_j =∑_a=0^m-2v_i^av_j^m-2-a. The sum over a is symmetric in the two coordinates. Hence twice the sum over i<j equals the sum over i≠ j. For each a, that ordered sum is s_as_m-2-a-s_m-2. There ","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-91ec47101c58ab98","tex":"s_n","aliases":"s_n","display":"s_n","kind":"use","excerpt":"The chain-rule metric and polynomial assembly. The same projected gradients give Γ_mn:=∇_Hs_m·∇_Hs_n =mn(s_m+n-2-frac15s_m-1s_n-1). Expanding the product in [source reference] gives the first power sum once and the coordinate-mean product with coefficient -1/5. This identity includes m,n≥1, with the evident zero tangential gradient for s_1.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-48c952833d010b8e","tex":"s_{n-1}","aliases":"s_n-1","display":"s_{n-1}","kind":"use","excerpt":"The chain-rule metric and polynomial assembly. The same projected gradients give Γ_mn:=∇_Hs_m·∇_Hs_n =mn(s_m+n-2-frac15s_m-1s_n-1). Expanding the product in [source reference] gives the first power sum once and the coordinate-mean product with coefficient -1/5. This identity includes m,n≥1, with the evident zero tangential gradient for s_1.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b37e9083bef6f614","tex":"S_r","aliases":"S_r","display":"S_r","kind":"use","excerpt":"Why strict localization and automatic valuation fail. Conserved sectors can preserve opposite signs. Partition a shell into sectors S_r, with masses t_r=∑_v∈ S_rW(v). Suppose every permitted correction is null inside every sector: ∑_v∈ S_rW(v)Q_j(v)=0. Then F=1+∑ a_jQ_j leaves each corrected sector mass equal to t_r. If two sector masses have oppos","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-18f5af16bd34d11c","tex":"\\sgn","aliases":"sgn","display":null,"kind":"use","excerpt":"figplaceholdercoefficients512The exact coefficients through 512 on a signed logarithmic scale. The plotted ordinate is sgn(τ(n))log_10(1+|τ(n)|). Sign changes and large values are visible, while every plotted coefficient is nonzero. This finite plot does not establish the conjecture.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-59d24711aca21962","tex":"\\mathrm{st}","aliases":"st mathrm","display":"\\mathrm{st}","kind":"use","excerpt":"The realization theorem and conventions. Colmez--Fontaine's theorem makes a weakly admissible filtered module admissible [source reference]. In the present setting K=K_0=Q_p and the base Frobenius is the identity. The theorem supplies a two-dimensional semistable Q_p-representation realizing the filtered module. Since N=0, the comparison is crystal","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"Newton reductions with zero first power sum. Let e_j be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is ∏_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5, because e_1=s_1=0. The first Newton identities give e_2 · =-frac12s_2, · e_3 · =frac13s_3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","excerpt":"Newton reductions with zero first power sum. Let e_j be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is ∏_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5, because e_1=s_1=0. The first Newton identities give e_2 · =-frac12s_2, · e_3 · =frac13s_3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"An explicit homogeneous interpolation basis. Let v_1,…,v_N represent the distinct J-orbits. For an integer t≥0, define the degree-10 symmetric polynomial E_t=s_4s_2^3+t s_3^2s_2^2+t^2s_3s_5s_2+t^3s_5^2. For two distinct orbits, E_t(v_i)-E_t(v_j) is a nonzero polynomial in t of degree at most 3, by Proposition~[source reference]. It has at most thre","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-aa1878106eeff5b6","tex":"T^5","aliases":"T^5","display":"T^5","kind":"use","excerpt":"Newton reductions with zero first power sum. Let e_j be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is ∏_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5, because e_1=s_1=0. The first Newton identities give e_2 · =-frac12s_2, · e_3 · =frac13s_3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b4fb2a948e5c67c2","tex":"T_{-1}","aliases":"T_-1","display":"T_{-1}","kind":"use","excerpt":"∑_r≥0R(p^r)X^r =frac1-cX^21-aX+bX^2. It follows either by multiplying the Hecke factor by the square-supported Möbius factor, or by reading the two possible square divisors with nonzero Möbius value at a prime power. In particular, R(1)=1, R(p)=a, R(p^r)=T_r-cT_r-2 (r≥2). The subtraction begins at r=2. No undefined T_-1 or T_-2 is needed.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e4cadb7cd54dc378","tex":"T_{-2}","aliases":"T_-2","display":"T_{-2}","kind":"use","excerpt":"local ∑_r≥0R(p^r)X^r =frac1-cX^21-aX+bX^2. It follows either by multiplying the Hecke factor by the square-supported Möbius factor, or by reading the two possible square divisors with nonzero Möbius value at a prime power. In particular, R(1)=1, R(p)=a, R(p^r)=T_r-cT_r-2 (r≥2). The subtraction begins at r=2. No undefined T_-1 or T_-2 is needed.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"At a prime p, the default local parameters are a=τ(p) and b=p^11. The scalar recurrence has T_0=1, T_1=a and T_r=aT_r-1-bT_r-2 for r≥2. Thus T_r=τ(p^r). A construction that permits an arbitrary integer a states that change explicitly. Such a recurrence need not be the local sequence of Δ.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"At a prime p, the default local parameters are a=τ(p) and b=p^11. The scalar recurrence has T_0=1, T_1=a and T_r=aT_r-1-bT_r-2 for r≥2. Thus T_r=τ(p^r). A construction that permits an arbitrary integer a states that change explicitly. Such a recurrence need not be the local sequence of Δ.","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"T_j≤ m a_j(9T_j-m)f_m-T_j. At m=3, only T_1=1, T_2=3 contribute. Its numerator is (-3)(9-3)(252)+5(27-3)(1)=-4536+120=-4416. Division by 3 again gives -1472. The direct product, divisor recurrence and triangular recurrence reach the same integer through different algebraic operations.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-14d0514739ae8d60","tex":"T_{2m}","aliases":"T_2m","display":"T_{2m}","kind":"use","excerpt":"uadτ(3)=252. Their quotients a^2/p^11 are not integers. Thus no prime power of 2 or 3 vanishes. We obtain the exact local criterion: τ(p^r)=0 Longleftrightarrow τ(p)=0 and r is odd. The forward implication is what we just proved. For the reverse implication and the even case, set a=0 in the recurrence: T_2m=(-p^11)^m, T_2m+1=0.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-978f84fb12a5cf85","tex":"T_{2m-2}","aliases":"T_2m-2","display":"T_{2m-2}","kind":"use","excerpt":"First suppose a≠0, and write t=v_p(a). In T_2m, the term indexed by j has valuation at least 2mt+j(11-2t). If t≤4, the j=0 term a^2m has strictly least valuation 2mt. Likewise T_2m-2 has valuation (2m-2)t. The correction p^10T_2m-2 has valuation 2mt+(10-2t)>2mt. Thus v_p(R(p^2m))=2mt, t≤4.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-48aeca4a8b30ba97","tex":"T_{2r+2}","aliases":"T_2r+2","display":"T_{2r+2}","kind":"use","excerpt":"The Eichler--Selberg trace identities give, for r≥1, ∑_j=0^r(-1)^r-jbinomr+jr-jp^r-js_2j(p) =-1-Tr T_2r+2(p), together with s_0(p)=p. Here T_k(p) is the Hecke operator at p on S_k(SL_2(Z)). It is distinct from the scalar recurrence notation T_r=τ(p^r) at a fixed prime. The trace normalization and Catalan inversion are checked against [source refere","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","excerpt":"The first local blocks are T_2=a^2-b, T_3=a^3-2ab, T_4=a^4-3a^2b+b^2. Even if a≠0, it is not formally obvious from these expressions that none vanishes. The reduction to primes needs a short arithmetic argument.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-64f17e7dedd6a141","tex":"T_4","aliases":"T_4","display":"T_4","kind":"use","excerpt":"The first local blocks are T_2=a^2-b, T_3=a^3-2ab, T_4=a^4-3a^2b+b^2. Even if a≠0, it is not formally obvious from these expressions that none vanishes. The reduction to primes needs a short arithmetic argument.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3f182b51ffbd833d","tex":"T_g","aliases":"T_g","display":"T_g","kind":"use","excerpt":"The strongest conceivable content fullness. Let the orbits be grouped by exact content and let T_g be the mass of each group. Suppose, as an unproved hypothesis, that all-degree universal null transfers span every vector with zero sum in each content class. Then a strictly positive corrected mass on every orbit exists exactly when every nonempty T_","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8428b354040d2207","tex":"t_H","aliases":"t_H","display":"t_H","kind":"use","excerpt":"Checking weak admissibility. For the chosen cohomological convention, define t_H(D)=∑_i idimgr^iD, t_N(D)=v_p(detφ). Weak admissibility requires t_H(D)=t_N(D) and t_H(D')≤ t_N(D') for every φ-stable subobject D', with the induced filtration. The total equality is t_H(D)=11=t_N(D).","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"use","excerpt":"Jacobi gives a third engine. Put T_j=j(j+1)/2 and a_j=(-1)^j(2j+1). The series A(q)=∏_k≥1(1-q^k)^3=1+∑_j≥1a_jq^T_j satisfies F=A^8. The identity AqF'=8FqA' yields m f_m=∑_substackj≥1","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-830c9c56b4f2bf21","tex":"t_k","aliases":"t_k","display":"t_k","kind":"use","excerpt":"Refinement and exact transfer spaces. Mass vectors and aggregation. Fix a shell, a finite rational family Q_1,…,Q_h of universally null corrections, and the occupied sectors at depth k. Let their number be s_k. Define t_k,r=∑_v∈ S_k,rW(v), (M_k)_rj=∑_v∈ S_k,rW(v)Q_j(v). The vector t_k contains sector masses. The columns of M_k are the effects of th","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4e56fe1bda71e196","tex":"T_{k+1}","aliases":"T_k+1","display":"T_{k+1}","kind":"use","excerpt":"Equation [source reference] gives A(T_k+1)=T_k, so aggregation induces a surjective map between the quotients. Its kernel is the quotient of ker A by T_k+1∩ker A. A transfer M_k+1a lies in ker A exactly when AM_k+1a=M_ka=0. Thus T_k+1∩ker A=M_k+1(ker M_k), which proves [source reference]. Finally dim E_k=s_k-1-r_k, giving [source reference].","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-221fdc610b5eb2ef","tex":"t_{k+1}","aliases":"t_k+1","display":"t_{k+1}","kind":"use","excerpt":"Every depth-k+1 sector belongs to exactly one parent at depth k. Let A be the corresponding 0--1 aggregation matrix. Summing child contributions gives the exact identities t_k=At_k+1, M_k=AM_k+1. Every occupied parent has a child. Each correction column has global total zero.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9a17cf0c7a34c61c","tex":"t_{k,r}","aliases":"t_k,r","display":"t_{k,r}","kind":"use","excerpt":"Refinement and exact transfer spaces. Mass vectors and aggregation. Fix a shell, a finite rational family Q_1,…,Q_h of universally null corrections, and the occupied sectors at depth k. Let their number be s_k. Define t_k,r=∑_v∈ S_k,rW(v), (M_k)_rj=∑_v∈ S_k,rW(v)Q_j(v). The vector t_k contains sector masses. The columns of M_k are the effects of th","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-163dd25fe6cb484d","tex":"t_N","aliases":"t_N","display":"t_N","kind":"use","excerpt":"Checking weak admissibility. For the chosen cohomological convention, define t_H(D)=∑_i idimgr^iD, t_N(D)=v_p(detφ). Weak admissibility requires t_H(D)=t_N(D) and t_H(D')≤ t_N(D') for every φ-stable subobject D', with the induced filtration. The total equality is t_H(D)=11=t_N(D).","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","excerpt":"Why strict localization and automatic valuation fail. Conserved sectors can preserve opposite signs. Partition a shell into sectors S_r, with masses t_r=∑_v∈ S_rW(v). Suppose every permitted correction is null inside every sector: ∑_v∈ S_rW(v)Q_j(v)=0. Then F=1+∑ a_jQ_j leaves each corrected sector mass equal to t_r. If two sector masses have oppos","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-df3835cdb008f051","tex":"T_{r-1}","aliases":"T_r-1","display":"T_{r-1}","kind":"use","excerpt":"At a prime p, the default local parameters are a=τ(p) and b=p^11. The scalar recurrence has T_0=1, T_1=a and T_r=aT_r-1-bT_r-2 for r≥2. Thus T_r=τ(p^r). A construction that permits an arbitrary integer a states that change explicitly. Such a recurrence need not be the local sequence of Δ.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-679db95f0575163f","tex":"T_{r-2}","aliases":"T_r-2","display":"T_{r-2}","kind":"use","excerpt":"At a prime p, the default local parameters are a=τ(p) and b=p^11. The scalar recurrence has T_0=1, T_1=a and T_r=aT_r-1-bT_r-2 for r≥2. Thus T_r=τ(p^r). A construction that permits an arbitrary integer a states that change explicitly. Such a recurrence need not be the local sequence of Δ.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr","display":null,"kind":"use","excerpt":"The Eichler--Selberg trace identities give, for r≥1, ∑_j=0^r(-1)^r-jbinomr+jr-jp^r-js_2j(p) =-1-Tr T_2r+2(p), together with s_0(p)=p. Here T_k(p) is the Hecke operator at p on S_k(SL_2(Z)). It is distinct from the scalar recurrence notation T_r=τ(p^r) at a fixed prime. The trace normalization and Catalan inversion are checked against [source refere","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"The chart Laplacian is consequently Δ_H=∑_a,b=0^3 (δ_ab-frac15)frac∂^2∂ x_a∂ x_b. The coordinates x_a are not orthonormal. Using only the four diagonal second derivatives would use a different metric. As a normalization check, s_2=x^TGx, so its Hessian is 2G and Δ_Hs_2=tr(G^-12G)=8. The rank of the Euclidean space is four, giving twice that rank. L","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"[3pt] 2, k≥2 · (-25M-5,10M+2,-15M+4,15M-4,15M+3) · (10M-5,10M+2,20M+4,-20M-4,-20M+3) [3pt] 0, k≥3 · (-20M-15,10M-3,10M+4,20M+1,-20M+13) · (15M-15,-25M-3,10M+4,-15M+1,15M+13) The appropriate powers of 2 make each entry divisible by 7. After division, each vector has the standard residues, sum zero and norm squared 10N_k. The norm identities before d","count":21},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e28cd1be4d9be5f9","tex":"U^2","aliases":"U^2","display":"U^2","kind":"use","excerpt":"G_499(1) · =-250854265203840. Reducing the normalized coefficients gives G_499(U) · ≡2+3U^2pmod5, G_499(U) · ≡2+8U^2+15U^3+10U^4pmod25. The first polynomial has the simple roots 1 and 4 modulo 5. At U=1, the exact sum is 10 modulo 25 and its derivative is 1 modulo 5. Consequently G_499(1+5h)≡10+5hpmod25. The five possibilities are 10,15,20,0,5 as h","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"a=p^5u with pnmid u, and set H_r(u)=p^-5rT_r. Equation [source reference] shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses is a monic polynomial F_m(x) in x=u^2. Its constant term and reduction modulo p are F_m(0)=","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-63f54b10a04f228a","tex":"u^{2m}","aliases":"u^2m","display":"u^{2m}","kind":"use","excerpt":"H_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses is a monic polynomial F_m(x) in x=u^2. Its constant term and reduction modulo p are F_m(0)=(-1)^mp^m-1(p+1), F_m(x)≡ x^m-1(x-1)pmod p. The leading term comes from u^2m. The constant term comes from H_2m(0)=(-p)^m and H_2m-2(0)=(-p)^m-1.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8e8dbfc6926d65a2","tex":"U^3","aliases":"U^3","display":"U^3","kind":"use","excerpt":"G_499(1) · =-250854265203840. Reducing the normalized coefficients gives G_499(U) · ≡2+3U^2pmod5, G_499(U) · ≡2+8U^2+15U^3+10U^4pmod25. The first polynomial has the simple roots 1 and 4 modulo 5. At U=1, the exact sum is 10 modulo 25 and its derivative is 1 modulo 5. Consequently G_499(1+5h)≡10+5hpmod25. The five possibilities are 10,15,20,0,5 as h","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4d0ee566c7488a3c","tex":"U^4","aliases":"U^4","display":"U^4","kind":"use","excerpt":"G_499(1) · =-250854265203840. Reducing the normalized coefficients gives G_499(U) · ≡2+3U^2pmod5, G_499(U) · ≡2+8U^2+15U^3+10U^4pmod25. The first polynomial has the simple roots 1 and 4 modulo 5. At U=1, the exact sum is 10 modulo 25 and its derivative is 1 modulo 5. Consequently G_499(1+5h)≡10+5hpmod25. The five possibilities are 10,15,20,0,5 as h","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a7864edc5f4fc8ab","tex":"U^5","aliases":"U^5","display":"U^5","kind":"use","excerpt":"G_16249(1) · =61339951998700724370240. The first equals 288τ(16249), and the second equals the first divided by 625. Modulo 5, the normalized polynomial is 3+U^2+U^3=(U-1)^2(U-2). Modulo 25, the table gives G_16249(U)≡13+15U+11U^2+11U^3+15U^5. Thus G_16249(1)≡15pmod25 while G'_16249(1)≡0pmod5. For every integer or 5-adic integer h, polynomial Taylo","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8ec935f0df46ab07","tex":"U^j","aliases":"U^j","display":"U^j","kind":"use","excerpt":"For a nonzero P_n, use the convention v_5(0)=+∞ and define μ_n=min_j(j+v_5(B_j)), G_n(U)=5^-μ_nP_n(5U)=∑_j=0^10C_jU^j. Every C_j=B_j5^j-μ_n is an integer. The power of 5 should be removed only after verifying divisibility for each coefficient. Since 288 is a 5-adic unit, v_5(τ(n))=μ_n+v_5(G_n(1)). All coefficients in the next two tables are exact i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"use","excerpt":"Take the prime indices (u_1,p_1)=(15,245759), (u_2,p_2)=(47,770047). The exact values are τ(245759) · =77900595298037436891536424960,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2f97c9046c1a4c7a","tex":"u_2","aliases":"u_2","display":"u_2","kind":"use","excerpt":"Take the prime indices (u_1,p_1)=(15,245759), (u_2,p_2)=(47,770047). The exact values are τ(245759) · =77900595298037436891536424960,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"The all-even class scales by 2. The bijection u_j=v_2jbmod5/2 has a coordinate permutation of sign -1, so A(q)=-2^10Δ(q^4). Using the Hecke recurrence at 2 gives B(q) · =-24Δ(q^2)-2^10Δ(q^4),","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"use","excerpt":"On each such prime shell, all packets have content one. The residual is therefore the full signed sum τ(p)=∑_v∈ V_pW(v). The summands are nonzero integers, and for p>20 they are units at the index prime. Neither property forbids their sum from being zero. The unresolved statement concerns the exact balance of their signed masses.","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5a29eeb0d9bdd44e","tex":"v_{(-i\\bmod5)}","aliases":"v_(-ibmod5) bmod","display":"v_{(-i\\bmod5)}","kind":"use","excerpt":"Symmetry and the harmonic theta mechanism. An involution that preserves the weight. The coordinate labels are residue labels, so negation alone does not preserve the standard shell. The correct involution is J(v)_i=-v_(-ibmod5). It preserves the residues, sum and norm. Its coordinate permutation is (1 4)(2 3), which is even. Negation contributes (-","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-cfc34f991109df65","tex":"v_{-i\\bmod5}","aliases":"v_-ibmod5 bmod","display":"v_{-i\\bmod5}","kind":"use","excerpt":"The corresponding five-core partition has size n-1. In particular, the coefficient at n=1 corresponds to the empty partition. The involution is J(v)_i=-v_-ibmod5. A J-orbit may contain one or two packets, and its mass is the sum of their weights.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"A complete shell enumerator. The shell is finite because every coordinate has absolute value at most sqrt10n. Enumeration can reduce its dimension without discarding any solution. Choose v_0,v_1,v_2 in their residue classes within the bound. The remaining coordinates x=v_3, y=v_4 must satisfy x+y=S=-v_0-v_1-v_2, x^2+y^2=T=10n-v_0^2-v_1^2-v_2^2. The","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9e047bdf76d2d575","tex":"v_0^2","aliases":"v_0^2","display":"v_0^2","kind":"use","excerpt":"A complete shell enumerator. The shell is finite because every coordinate has absolute value at most sqrt10n. Enumeration can reduce its dimension without discarding any solution. Choose v_0,v_1,v_2 in their residue classes within the bound. The remaining coordinates x=v_3, y=v_4 must satisfy x+y=S=-v_0-v_1-v_2, x^2+y^2=T=10n-v_0^2-v_1^2-v_2^2. The","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3565225c19d3bc52","tex":"v_1","aliases":"v_1","display":"v_1","kind":"use","excerpt":"A complete shell enumerator. The shell is finite because every coordinate has absolute value at most sqrt10n. Enumeration can reduce its dimension without discarding any solution. Choose v_0,v_1,v_2 in their residue classes within the bound. The remaining coordinates x=v_3, y=v_4 must satisfy x+y=S=-v_0-v_1-v_2, x^2+y^2=T=10n-v_0^2-v_1^2-v_2^2. The","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-77f5a84fa3a0612b","tex":"\\mathcal V_{13}","aliases":"V_13 mathcal","display":"\\mathcal V_{13}","kind":"use","excerpt":"Witnesses attaining the bound. At k=1, take v=(-5,-4,2,-2,9), w=(5,6,-8,-2,-1). They belong to V_13, have opposite weights and differ coordinatewise by multiples of 10. Thus they lie in one depth-one sector.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b40d434a1371fd93","tex":"v_1^2","aliases":"v_1^2","display":"v_1^2","kind":"use","excerpt":"A complete shell enumerator. The shell is finite because every coordinate has absolute value at most sqrt10n. Enumeration can reduce its dimension without discarding any solution. Choose v_0,v_1,v_2 in their residue classes within the bound. The remaining coordinates x=v_3, y=v_4 must satisfy x+y=S=-v_0-v_1-v_2, x^2+y^2=T=10n-v_0^2-v_1^2-v_2^2. The","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-803a8ea9ac9a6f78","tex":"\\mathcal V_2","aliases":"V_2 mathcal","display":"\\mathcal V_2","kind":"use","excerpt":"figplaceholdershell_coordinatesThe shell V_2 is a concrete intersection of a zero-sum hyperplane, an integral residue condition and a sphere. Coordinates are labeled by their residues modulo 5. A shell point is a packet, with its separate nonzero integral Vandermonde weight.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6113931409c46c4d","tex":"v_2^2","aliases":"v_2^2","display":"v_2^2","kind":"use","excerpt":"A complete shell enumerator. The shell is finite because every coordinate has absolute value at most sqrt10n. Enumeration can reduce its dimension without discarding any solution. Choose v_0,v_1,v_2 in their residue classes within the bound. The remaining coordinates x=v_3, y=v_4 must satisfy x+y=S=-v_0-v_1-v_2, x^2+y^2=T=10n-v_0^2-v_1^2-v_2^2. The","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-cef9e754a06642bc","tex":"v_{2j\\bmod5}","aliases":"v_2jbmod5 bmod","display":"v_{2j\\bmod5}","kind":"use","excerpt":"The all-even class scales by 2. The bijection u_j=v_2jbmod5/2 has a coordinate permutation of sign -1, so A(q)=-2^10Δ(q^4). Using the Hecke recurrence at 2 gives B(q) · =-24Δ(q^2)-2^10Δ(q^4),","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-26ab02814a81318e","tex":"v_3","aliases":"v_3","display":"v_3","kind":"use","excerpt":"A complete shell enumerator. The shell is finite because every coordinate has absolute value at most sqrt10n. Enumeration can reduce its dimension without discarding any solution. Choose v_0,v_1,v_2 in their residue classes within the bound. The remaining coordinates x=v_3, y=v_4 must satisfy x+y=S=-v_0-v_1-v_2, x^2+y^2=T=10n-v_0^2-v_1^2-v_2^2. The","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-046fdabe03f65bb7","tex":"v_4","aliases":"v_4","display":"v_4","kind":"use","excerpt":"The retained programs organize these same operations using sparse rational polynomials. A direct four-coordinate implementation can instead substitute v_4=-∑_a=0^3x_a into DQ and apply [source reference]. Agreement between these organizations checks the symbolic construction through different algebraic representations. The universal justification r","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-dce3dc9606f0a4f4","tex":"v_5","aliases":"v_5","display":"v_5","kind":"use","excerpt":"For a nonzero P_n, use the convention v_5(0)=+∞ and define μ_n=min_j(j+v_5(B_j)), G_n(U)=5^-μ_nP_n(5U)=∑_j=0^10C_jU^j. Every C_j=B_j5^j-μ_n is an integer. The power of 5 should be removed only after verifying divisibility for each coefficient. Since 288 is a 5-adic unit, v_5(τ(n))=μ_n+v_5(G_n(1)). All coefficients in the next two tables are exact i","count":18},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-66cebab14d391946","tex":"v_{gj\\bmod5}","aliases":"v_gjbmod5 bmod","display":"v_{gj\\bmod5}","kind":"use","excerpt":"The content is compatible with the shell in a precise way. If g=g(v), define u_j=fracv_gjbmod5g, j=0,…,4. The coordinate permutation restores the standard residues after division. The vector u has sum zero and gcd 1. Its norm squared is 10n/g^2. For every integral standard-residue vector of sum zero, the norm is divisible by 5, because its squared ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d539adf39feff28f","tex":"v_i^2","aliases":"v_i^2","display":"v_i^2","kind":"use","excerpt":"Packets, content and the coefficient shift. Coordinates have indices 0,1,2,3,4. A shell is Vn={v∈Z^5: v_i≡ ipmod5, ∑_i v_i=0, ∑_i v_i^2=10n}. The standard orientation is D(v)=∏_0≤ i<j≤4(v_i-v_j), W(v)=D(v)/288. The identity is τ(n)=∑_v∈VnW(v). No absolute values occur in that sum. Distinct residue classes ensure that no Vandermonde factor is zero. ","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-86693d05357f8c53","tex":"v_i^a","aliases":"v_i^a","display":"v_i^a","kind":"use","excerpt":"Δ_Hs_m · =frac45m(m-1)s_m-2. In the logarithmic term, the coordinate-mean subtraction vanishes because the logarithmic gradient has zero sum. Pair the ordered indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v_i-v_j. The divided difference is the finite polynomial fracv_i^m-1-v_j^m-1v_i-v_j =∑_a=0^m-2v_i^av_j^m-2-a. The sum","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-42486f1e833d8d72","tex":"v_i^{k-5}","aliases":"v_i^k-5","display":"v_i^{k-5}","kind":"use","excerpt":"Every coordinate satisfies its characteristic polynomial. Multiplying that equality by v_i^k-5 and summing, for k≥6, gives s_k=-e_2s_k-2+e_3s_k-3-e_4s_k-4+e_5s_k-5. This recurrence uses only lower indices. Substitution at k=6,7 yields s_6 · =frac34s_2s_4+frac13s_3^2-frac18s_2^3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ca2039617606e78e","tex":"v_i^m","aliases":"v_i^m","display":"v_i^m","kind":"use","excerpt":"Write s_m=∑_i=0^4v_i^m, with s_0=5 and s_1=0, and let D=∏_i<j(v_i-v_j). We distinguish the ordinary Laplacian Δ_H from the conjugated operator L defined by Δ_H(DQ)=D LQ. The factor D changes radial coefficients substantially. Confusing the two operators would invalidate the harmonicity test.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-15bf7e675efde753","tex":"v_i^{m-1}","aliases":"v_i^m-1","display":"v_i^{m-1}","kind":"use","excerpt":"Deriving the power-sum operator. For m≥2, differentiation and projection give (∇_Hs_m)_i · =m(v_i^m-1-frac15s_m-1), Δ_Hs_m · =frac45m(m-1)s_m-2. In the logarithmic term, the coordinate-mean subtraction vanishes because the logarithmic gradient has zero sum. Pair the ordered indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"Packets, content and the coefficient shift. Coordinates have indices 0,1,2,3,4. A shell is Vn={v∈Z^5: v_i≡ ipmod5, ∑_i v_i=0, ∑_i v_i^2=10n}. The standard orientation is D(v)=∏_0≤ i<j≤4(v_i-v_j), W(v)=D(v)/288. The identity is τ(n)=∑_v∈VnW(v). No absolute values occur in that sum. Distinct residue classes ensure that no Vandermonde factor is zero. ","count":18},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e3f0c47930b412df","tex":"v_j^2","aliases":"v_j^2","display":"v_j^2","kind":"use","excerpt":"The index-prime unit obstruction. If p>20 is prime and v∈ V_p, then for each pair 0<(v_i-v_j)^2≤2(v_i^2+v_j^2)≤20p<p^2. Thus p divides none of the differences and does not divide 288. We obtain For every prime p>20 and every v∈ V_p, one has v_p(W(v))=0. All packets are units at the index prime. There is no increasing termwise p-adic depth automatic","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9795549e5e7389ca","tex":"v_j^{m-1}","aliases":"v_j^m-1","display":"v_j^{m-1}","kind":"use","excerpt":"Δ_Hs_m · =frac45m(m-1)s_m-2. In the logarithmic term, the coordinate-mean subtraction vanishes because the logarithmic gradient has zero sum. Pair the ordered indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v_i-v_j. The divided difference is the finite polynomial fracv_i^m-1-v_j^m-1v_i-v_j =∑_a=0^m-2v_i^av_j^m-2-a. The sum","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-007dee41e4900a7a","tex":"v_j^{m-2-a}","aliases":"v_j^m-2-a","display":"v_j^{m-2-a}","kind":"use","excerpt":"Δ_Hs_m · =frac45m(m-1)s_m-2. In the logarithmic term, the coordinate-mean subtraction vanishes because the logarithmic gradient has zero sum. Pair the ordered indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v_i-v_j. The divided difference is the finite polynomial fracv_i^m-1-v_j^m-1v_i-v_j =∑_a=0^m-2v_i^av_j^m-2-a. The sum","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-04e3bef7aa63bf65","tex":"v_N","aliases":"v_N","display":"v_N","kind":"use","excerpt":"An explicit homogeneous interpolation basis. Let v_1,…,v_N represent the distinct J-orbits. For an integer t≥0, define the degree-10 symmetric polynomial E_t=s_4s_2^3+t s_3^2s_2^2+t^2s_3s_5s_2+t^3s_5^2. For two distinct orbits, E_t(v_i)-E_t(v_j) is a nonzero polynomial in t of degree at most 3, by Proposition~[source reference]. It has at most thre","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a9b01f62dbf3fb3c","tex":"\\mathcal V_p","aliases":"V_p mathcal","display":"\\mathcal V_p","kind":"use","excerpt":"On each such prime shell, all packets have content one. The residual is therefore the full signed sum τ(p)=∑_v∈ V_pW(v). The summands are nonzero integers, and for p>20 they are units at the index prime. Neither property forbids their sum from being zero. The unresolved statement concerns the exact balance of their signed masses.","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"use","excerpt":"For a nonzero actual coefficient, the same comparison gives v_p(τ(p))≤5. A hypothetical assertion v_p(τ(p))≤3 would be stronger than needed, and hence sufficient, but requires its own proof. One must not obtain such a valuation bound merely by rewriting the coefficient as another exact sum.","count":11},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-fa83faab1d44c512","tex":"v_\\ell","aliases":"v_ℓ ell","display":"v_\\ell","kind":"use","excerpt":"The valuation v_ℓ(x) is the exponent of ℓ in a nonzero rational number x, with v_ℓ(0)=∞. The quadratic character used in the radial extraction is chifive(d)= 0 · 5mid d,","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-50ce1a62663b77d9","tex":"\\Vert","aliases":"Vert","display":"\\Vert","kind":"use","excerpt":"The actual joint image. Factor M=∏_ℓ^eVert Mℓ^e and form the single joint finite map Ψ:G_Q⟶ (∏_ℓ^eVert MGL_2(Z/ℓ^eZ)) ×(Z/5MZ)^×. Its components are the integral representation reductions and the cyclotomic character modulo 5M. Let G be its actual image. The kernel defines a finite Galois extension of Q. The image need not be the whole displayed pr","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Packets, content and the coefficient shift. Coordinates have indices 0,1,2,3,4. A shell is Vn={v∈Z^5: v_i≡ ipmod5, ∑_i v_i=0, ∑_i v_i^2=10n}. The standard orientation is D(v)=∏_0≤ i<j≤4(v_i-v_j), W(v)=D(v)/288. The identity is τ(n)=∑_v∈VnW(v). No absolute values occur in that sum. Distinct residue classes ensure that no Vandermonde factor is zero. ","count":39},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Completeness retains the total. Suppose a shell has nonzero orbit masses w_1,…,w_N and total T. Even allow every rational correction vector q with ∑_iw_iq_i=0. Then ∃ q: w_i(1+q_i) have one weak sign and are not all zero Longleftrightarrow T≠0. Necessity follows from conservation. For sufficiency, if T≠0, take q_i=fracTNw_i-1. Every corrected mass ","count":26},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"Completeness retains the total. Suppose a shell has nonzero orbit masses w_1,…,w_N and total T. Even allow every rational correction vector q with ∑_iw_iq_i=0. Then ∃ q: w_i(1+q_i) have one weak sign and are not all zero Longleftrightarrow T≠0. Necessity follows from conservation. For sufficiency, if T≠0, take q_i=fracTNw_i-1. Every corrected mass ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"Completeness retains the total. Suppose a shell has nonzero orbit masses w_1,…,w_N and total T. Even allow every rational correction vector q with ∑_iw_iq_i=0. Then ∃ q: w_i(1+q_i) have one weak sign and are not all zero Longleftrightarrow T≠0. Necessity follows from conservation. For sufficiency, if T≠0, take q_i=fracTNw_i-1. Every corrected mass ","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"If d ^2≥6, [source reference] gives n≥frac154M^2>frac100M^2+10M+235, M≥2. The remaining case is d ^2=4. Then d has two coordinates 1, two -1 and one 0. Of the ten coordinate pairs, two have equal d entries and eight have unequal entries. For a pair with δ=d_i-d_j, (v_i-v_j)(w_i-w_j)=(m_i-m_j)^2-a^2δ^2. The two equal-entry pairs have positive produc","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2b7f78e3b57fec58","tex":"w_N","aliases":"w_N","display":"w_N","kind":"use","excerpt":"Completeness retains the total. Suppose a shell has nonzero orbit masses w_1,…,w_N and total T. Even allow every rational correction vector q with ∑_iw_iq_i=0. Then ∃ q: w_i(1+q_i) have one weak sign and are not all zero Longleftrightarrow T≠0. Necessity follows from conservation. For sufficiency, if T≠0, take q_i=fracTNw_i-1. Every corrected mass ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"Local and global conventions. For ℓ≠ p, the arithmetic-Frobenius convention in the integral ℓ-adic representation gives the characteristic polynomial X^2-τ(p)X+p^11. The cohomological conventions in the filtered-module discussion are stated there separately. An abstract trace-zero filtered module and a representation realizing it are local objects.","count":12},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"The valuation v_ℓ(x) is the exponent of ℓ in a nonzero rational number x, with v_ℓ(0)=∞. The quadratic character used in the radial extraction is chifive(d)= 0 · 5mid d,","count":42},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"Local and global conventions. For ℓ≠ p, the arithmetic-Frobenius convention in the integral ℓ-adic representation gives the characteristic polynomial X^2-τ(p)X+p^11. The cohomological conventions in the filtered-module discussion are stated there separately. An abstract trace-zero filtered module and a representation realizing it are local objects.","count":9},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"A complete shell enumerator. The shell is finite because every coordinate has absolute value at most sqrt10n. Enumeration can reduce its dimension without discarding any solution. Choose v_0,v_1,v_2 in their residue classes within the bound. The remaining coordinates x=v_3, y=v_4 must satisfy x+y=S=-v_0-v_1-v_2, x^2+y^2=T=10n-v_0^2-v_1^2-v_2^2. The","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4369b951ca08b0ee","tex":"x^{m-1}","aliases":"x^m-1","display":"x^{m-1}","kind":"use","excerpt":"refeq:trexpansion shows that H_r is an integral polynomial. Its recurrence is H_0=1, H_1=u, H_r=uH_r-1-pH_r-2. The primitive block becomes R(p^2m)=p^10m(H_2m(u)-H_2m-2(u)). The expression in parentheses is a monic polynomial F_m(x) in x=u^2. Its constant term and reduction modulo p are F_m(0)=(-1)^mp^m-1(p+1), F_m(x)≡ x^m-1(x-1)pmod p. The leading ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-14fb02863a861bee","tex":"X^r","aliases":"X^r","display":"X^r","kind":"use","excerpt":"For a fixed prime put T_r=τ(p^r) and b=p^11. The generating series is ∑_r≥0T_rX^r=frac11-aX+bX^2. Multiplication by the denominator gives T_0=1, T_1=a and the recurrence. Thus this is a formal identity over Z[[X]]. No analytic continuation in the prime index is involved.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0514a60dd6e60420","tex":"x^T","aliases":"x^T","display":"x^T","kind":"use","excerpt":"The chart Laplacian is consequently Δ_H=∑_a,b=0^3 (δ_ab-frac15)frac∂^2∂ x_a∂ x_b. The coordinates x_a are not orthonormal. Using only the four diagonal second derivatives would use a different metric. As a normalization check, s_2=x^TGx, so its Hessian is 2G and Δ_Hs_2=tr(G^-12G)=8. The rank of the Euclidean space is four, giving twice that rank. L","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-98391915cbb7346b","tex":"x_5","aliases":"x_5","display":"x_5","kind":"use","excerpt":"One parent gives a concrete view of this second stage. Parent 5 has fine children 5,6. Just after the coarse stage their values are x_5 · =-frac161598180629860124657534386744 23300823161076013875625, x_6 · =-frac1069691132209170737342448593256 23300823161076013875625. Their exact sum is -52843168. The detail stage adds -x_5 to child 5 and x_5 to ch","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3864fe3cacc7c90d","tex":"x_6","aliases":"x_6","display":"x_6","kind":"use","excerpt":"One parent gives a concrete view of this second stage. Parent 5 has fine children 5,6. Just after the coarse stage their values are x_5 · =-frac161598180629860124657534386744 23300823161076013875625, x_6 · =-frac1069691132209170737342448593256 23300823161076013875625. Their exact sum is -52843168. The detail stage adds -x_5 to child 5 and x_5 to ch","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9067dcf296a5b175","tex":"x_a","aliases":"x_a","display":"x_a","kind":"use","excerpt":"The chart Laplacian is consequently Δ_H=∑_a,b=0^3 (δ_ab-frac15)frac∂^2∂ x_a∂ x_b. The coordinates x_a are not orthonormal. Using only the four diagonal second derivatives would use a different metric. As a normalization check, s_2=x^TGx, so its Hessian is 2G and Δ_Hs_2=tr(G^-12G)=8. The rank of the Euclidean space is four, giving twice that rank. L","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-14c8085d57b478d9","tex":"x_b","aliases":"x_b","display":"x_b","kind":"use","excerpt":"The chart Laplacian is consequently Δ_H=∑_a,b=0^3 (δ_ab-frac15)frac∂^2∂ x_a∂ x_b. The coordinates x_a are not orthonormal. Using only the four diagonal second derivatives would use a different metric. As a normalization check, s_2=x^TGx, so its Hessian is 2G and Δ_Hs_2=tr(G^-12G)=8. The rank of the Euclidean space is four, giving twice that rank. L","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-cd70314149f91751","tex":"X_H","aliases":"X_H","display":"X_H","kind":"use","excerpt":"bibitemDHZ Maarten Derickx, Mark van Hoeij, and Jinxiang Zeng, Computing Galois representations and equations for modular curves X_H(ℓ), https://arxiv.org/pdf/1312.6819v2arXiv:1312.6819v2, 18 March 2014. Corollary~1.2, p.~2; its computational proof in S4, p.~11. The cited nonvanishing range has the strict endpoint n<816212624008487344127999. This i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"The coarse coefficient vector a∈Q^18 makes x=s_f+M_fa, Ax=(0,0,0,0,0,-52843168)^t. The certificate then chooses a coefficient vector d∈ker M_c. Its final fine vector y=x+M_fd is zero except at index 6, where it equals -52843168.","count":16},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"A complete shell enumerator. The shell is finite because every coordinate has absolute value at most sqrt10n. Enumeration can reduce its dimension without discarding any solution. Choose v_0,v_1,v_2 in their residue classes within the bound. The remaining coordinates x=v_3, y=v_4 must satisfy x+y=S=-v_0-v_1-v_2, x^2+y^2=T=10n-v_0^2-v_1^2-v_2^2. The","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"An exact obstruction in a specified smaller space. At n=31, the seven retained corrections through degree 16 do not admit a nontrivial common-sign certificate. The shell has sixteen J-orbits. Let ϵ_i=sgn W(v_i) for their representatives. The retained exact dual has strictly positive rational entries y_i satisfying ∑_i y_i=1, ∑_i y_iϵ_i=0, ∑_i y_iϵ_","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"Cores, harmonic theta series and valence. S02: Han's packet identity. Han's Definition~1.1 and Theorems~1.1--1.2 in the checked first arXiv version provide the five-core specialization and Vandermonde product [source reference]. The specialization t=5, z=25 gives the shell norm 10n after the coefficient shift. With the orientation used here, the no","count":11},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-db5094c3f826ba04","tex":"\\Z^5","aliases":"Z^5 Z","display":null,"kind":"use","excerpt":"Packets, content and the coefficient shift. Coordinates have indices 0,1,2,3,4. A shell is Vn={v∈Z^5: v_i≡ ipmod5, ∑_i v_i=0, ∑_i v_i^2=10n}. The standard orientation is D(v)=∏_0≤ i<j≤4(v_i-v_j), W(v)=D(v)/288. The identity is τ(n)=∑_v∈VnW(v). No absolute values occur in that sum. Distinct residue classes ensure that no Vandermonde factor is zero. ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e584afe354e47377","tex":"\\Z_5","aliases":"Z_5 Z","display":null,"kind":"use","excerpt":"=3 and reduction gives overline G_499(U)=2+3U^2=3(U-1)(U+1) in F_5[U]. At the relevant value U=1, G_499(1)≡10pmod25, G'_499(1)≡1pmod5. Substitute U=1+5d. The integral Taylor identity gives G_499(1+5d)≡10+5dpmod25. The root in this residue class is therefore ρ≡16pmod25. Hensel's lemma lifts it uniquely to Z_5. The divided difference (G(U)-G(ρ))/(U-ρ","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a926644fa7a9ec32","tex":"\\Z_\\ell","aliases":"Z_ℓ Z ell","display":null,"kind":"use","excerpt":"A continuous representation of the compact group G_Q has a stable Z_ℓ lattice. To see this directly, take an initial lattice L_0. Compactness bounds all its translates inside ℓ^-aL_0 for some a. The module L=∑_g∈ G_Qρ_ℓ(g)L_0 is contained in ℓ^-aL_0 and contains L_0. It is finitely generated, since a submodule of a finite free Z_ℓ module is finitel","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3d7f36918ca613a9","tex":"|_H","aliases":"|_H","display":"|_H","kind":"use","excerpt":"projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=0^4∂_i^2F -frac15(∑_i=0^4∂_i)^2F)|_H. Only tangentia","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"in one sector. Replace w by Jw if needed so that w-v=5Md, d∈ A_4∖{0}. Set a=5M/2 and m=(v+w)/2. Since M is even, both are integral. Equal norms give m· d=0,qquad10n= m ^2+a^2 d ^2. The squared norm of a nonzero integral zero-sum vector is a positive even integer. It cannot be 2 here. Such a vector is ±(e_i-e_j), and m· d=0 would force v_i-v_j=±5M. ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"The two roots and a root of unity. Let α,β be the complex roots of X^2-aX+p^11=0. They are nonzero because their product is p^11. They are distinct: equality would require a^2=4p^11, but the p-adic valuation of the right side is odd for odd p, and is 13 for p=2.","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ced9c73c684ad7cb","tex":"\\alpha^{r+1}","aliases":"α^r+1 alpha","display":"\\alpha^{r+1}","kind":"use","excerpt":"Solving the recurrence gives T_r=fracα^r+1-β^r+1α-β =∑_j=0^rα^r-jβ^j. If T_r=0 for some r≥1, then ζ=α/β is a root of unity with ζ^r+1=1. Consequently fraca^2p^11 =frac(α+β)^2αβ =2+ζ+ζ^-1. The left side is rational. The right side is an algebraic integer, because roots of unity are algebraic integers. A rational algebraic integer is an integer. It i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-bad9690669867e2e","tex":"\\alpha^{r-j}","aliases":"α^r-j alpha","display":"\\alpha^{r-j}","kind":"use","excerpt":"Solving the recurrence gives T_r=fracα^r+1-β^r+1α-β =∑_j=0^rα^r-jβ^j. If T_r=0 for some r≥1, then ζ=α/β is a root of unity with ζ^r+1=1. Consequently fraca^2p^11 =frac(α+β)^2αβ =2+ζ+ζ^-1. The left side is rational. The right side is an algebraic integer, because roots of unity are algebraic integers. A rational algebraic integer is an integer. It i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"The two roots and a root of unity. Let α,β be the complex roots of X^2-aX+p^11=0. They are nonzero because their product is p^11. They are distinct: equality would require a^2=4p^11, but the p-adic valuation of the right side is odd for odd p, and is 13 for p=2.","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f348fc1b72226af3","tex":"\\beta^j","aliases":"β^j beta","display":"\\beta^j","kind":"use","excerpt":"Solving the recurrence gives T_r=fracα^r+1-β^r+1α-β =∑_j=0^rα^r-jβ^j. If T_r=0 for some r≥1, then ζ=α/β is a root of unity with ζ^r+1=1. Consequently fraca^2p^11 =frac(α+β)^2αβ =2+ζ+ζ^-1. The left side is rational. The right side is an algebraic integer, because roots of unity are algebraic integers. A rational algebraic integer is an integer. It i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-50603a86fcb2cf81","tex":"\\beta^{r+1}","aliases":"β^r+1 beta","display":"\\beta^{r+1}","kind":"use","excerpt":"Solving the recurrence gives T_r=fracα^r+1-β^r+1α-β =∑_j=0^rα^r-jβ^j. If T_r=0 for some r≥1, then ζ=α/β is a root of unity with ζ^r+1=1. Consequently fraca^2p^11 =frac(α+β)^2αβ =2+ζ+ζ^-1. The left side is rational. The right side is an algebraic integer, because roots of unity are algebraic integers. A rational algebraic integer is an integer. It i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"The chain-rule metric and polynomial assembly. The same projected gradients give Γ_mn:=∇_Hs_m·∇_Hs_n =mn(s_m+n-2-frac15s_m-1s_n-1). Expanding the product in [source reference] gives the first power sum once and the coordinate-mean product with coefficient -1/5. This identity includes m,n≥1, with the evident zero tangential gradient for s_1.","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"The sharp threshold. Let k≥1, M=2^k, and set N_k=lceilfrac100M^2+10M+235rceil =frac20M^2+2M+γ_k7, where γ_k= 7 · k≡1pmod3,","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-49088f12c06b6867","tex":"\\Gamma_{22}","aliases":"Γ_22 Gamma","display":"\\Gamma_{22}","kind":"use","excerpt":"L(s_3^2) · =18s_4-frac185s_2^2. For the first identity, 2s_2 Ls_2+2Γ_22=96s_2+8s_2. For the mixed product, s_4 Ls_2+s_2 Ls_4+2Γ_24 gives the displayed coefficients. Each calculation is a check of the ordered-pair convention.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-18d4141c9dfc159a","tex":"\\Gamma_{24}","aliases":"Γ_24 Gamma","display":"\\Gamma_{24}","kind":"use","excerpt":"L(s_3^2) · =18s_4-frac185s_2^2. For the first identity, 2s_2 Ls_2+2Γ_22=96s_2+8s_2. For the mixed product, s_4 Ls_2+s_2 Ls_4+2Γ_24 gives the displayed coefficients. Each calculation is a check of the ordered-pair convention.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-530f5d7a3b27e60f","tex":"\\gamma_k","aliases":"γ_k gamma","display":"\\gamma_k","kind":"use","excerpt":"The sharp threshold. Let k≥1, M=2^k, and set N_k=lceilfrac100M^2+10M+235rceil =frac20M^2+2M+γ_k7, where γ_k= 7 · k≡1pmod3,","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-78467867966e2110","tex":"\\Gamma_{mn}","aliases":"Γ_mn Gamma","display":"\\Gamma_{mn}","kind":"use","excerpt":"The chain-rule metric and polynomial assembly. The same projected gradients give Γ_mn:=∇_Hs_m·∇_Hs_n =mn(s_m+n-2-frac15s_m-1s_n-1). Expanding the product in [source reference] gives the first power sum once and the coordinate-mean product with coefficient -1/5. This identity includes m,n≥1, with the evident zero tangential gradient for s_1.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"The chart Laplacian is consequently Δ_H=∑_a,b=0^3 (δ_ab-frac15)frac∂^2∂ x_a∂ x_b. The coordinates x_a are not orthonormal. Using only the four diagonal second derivatives would use a different metric. As a normalization check, s_2=x^TGx, so its Hessian is 2G and Δ_Hs_2=tr(G^-12G)=8. The rank of the Euclidean space is four, giving twice that rank. L","count":9},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"use","excerpt":"If d ^2≥6, [source reference] gives n≥frac154M^2>frac100M^2+10M+235, M≥2. The remaining case is d ^2=4. Then d has two coordinates 1, two -1 and one 0. Of the ten coordinate pairs, two have equal d entries and eight have unequal entries. For a pair with δ=d_i-d_j, (v_i-v_j)(w_i-w_j)=(m_i-m_j)^2-a^2δ^2. The two equal-entry pairs have positive produc","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1525b57dc165eb8a","tex":"\\delta_{ab}","aliases":"δ_ab delta","display":"\\delta_{ab}","kind":"use","excerpt":"The chart Laplacian is consequently Δ_H=∑_a,b=0^3 (δ_ab-frac15)frac∂^2∂ x_a∂ x_b. The coordinates x_a are not orthonormal. Using only the four diagonal second derivatives would use a different metric. As a normalization check, s_2=x^TGx, so its Hessian is 2G and Δ_Hs_2=tr(G^-12G)=8. The rank of the Euclidean space is four, giving twice that rank. L","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"use","excerpt":"For that t, set L_i(v)=∏_j≠ i fracE_t(v)-e_j(s_2(v)/C)^5e_i-e_j. Each factor is homogeneous of degree 10, and the denominators are nonzero rational numbers. The resulting polynomial is symmetric homogeneous of degree 10(N-1). On the shell, s_2/C=1, so L_i(v_j)=δ_ij. Consequently, any prescribed rational orbit values a_1,…,a_N are interpolated by Q=","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"An exact obstruction in a specified smaller space. At n=31, the seven retained corrections through degree 16 do not admit a nontrivial common-sign certificate. The shell has sixteen J-orbits. Let ϵ_i=sgn W(v_i) for their representatives. The retained exact dual has strictly positive rational entries y_i satisfying ∑_i y_i=1, ∑_i y_iϵ_i=0, ∑_i y_iϵ_","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"Suppose the construction proves a sign ε_p∈{-1,1} for which ε_p W(v)F_p(v)≥0 for every v∈ V_p, and proves that at least one of these finitely many quantities is strictly positive. Then summing [source reference] gives ε_pτ(p)>0. This would be a valid all-prime certificate theorem.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0773a19734ee0848","tex":"\\epsilon_i","aliases":"ϵ_i epsilon","display":"\\epsilon_i","kind":"use","excerpt":"An exact obstruction in a specified smaller space. At n=31, the seven retained corrections through degree 16 do not admit a nontrivial common-sign certificate. The shell has sixteen J-orbits. Let ϵ_i=sgn W(v_i) for their representatives. The retained exact dual has strictly positive rational entries y_i satisfying ∑_i y_i=1, ∑_i y_iϵ_i=0, ∑_i y_iϵ_","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3972b5f2efcbd900","tex":"\\varepsilon_p","aliases":"ε_p varepsilon","display":"\\varepsilon_p","kind":"use","excerpt":"Suppose the construction proves a sign ε_p∈{-1,1} for which ε_p W(v)F_p(v)≥0 for every v∈ V_p, and proves that at least one of these finitely many quantities is strictly positive. Then summing [source reference] gives ε_pτ(p)>0. This would be a valid all-prime certificate theorem.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"Solving the recurrence gives T_r=fracα^r+1-β^r+1α-β =∑_j=0^rα^r-jβ^j. If T_r=0 for some r≥1, then ζ=α/β is a root of unity with ζ^r+1=1. Consequently fraca^2p^11 =frac(α+β)^2αβ =2+ζ+ζ^-1. The left side is rational. The right side is an algebraic integer, because roots of unity are algebraic integers. A rational algebraic integer is an integer. It i","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a1810dd5bbc6aace","tex":"\\zeta^{-1}","aliases":"ζ^-1 zeta","display":"\\zeta^{-1}","kind":"use","excerpt":"Solving the recurrence gives T_r=fracα^r+1-β^r+1α-β =∑_j=0^rα^r-jβ^j. If T_r=0 for some r≥1, then ζ=α/β is a root of unity with ζ^r+1=1. Consequently fraca^2p^11 =frac(α+β)^2αβ =2+ζ+ζ^-1. The left side is rational. The right side is an algebraic integer, because roots of unity are algebraic integers. A rational algebraic integer is an integer. It i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d3f2d12c1dd70f92","tex":"\\zeta^{r+1}","aliases":"ζ^r+1 zeta","display":"\\zeta^{r+1}","kind":"use","excerpt":"Solving the recurrence gives T_r=fracα^r+1-β^r+1α-β =∑_j=0^rα^r-jβ^j. If T_r=0 for some r≥1, then ζ=α/β is a root of unity with ζ^r+1=1. Consequently fraca^2p^11 =frac(α+β)^2αβ =2+ζ+ζ^-1. The left side is rational. The right side is an algebraic integer, because roots of unity are algebraic integers. A rational algebraic integer is an integer. It i","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"More generally, if Q is symmetric and DQ is harmonic homogeneous of degree 10+d, its weighted shell theta series is Θ_Q(q)=∑_n≥1(∑_v∈VnW(v)Q(v))q^n. The harmonic theta theorem initially supplies a vector-valued modular form. Obtaining a scalar full-level cusp form requires the following lattice specialization, rather than an unexplained assertion t","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"Radial factors become derivatives. For an alternating harmonic h_ℓ, write θ_h(τ)=∑_substackv_i≡ i (5) ∑ v_i=0 h(v)exp(2π iτfrac v ^210). Chapter~[source reference] establishes that it is a scalar cusp form of weight ℓ+2. The normalization includes only a fixed scalar if one divides by 288.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7bb6de5d0df10117","tex":"\\theta_h","aliases":"θ_h theta","display":"\\theta_h","kind":"use","excerpt":"Radial factors become derivatives. For an alternating harmonic h_ℓ, write θ_h(τ)=∑_substackv_i≡ i (5) ∑ v_i=0 h(v)exp(2π iτfrac v ^210). Chapter~[source reference] establishes that it is a scalar cusp form of weight ℓ+2. The normalization includes only a fixed scalar if one divides by 288.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f770d5cb5d96521a","tex":"\\Theta_Q","aliases":"Θ_Q Theta","display":"\\Theta_Q","kind":"use","excerpt":"More generally, if Q is symmetric and DQ is harmonic homogeneous of degree 10+d, its weighted shell theta series is Θ_Q(q)=∑_n≥1(∑_v∈VnW(v)Q(v))q^n. The harmonic theta theorem initially supplies a vector-valued modular form. Obtaining a scalar full-level cusp form requires the following lattice specialization, rather than an unexplained assertion t","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-83d3a4a279a497ad","tex":"\\theta_{s_2^jh}","aliases":"θ_s_2^jh theta","display":"\\theta_{s_2^jh}","kind":"use","excerpt":"Each application of DQ=q d/dq multiplies the shell coefficient by v ^2/10. Therefore θ_s_2^jh=10^jDQ^jθ_h. Differentiating a modular transformation law shows that DQ^jf, for modular weight w, is quasimodular of weight w+2j and depth at most j. Combining [source reference] and [source reference], the homogeneous shell series F_d(τ)=∑_n≥1(∑_v∈VnW(v)Q","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"The Nekrasov--Okounkov identity in Han's formulation is ∏_m≥1(1-q^m)^z-1 =∑_λ q^|λ|∏_h∈ H(λ)(1-frac zh^2). At z=25, a partition having a hook of length 5 contributes zero. The surviving terms are the 5-cores. Multiplication by q shows that the coefficient τ(n) is the sum of the hook products over 5-cores of size n-1.","count":6},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-308a821eda6e7229","tex":"\\lambda^2","aliases":"λ^2 lambda","display":"\\lambda^2","kind":"use","excerpt":"There is no proper nonzero φ-stable Q_p-subspace. In dimension two such a subspace would be a line, and φ on it would have an eigenvalue λ∈Q_p satisfying λ^2=-p^11. This would require 2v_p(λ)=11, impossible for an integer-valued valuation on Q_p^×. Thus the proper-subobject inequalities are vacuous. The zero object imposes equality 0=0, and the who","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"R(n)=∑_d^2mid nμ(d)χ_5(d)d^10τ(n/d^2).","count":16},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-21d7089401a15fc8","tex":"\\mu_{16249}","aliases":"μ_16249 mu","display":"\\mu_{16249}","kind":"use","excerpt":"At 16249 the shell contains 16250 packets, with 8284 positive and 7966 negative weights. The exact coefficient is τ(16249)=133116215274958169206250. Here μ_16249=4. The full coefficient table is as follows.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-27db4055332c3f13","tex":"\\mu_{499}","aliases":"μ_499 mu","display":"\\mu_{499}","kind":"use","excerpt":"At n=499 the shell has 500 packets. Of their weights, 262 are positive and 238 negative. Their sum is τ(499)=-108877719272500. The minimum layer valuation is μ_499=3.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-625e48cfdf47c402","tex":"\\mu_n","aliases":"μ_n mu","display":"\\mu_n","kind":"use","excerpt":"For a nonzero P_n, use the convention v_5(0)=+∞ and define μ_n=min_j(j+v_5(B_j)), G_n(U)=5^-μ_nP_n(5U)=∑_j=0^10C_jU^j. Every C_j=B_j5^j-μ_n is an integer. The power of 5 should be removed only after verifying divisibility for each coefficient. Since 288 is a 5-adic unit, v_5(τ(n))=μ_n+v_5(G_n(1)). All coefficients in the next two tables are exact i","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"use","excerpt":"A double residual root with no lift. At p=16249, the shell has 16250 packets. The weighted coefficient valuations of P_p(5U) are (4,5,4,4,6,5,7,8,9,9,11), so μ_p=4. The reduction is overline G_p(U)=3+U^2+U^3=(U-1)^2(U-2)pmod5. The residual root at 1 is double. Exact coefficient reduction gives G_p(1)≡15pmod25, G'_p(1)≡0pmod5. For every y∈Z_5, polyn","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"S03: harmonic theta modularity. Ibukiyama and Williams, Section~3.4 of the checked version, give the vector-valued harmonic theta theorem with weight ν+r/2 [source reference]. Here the rank is four and the harmonic polynomial is alternating of degree 10+d. The scalar full-level transformation follows from the separate sign-line descent in Chapter~8","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"Radial factors become derivatives. For an alternating harmonic h_ℓ, write θ_h(τ)=∑_substackv_i≡ i (5) ∑ v_i=0 h(v)exp(2π iτfrac v ^210). Chapter~[source reference] establishes that it is a scalar cusp form of weight ℓ+2. The normalization includes only a fixed scalar if one divides by 288.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"luations are (3,5,3,4,4,5,7,11,8,9,12). Hence μ_499=3 and reduction gives overline G_499(U)=2+3U^2=3(U-1)(U+1) in F_5[U]. At the relevant value U=1, G_499(1)≡10pmod25, G'_499(1)≡1pmod5. Substitute U=1+5d. The integral Taylor identity gives G_499(1+5d)≡10+5dpmod25. The root in this residue class is therefore ρ≡16pmod25. Hensel's lemma lifts it uniqu","count":9},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b3424d76ff899efd","tex":"\\rho_\\ell","aliases":"ρ_ℓ rho ell","display":"\\rho_\\ell","kind":"use","excerpt":"Integral representations and complex conjugation. For each prime ℓ, the representation attached to Δ has the arithmetic-Frobenius normalization ρ_ℓ:G_Q⟶ GL_2(Q_ℓ), is continuous and unramified outside ℓ, and satisfies det(X-ρ_ℓ(Frob_p))=X^2-τ(p)X+p^11, p≠ℓ. Its determinant is the eleventh power of the ℓ-adic cyclotomic character. The original cohom","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5048c4b9123eb530","tex":"\\sigma_1","aliases":"σ_1 sigma","display":"\\sigma_1","kind":"use","excerpt":"The logarithmic derivative provides a different exact engine. With σ_1(j)=∑_dmid jd, comparison of coefficients in qfracF'(q)F(q)=-24∑_j≥1σ_1(j)q^j gives m f_m=-24∑_j=1^mσ_1(j)f_m-j (m≥1). For m=3, the divisor sums are 1,3,4. Hence f_3=-8(252+3(-24)+4)=-8·184=-1472. An implementation should check that the numerator in [source reference] is divisibl","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7062860612871e61","tex":"\\sigma_7","aliases":"σ_7 sigma","display":"\\sigma_7","kind":"use","excerpt":"These four coefficients were checked independently by the Eisenstein convolution. If σ_r(n)=∑_dmid nd^r, the coefficient of (E_4E_8-E_6^2)/1728 is obtained from 1728τ(n)= · 240σ_3(n)+480σ_7(n)+1008σ_5(n) · +115200∑_k=1^n-1σ_3(k)σ_7(n-k) · -254016∑_k=1^n-1σ_5(k)σ_5(n-k). This computes the selected integers without the triangular recurrence. The nume","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8de4fa9e7f9c5804","tex":"\\sigma_j","aliases":"σ_j sigma","display":"\\sigma_j","kind":"use","excerpt":"Exact coefficient engines. The product and a logarithmic recurrence. Let σ_j(n)=∑_dmid nd^j and write σ=σ_1. Logarithmic differentiation of the product gives DQΔ=(1-24∑_m≥1σ(m)q^m)Δ, DQ=qfracddq. Indeed, the contribution from the factor (1-q^r)^24 is -24fracrq^r1-q^r=-24∑_k≥1rq^rk. Collecting the terms with rk=m gives -24σ(m). The leading factor q ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f393e12f20aeb427","tex":"\\sigma_r","aliases":"σ_r sigma","display":"\\sigma_r","kind":"use","excerpt":"These four coefficients were checked independently by the Eisenstein convolution. If σ_r(n)=∑_dmid nd^r, the coefficient of (E_4E_8-E_6^2)/1728 is obtained from 1728τ(n)= · 240σ_3(n)+480σ_7(n)+1008σ_5(n)","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"For every prime p, there is a weakly admissible two-dimensional filtered φ-module over Q_p, with monodromy N=0, filtration jumps 0,11 and Frobenius characteristic polynomial X^2+p^11.","count":8},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6dd87aa4d0d3b436","tex":"\\varphi^2","aliases":"φ^2 varphi","display":"\\varphi^2","kind":"use","excerpt":"The matrix and filtration. Take D=Q_pe_1+Q_pe_2 and define φ(e_1)=e_2, φ(e_2)=-p^11e_1, N=0. In the displayed basis, φ=0 · -p^11 1 · 0, Trφ=0, detφ=p^11, φ^2=-p^11I. Choose any Q_p-line L⊂ D and put Fil^iD= D · i≤0,","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"R(n)=∑_d^2mid nμ(d)χ_5(d)d^10τ(n/d^2).","count":10},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-11fdaa25e327864a","tex":"\\chi^2","aliases":"χ^2 chi","display":"\\chi^2","kind":"use","excerpt":"A residue eliminates three prime classes. Suppose p≠5 and put χ=chifive(p). Modulo 5, χ≡ p^2, so the two recurrence roots may be taken as p and χ: p+χ≡ a, pχ≡ p^11pmod5. The complete homogeneous recurrence coefficient is T_r≡ C_r=∑_i=0^r p^r-iχ^ipmod5. Also χ p^10≡1pmod5. Since χ^2=1, for r≥2 the sum telescopes: R(p^r)≡ C_r-C_r-2 =p^r-1(p+χ)pmod5. ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-10572bc8a90f9fa6","tex":"\\chi^i","aliases":"χ^i chi","display":"\\chi^i","kind":"use","excerpt":"A residue eliminates three prime classes. Suppose p≠5 and put χ=chifive(p). Modulo 5, χ≡ p^2, so the two recurrence roots may be taken as p and χ: p+χ≡ a, pχ≡ p^11pmod5. The complete homogeneous recurrence coefficient is T_r≡ C_r=∑_i=0^r p^r-iχ^ipmod5. Also χ p^10≡1pmod5. Since χ^2=1, for r≥2 the sum telescopes: R(p^r)≡ C_r-C_r-2 =p^r-1(p+χ)pmod5. ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e0e0b5f27cb73991","tex":"\\chi_5","aliases":"χ_5 chi","display":"\\chi_5","kind":"use","excerpt":"R(n)=∑_d^2mid nμ(d)χ_5(d)d^10τ(n/d^2).","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"The actual joint image. Factor M=∏_ℓ^eVert Mℓ^e and form the single joint finite map Ψ:G_Q⟶ (∏_ℓ^eVert MGL_2(Z/ℓ^eZ)) ×(Z/5MZ)^×. Its components are the integral representation reductions and the cyclotomic character modulo 5M. Let G be its actual image. The kernel defines a finite Galois extension of Q. The image need not be the whole displayed pr","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"For direct symbolic computation, use four coordinates x_0,…,x_3 and substitute v=(x_0,x_1,x_2,x_3,-x_0-x_1-x_2-x_3). The coordinate tangent vectors are b_a=e_a-e_4. Their Gram matrix and its inverse are G=I_4+11^T, G^-1=I_4-frac1511^T, now with a four-entry vector 1. Matrix multiplication verifies the inverse. The factor 1/5 remains because the mis","count":9},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"The valuation v_ℓ(x) is the exponent of ℓ in a nonzero rational number x, with v_ℓ(0)=∞. The quadratic character used in the radial extraction is chifive(d)= 0 · 5mid d,","count":59},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0969800a2a6ee630","tex":"\\ell^{-a}","aliases":"ℓ^-a ell","display":"\\ell^{-a}","kind":"use","excerpt":"A continuous representation of the compact group G_Q has a stable Z_ℓ lattice. To see this directly, take an initial lattice L_0. Compactness bounds all its translates inside ℓ^-aL_0 for some a. The module L=∑_g∈ G_Qρ_ℓ(g)L_0 is contained in ℓ^-aL_0 and contains L_0. It is finitely generated, since a submodule of a finite free Z_ℓ module is finitel","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-78faf17a83e8bb07","tex":"\\ell^e","aliases":"ℓ^e ell","display":"\\ell^e","kind":"use","excerpt":"l lattice. To see this directly, take an initial lattice L_0. Compactness bounds all its translates inside ℓ^-aL_0 for some a. The module L=∑_g∈ G_Qρ_ℓ(g)L_0 is contained in ℓ^-aL_0 and contains L_0. It is finitely generated, since a submodule of a finite free Z_ℓ module is finitely generated, and it is invariant by construction. Reductions modulo ","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Put H_k=ker(1^T:Q^s_k→Q), T_k=im M_k, E_k=H_k/T_k. The quotient E_k is the vector space of total-preserving changes that are unavailable from the chosen family. It is not a count of failed sign certificates. A sign certificate may exist even when E_k is nonzero.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"The permutation j↦ gjbmod5 has sign chifive(g). For the four units modulo 5, multiplication by 1 is the identity, multiplication by 2 or 3 is a four-cycle on the nonzero residues, and multiplication by 4 is two transpositions. Scaling the degree-10 Vandermonde therefore gives W(v)=chifive(g)g^10W(u). The character is not an arbitrary normalization.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"The assertion is ∀ M ∃ a positive-density set of suitable p. It is not ∃ p ∀ M: Mmidτ(p). The surviving primes may change with M. No passage to an exact integer zero is available by reversing the quantifiers.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=0^4∂_i^2F -frac15(∑_i=0^4∂_i)^2F)|_H. Only tangentia","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9aa076a44aa46e0a","tex":"\\partial^2","aliases":"∂^2 partial","display":"\\partial^2","kind":"use","excerpt":"The chart Laplacian is consequently Δ_H=∑_a,b=0^3 (δ_ab-frac15)frac∂^2∂ x_a∂ x_b. The coordinates x_a are not orthonormal. Using only the four diagonal second derivatives would use a different metric. As a normalization check, s_2=x^TGx, so its Hessian is 2G and Δ_Hs_2=tr(G^-12G)=8. The rank of the Euclidean space is four, giving twice that rank. L","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c97379bf19e9e4d6","tex":"\\partial_i","aliases":"∂_i partial","display":"\\partial_i","kind":"use","excerpt":"projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=0^4∂_i^2F -frac15(∑_i=0^4∂_i)^2F)|_H. Only tangentia","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ae4b364ad30c6ac5","tex":"\\partial_i^2","aliases":"∂_i^2 partial","display":"\\partial_i^2","kind":"use","excerpt":"projection from R^5 to H is P=I_5-frac1511^T, where 1=(1,1,1,1,1)^T. Indeed, subtracting the coordinate mean leaves a zero coordinate sum. The matrix is symmetric, satisfies P^2=P, and has image H. For an ambient polynomial F, its gradient along H is P∇ F. Its Laplacian along H is Δ_H(F|_H) =(∑_i=0^4∂_i^2F -frac15(∑_i=0^4∂_i)^2F)|_H. Only tangentia","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"Completeness retains the total. Suppose a shell has nonzero orbit masses w_1,…,w_N and total T. Even allow every rational correction vector q with ∑_iw_iq_i=0. Then ∃ q: w_i(1+q_i) have one weak sign and are not all zero Longleftrightarrow T≠0. Necessity follows from conservation. For sufficiency, if T≠0, take q_i=fracTNw_i-1. Every corrected mass ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0187aedf63e21d96","tex":"\\prod_{0\\le i<j\\le4}","aliases":"∏_0≤ i<j≤4 prod le le","display":"\\prod_{0\\le i<j\\le4}","kind":"use","excerpt":"Packets, content and the coefficient shift. Coordinates have indices 0,1,2,3,4. A shell is Vn={v∈Z^5: v_i≡ ipmod5, ∑_i v_i=0, ∑_i v_i^2=10n}. The standard orientation is D(v)=∏_0≤ i<j≤4(v_i-v_j), W(v)=D(v)/288. The identity is τ(n)=∑_v∈VnW(v). No absolute values occur in that sum. Distinct residue classes ensure that no Vandermonde factor is zero. ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8b1a86572f392d65","tex":"\\prod_{h\\in H(\\lambda)}","aliases":"∏_h∈ H(λ) prod in lambda","display":"\\prod_{h\\in H(\\lambda)}","kind":"use","excerpt":"The Nekrasov--Okounkov identity in Han's formulation is ∏_m≥1(1-q^m)^z-1 =∑_λ q^|λ|∏_h∈ H(λ)(1-frac zh^2). At z=25, a partition having a hook of length 5 contributes zero. The surviving terms are the 5-cores. Multiplication by q shows that the coefficient τ(n) is the sum of the hook products over 5-cores of size n-1.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"Newton reductions with zero first power sum. Let e_j be the elementary symmetric polynomials in the five coordinates. Their characteristic polynomial is ∏_i(T-v_i)=T^5+e_2T^3-e_3T^2+e_4T-e_5, because e_1=s_1=0. The first Newton identities give e_2 · =-frac12s_2, · e_3 · =frac13s_3,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-260615486598a1a4","tex":"\\prod_{j=0}^4","aliases":"∏_j=0^4 prod","display":"\\prod_{j=0}^4","kind":"use","excerpt":"Integrality follows from an integer-valued basis. For every integer x, including negative integers, the generalized binomial coefficient binomxj is an integer. Its leading coefficient as a polynomial in x is 1/j!. Thus det[binomv_ij]_0≤ i,j≤4 =frac∏_i<j(v_j-v_i)∏_j=0^4j!. There are ten pairs, so reversing every difference changes the sign by (-1)^1","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-5e4c84160d4d61eb","tex":"\\prod_{j\\ne i}","aliases":"∏_j≠ i prod ne","display":"\\prod_{j\\ne i}","kind":"use","excerpt":"For that t, set L_i(v)=∏_j≠ i fracE_t(v)-e_j(s_2(v)/C)^5e_i-e_j. Each factor is homogeneous of degree 10, and the denominators are nonzero rational numbers. The resulting polynomial is symmetric homogeneous of degree 10(N-1). On the shell, s_2/C=1, so L_i(v_j)=δ_ij. Consequently, any prescribed rational orbit values a_1,…,a_N are interpolated by Q=","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-69d0131f364b5da2","tex":"\\prod_{k\\ge1}","aliases":"∏_k≥1 prod ge","display":"\\prod_{k\\ge1}","kind":"use","excerpt":"Write F(q)=fracΔ(q)q=∏_k≥1(1-q^k)^24 =∑_m≥0f_mq^m. Then f_0=1 and τ(n)=f_n-1. This shift is essential. A program that stores τ(0)=0 as a sentinel must not confuse that entry with the constant coefficient f_0.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8a091b441c96ffcc","tex":"\\prod_{m\\ge1}","aliases":"∏_m≥1 prod ge","display":"\\prod_{m\\ge1}","kind":"use","excerpt":"The coefficient and its local parameters. The normalized form is Δ(q)=q∏_m≥1(1-q^m)^24 =∑_n≥1τ(n)q^n, τ(1)=1. All coefficient indices in this book are positive integers. A stored value at index zero is a programming sentinel unless a displayed formula explicitly introduces the coefficients of Δ/q. For the latter coefficients, f_m=τ(m+1) and f_0=1.","count":5},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b9d5148b142926aa","tex":"\\prod_p","aliases":"∏_p prod","display":"\\prod_p","kind":"use","excerpt":"· and τ(p)=0. In particular, Lehmer's conjecture is equivalent to nonvanishing at every prime. Factor n=∏_p p^r_p. Multiplicativity expresses τ(n) as a finite product of local blocks. An integral product is zero exactly when one factor is zero. Apply [source reference] to each factor.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d03631915cf67339","tex":"\\prod_{p^{r_p}\\Vert n}","aliases":"∏_p^r_pVert n prod Vert","display":"\\prod_{p^{r_p}\\Vert n}","kind":"use","excerpt":"The full size bound survives. At primes other than q, the Hecke roots have absolute value p^11/2 by Deligne. Formula [source reference] therefore gives |τ(p^r)|≤(r+1)p^11r/2. At the modified prime, F_q,2m=(-q^11)^m, F_q,2m+1=0. These obey the same local bound at every exponent, including zero. Multiplying the local estimates gives |f(n)|≤∏_p^r_pVer","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6d53881b7a7a4f3c","tex":"\\prod_{\\ell^e\\Vert M}","aliases":"∏_ℓ^eVert M prod ell Vert","display":"\\prod_{\\ell^e\\Vert M}","kind":"use","excerpt":"The actual joint image. Factor M=∏_ℓ^eVert Mℓ^e and form the single joint finite map Ψ:G_Q⟶ (∏_ℓ^eVert MGL_2(Z/ℓ^eZ)) ×(Z/5MZ)^×. Its components are the integral representation reductions and the cyclotomic character modulo 5M. Let G be its actual image. The kernel defines a finite Galois extension of Q. The image need not be the whole displayed pr","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c4dd6744667e3c9a","tex":"\\sum_{a,b=0}^3","aliases":"∑_a,b=0^3 sum","display":"\\sum_{a,b=0}^3","kind":"use","excerpt":"The chart Laplacian is consequently Δ_H=∑_a,b=0^3 (δ_ab-frac15)frac∂^2∂ x_a∂ x_b. The coordinates x_a are not orthonormal. Using only the four diagonal second derivatives would use a different metric. As a normalization check, s_2=x^TGx, so its Hessian is 2G and Δ_Hs_2=tr(G^-12G)=8. The rank of the Euclidean space is four, giving twice that rank. L","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b5796fb421f244c4","tex":"\\sum_{a=0}^3","aliases":"∑_a=0^3 sum","display":"\\sum_{a=0}^3","kind":"use","excerpt":"The retained programs organize these same operations using sparse rational polynomials. A direct four-coordinate implementation can instead substitute v_4=-∑_a=0^3x_a into DQ and apply [source reference]. Agreement between these organizations checks the symbolic construction through different algebraic representations. The universal justification r","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-47c8502650fb169b","tex":"\\sum_{a=0}^{m-2}","aliases":"∑_a=0^m-2 sum","display":"\\sum_{a=0}^{m-2}","kind":"use","excerpt":"Δ_Hs_m · =frac45m(m-1)s_m-2. In the logarithmic term, the coordinate-mean subtraction vanishes because the logarithmic gradient has zero sum. Pair the ordered indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v_i-v_j. The divided difference is the finite polynomial fracv_i^m-1-v_j^m-1v_i-v_j =∑_a=0^m-2v_i^av_j^m-2-a. The sum","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-581e46df03c6e7a6","tex":"\\sum_d","aliases":"∑_d sum","display":"\\sum_d","kind":"use","excerpt":"Let Q be any finite symmetric polynomial over C, restricted to ∑_i v_i=0. Suppose ∑_v∈VnW(v)Q(v)=0 for every positive integer n. Write Q=∑_dQ_d in homogeneous degrees. Then every Q_d is separately universally null. Moreover, the weighted Q-sum is zero in each exact-content sector of every shell.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-88173d4630911f5f","tex":"\\sum_{d^2\\mid n}","aliases":"∑_d^2mid n sum mid","display":null,"kind":"use","excerpt":"R(n)=∑_d^2mid nμ(d)χ_5(d)d^10τ(n/d^2).","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-30d4ed40d797d0a1","tex":"\\sum_{d\\mid j}","aliases":"∑_dmid j sum mid","display":null,"kind":"use","excerpt":"The logarithmic derivative provides a different exact engine. With σ_1(j)=∑_dmid jd, comparison of coefficients in qfracF'(q)F(q)=-24∑_j≥1σ_1(j)q^j gives m f_m=-24∑_j=1^mσ_1(j)f_m-j (m≥1). For m=3, the divisor sums are 1,3,4. Hence f_3=-8(252+3(-24)+4)=-8·184=-1472. An implementation should check that the numerator in [source reference] is divisibl","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c5bab67784766df4","tex":"\\sum_{d\\mid m}","aliases":"∑_dmid m sum mid","display":null,"kind":"use","excerpt":"The square-supported arithmetic kernel in [source reference] has inverse d^2⟼μ(d)chifive(d)d^10. Indeed, at a square index m^2, the convolution of the two kernels is chifive(m)m^10∑_dmid mμ(d), which is 1 at m=1 and 0 otherwise. At nonsquare indices both kernels have no support. Consequently R(n)=∑_d^2mid nμ(d)chifive(d)d^10τ(n/d^2). Both formulas ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-deb9c97b11939ae2","tex":"\\sum_{d\\mid n}","aliases":"∑_dmid n sum mid","display":null,"kind":"use","excerpt":"S13: packet cardinality. Garvan, Kim and Stanton's Theorem~4 and equation~(5.3) give the five-core count [source reference]. In the present conventions this reads a_5(n-1)=∑_dmid nchifive(d)frac nd. For a prime different from 5, the shell cardinality is p+chifive(p). This is the number of packets, not their weighted sum.","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-76e758f1be5158a2","tex":"\\sum_g","aliases":"∑_g sum","display":"\\sum_g","kind":"use","excerpt":"The negative version is identical. For a weakly nonnegative certificate with at least one strict mass, the criterion is T_g≥0 for every class and ∑_gT_g>0.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b31f251b71fc5838","tex":"\\sum_{g^2\\mid n}","aliases":"∑_g^2mid n sum mid","display":null,"kind":"use","excerpt":"The content of a packet is the positive gcd of its five coordinates. Content-one packets are primitive. Multiplication followed by the prescribed residue reindexing gives τ(n)=∑_g^2mid nchifive(g)g^10R(n/g^2), R(n)=∑_d^2mid nμ(d)chifive(d)d^10τ(n/d^2). The boundary values are R(1)=1 and R(p)=τ(p). At a prime shell the only possible content is one, ","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7869a6b7878f39ca","tex":"\\sum_{g\\in G_{\\Q}}","aliases":"∑_g∈ G_Q sum in Q","display":null,"kind":"use","excerpt":"A continuous representation of the compact group G_Q has a stable Z_ℓ lattice. To see this directly, take an initial lattice L_0. Compactness bounds all its translates inside ℓ^-aL_0 for some a. The module L=∑_g∈ G_Qρ_ℓ(g)L_0 is contained in ℓ^-aL_0 and contains L_0. It is finitely generated, since a submodule of a finite free Z_ℓ module is finitel","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Packets, content and the coefficient shift. Coordinates have indices 0,1,2,3,4. A shell is Vn={v∈Z^5: v_i≡ ipmod5, ∑_i v_i=0, ∑_i v_i^2=10n}. The standard orientation is D(v)=∏_0≤ i<j≤4(v_i-v_j), W(v)=D(v)/288. The identity is τ(n)=∑_v∈VnW(v). No absolute values occur in that sum. Distinct residue classes ensure that no Vandermonde factor is zero. ","count":17},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8524806a0ba8de34","tex":"\\sum_{i<j}","aliases":"∑_i<j sum","display":"\\sum_{i<j}","kind":"use","excerpt":"Δ_Hs_m · =frac45m(m-1)s_m-2. In the logarithmic term, the coordinate-mean subtraction vanishes because the logarithmic gradient has zero sum. Pair the ordered indices i,j and j,i to obtain 2∇_Hlog D·∇_Hs_m =2m∑_i<jfracv_i^m-1-v_j^m-1v_i-v_j. The divided difference is the finite polynomial fracv_i^m-1-v_j^m-1v_i-v_j =∑_a=0^m-2v_i^av_j^m-2-a. The sum","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a5d3926ec8ff757e","tex":"\\sum_{i=0}^4","aliases":"∑_i=0^4 sum","display":"\\sum_{i=0}^4","kind":"use","excerpt":"Write s_m=∑_i=0^4v_i^m, with s_0=5 and s_1=0, and let D=∏_i<j(v_i-v_j). We distinguish the ordinary Laplacian Δ_H from the conjugated operator L defined by Δ_H(DQ)=D LQ. The factor D changes radial coefficients substantially. Confusing the two operators would invalidate the harmonicity test.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-e2705c1a4fc8894e","tex":"\\sum_{i=0}^r","aliases":"∑_i=0^r sum","display":"\\sum_{i=0}^r","kind":"use","excerpt":"A residue eliminates three prime classes. Suppose p≠5 and put χ=chifive(p). Modulo 5, χ≡ p^2, so the two recurrence roots may be taken as p and χ: p+χ≡ a, pχ≡ p^11pmod5. The complete homogeneous recurrence coefficient is T_r≡ C_r=∑_i=0^r p^r-iχ^ipmod5. Also χ p^10≡1pmod5. Since χ^2=1, for r≥2 the sum telescopes: R(p^r)≡ C_r-C_r-2 =p^r-1(p+χ)pmod5. ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"A sufficient sign input. A second precise route would construct permitted corrections with an independently proved sign. At each remaining prime, choose finitely many rational universal null polynomials Q_p,j and rational coefficients a_p,j, and put F_p(v)=1+∑_j a_p,jQ_p,j(v). The choice may depend on p. Universal nullness must hold after the choic","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-b269a512e2c4d1ab","tex":"\\sum_{j=0}^{24}","aliases":"∑_j=0^24 sum","display":"\\sum_{j=0}^{24}","kind":"use","excerpt":"For direct product expansion, terms of degree at most m depend only on factors with k≤ m. The exact binomial expansion (1-q^k)^24=∑_j=0^24(-1)^jbinom24jq^kj therefore supplies a finite algorithm for each desired coefficient. Three short calculations are f_1 · =-24,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-540954dab24fb4cd","tex":"\\sum_{j=0}^{\\lfloor r/2\\rfloor}","aliases":"∑_j=0^lfloor r/2rfloor sum lfloor rfloor","display":null,"kind":"use","excerpt":"The recurrence polynomial and untied valuations. For p≡4pmod5, χ=1 and a≡0pmod5. The exact recurrence expansion is T_r=∑_j=0^lfloor r/2rfloor (-1)^jbinomr-jjb^ja^r-2j. It follows by induction: multiplying the previous row by a and subtracting b times the row before it combines neighboring binomial coefficients. Equivalently, expand the rational gen","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9098bd3c42e16d5a","tex":"\\sum_{j=0}^r","aliases":"∑_j=0^r sum","display":"\\sum_{j=0}^r","kind":"use","excerpt":"Solving the recurrence gives T_r=fracα^r+1-β^r+1α-β =∑_j=0^rα^r-jβ^j. If T_r=0 for some r≥1, then ζ=α/β is a root of unity with ζ^r+1=1. Consequently fraca^2p^11 =frac(α+β)^2αβ =2+ζ+ζ^-1. The left side is rational. The right side is an algebraic integer, because roots of unity are algebraic integers. A rational algebraic integer is an integer. It i","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-55bfd4be1daf726b","tex":"\\sum_{j=1}^{m}","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^{m}","kind":"use","excerpt":"The logarithmic derivative provides a different exact engine. With σ_1(j)=∑_dmid jd, comparison of coefficients in qfracF'(q)F(q)=-24∑_j≥1σ_1(j)q^j gives m f_m=-24∑_j=1^mσ_1(j)f_m-j (m≥1). For m=3, the divisor sums are 1,3,4. Hence f_3=-8(252+3(-24)+4)=-8·184=-1472. An implementation should check that the numerator in [source reference] is divisibl","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-00a7ede556f767a5","tex":"\\sum_{j\\ne i}","aliases":"∑_j≠ i sum ne","display":"\\sum_{j\\ne i}","kind":"use","excerpt":"Translation invariance of D gives ∑_ifrac∂_iDD=0, frac∂_iDD=∑_j≠ ifrac1v_i-v_j. Its ambient logarithmic gradient is already tangential. This removes a potential extra projection term in the derivation below.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-49afb54019149b8f","tex":"\\sum_{j\\ge0}","aliases":"∑_j≥0 sum ge","display":"\\sum_{j\\ge0}","kind":"use","excerpt":"Jacobi's triangular series. Jacobi's identity is ∏_m≥1(1-q^m)^3 =∑_j≥0(-1)^j(2j+1)q^j(j+1)/2. Raising this series to the eighth power and multiplying by q gives Δ. The sparse triangular exponents make it useful for coefficient calculations. The source check in [source reference] also uses this identity to establish the parity criterion discussed be","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0c18bcf01edb7e76","tex":"\\sum_{j\\ge1}","aliases":"∑_j≥1 sum ge","display":"\\sum_{j\\ge1}","kind":"use","excerpt":"The logarithmic derivative provides a different exact engine. With σ_1(j)=∑_dmid jd, comparison of coefficients in qfracF'(q)F(q)=-24∑_j≥1σ_1(j)q^j gives m f_m=-24∑_j=1^mσ_1(j)f_m-j (m≥1). For m=3, the divisor sums are 1,3,4. Hence f_3=-8(252+3(-24)+4)=-8·184=-1472. An implementation should check that the numerator in [source reference] is divisibl","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"c · d]∈ SL_2(Z). Constants from the usual quasimodular convention can be absorbed into the coefficient functions. Assume ∑_kF_k=0 identically.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-9972412a61817be9","tex":"\\sum_{k=1}^{n-1}","aliases":"∑_k=1^n-1 sum","display":"\\sum_{k=1}^{n-1}","kind":"use","excerpt":"These four coefficients were checked independently by the Eisenstein convolution. If σ_r(n)=∑_dmid nd^r, the coefficient of (E_4E_8-E_6^2)/1728 is obtained from 1728τ(n)= · 240σ_3(n)+480σ_7(n)+1008σ_5(n) · +115200∑_k=1^n-1σ_3(k)σ_7(n-k) · -254016∑_k=1^n-1σ_5(k)σ_5(n-k). This computes the selected integers without the triangular recurrence. The nume","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-01f56a5e1c2b8b56","tex":"\\sum_{k\\ge1}","aliases":"∑_k≥1 sum ge","display":"\\sum_{k\\ge1}","kind":"use","excerpt":"oefficient engines The product and a logarithmic recurrence. Let σ_j(n)=∑_dmid nd^j and write σ=σ_1. Logarithmic differentiation of the product gives DQΔ=(1-24∑_m≥1σ(m)q^m)Δ, DQ=qfracddq. Indeed, the contribution from the factor (1-q^r)^24 is -24fracrq^r1-q^r=-24∑_k≥1rq^rk. Collecting the terms with rk=m gives -24σ(m). The leading factor q contribu","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-f0fa6a65524a6e77","tex":"\\sum_{m,n=2}^5","aliases":"∑_m,n=2^5 sum","display":"\\sum_{m,n=2}^5","kind":"use","excerpt":"If Q=F(s_2,s_3,s_4,s_5), the ordinary second-order chain rule and [source reference] yield LQ =∑_m=2^5frac∂ F∂ s_m Ls_m +∑_m,n=2^5 frac∂^2F∂ s_m∂ s_nΓ_mn. The second sum runs over ordered pairs. Thus a mixed derivative contributes twice when m≠ n. If one instead sums over m≤ n, the off-diagonal terms require a factor two. These are two equivalent c","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-8e83a85ca0594337","tex":"\\sum_{m=1}^{n-1}","aliases":"∑_m=1^n-1 sum","display":"\\sum_{m=1}^{n-1}","kind":"use","excerpt":"Comparing coefficients of q^n in [source reference], for n>1, yields (n-1)τ(n)=-24∑_m=1^n-1σ(m)τ(n-m), τ(1)=1. This recurrence uses only smaller indices. Division by n-1 is exact because the recurrence came from the integral product, although an implementation should still check divisibility rather than silently round a quotient. It is a convenient","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-eca8c65f73b8542d","tex":"\\sum_{m=2}^5","aliases":"∑_m=2^5 sum","display":"\\sum_{m=2}^5","kind":"use","excerpt":"If Q=F(s_2,s_3,s_4,s_5), the ordinary second-order chain rule and [source reference] yield LQ =∑_m=2^5frac∂ F∂ s_m Ls_m +∑_m,n=2^5 frac∂^2F∂ s_m∂ s_nΓ_mn. The second sum runs over ordered pairs. Thus a mixed derivative contributes twice when m≠ n. If one instead sums over m≤ n, the off-diagonal terms require a factor two. These are two equivalent c","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d328b95fa488fa30","tex":"\\sum_{m\\ge0}","aliases":"∑_m≥0 sum ge","display":"\\sum_{m\\ge0}","kind":"use","excerpt":"Write F(q)=fracΔ(q)q=∏_k≥1(1-q^k)^24 =∑_m≥0f_mq^m. Then f_0=1 and τ(n)=f_n-1. This shift is essential. A program that stores τ(0)=0 as a sentinel must not confuse that entry with the constant coefficient f_0.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a0a9cffbc383b69e","tex":"\\sum_{m\\ge1}","aliases":"∑_m≥1 sum ge","display":"\\sum_{m\\ge1}","kind":"use","excerpt":"Exact coefficient engines. The product and a logarithmic recurrence. Let σ_j(n)=∑_dmid nd^j and write σ=σ_1. Logarithmic differentiation of the product gives DQΔ=(1-24∑_m≥1σ(m)q^m)Δ, DQ=qfracddq. Indeed, the contribution from the factor (1-q^r)^24 is -24fracrq^r1-q^r=-24∑_k≥1rq^rk. Collecting the terms with rk=m gives -24σ(m). The leading factor q ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"Fix z in the upper half-plane. The polynomial P_z(c,d)=∑_k∑_r f_k,r(z)c^r(cz+d)^k-r vanishes at every relatively prime integer pair (c,d), because every such pair is the bottom row of a modular matrix. These pairs are Zariski dense. To see this without a geometric theorem, fix any positive prime c. Infinitely many integers d are coprime to it, so t","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-1cf3aa5b11ae4199","tex":"\\sum_{r=0}^{p_k}","aliases":"∑_r=0^p_k sum","display":"\\sum_{r=0}^{p_k}","kind":"use","excerpt":"Separating the quasimodular weights. We need only an elementary weight-separation statement. Suppose finitely many quasimodular functions F_k of distinct nonnegative integer weights satisfy F_k(γ z)=∑_r=0^p_kf_k,r(z)c^r(cz+d)^k-r, f_k,0=F_k, p_k≤ k, for γ=[a · b","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-6a4250883c4968da","tex":"\\sum_{r\\ge0}","aliases":"∑_r≥0 sum ge","display":"\\sum_{r\\ge0}","kind":"use","excerpt":"For a fixed prime put T_r=τ(p^r) and b=p^11. The generating series is ∑_r≥0T_rX^r=frac11-aX+bX^2. Multiplication by the denominator gives T_0=1, T_1=a and the recurrence. Thus this is a formal identity over Z[[X]]. No analytic continuation in the prime index is involved.","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-99c50d63caca97ea","tex":"\\sum_{\\substack{j\\ge1\\\\T_j\\le m}}","aliases":"∑_substackj≥1 T_j≤ m sum substack ge T le","display":null,"kind":"use","excerpt":"Jacobi gives a third engine. Put T_j=j(j+1)/2 and a_j=(-1)^j(2j+1). The series A(q)=∏_k≥1(1-q^k)^3=1+∑_j≥1a_jq^T_j satisfies F=A^8. The identity AqF'=8FqA' yields m f_m=∑_substackj≥1 T_j≤ m a_j(9T_j-m)f_m-T_j. At m=3, only T_1=1, T_2=3 contribute. Its numerator is (-3)(9-3)(252)+5(27-3)(1)=-4536+120=-4416. Division by 3 again gives -1472. The direc","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d852990fc3ac4bde","tex":"\\sum_{\\substack{v_i\\equiv i\\ (5)\\\\\\sum v_i=0}}","aliases":"∑_substackv_i≡ i (5) ∑ v_i=0 sum substack equiv sum","display":"\\sum_{\\substack{v_i\\equiv i\\ (5)\\\\\\sum v_i=0}}","kind":"use","excerpt":"Radial factors become derivatives. For an alternating harmonic h_ℓ, write θ_h(τ)=∑_substackv_i≡ i (5) ∑ v_i=0 h(v)exp(2π iτfrac v ^210). Chapter~[source reference] establishes that it is a scalar cusp form of weight ℓ+2. The normalization includes only a fixed scalar if one divides by 288.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-49d8198b61469bf8","tex":"\\sum_{\\substack{v\\in\\mathcal V_p\\\\g(v)=1}}","aliases":"∑_substackv∈ V_p g(v)=1 sum substack in mathcal g","display":null,"kind":"use","excerpt":"At a prime shell there is no imprimitive part, so the universal target is still ∑_substackv∈ V_p g(v)=1W(v) =R(p)=τ(p)≠0. The content geometry has clarified the radial arithmetic. The prime total still needs a global or uniform arithmetic mechanism excluding cancellation to zero.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d74a9ab2619d83bf","tex":"\\sum_{\\substack{v\\in\\Vn\\\\g(v)=1}}","aliases":"∑_substackv∈Vn g(v)=1 sum substack in Vn g","display":null,"kind":"use","excerpt":"The primitive residual and its inverse. Define R(n)=∑_substackv∈Vn g(v)=1W(v). Partitioning the packets by content and using [source reference] yields τ(n)=∑_g^2mid nchifive(g)g^10R(n/g^2). Terms with 5mid g have character zero and correspond to no actual content sector. They may be included in the formula without adding contributions.","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-3c0172ae3dc5e4c3","tex":"\\sum_{v\\in S_{k,r}}","aliases":"∑_v∈ S_k,r sum in","display":"\\sum_{v\\in S_{k,r}}","kind":"use","excerpt":"Refinement and exact transfer spaces. Mass vectors and aggregation. Fix a shell, a finite rational family Q_1,…,Q_h of universally null corrections, and the occupied sectors at depth k. Let their number be s_k. Define t_k,r=∑_v∈ S_k,rW(v), (M_k)_rj=∑_v∈ S_k,rW(v)Q_j(v). The vector t_k contains sector masses. The columns of M_k are the effects of th","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7d3467bf8a7de172","tex":"\\sum_{v\\in S_r}","aliases":"∑_v∈ S_r sum in","display":"\\sum_{v\\in S_r}","kind":"use","excerpt":"Why strict localization and automatic valuation fail. Conserved sectors can preserve opposite signs. Partition a shell into sectors S_r, with masses t_r=∑_v∈ S_rW(v). Suppose every permitted correction is null inside every sector: ∑_v∈ S_rW(v)Q_j(v)=0. Then F=1+∑ a_jQ_j leaves each corrected sector mass equal to t_r. If two sector masses have oppos","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-930def1bbe4663c3","tex":"\\sum_{v\\in\\mathcal V_p}","aliases":"∑_v∈ V_p sum in mathcal","display":"\\sum_{v\\in\\mathcal V_p}","kind":"use","excerpt":"On each such prime shell, all packets have content one. The residual is therefore the full signed sum τ(p)=∑_v∈ V_pW(v). The summands are nonzero integers, and for p>20 they are units at the index prime. Neither property forbids their sum from being zero. The unresolved statement concerns the exact balance of their signed masses.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ade7e6465dee6e86","tex":"\\sum_\\lambda","aliases":"∑_λ sum lambda","display":"\\sum_\\lambda","kind":"use","excerpt":"The Nekrasov--Okounkov identity in Han's formulation is ∏_m≥1(1-q^m)^z-1 =∑_λ q^|λ|∏_h∈ H(λ)(1-frac zh^2). At z=25, a partition having a hook of length 5 contributes zero. The surviving terms are the 5-cores. Multiplication by q shows that the coefficient τ(n) is the sum of the hook products over 5-cores of size n-1.","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"The geometric lower bound. Take opposite-sign packets v,w in one sector. Replace w by Jw if needed so that w-v=5Md, d∈ A_4∖{0}. Set a=5M/2 and m=(v+w)/2. Since M is even, both are integral. Equal norms give m· d=0,qquad10n= m ^2+a^2 d ^2. The squared norm of a nonzero integral zero-sum vector is a positive even integer. It cannot be 2 here. Such a ","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"The valuation v_ℓ(x) is the exponent of ℓ in a nonzero rational number x, with v_ℓ(0)=∞. The quadratic character used in the radial extraction is chifive(d)= 0 · 5mid d,","count":4},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"The regular discriminant cosets. Let A_4=H∩Z^5 and L=sqrt5 A_4. Its discriminant module is L^*/L≅ A_4^*/5A_4, |L^*/L|=5^5=3125. A representative of L^* can be written x=frac1sqrt5(a_i+t/5)_i=0^4, qquad0≤ t<5, ∑_i a_i=-t. If two a_i have the same residue modulo 5, the transposition of those coordinates fixes the discriminant coset. An alternating th","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"The regular discriminant cosets. Let A_4=H∩Z^5 and L=sqrt5 A_4. Its discriminant module is L^*/L≅ A_4^*/5A_4, |L^*/L|=5^5=3125. A representative of L^* can be written x=frac1sqrt5(a_i+t/5)_i=0^4, qquad0≤ t<5, ∑_i a_i=-t. If two a_i have the same residue modulo 5, the transposition of those coordinates fixes the discriminant coset. An alternating th","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"The valuation v_ℓ(x) is the exponent of ℓ in a nonzero rational number x, with v_ℓ(0)=∞. The quadratic character used in the radial extraction is chifive(d)= 0 · 5mid d, 1 · d≡1,4pmod5, -1 · d≡2,3pmod5. The character value and the tenth power in the scaling formula are both essential.","count":66},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"Packets, content and the coefficient shift. Coordinates have indices 0,1,2,3,4. A shell is Vn={v∈Z^5: v_i≡ ipmod5, ∑_i v_i=0, ∑_i v_i^2=10n}. The standard orientation is D(v)=∏_0≤ i<j≤4(v_i-v_j), W(v)=D(v)/288. The identity is τ(n)=∑_v∈VnW(v). No absolute values occur in that sum. Distinct residue classes ensure that no Vandermonde factor is zero. ","count":37},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"1 · 0, Trφ=0, detφ=p^11, φ^2=-p^11I. Choose any Q_p-line L⊂ D and put Fil^iD= D · i≤0,","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"Project e_i-e_j orthogonally to d^⊥. Its squared norm is 2-fracδ^24. Because m⊥ d, Cauchy's inequality gives m ^2≥frac(a|δ|+1)^22-δ^2/4. For |δ|=1 this is 4(a+1)^2/7. For |δ|=2 it is (2a+1)^2, a larger quantity. Therefore 10n≥4a^2+frac47(a+1)^2, which simplifies to the lower bound inside the ceiling in [source reference]. Taking the ceiling proves ","count":2},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"A fixed modular form and an exact question. The coefficient that must not disappear. For |q|<1 the product defining Δ converges. As a formal power series it is even simpler: the coefficient of q^N depends only on the first N product factors. Thus each τ(n) is an unambiguously defined integer. The opening terms are Δ(q)= · q-24q^2+252q^3-1472q^4+483","count":1},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"The exact refinement sequence. There is a short exact sequence 0⟶ fracker AM_k+1(ker M_k) ⟶ E_k+1⟶ E_k⟶0. Consequently, if r_k=rank M_k, dim E_k+1-dim E_k =(s_k+1-s_k)-(r_k+1-r_k). The aggregation map sends H_k+1 onto H_k. Given a zero-total parent vector, place each parent entry in one selected child and zero in its other children. The resulting c","count":3},{"work":"lehmer-reader","url":"notation/lehmer-reader.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"The square-supported arithmetic kernel in [source reference] has inverse d^2⟼μ(d)chifive(d)d^10. Indeed, at a square index m^2, the convolution of the two kernels is chifive(m)m^10∑_dmid mμ(d), which is 1 at m=1 and 0 otherwise. At nonsquare indices both kernels have no support. Consequently R(n)=∑_d^2mid nμ(d)chifive(d)d^10τ(n/d^2). Both formulas ","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"R=rad(abc) · Activated prime support of a primitive triple","count":453},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"notation-table","excerpt":"R=rad(abc) · Activated prime support of a primitive triple","count":245},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"notation-table","excerpt":"B_d(n) · Exact-order divisor-polyphase bandwidth packet","count":46},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"notation-table","excerpt":"R=rad(abc) · Activated prime support of a primitive triple","count":229},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7be573071ebda78d","tex":"B_d","aliases":"B_d","display":"B_d","kind":"notation-table","excerpt":"B_d(n) · Exact-order divisor-polyphase bandwidth packet","count":10},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"notation-table","excerpt":"C · Coherence quotient · Depth created by genuine simultaneous cancellation among the normalized cross-prime logarithmic channels","count":120},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"notation-table","excerpt":"R=rad(abc) · Activated prime support of a primitive triple","count":268},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-57d5f54bcad0aa19","tex":"\\mathrm{coh}","aliases":"coh mathrm","display":"\\mathrm{coh}","kind":"notation-table","excerpt":"Λ_coh · Kernel lattice of the coherence character","count":24},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"notation-table","excerpt":"B_d(n) · Exact-order divisor-polyphase bandwidth packet","count":27},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"notation-table","excerpt":"H(k) · Exact shortest height in obstruction coset k","count":42},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-753ac74d7d5e39c2","tex":"\\Jcal","aliases":"Jcal","display":null,"kind":"notation-table","excerpt":"Jcal_c · Filtered cusp-contact module or complex","count":12},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-db2a654890422563","tex":"\\Jcal_c","aliases":"Jcal_c Jcal","display":null,"kind":"notation-table","excerpt":"Jcal_c · Filtered cusp-contact module or complex","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","excerpt":"K · Kummer preload · Depth removed by a common power in the signed input valuation vector","count":43},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"notation-table","excerpt":"H(k) · Exact shortest height in obstruction coset k","count":78},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","excerpt":"L · Additive packet lattice","count":30},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"notation-table","excerpt":"L_0 · Wronskian-zero sublattice","count":9},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3755b02b61222ecc","tex":"\\Lcal","aliases":"Lcal","display":null,"kind":"notation-table","excerpt":"Lcal_P,Q · Cross-prime logarithmic filter matrix","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2b9d69e9d32c8f71","tex":"\\Lcal_{P,Q}","aliases":"Lcal_P,Q Lcal","display":null,"kind":"notation-table","excerpt":"Lcal_P,Q · Cross-prime logarithmic filter matrix","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"notation-table","excerpt":"M=c/R_c · Output powerful part or contact modulus","count":95},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"notation-table","excerpt":"B_d(n) · Exact-order divisor-polyphase bandwidth packet","count":83},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7fa9a05b52100c5d","tex":"\\overline H","aliases":"overline H overline","display":"\\overline H","kind":"notation-table","excerpt":"overline H · Stable real relaxation of the obstruction height","count":18},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"Lcal_P,Q · Cross-prime logarithmic filter matrix","count":39},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"Lcal_P,Q · Cross-prime logarithmic filter matrix","count":22},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"notation-table","excerpt":"R=rad(abc) · Activated prime support of a primitive triple","count":101},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-37f2bdf109707126","tex":"R_c","aliases":"R_c","display":"R_c","kind":"notation-table","excerpt":"M=c/R_c · Output powerful part or contact modulus","count":21},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-99bbe0f6af468b02","tex":"R_u","aliases":"R_u","display":"R_u","kind":"notation-table","excerpt":"R_u=rad(u) · Radical of one term","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-650ae3274f788b89","tex":"\\rad","aliases":"rad","display":null,"kind":"notation-table","excerpt":"R=rad(abc) · Activated prime support of a primitive triple","count":28},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"notation-table","excerpt":"R_u=rad(u) · Radical of one term","count":46},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"notation-table","excerpt":"W · Local logarithmic defect · Depth already present because every active logarithmic coefficient is exceptionally divisible","count":75},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-359d1d19067f6d53","tex":"\\Witt","aliases":"Witt","display":null,"kind":"notation-table","excerpt":"Witt_c · Product of local truncated Witt contact crystals","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-35efe35d98ac2fcc","tex":"\\Witt_c","aliases":"Witt_c Witt","display":null,"kind":"notation-table","excerpt":"Witt_c · Product of local truncated Witt contact crystals","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"notation-table","excerpt":"ϵ_2 · Dyadic sign layer · The separate weighted two-adic sign contribution","count":28},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c7ffdc6ef6b651ea","tex":"\\epsilon_2","aliases":"ϵ_2 epsilon","display":"\\epsilon_2","kind":"notation-table","excerpt":"ϵ_2 · Dyadic sign layer · The separate weighted two-adic sign contribution","count":24},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"notation-table","excerpt":"Θ · Coarse shortest transverse derivative packet","count":10},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"notation-table","excerpt":"κ_c · Global Kummer map carrying the contact packet","count":11},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3eaf6610cad1f3da","tex":"\\kappa_c","aliases":"κ_c kappa","display":"\\kappa_c","kind":"notation-table","excerpt":"κ_c · Global Kummer map carrying the contact packet","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"notation-table","excerpt":"Λ_coh · Kernel lattice of the coherence character","count":36},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-73a92e6394b23f10","tex":"\\Lambda_{\\mathrm{coh}}","aliases":"Λ_coh Lambda mathrm","display":"\\Lambda_{\\mathrm{coh}}","kind":"notation-table","excerpt":"Λ_coh · Kernel lattice of the coherence character","count":15},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"notation-table","excerpt":"Φ · Phase-refined shortest transverse packet. The same symbol denotes the shortest nonzero detector when the context is clear","count":13},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"notation-table","excerpt":"χ_coh · Dual cyclic coherence character","count":22},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-847a7aea863f53a5","tex":"\\chi_{\\mathrm{coh}}","aliases":"χ_coh chi mathrm","display":"\\chi_{\\mathrm{coh}}","kind":"notation-table","excerpt":"χ_coh · Dual cyclic coherence character","count":14},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ ","count":137},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ ","count":50},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d11f2eb675d387eb","tex":"a_{p,q}","aliases":"a_p,q","display":"a_{p,q}","kind":"introduction-passage","excerpt":"d d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row co","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1784da886e1aea15","tex":"\\bar v","aliases":"bar v bar","display":"\\bar v","kind":"introduction-passage","excerpt":"be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a87a1d8861d8d21d","tex":"\\bar v_p","aliases":"bar v_p bar","display":"\\bar v_p","kind":"introduction-passage","excerpt":"be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0efdddd015383ecb","tex":"d_p","aliases":"d_p","display":"d_p","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":139},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":12},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f0160f5b8a57192c","tex":"F_M","aliases":"F_M","display":"F_M","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5b67a225dc0d299e","tex":"h_{\\mathcal O(1)}","aliases":"h_ O(1) mathcal","display":"h_{\\mathcal O(1)}","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1b26238865ef07e5","tex":"h_{K_{F_M}}","aliases":"h_K_F_M","display":"h_{K_{F_M}}","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":18},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":52},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":14},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6832be6d7109cc9f","tex":"K_{F_M}","aliases":"K_F_M","display":"K_{F_M}","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b897ba65dbba23a0","tex":"K_{\\mathrm{Kum}}","aliases":"K_Kum mathrm","display":"K_{\\mathrm{Kum}}","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-febfa676c8169791","tex":"\\mathrm{Kum}","aliases":"Kum mathrm","display":"\\mathrm{Kum}","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":54},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"introduction-passage","excerpt":"me. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid","count":64},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"introduction-passage","excerpt":"ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p","count":20},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-22899074deab8ca2","tex":"\\min_{q\\mid ab}","aliases":"min_qmid ab min mid","display":null,"kind":"introduction-passage","excerpt":"be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":11},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f26013157a61075a","tex":"\\mathcal O_{F_M}","aliases":"O_F_M mathcal","display":"\\mathcal O_{F_M}","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":219},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b47e4ae8f60bef45","tex":"p^e","aliases":"p^e","display":"p^e","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":26},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-93d9a4bd78ce9dca","tex":"p^{r_p}","aliases":"p^r_p","display":"p^{r_p}","kind":"introduction-passage","excerpt":"rallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b49eb13eda2fd7da","tex":"p^{w_p}","aliases":"p^w_p","display":"p^{w_p}","kind":"introduction-passage","excerpt":"athrmKummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. Th","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a8b017d2e4c8ef87","tex":"p^{\\chi_p}","aliases":"p^χ_p chi","display":"p^{\\chi_p}","kind":"introduction-passage","excerpt":"athrmKummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. Th","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fe1b7ade23d1e9a1","tex":"P_M","aliases":"P_M","display":"P_M","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6f5181868d16d4ae","tex":"P_{r',s'}","aliases":"P_r',s'","display":"P_{r',s'}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4734da4707b22bcc","tex":"P_{r,s}","aliases":"P_r,s","display":"P_{r,s}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-474a8c74e28ce2e1","tex":"\\parallel","aliases":"parallel","display":null,"kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":17},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"d d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row co","count":87},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":65},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0d5a58c436e4715b","tex":"r_p","aliases":"r_p","display":"r_p","kind":"introduction-passage","excerpt":"ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":29},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5e769b89788d547a","tex":"s'","aliases":"s'","display":"s'","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"introduction-passage","excerpt":"Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p","count":82},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"introduction-passage","excerpt":"ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p","count":35},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a","count":18},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2b65a86b582ae995","tex":"w_p","aliases":"w_p","display":"w_p","kind":"introduction-passage","excerpt":"be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a","count":9},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":11},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z","display":null,"kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":97},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ ","count":28},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":26},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4ee2fc118c6d1327","tex":"\\delta_{r,r'}","aliases":"δ_r,r' delta","display":"\\delta_{r,r'}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-83ebf9d2f08acb0d","tex":"\\delta_{s,s'}","aliases":"δ_s,s' delta","display":"\\delta_{s,s'}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":13},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9f6e692e69afc404","tex":"\\zeta_M^{-ri-sj}","aliases":"ζ_M^-ri-sj zeta","display":"\\zeta_M^{-ri-sj}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":14},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":13},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b7ec60a68d5fe62e","tex":"\\mu_M^2","aliases":"μ_M^2 mu","display":"\\mu_M^2","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":10},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","excerpt":"Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ ","count":15},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6413311b725a828d","tex":"\\sigma^i","aliases":"σ^i sigma","display":"\\sigma^i","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7cc70e05d3b03d7a","tex":"\\sigma_a","aliases":"σ_a sigma","display":"\\sigma_a","kind":"introduction-passage","excerpt":"Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ ","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fc148a89e558c088","tex":"\\sigma_b","aliases":"σ_b sigma","display":"\\sigma_b","kind":"introduction-passage","excerpt":"Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ ","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ae00882e1fd7fcb3","tex":"\\sigma_c","aliases":"σ_c sigma","display":"\\sigma_c","kind":"introduction-passage","excerpt":"Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ ","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3174dcdc9ca749bd","tex":"\\tau^j","aliases":"τ^j tau","display":"\\tau^j","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-95fabc8465dd628e","tex":"\\chi_p","aliases":"χ_p chi","display":"\\chi_p","kind":"introduction-passage","excerpt":"be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a","count":9},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":36},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"introduction-passage","excerpt":"rallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the","count":36},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e89148161d5d541d","tex":"\\prod_{p\\mid c}","aliases":"∏_pmid c prod mid","display":null,"kind":"introduction-passage","excerpt":"rallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":38},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-997bb90cd39331b0","tex":"\\sum_{i,j=0}^{M-1}","aliases":"∑_i,j=0^M-1 sum","display":"\\sum_{i,j=0}^{M-1}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0031c656c3e56dba","tex":"\\sum_{r,s}","aliases":"∑_r,s sum","display":"\\sum_{r,s}","kind":"introduction-passage","excerpt":"Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":24},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ ","count":54},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":36},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","excerpt":"Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-cu","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Exact ratio semantics. Three ratios recur in the computation: fracΦoverline H, fracΦk_*overline H, fracH(1)Φ. They are all at least one by definition. A value greater than one is a discrete overhead, not an error. Exact rational strings should accompany decimal displays when the integers are large.","count":9},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4487f11646417749","tex":"\\mathbf1_{0\\le r<e_p}","aliases":"1_0≤ r<e_p mathbf le","display":"\\mathbf1_{0\\le r<e_p}","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4b82e0794ad5cb35","tex":"\\mathfrak A","aliases":"A mathfrak","display":"\\mathfrak A","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discrimina","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4fbf9458f594ce86","tex":"A^x","aliases":"A^x","display":"A^x","kind":"use","excerpt":"Beal-type equations. A primitive counterexample to A^x+B^y=C^z, x,y,z>2, would contain three deep contact packets on disjoint prime supports. This is an especially strong setting for prime-contact filtering.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"Complete-intersection control case. Let T=Z[X]/(F) and evaluate at a simple root a_0. Then the congruence ideal and cotangent module both have order |F'(a_0)|. The defect is one. Extra multiplicity is therefore tied to non-complete-intersection structure rather than being a normalization artifact.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c9f16574b6182941","tex":"A_{ij}","aliases":"A_ij","display":"A_{ij}","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bb2614930ea303a4","tex":"a_s","aliases":"a_s","display":"a_s","kind":"use","excerpt":"One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5118a3baa8303c7d","tex":"A_\\varepsilon","aliases":"A_ε varepsilon","display":"A_\\varepsilon","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-61598b6c6bd4aad0","tex":"A_{\\varepsilon'}","aliases":"A_ε' varepsilon","display":"A_{\\varepsilon'}","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d0a82faef8552fd4","tex":"A_\\varepsilon^2","aliases":"A_ε^2 varepsilon","display":"A_\\varepsilon^2","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a3a52d54c059cb53","tex":"a_\\ell","aliases":"a_ℓ ell","display":"a_\\ell","kind":"use","excerpt":"The automorphic eigenchannel. Modularity associates a weight-two newform f_E to the rational isogeny class of E. Its Hecke coefficients a_ℓ(E) form a global frequency signal. The conductor is the automorphic bandwidth. The newform, however, remembers the rational isogeny class rather than the marked integral representative.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"Filtered contact packets. For a prime power p^eparallel n_i, define the local filtered contact packet Jcal_i,p=pZ_p/p^eZ_p. Its order is absJcal_i,p=p^e-1. Its associated graded is grJcal_i,p≅(F_p)^e-1. The distinction is essential. The filtered packet is cyclic. The associated graded has p-rank e-1. The non-split extensions between successive grad","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0230e8bde40956b0","tex":"\\Acal","aliases":"Acal","display":null,"kind":"use","excerpt":"The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth pack","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ddb20c3d29efe669","tex":"\\Acal^{\\sharp}","aliases":"Acal^sharp Acal sharp","display":null,"kind":"use","excerpt":"The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth pack","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-99fa55a48bd5ebd6","tex":"\\mathrm{act}","aliases":"act mathrm","display":"\\mathrm{act}","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e1ab77241aa5e261","tex":"\\mathrm{add}","aliases":"add mathrm","display":"\\mathrm{add}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-59a706af3e94a284","tex":"\\mathrm{alg}","aliases":"alg mathrm","display":"\\mathrm{alg}","kind":"use","excerpt":"For the output source m=c/R_c, the corrected algebraic defect against a cyclic target of order η_f is δ_alg(c,f) = fracc/R_cgcd(c/R_c,η_f). Prime by prime, δ_alg(c,f) = ∏_p^eparallel c p^max(e-1-v_p(η_f),0). The problem is missing depth, not missing rank.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discrimina","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-709c8692f9d032bb","tex":"B^d","aliases":"B^d","display":"B^d","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-268673dd5127a3f5","tex":"B^y","aliases":"B^y","display":"B^y","kind":"use","excerpt":"Beal-type equations. A primitive counterexample to A^x+B^y=C^z, x,y,z>2, would contain three deep contact packets on disjoint prime supports. This is an especially strong setting for prime-contact filtering.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c3a6aadba82617b0","tex":"B_{k+1}","aliases":"B_k+1","display":"B_{k+1}","kind":"use","excerpt":"Bernoulli--Witt direction. The Bernoulli branch begins with the identity ∑_j=1^m-1j^k = fracB_k+1(m)-B_k+1k+1. The equation with m^k imposes strong congruences on Bernoulli values. A filtered prime-contact treatment should separate:","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5daa5707e80f7e4e","tex":"\\bar v_2","aliases":"bar v_2 bar","display":"\\bar v_2","kind":"use","excerpt":"Weighted dyadic row. For an odd input prime q, define ℓ_2,e(q) = frac14 log_2((-1)^(q-1)/2q) pmod2^e-2. With the analogous truncated valuation on Z/2^e-2Z, bar v_2(ℓ_2,e(q)) = min{e-2,v_2(q^2-1)-3}. A separate sign packet supplies the missing first dyadic layer. This is the weighted dyadic branch in its final logarithmic form.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5bd492e4ea78c5fe","tex":"\\bigsqcup","aliases":"bigsqcup","display":null,"kind":"use","excerpt":"Obstruction cosets. Choose a section x_0∈ L with ω(x_0)=1. Then L = bigsqcup_k∈Z (kx_0+L_0). For fixed k, the problem is a closest-vector problem to the moving target -kx_0 in the lattice L_0. The three-sector compression makes this search much more explicit.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ccbecb81e1a49e42","tex":"\\bigsqcup_{k\\in\\Z}","aliases":"bigsqcup_k∈Z bigsqcup in Z","display":null,"kind":"use","excerpt":"Obstruction cosets. Choose a section x_0∈ L with ω(x_0)=1. Then L = bigsqcup_k∈Z (kx_0+L_0). For fixed k, the problem is a closest-vector problem to the moving target -kx_0 in the lattice L_0. The three-sector compression makes this search much more explicit.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"Simultaneous Wieferich theorems. The edge weight w_p(q)=v_p(q^p-1-1)-1 measures local degeneracy. A first family of theorems could bound the number of pairs (p,q) with w_p(q)≥ k in a declared box. A more relevant target would control simultaneous rows: #{ (P,Q): rank( L_P,Qbmod p^k) is deficient }. The coherence lattice suggests studying exterior p","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2f40d444413ea162","tex":"\\C","aliases":"C","display":null,"kind":"use","excerpt":"Let I_N=ker(A_Nxrightarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-420668a5bdd9e0bf","tex":"\\mathbf C","aliases":"C mathbf","display":"\\mathbf C","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-aaafc3af610c8063","tex":"C'_\\eps","aliases":"C'_eps eps","display":null,"kind":"use","excerpt":"[Conditional filtered-transfer criterion] Suppose a canonical map H^0(Θ_f):Z/(c/R_c)Z⟶ H^0( C_f) satisfies, for every eps>0, |ker H^0(Θ_f)|≤ C_eps R^eps and |H^0( C_f)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","excerpt":"The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discrimina","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-54a1ea9db444fe2c","tex":"c^{3/2}","aliases":"c^3/2","display":"c^{3/2}","kind":"use","excerpt":"ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be w","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-33ec106c4d586319","tex":"c^k","aliases":"c^k","display":"c^k","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c86e48078fe8fd92","tex":"c^{o(1)}","aliases":"c^o(1)","display":"c^{o(1)}","kind":"use","excerpt":"Define the shortest transverse packet Θ(a,b,c) = min{ normx_∞: x∈ T_a,b,c, W_x(a,b)≠0 }. Then c ≤ fracRlog2Θ(a,b,c)log c. A subpolynomial estimate Θ(a,b,c)=c^o(1) would imply the strong abc inequality. Later chapters explain why this target is too close to the conjecture to be reached by generic lattice geometry alone.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-093e875f46256c76","tex":"C^z","aliases":"C^z","display":"C^z","kind":"use","excerpt":"Beal-type equations. A primitive counterexample to A^x+B^y=C^z, x,y,z>2, would contain three deep contact packets on disjoint prime supports. This is an especially strong setting for prime-contact filtering.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"use","excerpt":"Coherence-character composition. Suppose the two normalized coherence characters have orders C_1 and C_2. Their product character has order dividing lcm(C_1,C_2). If the shared term forces a compatibility relation, the actual order can be strictly smaller. Conversely, the combined system can expose a higher-rank determinant even when each individua","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"The Legendre parameter of the parent is λ=t^2, t=v/u. The degree-two Landen transformation gives λ ⟼ frac4sqrtλ(1+sqrtλ)^2. This is exactly the child ratio a_1/c_1. The transformation preserves the automorphic channel up to the corresponding two-isogeny while lowering the integral height. It proves that the integral representative inside an isogeny","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"Coherence-character composition. Suppose the two normalized coherence characters have orders C_1 and C_2. Their product character has order dividing lcm(C_1,C_2). If the shared term forces a compatibility relation, the actual order can be strictly smaller. Conversely, the combined system can expose a higher-rank determinant even when each individua","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e0c8a76d0e325cac","tex":"c_4","aliases":"c_4","display":"c_4","kind":"use","excerpt":"The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discrimina","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d4229df7b957b2e3","tex":"c_E","aliases":"c_E","display":"c_E","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f616adbba955d792","tex":"C_{e,N}","aliases":"C_e,N","display":"C_{e,N}","kind":"use","excerpt":"Finite polyphase rings. Fix an integer N≥2 and work in A_N=Z[T]/(T^N-1). The residue of V_e(T) records the number of valuation shells in each congruence class modulo N. Multiplication by V_e is represented by a circulant matrix C_e,N.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9bde2ad1acf6369e","tex":"c_E^2","aliases":"c_E^2","display":"c_E^2","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ebbcc4b3a230e297","tex":"C_\\eps","aliases":"C_eps eps","display":null,"kind":"use","excerpt":"Combined logarithmic--coherence conductor. For the actual rows constructed from the input primes and output-prime logarithms, prove WC≤ C_eps R_aR_bR^eps. The theorem must control both entrywise logarithmic degeneration and collective coherence and must exploit the addition law. On the subfamily W=1, it reduces to a pure bound for C. The correspond","count":13},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e57ad15f949be943","tex":"C_f","aliases":"C_f","display":"C_f","kind":"use","excerpt":"Hecke filter · A_ε, e_f, C_f · local signs and integral eigenline separation · output contact depth by itself","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b532a9b68d20fb27","tex":"\\mathbf C_f","aliases":"C_f mathbf","display":"\\mathbf C_f","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0f386ca9ef93bd37","tex":"C_{\\mathrm{glued}}","aliases":"C_glued mathrm","display":"C_{\\mathrm{glued}}","kind":"use","excerpt":"A useful first target is: C_glued ≤ fracC_1C_2rad(C) × subpolynomial correction. The displayed formula is schematic. It expresses the hope that the shared term cancels one full support layer. No such theorem is proved.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-374d4fe4488ce15e","tex":"C_p","aliases":"C_p","display":"C_p","kind":"use","excerpt":"The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b59c9966dcf6bcf5","tex":"C_{\\mathrm{red}}","aliases":"C_red mathrm","display":"C_{\\mathrm{red}}","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c6d1d8d8724e93f3","tex":"C_U","aliases":"C_U","display":"C_U","kind":"use","excerpt":"Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-073a71f814bdfa5f","tex":"\\coker","aliases":"coker","display":null,"kind":"use","excerpt":"Exact cyclic transducer. Let C(m)=[ZxrightarrowmZ] and C(η)=[ZxrightarrowηZ]. Set g=gcd(m,η). Multiplication by m/g in degree -1 and by η/g in degree 0 defines a chain map. On cohomology, |ker|=m/g, |im|=g, |coker|=η/g. The maximal transferable packet is the common prime-power depth.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6e6d0cbeeb71722a","tex":"\\operatorname{coker}","aliases":"coker operatorname","display":"\\operatorname{coker}","kind":"use","excerpt":"[Smith obstruction theorem] Let a+b=c be primitive, with at least two of a,b,c greater than one. Then SNF( M_a,b,c) = diag(1,Q). Consequently, coker( M_a,b,c)_tors ≅ Z/QZ. The exceptional triple (1,1,2) has only one nonempty prime sector, so the matrix has rank one and is excluded from the two-row statement.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"The phase packet y=(-8,8,3) satisfies D_y(3)=-8, D_y(125)=200, D_y(128)=192, and -8+200=192. Its Wronskian is also 1600. The phase height is eight. The phase inequality is sharp: 128 = 30(frac13+frac15)8. This illustrates that the exponent-free phase channel can sharpen the bound even when its packet norm is larger.","count":11},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e63d808025fba917","tex":"\\mathrm d","aliases":"d mathrm","display":"\\mathrm d","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":15},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d0d28795ad33b969","tex":"D^{1/d}","aliases":"D^1/d","display":"D^{1/d}","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-34ed1a97bb97d001","tex":"d^{d/2}","aliases":"d^d/2","display":"d^{d/2}","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-54ff0eb6a1af0d6f","tex":"D_f","aliases":"D_f","display":"D_f","kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally rep","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d444e4eb5807d954","tex":"\\mathrm d_x","aliases":"d_x mathrm","display":"\\mathrm d_x","kind":"use","excerpt":"A scalar packet is an integer vector x=(x_p)_p∈ S∈Z^S. Pairing with x gives the arithmetic derivative d_x(n) = n∑_pmid nfracv_p(n)p x_p. The Leibniz rule survives scalarization.","count":13},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-521b9139cdb95729","tex":"D_y","aliases":"D_y","display":"D_y","kind":"use","excerpt":"The phase packet y=(-8,8,3) satisfies D_y(3)=-8, D_y(125)=200, D_y(128)=192, and -8+200=192. Its Wronskian is also 1600. The phase height is eight. The phase inequality is sharp: 128 = 30(frac13+frac15)8. This illustrates that the exponent-free phase channel can sharpen the bound even when its packet norm is larger.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"A determinant viewpoint. When several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let L be a normalized cross-prime logarithm matrix. A nonzero r× r minor det L_I,J measures the failure of r input packets to cancel simultaneously across r output channels.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"Obstruction complex. The joint exponent-free phase--Wronskian matrix has Smith form diag(1,1600). The additive lattice maps onto 1600Z through the Wronskian. The primitive obstruction section exists algebraically. Its height remains the metric problem.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8f42a8e27ac6c866","tex":"\\operatorname{div}","aliases":"div operatorname","display":"\\operatorname{div}","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0645873febb68b5c","tex":"\\Downarrow","aliases":"Downarrow","display":null,"kind":"use","excerpt":"· primitive sparse relation · Downarrow · prime-packet differential and Wronskian · Downarrow · valuation-word and shell refinement · Downarrow · joint obstruction complex and exact search · Downarrow · prime-shell signal and Frey transform · Downarrow · divisor-polyphase persistence and Hecke filters · Downarrow · filtered cusp-contact and Witt pa","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Consider A+B=C and C+D=E. The term C is an output in the first relation and an input in the second. Its contact packet should not be duplicated independently. A functorial theory should identify the two copies and record how depth moves across the diagram.","count":25},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ef05a20d6dca20f4","tex":"e^{2\\pi i g\\cdot x/C}","aliases":"e^2π i g· x/C pi cdot","display":"e^{2\\pi i g\\cdot x/C}","kind":"use","excerpt":"The resulting row g=(g_q)_qmid ab is primitive modulo C because every local row is primitive. Its character χ_coh(x) = e^2π i g· x/C has exact order C. The coherence lattice is its kernel.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-79542e3d5fe27d48","tex":"\\mathcal E_A","aliases":"E_A mathcal","display":"\\mathcal E_A","kind":"use","excerpt":"E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8618d23a0187d108","tex":"E_{a,b,c}","aliases":"E_a,b,c","display":"E_{a,b,c}","kind":"use","excerpt":"The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discrimina","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-caf1c79da7ed3a79","tex":"\\mathcal E_C","aliases":"E_C mathcal","display":"\\mathcal E_C","kind":"use","excerpt":"E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-420aa7a6632049fa","tex":"e_f","aliases":"e_f","display":"e_f","kind":"use","excerpt":"The same-sign separation problem. After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let Λ be an integral Hecke lattice and let e_f be the rational projector onto the Frey eigenline. Define η_f = min{m≥1:m e_f(Λ)⊆Λ}. This is the integral eigenchannel separation denominator. It is an arithmetic analogue of ","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e9f62370ec92650a","tex":"\\mathcal E_K","aliases":"E_K mathcal","display":"\\mathcal E_K","kind":"use","excerpt":"E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1b09d4efdcdc1f39","tex":"e_p","aliases":"e_p","display":"e_p","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":13},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-504dc54aa7ab931b","tex":"e_U","aliases":"e_U","display":"e_U","kind":"use","excerpt":"Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9eb0405d53adf7cc","tex":"\\mathcal E_W","aliases":"E_W mathcal","display":"\\mathcal E_W","kind":"use","excerpt":"E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps","display":null,"kind":"use","excerpt":"[leftmargin=1.5em] item Construct a finite source of order comparable to c/R_c. item Map it into a global target with kernel at most R^eps. item Bound the target size by R_aR_bR^eps. item Multiply by R_c and absorb logarithmic factors.","count":29},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F","display":null,"kind":"use","excerpt":"Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-si","count":13},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted prod","count":23},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"Complete-intersection control case. Let T=Z[X]/(F) and evaluate at a simple root a_0. Then the congruence ideal and cotangent module both have order |F'(a_0)|. The defect is one. Extra multiplicity is therefore tied to non-complete-intersection structure rather than being a normalization artifact.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-101fbe8a1f9aa65c","tex":"\\F_2","aliases":"F_2 F","display":null,"kind":"use","excerpt":"Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-si","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b0133f9470b1d08c","tex":"F_{a,b,c}","aliases":"F_a,b,c","display":"F_{a,b,c}","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-dafda06c8efa3652","tex":"f_E","aliases":"f_E","display":"f_E","kind":"use","excerpt":"The automorphic eigenchannel. Modularity associates a weight-two newform f_E to the rational isogeny class of E. Its Hecke coefficients a_ℓ(E) form a global frequency signal. The conductor is the automorphic bandwidth. The newform, however, remembers the rational isogeny class rather than the marked integral representative.","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fda40c00a5457366","tex":"\\F_p","aliases":"F_p F","display":null,"kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.","count":11},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral fo","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounde","count":30},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f7af1d0b6531c887","tex":"\\mathcal G_{\\mathbf n}","aliases":"G_ n mathcal mathbf","display":"\\mathcal G_{\\mathbf n}","kind":"use","excerpt":"The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a74535153281e2c4","tex":"g_a","aliases":"g_a","display":"g_a","kind":"use","excerpt":"Three-sector compression. Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d230c9e7fd4e8bb2","tex":"g_b","aliases":"g_b","display":"g_b","kind":"use","excerpt":"Three-sector compression. Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4000f6722b507ccb","tex":"g_C","aliases":"g_C","display":"g_C","kind":"use","excerpt":"Thus Λ_C = {x:g_C· x≡0pmod C} has index C and contains the same fixed vector v. This proves that no bound on C can follow from the length or support of v alone.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a4676088e867baee","tex":"g_c","aliases":"g_c","display":"g_c","kind":"use","excerpt":"Three-sector compression. Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-09f3e72028e864cd","tex":"G_m","aliases":"G_m","display":"G_m","kind":"use","excerpt":"Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral fo","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f046713e329fd428","tex":"g_p","aliases":"g_p","display":"g_p","kind":"use","excerpt":"The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a2b4d1f7c8f514ef","tex":"g_{p,q}","aliases":"g_p,q","display":"g_{p,q}","kind":"use","excerpt":"CRT synthesis of the global row. The local residual moduli C_p are pairwise coprime. Let C=∏_pmid cC_p. For each input coordinate q, the Chinese remainder theorem gives a unique residue g_qpmod C whose reduction modulo C_p is the local coefficient g_p,q.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7eadc3987326373a","tex":"g_q","aliases":"g_q","display":"g_q","kind":"use","excerpt":"CRT synthesis of the global row. The local residual moduli C_p are pairwise coprime. Let C=∏_pmid cC_p. For each input coordinate q, the Chinese remainder theorem gives a unique residue g_qpmod C whose reduction modulo C_p is the local coefficient g_p,q.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f66bcd9a4f79a720","tex":"g_{\\mathrm{val}}","aliases":"g_val mathrm","display":"g_{\\mathrm{val}}","kind":"use","excerpt":"Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then ","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.","count":27},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f693946f87e25002","tex":"\\gcd_{p\\mid u}","aliases":"gcd_pmid u gcd mid","display":null,"kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c5d62edc1cdf57e7","tex":"\\mathrm{GL}","aliases":"GL mathrm","display":"\\mathrm{GL}","kind":"use","excerpt":"One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4fe38f804991b29b","tex":"\\mathrm{GL}_s","aliases":"GL_s mathrm","display":"\\mathrm{GL}_s","kind":"use","excerpt":"One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-dbaf3bbf176dd3a0","tex":"\\mathrm{glob}","aliases":"glob mathrm","display":"\\mathrm{glob}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5860ebdad5226973","tex":"\\mathrm{glued}","aliases":"glued mathrm","display":"\\mathrm{glued}","kind":"use","excerpt":"A useful first target is: C_glued ≤ fracC_1C_2rad(C) × subpolynomial correction. The displayed formula is schematic. It expresses the hope that the shared term cancels one full support layer. No such theorem is proved.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-70196007a27c98a9","tex":"\\gr","aliases":"gr","display":null,"kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4897511f6c74a73e","tex":"\\mathbf H","aliases":"H mathbf","display":"\\mathbf H","kind":"use","excerpt":"Open theorem C: filtered local-global transfer. The filtered contact construction proposes a map J_c ⟶ H_f from the cusp-contact complex into a Hecke, Selmer, or deformation complex.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) satisfying |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows after absorbing the logarithmic Kummer kern","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fa5a6573c905a228","tex":"H^0","aliases":"H^0","display":"H^0","kind":"use","excerpt":"[Filtered contact quotient] H^0( J_c) = R_cZ/cZ ≅ Z/(c/R_c)Z.","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4f5ceb3e1772c369","tex":"\\mathcal H_3","aliases":"H_3 mathcal","display":"\\mathcal H_3","kind":"use","excerpt":"Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) satisfying |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows after absorbing the logarithmic Kummer kern","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f95ce74b43c969c5","tex":"\\mathbf H_f","aliases":"H_f mathbf","display":"\\mathbf H_f","kind":"use","excerpt":"Open theorem C: filtered local-global transfer. The filtered contact construction proposes a map J_c ⟶ H_f from the cusp-contact complex into a Hecke, Selmer, or deformation complex.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-eff6aa27c552c5d2","tex":"I_a","aliases":"I_a","display":"I_a","kind":"use","excerpt":"[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2344f57325d4c16a","tex":"I_b","aliases":"I_b","display":"I_b","kind":"use","excerpt":"[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f9648a100dad73ed","tex":"I_c","aliases":"I_c","display":"I_c","kind":"use","excerpt":"[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":10},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"use","excerpt":"Why exact synthesis prevents free coordinate splitting. Let X and Y be normed spaces, and suppose J:X⟶ Y, S:Y⟶ X satisfy SJ=I_X. Then normx = normSJx ≤ normSnormJx. Therefore normJx ≥ fracnormxnormS. An analysis map may make coordinates smaller only by increasing the synthesis norm. This elementary inequality is the common source of several no-free","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im","display":null,"kind":"use","excerpt":"Exact cyclic transducer. Let C(m)=[ZxrightarrowmZ] and C(η)=[ZxrightarrowηZ]. Set g=gcd(m,η). Multiplication by m/g in degree -1 and by η/g in degree 0 defines a chain map. On cohomology, |ker|=m/g, |im|=g, |coker|=η/g. The maximal transferable packet is the common prime-power depth.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ef28ee4048cca398","tex":"\\mathrm{in}","aliases":"in mathrm","display":"\\mathrm{in}","kind":"use","excerpt":"Contact pushout. Let Jcal_C^out be the contact packet of C in the first equation and Jcal_C^in its packet in the second. Both are canonically rad(C)Z/CZ. The first gluing operation identifies them through the identity map. The total contact object is the pushout of the two relation diagrams along this common packet.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5c2455f4c0bb929a","tex":"\\inf_{k\\ge1}","aliases":"inf_k≥1 inf ge","display":"\\inf_{k\\ge1}","kind":"use","excerpt":"For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"A determinant viewpoint. When several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let L be a normalized cross-prime logarithm matrix. A nonzero r× r minor det L_I,J measures the failure of r input packets to cancel simultaneously across r output channels.","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4978cb2ce63e316b","tex":"\\mathbf J","aliases":"J mathbf","display":"\\mathbf J","kind":"use","excerpt":"section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2fe076b744e160c0","tex":"j^k","aliases":"j^k","display":"j^k","kind":"use","excerpt":"Bernoulli--Witt direction. The Bernoulli branch begins with the identity ∑_j=1^m-1j^k = fracB_k+1(m)-B_k+1k+1. The equation with m^k imposes strong congruences on Bernoulli values. A filtered prime-contact treatment should separate:","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f63cab9c738188a7","tex":"J_a","aliases":"J_a","display":"J_a","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-88fbeb34dbc66890","tex":"J_b","aliases":"J_b","display":"J_b","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-365e4afde3f07c22","tex":"J_c","aliases":"J_c","display":"J_c","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-528b7456d232f332","tex":"\\mathbf J_c","aliases":"J_c mathbf","display":"\\mathbf J_c","kind":"use","excerpt":"section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a620020ba505249b","tex":"J_n","aliases":"J_n","display":"J_n","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-040847a1e4e56477","tex":"\\Jcal_C^{\\mathrm{in}}","aliases":"Jcal_C^in Jcal mathrm","display":null,"kind":"use","excerpt":"Contact pushout. Let Jcal_C^out be the contact packet of C in the first equation and Jcal_C^in its packet in the second. Both are canonically rad(C)Z/CZ. The first gluing operation identifies them through the identity map. The total contact object is the pushout of the two relation diagrams along this common packet.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-be5311b40fcf43b9","tex":"\\Jcal_C^{\\mathrm{out}}","aliases":"Jcal_C^out Jcal mathrm","display":null,"kind":"use","excerpt":"Contact pushout. Let Jcal_C^out be the contact packet of C in the first equation and Jcal_C^in its packet in the second. Both are canonically rad(C)Z/CZ. The first gluing operation identifies them through the identity map. The total contact object is the pushout of the two relation diagrams along this common packet.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7a4f2561e4afa4fc","tex":"\\Jcal_{i,p}","aliases":"Jcal_i,p Jcal","display":null,"kind":"use","excerpt":"Filtered contact packets. For a prime power p^eparallel n_i, define the local filtered contact packet Jcal_i,p=pZ_p/p^eZ_p. Its order is absJcal_i,p=p^e-1. Its associated graded is grJcal_i,p≅(F_p)^e-1. The distinction is essential. The filtered packet is cyclic. The associated graded has p-rank e-1. The non-split extensions between successive grad","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-36071631df24a0f3","tex":"\\Jcal_{n_i}","aliases":"Jcal_n_i Jcal","display":null,"kind":"use","excerpt":"Globally, the contact packet of one term is Jcal_n_i = R_iZ/n_iZ ≅ Z/M_iZ.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3e498617a2adef3c","tex":"\\Jcal_{p,e}","aliases":"Jcal_p,e Jcal","display":null,"kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cd49a1788f8baddd","tex":"k_*","aliases":"k_*","display":"k_*","kind":"use","excerpt":"Exact ratio semantics. Three ratios recur in the computation: fracΦoverline H, fracΦk_*overline H, fracH(1)Φ. They are all at least one by definition. A value greater than one is a discrete overhead, not an error. Exact rational strings should accompany decimal displays when the integers are large.","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-544807fb1bbf53c0","tex":"K_{\\max}","aliases":"K_max max","display":"K_{\\max}","kind":"use","excerpt":"[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build ce","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b39e465f5dcb17d2","tex":"K_p","aliases":"K_p","display":"K_p","kind":"use","excerpt":"Start with the signed input exponent vector ν=(ν_q)_qmid ab, ν_q=v_q(b)-v_q(a). Let K_p be the p-primary part of the common Kummer factor. Divide the exponent vector and the local modulus by this common power. The normalized vector is primitive relative to the remaining modulus.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a77d87feaab21d87","tex":"K_\\beta","aliases":"K_β beta","display":"K_\\beta","kind":"use","excerpt":"Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) satisfying |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows after absorbing the logarithmic Kummer kern","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e08d6b60be0ec0b0","tex":"\\Ker","aliases":"Ker","display":null,"kind":"use","excerpt":"[First-order no-free-polyphase principle] Let S:(T_p,r)_p,r⟼ (∑_rT_p,r)_p be shell synthesis. The first-order additive map and first-order Wronskian factor through S. Hence Ker S is contained in the kernels of both first-order maps.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.","count":10},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-08093f17d5992ea3","tex":"\\mathbf L","aliases":"L mathbf","display":"\\mathbf L","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"Simultaneous Wieferich theorems. The edge weight w_p(q)=v_p(q^p-1-1)-1 measures local degeneracy. A first family of theorems could bound the number of pairs (p,q) with w_p(q)≥ k in a declared box. A more relevant target would control simultaneous rows: #{ (P,Q): rank( L_P,Qbmod p^k) is deficient }. The coherence lattice suggests studying exterior p","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-41311a6b38f229d0","tex":"\\mathcal L_{\\mathbf n}","aliases":"L_ n mathcal mathbf","display":"\\mathcal L_{\\mathbf n}","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1baf4f3c36fac5d7","tex":"\\mathbf L_{\\mathbb T_f}","aliases":"L_ T_f mathbf mathbb","display":"\\mathbf L_{\\mathbb T_f}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-aacfb5e5cd28e2b2","tex":"L_1","aliases":"L_1","display":"L_1","kind":"use","excerpt":"Logarithmic matrix composition. Let L_1 and L_2 be the cross-prime logarithmic matrices of the two relations. The shared term determines a common set of prime channels. The combined system has block form L_1 · 0","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b4a121d33c2f6cd4","tex":"L_2","aliases":"L_2","display":"L_2","kind":"use","excerpt":"Logarithmic matrix composition. Let L_1 and L_2 be the cross-prime logarithmic matrices of the two relations. The shared term determines a common set of prime channels. The combined system has block form L_1 · 0","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-07b75e59ba673a0e","tex":"L_{I,J}","aliases":"L_I,J","display":"L_{I,J}","kind":"use","excerpt":"A determinant viewpoint. When several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let L be a normalized cross-prime logarithm matrix. A nonzero r× r minor det L_I,J measures the failure of r input packets to cancel simultaneously across r output channels.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9a6fdfda10627dfd","tex":"L_{p,q}","aliases":"L_p,q","display":"L_{p,q}","kind":"use","excerpt":"Cross-prime logarithmic filters. Let Q be a set of input primes and P a set of output primes. At odd p∈ P, define L_p,q = frac1plog_p⟨ q⟩_p with the appropriate finite-depth truncation. At p=2, use the weighted principal-unit coordinate together with a separate sign bit.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5ff8e6b9da4f5e48","tex":"\\mathcal L_{P,Q}","aliases":"L_P,Q mathcal","display":"\\mathcal L_{P,Q}","kind":"use","excerpt":"Simultaneous Wieferich theorems. The edge weight w_p(q)=v_p(q^p-1-1)-1 measures local degeneracy. A first family of theorems could bound the number of pairs (p,q) with w_p(q)≥ k in a declared box. A more relevant target would control simultaneous rows: #{ (P,Q): rank( L_P,Qbmod p^k) is deficient }. The coherence lattice suggests studying exterior p","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c1f2591d69ff8561","tex":"\\mathbf L_p^{\\mathrm{st}}","aliases":"L_p^st mathbf mathrm","display":"\\mathbf L_p^{\\mathrm{st}}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a4d5ae1ee956514f","tex":"\\mathbf L_{R^{\\mathrm{glob}}}","aliases":"L_R^glob mathbf mathrm","display":"\\mathbf L_{R^{\\mathrm{glob}}}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d45687e755f7a395","tex":"\\Lcal_{\\mathbf n}","aliases":"Lcal_ n Lcal mathbf","display":null,"kind":"use","excerpt":"The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"For a scalar split t=y_1+⋯+y_e, the child sup norm can fall to lceilfrac|t|erceil. The synthesis map has norm e, so elceilfrac|t|erceil≥|t|. This explains why raw shell splitting cannot lower the intrinsic packet height.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3eda83bfb4bc77ed","tex":"\\lcm","aliases":"lcm","display":"\\lcm","kind":"use","excerpt":"Coherence-character composition. Suppose the two normalized coherence characters have orders C_1 and C_2. Their product character has order dividing lcm(C_1,C_2). If the shared term forces a compatibility relation, the actual order can be strictly smaller. Conversely, the combined system can expose a higher-rank determinant even when each individua","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3860d7e8f88507bb","tex":"\\operatorname{lcm}","aliases":"lcm operatorname","display":"\\operatorname{lcm}","kind":"use","excerpt":"The class [c]^k is trivial exactly when every prime valuation ke_i is divisible by M. The least positive such k is lcm_i fracMgcd(M,e_i) = fracMgcd(M,e_1,…,e_r). This is the order of the image. The kernel size follows.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4b914935d1b77c2b","tex":"\\operatorname{lcm}_i","aliases":"lcm_i operatorname","display":"\\operatorname{lcm}_i","kind":"use","excerpt":"The class [c]^k is trivial exactly when every prime valuation ke_i is divisible by M. The least positive such k is lcm_i fracMgcd(M,e_i) = fracMgcd(M,e_1,…,e_r). This is the order of the image. The kernel size follows.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build ce","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cb7455ea65e17be9","tex":"\\operatorname{Li}","aliases":"Li operatorname","display":"\\operatorname{Li}","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-dd11031bedecd830","tex":"\\operatorname{Li}_2","aliases":"Li_2 operatorname","display":"\\operatorname{Li}_2","kind":"use","excerpt":"Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β)","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-69d6f6e88da8767d","tex":"\\operatorname{Li}_3","aliases":"Li_3 operatorname","display":"\\operatorname{Li}_3","kind":"use","excerpt":"Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β) · = ζ(3) +frac16log^3z +fracπ^26log z -frac12log^2zlog(1-z). The balanced Kummer coordinate is therefore exactly the third a","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d138b1d18c96e445","tex":"\\operatorname{Li}_n","aliases":"Li_n operatorname","display":"\\operatorname{Li}_n","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bbf142bde13bd585","tex":"\\log^2","aliases":"log^2 log","display":"\\log^2","kind":"use","excerpt":"Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β) · = ζ(3) +frac16log^3z +fracπ^26log z -frac12log^2zlog(1-z). The balanced Kummer coordinate is therefore exactly the third a","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-34a700d5345e5b51","tex":"\\log^3","aliases":"log^3 log","display":"\\log^3","kind":"use","excerpt":"Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β) · = ζ(3) +frac16log^3z +fracπ^26log z -frac12log^2zlog(1-z). The balanced Kummer coordinate is therefore exactly the third a","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-53f303c8b1bf6b4c","tex":"\\log_p","aliases":"log_p log","display":"\\log_p","kind":"use","excerpt":"Local logarithmic layer. For odd p, the identity v_p(log_p(1+u))=v_p(u) holds when v_p(u)>0. This yields the exact output notch depth. The coefficient formula v_p(q^p-1-1)-1 comes from the principal-unit logarithm of the input prime.","count":9},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4a7e9f6a85466609","tex":"\\longleftrightarrow","aliases":"longleftrightarrow","display":null,"kind":"use","excerpt":"The Frey curve as a nonlinear signal transform. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). Its discriminant is Δ_E=16a^2b^2c^2. At odd primes dividing abc, the conductor records the activated channel while the minimal discriminant records the local depth. Thus the Frey transform converts the prime-shell split into a standard automorphic ","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-65b951b5219e3cf9","tex":"\\mathbf M","aliases":"M mathbf","display":"\\mathbf M","kind":"use","excerpt":"The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cc8b5dc0bd10c348","tex":"\\mathbf m","aliases":"m mathbf","display":"\\mathbf m","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-af9070f1457f1310","tex":"\\mathsf M","aliases":"M mathsf","display":"\\mathsf M","kind":"use","excerpt":"0, · pmid c. Consider M_a,b,c = Λ w : Z^S⟶Z^2. The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set Q=fracabcR.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"Pillai and Hall. For x^m-y^n=k, the residual support is fixed by k. The two perfect-power packets must cancel to high precision while producing a small fixed output. This is a natural two-sector coherence problem.","count":44},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d9fb956038d4d543","tex":"M^{1-n}","aliases":"M^1-n","display":"M^{1-n}","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-051c4a9c687cb954","tex":"M^3","aliases":"M^3","display":"M^3","kind":"use","excerpt":"The inverse synthesis gain at weight n is M^n-1. Through weights one, two, and three, the total gain is 1· M· M^2=M^3. The analytic branch count becomes fixed, but the integral transition matrix retains a large determinant.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-890e93043dfa23b3","tex":"m^k","aliases":"m^k","display":"m^k","kind":"use","excerpt":"Bernoulli--Witt and ErdHos--Moser. Faulhaber's formula rewrites 1^k+⋯+(m-1)^k=m^k as a Bernoulli-polynomial identity. The denominator primes of Bernoulli numbers, the finite-difference divisibility, and the separate 2-adic and odd-adic lift structures suggest a different contact object: Bernoulli contact crystal + denominator prime packet + finite-","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d21cae7cac832b3b","tex":"M^{n-1}","aliases":"M^n-1","display":"M^{n-1}","kind":"use","excerpt":"The inverse synthesis gain at weight n is M^n-1. Through weights one, two, and three, the total gain is 1· M· M^2=M^3. The analytic branch count becomes fixed, but the integral transition matrix retains a large determinant.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":"Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then ","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-54f0b300c9515311","tex":"m_1","aliases":"m_1","display":"m_1","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9f04231559aedbad","tex":"\\mathsf M_{a,b,c}","aliases":"M_a,b,c mathsf","display":"\\mathsf M_{a,b,c}","kind":"use","excerpt":"0, · pmid c. Consider M_a,b,c = Λ w : Z^S⟶Z^2. The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set Q=fracabcR.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-207e98afc142d0de","tex":"M_d","aliases":"M_d","display":"M_d","kind":"use","excerpt":"Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2d4e30878516d828","tex":"\\mathcal M_{\\mathrm{glob}}","aliases":"M_glob mathcal mathrm","display":"\\mathcal M_{\\mathrm{glob}}","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-df92634da3a3e068","tex":"M_i","aliases":"M_i","display":"M_i","kind":"use","excerpt":"Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-19afb20993a1d995","tex":"m_r","aliases":"m_r","display":"m_r","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build ce","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4bc014492ab92312","tex":"\\min_i","aliases":"min_i min","display":"\\min_i","kind":"use","excerpt":"Because g_c≤min_i e_i≤log_2 c, we obtain the universal near-injectivity estimate |kerκ_c|≤log_2c. This closes the global-occupancy problem for the contact packet up to logarithmic loss.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-54507a99799b42ce","tex":"\\min_{k\\neq0}","aliases":"min_k≠0 min neq","display":"\\min_{k\\neq0}","kind":"use","excerpt":"For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":34},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2aa34ef1611fca9c","tex":"\\mathbf n","aliases":"n mathbf","display":"\\mathbf n","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a87bbd35b7c2dadd","tex":"N_E","aliases":"N_E","display":"N_E","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"A general contact functor. The construction extends to a sparse relation ∑_j=1^rϵ_jn_j=0. Choose one or more output terms whose repeated depth is to be explained. The assignment ( n,ϵ) ⟼ A( n,ϵ) is canonical under permutations, sign reversal, primitive reduction, and primary decomposition. The optional geometric lift depends on the equation, but th","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-eb171c4fc1202d77","tex":"n_r","aliases":"n_r","display":"n_r","kind":"use","excerpt":"Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"Why exact synthesis prevents free coordinate splitting. Let X and Y be normed spaces, and suppose J:X⟶ Y, S:Y⟶ X satisfy SJ=I_X. Then normx = normSJx ≤ normSnormJx. Therefore normJx ≥ fracnormxnormS. An analysis map may make coordinates smaller only by increasing the synthesis norm. This elementary inequality is the common source of several no-free","count":23},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"Define the shortest transverse packet Θ(a,b,c) = min{ normx_∞: x∈ T_a,b,c, W_x(a,b)≠0 }. Then c ≤ fracRlog2Θ(a,b,c)log c. A subpolynomial estimate Θ(a,b,c)=c^o(1) would imply the strong abc inequality. Later chapters explain why this target is too close to the conjecture to be reached by generic lattice geometry alone.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3c84a1672e483df5","tex":"\\mathrm{obs}","aliases":"obs mathrm","display":"\\mathrm{obs}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ab3321d6961879fc","tex":"\\ord","aliases":"ord","display":null,"kind":"use","excerpt":"[Kummer order from the valuation vector] Assume x>0. Then ord([x]) = fracMgcd(M,{ν_q}_q∈ S). For a negative rational number and even M, one additional dyadic sign layer may occur.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-44a55b48f8dea448","tex":"\\mathrm{out}","aliases":"out mathrm","display":"\\mathrm{out}","kind":"use","excerpt":"Contact pushout. Let Jcal_C^out be the contact packet of C in the first equation and Jcal_C^in its packet in the second. Both are canonically rad(C)Z/CZ. The first gluing operation identifies them through the identity map. The total contact object is the pushout of the two relation diagrams along this common packet.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7c40cce8c99d5a97","tex":"\\overline{\\omega_E}","aliases":"overlineω_E overline omega","display":"\\overline{\\omega_E}","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ef958519c2ac9923","tex":"\\mathbb P","aliases":"P mathbb","display":"\\mathbb P","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split ","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c18367064df5a710","tex":"\\mathfrak p","aliases":"p mathfrak","display":"\\mathfrak p","kind":"use","excerpt":"persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retai","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fa15e5879160a296","tex":"\\mathbb P^1_\\Z","aliases":"P^1_Z mathbb Z","display":null,"kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a9301b691d286a0d","tex":"\\mathfrak p^2","aliases":"p^2 mathfrak","display":"\\mathfrak p^2","kind":"use","excerpt":"persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retai","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4712dba3d0909d07","tex":"p^{e-1}","aliases":"p^e-1","display":"p^{e-1}","kind":"use","excerpt":"Sequential normalization. Let p^eparallel c. The local output contact packet has order p^e-1. The final factorization separates this depth into distinct mechanisms. The separation must be sequential because the same prime-power depth can otherwise be counted more than once.","count":15},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3157e31c12869d3e","tex":"p^{e_p}","aliases":"p^e_p","display":"p^{e_p}","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-44ddf871c8e79a19","tex":"p^j","aliases":"p^j","display":"p^j","kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0069d6e7d8439e52","tex":"p^{j-1}","aliases":"p^j-1","display":"p^{j-1}","kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b7f360bb2799a8d5","tex":"p^k","aliases":"p^k","display":"p^k","kind":"use","excerpt":"Why the Kummer layer comes first. Suppose every input exponent is divisible by p^k. Then the balanced rational packet is already a p^kth power in the valuation lattice. The corresponding output contact depth should not also be attributed to a local logarithm defect.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3d4811701cdaca93","tex":"p^{\\max(e-1-v_p(\\eta_f),0)}","aliases":"p^max(e-1-v_p(η_f),0) max eta","display":"p^{\\max(e-1-v_p(\\eta_f),0)}","kind":"use","excerpt":"For the output source m=c/R_c, the corrected algebraic defect against a cyclic target of order η_f is δ_alg(c,f) = fracc/R_cgcd(c/R_c,η_f). Prime by prime, δ_alg(c,f) = ∏_p^eparallel c p^max(e-1-v_p(η_f),0). The problem is missing depth, not missing rank.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-460ce5afc42aa103","tex":"p^{\\max(e-2,0)}","aliases":"p^max(e-2,0) max","display":"p^{\\max(e-2,0)}","kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split ","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d3cdc9ecc001d6c4","tex":"p^n","aliases":"p^n","display":"p^n","kind":"use","excerpt":"bibitemAKN B. Antieau, A. Krause, and T. Nikolaus, ``On the K-theory of Z/p^n,'' arXiv:2405.04329, 2024.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6460407da427a490","tex":"P_{\\mathrm{act}}","aliases":"P_act mathrm","display":"P_{\\mathrm{act}}","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-48ff2d49e6a3d8a2","tex":"P_{\\mathrm{add}}","aliases":"P_add mathrm","display":"P_{\\mathrm{add}}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3270e71bf1ae54c5","tex":"P_c","aliases":"P_c","display":"P_c","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":10},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9a7264055f2d5e73","tex":"\\mathcal P_c","aliases":"P_c mathcal","display":"\\mathcal P_c","kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split ","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":11},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e4184e130d662de9","tex":"p_i^{e_i}","aliases":"p_i^e_i","display":"p_i^{e_i}","kind":"use","excerpt":"The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1a251b340cfd2fcc","tex":"p_i^{e_i-1}","aliases":"p_i^e_i-1","display":"p_i^{e_i-1}","kind":"use","excerpt":"The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-eb7979ea66ee31a5","tex":"p_m","aliases":"p_m","display":"p_m","kind":"use","excerpt":"Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-928c8c4569a33b2b","tex":"P_{\\mathrm{obs}}","aliases":"P_obs mathrm","display":"P_{\\mathrm{obs}}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7cd951ba120bfdb9","tex":"P_u","aliases":"P_u","display":"P_u","kind":"use","excerpt":"[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build ce","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-99acd602923b2989","tex":"\\mathrm{Pet}","aliases":"Pet mathrm","display":"\\mathrm{Pet}","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-35d9f25582f07702","tex":"\\mathrm{ph}","aliases":"ph mathrm","display":"\\mathrm{ph}","kind":"use","excerpt":"Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounde","count":19},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3f115f6cb4125cf9","tex":"\\Q","aliases":"Q","display":null,"kind":"use","excerpt":"Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"bibitemBCDT C. Breuil, B. Conrad, F. Diamond, and R. Taylor, ``On the modularity of elliptic curves over Q: wild 3-adic exercises,'' Journal of the American Mathematical Society 14 (2001), 843--939. https://doi.org/10.1090/S0894-0347-01-00370-8doi:10.1090/S0894-0347-01-00370-8.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e80a32868f710c6e","tex":"q^{e-1}","aliases":"q^e-1","display":"q^{e-1}","kind":"use","excerpt":"ce is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-q factor q^","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1dab4320f437abad","tex":"q^{p-1}","aliases":"q^p-1","display":"q^{p-1}","kind":"use","excerpt":"Local logarithmic layer. For odd p, the identity v_p(log_p(1+u))=v_p(u) holds when v_p(u)>0. This yields the exact output notch depth. The coefficient formula v_p(q^p-1-1)-1 comes from the principal-unit logarithm of the input prime.","count":9},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f7ec10b45d9b6e51","tex":"\\Q^\\times","aliases":"Q^× Q times","display":null,"kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-22fd5715c5058ee4","tex":"q^{\\nu_q}","aliases":"q^ν_q nu","display":"q^{\\nu_q}","kind":"use","excerpt":"Exact Kummer kernel for a rational packet. Let x=±∏_q∈ Sq^ν_q and consider its class modulo Mth powers. For positive x, the order is fracMgcd(M,{ν_q}). If x<0 and M is even, the sign contributes a separate element of order two. The exact order is therefore the least common multiple of the valuation order and the sign order.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-22f1dad501a06ad3","tex":"q_E","aliases":"q_E","display":"q_E","kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally rep","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-920b7535e6d3effa","tex":"q_p","aliases":"q_p","display":"q_p","kind":"use","excerpt":"The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth pack","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b31afc68647d37e3","tex":"\\mathbf R","aliases":"R mathbf","display":"\\mathbf R","kind":"use","excerpt":"The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-32fe36360874826c","tex":"\\mathfrak R","aliases":"R mathfrak","display":"\\mathfrak R","kind":"use","excerpt":"Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fb58157f40a3c154","tex":"R^\\eps","aliases":"R^eps eps","display":null,"kind":"use","excerpt":"[leftmargin=1.5em] item Construct a finite source of order comparable to c/R_c. item Map it into a global target with kernel at most R^eps. item Bound the target size by R_aR_bR^eps. item Multiply by R_c and absorb logarithmic factors.","count":22},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2eee3ce95fb6aa1d","tex":"R^{\\mathrm{glob}}","aliases":"R^glob mathrm","display":"R^{\\mathrm{glob}}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-33462a02fcce1a84","tex":"R_a","aliases":"R_a","display":"R_a","kind":"use","excerpt":"[leftmargin=1.5em] item Construct a finite source of order comparable to c/R_c. item Map it into a global target with kernel at most R^eps. item Bound the target size by R_aR_bR^eps. item Multiply by R_c and absorb logarithmic factors.","count":18},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fb14b1a407e1de8e","tex":"r_a","aliases":"r_a","display":"r_a","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-559e0bf6a6b7bef3","tex":"\\mathfrak R_{a,b,c}","aliases":"R_a,b,c mathfrak","display":"\\mathfrak R_{a,b,c}","kind":"use","excerpt":"Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-baf9ce2c27891e4e","tex":"R_b","aliases":"R_b","display":"R_b","kind":"use","excerpt":"[leftmargin=1.5em] item Construct a finite source of order comparable to c/R_c. item Map it into a global target with kernel at most R^eps. item Bound the target size by R_aR_bR^eps. item Multiply by R_c and absorb logarithmic factors.","count":18},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4bdf782a5897d64d","tex":"r_b","aliases":"r_b","display":"r_b","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-567e1106095a8a61","tex":"r_c","aliases":"r_c","display":"r_c","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"use","excerpt":"Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-82e3ceef747ab952","tex":"R_{\\mathrm{in}}","aliases":"R_in mathrm","display":"R_{\\mathrm{in}}","kind":"use","excerpt":"A model target is WC ≤ C_eps R_inR^eps, where R_in is the activated input bandwidth. For abc, R_in=R_aR_b.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7a780768179287f0","tex":"r_u","aliases":"r_u","display":"r_u","kind":"use","excerpt":"[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build ce","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e02867c400d9f5e2","tex":"\\mathcal R_\\infty","aliases":"R_∞ mathcal infty","display":"\\mathcal R_\\infty","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"Simultaneous Wieferich theorems. The edge weight w_p(q)=v_p(q^p-1-1)-1 measures local degeneracy. A first family of theorems could bound the number of pairs (p,q) with w_p(q)≥ k in a declared box. A more relevant target would control simultaneous rows: #{ (P,Q): rank( L_P,Qbmod p^k) is deficient }. The coherence lattice suggests studying exterior p","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"For a scalar split t=y_1+⋯+y_e, the child sup norm can fall to lceilfrac|t|erceil. The synthesis map has norm e, so elceilfrac|t|erceil≥|t|. This explains why raw shell splitting cannot lower the intrinsic packet height.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7fdf44047d9358ee","tex":"\\mathrm{red}","aliases":"red mathrm","display":"\\mathrm{red}","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build ce","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"Arithmetic endpoint. The arithmetic endpoint is the coherence character χ_coh: Z^S_ab→Z/CZ whose row is built from simultaneous logarithms of the actual input primes and whose kernel contains the normalized exponent vector forced by a+b=c.","count":34},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-785e812c9b93033d","tex":"S_a","aliases":"S_a","display":"S_a","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ba2c2b6914e4fdfc","tex":"S_{ab}","aliases":"S_ab","display":"S_{ab}","kind":"use","excerpt":"Arithmetic endpoint. The arithmetic endpoint is the coherence character χ_coh: Z^S_ab→Z/CZ whose row is built from simultaneous logarithms of the actual input primes and whose kernel contains the normalized exponent vector forced by a+b=c.","count":10},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-68387971995ff129","tex":"S_b","aliases":"S_b","display":"S_b","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8ec8b0faab9c7fd8","tex":"S_c","aliases":"S_c","display":"S_c","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5eac484be3cd94a9","tex":"\\mathrm{sh}","aliases":"sh mathrm","display":"\\mathrm{sh}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bd8be26957444f53","tex":"\\sharp","aliases":"sharp","display":null,"kind":"use","excerpt":"The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth pack","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"Its depth Fourier transform is a Dirichlet packet: widehatΨ_n,p(θ) = sqrtlog p exp(-fraci(e_p-1)θ2) fracsin(e_pθ/2)sin(θ/2). The zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fe9982bf1835e2aa","tex":"\\SNF","aliases":"SNF","display":null,"kind":"use","excerpt":"[Polyphase Smith normal form] If gcd(e,N)=1, then SNF(C_e,N) = diag(1,…,1,e).","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-327dcbd8f0058da9","tex":"\\Spec","aliases":"Spec","display":null,"kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-59d24711aca21962","tex":"\\mathrm{st}","aliases":"st mathrm","display":"\\mathrm{st}","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":26},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-adac703e9d252af9","tex":"\\mathbb T","aliases":"T mathbb","display":"\\mathbb T","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d589f35b0c8bc738","tex":"\\mathbf T","aliases":"T mathbf","display":"\\mathbf T","kind":"use","excerpt":"Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral fo","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","excerpt":"[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":49},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral fo","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-964fcf8c1f59b656","tex":"T^e","aliases":"T^e","display":"T^e","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted prod","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9b7f9d17ba2c44f5","tex":"T^{e-1}","aliases":"T^e-1","display":"T^{e-1}","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-aea3af576abad800","tex":"t^m","aliases":"t^m","display":"t^m","kind":"use","excerpt":"Homogeneous higher obstructions. Let O be homogeneous of degree m: O(tx)=t^m O(x). Then fracnormx | O(x)|^1/m is scale invariant. Replacing one large coordinate by several coordinates of size roughly the mth root does not improve a bound whose obstruction estimate raises those coordinates back to degree m.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ac72b989bba7a709","tex":"T^N","aliases":"T^N","display":"T^N","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4a7bd9c09ab323ef","tex":"T^{v_p(m)}","aliases":"T^v_p(m)","display":"T^{v_p(m)}","kind":"use","excerpt":"t by a valuation word The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4a9e32fb56cb7f6c","tex":"t_a","aliases":"t_a","display":"t_a","kind":"use","excerpt":"ionThree-sector compression Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_","count":15},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bc902e144c8dfe79","tex":"\\mathcal T_{a,b,c}","aliases":"T_a,b,c mathcal","display":"\\mathcal T_{a,b,c}","kind":"use","excerpt":"[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0a30bcd98cf55af8","tex":"t_a^{(0)}","aliases":"t_a^(0)","display":"t_a^{(0)}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","excerpt":"ionThree-sector compression Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_","count":14},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b4b7de788cf0c385","tex":"t_b^{(0)}","aliases":"t_b^(0)","display":"t_b^{(0)}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-04db525251126e68","tex":"t_c","aliases":"t_c","display":"t_c","kind":"use","excerpt":"ionThree-sector compression Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_","count":10},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8dc401ec678ef127","tex":"t_c^{(0)}","aliases":"t_c^(0)","display":"t_c^{(0)}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f00f567e4a81164c","tex":"\\mathbb T_f","aliases":"T_f mathbb","display":"\\mathbb T_f","kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-81549140e4fc02bc","tex":"t_p","aliases":"t_p","display":"t_p","kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8decad66f8014b7e","tex":"T_{p,r}","aliases":"T_p,r","display":"T_{p,r}","kind":"use","excerpt":"[First-order no-free-polyphase principle] Let S:(T_p,r)_p,r⟼ (∑_rT_p,r)_p be shell synthesis. The first-order additive map and first-order Wronskian factor through S. Hence Ker S is contained in the kernels of both first-order maps.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-df3f310abe33eba9","tex":"T_r","aliases":"T_r","display":"T_r","kind":"use","excerpt":"First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6d6c4f072fd1f994","tex":"T_s","aliases":"T_s","display":"T_s","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c9ae0c36264af704","tex":"t_u","aliases":"t_u","display":"t_u","kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8dd594b4bce51346","tex":"\\mathrm{tors}","aliases":"tors mathrm","display":"\\mathrm{tors}","kind":"use","excerpt":"[Smith obstruction theorem] Let a+b=c be primitive, with at least two of a,b,c greater than one. Then SNF( M_a,b,c) = diag(1,Q). Consequently, coker( M_a,b,c)_tors ≅ Z/QZ. The exceptional triple (1,1,2) has only one nonempty prime sector, so the matrix has rank one and is excluded from the two-row statement.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.","count":19},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0e88dc7b57e42bdd","tex":"u^M","aliases":"u^M","display":"u^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-76bfead452deff02","tex":"U_2","aliases":"U_2","display":"U_2","kind":"use","excerpt":"[Formal affine-space theorem] Let d be the number of shell variables. After a unimodular integral change of shell coordinates, the ( T)-adic completion of Z[ T]/(F_a,b,c) at the zero section is isomorphic to a formal power-series ring Z[[U_2,…,U_d]].","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1361a2562f51cd0c","tex":"U_{ab}","aliases":"U_ab","display":"U_{ab}","kind":"use","excerpt":"Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0b3db69758fee0b4","tex":"U_d","aliases":"U_d","display":"U_d","kind":"use","excerpt":"[Formal affine-space theorem] Let d be the number of shell variables. After a unimodular integral change of shell coordinates, the ( T)-adic completion of Z[ T]/(F_a,b,c) at the zero section is isomorphic to a formal power-series ring Z[[U_2,…,U_d]].","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"p_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gives one quotient equation ∑_i z_i=q(t). For a box radius H, each z_i lies in the interval ℓ_i(H) = lceilfrac-H-ξ_","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":18},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-87f0c3c75c88d8b5","tex":"v^M","aliases":"v^M","display":"v^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8c90cb56cd9a3747","tex":"v_2","aliases":"v_2","display":"v_2","kind":"use","excerpt":"Polylogarithmic path. Set z=frac3128, 1-z=frac125128, β=1-frac1z=-frac1253. The depth-three functional relation closes on these three arguments. At the output prime two, v_2(β-1) = v_2(-frac1283) =7. The unscaled principal logarithm has 2-adic valuation seven. After division by four, the weighted principal coordinate occupies five filtration levels","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f7fd62f7cd34cb10","tex":"V_e","aliases":"V_e","display":"V_e","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":15},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-74b99f2f5e5d09ac","tex":"V_{e+f}","aliases":"V_e+f","display":"V_{e+f}","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted prod","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4f979281d0af8ba8","tex":"V_f","aliases":"V_f","display":"V_f","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted prod","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-724afc19e77e7bfd","tex":"v_q","aliases":"v_q","display":"v_q","kind":"use","excerpt":"Start with the signed input exponent vector ν=(ν_q)_qmid ab, ν_q=v_q(b)-v_q(a). Let K_p be the p-primary part of the common Kummer factor. Divide the exponent vector and the local modulus by this common power. The normalized vector is primitive relative to the remaining modulus.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c07310b5bed07b57","tex":"v_s","aliases":"v_s","display":"v_s","kind":"use","excerpt":"Arbitrary-index congruence lattices. Let v∈Z^s be primitive. Complete v to a unimodular basis v,v_2,…,v_s. In the dual basis, choose the row that vanishes on v and takes value one on v_2. Reduce it modulo an arbitrary integer C. The resulting row is primitive modulo C and annihilates v.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-aee140b15026b684","tex":"V_{v_p(n)}","aliases":"V_v_p(n)","display":"V_{v_p(n)}","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted prod","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fa83faab1d44c512","tex":"v_\\ell","aliases":"v_ℓ ell","display":"v_\\ell","kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally rep","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-694bdc6e24088dc1","tex":"\\mathrm{val}","aliases":"val mathrm","display":"\\mathrm{val}","kind":"use","excerpt":"Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then ","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ce439056c74ce10f","tex":"\\mathbb W","aliases":"W mathbb","display":"\\mathbb W","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1c2a9b04e6197b99","tex":"\\mathsf w","aliases":"w mathsf","display":"\\mathsf w","kind":"use","excerpt":"A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-dfe882024b9c7f29","tex":"\\mathbb W_{\\mathbf n}","aliases":"W_ n mathbb mathbf","display":"\\mathbb W_{\\mathbf n}","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d01794b5f0c3ada0","tex":"W_1","aliases":"W_1","display":"W_1","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7705f9bd8924852f","tex":"W_6","aliases":"W_6","display":"W_6","kind":"use","excerpt":"The local Witt coordinate is W_6(F_2) ≅ Z/64Z. The weighted multiplicative dyadic coordinate is (1+4Z_2)/(1+2^8Z_2), which has the same order.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f38b6368252abbb5","tex":"W_{e-1}","aliases":"W_e-1","display":"W_{e-1}","kind":"use","excerpt":"Witt realization. The local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). The Verschiebung filtration matches multiplication by successive powers of p. This is the typed local object underlying the carry chain.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cc7e0d5f7e350ab2","tex":"w_i","aliases":"w_i","display":"w_i","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f1ace4a0d0c8f6df","tex":"w_i^{-1}","aliases":"w_i^-1","display":"w_i^{-1}","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ffbde798b9a7c58f","tex":"W_j","aliases":"W_j","display":"W_j","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0be23d285935109c","tex":"W_j^2","aliases":"W_j^2","display":"W_j^2","kind":"use","excerpt":"The Atkin--Lehner Walsh bank is a finite product of commuting involutions. Its identities are polynomial consequences of W_j^2=I.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a6dc04e23090a2a5","tex":"w_m","aliases":"w_m","display":"w_m","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8d7f1cc8c150d4e5","tex":"W_p","aliases":"W_p","display":"W_p","kind":"use","excerpt":"[leftmargin=1.5em] item Form the signed input exponent vector ν. item Remove the common Kummer preload K. item For every output prime, construct the finite logarithmic row. item Divide each row by its common coefficient depth to obtain the local defect W_p. item Remove the dyadic sign layer separately. item Combine primitive residual rows by CRT. i","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d97ea8d524e0d0e0","tex":"\\mathsf w_p","aliases":"w_p mathsf","display":"\\mathsf w_p","kind":"use","excerpt":"A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4a4f3028bb492bb2","tex":"W_r","aliases":"W_r","display":"W_r","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-36d884ad81c0ef6a","tex":"W_x","aliases":"W_x","display":"W_x","kind":"use","excerpt":"Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.","count":16},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-27826ef365351b0c","tex":"W_y","aliases":"W_y","display":"W_y","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cb66caadea9f666a","tex":"\\wedge","aliases":"wedge","display":"\\wedge","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b0d65d8f4ac93228","tex":"\\widehat T","aliases":"widehat T widehat","display":"\\widehat T","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0b1d837601c3d673","tex":"\\widehat T_{-j}","aliases":"widehat T_-j widehat","display":"\\widehat T_{-j}","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-28dd0e7bd21d34c5","tex":"\\widehat T_0","aliases":"widehat T_0 widehat","display":"\\widehat T_0","kind":"use","excerpt":"onSecond-order phase energy At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f387d54db2d29aed","tex":"\\widehat T_0^2","aliases":"widehat T_0^2 widehat","display":"\\widehat T_0^2","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-284097439a5a6f5c","tex":"\\widehat T_j","aliases":"widehat T_j widehat","display":"\\widehat T_j","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5d492261e9fa3949","tex":"\\widehat\\Psi","aliases":"widehatΨ widehat Psi","display":"\\widehat\\Psi","kind":"use","excerpt":"Its depth Fourier transform is a Dirichlet packet: widehatΨ_n,p(θ) = sqrtlog p exp(-fraci(e_p-1)θ2) fracsin(e_pθ/2)sin(θ/2). The zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0fd177bc0f0a8245","tex":"\\widehat\\Psi_{n,p}","aliases":"widehatΨ_n,p widehat Psi","display":"\\widehat\\Psi_{n,p}","kind":"use","excerpt":"Its depth Fourier transform is a Dirichlet packet: widehatΨ_n,p(θ) = sqrtlog p exp(-fraci(e_p-1)θ2) fracsin(e_pθ/2)sin(θ/2). The zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c77170c848a2e8b4","tex":"\\widetilde x","aliases":"widetilde x widetilde","display":"\\widetilde x","kind":"use","excerpt":"Odd logarithmic rows. For odd p^eparallel c, define the finite-depth row coefficient ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. The balanced relation gives ∑_qmid ab ν_qℓ_p,e(q) ≡0 pmodp^e-1. Define the truncated valuation on Z/p^e-1Z by bar v_p(0)=e-1, bar v_p(x)=min{e-1,v_p(widetilde x)} for any integer lift widetilde x of a nonzero residue. Then ba","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-85808aaccac2b99d","tex":"\\Witt_{\\mathbf n}","aliases":"Witt_ n Witt mathbf","display":null,"kind":"use","excerpt":"The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.","count":101},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9f0e345c449b459b","tex":"x^{(1)}","aliases":"x^(1)","display":"x^{(1)}","kind":"use","excerpt":"Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral fo","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5ad3765b02d5f921","tex":"x^{(2)}","aliases":"x^(2)","display":"x^{(2)}","kind":"use","excerpt":"Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral fo","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-793998b5f63ba05c","tex":"x^{(a)}","aliases":"x^(a)","display":"x^{(a)}","kind":"use","excerpt":"For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e777ea48e7254005","tex":"x^{(m)}","aliases":"x^(m)","display":"x^{(m)}","kind":"use","excerpt":"Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral fo","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ab6e16030ec3c9e5","tex":"x^{(m-1)}","aliases":"x^(m-1)","display":"x^{(m-1)}","kind":"use","excerpt":"Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral fo","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-10a75392a93dc503","tex":"X^3","aliases":"X^3","display":"X^3","kind":"use","excerpt":"[leftmargin=1.5em] item the 2-adic contact filtration of y^2 item the 3-adic contact filtration of x^3 item the fixed support of k item the depth-three polylogarithmic packet item the elliptic curve Y^2=X^3-k","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"use","excerpt":"Hall's equation x^3-y^2=k is particularly attractive because the dyadic and triadic branches are built into the exponents. The polylogarithmic weight-three resonance may be relevant, but a rigorous connection has not yet been established.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-96d5a47706807486","tex":"X^{64}","aliases":"X^64","display":"X^{64}","kind":"use","excerpt":"For the balanced ratio β=-frac1253, the signed exponent vector on the input primes is ν=(-1,3). Its Kummer order modulo 64 is full. Choosing 64th roots gives a point on X^64+Y^64=1. The coordinate height is fraclog12864. With the tensor-power metric induced from O(1), the full plane-model canonical scale is frac6164log128. For an arbitrary compatib","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-38ee95395ade7e66","tex":"X^M","aliases":"X^M","display":"X^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0ed7bdf4a651c709","tex":"x^m","aliases":"x^m","display":"x^m","kind":"use","excerpt":"Pillai and Hall. For x^m-y^n=k, the residual support is fixed by k. The two perfect-power packets must cancel to high precision while producing a small fixed output. This is a natural two-sector coherence problem.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4ab09af538752b5e","tex":"x^p","aliases":"x^p","display":"x^p","kind":"use","excerpt":"Generalized Fermat. For x^p+y^q=z^r, attach filtered contact packets to the three terms. The exponents multiply the local valuation depths. Balanced Kummer ratios and cross-prime logarithmic filters can eliminate many local patterns before a Frey or hypergeometric motive is attached.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"Obstruction cosets. Choose a section x_0∈ L with ω(x_0)=1. Then L = bigsqcup_k∈Z (kx_0+L_0). For fixed k, the problem is a closest-vector problem to the moving target -kx_0 in the lattice L_0. The three-sector compression makes this search much more explicit.","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounde","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-98391915cbb7346b","tex":"x_5","aliases":"x_5","display":"x_5","kind":"use","excerpt":"Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounde","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":9},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-53687bf254c09aec","tex":"x_p","aliases":"x_p","display":"x_p","kind":"use","excerpt":"A scalar packet is an integer vector x=(x_p)_p∈ S∈Z^S. Pairing with x gives the arithmetic derivative d_x(n) = n∑_pmid nfracv_p(n)p x_p. The Leibniz rule survives scalarization.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-si","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"The phase packet y=(-8,8,3) satisfies D_y(3)=-8, D_y(125)=200, D_y(128)=192, and -8+200=192. Its Wronskian is also 1600. The phase height is eight. The phase inequality is sharp: 128 = 30(frac13+frac15)8. This illustrates that the exponent-free phase channel can sharpen the bound even when its packet norm is larger.","count":32},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0e7659cd4ef8f947","tex":"Y^2","aliases":"Y^2","display":"Y^2","kind":"use","excerpt":"[leftmargin=1.5em] item the 2-adic contact filtration of y^2 item the 3-adic contact filtration of x^3 item the fixed support of k item the depth-three polylogarithmic packet item the elliptic curve Y^2=X^3-k","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-abdf161576a33b61","tex":"y^2","aliases":"y^2","display":"y^2","kind":"use","excerpt":"The marked Frey curve is y^2=x(x-3)(x+125). Its discriminant is 16·3^2·125^2·128^2. At the odd primes, the marked depths are two at 3 and six at 5. The dyadic depth requires the weighted branch.","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fd3001e9f1f0d1d2","tex":"Y^{64}","aliases":"Y^64","display":"Y^{64}","kind":"use","excerpt":"For the balanced ratio β=-frac1253, the signed exponent vector on the input primes is ν=(-1,3). Its Kummer order modulo 64 is full. Choosing 64th roots gives a point on X^64+Y^64=1. The coordinate height is fraclog12864. With the tensor-power metric induced from O(1), the full plane-model canonical scale is frac6164log128. For an arbitrary compatib","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-798bab1eeb3041a1","tex":"Y^M","aliases":"Y^M","display":"Y^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-de7ba1dbdac1562b","tex":"y^n","aliases":"y^n","display":"y^n","kind":"use","excerpt":"Pillai and Hall. For x^m-y^n=k, the residual support is fixed by k. The two perfect-power packets must cancel to high precision while producing a small fixed output. This is a natural two-sector coherence problem.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ce6d149a19650241","tex":"y^q","aliases":"y^q","display":"y^q","kind":"use","excerpt":"Generalized Fermat. For x^p+y^q=z^r, attach filtered contact packets to the three terms. The exponents multiply the local valuation depths. Balanced Kummer ratios and cross-prime logarithmic filters can eliminate many local patterns before a Frey or hypergeometric motive is attached.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"For a scalar split t=y_1+⋯+y_e, the child sup norm can fall to lceilfrac|t|erceil. The synthesis map has norm e, so elceilfrac|t|erceil≥|t|. This explains why raw shell splitting cannot lower the intrinsic packet height.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ff216c70e798e26f","tex":"y_e","aliases":"y_e","display":"y_e","kind":"use","excerpt":"For a scalar split t=y_1+⋯+y_e, the child sup norm can fall to lceilfrac|t|erceil. The synthesis map has norm e, so elceilfrac|t|erceil≥|t|. This explains why raw shell splitting cannot lower the intrinsic packet height.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7a4d74bf293c7c99","tex":"y_p","aliases":"y_p","display":"y_p","kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5d14d79ba80e05b7","tex":"y_{p,r}","aliases":"y_p,r","display":"y_{p,r}","kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"bibitemAKN B. Antieau, A. Krause, and T. Nikolaus, ``On the K-theory of Z/p^n,'' arXiv:2405.04329, 2024.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"Polylogarithmic path. Set z=frac3128, 1-z=frac125128, β=1-frac1z=-frac1253. The depth-three functional relation closes on these three arguments. At the output prime two, v_2(β-1) = v_2(-frac1283) =7. The unscaled principal logarithm has 2-adic valuation seven. After division by four, the weighted principal coordinate occupies five filtration levels","count":25},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ddecadd3df3a219e","tex":"\\Z^2","aliases":"Z^2 Z","display":null,"kind":"use","excerpt":"0, · pmid c. Consider M_a,b,c = Λ w : Z^S⟶Z^2. The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set Q=fracabcR.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-632795a97bd80b33","tex":"Z^M","aliases":"Z^M","display":"Z^M","kind":"use","excerpt":"The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8b55f569822030f0","tex":"\\Z^n","aliases":"Z^n Z","display":null,"kind":"use","excerpt":"For a primitive rank-one vector v∈Z^n, the orthogonal projector denominator is ⟨ v,v⟩, and the congruence quotient is cyclic of that order.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-65cfeb4bbc54304a","tex":"\\Z^Q","aliases":"Z^Q Z","display":null,"kind":"use","excerpt":"The coherence lattice. After removing common Kummer powers and local logarithmic image defects, the surviving rows combine by the Chinese remainder theorem into a primitive cyclic character χ_coh:Z^Q⟶Z/CZ. The coherence lattice is Λ_coh = Kerχ_coh. When the row is primitive, [Z^Q:Λ_coh]=C. The normalized exponent vector of the relation lies in this","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-10d0329c35407a66","tex":"z^r","aliases":"z^r","display":"z^r","kind":"use","excerpt":"Generalized Fermat. For x^p+y^q=z^r, attach filtered contact packets to the three terms. The exponents multiply the local valuation depths. Balanced Kummer ratios and cross-prime logarithmic filters can eliminate many local patterns before a Frey or hypergeometric motive is attached.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e67551e8d1a79560","tex":"\\Z^{r+1}","aliases":"Z^r+1 Z","display":null,"kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f4d3cabb397a3cc8","tex":"\\Z^S","aliases":"Z^S Z","display":null,"kind":"use","excerpt":"A scalar packet is an integer vector x=(x_p)_p∈ S∈Z^S. Pairing with x gives the arithmetic derivative d_x(n) = n∑_pmid nfracv_p(n)p x_p. The Leibniz rule survives scalarization.","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bc798b14e6021234","tex":"\\Z^s","aliases":"Z^s Z","display":null,"kind":"use","excerpt":"Arbitrary-index congruence lattices. Let v∈Z^s be primitive. Complete v to a unimodular basis v,v_2,…,v_s. In the dual basis, choose the row that vanishes on v and takes value one on v_2. Reduce it modulo an arbitrary integer C. The resulting row is primitive modulo C and annihilates v.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4ed5de31ab0a09ef","tex":"\\Z^{S_{ab}}","aliases":"Z^S_ab Z","display":null,"kind":"use","excerpt":"Arithmetic endpoint. The arithmetic endpoint is the coherence character χ_coh: Z^S_ab→Z/CZ whose row is built from simultaneous logarithms of the actual input primes and whose kernel contains the normalized exponent vector forced by a+b=c.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-78a80f694d80e6e7","tex":"\\Z_2","aliases":"Z_2 Z","display":null,"kind":"use","excerpt":"The local Witt coordinate is W_6(F_2) ≅ Z/64Z. The weighted multiplicative dyadic coordinate is (1+4Z_2)/(1+2^8Z_2), which has the same order.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-86298b26f7dd1fb4","tex":"z_i","aliases":"z_i","display":"z_i","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b910972bdb78c268","tex":"\\Z_p","aliases":"Z_p Z","display":null,"kind":"use","excerpt":"Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e7f32becee04ace5","tex":"|^{1/m}","aliases":"|^1/m","display":"|^{1/m}","kind":"use","excerpt":"Homogeneous higher obstructions. Let O be homogeneous of degree m: O(tx)=t^m O(x). Then fracnormx | O(x)|^1/m is scale invariant. Replacing one large coordinate by several coordinates of size roughly the mth root does not improve a bound whose obstruction estimate raises those coordinates back to degree m.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"onSecond-order phase energy At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"For the exponent-free phase obstruction matrix, every nonzero cross-sector minor is ±fracabcpq = ±fracabcRfracRpq. All such minors are divisible by abc/R. For each prime ℓmid R, choose one column with p=ℓ and one column from a different nonempty sector. The reduced minor R/(pq) is then not divisible by ℓ. Hence the gcd of the reduced minors is one ","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"A determinant viewpoint. When several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let L be a normalized cross-prime logarithm matrix. A nonzero r× r minor det L_I,J measures the failure of r input packets to cancel simultaneously across r output channels.","count":12},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The ","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-80f45a66a06b79f2","tex":"\\alpha_u","aliases":"α_u alpha","display":"\\alpha_u","kind":"use","excerpt":"Phase and coarse separation. The local target has the form t_u(k,n)=α_uk+r_un. For a local prime pmid r_u, t_u(k,n)≡α_ukpmod p. Thus the obstruction multiple k determines the finite phase packet. The coarse parameter n changes only the quotient load. The exact search is a moving polyphase mechanism rather than a generic mixed-integer program.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-18d68840d1fdac68","tex":"\\alpha_\\ell","aliases":"α_ℓ alpha ell","display":"\\alpha_\\ell","kind":"use","excerpt":"Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The ","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"For the balanced ratio β=-frac1253, the signed exponent vector on the input primes is ν=(-1,3). Its Kummer order modulo 64 is full. Choosing 64th roots gives a point on X^64+Y^64=1. The coordinate height is fraclog12864. With the tensor-power metric induced from O(1), the full plane-model canonical scale is frac6164log128. For an arbitrary compatib","count":18},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8ed4bfa870aa1639","tex":"\\beta^{1/K}","aliases":"β^1/K beta","display":"\\beta^{1/K}","kind":"use","excerpt":"Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1bac3c7ce6eb2507","tex":"\\beta_1","aliases":"β_1 beta","display":"\\beta_1","kind":"use","excerpt":"Kummer fiber product. Balanced Kummer ratios from the two equations define classes β_1=-B/A and β_2=-D/C. The shared term C creates a multiplicative compatibility. The natural global object is a fiber product of the two Kummer torsors over the Kummer class of C.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-05f6c598f663dcba","tex":"\\beta_2","aliases":"β_2 beta","display":"\\beta_2","kind":"use","excerpt":"Kummer fiber product. Balanced Kummer ratios from the two equations define classes β_1=-B/A and β_2=-D/C. The shared term C creates a multiplicative compatibility. The natural global object is a fiber product of the two Kummer torsors over the Kummer class of C.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"ve hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=overline H k_*Γ for the optimal obstruction multiple k_* and an integrality factor","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Since Q=abc/R, the exact minimum is overline H = fraccRΔ(a,b,c), where Δ = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. The real problem is therefore completely solved. The unresolved content lies in the integral departure from this stable width.","count":11},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bf830770d8c035f1","tex":"\\delta_a","aliases":"δ_a delta","display":"\\delta_a","kind":"use","excerpt":"The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9db778080ee52c01","tex":"\\delta_{\\mathrm{alg}}","aliases":"δ_alg delta mathrm","display":"\\delta_{\\mathrm{alg}}","kind":"use","excerpt":"For the output source m=c/R_c, the corrected algebraic defect against a cyclic target of order η_f is δ_alg(c,f) = fracc/R_cgcd(c/R_c,η_f). Prime by prime, δ_alg(c,f) = ∏_p^eparallel c p^max(e-1-v_p(η_f),0). The problem is missing depth, not missing rank.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1d6bb66bd4727edf","tex":"\\delta_c","aliases":"δ_c delta","display":"\\delta_c","kind":"use","excerpt":"The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0ba612a4176bddf5","tex":"\\Delta_E","aliases":"Δ_E Delta","display":"\\Delta_E","kind":"use","excerpt":"The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discrimina","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e171fd46e82d4877","tex":"\\boldsymbol\\epsilon","aliases":"ϵ boldsymbol epsilon","display":"\\boldsymbol\\epsilon","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2930352759bad3e1","tex":"\\varepsilon'","aliases":"ε' varepsilon","display":"\\varepsilon'","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-da012babef541427","tex":"\\epsilon_1","aliases":"ϵ_1 epsilon","display":"\\epsilon_1","kind":"use","excerpt":"Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8b15e8609cc3fb45","tex":"\\epsilon_{2,p}","aliases":"ϵ_2,p epsilon","display":"\\epsilon_{2,p}","kind":"use","excerpt":"The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0773a19734ee0848","tex":"\\epsilon_i","aliases":"ϵ_i epsilon","display":"\\epsilon_i","kind":"use","excerpt":"Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a14473f51c7df774","tex":"\\epsilon_j","aliases":"ϵ_j epsilon","display":"\\epsilon_j","kind":"use","excerpt":"A general contact functor. The construction extends to a sparse relation ∑_j=1^rϵ_jn_j=0. Choose one or more output terms whose repeated depth is to be explained. The assignment ( n,ϵ) ⟼ A( n,ϵ) is canonical under permutations, sign reversal, primitive reduction, and primary decomposition. The optional geometric lift depends on the equation, but th","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b3de5d9511fcd1b4","tex":"\\varepsilon_j","aliases":"ε_j varepsilon","display":"\\varepsilon_j","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-70bb170c716771c3","tex":"\\epsilon_r","aliases":"ϵ_r epsilon","display":"\\epsilon_r","kind":"use","excerpt":"Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0b5701ed0b92c9e3","tex":"\\zeta^e","aliases":"ζ^e zeta","display":"\\zeta^e","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7434f23c339ba4fb","tex":"\\zeta^N","aliases":"ζ^N zeta","display":"\\zeta^N","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5048985144717904","tex":"\\zeta_M","aliases":"ζ_M zeta","display":"\\zeta_M","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5d39f2d87876b475","tex":"\\zeta_M^j","aliases":"ζ_M^j zeta","display":"\\zeta_M^j","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c85cee5084c73471","tex":"\\eta_f","aliases":"η_f eta","display":"\\eta_f","kind":"use","excerpt":"The same-sign separation problem. After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let Λ be an integral Hecke lattice and let e_f be the rational projector onto the Frey eigenline. Define η_f = min{m≥1:m e_f(Λ)⊆Λ}. This is the integral eigenchannel separation denominator. It is an arithmetic analogue of ","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9359c9fe8689c79c","tex":"\\eta_U","aliases":"η_U eta","display":"\\eta_U","kind":"use","excerpt":"[Integral projector theorem] The least positive integer η_U satisfying η_U e_U(Λ)⊆Λ is the exponent of the finite quotient C_U.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"Its depth Fourier transform is a Dirichlet packet: widehatΨ_n,p(θ) = sqrtlog p exp(-fraci(e_p-1)θ2) fracsin(e_pθ/2)sin(θ/2). The zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9066618856182e83","tex":"\\Theta_3","aliases":"Θ_3 Theta","display":"\\Theta_3","kind":"use","excerpt":"Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) satisfying |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows after absorbing the logarithmic Kummer kern","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-071c8af44ea8db86","tex":"\\Theta_f","aliases":"Θ_f Theta","display":"\\Theta_f","kind":"use","excerpt":"[Conditional filtered-transfer criterion] Suppose a canonical map H^0(Θ_f):Z/(c/R_c)Z⟶ H^0( C_f) satisfies, for every eps>0, |ker H^0(Θ_f)|≤ C_eps R^eps and |H^0( C_f)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth pack","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3f9c758e6b928a59","tex":"\\kappa_{\\mathbf n}","aliases":"κ_ n kappa mathbf","display":"\\kappa_{\\mathbf n}","kind":"use","excerpt":"Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being b","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d9001eb0a3ef0a23","tex":"\\kappa_{128}","aliases":"κ_128 kappa","display":"\\kappa_{128}","kind":"use","excerpt":"Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.","count":13},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2c9dd55b67cbaf19","tex":"\\lambda^{\\mathsf T}","aliases":"λ^ T lambda mathsf","display":"\\lambda^{\\mathsf T}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f781fdf3ad4821b6","tex":"\\lambda^{(a)}","aliases":"λ^(a) lambda","display":"\\lambda^{(a)}","kind":"use","excerpt":"Three-sector compression. Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e8eed9b54e1d4e21","tex":"\\lambda^{\\mathrm{ph}}","aliases":"λ^ph lambda mathrm","display":"\\lambda^{\\mathrm{ph}}","kind":"use","excerpt":"Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7238fc50817cce55","tex":"\\lambda_0","aliases":"λ_0 lambda","display":"\\lambda_0","kind":"use","excerpt":"Their Legendre parameters are linked by the standard dyadic Landen transformation. If λ_0=(frac vu)^2, then the complementary child parameter is λ_1 = frac4uv(u+v)^2 = frac4sqrtλ_0(1+sqrtλ_0)^2. This is the exact two-isogeny refinement law behind the family.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"Their Legendre parameters are linked by the standard dyadic Landen transformation. If λ_0=(frac vu)^2, then the complementary child parameter is λ_1 = frac4uv(u+v)^2 = frac4sqrtλ_0(1+sqrtλ_0)^2. This is the exact two-isogeny refinement law behind the family.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fd56c486b0b1eabb","tex":"\\Lambda_C","aliases":"Λ_C Lambda","display":"\\Lambda_C","kind":"use","excerpt":"Thus Λ_C = {x:g_C· x≡0pmod C} has index C and contains the same fixed vector v. This proves that no bound on C can follow from the length or support of v alone.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a85265c85d543bbb","tex":"\\Lambda_E","aliases":"Λ_E Lambda","display":"\\Lambda_E","kind":"use","excerpt":"The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth pack","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7daa0edaa864bbe2","tex":"\\Lambda_p","aliases":"Λ_p Lambda","display":"\\Lambda_p","kind":"use","excerpt":"A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-fdc2dc9f687c22a4","tex":"\\lambda_p","aliases":"λ_p lambda","display":"\\lambda_p","kind":"use","excerpt":"The additive row. The condition d_x(a)+ d_x(b)= d_x(c) is one integer equation. Define λ_p= dfraca v_p(a)p, · pmid a, [6pt] dfracb v_p(b)p, · pmid b, [6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cf331a2bc1f6a864","tex":"\\Lambda_U","aliases":"Λ_U Lambda","display":"\\Lambda_U","kind":"use","excerpt":"Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c259e39137ef0548","tex":"\\Lambda_W","aliases":"Λ_W Lambda","display":"\\Lambda_W","kind":"use","excerpt":"Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1370487ea3007c46","tex":"\\mu_{64}","aliases":"μ_64 mu","display":"\\mu_{64}","kind":"use","excerpt":"Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0bf8701ccd3ba8f5","tex":"\\mu_M","aliases":"μ_M mu","display":"\\mu_M","kind":"use","excerpt":"The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"use","excerpt":"The arithmetic Wronskian. Define W_x(a,b) =a d_x(b)-b d_x(a). Writing μ_p= -dfracv_p(a)p, · pmid a, [6pt] dfracv_p(b)p, · pmid b, [6pt] 0, · pmid c, we obtain W_x(a,b)=ab μ· x. An additive packet is transverse when W_x(a,b)≠0. It is degenerate when the two input terms remain derivative-proportional.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e2458c5b6cb798e0","tex":"\\mu_\\perp","aliases":"μ_⊥ mu perp","display":"\\mu_\\perp","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-461d64059c8c1199","tex":"\\mu_\\perp^{\\mathsf T}","aliases":"μ_⊥^ T mu perp mathsf","display":"\\mu_\\perp^{\\mathsf T}","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"For the balanced ratio β=-frac1253, the signed exponent vector on the input primes is ν=(-1,3). Its Kummer order modulo 64 is full. Choosing 64th roots gives a point on X^64+Y^64=1. The coordinate height is fraclog12864. With the tensor-power metric induced from O(1), the full plane-model canonical scale is frac6164log128. For an arbitrary compatib","count":23},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-cd0b7ab5ccc033f7","tex":"\\nu_0","aliases":"ν_0 nu","display":"\\nu_0","kind":"use","excerpt":"ν_0 · normalized input exponent vector","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-898af8461f33e86a","tex":"\\nu_{a,b,c}","aliases":"ν_a,b,c nu","display":"\\nu_{a,b,c}","kind":"use","excerpt":"The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d5fba9031264fee8","tex":"\\nu_q","aliases":"ν_q nu","display":"\\nu_q","kind":"use","excerpt":"Start with the signed input exponent vector ν=(ν_q)_qmid ab, ν_q=v_q(b)-v_q(a). Let K_p be the p-primary part of the common Kummer factor. Divide the exponent vector and the local modulus by this common power. The normalized vector is primitive relative to the remaining modulus.","count":15},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-273ca48755e332d5","tex":"\\xi_i","aliases":"ξ_i xi","display":"\\xi_i","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ffb8af09a0c235fd","tex":"\\xi_p","aliases":"ξ_p xi","display":"\\xi_p","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"A fixed path packet is Π_≤3(z) = ( log z, log(1-z), L_2(z), Li_3(z), Li_3(1-z) ). Its coordinate count is independent of M.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.","count":10},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-76b8428882ee3f96","tex":"\\Pi_{\\le3}","aliases":"Π_≤3 Pi le","display":"\\Pi_{\\le3}","kind":"use","excerpt":"A fixed path packet is Π_≤3(z) = ( log z, log(1-z), L_2(z), Li_3(z), Li_3(1-z) ). Its coordinate count is independent of M.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9bf4e85984084b75","tex":"\\rho_a","aliases":"ρ_a rho","display":"\\rho_a","kind":"use","excerpt":"For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-153325643fdda58b","tex":"\\rho_b","aliases":"ρ_b rho","display":"\\rho_b","kind":"use","excerpt":"For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3b48fafc33330554","tex":"\\rho_c","aliases":"ρ_c rho","display":"\\rho_c","kind":"use","excerpt":"For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-37acf16cb7002188","tex":"\\rho_u^{\\mathrm{sh}}","aliases":"ρ_u^sh rho mathrm","display":"\\rho_u^{\\mathrm{sh}}","kind":"use","excerpt":"The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-98b0d6c9206da517","tex":"\\sigma_{a,b,c}","aliases":"σ_a,b,c sigma","display":"\\sigma_{a,b,c}","kind":"use","excerpt":"The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b17d423cde5b3e30","tex":"\\sigma_u","aliases":"σ_u sigma","display":"\\sigma_u","kind":"use","excerpt":"The stable real height. Relax the integral packet to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=ov","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":8},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bd1555616c800972","tex":"\\phi_E","aliases":"ϕ_E phi","display":"\\phi_E","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-afa3e839955395af","tex":"\\phi_E^*","aliases":"ϕ_E^* phi","display":"\\phi_E^*","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":11},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c152a6f955cd6980","tex":"\\Psi_a","aliases":"Ψ_a Psi","display":"\\Psi_a","kind":"use","excerpt":"For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1dc185857eee32ba","tex":"\\Psi_{a,b,c}","aliases":"Ψ_a,b,c Psi","display":"\\Psi_{a,b,c}","kind":"use","excerpt":"For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-48e24f5ca62767fd","tex":"\\Psi_b","aliases":"Ψ_b Psi","display":"\\Psi_b","kind":"use","excerpt":"For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3664f0ed22551ee1","tex":"\\Psi_c","aliases":"Ψ_c Psi","display":"\\Psi_c","kind":"use","excerpt":"For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-02cb516a6f66aaeb","tex":"\\Psi_n","aliases":"Ψ_n Psi","display":"\\Psi_n","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":9},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Odd-prime coefficient depth. Let q≠ p be prime. Write q=ω(q)⟨ q⟩_p, where ω(q) is the Teichm uller factor. Since ⟨ q⟩_p^p-1=q^p-1, we have log_p⟨ q⟩_p = frac1p-1log_p(q^p-1). If s=v_p(q^p-1-1)>0, then v_p(log_p(q^p-1))=s. Therefore v_p(frac1plog_p⟨ q⟩_p) =s-1. Truncating at the available contact depth gives v_p(ℓ_p,e(q)) = min{e-1,s-1}.","count":13},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5544658fd58cf8cf","tex":"\\omega_E","aliases":"ω_E omega","display":"\\omega_E","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-af8076f32fb2994d","tex":"\\Omega_S","aliases":"Ω_S Omega","display":"\\Omega_S","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.","count":20},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.","count":38},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9351643193ee272c","tex":"\\ell^{-1}","aliases":"ℓ^-1 ell","display":"\\ell^{-1}","kind":"use","excerpt":"No-go theorem: linear Hecke averaging. In the local Tate model, the ℓ+1 isogeny branches multiply contact depth by ℓ, underbraceℓ^-1,…,ℓ^-1_ℓ branches. Uniform averaging preserves the linear mean exactly. Typical logarithmic depth falls, but rare deep branches retain the mean. Linear Hecke averaging is redistribution rather than contraction.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3f72a4cc0972b04d","tex":"\\ell^{2k-n}","aliases":"ℓ^2k-n ell","display":"\\ell^{2k-n}","kind":"use","excerpt":"The mean-preserving Hecke cascade. At a Tate curve, an ℓ-isogeny packet contains one branch multiplying depth by ℓ and ℓ branches multiplying depth by 1/ℓ. Under uniform averaging, fracℓ+ℓ(1/ℓ)ℓ+1=1. After n independent refinements, if the deep branch is chosen k times, the normalized depth is ℓ^2k-n with multiplicity binom nkℓ^n-k. Therefore ∑_k=0","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-28c6765d59fee59f","tex":"\\ell^n","aliases":"ℓ^n ell","display":"\\ell^n","kind":"use","excerpt":"Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally rep","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b229d08eb00f960d","tex":"\\ell^{n-k}","aliases":"ℓ^n-k ell","display":"\\ell^{n-k}","kind":"use","excerpt":"The mean-preserving Hecke cascade. At a Tate curve, an ℓ-isogeny packet contains one branch multiplying depth by ℓ and ℓ branches multiplying depth by 1/ℓ. Under uniform averaging, fracℓ+ℓ(1/ℓ)ℓ+1=1. After n independent refinements, if the deep branch is chosen k times, the normalized depth is ℓ^2k-n with multiplicity binom nkℓ^n-k. Therefore ∑_k=0","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-943e031326adef72","tex":"\\ell^{\\alpha_\\ell}","aliases":"ℓ^α_ℓ ell alpha ell","display":"\\ell^{\\alpha_\\ell}","kind":"use","excerpt":"Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The ","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3e646a32ec34937c","tex":"\\ell_{2,e}","aliases":"ℓ_2,e ell","display":"\\ell_{2,e}","kind":"use","excerpt":"Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"p_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gives one quotient equation ∑_i z_i=q(t). For a box radius H, each z_i lies in the interval ℓ_i(H) = lceilfrac-H-ξ_","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b16d675453be8fea","tex":"\\ell_p","aliases":"ℓ_p ell","display":"\\ell_p","kind":"use","excerpt":"Simultaneous Wieferich and regulator theory. This is the closest immediate field because the cross-prime logarithmic matrix is already the main object. For finite sets of primes P and Q, define Lcal_P,Q = ( ℓ_p(q) )_p∈ P,q∈ Q. Natural theorem targets include: [leftmargin=1.5em] item a rank criterion for the matrix modulo prescribed prime powers ite","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c812626f1088bbe9","tex":"\\ell_{p,e}","aliases":"ℓ_p,e ell","display":"\\ell_{p,e}","kind":"use","excerpt":"For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-si","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted prod","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-2e558249cf09608d","tex":"\\partial_{p,T}","aliases":"∂_p,T partial","display":"\\partial_{p,T}","kind":"use","excerpt":"Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted prod","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f67711a953b0a584","tex":"\\prod_{d\\ge1}","aliases":"∏_d≥1 prod ge","display":"\\prod_{d\\ge1}","kind":"use","excerpt":"dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3ac28c105ca2b315","tex":"\\prod_{d\\ge2}","aliases":"∏_d≥2 prod ge","display":"\\prod_{d\\ge2}","kind":"use","excerpt":"dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ebf4e0c91e2a4804","tex":"\\prod_{i=1}^{m}","aliases":"∏_i=1^m prod","display":"\\prod_{i=1}^{m}","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9079021fd6762740","tex":"\\prod_{i=1}^{r}","aliases":"∏_i=1^r prod","display":"\\prod_{i=1}^{r}","kind":"use","excerpt":"The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-08e723782301ff4e","tex":"\\prod_{j=1}^{r}","aliases":"∏_j=1^r prod","display":"\\prod_{j=1}^{r}","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b9d5148b142926aa","tex":"\\prod_p","aliases":"∏_p prod","display":"\\prod_p","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-dac53b23d92ca567","tex":"\\prod_{p,r}","aliases":"∏_p,r prod","display":"\\prod_{p,r}","kind":"use","excerpt":"Gauge collapse. Within one sector, the product ∏_p,r(p+T_p,r) depends only on the aggregate product of its shell variables. Internal phase directions that preserve this product form a gauge torus. The full shell equation factors through three aggregate sector products A, B, C, with one linear relation aA+bB=cC. Thus all formal complexity collapses ","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c7ba725b7697f4ca","tex":"\\prod_{p^e\\parallel abc}","aliases":"∏_p^eparallel abc prod parallel","display":null,"kind":"use","excerpt":"The prime supports of a, b, and c are disjoint. Multiplying all local divisibilities gives ∏_p^eparallel abcp^e-1 = fracabcR mid W_x(a,b).","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-1d6641086f3edea1","tex":"\\prod_{p^e\\parallel c}","aliases":"∏_p^eparallel c prod parallel","display":null,"kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split ","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-26fdda4e62a4020e","tex":"\\prod_{p^e\\parallel n}","aliases":"∏_p^eparallel n prod parallel","display":null,"kind":"use","excerpt":"The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d1d8381ed64661dc","tex":"\\prod_{p\\mid n,\\ d\\mid e_p}","aliases":"∏_pmid n, dmid e_p prod mid mid","display":null,"kind":"use","excerpt":"Divisor-polyphase perfect reconstruction. For n=∏_pp^e_p, define B_d(n)=∏_pmid n, dmid e_pp. Then n=∏_d≥1B_d(n)^φ(d) and fracnrad(n) = ∏_d≥2B_d(n)^φ(d). This is an exact multiplicative polyphase decomposition of prime-power persistence.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bf2494072e7ab581","tex":"\\prod_{q\\mid ab}","aliases":"∏_qmid ab prod mid","display":null,"kind":"use","excerpt":"Writing β=-∏_qmid abq^ν_q, we obtain the cross-prime logarithmic filter relation log_pβ = ∑_qmid ab ν_qlog_p⟨ q⟩_p. The same depth-three path therefore has a small complex envelope and exact deep finite notches.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-80171e1b8351526c","tex":"\\prod_{q\\in S}","aliases":"∏_q∈ S prod in","display":"\\prod_{q\\in S}","kind":"use","excerpt":"Exact Kummer kernel for a rational packet. Let x=±∏_q∈ Sq^ν_q and consider its class modulo Mth powers. For positive x, the order is fracMgcd(M,{ν_q}). If x<0 and M is even, the sign contributes a separate element of order two. The exact order is therefore the least common multiple of the valuation order and the sign order.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-3b928adc47a853c8","tex":"\\prod_{r=0}^{e-1}","aliases":"∏_r=0^e-1 prod","display":"\\prod_{r=0}^{e-1}","kind":"use","excerpt":"First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-bbcd7f9ad13fce2b","tex":"\\prod_{\\substack{p\\mid n\\\\d\\mid e_p}}","aliases":"∏_substackpmid n dmid e_p prod substack mid d mid","display":null,"kind":"use","excerpt":"Exact-order divisor packets. Let n=∏_p p^e_p. For every d≥1, define B_d(n) = ∏_substackpmid n dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact c","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-eda8877465c42ed5","tex":"\\prod_{\\substack{\\zeta^N=1\\\\\\zeta\\neq1}}","aliases":"∏_substackζ^N=1 ζ≠1 prod substack zeta zeta neq","display":"\\prod_{\\substack{\\zeta^N=1\\\\\\zeta\\neq1}}","kind":"use","excerpt":"ghtarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1 ζ≠1 frac1-ζ^e1-ζ =1. Thus mu","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-10722bba9dd150f9","tex":"\\prod_{\\zeta^N=1}","aliases":"∏_ζ^N=1 prod zeta","display":"\\prod_{\\zeta^N=1}","kind":"use","excerpt":"Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefor","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ee654d78dddfa1e6","tex":"\\prod_{\\ell\\text{ odd}}","aliases":"∏_ℓ odd prod ell text","display":"\\prod_{\\ell\\text{ odd}}","kind":"use","excerpt":"Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The ","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ce043e53d6db1168","tex":"\\sum_{d\\mid e}","aliases":"∑_dmid e sum mid","display":null,"kind":"use","excerpt":"dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4989fc2730a6d003","tex":"\\sum_{d\\ge2}","aliases":"∑_d≥2 sum ge","display":"\\sum_{d\\ge2}","kind":"use","excerpt":"Quality as input-output persistence imbalance. For a primitive triple, q-1 = frac log(c/R_c)-log(R_aR_b) log R . Using the divisor packets, q-1 = frac ∑_d≥2φ(d)log B_d(c) - log(R_aR_b) log R. A high-quality triple has more repeated-depth persistence in the output than the input radicals supply as activated bandwidth.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d406db64a8c9cb22","tex":"\\sum_{j=0}^{M-1}","aliases":"∑_j=0^M-1 sum","display":"\\sum_{j=0}^{M-1}","kind":"use","excerpt":"Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-04a3e524463e2337","tex":"\\sum_{j=1}^{m-1}","aliases":"∑_j=1^m-1 sum","display":"\\sum_{j=1}^{m-1}","kind":"use","excerpt":"Bernoulli--Witt direction. The Bernoulli branch begins with the identity ∑_j=1^m-1j^k = fracB_k+1(m)-B_k+1k+1. The equation with m^k imposes strong congruences on Bernoulli values. A filtered prime-contact treatment should separate:","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6ce2ba87b3975aab","tex":"\\sum_{j=1}^{r}","aliases":"∑_j=1^r sum","display":"\\sum_{j=1}^{r}","kind":"use","excerpt":"A general contact functor. The construction extends to a sparse relation ∑_j=1^rϵ_jn_j=0. Choose one or more output terms whose repeated depth is to be explained. The assignment ( n,ϵ) ⟼ A( n,ϵ) is canonical under permutations, sign reversal, primitive reduction, and primary decomposition. The optional geometric lift depends on the equation, but th","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9914d30e7b62691b","tex":"\\sum_{j\\neq0}","aliases":"∑_j≠0 sum neq","display":"\\sum_{j\\neq0}","kind":"use","excerpt":"onSecond-order phase energy At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b7f7abb917a0d195","tex":"\\sum_{k=0}^n","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^n","kind":"use","excerpt":"Hecke cascade At a Tate curve, an ℓ-isogeny packet contains one branch multiplying depth by ℓ and ℓ branches multiplying depth by 1/ℓ. Under uniform averaging, fracℓ+ℓ(1/ℓ)ℓ+1=1. After n independent refinements, if the deep branch is chosen k times, the normalized depth is ℓ^2k-n with multiplicity binom nkℓ^n-k. Therefore ∑_k=0^n fracbinom nkℓ^n-k(","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-df09d859c509da59","tex":"\\sum_p","aliases":"∑_p sum","display":"\\sum_p","kind":"use","excerpt":"Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidt","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-9212aad30f9a37fa","tex":"\\sum_{p\\mid a}","aliases":"∑_pmid a sum mid","display":null,"kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-6280728bb9fdface","tex":"\\sum_{p\\mid b}","aliases":"∑_pmid b sum mid","display":null,"kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b8e5fc8aeac683b4","tex":"\\sum_{p\\mid c}","aliases":"∑_pmid c sum mid","display":null,"kind":"use","excerpt":"The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b0678753651fceb8","tex":"\\sum_{p\\mid n}","aliases":"∑_pmid n sum mid","display":null,"kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-98d653aa96458ba7","tex":"\\sum_{p\\mid u}","aliases":"∑_pmid u sum mid","display":null,"kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":4},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-499f779743314d49","tex":"\\sum_{q\\mid ab}","aliases":"∑_qmid ab sum mid","display":null,"kind":"use","excerpt":"Writing β=-∏_qmid abq^ν_q, we obtain the cross-prime logarithmic filter relation log_pβ = ∑_qmid ab ν_qlog_p⟨ q⟩_p. The same depth-three path therefore has a small complex envelope and exact deep finite notches.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-8586d754d52f2508","tex":"\\sum_{r<s}","aliases":"∑_r<s sum","display":"\\sum_{r<s}","kind":"use","excerpt":"Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-82fc846b089bd1f7","tex":"\\sum_{r=0}^{v_p(u)-1}","aliases":"∑_r=0^v_p(u)-1 sum","display":"\\sum_{r=0}^{v_p(u)-1}","kind":"use","excerpt":"Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-53017beba95d8b75","tex":"\\sum_\\varepsilon","aliases":"∑_ε sum varepsilon","display":"\\sum_\\varepsilon","kind":"use","excerpt":"Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign s","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"The master height inequality. For every positive n, | d_x(n)| · ≤ nnormx_∞ ∑_pmid nfracv_p(n)p. Since p≥2, frac1p≤frac12log2log p. Therefore ∑_pmid nfracv_p(n)p ≤ fraclog n2log2, and | d_x(n)| ≤ fracnlog n2log2normx_∞. Consequently, |W_x(a,b)| · ≤ a| d_x(b)|+b| d_x(a)|","count":12},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.","count":5},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4a810ebfdbc59ab3","tex":"\\int_{E(\\C)}","aliases":"∫_E(C) int C","display":null,"kind":"use","excerpt":"Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area fa","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-si","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Pillai and fixed perfect-power gaps. For x^m-y^n=k, k≠0 fixed, two deep contact packets collide through a residual channel whose prime support is fixed. This is more rigid than unrestricted abc because the output bandwidth is prescribed by rad(k).","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"use","excerpt":"Odd-prime coefficient depth. Let q≠ p be prime. Write q=ω(q)⟨ q⟩_p, where ω(q) is the Teichm uller factor. Since ⟨ q⟩_p^p-1=q^p-1, we have log_p⟨ q⟩_p = frac1p-1log_p(q^p-1). If s=v_p(q^p-1-1)>0, then v_p(log_p(q^p-1))=s. Therefore v_p(frac1plog_p⟨ q⟩_p) =s-1. Truncating at the available contact depth gives v_p(ℓ_p,e(q)) = min{e-1,s-1}.","count":12},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounde","count":12},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"lication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"The same-sign separation problem. After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let Λ be an integral Hecke lattice and let e_f be the rational projector onto the Frey eigenline. Define η_f = min{m≥1:m e_f(Λ)⊆Λ}. This is the integral eigenchannel separation denominator. It is an arithmetic analogue of ","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":6},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"The exact lattice extension. Define L=ker(Λ)∩Z^S and L_0=L∩ker( w). Because Λ is primitive, choose t∈Z^S with Λ(t)=1. Then Z^S=L⊕Z t. The image of M_a,b,c is generated by (1, w(t)) together with (0, w(L)). Its index in Z^2 is therefore the index of w(L) in Z. Theorem~[source reference] shows that this index is Q, so w(L)=QZ. Hence ω(x)=frac w· xQ i","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-61f317edc40cbb12","tex":"\\bigoplus_{i=1}^{r}","aliases":"⊕_i=1^r bigoplus","display":"\\bigoplus_{i=1}^{r}","kind":"use","excerpt":"A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity d","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b580a3a49114f560","tex":"\\bigoplus_{p^e\\parallel c}","aliases":"⊕_p^eparallel c bigoplus parallel","display":null,"kind":"use","excerpt":"Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split ","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-21bfcda8b3531dc2","tex":"\\bigoplus_{p^e\\parallel n}","aliases":"⊕_p^eparallel n bigoplus parallel","display":null,"kind":"use","excerpt":"[Exact contact quotient] For every positive integer n, rad(n)Z/nZ ≅ Z/(n/rad(n))Z. Its associated graded is gr(rad(n)Z/nZ) ≅ ⊕_p^eparallel n(F_p)^e-1.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-d797b415b8993005","tex":"\\bigoplus_{p\\mid c}","aliases":"⊕_pmid c bigoplus mid","display":null,"kind":"use","excerpt":"The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archim","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-407a0721917a90f9","tex":"\\bigoplus_{p\\in S}","aliases":"⊕_p∈ S bigoplus in","display":"\\bigoplus_{p\\in S}","kind":"use","excerpt":"The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an i","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation dir","count":3},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"Bernoulli--Witt and ErdHos--Moser. Faulhaber's formula rewrites 1^k+⋯+(m-1)^k=m^k as a Bernoulli-polynomial identity. The denominator primes of Bernoulli numbers, the finite-difference divisibility, and the separate 2-adic and odd-adic lift structures suggest a different contact object: Bernoulli contact crystal + denominator prime packet + finite-","count":9},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.","count":12},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.","count":12},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-25fb92dda41f7b3d","tex":"\\rangle_2","aliases":"⟩_2 rangle","display":"\\rangle_2","kind":"use","excerpt":"Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.","count":2},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-89db93c039cb2e59","tex":"\\rangle_2^2","aliases":"⟩_2^2 rangle","display":"\\rangle_2^2","kind":"use","excerpt":"Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-36c420d31e7a756e","tex":"\\rangle_p","aliases":"⟩_p rangle","display":"\\rangle_p","kind":"use","excerpt":"For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.","count":7},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-b13d0f69099cf79e","tex":"\\rangle_p^{p-1}","aliases":"⟩_p^p-1 rangle","display":"\\rangle_p^{p-1}","kind":"use","excerpt":"Odd-prime coefficient depth. Let q≠ p be prime. Write q=ω(q)⟨ q⟩_p, where ω(q) is the Teichm uller factor. Since ⟨ q⟩_p^p-1=q^p-1, we have log_p⟨ q⟩_p = frac1p-1log_p(q^p-1). If s=v_p(q^p-1-1)>0, then v_p(log_p(q^p-1))=s. Therefore v_p(frac1plog_p⟨ q⟩_p) =s-1. Truncating at the available contact depth gives v_p(ℓ_p,e(q)) = min{e-1,s-1}.","count":1},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"Why exact synthesis prevents free coordinate splitting. Let X and Y be normed spaces, and suppose J:X⟶ Y, S:Y⟶ X satisfy SJ=I_X. Then normx = normSJx ≤ normSnormJx. Therefore normJx ≥ fracnormxnormS. An analysis map may make coordinates smaller only by increasing the synthesis norm. This elementary inequality is the common source of several no-free","count":24},{"work":"prime-contact-coherence","url":"notation/prime-contact-coherence.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[First-order no-free-polyphase principle] Let S:(T_p,r)_p,r⟼ (∑_rT_p,r)_p be shell synthesis. The first-order additive map and first-order Wronskian factor through S. Hence Ker S is contained in the kernels of both first-order maps.","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"A · A Boolean algebra of observables, often sentences modulo provable equivalence.","count":26},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"notation-table","excerpt":"B_M · The Boolean algebra of values at one world.","count":20},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a79697eeeb533ed7","tex":"B_M","aliases":"B_M","display":"B_M","kind":"notation-table","excerpt":"B_M · The Boolean algebra of values at one world.","count":6},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"notation-table","excerpt":"C · A small category of typed world transformations.","count":8},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"notation-table","excerpt":"D_f · Additive defect between transported source evaluation and target evaluation.","count":23},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-54ff0eb6a1af0d6f","tex":"D_f","aliases":"D_f","display":"D_f","kind":"notation-table","excerpt":"D_f · Additive defect between transported source evaluation and target evaluation.","count":14},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"notation-table","excerpt":"E_Π · Weighted real orthogonal projection. Not a Boolean homomorphism.","count":16},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-b81ca8e6727fc293","tex":"E_\\Pi","aliases":"E_Π Pi","display":"E_\\Pi","kind":"notation-table","excerpt":"E_Π · Weighted real orthogonal projection. Not a Boolean homomorphism.","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"notation-table","excerpt":"D_f · Additive defect between transported source evaluation and target evaluation.","count":33},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-05c6ffe6831b15ec","tex":"\\Inv","aliases":"Inv I mathcal","display":"{\\mathcal I}","kind":"notation-table","excerpt":"I · Zero-defect observables. A subalgebra, generally not an ideal.","count":32},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"notation-table","excerpt":"K · Observables vanishing at every world. An ideal.","count":6},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"notation-table","excerpt":"Q_k · Real multiresolution detail projector for nested partitions.","count":17},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"notation-table","excerpt":"L_Π,U_Π · Canonical lower and upper invariant approximations.","count":19},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-b7c82805647052c3","tex":"L_\\Pi","aliases":"L_Π Pi","display":"L_\\Pi","kind":"notation-table","excerpt":"L_Π,U_Π · Canonical lower and upper invariant approximations.","count":11},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"notation-table","excerpt":"midrule Mmodels T · A supplied model, not an assertion that the ambient theory proves such a model exists.","count":58},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-7997c4f3be17c7ce","tex":"\\models","aliases":"models","display":null,"kind":"notation-table","excerpt":"midrule Mmodels T · A supplied model, not an assertion that the ambient theory proves such a model exists.","count":10},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"notation-table","excerpt":"P_s · Boolean retraction determined by a representative in each finite block.","count":22},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"notation-table","excerpt":"Δ_p,q · Typed discrepancy between parallel transport paths.","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-78aa8f72acca2b03","tex":"P_s","aliases":"P_s","display":"P_s","kind":"notation-table","excerpt":"P_s · Boolean retraction determined by a representative in each finite block.","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"notation-table","excerpt":"Q_k · Real multiresolution detail projector for nested partitions.","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"notation-table","excerpt":"Δ_p,q · Typed discrepancy between parallel transport paths.","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a9af1147d28b7af5","tex":"Q_k","aliases":"Q_k","display":"Q_k","kind":"notation-table","excerpt":"Q_k · Real multiresolution detail projector for nested partitions.","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"notation-table","excerpt":"P_s · Boolean retraction determined by a representative in each finite block.","count":13},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"notation-table","excerpt":"midrule Mmodels T · A supplied model, not an assertion that the ambient theory proves such a model exists.","count":14},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"notation-table","excerpt":"L_Π,U_Π · Canonical lower and upper invariant approximations.","count":17},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3731cf5cf895b337","tex":"U_\\Pi","aliases":"U_Π Pi","display":"U_\\Pi","kind":"notation-table","excerpt":"L_Π,U_Π · Canonical lower and upper invariant approximations.","count":11},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"notation-table","excerpt":"v_M · Semantic evaluation of observables.","count":16},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-20cd50e2bdb5b105","tex":"v_M","aliases":"v_M","display":"v_M","kind":"notation-table","excerpt":"v_M · Semantic evaluation of observables.","count":10},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"notation-table","excerpt":"Δ_p,q · Typed discrepancy between parallel transport paths.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-afa9ec6f446ca906","tex":"\\Delta_{p,q}","aliases":"Δ_p,q Delta","display":"\\Delta_{p,q}","kind":"notation-table","excerpt":"Δ_p,q · Typed discrepancy between parallel transport paths.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"notation-table","excerpt":"L_Π,U_Π · Canonical lower and upper invariant approximations.","count":30},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.","count":35},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.","count":100},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.","count":64},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6809700bed2de223","tex":"B_f","aliases":"B_f","display":"B_f","kind":"definition-passage","excerpt":"[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In ","count":8},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-162eee340db91ac5","tex":"B_N","aliases":"B_N","display":"B_N","kind":"definition-passage","excerpt":"[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In ","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-14dc0e60d226d94d","tex":"\\Bool","aliases":"Bool mathbf","display":"{\\mathbf{Bool}}","kind":"definition-passage","excerpt":"[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In ","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"definition-passage","excerpt":"[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In ","count":18},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-04e3bef7aa63bf65","tex":"v_N","aliases":"v_N","display":"v_N","kind":"definition-passage","excerpt":"[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In ","count":17},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Transport defect] For f:M→ N define D_f(a)=B_f(v_M(a))+v_N(a)∈ B_N. The terms on the right live in the same algebra. A proposed residual without this typing condition is not a mathematical definition.","count":34},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Resolution datum] A resolution datum consists of a small category C, a unital Boolean algebra A, a functor B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In ","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1f891a7f61f89a21","tex":"E_k","aliases":"E_k","display":"E_k","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-e9e6dade21e7f1b0","tex":"E_{k-1}","aliases":"E_k-1","display":"E_{k-1}","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-e749405022a66e6b","tex":"E_m","aliases":"E_m","display":"E_m","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-b5a58e2add722378","tex":"E_{\\min(i,j)}","aliases":"E_min(i,j) min","display":"E_{\\min(i,j)}","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-397b1dd304912323","tex":"E_{\\Pi_k}","aliases":"E_Π_k Pi","display":"E_{\\Pi_k}","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":20},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id operatorname","display":"{\\operatorname{id}}","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":7},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1d7d1fdf60d8fdac","tex":"\\neg","aliases":"neg","display":"\\neg","kind":"introduction-passage","excerpt":"B: C⟶Bool, and a unital Boolean homomorphism v_M:A→ B_M for each object M. Write B_f:B_M→ B_N for the map attached to f:M→ N. The category Bool here uses ordinary Boolean homomorphisms. It does not require preservation of arbitrary joins. In Boolean-ring notation, + is symmetric difference, multiplication is conjunction, and neg a=1+a. In particula","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-9a273618a8999b3a","tex":"Q_k^2","aliases":"Q_k^2","display":"Q_k^2","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4b23d09a909a4fdf","tex":"Q_l","aliases":"Q_l","display":"Q_l","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-db7c47d3a8fa6e3e","tex":"\\mathbb R^W","aliases":"R^W mathbb","display":"\\mathbb R^W","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":12},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a7db4623008ff11b","tex":"\\Pi_0","aliases":"Π_0 Pi","display":"\\Pi_0","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-74383ad386c1f57d","tex":"\\Pi_k","aliases":"Π_k Pi","display":"\\Pi_k","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-7b0f169a61793b84","tex":"\\Pi_m","aliases":"Π_m Pi","display":"\\Pi_m","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":6},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-374ba43ef897c062","tex":"\\sum_{k=1}^m","aliases":"∑_k=1^m sum","display":"\\sum_{k=1}^m","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"introduction-passage","excerpt":"[Nested resolution decomposition] Let Π_0,…,Π_m be successively finer partitions, using the same weights. Write E_k=E_Π_k and Q_k=E_k-E_k-1 for 1≤ k≤ m. Then E_iE_j=E_min(i,j), Q_k^2=Q_k, Q_kQ_l=0 (k≠ l), and every f∈ R^W has the orthogonal decomposition f=E_0f+∑_k=1^mQ_kf+(id-E_m)f. In particular the squared norm is the sum of the squared norms of","count":9},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forcing and the role of generic quotient models are developed in [source reference]. The completeness assumption here is internal to M. The map b↦1_b∈ G is a Boolean homomorphism but need not preserve every externall","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4d8d8e9c58052e6d","tex":"\\mathbf1_{b\\in G}","aliases":"1_b∈ G mathbf in","display":"\\mathbf1_{b\\in G}","kind":"use","excerpt":"φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forcing and the role of generic quotient models are developed in [source reference]. The completeness assumption here is internal to M. The map b↦1_b∈ G is a Boolean homomorphism but need not preserve every externall","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-ee91bc1f54b74071","tex":"\\mathbf1_{M[G]\\models\\varphi}","aliases":"1_M[G]modelsφ mathbf models varphi","display":null,"kind":"use","excerpt":"A changing observable and an invariant fragment. Taking two-valued fibers and an arrow M→ M[G] gives D_f(φ)=1_Mmodelsφ +1_M[G]modelsφ. For suitable forcing, CH can change, so it is not invariant under unrestricted forcing. Both CH and its negation are forceable in the relevant forcing setting [source reference]. This is not a contradiction: the qua","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-65e90b44a80ba110","tex":"\\mathbf1_{M\\models\\varphi}","aliases":"1_Mmodelsφ mathbf models varphi","display":null,"kind":"use","excerpt":"A changing observable and an invariant fragment. Taking two-valued fibers and an arrow M→ M[G] gives D_f(φ)=1_Mmodelsφ +1_M[G]modelsφ. For suitable forcing, CH can change, so it is not invariant under unrestricted forcing. Both CH and its negation are forceable in the relevant forcing setting [source reference]. This is not a contradiction: the qua","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A compon","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"With root value a_0=0 the labeling above is recovered. With root value a_0=1 the reconstruction is (1,0,1,0). The derivative cannot distinguish these two complementary labelings. That ambiguity is exactly the one-dimensional invariant sector of this connected graph.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty ","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty ","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty ","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-d9b71db518364d73","tex":"A_M","aliases":"A_M","display":"A_M","kind":"use","excerpt":"Transported syntax and parameters. For parameters or changing languages, replace A by a functor A_M with translation maps α_f:A_M→ A_N. The correct defect is D_f(a)=B_fv_M(a)+v_Nα_f(a), a∈ A_M. Assuming both transport systems are functorial, the composition law becomes D_gf(a)=B_gD_f(a)+D_g(α_f(a)). The same cancellation proves it. The fixed-A theo","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"Transported syntax and parameters. For parameters or changing languages, replace A by a functor A_M with translation maps α_f:A_M→ A_N. The correct defect is D_f(a)=B_fv_M(a)+v_Nα_f(a), a∈ A_M. Assuming both transport systems are functorial, the composition law becomes D_gf(a)=B_gD_f(a)+D_g(α_f(a)). The same cancellation proves it. The fixed-A theo","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-07112285aabb492d","tex":"a_N","aliases":"a_N","display":"a_N","kind":"use","excerpt":"Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty ","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4c8131608f522ec2","tex":"a_{n+1}","aliases":"a_n+1","display":"a_{n+1}","kind":"use","excerpt":"Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-fc85a7802b707b87","tex":"a_{s(C)}","aliases":"a_s(C)","display":"a_{s(C)}","kind":"use","excerpt":"[Finite resolution envelope] Every Boolean retraction P: F_2^W→ I_Π is obtained by choosing one representative s(C)∈ C per block and setting P_s(a)_w=a_s(C) (w∈ C). For each fixed observable a, its possible projected values are exactly {P_s(a):s chooses one point in each block} ={b∈ I_Π:L_Π(a)≤ b≤ U_Π(a)}. Consequently, L_Π(a)=bigwedge_sP_s(a), U_Π","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-84e075bd0dff7e4d","tex":"a_u","aliases":"a_u","display":"a_u","kind":"use","excerpt":"Finite integrability and reconstruction. Let G=(W,E) be a finite simple undirected graph. Edges record which worlds are directly compared. Choose an orientation only to index them. For a∈ F_2^W set (da)_{u,v}=a_u+a_v. Unlike the semantic algebra A, the space F_2^W contains every binary labeling. An actual family of models and a selected language ma","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-eae1dc41a7db82ef","tex":"a_v","aliases":"a_v","display":"a_v","kind":"use","excerpt":"Finite integrability and reconstruction. Let G=(W,E) be a finite simple undirected graph. Edges record which worlds are directly compared. Choose an orientation only to index them. For a∈ F_2^W set (da)_{u,v}=a_u+a_v. Unlike the semantic algebra A, the space F_2^W contains every binary labeling. An actual family of models and a selected language ma","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-7608774b4f52a149","tex":"\\mathcal A_\\Pi","aliases":"A_Π mathcal Pi","display":"\\mathcal A_\\Pi","kind":"use","excerpt":"[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A compon","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-187c26e46b9a87ad","tex":"\\mathbb B","aliases":"B mathbb","display":"\\mathbb B","kind":"use","excerpt":"Forcing as a conditional set-theoretic instance. Fix externally a countable transitive MmodelsZFC. Let B∈ M be a Boolean algebra that M regards as complete, and let G be an M-generic ultrafilter. For a sentence φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forci","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"[Defect calculus] Put u=B_f∘ v_M and w=v_N. Then D_f(0) · =D_f(1)=0, D_f(neg a)=D_f(a), D_f(a+b) · =D_f(a)+D_f(b), D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including ","count":14},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-d9845db9d41b34bd","tex":"B_{f_n\\cdots f_1}","aliases":"B_f_n⋯ f_1 cdots","display":"B_{f_n\\cdots f_1}","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8f434d227f864429","tex":"B_{f_n\\cdots f_{k+1}}","aliases":"B_f_n⋯ f_k+1 cdots","display":"B_{f_n\\cdots f_{k+1}}","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a0f42eb1ac146472","tex":"B_g","aliases":"B_g","display":"B_g","kind":"use","excerpt":"D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b).","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3112c801d2948784","tex":"B_{gf}","aliases":"B_gf","display":"B_{gf}","kind":"use","excerpt":"Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_f","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-163eaab4cc2384a2","tex":"B_{\\id_M}","aliases":"B_id_M id operatorname","display":"B_{{\\operatorname{id}}_M}","kind":"use","excerpt":"· =B_gfv_M(a)+v_P(a). The identity case follows from B_id_M=id.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-037c6283d361171d","tex":"b_M","aliases":"b_M","display":"b_M","kind":"use","excerpt":"Coherent sections and definability. The limit varprojlim_ CB consists of families (b_M)_M satisfying B_f(b_M)=b_N. Thus a∈ I precisely when Ev(a) is a coherent family. But not every coherent family need be represented by a sentence, and distinct sentences can have the same evaluations.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-9634f79e2739535a","tex":"b_N","aliases":"b_N","display":"b_N","kind":"use","excerpt":"Coherent sections and definability. The limit varprojlim_ CB consists of families (b_M)_M satisfying B_f(b_M)=b_N. Thus a∈ I precisely when Ev(a) is a coherent family. But not every coherent family need be represented by a sentence, and distinct sentences can have the same evaluations.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-dd6f9b6573e5306c","tex":"B_{\\operatorname{target}(f)}","aliases":"B_target(f) operatorname","display":"B_{\\operatorname{target}(f)}","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-b61c22b86277b610","tex":"\\bigvee","aliases":"bigvee","display":"\\bigvee","kind":"use","excerpt":"The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. ","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1b918b6bfd07eeba","tex":"\\bigvee_s","aliases":"bigvee_s bigvee","display":"\\bigvee_s","kind":"use","excerpt":"is obtained by choosing one representative s(C)∈ C per block and setting P_s(a)_w=a_s(C) (w∈ C). For each fixed observable a, its possible projected values are exactly {P_s(a):s chooses one point in each block} ={b∈ I_Π:L_Π(a)≤ b≤ U_Π(a)}. Consequently, L_Π(a)=bigwedge_sP_s(a), U_Π(a)=bigvee_sP_s(a). If a is mixed on exactly k blocks, there are exa","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8d8b466588eef281","tex":"\\bigvee_{u\\in C}","aliases":"bigvee_u∈ C bigvee in","display":"\\bigvee_{u\\in C}","kind":"use","excerpt":"The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. ","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-d13b1528b249ec85","tex":"\\bigwedge","aliases":"bigwedge","display":"\\bigwedge","kind":"use","excerpt":"The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. ","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-297a9cb27347b94a","tex":"\\bigwedge_s","aliases":"bigwedge_s bigwedge","display":"\\bigwedge_s","kind":"use","excerpt":"is obtained by choosing one representative s(C)∈ C per block and setting P_s(a)_w=a_s(C) (w∈ C). For each fixed observable a, its possible projected values are exactly {P_s(a):s chooses one point in each block} ={b∈ I_Π:L_Π(a)≤ b≤ U_Π(a)}. Consequently, L_Π(a)=bigwedge_sP_s(a), U_Π(a)=bigvee_sP_s(a). If a is mixed on exactly k blocks, there are exa","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-113961b6df199e09","tex":"\\bigwedge_{u\\in C}","aliases":"bigwedge_u∈ C bigwedge in","display":"\\bigwedge_{u\\in C}","kind":"use","excerpt":"The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. ","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1240e4448dd87a5e","tex":"\\bnorm","aliases":"bnorm","display":null,"kind":"use","excerpt":"res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-ff66021ec1d8e804","tex":"\\bot","aliases":"bot","display":"\\bot","kind":"use","excerpt":"There is an equally important distinction between transformation and information refinement. In the information order on {bot,0,1}, where bot≤0 and bot≤1 but 0 and 1 are incomparable, a monotone refinement cannot change a settled 0 into a settled 1. A truth-changing world transformation is therefore not a monotone refinement of a fixed partial trut","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-f157f71e0163a897","tex":"\\Box","aliases":"Box","display":"\\Box","kind":"use","excerpt":"On a reflexive accessibility relation, DiamondBoxφ says that a world can be reached from which every later accessible world satisfies φ. A path that happens to stay at a φ-world does not establish this. A two-state switch system with arrows in both directions and identity loops already separates the notions.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":14},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":7},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-e7e42689204088d5","tex":"C^0","aliases":"C^0","display":"C^0","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-bb4b39a0fa3e20d1","tex":"C^1","aliases":"C^1","display":"C^1","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-137bde843960a611","tex":"c_M","aliases":"c_M","display":"c_M","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-c1ecf2aa8f993509","tex":"c_N","aliases":"c_N","display":"c_N","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":17},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-63963f1cc71314e6","tex":"d^0","aliases":"d^0","display":"d^0","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-767b68bb8b3ef50e","tex":"D_{f_k}","aliases":"D_f_k","display":"D_{f_k}","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-9d9ed55d11b4efec","tex":"D_g","aliases":"D_g","display":"D_g","kind":"use","excerpt":"D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b).","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-b33683d18bcf6999","tex":"D_{gf}","aliases":"D_gf","display":"D_{gf}","kind":"use","excerpt":"D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b).","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8f43de6e8de26ef5","tex":"D_{\\id_M}","aliases":"D_id_M id operatorname","display":"D_{{\\operatorname{id}}_M}","kind":"use","excerpt":"D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b).","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-5136fc4246e7d497","tex":"\\Diamond","aliases":"Diamond","display":null,"kind":"use","excerpt":"On a reflexive accessibility relation, DiamondBoxφ says that a world can be reached from which every later accessible world satisfies φ. A path that happens to stay at a φ-world does not establish this. A two-state switch system with arrows in both directions and identity loops already separates the notions.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"m edges, and c connected components. item ker d consists exactly of the componentwise constant labelings and has dimension c. item An edge field r∈ F_2^E equals da for some a if and only if its sum around every cycle is zero. item Every integrable r has exactly 2^c reconstructions. Fixing one vertex value per component makes the reconstruction uniq","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"A two-world obstruction. [No symmetric Boolean projector] Let A= F_2^2 and I={(0,0),(1,1)}. There is no Boolean retraction P:A→ I that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto I. Write e_1=(1,0) and e_2=(0,1). Swapping coordinates fixes every element of I, so equivariance forces P(e_1","count":7},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"A two-world obstruction. [No symmetric Boolean projector] Let A= F_2^2 and I={(0,0),(1,1)}. There is no Boolean retraction P:A→ I that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto I. Write e_1=(1,0) and e_2=(0,1). Swapping coordinates fixes every element of I, so equivariance forces P(e_1","count":6},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"A two-world obstruction. [No symmetric Boolean projector] Let A= F_2^2 and I={(0,0),(1,1)}. There is no Boolean retraction P:A→ I that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto I. Write e_1=(1,0) and e_2=(0,1). Swapping coordinates fixes every element of I, so equivariance forces P(e_1","count":6},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-da6e0ead8ae0dd34","tex":"\\Ev","aliases":"Ev operatorname","display":"{\\operatorname{Ev}}","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F F_2 mathbb","display":"{\\mathbb F_2}","kind":"use","excerpt":"For sentences, A can be the Lindenbaum Boolean algebra of L-sentences modulo provable equivalence over T. An ordinary model gives B_M= F_2 and v_M([φ])=1 precisely when Mmodelsφ. The datum can also be studied without claiming that its valuations arise from models of ZFC.","count":21},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-7d5445169d32374e","tex":"\\F^{\\mathbb N}","aliases":"F^ N F_2^ N F mathbb mathbb mathbb","display":"{\\mathbb F_2}^{\\mathbb N}","kind":"use","excerpt":"For example, a countable sentence algebra cannot cover all coherent families over a countably infinite discrete diagram with B_M= F_2, since that limit is F_2^ N and is uncountable. The correct phrase is therefore definable coherent observables, not ``all global sections.'' No sheaf has yet been defined. A sheaf additionally requires a site and a g","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-587483d2f15c9ec1","tex":"\\F^2","aliases":"F^2 F_2^2 F mathbb","display":"{\\mathbb F_2}^2","kind":"use","excerpt":"A two-world obstruction. [No symmetric Boolean projector] Let A= F_2^2 and I={(0,0),(1,1)}. There is no Boolean retraction P:A→ I that commutes with swapping the two coordinates. There is not even a swap-equivariant additive retraction onto I. Write e_1=(1,0) and e_2=(0,1). Swapping coordinates fixes every element of I, so equivariance forces P(e_1","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-47d35de01c4f6dae","tex":"\\F^E","aliases":"F^E F_2^E F mathbb","display":"{\\mathbb F_2}^E","kind":"use","excerpt":"[Finite residual integrability] Suppose G has n vertices, m edges, and c connected components. item ker d consists exactly of the componentwise constant labelings and has dimension c. item An edge field r∈ F_2^E equals da for some a if and only if its sum around every cycle is zero. item Every integrable r has exactly 2^c reconstructions. Fixing on","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-274b080a08341fb5","tex":"\\F^W","aliases":"F^W F_2^W F mathbb","display":"{\\mathbb F_2}^W","kind":"use","excerpt":"Finite integrability and reconstruction. Let G=(W,E) be a finite simple undirected graph. Edges record which worlds are directly compared. Choose an orientation only to index them. For a∈ F_2^W set (da)_{u,v}=a_u+a_v. Unlike the semantic algebra A, the space F_2^W contains every binary labeling. An actual family of models and a selected language ma","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-2f6b5c0f84584144","tex":"f_k","aliases":"f_k","display":"f_k","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-0be3a2ded0c0a234","tex":"f_{k+1}","aliases":"f_k+1","display":"f_{k+1}","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-e3f4ba757b6d2cf8","tex":"f_n","aliases":"f_n","display":"f_n","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-73373fd911c8ae54","tex":"f_u","aliases":"f_u","display":"f_u","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a31d948777ad9761","tex":"f_w","aliases":"f_w","display":"f_w","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Finite integrability and reconstruction. Let G=(W,E) be a finite simple undirected graph. Edges record which worlds are directly compared. Choose an orientation only to index them. For a∈ F_2^W set (da)_{u,v}=a_u+a_v. Unlike the semantic algebra A, the space F_2^W contains every binary labeling. An actual family of models and a selected language ma","count":10},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b).","count":7},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4000f6722b507ccb","tex":"g_C","aliases":"g_C","display":"g_C","kind":"use","excerpt":"d E_Π is the unique orthogonal projector onto the real vector space of blockwise constant functions. It is positive, fixes 1, and is equivariant under weight-preserving permutations that preserve the partition. For binary a, L_Π(a)≤ E_Π(a)≤ U_Π(a). The formula is blockwise constant and fixes such functions. For every blockwise constant g, ⟨ f-E_Π f","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6b5c141258d79d03","tex":"g_w","aliases":"g_w","display":"g_w","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic re","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-ff6b0f89378f4410","tex":"h^{-1}","aliases":"h^-1","display":"h^{-1}","kind":"use","excerpt":"Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic relations are preserved by the definition of an isomorphism. Boolean connectives commute with the induction. For existential quantification, transport a witness through h in one direction an","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"A common presentation exists, but need not explain anything. [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ ","count":9},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1ce608af84df2a4c","tex":"\\id_M","aliases":"id_M id operatorname","display":"{\\operatorname{id}}_M","kind":"use","excerpt":"D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b).","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im operatorname","display":"{\\operatorname{im}}","kind":"use","excerpt":"[Invariant algebra and residual quotient] The set I=ker D=⋂_f{a:B_fv_M(a)=v_N(a)} is a unital Boolean subalgebra of A. There is a natural exact sequence of F_2-vector spaces 0⟶ I⟶ AxrightarrowDim D⟶0, and therefore a natural additive isomorphism A/ I ≅ im D. Two observables have the same residual profile exactly when their symmetric difference belo","count":7},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-5fd6bf12dd49680e","tex":"\\Inv_\\Pi","aliases":"Inv_Π I_Π Inv Pi mathcal Pi","display":"{\\mathcal I}_\\Pi","kind":"use","excerpt":"The finite resolution envelope theorem. Let Π be a partition of a finite nonempty set W. Let I_Π⊆ F_2^W be the Boolean algebra of functions constant on each block. For w∈ C∈Π define L_Π(a)_w=bigwedge_u∈ Ca_u, U_Π(a)_w=bigvee_u∈ Ca_u. These lower and upper approximations are the classical equivalence-relation rough-set operators [source reference]. ","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"If r is a Boolean retraction, its kernel is an ideal J and the additive decomposition is A= I⊕ J. In these coordinates multiplication is (i,x)(j,y)=(ij, iy+jx+xy). It is not generally a direct-product multiplication. Writing a direct sum symbol must not silently erase these cross terms.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"[Invariant algebra and residual quotient] The set I=ker D=⋂_f{a:B_fv_M(a)=v_N(a)} is a unital Boolean subalgebra of A. There is a natural exact sequence of F_2-vector spaces 0⟶ I⟶ AxrightarrowDim D⟶0, and therefore a natural additive isomorphism A/ I ≅ im D. Two observables have the same residual profile exactly when their symmetric difference belo","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3aef166dcb861ab6","tex":"L_{\\Pi_0}","aliases":"L_Π_0 Pi","display":"L_{\\Pi_0}","kind":"use","excerpt":"For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3e2f4f7a6ac57bb4","tex":"L_{\\Pi_1}","aliases":"L_Π_1 Pi","display":"L_{\\Pi_1}","kind":"use","excerpt":"For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic re","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"use","excerpt":"A common presentation exists, but need not explain anything. [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ ","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"use","excerpt":"Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-df92634da3a3e068","tex":"M_i","aliases":"M_i","display":"M_i","kind":"use","excerpt":"A common presentation exists, but need not explain anything. [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ ","count":8},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-e41b4efa8bb66cc1","tex":"\\Mor","aliases":"Mor operatorname","display":"{\\operatorname{Mor}}","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"For example, a countable sentence algebra cannot cover all coherent families over a countably infinite discrete diagram with B_M= F_2, since that limit is F_2^ N and is uncountable. The correct phrase is therefore definable coherent observables, not ``all global sections.'' No sheaf has yet been defined. A sheaf additionally requires a site and a g","count":5},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty ","count":13},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob operatorname","display":"{\\operatorname{Ob}}","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"[Additive splitting] Over a metatheory with choice, the inclusion I⊆ A always admits an F_2-linear retraction r:A→ I. For P=ι r and R=id+P, P^2=P, R^2=R, PR=RP=0, a=P(a)+R(a). This is a vector-space decomposition. The retraction need not preserve multiplication or be canonical. Choose a vector-space basis of I and extend it to a basis of A. Let r f","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-d692f9ac1c18626a","tex":"\\Pow","aliases":"Pow P mathcal","display":"{\\mathcal P}","kind":"use","excerpt":"A common presentation exists, but need not explain anything. [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ ","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"[Finite residual integrability] Suppose G has n vertices, m edges, and c connected components. item ker d consists exactly of the componentwise constant labelings and has dimension c. item An edge field r∈ F_2^E equals da for some a if and only if its sum around every cycle is zero. item Every integrable r has exactly 2^c reconstructions. Fixing on","count":12},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"[Additive splitting] Over a metatheory with choice, the inclusion I⊆ A always admits an F_2-linear retraction r:A→ I. For P=ι r and R=id+P, P^2=P, R^2=R, PR=RP=0, a=P(a)+R(a). This is a vector-space decomposition. The retraction need not preserve multiplication or be canonical. Choose a vector-space basis of I and extend it to a basis of A. Let r f","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"ommon presentation exists, but need not explain anything [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} r","count":9},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"[Nontrivial holonomy with an exact defect] Let the one-object category be the group with generator s and relation s^2=1. Put X= F_2^2 and let T_s swap coordinates. This is a valid nontrivial action. Its invariant vectors are (0,0) and (1,1). For x=(1,0), T_sx+x=(1,1)≠0, but T_s(T_sx+x)+(T_sx+x)=0. The second equation is the transported composition ","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-422d05ebd68c604a","tex":"\\Sat","aliases":"Sat operatorname","display":"{\\operatorname{Sat}}","kind":"use","excerpt":"[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A compon","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6b5f747d208bbe25","tex":"\\Sat_\\Pi","aliases":"Sat_Π Sat Pi operatorname Pi","display":"{\\operatorname{Sat}}_\\Pi","kind":"use","excerpt":"[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A compon","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1084f45e6b3e947e","tex":"\\supp","aliases":"supp operatorname","display":"{\\operatorname{supp}}","kind":"use","excerpt":"[Residual support determines ambiguity] Take Π to be the connected components of a finite graph G. Define A_Π(a)=U_Π(a)+L_Π(a). Then A_Π(a) is the indicator of the union of components containing a nonzero edge of da: supp A_Π(a)=Sat_Π(supp da). Here edge support is saturated by taking all vertices in components containing a supported edge. A compon","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4b7fbd7d66e3a28d","tex":"T_f","aliases":"T_f","display":"T_f","kind":"use","excerpt":"To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same mo","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-807e619b0b36adc6","tex":"T_p","aliases":"T_p","display":"T_p","kind":"use","excerpt":"To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same mo","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-79b896ccd56f4035","tex":"T_q","aliases":"T_q","display":"T_q","kind":"use","excerpt":"To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same mo","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6d6c4f072fd1f994","tex":"T_s","aliases":"T_s","display":"T_s","kind":"use","excerpt":"[Nontrivial holonomy with an exact defect] Let the one-object category be the group with generator s and relation s^2=1. Put X= F_2^2 and let T_s swap coordinates. This is a valid nontrivial action. Its invariant vectors are (0,0) and (1,1). For x=(1,0), T_sx+x=(1,1)≠0, but T_s(T_sx+x)+(T_sx+x)=0. The second equation is the transported composition ","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-89ca555db043e94d","tex":"\\operatorname{target}","aliases":"target operatorname","display":"\\operatorname{target}","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"[Defect calculus] Put u=B_f∘ v_M and w=v_N. Then D_f(0) · =D_f(1)=0, D_f(neg a)=D_f(a),","count":11},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"ommon presentation exists, but need not explain anything [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} r","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-85b4dd54a3e91835","tex":"U_{\\Pi_0}","aliases":"U_Π_0 Pi","display":"U_{\\Pi_0}","kind":"use","excerpt":"For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-ccc653d54ff821c1","tex":"U_{\\Pi_1}","aliases":"U_Π_1 Pi","display":"U_{\\Pi_1}","kind":"use","excerpt":"For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-278a332de5ba46a3","tex":"v_{M_0}","aliases":"v_M_0","display":"v_{M_0}","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-5ac96b70a75a21a9","tex":"v_{M_n}","aliases":"v_M_n","display":"v_{M_n}","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-93b491f1b04ab025","tex":"v_P","aliases":"v_P","display":"v_P","kind":"use","excerpt":"Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b). The mixed terms cancel in characteristic two. For composition, functoriality and additivity give B_g(D_f(a))+D_g(a) · =B_gB_f","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-81a5693cba0f984d","tex":"v_U","aliases":"v_U","display":"v_U","kind":"use","excerpt":"This example can occur as an invariant core of a resolution datum, rather than merely as an arbitrary subalgebra. Take worlds to be the nonprincipal ultrafilters on N, join every ordered pair by an arrow, and use v_U(S)=1 iff S∈ U. All these worlds agree on finite and cofinite sets. Every infinite coinfinite set belongs to some nonprincipal ultrafi","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"use","excerpt":"Coherent sections and definability. The limit varprojlim_ CB consists of families (b_M)_M satisfying B_f(b_M)=b_N. Thus a∈ I precisely when Ev(a) is a coherent family. But not every coherent family need be represented by a sentence, and distinct sentences can have the same evaluations.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3a5c0a4ba7d8145c","tex":"\\varprojlim_{\\mathcal C}","aliases":"varprojlim_ C varprojlim mathcal","display":null,"kind":"use","excerpt":"Coherent sections and definability. The limit varprojlim_ CB consists of families (b_M)_M satisfying B_f(b_M)=b_N. Thus a∈ I precisely when Ev(a) is a coherent family. But not every coherent family need be represented by a sentence, and distinct sentences can have the same evaluations.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8695764692c4bf3e","tex":"\\vec a","aliases":"vec a vec","display":"\\vec a","kind":"use","excerpt":"[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic re","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-5ccb84c570fcf376","tex":"\\vec x","aliases":"vec x vec","display":"\\vec x","kind":"use","excerpt":"[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic re","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-29eb290757438b56","tex":"\\vec x_i","aliases":"vec x_i vec","display":"\\vec x_i","kind":"use","excerpt":"res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8d4c6c49046bf7c0","tex":"\\vec\\tau","aliases":"vecτ vec tau","display":"\\vec\\tau","kind":"use","excerpt":"Forcing as a conditional set-theoretic instance. Fix externally a countable transitive MmodelsZFC. Let B∈ M be a Boolean algebra that M regards as complete, and let G be an M-generic ultrafilter. For a sentence φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forci","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-5d599659dd54a00c","tex":"\\vec\\tau_G","aliases":"vecτ_G vec tau","display":"\\vec\\tau_G","kind":"use","excerpt":"Forcing as a conditional set-theoretic instance. Fix externally a countable transitive MmodelsZFC. Let B∈ M be a Boolean algebra that M regards as complete, and let G be an M-generic ultrafilter. For a sentence φ with names vecτ, the standard forcing truth lemma gives M[G]modelsφ(vecτ_G) Longleftrightarrow bnormφ(vecτ)_ B^M∈ G. Boolean-valued forci","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"[Defect calculus] Put u=B_f∘ v_M and w=v_N. Then D_f(0) · =D_f(1)=0, D_f(neg a)=D_f(a),","count":15},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coor","count":6},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels","count":9},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-0597cb543b16b78f","tex":"x_i^2","aliases":"x_i^2","display":"x_i^2","kind":"use","excerpt":"Why the symbolic checks support universal identities. A Boolean polynomial is represented as a finite sum of squarefree monomials over F_2. Addition cancels duplicate monomials. Multiplication unions their variable supports, using x_i^2=x_i. These rules are valid in every Boolean ring. A polynomial identity whose difference reduces to the empty nor","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-10665c77b53ab88a","tex":"X_M","aliases":"X_M","display":"X_M","kind":"use","excerpt":"To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same mo","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-854a0c87655a5383","tex":"X_N","aliases":"X_N","display":"X_N","kind":"use","excerpt":"To retain information about structures rather than just truth values, give each world a fiber X_M and each arrow an actual transport map T_f:X_M→ X_N. Parallel paths p,q:M→ N can then be compared only because their source and target types agree. For linear fibers define the path discrepancy Δ_p,q=T_p-T_q. If the paths are declared to be the same mo","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"D_f(ab) · =D_f(a)u(b)+w(a)D_f(b). For composable arrows MxrightarrowfNxrightarrowgP, D_gf(a)=B_g(D_f(a))+D_g(a), D_id_M(a)=0. Both u and w preserve 0, 1, addition, and multiplication. The first identities follow immediately, including D_f(1+a)=(1+u(a))+(1+w(a))=D_f(a). For the product rule, expand (u(a)+w(a))u(b)+w(a)(u(b)+w(b)) =u(a)u(b)+w(a)w(b).","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels","count":8},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"res. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coordinatewise. Set bnormx=y={i:x_i=y_i}, bnormR(vec x)={i:M_imodels","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-e349851823caf578","tex":"\\ZFC","aliases":"ZFC mathsf","display":"{\\mathsf{ZFC}}","kind":"use","excerpt":"Worlds, presentations, and the ambient theory. Fix a first-order language L and an L-theory T. The intended case is the membership language with T=ZFC, but the algebraic results do not depend on that choice. Work in an external metatheory sufficient to discuss sets, categories, and satisfaction for set-sized structures.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coor","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-1c18740f8f5dc62c","tex":"|_{\\Inv}","aliases":"|_Inv |_ I Inv mathcal","display":"|_{{\\mathcal I}}","kind":"use","excerpt":"[The precise section statement] Let K=⋂_Mker v_M. Then K is a Boolean ideal contained in I, and I/K ≅ im(Ev|_ I) ⊆ varprojlim_ CB as Boolean algebras. The kernel of a Boolean-ring homomorphism is an ideal. An observable whose evaluations are all zero has zero defect, so K⊆ I. Restrict Ev to I and apply the Boolean-ring first isomorphism theorem.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"Transported syntax and parameters. For parameters or changing languages, replace A by a functor A_M with translation maps α_f:A_M→ A_N. The correct defect is D_f(a)=B_fv_M(a)+v_Nα_f(a), a∈ A_M. Assuming both transport systems are functorial, the composition law becomes D_gf(a)=B_gD_f(a)+D_g(α_f(a)). The same cancellation proves it. The fixed-A theo","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-0bad476b0101a5e4","tex":"\\alpha_f","aliases":"α_f alpha","display":"\\alpha_f","kind":"use","excerpt":"Transported syntax and parameters. For parameters or changing languages, replace A by a functor A_M with translation maps α_f:A_M→ A_N. The correct defect is D_f(a)=B_fv_M(a)+v_Nα_f(a), a∈ A_M. Assuming both transport systems are functorial, the composition law becomes D_gf(a)=B_gD_f(a)+D_g(α_f(a)). The same cancellation proves it. The fixed-A theo","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"[Generating transformations suffice] If a family Γ generates the arrows of C under composition, then I=⋂_f∈Γker D_f. Adding admissible transformations can only shrink the invariant core. The composition law propagates vanishing along finite composites. Identity defects vanish. The monotonicity statement is an immediate property of intersections.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"use","excerpt":"[Additive splitting] Over a metatheory with choice, the inclusion I⊆ A always admits an F_2-linear retraction r:A→ I. For P=ι r and R=id+P, P^2=P, R^2=R, PR=RP=0, a=P(a)+R(a). This is a vector-space decomposition. The retraction need not preserve multiplication or be canonical. Choose a vector-space basis of I and extend it to a basis of A. Let r f","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-13d997755df30de3","tex":"\\mu_u","aliases":"μ_u mu","display":"\\mu_u","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-cf5ce06964140e8a","tex":"\\mu_w","aliases":"μ_w mu","display":"\\mu_w","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-c3acc8569f115745","tex":"\\Pi_1","aliases":"Π_1 Pi","display":"\\Pi_1","kind":"use","excerpt":"For the original Boolean bounds, finer partitions instead give monotone information: L_Π_0(a)≤ L_Π_1(a)≤⋯≤ a ≤⋯≤ U_Π_1(a)≤ U_Π_0(a). The ambiguity support shrinks as blocks are refined. This is an order-theoretic resolution process, not an additive Boolean detail expansion.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic re","count":14},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Modulo U_i, equality is exactly equality of the ith coordinate. The map [x]_U_i↦ x_i is well-defined, injective, and surjective. It preserves functions and relations, so it is an isomorphism to M_i. Equation~[source reference] also gives Boolean value I to each axiom satisfied by every coordinate structure.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coor","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-e7ec69cdf8302a6c","tex":"\\prod_{f\\in\\Mor(\\mathcal C)}","aliases":"∏_f∈Mor( C) prod in Mor mathcal prod in operatorname mathcal","display":"\\prod_{f\\in{\\operatorname{Mor}}(\\mathcal C)}","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-34198ed3688237b7","tex":"\\prod_{i<k}","aliases":"∏_i<k prod","display":"\\prod_{i<k}","kind":"use","excerpt":"[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty ","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-bd57437bf263dc5d","tex":"\\prod_{i=1}^n","aliases":"∏_i=1^n prod","display":"\\prod_{i=1}^n","kind":"use","excerpt":"[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty ","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-5d5e9748fdaba01e","tex":"\\prod_{i>k}","aliases":"∏_i>k prod","display":"\\prod_{i>k}","kind":"use","excerpt":"[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty ","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-9c0ad644774e2f10","tex":"\\prod_{i\\in I}","aliases":"∏_i∈ I prod in","display":"\\prod_{i\\in I}","kind":"use","excerpt":"and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} recovers M_i up to isomorphism. If every M_imodels T, every axiom of T has Boolean value I in M. Take the domain X=∏_i∈ I|M_i|. Interpret function symbols coor","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-58265ad25c39e014","tex":"\\prod_{M\\in\\Ob(\\mathcal C)}","aliases":"∏_M∈Ob( C) prod in Ob mathcal prod in operatorname mathcal","display":"\\prod_{M\\in{\\operatorname{Ob}}(\\mathcal C)}","kind":"use","excerpt":"What the invariant core actually is. For the fixed-A datum set C^0=∏_M∈Ob( C)B_M, C^1=∏_f∈Mor( C)B_target(f). As additive groups these are vector spaces over F_2. Define (d^0c)_f=B_f(c_M)+c_N, Ev(a)=(v_M(a))_M, D=d^0Ev. Here Ev is a Boolean homomorphism, while d^0 is generally only additive.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-3ffe7c3f6b22ea54","tex":"\\sum_{C\\in\\Pi}","aliases":"∑_C∈Π sum in Pi","display":"\\sum_{C\\in\\Pi}","kind":"use","excerpt":"d E_Π is the unique orthogonal projector onto the real vector space of blockwise constant functions. It is positive, fixes 1, and is equivariant under weight-preserving permutations that preserve the partition. For binary a, L_Π(a)≤ E_Π(a)≤ U_Π(a). The formula is blockwise constant and fixes such functions. For every blockwise constant g, ⟨ f-E_Π f","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-dfd8567f7f80b3f0","tex":"\\sum_{k=1}^n","aliases":"∑_k=1^n sum","display":"\\sum_{k=1}^n","kind":"use","excerpt":"[Product telescope] For a_1,…,a_n∈ A, D_f(∏_i=1^n a_i) =∑_k=1^n(∏_i<kw(a_i)) D_f(a_k)(∏_i>ku(a_i)). Expand D_f(a_k)=u(a_k)+w(a_k). Every intermediate product occurs twice, once with transition index k and once with transition index k+1. They cancel, leaving the all-u product plus the all-w product. This also gives an induction on n, with the empty ","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-db564a7249f0c29a","tex":"\\sum_{u\\in C}","aliases":"∑_u∈ C sum in","display":"\\sum_{u\\in C}","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-b20dc6b3443297d7","tex":"\\sum_{w\\in C}","aliases":"∑_w∈ C sum in","display":"\\sum_{w\\in C}","kind":"use","excerpt":"d E_Π is the unique orthogonal projector onto the real vector space of blockwise constant functions. It is positive, fixes 1, and is equivariant under weight-preserving permutations that preserve the partition. For binary a, L_Π(a)≤ E_Π(a)≤ U_Π(a). The formula is blockwise constant and fixes such functions. For every blockwise constant g, ⟨ f-E_Π f","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-555b22e1f4e3d9a2","tex":"\\sum_{w\\in W}","aliases":"∑_w∈ W sum in","display":"\\sum_{w\\in W}","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"[Defect calculus] Put u=B_f∘ v_M and w=v_N. Then D_f(0) · =D_f(1)=0, D_f(neg a)=D_f(a),","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[Completeness of the ambient algebra is not enough] Let A= P( N) and let I be the finite-cofinite subalgebra. There is no Boolean retraction r:A→ I fixing I. Indeed, for every S⊆ N and n∈ N, r(S)∩{n}=r(S∩{n})=S∩{n}. Thus r(S) has exactly the same members as S, so r(S)=S. An infinite coinfinite S gives a contradiction.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"[Coordinate changes preserve truth] If h:M≅ N is an isomorphism of L-structures, then for every formula φ(vec x) and tuple vec a in M, Mmodelsφ(vec a) Longleftrightarrow Nmodelsφ(hvec a). Consequently, two worlds that disagree on an L-sentence are not related by a mere isomorphic change of coordinates in that language. Induct on formulas. Atomic re","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"Stabilization requires a path quantifier. For a path M_0→ M_1→⋯ with two-valued observations, write a_n=v_M_n(a). Along that path, eventual stability means ∃ N ∀ n≥ N a_n=a_N, equivalently a_n+a_n+1=0 eventually. This differs from requiring stability on every continuation from some reachable world.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"ommon presentation exists, but need not explain anything [Boolean product presentation] Assume the axiom of choice in the metatheory. Let I be a nonempty set and let (M_i)_i∈ I be a set-indexed family of nonempty L-structures. There is a full P(I)-valued structure M such that for each i∈ I, quotienting at the principal ultrafilter U_i={S⊆ I:i∈ S} r","count":7},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"If r is a Boolean retraction, its kernel is an ideal J and the additive decomposition is A= I⊕ J. In these coordinates multiplication is (i,x)(j,y)=(ij, iy+jx+xy). It is not generally a direct-product multiplication. Writing a direct sum symbol must not silently erase these cross terms.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"[Invariant algebra and residual quotient] The set I=ker D=⋂_f{a:B_fv_M(a)=v_N(a)} is a unital Boolean subalgebra of A. There is a natural exact sequence of F_2-vector spaces 0⟶ I⟶ AxrightarrowDim D⟶0, and therefore a natural additive isomorphism A/ I ≅ im D. Two observables have the same residual profile exactly when their symmetric difference belo","count":3},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-8ce0f73f833f418a","tex":"\\bigcap_f","aliases":"⋂_f bigcap","display":"\\bigcap_f","kind":"use","excerpt":"[Invariant algebra and residual quotient] The set I=ker D=⋂_f{a:B_fv_M(a)=v_N(a)} is a unital Boolean subalgebra of A. There is a natural exact sequence of F_2-vector spaces 0⟶ I⟶ AxrightarrowDim D⟶0, and therefore a natural additive isomorphism A/ I ≅ im D. Two observables have the same residual profile exactly when their symmetric difference belo","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-62e178a289abec44","tex":"\\bigcap_{f\\in\\Gamma}","aliases":"⋂_f∈Γ bigcap in Gamma","display":"\\bigcap_{f\\in\\Gamma}","kind":"use","excerpt":"[Generating transformations suffice] If a family Γ generates the arrows of C under composition, then I=⋂_f∈Γker D_f. Adding admissible transformations can only shrink the invariant core. The composition law propagates vanishing along finite composites. Identity defects vanish. The monotonicity statement is an immediate property of intersections.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-19a5d90019322d3c","tex":"\\bigcap_M","aliases":"⋂_M bigcap","display":"\\bigcap_M","kind":"use","excerpt":"[The precise section statement] Let K=⋂_Mker v_M. Then K is a Boolean ideal contained in I, and I/K ≅ im(Ev|_ I) ⊆ varprojlim_ CB as Boolean algebras. The kernel of a Boolean-ring homomorphism is an ideal. An observable whose evaluations are all zero has zero defect, so K⊆ I. Restrict Ev to I and apply the Boolean-ring first isomorphism theorem.","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"R(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An induction on formulas gives bnormφ(vec x)={i:M_imodelsφ(vec x_i)}. Only the quantifier step requires comment. A coordinate i belongs to ⋃_y∈ Xbnormφ(y,vec x) precisely when a witness ","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-d30b7e9b1b9abba4","tex":"\\bigcup_{y\\in X}","aliases":"⋃_y∈ X bigcup in","display":"\\bigcup_{y\\in X}","kind":"use","excerpt":"R(vec x)={i:M_imodels R(vec x_i)}. The equality and substitution axioms hold coordinatewise. Extend the semantics by complements, intersections, and joins in P(I). An induction on formulas gives bnormφ(vec x)={i:M_imodelsφ(vec x_i)}. Only the quantifier step requires comment. A coordinate i belongs to ⋃_y∈ Xbnormφ(y,vec x) precisely when a witness ","count":1},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"Reconstruction with nontrivial transport. For a path M_0xrightarrowf_1⋯xrightarrowf_nM_n, repeated use of [source reference] gives v_M_n(a)=B_f_n⋯ f_1v_M_0(a) +∑_k=1^n B_f_n⋯ f_k+1D_f_k(a), where the empty tail transport is the identity. This formula needs no inverse transport maps. It reconstructs a reachable target from a source value and the tra","count":4},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":2},{"work":"resolution-semantics","url":"notation/resolution-semantics.html#symbol-cb89ae5751eaa9b5","tex":"\\rangle_\\mu","aliases":"⟩_μ rangle mu","display":"\\rangle_\\mu","kind":"use","excerpt":"A real-valued realization of Komorebi multiresolution. The Boolean obstruction does not prohibit a canonical projector after extra structure and a different codomain are supplied. Let W be finite and assign positive weights μ_w. On R^W use ⟨ f,g⟩_μ=∑_w∈ Wμ_w f_wg_w. For a partition Π, define (E_Π f)_w= frac∑_u∈ Cμ_u f_u∑_u∈ Cμ_u (w∈ C∈Π).","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":129},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":441},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[One-step zoo operator] For A⊆ U, define Φ(A):= A ∪ mathscr T[A] ∪ mathscr C[A] ∪ mathscr L[A] ∪ mathscr D[A], where each term denotes all defined outputs of the corresponding constructors applied to inputs from A.","count":178},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly on","count":138},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-da62b0f97d94856e","tex":"A_r","aliases":"A_r","display":"A_r","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e50ca40bad47ed08","tex":"A_s","aliases":"A_s","display":"A_s","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e1ab77241aa5e261","tex":"\\mathrm{add}","aliases":"add mathrm","display":"\\mathrm{add}","kind":"definition-passage","excerpt":"[Additive two-axis system] An additive two-axis system over a small resolution category R is a functor F: R^op⟶ Cat^add. Thus every fiber F(r) is additive and every coarsening functor C_s→ r is additive.","count":20},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Interaction residual] The interaction residual of this specified square is Θ=Cofib(Cofib(A→ B)⟶ Cofib(C→ D)). The specified square, including its coherent homotopy, is part of the input.","count":96},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.","count":105},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2d4104b41ee90adc","tex":"b_{r\\to s}","aliases":"b_r→ s to","display":"b_{r\\to s}","kind":"definition-passage","excerpt":"[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":148},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"definition-passage","excerpt":"[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly on","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6db54b248f3844fc","tex":"C_b","aliases":"C_b","display":"C_b","kind":"definition-passage","excerpt":"[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly on","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"definition-passage","excerpt":"[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly on","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3bb0acec894ec095","tex":"C_{s\\to r}","aliases":"C_s→ r to","display":"C_{s\\to r}","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":33},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b370f7a2878491ab","tex":"C_{t\\to r}","aliases":"C_t→ r to","display":"C_{t\\to r}","kind":"definition-passage","excerpt":"[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-81522c35b81aa6de","tex":"C_{t\\to s}","aliases":"C_t→ s to","display":"C_{t\\to s}","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5b779703a5dd1742","tex":"\\Cat","aliases":"Cat mathbf","display":"{\\mathbf{Cat}}","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b5af1fa1d9752111","tex":"\\mathbf{Cat}","aliases":"Cat mathbf","display":"\\mathbf{Cat}","kind":"definition-passage","excerpt":"[Additive two-axis system] An additive two-axis system over a small resolution category R is a functor F: R^op⟶ Cat^add. Thus every fiber F(r) is additive and every coarsening functor C_s→ r is additive.","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ac5eb4afe01fa275","tex":"\\mathbf{Cat}^{\\mathrm{add}}","aliases":"Cat^add mathbf mathrm","display":"\\mathbf{Cat}^{\\mathrm{add}}","kind":"definition-passage","excerpt":"[Additive two-axis system] An additive two-axis system over a small resolution category R is a functor F: R^op⟶ Cat^add. Thus every fiber F(r) is additive and every coarsening functor C_s→ r is additive.","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8ab9f5fcc8340ebc","tex":"\\Cofib","aliases":"Cofib operatorname","display":"{\\operatorname{Cofib}}","kind":"definition-passage","excerpt":"[Stable mixed residual] Assume each T_r is a small stable ∞-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For r≤ s and X∈ T_s, define Δ^A_s,r(X) := Cofib(η^A_s,r,X) ∈ T_r.","count":30},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refineme","count":80},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7c6669ab098dda59","tex":"\\mathbb D","aliases":"D mathbb","display":"\\mathbb D","kind":"definition-passage","excerpt":"class of vertical generating 1-cells, interpreted as resolution generators; item designated generating 2-cells, interpreted as exchange or compatibility cells. Generator boundaries may be well-typed words, but no composition of the generators is identified with another generator at the signature level. An unlabelled realization of S in a strict dou","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"definition-passage","excerpt":"th] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i","count":15},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-014b3f233c470cd1","tex":"D_{\\mathrm{tot}}","aliases":"D_tot mathrm","display":"D_{\\mathrm{tot}}","kind":"definition-passage","excerpt":"th] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].","count":51},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"definition-passage","excerpt":"[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→","count":9},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-19439c241400d146","tex":"\\mathcal E_j","aliases":"E_j mathcal","display":"\\mathcal E_j","kind":"definition-passage","excerpt":"[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-555e836cd877fd04","tex":"\\mathcal E_{j-1}","aliases":"E_j-1 mathcal","display":"\\mathcal E_{j-1}","kind":"definition-passage","excerpt":"hain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, whi","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c12fd86ce71cc224","tex":"E_{T,C}","aliases":"E_T,C","display":"E_{T,C}","kind":"definition-passage","excerpt":"[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":53},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":57},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":17},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"definition-passage","excerpt":"[Operation orbit of the seeds] In a stable indexed system, let mathscr P be the designated exact transformations and coarsenings. For each r, let G_r contain every image at level r of a seed under a finite well-typed word in mathscr P, including the empty word. Define Z_r^thick:=Thick( G_r). Refinement is included here only when an actual exact ref","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.","count":83},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ba92c85b61fe7055","tex":"\\mathcal G_r","aliases":"G_r mathcal","display":"\\mathcal G_r","kind":"definition-passage","excerpt":"[Operation orbit of the seeds] In a stable indexed system, let mathscr P be the designated exact transformations and coarsenings. For each r, let G_r contain every image at level r of a seed under a finite well-typed word in mathscr P, including the empty word. Define Z_r^thick:=Thick( G_r). Refinement is included here only when an actual exact ref","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-54cfb11ff55cbff5","tex":"g_{\\Seeds_0}","aliases":"g_Seeds_0 g_Σ_0 Seeds Sigma","display":"g_{{\\Sigma}_0}","kind":"definition-passage","excerpt":"[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.","count":76},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"ymbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · ","count":77},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a32ef3542749d85b","tex":"H_{r\\to s}","aliases":"H_r→ s to","display":"H_{r\\to s}","kind":"definition-passage","excerpt":"[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Coherent exchange-event diagram] For a finite event poset P, a coherent exchange-event diagram is a functor X:NJ(P)⟶ S into a stable ∞-category. Here N denotes the nerve. Its value X_I is the semantic state after the ideal I of events has been completed. Values on cover arrows realize the elementary exchanges, and its higher simplices supply the c","count":116},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":139},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id operatorname","display":"{\\operatorname{id}}","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":15},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-73e09fa23cd75f65","tex":"\\mathrm{inv}","aliases":"inv mathrm","display":"\\mathrm{inv}","kind":"definition-passage","excerpt":"[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.","count":9},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"(h,j) · prec(i,j) · · when h>i and both events exist. These relations are acyclic: along a generating relation the transformation index decreases or the coarsening index increases, with at least one strict change. Let J(P_W) be the poset of order ideals, ordered by inclusion.","count":66},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-146cbf287aa37a5f","tex":"\\mathcal J","aliases":"J mathcal","display":"\\mathcal J","kind":"definition-passage","excerpt":"[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.","count":23},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly on","count":77},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d591dbdaa3d0a539","tex":"\\mathcal J_{\\mathscr M}","aliases":"J_mathscr M mathcal mathscr","display":"\\mathcal J_{\\mathscr M}","kind":"definition-passage","excerpt":"[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.","count":41},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"definition-passage","excerpt":"[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.","count":17},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"ymbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · ","count":65},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1741945a8587f585","tex":"\\mathcal K_{\\mathscr M}","aliases":"K_mathscr M mathcal mathscr","display":"\\mathcal K_{\\mathscr M}","kind":"definition-passage","excerpt":"[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.","count":15},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refineme","count":41},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"definition-passage","excerpt":"[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3920e880fa517994","tex":"\\Lift","aliases":"Lift operatorname","display":"{\\operatorname{Lift}}","kind":"definition-passage","excerpt":"[Lift set] For r≤ s and X∈ T_r, define the set of isomorphism classes of lifts Lift_r→ s(X) := { Y∈Ob( T_s):C_s→ r(Y)≅ X }/≅.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-de4f2a5f46a2e7f4","tex":"\\Lift_{r\\to s}","aliases":"Lift_r→ s Lift to operatorname to","display":"{\\operatorname{Lift}}_{r\\to s}","kind":"definition-passage","excerpt":"[Lift set] For r≤ s and X∈ T_r, define the set of isomorphism classes of lifts Lift_r→ s(X) := { Y∈Ob( T_s):C_s→ r(Y)≅ X }/≅.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"definition-passage","excerpt":"[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"definition-passage","excerpt":"[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly on","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"definition-passage","excerpt":"[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.","count":191},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":131},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"definition-passage","excerpt":"[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refineme","count":82},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"definition-passage","excerpt":"[Coherent exchange-event diagram] For a finite event poset P, a coherent exchange-event diagram is a functor X:NJ(P)⟶ S into a stable ∞-category. Here N denotes the nerve. Its value X_I is the semantic state after the ideal I of events has been completed. Values on cover arrows realize the elementary exchanges, and its higher simplices supply the c","count":58},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"definition-passage","excerpt":"[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.","count":19},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":340},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-282ab7db56b5f152","tex":"\\mathcal N_{\\mathscr M}","aliases":"N_mathscr M mathcal mathscr","display":"\\mathcal N_{\\mathscr M}","kind":"definition-passage","excerpt":"[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.","count":14},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ff513f63b59b94cf","tex":"\\mathrm{nf}","aliases":"nf mathrm","display":"\\mathrm{nf}","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"definition-passage","excerpt":"[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":16},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ad1c2358d828bc8a","tex":"\\mathcal O^{\\mathrm{inv}}_{\\mathscr M}","aliases":"O^inv_mathscr M mathcal mathrm mathscr","display":"\\mathcal O^{\\mathrm{inv}}_{\\mathscr M}","kind":"definition-passage","excerpt":"[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fbb7e4d4cad7cccb","tex":"\\mathcal O_{\\mathscr M}","aliases":"O_mathscr M mathcal mathscr","display":"\\mathcal O_{\\mathscr M}","kind":"definition-passage","excerpt":"[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob operatorname","display":"{\\operatorname{Ob}}","kind":"definition-passage","excerpt":"[Resolved object] A resolved object is a pair (r,X) with X∈Ob( T_r).","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e942b1c655fe6bde","tex":"\\Obs","aliases":"Obs operatorname","display":"{\\operatorname{Obs}}","kind":"definition-passage","excerpt":"[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fbff96c6e7e9b058","tex":"\\Obs_{\\mathscr M}","aliases":"Obs_mathscr M Obs mathscr operatorname mathscr","display":"{\\operatorname{Obs}}_{\\mathscr M}","kind":"definition-passage","excerpt":"[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"set Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as t","count":96},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":561},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0c723e0565a0c218","tex":"P_W","aliases":"P_W","display":"P_W","kind":"definition-passage","excerpt":"set Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as t","count":11},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-003001ebf5867324","tex":"\\prec","aliases":"prec","display":null,"kind":"definition-passage","excerpt":"ymbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly once. Define P_W as the partial order generated by (i,k) · prec(i,j) · ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"definition-passage","excerpt":"[Directed annihilator property] The constructor system mathscr M has the directed annihilator property if for every U,V∈mathscr M there exist P,Q,W∈mathscr M and natural equivalences W≃ P∘ U≃ Q∘ V. Thus any two constructor words admit a common postcomposition. With multiplication written as composition, P∘ U=Q∘ V is a common left multiple. The disp","count":39},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"[Stable obstruction category and obstruction class] The stable obstruction category is the Verdier quotient O_mathscr M := Z/ N_mathscr M, with quotient functor q: Z⟶ O_mathscr M. For X∈ Z, define its obstruction class Obs_mathscr M(X):=q(X).","count":65},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-14a6f6662724dcb7","tex":"q^{\\mathrm{inv}}","aliases":"q^inv mathrm","display":"q^{\\mathrm{inv}}","kind":"definition-passage","excerpt":"[Constructor-invariant thick saturation] For a small stable constructor system ( Z,mathscr M) as above, put J_mathscr M :=Thick{W(K):W∈mathscr M, K∈ K_mathscr M}. Its quotient is denoted q^inv: Z⟶ O^inv_mathscr M := Z/ J_mathscr M. The identity constructor ensures N_mathscr M⊆ J_mathscr M.","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.","count":148},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.","count":214},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"definition-passage","excerpt":"[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4bdf782a5897d64d","tex":"r_b","aliases":"r_b","display":"r_b","kind":"definition-passage","excerpt":"[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-53c7a7c85c8235b5","tex":"\\Rcat","aliases":"Rcat R mathbf","display":"{\\mathbf R}","kind":"definition-passage","excerpt":"[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.","count":20},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-898f9860d883e903","tex":"\\Rcat^{op}","aliases":"Rcat^op R^op Rcat mathbf","display":"{\\mathbf R}^{op}","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":61},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7013b2edec14a3ac","tex":"\\mathbb S","aliases":"S mathbb","display":"\\mathbb S","kind":"definition-passage","excerpt":"[Seeded two-axis double computad] A seeded two-axis double computad S consists of a small double computad together with: item a distinguished set Σ⊆ S_0 of seed objects; item a designated class of horizontal generating 1-cells, interpreted as transformation generators; item a designated class of vertical generating 1-cells, interpreted as resolutio","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":17},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b617b296bbfd924b","tex":"\\mathfrak S","aliases":"S mathfrak","display":"\\mathfrak S","kind":"definition-passage","excerpt":"[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refineme","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.","count":148},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e7567b294dabec8f","tex":"\\mathbb S_0","aliases":"S_0 mathbb","display":"\\mathbb S_0","kind":"definition-passage","excerpt":"[Seeded two-axis double computad] A seeded two-axis double computad S consists of a small double computad together with: item a distinguished set Σ⊆ S_0 of seed objects; item a designated class of horizontal generating 1-cells, interpreted as transformation generators; item a designated class of vertical generating 1-cells, interpreted as resolutio","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1f022a4acc79055b","tex":"\\mathcal S_j","aliases":"S_j mathcal","display":"\\mathcal S_j","kind":"definition-passage","excerpt":"[Strict chain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-10a32da074a2b0a3","tex":"\\Seeds","aliases":"Seeds Σ Sigma","display":"{\\Sigma}","kind":"definition-passage","excerpt":"[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refineme","count":20},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bdc5a55a707bf098","tex":"\\Seeds_0","aliases":"Seeds_0 Σ_0 Seeds Sigma","display":"{\\Sigma}_0","kind":"definition-passage","excerpt":"[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refineme","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f46d04844e1df5ff","tex":"\\Seeds_r","aliases":"Seeds_r Σ_r Seeds Sigma","display":"{\\Sigma}_r","kind":"definition-passage","excerpt":"[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e5fd5471d426d46d","tex":"\\operatorname{St}","aliases":"St operatorname","display":"\\operatorname{St}","kind":"definition-passage","excerpt":"[Universal stable mixed residual] For X∈St^add(F(s)), define Δ^A_s,r(X) := Cofib ( widehatη^A_s,r,X ) ∈ St^add(F(r)).","count":16},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-46c98c256fb0bcbe","tex":"\\operatorname{St}^{\\mathrm{add}}","aliases":"St^add operatorname mathrm","display":"\\operatorname{St}^{\\mathrm{add}}","kind":"definition-passage","excerpt":"[Universal stable mixed residual] For X∈St^add(F(s)), define Δ^A_s,r(X) := Cofib ( widehatη^A_s,r,X ) ∈ St^add(F(r)).","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5377c3e3a904ef57","tex":"\\operatorname{St}^{\\mathrm{add}}_{\\Rcat}","aliases":"St^add_Rcat St^add_ R operatorname mathrm Rcat operatorname mathrm mathbf","display":"\\operatorname{St}^{\\mathrm{add}}_{{\\mathbf R}}","kind":"definition-passage","excerpt":"[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo ","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refineme","count":61},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.","count":56},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6da3583735a2a7a0","tex":"T'","aliases":"T'","display":"T'","kind":"definition-passage","excerpt":"[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"definition-passage","excerpt":"[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly on","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d1c70721598d3d2e","tex":"T_a","aliases":"T_a","display":"T_a","kind":"definition-passage","excerpt":"[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly on","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-219b7412a89e1904","tex":"T_i","aliases":"T_i","display":"T_i","kind":"definition-passage","excerpt":"[Exchange event poset] Fix a finite word W in two types of labelled symbols. Number its transformation symbols T_1,…,T_a and its coarsening symbols C_1,…,C_b in their original left-to-right order. An exchange event is a pair (i,j)∈ E(W) Longleftrightarrow T_i occurs to the left of C_j in W. The event swaps the relative order of this pair exactly on","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-51a5a90f6276d9e3","tex":"T_{i,j}","aliases":"T_i,j","display":"T_{i,j}","kind":"definition-passage","excerpt":"hain-natural realization] Fix a finite resolution chain r_0≤⋯≤ r_b. At each level choose a category E_j of complexes and a functor L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, whi","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-409324e1ebce0322","tex":"T_{i,j-1}","aliases":"T_i,j-1","display":"T_{i,j-1}","kind":"definition-passage","excerpt":"or L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-89d63f757e34d418","tex":"\\Tcat","aliases":"Tcat T mathbf","display":"{\\mathbf T}","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":25},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-48eb15ee21ddb643","tex":"\\Tcat_r","aliases":"Tcat_r T_r Tcat mathbf","display":"{\\mathbf T}_r","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":15},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-708341d2d3e9f442","tex":"\\Tcat_s","aliases":"Tcat_s T_s Tcat mathbf","display":"{\\mathbf T}_s","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ed373149983c58b7","tex":"\\mathrm{thick}","aliases":"thick mathrm","display":"\\mathrm{thick}","kind":"definition-passage","excerpt":"[Operation orbit of the seeds] In a stable indexed system, let mathscr P be the designated exact transformations and coarsenings. For each r, let G_r contain every image at level r of a seed under a finite well-typed word in mathscr P, including the empty word. Define Z_r^thick:=Thick( G_r). Refinement is included here only when an actual exact ref","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4c98537cc9d89006","tex":"\\operatorname{Thick}","aliases":"Thick operatorname","display":"\\operatorname{Thick}","kind":"definition-passage","excerpt":"[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.","count":12},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-70a3fc170e8a1c43","tex":"\\mathrm{tot}","aliases":"tot mathrm","display":"\\mathrm{tot}","kind":"definition-passage","excerpt":"th] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Write D_i:=Cofib(f_i","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Seed signature] A seed signature on U is data S=(Σ_0,mathscr T,mathscr C,mathscr L,mathscr D), where: item Σ_0⊆ U is the initial seed set; item mathscr T is a collection of partial unary transformation constructors on U; item mathscr C is the collection of allowed coarsening constructors; item mathscr L assigns to an object its admissible refineme","count":71},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Directed annihilator property] The constructor system mathscr M has the directed annihilator property if for every U,V∈mathscr M there exist P,Q,W∈mathscr M and natural equivalences W≃ P∘ U≃ Q∘ V. Thus any two constructor words admit a common postcomposition. With multiplication written as composition, P∘ U=Q∘ V is a common left multiple. The disp","count":56},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"definition-passage","excerpt":"[Inversion number] For a word W, let ι(W) be the number of ordered pairs (T,C) in which a transformation symbol occurs to the left of a resolution symbol when the chosen normal form requires all resolution symbols first.","count":44},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-49ff1aed73cd067a","tex":"W_{\\mathrm{nf}}","aliases":"W_nf mathrm","display":"W_{\\mathrm{nf}}","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8d3781ce495edcf4","tex":"\\widehat A","aliases":"widehat A widehat","display":"\\widehat A","kind":"definition-passage","excerpt":"[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo ","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-95abc9f3c0f96458","tex":"\\widehat A_r","aliases":"widehat A_r widehat","display":"\\widehat A_r","kind":"definition-passage","excerpt":"[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo ","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7bc1f2bd77a0ea58","tex":"\\widehat C","aliases":"widehat C widehat","display":"\\widehat C","kind":"definition-passage","excerpt":"phstable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a84fc6e6cbe65f0f","tex":"\\widehat C_{s\\to r}","aliases":"widehat C_s→ r widehat to","display":"\\widehat C_{s\\to r}","kind":"definition-passage","excerpt":"phstable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9bc4ac46f9980428","tex":"\\widehat\\eta","aliases":"widehatη widehat eta","display":"\\widehat\\eta","kind":"definition-passage","excerpt":"[Universal stable mixed residual] For X∈St^add(F(s)), define Δ^A_s,r(X) := Cofib ( widehatη^A_s,r,X ) ∈ St^add(F(r)).","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-16da42d5dd5639c6","tex":"\\widehat\\eta^A_{s,r,X}","aliases":"widehatη^A_s,r,X widehat eta","display":"\\widehat\\eta^A_{s,r,X}","kind":"definition-passage","excerpt":"[Universal stable mixed residual] For X∈St^add(F(s)), define Δ^A_s,r(X) := Cofib ( widehatη^A_s,r,X ) ∈ St^add(F(r)).","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Resolved object] A resolved object is a pair (r,X) with X∈Ob( T_r).","count":185},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":9},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9a1069a254bccd17","tex":"X_I","aliases":"X_I","display":"X_I","kind":"definition-passage","excerpt":"[Coherent exchange-event diagram] For a finite event poset P, a coherent exchange-event diagram is a functor X:NJ(P)⟶ S into a stable ∞-category. Here N denotes the nerve. Its value X_I is the semantic state after the ideal I of events has been completed. Values on cover arrows realize the elementary exchanges, and its higher simplices supply the c","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f3be737b42076753","tex":"X_m","aliases":"X_m","display":"X_m","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":29},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.","count":61},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c656dc90ffbe53cd","tex":"\\mathcal Z","aliases":"Z mathcal","display":"\\mathcal Z","kind":"definition-passage","excerpt":"[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.","count":42},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5d9c7369ea0b6ebd","tex":"\\mathcal Z_r","aliases":"Z_r mathcal","display":"\\mathcal Z_r","kind":"definition-passage","excerpt":"[Stable seeded zoo] Fix seed sets Σ_r⊆Ob(F(r)). A stable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo ","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7f7016e50f3d9ad2","tex":"\\mathcal Z_r^{\\mathrm{thick}}","aliases":"Z_r^thick mathcal mathrm","display":"\\mathcal Z_r^{\\mathrm{thick}}","kind":"definition-passage","excerpt":"[Operation orbit of the seeds] In a stable indexed system, let mathscr P be the designated exact transformations and coarsenings. For each r, let G_r contain every image at level r of a seed under a finite well-typed word in mathscr P, including the empty word. Define Z_r^thick:=Thick( G_r). Refinement is included here only when an actual exact ref","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2a8630bce38ec51e","tex":"\\mathcal Z_s","aliases":"Z_s mathcal","display":"\\mathcal Z_s","kind":"definition-passage","excerpt":"phstable seeded sub-zoo of St^add_ RF is a family of full stable subcategories Z_r⊆St^add(F(r)) such that: item Z_r contains the images of all seeds in Σ_r; item widehat A_r( Z_r)⊆ Z_r for every designated transformation generator; item widehat C_s→ r( Z_s)⊆ Z_r for every r≤ s. The minimal stable zoo is the intersection of all such families.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b6da4fe71cad814c","tex":"\\Zoo","aliases":"Zoo Z mathcal","display":"{\\mathcal Z}","kind":"definition-passage","excerpt":"[Generated zoo] The least fixed point in [source reference] is denoted Z(Σ_0).","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Resolution branching number] When Lift_r→ s(X) is finite, define b_r→ s(X):=|Lift_r→ s(X)|. When b_r→ s(X)>0, its logarithm H_r→ s(X):=log b_r→ s(X) may be regarded as a crude resolution information measuring how much ambiguity is hidden by coarsening.","count":25},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":20},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"definition-passage","excerpt":"[Stable mixed residual] Assume each T_r is a small stable ∞-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For r≤ s and X∈ T_s, define Δ^A_s,r(X) := Cofib(η^A_s,r,X) ∈ T_r.","count":38},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"definition-passage","excerpt":"[First distinguishing scale] When finite, define δ(X,Y):=ν(X,Y)+1. This is the first resolution at which X and Y become distinguishable.","count":25},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-132d6c340df152b8","tex":"\\Delta^A_{s,r}","aliases":"Δ^A_s,r Delta","display":"\\Delta^A_{s,r}","kind":"definition-passage","excerpt":"[Stable mixed residual] Assume each T_r is a small stable ∞-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For r≤ s and X∈ T_s, define Δ^A_s,r(X) := Cofib(η^A_s,r,X) ∈ T_r.","count":18},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bdff22fc223e045b","tex":"\\Delta_{T,C}","aliases":"Δ_T,C Delta","display":"\\Delta_{T,C}","kind":"definition-passage","excerpt":"[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":26},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c3b034bc27677b65","tex":"\\eta^A_{r,r}","aliases":"η^A_r,r eta","display":"\\eta^A_{r,r}","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e17b9e82c2b65c32","tex":"\\eta^A_{s,r}","aliases":"η^A_s,r eta","display":"\\eta^A_{s,r}","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-acc8123f34db30bc","tex":"\\eta^A_{s,r,X}","aliases":"η^A_s,r,X eta","display":"\\eta^A_{s,r,X}","kind":"definition-passage","excerpt":"[Stable mixed residual] Assume each T_r is a small stable ∞-category and the coarsening and transformation functors are exact, with homotopy-coherent exchange data. For r≤ s and X∈ T_s, define Δ^A_s,r(X) := Cofib(η^A_s,r,X) ∈ T_r.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c6425cac68596df0","tex":"\\eta^A_{t,r}","aliases":"η^A_t,r eta","display":"\\eta^A_{t,r}","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1eacc73ffcd21420","tex":"\\eta^A_{t,s}","aliases":"η^A_t,s eta","display":"\\eta^A_{t,s}","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d5b3c430327aa61e","tex":"\\eta_X","aliases":"η_X eta","display":"\\eta_X","kind":"definition-passage","excerpt":"[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"definition-passage","excerpt":"[Interaction residual] The interaction residual of this specified square is Θ=Cofib(Cofib(A→ B)⟶ Cofib(C→ D)). The specified square, including its coherent homotopy, is part of the input.","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"definition-passage","excerpt":"[Inversion number] For a word W, let ι(W) be the number of ordered pairs (T,C) in which a transformation symbol occurs to the left of a resolution symbol when the chosen normal form requires all resolution symbols first.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"[One-step zoo operator] For A⊆ U, define Φ(A):= A ∪ mathscr T[A] ∪ mathscr C[A] ∪ mathscr L[A] ∪ mathscr D[A], where each term denotes all defined outputs of the corresponding constructors applied to inputs from A.","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"definition-passage","excerpt":"or L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0d0d671532562498","tex":"\\chi_{i,j}","aliases":"χ_i,j chi","display":"\\chi_{i,j}","kind":"definition-passage","excerpt":"or L_j: E_j→ S_j into a stable ∞-category. A model example is bounded complexes of finitely generated projective modules mapping to their derived category. Specify chain-level functors C_j: E_j→ E_j-1, T_i,j: E_j→ E_j, which induce exact functors in the chosen stable realizations, and strict natural transformations χ_i,j:T_i,j-1C_jLongrightarrow C_","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"definition-passage","excerpt":"[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-19a9e4e6c616bdbb","tex":"\\Psi^n","aliases":"Ψ^n Psi","display":"\\Psi^n","kind":"definition-passage","excerpt":"[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":29},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.","count":146},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"definition-passage","excerpt":"[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"definition-passage","excerpt":"[Operationally killable subcategory] Define K_mathscr M := {X∈ Z:∃ W∈mathscr M, W(X)≃0}. Let N_mathscr M := Thick( K_mathscr M) be the smallest thick stable subcategory containing all operationally killable objects.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"definition-passage","excerpt":"[Growth function] When each finite stage contains finitely many isomorphism classes, define g_Σ_0(n):=|Ψ^n(∅)|.","count":15},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":156},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"definition-passage","excerpt":"[Lax resolution-compatible transformation] A lax resolution-compatible transformation consists of endofunctors A_r: T_r→ T_r for every r∈ R, together with natural transformations η^A_s,r: A_rC_s→ r Longrightarrow C_s→ rA_s (r≤ s), satisfying identity and composition coherence: η^A_r,r=id and, for r≤ s≤ t, η^A_t,r = (C_s→ rη^A_t,s) ∘ (η^A_s,rC_t→ s)","count":14},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":22},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":14},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.","count":21},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0c14761dac85dc4d","tex":"\\sim_0","aliases":"∼_0 sim","display":"\\sim_0","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b2d1385478f04d67","tex":"\\sim_n","aliases":"∼_n sim","display":"\\sim_n","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9895b3ce95bf0424","tex":"\\sim_{n+1}","aliases":"∼_n+1 sim","display":"\\sim_{n+1}","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fdd9d81e23a10461","tex":"\\sim_r","aliases":"∼_r sim","display":"\\sim_r","kind":"definition-passage","excerpt":"[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"definition-passage","excerpt":"[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].","count":98},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"definition-passage","excerpt":"[Indistinguishability at resolution r] We write X∼_r Y when C_t→ r(X)≅ C_t→ r(Y) in T_r.","count":34},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"[Resolution category] A resolution category is a small poset (R,≤) regarded as a thin category R. An arrow r→ s exists exactly when r≤ s. We interpret s as at least as fine as r.","count":32},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Separation depth] Let S be a set of states, with equivalence relations ∼_n for every n∈ N={0,1,…}. Assume ∼_n+1 ⊆ ∼_n, ∼_0=S× S. Define ν(X,Y):=sup{n∈ N:X∼_nY}∈ N∪{∞}. The universal zeroth level makes this set nonempty. Assume separation, meaning that X∼_nY for every n implies X=Y. The states can, for example, be coherent towers equipped with the ","count":28},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"definition-passage","excerpt":"[Split residual exchange] A chosen exchange morphism η:TC⇒ CT' is split residual if it is naturally identified with the projection in a decomposition TC(X)≅ CT'(X)⊕ E_T,C(X). Here E_T,C is the split source-side remainder. It is not denoted by Δ, because in the stable convention of this paper Cofib(η_X)≃ E_T,C(X)[1], E_T,C(X)≃Δ_T,C(X)[-1].","count":29},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Normalization path] Let S be a stable ∞-category, or a stable enhancement with functorial cofibers. A normalization path for a mixed word W evaluated on s is a finite chain X_0=W(s)xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m=W_nf(s), where every f_i is obtained by applying an exact surrounding context to one elementary exchange morphism. Wri","count":11},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Indexed transformation system] An indexed transformation system over R is a strict functor F: R^op⟶Cat. For r∈ R, write T_r:=F(r). For r≤ s, write C_s→ r:=F(r≤ s): T_s⟶ T_r. We call C_s→ r the coarsening functor.","count":59},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"definition-passage","excerpt":"[Residual promotion] If X is a seed-generated object and Δ^A_s,r(X) is nonzero, the operation X⟼Δ^A_s,r(X) is called residual promotion when the residual is added to the active seed set.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"introduction-passage","excerpt":"Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I)","count":18},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-528d89ede77f7358","tex":"\\mathcal A_e","aliases":"A_e mathcal","display":"\\mathcal A_e","kind":"introduction-passage","excerpt":"Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I)","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-73c1bf580d5c1dbc","tex":"A_g","aliases":"A_g","display":"A_g","kind":"introduction-passage","excerpt":"Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-365d7ca82603f468","tex":"A_g^{-1}","aliases":"A_g^-1","display":"A_g^{-1}","kind":"introduction-passage","excerpt":"Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-402ce4d04a80ceff","tex":"A_{gh}^{-1}","aliases":"A_gh^-1","display":"A_{gh}^{-1}","kind":"introduction-passage","excerpt":"Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8da2f1b2bafe729c","tex":"A_h","aliases":"A_h","display":"A_h","kind":"introduction-passage","excerpt":"Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"introduction-passage","excerpt":"[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-inva","count":18},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"introduction-passage","excerpt":"For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].","count":15},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and th","count":54},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8f68aa627baa0435","tex":"C^0_{m,n}","aliases":"C^0_m,n","display":"C^0_{m,n}","kind":"introduction-passage","excerpt":"cular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. Th","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"introduction-passage","excerpt":"[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e4c4b465366a9288","tex":"C_{m,n}","aliases":"C_m,n","display":"C_{m,n}","kind":"introduction-passage","excerpt":"cular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. Th","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-95ec4c9d28187cde","tex":"\\mathbf{Cat}^{\\mathrm{st}}","aliases":"Cat^st mathbf mathrm","display":"\\mathbf{Cat}^{\\mathrm{st}}","kind":"introduction-passage","excerpt":"The residual layer becomes canonical after passing to stable categories. Let Cat^add denote the ∞-category of small additive ∞-categories and additive functors, and let Cat^st denote small stable ∞-categories and exact functors. Ordinary additive categories are included by their nerves. The forgetful functor retains the underlying additive ∞-catego","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6e6d0cbeeb71722a","tex":"\\operatorname{coker}","aliases":"coker operatorname","display":"\\operatorname{coker}","kind":"introduction-passage","excerpt":"Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and th","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and th","count":121},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f8b4bdd2d3f99db6","tex":"d^1","aliases":"d^1","display":"d^1","kind":"introduction-passage","excerpt":"Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and th","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9b51ab408210978e","tex":"D_0^n","aliases":"D_0^n","display":"D_0^n","kind":"introduction-passage","excerpt":"[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-inva","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3b858f3e8665ea65","tex":"D_e","aliases":"D_e","display":"D_e","kind":"introduction-passage","excerpt":"Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I)","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-39c1c7c6d86a3aa2","tex":"d_G","aliases":"d_G","display":"d_G","kind":"introduction-passage","excerpt":"Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and th","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"introduction-passage","excerpt":"Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I)","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I)","count":117},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a20a0c8f1552bcce","tex":"E_e","aliases":"E_e","display":"E_e","kind":"introduction-passage","excerpt":"Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I)","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0d78e47ce8171d5b","tex":"\\mathrm{ex}","aliases":"ex mathrm","display":"\\mathrm{ex}","kind":"introduction-passage","excerpt":"For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-428439cb1463872a","tex":"\\operatorname{Exc}","aliases":"Exc operatorname","display":"\\operatorname{Exc}","kind":"introduction-passage","excerpt":"[Principal-ideal reconstruction] Let P be finite and S stable. Write Exc_0(J(P), S) for the full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-44b5476a8fab972e","tex":"\\operatorname{Exc}_0","aliases":"Exc_0 operatorname","display":"\\operatorname{Exc}_0","kind":"introduction-passage","excerpt":"[Principal-ideal reconstruction] Let P be finite and S stable. Write Exc_0(J(P), S) for the full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6836861e0dab62cd","tex":"\\operatorname{Fun}","aliases":"Fun operatorname","display":"\\operatorname{Fun}","kind":"introduction-passage","excerpt":"elthm:principal-reconstruction Let P be finite and S stable. Write Exc_0(J(P), S) for the full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-98a707b257300ae7","tex":"\\operatorname{Fun}^{\\mathrm{ex}}","aliases":"Fun^ex operatorname mathrm","display":"\\operatorname{Fun}^{\\mathrm{ex}}","kind":"introduction-passage","excerpt":"For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7d6e4b4b124c46ee","tex":"\\operatorname{Fun}^{\\oplus}","aliases":"Fun^⊕ operatorname oplus","display":"\\operatorname{Fun}^{\\oplus}","kind":"introduction-passage","excerpt":"For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"introduction-passage","excerpt":"Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and th","count":106},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0fcc70536b44cfb8","tex":"\\operatorname{GL}","aliases":"GL operatorname","display":"\\operatorname{GL}","kind":"introduction-passage","excerpt":"Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p","count":31},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bca14ec6fc8c9c7a","tex":"\\operatorname{GL}_d","aliases":"GL_d operatorname","display":"\\operatorname{GL}_d","kind":"introduction-passage","excerpt":"Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p","count":26},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"introduction-passage","excerpt":"Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and th","count":18},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e0d53229e96ed0e9","tex":"\\operatorname{Hom}","aliases":"Hom operatorname","display":"\\operatorname{Hom}","kind":"introduction-passage","excerpt":"[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication tabl","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e2ef36ca26c82a63","tex":"\\mathcal J_n","aliases":"J_n mathcal","display":"\\mathcal J_n","kind":"introduction-passage","excerpt":"[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-inva","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e1be41e1676c5fbc","tex":"\\mathcal J_P","aliases":"J_P mathcal","display":"\\mathcal J_P","kind":"introduction-passage","excerpt":"[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quot","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-94c3d42bb63f228a","tex":"\\mathcal J_{\\square}","aliases":"J_square mathcal square","display":"\\mathcal J_{\\square}","kind":"introduction-passage","excerpt":"Let Z be a small stable category, let mathscr M be a coherently acting monoid of exact endofunctors, and let mathscr X be a small family of finite coherent event diagrams in Z. Denote by mathscr H the family of interaction residuals of all their elementary independent-event squares. Put J_square =Thick{T(H):T∈mathscr M, H∈mathscr H}. This notation ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-08093f17d5992ea3","tex":"\\mathbf L","aliases":"L mathbf","display":"\\mathbf L","kind":"introduction-passage","excerpt":"ification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its","count":24},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-578177ca609e29d3","tex":"L_s","aliases":"L_s","display":"L_s","kind":"introduction-passage","excerpt":"[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication tabl","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4a7e9f6a85466609","tex":"\\longleftrightarrow","aliases":"longleftrightarrow","display":null,"kind":"introduction-passage","excerpt":"[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication tabl","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3cc0ed3324bf0e59","tex":"M_m","aliases":"M_m","display":"M_m","kind":"introduction-passage","excerpt":"[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5db07d00340a9b15","tex":"M_m^{C_m}","aliases":"M_m^C_m","display":"M_m^{C_m}","kind":"introduction-passage","excerpt":"[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"introduction-passage","excerpt":"[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-718d406878f6c5a6","tex":"N_m","aliases":"N_m","display":"N_m","kind":"introduction-passage","excerpt":"C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO.","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"introduction-passage","excerpt":"Throughout the lattice part, fix a prime p, write O= Z_p and O_n=O/p^nO, and take n≥1. A lattice means a finite free O-module. The exponent in Q[p^n] denotes annihilation by p^n, not a quotient. We distinguish a residual module, a selected element of that module obstructing a particular lift, and a cone in a specified derived category.","count":35},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-93f77af3d4162a81","tex":"O_n","aliases":"O_n","display":"O_n","kind":"introduction-passage","excerpt":"Throughout the lattice part, fix a prime p, write O= Z_p and O_n=O/p^nO, and take n≥1. A lattice means a finite free O-module. The exponent in Q[p^n] denotes annihilation by p^n, not a quotient. We distinguish a residual module, a selected element of that module obstructing a particular lift, and a cone in a specified derived category.","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"introduction-passage","excerpt":"[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-inva","count":21},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fbe5654b944a7373","tex":"p^a","aliases":"p^a","display":"p^a","kind":"introduction-passage","excerpt":"Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and th","count":15},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0edaee8e5b48a0f0","tex":"p^h","aliases":"p^h","display":"p^h","kind":"introduction-passage","excerpt":"[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication tabl","count":14},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e58a21154cdfe2d8","tex":"p^{\\min(a,n)}","aliases":"p^min(a,n) min","display":"p^{\\min(a,n)}","kind":"introduction-passage","excerpt":"[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d3cdc9ecc001d6c4","tex":"p^n","aliases":"p^n","display":"p^n","kind":"introduction-passage","excerpt":"[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-inva","count":90},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-360f794da8094012","tex":"p^s","aliases":"p^s","display":"p^s","kind":"introduction-passage","excerpt":"[A finite catalogue of integral action classes] Let s=e+1 and h=2e+1. Denote by mathscr L_s the image of Hom(G,GL_d( Z/p^h)) ⟶ Hom(G,GL_d( Z/p^s)). Reduction gives a bijection fracHom(G,GL_d( Z_p)) GL_d( Z_p) longleftrightarrow fracmathscr L_sGL_d( Z/p^s). Consequently this class set is finite and can be determined from a finite multiplication tabl","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c50e47d926edc1d2","tex":"p^{s-n}","aliases":"p^s-n","display":"p^{s-n}","kind":"introduction-passage","excerpt":"[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-14f8900152d6716a","tex":"P_d","aliases":"P_d","display":"P_d","kind":"introduction-passage","excerpt":"There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-777921d6360282c7","tex":"\\operatorname{Perf}","aliases":"Perf operatorname","display":"\\operatorname{Perf}","kind":"introduction-passage","excerpt":"[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quot","count":19},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"introduction-passage","excerpt":")O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, the e","count":17},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"introduction-passage","excerpt":"There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a2dbd37e0ccec228","tex":"q_m","aliases":"q_m","display":"q_m","kind":"introduction-passage","excerpt":"cular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. Th","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4a77ffe628ff771a","tex":"q_n","aliases":"q_n","display":"q_n","kind":"introduction-passage","excerpt":"cal P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-c","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cd98a9c47ecfa404","tex":"\\mathbb Q_p","aliases":"Q_p mathbb","display":"\\mathbb Q_p","kind":"introduction-passage","excerpt":")O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplication by t-1. Its cokernel is O/(N_m(1))=O/mO. Over Q_p, the e","count":14},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0db94dc2cdc9bd9f","tex":"q_P","aliases":"q_P","display":"q_P","kind":"introduction-passage","excerpt":"[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quot","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6dbea45c8ccd5613","tex":"\\mathcal Q_\\Gamma","aliases":"Q_Γ mathcal Gamma","display":"\\mathcal Q_\\Gamma","kind":"introduction-passage","excerpt":"There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-afbd45e26f0e14e2","tex":"R_n^d","aliases":"R_n^d","display":"R_n^d","kind":"introduction-passage","excerpt":"There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5c93e964f5318448","tex":"\\rightleftarrows","aliases":"rightleftarrows","display":null,"kind":"introduction-passage","excerpt":"For a small additive category A, write St^add( A) for its free stable envelope. We use the established adjunction St^add: Cat^add rightleftarrows Cat^st:U, characterized by Fun^ex (St^add( A), C) ≃ Fun^⊕( A,U C) for every stable C; see Saunier--Winges~[source reference].","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2c0d7ac808569ec2","tex":"S_P","aliases":"S_P","display":"S_P","kind":"introduction-passage","excerpt":"[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quot","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fc5e3257a579763c","tex":"S_P^{-1}","aliases":"S_P^-1","display":"S_P^{-1}","kind":"introduction-passage","excerpt":"ification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d51be8ddc0d648b9","tex":"\\square","aliases":"square","display":"\\square","kind":"introduction-passage","excerpt":"Let Z be a small stable category, let mathscr M be a coherently acting monoid of exact endofunctors, and let mathscr X be a small family of finite coherent event diagrams in Z. Denote by mathscr H the family of interaction residuals of all their elementary independent-event squares. Put J_square =Thick{T(H):T∈mathscr M, H∈mathscr H}. This notation ","count":9},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-59d24711aca21962","tex":"\\mathrm{st}","aliases":"st mathrm","display":"\\mathrm{st}","kind":"introduction-passage","excerpt":"The residual layer becomes canonical after passing to stable categories. Let Cat^add denote the ∞-category of small additive ∞-categories and additive functors, and let Cat^st denote small stable ∞-categories and exact functors. Ordinary additive categories are included by their nerves. The forgetful functor retains the underlying additive ∞-catego","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cd3d080377149900","tex":"\\stackrel","aliases":"stackrel","display":null,"kind":"introduction-passage","excerpt":"Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","excerpt":"Write d^1(c)(g,u)=g c(u)-c(gu)+c(g). If a class [c] in cokerd_G is torsion, then p^a c=d_G(m) for some a. Applying d^1 gives p^a d^1(c)=0, whence d^1(c)=0 since the target is torsion-free. Thus its class lies in H^1. Conversely the preceding annihilation argument shows that every H^1 class is torsion. This proves the canonical identification and th","count":31},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ec61c8382b2d3710","tex":"U'","aliases":"U'","display":"U'","kind":"introduction-passage","excerpt":"There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.","count":9},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to","count":46},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"introduction-passage","excerpt":"[Norm-quotient lattice and transient invariants] For this action, M_m^C_m=0, H^1(C_m,M_m)≅ O/mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"introduction-passage","excerpt":"Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5435ba93fa0441bf","tex":"\\varprojlim_r","aliases":"varprojlim_r varprojlim","display":null,"kind":"introduction-passage","excerpt":"Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5b086edc8a88e35b","tex":"X_{I\\cup\\{e\\}}","aliases":"X_I∪{e} cup","display":"X_{I\\cup\\{e\\}}","kind":"introduction-passage","excerpt":"Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I)","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-10665c77b53ab88a","tex":"X_M","aliases":"X_M","display":"X_M","kind":"introduction-passage","excerpt":"Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its constructor action is defined, every q_PX_M is excisive, and its reduce","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8285a92cff098813","tex":"X_r","aliases":"X_r","display":"X_r","kind":"introduction-passage","excerpt":"Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bfd71ea41b57ecef","tex":"X_\\infty","aliases":"X_∞ infty","display":"X_\\infty","kind":"introduction-passage","excerpt":"Resolution completion. A resolution tower may admit an inverse limit X_∞≃varprojlim_r X_r. One can ask whether the seed zoo commutes with completion: Z(X_∞) stackrel?≃ varprojlim_r Z(X_r). This is only a proposed comparison. A common category and an actual comparison map must first be constructed before a residual is defined.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"introduction-passage","excerpt":"[Prime-overlap classification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quot","count":223},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8723c894d5150ea2","tex":"\\mathbb Z_p","aliases":"Z_p mathbb","display":"\\mathbb Z_p","kind":"introduction-passage","excerpt":"Throughout the lattice part, fix a prime p, write O= Z_p and O_n=O/p^nO, and take n≥1. A lattice means a finite free O-module. The exponent in Q[p^n] denotes annihilation by p^n, not a quotient. We distinguish a residual module, a selected element of that module obstructing a particular lift, and a cone in a specified derived category.","count":75},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"introduction-passage","excerpt":"Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p","count":27},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-191835d3d506f9ee","tex":"\\alpha_g","aliases":"α_g alpha","display":"\\alpha_g","kind":"introduction-passage","excerpt":"Let N=|G| and write e=v_p(N). For a matrix X over Q_p, the assertion v_p(X)≥ a means every entry lies in p^a Z_p. Invertible integral matrices and their inverses preserve this divisibility under multiplication. Let A:G→GL_d( Z_p) be a normalized map, not necessarily a homomorphism. Define E(g,h)=A_gA_hA_gh^-1-I, α_g(X)=A_gXA_g^-1. Suppose E(g,h)∈ p","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"introduction-passage","excerpt":"There is a necessary variance distinction. Let Q_Γ have open normal subgroups as objects and an arrow U→ U' when U'⊆ U. Write mathscr P_d(n,U) for the groupoid of rank-d representations of Γ/U on R_n^d, with invertible intertwining matrices as arrows.","count":34},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cc20a1f0376dac78","tex":"\\Theta_{e,f}","aliases":"Θ_e,f Theta","display":"\\Theta_{e,f}","kind":"introduction-passage","excerpt":"m X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I) for the interaction residual of their ele","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","excerpt":"mO. Consequently, with a=v_p(m), (M_m/p^nM_m)^C_m≅ O/p^min(a,n)O. Under the boundary identification with (O/mO)[p^n], reduction from level s to level n is multiplication by p^s-n. For a>0 the sharp uniform delay is a. Every nonzero finite invariant fails to lift integrally, although the action itself is defined over O. The operator σ-1 is multiplic","count":12},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"introduction-passage","excerpt":"Fix a finite poset P and a coherent diagram X:NJ(P)→ S. For e∈ P, put D_e={p∈ P:p<e}, downarrow e=D_e∪{e}. The eligible contexts for e form the interval of ideals A_e={I∈ J(P):D_e⊆ I, e∉ I}. Its initial object is D_e. Define the residual functor on this interval by E_e(I)=Cofib(X_I→ X_I∪{e}). For incomparable events e,f enabled at I, write Θ_e,f(I)","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"introduction-passage","excerpt":"[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-inva","count":17},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c75ab87dac99eabb","tex":"\\prod_{p\\in\\mathcal P\\setminus S}","aliases":"∏_p∈ P∖ S prod in mathcal setminus","display":"\\prod_{p\\in\\mathcal P\\setminus S}","kind":"introduction-passage","excerpt":"[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-inva","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-516d6ddd3d40ea65","tex":"\\prod_{p\\in S}","aliases":"∏_p∈ S prod in","display":"\\prod_{p\\in S}","kind":"introduction-passage","excerpt":"[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-inva","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"introduction-passage","excerpt":"[Prime-component quotient and tower compatibility] Under the Chinese remainder equivalence, J_n= ∏_p∈ SPerf( Z/p^n) ×∏_p∈ P∖ S0. In particular there is an equivalence of stable categories Perf(B_n)/ J_n ≃ ∏_p∈ P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-inva","count":14},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"introduction-passage","excerpt":"athcal P∖ SPerf( Z/p^n) ≃Perf( Z/D_0^n). The equivalence is realized by projection to the retained prime components. Its kernel is constructor-invariant and preserved by every coarsening. If q_n denotes projection, then q_nC_m,n≃ C^0_m,nq_m with the supplied base-change coherence. The quotient is nonzero exactly when S≠ P. When S= P, use the zero c","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"introduction-passage","excerpt":"Throughout the lattice part, fix a prime p, write O= Z_p and O_n=O/p^nO, and take n≥1. A lattice means a finite free O-module. The exponent in Q[p^n] denotes annihilation by p^n, not a quotient. We distinguish a residual module, a selected element of that module obstructing a particular lift, and a cone in a specified derived category.","count":40},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"introduction-passage","excerpt":"ification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its","count":32},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ba1ef6a3fbaaba1f","tex":"\\otimes^{\\mathbf L}_{\\mathbb Z}","aliases":"⊗^ L_ Z otimes mathbf mathbb","display":"\\otimes^{\\mathbf L}_{\\mathbb Z}","kind":"introduction-passage","excerpt":"ification of the excisive kernel] In Perf( Z), J_P= Thick{ Z/p:p∈ S_P}. Equivalently, J_P consists exactly of perfect complexes whose cohomology groups are finite abelian groups with prime divisors in S_P. For the stable Verdier quotient q_P, and every perfect complex M, q_P(M)≃0 Longleftrightarrow M⊗^ L_ Z Z[S_P^-1]≃0. The quotient is nonzero. Its","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Now let F be reduced and excisive. We prove the counit j_!j^*F→ F is an equivalence on an ideal I by induction on |I|. The empty ideal is the zero object on both sides. Choose a maximal element e∈ I. Put I'=I∖{e}. Then I=I'∪downarrow e, I'∩downarrow e=D_e. The ideal-excision lemma expresses F(I) as the pushout of F(I') and F(downarrow e) over F(D_e","count":20},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-776b5d9d1f9b084d","tex":"\\operatorname*","aliases":"* operatorname","display":"\\operatorname*","kind":"use","excerpt":"The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objectiv","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, p","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6b6c4ba66f09915d","tex":"A'_g","aliases":"A'_g","display":"A'_g","kind":"use","excerpt":"[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, p","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-405157a6dc3ce265","tex":"A'_{gh}","aliases":"A'_gh","display":"A'_{gh}","kind":"use","excerpt":"[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, p","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ce05f05a08a7c7b9","tex":"A'_h","aliases":"A'_h","display":"A'_h","kind":"use","excerpt":"[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, p","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-92312bec7dc0bc70","tex":"a'_{ij}","aliases":"a'_ij","display":"a'_{ij}","kind":"use","excerpt":"be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a]∈check H^1","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"3 · 1, G=C_3. Then A^3=I over R, I+A+A^2=3I, and M^G=3M. The invariant cochain complex is quasi-isomorphic to ( F_3)^2 in degree zero. Derived reduction of that invariant object has cohomology in every nonpositive degree, whereas taking derived invariants after reduction has cohomology in every nonnegative degree. The comparison cone has ( F_3)^2 i","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-06b373fe54fc27d1","tex":"A^3","aliases":"A^3","display":"A^3","kind":"use","excerpt":"For p=m=3, in the basis 1,t, A=0 · -1 1 · -1, A^3=I, det(A-I)=3. The fixed vectors modulo 3^n are exactly {b 3^n-1binom1-1:b∈{0,1,2}}. The three vectors at level n+1 all reduce to zero at level n. Thus a nonzero residual appears at every level, while its inverse system has zero limit. This is not persistent nonexistence of the lattice; it is the fa","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2028876d3a4cc668","tex":"A^a","aliases":"A^a","display":"A^a","kind":"use","excerpt":"[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint resid","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-38b93d262e026b72","tex":"A^b","aliases":"A^b","display":"A^b","kind":"use","excerpt":"Any alternative matrix lift is uniquely of the form (I+p^nb(g))A_g, for a normalized cochain b. Expanding its multiplicativity residual and discarding square-zero terms gives c_A^b=c_A+δ b. Thus the class is unchanged and the new matrices satisfy every group relation exactly if and only if δ b=-c_A. Such a cochain exists precisely when [c_A]=0. Inv","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1c343a1277890414","tex":"A^\\bullet","aliases":"A^bullet bullet","display":"A^\\bullet","kind":"use","excerpt":"[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudo","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bce8bfd89f58273e","tex":"a^d","aliases":"a^d","display":"a^d","kind":"use","excerpt":"_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-341faa653baa07cb","tex":"A^i","aliases":"A^i","display":"A^i","kind":"use","excerpt":"zero on H^0. Its functor-intrinsic cofiber satisfies H^iCofib(β)≅ k^d for every i∈ Z. Both composites defining β are exact. The nonzero residual is therefore a genuine defect between fully derived exact functors, not solely a degree-zero truncation effect. The binomial expansion gives (I+pJ)^p=I modulo p^2. Also A^i=I+ipJ modulo p^2, so N=pI+pfracp","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-339ddf229ead4fcf","tex":"A^m","aliases":"A^m","display":"A^m","kind":"use","excerpt":"[When an integral matrix has continuous procyclic powers] Let A∈GL_d( Z_p). The map m↦ A^m on Z extends uniquely to a continuous homomorphism Z_p→GL_d( Z_p) if and only if the reduction of A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a conti","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9a52b1bc77ae2b6d","tex":"A^n","aliases":"A^n","display":"A^n","kind":"use","excerpt":"[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudo","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fca57a1241a0dffd","tex":"A^p","aliases":"A^p","display":"A^p","kind":"use","excerpt":"[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0,","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e1c0ee83725f7d29","tex":"A^{p-1}","aliases":"A^p-1","display":"A^{p-1}","kind":"use","excerpt":"[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0,","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-229000b7d2d27346","tex":"A^{p^2}","aliases":"A^p^2","display":"A^{p^2}","kind":"use","excerpt":"[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint resid","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6b7002a2163facaa","tex":"A^{p^t}","aliases":"A^p^t","display":"A^{p^t}","kind":"use","excerpt":"a continuous homomorphism Z_p→GL_d( Z_p) if and only if the reduction of A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9c0e875b8dda74d2","tex":"A^{p^{t+j}}","aliases":"A^p^t+j","display":"A^{p^{t+j}}","kind":"use","excerpt":"p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows that A^p^t+j∈ I+p^j+1M_d( Z_p). Hence exponen","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0324c102d49c31af","tex":"A^q","aliases":"A^q","display":"A^q","kind":"use","excerpt":"[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solut","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-99155510890729d1","tex":"A^r","aliases":"A^r","display":"A^r","kind":"use","excerpt":"bda⊗_ Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4fbf9458f594ce86","tex":"A^x","aliases":"A^x","display":"A^x","kind":"use","excerpt":"delta u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_p^2 put b(x)=lfloorfracxprfloor Jpmod p (0≤ x<p^2). Writing each x as its low digit plus p times its high digit shows δ b=-Inf(c): the high-digit carry is the negative of the low-digit carry modulo p. The repaired map is exactly x↦ A^x. Thus inflation kills a genuin","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-24f07e60acb49c42","tex":"A_*","aliases":"A_*","display":"A_*","kind":"use","excerpt":"[Complete rank-two profinite lifting test] Let p be any prime and let ρ_n: Z_p^r→GL_2(R_n) be continuous, with finitely many topological generators as above. item If a=n, all the A_i are scalar and ρ_n has a continuous integral lift. item If a<n, choose any generator A_* with scalar depth a. Take λ_i to be an integer representative of (A_i)_11 and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id.","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"[Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other co","count":12},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b9fd26a5764aba96","tex":"A_{11}","aliases":"A_11","display":"A_{11}","kind":"use","excerpt":"For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d8afff31ce6342b8","tex":"A_{12}","aliases":"A_12","display":"A_{12}","kind":"use","excerpt":"For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-89b2e69cbc9c02e9","tex":"A_{21}","aliases":"A_21","display":"A_{21}","kind":"use","excerpt":"For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cb4536806c27fb55","tex":"A_{22}","aliases":"A_22","display":"A_{22}","kind":"use","excerpt":"For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint resid","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d225b16c6da9df70","tex":"A_g^b","aliases":"A_g^b","display":"A_g^b","kind":"use","excerpt":"heorem[Complete one-step relation obstruction] The residual c_A is a 2-cocycle. Its cohomology class o_n(ρ_n)=[c_A]∈ H^2(G,V) is independent of the chosen matrix lifts. Moreover, o_n(ρ_n)=0 Longleftrightarrow ρ_n lifts to GL_d(R_n+1). More precisely, all lifts with the fixed reduction have the form A_g^b=(I+p^n b(g))A_g, where representatives of b(","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S,","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"For a commutative ring R and a∈ R, set C_R(a)=[RxrightarrowaR], with the source in cohomological degree -1 and the target in degree 0. The tensor product ⊗_i C_R(a_i) is the Koszul complex in this cohomological convention. Its tensor, cone, and change-of-basis properties are classical; see [source reference]. We use them with an explicit sign conve","count":9},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0550d6e15e6a7c3d","tex":"A_i'","aliases":"A_i'","display":"A_i'","kind":"use","excerpt":"alent: ρ_n lifts over Z_p, ρ_n extends over R_n+a, Ω_a,*(ρ_n)=0. The extension remains a representation of the same Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the finite extension. Given a finite extension with commuting matrices A_i' modulo p^n+a, subtract the same λ_iI and divide by ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-daf800c8ae3d0681","tex":"a_{ij}","aliases":"a_ij","display":"a_{ij}","kind":"use","excerpt":"[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing ","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ae3cae372c62870b","tex":"a_{ik}","aliases":"a_ik","display":"a_{ik}","kind":"use","excerpt":"On a triple overlap, s_k=s_j+a_jk=s_i+a_ij+a_jk, while also s_k=s_i+a_ik. Freeness of the torsor action gives a_ij+a_jk=a_ik, which is the vCech cocycle identity.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9f9a558588be02af","tex":"a_{j+1}","aliases":"a_j+1","display":"a_{j+1}","kind":"use","excerpt":"Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ee0853f347f680bc","tex":"a_j^2","aliases":"a_j^2","display":"a_j^2","kind":"use","excerpt":"Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0b50cb77343ec39b","tex":"a_{jk}","aliases":"a_jk","display":"a_{jk}","kind":"use","excerpt":"On a triple overlap, s_k=s_j+a_jk=s_i+a_ij+a_jk, while also s_k=s_i+a_ik. Freeness of the torsor action gives a_ij+a_jk=a_ik, which is the vCech cocycle identity.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7c3425b798827ac6","tex":"A_{L,m}","aliases":"A_L,m","display":"A_{L,m}","kind":"use","excerpt":"For a fixed perfect integral complex L, put A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-aed92e8fef210908","tex":"A_{L,n}","aliases":"A_L,n","display":"A_{L,n}","kind":"use","excerpt":"For a fixed perfect integral complex L, put A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"For m=1 this is the definition. Taking cofibers in the last coordinate replaces its factor A_m→ B_m by Cofib(f_m) because tensoring by the other factors is exact. Repeat for the remaining coordinates. Interchanging finite colimits shows that this iterated construction is the cofiber of the punctured-cube colimit map. The universal properties give n","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"[Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other co","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"For a constructor output from ⋃_n A_n, its finitely many inputs all belong to a common A_N. The output therefore already belongs to Φ(A_N). This proves preservation of increasing unions. The union on the right is a fixed point because Ψ preserves that union. It contains Σ_0 and is contained in every fixed point by induction on n, hence is least.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"use","excerpt":"For a constructor output from ⋃_n A_n, its finitely many inputs all belong to a common A_N. The output therefore already belongs to Φ(A_N). This proves preservation of increasing unions. The union on the right is a fixed point because Ψ preserves that union. It contains Σ_0 and is contained in every fixed point by induction on n, hence is least.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2277836fd5789d74","tex":"A_R","aliases":"A_R","display":"A_R","kind":"use","excerpt":"A two-fiber exact model can be written entirely with endofunctors: set K_R=D(R[G])× D(R), A_R(M,N)=(0,Γ_R(M)), and let C_R,S apply derived coefficient change in both components. The second component of C_R,SA_R→ A_SC_R,S is precisely β. This places the example in a genuine two-axis exact system, rather than identifying differently typed functors by","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"use","excerpt":"Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f78009c8b1487c7d","tex":"A_S","aliases":"A_S","display":"A_S","kind":"use","excerpt":"A two-fiber exact model can be written entirely with endofunctors: set K_R=D(R[G])× D(R), A_R(M,N)=(0,Γ_R(M)), and let C_R,S apply derived coefficient change in both components. The second component of C_R,SA_R→ A_SC_R,S is precisely β. This places the example in a genuine two-axis exact system, rather than identifying differently typed functors by","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomolo","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3756eda5b09f3445","tex":"\\operatorname{Ad}","aliases":"Ad operatorname","display":"\\operatorname{Ad}","kind":"use","excerpt":"[Continuous reconstruction from unobstructed stages] If H^2_cts(Γ,Adbarρ)=0, every given ρ_n with that residual action extends to a continuous representation ρ:Γ⟶GL_d( Z_p). Without the cohomology-vanishing hypothesis, a compatible sequence of continuous lifts still reconstructs uniquely as a framed representation. If Γ is topologically finitely ge","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c1df4245c27dd848","tex":"\\amalg","aliases":"amalg","display":null,"kind":"use","excerpt":"[Interaction is the total cofiber] There are natural equivalences Θ≃Cofib(Bamalg_A C⟶ D) ≃Cofib(Cofib(A→ C)⟶ Cofib(B→ D)). Consequently Θ≃0 if and only if the square is a homotopy pushout, equivalently a homotopy pullback. An exact functor preserves these equivalences.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-402e657c1e624b2e","tex":"\\amalg_A","aliases":"amalg_A amalg","display":null,"kind":"use","excerpt":"[Interaction is the total cofiber] There are natural equivalences Θ≃Cofib(Bamalg_A C⟶ D) ≃Cofib(Cofib(A→ C)⟶ Cofib(B→ D)). Consequently Θ≃0 if and only if the square is a homotopy pushout, equivalently a homotopy pullback. An exact functor preserves these equivalences.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c0319ae1a9e60a24","tex":"\\mathrm{arith}","aliases":"arith mathrm","display":"\\mathrm{arith}","kind":"use","excerpt":"Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-62a0c6783e936d9c","tex":"\\arrow","aliases":"arrow","display":null,"kind":"use","excerpt":"The square is [column sep=large,row sep=large] A_rC_s→ r(X) arrow[r,\"η^A_s,r,X\"] · C_s→ rA_s(X). It compares transforming after coarsening with coarsening after transforming.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3d228675fe9781a5","tex":"\\mathrm{aut}","aliases":"aut mathrm","display":"\\mathrm{aut}","kind":"use","excerpt":"Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"[Idempotent projection model] Let Z= A× B for stable categories A, B, and let L(a,b)=(a,0). With constructor monoid generated by the idempotent exact functor L, an object is operationally killable exactly when it lies in 0× B. The obstruction quotient is equivalent to A, and Obs(a,b)≃ a. Thus the quotient literally retains the component that cannot","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-16636769fc160a5a","tex":"b^{-1}","aliases":"b^-1","display":"b^{-1}","kind":"use","excerpt":"s of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and let E_p=θ_p^-1(L_p) be the corresponding lattice in Q_p^r. Bounds on the entries of θ_p and its inverse give a common N≥1 such that, with b=D^N, b Z_p^r⊆ E_p⊆ b","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"Here is also an explicit finite construction proving the local identifications. Put F_p=bE_p, so b^2 Z_p^r⊆ F_p⊆ Z_p^r. The Chinese remainder isomorphism identifies Z^r/b^2 Z^r≅ ∏_p∈ P Z_p^r/b^2 Z_p^r. Let H⊆ Z^r be the inverse image of the product of the subgroups F_p/b^2 Z_p^r. Directly from the definitions, H=bΛ. The quotient Z^r/H is the produc","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b8bfd3a8b89585c7","tex":"B^\\bullet","aliases":"B^bullet bullet","display":"B^\\bullet","kind":"use","excerpt":"[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudo","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-64d65b0368d27e09","tex":"B^{d-1}","aliases":"B^d-1","display":"B^{d-1}","kind":"use","excerpt":"[Cyclic centralizers over a coefficient ring] Let R be a commutative ring, B∈ M_d(R), and suppose a vector v makes Q=(v,Bv,…,B^d-1v)∈GL_d(R). Then every matrix commuting with B is uniquely f(B) for a polynomial f∈ R[t] of degree less than d. The coefficients of f are the entries of Q^-1Mv. In particular, the centralizer is commutative.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f517e8c0c0edd7b5","tex":"b^e","aliases":"b^e","display":"b^e","kind":"use","excerpt":"_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9b3234202ef44211","tex":"B^j","aliases":"B^j","display":"B^j","kind":"use","excerpt":"Over R= Z/p^s, it is enough that bar B have a cyclic vector over F_p. Every nonscalar 2×2 matrix over a field has such a vector. There is a unique polynomial of degree less than d satisfying f(B)v=Mv, since the columns of Q are a basis. If MB=BM, then M-f(B) annihilates every B^jv, so it is zero. Conversely every polynomial in B commutes with B. Th","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-55668e3936108669","tex":"b^{l-m}","aliases":"b^l-m","display":"b^{l-m}","kind":"use","excerpt":"independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step cell has scalar b^m-nb^l-m=b^l-n. The associativity coher","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-565d3c4f290c2391","tex":"b^{l-n}","aliases":"b^l-n","display":"b^{l-n}","kind":"use","excerpt":"independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step cell has scalar b^m-nb^l-m=b^l-n. The associativity coher","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f5066e1a8c41bf20","tex":"b^{m-n}","aliases":"b^m-n","display":"b^{m-n}","kind":"use","excerpt":"ut A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use the cell η_L,b;m,n=b^m-nθ_L;m,n. For sca","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c2dedbbf3ad65db3","tex":"B^n","aliases":"B^n","display":"B^n","kind":"use","excerpt":"[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudo","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4e28441ccdc4e8e5","tex":"B^p","aliases":"B^p","display":"B^p","kind":"use","excerpt":"A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f87c7c154af9d919","tex":"B_*","aliases":"B_*","display":"B_*","kind":"use","excerpt":"d integer scalars λ_i such that A_i≡λ_i Ipmodp^a for every i. There is no restriction in this display when a=0. Define the normalized matrices B_i=fracA_i-λ_i Ip^a ∈ M_d(R_n-a). This is well-defined from the given finite data. It does not invert p inside R_n. Assume that, for some distinguished index *, the matrix B_* is cyclic modulo p. Define the","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-81840733386af53b","tex":"B_*'","aliases":"B_*'","display":"B_*'","kind":"use","excerpt":"thbb Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the finite extension. Given a finite extension with commuting matrices A_i' modulo p^n+a, subtract the same λ_iI and divide by p^a. The resulting matrices B_i' are defined modulo p^n and reduce to B_i modulo p^n-a. The equality p^2a[B_*',","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3ea71b67efc35822","tex":"B_0","aliases":"B_0","display":"B_0","kind":"use","excerpt":"mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"For a fixed multiplication table and residual action, deciding existence is a finite linear problem over k. A solution b is a positive certificate. A linear functional λ on normalized 2-cochains with λ∘δ=0, λ(c_A)≠0 is a negative certificate. If b_0 is one solution, every other solution is b_0+z with δ z=0. Conjugating the matrices by I+p^nX replac","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id.","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8efd426e3c3875e3","tex":"B_2","aliases":"B_2","display":"B_2","kind":"use","excerpt":"mar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation families A_0=P_2, A_1=A_2=id, B_0=B_1=P_1, B_2=id.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"use","excerpt":"[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S,","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-56dbd2ec9f2cd448","tex":"B_i'","aliases":"B_i'","display":"B_i'","kind":"use","excerpt":"Ω_a,*(ρ_n)=0. The extension remains a representation of the same Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the finite extension. Given a finite extension with commuting matrices A_i' modulo p^n+a, subtract the same λ_iI and divide by p^a. The resulting matrices B_i' are defined modulo","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"use","excerpt":"[What normalization reveals] The input relations only force p^2a[B_i,B_j]≡0pmodp^n. When a>0, this can conceal a nonzero commutator in R_n-a. The obstruction is therefore not just the original zero commutator re-labelled. It measures the additional commutativity required after removing a common scalar precision. Different permitted scalar choices c","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a]∈check H^1","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b2f9008f03849cc5","tex":"B_k","aliases":"B_k","display":"B_k","kind":"use","excerpt":"orem[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"use","excerpt":"For m=1 this is the definition. Taking cofibers in the last coordinate replaces its factor A_m→ B_m by Cofib(f_m) because tensoring by the other factors is exact. Repeat for the remaining coordinates. Interchanging finite colimits shows that this iterated construction is the cofiber of the punctured-cube colimit map. The universal properties give n","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d5ac1339f0e03c92","tex":"B_{p,n}","aliases":"B_p,n","display":"B_{p,n}","kind":"use","excerpt":"Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. Th","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f3c4cf0a4da83f47","tex":"b_{r_0\\to r_1}","aliases":"b_r_0→ r_1 to","display":"b_{r_0\\to r_1}","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-82ba26d2818eba31","tex":"b_{r_{n-1}\\to r_n}","aliases":"b_r_n-1→ r_n to","display":"b_{r_{n-1}\\to r_n}","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-edcd2529ad4f495e","tex":"B_X","aliases":"B_X","display":"B_X","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2d72109f1a3a7929","tex":"\\bar A","aliases":"bar A bar","display":"\\bar A","kind":"use","excerpt":"[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solut","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bc097b9a50d06903","tex":"\\bar A^i","aliases":"bar A^i bar","display":"\\bar A^i","kind":"use","excerpt":"[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solut","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d5f0560604825613","tex":"\\bar A^{q-1-i}","aliases":"bar A^q-1-i bar","display":"\\bar A^{q-1-i}","kind":"use","excerpt":"[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solut","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e15729f91b24285c","tex":"\\bar B","aliases":"bar B bar","display":"\\bar B","kind":"use","excerpt":"Over R= Z/p^s, it is enough that bar B have a cyclic vector over F_p. Every nonscalar 2×2 matrix over a field has such a vector. There is a unique polynomial of degree less than d satisfying f(B)v=Mv, since the columns of Q are a basis. If MB=BM, then M-f(B) annihilates every B^jv, so it is zero. Conversely every polynomial in B commutes with B. Th","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d0086c8c28eb495d","tex":"\\bar E","aliases":"bar E bar","display":"\\bar E","kind":"use","excerpt":"item every operationally killable object maps to zero; item q(X)≃0 if and only if X∈ N_mathscr M; item for every exact functor E: Z→ D into a stable ∞-category such that E(K)≃0 for every K∈ K_mathscr M, there is an exact factorization, unique up to contractible choice together with its comparison to E, Zxrightarrowq O_mathscr Mxrightarrowbar E D. T","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2686122cebe59f5e","tex":"\\bar m","aliases":"bar m bar","display":"\\bar m","kind":"use","excerpt":"[Cohomological identification of the symmetry residual] There is a natural exact sequence 0⟶ M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0172f00d6015ed24","tex":"\\bar T","aliases":"bar T bar","display":"\\bar T","kind":"use","excerpt":"Now q^invT annihilates J_mathscr M, so the quotient universal property gives bar T with bar T q^inv≃ q^invT. The full faithfulness of precomposition in the functor-category universal property descends any supplied unit, composition, and higher coherence data. It does not create action data absent from the original system.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0dadbf76b8edc04c","tex":"\\bar U","aliases":"bar U bar","display":"\\bar U","kind":"use","excerpt":"en its first and third components vanish. Indeed, a summand of a zero direct sum is zero. Consequently N_mathscr M= C× C×0, Z/ N_mathscr M≃ C. Nevertheless U(0,A,0)=(0,0,A) for every A∈ C. The input lies in N_mathscr M and the output generally does not. If U descended to an exact bar U on the quotient, then qU(0,A,0)≃bar Uq(0,A,0)≃0, a contradictio","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1784da886e1aea15","tex":"\\bar v","aliases":"bar v bar","display":"\\bar v","kind":"use","excerpt":"[Intrinsic kernel exchange and its boundary] There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a217ed9b9db0e555","tex":"\\bar\\Lambda","aliases":"barΛ bar Lambda","display":"\\bar\\Lambda","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-db1d2ae53aa18234","tex":"\\bar\\rho","aliases":"barρ bar rho","display":"\\bar\\rho","kind":"use","excerpt":"Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d13b1528b249ec85","tex":"\\bigwedge","aliases":"bigwedge","display":"\\bigwedge","kind":"use","excerpt":"The Euclidean algorithm produces a unimodular matrix U∈GL_m( Z) with (a_1,…,a_m)U=(g,0,…,0). The Koszul complex is the exterior algebra on a free module in degree -1, with differential the derivation determined by this row vector. Hence the maps bigwedge^jU give an isomorphism of complexes from K(g,0,…,0) to K(a_1,…,a_m).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-955f00f61f326f44","tex":"\\bigwedge^j","aliases":"bigwedge^j bigwedge","display":"\\bigwedge^j","kind":"use","excerpt":"The Euclidean algorithm produces a unimodular matrix U∈GL_m( Z) with (a_1,…,a_m)U=(g,0,…,0). The Koszul complex is the exterior algebra on a free module in degree -1, with differential the derivation determined by this row vector. Hence the maps bigwedge^jU give an isomorphism of complexes from K(g,0,…,0) to K(a_1,…,a_m).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8e3c0fce1b7a60ae","tex":"\\bullet","aliases":"bullet","display":"\\bullet","kind":"use","excerpt":"[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudo","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"A strong correspondence should include: Λ(TX) · ≃ T'Λ(X), Λ(CX) · ≃ C'Λ(X), Λ(Δ_T,CX) · ≃Δ_T',C'Λ(X), barΛ(Obs X) · ≃Obs'Λ(X).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4ae37711bd70b619","tex":"\\mathcal C^3","aliases":"C^3 mathcal","display":"\\mathcal C^3","kind":"use","excerpt":"[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0e58d04c97b6e684","tex":"C_{\\mathbb Z}","aliases":"C_ Z mathbb","display":"C_{\\mathbb Z}","kind":"use","excerpt":"For an enabled event e, E_e(I)≃ M⊗^ L_ ZC_ Z(n_e). The comparison after an unrelated event f is multiplication by n_f on that complex. Consequently Θ_e,f(I)≃ M⊗^ L_ Z C_ Z(n_e)⊗^ L_ ZC_ Z(n_f). The complexes C_ Z(n) are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead F_X_M(P)","count":16},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c691076ae0cfb7ce","tex":"C_{\\mathbb Z_p}","aliases":"C_ Z_p mathbb","display":"C_{\\mathbb Z_p}","kind":"use","excerpt":"put R_n= Z/p^n. Then H^0C_R_n(p^a)≅ Z/p^a, H^-1C_R_n(p^a)≅ Z/p^a. Under reduction from m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-250efa8cd80c68e5","tex":"C_{1\\to0}","aliases":"C_1→0 to","display":"C_{1\\to0}","kind":"use","excerpt":"On the seed x=(k[0],k^2[0]), the word W=A_0C_1→0B_1C_2→1 evaluates to zero. Exchanging its left inversion first yields P_1x. Exchanging its right inversion first yields P_2x. The second exchange in either path yields x. Both paths then perform the same identity exchange to reach the normal word C_1→0C_2→1A_2B_2. Thus the complete paths have residua","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"Consider a typed alphabet containing transformation symbols T and resolution symbols C. A mixed word is a composable expression such as W=T_1C_1T_2T_3C_2T_4. Assume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form TC ⇒ C T', where T' is the correctly reindexed transformati","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3dc1d8cf076031e1","tex":"C_{2\\to1}","aliases":"C_2→1 to","display":"C_{2\\to1}","kind":"use","excerpt":"On the seed x=(k[0],k^2[0]), the word W=A_0C_1→0B_1C_2→1 evaluates to zero. Exchanging its left inversion first yields P_1x. Exchanging its right inversion first yields P_2x. The second exchange in either path yields x. Both paths then perform the same identity exchange to reach the normal word C_1→0C_2→1A_2B_2. Thus the complete paths have residua","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"[The two-dimensional Z/9 instance] Take R= Z/9, A=1 · 3 3 · 1, G=C_3. Then A^3=I over R, I+A+A^2=3I, and M^G=3M. The invariant cochain complex is quasi-isomorphic to ( F_3)^2 in degree zero. Derived reduction of that invariant object has cohomology in every nonpositive degree, whereas taking derived invariants after reduction has cohomology in ever","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9b74ac7e4a926fd8","tex":"c_A","aliases":"c_A","display":"c_A","kind":"use","excerpt":"Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero ideal p^","count":18},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-43cbd7f2f884a2bf","tex":"c_{A^b}","aliases":"c_A^b","display":"c_{A^b}","kind":"use","excerpt":"Any alternative matrix lift is uniquely of the form (I+p^nb(g))A_g, for a normalized cochain b. Expanding its multiplicativity residual and discarding square-zero terms gives c_A^b=c_A+δ b. Thus the class is unchanged and the new matrices satisfy every group relation exactly if and only if δ b=-c_A. Such a cochain exists precisely when [c_A]=0. Inv","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-72266f70cc0ed3b6","tex":"C_{B_k}","aliases":"C_B_k","display":"C_{B_k}","kind":"use","excerpt":"orem[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7144b96281440a48","tex":"C_{B_n}","aliases":"C_B_n","display":"C_{B_n}","kind":"use","excerpt":"[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7b28a4355d9169a4","tex":"c_{\\bar m}","aliases":"c_bar m bar","display":"c_{\\bar m}","kind":"use","excerpt":"[Cohomological identification of the symmetry residual] There is a natural exact sequence 0⟶ M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"Before event (i,j) can occur, every initially intervening C_k with k<j must already have crossed T_i, and every initially intervening T_h with h>i must already have crossed C_j. Thus all predecessors of a performed event have already occurred. The performed events form an ideal.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f619694f192b2ca7","tex":"C_{l,m}","aliases":"C_l,m","display":"C_{l,m}","kind":"use","excerpt":"mathbb Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. These are the data of an enhanced indexed system. Literal","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-efb69ec6ea9609dd","tex":"C_{l,n}","aliases":"C_l,n","display":"C_{l,n}","kind":"use","excerpt":"mathbb Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. These are the data of an enhanced indexed system. Literal","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1f0efa0f2f69e6b7","tex":"C_{n,k}","aliases":"C_n,k","display":"C_{n,k}","kind":"use","excerpt":"orem[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8fb97f0d5eea62b2","tex":"C_{n\\to m}","aliases":"C_n→ m to","display":"C_{n\\to m}","kind":"use","excerpt":"Let the resolution level be n≥1 and take X_n= Z/p^n Z. For m≤ n, coarsening is reduction C_n→ m: Z/p^n Z→ Z/p^m Z. For any polynomial f∈ Z[x], the induced transformation satisfies C_n→ m(f(x)) = f(C_n→ m(x)). Thus the transformation-resolution square commutes exactly. This is a set-level equality, not a cofiber calculation. Regarding the sets as di","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-374d4fe4488ce15e","tex":"C_p","aliases":"C_p","display":"C_p","kind":"use","excerpt":"The perfectness hypothesis cannot be removed. Fix an odd prime p, let R= Z/p^2, k= F_p, G=C_p, and choose a matrix J∈ M_d( Z) whose reduction modulo p is invertible. Put M=R^d and let the chosen generator act by A=I+pJpmodp^2. For d=2, the choice J=(0 · 1","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b06d49525ec1dbc9","tex":"C_{p^2}","aliases":"C_p^2","display":"C_{p^2}","kind":"use","excerpt":"ion of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint residual action is trivial. For any normalized cochain u on C_p, ∑_t=0^p-1(δ u)(1,t)=0, ∑_t=0^p-1c(1,t)=J≠0. So the finite-quotient class is nonzero. On C_p^2 put b(x)=lfloorfracxprf","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-25b07618d510bdea","tex":"C_q","aliases":"C_q","display":"C_q","kind":"use","excerpt":"[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solut","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-557d8a93e9288da4","tex":"C_R","aliases":"C_R","display":"C_R","kind":"use","excerpt":"For a commutative ring R and a∈ R, set C_R(a)=[RxrightarrowaR], with the source in cohomological degree -1 and the target in degree 0. The tensor product ⊗_i C_R(a_i) is the Koszul complex in this cohomological convention. Its tensor, cone, and change-of-basis properties are classical; see [source reference]. We use them with an explicit sign conve","count":15},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5c05c55fef04bc52","tex":"C_{R,S}","aliases":"C_R,S","display":"C_{R,S}","kind":"use","excerpt":"A two-fiber exact model can be written entirely with endofunctors: set K_R=D(R[G])× D(R), A_R(M,N)=(0,Γ_R(M)), and let C_R,S apply derived coefficient change in both components. The second component of C_R,SA_R→ A_SC_R,S is precisely β. This places the example in a genuine two-axis exact system, rather than identifying differently typed functors by","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-04cd13ea628c4ebf","tex":"C_{r_{i+1}\\to r_i}","aliases":"C_r_i+1→ r_i to","display":"C_{r_{i+1}\\to r_i}","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6ba9ef27f6dd13d2","tex":"C_{R_n}","aliases":"C_R_n","display":"C_{R_n}","kind":"use","excerpt":"[Transient negative cohomology in the resolution direction] Fix p and an integer a≥1. For n≥ a put R_n= Z/p^n. Then H^0C_R_n(p^a)≅ Z/p^a, H^-1C_R_n(p^a)≅ Z/p^a. Under reduction from m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, a","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-db7a4ab3f2cf9608","tex":"C_{r\\to r}","aliases":"C_r→ r to","display":"C_{r\\to r}","kind":"use","excerpt":"Strictness gives C_r→ r=id_ T_r and, for r≤ s≤ t, C_t→ r=C_s→ r∘ C_t→ s. A pseudofunctor version is more natural in some applications. The strict version isolates the main structure without coherence bookkeeping.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6b993d279f02d1f2","tex":"C_{r\\to u}","aliases":"C_r→ u to","display":"C_{r\\to u}","kind":"use","excerpt":"This is the formal version of the two-axis statement mixed movement = within-resolution transformation + resolution movement. Categorically, π: K→ R is the Grothendieck fibration associated with F. Indeed, a map (u,Z)→(s,Y) above u≤ r≤ s is exactly a map Z→ C_s→ uY=C_r→ uC_s→ rY. This uniquely factors through κ_r,s,Y as a morphism to (r,C_s→ rY) ab","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0c5e2d447c6d879a","tex":"C_{s\\to u}","aliases":"C_s→ u to","display":"C_{s\\to u}","kind":"use","excerpt":"This is the formal version of the two-axis statement mixed movement = within-resolution transformation + resolution movement. Categorically, π: K→ R is the Grothendieck fibration associated with F. Indeed, a map (u,Z)→(s,Y) above u≤ r≤ s is exactly a map Z→ C_s→ uY=C_r→ uC_s→ rY. This uniquely factors through κ_r,s,Y as a morphism to (r,C_s→ rY) ab","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-05a182de0f8d54cc","tex":"C_{\\widehat B}","aliases":"C_widehat B widehat","display":"C_{\\widehat B}","kind":"use","excerpt":"The generic-fiber criterion also gives a positive case. For M=C_widehat B(u) with a nonzero integer u, its selected generic fibers vanish, because u becomes invertible in each Q_p. Thus this comparison is an equivalence on that torsion residual even though it is not an equivalence on the unit. Commutation depends on the object, not only on the exis","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-57e0b5931eada159","tex":"\\Cat_\\infty","aliases":"Cat_∞ Cat infty mathbf infty","display":"{\\mathbf{Cat}}_\\infty","kind":"use","excerpt":"[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudo","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-27b5a2e3b286b58f","tex":"\\mathsf{Cert}","aliases":"Cert mathsf","display":"\\mathsf{Cert}","kind":"use","excerpt":"The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objectiv","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c9436aea1bf2444f","tex":"\\check","aliases":"check","display":null,"kind":"use","excerpt":"). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech coboundary a'_ij=a_ij+b_j-b_i; item the class Ω(P;{U_i})=[a]∈check H^1({U_i}, A) is independent of the chosen local sections","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3185762aaf155509","tex":"\\colim","aliases":"colim *colim operatorname","display":"{\\operatorname*{colim}}","kind":"use","excerpt":"e full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j_!G)(I)=*colim_e∈ IG(e), (j_!G)(∅)=0. All colimits are finite. In particular, for an exc","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d0f019ffd6d53d4e","tex":"\\operatorname{colim}","aliases":"colim operatorname","display":"\\operatorname{colim}","kind":"use","excerpt":"and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1eef70c3305099e1","tex":"\\colim_{e\\in I}","aliases":"colim_e∈ I *colim_e∈ I colim in operatorname in","display":"{\\operatorname*{colim}}_{e\\in I}","kind":"use","excerpt":"e full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j_!G)(I)=*colim_e∈ IG(e), (j_!G)(∅)=0. All colimits are finite. In particular, for an exc","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7a9836780a612d50","tex":"\\operatorname{const}","aliases":"const operatorname","display":"\\operatorname{const}","kind":"use","excerpt":"[Criterion for context-independent event residuals] The following conditions are equivalent. item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every nor","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-028b45b9436627c5","tex":"\\operatorname{const}_{\\mathcal A_e}","aliases":"const_ A_e operatorname mathcal","display":"\\operatorname{const}_{\\mathcal A_e}","kind":"use","excerpt":"[Criterion for context-independent event residuals] The following conditions are equivalent. item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every nor","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8b9e658b5fed2443","tex":"\\mathrm{cts}","aliases":"cts mathrm","display":"\\mathrm{cts}","kind":"use","excerpt":"[Complete continuous one-step obstruction] The class o_n^cts(ρ_n)=[c_A]∈ H^2_cts(Γ,V) is independent of the continuous choices. It vanishes if and only if ρ_n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"[Universal stable obstruction quotient] The quotient q is exact and satisfies: item every operationally killable object maps to zero; item q(X)≃0 if and only if X∈ N_mathscr M; item for every exact functor E: Z→ D into a stable ∞-category such that E(K)≃0 for every K∈ K_mathscr M, there is an exact factorization, unique up to contractible choice to","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"tive certificate. If b_0 is one solution, every other solution is b_0+z with δ z=0. Conjugating the matrices by I+p^nX replaces b(g) by b(g)+X-g· X. These are exactly the negatives of the usual 0-coboundaries and yield the same quotient subgroup. Conjugation fixes the action precisely when X is invariant. The cochain spaces have dimensions (|G|-1)d","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-58207b6661c2302d","tex":"d^i","aliases":"d^i","display":"d^i","kind":"use","excerpt":"Rvarprojlim_n M_n. Finite tensor products, finite direct sums, shifts, and cones of compatible based maps commute with this reconstruction. In particular this holds for the scalar residual and interaction complexes with fixed integral labels. Every matrix entry is a compatible residue sequence and therefore determines a unique element of widehat B.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b0267738129b3cc1","tex":"d^{i+1}","aliases":"d^i+1","display":"d^{i+1}","kind":"use","excerpt":"Rvarprojlim_n M_n. Finite tensor products, finite direct sums, shifts, and cones of compatible based maps commute with this reconstruction. In particular this holds for the scalar residual and interaction complexes with fixed integral labels. Every matrix entry is a compatible residue sequence and therefore determines a unique element of widehat B.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8cc2ea12822603f3","tex":"D^m","aliases":"D^m","display":"D^m","kind":"use","excerpt":"Finally D^n is not zero modulo D^m, but is zero modulo D^n. Since Hom_D(B_m)(B_m,B_m)=B_m, this is genuine loss of morphism information, not just a change of representatives.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7a3a2aba376670a9","tex":"D^N","aliases":"D^N","display":"D^N","kind":"use","excerpt":"For surjectivity take (y_p)∈ G. Choose N large enough that D^Ny_p∈ Z_p for every p. For each p, choose the residue of D^Ny_p modulo p^N. The ordinary Chinese remainder theorem provides z∈ Z with all these residues. Set a=z/D^N∈ A. Then a-y_p∈ Z_p for every p, so (a,(a-y_p)_p) maps to (y_p)_p. This proves exactness.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-76b7929465487fe2","tex":"D^n","aliases":"D^n","display":"D^n","kind":"use","excerpt":"Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. Th","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e7d029151e6271a6","tex":"\\mathbb D_{\\mathbb S}","aliases":"D_ S mathbb mathbb","display":"\\mathbb D_{\\mathbb S}","kind":"use","excerpt":"[Free seeded two-axis double category] For every small seeded two-axis double computad S, there exists a strict double category D_ S and a canonical map of seeded double computads ι_ S: S⟶ U D_ S such that for every strict double category D, restriction along ι_ S induces a natural bijection DblCat( D_ S, D) ≅ DblCptd( S,U D), for the underlying co","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"Specify finitely many positive scalar labels and a collection I of allowed independent pairs. Include their unit-seed interaction diagrams as the designated tests. This hypothesis ensures that the minimal kernel below is not being inferred from templates that accidentally annihilate an interaction already. Put S={p∈ P:some (a,b)∈ I satisfies pmid a","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d3b4cd694374cac7","tex":"d_e","aliases":"d_e","display":"d_e","kind":"use","excerpt":"nenumerate item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every normalization history are the event-labelled objects d_e:=Cofib(X_D_e→ X_downarrow e)","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-54ff0eb6a1af0d6f","tex":"D_f","aliases":"D_f","display":"D_f","kind":"use","excerpt":"First, every pair of incomparable events occurs in an elementary square. Take the ideal D_e∪ D_f. It contains neither e nor f, contains their predecessors, and enables both. The square residual at the unit is K(n_e,n_f).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bb43cb56e33c522f","tex":"d_{i,1}","aliases":"d_i,1","display":"d_{i,1}","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-dc162657ce651e2d","tex":"d_{i,k}","aliases":"d_i,k","display":"d_{i,k}","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0, 0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. ","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3169db205c09d2b3","tex":"d_{i,r_i}","aliases":"d_i,r_i","display":"d_{i,r_i}","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-729dc40006cd96e4","tex":"d_{j,l}","aliases":"d_j,l","display":"d_{j,l}","kind":"use","excerpt":"If no allowed pair has two free summands, every generating tensor is torsion. A free summand tensored with a torsion cone preserves that torsion's prime support. Two torsion summands have tensor product K(d_i,k,d_j,l), whose thick support is the prime intersection by [source reference]. Taking finite direct sums gives exactly the listed sets S_ij. ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-79e4e0f34503c5b8","tex":"d_R","aliases":"d_R","display":"d_R","kind":"use","excerpt":"[Resolution ultrametric] Define d_R(X,Y):= 0, · X=Y, 2^-ν(X,Y), · X≠ Y. Then d_R is an ultrametric: d_R(X,Z)≤max{d_R(X,Y),d_R(Y,Z)}.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-13339c696aa18d25","tex":"\\dashv","aliases":"dashv","display":null,"kind":"use","excerpt":"Any adjunction Ldashv U induces an adjunction on functor categories by postcomposition. Concretely, a natural transformation α:L∘ F⇒ G has pointwise adjoint mates α_r^flat:F(r)⟶ U(G(r)). Naturality of the unit and counit implies that the family {α_r^flat}_r is natural in r. The mate construction is invertible and respects higher coherence. Applying","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-63472510beee817f","tex":"\\operatorname{DblCat}","aliases":"DblCat operatorname","display":"\\operatorname{DblCat}","kind":"use","excerpt":"category] For every small seeded two-axis double computad S, there exists a strict double category D_ S and a canonical map of seeded double computads ι_ S: S⟶ U D_ S such that for every strict double category D, restriction along ι_ S induces a natural bijection DblCat( D_ S, D) ≅ DblCptd( S,U D), for the underlying computadic data. If the target ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2923caec9a6c7205","tex":"\\operatorname{DblCptd}","aliases":"DblCptd operatorname","display":"\\operatorname{DblCptd}","kind":"use","excerpt":"category] For every small seeded two-axis double computad S, there exists a strict double category D_ S and a canonical map of seeded double computads ι_ S: S⟶ U D_ S such that for every strict double category D, restriction along ι_ S induces a natural bijection DblCat( D_ S, D) ≅ DblCptd( S,U D), for the underlying computadic data. If the target ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"[Normalized cyclic lifting criterion] Under the preceding hypotheses, ρ_n has a continuous integral lift Longleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"re calculation is visible: K(a,b)=[ Zxrightarrowbinom-ba Z^2xrightarrow(a b) Z ], degrees -2,-1,0. The product of the two differentials is zero. The kernel of (a b) is generated by (-b/g,a/g), and the incoming image is g times that generator. Both H^-1 and H^0 are Z/g. Equivalently, if au+bv=g, then U=u · -b/g v · a/g, det U=1, (a b)U=(g 0).","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"3 · 2pmod9, B_2=2I+4B_1pmod9. The vector e_2 is cyclic for B_1 modulo 3. The displayed integer representatives of A_1,A_2 do not commute exactly: their additive commutator is diag(-81,81). The criterion nevertheless passes. Retain B_1 integrally and replace B_2 by the exact polynomial 2I+4B_1. This constructs widehat A_1=1 · 3","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"nameGL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This c","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2551402eef77c95b","tex":"\\dim_k","aliases":"dim_k dim","display":"\\dim_k","kind":"use","excerpt":"nameGL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This c","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c7c26f6b3b085b5c","tex":"E'","aliases":"E'","display":"E'","kind":"use","excerpt":"C arrow[r,\"u'\"'] · D. Write E=Cofib(u) and E'=Cofib(u'). The square induces E→ E'.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f1285db44838b010","tex":"e_+","aliases":"e_+","display":"e_+","kind":"use","excerpt":"M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not lift as a finite free Z_p-representation of this fixed group. The two displayed matrices commute and satisfy the defining relations. Since 2 is invertible in R, the central orthogonal idempotents e_+=(1+τ)/2, e_-=(1-τ)/2 split R[G] into a prod","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3da4494053c2f25c","tex":"e_-","aliases":"e_-","display":"e_-","kind":"use","excerpt":"M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not lift as a finite free Z_p-representation of this fixed group. The two displayed matrices commute and satisfy the defining relations. Since 2 is invertible in R, the central orthogonal idempotents e_+=(1+τ)/2, e_-=(1-τ)/2 split R[G] into a prod","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4d60cc719878240d","tex":"\\mathcal E_0","aliases":"E_0 mathcal","display":"\\mathcal E_0","kind":"use","excerpt":"Coarsening symbols occur in their fixed order C_1,…,C_b. A transformation symbol with exactly j coarsening symbols to its left is interpreted as T_i,j. Hence every reachable word represents a functor from E_b to E_0. Moving T_i across C_j reindexes its level rather than creating a new untyped symbol. No rule in this subsection contracts coarsenings","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a6ee72f7ee077722","tex":"E_1","aliases":"E_1","display":"E_1","kind":"use","excerpt":"At step i, the split exchange and additivity of the surrounding context give X_i-1≅ X_i⊕ E_i(s). Substituting these decompositions successively yields X_0≅ X_m⊕ E_m(s)⊕⋯⊕ E_1(s). The direct sum is finite because the chosen principal path is finite. The identification with shifted cofibers follows from the cofiber of a split projection.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"Finally, if a 2×2 matrix has no cyclic vector, it takes each vector into its own span. On a basis e_1,e_2 it is diagonal, and applying this condition to e_1+e_2 makes the two diagonal entries equal. Thus it is scalar. This argument includes the field F_2.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f2b11b7b1856ea86","tex":"E_{12}","aliases":"E_12","display":"E_{12}","kind":"use","excerpt":"For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde ","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"Finally, if a 2×2 matrix has no cyclic vector, it takes each vector into its own span. On a basis e_1,e_2 it is diagonal, and applying this condition to e_1+e_2 makes the two diagonal entries equal. Thus it is scalar. This argument includes the field F_2.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-eebcf5feadee9573","tex":"E_{21}","aliases":"E_21","display":"E_{21}","kind":"use","excerpt":"For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde ","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-79542e3d5fe27d48","tex":"\\mathcal E_A","aliases":"E_A mathcal","display":"\\mathcal E_A","kind":"use","excerpt":"Fix X∈ T_s. First record the exact commutation profile E_A(X;s)={r≤ s:Δ^A_s,r(X)≃0}. No interval or monotonicity property is assumed. For R= N and finite s, a well-defined consecutive coarsening depth is κ_A(X;s):=max{k∈{0,…,s}:Δ^A_s,r(X)≃0 for all s-k≤ r≤ s}. The set is nonempty because Δ^A_s,s(X)≃0. A larger value means that more consecutive coar","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9c08bebdf527fc50","tex":"\\mathcal E_b","aliases":"E_b mathcal","display":"\\mathcal E_b","kind":"use","excerpt":"Coarsening symbols occur in their fixed order C_1,…,C_b. A transformation symbol with exactly j coarsening symbols to its left is interpreted as T_i,j. Hence every reachable word represents a functor from E_b to E_0. Moving T_i across C_j reindexes its level rather than creating a new untyped symbol. No rule in this subsection contracts coarsenings","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-58230f0d9aab671b","tex":"E_i","aliases":"E_i","display":"E_i","kind":"use","excerpt":"[Finite split residual expansion] Fix a finite principal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then W(s)≅ W_nf(s)⊕⊕_i=1^m E_i(s), where E_i(s) is the surrounding context applied to the source-side remainder of step i. In the","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e749405022a66e6b","tex":"E_m","aliases":"E_m","display":"E_m","kind":"use","excerpt":"At step i, the split exchange and additivity of the surrounding context give X_i-1≅ X_i⊕ E_i(s). Substituting these decompositions successively yields X_0≅ X_m⊕ E_m(s)⊕⋯⊕ E_1(s). The direct sum is finite because the chosen principal path is finite. The identification with shifted cofibers follows from the cofiber of a split projection.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0dd6372c1de2074b","tex":"E_p","aliases":"E_p","display":"E_p","kind":"use","excerpt":"bb Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and le","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-70a9953dd9bd9b22","tex":"\\operatorname{End}","aliases":"End operatorname","display":"\\operatorname{End}","kind":"use","excerpt":"[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d99b0089dd4f1347","tex":"\\operatorname{End}_{\\mathbb Z_p}","aliases":"End_ Z_p operatorname mathbb","display":"\\operatorname{End}_{\\mathbb Z_p}","kind":"use","excerpt":"[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5c2298535cc2b513","tex":"\\operatorname{End}_A","aliases":"End_A operatorname","display":"\\operatorname{End}_A","kind":"use","excerpt":"[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-533a108b23c10029","tex":"\\operatorname{End}_k","aliases":"End_k operatorname","display":"\\operatorname{End}_k","kind":"use","excerpt":"Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":". In particular, for m>n, the nonzero endomorphism D^n:B_m→ B_m becomes zero under C_m,n. Exactness follows by reducing bounded projective cone diagrams. For conservativity it suffices, by the Chinese remainder decomposition and composition with reduction to level one, to prove that a perfect complex over R= Z/p^m is zero if its derived reduction t","count":34},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"[Ideal-indexed residual diagram] A coherent exchange-event diagram determines a functor F_X:NJ(P)⟶ S, F_X(I)=Cofib(X_∅⟶ X_I), with the following properties: item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e","count":12},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c1f02f9950b99117","tex":"f_*","aliases":"f_*","display":"f_*","kind":"use","excerpt":"[Normalized cyclic lifting criterion] Under the preceding hypotheses, ρ_n has a continuous integral lift Longleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange resid","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange resid","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"Finally, if a 2×2 matrix has no cyclic vector, it takes each vector into its own span. On a basis e_1,e_2 it is diagonal, and applying this condition to e_1+e_2 makes the two diagonal entries equal. Thus it is scalar. This argument includes the field F_2.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-49ea7c696066b690","tex":"\\mathbb F_2^2","aliases":"F_2^2 mathbb","display":"\\mathbb F_2^2","kind":"use","excerpt":"icient pairs in the eight listed rings Z/q with q∈{2,3,4,5,8,9,16,25}. It enumerates kernels and images to check three cohomology degrees and verifies middle-cohomology elementary divisors by torsion counts. It also checks 160 annihilator-transition tuples, 1,728 exchange coherences, 72 prime-component projections, 2,304 positive-label pair instanc","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-54c7a07f66f1253f","tex":"\\mathbb F_3","aliases":"F_3 mathbb","display":"\\mathbb F_3","kind":"use","excerpt":"3 · 1, G=C_3. Then A^3=I over R, I+A+A^2=3I, and M^G=3M. The invariant cochain complex is quasi-isomorphic to ( F_3)^2 in degree zero. Derived reduction of that invariant object has cohomology in every nonpositive degree, whereas taking derived invariants after reduction has cohomology in every nonnegative degree. The comparison cone has ( F_3)^2 i","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4677bcfd85b4c82d","tex":"\\mathbb F_3^2","aliases":"F_3^2 mathbb","display":"\\mathbb F_3^2","kind":"use","excerpt":"eight listed rings Z/q with q∈{2,3,4,5,8,9,16,25}. It enumerates kernels and images to check three cohomology degrees and verifies middle-cohomology elementary divisors by torsion counts. It also checks 160 annihilator-transition tuples, 1,728 exchange coherences, 72 prime-component projections, 2,304 positive-label pair instances, all 30 pairs of ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f3ee6a8ba9575903","tex":"\\mathbb F_5","aliases":"F_5 mathbb","display":"\\mathbb F_5","kind":"use","excerpt":"Cohomology dimensions over F_2, F_3, F_5, F_7 · 29,856","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-23af4eb7208f9cad","tex":"\\mathbb F_7","aliases":"F_7 mathbb","display":"\\mathbb F_7","kind":"use","excerpt":"Cohomology dimensions over F_2, F_3, F_5, F_7 · 29,856","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-31d7232e6fa917d3","tex":"F_i","aliases":"F_i","display":"F_i","kind":"use","excerpt":"[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange resid","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-eb09c0fffadc0f11","tex":"F_{i-1}","aliases":"F_i-1","display":"F_{i-1}","kind":"use","excerpt":"[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange resid","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"use","excerpt":"0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"use","excerpt":"[Stable residual filtration along a normalization path] For every finite normalization path in a stable enhancement there is a canonical finite filtration 0=F_0⟶ F_1⟶⋯⟶ F_m=D_tot with cofiber sequences F_i-1⟶ F_i⟶ D_i. Consequently the associated graded pieces of the total normalization defect are precisely the transported elementary exchange resid","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6a27312454be2902","tex":"F_p","aliases":"F_p","display":"F_p","kind":"use","excerpt":"Here is also an explicit finite construction proving the local identifications. Put F_p=bE_p, so b^2 Z_p^r⊆ F_p⊆ Z_p^r. The Chinese remainder isomorphism identifies Z^r/b^2 Z^r≅ ∏_p∈ P Z_p^r/b^2 Z_p^r. Let H⊆ Z^r be the inverse image of the product of the subgroups F_p/b^2 Z_p^r. Directly from the definitions, H=bΛ. The quotient Z^r/H is the produc","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9da98cb9c6f0ae74","tex":"\\mathbb F_p","aliases":"F_p mathbb","display":"\\mathbb F_p","kind":"use","excerpt":". In particular, for m>n, the nonzero endomorphism D^n:B_m→ B_m becomes zero under C_m,n. Exactness follows by reducing bounded projective cone diagrams. For conservativity it suffices, by the Chinese remainder decomposition and composition with reduction to level one, to prove that a perfect complex over R= Z/p^m is zero if its derived reduction t","count":21},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4cf4c741d2045e51","tex":"\\mathbb F_p^\\times","aliases":"F_p^× mathbb times","display":"\\mathbb F_p^\\times","kind":"use","excerpt":"ting input. It does not require enumeration of deeper coefficient lifts. When it passes, the polynomial construction in [source reference] produces exact commuting integral generator matrices. If all generators are scalar, lift their scalar entries integrally. Their residues modulo p have p-power order and therefore equal 1, since F_p^× has order p","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e410957d33ed25bf","tex":"F_r^{(p)}","aliases":"F_r^(p)","display":"F_r^{(p)}","kind":"use","excerpt":"[Unrestricted coefficient lifting for a free pro-p source] Let F_r^(p) be the free pro-p group on finitely many generators. Every continuous F_r^(p)⟶GL_d(R_n) has a continuous lift to GL_d( Z_p). One may choose arbitrary integral lifts of its generator matrices. Construct F_r^(p) as the inverse limit of the finite p-group quotients of the abstract ","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-47567af3d7dd306d","tex":"f_s","aliases":"f_s","display":"f_s","kind":"use","excerpt":"[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e34327622e5e789e","tex":"f_v","aliases":"f_v","display":"f_v","kind":"use","excerpt":"[Two-axis factorization] Every morphism f:(r,X)⟶(s,Y) in K factors uniquely as (r,X) xrightarrowf_v (r,C_s→ rY) xrightarrowκ_r,s,Y (s,Y), where f_v lies entirely in the fiber T_r.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-76a3d72751b845de","tex":"\\mathcal F_X","aliases":"F_X mathcal","display":"\\mathcal F_X","kind":"use","excerpt":"[Ideal-indexed residual diagram] A coherent exchange-event diagram determines a functor F_X:NJ(P)⟶ S, F_X(I)=Cofib(X_∅⟶ X_I), with the following properties: item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e","count":11},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-febf5f4b74ad9419","tex":"\\mathcal F_{X_M}","aliases":"F_X_M mathcal","display":"\\mathcal F_{X_M}","kind":"use","excerpt":"comparison after an unrelated event f is multiplication by n_f on that complex. Consequently Θ_e,f(I)≃ M⊗^ L_ Z C_ Z(n_e)⊗^ L_ ZC_ Z(n_f). The complexes C_ Z(n) are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead F_X_M(P)≃ M⊗^ L_ ZC_ Z(N), N=∏_e∈ Pn_e. This is not the total c","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-507c1759c57b7764","tex":"\\operatorname{Fib}","aliases":"Fib operatorname","display":"\\operatorname{Fib}","kind":"use","excerpt":"For every perfect widehat B-complex M, the natural comparison χ_M:M⊗^ L_widehat Bwidehat B[1/T] ⟶ Rvarprojlim_n (M⊗^ L_widehat BB_n[1/T]) has fiber Fib(χ_M)≃ ⊕_p∈ SM⊗^ L_widehat B Q_p. Thus χ_M is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by [1] of that fiber. These statements take place in D(wideha","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f4a70ad89c34bacc","tex":"\\flat","aliases":"flat","display":null,"kind":"use","excerpt":"Any adjunction Ldashv U induces an adjunction on functor categories by postcomposition. Concretely, a natural transformation α:L∘ F⇒ G has pointwise adjoint mates α_r^flat:F(r)⟶ U(G(r)). Naturality of the unit and counit implies that the family {α_r^flat}_r is natural in r. The mate construction is invertible and respects higher coherence. Applying","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-459a3ba25f00f8ca","tex":"g_{p,1}","aliases":"g_p,1","display":"g_{p,1}","kind":"use","excerpt":"[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7037a322f98ef05b","tex":"g_{p,i}","aliases":"g_p,i","display":"g_{p,i}","kind":"use","excerpt":"[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-969c9ce2b9a20c5e","tex":"g_{p,s}","aliases":"g_p,s","display":"g_{p,s}","kind":"use","excerpt":"[Compatible operators glue with the lattice] In [source reference], suppose f_1,…,f_s∈End_A(V) and g_p,1,…,g_p,s∈End_ Z_p(L_p) satisfy the exact intertwining equations θ_p(f_i⊗1)=(g_p,i⊗1)θ_p. Then they induce unique integral endomorphisms of Λ. Every specified noncommutative polynomial relation with integer coefficients holding on V holds on Λ. If","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-077b802462413614","tex":"g_R","aliases":"g_R","display":"g_R","kind":"use","excerpt":"Instead of a single growth function, separate growth by generator type: g_T(n), g_R(n), g_Δ(n). This may distinguish theories in which most complexity comes from representation changes, resolution branching, or residual generation.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1f7f79bd913b02ee","tex":"\\mathcal G_s","aliases":"G_s mathcal","display":"\\mathcal G_s","kind":"use","excerpt":"[Orbit-generator test for an exchange obstruction] Fix an exchange η^A_s,r between exact functors, and an exact quotient q_r of the target fiber. If q_rΔ^A_s,r(g)≃0 for every g∈ G_s, then q_rΔ^A_s,r(X)≃0 for every X∈ Z_s^thick. The same assertion holds for q_r=id, in which case the exchange is an equivalence on the entire generated thick zoo.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-21e3d4b1c6847f32","tex":"g_T","aliases":"g_T","display":"g_T","kind":"use","excerpt":"Instead of a single growth function, separate growth by generator type: g_T(n), g_R(n), g_Δ(n). This may distinguish theories in which most complexity comes from representation changes, resolution branching, or residual generation.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a0b297ebe15ff2fd","tex":"g_\\Delta","aliases":"g_Δ Delta","display":"g_\\Delta","kind":"use","excerpt":"Instead of a single growth function, separate growth by generator type: g_T(n), g_R(n), g_Δ(n). This may distinguish theories in which most complexity comes from representation changes, resolution branching, or residual generation.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"[Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other co","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fd40d683c5451dff","tex":"\\operatorname{GL}_2","aliases":"GL_2 operatorname","display":"\\operatorname{GL}_2","kind":"use","excerpt":"[Complete rank-two profinite lifting test] Let p be any prime and let ρ_n: Z_p^r→GL_2(R_n) be continuous, with finitely many topological generators as above. item If a=n, all the A_i are scalar and ρ_n has a continuous integral lift. item If a<n, choose any generator A_* with scalar depth a. Take λ_i to be an integer representative of (A_i)_11 and ","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-301f280765554fc6","tex":"\\operatorname{GL}_m","aliases":"GL_m operatorname","display":"\\operatorname{GL}_m","kind":"use","excerpt":"The Euclidean algorithm produces a unimodular matrix U∈GL_m( Z) with (a_1,…,a_m)U=(g,0,…,0). The Koszul complex is the exterior algebra on a free module in degree -1, with differential the derivation determined by this row vector. Hence the maps bigwedge^jU give an isomorphism of complexes from K(g,0,…,0) to K(a_1,…,a_m).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c5c2a0a2da51d5b8","tex":"\\mathbf{Gpd}","aliases":"Gpd mathbf","display":"\\mathbf{Gpd}","kind":"use","excerpt":"[The source--coefficient diagram has mixed variance] Reduction of coefficients and inflation of source define a functor mathscr P_d: N_≥1^op× Q_Γ ⟶Gpd. Each source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at (m,U') is a compatible refinement of ρ_n at (n,U) precisely when its reduction to level n eq","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-daaa917c98a2902e","tex":"H^{-1}","aliases":"H^-1","display":"H^{-1}","kind":"use","excerpt":"For two events the entire calculation is visible: K(a,b)=[ Zxrightarrowbinom-ba Z^2xrightarrow(a b) Z ], degrees -2,-1,0. The product of the two differentials is zero. The kernel of (a b) is generated by (-b/g,a/g), and the incoming image is g times that generator. Both H^-1 and H^0 are Z/g. Equivalently, if au+bv=g, then U=u · -b/g","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a2043676220c271f","tex":"H^{-j}","aliases":"H^-j","display":"H^{-j}","kind":"use","excerpt":"Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other coh","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fa5a6573c905a228","tex":"H^0","aliases":"H^0","display":"H^0","kind":"use","excerpt":"Each zero entry contributes C_ Z(0)≅ Z⊕ Z[1] with zero differential. Distributing these m-1 factors gives the stated shifted copies, with the standard shift isomorphisms. Since g>0, the map g: Z→ Z is injective. Thus C_ Z(g) has only H^0= Z/g, proving the cohomology formula and the vanishing criterion.","count":14},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5028e979975b689b","tex":"H^0_{\\mathrm{cts}}","aliases":"H^0_cts mathrm","display":"H^0_{\\mathrm{cts}}","kind":"use","excerpt":"lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))A_g. When lifts exist, their framed set is a torsor for Z^1_cts(Γ,V). Strict-conjugacy classes and stabilizers are governed by H^1_cts and H^0_cts, respectively. The associativ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-27a65b26c5e90d8e","tex":"H^1_{\\mathrm{cts}}","aliases":"H^1_cts mathrm","display":"H^1_{\\mathrm{cts}}","kind":"use","excerpt":"n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))A_g. When lifts exist, their framed set is a torsor for Z^1_cts(Γ,V). Strict-conjugacy classes and stabilizers are governed by H^1_cts and H^0_cts, respectiv","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"This is a concrete relation obstruction for a prescribed deformation problem. It is not asserted to classify arbitrary deformation obstructions or to identify the entire unbounded exchange cone with an H^2 obstruction group. In contrast, the norm-quotient actions in [source reference] genuinely exist over Z_p at all precisions, even though some of ","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c09266a6ce791cef","tex":"H^2_{\\mathrm{cts}}","aliases":"H^2_cts mathrm","display":"H^2_{\\mathrm{cts}}","kind":"use","excerpt":"[Complete continuous one-step obstruction] The class o_n^cts(ρ_n)=[c_A]∈ H^2_cts(Γ,V) is independent of the continuous choices. It vanishes if and only if ρ_n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-92ad3ff2bdd044cf","tex":"H^i","aliases":"H^i","display":"H^i","kind":"use","excerpt":"[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0, 0 · i>0, H^i(V)≅0 · i<0, k^d · i≥0. The natural comparison β:U→ V is zero on H^0. Its functor-intrinsic cofiber satisfies H^i","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-25c4943486fc4389","tex":"H^{i+1}","aliases":"H^i+1","display":"H^{i+1}","kind":"use","excerpt":"The evaluation map on degree-zero invariants is the inclusion pM↪ M. On tensoring with k its degree-zero map is zero. Evaluation on H^0(V)=(k^d)^G=k^d is an isomorphism. Naturality of evaluation defining β therefore proves H^0(β)=0. The long exact cohomology sequence of its cofiber now gives k^d in every degree: the groups in negative degrees come ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1e49578129638f1e","tex":"H^j","aliases":"H^j","display":"H^j","kind":"use","excerpt":"and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e632673879a151df","tex":"H^j_{\\mathrm{cts}}","aliases":"H^j_cts mathrm","display":"H^j_{\\mathrm{cts}}","kind":"use","excerpt":"and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7d41730099edd284","tex":"h_G","aliases":"h_G","display":"h_G","kind":"use","excerpt":"[Finite horizon for fixed finite-group representations] Let G be a finite group, d,n≥1, and e=v_p(|G|). Set h_G(n)=max{n+e, 2e+1}. For every ρ_n:G→GL_d( Z/p^n), the following are equivalent: item There is a representation ρ:G→GL_d( Z_p) with reduction ρ_n. item There is a representation ρ_h:G→GL_d( Z/p^h_G(n)) with reduction ρ_n. The implication fr","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-480b3009a21bb6ea","tex":"\\operatorname{Hom}_{D(B_m)}","aliases":"Hom_D(B_m) operatorname","display":"\\operatorname{Hom}_{D(B_m)}","kind":"use","excerpt":"Finally D^n is not zero modulo D^m, but is zero modulo D^n. Since Hom_D(B_m)(B_m,B_m)=B_m, this is genuine loss of morphism information, not just a change of representatives.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cbb382faef7f21fa","tex":"\\operatorname{Hom}_O","aliases":"Hom_O operatorname","display":"\\operatorname{Hom}_O","kind":"use","excerpt":"[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a26446d8881c4cf4","tex":"\\operatorname{Hom}_{O[G]}","aliases":"Hom_O[G] operatorname","display":"\\operatorname{Hom}_{O[G]}","kind":"use","excerpt":"[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM) ⟶ · H^1(G,V)[p^n]⟶0 is exact and natural. A modular equivariant map lifts to an integral equivariant map if and only if its boundary class vanishes. If p^h annihilates the p-primary ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b7dcabd550bce83f","tex":"\\operatorname{Hom}_{O_n[G]}","aliases":"Hom_O_n[G] operatorname","display":"\\operatorname{Hom}_{O_n[G]}","kind":"use","excerpt":"[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM) ⟶ · H^1(G,V)[p^n]⟶0 is exact and natural. A modular equivariant map lifts to an integral equivariant map if and only if its boundary class vanishes. If p^h annihilates the p-primary ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-93c2db40448e9619","tex":"\\operatorname{Hom}_{R[G]}","aliases":"Hom_R[G] operatorname","display":"\\operatorname{Hom}_{R[G]}","kind":"use","excerpt":"For a commutative ring R, a finite group G, and M∈ D(R[G]), define Γ_R(M)= RHom_R[G](R,M)∈ D(R). This is the derived invariant functor. Derived categories in this section are the enhanced unbounded derived categories. They are stable. Their exact functors preserve finite cofiber sequences but are not asserted to preserve every infinite limit or col","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"use","excerpt":"[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-46cc9b01cd7f4804","tex":"I'","aliases":"I'","display":"I'","kind":"use","excerpt":"Now let F be reduced and excisive. We prove the counit j_!j^*F→ F is an equivalence on an ideal I by induction on |I|. The empty ideal is the zero object on both sides. Choose a maximal element e∈ I. Put I'=I∖{e}. Then I=I'∪downarrow e, I'∩downarrow e=D_e. The ideal-excision lemma expresses F(I) as the pushout of F(I') and F(downarrow e) over F(D_e","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8de7a19f20a0fec4","tex":"\\id_{C_{s\\to r}Y}","aliases":"id_C_s→ rY id to operatorname to","display":"{\\operatorname{id}}_{C_{s\\to r}Y}","kind":"use","excerpt":"For r≤ s and Y∈ T_s, define the canonical morphism κ_r,s,Y: (r,C_s→ rY)⟶(s,Y) represented by the identity map id_C_s→ rY.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-65624818fd62feb8","tex":"\\id_k","aliases":"id_k id operatorname","display":"{\\operatorname{id}}_k","kind":"use","excerpt":"The normal-form endpoint alone does not determine a residual. For example, the maps id_k and 0:k→ k have the same source and target, but cofibers 0 and k⊕ k[1]. Comparison maps therefore belong to the data.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1ce608af84df2a4c","tex":"\\id_M","aliases":"id_M id operatorname","display":"{\\operatorname{id}}_M","kind":"use","excerpt":"Fix a finite poset P and assign a positive integer n_e to every event e∈ P. For M∈Perf( Z) define X_M(I)=M, X_M(I⊆ J)=(∏_e∈ J∖ In_e)id_M. Empty products are 1. For nested ideals these products multiply exactly, so X_M is a strict chain-level diagram followed by the stable realization. When P=P_W, take identity transformation and coarsening functors","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5d0302fc76bc0ed4","tex":"\\id_{\\Tcat_r}","aliases":"id_Tcat_r id_ T_r id Tcat operatorname mathbf","display":"{\\operatorname{id}}_{{\\mathbf T}_r}","kind":"use","excerpt":"Strictness gives C_r→ r=id_ T_r and, for r≤ s≤ t, C_t→ r=C_s→ r∘ C_t→ s. A pseudofunctor version is more natural in some applications. The strict version isolates the main structure without coherence bookkeeping.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-de717c99c15e8582","tex":"\\id_X","aliases":"id_X id operatorname","display":"{\\operatorname{id}}_X","kind":"use","excerpt":"Identity at (r,X) is id_X:X→ C_r→ r(X)=X. For associativity, let (r,X)xrightarrowα(s,Y) xrightarrowβ(t,Z) xrightarrowγ(u,W). One composite gives C_t→ r(γ)∘ C_s→ r(β)∘α. The other gives the same expression by functoriality and C_t→ r=C_s→ rC_t→ s. Thus composition is associative.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"[When operational killability is already thick] If mathscr M consists of exact endofunctors and has the directed annihilator property, then K_mathscr M is a thick stable subcategory. Hence N_mathscr M= K_mathscr M and Obs_mathscr M(X)≃0 iff X is operationally killable.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-28c108f2e7526500","tex":"\\operatorname{Inf}","aliases":"Inf operatorname","display":"\\operatorname{Inf}","kind":"use","excerpt":"[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5a3d32242b747ca2","tex":"\\operatorname{Inf}_{\\Gamma/U}^{\\Gamma/U'}","aliases":"Inf_Γ/U^Γ/U' operatorname Gamma Gamma","display":"\\operatorname{Inf}_{\\Gamma/U}^{\\Gamma/U'}","kind":"use","excerpt":"[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2ba8441ab7c7d7ef","tex":"j^*","aliases":"j^*","display":"j^*","kind":"use","excerpt":"Now let F be reduced and excisive. We prove the counit j_!j^*F→ F is an equivalence on an ideal I by induction on |I|. The empty ideal is the zero object on both sides. Choose a maximal element e∈ I. Put I'=I∖{e}. Then I=I'∪downarrow e, I'∩downarrow e=D_e. The ideal-excision lemma expresses F(I) as the pushout of F(I') and F(downarrow e) over F(D_e","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-dd29352dd1dee06c","tex":"J^j","aliases":"J^j","display":"J^j","kind":"use","excerpt":"[A profinite lift need not preserve the finite quotient] Let p be odd and J∈ M_d( Z_p) be invertible modulo p. The action C_p→GL_d( Z/p^2) given by I+pJ has no integral lift as an action of C_p. Its pullback along Z_p↠ C_p, however, has a continuous integral lift ρ(x)=(I+pJ)^x =∑_j≥0binomxjp^jJ^j, x∈ Z_p. For each j, the polynomial binomxj is conti","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-614ae20ad64312e9","tex":"j_{\\mathcal B}","aliases":"j_ B mathcal","display":"j_{\\mathcal B}","kind":"use","excerpt":"The universal property above is an equivalence of functor ∞-categories, not only of their cores. This distinction is essential for noninvertible exchange maps. To extend η:f⇒ g between additive functors A→ B, compose it with j_ B: B→St^add( B) and lift the resulting morphism through Fun^ex(St^add( A),St^add( B)) ≃Fun^⊕( A,St^add( B)). Full faithful","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9a01c8801f265cf0","tex":"j_!","aliases":"j_!","display":"j_!","kind":"use","excerpt":"e full subcategory of coherent diagrams F:NJ(P)→ S such that F(∅)≃0 and every elementary independent-event square is a homotopy pushout. Restriction along j:P⟶ J(P), e⟼downarrow e induces an equivalence Exc_0(J(P), S) ≃Fun(NP, S). Its inverse sends G:NP→ S to (j_!G)(I)=*colim_e∈ IG(e), (j_!G)(∅)=0. All colimits are finite. In particular, for an exc","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a7e0c315ce3cf160","tex":"\\K","aliases":"K mathbf","display":"{\\mathbf K}","kind":"use","excerpt":"[Total Komorebi category] Given F: R^op→Cat, define the category K:=∫_ RF as follows.","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-53283f2f6027fba4","tex":"\\mathbb K","aliases":"K mathbb","display":"\\mathbb K","kind":"use","excerpt":"Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"Work in the stable category of perfect complexes over a field k. Put M=k[0] and N=k^2[0], and consider the square [column sep=large,row sep=large] 0 arrow[r] arrow[d] · M arrow[d,hook]","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e6c19a060d9e281f","tex":"k^d","aliases":"k^d","display":"k^d","kind":"use","excerpt":"[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0,","count":11},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7e3de3b4eb4b9742","tex":"\\mathbb K_{\\mathrm{arith}}","aliases":"K_arith mathbb mathrm","display":"\\mathbb K_{\\mathrm{arith}}","kind":"use","excerpt":"Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-afb7963eae355ad1","tex":"\\mathbb K_{\\mathrm{aut}}","aliases":"K_aut mathbb mathrm","display":"\\mathbb K_{\\mathrm{aut}}","kind":"use","excerpt":"Let K_arith and K_aut be arithmetic and automorphic seeded two-axis calculi.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2ea1f9d126d9af31","tex":"K_{B_n}","aliases":"K_B_n","display":"K_{B_n}","kind":"use","excerpt":"_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-46d21cf1bf7572c1","tex":"K_m","aliases":"K_m","display":"K_m","kind":"use","excerpt":"its boundary] There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently the cokernel of the natural co","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-84ab7f4d8b9d075b","tex":"\\mathcal K_{\\mathscr M_A}","aliases":"K_mathscr M_A mathcal mathscr","display":"\\mathcal K_{\\mathscr M_A}","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bb9a538f5e406f30","tex":"K_n","aliases":"K_n","display":"K_n","kind":"use","excerpt":"For a map d:V→ W of lattices, put K=ker d, Q=cokerd, K_n=ker(d⊗_O O_n). The module K is saturated in V: if p^n v∈ K, then p^n d(v)=0 in the torsion-free module W, so v∈ K. It is therefore a finite free direct summand as an O-module, although a splitting is not canonical. Reduction supplies a natural injection α_n:K/p^nK→ K_n.","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ed950e86be318df7","tex":"K_{n+h}","aliases":"K_n+h","display":"K_{n+h}","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1c27eeb827cbc2e0","tex":"K_R","aliases":"K_R","display":"K_R","kind":"use","excerpt":"_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These ","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-40428f0704f0846d","tex":"\\mathcal K_R","aliases":"K_R mathcal","display":"\\mathcal K_R","kind":"use","excerpt":"A two-fiber exact model can be written entirely with endofunctors: set K_R=D(R[G])× D(R), A_R(M,N)=(0,Γ_R(M)), and let C_R,S apply derived coefficient change in both components. The second component of C_R,SA_R→ A_SC_R,S is precisely β. This places the example in a genuine two-axis exact system, rather than identifying differently typed functors by","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"In an abelian fiber one may instead record Δ^-_s,r(X)=kerη^A_s,r,X, Δ^+_s,r(X)=cokerη^A_s,r,X. These vanish simultaneously exactly when the comparison is an isomorphism.","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"The square is [column sep=large,row sep=large] A_rC_s→ r(X) arrow[r,\"η^A_s,r,X\"] · C_s→ rA_s(X). It compares transforming after coarsening with coarsening after transforming.","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"Evaluate this ordinary functor on s, take its nerve, and compose with L_0. An ordinary functor gives all simplices of its nerve, so the output is a coherent diagram, not merely a diagram in a homotopy category. The prescribed vertices and edges are unchanged by this construction.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cba2774e16404a68","tex":"L_{\\bar A}","aliases":"L_bar A bar","display":"L_{\\bar A}","kind":"use","excerpt":"[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solut","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-32067f05b0fb68a7","tex":"L_I","aliases":"L_I","display":"L_I","kind":"use","excerpt":"For A=I+pJ modulo p^2 and q=p, the residual is c=J while L_I(B)=pB=0 in k. This recovers the one-step obstruction without exhaustive testing. At other inputs the Jacobian may have nonzero rank, so some nonzero initial relation residuals are removable rather than persistent.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5168d74b869524fd","tex":"L_p","aliases":"L_p","display":"L_p","kind":"use","excerpt":"[Effective arithmetic lattice gluing] Let V be a finite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the uniqu","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-345427490e88d80b","tex":"\\leadsto","aliases":"leadsto","display":null,"kind":"use","excerpt":"c diamond criterion] Let a rewrite graph have a height h in N that decreases on every edge, and suppose every vertex x has a unique reachable normal vertex n(x). Assign an object X_x in a stable ∞-category to each vertex and a morphism to each edge. Assume every elementary peak xxrightarrowey, xxrightarrowfz has joining paths a:yleadsto u and b:zle","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint resid","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudo","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-29c5057f9a8c7b85","tex":"\\lim_{[n]\\in\\Delta}","aliases":"lim_[n]∈Δ lim in Delta","display":"\\lim_{[n]\\in\\Delta}","kind":"use","excerpt":"[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudo","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7d196e2e6f214be1","tex":"M^G","aliases":"M^G","display":"M^G","kind":"use","excerpt":"Let G be a finite group acting O-linearly on a lattice M. The natural comparison is α_G,n:M^G/p^nM^G⟶(M/p^nM)^G. The first path takes invariant vectors and then reduces. The second reduces the representation and then takes invariant vectors. Its direction is fixed by reduction of an invariant vector; no arbitrary scalar enters this map.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-335b5e6f91a8441f","tex":"m_0","aliases":"m_0","display":"m_0","kind":"use","excerpt":"[ ρ:G⟶GL_d( Z_p) with ρ(g)≡ A_gpmodp^m-e. After j iterations the certified defect depth is m_j=2e+2^j(m-2e), and the next change has depth at least m_j-e. In particular, the sequence converges uniformly on G and all prescribed multiplication relations hold exactly in the limit. Set m_0=m and m_j+1=2m_j-2e. Induction gives the displayed formula, so ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"use","excerpt":"For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-63019db73ab92a7c","tex":"M_A","aliases":"M_A","display":"M_A","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-098ede2828d25010","tex":"M_B","aliases":"M_B","display":"M_B","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-207e98afc142d0de","tex":"M_d","aliases":"M_d","display":"M_d","kind":"use","excerpt":"The perfectness hypothesis cannot be removed. Fix an odd prime p, let R= Z/p^2, k= F_p, G=C_p, and choose a matrix J∈ M_d( Z) whose reduction modulo p is invertible. Put M=R^d and let the chosen generator act by A=I+pJpmodp^2. For d=2, the choice J=(0 · 1","count":23},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c879cfab2bd64d1b","tex":"m_j","aliases":"m_j","display":"m_j","kind":"use","excerpt":"[Convergent finite-group action repair] If the initial defect depth satisfies m>2e, the preceding averaging operation iterates to a representation ρ:G⟶GL_d( Z_p) with ρ(g)≡ A_gpmodp^m-e. After j iterations the certified defect depth is m_j=2e+2^j(m-2e), and the next change has depth at least m_j-e. In particular, the sequence converges uniformly on","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-255faff69831c228","tex":"m_{j+1}","aliases":"m_j+1","display":"m_{j+1}","kind":"use","excerpt":"ngrightarrowGL_d( Z_p) with ρ(g)≡ A_gpmodp^m-e. After j iterations the certified defect depth is m_j=2e+2^j(m-2e), and the next change has depth at least m_j-e. In particular, the sequence converges uniformly on G and all prescribed multiplication relations hold exactly in the limit. Set m_0=m and m_j+1=2m_j-2e. Induction gives the displayed formul","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"use","excerpt":"[Uniform based-complex tower reconstruction] Suppose M_n are based finite free cochain complexes over B_n, concentrated in a common finite interval with fixed ranks in each degree. Suppose their differential matrices reduce exactly from level m to level n for m≥ n. Then there is a unique based complex widehat M with those ranks over widehat B whose","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b428dd436b8c5725","tex":"M_n^i","aliases":"M_n^i","display":"M_n^i","kind":"use","excerpt":"Degreewise transition maps of the finite free modules are surjective. In the product model of the homotopy inverse limit, the map 1-shift:∏_n M_n^i⟶∏_n M_n^i is surjective in each degree: choose an initial entry and successively lift the required difference across the surjective transition map. Its kernel is the ordinary inverse limit. Thus the hom","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c29c15bb256c2fa4","tex":"\\operatorname{Map}","aliases":"Map operatorname","display":"\\operatorname{Map}","kind":"use","excerpt":"hbfCat^st), F⟼ St^add∘ F. It is left adjoint to pointwise forgetting U_ R: Fun( R^op,Cat^st) ⟶ Fun( R^op,Cat^add). Equivalently, for every additive two-axis system F and stable two-axis system G there is a natural equivalence Map (St^add_ RF,G) ≃ Map (F,U_ RG).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"2^-ν(X,Y), · X≠ Y. Then d_R is an ultrametric: d_R(X,Z)≤max{d_R(X,Y),d_R(Y,Z)}.","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"For the fixed labelled poset define the finite prime set S_P=⋃_eparallel f {p prime:pmidgcd(n_e,n_f)}, where eparallel f means incomparable. Let J_P be the invariant thick closure of the square interaction residuals for the unit seed Z.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4bc014492ab92312","tex":"\\min_i","aliases":"min_i min","display":"\\min_i","kind":"use","excerpt":"For A∈ M_2(R_n) define its scalar depth by a(A)=max{0≤ t≤ n:A is scalar modulo p^t}. Writing entries in R_n, this is the minimum of the capped valuations of A_12,A_21,A_11-A_22. It is unchanged by invertible conjugation. For a tuple put a=min_i a(A_i).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b9588331116f694e","tex":"n_e","aliases":"n_e","display":"n_e","kind":"use","excerpt":"Fix a finite poset P and assign a positive integer n_e to every event e∈ P. For M∈Perf( Z) define X_M(I)=M, X_M(I⊆ J)=(∏_e∈ J∖ In_e)id_M. Empty products are 1. For nested ideals these products multiply exactly, so X_M is a strict chain-level diagram followed by the stable realization. When P=P_W, take identity transformation and coarsening functors","count":9},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e434992e440b99a8","tex":"n_f","aliases":"n_f","display":"n_f","kind":"use","excerpt":"For an enabled event e, E_e(I)≃ M⊗^ L_ ZC_ Z(n_e). The comparison after an unrelated event f is multiplication by n_f on that complex. Consequently Θ_e,f(I)≃ M⊗^ L_ Z C_ Z(n_e)⊗^ L_ ZC_ Z(n_f). The complexes C_ Z(n) are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead F_X_M(P)","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d7c0e2c4594eec1b","tex":"n_m","aliases":"n_m","display":"n_m","kind":"use","excerpt":"quare's commutativity is not its excision. The second cofiber vanishes precisely for a homotopy pushout, not for every commuting square. item Context-independent event residuals do not identify ordered filtrations. Nor do they split extensions along dependent events: the Z/p^2 chain is an explicit counterexample. item The total normalization residu","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a7fd5345afb37e08","tex":"\\mathcal N_{\\mathscr M_A}","aliases":"N_mathscr M_A mathcal mathscr","display":"\\mathcal N_{\\mathscr M_A}","kind":"use","excerpt":"mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In that case obstruction vanishing is preserved an","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5da366b3908c86e4","tex":"\\mathcal N_{\\mathscr M_B}","aliases":"N_mathscr M_B mathcal mathscr","display":"\\mathcal N_{\\mathscr M_B}","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ae4704a81fdf75d2","tex":"n_t","aliases":"n_t","display":"n_t","kind":"use","excerpt":"[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2780b267623da921","tex":"\\mathbb N_{\\geq1}^{op}","aliases":"N_≥1^op mathbb geq","display":"\\mathbb N_{\\geq1}^{op}","kind":"use","excerpt":"[The source--coefficient diagram has mixed variance] Reduction of coefficients and inflation of source define a functor mathscr P_d: N_≥1^op× Q_Γ ⟶Gpd. Each source-inflation functor is fully faithful, and it commutes with coefficient reduction. An object at (m,U') is a compatible refinement of ρ_n at (n,U) precisely when its reduction to level n eq","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3af85118b5e0d756","tex":"\\nleq","aliases":"nleq","display":null,"kind":"use","excerpt":"For r≤ s, a morphism (r,X)⟶(s,Y) is a morphism α:X⟶ C_s→ r(Y) in T_r. If rnleq s, there is no such morphism in this directed version.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"For pnmid|G|, one has e=0 and h_G(n)=n. Every finite-level representation in this fixed finite-group problem then lifts. When e>0, the test is still finite, but its number of branches may be enormous. No polynomial-time or uniformly practical algorithm is claimed.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","excerpt":"ex:killable-not-thick Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)≃0 and V(Y)≃0, so bot","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-af825476050d513b","tex":"\\mathcal O_{\\mathscr M_A}","aliases":"O_mathscr M_A mathcal mathscr","display":"\\mathcal O_{\\mathscr M_A}","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4cd47839c7ee93da","tex":"\\mathcal O_{\\mathscr M_B}","aliases":"O_mathscr M_B mathcal mathscr","display":"\\mathcal O_{\\mathscr M_B}","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-02c0bdde04997545","tex":"o_n","aliases":"o_n","display":"o_n","kind":"use","excerpt":"[Complete one-step relation obstruction] The residual c_A is a 2-cocycle. Its cohomology class o_n(ρ_n)=[c_A]∈ H^2(G,V) is independent of the chosen matrix lifts. Moreover, o_n(ρ_n)=0 Longleftrightarrow ρ_n lifts to GL_d(R_n+1). More precisely, all lifts with the fixed reduction have the form A_g^b=(I+p^n b(g))A_g, where representatives of b(g) may","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a852bfa3f3775699","tex":"o_n^{\\mathrm{cts}}","aliases":"o_n^cts mathrm","display":"o_n^{\\mathrm{cts}}","kind":"use","excerpt":"[Complete continuous one-step obstruction] The class o_n^cts(ρ_n)=[c_A]∈ H^2_cts(Γ,V) is independent of the continuous choices. It vanishes if and only if ρ_n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d904eb5b0516c55b","tex":"\\Obs'","aliases":"Obs' Obs operatorname","display":"{\\operatorname{Obs}}'","kind":"use","excerpt":"A strong correspondence should include: Λ(TX) · ≃ T'Λ(X), Λ(CX) · ≃ C'Λ(X), Λ(Δ_T,CX) · ≃Δ_T',C'Λ(X), barΛ(Obs X) · ≃Obs'Λ(X).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-106a86b45be386c3","tex":"\\mathrm{open}","aliases":"open mathrm","display":"\\mathrm{open}","kind":"use","excerpt":"[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"For an antichain, each principal ideal is a singleton, D_e=∅, and a finite colimit over the discrete poset is a finite direct sum. For the negative statement, take P to be a two-element chain and a reduced diagram in Perf( Z) 0⟶ Z/pxrightarrow p Z/p^2 for a prime p, where the second map sends the class of x to the class of px. Both successive cofib","count":32},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-16bf1666da550802","tex":"p^{2a}","aliases":"p^2a","display":"p^{2a}","kind":"use","excerpt":"] Consequently E(g,h)+E(gh,t)-α_g E(h,t)-E(g,ht) ∈ p^2mM_d( Z_p). Sum over t and divide by N. The permutation t↦ ht identifies the last average with b(g), giving E(g,h)-b(g)-α_g b(h)+b(gh) ∈ p^2m-eM_d( Z_p). Put a=m-e>0. The repaired multiplicativity product is exactly (I-b(g))(I-α_g b(h))(I+E(g,h))(I-b(gh))^-1. Modulo p^2a its difference from I is","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-80a406874da00307","tex":"p^{2m}","aliases":"p^2m","display":"p^{2m}","kind":"use","excerpt":"Associativity gives an identity without assuming an exact adjoint action: (I+E(g,h))(I+E(gh,t)) = (I+α_g E(h,t))(I+E(g,ht)). Consequently E(g,h)+E(gh,t)-α_g E(h,t)-E(g,ht) ∈ p^2mM_d( Z_p). Sum over t and divide by N. The permutation t↦ ht identifies the last average with b(g), giving E(g,h)-b(g)-α_g b(h)+b(gh) ∈ p^2m-eM_d( Z_p). Put a=m-e>0. The re","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bceebdc8e7e0cc9f","tex":"p^{2m-2e}","aliases":"p^2m-2e","display":"p^{2m-2e}","kind":"use","excerpt":"[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, p","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3d6e85e445dca030","tex":"p^{2m-e}","aliases":"p^2m-e","display":"p^{2m-e}","kind":"use","excerpt":"Associativity gives an identity without assuming an exact adjoint action: (I+E(g,h))(I+E(gh,t)) = (I+α_g E(h,t))(I+E(g,ht)). Consequently E(g,h)+E(gh,t)-α_g E(h,t)-E(g,ht) ∈ p^2mM_d( Z_p). Sum over t and divide by N. The permutation t↦ ht identifies the last average with b(g), giving E(g,h)-b(g)-α_g b(h)+b(gh) ∈ p^2m-eM_d( Z_p). Put a=m-e>0. The re","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b61684520f621e62","tex":"p^{2n-1}","aliases":"p^2n-1","display":"p^{2n-1}","kind":"use","excerpt":"For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a10cb4d81dee2087","tex":"p^{2n-2}","aliases":"p^2n-2","display":"p^{2n-2}","kind":"use","excerpt":"For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-578b7099ecb3368e","tex":"p^3","aliases":"p^3","display":"p^3","kind":"use","excerpt":"[A one-step representation-lifting obstruction] In [source reference], the representation of C_p on ( Z/p^2)^d has no lift to a representation on ( Z/p^3)^d with that reduction. More explicitly, every matrix lift widetilde A obeys widetilde A^p-I≡ p^2Jnot≡0pmodp^3. This obstruction is unchanged by the choice of the lift. There is also no finite fre","count":12},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-187dacd3bbe0542a","tex":"p^{a_1}","aliases":"p^a_1","display":"p^{a_1}","kind":"use","excerpt":"Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ec7e5beb2030bcde","tex":"p^{a_i}","aliases":"p^a_i","display":"p^{a_i}","kind":"use","excerpt":"The residual system (Q[p^n],p^m-n) is pro-zero, and Kxrightarrow∼varprojlim_n K_n. Here pro-zero means that for each target level some later transition is the zero map. It does not mean that the individual residual modules vanish. In Smith coordinates a nonzero diagonal entry imposes p^a_ix_i=0 modulo p^n, giving min(a_i,n) units of O-length. Each ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-42fedaa1fd71be8d","tex":"p^{a_r}","aliases":"p^a_r","display":"p^{a_r}","kind":"use","excerpt":"Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ac938819e2e1908a","tex":"p^c","aliases":"p^c","display":"p^c","kind":"use","excerpt":"If c<h, choose an element of exact order p^h in Q_tors and take n≥ h. Surjectivity of δ_n+c supplies a preimage. Its reduction to level n has nonzero boundary after multiplication by p^c, so it is not the reduction of an exact solution. This proves sharpness. Multiplication by p^h also makes every transition with gap at least h zero on residuals.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ac0ad34a685eff8b","tex":"p^{\\dim_k\\ker L_{\\bar A}}","aliases":"p^dim_kker L_bar A dim ker bar","display":"p^{\\dim_k\\ker L_{\\bar A}}","kind":"use","excerpt":"nameGL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solution set is nonempty, it has p^dim_kker L_bar A elements. This c","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b47e4ae8f60bef45","tex":"p^e","aliases":"p^e","display":"p^e","kind":"use","excerpt":"M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^1(G,M) is finite and killed by |G|. In particular it is killed by p^e, where e=v_p(|G|), and ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-82b25c825d7b33e6","tex":"p^{h-e}","aliases":"p^h-e","display":"p^{h-e}","kind":"use","excerpt":", not necessarily every digit of the chosen level-h witness. Reduction proves (1) implies (2). Conversely, lift every matrix of ρ_h arbitrarily to an invertible integral matrix, keeping the identity equal to I. Its multiplicativity defects are divisible by p^h. Since h>2e, convergent repair applies and returns an action agreeing with these matrices","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-df033657795b2442","tex":"p^{h_G(n)}","aliases":"p^h_G(n)","display":"p^{h_G(n)}","kind":"use","excerpt":"group representations] Let G be a finite group, d,n≥1, and e=v_p(|G|). Set h_G(n)=max{n+e, 2e+1}. For every ρ_n:G→GL_d( Z/p^n), the following are equivalent: item There is a representation ρ:G→GL_d( Z_p) with reduction ρ_n. item There is a representation ρ_h:G→GL_d( Z/p^h_G(n)) with reduction ρ_n. The implication from (2) to (1) is constructive to ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-44ddf871c8e79a19","tex":"p^j","aliases":"p^j","display":"p^j","kind":"use","excerpt":"For pmid g, tensor C_ Z(g) by C_ Z(p). The gcd theorem identifies the result with C_ Z(p)⊕ C_ Z(p)[1]. Hence Z/p≃ C_ Z(p) belongs to the thick closure. Conversely, the Chinese remainder decomposition of Z/g and the short exact sequences building Z/p^j from copies of Z/p put C_ Z(g)≃ Z/g in the thick subcategory generated by its prime divisors. This","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-22605ace34e5a239","tex":"p^{j+1}","aliases":"p^j+1","display":"p^{j+1}","kind":"use","excerpt":"p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows that A^p^t+j∈ I+p^j+1M_d( Z_p). Hence exponen","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7441b5cfbaa2c0c0","tex":"p^m","aliases":"p^m","display":"p^m","kind":"use","excerpt":"vative on perfect complexes. They need not be faithful. In particular, for m>n, the nonzero endomorphism D^n:B_m→ B_m becomes zero under C_m,n. Exactness follows by reducing bounded projective cone diagrams. For conservativity it suffices, by the Chinese remainder decomposition and composition with reduction to level one, to prove that a perfect co","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5c5007fbacab25d5","tex":"p^{m-2e}","aliases":"p^m-2e","display":"p^{m-2e}","kind":"use","excerpt":"This is a classification of isomorphism classes, not an equivalence with a finite groupoid. Integral automorphism groups and noninvertible morphisms need not be determined by finite reduction. Likewise the averaging construction is not asserted to be a canonical framed lift. Two outputs agreeing modulo p^m-e are conjugate by a matrix congruent to I","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-478fe444a346cfcd","tex":"p^{m-a}","aliases":"p^m-a","display":"p^{m-a}","kind":"use","excerpt":". On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and identify Z/p^a with it by x↦ p^n-ax. Reduction of p^m-ax gives p^n-ap^m-nx. Thus for each target level n, ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-79b8bb76c63fb047","tex":"p^{m-e}","aliases":"p^m-e","display":"p^{m-e}","kind":"use","excerpt":"[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, p","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-824f2e0e75ce4855","tex":"p^{m-n}","aliases":"p^m-n","display":"p^{m-n}","kind":"use","excerpt":"[Transient negative cohomology in the resolution direction] Fix p and an integer a≥1. For n≥ a put R_n= Z/p^n. Then H^0C_R_n(p^a)≅ Z/p^a, H^-1C_R_n(p^a)≅ Z/p^a. Under reduction from m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, a","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-03907d548ea5e213","tex":"p^{\\min(a_i,n)}","aliases":"p^min(a_i,n) min","display":"p^{\\min(a_i,n)}","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d6c9282569aea3fa","tex":"p^N","aliases":"p^N","display":"p^N","kind":"use","excerpt":"For surjectivity take (y_p)∈ G. Choose N large enough that D^Ny_p∈ Z_p for every p. For each p, choose the residue of D^Ny_p modulo p^N. The ordinary Chinese remainder theorem provides z∈ Z with all these residues. Set a=z/D^N∈ A. Then a-y_p∈ Z_p for every p, so (a,(a-y_p)_p) maps to (y_p)_p. This proves exactness.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c020cb0886702b18","tex":"p^{n+1}","aliases":"p^n+1","display":"p^{n+1}","kind":"use","excerpt":"t directions: transformation: change of representation and resolution: change of distinguishability. The distinction is operational rather than rhetorical. A Fourier transform, a duality, or a change of coordinates may alter representation while preserving the level of information being resolved. Coarsening a filtration, passing from modulus p^n+1 ","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ce80fd2a612fea5d","tex":"p^{n+a}","aliases":"p^n+a","display":"p^{n+a}","kind":"use","excerpt":"o_n lifts over Z_p, ρ_n extends over R_n+a, Ω_a,*(ρ_n)=0. The extension remains a representation of the same Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the finite extension. Given a finite extension with commuting matrices A_i' modulo p^n+a, subtract the same λ_iI and divide by p^a. Th","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8ba26e75b932b011","tex":"p^{n-1}","aliases":"p^n-1","display":"p^{n-1}","kind":"use","excerpt":"For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a3b1f5849d426291","tex":"p^{n-a}","aliases":"p^n-a","display":"p^{n-a}","kind":"use","excerpt":"m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and identify Z/p^a with it by x↦ p^n-a","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-578ae0f100d049b6","tex":"p^{n-h}","aliases":"p^n-h","display":"p^{n-h}","kind":"use","excerpt":"Each zero entry contributes R⊕ R[1]. This proves the complex decomposition. Multiplication by p^h on R has cokernel Q_h and kernel p^n-hR, the latter isomorphic to Q_h by multiplication by p^n-h. Hence C_R(p^h) has Q_h in degrees -1 and 0. Counting the shifted copies and using Pascal's identity gives the formula. The unit case is contractible, and ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e4e5526bd8628ebc","tex":"p^{s+1}","aliases":"p^s+1","display":"p^{s+1}","kind":"use","excerpt":"A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b7a17992650d6f06","tex":"p^{s-e}","aliases":"p^s-e","display":"p^{s-e}","kind":"use","excerpt":"[Close integral actions are conjugate] Let ρ,σ:G→GL_d( Z_p) be representations satisfying σ(g)≡ρ(g)pmodp^s (g∈ G), s>e. Then T=frac1|G|∑_g∈ Gσ(g)ρ(g)^-1 ∈ I+p^s-eM_d( Z_p) is invertible and satisfies σ(g)T=Tρ(g), σ(g)=Tρ(g)T^-1. Every summand is congruent to I modulo p^s. Its average is therefore I modulo p^s-e, so it is integral and invertible. Fo","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-965ab417def97152","tex":"p^t","aliases":"p^t","display":"p^t","kind":"use","excerpt":"a continuous homomorphism Z_p→GL_d( Z_p) if and only if the reduction of A modulo p has p-power order. In particular a commuting tuple of such matrices determines a continuous representation of Z_p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-62f1e25270bb157d","tex":"p^{t+j}","aliases":"p^t+j","display":"p^{t+j}","kind":"use","excerpt":"p^r. Necessity follows because a continuous image of Z_p in a finite group is a cyclic p-group. Conversely suppose A^p^t∈ I+pM_d( Z_p). For B∈ I+p^sM_d( Z_p) with s≥1, binomial expansion shows B^p∈ I+p^s+1M_d( Z_p). For p=2 the quadratic term has depth 2s≥ s+1, so this estimate remains valid. It follows that A^p^t+j∈ I+p^j+1M_d( Z_p). Hence exponen","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0, 0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"gory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed pairs of S_ij= P_i∩ P_j · ϵ_i=ϵ_j=0, P_j · ϵ_i=1, ϵ_j=0, P_i · ϵ_i=0, ϵ_j=1. Localization at ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4d3243980b9ae21f","tex":"P_r","aliases":"P_r","display":"P_r","kind":"use","excerpt":"Resolution spectrum. For a fixed X∈ T_s and a specified detection predicate P_r at each level r≤ s, define: Spec_R(P;X) = {r≤ s:P_r(C_s→ rX)}. This could behave like a support or spectrum for distinguishability.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-474a8c74e28ce2e1","tex":"\\parallel","aliases":"parallel","display":null,"kind":"use","excerpt":"For the fixed labelled poset define the finite prime set S_P=⋃_eparallel f {p prime:pmidgcd(n_e,n_f)}, where eparallel f means incomparable. Let J_P be the invariant thick closure of the square interaction residuals for the unit seed Z.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"The perfectness hypothesis cannot be removed. Fix an odd prime p, let R= Z/p^2, k= F_p, G=C_p, and choose a matrix J∈ M_d( Z) whose reduction modulo p is invertible. Put M=R^d and let the chosen generator act by A=I+pJpmodp^2. For d=2, the choice J=(0 · 1","count":26},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f3003d8698b757c1","tex":"Q^{-1}","aliases":"Q^-1","display":"Q^{-1}","kind":"use","excerpt":"[Cyclic centralizers over a coefficient ring] Let R be a commutative ring, B∈ M_d(R), and suppose a vector v makes Q=(v,Bv,…,B^d-1v)∈GL_d(R). Then every matrix commuting with B is uniquely f(B) for a polynomial f∈ R[t] of degree less than d. The coefficients of f are the entries of Q^-1Mv. In particular, the centralizer is commutative.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"use","excerpt":"[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomolo","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-812a1fd77a9f35a3","tex":"\\mathbb Q_2","aliases":"Q_2 mathbb","display":"\\mathbb Q_2","kind":"use","excerpt":":repair-before-completion For P={p} and two independent labels divisible by p, every finite excisive quotient is zero, whereas (varprojlim_n Z/p^n)[1/p]= Q_p≠0. For a nonzero finite-level quotient, take D=30 and labels 6,10. Only the prime 2 is shared, so the finite repair is B_n[1/2]≅ Z/15^n. The two completed outputs are Q_2× Z_3× Z_5 and Z_3× Z_","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c8ef5ef2a4fa13a8","tex":"q_B","aliases":"q_B","display":"q_B","kind":"use","excerpt":"The hypothesis and exactness imply Λ( N_mathscr M_A)⊆ N_mathscr M_B, because exact functors preserve the operations used to generate a thick subcategory. Therefore q_BΛ kills N_mathscr M_A, so the universal property in [source reference] gives barΛ. If Λ is an equivalence identifying the two thick kernels, a quasi-inverse descends in the same way a","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d6a9b153d5eaddc8","tex":"Q_h","aliases":"Q_h","display":"Q_h","kind":"use","excerpt":"[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomolo","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-23b5aa6bde1ac779","tex":"Q_h^{\\oplus\\binom tj}","aliases":"Q_h^⊕binom tj oplus binom","display":"Q_h^{\\oplus\\binom tj}","kind":"use","excerpt":"rime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomology","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c95e041440f50ec9","tex":"Q_n","aliases":"Q_n","display":"Q_n","kind":"use","excerpt":"[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomolo","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9ef7a33166aed2e0","tex":"\\mathbb Q_p^r","aliases":"Q_p^r mathbb","display":"\\mathbb Q_p^r","kind":"use","excerpt":"side V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and let E_p=θ_p^-1(L_p) be the corresponding lat","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9666ac009c3e72f6","tex":"q_r","aliases":"q_r","display":"q_r","kind":"use","excerpt":"[Orbit-generator test for an exchange obstruction] Fix an exchange η^A_s,r between exact functors, and an exact quotient q_r of the target fiber. If q_rΔ^A_s,r(g)≃0 for every g∈ G_s, then q_rΔ^A_s,r(X)≃0 for every X∈ Z_s^thick. The same assertion holds for q_r=id, in which case the exchange is an equivalence on the entire generated thick zoo.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0e9f702d73daf943","tex":"\\mathcal Q_{\\square}","aliases":"Q_square mathcal square","display":"\\mathcal Q_{\\square}","kind":"use","excerpt":"[Universal constructor-compatible excisive quotient] The stable Verdier quotient q_square: Z⟶ Q_square := Z/ J_square has the following properties. item The constructors descend, with their supplied coherent action data. item Every diagram q_squareX, for X∈mathscr X, has context-independent event residuals and satisfies principal-ideal reconstructi","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c9e6efbf0880c247","tex":"q_{\\square}","aliases":"q_square square","display":"q_{\\square}","kind":"use","excerpt":"[Universal constructor-compatible excisive quotient] The stable Verdier quotient q_square: Z⟶ Q_square := Z/ J_square has the following properties. item The constructors descend, with their supplied coherent action data. item Every diagram q_squareX, for X∈mathscr X, has context-independent event residuals and satisfies principal-ideal reconstructi","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5a70b1ce4482c9d1","tex":"Q_{\\mathrm{tors}}","aliases":"Q_tors mathrm","display":"Q_{\\mathrm{tors}}","kind":"use","excerpt":"Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"Let a fine state be x=(x_1,x_2)∈ R^2 and let coarse resolution record only the average C(x)=fracx_1+x_22. Let the fine transformation be A_1(x_1,x_2)=(ax_1,bx_2) and the coarse transformation be multiplication by c: A_0(y)=cy. Then C(A_1x)-A_0(Cx) = frac(a-c)x_1+(b-c)x_22. Choosing c=fraca+b2 gives C(A_1x)-A_0(Cx) = fraca-b4(x_1-x_2). The defect de","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b31afc68647d37e3","tex":"\\mathbf R","aliases":"R mathbf","display":"\\mathbf R","kind":"use","excerpt":"For a commutative ring R, a finite group G, and M∈ D(R[G]), define Γ_R(M)= RHom_R[G](R,M)∈ D(R). This is the derived invariant functor. Derived categories in this section are the enhanced unbounded derived categories. They are stable. Their exact functors preserve finite cofiber sequences but are not asserted to preserve every infinite limit or col","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a3d472d51734fedb","tex":"\\mathbb R^2","aliases":"R^2 mathbb","display":"\\mathbb R^2","kind":"use","excerpt":"Let a fine state be x=(x_1,x_2)∈ R^2 and let coarse resolution record only the average C(x)=fracx_1+x_22. Let the fine transformation be A_1(x_1,x_2)=(ax_1,bx_2) and the coarse transformation be multiplication by c: A_0(y)=cy. Then C(A_1x)-A_0(Cx) = frac(a-c)x_1+(b-c)x_22. Choosing c=fraca+b2 gives C(A_1x)-A_0(Cx) = fraca-b4(x_1-x_2). The defect de","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fe9a73354b93f7b7","tex":"R^d","aliases":"R^d","display":"R^d","kind":"use","excerpt":"The perfectness hypothesis cannot be removed. Fix an odd prime p, let R= Z/p^2, k= F_p, G=C_p, and choose a matrix J∈ M_d( Z) whose reduction modulo p is invertible. Put M=R^d and let the chosen generator act by A=I+pJpmodp^2. For d=2, the choice J=(0 · 1","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2134bf64e3b53762","tex":"R^r","aliases":"R^r","display":"R^r","kind":"use","excerpt":"A finite projective module over R is free. Indeed, lift a basis modulo (p) to obtain a surjection R^r→ P. It is surjective because its cokernel Q satisfies Q=pQ, whence Q=p^mQ=0. The surjection splits, and its kernel has zero reduction modulo p, so the same argument kills the kernel. A perfect complex therefore has a bounded finite free representat","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9f7c520f44d5ce00","tex":"R^\\ast","aliases":"R^∗ ast","display":"R^\\ast","kind":"use","excerpt":"The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objectiv","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3e393fa27aa9dcb1","tex":"R_{2n-1}","aliases":"R_2n-1","display":"R_{2n-1}","kind":"use","excerpt":"[A sharp uniform rank-two horizon] For every p,n,r as above, ρ_n lifts continuously over Z_p Longleftrightarrow ρ_n extends continuously to R_2n-1. For n≥2 and r≥2, 2n-1 is optimal as a uniform coefficient level. For n=1, all continuous rank-two representations of Z_p^r over F_p lift. For a nonscalar tuple, a≤ n-1, so an extension to 2n-1 restricts","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c14a339292a2cf1a","tex":"r_{i+1}","aliases":"r_i+1","display":"r_{i+1}","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6561c0c615acf6e9","tex":"r_{m,n}","aliases":"r_m,n","display":"r_{m,n}","kind":"use","excerpt":"its boundary] There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently the cokernel of the natural co","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-47fcb4b6009bdd6c","tex":"R_n","aliases":"R_n","display":"R_n","kind":"use","excerpt":"[Transient negative cohomology in the resolution direction] Fix p and an integer a≥1. For n≥ a put R_n= Z/p^n. Then H^0C_R_n(p^a)≅ Z/p^a, H^-1C_R_n(p^a)≅ Z/p^a. Under reduction from m to n with m≥ n≥ a, the map on H^0 is the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, a","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a4ff961cd74d1869","tex":"r_n","aliases":"r_n","display":"r_n","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ce22c6bf2dddf077","tex":"R_{n+1}","aliases":"R_n+1","display":"R_{n+1}","kind":"use","excerpt":"Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero","count":9},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cec347f268fa126b","tex":"R_{n+a}","aliases":"R_n+a","display":"R_{n+a}","kind":"use","excerpt":"[A source-fixed finite horizon in the cyclic stratum] Under the hypotheses of [source reference], the following are equivalent: ρ_n lifts over Z_p, ρ_n extends over R_n+a, Ω_a,*(ρ_n)=0. The extension remains a representation of the same Z_p^r. When a=0, the current representation is already a sufficient certificate. An integral lift supplies the fi","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9d4ab28ea1385868","tex":"r_{n-1}","aliases":"r_n-1","display":"r_{n-1}","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-acac5622999dd4a9","tex":"R_{n-a}","aliases":"R_n-a","display":"R_{n-a}","kind":"use","excerpt":"Choose 0≤ a<n and integer scalars λ_i such that A_i≡λ_i Ipmodp^a for every i. There is no restriction in this display when a=0. Define the normalized matrices B_i=fracA_i-λ_i Ip^a ∈ M_d(R_n-a). This is well-defined from the given finite data. It does not invert p inside R_n. Assume that, for some distinguished index *, the matrix B_* is cyclic modu","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-41405362f32f8720","tex":"\\operatorname{rank}_O","aliases":"rank_O operatorname","display":"\\operatorname{rank}_O","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"[An explicit obstruction killed by refining the source] Let p be odd, J∈ M_d( Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint resid","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-17e6e62cdd37d68b","tex":"\\mathsf S","aliases":"S mathsf","display":"\\mathsf S","kind":"use","excerpt":"The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objectiv","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d6e7ed7d11f264e9","tex":"S_{\\mathcal I}","aliases":"S_ I mathcal","display":"S_{\\mathcal I}","kind":"use","excerpt":"[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-45008a9a244e9093","tex":"\\mathcal S_0","aliases":"S_0 mathcal","display":"\\mathcal S_0","kind":"use","excerpt":"[Strict semantic realization of the event diagram] For a typed pure word W in a strict chain-natural realization and a seed s∈ E_b, evaluation gives a functor X_W:NJ(P_W)⟶ S_0. Its vertices evaluate the reachable words and its cover arrows evaluate the whiskered exchange maps. Thus the residual diagram in [source reference] is constructed, rather t","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"Finite computation contract. All five exact regression scripts passed, including reruns of the four inherited suites. The new script uses integer and modular arithmetic only in mathematical checks. It examines eight specified cochain fixtures, including the nonabelian group S_3, with 593 cocycle triples and 56 changes of section. Positive solutions","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing ","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-dfbf9a1694a645b8","tex":"s_i'","aliases":"s_i'","display":"s_i'","kind":"use","excerpt":"e fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing the local sections by a 0-cochain b=(b_i), s_i'=s_i+b_i, changes the residual by the vCech co","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-765a006157afb823","tex":"S_{ij}","aliases":"S_ij","display":"S_{ij}","kind":"use","excerpt":"gory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed pairs of S_ij= P_i∩ P_j · ϵ_i=ϵ_j=0, P_j · ϵ_i=1, ϵ_j=0, P_i · ϵ_i=0, ϵ_j=1. Localization at ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"use","excerpt":"[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing ","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6e6201827cc32090","tex":"s_j'","aliases":"s_j'","display":"s_j'","kind":"use","excerpt":"If s_i'=s_i+b_i, then s_j'=s_j+b_j=s_i+a_ij+b_j =s_i'+(a_ij+b_j-b_i), so the residual changes by the coboundary δ b. Hence its cohomology class is independent of the chosen local sections.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b4ebcce0dc1a5c39","tex":"s_k","aliases":"s_k","display":"s_k","kind":"use","excerpt":"On a triple overlap, s_k=s_j+a_jk=s_i+a_ij+a_jk, while also s_k=s_i+a_ik. Freeness of the torsor action gives a_ij+a_jk=a_ik, which is the vCech cocycle identity.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-88ebc019776cb5b6","tex":"\\mathcal S_m","aliases":"S_m mathcal","display":"\\mathcal S_m","kind":"use","excerpt":"[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-687fefc36da68a9f","tex":"\\mathcal S_n","aliases":"S_n mathcal","display":"\\mathcal S_n","kind":"use","excerpt":"Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. Th","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-15b82495b9b6ea19","tex":"\\Seeds_1","aliases":"Seeds_1 Σ_1 Seeds Sigma","display":"{\\Sigma}_1","kind":"use","excerpt":"This turns the seed calculus into an explicit generative process: Σ_0 ⊆ Σ_1 ⊆ Σ_2 ⊆⋯, Σ_n+1=Ψ(Σ_n).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d0d4504114af9361","tex":"\\Seeds_2","aliases":"Seeds_2 Σ_2 Seeds Sigma","display":"{\\Sigma}_2","kind":"use","excerpt":"This turns the seed calculus into an explicit generative process: Σ_0 ⊆ Σ_1 ⊆ Σ_2 ⊆⋯, Σ_n+1=Ψ(Σ_n).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-96407c655322eaea","tex":"\\Seeds_n","aliases":"Seeds_n Σ_n Seeds Sigma","display":"{\\Sigma}_n","kind":"use","excerpt":"This turns the seed calculus into an explicit generative process: Σ_0 ⊆ Σ_1 ⊆ Σ_2 ⊆⋯, Σ_n+1=Ψ(Σ_n).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-43d8f42bb77eb798","tex":"\\Seeds_{n+1}","aliases":"Seeds_n+1 Σ_n+1 Seeds Sigma","display":"{\\Sigma}_{n+1}","kind":"use","excerpt":"This turns the seed calculus into an explicit generative process: Σ_0 ⊆ Σ_1 ⊆ Σ_2 ⊆⋯, Σ_n+1=Ψ(Σ_n).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6c79f030da824653","tex":"\\mathrm{shift}","aliases":"shift mathrm","display":"\\mathrm{shift}","kind":"use","excerpt":"Degreewise transition maps of the finite free modules are surjective. In the product model of the homotopy inverse limit, the map 1-shift:∏_n M_n^i⟶∏_n M_n^i is surjective in each degree: choose an initial entry and successively lift the required difference across the surjective transition map. Its kernel is the ordinary inverse limit. Thus the hom","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ae51e1c510f803eb","tex":"\\operatorname{Spec}","aliases":"Spec operatorname","display":"\\operatorname{Spec}","kind":"use","excerpt":"Resolution spectrum. For a fixed X∈ T_s and a specified detection predicate P_r at each level r≤ s, define: Spec_R(P;X) = {r≤ s:P_r(C_s→ rX)}. This could behave like a support or spectrum for distinguishability.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5b478d5d8c1da172","tex":"\\operatorname{Spec}_R","aliases":"Spec_R operatorname","display":"\\operatorname{Spec}_R","kind":"use","excerpt":"Resolution spectrum. For a fixed X∈ T_s and a specified detection predicate P_r at each level r≤ s, define: Spec_R(P;X) = {r≤ s:P_r(C_s→ rX)}. This could behave like a support or spectrum for distinguishability.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"use","excerpt":"[Close integral actions are conjugate] Let ρ,σ:G→GL_d( Z_p) be representations satisfying σ(g)≡ρ(g)pmodp^s (g∈ G), s>e. Then T=frac1|G|∑_g∈ Gσ(g)ρ(g)^-1 ∈ I+p^s-eM_d( Z_p) is invertible and satisfies σ(g)T=Tρ(g), σ(g)=Tρ(g)T^-1. Every summand is congruent to I modulo p^s. Its average is therefore I modulo p^s-e, so it is integral and invertible. Fo","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-aea3af576abad800","tex":"t^m","aliases":"t^m","display":"t^m","kind":"use","excerpt":"Let m≥2, and put N_m(t)=1+t+⋯+t^m-1, M_m=O[t]/(N_m(t)). This is free of rank m-1. Let a generator σ of C_m act by multiplication by t. Since (t-1)N_m(t)=t^m-1, this is a genuine cyclic-group action.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2ca79bce6422a743","tex":"t^{m-1}","aliases":"t^m-1","display":"t^{m-1}","kind":"use","excerpt":"Let m≥2, and put N_m(t)=1+t+⋯+t^m-1, M_m=O[t]/(N_m(t)). This is free of rank m-1. Let a generator σ of C_m act by multiplication by t. Since (t-1)N_m(t)=t^m-1, this is a genuine cyclic-group action.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-806c945e97add606","tex":"T^\\ast","aliases":"T^∗ ast","display":"T^\\ast","kind":"use","excerpt":"The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objectiv","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"Consider a typed alphabet containing transformation symbols T and resolution symbols C. A mixed word is a composable expression such as W=T_1C_1T_2T_3C_2T_4. Assume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form TC ⇒ C T', where T' is the correctly reindexed transformati","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ee58601ada9f30db","tex":"T_3","aliases":"T_3","display":"T_3","kind":"use","excerpt":"Consider a typed alphabet containing transformation symbols T and resolution symbols C. A mixed word is a composable expression such as W=T_1C_1T_2T_3C_2T_4. Assume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form TC ⇒ C T', where T' is the correctly reindexed transformati","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-64f17e7dedd6a141","tex":"T_4","aliases":"T_4","display":"T_4","kind":"use","excerpt":"Consider a typed alphabet containing transformation symbols T and resolution symbols C. A mixed word is a composable expression such as W=T_1C_1T_2T_3C_2T_4. Assume every well-typed adjacent transformation/coarsening inversion admits at least one length-preserving exchange rule of the form TC ⇒ C T', where T' is the correctly reindexed transformati","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0b61846000086dfe","tex":"T_h","aliases":"T_h","display":"T_h","kind":"use","excerpt":"Before event (i,j) can occur, every initially intervening C_k with k<j must already have crossed T_i, and every initially intervening T_h with h>i must already have crossed C_j. Thus all predecessors of a performed event have already occurred. The performed events form an ideal.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-779ead1f6fe29f5c","tex":"\\Tcat_0","aliases":"Tcat_0 T_0 Tcat mathbf","display":"{\\mathbf T}_0","kind":"use","excerpt":"[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-49d15bb91a343505","tex":"\\Tcat_1","aliases":"Tcat_1 T_1 Tcat mathbf","display":"{\\mathbf T}_1","kind":"use","excerpt":"[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a4c919ce0a4a3992","tex":"\\Tcat_2","aliases":"Tcat_2 T_2 Tcat mathbf","display":"{\\mathbf T}_2","kind":"use","excerpt":"[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-28bed0b6e695da2e","tex":"\\Tcat_{r_i}","aliases":"Tcat_r_i T_r_i Tcat mathbf","display":"{\\mathbf T}_{r_i}","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b9718cdc4925f9e1","tex":"\\Tcat_t","aliases":"Tcat_t T_t Tcat mathbf","display":"{\\mathbf T}_t","kind":"use","excerpt":"Let X,Y∈ T_t and let r≤ t.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-30a66bc953ef02ad","tex":"\\operatorname{tcofib}","aliases":"tcofib operatorname","display":"\\operatorname{tcofib}","kind":"use","excerpt":"A_i · i∉ S. Its total cofiber, obtained by taking the cofiber in every coordinate, satisfies tcofib(Y)≃ ⊗_i=1^mCofib(f_i). Equivalently it is the cofiber of the canonical map from the colimit of the punctured cube to its terminal vertex. The equivalences are natural in the arrows, with the coherent symmetries of the monoidal structure, rather than ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-faad188781a0fc33","tex":"\\operatorname{Tor}","aliases":"Tor operatorname","display":"\\operatorname{Tor}","kind":"use","excerpt":"stitutes] Over R= Z/p^2, the exact sequence 0→(p)→ R→ F_p→0 ceases to be left exact after ordinary tensor with F_p: the first map becomes zero. The module F_p has the infinite periodic free resolution with every differential multiplication by p. Tensoring that resolution with F_p gives zero differentials, hence Tor_i^R( F_p, F_p)= F_p for every i≥0","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-28db2200fd80ef49","tex":"\\operatorname{Tor}_i^R","aliases":"Tor_i^R operatorname","display":"\\operatorname{Tor}_i^R","kind":"use","excerpt":"stitutes] Over R= Z/p^2, the exact sequence 0→(p)→ R→ F_p→0 ceases to be left exact after ordinary tensor with F_p: the first map becomes zero. The module F_p has the infinite periodic free resolution with every differential multiplication by p. Tensoring that resolution with F_p gives zero differentials, hence Tor_i^R( F_p, F_p)= F_p for every i≥0","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8dd594b4bce51346","tex":"\\mathrm{tors}","aliases":"tors mathrm","display":"\\mathrm{tors}","kind":"use","excerpt":"Choose Smith data for d over the discrete valuation ring O. After invertible row and column operations its nonzero entries are p^a_1,…,p^a_r, with a_i≥0. Units have been absorbed into the basis changes. Let h=max{a_1,…,a_r,0}. Equivalently, h is the least integer with p^h Q_tors=0. If Q_tors=0, set h=0.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-65fa7282e2650c3e","tex":"\\triangleleft","aliases":"triangleleft","display":null,"kind":"use","excerpt":"[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-688574fd67068d08","tex":"\\triangleleft_{\\mathrm{open}}","aliases":"triangleleft_open triangleleft mathrm","display":null,"kind":"use","excerpt":"[Source refinement and inflation] Choose an open normal U⊆kerρ_n and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ec7d40a789aa2d83","tex":"u'","aliases":"u'","display":"u'","kind":"use","excerpt":"Let a specified coherent square in a stable ∞-category S be A arrow[r,\"u\"] arrow[d,\"v\"'] · B arrow[d,\"v'\"] C arrow[r,\"u'\"'] · D. Write E=Cofib(u) and E'=Cofib(u'). The square induces E→ E'.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e28cd1be4d9be5f9","tex":"U^2","aliases":"U^2","display":"U^2","kind":"use","excerpt":"[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing ","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"Otherwise every A_i is scalar modulo p^a, while B_* is nonscalar modulo p by the definition of a. It is cyclic by [source reference]. Thus every hypothesis of [source reference] holds. In rank two the polynomials have the form u_i+v_it and can be recovered from a two-column cyclic basis by solving one invertible 2×2 linear system modulo p^n-a. The ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4a960fc6b6aefe49","tex":"U_{ij}","aliases":"U_ij","display":"U_{ij}","kind":"use","excerpt":"[Torsor residual obstruction] Let A be a sheaf of abelian groups on a space or a site with the fiber products used below, let P be an A-torsor, and let {U_i→ U} be a cover on which P admits local sections s_i∈ P(U_i). On overlaps there are unique elements a_ij∈ A(U_ij) such that s_j=s_i+a_ij. Then: item a=(a_ij) is a vCech 1-cocycle; item changing ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9e470cd4c498b9bd","tex":"U_{\\Rcat}","aliases":"U_Rcat U_ R Rcat mathbf","display":"U_{{\\mathbf R}}","kind":"use","excerpt":"intwise-stable Let R be small. Postcomposition with St^add defines a functor St^add_ R: Fun( R^op,Cat^add) ⟶ Fun( R^op,Cat^st), F⟼ St^add∘ F. It is left adjoint to pointwise forgetting U_ R: Fun( R^op,Cat^st) ⟶ Fun( R^op,Cat^add). Equivalently, for every additive two-axis system F and stable two-axis system G there is a natural equivalence Map (St^","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bb73e4788f1d3199","tex":"\\underset","aliases":"underset","display":null,"kind":"use","excerpt":"and compute c_A on Q=Γ/U. Then o_n^cts(ρ_n)=0 Longleftrightarrow ∃ U'triangleleft_openΓ, U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"use","excerpt":"Any other lattice with the specified generic and local identifications has analogous bounds by powers of D. After a common scaling, its finite quotient is determined by the same primary subgroups in the Chinese remainder decomposition. It therefore equals Λ. A compatible A-linear map V→ V' sends every θ_p-integral element to a θ'_p-integral element","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4e7142ad42e2513e","tex":"v'","aliases":"v'","display":"v'","kind":"use","excerpt":"Let a specified coherent square in a stable ∞-category S be A arrow[r,\"u\"] arrow[d,\"v\"'] · B arrow[d,\"v'\"] C arrow[r,\"u'\"'] · D. Write E=Cofib(u) and E'=Cofib(u'). The square induces E→ E'.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-759397de55065787","tex":"V^2","aliases":"V^2","display":"V^2","kind":"use","excerpt":"[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"Otherwise every A_i is scalar modulo p^a, while B_* is nonscalar modulo p by the definition of a. It is cyclic by [source reference]. Thus every hypothesis of [source reference] holds. In rank two the polynomials have the form u_i+v_it and can be recovered from a two-column cyclic basis by solving one invertible 2×2 linear system modulo p^n-a. The ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e24b729afa00f95f","tex":"\\varprojlim_n","aliases":"varprojlim_n varprojlim","display":null,"kind":"use","excerpt":"Write widehat B=varprojlim_n B_n≅∏_p∈ P Z_p. The isomorphism follows by taking inverse limits of the compatible Chinese remainder decompositions. Here Z_p is the ring of compatible residue sequences modulo p^n, and Q_p= Z_p[1/p].","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"The square is [column sep=large,row sep=large] A_rC_s→ r(X) arrow[r,\"η^A_s,r,X\"] · C_s→ rA_s(X). It compares transforming after coarsening with coarsening after transforming.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e9fe8ee5ada9c8bf","tex":"W'","aliases":"W'","display":"W'","kind":"use","excerpt":"The last universal property follows directly from thick closure and Verdier localization. If E is equivariant and kills K, then E(WK)≃ W'E(K)≃ W'(0)≃0. Thus it kills all the generators and factors through the quotient.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3e5524065eabb0b8","tex":"\\widehat A_*","aliases":"widehat A_* widehat","display":"\\widehat A_*","kind":"use","excerpt":"at B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preserves the prescribed reduction modulo p^n and does not pass to an obstruction quotient. Suppose first that widehat A_i are any commuting integral lifts. Then widehat B_i=(widehat A_i-λ_iI)/p^a are integral. The identities 0=[widehat A_*,widehat A","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-891994962d96636a","tex":"\\widehat A_1","aliases":"widehat A_1 widehat","display":"\\widehat A_1","kind":"use","excerpt":"3 · 2pmod9, B_2=2I+4B_1pmod9. The vector e_2 is cyclic for B_1 modulo 3. The displayed integer representatives of A_1,A_2 do not commute exactly: their additive commutator is diag(-81,81). The criterion nevertheless passes. Retain B_1 integrally and replace B_2 by the exact polynomial 2I+4B_1. This constructs widehat A_1=1 · 3 9 · 1, widehat A_2=7 ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a3b78dafe2e204c1","tex":"\\widehat A_2","aliases":"widehat A_2 widehat","display":"\\widehat A_2","kind":"use","excerpt":"3 · 2pmod9, B_2=2I+4B_1pmod9. The vector e_2 is cyclic for B_1 modulo 3. The displayed integer representatives of A_1,A_2 do not commute exactly: their additive commutator is diag(-81,81). The criterion nevertheless passes. Retain B_1 integrally and replace B_2 by the exact polynomial 2I+4B_1. This constructs widehat A_1=1 · 3 9 · 1, widehat A_2=7 ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4895bc4181eddb45","tex":"\\widehat A_i","aliases":"widehat A_i widehat","display":"\\widehat A_i","kind":"use","excerpt":"ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. The","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-60add838d977758e","tex":"\\widehat A_i^{x_i}","aliases":"widehat A_i^x_i widehat","display":"\\widehat A_i^{x_i}","kind":"use","excerpt":"ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. The","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ef3ab629cdd6af07","tex":"\\widehat A_s","aliases":"widehat A_s widehat","display":"\\widehat A_s","kind":"use","excerpt":"cal A),St^add( B)) ≃Fun^⊕( A,St^add( B)). Full faithfulness transports the supplied composition coherences as well. Accordingly A_r, C_s→ r, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations widehat A_r, widehat C_s→ r, widehatη^A_s,r: widehat A_rwidehat C_","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1b3f66f942a5fe9a","tex":"\\widehat B","aliases":"widehat B widehat","display":"\\widehat B","kind":"use","excerpt":"Write widehat B=varprojlim_n B_n≅∏_p∈ P Z_p. The isomorphism follows by taking inverse limits of the compatible Chinese remainder decompositions. Here Z_p is the ring of compatible residue sequences modulo p^n, and Q_p= Z_p[1/p].","count":23},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-391de6b4be1aba29","tex":"\\widehat B_*","aliases":"widehat B_* widehat","display":"\\widehat B_*","kind":"use","excerpt":"at B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preserves the prescribed reduction modulo p^n and does not pass to an obstruction quotient. Suppose first that widehat A_i are any commuting integral lifts. Then widehat B_i=(widehat A_i-λ_iI)/p^a are integral. The identities 0=[widehat A_*,widehat A","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5e0efde6e8ac93ba","tex":"\\widehat B_i","aliases":"widehat B_i widehat","display":"\\widehat B_i","kind":"use","excerpt":"widehat B. Then put widehat A_i=λ_i I+p^awidehat f_i(widehat B), ρ(x_1,…,x_r)=∏_iwidehat A_i^x_i. For d=1, use the scalar matrices directly. The construction preserves the prescribed reduction modulo p^n and does not pass to an obstruction quotient. Suppose first that widehat A_i are any commuting integral lifts. Then widehat B_i=(widehat A_i-λ_iI)","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"ic-lift Under the preceding hypotheses, ρ_n has a continuous integral lift Longleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of t","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c6d54fe029f6fb4c","tex":"\\widehat f_i","aliases":"widehat f_i widehat","display":"\\widehat f_i","kind":"use","excerpt":"ic-lift Under the preceding hypotheses, ρ_n has a continuous integral lift Longleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of t","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a410e85313b42614","tex":"\\widehat M","aliases":"widehat M widehat","display":"\\widehat M","kind":"use","excerpt":"[Uniform based-complex tower reconstruction] Suppose M_n are based finite free cochain complexes over B_n, concentrated in a common finite interval with fixed ranks in each degree. Suppose their differential matrices reduce exactly from level m to level n for m≥ n. Then there is a unique based complex widehat M with those ranks over widehat B whose","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ce0c8ec45ad2de65","tex":"\\widehat{(-)}","aliases":"widehat(-) widehat","display":"\\widehat{(-)}","kind":"use","excerpt":"Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ab65f1f9f1561355","tex":"\\widehat{(-)}_R","aliases":"widehat(-)_R widehat","display":"\\widehat{(-)}_R","kind":"use","excerpt":"Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-83ec3437fa184e6c","tex":"\\widehat{(-)}_T","aliases":"widehat(-)_T widehat","display":"\\widehat{(-)}_T","kind":"use","excerpt":"Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-10cda2779e1cc351","tex":"\\widehat{\\widehat X_R}","aliases":"widehatwidehat X_R widehat widehat","display":"\\widehat{\\widehat X_R}","kind":"use","excerpt":"Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3b225b4e8fe15c27","tex":"\\widehat{\\widehat X_R}_T","aliases":"widehatwidehat X_R_T widehat widehat","display":"\\widehat{\\widehat X_R}_T","kind":"use","excerpt":"Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e7375ec8348dbfdb","tex":"\\widehat{\\widehat X_T}","aliases":"widehatwidehat X_T widehat widehat","display":"\\widehat{\\widehat X_T}","kind":"use","excerpt":"Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4b364a27543c172c","tex":"\\widehat{\\widehat X_T}_R","aliases":"widehatwidehat X_T_R widehat widehat","display":"\\widehat{\\widehat X_T}_R","kind":"use","excerpt":"Bi-completion. There are then two completion operations: widehat(-)_T, widehat(-)_R. After both completions and the comparison have been defined in a common category, the mixed question is whether widehatwidehat X_T_R ≃ widehatwidehat X_R_T. Given the specified comparison, its defect is a candidate completion-level mixed residual in a setting where","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-251ac6d15319669a","tex":"\\widehat\\eta^A_{s,r}","aliases":"widehatη^A_s,r widehat eta","display":"\\widehat\\eta^A_{s,r}","kind":"use","excerpt":"cal A),St^add( B)) ≃Fun^⊕( A,St^add( B)). Full faithfulness transports the supplied composition coherences as well. Accordingly A_r, C_s→ r, and their coherently supplied exchange cells extend, essentially uniquely relative to the input, to exact functors and natural transformations widehat A_r, widehat C_s→ r, widehatη^A_s,r: widehat A_rwidehat C_","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9a5c0f9dd0b0c63d","tex":"\\widetilde A","aliases":"widetilde A widetilde","display":"\\widetilde A","kind":"use","excerpt":"[A one-step representation-lifting obstruction] In [source reference], the representation of C_p on ( Z/p^2)^d has no lift to a representation on ( Z/p^3)^d with that reduction. More explicitly, every matrix lift widetilde A obeys widetilde A^p-I≡ p^2Jnot≡0pmodp^3. This obstruction is unchanged by the choice of the lift. There is also no finite fre","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-30e92e5d53064c51","tex":"\\widetilde A^p","aliases":"widetilde A^p widetilde","display":"\\widetilde A^p","kind":"use","excerpt":"[A one-step representation-lifting obstruction] In [source reference], the representation of C_p on ( Z/p^2)^d has no lift to a representation on ( Z/p^3)^d with that reduction. More explicitly, every matrix lift widetilde A obeys widetilde A^p-I≡ p^2Jnot≡0pmodp^3. This obstruction is unchanged by the choice of the lift. There is also no finite fre","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a292de809ab4a4a6","tex":"\\widetilde A^q","aliases":"widetilde A^q widetilde","display":"\\widetilde A^q","kind":"use","excerpt":"[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solut","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f596b26bb2804e7d","tex":"\\widetilde B","aliases":"widetilde B widetilde","display":"\\widetilde B","kind":"use","excerpt":"For sharpness fix n≥2 and put A=I+p^n-1E_12, B=I+p^n-1E_21. These matrices commute modulo p^2n-2 and generate a finite abelian p-group there, so they give a continuous extension of their level-n representation to level 2n-2. However every pair of lifts preserving level n has the form widetilde A=A+p^nP, widetilde B=B+p^nQ, and widetilde Awidetilde ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d74a2e10811483dc","tex":"\\widetilde f","aliases":"widetilde f widetilde","display":"\\widetilde f","kind":"use","excerpt":"ies existence of an integral one whose reduction at precision n is the specified map. The coarser bound v_p(|G|) is always valid. Finite freeness identifies V/p^nV with the full module of linear maps between the reduced lattices. Its G-invariants are exactly the equivariant maps. Apply [source reference]. Explicitly, for a lift widetilde f the boun","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation","count":52},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-27db5ab2afe7c2b0","tex":"X^j","aliases":"X^j","display":"X^j","kind":"use","excerpt":"hoice of the lift. There is also no finite free Z_p-representation with this reduction, even up to isomorphism of the reduced representation. Every lift has the form I+pJ+p^2B modulo p^3. Put X=pJ+p^2B and use the binomial identity for I+X. The linear term is pX=p^2J modulo p^3. For 2≤ j<p, the coefficient binom pj is divisible by p and X^j by p^j,","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8a46ff3f62bf1268","tex":"x^k","aliases":"x^k","display":"x^k","kind":"use","excerpt":"ve ring R and nilpotent x∈ R, the thick subcategory generated by C_R(x) is all of Perf(R). Therefore, over R= Z/p^n, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero. Choose N with x^N=0. For composable scalar multiplications, the octahedral cofiber relation builds C_R(x^k) from C_R","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-19c12121d9f6054e","tex":"x^{k-1}","aliases":"x^k-1","display":"x^{k-1}","kind":"use","excerpt":"nilpotent x∈ R, the thick subcategory generated by C_R(x) is all of Perf(R). Therefore, over R= Z/p^n, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero. Choose N with x^N=0. For composable scalar multiplications, the octahedral cofiber relation builds C_R(x^k) from C_R(x^k-1) and C_","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-12f1075149375120","tex":"x^N","aliases":"x^N","display":"x^N","kind":"use","excerpt":"t interaction forces excisive collapse] For any commutative ring R and nilpotent x∈ R, the thick subcategory generated by C_R(x) is all of Perf(R). Therefore, over R= Z/p^n, forcing the interaction of two nonunit scalar labels to vanish in an exact quotient forces that quotient to be zero. Choose N with x^N=0. For composable scalar multiplications,","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-baf58144b77f8bed","tex":"X^p","aliases":"X^p","display":"X^p","kind":"use","excerpt":"esentation with this reduction, even up to isomorphism of the reduced representation. Every lift has the form I+pJ+p^2B modulo p^3. Put X=pJ+p^2B and use the binomial identity for I+X. The linear term is pX=p^2J modulo p^3. For 2≤ j<p, the coefficient binom pj is divisible by p and X^j by p^j, so those terms vanish modulo p^3. The last term X^p als","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. The","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"Let a fine state be x=(x_1,x_2)∈ R^2 and let coarse resolution record only the average C(x)=fracx_1+x_22. Let the fine transformation be A_1(x_1,x_2)=(ax_1,bx_2) and the coarse transformation be multiplication by c: A_0(y)=cy. Then C(A_1x)-A_0(Cx) = frac(a-c)x_1+(b-c)x_22. Choosing c=fraca+b2 gives C(A_1x)-A_0(Cx) = fraca-b4(x_1-x_2). The defect de","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8eeb0a1bbb468424","tex":"X_3","aliases":"X_3","display":"X_3","kind":"use","excerpt":"[Why thick closure is genuinely necessary] Let C be any nonzero stable category and set Z= C^3. Define exact idempotent endofunctors U(X_1,X_2,X_3)=(0,0,X_2⊕ X_3), V(X_1,X_2,X_3)=(0,0,X_1⊕ X_3). They satisfy U^2≃ U, V^2≃ V, U∘ V≃ V, V∘ U≃ U, so mathscr M={id,U,V} is closed under composition. Choose 0not≃ A∈ C and set X=(A,0,0), Y=(0,A,0). Then U(X)","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2064dd2f33e53d94","tex":"X_{D_e}","aliases":"X_D_e","display":"X_{D_e}","kind":"use","excerpt":"nenumerate item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every normalization history are the event-labelled objects d_e:=Cofib(X_D_e→ X_downarrow e)","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5ee00465b1c4f104","tex":"X_{\\downarrow e}","aliases":"X_downarrow e downarrow","display":"X_{\\downarrow e}","kind":"use","excerpt":"nenumerate item Every elementary independent-event square of X is a homotopy pushout. item Every Θ_e,f(I) vanishes. item For each event e, the natural comparison const_ A_eE_e(D_e)⟶ E_e is an equivalence of diagrams. In this regime the successive cofibers of every normalization history are the event-labelled objects d_e:=Cofib(X_D_e→ X_downarrow e)","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"use","excerpt":"At step i, the split exchange and additivity of the surrounding context give X_i-1≅ X_i⊕ E_i(s). Substituting these decompositions successively yields X_0≅ X_m⊕ E_m(s)⊕⋯⊕ E_1(s). The direct sum is finite because the chosen principal path is finite. The identification with shifted cofibers follows from the cofiber of a split projection.","count":11},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"The residual system (Q[p^n],p^m-n) is pro-zero, and Kxrightarrow∼varprojlim_n K_n. Here pro-zero means that for each target level some later transition is the zero map. It does not mean that the individual residual modules vanish. In Smith coordinates a nonzero diagonal entry imposes p^a_ix_i=0 modulo p^n, giving min(a_i,n) units of O-length. Each ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-697a931a10a788b8","tex":"X_{i+1}","aliases":"X_i+1","display":"X_{i+1}","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8053762bbe02a919","tex":"X_{i-1}","aliases":"X_i-1","display":"X_{i-1}","kind":"use","excerpt":"At step i, the split exchange and additivity of the surrounding context give X_i-1≅ X_i⊕ E_i(s). Substituting these decompositions successively yields X_0≅ X_m⊕ E_m(s)⊕⋯⊕ E_1(s). The direct sum is finite because the chosen principal path is finite. The identification with shifted cofibers follows from the cofiber of a split projection.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8670d4aa5d0c39d6","tex":"X_{I\\cap J}","aliases":"X_I∩ J cap","display":"X_{I\\cap J}","kind":"use","excerpt":"[Elementary excision implies ideal excision] If every elementary independent-event square of X:NJ(P)→ S is a homotopy pushout, then for all ideals I,J the square X_I∩ Jarrow[r]arrow[d] · X_Iarrow[d] X_Jarrow[r] · X_I∪ J is a homotopy pushout.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-26bdfc254ee0f52d","tex":"X_{I\\cup J}","aliases":"X_I∪ J cup","display":"X_{I\\cup J}","kind":"use","excerpt":"[Elementary excision implies ideal excision] If every elementary independent-event square of X:NJ(P)→ S is a homotopy pushout, then for all ideals I,J the square X_I∩ Jarrow[r]arrow[d] · X_Iarrow[d] X_Jarrow[r] · X_I∪ J is a homotopy pushout.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ece2f99bd925957b","tex":"X_J","aliases":"X_J","display":"X_J","kind":"use","excerpt":"[Elementary excision implies ideal excision] If every elementary independent-event square of X:NJ(P)→ S is a homotopy pushout, then for all ideals I,J the square X_I∩ Jarrow[r]arrow[d] · X_Iarrow[d] X_Jarrow[r] · X_I∪ J is a homotopy pushout.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"use","excerpt":"[The first hidden commutator] Suppose n=2 and all generator matrices are A_i=I+pX_i modulo p^2. They always commute at that coefficient level. For rank two, [source reference] gives ρ_2 lifts over Z_p Longleftrightarrow [X_i,X_j]=0 in M_2( F_p) for all i,j. If some X_i is nonscalar, its centralizer is commutative and the displayed test agrees with ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"use","excerpt":"Let the resolution level be n≥1 and take X_n= Z/p^n Z. For m≤ n, coarsening is reduction C_n→ m: Z/p^n Z→ Z/p^m Z. For any polynomial f∈ Z[x], the induced transformation satisfies C_n→ m(f(x)) = f(C_n→ m(x)). Thus the transformation-resolution square commutes exactly. This is a set-level equality, not a cofiber calculation. Regarding the sets as di","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ae977da819b008be","tex":"X_{n-1}","aliases":"X_n-1","display":"X_{n-1}","kind":"use","excerpt":"For a chain r_0<r_1<⋯<r_n and chosen lifts X_i∈ T_r_i with C_r_i+1→ r_iX_i+1≅ X_i, record B_X=(b_r_0→ r_1(X_0),…,b_r_n-1→ r_n(X_n-1)). This profile measures how hidden multiplicity emerges under refinement.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-853cecd6231f3bad","tex":"X_P","aliases":"X_P","display":"X_P","kind":"use","excerpt":"ginenumerate item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e}) ⟶Cofib(X_I→ X_I∪{e}); item its terminal object is the total residual Cofib(X_∅→ X_P), independent of the linear extension used to reach it; i","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. The","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0613772511c49a53","tex":"X_W","aliases":"X_W","display":"X_W","kind":"use","excerpt":"[Strict semantic realization of the event diagram] For a typed pure word W in a strict chain-natural realization and a seed s∈ E_b, evaluation gives a functor X_W:NJ(P_W)⟶ S_0. Its vertices evaluate the reachable words and its cover arrows evaluate the whiskered exchange maps. Thus the residual diagram in [source reference] is constructed, rather t","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f375faba5416e7e7","tex":"X_x","aliases":"X_x","display":"X_x","kind":"use","excerpt":"[Semantic diamond criterion] Let a rewrite graph have a height h in N that decreases on every edge, and suppose every vertex x has a unique reachable normal vertex n(x). Assign an object X_x in a stable ∞-category to each vertex and a morphism to each edge. Assume every elementary peak xxrightarrowey, xxrightarrowfz has joining paths a:yleadsto u a","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-83a86bc483193312","tex":"X_{\\varnothing}","aliases":"X_∅ varnothing","display":"X_{\\varnothing}","kind":"use","excerpt":"[Ideal-indexed residual diagram] A coherent exchange-event diagram determines a functor F_X:NJ(P)⟶ S, F_X(I)=Cofib(X_∅⟶ X_I), with the following properties: item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[The counterexample occurs inside the two-axis grammar] The preceding square is not merely an unrelated diagram. Let the resolution base be 0<1<2, put T_0= T_1= T_2=Perf(k)×Perf(k), and take every coarsening functor to be the identity between labelled copies. Define the exact coordinate projections P_1(x,y)=(x,0), P_2(x,y)=(0,y). Use transformation","count":13},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7a4d74bf293c7c99","tex":"y_p","aliases":"y_p","display":"y_p","kind":"use","excerpt":"For surjectivity take (y_p)∈ G. Choose N large enough that D^Ny_p∈ Z_p for every p. For each p, choose the residue of D^Ny_p modulo p^N. The ordinary Chinese remainder theorem provides z∈ Z with all these residues. Set a=z/D^N∈ A. Then a-y_p∈ Z_p for every p, so (a,(a-y_p)_p) maps to (y_p)_p. This proves exactness.","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Given (r,X)xrightarrowα(s,Y) and (s,Y)xrightarrowβ(t,Z), define β∘_ Kα := C_s→ r(β)∘α: X→ C_t→ r(Z).","count":14},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"[Semantic diamond criterion] Let a rewrite graph have a height h in N that decreases on every edge, and suppose every vertex x has a unique reachable normal vertex n(x). Assign an object X_x in a stable ∞-category to each vertex and a morphism to each edge. Assume every elementary peak xxrightarrowey, xxrightarrowfz has joining paths a:yleadsto u a","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-544ac8aed4a2c573","tex":"Z^1","aliases":"Z^1","display":"Z^1","kind":"use","excerpt":"[Repair ambiguity, conjugacy, and finite certificates] If the one-step lift set is nonempty, its framed lifts form a torsor for Z^1(G,V). Conjugation by matrices I+p^nX, X∈ V, quotients this torsor by coboundaries. Hence the strict-conjugacy classes form a torsor for H^1(G,V), and the stabilizer of a lift under this conjugation is H^0(G,V) as an ad","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1b1776a6e72cb263","tex":"Z^1_{\\mathrm{cts}}","aliases":"Z^1_cts mathrm","display":"Z^1_{\\mathrm{cts}}","kind":"use","excerpt":"(Γ,V) is independent of the continuous choices. It vanishes if and only if ρ_n has a continuous lift to GL_d(R_n+1) with its specified reduction. More precisely, the repair equation is δ b=-c_A, b:Γ→ V continuous and normalized, and the repaired matrices are (I+p^nb(g))A_g. When lifts exist, their framed set is a torsor for Z^1_cts(Γ,V). Strict-con","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-af5f0843f8046d2c","tex":"\\mathbb Z^2","aliases":"Z^2 mathbb","display":"\\mathbb Z^2","kind":"use","excerpt":"For two events the entire calculation is visible: K(a,b)=[ Zxrightarrowbinom-ba Z^2xrightarrow(a b) Z ], degrees -2,-1,0. The product of the two differentials is zero. The kernel of (a b) is generated by (-b/g,a/g), and the incoming image is g times that generator. Both H^-1 and H^0 are Z/g. Equivalently, if au+bv=g, then U=u · -b/g","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6f22669574ee3af7","tex":"\\mathbb Z^{a_i}","aliases":"Z^a_i mathbb","display":"\\mathbb Z^{a_i}","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ddb351eae1c24766","tex":"\\mathbb Z^{a_i-r_i}","aliases":"Z^a_i-r_i mathbb","display":"\\mathbb Z^{a_i-r_i}","kind":"use","excerpt":"Changing bases in the source and target yields a chain isomorphism Cofib(f_i)≃ Z^b_i-r_i⊕ Z^a_i-r_i[1] ⊕⊕_k=1^r_iC_ Z(d_i,k). Unit diagonal entries contribute contractible complexes. Thus ϵ_i=1 means that a shift of the unit occurs as a direct summand. If two independent cones each have such a summand, their tensor product contains a shift of the u","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-684a71099ea13890","tex":"\\mathbb Z^{b_i}","aliases":"Z^b_i mathbb","display":"\\mathbb Z^{b_i}","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0,","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7cd790f45497a44d","tex":"\\mathbb Z^{b_i-r_i}","aliases":"Z^b_i-r_i mathbb","display":"\\mathbb Z^{b_i-r_i}","kind":"use","excerpt":"Changing bases in the source and target yields a chain isomorphism Cofib(f_i)≃ Z^b_i-r_i⊕ Z^a_i-r_i[1] ⊕⊕_k=1^r_iC_ Z(d_i,k). Unit diagonal entries contribute contractible complexes. Thus ϵ_i=1 means that a shift of the unit occurs as a direct summand. If two independent cones each have such a summand, their tensor product contains a shift of the u","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-23a056bd23b65ee2","tex":"\\mathbb Z^r","aliases":"Z^r mathbb","display":"\\mathbb Z^r","kind":"use","excerpt":"ic identifications. Choose a basis V=A^r, and let E_p=θ_p^-1(L_p) be the corresponding lattice in Q_p^r. Bounds on the entries of θ_p and its inverse give a common N≥1 such that, with b=D^N, b Z_p^r⊆ E_p⊆ b^-1 Z_p^r for all p. The intersection defining Λ therefore satisfies b Z^r⊆Λ⊆ b^-1 Z^r. The right inclusion uses the same integrality-of-denomin","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-370c87df70534fa4","tex":"\\mathbb Z_2","aliases":"Z_2 mathbb","display":"\\mathbb Z_2","kind":"use","excerpt":"Starting with the integer matrix a_0=3, the averaged repair becomes a_j+1=frac3-a_j^22 a_j. The first values are 3,-9,351, whose relation defects have valuations 3,4,6. The convergence theorem gives an exact 2-adic involution congruent to 3pmod4. In the integral domain Z_2, the equation a^2=1 has only the solutions 1,-1. Thus the repaired limit is ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9bb63c10f3cd1141","tex":"\\mathbb Z_3","aliases":"Z_3 mathbb","display":"\\mathbb Z_3","kind":"use","excerpt":":repair-before-completion For P={p} and two independent labels divisible by p, every finite excisive quotient is zero, whereas (varprojlim_n Z/p^n)[1/p]= Q_p≠0. For a nonzero finite-level quotient, take D=30 and labels 6,10. Only the prime 2 is shared, so the finite repair is B_n[1/2]≅ Z/15^n. The two completed outputs are Q_2× Z_3× Z_5 and Z_3× Z_","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-347d071dc857b815","tex":"\\mathbb Z_5","aliases":"Z_5 mathbb","display":"\\mathbb Z_5","kind":"use","excerpt":":repair-before-completion For P={p} and two independent labels divisible by p, every finite excisive quotient is zero, whereas (varprojlim_n Z/p^n)[1/p]= Q_p≠0. For a nonzero finite-level quotient, take D=30 and labels 6,10. Only the prime 2 is shared, so the finite repair is B_n[1/2]≅ Z/15^n. The two completed outputs are Q_2× Z_3× Z_5 and Z_3× Z_","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8c3d66bb2b40481d","tex":"\\mathcal Z_A","aliases":"Z_A mathcal","display":"\\mathcal Z_A","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4d5f0e25af5a5f7d","tex":"\\mathcal Z_B","aliases":"Z_B mathcal","display":"\\mathcal Z_B","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bc1f10031646b5b4","tex":"Z_i","aliases":"Z_i","display":"Z_i","kind":"use","excerpt":"[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S, A_i · i∉ S. Its total cofiber, obtained by taking the cofiber in every coordinate, satisfies tcofib(Y)≃ ⊗_i=1^mCofib","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7f3b8fae797e61c1","tex":"\\mathbb Z_p^2","aliases":"Z_p^2 mathbb","display":"\\mathbb Z_p^2","kind":"use","excerpt":"[X_i,X_j]=0 in M_2( F_p) for all i,j. If some X_i is nonscalar, its centralizer is commutative and the displayed test agrees with the distinguished-generator criterion; if all are scalar the lift is immediate. For X_1=E_12 and X_2=E_21 it fails. Each generator separately has a continuous integral procyclic lift, but the pair does not define an inte","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-66c5e4d90af97908","tex":"\\mathbb Z_p^r","aliases":"Z_p^r mathbb","display":"\\mathbb Z_p^r","kind":"use","excerpt":"For continuous representations of Z_p^r, removing a common scalar precision exposes normalized commutators. When one normalized generator is cyclic modulo p, their vanishing is necessary and sufficient for an integral lift, constructed by finite polynomial certificates. This condition covers every rank-two input. It yields a complete finite-matrix ","count":18},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9a01906ff34cc704","tex":"\\mathbb Z_p^\\times","aliases":"Z_p^× mathbb times","display":"\\mathbb Z_p^\\times","kind":"use","excerpt":"define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, proving the bound on b. Normalization gives b(1)=0. Because m-","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-53acb0f65622312d","tex":"\\mathcal Z_s^{\\mathrm{thick}}","aliases":"Z_s^thick mathcal mathrm","display":"\\mathcal Z_s^{\\mathrm{thick}}","kind":"use","excerpt":"[Orbit-generator test for an exchange obstruction] Fix an exchange η^A_s,r between exact functors, and an exact quotient q_r of the target fiber. If q_rΔ^A_s,r(g)≃0 for every g∈ G_s, then q_rΔ^A_s,r(X)≃0 for every X∈ Z_s^thick. The same assertion holds for q_r=id, in which case the exchange is an equivalence on the entire generated thick zoo.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-50f1d51fcf879c34","tex":"|_I","aliases":"|_I","display":"|_I","kind":"use","excerpt":"For arbitrary ideals I,J, every chain in I∪ J lies wholly in I or wholly in J. Its maximal element lies in one ideal and all its predecessors lie there too. Hence the nerve of their union is the union of their nerves along the intersection. Cocones on G|_I∪ J are precisely compatible cocones on G|_I and G|_J. The resulting colimit is the pushout of","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-36f7024eb39dd2e6","tex":"|_{I\\cup J}","aliases":"|_I∪ J cup","display":"|_{I\\cup J}","kind":"use","excerpt":"For arbitrary ideals I,J, every chain in I∪ J lies wholly in I or wholly in J. Its maximal element lies in one ideal and all its predecessors lie there too. Hence the nerve of their union is the union of their nerves along the intersection. Cocones on G|_I∪ J are precisely compatible cocones on G|_I and G|_J. The resulting colimit is the pushout of","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-81c484949929c99e","tex":"|_J","aliases":"|_J","display":"|_J","kind":"use","excerpt":"For arbitrary ideals I,J, every chain in I∪ J lies wholly in I or wholly in J. Its maximal element lies in one ideal and all its predecessors lie there too. Hence the nerve of their union is the union of their nerves along the intersection. Cocones on G|_I∪ J are precisely compatible cocones on G|_I and G|_J. The resulting colimit is the pushout of","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4a930ea4e8f20684","tex":"|_{U_{ij}}","aliases":"|_U_ij","display":"|_{U_{ij}}","kind":"use","excerpt":"Given a cover {U_i→ U}, local objects X_i∈mathscr K(U_i) carry descent isomorphisms ϕ_ij:X_i|_U_ijxrightarrow∼X_j|_U_ij satisfying the cocycle condition on triple overlaps.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5c01dba2efee97a6","tex":"\\alpha_{G,n}","aliases":"α_G,n alpha","display":"\\alpha_{G,n}","kind":"use","excerpt":"Let G be a finite group acting O-linearly on a lattice M. The natural comparison is α_G,n:M^G/p^nM^G⟶(M/p^nM)^G. The first path takes invariant vectors and then reduces. The second reduces the representation and then takes invariant vectors. Its direction is fixed by reduction of an invariant vector; no arbitrary scalar enters this map.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a823753ea2193233","tex":"\\alpha_n","aliases":"α_n alpha","display":"\\alpha_n","kind":"use","excerpt":"For a map d:V→ W of lattices, put K=ker d, Q=cokerd, K_n=ker(d⊗_O O_n). The module K is saturated in V: if p^n v∈ K, then p^n d(v)=0 in the torsion-free module W, so v∈ K. It is therefore a finite free direct summand as an O-module, although a splitting is not canonical. Reduction supplies a natural injection α_n:K/p^nK→ K_n.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a99ae1b6c0cfda2b","tex":"\\alpha_r^{\\flat}","aliases":"α_r^flat alpha flat","display":null,"kind":"use","excerpt":"Any adjunction Ldashv U induces an adjunction on functor categories by postcomposition. Concretely, a natural transformation α:L∘ F⇒ G has pointwise adjoint mates α_r^flat:F(r)⟶ U(G(r)). Naturality of the unit and counit implies that the family {α_r^flat}_r is natural in r. The mate construction is invertible and respects higher coherence. Applying","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"Given (r,X)xrightarrowα(s,Y) and (s,Y)xrightarrowβ(t,Z), define β∘_ Kα := C_s→ r(β)∘α: X→ C_t→ r(Z).","count":24},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1b25d223e7553f00","tex":"\\beta_{O,R,N}","aliases":"β_O,R,N beta","display":"\\beta_{O,R,N}","kind":"use","excerpt":"For O= Z_p, R= Z/p^m, and S= Z/p^n with m≥ n≥1, this applies at every pair of levels. It applies in particular to reductions of a supplied integral G-lattice. The canonical base-change comparisons obey transitivity. After the tensor associativities, the composite Γ_O(N)⊗_O^ LS · xrightarrowβ_O,R,N⊗_R S Γ_R(N⊗_O^ LR)⊗_R^ LS · xrightarrowβ_R,S,M Γ_S(","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-abc5fc0ec31b1306","tex":"\\beta_{O,S,N}","aliases":"β_O,S,N beta","display":"\\beta_{O,S,N}","kind":"use","excerpt":"· xrightarrowβ_R,S,M Γ_S(N⊗_O^ LS) is β_O,S,N. To check this equality with its natural coherence, take the adjoints defining the comparisons: both are base change of the same evaluation map from homotopy invariants to N. The coefficient extension adjunctions compose by derived tensor associativity. Their mates therefore agree. No arbitrary homotopy","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-245e48d480c30eae","tex":"\\beta_{R,k,M}","aliases":"β_R,k,M beta","display":"\\beta_{R,k,M}","kind":"use","excerpt":"[A central sign hides the obstruction from derived invariants] Let p be odd, R= Z/p^2, and let J∈ M_d( Z) be invertible modulo p. On M=R^d let G=C_p× C_2=⟨σ,τ⟩, σ↦ I+pJ, τ↦-I. For the canonical comparison of [source reference], Γ_R(M)≃0, Γ_k(M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not l","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2bb15f46b12763d8","tex":"\\beta_{R,S,-}","aliases":"β_R,S,- beta","display":"\\beta_{R,S,-}","kind":"use","excerpt":"The cofiber of β_R,S,- is an exact functor. Its vanishing class is therefore thick and contains every coefficient-lifted input, as well as the thick closure of those inputs. The example in [source reference] lies outside that class. This is a functor-intrinsic necessary lifting obstruction, not merely a failed scalar equality.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-cfe99f075fe5f2ac","tex":"\\beta_{R,S,M}","aliases":"β_R,S,M beta","display":"\\beta_{R,S,M}","kind":"use","excerpt":"For a ring map R→ S there is a canonical comparison β_R,S,M:Γ_R(M)⊗_R^ LS ⟶Γ_S(M⊗_R^ LS). It is adjoint to base change of the evaluation map from the object of homotopy invariants, regarded as carrying the trivial group action, to M. Its construction is natural in M and uses the actual coefficient map. No exchange weight is selected.","count":6},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"Identity at (r,X) is id_X:X→ C_r→ r(X)=X. For associativity, let (r,X)xrightarrowα(s,Y) xrightarrowβ(t,Z) xrightarrowγ(u,W). One composite gives C_t→ r(γ)∘ C_s→ r(β)∘α. The other gives the same expression by functoriality and C_t→ r=C_s→ rC_t→ s. Thus composition is associative.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ff9a1064ad5a9b6b","tex":"\\Gamma^j","aliases":"Γ^j Gamma","display":"\\Gamma^j","kind":"use","excerpt":"U'⊆ U: Inf_Γ/U^Γ/U'[c_A]=0. Thus a continuous next-level lift exists exactly when a repair exists at some finer source quotient, not necessarily the quotient used initially. For every j≥0, H^j_cts(Γ,V) ≅undersetU'⊆ Ucolim H^j(Γ/U',V). Every continuous map Γ^j→ V is locally constant. Cover its compact domain by finitely many products of cosets on wh","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8335fc4d7b413b19","tex":"\\Gamma_k","aliases":"Γ_k Gamma","display":"\\Gamma_k","kind":"use","excerpt":"[Unbounded intrinsic derived exchange defect] For the preceding data, A^p=I and the norm is N=I+A+⋯+A^p-1=pI. Moreover, Γ_R(M)≃ pM[0]≅ k^d[0]. Let U=Γ_R(M)⊗_R^ Lk and V=Γ_k(M⊗_R^ Lk). Their cohomology is H^i(U)congk^d · i≤0,","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-694ab2d9f0794d2d","tex":"\\Gamma_O","aliases":"Γ_O Gamma","display":"\\Gamma_O","kind":"use","excerpt":"For O= Z_p, R= Z/p^m, and S= Z/p^n with m≥ n≥1, this applies at every pair of levels. It applies in particular to reductions of a supplied integral G-lattice. The canonical base-change comparisons obey transitivity. After the tensor associativities, the composite Γ_O(N)⊗_O^ LS · xrightarrowβ_O,R,N⊗_R S Γ_R(N⊗_O^ LR)⊗_R^ LS · xrightarrowβ_R,S,M Γ_S(","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3c86d4a426c471d1","tex":"\\Gamma_R","aliases":"Γ_R Gamma","display":"\\Gamma_R","kind":"use","excerpt":"For a commutative ring R, a finite group G, and M∈ D(R[G]), define Γ_R(M)= RHom_R[G](R,M)∈ D(R). This is the derived invariant functor. Derived categories in this section are the enhanced unbounded derived categories. They are stable. Their exact functors preserve finite cofiber sequences but are not asserted to preserve every infinite limit or col","count":11},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-693cfb4a6a0cb28a","tex":"\\Gamma_S","aliases":"Γ_S Gamma","display":"\\Gamma_S","kind":"use","excerpt":"For a ring map R→ S there is a canonical comparison β_R,S,M:Γ_R(M)⊗_R^ LS ⟶Γ_S(M⊗_R^ LS). It is adjoint to base change of the evaluation map from the object of homotopy invariants, regarded as carrying the trivial group action, to M. Its construction is natural in M and uses the actual coefficient map. No exchange weight is selected.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e65805130c408e92","tex":"\\Delta^+_{s,r}","aliases":"Δ^+_s,r Delta","display":"\\Delta^+_{s,r}","kind":"use","excerpt":"In an abelian fiber one may instead record Δ^-_s,r(X)=kerη^A_s,r,X, Δ^+_s,r(X)=cokerη^A_s,r,X. These vanish simultaneously exactly when the comparison is an isomorphism.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-48a0cf544833bcad","tex":"\\Delta^-_{s,r}","aliases":"Δ^-_s,r Delta","display":"\\Delta^-_{s,r}","kind":"use","excerpt":"In an abelian fiber one may instead record Δ^-_s,r(X)=kerη^A_s,r,X, Δ^+_s,r(X)=cokerη^A_s,r,X. These vanish simultaneously exactly when the comparison is an isomorphism.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6c25d964f1c23310","tex":"\\Delta^A_{s,s}","aliases":"Δ^A_s,s Delta","display":"\\Delta^A_{s,s}","kind":"use","excerpt":"Fix X∈ T_s. First record the exact commutation profile E_A(X;s)={r≤ s:Δ^A_s,r(X)≃0}. No interval or monotonicity property is assumed. For R= N and finite s, a well-defined consecutive coarsening depth is κ_A(X;s):=max{k∈{0,…,s}:Δ^A_s,r(X)≃0 for all s-k≤ r≤ s}. The set is nonempty because Δ^A_s,s(X)≃0. A larger value means that more consecutive coar","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-101e54c630828bcd","tex":"\\Delta^B_{s,r}","aliases":"Δ^B_s,r Delta","display":"\\Delta^B_{s,r}","kind":"use","excerpt":"mathrmadd_ RF⟶ G be an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every r≤ s and every X there is a canonical equivalence Φ_r(Δ^A_s,r(X)) ≃ Cofib ( Φ_r(widehatη^A_s,r,X) ). In particular, if the exchange data in G are denoted by η^B, then Φ_r(Δ^A_s,r(X)) ≃ Δ^B_s,r(Φ_sX).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2bb168c166c79f1d","tex":"\\Delta_1","aliases":"Δ_1 Delta","display":"\\Delta_1","kind":"use","excerpt":"Residual genealogy. Residual promotion creates a directed graph s→Δ_1→Δ_2→⋯. Classify whether genealogies terminate, become periodic, stabilize, or grow indefinitely. A terminating genealogy would correspond to finite obstruction resolution. Periodic genealogy may encode a persistent but structured defect.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6c0b4ab8dd399bba","tex":"\\Delta_2","aliases":"Δ_2 Delta","display":"\\Delta_2","kind":"use","excerpt":"Residual genealogy. Residual promotion creates a directed graph s→Δ_1→Δ_2→⋯. Classify whether genealogies terminate, become periodic, stabilize, or grow indefinitely. A terminating genealogy would correspond to finite obstruction resolution. Periodic genealogy may encode a persistent but structured defect.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-23fe82c4424f67f6","tex":"\\delta_m","aliases":"δ_m delta","display":"\\delta_m","kind":"use","excerpt":"ockstein There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently the cokernel of the natural compari","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ae5d1ee4ca82d29f","tex":"\\delta_n","aliases":"δ_n delta","display":"\\delta_n","kind":"use","excerpt":"[Intrinsic kernel exchange and its boundary] There is a natural short exact sequence 0⟶ K/p^nKxrightarrowα_nK_n xrightarrowδ_nQ[p^n]⟶0. For bar v∈ K_n, choose a lift v∈ V. Since d(v)∈ p^nW, set δ_n(bar v)=[fracd(v)p^n]∈ Q. This formula is independent of the lift. For m≥ n, ordinary reduction r_m,n:K_m→ K_n satisfies δ_n r_m,n=p^m-nδ_m. Consequently","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d229ccb7d9f7b4f6","tex":"\\delta_{n+c}","aliases":"δ_n+c delta","display":"\\delta_{n+c}","kind":"use","excerpt":"If c<h, choose an element of exact order p^h in Q_tors and take n≥ h. Surjectivity of δ_n+c supplies a preimage. Its reduction to level n has nonzero boundary after multiplication by p^c, so it is not the reduction of an exact solution. This proves sharpness. Multiplication by p^h also makes every transition with gap at least h zero on residuals.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3af151c5bc892c0c","tex":"\\delta_{n+h}","aliases":"δ_n+h delta","display":"\\delta_{n+h}","kind":"use","excerpt":"By [source reference], the image of an element of K_n+h has boundary p^hδ_n+h=0, so lies in the image of K/p^nK. The reverse inclusion follows by reducing any exact lift at level n+h.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-39602252d945435e","tex":"\\Delta_{T',C'}","aliases":"Δ_T',C' Delta","display":"\\Delta_{T',C'}","kind":"use","excerpt":"A strong correspondence should include: Λ(TX) · ≃ T'Λ(X), Λ(CX) · ≃ C'Λ(X), Λ(Δ_T,CX) · ≃Δ_T',C'Λ(X), barΛ(Obs X) · ≃Obs'Λ(X).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9962079355edd4fc","tex":"\\Delta_\\eta","aliases":"Δ_η Delta eta","display":"\\Delta_\\eta","kind":"use","excerpt":"[Residual formation is exact in its input] Let P,Q: C→ D be exact functors between small stable ∞-categories, and let η:P⇒ Q be a natural transformation. Then Δ_η(X):=Cofib(η_X) is an exact functor C→ D. If q: D→ O is exact, the subcategory {X:qΔ_η(X)≃0} is thick.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0, 0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. ","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0773a19734ee0848","tex":"\\epsilon_i","aliases":"ϵ_i epsilon","display":"\\epsilon_i","kind":"use","excerpt":"[Finite integral matrix collapse criterion] Let f_i: Z^a_i→ Z^b_i be finitely many integer matrices. Specify which pairs of directions may occur independently. For each matrix fix a Smith normal form, with rank r_i and positive nonzero diagonal entries d_i,1,…,d_i,r_i. Put ϵ_i= 1 · a_i+b_i-2r_i>0, 0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. ","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-a14473f51c7df774","tex":"\\epsilon_j","aliases":"ϵ_j epsilon","display":"\\epsilon_j","kind":"use","excerpt":"0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4ea280c2275adb2f","tex":"\\eta^A","aliases":"η^A eta","display":"\\eta^A","kind":"use","excerpt":"A morphism (r,X)→(s,Y) in K is represented by α:X→ C_s→ rY. Define widehat A(α) to be A_rX xrightarrowA_r(α) A_rC_s→ rY xrightarrowη^A_s,r,Y C_s→ rA_sY. Identity preservation follows from η^A_r,r=id. Composition preservation is exactly the coherence condition in [source reference], together with naturality of η^A.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-235e9068fcfdb970","tex":"\\eta^A_{s,r,Y}","aliases":"η^A_s,r,Y eta","display":"\\eta^A_{s,r,Y}","kind":"use","excerpt":"A morphism (r,X)→(s,Y) in K is represented by α:X→ C_s→ rY. Define widehat A(α) to be A_rX xrightarrowA_r(α) A_rC_s→ rY xrightarrowη^A_s,r,Y C_s→ rA_sY. Identity preservation follows from η^A_r,r=id. Composition preservation is exactly the coherence condition in [source reference], together with naturality of η^A.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b3dc580bd88a9a6f","tex":"\\eta^B","aliases":"η^B eta","display":"\\eta^B","kind":"use","excerpt":"ratornameSt^add_ RF⟶ G be an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every r≤ s and every X there is a canonical equivalence Φ_r(Δ^A_s,r(X)) ≃ Cofib ( Φ_r(widehatη^A_s,r,X) ). In particular, if the exchange data in G are denoted by η^B, then Φ_r(Δ^A_s,r(X)) ≃ Δ^B_s,r(Φ_sX).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ba7a4cdb1674dfac","tex":"\\eta_{L,b;m,n}","aliases":"η_L,b;m,n eta","display":"\\eta_{L,b;m,n}","kind":"use","excerpt":"ut A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use the cell η_L,b;m,n=b^m-nθ_L;m,n. For sca","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-656b78391b3ddd4e","tex":"\\eta_{L,b;m,n,M}","aliases":"η_L,b;m,n,M eta","display":"\\eta_{L,b;m,n,M}","kind":"use","excerpt":"[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"0, |Lift_r→ s(X)|=1, |Lift_r→ s(X)|>1. They correspond respectively to obstruction to refinement, uniqueness of the fine object's isomorphism class, and hidden fine-scale multiplicity. The middle case does not imply a contractible space of refinements. It forgets the identification C_s→ rY≅ X and all automorphisms. When these matter, use the groupo","count":17},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2f036853692243d7","tex":"\\theta'_p","aliases":"θ'_p theta","display":"\\theta'_p","kind":"use","excerpt":"Any other lattice with the specified generic and local identifications has analogous bounds by powers of D. After a common scaling, its finite quotient is determined by the same primary subgroups in the Chinese remainder decomposition. It therefore equals Λ. A compatible A-linear map V→ V' sends every θ_p-integral element to a θ'_p-integral element","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f035e1cc3facb7c1","tex":"\\theta_{L;l,n}","aliases":"θ_L;l,n theta","display":"\\theta_{L;l,n}","kind":"use","excerpt":"B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These formulas also hold with compatible tensor templates absorbed into N. Naturality follows from central scalar multiplication. For l≥ m≥ n, the two-step cell has scalar b^m-nb^l-m=b^l-n. The associativity coherence of derived tensor identifies its underlying θ with θ_L;l,n. Identity intervals have","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4aea8d15c738ba27","tex":"\\theta_{L;m,n}","aliases":"θ_L;m,n theta","display":"\\theta_{L;m,n}","kind":"use","excerpt":"For a fixed perfect integral complex L, put A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c4bf4a142569c578","tex":"\\theta_p","aliases":"θ_p theta","display":"\\theta_p","kind":"use","excerpt":"[Effective arithmetic lattice gluing] Let V be a finite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the uniqu","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-107c98e27eeb3890","tex":"\\theta_p^{-1}","aliases":"θ_p^-1 theta","display":"\\theta_p^{-1}","kind":"use","excerpt":"bb Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these specified embeddings. Compatible maps of gluing data restrict uniquely to maps of lattices. Hence finite free integral modules are equivalent to this category of localized free modules, local lattices, and specified generic identifications. Choose a basis V=A^r, and le","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2632b408a8f1e59a","tex":"\\iota_{\\mathbb S}","aliases":"ι_ S iota mathbb","display":"\\iota_{\\mathbb S}","kind":"use","excerpt":"[Free seeded two-axis double category] For every small seeded two-axis double computad S, there exists a strict double category D_ S and a canonical map of seeded double computads ι_ S: S⟶ U D_ S such that for every strict double category D, restriction along ι_ S induces a natural bijection DblCat( D_ S, D) ≅ DblCptd( S,U D), for the underlying co","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"For r≤ s and Y∈ T_s, define the canonical morphism κ_r,s,Y: (r,C_s→ rY)⟶(s,Y) represented by the identity map id_C_s→ rY.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-729285a42e26eda6","tex":"\\kappa_A","aliases":"κ_A kappa","display":"\\kappa_A","kind":"use","excerpt":"Fix X∈ T_s. First record the exact commutation profile E_A(X;s)={r≤ s:Δ^A_s,r(X)≃0}. No interval or monotonicity property is assumed. For R= N and finite s, a well-defined consecutive coarsening depth is κ_A(X;s):=max{k∈{0,…,s}:Δ^A_s,r(X)≃0 for all s-k≤ r≤ s}. The set is nonempty because Δ^A_s,s(X)≃0. A larger value means that more consecutive coar","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-70c7f40be1bf1990","tex":"\\kappa_{r,s,Y}","aliases":"κ_r,s,Y kappa","display":"\\kappa_{r,s,Y}","kind":"use","excerpt":"For r≤ s and Y∈ T_s, define the canonical morphism κ_r,s,Y: (r,C_s→ rY)⟶(s,Y) represented by the identity map id_C_s→ rY.","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"[Functorial transport of obstruction] Let ( Z_A,mathscr M_A) and ( Z_B,mathscr M_B) be stable obstruction systems, and let Λ: Z_A⟶ Z_B be exact. If Λ( K_mathscr M_A)⊆ N_mathscr M_B, then Λ induces an exact functor barΛ: O_mathscr M_A⟶ O_mathscr M_B. If Λ is an equivalence and carries N_mathscr M_A onto N_mathscr M_B, then barΛ is an equivalence. In","count":27},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"For a fixed multiplication table and residual action, deciding existence is a finite linear problem over k. A solution b is a positive certificate. A linear functional λ on normalized 2-cochains with λ∘δ=0, λ(c_A)≠0 is a negative certificate. If b_0 is one solution, every other solution is b_0+z with δ z=0. Conjugating the matrices by I+p^nX replac","count":10},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b20ef177d3714eb3","tex":"\\Lambda'","aliases":"Λ' Lambda","display":"\\Lambda'","kind":"use","excerpt":"ther lattice with the specified generic and local identifications has analogous bounds by powers of D. After a common scaling, its finite quotient is determined by the same primary subgroups in the Chinese remainder decomposition. It therefore equals Λ. A compatible A-linear map V→ V' sends every θ_p-integral element to a θ'_p-integral element and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f997f2c3c2ddc790","tex":"\\Lambda^\\bullet","aliases":"Λ^bullet Lambda bullet","display":"\\Lambda^\\bullet","kind":"use","excerpt":"[Descent transfer lemma] Let A^bullet,B^bullet:Δ→Cat_∞ be small vCech descent diagrams for a fixed cover. Suppose Λ^bullet:A^bullet→ B^bullet is a natural transformation that is an equivalence at every degree, including every overlap level. Then lim_[n]∈Δ A^n≃lim_[n]∈Δ B^n. For ordinary category-valued descent, use the corresponding coherent pseudo","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6559d219a1ff939b","tex":"\\Lambda_*","aliases":"Λ_* Lambda","display":"\\Lambda_*","kind":"use","excerpt":"Interpretation. This reframes a possible route to local-global reciprocity as: identify the globalization obstruction and prove that local correspondence transports it. If Λ_*(Ω_arith)=Ω_aut, and Λ_* preserves and reflects the zero object, then vanishing of one obstruction is equivalent to vanishing of the other. An equivalence, or a conservative e","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-326f3d290f6131a7","tex":"\\lambda_1","aliases":"λ_1 lambda","display":"\\lambda_1","kind":"use","excerpt":"9 · 7pmod27. They commute modulo 27 and both are the identity modulo 3. Choose a=1, λ_1=λ_2=1. Then B_1=0 · 1","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-975e1a0c6d000741","tex":"\\lambda_2","aliases":"λ_2 lambda","display":"\\lambda_2","kind":"use","excerpt":"9 · 7pmod27. They commute modulo 27 and both are the identity modulo 3. Choose a=1, λ_1=λ_2=1. Then B_1=0 · 1","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"use","excerpt":"Choose 0≤ a<n and integer scalars λ_i such that A_i≡λ_i Ipmodp^a for every i. There is no restriction in this display when a=0. Define the normalized matrices B_i=fracA_i-λ_i Ip^a ∈ M_d(R_n-a). This is well-defined from the given finite data. It does not invert p inside R_n. Assume that, for some distinguished index *, the matrix B_* is cyclic modu","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"Resolution system: · F: R^op→Cat. [0.4em] Fiber: · T_r=F(r). [0.4em] Coarsening: · C_s→ r: T_s→ T_r. [0.4em] Total category: · K=∫_ RF. [0.4em] Resolution equivalence: · X∼_rYiff C_t→ rX≅ C_t→ rY. [0.4em] Lift set: · Lift_r→ s(X)={Y:C_s→ rY≅ X}/≅. [0.4em] Exchange map: · η^A_s,r:A_rC_s→ r⇒ C_s→ rA_s. [0.4em] Stable residual: · Δ^A_s,r(X)=Cofib(η^A_","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"Define π: K→ R, π(r,X)=r. The fiber over r is canonically equivalent to T_r.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)","count":52},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d7ba0ffb2518dd65","tex":"\\rho_2","aliases":"ρ_2 rho","display":"\\rho_2","kind":"use","excerpt":"[The first hidden commutator] Suppose n=2 and all generator matrices are A_i=I+pX_i modulo p^2. They always commute at that coefficient level. For rank two, [source reference] gives ρ_2 lifts over Z_p Longleftrightarrow [X_i,X_j]=0 in M_2( F_p) for all i,j. If some X_i is nonscalar, its centralizer is commutative and the displayed test agrees with ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3c51f5238630cd1f","tex":"\\rho_h","aliases":"ρ_h rho","display":"\\rho_h","kind":"use","excerpt":"group representations] Let G be a finite group, d,n≥1, and e=v_p(|G|). Set h_G(n)=max{n+e, 2e+1}. For every ρ_n:G→GL_d( Z/p^n), the following are equivalent: item There is a representation ρ:G→GL_d( Z_p) with reduction ρ_n. item There is a representation ρ_h:G→GL_d( Z/p^h_G(n)) with reduction ρ_n. The implication from (2) to (1) is constructive to ","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f6c67cd7deda6b91","tex":"\\rho_L","aliases":"ρ_L rho","display":"\\rho_L","kind":"use","excerpt":"[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-df0e4853e5c546e4","tex":"\\rho_M","aliases":"ρ_M rho","display":"\\rho_M","kind":"use","excerpt":"[Certified lifting of equivariant maps] Let L and M be G-lattices and put V=Hom_O(L,M), (g· f)=ρ_M(g)fρ_L(g)^-1. Then 0⟶ · Hom_O[G](L,M)/p^n ⟶Hom_O_n[G](L/p^nL,M/p^nM)","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-18406af98fd43046","tex":"\\rho_m","aliases":"ρ_m rho","display":"\\rho_m","kind":"use","excerpt":"f Γ is topologically finitely generated, existence of some extension of the fixed ρ_n to every coefficient level is enough to obtain a compatible sequence and hence an integral lift. The residual adjoint module is unchanged by each coefficient lift. Under the vanishing hypothesis, [source reference] permits recursive choice of compatible lifts ρ_m ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-c03851211fa61002","tex":"\\rho_n","aliases":"ρ_n rho","display":"\\rho_n","kind":"use","excerpt":"Put R_n= Z/p^n, k= F_p, and fix ρ_n:G⟶GL_d(R_n), n≥1. Let V=End_k(k^d) with action g· X=barρ(g)Xbarρ(g)^-1, barρ=ρ_nbmod p. Choose matrices A_g∈GL_d(R_n+1) lifting ρ_n(g), with A_1=I. They need not constitute a representation. Define the normalized relation residual c_A(g,h)=fracA_gA_hA_gh^-1-Ip^npmod p ∈ V. Division here identifies the square-zero","count":32},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-623c596d9e288ed4","tex":"\\sigma^p","aliases":"σ^p sigma","display":"\\sigma^p","kind":"use","excerpt":"has the form I+pJ+p^2B modulo p^3. Put X=pJ+p^2B and use the binomial identity for I+X. The linear term is pX=p^2J modulo p^3. For 2≤ j<p, the coefficient binom pj is divisible by p and X^j by p^j, so those terms vanish modulo p^3. The last term X^p also vanishes because p≥3. Since J is invertible modulo p, the surviving term is nonzero. Hence the ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"[A central sign hides the obstruction from derived invariants] Let p be odd, R= Z/p^2, and let J∈ M_d( Z) be invertible modulo p. On M=R^d let G=C_p× C_2=⟨σ,τ⟩, σ↦ I+pJ, τ↦-I. For the canonical comparison of [source reference], Γ_R(M)≃0, Γ_k(M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not l","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"Given a cover {U_i→ U}, local objects X_i∈mathscr K(U_i) carry descent isomorphisms ϕ_ij:X_i|_U_ijxrightarrow∼X_j|_U_ij satisfying the cocycle condition on triple overlaps.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-012e4b88d8b99f5f","tex":"\\phi_{ij}","aliases":"ϕ_ij phi","display":"\\phi_{ij}","kind":"use","excerpt":"Given a cover {U_i→ U}, local objects X_i∈mathscr K(U_i) carry descent isomorphisms ϕ_ij:X_i|_U_ijxrightarrow∼X_j|_U_ij satisfying the cocycle condition on triple overlaps.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-00bfe1f837258b4d","tex":"\\Phi_r","aliases":"Φ_r Phi","display":"\\Phi_r","kind":"use","excerpt":"niversal residual preservation] Let G: R^op→Cat^st be a stable two-axis system and let Φ:St^add_ RF⟶ G be an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every r≤ s and every X there is a canonical equivalence Φ_r(Δ^A_s,r(X)) ≃ Cofib ( Φ_r(widehatη^A_s,r,X) ). In particular, if the e","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d0f231c57bed48c8","tex":"\\Phi_s","aliases":"Φ_s Phi","display":"\\Phi_s","kind":"use","excerpt":"mathrmadd_ RF⟶ G be an exact natural transformation which intertwines the chosen transformation families and exchange cells. Then for every r≤ s and every X there is a canonical equivalence Φ_r(Δ^A_s,r(X)) ≃ Cofib ( Φ_r(widehatη^A_s,r,X) ). In particular, if the exchange data in G are denoted by η^B, then Φ_r(Δ^A_s,r(X)) ≃ Δ^B_s,r(Φ_sX).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2e32874fc399e98b","tex":"\\chi_M","aliases":"χ_M chi","display":"\\chi_M","kind":"use","excerpt":"For every perfect widehat B-complex M, the natural comparison χ_M:M⊗^ L_widehat Bwidehat B[1/T] ⟶ Rvarprojlim_n (M⊗^ L_widehat BB_n[1/T]) has fiber Fib(χ_M)≃ ⊕_p∈ SM⊗^ L_widehat B Q_p. Thus χ_M is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by [1] of that fiber. These statements take place in D(wideha","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.","count":15},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"[Finitary iteration] Assume each constructor is given by a fixed input/output relation with finitely many inputs, independent of the stage A. Then Φ preserves increasing unions of ω-chains. Then Z(Σ_0) = ⋃_n<ωΨ^n(∅).","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b790219aab282d57","tex":"\\Omega_{a,*}","aliases":"Ω_a,* Omega","display":"\\Omega_{a,*}","kind":"use","excerpt":"modp^a for every i. There is no restriction in this display when a=0. Define the normalized matrices B_i=fracA_i-λ_i Ip^a ∈ M_d(R_n-a). This is well-defined from the given finite data. It does not invert p inside R_n. Assume that, for some distinguished index *, the matrix B_* is cyclic modulo p. Define the finite obstruction record Ω_a,*(ρ_n)= ([B","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f9504e3b7e0fef2c","tex":"\\Omega_{\\mathrm{arith}}","aliases":"Ω_arith Omega mathrm","display":"\\Omega_{\\mathrm{arith}}","kind":"use","excerpt":"Interpretation. This reframes a possible route to local-global reciprocity as: identify the globalization obstruction and prove that local correspondence transports it. If Λ_*(Ω_arith)=Ω_aut, and Λ_* preserves and reflects the zero object, then vanishing of one obstruction is equivalent to vanishing of the other. An equivalence, or a conservative e","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-60a1fec64d4d80ad","tex":"\\Omega_{\\mathrm{aut}}","aliases":"Ω_aut Omega mathrm","display":"\\Omega_{\\mathrm{aut}}","kind":"use","excerpt":"Interpretation. This reframes a possible route to local-global reciprocity as: identify the globalization obstruction and prove that local correspondence transports it. If Λ_*(Ω_arith)=Ω_aut, and Λ_* preserves and reflects the zero object, then vanishing of one obstruction is equivalent to vanishing of the other. An equivalence, or a conservative e","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-730cdf1a7eedec10","tex":"\\Omega_G","aliases":"Ω_G Omega","display":"\\Omega_G","kind":"use","excerpt":"A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-acf627bb291a6421","tex":"\\Omega_R","aliases":"Ω_R Omega","display":"\\Omega_R","kind":"use","excerpt":"A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-56844c9c83b9ec76","tex":"\\Omega_T","aliases":"Ω_T Omega","display":"\\Omega_T","kind":"use","excerpt":"A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-95df3c3efc737d6a","tex":"\\Omega_{TR}","aliases":"Ω_TR Omega","display":"\\Omega_{TR}","kind":"use","excerpt":"A first taxonomy is: Ω=(Ω_R,Ω_T,Ω_TR,Ω_G) with: item Ω_R: no admissible refinement exists; item Ω_T: a required within-resolution transformation does not exist; item Ω_TR: exchange cannot be made coherent or its residual persists; item Ω_G: locally compatible data fail to reconstruct globally.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-53c979919223766b","tex":"\\ell_O","aliases":"ℓ_O ell","display":"\\ell_O","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9ec711e428117ead","tex":"\\twoheadrightarrow","aliases":"↠ twoheadrightarrow","display":"\\twoheadrightarrow","kind":"use","excerpt":"[A profinite lift need not preserve the finite quotient] Let p be odd and J∈ M_d( Z_p) be invertible modulo p. The action C_p→GL_d( Z/p^2) given by I+pJ has no integral lift as an action of C_p. Its pullback along Z_p↠ C_p, however, has a continuous integral lift ρ(x)=(I+pJ)^x =∑_j≥0binomxjp^jJ^j, x∈ Z_p. For each j, the polynomial binomxj is conti","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"the identity in these coordinates. On H^-1 it is multiplication by p^m-n. The negative-cohomology system is therefore pro-zero, although every one of its terms is nonzero. The completed complex is C_ Z_p(p^a), H^-1=0, H^0= Z/p^a. Represent H^-1 by the annihilator p^n-aR_n and identify Z/p^a with it by x↦ p^n-ax. Reduction of p^m-ax gives p^n-ap^m-n","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"The evaluation map on degree-zero invariants is the inclusion pM↪ M. On tensoring with k its degree-zero map is zero. Evaluation on H^0(V)=(k^d)^G=k^d is an isomorphism. Naturality of evaluation defining β therefore proves H^0(β)=0. The long exact cohomology sequence of its cofiber now gives k^d in every degree: the groups in negative degrees come ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"[Cohomological identification of the symmetry residual] There is a natural exact sequence 0⟶ M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d8081390594d54f0","tex":"\\partial_n","aliases":"∂_n partial","display":"\\partial_n","kind":"use","excerpt":"[Cohomological identification of the symmetry residual] There is a natural exact sequence 0⟶ M^G/p^nM^G⟶(M/p^nM)^G xrightarrow∂_nH^1(G,M)[p^n]⟶0. For a fixed residue class bar m and a lift m∈ M, the obstruction is represented by the crossed homomorphism c_bar m(g)=fracgm-mp^n. It vanishes exactly when bar m has a G-invariant lift in M. The group H^","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-50d049012fa75c98","tex":"\\prod_{e\\in J\\setminus I}","aliases":"∏_e∈ J∖ I prod in setminus","display":"\\prod_{e\\in J\\setminus I}","kind":"use","excerpt":"Fix a finite poset P and assign a positive integer n_e to every event e∈ P. For M∈Perf( Z) define X_M(I)=M, X_M(I⊆ J)=(∏_e∈ J∖ In_e)id_M. Empty products are 1. For nested ideals these products multiply exactly, so X_M is a strict chain-level diagram followed by the stable realization. When P=P_W, take identity transformation and coarsening functors","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e11c0489ef159380","tex":"\\prod_{e\\in P}","aliases":"∏_e∈ P prod in","display":"\\prod_{e\\in P}","kind":"use","excerpt":"comparison after an unrelated event f is multiplication by n_f on that complex. Consequently Θ_e,f(I)≃ M⊗^ L_ Z C_ Z(n_e)⊗^ L_ ZC_ Z(n_f). The complexes C_ Z(n) are bounded and free, so ordinary tensor complexes compute these derived tensors. The total normalization residual is instead F_X_M(P)≃ M⊗^ L_ ZC_ Z(N), N=∏_e∈ Pn_e. This is not the total c","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ae5d80b648ebc895","tex":"\\prod_{g\\in G}","aliases":"∏_g∈ G prod in","display":"\\prod_{g\\in G}","kind":"use","excerpt":"More precisely, for d_G:M⟶∏_g∈ GM, d_G(m)=(gm-m)_g, the torsion subgroup of cokerd_G is canonically H^1(G,M). Therefore the sharp bound is computed by [source reference], and can be strictly smaller than e. The formula for c satisfies c(gh)=g c(h)+c(g), because this identity holds before division by p^n in the torsion-free module M. Changing m by p","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"ngleftrightarrow Ω_a,*(ρ_n)=0. If the obstruction vanishes, an integral lift is constructed from finite polynomial certificates. Express B_i=f_i(B_*) in R_n-a, with deg f_i<d and f_*(t)=t when d>1. Choose an integral lift widehat B of B_* and integer lifts widehat f_i of these polynomials, taking the distinguished generator itself to widehat B. The","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-849be21d5ad10175","tex":"\\prod_n","aliases":"∏_n prod","display":"\\prod_n","kind":"use","excerpt":"Degreewise transition maps of the finite free modules are surjective. In the product model of the homotopy inverse limit, the map 1-shift:∏_n M_n^i⟶∏_n M_n^i is surjective in each degree: choose an initial entry and successively lift the required difference across the surjective transition map. Its kernel is the ordinary inverse limit. Thus the hom","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-92099f991bb443f2","tex":"\\prod_{p\\in\\mathcal P}","aliases":"∏_p∈ P prod in mathcal","display":"\\prod_{p\\in\\mathcal P}","kind":"use","excerpt":"Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. Th","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-76e758f1be5158a2","tex":"\\sum_g","aliases":"∑_g sum","display":"\\sum_g","kind":"use","excerpt":"vρ(g)pmodp^s (g∈ G), s>e. Then T=frac1|G|∑_g∈ Gσ(g)ρ(g)^-1 ∈ I+p^s-eM_d( Z_p) is invertible and satisfies σ(g)T=Tρ(g), σ(g)=Tρ(g)T^-1. Every summand is congruent to I modulo p^s. Its average is therefore I modulo p^s-e, so it is integral and invertible. For h∈ G, σ(h)Tρ(h)^-1 =frac1|G|∑_gσ(hg)ρ(hg)^-1=T, by reindexing the finite sum. This is the re","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6d39fe36bb310cf9","tex":"\\sum_{g\\in G}","aliases":"∑_g∈ G sum in","display":"\\sum_{g\\in G}","kind":"use","excerpt":"For completeness, let c:G→ M be a cocycle and b=∑_g∈ Gc(g). For u∈ G, ub=b-|G|c(u), |G|c(u)=u(-b)-(-b). Thus |G|[c]=0. The cocycle and coboundary modules are subquotients of finite free O-modules, so H^1 is finitely generated and hence finite when annihilated by this nonzero integer.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-07bc3bef47a9c2f3","tex":"\\sum_{i=0}^{q-1}","aliases":"∑_i=0^q-1 sum","display":"\\sum_{i=0}^{q-1}","kind":"use","excerpt":"[Cyclic relation Jacobian] Let G=C_q for an integer q≥1, and let A∈GL_d( Z/p^n) satisfy A^q=I. Choose a lift widetilde A modulo p^n+1 and set c=fracwidetilde A^q-Ip^npmod p, L_bar A(B)=∑_i=0^q-1bar A^iBbar A^q-1-i. Then the lifts of the cyclic action to level n+1 are exactly the matrices widetilde A+p^nB such that L_bar A(B)=-c. If the affine solut","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6182c3bad2d01e16","tex":"\\sum_{i=1}^r","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^r","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-89330e1a143912f2","tex":"\\sum_{j\\geq0}","aliases":"∑_j≥0 sum geq","display":"\\sum_{j\\geq0}","kind":"use","excerpt":"[A profinite lift need not preserve the finite quotient] Let p be odd and J∈ M_d( Z_p) be invertible modulo p. The action C_p→GL_d( Z/p^2) given by I+pJ has no integral lift as an action of C_p. Its pullback along Z_p↠ C_p, however, has a continuous integral lift ρ(x)=(I+pJ)^x =∑_j≥0binomxjp^jJ^j, x∈ Z_p. For each j, the polynomial binomxj is conti","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-92c8f645305eec8c","tex":"\\sum_{t=0}^{p-1}","aliases":"∑_t=0^p-1 sum","display":"\\sum_{t=0}^{p-1}","kind":"use","excerpt":"Z) nonzero modulo p, and A=I+pJ modulo p^3. Then A^p=I+p^2J and A^p^2=I modulo p^3. The representation of C_p modulo p^2 uses powers of A. For representatives 0≤ a,b<p, choose A_a=A^a at the next level. Its cocycle is c(a,b)=lfloorfraca+bprfloor Jpmod p. The adjoint residual action is trivial. For any normalized cochain u on C_p, ∑_t=0^p-1(δ u)(1,t","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-760b54cd8399597d","tex":"\\sum_{t\\in G}","aliases":"∑_t∈ G sum in","display":"\\sum_{t\\in G}","kind":"use","excerpt":"[Averaged correction with controlled precision loss] If m>e, define in M_d( Q_p) b(g)=frac1N∑_t∈ GE(g,t), A'_g=(I-b(g))A_g. Then b(g)∈ p^m-eM_d( Z_p), b(1)=0, and A' is a normalized map into GL_d( Z_p). Its defect satisfies A'_gA'_h(A'_gh)^-1-I∈ p^2m-2eM_d( Z_p). Also A'_g-A_g∈ p^m-eM_d( Z_p). Division by N=p^eu, u∈ Z_p^×, loses at most e digits, p","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6f333436e7d549b7","tex":"\\ast","aliases":"∗ ast","display":"\\ast","kind":"use","excerpt":"The seed calculus can be interpreted as a search grammar. A machine state may contain S= (Σ,mathscr T,mathscr R,Δ,Ω,Cert). For a target problem P, search can occur jointly over transformation and resolution: (T^∗,R^∗) ∈ *arg min_T,R C(Ω(T,R;P)), where C is a problem-dependent residual or obstruction cost. This expression describes a search objectiv","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ff5308563a6a5576","tex":"\\circ_{\\K}","aliases":"∘_K ∘_ K circ K circ mathbf","display":"\\circ_{{\\mathbf K}}","kind":"use","excerpt":"Given (r,X)xrightarrowα(s,Y) and (s,Y)xrightarrowβ(t,Z), define β∘_ Kα := C_s→ r(β)∘α: X→ C_t→ r(Z).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"[Elementary excision implies ideal excision] If every elementary independent-event square of X:NJ(P)→ S is a homotopy pushout, then for all ideals I,J the square X_I∩ Jarrow[r]arrow[d] · X_Iarrow[d] X_Jarrow[r] · X_I∪ J is a homotopy pushout.","count":5},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"[Total Komorebi category] Given F: R^op→Cat, define the category K:=∫_ RF as follows.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-aefc0891cbb293fc","tex":"\\int_{\\Rcat}","aliases":"∫_Rcat ∫_ R int Rcat int mathbf","display":"\\int_{{\\mathbf R}}","kind":"use","excerpt":"[Total Komorebi category] Given F: R^op→Cat, define the category K:=∫_ RF as follows.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6893de4c4fc1bc6a","tex":"\\sim_s","aliases":"∼_s sim","display":"\\sim_s","kind":"use","excerpt":"[Resolution monotonicity] Let r≤ s≤ t and X,Y∈ T_t. If X∼_sY, then X∼_rY.","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"[A one-step representation-lifting obstruction] In [source reference], the representation of C_p on ( Z/p^2)^d has no lift to a representation on ( Z/p^3)^d with that reduction. More explicitly, every matrix lift widetilde A obeys widetilde A^p-I≡ p^2Jnot≡0pmodp^3. This obstruction is unchanged by the choice of the lift. There is also no finite fre","count":9},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"rime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomology","count":11},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"Let m=min{ν(X,Y),ν(Y,Z)}. For every n≤ m, we have X∼_nY and Y∼_nZ. Since ∼_n is an equivalence relation, X∼_nZ. Hence ν(X,Z)≥ m. Therefore 2^-ν(X,Z) ≤ 2^-m = max{2^-ν(X,Y),2^-ν(Y,Z)}. The identity and symmetry axioms are immediate, and separation gives positivity for distinct states. Without separation the same formula, with 2^-∞=0, is an ultrapseu","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"coherent exchange-event diagram determines a functor F_X:NJ(P)⟶ S, F_X(I)=Cofib(X_∅⟶ X_I), with the following properties: item F_X(∅)≃0; item along every maximal chain of ideals it restricts to the residual filtration of [source reference]; item for every cover I⊂ I∪{e} there is a cofiber sequence F_X(I)⟶ F_X(I∪{e}) ⟶Cofib(X_I→ X_I∪{e}); item its t","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"[Finite split residual expansion] Fix a finite principal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then W(s)≅ W_nf(s)⊕⊕_i=1^m E_i(s), where E_i(s) is the surrounding context applied to the source-side remainder of step i. In the","count":7},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-283e464cfc941be2","tex":"\\bigoplus_{e\\in P}","aliases":"⊕_e∈ P bigoplus in","display":"\\bigoplus_{e\\in P}","kind":"use","excerpt":"[Independent events split, dependencies retain extensions] Under [source reference], if P is an antichain then F_X(P)≃⊕_e∈ Pd_e. For a general event poset, a direct-sum formula does not follow.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-15ff689d76e6fe95","tex":"\\bigoplus_{i=1}^m","aliases":"⊕_i=1^m bigoplus","display":"\\bigoplus_{i=1}^m","kind":"use","excerpt":"[Finite split residual expansion] Fix a finite principal normalization path for a well-typed mixed word. Assume every step has a chosen split residual exchange and every surrounding operator preserves finite direct sums. Then W(s)≅ W_nf(s)⊕⊕_i=1^m E_i(s), where E_i(s) is the surrounding context applied to the source-side remainder of step i. In the","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5c779b0b534e7ac8","tex":"\\bigoplus_{i=1}^r","aliases":"⊕_i=1^r bigoplus","display":"\\bigoplus_{i=1}^r","kind":"use","excerpt":"[Residual profile and sharp finite-resolution lifting] Let v=rank_O V. Then Q[p^n]≅⊕_i=1^r O/p^min(a_i,n)O, ℓ_O(K_n)=n(v-r)+∑_i=1^rmin(a_i,n). The splittings require chosen Smith bases and are not canonical. For every n≥1, im(K_n+h⟶ K_n) =im(K/p^nK⟶ K_n). Thus a particular solution at precision n is the reduction of an exact p-adic solution if and ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-43b97d47b6ecc1ba","tex":"\\bigoplus_{j=0}^{m-1}","aliases":"⊕_j=0^m-1 bigoplus","display":"\\bigoplus_{j=0}^{m-1}","kind":"use","excerpt":"[Integral interaction is supported at the gcd] For positive integers a_1,…,a_m, m≥1, put g=gcd(a_1,…,a_m). Then K(a_1,…,a_m):= ⊗_i=1^m C_ Z(a_i) ≃⊕_j=0^m-1 C_ Z(g)[j]^⊕binomm-1j. The displayed splitting depends on a unimodular change of basis and is not claimed canonical. Therefore H^-jK(a_1,…,a_m)≅ ( Z/g)^⊕binomm-1j (0≤ j≤ m-1), and every other co","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d19496472fff50fc","tex":"\\bigoplus_{j=0}^{t-1}","aliases":"⊕_j=0^t-1 bigoplus","display":"\\bigoplus_{j=0}^{t-1}","kind":"use","excerpt":"[Prime-power interaction cohomology] Let R= Z/p^n, let a_1,…,a_t be positive integers with t≥1, and put h=min{n,v_p(a_1),…,v_p(a_t)}, Q_h=R/(p^h). Here Q_0=0 and Q_n=R. There is a generally noncanonical chain isomorphism K_R(a_1,…,a_t)≅ ⊕_j=0^t-1C_R(p^h)[j]^⊕binomt-1j. Consequently H^-jK_R(a_1,…,a_t)≅ Q_h^⊕binom tj (0≤ j≤ t), and all other cohomolo","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-88eb8868ad67b0a3","tex":"\\bigoplus_{k=1}^{r_i}","aliases":"⊕_k=1^r_i bigoplus","display":"\\bigoplus_{k=1}^{r_i}","kind":"use","excerpt":"Changing bases in the source and target yields a chain isomorphism Cofib(f_i)≃ Z^b_i-r_i⊕ Z^a_i-r_i[1] ⊕⊕_k=1^r_iC_ Z(d_i,k). Unit diagonal entries contribute contractible complexes. Thus ϵ_i=1 means that a shift of the unit occurs as a direct summand. If two independent cones each have such a summand, their tensor product contains a shift of the u","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-407a0721917a90f9","tex":"\\bigoplus_{p\\in S}","aliases":"⊕_p∈ S bigoplus in","display":"\\bigoplus_{p\\in S}","kind":"use","excerpt":"For every perfect widehat B-complex M, the natural comparison χ_M:M⊗^ L_widehat Bwidehat B[1/T] ⟶ Rvarprojlim_n (M⊗^ L_widehat BB_n[1/T]) has fiber Fib(χ_M)≃ ⊕_p∈ SM⊗^ L_widehat B Q_p. Thus χ_M is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by [1] of that fiber. These statements take place in D(wideha","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-5aaaf24677ceb564","tex":"\\bigotimes","aliases":"⊗ bigotimes","display":"\\bigotimes","kind":"use","excerpt":"[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S, A_i · i∉ S. Its total cofiber, obtained by taking the cofiber in every coordinate, satisfies tcofib(Y)≃ ⊗_i=1^mCofib","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-bfbca902edd26f22","tex":"\\otimes^{\\mathbf L}","aliases":"⊗^ L otimes mathbf","display":"\\otimes^{\\mathbf L}","kind":"use","excerpt":"0 · a_i+b_i-2r_i=0, P_i={p:pmid d_i,k for some k}. Let J be the tensor-invariant thick subcategory generated by Cofib(f_i)⊗^ LCofib(f_j) over the allowed independent pairs. item If an allowed pair has ϵ_i=ϵ_j=1, then J=Perf( Z) and its quotient is zero. item Otherwise the quotient is nonzero and J=Thick{ Z/p:p∈ S}, where S is the union over allowed","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-3bfd19754746e56e","tex":"\\otimes^{\\mathbf L}_{B_k}","aliases":"⊗^ L_B_k otimes mathbf","display":"\\otimes^{\\mathbf L}_{B_k}","kind":"use","excerpt":"orem[Coherent weighted exchange and residual transport] The cells η_L,b;m,n are natural and coherent with composition of resolution changes. Their residual at M∈ S_m is Cofib(η_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-53035163f63e3fbd","tex":"\\otimes^{\\mathbf L}_{B_m}","aliases":"⊗^ L_B_m otimes mathbf","display":"\\otimes^{\\mathbf L}_{B_m}","kind":"use","excerpt":"Fix a finite nonempty set P of primes and put D=∏_p∈ Pp, B_n= Z/D^n Z (n≥1). Write B_p,n= Z/p^n Z. The Chinese remainder isomorphisms B_n≅∏_p∈ PB_p,n are compatible with reduction. Use the enhanced category S_n=Perf(B_n) and the coarsening C_m,n(M)=M⊗^ L_B_mB_n, m≥ n. Derived tensor associativity supplies coherent equivalences C_l,n≃ C_m,nC_l,m. Th","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6547ce75ab4f7558","tex":"\\otimes^{\\mathbf L}_{B_n}","aliases":"⊗^ L_B_n otimes mathbf","display":"\\otimes^{\\mathbf L}_{B_n}","kind":"use","excerpt":"For a fixed perfect integral complex L, put A_L,n(M)=(L⊗^ L_ ZB_n)⊗^ L_B_nM. Base change gives a natural equivalence θ_L;m,n:A_L,nC_m,n≃ C_m,nA_L,m. Thus the canonical tensor/base-change exchange already has zero residual. The tower alone does not force a defect. To test a noninvertible exchange, we explicitly add a positive integer label b and use","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-d6673373de86e1b6","tex":"\\otimes^{\\mathbf L}_R","aliases":"⊗^ L_R otimes mathbf","display":"\\otimes^{\\mathbf L}_R","kind":"use","excerpt":"_L,b;m,n,M)≃ A_L,nC_m,n(M)⊗^ L_B_nC_B_n(b^m-n). For a fixed scalar a, subsequent base change transports its cone exactly: C_n,k(N⊗^ L_B_nC_B_n(a)) ≃ C_n,k(N)⊗^ L_B_kC_B_k(a). At a common target level n, two independent scalar events of positive lengths d,e and labels a,b have interaction N⊗^ L_B_nK_B_n(a^d,b^e), K_R(u,v)=C_R(u)⊗^ L_R C_R(v). These ","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-602c45e2471c8637","tex":"\\otimes^{\\mathbf L}_{\\widehat B}","aliases":"⊗^ L_widehat B otimes mathbf widehat","display":"\\otimes^{\\mathbf L}_{\\widehat B}","kind":"use","excerpt":"For every perfect widehat B-complex M, the natural comparison χ_M:M⊗^ L_widehat Bwidehat B[1/T] ⟶ Rvarprojlim_n (M⊗^ L_widehat BB_n[1/T]) has fiber Fib(χ_M)≃ ⊕_p∈ SM⊗^ L_widehat B Q_p. Thus χ_M is an equivalence exactly when every displayed generic fiber is zero. Its cofiber is the shift by [1] of that fiber. These statements take place in D(wideha","count":2},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-40aa46a044ccf15d","tex":"\\otimes_{\\mathbb Z}","aliases":"⊗_ Z otimes mathbb","display":"\\otimes_{\\mathbb Z}","kind":"use","excerpt":"ite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the unique lattice inside V with these properties and these s","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-752b5dc4f3049dd1","tex":"\\otimes_{\\mathbb Z_p}","aliases":"⊗_ Z_p otimes mathbb","display":"\\otimes_{\\mathbb Z_p}","kind":"use","excerpt":"[Effective arithmetic lattice gluing] Let V be a finite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the uniqu","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f9b87410270e30bb","tex":"\\otimes_A","aliases":"⊗_A otimes","display":"\\otimes_A","kind":"use","excerpt":"[Effective arithmetic lattice gluing] Let V be a finite free A= Z[1/D]-module of rank r. For each p∈ P let L_p be a finite free Z_p-module, with an isomorphism θ_p:V⊗_A Q_pxrightarrow∼ L_p⊗_ Z_p Q_p. Then Λ={v∈ V:θ_p(v)∈ L_p for every p∈ P} is a finite free Z-module of rank r, and the natural maps identify Λ[1/D]≅ V, Λ⊗_ Z Z_p≅ L_p. It is the uniqu","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-61195f8a22af8d91","tex":"\\bigotimes_i","aliases":"⊗_i bigotimes","display":"\\bigotimes_i","kind":"use","excerpt":"For a commutative ring R and a∈ R, set C_R(a)=[RxrightarrowaR], with the source in cohomological degree -1 and the target in degree 0. The tensor product ⊗_i C_R(a_i) is the Koszul complex in this cohomological convention. Its tensor, cone, and change-of-basis properties are classical; see [source reference]. We use them with an explicit sign conve","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-aac07282d0589849","tex":"\\bigotimes_{i=1}^m","aliases":"⊗_i=1^m bigotimes","display":"\\bigotimes_{i=1}^m","kind":"use","excerpt":"[Tensor-cube residual formula] Let S be a symmetric monoidal stable category whose tensor product is exact separately in each variable. For m≥1 and arrows f_i:A_i→ B_i, form the coherent m-cube Y(S)=⊗_i=1^m Z_i(S), Z_i(S)=B_i · i∈ S, A_i · i∉ S. Its total cofiber, obtained by taking the cofiber in every coordinate, satisfies tcofib(Y)≃ ⊗_i=1^mCofib","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-b17ecb191f331b32","tex":"\\otimes_O","aliases":"⊗_O otimes","display":"\\otimes_O","kind":"use","excerpt":"For a map d:V→ W of lattices, put K=ker d, Q=cokerd, K_n=ker(d⊗_O O_n). The module K is saturated in V: if p^n v∈ K, then p^n d(v)=0 in the torsion-free module W, so v∈ K. It is therefore a finite free direct summand as an O-module, although a splitting is not canonical. Reduction supplies a natural injection α_n:K/p^nK→ K_n.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-2001d6605dfd2c6d","tex":"\\otimes_O^{\\mathbf L}","aliases":"⊗_O^ L otimes mathbf","display":"\\otimes_O^{\\mathbf L}","kind":"use","excerpt":"[Lifted-input exchange and a necessary obstruction to lifting] Let O→ R→ S be maps of commutative rings. Assume R and S are perfect as O-modules. For every N∈ D(O[G]), put M=N⊗_O^ LR. Then β_R,S,M:Γ_R(M)⊗_R^ LS xrightarrow∼Γ_S(M⊗_R^ LS). No perfectness of S over R is required. Consequently, if Cofib(β_R,S,M) is nonzero, there is no N∈ D(O[G]) toget","count":3},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-10d64144ae124a95","tex":"\\otimes_R","aliases":"⊗_R otimes","display":"\\otimes_R","kind":"use","excerpt":"For O= Z_p, R= Z/p^m, and S= Z/p^n with m≥ n≥1, this applies at every pair of levels. It applies in particular to reductions of a supplied integral G-lattice. The canonical base-change comparisons obey transitivity. After the tensor associativities, the composite Γ_O(N)⊗_O^ LS · xrightarrowβ_O,R,N⊗_R S Γ_R(N⊗_O^ LR)⊗_R^ LS · xrightarrowβ_R,S,M Γ_S(","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f92c003c879d28f9","tex":"\\otimes_R^{\\mathbf L}","aliases":"⊗_R^ L otimes mathbf","display":"\\otimes_R^{\\mathbf L}","kind":"use","excerpt":"For a ring map R→ S there is a canonical comparison β_R,S,M:Γ_R(M)⊗_R^ LS ⟶Γ_S(M⊗_R^ LS). It is adjoint to base change of the evaluation map from the object of homotopy invariants, regarded as carrying the trivial group action, to M. Its construction is natural in M and uses the actual coefficient map. No exchange weight is selected.","count":8},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"Finally a compatible family of vectors modulo p^n has a unique lift to V, since V is finite free and complete. If every reduction solves the equation, its image under d lies in ⋂_n p^nW=0. This proves the inverse-limit assertion directly, without interchanging an arbitrary kernel and an unspecified completion.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-9c20590cf5841569","tex":"\\bigcap_n","aliases":"⋂_n bigcap","display":"\\bigcap_n","kind":"use","excerpt":"Finally a compatible family of vectors modulo p^n has a unique lift to V, since V is finite free and complete. If every reduction solves the equation, its image under d lies in ⋂_n p^nW=0. This proves the inverse-limit assertion directly, without interchanging an arbitrary kernel and an unspecified completion.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"[Finitary iteration] Assume each constructor is given by a fixed input/output relation with finitely many inputs, independent of the stage A. Then Φ preserves increasing unions of ω-chains. Then Z(Σ_0) = ⋃_n<ωΨ^n(∅).","count":4},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-f000d83f6a26989f","tex":"\\bigcup_{(i,j)\\in\\mathcal I}","aliases":"⋃_(i,j)∈ I bigcup in mathcal","display":"\\bigcup_{(i,j)\\in\\mathcal I}","kind":"use","excerpt":"[Finite-signature arithmetic certification] Fix finitely many positive integer exchange labels n_1,…,n_t and a specified symmetric relation I of label pairs allowed to occur on incomparable events. The relation may contain a diagonal pair. Put S_ I=⋃_(i,j)∈ I {p:pmidgcd(n_i,n_j)}. Killing the finitely many unit-seed interactions K(n_i,n_j) for (i,j","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-be4b537907f424fe","tex":"\\bigcup_{e\\parallel f}","aliases":"⋃_eparallel f bigcup parallel","display":null,"kind":"use","excerpt":"For the fixed labelled poset define the finite prime set S_P=⋃_eparallel f {p prime:pmidgcd(n_e,n_f)}, where eparallel f means incomparable. Let J_P be the invariant thick closure of the square interaction residuals for the unit seed Z.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-6516d4db9b1af250","tex":"\\bigcup_n","aliases":"⋃_n bigcup","display":"\\bigcup_n","kind":"use","excerpt":"For a constructor output from ⋃_n A_n, its finitely many inputs all belong to a common A_N. The output therefore already belongs to Φ(A_N). This proves preservation of increasing unions. The union on the right is a fixed point because Ψ preserves that union. It contains Σ_0 and is contained in every fixed point by induction on n, hence is least.","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7ab7a6432078c75e","tex":"\\bigcup_{n<\\omega}","aliases":"⋃_n<ω bigcup omega","display":"\\bigcup_{n<\\omega}","kind":"use","excerpt":"[Finitary iteration] Assume each constructor is given by a fixed input/output relation with finitely many inputs, independent of the stage A. Then Φ preserves increasing unions of ω-chains. Then Z(Σ_0) = ⋃_n<ωΨ^n(∅).","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"[A central sign hides the obstruction from derived invariants] Let p be odd, R= Z/p^2, and let J∈ M_d( Z) be invertible modulo p. On M=R^d let G=C_p× C_2=⟨σ,τ⟩, σ↦ I+pJ, τ↦-I. For the canonical comparison of [source reference], Γ_R(M)≃0, Γ_k(M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not l","count":1},{"work":"seeded-two-axis","url":"notation/seeded-two-axis.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"[A central sign hides the obstruction from derived invariants] Let p be odd, R= Z/p^2, and let J∈ M_d( Z) be invertible modulo p. On M=R^d let G=C_p× C_2=⟨σ,τ⟩, σ↦ I+pJ, τ↦-I. For the canonical comparison of [source reference], Γ_R(M)≃0, Γ_k(M⊗_R^ Lk)≃0, Cofib(β_R,k,M)≃0. Nevertheless this representation does not lift to Z/p^3, and hence does not l","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Compute P_aP_b=V^-1E_aE_bV=δ_abP_a and ∑_aP_a=V^-1IV=I. In the unitary case the pulled-back projections are orthogonal: P_a^*=P_a. Their images are mutually orthogonal, and SG-01 writes x as their finite orthogonal sum. The Pythagorean identity gives the energy formula. Without unitarity and orthogonality the algebraic identities remain valid while","count":10},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]","count":135},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3b086317f3e9a76c","tex":"\\mathbf1_{[-a,a)}","aliases":"1_[-a,a) mathbf","display":"\\mathbf1_{[-a,a)}","kind":"use","excerpt":"smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-97df433d6f083dab","tex":"\\mathbf1_{[0,T/2)}","aliases":"1_[0,T/2) mathbf","display":"\\mathbf1_{[0,T/2)}","kind":"use","excerpt":"G_T _1=T/4. For a shift by d∈[0,T/2], on an interval of length T-d the difference has magnitude d/T, and on an interval of length d it has magnitude 1-d/T. Hence its L^1 norm is (T-d)d/T+d(1-d/T)=2d(1-d/T). Apply the triangle inequality term by term. Periodic translation invariance makes the calculation depend only on circular shift distance. To re","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8b8ebc68ebab5bbf","tex":"\\mathbf1_{[T/4,T/2)}","aliases":"1_[T/4,T/2) mathbf","display":"\\mathbf1_{[T/4,T/2)}","kind":"use","excerpt":"G_T _1=T/4. For a shift by d∈[0,T/2], on an interval of length T-d the difference has magnitude d/T, and on an interval of length d it has magnitude 1-d/T. Hence its L^1 norm is (T-d)d/T+d(1-d/T)=2d(1-d/T). Apply the triangle inequality term by term. Periodic translation invariance makes the calculation depend only on circular shift distance. To re","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":270},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]","count":35},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.","count":8},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":98},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0014cfc2591bdee6","tex":"a_*","aliases":"a_*","display":"a_*","kind":"use","excerpt":"smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-cf049ba227ada547","tex":"a_{c\\ell}","aliases":"a_cℓ ell","display":"a_{c\\ell}","kind":"use","excerpt":"smallDomain and hypotheses. A finite collection of exponential moment channels. Let m_n^(c)=∑_ℓ=1^ra_cℓz_ℓ^n, with distinct common unit-modulus nodes. At each node at least one channel amplitude is nonzero. Form the stacked linear recurrence system for a monic polynomial of degree r and assume its coefficient matrix has column rank r. [SG-39 Shared","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0bdb6564f1e0d191","tex":"a_{k-1}","aliases":"a_k-1","display":"a_{k-1}","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d9b71db518364d73","tex":"A_M","aliases":"A_M","display":"A_M","kind":"use","excerpt":"smallDomain and hypotheses. A normed signal space. Let f=s+j, let widehat j belong to the same space, and let A_M be a bounded linear operator with norm at most C_M. Define widehat f_M=A_M(f-widehat j)+widehat j. [SG-36 Resolved synthesis splits the two error sources]","count":9},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0aa00d546f91ab74","tex":"\\mathcal A_P","aliases":"A_P mathcal","display":"\\mathcal A_P","kind":"use","excerpt":"P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-9ecba2f3c7ea58ca","tex":"\\mathcal A_Q","aliases":"A_Q mathcal","display":"\\mathcal A_Q","kind":"use","excerpt":"P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a3a52d54c059cb53","tex":"a_\\ell","aliases":"a_ℓ ell","display":"a_\\ell","kind":"use","excerpt":"Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero act","count":20},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-27029f09b14e6d27","tex":"\\arcsin","aliases":"arcsin","display":null,"kind":"use","excerpt":"ly the triangle inequality to the coefficient error polynomial and use |z|≤|z_ℓ|+ρ. On the circle, |z-z_ℓ|=ρ and |z-z_j|≥|z_ℓ-z_j|-ρ, giving the lower product bound. For unit true nodes and a recovered root inside a disk with ρ<1, radial normalization changes its chord error to at most 2ρ by the triangle inequality. Its angular location error is th","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4e2b11a7001e1904","tex":"\\arg","aliases":"arg","display":"\\arg","kind":"use","excerpt":"The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"Insert I=∑_aP_a on both sides of T to obtain reconstruction. Insert that identity between S and T to obtain the product formula. Equivalently, the block map is T↦ A T S; SG-01 shows that its inverse is B↦ S B A and that it preserves multiplication. The path formula follows by inserting the same identity between every successive pair of operators. T","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":48},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors. For the amplitude use widehat a=widehat b_0 overl","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","excerpt":"Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-037c6283d361171d","tex":"b_M","aliases":"b_M","display":"b_M","kind":"use","excerpt":"smallDomain and hypotheses. The rate family of SG-35 with T=2π. Assume a fixed finite jump count, bounded jump amplitudes, the hypotheses of SG-35, and a bounded linear smooth reconstruction A_M acting on L^1( T). Require widehat j to be the canonical packet of real matched estimated amplitudes whose sum is zero. Let b_M= s-A_Ms _1 and C_M= A_M _L^","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-87d9443f3d2b57ff","tex":"b_{\\min}","aliases":"b_min min","display":"b_{\\min}","kind":"use","excerpt":"smallDomain and hypotheses. A selected sequence of decimation scales. Let p≥0, fix q_0 and the finite number of channel samples and offsets, and take a sequence N→∞. Assume distinct aliased nodes, uniform rank-threshold gaps/local inverse bounds for the aligned Prony and coefficient fits, |b_ℓ,0|≥ b_min>0, uniformly bounded amplitudes, and the stri","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f4514ffb4ecf9484","tex":"b_{\\ell,0}","aliases":"b_ℓ,0 ell","display":"b_{\\ell,0}","kind":"use","excerpt":"All high indices exceed K, so the derivative moments have no smooth defect. The base channel is an exponential sum in the distinct u_ℓ with coefficients b_ℓ,0=a_ℓ u_ℓ^q_0≠0. Its 2L_max+1 moments form an (L_max+1) square Hankel matrix; SG-19 gives its rank r. If r=0, the edge is zero and the low modes finish the recovery. Otherwise SG-20 recovers th","count":8},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-aa33051e8b53fedf","tex":"b_{\\ell,1}","aliases":"b_ℓ,1 ell","display":"b_{\\ell,1}","kind":"use","excerpt":"0≠0. Its 2L_max+1 moments form an (L_max+1) square Hankel matrix; SG-19 gives its rank r. If r=0, the edge is zero and the low modes finish the recovery. Otherwise SG-20 recovers the u_ℓ and then the base coefficients. The adjacent channel supplies at least r moments of ∑_ℓ b_ℓ,0z_ℓ u_ℓ^q. Using the same recovered nodes, its Vandermonde system reco","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-eeeb318e1a4d1021","tex":"b_{\\ell,r}","aliases":"b_ℓ,r ell","display":"b_{\\ell,r}","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fc4f303c22e1fb30","tex":"b_{\\ell,r+1}","aliases":"b_ℓ,r+1 ell","display":"b_{\\ell,r+1}","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.","count":29},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":33},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2fe8f48453df0204","tex":"\\mathbb C^r","aliases":"C^r mathbb","display":"\\mathbb C^r","kind":"use","excerpt":"Expand a matrix entry to obtain the factorization. The leading r× r Vandermonde submatrix has determinant ∏_ℓ<k(z_k-z_ℓ), which is nonzero. Thus both rectangular Vandermonde factors have full column rank, and their transposes have full row rank. The diagonal factor is invertible. Its product with V_w^T is surjective onto C^r, and subsequent multipl","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"S(c)=R|b|^2z≠0, Φ(c):=S(c)/|S(c)|=z. Two channels already give z=c_1/c_0 exactly. More channels provide repeated adjacent relations, without an unconditional deterministic averaging gain.","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"S(c)=R|b|^2z≠0, Φ(c):=S(c)/|S(c)|=z. Two channels already give z=c_1/c_0 exactly. More channels provide repeated adjacent relations, without an unconditional deterministic averaging gain.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-62ce335ae3e26f33","tex":"C_M","aliases":"C_M","display":"C_M","kind":"use","excerpt":"smallDomain and hypotheses. A normed signal space. Let f=s+j, let widehat j belong to the same space, and let A_M be a bounded linear operator with norm at most C_M. Define widehat f_M=A_M(f-widehat j)+widehat j. [SG-36 Resolved synthesis splits the two error sources]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-25b07618d510bdea","tex":"C_q","aliases":"C_q","display":"C_q","kind":"use","excerpt":"For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"use","excerpt":"smallDomain and hypotheses. One matched unit-circle coefficient progression. Let R≥1 be an integer counting adjacent products, and let c_r=bz^r for 0≤ r≤ R, with b≠0 and |z|=1. Define S(c)=∑_r=0^R-1overlinec_rc_r+1. [SG-29 Exact phase from adjacent offset products]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c53ca1b04cb9878c","tex":"c_{r+1}","aliases":"c_r+1","display":"c_{r+1}","kind":"use","excerpt":"smallDomain and hypotheses. One matched unit-circle coefficient progression. Let R≥1 be an integer counting adjacent products, and let c_r=bz^r for 0≤ r≤ R, with b≠0 and |z|=1. Define S(c)=∑_r=0^R-1overlinec_rc_r+1. [SG-29 Exact phase from adjacent offset products]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-307d2c7e46d5ba73","tex":"C_s","aliases":"C_s","display":"C_s","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8ec5cf28b1acacd8","tex":"c_\\ell","aliases":"c_ℓ ell","display":"c_\\ell","kind":"use","excerpt":"smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]","count":28},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":51},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5f1d3c3e85a0d9d7","tex":"d_{\\mathbb T}","aliases":"d_ T mathbb","display":"d_{\\mathbb T}","kind":"use","excerpt":"identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord di","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8ccc55c01d02322b","tex":"D_F","aliases":"D_F","display":"D_F","kind":"use","excerpt":"smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-34e87a8bf13b452f","tex":"D_F^{-1}","aliases":"D_F^-1","display":"D_F^{-1}","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-da868d3c0ad1089a","tex":"D_F^{-1/2}","aliases":"D_F^-1/2","display":"D_F^{-1/2}","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f1be8e49b81c9b1e","tex":"D_M","aliases":"D_M","display":"D_M","kind":"use","excerpt":"Direct integration gives widehat j(0)=a/π and widehat j(n)=sin(na)/(π n) for n≠0. The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by sum","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d57ec070569225e9","tex":"d_T","aliases":"d_T","display":"d_T","kind":"use","excerpt":"smallDomain and hypotheses. Mean-zero periodic packets of period T>0. Let j,widehat j have matched real jump data (a_ℓ,τ_ℓ) and (widehat a_ℓ,widehatτ_ℓ), with both amplitude sums zero. Use their canonical representatives and let d_ℓ=d_T(τ_ℓ,widehatτ_ℓ)≤ T/2. [SG-37 Sharp additive L1 control for normalized jump packets]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0a1ac0d65e3855ed","tex":"d_\\ell","aliases":"d_ℓ ell","display":"d_\\ell","kind":"use","excerpt":"smallDomain and hypotheses. Mean-zero periodic packets of period T>0. Let j,widehat j have matched real jump data (a_ℓ,τ_ℓ) and (widehat a_ℓ,widehatτ_ℓ), with both amplitude sums zero. Use their canonical representatives and let d_ℓ=d_T(τ_ℓ,widehatτ_ℓ)≤ T/2. [SG-37 Sharp additive L1 control for normalized jump packets]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"smallDomain and hypotheses. Polynomial-exponential sequences. Let m_n=∑_ℓ=1^rz_ℓ^nP_ℓ(n) with distinct nonzero nodes and deg P_ℓ≤ d. Let E be the forward shift. For an offset ratio assume N≥1 and the leading coefficient of P_ℓ at degree d is nonzero. [SG-40 Confluent annihilation and the leading offset phase]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"The singular-value perturbation inequality [source reference] gives |σ_j(H)-σ_j(H_0)|≤ E _2 for 1≤ j≤ k. It holds equally over the complex field: in the min--max characterization σ_j(A)=min_dim V=w-j+1max_x∈ V, x =1 Ax . the triangle inequality bounds each restricted maximum for A+E by that for A plus E _2. Minimize and then interchange the two mat","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-6bdeb30989e12919","tex":"\\operatorname{dist}","aliases":"dist operatorname","display":"\\operatorname{dist}","kind":"use","excerpt":"For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces with an invertible transform. Let V:X→ Y be an isomorphism and (E_a) a finite complete packet family on Y. [SG-04 Pulled-back transform packets]","count":26},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":38},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-48a4c36be6fe9d66","tex":"e^{-inx}","aliases":"e^-inx","display":"e^{-inx}","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-38b24c572b5b52a1","tex":"e^{-in\\delta}","aliases":"e^-inδ delta","display":"e^{-in\\delta}","kind":"use","excerpt":"Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c2f4d3b9fc71f81b","tex":"e^{-in\\pi}","aliases":"e^-inπ pi","display":"e^{-in\\pi}","kind":"use","excerpt":"Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e33dd075b7e56c15","tex":"e^{-in\\tau_\\ell}","aliases":"e^-inτ_ℓ tau ell","display":"e^{-in\\tau_\\ell}","kind":"use","excerpt":"For a smooth periodic test function φ, integrate by parts over each constant segment of j. Boundary terms telescope at each jump to a_ℓφ(τ_ℓ), which is exactly the displayed distribution. Evaluate the distribution on e^-inx: ⟨ Dj,e^-inx⟩ =-∫ j(x)(-in)e^-inx dx =2π inwidehat j(n). The Dirac sum gives ∑_ℓ a_ℓ e^-inτ_ℓ. For n>K, the smooth polynomial ","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a01743648714134d","tex":"e^{-it}","aliases":"e^-it","display":"e^{-it}","kind":"use","excerpt":"Every circular distance from a point of {0,δ} to a point of {π,π+δ} is at least π-δ. This bounds every matching. The moments of the first packet are 1-e^-inδ; translating by π multiplies them by e^-inπ. Subtraction gives the formula. The elementary inequality |1-e^-it|≤|t| proves the bound. Every fixed finite norm of the measurement difference tend","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-414c1b7819d9bdd4","tex":"e^{-i\\tau_\\ell}","aliases":"e^-iτ_ℓ tau ell","display":"e^{-i\\tau_\\ell}","kind":"use","excerpt":"Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero act","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-03f1748d577b13ea","tex":"e^{-\\rho}","aliases":"e^-ρ rho","display":"e^{-\\rho}","kind":"use","excerpt":"The reconstruction error is exactly f- R_Mf=s-S_Ms, and f- R_Mf _∞ ≤frac2Ce^-ρ(M+1)1-e^-ρ. The bound is uniform across the jumps of j.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f56848451617b887","tex":"e^{-\\rho n}","aliases":"e^-ρ n rho","display":"e^{-\\rho n}","kind":"use","excerpt":"Cancel j from the defining identity. Absolute summability of the Fourier coefficients gives an everywhere uniformly convergent series for s. The omitted positive and negative frequencies are bounded by 2C∑_n>Me^-ρ n, which is the displayed geometric series. No limiting trigonometric approximation of j occurs in this estimate.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-7e3b3edf7d4da5f0","tex":"e^{-\\rho(M+1)}","aliases":"e^-ρ(M+1) rho","display":"e^{-\\rho(M+1)}","kind":"use","excerpt":"The reconstruction error is exactly f- R_Mf=s-S_Ms, and f- R_Mf _∞ ≤frac2Ce^-ρ(M+1)1-e^-ρ. The bound is uniform across the jumps of j.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fcb4e80d5433e71d","tex":"e^{-\\rho|n|}","aliases":"e^-ρ|n| rho","display":"e^{-\\rho|n|}","kind":"use","excerpt":"smallDomain and hypotheses. Periodic signals with a supplied exact edge packet. Let f=s+j with |widehat s(n)|≤ Ce^-ρ|n|, where C≥0 and ρ>0. Use the canonical uniformly convergent Fourier-series representative of s. Set R_Mf=S_Ms+j. [SG-11 Resolved error for an analytic background]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c1978f8f0edde835","tex":"e^{2ia}","aliases":"e^2ia","display":"e^{2ia}","kind":"use","excerpt":"_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia);","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d96c3edc7cd9e445","tex":"e^{2ina}","aliases":"e^2ina","display":"e^{2ina}","kind":"use","excerpt":"The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-bd2f094fd41a0b0d","tex":"e^{ikt}","aliases":"e^ikt","display":"e^{ikt}","kind":"use","excerpt":"The averaging kernel is F_M(t)=frac1M+1|∑_k=0^Me^ikt|^2≥0, frac12π∫_0^2πF_M(t) dt=1. Its convolution with j is therefore a weighted mean of numbers in [0,1]. The resulting finite trigonometric polynomial is continuous. A nonconstant interval indicator is discontinuous, so the mean cannot equal its selected representative at all points or have its e","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-69658308338b50e3","tex":"e_{(q_0+q)N+r}","aliases":"e_(q_0+q)N+r","display":"e_{(q_0+q)N+r}","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d9b6193b1833f637","tex":"E_a","aliases":"E_a","display":"E_a","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces with an invertible transform. Let V:X→ Y be an isomorphism and (E_a) a finite complete packet family on Y. [SG-04 Pulled-back transform packets]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-9a603588606ce810","tex":"E_b","aliases":"E_b","display":"E_b","kind":"use","excerpt":"Compute P_aP_b=V^-1E_aE_bV=δ_abP_a and ∑_aP_a=V^-1IV=I. In the unitary case the pulled-back projections are orthogonal: P_a^*=P_a. Their images are mutually orthogonal, and SG-01 writes x as their finite orthogonal sum. The Pythagorean identity gives the energy formula. Without unitarity and orthogonality the algebraic identities remain valid while","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"The first statement is the defining subtraction identity. Retaining r can carry every component omitted from q, so its data cost is not bounded by a small number of packet coefficients. For an image f∈ R^m, the standard pixel vectors satisfy f=∑_j=1^mf_je_j and form a complete finite basis. Those coefficients are all the pixels themselves. A progra","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"Dj=∑_ℓ a_ℓδ_τ_ℓ, m_n:=2π inwidehat f(n)=∑_ℓ=1^ra_ℓ z_ℓ^n. For a non-bandlimited integrable background the same formula holds with an additive defect e_n=2π inwidehat s(n).","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-dd4edbcb1e1d0e3d","tex":"E_Q","aliases":"E_Q","display":"E_Q","kind":"use","excerpt":"smallDomain and hypotheses. Monic complex polynomials of the same degree. Let r≥1 and Q(z)=∏_ℓ=1^r(z-z_ℓ) have distinct roots and widetilde Q=Q+E_Q. Choose 0<ρ<tfrac12min_ℓ≠ k|z_ℓ-z_k| when r≥2, or any ρ>0 when r=1. [SG-25 Conditional simple-root disk localization]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"smallDomain and hypotheses. A noisy progression as in SG-29. Let c_r=bz^r+e_r for 0≤ r≤ R, with R≥1, b≠0, |z|=1, and |e_r|≤ϵ. Put A=R|b|^2 and Δ=R(2|b|ϵ+ϵ^2). Assume Δ<A. [SG-30 Deterministic offset-phase perturbation]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-7b243e274b6d1702","tex":"e_{r+1}","aliases":"e_r+1","display":"e_{r+1}","kind":"use","excerpt":"Expand each product: overlinec_rc_r+1=|b|^2z +overlinebz^re_r+1+overlinee_rbz^r+1 +overlinee_re_r+1. The three error terms have total magnitude at most 2|b|ϵ+ϵ^2. Summing gives S(c)=Az+E with |E|≤Δ. Therefore |S(c)|≥ A-Δ>0. For any nonzero v and unit z, |frac v|v|-z| ≤|frac v|v|-frac vA|+|frac vA-z| =frac|A-|v||A+frac|v-Az|A ≤frac2|v-Az|A. Apply th","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-70a9953dd9bd9b22","tex":"\\operatorname{End}","aliases":"End operatorname","display":"\\operatorname{End}","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]","count":17},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":32},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-bb2cd4a0f81f2dea","tex":"\\mathbb F^m","aliases":"F^m mathbb","display":"\\mathbb F^m","kind":"use","excerpt":"smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-9d8d0e639654813f","tex":"\\mathbb F^{m\\times d}","aliases":"F^m× d mathbb times","display":"\\mathbb F^{m\\times d}","kind":"use","excerpt":"smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","excerpt":"If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-54b44dec26d2ba54","tex":"F_b","aliases":"F_b","display":"F_b","kind":"use","excerpt":"P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"use","excerpt":"The first statement is the defining subtraction identity. Retaining r can carry every component omitted from q, so its data cost is not bounded by a small number of packet coefficients. For an image f∈ R^m, the standard pixel vectors satisfy f=∑_j=1^mf_je_j and form a complete finite basis. Those coefficients are all the pixels themselves. A progra","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f0160f5b8a57192c","tex":"F_M","aliases":"F_M","display":"F_M","kind":"use","excerpt":"The averaging kernel is F_M(t)=frac1M+1|∑_k=0^Me^ikt|^2≥0, frac12π∫_0^2πF_M(t) dt=1. Its convolution with j is therefore a weighted mean of numbers in [0,1]. The resulting finite trigonometric polynomial is continuous. A nonconstant interval indicator is discontinuous, so the mean cannot equal its selected representative at all points or have its e","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….","count":8},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-07ae6e95d4df7899","tex":"G_T","aliases":"G_T","display":"G_T","kind":"use","excerpt":"smallDomain and hypotheses. A period of length T>0. Let distinct locations τ_ℓ∈ R/(T Z) and real amplitudes a_ℓ satisfy ∑_ℓ a_ℓ=0. Define the mean-zero periodic sawtooth G_T(x)=1/2-x/T for 0<x<T, with the right-continuous endpoint value. [SG-21 Canonical periodic integration of jump data]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]","count":19},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]","count":29},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-daaa917c98a2902e","tex":"H^{-1}","aliases":"H^-1","display":"H^{-1}","kind":"use","excerpt":"smallDomain and hypotheses. The invertible square Prony system of SG-20. Write Hq=-v, and suppose widetilde H=H+E, widetilde v=v+e. Assume H^-1 _2 E _2<1. [SG-24 Annihilator coefficient perturbation]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"smallDomain and hypotheses. Finite rectangular complex matrices. Let H=H_0+E, with H_0 of rank r≥1, k=min(h,w)≥ r, and ϵ= E _2<σ_r(H_0)/2. Supply a threshold t with ϵ<t<σ_r(H_0)-ϵ. [SG-28 Order recovery with a supplied singular threshold]","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ef07b8ee9bc1051b","tex":"h_{ba}","aliases":"h_ba","display":"h_{ba}","kind":"use","excerpt":"Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-bd222d4640de27ae","tex":"H_{h,w}","aliases":"H_h,w","display":"H_{h,w}","kind":"use","excerpt":"smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2a85f2d2d6dcdd7a","tex":"H_{L,L}","aliases":"H_L,L","display":"H_{L,L}","kind":"use","excerpt":"Use SG-18 to form 2L exact high derivative moments. With n_0=K+1, form H_L,L using the first 2L-1 of them. SG-19 says its rank is the actual r. If r=0, j=0 and the low coefficients directly give s. Otherwise the first 2r moments are available. SG-20 recovers the nodes and amplitudes. SG-21 constructs the unique normalized edge. For each |n|≤ K, sub","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1cbe10475e12705d","tex":"H_{r,r}","aliases":"H_r,r","display":"H_{r,r}","kind":"use","excerpt":"The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":19},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"use","excerpt":"The Fourier coefficients at the high indices I_0={(q_0+q)N:0≤ q≤2L_max}, I_1={(q_0+q)N+1:0≤ q<L_max} determine the unknown edge order, every physical node, and every jump amplitude exactly. The high-frequency budget is 3L_max+1. Low modes -K≤ n≤ K then give the complete signal. No supplied jump neighborhood is required.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":27},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ec0a583d08a29cb3","tex":"i^n","aliases":"i^n","display":"i^n","kind":"use","excerpt":"Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fb0093a6af8e437a","tex":"\\mathcal I_0","aliases":"I_0 mathcal","display":"\\mathcal I_0","kind":"use","excerpt":"The Fourier coefficients at the high indices I_0={(q_0+q)N:0≤ q≤2L_max}, I_1={(q_0+q)N+1:0≤ q<L_max} determine the unknown edge order, every physical node, and every jump amplitude exactly. The high-frequency budget is 3L_max+1. Low modes -K≤ n≤ K then give the complete signal. No supplied jump neighborhood is required.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-b8464e1fbb4674b3","tex":"\\mathcal I_1","aliases":"I_1 mathcal","display":"\\mathcal I_1","kind":"use","excerpt":"The Fourier coefficients at the high indices I_0={(q_0+q)N:0≤ q≤2L_max}, I_1={(q_0+q)N+1:0≤ q<L_max} determine the unknown edge order, every physical node, and every jump amplitude exactly. The high-frequency budget is 3L_max+1. Low modes -K≤ n≤ K then give the complete signal. No supplied jump neighborhood is required.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-429060ff5e4e8861","tex":"\\inf_{q\\ne q_*}","aliases":"inf_q≠ q_* inf ne","display":"\\inf_{q\\ne q_*}","kind":"use","excerpt":"For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.","count":14},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-146cbf287aa37a5f","tex":"\\mathcal J","aliases":"J mathcal","display":"\\mathcal J","kind":"use","excerpt":"smallDomain and hypotheses. An arbitrary measurement model and linear packet classes. Let M:X→ Y be any map and let an edge assignment J:X→ J be specified. Separately, let S, J be linear subspaces of a common vector space. [SG-07 Measurement and decomposition obstructions]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":61},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]","count":26},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":29},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"Every vector α with α _0≤ s is uniquely determined by Aα if and only if ker A∩{h: h _0≤min(2s,d)}={0}. Equivalently every set of at most min(2s,d) columns is independent. With the convention spark(A)=d+1 when all d columns are independent, the equivalent condition is spark(A)>min(2s,d). For s=0 uniqueness is automatic.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"ments determine an unknown finite jump count through Hankel rank, followed by an annihilating polynomial and Vandermonde recovery. Matched adjacent offsets recover the physical phase lost by decimation. Noisy order, root, and branch conclusions retain explicit conditioning and margin gates, with conditional asymptotic rates. A corrected periodic me","count":32},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-9d0fbf856fe55b03","tex":"L^1","aliases":"L^1","display":"L^1","kind":"use","excerpt":"ments determine an unknown finite jump count through Hankel rank, followed by an annihilating polynomial and Vandermonde recovery. Matched adjacent offsets recover the physical phase lost by decimation. Noisy order, root, and branch conclusions retain explicit conditioning and margin gates, with conditional asymptotic rates. A corrected periodic me","count":13},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fc686f6bdce3b5ca","tex":"L_{\\max}","aliases":"L_max max","display":"L_{\\max}","kind":"use","excerpt":"smallDomain and hypotheses. Bandlimited background with a bounded number of jumps. Let f∈ U_K,L_max, with L_max≥1. Supply N≥2, q_0≥1 with q_0N>K, and assume the active aliased nodes u_ℓ=z_ℓ^N are distinct. [SG-34 An explicit exact self-calibration budget]","count":9},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"For x∈ X choose x_n∈ D with x_n→ x. Continuity gives Ax=lim_nAx_n=lim_nBx_n=Bx. This proves uniqueness of bounded extensions. For an infinite packet sum one would need partial sums to converge to the declared operator, in a stated operator or strong topology. Neither density nor the finite identity by itself proves such a limit. In SG-42 density of","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-53f3164e2d42bc85","tex":"\\lim_n","aliases":"lim_n lim","display":"\\lim_n","kind":"use","excerpt":"For x∈ X choose x_n∈ D with x_n→ x. Continuity gives Ax=lim_nAx_n=lim_nBx_n=Bx. This proves uniqueness of bounded extensions. For an infinite packet sum one would need partial sums to converge to the declared operator, in a stated operator or strong topology. Neither density nor the finite identity by itself proves such a limit. In SG-42 density of","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia);","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"smallDomain and hypotheses. An arbitrary measurement model and linear packet classes. Let M:X→ Y be any map and let an edge assignment J:X→ J be specified. Separately, let S, J be linear subspaces of a common vector space. [SG-07 Measurement and decomposition obstructions]","count":50},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":38},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a2127b3e9668951d","tex":"M^{\\mathsf T}","aliases":"M^ T mathsf","display":"M^{\\mathsf T}","kind":"use","excerpt":"The minimizer is unique and (M^ TM+λ I)c=M^ Ty, M^ Tr=λ c. Adding candidate columns cannot increase the minimized ridge objective, but exact orthogonality M^ Tr=0 is not the ridge stationarity law. Even a complete pixel basis need not give zero residual when λ>0.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4087b32383a0ffb7","tex":"M^{-p-1}","aliases":"M^-p-1","display":"M^{-p-1}","kind":"use","excerpt":"j-widehat j _1=O(N^-p-1), f-widehat f_M _1≤ b_M+O((1+C_M)N^-p-1). For Fourier truncation, C_M=O(1+log(M+1)). If also |widehat s(n)|≤ C_s(1+|n|)^-p-2, its smooth remainder satisfies b_M=O(M^-p-1). Taking M comparable to N yields O((1+log N)N^-p-1).","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8868d63653f7b040","tex":"M^T","aliases":"M^T","display":"M^T","kind":"use","excerpt":"The Hessian is twice M^TM+λ I, which is positive definite because v^T(M^TM+λ I)v= Mv ^2+λ v ^2>0 for nonzero v. Setting the gradient to zero gives the unique minimizer and rearrangement gives M^Tr=λ c. When columns are added, extending the previous coefficient vector by zero is a feasible point with exactly the previous objective, so the new optimu","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ca9d35efd7c047b3","tex":"m_{(q_0+q)N+r}","aliases":"m_(q_0+q)N+r","display":"m_{(q_0+q)N+r}","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-49eabf57c1d4d69c","tex":"m_{(q_0+q)N+v}","aliases":"m_(q_0+q)N+v","display":"m_{(q_0+q)N+v}","kind":"use","excerpt":"∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leadin","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1fad90ca8a3f64d8","tex":"m_{4q}","aliases":"m_4q","display":"m_{4q}","kind":"use","excerpt":"Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ceff9d1900ba1281","tex":"m_{4q+1}","aliases":"m_4q+1","display":"m_{4q+1}","kind":"use","excerpt":"Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ee24f5a172df72e7","tex":"m_{4q+2}","aliases":"m_4q+2","display":"m_{4q+2}","kind":"use","excerpt":"Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-721c71fb160021e6","tex":"m_{4q+3}","aliases":"m_4q+3","display":"m_{4q+3}","kind":"use","excerpt":"Take a bandlimited background s(x)=frac14+frac25cos(2x)-frac13sin(3x)+frac15cos(5x) and a mean-zero edge with locations and jumps (τ_ℓ)=(0,π/2,π,3π/2), (a_ℓ)=(1,-2,-1,2). The physical nodes are (1,-i,-1,i). Their derivative moments are m_n=1-2(-i)^n-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ce78c2a469d42dfa","tex":"m_n","aliases":"m_n","display":"m_n","kind":"use","excerpt":"Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ce84ad5d82a6228e","tex":"m_{n+4}","aliases":"m_n+4","display":"m_{n+4}","kind":"use","excerpt":"-(-1)^n+2i^n, so in residues modulo four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hankel factorization has four distinct nodes and four nonzero amplitudes, hence rank four. Its recurren","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5ebba539f5d73f07","tex":"m_{n+k}","aliases":"m_n+k","display":"m_{n+k}","kind":"use","excerpt":"For every available recurrence index, ∑_k=0^rq_km_n+k =∑_ℓ c_ℓ z_ℓ^n-n_0Q(z_ℓ)=0, q_r=1. SG-19 makes H_r,r invertible. Thus the system has one solution and the true polynomial supplies it. Factoring the resulting polynomial recovers the distinct nodes up to a permutation. The amplitude matrix for consecutive moments is a square Vandermonde multipli","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-59a732086fba0ea1","tex":"m_n^{(c)}","aliases":"m_n^(c)","display":"m_n^{(c)}","kind":"use","excerpt":"smallDomain and hypotheses. A finite collection of exponential moment channels. Let m_n^(c)=∑_ℓ=1^ra_cℓz_ℓ^n, with distinct common unit-modulus nodes. At each node at least one channel amplitude is nonzero. Form the stacked linear recurrence system for a monic polynomial of degree r and assume its coefficient matrix has column rank r. [SG-39 Shared","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e65cca66a191c71c","tex":"m_{n_0}","aliases":"m_n_0","display":"m_{n_0}","kind":"use","excerpt":"smallDomain and hypotheses. The nonzero-order moment model of SG-19. Assume r≥1 and 2r consecutive moments m_n_0,…,m_n_0+2r-1 are known. [SG-20 Exact Prony annihilation and location recovery]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e14710e0615737e2","tex":"m_{n_0+2r-1}","aliases":"m_n_0+2r-1","display":"m_{n_0+2r-1}","kind":"use","excerpt":"smallDomain and hypotheses. The nonzero-order moment model of SG-19. Assume r≥1 and 2r consecutive moments m_n_0,…,m_n_0+2r-1 are known. [SG-20 Exact Prony annihilation and location recovery]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-690005fb4ce90d2d","tex":"m_{n_0+i+j}","aliases":"m_n_0+i+j","display":"m_{n_0+i+j}","kind":"use","excerpt":"smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-56bdb1575c379f30","tex":"m_{n_0+k}","aliases":"m_n_0+k","display":"m_{n_0+k}","kind":"use","excerpt":"smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-74640893b4d1b63d","tex":"m_{n_0+r+i}","aliases":"m_n_0+r+i","display":"m_{n_0+r+i}","kind":"use","excerpt":"The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"ent, the distance to any nonempty closed set is 1-Lipschitz by the triangle inequality. The true ray has perturbed distance at most e _2, and every wrong ray has distance at least Γ- e _2> e _2. Thus the true ray wins strictly. A squared objective gap must be square-rooted before it is used in this norm-distance condition. For positive constraints ","count":14},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e84f13110596fa9c","tex":"\\max_{0\\le i<h,\\ 0\\le j<w}","aliases":"max_0≤ i<h, 0≤ j<w max le le","display":"\\max_{0\\le i<h,\\ 0\\le j<w}","kind":"use","excerpt":"E _2≤ E _F ≤sqrthwmax_0≤ i<h, 0≤ j<w|ϵ_n_0+i+j|. In particular, an error bound |η_n|≤η implies E _2≤2πsqrthwηmax_i,j|n_0+i+j|.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c58416f845294e63","tex":"\\max_{i,j}","aliases":"max_i,j max","display":"\\max_{i,j}","kind":"use","excerpt":"E _2≤ E _F ≤sqrthwmax_0≤ i<h, 0≤ j<w|ϵ_n_0+i+j|. In particular, an error bound |η_n|≤η implies E _2≤2πsqrthwηmax_i,j|n_0+i+j|.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3a5543358a2b99f3","tex":"\\max_{x\\in V,\\,\\|x\\|=1}","aliases":"max_x∈ V, x =1 max in","display":"\\max_{x\\in V,\\,\\|x\\|=1}","kind":"use","excerpt":"The singular-value perturbation inequality [source reference] gives |σ_j(H)-σ_j(H_0)|≤ E _2 for 1≤ j≤ k. It holds equally over the complex field: in the min--max characterization σ_j(A)=min_dim V=w-j+1max_x∈ V, x =1 Ax . the triangle inequality bounds each restricted maximum for A+E by that for A plus E _2. Minimize and then interchange the two mat","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"Every vector α with α _0≤ s is uniquely determined by Aα if and only if ker A∩{h: h _0≤min(2s,d)}={0}. Equivalently every set of at most min(2s,d) columns is independent. With the convention spark(A)=d+1 when all d columns are independent, the equivalent condition is spark(A)>min(2s,d). For s=0 uniqueness is automatic.","count":15},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-798cc2cbf6d4fe03","tex":"\\min_{\\dim V=w-j+1}","aliases":"min_dim V=w-j+1 min dim","display":"\\min_{\\dim V=w-j+1}","kind":"use","excerpt":"The singular-value perturbation inequality [source reference] gives |σ_j(H)-σ_j(H_0)|≤ E _2 for 1≤ j≤ k. It holds equally over the complex field: in the min--max characterization σ_j(A)=min_dim V=w-j+1max_x∈ V, x =1 Ax . the triangle inequality bounds each restricted maximum for A+E by that for A plus E _2. Minimize and then interchange the two mat","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-87e9692dc8e5a327","tex":"\\min_{\\substack{\\|h\\|_2=1\\\\\\|h\\|_0\\le\\min(2s,d)}}","aliases":"min_substack h _2=1 h _0≤min(2s,d) min substack le min","display":"\\min_{\\substack{\\|h\\|_2=1\\\\\\|h\\|_0\\le\\min(2s,d)}}","kind":"use","excerpt":"Define γ=min_substack h _2=1 h _0≤min(2s,d) Ah _2>0. Then widehatα-α_0 _2≤2ϵ/γ, D(widehatα-α_0) _2 ≤2 D _2ϵ/γ. If every nonzero true coefficient has magnitude greater than 4ϵ/γ, thresholding at 2ϵ/γ with a strict greater-than rule recovers the true support.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-54207576dcfe7b5d","tex":"\\min_{|z-z_\\ell|=\\rho}","aliases":"min_|z-z_ℓ|=ρ min ell rho","display":"\\min_{|z-z_\\ell|=\\rho}","kind":"use","excerpt":"If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1ce99d3ca4a35509","tex":"\\min_{\\ell\\ne k}","aliases":"min_ℓ≠ k min ell ne","display":"\\min_{\\ell\\ne k}","kind":"use","excerpt":"smallDomain and hypotheses. Monic complex polynomials of the same degree. Let r≥1 and Q(z)=∏_ℓ=1^r(z-z_ℓ) have distinct roots and widetilde Q=Q+E_Q. Choose 0<ρ<tfrac12min_ℓ≠ k|z_ℓ-z_k| when r≥2, or any ρ>0 when r=1. [SG-25 Conditional simple-root disk localization]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]","count":61},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":66},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-213d86a5e6b34393","tex":"N^{-1}","aliases":"N^-1","display":"N^{-1}","kind":"use","excerpt":"verse bounds transfer the moment error to aliased node and matched coefficient errors of order N^-p-1. The supplied rank gap and threshold select the correct order by SG-28. Nonvanishing base coefficients allow SG-30 to give the same order for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the alias","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a3fe83ae353c7377","tex":"N^{-p-1}","aliases":"N^-p-1","display":"N^{-p-1}","kind":"use","excerpt":"cted sequence of decimation scales. Let p≥0, fix q_0 and the finite number of channel samples and offsets, and take a sequence N→∞. Assume distinct aliased nodes, uniform rank-threshold gaps/local inverse bounds for the aligned Prony and coefficient fits, |b_ℓ,0|≥ b_min>0, uniformly bounded amplitudes, and the strict branch margin of SG-32. Suppose","count":7},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3734b569b3946185","tex":"N^{-p-2}","aliases":"N^-p-2","display":"N^{-p-2}","kind":"use","excerpt":"The threshold-selected order is eventually correct. The aligned lifted nodes and locations satisfy |widehat z_ℓ-z_ℓ|=O(N^-p-2), d_ T(widehatτ_ℓ,τ_ℓ)=O(N^-p-2). The recovered amplitudes satisfy |widehat a_ℓ-a_ℓ|=O(N^-p-1).","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1fea4657810bf7e4","tex":"N^d","aliases":"N^d","display":"N^d","kind":"use","excerpt":"For one term, (E-z)(z^nP(n))=z^n+1(P(n+1)-P(n)). A finite difference lowers a nonconstant polynomial's degree by one, and annihilates a constant. After d+1 differences the term vanishes. The shift factors commute, so the product annihilates every term in the sum. If P_ℓ(n) has leading coefficient c_ℓ≠0 at degree d, substitution of (q_0+q)N+v has le","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-b3f6ba5bad3f6071","tex":"n_0","aliases":"n_0","display":"n_0","kind":"use","excerpt":"smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]","count":11},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"cted sequence of decimation scales. Let p≥0, fix q_0 and the finite number of channel samples and offsets, and take a sequence N→∞. Assume distinct aliased nodes, uniform rank-threshold gaps/local inverse bounds for the aligned Prony and coefficient fits, |b_ℓ,0|≥ b_min>0, uniformly bounded amplitudes, and the strict branch margin of SG-32. Suppose","count":8},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-b3f4c252ebea45ab","tex":"\\overline u","aliases":"overline u overline","display":"\\overline u","kind":"use","excerpt":"smallDomain and hypotheses. Nth roots of a unit-modulus estimate. Let N≥2, |z|=1, u=z^N, and overline u=widehat u/|widehat u| with widehat u≠0. Let ρ be a root of overline u continuously associated with z, and widetilde z a coarse phase estimate. [SG-32 Unique branch selection under a strict margin]","count":8},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3003e43c77eb6a52","tex":"\\overline u^{-q_0}","aliases":"overline u^-q_0 overline","display":"\\overline u^{-q_0}","kind":"use","excerpt":"se coefficients allow SG-30 to give the same order for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c5a8ad59a6d25307","tex":"\\overline{bz^r}","aliases":"overlinebz^r overline","display":"\\overline{bz^r}","kind":"use","excerpt":"Each summand is overlinebz^r bz^r+1=|b|^2z because |z|=1. There are R summands and R≥1, so normalization returns z. For two channels direct division cancels b. No statistical independence is assumed, so one cannot improve a worst-case noise order merely by increasing R; the next bound shows the cancellation of the number of products.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0a1d3ba89e74d253","tex":"\\overline{c_r}","aliases":"overlinec_r overline","display":"\\overline{c_r}","kind":"use","excerpt":"smallDomain and hypotheses. One matched unit-circle coefficient progression. Let R≥1 be an integer counting adjacent products, and let c_r=bz^r for 0≤ r≤ R, with b≠0 and |z|=1. Define S(c)=∑_r=0^R-1overlinec_rc_r+1. [SG-29 Exact phase from adjacent offset products]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8e028b88d3ec902e","tex":"\\overline{e_r}","aliases":"overlinee_r overline","display":"\\overline{e_r}","kind":"use","excerpt":"Expand each product: overlinec_rc_r+1=|b|^2z +overlinebz^re_r+1+overlinee_rbz^r+1 +overlinee_re_r+1. The three error terms have total magnitude at most 2|b|ϵ+ϵ^2. Summing gives S(c)=Az+E with |E|≤Δ. Therefore |S(c)|≥ A-Δ>0. For any nonzero v and unit z, |frac v|v|-z| ≤|frac v|v|-frac vA|+|frac vA-z| =frac|A-|v||A+frac|v-Az|A ≤frac2|v-Az|A. Apply th","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-83f3fb3a73c7dd8f","tex":"\\overline{\\widehat m(n)}","aliases":"overlinewidehat m(n) overline widehat","display":"\\overline{\\widehat m(n)}","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":44},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":18},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":17},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a7abbe02c9f0ed6a","tex":"P_a^*","aliases":"P_a^*","display":"P_a^*","kind":"use","excerpt":"Compute P_aP_b=V^-1E_aE_bV=δ_abP_a and ∑_aP_a=V^-1IV=I. In the unitary case the pulled-back projections are orthogonal: P_a^*=P_a. Their images are mutually orthogonal, and SG-01 writes x as their finite orthogonal sum. The Pythagorean identity gives the energy formula. Without unitarity and orthogonality the algebraic identities remain valid while","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a3db31f834a299d9","tex":"P_{a_0}","aliases":"P_a_0","display":"P_{a_0}","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2693c2c3c4636ad8","tex":"P_{a_1}","aliases":"P_a_1","display":"P_{a_1}","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d6a1c0e25521f84c","tex":"P_{a_k}","aliases":"P_a_k","display":"P_{a_k}","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8d7bfa3a46319578","tex":"P_{a_{k-1}}","aliases":"P_a_k-1","display":"P_{a_{k-1}}","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":8},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-7e32786a5b15df61","tex":"P_J","aliases":"P_J","display":"P_J","kind":"use","excerpt":"smallDomain and hypotheses. The model space of SG-09. Let P_Jg=ℓ_F(g)m, P_S=I-P_J, and let S_M be Fourier truncation of degree M≥ K. [SG-10 The singularity-exact block operator]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fcefe6d7d0529a3b","tex":"P_J^2","aliases":"P_J^2","display":"P_J^2","kind":"use","excerpt":"P_J^2=P_J, P_S^2=P_S, P_SP_J=P_JP_S=0, P_S+P_J=I_X, and the resolved reconstruction obeys R_M=S_MP_S+P_J=I_X.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d4737801d4999590","tex":"P_k","aliases":"P_k","display":"P_k","kind":"use","excerpt":"smallDomain and hypotheses. A Hilbert space with finite selected packet spans. Let V_1⊆ V_2⊆⋯ be finite-dimensional subspaces. Let P_k be the orthogonal projector onto V_k. [SG-42 Ideal nested-span refitting decreases residual]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-25fb50d41b9f13de","tex":"P_{k+1}","aliases":"P_k+1","display":"P_{k+1}","kind":"use","excerpt":"f-P_k+1f ≤ f-P_kf , f-P_kf⊥ V_k. If ⋃_kV_k is dense, then P_kf→ f in norm.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-49f3b852810c6d4e","tex":"P_m","aliases":"P_m","display":"P_m","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-010f131ed455d9fc","tex":"P_S","aliases":"P_S","display":"P_S","kind":"use","excerpt":"smallDomain and hypotheses. The model space of SG-09. Let P_Jg=ℓ_F(g)m, P_S=I-P_J, and let S_M be Fourier truncation of degree M≥ K. [SG-10 The singularity-exact block operator]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e3d9206a2554a848","tex":"P_S^2","aliases":"P_S^2","display":"P_S^2","kind":"use","excerpt":"P_J^2=P_J, P_S^2=P_S, P_SP_J=P_JP_S=0, P_S+P_J=I_X, and the resolved reconstruction obeys R_M=S_MP_S+P_J=I_X.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e011202aec8a0e8b","tex":"P_\\ell","aliases":"P_ℓ ell","display":"P_\\ell","kind":"use","excerpt":"smallDomain and hypotheses. Polynomial-exponential sequences. Let m_n=∑_ℓ=1^rz_ℓ^nP_ℓ(n) with distinct nonzero nodes and deg P_ℓ≤ d. Let E be the forward shift. For an offset ratio assume N≥1 and the leading coefficient of P_ℓ at degree d is nonzero. [SG-40 Confluent annihilation and the leading offset phase]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]","count":16},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"use","excerpt":"smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]","count":57},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1092a5158c3806e4","tex":"q^h","aliases":"q^h","display":"q^h","kind":"use","excerpt":"∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leadin","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4e4bfa4de54508e8","tex":"q_*","aliases":"q_*","display":"q_*","kind":"use","excerpt":"smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]","count":7},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","excerpt":"smallDomain and hypotheses. Derivative moments with an additive defect. Fix integers N≥2, q_0≥1, and an offset r≥0. Suppose m_n=∑_ℓ a_ℓ z_ℓ^n+e_n, with |z_ℓ|=1. [SG-27 Shared-node offset channels]","count":20},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e15bd658e05b8d9b","tex":"q_k","aliases":"q_k","display":"q_k","kind":"use","excerpt":"The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-9666ac009c3e72f6","tex":"q_r","aliases":"q_r","display":"q_r","kind":"use","excerpt":"For every available recurrence index, ∑_k=0^rq_km_n+k =∑_ℓ c_ℓ z_ℓ^n-n_0Q(z_ℓ)=0, q_r=1. SG-19 makes H_r,r invertible. Thus the system has one solution and the true polynomial supplies it. Factoring the resulting polynomial recovers the distinct nodes up to a permutation. The amplitude matrix for consecutive moments is a square Vandermonde multipli","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c4b16a8e12745771","tex":"Q_\\rho","aliases":"Q_ρ rho","display":"Q_\\rho","kind":"use","excerpt":"smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c37d1813ec5d2f33","tex":"Q_\\sigma","aliases":"Q_σ sigma","display":"Q_\\sigma","kind":"use","excerpt":"P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.","count":33},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"P_J^2=P_J, P_S^2=P_S, P_SP_J=P_JP_S=0, P_S+P_J=I_X, and the resolved reconstruction obeys R_M=S_MP_S+P_J=I_X.","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":83},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-6d573f7fcc035735","tex":"\\mathbb R^m","aliases":"R^m mathbb","display":"\\mathbb R^m","kind":"use","excerpt":"smallDomain and hypotheses. Finite-dimensional real packet designs. Let M∈ R^m× d, y∈ R^m, and λ>0. Let c minimize y-Mc _2^2+λ c _2^2 and set r=y-Mc. [SG-43 Ridge refitting has a different stationarity law]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-b0b19a2fe9a099a9","tex":"\\mathbb R^{m\\times d}","aliases":"R^m× d mathbb times","display":"\\mathbb R^{m\\times d}","kind":"use","excerpt":"smallDomain and hypotheses. Finite-dimensional real packet designs. Let M∈ R^m× d, y∈ R^m, and λ>0. Let c minimize y-Mc _2^2+λ c _2^2 and set r=y-Mc. [SG-43 Ridge refitting has a different stationarity law]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e3b6ec511215a68d","tex":"r^n","aliases":"r^n","display":"r^n","kind":"use","excerpt":"_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia);","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d8079124f7b34204","tex":"\\mathcal R_M","aliases":"R_M mathcal","display":"\\mathcal R_M","kind":"use","excerpt":"P_J^2=P_J, P_S^2=P_S, P_SP_J=P_JP_S=0, P_S+P_J=I_X, and the resolved reconstruction obeys R_M=S_MP_S+P_J=I_X.","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]","count":21},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.","count":20},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.","count":61},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-b60dd8e5854d9e87","tex":"s_0","aliases":"s_0","display":"s_0","kind":"use","excerpt":"-h on the second set. Then h=α-β, and Aα=Aβ contradicts uniqueness. Column dependence is exactly the existence of a nonzero kernel vector supported on the selected columns. The spark convention then gives the stated capped inequality, including a fully independent matrix when 2s>d. When s=0, the only admissible vector is zero and the kernel test co","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-237016f1327ba6a3","tex":"S_{cb}","aliases":"S_cb","display":"S_{cb}","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-deef7bca5ee0eda2","tex":"S_k","aliases":"S_k","display":"S_k","kind":"use","excerpt":"smallDomain and hypotheses. Periodic bounded real signals. Let σ_M=(M+1)^-1∑_k=0^MS_k be the Fejer mean. Suppose 0≤ j≤1 almost everywhere. [SG-13 Positive Fourier averaging preserves range]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e5038d774b6f0114","tex":"\\mathcal S_K","aliases":"S_K mathcal","display":"\\mathcal S_K","kind":"use","excerpt":"smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c62b42c7b40a5a7d","tex":"S_M","aliases":"S_M","display":"S_M","kind":"use","excerpt":"smallDomain and hypotheses. The model space of SG-09. Let P_Jg=ℓ_F(g)m, P_S=I-P_J, and let S_M be Fourier truncation of degree M≥ K. [SG-10 The singularity-exact block operator]","count":9},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a38eec2859897799","tex":"\\mathcal S_P","aliases":"S_P mathcal","display":"\\mathcal S_P","kind":"use","excerpt":"P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a7e6148c104d684f","tex":"\\mathcal S_Q","aliases":"S_Q mathcal","display":"\\mathcal S_Q","kind":"use","excerpt":"P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8b8011d9dfc22b9d","tex":"\\operatorname{Si}","aliases":"Si operatorname","display":"\\operatorname{Si}","kind":"use","excerpt":"Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….","count":15},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3c1625bf8c56638e","tex":"\\operatorname{spark}","aliases":"spark operatorname","display":"\\operatorname{spark}","kind":"use","excerpt":"Every vector α with α _0≤ s is uniquely determined by Aα if and only if ker A∩{h: h _0≤min(2s,d)}={0}. Equivalently every set of at most min(2s,d) columns is independent. With the convention spark(A)=d+1 when all d columns are independent, the equivalent condition is spark(A)>min(2s,d). For s=0 uniqueness is automatic.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-cd7ce59e0c209271","tex":"\\sup_{|z-z_\\ell|=\\rho}","aliases":"sup_|z-z_ℓ|=ρ sup ell rho","display":"\\sup_{|z-z_\\ell|=\\rho}","kind":"use","excerpt":"If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0b7cc491b1ee030b","tex":"\\operatorname{supp}","aliases":"supp operatorname","display":"\\operatorname{supp}","kind":"use","excerpt":"aq-widetilde awidetilde q _2^2 · =|a-widetilde a|^2|suppq∩suppwidetilde q| · +|a|^2|suppq∖suppwidetilde q| +|widetilde a|^2|suppwidetilde q∖suppq|. For a=widetilde a=1 the squared norm is the cardinality of the symmetric support difference.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]","count":36},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-adac703e9d252af9","tex":"\\mathbb T","aliases":"T mathbb","display":"\\mathbb T","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":7},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.","count":19},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d83472cd546ed2e5","tex":"T^{-1}","aliases":"T^-1","display":"T^{-1}","kind":"use","excerpt":"The sawtooth derivative is DG_T=δ_0-T^-1dx: its slope is -1/T away from the endpoint and its endpoint jump is one. Summing translated copies gives the prescribed Dirac masses and a constant-density term -T^-1∑ a_ℓ, which vanishes. Consequently j is constant between the jump points. Each translated sawtooth has mean zero, so j does too. Any other pa","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8eaeac84691b5c3b","tex":"T^k","aliases":"T^k","display":"T^k","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-bd76f79f8f6e6b19","tex":"T_{ba}","aliases":"T_ba","display":"T_{ba}","kind":"use","excerpt":"smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"use","excerpt":"Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero act","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":20},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-aa7d9513e8040fb5","tex":"u^{-q_0}","aliases":"u^-q_0","display":"u^{-q_0}","kind":"use","excerpt":"for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors. For the amplitude use widehat a=widehat b_0 overl","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d06980367bb55173","tex":"u^c","aliases":"u^c","display":"u^c","kind":"use","excerpt":"Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-6db9cae8100ce412","tex":"\\mathcal U_{K,L}","aliases":"U_K,L mathcal","display":"\\mathcal U_{K,L}","kind":"use","excerpt":"Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero act","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1eaa502e0f05f3c6","tex":"\\mathcal U_{K,L_{\\max}}","aliases":"U_K,L_max mathcal max","display":"\\mathcal U_{K,L_{\\max}}","kind":"use","excerpt":"smallDomain and hypotheses. Bandlimited background with a bounded number of jumps. Let f∈ U_K,L_max, with L_max≥1. Supply N≥2, q_0≥1 with q_0N>K, and assume the active aliased nodes u_ℓ=z_ℓ^N are distinct. [SG-34 An explicit exact self-calibration budget]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f72e3b663bbd288a","tex":"u_\\ell","aliases":"u_ℓ ell","display":"u_\\ell","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fb3d0d24adc570be","tex":"u_\\ell^{-q_0}","aliases":"u_ℓ^-q_0 ell","display":"u_\\ell^{-q_0}","kind":"use","excerpt":"-19 gives its rank r. If r=0, the edge is zero and the low modes finish the recovery. Otherwise SG-20 recovers the u_ℓ and then the base coefficients. The adjacent channel supplies at least r moments of ∑_ℓ b_ℓ,0z_ℓ u_ℓ^q. Using the same recovered nodes, its Vandermonde system recovers b_ℓ,1=b_ℓ,0z_ℓ. Division gives z_ℓ=b_ℓ,1/b_ℓ,0, and a_ℓ=b_ℓ,0u_","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-01b53abd042200f7","tex":"u_\\ell^q","aliases":"u_ℓ^q ell","display":"u_\\ell^q","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-98193535a98d5c35","tex":"u_\\ell^{q_0}","aliases":"u_ℓ^q_0 ell","display":"u_\\ell^{q_0}","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces with an invertible transform. Let V:X→ Y be an isomorphism and (E_a) a finite complete packet family on Y. [SG-04 Pulled-back transform packets]","count":26},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to t","count":18},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a8ee3eea11f4b48b","tex":"V^{-1}","aliases":"V^-1","display":"V^{-1}","kind":"use","excerpt":"The operators P_a=V^-1E_aV are a complete packet family on X. For a unitary V and orthogonal coordinate selectors E_a in a Hilbert space, x ^2=∑_a P_ax ^2.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f02bee6087e6f2d3","tex":"v^d","aliases":"v^d","display":"v^d","kind":"use","excerpt":"If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-bf5d1376781bcc57","tex":"v^N","aliases":"v^N","display":"v^N","kind":"use","excerpt":"If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f9896e0d72ef478c","tex":"v^{r_1}","aliases":"v^r_1","display":"v^{r_1}","kind":"use","excerpt":"If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-aaed6024ec4d2ac4","tex":"v^{r_k}","aliases":"v^r_k","display":"v^{r_k}","kind":"use","excerpt":"If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to t","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-36077cb07719c8a1","tex":"v^T","aliases":"v^T","display":"v^T","kind":"use","excerpt":"The Hessian is twice M^TM+λ I, which is positive definite because v^T(M^TM+λ I)v= Mv ^2+λ v ^2>0 for nonzero v. Setting the gradient to zero gives the unique minimizer and rearrangement gives M^Tr=λ c. When columns are added, extending the previous coefficient vector by zero is a feasible point with exactly the previous objective, so the new optimu","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"smallDomain and hypotheses. A Hilbert space with finite selected packet spans. Let V_1⊆ V_2⊆⋯ be finite-dimensional subspaces. Let P_k be the orthogonal projector onto V_k. [SG-42 Ideal nested-span refitting decreases residual]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c5bc961c4206a922","tex":"V_2","aliases":"V_2","display":"V_2","kind":"use","excerpt":"smallDomain and hypotheses. A Hilbert space with finite selected packet spans. Let V_1⊆ V_2⊆⋯ be finite-dimensional subspaces. Let P_k be the orthogonal projector onto V_k. [SG-42 Ideal nested-span refitting decreases residual]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0b837e89a1e8104e","tex":"V_h","aliases":"V_h","display":"V_h","kind":"use","excerpt":"Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-616a0b138220d2cb","tex":"V_K","aliases":"V_K","display":"V_K","kind":"use","excerpt":"Orthogonal projection minimizes distance to its subspace. Since P_kf belongs to the larger subspace V_k+1, its residual is a feasible upper bound on the next minimum. Orthogonality follows from the defining projection identity, or by varying the quadratic norm along every vector in V_k. For convergence, fix ϵ>0 and choose v∈ V_K within ϵ of f by de","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fa886686fadc3c64","tex":"V_k","aliases":"V_k","display":"V_k","kind":"use","excerpt":"smallDomain and hypotheses. A Hilbert space with finite selected packet spans. Let V_1⊆ V_2⊆⋯ be finite-dimensional subspaces. Let P_k be the orthogonal projector onto V_k. [SG-42 Ideal nested-span refitting decreases residual]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-cdebe96f7a1525e6","tex":"V_{k+1}","aliases":"V_k+1","display":"V_{k+1}","kind":"use","excerpt":"Orthogonal projection minimizes distance to its subspace. Since P_kf belongs to the larger subspace V_k+1, its residual is a feasible upper bound on the next minimum. Orthogonality follows from the defining projection identity, or by varying the quadratic norm along every vector in V_k. For convergence, fix ϵ>0 and choose v∈ V_K within ϵ of f by de","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-7ee40afe580bc8e6","tex":"V_w","aliases":"V_w","display":"V_w","kind":"use","excerpt":"Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3ff1b9db7af7f5bc","tex":"V_w^{\\mathsf T}","aliases":"V_w^ T mathsf","display":"V_w^{\\mathsf T}","kind":"use","excerpt":"Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fa3357a7e55dffff","tex":"V_w^T","aliases":"V_w^T","display":"V_w^T","kind":"use","excerpt":"Expand a matrix entry to obtain the factorization. The leading r× r Vandermonde submatrix has determinant ∏_ℓ<k(z_k-z_ℓ), which is nonzero. Thus both rectangular Vandermonde factors have full column rank, and their transposes have full row rank. The diagonal factor is invertible. Its product with V_w^T is surjective onto C^r, and subsequent multipl","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"t_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the","count":14},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3dd666fecdd8fcbb","tex":"w_j^{b_j}","aliases":"w_j^b_j","display":"w_j^{b_j}","kind":"use","excerpt":"Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_","count":12},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-61fa5d63cf733a4e","tex":"\\widehat a_\\ell","aliases":"widehat a_ℓ widehat ell","display":"\\widehat a_\\ell","kind":"use","excerpt":"The threshold-selected order is eventually correct. The aligned lifted nodes and locations satisfy |widehat z_ℓ-z_ℓ|=O(N^-p-2), d_ T(widehatτ_ℓ,τ_ℓ)=O(N^-p-2). The recovered amplitudes satisfy |widehat a_ℓ-a_ℓ|=O(N^-p-1).","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ea83107951d84fac","tex":"\\widehat b","aliases":"widehat b widehat","display":"\\widehat b","kind":"use","excerpt":"se coefficients allow SG-30 to give the same order for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2244a6da807d2026","tex":"\\widehat b_0","aliases":"widehat b_0 widehat","display":"\\widehat b_0","kind":"use","excerpt":"se coefficients allow SG-30 to give the same order for the coarse phase error. The strict branch margin selects the associated root by SG-32. Normalize the aliased estimate and use SG-33: its root error is N^-1 times an error of order N^-p-1, giving N^-p-2. On the unit circle, d_ T=2arcsin(|widehat z-z|/2), which has the same order for small errors","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":7},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-41c90d716c784808","tex":"\\widehat f_M","aliases":"widehat f_M widehat","display":"\\widehat f_M","kind":"use","excerpt":"smallDomain and hypotheses. A normed signal space. Let f=s+j, let widehat j belong to the same space, and let A_M be a bounded linear operator with norm at most C_M. Define widehat f_M=A_M(f-widehat j)+widehat j. [SG-36 Resolved synthesis splits the two error sources]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-45e216c04187fe54","tex":"\\widehat g","aliases":"widehat g widehat","display":"\\widehat g","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-6b108551467ed661","tex":"\\widehat j","aliases":"widehat j widehat","display":"\\widehat j","kind":"use","excerpt":"Direct integration gives widehat j(0)=a/π and widehat j(n)=sin(na)/(π n) for n≠0. The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by sum","count":15},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-77fd09daba007b85","tex":"\\widehat m","aliases":"widehat m widehat","display":"\\widehat m","kind":"use","excerpt":"smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-7d3704d4c4eab86e","tex":"\\widehat s","aliases":"widehat s widehat","display":"\\widehat s","kind":"use","excerpt":"smallDomain and hypotheses. Periodic signals with a supplied exact edge packet. Let f=s+j with |widehat s(n)|≤ Ce^-ρ|n|, where C≥0 and ρ>0. Use the canonical uniformly convergent Fourier-series representative of s. Set R_Mf=S_Ms+j. [SG-11 Resolved error for an analytic background]","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2c8131c596a2a289","tex":"\\widehat u","aliases":"widehat u widehat","display":"\\widehat u","kind":"use","excerpt":"smallDomain and hypotheses. Nth roots of a unit-modulus estimate. Let N≥2, |z|=1, u=z^N, and overline u=widehat u/|widehat u| with widehat u≠0. Let ρ be a root of overline u continuously associated with z, and widetilde z a coarse phase estimate. [SG-32 Unique branch selection under a strict margin]","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c88f3c79e4ba1c43","tex":"\\widehat z","aliases":"widehat z widehat","display":"\\widehat z","kind":"use","excerpt":"widehat a-a _2 ≤frac e _2+ Δ V _2 a _2 σ_min(V)- Δ V _2. For unit nodes in rows with nonnegative powers n, matched chord error |widehat z_ℓ-z_ℓ|≤δ gives the entry bound |widehat z_ℓ^n-z_ℓ^n|≤ nδ.","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8fab4f601a2cb18c","tex":"\\widehat z^n","aliases":"widehat z^n widehat","display":"\\widehat z^n","kind":"use","excerpt":"The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. Th","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-13b4afcd556c6b11","tex":"\\widehat z^{n-1-k}","aliases":"widehat z^n-1-k widehat","display":"\\widehat z^{n-1-k}","kind":"use","excerpt":"The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. Th","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ffb97250966c5513","tex":"\\widehat z_\\ell","aliases":"widehat z_ℓ widehat ell","display":"\\widehat z_\\ell","kind":"use","excerpt":"widehat a-a _2 ≤frac e _2+ Δ V _2 a _2 σ_min(V)- Δ V _2. For unit nodes in rows with nonnegative powers n, matched chord error |widehat z_ℓ-z_ℓ|≤δ gives the entry bound |widehat z_ℓ^n-z_ℓ^n|≤ nδ.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f531145d0de1d982","tex":"\\widehat z_\\ell^n","aliases":"widehat z_ℓ^n widehat ell","display":"\\widehat z_\\ell^n","kind":"use","excerpt":"widehat a-a _2 ≤frac e _2+ Δ V _2 a _2 σ_min(V)- Δ V _2. For unit nodes in rows with nonnegative powers n, matched chord error |widehat z_ℓ-z_ℓ|≤δ gives the entry bound |widehat z_ℓ^n-z_ℓ^n|≤ nδ.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1d36ebc29fcba0ad","tex":"\\widehat\\alpha","aliases":"widehatα widehat alpha","display":"\\widehat\\alpha","kind":"use","excerpt":"smallDomain and hypotheses. The Euclidean model of SG-14. Assume s≥1, d≥1, and the injectivity condition of SG-14. Let y=s_0+Dα_0+η, s_0∈ S, α_0 _0≤ s, and Rη _2≤ϵ. Let widehatα be s-sparse with Awidehatα-Ry _2≤ϵ. [SG-15 Sparse noise stability and support thresholding]","count":9},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ad8c5d8ce79e0069","tex":"\\widehat\\tau","aliases":"widehatτ widehat tau","display":"\\widehat\\tau","kind":"use","excerpt":"The threshold-selected order is eventually correct. The aligned lifted nodes and locations satisfy |widehat z_ℓ-z_ℓ|=O(N^-p-2), d_ T(widehatτ_ℓ,τ_ℓ)=O(N^-p-2). The recovered amplitudes satisfy |widehat a_ℓ-a_ℓ|=O(N^-p-1).","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-373159b90f06e4ed","tex":"\\widehat\\tau_\\ell","aliases":"widehatτ_ℓ widehat tau ell","display":"\\widehat\\tau_\\ell","kind":"use","excerpt":"The threshold-selected order is eventually correct. The aligned lifted nodes and locations satisfy |widehat z_ℓ-z_ℓ|=O(N^-p-2), d_ T(widehatτ_ℓ,τ_ℓ)=O(N^-p-2). The recovered amplitudes satisfy |widehat a_ℓ-a_ℓ|=O(N^-p-1).","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2b26d57eb5a9595f","tex":"\\widetilde a","aliases":"widetilde a widetilde","display":"\\widetilde a","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-70fad85f996380ab","tex":"\\widetilde H","aliases":"widetilde H widetilde","display":"\\widetilde H","kind":"use","excerpt":"smallDomain and hypotheses. The invertible square Prony system of SG-20. Write Hq=-v, and suppose widetilde H=H+E, widetilde v=v+e. Assume H^-1 _2 E _2<1. [SG-24 Annihilator coefficient perturbation]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a43c2012e6f47f4e","tex":"\\widetilde H^{-1}","aliases":"widetilde H^-1 widetilde","display":"\\widetilde H^{-1}","kind":"use","excerpt":"Factor widetilde H=H(I+H^-1E). The Neumann series converges in operator norm under the supplied inequality, so widetilde H^-1 _2≤ H^-1 _2/(1- H^-1 _2 E _2). Subtract the equations to obtain widetilde H(widetilde q-q)=-e-Eq. Apply the inverse bound and the triangle inequality. The estimate assumes that the same true order is used; it does not itself","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5f43b2813d6b7cc9","tex":"\\widetilde m","aliases":"widetilde m widetilde","display":"\\widetilde m","kind":"use","excerpt":"Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ea7b5113dcaccd20","tex":"\\widetilde m_n","aliases":"widetilde m_n widetilde","display":"\\widetilde m_n","kind":"use","excerpt":"Their unordered location sets have permutation-matched bottleneck distance at least π-δ, while their measured derivative moments satisfy |m_n-widetilde m_n| =|(1-e^-inπ)(1-e^-inδ)| ≤2|n|δ (n∈ F). There is no Lipschitz inverse for locations with one constant on all distinct two-jump configurations, even at this fixed order and with known amplitudes.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ed8c63057cfee2e2","tex":"\\widetilde Q","aliases":"widetilde Q widetilde","display":"\\widetilde Q","kind":"use","excerpt":"smallDomain and hypotheses. Monic complex polynomials of the same degree. Let r≥1 and Q(z)=∏_ℓ=1^r(z-z_ℓ) have distinct roots and widetilde Q=Q+E_Q. Choose 0<ρ<tfrac12min_ℓ≠ k|z_ℓ-z_k| when r≥2, or any ρ>0 when r=1. [SG-25 Conditional simple-root disk localization]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-713eeaa785c879b7","tex":"\\widetilde q","aliases":"widetilde q widetilde","display":"\\widetilde q","kind":"use","excerpt":"smallDomain and hypotheses. Finite binary masks. Let q,widetilde q∈{0,1}^m and a,widetilde a∈ R. [SG-17 Mask error is a support geometry calculation]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a370d18aaa4f9f62","tex":"\\widetilde q_k","aliases":"widetilde q_k widetilde","display":"\\widetilde q_k","kind":"use","excerpt":"If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-60cd197d59c08b6e","tex":"\\widetilde V","aliases":"widetilde V widetilde","display":"\\widetilde V","kind":"use","excerpt":"smallDomain and hypotheses. A square or overdetermined Vandermonde amplitude fit. Let V have full column rank, m=Va, widetilde V=V+Δ V, and widetilde m=m+e. Assume σ_min(V)> Δ V _2. Let widehat a minimize widetilde Vb-widetilde m _2. [SG-26 Amplitude stability with perturbed nodes]","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3323d1822e9c5f0f","tex":"\\widetilde v","aliases":"widetilde v widetilde","display":"\\widetilde v","kind":"use","excerpt":"smallDomain and hypotheses. The invertible square Prony system of SG-20. Write Hq=-v, and suppose widetilde H=H+E, widetilde v=v+e. Assume H^-1 _2 E _2<1. [SG-24 Annihilator coefficient perturbation]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d1bf38ed91f40117","tex":"\\widetilde V^\\dagger","aliases":"widetilde V^† widetilde dagger","display":"\\widetilde V^\\dagger","kind":"use","excerpt":"The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. Th","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-86c1d03c1a864e81","tex":"\\widetilde z","aliases":"widetilde z widetilde","display":"\\widetilde z","kind":"use","excerpt":"smallDomain and hypotheses. Nth roots of a unit-modulus estimate. Let N≥2, |z|=1, u=z^N, and overline u=widehat u/|widehat u| with widehat u≠0. Let ρ be a root of overline u continuously associated with z, and widetilde z a coarse phase estimate. [SG-32 Unique branch selection under a strict margin]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":20},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":39},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-51e189ae43085a7d","tex":"X_a","aliases":"X_a","display":"X_a","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-9067dcf296a5b175","tex":"x_a","aliases":"x_a","display":"x_a","kind":"use","excerpt":"The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.","count":8},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-73a4e2e74cee4766","tex":"X_b","aliases":"X_b","display":"X_b","kind":"use","excerpt":"smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-14c8085d57b478d9","tex":"x_b","aliases":"x_b","display":"x_b","kind":"use","excerpt":"Completeness gives S A x=x. If x_a∈ X_a, then P_ax_a=x_a and P_bx_a=0 for b≠ a. Thus the bth component of A S(x_a)_a is P_b∑_ax_a=x_b. Both inverse identities follow. For uniqueness, apply P_b to any proposed sum of components. This proof uses pairwise annihilation, not orthogonality; the packet projections can be oblique.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-adacda432da0fd15","tex":"x_M","aliases":"x_M","display":"x_M","kind":"use","excerpt":"smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"For x∈ X choose x_n∈ D with x_n→ x. Continuity gives Ax=lim_nAx_n=lim_nBx_n=Bx. This proves uniqueness of bounded extensions. For an infinite packet sum one would need partial sums to converge to the declared operator, in a stated operator or strong topology. Neither density nor the finite identity by itself proves such a limit. In SG-42 density of","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces with an invertible transform. Let V:X→ Y be an isomorphism and (E_a) a finite complete packet family on Y. [SG-04 Pulled-back transform packets]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]","count":28},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d65dae5c5b4349f7","tex":"y_b","aliases":"y_b","display":"y_b","kind":"use","excerpt":"smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"t_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the","count":89},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1479ca52dd340112","tex":"z'","aliases":"z'","display":"z'","kind":"use","excerpt":"If z' gives the same data, the quotient v=z'/z satisfies v^N=v^r_1=⋯=v^r_k=1. Bezout's identity gives v^d=1. Conversely d divides every supplied exponent, so every dth root of unity has all those powers equal to one. Therefore the fiber consists exactly of z multiplied by the d distinct dth roots of unity. Applying the Bezout identity directly to t","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-34d4b252430fe95b","tex":"z^4","aliases":"z^4","display":"z^4","kind":"use","excerpt":"four they are m_4q=0, m_4q+1=2+4i, m_4q+2=0, m_4q+3=2-4i. Since the smooth degree is five, the first clean consecutive moment is n_0=6. With an order bound four, coefficients at n=6,…,13 determine the edge. The Hankel factorization has four distinct nodes and four nonzero amplitudes, hence rank four. Its recurrence is m_n+4-m_n=0, with monic polyno","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-54b855e49989865e","tex":"z^d","aliases":"z^d","display":"z^d","kind":"use","excerpt":"Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","excerpt":"The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-afc30c788aac6f77","tex":"z^N","aliases":"z^N","display":"z^N","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"use","excerpt":"The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. Th","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-33fb1cf2cddde12c","tex":"z^{n+1}","aliases":"z^n+1","display":"z^{n+1}","kind":"use","excerpt":"For one term, (E-z)(z^nP(n))=z^n+1(P(n+1)-P(n)). A finite difference lowers a nonconstant polynomial's degree by one, and annihilates a constant. After d+1 differences the term vanishes. The shift factors commute, so the product annihilates every term in the sum. If P_ℓ(n) has leading coefficient c_ℓ≠0 at degree d, substitution of (q_0+q)N+v has le","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-10d0329c35407a66","tex":"z^r","aliases":"z^r","display":"z^r","kind":"use","excerpt":"The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f302306eb76a618b","tex":"z^{r+1}","aliases":"z^r+1","display":"z^{r+1}","kind":"use","excerpt":"Each summand is overlinebz^r bz^r+1=|b|^2z because |z|=1. There are R summands and R≥1, so normalization returns z. For two channels direct division cancels b. No statistical independence is assumed, so one cannot improve a worst-case noise order merely by increasing R; the next bound shows the cancellation of the number of products.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-90767a4e71c6813f","tex":"z^{r_j}","aliases":"z^r_j","display":"z^{r_j}","kind":"use","excerpt":"smallDomain and hypotheses. Unit-modulus exponential nodes. Let z∈ C, |z|=1, and suppose u=z^N and w_j=z^r_j are known, with N≥1 and integer offsets r_j. Let d=gcd(N,r_1,…,r_k). [SG-06 The arithmetic of residual alias ambiguity]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ae33ff0c9a9cfbb4","tex":"z_j","aliases":"z_j","display":"z_j","kind":"use","excerpt":"If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-74e0eb883b673576","tex":"z_k","aliases":"z_k","display":"z_k","kind":"use","excerpt":"Expand a matrix entry to obtain the factorization. The leading r× r Vandermonde submatrix has determinant ∏_ℓ<k(z_k-z_ℓ), which is nonzero. Thus both rectangular Vandermonde factors have full column rank, and their transposes have full row rank. The diagonal factor is invertible. Its product with V_w^T is surjective onto C^r, and subsequent multipl","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-ed97dbea1d4aa8a8","tex":"z_\\ell","aliases":"z_ℓ ell","display":"z_\\ell","kind":"use","excerpt":"Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero act","count":27},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fd00cdc2b49367ea","tex":"z_\\ell^6","aliases":"z_ℓ^6 ell","display":"z_\\ell^6","kind":"use","excerpt":"The same object demonstrates internal phase recovery. Choose N=3 and q_0=2. The nodes are permuted by cubing, so the aliased nodes remain distinct. The sufficient self-calibration sets for L_max=4 are I_0={6,9,12,15,18,21,24,27,30}, I_1={7,10,13,16}. For each matched term the base coefficient is b_ℓ,0=a_ℓ z_ℓ^6, and the adjacent coefficient is b_ℓ,","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d7f4cd3b4cc1d7ef","tex":"z_\\ell^i","aliases":"z_ℓ^i ell","display":"z_\\ell^i","kind":"use","excerpt":"Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5a03f87b36ae5c02","tex":"z_\\ell^j","aliases":"z_ℓ^j ell","display":"z_\\ell^j","kind":"use","excerpt":"Writing V_h=(z_ℓ^i)_i,ℓ and V_w=(z_ℓ^j)_j,ℓ, H_h,w=V_hdiag(c_ℓ)V_w^ T, rankH_h,w=r. For r=0 the Hankel matrix is zero. The transpose is algebraic and is not a conjugate transpose.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-824dc780a2a10c0e","tex":"z_\\ell^k","aliases":"z_ℓ^k ell","display":"z_\\ell^k","kind":"use","excerpt":"smallDomain and hypotheses. Finite exponential moments over C. Let m_n_0+k=∑_ℓ=1^r c_ℓ z_ℓ^k, with distinct nonzero nodes and nonzero coefficients. Let H_h,w=(m_n_0+i+j)_0≤ i<h, 0≤ j<w, with h,w≥ r. [SG-19 Hankel rank detects the actual order]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2c0d8c9942f6dd27","tex":"z_\\ell^N","aliases":"z_ℓ^N ell","display":"z_\\ell^N","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-49d0429d56495b66","tex":"z_\\ell^n","aliases":"z_ℓ^n ell","display":"z_\\ell^n","kind":"use","excerpt":"Dj=∑_ℓ a_ℓδ_τ_ℓ, m_n:=2π inwidehat f(n)=∑_ℓ=1^ra_ℓ z_ℓ^n. For a non-bandlimited integrable background the same formula holds with an additive defect e_n=2π inwidehat s(n).","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-03e04fa74a6cba17","tex":"z_\\ell^{n-n_0}","aliases":"z_ℓ^n-n_0 ell","display":"z_\\ell^{n-n_0}","kind":"use","excerpt":"For every available recurrence index, ∑_k=0^rq_km_n+k =∑_ℓ c_ℓ z_ℓ^n-n_0Q(z_ℓ)=0, q_r=1. SG-19 makes H_r,r invertible. Thus the system has one solution and the true polynomial supplies it. Factoring the resulting polynomial recovers the distinct nodes up to a permutation. The amplitude matrix for consecutive moments is a square Vandermonde multipli","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e006e21561aca3f0","tex":"z_\\ell^{q_0N+v}","aliases":"z_ℓ^q_0N+v ell","display":"z_\\ell^{q_0N+v}","kind":"use","excerpt":")). A finite difference lowers a nonconstant polynomial's degree by one, and annihilates a constant. After d+1 differences the term vanishes. The shift factors commute, so the product annihilates every term in the sum. If P_ℓ(n) has leading coefficient c_ℓ≠0 at degree d, substitution of (q_0+q)N+v has leading q coefficient c_ℓ N^d. The remaining ex","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2bfb7ff95e6bcda6","tex":"z_\\ell^r","aliases":"z_ℓ^r ell","display":"z_\\ell^r","kind":"use","excerpt":"m_(q_0+q)N+r=∑_ℓ b_ℓ,ru_ℓ^q+e_(q_0+q)N+r, u_ℓ=z_ℓ^N, b_ℓ,r=a_ℓ u_ℓ^q_0z_ℓ^r. For each matched edge b_ℓ,r+1=z_ℓ b_ℓ,r. If |widehat s(n)|≤ C_s(1+|n|)^-p-2 with p≥0, then for nonnegative q,r |e_(q_0+q)N+r|≤2π C_s(q_0N)^-p-1.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"t_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the","count":71},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]","count":10},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"SG-21 writes both packets as translated sawtooth sums. Subtract in the form j-widehat j= ∑_ℓ(a_ℓ-widehat a_ℓ)G_T(·-τ_ℓ) +∑_ℓwidehat a_ℓ [G_T(·-τ_ℓ)-G_T(·-widehatτ_ℓ)]. Direct integration gives G_T _1=T/4. For a shift by d∈[0,T/2], on an interval of length T-d the difference has magnitude d/T, and on an interval of length d it has magnitude 1-d/T. H","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"Every vector α with α _0≤ s is uniquely determined by Aα if and only if ker A∩{h: h _0≤min(2s,d)}={0}. Equivalently every set of at most min(2s,d) columns is independent. With the convention spark(A)=d+1 when all d columns are independent, the equivalent condition is spark(A)>min(2s,d). For s=0 uniqueness is automatic.","count":17},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-59ce46150d0cfe77","tex":"\\alpha_0","aliases":"α_0 alpha","display":"\\alpha_0","kind":"use","excerpt":"smallDomain and hypotheses. The Euclidean model of SG-14. Assume s≥1, d≥1, and the injectivity condition of SG-14. Let y=s_0+Dα_0+η, s_0∈ S, α_0 _0≤ s, and Rη _2≤ϵ. Let widehatα be s-sparse with Awidehatα-Ry _2≤ϵ. [SG-15 Sparse noise stability and support thresholding]","count":9},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"The difference of two s-sparse vectors has support of size at most 2s and at most d. Thus the kernel condition implies uniqueness. Conversely, if a nonzero kernel vector h has support size at most 2s, partition that support into two sets of at most s indices. Let α equal h on the first set and zero elsewhere; let β equal -h on the second set. Then ","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"Define γ=min_substack h _2=1 h _0≤min(2s,d) Ah _2>0. Then widehatα-α_0 _2≤2ϵ/γ, D(widehatα-α_0) _2 ≤2 D _2ϵ/γ. If every nonzero true coefficient has magnitude greater than 4ϵ/γ, thresholding at 2ϵ/γ with a strict greater-than rule recovers the true support.","count":13},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8baf46bc7fda8c08","tex":"\\gamma_{\\ell,0,d}","aliases":"γ_ℓ,0,d gamma ell","display":"\\gamma_{\\ell,0,d}","kind":"use","excerpt":"∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leadin","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-be5d39271d254726","tex":"\\gamma_{\\ell,1,d}","aliases":"γ_ℓ,1,d gamma ell","display":"\\gamma_{\\ell,1,d}","kind":"use","excerpt":"∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leadin","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-28d86dd2a6c85554","tex":"\\gamma_{\\ell,v,d}","aliases":"γ_ℓ,v,d gamma ell","display":"\\gamma_{\\ell,v,d}","kind":"use","excerpt":"lowers a nonconstant polynomial's degree by one, and annihilates a constant. After d+1 differences the term vanishes. The shift factors commute, so the product annihilates every term in the sum. If P_ℓ(n) has leading coefficient c_ℓ≠0 at degree d, substitution of (q_0+q)N+v has leading q coefficient c_ℓ N^d. The remaining exponential prefactor is z","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-eaa8037f73e07539","tex":"\\gamma_{\\ell,v,h}","aliases":"γ_ℓ,v,h gamma ell","display":"\\gamma_{\\ell,v,h}","kind":"use","excerpt":"∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leadin","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"smallDomain and hypotheses. A square or overdetermined Vandermonde amplitude fit. Let V have full column rank, m=Va, widetilde V=V+Δ V, and widetilde m=m+e. Assume σ_min(V)> Δ V _2. Let widehat a minimize widetilde Vb-widetilde m _2. [SG-26 Amplitude stability with perturbed nodes]","count":11},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":20},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"The sawtooth derivative is DG_T=δ_0-T^-1dx: its slope is -1/T away from the endpoint and its endpoint jump is one. Summing translated copies gives the prescribed Dirac masses and a constant-density term -T^-1∑ a_ℓ, which vanishes. Consequently j is constant between the jump points. Each translated sawtooth has mean zero, so j does too. Any other pa","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1525b57dc165eb8a","tex":"\\delta_{ab}","aliases":"δ_ab delta","display":"\\delta_{ab}","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-78ebdc874668af02","tex":"\\delta_{\\tau_\\ell}","aliases":"δ_τ_ℓ delta tau ell","display":"\\delta_{\\tau_\\ell}","kind":"use","excerpt":"Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero act","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"smallDomain and hypotheses. The Euclidean model of SG-14. Assume s≥1, d≥1, and the injectivity condition of SG-14. Let y=s_0+Dα_0+η, s_0∈ S, α_0 _0≤ s, and Rη _2≤ϵ. Let widehatα be s-sparse with Awidehatα-Ry _2≤ϵ. [SG-15 Sparse noise stability and support thresholding]","count":29},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fb43c73461568514","tex":"\\epsilon^2","aliases":"ϵ^2 epsilon","display":"\\epsilon^2","kind":"use","excerpt":"smallDomain and hypotheses. A noisy progression as in SG-29. Let c_r=bz^r+e_r for 0≤ r≤ R, with R≥1, b≠0, |z|=1, and |e_r|≤ϵ. Put A=R|b|^2 and Δ=R(2|b|ϵ+ϵ^2). Assume Δ<A. [SG-30 Deterministic offset-phase perturbation]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-51399ec8ea079f5f","tex":"\\epsilon_n","aliases":"ϵ_n epsilon","display":"\\epsilon_n","kind":"use","excerpt":"smallDomain and hypotheses. A rectangular Hankel matrix of derivative moments. Let measured Fourier coefficients have errors η_n and moment errors ϵ_n=2π inη_n. Let E=(ϵ_n_0+i+j) have h≥1 rows and w≥1 columns. [SG-23 Hankel perturbation from Fourier noise]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-24ed22ccc61271b0","tex":"\\epsilon_{n_0+i+j}","aliases":"ϵ_n_0+i+j epsilon","display":"\\epsilon_{n_0+i+j}","kind":"use","excerpt":"smallDomain and hypotheses. A rectangular Hankel matrix of derivative moments. Let measured Fourier coefficients have errors η_n and moment errors ϵ_n=2π inη_n. Let E=(ϵ_n_0+i+j) have h≥1 rows and w≥1 columns. [SG-23 Hankel perturbation from Fourier noise]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"smallDomain and hypotheses. Finite complex offset data. Let R≥1, c_0≠0, and |ζ|=1. Define ν(c,ζ)=(∑_r=0^R-1|c_r+1-ζ c_r|^2)^1/2/(∑_r=0^R-1|c_r|^2)^1/2. [SG-31 A zero offset residual tests a progression]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8b66fbb722c79992","tex":"\\zeta^r","aliases":"ζ^r zeta","display":"\\zeta^r","kind":"use","excerpt":"Then ν(c,ζ)=0 if and only if c_r=c_0ζ^r for 0≤ r≤ R.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":17},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5ed478059bcd3710","tex":"\\eta_n","aliases":"η_n eta","display":"\\eta_n","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":6},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c45faae0260eabc9","tex":"\\theta_*","aliases":"θ_* theta","display":"\\theta_*","kind":"use","excerpt":"smallDomain and hypotheses. Finite real dictionaries with positive amplitudes. Let B have full column rank, P be the orthogonal projector onto its range, R=I-P, and let Q be a finite mask library with Rq≠0 for every q. Suppose y=Bθ_*+a_*q_*, with a_*>0 and q_*∈ Q. [SG-16 Profiled one-packet discovery and its finite gap]","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]","count":15},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-cafd172fec6b2b6e","tex":"\\lambda_M","aliases":"λ_M lambda","display":"\\lambda_M","kind":"use","excerpt":"smallDomain and hypotheses. The interval packet j=1_[-a,a) on T. Assume 0<a<π, let λ_M=M+1/2, and use the right edge coordinate x_M(z)=a-z/λ_M. [SG-12 The scaled interval-edge Gibbs profile]","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"use","excerpt":"smallDomain and hypotheses. Finite complex offset data. Let R≥1, c_0≠0, and |ζ|=1. Define ν(c,ζ)=(∑_r=0^R-1|c_r+1-ζ c_r|^2)^1/2/(∑_r=0^R-1|c_r|^2)^1/2. [SG-31 A zero offset residual tests a progression]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":46},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]","count":27},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-a9fc1ff1221962c3","tex":"\\rho'","aliases":"ρ' rho","display":"\\rho'","kind":"use","excerpt":"The N candidates lie on the unit circle at angles spaced by 2π/N. The smallest distance between distinct candidates is 2sin(π/N). The triangle inequality gives |widetilde z-ρ|≤|widetilde z-z|+|z-ρ|<sin(π/N). For any other candidate ρ', |widetilde z-ρ'|≥|ρ'-ρ|-|ρ-widetilde z| >sin(π/N). Thus the nearest candidate is unique. The same spacing argument","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord di","count":15},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8de4fa9e7f9c5804","tex":"\\sigma_j","aliases":"σ_j sigma","display":"\\sigma_j","kind":"use","excerpt":"The singular-value perturbation inequality [source reference] gives |σ_j(H)-σ_j(H_0)|≤ E _2 for 1≤ j≤ k. It holds equally over the complex field: in the min--max characterization σ_j(A)=min_dim V=w-j+1max_x∈ V, x =1 Ax . the triangle inequality bounds each restricted maximum for A+E by that for A plus E _2. Minimize and then interchange the two mat","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2b44b17ec5f59013","tex":"\\sigma_{k+1}","aliases":"σ_k+1 sigma","display":"\\sigma_{k+1}","kind":"use","excerpt":"Exactly r singular values of H exceed t. If r=0, a supplied threshold t> E _2 selects zero. If r=k, all k singular values exceed the threshold and no nonexistent σ_k+1 is used.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-68244408fe744f80","tex":"\\sigma_M","aliases":"σ_M sigma","display":"\\sigma_M","kind":"use","excerpt":"smallDomain and hypotheses. Periodic bounded real signals. Let σ_M=(M+1)^-1∑_k=0^MS_k be the Fejer mean. Suppose 0≤ j≤1 almost everywhere. [SG-13 Positive Fourier averaging preserves range]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-22301d4bf1c931e5","tex":"\\sigma_{\\min}","aliases":"σ_min sigma min","display":"\\sigma_{\\min}","kind":"use","excerpt":"smallDomain and hypotheses. A square or overdetermined Vandermonde amplitude fit. Let V have full column rank, m=Va, widetilde V=V+Δ V, and widetilde m=m+e. Assume σ_min(V)> Δ V _2. Let widehat a minimize widetilde Vb-widetilde m _2. [SG-26 Amplitude stability with perturbed nodes]","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f393e12f20aeb427","tex":"\\sigma_r","aliases":"σ_r sigma","display":"\\sigma_r","kind":"use","excerpt":"smallDomain and hypotheses. Finite rectangular complex matrices. Let H=H_0+E, with H_0 of rank r≥1, k=min(h,w)≥ r, and ϵ= E _2<σ_r(H_0)/2. Supply a threshold t with ϵ<t<σ_r(H_0)-ϵ. [SG-28 Order recovery with a supplied singular threshold]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"use","excerpt":"identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an L^1 norm occurs it is the unnormalized integral over a period. On the unit circle we use chord di","count":15},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-69258b952eb9362e","tex":"\\tau_\\ell","aliases":"τ_ℓ tau ell","display":"\\tau_\\ell","kind":"use","excerpt":"Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero act","count":14},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"S(c)=R|b|^2z≠0, Φ(c):=S(c)/|S(c)|=z. Two channels already give z=c_1/c_0 exactly. More channels provide repeated adjacent relations, without an unconditional deterministic averaging gain.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"For a smooth periodic test function φ, integrate by parts over each constant segment of j. Boundary terms telescope at each jump to a_ℓφ(τ_ℓ), which is exactly the displayed distribution. Evaluate the distribution on e^-inx: ⟨ Dj,e^-inx⟩ =-∫ j(x)(-in)e^-inx dx =2π inwidehat j(n). The Dirac sum gives ∑_ℓ a_ℓ e^-inτ_ℓ. For n>K, the smooth polynomial ","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"use","excerpt":"The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. Th","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":8},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]","count":76},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f07e36f47bcd4628","tex":"\\ell^1","aliases":"ℓ^1 ell","display":"\\ell^1","kind":"use","excerpt":"smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-35d9c98da5fdc9e5","tex":"\\ell_F","aliases":"ℓ_F ell","display":"\\ell_F","kind":"use","excerpt":"On X= S_K+span{m}, the functional ℓ_F(g)=D_F^-1∑_n∈ Foverlinewidehat m(n)widehat g(n) satisfies ℓ_F(s+am)=a. The decomposition is direct. If the measured coefficients have errors η_n, then |widetilde a-a|≤ D_F^-1/2(∑_n∈ F|η_n|^2)^1/2.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"smallDomain and hypotheses. The packet space of SG-01. Let S,T∈End(X). Set T_ba=P_bTP_a, viewed as a map X_a→ X_b. [SG-02 Transfer algebra and packet paths]","count":11},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Insert I=∑_aP_a on both sides of T to obtain reconstruction. Insert that identity between S and T to obtain the product formula. Equivalently, the block map is T↦ A T S; SG-01 shows that its inverse is B↦ S B A and that it preserves multiplication. The path formula follows by inserting the same identity between every successive pair of operators. T","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"ehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_q≠ q_*dist(Ry,C_q)>0, with the convention inf∅=+∞, then every perturb","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":34},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.","count":7},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2d3db36262598bcd","tex":"\\prod_j","aliases":"∏_j prod","display":"\\prod_j","kind":"use","excerpt":"Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-b2651115d5545c6f","tex":"\\prod_{j\\ne\\ell}","aliases":"∏_j≠ℓ prod ne ell","display":"\\prod_{j\\ne\\ell}","kind":"use","excerpt":"If for every ℓ sup_|z-z_ℓ|=ρ|E_Q(z)| <min_|z-z_ℓ|=ρ|Q(z)|, then widetilde Q has exactly one root, counted with multiplicity, in each disk |z-z_ℓ|<ρ. All its roots are thereby matched with error less than ρ. A sufficient explicit bound on each circle is ∑_k=0^r-1|widetilde q_k-q_k|(|z_ℓ|+ρ)^k <ρ∏_j≠ℓ(|z_ℓ-z_j|-ρ).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-058f6157633d37d5","tex":"\\prod_\\ell","aliases":"∏_ℓ prod ell","display":"\\prod_\\ell","kind":"use","excerpt":"The stacked system uniquely recovers the common annihilator Q(z)=∏_ℓ(z-z_ℓ). Its roots recover the common locations. With r consecutive moments per channel, a channelwise Vandermonde solve recovers every amplitude, including amplitudes that vanish in that channel.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-97674100d9dc2c04","tex":"\\prod_{\\ell<k}","aliases":"∏_ℓ<k prod ell","display":"\\prod_{\\ell<k}","kind":"use","excerpt":"Expand a matrix entry to obtain the factorization. The leading r× r Vandermonde submatrix has determinant ∏_ℓ<k(z_k-z_ℓ), which is nonzero. Thus both rectangular Vandermonde factors have full column rank, and their transposes have full row rank. The diagonal factor is invertible. Its product with V_w^T is surjective onto C^r, and subsequent multipl","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-cf9003aa14c53f1b","tex":"\\prod_{\\ell=1}^r","aliases":"∏_ℓ=1^r prod ell","display":"\\prod_{\\ell=1}^r","kind":"use","excerpt":"The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":47},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"smallDomain and hypotheses. Vector spaces over a field. Let P_1,…,P_m∈End(X) satisfy P_aP_b=δ_abP_a and ∑_aP_a=I_X. Set X_a=P_aX. [SG-01 Complete packet analysis and synthesis]","count":7},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-7684dda7bca2d3dd","tex":"\\sum_{a,b}","aliases":"∑_a,b sum","display":"\\sum_{a,b}","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-00042d3fcfa534d2","tex":"\\sum_{a=0}^{N-1}","aliases":"∑_a=0^N-1 sum","display":"\\sum_{a=0}^{N-1}","kind":"use","excerpt":"Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-dabbb4a95f09b94d","tex":"\\sum_{a_1,\\ldots,a_{k-1}}","aliases":"∑_a_1,…,a_k-1 sum ldots","display":"\\sum_{a_1,\\ldots,a_{k-1}}","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-159e0bfa6d438a0f","tex":"\\sum_{h=0}^d","aliases":"∑_h=0^d sum","display":"\\sum_{h=0}^d","kind":"use","excerpt":"∏_ℓ=1^r(E-z_ℓ)^d+1m=0. For offset v≥0, writing a decimated channel as m_(q_0+q)N+v=∑_ℓ u_ℓ^q ∑_h=0^dγ_ℓ,v,hq^h, u_ℓ=z_ℓ^N, the highest-degree coefficients satisfy γ_ℓ,1,d/γ_ℓ,0,d=z_ℓ. This ratio applies to separated, matched confluent terms with nonzero leading coefficient. The displayed annihilator is not asserted to have minimal order when leadin","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Exactly d unit-modulus nodes satisfy the same data. Hence the physical node is unique precisely when d=1. If integers c,b_1,…,b_k obey Nc+∑_jr_jb_j=d, then u^c∏_jw_j^b_j=z^d. For d=1 this is an explicit exact de-aliasing formula.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-24dfb51124d8ae91","tex":"\\sum_{j=1}^m","aliases":"∑_j=1^m sum","display":"\\sum_{j=1}^m","kind":"use","excerpt":"The first statement is the defining subtraction identity. Retaining r can carry every component omitted from q, so its data cost is not bounded by a small number of packet coefficients. For an image f∈ R^m, the standard pixel vectors satisfy f=∑_j=1^mf_je_j and form a complete finite basis. Those coefficients are all the pixels themselves. A progra","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0c6cb14840b1467d","tex":"\\sum_{k=0}^M","aliases":"∑_k=0^M sum","display":"\\sum_{k=0}^M","kind":"use","excerpt":"smallDomain and hypotheses. Periodic bounded real signals. Let σ_M=(M+1)^-1∑_k=0^MS_k be the Fejer mean. Suppose 0≤ j≤1 almost everywhere. [SG-13 Positive Fourier averaging preserves range]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-fec10602e4575bf9","tex":"\\sum_{k=0}^{n-1}","aliases":"∑_k=0^n-1 sum","display":"\\sum_{k=0}^{n-1}","kind":"use","excerpt":"The minimum singular value of widetilde V is at least σ_min(V)- Δ V _2>0. Thus its least-squares inverse exists. Since widetilde V a=m+Δ V a, widehat a-a=widetilde V^†(e-Δ V a). Apply widetilde V^† _2=1/σ_min(widetilde V). The power bound follows by factoring widehat z^n-z^n=(widehat z-z)∑_k=0^n-1widehat z^n-1-kz^k; each summand has modulus one. Th","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-6cef2056df2f0373","tex":"\\sum_{k=0}^r","aliases":"∑_k=0^r sum","display":"\\sum_{k=0}^r","kind":"use","excerpt":"For every available recurrence index, ∑_k=0^rq_km_n+k =∑_ℓ c_ℓ z_ℓ^n-n_0Q(z_ℓ)=0, q_r=1. SG-19 makes H_r,r invertible. Thus the system has one solution and the true polynomial supplies it. Factoring the resulting polynomial recovers the distinct nodes up to a permutation. The amplitude matrix for consecutive moments is a square Vandermonde multipli","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-55391d8e2ecb990e","tex":"\\sum_{k=0}^{r-1}","aliases":"∑_k=0^r-1 sum","display":"\\sum_{k=0}^{r-1}","kind":"use","excerpt":"The unique monic annihilator Q(z)=∏_ℓ=1^r(z-z_ℓ)=∑_k=0^r-1q_kz^k+z^r is recovered from H_r,rq=-v, v=(m_n_0+r+i)_i=0^r-1. Its roots are exactly the nodes. Once the nodes are known, r consecutive moments determine the coefficients uniquely by a Vandermonde solve. For unit-modulus nodes the corresponding locations are τ_ℓ=-arg z_ℓpmod2π.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-be822b230183da9b","tex":"\\sum_{n=1}^M","aliases":"∑_n=1^M sum","display":"\\sum_{n=1}^M","kind":"use","excerpt":"Direct integration gives widehat j(0)=a/π and widehat j(n)=sin(na)/(π n) for n≠0. The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by sum","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-615730d3b8a08359","tex":"\\sum_{n>M}","aliases":"∑_n>M sum","display":"\\sum_{n>M}","kind":"use","excerpt":"Cancel j from the defining identity. Absolute summability of the Fourier coefficients gives an everywhere uniformly convergent series for s. The omitted positive and negative frequencies are bounded by 2C∑_n>Me^-ρ n, which is the displayed geometric series. No limiting trigonometric approximation of j occurs in this estimate.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-41a5548d37bd81ad","tex":"\\sum_{n\\in F}","aliases":"∑_n∈ F sum in","display":"\\sum_{n\\in F}","kind":"use","excerpt":"smallDomain and hypotheses. Periodic integrable signals with one known edge shape. Let S_K be the trigonometric polynomials of degree at most K. Let m∈ L^1( T) and a finite nonempty set F⊂{n:|n|>K} satisfy D_F=∑_n∈ F|widehat m(n)|^2>0. [SG-09 Out-of-band matched identification]","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by summation by parts from bounded partial sums of e^2ina and by taking radial limits in ∑_n≥1r^ne^2ina/n=-log(1-re^2ia);","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-09ea8e176c2c6916","tex":"\\sum_{r=0}^{R-1}","aliases":"∑_r=0^R-1 sum","display":"\\sum_{r=0}^{R-1}","kind":"use","excerpt":"smallDomain and hypotheses. One matched unit-circle coefficient progression. Let R≥1 be an integer counting adjacent products, and let c_r=bz^r for 0≤ r≤ R, with b≠0 and |z|=1. Define S(c)=∑_r=0^R-1overlinec_rc_r+1. [SG-29 Exact phase from adjacent offset products]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-08dda244c4a2816c","tex":"\\sum_{t\\in\\mathbb Z}","aliases":"∑_t∈ Z sum in mathbb","display":"\\sum_{t\\in\\mathbb Z}","kind":"use","excerpt":"Interleaving the N channels recovers x exactly. With h_ba[k]=h[Nk+b-a], the output channels satisfy y_b[q]=∑_a=0^N-1∑_t∈ Zh_ba[q-t]x_a[t]. Thus convolution becomes an N× N transfer-kernel system.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-6999be63b2d4b543","tex":"\\sum_{\\rho\\in F_a}","aliases":"∑_ρ∈ F_a sum rho in","display":"\\sum_{\\rho\\in F_a}","kind":"use","excerpt":"smallDomain and hypotheses. Finite complete packet systems. Let (Q_ρ) refine (P_a), with a partition of fine labels into F_a and P_a=∑_ρ∈ F_aQ_ρ. [SG-03 Refinement preserves represented operators]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0c35b8475c6ded66","tex":"\\sum_{\\sigma\\in F_b}","aliases":"∑_σ∈ F_b sum sigma in","display":"\\sum_{\\sigma\\in F_b}","kind":"use","excerpt":"P_bTP_a=∑_σ∈ F_b∑_ρ∈ F_aQ_σ TQ_ρ. The coordinate changes J= A_Q S_P and C= A_P S_Q satisfy CJ=I and JC=I on their packet coordinate spaces. Refinement is exact before any child packet is discarded.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e3116f8319e2684e","tex":"\\sum_\\ell","aliases":"∑_ℓ sum ell","display":"\\sum_\\ell","kind":"use","excerpt":"Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero act","count":14},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c3e1d7ea3e919935","tex":"\\sum_{\\ell=1}^r","aliases":"∑_ℓ=1^r sum ell","display":"\\sum_{\\ell=1}^r","kind":"use","excerpt":"Let U_K,L consist of real periodic signals f=s+j, where s∈ S_K and j is right-continuous, piecewise constant, mean zero, and has at most L distinct active jumps. An active jump has nonzero amplitude. For an actual jump count r, write Dj=∑_ℓ=1^r a_ℓδ_τ_ℓ, z_ℓ=e^-iτ_ℓ, ∑_ℓ a_ℓ=0. The sum-zero condition is required by periodicity. A single nonzero act","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"aq-widetilde awidetilde q _2^2 · =|a-widetilde a|^2|suppq∩suppwidetilde q| · +|a|^2|suppq∖suppwidetilde q| +|widetilde a|^2|suppwidetilde q∖suppq|. For a=widetilde a=1 the squared norm is the cardinality of the symmetric support difference.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"The reconstruction error is exactly f- R_Mf=s-S_Ms, and f- R_Mf _∞ ≤frac2Ce^-ρ(M+1)1-e^-ρ. The bound is uniform across the jumps of j.","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"If Mf_0=Mf_1 but J(f_0)≠ J(f_1), no map R:Y→ J recovers the correct edge for both. If S∩ J≠{0}, the sum decomposition f=s+j is not unique. In particular, finitely many point samples on a periodic interval cannot recover every arbitrary interval packet.","count":4},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":5},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-dc1f0583051176b4","tex":"\\int_{-a}^{a}","aliases":"∫_-a^a int","display":"\\int_{-a}^{a}","kind":"use","excerpt":"Direct integration gives widehat j(0)=a/π and widehat j(n)=sin(na)/(π n) for n≠0. The Dirichlet kernel D_M(t)=sin(λ_Mt)/sin(t/2) yields S_Mj(x)=frac12π∫_-a^aD_M(x-t) dt, (S_Mj)'(x)=fracD_M(x+a)-D_M(x-a)2π. At x=a, the coefficient formula gives S_Mj(a)=frac aπ+frac1π∑_n=1^Mfracsin(2na)n ⟶frac12. For completeness, the sine-series limit follows by sum","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-2547054a0e887b21","tex":"\\int_0^{2\\pi}","aliases":"∫_0^2π int pi","display":"\\int_0^{2\\pi}","kind":"use","excerpt":"The default periodic domain is T= R/(2π Z), with widehat f(n)=frac12π∫_0^2πf(x)e^-inx dx. Signal identities inferred from Fourier coefficients are identities of L^1 classes. For a piecewise-constant edge we select the right-continuous representative. This convention fixes the values at jumps, which Fourier integrals alone cannot determine. When an ","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-8bcd55715febb3c0","tex":"\\int_0^z","aliases":"∫_0^z int","display":"\\int_0^z","kind":"use","excerpt":"Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Completeness gives S A x=x. If x_a∈ X_a, then P_ax_a=x_a and P_bx_a=0 for b≠ a. Thus the bth component of A S(x_a)_a is P_b∑_ax_a=x_b. Both inverse identities follow. For uniqueness, apply P_b to any proposed sum of components. This proof uses pairwise annihilation, not orthogonality; the packet projections can be oblique.","count":16},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"smallDomain and hypotheses. Absolutely summable sequences on Z. Fix an integer N≥1. Let h,x∈ℓ^1( Z) and y=h*x. Define x_a[q]=x[Nq+a] and y_b[q]=y[Nq+b], for 0≤ a,b<N. [SG-05 Polyphase analysis and convolution kernels]","count":71},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":48},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"smallDomain and hypotheses. Two-jump periodic packets of fixed known amplitudes. Fix a finite set F⊂ Z of derivative-moment indices. For 0<δ<π/4, compare locations (0,δ) and (π,π+δ), both with amplitudes (1,-1). [SG-08 No uniform location stability without separation]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"smallDomain and hypotheses. Finite real or complex Euclidean spaces. Let S⊆ F^m be a subspace, P its orthogonal projector, R=I-P, D∈ F^m× d, and A=RD. Let s≥0 be an integer. [SG-14 Residualized sparse injectivity]","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"use","excerpt":"The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-50db95a0004bcd02","tex":"\\bigoplus_a","aliases":"⊕_a bigoplus","display":"\\bigoplus_a","kind":"use","excerpt":"The maps A x=(P_ax)_a, S(x_a)_a=∑_ax_a are inverse isomorphisms between X and ⊕_aX_a. In particular, x=∑_aP_ax uniquely.","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"f-P_k+1f ≤ f-P_kf , f-P_kf⊥ V_k. If ⋃_kV_k is dense, then P_kf→ f in norm.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"f-P_k+1f ≤ f-P_kf , f-P_kf⊥ V_k. If ⋃_kV_k is dense, then P_kf→ f in norm.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-21851f6fd846dcf7","tex":"\\bigcup_k","aliases":"⋃_k bigcup","display":"\\bigcup_k","kind":"use","excerpt":"f-P_k+1f ≤ f-P_kf , f-P_kf⊥ V_k. If ⋃_kV_k is dense, then P_kf→ f in norm.","count":1},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"T=∑_a,bP_bTP_a, (ST)_ca=∑_bS_cbT_ba. The block representation is an algebra isomorphism to End(⊕_aX_a), with multiplication by contraction of intermediate packets. For every integer k≥1, iterating the product gives P_a_kT^kP_a_0 =∑_a_1,…,a_k-1 (P_a_kTP_a_k-1)⋯(P_a_1TP_a_0).","count":3},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"For an unrestricted scalar amplitude the least-squares profile is widehat a(q)=frac⟨ Rq,Ry⟩ Rq _2^2, J(q)= Ry-widehat a(q)Rq _2^2. The true mask has widehat a(q_*)=a_* and J(q_*)=0. It is the unique zero-residual positive-amplitude mask exactly when no other residualized mask lies on its positive ray. Define the closed rays C_q={aRq:a≥0}. If Γ=inf_","count":2},{"work":"signal-reconstruction","url":"notation/signal-reconstruction.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"Uniformly for z in compact real intervals, S_Mj(x_M(z))⟶frac12+frac1πSi(z), Si(z)=∫_0^zfracsin tt dt. The first positive maximum of the limiting profile is 1+G, where G=fracSi(π)π-frac12 =0.089489872236….","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"definition-passage","excerpt":"d ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such as Banach completion, a C^*-involution, a multiplier completion, or a m","count":38},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":332},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5371d95d86e6bf12","tex":"\\mathbf{1}","aliases":"1 mathbf","display":"\\mathbf{1}","kind":"definition-passage","excerpt":"[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":1096},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":253},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"definition-passage","excerpt":"[Linear grammar category] For a finite linear resolved presentation K_rm lin=(Q,I, P,Ref), let A_ K:=𝕜 Q/I. This is the 𝕜-linear category whose objects are the vertices of Q and whose morphisms are 𝕜-linear combinations of typed paths modulo the declared relations.","count":36},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4b82e0794ad5cb35","tex":"\\mathfrak A","aliases":"A mathfrak","display":"\\mathfrak A","kind":"definition-passage","excerpt":"[Certified Global synthesis atlas atlas] Fix a prime p, a characteristic-zero field K, and a finite Galois extension L/K for the Galois component. The certified atlas A_13 is the finite coproduct of the following five connected profiled atlases.","count":17},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-64209c7327ae53ce","tex":"\\mathfrak a","aliases":"a mathfrak","display":"\\mathfrak a","kind":"definition-passage","excerpt":"[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-36e242206d1129a8","tex":"\\mathsf A","aliases":"A mathsf","display":"\\mathsf A","kind":"definition-passage","excerpt":"[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interprete","count":500},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cafa6dff58b9e731","tex":"A^r_{U,i}","aliases":"A^r_U,i","display":"A^r_{U,i}","kind":"definition-passage","excerpt":"For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrigh","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3a88ad49c8dee806","tex":"A^r_{V,\\lambda^A_{\\gamma,r}(i)}","aliases":"A^r_V,λ^A_γ,r(i) lambda gamma","display":"A^r_{V,\\lambda^A_{\\gamma,r}(i)}","kind":"definition-passage","excerpt":"ject A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the ad","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3f481513c595a7d7","tex":"\\mathcal A_{\\mathbb K}","aliases":"A_ K mathcal mathbb","display":"\\mathcal A_{\\mathbb K}","kind":"definition-passage","excerpt":"[Linear grammar category] For a finite linear resolved presentation K_rm lin=(Q,I, P,Ref), let A_ K:=𝕜 Q/I. This is the 𝕜-linear category whose objects are the vertices of Q and whose morphisms are 𝕜-linear combinations of typed paths modulo the declared relations.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"definition-passage","excerpt":"[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"definition-passage","excerpt":"[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b6849b039ee0d0ce","tex":"\\mathfrak A_{13}","aliases":"A_13 mathfrak","display":"\\mathfrak A_{13}","kind":"definition-passage","excerpt":"[Certified Global synthesis atlas atlas] Fix a prime p, a characteristic-zero field K, and a finite Galois extension L/K for the Galois component. The certified atlas A_13 is the finite coproduct of the following five connected profiled atlases.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f1938222efb486e3","tex":"a_{[u]_k+k^{|u|}[v]_k}","aliases":"a_[u]_k+k^|u|[v]_k","display":"a_{[u]_k+k^{|u|}[v]_k}","kind":"definition-passage","excerpt":"[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-18e745043bd4a8c1","tex":"a_{[w]_k}","aliases":"a_[w]_k","display":"a_{[w]_k}","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-08eb65adb1a856b8","tex":"A_b","aliases":"A_b","display":"A_b","kind":"definition-passage","excerpt":"[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3581b621d385d761","tex":"A_d","aliases":"A_d","display":"A_d","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c5b197d46eb7050e","tex":"A_{d_0}","aliases":"A_d_0","display":"A_{d_0}","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1d69a4c99ff98ae1","tex":"A_{d_{\\ell-1}}","aliases":"A_d_ℓ-1 ell","display":"A_{d_{\\ell-1}}","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bddd52c978a480a2","tex":"\\mathfrak A_{\\mathrm{DRLM},p}","aliases":"A_DRLM,p mathfrak mathrm","display":"\\mathfrak A_{\\mathrm{DRLM},p}","kind":"definition-passage","excerpt":"[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prim","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3205cc18ccaeff8a","tex":"\\mathfrak A_{\\mathrm{EM}}","aliases":"A_EM mathfrak mathrm","display":"\\mathfrak A_{\\mathrm{EM}}","kind":"definition-passage","excerpt":"[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prim","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f5cf7dd838e42b29","tex":"\\mathfrak A_{\\mathrm{Fuss}}","aliases":"A_Fuss mathfrak mathrm","display":"\\mathfrak A_{\\mathrm{Fuss}}","kind":"definition-passage","excerpt":"[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prim","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d05e640e689df6cd","tex":"\\mathfrak A_{\\mathrm{Gal},L/K}","aliases":"A_Gal,L/K mathfrak mathrm","display":"\\mathfrak A_{\\mathrm{Gal},L/K}","kind":"definition-passage","excerpt":"component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c0307cd43e859d23","tex":"A_j","aliases":"A_j","display":"A_j","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-da2a817868989569","tex":"A_{k-1}","aliases":"A_k-1","display":"A_{k-1}","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a93127a25ff5f1b7","tex":"\\mathfrak A_{\\mathrm{Loc},p}","aliases":"A_Loc,p mathfrak mathrm","display":"\\mathfrak A_{\\mathrm{Loc},p}","kind":"definition-passage","excerpt":"[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prim","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ce233bc7fb0cd555","tex":"\\mathfrak a_N","aliases":"a_N mathfrak","display":"\\mathfrak a_N","kind":"definition-passage","excerpt":"[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"definition-passage","excerpt":"[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.","count":29},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2a5031d697e77f22","tex":"A_U","aliases":"A_U","display":"A_U","kind":"definition-passage","excerpt":"For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrigh","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d511697a146393f2","tex":"A_w","aliases":"A_w","display":"A_w","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c99bb6126c907964","tex":"a_{\\sigma-1}","aliases":"a_σ-1 sigma","display":"a_{\\sigma-1}","kind":"definition-passage","excerpt":"[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b9711ee2915de9c0","tex":"\\Add","aliases":"Add operatorname","display":"{\\operatorname{Add}}","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8027e5b7e2962147","tex":"\\Analysis","aliases":"Analysis Analysis A mathcal","display":null,"kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f946bf9346c8fc8a","tex":"\\Analysis_{\\mathcal P}","aliases":"Analysis_ P A_ P Analysis mathcal mathcal mathcal","display":"{\\mathcal A}_{\\mathcal P}","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-87708a8e37000af2","tex":"\\operatorname{Ann}","aliases":"Ann operatorname","display":"\\operatorname{Ann}","kind":"definition-passage","excerpt":"[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-136beb0587bed266","tex":"\\operatorname{Ann}_{\\mathcal D}","aliases":"Ann_ D operatorname mathcal","display":"\\operatorname{Ann}_{\\mathcal D}","kind":"definition-passage","excerpt":"[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c8353bcb1d523db5","tex":"\\operatorname{Ann}_{\\rm tail}","aliases":"Ann_rm tail operatorname rm","display":"\\operatorname{Ann}_{\\rm tail}","kind":"definition-passage","excerpt":"[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":152},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-58f9b1ca3c57804b","tex":"\\mathfrak b","aliases":"b mathfrak","display":"\\mathfrak b","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d1e4f21b86ad3668","tex":"\\mathsf B","aliases":"B mathsf","display":"\\mathsf B","kind":"definition-passage","excerpt":"[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":175},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-645d4a0021852301","tex":"B_\\mathfrak","aliases":"B_ mathfrak","display":"B_\\mathfrak","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f56251a6f87c66b1","tex":"b_d","aliases":"b_d","display":"b_d","kind":"definition-passage","excerpt":"[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c5d91194cc38edd2","tex":"B_i","aliases":"B_i","display":"B_i","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"definition-passage","excerpt":"[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fc9ad75de14545d5","tex":"b_m","aliases":"b_m","display":"b_m","kind":"definition-passage","excerpt":"[Euler-coordinate map] For b=(b_m)_m≥1∈ E_ Z, define E(b)(z) = ∏_m≥1(1-z^m)^-b_m. The product is coefficientwise filtered because the mth factor is congruent to 1 modulo z^m.","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-37cfe55080a4bdf4","tex":"\\bbK","aliases":"bbK K mathbb","display":"{\\mathbb K}","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-891c070aa0fd54dd","tex":"\\Beh","aliases":"Beh","display":null,"kind":"definition-passage","excerpt":"lved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object i","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d10f1cc7616e4874","tex":"\\Beh_{\\mathsf S}","aliases":"Beh_ S Beh mathsf","display":null,"kind":"definition-passage","excerpt":"lved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object i","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"definition-passage","excerpt":"[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a16d7db299007129","tex":"\\operatorname{Br}","aliases":"Br operatorname","display":"\\operatorname{Br}","kind":"definition-passage","excerpt":"[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F},","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-326f3e1a7b217765","tex":"\\operatorname{Br}_0","aliases":"Br_0 operatorname","display":"\\operatorname{Br}_0","kind":"definition-passage","excerpt":"[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F},","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8e3c0fce1b7a60ae","tex":"\\bullet","aliases":"bullet","display":"\\bullet","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"d ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such as Banach completion, a C^*-involution, a multiplier completion, or a m","count":179},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"definition-passage","excerpt":"[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interprete","count":54},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ab8bde9a479f56a0","tex":"\\mathfrak C","aliases":"C mathfrak","display":"\\mathfrak C","kind":"definition-passage","excerpt":"[Constructor universe] Fix a declared universe C of constructors and universal properties. Typical entries are 0, 1, finite sums, composition, scalar action, biproducts,","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bbc9ba3c4c4e2bc2","tex":"\\mathfrak c","aliases":"c mathfrak","display":"\\mathfrak c","kind":"definition-passage","excerpt":"[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-52f862c0486a95ce","tex":"\\mathsf C","aliases":"C mathsf","display":"\\mathsf C","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"definition-passage","excerpt":"[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.","count":160},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6023b95954c36f1d","tex":"C^*","aliases":"C^*","display":"C^*","kind":"definition-passage","excerpt":"d ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a list of optional completion tags. A tag names a doctrine to be added later, such as Banach completion, a C^*-involution, a multiplier completion, or a m","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6bfba5cbf12e1727","tex":"C^\\bullet","aliases":"C^bullet bullet","display":"C^\\bullet","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-70f1ea36ff493278","tex":"C^{n+1}","aliases":"C^n+1","display":"C^{n+1}","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3757a2a1d626242a","tex":"\\mathfrak c^{\\rm tuple}_{N,j}","aliases":"c^rm tuple_N,j mathfrak rm","display":"\\mathfrak c^{\\rm tuple}_{N,j}","kind":"definition-passage","excerpt":"[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-227ceb7baf501818","tex":"c_d","aliases":"c_d","display":"c_d","kind":"definition-passage","excerpt":"[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1f176461b1ea53f6","tex":"C_E","aliases":"C_E","display":"C_E","kind":"definition-passage","excerpt":"[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"definition-passage","excerpt":"[Integral saturation defect] The finite abelian group C_k(a) = L_k(a)/M_k(a) is the integral saturation defect. Its order δ_k(a) = [L_k(a):M_k(a)] is the saturation index.","count":20},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-475a2c7034fe4719","tex":"\\mathsf C_k","aliases":"C_k mathsf","display":"\\mathsf C_k","kind":"definition-passage","excerpt":"[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"definition-passage","excerpt":"[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e9ef8a18e5113bfa","tex":"\\mathcal C_{N,j}","aliases":"C_N,j mathcal","display":"\\mathcal C_{N,j}","kind":"definition-passage","excerpt":"[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7d740b1ea82be06f","tex":"\\mathfrak c_{N,j}","aliases":"c_N,j mathfrak","display":"\\mathfrak c_{N,j}","kind":"definition-passage","excerpt":"[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-61379e40b103d72c","tex":"\\mathsf C_p","aliases":"C_p mathsf","display":"\\mathsf C_p","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":17},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-af557f004b3de499","tex":"\\mathcal C_U","aliases":"C_U mathcal","display":"\\mathcal C_U","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0b87a98bdf27d6a5","tex":"\\mathcal C_w","aliases":"C_w mathcal","display":"\\mathcal C_w","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cdb4e1b958753681","tex":"\\mathcal C_\\alpha","aliases":"C_α mathcal alpha","display":"\\mathcal C_\\alpha","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5b779703a5dd1742","tex":"\\Cat","aliases":"Cat mathbf","display":"{\\mathbf{Cat}}","kind":"definition-passage","excerpt":"[Finite profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap ca","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ccab83dfdb306d04","tex":"\\operatorname{cdim}","aliases":"cdim operatorname","display":"\\operatorname{cdim}","kind":"definition-passage","excerpt":"[Address-preserving cyclic dimension] The cyclic packet dimension of the marked system is cdim_ P(Φ) = ∑_C cyclic SCC|C|. tag11.7 It measures the number of packet variables that remain in coupled nonlinear components after exact elimination of the acyclic singleton components.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-819aa35c739f8ca3","tex":"\\operatorname{cdim}_{\\mathcal P}","aliases":"cdim_ P operatorname mathcal","display":"\\operatorname{cdim}_{\\mathcal P}","kind":"definition-passage","excerpt":"[Address-preserving cyclic dimension] The cyclic packet dimension of the marked system is cdim_ P(Φ) = ∑_C cyclic SCC|C|. tag11.7 It measures the number of packet variables that remain in coupled nonlinear components after exact elimination of the acyclic singleton components.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-27b5a2e3b286b58f","tex":"\\mathsf{Cert}","aliases":"Cert mathsf","display":"\\mathsf{Cert}","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c9436aea1bf2444f","tex":"\\check","aliases":"check","display":null,"kind":"definition-passage","excerpt":"[Finite profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap ca","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-80bb66e28ec49c6d","tex":"\\Cl","aliases":"Cl","display":null,"kind":"definition-passage","excerpt":"[Extensional task] An extensional task is a closed query theory, equivalently a query family modulo equality of the closure Cl. When the admitted query universe separates the desired quotients, an extensional task can equivalently be represented by its observationally definable congruence.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c01648ee0438f528","tex":"\\operatorname{Coim}","aliases":"Coim operatorname","display":"\\operatorname{Coim}","kind":"definition-passage","excerpt":"with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object is the coimage Q_ S:=Coim(Beh_ S) whenever that coimage exists.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-073a71f814bdfa5f","tex":"\\coker","aliases":"coker coker operatorname","display":null,"kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e6d0cbeeb71722a","tex":"\\operatorname{coker}","aliases":"coker operatorname","display":"\\operatorname{coker}","kind":"definition-passage","excerpt":"[Ghost and missing reduction modules] Define G_k,m(a) = ker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ) and Q_k,m(a) = coker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). The first records raw residue coordinates that represent the zero semantic sequence. The second records saturated residue classes not reached by raw sections.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-67df59e7be0df569","tex":"\\operatorname{Coll}","aliases":"Coll operatorname","display":"\\operatorname{Coll}","kind":"definition-passage","excerpt":"[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The f","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3a573fe09d58c3a7","tex":"\\operatorname{Coll}_B","aliases":"Coll_B operatorname","display":"\\operatorname{Coll}_B","kind":"definition-passage","excerpt":"[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The f","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-593c8651c42f135c","tex":"\\mathsf{Comp}","aliases":"Comp mathsf","display":"\\mathsf{Comp}","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8d94488f5df4843b","tex":"\\Cone","aliases":"Cone operatorname","display":"{\\operatorname{Cone}}","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fdff86c47213d208","tex":"\\Core","aliases":"Core operatorname","display":"{\\operatorname{Core}}","kind":"definition-passage","excerpt":"realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theorem is stated. item Sem_α: M_α⟶ O_α is the native semantics functor. item I_α:Core( N_α)⟶ D_α is a declared invariant functor, when one has ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d78d120bcb436543","tex":"\\operatorname{csupp}","aliases":"csupp operatorname","display":"\\operatorname{csupp}","kind":"definition-passage","excerpt":"[Proof-support certificate] A proof-support certificate for a finite proof P is a finite subset csupp( P)⊆ C containing every constructor and universal property used in the proof. The certificate need not be minimal. It must be sufficient and auditable.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":137},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-eae54bb0ec4239ae","tex":"\\mathbf d","aliases":"d mathbf","display":"\\mathbf d","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"definition-passage","excerpt":"[Verified preservation profile] A verified preservation profile Π_F for a functor F: C→ D consists of a subset of C together with a proof or declared comparison isomorphism showing that F preserves each listed constructor. We write c∈Π_F only after the corresponding preservation claim has been verified.","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Karoubi envelope] The Karoubi envelope Kar(Add( A_ K)) has pairs (A,e), where e:A→ A is idempotent, as objects. A morphism f:(A,e)⟶(B,d) is a morphism f:A→ B satisfying f=dfe. The identity on (A,e) is e.","count":225},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"definition-passage","excerpt":"[Doctrine-indexed task fiber] For a doctrine D and represented object X, let Task_D(X) denote the ordered family of continuation-stable semantic quotients admitted by the declared query universe. Inside one fiber, kernel or congruence calculus governs task change. A change of doctrine D⟶ D' requires a bridge passport specifying carrier transport, p","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-87a4659e9ddda329","tex":"D^j","aliases":"D^j","display":"D^j","kind":"definition-passage","excerpt":"[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1b655d23a365d7e4","tex":"d^N","aliases":"d^N","display":"d^N","kind":"definition-passage","excerpt":"[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.","count":23},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-09e71d50145ecad3","tex":"D^r","aliases":"D^r","display":"D^r","kind":"definition-passage","excerpt":"[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5c308718fcbd3df8","tex":"d_C","aliases":"d_C","display":"d_C","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-eb69ef05a69fcd3e","tex":"d_{\\Cone}","aliases":"d_Cone Cone operatorname","display":"d_{{\\operatorname{Cone}}}","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a3008bb1b6de0e7c","tex":"d_E","aliases":"d_E","display":"d_E","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7277d5a17e7916e9","tex":"\\mathcal D_K","aliases":"D_K mathcal","display":"\\mathcal D_K","kind":"definition-passage","excerpt":"[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aafbdea62a4e791c","tex":"d_X","aliases":"d_X","display":"d_X","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6f05263dc9eb489b","tex":"\\mathcal D_\\alpha","aliases":"D_α mathcal alpha","display":"\\mathcal D_\\alpha","kind":"definition-passage","excerpt":"realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theorem is stated. item Sem_α: M_α⟶ O_α is the native semantics functor. item I_α:Core( N_α)⟶ D_α is a declared invariant functor, when one has ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ed6ff526c435c5c5","tex":"d_\\rho","aliases":"d_ρ rho","display":"d_\\rho","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f7fb4a3491022b72","tex":"d_{\\ell-1}","aliases":"d_ℓ-1 ell","display":"d_{\\ell-1}","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-26d83baff19c71bc","tex":"\\Desc","aliases":"Desc Desc operatorname","display":null,"kind":"definition-passage","excerpt":"[Descent category] The category Desc( R) has pseudocones over R as objects and modifications as morphisms. Concretely, an object is a compatible family (M_α,θ_αβ) with invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.","count":22},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-337441eb94c64c38","tex":"\\dim_{\\mathbb F_p}","aliases":"dim_ F_p dim mathbb","display":"\\dim_{\\mathbb F_p}","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-98d940b7f3ffe088","tex":"\\dim_K","aliases":"dim_K dim","display":"\\dim_K","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c2a7d3d0f0d1ca66","tex":"\\mathfrak{Dir}","aliases":"Dir mathfrak","display":"\\mathfrak{Dir}","kind":"definition-passage","excerpt":"[Typed directed transformation category] The directed transformation category Dir( K) has pairs (α,M), with M∈ M_α, as objects. A morphism (α,M)⟶(β,N) is an admitted bridge instance equipped with a bridge passport as in [source reference]. Composition is defined only when the codomain profile of the first bridge satisfies the domain requirements of","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b7a268c3b16fa034","tex":"\\mathrm{DRLM}","aliases":"DRLM mathrm","display":"\\mathrm{DRLM}","kind":"definition-passage","excerpt":"[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prim","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.","count":138},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"definition-passage","excerpt":"[Euler-coordinate map] For b=(b_m)_m≥1∈ E_ Z, define E(b)(z) = ∏_m≥1(1-z^m)^-b_m. The product is coefficientwise filtered because the mth factor is congruent to 1 modulo z^m.","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Karoubi envelope] The Karoubi envelope Kar(Add( A_ K)) has pairs (A,e), where e:A→ A is idempotent, as objects. A morphism f:(A,e)⟶(B,d) is a morphism f:A→ B satisfying f=dfe. The identity on (A,e) is e.","count":159},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f06630fc4bbd48b8","tex":"E^\\bullet","aliases":"E^bullet bullet","display":"E^\\bullet","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e40ba00de6234ae1","tex":"E^n","aliases":"E^n","display":"E^n","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-53f96d10187d14f0","tex":"\\mathcal E_{\\mathbb Z}","aliases":"E_ Z mathcal mathbb","display":"\\mathcal E_{\\mathbb Z}","kind":"definition-passage","excerpt":"[Euler-coordinate map] For b=(b_m)_m≥1∈ E_ Z, define E(b)(z) = ∏_m≥1(1-z^m)^-b_m. The product is coefficientwise filtered because the mth factor is congruent to 1 modulo z^m.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6f6546bd1d109f4f","tex":"\\mathrm{EM}","aliases":"EM mathrm","display":"\\mathrm{EM}","kind":"definition-passage","excerpt":"[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prim","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End End operatorname","display":null,"kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-70a9953dd9bd9b22","tex":"\\operatorname{End}","aliases":"End operatorname","display":"\\operatorname{End}","kind":"definition-passage","excerpt":"[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e2e963e279b9f760","tex":"\\End_{\\mathcal C}","aliases":"End_ C End mathcal","display":null,"kind":"definition-passage","excerpt":"[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, togeth","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-15474d1cea1e947b","tex":"\\End_K","aliases":"End_K End operatorname","display":"{\\operatorname{End}}_K","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b9a7980adbd81d35","tex":"\\operatorname{EssIm}","aliases":"EssIm operatorname","display":"\\operatorname{EssIm}","kind":"definition-passage","excerpt":"[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The f","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5ab4bba0c680ebe5","tex":"\\mathbf{EulMult}","aliases":"EulMult mathbf","display":"\\mathbf{EulMult}","kind":"definition-passage","excerpt":"[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f258b59898cd7bc5","tex":"\\mathbf{EulMult}_{\\mathbb Z}","aliases":"EulMult_ Z mathbf mathbb","display":"\\mathbf{EulMult}_{\\mathbb Z}","kind":"definition-passage","excerpt":"[Multiplicative Euler-overlap category] Let EulMult_ Z be the discrete category of multiplicative exponent functions b: N_≥1→ Z, b(1)=1. Associate to b the four normalized charts B(b)=b, A(b)=A_b(z), C(b)= G(b), P(b)=(b(p^e),c(p^e))_p,e.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e84b1a98b5a23eb6","tex":"\\operatorname{ev}","aliases":"ev operatorname","display":"\\operatorname{ev}","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7caa90106da4f488","tex":"\\operatorname{ev}_0","aliases":"ev_0 operatorname","display":"\\operatorname{ev}_0","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.","count":277},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":61},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":186},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-49ddaa56c96d4e05","tex":"f^{-1}","aliases":"f^-1","display":"f^{-1}","kind":"definition-passage","excerpt":"[Task-compatible bridge] The bridge is task compatible when f(N_X)⊆ N_Y. Equivalently, N_X⊆ f^-1(N_Y). Then there is a unique induced map overline f:X/N_X⟶ Y/N_Y.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3053e1f607009b01","tex":"f_{ai}","aliases":"f_ai","display":"f_{ai}","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-50978a3c48982e30","tex":"F_{f,p}","aliases":"F_f,p","display":"F_{f,p}","kind":"definition-passage","excerpt":"[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8e49021145d6ab04","tex":"F_m","aliases":"F_m","display":"F_m","kind":"definition-passage","excerpt":"[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9da98cb9c6f0ae74","tex":"\\mathbb F_p","aliases":"F_p mathbb","display":"\\mathbb F_p","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":31},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ae298805dda59c8e","tex":"F_\\gamma","aliases":"F_γ gamma","display":"F_\\gamma","kind":"definition-passage","excerpt":"ject A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the ad","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-48e3fa1a1d9f9cd0","tex":"\\operatorname{Fitt}","aliases":"Fitt operatorname","display":"\\operatorname{Fitt}","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-956f20e497452b52","tex":"\\operatorname{Fitt}_0","aliases":"Fitt_0 operatorname","display":"\\operatorname{Fitt}_0","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cb7f1a6b37de83fe","tex":"\\operatorname{Fitt}_{\\rho-1}","aliases":"Fitt_ρ-1 operatorname rho","display":"\\operatorname{Fitt}_{\\rho-1}","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4a5a5feeeb6fd3dd","tex":"\\Fun","aliases":"Fun operatorname","display":"{\\operatorname{Fun}}","kind":"definition-passage","excerpt":"[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functor","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5cca02d51208e31e","tex":"\\Fun_\\Pi","aliases":"Fun_Π Fun Pi operatorname Pi","display":"{\\operatorname{Fun}}_\\Pi","kind":"definition-passage","excerpt":"[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functor","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-31c6e76be1521c47","tex":"\\mathrm{Fuss}","aliases":"Fuss mathrm","display":"\\mathrm{Fuss}","kind":"definition-passage","excerpt":"[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prim","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":144},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"definition-passage","excerpt":"[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3a9e648245c571b5","tex":"\\mathfrak G","aliases":"G mathfrak","display":"\\mathfrak G","kind":"definition-passage","excerpt":"[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The f","count":106},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-514f8a70862bdc2d","tex":"G_{\\mathbf d}","aliases":"G_ d mathbf","display":"G_{\\mathbf d}","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7e44c8e966347dc0","tex":"\\mathcal G_{\\mathbb Z}","aliases":"G_ Z mathcal mathbb","display":"\\mathcal G_{\\mathbb Z}","kind":"definition-passage","excerpt":"[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f49b4464b07d1b5d","tex":"G_{k,m}","aliases":"G_k,m","display":"G_{k,m}","kind":"definition-passage","excerpt":"[Ghost and missing reduction modules] Define G_k,m(a) = ker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ) and Q_k,m(a) = coker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). The first records raw residue coordinates that represent the zero semantic sequence. The second records saturated residue classes not reached by raw sections.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6eb7115d056840da","tex":"G_{N,j}^{\\lambda,\\alpha}","aliases":"G_N,j^λ,α lambda alpha","display":"G_{N,j}^{\\lambda,\\alpha}","kind":"definition-passage","excerpt":"[Marked binomial--Fuchsian packet] A marked packet is a tuple b=(α,λ,N,j;G), where α≠0, N≥1, 0≤ j<N, and G(x)=G_N,j^λ,α(x). The parameters (α,λ,N,j) are part of the semantic object rather than metadata to be forgotten.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c28a753638b31e36","tex":"\\mathrm{Gal}","aliases":"Gal mathrm","display":"\\mathrm{Gal}","kind":"definition-passage","excerpt":"component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prime-uniform multiplicative component; item A_Gal,L/K, the finite semisimple","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.","count":152},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.","count":53},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-047a2d7f6191ab3d","tex":"H_a","aliases":"H_a","display":"H_a","kind":"definition-passage","excerpt":"[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e2d25bbe861bf735","tex":"h_{a i}","aliases":"h_a i","display":"h_{a i}","kind":"definition-passage","excerpt":"y Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, as morphisms. Composition is matrix contract","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-24d533e191d3b078","tex":"h_{ai}","aliases":"h_ai","display":"h_{ai}","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ce41b67c7e8dafd4","tex":"h_U","aliases":"h_U","display":"h_U","kind":"definition-passage","excerpt":"⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a049a97dc219bcd6","tex":"h_V","aliases":"h_V","display":"h_V","kind":"definition-passage","excerpt":"⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) (β^X_γ,r,i)^-1. The group of compatible sections is Γ (mathscr G,mathscr M^u,r_Y,X) := {(h_U)_U:γ_*h_U=h_V for every γ:U→ V}.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-166165104e6a5a12","tex":"\\mathbf{Hol}","aliases":"Hol mathbf","display":"\\mathbf{Hol}","kind":"definition-passage","excerpt":"[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fc33878dfb7b79e0","tex":"\\mathbf{Hol}^{\\rm ord}_K","aliases":"Hol^rm ord_K mathbf rm","display":"\\mathbf{Hol}^{\\rm ord}_K","kind":"definition-passage","excerpt":"[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom Hom operatorname","display":null,"kind":"definition-passage","excerpt":"[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The f","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cd3e7429be54f279","tex":"\\Hom_{\\mathcal C_U}","aliases":"Hom_ C_U Hom mathcal","display":null,"kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1fc8a506836e4143","tex":"\\Hom_{\\mathcal U}","aliases":"Hom_ U Hom mathcal operatorname mathcal","display":"{\\operatorname{Hom}}_{\\mathcal U}","kind":"definition-passage","excerpt":"[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The f","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":99},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":177},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-83886be80b02e984","tex":"I_{\\mathcal P}","aliases":"I_ P mathcal","display":"I_{\\mathcal P}","kind":"definition-passage","excerpt":"[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9a85b9142fc7596f","tex":"I_{\\mathcal Q}","aliases":"I_ Q mathcal","display":"I_{\\mathcal Q}","kind":"definition-passage","excerpt":"[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d64662fd360cdfa7","tex":"I_{A,U,r}","aliases":"I_A,U,r","display":"I_{A,U,r}","kind":"definition-passage","excerpt":"For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrigh","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e00841bb1b56a954","tex":"I_{A,V,r}","aliases":"I_A,V,r","display":"I_{A,V,r}","kind":"definition-passage","excerpt":"For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrigh","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4ca4bbde7d138543","tex":"\\mathcal I_k","aliases":"I_k mathcal","display":"\\mathcal I_k","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e9855187f190ff70","tex":"I_r","aliases":"I_r","display":"I_r","kind":"definition-passage","excerpt":"[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-111b6ee2c5f8dbd2","tex":"I_s","aliases":"I_s","display":"I_s","kind":"definition-passage","excerpt":"[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"definition-passage","excerpt":"[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a8bd503d741eaa3","tex":"I_{X,r}","aliases":"I_X,r","display":"I_{X,r}","kind":"definition-passage","excerpt":"ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accomp","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ef0567be6068f3e2","tex":"I_{X,s}","aliases":"I_X,s","display":"I_{X,s}","kind":"definition-passage","excerpt":"ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accomp","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fe329eeb562326b5","tex":"I_{X,U,r}","aliases":"I_X,U,r","display":"I_{X,U,r}","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ab3ef27be5e3d7f7","tex":"I_Y","aliases":"I_Y","display":"I_Y","kind":"definition-passage","excerpt":"[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-601d52e57eecfbf3","tex":"I_{Y,U,u}","aliases":"I_Y,U,u","display":"I_{Y,U,u}","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bdedc0251851d395","tex":"I_Z","aliases":"I_Z","display":"I_Z","kind":"definition-passage","excerpt":"[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54e29c60c9a464b5","tex":"\\mathcal I_\\alpha","aliases":"I_α mathcal alpha","display":"\\mathcal I_\\alpha","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-da49f71a0a4e2077","tex":"I_{\\lambda,a}","aliases":"I_λ,a lambda","display":"I_{\\lambda,a}","kind":"definition-passage","excerpt":"[Coherent invariant family] The family (I_λ,a,κ_ab) is coherent when the two induced comparison maps from a to c agree on every triple overlap and the corresponding quadruple-overlap equation holds.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id mathrm","display":"{\\mathrm{id}}","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b09c3d3928fe2c8b","tex":"\\operatorname{id}","aliases":"id operatorname","display":"\\operatorname{id}","kind":"definition-passage","excerpt":"[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im im operatorname","display":null,"kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":30},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"definition-passage","excerpt":"[Class-relative defect] Let X be a represented system, let Γ be a declared class of admissible auxiliary structures, and let d(X,g)∈[0,∞] be a defect functional. Define δ_Γ(X):=inf_g∈Γd(X,g). The pair consisting of the infimum and its attainment status is the class-relative defect profile.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d7a2b4165f603ed3","tex":"\\inf_{g\\in\\Gamma}","aliases":"inf_g∈Γ inf in Gamma","display":"\\inf_{g\\in\\Gamma}","kind":"definition-passage","excerpt":"[Class-relative defect] Let X be a represented system, let Γ be a declared class of admissible auxiliary structures, and let d(X,g)∈[0,∞] be a defect functional. Define δ_Γ(X):=inf_g∈Γd(X,g). The pair consisting of the infimum and its attainment status is the class-relative defect profile.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-55466b76f7f0c090","tex":"\\mathrm{int}","aliases":"int mathrm","display":"\\mathrm{int}","kind":"definition-passage","excerpt":"[Three effectivity obstructions] Let x∈ M_k,p(a). item The p-adic effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The s","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"definition-passage","excerpt":"abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normal","count":61},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-146cbf287aa37a5f","tex":"\\mathcal J","aliases":"J mathcal","display":"\\mathcal J","kind":"definition-passage","excerpt":"[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).","count":27},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the r","count":257},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-065e5bf0ea891f14","tex":"\\mathcal J_{k,K}^{\\mathrm{reg}}","aliases":"J_k,K^reg mathcal mathrm","display":"\\mathcal J_{k,K}^{\\mathrm{reg}}","kind":"definition-passage","excerpt":"[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aabbef4d5adca365","tex":"J_{\\rm alg}","aliases":"J_rm alg rm","display":"J_{\\rm alg}","kind":"definition-passage","excerpt":"[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4ea6dd870b386296","tex":"J_{s,t}","aliases":"J_s,t","display":"J_{s,t}","kind":"definition-passage","excerpt":"[Joint semantic completion] Define j=(s,t): R⟶ S× T, J_s,t:=im(j). The projection p_s:J_s,t⟶im(s) is the relative completion map from the joint semantics to the source semantics.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-439a48391cf755cb","tex":"J_{\\alpha\\beta}","aliases":"J_αβ alpha beta","display":"J_{\\alpha\\beta}","kind":"definition-passage","excerpt":"abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normal","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cdbeca40dd7bbf39","tex":"\\jmath","aliases":"jmath","display":"\\jmath","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a718b4b51013798e","tex":"\\jmath_a","aliases":"jmath_a jmath","display":"\\jmath_a","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":208},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-53283f2f6027fba4","tex":"\\mathbb K","aliases":"K mathbb","display":"\\mathbb K","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":20},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":23},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":284},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b6162b817c961630","tex":"k^{|u|}","aliases":"k^|u|","display":"k^{|u|}","kind":"definition-passage","excerpt":"[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e9bdb3c243e0297e","tex":"\\mathcal K_k","aliases":"K_k mathcal","display":"\\mathcal K_k","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6bc423b107b319be","tex":"\\mathcal K_{k,p^r}^{\\mathrm{sem}}","aliases":"K_k,p^r^sem mathcal mathrm","display":"\\mathcal K_{k,p^r}^{\\mathrm{sem}}","kind":"definition-passage","excerpt":"[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-21aa3737dc1dd223","tex":"\\mathbb K_{\\rm fin}","aliases":"K_rm fin mathbb rm","display":"\\mathbb K_{\\rm fin}","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-48b1a5fe6f839910","tex":"\\mathbb K_{\\rm lin}","aliases":"K_rm lin mathbb rm","display":"\\mathbb K_{\\rm lin}","kind":"definition-passage","excerpt":"[Linear grammar category] For a finite linear resolved presentation K_rm lin=(Q,I, P,Ref), let A_ K:=𝕜 Q/I. This is the 𝕜-linear category whose objects are the vertices of Q and whose morphisms are 𝕜-linear combinations of typed paths modulo the declared relations.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-071eab114cf0c0a7","tex":"\\Kar","aliases":"Kar operatorname","display":"{\\operatorname{Kar}}","kind":"definition-passage","excerpt":"[Karoubi envelope] The Karoubi envelope Kar(Add( A_ K)) has pairs (A,e), where e:A→ A is idempotent, as objects. A morphism f:(A,e)⟶(B,d) is a morphism f:A→ B satisfying f=dfe. The identity on (A,e) is e.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"definition-passage","excerpt":"[Ghost and missing reduction modules] Define G_k,m(a) = ker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ) and Q_k,m(a) = coker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). The first records raw residue coordinates that represent the zero semantic sequence. The second records saturated residue classes not reached by raw sections.","count":33},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a4f733cd50dc9a2c","tex":"\\kk","aliases":"kk kk 𝕜 Bbbk","display":null,"kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":36},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":191},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"definition-passage","excerpt":"[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, togeth","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2aab5edbc5f3efff","tex":"\\mathsf L","aliases":"L mathsf","display":"\\mathsf L","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"definition-passage","excerpt":"[Linear grammar category] For a finite linear resolved presentation K_rm lin=(Q,I, P,Ref), let A_ K:=𝕜 Q/I. This is the 𝕜-linear category whose objects are the vertices of Q and whose morphisms are 𝕜-linear combinations of typed paths modulo the declared relations.","count":22},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-384563b34b305544","tex":"\\mathcal L_{\\mathsf R}","aliases":"L_ R mathcal mathsf","display":"\\mathcal L_{\\mathsf R}","kind":"definition-passage","excerpt":"[Lane matrix category at a fixed resolution] Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, a","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-af99c822e350a366","tex":"L_{d_X}","aliases":"L_d_X","display":"L_{d_X}","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8285afe9914e471a","tex":"L_F","aliases":"L_F","display":"L_F","kind":"definition-passage","excerpt":"[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":31},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-95c1d3752ed52e36","tex":"\\mathcal L_X","aliases":"L_X mathcal","display":"\\mathcal L_X","kind":"definition-passage","excerpt":"[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, togeth","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d2dae003651f7970","tex":"\\mathrm{Loc}","aliases":"Loc mathrm","display":"\\mathrm{Loc}","kind":"definition-passage","excerpt":"[label=textup(Carabic*)] item A_DRLM,p, the fixed-prime digital--regular component with rational functional-Mahler and periodic linear subcharts; item A_Loc,p, the continuous Newton--Mahler and complete clopen-branch component; item A_Fuss, the marked Fuss--Catalan nonlinear--algebraic--holonomic component; item A_EM, the fixed-Euler-basis and prim","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"definition-passage","excerpt":"[Complete invariant on a declared target] An invariant I_α is complete on N_α when I_α(X)≅ I_α(Y) Longleftrightarrow X≅ Y for all X,Y∈ N_α. This statement classifies the objects of N_α up to the isomorphisms allowed there. It does not classify arbitrary presentations outside that subcategory.","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"definition-passage","excerpt":"[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic ","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"definition-passage","excerpt":"[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interprete","count":189},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6c6a514803a6f591","tex":"\\mathcal M","aliases":"M mathcal","display":"\\mathcal M","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":30},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":164},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4545e80268433a54","tex":"M^{u,r}_{Y,X}","aliases":"M^u,r_Y,X","display":"M^{u,r}_{Y,X}","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c887199df3da3b63","tex":"M_a","aliases":"M_a","display":"M_a","kind":"definition-passage","excerpt":"[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interprete","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c9f29e7d9e7a175b","tex":"M_k","aliases":"M_k","display":"M_k","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":37},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7809af6b4040eb17","tex":"M_{k,p}","aliases":"M_k,p","display":"M_{k,p}","kind":"definition-passage","excerpt":"[Three effectivity obstructions] Let x∈ M_k,p(a). item The p-adic effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The s","count":14},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e8b86078eb6fe959","tex":"\\mathcal M_p","aliases":"M_p mathcal","display":"\\mathcal M_p","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":14},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-00c73b19032652dd","tex":"M_{s(a)}","aliases":"M_s(a)","display":"M_{s(a)}","kind":"definition-passage","excerpt":"[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interprete","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d64adad72788fa70","tex":"M_{t(a)}","aliases":"M_t(a)","display":"M_{t(a)}","kind":"definition-passage","excerpt":"[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interprete","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-03acbc0f9f818256","tex":"\\mathcal M_w","aliases":"M_w mathcal","display":"\\mathcal M_w","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-88849f84a35be77a","tex":"M_X","aliases":"M_X","display":"M_X","kind":"definition-passage","excerpt":"[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interprete","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cc2f5f4bfec5fb41","tex":"M_\\alpha","aliases":"M_α alpha","display":"M_\\alpha","kind":"definition-passage","excerpt":"[Descent category] The category Desc( R) has pseudocones over R as objects and modifications as morphisms. Concretely, an object is a compatible family (M_α,θ_αβ) with invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-967ea86ebf866388","tex":"\\mathcal M_\\alpha","aliases":"M_α mathcal alpha","display":"\\mathcal M_\\alpha","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c49e946f6528c43f","tex":"\\mathcal M_{\\alpha\\beta}","aliases":"M_αβ mathcal alpha beta","display":"\\mathcal M_{\\alpha\\beta}","kind":"definition-passage","excerpt":"[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized w","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3955726942690d93","tex":"\\mathcal M_\\beta","aliases":"M_β mathcal beta","display":"\\mathcal M_\\beta","kind":"definition-passage","excerpt":"from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records preserved constructors and universal p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-55da71ccb79eafd2","tex":"\\mathcal M_{\\beta\\alpha}","aliases":"M_βα mathcal beta alpha","display":"\\mathcal M_{\\beta\\alpha}","kind":"definition-passage","excerpt":"[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized w","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2c7620add7ce256d","tex":"\\operatorname{Mat}","aliases":"Mat operatorname","display":"\\operatorname{Mat}","kind":"definition-passage","excerpt":"[Lane matrix category at a fixed resolution] Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, a","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9ac2694c7dab3f81","tex":"\\operatorname{Mat}_r","aliases":"Mat_r operatorname","display":"\\operatorname{Mat}_r","kind":"definition-passage","excerpt":"[Lane matrix category at a fixed resolution] Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, a","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2228b8fc43792a93","tex":"\\operatorname{Mat}_s","aliases":"Mat_s operatorname","display":"\\operatorname{Mat}_s","kind":"definition-passage","excerpt":"For ssucceq r, define Γ_sr:Mat_s( L_ R) ⟶Mat_r( L_ R) by replacing each object tag with its r-tag and sending a fine matrix to the certified source-and-target coordinate change of [source reference].","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-51b7ccf8b09eaeca","tex":"\\mathbin","aliases":"mathbin","display":null,"kind":"definition-passage","excerpt":"[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).","count":14},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1a4ee77b1419e30a","tex":"\\mathscr","aliases":"mathscr","display":"\\mathscr","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":21},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"definition-passage","excerpt":"[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F}, [3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-019f2dd2fc382aac","tex":"\\max_{G\\in\\operatorname{Br}_0(P_F),\\,G\\neq F}","aliases":"max_G∈Br_0(P_F), G≠ F max in operatorname neq","display":"\\max_{G\\in\\operatorname{Br}_0(P_F),\\,G\\neq F}","kind":"definition-passage","excerpt":"[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F}, [3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"definition-passage","excerpt":"[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).","count":41},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-91e461eca7d576c7","tex":"\\Mod","aliases":"Mod operatorname","display":"{\\operatorname{Mod}}","kind":"definition-passage","excerpt":"[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functor","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c269e7fe13707a59","tex":"\\mathbf{Mod}","aliases":"Mod mathbf","display":"\\mathbf{Mod}","kind":"definition-passage","excerpt":"athcal V, O) consisting of: [label=textup(roman*)] item a carrier doctrine A, which specifies how alternatives, routes, or packets combine; item a continuation grammar G, presented here by a monoid M or by a chosen generating family of admissible endomorphisms; item an observation category V, such as Set, R-Mod, or K-Vect; item a declared family O=","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6dac2a7a617678e1","tex":"\\Mod_\\Pi","aliases":"Mod_Π Mod Pi operatorname Pi","display":"{\\operatorname{Mod}}_\\Pi","kind":"definition-passage","excerpt":"[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functor","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e88e363b3ae7e0bc","tex":"\\operatorname{MorKer}","aliases":"MorKer operatorname","display":"\\operatorname{MorKer}","kind":"definition-passage","excerpt":"[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The f","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a4e88c64bb34cec1","tex":"\\operatorname{MorKer}_B","aliases":"MorKer_B operatorname","display":"\\operatorname{MorKer}_B","kind":"definition-passage","excerpt":"[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The f","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-156f8dae98d42125","tex":"\\mathbf{Mult}","aliases":"Mult mathbf","display":"\\mathbf{Mult}","kind":"definition-passage","excerpt":"[Integral multiplicative category] Let Mult_ Z be the discrete category whose objects are multiplicative arithmetic functions f: N_≥1→ Z satisfying f(1)=1, f(mn)=f(m)f(n) when (m,n)=1. The monoidal operation is Dirichlet convolution.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-43cb43115203d177","tex":"\\mathbf{Mult}_{\\mathbb Z}","aliases":"Mult_ Z mathbf mathbb","display":"\\mathbf{Mult}_{\\mathbb Z}","kind":"definition-passage","excerpt":"[Integral multiplicative category] Let Mult_ Z be the discrete category whose objects are multiplicative arithmetic functions f: N_≥1→ Z satisfying f(1)=1, f(mn)=f(m)f(n) when (m,n)=1. The monoidal operation is Dirichlet convolution.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"definition-passage","excerpt":"[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic ","count":413},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":258},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-91fc30aba54cfd47","tex":"N_{\\mathsf S}","aliases":"N_ S mathsf","display":"N_{\\mathsf S}","kind":"definition-passage","excerpt":"[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-efe7f0a6403a8266","tex":"N_X","aliases":"N_X","display":"N_X","kind":"definition-passage","excerpt":"[Task-compatible bridge] The bridge is task compatible when f(N_X)⊆ N_Y. Equivalently, N_X⊆ f^-1(N_Y). Then there is a unique induced map overline f:X/N_X⟶ Y/N_Y.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-05df5fdce379b495","tex":"N_Y","aliases":"N_Y","display":"N_Y","kind":"definition-passage","excerpt":"[Task-compatible bridge] The bridge is task compatible when f(N_X)⊆ N_Y. Equivalently, N_X⊆ f^-1(N_Y). Then there is a unique induced map overline f:X/N_X⟶ Y/N_Y.","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8ae0249a2fe4cccf","tex":"N_\\alpha","aliases":"N_α alpha","display":"N_\\alpha","kind":"definition-passage","excerpt":"[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f965b6cd88a9272b","tex":"\\mathcal N_\\alpha","aliases":"N_α mathcal alpha","display":"\\mathcal N_\\alpha","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a6acfb5cd2203413","tex":"\\mathbb N_{\\geq1}","aliases":"N_≥1 mathbb geq","display":"\\mathbb N_{\\geq1}","kind":"definition-passage","excerpt":"[Integral multiplicative category] Let Mult_ Z be the discrete category whose objects are multiplicative arithmetic functions f: N_≥1→ Z satisfying f(1)=1, f(mn)=f(m)f(n) when (m,n)=1. The monoidal operation is Dirichlet convolution.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"definition-passage","excerpt":"tegory in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theorem is stated. item Sem_α: M_α⟶ O_α is the n","count":38},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ef90ac3fea9a395b","tex":"\\mathfrak O","aliases":"O mathfrak","display":"\\mathfrak O","kind":"definition-passage","excerpt":"hcal A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category R(U); item coherent equivalences on the reversible overlaps and separately typed directional transformations; item a pseudonatural semantics Sem: R⇒ O into a com","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"definition-passage","excerpt":"[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.","count":49},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-96526df066f36358","tex":"o_j","aliases":"o_j","display":"o_j","kind":"definition-passage","excerpt":"item a carrier doctrine A, which specifies how alternatives, routes, or packets combine; item a continuation grammar G, presented here by a monoid M or by a chosen generating family of admissible endomorphisms; item an observation category V, such as Set, R-Mod, or K-Vect; item a declared family O=(o_j)_j∈ J of observations in V. The slice belongs ","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-503974c7d3a08ed5","tex":"\\mathcal O_k","aliases":"O_k mathcal","display":"\\mathcal O_k","kind":"definition-passage","excerpt":"effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The section-orbit obstruction is membership in M_k(a)∖ O_k(a). Even aft","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-767a0592cb5c5b4b","tex":"\\mathcal O_\\alpha","aliases":"O_α mathcal alpha","display":"\\mathcal O_\\alpha","kind":"definition-passage","excerpt":"tegory in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible models of the declared resolved syntax. item N_α⊆ M_α is a full replete normalized subcategory on which the intended inverse or classification theorem is stated. item Sem_α: M_α⟶ O_α is the n","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob operatorname","display":"{\\operatorname{Ob}}","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-480cac95eb168013","tex":"\\operatorname{Ob}","aliases":"Ob operatorname","display":"\\operatorname{Ob}","kind":"definition-passage","excerpt":"[Information-loss profile] For a functor B: U→ V, its information-loss profile contains at least: MorKer_B(X,Y) · = {(f,g)∈Hom_ U(X,Y)^2:Bf=Bg}, Coll_B · = {(X,Y)∈Ob( U)^2:BX≅ BY}, EssIm(B) · ⊆ V, together with the declared constructors, packet labels, residuals, certificates, branch data, topology, or arithmetic structure not preserved by B. The f","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-46c1284bb2e365e3","tex":"\\Ob_D","aliases":"Ob_D Ob operatorname","display":"{\\operatorname{Ob}}_D","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f814b60f967151c4","tex":"\\mathrm{op}","aliases":"op mathrm","display":"\\mathrm{op}","kind":"definition-passage","excerpt":"[Finite profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap ca","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fd71dfd389a79ac5","tex":"\\operatorname{ord}","aliases":"ord operatorname","display":"\\operatorname{ord}","kind":"definition-passage","excerpt":"[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2029d5686dea49d7","tex":"\\operatorname{ord}_D","aliases":"ord_D operatorname","display":"\\operatorname{ord}_D","kind":"definition-passage","excerpt":"[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-235862a14fa14aa7","tex":"\\operatorname{ord}_z","aliases":"ord_z operatorname","display":"\\operatorname{ord}_z","kind":"definition-passage","excerpt":"[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F}, [3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e83fbd63131abb69","tex":"\\overline f","aliases":"overline f overline","display":"\\overline f","kind":"definition-passage","excerpt":"[Task-compatible bridge] The bridge is task compatible when f(N_X)⊆ N_Y. Equivalently, N_X⊆ f^-1(N_Y). Then there is a unique induced map overline f:X/N_X⟶ Y/N_Y.","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9d8a1921f2a37e0a","tex":"\\overline M","aliases":"overline M overline","display":"\\overline M","kind":"definition-passage","excerpt":"[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bb4528c19948f571","tex":"\\overline M_{k,p^r}","aliases":"overline M_k,p^r overline","display":"\\overline M_{k,p^r}","kind":"definition-passage","excerpt":"[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0e517349dec3b6ba","tex":"\\overset","aliases":"overset","display":null,"kind":"definition-passage","excerpt":"[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized w","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F},","count":99},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"definition-passage","excerpt":"[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.","count":39},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2e99a99588c03ac6","tex":"\\mathsf P","aliases":"P mathsf","display":"\\mathsf P","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorp","count":547},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f24d8467a0d7b522","tex":"p'_{B(X),a}","aliases":"p'_B(X),a","display":"p'_{B(X),a}","kind":"definition-passage","excerpt":"cture used downstream: [label=textup(roman*)] item native semantics; item packet analysis and synthesis; item refinement and coarsening maps; item selected-support compression and named residuals; item proof or computation certificates; item normalization and invariant comparison maps. For example, packet naturality requires comparison maps under w","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-663b3f359a5f1a61","tex":"p'_{B(Y),b}","aliases":"p'_B(Y),b","display":"p'_{B(Y),b}","kind":"definition-passage","excerpt":"cture used downstream: [label=textup(roman*)] item native semantics; item packet analysis and synthesis; item refinement and coarsening maps; item selected-support compression and named residuals; item proof or computation certificates; item normalization and invariant comparison maps. For example, packet naturality requires comparison maps under w","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b47e4ae8f60bef45","tex":"p^e","aliases":"p^e","display":"p^e","kind":"definition-passage","excerpt":"[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].","count":17},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a7f614e80bed5fdb","tex":"\\mathcal P^r","aliases":"P^r mathcal","display":"\\mathcal P^r","kind":"definition-passage","excerpt":"[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6f0ea21bbfbf69fb","tex":"p^r","aliases":"p^r","display":"p^r","kind":"definition-passage","excerpt":"[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).","count":30},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-13e91aac693e21c7","tex":"p^{r}_{A,U,i}","aliases":"p^r_A,U,i","display":"p^{r}_{A,U,i}","kind":"definition-passage","excerpt":"For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrigh","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7e84eb2bbd52aac2","tex":"p^r_i","aliases":"p^r_i","display":"p^r_i","kind":"definition-passage","excerpt":"[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9e8498585285c9cc","tex":"\\mathcal P^s","aliases":"P^s mathcal","display":"\\mathcal P^s","kind":"definition-passage","excerpt":"[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0efdb214bb3d9bcd","tex":"p^s_j","aliases":"p^s_j","display":"p^s_j","kind":"definition-passage","excerpt":"[Finite refinement] A packet system P^s=(p^s_j)_j∈ I_s refines P^r=(p^r_i)_i∈ I_r when there is a surjection ρ_sr:I_s↠ I_r such that p^r_i = ∑_ρ_sr(j)=ip^s_j. The map ρ_sr records the address of each fine packet inside its coarse ancestor.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e9d48e107ca79b17","tex":"P_F","aliases":"P_F","display":"P_F","kind":"definition-passage","excerpt":"[Branch-separation index] For F∈Br_0(P_F), define σ(P_F,F) = 0, · Br_0(P_F)={F},","count":27},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"definition-passage","excerpt":"[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-204484e64d4bc936","tex":"p_i^2","aliases":"p_i^2","display":"p_i^2","kind":"definition-passage","excerpt":"[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-85687051228b1e6b","tex":"p_j","aliases":"p_j","display":"p_j","kind":"definition-passage","excerpt":"[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-be8331cd1f9921b0","tex":"p_s","aliases":"p_s","display":"p_s","kind":"definition-passage","excerpt":"[Joint semantic completion] Define j=(s,t): R⟶ S× T, J_s,t:=im(j). The projection p_s:J_s,t⟶im(s) is the relative completion map from the joint semantics to the source semantics.","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ad298f8ec8a9c512","tex":"p_{X,a}","aliases":"p_X,a","display":"p_{X,a}","kind":"definition-passage","excerpt":"cture used downstream: [label=textup(roman*)] item native semantics; item packet analysis and synthesis; item refinement and coarsening maps; item selected-support compression and named residuals; item proof or computation certificates; item normalization and invariant comparison maps. For example, packet naturality requires comparison maps under w","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b08062da8805bdf0","tex":"p_{X,i}^r","aliases":"p_X,i^r","display":"p_{X,i}^r","kind":"definition-passage","excerpt":"[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, togeth","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b58b1474c50a261a","tex":"p_{X,j}^r","aliases":"p_X,j^r","display":"p_{X,j}^r","kind":"definition-passage","excerpt":"block15-finite-profiled-lane-system Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a027049060f1f7a8","tex":"p_{X,j}^s","aliases":"p_X,j^s","display":"p_{X,j}^s","kind":"definition-passage","excerpt":"p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address set","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fc48804947dd6caf","tex":"p_{X,r,i}","aliases":"p_X,r,i","display":"p_{X,r,i}","kind":"definition-passage","excerpt":"item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorp","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6fa185cff1cbb941","tex":"p_{X,r,j}","aliases":"p_X,r,j","display":"p_{X,r,j}","kind":"definition-passage","excerpt":"nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the r","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1f87acfa5fe35545","tex":"p_{X,s,j}","aliases":"p_X,s,j","display":"p_{X,s,j}","kind":"definition-passage","excerpt":"is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a li","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dc69c0bea1784ddc","tex":"\\mathcal P_X^r","aliases":"P_X^r mathcal","display":"\\mathcal P_X^r","kind":"definition-passage","excerpt":"[Finite profiled lane system] Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, togeth","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b266f62d11b04309","tex":"p_{Y,b}","aliases":"p_Y,b","display":"p_{Y,b}","kind":"definition-passage","excerpt":"cture used downstream: [label=textup(roman*)] item native semantics; item packet analysis and synthesis; item refinement and coarsening maps; item selected-support compression and named residuals; item proof or computation certificates; item normalization and invariant comparison maps. For example, packet naturality requires comparison maps under w","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1efb91558197e2bb","tex":"\\mathrm{pres}","aliases":"pres mathrm","display":"\\mathrm{pres}","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":110},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":32},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":14},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":167},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f0c55bdfd19c18ba","tex":"\\mathbb Q^{\\mathbb N}","aliases":"Q^ N mathbb mathbb","display":"\\mathbb Q^{\\mathbb N}","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a6eb4f5fbb2cd586","tex":"Q_{\\mathsf S}","aliases":"Q_ S mathsf","display":"Q_{\\mathsf S}","kind":"definition-passage","excerpt":"with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object is the coimage Q_ S:=Coim(Beh_ S) whenever that coimage exists.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-190c2533d86dd7b7","tex":"Q_1","aliases":"Q_1","display":"Q_1","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-55d4600438de195a","tex":"q_a","aliases":"q_a","display":"q_a","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c2f20a016f11899a","tex":"Q_{k,m}","aliases":"Q_k,m","display":"Q_{k,m}","kind":"definition-passage","excerpt":"[Ghost and missing reduction modules] Define G_k,m(a) = ker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ) and Q_k,m(a) = coker ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). The first records raw residue coordinates that represent the zero semantic sequence. The second records saturated residue classes not reached by raw sections.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.","count":103},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"definition-passage","excerpt":"[Joint semantic completion] Define j=(s,t): R⟶ S× T, J_s,t:=im(j). The projection p_s:J_s,t⟶im(s) is the relative completion map from the joint semantics to the source semantics.","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-32fe36360874826c","tex":"\\mathfrak R","aliases":"R mathfrak","display":"\\mathfrak R","kind":"definition-passage","excerpt":"[Finite profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap ca","count":30},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-02f6e210c4b44c83","tex":"\\mathsf R","aliases":"R mathsf","display":"\\mathsf R","kind":"definition-passage","excerpt":"[Finite resolution shape] A finite resolution shape is a finite partially ordered set R. We write ssucceq r when s is finer than r. The shape contains no address sets: object-specific addresses and their refinement maps are declared in the lane system below.","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorp","count":397},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-33462a02fcce1a84","tex":"R_a","aliases":"R_a","display":"R_a","kind":"definition-passage","excerpt":"[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-001ad2900a7bd426","tex":"r_D","aliases":"r_D","display":"r_D","kind":"definition-passage","excerpt":"[Minimal differential annihilator] For a nonzero D-finite series F∈ K[[z]], define Ann_ D(F) = {L∈ D_K:LF=0}. Its unique monic generator is denoted L_F. The differential order r_D(F)=ord_D L_F is the minimal scalar differential order of F over K(z).","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8acad0d31da451a8","tex":"R_E","aliases":"R_E","display":"R_E","kind":"definition-passage","excerpt":"[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":24},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3e66ab2852645abf","tex":"r_{N,j}^{\\rm char}","aliases":"r_N,j^rm char rm","display":"r_{N,j}^{\\rm char}","kind":"definition-passage","excerpt":"[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a7501108d19c7630","tex":"r_N^{\\rm orb}","aliases":"r_N^rm orb rm","display":"r_N^{\\rm orb}","kind":"definition-passage","excerpt":"[Root-filter alias and address-cancellation defects] Define a_N(F)=Nr-r_N^rm orb, c_N,j(F)=r_N^rm orb-r_N,j^rm char. The first quantity measures dependence already present among the rotated holonomic modules. The second measures the additional loss caused by the jth character projector after the rotated orbit has been assembled.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0d5a58c436e4715b","tex":"r_p","aliases":"r_p","display":"r_p","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-34c3b4fc296bd55d","tex":"R_{T\\mid S}","aliases":"R_Tmid S mid","display":null,"kind":"definition-passage","excerpt":"[Relative semantic residual] The information needed to augment S so that it also retains T is R_Tmid S := ker(J→ S) ≅ N/(N∩ M) ≅ (N+M)/M.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bb26d33010c8217d","tex":"\\mathsf{Ref}","aliases":"Ref mathsf","display":"\\mathsf{Ref}","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c595c4c8f5a6614e","tex":"\\mathrm{refl}","aliases":"refl mathrm","display":"\\mathrm{refl}","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-89be60020e6ac1f8","tex":"\\mathbf{Reg}","aliases":"Reg mathbf","display":"\\mathbf{Reg}","kind":"definition-passage","excerpt":"ular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bb3491ad026a6506","tex":"\\mathrm{reg}","aliases":"reg mathrm","display":"\\mathrm{reg}","kind":"definition-passage","excerpt":"[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4a3066420777ee91","tex":"\\mathbf{Reg}_k","aliases":"Reg_k mathbf","display":"\\mathbf{Reg}_k","kind":"definition-passage","excerpt":"ular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-977b3a1a8ec8c8d2","tex":"\\Rep","aliases":"Rep operatorname","display":"{\\operatorname{Rep}}","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5688889a83c48b8b","tex":"\\Rep_{\\mathbf d}","aliases":"Rep_ d Rep mathbf operatorname mathbf","display":"{\\operatorname{Rep}}_{\\mathbf d}","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":56},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Relative semantic residual] The information needed to augment S so that it also retains T is R_Tmid S := ker(J→ S) ≅ N/(N∩ M) ≅ (N+M)/M.","count":67},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e7c7694843a64055","tex":"\\mathbf S","aliases":"S mathbf","display":"\\mathbf S","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"definition-passage","excerpt":"[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6d13875e4e83a846","tex":"\\mathfrak s","aliases":"s mathfrak","display":"\\mathfrak s","kind":"definition-passage","excerpt":"[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-17e6e62cdd37d68b","tex":"\\mathsf S","aliases":"S mathsf","display":"\\mathsf S","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":25},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":211},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ea87189622cbbdcf","tex":"\\mathfrak s^{\\rm Gal}_{N,j}","aliases":"s^rm Gal_N,j mathfrak rm","display":"\\mathfrak s^{\\rm Gal}_{N,j}","kind":"definition-passage","excerpt":"[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e5038d774b6f0114","tex":"\\mathcal S_K","aliases":"S_K mathcal","display":"\\mathcal S_K","kind":"definition-passage","excerpt":"[Minimal tail recurrence] For a sequence a=(a_n)_n≥0, define its tail annihilator by Ann_rm tail(a) = {R∈ S_K:(Ra)_n=0 for all sufficiently large n}. For a non-eventually-zero P-recursive sequence, its unique monic generator is denoted R_a. If the sequence is eventually zero, the tail generator is 1 and has order zero.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-64ac4e27d63a5e27","tex":"\\mathbf S_\\alpha","aliases":"S_α mathbf alpha","display":"\\mathbf S_\\alpha","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6bcff29c6f715bc2","tex":"\\Sem","aliases":"Sem operatorname","display":"{\\operatorname{Sem}}","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-931e712f6fd8a49c","tex":"\\mathrm{sem}","aliases":"sem mathrm","display":"\\mathrm{sem}","kind":"definition-passage","excerpt":"[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0c8b1379b205e5a2","tex":"\\Sem_\\alpha","aliases":"Sem_α Sem alpha operatorname alpha","display":"{\\operatorname{Sem}}_\\alpha","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5d5374f2a88be9ef","tex":"\\Sem_\\beta","aliases":"Sem_β Sem beta operatorname beta","display":"{\\operatorname{Sem}}_\\beta","kind":"definition-passage","excerpt":"abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normal","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-00c6581179a51c65","tex":"\\mathbf{Set}","aliases":"Set mathbf","display":"\\mathbf{Set}","kind":"definition-passage","excerpt":"al A, G, V, O) consisting of: [label=textup(roman*)] item a carrier doctrine A, which specifies how alternatives, routes, or packets combine; item a continuation grammar G, presented here by a monoid M or by a chosen generating family of admissible endomorphisms; item an observation category V, such as Set, R-Mod, or K-Vect; item a declared family ","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9b98b13e586e1933","tex":"\\mathsf{SF}","aliases":"SF mathsf","display":"\\mathsf{SF}","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ccf840fc9f7861e3","tex":"\\mathsf{SF}_k","aliases":"SF_k mathsf","display":"\\mathsf{SF}_k","kind":"definition-passage","excerpt":"[Canonical integral regular record] The canonical integral record of a is I_k(a) = ( V_ Q(a), M_k(a), L_k(a), C_k(a), (Λ_d)_0≤ d<k, a, ev_0 ). Its Smith--Fitting profile is SF_k(a) = ( d_1,…,d_ρ; Fitt_0(C_k(a)),…, Fitt_ρ-1(C_k(a)) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a6b6a76c6b1eda71","tex":"\\mathsf{SL}","aliases":"SL mathsf","display":"\\mathsf{SL}","kind":"definition-passage","excerpt":"[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0d3b20eb53a6de97","tex":"\\mathsf{SL}_{k,K}","aliases":"SL_k,K mathsf","display":"\\mathsf{SL}_{k,K}","kind":"definition-passage","excerpt":"[Fixed-radix regular invariant] The normalized fixed-radix invariant is J_k,K^reg(a) = ( r_k(a), C_k(a), SL_k,K(a), [H_a[U,V]] ), where [H_a[U,V]] denotes any recorded nonsingular rank certificate together with its word addresses. The strict normal form makes this package complete on Reg_k(K).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aa02db409e6526a5","tex":"\\Span","aliases":"Span Span span operatorname","display":null,"kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-02a9a108c88a2b24","tex":"\\operatorname{span}_{\\mathbb Z}","aliases":"span_ Z operatorname mathbb","display":"\\operatorname{span}_{\\mathbb Z}","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-32f1303b9491073a","tex":"\\Span_K","aliases":"Span_K Span Span_K span_K Span operatorname","display":null,"kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"definition-passage","excerpt":"ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-17b391d057c23c7b","tex":"\\Syn","aliases":"Syn operatorname","display":"{\\operatorname{Syn}}","kind":"definition-passage","excerpt":"[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functor","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-46994a8abe30795b","tex":"\\Syn_\\Pi","aliases":"Syn_Π Syn Pi operatorname Pi","display":"{\\operatorname{Syn}}_\\Pi","kind":"definition-passage","excerpt":"[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functor","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1f3c93360ea50221","tex":"\\Synthesis","aliases":"Synthesis Synthesis S mathcal","display":null,"kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-61e3762cb0d9b42f","tex":"\\Synthesis_{\\mathcal P}","aliases":"Synthesis_ P S_ P Synthesis mathcal mathcal mathcal","display":"{\\mathcal S}_{\\mathcal P}","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].","count":127},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"definition-passage","excerpt":"[Finite resolved signature] Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-s","count":97},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-964fcf8c1f59b656","tex":"T^e","aliases":"T^e","display":"T^e","kind":"definition-passage","excerpt":"[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8b7dd8e4d9d0fe5f","tex":"\\mathsf{Task}","aliases":"Task mathsf","display":"\\mathsf{Task}","kind":"definition-passage","excerpt":"[Doctrine-indexed task fiber] For a doctrine D and represented object X, let Task_D(X) denote the ordered family of continuation-stable semantic quotients admitted by the declared query universe. Inside one fiber, kernel or congruence calculus governs task change. A change of doctrine D⟶ D' requires a bridge passport specifying carrier transport, p","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c1a5d460fbcd6aa7","tex":"\\mathsf{Task}_D","aliases":"Task_D mathsf","display":"\\mathsf{Task}_D","kind":"definition-passage","excerpt":"[Doctrine-indexed task fiber] For a doctrine D and represented object X, let Task_D(X) denote the ordered family of continuation-stable semantic quotients admitted by the declared query universe. Inside one fiber, kernel or congruence calculus governs task change. A change of doctrine D⟶ D' requires a bridge passport specifying carrier transport, p","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-65fa7282e2650c3e","tex":"\\triangleleft","aliases":"triangleleft","display":null,"kind":"definition-passage","excerpt":"labeldef:finite-resolved-signature Fix a commutative coefficient ring 𝕜. A finite resolved signature is a tuple K_rm fin = (Q,I, P,Ref,Comp,Cert) with the following data. [label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generate","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":167},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2a4a57b7bee5acb6","tex":"\\mathcal U","aliases":"U mathcal","display":"\\mathcal U","kind":"definition-passage","excerpt":"[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic ","count":14},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":119},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2a24ca9966c46cfc","tex":"\\mathcal U_\\mathfrak","aliases":"U_ mathcal mathfrak","display":"\\mathcal U_\\mathfrak","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2308f29635998d1d","tex":"u_1","aliases":"u_1","display":"u_1","kind":"definition-passage","excerpt":"[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ec4379f6b5fb17bc","tex":"U_a","aliases":"U_a","display":"U_a","kind":"definition-passage","excerpt":"[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"definition-passage","excerpt":"[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"definition-passage","excerpt":"[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.","count":19},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a7813abebbece0ac","tex":"u_{r_k(a)}","aliases":"u_r_k(a)","display":"u_{r_k(a)}","kind":"definition-passage","excerpt":"[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-82db4f20a4afe761","tex":"\\mathcal U_\\alpha","aliases":"U_α mathcal alpha","display":"\\mathcal U_\\alpha","kind":"definition-passage","excerpt":"[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic ","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Linear correction datum] A linear correction datum is a map D:U⟶ V and a defect r∈ V. A correction is u∈ U satisfying Du=-r. The relative obstruction class is Ob_D(r)=[r]∈coker D=V/im D.","count":104},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":39},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":92},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-da8df5a6b3abb77c","tex":"\\mathcal V_\\mathfrak","aliases":"V_ mathcal mathfrak","display":"\\mathcal V_\\mathfrak","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-df3d3da7aac859c8","tex":"V_{\\mathbb Q}","aliases":"V_ Q mathbb","display":"V_{\\mathbb Q}","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bef331a46a8ed423","tex":"\\mathcal V_k","aliases":"V_k mathcal","display":"\\mathcal V_k","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a9b01f62dbf3fb3c","tex":"\\mathcal V_p","aliases":"V_p mathcal","display":"\\mathcal V_p","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":16},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6d81830a72862e16","tex":"\\mathbf{Vect}","aliases":"Vect mathbf","display":"\\mathbf{Vect}","kind":"definition-passage","excerpt":"ting of: [label=textup(roman*)] item a carrier doctrine A, which specifies how alternatives, routes, or packets combine; item a continuation grammar G, presented here by a monoid M or by a chosen generating family of admissible endomorphisms; item an observation category V, such as Set, R-Mod, or K-Vect; item a declared family O=(o_j)_j∈ J of obser","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":31},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":74},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-73ffecb5618fb3cc","tex":"W^*","aliases":"W^*","display":"W^*","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"numerate[label=textup(roman*)] item Q=(Q_0,Q_1,s,t) is a finite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration o","count":216},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Three effectivity obstructions] Let x∈ M_k,p(a). item The p-adic effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The s","count":180},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1a115ce512d7ca2a","tex":"X^r_{U,i}","aliases":"X^r_U,i","display":"X^r_{U,i}","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4ae6c97e64ef49dd","tex":"X_i^r","aliases":"X_i^r","display":"X_i^r","kind":"definition-passage","excerpt":"finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"definition-passage","excerpt":"finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^","count":25},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":145},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"definition-passage","excerpt":"[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.","count":24},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a21672b2a6fe1ef6","tex":"Y^u_{U,a}","aliases":"Y^u_U,a","display":"Y^u_{U,a}","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-536459ef72a0484d","tex":"Y_a","aliases":"Y_a","display":"Y_a","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5475a4d1954cb6f3","tex":"Y_a^r","aliases":"Y_a^r","display":"Y_a^r","kind":"definition-passage","excerpt":"y Let L_ R be the category whose objects are objects of C equipped with full finite profiled lane systems over R, with all underlying C-morphisms allowed. For r∈ R, define Mat_r( L_ R) to have the tagged finite families [X;(X_i^r)_i∈ I_X,r] as objects and rectangular matrices (h_a i), h_a i:X_i^r→ Y_a^r, as morphisms. Composition is matrix contract","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-27bd33e3fbb9a571","tex":"Y_j","aliases":"Y_j","display":"Y_j","kind":"definition-passage","excerpt":"[Resolved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral obj","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"definition-passage","excerpt":"[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":136},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.","count":265},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-44b73d22a81276d2","tex":"\\mathbb Z^{\\mathbb N}","aliases":"Z^ N mathbb mathbb","display":"\\mathbb Z^{\\mathbb N}","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-56835621b2f38901","tex":"z^m","aliases":"z^m","display":"z^m","kind":"definition-passage","excerpt":"[Euler-coordinate map] For b=(b_m)_m≥1∈ E_ Z, define E(b)(z) = ∏_m≥1(1-z^m)^-b_m. The product is coefficientwise filtered because the mth factor is congruent to 1 modulo z^m.","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-afc30c788aac6f77","tex":"z^N","aliases":"z^N","display":"z^N","kind":"definition-passage","excerpt":"[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-284681a0009d141e","tex":"z^n","aliases":"z^n","display":"z^n","kind":"definition-passage","excerpt":"[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.","count":24},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cb9bf5755df6cc8e","tex":"z^{w(m)}","aliases":"z^w(m)","display":"z^{w(m)}","kind":"definition-passage","excerpt":"[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8723c894d5150ea2","tex":"\\mathbb Z_p","aliases":"Z_p mathbb","display":"\\mathbb Z_p","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":44},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-180c17fa6495e993","tex":"\\mathbb Z_{\\geq0}","aliases":"Z_≥0 mathbb geq","display":"\\mathbb Z_{\\geq0}","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic ","count":45},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3c984b82bd896807","tex":"|_{\\mathcal U_\\alpha}","aliases":"|_ U_α mathcal alpha","display":"|_{\\mathcal U_\\alpha}","kind":"definition-passage","excerpt":"[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9db02889931f8d23","tex":"|_{\\mathcal V_p(F)}","aliases":"|_ V_p(F) mathcal","display":"|_{\\mathcal V_p(F)}","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"definition-passage","excerpt":"[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.","count":68},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":125},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"definition-passage","excerpt":"[Typed directed transformation category] The directed transformation category Dir( K) has pairs (α,M), with M∈ M_α, as objects. A morphism (α,M)⟶(β,N) is an admitted bridge instance equipped with a bridge passport as in [source reference]. Composition is defined only when the codomain profile of the first bridge satisfies the domain requirements of","count":51},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-40a0aa859d9e7386","tex":"\\beta^A_{\\gamma,r,i}","aliases":"β^A_γ,r,i beta gamma","display":"\\beta^A_{\\gamma,r,i}","kind":"definition-passage","excerpt":"ject A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the ad","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4035a1dd401d64d6","tex":"\\beta^X_{\\gamma,r,i}","aliases":"β^X_γ,r,i beta gamma","display":"\\beta^X_{\\gamma,r,i}","kind":"definition-passage","excerpt":"atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-82bf002ec4f697f4","tex":"\\beta^Y_{\\gamma,u,a}","aliases":"β^Y_γ,u,a beta gamma","display":"\\beta^Y_{\\gamma,u,a}","kind":"definition-passage","excerpt":"atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"definition-passage","excerpt":"[Algebraic packet collision defects] Define c^rm tuple_N,j = d^N-| C_N,j|, and s^rm Gal_N,j = | C_N,j|-|Γ· U_j|. The first records collisions caused by projection from complete branch tuples to one packet coordinate. The second records the remaining loss caused by the Galois stabilizer of the selected coordinate.","count":44},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":58},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4c4d36aaaeafb01d","tex":"\\gamma_*","aliases":"γ_* gamma","display":"\\gamma_*","kind":"definition-passage","excerpt":"atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) ","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6167e8114e9baabc","tex":"\\Gamma_{sr}","aliases":"Γ_sr Gamma","display":"\\Gamma_{sr}","kind":"definition-passage","excerpt":"For ssucceq r, define Γ_sr:Mat_s( L_ R) ⟶Mat_r( L_ R) by replacing each object tag with its r-tag and sending a fine matrix to the certified source-and-target coordinate change of [source reference].","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"definition-passage","excerpt":"nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the r","count":38},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a752f44c1655724a","tex":"\\delta_{ij}","aliases":"δ_ij delta","display":"\\delta_{ij}","kind":"definition-passage","excerpt":"nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the r","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9601f08c373fe8e4","tex":"\\delta_k","aliases":"δ_k delta","display":"\\delta_k","kind":"definition-passage","excerpt":"[Integral saturation defect] The finite abelian group C_k(a) = L_k(a)/M_k(a) is the integral saturation defect. Its order δ_k(a) = [L_k(a):M_k(a)] is the saturation index.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-844570e10eea2d2c","tex":"\\delta_\\Gamma","aliases":"δ_Γ delta Gamma","display":"\\delta_\\Gamma","kind":"definition-passage","excerpt":"[Class-relative defect] Let X be a represented system, let Γ be a declared class of admissible auxiliary structures, and let d(X,g)∈[0,∞] be a defect functional. Define δ_Γ(X):=inf_g∈Γd(X,g). The pair consisting of the infimum and its attainment status is the class-relative defect profile.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-890e777d8bea7959","tex":"\\epsilon_0","aliases":"ϵ_0 epsilon","display":"\\epsilon_0","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e1b53138638199f9","tex":"\\epsilon_{\\mathrm{int}}","aliases":"ϵ_int epsilon mathrm","display":"\\epsilon_{\\mathrm{int}}","kind":"definition-passage","excerpt":"[Three effectivity obstructions] Let x∈ M_k,p(a). item The p-adic effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The s","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-881f5922d6ff0164","tex":"\\eta_{\\alpha\\beta\\gamma}","aliases":"η_αβγ eta alpha beta gamma","display":"\\eta_{\\alpha\\beta\\gamma}","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-52519fa4abe3008e","tex":"\\eta_{\\alpha\\beta\\delta}","aliases":"η_αβδ eta alpha beta delta","display":"\\eta_{\\alpha\\beta\\delta}","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-17b97af57c3acb99","tex":"\\eta_{\\alpha\\gamma\\delta}","aliases":"η_αγδ eta alpha gamma delta","display":"\\eta_{\\alpha\\gamma\\delta}","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c1fc4ed4539669c0","tex":"\\eta_{\\beta\\gamma\\delta}","aliases":"η_βγδ eta beta gamma delta","display":"\\eta_{\\beta\\gamma\\delta}","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"definition-passage","excerpt":"[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.","count":33},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"definition-passage","excerpt":"[Descent category] The category Desc( R) has pseudocones over R as objects and modifications as morphisms. Concretely, an object is a compatible family (M_α,θ_αβ) with invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-99ad2a3c66a9865a","tex":"\\theta^A_\\gamma","aliases":"θ^A_γ theta gamma","display":"\\theta^A_\\gamma","kind":"definition-passage","excerpt":"profiled-lane-system. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-47cb2bdd631c1f94","tex":"\\Theta_{\\mathsf S}","aliases":"Θ_ S Theta mathsf","display":"\\Theta_{\\mathsf S}","kind":"definition-passage","excerpt":"[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8bbfe6d49fff3168","tex":"\\theta_{\\alpha\\beta}","aliases":"θ_αβ theta alpha beta","display":"\\theta_{\\alpha\\beta}","kind":"definition-passage","excerpt":"[Descent category] The category Desc( R) has pseudocones over R as objects and modifications as morphisms. Concretely, an object is a compatible family (M_α,θ_αβ) with invertible overlap comparisons satisfying the cocycle law. A morphism is a compatible family of chart morphisms.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-95d58d27f67c1fbd","tex":"\\iota","aliases":"ι iota","display":"\\iota","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":28},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e0a1376ad0b8f2dc","tex":"\\iota^r_{A,U,i}","aliases":"ι^r_A,U,i iota","display":"\\iota^r_{A,U,i}","kind":"definition-passage","excerpt":"For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrigh","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-abafb816d9b3abc7","tex":"\\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}","aliases":"ι^r_A,V,λ^A_γ,r(i) iota lambda gamma","display":"\\iota^r_{A,V,\\lambda^A_{\\gamma,r}(i)}","kind":"definition-passage","excerpt":"profiled-lane-system. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-add99bd52202dbca","tex":"\\iota_i","aliases":"ι_i iota","display":"\\iota_i","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-99fc23a31bb6e249","tex":"\\iota_{X,i}^r","aliases":"ι_X,i^r iota","display":"\\iota_{X,i}^r","kind":"definition-passage","excerpt":"finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"definition-passage","excerpt":"nd its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":37},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c6c5422fa482b46e","tex":"\\kappa_{ab}","aliases":"κ_ab kappa","display":"\\kappa_{ab}","kind":"definition-passage","excerpt":"[Coherent invariant family] The family (I_λ,a,κ_ab) is coherent when the two induced comparison maps from a to c agree on every triple overlap and the corresponding quadruple-overlap equation holds.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aa40a51834232b37","tex":"\\kappa_k","aliases":"κ_k kappa","display":"\\kappa_k","kind":"definition-passage","excerpt":"nd its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b6d8daf4f26095b4","tex":"\\kappa_{k,p^r}^{\\mathrm{sem}}","aliases":"κ_k,p^r^sem kappa mathrm","display":"\\kappa_{k,p^r}^{\\mathrm{sem}}","kind":"definition-passage","excerpt":"[Prime-power section automaton] The semantic p^r-kernel is K_k,p^r^sem(a) = { Λ_w abmod p^rL_k(a): w∈Σ_k^* } ⊆ overline M_k,p^r(a). Its cardinality is denoted κ_k,p^r^sem(a).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":59},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":94},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c89b70e4fd524363","tex":"\\lambda^A_{\\gamma,r}","aliases":"λ^A_γ,r lambda gamma","display":"\\lambda^A_{\\gamma,r}","kind":"definition-passage","excerpt":"For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrigh","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f961429a87a3d3d2","tex":"\\lambda^A_{\\gamma,s}","aliases":"λ^A_γ,s lambda gamma","display":"\\lambda^A_{\\gamma,s}","kind":"definition-passage","excerpt":"V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refinement maps are equivariant: ρ^A,V_srλ^A_γ,s = λ^A_γ,rρ^A,U_sr. No global address set and no global trivialization of th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4ead150c2ca618e8","tex":"\\lambda^X_{\\gamma,r}","aliases":"λ^X_γ,r lambda gamma","display":"\\lambda^X_{\\gamma,r}","kind":"definition-passage","excerpt":"atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-508981dc8ceb6176","tex":"\\lambda^Y_{\\gamma,u}","aliases":"λ^Y_γ,u lambda gamma","display":"\\lambda^Y_{\\gamma,u}","kind":"definition-passage","excerpt":"atrix-local-system Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_ai) ","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b41a6fa965684516","tex":"\\Lambda_\\mathfrak","aliases":"Λ_ Lambda mathfrak","display":"\\Lambda_\\mathfrak","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-44ef144f9567bd0e","tex":"\\Lambda_d","aliases":"Λ_d Lambda","display":"\\Lambda_d","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":23},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-73a733514251f5f8","tex":"\\Lambda_u","aliases":"Λ_u Lambda","display":"\\Lambda_u","kind":"definition-passage","excerpt":"[Word-Hankel matrix] The fixed-radix word-Hankel matrix of a is the infinite matrix H_a(u,v) = a_[u]_k+k^|u|[v]_k = (Λ_u a)([v]_k), u,v∈Σ_k^*.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b0b97675a09af039","tex":"\\Lambda_{u_i}","aliases":"Λ_u_i Lambda","display":"\\Lambda_{u_i}","kind":"definition-passage","excerpt":"[Greedy shortlex kernel basis] For a regular sequence a, let U_a=(u_1,…,u_r_k(a)) be the unique ordered list produced by the greedy shortlex rule, and put b_i=Λ_u_ia. For the zero sequence, the list is empty.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d803f773c83f5d83","tex":"\\Lambda_w","aliases":"Λ_w Lambda","display":"\\Lambda_w","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":21},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"definition-passage","excerpt":"[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).","count":38},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-84eb2b123d1102b8","tex":"\\nu","aliases":"ν nu","display":"\\nu","kind":"definition-passage","excerpt":"[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic ","count":30},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0857eb1e10d64876","tex":"\\nu_\\mathfrak","aliases":"ν_ nu mathfrak","display":"\\nu_\\mathfrak","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c7463eebe44dde97","tex":"\\nu_f","aliases":"ν_f nu","display":"\\nu_f","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dfeb211077c1be9a","tex":"\\nu_\\alpha","aliases":"ν_α nu alpha","display":"\\nu_\\alpha","kind":"definition-passage","excerpt":"[Normalization witness] A normalization witness on a full subcategory U_α⊆ M_α consists of a functor N_α: U_α⟶ N_α and a natural isomorphism ν_α:Sem_α N_α Longrightarrow Sem_α|_ U_α. It is classification-normalizing when every object of U_α is semantically isomorphic to its image, and two normalized objects with isomorphic semantics are isomorphic ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"definition-passage","excerpt":"[Verified preservation profile] A verified preservation profile Π_F for a functor F: C→ D consists of a subset of C together with a proof or declared comparison isomorphism showing that F preserves each listed constructor. We write c∈Π_F only after the corresponding preservation claim has been verified.","count":19},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":43},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cb28cf0122b8897d","tex":"\\varpi","aliases":"ϖ varpi","display":"\\varpi","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a29426a92b821f5","tex":"\\Pi^{\\mathrm{pres}}_\\mathfrak","aliases":"Π^pres_ Pi mathrm mathfrak","display":"\\Pi^{\\mathrm{pres}}_\\mathfrak","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f690773b893e2459","tex":"\\pi^r_{A,U,i}","aliases":"π^r_A,U,i pi","display":"\\pi^r_{A,U,i}","kind":"definition-passage","excerpt":"For every chart U and resolution r∈ R, the object A_U has a finite profiled lane system (I_A,U,r,p^r_A,U,i,A^r_U,i, ι^r_A,U,i,π^r_A,U,i)_i∈ I_A,U,r in the sense of [source reference]. For every arrow γ:U→ V, there is an address bijection λ^A_γ,r:I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrigh","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3f642a72968dc897","tex":"\\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}","aliases":"π^r_A,V,λ^A_γ,r(i) pi lambda gamma","display":"\\pi^r_{A,V,\\lambda^A_{\\gamma,r}(i)}","kind":"definition-passage","excerpt":"I_A,U,rxrightarrow∼I_A,V,r and, for every address i, a splitting comparison β^A_γ,r,i: F_γ(A^r_U,i)xrightarrow∼ A^r_V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refineme","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b8219ed897dbecfb","tex":"\\Pi^{\\mathrm{refl}}_\\mathfrak","aliases":"Π^refl_ Pi mathrm mathfrak","display":"\\Pi^{\\mathrm{refl}}_\\mathfrak","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8dd10f4f67c70fb5","tex":"\\varpi_a","aliases":"ϖ_a varpi","display":"\\varpi_a","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8bb5b5bb02010e64","tex":"\\Pi_F","aliases":"Π_F Pi","display":"\\Pi_F","kind":"definition-passage","excerpt":"[Verified preservation profile] A verified preservation profile Π_F for a functor F: C→ D consists of a subset of C together with a proof or declared comparison isomorphism showing that F preserves each listed constructor. We write c∈Π_F only after the corresponding preservation claim has been verified.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-403b4646d62f0c8b","tex":"\\pi_i","aliases":"π_i pi","display":"\\pi_i","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4187a876dac95e6a","tex":"\\Pi_U","aliases":"Π_U Pi","display":"\\Pi_U","kind":"definition-passage","excerpt":"e profiled resolved atlas] Fix a small resolved theory K. A finite profiled resolved atlas consists of: [label=textup(Aarabic*)] item a finite indexing category check A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-31d9ac6039702572","tex":"\\pi_{X,i}^r","aliases":"π_X,i^r pi","display":"\\pi_{X,i}^r","kind":"definition-passage","excerpt":"finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-23f22de9e2bf6ced","tex":"\\Pi_\\alpha","aliases":"Π_α Pi alpha","display":"\\Pi_\\alpha","kind":"definition-passage","excerpt":"[Profiled stratum passport] A profiled Komorebi stratum is a record S_α = ( C_α,Π_α, M_α, N_α,Sem_α, I_α), with the following fields. [label=textup(roman*)] item C_α is the ambient semantic category in which the realized objects live. item Π_α is the verified preservation profile of the realization. item M_α is a full replete category of admissible","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accomp","count":74},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-93737afe65ce6325","tex":"\\rho^{A,U}_{sr}","aliases":"ρ^A,U_sr rho","display":"\\rho^{A,U}_{sr}","kind":"definition-passage","excerpt":"V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refinement maps are equivariant: ρ^A,V_srλ^A_γ,s = λ^A_γ,rρ^A,U_sr. No global address set and no global trivialization of th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b66a611263a0554f","tex":"\\rho^{A,V}_{sr}","aliases":"ρ^A,V_sr rho","display":"\\rho^{A,V}_{sr}","kind":"definition-passage","excerpt":"V,λ^A_γ,r(i). They satisfy the address and splitting cocycle laws and the two compatibility identities θ^A_γ F_γ(ι^r_A,U,i) = ι^r_A,V,λ^A_γ,r(i) β^A_γ,r,i, β^A_γ,r,iF_γ(π^r_A,U,i) = π^r_A,V,λ^A_γ,r(i)θ^A_γ. For ssucceq r, the refinement maps are equivariant: ρ^A,V_srλ^A_γ,s = λ^A_γ,rρ^A,U_sr. No global address set and no global trivialization of th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-075516250dd62def","tex":"\\rho^X_{rr}","aliases":"ρ^X_rr rho","display":"\\rho^X_{rr}","kind":"definition-passage","excerpt":"ittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address sets and refinement maps are declared data; they are not inferred from thei","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e7d83795ad4c5c54","tex":"\\rho^X_{sr}","aliases":"ρ^X_sr rho","display":"\\rho^X_{sr}","kind":"definition-passage","excerpt":"_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Her","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8a4a0de6d84e2f17","tex":"\\rho^X_{tr}","aliases":"ρ^X_tr rho","display":"\\rho^X_{tr}","kind":"definition-passage","excerpt":"p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address set","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-090e008d3f404d54","tex":"\\rho^X_{ts}","aliases":"ρ^X_ts rho","display":"\\rho^X_{ts}","kind":"definition-passage","excerpt":"p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address set","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0e302f8c924f0c53","tex":"\\rho_{sr}","aliases":"ρ_sr rho","display":"\\rho_{sr}","kind":"definition-passage","excerpt":"ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accomp","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":19},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normal","count":34},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-462d10e826ee6d79","tex":"\\Sigma_k^*","aliases":"Σ_k^* Sigma","display":"\\Sigma_k^*","kind":"definition-passage","excerpt":"[Fixed-radix kernel and section module] The k-kernel and its linear span are K_k(a) = {Λ_w a:w∈Σ_k^*}, V_k(a) = span_K K_k(a). The sequence is k-regular over K when V_k(a) is finite dimensional. Its regular rank is r_k(a)=dim_K V_k(a). It is k-automatic in the present reading convention when K_k(a) is finite. In that case set κ_k(a)=| K_k(a)|.","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0ee6bcdb8ab619ac","tex":"\\sigma_{\\alpha\\beta}","aliases":"σ_αβ sigma alpha beta","display":"\\sigma_{\\alpha\\beta}","kind":"definition-passage","excerpt":"abeldef:typed-overlap A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normal","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":42},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4f43fbaed6705586","tex":"\\tau_\\mathfrak","aliases":"τ_ tau mathfrak","display":"\\tau_\\mathfrak","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized w","count":50},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-93d3c782b6f664fd","tex":"\\Phi_{\\alpha\\beta}","aliases":"Φ_αβ Phi alpha beta","display":"\\Phi_{\\alpha\\beta}","kind":"definition-passage","excerpt":"[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized w","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-372147438bc34d7d","tex":"\\Phi_{\\alpha\\delta}","aliases":"Φ_αδ Phi alpha delta","display":"\\Phi_{\\alpha\\delta}","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cdc1f0d19860bb24","tex":"\\chi_\\mathfrak","aliases":"χ_ chi mathfrak","display":"\\chi_\\mathfrak","kind":"definition-passage","excerpt":"[Bridge passport] A certified bridge from stratum α to stratum β is a record b = ( U_ b, V_ b, B_ b,τ_ b, Π^pres_ b, Π^refl_ b, Λ_ b, ν_ b,χ_ b), where: [label=textup(roman*)] item U_ b⊆ M_α and V_ b⊆ M_β are the declared domain and codomain subcategories; item B_ b: U_ b→ V_ b is the functor; item τ_ b is its bridge class; item Π^pres_ b records p","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8333cde4586cf4f8","tex":"\\Omega_{\\alpha\\beta\\gamma\\delta}","aliases":"Ω_αβγδ Omega alpha beta gamma delta","display":"\\Omega_{\\alpha\\beta\\gamma\\delta}","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1e05c28e5f477db7","tex":"\\dots","aliases":"… dots","display":"\\dots","kind":"definition-passage","excerpt":"[Finite additive hull] The category Add( A_ K) has finite tuples A=(A_1,…,A_m) as objects. A morphism A→ B=(B_1,…,B_n) is an n× m matrix with entries in A_ K(A_j,B_i). Composition is matrix multiplication using composition and addition in A_ K.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":78},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":50},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8943b22d90e5e02b","tex":"\\ell_j","aliases":"ℓ_j ell","display":"\\ell_j","kind":"definition-passage","excerpt":"[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Mixed transfer blocks] Let P=(p_i) be a packet system on X, let Q=(q_a) be a packet system on Y, and choose splittings (ι_i,π_i) and (jmath_a,ϖ_a). For f:X→ Y, define the packet-to-packet map f_ai:=ϖ_a fι_i:X_i⟶ Y_a. Its ambient corner is q_afp_i=jmath_a f_aiπ_i.","count":106},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9ec711e428117ead","tex":"\\twoheadrightarrow","aliases":"↠ twoheadrightarrow","display":"\\twoheadrightarrow","kind":"definition-passage","excerpt":"ar combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accomp","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"definition-passage","excerpt":"hcal A, retained through the overlap degrees needed for the cocycle law; item a pseudofunctor R:check A^op→Cat; item a declared preservation profile Π_U on every chart and overlap category R(U); item coherent equivalences on the reversible overlaps and separately typed directional transformations; item a pseudonatural semantics Sem: R⇒ O into a com","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interprete","count":280},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":37},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0930362556ae288b","tex":"\\prod_{(m,j)\\in M\\times J}","aliases":"∏_(m,j)∈ M× J prod in times","display":"\\prod_{(m,j)\\in M\\times J}","kind":"definition-passage","excerpt":"lved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object i","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a49212b8a1084867","tex":"\\prod_{m\\geq1}","aliases":"∏_m≥1 prod geq","display":"\\prod_{m\\geq1}","kind":"definition-passage","excerpt":"[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-16115065fd014436","tex":"\\prod_{X\\in\\mathsf S}","aliases":"∏_X∈ S prod in mathsf","display":"\\prod_{X\\in\\mathsf S}","kind":"definition-passage","excerpt":"[Finite coordinate action groupoid] Fix a dimension vector d=(d_X)_X∈ S. Let G_ d=∏_X∈ SGL_d_X(𝕜) act on the finite-dimensional representation space Rep_ d( K) by base change. The action groupoid [Rep_ d( K)/G_ d] has representations as objects and coordinate changes carrying one representation to another as morphisms.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the r","count":164},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-19f15b31c1195b7f","tex":"\\sum_{(a,i)\\in E}","aliases":"∑_(a,i)∈ E sum in","display":"\\sum_{(a,i)\\in E}","kind":"definition-passage","excerpt":"[Support compression and residual] For packet systems P on X and Q on Y, let E⊆ I_ Q× I_ P. Define C_E(f):=∑_(a,i)∈ Eq_afp_i, R_E(f):=f-C_E(f). The first term is the retained transfer support and the second is its named residual.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-263a48a6852fd440","tex":"\\sum_{C\\text{ cyclic SCC}}","aliases":"∑_C cyclic SCC sum text","display":"\\sum_{C\\text{ cyclic SCC}}","kind":"definition-passage","excerpt":"[Address-preserving cyclic dimension] The cyclic packet dimension of the marked system is cdim_ P(Φ) = ∑_C cyclic SCC|C|. tag11.7 It measures the number of packet variables that remain in coupled nonlinear components after exact elimination of the acyclic singleton components.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-deb9c97b11939ae2","tex":"\\sum_{d\\mid n}","aliases":"∑_dmid n sum mid","display":null,"kind":"definition-passage","excerpt":"[Euler ghost coordinates] For b∈ E_ Z, define c_n(b)=∑_dmid nd b_d, n≥1. Let G_ Z be the set of integer sequences c=(c_n) satisfying frac1n∑_dmid nμ (frac nd)c_d∈ Z (n≥1).","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-911e210cde7fc6ad","tex":"\\sum_{e\\geq0}","aliases":"∑_e≥0 sum geq","display":"\\sum_{e\\geq0}","kind":"definition-passage","excerpt":"[Prime-power seed series] For a multiplicative function f and a prime p, define F_f,p(T) = ∑_e≥0f(p^e)T^e ∈1+T Z[[T]].","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"definition-passage","excerpt":"nite quiver. Its vertices are sorts and its arrows are typed primitive operations. item Itriangleleft 𝕜 Q is a finitely generated two-sided ideal generated by 𝕜-linear combinations of paths with common source and common target. item P is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the r","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7fec58bf80198971","tex":"\\sum_{i\\in I}","aliases":"∑_i∈ I sum in","display":"\\sum_{i\\in I}","kind":"definition-passage","excerpt":"[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8b08c7b7ae33ed2f","tex":"\\sum_{i\\in I_{X,r}}","aliases":"∑_i∈ I_X,r sum in","display":"\\sum_{i\\in I_{X,r}}","kind":"definition-passage","excerpt":"block15-finite-profiled-lane-system Let C be an additive idempotent-complete category, let X∈ C, and let R be a finite resolution shape. A finite profiled lane system L_X consists, for every r∈ R, of a finite address set I_X,r and a family P_X^r=(p_X,i^r)_i∈ I_X,r ⊆End_ C(X) with (p_X,i^r)^2=p_X,i^r, p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d0b76425559f0527","tex":"\\sum_{j=0}^{r-1}","aliases":"∑_j=0^r-1 sum","display":"\\sum_{j=0}^{r-1}","kind":"definition-passage","excerpt":"[Ordinary-point locus] A nonzero D-finite series F∈ K[[z]] belongs to Hol^rm ord_K when its monic minimal operator can be written L_F = D^r+∑_j=0^r-1ℓ_j(z)D^j with every ℓ_j regular at z=0.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-441a6af1afa122e0","tex":"\\sum_{n\\geq0}","aliases":"∑_n≥0 sum geq","display":"\\sum_{n\\geq0}","kind":"definition-passage","excerpt":"[3pt] 1+displaystylemax_G∈Br_0(P_F), G≠ F ord_z(F-G), · otherwise. The corresponding branch selector is the finite coefficient jet J_rm alg(F) = (a_0,a_1,…,a_σ-1), F(z)=∑_n≥0a_nz^n. When σ=0, the selector is empty.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d12397dfd76965fd","tex":"\\sum_{\\rho^X_{sr}(j)=i}","aliases":"∑_ρ^X_sr(j)=i sum rho","display":"\\sum_{\\rho^X_{sr}(j)=i}","kind":"definition-passage","excerpt":"p_X,i^rp_X,j^r=0 (i≠ j), ∑_i∈ I_X,rp_X,i^r=1_X, together with chosen splittings X_i^rxrightarrowι_X,i^rX xrightarrowπ_X,i^rX_i^r, ι_X,i^rπ_X,i^r=p_X,i^r, π_X,i^rι_X,i^r=1_X_i^r. For ssucceq r, the system also contains a surjection ρ^X_sr:I_X,s↠ I_X,r such that p_X,i^r = ∑_ρ^X_sr(j)=ip_X,j^s, ρ^X_tr=ρ^X_srρ^X_ts. Here ρ^X_rr=1_I_X,r. The address set","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8fb7e40c2afb45a0","tex":"\\sum_{\\rho_{sr}(j)=i}","aliases":"∑_ρ_sr(j)=i sum rho","display":"\\sum_{\\rho_{sr}(j)=i}","kind":"definition-passage","excerpt":"is a finite list of packet declarations. A declaration on a sort X consists of endomorphism words p_X,r,i and the relations p_X,r,ip_X,r,j=δ_ijp_X,r,i, ∑_i p_X,r,i=1_X. item Ref is a finite list of refinement declarations. For ssucceq r, an address map ρ_sr:I_X,s↠ I_X,r is accompanied by the equations p_X,r,i = ∑_ρ_sr(j)=ip_X,s,j. item Comp is a li","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"definition-passage","excerpt":"effectivity condition is simply x∈ M_k,p(a); every compatible full semantic module tower satisfies it by [source reference]. item The integral lattice obstruction is ϵ_int(x) = [x]∈ M_k,p(a)/M_k(a). It vanishes exactly when x is represented by one integral lattice element. item The section-orbit obstruction is membership in M_k(a)∖ O_k(a). Even aft","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6f333436e7d549b7","tex":"\\ast","aliases":"∗ ast","display":"\\ast","kind":"definition-passage","excerpt":"[Quadruple-overlap coherence defect] Write horizontal composition of natural transformations as ∗. Define Ω_αβγδ = [ η_αγδ ∘ (id∗η_αβγ) ] ∘ [ η_αβδ ∘ (η_βγδ∗id) ]^-1. It is a natural automorphism of Φ_αδ on the quadruple overlap. The atlas is coherent there exactly when Ω_αβγδ=id.","count":17},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cdb50684e393a85a","tex":"\\ast_D","aliases":"∗_D ast","display":"\\ast_D","kind":"definition-passage","excerpt":"[Divisor--ghost transform] For an arithmetic exponent function b: N_≥1→ Z, define G(b) = 1mathbin∗_D(id· b), so that G(b)(n) = ∑_dmid nd b(d).","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"definition-passage","excerpt":"[Relational support product] For E⊆ I_Y× I_X and F⊆ I_Z× I_Y, define F∘ E := {(c,i):there exists a∈ I_Y with (c,a)∈ F and (a,i)∈ E}.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"definition-passage","excerpt":"[Raw and saturated section lattices] The raw section lattice and the saturated section lattice are M_k(a) = span_ Z K_k(a), L_k(a) = V_ Q(a)∩ Z^ N. The intersection defining L_k(a) is taken inside Q^ N.","count":21},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"definition-passage","excerpt":"[Exact valuation profile and certified class] For f= M_p(c), define ν_f(r)=v_p(c_r)∈ Z_≥0∪{∞}. If w: N→ Z_≥0 is nondecreasing and tends to infinity, put M_w = {c∈ c_0( Z_p):v_p(c_r)≥ w(r)}, C_w= M_p( M_w). The function w is a declared precision certificate. The class C_w is not asserted to be a characterization of every possible notion of local ana","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"[Typed overlap] A reversible overlap between strata α and β consists of full replete subcategories M_αβ⊆ M_α, M_βα⊆ M_β, an equivalence Φ_αβ: M_αβ overset∼⟶ M_βα, and a semantic natural isomorphism σ_αβ: J_αβSem_α Longrightarrow Sem_βΦ_αβ for an explicitly declared comparison functor J_αβ between the observable doctrines. An overlap is normalized w","count":16},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"definition-passage","excerpt":"[Profile-indexed resolved envelope] For a preservation profile Π, a profile-indexed resolved envelope Syn_Π( K), when it exists, is a universal source satisfying Mod_Π( K, C) ≃ Fun_Π(Syn_Π( K), C) for every target category C carrying the structures named by Π. The equivalence must preserve the chosen comparison data, not only the underlying functor","count":16},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"definition-passage","excerpt":"[Complete invariant on a declared target] An invariant I_α is complete on N_α when I_α(X)≅ I_α(Y) Longleftrightarrow X≅ Y for all X,Y∈ N_α. This statement classifies the objects of N_α up to the isomorphisms allowed there. It does not classify arbitrary presentations outside that subcategory.","count":45},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"definition-passage","excerpt":"[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"definition-passage","excerpt":"[Marked binomial--Fuchsian packet] A marked packet is a tuple b=(α,λ,N,j;G), where α≠0, N≥1, 0≤ j<N, and G(x)=G_N,j^λ,α(x). The parameters (α,λ,N,j) are part of the semantic object rather than metadata to be forgotten.","count":22},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"definition-passage","excerpt":"[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.","count":19},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8e30d8d58a04592c","tex":"\\equiv_{\\mathsf S}","aliases":"≡_ S equiv mathsf","display":"\\equiv_{\\mathsf S}","kind":"definition-passage","excerpt":"[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"definition-passage","excerpt":"[Canonical Cartier module and Cartier rank] The canonical Cartier module of F is C_p(F) = ( V_p(F), (Λ_d|_ V_p(F))_0≤ d<p, F, ϵ_0 ). Its dimension r_p(F)=dim_ F_p V_p(F) is the Cartier rank.","count":86},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Coefficientwise filtered product] Let R be a commutative ring. A filtered factor family is a sequence F_m(z)∈ 1+z^w(m)R[[z]], w(m)⟶∞. Its coefficientwise product is ∏_m≥1F_m(z), where the coefficient of z^N is computed from the finitely many factors satisfying w(m)≤ N.","count":127},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"definition-passage","excerpt":"[Finite packet system] Let X be an object of a preadditive category. A finite packet system on X is a family P=(p_i)_i∈ I⊆End(X) satisfying p_i^2=p_i, p_ip_j=0 (i≠ j), ∑_i∈ Ip_i=1_X. No self-adjointness or metric orthogonality is assumed unless stated separately.","count":61},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":34},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"definition-passage","excerpt":"[Mapping cone] For a cochain map D:C^bullet→ E^bullet, define Cone(D)^n=E^n⊕ C^n+1, d_Cone(e,c)=(d_Ee+Dc,-d_Cc).","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ddeace29098c3395","tex":"\\bigoplus_{a\\in I_{Y,U,u}}","aliases":"⊕_a∈ I_Y,U,u bigoplus in","display":"\\bigoplus_{a\\in I_{Y,U,u}}","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fa0b3e595fc9f5b4","tex":"\\bigoplus_{i\\in I}","aliases":"⊕_i∈ I bigoplus in","display":"\\bigoplus_{i\\in I}","kind":"definition-passage","excerpt":"[Packet analysis and synthesis] The packet analysis and synthesis maps are A_ P := (π_i)_i∈ I: X⟶⊕_i∈ IX_i, S_ P := (ι_i)_i∈ I: ⊕_i∈ IX_i⟶ X.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5b3f3182fc77c747","tex":"\\bigoplus_{i\\in I_{X,U,r}}","aliases":"⊕_i∈ I_X,U,r bigoplus in","display":"\\bigoplus_{i\\in I_{X,U,r}}","kind":"definition-passage","excerpt":"[Matrix local system] Let X,Y carry holonomy-profiled lane data. At source resolution r and target resolution u, define the matrix group over a chart U by mathscr M^u,r_Y,X(U) := ⊕_a∈ I_Y,U,u ⊕_i∈ I_X,U,r Hom_ C_U(X^r_U,i,Y^u_U,a). For h=(h_ai)∈mathscr M^u,r_Y,X(U), define transport along γ:U→ V by (γ_*h)_ λ^Y_γ,u(a),λ^X_γ,r(i) := β^Y_γ,u,a F_γ(h_a","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"definition-passage","excerpt":"[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b03160646378fb36","tex":"\\bigcap_{j\\in J}","aliases":"⋂_j∈ J bigcap in","display":"\\bigcap_{j\\in J}","kind":"definition-passage","excerpt":"[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-42b36b0f6b095bcb","tex":"\\bigcap_{m\\in M}","aliases":"⋂_m∈ M bigcap in","display":"\\bigcap_{m\\in M}","kind":"definition-passage","excerpt":"[Behavioral congruence and linear behavioral kernel] In Set or a variety of algebras, put x≡_ Sy Longleftrightarrow o_j(ρ(m)x)=o_j(ρ(m)y) for every m∈ M, j∈ J. Write Θ_ S for this relation. In R-Mod, define N_ S := ⋂_m∈ M⋂_j∈ J ker(o_jρ(m)). Then Q_ S=X/N_ S.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"definition-passage","excerpt":"[Digit-linear realization] A digit-linear realization of a is a tuple L=(W,A_0,…,A_k-1,v,λ), where W is a finite-dimensional K-vector space, A_d∈End_K(W), v∈ W, and λ∈ W^*, such that for every word w=d_0⋯ d_ℓ-1, a_[w]_k = λ A_wv, A_w=A_d_ℓ-1⋯ A_d_0. Because different words can represent the same integer after terminal high zeros are appended, equat","count":38},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Realization of a finite resolved signature] Let C be a 𝕜-linear category. A realization of K_rm fin in C assigns an object M_X to every vertex X∈ Q_0 and a morphism M_a:M_s(a)⟶ M_t(a) to every arrow a∈ Q_1, such that all relations in I, all packet equations, and all declared refinement equations hold. Completion and certificate tags are interprete","count":103},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"definition-passage","excerpt":"lved observational system] Let X be an object equipped with an action ρ:M⟶ End(X) of a continuation monoid and with observations o_j:X⟶ Y_j (j∈ J). The tuple S=(X,M,ρ,(o_j)_j∈ J) is a resolved observational system. When the indicated product exists, its behavior map is Beh_ S:X⟶ ∏_(m,j)∈ M× JY_j, x⟼(o_j(ρ(m)x))_m,j. The relative behavioral object i","count":20},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-68421187e198542c","tex":"a_c","aliases":"a_c","display":"a_c","kind":"introduction-passage","excerpt":"Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ca4742dbee826e08","tex":"\\mathfrak A_{\\mathrm{cert}}","aliases":"A_cert mathfrak mathrm","display":"\\mathfrak A_{\\mathrm{cert}}","kind":"introduction-passage","excerpt":"Let A_cert:= A_13 denote the finite certified atlas constructed in [source reference]. Its reversible core is the disjoint union of five connected semantic components, while its directional satellites retain explicit loss or obstruction passports.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c2e6b50202ef016e","tex":"A_F","aliases":"A_F","display":"A_F","kind":"introduction-passage","excerpt":"Let A_F=K(z)[Y]/(P_F), and let bar Y denote the residue class of Y. In the basis 1,bar Y,…,bar Y^d-1, multiplication by bar Y is represented by the companion matrix M_F. Its characteristic and minimal polynomials are both P_F.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-40166d2eb0bbd2d6","tex":"a_{n+T}","aliases":"a_n+T","display":"a_{n+T}","kind":"introduction-passage","excerpt":"Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-105e51e805a184da","tex":"a_r","aliases":"a_r","display":"a_r","kind":"introduction-passage","excerpt":"p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c7f299f46e2f11df","tex":"a_{T(a)-1}","aliases":"a_T(a)-1","display":"a_{T(a)-1}","kind":"introduction-passage","excerpt":"Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f3b515667bb54896","tex":"\\bar Y","aliases":"bar Y bar","display":"\\bar Y","kind":"introduction-passage","excerpt":"Let A_F=K(z)[Y]/(P_F), and let bar Y denote the residue class of Y. In the basis 1,bar Y,…,bar Y^d-1, multiplication by bar Y is represented by the companion matrix M_F. Its characteristic and minimal polynomials are both P_F.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2ebdca52d632fc21","tex":"\\bar Y^{d-1}","aliases":"bar Y^d-1 bar","display":"\\bar Y^{d-1}","kind":"introduction-passage","excerpt":"Let A_F=K(z)[Y]/(P_F), and let bar Y denote the residue class of Y. In the basis 1,bar Y,…,bar Y^d-1, multiplication by bar Y is represented by the companion matrix M_F. Its characteristic and minimal polynomials are both P_F.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ca71bbaea6ea0f3","tex":"\\mathrm{cert}","aliases":"cert mathrm","display":"\\mathrm{cert}","kind":"introduction-passage","excerpt":"Let A_cert:= A_13 denote the finite certified atlas constructed in [source reference]. Its reversible core is the disjoint union of five connected semantic components, while its directional satellites retain explicit loss or obstruction passports.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ddc7e0f1c8697a47","tex":"\\mathsf D","aliases":"D mathsf","display":"\\mathsf D","kind":"introduction-passage","excerpt":"parbox0.96linewidthscriptsize Legend. E: equivalence; FF: fully faithful inclusion; D: information-losing directional functor; P: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts per, rm, and Fuss denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell o","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5a3fc062502fca22","tex":"\\mathsf E","aliases":"E mathsf","display":"\\mathsf E","kind":"introduction-passage","excerpt":"parbox0.96linewidthscriptsize Legend. E: equivalence; FF: fully faithful inclusion; D: information-losing directional functor; P: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts per, rm, and Fuss denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell o","count":23},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a09a14226d28fe19","tex":"\\mathsf E_{\\mathrm{Fuss}}","aliases":"E_Fuss mathsf mathrm","display":"\\mathsf E_{\\mathrm{Fuss}}","kind":"introduction-passage","excerpt":"E: equivalence; FF: fully faithful inclusion; D: information-losing directional functor; P: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts per, rm, and Fuss denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form E_Fuss/ D means equivalence","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7c797e2a6c426ad6","tex":"\\mathbb F_p^{\\mathbb N}","aliases":"F_p^ N mathbb mathbb","display":"\\mathbb F_p^{\\mathbb N}","kind":"introduction-passage","excerpt":"Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5be067b8d2d5399c","tex":"\\mathsf{FF}","aliases":"FF mathsf","display":"\\mathsf{FF}","kind":"introduction-passage","excerpt":"parbox0.96linewidthscriptsize Legend. E: equivalence; FF: fully faithful inclusion; D: information-losing directional functor; P: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts per, rm, and Fuss denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell o","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-08dc6e54edbcd212","tex":"K_c","aliases":"K_c","display":"K_c","kind":"introduction-passage","excerpt":"Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4b91ccdd1e345e08","tex":"M_F","aliases":"M_F","display":"M_F","kind":"introduction-passage","excerpt":"Let A_F=K(z)[Y]/(P_F), and let bar Y denote the residue class of Y. In the basis 1,bar Y,…,bar Y^d-1, multiplication by bar Y is represented by the companion matrix M_F. Its characteristic and minimal polynomials are both P_F.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"introduction-passage","excerpt":"Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not","count":25},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4f7d45b3cbc98ac7","tex":"\\min_{g\\in K_c}","aliases":"min_g∈ K_c min in","display":"\\min_{g\\in K_c}","kind":"introduction-passage","excerpt":"Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fd3a8f4d3f154243","tex":"\\Per","aliases":"Per mathbf","display":"{\\mathbf{Per}}","kind":"introduction-passage","excerpt":"Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-827f365cc607b14a","tex":"\\mathrm{per}","aliases":"per mathrm","display":"\\mathrm{per}","kind":"introduction-passage","excerpt":"Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-","count":24},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3814eb035dca537f","tex":"\\Per_p^{\\times,\\mathrm{per}}","aliases":"Per_p^×,per Per times mathrm mathbf times mathrm","display":"{\\mathbf{Per}}_p^{\\times,\\mathrm{per}}","kind":"introduction-passage","excerpt":"Fix a prime p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bcf57c4f7b31467d","tex":"\\mathfrak R_p^{\\mathrm{per}}","aliases":"R_p^per mathfrak mathrm","display":"\\mathfrak R_p^{\\mathrm{per}}","kind":"introduction-passage","excerpt":"Let R_p^per denote the four-chart pseudofunctor defined by the preceding passports and comparison cells.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-012cb3d55b967600","tex":"\\mathfrak R_p^{\\mathrm{rm}}","aliases":"R_p^rm mathfrak mathrm","display":"\\mathfrak R_p^{\\mathrm{rm}}","kind":"introduction-passage","excerpt":"Let R_p^rm denote the three-chart pseudofunctor defined by the passports above.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-560207b790a4df40","tex":"\\mathrm{rm}","aliases":"rm mathrm","display":"\\mathrm{rm}","kind":"introduction-passage","excerpt":"Let R_p^rm denote the three-chart pseudofunctor defined by the passports above.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"introduction-passage","excerpt":"Let T_M,s denote this coefficient operator. Conjugating the clopen packet theorem by M_p gives the following normalized coefficient atlas.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4ff8fe4a7eb22523","tex":"\\mathcal T_{M,s}","aliases":"T_M,s mathcal","display":"\\mathcal T_{M,s}","kind":"introduction-passage","excerpt":"Let T_M,s denote this coefficient operator. Conjugating the clopen packet theorem by M_p gives the following normalized coefficient atlas.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0289b7ef692ee27e","tex":"W_a","aliases":"W_a","display":"W_a","kind":"introduction-passage","excerpt":"p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-10d0329c35407a66","tex":"z^r","aliases":"z^r","display":"z^r","kind":"introduction-passage","excerpt":"p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The ","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3027806ef1af3a82","tex":"\\Gamma_c","aliases":"Γ_c Gamma","display":"\\Gamma_c","kind":"introduction-passage","excerpt":"Fix a declared conditioning bound c, write Γ_c for the parameters satisfying it, and let K_c={g∈Γ_c:d(X,g)≤ 1}. By hypothesis K_c is compact. If K_c is empty, every near-minimizing sequence eventually has defect below 1 and therefore lies outside Γ_c. Otherwise continuity gives an attained minimum a_c=min_g∈ K_cd(X,g). Since the zero infimum is not","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a8df13ded83b21f","tex":"\\sum_{r=0}^{T(a)-1}","aliases":"∑_r=0^T(a)-1 sum","display":"\\sum_{r=0}^{T(a)-1}","kind":"introduction-passage","excerpt":"p. Let Per_p^×,per denote the discrete category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"Constant-one function 1 · β_1=1 · 1,1,1,1,1,1,… · 1,1,1,1,1,1,…","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e0a61d58c33b3ed4","tex":"\\mathbf a","aliases":"a mathbf","display":"\\mathbf a","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"[Ghost-coordinate equivalence] The divisor ghost map W: E_ Z ⟶ G_ Z, b⟼ c(b), is an additive group isomorphism. Its inverse is b_n = frac1n∑_dmid n μ (frac nd)c_d. If A(z)= E(b)(z), then fraczA'(z)A(z) = ∑_n≥1c_nz^n.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e0e42505360479a3","tex":"a'_t","aliases":"a'_t","display":"a'_t","kind":"use","excerpt":"ed-observer Let S be a finite continuation-state set, let (A,⊕,0) be a commutative monoid, let B be a finite branch alphabet, and let δ:S× B⟶ S be a state transition. Suppose every branch b∈ B acts by a monoid homomorphism L_b:A⟶ A. Take an empty aggregation to be 0. For a state packet a=(a_s)_s∈ S, define a'_t := ⊕_δ(s,b)=tL_b(a_s). If a finite fa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54034542483dca3c","tex":"A^{\\mathsf T}","aliases":"A^ T mathsf","display":"A^{\\mathsf T}","kind":"use","excerpt":"For a linear map A, h_A^ TC(x)=h_C(Ax). If the incoming fiber is empty, both sides of the formula are -∞. Otherwise, the support function of the convex hull of a finite union is the maximum of the component support functions, with the same conventions for empty components. These identities give the formula. Finite unions of compact sets have compac","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fd5fce2690e249bf","tex":"\\mathfrak a^{u,r}_{X,Y}","aliases":"a^u,r_X,Y mathfrak","display":"\\mathfrak a^{u,r}_{X,Y}","kind":"use","excerpt":"[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contrib","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-55d5dcaf90371b0c","tex":"\\mathfrak A^{u,r}_{Y,X}","aliases":"A^u,r_Y,X mathfrak","display":"\\mathfrak A^{u,r}_{Y,X}","kind":"use","excerpt":"item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix rec","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e10676711805547b","tex":"\\mathfrak A^{v,r}_{Z,X}","aliases":"A^v,r_Z,X mathfrak","display":"\\mathfrak A^{v,r}_{Z,X}","kind":"use","excerpt":"item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precis","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-89da852a7b74099c","tex":"\\mathfrak a^{v,s}_{X,Y}","aliases":"a^v,s_X,Y mathfrak","display":"\\mathfrak a^{v,s}_{X,Y}","kind":"use","excerpt":"For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8c04ee1b67aee31b","tex":"\\mathfrak A^{v,u}_{Z,Y}","aliases":"A^v,u_Z,Y mathfrak","display":"\\mathfrak A^{v,u}_{Z,Y}","kind":"use","excerpt":"item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precis","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2b083e98dcc30f95","tex":"\\mathbf a_0","aliases":"a_0 mathbf","display":"\\mathbf a_0","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"use","excerpt":"[The selected object is a marked system] Two different polynomial systems can have the same fixed point. Conversely, a single polynomial system can have several fixed points outside the contraction ball. The normalized object in this section is therefore (s,A_2,…,A_D; P;certificate), not an unmarked semantic vector u.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d325da4f9b58928d","tex":"A_3","aliases":"A_3","display":"A_3","kind":"use","excerpt":"[The S_3 nonnormal atlas] Let G=S_3 and let H be generated by one transposition. There are three conjugate order-two subgroups and hence three conjugate cubic fixed-field charts. Their subgroup core is trivial, while their normal closure is all of S_3. By contrast, the order-three subgroup A_3 is normal and its fixed chart carries the quotient symm","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d84b5a24a5037067","tex":"A_4","aliases":"A_4","display":"A_4","kind":"use","excerpt":"The Galois-Tannakian atlas verifier uses exact rational group-algebra arithmetic, exact finite permutation combinatorics, and exact Gaussian-rational matrix algebra. It audits cyclic groups, V_4, S_3, D_4, and A_4; every subgroup Reynolds idempotent; subgroup inclusion and projector order; normality and centrality; core and normal closure; commutin","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-392e14c08f650331","tex":"A_b'","aliases":"A_b'","display":"A_b'","kind":"use","excerpt":"heorem[Exact divisor--ghost bridge] The map b↦ c= G(b) is injective, with inverse b(n) = frac1n (μmathbin∗_Dc)(n). Its integral image consists exactly of the arithmetic functions c satisfying nmid(μmathbin∗_Dc)(n) (n≥1). For the Euler product A_b(z)=∏_n≥1(1-z^n)^-b(n), one has fraczA_b'(z)A_b(z) = ∑_n≥1c(n)z^n.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-875f61e7844f5048","tex":"A_b^{\\mathsf T}","aliases":"A_b^ T mathsf","display":"A_b^{\\mathsf T}","kind":"use","excerpt":"Two important realizations have the same state skeleton but different carriers. item For convex value semantics, A is the family of compact convex sets, including the empty set, ⊕ is convex hull of union with identity ∅, and L_b(C)=A_b^ TC. item For linear polyphase routing, A is an additive convolution algebra, ⊕ is coefficient addition, and L_b m","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ecc1467eaa80d3e9","tex":"A_D","aliases":"A_D","display":"A_D","kind":"use","excerpt":"[The selected object is a marked system] Two different polynomial systems can have the same fixed point. Conversely, a single polynomial system can have several fixed points outside the contraction ball. The normalized object in this section is therefore (s,A_2,…,A_D; P;certificate), not an unmarked semantic vector u.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2b082c7cf8e051da","tex":"a_{d-1}","aliases":"a_d-1","display":"a_{d-1}","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-735f75f6033b9428","tex":"A_{\\operatorname{dig}_k(n)}","aliases":"A_dig_k(n) operatorname","display":"A_{\\operatorname{dig}_k(n)}","kind":"use","excerpt":"Conversely, suppose a has a realization (W,A_d,v,λ). For x∈ W, define its represented behavior by B(x)(n) = λ A_dig_k(n)x, where A_dig_k(n) is the product associated with the canonical least-significant-digit-first expansion of n. Equation [source reference] and concatenation of digit words give Λ_w a = B(A_wv). Hence every kernel section belongs t","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-41ae46808180b866","tex":"a_e","aliases":"a_e","display":"a_e","kind":"use","excerpt":"[Observer base-change exact sequence] The evaluation map π:K^(E)⟶Span_K(E), ∑_e∈ Ea_e[e] ⟼ ∑_e∈ Ea_e e is surjective and fits into the exact sequence 0⟶Rel_K(E) ⟶ K^(E) xrightarrowπSpan_K(E) ⟶0, where Rel_K(E):=kerπ is the module of linear relations among distinct discrete residual states. Hence Set minimization commutes with free linearization exa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-543fc839494fb355","tex":"A_{h j,h i}","aliases":"A_h j,h i","display":"A_{h j,h i}","kind":"use","excerpt":"Write an arbitrary kk-linear map as a matrix A=(A_ji). The intertwining equations for the permutation actions are equivalent to A_h j,h i=A_j i (h∈ H). Hence the entries are constant on diagonal orbits. The orbit-indicator matrices have disjoint supports, so they are linearly independent over every commutative ring and span all invariant matrices.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4f6afa45de73ec0e","tex":"a_{H_ax+j}","aliases":"a_H_ax+j","display":"a_{H_ax+j}","kind":"use","excerpt":"[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a88ad33e388dbded","tex":"a_{Hx+j}","aliases":"a_Hx+j","display":"a_{Hx+j}","kind":"use","excerpt":"[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d8504a2c70663cb6","tex":"A_{j i}","aliases":"A_j i","display":"A_{j i}","kind":"use","excerpt":"Write an arbitrary kk-linear map as a matrix A=(A_ji). The intertwining equations for the permutation actions are equivalent to A_h j,h i=A_j i (h∈ H). Hence the entries are constant on diagonal orbits. The orbit-indicator matrices have disjoint supports, so they are linearly independent over every commutative ring and span all invariant matrices.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fafe0eee210a28d6","tex":"A_{ji}","aliases":"A_ji","display":"A_{ji}","kind":"use","excerpt":"Write an arbitrary kk-linear map as a matrix A=(A_ji). The intertwining equations for the permutation actions are equivalent to A_h j,h i=A_j i (h∈ H). Hence the entries are constant on diagonal orbits. The orbit-indicator matrices have disjoint supports, so they are linearly independent over every commutative ring and span all invariant matrices.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2b57d0a64273a5ab","tex":"a_{kn+d}","aliases":"a_kn+d","display":"a_{kn+d}","kind":"use","excerpt":"[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coe","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"use","excerpt":"For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c6329d08fd9350ec","tex":"A_{M,s}^{(d)}","aliases":"A_M,s^(d)","display":"A_{M,s}^{(d)}","kind":"use","excerpt":"The same integer kernel matrix has different categorical status after changing the coefficient doctrine. Fix d≥0 and write M_d(R)=R^d+1 for the formal truncated Mahler coefficient module. Let A_M,s^(d) =[Θ_M,s(r,k)]_0≤ k,r≤ d, where the target degree k is the row index.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6db553fa69594bd3","tex":"A_{m,s}^{(d)}","aliases":"A_m,s^(d)","display":"A_{m,s}^{(d)}","kind":"use","excerpt":"Fix d≥0. Over characteristic zero, the affine Mahler coefficient maps A_m,s^(d) are invertible and satisfy A_n,t^(d)A_m,s^(d)=A_mn,mt+s^(d). For coprime m,n, the two paths around every declared Chinese-remainder diamond are equal [source reference]. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with tri","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f6cd2bab542d5668","tex":"A_{m;i\\mid j_1,\\ldots,j_m}","aliases":"A_m;imid j_1,…,j_m mid ldots","display":null,"kind":"use","excerpt":"For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d845b5781e37802e","tex":"A_{mn,mt+s}^{(d)}","aliases":"A_mn,mt+s^(d)","display":"A_{mn,mt+s}^{(d)}","kind":"use","excerpt":"Fix d≥0. Over characteristic zero, the affine Mahler coefficient maps A_m,s^(d) are invertible and satisfy A_n,t^(d)A_m,s^(d)=A_mn,mt+s^(d). For coprime m,n, the two paths around every declared Chinese-remainder diamond are equal [source reference]. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with tri","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"Define the alias and cancellation deficits by A_N(λ)=N-r_rm orb(N;λ), C_N,j(λ) = r_rm orb(N;λ) -r_D(N,j;λ). Then N-r_D(N,j;λ) = A_N(λ)+C_N,j(λ).","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-07112285aabb492d","tex":"a_N","aliases":"a_N","display":"a_N","kind":"use","excerpt":"[Finite-prefix triangularity] For each N, the coefficient prefix (a_1,…,a_N) of A(z)=1+∑_n≥1a_nz^n and the exponent prefix (b_1,…,b_N) determine one another by integral triangular recursions. No exponent b_m with m>N affects the coefficient prefix through degree N.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4c8131608f522ec2","tex":"a_{n+1}","aliases":"a_n+1","display":"a_{n+1}","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8391cb54fff16235","tex":"a_{n+d}","aliases":"a_n+d","display":"a_{n+d}","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3ad04bec4bcc6402","tex":"a_{n+d-1}","aliases":"a_n+d-1","display":"a_{n+d-1}","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bb23b5c0c1daeae2","tex":"a_{n+s}","aliases":"a_n+s","display":"a_{n+s}","kind":"use","excerpt":"Conversely, suppose ∑_s=0^qp_s(n)a_n+s=0 (n≥ N), where p_s∈ K[n], and put θ=zD.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5f3deb161f4183b8","tex":"A_{N,j}","aliases":"A_N,j","display":"A_{N,j}","kind":"use","excerpt":"Let Q^rm cl_N,j be an annihilator produced by a declared closure or elimination pipeline, and let L_N,j be the monic minimal operator of the section. Since every annihilator belongs to the principal left ideal generated by L_N,j, Q^rm cl_N,j=A_N,jL_N,j for some Ore operator A_N,j. The nonnegative quantity δ^rm cl_N,j = ordQ^rm cl_N,j - ordL_N,j mea","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b5add331fb205c25","tex":"A_{n,t}^{(d)}","aliases":"A_n,t^(d)","display":"A_{n,t}^{(d)}","kind":"use","excerpt":"Fix d≥0. Over characteristic zero, the affine Mahler coefficient maps A_m,s^(d) are invertible and satisfy A_n,t^(d)A_m,s^(d)=A_mn,mt+s^(d). For coprime m,n, the two paths around every declared Chinese-remainder diamond are equal [source reference]. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with tri","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0ba8ef421f876d55","tex":"a_{n-j}","aliases":"a_n-j","display":"a_{n-j}","kind":"use","excerpt":"[Exact coefficient recurrence] Let A(z)=1+∑_n≥1a_nz^n and let c_n be its Euler ghost coordinates. Then na_n = ∑_j=1^nc_j a_n-j (n≥1), with a_0=1. Conversely, any c∈ G_ Z and this recurrence reconstruct the integral unit series corresponding to its Euler exponents.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4aa363ca1be4278b","tex":"a_{n-m+j}","aliases":"a_n-m+j","display":"a_{n-m+j}","kind":"use","excerpt":"[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-50feb1ecb3e9023f","tex":"a_{N_a+s-1}","aliases":"a_N_a+s-1","display":"a_{N_a+s-1}","kind":"use","excerpt":"Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite pre","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c184f4d518178830","tex":"a_{Nm+j}","aliases":"a_Nm+j","display":"a_{Nm+j}","kind":"use","excerpt":"Over characteristic zero, write the unique active expansion over an algebraic closure as a_n=∑_λ∈ΛP_λ(n)λ^n. For each sampling modulus N and address j, the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of a_Nm+j [so","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-097a2940caa684dc","tex":"a_{Nn+j}","aliases":"a_Nn+j","display":"a_{Nn+j}","kind":"use","excerpt":"[Typed polyphase realization] Let K be a field, let N∈ Z with N≥1 be invertible in K, and suppose K contains a primitive N-th root of unity. For F(z)=∑_n≥0a_nz^n∈ K[[z]], set F_j(w):=∑_n≥0a_Nn+jw^n. Then F(z)=∑_j=0^N-1z^jF_j(z^N) is exact analysis and synthesis. The Fourier residue projectors and the polyphase components satisfy the explicit coordi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c02d97deddf86559","tex":"\\mathcal A_p","aliases":"A_p mathcal","display":"\\mathcal A_p","kind":"use","excerpt":"For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a),","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-561015bea377ac4c","tex":"a_{p^\\ell n+r}","aliases":"a_p^ℓ n+r ell","display":"a_{p^\\ell n+r}","kind":"use","excerpt":"The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9a2d6ad5301515ad","tex":"a_{pn+d}","aliases":"a_pn+d","display":"a_{pn+d}","kind":"use","excerpt":"For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9e9def832b7eb98b","tex":"A_q","aliases":"A_q","display":"A_q","kind":"use","excerpt":"Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bb2614930ea303a4","tex":"a_s","aliases":"a_s","display":"a_s","kind":"use","excerpt":"belthm:block18-carrier-indexed-observer Let S be a finite continuation-state set, let (A,⊕,0) be a commutative monoid, let B be a finite branch alphabet, and let δ:S× B⟶ S be a state transition. Suppose every branch b∈ B acts by a monoid homomorphism L_b:A⟶ A. Take an empty aggregation to be 0. For a state packet a=(a_s)_s∈ S, define a'_t := ⊕_δ(s,","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4f1eee202dc651a9","tex":"A_u","aliases":"A_u","display":"A_u","kind":"use","excerpt":"For an arbitrary realization (W,A_d,v_in,λ), concatenation gives the factorization H_a(u,v) = λ A_vA_uv_in. Thus H_a factors through W, so rank H_a≤dim W. The canonical realization has dimension r_k(a), proving the minimality formula.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-79d5c9fff39cf939","tex":"A_V","aliases":"A_V","display":"A_V","kind":"use","excerpt":"Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucce","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0083f442b19187bb","tex":"A_v","aliases":"A_v","display":"A_v","kind":"use","excerpt":"For an arbitrary realization (W,A_d,v_in,λ), concatenation gives the factorization H_a(u,v) = λ A_vA_uv_in. Thus H_a factors through W, so rank H_a≤dim W. The canonical realization has dimension r_k(a), proving the minimality formula.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-630af6a54b3ef5a2","tex":"A_\\sigma","aliases":"A_σ sigma","display":"A_\\sigma","kind":"use","excerpt":"A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8c2d6535cdc12eae","tex":"a_\\sigma","aliases":"a_σ sigma","display":"a_\\sigma","kind":"use","excerpt":"[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of th","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-82f9b8f3cf0e88aa","tex":"a_{\\sigma\\tau}","aliases":"a_στ sigma tau","display":"a_{\\sigma\\tau}","kind":"use","excerpt":"[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of th","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9b27880c1edeca32","tex":"A_{\\sigma\\tau}^{-1}","aliases":"A_στ^-1 sigma tau","display":"A_{\\sigma\\tau}^{-1}","kind":"use","excerpt":"A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3cc4428162985eac","tex":"A_\\tau","aliases":"A_τ tau","display":"A_\\tau","kind":"use","excerpt":"A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-57c96dc06711c775","tex":"a_\\tau","aliases":"a_τ tau","display":"a_\\tau","kind":"use","excerpt":"[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of th","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-452fd38454387c12","tex":"\\acute","aliases":"acute","display":null,"kind":"use","excerpt":"cal descent, profinite refinement, and the tensor-fiber boundary at which a symmetry group can be reconstructed. The strongest Galois source already contains Reynolds projectors, the subgroup--fixed-field correspondence, normality and quotient compression, conjugate projector atlases, Mackey double-coset packets, ramification details, Haar projecto","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5d18d0988cdd2e54","tex":"\\mathrm{ak}","aliases":"ak mathrm","display":"\\mathrm{ak}","kind":"use","excerpt":"[Finite additive-Karoubi resolved envelope] The category Syn_ak( K_rm lin) := Kar (Add(𝕜 Q/I)) is small, additive, idempotent complete, and 𝕜-linear. For every additive idempotent-complete 𝕜-linear category C, restriction to the generating quiver induces an equivalence Fun_𝕜^⊕ (Syn_ak( K_rm lin), C) ≃ Mod_ak( K_rm lin, C), where the right side is t","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-10a84b93033591b1","tex":"\\Analysis_{p^a}","aliases":"Analysis_p^a Analysis","display":null,"kind":"use","excerpt":"[Typed local clopen realization] Fix a prime p and an integer a≥0. For f∈ C( Z_p, Z_p), with q∈ Z_p the branch coordinate, define Analysis_p^a(f) := (q↦ f(p^aq+s))_0≤ s<p^a. Piecewise synthesis on the disjoint clopen cosets s+p^a Z_p is its inverse. Refining from p^a to p^a+b, for every integer b≥0, gives the exact child-address and mixed-radix com","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ba484e7138f7dfb2","tex":"\\Analysis_X^r","aliases":"Analysis_X^r Analysis","display":null,"kind":"use","excerpt":"At resolution r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-01478c4149c7701c","tex":"\\Aut","aliases":"Aut Aut operatorname","display":null,"kind":"use","excerpt":"Intrinsic symmetry · Aut_ M_α(M) · automorphisms of one resolved object inside the stratum · none beyond the native category","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5dd6a9f6e4fd9e0a","tex":"\\mathbf{Aut}","aliases":"Aut mathbf","display":"\\mathbf{Aut}","kind":"use","excerpt":"Minimal digital chart. An object of Aut_p^min,rat is a reachable minimal least-significant-digit-first p-DFAO whose coefficient series belongs to RMah_p^×. Its canonical representative has state set K_p(F), transition G↦Λ_dG, start state F, and output G↦ G(0).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1fc76a250d22a8eb","tex":"\\operatorname{Aut}","aliases":"Aut operatorname","display":"\\operatorname{Aut}","kind":"use","excerpt":"[Neutral Tannakian reconstruction] Let C be a neutral Tannakian category over k, and let ω: C⟶Vect_k be a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) tag12.20 is an affine group scheme, and C≃Rep_k( G) tag12.21 as tensor categories. Declared packet idempotents and refinement maps preserved by ω become tensor-compatible concrete projector r","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3bb9225404bf13a5","tex":"\\Aut^{\\otimes}","aliases":"Aut^⊗ Aut otimes operatorname otimes","display":"{\\operatorname{Aut}}^{\\otimes}","kind":"use","excerpt":"Reconstructed symmetry · Aut^⊗(ω) · tensor automorphisms of a fiber functor and the corresponding affine group scheme · faithful exact tensor fiber functor","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e5026cad2f7a947c","tex":"\\operatorname{Aut}^{\\otimes}","aliases":"Aut^⊗ operatorname otimes","display":"\\operatorname{Aut}^{\\otimes}","kind":"use","excerpt":"[Neutral Tannakian reconstruction] Let C be a neutral Tannakian category over k, and let ω: C⟶Vect_k be a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) tag12.20 is an affine group scheme, and C≃Rep_k( G) tag12.21 as tensor categories. Declared packet idempotents and refinement maps preserved by ω become tensor-compatible concrete projector r","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b8bbd6786751c34f","tex":"\\Aut_{\\mathcal M_\\alpha}","aliases":"Aut_ M_α Aut mathcal alpha operatorname mathcal alpha","display":"{\\operatorname{Aut}}_{\\mathcal M_\\alpha}","kind":"use","excerpt":"Intrinsic symmetry · Aut_ M_α(M) · automorphisms of one resolved object inside the stratum · none beyond the native category","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6660bab5d83a11c4","tex":"\\mathbf{Aut}_k","aliases":"Aut_k mathbf","display":"\\mathbf{Aut}_k","kind":"use","excerpt":"k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1164dc628ce8f346","tex":"\\Aut_{\\mathscr G}","aliases":"Aut_mathscr G Aut mathscr","display":null,"kind":"use","excerpt":"[Base-fiber invariant normal form] Fix a base chart U_0. Evaluation at U_0 induces an isomorphism Γ (mathscr G,mathscr M^u,r_Y,X) xrightarrow∼ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0), where the right-hand side is the subgroup fixed by every loop-transport map. Consequently, Hom_Desc(mathscr G)(X,Y) ≅ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0). Aft","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cea2acae56660d10","tex":"\\mathbf{Aut}_p^{\\mathrm{min,rat}}","aliases":"Aut_p^min,rat mathbf mathrm","display":"\\mathbf{Aut}_p^{\\mathrm{min,rat}}","kind":"use","excerpt":"Minimal digital chart. An object of Aut_p^min,rat is a reachable minimal least-significant-digit-first p-DFAO whose coefficient series belongs to RMah_p^×. Its canonical representative has state set K_p(F), transition G↦Λ_dG, start state F, and output G↦ G(0).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4157522265c5d142","tex":"\\Aut_p^{\\times,\\mathrm{per}}","aliases":"Aut_p^×,per Aut times mathrm operatorname times mathrm","display":"{\\operatorname{Aut}}_p^{\\times,\\mathrm{per}}","kind":"use","excerpt":"Let Aut_p^×,per be the groupoid of reachable minimal least-significant-digit-first p-DFAOs whose output sequence lies in Per_p^×,per. Morphisms are automaton isomorphisms preserving the start state, transitions, and outputs. Reachability forces every automorphism to be the identity.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dda3f4b7df54c019","tex":"\\mathbf b","aliases":"b mathbf","display":"\\mathbf b","kind":"use","excerpt":"K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"Conversely, suppose a has a realization (W,A_d,v,λ). For x∈ W, define its represented behavior by B(x)(n) = λ A_dig_k(n)x, where A_dig_k(n) is the product associated with the canonical least-significant-digit-first expansion of n. Equation [source reference] and concatenation of digit words give Λ_w a = B(A_wv). Hence every kernel section belongs t","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"use","excerpt":"Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preced","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of th","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9584baaf6d3834d4","tex":"B^{p-2}","aliases":"B^p-2","display":"B^{p-2}","kind":"use","excerpt":"Let R(z)=fracA(z)B(z), A(0)=B(0)=1, with a=deg A, b=deg B, c=a+(p-2)b, and define D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source refer","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-acb79e87d28a764d","tex":"B^r","aliases":"B^r","display":"B^r","kind":"use","excerpt":"[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fe3981a3f06083fb","tex":"\\mathsf B^{u,r}","aliases":"B^u,r mathsf","display":"\\mathsf B^{u,r}","kind":"use","excerpt":"For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b40f7db64f2b4cf6","tex":"\\mathsf B^{v,s}","aliases":"B^v,s mathsf","display":"\\mathsf B^{v,s}","kind":"use","excerpt":"For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2ee417144c193688","tex":"B_{\\mathcal O}","aliases":"B_ O mathcal","display":"B_{\\mathcal O}","kind":"use","excerpt":"[Permutation-module orbital lane calculus] Let kk be a commutative ring, let a finite group H act on finite sets I,J,K, and let kk[I],kk[J],kk[K] be the corresponding permutation modules. For an orbit O⊆ J× I under the diagonal action, let B_ O:kk[I]→kk[J] be the matrix whose (j,i)-entry is one when (j,i)∈ O and zero otherwise. Then Hom_kk H(kk[I],","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6eaeb9b43de87ec7","tex":"B_{\\mathcal P}","aliases":"B_ P mathcal","display":"B_{\\mathcal P}","kind":"use","excerpt":"Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-07c560986fc64084","tex":"B_{\\mathcal Q}","aliases":"B_ Q mathcal","display":"B_{\\mathcal Q}","kind":"use","excerpt":"Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9db9f91e80e4a17d","tex":"B_{\\mathcal R}","aliases":"B_ R mathcal","display":"B_{\\mathcal R}","kind":"use","excerpt":"Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"e p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b is b_0 and (Λ_0b)_0=b_0.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preced","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-75226eb7ab826145","tex":"\\mathcal B_2","aliases":"B_2 mathcal","display":"\\mathcal B_2","kind":"use","excerpt":"[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1a4c2e9cb4075d24","tex":"B_a","aliases":"B_a","display":"B_a","kind":"use","excerpt":"[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6cb522914c85fa87","tex":"\\mathcal B_a","aliases":"B_a mathcal","display":"\\mathcal B_a","kind":"use","excerpt":"For a≥1 and 0≤ s<p^a, define the branch B_a,sf(q)=f(p^a q+s), q∈ Z_p, and the complete packet B_a f=( B_a,sf)_0≤ s<p^a.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-83d6608b4d1f0c95","tex":"\\mathcal B_{a,s}","aliases":"B_a,s mathcal","display":"\\mathcal B_{a,s}","kind":"use","excerpt":"For a≥1 and 0≤ s<p^a, define the branch B_a,sf(q)=f(p^a q+s), q∈ Z_p, and the complete packet B_a f=( B_a,sf)_0≤ s<p^a.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7be573071ebda78d","tex":"B_d","aliases":"B_d","display":"B_d","kind":"use","excerpt":"Choose Smith bases for M_k(a) and L_k(a), and let D=diag(d_1,…,d_ρ) be the inclusion matrix. Let A_d and B_d be the integer matrices of Λ_d on the raw and saturated lattices.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-66c8d0359e712fa2","tex":"B_e","aliases":"B_e","display":"B_e","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b6d434243bc26c4c","tex":"b_e","aliases":"b_e","display":"b_e","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2a1cc1a422a2528f","tex":"B_{e'}","aliases":"B_e'","display":"B_{e'}","kind":"use","excerpt":"Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-52e721b671883513","tex":"b_{e'}","aliases":"b_e'","display":"b_{e'}","kind":"use","excerpt":"Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-162eee340db91ac5","tex":"B_N","aliases":"B_N","display":"B_N","kind":"use","excerpt":"For a fixed modulus N≥2, set B_N(M) = max({1}∪ {p^v_p(M):pmidgcd(M,N)}). The least packet size for a left inverse over Z/N Z is L_N(M,d)=min(d+1,B_N(M)). The exponents v_p(N) do not change the minimum; the first residue layer already detects the obstruction.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f868f37df68e6bb7","tex":"\\mathcal B_N","aliases":"B_N mathcal","display":"\\mathcal B_N","kind":"use","excerpt":"Let Ω be a normal splitting field of the generic packet scheme over K(x), and let Γ=Gal(Ω/K(x)). After splitting E_N, the geometric packet branches form a set B_N of cardinality d^N. Let π_j: B_N→Ω be the jth packet coordinate, and set C_N,j=π_j( B_N). For the selected semantic packet U_j, separability gives [K(x)(U_j):K(x)] = |Γ· U_j|.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9634f79e2739535a","tex":"b_N","aliases":"b_N","display":"b_N","kind":"use","excerpt":"[Finite-prefix triangularity] For each N, the coefficient prefix (a_1,…,a_N) of A(z)=1+∑_n≥1a_nz^n and the exponent prefix (b_1,…,b_N) determine one another by integral triangular recursions. No exponent b_m with m>N affects the coefficient prefix through degree N.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","excerpt":"Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preced","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-748f20d0c5f8b555","tex":"b_n'","aliases":"b_n'","display":"b_n'","kind":"use","excerpt":"iv 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preceding operations occur in Z[[z]]. Induction produces one exponent sequence with the required product. If two exponent sequences differed first at n, their quotient would have nonzero coefficien","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9a8a3a68291c0c9c","tex":"B_{N,F}","aliases":"B_N,F","display":"B_{N,F}","kind":"use","excerpt":"Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f0e1a295abce9295","tex":"B_{N,j}","aliases":"B_N,j","display":"B_{N,j}","kind":"use","excerpt":"Its coefficients are a_n = frac(λ)_nn!α^n. For 0≤ j<N, put G_N,j^λ,α(x) = ∑_m≥0 frac(λ)_Nm+j(Nm+j)! α^Nm+jx^m. Define the upper and lower parameter packets A_N,j(λ) = { fracλ+j+rN:0≤ r<N }, B_N,j = { fracj+1+rN:0≤ r<N }∖{1}.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dcab43dab13d684b","tex":"\\mathbf b_{N,j}","aliases":"b_N,j mathbf","display":"\\mathbf b_{N,j}","kind":"use","excerpt":"K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0820598b17cf5fd9","tex":"b_{n-1}","aliases":"b_n-1","display":"b_{n-1}","kind":"use","excerpt":"Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preced","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-305ce1517344aaba","tex":"B_Q","aliases":"B_Q","display":"B_Q","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-62b71ebe8e02f7b7","tex":"B_q","aliases":"B_q","display":"B_q","kind":"use","excerpt":"x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is t","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cd086233ebcc43cb","tex":"b_{u,r}","aliases":"b_u,r","display":"b_{u,r}","kind":"use","excerpt":"[Finite periodic kernel and exact state model] Let a∈Per_p^×,per have minimal period T and period word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-17147eee331647c0","tex":"\\mathcal B_W","aliases":"B_W mathcal","display":"\\mathcal B_W","kind":"use","excerpt":"An infinite matrix of finite rank r_k(a) has a nonsingular finite r_k(a)× r_k(a) minor, which gives the first assertion. For the remaining assertions, define the behavior map of any realization by B_W(x)(n)=λ A_dig_k(n)x. Observability makes this map injective. Reachability and equality of the represented sequence make its image equal to V_k(a). In","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b72211184c9a6c41","tex":"B_\\Delta","aliases":"B_Δ Delta","display":"B_\\Delta","kind":"use","excerpt":"0 · 1. The individual projectors do not descend because Sp_1S^-1=p_2, while their orbit sum p_1+p_2=1 does descend. The pair-address set has two orbits, Δ={(1,1),(2,2)}, Ω={(1,2),(2,1)}. Accordingly, End_kk C_2(kk^2) = { a · b b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivarian","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ff6964aeddd8b24","tex":"B_\\Omega","aliases":"B_Ω Omega","display":"B_\\Omega","kind":"use","excerpt":"0 · 1. The individual projectors do not descend because Sp_1S^-1=p_2, while their orbit sum p_1+p_2=1 does descend. The pair-address set has two orbits, Δ={(1,1),(2,2)}, Ω={(1,2),(2,1)}. Accordingly, End_kk C_2(kk^2) = { a · b b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivarian","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-58af0f264fe7ee38","tex":"\\backslash","aliases":"backslash","display":null,"kind":"use","excerpt":"Restriction and induction connect subgroup charts that need not commute. Their exact address set is the double-coset space Kbackslash G/H.","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e2ec3d8de7f39249","tex":"\\bar a","aliases":"bar a bar","display":"\\bar a","kind":"use","excerpt":"A residue sequence (bar a_n) admits many integral lifts. If (a_n) is one lift, then (a_n+pn) is another with the same reduction modulo p. Their Newton--Mahler coefficients differ by the nonzero coefficient sequence of pn. Thus the residue-series or functional-mask presentation cannot select one local interpolation without a declared lifting rule.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d2714d56b2692c14","tex":"\\bar a_n","aliases":"bar a_n bar","display":"\\bar a_n","kind":"use","excerpt":"A residue sequence (bar a_n) admits many integral lifts. If (a_n) is one lift, then (a_n+pn) is another with the same reduction modulo p. Their Newton--Mahler coefficients differ by the nonzero coefficient sequence of pn. Thus the residue-series or functional-mask presentation cannot select one local interpolation without a declared lifting rule.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0d10bedd52e37f8f","tex":"\\bar e","aliases":"bar e bar","display":"\\bar e","kind":"use","excerpt":"Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b4d7ee29d2029812","tex":"\\bar x","aliases":"bar x bar","display":"\\bar x","kind":"use","excerpt":"For a connected scheme X with geometric point bar x, the fiber functor on finite acuteetale covers identifies that category with finite continuous π_1^acuteet(X,bar x)-sets. Every finite fiber S produces a permutation resolution on ℓ^2(S): coordinate projectors are indexed by points, orbit projectors by transitive components, and covering morphisms","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-729fb4bd4680d0cf","tex":"\\BlockFifteenValidationAssertions","aliases":"BlockFifteenValidationAssertions","display":null,"kind":"use","excerpt":"ted replay assertions already counted in the preceding records and contribute BlockThirteenNewChecks new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is BlockThirteenCumulative exact assertions with no failures; the raw executed total, including replay, is BlockThirteenCumulativeExecuted. The Block","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-37f929b5816ca10e","tex":"\\BlockSixteenValidationAssertions","aliases":"BlockSixteenValidationAssertions","display":null,"kind":"use","excerpt":"kThirteenCumulative exact assertions with no failures; the raw executed total, including replay, is BlockThirteenCumulativeExecuted. The Block 15 verifier separately executes BlockFifteenValidationAssertions exact finite regression assertions for selected lane-analysis, block-composition, refinement, fold, support, and factorial implementation clau","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4e46ddbbf63dffb1","tex":"\\BlockSixteenValidationFailures","aliases":"BlockSixteenValidationFailures","display":null,"kind":"use","excerpt":"ne-analysis, block-composition, refinement, fold, support, and factorial implementation clauses. The Block 16 verifier executes BlockSixteenValidationAssertions exact assertions for finite group actions, object and pair orbits, orbital structure constants, compatible and incompatible selectors, equivariant refinement, rational cyclotomic orbit proj","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-350d418cf1b3405e","tex":"\\BlockThirteenBridgeChecks","aliases":"BlockThirteenBridgeChecks","display":null,"kind":"use","excerpt":"or fixed-radix regular systems, 697,958 for integral saturation and prime-power towers, 595,964 for Euler-product and multiplicative coordinates, 229,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-725e13adf052496b","tex":"\\BlockThirteenChecks","aliases":"BlockThirteenChecks","display":null,"kind":"use","excerpt":"The two suites execute BlockThirteenChecks exact assertions with no failures. Of these, BlockThirteenRegressionChecks authenticate or replay assertions already represented in the preceding cumulative record. The global synthesis therefore contributes BlockThirteenNewChecks nonduplicative comparison, coherence, packet, and loss-control assertions. T","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dfed7178e126d4ec","tex":"\\BlockThirteenCumulative","aliases":"BlockThirteenCumulative","display":null,"kind":"use","excerpt":"BlockThirteenBridgeChecks bridge assertions and BlockThirteenInvariantChecks weighted Fuss and tagged-invariant assertions. They include BlockThirteenRegressionChecks authenticated replay assertions already counted in the preceding records and contribute BlockThirteenNewChecks new comparison, coherence, packet, and loss-control assertions. The nond","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fc3009aef2a51efb","tex":"\\BlockThirteenCumulativeExecuted","aliases":"BlockThirteenCumulativeExecuted","display":null,"kind":"use","excerpt":"invariant assertions. They include BlockThirteenRegressionChecks authenticated replay assertions already counted in the preceding records and contribute BlockThirteenNewChecks new comparison, coherence, packet, and loss-control assertions. The nonduplicated cumulative record is BlockThirteenCumulative exact assertions with no failures; the raw exec","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-44ea6748820e27f2","tex":"\\BlockThirteenInvariantChecks","aliases":"BlockThirteenInvariantChecks","display":null,"kind":"use","excerpt":"saturation and prime-power towers, 595,964 for Euler-product and multiplicative coordinates, 229,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute Block","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1914e1779ca29386","tex":"\\BlockThirteenNewChecks","aliases":"BlockThirteenNewChecks","display":null,"kind":"use","excerpt":"ts, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute BlockThirteenBridgeChecks bridge assertions and BlockThirteenInvariantChecks weighted Fuss and tagged-invariant assertions. They include BlockThirteenRegressionChecks authenticated replay assertions already counted in the ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6eef8e64995bbc35","tex":"\\BlockThirteenRegressionChecks","aliases":"BlockThirteenRegressionChecks","display":null,"kind":"use","excerpt":"29,743 for local Newton--Mahler and polyphase transport, 174,940 for holonomic packets, 73,088 for selected algebraic packets, 115,721 for nonlinear tree systems, and 11,195 for finite Galois and tensor models. The two global-synthesis suites execute BlockThirteenBridgeChecks bridge assertions and BlockThirteenInvariantChecks weighted Fuss and tagg","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-51dfbcc72d139cd4","tex":"\\Br","aliases":"Br operatorname","display":"{\\operatorname{Br}}","kind":"use","excerpt":"[Brauer obstruction to projective packet descent] Projective semilinear descent data define a class α∈ H^1(G,PGL_n(L)). The connecting map for 1⟶ L^× ⟶ GL_n(L) ⟶ PGL_n(L) ⟶1 produces δ(α) ∈ H^2(G,L^×) = Br(L/K). The projective atlas lifts to linear semilinear descent data exactly when δ(α)=0. When the class is nonzero, the descended object is natur","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fcc059c4712631ce","tex":"\\mathbf{Branch}","aliases":"Branch mathbf","display":"\\mathbf{Branch}","kind":"use","excerpt":"Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding bran","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ec1b918a6e276534","tex":"\\mathbf{Branch}^{\\mathrm{cts}}_{p,a}","aliases":"Branch^cts_p,a mathbf mathrm","display":"\\mathbf{Branch}^{\\mathrm{cts}}_{p,a}","kind":"use","excerpt":"Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding bran","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-63ed804fe0989006","tex":"\\mathbf{BranchMah}","aliases":"BranchMah mathbf","display":"\\mathbf{BranchMah}","kind":"use","excerpt":"Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding bran","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-db86e4bb0de26157","tex":"\\mathbf{BranchMah}^{c_0}_{p,a}","aliases":"BranchMah^c_0_p,a mathbf","display":"\\mathbf{BranchMah}^{c_0}_{p,a}","kind":"use","excerpt":"Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding bran","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2e12e550b72be021","tex":"\\mathbb C","aliases":"C mathbb","display":"\\mathbb C","kind":"use","excerpt":"Throughout this section, K is a characteristic-zero field. When local monodromy is discussed, K is embedded in C and all local exponents are taken over an algebraic closure.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-836573079e170785","tex":"C'_t","aliases":"C'_t","display":"C'_t","kind":"use","excerpt":"ite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-936d78089fa24553","tex":"c^{\\mathcal R}_{\\mathcal Q,\\mathcal P}","aliases":"c^ R_ Q, P mathcal mathcal mathcal","display":"c^{\\mathcal R}_{\\mathcal Q,\\mathcal P}","kind":"use","excerpt":"Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9d46643740805c1d","tex":"C^{12}","aliases":"C^12","display":"C^{12}","kind":"use","excerpt":"1 · 0, one has C^12 =I+829 · 18 18 · 11. Hence Δ_12·2^a^ rF_n ≡0pmod2^(a+3)r.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4a31c02ba395868e","tex":"C^2","aliases":"C^2","display":"C^2","kind":"use","excerpt":"Substitute C=E+zO into C=1+zC^2 and separate even and odd powers. The two resultants give the displayed quartics.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6c8483926602e3e3","tex":"c^2","aliases":"c^2","display":"c^2","kind":"use","excerpt":"Write Q=a · c c · b, d=ab-c^2>0. The squared induced norms are u:= A_1 _Q^2 =fracb(a+b+2c)d, v:= A_2 _Q^2 =fraca(a+b-2c)d. A direct calculation gives ab((a+b)^2-4c^2)-4(ab-c^2)^2 =ab(a-b)^2+4c^2(ab-c^2)≥0. Thus uv≥4, and therefore max{sqrt u,sqrt v}≥(uv)^1/4≥sqrt2. Equality forces a=b and c=0, and then u=v=2.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ce7e7a918f6fa28e","tex":"C^H","aliases":"C^H","display":"C^H","kind":"use","excerpt":"[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bec59126aabbbf80","tex":"C^{Hx+j}","aliases":"C^Hx+j","display":"C^{Hx+j}","kind":"use","excerpt":"Use C^Hx+j=C^j(I+p^LB)^x and expand the principal-unit matrix power by the convergent binomial series. One forward difference in x contributes the factor p^LB.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-04ebc96f0d8e8891","tex":"C^j","aliases":"C^j","display":"C^j","kind":"use","excerpt":"[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fc4ec113b29e6ee4","tex":"C^n","aliases":"C^n","display":"C^n","kind":"use","excerpt":"The class of (e,0) is a boundary exactly when there are e'∈ E^n-1 and c∈ C^n with e=d_Ee'+Dc, d_Cc=0. This is equivalent to [e]=H^n(D)[c]. Concentration in degree zero reduces the cone quotient to E^0/im D.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5fb559d0d8008ab5","tex":"C^{p-1}","aliases":"C^p-1","display":"C^{p-1}","kind":"use","excerpt":"Conversely, suppose all valuation residues vanish. Since A and B are coprime, every irreducible exponent in each polynomial is divisible by p-1. Factoring into monic irreducibles therefore gives A=C^p-1, B=D^p-1 for polynomials C,D after a constant normalization. The unit conditions A(0)=B(0)=1 remove the remaining scalar obstruction. Thus B/A=(D/C","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a429940adeec0b30","tex":"C^q","aliases":"C^q","display":"C^q","kind":"use","excerpt":"Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2a019f88cc8776b9","tex":"\\mathfrak c^{\\rm tuple}_{2,0}","aliases":"c^rm tuple_2,0 mathfrak rm","display":"\\mathfrak c^{\\rm tuple}_{2,0}","kind":"use","excerpt":"[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-80c7fb6424224879","tex":"\\mathfrak c^{\\rm tuple}_{2,1}","aliases":"c^rm tuple_2,1 mathfrak rm","display":"\\mathfrak c^{\\rm tuple}_{2,1}","kind":"use","excerpt":"r:block10-antisymmetric-collision If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-62d6dd9a37e47fe4","tex":"\\mathbb C^\\times","aliases":"C^× mathbb times","display":"\\mathbb C^\\times","kind":"use","excerpt":"-1 · 0 . Then A_σσ(A_σ)=-I. The projective cocycle condition holds because -I is scalar. Its boundary is represented by -1 in R^×/N_ C/ R( C^×). Every complex norm is positive, so -1 is nontrivial. The descended central simple algebra is the Hamilton quaternion algebra. Replacing A_σ by 0 · 1","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5db3f118c8bf4e41","tex":"C_{(p)}","aliases":"C_(p)","display":"C_{(p)}","kind":"use","excerpt":"The Z-module Z_p is torsion free and hence flat. Tensor the integral short exact sequence with Z_p. For a finite abelian group C, C⊗_ Z Z_p≅ C_(p). The Smith-coordinate formula follows by tensoring the diagonal inclusion.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"[The S_3 nonnormal atlas] Let G=S_3 and let H be generated by one transposition. There are three conjugate order-two subgroups and hence three conjugate cubic fixed-field charts. Their subgroup core is trivial, while their normal closure is all of S_3. By contrast, the order-three subgroup A_3 is normal and its fixed chart carries the quotient symm","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"For f(x)=x^2, the source Mahler vector is (c_0,c_1,c_2)=(0,1,2). The two dyadic branches are f(2q)=4binom q1+8binom q2, and f(2q+1)=1+8binom q1+8binom q2. This example exhibits exact finite-window coefficient transport.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6ef64106c8bd8542","tex":"\\mathcal C_{2,0}","aliases":"C_2,0 mathcal","display":"\\mathcal C_{2,0}","kind":"use","excerpt":"[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-504c66f14748f262","tex":"\\mathcal C_{2,1}","aliases":"C_2,1 mathcal","display":"\\mathcal C_{2,1}","kind":"use","excerpt":"[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b6010785cb6069e5","tex":"c_{a,j,r}","aliases":"c_a,j,r","display":"c_{a,j,r}","kind":"use","excerpt":"[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dff8e8535d2ec4f4","tex":"c_{d-1}","aliases":"c_d-1","display":"c_{d-1}","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d36b2bcde94eaa1d","tex":"C_E^2","aliases":"C_E^2","display":"C_E^2","kind":"use","excerpt":"[Boolean algebra of block selectors] As endomorphisms of the 𝕜-module Hom(X,Y), the selectors satisfy C_E^2=C_E, C_EC_F=C_E∩ F, R_E=I-C_E. In particular, C_ER_E=R_EC_E=0, f=C_E(f)+R_E(f).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b509f30fff306753","tex":"C_E^{u,r}","aliases":"C_E^u,r","display":"C_E^{u,r}","kind":"use","excerpt":"item Selectors and named residuals. For E⊆ I_Y,u× I_X,r, put C_E^u,r(f) :=∑_(a,i)∈ Ep_Y,a^ufp_X,i^r, R_E^u,r(f):=f-C_E^u,r(f). As additive endomorphisms of Hom_ C(X,Y), C_E^2=C_E, C_EC_F=C_E∩ F, R_E=1-C_E, f=C_E(f)+R_E(f). Thus deletion of a lane block is never silent. Its complement is an exact, named residual.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-85696a5126da9c88","tex":"C_{E_U}","aliases":"C_E_U","display":"C_{E_U}","kind":"use","excerpt":"item Holonomy-compatible selectors and residuals. Let E_U⊆ I_Y,U,u× I_X,U,r be a family satisfying E_V= (λ^Y_γ,u×λ^X_γ,r)(E_U) for every arrow γ:U→ V. Then the local selectors C_E_U(h_U)_ai = h_U,ai, · (a,i)∈ E_U, 0, · (a,i)∉ E_U assemble to an idempotent endomorphism C_E of the section group. For two compatible support families E,F, C_EC_F=C_E∩ F,","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7b88518f08cd936f","tex":"C_{E_V}","aliases":"C_E_V","display":"C_{E_V}","kind":"use","excerpt":"For (H5), support compatibility implies γ_*C_E_U=C_E_Vγ_*, so the selectors preserve compatible sections. Every Boolean and residual identity is then checked chartwise.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f68b8522bceae2df","tex":"C_{E\\cap F}","aliases":"C_E∩ F cap","display":"C_{E\\cap F}","kind":"use","excerpt":"[Boolean algebra of block selectors] As endomorphisms of the 𝕜-module Hom(X,Y), the selectors satisfy C_E^2=C_E, C_EC_F=C_E∩ F, R_E=I-C_E. In particular, C_ER_E=R_EC_E=0, f=C_E(f)+R_E(f).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0fa0207652600c09","tex":"C_F","aliases":"C_F","display":"C_F","kind":"use","excerpt":"[Boolean algebra of block selectors] As endomorphisms of the 𝕜-module Hom(X,Y), the selectors satisfy C_E^2=C_E, C_EC_F=C_E∩ F, R_E=I-C_E. In particular, C_ER_E=R_EC_E=0, f=C_E(f)+R_E(f).","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b879feff07e5a55b","tex":"C_{F\\circ E}","aliases":"C_F∘ E circ","display":"C_{F\\circ E}","kind":"use","excerpt":"[Support propagation under composition] If f=C_E(f) and g=C_F(g), then gf=C_F∘ E(gf). Thus exact zero patterns propagate by relational composition.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"[Exact coefficient recurrence] Let A(z)=1+∑_n≥1a_nz^n and let c_n be its Euler ghost coordinates. Then na_n = ∑_j=1^nc_j a_n-j (n≥1), with a_0=1. Conversely, any c∈ G_ Z and this recurrence reconstruct the integral unit series corresponding to its Euler exponents.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0ce23018e3603d00","tex":"c_{j,r}","aliases":"c_j,r","display":"c_{j,r}","kind":"use","excerpt":"[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1e15d2e27313753c","tex":"C_N","aliases":"C_N","display":"C_N","kind":"use","excerpt":"[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c1ecf2aa8f993509","tex":"c_N","aliases":"c_N","display":"c_N","kind":"use","excerpt":"[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordina","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f0deb10def160a59","tex":"c_{n+1}","aliases":"c_n+1","display":"c_{n+1}","kind":"use","excerpt":"ssed Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is the normalized first-order tail recurrence","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-04be08fb409b7209","tex":"c_{n+r}","aliases":"c_n+r","display":"c_{n+r}","kind":"use","excerpt":"Write F(z)=∑ c_nz^n. At coefficient degree n, the monic term D^rF contributes frac(n+r)!n!c_n+r, while every lower derivative involves only c_0,…,c_n+r-1. Characteristic zero makes the displayed coefficient invertible. The equation therefore determines c_n+r recursively from the jet.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-945f1ba8fb8b4fce","tex":"c_{n+r-1}","aliases":"c_n+r-1","display":"c_{n+r-1}","kind":"use","excerpt":"Write F(z)=∑ c_nz^n. At coefficient degree n, the monic term D^rF contributes frac(n+r)!n!c_n+r, while every lower derivative involves only c_0,…,c_n+r-1. Characteristic zero makes the displayed coefficient invertible. The equation therefore determines c_n+r recursively from the jet.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-40591b7fff4431b2","tex":"C_{N,j}","aliases":"C_N,j","display":"C_{N,j}","kind":"use","excerpt":"Define the alias and cancellation deficits by A_N(λ)=N-r_rm orb(N;λ), C_N,j(λ) = r_rm orb(N;λ) -r_D(N,j;λ). Then N-r_D(N,j;λ) = A_N(λ)+C_N,j(λ).","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-233870d5c6c2233b","tex":"C_{n,M}","aliases":"C_n,M","display":"C_{n,M}","kind":"use","excerpt":"[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4c79fafc541a02b5","tex":"C_o","aliases":"C_o","display":"C_o","kind":"use","excerpt":"a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c9e4809059f5ef17","tex":"\\mathcal C_p","aliases":"C_p mathcal","display":"\\mathcal C_p","kind":"use","excerpt":"This section treats the second channel. Fix a prime p. Let C_p=C( Z_p, Z_p) with the uniform norm, and let c_0( Z_p) = {c=(c_r)_r≥0:c_r∈ Z_p, c_r⟶0} with the coefficient sup norm.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a7f55c1da8911f74","tex":"c_{p^ak}","aliases":"c_p^ak","display":"c_{p^ak}","kind":"use","excerpt":"[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-be57c172ad4ba898","tex":"c_{p^ak+u}","aliases":"c_p^ak+u","display":"c_{p^ak+u}","kind":"use","excerpt":"[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e8b9f48bd6d09aed","tex":"c_q","aliases":"c_q","display":"c_q","kind":"use","excerpt":", where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-650ab424b2beee12","tex":"C_{q,\\alpha}","aliases":"C_q,α alpha","display":"C_{q,\\alpha}","kind":"use","excerpt":"Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-307d2c7e46d5ba73","tex":"C_s","aliases":"C_s","display":"C_s","kind":"use","excerpt":"Let S be a finite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-433d97a8fbb31f65","tex":"C_t","aliases":"C_t","display":"C_t","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a7d678072fed2039","tex":"C_v","aliases":"C_v","display":"C_v","kind":"use","excerpt":"a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-85b4eefe3cd8f404","tex":"\\mathcal C_V","aliases":"C_V mathcal","display":"\\mathcal C_V","kind":"use","excerpt":"Let mathscr G be a finite connected groupoid of charts. For an arrow γ:U→ V, let F_γ: C_U⟶ C_V be an additive equivalence between additive idempotent-complete categories. The transition functors carry specified identity and composition comparisons. To keep the formulas readable, those comparison isomorphisms are suppressed; every displayed cocycle ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a6b159ee119659b3","tex":"C_\\Delta","aliases":"C_Δ Delta","display":"C_\\Delta","kind":"use","excerpt":"b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivariant identity matrix to p_1, which is not equivariant. The orbit selectors C_Δ and C_Ω are compatible and split every equivariant endomorphism into its diagonal-orbit and off-diagonal-orbit parts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-43a6945757cd6afe","tex":"C_{\\kappa(e)}","aliases":"C_κ(e) kappa","display":"C_{\\kappa(e)}","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-abb15f2c90918da8","tex":"\\mathcal C_{\\lambda,a}","aliases":"C_λ,a mathcal lambda","display":"\\mathcal C_{\\lambda,a}","kind":"use","excerpt":"Every connected component in [source reference] has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-969e7d80631775c7","tex":"C_\\Omega","aliases":"C_Ω Omega","display":"C_\\Omega","kind":"use","excerpt":"b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivariant identity matrix to p_1, which is not equivariant. The orbit selectors C_Δ and C_Ω are compatible and split every equivariant endomorphism into its diagonal-orbit and off-diagonal-orbit parts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-20a0dce68cfe40d6","tex":"c_\\infty","aliases":"c_∞ infty","display":"c_\\infty","kind":"use","excerpt":"For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordinary integer. Thus s_k+c_∞ is p-adically ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-21355a3045e08010","tex":"\\mathrm{can}","aliases":"can mathrm","display":"\\mathrm{can}","kind":"use","excerpt":"[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2cb31e5909f9be5e","tex":"\\mathbf{Cart}","aliases":"Cart mathbf","display":"\\mathbf{Cart}","kind":"use","excerpt":"Minimal Cartier-module chart. An object of Cart_p^min,rat is a finite-dimensional tuple (V,(L_d)_d=0^p-1,v,λ) that is reachable and observable, whose digit coefficient function belongs to RMah_p^×. Morphisms are similarities preserving every L_d, the initial vector, and the output functional. Its normalized representative is C_p(F).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0b6630d408e00a6e","tex":"\\mathbf{Cart}_p^{\\mathrm{min,rat}}","aliases":"Cart_p^min,rat mathbf mathrm","display":"\\mathbf{Cart}_p^{\\mathrm{min,rat}}","kind":"use","excerpt":"Minimal Cartier-module chart. An object of Cart_p^min,rat is a finite-dimensional tuple (V,(L_d)_d=0^p-1,v,λ) that is reachable and observable, whose digit coefficient function belongs to RMah_p^×. Morphisms are similarities preserving every L_d, the initial vector, and the output functional. Its normalized representative is C_p(F).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bdb62e64cd77e5ef","tex":"\\operatorname{codim}","aliases":"codim operatorname","display":"\\operatorname{codim}","kind":"use","excerpt":"The classical conductor formula uses codimV^G_i. Since P_i projects onto V^G_i, this codimension is Tr(I-P_i). Also codimV^G_i=∑_j≥ id_j. Reversing the finite sums gives the detail formula.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b286810fbe19cc0e","tex":"\\mathrm{col}","aliases":"col mathrm","display":"\\mathrm{col}","kind":"use","excerpt":"The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6b25eb57dad6be5b","tex":"\\operatorname{Coll}_{CB}","aliases":"Coll_CB operatorname","display":"\\operatorname{Coll}_{CB}","kind":"use","excerpt":"position law for bridge loss] Let UxrightarrowB VxrightarrowC W be certified bridges. [label=textup(roman*)] item Every constructor or universal property preserved by both B and C is preserved by CB. item MorKer_B(X,Y) ⊆ MorKer_CB(X,Y). item Coll_B ⊆ Coll_CB. item EssIm(CB) ⊆ EssIm(C). item If both bridges are conservative, then their composite is ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-491630d7018dba28","tex":"\\operatorname{conv}","aliases":"conv operatorname","display":"\\operatorname{conv}","kind":"use","excerpt":"Let S be a finite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ffdeb948c6faf1b4","tex":"\\operatorname{Core}","aliases":"Core operatorname","display":"\\operatorname{Core}","kind":"use","excerpt":"If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f32ef004f2d9a9a1","tex":"\\operatorname{Core}_G","aliases":"Core_G operatorname","display":"\\operatorname{Core}_G","kind":"use","excerpt":"If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"use","excerpt":"For 2cosh z, the even packet is nonzero and the odd packet is zero. There is no finite singularity, so the cancellation occurs at the level of the character-projected solution rather than by finite singularity collision.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a1ec1e6831631fdb","tex":"\\cr","aliases":"cr","display":null,"kind":"use","excerpt":"The proposed architecture has six layers: resolved syntax⟶normalized strata ⟶typed overlaps⟶coherencecr ⟶effective descent ⟶compatible complete invariants. Each arrow carries a proof obligation. A semantic analogy is not a bridge. A bridge is not reversible merely because formulas exist in both directions. Local completeness is not global completen","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d565d6a34f41b3b9","tex":"\\cref","aliases":"cref","display":null,"kind":"use","excerpt":"[4pt] Reduced OGF ↔ minimal DFAO · equivalence on the periodic essential image · unit purely periodic series and reachable minimal LSD-first p-DFAOs · coefficient semantics, minimal period, canonical p-kernel, output behavior · equality of represented sequences · a nonminimal residue-pair automaton is quotiented before entering the normalized chart","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8b9e658b5fed2443","tex":"\\mathrm{cts}","aliases":"cts mathrm","display":"\\mathrm{cts}","kind":"use","excerpt":"Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding bran","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c3d74137db38f778","tex":"D^2","aliases":"D^2","display":"D^2","kind":"use","excerpt":"[Derivative-module rank] For a nonzero D-finite series F, r_D(F) = dim_K(z)span_K(z) {F,DF,D^2F,…}.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-03223db37cda3c62","tex":"D^{k(\\mu)}","aliases":"D^k(μ) mu","display":"D^{k(\\mu)}","kind":"use","excerpt":"[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-76b7929465487fe2","tex":"D^n","aliases":"D^n","display":"D^n","kind":"use","excerpt":"[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-08ad6e3172b85bbb","tex":"D^{N+q}","aliases":"D^N+q","display":"D^{N+q}","kind":"use","excerpt":"[Recurrence residual as an Euler-operator polynomial] Define Q = ∑_s=0^q z^q-sp_s(θ-s). Then [z^n+q]QF = ∑_s=0^qp_s(n)a_n+s. Hence QF is a polynomial of degree less than N+q, and D^N+qQF=0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-36e9f7eabebb313a","tex":"d^{N/g}","aliases":"d^N/g","display":"d^{N/g}","kind":"use","excerpt":"If, in addition, their distinct Kummer classes generate a subgroup isomorphic to ( Z/d Z)^N/g in the Kummer quotient K(z)^×/K(z)^× d, then the compositum of the corresponding radical branch fields has degree d^N/g over K(z). Without that independence hypothesis, the lane orbit count remains exact but the degree formula is only an upper bound.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3cc0f79659093353","tex":"D^{p-1}","aliases":"D^p-1","display":"D^{p-1}","kind":"use","excerpt":"Conversely, suppose all valuation residues vanish. Since A and B are coprime, every irreducible exponent in each polynomial is divisible by p-1. Factoring into monic irreducibles therefore gives A=C^p-1, B=D^p-1 for polynomials C,D after a constant normalization. The unit conditions A(0)=B(0)=1 remove the remaining scalar obstruction. Thus B/A=(D/C","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d51565e421569b93","tex":"d^{r+1}","aliases":"d^r+1","display":"d^{r+1}","kind":"use","excerpt":"[Power bridge] For b(n)=n^r, which is completely multiplicative, the divisor ghost is c(n)=∑_dmid nd^r+1=σ_r+1(n). At a prime power, σ_r+1(p^e) = 1+p^r+1+⋯+p^e(r+1). Thus one simple multiplicative exponent rule simultaneously produces a filtered ordinary product, a multiplicative divisor-sum ghost, and a one-dimensional geometric local rule at ever","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e1b78d160493a87d","tex":"d_{\\mathcal A}","aliases":"d_ A mathcal","display":"d_{\\mathcal A}","kind":"use","excerpt":"Define the arity period d_ A = gcd{m-1:m∈ A}. tag11.9 Every tree with internal vertices v has leaf count ℓ(T)=1+∑_v(deg(v)-1).","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-737f63af45789c25","tex":"d_{0,k}","aliases":"d_0,k","display":"d_{0,k}","kind":"use","excerpt":"[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"use","excerpt":"By the Smith normal form, there are uniquely determined integers 1≤ d_1(a)mid d_2(a)mid⋯mid d_ρ(a) such that C_k(a) ≅ ⊕_i=1^ρ Z/d_i(a) Z. Factors with d_i=1 are harmless but are retained so that the list has length ρ.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ca8301964f9bd376","tex":"D_4","aliases":"D_4","display":"D_4","kind":"use","excerpt":"The Galois-Tannakian atlas verifier uses exact rational group-algebra arithmetic, exact finite permutation combinatorics, and exact Gaussian-rational matrix algebra. It audits cyclic groups, V_4, S_3, D_4, and A_4; every subgroup Reynolds idempotent; subgroup inclusion and projector order; normality and centrality; core and normal closure; commutin","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7903c81010eebdd6","tex":"D_a","aliases":"D_a","display":"D_a","kind":"use","excerpt":"e finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summands and D is block diagonal, this becomes coker D≅⊕_acoker D_a. Off-diagona","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-89cdebe9be220869","tex":"d_{a-1}","aliases":"d_a-1","display":"d_{a-1}","kind":"use","excerpt":"elprop:block15-cartier-specialization Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It s","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ee9c69896c4d1a6d","tex":"D_c","aliases":"D_c","display":"D_c","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54ff0eb6a1af0d6f","tex":"D_f","aliases":"D_f","display":"D_f","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c014f2db4f1db6d7","tex":"D_i","aliases":"D_i","display":"D_i","kind":"use","excerpt":"Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"use","excerpt":"For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b ","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f951e623c1ba0b72","tex":"d_j","aliases":"d_j","display":"d_j","kind":"use","excerpt":"The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-61f0cb69c9e8e1d3","tex":"d_k","aliases":"d_k","display":"d_k","kind":"use","excerpt":"(1+2)^x, branch 2q · c_r=2^r, 0≤ r≤4 · d_k=8^k, 0≤ k≤4","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b1d9b7ac0d7ce252","tex":"D_Q","aliases":"D_Q","display":"D_Q","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7ae4bace2223e197","tex":"D_{Q_s}","aliases":"D_Q_s","display":"D_{Q_s}","kind":"use","excerpt":"0 · 1. There is no positive-definite metric Q with H^*QH=Q. Nevertheless, for Q_s=diag(s^2,1), we have D_Q_s(H)⟶1 (sdownarrow0), while κ_2(Q_s)=s^-2⟶∞. Thus the metric-completion defect has zero infimum but is not attained.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b22090db332ee7ef","tex":"d_{s,k}","aliases":"d_s,k","display":"d_{s,k}","kind":"use","excerpt":"[Affine Newton--Mahler transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion and put g_s(q)=f(Mq+s). Then g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k =∑_r=k^Mk+sc_rΘ_M,s(r,k). Every output coefficient depends on a finite source window.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"use","excerpt":"The root filter also gives a construction-independent order ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8f767f66911f09a3","tex":"D_x","aliases":"D_x","display":"D_x","kind":"use","excerpt":"_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d769fd16b8435684","tex":"d_{\\rho}","aliases":"d_ρ rho","display":"d_{\\rho}","kind":"use","excerpt":"By the Smith normal form, there are uniquely determined integers 1≤ d_1(a)mid d_2(a)mid⋯mid d_ρ(a) such that C_k(a) ≅ ⊕_i=1^ρ Z/d_i(a) Z. Factors with d_i=1 are harmless but are retained so that the list has length ρ.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-115177c498412b4f","tex":"d_{\\rho-j}","aliases":"d_ρ-j rho","display":"d_{\\rho-j}","kind":"use","excerpt":"[Fitting-ideal certificate] For 0≤ j<ρ, Fitt_j(C_k(a)) = ( d_1(a)d_2(a)⋯ d_ρ-j(a) ), and Fitt_j(C_k(a))= Z (j≥ρ). Thus the Smith profile and the Fitting filtration contain the same reduction-defect data at different resolutions.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f248e9e31b983389","tex":"\\dashrightarrow","aliases":"dashrightarrow","display":null,"kind":"use","excerpt":"[4pt] Reduced OGF ↔ minimal DFAO · equivalence on the periodic essential image · unit purely periodic series and reachable minimal LSD-first p-DFAOs · coefficient semantics, minimal period, canonical p-kernel, output behavior · equality of represented sequences · a nonminimal residue-pair automaton is quotiented before entering the normalized chart","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-625573dcacdf7fff","tex":"\\Def","aliases":"Def","display":null,"kind":"use","excerpt":"Fix a universe mathscr Q(X) of admissible queries on X. For Q⊆mathscr Q(X), define Ind( Q) := ⋂_q∈ Qkerpair(q). For an equivalence relation or congruence R, define Def(R) := {q∈mathscr Q(X):R⊆kerpair(q)}.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.","count":17},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0d469a888ac8840b","tex":"\\deg_T","aliases":"deg_T deg","display":"\\deg_T","kind":"use","excerpt":"[Constructive packet eliminant] For each j, one can compute an annihilating polynomial Q_N,j(x,T)∈ K(x)[T] with deg_TQ_N,j≤ d^N by multiplication matrices, resultants, or a Gr obner elimination of the exact packet system.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7f47da7c88fef2d2","tex":"\\deg_Y","aliases":"deg_Y deg","display":"\\deg_Y","kind":"use","excerpt":"Let F∈ K[[z]] be algebraic over K(z). Its monic irreducible polynomial is P_F(z,Y)∈ K(z)[Y], P_F(z,F(z))=0. Write d(F)=deg_Y P_F. After multiplying by a common denominator and dividing by the greatest common divisor of the polynomial coefficients, one obtains a primitive representative in K[z,Y], unique up to a nonzero scalar. We normalize that sca","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"[Typed affine-channel ladder] For every M≥1, s≥0, and d≥0: item The matrix A_M,s^(d) is upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel h","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e4eca5036789dcf0","tex":"\\mathbf{DFAO}","aliases":"DFAO mathbf","display":"\\mathbf{DFAO}","kind":"use","excerpt":"k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c39cf04ea934b07c","tex":"\\mathbf{DFAO}_{k,K}^{\\min}","aliases":"DFAO_k,K^min mathbf min","display":"\\mathbf{DFAO}_{k,K}^{\\min}","kind":"use","excerpt":"k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a33d196df6fa5e19","tex":"\\mathrm{DG}","aliases":"DG mathrm","display":"\\mathrm{DG}","kind":"use","excerpt":"[Effective descent on the divisor--ghost overlap] The exponent, fixed Euler-product, multiplicative ghost, and prime-power seed charts form a finite normalized atlas R_DG. Every bridge is reversible on the declared overlap, every multiple-overlap comparison is the unique identity over the represented exponent function, and Desc( R_DG) ≃ EulMult_ Z.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"Choose Smith bases in which the inclusion M_k(a)↪ L_k(a) is the diagonal matrix D=diag(d_1,…,d_ρ). Modulo m, the kernel and cokernel of multiplication by d_i are both cyclic of order gcd(m,d_i), and the image has order m/gcd(m,d_i). The map is invertible in every coordinate exactly when every d_i is a unit modulo m, which is equivalent to gcd(m,d_ρ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-10f439949d43bb81","tex":"\\operatorname{dig}","aliases":"dig operatorname","display":"\\operatorname{dig}","kind":"use","excerpt":"Conversely, suppose a has a realization (W,A_d,v,λ). For x∈ W, define its represented behavior by B(x)(n) = λ A_dig_k(n)x, where A_dig_k(n) is the product associated with the canonical least-significant-digit-first expansion of n. Equation [source reference] and concatenation of digit words give Λ_w a = B(A_wv). Hence every kernel section belongs t","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2f13fbcc1aa6c587","tex":"\\operatorname{dig}_k","aliases":"dig_k operatorname","display":"\\operatorname{dig}_k","kind":"use","excerpt":"Conversely, suppose a has a realization (W,A_d,v,λ). For x∈ W, define its represented behavior by B(x)(n) = λ A_dig_k(n)x, where A_dig_k(n) is the product associated with the canonical least-significant-digit-first expansion of n. Equation [source reference] and concatenation of digit words give Λ_w a = B(A_wv). Hence every kernel section belongs t","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d6bb63cf0ad415a0","tex":"\\dim_{\\mathbb Q}","aliases":"dim_ Q dim mathbb","display":"\\dim_{\\mathbb Q}","kind":"use","excerpt":"Throughout this section, k≥2 is fixed and a: N⟶ Z is an integer-valued sequence whose rational section space V_ Q(a) = span_ Q K_k(a) ⊆ Q^ N has finite dimension ρ_k(a)=dim_ QV_ Q(a). Thus a is k-regular after extension of scalars to Q.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bf09fc45b861ed9d","tex":"\\dim_{K(z)}","aliases":"dim_K(z) dim","display":"\\dim_{K(z)}","kind":"use","excerpt":"[Derivative-module rank] For a nonzero D-finite series F, r_D(F) = dim_K(z)span_K(z) {F,DF,D^2F,…}.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"rem[Cokernel criterion, universality, and naturality] For a correction datum (D,r) [source reference]: [label=textup(roman*)] item a correction exists if and only if Ob_D(r)=0; item every linear obstruction test φ:V→ W satisfying φ D=0 factors uniquely through V→coker D; item for every commuting square ccc U · xrightarrowD · V downarrow a · · downa","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-64fbb1a3212deefd","tex":"e'","aliases":"e'","display":"e'","kind":"use","excerpt":"The class of (e,0) is a boundary exactly when there are e'∈ E^n-1 and c∈ C^n with e=d_Ee'+Dc, d_Cc=0. This is equivalent to [e]=H^n(D)[c]. Concentration in degree zero reduces the cone quotient to E^0/im D.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-46e5e76543623d92","tex":"e^{-2\\pi i\\lambda}","aliases":"e^-2π iλ pi lambda","display":"e^{-2\\pi i\\lambda}","kind":"use","excerpt":"item the monodromy eigenvalues at zero form the complete group of Nth roots of unity; item the ratios of monodromy eigenvalues at infinity are Nth roots of unity; item the distinguished nontrivial eigenvalue at α^-N is e^-2π iλ.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-99e4c97cf0f019b5","tex":"E^0","aliases":"E^0","display":"E^0","kind":"use","excerpt":"The class of (e,0) is a boundary exactly when there are e'∈ E^n-1 and c∈ C^n with e=d_Ee'+Dc, d_Cc=0. This is equivalent to [e]=H^n(D)[c]. Concentration in degree zero reduces the cone quotient to E^0/im D.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b87f37426ee94627","tex":"E^2","aliases":"E^2","display":"E^2","kind":"use","excerpt":"[Split bridge rectification] Let A be an additive idempotent-complete category and suppose XxrightarrowAYxrightarrowLX satisfies LA=id_X. Put E=AL. Then: [label=textup(roman*)] item E^2=E; item Y≅im E⊕im(1-E); item A:X→im E is an isomorphism with inverse induced by L; item L(1-E)=0, so im(1-E) is a named residual invisible to the decoder; item the ","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5171c82885ec0f71","tex":"e^{2\\pi i a\\ell/q}","aliases":"e^2π i aℓ/q pi ell","display":"e^{2\\pi i a\\ell/q}","kind":"use","excerpt":"Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q gcd(a,q)=1 e^2π i aℓ/q = ∑_dmidgcd(q,ℓ)d μ(q/d) ∈ Z is the Ramanujan sum. The projec","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6b229b436cc51b6f","tex":"e^{2\\pi i\\rho}","aliases":"e^2π iρ pi rho","display":"e^{2\\pi i\\rho}","kind":"use","excerpt":"finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences between ex","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f159b97c629a9ee0","tex":"E^4","aliases":"E^4","display":"E^4","kind":"use","excerpt":"[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b2547c257148d58b","tex":"E^N","aliases":"E^N","display":"E^N","kind":"use","excerpt":"i). The orbit of v under an invertible map of the finite set F_p^d is a cycle containing its initial state, so (ii) ⇒ (iii). A sequence of period T is annihilated by E^T-I, where E is the forward shift, giving (iii) ⇒ (i). The minimal annihilator divides every annihilator, hence m_amid x^T-1. Conversely, if m_amid x^N-1, then E^N-I annihilates the ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1009899abf2147a9","tex":"E^{n-1}","aliases":"E^n-1","display":"E^{n-1}","kind":"use","excerpt":"The class of (e,0) is a boundary exactly when there are e'∈ E^n-1 and c∈ C^n with e=d_Ee'+Dc, d_Cc=0. This is equivalent to [e]=H^n(D)[c]. Concentration in degree zero reduces the cone quotient to E^0/im D.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4c3e6618689429e9","tex":"E^T","aliases":"E^T","display":"E^T","kind":"use","excerpt":"A recurrence with nonzero constant term has an invertible companion matrix, giving (i) ⇒ (ii). The orbit of v under an invertible map of the finite set F_p^d is a cycle containing its initial state, so (ii) ⇒ (iii). A sequence of period T is annihilated by E^T-I, where E is the forward shift, giving (iii) ⇒ (i). The minimal annihilator divides ever","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8611063e24d14e09","tex":"e^z","aliases":"e^z","display":"e^z","kind":"use","excerpt":"uppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. I","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4e2e7ebf9d63a99b","tex":"E_{\\mathbf n}","aliases":"E_ n mathbf","display":"E_{\\mathbf n}","kind":"use","excerpt":"[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-535f17963fb5c472","tex":"E_0","aliases":"E_0","display":"E_0","kind":"use","excerpt":"For λ∉ Z, the hypergeometric operator in [source reference] is Fuchsian with singularities 0, α^-N, ∞. Its exponent packets are E_0 = {0}∪{1-b:b∈ B_N,j}, E_α^-N = {0,1,…,N-2,-λ}, and E_∞ = A_N,j(λ). Modulo integers, E_0 is the complete N-torsion set {0,frac1N,…,fracN-1N}, and the pairwise differences in E_∞ are also N-torsion.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-972c0aada81247ef","tex":"e_0^{\\mathsf T}","aliases":"e_0^ T mathsf","display":"e_0^{\\mathsf T}","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9cfec4d8a3402fa0","tex":"e_{aj}","aliases":"e_aj","display":"e_{aj}","kind":"use","excerpt":"rier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j be","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f8b63d55908fc609","tex":"E_f","aliases":"E_f","display":"E_f","kind":"use","excerpt":"t sequence] For every task-compatible linear bridge, there is an exact sequence 0⟶ fracf^-1(N_Y)N_X ⟶ fracXN_X xrightarrowoverline f fracYN_Y ⟶ fracYN_Y+f(X) ⟶0. Thus L_f:=fracf^-1(N_Y)N_X measures source distinctions collapsed by the bridge, while E_f:=fracYN_Y+f(X) measures target semantic states outside the essential image.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dafb044ef50096de","tex":"e_{gHg^{-1}}","aliases":"e_gHg^-1","display":"e_{gHg^{-1}}","kind":"use","excerpt":"Conjugation sends e_H to e_gHg^-1. Centrality is therefore equivalent to normality. The represented identity U_gP_HU_g^-1=P_gHg^-1 gives the operator conditions. The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b576b4aed1362b85","tex":"e_H","aliases":"e_H","display":"e_H","kind":"use","excerpt":"In the faithful regular realization, or in the field realization supplied by a normal basis, the projector determines the subgroup. Indeed, the support of the group-algebra idempotent e_H=frac1|H|∑_h∈ Hh is exactly H.","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"[p-adic saturation sequence] There is a digit-equivariant short exact sequence 0 ⟶ M_k,p(a) ⟶ L_k,p(a) ⟶ C_k(a)_(p) ⟶ 0, where C_k(a)_(p) is the p-primary subgroup of C_k(a). In Smith coordinates, M_k,p(a) = ⊕_i=1^ρp^ν_i Z_p e_i ⊆ ⊕_i=1^ρ Z_p e_i = L_k,p(a).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ba3213d13f30be4f","tex":"e_J","aliases":"e_J","display":"e_J","kind":"use","excerpt":"If HJ=JH, every element of HJ has |H∩ J| representations as hj. Multiplying the two group averages therefore gives the average over HJ. Conversely, the support of e_He_J is HJ, while the support of e_Je_H is JH. Equality in the regular group algebra forces equality of the product sets.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":"[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cf85306cf8e31b85","tex":"e_{j+1}","aliases":"e_j+1","display":"e_{j+1}","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b6fe64bcd173562e","tex":"E_{\\mathrm{mark}}","aliases":"E_mark mathrm","display":"E_{\\mathrm{mark}}","kind":"use","excerpt":"The entries E_per, E_rm, and E_mark refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap. In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e466c5eab5f55a38","tex":"E_N","aliases":"E_N","display":"E_N","kind":"use","excerpt":"Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e3ca2ec43cc1e4a1","tex":"E_{N,j}","aliases":"E_N,j","display":"E_{N,j}","kind":"use","excerpt":"uppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. I","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-19911032e8985ebc","tex":"e_{N-1}","aliases":"e_N-1","display":"e_{N-1}","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1449f187e2019d37","tex":"e_{N-k}^{\\mathsf T}","aliases":"e_N-k^ T mathsf","display":"e_{N-k}^{\\mathsf T}","kind":"use","excerpt":"Factor H=OC, where row i of O is e_0^ TM_N^i, and column j of C is M_N^j e_0. For j<N, the binomial expansion of (I+S_z)^j e_0 is a unit upper-triangular change from (e_0,…,e_N-1), hence det C=1. The rows e_0^ TS_z^k, 0≤ k<N, are e_0^ T followed by z e_N-k^ T, so their determinant is ± z^N-1. The binomial row change is also unit triangular, proving","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-603e4dc0eaf54bf8","tex":"E_{\\mathrm{per}}","aliases":"E_per mathrm","display":"E_{\\mathrm{per}}","kind":"use","excerpt":"The entries E_per, E_rm, and E_mark refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap. In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b3f1fc86e79071cb","tex":"E_q","aliases":"E_q","display":"E_q","kind":"use","excerpt":"ll=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = fr","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0bbcf87df6a4ea83","tex":"E_{q'}","aliases":"E_q'","display":"E_{q'}","kind":"use","excerpt":"are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-540d75a0692bf27e","tex":"E_{\\mathrm{rm}}","aliases":"E_rm mathrm","display":"E_{\\mathrm{rm}}","kind":"use","excerpt":"The entries E_per, E_rm, and E_mark refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap. In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-38587d825b196d79","tex":"\\mathsf E_{\\rm Fuss}","aliases":"E_rm Fuss mathsf rm","display":"\\mathsf E_{\\rm Fuss}","kind":"use","excerpt":"HOL · P · E · -- · E_rm Fuss/ P · E_rm Fuss/ P · -- · -- · --","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b8b743ab50d39362","tex":"\\mathsf E_{\\rm per}","aliases":"E_rm per mathsf rm","display":"\\mathsf E_{\\rm per}","kind":"use","excerpt":"LIN · E · FF · E_rm per · -- · -- · -- · E_rm per · --","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-621e213d8adce085","tex":"\\mathsf E_{\\rm rm}","aliases":"E_rm rm mathsf rm","display":"\\mathsf E_{\\rm rm}","kind":"use","excerpt":"DIG · E_rm per · P · E · P · -- · -- · E_rm rm · --","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5da6126b8f56e09d","tex":"E_U","aliases":"E_U","display":"E_U","kind":"use","excerpt":"item Holonomy-compatible selectors and residuals. Let E_U⊆ I_Y,U,u× I_X,U,r be a family satisfying E_V= (λ^Y_γ,u×λ^X_γ,r)(E_U) for every arrow γ:U→ V. Then the local selectors C_E_U(h_U)_ai = h_U,ai, · (a,i)∈ E_U,","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ac7db383a5fa1fb7","tex":"E_V","aliases":"E_V","display":"E_V","kind":"use","excerpt":"item Holonomy-compatible selectors and residuals. Let E_U⊆ I_Y,U,u× I_X,U,r be a family satisfying E_V= (λ^Y_γ,u×λ^X_γ,r)(E_U) for every arrow γ:U→ V. Then the local selectors C_E_U(h_U)_ai = h_U,ai, · (a,i)∈ E_U,","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5545430c4956d174","tex":"e_X","aliases":"e_X","display":"e_X","kind":"use","excerpt":"[Typed realization and module equivalence, imported] Let Λ_ K:=𝕜 Q/I. In 𝕜-modules, the category of typed realizations of the linear grammar (Q,I) is equivalent to the category of unital left Λ_ K-modules. Under the equivalence, the vertex idempotent e_X acts as the projector onto the component of sort X, and each arrow acts in its declared source-","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ee9f84268f2af722","tex":"e_Y","aliases":"e_Y","display":"e_Y","kind":"use","excerpt":"A typed realization defines the action of a path by composition and then extends 𝕜-linearly. Every element of I acts by zero, so the action factors through Λ_ K. Conversely, for a unital module M, the pairwise orthogonal vertex idempotents give M=⊕_X∈ Q_0e_XM, and an arrow a:X→ Y maps e_XM into e_YM. These constructions are inverse up to their cano","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7bfb4afe5cc6b2f6","tex":"E_{\\alpha^{-N}}","aliases":"E_α^-N alpha","display":"E_{\\alpha^{-N}}","kind":"use","excerpt":"For λ∉ Z, the hypergeometric operator in [source reference] is Fuchsian with singularities 0, α^-N, ∞. Its exponent packets are E_0 = {0}∪{1-b:b∈ B_N,j}, E_α^-N = {0,1,…,N-2,-λ}, and E_∞ = A_N,j(λ). Modulo integers, E_0 is the complete N-torsion set {0,frac1N,…,fracN-1N}, and the pairwise differences in E_∞ are also N-torsion.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9ea5301de7516231","tex":"e_\\chi","aliases":"e_χ chi","display":"e_\\chi","kind":"use","excerpt":"[Typed semisimple Galois realization] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a14da4e5535e89e4","tex":"E_\\infty","aliases":"E_∞ infty","display":"E_\\infty","kind":"use","excerpt":"For λ∉ Z, the hypergeometric operator in [source reference] is Fuchsian with singularities 0, α^-N, ∞. Its exponent packets are E_0 = {0}∪{1-b:b∈ B_N,j}, E_α^-N = {0,1,…,N-2,-λ}, and E_∞ = A_N,j(λ). Modulo integers, E_0 is the complete N-torsion set {0,frac1N,…,fracN-1N}, and the pairwise differences in E_∞ are also N-torsion.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-033dc5a65f23766b","tex":"\\End_{\\kk C_2}","aliases":"End_kk C_2 End kk","display":null,"kind":"use","excerpt":"0 · 1. The individual projectors do not descend because Sp_1S^-1=p_2, while their orbit sum p_1+p_2=1 does descend. The pair-address set has two orbits, Δ={(1,1),(2,2)}, Ω={(1,2),(2,1)}. Accordingly, End_kk C_2(kk^2) = { a · b b · a:a,b∈kk } = kk B_Δ⊕kk B_Ω. The singleton endpoint selector {(1,1)} is not holonomy compatible and sends the equivarian","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ac231773af7a5e8f","tex":"\\End_{\\kk H}","aliases":"End_kk H End kk","display":null,"kind":"use","excerpt":"[Orbital endomorphism algebra] For one finite H-set I, the pair-orbit matrices form a basis of End_kk H(kk[I]). Matrix multiplication in this basis is exactly the finite compatible-path contraction of [source reference], with the preceding intersection numbers as structure constants.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b6bb9a2af76d66fc","tex":"\\mathbf F","aliases":"F mathbf","display":"\\mathbf F","kind":"use","excerpt":"item A selected algebraic branch can produce a D-finite annihilator, but the operator may forget the selected branch and its minimal polynomial. The bridge is directional unless that data are retained. item A rational functional-Mahler mask over F_p produces a Cartier representation and finite automaton under the stated hypotheses, but this does no","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"The normalized jet is J_0(F) = (F(0),F'(0),…,F^(r-1)(0)).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0c767c7d70dbba4a","tex":"F^{(d)}","aliases":"F^(d)","display":"F^{(d)}","kind":"use","excerpt":"Differentiate P_F(z,F(z))=0. Separability makes ∂_YP_F(z,F) nonzero in the function field. The field is closed under the induced derivation. The d+1 elements F,F',…,F^(d) are linearly dependent over K(z).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-67012604beb21a70","tex":"F^{(j)}","aliases":"F^(j)","display":"F^{(j)}","kind":"use","excerpt":"[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-696d6e9cd0554878","tex":"F^{(r-1)}","aliases":"F^(r-1)","display":"F^{(r-1)}","kind":"use","excerpt":"The normalized jet is J_0(F) = (F(0),F'(0),…,F^(r-1)(0)).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b23cbbeba59d5f4e","tex":"f^{-1_D}","aliases":"f^-1_D","display":"f^{-1_D}","kind":"use","excerpt":"[Dirichlet inverse and multiplicative closure] If f(1)=1, then f has a unique integral Dirichlet inverse. If f and g are multiplicative, then fmathbin∗_Dg and f^-1_D are multiplicative.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-994c5e5bb48190c6","tex":"F^{p-1}","aliases":"F^p-1","display":"F^{p-1}","kind":"use","excerpt":"[Frobenius-radical characterization] Let F∈1+z F_p[[z]]. Then R_F(z)=F(z)^1-p. Consequently, the following are equivalent. [label=textup(roman*)] item The canonical mask R_F is rational. item The power F^p-1 belongs to F_p(z). item There are coprime polynomials A,B∈ F_p[z] with A(0)=B(0)=1 and B(z)F(z)=A(z)F(z^p). For the reduced mask R_F=A/B, one ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0e4763978622e187","tex":"f_1","aliases":"f_1","display":"f_1","kind":"use","excerpt":"[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"Even indicator over F_2. For the word 10, T=2, m(x)=x^2+1=(x+1)^2, F(z)=frac11+z^2, R(z)=1+z^2. The repeated recurrence root is compatible with pure periodicity because the field has characteristic two. The canonical 2-kernel has three states: the original sequence, the constant-one sequence, and the constant-zero sequence. The rational-mask compil","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-98550483d35bb33d","tex":"f_2","aliases":"f_2","display":"f_2","kind":"use","excerpt":"[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54c7a07f66f1253f","tex":"\\mathbb F_3","aliases":"F_3 mathbb","display":"\\mathbb F_3","kind":"use","excerpt":"a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] need not be reduced. Cancellation is part of the normalized rational chart rather than an error. For example, over F_3, the period word 12 has frac1+2z1-z^2=frac11+z.","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"screte category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] n","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-88ed2cdafda7b51a","tex":"f_{a+b,s+p^a t}","aliases":"f_a+b,s+p^a t","display":"f_{a+b,s+p^a t}","kind":"use","excerpt":"[Strict nested-branch law] For a,b≥1, 0≤ s<p^a, and 0≤ t<p^b, (f_a,s)_b,t=f_a+b,s+p^a t. All multiple-overlap comparison cells for the nested p-power branch atlas are identities. Hence its higher coherence is strict rather than merely up to a chosen natural isomorphism.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-45843ed5f03c5c85","tex":"f_{a,s}","aliases":"f_a,s","display":"f_{a,s}","kind":"use","excerpt":"[Strict nested-branch law] For a,b≥1, 0≤ s<p^a, and 0≤ t<p^b, (f_a,s)_b,t=f_a+b,s+p^a t. All multiple-overlap comparison cells for the nested p-power branch atlas are identities. Hence its higher coherence is strict rather than merely up to a chosen natural isomorphism.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0b8eb6ad7f8b7acf","tex":"f_b","aliases":"f_b","display":"f_b","kind":"use","excerpt":"Let A_b(z)=∏_m≥1(1-z^m)^-b_m be an integral unit series in the fixed Euler basis, and let c_n=∑_dmid nd b_d be its divisor-ghost sequence. Define the prime-uniform multiplicative function f_b(n)=∏_p^eparallel n[z^e]A_b(z). Every prime has the same local factor A_b(T). By [source reference], the coefficient, exponent, ghost, and prime-uniform multip","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dafda06c8efa3652","tex":"f_E","aliases":"f_E","display":"f_E","kind":"use","excerpt":"[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e1bc7ddf293c6557","tex":"F_{f,p}'","aliases":"F_f,p'","display":"F_{f,p}'","kind":"use","excerpt":"[Valuation logarithmic derivative] Fix a prime p. Let Γ_f,p be the arithmetic function supported on powers of p with Γ_f,p(1)=0, Γ_f,p(p^e)=γ_p,e(f) (e≥1). Then (v_p· f)(n) = (Γ_f,pmathbin∗_Df)(n) (n≥1). Equivalently, fracT F_f,p'(T)F_f,p(T) = ∑_e≥1γ_p,e(f)T^e.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4f0594fb73dc48e0","tex":"F_{f\\ast_D g,p}","aliases":"F_f∗_D g,p ast","display":"F_{f\\ast_D g,p}","kind":"use","excerpt":"[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-51e7a1534ed7ce31","tex":"F_{g,p}","aliases":"F_g,p","display":"F_{g,p}","kind":"use","excerpt":"[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-74cb89ef83acdc31","tex":"F_j","aliases":"F_j","display":"F_j","kind":"use","excerpt":"[Typed polyphase realization] Let K be a field, let N∈ Z with N≥1 be invertible in K, and suppose K contains a primitive N-th root of unity. For F(z)=∑_n≥0a_nz^n∈ K[[z]], set F_j(w):=∑_n≥0a_Nn+jw^n. Then F(z)=∑_j=0^N-1z^jF_j(z^N) is exact analysis and synthesis. The Fourier residue projectors and the polyphase components satisfy the explicit coordi","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"use","excerpt":"[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c57cafb013c6775f","tex":"F_n","aliases":"F_n","display":"F_n","kind":"use","excerpt":"Fibonacci F_n · x^2-x-1 · z/(1-z-z^2) · 2 · The root ratio is not torsion. Every arithmetic subsequence retains rank 2.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-43135e362a6370f0","tex":"f_N","aliases":"f_N","display":"f_N","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b9a66bfb8f08c990","tex":"F_{N-1}","aliases":"F_N-1","display":"F_{N-1}","kind":"use","excerpt":"[Exact hypergeometric packet form] If (λ)_j≠0, then G_N,j^λ,α(x) = frac(λ)_jα^jj! _NF_N-1 ( A_N,j(λ) B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5fa481e02231319b","tex":"F_{n-1}","aliases":"F_n-1","display":"F_{n-1}","kind":"use","excerpt":"bibitemBeukersHeckman1989 F. Beukers and G. Heckman, ``Monodromy for the hypergeometric function _nF_n-1,'' Inventiones Mathematicae 95 (1989), 325--354.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ccdb1c456beed560","tex":"F_o","aliases":"F_o","display":"F_o","kind":"use","excerpt":"Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2be6cf11bf912bc7","tex":"\\mathbf F_p","aliases":"F_p mathbf","display":"\\mathbf F_p","kind":"use","excerpt":"item A selected algebraic branch can produce a D-finite annihilator, but the operator may forget the selected branch and its minimal polynomial. The bridge is directional unless that data are retained. item A rational functional-Mahler mask over F_p produces a Cartier representation and finite automaton under the stated hypotheses, but this does no","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2d6d5a8ba40d633c","tex":"\\mathbb F_p^d","aliases":"F_p^d mathbb","display":"\\mathbb F_p^d","kind":"use","excerpt":"A recurrence with nonzero constant term has an invertible companion matrix, giving (i) ⇒ (ii). The orbit of v under an invertible map of the finite set F_p^d is a cycle containing its initial state, so (ii) ⇒ (iii). A sequence of period T is annihilated by E^T-I, where E is the forward shift, giving (iii) ⇒ (i). The minimal annihilator divides ever","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-38441d9404357963","tex":"\\mathbb F_p^{D+1}","aliases":"F_p^D+1 mathbb","display":"\\mathbb F_p^{D+1}","kind":"use","excerpt":"a=deg A, b=deg B, c=a+(p-2)b, and define D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source reference]. It computes the coefficient sequen","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-666cac5565f15579","tex":"F_q","aliases":"F_q","display":"F_q","kind":"use","excerpt":"The root filter also gives a construction-independent order ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-86f19f5a64cb25fa","tex":"\\mathbb F_q","aliases":"F_q mathbb","display":"\\mathbb F_q","kind":"use","excerpt":"a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K^min. If K= F_q is finite, every k-regular K-valued sequence is automatic.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4a14d2ab0a464e3b","tex":"F_u","aliases":"F_u","display":"F_u","kind":"use","excerpt":"Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4a60f87925a8ea8b","tex":"f_U","aliases":"f_U","display":"f_U","kind":"use","excerpt":"s whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z ","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6d255d2671bc9fd0","tex":"F_{UV}","aliases":"F_UV","display":"F_{UV}","kind":"use","excerpt":"[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms sa","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-78b29d109052da32","tex":"F_v","aliases":"F_v","display":"F_v","kind":"use","excerpt":"Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a85f7bd78266c271","tex":"f_V","aliases":"f_V","display":"f_V","kind":"use","excerpt":"A descent object A=(A_U,θ^A_γ) consists of objects A_U∈ C_U and isomorphisms θ^A_γ:F_γ(A_U)xrightarrow∼A_V satisfying the identity and cocycle equations. A descended morphism f:X→ Y is a compatible family (f_U)_U satisfying θ^Y_γ F_γ(f_U)=f_Vθ^X_γ.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a2d65fee6938939b","tex":"F_{\\alpha,\\alpha}","aliases":"F_α,α alpha alpha","display":"F_{\\alpha,\\alpha}","kind":"use","excerpt":"If α=β, the quadratic equation Y^2-(1-α z)^2=0 is not the normalized source polynomial. The selected branch is rational: F_α,α(z)=1-α z, so the correct source degree is one. This is a negative control against using a nonminimal defining equation in the degree ledger.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e4a88c6b52e0e670","tex":"F_{\\alpha,\\beta}","aliases":"F_α,β alpha beta","display":"F_{\\alpha,\\beta}","kind":"use","excerpt":"Let F_α,β(z) = sqrt(1-α z)(1-β z), F_α,β(0)=1, and assume first that α≠β, so the normalized source degree is two. Write F_α,β(z)=E(x)+zO(x), x=z^2, and put s=α+β, p=αβ.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a94ca35a26e72bce","tex":"F_{\\lambda,\\alpha}","aliases":"F_λ,α lambda alpha","display":"F_{\\lambda,\\alpha}","kind":"use","excerpt":"The general minimal multisection problem remains open. The family F_λ,α(z) = (1-α z)^-λ provides a nontrivial exact model that contains irreducible Fuchsian packets, rational collapse, finite-support aliasing, and zero-address cancellation in one parameter space.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ae4e9af5b31b6091","tex":"\\mathbf{FinOrbLin}","aliases":"FinOrbLin mathbf","display":"\\mathbf{FinOrbLin}","kind":"use","excerpt":"k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-879c879b41c0b5fd","tex":"\\mathbf{FinOrbLin}_{k,K}^{\\mathrm{sl}}","aliases":"FinOrbLin_k,K^sl mathbf mathrm","display":"\\mathbf{FinOrbLin}_{k,K}^{\\mathrm{sl}}","kind":"use","excerpt":"k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-25586c9c82f252a7","tex":"\\mathbf{FinOrbSec}","aliases":"FinOrbSec mathbf","display":"\\mathbf{FinOrbSec}","kind":"use","excerpt":"k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cc69fb680cd72839","tex":"\\mathbf{FinOrbSec}_k","aliases":"FinOrbSec_k mathbf","display":"\\mathbf{FinOrbSec}_k","kind":"use","excerpt":"k-regular sequence a, the following are equivalent: item a is k-automatic; item the orbit {Λ_wa:w∈Σ_k^*} of the initial vector in C_k(a) is finite; item the orbit of v(a) under the strict shortlex digit matrices is finite. On this locus, the minimal DFAO is recovered as the orbit itself. Hence Aut_k(K) ≃ FinOrbSec_k(K) ≃ FinOrbLin_k,K^sl ≃ DFAO_k,K","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fe86a8cfc69bcf10","tex":"\\operatorname{Fitt}_j","aliases":"Fitt_j operatorname","display":"\\operatorname{Fitt}_j","kind":"use","excerpt":"[Fitting-ideal certificate] For 0≤ j<ρ, Fitt_j(C_k(a)) = ( d_1(a)d_2(a)⋯ d_ρ-j(a) ), and Fitt_j(C_k(a))= Z (j≥ρ). Thus the Smith profile and the Fitting filtration contain the same reduction-defect data at different resolutions.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-df668eb6def3fe91","tex":"\\Fix","aliases":"Fix operatorname","display":"{\\operatorname{Fix}}","kind":"use","excerpt":"a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cbc27ab00443902b","tex":"\\Fun_{\\kk}^{\\oplus}","aliases":"Fun_kk^⊕ Fun_𝕜^⊕ Fun kk oplus operatorname Bbbk oplus","display":"{\\operatorname{Fun}}_{{\\Bbbk}}^{\\oplus}","kind":"use","excerpt":"gintheorem[Finite additive-Karoubi resolved envelope] The category Syn_ak( K_rm lin) := Kar (Add(𝕜 Q/I)) is small, additive, idempotent complete, and 𝕜-linear. For every additive idempotent-complete 𝕜-linear category C, restriction to the generating quiver induces an equivalence Fun_𝕜^⊕ (Syn_ak( K_rm lin), C) ≃ Mod_ak( K_rm lin, C), where the right","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aa45a7ad376e6ef9","tex":"g^{-1}","aliases":"g^-1","display":"g^{-1}","kind":"use","excerpt":"Conjugation sends e_H to e_gHg^-1. Centrality is therefore equivalent to normality. The represented identity U_gP_HU_g^-1=P_gHg^-1 gives the operator conditions. The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-886996535a9d2ef0","tex":"G^{p-1}","aliases":"G^p-1","display":"G^{p-1}","kind":"use","excerpt":"iterion] Let F= M_p[A/B]. The following are equivalent. [label=textup(roman*)] item F∈ F_p(z). item B/A is a (p-1)-st power in F_p(z). item ω_p-1(A/B)=0. When these conditions hold, there is a unique rational function G∈ F_p(z) with G(0)=1 and G^p-1=B/A, and F=G.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7ae44fdabdf29cc3","tex":"g^\\ell","aliases":"g^ℓ ell","display":"g^\\ell","kind":"use","excerpt":"[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-575d86b85365c7b4","tex":"G_0","aliases":"G_0","display":"G_0","kind":"use","excerpt":"Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"use","excerpt":"Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3b9dcaf46a46092c","tex":"G_2","aliases":"G_2","display":"G_2","kind":"use","excerpt":"Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-32bdff61f9c93d06","tex":"g_{ca}","aliases":"g_ca","display":"g_{ca}","kind":"use","excerpt":"[Exact transfer reconstruction and block composition, imported] Every morphism f:X→ Y has the exact expansion f = ∑_a,iq_afp_i = ∑_a,ijmath_a f_aiπ_i. If g:Y→ Z is resolved by a packet system R=(r_c), then (gf)_ci = ∑_a g_caf_ai.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-33f3ee3e874a2b24","tex":"g_F","aliases":"g_F","display":"g_F","kind":"use","excerpt":"[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cd7038b72872679a","tex":"G_i","aliases":"G_i","display":"G_i","kind":"use","excerpt":"Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"use","excerpt":"[Exact Smith formulas] Let g_i(m)=gcd(m,d_i(a)). Then G_k,m(a) ≅ Q_k,m(a) ≅ ⊕_i=1^ρ Z/g_i(m) Z, and |overline M_k,m(a)| = ∏_i=1^ρ fracmg_i(m). The reduction map is an isomorphism if and only if gcd(m,d_ρ(a))=1. Equivalently, every prime dividing a reduction defect divides the saturation index.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fb1ccfdcfa4d6752","tex":"G_{k,p^r}","aliases":"G_k,p^r","display":"G_{k,p^r}","kind":"use","excerpt":"[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3632ec9c2de4f4a3","tex":"g_N","aliases":"g_N","display":"g_N","kind":"use","excerpt":"[Finite tree-address signature] Fix a packet width N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-605af900ba9b414b","tex":"G_{N,j}","aliases":"G_N,j","display":"G_{N,j}","kind":"use","excerpt":"The root filter also gives a construction-independent order ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c4b1df9b697b122a","tex":"G_{N,j}^{-M,\\alpha}","aliases":"G_N,j^-M,α alpha","display":"G_{N,j}^{-M,\\alpha}","kind":"use","excerpt":"item If λ∉ Z, then r_D(N,j;λ)=N. item If λ=L∈ Z_>0, then every packet is a nonzero rational function with exact denominator (1-α^Nx)^L, and r_D(N,j;L)=1. item If λ=-M∈ Z_≤0, then G_N,j^-M,α is zero when j>M. Otherwise it is a nonzero polynomial of degree lfloorfracM-jNrfloor and r_D(N,j;-M) = 1, · j≤ M,","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8ebee069f041f580","tex":"g_{r,\\gamma}","aliases":"g_r,γ gamma","display":"g_{r,\\gamma}","kind":"use","excerpt":"Let F:Y→ Z and G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-66816f2f34b0eaea","tex":"g_s","aliases":"g_s","display":"g_s","kind":"use","excerpt":"[Complete clopen packet equivalence] The analysis map B_a: C_p ⟶ ∏_s=0^p^a-1 C_p is an isometric isomorphism. Its inverse is the piecewise synthesis ( S_a g)(x) =g_s (fracx-sp^a) for x∈ s+p^a Z_p. Thus the clopen residue cover of Z_p has effective descent for both objects and equality morphisms.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f419c722fddc9256","tex":"G_u","aliases":"G_u","display":"G_u","kind":"use","excerpt":"el in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-169c77a4f93260fb","tex":"g_U","aliases":"g_U","display":"g_U","kind":"use","excerpt":"s whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms satisfy the identity and cocycle equations. Let f=(f_U):X⟶ Y, g=(g_U):Y⟶ Z ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8bea312cd06acfdb","tex":"G_v","aliases":"G_v","display":"G_v","kind":"use","excerpt":"Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-008f73407822c980","tex":"G_v^{-1}","aliases":"G_v^-1","display":"G_v^{-1}","kind":"use","excerpt":"el in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-639e6eca63f09513","tex":"G_\\Phi","aliases":"G_Φ Phi","display":"G_\\Phi","kind":"use","excerpt":"For the packet system textup(11.4), define the directed dependency graph G_Φ on I by placing an edge j⟶ i when some active block contributing to the equation for u_i contains u_j as an input. Let SCC(G_Φ) be its strongly connected components. A component is called cyclic when it contains more than one address or is a singleton carrying a self-loop.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-21cf47c47bcf5c41","tex":"\\Gal","aliases":"Gal","display":null,"kind":"use","excerpt":"[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-efa0a3eea83b0f61","tex":"\\operatorname{Gal}","aliases":"Gal operatorname","display":"\\operatorname{Gal}","kind":"use","excerpt":"Let Ω be a normal splitting field of the generic packet scheme over K(x), and let Γ=Gal(Ω/K(x)). After splitting E_N, the geometric packet branches form a set B_N of cardinality d^N. Let π_j: B_N→Ω be the jth packet coordinate, and set C_N,j=π_j( B_N). For the selected semantic packet U_j, separability gives [K(x)(U_j):K(x)] = |Γ· U_j|.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-09c99fb5d24e4265","tex":"\\gcd","aliases":"gcd","display":"\\gcd","kind":"use","excerpt":"An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.","count":23},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0fcc70536b44cfb8","tex":"\\operatorname{GL}","aliases":"GL operatorname","display":"\\operatorname{GL}","kind":"use","excerpt":"The image is a compact totally disconnected subgroup of GL_n( C). After conjugation it lies in a compact Lie group. A compact totally disconnected Lie group is finite.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8df4bc39535de40c","tex":"\\operatorname{GL}_n","aliases":"GL_n operatorname","display":"\\operatorname{GL}_n","kind":"use","excerpt":"The image is a compact totally disconnected subgroup of GL_n( C). After conjugation it lies in a compact Lie group. A compact totally disconnected Lie group is finite.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c3d50e4b12d0681c","tex":"H^*","aliases":"H^*","display":"H^*","kind":"use","excerpt":"0 · 1. There is no positive-definite metric Q with H^*QH=Q. Nevertheless, for Q_s=diag(s^2,1), we have D_Q_s(H)⟶1 (sdownarrow0), while κ_2(Q_s)=s^-2⟶∞. Thus the metric-completion defect has zero infimum but is not attained.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fa5a6573c905a228","tex":"H^0","aliases":"H^0","display":"H^0","kind":"use","excerpt":"[Cohomological correction criterion] For a cocycle e∈ E^n, the class of (e,0) in H^n(Cone(D)) vanishes exactly when [e] lies in the image of H^n(D). When the complexes are concentrated in degree zero, H^0(Cone(D))≅coker D.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"[Brauer obstruction to projective packet descent] Projective semilinear descent data define a class α∈ H^1(G,PGL_n(L)). The connecting map for 1⟶ L^× ⟶ GL_n(L) ⟶ PGL_n(L) ⟶1 produces δ(α) ∈ H^2(G,L^×) = Br(L/K). The projective atlas lifts to linear semilinear descent data exactly when δ(α)=0. When the class is nonzero, the descended object is natur","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"[Brauer obstruction to projective packet descent] Projective semilinear descent data define a class α∈ H^1(G,PGL_n(L)). The connecting map for 1⟶ L^× ⟶ GL_n(L) ⟶ PGL_n(L) ⟶1 produces δ(α) ∈ H^2(G,L^×) = Br(L/K). The projective atlas lifts to linear semilinear descent data exactly when δ(α)=0. When the class is nonzero, the descended object is natur","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2260d36a8935245e","tex":"\\mathcal H^H","aliases":"H^H mathcal","display":"\\mathcal H^H","kind":"use","excerpt":"Let G be a compact group and let U:G→ U( H) be strongly continuous. For a closed subgroup H, normalized Haar measure defines P_H=∫_HU_h dμ_H(h). tag12.14 The same averaging proof as in the finite case shows that P_H is the orthogonal projector onto H^H.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7c6fa035c4defd46","tex":"H^n","aliases":"H^n","display":"H^n","kind":"use","excerpt":"[Cohomological correction criterion] For a cocycle e∈ E^n, the class of (e,0) in H^n(Cone(D)) vanishes exactly when [e] lies in the image of H^n(D). When the complexes are concentrated in degree zero, H^0(Cone(D))≅coker D.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":",mathscr M^u,r_Y,X) xrightarrow∼ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0), where the right-hand side is the subgroup fixed by every loop-transport map. Consequently, Hom_Desc(mathscr G)(X,Y) ≅ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0). After choosing one arrow τ_U:U_0→ U for each chart, the inverse sends a fixed base matrix h_0 to the section h_U=","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6b6f4ce8a788a1f1","tex":"h_{A^{\\mathsf T}C}","aliases":"h_A^ TC mathsf","display":"h_{A^{\\mathsf T}C}","kind":"use","excerpt":"For a linear map A, h_A^ TC(x)=h_C(Ax). If the incoming fiber is empty, both sides of the formula are -∞. Otherwise, the support function of the convex hull of a finite union is the maximum of the component support functions, with the same conventions for empty components. These identities give the formula. Finite unions of compact sets have compac","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-642347304bc7ab54","tex":"H_A^r","aliases":"H_A^r","display":"H_A^r","kind":"use","excerpt":"Fix a base chart U_0. Every loop γ∈Aut_mathscr G(U_0) acts on an address set by i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is deno","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cdaad3bc29f7c190","tex":"h_{bj}","aliases":"h_bj","display":"h_{bj}","kind":"use","excerpt":"[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contrib","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-38becd0552b037e7","tex":"h_C","aliases":"h_C","display":"h_C","kind":"use","excerpt":"For a linear map A, h_A^ TC(x)=h_C(Ax). If the incoming fiber is empty, both sides of the formula are -∞. Otherwise, the support function of the convex hull of a finite union is the maximum of the component support functions, with the same conventions for empty components. These identities give the formula. Finite unions of compact sets have compac","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fca539778b169102","tex":"h_{C'_t}","aliases":"h_C'_t","display":"h_{C'_t}","kind":"use","excerpt":"[Observer-conditioned convex packet transport] For every state t∈ S and direction x, h_C'_t(x) = max_δ(s,b)=th_C_s(A_bx). Repeated updates compose exactly with branch words. Within each fixed observer state, replacing a finite atom family by any other finite generating family of the same convex hull preserves all future value queries generated by l","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8935bf5905716354","tex":"h_{C_s}","aliases":"h_C_s","display":"h_{C_s}","kind":"use","excerpt":"[Observer-conditioned convex packet transport] For every state t∈ S and direction x, h_C'_t(x) = max_δ(s,b)=th_C_s(A_bx). Repeated updates compose exactly with branch words. Within each fixed observer state, replacing a finite atom family by any other finite generating family of the same convex hull preserves all future value queries generated by l","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5d724fab1c00df76","tex":"H_e","aliases":"H_e","display":"H_e","kind":"use","excerpt":"el in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f10380c142f70a49","tex":"H_e^*","aliases":"H_e^*","display":"H_e^*","kind":"use","excerpt":"Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible fa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f74f6a57337ee481","tex":"H_{ij}","aliases":"H_ij","display":"H_{ij}","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b28d6c4b94c5afd8","tex":"h_N","aliases":"h_N","display":"h_N","kind":"use","excerpt":"[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arit","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a7e9d829aea21955","tex":"\\mathcal H_{N,j}^{\\lambda,\\alpha}","aliases":"H_N,j^λ,α mathcal lambda alpha","display":"\\mathcal H_{N,j}^{\\lambda,\\alpha}","kind":"use","excerpt":"B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-577c3b3594f44970","tex":"h_{U,ai}","aliases":"h_U,ai","display":"h_{U,ai}","kind":"use","excerpt":"item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix rec","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-07cec1aa3286340f","tex":"H_{Y,X}^{u,r}","aliases":"H_Y,X^u,r","display":"H_{Y,X}^{u,r}","kind":"use","excerpt":"[ H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is denoted H_Y,X^u,r. This is the simultaneous monodromy image; it need not be the direct product of the ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-61d72da262c77371","tex":"h_{\\varnothing}","aliases":"h_∅ varnothing","display":"h_{\\varnothing}","kind":"use","excerpt":"Let S be a finite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plan","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e0d53229e96ed0e9","tex":"\\operatorname{Hom}","aliases":"Hom operatorname","display":"\\operatorname{Hom}","kind":"use","excerpt":"[Universal mapping property of packet coordinates] For every object Z, restriction to the packet objects gives natural bijections Hom(X,Z) ≅ ∏_iHom(X_i,Z), Hom(Z,X) ≅ ∏_iHom(Z,X_i), where the products are finite. Packetwise maps therefore assemble uniquely into a global map.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-56f7ec20111f5a0c","tex":"\\Hom_{\\mathcal C}","aliases":"Hom_ C Hom mathcal","display":null,"kind":"use","excerpt":"item Unique lane matrix. Every morphism f:X→ Y is recovered uniquely from its lane blocks: f = ∑_a,ip_Y,a^u f p_X,i^r = ∑_a,iι_Y,a^u [f]^u,r_a iπ_X,i^r. Equivalently, the assignment f⟼ ([f]^u,r_a i)_a,i is a bijection from Hom_ C(X,Y) to the group of rectangular matrices with entries in Hom_ C(X_i^r,Y_a^u).","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bf9d8f5ac9aec205","tex":"\\Hom_{\\Desc(\\mathscr G)}","aliases":"Hom_Desc(mathscr G) Hom Desc mathscr","display":null,"kind":"use","excerpt":"item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix rec","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c31b8eb8e8bdf6c9","tex":"\\Hom_{\\kk H}","aliases":"Hom_kk H Hom kk","display":null,"kind":"use","excerpt":"ion-orbital-calculus Let kk be a commutative ring, let a finite group H act on finite sets I,J,K, and let kk[I],kk[J],kk[K] be the corresponding permutation modules. For an orbit O⊆ J× I under the diagonal action, let B_ O:kk[I]→kk[J] be the matrix whose (j,i)-entry is one when (j,i)∈ O and zero otherwise. Then Hom_kk H(kk[I],kk[J]) = ⊕_ O∈ Hbacksl","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e3bae9b717557c2b","tex":"\\mathrm{HS}","aliases":"HS mathrm","display":"\\mathrm{HS}","kind":"use","excerpt":"[Hilbert--Schmidt optimality, imported] Assume X,Y are Hilbert spaces, the packet idempotents are orthogonal projectors, and f:X→ Y is Hilbert--Schmidt. Then C_E is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and R_E(f) _HS^2 = ∑_(a,i)∉ E q_afp_i _HS^2. Consequently, C_E(f) is the unique best Hilbert--Schmidt approxi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8da0f8726aa71190","tex":"i'","aliases":"i'","display":"i'","kind":"use","excerpt":"The (k,i)-entry of B_ QB_ P is the displayed cardinality. If (k',i')=(hk,hi), then j↦ hj is a bijection between the two sets being counted. The entry is therefore constant on R, which gives the orbit-basis expansion.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"use","excerpt":"[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of mult","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-24d36966d52a24a5","tex":"\\mathcal I_{13}","aliases":"I_13 mathcal","display":"\\mathcal I_{13}","kind":"use","excerpt":"nvariant-descent Let A_13 be the atlas of [source reference]. Suppose that, in every connected component: item the local invariant is complete on every normalized chart; item the bridge comparisons are coherent; item invariant-compatible local isomorphisms satisfy effective descent. Then the local invariants induce a functor I_13: Desc( A_13) ⟶ Des","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ffab190d301c243e","tex":"I_{A,r}","aliases":"I_A,r","display":"I_{A,r}","kind":"use","excerpt":"V^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrig","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d236c16abf3a2ebe","tex":"I_{A,s}","aliases":"I_A,s","display":"I_{A,s}","kind":"use","excerpt":"I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fd0d10a990a7b15e","tex":"I_{A,U_0,r}","aliases":"I_A,U_0,r","display":"I_{A,U_0,r}","kind":"use","excerpt":"Fix a base chart U_0. Every loop γ∈Aut_mathscr G(U_0) acts on an address set by i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is deno","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-eeb475be6c821cde","tex":"I_{A_U,r}","aliases":"I_A_U,r","display":"I_{A_U,r}","kind":"use","excerpt":"Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucce","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d2ab55151f4c3a08","tex":"I_{A_V,r}","aliases":"I_A_V,r","display":"I_{A_V,r}","kind":"use","excerpt":"Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucce","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-23757f00da73f2f9","tex":"i_d","aliases":"i_d","display":"i_d","kind":"use","excerpt":"[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of mult","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-03868a4612515256","tex":"I_{X,U_0,r}","aliases":"I_X,U_0,r","display":"I_{X,U_0,r}","kind":"use","excerpt":"i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is denoted H_Y,X^u,r. This is the simultaneous monodromy image; it need not be the dire","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-37b6ffe075592c00","tex":"I_{Y,u}","aliases":"I_Y,u","display":"I_{Y,u}","kind":"use","excerpt":"item Composition and path law. If f:X→ Y and g:Y→ Z, with the same resolution u used on the intermediate object, then [gf]^v,r_c i = ∑_a∈ I_Y,u[g]^v,u_c a[f]^u,r_a i. More generally, a block of a composable string is the finite sum of the products along all compatible intermediate address paths.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1c1e862b140c097e","tex":"I_{Y,U_0,u}","aliases":"I_Y,U_0,u","display":"I_{Y,U_0,u}","kind":"use","excerpt":"i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is denoted H_Y,X^u,r. This is the simultaneous monodromy image; it need not be the dire","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-45b03aa0febc4030","tex":"I_{Y,v}","aliases":"I_Y,v","display":"I_{Y,v}","kind":"use","excerpt":"For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0baa9a9d384779c4","tex":"I_{\\alpha,L}","aliases":"I_α,L alpha","display":"I_{\\alpha,L}","kind":"use","excerpt":"Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-998362834823bc38","tex":"I_{\\lambda,b}","aliases":"I_λ,b lambda","display":"I_{\\lambda,b}","kind":"use","excerpt":"For every chart C_λ,a, let I_λ,a: C_λ,a⟶ V_λ,a be its native normalized invariant. A bridge comparison is a natural isomorphism κ_ab: J_abI_λ,a Longrightarrow I_λ,bΦ_ab, where J_ab transports invariant coordinates between their chart-specific target categories.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a7c524f58e198b86","tex":"I_{\\xi,L_G}","aliases":"I_ξ,L_G xi","display":"I_{\\xi,L_G}","kind":"use","excerpt":"ther with its finite initial record; item the monic minimal differential operator from [source reference], together with the distinguished power-series solution selected by the marked coefficient record; item the hypergeometric parameter packet (A_N,j(λ),B_N,j;α^N) and leading coefficient; item the primitive minimal-operator singular and indicial r","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-97fa69c0b29da91b","tex":"i_\\ell","aliases":"i_ℓ ell","display":"i_\\ell","kind":"use","excerpt":"Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-de717c99c15e8582","tex":"\\id_X","aliases":"id_X id mathrm","display":"{\\mathrm{id}}_X","kind":"use","excerpt":"[Split bridge rectification] Let A be an additive idempotent-complete category and suppose XxrightarrowAYxrightarrowLX satisfies LA=id_X. Put E=AL. Then: [label=textup(roman*)] item E^2=E; item Y≅im E⊕im(1-E); item A:X→im E is an isomorphism with inverse induced by L; item L(1-E)=0, so im(1-E) is a named residual invisible to the decoder; item the ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"For an integer m≥2, the raw coefficient reduction is M_k(a)/mM_k(a), while the semantic reduction inside the saturated lattice is overline M_k,m(a) = im ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). These need not agree.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ef28ee4048cca398","tex":"\\mathrm{in}","aliases":"in mathrm","display":"\\mathrm{in}","kind":"use","excerpt":"For an arbitrary realization (W,A_d,v_in,λ), concatenation gives the factorization H_a(u,v) = λ A_vA_uv_in. Thus H_a factors through W, so rank H_a≤dim W. The canonical realization has dimension r_k(a), proving the minimality formula.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-59f64376d60523e7","tex":"\\Ind","aliases":"Ind","display":null,"kind":"use","excerpt":"Fix a universe mathscr Q(X) of admissible queries on X. For Q⊆mathscr Q(X), define Ind( Q) := ⋂_q∈ Qkerpair(q). For an equivalence relation or congruence R, define Def(R) := {q∈mathscr Q(X):R⊆kerpair(q)}.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ad8e02273a5aecf9","tex":"\\operatorname{Ind}","aliases":"Ind operatorname","display":"\\operatorname{Ind}","kind":"use","excerpt":"[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-26f8de888a6f6bae","tex":"\\operatorname{Ind}_H^G","aliases":"Ind_H^G operatorname","display":"\\operatorname{Ind}_H^G","kind":"use","excerpt":"[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1c28aae22777e7c2","tex":"\\operatorname{Ind}_{K\\cap xHx^{-1}}^K","aliases":"Ind_K∩ xHx^-1^K operatorname cap","display":"\\operatorname{Ind}_{K\\cap xHx^{-1}}^K","kind":"use","excerpt":"[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-21289cd0bdc8c49b","tex":"\\mathbf{Inv}","aliases":"Inv mathbf","display":"\\mathbf{Inv}","kind":"use","excerpt":"[Invariant descent and global completeness on the periodic atlas] The native invariants in the recurrence, rational, digital, and mask charts admit coherent comparison maps to J_p. These maps descend to J_p: Desc( R_p^per) ⟶ Inv_p^per. The descended invariant is complete on the declared semantic category.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ff95be24237f2029","tex":"\\mathbf{Inv}_p^{\\mathrm{per}}","aliases":"Inv_p^per mathbf mathrm","display":"\\mathbf{Inv}_p^{\\mathrm{per}}","kind":"use","excerpt":"[Invariant descent and global completeness on the periodic atlas] The native invariants in the recurrence, rational, digital, and mask charts admit coherent comparison maps to J_p. These maps descend to J_p: Desc( R_p^per) ⟶ Inv_p^per. The descended invariant is complete on the declared semantic category.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-597f9096bfa5543e","tex":"\\operatorname{Irr}","aliases":"Irr operatorname","display":"\\operatorname{Irr}","kind":"use","excerpt":"[Typed semisimple Galois realization] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0ac1e2c7bfcd4e51","tex":"\\operatorname{Irr}_K","aliases":"Irr_K operatorname","display":"\\operatorname{Irr}_K","kind":"use","excerpt":"[Typed semisimple Galois realization] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-485db41ee683b54c","tex":"J_0","aliases":"J_0","display":"J_0","kind":"use","excerpt":"The normalized jet is J_0(F) = (F(0),F'(0),…,F^(r-1)(0)).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0e97a80aa927d24d","tex":"j_1","aliases":"j_1","display":"j_1","kind":"use","excerpt":"For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b0b448937a1e36f2","tex":"\\mathcal J_{13}","aliases":"J_13 mathcal","display":"\\mathcal J_{13}","kind":"use","excerpt":"Let Λ_13 = {DRLM,Loc,Fuss,EM,Gal} be the component-tag set. Define J_13(X) = (λ(X), J_λ(X)(X)).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e268ef5e9ac6d864","tex":"J_{ab}","aliases":"J_ab","display":"J_{ab}","kind":"use","excerpt":"For every chart C_λ,a, let I_λ,a: C_λ,a⟶ V_λ,a be its native normalized invariant. A bridge comparison is a natural isomorphism κ_ab: J_abI_λ,a Longrightarrow I_λ,bΦ_ab, where J_ab transports invariant coordinates between their chart-specific target categories.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-87cd152cd06e351a","tex":"\\mathcal J_{\\mathrm{cert}}","aliases":"J_cert mathcal mathrm","display":"\\mathcal J_{\\mathrm{cert}}","kind":"use","excerpt":"whose constructor support lies in the common verified preservation profile transports to compatible local theorems and hence to the corresponding global resolved object. item The semantic normalizers induce the comparison 2-cells on all nonempty triple overlaps and force the quadruple-overlap coherence equation. item The compatible native invariant","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-37d9c00a511f4d36","tex":"\\mathcal J_{\\mathrm{DG}}","aliases":"J_DG mathcal mathrm","display":"\\mathcal J_{\\mathrm{DG}}","kind":"use","excerpt":"[Complete descended prime-power invariant] The prime-power package J_DG(b) = ( b(p^e), c(p^e) )_p prime, e≥0 subject to p^e b(p^e)=c(p^e)-c(p^e-1) (e≥1) is complete on EulMult_ Z. It reconstructs b, c, the Euler product A_b, and the coefficient recurrence of A_b.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ed585fecc1799410","tex":"\\mathcal J_{\\mathrm{DRLM},p}","aliases":"J_DRLM,p mathcal mathrm","display":"\\mathcal J_{\\mathrm{DRLM},p}","kind":"use","excerpt":"For a∈ O_DRLM, define J_DRLM,p(a) = ( SL_p, F_p(a), P_rm(a), P_per(a) ), where SL_p, F_p(a) is the strict shortlex record, P_rm(a) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and P_per(a) is empty off the purely periodic locus and otherwise co","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6a41a228032513c9","tex":"\\mathcal J_{\\mathrm{EM}}","aliases":"J_EM mathcal mathrm","display":"\\mathcal J_{\\mathrm{EM}}","kind":"use","excerpt":"[Euler--multiplicative invariant comparison] The exponent sequence J_EM(A_b)=b is a complete invariant of the certified Euler--multiplicative component. In every chart it is recovered by the same M obius formula b_n = frac1n ∑_dmid nμ(n/d)c_d, and Dirichlet convolution corresponds to exponent addition.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-080579346daa8b62","tex":"\\mathcal J_{\\mathrm{Fuss}}","aliases":"J_Fuss mathcal mathrm","display":"\\mathcal J_{\\mathrm{Fuss}}","kind":"use","excerpt":"Define J_Fuss(q,α) = ( q, α, P_q,α, {c_n(q,α)}_n≥0, {g_N}_N≥1, {fracq-1g_N}_N≥1 ).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8b02946d61d30bf1","tex":"\\mathcal J_{\\mathrm{Gal}}","aliases":"J_Gal mathcal mathrm","display":"\\mathcal J_{\\mathrm{Gal}}","kind":"use","excerpt":"Fix a finite Galois extension L/K in the semisimple setting. For each subgroup H≤ G, define J_Gal(H) = ( H, L^H, P_H, Core_G(H), Ncl_G(H), {Jbackslash G/H}_J≤ G ).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-379f4e9dc1d9aae3","tex":"\\mathcal J_{L/K}","aliases":"J_L/K mathcal","display":"\\mathcal J_{L/K}","kind":"use","excerpt":"A complete chart record on the finite subgroup locus is J_L/K(H) = ( H, L^H, P_H, Core_G(H), Ncl_G(H), {Kbackslash G/H}_K≤ G ). The first three entries already classify the normalized object, while the remaining entries record its atlas position and transition structure. For a framed semilinear vector object, the descended K-space with its frame is","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-33badb8524213b92","tex":"\\mathcal J_{\\mathrm{Loc},p}","aliases":"J_Loc,p mathcal mathrm","display":"\\mathcal J_{\\mathrm{Loc},p}","kind":"use","excerpt":"[Continuous local invariant comparison] The Mahler coefficient sequence J_Loc,p(f) = (c_r(f))_r≥0 is a complete invariant of A_Loc,p. Complete branch packets, branchwise Mahler coordinates, affine coefficient kernels, and nested refinements admit coherent comparison maps to J_Loc,p.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-416b4c70e8031735","tex":"j_m","aliases":"j_m","display":"j_m","kind":"use","excerpt":"For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a11da8ada0dbbc8c","tex":"J_{N,j}","aliases":"J_N,j","display":"J_{N,j}","kind":"use","excerpt":"[Nonautonomous state jump] Suppose a P-recursive sequence is represented, beyond its singular prefix, by v_n+1=C(n)v_n, a_n=λ(n)v_n, where v_n∈ K^s and C(n) is rational. Then the jth N-section has the jump system w_m+1=J_N,j(m)w_m, where J_N,j(m) = C(Nm+j+N-1)⋯ C(Nm+j). After denominator clearing, every scalar output has a polynomial recurrence of ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a9a426706cf5b33c","tex":"\\mathcal J_p","aliases":"J_p mathcal","display":"\\mathcal J_p","kind":"use","excerpt":"For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a),","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-abb3222d614ef4e9","tex":"\\mathcal J_p^{\\mathrm{rm}}","aliases":"J_p^rm mathcal mathrm","display":"\\mathcal J_p^{\\mathrm{rm}}","kind":"use","excerpt":"For F∈RMah_p^×, define J_p^rm(F) = ( R_F, ω_p-1(R_F), D(R_F), V_p(F), r_p(F), K_p(F), κ_p(F), A_p(F) ).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e777d061fa580170","tex":"\\mathcal J_{\\rm BF}","aliases":"J_rm BF mathcal rm","display":"\\mathcal J_{\\rm BF}","kind":"use","excerpt":"[Effective marked binomial--Fuchsian atlas] The five normalized presentations above determine one another on the declared marked family. With equality morphisms they form equivalent discrete groupoids, every normalized fiber is contractible, and objects and morphisms descend effectively. The package J_rm BF = ( α,λ,N,j, R_N,j,L_N,j, A_N,j(λ),B_N,j,","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8999a195007514c5","tex":"j_t","aliases":"j_t","display":"j_t","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cf9ed6374d3b04fe","tex":"j_{\\kappa(e')}","aliases":"j_κ(e') kappa","display":"j_{\\kappa(e')}","kind":"use","excerpt":"Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f7a80ea206ddc084","tex":"j_{\\kappa(e)}","aliases":"j_κ(e) kappa","display":"j_{\\kappa(e)}","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7ac9ec4bf56192f2","tex":"\\mathcal J_{\\lambda(X)}","aliases":"J_λ(X) mathcal lambda","display":"\\mathcal J_{\\lambda(X)}","kind":"use","excerpt":"Let Λ_13 = {DRLM,Loc,Fuss,EM,Gal} be the component-tag set. Define J_13(X) = (λ(X), J_λ(X)(X)).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a0a635a73cee305","tex":"k'","aliases":"k'","display":"k'","kind":"use","excerpt":"The (k,i)-entry of B_ QB_ P is the displayed cardinality. If (k',i')=(hk,hi), then j↦ hj is a bijection between the two sets being counted. The entry is therefore constant on R, which gives the orbit-basis expansion.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f98449f8a9e4db73","tex":"K^{\\mathbb N}","aliases":"K^ N mathbb","display":"K^{\\mathbb N}","kind":"use","excerpt":"For a sequence a=(a_n)_n≥0∈ K^ N, define the digit-section operators (Λ_d a)(n)=a(kn+d), 0≤ d<k. For a word w=d_0⋯ d_ℓ-1, put Λ_w = Λ_d_ℓ-1⋯Λ_d_1Λ_d_0. Then (Λ_w a)(n) = a(k^|w|n+[w]_k).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-458b8bf15ea1cd1a","tex":"K^{(E)}","aliases":"K^(E)","display":"K^{(E)}","kind":"use","excerpt":"Let E⊆ K^M be the set of distinct residual future functions of a scalar series or observational system. Set minimization retains the elements of E as distinct states. Free linearization produces the vector space K^(E), while linear minimization retains only their span.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4b779dcac03cb4e5","tex":"K^2","aliases":"K^2","display":"K^2","kind":"use","excerpt":"[Endpoint selection does not implement refinement] Let K be a field and let X=K^2. At the coarse resolution use the single projector p=1_X. At the fine resolution use p_1= 1 · 0","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-636078349a523738","tex":"k^a","aliases":"k^a","display":"k^a","kind":"use","excerpt":"it-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-243541f20670459d","tex":"k^{a-1}","aliases":"k^a-1","display":"k^{a-1}","kind":"use","excerpt":"it-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_0+kd_1+⋯+k^a-1d_a-1. Concatenation gives strict radix refinement and the lane path ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-25e27fdec946575c","tex":"K^d","aliases":"K^d","display":"K^d","kind":"use","excerpt":"An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ba8794eccc724e55","tex":"k^j","aliases":"k^j","display":"k^j","kind":"use","excerpt":"Fix an integer radix k≥2 and a field K. Let Σ_k={0,1,…,k-1}. Words are read from least significant digit to most significant digit. For w=d_0d_1⋯ d_ℓ-1∈Σ_k^*, define |w|=ℓ, [w]_k=∑_j=0^ℓ-1d_jk^j. The empty word ε has length zero and value zero.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-de52ccc7804b5bed","tex":"K^M","aliases":"K^M","display":"K^M","kind":"use","excerpt":"Let E⊆ K^M be the set of distinct residual future functions of a scalar series or observational system. Set minimization retains the elements of E as distinct states. Free linearization produces the vector space K^(E), while linear minimization retains only their span.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2cee7e10b8e55a15","tex":"K^N","aliases":"K^N","display":"K^N","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4d2386e0af951739","tex":"K^r","aliases":"K^r","display":"K^r","kind":"use","excerpt":"[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b954b6d761a57d12","tex":"K^s","aliases":"K^s","display":"K^s","kind":"use","excerpt":"[Nonautonomous state jump] Suppose a P-recursive sequence is represented, beyond its singular prefix, by v_n+1=C(n)v_n, a_n=λ(n)v_n, where v_n∈ K^s and C(n) is rational. Then the jth N-section has the jump system w_m+1=J_N,j(m)w_m, where J_N,j(m) = C(Nm+j+N-1)⋯ C(Nm+j). After denominator clearing, every scalar output has a polynomial recurrence of ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-44467e65a69f18a5","tex":"k^{|w|}","aliases":"k^|w|","display":"k^{|w|}","kind":"use","excerpt":"For a sequence a=(a_n)_n≥0∈ K^ N, define the digit-section operators (Λ_d a)(n)=a(kn+d), 0≤ d<k. For a word w=d_0⋯ d_ℓ-1, put Λ_w = Λ_d_ℓ-1⋯Λ_d_1Λ_d_0. Then (Λ_w a)(n) = a(k^|w|n+[w]_k).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ac62d9119015a5fb","tex":"K^\\times","aliases":"K^× times","display":"K^\\times","kind":"use","excerpt":"Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e36d52787f217bd","tex":"k^\\ell","aliases":"k^ℓ ell","display":"k^\\ell","kind":"use","excerpt":"Choose ℓ with k^ℓ>m and consider the m+1 sections P(k^ℓ n+r), 0≤ r≤ m. When expanded in the basis 1,n,…,n^m, their coefficient functions as functions of r have successive degrees m,m-1,…,0, with nonzero leading coefficients. Evaluation at the distinct points 0,1,…,m therefore gives an invertible generalized Vandermonde matrix. These sections are li","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"kappa(e)=tτ_e(b_e) ∈ C_t, and put supp(overline b) :={t∈ T:overline b_t≠0}, K_0:={t∈Γ:overline b_t=0}. Then |S_0|-|supp(overline b)| = |A_0| +(|S_1|-|Γ|) +|K_0|. The three terms record, respectively, annihilation by the local maps τ_e, collision of surviving lane addresses under κ, and cancellation among folded values at a common address. If every ","count":14},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8193613aedbeb13a","tex":"\\mathcal K_{k,p^{r+1}}^{\\mathrm{sem}}","aliases":"K_k,p^r+1^sem mathcal mathrm","display":"\\mathcal K_{k,p^{r+1}}^{\\mathrm{sem}}","kind":"use","excerpt":"ower-automata For every r≥1, the semantic p^r-kernel is finite and is the state set of the unique minimal least-significant-digit-first deterministic output automaton for the coefficient sequence abmod p^r. Moreover, κ_k,p^r^sem(a) ≤ |overline M_k,p^r(a)| = p^∑_i(r-min(r,ν_i)). Reduction induces surjective digit-equivariant maps K_k,p^r+1^sem(a) ↠ ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e9a37605986fa4ad","tex":"\\mathcal K_p","aliases":"K_p mathcal","display":"\\mathcal K_p","kind":"use","excerpt":"For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b ","count":12},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3a2bb79463de260b","tex":"K_U","aliases":"K_U","display":"K_U","kind":"use","excerpt":"quare remains a projector-sum refinement square in the descent category; item every fold whose source objects, target objects, and structure maps are compatible descent data defines a fold in Desc( R). Compatible selectors and left inverses descend, so a split-losslessness certificate descends directly from its left-inverse identity. Moreover, supp","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d4a5084779c12538","tex":"k_U","aliases":"k_U","display":"k_U","kind":"use","excerpt":"For (H3), the local path law gives the displayed contraction formula in every chart. Applying γ_* to a contracted block distributes over the finite sum and carries each product to the product of the transported blocks. Since λ^Y_γ,u is a bijection, reindexing the intermediate address shows γ_*(k_U⋆ h_U) = (γ_*k_U)⋆(γ_*h_U) = k_V⋆ h_V. The identity ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c4856b4c3a37fedf","tex":"k_{U,ca}","aliases":"k_U,ca","display":"k_{U,ca}","kind":"use","excerpt":"item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precis","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5dd4d8b7ad7c3feb","tex":"K_V","aliases":"K_V","display":"K_V","kind":"use","excerpt":"et objects, and structure maps are compatible descent data defines a fold in Desc( R). Compatible selectors and left inverses descend, so a split-losslessness certificate descends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f4315e66d6b9cbcb","tex":"k_V","aliases":"k_V","display":"k_V","kind":"use","excerpt":"For (H3), the local path law gives the displayed contraction formula in every chart. Applying γ_* to a contracted block distributes over the finite sum and carries each product to the product of the transported blocks. Since λ^Y_γ,u is a bijection, reindexing the intermediate address shows γ_*(k_U⋆ h_U) = (γ_*k_U)⋆(γ_*h_U) = k_V⋆ h_V. The identity ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-109c579b6e568899","tex":"\\ker_{\\mathfrak R(U)}","aliases":"ker_ R(U) ker mathfrak","display":"\\ker_{\\mathfrak R(U)}","kind":"use","excerpt":"cends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent comparison isomorphisms η_UV:F_UV(K_U)xrightarrow∼K_V compatible with the kernel inclusions and satisfying the triple-overlap cocycle. Then the global kernel exists and is co","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-131957a290a4156f","tex":"\\ker_{\\Desc(\\mathfrak R)}","aliases":"ker_Desc( R) ker Desc mathfrak","display":null,"kind":"use","excerpt":"cends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent comparison isomorphisms η_UV:F_UV(K_U)xrightarrow∼K_V compatible with the kernel inclusions and satisfying the triple-overlap cocycle. Then the global kernel exists and is co","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-832e1fdffb23b3fd","tex":"\\kerpair","aliases":"kerpair","display":null,"kind":"use","excerpt":"Let K_0:=⋂_jkerpair(o_j) in the relational setting and N_0:=⋂_jker(o_j) in the linear setting.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-97491b80a0eec919","tex":"\\kk^2","aliases":"kk^2 kk kk^2 𝕜^2 kk Bbbk","display":null,"kind":"use","excerpt":"0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3fb867e1e5112d49","tex":"L^{\\operatorname{Core}_G(H)}","aliases":"L^Core_G(H) operatorname","display":"L^{\\operatorname{Core}_G(H)}","kind":"use","excerpt":"[Canonical envelopes of a nonnormal Galois atlas] Let M=L^H. Then L^Core_G(H) =∏_g∈ Gg(M) tag12.6 is the smallest Galois extension inside L containing every conjugate chart, while L^Ncl_G(H) =⋂_g∈ Gg(M) tag12.7 is the largest Galois subfield contained in every chart. The core and normal-closure projectors are the two canonical global envelopes of t","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c85018018cbe41e1","tex":"L^{\\operatorname{Gal}(L/E)}","aliases":"L^Gal(L/E) operatorname","display":"L^{\\operatorname{Gal}(L/E)}","kind":"use","excerpt":"The finite Galois correspondence gives H=Gal(L/L^H) and E=L^Gal(L/E). The range of P_H is L^H after faithful scalar extension, and the support of e_H recovers H. All three assignments are therefore mutually inverse on the declared charts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8359725b04f5ac32","tex":"L^H","aliases":"L^H","display":"L^H","kind":"use","excerpt":"[Normalized finite Galois chart] Fix the finite Galois extension L/K. The following data determine one another exactly: H longleftrightarrow L^H longleftrightarrow P_H tag12.3 where the projector is taken in a faithful field or regular realization. The correspondence is order reversing: H≤ J Longleftrightarrow L^H⊇ L^J Longleftrightarrow P_H≥ P_J. ","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9e7e8bdc908597d7","tex":"L^J","aliases":"L^J","display":"L^J","kind":"use","excerpt":"intheorem[Normalized finite Galois chart] Fix the finite Galois extension L/K. The following data determine one another exactly: H longleftrightarrow L^H longleftrightarrow P_H tag12.3 where the projector is taken in a faithful field or regular realization. The correspondence is order reversing: H≤ J Longleftrightarrow L^H⊇ L^J Longleftrightarrow P","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-478acc5f340c0573","tex":"L^n","aliases":"L^n","display":"L^n","kind":"use","excerpt":"[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of th","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0be6f2453d9d6dec","tex":"L^{\\operatorname{Ncl}_G(H)}","aliases":"L^Ncl_G(H) operatorname","display":"L^{\\operatorname{Ncl}_G(H)}","kind":"use","excerpt":"[Canonical envelopes of a nonnormal Galois atlas] Let M=L^H. Then L^Core_G(H) =∏_g∈ Gg(M) tag12.6 is the smallest Galois extension inside L containing every conjugate chart, while L^Ncl_G(H) =⋂_g∈ Gg(M) tag12.7 is the largest Galois subfield contained in every chart. The core and normal-closure projectors are the two canonical global envelopes of t","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-98a2affcd1da416b","tex":"L^\\times","aliases":"L^× times","display":"L^\\times","kind":"use","excerpt":"[Brauer obstruction to projective packet descent] Projective semilinear descent data define a class α∈ H^1(G,PGL_n(L)). The connecting map for 1⟶ L^× ⟶ GL_n(L) ⟶ PGL_n(L) ⟶1 produces δ(α) ∈ H^2(G,L^×) = Br(L/K). The projective atlas lifts to linear semilinear descent data exactly when δ(α)=0. When the class is nonzero, the descended object is natur","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3e7068b92dc3e43d","tex":"L_{\\mathbb Z}","aliases":"L_ Z mathbb","display":"L_{\\mathbb Z}","kind":"use","excerpt":"Addendum 04 proves that the prefix S_*={0,1,…,L_ Z(m,d)-1} with L_ Z(m,d)=min(d+1,B(m)), B(m)=max({1}∪{p^v_p(m):pmid m}), always admits an integer decoder and is cardinality-minimal [source reference].","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"der ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ord","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-97447770a96e00b0","tex":"L_b","aliases":"L_b","display":"L_b","kind":"use","excerpt":"[Carrier-indexed observer packet theorem] Let S be a finite continuation-state set, let (A,⊕,0) be a commutative monoid, let B be a finite branch alphabet, and let δ:S× B⟶ S be a state transition. Suppose every branch b∈ B acts by a monoid homomorphism L_b:A⟶ A. Take an empty aggregation to be 0. For a state packet a=(a_s)_s∈ S, define a'_t := ⊕_δ(","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2752bc7b3d7d6299","tex":"L_d","aliases":"L_d","display":"L_d","kind":"use","excerpt":"[Finite-field persistence equals pure periodicity] For a sequence over F_p, the following are equivalent. [label=textup(roman*)] item The sequence is C-finite and its minimal recurrence has nonzero constant term. item The sequence has a finite-dimensional state realization a_n=λ C^nv with C∈ GL_d( F_p). item The sequence is purely periodic from n=0","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ec6b4ea000366681","tex":"L_f","aliases":"L_f","display":"L_f","kind":"use","excerpt":"[Bridge-loss exact sequence] For every task-compatible linear bridge, there is an exact sequence 0⟶ fracf^-1(N_Y)N_X ⟶ fracXN_X xrightarrowoverline f fracYN_Y ⟶ fracYN_Y+f(X) ⟶0. Thus L_f:=fracf^-1(N_Y)N_X measures source distinctions collapsed by the bridge, while E_f:=fracYN_Y+f(X) measures target semantic states outside the essential image.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9ac358794339c560","tex":"L_G","aliases":"L_G","display":"L_G","kind":"use","excerpt":"ther with its finite initial record; item the monic minimal differential operator from [source reference], together with the distinguished power-series solution selected by the marked coefficient record; item the hypergeometric parameter packet (A_N,j(λ),B_N,j;α^N) and leading coefficient; item the primitive minimal-operator singular and indicial r","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ff93f51daddcc717","tex":"L_g","aliases":"L_g","display":"L_g","kind":"use","excerpt":"Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0971e95674fa209c","tex":"L_{gf}","aliases":"L_gf","display":"L_{gf}","kind":"use","excerpt":"Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9aed76a21185e438","tex":"L_{k,p}","aliases":"L_k,p","display":"L_{k,p}","kind":"use","excerpt":"Define M_k,p(a)=M_k(a)⊗_ Z Z_p, L_k,p(a)=L_k(a)⊗_ Z Z_p.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c91dabfc8b700b3b","tex":"L_N","aliases":"L_N","display":"L_N","kind":"use","excerpt":"For a fixed modulus N≥2, set B_N(M) = max({1}∪ {p^v_p(M):pmidgcd(M,N)}). The least packet size for a left inverse over Z/N Z is L_N(M,d)=min(d+1,B_N(M)). The exponents v_p(N) do not change the minimum; the first residue layer already detects the obstruction.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ed76d6b3d182baf7","tex":"L_n","aliases":"L_n","display":"L_n","kind":"use","excerpt":"[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of th","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f521d2e4e9296672","tex":"L_{N,j}","aliases":"L_N,j","display":"L_{N,j}","kind":"use","excerpt":"Let Q^rm cl_N,j be an annihilator produced by a declared closure or elimination pipeline, and let L_N,j be the monic minimal operator of the section. Since every annihilator belongs to the principal left ideal generated by L_N,j, Q^rm cl_N,j=A_N,jL_N,j for some Ore operator A_N,j. The nonnegative quantity δ^rm cl_N,j = ordQ^rm cl_N,j - ordL_N,j mea","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ebf15945a4467eee","tex":"L_{N,j}^{\\rm char}","aliases":"L_N,j^rm char rm","display":"L_{N,j}^{\\rm char}","kind":"use","excerpt":"qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ordL_N^rm orb, and, when G_N,j≠0, let L_N,j^rm char be its monic minimal annihilator with order r_N,j^rm char. For a zero char","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2c4fa1a2b4dd2b87","tex":"L_{N-1}","aliases":"L_N-1","display":"L_{N-1}","kind":"use","excerpt":"der ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ord","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e229b7d53b307192","tex":"L_N^{\\rm orb}","aliases":"L_N^rm orb rm","display":"L_N^{\\rm orb}","kind":"use","excerpt":"der ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ord","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ed5212305c5c9989","tex":"L_q","aliases":"L_q","display":"L_q","kind":"use","excerpt":"The root filter also gives a construction-independent order ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d8ade83d2dcc64e3","tex":"L_w","aliases":"L_w","display":"L_w","kind":"use","excerpt":"For uniqueness, let (V,(L_d),v,λ) be another reachable-observable realization of the same digit function. Send the reachable vector L_wv to Λ_wF. If a linear combination of reachable vectors vanishes, applying every continuation and then λ shows that the corresponding combination of Cartier sections has every coefficient zero. Observability makes t","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"Let R(z)=fracA(z)B(z), A(0)=B(0)=1, with a=deg A, b=deg B, c=a+(p-2)b, and define D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source refer","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-305626df8ca384d4","tex":"\\operatorname{LCLM}","aliases":"LCLM operatorname","display":"\\operatorname{LCLM}","kind":"use","excerpt":"der ledger before the algebraic descent from t to x=t^N. Work over a finite extension E/K containing ω_N. Put F_q(t)=F(ω_N^qt), G_N,j(t)=frac1N∑_q=0^N-1ω_N^-jqF_q(t). Let L_q∈ E(t)⟨ D_t⟩ be the monic minimal annihilator of F_q. Rotation is an Ore-algebra automorphism, so every L_q has order r=ordL_F. Let L_N^rm orb=LCLM(L_0,…,L_N-1), r_N^rm orb=ord","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":"If deg P≤ D, then deg(PH) ≤ D+a+(p-2)b. Cartier selection divides the retained degrees by p. The definition of D gives deg T_d(P) ≤ lfloorfracD+a+(p-2)bprfloor ≤ D. This proves (i).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a60a3fa2f483385b","tex":"\\mathbf{Lin}","aliases":"Lin mathbf","display":"\\mathbf{Lin}","kind":"use","excerpt":"[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9aff2b472a24ffcd","tex":"\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}","aliases":"Lin_k,K^sl mathbf mathrm","display":"\\mathbf{Lin}_{k,K}^{\\mathrm{sl}}","kind":"use","excerpt":"[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-548fbea474a2c12e","tex":"\\linrank","aliases":"linrank operatorname","display":"{\\operatorname{linrank}}","kind":"use","excerpt":"For a persistent C-finite sequence, the minimal dimension ρ(a)=linrank(a)=rank H(a) is simultaneously the minimal recurrence order, the reduced denominator degree, and the dimension of a minimal reachable-observable state.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bba1a392427b2f31","tex":"\\mathbf{Loc}","aliases":"Loc mathbf","display":"\\mathbf{Loc}","kind":"use","excerpt":"Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding bran","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4900d346de512a89","tex":"\\mathbf{Loc}^{\\mathrm{cts}}_p","aliases":"Loc^cts_p mathbf mathrm","display":"\\mathbf{Loc}^{\\mathrm{cts}}_p","kind":"use","excerpt":"Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding bran","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4a7e9f6a85466609","tex":"\\longleftrightarrow","aliases":"longleftrightarrow","display":null,"kind":"use","excerpt":"[Normalized finite Galois chart] Fix the finite Galois extension L/K. The following data determine one another exactly: H longleftrightarrow L^H longleftrightarrow P_H tag12.3 where the projector is taken in a faithful field or regular realization. The correspondence is order reversing: H≤ J Longleftrightarrow L^H⊇ L^J Longleftrightarrow P_H≥ P_J. ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-25d467671d09a670","tex":"\\lVert","aliases":"lVert","display":"\\lVert","kind":"use","excerpt":"Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f0042c6cf9ee0b72","tex":"\\mathfrak M","aliases":"M mathfrak","display":"\\mathfrak M","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-76a8c8c02bf83b7a","tex":"M'","aliases":"M'","display":"M'","kind":"use","excerpt":"[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4c3503e879b74874","tex":"\\mathcal M^{-1}","aliases":"M^-1 mathcal","display":"\\mathcal M^{-1}","kind":"use","excerpt":"s inverse. Refining from p^a to p^a+b, for every integer b≥0, gives the exact child-address and mixed-radix composition law. Let M:= M_p^-1:C( Z_p, Z_p) xrightarrow∼c_0( Z_p) be Mahler analysis from [source reference], and write R_p^a,sf(q):=f(p^aq+s). Conjugation gives the exact coefficient operator T_p^a,s := M R_p^a,s M^-1: c_0( Z_p)⟶ c_0( Z_p),","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","excerpt":"[Typed affine-channel ladder] For every M≥1, s≥0, and d≥0: item The matrix A_M,s^(d) is upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel h","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a591ad2f94f5ddd","tex":"M^d","aliases":"M^d","display":"M^d","kind":"use","excerpt":"[Typed affine-channel ladder] For every M≥1, s≥0, and d≥0: item The matrix A_M,s^(d) is upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel h","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ef20dff857eefea8","tex":"M^{d(d+1)/2}","aliases":"M^d(d+1)/2","display":"M^{d(d+1)/2}","kind":"use","excerpt":"[Typed affine-channel ladder] For every M≥1, s≥0, and d≥0: item The matrix A_M,s^(d) is upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel h","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-05e170b594a4705a","tex":"M^k","aliases":"M^k","display":"M^k","kind":"use","excerpt":"[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-84ac411db65b3fe4","tex":"M^{r_B,r_A}_{B,A}","aliases":"M^r_B,r_A_B,A","display":"M^{r_B,r_A}_{B,A}","kind":"use","excerpt":"_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, mathscr M^r_B,r_A_B,A), an","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ddae4ab24947e613","tex":"M^{v,r}_{Z,X}","aliases":"M^v,r_Z,X","display":"M^{v,r}_{Z,X}","kind":"use","excerpt":"item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precis","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b1dfff4a63d7a1ee","tex":"M^{v,u}_{Z,Y}","aliases":"M^v,u_Z,Y","display":"M^{v,u}_{Z,Y}","kind":"use","excerpt":"item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precis","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a66db8ec4f89a9e7","tex":"M_0","aliases":"M_0","display":"M_0","kind":"use","excerpt":"[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bc4c24fa42db60db","tex":"m_a","aliases":"m_a","display":"m_a","kind":"use","excerpt":"nt-periodic For a sequence over F_p, the following are equivalent. [label=textup(roman*)] item The sequence is C-finite and its minimal recurrence has nonzero constant term. item The sequence has a finite-dimensional state realization a_n=λ C^nv with C∈ GL_d( F_p). item The sequence is purely periodic from n=0. If m_a(x) is the monic minimal recurr","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-207e98afc142d0de","tex":"M_d","aliases":"M_d","display":"M_d","kind":"use","excerpt":"An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c3792dafeb7b2d2e","tex":"\\mathcal M_d","aliases":"M_d mathcal","display":"\\mathcal M_d","kind":"use","excerpt":"The same integer kernel matrix has different categorical status after changing the coefficient doctrine. Fix d≥0 and write M_d(R)=R^d+1 for the formal truncated Mahler coefficient module. Let A_M,s^(d) =[Θ_M,s(r,k)]_0≤ k,r≤ d, where the target degree k is the row index.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1eb8b7d5ce216250","tex":"M_{k-1}","aliases":"M_k-1","display":"M_{k-1}","kind":"use","excerpt":"[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0901f1f56ba2bc5e","tex":"M_N","aliases":"M_N","display":"M_N","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"use","excerpt":"A useful constructive form is obtained by choosing any matrix C∈ M_n(L) and setting T(C)=∑_σ∈ Ga_σσ(C). Then a_ττ(T(C))=T(C). Whenever T(C) is invertible, B=T(C)^-1 is a Hilbert 90 trivialization. Standard descent guarantees that such a C exists.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ba92d7175bacfc4","tex":"\\mathfrak M_{N,j}","aliases":"M_N,j mathfrak","display":"\\mathfrak M_{N,j}","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6357ecbc78f374e8","tex":"M_N^i","aliases":"M_N^i","display":"M_N^i","kind":"use","excerpt":"Factor H=OC, where row i of O is e_0^ TM_N^i, and column j of C is M_N^j e_0. For j<N, the binomial expansion of (I+S_z)^j e_0 is a unit upper-triangular change from (e_0,…,e_N-1), hence det C=1. The rows e_0^ TS_z^k, 0≤ k<N, are e_0^ T followed by z e_N-k^ T, so their determinant is ± z^N-1. The binomial row change is also unit triangular, proving","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5d216fafea89baa9","tex":"M_N^j","aliases":"M_N^j","display":"M_N^j","kind":"use","excerpt":"Factor H=OC, where row i of O is e_0^ TM_N^i, and column j of C is M_N^j e_0. For j<N, the binomial expansion of (I+S_z)^j e_0 is a unit upper-triangular change from (e_0,…,e_N-1), hence det C=1. The rows e_0^ TS_z^k, 0≤ k<N, are e_0^ T followed by z e_N-k^ T, so their determinant is ± z^N-1. The binomial row change is also unit triangular, proving","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8ca1a211281a4b30","tex":"M_N^n","aliases":"M_N^n","display":"M_N^n","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-68a06b745f2e5895","tex":"\\mathcal M_p^{-1}","aliases":"M_p^-1 mathcal","display":"\\mathcal M_p^{-1}","kind":"use","excerpt":"). For f∈ C( Z_p, Z_p), with q∈ Z_p the branch coordinate, define Analysis_p^a(f) := (q↦ f(p^aq+s))_0≤ s<p^a. Piecewise synthesis on the disjoint clopen cosets s+p^a Z_p is its inverse. Refining from p^a to p^a+b, for every integer b≥0, gives the exact child-address and mixed-radix composition law. Let M:= M_p^-1:C( Z_p, Z_p) xrightarrow∼c_0( Z_p) ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-03e0d5903db86e3c","tex":"\\mathfrak M_r","aliases":"M_r mathfrak","display":"\\mathfrak M_r","kind":"use","excerpt":"[Fixed-resolution matrix-category equivalence] The functor M_r: L_ R⟶ Mat_r( L_ R), X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq s","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-19ddedb9073634ad","tex":"\\mathfrak M_s","aliases":"M_s mathfrak","display":"\\mathfrak M_s","kind":"use","excerpt":"L_ R⟶ Mat_r( L_ R), X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq ssucceq r, they also satisfy Γ_tr=Γ_srΓ_ts. Thus the equivalences","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e3e697d6b6f9bc2b","tex":"M_w","aliases":"M_w","display":"M_w","kind":"use","excerpt":"[Strict fixed-radix normal form] The record SL_k,K(a) is uniquely determined by a and reconstructs a exactly through a_[w]_k = ℓ(a)M_w(a)v(a). It has dimension r_k(a). Two regular sequences have the same strict shortlex record if and only if they are equal.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ab3a6975ed9c0370","tex":"M_Y","aliases":"M_Y","display":"M_Y","kind":"use","excerpt":"[Two-sort transfer grammar] Take two sorts X,Y, one arrow a:X→ Y, and no nontrivial relation. A realization is a single typed map M_XxrightarrowM_aM_Y. After collapsing to M_X⊕ M_Y, the vertex idempotents are coordinate packets and M_a is one off-diagonal transfer block. The smallest typed grammar therefore already contains the distinction between ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-134118eb61fe8c84","tex":"\\mathbf{Mah}","aliases":"Mah mathbf","display":"\\mathbf{Mah}","kind":"use","excerpt":"Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding bran","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a069692ea27383be","tex":"\\mathbf{Mah}^{c_0}_p","aliases":"Mah^c_0_p mathbf","display":"\\mathbf{Mah}^{c_0}_p","kind":"use","excerpt":"Normalized local atlas. For every fixed prime p and depth a, the following normalized charts are equivalent: Loc^cts_p ≃ Mah^c_0_p ≃ Branch^cts_p,a ≃ BranchMah^c_0_p,a. The first chart contains continuous functions on Z_p, the second their unique Mahler coefficients, the third the complete clopen branch packet, and the fourth the corresponding bran","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-66f758abe439c84e","tex":"\\mathrm{mark}","aliases":"mark mathrm","display":"\\mathrm{mark}","kind":"use","excerpt":"The entries E_per, E_rm, and E_mark refer, respectively, to the unit-periodic overlap, the rational functional-Mahler overlap, and the marked polynomial fixed-point overlap. In particular, the functional-Mahler and Newton--Mahler languages remain separate outside their explicitly declared interfaces, and faithful corner realization remains separate","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1b4690bcefadabd7","tex":"\\Mask","aliases":"Mask mathbf","display":"{\\mathbf{Mask}}","kind":"use","excerpt":"Define Mask_p^×,per to be the groupoid of such reduced rational masks whose normalized product solution [source reference] has purely periodic coefficients. Morphisms preserve the reduced mask.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-91aae676b6d463f1","tex":"\\Mask_2^{\\times,\\mathrm{per}}","aliases":"Mask_2^×,per Mask times mathrm mathbf times mathrm","display":"{\\mathbf{Mask}}_2^{\\times,\\mathrm{per}}","kind":"use","excerpt":"Negative control: a nonunit periodic sequence. Over F_2, the word 01 gives F(z)=fracz1+z^2. Then fracF(z)F(z^2) = frac1+z^2z, which is Laurent rather than regular at the origin. The sequence remains purely periodic, C-finite, and automatic, but it is not an object of Mask_2^×,per. This verifies that the unit overlap condition is structural rather t","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c0aa1d61d3611c6c","tex":"\\Mask_p^{\\mathrm{rat}}","aliases":"Mask_p^rat Mask mathrm mathbf mathrm","display":"{\\mathbf{Mask}}_p^{\\mathrm{rat}}","kind":"use","excerpt":"Reduced mask chart. An object of Mask_p^rat is a reduced rational function R(z)=fracA(z)B(z), A(0)=B(0)=1. Morphisms preserve the rational function. The semantics is R↦ M_p[R]. The inverse chart map is the canonical ratio F↦ F/F(z^p).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4f9e1b9e842d1e11","tex":"\\Mask_p^{\\times,\\mathrm{per}}","aliases":"Mask_p^×,per Mask times mathrm mathbf times mathrm","display":"{\\mathbf{Mask}}_p^{\\times,\\mathrm{per}}","kind":"use","excerpt":"Define Mask_p^×,per to be the groupoid of such reduced rational masks whose normalized product solution [source reference] has purely periodic coefficients. Morphisms preserve the reduced mask.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-384202a4833e8e0b","tex":"\\operatorname{Mat}_d","aliases":"Mat_d operatorname","display":"\\operatorname{Mat}_d","kind":"use","excerpt":"[Typed holonomic realization] Let K be a characteristic-zero field and let the K-valued sequence (u_n) satisfy a linear recurrence with polynomial coefficients. Beyond a finite singular prefix, write the recurrence as a finite-dimensional rational nonautonomous state model v_n+1=A(n)v_n, A(n)∈Mat_d(K(n)). For a fixed integer N≥1 and residue 0≤ j<N,","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5e113fc6ae0af9bc","tex":"\\max_{\\delta(s,b)=t}","aliases":"max_δ(s,b)=t max delta","display":"\\max_{\\delta(s,b)=t}","kind":"use","excerpt":"[Observer-conditioned convex packet transport] For every state t∈ S and direction x, h_C'_t(x) = max_δ(s,b)=th_C_s(A_bx). Repeated updates compose exactly with branch words. Within each fixed observer state, replacing a finite atom family by any other finite generating family of the same convex hull preserves all future value queries generated by l","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bd28c569d090a3e3","tex":"\\max_{\\lambda\\in\\operatorname{spec}(H)}","aliases":"max_λ∈spec(H) max lambda in operatorname","display":"\\max_{\\lambda\\in\\operatorname{spec}(H)}","kind":"use","excerpt":"[Coordinate covariance and composition] Simultaneous coordinate transport of T,Q_s,Q_t leaves D_Q(T) invariant. For composable invertible maps, D_Q(T_2T_1) ≤ D_Q(T_2)D_Q(T_1), so Δ_Q(T_2T_1) ≤ Δ_Q(T_2)+Δ_Q(T_1). For a closed word H, D_Q(H) ≥ fracmax_λ∈spec(H)|λ| min_λ∈spec(H)|λ|.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ce0891f5c72d7611","tex":"\\mathrm{min}","aliases":"min mathrm","display":"\\mathrm{min}","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-eb53f6a47d0f4feb","tex":"\\mathrm{min,rat}","aliases":"min,rat mathrm","display":"\\mathrm{min,rat}","kind":"use","excerpt":"Minimal Cartier-module chart. An object of Cart_p^min,rat is a finite-dimensional tuple (V,(L_d)_d=0^p-1,v,λ) that is reachable and observable, whose digit coefficient function belongs to RMah_p^×. Morphisms are similarities preserving every L_d, the initial vector, and the output functional. Its normalized representative is C_p(F).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-60698b1979c563d1","tex":"\\min_j","aliases":"min_j min","display":"\\min_j","kind":"use","excerpt":"Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b0386100c5276882","tex":"\\min_{k\\leq r\\leq Mk+s}","aliases":"min_k≤ r≤ Mk+s min leq leq","display":"\\min_{k\\leq r\\leq Mk+s}","kind":"use","excerpt":"[Exact branchwise valuation ledger] For every refined coefficient, v_p(d_s,k) ≥ min_k≤ r≤ Mk+s (v_p(c_r)+v_p(Θ_M,s(r,k))). If v_p(c_r)≥ w(r) for a nondecreasing weight w, then v_p(d_s,k)≥ w(k). Thus every declared monotone Mahler precision ladder survives affine polyphase refinement.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f0ef0b34a5d22f7d","tex":"\\min_{\\lambda\\in\\operatorname{spec}(H)}","aliases":"min_λ∈spec(H) min lambda in operatorname","display":"\\min_{\\lambda\\in\\operatorname{spec}(H)}","kind":"use","excerpt":"[Coordinate covariance and composition] Simultaneous coordinate transport of T,Q_s,Q_t leaves D_Q(T) invariant. For composable invertible maps, D_Q(T_2T_1) ≤ D_Q(T_2)D_Q(T_1), so Δ_Q(T_2T_1) ≤ Δ_Q(T_2)+Δ_Q(T_1). For a closed word H, D_Q(H) ≥ fracmax_λ∈spec(H)|λ| min_λ∈spec(H)|λ|.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f7ed41372efc24c8","tex":"\\mathrm{mod}","aliases":"mod mathrm","display":"\\mathrm{mod}","kind":"use","excerpt":"[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the sou","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e1762df34a33ed0b","tex":"\\Mod_{\\mathrm{ak}}","aliases":"Mod_ak Mod mathrm operatorname mathrm","display":"{\\operatorname{Mod}}_{\\mathrm{ak}}","kind":"use","excerpt":"gintheorem[Finite additive-Karoubi resolved envelope] The category Syn_ak( K_rm lin) := Kar (Add(𝕜 Q/I)) is small, additive, idempotent complete, and 𝕜-linear. For every additive idempotent-complete 𝕜-linear category C, restriction to the generating quiver induces an equivalence Fun_𝕜^⊕ (Syn_ak( K_rm lin), C) ≃ Mod_ak( K_rm lin, C), where the right","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-101739049ac8fb45","tex":"\\operatorname{MorKer}_{CB}","aliases":"MorKer_CB operatorname","display":"\\operatorname{MorKer}_{CB}","kind":"use","excerpt":"[Composition law for bridge loss] Let UxrightarrowB VxrightarrowC W be certified bridges. [label=textup(roman*)] item Every constructor or universal property preserved by both B and C is preserved by CB. item MorKer_B(X,Y) ⊆ MorKer_CB(X,Y). item Coll_B ⊆ Coll_CB. item EssIm(CB) ⊆ EssIm(C). item If both bridges are conservative, then their composite","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2aa34ef1611fca9c","tex":"\\mathbf n","aliases":"n mathbf","display":"\\mathbf n","kind":"use","excerpt":"[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6c132ef0f02f8d2c","tex":"N^{(0)}","aliases":"N^(0)","display":"N^{(0)}","kind":"use","excerpt":"[Finite certification] For finite X, descending partition refinement stabilizes after finitely many strict refinements. For finite-dimensional X over a field, the chain N^(0)=N_0, N^(r+1)= N_0∩⋂_a∈ Aρ(a)^-1(N^(r)) stabilizes after at most dim X strict dimension drops. At stabilization it equals N_ S.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dedab168d428b9f4","tex":"N^{(r)}","aliases":"N^(r)","display":"N^{(r)}","kind":"use","excerpt":"[Finite certification] For finite X, descending partition refinement stabilizes after finitely many strict refinements. For finite-dimensional X over a field, the chain N^(0)=N_0, N^(r+1)= N_0∩⋂_a∈ Aρ(a)^-1(N^(r)) stabilizes after at most dim X strict dimension drops. At stabilization it equals N_ S.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fc83f453c99df61d","tex":"N^{(r+1)}","aliases":"N^(r+1)","display":"N^{(r+1)}","kind":"use","excerpt":"[Finite certification] For finite X, descending partition refinement stabilizes after finitely many strict refinements. For finite-dimensional X over a field, the chain N^(0)=N_0, N^(r+1)= N_0∩⋂_a∈ Aρ(a)^-1(N^(r)) stabilizes after at most dim X strict dimension drops. At stabilization it equals N_ S.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"For a_n=n, one has r_k(a)=2 and exactly k^ℓ distinct sections at depth ℓ. For a_n=n^2, one has r_k(a)=3 and again exactly k^ℓ distinct sections at depth ℓ.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9751435aeea616f0","tex":"n^m","aliases":"n^m","display":"n^m","kind":"use","excerpt":"Choose ℓ with k^ℓ>m and consider the m+1 sections P(k^ℓ n+r), 0≤ r≤ m. When expanded in the basis 1,n,…,n^m, their coefficient functions as functions of r have successive degrees m,m-1,…,0, with nonzero leading coefficients. Evaluation at the distinct points 0,1,…,m therefore gives an invertible generalized Vandermonde matrix. These sections are li","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b706167f987bdce5","tex":"N^N","aliases":"N^N","display":"N^N","kind":"use","excerpt":"P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. If b_N,j := { fracj+rN:1≤ r≤ N, r≠ N-j }, then E_N,j(z) = fracz^jj! _0F_N-1 ( - b_N,j; fracz^NN^N ). Thus coefficient-residue, root-of-unity, ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-821e1514a603c20a","tex":"N^{Nn}","aliases":"N^Nn","display":"N^{Nn}","kind":"use","excerpt":"The projector laws follow coefficient by coefficient. The residue classes modulo NM lying over jpmod N are precisely j,j+N,…,j+(M-1)N, which proves refinement. Character orthogonality gives the Fourier formula. Finally, (Nn+j)! = j! N^Nn ∏_r=1^N(fracj+rN)_n. Exactly one factor in the product is (1)_n=n!. Removing it leaves the parameter set b_N,j a","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f652639badcf7db2","tex":"n^r","aliases":"n^r","display":"n^r","kind":"use","excerpt":"[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plan","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-76932fedc2d9180a","tex":"N_{\\mathbb C/\\mathbb R}","aliases":"N_ C/ R mathbb mathbb","display":"N_{\\mathbb C/\\mathbb R}","kind":"use","excerpt":"-1 · 0 . Then A_σσ(A_σ)=-I. The projective cocycle condition holds because -I is scalar. Its boundary is represented by -1 in R^×/N_ C/ R( C^×). Every complex norm is positive, so -1 is nontrivial. The descended central simple algebra is the Hamilton quaternion algebra. Replacing A_σ by 0 · 1","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ebc65dd86b43b2d4","tex":"N_0","aliases":"N_0","display":"N_0","kind":"use","excerpt":"Let K_0:=⋂_jkerpair(o_j) in the relational setting and N_0:=⋂_jker(o_j) in the linear setting.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ffaa0ad23479bca8","tex":"N_1","aliases":"N_1","display":"N_1","kind":"use","excerpt":"[Composition of task loss] For N_1⊆ N_2⊆ N_3, there is an exact sequence 0⟶ N_2/N_1 ⟶ N_3/N_1 ⟶ N_3/N_2 ⟶0. Hence task loss composes as extension data rather than as a canonically split sum.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-583292a11ba24381","tex":"n_1","aliases":"n_1","display":"n_1","kind":"use","excerpt":"[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-72ec741403c87a9e","tex":"N_2","aliases":"N_2","display":"N_2","kind":"use","excerpt":"[Composition of task loss] For N_1⊆ N_2⊆ N_3, there is an exact sequence 0⟶ N_2/N_1 ⟶ N_3/N_1 ⟶ N_3/N_2 ⟶0. Hence task loss composes as extension data rather than as a canonically split sum.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bcbc010f58d53ce9","tex":"N_3","aliases":"N_3","display":"N_3","kind":"use","excerpt":"[Composition of task loss] For N_1⊆ N_2⊆ N_3, there is an exact sequence 0⟶ N_2/N_1 ⟶ N_3/N_1 ⟶ N_3/N_2 ⟶0. Hence task loss composes as extension data rather than as a canonically split sum.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d583dfc77954957f","tex":"N_a","aliases":"N_a","display":"N_a","kind":"use","excerpt":"Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite pre","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"The coordinate functionals on Q^ N separate points of V. Choose n_1 for which some element of V has nonzero value. If n_1,…,n_j have been chosen and the joint evaluation map has rank j<ρ, its kernel contains a nonzero f. Choose n_j+1 with f(n_j+1)≠0. The rank increases. After ρ steps, the evaluation map has full rank.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c6ac5aa30621bef6","tex":"n_{j+1}","aliases":"n_j+1","display":"n_{j+1}","kind":"use","excerpt":"The coordinate functionals on Q^ N separate points of V. Choose n_1 for which some element of V has nonzero value. If n_1,…,n_j have been chosen and the joint evaluation map has rank j<ρ, its kernel contains a nonzero f. Choose n_j+1 with f(n_j+1)≠0. The rank increases. After ρ steps, the evaluation map has full rank.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-62782243b1cfc0cb","tex":"N_{n,M}","aliases":"N_n,M","display":"N_{n,M}","kind":"use","excerpt":"[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2f56096dd29a0f85","tex":"N_s","aliases":"N_s","display":"N_s","kind":"use","excerpt":"[Task-change exact sequence] Let a stronger task have behavioral kernel N_s, and let a weaker task have behavioral kernel N_w, with N_s⊆ N_w. Then 0⟶ N_w/N_s ⟶ X/N_s ⟶ X/N_w ⟶0 is exact. Thus N_w/N_s is the exact semantic information discarded when the task is weakened.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3eb2b6b7b8cb481a","tex":"N_{\\mathrm{strong}}","aliases":"N_strong mathrm","display":"N_{\\mathrm{strong}}","kind":"use","excerpt":"Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a794867609ad50e5","tex":"N_w","aliases":"N_w","display":"N_w","kind":"use","excerpt":"[Task-change exact sequence] Let a stronger task have behavioral kernel N_s, and let a weaker task have behavioral kernel N_w, with N_s⊆ N_w. Then 0⟶ N_w/N_s ⟶ X/N_s ⟶ X/N_w ⟶0 is exact. Thus N_w/N_s is the exact semantic information discarded when the task is weakened.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0ed2cd3231f265fb","tex":"N_{\\mathrm{weak}}","aliases":"N_weak mathrm","display":"N_{\\mathrm{weak}}","kind":"use","excerpt":"Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6a94c7b9707a1f69","tex":"N_{\\lambda,a}","aliases":"N_λ,a lambda","display":"N_{\\lambda,a}","kind":"use","excerpt":"Every connected component in [source reference] has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0372214b6973af12","tex":"n_\\rho","aliases":"n_ρ rho","display":"n_\\rho","kind":"use","excerpt":"[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c65b8526c5f04fd9","tex":"\\operatorname{Ncl}","aliases":"Ncl operatorname","display":"\\operatorname{Ncl}","kind":"use","excerpt":"If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-818c1dee9cf63335","tex":"\\operatorname{Ncl}_G","aliases":"Ncl_G operatorname","display":"\\operatorname{Ncl}_G","kind":"use","excerpt":"If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6933c4f620588aa2","tex":"\\mathrm{ncol}","aliases":"ncol mathrm","display":"\\mathrm{ncol}","kind":"use","excerpt":"The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"[Good-prime transfer] If pnmidδ_k(a), then for every r≥1, M_k(a)/p^rM_k(a) overset∼⟶ overline M_k,p^r(a) = L_k(a)/p^rL_k(a). At a good prime, the raw and saturated prime-power towers agree exactly.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e16b60493b1a650b","tex":"\\not","aliases":"not","display":null,"kind":"use","excerpt":"Take u=z+u^q, q≥2. tag11.14 Lagrange inversion gives the exact coefficient formula [z^n]u = displaystyle frac1(q-1)m+1binomqmm, · n=1+(q-1)m, [9pt] 0, · nnot≡1pmodq-1. tag11.15 The nonzero coefficients are the Fuss--Catalan numbers of full plane q-ary trees.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[4pt] Reduced OGF ↔ minimal DFAO · equivalence on the periodic essential image · unit purely periodic series and reachable minimal LSD-first p-DFAOs · coefficient semantics, minimal period, canonical p-kernel, output behavior · equality of represented sequences · a nonminimal residue-pair automaton is quotiented before entering the normalized chart","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b9dd02171b5ea5b5","tex":"\\mathcal O'","aliases":"O' mathcal","display":"\\mathcal O'","kind":"use","excerpt":"[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2c57b807d517006f","tex":"O^2","aliases":"O^2","display":"O^2","kind":"use","excerpt":"[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ff82bed0260b60e2","tex":"O^3","aliases":"O^3","display":"O^3","kind":"use","excerpt":"[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ccff27e25e35506a","tex":"O^4","aliases":"O^4","display":"O^4","kind":"use","excerpt":"[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a1af14704e9363e1","tex":"\\mathcal O^{\\mathrm{raw}}_{k,p^r}","aliases":"O^raw_k,p^r mathcal mathrm","display":"\\mathcal O^{\\mathrm{raw}}_{k,p^r}","kind":"use","excerpt":"Let O_k(a)={Λ_w a:w∈Σ_k^*}⊆ M_k(a) be the actual section orbit, and let O^raw_k,p^r(a) be its image in M_k(a)/p^rM_k(a).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9f2cde86f326da21","tex":"\\mathcal O^{\\mathrm{raw}}_{k,p^{r+1}}","aliases":"O^raw_k,p^r+1 mathcal mathrm","display":"\\mathcal O^{\\mathrm{raw}}_{k,p^{r+1}}","kind":"use","excerpt":"[Raw orbit-closure descent] The reduction maps O^raw_k,p^r+1(a) ↠ O^raw_k,p^r(a) are surjective, and overline O_k(a)^ p overset∼⟶ varprojlim_r O^raw_k,p^r(a), where the closure is taken in M_k,p(a).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bf1769ae81042a61","tex":"\\mathcal O_{13}","aliases":"O_13 mathcal","display":"\\mathcal O_{13}","kind":"use","excerpt":"The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyp","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-240d1123fe170569","tex":"\\mathcal O_{\\mathrm{DRLM}}","aliases":"O_DRLM mathcal mathrm","display":"\\mathcal O_{\\mathrm{DRLM}}","kind":"use","excerpt":"The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyp","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b4ed21d9825eb3d8","tex":"\\mathcal O_{\\mathrm{EM}}","aliases":"O_EM mathcal mathrm","display":"\\mathcal O_{\\mathrm{EM}}","kind":"use","excerpt":"The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyp","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7f2e36809a710027","tex":"\\mathcal O_{\\mathrm{Fuss}}","aliases":"O_Fuss mathcal mathrm","display":"\\mathcal O_{\\mathrm{Fuss}}","kind":"use","excerpt":"The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyp","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8530380ace28d75d","tex":"\\mathcal O_{\\mathrm{Gal}}","aliases":"O_Gal mathcal mathrm","display":"\\mathcal O_{\\mathrm{Gal}}","kind":"use","excerpt":"The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyp","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-664d5fd5c242c148","tex":"\\mathcal O_{\\mathrm{Loc}}","aliases":"O_Loc mathcal mathrm","display":"\\mathcal O_{\\mathrm{Loc}}","kind":"use","excerpt":"The semantic category of the total atlas is O_13 = O_DRLM sqcup O_Loc sqcup O_Fuss sqcup O_EM sqcup O_Gal. The coproduct symbol is literal. It does not assert that a finite-field regular sequence, a continuous p-adic function, a characteristic-zero marked tree branch, an integral Euler template, and a finite Galois subgroup are objects of one untyp","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dfd7873e0e65ecf4","tex":"\\mathcal O_N","aliases":"O_N mathcal","display":"\\mathcal O_N","kind":"use","excerpt":"Let ω_N be primitive and consider the rotated orbit O_N(F_λ,α) = {(1-αω_N^rt)^-λ:0≤ r<N}. Let r_rm orb be its dimension over the constant field.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3fa1bad690018d5e","tex":"\\mathcal O_q","aliases":"O_q mathcal","display":"\\mathcal O_q","kind":"use","excerpt":"=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-63da609e48d4a967","tex":"\\mathcal O_U","aliases":"O_U mathcal","display":"\\mathcal O_U","kind":"use","excerpt":"[label=textup(Oarabic*)] item For every orbit O⊆ I_A,U_0,r of H_A^r, path transport defines a path-independent orbit subset O_U⊆ I_A,U,r in every chart. The local projector sums P^r_A,U, O := ∑_i∈ O_Up^r_A,U,i descend to a split idempotent on A. As O ranges over the monodromy orbits, these projectors are pairwise annihilating and sum to 1_A.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d27490cf98bd66ea","tex":"\\mathcal O_\\lambda","aliases":"O_λ mathcal lambda","display":"\\mathcal O_\\lambda","kind":"use","excerpt":"Every connected component in [source reference] has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7acafa494704338e","tex":"\\Ob_{D'}","aliases":"Ob_D' Ob operatorname","display":"{\\operatorname{Ob}}_{D'}","kind":"use","excerpt":"U' · xrightarrowD' · V', bD=D'a, there is a unique induced map overline b:coker D⟶coker D' with overline b Ob_D(r)=Ob_D'(br).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-39d73ca4e8f7cf7a","tex":"\\operatorname{ord}_1","aliases":"ord_1 operatorname","display":"\\operatorname{ord}_1","kind":"use","excerpt":"The pair model gives a useful finite bound. Write T=p^ν M, gcd(M,p)=1, and set ord_1(p)=1.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-654ddb14d2b79b9e","tex":"\\operatorname{ord}_M","aliases":"ord_M operatorname","display":"\\operatorname{ord}_M","kind":"use","excerpt":"[Exact multiplier-orbit size] The set U_p(T) has cardinality |U_p(T)| = ν+ord_M(p). Consequently, | K_p(a)| ≤ T(ν+ord_M(p)).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-842e4a37631a1f1f","tex":"\\operatorname{ord}_{M}","aliases":"ord_M operatorname","display":"\\operatorname{ord}_{M}","kind":"use","excerpt":"For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a), T(a) · =min{N:m_amid x^N-1} =min{N:Q_amid 1-z^N}, | A_p(a)| · =| K_p(a)|","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-01eecdf5cc4bca8b","tex":"\\overline A","aliases":"overline A overline","display":"\\overline A","kind":"use","excerpt":"e det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d)","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0900d3a9c42eb699","tex":"\\overline A_{M,s}^{(d)}","aliases":"overline A_M,s^(d) overline","display":"\\overline A_{M,s}^{(d)}","kind":"use","excerpt":"e det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d)","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54037e9a059bedc5","tex":"\\overline B","aliases":"overline B overline","display":"\\overline B","kind":"use","excerpt":"[Selected polynomial fixed point and tree expansion] Assume a+φ(R)≤ R, κ_R<1. Then equation textup(11.1) has a unique solution in the closed ball overline B_R. Moreover, that solution is the absolutely convergent sum u=∑_T∈ T_ Aev(T), tag11.2 where A={m:A_m≠0}, the leaves of T evaluate to s, and an internal vertex of arity m evaluates by A_m applie","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7da834df65459b68","tex":"\\overline b","aliases":"overline b overline","display":"\\overline b","kind":"use","excerpt":"U' · xrightarrowD' · V', bD=D'a, there is a unique induced map overline b:coker D⟶coker D' with overline b Ob_D(r)=Ob_D'(br).","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ce8712e6c17bb593","tex":"\\overline B_R","aliases":"overline B_R overline","display":"\\overline B_R","kind":"use","excerpt":"[Selected polynomial fixed point and tree expansion] Assume a+φ(R)≤ R, κ_R<1. Then equation textup(11.1) has a unique solution in the closed ball overline B_R. Moreover, that solution is the absolutely convergent sum u=∑_T∈ T_ Aev(T), tag11.2 where A={m:A_m≠0}, the leaves of T evaluate to s, and an internal vertex of arity m evaluates by A_m applie","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a932516694d0bfa8","tex":"\\overline b_t","aliases":"overline b_t overline","display":"\\overline b_t","kind":"use","excerpt":"[Exact visible-support loss ledger] Use the finite fold data of (L8) and fix an input family b=(b_e)_e∈ E. Define S_0:={e∈ E:b_e≠0}, A_0:={e∈ S_0:τ_e(b_e)=0}, S_1:=S_0∖ A_0, and let Γ:=κ(S_1)⊆ T. For t∈ T, write the folded coordinate as overline b_t := ∑_substacke∈ S_1 κ(e)=tτ_e(b_e) ∈ C_t, and put supp(overline b) :={t∈ T:overline b_t≠0}, K_0:={t∈","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5f51cdcfd251a311","tex":"\\overline g","aliases":"overline g overline","display":"\\overline g","kind":"use","excerpt":"[Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b7a5b96653c1dfce","tex":"\\overline K","aliases":"overline K overline","display":"\\overline K","kind":"use","excerpt":"Throughout this section, K is a characteristic-zero field and overline K is an algebraic closure. Algebraic degree always means degree over the displayed rational-function field. All assertions about formal branches are made inside overline K[[z]] unless a different local field is named.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fe20cd14df4daa80","tex":"\\overline M_{k,m}","aliases":"overline M_k,m overline","display":"\\overline M_{k,m}","kind":"use","excerpt":"For an integer m≥2, the raw coefficient reduction is M_k(a)/mM_k(a), while the semantic reduction inside the saturated lattice is overline M_k,m(a) = im ( M_k(a)/mM_k(a) ⟶ L_k(a)/mL_k(a) ). These need not agree.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54c0bf49965f0248","tex":"\\overline{\\mathcal O_k(a)}","aliases":"overline O_k(a) overline mathcal","display":"\\overline{\\mathcal O_k(a)}","kind":"use","excerpt":"[Raw orbit-closure descent] The reduction maps O^raw_k,p^r+1(a) ↠ O^raw_k,p^r(a) are surjective, and overline O_k(a)^ p overset∼⟶ varprojlim_r O^raw_k,p^r(a), where the closure is taken in M_k,p(a).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f0f0466b171e4f07","tex":"\\overline{\\mathcal O_k(a)}^{\\,p}","aliases":"overline O_k(a)^ p overline mathcal","display":"\\overline{\\mathcal O_k(a)}^{\\,p}","kind":"use","excerpt":"[Raw orbit-closure descent] The reduction maps O^raw_k,p^r+1(a) ↠ O^raw_k,p^r(a) are surjective, and overline O_k(a)^ p overset∼⟶ varprojlim_r O^raw_k,p^r(a), where the closure is taken in M_k,p(a).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e6f2a96deede12c4","tex":"\\overline{\\mathcal O_k(s_k)}","aliases":"overline O_k(s_k) overline mathcal","display":"\\overline{\\mathcal O_k(s_k)}","kind":"use","excerpt":"[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordina","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e0cb5dea8eff7b75","tex":"\\overline{\\mathcal O_k(s_k)}^{\\,p}","aliases":"overline O_k(s_k)^ p overline mathcal","display":"\\overline{\\mathcal O_k(s_k)}^{\\,p}","kind":"use","excerpt":"[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordina","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-42c6158ae78541bc","tex":"\\overline Q","aliases":"overline Q overline","display":"\\overline Q","kind":"use","excerpt":"[Packet coordinates on the obstruction quotient] Suppose (Q_a) is a complete finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summa","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-eb01f03470ab7fcb","tex":"\\overline q","aliases":"overline q overline","display":"\\overline q","kind":"use","excerpt":"[Universal behavioral coimage] Assume the relevant effective quotient or module quotient exists. Then Q_ S preserves every declared future observation. Moreover, if q:X↠ Z is any surjective representation through which every map o_jρ(m) factors, there is a unique map overline q:Z⟶ Q_ S with overline q q:X→ Q_ S equal to the canonical quotient map. ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f788ebf4718ce05f","tex":"\\overline Q_a","aliases":"overline Q_a overline","display":"\\overline Q_a","kind":"use","excerpt":"[Packet coordinates on the obstruction quotient] Suppose (Q_a) is a complete finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summa","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordinary integer. Thus s_k+c_∞ is p-adically ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-131338dff4145d42","tex":"p^{2j}","aliases":"p^2j","display":"p^{2j}","kind":"use","excerpt":"[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordina","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fbe5654b944a7373","tex":"p^a","aliases":"p^a","display":"p^a","kind":"use","excerpt":"For a≥1 and 0≤ s<p^a, define the branch B_a,sf(q)=f(p^a q+s), q∈ Z_p, and the complete packet B_a f=( B_a,sf)_0≤ s<p^a.","count":32},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ca74460520851a88","tex":"p^{a+b}","aliases":"p^a+b","display":"p^{a+b}","kind":"use","excerpt":"Direct substitution gives f_a,s(p^bq+t) =f(p^a(p^bq+t)+s) =f(p^a+bq+s+p^a t). The address is uniquely determined in the interval 0≤ s+p^a t<p^a+b.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7f4968674b99b5fb","tex":"p^b","aliases":"p^b","display":"p^b","kind":"use","excerpt":"[Strict nested-branch law] For a,b≥1, 0≤ s<p^a, and 0≤ t<p^b, (f_a,s)_b,t=f_a+b,s+p^a t. All multiple-overlap comparison cells for the nested p-power branch atlas are identities. Hence its higher coherence is strict rather than merely up to a chosen natural isomorphism.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-725d03dd8cead47e","tex":"p^{D(A,B)+1}","aliases":"p^D(A,B)+1","display":"p^{D(A,B)+1}","kind":"use","excerpt":"[Exact state-complexity hierarchy] For a reduced rational mask A/B, r_p(F) ≤ κ_p(F) ≤ p^r_p(F) ≤ p^D(A,B)+1. Moreover, r_p(F)=dim W(A,B), κ_p(F)=| R(A,B)|.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-835ca25ac48774fc","tex":"p^{D+1}","aliases":"p^D+1","display":"p^{D+1}","kind":"use","excerpt":"n D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source reference]. It computes the coefficient sequence exactly.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cbe775442cdd2edf","tex":"p^{e(r+1)}","aliases":"p^e(r+1)","display":"p^{e(r+1)}","kind":"use","excerpt":"[Power bridge] For b(n)=n^r, which is completely multiplicative, the divisor ghost is c(n)=∑_dmid nd^r+1=σ_r+1(n). At a prime power, σ_r+1(p^e) = 1+p^r+1+⋯+p^e(r+1). Thus one simple multiplicative exponent rule simultaneously produces a filtered ordinary product, a multiplicative divisor-sum ghost, and a one-dimensional geometric local rule at ever","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4712dba3d0909d07","tex":"p^{e-1}","aliases":"p^e-1","display":"p^{e-1}","kind":"use","excerpt":"theorem[Multiplicativity is preserved and reflected] Assume b(1)=1 and c= G(b). Then b is multiplicative Longleftrightarrow c is multiplicative. On this locus, both b and c are reconstructed from prime-power seeds. For every prime p and e≥1, c(p^e)=∑_j=0^ep^j b(p^j), b(p^e)=fracc(p^e)-c(p^e-1)p^e.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e4a9fe0cfd379dc3","tex":"p^{e-j}","aliases":"p^e-j","display":"p^{e-j}","kind":"use","excerpt":"The local logarithmic-derivative recurrence gives e f(p^e) = ∑_j=1^eγ_p,j(f)f(p^e-j). Write n=p^e u with (u,p)=1 and multiply by f(u). Since Γ_f,p is supported on p-powers, the resulting sum is exactly the Dirichlet convolution at n.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4e435a0aa166d5e2","tex":"p^{e-r}","aliases":"p^e-r","display":"p^{e-r}","kind":"use","excerpt":"Unique factorization writes every positive integer as a finite product of pairwise coprime prime powers, so multiplicativity gives the reconstruction formula. Conversely, an arbitrary family of local unit series defines a multiplicative function by the same finite product. For n=p^e, Dirichlet convolution becomes (fmathbin∗_Dg)(p^e) = ∑_r=0^ef(p^r)","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-44ddf871c8e79a19","tex":"p^j","aliases":"p^j","display":"p^j","kind":"use","excerpt":"The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-22605ace34e5a239","tex":"p^{j+1}","aliases":"p^j+1","display":"p^{j+1}","kind":"use","excerpt":"Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-704b220b0886246a","tex":"p^L","aliases":"p^L","display":"p^L","kind":"use","excerpt":"[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-09c8b25bb1d743c8","tex":"p^{L+a}","aliases":"p^L+a","display":"p^{L+a}","kind":"use","excerpt":"[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7962ecdd80d0a694","tex":"p^{L+a+1}","aliases":"p^L+a+1","display":"p^{L+a+1}","kind":"use","excerpt":"Induct on a. If X_a=I+p^L+aB_a, the binomial expansion of X_a^p has linear term divisible by p^L+a+1, and every higher term has at least the same valuation. Apply [source reference] at the refined jump.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5ed6db9428239d9e","tex":"p^{Lr}","aliases":"p^Lr","display":"p^{Lr}","kind":"use","excerpt":"[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-422aac798eb70b6d","tex":"p^{\\min(r,\\nu_i)}","aliases":"p^min(r,ν_i) min nu","display":"p^{\\min(r,\\nu_i)}","kind":"use","excerpt":"[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-01d645aa75670391","tex":"p^{N+1}","aliases":"p^N+1","display":"p^{N+1}","kind":"use","excerpt":"Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9b13adfda2839132","tex":"p^{r+1}","aliases":"p^r+1","display":"p^{r+1}","kind":"use","excerpt":"ower-automata For every r≥1, the semantic p^r-kernel is finite and is the state set of the unique minimal least-significant-digit-first deterministic output automaton for the coefficient sequence abmod p^r. Moreover, κ_k,p^r^sem(a) ≤ |overline M_k,p^r(a)| = p^∑_i(r-min(r,ν_i)). Reduction induces surjective digit-equivariant maps K_k,p^r+1^sem(a) ↠ ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f2bee4fda516a276","tex":"P^r_{A,\\mathcal O}","aliases":"P^r_A, O mathcal","display":"P^r_{A,\\mathcal O}","kind":"use","excerpt":"item If ssucceq r, equivariance of the refinement map sends every fine monodromy orbit onto a coarse monodromy orbit. For a coarse orbit O, P^r_A, O = ∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) P^s_A, Q. Thus orbit descent and projector-sum refinement commute exactly.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ea550dedf3c93a6d","tex":"P^r_{A,U,\\mathcal O}","aliases":"P^r_A,U, O mathcal","display":"P^r_{A,U,\\mathcal O}","kind":"use","excerpt":"[label=textup(Oarabic*)] item For every orbit O⊆ I_A,U_0,r of H_A^r, path transport defines a path-independent orbit subset O_U⊆ I_A,U,r in every chart. The local projector sums P^r_A,U, O := ∑_i∈ O_Up^r_A,U,i descend to a split idempotent on A. As O ranges over the monodromy orbits, these projectors are pairwise annihilating and sum to 1_A.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-05afee2accd1832a","tex":"p^r_{A,U,i}","aliases":"p^r_A,U,i","display":"p^r_{A,U,i}","kind":"use","excerpt":"[label=textup(Oarabic*)] item For every orbit O⊆ I_A,U_0,r of H_A^r, path transport defines a path-independent orbit subset O_U⊆ I_A,U,r in every chart. The local projector sums P^r_A,U, O := ∑_i∈ O_Up^r_A,U,i descend to a split idempotent on A. As O ranges over the monodromy orbits, these projectors are pairwise annihilating and sum to 1_A.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-35092aa92b49e62f","tex":"P^r_{A,V,\\mathcal O}","aliases":"P^r_A,V, O mathcal","display":"P^r_{A,V,\\mathcal O}","kind":"use","excerpt":"Choose a path from U_0 to U and transport O along its address bijection. Two path choices differ by a loop at U_0, which preserves O setwise. Hence O_U is path independent. Projector compatibility gives θ^A_γ F_γ(P^r_A,U, O) = P^r_A,V, Oθ^A_γ, so the orbit sums descend. Their local splittings are the finite direct sums ⊕_i∈ O_UA^r_U,i; the direct s","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3ff2a4a55c8672ff","tex":"p^{r_p(F)}","aliases":"p^r_p(F)","display":"p^{r_p(F)}","kind":"use","excerpt":"[Exact state-complexity hierarchy] For a reduced rational mask A/B, r_p(F) ≤ κ_p(F) ≤ p^r_p(F) ≤ p^D(A,B)+1. Moreover, r_p(F)=dim W(A,B), κ_p(F)=| R(A,B)|.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5d6a6594e3f48602","tex":"P^s_{A,\\mathcal Q}","aliases":"P^s_A, Q mathcal","display":"P^s_{A,\\mathcal Q}","kind":"use","excerpt":"item If ssucceq r, equivariance of the refinement map sends every fine monodromy orbit onto a coarse monodromy orbit. For a coarse orbit O, P^r_A, O = ∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) P^s_A, Q. Thus orbit descent and projector-sum refinement commute exactly.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a99c32f598302013","tex":"p^{v_p(M)}","aliases":"p^v_p(M)","display":"p^{v_p(M)}","kind":"use","excerpt":"upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction m","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8445b6dc01a48a5c","tex":"p^{v_p(m)}","aliases":"p^v_p(m)","display":"p^{v_p(m)}","kind":"use","excerpt":"Addendum 04 proves that the prefix S_*={0,1,…,L_ Z(m,d)-1} with L_ Z(m,d)=min(d+1,B(m)), B(m)=max({1}∪{p^v_p(m):pmid m}), always admits an integer decoder and is cardinality-minimal [source reference].","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e33bcfb953873709","tex":"p^\\nu","aliases":"p^ν nu","display":"p^\\nu","kind":"use","excerpt":"The pair model gives a useful finite bound. Write T=p^ν M, gcd(M,p)=1, and set ord_1(p)=1.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-36a54396670b8bb6","tex":"p^{\\nu_i}","aliases":"p^ν_i nu","display":"p^{\\nu_i}","kind":"use","excerpt":"[p-adic saturation sequence] There is a digit-equivariant short exact sequence 0 ⟶ M_k,p(a) ⟶ L_k,p(a) ⟶ C_k(a)_(p) ⟶ 0, where C_k(a)_(p) is the p-primary subgroup of C_k(a). In Smith coordinates, M_k,p(a) = ⊕_i=1^ρp^ν_i Z_p e_i ⊆ ⊕_i=1^ρ Z_p e_i = L_k,p(a).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-089fb8c09d03662b","tex":"p^{\\nu_p(T(a))}","aliases":"p^ν_p(T(a)) nu","display":"p^{\\nu_p(T(a))}","kind":"use","excerpt":"For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a), T(a) · =min{N:m_amid x^N-1} =min{N:Q_amid 1-z^N}, | A_p(a)| · =| K_p(a)|","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-20094bff87ea1f75","tex":"p^\\ell","aliases":"p^ℓ ell","display":"p^\\ell","kind":"use","excerpt":"The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f150917365c9af4d","tex":"p^{\\ell-1}","aliases":"p^ℓ-1 ell","display":"p^{\\ell-1}","kind":"use","excerpt":"m the linear map Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d85fb9fd2ef3bc4d","tex":"p^{\\sum_i(r-\\min(r,\\nu_i))}","aliases":"p^∑_i(r-min(r,ν_i)) sum min nu","display":"p^{\\sum_i(r-\\min(r,\\nu_i))}","kind":"use","excerpt":"[Finite prime-power automata] For every r≥1, the semantic p^r-kernel is finite and is the state set of the unique minimal least-significant-digit-first deterministic output automaton for the coefficient sequence abmod p^r. Moreover, κ_k,p^r^sem(a) ≤ |overline M_k,p^r(a)| = p^∑_i(r-min(r,ν_i)). Reduction induces surjective digit-equivariant maps K_k","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5e043fa523ded0e1","tex":"p^{\\sum_{i=1}^{\\rho}(r-\\min(r,\\nu_i))}","aliases":"p^∑_i=1^ρ(r-min(r,ν_i)) sum rho min nu","display":"p^{\\sum_{i=1}^{\\rho}(r-\\min(r,\\nu_i))}","kind":"use","excerpt":"[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e50b5dbfe7add137","tex":"P_0","aliases":"P_0","display":"P_0","kind":"use","excerpt":"d preserves P_D; item the linear map Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"[Exact oblique packets] Over any field, set p_1= 1 · 1 0 · 0 , p_2= 0 · -1 0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5bbc555bd1de9b42","tex":"p_1^2","aliases":"p_1^2","display":"p_1^2","kind":"use","excerpt":"0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"[Exact oblique packets] Over any field, set p_1= 1 · 1 0 · 0 , p_2= 0 · -1 0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-913c827df47c97f1","tex":"p_2^2","aliases":"p_2^2","display":"p_2^2","kind":"use","excerpt":"0 · 1 . Then p_1^2=p_1, p_2^2=p_2, p_1p_2=p_2p_1=0, and p_1+p_2=I. The decomposition of 𝕜^2 is exact, although the packets are not orthogonal for the standard inner product. This separates the algebraic reconstruction theorem from later metric statements about energy or best approximation.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"Let F_a(z)=fracP_a(z)Q_a(z) be reduced with P_a(0)=Q_a(0)=1. If d=deg m_a, then Q_a(z)=z^dm_a(z^-1), with the evident coefficient normalization. Taking reciprocal polynomials in [source reference] gives Q_a(z)mid 1-z^T, T=min{N≥ 1:Q_a(z)mid 1-z^N}. Thus pure periodicity is visible both in the linear annihilator and in the reduced rational denominat","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c261d0404352ff34","tex":"p_a","aliases":"p_a","display":"p_a","kind":"use","excerpt":"The displayed matrix multiplication gives the two values immediately. The full refinement identity recovers f=∑_a,i∈{1,2}p_afp_i, whose only nonzero term here is p_2fp_2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aec0d725949e5720","tex":"p_{A_U,i}^r","aliases":"p_A_U,i^r","display":"p_{A_U,i}^r","kind":"use","excerpt":"there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U,i^r) = p_A_V,λ_UV^A,r(i)^r θ^A_UV, and that the ve","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7b2609c526143f58","tex":"p_{A_V,\\lambda_{UV}^{A,r}(i)}^r","aliases":"p_A_V,λ_UV^A,r(i)^r lambda","display":"p_{A_V,\\lambda_{UV}^{A,r}(i)}^r","kind":"use","excerpt":"there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U,i^r) = p_A_V,λ_UV^A,r(i)^r θ^A_UV, and that the ve","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-358af0e9624bf6aa","tex":"\\mathcal P_D","aliases":"P_D mathcal","display":"\\mathcal P_D","kind":"use","excerpt":"Write a=deg A, b=deg B, H(z)=A(z)B(z)^p-2, and define D(A,B) = max{ b, lceilfraca+(p-2)bp-1rceil }. Let P_D= F_p[z]_≤ D. For 0≤ d<p, define T_d(P)=Λ_d(PH), where Λ_d selects the coefficients in degrees congruent to d modulo p.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0c3f41d8ee9a8c79","tex":"P_{gHg^{-1}}","aliases":"P_gHg^-1","display":"P_{gHg^{-1}}","kind":"use","excerpt":"Conjugation sends e_H to e_gHg^-1. Centrality is therefore equivalent to normality. The represented identity U_gP_HU_g^-1=P_gHg^-1 gives the operator conditions. The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b9e70c157342a064","tex":"P_H","aliases":"P_H","display":"P_H","kind":"use","excerpt":"Let L/K be a finite Galois extension with group G=Gal(L/K). Let V be a finite-dimensional representation of G over a characteristic-zero field, or a unitary complex realization obtained after scalar extension. For every subgroup H≤ G, define P_H=frac1|H|∑_h∈ HU_h. tag12.1","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b64cf0676b9b02b3","tex":"P_H^2","aliases":"P_H^2","display":"P_H^2","kind":"use","excerpt":"The average is invariant under left and right multiplication by elements of H. Therefore P_H^2=P_H, and its range is exactly the fixed subspace. If H≤ J, then V^J⊆ V^H. The two projections are nested and multiply to the projector with smaller range.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-07d7277bad4bdeed","tex":"P_{HJ}","aliases":"P_HJ","display":"P_{HJ}","kind":"use","excerpt":"[Permutable subgroups and exact common refinement] In a faithful regular realization, the following are equivalent: item HJ=JH; item HJ is a subgroup of G; item P_HP_J=P_JP_H. When these conditions hold, P_HP_J=P_HJ. tag12.4 If they fail, P_HP_J and P_JP_H are mixed atlas corners rather than projectors.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"Throughout the analytic part, X is a Banach space over R or C. A finite packet system on X is a family of bounded projections P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. All sums over packet addresses are finite. The formal-series part is valid over any characteristic-zero field unless positivity or singularity analysis is explicitly invoked.","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f79a3358d40cf4c5","tex":"P_{i+1}","aliases":"P_i+1","display":"P_{i+1}","kind":"use","excerpt":"Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7e32786a5b15df61","tex":"P_J","aliases":"P_J","display":"P_J","kind":"use","excerpt":"[Reynolds fixed packet and reverse order] The operator P_H is the projector onto the fixed packet V^H={v:U_hv=v for all h∈ H}. If H≤ J, then P_HP_J=P_JP_H=P_J. tag12.2 Consequently, subgroup inclusion reverses the fixed-projector order.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"Throughout the analytic part, X is a Banach space over R or C. A finite packet system on X is a family of bounded projections P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. All sums over packet addresses are finite. The formal-series part is valid over any characteristic-zero field unless positivity or singularity analysis is explicitly invoked.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-81147b115446999c","tex":"P_{j_1}","aliases":"P_j_1","display":"P_{j_1}","kind":"use","excerpt":"For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-234598544176e608","tex":"P_{j_m}","aliases":"P_j_m","display":"P_{j_m}","kind":"use","excerpt":"For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c018197aa5eed3ed","tex":"P_{m,S}^{(d)}","aliases":"P_m,S^(d)","display":"P_{m,S}^{(d)}","kind":"use","excerpt":"Let A=P_m,S^(d): Z^d+1→ Z^|S|(d+1) be a stacked affine-Mahler packet analysis map. Whenever an integer left inverse L exists, LA=I, Π=AL, Π^2=Π, and Z^|S|(d+1)=im A⊕ker L. Thus the source chart and the normalized packet core im A are equivalent, while ker L is a named residual rather than discarded information.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-eb35d49ac0429279","tex":"P_N","aliases":"P_N","display":"P_N","kind":"use","excerpt":"12-shrinking-subgroups Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f0c02aa550cfc1b1","tex":"\\mathcal P_{N,j}","aliases":"P_N,j mathcal","display":"\\mathcal P_{N,j}","kind":"use","excerpt":"For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0 m≡ j (mod N)a_mz^m, 0≤ j<N.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9deb55554193ff03","tex":"\\mathcal P_{N,j}^2","aliases":"P_N,j^2 mathcal","display":"\\mathcal P_{N,j}^2","kind":"use","excerpt":"ion] Let N,c,n∈ Z with N≥1, c≥0, and n≥0. Then frac(Nn+c)!c! = ∏_r=c+1^c+N(r)_n;N. Each integer factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1, their exact refinement law is P_","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-90fba425db5ca238","tex":"P_{N,k}","aliases":"P_N,k","display":"P_{N,k}","kind":"use","excerpt":"[Multiplicative residue-packet law] For N≥1, let P_N,rf be the arithmetic function equal to f(n) when n≡ rpmod N and zero otherwise. Then P_N,k(fmathbin∗_Dg) = ∑_rs≡ k (N) (P_N,rf)mathbin∗_D(P_N,sg). Thus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid Z/N Z, including its zero divisors.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ee990001b15a4c67","tex":"\\mathcal P_{N,k}","aliases":"P_N,k mathcal","display":"\\mathcal P_{N,k}","kind":"use","excerpt":"ion] Let N,c,n∈ Z with N≥1, c≥0, and n≥0. Then frac(Nn+c)!c! = ∏_r=c+1^c+N(r)_n;N. Each integer factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1, their exact refinement law is P_","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2bd7ecfa07c32400","tex":"P_{N,r}","aliases":"P_N,r","display":"P_{N,r}","kind":"use","excerpt":"[Multiplicative residue-packet law] For N≥1, let P_N,rf be the arithmetic function equal to f(n) when n≡ rpmod N and zero otherwise. Then P_N,k(fmathbin∗_Dg) = ∑_rs≡ k (N) (P_N,rf)mathbin∗_D(P_N,sg). Thus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid Z/N Z, including its zero divisors.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-09523592e6cc4b12","tex":"P_{N,s}","aliases":"P_N,s","display":"P_{N,s}","kind":"use","excerpt":"[Multiplicative residue-packet law] For N≥1, let P_N,rf be the arithmetic function equal to f(n) when n≡ rpmod N and zero otherwise. Then P_N,k(fmathbin∗_Dg) = ∑_rs≡ k (N) (P_N,rf)mathbin∗_D(P_N,sg). Thus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid Z/N Z, including its zero divisors.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c105a464fb76d0f8","tex":"\\mathcal P_{NM,j+\\ell N}","aliases":"P_NM,j+ℓ N mathcal ell","display":"\\mathcal P_{NM,j+\\ell N}","kind":"use","excerpt":"factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1, their exact refinement law is P_N,j = ∑_ℓ=0^M-1 P_NM,j+ℓ N.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-71ddce08f50bf591","tex":"\\mathcal P_{\\mathrm{per}}","aliases":"P_per mathcal mathrm","display":"\\mathcal P_{\\mathrm{per}}","kind":"use","excerpt":"For a∈ O_DRLM, define J_DRLM,p(a) = ( SL_p, F_p(a), P_rm(a), P_per(a) ), where SL_p, F_p(a) is the strict shortlex record, P_rm(a) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and P_per(a) is empty off the purely periodic locus and otherwise co","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7452ea4fa6c0355b","tex":"P_{q,\\alpha}","aliases":"P_q,α alpha","display":"P_{q,\\alpha}","kind":"use","excerpt":"Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-59ebd53ce40c280c","tex":"\\mathcal P_{\\mathrm{rm}}","aliases":"P_rm mathcal mathrm","display":"\\mathcal P_{\\mathrm{rm}}","kind":"use","excerpt":"For a∈ O_DRLM, define J_DRLM,p(a) = ( SL_p, F_p(a), P_rm(a), P_per(a) ), where SL_p, F_p(a) is the strict shortlex record, P_rm(a) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and P_per(a) is empty off the purely periodic locus and otherwise co","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-afe01bfeb11d338e","tex":"p_s^{-1}","aliases":"p_s^-1","display":"p_s^{-1}","kind":"use","excerpt":"The pair (a,b) factors j through c, and the image universal property gives the map to J_s,t. If t=φ s, then J_s,t is the graph of φ, so p_s is invertible. Conversely, if p_s is invertible, define φ=p_t p_s^-1.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d58992a0c0c08b27","tex":"p_t","aliases":"p_t","display":"p_t","kind":"use","excerpt":"The pair (a,b) factors j through c, and the image universal property gives the map to J_s,t. If t=φ s, then J_s,t is the graph of φ, so p_s is invertible. Conversely, if p_s is invertible, define φ=p_t p_s^-1.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3ac110d4dfecb888","tex":"P_{U_n}","aliases":"P_U_n","display":"P_{U_n}","kind":"use","excerpt":"[Shrinking-subgroup reconstruction] Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ceb5bad97632be4d","tex":"p_{Y,a}","aliases":"p_Y,a","display":"p_{Y,a}","kind":"use","excerpt":"Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3cc90ed907afa576","tex":"p_{Y,a}^u","aliases":"p_Y,a^u","display":"p_{Y,a}^u","kind":"use","excerpt":"ion r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-561eb4c74d0535ec","tex":"p_{Y,b}^v","aliases":"p_Y,b^v","display":"p_{Y,b}^v","kind":"use","excerpt":"item Projector-sum refinement. If ssucceq r on X and vsucceq u on Y, then p_Y,a^u f p_X,i^r = ∑_ρ^Y_vu(b)=a ∑_ρ^X_sr(j)=i p_Y,b^v f p_X,j^s. Coarsening a fine composite gives the same ambient morphism as composing at the coarse level.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aad7607e608d0052","tex":"P_\\beta","aliases":"P_β beta","display":"P_\\beta","kind":"use","excerpt":"Let F:Y→ Z and G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-17417a9415875e48","tex":"P_\\delta","aliases":"P_δ delta","display":"P_\\delta","kind":"use","excerpt":"[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5ed30585279498bf","tex":"P_\\lambda","aliases":"P_λ lambda","display":"P_\\lambda","kind":"use","excerpt":"Over characteristic zero, write the unique active expansion over an algebraic closure as a_n=∑_λ∈ΛP_λ(n)λ^n. For each sampling modulus N and address j, the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of a_Nm+j [so","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4701c7d0320f6cc9","tex":"p_{\\ell,i_\\ell}","aliases":"p_ℓ,i_ℓ ell ell","display":"p_{\\ell,i_\\ell}","kind":"use","excerpt":"Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-474a8c74e28ce2e1","tex":"\\parallel","aliases":"parallel","display":null,"kind":"use","excerpt":"[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c1b5371e3a9d5881","tex":"\\mathrm{pers}","aliases":"pers mathrm","display":"\\mathrm{pers}","kind":"use","excerpt":"Let Seq_K^pers be the discrete groupoid of persistent C-finite sequences over K.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e988c4dae5aed27d","tex":"\\operatorname{PGL}","aliases":"PGL operatorname","display":"\\operatorname{PGL}","kind":"use","excerpt":"A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-661d27ea0e83e45c","tex":"\\operatorname{PGL}_n","aliases":"PGL_n operatorname","display":"\\operatorname{PGL}_n","kind":"use","excerpt":"A projective cocycle takes values in PGL_n(L). Choose lifts A_σ∈GL_n(L). Their cocycle failure is scalar: c(σ,τ) =A_σσ(A_τ)A_στ^-1 ∈ L^×. tag12.18 Changing the lifts alters c by a scalar coboundary.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"belprop:periodic-kernel Let a∈Per_p^×,per have minimal period T and period word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives by equality of the seq","count":17},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"Let D_K=K(z)⟨ D⟩, Dq=qD+q', be the differential Ore algebra, and let S_K=K(n)⟨ S⟩, Sq(n)=q(n+1)S, be the forward-shift Ore algebra. Both are Euclidean skew-polynomial rings in their operator variable, so their nonzero left ideals have unique monic generators [source reference].","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-deb8730cc86fd5ef","tex":"q^{-1/(q-1)}","aliases":"q^-1/(q-1)","display":"q^{-1/(q-1)}","kind":"use","excerpt":"r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,N/g,h). tag11.17","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-019e18bf385913a1","tex":"q^{-q/(q-1)}","aliases":"q^-q/(q-1)","display":"q^{-q/(q-1)}","kind":"use","excerpt":"r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the source dominant orbit has q-1 points, while the complete packet orbit has exactly h points. The phase is therefore determined by the finite tuple (q,N,g,N/g,h). tag11.17","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c45fb5bc9791a996","tex":"Q^2","aliases":"Q^2","display":"Q^2","kind":"use","excerpt":"c · b, d=ab-c^2>0. The squared induced norms are u:= A_1 _Q^2 =fracb(a+b+2c)d, v:= A_2 _Q^2 =fraca(a+b-2c)d. A direct calculation gives ab((a+b)^2-4c^2)-4(ab-c^2)^2 =ab(a-b)^2+4c^2(ab-c^2)≥0. Thus uv≥4, and therefore max{sqrt u,sqrt v}≥(uv)^1/4≥sqrt2. Equality forces a=b and c=0, and then u=v=2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-981774e492259150","tex":"q^{r_k(a)}","aliases":"q^r_k(a)","display":"q^{r_k(a)}","kind":"use","excerpt":"statement written in semantic, intrinsic-module, and coordinate form. The kernel automaton in [source reference] recovers the deterministic chart, while its state-to-section map recovers the finite orbit inside the canonical module. These constructions are inverse on normalized objects. If K= F_q and r_k(a)<∞, the vector space V_k(a) has only q^r_k","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ffb8d3f447fa1db7","tex":"Q^{\\rm cl}_{N,j}","aliases":"Q^rm cl_N,j rm","display":"Q^{\\rm cl}_{N,j}","kind":"use","excerpt":"Let Q^rm cl_N,j be an annihilator produced by a declared closure or elimination pipeline, and let L_N,j be the monic minimal operator of the section. Since every annihilator belongs to the principal left ideal generated by L_N,j, Q^rm cl_N,j=A_N,jL_N,j for some Ore operator A_N,j. The nonnegative quantity δ^rm cl_N,j = ordQ^rm cl_N,j - ordL_N,j mea","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-013e6363db462052","tex":"q^u_a","aliases":"q^u_a","display":"q^u_a","kind":"use","excerpt":"[Exact refinement law, imported] Let p^r_i=∑_ρ_sr(j)=ip^s_j and q^u_a=∑_σ_vu(b)=aq^v_b. For every morphism f:X→ Y, q^u_afp^r_i = ∑_σ_vu(b)=a ∑_ρ_sr(j)=i q^v_bfp^s_j. Coarsening the fine block product of two composable maps gives the coarse block product.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-84d500fc0153289c","tex":"q^v_b","aliases":"q^v_b","display":"q^v_b","kind":"use","excerpt":"[Exact refinement law, imported] Let p^r_i=∑_ρ_sr(j)=ip^s_j and q^u_a=∑_σ_vu(b)=aq^v_b. For every morphism f:X→ Y, q^u_afp^r_i = ∑_σ_vu(b)=a ∑_ρ_sr(j)=i q^v_bfp^s_j. Coarsening the fine block product of two composable maps gives the coarse block product.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6df5f97ad5683166","tex":"\\mathbb Q^\\rho","aliases":"Q^ρ mathbb rho","display":"\\mathbb Q^\\rho","kind":"use","excerpt":"[Finite coordinate detection] Let V⊆ Q^ N be a vector space of finite dimension ρ. There exist indices 0≤ n_1<⋯<n_ρ such that the evaluation map E_ n:V⟶ Q^ρ, f⟼ (f(n_1),…,f(n_ρ)) is an isomorphism.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5ca75c044055616d","tex":"Q_a","aliases":"Q_a","display":"Q_a","kind":"use","excerpt":"[Packet coordinates on the obstruction quotient] Suppose (Q_a) is a complete finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summa","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-91837d3357d7ad8e","tex":"q_b","aliases":"q_b","display":"q_b","kind":"use","excerpt":"Applying C_E to a block q_bfp_j retains it exactly when (b,j)∈ E. Pairwise annihilation removes every mismatched source or target index. The formulas follow block by block.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-42d66614fe89f142","tex":"Q_e","aliases":"Q_e","display":"Q_e","kind":"use","excerpt":"Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d701f4f865b9980e","tex":"q_e","aliases":"q_e","display":"q_e","kind":"use","excerpt":"[Abelian fold-loss exact sequence] Let A be an abelian category. Suppose a finite family of lane objects (B_e)_e∈ E, target objects (C_t)_t∈ T, and morphisms τ_e:B_e→ C_κ(e) is given. Let q_e:⊕_e'∈ EB_e'→ B_e be the coordinate projection and define Φ:=∑_e∈ Ej_κ(e)τ_eq_e: ⊕_e∈ EB_e⟶ ⊕_t∈ TC_t. Then there are canonical short exact sequences 0⟶kerΦ ⟶⊕","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ecfc6d63a3b68fb8","tex":"q_j","aliases":"q_j","display":"q_j","kind":"use","excerpt":"Let F(z)=∑_n≥0a_nz^n and L = ∑_j=0^rq_j(z)D^j, q_j(z)=∑_m=0^d_jq_j,mz^m.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-97a6b1d04a0e9f82","tex":"q_{j,m}","aliases":"q_j,m","display":"q_{j,m}","kind":"use","excerpt":"Let F(z)=∑_n≥0a_nz^n and L = ∑_j=0^rq_j(z)D^j, q_j(z)=∑_m=0^d_jq_j,mz^m.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8aecabfd6bc33fed","tex":"q_{j,v_j}","aliases":"q_j,v_j","display":"q_{j,v_j}","kind":"use","excerpt":"Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b734e028c4b3e98e","tex":"Q_{k,p^r}","aliases":"Q_k,p^r","display":"Q_{k,p^r}","kind":"use","excerpt":"[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9062c6d844288cbe","tex":"Q_{M',\\mathcal O'}","aliases":"Q_M', O' mathcal","display":"Q_{M',\\mathcal O'}","kind":"use","excerpt":"[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e10040de05014656","tex":"Q_{M,\\mathcal O}","aliases":"Q_M, O mathcal","display":"Q_{M,\\mathcal O}","kind":"use","excerpt":"[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d35fdff78185ee8e","tex":"Q_{N,j}","aliases":"Q_N,j","display":"Q_{N,j}","kind":"use","excerpt":"[Constructive packet eliminant] For each j, one can compute an annihilating polynomial Q_N,j(x,T)∈ K(x)[T] with deg_TQ_N,j≤ d^N by multiplication matrices, resultants, or a Gr obner elimination of the exact packet system.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e69d7d4f472a6cd9","tex":"Q_o","aliases":"Q_o","display":"Q_o","kind":"use","excerpt":"A compatible family makes every closed root holonomy H conformal for the root metric Q_o. Taking determinants fixes its positive scale as |det H|^1/n, so widehat H preserves Q_o.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9666ac009c3e72f6","tex":"q_r","aliases":"q_r","display":"q_r","kind":"use","excerpt":"Multiply L_F by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic. This gives a unique primitive polynomial representative widetilde L_F = ∑_j=0^rq_j(z)D^j. Its operator singular support is Σ_rm op(F) = {q_r(z)=0}∪{∞}. Apparent singularities remain part of th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d70d66d45c49fb6a","tex":"Q_s","aliases":"Q_s","display":"Q_s","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-53c2d6d15f1b4102","tex":"Q_{s(e)}","aliases":"Q_s(e)","display":"Q_{s(e)}","kind":"use","excerpt":"raph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge s","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0066a3bbb775672c","tex":"Q_{\\mathrm{strong}}","aliases":"Q_strong mathrm","display":"Q_{\\mathrm{strong}}","kind":"use","excerpt":"Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6fd3920b0502031c","tex":"Q_t","aliases":"Q_t","display":"Q_t","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9768261cdb0554c9","tex":"Q_{t(e)}","aliases":"Q_t(e)","display":"Q_{t(e)}","kind":"use","excerpt":"raph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge s","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ddc0eef4732461d5","tex":"Q_U","aliases":"Q_U","display":"Q_U","kind":"use","excerpt":"item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-baed31a30986ab0a","tex":"Q_u","aliases":"Q_u","display":"Q_u","kind":"use","excerpt":"Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible fa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8141a49faa601213","tex":"Q_v","aliases":"Q_v","display":"Q_v","kind":"use","excerpt":"Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible fa","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-af6d70e435d4204c","tex":"Q_{\\mathrm{weak}}","aliases":"Q_weak mathrm","display":"Q_{\\mathrm{weak}}","kind":"use","excerpt":"Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b55754320a8637cb","tex":"Q_X","aliases":"Q_X","display":"Q_X","kind":"use","excerpt":"[Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-79fe9e26e1a59839","tex":"Q_Y","aliases":"Q_Y","display":"Q_Y","kind":"use","excerpt":"[Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d591f47f078d0a89","tex":"Q_Z","aliases":"Q_Z","display":"Q_Z","kind":"use","excerpt":"[Loss-composition extension] For induced quotient maps Q_Xxrightarrowoverline fQ_Yxrightarrowoverline gQ_Z, there is a natural exact sequence 0⟶keroverline f ⟶ker(overline g overline f) ⟶keroverline g∩imoverline f ⟶0. If overline f is surjective, this reduces to 0⟶ L_f ⟶ L_gf ⟶ L_g ⟶0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5a423e1823849ed7","tex":"Q_\\gamma","aliases":"Q_γ gamma","display":"Q_\\gamma","kind":"use","excerpt":"Let F:Y→ Z and G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-400155c5c91e095f","tex":"Q_{\\lambda,a}","aliases":"Q_λ,a lambda","display":"Q_{\\lambda,a}","kind":"use","excerpt":"Every connected component in [source reference] has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ea38efd4e301edaf","tex":"Q_{\\lambda,b}","aliases":"Q_λ,b lambda","display":"Q_{\\lambda,b}","kind":"use","excerpt":"has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"Throughout the analytic part, X is a Banach space over R or C. A finite packet system on X is a family of bounded projections P_iP_j=δ_ijP_i, ∑_i∈ IP_i=I_X. All sums over packet addresses are finite. The formal-series part is valid over any characteristic-zero field unless positivity or singularity analysis is explicitly invoked.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a3d472d51734fedb","tex":"\\mathbb R^2","aliases":"R^2 mathbb","display":"\\mathbb R^2","kind":"use","excerpt":"[Sharp geometry-class separation] The norm x _∞ satisfies max{ A_1x _∞, A_2x _∞} = x _∞ for every x∈ R^2. In contrast, for every positive-definite quadratic metric Q, max{ A_1 _Q, A_2 _Q}≥sqrt2. Equality holds for scalar multiples of the Euclidean metric. Hence the exact optimal convex/Finsler bound is 1, while the exact optimal quadratic bound is ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-855b51431c459bf3","tex":"\\mathbb R^d","aliases":"R^d mathbb","display":"\\mathbb R^d","kind":"use","excerpt":"Let S be a finite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c055d5f42c0b87dd","tex":"R^{d+1}","aliases":"R^d+1","display":"R^{d+1}","kind":"use","excerpt":"The same integer kernel matrix has different categorical status after changing the coefficient doctrine. Fix d≥0 and write M_d(R)=R^d+1 for the formal truncated Mahler coefficient module. Let A_M,s^(d) =[Θ_M,s(r,k)]_0≤ k,r≤ d, where the target degree k is the row index.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c4179ac2053e2353","tex":"R^m","aliases":"R^m","display":"R^m","kind":"use","excerpt":"The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1381075c5bc7ca5f","tex":"R^{m-1}","aliases":"R^m-1","display":"R^{m-1}","kind":"use","excerpt":"Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-681be4307d7012ce","tex":"\\mathbb R^\\times","aliases":"R^× mathbb times","display":"\\mathbb R^\\times","kind":"use","excerpt":"-1 · 0 . Then A_σσ(A_σ)=-I. The projective cocycle condition holds because -I is scalar. Its boundary is represented by -1 in R^×/N_ C/ R( C^×). Every complex norm is positive, so -1 is nontrivial. The descended central simple algebra is the Hamilton quaternion algebra. Replacing A_σ by 0 · 1","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7c8009958676522c","tex":"\\mathfrak R^{\\simeq}","aliases":"R^≃ mathfrak simeq","display":"\\mathfrak R^{\\simeq}","kind":"use","excerpt":"[label=textup(roman*)] item Desc( R) exists and has the universal property of the finite pseudolimit of R. item There is a canonical equivalence Core(Desc( R)) ≃ Desc( R^≃). item Every pseudonatural transformation T: R⇒ S induces a functor Desc(T):Desc( R)→Desc( S). If every T_U is an equivalence, then Desc(T) is an equivalence. A pointwise noninve","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the sou","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d5c7051b56820f95","tex":"r_3","aliases":"r_3","display":"r_3","kind":"use","excerpt":"A genuine denominator example. Over F_3, take R(z)=frac11+z. Then F(z)^2=1+z. The reduced denominator remains fixed under every Cartier section. The ambient compiler has dimension two, its reachable numerator set is {0,1,2,1+z}, and its reachable span has dimension two. Hence r_3(F)=2, κ_3(F)=4. The nonzero collapse obstruction again proves that th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e93838ea3bf8d2c","tex":"r_A","aliases":"r_A","display":"r_A","kind":"use","excerpt":"v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category wit","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fb14b1a407e1de8e","tex":"r_a","aliases":"r_a","display":"r_a","kind":"use","excerpt":"[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the sou","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c6016f5cf675a3ae","tex":"r_B","aliases":"r_B","display":"r_B","kind":"use","excerpt":"_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precisely, fix one declared resolution r_A for each lane-decorated descent object A. The category with the same objects, morphism groups Γ (mathscr G, mathscr M^r_B,r_A_B,A), an","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-567e1106095a8a61","tex":"r_c","aliases":"r_c","display":"r_c","kind":"use","excerpt":"[Exact transfer reconstruction and block composition, imported] Every morphism f:X→ Y has the exact expansion f = ∑_a,iq_afp_i = ∑_a,ijmath_a f_aiπ_i. If g:Y→ Z is resolved by a packet system R=(r_c), then (gf)_ci = ∑_a g_caf_ai.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ea6f837add16d1a3","tex":"\\mathfrak R_{\\mathrm{DG}}","aliases":"R_DG mathfrak mathrm","display":"\\mathfrak R_{\\mathrm{DG}}","kind":"use","excerpt":"[Effective descent on the divisor--ghost overlap] The exponent, fixed Euler-product, multiplicative ghost, and prime-power seed charts form a finite normalized atlas R_DG. Every bridge is reversible on the declared overlap, every multiple-overlap comparison is the unique identity over the represented exponent function, and Desc( R_DG) ≃ EulMult_ Z.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b3975fa3c319f1d3","tex":"r_e","aliases":"r_e","display":"r_e","kind":"use","excerpt":"Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2834367885f3bd25","tex":"R_E^{u,r}","aliases":"R_E^u,r","display":"R_E^{u,r}","kind":"use","excerpt":"item Selectors and named residuals. For E⊆ I_Y,u× I_X,r, put C_E^u,r(f) :=∑_(a,i)∈ Ep_Y,a^ufp_X,i^r, R_E^u,r(f):=f-C_E^u,r(f). As additive endomorphisms of Hom_ C(X,Y), C_E^2=C_E, C_EC_F=C_E∩ F, R_E=1-C_E, f=C_E(f)+R_E(f). Thus deletion of a lane block is never silent. Its complement is an exact, named residual.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c9377ce9ef801522","tex":"R_F","aliases":"R_F","display":"R_F","kind":"use","excerpt":"[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9bb48203991c2d2f","tex":"r_f","aliases":"r_f","display":"r_f","kind":"use","excerpt":"[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-80744e0064457114","tex":"R_F^{-1}","aliases":"R_F^-1","display":"R_F^{-1}","kind":"use","excerpt":"In characteristic p, Frobenius gives F(z^p)=F(z)^p. Substitution into [source reference] gives [source reference]. Thus R_F is rational exactly when F^p-1=R_F^-1 is rational. Writing the reduced rational mask as A/B gives [source reference] and [source reference].","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ade69a4d18e62b84","tex":"r_g","aliases":"r_g","display":"r_g","kind":"use","excerpt":"[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-455a3f2c7c575c0e","tex":"r_h","aliases":"r_h","display":"r_h","kind":"use","excerpt":"Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-91bc078310ea7078","tex":"r_{h+1}","aliases":"r_h+1","display":"r_{h+1}","kind":"use","excerpt":"Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite pre","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a4ff961cd74d1869","tex":"r_n","aliases":"r_n","display":"r_n","kind":"use","excerpt":"Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preced","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-20c124b660a6606c","tex":"R_{N,j}","aliases":"R_N,j","display":"R_{N,j}","kind":"use","excerpt":"[Effective marked binomial--Fuchsian atlas] The five normalized presentations above determine one another on the declared marked family. With equality morphisms they form equivalent discrete groupoids, every normalized fiber is contractible, and objects and morphisms descend effectively. The package J_rm BF = ( α,λ,N,j, R_N,j,L_N,j, A_N,j(λ),B_N,j,","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-69210dd38fa5d69c","tex":"R_{p^a,s}","aliases":"R_p^a,s","display":"R_{p^a,s}","kind":"use","excerpt":"Piecewise synthesis on the disjoint clopen cosets s+p^a Z_p is its inverse. Refining from p^a to p^a+b, for every integer b≥0, gives the exact child-address and mixed-radix composition law. Let M:= M_p^-1:C( Z_p, Z_p) xrightarrow∼c_0( Z_p) be Mahler analysis from [source reference], and write R_p^a,sf(q):=f(p^aq+s). Conjugation gives the exact coef","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-05eb726555e3a980","tex":"r_q","aliases":"r_q","display":"r_q","kind":"use","excerpt":"[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the sou","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7e4cd8f652e0c5ee","tex":"r_{\\rm orb}","aliases":"r_rm orb rm","display":"r_{\\rm orb}","kind":"use","excerpt":"Let ω_N be primitive and consider the rotated orbit O_N(F_λ,α) = {(1-αω_N^rt)^-λ:0≤ r<N}. Let r_rm orb be its dimension over the constant field.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-418d72779ec9fbe9","tex":"r_{\\rm tail}","aliases":"r_rm tail rm","display":"r_{\\rm tail}","kind":"use","excerpt":"[htbp] centering small captionRepresentative exact order profiles in the binomial--Fuchsian packet family. textwidth@L0.17textwidthL0.13textwidthcccccY@ Source · Packet · r_rm tail · r_D · r_rm orb · Alias · Cancel · Interpretation","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f7b985b4cff51815","tex":"R_s","aliases":"R_s","display":"R_s","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1e7e11a6c76c6fc5","tex":"R_s^*","aliases":"R_s^*","display":"R_s^*","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-874bcff99829ac7b","tex":"R_s^{-1}","aliases":"R_s^-1","display":"R_s^{-1}","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-49121c793f36cbb5","tex":"R_t","aliases":"R_t","display":"R_t","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-25518eedee22081f","tex":"R_t^*","aliases":"R_t^*","display":"R_t^*","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6621036c9088e172","tex":"\\rank","aliases":"rank operatorname","display":"{\\operatorname{rank}}","kind":"use","excerpt":"; item (C_v)_v is path-flat, meaning that all path transports with the same endpoints agree on the core; item it is the unique greatest path-flat subfamily; item it is independent of the chosen spanning tree and equals the common fixed space of all closed-path holonomies; item for Δ_S:F_o⟶⊕_e∉ SF_o, Δ_Sx=((H_e-I)x)_e∉ S, one has C_o=kerΔ_S, dim C_o","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-50bd15a8cca9ce57","tex":"\\rank_{\\mathbb F_p}","aliases":"rank_ F_p rank mathbb operatorname mathbb","display":"{\\operatorname{rank}}_{\\mathbb F_p}","kind":"use","excerpt":"e det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction modulo p has exact rank rank_ F_poverline A_M,s^(d)","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8aa5fa60125a2264","tex":"\\rank_K","aliases":"rank_K rank operatorname","display":"{\\operatorname{rank}}_K","kind":"use","excerpt":"[Regular rank theorem] Let a be k-regular over K. Then r_k(a) = rank_K H_a = min{dim_K W:W supports a digit-linear realization of a}. Moreover: item there exist finite word sets U,V⊂Σ_k^*, each of cardinality r_k(a), for which the minor H_a[U,V] is nonsingular; item every reachable-observable realization of dimension r_k(a) is isomorphic to C_k(a);","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-09c3a4e1bbdbe5bf","tex":"\\Rat","aliases":"Rat mathbf","display":"{\\mathbf{Rat}}","kind":"use","excerpt":"An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-12df37181a52fbb8","tex":"\\mathrm{rat}","aliases":"rat mathrm","display":"\\mathrm{rat}","kind":"use","excerpt":"Reduced mask chart. An object of Mask_p^rat is a reduced rational function R(z)=fracA(z)B(z), A(0)=B(0)=1. Morphisms preserve the rational function. The semantics is R↦ M_p[R]. The inverse chart map is the canonical ratio F↦ F/F(z^p).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6d43a42603ccbb77","tex":"\\Rat_K^{\\mathrm{red}}","aliases":"Rat_K^red Rat mathrm mathbf mathrm","display":"{\\mathbf{Rat}}_K^{\\mathrm{red}}","kind":"use","excerpt":"An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f16dd4c163a7bf4a","tex":"\\Rat_p^{\\times,\\mathrm{per}}","aliases":"Rat_p^×,per Rat times mathrm mathbf times mathrm","display":"{\\mathbf{Rat}}_p^{\\times,\\mathrm{per}}","kind":"use","excerpt":"An object of Rat_p^×,per is a reduced fraction F(z)=fracP(z)Q(z)∈ F_p(z), P(0)=Q(0)=1, whose coefficient sequence is purely periodic. Equivalently, Qmid 1-z^T for some T and the reduced coefficient sequence has no transient prefix. Morphisms preserve the rational function. Reducedness and the constant-term normalization make this chart rigid.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bc042a1db2b5b7bd","tex":"\\mathrm{raw}","aliases":"raw mathrm","display":"\\mathrm{raw}","kind":"use","excerpt":"Let O_k(a)={Λ_w a:w∈Σ_k^*}⊆ M_k(a) be the actual section orbit, and let O^raw_k,p^r(a) be its image in M_k(a)/p^rM_k(a).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"Let R(z)=fracA(z)B(z), A(0)=B(0)=1, with a=deg A, b=deg B, c=a+(p-2)b, and define D=max{b,lceilfraccp-1rceil}. The rational-mask Cartier theorem gives an invariant numerator space of degree at most D under P⟼Λ_d(PAB^p-2), 0≤ d<p. The resulting linear state vector belongs to F_p^D+1, so the raw output automaton has at most p^D+1 states [source refer","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-38ac57489dd0edfc","tex":"\\Rec","aliases":"Rec mathbf","display":"{\\mathbf{Rec}}","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cab7e81d98ffd500","tex":"\\Rec_K^{\\mathrm{min}}","aliases":"Rec_K^min Rec mathrm mathbf mathrm","display":"{\\mathbf{Rec}}_K^{\\mathrm{min}}","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8373be9e814a3141","tex":"\\Rec_p^{\\times,\\mathrm{per}}","aliases":"Rec_p^×,per Rec times mathrm mathbf times mathrm","display":"{\\mathbf{Rec}}_p^{\\times,\\mathrm{per}}","kind":"use","excerpt":"An object of Rec_p^×,per is the normalized minimal recurrence polynomial m(x)=x^d+c_1x^d-1+⋯+c_d, c_d≠ 0, together with the initial vector (a_0,…,a_d-1), subject to a_0=1 and the requirement that the generated sequence is purely periodic. Minimality means that no monic recurrence of lower degree annihilates the sequence. Morphisms are recurrence-pr","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7fdf44047d9358ee","tex":"\\mathrm{red}","aliases":"red mathrm","display":"\\mathrm{red}","kind":"use","excerpt":"An object of Rat_K^red is a reduced rational power series A(z)=fracP(z)Q(z), Q(0)=1, gcd(P,Q)=1, deg P<deg Q, whose reciprocal characteristic polynomial has nonzero constant term. Again the normalized reduced form is unique, so Rat_K^red is discrete.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-42129867117682a5","tex":"\\mathbf{Reg}_p","aliases":"Reg_p mathbf","display":"\\mathbf{Reg}_p","kind":"use","excerpt":"Fix the prime p and let O_DRLM =Reg_p( F_p) be the discrete semantic category of p-regular sequences over F_p. By the finite-field fixed-radix theorem, every object is also p-automatic. Its globally defined normalized charts are the canonical section module, the strict shortlex pointed digit-linear record, and the minimal least-significant-digit-fi","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0967943add145e14","tex":"\\Rel","aliases":"Rel","display":null,"kind":"use","excerpt":"[Observer base-change exact sequence] The evaluation map π:K^(E)⟶Span_K(E), ∑_e∈ Ea_e[e] ⟼ ∑_e∈ Ea_e e is surjective and fits into the exact sequence 0⟶Rel_K(E) ⟶ K^(E) xrightarrowπSpan_K(E) ⟶0, where Rel_K(E):=kerπ is the module of linear relations among distinct discrete residual states. Hence Set minimization commutes with free linearization exa","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e238c707f931f548","tex":"\\Rel_K","aliases":"Rel_K Rel","display":null,"kind":"use","excerpt":"[Observer base-change exact sequence] The evaluation map π:K^(E)⟶Span_K(E), ∑_e∈ Ea_e[e] ⟼ ∑_e∈ Ea_e e is surjective and fits into the exact sequence 0⟶Rel_K(E) ⟶ K^(E) xrightarrowπSpan_K(E) ⟶0, where Rel_K(E):=kerπ is the module of linear relations among distinct discrete residual states. Hence Set minimization commutes with free linearization exa","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f582ccd1037c1131","tex":"\\mathfrak{Rep}","aliases":"Rep mathfrak","display":"\\mathfrak{Rep}","kind":"use","excerpt":"For every stratum, the invertible part is the maximal subgroupoid Rep_α( K) = Core( M_α). This core records changes of presentation that preserve all structure required by the overlap. Quotients, truncations, completions, specializations, differential annihilator maps, and compression maps generally remain outside it.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c149f9fc50e0dfdb","tex":"\\operatorname{Rep}","aliases":"Rep operatorname","display":"\\operatorname{Rep}","kind":"use","excerpt":"[Tannakian boundary] Let T be a neutral Tannakian category over k and ω: T⟶Vect_k a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) is an affine group scheme and T≃Rep_k( G) as tensor categories. A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary [source reference].","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-151c24aa3109de7d","tex":"\\operatorname{Rep}_k","aliases":"Rep_k operatorname","display":"\\operatorname{Rep}_k","kind":"use","excerpt":"[Tannakian boundary] Let T be a neutral Tannakian category over k and ω: T⟶Vect_k a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) is an affine group scheme and T≃Rep_k( G) as tensor categories. A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary [source reference].","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7079ca26521c500e","tex":"\\mathfrak{Rep}_\\alpha","aliases":"Rep_α mathfrak alpha","display":"\\mathfrak{Rep}_\\alpha","kind":"use","excerpt":"For every stratum, the invertible part is the maximal subgroupoid Rep_α( K) = Core( M_α). This core records changes of presentation that preserve all structure required by the overlap. Quotients, truncations, completions, specializations, differential annihilator maps, and compression maps generally remain outside it.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ecf96b50772dd74","tex":"\\operatorname{Res}","aliases":"Res operatorname","display":"\\operatorname{Res}","kind":"use","excerpt":"Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e8877f478fbadf74","tex":"\\operatorname{Res}_{E_N/R}","aliases":"Res_E_N/R operatorname","display":"\\operatorname{Res}_{E_N/R}","kind":"use","excerpt":"Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2ea2d1c18cca3e20","tex":"\\operatorname{Res}_{H\\cap x^{-1}Kx}^H","aliases":"Res_H∩ x^-1Kx^H operatorname cap","display":"\\operatorname{Res}_{H\\cap x^{-1}Kx}^H","kind":"use","excerpt":"[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7bce2162bd53d859","tex":"\\operatorname{Res}_K^G","aliases":"Res_K^G operatorname","display":"\\operatorname{Res}_K^G","kind":"use","excerpt":"[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":"If deg P≤ D, then deg(PH) ≤ D+a+(p-2)b. Cartier selection divides the retained degrees by p. The definition of D gives deg T_d(P) ≤ lfloorfracD+a+(p-2)bprfloor ≤ D. This proves (i).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f9c0392cebf6b641","tex":"\\RMah","aliases":"RMah RMah mathbf","display":null,"kind":"use","excerpt":"Fix a prime p. Let RMah_p^× = { F(z)∈ 1+z F_p[[z]]: fracF(z)F(z^p)∈ F_p(z) } be the discrete category of unit formal series with rational canonical functional-Mahler mask. This category contains the periodic objects of [source reference], but for odd p it also contains genuinely nonrational and non-C-finite automatic series.","count":14},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3be9a937e3e90337","tex":"\\RMah_{p,\\mathrm{col}}^{\\times}","aliases":"RMah_p,col^× RMah mathrm times mathbf mathrm times","display":"{\\mathbf{RMah}}_{p,\\mathrm{col}}^{\\times}","kind":"use","excerpt":"The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9baac0c4c7fa8bfb","tex":"\\RMah_{p,\\mathrm{ncol}}^{\\times}","aliases":"RMah_p,ncol^× RMah mathrm times mathbf mathrm times","display":"{\\mathbf{RMah}}_{p,\\mathrm{ncol}}^{\\times}","kind":"use","excerpt":"The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ef5f15a1b432fd34","tex":"\\RMah_p^{\\times}","aliases":"RMah_p^× RMah times RMah_p^× RMah times mathbf times","display":null,"kind":"use","excerpt":"Fix a prime p. Let RMah_p^× = { F(z)∈ 1+z F_p[[z]]: fracF(z)F(z^p)∈ F_p(z) } be the discrete category of unit formal series with rational canonical functional-Mahler mask. This category contains the periodic objects of [source reference], but for odd p it also contains genuinely nonrational and non-C-finite automatic series.","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-61f9609b70de73a0","tex":"\\rVert","aliases":"rVert","display":"\\rVert","kind":"use","excerpt":"Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b617b296bbfd924b","tex":"\\mathfrak S","aliases":"S mathfrak","display":"\\mathfrak S","kind":"use","excerpt":"[label=textup(roman*)] item Desc( R) exists and has the universal property of the finite pseudolimit of R. item There is a canonical equivalence Core(Desc( R)) ≃ Desc( R^≃). item Every pseudonatural transformation T: R⇒ S induces a functor Desc(T):Desc( R)→Desc( S). If every T_U is an equivalence, then Desc(T) is an equivalence. A pointwise noninve","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cf8c7f65df694c86","tex":"S^{-1}","aliases":"S^-1","display":"S^{-1}","kind":"use","excerpt":"An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2b70d7597172d6b7","tex":"s^{-1}","aliases":"s^-1","display":"s^{-1}","kind":"use","excerpt":"c · b, the equation H^*QH=Q forces a=0, contradicting positive definiteness. For Q_s, the metric-normalized matrix is s · 0 0 · 1 H s^-1 · 0 0 · 1 = 1 · s 0 · 1, whose singular-value condition number tends to one. The conditioning formula is immediate.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-44c4e75467466aba","tex":"s^{-2}","aliases":"s^-2","display":"s^{-2}","kind":"use","excerpt":"0 · 1. There is no positive-definite metric Q with H^*QH=Q. Nevertheless, for Q_s=diag(s^2,1), we have D_Q_s(H)⟶1 (sdownarrow0), while κ_2(Q_s)=s^-2⟶∞. Thus the metric-completion defect has zero infimum but is not attained.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"[Quadratic-radical parity packets] The packets satisfy E^2+xO^2=1+px, 2EO=-s. Their eliminants are E^4-(1+px)E^2+fracs^2x4=0 and xO^4-(1+px)O^2+fracs^24=0. If s≠0, α^2≠β^2, then both selected packets have degree four over K(x).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-49dc1e8e164f8d01","tex":"S^j","aliases":"S^j","display":"S^j","kind":"use","excerpt":"Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite pre","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a14d09587ee9543a","tex":"\\mathfrak s^{\\rm Gal}_{2,0}","aliases":"s^rm Gal_2,0 mathfrak rm","display":"\\mathfrak s^{\\rm Gal}_{2,0}","kind":"use","excerpt":"[Antisymmetric branch collision] If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-189f8b61fe576c2b","tex":"\\mathfrak s^{\\rm Gal}_{2,1}","aliases":"s^rm Gal_2,1 mathfrak rm","display":"\\mathfrak s^{\\rm Gal}_{2,1}","kind":"use","excerpt":"r:block10-antisymmetric-collision If β=-α≠0, then O=0, E^2=1-α^2x. For the even coordinate, | B_2|=4, | C_2,0|=3, |Γ· E|=2. Thus c^rm tuple_2,0=1, s^rm Gal_2,0=1. For the selected odd coordinate, | C_2,1|=3, |Γ· O|=1, so c^rm tuple_2,1=1, s^rm Gal_2,1=2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e57ae220c906bafc","tex":"S^s","aliases":"S^s","display":"S^s","kind":"use","excerpt":"Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite pre","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ee3cd44ebbba07c1","tex":"S_*","aliases":"S_*","display":"S_*","kind":"use","excerpt":"Addendum 04 proves that the prefix S_*={0,1,…,L_ Z(m,d)-1} with L_ Z(m,d)=min(d+1,B(m)), B(m)=max({1}∪{p^v_p(m):pmid m}), always admits an integer decoder and is cardinality-minimal [source reference].","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f1e5183bd7411038","tex":"S_0","aliases":"S_0","display":"S_0","kind":"use","excerpt":"[Exact visible-support loss ledger] Use the finite fold data of (L8) and fix an input family b=(b_e)_e∈ E. Define S_0:={e∈ E:b_e≠0}, A_0:={e∈ S_0:τ_e(b_e)=0}, S_1:=S_0∖ A_0, and let Γ:=κ(S_1)⊆ T. For t∈ T, write the folded coordinate as overline b_t := ∑_substacke∈ S_1","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"use","excerpt":"[Exact visible-support loss ledger] Use the finite fold data of (L8) and fix an input family b=(b_e)_e∈ E. Define S_0:={e∈ E:b_e≠0}, A_0:={e∈ S_0:τ_e(b_e)=0}, S_1:=S_0∖ A_0, and let Γ:=κ(S_1)⊆ T. For t∈ T, write the folded coordinate as overline b_t := ∑_substacke∈ S_1","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-549ae8b0f114acc4","tex":"s_2","aliases":"s_2","display":"s_2","kind":"use","excerpt":"2 be the parity of the binary digit sum of n, and put u_n=1+t_n. The sequence u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq with q odd, the binary digit count of 2^m-T has parity m-ν-s_2(q-1) for ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"[The S_3 nonnormal atlas] Let G=S_3 and let H be generated by one transposition. There are three conjugate order-two subgroups and hence three conjugate cubic fixed-field charts. Their subgroup core is trivial, while their normal closure is all of S_3. By contrast, the order-three subgroup A_3 is normal and its fixed chart carries the quotient symm","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d50bfc3cab84f7ec","tex":"\\mathcal S_a","aliases":"S_a mathcal","display":"\\mathcal S_a","kind":"use","excerpt":"[Complete clopen packet equivalence] The analysis map B_a: C_p ⟶ ∏_s=0^p^a-1 C_p is an isometric isomorphism. Its inverse is the piecewise synthesis ( S_a g)(x) =g_s (fracx-sp^a) for x∈ s+p^a Z_p. Thus the clopen residue cover of Z_p has effective descent for both objects and equality morphisms.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-508e9a069e81ec8b","tex":"S_d","aliases":"S_d","display":"S_d","kind":"use","excerpt":"[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coe","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-45333b1806be2764","tex":"S_e","aliases":"S_e","display":"S_e","kind":"use","excerpt":"[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coe","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b4ebcce0dc1a5c39","tex":"s_k","aliases":"s_k","display":"s_k","kind":"use","excerpt":"[Digit-sum regularity] Let s_k(n) be the sum of the base-k digits of n, regarded as a sequence over a characteristic-zero field. Then r_k(s_k)=2, and K_k(s_k) = {s_k+c:c∈ Z_≥0}. At exact depth ℓ, the number of distinct sections is (k-1)ℓ+1. Thus s_k is k-regular but not k-automatic over a characteristic-zero field.","count":25},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-27ec07552f148ce9","tex":"s_{\\max}","aliases":"s_max max","display":"s_{\\max}","kind":"use","excerpt":"First, it exhaustively enumerates bounded deterministic automata in three configurations: (k,q,s_max)=(2,2,3), (2,3,3), (3,2,2). Every candidate is restricted to reachable states, minimized by behavioral equivalence, and filtered by the terminal-high-zero consistency required by integer semantics. For every resulting normalized automaton the verifi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-640c00c7c9a35171","tex":"S_U","aliases":"S_U","display":"S_U","kind":"use","excerpt":"item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-427ad78ec70de65d","tex":"S_{U_0}","aliases":"S_U_0","display":"S_{U_0}","kind":"use","excerpt":"item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cb79e1a36c0e4809","tex":"S_V","aliases":"S_V","display":"S_V","kind":"use","excerpt":"item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-87a2b7a1a86d2b97","tex":"S_z","aliases":"S_z","display":"S_z","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3e03c4abc215f7cb","tex":"S_z^k","aliases":"S_z^k","display":"S_z^k","kind":"use","excerpt":"Factor H=OC, where row i of O is e_0^ TM_N^i, and column j of C is M_N^j e_0. For j<N, the binomial expansion of (I+S_z)^j e_0 is a unit upper-triangular change from (e_0,…,e_N-1), hence det C=1. The rows e_0^ TS_z^k, 0≤ k<N, are e_0^ T followed by z e_N-k^ T, so their determinant is ± z^N-1. The binomial row change is also unit triangular, proving","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0a3460b899a13ac7","tex":"S_z^N","aliases":"S_z^N","display":"S_z^N","kind":"use","excerpt":"[Exact polyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-145837c99a5c8f54","tex":"\\operatorname{Sat}","aliases":"Sat operatorname","display":"\\operatorname{Sat}","kind":"use","excerpt":"[Task--congruence Galois connection] For every query family Q and relation R, Q⊆Def(R) Longleftrightarrow R⊆Ind( Q). Consequently, Cl( Q):=Def(Ind( Q)) is a closure operator on query families, and Sat(R):=Ind(Def(R)) is the observational saturation of R relative to the admitted query universe.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-17f4571694a5d7ee","tex":"\\operatorname{SCC}","aliases":"SCC operatorname","display":"\\operatorname{SCC}","kind":"use","excerpt":"For the packet system textup(11.4), define the directed dependency graph G_Φ on I by placing an edge j⟶ i when some active block contributing to the equation for u_i contains u_j as an input. Let SCC(G_Φ) be its strongly connected components. A component is called cyclic when it contains more than one address or is a singleton carrying a self-loop.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1fa284989c5acecb","tex":"\\mathbf{Sec}","aliases":"Sec mathbf","display":"\\mathbf{Sec}","kind":"use","excerpt":"[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-04917db0ff97f6e0","tex":"\\mathbf{Sec}_k^{\\mathrm{can}}","aliases":"Sec_k^can mathbf mathrm","display":"\\mathbf{Sec}_k^{\\mathrm{can}}","kind":"use","excerpt":"[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-90afe1bed224581d","tex":"\\operatorname{Semilin}","aliases":"Semilin operatorname","display":"\\operatorname{Semilin}","kind":"use","excerpt":"Let L/K be finite Galois with group G. A semilinear G-object is an L-vector space V with maps ρ_σ:V→ V, ρ_σ(λ v)=σ(λ)ρ_σ(v), satisfying the group law. Scalar extension and fixed vectors define functors L⊗_K-:Vect_K⟶Semilin_G(L), (-)^G:Semilin_G(L)⟶Vect_K.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-76fab61f39af6e92","tex":"\\operatorname{Semilin}_G","aliases":"Semilin_G operatorname","display":"\\operatorname{Semilin}_G","kind":"use","excerpt":"Let L/K be finite Galois with group G. A semilinear G-object is an L-vector space V with maps ρ_σ:V→ V, ρ_σ(λ v)=σ(λ)ρ_σ(v), satisfying the group law. Scalar extension and fixed vectors define functors L⊗_K-:Vect_K⟶Semilin_G(L), (-)^G:Semilin_G(L)⟶Vect_K.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1689aab20eb2fec0","tex":"\\Seq","aliases":"Seq mathbf","display":"{\\mathbf{Seq}}","kind":"use","excerpt":"Let Seq_K^pers be the discrete groupoid of persistent C-finite sequences over K.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0209eab911c086eb","tex":"\\Seq_K^{\\mathrm{pers}}","aliases":"Seq_K^pers Seq mathrm mathbf mathrm","display":"{\\mathbf{Seq}}_K^{\\mathrm{pers}}","kind":"use","excerpt":"Let Seq_K^pers be the discrete groupoid of persistent C-finite sequences over K.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-89922a6778cee948","tex":"\\mathrm{sl}","aliases":"sl mathrm","display":"\\mathrm{sl}","kind":"use","excerpt":"[Fixed-radix regular normalized atlas] Let Reg_k(K) be the discrete semantic category of K-valued k-regular sequences. Let Sec_k^can(K) be the category of canonical section-module records [source reference], and let Lin_k,K^sl be the category of strict shortlex records [source reference], with morphisms required to preserve the represented sequence","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f5a8e0e447dd2d94","tex":"\\mathsf{SL}_{p,\\mathbb F_p}","aliases":"SL_p, F_p mathsf mathbb","display":"\\mathsf{SL}_{p,\\mathbb F_p}","kind":"use","excerpt":"For a∈ O_DRLM, define J_DRLM,p(a) = ( SL_p, F_p(a), P_rm(a), P_per(a) ), where SL_p, F_p(a) is the strict shortlex record, P_rm(a) is empty off the rational-mask locus and otherwise contains the reduced mask, Cartier module, collapse obstruction, and minimal automaton coordinates, and P_per(a) is empty off the purely periodic locus and otherwise co","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9353bd8b1dba9093","tex":"\\operatorname{span}_{\\mathbb F_p}","aliases":"span_ F_p operatorname mathbb","display":"\\operatorname{span}_{\\mathbb F_p}","kind":"use","excerpt":"For a digit word w=d_0⋯ d_ℓ-1, read least significant digit first and write Λ_w = Λ_d_ℓ-1⋯Λ_d_0. Define the Cartier kernel and its linear span by K_p(F) = {Λ_wF:w∈{0,…,p-1}^*}, V_p(F) = span_ F_p K_p(F).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54290d6da64dd36b","tex":"\\operatorname{span}_{\\mathbb Q}","aliases":"span_ Q operatorname mathbb","display":"\\operatorname{span}_{\\mathbb Q}","kind":"use","excerpt":"Throughout this section, k≥2 is fixed and a: N⟶ Z is an integer-valued sequence whose rational section space V_ Q(a) = span_ Q K_k(a) ⊆ Q^ N has finite dimension ρ_k(a)=dim_ QV_ Q(a). Thus a is k-regular after extension of scalars to Q.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-546e1d79f916cfb6","tex":"\\operatorname{span}_{K(z)}","aliases":"span_K(z) operatorname","display":"\\operatorname{span}_{K(z)}","kind":"use","excerpt":"[Derivative-module rank] For a nonzero D-finite series F, r_D(F) = dim_K(z)span_K(z) {F,DF,D^2F,…}.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ae51e1c510f803eb","tex":"\\operatorname{Spec}","aliases":"Spec operatorname","display":"\\operatorname{Spec}","kind":"use","excerpt":"Put R=K(x), E_N=R[z]/(z^N-x). Characteristic zero makes z^N-x separable over R. Interpret the monic polynomial P_F(z,Y) as a polynomial in E_N[Y], and define B_N,F=E_N[Y]/(P_F). The packet polynomial system is the coordinate presentation of the finite Weil restriction Res_E_N/RSpecB_N,F.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-59a4b2862212f500","tex":"\\operatorname{spec}","aliases":"spec operatorname","display":"\\operatorname{spec}","kind":"use","excerpt":"[Coordinate covariance and composition] Simultaneous coordinate transport of T,Q_s,Q_t leaves D_Q(T) invariant. For composable invertible maps, D_Q(T_2T_1) ≤ D_Q(T_2)D_Q(T_1), so Δ_Q(T_2T_1) ≤ Δ_Q(T_2)+Δ_Q(T_1). For a closed word H, D_Q(H) ≥ fracmax_λ∈spec(H)|λ| min_λ∈spec(H)|λ|.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"use","excerpt":"The obstruction ω_p-1 divides the semantic category into two exact loci: RMah_p^× = RMah_p,col^× sqcup RMah_p,ncol^×. The collapse locus consists of rational, hence ultimately periodic, series. The noncollapse locus consists of non-C-finite automatic series. The unit-periodic atlas of Periodic atlas is the persistent, zero-transient subatlas inside","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a72b2476e977b00d","tex":"\\State","aliases":"State mathbf","display":"{\\mathbf{State}}","kind":"use","excerpt":"An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e6fb2949ad440310","tex":"\\State_K^{\\mathrm{min}}","aliases":"State_K^min State mathrm mathbf mathrm","display":"{\\mathbf{State}}_K^{\\mathrm{min}}","kind":"use","excerpt":"An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-49eb10e5997665b3","tex":"\\State_{\\min}","aliases":"State_min State min mathbf min","display":"{\\mathbf{State}}_{\\min}","kind":"use","excerpt":"For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a), T(a) · =min{N:m_amid x^N-1} =min{N:Q_amid 1-z^N}, | A_p(a)| · =| K_p(a)|","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7df451bef70a0ea5","tex":"\\mathrm{strong}","aliases":"strong mathrm","display":"\\mathrm{strong}","kind":"use","excerpt":"Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9234b3b715674c15","tex":"\\subsubsection","aliases":"subsubsection","display":null,"kind":"use","excerpt":"[4pt] Reduced OGF ↔ minimal DFAO · equivalence on the periodic essential image · unit purely periodic series and reachable minimal LSD-first p-DFAOs · coefficient semantics, minimal period, canonical p-kernel, output behavior · equality of represented sequences · a nonminimal residue-pair automaton is quotiented before entering the normalized chart","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"use","excerpt":"For each v∈ H, strong continuity gives sup_u∈ U_n U_uv-v ⟶0. Averaging this estimate over U_n proves the first statement. Apply it to the quotient action on the N-fixed packet for the second.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-15da85d6697dd325","tex":"\\sup_{u\\in U_n}","aliases":"sup_u∈ U_n sup in","display":"\\sup_{u\\in U_n}","kind":"use","excerpt":"For each v∈ H, strong continuity gives sup_u∈ U_n U_uv-v ⟶0. Averaging this estimate over U_n proves the first statement. Apply it to the quotient action on the N-fixed packet for the second.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9498acc656e68b9f","tex":"\\operatorname{Supp}","aliases":"Supp operatorname","display":"\\operatorname{Supp}","kind":"use","excerpt":"N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0b7cc491b1ee030b","tex":"\\operatorname{supp}","aliases":"supp operatorname","display":"\\operatorname{supp}","kind":"use","excerpt":"kappa(e)=tτ_e(b_e) ∈ C_t, and put supp(overline b) :={t∈ T:overline b_t≠0}, K_0:={t∈Γ:overline b_t=0}. Then |S_0|-|supp(overline b)| = |A_0| +(|S_1|-|Γ|) +|K_0|. The three terms record, respectively, annihilation by the local maps τ_e, collision of surviving lane addresses under κ, and cancellation among folded values at a common address. If every ","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8ca3f3c55d3c22be","tex":"\\operatorname{Supp}_N","aliases":"Supp_N operatorname","display":"\\operatorname{Supp}_N","kind":"use","excerpt":"N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ad3fefc79ffc997c","tex":"\\operatorname{Sw}","aliases":"Sw operatorname","display":"\\operatorname{Sw}","kind":"use","excerpt":"[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-580023d673e19885","tex":"\\operatorname{Sym}","aliases":"Sym operatorname","display":"\\operatorname{Sym}","kind":"use","excerpt":"Fix a base chart U_0. Every loop γ∈Aut_mathscr G(U_0) acts on an address set by i↦λ^A_γ,r(i). Write H_A^r := im ( Aut_mathscr G(U_0) ⟶ Sym(I_A,U_0,r) ) for the resulting finite monodromy group. For a source-target pair, the same loop acts diagonally by (a,i)⟼ (λ^Y_γ,u(a), λ^X_γ,r(i)). Its image in the symmetric group of I_Y,U_0,u× I_X,U_0,r is deno","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fc34b95840f12e94","tex":"\\Syn_{\\mathrm{ak}}","aliases":"Syn_ak Syn mathrm operatorname mathrm","display":"{\\operatorname{Syn}}_{\\mathrm{ak}}","kind":"use","excerpt":"[Finite additive-Karoubi resolved envelope] The category Syn_ak( K_rm lin) := Kar (Add(𝕜 Q/I)) is small, additive, idempotent complete, and 𝕜-linear. For every additive idempotent-complete 𝕜-linear category C, restriction to the generating quiver induces an equivalence Fun_𝕜^⊕ (Syn_ak( K_rm lin), C) ≃ Mod_ak( K_rm lin, C), where the right side is t","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8a075c4b2495703c","tex":"\\Synthesis_X^r","aliases":"Synthesis_X^r Synthesis","display":null,"kind":"use","excerpt":"At resolution r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-da4fb737a4a4bc72","tex":"\\mathfrak T","aliases":"T mathfrak","display":"\\mathfrak T","kind":"use","excerpt":"[Finite bottom-up residue automaton] Label every leaf by 1∈ Z/N Z and every m-ary vertex by the sum of its child labels. The reachable state set is exactly {j:j≡1pmodg_N}. Thus the finite pair T_N( A) = (d_ A,g_N,N/g_N) tag11.11 classifies packet support for the marked simply generated family.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"use","excerpt":"Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"For the even packet, put T=E^2. Then 4xT^2-T+1=0. Its discriminant 1-16x is not a square in K(x), so [K(x)(T):K(x)]=2. The norm of T from this quadratic field is 1/(4x), which is not a square in K(x). Therefore T is not a square in its quadratic field and E has degree four.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-68c16f5dbc93324c","tex":"t^j","aliases":"t^j","display":"t^j","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54815db7d2265570","tex":"T^m","aliases":"T^m","display":"T^m","kind":"use","excerpt":"Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aea3af576abad800","tex":"t^m","aliases":"t^m","display":"t^m","kind":"use","excerpt":"Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8fa21937d664fe3d","tex":"t^N","aliases":"t^N","display":"t^N","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-82d87f0c09c0e190","tex":"t^n","aliases":"t^n","display":"t^n","kind":"use","excerpt":"If λ=-M, expand (1-αω_N^rt)^M = ∑_n=0^M binom Mn(-α)^nω_N^rnt^n. The rotation matrix is a truncated discrete Fourier matrix. Its rank is the number of available residue classes, namely min(N,M+1).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a56f11e5365cd5f6","tex":"\\mathcal T_{\\mathcal A}","aliases":"T_ A mathcal mathcal","display":"\\mathcal T_{\\mathcal A}","kind":"use","excerpt":"[Selected polynomial fixed point and tree expansion] Assume a+φ(R)≤ R, κ_R<1. Then equation textup(11.1) has a unique solution in the closed ball overline B_R. Moreover, that solution is the absolutely convergent sum u=∑_T∈ T_ Aev(T), tag11.2 where A={m:A_m≠0}, the leaves of T evaluate to s, and an internal vertex of arity m evaluates by A_m applie","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e14cb192aabec81c","tex":"T_0","aliases":"T_0","display":"T_0","kind":"use","excerpt":"Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expa","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9bbeb7d12eb04164","tex":"T_1","aliases":"T_1","display":"T_1","kind":"use","excerpt":"Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expa","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c462c3acfcc92ec3","tex":"T_2","aliases":"T_2","display":"T_2","kind":"use","excerpt":"Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expa","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6ccea74ce6b4ba7d","tex":"t_{2^k+n}","aliases":"t_2^k+n","display":"t_{2^k+n}","kind":"use","excerpt":"Negative control: automatic does not imply rational mask. Let t_n∈ F_2 be the parity of the binary digit sum of n, and put u_n=1+t_n. The sequence u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq wit","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d0ee7cbc371f28e8","tex":"t_{2^m}","aliases":"t_2^m","display":"t_{2^m}","kind":"use","excerpt":"(u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq with q odd, the binary digit count of 2^m-T has parity m-ν-s_2(q-1) for all sufficiently large m. Choosing the parity of m makes t_2^m-T≠ t_2^m, cont","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ad17c3a6f0f02a0","tex":"t_{2^m-T}","aliases":"t_2^m-T","display":"t_{2^m-T}","kind":"use","excerpt":"(u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq with q odd, the binary digit count of 2^m-T has parity m-ν-s_2(q-1) for all sufficiently large m. Choosing the parity of m makes t_2^m-T≠ t_2^m, cont","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-60c14b77986170c6","tex":"\\mathcal T_a","aliases":"T_a mathcal","display":"\\mathcal T_a","kind":"use","excerpt":"[Complete coefficient-packet equivalence] For M=p^a, the map T_a:c_0( Z_p) ⟶ ∏_s=0^p^a-1c_0( Z_p), c⟼( T_p^a,sc)_s, is a topological isomorphism. It is the coefficient conjugate of complete clopen analysis.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-775a705646f9a2a5","tex":"T_{a;i_1,\\ldots,i_d}","aliases":"T_a;i_1,…,i_d ldots","display":"T_{a;i_1,\\ldots,i_d}","kind":"use","excerpt":"[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of mult","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ac92b2cd0783d43a","tex":"T_{\\bar e}","aliases":"T_bar e bar","display":"T_{\\bar e}","kind":"use","excerpt":"Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a48ad39b3e57f130","tex":"T_D","aliases":"T_D","display":"T_D","kind":"use","excerpt":"Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e8969a358fc5266","tex":"T_d","aliases":"T_d","display":"T_d","kind":"use","excerpt":"Write a=deg A, b=deg B, H(z)=A(z)B(z)^p-2, and define D(A,B) = max{ b, lceilfraca+(p-2)bp-1rceil }. Let P_D= F_p[z]_≤ D. For 0≤ d<p, define T_d(P)=Λ_d(PH), where Λ_d selects the coefficients in degrees congruent to d modulo p.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-03f659a72f2fc016","tex":"T_{d_0}","aliases":"T_d_0","display":"T_{d_0}","kind":"use","excerpt":"Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-629be28e4a453233","tex":"T_{d_{\\ell-1}}","aliases":"T_d_ℓ-1 ell","display":"T_{d_{\\ell-1}}","kind":"use","excerpt":"Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bb1ad3b8487cb395","tex":"T_e","aliases":"T_e","display":"T_e","kind":"use","excerpt":"Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-021991eaf5ecd844","tex":"T_e^*","aliases":"T_e^*","display":"T_e^*","kind":"use","excerpt":"raph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge s","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3a0db745a91a40a2","tex":"T_e^{-1}","aliases":"T_e^-1","display":"T_e^{-1}","kind":"use","excerpt":"Finite graph transport gives the sharpest reversible gluing model in the Library. Let Γ=(V,E) be a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-02d6c65a61445f1b","tex":"\\mathcal T_{MN,Mt+s}","aliases":"T_MN,Mt+s mathcal","display":"\\mathcal T_{MN,Mt+s}","kind":"use","excerpt":"[Mixed-radix kernel composition] For M,N≥1, ∑_k≥0 Θ_M,s(r,k)Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). Equivalently, T_N,t T_M,s = T_MN,Mt+s. When gcd(M,N)=1, the two Chinese-remainder refinement paths to a common address agree exactly. Their holonomy is the identity.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-64afaf0ed6cbcadc","tex":"T_n","aliases":"T_n","display":"T_n","kind":"use","excerpt":"Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expa","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-426c2e2ef16a6555","tex":"\\mathfrak T_N","aliases":"T_N mathfrak","display":"\\mathfrak T_N","kind":"use","excerpt":"[Finite bottom-up residue automaton] Label every leaf by 1∈ Z/N Z and every m-ary vertex by the sum of its child labels. The reachable state set is exactly {j:j≡1pmodg_N}. Thus the finite pair T_N( A) = (d_ A,g_N,N/g_N) tag11.11 classifies packet support for the marked simply generated family.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c2ed7a9fa236bbf0","tex":"t_n","aliases":"t_n","display":"t_n","kind":"use","excerpt":"Negative control: automatic does not imply rational mask. Let t_n∈ F_2 be the parity of the binary digit sum of n, and put u_n=1+t_n. The sequence u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq wit","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e6258c6235b62419","tex":"\\mathcal T_{N,t}","aliases":"T_N,t mathcal","display":"\\mathcal T_{N,t}","kind":"use","excerpt":"[Mixed-radix kernel composition] For M,N≥1, ∑_k≥0 Θ_M,s(r,k)Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). Equivalently, T_N,t T_M,s = T_MN,Mt+s. When gcd(M,N)=1, the two Chinese-remainder refinement paths to a common address agree exactly. Their holonomy is the identity.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1265a9ce41e50509","tex":"\\mathcal T_{p^a,s}","aliases":"T_p^a,s mathcal","display":"\\mathcal T_{p^a,s}","kind":"use","excerpt":"[Complete coefficient-packet equivalence] For M=p^a, the map T_a:c_0( Z_p) ⟶ ∏_s=0^p^a-1c_0( Z_p), c⟼( T_p^a,sc)_s, is a topological isomorphism. It is the coefficient conjugate of complete clopen analysis.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6b2aa248182f38a8","tex":"T_U","aliases":"T_U","display":"T_U","kind":"use","excerpt":"Desc( R) exists and has the universal property of the finite pseudolimit of R. item There is a canonical equivalence Core(Desc( R)) ≃ Desc( R^≃). item Every pseudonatural transformation T: R⇒ S induces a functor Desc(T):Desc( R)→Desc( S). If every T_U is an equivalence, then Desc(T) is an equivalence. A pointwise noninvertible T remains directional","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d1272d40e795db7e","tex":"T_w","aliases":"T_w","display":"T_w","kind":"use","excerpt":"For every word w, repeated use of [source reference] gives Θ_A,B(T_wB)=Λ_wF. Thus Θ_A,B maps the reachable numerator set onto the Cartier kernel. Injectivity of Θ_A,B makes the map bijective, and taking linear spans gives [source reference].","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-faad188781a0fc33","tex":"\\operatorname{Tor}","aliases":"Tor operatorname","display":"\\operatorname{Tor}","kind":"use","excerpt":"Both M_k(a) and L_k(a) are free abelian. The long exact Tor sequence therefore begins 0⟶ Tor_1^ Z (C_k(a), Z/m Z) ⟶ M_k(a)⊗ Z/m Z ⟶ L_k(a)⊗ Z/m Z. For any abelian group C, Tor_1^ Z(C, Z/m Z) ≅ C[m], C⊗ Z/m Z ≅ C/mC. The digit-section maps preserve the original short exact sequence, so naturality of Tor makes the five-term sequence digit-equivariant","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e1414af246c00450","tex":"\\operatorname{Tor}_1^{\\mathbb Z}","aliases":"Tor_1^ Z operatorname mathbb","display":"\\operatorname{Tor}_1^{\\mathbb Z}","kind":"use","excerpt":"Both M_k(a) and L_k(a) are free abelian. The long exact Tor sequence therefore begins 0⟶ Tor_1^ Z (C_k(a), Z/m Z) ⟶ M_k(a)⊗ Z/m Z ⟶ L_k(a)⊗ Z/m Z. For any abelian group C, Tor_1^ Z(C, Z/m Z) ≅ C[m], C⊗ Z/m Z ≅ C/mC. The digit-section maps preserve the original short exact sequence, so naturality of Tor makes the five-term sequence digit-equivariant","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1303203f6195020a","tex":"\\operatorname{Tr}","aliases":"Tr operatorname","display":"\\operatorname{Tr}","kind":"use","excerpt":"[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b66b6a2cb286718a","tex":"\\mathbf U","aliases":"U mathbf","display":"\\mathbf U","kind":"use","excerpt":"Let P(z,Y)∈ K[z,Y] be the normalized primitive representative of P_F. Introduce indeterminates U=(U_0,…,U_N-1) and substitute Y=∑_j=0^N-1z^jU_j. Every polynomial in z has a unique decomposition modulo the relation z^N=x: P (z,∑_j=0^N-1z^jU_j) = ∑_r=0^N-1z^rΦ_N,r(x, U). The polynomials Φ_N,r(x, U)∈ K[x, U] are the algebraic packet equations.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ec61c8382b2d3710","tex":"U'","aliases":"U'","display":"U'","kind":"use","excerpt":"rem[Cokernel criterion, universality, and naturality] For a correction datum (D,r) [source reference]: [label=textup(roman*)] item a correction exists if and only if Ob_D(r)=0; item every linear obstruction test φ:V→ W satisfying φ D=0 factors uniquely through V→coker D; item for every commuting square ccc U · xrightarrowD · V downarrow a · · downa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f917d7010564bfa7","tex":"u^{(0)}","aliases":"u^(0)","display":"u^{(0)}","kind":"use","excerpt":"Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-adb6b8e298cab5d1","tex":"u^{(h)}","aliases":"u^(h)","display":"u^{(h)}","kind":"use","excerpt":"Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ebccc040c39ea352","tex":"u^{(h+1)}","aliases":"u^(h+1)","display":"u^{(h+1)}","kind":"use","excerpt":"Let u^(0)=0 and u^(h+1)=Ψ(u^(h)). Expanding recursively shows that u^(h) is the sum of the evaluations of all allowed plane rooted trees of height at most h-1. If r_0=0 and r_h+1=a+φ(r_h), then the sum of the norms of those tree evaluations is at most r_h. The scalar iterates increase and remain bounded by R, hence converge. Monotone convergence of","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-da5e436b748b454f","tex":"u^p","aliases":"u^p","display":"u^p","kind":"use","excerpt":"For a principal unit u=1+p, u^x=∑_r≥0p^rbinom xr. The branch x=pq has coefficients (u^p-1)^k, whose valuation is at least 2k. The branch therefore improves the original slope-one certificate.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dcb9d4ba9bfde464","tex":"U^q","aliases":"U^q","display":"U^q","kind":"use","excerpt":"Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3fd5c0aabd378aa4","tex":"u^q","aliases":"u^q","display":"u^q","kind":"use","excerpt":"Take u=z+u^q, q≥2. tag11.14 Lagrange inversion gives the exact coefficient formula [z^n]u = displaystyle frac1(q-1)m+1binomqmm, · n=1+(q-1)m,","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-352c1df03303fdc3","tex":"u^{q-1}","aliases":"u^q-1","display":"u^{q-1}","kind":"use","excerpt":"Formula textup(11.15) follows from Lagrange inversion applied to z=u(1-u^q-1). The support and active-address statements are [source reference] with d_ A=q-1. Expanding u^q in complete packets and collecting the exponent carry r_1+⋯+r_q=j+Nc gives textup(11.16). The critical equations qτ^q-1=1, ρ=τ-τ^q give the printed values, and [source reference","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b738a066d3a87bde","tex":"u^x","aliases":"u^x","display":"u^x","kind":"use","excerpt":"For a principal unit u=1+p, u^x=∑_r≥0p^rbinom xr. The branch x=pq has coefficients (u^p-1)^k, whose valuation is at least 2k. The branch therefore improves the original slope-one certificate.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8bf706dc6c2686ba","tex":"\\mathcal U_{\\mathbb Z}","aliases":"U_ Z mathcal mathbb","display":"\\mathcal U_{\\mathbb Z}","kind":"use","excerpt":"Let U_ Z=1+z Z[[z]] be the multiplicative group of integral unit series, and let E_ Z= Z^ N_≥1 be the additive group of integer exponent sequences.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0","count":22},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-98de45ac415e9b4a","tex":"U_1","aliases":"U_1","display":"U_1","kind":"use","excerpt":"[Shrinking-subgroup reconstruction] Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-76bfead452deff02","tex":"U_2","aliases":"U_2","display":"U_2","kind":"use","excerpt":"[Shrinking-subgroup reconstruction] Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-65fa973c459a6b3b","tex":"U_c","aliases":"U_c","display":"U_c","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ada27fd6e08006d5","tex":"U_f","aliases":"U_f","display":"U_f","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9f5988962b4f6877","tex":"U_g","aliases":"U_g","display":"U_g","kind":"use","excerpt":"[Normality and quotient compression] For a subgroup H≤ G, the following conditions are equivalent: item Htriangleleft G; item e_H is central in the group algebra; item P_HU_g=U_gP_H for every g∈ G; item the fixed packet V^H is G-invariant; item the fixed field L^H is Galois over K. When they hold, the compressed action on V^H factors through G/H, a","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2db86e8af5a3bba1","tex":"U_g^{-1}","aliases":"U_g^-1","display":"U_g^{-1}","kind":"use","excerpt":"Conjugation sends e_H to e_gHg^-1. Centrality is therefore equivalent to normality. The represented identity U_gP_HU_g^-1=P_gHg^-1 gives the operator conditions. The field and quotient statements are the normal-subgroup part of the finite Galois correspondence.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8a830c0eacd44ea9","tex":"U_h","aliases":"U_h","display":"U_h","kind":"use","excerpt":"Let L/K be a finite Galois extension with group G=Gal(L/K). Let V be a finite-dimensional representation of G over a characteristic-zero field, or a unitary complex realization obtained after scalar extension. For every subgroup H≤ G, define P_H=frac1|H|∑_h∈ HU_h. tag12.1","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"For the packet system textup(11.4), define the directed dependency graph G_Φ on I by placing an edge j⟶ i when some active block contributing to the equation for u_i contains u_j as an input. Let SCC(G_Φ) be its strongly connected components. A component is called cyclic when it contains more than one address or is a singleton carrying a self-loop.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c63ed26b7a73d343","tex":"u_{j_1}","aliases":"u_j_1","display":"u_{j_1}","kind":"use","excerpt":"I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f307291586d985d1","tex":"u_{j_m}","aliases":"u_j_m","display":"u_{j_m}","kind":"use","excerpt":"I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8d86088ae482ad75","tex":"u_m","aliases":"u_m","display":"u_m","kind":"use","excerpt":"B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-30d31e43294cdc7c","tex":"u_{m+1}","aliases":"u_m+1","display":"u_{m+1}","kind":"use","excerpt":"B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-167a8b5df4139bcf","tex":"U_n","aliases":"U_n","display":"U_n","kind":"use","excerpt":"[Shrinking-subgroup reconstruction] Let G be profinite and let U_1⊇ U_2⊇⋯ be a neighborhood basis of the identity consisting of open normal subgroups. Then P_U_n⟶ I tag12.15 strongly in every strongly continuous unitary representation. More generally, if Ntriangleleft G is closed and U_n/N shrinks to the identity in G/N, then P_U_n→ P_N strongly.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"use","excerpt":"Negative control: automatic does not imply rational mask. Let t_n∈ F_2 be the parity of the binary digit sum of n, and put u_n=1+t_n. The sequence u is unit and 2-automatic. It is not eventually periodic. Indeed, eventual period T would propagate backwards through the identities t_2^k+n=1+t_n (0≤ n<2^k) and force a pure period T. Writing T=2^νq wit","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9c883dfc50a46664","tex":"U_{N-1}","aliases":"U_N-1","display":"U_{N-1}","kind":"use","excerpt":"Let P(z,Y)∈ K[z,Y] be the normalized primitive representative of P_F. Introduce indeterminates U=(U_0,…,U_N-1) and substitute Y=∑_j=0^N-1z^jU_j. Every polynomial in z has a unique decomposition modulo the relation z^N=x: P (z,∑_j=0^N-1z^jU_j) = ∑_r=0^N-1z^rΦ_N,r(x, U). The polynomials Φ_N,r(x, U)∈ K[x, U] are the algebraic packet equations.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d17a28f5e5468c9f","tex":"U_p","aliases":"U_p","display":"U_p","kind":"use","excerpt":"[Finite periodic kernel and exact state model] Let a∈Per_p^×,per have minimal period T and period word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d42ca9e4eff519f2","tex":"U_q","aliases":"U_q","display":"U_q","kind":"use","excerpt":"t [z^1+(q-1)n]U =frac1(q-1)n+1binomqnnα^n ≠0 (n≥0), and every other coefficient is zero. Thus its support is exactly the set of degrees congruent to 1pmodq-1. For an integer packet width N≥1, set g=gcd(N,q-1). Exactly N/g residue addresses are active, namely the congruence class 1pmod g. Packetizing the specified multilinear map (U_1,…,U_q)↦ U_1⋯ U","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-47d1799755561eb2","tex":"U_{q,\\alpha}","aliases":"U_q,α alpha","display":"U_{q,\\alpha}","kind":"use","excerpt":"Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ed288c7acdf327f","tex":"U_{r_a}","aliases":"U_r_a","display":"U_{r_a}","kind":"use","excerpt":"[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the sou","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ab56dd36935ce891","tex":"U_u","aliases":"U_u","display":"U_u","kind":"use","excerpt":"For each v∈ H, strong continuity gives sup_u∈ U_n U_uv-v ⟶0. Averaging this estimate over U_n proves the first statement. Apply it to the quotient action on the N-fixed packet for the second.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-97127b69fb3aca55","tex":"\\underline j","aliases":"underline j underline","display":"\\underline j","kind":"use","excerpt":"[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ff1eefe4ba5ba868","tex":"\\underline{j}","aliases":"underlinej underline","display":"\\underline{j}","kind":"use","excerpt":"[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4ad80e13418e07d2","tex":"\\mathfrak V","aliases":"V mathfrak","display":"\\mathfrak V","kind":"use","excerpt":"n equivalence. A pointwise noninvertible T remains directional. item If two global morphism expressions use only constructors in the common verified preservation profile and their equality is proved in the resolved syntax, then, after every domain-specific hypothesis is verified locally, their realizations are equal in Desc( R). item A coherent pse","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"use","excerpt":"rem[Cokernel criterion, universality, and naturality] For a correction datum (D,r) [source reference]: [label=textup(roman*)] item a correction exists if and only if Ob_D(r)=0; item every linear obstruction test φ:V→ W satisfying φ D=0 factors uniquely through V→coker D; item for every commuting square ccc U · xrightarrowD · V downarrow a · · downa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4e7142ad42e2513e","tex":"v'","aliases":"v'","display":"v'","kind":"use","excerpt":"An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8adc12fd10486b0b","tex":"V^G","aliases":"V^G","display":"V^G","kind":"use","excerpt":"[Effective finite Galois vector descent] The two functors above are quasi-inverse equivalences: Vect_K ≃ Semilin_G(L). tag12.16 For every semilinear object V, the fixed packet W=V^G satisfies W⊗_KL≅ V. tag12.17 The equivalence is effective on both objects and morphisms.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d29d15600febd890","tex":"V^{G_0}","aliases":"V^G_0","display":"V^{G_0}","kind":"use","excerpt":"[Orthogonal ramification details] The operators D_i are mutually orthogonal projectors, and V=V^G_0⊕⊕_i D_iV. tag12.10 Every detail packet is stable under the quotient action permitted by the corresponding normal subgroup.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3104ed7defc543ec","tex":"V^{G_i}","aliases":"V^G_i","display":"V^{G_i}","kind":"use","excerpt":"The classical conductor formula uses codimV^G_i. Since P_i projects onto V^G_i, this codimension is Tr(I-P_i). Also codimV^G_i=∑_j≥ id_j. Reversing the finite sums gives the detail formula.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a6544df8b60d670","tex":"V^H","aliases":"V^H","display":"V^H","kind":"use","excerpt":"[Reynolds fixed packet and reverse order] The operator P_H is the projector onto the fixed packet V^H={v:U_hv=v for all h∈ H}. If H≤ J, then P_HP_J=P_JP_H=P_J. tag12.2 Consequently, subgroup inclusion reverses the fixed-projector order.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7d1eb789a7676fd8","tex":"V^J","aliases":"V^J","display":"V^J","kind":"use","excerpt":"The average is invariant under left and right multiplication by elements of H. Therefore P_H^2=P_H, and its range is exactly the fixed subspace. If H≤ J, then V^J⊆ V^H. The two projections are nested and multiply to the projector with smaller range.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-76983b8d3a059b95","tex":"V^{\\langle H,J\\rangle}","aliases":"V^⟨ H,J⟩ langle rangle","display":"V^{\\langle H,J\\rangle}","kind":"use","excerpt":"For two subgroups H,J≤ G, the common fixed packet is always V^H∩ V^J=V^⟨ H,J⟩. The two averaging projectors need not commute.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0f1a455a83b273b0","tex":"v_0","aliases":"v_0","display":"v_0","kind":"use","excerpt":"Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9e3ebcdaba6dff07","tex":"\\mathfrak V_{13}","aliases":"V_13 mathfrak","display":"\\mathfrak V_{13}","kind":"use","excerpt":"nvariant-descent Let A_13 be the atlas of [source reference]. Suppose that, in every connected component: item the local invariant is complete on every normalized chart; item the bridge comparisons are coherent; item invariant-compatible local isomorphisms satisfy effective descent. Then the local invariants induce a functor I_13: Desc( A_13) ⟶ Des","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-641dfc399b115eba","tex":"V_4","aliases":"V_4","display":"V_4","kind":"use","excerpt":"The Galois-Tannakian atlas verifier uses exact rational group-algebra arithmetic, exact finite permutation combinatorics, and exact Gaussian-rational matrix algebra. It audits cyclic groups, V_4, S_3, D_4, and A_4; every subgroup Reynolds idempotent; subgroup inclusion and projector order; normality and centrality; core and normal closure; commutin","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ef50ddb6014d940d","tex":"V_c","aliases":"V_c","display":"V_c","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e98e9295289e8404","tex":"\\mathcal V_{\\mathrm{cert}}","aliases":"V_cert mathcal mathrm","display":"\\mathcal V_{\\mathrm{cert}}","kind":"use","excerpt":"whose constructor support lies in the common verified preservation profile transports to compatible local theorems and hence to the corresponding global resolved object. item The semantic normalizers induce the comparison 2-cells on all nonempty triple overlaps and force the quadruple-overlap coherence equation. item The compatible native invariant","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4f979281d0af8ba8","tex":"V_f","aliases":"V_f","display":"V_f","kind":"use","excerpt":"[Relative obstruction exact sequence] Suppose the diagram ccccccccc 0 · → · U_0 · → · U_f · → · U_c · → · 0 · · downarrow D_0 · · downarrow D_f · · downarrow D_c 0 · → · V_0 · → · V_f · → · V_c · → · 0 commutes and has exact rows. Then there is a natural exact sequence 0 · ⟶ker D_0 ⟶ker D_f ⟶ker D_c xrightarrowδcoker D_0","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-23dfe8b37c1f5d9f","tex":"v_i","aliases":"v_i","display":"v_i","kind":"use","excerpt":"[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-824b6808db63e6ce","tex":"v_{\\mathrm{in}}","aliases":"v_in mathrm","display":"v_{\\mathrm{in}}","kind":"use","excerpt":"For an arbitrary realization (W,A_d,v_in,λ), concatenation gives the factorization H_a(u,v) = λ A_vA_uv_in. Thus H_a factors through W, so rank H_a≤dim W. The canonical realization has dimension r_k(a), proving the minimality formula.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"use","excerpt":"Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7fd7bfa7519bcf67","tex":"v_{n+1}","aliases":"v_n+1","display":"v_{n+1}","kind":"use","excerpt":"Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-07c7f30b23f99da1","tex":"v_{Nm+j}","aliases":"v_Nm+j","display":"v_{Nm+j}","kind":"use","excerpt":"n Let K be a characteristic-zero field and let the K-valued sequence (u_n) satisfy a linear recurrence with polynomial coefficients. Beyond a finite singular prefix, write the recurrence as a finite-dimensional rational nonautonomous state model v_n+1=A(n)v_n, A(n)∈Mat_d(K(n)). For a fixed integer N≥1 and residue 0≤ j<N, the lane state w_m=v_Nm+j e","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c3a76197560b59e1","tex":"v_\\alpha","aliases":"v_α alpha","display":"v_\\alpha","kind":"use","excerpt":"Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a9d67ddc851a2ede","tex":"\\mathcal V_{\\lambda,a}","aliases":"V_λ,a mathcal lambda","display":"\\mathcal V_{\\lambda,a}","kind":"use","excerpt":"For every chart C_λ,a, let I_λ,a: C_λ,a⟶ V_λ,a be its native normalized invariant. A bridge comparison is a natural isomorphism κ_ab: J_abI_λ,a Longrightarrow I_λ,bΦ_ab, where J_ab transports invariant coordinates between their chart-specific target categories.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f4b68e58aab2e0d5","tex":"v_{\\pi}","aliases":"v_π pi","display":"v_{\\pi}","kind":"use","excerpt":"Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"use","excerpt":"[Raw and semantic inverse limits] There are canonical digit-equivariant identifications varprojlim_r M_k(a)/p^rM_k(a) ≅ M_k,p(a) and varprojlim_r overline M_k,p^r(a) ≅ M_k,p(a) ⊆ L_k,p(a). Furthermore, varprojlim_r G_k,p^r(a)=0, varprojlim_r Q_k,p^r(a) ≅ C_k(a)_(p).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9e2d9dcc048eb142","tex":"\\varprojlim_m","aliases":"varprojlim_m varprojlim","display":null,"kind":"use","excerpt":"Let widehatM_k(a) = varprojlim_m M_k(a)/mM_k(a) be the profinite completion over divisibility of moduli.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5435ba93fa0441bf","tex":"\\varprojlim_r","aliases":"varprojlim_r varprojlim","display":null,"kind":"use","excerpt":"[Raw and semantic inverse limits] There are canonical digit-equivariant identifications varprojlim_r M_k(a)/p^rM_k(a) ≅ M_k,p(a) and varprojlim_r overline M_k,p^r(a) ≅ M_k,p(a) ⊆ L_k,p(a). Furthermore, varprojlim_r G_k,p^r(a)=0, varprojlim_r Q_k,p^r(a) ≅ C_k(a)_(p).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7929adab1eaacd20","tex":"\\operatorname{Vect}","aliases":"Vect operatorname","display":"\\operatorname{Vect}","kind":"use","excerpt":"[Tannakian boundary] Let T be a neutral Tannakian category over k and ω: T⟶Vect_k a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) is an affine group scheme and T≃Rep_k( G) as tensor categories. A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary [source reference].","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3e8398c2dc335e37","tex":"\\operatorname{Vect}_K","aliases":"Vect_K operatorname","display":"\\operatorname{Vect}_K","kind":"use","excerpt":"Let L/K be finite Galois with group G. A semilinear G-object is an L-vector space V with maps ρ_σ:V→ V, ρ_σ(λ v)=σ(λ)ρ_σ(v), satisfying the group law. Scalar extension and fixed vectors define functors L⊗_K-:Vect_K⟶Semilin_G(L), (-)^G:Semilin_G(L)⟶Vect_K.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5c93295ece87b2e2","tex":"\\operatorname{Vect}_k","aliases":"Vect_k operatorname","display":"\\operatorname{Vect}_k","kind":"use","excerpt":"[Tannakian boundary] Let T be a neutral Tannakian category over k and ω: T⟶Vect_k a faithful exact tensor fiber functor. Then G=Aut^⊗(ω) is an affine group scheme and T≃Rep_k( G) as tensor categories. A faithful projector-corner representation without verified exact strong monoidal structure does not cross this boundary [source reference].","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a675678fa6689935","tex":"\\vee","aliases":"vee","display":"\\vee","kind":"use","excerpt":"[Combined and shared semantics] For semantic quotients with kernels N and M: [label=textup(roman*)] item the least common refinement has kernel N∩ M; item the greatest common coarsening has kernel the generated congruence join Nvee M; item in a module category, Nvee M=N+M.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e83c58309eb2060b","tex":"\\mathcal W","aliases":"W mathcal","display":"\\mathcal W","kind":"use","excerpt":"[Composition law for bridge loss] Let UxrightarrowB VxrightarrowC W be certified bridges. [label=textup(roman*)] item Every constructor or universal property preserved by both B and C is preserved by CB. item MorKer_B(X,Y) ⊆ MorKer_CB(X,Y). item Coll_B ⊆ Coll_CB. item EssIm(CB) ⊆ EssIm(C). item If both bridges are conservative, then their composite","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c57da8189a7c7cb4","tex":"\\mathfrak W","aliases":"W mathfrak","display":"\\mathfrak W","kind":"use","excerpt":"Define the min-plus certificate transform ( W_M,sw)(k) = min_k≤ r≤ Mk+s (w(r)+v_p(Θ_M,s(r,k))). This transform is a certified lower bound. It need not be the exact output valuation, because cancellation can increase the valuation beyond the termwise minimum.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-00883947feeb7d79","tex":"w^{\\mathsf T}","aliases":"w^ T mathsf","display":"w^{\\mathsf T}","kind":"use","excerpt":"Finite modular state dynamics provide a directional constructor into the local Mahler chart. Let v_n+1=Cv_n, a_n=w^ TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e6f16f2e02ae0bf2","tex":"w^n","aliases":"w^n","display":"w^n","kind":"use","excerpt":"[Typed polyphase realization] Let K be a field, let N∈ Z with N≥1 be invertible in K, and suppose K contains a primitive N-th root of unity. For F(z)=∑_n≥0a_nz^n∈ K[[z]], set F_j(w):=∑_n≥0a_Nn+jw^n. Then F(z)=∑_j=0^N-1z^jF_j(z^N) is exact analysis and synthesis. The Fourier residue projectors and the polyphase components satisfy the explicit coordi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-777ef1a9e62bb9fe","tex":"w_{(pu)n+(r+ud)}","aliases":"w_(pu)n+(r+ud)","display":"w_{(pu)n+(r+ud)}","kind":"use","excerpt":"After ℓ digits, the resulting subsequence is n↦ a_p^ℓ n+r, which depends only on (p^ℓbmod T,rbmod T). Applying Λ_d gives Λ_db_u,r(n) =b_u,r(pn+d) =w_(pu)n+(r+ud), proving [source reference]. The pair set is finite, so the kernel is finite. Every kernel state is reachable by construction. If two kernel sequences differ, they differ at some n; readin","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a6dc04e23090a2a5","tex":"w_m","aliases":"w_m","display":"w_m","kind":"use","excerpt":"[Nonautonomous state jump] Suppose a P-recursive sequence is represented, beyond its singular prefix, by v_n+1=C(n)v_n, a_n=λ(n)v_n, where v_n∈ K^s and C(n) is rational. Then the jth N-section has the jump system w_m+1=J_N,j(m)w_m, where J_N,j(m) = C(Nm+j+N-1)⋯ C(Nm+j). After denominator clearing, every scalar output has a polynomial recurrence of ","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4d9d1dce07ef9bae","tex":"w_{m+1}","aliases":"w_m+1","display":"w_{m+1}","kind":"use","excerpt":"[Nonautonomous state jump] Suppose a P-recursive sequence is represented, beyond its singular prefix, by v_n+1=C(n)v_n, a_n=λ(n)v_n, where v_n∈ K^s and C(n) is rational. Then the jth N-section has the jump system w_m+1=J_N,j(m)w_m, where J_N,j(m) = C(Nm+j+N-1)⋯ C(Nm+j). After denominator clearing, every scalar output has a polynomial recurrence of ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-08d2a928fa01f44f","tex":"\\mathfrak W_{M,s}","aliases":"W_M,s mathfrak","display":"\\mathfrak W_{M,s}","kind":"use","excerpt":"Define the min-plus certificate transform ( W_M,sw)(k) = min_k≤ r≤ Mk+s (w(r)+v_p(Θ_M,s(r,k))). This transform is a certified lower bound. It need not be the exact output valuation, because cancellation can increase the valuation beyond the termwise minimum.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dfa15e280d3611a8","tex":"w_r","aliases":"w_r","display":"w_r","kind":"use","excerpt":"od word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives by equality of the sequences in [source reference] gives the canonical minimal least-significa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-40d7f305e6bf401c","tex":"w_{un+r\\bmod T}","aliases":"w_un+rbmod T bmod","display":"w_{un+r\\bmod T}","kind":"use","excerpt":"[Finite periodic kernel and exact state model] Let a∈Per_p^×,per have minimal period T and period word w. Every kernel state has the form b_u,r(n)=w_un+rbmod T, u∈ U_p(T), r∈ Z/T Z, where U_p(T)={p^ebmod T:e≥ 0}. The digit transition on pair representatives is (u,r)xrightarrowd(pu,r+ud)pmod T, and the output is w_r. Quotienting pair representatives","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-56d7af085245cb80","tex":"\\mathrm{weak}","aliases":"weak mathrm","display":"\\mathrm{weak}","kind":"use","excerpt":"Task changes and admissibility changes have opposite-looking effects. If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information: N_strong⊆ N_weak, Q_strong↠ Q_weak. If an admissible realization class is enlarged, the optimized defect can only decrease: Γ⊆Γ' Longrightarrow δ_Γ'≤δ_Γ. These are different orders","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5c577d5457da31ff","tex":"\\widehat H","aliases":"widehat H widehat","display":"\\widehat H","kind":"use","excerpt":"[Conformal metric-completion criterion] Let a connected finite graph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy repr","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-657478f81580a29e","tex":"\\widehat{\\mathbb Z}","aliases":"widehat Z widehat mathbb","display":"\\widehat{\\mathbb Z}","kind":"use","excerpt":"Finite free abelian groups have profinite completion equal to the product of their p-adic completions. In Smith coordinates, the inclusion is multiplication by d_i in each profinite coordinate. Its cokernel is widehat Z/d_iwidehat Z ≅ Z/d_i Z. Taking the direct sum over i gives C_k(a). The final assertion is the definition of belonging to the diago","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-62d17281aad0e0e9","tex":"\\widehat{L_k(a)}","aliases":"widehatL_k(a) widehat","display":"\\widehat{L_k(a)}","kind":"use","excerpt":"[Profinite saturation sequence] There are canonical digit-equivariant identifications widehatM_k(a) ≅ ∏_p M_k,p(a), widehatL_k(a) ≅ ∏_p L_k,p(a), and a short exact sequence 0 ⟶ widehatM_k(a) ⟶ widehatL_k(a) ⟶ C_k(a) ⟶ 0. A compatible all-moduli raw tower has an ordinary integral lift exactly when its class in widehatM_k(a)/M_k(a) vanishes.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2df00f1b9f726c87","tex":"\\widehat{M_k(a)}","aliases":"widehatM_k(a) widehat","display":"\\widehat{M_k(a)}","kind":"use","excerpt":"Let widehatM_k(a) = varprojlim_m M_k(a)/mM_k(a) be the profinite completion over divisibility of moduli.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-723998d1f368812a","tex":"\\widehat{x}","aliases":"widehatx widehat","display":"\\widehat{x}","kind":"use","excerpt":"[Local-global separation] Every compatible prime-power tower is effective in a p-adic completion. A compatible family over all primes is effective in the profinite completion. Neither statement alone implies an ordinary integral lift. Integral effectivity is the separate diagonal condition widehatx∈ M_k(a)⊆widehatM_k(a).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f596b26bb2804e7d","tex":"\\widetilde B","aliases":"widetilde B widetilde","display":"\\widetilde B","kind":"use","excerpt":"[Forgetful descent and enhancement lifting] A rich bridge widetilde B:widetilde C→widetilde D automatically determines an underlying bridge Vwidetilde B:widetilde C→ D. Given only a bare bridge B: C→ D, constructing widetilde B together with a comparison Vwidetilde B≅ BU is a lifting problem. Its existence, uniqueness, and residual automorphisms ar","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-eefe24abae0ff118","tex":"\\widetilde{\\mathcal C}","aliases":"widetilde C widetilde mathcal","display":"\\widetilde{\\mathcal C}","kind":"use","excerpt":"Suppose richer categories widetilde C and widetilde D carry forgetful functors U:widetilde C→ C, V:widetilde D→ D.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0c9b51f69c8fee02","tex":"\\widetilde{\\mathcal D}","aliases":"widetilde D widetilde mathcal","display":"\\widetilde{\\mathcal D}","kind":"use","excerpt":"Suppose richer categories widetilde C and widetilde D carry forgetful functors U:widetilde C→ C, V:widetilde D→ D.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-468630464758f8a1","tex":"\\widetilde L","aliases":"widetilde L widetilde","display":"\\widetilde L","kind":"use","excerpt":"Multiply L_F by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic. This gives a unique primitive polynomial representative widetilde L_F = ∑_j=0^rq_j(z)D^j. Its operator singular support is Σ_rm op(F) = {q_r(z)=0}∪{∞}. Apparent singularities remain part of th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8560bfc6b7368f93","tex":"\\widetilde L_F","aliases":"widetilde L_F widetilde","display":"\\widetilde L_F","kind":"use","excerpt":"Multiply L_F by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic. This gives a unique primitive polynomial representative widetilde L_F = ∑_j=0^rq_j(z)D^j. Its operator singular support is Σ_rm op(F) = {q_r(z)=0}∪{∞}. Apparent singularities remain part of th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3c8e1dd7784a5fd7","tex":"X^{(1)}","aliases":"X^(1)","display":"X^{(1)}","kind":"use","excerpt":"[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of mult","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dc4c9ec5f00e4f88","tex":"X^{(d)}","aliases":"X^(d)","display":"X^{(d)}","kind":"use","excerpt":"[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of mult","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0ac006294291421f","tex":"x^{(r_1+\\cdots+r_q-j)/N}","aliases":"x^(r_1+⋯+r_q-j)/N cdots","display":"x^{(r_1+\\cdots+r_q-j)/N}","kind":"use","excerpt":"[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the sou","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-65e8ec76ae768d6c","tex":"x^{-1}","aliases":"x^-1","display":"x^{-1}","kind":"use","excerpt":"[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-43bb131db36a9433","tex":"x^{1/N}","aliases":"x^1/N","display":"x^{1/N}","kind":"use","excerpt":"Away from Σ^[N] and zero, choose a local analytic branch t=x^1/N. Every rotated argument ω_N^qt avoids Σ, so the root-filter expression is analytic. A singularity can therefore occur only when one rotated preimage meets Σ.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Fibonacci F_n · x^2-x-1 · z/(1-z-z^2) · 2 · The root ratio is not torsion. Every arithmetic subsequence retains rank 2.","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0bb5396bfe68a547","tex":"x^3","aliases":"x^3","display":"x^3","kind":"use","excerpt":"[Exact Catalan packet system] The parity packets satisfy E=1+2xEO, O=E^2+xO^2. Elimination gives 4xE^4-E^2+1=0 and 4x^3O^4-8x^2O^3+5xO^2-O+1=0. Both packets have algebraic degree four over K(x). Hence the universal bound d^N=2^2=4 is sharp.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b8bbaae1ffe15b3b","tex":"x^d","aliases":"x^d","display":"x^d","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-365e3cd3855c68ff","tex":"x^{d-1}","aliases":"x^d-1","display":"x^{d-1}","kind":"use","excerpt":"An object of Rec_K^min is a pair (χ, a_0), where χ(x)=x^d-c_d-1x^d-1-⋯-c_1x-c_0, c_0≠0, is the monic minimal recurrence polynomial and a_0=(a_0,…,a_d-1) is the initial vector. It generates the sequence a_n+d=c_d-1a_n+d-1+⋯+c_1a_n+1+c_0a_n. Because the recurrence is normalized and minimal, two objects representing the same sequence are equal. Thus R","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7272f222c85749d6","tex":"X^m","aliases":"X^m","display":"X^m","kind":"use","excerpt":"Let A_m:X^m⟶ X, 2≤ m≤ D, be bounded symmetric multilinear maps, and let s∈ X. Consider the marked equation u=s+Φ(u), Φ(u)=∑_m=2^DA_m[u,…,u]. tag11.1 The word marked means that the source term, every multilinear operation, the packet system, and the selected convergence or filtration certificate are retained as part of the presentation. The fixed po","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0ed7bdf4a651c709","tex":"x^m","aliases":"x^m","display":"x^m","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cbfffa4fd298f447","tex":"x^{m-r}","aliases":"x^m-r","display":"x^{m-r}","kind":"use","excerpt":"The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e189c65b70ed9178","tex":"X^n","aliases":"X^n","display":"X^n","kind":"use","excerpt":"Separately, let X,Y be kk-modules with finite split lane systems and suppose a polynomial map is presented with specified data P(x)=T_0+T_1(x)+T_2(x,x)+⋯+T_D(x,…,x), where T_0∈ Y, T_n:X^n⟶ Y (1≤ n≤ D) and every T_n is a supplied multilinear map. Then T_0=∑_aι_Y,aπ_Y,a(T_0), and adding this degree-zero decomposition to the displayed multilinear expa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-12f1075149375120","tex":"x^N","aliases":"x^N","display":"x^N","kind":"use","excerpt":"umerate[label=textup(roman*)] item The sequence is C-finite and its minimal recurrence has nonzero constant term. item The sequence has a finite-dimensional state realization a_n=λ C^nv with C∈ GL_d( F_p). item The sequence is purely periodic from n=0. If m_a(x) is the monic minimal recurrence polynomial and T=T(a), then m_a(x)mid x^T-1, T=min{N≥ 1","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d8c5e1feefd58efd","tex":"x^n","aliases":"x^n","display":"x^n","kind":"use","excerpt":"[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0648303074e446a8","tex":"X^{p^a}","aliases":"X^p^a","display":"X^{p^a}","kind":"use","excerpt":"[Principal-unit precision amplification] If X=I+p^LB, then for every a≥0 there exists B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a. Consequently, for H_a=Hp^a, every branch x↦ a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Each additional p-power refinement contributes at least one further digit of precision per Mahler depth.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0cc6c23e58081877","tex":"x^r","aliases":"x^r","display":"x^r","kind":"use","excerpt":"[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0514a60dd6e60420","tex":"x^T","aliases":"x^T","display":"x^T","kind":"use","excerpt":"umerate[label=textup(roman*)] item The sequence is C-finite and its minimal recurrence has nonzero constant term. item The sequence has a finite-dimensional state realization a_n=λ C^nv with C∈ GL_d( F_p). item The sequence is purely periodic from n=0. If m_a(x) is the monic minimal recurrence polynomial and T=T(a), then m_a(x)mid x^T-1, T=min{N≥ 1","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-36de5efcc886436e","tex":"x^{\\underline j}","aliases":"x^underline j underline","display":"x^{\\underline j}","kind":"use","excerpt":"[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"[Haar cancellation] For the normalized Haar bank, ℓ=fracx_0+x_1sqrt2, d=fracx_0-x_1sqrt2, and x_0=fracℓ+dsqrt2, x_1=fracℓ-dsqrt2. At x=(1,0), the two route contributions to x_0 are 1/2,1/2, while those to x_1 are 1/2,-1/2. Additive reconstruction returns (1,0). Replacing route addition by maximum returns (1/2,1/2). The convex/max carrier is therefo","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps i","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"The two images are A_1(x_1,x_2)=(x_1,x_1), A_2(x_1,x_2)=(x_2,-x_2), so the first identity is immediate.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-51e189ae43085a7d","tex":"X_a","aliases":"X_a","display":"X_a","kind":"use","excerpt":"Induct on a. If X_a=I+p^L+aB_a, the binomial expansion of X_a^p has linear term divisible by p^L+a+1, and every higher term has at least the same valuation. Apply [source reference] at the refined jump.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1fb453060f4388fc","tex":"X_a^p","aliases":"X_a^p","display":"X_a^p","kind":"use","excerpt":"Induct on a. If X_a=I+p^L+aB_a, the binomial expansion of X_a^p has linear term divisible by p^L+a+1, and every higher term has at least the same valuation. Apply [source reference] at the refined jump.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f4370da509655e62","tex":"x_d","aliases":"x_d","display":"x_d","kind":"use","excerpt":"inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps i","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Expand the raw history update and group all terms by their predecessor state and new branch. Associativity and commutativity permit the regrouping. The identity L_b (⊕_i x_i) = ⊕_i L_b(x_i) then moves aggregation through the branch action. The displayed formula is the resulting statewise sum.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"use","excerpt":"[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c597f6bebee232c6","tex":"X_j^s","aliases":"X_j^s","display":"X_j^s","kind":"use","excerpt":"For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f3be737b42076753","tex":"X_m","aliases":"X_m","display":"X_m","kind":"use","excerpt":"[Path expansion] For a composable string X_0xrightarrowf_1X_1xrightarrowf_2⋯xrightarrowf_mX_m with packet systems on every X_j, each source-target block of f_m⋯ f_1 is the sum over all compatible intermediate packet paths of the corresponding block products.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"A section word at level r is lifted by the same word at level r+1, proving surjectivity. Every p-adic limit of orbit points gives a compatible tower. Conversely, choose for each compatible residue state an orbit representative x_r. Compatibility gives x_r+1-x_r∈ p^rM_k(a), so (x_r) is Cauchy in M_k,p(a). Its limit lies in the orbit closure and maps","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1e9d35fe4d36c66f","tex":"x_{r+1}","aliases":"x_r+1","display":"x_{r+1}","kind":"use","excerpt":"A section word at level r is lifted by the same word at level r+1, proving surjectivity. Every p-adic limit of orbit points gives a compatible tower. Conversely, choose for each compatible residue state an orbit representative x_r. Compatibility gives x_r+1-x_r∈ p^rM_k(a), so (x_r) is Cauchy in M_k,p(a). Its limit lies in the orbit closure and maps","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2cc214a50d4b2e13","tex":"X_U","aliases":"X_U","display":"X_U","kind":"use","excerpt":"[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms sa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"use","excerpt":"For x∈ C_o, set x_v=G_vx. Tree edges satisfy the parallel relation by construction, and a non-tree edge satisfies it because H_ex=x. Conversely, any global section is determined along the tree by its root value; every remaining edge forces that value to be fixed by the corresponding H_e. This proves the section classification. Closed paths are word","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4ec4142068739e9a","tex":"x_\\ell","aliases":"x_ℓ ell","display":"x_\\ell","kind":"use","excerpt":"Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0e7659cd4ef8f947","tex":"Y^2","aliases":"Y^2","display":"Y^2","kind":"use","excerpt":"If α=β, the quadratic equation Y^2-(1-α z)^2=0 is not the normalized source polynomial. The selected branch is rational: F_α,α(z)=1-α z, so the correct source degree is one. This is a negative control against using a nonminimal defining equation in the degree ledger.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-560f4e5df768a9d8","tex":"y^m","aliases":"y^m","display":"y^m","kind":"use","excerpt":"The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5db6717e646243d3","tex":"Y^q","aliases":"Y^q","display":"Y^q","kind":"use","excerpt":"Let K be a characteristic-zero field. For an integer q≥2 and α∈ K^×, let U_q,α(z)∈ zK[[z]] be the unique selected branch satisfying U=z+α U^q. The derivative of P_q,α(z,Y)=Y-z-α Y^q with respect to Y is one at (0,0), so the branch is simple and selected by U(0)=0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6aee2e71bed537cd","tex":"y^{r-1}","aliases":"y^r-1","display":"y^{r-1}","kind":"use","excerpt":"The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-68e7992fcc95824e","tex":"Y_a^u","aliases":"Y_a^u","display":"Y_a^u","kind":"use","excerpt":"At resolution r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0d79a34602d32266","tex":"Y_b^v","aliases":"Y_b^v","display":"Y_b^v","kind":"use","excerpt":"For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e3d4192e022af00a","tex":"y_j","aliases":"y_j","display":"y_j","kind":"use","excerpt":"[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-53ed6f023dce8cbe","tex":"y_{r-1}","aliases":"y_r-1","display":"y_{r-1}","kind":"use","excerpt":"[Ordinary-point reconstruction] Let L=D^r+∑_j=0^r-1ℓ_j(z)D^j with all ℓ_j regular at zero. For every jet (y_0,…,y_r-1)∈ K^r there is a unique formal series F∈ K[[z]] satisfying LF=0, F^(j)(0)=y_j.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5d9e3361a96f40b3","tex":"Y_U","aliases":"Y_U","display":"Y_U","kind":"use","excerpt":"[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms sa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c17b4c6a44980c6c","tex":"\\mathbb Z^{\\mathbb N_{\\geq1}}","aliases":"Z^ N_≥1 mathbb mathbb geq","display":"\\mathbb Z^{\\mathbb N_{\\geq1}}","kind":"use","excerpt":"Let U_ Z=1+z Z[[z]] be the multiplicative group of integral unit series, and let E_ Z= Z^ N_≥1 be the additive group of integer exponent sequences.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-449753faf1f7a481","tex":"z^{-1}","aliases":"z^-1","display":"z^{-1}","kind":"use","excerpt":"The recurrence-to-OGF bridge is explicit. From χ(x)=x^d-c_d-1x^d-1-⋯-c_0 set Q(z)=z^dχ(z^-1) =1-c_d-1z-⋯-c_0z^d. The numerator is the degree-<d truncation of Q(z)∑_n≥0a_nz^n. Conversely, clearing Q recovers the recurrence and the initial vector. Minimality is equivalent to reducedness.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-793456d5c29ba7d9","tex":"z^0","aliases":"z^0","display":"z^0","kind":"use","excerpt":"rtwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is [z^n]F = ϵ_0 ( Θ_A,B (T_d_ℓ-1⋯ T_d_0B) ), where ϵ_0(G)=[z^0]G.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-177a047771fa9333","tex":"z^{1+(q-1)n}","aliases":"z^1+(q-1)n","display":"z^{1+(q-1)n}","kind":"use","excerpt":"[Typed Fuss--Catalan realization] Let K be a characteristic-zero field, let q∈ Z with q≥2, and let α∈ K^×. The uniquely marked formal solution of U=z+α U^q, U(0)=0, has support [z^1+(q-1)n]U =frac1(q-1)n+1binomqnnα^n ≠0 (n≥0), and every other coefficient is zero. Thus its support is exactly the set of degrees congruent to 1pmodq-1. For an integer p","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"Fibonacci F_n · x^2-x-1 · z/(1-z-z^2) · 2 · The root ratio is not torsion. Every arithmetic subsequence retains rank 2.","count":26},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a87f7202de83de58","tex":"z^{2m}","aliases":"z^2m","display":"z^{2m}","kind":"use","excerpt":"[Distinct parts and normalization] The product ∏_m≥1(1+z^m) = ∏_m≥1frac1-z^2m1-z^m has normalized Euler exponent b_n= 1, · n odd,","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4140de75758a272d","tex":"z^3","aliases":"z^3","display":"z^3","kind":"use","excerpt":"Alternating unit over F_3. For the word 12, T=2, m(x)=x+1, F(z)=frac11+z, R(z)=frac1+z^31+z=1+2z+z^2. The recurrence order is one although the minimal period is two. The 3-kernel consists of the sequence and its negative, giving two minimal digital states. The compiler bound is nine.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-cff190730fb88c49","tex":"z^{3^j}","aliases":"z^3^j","display":"z^{3^j}","kind":"use","excerpt":"Cantor-digit indicator. Over F_3, let R(z)=1+z. Then F(z)=∏_j≥0(1+z^3^j) has coefficient one exactly when the ternary expansion of the index uses only the digits zero and one. The functional equation is F(z)=(1+z)F(z^3), F(z)^2=frac11+z. The factor 1+z has odd divisor multiplicity, so the collapse obstruction is nonzero. Thus the sequence is automa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-54b855e49989865e","tex":"z^d","aliases":"z^d","display":"z^d","kind":"use","excerpt":"The recurrence-to-OGF bridge is explicit. From χ(x)=x^d-c_d-1x^d-1-⋯-c_0 set Q(z)=z^dχ(z^-1) =1-c_d-1z-⋯-c_0z^d. The numerator is the degree-<d truncation of Q(z)∑_n≥0a_nz^n. Conversely, clearing Q recovers the recurrence and the initial vector. Minimality is equivalent to reducedness.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9b9ff0ac28ecd014","tex":"\\mathbb Z^{d+1}","aliases":"Z^d+1 mathbb","display":"\\mathbb Z^{d+1}","kind":"use","excerpt":"Let A=P_m,S^(d): Z^d+1→ Z^|S|(d+1) be a stacked affine-Mahler packet analysis map. Whenever an integer left inverse L exists, LA=I, Π=AL, Π^2=Π, and Z^|S|(d+1)=im A⊕ker L. Thus the source chart and the normalized packet core im A are equivalent, while ker L is a named residual rather than discarded information.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-63ed4cae9140e4ad","tex":"z^{d_{\\mathcal A}}","aliases":"z^d_ A mathcal","display":"z^{d_{\\mathcal A}}","kind":"use","excerpt":"By [source reference], the coefficient support is contained in 1+d_ A N and the reduced exponent set is aperiodic by maximality of the gcd. Hence u(z)=zV(z^d_ A) with V aperiodic. The smooth implicit schema gives one positive dominant singularity for V, and pulling back through z^d_ A gives textup(11.12). The map z↦ z^N sends μ_d_ A onto a root gro","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-15064ddd2f735b13","tex":"z^e","aliases":"z^e","display":"z^e","kind":"use","excerpt":"Let A_b(z)=∏_m≥1(1-z^m)^-b_m be an integral unit series in the fixed Euler basis, and let c_n=∑_dmid nd b_d be its divisor-ghost sequence. Define the prime-uniform multiplicative function f_b(n)=∏_p^eparallel n[z^e]A_b(z). Every prime has the same local factor A_b(T). By [source reference], the coefficient, exponent, ghost, and prime-uniform multip","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"use","excerpt":"The logarithmic-derivative identity is equivalent to zA'(z)=(∑_j≥1c_jz^j)A(z). Comparing coefficients gives the recurrence. Conversely, the recurrence uniquely determines the coefficient sequence over Q. Since c∈ G_ Z reconstructs an integral exponent sequence, the resulting series is the Euler product from [source reference] and therefore has inte","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ea4119547760ca2f","tex":"z^k","aliases":"z^k","display":"z^k","kind":"use","excerpt":"The operator p_s(θ-s) multiplies the coefficient of z^k by p_s(k-s). After multiplication by z^q-s, the coefficient at z^n+q is p_s(n)a_n+s. The recurrence kills every coefficient from degree N+q onward. Differentiating more times than the residual degree gives a homogeneous annihilator.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e198fb464f0d1114","tex":"z^{mr}","aliases":"z^mr","display":"z^{mr}","kind":"use","excerpt":"Formal logarithmic differentiation gives fraczA'A = ∑_m≥1b_mfracmz^m1-z^m = ∑_m≥1∑_r≥1m b_m z^mr. The coefficient at z^n is ∑_mmid nm b_m=c_n. M obius inversion gives the displayed inverse formula. The definition of G_ Z is exactly the integrality condition for that inverse.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dbd2010dd9658c4d","tex":"z^{N+1}","aliases":"z^N+1","display":"z^{N+1}","kind":"use","excerpt":"[Filtered product completion] Every filtered factor family has a unique coefficientwise product in 1+zR[[z]]. For every N, the truncation modulo z^N+1 is equal to the product of the finitely many factors with w(m)≤ N.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-33fb1cf2cddde12c","tex":"z^{n+1}","aliases":"z^n+1","display":"z^{n+1}","kind":"use","excerpt":"Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preced","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-53a2dc722026e71b","tex":"z^{n+q}","aliases":"z^n+q","display":"z^{n+q}","kind":"use","excerpt":"[Recurrence residual as an Euler-operator polynomial] Define Q = ∑_s=0^q z^q-sp_s(θ-s). Then [z^n+q]QF = ∑_s=0^qp_s(n)a_n+s. Hence QF is a polynomial of degree less than N+q, and D^N+qQF=0.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4575bcbd3a963aec","tex":"z^{N-1}","aliases":"z^N-1","display":"z^{N-1}","kind":"use","excerpt":"olyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e9600a5c626ea917","tex":"z^{n-m}","aliases":"z^n-m","display":"z^{n-m}","kind":"use","excerpt":"The coefficient of z^n-m in D^jF is (n-m+j)^underline ja_n-m+j. Multiplication by q_j,mz^m shifts that contribution to degree n. Summing over j and m gives the formula.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f9f1f546f3ae0830","tex":"z^{Nn+j}","aliases":"z^Nn+j","display":"z^{Nn+j}","kind":"use","excerpt":"uppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. I","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5f2286bd098158c9","tex":"z^{Nr+j}","aliases":"z^Nr+j","display":"z^{Nr+j}","kind":"use","excerpt":"[Finite tree-address signature] Fix a packet width N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6894c7248320dbce","tex":"z^p","aliases":"z^p","display":"z^p","kind":"use","excerpt":"[Why the unit condition is retained] The functional-Mahler bridge uses the formal quotient R_a(z)=fracF_a(z)F_a(z^p). It is regular at the origin exactly when F_a(0)≠ 0. The normalization a_0=1 removes scalar ambiguity and gives R_a(0)=1. Purely periodic sequences with a_0=0 remain valid C-finite and automatic objects, but they need not lie in this","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e765b594366f3e90","tex":"z^{p^a}","aliases":"z^p^a","display":"z^{p^a}","kind":"use","excerpt":"In characteristic p, (1+z)^p^a-1≡ z^p^a. Substitute this into the generating definition of Θ_p^a,s and compare coefficients.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a91f3ff2ba732c63","tex":"z^{p^j}","aliases":"z^p^j","display":"z^{p^j}","kind":"use","excerpt":"For a unit formal series F(0)=1, define its local functional-Mahler mask by M_p(F)(z) = fracF(z)F(z^p). Then F(z)= M_p(F)(z)F(z^p). Iterating [source reference] gives the coefficientwise product F(z) = ∏_j≥0 M_p(F)(z^p^j). Only finitely many factors contribute to each fixed coefficient.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1a1ad11ff428d7e6","tex":"z^{p^{j+1}}","aliases":"z^p^j+1","display":"z^{p^{j+1}}","kind":"use","excerpt":"Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ef6e68e8044bc078","tex":"z^{p^{N+1}}","aliases":"z^p^N+1","display":"z^{p^{N+1}}","kind":"use","excerpt":"Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-de29e5d648115a0b","tex":"z^q","aliases":"z^q","display":"z^q","kind":"use","excerpt":"The examples show that the finite invariant retains the weight parameter as well as the arity. Support period alone recovers only q-1; the coefficient [z^q]U=α supplies the second coordinate.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5159eb9ea3b12cf9","tex":"z^{q-1}","aliases":"z^q-1","display":"z^{q-1}","kind":"use","excerpt":"Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f93722a4d932b76b","tex":"z^{q-s}","aliases":"z^q-s","display":"z^{q-s}","kind":"use","excerpt":"[Recurrence residual as an Euler-operator polynomial] Define Q = ∑_s=0^q z^q-sp_s(θ-s). Then [z^n+q]QF = ∑_s=0^qp_s(n)a_n+s. Hence QF is a polynomial of degree less than N+q, and D^N+qQF=0.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-192c9c4573735e27","tex":"z^T","aliases":"z^T","display":"z^T","kind":"use","excerpt":"Let F_a(z)=fracP_a(z)Q_a(z) be reduced with P_a(0)=Q_a(0)=1. If d=deg m_a, then Q_a(z)=z^dm_a(z^-1), with the evident coefficient normalization. Taking reciprocal polynomials in [source reference] gives Q_a(z)mid 1-z^T, T=min{N≥ 1:Q_a(z)mid 1-z^N}. Thus pure periodicity is visible both in the linear annihilator and in the reduced rational denominat","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-95957698a5184509","tex":"z^{T(a)}","aliases":"z^T(a)","display":"z^{T(a)}","kind":"use","excerpt":"screte category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] n","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ba58311d45bbdddb","tex":"\\mathbb Z^{|S|(d+1)}","aliases":"Z^|S|(d+1) mathbb","display":"\\mathbb Z^{|S|(d+1)}","kind":"use","excerpt":"Let A=P_m,S^(d): Z^d+1→ Z^|S|(d+1) be a stacked affine-Mahler packet analysis map. Whenever an integer left inverse L exists, LA=I, Π=AL, Π^2=Π, and Z^|S|(d+1)=im A⊕ker L. Thus the source chart and the normalized packet core im A are equivalent, while ker L is a named residual rather than discarded information.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d0a9f7f8a3a88915","tex":"\\mathbb Z^\\rho","aliases":"Z^ρ mathbb rho","display":"\\mathbb Z^\\rho","kind":"use","excerpt":"Choose a coordinate-detection set n by [source reference]. Evaluation embeds L_k(a) into Z^ρ, so L_k(a) is a finitely generated free abelian group of rank at most ρ. The raw lattice M_k(a) is contained in L_k(a), and its rational span is all of V_ Q(a). Hence M_k(a) has rank ρ, and therefore L_k(a) also has rank ρ. A subgroup of a finitely generate","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-57105cdc4119a19b","tex":"\\mathbb Z_{>0}","aliases":"Z_>0 mathbb","display":"\\mathbb Z_{>0}","kind":"use","excerpt":"item If λ∉ Z, then r_D(N,j;λ)=N. item If λ=L∈ Z_>0, then every packet is a nonzero rational function with exact denominator (1-α^Nx)^L, and r_D(N,j;L)=1. item If λ=-M∈ Z_≤0, then G_N,j^-M,α is zero when j>M. Otherwise it is a nonzero polynomial of degree lfloorfracM-jNrfloor and r_D(N,j;-M) = 1, · j≤ M,","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e51336bbaf29757c","tex":"Z_U","aliases":"Z_U","display":"Z_U","kind":"use","excerpt":"[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms sa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a9d303343a845470","tex":"\\mathbb Z_{\\leq0}","aliases":"Z_≤0 mathbb leq","display":"\\mathbb Z_{\\leq0}","kind":"use","excerpt":"item If λ∉ Z, then r_D(N,j;λ)=N. item If λ=L∈ Z_>0, then every packet is a nonzero rational function with exact denominator (1-α^Nx)^L, and r_D(N,j;L)=1. item If λ=-M∈ Z_≤0, then G_N,j^-M,α is zero when j>M. Otherwise it is a nonzero polynomial of degree lfloorfracM-jNrfloor and r_D(N,j;-M) = 1, · j≤ M,","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-76b52b898df4bdc1","tex":"\\mathbb Z_{\\ge1}","aliases":"Z_≥1 mathbb ge","display":"\\mathbb Z_{\\ge1}","kind":"use","excerpt":"[Typed cyclic-radical realization] Let K be a characteristic-zero field containing μ_d and a primitive N-th root of unity ω, let d,m,N∈ Z_≥1, and choose the formal branch F(z)=(1-α z^m)^-1/d, α∈ K^×. With g=gcd(N,m), its nonzero coefficients occupy exactly the subgroup of Z/N Z generated by m, which has N/g addresses. The rotated radicands 1-αω^jmz","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-340cfaf9a3edda4b","tex":"|^{-1/n}","aliases":"|^-1/n","display":"|^{-1/n}","kind":"use","excerpt":"[Conformal metric-completion criterion] Let a connected finite graph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy repr","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e86777b436179d44","tex":"|^{1/n}","aliases":"|^1/n","display":"|^{1/n}","kind":"use","excerpt":"A compatible family makes every closed root holonomy H conformal for the root metric Q_o. Taking determinants fixes its positive scale as |det H|^1/n, so widehat H preserves Q_o.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2bd7dc869f88a3da","tex":"|^{2/n}","aliases":"|^2/n","display":"|^{2/n}","kind":"use","excerpt":"Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible fa","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e6dd4284b98c5bb7","tex":"|_{\\mathcal N_\\alpha}","aliases":"|_ N_α mathcal alpha","display":"|_{\\mathcal N_\\alpha}","kind":"use","excerpt":"[Semantic conservativity of a normalized chart] If N_α is classification-normalizing, then Sem_α|_ N_α reflects isomorphisms. Consequently, a complete invariant on N_α can be transported through a reversible overlap only after the overlap is shown to preserve the normalization and reflect the relevant isomorphisms.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e3ee31e9cde749cf","tex":"|_{\\im A}","aliases":"|_im A im operatorname","display":"|_{{\\operatorname{im}} A}","kind":"use","excerpt":"The identities LA=I and Π=AL show that A identifies the source with im A, with inverse L|_im A. For any packet vector y, y=ALy+(y-ALy), where ALy∈im A and y-ALy∈ker L. The intersection is zero because Ax∈ker L implies x=LAx=0. Tensoring LA=I with any coefficient ring preserves the split identity. Intertwining maps preserve the image and kernel deco","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"olyphase separation] Fix N≥1 and z≠0. Let S_z be the cyclic shift on K^N satisfying S_z e_j=e_j+1 (j<N-1), S_z e_N-1=z e_0, so S_z^N=zI. Put M_N=I+S_z, f_N(n)=e_0^ TM_N^n e_0. Then the N× N Hankel block H_ij=f_N(i+j), 0≤ i,j<N, has determinant det H=± z^N-1. Consequently its linear residual space has dimension N. If z>0, the residual functions k↦ f","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-62547ed5692a28f7","tex":"\\times_C","aliases":"×_C times","display":"\\times_C","kind":"use","excerpt":"S @>>> C is a pullback in R-Mod. Equivalently, J≅ S×_C T. There is also a canonical short exact sequence 0⟶ X/(N∩ M) xrightarrowα X/N⊕ X/M xrightarrowβ X/(N+M) ⟶0, where α(x+N∩ M)=(x+N,x+M) and β(a+N,b+M)=a-b+(N+M).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0007afcbb2220293","tex":"\\alpha^{-N}","aliases":"α^-N alpha","display":"\\alpha^{-N}","kind":"use","excerpt":"For λ∉ Z, the hypergeometric operator in [source reference] is Fuchsian with singularities 0, α^-N, ∞. Its exponent packets are E_0 = {0}∪{1-b:b∈ B_N,j}, E_α^-N = {0,1,…,N-2,-λ}, and E_∞ = A_N,j(λ). Modulo integers, E_0 is the complete N-torsion set {0,frac1N,…,fracN-1N}, and the pairwise differences in E_∞ are also N-torsion.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d830d9bc7e9c803c","tex":"\\alpha^2","aliases":"α^2 alpha","display":"\\alpha^2","kind":"use","excerpt":"[Quadratic-radical parity packets] The packets satisfy E^2+xO^2=1+px, 2EO=-s. Their eliminants are E^4-(1+px)E^2+fracs^2x4=0 and xO^4-(1+px)O^2+fracs^24=0. If s≠0, α^2≠β^2, then both selected packets have degree four over K(x).","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4773ad13281af1e0","tex":"\\alpha^j","aliases":"α^j alpha","display":"\\alpha^j","kind":"use","excerpt":"[Exact hypergeometric packet form] If (λ)_j≠0, then G_N,j^λ,α(x) = frac(λ)_jα^jj! _NF_N-1 ( A_N,j(λ) B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b6ac1774de8bd95d","tex":"\\alpha^N","aliases":"α^N alpha","display":"\\alpha^N","kind":"use","excerpt":"[Singularity orbit under multisection] Assume F is an analytic germ at zero with analytic continuation away from a finite set Σ. Then every finite nonzero analytic singularity of M_N,jF lies in Σ^[N] = {α^N:α∈Σ}. Distinct singularities in one fiber of α↦α^N are aliased by the multisection map. A character projection may remove some aliased local co","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-34a0d2bb0218af80","tex":"\\alpha^n","aliases":"α^n alpha","display":"\\alpha^n","kind":"use","excerpt":"Its coefficients are a_n = frac(λ)_nn!α^n. For 0≤ j<N, put G_N,j^λ,α(x) = ∑_m≥0 frac(λ)_Nm+j(Nm+j)! α^Nm+jx^m. Define the upper and lower parameter packets A_N,j(λ) = { fracλ+j+rN:0≤ r<N }, B_N,j = { fracj+1+rN:0≤ r<N }∖{1}.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-269f818446b8a8bc","tex":"\\alpha^{Nm+j}","aliases":"α^Nm+j alpha","display":"\\alpha^{Nm+j}","kind":"use","excerpt":"Its coefficients are a_n = frac(λ)_nn!α^n. For 0≤ j<N, put G_N,j^λ,α(x) = ∑_m≥0 frac(λ)_Nm+j(Nm+j)! α^Nm+jx^m. Define the upper and lower parameter packets A_N,j(λ) = { fracλ+j+rN:0≤ r<N }, B_N,j = { fracj+1+rN:0≤ r<N }∖{1}.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ab15d1c00936ee24","tex":"\\beta^2","aliases":"β^2 beta","display":"\\beta^2","kind":"use","excerpt":"[Quadratic-radical parity packets] The packets satisfy E^2+xO^2=1+px, 2EO=-s. Their eliminants are E^4-(1+px)E^2+fracs^2x4=0 and xO^4-(1+px)O^2+fracs^24=0. If s≠0, α^2≠β^2, then both selected packets have degree four over K(x).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c2b40d4e80bb0dae","tex":"\\beta_{1}","aliases":"β_1 beta","display":"\\beta_{1}","kind":"use","excerpt":"Constant-one function 1 · β_1=1 · 1,1,1,1,1,1,… · 1,1,1,1,1,1,…","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ba038883f9ac5b81","tex":"\\beta_{2}","aliases":"β_2 beta","display":"\\beta_{2}","kind":"use","excerpt":"Liouville λ · β_1=-1, β_2=1 · 1,-1,1,-1,1,-1,… · -1,1,-1,1,-1,1,…","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-021b7991697753e4","tex":"\\beta_p","aliases":"β_p beta","display":"\\beta_p","kind":"use","excerpt":"upper triangular with diagonal 1,M,M^2,…,M^d. Hence det A_M,s^(d)=M^d(d+1)/2. Every characteristic-zero polynomial chart is invertible. item On the integral lattice Z^d+1, one channel is injective. If M>1 and d≥1, its image has index M^d(d+1)/2, so the channel has no integer linear left inverse. item Let p be prime and put β_p=p^v_p(M). Reduction m","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8e789d271780348c","tex":"\\beta_{p,d}","aliases":"β_p,d beta","display":"\\beta_{p,d}","kind":"use","excerpt":"Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bff30d0fad3c556f","tex":"\\beta_{p,m}","aliases":"β_p,m beta","display":"\\beta_{p,m}","kind":"use","excerpt":"Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9c815a51ca56e21f","tex":"\\Gamma'","aliases":"Γ' Gamma","display":"\\Gamma'","kind":"use","excerpt":"[Admissibility monotonicity] If Γ⊆Γ', then δ_Γ'(X)≤δ_Γ(X). Thus an obstruction in a restricted realization class need not remain an obstruction after the class is enlarged.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d1fb4b7a187b28f8","tex":"\\Gamma^{(u,r)}_{(v,s)}","aliases":"Γ^(u,r)_(v,s) Gamma","display":"\\Gamma^{(u,r)}_{(v,s)}","kind":"use","excerpt":"[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contrib","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-37edc912e92ad0b2","tex":"\\Gamma_{f,p}","aliases":"Γ_f,p Gamma","display":"\\Gamma_{f,p}","kind":"use","excerpt":"[Valuation logarithmic derivative] Fix a prime p. Let Γ_f,p be the arithmetic function supported on powers of p with Γ_f,p(1)=0, Γ_f,p(p^e)=γ_p,e(f) (e≥1). Then (v_p· f)(n) = (Γ_f,pmathbin∗_Df)(n) (n≥1). Equivalently, fracT F_f,p'(T)F_f,p(T) = ∑_e≥1γ_p,e(f)T^e.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ab5518db2ef46fc","tex":"\\gamma_{p,d}","aliases":"γ_p,d gamma","display":"\\gamma_{p,d}","kind":"use","excerpt":"l(β_p,m(f))_p,m, (γ_p,e(f))_p,e. Dirichlet convolution is linearized by the last two charts: β_p,m(fmathbin∗_Dg) = β_p,m(f)+β_p,m(g), γ_p,e(fmathbin∗_Dg) = γ_p,e(f)+γ_p,e(g). The ghost chart consists exactly of the prime-indexed sequences satisfying the local M obius divisibility conditions frac1e∑_dmid eμ (frac ed)γ_p,d∈ Z.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f8a608803709ed75","tex":"\\gamma_{p,e}","aliases":"γ_p,e gamma","display":"\\gamma_{p,e}","kind":"use","excerpt":"Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ccd1aa31c701086","tex":"\\gamma_{p,j}","aliases":"γ_p,j gamma","display":"\\gamma_{p,j}","kind":"use","excerpt":"The local logarithmic-derivative recurrence gives e f(p^e) = ∑_j=1^eγ_p,j(f)f(p^e-j). Write n=p^e u with (u,p)=1 and multiply by f(u). Since Γ_f,p is supported on p-powers, the resulting sum is exactly the Dirichlet convolution at n.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-11f0da0c0a47777b","tex":"\\Gamma_{tr}","aliases":"Γ_tr Gamma","display":"\\Gamma_{tr}","kind":"use","excerpt":"X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq ssucceq r, they also satisfy Γ_tr=Γ_srΓ_ts. Thus the equivalences are compatible with","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-086df5d7b47c88b8","tex":"\\Gamma_{ts}","aliases":"Γ_ts Gamma","display":"\\Gamma_{ts}","kind":"use","excerpt":"X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq ssucceq r, they also satisfy Γ_tr=Γ_srΓ_ts. Thus the equivalences are compatible with","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"extup(roman*)] item evaluation at the root identifies the space of global parallel sections with C_o; item (C_v)_v is path-flat, meaning that all path transports with the same endpoints agree on the core; item it is the unique greatest path-flat subfamily; item it is independent of the chosen spanning tree and equals the common fixed space of all c","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0d6cdab24a850778","tex":"\\Delta^r","aliases":"Δ^r Delta","display":"\\Delta^r","kind":"use","excerpt":"[Mahler coordinate theorem] The synthesis map M_p:c_0( Z_p)⟶ C_p, c⟼ (x⟼∑_r≥0c_rbinom xr), is an isometric isomorphism of Z_p-modules. Its inverse is c_r=Δ^r f(0). In particular, the Newton--Mahler coefficient sequence is a canonical complete coordinate for a continuous Z_p-valued function.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fbf1bc753c2507e3","tex":"\\delta^{\\rm cl}_{N,j}","aliases":"δ^rm cl_N,j delta rm","display":"\\delta^{\\rm cl}_{N,j}","kind":"use","excerpt":"Let Q^rm cl_N,j be an annihilator produced by a declared closure or elimination pipeline, and let L_N,j be the monic minimal operator of the section. Since every annihilator belongs to the principal left ideal generated by L_N,j, Q^rm cl_N,j=A_N,jL_N,j for some Ore operator A_N,j. The nonnegative quantity δ^rm cl_N,j = ordQ^rm cl_N,j - ordL_N,j mea","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fad4cf6b33469381","tex":"\\delta_*","aliases":"δ_* delta","display":"\\delta_*","kind":"use","excerpt":"For (H1), additivity and invertibility follow from the corresponding properties of F_γ and the splitting comparisons β. The address and splitting cocycle laws give (δγ)_*=δ_*γ_* after the declared transition comparisons.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-49a63bc6da6e2534","tex":"\\Delta_{12\\cdot2^a}^{\\,r}","aliases":"Δ_12·2^a^ r Delta cdot","display":"\\Delta_{12\\cdot2^a}^{\\,r}","kind":"use","excerpt":"18 · 11. Hence Δ_12·2^a^ rF_n ≡0pmod2^(a+3)r.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-37de6f15404aec98","tex":"\\delta_d","aliases":"δ_d delta","display":"\\delta_d","kind":"use","excerpt":"The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9437cea347b38d28","tex":"\\delta_{de}","aliases":"δ_de delta","display":"\\delta_{de}","kind":"use","excerpt":"[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coe","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-110a913431867dcf","tex":"\\delta_{e,e'}","aliases":"δ_e,e' delta","display":"\\delta_{e,e'}","kind":"use","excerpt":"Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aa7d5fad49c50597","tex":"\\Delta_H^r","aliases":"Δ_H^r Delta","display":"\\Delta_H^r","kind":"use","excerpt":"[Matrix Carmichael--Mahler constructor] Assume C^H=I+p^LB, L≥1. For every address j, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^ TC^jB^rv_0, where the series converges for x∈ Z_p. Its Mahler coefficients satisfy v_p(c_j,r)≥ Lr, and Δ_H^r a_n≡0pmodp^Lr.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-23d5cb86eddc6ee5","tex":"\\delta_{j,1}","aliases":"δ_j,1 delta","display":"\\delta_{j,1}","kind":"use","excerpt":"[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the sou","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9e83e11e41159cf8","tex":"\\Delta_Q","aliases":"Δ_Q Delta","display":"\\Delta_Q","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-78cebc60c8a61495","tex":"\\delta_{q,q'}","aliases":"δ_q,q' delta","display":"\\delta_{q,q'}","kind":"use","excerpt":"are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2d0446810a52779a","tex":"\\Delta_S","aliases":"Δ_S Delta","display":"\\Delta_S","kind":"use","excerpt":"extup(roman*)] item evaluation at the root identifies the space of global parallel sections with C_o; item (C_v)_v is path-flat, meaning that all path transports with the same endpoints agree on the core; item it is the unique greatest path-flat subfamily; item it is independent of the chosen spanning tree and equals the common fixed space of all c","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b5246be98c405b5f","tex":"\\delta_{\\Gamma'}","aliases":"δ_Γ' delta Gamma","display":"\\delta_{\\Gamma'}","kind":"use","excerpt":"[Admissibility monotonicity] If Γ⊆Γ', then δ_Γ'(X)≤δ_Γ(X). Thus an obstruction in a restricted realization class need not remain an obstruction after the class is enlarged.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"Fix an integer radix k≥2 and a field K. Let Σ_k={0,1,…,k-1}. Words are read from least significant digit to most significant digit. For w=d_0d_1⋯ d_ℓ-1∈Σ_k^*, define |w|=ℓ, [w]_k=∑_j=0^ℓ-1d_jk^j. The empty word ε has length zero and value zero.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"use","excerpt":"[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}.","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6a3433261c28571a","tex":"\\zeta_N","aliases":"ζ_N zeta","display":"\\zeta_N","kind":"use","excerpt":"[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8a71f72790f27037","tex":"\\zeta_N^{-j\\ell}","aliases":"ζ_N^-jℓ zeta ell","display":"\\zeta_N^{-j\\ell}","kind":"use","excerpt":"[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b4dd2da0a9b820fd","tex":"\\zeta_N^a","aliases":"ζ_N^a zeta","display":"\\zeta_N^a","kind":"use","excerpt":"ional descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. The","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f362658753c0c7bf","tex":"\\zeta_q","aliases":"ζ_q zeta","display":"\\zeta_q","kind":"use","excerpt":"gcd(a,q)=1 e^2π i aℓ/q = ∑_dmidgcd(q,ℓ)d μ(q/d) ∈ Z is the Ramanujan sum. The projectors (E_q)_qmid N form the finest Q-rational coarsening obtainable by summing the split Fourier idempotents, and Q[C_N]E_q≅ Q(ζ_q).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d7aaaf906dea3501","tex":"\\zeta_q^{-a\\ell}","aliases":"ζ_q^-aℓ zeta ell","display":"\\zeta_q^{-a\\ell}","kind":"use","excerpt":"Write every j∈ O_q uniquely as j=(N/q)a with a∈( Z/q Z)^×. The coefficient of g^ℓ in E_q is then frac1N ∑_a∈( Z/q Z)^× ζ_q^-aℓ = fracc_q(ℓ)N, where the last equality uses the integrality and evenness of the Ramanujan sum. Hence E_q∈ Q[C_N]. Orthogonality and completeness follow by grouping the primitive e_j by their disjoint Galois orbits. Any rati","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-61f8915ac0e5c978","tex":"\\eta_{abc}","aliases":"η_abc eta","display":"\\eta_{abc}","kind":"use","excerpt":"[Normalizer-induced coherence] Assume that every normalized fiber of N_λ,a is contractible. Then the comparison isomorphism η_abc: Φ_bcΦ_ab Longrightarrow Φ_ac is unique. Every triple-overlap diagram commutes, every quadruple-overlap coherence equation holds, and the composite around every finite bridge cycle is the identity transformation of the n","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-79dd9729e049d967","tex":"\\eta_{UV}","aliases":"η_UV eta","display":"\\eta_{UV}","kind":"use","excerpt":"et objects, and structure maps are compatible descent data defines a fold in Desc( R). Compatible selectors and left inverses descend, so a split-losslessness certificate descends directly from its left-inverse identity. Moreover, suppose the relevant local kernels K_U=ker(Φ_U) exist, every transition functor preserves them, and there are coherent ","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-99c6eec1fab7ee66","tex":"\\theta^A_{UV}","aliases":"θ^A_UV theta","display":"\\theta^A_{UV}","kind":"use","excerpt":"there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also that the transition isomorphisms intertwine the local projectors: θ^A_UV F_UV(p_A_U,i^r) = p_A_V,λ_UV^A,r(i)^r θ^A_UV, and that the ve","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c52b392f011cef82","tex":"\\theta^X_{UV}","aliases":"θ^X_UV theta","display":"\\theta^X_{UV}","kind":"use","excerpt":"[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms sa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2eed85328a2df168","tex":"\\theta^X_\\gamma","aliases":"θ^X_γ theta gamma","display":"\\theta^X_\\gamma","kind":"use","excerpt":"A descent object A=(A_U,θ^A_γ) consists of objects A_U∈ C_U and isomorphisms θ^A_γ:F_γ(A_U)xrightarrow∼A_V satisfying the identity and cocycle equations. A descended morphism f:X→ Y is a compatible family (f_U)_U satisfying θ^Y_γ F_γ(f_U)=f_Vθ^X_γ.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-59f21ecefbebe879","tex":"\\theta^Y_{UV}","aliases":"θ^Y_UV theta","display":"\\theta^Y_{UV}","kind":"use","excerpt":"[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms sa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dc31a2484f1dcab7","tex":"\\theta^Y_\\gamma","aliases":"θ^Y_γ theta gamma","display":"\\theta^Y_\\gamma","kind":"use","excerpt":"A descent object A=(A_U,θ^A_γ) consists of objects A_U∈ C_U and isomorphisms θ^A_γ:F_γ(A_U)xrightarrow∼A_V satisfying the identity and cocycle equations. A descended morphism f:X→ Y is a compatible family (f_U)_U satisfying θ^Y_γ F_γ(f_U)=f_Vθ^X_γ.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-60f70cc6e61fdab5","tex":"\\theta^Z_{UV}","aliases":"θ^Z_UV theta","display":"\\theta^Z_{UV}","kind":"use","excerpt":"[Finite atlas descent for lane transport] Let R be a finite profiled atlas whose participating chart categories are additive and idempotent-complete, with additive transition functors F_UV: R(U)→ R(V) on its declared overlaps. Let X=(X_U,θ^X_UV), Y=(Y_U,θ^Y_UV), Z=(Z_U,θ^Z_UV) be three compatible descent objects. Thus the transition isomorphisms sa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c62238b127faaf97","tex":"\\Theta_{A,B}","aliases":"Θ_A,B Theta","display":"\\Theta_{A,B}","kind":"use","excerpt":"[Fixed-denominator Cartier realization] Let F= M_p[A/B] and D=D(A,B). Then: [label=textup(roman*)] item every T_d preserves P_D; item the linear map Θ_A,B: P_D⟶ F_p[[z]], Θ_A,B(P)=fracPBF, is injective; item the transitions intertwine exactly: Λ_dΘ_A,B(P) = Θ_A,B(T_d(P)); item starting from P_0=B, the coefficient of index n=d_0+d_1p+⋯+d_ℓ-1p^ℓ-1 is","count":9},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-27f4d24e4ff91e81","tex":"\\Theta_{M',\\mathcal O'}","aliases":"Θ_M', O' Theta mathcal","display":"\\Theta_{M',\\mathcal O'}","kind":"use","excerpt":"[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-27edd9599acaca5a","tex":"\\Theta_{M,\\mathcal O}","aliases":"Θ_M, O Theta mathcal","display":"\\Theta_{M,\\mathcal O}","kind":"use","excerpt":"[Task strengthening] Adding continuations or observations can only refine behavioral indistinguishability. Thus M⊆ M', O⊆ O' Longrightarrow Θ_M', O'⊆Θ_M, O, and there is a canonical forgetting map Q_M', O'⟶ Q_M, O.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-017adcf52ee20d92","tex":"\\Theta_{M,s}","aliases":"Θ_M,s Theta","display":"\\Theta_{M,s}","kind":"use","excerpt":"For M≥1, s≥0, and r,k≥0, define Φ_M(z)=(1+z)^M-1 and Θ_M,s(r,k) =[z^r](1+z)^sΦ_M(z)^k. Equivalently, Θ_M,s(r,k) =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. The support is finite: Θ_M,s(r,k)=0 unless k≤ r≤ Mk+s.","count":10},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6b4a0b405d0fc12a","tex":"\\Theta_{MN,Mt+s}","aliases":"Θ_MN,Mt+s Theta","display":"\\Theta_{MN,Mt+s}","kind":"use","excerpt":"[Mixed-radix kernel composition] For M,N≥1, ∑_k≥0 Θ_M,s(r,k)Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). Equivalently, T_N,t T_M,s = T_MN,Mt+s. When gcd(M,N)=1, the two Chinese-remainder refinement paths to a common address agree exactly. Their holonomy is the identity.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-78b975269dab6a12","tex":"\\Theta_{N,t}","aliases":"Θ_N,t Theta","display":"\\Theta_{N,t}","kind":"use","excerpt":"[Mixed-radix kernel composition] For M,N≥1, ∑_k≥0 Θ_M,s(r,k)Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). Equivalently, T_N,t T_M,s = T_MN,Mt+s. When gcd(M,N)=1, the two Chinese-remainder refinement paths to a common address agree exactly. Their holonomy is the identity.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ca492aad33d122fe","tex":"\\Theta_{p^a,s}","aliases":"Θ_p^a,s Theta","display":"\\Theta_{p^a,s}","kind":"use","excerpt":"[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-be1413dca890d698","tex":"\\iota^r_{X,U,i}","aliases":"ι^r_X,U,i iota","display":"\\iota^r_{X,U,i}","kind":"use","excerpt":"For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6de902212c61cdd9","tex":"\\iota^r_{X,V,\\lambda^X(i)}","aliases":"ι^r_X,V,λ^X(i) iota lambda","display":"\\iota^r_{X,V,\\lambda^X(i)}","kind":"use","excerpt":"For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d79d4f252e8fd7ed","tex":"\\iota^u_{Y,U,a}","aliases":"ι^u_Y,U,a iota","display":"\\iota^u_{Y,U,a}","kind":"use","excerpt":"item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix rec","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e58c313ef5f4baf","tex":"\\iota_{1,i_1}","aliases":"ι_1,i_1 iota","display":"\\iota_{1,i_1}","kind":"use","excerpt":"[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of mult","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-069e07007ede1cc9","tex":"\\iota_{d,i_d}","aliases":"ι_d,i_d iota","display":"\\iota_{d,i_d}","kind":"use","excerpt":"[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of mult","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-be2465856ff280ea","tex":"\\iota_j","aliases":"ι_j iota","display":"\\iota_j","kind":"use","excerpt":"Assume now that C is additive and idempotent complete. Choose a splitting X_ixrightarrowι_iXxrightarrowπ_iX_i, ι_iπ_i=p_i, π_iι_i=1_X_i. The packet equations imply π_iι_j=0 (i≠ j).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-738597aa47cb5523","tex":"\\iota_{X,j}^r","aliases":"ι_X,j^r iota","display":"\\iota_{X,j}^r","kind":"use","excerpt":"For (L1), the first composite is ∑_iι_X,i^rπ_X,i^r =∑_i p_X,i^r=1_X. The (i,j)-entry of the other composite is π_X,i^rι_X,j^r. It is the identity when i=j, and it is zero otherwise because the corresponding idempotents annihilate one another.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9021615a7ca7b785","tex":"\\iota_{Y,a}","aliases":"ι_Y,a iota","display":"\\iota_{Y,a}","kind":"use","excerpt":"inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps i","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1d0b35b26fa16cc2","tex":"\\iota_{Y,a}^r","aliases":"ι_Y,a^r iota","display":"\\iota_{Y,a}^r","kind":"use","excerpt":"[Fixed-resolution matrix-category equivalence] The functor M_r: L_ R⟶ Mat_r( L_ R), X⟼[X;(X_i^r)_i], f⟼([f]_a i^r,r), is an equivalence. A quasi-inverse returns the lane-decorated object named by the tag and sends a matrix to ∑_a,iι_Y,a^r h_a iπ_X,i^r. For every ssucceq r, the coordinate changes are functors and satisfy Γ_sr M_s= M_r. For tsucceq s","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-277e0a4bd7d4be01","tex":"\\iota_{Y,a}^u","aliases":"ι_Y,a^u iota","display":"\\iota_{Y,a}^u","kind":"use","excerpt":"ion r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-96f19cfd5c9aadf5","tex":"\\iota_{Y,b}^v","aliases":"ι_Y,b^v iota","display":"\\iota_{Y,b}^v","kind":"use","excerpt":"[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contrib","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-93055c50da4a404a","tex":"\\kappa_2","aliases":"κ_2 kappa","display":"\\kappa_2","kind":"use","excerpt":"0 · 1. There is no positive-definite metric Q with H^*QH=Q. Nevertheless, for Q_s=diag(s^2,1), we have D_Q_s(H)⟶1 (sdownarrow0), while κ_2(Q_s)=s^-2⟶∞. Thus the metric-completion defect has zero infimum but is not attained.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-545253b67e7a1373","tex":"\\kappa_3","aliases":"κ_3 kappa","display":"\\kappa_3","kind":"use","excerpt":"A genuine denominator example. Over F_3, take R(z)=frac11+z. Then F(z)^2=1+z. The reduced denominator remains fixed under every Cartier section. The ambient compiler has dimension two, its reachable numerator set is {0,1,2,1+z}, and its reachable span has dimension two. Hence r_3(F)=2, κ_3(F)=4. The nonzero collapse obstruction again proves that th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-810ceeda3892b7a8","tex":"\\kappa_p","aliases":"κ_p kappa","display":"\\kappa_p","kind":"use","excerpt":"Let κ_p(F)=| K_p(F)|.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c08cad81c2ea02b9","tex":"\\kappa_R","aliases":"κ_R kappa","display":"\\kappa_R","kind":"use","excerpt":"Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-11c22a9994059ca8","tex":"\\lambda'","aliases":"λ' lambda","display":"\\lambda'","kind":"use","excerpt":"An object of State_K^min is a triple (C,v,λ), C∈ M_d(K), v∈ K^d, λ∈(K^d)^*, with C invertible, producing a_n=λ C^nv, and satisfying reachability and observability. A morphism S:(C,v,λ)⟶(C',v',λ') is a similarity satisfying C'=SCS^-1, v'=Sv, λ'=λ S^-1. The semantic functor sends a state to its coefficient sequence.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-426d72493ed0249b","tex":"\\lambda^n","aliases":"λ^n lambda","display":"\\lambda^n","kind":"use","excerpt":"Over characteristic zero, write the unique active expansion over an algebraic closure as a_n=∑_λ∈ΛP_λ(n)λ^n. For each sampling modulus N and address j, the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of a_Nm+j [so","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dfdfa2caaa380bf7","tex":"\\lambda^X","aliases":"λ^X lambda","display":"\\lambda^X","kind":"use","excerpt":"For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d07a79cf101a2eb4","tex":"\\lambda^Y","aliases":"λ^Y lambda","display":"\\lambda^Y","kind":"use","excerpt":"For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6a7b56c3c1ef1d21","tex":"\\Lambda_{\\mathbb K}","aliases":"Λ_ K Lambda mathbb","display":"\\Lambda_{\\mathbb K}","kind":"use","excerpt":"[Typed realization and module equivalence, imported] Let Λ_ K:=𝕜 Q/I. In 𝕜-modules, the category of typed realizations of the linear grammar (Q,I) is equivalent to the category of unital left Λ_ K-modules. Under the equivalence, the vertex idempotent e_X acts as the projector onto the component of sort X, and each arrow acts in its declared source-","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6ede142bbc0a6d7e","tex":"\\Lambda_0","aliases":"Λ_0 Lambda","display":"\\Lambda_0","kind":"use","excerpt":"el is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b is b_0 and (Λ_0b)_0=b_0.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8158b283e39d416e","tex":"\\Lambda_{13}","aliases":"Λ_13 Lambda","display":"\\Lambda_{13}","kind":"use","excerpt":"Let Λ_13 = {DRLM,Loc,Fuss,EM,Gal} be the component-tag set. Define J_13(X) = (λ(X), J_λ(X)(X)).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f5edfb085a6f5e0b","tex":"\\Lambda_{d_0}","aliases":"Λ_d_0 Lambda","display":"\\Lambda_{d_0}","kind":"use","excerpt":"For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b ","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-78c46d03c0d19f14","tex":"\\Lambda_{d_1}","aliases":"Λ_d_1 Lambda","display":"\\Lambda_{d_1}","kind":"use","excerpt":"For a sequence a=(a_n)_n≥0∈ K^ N, define the digit-section operators (Λ_d a)(n)=a(kn+d), 0≤ d<k. For a word w=d_0⋯ d_ℓ-1, put Λ_w = Λ_d_ℓ-1⋯Λ_d_1Λ_d_0. Then (Λ_w a)(n) = a(k^|w|n+[w]_k).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-65fc2d4695a42747","tex":"\\Lambda_{d_{a-1}}","aliases":"Λ_d_a-1 Lambda","display":"\\Lambda_{d_{a-1}}","kind":"use","excerpt":"eld and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coefficients with index k^an+[w]_k, [w]_k=d_","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ab3ba0e515df3de9","tex":"\\Lambda_{d_{\\ell-1}}","aliases":"Λ_d_ℓ-1 Lambda ell","display":"\\Lambda_{d_{\\ell-1}}","kind":"use","excerpt":"For 0≤ d<p, define the Cartier section of a sequence by (Λ_da)_n=a_pn+d. The p-kernel is K_p(a) = { Λ_d_ℓ-1⋯Λ_d_0a: ℓ≥ 0, 0≤ d_i<p }. We use least-significant-digit-first words. Every finite digit word is allowed. Appending high zero digits may change the internal state but never changes the represented coefficient, because the output of a state b ","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c8f42b6771ecc9c8","tex":"\\lambda_e","aliases":"λ_e lambda","display":"\\lambda_e","kind":"use","excerpt":"Conversely, suppose every normalized root holonomy preserves Q_o. Choose spanning-tree maps G_v from the root to each vertex and put Q_v=(G_v^-1)^*Q_oG_v^-1. For an edge e:u→ v, the root word H_e=G_v^-1T_eG_u satisfies H_e^*Q_oH_e=|det H_e|^2/nQ_o. Consequently T_e^*Q_vT_e=|det H_e|^2/nQ_u, λ_e=|det H_e|^1/n>0. This gives the required compatible fa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8030f7a3a607b76a","tex":"\\lambda_e^2","aliases":"λ_e^2 lambda","display":"\\lambda_e^2","kind":"use","excerpt":"raph carry real or complex vector spaces of common dimension n≥1 and invertible edge transports T_e. Reverse traversal of an edge uses T_e^-1. At a chosen root, determinant-normalized closed holonomy words are widehat H:=|det H|^-1/nH. They form the determinant-normalized holonomy representation. Positive-definite vertex metrics and positive edge s","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-dd94824bf4325f68","tex":"\\Lambda_{k-1}","aliases":"Λ_k-1 Lambda","display":"\\Lambda_{k-1}","kind":"use","excerpt":"[Section-module realization theorem] For a sequence a∈ K^ N, the following are equivalent: item a is k-regular over K; item a admits a finite-dimensional digit-linear realization. When these conditions hold, the tuple C_k(a) = ( V_k(a), Λ_0,…,Λ_k-1, a, ϵ_0 ), ϵ_0(f)=f(0), is a canonical realization of dimension r_k(a).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bf871da91f82e1bb","tex":"\\lambda_{UT}^{A,r}","aliases":"λ_UT^A,r lambda","display":"\\lambda_{UT}^{A,r}","kind":"use","excerpt":"Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucce","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-74995e757386c4ca","tex":"\\lambda_{UU}^{A,r}","aliases":"λ_UU^A,r lambda","display":"\\lambda_{UU}^{A,r}","kind":"use","excerpt":"Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucce","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5f3e526d5aaef701","tex":"\\lambda_{UV}^{A,r}","aliases":"λ_UV^A,r lambda","display":"\\lambda_{UV}^{A,r}","kind":"use","excerpt":"Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucce","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a5a0fe6f3b59cc36","tex":"\\lambda_{UV}^{A,s}","aliases":"λ_UV^A,s lambda","display":"\\lambda_{UV}^{A,s}","kind":"use","excerpt":"carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for ev","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fc49fd558a416e05","tex":"\\lambda_{VT}^{A,r}","aliases":"λ_VT^A,r lambda","display":"\\lambda_{VT}^{A,r}","kind":"use","excerpt":"Fix one finite resolution shape R. Suppose every local object A_U, for A∈{X,Y,Z}, carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucce","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-595cb14f66316f0e","tex":"\\mu^2","aliases":"μ^2 mu","display":"\\mu^2","kind":"use","excerpt":"Squarefree indicator μ^2 · β_1=1, β_2=-1 · 1,1,0,0,0,0,… · 1,-1,1,-1,1,-1,…","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ac6cb066658534c5","tex":"\\mu_1","aliases":"μ_1 mu","display":"\\mu_1","kind":"use","excerpt":"[One operator and primary packets] Take one sort, one endomorphism T, and one relation μ(T)=0. The linear grammar algebra is 𝕜[x]/(μ). When μ=μ_1⋯μ_r is a product of pairwise comaximal factors, the Chinese remainder idempotents define packet objects for the primary summands. The envelope splits those idempotents formally. Over a realization where t","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0a4a83311794d38e","tex":"\\mu_d","aliases":"μ_d mu","display":"\\mu_d","kind":"use","excerpt":"[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arit","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-96de71b6d2f7dbf6","tex":"\\mu_{d_{\\mathcal A}}","aliases":"μ_d_ A mu mathcal","display":"\\mu_{d_{\\mathcal A}}","kind":"use","excerpt":"[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arit","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-25b9b80beb1d5761","tex":"\\mu_H","aliases":"μ_H mu","display":"\\mu_H","kind":"use","excerpt":"Let G be a compact group and let U:G→ U( H) be strongly continuous. For a closed subgroup H, normalized Haar measure defines P_H=∫_HU_h dμ_H(h). tag12.14 The same averaging proof as in the finite case shows that P_H is the orthogonal projector onto H^H.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e5715295ee0150ed","tex":"\\mu_{h_N}","aliases":"μ_h_N mu","display":"\\mu_{h_N}","kind":"use","excerpt":"[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arit","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ccd90bd341db9817","tex":"\\mu_{p^r}","aliases":"μ_p^r mu","display":"\\mu_{p^r}","kind":"use","excerpt":"[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c27be89125069de6","tex":"\\mu_{q-1}","aliases":"μ_q-1 mu","display":"\\mu_{q-1}","kind":"use","excerpt":"power map used for packet synthesis has fracq-1gcd(N,q-1) distinct images. If a fixed embedding K↪ C sends α to a positive real number and the marked branch is interpreted in the smooth implicit analytic schema, its critical points and values satisfy qατ^q-1=1, ρ=τ-ατ^q=fracq-1qτ. The resulting orbit ρμ_q-1 is precisely the dominant singularity orb","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-22d0564d0934fa50","tex":"\\mu_r","aliases":"μ_r mu","display":"\\mu_r","kind":"use","excerpt":"[One operator and primary packets] Take one sort, one endomorphism T, and one relation μ(T)=0. The linear grammar algebra is 𝕜[x]/(μ). When μ=μ_1⋯μ_r is a product of pairwise comaximal factors, the Chinese remainder idempotents define packet objects for the primary summands. The envelope splits those idempotents formally. Over a realization where t","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f14a00361f68e31a","tex":"\\mu_{r,\\gamma}","aliases":"μ_r,γ mu gamma","display":"\\mu_{r,\\gamma}","kind":"use","excerpt":"Let F:Y→ Z and G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4a7fb58e6da3d5f9","tex":"\\mu_U^{A,r}","aliases":"μ_U^A,r mu","display":"\\mu_U^{A,r}","kind":"use","excerpt":"UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-201512ba4e1f3a60","tex":"\\mu_U^{A,s}","aliases":"μ_U^A,s mu","display":"\\mu_U^{A,s}","kind":"use","excerpt":"iple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6490ea5f6229cad8","tex":"\\mu_V^{A,r}","aliases":"μ_V^A,r mu","display":"\\mu_V^{A,r}","kind":"use","excerpt":"iple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_V^A,rλ_UV^A,r=μ_U^A,r, μ_U^A,rρ_sr^A_U =ρ_sr^Aμ_U^A,s. Assume also th","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a1682ede43244542","tex":"\\nu_a","aliases":"ν_a nu","display":"\\nu_a","kind":"use","excerpt":"Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite pre","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-519aa278b83d7616","tex":"\\nu_i","aliases":"ν_i nu","display":"\\nu_i","kind":"use","excerpt":"Fix a prime p and write ν_i=ν_p,i(a)=v_p(d_i(a)).","count":13},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-62d0d50e0643749d","tex":"\\nu_p","aliases":"ν_p nu","display":"\\nu_p","kind":"use","excerpt":"For a∈Per_p^×,per, define the periodic resolved invariant package J_p(a) = ( T(a), w(a), m_a(x), P_a(z)/Q_a(z), K_p(a), A_p(a), R_a(z) ), where A_p(a) is the canonical minimal digital automaton. Its comparison identities include deg m_a · =deg Q_a =rank H(a) =dim State_min(a), T(a) · =min{N:m_amid x^N-1} =min{N:Q_amid 1-z^N}, | A_p(a)| · =| K_p(a)|","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6cbe06c3ea3d8f3a","tex":"\\nu_{p,i}","aliases":"ν_p,i nu","display":"\\nu_{p,i}","kind":"use","excerpt":"Fix a prime p and write ν_i=ν_p,i(a)=v_p(d_i(a)).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"use","excerpt":"ther with its finite initial record; item the monic minimal differential operator from [source reference], together with the distinguished power-series solution selected by the marked coefficient record; item the hypergeometric parameter packet (A_N,j(λ),B_N,j;α^N) and leading coefficient; item the primitive minimal-operator singular and indicial r","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f6c04fa92bb658c0","tex":"\\Pi^2","aliases":"Π^2 Pi","display":"\\Pi^2","kind":"use","excerpt":"Let A=P_m,S^(d): Z^d+1→ Z^|S|(d+1) be a stacked affine-Mahler packet analysis map. Whenever an integer left inverse L exists, LA=I, Π=AL, Π^2=Π, and Z^|S|(d+1)=im A⊕ker L. Thus the source chart and the normalized packet core im A are equivalent, while ker L is a named residual rather than discarded information.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0dbdbbd8881c1f6b","tex":"\\pi^r_{X,U,i}","aliases":"π^r_X,U,i pi","display":"\\pi^r_{X,U,i}","kind":"use","excerpt":"item Exact twisted analysis and synthesis. Lane analysis induces an isomorphism A^u,r_Y,X: Hom_Desc(mathscr G)(X,Y) xrightarrow∼ Γ (mathscr G,mathscr M^u,r_Y,X), f⟼ (([f_U]^u,r_ai)_a,i)_U. Its inverse is chartwise lane synthesis: (h_U)_U⟼ ( ∑_a,i ι^u_Y,U,ah_U,aiπ^r_X,U,i )_U. Thus nontrivial address holonomy does not destroy the complete matrix rec","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-53b21d9e927e4ae4","tex":"\\pi^u_{Y,U,a}","aliases":"π^u_Y,U,a pi","display":"\\pi^u_{Y,U,a}","kind":"use","excerpt":"For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-be960e5058eac294","tex":"\\pi^u_{Y,V,\\lambda^Y(a)}","aliases":"π^u_Y,V,λ^Y(a) pi lambda","display":"\\pi^u_{Y,V,\\lambda^Y(a)}","kind":"use","excerpt":"For (H2), let f=(f_U)_U be descended. For a transition γ:U→ V, the transported (a,i)-block is · β^Y_γ,u,a F_γ ( π^u_Y,U,af_Uι^r_X,U,i ) (β^X_γ,r,i)^-1 · = π^u_Y,V,λ^Y(a) θ^Y_γ F_γ(f_U)(θ^X_γ)^-1 ι^r_X,V,λ^X(i) · = π^u_Y,V,λ^Y(a) f_V ι^r_X,V,λ^X(i), which is the corresponding block of f_V. Thus analysis produces a compatible section.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-222fb49e971fef73","tex":"\\pi_1","aliases":"π_1 pi","display":"\\pi_1","kind":"use","excerpt":"Finite transport holonomy · connected graph, invertible edge transports, chosen structure group · fundamental cycle holonomies or the conjugacy class of π_1(Γ)→ G · one global frame on the full fibers; on a fixed subspace, path-independent transport · path dependence; possibly a smaller nonzero fixed core","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b4bd26a1214b5b6f","tex":"\\pi_{1,i_1}","aliases":"π_1,i_1 pi","display":"\\pi_{1,i_1}","kind":"use","excerpt":"inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps i","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-41b82089b42bb440","tex":"\\pi_1^{\\acute{e}t}","aliases":"π_1^acuteet pi acute","display":null,"kind":"use","excerpt":"For a connected scheme X with geometric point bar x, the fiber functor on finite acuteetale covers identifies that category with finite continuous π_1^acuteet(X,bar x)-sets. Every finite fiber S produces a permutation resolution on ℓ^2(S): coordinate projectors are indexed by points, orbit projectors by transitive components, and covering morphisms","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-384c4c16a2a53fc8","tex":"\\pi_{d,i_d}","aliases":"π_d,i_d pi","display":"\\pi_{d,i_d}","kind":"use","excerpt":"inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps i","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8f322ee5409feacc","tex":"\\pi_j","aliases":"π_j pi","display":"\\pi_j","kind":"use","excerpt":"Let Ω be a normal splitting field of the generic packet scheme over K(x), and let Γ=Gal(Ω/K(x)). After splitting E_N, the geometric packet branches form a set B_N of cardinality d^N. Let π_j: B_N→Ω be the jth packet coordinate, and set C_N,j=π_j( B_N). For the selected semantic packet U_j, separability gives [K(x)(U_j):K(x)] = |Γ· U_j|.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-131b077bd90df974","tex":"\\Pi_{\\rm common}","aliases":"Π_rm common Pi rm","display":"\\Pi_{\\rm common}","kind":"use","excerpt":"[Common profile and composition] Let F: C→ D and G: D→ E. Every constructor preserved by both functors, with compatible chosen comparison data, is preserved by GF. For a finite diagram of realizations and bridges, the intersection Π_rm common:=⋂_FΠ_F is therefore a valid common profile for every proof transported through the whole diagram.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7555eb76851a882c","tex":"\\pi_{X,j}^s","aliases":"π_X,j^s pi","display":"\\pi_{X,j}^s","kind":"use","excerpt":"[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contrib","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-50ff986896f3fa59","tex":"\\pi_{Y,a}","aliases":"π_Y,a pi","display":"\\pi_{Y,a}","kind":"use","excerpt":"[Module multilinear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of mult","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a3bd55dd0334eb19","tex":"\\pi_{Y,a}^u","aliases":"π_Y,a^u pi","display":"\\pi_{Y,a}^u","kind":"use","excerpt":"At resolution r, write Analysis_X^r :=(π_X,i^r)_i: X⟶⊕_iX_i^r, Synthesis_X^r :=(ι_X,i^r)_i: ⊕_iX_i^r⟶ X. For profiled lane systems on X and Y, define the coordinate block of f:X→ Y at resolutions r,u by [f]^u,r_a i := π_Y,a^u fι_X,i^r: X_i^r⟶ Y_a^u. Its ambient corner is p_Y,a^u f p_X,i^r = ι_Y,a^u[f]^u,r_a iπ_X,i^r.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bdfeea1824c7d599","tex":"\\rho^A_{sr}","aliases":"ρ^A_sr rho","display":"\\rho^A_{sr}","kind":"use","excerpt":"item If ssucceq r, equivariance of the refinement map sends every fine monodromy orbit onto a coarse monodromy orbit. For a coarse orbit O, P^r_A, O = ∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) P^s_A, Q. Thus orbit descent and projector-sum refinement commute exactly.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d4480c8bdf9c98bc","tex":"\\rho^N","aliases":"ρ^N rho","display":"\\rho^N","kind":"use","excerpt":"[Complete dominant-orbit signature] Under the smooth implicit schema, the dominant singularities of u(z) form ρ μ_d_ A, tag11.12 where μ_d is the group of dth roots of unity. After packetization with x=z^N, the union of dominant singularities of the complete packet tuple is ρ^N μ_h_N, h_N=fracd_ Agcd(N,d_ A). tag11.13 Therefore the same finite arit","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-91460788b7a82706","tex":"\\rho^{\\underline j}","aliases":"ρ^underline j rho underline","display":"\\rho^{\\underline j}","kind":"use","excerpt":"Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5e64ae5344d5eb86","tex":"\\rho^Y_{vu}","aliases":"ρ^Y_vu rho","display":"\\rho^Y_{vu}","kind":"use","excerpt":"item Projector-sum refinement. If ssucceq r on X and vsucceq u on Y, then p_Y,a^u f p_X,i^r = ∑_ρ^Y_vu(b)=a ∑_ρ^X_sr(j)=i p_Y,b^v f p_X,j^s. Coarsening a fine composite gives the same ambient morphism as composing at the coarse level.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aaa4c9bd206e21e1","tex":"\\rho_k","aliases":"ρ_k rho","display":"\\rho_k","kind":"use","excerpt":"Throughout this section, k≥2 is fixed and a: N⟶ Z is an integer-valued sequence whose rational section space V_ Q(a) = span_ Q K_k(a) ⊆ Q^ N has finite dimension ρ_k(a)=dim_ QV_ Q(a). Thus a is k-regular after extension of scalars to Q.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3e691508b0d59ac7","tex":"\\rho_{sr}^{A}","aliases":"ρ_sr^A rho","display":"\\rho_{sr}^{A}","kind":"use","excerpt":"I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for every A and r, there is a global finite address set I_A,r, a global refinement map ρ_sr^A:I_A,s↠ I_A,r, and bijections μ_U^A,r:I_A_U,rxrightarrow∼I_A,r such that μ_","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0d0d24000fa09685","tex":"\\rho_{sr}^{A_U}","aliases":"ρ_sr^A_U rho","display":"\\rho_{sr}^{A_U}","kind":"use","excerpt":"carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for ev","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d656aa14bf1e9356","tex":"\\rho_{sr}^{A_V}","aliases":"ρ_sr^A_V rho","display":"\\rho_{sr}^{A_V}","kind":"use","excerpt":"carries a finite profiled lane system over R. For every overlap, every r∈ R, and every such A, suppose there is an address bijection λ_UV^A,r: I_A_U,rxrightarrow∼I_A_V,r such that λ_UU^A,r=1, λ_UT^A,r =λ_VT^A,rλ_UV^A,r, on every triple overlap (U,V,T), and, for ssucceq r, ρ_sr^A_Vλ_UV^A,s = λ_UV^A,rρ_sr^A_U. Require trivial address holonomy: for ev","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-586e4bb36a3de8dc","tex":"\\rho_{tr}","aliases":"ρ_tr rho","display":"\\rho_{tr}","kind":"use","excerpt":"[Aggregation and transitivity] A finite aggregation of pairwise annihilating packets is again a packet system. If tsucceq ssucceq r, then refinement is transitive with address map ρ_tr=ρ_sr∘ρ_ts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-681cd151da36be84","tex":"\\rho_{ts}","aliases":"ρ_ts rho","display":"\\rho_{ts}","kind":"use","excerpt":"[Aggregation and transitivity] A finite aggregation of pairwise annihilating packets is again a packet system. If tsucceq ssucceq r, then refinement is transitive with address map ρ_tr=ρ_sr∘ρ_ts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ecd16e1cf56c4339","tex":"\\rho_\\sigma","aliases":"ρ_σ rho sigma","display":"\\rho_\\sigma","kind":"use","excerpt":"Let L/K be finite Galois with group G. A semilinear G-object is an L-vector space V with maps ρ_σ:V→ V, ρ_σ(λ v)=σ(λ)ρ_σ(v), satisfying the group law. Scalar extension and fixed vectors define functors L⊗_K-:Vect_K⟶Semilin_G(L), (-)^G:Semilin_G(L)⟶Vect_K.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-18e4292450c4f350","tex":"\\Sigma^{[N]}","aliases":"Σ^[N] Sigma","display":"\\Sigma^{[N]}","kind":"use","excerpt":"[Singularity orbit under multisection] Assume F is an analytic germ at zero with analytic continuation away from a finite set Σ. Then every finite nonzero analytic singularity of M_N,jF lies in Σ^[N] = {α^N:α∈Σ}. Distinct singularities in one fiber of α↦α^N are aliased by the multisection map. A character projection may remove some aliased local co","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5048c4b9123eb530","tex":"\\sigma_1","aliases":"σ_1 sigma","display":"\\sigma_1","kind":"use","excerpt":"[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plan","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-15fde97ab40a2e94","tex":"\\sigma_2","aliases":"σ_2 sigma","display":"\\sigma_2","kind":"use","excerpt":"[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plan","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7cc70e05d3b03d7a","tex":"\\sigma_a","aliases":"σ_a sigma","display":"\\sigma_a","kind":"use","excerpt":"[Rational descent of split cyclic Fourier lanes] Let N≥1, let C_N=⟨ g⟩, and let K= Q(ζ_N). In K[C_N], define e_j := frac1N∑_ℓ=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3fec71360aadd761","tex":"\\Sigma_k","aliases":"Σ_k Sigma","display":"\\Sigma_k","kind":"use","excerpt":"Fix an integer radix k≥2 and a field K. Let Σ_k={0,1,…,k-1}. Words are read from least significant digit to most significant digit. For w=d_0d_1⋯ d_ℓ-1∈Σ_k^*, define |w|=ℓ, [w]_k=∑_j=0^ℓ-1d_jk^j. The empty word ε has length zero and value zero.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e7fc48d0319157d","tex":"\\sigma_{\\max}","aliases":"σ_max sigma max","display":"\\sigma_{\\max}","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-22301d4bf1c931e5","tex":"\\sigma_{\\min}","aliases":"σ_min sigma min","display":"\\sigma_{\\min}","kind":"use","excerpt":"Let T:(V,Q_s)→(W,Q_t) be invertible between real or complex spaces of common finite dimension n≥1 with positive-definite metrics. Choose factors Q_s=R_s^*R_s and Q_t=R_t^*R_t, and put B_Q(T):=R_tTR_s^-1. Define D_Q(T) := fracσ_max(B_Q(T))σ_min(B_Q(T)) ≥1, Δ_Q(T):=log D_Q(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e1ee5d0e033b41f4","tex":"\\sigma_{r+1}","aliases":"σ_r+1 sigma","display":"\\sigma_{r+1}","kind":"use","excerpt":"[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plan","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ae30ce77fb02391b","tex":"\\Sigma_{\\rm op}","aliases":"Σ_rm op Sigma rm","display":"\\Sigma_{\\rm op}","kind":"use","excerpt":"Multiply L_F by a common denominator, divide the polynomial coefficients by their greatest common divisor, and normalize the leading coefficient to be monic. This gives a unique primitive polynomial representative widetilde L_F = ∑_j=0^rq_j(z)D^j. Its operator singular support is Σ_rm op(F) = {q_r(z)=0}∪{∞}. Apparent singularities remain part of th","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c12c3bc0a6f4a7b5","tex":"\\sigma_{vu}","aliases":"σ_vu sigma","display":"\\sigma_{vu}","kind":"use","excerpt":"[Exact refinement law, imported] Let p^r_i=∑_ρ_sr(j)=ip^s_j and q^u_a=∑_σ_vu(b)=aq^v_b. For every morphism f:X→ Y, q^u_afp^r_i = ∑_σ_vu(b)=a ∑_ρ_sr(j)=i q^v_bfp^s_j. Coarsening the fine block product of two composable maps gives the coarse block product.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a86a467fd47d0f11","tex":"\\tau^q","aliases":"τ^q tau","display":"\\tau^q","kind":"use","excerpt":"Formula textup(11.15) follows from Lagrange inversion applied to z=u(1-u^q-1). The support and active-address statements are [source reference] with d_ A=q-1. Expanding u^q in complete packets and collecting the exponent carry r_1+⋯+r_q=j+Nc gives textup(11.16). The critical equations qτ^q-1=1, ρ=τ-τ^q give the printed values, and [source reference","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-15c8b38106861fcf","tex":"\\tau^{q-1}","aliases":"τ^q-1 tau","display":"\\tau^{q-1}","kind":"use","excerpt":"Formula textup(11.15) follows from Lagrange inversion applied to z=u(1-u^q-1). The support and active-address statements are [source reference] with d_ A=q-1. Expanding u^q in complete packets and collecting the exponent carry r_1+⋯+r_q=j+Nc gives textup(11.16). The critical equations qτ^q-1=1, ρ=τ-τ^q give the printed values, and [source reference","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e61935ffdc30b2b8","tex":"\\tau_3","aliases":"τ_3 tau","display":"\\tau_3","kind":"use","excerpt":"Three-fold divisor function τ_3 · β_1=3 · 1,3,6,10,15,21,… · 3,3,3,3,3,3,…","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-105fa42a83c8f65f","tex":"\\tau_e","aliases":"τ_e tau","display":"\\tau_e","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":18},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-af174b012382e385","tex":"\\tau_{e'}","aliases":"τ_e' tau","display":"\\tau_{e'}","kind":"use","excerpt":"Suppose, more strongly, that every τ_e is injective and that there are globally typed selectors Q_e∈End(C) satisfying, for every e,e'∈ E and every b_e'∈ B_e', Q_ej_κ(e')τ_e'(b_e') = δ_e,e'j_κ(e')τ_e'(b_e'). Then Φ_κ,τ is injective. If there are also homomorphisms r_e:C⟶ B_e, r_ej_κ(e)τ_e=1_B_e, then L:=(r_eQ_e)_e∈ E:C⟶⊕_e∈ EB_e is a left inverse of","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6325af2ecbecf4ff","tex":"\\tau_U","aliases":"τ_U tau","display":"\\tau_U","kind":"use","excerpt":"Evaluation at U_0 induces an isomorphism Γ (mathscr G,mathscr M^u,r_Y,X) xrightarrow∼ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0), where the right-hand side is the subgroup fixed by every loop-transport map. Consequently, Hom_Desc(mathscr G)(X,Y) ≅ mathscr M^u,r_Y,X(U_0)^Aut_mathscr G(U_0). After choosing one arrow τ_U:U_0→ U for each chart, the inve","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e5e6e36da01d2e6","tex":"\\tau_V","aliases":"τ_V tau","display":"\\tau_V","kind":"use","excerpt":"A compatible section evaluates to a loop-fixed element at U_0. Conversely, for an arrow γ:U→ V, the two arrows γτ_U and τ_V differ by a loop at U_0; loop invariance therefore gives γ_*h_U=h_V. Evaluation and this reconstruction are mutually inverse. Compose with the analysis isomorphism in (H2).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-959f488598ceedf4","tex":"\\varphi","aliases":"φ varphi","display":"\\varphi","kind":"use","excerpt":"[Cokernel criterion, universality, and naturality] For a correction datum (D,r) [source reference]: [label=textup(roman*)] item a correction exists if and only if Ob_D(r)=0; item every linear obstruction test φ:V→ W satisfying φ D=0 factors uniquely through V→coker D; item for every commuting square ccc U · xrightarrowD · V","count":15},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c89e189652ea7cb2","tex":"\\Phi'","aliases":"Φ' Phi","display":"\\Phi'","kind":"use","excerpt":"Assume now that the coefficients w_m are positive real numbers. The classical smooth implicit-function schema for simply generated trees applies when there is a positive solution τ of Φ'(τ)=1 and ρ=τ-Φ(τ)>0. Under its standard nondegeneracy hypotheses, the reduced aperiodic series has a square-root singularity at its positive radius [source referen","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2c08a3f3921a7de9","tex":"\\varphi'","aliases":"φ' varphi","display":"\\varphi'","kind":"use","excerpt":"Put a=lVert srVert, b_m=lVert A_mrVert, φ(t)=∑_m=2^Db_mt^m. For a radius R>0, define κ_R=φ'(R) =∑_m=2^Dm b_mR^m-1.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-43c933e54ba0a724","tex":"\\Phi_{ab}","aliases":"Φ_ab Phi","display":"\\Phi_{ab}","kind":"use","excerpt":"has a canonical semantic normalizer. For a chart C_λ,a in component λ, write N_λ,a: C_λ,a⟶ O_λ for the semantic normalization functor and choose its canonical inverse chart Q_λ,a: O_λ⟶ C_λ,a. The certified bridge from chart a to chart b is Φ_ab=Q_λ,bN_λ,a.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-01a6e20baab67409","tex":"\\Phi_{ac}","aliases":"Φ_ac Phi","display":"\\Phi_{ac}","kind":"use","excerpt":"[Normalizer-induced coherence] Assume that every normalized fiber of N_λ,a is contractible. Then the comparison isomorphism η_abc: Φ_bcΦ_ab Longrightarrow Φ_ac is unique. Every triple-overlap diagram commutes, every quadruple-overlap coherence equation holds, and the composite around every finite bridge cycle is the identity transformation of the n","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-09ba3bc3638dca2e","tex":"\\Phi_{bc}","aliases":"Φ_bc Phi","display":"\\Phi_{bc}","kind":"use","excerpt":"[Normalizer-induced coherence] Assume that every normalized fiber of N_λ,a is contractible. Then the comparison isomorphism η_abc: Φ_bcΦ_ab Longrightarrow Φ_ac is unique. Every triple-overlap diagram commutes, every quadruple-overlap coherence equation holds, and the composite around every finite bridge cycle is the identity transformation of the n","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-569ae08551ce0863","tex":"\\Phi_M","aliases":"Φ_M Phi","display":"\\Phi_M","kind":"use","excerpt":"For M≥1, s≥0, and r,k≥0, define Φ_M(z)=(1+z)^M-1 and Θ_M,s(r,k) =[z^r](1+z)^sΦ_M(z)^k. Equivalently, Θ_M,s(r,k) =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. The support is finite: Θ_M,s(r,k)=0 unless k≤ r≤ Mk+s.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f31917a46539915d","tex":"\\Phi_{N,r}","aliases":"Φ_N,r Phi","display":"\\Phi_{N,r}","kind":"use","excerpt":"Let P(z,Y)∈ K[z,Y] be the normalized primitive representative of P_F. Introduce indeterminates U=(U_0,…,U_N-1) and substitute Y=∑_j=0^N-1z^jU_j. Every polynomial in z has a unique decomposition modulo the relation z^N=x: P (z,∑_j=0^N-1z^jU_j) = ∑_r=0^N-1z^rΦ_N,r(x, U). The polynomials Φ_N,r(x, U)∈ K[x, U] are the algebraic packet equations.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d12125f74d570246","tex":"\\Phi_q","aliases":"Φ_q Phi","display":"\\Phi_q","kind":"use","excerpt":"= fracc_q(ℓ)N, where the last equality uses the integrality and evenness of the Ramanujan sum. Hence E_q∈ Q[C_N]. Orthogonality and completeness follow by grouping the primitive e_j by their disjoint Galois orbits. Any rational sum of split primitive idempotents must be Galois invariant, so its support is a union of these orbits; this proves finene","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-28f1c40fb2bc6244","tex":"\\Phi_U","aliases":"Φ_U Phi","display":"\\Phi_U","kind":"use","excerpt":"quare remains a projector-sum refinement square in the descent category; item every fold whose source objects, target objects, and structure maps are compatible descent data defines a fold in Desc( R). Compatible selectors and left inverses descend, so a split-losslessness certificate descends directly from its left-inverse identity. Moreover, supp","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5857360c047ef704","tex":"\\Phi_{\\alpha\\gamma}","aliases":"Φ_αγ Phi alpha gamma","display":"\\Phi_{\\alpha\\gamma}","kind":"use","excerpt":"On triple overlaps, pairwise equivalences are not enough. One needs comparison cells η_αβγ: Φ_βγΦ_αβ Longrightarrow Φ_αγ. For four charts, there are two composites from Φ_γδΦ_βγΦ_αβ to Φ_αδ. Their ratio is the higher coherence defect.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e5c6b4925be9156e","tex":"\\Phi_{\\beta\\gamma}","aliases":"Φ_βγ Phi beta gamma","display":"\\Phi_{\\beta\\gamma}","kind":"use","excerpt":"On triple overlaps, pairwise equivalences are not enough. One needs comparison cells η_αβγ: Φ_βγΦ_αβ Longrightarrow Φ_αγ. For four charts, there are two composites from Φ_γδΦ_βγΦ_αβ to Φ_αδ. Their ratio is the higher coherence defect.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a9ae5d0fdd6b65f4","tex":"\\Phi_{\\gamma\\delta}","aliases":"Φ_γδ Phi gamma delta","display":"\\Phi_{\\gamma\\delta}","kind":"use","excerpt":"On triple overlaps, pairwise equivalences are not enough. One needs comparison cells η_αβγ: Φ_βγΦ_αβ Longrightarrow Φ_αγ. For four charts, there are two composites from Φ_γδΦ_βγΦ_αβ to Φ_αδ. Their ratio is the higher coherence defect.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-98d9f615bdab7b1d","tex":"\\Phi_{\\kappa,\\tau}","aliases":"Φ_κ,τ Phi kappa tau","display":"\\Phi_{\\kappa,\\tau}","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"Reconstructed symmetry · Aut^⊗(ω) · tensor automorphisms of a fiber functor and the corresponding affine group scheme · faithful exact tensor fiber functor","count":27},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f12b1816b314c06d","tex":"\\omega^{-j\\ell}","aliases":"ω^-jℓ omega ell","display":"\\omega^{-j\\ell}","kind":"use","excerpt":"Suppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a8e0712f7cf91f55","tex":"\\omega^{jm}","aliases":"ω^jm omega","display":"\\omega^{jm}","kind":"use","excerpt":"characteristic-zero field containing μ_d and a primitive N-th root of unity ω, let d,m,N∈ Z_≥1, and choose the formal branch F(z)=(1-α z^m)^-1/d, α∈ K^×. With g=gcd(N,m), its nonzero coefficients occupy exactly the subgroup of Z/N Z generated by m, which has N/g addresses. The rotated radicands 1-αω^jmz^m, 0≤ j<N, also have exactly N/g distinct val","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8788597d7fe50fbe","tex":"\\omega^\\ell","aliases":"ω^ℓ omega ell","display":"\\omega^\\ell","kind":"use","excerpt":"Suppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a87553bb10c05ec","tex":"\\omega_N","aliases":"ω_N omega","display":"\\omega_N","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9381590dfa8ec0c2","tex":"\\omega_N^{-jq}","aliases":"ω_N^-jq omega","display":"\\omega_N^{-jq}","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-59e6e879fd86d31e","tex":"\\omega_N^q","aliases":"ω_N^q omega","display":"\\omega_N^q","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ced06e93882353e3","tex":"\\omega_N^r","aliases":"ω_N^r omega","display":"\\omega_N^r","kind":"use","excerpt":"Let ω_N be primitive and consider the rotated orbit O_N(F_λ,α) = {(1-αω_N^rt)^-λ:0≤ r<N}. Let r_rm orb be its dimension over the constant field.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-320b46771e8024be","tex":"\\omega_N^{rn}","aliases":"ω_N^rn omega","display":"\\omega_N^{rn}","kind":"use","excerpt":"If λ=-M, expand (1-αω_N^rt)^M = ∑_n=0^M binom Mn(-α)^nω_N^rnt^n. The rotation matrix is a truncated discrete Fourier matrix. Its rank is the number of available residue classes, namely min(N,M+1).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6cede2de03a92762","tex":"\\omega_{p-1}","aliases":"ω_p-1 omega","display":"\\omega_{p-1}","kind":"use","excerpt":"Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a9a9918440d8a7b0","tex":"\\omega_v","aliases":"ω_v omega","display":"\\omega_v","kind":"use","excerpt":"Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-56ca21ef81614b09","tex":"\\omega_{\\pi}","aliases":"ω_π omega pi","display":"\\omega_{\\pi}","kind":"use","excerpt":"Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-185aeab83c3ff804","tex":"\\omega_{\\infty}","aliases":"ω_∞ omega infty","display":"\\omega_{\\infty}","kind":"use","excerpt":"Let R=A/B be reduced and normalized. For each monic irreducible polynomial π∈ F_p[z], define ω_π(R) = (v_π(B)-v_π(A))bmod(p-1). Include the valuation at infinity ω_∞(R) = (deg A-deg B)bmod(p-1). The finite-support family ω_p-1(R) = (ω_v(R))_v is the rational-collapse obstruction.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"For a connected scheme X with geometric point bar x, the fiber functor on finite acuteetale covers identifies that category with finite continuous π_1^acuteet(X,bar x)-sets. Every finite fiber S produces a permutation resolution on ℓ^2(S): coordinate projectors are indexed by points, orbit projectors by transitive components, and covering morphisms","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a72bd9c889713ba","tex":"\\leftrightarrow","aliases":"↔ leftrightarrow","display":"\\leftrightarrow","kind":"use","excerpt":"item constant recurrence ↔ rational ordinary generating function ↔ finite linear state; item P-recursive ↔ D-finite coefficient translation; item finite digit realization ↔ finite k-kernel; item rational functional-Mahler mask → Cartier representation → finite automaton; item finite-mask Cartier transitions ↔ carry matrices; item selected algebraic","count":22},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"The invariance of im D makes [v]↦[Q_av] well defined. Idempotence, annihilation, and reconstruction descend to the quotient. In the block-diagonal case the image of D is the direct sum of the images of the restrictions.","count":17},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"use","excerpt":"Choose Smith bases in which the inclusion M_k(a)↪ L_k(a) is the diagonal matrix D=diag(d_1,…,d_ρ). Modulo m, the kernel and cokernel of multiplication by d_i are both cyclic of order gcd(m,d_i), and the image has order m/gcd(m,d_i). The map is invertible in every coordinate exactly when every d_i is a unit modulo m, which is equivalent to gcd(m,d_ρ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"[Simple-branch normalization] If P_F(0,a_0)=0, ∂_Y P_F(0,a_0)≠0, then a_0 alone selects a unique formal branch. Equivalently, σ(P_F,F)≤1.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-26ce661d61492b7c","tex":"\\partial_Y","aliases":"∂_Y partial","display":"\\partial_Y","kind":"use","excerpt":"[Simple-branch normalization] If P_F(0,a_0)=0, ∂_Y P_F(0,a_0)≠0, then a_0 alone selects a unique formal branch. Equivalently, σ(P_F,F)≤1.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ece38486b6c3c8e1","tex":"\\partial_z","aliases":"∂_z partial","display":"\\partial_z","kind":"use","excerpt":"[Algebraic derivative closure] Let F have separable minimal polynomial P_F of degree d. Then F' = -frac∂_zP_F(z,F)∂_YP_F(z,F) ∈ K(z,F). Every higher derivative lies in the same degree-d function field. Consequently, F has a differential annihilator of order at most d.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"Write E, FF, D, P, ∅ for an equivalence on a named normalized overlap, a fully faithful inclusion, an information-losing directional functor, a partial or hypothesis-dependent bridge, and the absence of a certified broad bridge, respectively. The matrix below records only relations already established in the preceding sections or in the audited sou","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"[Hilbert--Schmidt optimality, imported] Assume X,Y are Hilbert spaces, the packet idempotents are orthogonal projectors, and f:X→ Y is Hilbert--Schmidt. Then C_E is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and R_E(f) _HS^2 = ∑_(a,i)∉ E q_afp_i _HS^2. Consequently, C_E(f) is the unique best Hilbert--Schmidt approxi","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5db54251682b18a0","tex":"\\prod_{a=1}^{q}","aliases":"∏_a=1^q prod","display":"\\prod_{a=1}^{q}","kind":"use","excerpt":"[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the sou","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-be227de3146f3cde","tex":"\\prod_{a\\in A_{N,j}(\\lambda)}","aliases":"∏_a∈ A_N,j(λ) prod in lambda","display":"\\prod_{a\\in A_{N,j}(\\lambda)}","kind":"use","excerpt":"B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fee581af9534af41","tex":"\\prod_{b\\in B_{N,j}}","aliases":"∏_b∈ B_N,j prod in","display":"\\prod_{b\\in B_{N,j}}","kind":"use","excerpt":"B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ae5d80b648ebc895","tex":"\\prod_{g\\in G}","aliases":"∏_g∈ G prod in","display":"\\prod_{g\\in G}","kind":"use","excerpt":"[Canonical envelopes of a nonnormal Galois atlas] Let M=L^H. Then L^Core_G(H) =∏_g∈ Gg(M) tag12.6 is the smallest Galois extension inside L containing every conjugate chart, while L^Ncl_G(H) =⋂_g∈ Gg(M) tag12.7 is the largest Galois subfield contained in every chart. The core and normal-closure projectors are the two canonical global envelopes of t","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"[Universal mapping property of packet coordinates] For every object Z, restriction to the packet objects gives natural bijections Hom(X,Z) ≅ ∏_iHom(X_i,Z), Hom(Z,X) ≅ ∏_iHom(Z,X_i), where the products are finite. Packetwise maps therefore assemble uniquely into a global map.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c4fd4593936e0e85","tex":"\\prod_{i=1}^{\\rho}","aliases":"∏_i=1^ρ prod rho","display":"\\prod_{i=1}^{\\rho}","kind":"use","excerpt":"[Exact Smith formulas] Let g_i(m)=gcd(m,d_i(a)). Then G_k,m(a) ≅ Q_k,m(a) ≅ ⊕_i=1^ρ Z/g_i(m) Z, and |overline M_k,m(a)| = ∏_i=1^ρ fracmg_i(m). The reduction map is an isomorphism if and only if gcd(m,d_ρ(a))=1. Equivalently, every prime dividing a reduction defect divides the saturation index.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-952b949284fd75bd","tex":"\\prod_{j=0}^{N}","aliases":"∏_j=0^N prod","display":"\\prod_{j=0}^{N}","kind":"use","excerpt":"Starting with F, the finite product through level N telescopes: ∏_j=0^NfracF(z^p^j)F(z^p^j+1) = fracF(z)F(z^p^N+1). As N→∞, the denominator converges coefficientwise to F(0)=1, proving that the product recovers F. Conversely, the product solution satisfies F(z)=R(z)F(z^p), so R=F/F(z^p). Reduced normalization makes the rational mask unique.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a46169806cf6cd48","tex":"\\prod_{j\\geq 0}","aliases":"∏_j≥ 0 prod geq","display":"\\prod_{j\\geq 0}","kind":"use","excerpt":"For a unit formal series, define its canonical local mask by R_F(z) = fracF(z)F(z^p). The coefficientwise infinite product M_p[R](z) = ∏_j≥ 0R (z^p^j) is well defined for every R∈1+z F_p[[z]], because only finitely many factors contribute to a fixed coefficient.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7aac306ff05814ee","tex":"\\prod_{j\\geq0}","aliases":"∏_j≥0 prod geq","display":"\\prod_{j\\geq0}","kind":"use","excerpt":"For a unit formal series F(0)=1, define its local functional-Mahler mask by M_p(F)(z) = fracF(z)F(z^p). Then F(z)= M_p(F)(z)F(z^p). Iterating [source reference] gives the coefficientwise product F(z) = ∏_j≥0 M_p(F)(z^p^j). Only finitely many factors contribute to each fixed coefficient.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ff3b5ab046b2a71a","tex":"\\prod_{k=0}^{n-1}","aliases":"∏_k=0^n-1 prod","display":"\\prod_{k=0}^{n-1}","kind":"use","excerpt":"For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5172a8f75046abf3","tex":"\\prod_{m<n}","aliases":"∏_m<n prod","display":"\\prod_{m<n}","kind":"use","excerpt":"Filtered multiplication gives E(b+b')= E(b) E(b'). To prove surjectivity and uniqueness, construct the exponents recursively. Suppose b_1,…,b_n-1 have been chosen so that A(z)∏_m<n(1-z^m)^b_m ≡ 1+r_nz^npmodz^n+1. Because (1-z^n)^-r_n≡1+r_nz^npmodz^n+1, the unique choice b_n=r_n cancels the degree-n residual. Every r_n is integral because all preced","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-171192b9a8736142","tex":"\\prod_{n\\geq1}","aliases":"∏_n≥1 prod geq","display":"\\prod_{n\\geq1}","kind":"use","excerpt":"[Partition and power families] The following fixed-basis exponent functions give familiar filtered products: c|c|c b_n · A_b(z) · c_n 1 · displaystyle∏_n≥1(1-z^n)^-1 · σ_1(n) [4pt] n · displaystyle∏_n≥1(1-z^n)^-n · σ_2(n) [4pt] n^r · displaystyle∏_n≥1(1-z^n)^-n^r · σ_r+1(n). The first row is the ordinary partition series, and the second is the plan","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b9d5148b142926aa","tex":"\\prod_p","aliases":"∏_p prod","display":"\\prod_p","kind":"use","excerpt":"[Profinite saturation sequence] There are canonical digit-equivariant identifications widehatM_k(a) ≅ ∏_p M_k,p(a), widehatL_k(a) ≅ ∏_p L_k,p(a), and a short exact sequence 0 ⟶ widehatM_k(a) ⟶ widehatL_k(a) ⟶ C_k(a) ⟶ 0. A compatible all-moduli raw tower has an ordinary integral lift exactly when its class in widehatM_k(a)/M_k(a) vanishes.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ebbc0cddfcac28e3","tex":"\\prod_{p\\text{ prime}}","aliases":"∏_p prime prod text","display":"\\prod_{p\\text{ prime}}","kind":"use","excerpt":"[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-26fdda4e62a4020e","tex":"\\prod_{p^e\\parallel n}","aliases":"∏_p^eparallel n prod parallel","display":null,"kind":"use","excerpt":"[Prime-power descent equivalence] The assignment f⟼(F_f,p)_p is a group isomorphism (Mult_ Z,mathbin∗_D) ≃ ∏_p prime (1+T Z[[T]],·). The inverse reconstructs f(n)=∏_p^eparallel nf(p^e). Under this correspondence, F_f∗_D g,p(T) = F_f,p(T)F_g,p(T).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0b57ac1ebad43f9f","tex":"\\prod_{q\\mid N}","aliases":"∏_qmid N prod mid","display":null,"kind":"use","excerpt":"= fracc_q(ℓ)N, where the last equality uses the integrality and evenness of the Ramanujan sum. Hence E_q∈ Q[C_N]. Orthogonality and completeness follow by grouping the primitive e_j by their disjoint Galois orbits. Any rational sum of split primitive idempotents must be Galois invariant, so its support is a union of these orbits; this proves finene","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-897e488cb5fd7cfe","tex":"\\prod_{r,\\gamma}","aliases":"∏_r,γ prod gamma","display":"\\prod_{r,\\gamma}","kind":"use","excerpt":"[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a2217d433f664c6f","tex":"\\prod_{r=0}^{N-1}","aliases":"∏_r=0^N-1 prod","display":"\\prod_{r=0}^{N-1}","kind":"use","excerpt":"B_N,j ;α^Nx ). Its coefficients satisfy the first-order polynomial recurrence ∏_r=0^N-1(Nm+j+1+r)u_m+1 = α^N ∏_r=0^N-1(Nm+j+λ+r)u_m. The series is annihilated by H_N,j^λ,α = θ ∏_b∈ B_N,j(θ+b-1) - α^Nx ∏_a∈ A_N,j(λ)(θ+a), where θ=xD_x.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-deff9d873c3e9fc2","tex":"\\prod_{r=1}^{N}","aliases":"∏_r=1^N prod","display":"\\prod_{r=1}^{N}","kind":"use","excerpt":"The projector laws follow coefficient by coefficient. The residue classes modulo NM lying over jpmod N are precisely j,j+N,…,j+(M-1)N, which proves refinement. Character orthogonality gives the Fourier formula. Finally, (Nn+j)! = j! N^Nn ∏_r=1^N(fracj+rN)_n. Exactly one factor in the product is (1)_n=n!. Removing it leaves the parameter set b_N,j a","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bcb8c1cb4a1005ab","tex":"\\prod_{r=1}^{q}","aliases":"∏_r=1^q prod","display":"\\prod_{r=1}^{q}","kind":"use","excerpt":"x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this is t","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2b48d9bb19dadea5","tex":"\\prod_{r=2}^{q}","aliases":"∏_r=2^q prod","display":"\\prod_{r=2}^{q}","kind":"use","excerpt":"Put x=z^q-1, U_q,α(z)=zC_q,α(x). Then C=1+α xC^q. Lagrange inversion gives [x^n]C_q,α(x) = c_n(q,α) = fracα^n(q-1)n+1 binomqnn. Define A_q(n) =(n+1)∏_r=2^q((q-1)n+r), B_q(n) =∏_r=1^q(qn+r). Then the coefficient sequence satisfies the first-order polynomial recurrence A_q(n)c_n+1 = α B_q(n)c_n. After dividing the polynomial gcd of A_q and B_q, this ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9908ddfbb7d4bb3d","tex":"\\prod_{r=c+1}^{c+N}","aliases":"∏_r=c+1^c+N prod","display":"\\prod_{r=c+1}^{c+N}","kind":"use","excerpt":"[Exact factorial--residue realization] Let N,c,n∈ Z with N≥1, c≥0, and n≥0. Then frac(Nn+c)!c! = ∏_r=c+1^c+N(r)_n;N. Each integer factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-fdc1e09c5ae4d508","tex":"\\prod_{r\\ge1}","aliases":"∏_r≥1 prod ge","display":"\\prod_{r\\ge1}","kind":"use","excerpt":"[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-75e0eb0ccee7ee4c","tex":"\\prod_{s=0}^{p^a-1}","aliases":"∏_s=0^p^a-1 prod","display":"\\prod_{s=0}^{p^a-1}","kind":"use","excerpt":"[Complete clopen packet equivalence] The analysis map B_a: C_p ⟶ ∏_s=0^p^a-1 C_p is an isometric isomorphism. Its inverse is the piecewise synthesis ( S_a g)(x) =g_s (fracx-sp^a) for x∈ s+p^a Z_p. Thus the clopen residue cover of Z_p has effective descent for both objects and equality morphisms.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8014de08eec3c9db","tex":"\\sum_{\\mathcal R\\in H\\backslash(K\\times I)}","aliases":"∑_ R∈ Hbackslash(K× I) sum mathcal in backslash times","display":null,"kind":"use","excerpt":"Let P⊆ J× I, Q⊆ K× J, R⊆ K× I be diagonal H-orbits. For any (k,i)∈ R, define c^ R_ Q, P := #{ j∈ J:(k,j)∈ Q, (j,i)∈ P }. This integer is independent of the chosen (k,i)∈ R, and B_ QB_ P = ∑_ R∈ Hbackslash(K× I) c^ R_ Q, PB_ R. Thus the lane path law descends to an exact orbit basis with nonnegative integer structure constants.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f43fdc847d37c899","tex":"\\sum_{(a,i)\\notin E}","aliases":"∑_(a,i)∉ E sum notin","display":"\\sum_{(a,i)\\notin E}","kind":"use","excerpt":"[Hilbert--Schmidt optimality, imported] Assume X,Y are Hilbert spaces, the packet idempotents are orthogonal projectors, and f:X→ Y is Hilbert--Schmidt. Then C_E is the Hilbert--Schmidt orthogonal projection onto the selected block subspace and R_E(f) _HS^2 = ∑_(a,i)∉ E q_afp_i _HS^2. Consequently, C_E(f) is the unique best Hilbert--Schmidt approxi","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"[Exact transfer reconstruction and block composition, imported] Every morphism f:X→ Y has the exact expansion f = ∑_a,iq_afp_i = ∑_a,ijmath_a f_aiπ_i. If g:Y→ Z is resolved by a packet system R=(r_c), then (gf)_ci = ∑_a g_caf_ai.","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-68a980c436fad9b7","tex":"\\sum_{a,i}","aliases":"∑_a,i sum","display":"\\sum_{a,i}","kind":"use","excerpt":"[Exact transfer reconstruction and block composition, imported] Every morphism f:X→ Y has the exact expansion f = ∑_a,iq_afp_i = ∑_a,ijmath_a f_aiπ_i. If g:Y→ Z is resolved by a packet system R=(r_c), then (gf)_ci = ∑_a g_caf_ai.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5d6a7db697f3606e","tex":"\\sum_{a,i_1,\\ldots,i_d}","aliases":"∑_a,i_1,…,i_d sum ldots","display":"\\sum_{a,i_1,\\ldots,i_d}","kind":"use","excerpt":"inear lane expansion] Let kk be a commutative ring. Let X^(1),…,X^(d),Y be kk-modules with finite split lane systems, and let T:X^(1)×⋯× X^(d)⟶ Y be a specified kk-multilinear map. Define T_a;i_1,…,i_d := π_Y,aT (ι_1,i_1-,…,ι_d,i_d-). Then T(x_1,…,x_d) = ∑_a,i_1,…,i_d ι_Y,a T_a;i_1,…,i_d (π_1,i_1x_1,…,π_d,i_dx_d). Substitution of multilinear maps i","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2237647acab1b6a1","tex":"\\sum_{a,i\\in\\{1,2\\}}","aliases":"∑_a,i∈{1,2} sum in","display":"\\sum_{a,i\\in\\{1,2\\}}","kind":"use","excerpt":"The displayed matrix multiplication gives the two values immediately. The full refinement identity recovers f=∑_a,i∈{1,2}p_afp_i, whose only nonzero term here is p_2fp_2.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-49ce60fdcf83013f","tex":"\\sum_{a\\in I_{Y,u}}","aliases":"∑_a∈ I_Y,u sum in","display":"\\sum_{a\\in I_{Y,u}}","kind":"use","excerpt":"item Composition and path law. If f:X→ Y and g:Y→ Z, with the same resolution u used on the intermediate object, then [gf]^v,r_c i = ∑_a∈ I_Y,u[g]^v,u_c a[f]^u,r_a i. More generally, a block of a composable string is the finite sum of the products along all compatible intermediate address paths.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e65fc9b117e52f96","tex":"\\sum_{a\\in I_{Y,U,u}}","aliases":"∑_a∈ I_Y,U,u sum in","display":"\\sum_{a\\in I_{Y,U,u}}","kind":"use","excerpt":"item Twisted matrix composition. For h∈Γ(mathscr G,mathscr M^u,r_Y,X), k∈Γ(mathscr G,mathscr M^v,u_Z,Y), define chartwise contraction by (k⋆ h)_U,ci := ∑_a∈ I_Y,U,uk_U,cah_U,ai. Then k⋆ h is a compatible section of mathscr M^v,r_Z,X, and A^v,r_Z,X(gf) = A^v,u_Z,Y(g) ⋆ A^u,r_Y,X(f). The diagonal identity matrices are compatible sections. More precis","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b4f5a254f467262d","tex":"\\sum_{a\\in(\\mathbb Z/q\\mathbb Z)^\\times}","aliases":"∑_a∈( Z/q Z)^× sum in mathbb mathbb times","display":"\\sum_{a\\in(\\mathbb Z/q\\mathbb Z)^\\times}","kind":"use","excerpt":"Write every j∈ O_q uniquely as j=(N/q)a with a∈( Z/q Z)^×. The coefficient of g^ℓ in E_q is then frac1N ∑_a∈( Z/q Z)^× ζ_q^-aℓ = fracc_q(ℓ)N, where the last equality uses the integrality and evenness of the Ramanujan sum. Hence E_q∈ Q[C_N]. Orthogonality and completeness follow by grouping the primitive e_j by their disjoint Galois orbits. Any rati","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e2172593035b72e","tex":"\\sum_{b,j}","aliases":"∑_b,j sum","display":"\\sum_{b,j}","kind":"use","excerpt":"[Certified coarsening is an exact coordinate change] The change from fine to coarse block coordinates, canonical relative to the chosen splittings, is Γ^(u,r)_(v,s) := a^u,r_X,Y ( a^v,s_X,Y)^-1. For a fine matrix h=(h_bj), its (a,i)-entry is (Γ^(u,r)_(v,s)h)_a i = ∑_b,j (π_Y,a^uι_Y,b^v) h_bj (π_X,j^sι_X,i^r). Only descendants of a and i can contrib","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7f41bc7926840c39","tex":"\\sum_{d=0}^{k-1}","aliases":"∑_d=0^k-1 sum","display":"\\sum_{d=0}^{k-1}","kind":"use","excerpt":"[Typed Cartier realization] Let K be a field and let k≥2. Work with K-valued formal series and fix the least-significant-digit-first convention. Define Λ_d(∑_n≥0a_nz^n) :=∑_n≥0a_kn+dz^n, S_d(G):=z^dG(z^k), 0≤ d<k. Then Λ_dS_e=δ_de1, ∑_d=0^k-1S_dΛ_d=1. For a digit word w=d_0d_1⋯ d_a-1, the iterated Cartier lane is Λ_d_a-1⋯Λ_d_1Λ_d_0F. It selects coe","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ce043e53d6db1168","tex":"\\sum_{d\\mid e}","aliases":"∑_dmid e sum mid","display":null,"kind":"use","excerpt":"Apply the Euler-coordinate theorem independently at each prime: F_f,p(T) = ∏_m≥1(1-T^m)^-β_p,m(f). Define local ghosts γ_p,e(f) = ∑_dmid ed β_p,d(f).","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-48885cdcf9497481","tex":"\\sum_{d\\mid\\gcd(q,\\ell)}","aliases":"∑_dmidgcd(q,ℓ) sum mid gcd ell","display":null,"kind":"use","excerpt":"Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q gcd(a,q)=1 e^2π i aℓ/q = ∑_dmidgcd(q,ℓ)d μ(q/d) ∈ Z is the Ramanujan sum. The projec","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6d1546afc3ef11e5","tex":"\\sum_{e\\in E}","aliases":"∑_e∈ E sum in","display":"\\sum_{e\\in E}","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-43bb9ad90f6d063e","tex":"\\sum_{e\\geq1}","aliases":"∑_e≥1 sum geq","display":"\\sum_{e\\geq1}","kind":"use","excerpt":"[Valuation logarithmic derivative] Fix a prime p. Let Γ_f,p be the arithmetic function supported on powers of p with Γ_f,p(1)=0, Γ_f,p(p^e)=γ_p,e(f) (e≥1). Then (v_p· f)(n) = (Γ_f,pmathbin∗_Df)(n) (n≥1). Equivalently, fracT F_f,p'(T)F_f,p(T) = ∑_e≥1γ_p,e(f)T^e.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6d39fe36bb310cf9","tex":"\\sum_{g\\in G}","aliases":"∑_g∈ G sum in","display":"\\sum_{g\\in G}","kind":"use","excerpt":"[Typed semisimple Galois realization] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8b9dfe0ef547944e","tex":"\\sum_{g\\in G_i}","aliases":"∑_g∈ G_i sum in","display":"\\sum_{g\\in G_i}","kind":"use","excerpt":"Let L/K be a finite Galois extension of local fields. Write G_0⊇ G_1⊇ G_2⊇⋯ for the lower ramification filtration after inertia, and let V be a finite-dimensional complex representation of G. Define P_i=frac1|G_i|∑_g∈ G_iU_g. The ramification groups are nested and normal, so P_0≤ P_1≤ P_2≤⋯≤ I. Put D_i=P_i+1-P_i.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e177bb3bef4abaec","tex":"\\sum_{h\\in H}","aliases":"∑_h∈ H sum in","display":"\\sum_{h\\in H}","kind":"use","excerpt":"Let L/K be a finite Galois extension with group G=Gal(L/K). Let V be a finite-dimensional representation of G over a characteristic-zero field, or a unitary complex realization obtained after scalar extension. For every subgroup H≤ G, define P_H=frac1|H|∑_h∈ HU_h. tag12.1","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2ee5b079d386ca9e","tex":"\\sum_{i,j_1,\\ldots,j_m}","aliases":"∑_i,j_1,…,j_m sum ldots","display":"\\sum_{i,j_1,\\ldots,j_m}","kind":"use","excerpt":"For i,j_1,…,j_m∈ I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a5e5c4e54e8bc71b","tex":"\\sum_{i=0}^{j}","aliases":"∑_i=0^j sum","display":"\\sum_{i=0}^{j}","kind":"use","excerpt":"[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-287f8bab38983785","tex":"\\sum_{i=1}^{r_k(a)}","aliases":"∑_i=1^r_k(a) sum","display":"\\sum_{i=1}^{r_k(a)}","kind":"use","excerpt":"[Strict shortlex realization] Write Λ_db_j = ∑_i=1^r_k(a)(M_d(a))_ijb_i, a=∑_i v_i(a)b_i, ℓ_i(a)=b_i(0). The strict shortlex record is SL_k,K(a) = ( U_a, M_0(a),…,M_k-1(a), v(a), ℓ(a) ).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-28fb3af466657215","tex":"\\sum_{i=1}^{\\rho}","aliases":"∑_i=1^ρ sum rho","display":"\\sum_{i=1}^{\\rho}","kind":"use","excerpt":"[Prime-power defect profile] For r≥1, G_k,p^r(a) ≅ Q_k,p^r(a) ≅ ⊕_i=1^ρ Z/p^min(r,ν_i) Z, and |overline M_k,p^r(a)| = p^∑_i=1^ρ(r-min(r,ν_i)). The minimal number of generators of overline M_k,p^r(a) as a Z/p^r Z-module is μ_p^r(a) = #{i:ν_i<r}. In particular, p is a good reduction prime exactly when every ν_i=0.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-776050cd64f58868","tex":"\\sum_{i_\\ell}","aliases":"∑_i_ℓ sum ell","display":"\\sum_{i_\\ell}","kind":"use","excerpt":"Insert x_ℓ=∑_i_ℓp_ℓ,i_ℓx_ℓ in every input, use multilinearity to distribute the finite sums, and insert 1_Y=∑_a p_Y,a in the output. The coordinate formula follows from the chosen splittings. Repeating the calculation after substitution gives the contraction law. The degree-zero identity is output synthesis applied to T_0; the polynomial statement ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-da975bf219d53257","tex":"\\sum_{i\\in\\mathcal O_U}","aliases":"∑_i∈ O_U sum in mathcal","display":"\\sum_{i\\in\\mathcal O_U}","kind":"use","excerpt":"[label=textup(Oarabic*)] item For every orbit O⊆ I_A,U_0,r of H_A^r, path transport defines a path-independent orbit subset O_U⊆ I_A,U,r in every chart. The local projector sums P^r_A,U, O := ∑_i∈ O_Up^r_A,U,i descend to a split idempotent on A. As O ranges over the monodromy orbits, these projectors are pairwise annihilating and sum to 1_A.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f3b66c702f5b785d","tex":"\\sum_{i\\in S_U}","aliases":"∑_i∈ S_U sum in","display":"\\sum_{i\\in S_U}","kind":"use","excerpt":"item The orbit projectors form the finest ordinary globally labeled coarsening whose local projectors are sums of the original lane projectors. More precisely, any globally labeled projector of the form Q_U=∑_i∈ S_Up^r_A,U,i with S_V=λ^A_γ,r(S_U) and with the same global label preserved around every loop has S_U_0 equal to a union of H_A^r-orbits.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f727d6dacb4f1062","tex":"\\sum_{i\\geq0}","aliases":"∑_i≥0 sum geq","display":"\\sum_{i\\geq0}","kind":"use","excerpt":"[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e4ee3eaa6bf9adf","tex":"\\sum_{i\\geq1}","aliases":"∑_i≥1 sum geq","display":"\\sum_{i\\geq1}","kind":"use","excerpt":"[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"[Artin conductor as weighted unresolved rank] The Artin and Swan conductors satisfy a(V)=∑_i≥0frac|G_i||G_0|Tr(I-P_i), tag12.11 Sw(V)=∑_i≥1frac|G_i||G_0|Tr(I-P_i). tag12.12 If d_j=rankD_j, then a(V)=∑_jd_j∑_i=0^jfrac|G_i||G_0|. tag12.13 Thus the conductor is the weighted total rank that remains unresolved at each ramification scale.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4acd39d1b80c5dbd","tex":"\\sum_{j=0}^{e}","aliases":"∑_j=0^e sum","display":"\\sum_{j=0}^{e}","kind":"use","excerpt":"[Multiplicativity is preserved and reflected] Assume b(1)=1 and c= G(b). Then b is multiplicative Longleftrightarrow c is multiplicative. On this locus, both b and c are reconstructed from prime-power seeds. For every prime p and e≥1, c(p^e)=∑_j=0^ep^j b(p^j), b(p^e)=fracc(p^e)-c(p^e-1)p^e.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-48fa0e3108a03d2c","tex":"\\sum_{j=0}^{k}","aliases":"∑_j=0^k sum","display":"\\sum_{j=0}^{k}","kind":"use","excerpt":"For M≥1, s≥0, and r,k≥0, define Φ_M(z)=(1+z)^M-1 and Θ_M,s(r,k) =[z^r](1+z)^sΦ_M(z)^k. Equivalently, Θ_M,s(r,k) =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. The support is finite: Θ_M,s(r,k)=0 unless k≤ r≤ Mk+s.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b9a7896e38144f2e","tex":"\\sum_{j=0}^{N}","aliases":"∑_j=0^N sum","display":"\\sum_{j=0}^{N}","kind":"use","excerpt":"[Digit sum separates all three levels] For the base-k digit sum s_k, O_k(s_k) = {s_k+c:c∈ Z_≥0}, and M_k(s_k)=L_k(s_k)= Zs_k⊕ Z1. Hence overline O_k(s_k)^ p = {s_k+c:c∈ Z_p}. The partial sums c_N=∑_j=0^Np^2j are nonnegative integers and therefore produce actual section states s_k+c_N. They converge in Z_p to c_∞ = frac11-p^2, which is not an ordina","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-586bc3571b381e64","tex":"\\sum_{j=0}^{N-1}","aliases":"∑_j=0^N-1 sum","display":"\\sum_{j=0}^{N-1}","kind":"use","excerpt":"Fix N≥1. For F(z)=∑_n≥0a_nz^n, define the arithmetic packets U_j(x) = ∑_m≥0a_Nm+jx^m, 0≤ j<N. They satisfy the exact synthesis identity F(z) = ∑_j=0^N-1z^jU_j(z^N).","count":7},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-47306c517d6a1276","tex":"\\sum_{j=0}^{r}","aliases":"∑_j=0^r sum","display":"\\sum_{j=0}^{r}","kind":"use","excerpt":"Let F(z)=∑_n≥0a_nz^n and L = ∑_j=0^rq_j(z)D^j, q_j(z)=∑_m=0^d_jq_j,mz^m.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1c0becd0b6c8fcd0","tex":"\\sum_{j=0}^{s-1}","aliases":"∑_j=0^s-1 sum","display":"\\sum_{j=0}^{s-1}","kind":"use","excerpt":"Write R_a = S^s+∑_j=0^s-1r_j(n)S^j. Let ν_a be the least index from which R_a annihilates the sequence, and define N_a to be the least integer N≥ν_a larger than every nonnegative pole of the rational functions r_j(n). Then R(a) = (R_a,N_a,a_0,…,a_N_a+s-1) is a canonical complete recurrence chart. The operator controls the tail, while the finite pre","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-12395d43dc1c3e4a","tex":"\\sum_{j=0}^{\\ell-1}","aliases":"∑_j=0^ℓ-1 sum ell","display":"\\sum_{j=0}^{\\ell-1}","kind":"use","excerpt":"The canonical Moore automaton has state set K_p(a), start state a, transition δ_d(b)=Λ_db, and output o(b)=b_0. After reading d_0,…,d_ℓ-1, its state is Λ_d_ℓ-1⋯Λ_d_0a = (n⟼ a_p^ℓ n+r), r=∑_j=0^ℓ-1d_jp^j.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-67a1ec685c2361ce","tex":"\\sum_{j=1}^{e}","aliases":"∑_j=1^e sum","display":"\\sum_{j=1}^{e}","kind":"use","excerpt":"The local logarithmic-derivative recurrence gives e f(p^e) = ∑_j=1^eγ_p,j(f)f(p^e-j). Write n=p^e u with (u,p)=1 and multiply by f(u). Since Γ_f,p is supported on p-powers, the resulting sum is exactly the Dirichlet convolution at n.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ef75e15ec0684e46","tex":"\\sum_{j=1}^{n}","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^{n}","kind":"use","excerpt":"[Exact coefficient recurrence] Let A(z)=1+∑_n≥1a_nz^n and let c_n be its Euler ghost coordinates. Then na_n = ∑_j=1^nc_j a_n-j (n≥1), with a_0=1. Conversely, any c∈ G_ Z and this recurrence reconstruct the integral unit series corresponding to its Euler exponents.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8f481d92dc12027e","tex":"\\sum_{j_1,\\ldots,j_m}","aliases":"∑_j_1,…,j_m sum ldots","display":"\\sum_{j_1,\\ldots,j_m}","kind":"use","excerpt":"I, define the resolved multilinear block A_m;imid j_1,…,j_m = P_iA_m(P_j_1 ·,…,P_j_m ·). Completeness of the packet family gives the exact reconstruction A_m = ∑_i,j_1,…,j_m A_m;imid j_1,…,j_m. tag11.3 Write u_i=P_iu, s_i=P_is. Then equation textup(11.1) is equivalent to the complete packet system u_i = s_i+ ∑_m=2^D ∑_j_1,…,j_m A_m;imid j_1,…,j_m [","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8edd7a1fc272a237","tex":"\\sum_{j\\in\\mathcal O_q}","aliases":"∑_j∈ O_q sum in mathcal","display":"\\sum_{j\\in\\mathcal O_q}","kind":"use","excerpt":"ll=0^N-1 ζ_N^-jℓg^ℓ, j∈ Z/N Z. These are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = fr","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2eaf35ea95c563af","tex":"\\sum_{j\\geq i}","aliases":"∑_j≥ i sum geq","display":"\\sum_{j\\geq i}","kind":"use","excerpt":"The classical conductor formula uses codimV^G_i. Since P_i projects onto V^G_i, this codimension is Tr(I-P_i). Also codimV^G_i=∑_j≥ id_j. Reversing the finite sums gives the detail formula.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0c704ad6b0f809b5","tex":"\\sum_{j\\geq1}","aliases":"∑_j≥1 sum geq","display":"\\sum_{j\\geq1}","kind":"use","excerpt":"The logarithmic-derivative identity is equivalent to zA'(z)=(∑_j≥1c_jz^j)A(z). Comparing coefficients gives the recurrence. Conversely, the recurrence uniquely determines the coefficient sequence over Q. Since c∈ G_ Z reconstructs an integral exponent sequence, the resulting series is the Euler product from [source reference] and therefore has inte","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1f3736c162765e82","tex":"\\sum_{k=1}^{n}","aliases":"∑_k=1^n sum","display":"\\sum_{k=1}^{n}","kind":"use","excerpt":"[Colored integer-partition term count] If Y has M intermediate packets, the number of compressed diagonal profiles at order n is C_n,M =[x^n]∏_r≥1(1-x^r)^-M. tag11.6 Before compression, the number of packet-labeled set-partition terms is N_n,M=∑_k=1^nS(n,k)M^k.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a334e226daa86a9f","tex":"\\sum_{k\\geq0}","aliases":"∑_k≥0 sum geq","display":"\\sum_{k\\geq0}","kind":"use","excerpt":"[Affine Newton--Mahler transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion and put g_s(q)=f(Mq+s). Then g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k =∑_r=k^Mk+sc_rΘ_M,s(r,k). Every output coefficient depends on a finite source window.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ed2c2d2d4a89d819","tex":"\\sum_{m=0}^{d_j}","aliases":"∑_m=0^d_j sum","display":"\\sum_{m=0}^{d_j}","kind":"use","excerpt":"Let F(z)=∑_n≥0a_nz^n and L = ∑_j=0^rq_j(z)D^j, q_j(z)=∑_m=0^d_jq_j,mz^m.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a56e4ebd77c058f4","tex":"\\sum_{m=0}^{\\min(d_j,n)}","aliases":"∑_m=0^min(d_j,n) sum min","display":"\\sum_{m=0}^{\\min(d_j,n)}","kind":"use","excerpt":"[Differential-to-recurrence coefficient formula] For every n≥0, [z^n]LF = ∑_j=0^r ∑_m=0^min(d_j,n) q_j,m (n-m+j)^underlinej a_n-m+j, where x^underline j=x(x-1)⋯(x-j+1). Consequently, every polynomial-coefficient differential equation yields an explicit polynomial recurrence after a finite reindexing.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-343a53ddf9200253","tex":"\\sum_{m=2}^{D}","aliases":"∑_m=2^D sum","display":"\\sum_{m=2}^{D}","kind":"use","excerpt":"Let A_m:X^m⟶ X, 2≤ m≤ D, be bounded symmetric multilinear maps, and let s∈ X. Consider the marked equation u=s+Φ(u), Φ(u)=∑_m=2^DA_m[u,…,u]. tag11.1 The word marked means that the source term, every multilinear operation, the packet system, and the selected convergence or filtration certificate are retained as part of the presentation. The fixed po","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d8e7aa4bba69b4ae","tex":"\\sum_{m\\mid n}","aliases":"∑_mmid n sum mid","display":null,"kind":"use","excerpt":"Formal logarithmic differentiation gives fraczA'A = ∑_m≥1b_mfracmz^m1-z^m = ∑_m≥1∑_r≥1m b_m z^mr. The coefficient at z^n is ∑_mmid nm b_m=c_n. M obius inversion gives the displayed inverse formula. The definition of G_ Z is exactly the integrality condition for that inverse.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-be13c25b5edbaee3","tex":"\\sum_{m\\in\\mathcal A}","aliases":"∑_m∈ A sum in mathcal","display":"\\sum_{m\\in\\mathcal A}","kind":"use","excerpt":"Consider the scalar marked equation u(z)=z+Φ(u(z)), Φ(y)=∑_m∈ Aw_my^m, tag11.8 where A⊆{2,3,…} is finite and every w_m is a positive integer. Its unique formal solution belongs to z Z[[z]] and counts plane rooted trees whose internal arities lie in A, weighted by the w_m.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d328b95fa488fa30","tex":"\\sum_{m\\ge0}","aliases":"∑_m≥0 sum ge","display":"\\sum_{m\\ge0}","kind":"use","excerpt":"For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d1cabd23a4e71916","tex":"\\sum_{m\\geq0}","aliases":"∑_m≥0 sum geq","display":"\\sum_{m\\geq0}","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-97951b3ae8950fca","tex":"\\sum_{m\\geq1}","aliases":"∑_m≥1 sum geq","display":"\\sum_{m\\geq1}","kind":"use","excerpt":"Formal logarithmic differentiation gives fraczA'A = ∑_m≥1b_mfracmz^m1-z^m = ∑_m≥1∑_r≥1m b_m z^mr. The coefficient at z^n is ∑_mmid nm b_m=c_n. M obius inversion gives the displayed inverse formula. The definition of G_ Z is exactly the integrality condition for that inverse.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-044b8729aa25c54b","tex":"\\sum_{n=0}^{M}","aliases":"∑_n=0^M sum","display":"\\sum_{n=0}^{M}","kind":"use","excerpt":"If λ=-M, expand (1-αω_N^rt)^M = ∑_n=0^M binom Mn(-α)^nω_N^rnt^n. The rotation matrix is a truncated discrete Fourier matrix. Its rank is the number of available residue classes, namely min(N,M+1).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-657057157b7882f4","tex":"\\sum_{n\\geq 0}","aliases":"∑_n≥ 0 sum geq","display":"\\sum_{n\\geq 0}","kind":"use","excerpt":"screte category whose objects are sequences a=(a_n)_n≥ 0∈ F_p^ N satisfying a_0=1 and a_n+T=a_n (n≥ 0) for some T≥ 1. The least such positive integer is the minimal period T(a). Write w(a)=(a_0,…,a_T(a)-1) for the primitive period word and W_a(z)=∑_r=0^T(a)-1a_rz^r. Then F_a(z)=∑_n≥ 0a_nz^n = fracW_a(z)1-z^T(a). The fraction in [source reference] n","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"use","excerpt":"The recurrence-to-OGF bridge is explicit. From χ(x)=x^d-c_d-1x^d-1-⋯-c_0 set Q(z)=z^dχ(z^-1) =1-c_d-1z-⋯-c_0z^d. The numerator is the degree-<d truncation of Q(z)∑_n≥0a_nz^n. Conversely, clearing Q recovers the recurrence and the initial vector. Minimality is equivalent to reducedness.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f0d1bc9b91545210","tex":"\\sum_{n\\geq1}","aliases":"∑_n≥1 sum geq","display":"\\sum_{n\\geq1}","kind":"use","excerpt":"[Finite-prefix triangularity] For each N, the coefficient prefix (a_1,…,a_N) of A(z)=1+∑_n≥1a_nz^n and the exponent prefix (b_1,…,b_N) determine one another by integral triangular recursions. No exponent b_m with m>N affects the coefficient prefix through degree N.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c206a6fd89e90837","tex":"\\sum_{q=0}^{N-1}","aliases":"∑_q=0^N-1 sum","display":"\\sum_{q=0}^{N-1}","kind":"use","excerpt":"For N≥1 and 0≤ j<N, define M_N,jF(x) = ∑_m≥0a_Nm+jx^m. If ω_N is a primitive Nth root of unity and x=t^N, then t^j M_N,jF(t^N) = frac1N ∑_q=0^N-1 ω_N^-jqF(ω_N^qt). This is the character projector for the cyclic rotation action.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6e06df256e0c3f1f","tex":"\\sum_{q\\mid N}","aliases":"∑_qmid N sum mid","display":null,"kind":"use","excerpt":"are the primitive split Fourier idempotents. For σ_a∈Gal(K/ Q), where σ_a(ζ_N)=ζ_N^a, σ_a(e_j)=e_aj. The Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"[Functional Mahler is not Newton-Mahler] The mask equation in this section is F(z)=R(z)F(z^p). It is a digit-section and automaton construction. It is not the Newton-Mahler interpolation expansion f(x)=∑_rc_rbinomxr. Both occur elsewhere in the Library, but this atlas uses only the functional-Mahler meaning.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-23fea3c803c85b04","tex":"\\sum_{r,\\gamma}","aliases":"∑_r,γ sum gamma","display":"\\sum_{r,\\gamma}","kind":"use","excerpt":"G:X→ Y be n times differentiable. Fix an input packet β, an output packet δ, a point x, and a direction h∈ P_β X. For every intermediate packet γ and every r≥1, put g_r,γ = Q_γ D^rG(x)[h,…,h]. A colored partition profile is a finite family μ=(μ_r,γ)_r≥1, γ∈Γ of nonnegative integers satisfying ∑_r,γrμ_r,γ=n. Let k(μ)=∑_r,γμ_r,γ.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-93cad0b3282d6977","tex":"\\sum_{r=0}^{e}","aliases":"∑_r=0^e sum","display":"\\sum_{r=0}^{e}","kind":"use","excerpt":"Unique factorization writes every positive integer as a finite product of pairwise coprime prime powers, so multiplicativity gives the reconstruction formula. Conversely, an arbitrary family of local unit series defines a multiplicative function by the same finite product. For n=p^e, Dirichlet convolution becomes (fmathbin∗_Dg)(p^e) = ∑_r=0^ef(p^r)","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-07950c38fab5b77a","tex":"\\sum_{r=0}^{\\lfloor n/N\\rfloor}","aliases":"∑_r=0^lfloor n/Nrfloor sum lfloor rfloor","display":null,"kind":"use","excerpt":"For positive z, f_N(n)=∑_r=0^lfloor n/Nrfloorbinom nNrz^r. The sequence is strictly increasing from index N-1 onward. Prefixes below N are distinguished by choosing the first suffix that crosses index N, and later prefixes differ already at suffix zero. Thus all residual functions are distinct.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-029134516e4be179","tex":"\\sum_{r=0}^{N-1}","aliases":"∑_r=0^N-1 sum","display":"\\sum_{r=0}^{N-1}","kind":"use","excerpt":"Let P(z,Y)∈ K[z,Y] be the normalized primitive representative of P_F. Introduce indeterminates U=(U_0,…,U_N-1) and substitute Y=∑_j=0^N-1z^jU_j. Every polynomial in z has a unique decomposition modulo the relation z^N=x: P (z,∑_j=0^N-1z^jU_j) = ∑_r=0^N-1z^rΦ_N,r(x, U). The polynomials Φ_N,r(x, U)∈ K[x, U] are the algebraic packet equations.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-bf6132b8c151067a","tex":"\\sum_{r=1}^{m}","aliases":"∑_r=1^m sum","display":"\\sum_{r=1}^{m}","kind":"use","excerpt":"The map Ψ(x)=s+Φ(x) preserves overline B_R because lVertΨ(x)rVert ≤ a+∑_m=2^Db_mR^m ≤ R. For x,y∈overline B_R, multilinearity gives A_m[x^m]-A_m[y^m] = ∑_r=1^m A_m[y^r-1,x-y,x^m-r], so lVertΨ(x)-Ψ(y)rVert ≤ κ_RlVert x-yrVert. The contraction theorem gives the unique fixed point.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b2e9009509f5bdd9","tex":"\\sum_{r=k}^{Mk+s}","aliases":"∑_r=k^Mk+s sum","display":"\\sum_{r=k}^{Mk+s}","kind":"use","excerpt":"[Affine Newton--Mahler transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion and put g_s(q)=f(Mq+s). Then g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k =∑_r=k^Mk+sc_rΘ_M,s(r,k). Every output coefficient depends on a finite source window.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-c13087ece793d569","tex":"\\sum_{rs\\equiv k\\, (N)}","aliases":"∑_rs≡ k (N) sum equiv","display":"\\sum_{rs\\equiv k\\, (N)}","kind":"use","excerpt":"[Multiplicative residue-packet law] For N≥1, let P_N,rf be the arithmetic function equal to f(n) when n≡ rpmod N and zero otherwise. Then P_N,k(fmathbin∗_Dg) = ∑_rs≡ k (N) (P_N,rf)mathbin∗_D(P_N,sg). Thus the packet address algebra for Dirichlet convolution is the finite multiplicative monoid Z/N Z, including its zero divisors.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6a4250883c4968da","tex":"\\sum_{r\\ge0}","aliases":"∑_r≥0 sum ge","display":"\\sum_{r\\ge0}","kind":"use","excerpt":"[Finite tree-address signature] Fix a packet width N and put g_N=gcd(N,d_ A). For the arithmetic packets U_j(x)=∑_r≥0[z^Nr+j]u(z) x^r, the following are equivalent: item U_j≠0; item j-1 belongs to the subgroup of Z/N Z generated by {m-1:m∈ A}; item j≡1pmodg_N. Hence the exact number of active packet addresses is #Supp_N(u)=fracNg_N. tag11.10","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-6b4cbee70b31b1b1","tex":"\\sum_{r\\geq0}","aliases":"∑_r≥0 sum geq","display":"\\sum_{r\\geq0}","kind":"use","excerpt":"[Functional Mahler versus Newton-Mahler] The word ``Mahler'' continues to name two distinct constructions. This section concerns the functional equation F(z)=R(z)F(z^p). It does not identify rational masks with Newton-Mahler expansions f(x)=∑_r≥0c_rbinom xr or with p-adic analytic interpolation. Any bridge to that local analytic chart still require","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-28fc50db4cc325e2","tex":"\\sum_{r\\geq1}","aliases":"∑_r≥1 sum geq","display":"\\sum_{r\\geq1}","kind":"use","excerpt":"Formal logarithmic differentiation gives fraczA'A = ∑_m≥1b_mfracmz^m1-z^m = ∑_m≥1∑_r≥1m b_m z^mr. The coefficient at z^n is ∑_mmid nm b_m=c_n. M obius inversion gives the displayed inverse formula. The definition of G_ Z is exactly the integrality condition for that inverse.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-aa2f936466c8fe7a","tex":"\\sum_{s=0}^{q}","aliases":"∑_s=0^q sum","display":"\\sum_{s=0}^{q}","kind":"use","excerpt":"Conversely, suppose ∑_s=0^qp_s(n)a_n+s=0 (n≥ N), where p_s∈ K[n], and put θ=zD.","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-10c83533a88b2696","tex":"\\sum_{\\substack{\\mathcal Q\\text{ a fine orbit}\\\\\n \\mathcal Q\\subseteq(\\rho^A_{sr})^{-1}(\\mathcal O)}}","aliases":"∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) sum substack mathcal text mathcal subseteq rho mathcal","display":"\\sum_{\\substack{\\mathcal Q\\text{ a fine orbit}\\\\\n \\mathcal Q\\subseteq(\\rho^A_{sr})^{-1}(\\mathcal O)}}","kind":"use","excerpt":"item If ssucceq r, equivariance of the refinement map sends every fine monodromy orbit onto a coarse monodromy orbit. For a coarse orbit O, P^r_A, O = ∑_substack Q a fine orbit Q⊆(ρ^A_sr)^-1( O) P^s_A, Q. Thus orbit descent and projector-sum refinement commute exactly.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ed8fee2db4ccd62c","tex":"\\sum_{\\substack{0\\le r_1,\\ldots,r_q<N\\\\\n r_1+\\cdots+r_q\\equiv j\\ (\\mathrm{mod}\\ N)}}","aliases":"∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) sum substack le ldots cdots equiv mathrm","display":"\\sum_{\\substack{0\\le r_1,\\ldots,r_q<N\\\\\n r_1+\\cdots+r_q\\equiv j\\ (\\mathrm{mod}\\ N)}}","kind":"use","excerpt":"[Exact q-ary packet phase theorem] Put g=gcd(N,q-1), h=fracq-1g. Then: item the active packet addresses are exactly j≡1pmod g; item the complete packet system is U_j(x) = δ_j,1 + ∑_substack0≤ r_1,…,r_q<N r_1+⋯+r_q≡ j (mod N) x^(r_1+⋯+r_q-j)/N ∏_a=1^qU_r_a(x); tag11.16 item the smooth critical values are τ=q^-1/(q-1), ρ=(q-1)q^-q/(q-1); item the sou","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-78328f740bc05a89","tex":"\\sum_{\\substack{1\\le a\\le q\\\\\\gcd(a,q)=1}}","aliases":"∑_substack1≤ a≤ q gcd(a,q)=1 sum substack le le gcd","display":"\\sum_{\\substack{1\\le a\\le q\\\\\\gcd(a,q)=1}}","kind":"use","excerpt":"Galois orbits are indexed by divisors qmid N, with O_q = {j∈ Z/N Z: fracNgcd(N,j)=q}. Their orbit sums E_q:=∑_j∈ O_qe_j belong to Q[C_N] and satisfy E_qE_q'=δ_q,q'E_q, ∑_qmid NE_q=1. More explicitly, E_q = frac1N∑_ℓ=0^N-1c_q(ℓ)g^ℓ, where c_q(ℓ) := ∑_substack1≤ a≤ q gcd(a,q)=1 e^2π i aℓ/q = ∑_dmidgcd(q,ℓ)d μ(q/d) ∈ Z is the Ramanujan sum. The projec","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a89c55b129124a9","tex":"\\sum_{\\substack{d\\mid n\\\\d<n}}","aliases":"∑_substackdmid n d<n sum substack mid d","display":null,"kind":"use","excerpt":"The coefficient equation at n is triangular: g(1)=1, g(n)=-∑_substackdmid n d<nf(n/d)g(d). Hence the inverse exists integrally and uniquely. The standard coprime-divisor factorization of a divisor of mn proves closure of multiplicative functions under convolution; uniqueness of the inverse then proves multiplicativity of f^-1_D.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3026f36a5463dd29","tex":"\\sum_{\\substack{e\\in S_1\\\\\\kappa(e)=t}}","aliases":"∑_substacke∈ S_1 κ(e)=t sum substack in kappa","display":"\\sum_{\\substack{e\\in S_1\\\\\\kappa(e)=t}}","kind":"use","excerpt":"[Exact visible-support loss ledger] Use the finite fold data of (L8) and fix an input family b=(b_e)_e∈ E. Define S_0:={e∈ E:b_e≠0}, A_0:={e∈ S_0:τ_e(b_e)=0}, S_1:=S_0∖ A_0, and let Γ:=κ(S_1)⊆ T. For t∈ T, write the folded coordinate as overline b_t := ∑_substacke∈ S_1 κ(e)=tτ_e(b_e) ∈ C_t, and put supp(overline b) :={t∈ T:overline b_t≠0}, K_0:={t∈","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-37ae5f0711a1a238","tex":"\\sum_{\\substack{m\\ge0\\\\m\\equiv j\\ (\\mathrm{mod}\\ N)}}","aliases":"∑_substackm≥0 m≡ j (mod N) sum substack ge m equiv mathrm","display":null,"kind":"use","excerpt":"For integers N≥1 and n≥0, define the step factorial (a)_n;N:=∏_k=0^n-1(a+Nk), (a)_0;N:=1. For a commutative ring A and a formal series F(z)=∑_m≥0a_mz^m∈ A[[z]], define its residue lane P_N,jF(z) := ∑_substackm≥0 m≡ j (mod N)a_mz^m, 0≤ j<N.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-8e5c03e21aefbb67","tex":"\\sum_{T\\in\\mathcal T_{\\mathcal A}}","aliases":"∑_T∈ T_ A sum in mathcal mathcal","display":"\\sum_{T\\in\\mathcal T_{\\mathcal A}}","kind":"use","excerpt":"[Selected polynomial fixed point and tree expansion] Assume a+φ(R)≤ R, κ_R<1. Then equation textup(11.1) has a unique solution in the closed ball overline B_R. Moreover, that solution is the absolutely convergent sum u=∑_T∈ T_ Aev(T), tag11.2 where A={m:A_m≠0}, the leaves of T evaluate to s, and an internal vertex of arity m evaluates by A_m applie","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3cdd7357c745c72c","tex":"\\sum_{u=0}^{s}","aliases":"∑_u=0^s sum","display":"\\sum_{u=0}^{s}","kind":"use","excerpt":"[Prime-power Frobenius sparsification] For M=p^a, Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside 0≤ r-p^ak≤ s. Therefore d_s,k ≡ ∑_u=0^sbinom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2d0d50d4f2fa523e","tex":"\\sum_{v}","aliases":"∑_v sum","display":"\\sum_{v}","kind":"use","excerpt":"Define the arity period d_ A = gcd{m-1:m∈ A}. tag11.9 Every tree with internal vertices v has leaf count ℓ(T)=1+∑_v(deg(v)-1).","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-42a3abd494976575","tex":"\\sum_{v_j-j=\\mu}","aliases":"∑_v_j-j=μ sum mu","display":"\\sum_{v_j-j=\\mu}","kind":"use","excerpt":"Let α be a finite point and put v_j=v_α(q_j), μ=min_j(v_j-j). Write the first nonzero local coefficient of q_j as q_j,v_j. The indicial polynomial is I_α,L(ρ) = ∑_v_j-j=μ q_j,v_jρ^underline j. At a regular singular point, its roots form the local exponent packet. Over C, the corresponding local monodromy eigenvalues are e^2π iρ. Integer differences","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-af98395fff9c5326","tex":"\\sum_{x\\in K\\backslash G/H}","aliases":"∑_x∈ Kbackslash G/H sum in backslash","display":null,"kind":"use","excerpt":"The dimension identity for the trivial H-representation is [G:H] = ∑_x∈ Kbackslash G/H [K:K∩ xHx^-1]. tag12.9 This is the finite packet-count law audited by the Galois-Tannakian atlas verifier.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ef8abb451a5abafd","tex":"\\sum_{\\kappa(e')=t}","aliases":"∑_κ(e')=t sum kappa","display":"\\sum_{\\kappa(e')=t}","kind":"use","excerpt":"Injectivity of τ_e makes τ_e(b_e)≠0 for every e∈ S_0, so there is no annihilation. For strict selectors, suppose t∈ K_0 and choose e∈ S_1 with κ(e)=t. Applying Q_e to j_toverline b_t =∑_κ(e')=tj_tτ_e'(b_e')=0 leaves j_tτ_e(b_e)=0, contrary to e∈ S_1. Hence K_0=∅.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e1cda44bbae95d20","tex":"\\sum_{\\lambda\\in\\Lambda}","aliases":"∑_λ∈Λ sum lambda in Lambda","display":"\\sum_{\\lambda\\in\\Lambda}","kind":"use","excerpt":"Over characteristic zero, write the unique active expansion over an algebraic closure as a_n=∑_λ∈ΛP_λ(n)λ^n. For each sampling modulus N and address j, the spectral torsion-Fourier theorem computes the exact minimal recurrence, reduced denominator, Hankel rank, state dimension, rank-drop mechanism, and finite periodic packet signature of a_Nm+j [so","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f4e68ab10941d1c5","tex":"\\sum_{\\mu}","aliases":"∑_μ sum mu","display":"\\sum_{\\mu}","kind":"use","excerpt":"[Compressed diagonal packet Fa a di Bruno formula] With the notation above, · P_δ D^n(F∘ G)(x)[h,…,h] · =∑_μ fracn! displaystyle∏_r,γ (r!)^μ_r,γμ_r,γ! P_δ D^k(μ)F(Gx) [ (g_r,γ)^μ_r,γ ]_r,γ. tag11.5 Here the last line means that g_r,γ is inserted μ_r,γ times; symmetry makes the insertion order irrelevant.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-063a16e4b52f2ff2","tex":"\\sum_{\\rho^Y_{vu}(b)=a}","aliases":"∑_ρ^Y_vu(b)=a sum rho","display":"\\sum_{\\rho^Y_{vu}(b)=a}","kind":"use","excerpt":"item Projector-sum refinement. If ssucceq r on X and vsucceq u on Y, then p_Y,a^u f p_X,i^r = ∑_ρ^Y_vu(b)=a ∑_ρ^X_sr(j)=i p_Y,b^v f p_X,j^s. Coarsening a fine composite gives the same ambient morphism as composing at the coarse level.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-94e9e6125de3522b","tex":"\\sum_{\\sigma_{vu}(b)=a}","aliases":"∑_σ_vu(b)=a sum sigma","display":"\\sum_{\\sigma_{vu}(b)=a}","kind":"use","excerpt":"[Exact refinement law, imported] Let p^r_i=∑_ρ_sr(j)=ip^s_j and q^u_a=∑_σ_vu(b)=aq^v_b. For every morphism f:X→ Y, q^u_afp^r_i = ∑_σ_vu(b)=a ∑_ρ_sr(j)=i q^v_bfp^s_j. Coarsening the fine block product of two composable maps gives the coarse block product.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-740fcc1ef274f00a","tex":"\\sum_{\\sigma\\in G}","aliases":"∑_σ∈ G sum sigma in","display":"\\sum_{\\sigma\\in G}","kind":"use","excerpt":"A useful constructive form is obtained by choosing any matrix C∈ M_n(L) and setting T(C)=∑_σ∈ Ga_σσ(C). Then a_ττ(T(C))=T(C). Whenever T(C) is invertible, B=T(C)^-1 is a Hilbert 90 trivialization. Standard descent guarantees that such a C exists.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-4e96d3bac74365f7","tex":"\\sum_{\\ell=0}^{M-1}","aliases":"∑_ℓ=0^M-1 sum ell","display":"\\sum_{\\ell=0}^{M-1}","kind":"use","excerpt":"factor from c+1 through c+Nn occurs in exactly one lane. For every commutative ring A, on the A-module A[[z]] the operators ( P_N,j)_0≤ j<N satisfy P_N,j^2= P_N,j, P_N,j P_N,k=0 (j≠ k), ∑_j=0^N-1 P_N,j=1. If M∈ Z and M≥1, their exact refinement law is P_N,j = ∑_ℓ=0^M-1 P_NM,j+ℓ N.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b98eb4d33c4dd25d","tex":"\\sum_{\\ell=0}^{N-1}","aliases":"∑_ℓ=0^N-1 sum ell","display":"\\sum_{\\ell=0}^{N-1}","kind":"use","excerpt":"Suppose further that K is a field, that N is invertible in K, and that K contains a primitive N-th root of unity ω. Then for every F∈ K[[z]], P_N,jF(z) = frac1N∑_ℓ=0^N-1 ω^-jℓF(ω^ℓ z). If, in addition, K has characteristic zero, the exponential and hypergeometric series below are defined in K[[z]]. Put E_N,j(z) := P_N,je^z =∑_n≥0fracz^Nn+j(Nn+j)!. ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"Let G be a compact group and let U:G→ U( H) be strongly continuous. For a closed subgroup H, normalized Haar measure defines P_H=∫_HU_h dμ_H(h). tag12.14 The same averaging proof as in the finite case shows that P_H is the orthogonal projector onto H^H.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0b1c8888965e3158","tex":"\\int_H","aliases":"∫_H int","display":"\\int_H","kind":"use","excerpt":"Let G be a compact group and let U:G→ U( H) be strongly continuous. For a closed subgroup H, normalized Haar measure defines P_H=∫_HU_h dμ_H(h). tag12.14 The same averaging proof as in the finite case shows that P_H is the orthogonal projector onto H^H.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Exact residual product identity] Let f_E=C_E(f), r_f=R_E(f), g_F=C_F(g), r_g=R_F(g). Then gf-g_Ff_E = r_gf_E+g_Fr_f+r_gr_f. If the morphisms lie in a normed operator algebra with a submultiplicative norm, then gf-g_Ff_E ≤ r_g f_E + g_F r_f + r_g r_f .","count":20},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"Fix d≥0. Over characteristic zero, the affine Mahler coefficient maps A_m,s^(d) are invertible and satisfy A_n,t^(d)A_m,s^(d)=A_mn,mt+s^(d). For coprime m,n, the two paths around every declared Chinese-remainder diamond are equal [source reference]. Hence every finite subdiagram generated by these charts is a coherent finite chart groupoid with tri","count":33},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"[Regular rank theorem] Let a be k-regular over K. Then r_k(a) = rank_K H_a = min{dim_K W:W supports a digit-linear realization of a}. Moreover: item there exist finite word sets U,V⊂Σ_k^*, each of cardinality r_k(a), for which the minor H_a[U,V] is nonsingular; item every reachable-observable realization of dimension r_k(a) is isomorphic to C_k(a);","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9f55fb1c51ddcd12","tex":"\\supseteq","aliases":"⊇ supseteq","display":"\\supseteq","kind":"use","excerpt":"intheorem[Normalized finite Galois chart] Fix the finite Galois extension L/K. The following data determine one another exactly: H longleftrightarrow L^H longleftrightarrow P_H tag12.3 where the projector is taken in a faithful field or regular realization. The correspondence is order reversing: H≤ J Longleftrightarrow L^H⊇ L^J Longleftrightarrow P","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ed89b38d9fe9644f","tex":"\\bigoplus_{\\mathcal O\\in H\\backslash(J\\times I)}","aliases":"⊕_ O∈ Hbackslash(J× I) bigoplus mathcal in backslash times","display":null,"kind":"use","excerpt":"ion-orbital-calculus Let kk be a commutative ring, let a finite group H act on finite sets I,J,K, and let kk[I],kk[J],kk[K] be the corresponding permutation modules. For an orbit O⊆ J× I under the diagonal action, let B_ O:kk[I]→kk[J] be the matrix whose (j,i)-entry is one when (j,i)∈ O and zero otherwise. Then Hom_kk H(kk[I],kk[J]) = ⊕_ O∈ Hbacksl","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-50db95a0004bcd02","tex":"\\bigoplus_a","aliases":"⊕_a bigoplus","display":"\\bigoplus_a","kind":"use","excerpt":"[Packet coordinates on the obstruction quotient] Suppose (Q_a) is a complete finite family of pairwise annihilating idempotents on V and Q_a(im D)⊆im D for every a. Then the Q_a descend to quotient idempotents overline Q_a on coker D, coker D ≅ ⊕_aoverline Q_a(coker D), overline Q_aOb_D(r)=[Q_ar]. If U and V already split into matching packet summa","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-59d45b85e0a978c4","tex":"\\bigoplus_{b\\in I_{Y,v}}","aliases":"⊕_b∈ I_Y,v bigoplus in","display":"\\bigoplus_{b\\in I_{Y,v}}","kind":"use","excerpt":"For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-38ab6f84a4912d75","tex":"\\bigoplus_e","aliases":"⊕_e bigoplus","display":"\\bigoplus_e","kind":"use","excerpt":"Suppose a finite family of lane objects (B_e)_e∈ E, target objects (C_t)_t∈ T, and morphisms τ_e:B_e→ C_κ(e) is given. Let q_e:⊕_e'∈ EB_e'→ B_e be the coordinate projection and define Φ:=∑_e∈ Ej_κ(e)τ_eq_e: ⊕_e∈ EB_e⟶ ⊕_t∈ TC_t. Then there are canonical short exact sequences 0⟶kerΦ ⟶⊕_eB_e ⟶imΦ ⟶0 and 0⟶imΦ ⟶⊕_tC_t ⟶cokerΦ ⟶0. The first sequence is","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2052911acb88cf20","tex":"\\bigoplus_{e'\\in E}","aliases":"⊕_e'∈ E bigoplus in","display":"\\bigoplus_{e'\\in E}","kind":"use","excerpt":"[Abelian fold-loss exact sequence] Let A be an abelian category. Suppose a finite family of lane objects (B_e)_e∈ E, target objects (C_t)_t∈ T, and morphisms τ_e:B_e→ C_κ(e) is given. Let q_e:⊕_e'∈ EB_e'→ B_e be the coordinate projection and define Φ:=∑_e∈ Ej_κ(e)τ_eq_e: ⊕_e∈ EB_e⟶ ⊕_t∈ TC_t. Then there are canonical short exact sequences 0⟶kerΦ ⟶⊕","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-9e230cb842b55b7b","tex":"\\bigoplus_{e\\in E}","aliases":"⊕_e∈ E bigoplus in","display":"\\bigoplus_{e\\in E}","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d876683bc5214221","tex":"\\bigoplus_{e\\notin S}","aliases":"⊕_e∉ S bigoplus notin","display":"\\bigoplus_{e\\notin S}","kind":"use","excerpt":"extup(roman*)] item evaluation at the root identifies the space of global parallel sections with C_o; item (C_v)_v is path-flat, meaning that all path transports with the same endpoints agree on the core; item it is the unique greatest path-flat subfamily; item it is independent of the chosen spanning tree and equals the common fixed space of all c","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","excerpt":"[Declared packets become literal objects] Every declared idempotent p:X→ X defines an object (X,p) of Syn_ak( K_rm lin). A complete packet family (p_i) gives the canonical isomorphism (X,1_X)≅⊕_i(X,p_i).","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3709e962e547d707","tex":"\\oplus_i","aliases":"⊕_i oplus","display":"\\oplus_i","kind":"use","excerpt":"[Exact packet-coordinate isomorphism, imported] For every finite packet system in an additive idempotent-complete category, S_ P A_ P=1_X, A_ P S_ P =1_⊕_iX_i. Consequently, X≅⊕_i∈ IX_i.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-276b56ea53a79483","tex":"\\bigoplus_{i=1}^{\\rho}","aliases":"⊕_i=1^ρ bigoplus rho","display":"\\bigoplus_{i=1}^{\\rho}","kind":"use","excerpt":"By the Smith normal form, there are uniquely determined integers 1≤ d_1(a)mid d_2(a)mid⋯mid d_ρ(a) such that C_k(a) ≅ ⊕_i=1^ρ Z/d_i(a) Z. Factors with d_i=1 are harmless but are retained so that the list has length ρ.","count":4},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e6f12a971c5ff350","tex":"\\bigoplus_{i\\in\\mathcal O_U}","aliases":"⊕_i∈ O_U bigoplus in mathcal","display":"\\bigoplus_{i\\in\\mathcal O_U}","kind":"use","excerpt":"(U and transport O along its address bijection. Two path choices differ by a loop at U_0, which preserves O setwise. Hence O_U is path independent. Projector compatibility gives θ^A_γ F_γ(P^r_A,U, O) = P^r_A,V, Oθ^A_γ, so the orbit sums descend. Their local splittings are the finite direct sums ⊕_i∈ O_UA^r_U,i; the direct sums of the β-maps, with t","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2f8694440e8ff713","tex":"\\bigoplus_{j\\in I_{X,s}}","aliases":"⊕_j∈ I_X,s bigoplus in","display":"\\bigoplus_{j\\in I_{X,s}}","kind":"use","excerpt":"For resolutions ssucceq r on X and vsucceq u on Y, let B^v,s(X,Y) := ⊕_b∈ I_Y,v ⊕_j∈ I_X,s Hom_ C(X_j^s,Y_b^v), and define B^u,r(X,Y) similarly. Analysis of a morphism gives an isomorphism a^v,s_X,Y: Hom_ C(X,Y)xrightarrow∼ B^v,s(X,Y), whose inverse is lane synthesis.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-1ccfac60a0997553","tex":"\\bigoplus_t","aliases":"⊕_t bigoplus","display":"\\bigoplus_t","kind":"use","excerpt":"[Why the fold selectors live in one group] The endomorphisms Q_e in (L8) all act on the single group C=⊕_tC_t. Writing separate selectors on unrelated codomains would make the Kronecker relation ill-typed and would not prove injectivity of the combined fold.","count":5},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-58b685f3c2641fa3","tex":"\\bigoplus_{t\\in T}","aliases":"⊕_t∈ T bigoplus in","display":"\\bigoplus_{t\\in T}","kind":"use","excerpt":"item Exact fold loss. Let E be a finite set of retained lane positions and set B_e:=Hom_ C(X_i^r,Y_a^u) for e=(a,i)∈ E. Let T be a finite set, let κ:E→ T, let C_t be abelian groups, and let τ_e:B_e→ C_κ(e) be homomorphisms. With C=⊕_t∈ TC_t and coordinate inclusions j_t:C_t→ C, the fold Φ_κ,τ: ⊕_e∈ EB_e⟶ C, (b_e)_e⟼ ∑_e∈ Ej_κ(e)τ_e(b_e) has exact i","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-2cb3723528e5d0be","tex":"\\bigoplus_{x\\in K\\backslash G/H}","aliases":"⊕_x∈ Kbackslash G/H bigoplus in backslash","display":null,"kind":"use","excerpt":"[Mackey transition-packet decomposition] Let H,K≤ G, let W be a finite-dimensional representation of H, and choose representatives x for Kbackslash G/H. Then Res_K^GInd_H^GW ≅ ⊕_x∈ Kbackslash G/H Ind_K∩ xHx^-1^K (^xRes_H∩ x^-1Kx^HW). tag12.8 Each double coset is one exact transfer packet between the H- and K-charts.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-3241cc9716f694aa","tex":"\\bigoplus_{X\\in Q_0}","aliases":"⊕_X∈ Q_0 bigoplus in","display":"\\bigoplus_{X\\in Q_0}","kind":"use","excerpt":"A typed realization defines the action of a path by composition and then extends 𝕜-linearly. Every element of I acts by zero, so the action factors through Λ_ K. Conversely, for a unital module M, the pairwise orthogonal vertex idempotents give M=⊕_X∈ Q_0e_XM, and an arrow a:X→ Y maps e_XM into e_YM. These constructions are inverse up to their cano","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ff4ce89ac3b2d724","tex":"\\bigoplus_{\\delta(s,b)=t}","aliases":"⊕_δ(s,b)=t bigoplus delta","display":"\\bigoplus_{\\delta(s,b)=t}","kind":"use","excerpt":"ed-observer Let S be a finite continuation-state set, let (A,⊕,0) be a commutative monoid, let B be a finite branch alphabet, and let δ:S× B⟶ S be a state transition. Suppose every branch b∈ B acts by a monoid homomorphism L_b:A⟶ A. Take an empty aggregation to be 0. For a state packet a=(a_s)_s∈ S, define a'_t := ⊕_δ(s,b)=tL_b(a_s). If a finite fa","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-5bc24e1dc00a5f90","tex":"\\bigoplus_{\\chi\\in\\operatorname{Irr}_K(G)}","aliases":"⊕_χ∈Irr_K(G) bigoplus chi in operatorname","display":"\\bigoplus_{\\chi\\in\\operatorname{Irr}_K(G)}","kind":"use","excerpt":"ation] Let G be finite and let K be a characteristic-zero splitting field for G. The primitive central idempotents e_χ = fracχ(1)|G| ∑_g∈ Gχ(g^-1)g ∈ K[G] for χ∈Irr_K(G) form a complete pairwise-annihilating lane system on every finite-dimensional semisimple K[G]-module V: V=⊕_χ∈Irr_K(G) e_χ V. Every G-equivariant map is block diagonal in these iso","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"use","excerpt":"Reconstructed symmetry · Aut^⊗(ω) · tensor automorphisms of a fiber functor and the corresponding affine group scheme · faithful exact tensor fiber functor","count":11},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-40aa46a044ccf15d","tex":"\\otimes_{\\mathbb Z}","aliases":"⊗_ Z otimes mathbb","display":"\\otimes_{\\mathbb Z}","kind":"use","excerpt":"Define M_k,p(a)=M_k(a)⊗_ Z Z_p, L_k,p(a)=L_k(a)⊗_ Z Z_p.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-e4b0e5b9899a9e51","tex":"\\otimes_K","aliases":"⊗_K otimes","display":"\\otimes_K","kind":"use","excerpt":"vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of this form is ","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7173200b01de0ce2","tex":"\\bigcap_{a\\in A}","aliases":"⋂_a∈ A bigcap in","display":"\\bigcap_{a\\in A}","kind":"use","excerpt":"Assume M is generated by a family A. On congruences or submodules define Φ(R) := K_0∩⋂_a∈ Aρ(a)^-1(R), with the corresponding submodule formula in the linear setting.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a1b848de6c851a88","tex":"\\bigcap_{e\\notin S}","aliases":"⋂_e∉ S bigcap notin","display":"\\bigcap_{e\\notin S}","kind":"use","excerpt":"a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-15e0c0ed7f2caa43","tex":"\\bigcap_F","aliases":"⋂_F bigcap","display":"\\bigcap_F","kind":"use","excerpt":"[Common profile and composition] Let F: C→ D and G: D→ E. Every constructor preserved by both functors, with compatible chosen comparison data, is preserved by GF. For a finite diagram of realizations and bridges, the intersection Π_rm common:=⋂_FΠ_F is therefore a valid common profile for every proof transported through the whole diagram.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-03d39d81eaa6ba3e","tex":"\\bigcap_{g\\in G}","aliases":"⋂_g∈ G bigcap in","display":"\\bigcap_{g\\in G}","kind":"use","excerpt":"If H is not normal, its conjugates form a genuine atlas rather than one quotient chart. Put Core_G(H)=⋂_g∈ GgHg^-1, Ncl_G(H)=⟨ gHg^-1:g∈ G⟩.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d42ab8c0624e1096","tex":"\\bigcap_j","aliases":"⋂_j bigcap","display":"\\bigcap_j","kind":"use","excerpt":"Let K_0:=⋂_jkerpair(o_j) in the relational setting and N_0:=⋂_jker(o_j) in the linear setting.","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-f16804592bbf32c1","tex":"\\bigcap_{m,j}","aliases":"⋂_m,j bigcap","display":"\\bigcap_{m,j}","kind":"use","excerpt":"In the relational setting, the kernel pair of q is contained in Θ_ S, because equality under q forces equality under every factorized future observation. The universal property of the quotient by Θ_ S gives the unique map. In the module setting, ker q⊆ ⋂_m,jker(o_jρ(m))=N_ S, so the ordinary quotient universal property gives the same conclusion.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-b9acce69444411ea","tex":"\\bigcap_{q\\in\\mathcal Q}","aliases":"⋂_q∈ Q bigcap in mathcal","display":"\\bigcap_{q\\in\\mathcal Q}","kind":"use","excerpt":"Fix a universe mathscr Q(X) of admissible queries on X. For Q⊆mathscr Q(X), define Ind( Q) := ⋂_q∈ Qkerpair(q). For an equivalence relation or congruence R, define Def(R) := {q∈mathscr Q(X):R⊆kerpair(q)}.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-a3a10b829deb2cca","tex":"\\bigcap_{w\\in\\Sigma_k^*}","aliases":"⋂_w∈Σ_k^* bigcap in Sigma","display":"\\bigcap_{w\\in\\Sigma_k^*}","kind":"use","excerpt":"[Reachability and observability] The canonical section realization C_k(a) is linearly reachable and observable: span_K{Λ_wa:w∈Σ_k^*} = V_k(a), and ⋂_w∈Σ_k^* ker(ϵ_0Λ_w) = {0}.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"use","excerpt":"ite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-d3feba249ce10006","tex":"\\bigcup_{\\delta(s,b)=t}","aliases":"⋃_δ(s,b)=t bigcup delta","display":"\\bigcup_{\\delta(s,b)=t}","kind":"use","excerpt":"ite observer-state set, let B be a finite branch alphabet, and let C_s⊆ R^d be a compact convex packet for each state, with empty packets allowed. Use the conventions h_∅(x)=-∞, conv(∅)=∅, max∅=-∞. For branch matrices (A_b)_b∈ B and a transition δ:S× B→ S, put C'_t = conv ⋃_δ(s,b)=tA_b^ TC_s.","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"use","excerpt":"[Effective vector descent by Hilbert 90] Let L/K be finite Galois with group G. Suppose a_στ=a_σ σ(a_τ), a_σ∈ GL_n(L), defines a semilinear packet cocycle. Then there is B∈ GL_n(L) such that a_σ=B^-1σ(B). The twisted fixed packet W=(L^n)^G,⋆ is an n-dimensional K-vector space and W⊗_KLxrightarrow∼L^n. Thus every finite linear semilinear atlas of th","count":6},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"use","excerpt":"a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"a finite connected graph. Attach a finite-dimensional 𝕜-vector space F_v to each vertex and an isomorphism T_e:F_u→ F_v to each oriented edge e:u→ v, with T_bar e=T_e^-1. Choose a root o and spanning tree S. If G_v:F_o→ F_v is tree transport, then every non-tree edge e:u→ v has fundamental holonomy H_e=G_v^-1T_eG_u∈ GL(F_o). Define C_o = ⋂_e∉ Sker(","count":8},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-63972c25f4a6a948","tex":"\\rangle_{\\mathbb N}","aliases":"⟩_ N rangle mathbb","display":"\\rangle_{\\mathbb N}","kind":"use","excerpt":"Replacing one leaf by an m-ary vertex increases the leaf count by m-1. Starting from the one-leaf tree, every nonnegative sum of allowed increments is realized by a sequence of leaf replacements. Positivity of the weights prevents cancellation, so the coefficient support is exactly 1+⟨ m-1:m∈ A⟩_ N. Modulo N, the monoid generated by the increments ","count":1},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-0da92c8c15988310","tex":"\\mathbb{F}","aliases":"𝔽 mathbb","display":"\\mathbb{F}","kind":"use","excerpt":"𝔽_2/ word 10 · 2 · 2 · 2 · 3 · 2","count":3},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-58f84cd92fad8b3a","tex":"\\mathbb{F}_2","aliases":"𝔽_2 mathbb","display":"\\mathbb{F}_2","kind":"use","excerpt":"𝔽_2/ word 10 · 2 · 2 · 2 · 3 · 2","count":2},{"work":"stratified-universality","url":"notation/stratified-universality.html#symbol-ce368b6a74c41a5d","tex":"\\mathbb{F}_3","aliases":"𝔽_3 mathbb","display":"\\mathbb{F}_3","kind":"use","excerpt":"𝔽_3/ word 12 · 2 · 1 · 1 · 2 · 1","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":30},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":133},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":54},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a53ead434f8754af","tex":"A^\\top","aliases":"A^top top","display":"A^\\top","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":21},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":30},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":35},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-401361282589c1a2","tex":"\\mathcal B_{\\mathcal O}","aliases":"B_ O mathcal mathcal","display":"\\mathcal B_{\\mathcal O}","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-0bb7ba7e58f63875","tex":"\\mathcal B_{gh}","aliases":"B_gh mathcal","display":"\\mathcal B_{gh}","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-557f88fb6cda082f","tex":"B_h","aliases":"B_h","display":"B_h","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-12cb07ad60774339","tex":"\\mathcal B_{hg}","aliases":"B_hg mathcal","display":"\\mathcal B_{hg}","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":25},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2a62ce5b7390bdd7","tex":"c_e","aliases":"c_e","display":"c_e","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":28},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":28},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":70},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":40},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":22},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-9cde3c700d58d43c","tex":"J_{h1}","aliases":"J_h1","display":"J_{h1}","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-3f2d06bce4bd4ff2","tex":"J_{hB_h}","aliases":"J_hB_h","display":"J_{hB_h}","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":52},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":20},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":8},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d9d706ee857ada00","tex":"\\max_{x_e\\in\\{0,1\\}}","aliases":"max_x_e∈{0,1} max in","display":"\\max_{x_e\\in\\{0,1\\}}","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":14},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":28},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-ecdf2ff2033f302f","tex":"\\mathcal P_1","aliases":"P_1 mathcal","display":"\\mathcal P_1","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-45c4f17ed76d05ef","tex":"P_g","aliases":"P_g","display":"P_g","kind":"introduction-passage","excerpt":"Storage refinement changes only the order and grouping of finite sums. By partition refinement invariance, the streamed products reconstruct the same A, A^top, and therefore the same H=A^top A. Since each P_h is fixed by the structural phase partition, every block P_hHP_g is unchanged. The phase graph is defined only by these blocks, so its path st","count":17},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-ea63928309796eee","tex":"P_h","aliases":"P_h","display":"P_h","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":23},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-b9ca4be0e9bd233a","tex":"\\mathcal P_H","aliases":"P_H mathcal","display":"\\mathcal P_H","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-5627e8b6e58a3075","tex":"\\mathcal P_h","aliases":"P_h mathcal","display":"\\mathcal P_h","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":14},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-7fe36e042c4ebce0","tex":"\\sqcup","aliases":"sqcup","display":"\\sqcup","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":56},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-48423f1cb94c99be","tex":"\\widetilde\\sigma","aliases":"widetildeσ widetilde sigma","display":"\\widetilde\\sigma","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-0ef696fbdaa82fc8","tex":"\\widetilde\\sigma_i","aliases":"widetildeσ_i widetilde sigma","display":"\\widetilde\\sigma_i","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":10},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c69d923bf6621481","tex":"x_e","aliases":"x_e","display":"x_e","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":12},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2442980e628749af","tex":"\\beta_e^{(k)}","aliases":"β_e^(k) beta","display":"\\beta_e^{(k)}","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c5b085f18c150ac1","tex":"\\Delta_e","aliases":"Δ_e Delta","display":"\\Delta_e","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":10},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-1f868fca37586240","tex":"\\lambda_i","aliases":"λ_i lambda","display":"\\lambda_i","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-0ed845e79c8d8e41","tex":"\\sigma_i","aliases":"σ_i sigma","display":"\\sigma_i","kind":"introduction-passage","excerpt":"[Range-core singular-value certificate] Let the singular values of A be σ_i(A), and let widetildeσ_i(B) denote the singular values of B padded by zeros to the ambient spectral dimension. Then | σ_i(A) - widetildeσ_i(B) | ≤ δ. Consequently, max{0,widetildeσ_i(B)-δ}^2 ≤ λ_i(AA^top) ≤ ( widetildeσ_i(B)+δ )^2.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":10},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":21},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":38},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-ec1616068a6ee172","tex":"\\sum_e","aliases":"∑_e sum","display":"\\sum_e","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c373be36458d8e51","tex":"\\sum_{e\\in\\mathcal O}","aliases":"∑_e∈ O sum in mathcal","display":"\\sum_{e\\in\\mathcal O}","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":7},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":35},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"Let c_e≥0 denote the communication or storage cost saved by omitting pair e. Under the scalar additive certificate, the exact discrete selection problem is max_x_e∈{0,1} ∑_e c_ex_e subject to ∑_eβ_e^(k)x_e≤τ. This is the 0--1 knapsack problem. Hence unequal-cost scheduling is combinatorial in general. The implementation supplies a safe cost-to-risk","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"introduction-passage","excerpt":"Let E denote the set of undirected off-diagonal phase pairs. Because H is self-adjoint, B is symmetric. For a phase pair e={h,g} define the nonnegative matrix Δ_e by (Δ_e)_hg= B_hg, (Δ_e)_gh= B_gh, with every other entry zero. If O⊆ E is the set of omitted pairs, define B_ O = B-∑_e∈ OΔ_e. The diagonal is always retained.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"Fix a structural phase partition {1,…,m}= P_1sqcup⋯sqcup P_H and let each phase be subdivided into arbitrary storage blocks P_h = J_h1sqcup⋯sqcup J_hB_h. Let S denote this physical storage subdivision. The phase projectors P_h are defined by the structural phases, not by S.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":7},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-6d63899ef719c388","tex":"A_1^\\top","aliases":"A_1^top top","display":"A_1^\\top","kind":"use","excerpt":"Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-6c1908f93f8dee44","tex":"A_B","aliases":"A_B","display":"A_B","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-08eb65adb1a856b8","tex":"A_b","aliases":"A_b","display":"A_b","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d7439f4f816d2a74","tex":"A_{b,I_r}","aliases":"A_b,I_r","display":"A_{b,I_r}","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-938e7d9a4edb4554","tex":"A_{b,I_s}^\\top","aliases":"A_b,I_s^top top","display":"A_{b,I_s}^\\top","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-08e363ae4221d772","tex":"A_B^\\top","aliases":"A_B^top top","display":"A_B^\\top","kind":"use","excerpt":"Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-7347e27bcb45136f","tex":"A_b^\\top","aliases":"A_b^top top","display":"A_b^\\top","kind":"use","excerpt":"[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-5a8e57a002dc32f4","tex":"a_d","aliases":"a_d","display":"a_d","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-410901667e22470e","tex":"A_J","aliases":"A_J","display":"A_J","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-88149fe9acdfcaf6","tex":"A_{J'}","aliases":"A_J'","display":"A_{J'}","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-33fae3e04a3d5f9f","tex":"A_{J'}^\\top","aliases":"A_J'^top top","display":"A_{J'}^\\top","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-9227a949b8e25080","tex":"A_J^\\top","aliases":"A_J^top top","display":"A_J^\\top","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":32},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-368d1624fb84e4fe","tex":"\\mathcal B^j","aliases":"B^j mathcal","display":"\\mathcal B^j","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-280215750dd88be1","tex":"\\mathcal B^k","aliases":"B^k mathcal","display":"\\mathcal B^k","kind":"use","excerpt":"[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).","count":16},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-df15704d2f586b97","tex":"\\mathcal B^{k-1-j}","aliases":"B^k-1-j mathcal","display":"\\mathcal B^{k-1-j}","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8b99783075cb661f","tex":"B^\\top","aliases":"B^top top","display":"B^\\top","kind":"use","excerpt":"[Exact range core] The nonzero eigenvalues of K=AA^top are exactly the eigenvalues of BB^top.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2da2815025407827","tex":"\\mathcal B_{\\mathcal O}^j","aliases":"B_ O^j mathcal mathcal","display":"\\mathcal B_{\\mathcal O}^j","kind":"use","excerpt":"The first inequality in the second display is Theorem~[source reference]. For the transfer matrices, the exact noncommutative telescoping identity is B^k- B_ O^k = ∑_j=0^k-1 B^k-1-j ( B- B_ O) B_ O^j. All matrices and ρ_0 are nonnegative, and 0≤ B_ O≤ B entrywise. Hence B_ O^jρ_0 ≤ B^jρ_0. Substituting B- B_ O = ∑_e∈ OΔ_e and distributing the sum g","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-b7349162311a072a","tex":"\\mathcal B_{\\mathcal O}^k","aliases":"B_ O^k mathcal mathcal","display":"\\mathcal B_{\\mathcal O}^k","kind":"use","excerpt":"[Edge-additive leakage majorant] For every omitted set O⊆ E, 0≤ ( B^k- B_ O^k)ρ_0 ≤ ∑_e∈ Oq_e^(k) componentwise. Consequently, lVert(H^k-H_ O^k)XrVert_F ≤ lVert ( B^k- B_ O^k)ρ_0 rVert_2 ≤ ∑_e∈ Oβ_e^(k).","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-56e5c919eaefa876","tex":"\\mathcal B_0","aliases":"B_0 mathcal","display":"\\mathcal B_0","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0, 0, · otherwise, R= B- B_0. Symmetric pair retention preserves self-adjointness of H_0, although the following error theorem does not require positive semidefiniteness of H_0.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-5e42e638d2a47b03","tex":"\\mathcal B_0^j","aliases":"B_0^j mathcal","display":"\\mathcal B_0^j","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d307cf0f04802c95","tex":"\\mathcal B_0^k","aliases":"B_0^k mathcal","display":"\\mathcal B_0^k","kind":"use","excerpt":"[Thresholded-power certificate] For every k≥1, ρ((H^k-H_0^k)X) ≤ ( B^k- B_0^k)ρ(X). In particular, lVert(H^k-H_0^k)XrVert_F ≤ lVert( B^k- B_0^k)ρ(X)rVert_2.","count":8},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8d19b306c01e6077","tex":"\\bar G","aliases":"bar G bar","display":"\\bar G","kind":"use","excerpt":"G_b → G_b-bar G_b → variance scaling → M_X G_b → W_b^1/2 → A_bA_b^top,","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d526da74c961bf64","tex":"\\bar G_b","aliases":"bar G_b bar","display":"\\bar G_b","kind":"use","excerpt":"G_b → G_b-bar G_b → variance scaling → M_X G_b → W_b^1/2 → A_bA_b^top,","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.","count":10},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-db3c7ac2786221e9","tex":"\\mathcal C","aliases":"C mathcal","display":"\\mathcal C","kind":"use","excerpt":"Let E=H-H_0 be the self-adjoint error introduced by phase pruning. Using the same orthogonal phase projectors P_h, define C_hg = P_hEP_g _2.","count":7},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a1a3e5ac585d7cf0","tex":"\\mathcal C_{\\mathcal O}","aliases":"C_ O mathcal mathcal","display":"\\mathcal C_{\\mathcal O}","kind":"use","excerpt":"A production schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c609f244935cbee4","tex":"c_B","aliases":"c_B","display":"c_B","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e78a957e4b60188f","tex":"c_b","aliases":"c_b","display":"c_b","kind":"use","excerpt":"The mathematical streaming identities do not require transfers and GEMMs to be serial. Consider B storage blocks. Let t_b be the transfer time for block b and c_b its device compute time.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-201668608ff331e6","tex":"\\mathcal C_{hg}","aliases":"C_hg mathcal","display":"\\mathcal C_{hg}","kind":"use","excerpt":"Let E=H-H_0 be the self-adjoint error introduced by phase pruning. Using the same orthogonal phase projectors P_h, define C_hg = P_hEP_g _2.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d5b7a071b82101f6","tex":"\\mathcal C_{\\rm complete}","aliases":"C_rm complete mathcal rm","display":"\\mathcal C_{\\rm complete}","kind":"use","excerpt":"D . Consequently, the nonzero singular values of A are exactly the singular values of the stacked completion core C_rm complete = B D .","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8ca82c16234f1275","tex":"\\mathcal D","aliases":"D mathcal","display":"\\mathcal D","kind":"use","excerpt":"Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structur","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f6452e49beb78ef8","tex":"\\mathcal D_{\\rm phase}","aliases":"D_rm phase mathcal rm","display":"\\mathcal D_{\\rm phase}","kind":"use","excerpt":"Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structur","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8b00bd7034dc3ea0","tex":"\\mathcal D_{\\rm rank}","aliases":"D_rm rank mathcal rm","display":"\\mathcal D_{\\rm rank}","kind":"use","excerpt":"Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structur","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2ec52d3af23b660c","tex":"\\mathcal D_{\\rm spectrum}","aliases":"D_rm spectrum mathcal rm","display":"\\mathcal D_{\\rm spectrum}","kind":"use","excerpt":"Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structur","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-db716c254fd2b459","tex":"\\mathcal D_{\\rm storage}","aliases":"D_rm storage mathcal rm","display":"\\mathcal D_{\\rm storage}","kind":"use","excerpt":"Accordingly, the contribution claimed here is not the invention of blocking, streaming, matrix-free multiplication, or Krylov iteration. The candidate contribution is the exact composition of four independent representations: D = D_rm storage ∘ D_rm phase ∘ D_rm rank ∘ D_rm spectrum. The first controls device residency, the second controls structur","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"The resulting production logic is not ``approximate forever.'' It is a certified handoff between two regimes: c streamed / pruned Krylov [2mm] downarrow [-1mm] residual moments (t,v,ρ) [2mm] downarrow [-1mm] continue Krylov, · residual range still large, exact streamed residual completion, · completion rank affordable.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":8},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8687cbd65af8dafc","tex":"e^\\top","aliases":"e^top top","display":"e^\\top","kind":"use","excerpt":"item directly through the factor, Q=frac12 A^top e _2^2; item through the sample kernel, Q=frac12 e^top Ke; item through the eigencoordinates of K.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4d60cc719878240d","tex":"\\mathcal E_0","aliases":"E_0 mathcal","display":"\\mathcal E_0","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":29},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","excerpt":"This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form; the second records how concentrated the r","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-64863d8dd06c95a8","tex":"F^\\top","aliases":"F^top top","display":"F^\\top","kind":"use","excerpt":"[Exact core--residual decomposition] A^top A = B^top B+F^top F. In particular the unresolved spectral term R=F^top F is positive semidefinite.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-575255af81659400","tex":"F_0","aliases":"F_0","display":"F_0","kind":"use","excerpt":"[Exact frontier-to-I/O reduction] Suppose the exact phase graph of H is used and the current Krylov block is supported on a phase set F_0. At depth k, let F_k be the graph k-neighborhood of F_0. Then no storage block lying wholly in a phase outside F_k can affect the k-step Krylov vector. Such a block need not be loaded during that step.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-7e04443bf0b17f7b","tex":"F_k","aliases":"F_k","display":"F_k","kind":"use","excerpt":"[Exact frontier-to-I/O reduction] Suppose the exact phase graph of H is used and the current Krylov block is supported on a phase set F_0. At depth k, let F_k be the graph k-neighborhood of F_0. Then no storage block lying wholly in a phase outside F_k can affect the k-step Krylov vector. Such a block need not be loaded during that step.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":20},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-b355be6ee30169aa","tex":"G_{[:,J_b]}","aliases":"G_[:,J_b]","display":"G_{[:,J_b]}","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-76b53653bff85b9e","tex":"G_b","aliases":"G_b","display":"G_b","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e74ab3b7f0014aa3","tex":"G_b^\\top","aliases":"G_b^top top","display":"G_b^\\top","kind":"use","excerpt":"[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-425652a73ca954b2","tex":"H^k","aliases":"H^k","display":"H^k","kind":"use","excerpt":"[Polyphase light cone] If the graph distance from phase g to phase h exceeds k, then P_hH^kP_g=0.","count":14},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-7f0aa7666470b126","tex":"H^{k-1-j}","aliases":"H^k-1-j","display":"H^{k-1-j}","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-dbdd7c1a3a1ed831","tex":"H_{\\mathcal O}","aliases":"H_ O mathcal","display":"H_{\\mathcal O}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d0c3eb31a2b52ebb","tex":"H_{\\mathcal O}^k","aliases":"H_ O^k mathcal","display":"H_{\\mathcal O}^k","kind":"use","excerpt":"[Edge-additive leakage majorant] For every omitted set O⊆ E, 0≤ ( B^k- B_ O^k)ρ_0 ≤ ∑_e∈ Oq_e^(k) componentwise. Consequently, lVert(H^k-H_ O^k)XrVert_F ≤ lVert ( B^k- B_ O^k)ρ_0 rVert_2 ≤ ∑_e∈ Oβ_e^(k).","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,","count":12},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8eee6f57e815c963","tex":"H_0^j","aliases":"H_0^j","display":"H_0^j","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-defcfc919ed5c175","tex":"H_0^k","aliases":"H_0^k","display":"H_0^k","kind":"use","excerpt":"[Thresholded-power certificate] For every k≥1, ρ((H^k-H_0^k)X) ≤ ( B^k- B_0^k)ρ(X). In particular, lVert(H^k-H_0^k)XrVert_F ≤ lVert( B^k- B_0^k)ρ(X)rVert_2.","count":7},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-097615d1cce1b4b2","tex":"H_b","aliases":"H_b","display":"H_b","kind":"use","excerpt":"A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-766c113d1bfcf920","tex":"H_{b,I_r}","aliases":"H_b,I_r","display":"H_{b,I_r}","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":16},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-37475f8aeaec6cce","tex":"I_1","aliases":"I_1","display":"I_1","kind":"use","excerpt":"A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-5dc7ccbeace33e78","tex":"I_R","aliases":"I_R","display":"I_R","kind":"use","excerpt":"A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e9855187f190ff70","tex":"I_r","aliases":"I_r","display":"I_r","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-111b6ee2c5f8dbd2","tex":"I_s","aliases":"I_s","display":"I_s","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e7d6fe2c83275eaa","tex":"\\inf_{0<2\\theta(c+\\rho)<1}","aliases":"inf_0<2θ(c+ρ)<1 inf theta rho","display":"\\inf_{0<2\\theta(c+\\rho)<1}","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-146cbf287aa37a5f","tex":"\\mathcal J","aliases":"J mathcal","display":"\\mathcal J","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":13},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8d6308571f33e418","tex":"J'","aliases":"J'","display":"J'","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-37959be93056eb7b","tex":"\\mathcal J'","aliases":"J' mathcal","display":"\\mathcal J'","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-08e93c501f7d1688","tex":"J_1","aliases":"J_1","display":"J_1","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-fd3bc8d8a8eb6f8f","tex":"J_B","aliases":"J_B","display":"J_B","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-88fbeb34dbc66890","tex":"J_b","aliases":"J_b","display":"J_b","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":21},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-bfdad9428e6591e0","tex":"K_b","aliases":"K_b","display":"K_b","kind":"use","excerpt":"[H] centering includegraphics[width=.92linewidth]figure_02_exact_block_kernel_assembly.png captionExact block-kernel assembly. The sample interaction kernel carried by each resident variant block is K_b=A_bA_b^top, and the full kernel is their exact sum. Streaming changes the order of evaluation, not the operator.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d7448085f0c3c2e1","tex":"K_C","aliases":"K_C","display":"K_C","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-830b265e1ed762be","tex":"K_H","aliases":"K_H","display":"K_H","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-0d5b2978ec912e70","tex":"K_{I_r,I_s}","aliases":"K_I_r,I_s","display":"K_{I_r,I_s}","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"For an explicitly PSD polyphase operator H=L^top L with block-bidiagonal L, the smallest eigenvalue was approximately 2.86, and the largest norm observed outside the exact k-step light cone was 0.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d750be54985836b3","tex":"L^\\top","aliases":"L^top top","display":"L^\\top","kind":"use","excerpt":"For an explicitly PSD polyphase operator H=L^top L with block-bidiagonal L, the smallest eigenvalue was approximately 2.86, and the largest norm observed outside the exact k-step light cone was 0.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[Residual-rank bracket] lceil max{ fractρ, fract^2v } rceil ≤ rankF ≤ n-r. The zero-residual case is interpreted separately as rank zero.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-973977200963bdff","tex":"\\log_{10}","aliases":"log_10 log","display":"\\log_{10}","kind":"use","excerpt":"log_10p=-457.973916.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Let P_h project onto variant phase h, and again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-25d467671d09a670","tex":"\\lVert","aliases":"lVert","display":"\\lVert","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":15},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":25},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-66fd7e806d93b854","tex":"M^{k-1}","aliases":"M^k-1","display":"M^{k-1}","kind":"use","excerpt":"[Scalar threshold law] For any induced matrix norm, let M=max{lVert BrVert,lVert B_0rVert}. Then lVert B^k- B_0^krVert ≤ kM^k-1lVert RrVert. Hence weak discarded coupling produces a first-order error bound proportional to the discarded transfer norm and to the Krylov depth.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-12fcf6e1ba955880","tex":"M_{\\rm block}","aliases":"M_rm block rm","display":"M_{\\rm block}","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-1a33dd4e78d90d51","tex":"M_{\\rm double}","aliases":"M_rm double rm","display":"M_{\\rm double}","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f89f923e4f377d21","tex":"M_{\\rm fixed}","aliases":"M_rm fixed rm","display":"M_{\\rm fixed}","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a36159e6a5ce2ba9","tex":"M_{\\rm full}","aliases":"M_rm full rm","display":"M_{\\rm full}","kind":"use","excerpt":"For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-17c2d80fc3832b34","tex":"M_{\\rm single}","aliases":"M_rm single rm","display":"M_{\\rm single}","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a41dabcb081dfd4c","tex":"M_{\\rm stream}","aliases":"M_rm stream rm","display":"M_{\\rm stream}","kind":"use","excerpt":"For fixed n,p,s,c, fracM_rm streamM_rm full = O ( fraccm + fracsm + fracpm ), so device memory remains bounded as total variant count grows.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-88849f84a35be77a","tex":"M_X","aliases":"M_X","display":"M_X","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":19},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e37c0b4654d9fb2f","tex":"M_X^2","aliases":"M_X^2","display":"M_X^2","kind":"use","excerpt":"A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-171a34cbb1f0c1b6","tex":"M_X^\\top","aliases":"M_X^top top","display":"M_X^\\top","kind":"use","excerpt":"A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Suppose each retained phase pair has the same communication or scheduling cost. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":21},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[Working memory independent of total variant count] If at most c variants are resident and the block-vector width is s, then the algebraic working set is O(nc+ns+cs+np), independent of total variant count m.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":21},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d1641bdc80ea783c","tex":"P_1","aliases":"P_1","display":"P_1","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"Insert the projector resolution ∑_jP_j=I between successive factors of H. Every nonzero term in P_hH^kP_g corresponds to a phase path of length k. If no such path exists, every term vanishes.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d61b96062b482c72","tex":"P_q","aliases":"P_q","display":"P_q","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-6c033af8552f2dad","tex":"\\Pr","aliases":"Pr","display":"\\Pr","kind":"use","excerpt":"[Stochastic tail enclosure] For every threshold q, Pr(Q_-≥ q) ≤ Pr(Q≥ q) ≤ Pr(Q_+≥ q).","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":20},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2e7d480dd8714698","tex":"Q^\\top","aliases":"Q^top top","display":"Q^\\top","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c364e9f084300ad4","tex":"Q_+","aliases":"Q_+","display":"Q_+","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-22244099d34fb3a0","tex":"Q_-","aliases":"Q_-","display":"Q_-","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-78e11e0a9b0798b5","tex":"q_e^{(d)}","aliases":"q_e^(d)","display":"q_e^{(d)}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-0d15eef406611064","tex":"q_e^{(k)}","aliases":"q_e^(k)","display":"q_e^{(k)}","kind":"use","excerpt":"For depth k≥1 and starting phase profile ρ_0=ρ(X), define the edge risk vector q_e^(k) = ∑_j=0^k-1 B^k-1-jΔ_e B^jρ_0 and its scalar risk β_e^(k)=lVert q_e^(k)rVert_2. Unlike the raw block norm B_hg, this quantity depends on the start profile and on Krylov depth.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-0d86ba69a441f990","tex":"q_e^{(p)}","aliases":"q_e^(p)","display":"q_e^{(p)}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a4210e21e79e5611","tex":"Q_r","aliases":"Q_r","display":"Q_r","kind":"use","excerpt":"Suppose an orthonormal basis Q_r captures the factor range: range(A)⊆range(Q_r). Then A=Q_rB, B=Q_r^top A.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-46a25eef05df8c77","tex":"Q_r^\\top","aliases":"Q_r^top top","display":"Q_r^\\top","kind":"use","excerpt":"Suppose an orthonormal basis Q_r captures the factor range: range(A)⊆range(Q_r). Then A=Q_rB, B=Q_r^top A.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-b55754320a8637cb","tex":"Q_X","aliases":"Q_X","display":"Q_X","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-138c7f7e777f2927","tex":"Q_{X,I_r}","aliases":"Q_X,I_r","display":"Q_{X,I_r}","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-df947929914c1caa","tex":"Q_{X,I_r}^\\top","aliases":"Q_X,I_r^top top","display":"Q_{X,I_r}^\\top","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8843b5276a2a8f90","tex":"Q_X^\\top","aliases":"Q_X^top top","display":"Q_X^\\top","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-79e670199e387f36","tex":"Q_\\perp","aliases":"Q_⊥ perp","display":"Q_\\perp","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"A variant block may itself exceed device memory. Let H_b=G_bW_b^1/2 and partition samples into row tiles I_1,…,I_R.","count":20},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0, 0, · otherwise, R= B- B_0. Symmetric pair retention preserves self-adjointness of H_0, although the following error theorem does not require positive semidefiniteness of H_0.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":34},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2171bce1de561fa0","tex":"R^2","aliases":"R^2","display":"R^2","kind":"use","excerpt":"This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form; the second records how concentrated the r","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-55271cfa55ee85c8","tex":"r^2","aliases":"r^2","display":"r^2","kind":"use","excerpt":"[H] centering captionRealistic rare-variant simulation scenarios. Adjacent r^2 is computed within the simulated LD blocks after variant standardization. lrrrrr scenario · n · m · rank · median adjacent r^2 · effective rank","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-3a7615ff2031245c","tex":"\\mathbb R^{m\\times s}","aliases":"R^m× s mathbb times","display":"\\mathbb R^{m\\times s}","kind":"use","excerpt":"Let V∈ R^m× s be partitioned conformally: V= V_1","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-772c490b66e2c0ea","tex":"\\mathbb R^{n\\times m}","aliases":"R^n× m mathbb times","display":"\\mathbb R^{n\\times m}","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-dbebc888329e8fe1","tex":"\\mathbb R^{n\\times p}","aliases":"R^n× p mathbb times","display":"\\mathbb R^{n\\times p}","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-25213bd56a1923f1","tex":"\\mathbb R^{n\\times r}","aliases":"R^n× r mathbb times","display":"\\mathbb R^{n\\times r}","kind":"use","excerpt":"Let U∈ R^n× r have orthonormal columns and define B=U^top A, F=(I-UU^top)A. Then A=UB+F, U^top F=0.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c06cbcc9be02a47a","tex":"\\mathbb R^{n\\times s}","aliases":"R^n× s mathbb times","display":"\\mathbb R^{n\\times s}","kind":"use","excerpt":"[One-pass sample-space streaming action] For every block vector V∈ R^n× s, KV = M_X ∑_b=1^B G_bW_bG_b^top M_XV.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-baf9ce2c27891e4e","tex":"R_b","aliases":"R_b","display":"R_b","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-ade69a4d18e62b84","tex":"r_g","aliases":"r_g","display":"r_g","kind":"use","excerpt":"For any vector x, let r_g= P_gx _2. Then P_hEx _2 ≤ ∑_g P_hEP_g _2 P_gx _2 = ( C r)_h. Orthogonality of the phase decomposition gives Ex _2^2 = ∑_h P_hEx _2^2 ≤ C r _2^2 ≤ C _2^2 r _2^2 = C _2^2 x _2^2. Taking the supremum over unit vectors proves the claim.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-b8db6823423681b3","tex":"R_h","aliases":"R_h","display":"R_h","kind":"use","excerpt":"Let R be a newly generated Krylov residual and R_h=P_hR.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-455a3f2c7c575c0e","tex":"r_h","aliases":"r_h","display":"r_h","kind":"use","excerpt":"[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d19e0f8ac22a9168","tex":"r_{hg}","aliases":"r_hg","display":"r_{hg}","kind":"use","excerpt":"If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d0a12b6c6c489d99","tex":"r_i^2","aliases":"r_i^2","display":"r_i^2","kind":"use","excerpt":"The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-0876bfe73de67b89","tex":"r_s","aliases":"r_s","display":"r_s","kind":"use","excerpt":"The same invariants determine when it is better to stop approximating the missing spectrum and complete it exactly. Let s=rankF=rankR. If the nonzero eigenvalues of R are r_1,…,r_s, then t=∑_i r_i, v=∑_i r_i^2, r_i≤ρ.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-268aa2b4269fd3ac","tex":"R_\\perp","aliases":"R_⊥ perp","display":"R_\\perp","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-9bf28df79e66226e","tex":"\\operatorname{range}","aliases":"range operatorname","display":"\\operatorname{range}","kind":"use","excerpt":"[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"[Minimal exact phase width] Let r_h=rank(R_h). Every phase-supported basis spanning range(R_h) has at least r_h vectors, and a rank-revealing QR or compact SVD attains exactly r_h.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[Residual-rank bracket] lceil max{ fractρ, fract^2v } rceil ≤ rankF ≤ n-r. The zero-residual case is interpreted separately as rank zero.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-61f9609b70de73a0","tex":"\\rVert","aliases":"rVert","display":"\\rVert","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":15},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c1e5fffeb9eae9df","tex":"\\rVert_2","aliases":"rVert_2 rVert","display":"\\rVert_2","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-897e5303d2e01253","tex":"\\rVert_F","aliases":"rVert_F rVert","display":"\\rVert_F","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-ca9d46208a25ce8a","tex":"\\rVert_F^2","aliases":"rVert_F^2 rVert","display":"\\rVert_F^2","kind":"use","excerpt":"ing Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"[One-pass sample-space streaming action] For every block vector V∈ R^n× s, KV = M_X ∑_b=1^B G_bW_bG_b^top M_XV.","count":22},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"[Storage-phase commutation theorem] Let A=M_XGW^1/2, H=A^top A, and let S and S' be any two exact storage refinements that respect the same structural phase partition. Then: [label=(roman*)] item the exact operator H is identical under S and S'; item every phase coupling P_hHP_g is identical under S and S'; item the exact phase graph and every k-st","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-3f7a7b0d9ff643c3","tex":"\\succeq","aliases":"succeq","display":"\\succeq","kind":"use","excerpt":"Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"With sample tile height t and resident variant width c, the dominant working storage is O(tc+pc+t^2).","count":23},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"Factor the determinant through the known core: I-2θ(C+R) = (I-2θ C)^1/2 (I-2θ T) (I-2θ C)^1/2, where T=(I-2θ C)^-1/2 R (I-2θ C)^-1/2. Since (I-2θ C)^-1/2 _2^2=α, we have trT≤α t, tr(T^2)≤α^2v, T _2≤αρ. For every eigenvalue x of T, -frac12log(1-2θ x) ≤ θ x + fracθ^2x^2 1-2θαρ. The scalar inequality follows from the power series of -log(1-u) and a ge","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"use","excerpt":"With sample tile height t and resident variant width c, the dominant working storage is O(tc+pc+t^2).","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f544e1fddefb59a0","tex":"t^d","aliases":"t^d","display":"t^d","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-68c16f5dbc93324c","tex":"t^j","aliases":"t^j","display":"t^j","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-be8dcefb61dddd9f","tex":"t_1","aliases":"t_1","display":"t_1","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-cefa01be96058c3e","tex":"T_b","aliases":"T_b","display":"T_b","kind":"use","excerpt":"[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","excerpt":"The mathematical streaming identities do not require transfers and GEMMs to be serial. Consider B storage blocks. Let t_b be the transfer time for block b and c_b its device compute time.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e0d4b679b4a2b87a","tex":"t_{b+1}","aliases":"t_b+1","display":"t_{b+1}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-529849886462f9b6","tex":"T_{\\rm pipe}","aliases":"T_rm pipe rm","display":"T_{\\rm pipe}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8b06363b18c1a84d","tex":"T_{\\rm serial}","aliases":"T_rm serial rm","display":"T_{\\rm serial}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-6e51d2fa53a6aa7c","tex":"\\operatorname{tr}","aliases":"tr operatorname","display":"\\operatorname{tr}","kind":"use","excerpt":"This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form; the second records how concentrated the r","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.","count":30},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"If one resident block uses M_rm block bytes and all other fixed working state uses M_rm fixed bytes, the corresponding memory models are M_rm single=M_rm fixed+M_rm block, M_rm double=M_rm fixed+2M_rm block. The proposition is a scheduling model, not a hardware timing claim. PCIe or NVLink contention, kernel launch overhead, synchronization, page p","count":7},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d813568f15ad058f","tex":"U^\\top","aliases":"U^top top","display":"U^\\top","kind":"use","excerpt":"There is a second route that does not use the pruned operator as the final spectral approximation. Let U be any orthonormal sample-space basis, possibly generated cheaply from a pruned or approximate Krylov process, and define B=U^top A, F=(I-UU^top)A, δ= F _2.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-685bf0a5c43d8ed4","tex":"U_{hg}","aliases":"U_hg","display":"U_{hg}","kind":"use","excerpt":"If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-92359e5cbd8a18d0","tex":"u_i","aliases":"u_i","display":"u_i","kind":"use","excerpt":"[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"[One-pass sample-space streaming action] For every block vector V∈ R^n× s, KV = M_X ∑_b=1^B G_bW_bG_b^top M_XV.","count":22},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"This changes the certification problem. Instead of an arbitrary Hermitian error, the missing term is a positive residual with measurable spectral moments. Define t=trR, v=tr(R^2), ρ= R _2. Then t= F _F^2, v= FF^top _F^2, ρ= F _2^2. The first quantity is the exact missing mean of the Gaussian quadratic form; the second records how concentrated the r","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"Let V∈ R^m× s be partitioned conformally: V= V_1 vdots V_B .","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-9f5cc10c11d8d9e7","tex":"V_B","aliases":"V_B","display":"V_B","kind":"use","excerpt":"Let V∈ R^m× s be partitioned conformally: V= V_1 vdots V_B .","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2d8b529eb2a0a517","tex":"V_b","aliases":"V_b","display":"V_b","kind":"use","excerpt":"[Streamed AV] AV = M_X ∑_b=1^B G_bW_b^1/2V_b.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2199b487ebbba079","tex":"V_{hg}^\\top","aliases":"V_hg^top top","display":"V_{hg}^\\top","kind":"use","excerpt":"If an inter-phase coupling has exact factorization P_hHP_g = U_hgV_hg^top of rank r_hg, then P_hHP_gx = U_hg(V_hg^top x).","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-44cf5118c3be9cba","tex":"\\vdots","aliases":"vdots","display":"\\vdots","kind":"use","excerpt":"Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":23},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-148db1d69b066230","tex":"W^{1/2}","aliases":"W^1/2","display":"W^{1/2}","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8e80dcfdce0dc44d","tex":"W_b","aliases":"W_b","display":"W_b","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-56f8bedcef5fce45","tex":"W_b^{1/2}","aliases":"W_b^1/2","display":"W_b^{1/2}","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":7},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f1e02a32b6b73575","tex":"W_{J_b,J_b}","aliases":"W_J_b,J_b","display":"W_{J_b,J_b}","kind":"use","excerpt":"Partition the variant indices into disjoint blocks {1,…,m} = J_1sqcup⋯sqcup J_B. Let G_b=G_[:,J_b], W_b=W_J_b,J_b, A_b=M_XG_bW_b^1/2. Then A=[A_1 ⋯ A_B].","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a6dc04e23090a2a5","tex":"w_m","aliases":"w_m","display":"w_m","kind":"use","excerpt":"Let G∈ R^n× m, W=diag(w_1,…,w_m), w_j≥ 0, and let Q_X∈ R^n× p have orthonormal columns spanning the null-model covariates. Define M_X=I-Q_XQ_X^top, A=M_XGW^1/2, K=AA^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-5450287e32dd956c","tex":"\\widehat K","aliases":"widehat K widehat","display":"\\widehat K","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-1ec78fa50c736ccb","tex":"\\widehat K_H","aliases":"widehat K_H widehat","display":"\\widehat K_H","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":48},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"ginproof Factor the determinant through the known core: I-2θ(C+R) = (I-2θ C)^1/2 (I-2θ T) (I-2θ C)^1/2, where T=(I-2θ C)^-1/2 R (I-2θ C)^-1/2. Since (I-2θ C)^-1/2 _2^2=α, we have trT≤α t, tr(T^2)≤α^2v, T _2≤αρ. For every eigenvalue x of T, -frac12log(1-2θ x) ≤ θ x + fracθ^2x^2 1-2θαρ. The scalar inequality follows from the power series of -log(1-u)","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.","count":6},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":7},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-532408f4f67397a2","tex":"Z^\\top","aliases":"Z^top top","display":"Z^\\top","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-bc1f10031646b5b4","tex":"Z_i","aliases":"Z_i","display":"Z_i","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-86b443a234736cf2","tex":"Z_i^2","aliases":"Z_i^2","display":"Z_i^2","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"Sequence kernel association testing (SKAT) is a variance-component framework for set-based rare-variant association testing [source reference]. Large rare-variant and kernel tests often begin with a weighted factor A=M_XGW^1/2, K=AA^top, where G∈ R^n× m is a genotype or feature matrix, W is diagonal and nonnegative, and M_X projects away null-model","count":34},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"Set C=B^top B, H=C+R, c=λ_max(C). For 0<2θ(c+ρ)<1, put α=(1-2θ c)^-1.","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d830d9bc7e9c803c","tex":"\\alpha^2","aliases":"α^2 alpha","display":"\\alpha^2","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-eabc276ff3dd2e25","tex":"\\beta_{(1)}","aliases":"β_(1) beta","display":"\\beta_{(1)}","kind":"use","excerpt":"Suppose each retained phase pair has the same communication or scheduling cost. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f889f38b64a54aed","tex":"\\beta_{(2)}","aliases":"β_(2) beta","display":"\\beta_{(2)}","kind":"use","excerpt":"Suppose each retained phase pair has the same communication or scheduling cost. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-fd8f34ecec6238ae","tex":"\\beta_{(i)}","aliases":"β_(i) beta","display":"\\beta_{(i)}","kind":"use","excerpt":"[Cardinality-optimal certified pruning] Let r be the largest integer satisfying ∑_i=1^rβ_(i)≤τ. Then omitting the r least-risk phase pairs is feasible, and no feasible omitted set can contain more than r pairs under the scalar additive certificate.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-3138ed3758d4a624","tex":"\\beta_{(N)}","aliases":"β_(N) beta","display":"\\beta_{(N)}","kind":"use","excerpt":"Suppose each retained phase pair has the same communication or scheduling cost. Under the additive certificate, a set of omitted pairs is feasible if ∑_e∈ Oβ_e^(k)≤τ. Sort the risks as β_(1)≤β_(2)≤⋯≤β_(N).","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-3cf40acf5b62ff4c","tex":"\\beta_e^{(p)}","aliases":"β_e^(p) beta","display":"\\beta_e^{(p)}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-74cc4a0ac89e6235","tex":"\\beta_{hg}^{(5)}","aliases":"β_hg^(5) beta","display":"\\beta_{hg}^{(5)}","kind":"use","excerpt":"[H] centering includegraphics[width=.82linewidth]figure_13_edge_risk_vs_block_norm.png captionRaw phase-block magnitude and finite-depth query risk are different quantities. The edge score β_hg^(5) incorporates the start profile and every depth-five transfer path through the phase graph; the block norm B_hg does not.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"Set C=B^top B, H=C+R, c=λ_max(C). For 0<2θ(c+ρ)<1, put α=(1-2θ c)^-1.","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2b3930db632609a8","tex":"\\theta^2","aliases":"θ^2 theta","display":"\\theta^2","kind":"use","excerpt":"[Trace--Frobenius--cap log-MGF certificate] For the Gaussian quadratic form with coefficient matrix H, K_H(θ) =-frac12logdet(I-2θ H) satisfies K_H(θ) ≤ K_C(θ) + θα t + fracθ^2α^2v 1-2θαρ.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8b63c02e309e1522","tex":"\\lambda_j","aliases":"λ_j lambda","display":"\\lambda_j","kind":"use","excerpt":"The matrix packets above operate before eigendecomposition. They should be distinguished from the spectral packets used after the eigenvalues λ_j are known.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-22452671e06f3b7f","tex":"\\lambda_{\\max}","aliases":"λ_max lambda max","display":"\\lambda_{\\max}","kind":"use","excerpt":"Set C=B^top B, H=C+R, c=λ_max(C). For 0<2θ(c+ρ)<1, put α=(1-2θ c)^-1.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-252454ad397ec5a8","tex":"\\mu_i","aliases":"μ_i mu","display":"\\mu_i","kind":"use","excerpt":"Assume Hsucceq0 and let the eigenvalues of H and H_0, in decreasing order, be λ_i and μ_i. If H-H_0 _2≤ε, Weyl's inequality gives |λ_i-μ_i|≤ε.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"Let the phase projectors be P_1,…,P_q, and let X be a vector or block vector. Define its phase-norm profile ρ(X)_g=lVert P_gXrVert_F, and define the nonnegative phase-transfer matrix B_hg=lVert P_hHP_grVert_2. The matrix B records the strength of every one-step phase transfer without assuming that weak couplings vanish.","count":35},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","excerpt":"For depth k≥1 and starting phase profile ρ_0=ρ(X), define the edge risk vector q_e^(k) = ∑_j=0^k-1 B^k-1-jΔ_e B^jρ_0 and its scalar risk β_e^(k)=lVert q_e^(k)rVert_2. Unlike the raw block norm B_hg, this quantity depends on the start profile and on Krylov depth.","count":9},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4a895ee28e0d67c3","tex":"\\tau_K","aliases":"τ_K tau","display":"\\tau_K","kind":"use","excerpt":"A production schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-7a18b4081386ab4c","tex":"\\tau_\\lambda","aliases":"τ_λ tau lambda","display":"\\tau_\\lambda","kind":"use","excerpt":"A production schedule should therefore enforce ∑_e∈ Oβ_e^(k) ≤ τ_K, C_ O _2 ≤ τ_λ.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"er composes with an exact polyphase light-cone theorem, minimal phase-local rank allocation, low-rank interface messages, and exact QR-core updates. When phase couplings are weak rather than zero, a nonnegative transfer matrix gives certified phasewise leakage bounds for operator powers and complete Krylov polynomials. A dual query/spectral budget ","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-11fdaa25e327864a","tex":"\\chi^2","aliases":"χ^2 chi","display":"\\chi^2","kind":"use","excerpt":"er composes with an exact polyphase light-cone theorem, minimal phase-local rank allocation, low-rank interface messages, and exact QR-core updates. When phase couplings are weak rather than zero, a nonnegative transfer matrix gives certified phasewise leakage bounds for operator powers and complete Krylov polynomials. A dual query/spectral budget ","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.","count":4},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f71176b4c504c459","tex":"\\ell_2","aliases":"ℓ_2 ell","display":"\\ell_2","kind":"use","excerpt":"The QR-core update reproduced the full singular spectrum with relative ℓ_2 discrepancy 5.89×10^-16.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f9c5c5b39e3639f2","tex":"\\ell_i","aliases":"ℓ_i ell","display":"\\ell_i","kind":"use","excerpt":"[Positive spectral interval] For every i, ℓ_i = max{0,μ_i-ε} ≤ λ_i ≤ u_i = max{0,μ_i+ε}.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"use","excerpt":"[Streamed KV] [label=arabic*.] item Compute U=M_XV. item Set Y=0. item For each resident block b, compute T_b=G_b^top U, Y← Y+G_b(W_bT_b). item Return M_XY.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Let P_h project onto variant phase h, and again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"For every phase h∉ F_k and every starting phase g∈ F_0, the polyphase light-cone theorem gives P_hH^kP_g=0. Storage-phase commutation allows each active phase operator to be evaluated through arbitrary exact storage blocks. Therefore blocks outside the active structural frontier contribute identically zero to the requested iterate and may be omitte","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2f0d826a23cff58c","tex":"\\sum_{(h,g)\\in\\mathcal E_0}","aliases":"∑_(h,g)∈ E_0 sum in mathcal","display":"\\sum_{(h,g)\\in\\mathcal E_0}","kind":"use","excerpt":"Now choose a symmetric set of retained phase blocks and define H_0=∑_(h,g)∈ E_0P_hHP_g. Let ( B_0)_hg= B_hg, · (h,g)∈ E_0,","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4b1e8504628c9f8f","tex":"\\sum_b","aliases":"∑_b sum","display":"\\sum_b","kind":"use","excerpt":"Horizontal concatenation gives [A_1 ⋯ A_B] A_1^top vdots A_B^top = ∑_bA_bA_b^top. The final form follows from A_b=M_XG_bW_b^1/2 and M_X^top=M_X=M_X^2.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-d724b548ac80a603","tex":"\\sum_{b=1}^B","aliases":"∑_b=1^B sum","display":"\\sum_{b=1}^B","kind":"use","excerpt":"[Additive block Gram identity] K = AA^top = ∑_b=1^B A_bA_b^top = M_X ( ∑_b=1^B G_bW_bG_b^top ) M_X.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4e3e65f97e51859c","tex":"\\sum_{b=1}^{B}","aliases":"∑_b=1^B sum","display":"\\sum_{b=1}^{B}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-ce18b6ca08050486","tex":"\\sum_{b=1}^{B-1}","aliases":"∑_b=1^B-1 sum","display":"\\sum_{b=1}^{B-1}","kind":"use","excerpt":"[Ideal two-buffer pipeline] In a model with perfect asynchronous overlap, transfer of block b+1 may run while block b is being processed. The serial one-buffer time is T_rm serial=∑_b=1^B(t_b+c_b), whereas the two-buffer pipeline time is T_rm pipe = t_1 + ∑_b=1^B-1max{c_b,t_b+1} + c_B. For uniform times t_b=t and c_b=c, T_rm pipe=t+c+(B-1)max{t,c}.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2fb84539c83f1f39","tex":"\\sum_{d=0}^{D}","aliases":"∑_d=0^D sum","display":"\\sum_{d=0}^{D}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-06344775b94820a0","tex":"\\sum_{d=1}^{D}","aliases":"∑_d=1^D sum","display":"\\sum_{d=1}^{D}","kind":"use","excerpt":"[Polynomial edge-risk certificate] Let p(t)=∑_d=0^Da_dt^d. Define q_e^(p) = ∑_d=1^D|a_d|q_e^(d), β_e^(p)=lVert q_e^(p)rVert_2. Then lVert(p(H)-p(H_ O))XrVert_F ≤ ∑_e∈ Oβ_e^(p). Thus the same scheduling machinery applies to finite Krylov combinations and polynomial filters.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-76e758f1be5158a2","tex":"\\sum_g","aliases":"∑_g sum","display":"\\sum_g","kind":"use","excerpt":"Using ∑_gP_g=I and submultiplicativity, lVert P_hHXrVert_F · =lVert∑_gP_hHP_gXrVert_F","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-20794c41f8dd0413","tex":"\\sum_h","aliases":"∑_h sum","display":"\\sum_h","kind":"use","excerpt":"ing Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonality of the phase projectors gives lVert YrVert_F^2=∑_hlVert P_hYrVert_F^2, which proves the global bound.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"For independent standard normal variables Z_i, define Q_-=∑_iℓ_iZ_i^2, Q=∑_iλ_iZ_i^2, Q_+=∑_i u_iZ_i^2.","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2874dc45e47106fd","tex":"\\sum_{i=1}^{r}","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^{r}","kind":"use","excerpt":"[Cardinality-optimal certified pruning] Let r be the largest integer satisfying ∑_i=1^rβ_(i)≤τ. Then omitting the r least-risk phase pairs is feasible, and no feasible omitted set can contain more than r pairs under the scalar additive certificate.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-e4c6762a80e52f72","tex":"\\sum_{i=1}^{r+1}","aliases":"∑_i=1^r+1 sum","display":"\\sum_{i=1}^{r+1}","kind":"use","excerpt":"Among all subsets of cardinality r+1, the smallest possible sum of risks is ∑_i=1^r+1β_(i). By maximality of r, that sum exceeds τ. Hence no set of r+1 omitted pairs is feasible. The prefix of length r is feasible by construction.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-15292da14c260471","tex":"\\sum_{i=1}^s","aliases":"∑_i=1^s sum","display":"\\sum_{i=1}^s","kind":"use","excerpt":"The inequality t≤ sρ gives s≥ t/ρ. Cauchy--Schwarz gives t^2 = (∑_i=1^s r_i)^2 ≤ s∑_i=1^s r_i^2 =sv, so s≥ t^2/v. Finally, every column of F lies in range(U)^⊥, whose dimension is n-r.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Insert the projector resolution ∑_jP_j=I between successive factors of H. Every nonzero term in P_hH^kP_g corresponds to a phase path of length k. If no such path exists, every term vanishes.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-b98158348cda32a4","tex":"\\sum_{J'\\in\\mathcal J'}","aliases":"∑_J'∈ J' sum in mathcal","display":"\\sum_{J'\\in\\mathcal J'}","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-b845f4dcb81f67c5","tex":"\\sum_{j=0}^d","aliases":"∑_j=0^d sum","display":"\\sum_{j=0}^d","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-02998aeb731bff1f","tex":"\\sum_{j=0}^{k-1}","aliases":"∑_j=0^k-1 sum","display":"\\sum_{j=0}^{k-1}","kind":"use","excerpt":"Write E=H-H_0. The telescoping identity gives H^k-H_0^k = ∑_j=0^k-1H^k-1-jEH_0^j. The block-norm matrix of E is R. Applying Theorem~[source reference] successively to each term gives ρ((H^k-H_0^k)X) ≤ ∑_j=0^k-1 B^k-1-j R B_0^jρ(X). The matrix telescoping identity B^k- B_0^k = ∑_j=0^k-1 B^k-1-j( B- B_0) B_0^j yields the phasewise formula. Orthogonal","count":5},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-dec62b92295c8c6d","tex":"\\sum_{j=1}^d","aliases":"∑_j=1^d sum","display":"\\sum_{j=1}^d","kind":"use","excerpt":"[Krylov-polynomial certificate] Let p(t)=∑_j=0^d a_jt^j. Then ρ((p(H)-p(H_0))X) ≤ ∑_j=1^d |a_j|( B^j- B_0^j)ρ(X).","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-bea1c4ac9cd0066f","tex":"\\sum_{J\\in\\mathcal J}","aliases":"∑_J∈ J sum in mathcal","display":"\\sum_{J\\in\\mathcal J}","kind":"use","excerpt":"[Partition-refinement invariance] Let J be any partition of the variants and let J' refine J. Then ∑_J∈ JA_JA_J^top = ∑_J'∈ J'A_J'A_J'^top = AA^top. The analogous streamed operator action is also invariant.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-a96ab7f55c668d6e","tex":"\\sum_r","aliases":"∑_r sum","display":"\\sum_r","kind":"use","excerpt":"[Two-pass residualized tile construction] Define R_b=Q_X^top H_b. Then R_b = ∑_r Q_X,I_r^top H_b,I_r, and after R_b is known, A_b,I_r = H_b,I_r-Q_X,I_rR_b. Consequently, K_I_r,I_s = ∑_b A_b,I_rA_b,I_s^top.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"The central implementation difficulty is not the algebraic size of the model but the mismatch between the total factor size n× m and the working memory of an accelerator. The key point of this manuscript is that several decompositions act on different mathematical coordinates and therefore compose: storage blocks ∘ structural phases ∘ Krylov/range ","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"[Trace-aware Chernoff upper tail] For Q=Z^top HZ with Z∼ N(0,I), Pr(Q≥ q) ≤ inf_0<2θ(c+ρ)<1 exp{widehat K_H(θ)-θ q}, where widehat K_H is the preceding MGF upper bound.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Let P_h project onto variant phase h, and again set H=A^top A. Define the exact phase graph by g→ h Longleftrightarrow P_hHP_g≠0.","count":1},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).","count":10},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"[Weighted phase-transfer envelope] For every block vector X, ρ(HX)≤ Bρ(X) componentwise. Consequently, for every integer k≥0, ρ(H^kX)≤ B^kρ(X).","count":2},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"use","excerpt":"Suppose A=QR, Q^top Q=I, and append a new block U. Set C=Q^top U, F=U-QC, F=Q_⊥ R_⊥.","count":3},{"work":"streaming-skat","url":"notation/streaming-skat.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"This is the key composition law: mathematical locality ⟶ active phases ⟶ active storage blocks.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":257},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.","count":80},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":33},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0083f442b19187bb","tex":"A_v","aliases":"A_v","display":"A_v","kind":"definition-passage","excerpt":"[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-10f7880ac19fa1db","tex":"\\Adele","aliases":"Adele","display":null,"kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f3c47f030f4f5ce9","tex":"\\Adele_E","aliases":"Adele_E Adele","display":null,"kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ec1d896b92ecf03e","tex":"\\Adele_F","aliases":"Adele_F Adele","display":null,"kind":"definition-passage","excerpt":"[Finite Satake cusp form] A cuspidal representation of GL_n(Adele_F) is finite Satake when, outside one finite exceptional set, its normalized unramified Satake roots lie in one finite subgroup of CC^×. The exceptional set is fixed, not allowed to change separately for each coefficient or prime.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"definition-passage","excerpt":"[Universal bounded cyclotomic alphabet] For positive integers n and B, let calQ_n,B be the finite set of monic degree n polynomials whose roots are roots of unity of order at most B.","count":31},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":42},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-66de8b9acff08828","tex":"C_{\\calE}","aliases":"C_calE calE","display":null,"kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-63c53ef95ec79d76","tex":"\\calA","aliases":"calA","display":null,"kind":"definition-passage","excerpt":"[Finite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.","count":10},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ee14ca9c20d81274","tex":"\\calA_{\\pi}","aliases":"calA_π calA pi","display":null,"kind":"definition-passage","excerpt":"[Finite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-489cce3969fff893","tex":"\\calA_{\\rho}","aliases":"calA_ρ calA rho","display":null,"kind":"definition-passage","excerpt":"[Finite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-69664f5987ec91b4","tex":"\\calE","aliases":"calE","display":null,"kind":"definition-passage","excerpt":"[Entire packet set] An entire packet set calE is a collection of characters of subgroups of G whose Artin L functions are known to be entire.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ff241bb823072cbc","tex":"\\calJ","aliases":"calJ","display":null,"kind":"definition-passage","excerpt":"[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-38e464c9f261a0e0","tex":"\\calJ_H","aliases":"calJ_H calJ","display":null,"kind":"definition-passage","excerpt":"[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a032723c2a4de34a","tex":"\\calJ_v","aliases":"calJ_v calJ","display":null,"kind":"definition-passage","excerpt":"[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-645c21011e746ec3","tex":"\\calQ","aliases":"calQ","display":null,"kind":"definition-passage","excerpt":"[Universal bounded cyclotomic alphabet] For positive integers n and B, let calQ_n,B be the finite set of monic degree n polynomials whose roots are roots of unity of order at most B.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6d4648085a7141b8","tex":"\\calQ_{n,B}","aliases":"calQ_n,B calQ","display":null,"kind":"definition-passage","excerpt":"[Universal bounded cyclotomic alphabet] For positive integers n and B, let calQ_n,B be the finite set of monic degree n polynomials whose roots are roots of unity of order at most B.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-167caac527a0def7","tex":"\\CC","aliases":"CC","display":null,"kind":"definition-passage","excerpt":"[Finite Satake cusp form] A cuspidal representation of GL_n(Adele_F) is finite Satake when, outside one finite exceptional set, its normalized unramified Satake roots lie in one finite subgroup of CC^×. The exceptional set is fixed, not allowed to change separately for each coefficient or prime.","count":14},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ce2f5577f99d3780","tex":"\\CC^\\times","aliases":"CC^× CC times","display":null,"kind":"definition-passage","excerpt":"[Finite Satake cusp form] A cuspidal representation of GL_n(Adele_F) is finite Satake when, outside one finite exceptional set, its normalized unramified Satake roots lie in one finite subgroup of CC^×. The exceptional set is fixed, not allowed to change separately for each coefficient or prime.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":45},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"definition-passage","excerpt":"[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.","count":34},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-542812c54fe5f147","tex":"e_k","aliases":"e_k","display":"e_k","kind":"definition-passage","excerpt":"[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":69},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-78b29d109052da32","tex":"F_v","aliases":"F_v","display":"F_v","kind":"definition-passage","excerpt":"Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v ","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":97},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"definition-passage","excerpt":"[Stable and moving types] An irreducible representation τ∈Irr(H) is stable when τ^g≅τ for every g∈ G. Otherwise its orbit has size p and is called moving.","count":66},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4855ea42dd65b6f9","tex":"G_E","aliases":"G_E","display":"G_E","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":10},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2f3f7861c9f87c46","tex":"G_F","aliases":"G_F","display":"G_F","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":17},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f9b55a29a191e028","tex":"\\GL","aliases":"GL","display":null,"kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":27},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5e917386367b6fdc","tex":"\\GL_n","aliases":"GL_n GL","display":null,"kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":26},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"definition-passage","excerpt":"[Stable and moving types] An irreducible representation τ∈Irr(H) is stable when τ^g≅τ for every g∈ G. Otherwise its orbit has size p and is called moving.","count":85},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom","display":null,"kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4671b65256411ac1","tex":"\\Hom_H","aliases":"Hom_H Hom","display":null,"kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"definition-passage","excerpt":"[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.","count":39},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-36282b37d83ec448","tex":"\\Irr","aliases":"Irr","display":null,"kind":"definition-passage","excerpt":"[Stable and moving types] An irreducible representation τ∈Irr(H) is stable when τ^g≅τ for every g∈ G. Otherwise its orbit has size p and is called moving.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"definition-passage","excerpt":"[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.","count":60},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"definition-passage","excerpt":"[Separating coefficient prime] A finite place λ of K is separating for calA when reduction modulo λ is injective on calA.","count":30},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.","count":37},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"definition-passage","excerpt":"[Entire packet set] An entire packet set calE is a collection of characters of subgroups of G whose Artin L functions are known to be entire.","count":31},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":38},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":30},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4905688d877152a0","tex":"M_\\tau","aliases":"M_τ tau","display":"M_\\tau","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e201b5639026c7b4","tex":"m_{\\tau,a}","aliases":"m_τ,a tau","display":"m_{\\tau,a}","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":114},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"definition-passage","excerpt":"[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and it","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3126ea25a64cf272","tex":"\\overline k","aliases":"overline k overline","display":"\\overline k","kind":"definition-passage","excerpt":"[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e2fc6e5e5408b005","tex":"\\overline k_{\\lambda}","aliases":"overline k_λ overline lambda","display":"\\overline k_{\\lambda}","kind":"definition-passage","excerpt":"[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-41dacfd07761cf94","tex":"\\overline r","aliases":"overline r overline","display":"\\overline r","kind":"definition-passage","excerpt":"[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.","count":10},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-bc83ef9f3af83949","tex":"\\overline r_{\\pi,\\lambda}","aliases":"overline r_π,λ overline pi lambda","display":"\\overline r_{\\pi,\\lambda}","kind":"definition-passage","excerpt":"[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":29},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":162},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e9f8c83a17d2d639","tex":"P^{[p]}","aliases":"P^[p]","display":"P^{[p]}","kind":"definition-passage","excerpt":"[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b42faa8df7273d1f","tex":"P_{\\tau,V}","aliases":"P_τ,V tau","display":"P_{\\tau,V}","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":8},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-15af35cf21e3082c","tex":"P_{\\tau,W}","aliases":"P_τ,W tau","display":"P_{\\tau,W}","kind":"definition-passage","excerpt":"[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-221aa401f4b73008","tex":"\\QQ","aliases":"QQ","display":null,"kind":"definition-passage","excerpt":"[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and it","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-26bae0d86c26f15b","tex":"\\QQ_\\ell","aliases":"QQ_ℓ QQ ell","display":null,"kind":"definition-passage","excerpt":"[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and it","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"definition-passage","excerpt":"[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.","count":29},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)","count":122},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-257791562a17d424","tex":"R_v","aliases":"R_v","display":"R_v","kind":"definition-passage","excerpt":"[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f997e5b1308ec8c8","tex":"\\rec","aliases":"rec","display":null,"kind":"definition-passage","excerpt":"Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v ","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3b8736a7982da0f6","tex":"\\rec_{F_v}","aliases":"rec_F_v rec","display":null,"kind":"definition-passage","excerpt":"Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v ","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"definition-passage","excerpt":"[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.","count":18},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-145837c99a5c8f54","tex":"\\operatorname{Sat}","aliases":"Sat operatorname","display":"\\operatorname{Sat}","kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"definition-passage","excerpt":"[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.","count":46},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":66},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"definition-passage","excerpt":"Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v ","count":71},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"definition-passage","excerpt":"Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v ","count":35},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-12bdcbd357b105f0","tex":"W_{F_v}","aliases":"W_F_v","display":"W_{F_v}","kind":"definition-passage","excerpt":"Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v ","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-08422a7c74004217","tex":"\\widehat P","aliases":"widehat P widehat","display":"\\widehat P","kind":"definition-passage","excerpt":"[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-47843d13cdd41fbb","tex":"\\widehat P_{\\tau,V}","aliases":"widehat P_τ,V widehat tau","display":"\\widehat P_{\\tau,V}","kind":"definition-passage","excerpt":"[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-74b954f54a6a58b7","tex":"\\widehat P_{\\tau,W}","aliases":"widehat P_τ,W widehat tau","display":"\\widehat P_{\\tau,W}","kind":"definition-passage","excerpt":"[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-afe8e53205f4d68b","tex":"\\widetilde\\tau","aliases":"widetildeτ widetilde tau","display":"\\widetilde\\tau","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and it","count":28},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and it","count":37},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":24},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-00a09b294e775bb0","tex":"z^a","aliases":"z^a","display":"z^a","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-38e30a14620f14a0","tex":"z^j","aliases":"z^j","display":"z^j","kind":"definition-passage","excerpt":"[Alignment set] For V,W with the same restriction to H, define calJ_H(V,W) = {j∈ZZ/pZZ: P_τ,W=z^jP_τ,V for every stable τ}.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6894c7248320dbce","tex":"z^p","aliases":"z^p","display":"z^p","kind":"definition-passage","excerpt":"m polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6a98dd140a9c1336","tex":"\\ZZ","aliases":"ZZ","display":null,"kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":21},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-076c5702a922ca59","tex":"\\ZZ_{\\geq0}","aliases":"ZZ_≥0 ZZ geq","display":null,"kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":14},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c08adefef1d96f51","tex":"|_{G_E}","aliases":"|_G_E","display":"|_{G_E}","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d490f817f5e15003","tex":"|_{W_{F_v}}","aliases":"|_W_F_v","display":"|_{W_{F_v}}","kind":"definition-passage","excerpt":"Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v ","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"definition-passage","excerpt":"[Finite Satake cusp form] A cuspidal representation of GL_n(Adele_F) is finite Satake when, outside one finite exceptional set, its normalized unramified Satake roots lie in one finite subgroup of CC^×. The exceptional set is fixed, not allowed to change separately for each coefficient or prime.","count":8},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","excerpt":"[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f320d7b77ed8bff0","tex":"\\alpha_i","aliases":"α_i alpha","display":"\\alpha_i","kind":"definition-passage","excerpt":"[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-689041537ca5b553","tex":"\\alpha_i^p","aliases":"α_i^p alpha","display":"\\alpha_i^p","kind":"definition-passage","excerpt":"[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"definition-passage","excerpt":"[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.","count":18},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-74ecfbaf0fda4ff7","tex":"\\epsilon^j","aliases":"ϵ^j epsilon","display":"\\epsilon^j","kind":"definition-passage","excerpt":"nite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cae4149f9b41c826","tex":"\\epsilon_v^j","aliases":"ϵ_v^j epsilon","display":"\\epsilon_v^j","kind":"definition-passage","excerpt":"[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d8696536d4d7a2d6","tex":"\\zeta","aliases":"ζ zeta","display":"\\zeta","kind":"definition-passage","excerpt":"[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)","count":18},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-286538b6523eabf8","tex":"\\zeta^{jr}","aliases":"ζ^jr zeta","display":"\\zeta^{jr}","kind":"definition-passage","excerpt":"[Local spectral obstruction] The local spectral obstruction vanishes when, for every stable type τ, the zero Fourier modes of the two multiplicity spectra occur at the same indices and all ratios at the remaining indices satisfy widehat P_τ,W(r)/widehat P_τ,V(r)=ζ^jr for one common j. When widehat P_τ,V(r)=0, the condition requires widehat P_τ,W(r)","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1700b9c704f1fbba","tex":"\\Theta_1","aliases":"Θ_1 Theta","display":"\\Theta_1","kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3ce6dd6749b3ff4d","tex":"\\Theta_i","aliases":"Θ_i Theta","display":"\\Theta_i","kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b904ce629563a347","tex":"\\Theta_r","aliases":"Θ_r Theta","display":"\\Theta_r","kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"definition-passage","excerpt":"[Separating coefficient prime] A finite place λ of K is separating for calA when reduction modulo λ is injective on calA.","count":33},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-70c52efae8978345","tex":"\\xi","aliases":"ξ xi","display":"\\xi","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":15},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3b20e380f56b440c","tex":"\\xi^a","aliases":"ξ^a xi","display":"\\xi^a","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":21},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":84},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-39c80c5d717063a3","tex":"\\pi_v","aliases":"π_v pi","display":"\\pi_v","kind":"definition-passage","excerpt":"Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v ","count":8},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":57},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v","count":69},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5cc2da629590c491","tex":"\\tau^g","aliases":"τ^g tau","display":"\\tau^g","kind":"definition-passage","excerpt":"[Stable and moving types] An irreducible representation τ∈Irr(H) is stable when τ^g≅τ for every g∈ G. Otherwise its orbit has size p and is called moving.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"definition-passage","excerpt":"[Pseudocharacter family gate] A pseudocharacter family gate for a finite Satake descent π consists of a reduced p adic analytic space X, a closed point x valued in a finite extension of QQ_ℓ, and a continuous dimension n characteristic zero pseudorepresentation over O(X). It factors through the Galois group unramified outside one finite set, and it","count":27},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.","count":58},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"definition-passage","excerpt":"[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f486d8e45d3686cb","tex":"\\prod_{i=1}^{n}","aliases":"∏_i=1^n prod","display":"\\prod_{i=1}^{n}","kind":"definition-passage","excerpt":"[Frobenius transform of a polynomial] If P(T)=∏_i=1^n(T-α_i), define P^[p](T) = ∏_i=1^n(T-α_i^p). This is a polynomial expression in the coefficients of P.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":12},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ad8f486a66e260a5","tex":"\\sum_{a\\in\\ZZ/p\\ZZ}","aliases":"∑_a∈ZZ/pZZ sum in ZZ ZZ","display":null,"kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6182c3bad2d01e16","tex":"\\sum_{i=1}^r","aliases":"∑_i=1^r sum","display":"\\sum_{i=1}^r","kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"definition-passage","excerpt":"[Finite polynomial alphabet] The finite polynomial alphabet of the cyclic attachment problem is the union calA = calA_π ∪ calA_ρ, where calA_π is the set of normalized Hecke polynomials occurring for π outside the fixed exceptional set, and calA_ρ is the set of characteristic polynomials occurring for the finitely many twists ρ⊗ϵ^j, j∈ZZ/pZZ.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"definition-passage","excerpt":"Attached cyclic descent] Let E/F be cyclic of prime degree p, and let ρ be an irreducible finite image representation of G_F whose restriction τ=ρ|_G_E is irreducible. Suppose τ is automorphic with cuspidal representation Π of GL_n(Adele_E). An attached cyclic descent is a cuspidal descent π of Π such that rec_F_v(π_v) ≅ ρ|_W_F_v for every place v ","count":33},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c584776d69beecf7","tex":"\\neq","aliases":"≠ neq","display":"\\neq","kind":"definition-passage","excerpt":"[Active coefficient degrees] Define S(A)={k∈{1,…,n}:e_k(A)≠0}.","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"definition-passage","excerpt":"[Entire induction cone] Fix finitely many known entire induced packets Θ_1,…,Θ_r in the character lattice of G. Define C_calE = { ∑_i=1^ra_iΘ_i: a_i∈ZZ_≥0 }. Its saturation is Sat(C_calE) = {x:some m>0 has mx∈ C_calE}.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-50af2c348b6b2fe6","tex":"\\bigoplus","aliases":"⊕ bigoplus","display":"\\bigoplus","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-66dc5bf949528954","tex":"\\bigoplus_{a\\in\\ZZ/p\\ZZ}","aliases":"⊕_a∈ZZ/pZZ bigoplus in ZZ ZZ","display":null,"kind":"definition-passage","excerpt":"[Extension spectrum polynomial] For a stable type τ, the multiplicity space M_τ(V)=Hom_H(τ,V) becomes a representation of G/H after comparing the action on V with widetildeτ. Write M_τ(V) ≅ ⊕_a∈ZZ/pZZ (ξ^a)^⊕ m_τ,a(V). Define P_τ,V(z) = ∑_a∈ZZ/pZZ m_τ,a(V)z^a in ZZ[z]/(z^p-1).","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a56a245d78482165","tex":"\\otimes","aliases":"⊗ otimes","display":"\\otimes","kind":"definition-passage","excerpt":"[Local attachment set] When the local quotient has order p, define calJ_v = {j∈ZZ/pZZ: A_v⊗ϵ_v^j≅ R_v}. At a split place, set calJ_v=ZZ/pZZ.","count":34},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Residual realization of a descent] Let λ be a finite place of the coefficient field. A residual realization of π at λ is a continuous semisimple representation overline r_π,λ:G_F⟶GL_n(overline k_λ) whose characteristic polynomial at every unramified place outside a finite set is the reduction of the normalized Hecke polynomial of π.","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.","count":61},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3a342a2e332eabef","tex":"C_e","aliases":"C_e","display":"C_e","kind":"introduction-passage","excerpt":"Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"introduction-passage","excerpt":"At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has th","count":14},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-bf6efda13a9cd474","tex":"\\Frob","aliases":"Frob","display":null,"kind":"introduction-passage","excerpt":"At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has th","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7354d07b2ecd363e","tex":"\\Frob_v","aliases":"Frob_v Frob","display":null,"kind":"introduction-passage","excerpt":"At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has th","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"introduction-passage","excerpt":"At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has th","count":25},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-95d680fad2ccb816","tex":"I_n","aliases":"I_n","display":"I_n","kind":"introduction-passage","excerpt":"At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has th","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"introduction-passage","excerpt":"Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.","count":30},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-20881f05d1acb267","tex":"P_{\\pi,v}","aliases":"P_π,v pi","display":"P_{\\pi,v}","kind":"introduction-passage","excerpt":"At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has th","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-dc8741c688aee2d3","tex":"P_{\\rho,v}","aliases":"P_ρ,v rho","display":"P_{\\rho,v}","kind":"introduction-passage","excerpt":"At a finite place where ρ is unramified, fix the Frobenius convention used by the normalized local Langlands correspondence. Write P_ρ,v(T)=det(TI_n-ρ(Frob_v)). For an unramified automorphic local component π_v, write P_π,v(T) for the monic polynomial of its normalized Satake conjugacy class. The convention is chosen so that an attached pair has th","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"introduction-passage","excerpt":"Let K/F be finite Galois with group G. For a complex character θ of a subgroup, write L(s,θ) for the corresponding Artin L function.","count":25},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-964fcf8c1f59b656","tex":"T^e","aliases":"T^e","display":"T^e","kind":"introduction-passage","excerpt":"Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9b7f9d17ba2c44f5","tex":"T^{e-1}","aliases":"T^e-1","display":"T^{e-1}","kind":"introduction-passage","excerpt":"Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"introduction-passage","excerpt":"Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.","count":18},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"introduction-passage","excerpt":"Let K/F be finite Galois with group G. For a complex character θ of a subgroup, write L(s,θ) for the corresponding Artin L function.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","excerpt":"Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","excerpt":"Let C_e=⟨γ⟩ be a finite cyclic group of order e, acting linearly on a module M. Write T for the action of γ and set N=1+T+⋯+T^e-1. The relation T^e=1 implies N(T-1)=(T-1)N=0. These identities explain why the image of T-1 lies in the kernel of N, but they do not identify the two spaces.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is dete","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-677eab50351e7675","tex":"A^{-1}","aliases":"A^-1","display":"A^{-1}","kind":"use","excerpt":"Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"use","excerpt":"For degree three, the linear term is D_A,3(h)=A^2h+AhA+hA^2, and the nonlinear remainder is R_A,3(h)=Ah^2+hAh+h^2A+h^3. The middle word hAh cannot be absorbed into Ah^2 in a matrix ring. The remainder estimate keeps all four words and bounds each by its norm, so it respects the ordered expansion. This also explains why the estimate is not an assert","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-96c2916d997da17a","tex":"A^d","aliases":"A^d","display":"A^d","kind":"use","excerpt":"Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matt","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fdbf1e7dfaffd887","tex":"A^{d-1}","aliases":"A^d-1","display":"A^{d-1}","kind":"use","excerpt":"R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matters. Replacing this operator by dA^d-1h requires A and h to commute. The replacement is invalid for general matrices.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3dabb837de1fb9df","tex":"A^{d-1-i}","aliases":"A^d-1-i","display":"A^{d-1-i}","kind":"use","excerpt":"Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matt","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-341faa653baa07cb","tex":"A^i","aliases":"A^i","display":"A^i","kind":"use","excerpt":"Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matt","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fca57a1241a0dffd","tex":"A^p","aliases":"A^p","display":"A^p","kind":"use","excerpt":"Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"There is a genuinely singular contrast. Let A_0=diag(1,-1) over Q_p. Then D_A_0,2(E_12)=0, although E_12≠0. Thus this derivative has no inverse over the fraction field, including when p is odd and 2 is a unit. The certificate theorem cannot cover this center. A universal claim that invertible centers always have invertible power derivatives would b","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d325da4f9b58928d","tex":"A_3","aliases":"A_3","display":"A_3","kind":"use","excerpt":"[A moving type and self twist] Let G=S_3, let H=A_3, and let V be the standard two dimensional representation. Its restriction is Res_HV=τ⊕τ^-1 for the two nontrivial characters of C_3. These characters form one moving orbit under G/H≅ C_2. Hence there is no stable extension spectrum, and V⊗sgn≅ V.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-82f421ce47ab25e5","tex":"A_{\\mathrm{bal}}","aliases":"A_bal mathrm","display":"A_{\\mathrm{bal}}","kind":"use","excerpt":"Now compare the balanced matrix A_bal=diag(1,ω,ω^2). Its polynomial is T^3-1. Multiplying all eigenvalues by ω merely permutes their multiset. Every lower coefficient vanishes, and the determinant is fixed. Its full characteristic polynomial is therefore blind to the label. This is an actual local rotational stabilizer. A claim that one characteris","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":8},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-88a16ff39d69e5a7","tex":"\\Adele_\\QQ","aliases":"Adele_QQ Adele QQ","display":null,"kind":"use","excerpt":"[Conditional construction of Kim and Yamauchi] Kim and Yamauchi's construction for GL_n(Adele_QQ) uses five declared inputs: a finite Hecke field, integrality with a common denominator, cuspidal Galois conjugates, residual Galois realizations for all sufficiently large coefficient primes, and a Rankin Selberg coefficient bound obtainable from the r","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1d6437e0973a304f","tex":"\\AI","aliases":"AI","display":null,"kind":"use","excerpt":"ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the esta","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-64dc21fd3921bfe3","tex":"\\AI_{E/F}","aliases":"AI_E/F AI","display":null,"kind":"use","excerpt":"ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the esta","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":") whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put ","count":33},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1bdcbb0ff59dbc74","tex":"b^p","aliases":"b^p","display":"b^p","kind":"use","excerpt":"G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(g","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"Suppose an absolute two-cocycle has been cancelled by a continuous one-cochain b_0. Prescribed local lifts need not agree with the resulting global lift. Their remaining differences are local one-cocycles. Changing b_0 to another absolute primitive changes these differences by the restriction of a global one-cocycle. Conjugating the local lift at i","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"To see the criterion, let a_i be the local mismatch after b_0. A match up to the allowed local conjugations exists exactly when a_i=res_i(z)+δ x_i for a global continuous one-cocycle z and kernel elements x_i. This is exactly the assertion that the image of (a_i)_i in [source reference] is zero. If b_1 is another absolute primitive, then b_1-b_0 is","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c48ea07d659ad27e","tex":"B_x","aliases":"B_x","display":"B_x","kind":"use","excerpt":"The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with ","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ab5eb70439d279a3","tex":"\\mathrm{bad}","aliases":"bad mathrm","display":"\\mathrm{bad}","kind":"use","excerpt":"For a negative control, take U_bad=1 · 10 0 · 1. The difference from A^2 is (0,8,0,0) in entry coordinates, and B sends it to (0,16,0,0). This is nonzero modulo 32, so no square root of U_bad lies in that ball. This is a failure for the specified prefix. It is not a claim that the target has no square root anywhere in a larger coefficient field.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1cf2aff443cfe7db","tex":"\\mathrm{bal}","aliases":"bal mathrm","display":"\\mathrm{bal}","kind":"use","excerpt":"Now compare the balanced matrix A_bal=diag(1,ω,ω^2). Its polynomial is T^3-1. Multiplying all eigenvalues by ω merely permutes their multiset. Every lower coefficient vanishes, and the determinant is fixed. Its full characteristic polynomial is therefore blind to the label. This is an actual local rotational stabilizer. A claim that one characteris","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fa2c06158fa4c5b1","tex":"\\BC","aliases":"BC","display":null,"kind":"use","excerpt":"ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the esta","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cda094665542bbc8","tex":"\\BC_{E/F}","aliases":"BC_E/F BC","display":null,"kind":"use","excerpt":"ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the esta","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d13b1528b249ec85","tex":"\\bigwedge","aliases":"bigwedge","display":"\\bigwedge","kind":"use","excerpt":"The coefficient e_k(A) is the trace of bigwedge^k A. Scalar multiplication by ζ acts on bigwedge^kCC^n by ζ^k.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3c02c22f4e0c0340","tex":"\\bigwedge^k","aliases":"bigwedge^k bigwedge","display":"\\bigwedge^k","kind":"use","excerpt":"The coefficient e_k(A) is the trace of bigwedge^k A. Scalar multiplication by ζ acts on bigwedge^kCC^n by ζ^k.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-122842ce4df55b07","tex":"\\boxplus","aliases":"boxplus","display":null,"kind":"use","excerpt":"ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the esta","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-","count":17},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","excerpt":"Change the chosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the se","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4db3245f7b23bd25","tex":"c^{-1}","aliases":"c^-1","display":"c^{-1}","kind":"use","excerpt":"G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(g","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"use","excerpt":"[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b257aef99d8016bf","tex":"C_3^2","aliases":"C_3^2","display":"C_3^2","kind":"use","excerpt":"The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together w","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-dcb4e0ba5e2088bf","tex":"C_6","aliases":"C_6","display":"C_6","kind":"use","excerpt":"Let instead the generator of C_6 act by the same T. On the diagonal space, its norm is multiplication by 6=0 in F_3. On the off-diagonal space, the alternating six-term sum is also zero. Now N=0 on all of M, while the image of T-1 is still only the off-diagonal space. Therefore H^1(C_6,M) has dimension two. The generator matrix alone does not deter","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-374d4fe4488ce15e","tex":"C_p","aliases":"C_p","display":"C_p","kind":"use","excerpt":"The finite representation arguments have a parallel shape. If Htriangleleft G and G/H≅ C_p, an irreducible representation either restricts to a full orbit of p subgroup types or restricts irreducibly to one invariant type. In the first case induction reconstructs the representation. In the second case an extension must be selected. The second case ","count":11},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8b13809e7734a359","tex":"\\calP","aliases":"calP","display":null,"kind":"use","excerpt":"[One large residual realization gives an Artin representation] Let K be a number field, let S be a finite set of places of F, and let calP⊆OO_K[T] be a nonempty finite set of monic degree n polynomials. Assume all their roots are roots of unity. For v∉ S, choose P_v∈calP.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cf090df8943be9f2","tex":"\\CC^n","aliases":"CC^n CC","display":null,"kind":"use","excerpt":"The coefficient e_k(A) is the trace of bigwedge^k A. Scalar multiplication by ζ acts on bigwedge^kCC^n by ζ^k.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6e6d0cbeeb71722a","tex":"\\operatorname{coker}","aliases":"coker operatorname","display":"\\operatorname{coker}","kind":"use","excerpt":"After taking the local coboundary quotients, the remaining class belongs to coker( H^1_cont(G,I)⟶ ∏_i H^1_cont(G_i,I)). Here the local groups, their maps into G, and the precise continuous coefficient action must be supplied. The recovered source constructs the corresponding quotients directly in its square-zero data.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9563ef64b7b5663e","tex":"\\mathrm{cont}","aliases":"cont mathrm","display":"\\mathrm{cont}","kind":"use","excerpt":"After taking the local coboundary quotients, the remaining class belongs to coker( H^1_cont(G,I)⟶ ∏_i H^1_cont(G_i,I)). Here the local groups, their maps into G, and the precise continuous coefficient action must be supplied. The recovered source constructs the corresponding quotients directly in its square-zero data.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"Let t≥1 be the number of trace values in the source alphabet and put B=t^n^2. First fix a finite normal extension L/QQ containing K and the roots of unity of orders dividing D_B=lcm(1,2,…,B). Every polynomial in calQ_n,B, the bounded cyclotomic alphabet, has coefficients in L. Form the union of all Galois conjugates in L of calP∪calQ_n,B. This is a","count":20},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"Let E/F be a finite Galois extension of degree d. Suppose π base changes to an automorphic representation Π attached to a finite image Artin representation τ. Let M be the exponent of the finite image of τ. Fix a finite set S containing the exceptional comparison places and all ramification of the extension and the representations. The unramified c","count":49},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-26205c0484c30f37","tex":"D_{A,2}","aliases":"D_A,2","display":"D_{A,2}","kind":"use","excerpt":"c · e, the ordered derivative is D_A,2(X)=AX+XA =2a+c · a+2b+e 2c · c+2e. Using the entry order (a,b,c,e), its matrix and a scaled inverse are J= 2 · 0 · 1 · 0","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-13d2a4bbe46dc1c1","tex":"D_{A,3}","aliases":"D_A,3","display":"D_{A,3}","kind":"use","excerpt":"For degree three, the linear term is D_A,3(h)=A^2h+AhA+hA^2, and the nonlinear remainder is R_A,3(h)=Ah^2+hAh+h^2A+h^3. The middle word hAh cannot be absorbed into Ah^2 in a matrix ring. The remainder estimate keeps all four words and bounds each by its norm, so it respects the ordered expansion. This also explains why the estimate is not an assert","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1211cb43b956f107","tex":"D_{A,d}","aliases":"D_A,d","display":"D_{A,d}","kind":"use","excerpt":"Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matt","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-733b123de6b80a7d","tex":"D_{A,d+1}","aliases":"D_A,d+1","display":"D_{A,d+1}","kind":"use","excerpt":"The recursive identities are D_A,d+1(h) · =D_A,d(h)A+A^dh, R_A,d+1(h) · =R_A,d(h)(A+h)+D_A,d(h)h. They follow by multiplying (A+h)^d on the right by A+h and collecting the terms with zero, one, and at least two occurrences of h. This derivation does not rearrange any noncommuting factors.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-892f514366f204df","tex":"D_{A_0,2}","aliases":"D_A_0,2","display":"D_{A_0,2}","kind":"use","excerpt":"There is a genuinely singular contrast. Let A_0=diag(1,-1) over Q_p. Then D_A_0,2(E_12)=0, although E_12≠0. Thus this derivative has no inverse over the fraction field, including when p is odd and 2 is a unit. The certificate theorem cannot cover this center. A universal claim that invertible centers always have invertible power derivatives would b","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ee5bb7d36741efec","tex":"D_B","aliases":"D_B","display":"D_B","kind":"use","excerpt":"Let t≥1 be the number of trace values in the source alphabet and put B=t^n^2. First fix a finite normal extension L/QQ containing K and the roots of unity of orders dividing D_B=lcm(1,2,…,B). Every polynomial in calQ_n,B, the bounded cyclotomic alphabet, has coefficients in L. Form the union of all Galois conjugates in L of calP∪calQ_n,B. This is a","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-18b312b01ad13df4","tex":"D_d","aliases":"D_d","display":"D_d","kind":"use","excerpt":"Induct on d for the second estimate. The case d=0 is immediate. Write D_d=D_A,d and R_d=R_A,d. Subtracting the remainder recursions gives R_d+1(h)-R_d+1(k) = · (R_d(h)-R_d(k))(A+h)","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5cdd755c6c76898a","tex":"d_i","aliases":"d_i","display":"d_i","kind":"use","excerpt":"oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9ddc6686093e1bb0","tex":"\\det^a","aliases":"det^a det","display":"\\det^a","kind":"use","excerpt":"Every algebraic character of GL_n is an integral power of the determinant. Thus det∘ r=det^a for one integer a. Evaluate both sides at zI_n. The left side is z^wN, while the right side is z^an.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"[The same generator action gives different cyclic cohomology] Over F_3, let M=Mat_2( F_3) and let T be conjugation by F=diag(1,-1). The diagonal entries of a matrix form the +1 eigenspace of T. The off-diagonal entries form its -1 eigenspace. Both spaces have dimension two.","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"[Highly ramified epsilon twists are blind] Let V' = V⊗χ, where χ has order p, and assume pmiddim V. Then V and V' have the same determinant. For every sufficiently ramified auxiliary character η, ϵ(V⊗η,Ψ) = ϵ(V'⊗η,Ψ).","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2551402eef77c95b","tex":"\\dim_k","aliases":"dim_k dim","display":"\\dim_k","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-64fbb1a3212deefd","tex":"e'","aliases":"e'","display":"e'","kind":"use","excerpt":"The normalization by π^r deserves care. Writing F(x)=π^r e need not determine e uniquely in a ring with torsion. If e' is another choice, then π^r(e-e')=0. The faithful leading layer used by the theorem ensures that e-e' has zero residue below the terminal cutoff. Thus the residue correction problem is independent of the chosen normalized error. Th","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c2349512e48c7e97","tex":"e^{-1}","aliases":"e^-1","display":"e^{-1}","kind":"use","excerpt":"Suppose now that e is invertible in the coefficient ring. Define S=∑_j=0^e-1jT^j. A direct coefficient calculation gives (T-1)S=e-N. For x∈ker N, take y=e^-1Sx. Then (T-1)y=x, so every cocycle is a coboundary and H^1(C_e,M)=0. The averaging argument uses division by e at its decisive step. In residue characteristic dividing e, the same display rema","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the th","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f2b11b7b1856ea86","tex":"E_{12}","aliases":"E_12","display":"E_{12}","kind":"use","excerpt":"There is a genuinely singular contrast. Let A_0=diag(1,-1) over Q_p. Then D_A_0,2(E_12)=0, although E_12≠0. Thus this derivative has no inverse over the fraction field, including when p is odd and 2 is a unit. The certificate theorem cannot cover this center. A universal claim that invertible centers always have invertible power derivatives would b","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the th","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the th","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-238a2f0e8e975441","tex":"E_{ab}","aliases":"E_ab","display":"E_{ab}","kind":"use","excerpt":"dule. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) restricts on the ith block to m_iTr(XY). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units E_ab with E_ba. Each m_i is nonzero ","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2bfe92b6c0e7f510","tex":"E_{ba}","aliases":"E_ba","display":"E_{ba}","kind":"use","excerpt":"efore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) restricts on the ith block to m_iTr(XY). Matrix trace pairing is nondegenerate, as is seen by pairing matrix units E_ab with E_ba. Each m_i is nonzero in k: this is au","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ca7e75161b6bbf35","tex":"E_{ij}","aliases":"E_ij","display":"E_{ij}","kind":"use","excerpt":"There is a finite exact certificate for the inverse. For every matrix unit E_ij check QD(E_ij)=E_ij, DQ(E_ij)=E_ij. Linearity and the matrix-unit basis extend both identities to every matrix over Q_p. These equalities are exact in the coefficient field. They are not merely congruences modulo p. In particular, an operator singular over Q_p cannot sa","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4c336ec08d156ec7","tex":"E_w","aliases":"E_w","display":"E_w","kind":"use","excerpt":"For each place v of F, put R_v=ρ|_W_F_v, A_v=rec_F_v(π_0,v). Cyclic base change compatibility gives Res_W_E_wR_v ≅ Res_W_E_wA_v.","count":8},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"bibitemKimYamauchi2016 H. H. Kim and T. Yamauchi, A uniform structure on subgroups of GL_n( F_q) and its application to a conditional construction of Artin representations of GL_n, Journal of the Ramanujan Mathematical Society 32(1) (2017), 75--99. Checked preprint https://arxiv.org/abs/1403.5535arXiv:1403.5535v3, 7 April 2016. Theorem 1.1 is condi","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-08c5fee29bc1411d","tex":"F^{-1}","aliases":"F^-1","display":"F^{-1}","kind":"use","excerpt":"In Frobenius coordinates the relevant action is often T(X)=FXF^-1. The images of T-1 and 1-T coincide, since multiplication of a source vector by -1 exchanges the signs. This is the bridge to the concrete quotient by Frobenius coboundaries. The recovered v0.18 source proves that bridge under the explicit zero-norm hypothesis. It does not claim that","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-62f1baf4056016b8","tex":"F^2","aliases":"F^2","display":"F^2","kind":"use","excerpt":"Choose m=3, so the local root ball is A+8Mat_2( Z_2) and the finite target test is modulo 2^m+s=32. Consider F=1 · 1 8 · 1, U=F^2=9 · 2 16 · 9. Here AF=9 · 2","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-25665deaa55a5329","tex":"F^3","aliases":"F^3","display":"F^3","kind":"use","excerpt":"Nor does a successful shallow node necessarily extend. For example, the scalar prefix F=4 modulo 9 satisfies F^3=1 there. Its lifts modulo 27 are 4,13,22, all with cube 10. That branch stops at the next level. Merely checking a finite number of successful levels gives no all-depth conclusion without another theorem.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6e3ed25c824c19b9","tex":"F^d","aliases":"F^d","display":"F^d","kind":"use","excerpt":"The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d44b7273f4df5ad0","tex":"F^{d-1}","aliases":"F^d-1","display":"F^{d-1}","kind":"use","excerpt":"The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-54c7a07f66f1253f","tex":"\\mathbb F_3","aliases":"F_3 mathbb","display":"\\mathbb F_3","kind":"use","excerpt":"[The same generator action gives different cyclic cohomology] Over F_3, let M=Mat_2( F_3) and let T be conjugation by F=diag(1,-1). The diagonal entries of a matrix form the +1 eigenspace of T. The off-diagonal entries form its -1 eigenspace. Both spaces have dimension two.","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-86f19f5a64cb25fa","tex":"\\mathbb F_q","aliases":"F_q mathbb","display":"\\mathbb F_q","kind":"use","excerpt":"bibitemKimYamauchi2016 H. H. Kim and T. Yamauchi, A uniform structure on subgroups of GL_n( F_q) and its application to a conditional construction of Artin representations of GL_n, Journal of the Ramanujan Mathematical Society 32(1) (2017), 75--99. Checked preprint https://arxiv.org/abs/1403.5535arXiv:1403.5535v3, 7 April 2016. Theorem 1.1 is condi","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f7110079fe13a85a","tex":"F_r","aliases":"F_r","display":"F_r","kind":"use","excerpt":"This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predi","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4f75382573b44170","tex":"F_r^{d_i}","aliases":"F_r^d_i","display":"F_r^{d_i}","kind":"use","excerpt":"This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predi","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4d7b4480e3e06144","tex":"\\operatorname{Frob}","aliases":"Frob operatorname","display":"\\operatorname{Frob}","kind":"use","excerpt":"[Frobenius transform after reduction] Let P have integral coefficients at a place of residue characteristic p. Then overlineP^[p](T) = Frob(overline P(T)), where Frobenius raises every coefficient to its pth power. In particular, the map overline P⟼overlineP^[p] is injective over the residue field.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-25283e4418929cd5","tex":"\\Frob_w","aliases":"Frob_w Frob","display":null,"kind":"use","excerpt":"Restrict the residual realization to G_E. At every unramified place w of E above a good place v of F, cyclic base change identifies the characteristic polynomial of the restriction with the reduction of the Hecke polynomial of Π_w. Since Π is attached to τ, this is the characteristic polynomial of overlineτ(Frob_w). Chebotarev density and Brauer Ne","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-aa45a7ad376e6ef9","tex":"g^{-1}","aliases":"g^-1","display":"g^{-1}","kind":"use","excerpt":"Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3c040bd92821cda8","tex":"g^p","aliases":"g^p","display":"g^p","kind":"use","excerpt":"Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-08e49831f95ffbe1","tex":"g^r","aliases":"g^r","display":"g^r","kind":"use","excerpt":"[Fourier trace interpretation] Let g be a lift of a generator of G/H. For a stable type τ, widehat P_τ,V(r) = Tr(g^rmid M_τ(V)). If V is twisted by ξ^j, then widehat P_τ,V⊗ξ^j(r) = ζ^jrwidehat P_τ,V(r).","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-bf173953a9e841d7","tex":"g_1","aliases":"g_1","display":"g_1","kind":"use","excerpt":"Absolute irreducibility and Burnside's theorem give group elements g_1,…,g_n^2 whose matrices form a K basis of M_n(K). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7ccf548c96ad41ac","tex":"g_d","aliases":"g_d","display":"g_d","kind":"use","excerpt":"The algebraic pseudocharacter realization theorem supplies a semisimple representation r with trace T. Its continuity deserves a separate argument. Let A be the span of its image. The trace pairing on A is nondegenerate, by the characteristic zero semisimple algebra argument in [source reference]. Choose a basis r(g_1),…,r(g_d) from the image. The ","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cd7038b72872679a","tex":"G_i","aliases":"G_i","display":"G_i","kind":"use","excerpt":"After taking the local coboundary quotients, the remaining class belongs to coker( H^1_cont(G,I)⟶ ∏_i H^1_cont(G_i,I)). Here the local groups, their maps into G, and the precise continuous coefficient action must be supplied. The recovered source constructs the corresponding quotients directly in its square-zero data.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-36c99c0bb63416fc","tex":"g_i","aliases":"g_i","display":"g_i","kind":"use","excerpt":"algebraic pseudocharacter realization theorem supplies a semisimple representation r with trace T. Its continuity deserves a separate argument. Let A be the span of its image. The trace pairing on A is nondegenerate, by the characteristic zero semisimple algebra argument in [source reference]. Choose a basis r(g_1),…,r(g_d) from the image. The func","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-912a8305e3232f09","tex":"g_{n^2}","aliases":"g_n^2","display":"g_{n^2}","kind":"use","excerpt":"Absolute irreducibility and Burnside's theorem give group elements g_1,…,g_n^2 whose matrices form a K basis of M_n(K). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f046713e329fd428","tex":"g_p","aliases":"g_p","display":"g_p","kind":"use","excerpt":"For g∈ G, write its commuting primary decomposition as g=g_p g_s, where g_p has p power order and g_s has order prime to p. The image of g_s in C_p is trivial, so g_s∈ H. In residue characteristic p, the matrix of g_s is diagonalizable after extending the field, and the matrix of g_p is unipotent. They commute, so the latter preserves each eigenspa","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-66816f2f34b0eaea","tex":"g_s","aliases":"g_s","display":"g_s","kind":"use","excerpt":"For g∈ G, write its commuting primary decomposition as g=g_p g_s, where g_p has p power order and g_s has order prime to p. The image of g_s in C_p is trivial, so g_s∈ H. In residue characteristic p, the matrix of g_s is diagonalizable after extending the field, and the matrix of g_p is unipotent. They commute, so the latter preserves each eigenspa","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-21cf47c47bcf5c41","tex":"\\Gal","aliases":"Gal","display":null,"kind":"use","excerpt":"Let F be a number field, let G_F=Gal(overline F/F), and let ρ:G_F⟶GL_n(CC) be a continuous representation with finite image. Continuity is taken for the usual complex topology. Such a representation factors through the Galois group of a finite extension. Throughout the conjecture discussion, ρ is irreducible. The trivial one dimensional representat","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0fcc70536b44cfb8","tex":"\\operatorname{GL}","aliases":"GL operatorname","display":"\\operatorname{GL}","kind":"use","excerpt":"The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-14cd8aadb7f36d16","tex":"\\GL_3","aliases":"GL_3 GL","display":null,"kind":"use","excerpt":"The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together w","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-665db19fdc2e1b41","tex":"\\GL_N","aliases":"GL_N GL","display":null,"kind":"use","excerpt":"[Determinant character of an algebraic representation] Let r:GL_n(CC)⟶GL_N(CC) be an irreducible algebraic representation. Suppose the center acts with weight w, so r(zI_n)=z^wI_N. There is an integer a such that det r(g)=det(g)^a and wN=an.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8df4bc39535de40c","tex":"\\operatorname{GL}_n","aliases":"GL_n operatorname","display":"\\operatorname{GL}_n","kind":"use","excerpt":"The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fdacfbe74d9c216c","tex":"\\mathrm{GO}","aliases":"GO mathrm","display":"\\mathrm{GO}","kind":"use","excerpt":"The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together w","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"Choose g∈ G whose class generates G/H, so g^p∈ H. Stability gives an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-","count":61},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a9252825d2bf67d2","tex":"H^1","aliases":"H^1","display":"H^1","kind":"use","excerpt":"For a zero-cochain y∈ M, its coboundary at γ is (T-1)y. Taking the quotient by these changes gives the precise presentation H^1(C_e,M)≃ ker N/im(T-1). If the norm operator is identically zero, this reduces to M/im(T-1). That reduction has a hypothesis. Dropping the kernel from the general formula introduces classes that cannot satisfy the cyclic re","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cf294df70d9372ac","tex":"H^1_{\\mathrm{cont}}","aliases":"H^1_cont mathrm","display":"H^1_{\\mathrm{cont}}","kind":"use","excerpt":"After taking the local coboundary quotients, the remaining class belongs to coker( H^1_cont(G,I)⟶ ∏_i H^1_cont(G_i,I)). Here the local groups, their maps into G, and the precise continuous coefficient action must be supplied. The recovered source constructs the corresponding quotients directly in its square-zero data.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"hosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the section.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":"inear system F(x)=0, a relation among the residual Jacobian rows is only a first-order identity. Consider F(x)=(x,x^2), J_x(h)=(h,2xh). At the origin the derivative has horizontal image. The functional selecting the second coordinate is a residual syzygy, but it does not annihilate F(x) for general x. The exact relation is the moving relation (-x,1","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fc3c9dc938c10528","tex":"h^3","aliases":"h^3","display":"h^3","kind":"use","excerpt":"For degree three, the linear term is D_A,3(h)=A^2h+AhA+hA^2, and the nonlinear remainder is R_A,3(h)=Ah^2+hAh+h^2A+h^3. The middle word hAh cannot be absorbed into Ah^2 in a matrix ring. The remainder estimate keeps all four words and bounds each by its norm, so it respects the ordered expansion. This also explains why the estimate is not an assert","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d3cba85783189f4d","tex":"H_i","aliases":"H_i","display":"H_i","kind":"use","excerpt":"[Positive saturation criterion] Let χ be a nontrivial irreducible character of G. Suppose there are a positive integer m, nonnegative integers a_i, subgroups H_i, and characters θ_i∈calE such that mχ = ∑_i a_iInd_H_i^Gθ_i. Then L(s,χ) is entire.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"use","excerpt":"The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a1c5ac132799d427","tex":"h_{j+1}","aliases":"h_j+1","display":"h_{j+1}","kind":"use","excerpt":"The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-480c877d455d2c2b","tex":"h_t","aliases":"h_t","display":"h_t","kind":"use","excerpt":"The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0745ce822383e740","tex":"I^2","aliases":"I^2","display":"I^2","kind":"use","excerpt":"Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_g","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e2ed4b2197fa2598","tex":"I_2","aliases":"I_2","display":"I_2","kind":"use","excerpt":"The residual derivative is singular. In characteristic two, D_A,2(I_2)=2A=0, while I_2≠0. Nevertheless J is invertible over Q_2, with determinant 16. Residual singularity and singularity over the fraction field are different tests.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-78fd137c582a5e5f","tex":"I_4","aliases":"I_4","display":"I_4","kind":"use","excerpt":"0 · 0 · -1 · 2 . Direct multiplication gives BJ=JB=4I_4. Therefore Q=tfrac14 B is the exact inverse and s=2. The entry tfrac14 in this inverse prevents clearing its denominators with only one factor of 2.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b1cb24ac6e8394e8","tex":"I_N","aliases":"I_N","display":"I_N","kind":"use","excerpt":"[Determinant character of an algebraic representation] Let r:GL_n(CC)⟶GL_N(CC) be an irreducible algebraic representation. Suppose the center acts with weight w, so r(zI_n)=z^wI_N. There is an integer a such that det r(g)=det(g)^a and wN=an.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im","display":null,"kind":"use","excerpt":"[Uniform bound for a residual Galois image] Let overline r:G_F⟶GL_n(overlineFF_ℓ) be continuous and semisimple, with ℓ>n. If its Frobenius traces belong to a set of size at most t on a Chebotarev dense set, then absimoverline r≤ t^n^2.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-73d2ac313bb1a012","tex":"\\operatorname{im}","aliases":"im operatorname","display":"\\operatorname{im}","kind":"use","excerpt":"For a zero-cochain y∈ M, its coboundary at γ is (T-1)y. Taking the quotient by these changes gives the precise presentation H^1(C_e,M)≃ ker N/im(T-1). If the norm operator is identically zero, this reduces to M/im(T-1). That reduction has a hypothesis. Dropping the kernel from the general formula introduces classes that cannot satisfy the cyclic re","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-59f64376d60523e7","tex":"\\Ind","aliases":"Ind","display":null,"kind":"use","excerpt":"[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) ","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cced512be525f998","tex":"\\Ind_{G_E}^{G_F}","aliases":"Ind_G_E^G_F Ind","display":null,"kind":"use","excerpt":"The automorphic induction of the attached representation over E is already attached to Ind_G_E^G_Fτ ≅ ⊕_j=0^p-1 ρ⊗ϵ^j. It is also the isobaric sum of all descents. By [source reference], this total object is invariant under rotating either torsor. It proves the direct sum correspondence but cannot identify one constituent with the target extension.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-98888f9bd3780aad","tex":"\\Ind_H^G","aliases":"Ind_H^G Ind","display":null,"kind":"use","excerpt":"[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) ","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fc9593a21b07933f","tex":"\\Ind_{H_i}^G","aliases":"Ind_H_i^G Ind","display":null,"kind":"use","excerpt":"[Positive saturation criterion] Let χ be a nontrivial irreducible character of G. Suppose there are a positive integer m, nonnegative integers a_i, subgroups H_i, and characters θ_i∈calE such that mχ = ∑_i a_iInd_H_i^Gθ_i. Then L(s,χ) is entire.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is dete","count":16},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8d6308571f33e418","tex":"J'","aliases":"J'","display":"J'","kind":"use","excerpt":"If a third variable t is added so that the derivative becomes J'(u,v,t)=(u,v,u+v+t), the enlarged map is surjective. It cancels the second error with (u,v,t)=(0,0,-1). But the nonzero value of t records exactly the failed original relation. This enlarged solution cannot be declared a solution of the original equations by forgetting t. One must show","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-35b3ffd4dcaf66dc","tex":"J^*","aliases":"J^*","display":"J^*","kind":"use","excerpt":"Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is dete","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6affb9051b0cf580","tex":"j_0","aliases":"j_0","display":"j_0","kind":"use","excerpt":"If j_0 is one solution, every other solution differs from it by a rotation stabilizing every spectrum polynomial. The converse is immediate.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c39eabd056948ab2","tex":"j_{\\det}","aliases":"j_det det","display":"j_{\\det}","kind":"use","excerpt":"[Determinant chosen candidate] If pnmid n, determinant and central character select one candidate j_det. The cyclic Strong Artin step is then equivalent to j_det∈calJ_v for every v.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8af4e249618e23ea","tex":"J_x","aliases":"J_x","display":"J_x","kind":"use","excerpt":"For a nonlinear system F(x)=0, a relation among the residual Jacobian rows is only a first-order identity. Consider F(x)=(x,x^2), J_x(h)=(h,2xh). At the origin the derivative has horizontal image. The functional selecting the second coordinate is a residual syzygy, but it does not annihilate F(x) for general x. The exact relation is the moving rela","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-725ba61f1fb98c98","tex":"j_\\lambda","aliases":"j_λ lambda","display":"j_\\lambda","kind":"use","excerpt":"[Only finitely many local checks remain] Under [source reference], all local attachment to ρ is equivalent to checking the label -j_λ at every exceptional finite place excluded from the unramified comparison and at every archimedean place.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3a79eac863d7177c","tex":"j_{\\lambda}","aliases":"j_λ lambda","display":"j_{\\lambda}","kind":"use","excerpt":"labelthm:one_prime_attachment Use the cyclic Artin setup of [source reference]. For this unramified theorem it is enough that the lower cuspidal representation Π is weakly attached to τ, with π a cuspidal cyclic descent. Let λ be a good separating prime. Suppose π has a residual realization overline r_π,λ. Then there is a unique index j_λ∈ZZ/pZZ su","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8c9a5bd1e738f364","tex":"j_\\tau","aliases":"j_τ tau","display":"j_\\tau","kind":"use","excerpt":"ma, the first block demands shift one and the second demands shift two. The two restrictions to H have the same multiplicities. Each block separately is a quotient twist of the matching source block. The full representation is not one quotient twist, because the required shifts disagree. This is the reason the spectrum criterion quantifies over one","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, sinc","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-146b7f435ba6dc91","tex":"k^3","aliases":"k^3","display":"k^3","kind":"use","excerpt":"[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, sinc","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7e6bfa3224000497","tex":"k^{d_i}","aliases":"k^d_i","display":"k^{d_i}","kind":"use","excerpt":"oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-045f054126c48eb2","tex":"k^n","aliases":"k^n","display":"k^n","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-90285d389d0a9f56","tex":"\\ker","aliases":"ker","display":"\\ker","kind":"use","excerpt":"If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a translate of Nx, which is zero. Thus evaluating at the chosen generator identifies cocycles with ke","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c7f6796e75b9a514","tex":"L_3","aliases":"L_3","display":"L_3","kind":"use","excerpt":"bibitemWang2026 J. Wang, Strong Artin conjecture for generalized octahedral representations in GL_3, https://arxiv.org/abs/2609.38231arXiv:2609.38231v1, 28 September 2026. Theorem 1.1 and Corollary 5.4 state the special three and six dimensional cases mentioned here. Statements checked. The full proof has not been independently audited for this rea","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ed76d6b3d182baf7","tex":"L_n","aliases":"L_n","display":"L_n","kind":"use","excerpt":"bibitemKimYamauchi2016 H. H. Kim and T. Yamauchi, A uniform structure on subgroups of GL_n( F_q) and its application to a conditional construction of Artin representations of GL_n, Journal of the Ramanujan Mathematical Society 32(1) (2017), 75--99. Checked preprint https://arxiv.org/abs/1403.5535arXiv:1403.5535v3, 7 April 2016. Theorem 1.1 is condi","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3860d7e8f88507bb","tex":"\\operatorname{lcm}","aliases":"lcm operatorname","display":"\\operatorname{lcm}","kind":"use","excerpt":"Let t≥1 be the number of trace values in the source alphabet and put B=t^n^2. First fix a finite normal extension L/QQ containing K and the roots of unity of orders dividing D_B=lcm(1,2,…,B). Every polynomial in calQ_n,B, the bounded cyclotomic alphabet, has coefficients in L. Form the union of all Galois conjugates in L of calP∪calQ_n,B. This is a","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag · Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a ro","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1633dfbf62528f10","tex":"\\lvert","aliases":"lvert","display":"\\lvert","kind":"use","excerpt":"from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an X holomorphic limit of discrete series [source reference]. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the dens","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-bc4c24fa42db60db","tex":"m_a","aliases":"m_a","display":"m_a","kind":"use","excerpt":"Fix a primitive pth root ζ. For P(z)=∑_a∈ZZ/pZZm_az^a, define widehat P(r) = ∑_a∈ZZ/pZZm_aζ^ar, r∈ZZ/pZZ.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e5a8e8a33dcbaa8c","tex":"M_{d_i}","aliases":"M_d_i","display":"M_{d_i}","kind":"use","excerpt":"oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"use","excerpt":"oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"use","excerpt":"Absolute irreducibility and Burnside's theorem give group elements g_1,…,g_n^2 whose matrices form a K basis of M_n(K). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2c7620add7ce256d","tex":"\\operatorname{Mat}","aliases":"Mat operatorname","display":"\\operatorname{Mat}","kind":"use","excerpt":"[The same generator action gives different cyclic cohomology] Over F_3, let M=Mat_2( F_3) and let T be conjugation by F=diag(1,-1). The diagonal entries of a matrix form the +1 eigenspace of T. The off-diagonal entries form its -1 eigenspace. Both spaces have dimension two.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-048003adc2e71133","tex":"\\operatorname{Mat}_2","aliases":"Mat_2 operatorname","display":"\\operatorname{Mat}_2","kind":"use","excerpt":"[The same generator action gives different cyclic cohomology] Over F_3, let M=Mat_2( F_3) and let T be conjugation by F=diag(1,-1). The diagonal entries of a matrix form the +1 eigenspace of T. The off-diagonal entries form its -1 eigenspace. Both spaces have dimension two.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-47c6df3af655c448","tex":"\\operatorname{Mat}_n","aliases":"Mat_n operatorname","display":"\\operatorname{Mat}_n","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-75ef01eda1abe09b","tex":"\\mathop","aliases":"mathop","display":null,"kind":"use","excerpt":"ced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) inside the esta","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"polynomial in calQ_n,B, the bounded cyclotomic alphabet, has coefficients in L. Form the union of all Galois conjugates in L of calP∪calQ_n,B. This is a fixed finite set of integral polynomials. For each distinct pair choose one nonzero coefficient difference. Exclude the rational primes below the prime divisors of all these differences, and exclud","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cd7ba0f5e9cf0737","tex":"\\max_t","aliases":"max_t max","display":"\\max_t","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"By [source reference], the stabilizer of the characteristic polynomial is {ζ∈μ_p:ζ^k=1 for every k∈ S(A)}. Since p is prime, a nontrivial pth root satisfies ζ^k=1 exactly when pmid k.","count":11},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3f620354ed5db565","tex":"\\mathbb N","aliases":"N mathbb","display":"\\mathbb N","kind":"use","excerpt":"For integral matrices suppose the inverse has the certified form Q=p^-sB, s∈ N, where B is an integral linear operator on the matrix entries. In a fixed ordering of the n^2 entries it is an n^2 by n^2 matrix over Z_p. This is an operator on the matrix space, not ordinary left multiplication by an n by n matrix. Integral coefficients imply normBz≤no","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"Absolute irreducibility and Burnside's theorem give group elements g_1,…,g_n^2 whose matrices form a K basis of M_n(K). Vectorize these matrices in the fixed basis. Their determinant is nonzero. Exclude places at which that determinant is zero or has a denominator, together with denominators of the finitely many group matrices. At a remaining place","count":10},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-da3577d9d5817603","tex":"N_v","aliases":"N_v","display":"N_v","kind":"use","excerpt":"Fix v and wmid v, and write the local Weil Deligne parameter as (ϕ_v,N_v). Base change restricts ϕ_v to W_E_w and keeps the monodromy operator. Since the parameter of Π_w has zero monodromy, N_v=0.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"use","excerpt":"[Trace and determinant selectors] A nonzero trace separates all cyclic twists. The determinant separates all cyclic twists when pnmid n.","count":11},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i,","count":25},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"bibitemRamakrishnan2002 D. Ramakrishnan, Modularity of solvable Artin representations of GO(4) type, International Mathematics Research Notices 2002(1), 1--54. DOI: https://doi.org/10.1155/S107379280200001610.1155/S1073792802000016.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f35ce0db28d3fb4f","tex":"\\OO","aliases":"OO","display":null,"kind":"use","excerpt":"Let K be a number field containing the normalized unramified Hecke coefficients of a finite Satake descent, the character values of the target Artin representation, and the values of the quotient character. Enlarge it to a splitting field in which the target and its restriction have matrix realizations. Let OO_K be its ring of integers.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-83c44e1ef141b100","tex":"\\OO_K","aliases":"OO_K OO","display":null,"kind":"use","excerpt":"Let K be a number field containing the normalized unramified Hecke coefficients of a finite Satake descent, the character values of the target Artin representation, and the values of the quotient character. Enlarge it to a splitting field in which the target and its restriction have matrix realizations. Let OO_K be its ring of integers.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-54037e9a059bedc5","tex":"\\overline B","aliases":"overline B overline","display":"\\overline B","kind":"use","excerpt":"The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with ","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7964cc9395c719a8","tex":"\\overline e","aliases":"overline e overline","display":"\\overline e","kind":"use","excerpt":"The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with ","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d472066b913c5d84","tex":"\\overline F","aliases":"overline F overline","display":"\\overline F","kind":"use","excerpt":"Let F be a number field, let G_F=Gal(overline F/F), and let ρ:G_F⟶GL_n(CC) be a continuous representation with finite image. Continuity is taken for the usual complex topology. Such a representation factors through the Galois group of a finite extension. Throughout the conjecture discussion, ρ is irreducible. The trivial one dimensional representat","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-712a4047f09e3ce3","tex":"\\overline J","aliases":"overline J overline","display":"\\overline J","kind":"use","excerpt":"The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with ","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ffdd263bffb70d69","tex":"\\overline P","aliases":"overline P overline","display":"\\overline P","kind":"use","excerpt":"[Frobenius transform after reduction] Let P have integral coefficients at a place of residue characteristic p. Then overlineP^[p](T) = Frob(overline P(T)), where Frobenius raises every coefficient to its pth power. In particular, the map overline P⟼overlineP^[p] is injective over the residue field.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a392a5969bc5b657","tex":"\\overline r_\\lambda","aliases":"overline r_λ overline lambda","display":"\\overline r_\\lambda","kind":"use","excerpt":"There is a finite set Σ(calP) of rational primes, fixed from the alphabet before any realization is chosen, with the following property. Let λ be a coefficient place whose residue characteristic ℓ is outside Σ(calP). Suppose a continuous semisimple representation overline r_λ:G_F⟶GL_n(overlineFF_ℓ) is unramified outside S and the places above ℓ, an","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b3f4c252ebea45ab","tex":"\\overline u","aliases":"overline u overline","display":"\\overline u","kind":"use","excerpt":"on B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and p","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ad21f4f96eb368f9","tex":"\\overline V","aliases":"overline V overline","display":"\\overline V","kind":"use","excerpt":"llapse of cyclic extension labels] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V,W be semisimple representations over a number field K with isomorphic restrictions to H. Choose stable lattices at a coefficient place above p. Over an algebraic closure of the residue field their semisimplified reductions satisfy overline V^ss≅overline","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d2e0f008aa7e4226","tex":"\\overline V^{\\mathrm{ss}}","aliases":"overline V^ss overline mathrm","display":"\\overline V^{\\mathrm{ss}}","kind":"use","excerpt":"llapse of cyclic extension labels] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V,W be semisimple representations over a number field K with isomorphic restrictions to H. Choose stable lattices at a coefficient place above p. Over an algebraic closure of the residue field their semisimplified reductions satisfy overline V^ss≅overline","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-70becec0d32f5b1e","tex":"\\overline W","aliases":"overline W overline","display":"\\overline W","kind":"use","excerpt":"llapse of cyclic extension labels] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V,W be semisimple representations over a number field K with isomorphic restrictions to H. Choose stable lattices at a coefficient place above p. Over an algebraic closure of the residue field their semisimplified reductions satisfy overline V^ss≅overline","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-787742bf72378d1a","tex":"\\overline W^{\\mathrm{ss}}","aliases":"overline W^ss overline mathrm","display":"\\overline W^{\\mathrm{ss}}","kind":"use","excerpt":"llapse of cyclic extension labels] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V,W be semisimple representations over a number field K with isomorphic restrictions to H. Choose stable lattices at a coefficient place above p. Over an algebraic closure of the residue field their semisimplified reductions satisfy overline V^ss≅overline","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9a7bbab7f9578bc9","tex":"\\overline\\FF","aliases":"overlineFF overline FF","display":null,"kind":"use","excerpt":"[Uniform bound for a residual Galois image] Let overline r:G_F⟶GL_n(overlineFF_ℓ) be continuous and semisimple, with ℓ>n. If its Frobenius traces belong to a set of size at most t on a Chebotarev dense set, then absimoverline r≤ t^n^2.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-85d6c82d4049e502","tex":"\\overline\\FF_\\ell","aliases":"overlineFF_ℓ overline FF ell","display":null,"kind":"use","excerpt":"There is a finite set Σ(calP) of rational primes, fixed from the alphabet before any realization is chosen, with the following property. Let λ be a coefficient place whose residue characteristic ℓ is outside Σ(calP). Suppose a continuous semisimple representation overline r_λ:G_F⟶GL_n(overlineFF_ℓ) is unramified outside S and the places above ℓ, an","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c40a1d9dd849fe23","tex":"\\overline\\FF_{\\ell}","aliases":"overlineFF_ℓ overline FF ell","display":null,"kind":"use","excerpt":"[Uniform bound for a residual Galois image] Let overline r:G_F⟶GL_n(overlineFF_ℓ) be continuous and semisimple, with ℓ>n. If its Frobenius traces belong to a set of size at most t on a Chebotarev dense set, then absimoverline r≤ t^n^2.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f2ea2b15782e3f37","tex":"\\overline{P^{[p]}}","aliases":"overlineP^[p] overline","display":"\\overline{P^{[p]}}","kind":"use","excerpt":"[Frobenius transform after reduction] Let P have integral coefficients at a place of residue characteristic p. Then overlineP^[p](T) = Frob(overline P(T)), where Frobenius raises every coefficient to its pth power. In particular, the map overline P⟼overlineP^[p] is injective over the residue field.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7c088714b77c8097","tex":"\\overline{P_v}","aliases":"overlineP_v overline","display":"\\overline{P_v}","kind":"use","excerpt":"from the alphabet before any realization is chosen, with the following property. Let λ be a coefficient place whose residue characteristic ℓ is outside Σ(calP). Suppose a continuous semisimple representation overline r_λ:G_F⟶GL_n(overlineFF_ℓ) is unramified outside S and the places above ℓ, and has characteristic polynomial overlineP_v at every rem","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-21cc0a3a6796152e","tex":"\\overline{P_{\\pi,v}}","aliases":"overlineP_π,v overline pi","display":"\\overline{P_{\\pi,v}}","kind":"use","excerpt":"gincorollary[A normalized torsion realization closes the weak step] Use the cyclic Artin setup and a good separating prime λ. Suppose the reduced eigensystem of a descent occurs in the cohomology range of [source reference], with all source hypotheses met, and an explicit normalization comparison identifies the source Frobenius polynomials with ove","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1f14e8def8f1a7a2","tex":"\\overline\\QQ","aliases":"overlineQQ overline QQ","display":null,"kind":"use","excerpt":"[One semisimple l adic realization is enough] Let π be a finite Satake cuspidal representation with algebraic unramified Hecke polynomials. Fix an embedding of their number field into overlineQQ_ℓ. Suppose there is a continuous semisimple representation r:G_F→GL_n(L), for a finite extension L/QQ_ℓ, unramified outside a finite set, whose Frobenius p","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-abeb7005614280fe","tex":"\\overline\\QQ_\\ell","aliases":"overlineQQ_ℓ overline QQ ell","display":null,"kind":"use","excerpt":"[One semisimple l adic realization is enough] Let π be a finite Satake cuspidal representation with algebraic unramified Hecke polynomials. Fix an embedding of their number field into overlineQQ_ℓ. Suppose there is a continuous semisimple representation r:G_F→GL_n(L), for a finite extension L/QQ_ℓ, unramified outside a finite set, whose Frobenius p","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2fbddbaa9303e3f3","tex":"\\overline\\alpha","aliases":"overlineα overline alpha","display":"\\overline\\alpha","kind":"use","excerpt":"Over an algebraic closure of the residue field, overlineP^[p](T) = ∏_i(T-overlineα_i^p). The elementary symmetric functions of the pth powers are the pth powers of the elementary symmetric functions in characteristic p. Finite residue fields are perfect, so Frobenius is injective.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0c522ef5feeb0ed5","tex":"\\overline\\alpha_i^p","aliases":"overlineα_i^p overline alpha","display":"\\overline\\alpha_i^p","kind":"use","excerpt":"Over an algebraic closure of the residue field, overlineP^[p](T) = ∏_i(T-overlineα_i^p). The elementary symmetric functions of the pth powers are the pth powers of the elementary symmetric functions in characteristic p. Finite residue fields are perfect, so Frobenius is injective.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-91424c61d93b987b","tex":"\\overline\\epsilon","aliases":"overlineϵ overline epsilon","display":"\\overline\\epsilon","kind":"use","excerpt":"The representations overline r_π,λ and overlineρ are therefore two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5b1e617678ada53d","tex":"\\overline\\epsilon^{j_{\\lambda}}","aliases":"overlineϵ^j_λ overline epsilon lambda","display":"\\overline\\epsilon^{j_{\\lambda}}","kind":"use","excerpt":"The representations overline r_π,λ and overlineρ are therefore two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-881e84403d3704c7","tex":"\\overline\\rho","aliases":"overlineρ overline rho","display":"\\overline\\rho","kind":"use","excerpt":"The representations overline r_π,λ and overlineρ are therefore two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-63e70752eccb080b","tex":"\\overline{\\rho\\otimes\\epsilon^j}","aliases":"overlineρ⊗ϵ^j overline rho otimes epsilon","display":"\\overline{\\rho\\otimes\\epsilon^j}","kind":"use","excerpt":"[All cyclic twists collapse residually] For every j∈ZZ/pZZ, overlineρ⊗ϵ^j ≅ overlineρ at the semisimple character level modulo λ_p. Likewise, all automorphic descents have the same mod λ_p unramified Hecke eigensystem.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0f205f23f823611e","tex":"\\overline\\tau","aliases":"overlineτ overline tau","display":"\\overline\\tau","kind":"use","excerpt":"Restrict the residual realization to G_E. At every unramified place w of E above a good place v of F, cyclic base change identifies the characteristic polynomial of the restriction with the reduction of the Hecke polynomial of Π_w. Since Π is attached to τ, this is the characteristic polynomial of overlineτ(Frob_w). Chebotarev density and Brauer Ne","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2fdcc127f0252c77","tex":"p^{-(m+s)}","aliases":"p^-(m+s)","display":"p^{-(m+s)}","kind":"use","excerpt":"Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag · Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a ro","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e57276c69cbfb04f","tex":"p^{-1}","aliases":"p^-1","display":"p^{-1}","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8a1b7e2cb35dff5a","tex":"p^{-m}","aliases":"p^-m","display":"p^{-m}","kind":"use","excerpt":"The proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-acf97391a5aa4604","tex":"p^{-s}","aliases":"p^-s","display":"p^{-s}","kind":"use","excerpt":"For integral matrices suppose the inverse has the certified form Q=p^-sB, s∈ N, where B is an integral linear operator on the matrix entries. In a fixed ordering of the n^2 entries it is an n^2 by n^2 matrix over Z_p. This is an operator on the matrix space, not ordinary left multiplication by an n by n matrix. Integral coefficients imply normBz≤no","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7441b5cfbaa2c0c0","tex":"p^m","aliases":"p^m","display":"p^m","kind":"use","excerpt":"· Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a root in A+p^mMat_n( Z_p) by one finite target test, once the exact inverse certificate is supplied.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5edd1e60150c6c1b","tex":"p^{m+s}","aliases":"p^m+s","display":"p^{m+s}","kind":"use","excerpt":"Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag · Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a ro","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6f0ea21bbfbf69fb","tex":"p^r","aliases":"p^r","display":"p^r","kind":"use","excerpt":"This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predi","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-360f794da8094012","tex":"p^s","aliases":"p^s","display":"p^s","kind":"use","excerpt":"he proof gives a controlled approximation process. Start at h_0=0 and iterate h_j+1=Φ(h_j). Then h_1=QΔ, and normh_j+1-h_j≤ (Kr)^jnormh_1. For t>j, ultrametricity bounds normh_t-h_j by the largest of the intervening increments, which is at most (Kr)^jnormh_1. The limit therefore satisfies the same bound for its distance from h_j. With r=p^-m and K=","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4df24df273370dad","tex":"p^{s-m}","aliases":"p^s-m","display":"p^{s-m}","kind":"use","excerpt":"Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-690173a5c81110fe","tex":"p_p","aliases":"p_p","display":"p_p","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4d3243980b9ae21f","tex":"P_r","aliases":"P_r","display":"P_r","kind":"use","excerpt":"Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e70fd9df7027e2bf","tex":"P_v","aliases":"P_v","display":"P_v","kind":"use","excerpt":"[One large residual realization gives an Artin representation] Let K be a number field, let S be a finite set of places of F, and let calP⊆OO_K[T] be a nonempty finite set of monic degree n polynomials. Assume all their roots are roots of unity. For v∉ S, choose P_v∈calP.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3e8303224d02c139","tex":"P_{\\pi,v}^{[p]}","aliases":"P_π,v^[p] pi","display":"P_{\\pi,v}^{[p]}","kind":"use","excerpt":"Assume that v is unramified and nonsplit. Since the extension has prime degree, its residue degree is p. Unramified base change raises the Satake conjugacy class of π_v to its pth power. Restriction of ρ_v to W_E_w does the same to its Frobenius class. Since Π_w is attached to ρ|_W_E_w, P_π,v^[p](T) = P_ρ,v^[p](T). Reduce modulo λ_p and use [source","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-aebd035657016071","tex":"P_{\\rho,v}^{[p]}","aliases":"P_ρ,v^[p] rho","display":"P_{\\rho,v}^{[p]}","kind":"use","excerpt":"Assume that v is unramified and nonsplit. Since the extension has prime degree, its residue degree is p. Unramified base change raises the Satake conjugacy class of π_v to its pth power. Restriction of ρ_v to W_E_w does the same to its Frobenius class. Since Π_w is attached to ρ|_W_E_w, P_π,v^[p](T) = P_ρ,v^[p](T). Reduce modulo λ_p and use [source","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4c498dbe21691fc7","tex":"P_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}","aliases":"P_ρ⊗ϵ^j_λ,v rho otimes epsilon lambda","display":"P_{\\rho\\otimes\\epsilon^{j_{\\lambda}},v}","kind":"use","excerpt":"cyclic Artin setup of [source reference]. For this unramified theorem it is enough that the lower cuspidal representation Π is weakly attached to τ, with π a cuspidal cyclic descent. Let λ be a good separating prime. Suppose π has a residual realization overline r_π,λ. Then there is a unique index j_λ∈ZZ/pZZ such that P_π,v(T) = P_ρ⊗ϵ^j_λ,v(T) for ","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-76299a96a93c97e0","tex":"P_{\\tau,V_0}","aliases":"P_τ,V_0 tau","display":"P_{\\tau,V_0}","kind":"use","excerpt":"[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8aa1e974efda4bcd","tex":"P_{\\tau,V_1}","aliases":"P_τ,V_1 tau","display":"P_{\\tau,V_1}","kind":"use","excerpt":"[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b7ed02d87cfc62ef","tex":"P_{\\tau,V\\otimes\\xi^j}","aliases":"P_τ,V⊗ξ^j tau otimes xi","display":"P_{\\tau,V\\otimes\\xi^j}","kind":"use","excerpt":"[Effect of quotient twisting] For every j∈ZZ/pZZ, P_τ,V⊗ξ^j(z) = z^jP_τ,V(z). Moving orbit blocks are unchanged up to isomorphism by quotient twisting.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ. At every good unramified place, P_π,v ≡ P_ρ⊗ϵ^j_λ,v pmodλ. Bo","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"For every ordered pair of distinct polynomials P,Q∈calA, choose one nonzero coefficient difference. Only finitely many finite places divide one of these finitely many nonzero algebraic integers. Outside their union, no two polynomials have the same reduction.","count":17},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cc70f65000acd90e","tex":"\\mathbb Q","aliases":"Q mathbb","display":"\\mathbb Q","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"bibitemKimYamauchi2016 H. H. Kim and T. Yamauchi, A uniform structure on subgroups of GL_n( F_q) and its application to a conditional construction of Artin representations of GL_n, Journal of the Ramanujan Mathematical Society 32(1) (2017), 75--99. Checked preprint https://arxiv.org/abs/1403.5535arXiv:1403.5535v3, 7 April 2016. Theorem 1.1 is condi","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-812a1fd77a9f35a3","tex":"\\mathbb Q_2","aliases":"Q_2 mathbb","display":"\\mathbb Q_2","kind":"use","excerpt":"The residual derivative is singular. In characteristic two, D_A,2(I_2)=2A=0, while I_2≠0. Nevertheless J is invertible over Q_2, with determinant 16. Residual singularity and singularity over the fraction field are different tests.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cd98a9c47ecfa404","tex":"\\mathbb Q_p","aliases":"Q_p mathbb","display":"\\mathbb Q_p","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-44097bbb5591bf32","tex":"q_v^{i(n+1)/2}","aliases":"q_v^i(n+1)/2","display":"q_v^{i(n+1)/2}","kind":"use","excerpt":"t[A torsion cohomology realization range] Scholze's Corollary 5.4.3 supplies a unique continuous semisimple residual representation for a Hecke eigensystem actually occurring in the stated cohomology of a locally symmetric space, under the field, level, coefficient, and theorem dependencies of that source [source reference]. The specified polynomia","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"use","excerpt":"Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_g","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2a7405e4d1e4acc5","tex":"r'_g","aliases":"r'_g","display":"r'_g","kind":"use","excerpt":"Change the chosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the se","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b59724bb8a759a2a","tex":"r'_{gh}","aliases":"r'_gh","display":"r'_{gh}","kind":"use","excerpt":"Change the chosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the se","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-bdc38a8e76818a36","tex":"r'_h","aliases":"r'_h","display":"r'_h","kind":"use","excerpt":"Change the chosen lifts to r'_g=(1+b(g))r_g, where b:G→ I and b(1)=0. Expanding without commuting the factors gives r'_gr'_h =(1+b(g)+g· b(h)+c(g,h))r_gh. To compare with r'_gh, multiply by 1-b(gh). The new defect is c'(g,h)=c(g,h)+g· b(h)-b(gh)+b(g). Thus changing the section adds a two-coboundary. The class [c]∈ H^2(G,I) does not depend on the se","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-acf48a69456993a9","tex":"R^{-1}","aliases":"R^-1","display":"R^{-1}","kind":"use","excerpt":"ves an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These ar","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-aff4a4490ea52474","tex":"r^{-1}","aliases":"r^-1","display":"r^{-1}","kind":"use","excerpt":"The Lie group GL_n(CC) has a neighborhood U of the identity that contains no nontrivial subgroup. Continuity gives a neighborhood r^-1(U) of the identity in G. A profinite group has a basis of open normal subgroups, so one such subgroup N lies inside r^-1(U). Its image is a subgroup contained in U, hence is trivial. Therefore r factors through the ","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-efe0bc4f1e105018","tex":"R^p","aliases":"R^p","display":"R^p","kind":"use","excerpt":"ves an invertible intertwiner A with Aτ(h)A^-1=τ(ghg^-1) (h∈ H). Iterate this identity p times. The operator τ(g^p)^-1A^p commutes with τ(H), so Schur's lemma makes it a nonzero scalar cI. Also A commutes with τ(g^p), by the same intertwining identity at h=g^p. Choose b∈CC^× with b^p=c^-1 and put R=bA. Then Rτ(h)R^-1=τ(ghg^-1), R^p=τ(g^p). These ar","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1752f9e9c50e0ce5","tex":"R^\\times","aliases":"R^× times","display":"R^\\times","kind":"use","excerpt":"Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_g","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_g","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-49fcbff1ee511b84","tex":"R_{A,3}","aliases":"R_A,3","display":"R_{A,3}","kind":"use","excerpt":"For degree three, the linear term is D_A,3(h)=A^2h+AhA+hA^2, and the nonlinear remainder is R_A,3(h)=Ah^2+hAh+h^2A+h^3. The middle word hAh cannot be absorbed into Ah^2 in a matrix ring. The remainder estimate keeps all four words and bounds each by its norm, so it respects the ordered expansion. This also explains why the estimate is not an assert","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9cf7b90f63cc0c07","tex":"R_{A,d}","aliases":"R_A,d","display":"R_{A,d}","kind":"use","excerpt":"Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matt","count":11},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-30553085a3ccaa3c","tex":"R_{A,d+1}","aliases":"R_A,d+1","display":"R_{A,d+1}","kind":"use","excerpt":"The recursive identities are D_A,d+1(h) · =D_A,d(h)A+A^dh, R_A,d+1(h) · =R_A,d(h)(A+h)+D_A,d(h)h. They follow by multiplying (A+h)^d on the right by A+h and collecting the terms with zero, one, and at least two occurrences of h. This derivation does not rearrange any noncommuting factors.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-71e0f18ed7179e19","tex":"R_d","aliases":"R_d","display":"R_d","kind":"use","excerpt":"Induct on d for the second estimate. The case d=0 is immediate. Write D_d=D_A,d and R_d=R_A,d. Subtracting the remainder recursions gives R_d+1(h)-R_d+1(k) = · (R_d(h)-R_d(k))(A+h)","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5a7ee9546a2ddb0b","tex":"R_{d+1}","aliases":"R_d+1","display":"R_{d+1}","kind":"use","excerpt":"Induct on d for the second estimate. The case d=0 is immediate. Write D_d=D_A,d and R_d=R_A,d. Subtracting the remainder recursions gives R_d+1(h)-R_d+1(k) = · (R_d(h)-R_d(k))(A+h) · +R_d(k)(h-k) · +(D_d(h)-D_d(k))h+D_d(k)(h-k). The first term has norm at most rnormh-k by induction and normA+h≤1. Taking h=k and the other argument zero in the induct","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ade69a4d18e62b84","tex":"r_g","aliases":"r_g","display":"r_g","kind":"use","excerpt":"Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_g","count":12},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6ac1ea8c38f7651b","tex":"r_g^{-1}","aliases":"r_g^-1","display":"r_g^{-1}","kind":"use","excerpt":"The action of G on I is g· x=r_gxr_g^-1. It is independent of the chosen unit lift. Indeed, any other unit lift has the form (1+b)r_g with b∈ I. Since (1+b)^-1=1-b and products of two elements of I vanish, conjugating x∈ I by 1+b leaves x unchanged. The same observation shows that the defect in the multiplication of the r_g does not prevent these c","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-27204dac11744c0b","tex":"r_{gh}","aliases":"r_gh","display":"r_{gh}","kind":"use","excerpt":"Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_g","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2aabb4c9ba5f3f33","tex":"r_{gh}^{-1}","aliases":"r_gh^-1","display":"r_{gh}^{-1}","kind":"use","excerpt":"Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_g","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-56d9445f743a055e","tex":"r_{gh\\ell}","aliases":"r_ghℓ ell","display":"r_{gh\\ell}","kind":"use","excerpt":"Compute (r_gr_h)r_ℓ and r_g(r_hr_ℓ). After expressing both in terms of r_ghℓ, the first is (1+c(g,h)+c(gh,ℓ))r_ghℓ, while the second is (1+g· c(h,ℓ)+c(g,hℓ))r_ghℓ. All omitted cross products belong to I^2. Associativity therefore yields g· c(h,ℓ)-c(gh,ℓ)+c(g,hℓ)-c(g,h)=0. This is the usual inhomogeneous two-cocycle identity. Normalization at the id","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-455a3f2c7c575c0e","tex":"r_h","aliases":"r_h","display":"r_h","kind":"use","excerpt":"Let q:R'→ R be a ring map with kernel I, and assume I^2=0. Let ρ:G→ R^× be a group representation. Choose unit-valued lifts r_g∈ (R')^× with q(r_g)=ρ(g) and r_1=1. Such a choice is an input to this calculation. It is not already a multiplicative lift. Define its normalized defect by c(g,h)=(r_gr_h-r_gh)r_gh^-1∈ I. Equivalently, r_gr_h=(1+c(g,h))r_g","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1d2c80661490944e","tex":"r_\\gamma","aliases":"r_γ gamma","display":"r_\\gamma","kind":"use","excerpt":"The defect can also be seen without an existence argument. Choose the section r_1=1, r_γ=4, and r_γ^2=16 modulo 27. These reduce to the required units modulo 9. The products of consecutive powers have no defect until the exponent passes through three. At that point 4^3=10=1+9 modulo 27. Consequently the normalized defect is 9 when the exponents wra","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0c407c3b148cee23","tex":"r_{\\gamma^2}","aliases":"r_γ^2 gamma","display":"r_{\\gamma^2}","kind":"use","excerpt":"The defect can also be seen without an existence argument. Choose the section r_1=1, r_γ=4, and r_γ^2=16 modulo 27. These reduce to the required units modulo 9. The products of consecutive powers have no defect until the exponent passes through three. At that point 4^3=10=1+9 modulo 27. Consequently the normalized defect is 9 when the exponents wra","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cf15fb9e7a0ffdaf","tex":"r_\\ell","aliases":"r_ℓ ell","display":"r_\\ell","kind":"use","excerpt":"Compute (r_gr_h)r_ℓ and r_g(r_hr_ℓ). After expressing both in terms of r_ghℓ, the first is (1+c(g,h)+c(gh,ℓ))r_ghℓ, while the second is (1+g· c(h,ℓ)+c(g,hℓ))r_ghℓ. All omitted cross products belong to I^2. Associativity therefore yields g· c(h,ℓ)-c(gh,ℓ)+c(g,hℓ)-c(g,h)=0. This is the usual inhomogeneous two-cocycle identity. Normalization at the id","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-06b25b660a3b775f","tex":"\\Res","aliases":"Res","display":null,"kind":"use","excerpt":"[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) ","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8db5db648c850430","tex":"\\operatorname{res}","aliases":"res operatorname","display":"\\operatorname{res}","kind":"use","excerpt":"To see the criterion, let a_i be the local mismatch after b_0. A match up to the allowed local conjugations exists exactly when a_i=res_i(z)+δ x_i for a global continuous one-cocycle z and kernel elements x_i. This is exactly the assertion that the image of (a_i)_i in [source reference] is zero. If b_1 is another absolute primitive, then b_1-b_0 is","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6fe0a2b95bd926cf","tex":"\\Res_H","aliases":"Res_H Res","display":null,"kind":"use","excerpt":"[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) ","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-feed6c2e1f34a429","tex":"\\operatorname{res}_i","aliases":"res_i operatorname","display":"\\operatorname{res}_i","kind":"use","excerpt":"To see the criterion, let a_i be the local mismatch after b_0. A match up to the allowed local conjugations exists exactly when a_i=res_i(z)+δ x_i for a global continuous one-cocycle z and kernel elements x_i. This is exactly the assertion that the image of (a_i)_i in [source reference] is zero. If b_1 is another absolute primitive, then b_1-b_0 is","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5fa25adb3ead8c85","tex":"\\Res_{W_{E_w}}","aliases":"Res_W_E_w Res","display":null,"kind":"use","excerpt":"For each place v of F, put R_v=ρ|_W_F_v, A_v=rec_F_v(π_0,v). Cyclic base change compatibility gives Res_W_E_wR_v ≅ Res_W_E_wA_v.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fbfea48d83e1bf18","tex":"\\rtimes","aliases":"rtimes","display":null,"kind":"use","excerpt":"The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together w","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-226626f51f7539ed","tex":"\\rvert","aliases":"rvert","display":"\\rvert","kind":"use","excerpt":"from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an X holomorphic limit of discrete series [source reference]. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the dens","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ccab82c649c12498","tex":"\\rvert^{(1-n)/2}","aliases":"rvert^(1-n)/2 rvert","display":"\\rvert^{(1-n)/2}","kind":"use","excerpt":"from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an X holomorphic limit of discrete series [source reference]. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the dens","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"[A moving type and self twist] Let G=S_3, let H=A_3, and let V be the standard two dimensional representation. Its restriction is Res_HV=τ⊕τ^-1 for the two nontrivial characters of C_3. These characters form one moving orbit under G/H≅ C_2. Hence there is no stable extension spectrum, and V⊗sgn≅ V.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-28f39655338ed57d","tex":"\\operatorname{sgn}","aliases":"sgn operatorname","display":"\\operatorname{sgn}","kind":"use","excerpt":"[A moving type and self twist] Let G=S_3, let H=A_3, and let V be the standard two dimensional representation. Its restriction is Res_HV=τ⊕τ^-1 for the two nontrivial characters of C_3. These characters form one moving orbit under G/H≅ C_2. Hence there is no stable extension spectrum, and V⊗sgn≅ V.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cdec095fec211ac6","tex":"\\mathrm{SL}","aliases":"SL mathrm","display":"\\mathrm{SL}","kind":"use","excerpt":"The solvable two dimensional sectors treated by Langlands and Tunnell, further examples such as the solvable GO(4) range, and other established cases show that the conjecture is far from an empty expectation [source reference]. A recent preprint of Wang states the generalized octahedral GL_3 case with projective image C_3^2rtimesSL(2,3), together w","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e9c1c42c5931077f","tex":"\\operatorname{span}_k","aliases":"span_k operatorname","display":"\\operatorname{span}_k","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-348ccb55d8cc92e4","tex":"\\mathrm{ss}","aliases":"ss mathrm","display":"\\mathrm{ss}","kind":"use","excerpt":"e residual realization to G_E. At every unramified place w of E above a good place v of F, cyclic base change identifies the characteristic polynomial of the restriction with the reduction of the Hecke polynomial of Π_w. Since Π is attached to τ, this is the characteristic polynomial of overlineτ(Frob_w). Chebotarev density and Brauer Nesbitt give ","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":27},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c0614079049a7773","tex":"T^2","aliases":"T^2","display":"T^2","kind":"use","excerpt":"Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the th","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-87bd5948a28faad5","tex":"T^3","aliases":"T^3","display":"T^3","kind":"use","excerpt":"Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the th","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4d76eacfb2ced296","tex":"t^{\\dim_k A}","aliases":"t^dim_k A dim","display":"t^{\\dim_k A}","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9ea7696d8ec4186b","tex":"T^j","aliases":"T^j","display":"T^j","kind":"use","excerpt":"A one-cocycle is a function a:C_e→ M satisfying a(gh)=a(g)+g· a(h). Putting g=h=1 gives a(1)=0. If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a tra","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-dcaceb4d60dad5f5","tex":"T^{n-k}","aliases":"T^n-k","display":"T^{n-k}","kind":"use","excerpt":"Let ξ be a generator of X, and identify its values with pth roots of unity. For a matrix A∈GL_n(CC), write det(TI-A) = ∑_k=0^n(-1)^k e_k(A)T^n-k.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0d97ae22ba78aa56","tex":"t^{n^2}","aliases":"t^n^2","display":"t^{n^2}","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-645a5cbe71427f86","tex":"t^r","aliases":"t^r","display":"t^r","kind":"use","excerpt":"t to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t^r. This proves finiteness as well as the bound. Prior finiteness of Γ was not needed in this arg","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2a99a3c1947195ac","tex":"T_{i,v}","aliases":"T_i,v","display":"T_{i,v}","kind":"use","excerpt":"t[A torsion cohomology realization range] Scholze's Corollary 5.4.3 supplies a unique continuous semisimple residual representation for a Hecke eigensystem actually occurring in the stated cohomology of a locally symmetric space, under the field, level, coefficient, and theorem dependencies of that source [source reference]. The specified polynomia","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr","display":null,"kind":"use","excerpt":"[Fourier trace interpretation] Let g be a lift of a generator of G/H. For a stable type τ, widehat P_τ,V(r) = Tr(g^rmid M_τ(V)). If V is twisted by ξ^j, then widehat P_τ,V⊗ξ^j(r) = ζ^jrwidehat P_τ,V(r).","count":13},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-edbc8a146250e801","tex":"\\Tr_{k^n}","aliases":"Tr_k^n Tr","display":null,"kind":"use","excerpt":"le. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) res","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-65fa7282e2650c3e","tex":"\\triangleleft","aliases":"triangleleft","display":null,"kind":"use","excerpt":"The finite representation arguments have a parallel shape. If Htriangleleft G and G/H≅ C_p, an irreducible representation either restricts to a full orbit of p subgroup types or restricts irreducibly to one invariant type. In the first case induction reconstructs the representation. In the second case an extension must be selected. The second case ","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"The Lie group GL_n(CC) has a neighborhood U of the identity that contains no nontrivial subgroup. Continuity gives a neighborhood r^-1(U) of the identity in G. A profinite group has a basis of open normal subgroups, so one such subgroup N lies inside r^-1(U). Its image is a subgroup contained in U, hence is trivial. Therefore r factors through the ","count":22},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is dete","count":8},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8b35b3f63eb8e5bc","tex":"U^{-1}","aliases":"U^-1","display":"U^{-1}","kind":"use","excerpt":"The root constructed over Q_p is integral. Its center is integral, and every correction entry has norm at most p^-m≤1. If d>0 and U is a unit, the root F is also a unit. Indeed, F commutes with U=F^d and hence with U^-1, and F^d-1U^-1 is a two-sided inverse of F. Thus the unit version has an actual output in GL_n( Z_p).","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-678996c62e00979a","tex":"U_{\\mathrm{bad}}","aliases":"U_bad mathrm","display":"U_{\\mathrm{bad}}","kind":"use","excerpt":"For a negative control, take U_bad=1 · 10 0 · 1. The difference from A^2 is (0,8,0,0) in entry coordinates, and B sends it to (0,16,0,0). This is nonzero modulo 32, so no square root of U_bad lies in that ball. This is a failure for the specified prefix. It is not a claim that the target has no square root anywhere in a larger coefficient field.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predi","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f238e04705656734","tex":"V'","aliases":"V'","display":"V'","kind":"use","excerpt":"[Stable equality of epsilon factors] Let V,V' be finite dimensional complex Weil representations with finite inertia image, of equal dimension and determinant. For every sufficiently ramified auxiliary character η, ϵ(V⊗η,Ψ)=ϵ(V'⊗η,Ψ). Here Ψ is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henni","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4b270ddb72d4b503","tex":"V^*","aliases":"V^*","display":"V^*","kind":"use","excerpt":"Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is dete","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ae3df08d949531da","tex":"V_0","aliases":"V_0","display":"V_0","kind":"use","excerpt":"[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-af5dd21930308f17","tex":"V_1","aliases":"V_1","display":"V_1","kind":"use","excerpt":"[A stable cyclic type] Let G=C_3× C_2, H=C_3. Fix a nontrivial character τ of H. It is stable under G. After choosing the extension with trivial C_2 coordinate, the two one dimensional extensions have spectra P_τ,V_0(z)=1, P_τ,V_1(z)=z. The alignment set is the singleton containing the unique quotient twist carrying one extension to the other.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2c003b9cf5057fca","tex":"v_p","aliases":"v_p","display":"v_p","kind":"use","excerpt":"The value of s measures denominators of the whole inverse derivative. It is not automatically v_p(d). The proof makes no statement about shallow balls with m≤ s, where its contraction estimate is insufficient. It also does not provide automatic certificate discovery for arbitrary p-adic input. Exact rational examples can supply certificates by ordi","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f7cc81212c08d083","tex":"\\operatorname{vec}","aliases":"vec operatorname","display":"\\operatorname{vec}","kind":"use","excerpt":"Take r=p^-m with m>s. Then Kr=p^s-m<1, and normQ(U-A^d)≤ p^-m · Longleftrightarrow normB(U-A^d)≤ p^-(m+s)notag · Longleftrightarrow Bvec(U-A^d)=0pmod p^m+s. The last congruence is entrywise. It depends only on the reductions of the target, center, and integral operator modulo p^m+s. The theorem therefore decides the existence and uniqueness of a ro","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"[Determinant character of an algebraic representation] Let r:GL_n(CC)⟶GL_N(CC) be an irreducible algebraic representation. Suppose the center acts with weight w, so r(zI_n)=z^wI_N. There is an integer a such that det r(g)=det(g)^a and wN=an.","count":31},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-73ffecb5618fb3cc","tex":"W^*","aliases":"W^*","display":"W^*","kind":"use","excerpt":"Let J:V→ W be a linear map over a field k. The correction equation Ju+e=0 has a solution precisely when e∈imJ. A useful description of this image uses all the dual relations among the equations. Let S=ker(J^*:W^*→ V^*), C={w∈ W:ϕ(w)=0 for every ϕ∈ S}. Every Ju belongs to C. Conversely, if w does not belong to imJ, its nonzero class in W/imJ is dete","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-438757f12dd8c3fc","tex":"w_1","aliases":"w_1","display":"w_1","kind":"use","excerpt":"[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, sinc","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3106fac3e3c8f992","tex":"w_2","aliases":"w_2","display":"w_2","kind":"use","excerpt":"[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, sinc","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9763a163e7ff6591","tex":"w_3","aliases":"w_3","display":"w_3","kind":"use","excerpt":"[A reachable target with a visible failed correction] Let V=k^2, W=k^3, and J(u,v)=(u,v,u+v). Its image is the plane w_3=w_1+w_2. The dual kernel is generated by the functional ϕ(w_1,w_2,w_3)=w_3-w_1-w_2. The error (1,2,3) is compatible in every characteristic and is cancelled by (u,v)=(-1,-2). The error (0,0,1) is incompatible in every field, sinc","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fbb08df020898d1b","tex":"W_{E_w}","aliases":"W_E_w","display":"W_{E_w}","kind":"use","excerpt":"For each place v of F, put R_v=ρ|_W_F_v, A_v=rec_F_v(π_0,v). Cyclic base change compatibility gives Res_W_E_wR_v ≅ Res_W_E_wA_v.","count":8},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0814e679b4a9f66d","tex":"\\widehat P_{\\tau,V\\otimes\\xi^j}","aliases":"widehat P_τ,V⊗ξ^j widehat tau otimes xi","display":"\\widehat P_{\\tau,V\\otimes\\xi^j}","kind":"use","excerpt":"[Fourier trace interpretation] Let g be a lift of a generator of G/H. For a stable type τ, widehat P_τ,V(r) = Tr(g^rmid M_τ(V)). If V is twisted by ξ^j, then widehat P_τ,V⊗ξ^j(r) = ζ^jrwidehat P_τ,V(r).","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e81b61047eca0739","tex":"x'","aliases":"x'","display":"x'","kind":"use","excerpt":"J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"For a nonlinear system F(x)=0, a relation among the residual Jacobian rows is only a first-order identity. Consider F(x)=(x,x^2), J_x(h)=(h,2xh). At the origin the derivative has horizontal image. The functional selecting the second coordinate is a residual syzygy, but it does not annihilate F(x) for general x. The exact relation is the moving rela","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Suppose an absolute two-cocycle has been cancelled by a continuous one-cochain b_0. Prescribed local lifts need not agree with the resulting global lift. Their remaining differences are local one-cocycles. Changing b_0 to another absolute primitive changes these differences by the restriction of a global one-cocycle. Conjugating the local lift at i","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e6237e701c3d5270","tex":"X_{it}","aliases":"X_it","display":"X_{it}","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8285a92cff098813","tex":"X_r","aliases":"X_r","display":"X_r","kind":"use","excerpt":"Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8981b26b3b446b1c","tex":"x_r","aliases":"x_r","display":"x_r","kind":"use","excerpt":"Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"use","excerpt":"Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1029cb8ad51ed272","tex":"x_t","aliases":"x_t","display":"x_t","kind":"use","excerpt":"Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) restricts on the ith block to m_iTr(XY). Matrix trace pairing ","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"le. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. The trace pairing B(x,y)=Tr_k^n(xy) res","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d1b488be8bfdfa08","tex":"Y_{tj}","aliases":"Y_tj","display":"Y_{tj}","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-22ca0cbe9349b6f1","tex":"\\mathbb Z","aliases":"Z mathbb","display":"\\mathbb Z","kind":"use","excerpt":"[An obstruction that really does not vanish] Reduce Z/27 Z to Z/9 Z. Its kernel is 9 Z/27 Z, whose square is zero. Map a generator of C_3 to the unit 4 modulo 9. This defines a representation because 4^3=1 modulo 9.","count":10},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"Take G=C_2× C_3, let H=C_2, and let ξ be the quotient character taking a generator of C_3 to ω. Let τ be the nontrivial character of H, extended with trivial quotient action. Consider a semisimple representation with spectrum P_τ,V(z)=2+z. Its quotient multiplicity vector is (2,1,0). Twisting by ξ gives zP_τ,V(z)=2z+z^2, whose vector is (0,2,1). Tw","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7e25859a8ab59d67","tex":"z^{an}","aliases":"z^an","display":"z^{an}","kind":"use","excerpt":"Every algebraic character of GL_n is an integral power of the determinant. Thus det∘ r=det^a for one integer a. Evaluate both sides at zI_n. The left side is z^wN, while the right side is z^an.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c1ea6dc7e25e9163","tex":"z^w","aliases":"z^w","display":"z^w","kind":"use","excerpt":"[Determinant character of an algebraic representation] Let r:GL_n(CC)⟶GL_N(CC) be an irreducible algebraic representation. Suppose the center acts with weight w, so r(zI_n)=z^wI_N. There is an integer a such that det r(g)=det(g)^a and wN=an.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-de1511d5f26cf71c","tex":"z^{wN}","aliases":"z^wN","display":"z^{wN}","kind":"use","excerpt":"Every algebraic character of GL_n is an integral power of the determinant. Thus det∘ r=det^a for one integer a. Evaluate both sides at zI_n. The left side is z^wN, while the right side is z^an.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-370c87df70534fa4","tex":"\\mathbb Z_2","aliases":"Z_2 mathbb","display":"\\mathbb Z_2","kind":"use","excerpt":"Work over Z_2 and take A=1 · 1","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a82161455c4d6fea","tex":"z_{i,r}","aliases":"z_i,r","display":"z_{i,r}","kind":"use","excerpt":"This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predi","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c494253c0c6e8c91","tex":"z_{i,r}^{-1}","aliases":"z_i,r^-1","display":"z_{i,r}^{-1}","kind":"use","excerpt":"This matters for simultaneous matrix prescriptions. At level r, a node can contain a shared matrix F_r together with one local conjugator z_i,r for each member of a finite family. The equations are z_i,rF_r^d_iz_i,r^-1=U_ipmod p^r, with the prescribed prefix of F_r and prescribed residual identity for each conjugator. These are a single joint predi","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8723c894d5150ea2","tex":"\\mathbb Z_p","aliases":"Z_p mathbb","display":"\\mathbb Z_p","kind":"use","excerpt":"For integral matrices suppose the inverse has the certified form Q=p^-sB, s∈ N, where B is an integral linear operator on the matrix entries. In a fixed ordering of the n^2 entries it is an n^2 by n^2 matrix over Z_p. This is an operator on the matrix space, not ordinary left multiplication by an n by n matrix. Integral coefficients imply normBz≤no","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3d7f36918ca613a9","tex":"|_H","aliases":"|_H","display":"|_H","kind":"use","excerpt":"[The two prime cyclic cases] Let G be finite, let Htriangleleft G with G/H≅ C_p, and let V be irreducible over CC. Exactly one of the following occurs: item V|_H is irreducible and invariant under G, and V is one of its p quotient character extensions. item V|_H is a sum of p pairwise nonisomorphic conjugate irreducible representations, and V is in","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ae8ddb0b385adcc1","tex":"|_p","aliases":"|_p","display":"|_p","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9e63cc6587f37669","tex":"|_{W_{E_w}}","aliases":"|_W_E_w","display":"|_{W_{E_w}}","kind":"use","excerpt":"Assume that v is unramified and nonsplit. Since the extension has prime degree, its residue degree is p. Unramified base change raises the Satake conjugacy class of π_v to its pth power. Restriction of ρ_v to W_E_w does the same to its Frobenius class. Since Π_w is attached to ρ|_W_E_w, P_π,v^[p](T) = P_ρ,v^[p](T). Reduce modulo λ_p and use [source","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"use","excerpt":"from a simple unitary Kottwitz datum and a cuspidal representation whose infinity component is an X holomorphic limit of discrete series [source reference]. At a coefficient prime satisfying the good reduction and reflex field conditions, it supplies a semisimple l adic representation over the CM field. Its local compatibility is stated on the dens","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d3ec1c46c6686070","tex":"\\alpha_1","aliases":"α_1 alpha","display":"\\alpha_1","kind":"use","excerpt":"Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-242a27e1f9b3b6e8","tex":"\\alpha_i^f","aliases":"α_i^f alpha","display":"\\alpha_i^f","kind":"use","excerpt":"Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a823753ea2193233","tex":"\\alpha_n","aliases":"α_n alpha","display":"\\alpha_n","kind":"use","excerpt":"Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":20},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f42d1a795468f2ce","tex":"\\gamma'","aliases":"γ' gamma","display":"\\gamma'","kind":"use","excerpt":"Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4569df46706329ae","tex":"\\gamma^2","aliases":"γ^2 gamma","display":"\\gamma^2","kind":"use","excerpt":"The defect can also be seen without an existence argument. Choose the section r_1=1, r_γ=4, and r_γ^2=16 modulo 27. These reduce to the required units modulo 9. The products of consecutive powers have no defect until the exponent passes through three. At that point 4^3=10=1+9 modulo 27. Consequently the normalized defect is 9 when the exponents wra","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9a52374df6059c66","tex":"\\gamma^e","aliases":"γ^e gamma","display":"\\gamma^e","kind":"use","excerpt":"A one-cocycle is a function a:C_e→ M satisfying a(gh)=a(g)+g· a(h). Putting g=h=1 gives a(1)=0. If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a tra","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-bf7b715a29421879","tex":"\\gamma^i","aliases":"γ^i gamma","display":"\\gamma^i","kind":"use","excerpt":"For a proposed correction b, sum the equations c(γ^i,γ)=b(γ)-b(γ^i+1)+b(γ^i) for i=0,1,2. The right side telescopes to 3b(γ)=0, using b(1)=0 and the trivial action. The left side is 9≠0 in I. There is therefore no primitive. This explicit calculation recovers the nonzero obstruction class directly from its normalized representative.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5c211a0b07887935","tex":"\\gamma^{i+1}","aliases":"γ^i+1 gamma","display":"\\gamma^{i+1}","kind":"use","excerpt":"For a proposed correction b, sum the equations c(γ^i,γ)=b(γ)-b(γ^i+1)+b(γ^i) for i=0,1,2. The right side telescopes to 3b(γ)=0, using b(1)=0 and the trivial action. The left side is 9≠0 in I. There is therefore no primitive. This explicit calculation recovers the nonzero obstruction class directly from its normalized representative.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-eea1351e5f7cb3da","tex":"\\gamma^r","aliases":"γ^r gamma","display":"\\gamma^r","kind":"use","excerpt":"A one-cocycle is a function a:C_e→ M satisfying a(gh)=a(g)+g· a(h). Putting g=h=1 gives a(1)=0. If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a tra","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2771b61f3f3f5974","tex":"\\gamma_0","aliases":"γ_0 gamma","display":"\\gamma_0","kind":"use","excerpt":"Let Γ={diag(ω^a,ω^-a):a=0,1,2}. The trace alphabet is {2,-1}. The image algebra has dimension two, with basis the identity and γ_1=diag(ω,ω^2). The trace coordinate map from [source reference] gives Φ(γ_0)=(2,-1), Φ(γ_1)=(-1,-1), Φ(γ_2)=(-1,2). The tuples are distinct. The sharper algebra bound is 3≤2^2=4, while the rank only bound is 3≤2^2^2=16. T","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d2e737aa54abf669","tex":"\\gamma_1","aliases":"γ_1 gamma","display":"\\gamma_1","kind":"use","excerpt":"Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2f08cea1939265ee","tex":"\\gamma_2","aliases":"γ_2 gamma","display":"\\gamma_2","kind":"use","excerpt":"Let Γ={diag(ω^a,ω^-a):a=0,1,2}. The trace alphabet is {2,-1}. The image algebra has dimension two, with basis the identity and γ_1=diag(ω,ω^2). The trace coordinate map from [source reference] gives Φ(γ_0)=(2,-1), Φ(γ_1)=(-1,-1), Φ(γ_2)=(-1,2). The tuples are distinct. The sharper algebra bound is 3≤2^2=4, while the rank only bound is 3≤2^2^2=16. T","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ae2e30306bc89e00","tex":"\\gamma_i","aliases":"γ_i gamma","display":"\\gamma_i","kind":"use","excerpt":"Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-788db25068d804f5","tex":"\\gamma_r","aliases":"γ_r gamma","display":"\\gamma_r","kind":"use","excerpt":"Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Assume that D=D_A,d has an additive two-sided inverse Q, and that normQz≤ Knorm z, with K≥0. Choose 0≤ r≤1 with Kr<1. For a target U, set Δ=U-A^d. The proposed correction map is Φ(h)=Q(Δ-R_A,d(h)). Both inverse identities will be used. DQ=1 turns a fixed point into a root. QD=1 proves the necessary condition from an existing root.","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"The base change of ω_π_0 is the central character of Π. Since Π is attached to τ=ρ|_G_E, that character equals detτ. The base change of detρ is also detτ. Hence δ=(detρ)ω_π_0^-1 is a character killed by base change to E, so δ∈ X. We need χ^n=δ. Since X has order p and pnmid n, the nth power map on X is bijective. Thus there is one and only one such","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-eb80c7e7bdbf53f3","tex":"\\epsilon^{-j_{\\lambda}}","aliases":"ϵ^-j_λ epsilon lambda","display":"\\epsilon^{-j_{\\lambda}}","kind":"use","excerpt":"is weakly attached to τ, with π a cuspidal cyclic descent. Let λ be a good separating prime. Suppose π has a residual realization overline r_π,λ. Then there is a unique index j_λ∈ZZ/pZZ such that P_π,v(T) = P_ρ⊗ϵ^j_λ,v(T) for every finite place v outside a finite set. Consequently, π⊗ϵ^-j_λ is weakly attached to ρ.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ac4f0a25805b4746","tex":"\\epsilon^{j_{\\lambda}}","aliases":"ϵ^j_λ epsilon lambda","display":"\\epsilon^{j_{\\lambda}}","kind":"use","excerpt":"cyclic Artin setup of [source reference]. For this unramified theorem it is enough that the lower cuspidal representation Π is weakly attached to τ, with π a cuspidal cyclic descent. Let λ be a good separating prime. Suppose π has a residual realization overline r_π,λ. Then there is a unique index j_λ∈ZZ/pZZ such that P_π,v(T) = P_ρ⊗ϵ^j_λ,v(T) for ","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f6991c3ad9fdd14c","tex":"\\epsilon_v","aliases":"ϵ_v epsilon","display":"\\epsilon_v","kind":"use","excerpt":"cient separator] Let E/F be cyclic of prime degree p. Let π be cuspidal and assume its base change to E is cuspidal. Let ϵ be the quotient character. Then there is an unramified finite place v and an index k with pnmid k such that the degree k coefficient of the normalized Hecke polynomial of π_v is nonzero and ϵ_v(Frob_v)≠1.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5897adbee959fca1","tex":"\\zeta^a","aliases":"ζ^a zeta","display":"\\zeta^a","kind":"use","excerpt":"The quotient generator acts on the ξ^a summand by ζ^a. Taking the trace of its rth power gives the Fourier sum. The twist multiplies the action of g^r by ζ^jr.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e86f8a0fa9550f3d","tex":"\\zeta^{ar}","aliases":"ζ^ar zeta","display":"\\zeta^{ar}","kind":"use","excerpt":"Fix a primitive pth root ζ. For P(z)=∑_a∈ZZ/pZZm_az^a, define widehat P(r) = ∑_a∈ZZ/pZZm_aζ^ar, r∈ZZ/pZZ.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7f51439841d6ed8d","tex":"\\zeta^k","aliases":"ζ^k zeta","display":"\\zeta^k","kind":"use","excerpt":"[Coefficient rotation] For ζ∈CC^×, e_k(ζ A)=ζ^k e_k(A).","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"[Stable equality of epsilon factors] Let V,V' be finite dimensional complex Weil representations with finite inertia image, of equal dimension and determinant. For every sufficiently ramified auxiliary character η, ϵ(V⊗η,Ψ)=ϵ(V'⊗η,Ψ). Here Ψ is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henni","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a721963f725aae53","tex":"\\theta_i","aliases":"θ_i theta","display":"\\theta_i","kind":"use","excerpt":"[Positive saturation criterion] Let χ be a nontrivial irreducible character of G. Suppose there are a positive integer m, nonnegative integers a_i, subgroups H_i, and characters θ_i∈calE such that mχ = ∑_i a_iInd_H_i^Gθ_i. Then L(s,χ) is entire.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fdc2dc9f687c22a4","tex":"\\lambda_p","aliases":"λ_p lambda","display":"\\lambda_p","kind":"use","excerpt":"Let K be a number field containing the values of the order p quotient characters, let λ_p be a place above p, and let OO be the corresponding valuation ring.","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"[One polynomial separator] Let ζ range over μ_p. The twists ζ A have pairwise distinct characteristic polynomials exactly when S(A) contains a degree not divisible by p.","count":16},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c2c2df7568d8e9c7","tex":"\\mu_{dM}","aliases":"μ_dM mu","display":"\\mu_{dM}","kind":"use","excerpt":"[Finite Satake roots] At every finite place v where E/F, π, and Π are unramified and the stated Artin polynomial comparison holds above v, every normalized Satake root of π_v belongs to μ_dM.","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2d061a1725819087","tex":"\\mu_{fM}","aliases":"μ_fM mu","display":"\\mu_{fM}","kind":"use","excerpt":"Choose wmid v and let f=f(w/v). If the normalized Satake roots of π_v are α_1,…,α_n, unramified base change replaces the Satake conjugacy class by its fth power. Hence the roots of Π_w are α_i^f. Since Π_w is attached to a representation of exponent dividing M, (α_i^f)^M=1. Thus α_i∈μ_fM. Since fmid d, this group is contained in μ_dM.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-6a4a106be5ac6c91","tex":"\\mu_{\\operatorname{lcm}(1,\\ldots,B)}","aliases":"μ_lcm(1,…,B) mu operatorname ldots","display":"\\mu_{\\operatorname{lcm}(1,\\ldots,B)}","kind":"use","excerpt":"The large residual theorem uses two alphabets. One is the supplied automorphic alphabet. The other contains every degree n polynomial whose roots have orders at most B=t^n^2. A field containing μ_lcm(1,…,B) contains all their coefficients. Both the field and the exceptional coefficient primes are fixed before choosing a residual realization. This o","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7781ad48096ae3ff","tex":"\\mu_N","aliases":"μ_N mu","display":"\\mu_N","kind":"use","excerpt":"A polynomial is determined by the multiset of its n Satake roots. Those roots lie in the N element set μ_N. The number of multisets of size n selected from N elements is the displayed binomial coefficient.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f2979a72de73859e","tex":"\\mu_p","aliases":"μ_p mu","display":"\\mu_p","kind":"use","excerpt":"[One polynomial separator] Let ζ range over μ_p. The twists ζ A have pairwise distinct characteristic polynomials exactly when S(A) contains a degree not divisible by p.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e8149ec697a9ee64","tex":"\\xi^2","aliases":"ξ^2 xi","display":"\\xi^2","kind":"use","excerpt":"d let ξ be the quotient character taking a generator of C_3 to ω. Let τ be the nontrivial character of H, extended with trivial quotient action. Consider a semisimple representation with spectrum P_τ,V(z)=2+z. Its quotient multiplicity vector is (2,1,0). Twisting by ξ gives zP_τ,V(z)=2z+z^2, whose vector is (0,2,1). Twisting by ξ^2 gives (1,0,2). T","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1b835423857234f4","tex":"\\xi^j","aliases":"ξ^j xi","display":"\\xi^j","kind":"use","excerpt":"[Effect of quotient twisting] For every j∈ZZ/pZZ, P_τ,V⊗ξ^j(z) = z^jP_τ,V(z). Moving orbit blocks are unchanged up to isomorphism by quotient twisting.","count":8},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-851fb1347ebfaaed","tex":"\\pi^{2r}","aliases":"π^2r pi","display":"\\pi^{2r}","kind":"use","excerpt":"equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomial systems the remainder contains at least two copies of π^r, so it is divisible by π^2r an","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7bd0c0339aa31283","tex":"\\pi^N","aliases":"π^N pi","display":"\\pi^N","kind":"use","excerpt":"Over a suitable truncated principal coefficient ring, iteration terminates at π^N=0 and produces a root of the original system. The recovered v0.25 modules derive the necessary faithful-layer property under their stated principal residue and nilpotence hypotheses. They still require the moving identity and exactness. Equality of selected dimensions","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-edaf85362ef6e23c","tex":"\\pi^r","aliases":"π^r pi","display":"\\pi^r","kind":"use","excerpt":"The graded correction argument retains an exact moving relation B_xF(x)=0 and a residual exactness statement imoverline J=keroverline B. If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with ","count":9},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cfae670f1a208a98","tex":"\\pi^{r+1}","aliases":"π^r+1 pi","display":"\\pi^{r+1}","kind":"use","excerpt":"If F(x)=π^r e, the moving identity gives π^r B_xe=0. A faithful leading layer allows this equality to imply overline B overline e=0. Equation [source reference] then supplies a residual correction overline u with overline J overline u=-overline e. Lift overline u to u and put x'=x+π^r u. The linear part of F(x') is divisible by π^r+1. For polynomia","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-2d28a27ac7821632","tex":"\\pi_0","aliases":"π_0 pi","display":"\\pi_0","kind":"use","excerpt":"[Unique determinant matched descent] Use the cyclic Artin setup of [source reference], and assume pnmid n. Let π_0 be any cuspidal descent of Π. Then there is a unique quotient character χ∈ X such that ω_π_0⊗χ = detρ under global class field theory.","count":6},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8097f07f71855deb","tex":"\\pi_{0,v}","aliases":"π_0,v pi","display":"\\pi_{0,v}","kind":"use","excerpt":"For each place v of F, put R_v=ρ|_W_F_v, A_v=rec_F_v(π_0,v). Cyclic base change compatibility gives Res_W_E_wR_v ≅ Res_W_E_wA_v.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5b07bc7670575aac","tex":"\\pi_{r,t}","aliases":"π_r,t pi","display":"\\pi_{r,t}","kind":"use","excerpt":"Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d5c58b5026faa28f","tex":"\\Pi_w","aliases":"Π_w Pi","display":"\\Pi_w","kind":"use","excerpt":"Fix v and wmid v, and write the local Weil Deligne parameter as (ϕ_v,N_v). Base change restricts ϕ_v to W_E_w and keeps the monodromy operator. Since the parameter of Π_w has zero monodromy, N_v=0.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a9fc1ff1221962c3","tex":"\\rho'","aliases":"ρ' rho","display":"\\rho'","kind":"use","excerpt":"Fix one extension ρ. Every χ∈ X is trivial on H, so ρ⊗χ restricts to τ. Conversely, let ρ' be another extension. Since both restrictions are irreducible and isomorphic, Schur's lemma shows that after one choice of an H intertwiner, the ratio between the two actions is scalar on each coset. Those scalars define a character of G/H, so ρ'≅ρ⊗χ.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e92c058fa2a15cfa","tex":"\\rho_v","aliases":"ρ_v rho","display":"\\rho_v","kind":"use","excerpt":"Local class field theory identifies a quotient character χ∈ X with the corresponding Hecke character. Local Langlands is compatible with character twisting, [source reference]. Hence rec(π_v⊗χ_v) ≅ rec(π_v)⊗χ_v ≅ ρ_v⊗χ_v. The result holds at every place.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"There is a finite set Σ(calP) of rational primes, fixed from the alphabet before any realization is chosen, with the following property. Let λ be a coefficient place whose residue characteristic ℓ is outside Σ(calP). Suppose a continuous semisimple representation overline r_λ:G_F⟶GL_n(overlineFF_ℓ) is unramified outside S and the places above ℓ, an","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"Suppose a second stable H type σ occurs with spectrum one. If the target has spectra 2z+z^2 for τ and z^2 for σ, the first block demands shift one and the second demands shift two. The two restrictions to H have the same multiplicities. Each block separately is a quotient twist of the matching source block. The full representation is not one quotie","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-cff2c5f7d8b191ec","tex":"\\tau^{-1}","aliases":"τ^-1 tau","display":"\\tau^{-1}","kind":"use","excerpt":"[A moving type and self twist] Let G=S_3, let H=A_3, and let V be the standard two dimensional representation. Its restriction is Res_HV=τ⊕τ^-1 for the two nontrivial characters of C_3. These characters form one moving orbit under G/H≅ C_2. Hence there is no stable extension spectrum, and V⊗sgn≅ V.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"Choose a basis γ_1,…,γ_r of A from the spanning set Γ, where r=dim_k A≤ n^2. Send an image element to Φ(γ)=(Tr(γγ_1),…, Tr(γγ_r)). All coordinates lie in the trace set, because γγ_i∈Γ. If two tuples agree, then γ-γ' pairs to zero with every basis element, hence every element of A. Nondegeneracy gives γ=γ'. Therefore Φ injects Γ into a set of size t","count":7},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"Fix v and wmid v, and write the local Weil Deligne parameter as (ϕ_v,N_v). Base change restricts ϕ_v to W_E_w and keeps the monodromy operator. Since the parameter of Π_w has zero monodromy, N_v=0.","count":8},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-626b7403871925ac","tex":"\\phi_v","aliases":"ϕ_v phi","display":"\\phi_v","kind":"use","excerpt":"Fix v and wmid v, and write the local Weil Deligne parameter as (ϕ_v,N_v). Base change restricts ϕ_v to W_E_w and keeps the monodromy operator. Since the parameter of Π_w has zero monodromy, N_v=0.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"Fix one extension ρ. Every χ∈ X is trivial on H, so ρ⊗χ restricts to τ. Conversely, let ρ' be another extension. Since both restrictions are irreducible and isomorphic, Schur's lemma shows that after one choice of an H intertwiner, the ratio between the two actions is scalar on each coset. Those scalars define a character of G/H, so ρ'≅ρ⊗χ.","count":33},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c88de245e41e247a","tex":"\\chi^{\\dim V}","aliases":"χ^dim V chi dim","display":"\\chi^{\\dim V}","kind":"use","excerpt":"The determinant changes by χ^dim V=1. The two representations have the same dimension and determinant. Apply [source reference].","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0dd74aa8321089da","tex":"\\chi^n","aliases":"χ^n chi","display":"\\chi^n","kind":"use","excerpt":"For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-25e629e0699f6cd8","tex":"\\chi^w","aliases":"χ^w chi","display":"\\chi^w","kind":"use","excerpt":"The scalar twist acts on r(ρ) by χ^w. Hence its determinant changes by χ^wN. By [source reference], wN=an, which is divisible by p. Thus χ^wN=1. Apply the argument to each irreducible summand.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e9a278f712d13f40","tex":"\\chi^{wN}","aliases":"χ^wN chi","display":"\\chi^{wN}","kind":"use","excerpt":"The scalar twist acts on r(ρ) by χ^w. Hence its determinant changes by χ^wN. By [source reference], wN=an, which is divisible by p. Thus χ^wN=1. Apply the argument to each irreducible summand.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-c5fb43cac0e529ac","tex":"\\chi_0","aliases":"χ_0 chi","display":"\\chi_0","kind":"use","excerpt":"The restriction of every quotient character to H is trivial. The regular representation of G/H decomposes as the direct sum of all characters in X. Hence Ind_H^GRes_HV ≅ V⊗CC[G/H] ≅ ⊕_χ∈ XV⊗χ. Replacing V by V⊗χ_0 only permutes the summands. The automorphic identity is the cyclic automorphic induction and base change analogue. It has the same permu","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e2a9889efd225991","tex":"\\chi_v","aliases":"χ_v chi","display":"\\chi_v","kind":"use","excerpt":"Local class field theory identifies a quotient character χ∈ X with the corresponding Hecke character. Local Langlands is compatible with character twisting, [source reference]. Hence rec(π_v⊗χ_v) ≅ rec(π_v)⊗χ_v ≅ ρ_v⊗χ_v. The result holds at every place.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"[Stable equality of epsilon factors] Let V,V' be finite dimensional complex Weil representations with finite inertia image, of equal dimension and determinant. For every sufficiently ramified auxiliary character η, ϵ(V⊗η,Ψ)=ϵ(V'⊗η,Ψ). Here Ψ is a fixed nontrivial additive character and the same normalization is used on both sides. Deligne and Henni","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.","count":23},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f002f795e1ea0617","tex":"\\omega^{-a}","aliases":"ω^-a omega","display":"\\omega^{-a}","kind":"use","excerpt":"Let Γ={diag(ω^a,ω^-a):a=0,1,2}. The trace alphabet is {2,-1}. The image algebra has dimension two, with basis the identity and γ_1=diag(ω,ω^2). The trace coordinate map from [source reference] gives Φ(γ_0)=(2,-1), Φ(γ_1)=(-1,-1), Φ(γ_2)=(-1,2). The tuples are distinct. The sharper algebra bound is 3≤2^2=4, while the rank only bound is 3≤2^2^2=16. T","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-f2f53155f7e59bd3","tex":"\\omega^2","aliases":"ω^2 omega","display":"\\omega^2","kind":"use","excerpt":"Now compare the balanced matrix A_bal=diag(1,ω,ω^2). Its polynomial is T^3-1. Multiplying all eigenvalues by ω merely permutes their multiset. Every lower coefficient vanishes, and the determinant is fixed. Its full characteristic polynomial is therefore blind to the label. This is an actual local rotational stabilizer. A claim that one characteris","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a6860fcf52898ea2","tex":"\\omega^{2j}","aliases":"ω^2j omega","display":"\\omega^{2j}","kind":"use","excerpt":"Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the th","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-de405089c69b448d","tex":"\\omega^{3j}","aliases":"ω^3j omega","display":"\\omega^{3j}","kind":"use","excerpt":"Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the th","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-70900ee358b87ffe","tex":"\\omega^4","aliases":"ω^4 omega","display":"\\omega^4","kind":"use","excerpt":"For a balanced multiplicity spectrum 1+z+z^2, both nonzero Fourier modes vanish: 1+ω+ω^2=0, 1+ω^2+ω^4=0. The spectrum 3 has the same total multiplicity, but both nonzero modes equal three. Comparing only ratios for which the source mode is nonzero would omit both decisive comparisons. The zero support clause in [source reference] is therefore neces","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-52b468d21e22f93a","tex":"\\omega^a","aliases":"ω^a omega","display":"\\omega^a","kind":"use","excerpt":"Let Γ={diag(ω^a,ω^-a):a=0,1,2}. The trace alphabet is {2,-1}. The image algebra has dimension two, with basis the identity and γ_1=diag(ω,ω^2). The trace coordinate map from [source reference] gives Φ(γ_0)=(2,-1), Φ(γ_1)=(-1,-1), Φ(γ_2)=(-1,2). The tuples are distinct. The sharper algebra bound is 3≤2^2=4, while the rank only bound is 3≤2^2^2=16. T","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-80ee3a3b341691a6","tex":"\\omega^b","aliases":"ω^b omega","display":"\\omega^b","kind":"use","excerpt":"At quotient prime three and rank two, the determinant changes by ω^2j. If a required determinant ratio is ω^b, the unique candidate satisfies 2j=b modulo three, so j=2b. This solves a finite selection problem. It still does not show that every other coefficient of the candidate matches the target.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-a27b633daa590319","tex":"\\omega^j","aliases":"ω^j omega","display":"\\omega^j","kind":"use","excerpt":"Let ω be a primitive cube root of unity and put A=diag(1,1,ω). Expanding directly gives det(TI-A) =T^3-(2+ω)T^2+(1+2ω)T-ω. The three elementary symmetric coefficients are e_1=2+ω, e_2=1+2ω, e_3=ω. For scalar twist ω^jA, they become ω^j(2+ω), ω^2j(1+2ω), ω^3jω=ω. The determinant is constant for j=0,1,2, but the first coefficient distinguishes the th","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-03c0ea7172492518","tex":"\\omega_\\pi","aliases":"ω_π omega pi","display":"\\omega_\\pi","kind":"use","excerpt":"For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b1c5a4dcc1c6e1a4","tex":"\\omega_\\pi^{dM}","aliases":"ω_π^dM omega pi","display":"\\omega_\\pi^{dM}","kind":"use","excerpt":"Outside the fixed exceptional set, every unramified determinant value lies in μ_dM. The Hecke character ω_π^dM is therefore trivial at every unramified place outside a finite set. The one dimensional strong multiplicity one statement, or equivalently uniqueness of a Hecke character from its almost everywhere unramified values, makes it the trivial ","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0146061874d89d11","tex":"\\omega_{\\pi_0}","aliases":"ω_π_0 omega pi","display":"\\omega_{\\pi_0}","kind":"use","excerpt":"The base change of ω_π_0 is the central character of Π. Since Π is attached to τ=ρ|_G_E, that character equals detτ. The base change of detρ is also detτ. Hence δ=(detρ)ω_π_0^-1 is a character killed by base change to E, so δ∈ X. We need χ^n=δ. Since X has order p and pnmid n, the nth power map on X is bijective. Thus there is one and only one such","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d486dbdd61bfee4d","tex":"\\omega_{\\pi_0}^{-1}","aliases":"ω_π_0^-1 omega pi","display":"\\omega_{\\pi_0}^{-1}","kind":"use","excerpt":"The base change of ω_π_0 is the central character of Π. Since Π is attached to τ=ρ|_G_E, that character equals detτ. The base change of detρ is also detτ. Hence δ=(detρ)ω_π_0^-1 is a character killed by base change to E, so δ∈ X. We need χ^n=δ. Since X has order p and pnmid n, the nth power map on X is bijective. Thus there is one and only one such","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-e2f59fa8d4077552","tex":"\\omega_{\\pi_0\\otimes\\chi}","aliases":"ω_π_0⊗χ omega pi otimes chi","display":"\\omega_{\\pi_0\\otimes\\chi}","kind":"use","excerpt":"[Unique determinant matched descent] Use the cyclic Artin setup of [source reference], and assume pnmid n. Let π_0 be any cuspidal descent of Π. Then there is a unique quotient character χ∈ X such that ω_π_0⊗χ = detρ under global class field theory.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5ecbb9401ee325c3","tex":"\\omega_{\\pi\\otimes\\chi}","aliases":"ω_π⊗χ omega pi otimes chi","display":"\\omega_{\\pi\\otimes\\chi}","kind":"use","excerpt":"For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Use the cross characteristic lifting statement in [source reference], in a splitting modular system compatible with the chosen coefficient place. It gives an ordinary characteristic zero representation of Γ whose reduction is the prescribed residual representation. Compose it with G_F→Γ. Each lifted eigenvalue has order dividing the order of an ele","count":11},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"For a quotient character χ, det(ρ⊗χ)=det(ρ)χ^n. The central character satisfies the identical formula ω_π⊗χ=ω_πχ^n. Since X has prime order p and pnmid n, the map χ↦χ^n is an automorphism of X.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"use","excerpt":"Let X_r be finite sets with genuine reduction maps π_r,t:X_t→ X_r for r≤ t. Assume identity and composition laws for these maps. Let P_r⊆ X_r be solution sets, and assume reduction preserves solutions. Then (∀ r, P_r≠∅) Longleftrightarrow (∃ (x_r)_r, x_r∈ P_r and π_r,t(x_t)=x_r for r≤ t). The nontrivial implication is finite branch compactness. For","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"use","excerpt":"[One large residual realization gives an Artin representation] Let K be a number field, let S be a finite set of places of F, and let calP⊆OO_K[T] be a nonempty finite set of monic degree n polynomials. Assume all their roots are roots of unity. For v∉ S, choose P_v∈calP.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b7027dbff3dd2bf8","tex":"\\prod_i","aliases":"∏_i prod","display":"\\prod_i","kind":"use","excerpt":"Over an algebraic closure of the residue field, overlineP^[p](T) = ∏_i(T-overlineα_i^p). The elementary symmetric functions of the pth powers are the pth powers of the elementary symmetric functions in characteristic p. Finite residue fields are perfect, so Frobenius is injective.","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-5c21a30e63e2813a","tex":"\\sum_{i=0}^{d-1}","aliases":"∑_i=0^d-1 sum","display":"\\sum_{i=0}^{d-1}","kind":"use","excerpt":"Let E be a complete normed ring with norm1=1, a submultiplicative norm, and the ultrametric inequality normx+y≤max(normx,normy). Assume norm A≤1. For a nonnegative integer d, define D_A,d(h) · =∑_i=0^d-1A^ihA^d-1-i, R_A,d(h) · =(A+h)^d-A^d-D_A,d(h). For d=0 the sum is empty and both maps are zero. The order of the factors in [source reference] matt","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b26eca2dd8854316","tex":"\\sum_{j=0}^{e-1}","aliases":"∑_j=0^e-1 sum","display":"\\sum_{j=0}^{e-1}","kind":"use","excerpt":"Suppose now that e is invertible in the coefficient ring. Define S=∑_j=0^e-1jT^j. A direct coefficient calculation gives (T-1)S=e-N. For x∈ker N, take y=e^-1Sx. Then (T-1)y=x, so every cocycle is a coboundary and H^1(C_e,M)=0. The averaging argument uses division by e at its decisive step. In residue characteristic dividing e, the same display rema","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d0b76425559f0527","tex":"\\sum_{j=0}^{r-1}","aliases":"∑_j=0^r-1 sum","display":"\\sum_{j=0}^{r-1}","kind":"use","excerpt":"A one-cocycle is a function a:C_e→ M satisfying a(gh)=a(g)+g· a(h). Putting g=h=1 gives a(1)=0. If x=a(γ), repeated application of the cocycle equation gives a(γ^r)=∑_j=0^r-1T^j x (1≤ r≤ e). The group relation γ^e=1 therefore forces Nx=0. Conversely, if Nx=0, this formula defines a cocycle. Whenever exponents wrap around e, the discrepancy is a tra","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-41e0c211ba80d17c","tex":"\\sum_{k=0}^{n}","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^{n}","kind":"use","excerpt":"Let ξ be a generator of X, and identify its values with pth roots of unity. For a matrix A∈GL_n(CC), write det(TI-A) = ∑_k=0^n(-1)^k e_k(A)T^n-k.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-90c396c1f06e154a","tex":"\\sum_t","aliases":"∑_t sum","display":"\\sum_t","kind":"use","excerpt":"For E=Mat_n( Q_p) use the entrywise supremum norm and normalize abs p_p=p^-1. Matrix multiplication satisfies abs(XY)_ij_p =|∑_tX_itY_tj|_p ≤max_tabsX_it_pabsY_tj_p ≤norm Xnorm Y. Taking the maximum over the finite set of entries proves submultiplicativity. The same ultrametric calculation removes a matrix dimension factor. Completeness is inherite","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"[Trace separator for irreducible extensions] Let Htriangleleft G with G/H≅ C_p, and let ρ be an irreducible extension of an irreducible G invariant representation of H. Then there exists g∈ G∖ H such that Trρ(g)≠0. Consequently, the p quotient twists of ρ are distinguished by their traces at g.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"use","excerpt":"Every algebraic character of GL_n is an integral power of the determinant. Thus det∘ r=det^a for one integer a. Evaluate both sides at zI_n. The left side is z^wN, while the right side is z^an.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"For a zero-cochain y∈ M, its coboundary at γ is (T-1)y. Taking the quotient by these changes gives the precise presentation H^1(C_e,M)≃ ker N/im(T-1). If the norm operator is identically zero, this reduces to M/im(T-1). That reduction has a hypothesis. Dropping the kernel from the general formula introduces classes that cannot satisfy the cyclic re","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"For a proposed correction b, sum the equations c(γ^i,γ)=b(γ)-b(γ^i+1)+b(γ^i) for i=0,1,2. The right side telescopes to 3b(γ)=0, using b(1)=0 and the trivial action. The left side is 9≠0 in I. There is therefore no primitive. This explicit calculation recovers the nonzero obstruction class directly from its normalized representative.","count":5},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"two extensions of the same absolutely irreducible representation of G_E. Their ratio is a character of G_F/G_E. Since the residue characteristic is different from p, the reduced quotient character has order p, and there is a unique index j_λ with overline r_π,λ ≅ overlineρ⊗overlineϵ^j_λ. At every good unramified place, P_π,v ≡ P_ρ⊗ϵ^j_λ,v pmodλ. Bo","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":36},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"[Uniform bound for a residual Galois image] Let overline r:G_F⟶GL_n(overlineFF_ℓ) be continuous and semisimple, with ℓ>n. If its Frobenius traces belong to a set of size at most t on a Chebotarev dense set, then absimoverline r≤ t^n^2.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"odule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple algebra has the form A≅⊕_iM_d_i(k), k^n≅⊕_i(k^d_i)^⊕ m_i, ∑_i d_im_i=n. Faithfulness gives m_i≥1 for every block. T","count":4},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"[Finite trace image bound] Let k be algebraically closed of characteristic zero or characteristic ℓ>n. Let Γ⊆GL_n(k) be a subgroup, and assume its defining module k^n is semisimple. Suppose its trace set is finite, with t=#{Tr(γ):γ∈Γ}. Then Γ is finite and absΓ≤ t^dim_k A≤ t^n^2, A=span_kΓ. In particular this bounds a finite semisimple residual mat","count":3},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-3c62641491514988","tex":"\\bigoplus_i","aliases":"⊕_i bigoplus","display":"\\bigoplus_i","kind":"use","excerpt":"oof Because products of group matrices are group matrices, A is a finite dimensional subalgebra of M_n(k). An A submodule is the same thing as a Γ submodule. The given module is semisimple and faithful for A. The Jacobson radical annihilates every simple constituent, hence annihilates this faithful module. It is therefore zero. The split semisimple","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-050a45cb4caa57c9","tex":"\\bigoplus_{j=0}^{p-1}","aliases":"⊕_j=0^p-1 bigoplus","display":"\\bigoplus_{j=0}^{p-1}","kind":"use","excerpt":"The automorphic induction of the attached representation over E is already attached to Ind_G_E^G_Fτ ≅ ⊕_j=0^p-1 ρ⊗ϵ^j. It is also the isobaric sum of all descents. By [source reference], this total object is invariant under rotating either torsor. It proves the direct sum correspondence but cannot identify one constituent with the target extension.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-43e6f080801d0972","tex":"\\bigoplus_{\\chi\\in X}","aliases":"⊕_χ∈ X bigoplus chi in","display":"\\bigoplus_{\\chi\\in X}","kind":"use","excerpt":"[The total induced object is torsor blind] Let Htriangleleft G with cyclic quotient character group X, and let V be an extension of an H representation. Then ⊕_χ∈ X V⊗χ ≅ Ind_H^GRes_HV. The left side is unchanged when V is replaced by another member of its extension torsor. Likewise, for cyclic base change, mathopboxplus_χ∈ X π⊗χ ≅ AI_E/F(BC_E/Fπ) ","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"use","excerpt":"[Structure of the alignment set] The alignment set is empty or a coset of the common rotational stabilizer ⋂_τ {j:z^jP_τ,V=P_τ,V}. In particular, one stable type with aperiodic spectrum makes the alignment index unique whenever it exists.","count":2},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-63c03a011ef19538","tex":"\\bigcap_v","aliases":"⋂_v bigcap","display":"\\bigcap_v","kind":"use","excerpt":"[Global cyclic attachment criterion] With the preceding notation, the descent π_0⊗ϵ^j is attached to ρ if and only if j∈⋂_vcalJ_v. Consequently, the cyclic Strong Artin step succeeds exactly when this intersection is nonempty.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-eaf0c3abbbfc3011","tex":"\\bigcap_\\tau","aliases":"⋂_τ bigcap tau","display":"\\bigcap_\\tau","kind":"use","excerpt":"[Structure of the alignment set] The alignment set is empty or a coset of the common rotational stabilizer ⋂_τ {j:z^jP_τ,V=P_τ,V}. In particular, one stable type with aperiodic spectrum makes the alignment index unique whenever it exists.","count":1},{"work":"strong-artin-reader","url":"notation/strong-artin-reader.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"[Frobenius transform after reduction] Let P have integral coefficients at a place of residue characteristic p. Then overlineP^[p](T) = Frob(overline P(T)), where Frobenius raises every coefficient to its pth power. In particular, the map overline P⟼overlineP^[p] is injective over the residue field.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.","count":92},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of","count":12},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"definition-passage","excerpt":"[Elementary deformation law] An elementary deformation law is a map Φ: D⟶ X with the understanding that d∈ D_x starts at x and ends at Φ(d). We also write x⋆ d=Φ(d).","count":42},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-61f0cb69c9e8e1d3","tex":"d_k","aliases":"d_k","display":"d_k","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-66af625e26b38735","tex":"\\Dcal","aliases":"Dcal D mathcal","display":"{\\mathcal D}","kind":"definition-passage","excerpt":"[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.","count":18},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f5c512f53f4b26b9","tex":"\\Dcal_x","aliases":"Dcal_x D_x Dcal mathcal","display":"{\\mathcal D}_x","kind":"definition-passage","excerpt":"[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.","count":11},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4465667a9120b31e","tex":"\\Dcal_{x_{k-1}}","aliases":"Dcal_x_k-1 D_x_k-1 Dcal mathcal","display":"{\\mathcal D}_{x_{k-1}}","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c8f21276e6dfefea","tex":"\\ev","aliases":"ev operatorname","display":"{\\operatorname{ev}}","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":6},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"definition-passage","excerpt":"[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of","count":11},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c4f568dc946ddd98","tex":"\\Fcal","aliases":"Fcal F mathcal","display":"{\\mathcal F}","kind":"definition-passage","excerpt":"[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-421392caf0a65188","tex":"\\mathrm{fin}","aliases":"fin mathrm","display":"\\mathrm{fin}","kind":"definition-passage","excerpt":"[Quotient deformation graph] The quotient deformation graph is O(x_0)= T_fin(x_0)/∼. Its vertices are relation classes of paths. An edge is induced by one admissible extension.","count":6},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-8c918c284a1665cd","tex":"\\Gcal","aliases":"Gcal G mathcal","display":"{\\mathcal G}","kind":"definition-passage","excerpt":"[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e6884a0c40dcac51","tex":"\\mathcal I","aliases":"I mathcal","display":"\\mathcal I","kind":"definition-passage","excerpt":"[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":39},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"definition-passage","excerpt":"[Direction-complete expansion] An expansion rule is direction-complete relative to G at x when it exposes every element of D_x. It is direction-complete through depth n when this holds at every endpoint of every admissible path of length less than n.","count":83},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-295577f481743cf7","tex":"\\Ocal","aliases":"Ocal O mathcal","display":"{\\mathcal O}","kind":"definition-passage","excerpt":"[Quotient deformation graph] The quotient deformation graph is O(x_0)= T_fin(x_0)/∼. Its vertices are relation classes of paths. An edge is induced by one admissible extension.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"definition-passage","excerpt":"[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"definition-passage","excerpt":"[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"definition-passage","excerpt":"[Operadic perturbation hyperforest] The operadic perturbation hyperforest generated by a seed family is the set of all finite rooted typed trees whose leaves are seeds and whose internal vertices are labeled by admissible operations Φ or Ψ_r. Evaluation proceeds from leaves to root.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"definition-passage","excerpt":"[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-712d027743d7bde8","tex":"\\Tcal","aliases":"Tcal T mathcal","display":"{\\mathcal T}","kind":"definition-passage","excerpt":"[Quotient deformation graph] The quotient deformation graph is O(x_0)= T_fin(x_0)/∼. Its vertices are relation classes of paths. An edge is induced by one admissible extension.","count":14},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-33859b84a261e163","tex":"\\Tcal_{\\mathrm{fin}}","aliases":"Tcal_fin T_fin Tcal mathrm mathcal mathrm","display":"{\\mathcal T}_{\\mathrm{fin}}","kind":"definition-passage","excerpt":"[Quotient deformation graph] The quotient deformation graph is O(x_0)= T_fin(x_0)/∼. Its vertices are relation classes of paths. An edge is induced by one admissible extension.","count":6},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"definition-passage","excerpt":"[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f6e0d67093f9c9dc","tex":"\\widehat\\Xcal","aliases":"widehatXcal widehat X widehat Xcal widehat mathcal","display":"\\widehat{\\mathcal X}","kind":"definition-passage","excerpt":"[Convergent route] A route ω is convergent in a metric realization when its endpoint sequence x_0,x_1,x_2,… converges in the declared completion widehat X.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"definition-passage","excerpt":"[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.","count":90},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":29},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"definition-passage","excerpt":"[Convergent route] A route ω is convergent in a metric realization when its endpoint sequence x_0,x_1,x_2,… converges in the declared completion widehat X.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"definition-passage","excerpt":"[Convergent route] A route ω is convergent in a metric realization when its endpoint sequence x_0,x_1,x_2,… converges in the declared completion widehat X.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e60d53cec3f4568f","tex":"x_{k-1}","aliases":"x_k-1","display":"x_{k-1}","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":8},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-57a532a57d9c63d2","tex":"\\Xcal","aliases":"Xcal X mathcal","display":"{\\mathcal X}","kind":"definition-passage","excerpt":"[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.","count":9},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"definition-passage","excerpt":"begindefinition[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction ite","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":20},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"definition-passage","excerpt":"[Extension-compatible path relation] An equivalence relation ∼ on finite paths is extension-compatible when the following implication holds. If γ∼η and a direction label d is admissible after both paths with compatible meaning, then γ d∼η d. The relation may also identify endpoints under a declared symmetry group, rewrite system, canonical form, ga","count":8},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"definition-passage","excerpt":"[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"definition-passage","excerpt":"begindefinition[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction ite","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"definition-passage","excerpt":"[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"definition-passage","excerpt":"[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-6a331de74ff475c9","tex":"\\pi^{-1}","aliases":"π^-1 pi","display":"\\pi^{-1}","kind":"definition-passage","excerpt":"[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"definition-passage","excerpt":"[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of","count":8},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-58c1a4cd175eaace","tex":"\\sigma_d","aliases":"σ_d sigma","display":"\\sigma_d","kind":"definition-passage","excerpt":"[Generated family certificate] A generated family certificate is a tuple F=(Θ,F,{σ_d}_d∈ A, Q, I), where [label=(alph*)] item Θ is a parameter space item F:Θ→ O(x_0) assigns an object class to every parameter item A is a collection of closure directions item σ_d:Θ→Θ satisfies F(σ_d(θ))=d· F(θ) for every declared closure direction item Q is a set of","count":6},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"definition-passage","excerpt":"[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"definition-passage","excerpt":"[Elementary deformation law] An elementary deformation law is a map Φ: D⟶ X with the understanding that d∈ D_x starts at x and ends at Φ(d). We also write x⋆ d=Φ(d).","count":8},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"definition-passage","excerpt":"[Operadic perturbation hyperforest] The operadic perturbation hyperforest generated by a seed family is the set of all finite rooted typed trees whose leaves are seeds and whose internal vertices are labeled by admissible operations Φ or Ψ_r. Evaluation proceeds from leaves to root.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-36c3f158017444a2","tex":"\\Psi_r","aliases":"Ψ_r Psi","display":"\\Psi_r","kind":"definition-passage","excerpt":"[Operadic perturbation hyperforest] The operadic perturbation hyperforest generated by a seed family is the set of all finite rooted typed trees whose leaves are seeds and whose internal vertices are labeled by admissible operations Φ or Ψ_r. Evaluation proceeds from leaves to root.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"definition-passage","excerpt":"[Convergent route] A route ω is convergent in a metric realization when its endpoint sequence x_0,x_1,x_2,… converges in the declared completion widehat X.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":8},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"definition-passage","excerpt":"[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.","count":6},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Elementary deformation law] An elementary deformation law is a map Φ: D⟶ X with the understanding that d∈ D_x starts at x and ends at Φ(d). We also write x⋆ d=Φ(d).","count":14},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"definition-passage","excerpt":"[Typed generative deformation datum] A typed generative deformation datum is a tuple G= ( T, X,π: D→ X,Φ, R,τ,κ, V), where R is a path relation system, τ is a completion structure, κ: D→[0,∞) is a direction cost, and V is a validity predicate.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"[Extension-compatible path relation] An equivalence relation ∼ on finite paths is extension-compatible when the following implication holds. If γ∼η and a direction label d is admissible after both paths with compatible meaning, then γ d∼η d. The relation may also identify endpoints under a declared symmetry group, rewrite system, canonical form, ga","count":9},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"definition-passage","excerpt":"[Admissible path] An admissible path of length n from x_0 is a sequence γ=(d_1,…,d_n) for which x_k=x_k-1⋆ d_k, d_k∈ D_x_k-1, 1≤ k≤ n. Its endpoint is ev(γ)=x_n. The empty path ∅ has endpoint x_0.","count":21},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"definition-passage","excerpt":"[Elementary deformation law] An elementary deformation law is a map Φ: D⟶ X with the understanding that d∈ D_x starts at x and ends at Φ(d). We also write x⋆ d=Φ(d).","count":7},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[Direction bundle] A direction bundle over X is a map π: D⟶ X. The fiber D_x=π^-1(x) is the complete set of directions admitted by the declared grammar at x.","count":7},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-cef2f13a7966bbb0","tex":"\\coprod","aliases":"coprod","display":null,"kind":"introduction-passage","excerpt":"Write T_n(x_0) for the set of length-n paths and T_fin(x_0)=coprod_n≥0 T_n(x_0). Every nonempty path has a unique parent obtained by deleting its final direction. Therefore T_fin(x_0) is a rooted tree even when different paths have the same endpoint.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c16bac87f8d3a526","tex":"\\coprod_{n\\ge0}","aliases":"coprod_n≥0 coprod ge","display":null,"kind":"introduction-passage","excerpt":"Write T_n(x_0) for the set of length-n paths and T_fin(x_0)=coprod_n≥0 T_n(x_0). Every nonempty path has a unique parent obtained by deleting its final direction. Therefore T_fin(x_0) is a rooted tree even when different paths have the same endpoint.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-2bae6a4c91e98f36","tex":"\\Tcal_n","aliases":"Tcal_n T_n Tcal mathcal","display":"{\\mathcal T}_n","kind":"introduction-passage","excerpt":"Write T_n(x_0) for the set of length-n paths and T_fin(x_0)=coprod_n≥0 T_n(x_0). Every nonempty path has a unique parent obtained by deleting its final direction. Therefore T_fin(x_0) is a rooted tree even when different paths have the same endpoint.","count":8},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"introduction-passage","excerpt":"Write T_n(x_0) for the set of length-n paths and T_fin(x_0)=coprod_n≥0 T_n(x_0). Every nonempty path has a unique parent obtained by deleting its final direction. Therefore T_fin(x_0) is a rooted tree even when different paths have the same endpoint.","count":10},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"The statement is immediate at depth zero. Assume it holds at depth n. Every path of length n+1 is a path of length n followed by one direction from the fiber over its endpoint. Thus its endpoint lies in the (n+1)-step reachable set. Conversely, every object in the (n+1)-step reachable set is produced by an n-step reachable object and one admissible","count":44},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"[Summable-step convergence] Suppose widehat X is complete and d_X(x_n+1,x_n)≤ a_n with ∑_n≥0a_n<∞. Then (x_n) is Cauchy and the route has a unique limit in widehat X.","count":16},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-54034542483dca3c","tex":"A^{\\mathsf T}","aliases":"A^ T mathsf","display":"A^{\\mathsf T}","kind":"use","excerpt":"0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"Let A_0=1 · 1 0 · 1. The declared matrix directions are transpose, negation, squaring, conjugation by the coordinate swap, symmetrization, commutation with J=0 · -1","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-87eebef5df411d47","tex":"A_0^{\\mathsf T}","aliases":"A_0^ T mathsf","display":"A_0^{\\mathsf T}","kind":"use","excerpt":"The first calculations are explicit: A_0^ T=1 · 0 1 · 1, A_0^2=1 · 2 0 · 1, adj(A_0)=1 · -1 0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-74825ea0ef74bd59","tex":"A_0^2","aliases":"A_0^2","display":"A_0^2","kind":"use","excerpt":"The first calculations are explicit: A_0^ T=1 · 0 1 · 1, A_0^2=1 · 2 0 · 1, adj(A_0)=1 · -1 0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"The executable atlas starts from the eight-term Fibonacci prefix a=(0,1,1,2,3,5,8,13). Its fixed grammar contains first difference, partial summation, the binomial transform, sign alternation, and reversal. For a finite sequence a=(a_0,…,a_N-1), define (Da)_0=a_0, (Da)_n=a_n-a_n-1, (Ca)_n=∑_j=0^na_j, (Ba)_n=∑_j=0^nbinomnja_j. The two triangular ope","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1b08ed69f17703a9","tex":"a_k","aliases":"a_k","display":"a_k","kind":"use","excerpt":"For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"[Summable-step convergence] Suppose widehat X is complete and d_X(x_n+1,x_n)≤ a_n with ∑_n≥0a_n<∞. Then (x_n) is Cauchy and the route has a unique limit in widehat X.","count":9},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-7efa4a157d1ebbef","tex":"a_{N-1}","aliases":"a_N-1","display":"a_{N-1}","kind":"use","excerpt":"The executable atlas starts from the eight-term Fibonacci prefix a=(0,1,1,2,3,5,8,13). Its fixed grammar contains first difference, partial summation, the binomial transform, sign alternation, and reversal. For a finite sequence a=(a_0,…,a_N-1), define (Da)_0=a_0, (Da)_n=a_n-a_n-1, (Ca)_n=∑_j=0^na_j, (Ba)_n=∑_j=0^nbinomnja_j. The two triangular ope","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a2cfe3ae1e3afede","tex":"a_{n-1}","aliases":"a_n-1","display":"a_{n-1}","kind":"use","excerpt":"For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1b51e95719a5480d","tex":"a_{n-r}","aliases":"a_n-r","display":"a_{n-r}","kind":"use","excerpt":"For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-5498b3bc9fb8719d","tex":"A_{\\beta\\alpha}","aliases":"A_βα beta alpha","display":"A_{\\beta\\alpha}","kind":"use","excerpt":"Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"[Exact cylinder refinement] For every prefix α and every depth n>absα, P^(n)_α = ∑_d∈ D_ev(α)P^(n)_α d, with strong convergence when the child set is countable.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-74fd5e15b7fae5b2","tex":"\\operatorname{adj}","aliases":"adj operatorname","display":"\\operatorname{adj}","kind":"use","excerpt":"The first calculations are explicit: A_0^ T=1 · 0 1 · 1, A_0^2=1 · 2 0 · 1, adj(A_0)=1 · -1 0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.","count":13},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"With four ordered leaves and one binary operation, the free operadic layer contains the five Catalan parenthesizations ((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), a(b(cd)). They are different syntax trees. If the operation is associative, all five evaluate to one quotient object. If it is also commutative, the 4! permutations of the leaves are identifi","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c2dedbbf3ad65db3","tex":"B^n","aliases":"B^n","display":"B^n","kind":"use","excerpt":"[Geometric convergence bound] Suppose every vertex has at most B children and every direction costs at least c>0. Then Z_x_0(s) ≤ ∑_n≥1B^ne^-scn. Consequently, Z_x_0(s) converges whenever sc>log B. In a full regular B-ary tree with constant cost c, it diverges whenever sc≤log B.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e668127427bd9374","tex":"B_{\\gamma\\beta}","aliases":"B_γβ gamma beta","display":"B_{\\gamma\\beta}","kind":"use","excerpt":"Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.","count":10},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"[Geometric convergence bound] Suppose every vertex has at most B children and every direction costs at least c>0. Then Z_x_0(s) ≤ ∑_n≥1B^ne^-scn. Consequently, Z_x_0(s) converges whenever sc>log B. In a full regular B-ary tree with constant cost c, it diverges whenever sc≤log B.","count":12},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-5cee0473087cadf5","tex":"C^k","aliases":"C^k","display":"C^k","kind":"use","excerpt":"A second, closed subfamily starts from the impulse δ=(1,0,0,…). Repeated summation gives (C^kδ)_n=binomn+k-1k-1 (k≥1), by induction and the hockey-stick identity. This provides a genuine family certificate: perturbation depth k is the parameter, C advances the parameter, D reverses it away from the boundary, and the displayed coefficient formula is","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-cfbf41dac773cbfc","tex":"C_n","aliases":"C_n","display":"C_n","kind":"use","excerpt":"Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-0061e0d517664876","tex":"C_{n+1}","aliases":"C_n+1","display":"C_{n+1}","kind":"use","excerpt":"[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a56806078c22867e","tex":"C_{n-k}","aliases":"C_n-k","display":"C_{n-k}","kind":"use","excerpt":"[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-5f900eab61babb52","tex":"c_r","aliases":"c_r","display":"c_r","kind":"use","excerpt":"For a sequence a_0,a_1,…, the implementation can search for a constant-coefficient relation a_n=c_1a_n-1+⋯+c_ra_n-r using exact rational linear algebra. A proposed recurrence is accepted only when it reproduces every supplied term exactly. The built-in Fibonacci example recovers (c_1,c_2)=(1,1). This is a small example of the certificate principle.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a65443dd022b38f9","tex":"\\coprod_{t\\in\\mathsf T}","aliases":"coprod_t∈ T coprod in mathsf","display":null,"kind":"use","excerpt":"Let T be a set of types and let X=coprod_t∈ TX_t be the disjoint union of the corresponding object spaces.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e77ac3fd3658acb1","tex":"\\cosh","aliases":"cosh","display":"\\cosh","kind":"use","excerpt":"Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, p","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e1a4d4510c259fa0","tex":"\\cost","aliases":"cost operatorname","display":"{\\operatorname{cost}}","kind":"use","excerpt":"The cost of a finite path is cost(γ)=∑_k=1^nκ(d_k). A path record may also store error bounds, provenance, validity certificates, symbolic assumptions, and the first place at which an invariant changed. None of these fields alters reachability. They determine how a finite explorer prioritizes and audits the infinite object.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-04ea05958a7c1391","tex":"\\coth","aliases":"coth","display":null,"kind":"use","excerpt":"Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, p","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f22409745028e891","tex":"\\operatorname{csch}","aliases":"csch operatorname","display":"\\operatorname{csch}","kind":"use","excerpt":"Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, p","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.","count":10},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"use","excerpt":"An infinite route is a sequence ω=(d_1,d_2,…) whose every finite prefix is admissible. The set of infinite routes is the path boundary ∂ T(x_0).","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-643de7b1c3192f91","tex":"d_n^{\\pm}","aliases":"d_n^± pm","display":"d_n^{\\pm}","kind":"use","excerpt":"The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The sel","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-aafbdea62a4e791c","tex":"d_X","aliases":"d_X","display":"d_X","kind":"use","excerpt":"Assume the object space carries a metric d_X.","count":7},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-356e2ae3435dcc83","tex":"\\Dcal^{(r)}","aliases":"Dcal^(r) D^(r) Dcal mathcal","display":"{\\mathcal D}^{(r)}","kind":"use","excerpt":"For every r≥2, let Ψ_r: X^r× D^(r) ⟶ X be a typed r-input deformation law.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-37289ad141359639","tex":"\\Dcal_{\\ev(\\alpha)}","aliases":"Dcal_ev(α) D_ev(α) Dcal ev alpha mathcal operatorname alpha","display":"{\\mathcal D}_{{\\operatorname{ev}}(\\alpha)}","kind":"use","excerpt":"[Exact cylinder refinement] For every prefix α and every depth n>absα, P^(n)_α = ∑_d∈ D_ev(α)P^(n)_α d, with strong convergence when the child set is countable.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f1a41b1f75411012","tex":"\\det","aliases":"det","display":"\\det","kind":"use","excerpt":"0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":14},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1b7d51060bc9b459","tex":"e^{-sc}","aliases":"e^-sc","display":"e^{-sc}","kind":"use","excerpt":"There are at most B^n paths of length n, and each has cost at least cn. The upper bound is a geometric series with ratio Be^-sc. In the regular constant-cost case, equality holds at every depth, giving the divergence statement.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-dfe829645f137239","tex":"e^{-sC([x])}","aliases":"e^-sC([x])","display":"e^{-sC([x])}","kind":"use","excerpt":"A path zeta function measures construction complexity. A quotient zeta function measures distinct object complexity. Depending on the application, one may define Z_obj(s) = ∑_[x]∈ O(x_0)e^-sC([x]), where C([x]) is the least path cost reaching the class. The two functions answer different questions and should be stored together.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-0b474d1d6cb45d1b","tex":"e^{-scn}","aliases":"e^-scn","display":"e^{-scn}","kind":"use","excerpt":"[Geometric convergence bound] Suppose every vertex has at most B children and every direction costs at least c>0. Then Z_x_0(s) ≤ ∑_n≥1B^ne^-scn. Consequently, Z_x_0(s) converges whenever sc>log B. In a full regular B-ary tree with constant cost c, it diverges whenever sc≤log B.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-51216dc08de0ed5b","tex":"e^{-s\\cost(\\gamma)}","aliases":"e^-scost(γ) cost gamma operatorname gamma","display":"e^{-s{\\operatorname{cost}}(\\gamma)}","kind":"use","excerpt":"Path partition function · Z_x_0(s)=∑_γ≠∅e^-scost(γ)","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-3c91647003842c2f","tex":"e_0","aliases":"e_0","display":"e_0","kind":"use","excerpt":"[Recursive deformation stability] If the direction mismatch at step k is at most ϵ_k, then the endpoint error satisfies e_n ≤ (∏_k=1^nL_k)e_0 + ∑_j=1^n (M_jϵ_j+η_j) ∏_k=j+1^nL_k.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-542812c54fe5f147","tex":"e_k","aliases":"e_k","display":"e_k","kind":"use","excerpt":"Set e_k=d_X(x_k,widehat x_k). The hypothesis gives e_k≤ L_ke_k-1+M_kϵ_k. Repeated substitution yields the displayed formula.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-9bcd0c3373538abc","tex":"e_{k-1}","aliases":"e_k-1","display":"e_{k-1}","kind":"use","excerpt":"Set e_k=d_X(x_k,widehat x_k). The hypothesis gives e_k≤ L_ke_k-1+M_kϵ_k. Repeated substitution yields the displayed formula.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"[Recursive deformation stability] If the direction mismatch at step k is at most ϵ_k, then the endpoint error satisfies e_n ≤ (∏_k=1^nL_k)e_0 + ∑_j=1^n (M_jϵ_j+η_j) ∏_k=j+1^nL_k.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c470de86d3aa40db","tex":"\\Ecal","aliases":"Ecal E mathcal","display":"{\\mathcal E}","kind":"use","excerpt":"Sampling is an execution method, not the definition of the continuous forest. A continuous family is stored symbolically as an edge bundle E_x = {(d,x⋆ d):d∈ D_x}. Finite nets approximate this bundle. Refinement replaces one coarse parameter cell by smaller cells. The cell itself remains a first-class object with a radius and a coverage certificate","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-d8797baa513b5d15","tex":"\\Ecal_x","aliases":"Ecal_x E_x Ecal mathcal","display":"{\\mathcal E}_x","kind":"use","excerpt":"Sampling is an execution method, not the definition of the continuous forest. A continuous family is stored symbolically as an edge bundle E_x = {(d,x⋆ d):d∈ D_x}. Finite nets approximate this bundle. Refinement replaces one coarse parameter cell by smaller cells. The cell itself remains a first-class object with a radius and a coverage certificate","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, p","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","excerpt":"Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, p","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"The implementation works directly after quotienting global phase. A state is one of the six exact Bloch-axis vectors (0,0,±1), (±1,0,0), (0,±1,0), representing the Pauli eigenstates |0⟩, |1⟩, |+⟩, |-⟩, |+i⟩, and |-i⟩. The primitive grammar contains H,S,X,Z, acting as signed permutations of the Bloch coordinates: H(x,y,z)=(z,-y,x), S(x,y,z)=(-y,x,z)","count":7},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a03ffaed013c4d19","tex":"\\mathcal H","aliases":"H mathcal","display":"\\mathcal H","kind":"use","excerpt":"For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"Starting from |0⟩, one has |0⟩xrightarrowH|+⟩ xrightarrowS|+i⟩ xrightarrowS|-⟩, while H^2=X^2=Z^2=I on the projective orbit and S^4=I. The number of raw gate words grows as 4^n, but the quotient orbit saturates at six objects.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-99ae55ab4be5fb33","tex":"\\mathcal H_n","aliases":"H_n mathcal","display":"\\mathcal H_n","kind":"use","excerpt":"For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-fa69cf9f2df553c5","tex":"h_{\\mathrm{obj}}","aliases":"h_obj mathrm","display":"h_{\\mathrm{obj}}","kind":"use","excerpt":"The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-29b2f7e1ee572ba1","tex":"h_{\\mathrm{raw}}","aliases":"h_raw mathrm","display":"h_{\\mathrm{raw}}","kind":"use","excerpt":"The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"A complete cut is a prefix-free set Γ meeting every infinite route exactly once. Its projectors satisfy ∑_α∈ΓP_α=I. Thus one branch may be resolved deeply while another stops early. Adaptive exploration can be represented without discarding the unresolved region.","count":6},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"The implementation works directly after quotienting global phase. A state is one of the six exact Bloch-axis vectors (0,0,±1), (±1,0,0), (0,±1,0), representing the Pauli eigenstates |0⟩, |1⟩, |+⟩, |-⟩, |+i⟩, and |-i⟩. The primitive grammar contains H,S,X,Z, acting as signed permutations of the Bloch coordinates: H(x,y,z)=(z,-y,x), S(x,y,z)=(-y,x,z)","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"0 · 1. The declared matrix directions are transpose, negation, squaring, conjugation by the coordinate swap, symmetrization, commutation with J=0 · -1 1 · 0, and adjugation. Every result is stored as an exact integer matrix. The metadata include trace, determinant, rank, symmetry, and squared Frobenius norm.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"A computational realization uses a canonical key K: X⟶ K with K(x)=K(y) Longleftrightarrow x∼ y for the declared finite example. Exact polynomial coefficients, reduced rational numbers, canonical symbolic expressions, and normalized matrices are suitable examples. Floating point rounding alone is not an exact quotient relation.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"A computational realization uses a canonical key K: X⟶ K with K(x)=K(y) Longleftrightarrow x∼ y for the declared finite example. Exact polynomial coefficients, reduced rational numbers, canonical symbolic expressions, and normalized matrices are suitable examples. Floating point rounding alone is not an exact quotient relation.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"Take the rational triangle K_0={(0,0),(2,0),(1,frac32)}. The grammar contains quarter turns, reflection in the x axis, horizontal shear, scaling by two, and unit translations in the two coordinate directions. Each descendant is again a finite rational point configuration, so the grammar is queried recursively at every child.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f940ebf823a5a621","tex":"\\operatorname{key}","aliases":"key operatorname","display":"\\operatorname{key}","kind":"use","excerpt":"For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":12},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-dfc293305a0744ec","tex":"L_k","aliases":"L_k","display":"L_k","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":9},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-3a6c7a18f061eb44","tex":"\\limsup","aliases":"limsup","display":"\\limsup","kind":"use","excerpt":"The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-3e2f995f3f54e413","tex":"\\limsup_{n\\to\\infty}","aliases":"limsup_n→∞ limsup to infty","display":"\\limsup_{n\\to\\infty}","kind":"use","excerpt":"The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.","count":9},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-68a67f309b99ec73","tex":"\\longleftarrow","aliases":"longleftarrow","display":null,"kind":"use","excerpt":"For x^2=1 over powers of 3, the two simple roots lift uniquely: 1pmod 3longleftarrow1pmod 9longleftarrow1pmod27longleftarrow⋯, -1pmod 3longleftarrow-1pmod 9longleftarrow-1pmod27longleftarrow⋯. For the singular equation x^2=0 over powers of 2, some nodes have several valid next digits and some branches later die. Thus branching number is an object p","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"A computational realization uses a canonical key K: X⟶ K with K(x)=K(y) Longleftrightarrow x∼ y for the declared finite example. Exact polynomial coefficients, reduced rational numbers, canonical symbolic expressions, and normalized matrices are suitable examples. Floating point rounding alone is not an exact quotient relation.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1633dfbf62528f10","tex":"\\lvert","aliases":"lvert","display":"\\lvert","kind":"use","excerpt":"For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":10},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.","count":12},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-bf2227566ef624f5","tex":"M_j","aliases":"M_j","display":"M_j","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c9f29e7d9e7a175b","tex":"M_k","aliases":"M_k","display":"M_k","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":6},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).","count":11},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-b75f3a902c5f7e60","tex":"n^{-1}","aliases":"n^-1","display":"n^{-1}","kind":"use","excerpt":"Raw entropy · h_raw=limsup n^-1log a_n","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-6b13294d64eb5108","tex":"N_\\epsilon","aliases":"N_ϵ epsilon","display":"N_\\epsilon","kind":"use","excerpt":"Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-2c546fc308b870f3","tex":"\\mathrm{obj}","aliases":"obj mathrm","display":"\\mathrm{obj}","kind":"use","excerpt":"The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a64d0ffa80003a21","tex":"\\overline{P_3}","aliases":"overlineP_3 overline","display":"\\overline{P_3}","kind":"use","excerpt":"For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"For a prefix α of length k≤ n, define the cylinder projector P^(n)_α lvertγ⟩ = lvertγ⟩, · α is a prefix of γ, 0, · otherwise.","count":12},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.","count":9},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-171e10e212c438c2","tex":"p'","aliases":"p'","display":"p'","kind":"use","excerpt":"Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-7277e8d0197eae94","tex":"P^{(n)}_\\alpha","aliases":"P^(n)_α alpha","display":"P^{(n)}_\\alpha","kind":"use","excerpt":"For a prefix α of length k≤ n, define the cylinder projector P^(n)_α lvertγ⟩ = lvertγ⟩, · α is a prefix of γ, 0, · otherwise.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-0348ca9eb7850083","tex":"P^{(n)}_{\\alpha d}","aliases":"P^(n)_α d alpha","display":"P^{(n)}_{\\alpha d}","kind":"use","excerpt":"[Exact cylinder refinement] For every prefix α and every depth n>absα, P^(n)_α = ∑_d∈ D_ev(α)P^(n)_α d, with strong convergence when the child set is countable.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-7441b5cfbaa2c0c0","tex":"p^m","aliases":"p^m","display":"p^m","kind":"use","excerpt":"For a prime p, a state at level m is a residue class modulo p^m satisfying a declared congruence. Its direction fiber is the finite set of next base-p digits that lift the current solution to level m+1.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c5e705958e40c3bd","tex":"P_2","aliases":"P_2","display":"P_2","kind":"use","excerpt":"For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f0242a419961441b","tex":"P_3","aliases":"P_3","display":"P_3","kind":"use","excerpt":"Begin with the path graph P_3 on three vertices. The finite grammar contains graph complement, line-graph formation, leaf attachment, edge subdivision, and a relabeling direction. Raw states are labeled adjacency matrices. For a graph with n≤6 vertices, the quotient key is the lexicographically least adjacency bit string over all n! vertex permutat","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e5a0ea50bd109dfc","tex":"P_4","aliases":"P_4","display":"P_4","kind":"use","excerpt":"For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-33a5f2682723cba6","tex":"P_\\alpha","aliases":"P_α alpha","display":"P_\\alpha","kind":"use","excerpt":"A complete cut is a prefix-free set Γ meeting every infinite route exactly once. Its projectors satisfy ∑_α∈ΓP_α=I. Thus one branch may be resolved deeply while another stops early. Adaptive exploration can be represented without discarding the unresolved region.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-cc222a57bc711687","tex":"P_{\\alpha d}","aliases":"P_α d alpha","display":"P_{\\alpha d}","kind":"use","excerpt":"Projector refinement · P_α=∑_dP_α d","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"use","excerpt":"For x^2=1 over powers of 3, the two simple roots lift uniquely: 1pmod 3longleftarrow1pmod 9longleftarrow1pmod27longleftarrow⋯, -1pmod 3longleftarrow-1pmod 9longleftarrow-1pmod27longleftarrow⋯. For the singular equation x^2=0 over powers of 2, some nodes have several valid next digits and some branches later die. Thus branching number is an object p","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-71a98ec1e2bb9f0a","tex":"\\preceq","aliases":"preceq","display":"\\preceq","kind":"use","excerpt":"Cylinder projector · P^(n)_αlvertγ⟩=1_αpreceqγlvertγ⟩","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"Define a_n=# T_n(x_0) for raw paths and q_n=#(ev( T_n(x_0))/∼) for distinct quotient endpoints at exact depth n.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4a77ffe628ff771a","tex":"q_n","aliases":"q_n","display":"q_n","kind":"use","excerpt":"Define a_n=# T_n(x_0) for raw paths and q_n=#(ev( T_n(x_0))/∼) for distinct quotient endpoints at exact depth n.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-d2f8225af58ea4bd","tex":"Q_\\beta","aliases":"Q_β beta","display":"Q_\\beta","kind":"use","excerpt":"Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, p","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-78fc220bbdbf06ff","tex":"R^4","aliases":"R^4","display":"R^4","kind":"use","excerpt":"For example, R(x,y)=(-y,x), H(x,y)=(x+y,y), S_2(x,y)=(2x,2y). The relation R^4=I closes one route loop exactly. Reflection is an involution. Rotation preserves signed area, reflection reverses its sign, shear preserves it, and scaling by two multiplies it by four. These invariants separate visually similar but mathematically different descendants.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-bc042a1db2b5b7bd","tex":"\\mathrm{raw}","aliases":"raw mathrm","display":"\\mathrm{raw}","kind":"use","excerpt":"The raw branching entropy and object entropy are h_raw = limsup_n→∞frac1nlog a_n, h_obj = limsup_n→∞frac1nlog q_n. Always h_obj≤ h_raw. The gap measures how much formal construction redundancy is removed by identities and symmetries.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-607edbc5319584b2","tex":"\\Reach","aliases":"Reach operatorname","display":"{\\operatorname{Reach}}","kind":"use","excerpt":"[Exact reachability by depth] For every n≥0, ev( T_n(x_0)) = Reach_n(x_0), where Reach_n(x_0) is the set obtained by applying exactly n successive admissible directions.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c50bc24e585d6ef3","tex":"\\Reach_n","aliases":"Reach_n Reach operatorname","display":"{\\operatorname{Reach}}_n","kind":"use","excerpt":"[Exact reachability by depth] For every n≥0, ev( T_n(x_0)) = Reach_n(x_0), where Reach_n(x_0) is the set obtained by applying exactly n successive admissible directions.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"0 · 1. They satisfy (A^ T)^ T=A, Aadj(A)=det(A)I, det(SAS)=det(A) for the swap matrix S. Thus involution, adjugate, and conjugacy relations generate quotient identifications, while squaring and commutation usually move to new invariant values.","count":9},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.","count":11},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-994ff0da8ea0da61","tex":"S^4","aliases":"S^4","display":"S^4","kind":"use","excerpt":"Starting from |0⟩, one has |0⟩xrightarrowH|+⟩ xrightarrowS|+i⟩ xrightarrowS|-⟩, while H^2=X^2=Z^2=I on the projective orbit and S^4=I. The number of raw gate words grows as 4^n, but the quotient orbit saturates at six objects.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-db9af401ac382039","tex":"S_2","aliases":"S_2","display":"S_2","kind":"use","excerpt":"For example, R(x,y)=(-y,x), H(x,y)=(x+y,y), S_2(x,y)=(2x,2y). The relation R^4=I closes one route loop exactly. Reflection is an involution. Rotation preserves signed area, reflection reverses its sign, shear preserves it, and scaling by two multiplies it by four. These invariants separate visually similar but mathematically different descendants.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-2d39429b9e78430d","tex":"\\operatorname{sech}","aliases":"sech operatorname","display":"\\operatorname{sech}","kind":"use","excerpt":"Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, p","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-21670bd4e275805e","tex":"\\sinh","aliases":"sinh","display":"\\sinh","kind":"use","excerpt":"Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, p","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"Let T be a set of types and let X=coprod_t∈ TX_t be the disjoint union of the corresponding object spaces.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-978e87e4f3d831a3","tex":"\\tanh","aliases":"tanh","display":"\\tanh","kind":"use","excerpt":"Start from f_0(x)=cosh x. The declared directions are differentiation, reciprocal formation, logarithmic differentiation, squaring, even projection, odd projection, and a restricted argument-doubling direction. The first three give D(cosh x)=sinh x, R(cosh x)=sechx, L(cosh x)=tanh x. Applying the same grammar again produces cschx, coth x, powers, p","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"Represent a one-dimensional Gaussian law by the exact parameter pair (μ,v), where v=σ^2>0. The grammar contains shifts by ±1, reflection, convolution with N(0,1), centering, scaling by two, and standardization. Their parameter actions are (μ,v) · ↦(μ±1,v), · (μ,v) · ↦(-μ,v),","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-5993af1a96a90d09","tex":"\\widehat d","aliases":"widehat d widehat","display":"\\widehat d","kind":"use","excerpt":"Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f0a95732dd6dfe79","tex":"\\widehat d_k","aliases":"widehat d_k widehat","display":"\\widehat d_k","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e8d6f1354dafb866","tex":"\\widehat x","aliases":"widehat x widehat","display":"\\widehat x","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-52a834738eb29cba","tex":"\\widehat x_0","aliases":"widehat x_0 widehat","display":"\\widehat x_0","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e448fa420e31ed83","tex":"\\widehat x_k","aliases":"widehat x_k widehat","display":"\\widehat x_k","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f182a1b5a8b7f15a","tex":"\\widehat x_{k-1}","aliases":"widehat x_k-1 widehat","display":"\\widehat x_{k-1}","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-63f5ccf6df462987","tex":"\\widehat x_n","aliases":"widehat x_n widehat","display":"\\widehat x_n","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-05da05e4a35ed5a5","tex":"\\widetilde\\Phi","aliases":"widetildeΦ widetilde Phi","display":"\\widetilde\\Phi","kind":"use","excerpt":"Let Φ_k and widetildeΦ_k be exact and perturbed deformation laws along corresponding routes. Suppose d_X(Φ_k(x,d),widetildeΦ_k(y,e)) ≤ L_kd_X(x,y)+M_kρ(d,e)+η_k, where η_k measures model error.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-020a1fe2af07d5a5","tex":"\\widetilde\\Phi_k","aliases":"widetildeΦ_k widetilde Phi","display":"\\widetilde\\Phi_k","kind":"use","excerpt":"Let Φ_k and widetildeΦ_k be exact and perturbed deformation laws along corresponding routes. Suppose d_X(Φ_k(x,d),widetildeΦ_k(y,e)) ≤ L_kd_X(x,y)+M_kρ(d,e)+η_k, where η_k measures model error.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"Let T be a set of types and let X=coprod_t∈ TX_t be the disjoint union of the corresponding object spaces.","count":12},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"Starting from |0⟩, one has |0⟩xrightarrowH|+⟩ xrightarrowS|+i⟩ xrightarrowS|-⟩, while H^2=X^2=Z^2=I on the projective orbit and S^4=I. The number of raw gate words grows as 4^n, but the quotient orbit saturates at six objects.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.","count":7},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"Metric completion is not the only possibility. In arithmetic deformation, a branch may be a compatible tower x_1← x_2← x_3←⋯ whose object is an inverse limit. In nested function spaces, a branch may enter a direct limit. In weak probability or operator topologies, convergence may be defined only against a test class.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c7df6d1ed132831f","tex":"x_{k+1}","aliases":"x_k+1","display":"x_{k+1}","kind":"use","excerpt":"For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","excerpt":"For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-ec84613b712fe298","tex":"x_N","aliases":"x_N","display":"x_N","kind":"use","excerpt":"The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The sel","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4715e03ab2461874","tex":"x_{n+1}","aliases":"x_n+1","display":"x_{n+1}","kind":"use","excerpt":"[Summable-step convergence] Suppose widehat X is complete and d_X(x_n+1,x_n)≤ a_n with ∑_n≥0a_n<∞. Then (x_n) is Cauchy and the route has a unique limit in widehat X.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-ab5f283acc8ce60c","tex":"X_t","aliases":"X_t","display":"X_t","kind":"use","excerpt":"Let T be a set of types and let X=coprod_t∈ TX_t be the disjoint union of the corresponding object spaces.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e9cb7b3bcd969244","tex":"\\Xcal^r","aliases":"Xcal^r X^r Xcal mathcal","display":"{\\mathcal X}^r","kind":"use","excerpt":"For every r≥2, let Ψ_r: X^r× D^(r) ⟶ X be a typed r-input deformation law.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"use","excerpt":"Let p(x)=x^2+1, (Tp)(x)=p(x+1), (Dp)(x)=p'(x). The two primitive directions T and D satisfy D(Tp)=T(Dp)=2x+2. The raw path tree therefore contains two length-two routes, pxrightarrowTTpxrightarrowDDTp, pxrightarrowDDpxrightarrowTTDp, but the quotient graph contains one endpoint class because DTp=TDp exactly.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"[Free rooted path property] Let Y be a rooted directed graph with root y_0. Suppose each admissible pair (x,d) is assigned an edge operation on Y in a way that depends only on the current image vertex and the label d. Then there is a unique root-preserving map F: T_fin(x_0)⟶ Y that follows those assigned edges along every path.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"Given any two objects x and y, one can invent a one-step map that sends x to y. Such a map does not reveal a generative structure. It hides the target inside the operation.","count":10},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"use","excerpt":"[Free rooted path property] Let Y be a rooted directed graph with root y_0. Suppose each admissible pair (x,d) is assigned an edge operation on Y in a way that depends only on the current image vertex and the label d. Then there is a unique root-preserving map F: T_fin(x_0)⟶ Y that follows those assigned edges along every path.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.","count":9},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z mathbb","display":"{\\mathbb Z}","kind":"use","excerpt":"Let the root be 0∈ Z. Every integer has two directions, +1, -1. A path of length n is a word in these two labels.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"The implementation works directly after quotienting global phase. A state is one of the six exact Bloch-axis vectors (0,0,±1), (±1,0,0), (0,±1,0), representing the Pauli eigenstates |0⟩, |1⟩, |+⟩, |-⟩, |+i⟩, and |-i⟩. The primitive grammar contains H,S,X,Z, acting as signed permutations of the Bloch coordinates: H(x,y,z)=(z,-y,x), S(x,y,z)=(-y,x,z)","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1a49e6bec04c7b56","tex":"Z^2","aliases":"Z^2","display":"Z^2","kind":"use","excerpt":"Starting from |0⟩, one has |0⟩xrightarrowH|+⟩ xrightarrowS|+i⟩ xrightarrowS|-⟩, while H^2=X^2=Z^2=I on the projective orbit and S^4=I. The number of raw gate words grows as 4^n, but the quotient orbit saturates at six objects.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4b3ed10343bd692c","tex":"Z_{\\mathrm{obj}}","aliases":"Z_obj mathrm","display":"Z_{\\mathrm{obj}}","kind":"use","excerpt":"A path zeta function measures construction complexity. A quotient zeta function measures distinct object complexity. Depending on the application, one may define Z_obj(s) = ∑_[x]∈ O(x_0)e^-sC([x]), where C([x]) is the least path cost reaching the class. The two functions answer different questions and should be stored together.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e977d8f942d6173e","tex":"Z_{x_0}","aliases":"Z_x_0","display":"Z_{x_0}","kind":"use","excerpt":"Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"The implementation works directly after quotienting global phase. A state is one of the six exact Bloch-axis vectors (0,0,±1), (±1,0,0), (0,±1,0), representing the Pauli eigenstates |0⟩, |1⟩, |+⟩, |-⟩, |+i⟩, and |-i⟩. The primitive grammar contains H,S,X,Z, acting as signed permutations of the Bloch coordinates: H(x,y,z)=(z,-y,x), S(x,y,z)=(-y,x,z)","count":10},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"Starting at 0, let the directions at depth n be ±2^-(n+1). Every infinite sign route converges because ∑_n≥02^-(n+1)=1. The boundary therefore maps into [-1,1]. Different sign sequences can represent the same endpoint only at the usual binary-expansion ambiguity, which can be handled by a path relation.","count":5},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"For every r≥2, let Ψ_r: X^r× D^(r) ⟶ X be a typed r-input deformation law.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"For a prefix α of length k≤ n, define the cylinder projector P^(n)_α lvertγ⟩ = lvertγ⟩, · α is a prefix of γ,","count":15},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.","count":6},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-57885e4c75965b23","tex":"\\Gamma","aliases":"Γ Gamma","display":"\\Gamma","kind":"use","excerpt":"A complete cut is a prefix-free set Γ meeting every infinite route exactly once. Its projectors satisfy ∑_α∈ΓP_α=I. Thus one branch may be resolved deeply while another stops early. Adaptive exploration can be represented without discarding the unresolved region.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"A second, closed subfamily starts from the impulse δ=(1,0,0,…). Repeated summation gives (C^kδ)_n=binomn+k-1k-1 (k≥1), by induction and the hockey-stick identity. This provides a genuine family certificate: perturbation depth k is the parameter, C advances the parameter, D reverses it away from the boundary, and the displayed coefficient formula is","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-3fb1097541ef7948","tex":"\\delta_0","aliases":"δ_0 delta","display":"\\delta_0","kind":"use","excerpt":"Sequences · A Fibonacci prefix and the impulse δ_0 with difference, summation, binomial transform, sign alternation, and reversal · Exact transform relations generate structured integer families rather than one undifferentiated list of examples.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).","count":14},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The sel","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-a14473f51c7df774","tex":"\\epsilon_j","aliases":"ϵ_j epsilon","display":"\\epsilon_j","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-05d3afd272ede960","tex":"\\epsilon_k","aliases":"ϵ_k epsilon","display":"\\epsilon_k","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":7},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-dc06eebc88a8b163","tex":"\\varepsilon_n","aliases":"ε_n varepsilon","display":"\\varepsilon_n","kind":"use","excerpt":"The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The sel","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1dd0a87bdefa7845","tex":"\\eta_j","aliases":"η_j eta","display":"\\eta_j","kind":"use","excerpt":"[Recursive deformation stability] If the direction mismatch at step k is at most ϵ_k, then the endpoint error satisfies e_n ≤ (∏_k=1^nL_k)e_0 + ∑_j=1^n (M_jϵ_j+η_j) ∏_k=j+1^nL_k.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-fc03a685a0a01ebc","tex":"\\eta_k","aliases":"η_k eta","display":"\\eta_k","kind":"use","excerpt":"Let Φ_k and widetildeΦ_k be exact and perturbed deformation laws along corresponding routes. Suppose d_X(Φ_k(x,d),widetildeΦ_k(y,e)) ≤ L_kd_X(x,y)+M_kρ(d,e)+η_k, where η_k measures model error.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c32bc1bd750febf9","tex":"\\theta_0","aliases":"θ_0 theta","display":"\\theta_0","kind":"use","excerpt":"[Inductive soundness of a family certificate] Suppose the seed member F(θ_0) is valid, every closure identity is proved, and every relation in Q is preserved by the parameter actions. Then every finite parameter word in the maps σ_d produces a valid member of the certified family.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"Consider u=s+λ u^2. Repeated substitution expands u into rooted binary trees. If only shape is counted, the number of binary trees with n internal vertices is the Catalan number C_n.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"Represent a one-dimensional Gaussian law by the exact parameter pair (μ,v), where v=σ^2>0. The grammar contains shifts by ±1, reflection, convolution with N(0,1), centering, scaling by two, and standardization. Their parameter actions are (μ,v) · ↦(μ±1,v), · (μ,v) · ↦(-μ,v),","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).","count":6},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-19765190fb4007c9","tex":"\\rho^{-1}","aliases":"ρ^-1 rho","display":"\\rho^{-1}","kind":"use","excerpt":"For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-d903684d8cb76c26","tex":"\\rho_x","aliases":"ρ_x rho","display":"\\rho_x","kind":"use","excerpt":"Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"Represent a one-dimensional Gaussian law by the exact parameter pair (μ,v), where v=σ^2>0. The grammar contains shifts by ±1, reflection, convolution with N(0,1), centering, scaling by two, and standardization. Their parameter actions are (μ,v) · ↦(μ±1,v), · (μ,v) · ↦(-μ,v),","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4774aaf9cf585ef8","tex":"\\Phi_k","aliases":"Φ_k Phi","display":"\\Phi_k","kind":"use","excerpt":"Let Φ_k and widetildeΦ_k be exact and perturbed deformation laws along corresponding routes. Suppose d_X(Φ_k(x,d),widetildeΦ_k(y,e)) ≤ L_kd_X(x,y)+M_kρ(d,e)+η_k, where η_k measures model error.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"use","excerpt":"For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-ee59dabb8327b62f","tex":"\\leftarrow","aliases":"← leftarrow","display":"\\leftarrow","kind":"use","excerpt":"Metric completion is not the only possibility. In arithmetic deformation, a branch may be a compatible tower x_1← x_2← x_3←⋯ whose object is an inverse limit. In nested function spaces, a branch may enter a direct limit. In weak probability or operator topologies, convergence may be defined only against a test class.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"Represent a one-dimensional Gaussian law by the exact parameter pair (μ,v), where v=σ^2>0. The grammar contains shifts by ±1, reflection, convolution with N(0,1), centering, scaling by two, and standardization. Their parameter actions are (μ,v) · ↦(μ±1,v), · (μ,v) · ↦(-μ,v), (μ,v) · ↦(μ,v+1), · (μ,v) · ↦(0,v), (μ,v) · ↦(2μ,4v), · (μ,v) · ↦(0,1). St","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-196f36799a027983","tex":"\\partial","aliases":"∂ partial","display":"\\partial","kind":"use","excerpt":"An infinite route is a sequence ω=(d_1,d_2,…) whose every finite prefix is admissible. The set of infinite routes is the path boundary ∂ T(x_0).","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-c9955fd0d38cf7cb","tex":"\\prod","aliases":"∏ prod","display":"\\prod","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":3},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-badeefcf2b599837","tex":"\\prod_{k=1}^n","aliases":"∏_k=1^n prod","display":"\\prod_{k=1}^n","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-8044eafcf18f2498","tex":"\\prod_{k=j+1}^n","aliases":"∏_k=j+1^n prod","display":"\\prod_{k=j+1}^n","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-d023b6a1042bdc7b","tex":"\\prod_{k>j}","aliases":"∏_k>j prod","display":"\\prod_{k>j}","kind":"use","excerpt":"Cover closed bound · e_n≤ (∏ L_k)e_0+∑_jM_jϵ_j∏_k>jL_k","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"The cost of a finite path is cost(γ)=∑_k=1^nκ(d_k). A path record may also store error bounds, provenance, validity certificates, symbolic assumptions, and the first place at which an invariant changed. None of these fields alters reachability. They determine how a finite explorer prioritizes and audits the infinite object.","count":25},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-9b5ba91e2a7d0fa6","tex":"\\sum_{[x]\\in\\Ocal(x_0)}","aliases":"∑_[x]∈Ocal(x_0) ∑_[x]∈ O(x_0) sum in Ocal sum in mathcal","display":"\\sum_{[x]\\in{\\mathcal O}(x_0)}","kind":"use","excerpt":"A path zeta function measures construction complexity. A quotient zeta function measures distinct object complexity. Depending on the application, one may define Z_obj(s) = ∑_[x]∈ O(x_0)e^-sC([x]), where C([x]) is the least path cost reaching the class. The two functions answer different questions and should be stored together.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-581e46df03c6e7a6","tex":"\\sum_d","aliases":"∑_d sum","display":"\\sum_d","kind":"use","excerpt":"Projector refinement · P_α=∑_dP_α d","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-dd8808d67913fb3f","tex":"\\sum_{d\\in\\Dcal_{\\ev(\\alpha)}}","aliases":"∑_d∈Dcal_ev(α) ∑_d∈ D_ev(α) sum in Dcal ev alpha sum in mathcal operatorname alpha","display":"\\sum_{d\\in{\\mathcal D}_{{\\operatorname{ev}}(\\alpha)}}","kind":"use","excerpt":"[Exact cylinder refinement] For every prefix α and every depth n>absα, P^(n)_α = ∑_d∈ D_ev(α)P^(n)_α d, with strong convergence when the child set is countable.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Cover closed bound · e_n≤ (∏ L_k)e_0+∑_jM_jϵ_j∏_k>jL_k","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-ee6b2fc7cf7ca7f4","tex":"\\sum_{j=0}^{n}","aliases":"∑_j=0^n sum","display":"\\sum_{j=0}^{n}","kind":"use","excerpt":"The executable atlas starts from the eight-term Fibonacci prefix a=(0,1,1,2,3,5,8,13). Its fixed grammar contains first difference, partial summation, the binomial transform, sign alternation, and reversal. For a finite sequence a=(a_0,…,a_N-1), define (Da)_0=a_0, (Da)_n=a_n-a_n-1, (Ca)_n=∑_j=0^na_j, (Ba)_n=∑_j=0^nbinomnja_j. The two triangular ope","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-db50c77870a75686","tex":"\\sum_{j=1}^n","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^n","kind":"use","excerpt":"[Pathwise cover error] Consider an exact route x_k=x_k-1⋆ d_k and a sampled route widehat x_k=widehat x_k-1⋆widehat d_k. Suppose at step k, d_X(x⋆ d,y⋆ e) ≤ L_k d_X(x,y)+M_kρ(d,e), and ρ(d_k,widehat d_k)≤ϵ_k. Then d_X(x_n,widehat x_n) ≤ (∏_k=1^nL_k)d_X(x_0,widehat x_0) + ∑_j=1^n M_jϵ_j ∏_k=j+1^nL_k.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-9b39a1c18171b88a","tex":"\\sum_k","aliases":"∑_k sum","display":"\\sum_k","kind":"use","excerpt":"Path cost · cost(γ)=∑_kκ(d_k)","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-b7f7abb917a0d195","tex":"\\sum_{k=0}^n","aliases":"∑_k=0^n sum","display":"\\sum_{k=0}^n","kind":"use","excerpt":"[Catalan interaction growth] The counts satisfy C_0=1, C_n+1=∑_k=0^nC_kC_n-k.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-dfd8567f7f80b3f0","tex":"\\sum_{k=1}^n","aliases":"∑_k=1^n sum","display":"\\sum_{k=1}^n","kind":"use","excerpt":"The cost of a finite path is cost(γ)=∑_k=1^nκ(d_k). A path record may also store error bounds, provenance, validity certificates, symbolic assumptions, and the first place at which an invariant changed. None of these fields alters reachability. They determine how a finite explorer prioritizes and audits the infinite object.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-baa8f7877711f9a7","tex":"\\sum_{k=n}^{m-1}","aliases":"∑_k=n^m-1 sum","display":"\\sum_{k=n}^{m-1}","kind":"use","excerpt":"For m>n, d_X(x_m,x_n) ≤ ∑_k=n^m-1d_X(x_k+1,x_k) ≤ ∑_k=n^m-1a_k. The tail tends to zero. Completeness supplies a unique limit.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-01f56a5e1c2b8b56","tex":"\\sum_{k\\ge1}","aliases":"∑_k≥1 sum ge","display":"\\sum_{k\\ge1}","kind":"use","excerpt":"[Convergent direction covers] If L_k≤1, M_k≤ M, and ∑_k≥1ϵ_k<∞, then the total endpoint error of the sampled route is bounded uniformly in depth by M∑_k≥1ϵ_k when the roots coincide.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-0fbc71e3cef7af4e","tex":"\\sum_{n=0}^{N-1}","aliases":"∑_n=0^N-1 sum","display":"\\sum_{n=0}^{N-1}","kind":"use","excerpt":"The executable completion example starts from x_0=0 and exposes the two depth-dependent directions d_n^±=±2^-(n+1). A finite route with signs ε_n∈{-1,+1} reaches x_N=∑_n=0^N-1ε_n2^-(n+1). Every infinite sign route is Cauchy because |x_m-x_n| ≤ ∑_k=n^m-12^-(k+1) <2^-n. The completed object therefore contains a limit for every infinite route. The sel","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-f6ef3ac4ec7b2013","tex":"\\sum_{n\\ge0}","aliases":"∑_n≥0 sum ge","display":"\\sum_{n\\ge0}","kind":"use","excerpt":"[Summable-step convergence] Suppose widehat X is complete and d_X(x_n+1,x_n)≤ a_n with ∑_n≥0a_n<∞. Then (x_n) is Cauchy and the route has a unique limit in widehat X.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"[Geometric convergence bound] Suppose every vertex has at most B children and every direction costs at least c>0. Then Z_x_0(s) ≤ ∑_n≥1B^ne^-scn. Consequently, Z_x_0(s) converges whenever sc>log B. In a full regular B-ary tree with constant cost c, it diverges whenever sc≤log B.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","excerpt":"Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-647f67659ece0b3f","tex":"\\sum_{\\alpha\\in\\Gamma}","aliases":"∑_α∈Γ sum alpha in Gamma","display":"\\sum_{\\alpha\\in\\Gamma}","kind":"use","excerpt":"A complete cut is a prefix-free set Γ meeting every infinite route exactly once. Its projectors satisfy ∑_α∈ΓP_α=I. Thus one branch may be resolved deeply while another stops early. Adaptive exploration can be represented without discarding the unresolved region.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-dd5a8bdc2b931502","tex":"\\sum_\\beta","aliases":"∑_β sum beta","display":"\\sum_\\beta","kind":"use","excerpt":"Let A: H→ K be a bounded operator and let {P_α} and {Q_β} be complete route resolutions. Define A_βα=Q_β A P_α. Then A=∑_α,βA_βα. If B follows A, then (BA)_γα = ∑_β B_γβA_βα. Every operator composition becomes a sum over compatible intermediate perturbation packets.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-5b7d14c98fe44904","tex":"\\sum_{\\gamma\\ne\\varnothing}","aliases":"∑_γ≠∅ sum gamma ne varnothing","display":"\\sum_{\\gamma\\ne\\varnothing}","kind":"use","excerpt":"Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-6f333436e7d549b7","tex":"\\ast","aliases":"∗ ast","display":"\\ast","kind":"use","excerpt":"One explicit route is (0,1)xrightarrowshift +1(1,1) xrightarrow∗ N(0,1)(1,2) xrightarrowscale 2(2,8) xrightarrowstandardize(0,1). The final edge is a many-to-one canonicalization direction. It produces a strong quotient collapse without erasing the route that created the noncanonical law. The differential entropy h(N(μ,v))=frac12log(2π e v) is unaf","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"use","excerpt":"For the seed P_3, the first descendants include overlineP_3, L(P_3)≅ P_2, P_3 with one attached leaf, P_4 obtained by subdivision. If ρ reverses the vertex labels, then key(G)=key(ρ Gρ^-1) exactly, although the two raw paths and raw adjacency matrices remain distinct.","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"Define Z_x_0(s) = ∑_γ≠∅ exp(-scost(γ)). This is a path partition function. It weights short, inexpensive routes more strongly than long, expensive routes.","count":2},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"Let each fiber D_x carry a metric ρ_x. A finite set N_ϵ(x)⊆ D_x is an ϵ-net when every d∈ D_x lies within ϵ of some widehat d∈ N_ϵ(x).","count":1},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"use","excerpt":"Metric completion is not the only possibility. In arithmetic deformation, a branch may be a compatible tower x_1← x_2← x_3←⋯ whose object is an inverse limit. In nested function spaces, a branch may enter a direct limit. In weak probability or operator topologies, convergence may be defined only against a test class.","count":4},{"work":"universal-generative-deformation","url":"notation/universal-generative-deformation.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"use","excerpt":"For a finite or countable depth layer, define H_n=ℓ^2( T_n(x_0)). The canonical basis vector associated with a path γ is written lvertγ⟩.","count":12},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":51},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":32},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ea0b1d9a333c4d54","tex":"\\mathfrak B","aliases":"B mathfrak","display":"\\mathfrak B","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ab8bde9a479f56a0","tex":"\\mathfrak C","aliases":"C mathfrak","display":"\\mathfrak C","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":11},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a11d22a99cb376b1","tex":"\\mathfrak E","aliases":"E mathfrak","display":"\\mathfrak E","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d35f3661d97c3847","tex":"E_{a,\\alpha}","aliases":"E_a,α alpha","display":"E_{a,\\alpha}","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":5},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3a9e648245c571b5","tex":"\\mathfrak G","aliases":"G mathfrak","display":"\\mathfrak G","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-1bf12537a68f2d94","tex":"\\mathfrak H","aliases":"H mathfrak","display":"\\mathfrak H","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-e3d8f1ac9931ea94","tex":"\\mathfrak L","aliases":"L mathfrak","display":"\\mathfrak L","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"definition-passage","excerpt":"model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed","count":12},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-abf2236063a03f66","tex":"\\mathfrak P","aliases":"P mathfrak","display":"\\mathfrak P","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-1a59384e82a4f874","tex":"P_{a,\\alpha}","aliases":"P_a,α alpha","display":"P_{a,\\alpha}","kind":"definition-passage","excerpt":"model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of statistical states; item G is a typed","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-32fe36360874826c","tex":"\\mathfrak R","aliases":"R mathfrak","display":"\\mathfrak R","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":10},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-587412a39e90b8d4","tex":"\\mathfrak U","aliases":"U mathfrak","display":"\\mathfrak U","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ec4379f6b5fb17bc","tex":"U_a","aliases":"U_a","display":"U_a","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":8},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":12},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-df0285eec4b01028","tex":"\\mathfrak Z","aliases":"Z mathfrak","display":"\\mathfrak Z","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":13},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"definition-passage","excerpt":"[Complete pruning] A set Λ of packet-tree nodes is a complete pruning when every root-to-leaf path meets Λ exactly once.","count":9},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"definition-passage","excerpt":"[Projector-resolved large signal model] A finite PR-LASM instance is a tuple L= (X, U, E, P, Z, G, B, H, R, C), where: item X is the observation space; item U={U_a}_a∈ A is a family of exact coordinate atlases; item E={E_a,α} is the family of coordinate selectors; item P={P_a,α} is the pulled-back projector family; item Z is a nested family of stat","count":10},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":12},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":16},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-e2e02e56d4929ee2","tex":"\\mathbf1_{\\mathrm{changed}}","aliases":"1_changed mathbf mathrm","display":"\\mathbf1_{\\mathrm{changed}}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-531c116a81dd99e4","tex":"\\mathbf1_{\\mathrm{loss}}","aliases":"1_loss mathbf mathrm","display":"\\mathbf1_{\\mathrm{loss}}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-2d333d14ee1fe742","tex":"\\mathbf1_{m\\equiv r\\ (N)}","aliases":"1_m≡ r (N) mathbf equiv","display":"\\mathbf1_{m\\equiv r\\ (N)}","kind":"use","excerpt":"For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-6ddae0530ebcd696","tex":"\\mathbf1_{n\\equiv s\\ (N)}","aliases":"1_n≡ s (N) mathbf equiv","display":"\\mathbf1_{n\\equiv s\\ (N)}","kind":"use","excerpt":"For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-9aac36f73db150ba","tex":"\\mathbf1_{\\mathrm{untried}}","aliases":"1_untried mathbf mathrm","display":"\\mathbf1_{\\mathrm{untried}}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-e50cbb7db79d491b","tex":"\\mathbf1_{\\mathrm{win}}","aliases":"1_win mathbf mathrm","display":"\\mathbf1_{\\mathrm{win}}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-4d9dddd82afcfe91","tex":"\\mathbf1_{x[m,n]=j}","aliases":"1_x[m,n]=j mathbf","display":"\\mathbf1_{x[m,n]=j}","kind":"use","excerpt":"A finite-valued frame can be encoded by the one-hot coordinates C_j[m,n]= 1_x[m,n]=j. The original frame is recovered by the unique active color at each address. This reconstruction is categorical rather than linear over the original scalar encoding, so it is kept as a separately typed exact atlas.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-0230e8bde40956b0","tex":"\\Acal","aliases":"Acal A mathcal","display":"{\\mathcal A}","kind":"use","excerpt":"Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-fa986b9c4f40b1a7","tex":"\\Acal_N","aliases":"Acal_N A_N Acal mathcal","display":"{\\mathcal A}_N","kind":"use","excerpt":"Polyphase analysis and synthesis. For modulus N, ( A_Nx)_r,s[q,t]=x[Nq+r,Nt+s]. The synthesis operator places each channel value back at its unique original address. Therefore S_N A_N=I.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-22453f646578b304","tex":"\\appendix","aliases":"appendix","display":null,"kind":"use","excerpt":"[4pt] → memory and transition graph → active action → residual → selective refinement → exact reconstruction Within its finite declared scope, that loop is complete. appendix Exact transform formulas. Haar analysis and synthesis. For one pair, ℓ=fraca+bsqrt2, h=fraca-bsqrt2, and a=fracℓ+hsqrt2, b=fracℓ-hsqrt2. Applying these identities separably an","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-465e1a0d3cb61668","tex":"B_g^\\top","aliases":"B_g^top top","display":"B_g^\\top","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"use","excerpt":"The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-292e8005afbb88ea","tex":"\\bar q_t","aliases":"bar q_t bar","display":"\\bar q_t","kind":"use","excerpt":"The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-40894f114ea8a526","tex":"\\bar\\mu","aliases":"barμ bar mu","display":"\\bar\\mu","kind":"use","excerpt":"If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-e76ca40698ecc64e","tex":"\\bfseries","aliases":"bfseries","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"For modulus N, the phase program appends the one-hot phase e_tbmod N and interactions z_t,ie_tbmod N. Alternating, cyclic, or clocked dynamics therefore become linear in an expanded polyphase state.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[Exact atlas residuals] Across the tested frames, the maximum observed residuals were lr atlas · maximum absolute residual spatial · 0 Fourier · 3.55×10^-15 Haar · 2.49×10^-14 polyphase-2 · 0 polyphase-4 · 0 categorical color · 0 weighted multi-atlas frame · 5.33×10^-15 The nonzero values are double-precision roundoff.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":9},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d9494e21b365d43b","tex":"c_a","aliases":"c_a","display":"c_a","kind":"use","excerpt":"Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-134f1c50f630c94b","tex":"C_j","aliases":"C_j","display":"C_j","kind":"use","excerpt":"A finite-valued frame can be encoded by the one-hot coordinates C_j[m,n]= 1_x[m,n]=j. The original frame is recovered by the unique active color at each address. This reconstruction is categorical rather than linear over the original scalar encoding, so it is kept as a separately typed exact atlas.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a95360f059e69965","tex":"\\caption","aliases":"caption","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-444090bde130ba27","tex":"\\centering","aliases":"centering","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d0fe2f8238bcf56d","tex":"\\operatorname{ch}","aliases":"ch operatorname","display":"\\operatorname{ch}","kind":"use","excerpt":"Let c(v) be any finite additive node cost. Define recursively V(v)= min{c(v),∑_u∈ch(v)V(u)}.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-e4be53ae2a9b983e","tex":"\\mathrm{changed}","aliases":"changed mathrm","display":"\\mathrm{changed}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-450d165bfeb0ca8d","tex":"\\clearpage","aliases":"clearpage","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-581d8c822b2d8c56","tex":"\\operatorname{Complexity}","aliases":"Complexity operatorname","display":"\\operatorname{Complexity}","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-59c0124e24487c12","tex":"d_g","aliases":"d_g","display":"d_g","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-665b42b33f87687b","tex":"d_g^2","aliases":"d_g^2","display":"d_g^2","kind":"use","excerpt":"For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n), Haar atlas · O(n), polyphase atlas · O(n), feature projection · O(dk), one regression fit · O(md_g^2+d_g^3), all program fits · displaystyle∑_g=1^GO(md_g^2+d_g^3)","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-037c5997a1f4148e","tex":"d_g^3","aliases":"d_g^3","display":"d_g^3","kind":"use","excerpt":"For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n), Haar atlas · O(n), polyphase atlas · O(n), feature projection · O(dk), one regression fit · O(md_g^2+d_g^3), all program fits · displaystyle∑_g=1^GO(md_g^2+d_g^3)","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-140ecc80082313b5","tex":"e''","aliases":"e''","display":"e''","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d8ac034f27905dad","tex":"e^{-2\\pi i(km/H+\\ell n/W)}","aliases":"e^-2π i(km/H+ℓ n/W) pi ell","display":"e^{-2\\pi i(km/H+\\ell n/W)}","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-997eae91d3ca2403","tex":"e^{-4\\bar q_t}","aliases":"e^-4bar q_t bar","display":"e^{-4\\bar q_t}","kind":"use","excerpt":"The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a093566fc4e49e16","tex":"e^{W_g}","aliases":"e^W_g","display":"e^{W_g}","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-41e6fd4a7fcb4798","tex":"e^{W_h}","aliases":"e^W_h","display":"e^{W_h}","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-6272a68e19dc7d15","tex":"E_{a,\\beta}","aliases":"E_a,β beta","display":"E_{a,\\beta}","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8c03467ab8d8fd33","tex":"E_B","aliases":"E_B","display":"E_B","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-c76ccdfb56fe4fb6","tex":"e_{t\\bmod N}","aliases":"e_tbmod N bmod","display":"e_{t\\bmod N}","kind":"use","excerpt":"For modulus N, the phase program appends the one-hot phase e_tbmod N and interactions z_t,ie_tbmod N. Alternating, cyclic, or clocked dynamics therefore become linear in an expanded polyphase state.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-51748184895fafdf","tex":"\\mathrm{exp}","aliases":"exp mathrm","display":"\\mathrm{exp}","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a3382034c436fa8d","tex":"F^*","aliases":"F^*","display":"F^*","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f063dc7c1c3b3019","tex":"\\fontsize","aliases":"fontsize","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n),","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":16},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":10},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-eff6aa27c552c5d2","tex":"I_a","aliases":"I_a","display":"I_a","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d25b0ba4c91945ce","tex":"I_X","aliases":"I_X","display":"I_X","kind":"use","excerpt":"[Per-atlas packet reconstruction] For every atlas a, the operators P_a,α are orthogonal projectors and ∑_α∈ I_aP_a,α=I_X. Consequently, x=∑_αP_a,αx, T=∑_α,βP_a,βTP_a,α.","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-c2ad01df50f1bc45","tex":"I_{Y_a}","aliases":"I_Y_a","display":"I_{Y_a}","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-9a02f588c4e95e5a","tex":"\\includegraphics","aliases":"includegraphics","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ec4146e536630f36","tex":"\\operatorname{Information}","aliases":"Information operatorname","display":"\\operatorname{Information}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-fe2d7f6496f8d321","tex":"\\operatorname{InvGamma}","aliases":"InvGamma operatorname","display":"\\operatorname{InvGamma}","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-1bdfdabfb7ccb3ec","tex":"k_1","aliases":"k_1","display":"k_1","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a8cd867fa822df3a","tex":"k_2","aliases":"k_2","display":"k_2","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a57e3beb8044c054","tex":"k_3","aliases":"k_3","display":"k_3","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-49e6cb003fe8b536","tex":"L_t","aliases":"L_t","display":"L_t","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-bda22e23ac42a073","tex":"\\Large","aliases":"Large","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3b3d7f267bfbab0b","tex":"\\large","aliases":"large","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-5c66bb4ce7ecf070","tex":"\\mathrm{lin}","aliases":"lin mathrm","display":"\\mathrm{lin}","kind":"use","excerpt":"The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-45e90aa85790529d","tex":"\\linewidth","aliases":"linewidth","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n), Haar atlas · O(n), polyphase atlas · O(n), feature projection · O(dk), one regression fit · O(md_g^2+d_g^3), all program fits · displaystyle∑_g=1^GO(md_g^2+d_g^3)","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-7e6de63f4b8b266d","tex":"\\mathrm{loss}","aliases":"loss mathrm","display":"\\mathrm{loss}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":11},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-bc4c24fa42db60db","tex":"m_a","aliases":"m_a","display":"m_a","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-6a2eba51b6026fde","tex":"M_n","aliases":"M_n","display":"M_n","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-6f3ae7d6b37bdd61","tex":"m_t","aliases":"m_t","display":"m_t","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-cf19f32ef13753ab","tex":"m_t^{\\mathrm{exp}}","aliases":"m_t^exp mathrm","display":"m_t^{\\mathrm{exp}}","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-0b0f476597ac3de9","tex":"m_t^{\\mathrm{pow}}","aliases":"m_t^pow mathrm","display":"m_t^{\\mathrm{pow}}","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-fafdc58adc238007","tex":"\\mathrm{macro}","aliases":"macro mathrm","display":"\\mathrm{macro}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-16c3b6acce4190d9","tex":"\\operatorname{Mechanisms}","aliases":"Mechanisms operatorname","display":"\\operatorname{Mechanisms}","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-93b5f92d0dc913b0","tex":"\\mathrm{meso}","aliases":"meso mathrm","display":"\\mathrm{meso}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-9acd91ca3d20de49","tex":"\\mathrm{micro}","aliases":"micro mathrm","display":"\\mathrm{micro}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-19f3073c143a3a98","tex":"\\mid","aliases":"mid","display":null,"kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3f9db2ae5742966b","tex":"\\midrule","aliases":"midrule","display":null,"kind":"use","excerpt":"[Exact atlas residuals] Across the tested frames, the maximum observed residuals were lr atlas · maximum absolute residual spatial · 0 Fourier · 3.55×10^-15 Haar · 2.49×10^-14 polyphase-2 · 0 polyphase-4 · 0 categorical color · 0 weighted multi-atlas frame · 5.33×10^-15 The nonzero values are double-precision roundoff.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"Let c(v) be any finite additive node cost. Define recursively V(v)= min{c(v),∑_u∈ch(v)V(u)}.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":12},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":13},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"use","excerpt":"[Exact multi-atlas tight-frame reconstruction] The analysis and synthesis maps satisfy S A=I_X. Moreover, norm A x^2=cnormx^2.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-5526ec988ef53237","tex":"\\operatorname{Novelty}","aliases":"Novelty operatorname","display":"\\operatorname{Novelty}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":5},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d3c6dcd3c80aee0e","tex":"P_{a,\\beta}","aliases":"P_a,β beta","display":"P_{a,\\beta}","kind":"use","excerpt":"[Per-atlas packet reconstruction] For every atlas a, the operators P_a,α are orthogonal projectors and ∑_α∈ I_aP_a,α=I_X. Consequently, x=∑_αP_a,αx, T=∑_α,βP_a,βTP_a,α.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a5c35e870d97f9c4","tex":"P_B","aliases":"P_B","display":"P_B","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3a8ac723dc34a3dc","tex":"P_B^*","aliases":"P_B^*","display":"P_B^*","kind":"use","excerpt":"Fourier selector projectors. For any diagonal zero-one mask E_B, P_B=F^*E_BF, so P_B^2=P_B, P_B^*=P_B. Disjoint masks annihilate, and a complete mask partition sums to the identity.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-696a6127a0154036","tex":"P_B^2","aliases":"P_B^2","display":"P_B^2","kind":"use","excerpt":"Fourier selector projectors. For any diagonal zero-one mask E_B, P_B=F^*E_BF, so P_B^2=P_B, P_B^*=P_B. Disjoint masks annihilate, and a complete mask partition sums to the identity.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-2dd4810c8ecd9090","tex":"p_g","aliases":"p_g","display":"p_g","kind":"use","excerpt":"If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-68959ba0b4605625","tex":"P_{N,r,s}","aliases":"P_N,r,s","display":"P_{N,r,s}","kind":"use","excerpt":"For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-bc660fe2b7285e52","tex":"\\par","aliases":"par","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ed07df38becbad36","tex":"\\mathrm{pow}","aliases":"pow mathrm","display":"\\mathrm{pow}","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-28156301964cd09e","tex":"\\mathrm{pred}","aliases":"pred mathrm","display":"\\mathrm{pred}","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-4ae81572f06e1b88","tex":"Q","aliases":"Q","display":"Q","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":8},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-1fe5dd9f841d9595","tex":"Q_{1:k_1}","aliases":"Q_1:k_1","display":"Q_{1:k_1}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f76535969d048f1f","tex":"Q_{1:k_2}","aliases":"Q_1:k_2","display":"Q_{1:k_2}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3574ead0f39a344e","tex":"Q_{1:k_3}","aliases":"Q_1:k_3","display":"Q_{1:k_3}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-13843ad0837f9602","tex":"q_{t,i}","aliases":"q_t,i","display":"q_{t,i}","kind":"use","excerpt":"The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f9318f1b01d5ffc8","tex":"q_{t,j}","aliases":"q_t,j","display":"q_{t,j}","kind":"use","excerpt":"The quadratic program adds z_t,iz_t,j, q_t,iq_t,j, i≤ j, within a controlled rank. This is a finite packet jet, not an unrestricted polynomial expansion.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R mathbb","display":"{\\mathbb R}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":10},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-6e42d74a72040d0b","tex":"\\R^d","aliases":"R^d R mathbb","display":"{\\mathbb R}^d","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-9c10ef42be4e8cfe","tex":"\\R^{d_g}","aliases":"R^d_g R mathbb","display":"{\\mathbb R}^{d_g}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-0075521e479eb43d","tex":"r_l","aliases":"r_l","display":"r_l","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-49121c793f36cbb5","tex":"R_t","aliases":"R_t","display":"R_t","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-b0d5e932fa38960c","tex":"\\operatorname{RepeatCount}","aliases":"RepeatCount operatorname","display":"\\operatorname{RepeatCount}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-b4c709771f9318f7","tex":"\\operatorname{ResolutionPenalty}","aliases":"ResolutionPenalty operatorname","display":"\\operatorname{ResolutionPenalty}","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3a6dc709e846f3e9","tex":"\\operatorname{RoleHistory}","aliases":"RoleHistory operatorname","display":"\\operatorname{RoleHistory}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-5e769b89788d547a","tex":"s'","aliases":"s'","display":"s'","kind":"use","excerpt":"Each resolved frame receives a deterministic hash of its complete normalized array. An observed public action defines a graph edge (s,a)⟼ s'. Repeated observations accumulate reward, progress, win, loss, and visit counts.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-b764f1cbd00009f2","tex":"\\Scal","aliases":"Scal S mathcal","display":"{\\mathcal S}","kind":"use","excerpt":"Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f41d5a45a0059881","tex":"\\Scal_N","aliases":"Scal_N S_N Scal mathcal","display":"{\\mathcal S}_N","kind":"use","excerpt":"Polyphase analysis and synthesis. For modulus N, ( A_Nx)_r,s[q,t]=x[Nq+r,Nt+s]. The synthesis operator places each channel value back at its unique original address. Therefore S_N A_N=I.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f93687b75635f48a","tex":"\\section","aliases":"section","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-7c6199f1c7d32ba5","tex":"\\selectfont","aliases":"selectfont","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-25cdfc02207944fc","tex":"\\StatusBox","aliases":"StatusBox","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-848c3c3d231100c2","tex":"\\subsection","aliases":"subsection","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":21},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a9a485c98291d899","tex":"\\tableofcontents","aliases":"tableofcontents","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3000c3bd47e8007c","tex":"\\toprule","aliases":"toprule","display":null,"kind":"use","excerpt":"[Exact atlas residuals] Across the tested frames, the maximum observed residuals were lr atlas · maximum absolute residual spatial · 0 Fourier · 3.55×10^-15 Haar · 2.49×10^-14 polyphase-2 · 0 polyphase-4 · 0 categorical color · 0 weighted multi-atlas frame · 5.33×10^-15 The nonzero values are double-precision roundoff.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3e3becea69293024","tex":"\\mathrm{total}","aliases":"total mathrm","display":"\\mathrm{total}","kind":"use","excerpt":"If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8da22c10f7ad4a33","tex":"U_a^*","aliases":"U_a^*","display":"U_a^*","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":5},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a6ab3f6998c3dac0","tex":"u_t","aliases":"u_t","display":"u_t","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-83a0109bee973ca0","tex":"u_{t,j}","aliases":"u_t,j","display":"u_{t,j}","kind":"use","excerpt":"The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-c5acac65fc3cb090","tex":"\\mathrm{untried}","aliases":"untried mathrm","display":"\\mathrm{untried}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"Let c(v) be any finite additive node cost. Define recursively V(v)= min{c(v),∑_u∈ch(v)V(u)}.","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-9151633891e8e3e1","tex":"V_g","aliases":"V_g","display":"V_g","kind":"use","excerpt":"If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-3aa643447b38e662","tex":"V_{\\mathrm{total}}","aliases":"V_total mathrm","display":"V_{\\mathrm{total}}","kind":"use","excerpt":"If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-c7aebdde9d3e22d0","tex":"\\operatorname{Variance}","aliases":"Variance operatorname","display":"\\operatorname{Variance}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-0fef838539965a6b","tex":"\\vfill","aliases":"vfill","display":null,"kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":7},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-9012166380bed7e0","tex":"w_a","aliases":"w_a","display":"w_a","kind":"use","excerpt":"Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ac791e77e5ab00c5","tex":"w_a^2","aliases":"w_a^2","display":"w_a^2","kind":"use","excerpt":"Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-cf3773aab702a0f6","tex":"W_g","aliases":"W_g","display":"W_g","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-db94573662ebfd05","tex":"W_h","aliases":"W_h","display":"W_h","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-2a584b32cb183cf3","tex":"\\widehat p","aliases":"widehat p widehat","display":"\\widehat p","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-900db838f633cdb8","tex":"\\widehat p_{\\mathrm{loss}}","aliases":"widehat p_loss widehat mathrm","display":"\\widehat p_{\\mathrm{loss}}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-b90fb50d2eaab2dd","tex":"\\widehat p_{\\mathrm{win}}","aliases":"widehat p_win widehat mathrm","display":"\\widehat p_{\\mathrm{win}}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-967c63ebf4cd78be","tex":"\\widehat R","aliases":"widehat R widehat","display":"\\widehat R","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-799ccc4c3c03a7f0","tex":"\\widehat{\\Delta L}","aliases":"widehatΔ L widehat Delta","display":"\\widehat{\\Delta L}","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-352b3ac4a4067415","tex":"\\widetilde P","aliases":"widetilde P widetilde","display":"\\widetilde P","kind":"use","excerpt":"The normalized positive packets widetilde P_a,α =fracw_a^2cP_a,α satisfy ∑_a,αwidetilde P_a,α=I_X. They need not be mutually annihilating across different atlases. They are frame packets, not one enlarged orthogonal projector family. This distinction prevents the simultaneous use of several atlases from being mistaken for a direct-sum decomposition","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-b77ef49da9a096b8","tex":"\\widetilde P_{a,\\alpha}","aliases":"widetilde P_a,α widetilde alpha","display":"\\widetilde P_{a,\\alpha}","kind":"use","excerpt":"The normalized positive packets widetilde P_a,α =fracw_a^2cP_a,α satisfy ∑_a,αwidetilde P_a,α=I_X. They need not be mutually annihilating across different atlases. They are frame packets, not one enlarged orthogonal projector family. This distinction prevents the simultaneous use of several atlases from being mistaken for a direct-sum decomposition","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-91bffb1257de314e","tex":"\\mathrm{win}","aliases":"win mathrm","display":"\\mathrm{win}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":17},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-bb0e8953f025926b","tex":"X^\\top","aliases":"X^top top","display":"X^\\top","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-14a3332b34948231","tex":"x_*","aliases":"x_*","display":"x_*","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-2748c024609990ec","tex":"x_*^\\top","aliases":"x_*^top top","display":"x_*^\\top","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-75cf6c5142029723","tex":"X_g","aliases":"X_g","display":"X_g","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-65ed5da1c7a5991e","tex":"X_g^\\top","aliases":"X_g^top top","display":"X_g^\\top","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-655d3c9082b6bb48","tex":"x_{m,n}","aliases":"x_m,n","display":"x_{m,n}","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-df8851405c26a452","tex":"x_{r,s}","aliases":"x_r,s","display":"x_{r,s}","kind":"use","excerpt":"For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":5},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-1a532ad0b7c54d4f","tex":"Y^\\top","aliases":"Y^top top","display":"Y^\\top","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-57b7797a1c04a101","tex":"y^\\top","aliases":"y^top top","display":"y^\\top","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-536459ef72a0484d","tex":"Y_a","aliases":"Y_a","display":"Y_a","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-13077a4784c9a14d","tex":"y_t^{(g)}","aliases":"y_t^(g)","display":"y_t^{(g)}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":14},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-085b5b810fa92dba","tex":"z^{\\mathrm{macro}}","aliases":"z^macro mathrm","display":"z^{\\mathrm{macro}}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d9f38cdbb24de725","tex":"z^{\\mathrm{meso}}","aliases":"z^meso mathrm","display":"z^{\\mathrm{meso}}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-6dcb0b766f9058f1","tex":"z^{\\mathrm{micro}}","aliases":"z^micro mathrm","display":"z^{\\mathrm{micro}}","kind":"use","excerpt":"Let r(x)∈ R^d be the concatenated feature vector. A deterministic seeded QR factorization produces an orthonormal row system Q. The nested states are z^macro=Q_1:k_1r(x), z^meso=Q_1:k_2r(x), z^micro=Q_1:k_3r(x), with k_1<k_2<k_3<d. The supplied dimensions are 48, 96, and 192.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)","count":5},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d100944b4a5fa710","tex":"z_{t+1}^{(g)}","aliases":"z_t+1^(g)","display":"z_{t+1}^{(g)}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-b5c858416567dcbb","tex":"z_{t,i}","aliases":"z_t,i","display":"z_{t,i}","kind":"use","excerpt":"The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-5f8434165e598723","tex":"z_{t,j}","aliases":"z_t,j","display":"z_{t,j}","kind":"use","excerpt":"The quadratic program adds z_t,iz_t,j, q_t,iq_t,j, i≤ j, within a controlled rank. This is a finite packet jet, not an unrestricted polynomial expansion.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-9f2c528ad5400157","tex":"z_{t-1}","aliases":"z_t-1","display":"z_{t-1}","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8cdd068af3528ef3","tex":"z_{t-\\ell}","aliases":"z_t-ℓ ell","display":"z_{t-\\ell}","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-b6ba1645ab94ba03","tex":"z_t^{(g)}","aliases":"z_t^(g)","display":"z_t^{(g)}","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"[3pt]Large Algebraic Signal Model fontsize1721selectfontcolorMuted Exact Multi-Atlas Coordinates, Bayesian Operator Programs, Residual-Driven Refinement, Memory, and Active Planning 0.80linewidth large A signal-native world model in which spatial, Fourier, wavelet, polyphase, and categorical coordinates coexist inside one exact reconstruction syste","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f3f3804480e8551a","tex":"\\beta","aliases":"β beta","display":"\\beta","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d72a8a4f1d324e17","tex":"\\gamma_t","aliases":"γ_t gamma","display":"\\gamma_t","kind":"use","excerpt":"The gated program uses a smooth gate γ_t=frac11+e^-4bar q_t and appends γ_tz_t and (1-γ_t)z_t. The hybrid program contains a controlled combination of quadratic, delayed, power-memory, exponential-memory, and interaction terms. Its higher prior penalty prevents it from winning merely because it has more columns.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g) R_t/100 Δ L_t 1_win 1_loss 1_changed .","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-0fcb1c3cf5e3bf16","tex":"\\delta_{\\alpha\\beta}","aliases":"δ_αβ delta alpha beta","display":"\\delta_{\\alpha\\beta}","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"A large signal model should not hide all mechanism uncertainty inside one dense matrix. PR-LASM uses a typed program g=(a,r,f,κ), where a is an atlas, r is a resolution, f is an operator family, and κ contains family-specific ranks or periods.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-6ede142bbc0a6d7e","tex":"\\Lambda_0","aliases":"Λ_0 Lambda","display":"\\Lambda_0","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-7238fc50817cce55","tex":"\\lambda_0","aliases":"λ_0 lambda","display":"\\lambda_0","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-7ac40b6c2ce6f403","tex":"\\Lambda_0^{-1}","aliases":"Λ_0^-1 Lambda","display":"\\Lambda_0^{-1}","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ac5e3f20f76e9659","tex":"\\lambda_D","aliases":"λ_D lambda","display":"\\lambda_D","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-102becacb39a8e47","tex":"\\lambda_F","aliases":"λ_F lambda","display":"\\lambda_F","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-189e262e53188c4d","tex":"\\lambda_H","aliases":"λ_H lambda","display":"\\lambda_H","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ad8dd6c7c1bca510","tex":"\\lambda_I","aliases":"λ_I lambda","display":"\\lambda_I","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-71947d204642f781","tex":"\\lambda_L","aliases":"λ_L lambda","display":"\\lambda_L","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-62ed57f1458fb941","tex":"\\Lambda_n","aliases":"Λ_n Lambda","display":"\\Lambda_n","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-57d6bd6085b620be","tex":"\\lambda_N","aliases":"λ_N lambda","display":"\\lambda_N","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ae86bdce3f9ba2ea","tex":"\\Lambda_n^{-1}","aliases":"Λ_n^-1 Lambda","display":"\\Lambda_n^{-1}","kind":"use","excerpt":"For one program g, write Y=X_gB_g^top+ε, with independent output variances and a normal-inverse-gamma prior. If the prior precision is Λ_0=λ_0I, then Λ_n=Λ_0+X_g^top X_g, M_n=Y^top X_gΛ_n^-1. The posterior coefficient covariance is Λ_n^-1, and each output variance has an inverse-gamma posterior.","count":4},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-4c2cf3052bd12d0f","tex":"\\lambda_Q","aliases":"λ_Q lambda","display":"\\lambda_Q","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-412328d9c2d6574f","tex":"\\lambda_V","aliases":"λ_V lambda","display":"\\lambda_V","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-bb89ba50defcc109","tex":"\\lambda_W","aliases":"λ_W lambda","display":"\\lambda_W","kind":"use","excerpt":"For candidate action a, the planner forms a utility U(a)= · widehat R(a) +λ_LwidehatΔ L(a) +λ_Wwidehat p_win(a) -λ_Dwidehat p_loss(a) · +λ_NNovelty(a) +λ_IInformation(a) -λ_VVariance(a) · +λ_HRoleHistory(a) -λ_QRepeatCount(a)^2 +λ_F1_untried. Exact probe outcomes replace the corresponding predicted terms. Known graph wins and progress receive domin","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-603422880677d2e6","tex":"\\mu_g","aliases":"μ_g mu","display":"\\mu_g","kind":"use","excerpt":"If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-625e48cfdf47c402","tex":"\\mu_n","aliases":"μ_n mu","display":"\\mu_n","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-31335c0308b27e72","tex":"\\mu_n^\\top","aliases":"μ_n^top mu top","display":"\\mu_n^\\top","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":3},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8b857fa2a2cca622","tex":"\\rho^{\\ell-1}","aliases":"ρ^ℓ-1 rho ell","display":"\\rho^{\\ell-1}","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-c46ba1db118326e5","tex":"\\phi_g","aliases":"ϕ_g phi","display":"\\phi_g","kind":"use","excerpt":"For a packet state z_t, atlas features q_t, action features u_t, and memory state m_t, every program defines a design map ϕ_g(z_t,q_t,u_t,m_t,t)∈ R^d_g. The target is y_t^(g)= z_t+1^(g)-z_t^(g)","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-f8e733b0c123b84c","tex":"\\phi_{\\mathrm{lin}}","aliases":"ϕ_lin phi mathrm","display":"\\phi_{\\mathrm{lin}}","kind":"use","excerpt":"The basic linear program uses ϕ_lin=[1,z_t,q_t,u_t]. The action-bilinear extension adds products z_t,iu_t,j, q_t,iu_t,j, for a declared low interaction rank.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-b3496883c3d6e922","tex":"\\ell^{-\\alpha}","aliases":"ℓ^-α ell alpha","display":"\\ell^{-\\alpha}","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-711f0b3dcc0ebd21","tex":"\\ell_{\\mathrm{pred}}","aliases":"ℓ_pred ell mathrm","display":"\\ell_{\\mathrm{pred}}","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":6},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":15},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-9314650eb3dee5bd","tex":"\\sum_a","aliases":"∑_a sum","display":"\\sum_a","kind":"use","excerpt":"By unitarity, S A x =frac1c∑_a w_aU_a^*(w_aU_ax) =frac1c∑_aw_a^2x=x. Similarly, norm A x^2 =∑_aw_a^2normU_ax^2 =cnormx^2.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-1f17f6565da18c91","tex":"\\sum_{a,\\alpha}","aliases":"∑_a,α sum alpha","display":"\\sum_{a,\\alpha}","kind":"use","excerpt":"The normalized positive packets widetilde P_a,α =fracw_a^2cP_a,α satisfy ∑_a,αwidetilde P_a,α=I_X. They need not be mutually annihilating across different atlases. They are frame packets, not one enlarged orthogonal projector family. This distinction prevents the simultaneous use of several atlases from being mistaken for a direct-sum decomposition","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ad151dded7ef7d37","tex":"\\sum_{a\\in A}","aliases":"∑_a∈ A sum in","display":"\\sum_{a\\in A}","kind":"use","excerpt":"Choose weights w_a>0 and define A x=(w_aU_ax)_a∈ A, with synthesis S(c_a)_a∈ A = frac1c ∑_a∈ Aw_aU_a^*c_a, c=∑_a∈ Aw_a^2.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-76e758f1be5158a2","tex":"\\sum_g","aliases":"∑_g sum","display":"\\sum_g","kind":"use","excerpt":"If program g predicts scalar vector μ_g with variance V_g, then barμ=∑_gp_gμ_g, and the total predictive variance is V_total = underbrace∑_gp_gV_g_within-program + underbrace∑_gp_g(μ_g-barμ)^2_between-program. The second term is model disagreement and drives information-seeking behavior.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-8f048ab9746b40a7","tex":"\\sum_{g=1}^G","aliases":"∑_g=1^G sum","display":"\\sum_{g=1}^G","kind":"use","excerpt":"For n signal coordinates, A exact unitary atlases, projected state dimension k, G candidate programs, and m observed transitions, the dominant modeled costs are: ll FFT atlas · O(nlog n), Haar atlas · O(n), polyphase atlas · O(n), feature projection · O(dk), one regression fit · O(md_g^2+d_g^3), all program fits · displaystyle∑_g=1^GO(md_g^2+d_g^3)","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-20794c41f8dd0413","tex":"\\sum_h","aliases":"∑_h sum","display":"\\sum_h","kind":"use","excerpt":"The implementation computes held-out Student predictive likelihood when enough rows are available. The program log weight is W_g= ℓ_pred(g) -λ Complexity(g) -η Mechanisms(g) -ρ ResolutionPenalty(g). The posterior is p(gmid D)=frace^W_g∑_he^W_h.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-af320b0c1156522d","tex":"\\sum_{m=0}^{H-1}","aliases":"∑_m=0^H-1 sum","display":"\\sum_{m=0}^{H-1}","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-daa2b26da94bb194","tex":"\\sum_{n=0}^{W-1}","aliases":"∑_n=0^W-1 sum","display":"\\sum_{n=0}^{W-1}","kind":"use","excerpt":"For an H× W array, the normalized two-dimensional transform is (Fx)_k,ℓ =frac1sqrtHW ∑_m=0^H-1∑_n=0^W-1 x_m,ne^-2π i(km/H+ℓ n/W). The inverse is F^*. Disjoint frequency masks E_B produce projectors P_B=F^*E_BF. Fourier space therefore appears inside the same packet language as any other unitary atlas.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-46a1bb46e6353ae5","tex":"\\sum_{u\\in\\operatorname{ch}(v)}","aliases":"∑_u∈ch(v) sum in operatorname","display":"\\sum_{u\\in\\operatorname{ch}(v)}","kind":"use","excerpt":"Let c(v) be any finite additive node cost. Define recursively V(v)= min{c(v),∑_u∈ch(v)V(u)}.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ecf317dfca855f25","tex":"\\sum_{\\alpha}","aliases":"∑_α sum alpha","display":"\\sum_{\\alpha}","kind":"use","excerpt":"[Per-atlas packet reconstruction] For every atlas a, the operators P_a,α are orthogonal projectors and ∑_α∈ I_aP_a,α=I_X. Consequently, x=∑_αP_a,αx, T=∑_α,βP_a,βTP_a,α.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-722d121a5352a8a2","tex":"\\sum_{\\alpha,\\beta}","aliases":"∑_α,β sum alpha beta","display":"\\sum_{\\alpha,\\beta}","kind":"use","excerpt":"[Per-atlas packet reconstruction] For every atlas a, the operators P_a,α are orthogonal projectors and ∑_α∈ I_aP_a,α=I_X. Consequently, x=∑_αP_a,αx, T=∑_α,βP_a,βTP_a,α.","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-7c28cc4f649b8c72","tex":"\\sum_{\\alpha\\in I_a}","aliases":"∑_α∈ I_a sum alpha in","display":"\\sum_{\\alpha\\in I_a}","kind":"use","excerpt":"Let X be a finite-dimensional Hilbert space. For every atlas a∈ A, let U_a:X→ Y_a be unitary, and let {E_a,α}_α∈ I_a be orthogonal coordinate projectors satisfying E_a,αE_a,β=δ_αβE_a,α, ∑_α∈ I_aE_a,α=I_Y_a. Define the object-space packets P_a,α=U_a^*E_a,αU_a.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-71e2b91c820bc153","tex":"\\sum_{\\ell=1}^{H}","aliases":"∑_ℓ=1^H sum ell","display":"\\sum_{\\ell=1}^{H}","kind":"use","excerpt":"The delayed program appends z_t-1. The power-memory packet is m_t^pow = fracdisplaystyle∑_ℓ=1^Hℓ^-αz_t-ℓ displaystyle∑_ℓ=1^Hℓ^-α, while the exponential packet is m_t^exp = fracdisplaystyle∑_ℓ=1^Hρ^ℓ-1z_t-ℓ displaystyle∑_ℓ=1^Hρ^ℓ-1. The supplied defaults are α=0.68 and ρ=0.82.","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"For one output y and design matrix X, take βmidσ^2∼ N(0,σ^2Λ_0^-1), σ^2∼InvGamma(a_0,b_0). Then Λ_n=Λ_0+X^top X, μ_n=Λ_n^-1X^top y, a_n=a_0+frac n2, b_n=b_0+frac12(y^top y-μ_n^topΛ_nμ_n). For a feature row x_*, the predictive mean is x_*^topμ_n and the Student predictive scale contains 1+x_*^topΛ_n^-1x_*. The implementation applies this independent","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-ea7ef77bf7563068","tex":"\\equiv","aliases":"≡ equiv","display":"\\equiv","kind":"use","excerpt":"For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].","count":1},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"For modulus N, define x_r,s[q,t]=x[Nq+r,Nt+s], 0≤ r,s<N. The complete channel family is exactly inverted by interleaving. In object space, the phase projector is (P_N,r,sx)[m,n] = 1_m≡ r (N) 1_n≡ s (N)x[m,n].","count":2},{"work":"large-signal-model","url":"notation/large-signal-model.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"use","excerpt":"The separable Haar transform repeatedly applies (a,b)⟼ (fraca+bsqrt2,fraca-bsqrt2) along columns and rows. The inverse reverses those operations. Scale-location selectors in the packed transform pull back to exact wavelet projectors.","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"ng widehat R has regret at most 2η relative to a minimizer of R when the action set is finite and nonempty. Insert the intermediate expectation ∑widehatμ_ωℓ(a,y_ω). The first difference is at most e because the integrand is in [0,1]. The second is at most Kd_0. Risks themselves lie in [0,1], giving the cap. If widehat a and a^* minimize the approxi","count":5},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"use","excerpt":"A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the vali","count":33},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"use","excerpt":"For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return thr","count":6},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-09b0b2df35128655","tex":"\\mathcal A","aliases":"A mathcal","display":"\\mathcal A","kind":"use","excerpt":"A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the vali","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":14},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-3c08d2cf77d38259","tex":"a^*","aliases":"a^*","display":"a^*","kind":"use","excerpt":"ng widehat R has regret at most 2η relative to a minimizer of R when the action set is finite and nonempty. Insert the intermediate expectation ∑widehatμ_ωℓ(a,y_ω). The first difference is at most e because the integrand is in [0,1]. The second is at most Kd_0. Risks themselves lie in [0,1], giving the cap. If widehat a and a^* minimize the approxi","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"One common quotient across uncertain dynamics. Consider two possible linear realizations on R^3, A_0=1 · 1 · 0 0 · 1 · 0 0 · 0 · 2, A_1=1 · 1 · 0 0 · 1 · 0 1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"One common quotient across uncertain dynamics. Consider two possible linear realizations on R^3, A_0=1 · 1 · 0 0 · 1 · 0 0 · 0 · 2, A_1=1 · 1 · 0 0 · 1 · 0 1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-574cc7908da88025","tex":"A_i","aliases":"A_i","display":"A_i","kind":"use","excerpt":"1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1 0 · 1. Consequently the output after k operations is x_1+kx_2 for either fixed model. In this particular example it also holds for arbitrary switching, since the same local intertwining equation holds for both operations. The uncertain th","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"use","excerpt":"For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return thr","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-b3b3732c9a3a550c","tex":"\\mathcal B","aliases":"B mathcal","display":"\\mathcal B","kind":"use","excerpt":"A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the vali","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i su","count":16},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-4f135c4d5c93e6a2","tex":"\\bar q","aliases":"bar q bar","display":"\\bar q","kind":"use","excerpt":"[Two local bridges compose over finite continuations] Assume the domains used by both equations are preserved under all admitted operations and that a protected output has the form q=bar q Tr. Then, for every admitted finite word w, TrF_w=H_wTr, qF_w=bar q H_wTr. For one operation, TrF_a=TG_ar=H_aTr. The empty word gives the identity. Applying the ","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return thr","count":5},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-ab8bde9a479f56a0","tex":"\\mathfrak C","aliases":"C mathfrak","display":"\\mathfrak C","kind":"use","excerpt":"A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the vali","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":5},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have bee","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the vali","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":7},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-366584274fbf8700","tex":"\\F","aliases":"F mathbb","display":"{\\mathbb F}","kind":"use","excerpt":"Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i su","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-62198a97093af5d0","tex":"\\F_2^3","aliases":"F_2^3 F mathbb","display":"{\\mathbb F}_2^3","kind":"use","excerpt":"Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i su","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-ff53476e6fc124f9","tex":"F_a","aliases":"F_a","display":"F_a","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1 0 · 1. Consequently the output after k operations is x_1+kx_2 for either fixed model. In this particular example it also holds for arbitrary switching, since the same local intertwining equation holds for both operations. The uncertain th","count":5},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-8d428b98976a03b3","tex":"G_a","aliases":"G_a","display":"G_a","kind":"use","excerpt":"For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"use","excerpt":"A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-047a2d7f6191ab3d","tex":"H_a","aliases":"H_a","display":"H_a","kind":"use","excerpt":"For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-480c877d455d2c2b","tex":"h_t","aliases":"h_t","display":"h_t","kind":"use","excerpt":"A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-bc0529a14c9530dd","tex":"H_w","aliases":"H_w","display":"H_w","kind":"use","excerpt":"[Two local bridges compose over finite continuations] Assume the domains used by both equations are preserved under all admitted operations and that a protected output has the form q=bar q Tr. Then, for every admitted finite word w, TrF_w=H_wTr, qF_w=bar q H_wTr. For one operation, TrF_a=TG_ar=H_aTr. The empty word gives the identity. Applying the ","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"[Evidence must initialize the predictive class] Let p:X→ X/∼ be the behavioral quotient above. There exists an initializer i:r(X)→ X/∼ with i(r(x))=p(x) for every x if and only if r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because ","count":6},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"0 · 1. Consequently the output after k operations is x_1+kx_2 for either fixed model. In this particular example it also holds for arbitrary switching, since the same local intertwining equation holds for both operations. The uncertain third-coordinate dynamics remain irrelevant to the protected output.","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-6f4dd664f5dbbd2b","tex":"\\kerop","aliases":"kerop ker operatorname","display":"{\\operatorname{ker}}","kind":"use","excerpt":"For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return thr","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"use","excerpt":"[Evidence must initialize the predictive class] Let p:X→ X/∼ be the behavioral quotient above. There exists an initializer i:r(X)→ X/∼ with i(r(x))=p(x) for every x if and only if r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because ","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-4d95ab92bdb1dbe9","tex":"\\min","aliases":"min","display":"\\min","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-d472066b913c5d84","tex":"\\overline F","aliases":"overline F overline","display":"\\overline F","kind":"use","excerpt":"r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representative x. The quotient update overl","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-cd1227e8872d5282","tex":"\\overline F_a","aliases":"overline F_a overline","display":"\\overline F_a","kind":"use","excerpt":"r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representative x. The quotient update overl","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-6d0e1a1d5f7803a0","tex":"\\overline F_w","aliases":"overline F_w overline","display":"\\overline F_w","kind":"use","excerpt":"induced updates. Necessity follows because a function has one value on an equal input. For sufficiency, assign i(r(x))=p(x); the hypothesis makes the assignment independent of the representative x. The quotient update overline F_a(p(x))=p(F_ax) is well-defined because behavioral equivalence is preserved by prefixing an action. Induction on word len","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"[Evidence must initialize the predictive class] Let p:X→ X/∼ be the behavioral quotient above. There exists an initializer i:r(X)→ X/∼ with i(r(x))=p(x) for every x if and only if r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because ","count":8},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-ffc41bdcb0951586","tex":"\\mathcal Q","aliases":"Q mathcal","display":"\\mathcal Q","kind":"use","excerpt":"A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the vali","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":5},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":9},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-97a4ded34746a4f6","tex":"\\mathbb R","aliases":"R mathbb","display":"\\mathbb R","kind":"use","excerpt":"One common quotient across uncertain dynamics. Consider two possible linear realizations on R^3, A_0=1 · 1 · 0","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"The missing link is initialization. Let r(x) be the observer's actual initial record. A quotient may exist mathematically while a decoder from r(x) does not.","count":12},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-15bd74f3637fba8d","tex":"\\mathbb R^3","aliases":"R^3 mathbb","display":"\\mathbb R^3","kind":"use","excerpt":"One common quotient across uncertain dynamics. Consider two possible linear realizations on R^3, A_0=1 · 1 · 0","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-855b51431c459bf3","tex":"\\mathbb R^d","aliases":"R^d mathbb","display":"\\mathbb R^d","kind":"use","excerpt":"[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-39ee75563d2546f5","tex":"\\mathbb R^d_{\\geq0}","aliases":"R^d_≥0 mathbb geq","display":"\\mathbb R^d_{\\geq0}","kind":"use","excerpt":"[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-236f80d54e55c576","tex":"R_*","aliases":"R_*","display":"R_*","kind":"use","excerpt":"Revision can change the proof obligation. Suppose a newly admitted repair R_* succeeds for either value of b. Then the task no longer requires identifying b before acting. The old statement that both information channels are necessary applied to the previous repair language. It does not apply after R_* is introduced.","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-baf9ce2c27891e4e","tex":"R_b","aliases":"R_b","display":"R_b","kind":"use","excerpt":"Second: the repair question changes the required distinction. When the task becomes selecting R_b, the old display is insufficient. Worlds (0,0,u) and (1,1,u) have the same display and different successful repairs. Repeating the normal reading changes no information. Calibration supplies the missing convention, after which b=s⊕θ. The refinement is ","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"use","excerpt":"the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i succee","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i su","count":6},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.","count":8},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"use","excerpt":"A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer","count":8},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-985b4ee40a8bb177","tex":"\\top","aliases":"top","display":"\\top","kind":"use","excerpt":"[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-b3f5e05f76a79869","tex":"\\TV","aliases":"TV d_TV mathrm","display":"{d_{\\mathrm{TV}}}","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution","count":9},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-a6ab3f6998c3dac0","tex":"u_t","aliases":"u_t","display":"u_t","kind":"use","excerpt":"This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-bb8ce9f4ae8d655a","tex":"u_{t+1}","aliases":"u_t+1","display":"u_{t+1}","kind":"use","excerpt":"This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have bee","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"use","excerpt":"A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the vali","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"For a common invariant domain, let r:X→ Z encode the relevant state, let G_a act on Z, and let T:Z→ W translate that state into another representation with operations H_a. The desired equations are rF_a=G_ar, TG_a=H_aT. The first connects the source process to its representation. The second connects that representation to the recipient interface.","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":6},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-25a84a8fd44f7f54","tex":"w^\\top","aliases":"w^top top","display":"w^\\top","kind":"use","excerpt":"[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-1b7b2107b59db7f3","tex":"\\widehat a","aliases":"widehat a widehat","display":"\\widehat a","kind":"use","excerpt":"n action minimizing widehat R has regret at most 2η relative to a minimizer of R when the action set is finite and nonempty. Insert the intermediate expectation ∑widehatμ_ωℓ(a,y_ω). The first difference is at most e because the integrand is in [0,1]. The second is at most Kd_0. Risks themselves lie in [0,1], giving the cap. If widehat a and a^* min","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-967c63ebf4cd78be","tex":"\\widehat R","aliases":"widehat R widehat","display":"\\widehat R","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-51fdd71f6572f7c6","tex":"\\widehat y","aliases":"widehat y widehat","display":"\\widehat y","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-e1c6d4c93a294c7e","tex":"\\widehat y_\\omega","aliases":"widehat y_ω widehat omega","display":"\\widehat y_\\omega","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-b186d96cd6172b72","tex":"\\widehat\\mu","aliases":"widehatμ widehat mu","display":"\\widehat\\mu","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-dac00a99d5864526","tex":"\\widehat\\mu_\\omega","aliases":"widehatμ_ω widehat mu omega","display":"\\widehat\\mu_\\omega","kind":"use","excerpt":"res on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(a,y_ω) and widehat R(a)=∑_ωwidehatμ_ωℓ(","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-4b68ab3847feda7d","tex":"X","aliases":"X","display":"X","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":6},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer","count":13},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"1 · 0 · 3. Let p(x_1,x_2,x_3)=(x_1,x_2) and let the output be q(x)=x_1. Both operators satisfy pA_i=Gp, G=1 · 1","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-1029cb8ad51ed272","tex":"x_t","aliases":"x_t","display":"x_t","kind":"use","excerpt":"A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":6},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-38f55764f5bb6924","tex":"y_\\omega","aliases":"y_ω omega","display":"y_\\omega","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"For a linear map C:X→ Z and a known operation A:X→ X, an induced operation B satisfying CA=BC exists precisely when ker C is invariant under A, with the codomain restricted to the attained image if necessary. The condition says that two inputs collapsed by C remain collapsed after the operation. If it fails, the discarded distinction can return thr","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-8b24cd3e78c875c1","tex":"z_t","aliases":"z_t","display":"z_t","kind":"use","excerpt":"A realization of the contract may carry physical or computational state x_t, an observation history h_t, a retained predictive state z_t, a candidate set or probability record, an operator library, and applicable certificates. An institution adds offices, permissions, claims, resource rights, and revision procedures. The representation of one layer","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-8e5d0000abb9cf07","tex":"z_{t+1}","aliases":"z_t+1","display":"z_{t+1}","kind":"use","excerpt":"This failure has a concrete block form. In coordinates separating retained and hidden state, z_t+1 u_t+1 = A · B C · D z_t u_t. Even if u_t has small norm, the effect Bu_t may be large. Retaining less state is therefore not justified by omitted magnitude alone. A predictive error bound needs a sensitivity or propagation bound. Sufficient Resolution","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i su","count":13},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-3e08c12818fdd261","tex":"\\mu_\\omega","aliases":"μ_ω mu omega","display":"\\mu_\\omega","kind":"use","excerpt":"Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-09bfc832719222a5","tex":"\\Pi","aliases":"Π Pi","display":"\\Pi","kind":"use","excerpt":"A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the vali","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-d65d75b1e6562713","tex":"\\Sigma","aliases":"Σ Sigma","display":"\\Sigma","kind":"use","excerpt":"For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have bee","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"A common contract, not a universal architecture. For editorial purposes, write a task contract as C=(Ω,Π, Q, A, E, B, V). Here Ω is the declared family of possible worlds or models, Π the permitted observation protocols, Q the protected questions, A the admitted actions or operations, E the evidence language, B the resource envelope, and V the vali","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":5},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-1e6eb42c12591791","tex":"\\to","aliases":"→ to","display":"\\to","kind":"use","excerpt":"[Evidence must initialize the predictive class] Let p:X→ X/∼ be the behavioral quotient above. There exists an initializer i:r(X)→ X/∼ with i(r(x))=p(x) for every x if and only if r(x)=r(y) Longrightarrow x∼ y. When this holds, the initialized quotient predicts every declared finite continuation using its induced updates. Necessity follows because ","count":6},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"use","excerpt":"For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have bee","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-729f2cd8398e9960","tex":"\\in","aliases":"∈ in","display":"\\in","kind":"use","excerpt":"For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have bee","count":5},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-ea8a48c6147135b3","tex":"\\sum_\\omega","aliases":"∑_ω sum omega","display":"\\sum_\\omega","kind":"use","excerpt":"Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+Kd_0}, where R(a)=∑_ωμ_ωℓ(","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"use","excerpt":"For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have bee","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"Suppose a deterministic state space X has admitted transformations F_a and protected output q. Define x∼ y Longleftrightarrow q(F_wx)=q(F_wy) for every admitted finite word w. The empty word is included. When the operation domain is invariant, this equivalence supports well-defined quotient updates. It identifies which distinctions are needed for a","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-b92a69b030a52379","tex":"\\leq","aliases":"≤ leq","display":"\\leq","kind":"use","excerpt":"[A common decision bound] Let μ and widehatμ be probability measures on the same finite candidate set with d_TV(μ,widehatμ)≤ e. Suppose true and approximate responses satisfy d(y_ω,widehat y_ω)≤ d_0 for all retained candidates. If ℓ(a,y)∈[0,1] is K-Lipschitz in its second argument, then for every admissible action a, |R(a)-widehat R(a)|≤η:=min{1,e+","count":4},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-2dd02f064b00c6d5","tex":"\\geq","aliases":"≥ geq","display":"\\geq","kind":"use","excerpt":"[Resource gains belong to a declared ledger] For one fixed workflow and workload, let Δ∈ R^d be old-minus-new event counts after all charged setup and maintenance. Then w^topΔ≥0 for every w∈ R^d_≥0 if and only if every coordinate of Δ is nonnegative. For prices restricted to the convex hull of a finite set, the smallest priced saving is attained at","count":2},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-39c3d5a5ec4b06af","tex":"\\subseteq","aliases":"⊆ subseteq","display":"\\subseteq","kind":"use","excerpt":"For exact candidate elimination, a current set V⊆Ω is corrected by V↦ V∩ E. If the target observation is t(V), a future-correction signature records Σ(V)=(t(V∩ E))_E∈ E. Under the source's closure hypotheses, this captures the distinctions needed to answer after every permitted filter. The empty set matters: recognizing that all candidates have bee","count":1},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"Let the possible world be ω=(b,θ,u)∈ F_2^3. The bit b determines which of two repairs is appropriate. The bit θ is a sensor convention. The nuisance bit u affects none of the admitted observations, transitions, or repairs. A normal reading returns s=b⊕θ. A calibration returns θ. A public operation flips b, leaving θ and u unchanged. A repair R_i su","count":3},{"work":"machine-theory-compendium","url":"notation/machine-theory-compendium.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"use","excerpt":"The combined lesson is neither that every representation change speeds computation nor that compression is futile. It is that the complete path must be accounted for: acquire ⟶ represent ⟶ select ⟶ verify ⟶ execute ⟶ maintain. The arrows can have different units. Model observations, interpreter visits, physical time, peak memory, and stored records","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":16},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":51},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":18},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-8838c457f7756577","tex":"A^2","aliases":"A^2","display":"A^2","kind":"introduction-passage","excerpt":"ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this sectio","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"introduction-passage","excerpt":"ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this sectio","count":7},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd d mathrm","display":"{\\mathrm d}","kind":"introduction-passage","excerpt":"d τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this sectio","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-8b6f5031e4dc5e6e","tex":"\\eps","aliases":"eps ε varepsilon","display":"{\\varepsilon}","kind":"introduction-passage","excerpt":"ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this sectio","count":20},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":12},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":25},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":15},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":14},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-4c9797d8504bc641","tex":"n_j'","aliases":"n_j'","display":"n_j'","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":21},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":9},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":4},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"introduction-passage","excerpt":"ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this sectio","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-aee6c380cd411ada","tex":"S_E","aliases":"S_E","display":"S_E","kind":"introduction-passage","excerpt":"ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this sectio","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":7},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":6},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-7cd99be1a637db1a","tex":"u_j'","aliases":"u_j'","display":"u_j'","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":10},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"introduction-passage","excerpt":"ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this sectio","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":4},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-2a461142fc2f8e7a","tex":"v_j'","aliases":"v_j'","display":"v_j'","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":7},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-ffbde798b9a7c58f","tex":"W_j","aliases":"W_j","display":"W_j","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":7},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":10},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-7ca0813f9e0997a7","tex":"y_h^2","aliases":"y_h^2","display":"y_h^2","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":10},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-0fc1259c08b990e9","tex":"\\kappa^2","aliases":"κ^2 kappa","display":"\\kappa^2","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"introduction-passage","excerpt":"ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this sectio","count":16},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this sectio","count":14},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":4},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-11baa595827a4e0f","tex":"\\omega","aliases":"ω omega","display":"\\omega","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":12},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-5b95cb77ad7d5e57","tex":"\\omega_{h\\kappa}","aliases":"ω_hκ omega kappa","display":"\\omega_{h\\kappa}","kind":"introduction-passage","excerpt":"A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section den","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-152412184ef94979","tex":"\\omega_j","aliases":"ω_j omega","display":"\\omega_j","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":8},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-d66efea961193266","tex":"\\omega_{j,q}","aliases":"ω_j,q omega","display":"\\omega_{j,q}","kind":"introduction-passage","excerpt":"For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are c","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"use","excerpt":"ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The orig","count":13},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-6a4d1b76a4ccae86","tex":"A^4","aliases":"A^4","display":"A^4","kind":"use","excerpt":"Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-993220b6cfa76019","tex":"A_0","aliases":"A_0","display":"A_0","kind":"use","excerpt":"ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The orig","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The orig","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-92d9f5662cb19c20","tex":"A_0^4","aliases":"A_0^4","display":"A_0^4","kind":"use","excerpt":"[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupatio","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-44038b7af84230e7","tex":"a_0^4","aliases":"a_0^4","display":"a_0^4","kind":"use","excerpt":"[Invariant curvature power counting] At fixed ρ_ V and γ, the family [source reference] leaves N_H unchanged for each fixed depth: N_H =fracγη_0 a_0^416π^2∑_h=1^H d_hy_h^4 =fracγ64π^2ρ_ V∑_h=1^H d_hm_h^4. The factor λ(A/μ)^4 equals η_0a_0^4 by [source reference]. Equivalently, substitute μ^4=4λρ_ V into [source reference]. The physical masses are f","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-7435dd184fb65352","tex":"a_j","aliases":"a_j","display":"a_j","kind":"use","excerpt":"Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-a03621dbecc26bb7","tex":"a_j'","aliases":"a_j'","display":"a_j'","kind":"use","excerpt":"Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-584bf2656ae84301","tex":"a_j''","aliases":"a_j''","display":"a_j''","kind":"use","excerpt":"Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-7468d04a44be005b","tex":"a_\\star","aliases":"a_⋆ star","display":"a_\\star","kind":"use","excerpt":"There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-20c762a5ffdac39b","tex":"C_H","aliases":"C_H","display":"C_H","kind":"use","excerpt":"n Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At large |q|, δ C_H/C for ε=0.01 is approximately","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-bf1b37e9c699a1d1","tex":"d_h","aliases":"d_h","display":"d_h","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-5c8b9d31659a8bf5","tex":"\\dot\\phi","aliases":"dotϕ dot phi","display":"\\dot\\phi","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-206bb34091ccad2d","tex":"\\dot\\phi_\\star","aliases":"dotϕ_⋆ dot phi star","display":"\\dot\\phi_\\star","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-060ae96635fef1f6","tex":"\\mathcal E","aliases":"E mathcal","display":"\\mathcal E","kind":"use","excerpt":"onvention For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtrac","count":6},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-bd7df92c242c57f1","tex":"\\mathcal E^{(2)}_H","aliases":"E^(2)_H mathcal","display":"\\mathcal E^{(2)}_H","kind":"use","excerpt":"Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-6fa839e093c96e89","tex":"\\eps^{-2}","aliases":"eps^-2 ε^-2 eps varepsilon","display":"{\\varepsilon}^{-2}","kind":"use","excerpt":"mily satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupation multiplier is used in this construction. At fixed ρ_ V, μ^4=4λρ_ V, so widetildeρ=4ρ_ Vλ(A/μ)^4S_E. Substitute [source reference]. The factors ε^2 and ε^-2 cancel, leaving η_0 A_0^4S_E. Equation [source reference] gives K=K_0/ε. Division proves the final identity.","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-2cb803c0b9113454","tex":"\\eps^2","aliases":"eps^2 ε^2 eps varepsilon","display":"{\\varepsilon}^2","kind":"use","excerpt":"[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupatio","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-a8adfd562576492b","tex":"\\exp","aliases":"exp","display":"\\exp","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-1551bc759aa860fc","tex":"f_h","aliases":"f_h","display":"f_h","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-cd0aa9856147b6c5","tex":"g","aliases":"g","display":"g","kind":"use","excerpt":"The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-6ba4cd8d81889283","tex":"g_h","aliases":"g_h","display":"g_h","kind":"use","excerpt":"The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-1893fbda262d81ba","tex":"g_{\\rm peak}","aliases":"g_rm peak rm","display":"g_{\\rm peak}","kind":"use","excerpt":"The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"use","excerpt":"Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms an","count":6},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"The reservoir cannot be ignored. [Residual-energy obligation] Suppose an event begins and ends at the same scalar potential value, expansion and external losses are negligible during that event, and the receiving sector gains ε K_i. Energy conservation requires K_f=(1-ε)K_i. In particular, weak transfer in that event cannot also bring the backgroun","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,","count":10},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-f3c143179694d5bc","tex":"K_0","aliases":"K_0","display":"K_0","kind":"use","excerpt":"ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The orig","count":6},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-07baee216908c2f6","tex":"K_f","aliases":"K_f","display":"K_f","kind":"use","excerpt":"The reservoir cannot be ignored. [Residual-energy obligation] Suppose an event begins and ends at the same scalar potential value, expansion and external losses are negligible during that event, and the receiving sector gains ε K_i. Energy conservation requires K_f=(1-ε)K_i. In particular, weak transfer in that event cannot also bring the backgroun","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-0c7bd880455e7fb0","tex":"K_i","aliases":"K_i","display":"K_i","kind":"use","excerpt":"The reservoir cannot be ignored. [Residual-energy obligation] Suppose an event begins and ends at the same scalar potential value, expansion and external losses are negligible during that event, and the receiving sector gains ε K_i. Energy conservation requires K_f=(1-ε)K_i. In particular, weak transfer in that event cannot also bring the backgroun","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-62c66a7a5dd70c31","tex":"m","aliases":"m","display":"m","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":7},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","excerpt":"iently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-fc2f399c21f77ade","tex":"m_h","aliases":"m_h","display":"m_h","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-69377fbc0b4b52f2","tex":"M_h^2","aliases":"M_h^2","display":"M_h^2","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-8e2a80b8199fdbc9","tex":"m_h^2","aliases":"m_h^2","display":"m_h^2","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-0b749acf259c8843","tex":"m_h^4","aliases":"m_h^4","display":"m_h^4","kind":"use","excerpt":"[Invariant curvature power counting] At fixed ρ_ V and γ, the family [source reference] leaves N_H unchanged for each fixed depth: N_H =fracγη_0 a_0^416π^2∑_h=1^H d_hy_h^4 =fracγ64π^2ρ_ V∑_h=1^H d_hm_h^4. The factor λ(A/μ)^4 equals η_0a_0^4 by [source reference]. Equivalently, substitute μ^4=4λρ_ V into [source reference]. The physical masses are f","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"The DOP853 integration uses relative tolerance 2×10^-10 and absolute tolerance 2×10^-13. The largest total-energy residual in the principal run is max_τ|frac E(τ) E(0)-1| =7.34×10^-11. The maximum hyperboloid residual is 3.73×10^-13. The largest instantaneous mode-energy fraction is 0.648634, and the endpoint fraction is 0.126747. The final positio","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-cf1aa9503aec04fb","tex":"\\max_\\tau","aliases":"max_τ max tau","display":"\\max_\\tau","kind":"use","excerpt":"The DOP853 integration uses relative tolerance 2×10^-10 and absolute tolerance 2×10^-13. The largest total-energy residual in the principal run is max_τ|frac E(τ) E(0)-1| =7.34×10^-11. The maximum hyperboloid residual is 3.73×10^-13. The largest instantaneous mode-energy fraction is 0.648634, and the endpoint fraction is 0.126747. The final positio","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-6e9e0a3f9b6130ca","tex":"\\mathcal N","aliases":"N mathcal","display":"\\mathcal N","kind":"use","excerpt":"Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms an","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-a0305144eb5a1f22","tex":"n^{(2)}","aliases":"n^(2)","display":"n^{(2)}","kind":"use","excerpt":"Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-b3ba9738598b9443","tex":"\\mathcal N_H","aliases":"N_H mathcal","display":"\\mathcal N_H","kind":"use","excerpt":"Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms an","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-1dbe74185e6b4e59","tex":"\\overline{a_j}","aliases":"overlinea_j overline","display":"\\overline{a_j}","kind":"use","excerpt":"Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-3f5f7b6e4a773c81","tex":"\\overline{a_j'}","aliases":"overlinea_j' overline","display":"\\overline{a_j'}","kind":"use","excerpt":"Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-8cde4d454bfecb76","tex":"q''","aliases":"q''","display":"q''","kind":"use","excerpt":"The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The orig","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The orig","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-20859e8ffc4824b7","tex":"R_0^{-1}","aliases":"R_0^-1","display":"R_0^{-1}","kind":"use","excerpt":"R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove this energy from the evolving scalar.","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"use","excerpt":"Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-113da7f67a5c92e1","tex":"\\rhoV","aliases":"rhoV ρ_ V rho mathrm","display":"{\\rho_{\\mathrm V}}","kind":"use","excerpt":"Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,","count":9},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupatio","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-c00e0696f87fecf4","tex":"u_j^2","aliases":"u_j^2","display":"u_j^2","kind":"use","excerpt":"[Finite energy conservation and positivity] Assume C>0, w_j>0, y_j>0, and V≥0 is smooth. Start the modes in the instantaneous vacuum n_j=u_j=v_j=0. Equations [source reference]--[source reference] preserve (1+2n_j)^2-u_j^2-v_j^2=1, and conserve E=frac C2(q')^2+ V(q) +∑_jw_jω_j n_j. Every occupation remains nonnegative. Consequently 0≤∑_jw_jω_j n_j≤","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-6c8c1adb5fb270ca","tex":"\\mathcal V","aliases":"V mathcal","display":"\\mathcal V","kind":"use","excerpt":"The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-9bbcb8e590dbd641","tex":"\\mathrm V","aliases":"V mathrm","display":"\\mathrm V","kind":"use","excerpt":"There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-d2449baf83cf5e7e","tex":"\\mathcal V'","aliases":"V' mathcal","display":"\\mathcal V'","kind":"use","excerpt":"The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-e3f0c47930b412df","tex":"v_j^2","aliases":"v_j^2","display":"v_j^2","kind":"use","excerpt":"[Finite energy conservation and positivity] Assume C>0, w_j>0, y_j>0, and V≥0 is smooth. Start the modes in the instantaneous vacuum n_j=u_j=v_j=0. Equations [source reference]--[source reference] preserve (1+2n_j)^2-u_j^2-v_j^2=1, and conserve E=frac C2(q')^2+ V(q) +∑_jw_jω_j n_j. Every occupation remains nonnegative. Consequently 0≤∑_jw_jω_j n_j≤","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-16cd7a5ab2eeeb5c","tex":"\\widetilde\\rho","aliases":"widetildeρ widetilde rho","display":"\\widetilde\\rho","kind":"use","excerpt":"Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,","count":4},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-f39fc6268336557a","tex":"\\widetilde\\rho_0","aliases":"widetildeρ_0 widetilde rho","display":"\\widetilde\\rho_0","kind":"use","excerpt":"ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The orig","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":4},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-ad46065620a305e3","tex":"y^4","aliases":"y^4","display":"y^4","kind":"use","excerpt":"Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-3b627a1e8225f041","tex":"y_h","aliases":"y_h","display":"y_h","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-72f0f3cca9812ced","tex":"y_h^4","aliases":"y_h^4","display":"y_h^4","kind":"use","excerpt":"[Invariant curvature power counting] At fixed ρ_ V and γ, the family [source reference] leaves N_H unchanged for each fixed depth: N_H =fracγη_0 a_0^416π^2∑_h=1^H d_hy_h^4 =fracγ64π^2ρ_ V∑_h=1^H d_hm_h^4. The factor λ(A/μ)^4 equals η_0a_0^4 by [source reference]. Equivalently, substitute μ^4=4λρ_ V into [source reference]. The physical masses are f","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-e3d4192e022af00a","tex":"y_j","aliases":"y_j","display":"y_j","kind":"use","excerpt":"[Finite energy conservation and positivity] Assume C>0, w_j>0, y_j>0, and V≥0 is smooth. Start the modes in the instantaneous vacuum n_j=u_j=v_j=0. Equations [source reference]--[source reference] preserve (1+2n_j)^2-u_j^2-v_j^2=1, and conserve E=frac C2(q')^2+ V(q) +∑_jw_jω_j n_j. Every occupation remains nonnegative. Consequently 0≤∑_jw_jω_j n_j≤","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"Numerical results and tests that failed. The principal run takes ε=0.01, depth 16, 48 Gauss--Legendre momentum nodes per level, and κ=sqrty_h z, 0≤ z≤3.5, q(0)=-10, q'(0)=1, 0≤τ≤20. Here A is the nominal reference scale used to set the incoming speed. The actual speed at q=0 is an output of the coupled solution, not an imposed crossing condition. T","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-2657f47b6d71f248","tex":"z_{\\max}","aliases":"z_max max","display":"z_{\\max}","kind":"use","excerpt":"At depth 16, changing z_max from 2.5 to 3.5 to 4.5 to 6 changes the maximum mode-energy fraction from approximately 0.476 to 0.649 to 0.762 to 0.871. Moving the initial state from q=-10 to -15 or -20 also changes the endpoint result. Thus the failed scientific gate is genuine. Conservation has passed; regulator and state robustness have not.","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":6},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"sCopperrule1.25in1pt The crossing-energy obstruction, an abundance-preserving parameter family, and a quantum-mode calculation with explicit energy exchange. Komorebi P.hfill October 2026 Result and boundary. The original prescribed-background benchmark exceeds its crossing reservoir by 2.6272875374×10^20. This study constructs a finite dynamical s","count":20},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-d203ba01eef4198c","tex":"\\delta","aliases":"δ delta","display":"\\delta","kind":"use","excerpt":"n Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At large |q|, δ C_H/C for ε=0.01 is approximately","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove this energy from the evolving scalar.","count":8},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-62ecbfb276c1f894","tex":"\\eta_0","aliases":"η_0 eta","display":"\\eta_0","kind":"use","excerpt":"R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove this energy from the evolving scalar.","count":7},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-11a4b66cda93897f","tex":"\\eta_0^{1/4}","aliases":"η_0^1/4 eta","display":"\\eta_0^{1/4}","kind":"use","excerpt":"What is and is not preserved. The density and the normalized distributions in m/A and k/A are preserved. The physical mass scale is not the old one: fracμμ_0=η_0^1/4sqrtε, frac AA_0=fracΛΛ_0=η_0^1/4 =7.85458237208×10^-6. Thus the physical tower masses are independent of ε along this family, but differ from the original projected benchmark. Number d","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-4e73a24d6dbb26ff","tex":"\\Lambda","aliases":"Λ Lambda","display":"\\Lambda","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-6ede142bbc0a6d7e","tex":"\\Lambda_0","aliases":"Λ_0 Lambda","display":"\\Lambda_0","kind":"use","excerpt":"What is and is not preserved. The density and the normalized distributions in m/A and k/A are preserved. The physical mass scale is not the old one: fracμμ_0=η_0^1/4sqrtε, frac AA_0=fracΛΛ_0=η_0^1/4 =7.85458237208×10^-6. Thus the physical tower masses are independent of ε along this family, but differ from the original projected benchmark. Number d","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-7238fc50817cce55","tex":"\\lambda_0","aliases":"λ_0 lambda","display":"\\lambda_0","kind":"use","excerpt":"es as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-595cb14f66316f0e","tex":"\\mu^2","aliases":"μ^2 mu","display":"\\mu^2","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-01d3c6534e890ca5","tex":"\\mu^4","aliases":"μ^4 mu","display":"\\mu^4","kind":"use","excerpt":"Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-3b453de64aef4dc5","tex":"\\mu_0","aliases":"μ_0 mu","display":"\\mu_0","kind":"use","excerpt":"ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The orig","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"use","excerpt":"Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms an","count":4},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupatio","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-7e425697f7c81595","tex":"\\rho_r","aliases":"ρ_r rho","display":"\\rho_r","kind":"use","excerpt":"There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-2bdd558d975d62ca","tex":"\\rho_T","aliases":"ρ_T rho","display":"\\rho_T","kind":"use","excerpt":"[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupatio","count":2},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-d73ae6dc3936033e","tex":"\\phi^2","aliases":"ϕ^2 phi","display":"\\phi^2","kind":"use","excerpt":"The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the s","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-ab8c353106812d45","tex":"\\phi^4","aliases":"ϕ^4 phi","display":"\\phi^4","kind":"use","excerpt":"instant G. Felder, L. Kofman, and A. Linde. Instant Preheating. Physical Review D 59, 123523 (1999). arXiv:hep-ph/9812289. bibitemqke A. M. Fedotov, E. G. Gelfer, K. Yu. Korolev, and S. A. Smolyansky. On the kinetic equation approach to pair production by time-dependent electric field. arXiv:1008.2098 (2010). bibitemmpr C. Molina-Par is, P. R. Ande","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-efdf8990a98fa39d","tex":"\\omega'","aliases":"ω' omega","display":"\\omega'","kind":"use","excerpt":"Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-70900ee358b87ffe","tex":"\\omega^4","aliases":"ω^4 omega","display":"\\omega^4","kind":"use","excerpt":"Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-97940e0d1742e0e6","tex":"\\omega^5","aliases":"ω^5 omega","display":"\\omega^5","kind":"use","excerpt":"Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-ff1db34a200d7b94","tex":"\\Omega_{\\mathrm V}","aliases":"Ω_ V Omega mathrm","display":"\\Omega_{\\mathrm V}","kind":"use","excerpt":"There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-b942a5f2f22c3ea7","tex":"\\omega_j^2","aliases":"ω_j^2 omega","display":"\\omega_j^2","kind":"use","excerpt":"Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-bd7a39da506badf5","tex":"\\Omega_{r0}","aliases":"Ω_r0 Omega","display":"\\Omega_{r0}","kind":"use","excerpt":"There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"use","excerpt":"The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].","count":8},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-781753df207abfe0","tex":"\\sum_{h=1}^H","aliases":"∑_h=1^H sum","display":"\\sum_{h=1}^H","kind":"use","excerpt":"Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms an","count":4},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].","count":4},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-db8e84427c2e1e02","tex":"\\infty","aliases":"∞ infty","display":"\\infty","kind":"use","excerpt":"iently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"iently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"iently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^","count":1},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupatio","count":5},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"[Finite energy conservation and positivity] Assume C>0, w_j>0, y_j>0, and V≥0 is smooth. Start the modes in the instantaneous vacuum n_j=u_j=v_j=0. Equations [source reference]--[source reference] preserve (1+2n_j)^2-u_j^2-v_j^2=1, and conserve E=frac C2(q')^2+ V(q) +∑_jw_jω_j n_j. Every occupation remains nonnegative. Consequently 0≤∑_jw_jω_j n_j≤","count":3},{"work":"physics-energy-backreaction","url":"notation/physics-energy-backreaction.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"use","excerpt":"For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent re","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":98},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"introduction-passage","excerpt":"ber of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with","count":51},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-6a4d1b76a4ccae86","tex":"A^4","aliases":"A^4","display":"A^4","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7468d04a44be005b","tex":"a_\\star","aliases":"a_⋆ star","display":"a_\\star","kind":"introduction-passage","excerpt":"tion y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the co","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":21},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-762426111124f19d","tex":"B_L","aliases":"B_L","display":"B_L","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":19},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-18ac3e7343f01689","tex":"d","aliases":"d","display":"d","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-bf1b37e9c699a1d1","tex":"d_h","aliases":"d_h","display":"d_h","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-45e8f07731ac2d46","tex":"\\dd","aliases":"dd d mathrm","display":"{\\mathrm d}","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":6},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-44bd7ae60f478fae","tex":"H","aliases":"H","display":"H","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":28},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":8},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-72dfcfb0c470ac25","tex":"L","aliases":"L","display":"L","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":28},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-e3b98a4da31a127d","tex":"t","aliases":"t","display":"t","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-2d711642b726b044","tex":"x","aliases":"x","display":"x","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-a1fce4363854ff88","tex":"y","aliases":"y","display":"y","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":18},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-3b627a1e8225f041","tex":"y_h","aliases":"y_h","display":"y_h","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-72f0f3cca9812ced","tex":"y_h^4","aliases":"y_h^4","display":"y_h^4","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f1cf04093d08b6dc","tex":"\\pi","aliases":"π pi","display":"\\pi","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-62c285b6b8f8d397","tex":"\\pi^2","aliases":"π^2 pi","display":"\\pi^2","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":9},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-9fc65441b304fa9d","tex":"\\sum_{h=1}^{L}","aliases":"∑_h=1^L sum","display":"\\sum_{h=1}^{L}","kind":"introduction-passage","excerpt":"Fixed model and units. Let L denote the number of physical tower levels. This avoids confusing tower depth with the Hubble parameter H. Define y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale fac","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-362138992adfe804","tex":"\\star","aliases":"⋆ star","display":"\\star","kind":"introduction-passage","excerpt":"tion y_h=x2^h, d_h=2^h-1, x=10^-4, B_L=frac164π^2∑_h=1^Ld_hy_h^4. Time is τ=At, energies are densities divided by A^4, and a dot below denotes d/ dτ after setting the lapse to one. The initial numerical scale factor is a=1. In conditional comparisons with the stored cosmology, this corresponds to the historical production scale a_⋆=10^-6, so the co","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"use","excerpt":"[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s],","count":6},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":44},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-280e15d5f806ea95","tex":"a^{-2}","aliases":"a^-2","display":"a^{-2}","kind":"use","excerpt":"Write a nonzero trace summand as T_j(a)=w_jy_j^2f_j a^-2(k_j^2+a^2y_j^2)^-1/2. Then frac dlog T_j dlog a=-2-fraca^2y_j^2k_j^2+a^2y_j^2. This is strictly negative and lies in [-3,-2]. The finite sum is continuous, strictly decreases, and tends to zero. The intermediate value theorem gives existence and strict monotonicity gives uniqueness of the thr","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-4f409aebfdb56b59","tex":"a^{-3}","aliases":"a^-3","display":"a^{-3}","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-6b955e93845e89ca","tex":"a^{-4}","aliases":"a^-4","display":"a^{-4}","kind":"use","excerpt":"· +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for the physical conversion. The radiation term has density ρ_r=ρ_r0a^-4. The last mode term specifies an instantaneous finite subtraction. The static matched potential is already included in U.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-48179e33d77ba589","tex":"a^{-5}","aliases":"a^-5","display":"a^{-5}","kind":"use","excerpt":"For a≥ a_i the slope bounds also give T(a_i)(a_i/a)^3≤ T(a)≤ T(a_i)(a_i/a)^2. Thus the threshold lies between the cube-root and square-root dilution estimates based on the initial trace. For a finite set of massive modes the cold limit is T(a)=a^-3∑_jw_jy_jf_j+O(a^-5). The pressure correction therefore matters before the distribution becomes cold.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ","count":11},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7b48479868582d73","tex":"a^3","aliases":"a^3","display":"a^3","kind":"use","excerpt":"Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full ","count":11},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8da2f1b2bafe729c","tex":"A_h","aliases":"A_h","display":"A_h","kind":"use","excerpt":"The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient ","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-6d7097da4387177a","tex":"a_i","aliases":"a_i","display":"a_i","kind":"use","excerpt":"_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local cur","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-a28a764d1ce4e13a","tex":"a_{\\rm rel}","aliases":"a_rm rel rm","display":"a_{\\rm rel}","kind":"use","excerpt":"[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"use","excerpt":"Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-b5a44ae5838838af","tex":"B_{12}","aliases":"B_12","display":"B_{12}","kind":"use","excerpt":"For an explicit comparison, truncating to L=12 retains about 20.0724% of the old Gaussian production energy. Recalibrating that stipulated family gives E_0=43.8579 and B_12K(S)=9.02826. Choosing Δ=0.3E_0=13.1574 satisfies the window. This changes the species content, physical mass scale, and background matching. It is a counterfactual model, not a ","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-bfb601bd20166e6a","tex":"B_{15}","aliases":"B_15","display":"B_{15}","kind":"use","excerpt":"D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fi","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-6b23c0d5f35d1b11","tex":"C","aliases":"C","display":"C","kind":"use","excerpt":"D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fi","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-2e7d2c03a9507ae2","tex":"c","aliases":"c","display":"c","kind":"use","excerpt":"Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full ","count":8},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^","count":6},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-3f39d5c348e5b79d","tex":"D","aliases":"D","display":"D","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":12},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-bc156ea315fde667","tex":"\\ddot f","aliases":"ddot f ddot","display":"\\ddot f","kind":"use","excerpt":"After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12,","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-71bb8ebf5c549359","tex":"\\ddot f_j","aliases":"ddot f_j ddot","display":"\\ddot f_j","kind":"use","excerpt":"After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12,","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-842d40d677280379","tex":"\\ddot q","aliases":"ddot q ddot","display":"\\ddot q","kind":"use","excerpt":"F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 wit","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-6f9f83efac22f7f6","tex":"\\dot a","aliases":"dot a dot","display":"\\dot a","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c6493f97bdd006de","tex":"\\dot a^2","aliases":"dot a^2 dot","display":"\\dot a^2","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c81c88dde70f3786","tex":"\\dot f","aliases":"dot f dot","display":"\\dot f","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7a16efda7a5b8ccd","tex":"\\dot f_j","aliases":"dot f_j dot","display":"\\dot f_j","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-5726de1775de768d","tex":"\\dot f_j^*","aliases":"dot f_j^* dot","display":"\\dot f_j^*","kind":"use","excerpt":"After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12,","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d576368deebdc90f","tex":"\\dot H","aliases":"dot H dot","display":"\\dot H","kind":"use","excerpt":"dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the Friedmann constraint remains satisfied. In addition, a^3(τ)ρ(τ)+∫_0^τ3Ha^3p d t=ρ(0) when a(0)=1. Differentiate [source reference]. Insert dot n_j=W_ju_j/2 and dotΩ_j=(y","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-1701dbe4017fbbe7","tex":"\\dot n","aliases":"dot n dot","display":"\\dot n","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-89f4aee6e62b2ed8","tex":"\\dot n_j","aliases":"dot n_j dot","display":"\\dot n_j","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-32d3a4a4c8991fbd","tex":"\\dot q","aliases":"dot q dot","display":"\\dot q","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":16},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-4c606e269e33ef93","tex":"\\dot q^2","aliases":"dot q^2 dot","display":"\\dot q^2","kind":"use","excerpt":"F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 wit","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-51f3bbd1503295c5","tex":"\\dot u","aliases":"dot u dot","display":"\\dot u","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f6a673c40e5d91a7","tex":"\\dot u_j","aliases":"dot u_j dot","display":"\\dot u_j","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-3ec2563613dd0ed8","tex":"\\dot v","aliases":"dot v dot","display":"\\dot v","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-458452ed36c03a28","tex":"\\dot v_j","aliases":"dot v_j dot","display":"\\dot v_j","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-826d719bf280601a","tex":"\\dot\\rho","aliases":"dotρ dot rho","display":"\\dot\\rho","kind":"use","excerpt":"[Finite exchange and the gravitational constraint] On any regular solution of [source reference] and [source reference] with a>0 and Z>0, dotρ_χ+3H(ρ_χ+p_χ) · =dot q F, dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the ","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-325ccc655cc23049","tex":"\\dot\\rho_q","aliases":"dotρ_q dot rho","display":"\\dot\\rho_q","kind":"use","excerpt":"[Finite exchange and the gravitational constraint] On any regular solution of [source reference] and [source reference] with a>0 and Z>0, dotρ_χ+3H(ρ_χ+p_χ) · =dot q F, dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the ","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f3a906e5b764ea61","tex":"\\dot\\rho_\\chi","aliases":"dotρ_χ dot rho chi","display":"\\dot\\rho_\\chi","kind":"use","excerpt":"[Finite exchange and the gravitational constraint] On any regular solution of [source reference] and [source reference] with a>0 and Z>0, dotρ_χ+3H(ρ_χ+p_χ) · =dot q F, dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the ","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8052e1ac0acaea2d","tex":"\\dot\\Omega","aliases":"dotΩ dot Omega","display":"\\dot\\Omega","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-e60dda21ac11f4f7","tex":"\\dot\\Omega_j","aliases":"dotΩ_j dot Omega","display":"\\dot\\Omega_j","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"use","excerpt":"D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fi","count":10},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-3f79bb7b435b0532","tex":"e","aliases":"e","display":"e","kind":"use","excerpt":"[Local locking criterion and dilution] For the above gas, T=ρ_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-fd8ebda1c7c09855","tex":"e^{2iE_jt}","aliases":"e^2iE_jt","display":"e^{2iE_jt}","kind":"use","excerpt":"For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-ba526470e4ff9e1e","tex":"E_j","aliases":"E_j","display":"E_j","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-204f9ab8a5b56738","tex":"E_j^2","aliases":"E_j^2","display":"E_j^2","kind":"use","excerpt":"oxedκ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from t","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-4fc6710adf730065","tex":"\\Ezero","aliases":"Ezero E_0","display":"{E_0}","kind":"use","excerpt":"D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fi","count":12},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f67ab10ad4e4c531","tex":"F","aliases":"F","display":"F","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"Force and pressure. The same action gives ρ_χ · =a^-3∑_jw_jΩ_jn_j, p_χ · =a^-3∑_jw_j[frack_j^2n_j3a^2Ω_j -(fracΩ_j2-frack_j^26a^2Ω_j)u_j], F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its densit","count":6},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-252f10c83610ebca","tex":"f","aliases":"f","display":"f","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,","count":11},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-079c3e1846c8a9e9","tex":"F'","aliases":"F'","display":"F'","kind":"use","excerpt":"For the current hierarchy D/B_L is extremely small. Expanding the stationary equation gives s_*∼sqrtfrac2DB_LS, 1-fracV_R(sqrts_*)D∼frac4s_*3S. These estimates are supported by simple bounds. With α=2D/(B_LS), the inequality s^2/(1+s)≤ F'(s)≤ s^2 implies that s_* lies between the positive roots of s^2+(α/S)s-α=0 and s^2-α s-α=0.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,","count":9},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-62474a607550ca4c","tex":"f_j^*","aliases":"f_j^*","display":"f_j^*","kind":"use","excerpt":"After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12,","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-333e0a1e27815d0c","tex":"G","aliases":"G","display":"G","kind":"use","excerpt":"Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"er, curvature, and origin restriction] Every such quartic-matched potential satisfies Δ=fracSκ8-B_LJ(S), U_qq(0)=frac4S[B_LK(S)-Δ]. Consequently, retaining an origin maximum and a stationary vacuum at sqrt S requires Δ>B_LK(S). Stationarity gives b_0=-2c_0S-B_LG'(S). Substitute this into Δ=-b_0S-c_0S^2+B_L[G(0)-G(S)] and κ=4S[2c_0+B_LG''(S)]. Elimi","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-2b23da150b06fbef","tex":"G''","aliases":"G''","display":"G''","kind":"use","excerpt":"potential satisfies Δ=fracSκ8-B_LJ(S), U_qq(0)=frac4S[B_LK(S)-Δ]. Consequently, retaining an origin maximum and a stationary vacuum at sqrt S requires Δ>B_LK(S). Stationarity gives b_0=-2c_0S-B_LG'(S). Substitute this into Δ=-b_0S-c_0S^2+B_L[G(0)-G(S)] and κ=4S[2c_0+B_LG''(S)]. Eliminating c_0 gives the first identity. Substitution into U_qq(0)=2[b","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the Friedmann constraint remains satisfied. In addition, a^3(τ)ρ(τ)+∫_0^τ3Ha^3p d t=ρ(0) when a(0)=1. Differentiate [source reference]. Insert dot n_j=W_ju_j/2 and dotΩ_j=(y","count":6},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d3cba85783189f4d","tex":"H_i","aliases":"H_i","display":"H_i","kind":"use","excerpt":"The actual microscopic expansion scale. With the stored cosmological conversion inputs, the origin branch has A≃49.2467 eV and fracH_iA=3.35456×10^-25. Here H_i in the numerator is the dimensionful physical rate. The dimensionless solver uses that ratio. Over 0≤τ≤12, the accumulated log a is about 4.03×10^-24. The largest background-position differ","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7c78a383bd29d847","tex":"H_i^2","aliases":"H_i^2","display":"H_i^2","kind":"use","excerpt":"The stress test sets the initial dimensionless Hubble rate to 0.02 by changing the gravitational coupling at fixed initial densities. It is deliberately not the inherited physical cosmology. A representative L=15, z_max=16, 128-node run gives max_τ|fraca^3ρ+W_pρ(0)-1| · =4.75×10^-14, max_τ|fracH^2H_i^2-fracρρ(0)| · =3.77×10^-14, where W_p=∫3Ha^3p d","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-de7d1b721a1e0632","tex":"i","aliases":"i","display":"i","kind":"use","excerpt":"_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local cur","count":8},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-6da43b944e494e88","tex":"J","aliases":"J","display":"J","kind":"use","excerpt":"and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^2-S+log(1+","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-189f40034be7a199","tex":"j","aliases":"j","display":"j","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,","count":40},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-86be9a55762d316a","tex":"K","aliases":"K","display":"K","kind":"use","excerpt":"[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel","count":9},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,","count":12},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-72cd054376fe5951","tex":"K_h^2","aliases":"K_h^2","display":"K_h^2","kind":"use","excerpt":"The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient ","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-bcf3911a4b4214e2","tex":"k_j","aliases":"k_j","display":"k_j","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f9bf2ea0ac3be3ff","tex":"k_j^2","aliases":"k_j^2","display":"k_j^2","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ","count":10},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-5ce00ec03d085e0b","tex":"K_q","aliases":"K_q","display":"K_q","kind":"use","excerpt":"[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-38b80d70217119ea","tex":"k_{\\rm phys}","aliases":"k_rm phys rm","display":"k_{\\rm phys}","kind":"use","excerpt":"At large physical momentum and dot q=0, W≃2H. With the continuum measure k_rm phys^2 d k_rm phys/(2π^2), the leading contribution per species behaves as ρ^(2)∼ fracH^2K_rm phys^216π^2. This asymptotic diagnostic is not inserted as an isolated numerical subtraction. A consistent subtraction must also determine pressure, force, and gravitational term","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-52949bb89df6f388","tex":"K_{\\rm phys}^2","aliases":"K_rm phys^2 rm","display":"K_{\\rm phys}^2","kind":"use","excerpt":"At large physical momentum and dot q=0, W≃2H. With the continuum measure k_rm phys^2 d k_rm phys/(2π^2), the leading contribution per species behaves as ρ^(2)∼ fracH^2K_rm phys^216π^2. This asymptotic diagnostic is not inserted as an isolated numerical subtraction. A consistent subtraction must also determine pressure, force, and gravitational term","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-9521e6f332eae000","tex":"k_{\\rm phys}^2","aliases":"k_rm phys^2 rm","display":"k_{\\rm phys}^2","kind":"use","excerpt":"At large physical momentum and dot q=0, W≃2H. With the continuum measure k_rm phys^2 d k_rm phys/(2π^2), the leading contribution per species behaves as ρ^(2)∼ fracH^2K_rm phys^216π^2. This asymptotic diagnostic is not inserted as an isolated numerical subtraction. A consistent subtraction must also determine pressure, force, and gravitational term","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":6},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-08f271887ce94707","tex":"M","aliases":"M","display":"M","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":6},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":6},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-73172b3e9a37f4ac","tex":"M_{\\rm Pl}","aliases":"M_rm Pl rm","display":"M_{\\rm Pl}","kind":"use","excerpt":"· +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for the physical conversion. The radiation term has density ρ_r=ρ_r0a^-4. The last mode term specifies an instantaneous finite subtraction. The static matched potential is already included in U.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient ","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-cf1aa9503aec04fb","tex":"\\max_\\tau","aliases":"max_τ max tau","display":"\\max_\\tau","kind":"use","excerpt":"The stress test sets the initial dimensionless Hubble rate to 0.02 by changing the gravitational coupling at fixed initial densities. It is deliberately not the inherited physical cosmology. A representative L=15, z_max=16, 128-node run gives max_τ|fraca^3ρ+W_pρ(0)-1| · =4.75×10^-14, max_τ|fracH^2H_i^2-fracρρ(0)| · =3.77×10^-14, where W_p=∫3Ha^3p d","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8ce86a6ae65d3692","tex":"N","aliases":"N","display":"N","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-1b16b1df538ba12d","tex":"n","aliases":"n","display":"n","kind":"use","excerpt":"Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full ","count":12},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-3680c2fa0f185a98","tex":"n_j","aliases":"n_j","display":"n_j","kind":"use","excerpt":"Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full ","count":11},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"een -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from the origin is obtained by replacing E_j with [k_j^2/a^2+y_j^2(1+q^2)]^1/2. Its first q derivative vanishes at zero and its second derivative is y_j^2/E_j. Also F","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-65c74c15a686187b","tex":"o","aliases":"o","display":"o","kind":"use","excerpt":"Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full ","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-5c62e091b8c0565f","tex":"P","aliases":"P","display":"P","kind":"use","excerpt":"· +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for the physical conversion. The radiation term has density ρ_r=ρ_r0a^-4. The last mode term specifies an instantaneous finite subtraction. The static matched potential is already included in U.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-148de9c5a7a44d19","tex":"p","aliases":"p","display":"p","kind":"use","excerpt":"ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussia","count":18},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7c62188858e5960d","tex":"p_q","aliases":"p_q","display":"p_q","kind":"use","excerpt":"F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 wit","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-528374d34056b371","tex":"p_\\chi","aliases":"p_χ chi","display":"p_\\chi","kind":"use","excerpt":"ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussia","count":8},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8e35c2cd3bf6641b","tex":"q","aliases":"q","display":"q","kind":"use","excerpt":"The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are","count":41},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":9},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-6e535c3600229c66","tex":"q^4","aliases":"q^4","display":"q^4","kind":"use","excerpt":"Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-ddca2fd3ea7fd7ac","tex":"q^6","aliases":"q^6","display":"q^6","kind":"use","excerpt":"een -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from the origin is obtained by replacing E_j with [k_j^2/a^2+y_j^2(1+q^2)]^1/2. Its first q derivative vanishes at zero and its second derivative is y_j^2/E_j. Also F","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7ea2aec5575a954e","tex":"q^8","aliases":"q^8","display":"q^8","kind":"use","excerpt":"een -3 and -2. If T(a_i)>4D/S, there is exactly one later a at which the local curvature vanishes. The identity follows term by term from E_j^2-k_j^2/a^2=y_j^2. The matter energy away from the origin is obtained by replacing E_j with [k_j^2/a^2+y_j^2(1+q^2)]^1/2. Its first q derivative vanishes at zero and its second derivative is y_j^2/E_j. Also F","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-4e4bfa4de54508e8","tex":"q_*","aliases":"q_*","display":"q_*","kind":"use","excerpt":"A 70-digit stationary-point calculation for L=15 gives q_* · =3.70470008878×10^-6, fracV_R(q_*)D · =0.9999999999988562664, fracD-V_R(q_*)D=1.14373356231×10^-12. Extra arithmetic digits do not express physical parameter accuracy. They prevent cancellation from obscuring the very small difference of two potential energies.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8c2574892063f995","tex":"R","aliases":"R","display":"R","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-454349e422f05297","tex":"r","aliases":"r","display":"r","kind":"use","excerpt":"[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c1e0d6344f9474d8","tex":"R_S","aliases":"R_S","display":"R_S","kind":"use","excerpt":"A constructive but altered branch. The restriction also gives a constructive family. Let R_S(s) · =G(s)-G(S)-G'(S)(s-S)-tfrac12G''(S)(s-S)^2, U_Δ(s) · =fracΔ+B_LJ(S)S^2(s-S)^2+B_LR_S(s). Then U_Δ(S)=U_Δ,s(S)=0 and U_Δ(0)=Δ.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-1002ed943b3a0c5e","tex":"\\operatorname{Re}","aliases":"Re operatorname","display":"\\operatorname{Re}","kind":"use","excerpt":"After variation and N=1, a mode obeys ddot f_j+3Hdot f_j+Ω_j^2f_j=0, H=dot a/a, with preserved normalization a^3(f_jdot f_j^*-f_j^*dot f_j)=i. Use real variables n_j · =fraca^3(|dot f_j|^2+Ω_j^2|f_j|^2)2Ω_j-frac12, u_j · =a^3(Ω_j|f_j|^2-frac|dot f_j|^2Ω_j), v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot ","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-e402978cefc6e89c","tex":"\\rm","aliases":"rm","display":null,"kind":"use","excerpt":"[Local locking criterion and dilution] For the above gas, T=ρ_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a","count":8},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8de0b3c47f112c59","tex":"S","aliases":"S","display":"S","kind":"use","excerpt":"The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are","count":32},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-043a718774c572bd","tex":"s","aliases":"s","display":"s","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":29},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c8d0b37b4af3bfcc","tex":"S^2","aliases":"S^2","display":"S^2","kind":"use","excerpt":"[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s], V_ss · =frac2DS^2+2B_Llog(1+s)>0. Hence V_s is strictly increasing, starts negative, and","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7a7f0526c9c4d2c8","tex":"s_*","aliases":"s_*","display":"s_*","kind":"use","excerpt":"[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s],","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-aee6c380cd411ada","tex":"S_E","aliases":"S_E","display":"S_E","kind":"use","excerpt":"D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fi","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-e632b7095b0bf32c","tex":"T","aliases":"T","display":"T","kind":"use","excerpt":"Write a nonzero trace summand as T_j(a)=w_jy_j^2f_j a^-2(k_j^2+a^2y_j^2)^-1/2. Then frac dlog T_j dlog a=-2-fraca^2y_j^2k_j^2+a^2y_j^2. This is strictly negative and lies in [-3,-2]. The finite sum is continuous, strictly decreases, and tends to zero. The intermediate value theorem gives existence and strict monotonicity gives uniqueness of the thr","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-b48a351441abfbe5","tex":"T_j","aliases":"T_j","display":"T_j","kind":"use","excerpt":"Write a nonzero trace summand as T_j(a)=w_jy_j^2f_j a^-2(k_j^2+a^2y_j^2)^-1/2. Then frac dlog T_j dlog a=-2-fraca^2y_j^2k_j^2+a^2y_j^2. This is strictly negative and lies in [-3,-2]. The finite sum is continuous, strictly decreases, and tends to zero. The intermediate value theorem gives existence and strict monotonicity gives uniqueness of the thr","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f5e43e7ddc7d42d4","tex":"\\Tr","aliases":"Tr T mathcal","display":"{\\mathcal T}","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-a25513c7e0f6eaa8","tex":"U","aliases":"U","display":"U","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":24},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-0bfe935e70c321c7","tex":"u","aliases":"u","display":"u","kind":"use","excerpt":"Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full ","count":12},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"The first failed gate is momentum-cutoff acceptance in the expansion stress test. The second is absolute preservation of the mode hyperboloid in highly occupied altered-model runs. The tight altered-model run reaches occupation of order 10^5. Evaluating (1+2n)^2-u^2-v^2-1 then subtracts large terms to resolve a value near zero. Its maximum absolute","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full ","count":10},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c00e0696f87fecf4","tex":"u_j^2","aliases":"u_j^2","display":"u_j^2","kind":"use","excerpt":"The mode equations also preserve (1+2n_j)^2-u_j^2-v_j^2=1. For a normalized pure initial state on the positive sheet, continuity keeps 1+2n_j≥1, so n_j≥0. This proves positivity in the exact finite equations. Floating-point preservation is a separate numerical check.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d42ca9e4eff519f2","tex":"U_q","aliases":"U_q","display":"U_q","kind":"use","excerpt":"F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 wit","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-1940d9080cc2890f","tex":"U_{qq}","aliases":"U_qq","display":"U_{qq}","kind":"use","excerpt":"Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-44229cca67dc816c","tex":"U_s","aliases":"U_s","display":"U_s","kind":"use","excerpt":"Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-95b45958028681d3","tex":"U_{ss}","aliases":"U_ss","display":"U_{ss}","kind":"use","excerpt":"[An accessible double well under changed matching] If B_LK(S)<Δ<E_0, U_Δ(q^2) is nonnegative, has its only global minima at q=±sqrt S, and has an origin maximum. The origin barrier lies within the declared budget. Theorem~[source reference] gives U_s(0)<0. Also U_s(S)=0 and U_ss(S)=2[Δ+B_LJ(S)]/S^2>0. The function U_s is strictly convex because U_s","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-db2c54170241a256","tex":"U_{sss}","aliases":"U_sss","display":"U_{sss}","kind":"use","excerpt":"uble well under changed matching] If B_LK(S)<Δ<E_0, U_Δ(q^2) is nonnegative, has its only global minima at q=±sqrt S, and has an origin maximum. The origin barrier lies within the declared budget. Theorem~[source reference] gives U_s(0)<0. Also U_s(S)=0 and U_ss(S)=2[Δ+B_LJ(S)]/S^2>0. The function U_s is strictly convex because U_sss=2B_L/(1+s)>0. ","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7a18b66dd81c7e4a","tex":"U_\\Delta","aliases":"U_Δ Delta","display":"U_\\Delta","kind":"use","excerpt":"A constructive but altered branch. The restriction also gives a constructive family. Let R_S(s) · =G(s)-G(S)-G'(S)(s-S)-tfrac12G''(S)(s-S)^2, U_Δ(s) · =fracΔ+B_LJ(S)S^2(s-S)^2+B_LR_S(s). Then U_Δ(S)=U_Δ,s(S)=0 and U_Δ(0)=Δ.","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-a80e9efa4a67e5f4","tex":"U_{\\Delta,s}","aliases":"U_Δ,s Delta","display":"U_{\\Delta,s}","kind":"use","excerpt":"U_Δ(s) · =fracΔ+B_LJ(S)S^2(s-S)^2+B_LR_S(s). Then U_Δ(S)=U_Δ,s(S)=0 and U_Δ(0)=Δ.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-de5a6f78116eca62","tex":"V","aliases":"V","display":"V","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-4c94485e0c21ae6c","tex":"v","aliases":"v","display":"v","kind":"use","excerpt":"The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are","count":9},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"The first failed gate is momentum-cutoff acceptance in the expansion stress test. The second is absolute preservation of the mode hyperboloid in highly occupied altered-model runs. The tight altered-model run reaches occupation of order 10^5. Evaluating (1+2n)^2-u^2-v^2-1 then subtracts large terms to resolve a value near zero. Its maximum absolute","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-b700e39236b5b68f","tex":"v_j","aliases":"v_j","display":"v_j","kind":"use","excerpt":"For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-e3f0c47930b412df","tex":"v_j^2","aliases":"v_j^2","display":"v_j^2","kind":"use","excerpt":"The mode equations also preserve (1+2n_j)^2-u_j^2-v_j^2=1. For a normalized pure initial state on the positive sheet, continuity keeps 1+2n_j≥1, so n_j≥0. This proves positivity in the exact finite equations. Floating-point preservation is a separate numerical check.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7d6de4bafa8b9ab9","tex":"V_R","aliases":"V_R","display":"V_R","kind":"use","excerpt":"The inherited origin-matched potential is V_R(q) · =D(1-q^2/S)^2+B_LF(q^2), F(s) · =(1+s)^2log(1+s)-s-tfrac32s^2, D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-5df6cd07cd936c95","tex":"V_s","aliases":"V_s","display":"V_s","kind":"use","excerpt":"[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s], V_ss · =frac2DS^2+2B_Llog(1+s)>0. Hence V_s is strictly increasing, starts negative, and","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-0ab69d28e8e685ec","tex":"V_{ss}","aliases":"V_ss","display":"V_{ss}","kind":"use","excerpt":"[Unique positive-radius vacuum] For D,B_L,S>0, the potential [source reference] has exactly one minimum in s=q^2>0. It is the global minimum on s≥0. The two background minima are q=±sqrts_*. As a function of s, V_s · =-frac2DS+frac2DsS^2 +2B_L[(1+s)log(1+s)-s], V_ss · =frac2DS^2+2B_Llog(1+s)>0. Hence V_s is strictly increasing, starts negative, and","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-fcb5f40df9be6bae","tex":"W","aliases":"W","display":"W","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":8},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,","count":11},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-ffbde798b9a7c58f","tex":"W_j","aliases":"W_j","display":"W_j","kind":"use","excerpt":"v_j · =-2a^3Re(f_j^*dot f_j). Direct differentiation yields dot n_j · =tfrac12W_ju_j, · dot u_j · =W_j(1+2n_j)-2Ω_jv_j, · dot v_j · =2Ω_ju_j, W_j · =3H+fracdotΩ_jΩ_j =3H+fracy_j^2qdot q-Hk_j^2/a^2Ω_j^2. The 3H term follows from the expanding mode normalization. Dropping it while leaving the rest of the system unchanged is an incorrect finite model.","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-af7bc77c629d2fcb","tex":"w_j","aliases":"w_j","display":"w_j","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,","count":11},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-0be23d285935109c","tex":"W_j^2","aliases":"W_j^2","display":"W_j^2","kind":"use","excerpt":"A diagnostic for missing terms. The leading adiabatic response of [source reference] has v_j≃ W_j/(2Ω_j) and n_j≃ W_j^2/(16Ω_j^2), with u_j at higher derivative order. Thus the corresponding second-order mode density is ρ_χ^(2)≃frac116a^3∑_jfracw_jΩ_j [fracΩ_j,qΩ_jdot q+ (3-frack_j^2a^2Ω_j^2)H]^2. This contains terms proportional to dot q^2, Hdot q","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8d7f1cc8c150d4e5","tex":"W_p","aliases":"W_p","display":"W_p","kind":"use","excerpt":"The stress test sets the initial dimensionless Hubble rate to 0.02 by changing the gravitational coupling at fixed initial densities. It is deliberately not the inherited physical cosmology. A representative L=15, z_max=16, 128-node run gives max_τ|fraca^3ρ+W_pρ(0)-1| · =4.75×10^-14, max_τ|fracH^2H_i^2-fracρρ(0)| · =3.77×10^-14, where W_p=∫3Ha^3p d","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7ca0813f9e0997a7","tex":"y_h^2","aliases":"y_h^2","display":"y_h^2","kind":"use","excerpt":"The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient ","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-e3d4192e022af00a","tex":"y_j","aliases":"y_j","display":"y_j","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2,","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-05636e347f1fcac3","tex":"y_j^2","aliases":"y_j^2","display":"y_j^2","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ","count":11},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":11},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"use","excerpt":"The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient ","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-1c90720ad505f551","tex":"Z_a","aliases":"Z_a","display":"Z_a","kind":"use","excerpt":"F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 wit","count":5},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-2657f47b6d71f248","tex":"z_{\\max}","aliases":"z_max max","display":"z_{\\max}","kind":"use","excerpt":"The stress test sets the initial dimensionless Hubble rate to 0.02 by changing the gravitational coupling at fixed initial densities. It is deliberately not the inherited physical cosmology. A representative L=15, z_max=16, 128-node run gives max_τ|fraca^3ρ+W_pρ(0)-1| · =4.75×10^-14,","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d19728dce419085d","tex":"z_{\\max}^2","aliases":"z_max^2 max","display":"z_{\\max}^2","kind":"use","excerpt":"The kinetic tail and what expansion adds. The retained term. The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient ","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f0fa5f2a39dbffe4","tex":"Z_q","aliases":"Z_q","display":"Z_q","kind":"use","excerpt":"F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 wit","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-cbe5cfdf7c2118a9","tex":"|","aliases":"|","display":"|","kind":"use","excerpt":"For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussian potential retains that condition, but its empty-sector minima release only about 1.14×10^-12 of the refe","count":18},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-4893e9df8b5496eb","tex":"\\alpha","aliases":"α alpha","display":"\\alpha","kind":"use","excerpt":"For the current hierarchy D/B_L is extremely small. Expanding the stationary equation gives s_*∼sqrtfrac2DB_LS, 1-fracV_R(sqrts_*)D∼frac4s_*3S. These estimates are supported by simple bounds. With α=2D/(B_LS), the inequality s^2/(1+s)≤ F'(s)≤ s^2 implies that s_* lies between the positive roots of s^2+(α/S)s-α=0 and s^2-α s-α=0.","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"use","excerpt":"The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"Set s=q^2 and G(s)=(1+s)^2log(1+s). The most general static potential with this Gaussian determinant and polynomial counterterms through q^4 is U(s)=a_0+b_0s+c_0s^2+B_LG(s). Here a_0,b_0,c_0 are polynomial coefficients, not the scale factor or the stored crossing parameter. Suppose U_s(S)=0. Let Δ=U(0)-U(S) and κ=U_qq(sqrt S). Define J(S)=tfrac12S^","count":12},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"D · =4.65190089919765×10^-18. The nominal Gaussian family sets C=2S_E/ϵ, E_0=C/2, S_E=2.18498795564. At ϵ=0.01, C=436.997591128 and E_0=218.498795564. The parameter ϵ does not assert an actual conversion efficiency. The main branch uses L=15, for which B_15=3086.94039766. The input parameters are inherited reference data, not a new observational fi","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-055de4ec3d8e4886","tex":"\\kappa","aliases":"κ kappa","display":"\\kappa","kind":"use","excerpt":"[Local locking criterion and dilution] For the above gas, T=ρ_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d0054acf186dab30","tex":"\\kappa_{\\rm coh}","aliases":"κ_rm coh kappa rm","display":"\\kappa_{\\rm coh}","kind":"use","excerpt":"Why coherence cannot be omitted. In the quantum kinetic variables used below, the exact force contains n_j+u_j/2, not just n_j. Linearizing its explicit q dependence at the origin adds κ_rm coh=frac12a^3∑_jw_jfracy_j^2E_ju_j to the occupation contribution. The trace criterion in Theorem~[source reference] therefore does not substitute for the full ","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-bf31761201c1d62b","tex":"\\kappa_{\\rm eff}","aliases":"κ_rm eff kappa rm","display":"\\kappa_{\\rm eff}","kind":"use","excerpt":"[Local locking criterion and dilution] For the above gas, T=ρ_χ-3p_χ. At fixed occupations and fixed a, the curvature of the effective background potential at the origin is κ_rm eff(a)=ρ_χ(a)-3p_χ(a)-frac4DS. If at least one w_jf_j is positive, T(a) is strictly decreasing to zero. Each nonzero summand has logarithmic slope between -3 and -2. If T(a","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"use","excerpt":"55--56 for conditional locking and release. bibitemenergy Komorebi P. Energy-Conserving Production: A finite-depth backreaction study. Companion research manuscript. bibitemmatched Komorebi P. Vacuum Matching and Dynamical Energy Transfer. Companion research manuscript and numerical source package. bibitemmolina C. Molina-Par is, P. R. Anderson, an","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"use","excerpt":"ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussia","count":24},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-675246b7c7527b5d","tex":"\\rho^{(2)}","aliases":"ρ^(2) rho","display":"\\rho^{(2)}","kind":"use","excerpt":"At large physical momentum and dot q=0, W≃2H. With the continuum measure k_rm phys^2 d k_rm phys/(2π^2), the leading contribution per species behaves as ρ^(2)∼ fracH^2K_rm phys^216π^2. This asymptotic diagnostic is not inserted as an isolated numerical subtraction. A consistent subtraction must also determine pressure, force, and gravitational term","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-037b39a7fd3c33a4","tex":"\\rho_q","aliases":"ρ_q rho","display":"\\rho_q","kind":"use","excerpt":"F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its density and pressure are ρ_q=frac12Zdot q^2+U, p_q=frac12Zdot q^2-U+frac16aZ_adot q^2. The last term follows by differentiating a^3Zdot q^2/2 wit","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7e425697f7c81595","tex":"\\rho_r","aliases":"ρ_r rho","display":"\\rho_r","kind":"use","excerpt":"· +∑_jw_j[fraca^32N|dot f_j|^2 -fracNa^3Ω_j^22|f_j|^2+fracNΩ_j2]. Here N is the lapse and M^2=(M_rm Pl/A)^2 for the physical conversion. The radiation term has density ρ_r=ρ_r0a^-4. The last mode term specifies an instantaneous finite subtraction. The static matched potential is already included in U.","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7a103aae75d10702","tex":"\\rho_{r0}","aliases":"ρ_r0 rho","display":"\\rho_{r0}","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-51aed110a5c3a870","tex":"\\rho_\\chi","aliases":"ρ_χ rho chi","display":"\\rho_\\chi","kind":"use","excerpt":"ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussia","count":10},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d0f008a8b4d0e2b5","tex":"\\rho_{\\chi,0}","aliases":"ρ_χ,0 rho chi","display":"\\rho_{\\chi,0}","kind":"use","excerpt":"[htbp]centeringsmall @rrrrr@ Crossing · ρ_χ/E_0 · K_q/E_0 · conditional ρ_χ,0/D · conditional a_rm rel","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-4d7c645cc511721d","tex":"\\rho_\\chi^{(2)}","aliases":"ρ_χ^(2) rho chi","display":"\\rho_\\chi^{(2)}","kind":"use","excerpt":"A diagnostic for missing terms. The leading adiabatic response of [source reference] has v_j≃ W_j/(2Ω_j) and n_j≃ W_j^2/(16Ω_j^2), with u_j at higher derivative order. Thus the corresponding second-order mode density is ρ_χ^(2)≃frac116a^3∑_jfracw_jΩ_j [fracΩ_j,qΩ_jdot q+ (3-frack_j^2a^2Ω_j^2)H]^2. This contains terms proportional to dot q^2, Hdot q","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-10ce0200b450ea95","tex":"\\phi","aliases":"ϕ phi","display":"\\phi","kind":"use","excerpt":"The physical question and its correct location. The source cosmological model proposes density locking of the average scalar at ϕ=0 [source reference]. The minima ϕ=± v belong instead to its empty-sector tree potential. With the dimensionless coordinate q=sqrtγ ϕ/v, γ=S=16, the locked point is q=0 and the empty-sector tree vacua are q=±4. These are","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-ab8c353106812d45","tex":"\\phi^4","aliases":"ϕ^4 phi","display":"\\phi^4","kind":"use","excerpt":"55--56 for conditional locking and release. bibitemenergy Komorebi P. Energy-Conserving Production: A finite-depth backreaction study. Companion research manuscript. bibitemmatched Komorebi P. Vacuum Matching and Dynamical Energy Transfer. Companion research manuscript and numerical source package. bibitemmolina C. Molina-Par is, P. R. Anderson, an","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8d50b5beb7fe1e79","tex":"\\chi","aliases":"χ chi","display":"\\chi","kind":"use","excerpt":"ar A study of the distinction between density locking, empty-sector vacuum matching, and conserved energy transfer in a regulated quantum model. Komorebi P. vfill Results and scope. The density-locked branch of the source theory lies at the origin. Its local incoherent-matter stability condition is controlled by ρ_χ-3p_χ. The origin-matched Gaussia","count":12},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":10},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7ed617ddc8339555","tex":"\\Omega_j","aliases":"Ω_j Omega","display":"\\Omega_j","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-716b7b36e2fb8fd7","tex":"\\Omega_{j,q}","aliases":"Ω_j,q Omega","display":"\\Omega_{j,q}","kind":"use","excerpt":"Force and pressure. The same action gives ρ_χ · =a^-3∑_jw_jΩ_jn_j, p_χ · =a^-3∑_jw_j[frack_j^2n_j3a^2Ω_j -(fracΩ_j2-frack_j^26a^2Ω_j)u_j], F · =a^-3∑_jw_jΩ_j,q(n_j+u_j/2). The coherent term in pressure is as necessary as the coherent term in the reaction force. The background equation is Zddot q+H(3Z+aZ_a)dot q+frac12Z_qdot q^2+U_q+ F=0. Its densit","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-8aac5bb1267ad783","tex":"\\Omega_j^2","aliases":"Ω_j^2 Omega","display":"\\Omega_j^2","kind":"use","excerpt":"An expanding finite system from one action. Regulator, lapse, and subtraction. Choose a finite comoving momentum quadrature with positive weights w_j. For each mode, Ω_j^2(a,q)=k_j^2/a^2+y_j^2(1+q^2). Let Z(a,q)>0 be a specified kinetic coefficient. The reduced homogeneous action per unit comoving volume is L= · -frac3M^2adot a^2N +Na^3[fracZ2(frac","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-1ae839f007058732","tex":"\\sum_{h=1}^L","aliases":"∑_h=1^L sum","display":"\\sum_{h=1}^L","kind":"use","excerpt":"tained term The flat-space calculation retained the leading kinetic response of momentum modes above the explicit cutoff. The present finite action carries that same term at the evolving physical cutoff. For level h, let K_h^2(a)=y_hz_max^2/a^2, A_h=[fracK_h^2K_h^2+y_h^2(1+q^2)]^3/2. The coefficient used in the solver is Z(a,q)=C+∑_h=1^Lfracd_hy_h^","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-f9e2dbe88f36e12f","tex":"\\sum_j","aliases":"∑_j sum","display":"\\sum_j","kind":"use","excerpt":"Origin locking and the stress trace. A conditional incoherent distribution. Fix finitely many positive weights w_j, masses y_j>0, comoving momenta k_j, and nonnegative conserved occupations f_j. The weights include degeneracies and momentum quadrature. Assume an incoherent or phase-averaged gas at the origin. Define E_j(a) · =sqrtk_j^2/a^2+y_j^2, ρ","count":7},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet","count":3},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-a8bae13bbcececb2","tex":"\\int_0^T","aliases":"∫_0^T int","display":"\\int_0^T","kind":"use","excerpt":"For a held origin and a held scale factor, u_j(t)=u_j(0)cos(2E_jt)-v_j(0)sin(2E_jt). Direct integration gives the useful bound |frac1T∫_0^Tκ_rm coh(t) d t| ≤frac12a^3T∑_jw_jfracy_j^2E_j^2sqrtu_j(0)^2+v_j(0)^2. This follows because the modulus of the integral of e^2iE_jt is at most 1/E_j. It is a fixed-background averaging bound. It is not a Floquet","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-c07a291623b52b17","tex":"\\int_0^\\tau","aliases":"∫_0^τ int tau","display":"\\int_0^\\tau","kind":"use","excerpt":"dotρ_q+3H(ρ_q+p_q) · =-dot q F. After adding separately conserved radiation, the total density and pressure obey dotρ+3H(ρ+p)=0. If dot H=-(ρ+p)/(2M^2) and 3M^2H^2=ρ initially, the Friedmann constraint remains satisfied. In addition, a^3(τ)ρ(τ)+∫_0^τ3Ha^3p d t=ρ(0) when a(0)=1. Differentiate [source reference]. Insert dot n_j=W_ju_j/2 and dotΩ_j=(y","count":1},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"use","excerpt":"For the current hierarchy D/B_L is extremely small. Expanding the stationary equation gives s_*∼sqrtfrac2DB_LS, 1-fracV_R(sqrts_*)D∼frac4s_*3S. These estimates are supported by simple bounds. With α=2D/(B_LS), the inequality s^2/(1+s)≤ F'(s)≤ s^2 implies that s_* lies between the positive roots of s^2+(α/S)s-α=0 and s^2-α s-α=0.","count":2},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-dfb4f746dd2f3b1b","tex":"\\simeq","aliases":"≃ simeq","display":"\\simeq","kind":"use","excerpt":"A diagnostic for missing terms. The leading adiabatic response of [source reference] has v_j≃ W_j/(2Ω_j) and n_j≃ W_j^2/(16Ω_j^2), with u_j at higher derivative order. Thus the corresponding second-order mode density is ρ_χ^(2)≃frac116a^3∑_jfracw_jΩ_j [fracΩ_j,qΩ_jdot q+ (3-frack_j^2a^2Ω_j^2)H]^2. This contains terms proportional to dot q^2, Hdot q","count":6},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-7a03384e6e8b8519","tex":"\\le","aliases":"≤ le","display":"\\le","kind":"use","excerpt":"For a≥ a_i the slope bounds also give T(a_i)(a_i/a)^3≤ T(a)≤ T(a_i)(a_i/a)^2. Thus the threshold lies between the cube-root and square-root dilution estimates based on the initial trace. For a finite set of massive modes the cold limit is T(a)=a^-3∑_jw_jy_jf_j+O(a^-5). The pressure correction therefore matters before the distribution becomes cold.","count":4},{"work":"physics-origin-locking","url":"notation/physics-origin-locking.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"use","excerpt":"For a≥ a_i the slope bounds also give T(a_i)(a_i/a)^3≤ T(a)≤ T(a_i)(a_i/a)^2. Thus the threshold lies between the cube-root and square-root dilution estimates based on the initial trace. For a finite set of massive modes the cold limit is T(a)=a^-3∑_jw_jy_jf_j+O(a^-5). The pressure correction therefore matters before the distribution becomes cold.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-684888c0ebb17f37","tex":"*","aliases":"*","display":"*","kind":"notation-table","excerpt":"midrule V,F_v,f^* · A declared view category, its fibers, and contravariant restrictions.","count":131},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a318c24216defe20","tex":"+","aliases":"+","display":"+","kind":"notation-table","excerpt":"S_A · The prime set {p_2n+ 1_A(n):n∈ N}.","count":436},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-380918b946a52664","tex":"=","aliases":"=","display":"=","kind":"notation-table","excerpt":"χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.","count":1425},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-559aead08264d579","tex":"A","aliases":"A","display":"A","kind":"notation-table","excerpt":"χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.","count":498},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ca978112ca1bbdca","tex":"a","aliases":"a","display":"a","kind":"notation-table","excerpt":"c_f,a_v · Coherent nonabelian transition labels and gauge values.","count":449},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-eae1dc41a7db82ef","tex":"a_v","aliases":"a_v","display":"a_v","kind":"notation-table","excerpt":"c_f,a_v · Coherent nonabelian transition labels and gauge values.","count":15},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-34b9981d517fb6b9","tex":"a_\\alpha","aliases":"a_α alpha","display":"a_\\alpha","kind":"notation-table","excerpt":"β_T,r(M),a_α · Minimum residual at correction budget M, and the quadratic regularized correction.","count":60},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-df7e70e5021544f4","tex":"B","aliases":"B","display":"B","kind":"notation-table","excerpt":"χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.","count":341},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3e23e8160039594a","tex":"b","aliases":"b","display":"b","kind":"notation-table","excerpt":"E_b(R) · Realizations equipped with an actual identification of their boundary with b.","count":191},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-558a77c65edfbe0c","tex":"b_n","aliases":"b_n","display":"b_n","kind":"notation-table","excerpt":"ω,Q_n,C_n,b_n · A residual, its prefix-product rate, movement coefficient, and summable step budget. 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An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-06ddbea4003ffa9b","tex":"c_{1_v}","aliases":"c_1_v","display":"c_{1_v}","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-59651fe6df7d2f0f","tex":"c_g","aliases":"c_g","display":"c_g","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c04b92df09f891e3","tex":"c_{g\\circ f}","aliases":"c_g∘ f circ","display":"c_{g\\circ f}","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a2b16f8c547b88cb","tex":"\\End","aliases":"End operatorname","display":"{\\operatorname{End}}","kind":"definition-passage","excerpt":"[Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed add","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e84b1a98b5a23eb6","tex":"\\operatorname{ev}","aliases":"ev operatorname","display":"\\operatorname{ev}","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-db44ffded3c6918e","tex":"\\operatorname{ev}_v","aliases":"ev_v operatorname","display":"\\operatorname{ev}_v","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-52bb9700d630070f","tex":"F_w","aliases":"F_w","display":"F_w","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-29f81858f6292043","tex":"g^*","aliases":"g^*","display":"g^*","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8bea312cd06acfdb","tex":"G_v","aliases":"G_v","display":"G_v","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aaa9402664f1a41f","tex":"h","aliases":"h","display":"h","kind":"definition-passage","excerpt":"[Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.","count":167},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aedb0548bf0759b9","tex":"\\id","aliases":"id","display":null,"kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c6b61b46dee37794","tex":"\\inf","aliases":"inf","display":"\\inf","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1633dfbf62528f10","tex":"\\lvert","aliases":"lvert","display":"\\lvert","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6ca61f3078014d2e","tex":"\\mathcal O","aliases":"O mathcal","display":"\\mathcal O","kind":"definition-passage","excerpt":"[Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed add","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6c1f80c69718081b","tex":"\\Ob","aliases":"Ob","display":null,"kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f814b60f967151c4","tex":"\\mathrm{op}","aliases":"op mathrm","display":"\\mathrm{op}","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1b983f28fc05d3a0","tex":"\\operatorname{Realize}","aliases":"Realize operatorname","display":"\\operatorname{Realize}","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d2e3ad5bca4f700e","tex":"\\operatorname{Realize}_{\\mathfrak G}","aliases":"Realize_ G operatorname mathfrak","display":"\\operatorname{Realize}_{\\mathfrak G}","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-226626f51f7539ed","tex":"\\rvert","aliases":"rvert","display":"\\rvert","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2d2757f1a3838f81","tex":"\\Set","aliases":"Set","display":null,"kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0692c76d8494573e","tex":"T_c","aliases":"T_c","display":"T_c","kind":"definition-passage","excerpt":"[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a sectio","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0dcc526e2e33457d","tex":"\\trank","aliases":"trank operatorname","display":"{\\operatorname{trank}}","kind":"definition-passage","excerpt":"[Topological generation] For a topological group M, a subset S topologically generates M when the closure of the subgroup generated by S is all of M. Its least possible size is denoted trank(M). An operator version takes the closure of the operator hull instead. The closure is taken in the stated topology. A change of topology is a change of contra","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4ac0da71d791b586","tex":"\\V^{\\mathrm{op}}","aliases":"V^op V mathrm","display":null,"kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-50e721e49c013f00","tex":"w","aliases":"w","display":"w","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":95},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-67383509ca3cbfa6","tex":"x_v","aliases":"x_v","display":"x_v","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":17},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9d5ae088e190bcb8","tex":"x_w","aliases":"x_w","display":"x_w","kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c8031fd0bb708709","tex":"\\xrightarrow","aliases":"xrightarrow","display":null,"kind":"definition-passage","excerpt":"[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f301f970eb09c336","tex":"\\psi","aliases":"ψ psi","display":"\\psi","kind":"definition-passage","excerpt":"[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.","count":31},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ba20de831f2d9fe6","tex":"\\prod_{v\\in\\Ob\\V}","aliases":"∏_v∈ObV prod in Ob V","display":null,"kind":"definition-passage","excerpt":"[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonempti","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fcc886982cee8077","tex":"\\circ","aliases":"∘ circ","display":"\\circ","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-795a720997b72208","tex":"\\cap","aliases":"∩ cap","display":"\\cap","kind":"definition-passage","excerpt":"[Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-616522c60d7b122c","tex":"\\sim","aliases":"∼ sim","display":"\\sim","kind":"definition-passage","excerpt":"[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-eed4f7039b9ba49f","tex":"\\cong","aliases":"≅ cong","display":"\\cong","kind":"definition-passage","excerpt":"[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely","count":20},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d38f53421f824a97","tex":"\\ge","aliases":"≥ ge","display":"\\ge","kind":"definition-passage","excerpt":"[Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.","count":103},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-942faac63e3ce23f","tex":"\\longrightarrow","aliases":"⟶ longrightarrow","display":"\\longrightarrow","kind":"definition-passage","excerpt":"[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views ","count":31},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0014cfc2591bdee6","tex":"a_*","aliases":"a_*","display":"a_*","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f75dd322f11ad636","tex":"A_2","aliases":"A_2","display":"A_2","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-02882a5a91476024","tex":"a_m","aliases":"a_m","display":"a_m","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-beb14de5f7e22139","tex":"A_n","aliases":"A_n","display":"A_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":13},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-046b4724fc2febf0","tex":"a_n","aliases":"a_n","display":"a_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":49},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0083f442b19187bb","tex":"A_v","aliases":"A_v","display":"A_v","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8bed5507d096707b","tex":"B^{\\mathsf T}","aliases":"B^ T mathsf","display":"B^{\\mathsf T}","kind":"introduction-passage","excerpt":"The residual of the coordinate equation is a covector. In finite dimensions, use orthonormal coordinates in the ambient candidate and residual spaces. Let the columns of B form an arbitrary basis of H=ker p, not necessarily an orthonormal one. Write a=Bc, S=dB, and G=B^ TB. As before, G is positive definite. For an inexact coefficient vector c, the","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-16636769fc160a5a","tex":"b^{-1}","aliases":"b^-1","display":"b^{-1}","kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a5f1f01d70569d04","tex":"B_j","aliases":"B_j","display":"B_j","kind":"introduction-passage","excerpt":"For each j whose output lies in the family, choose B_j⊆ N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1a59e813e505c83b","tex":"B_U","aliases":"B_U","display":"B_U","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":15},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-69b376b6bbe4d3ac","tex":"b_v","aliases":"b_v","display":"b_v","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-edcd2529ad4f495e","tex":"B_X","aliases":"B_X","display":"B_X","kind":"introduction-passage","excerpt":"emma[Continuous inverse for Banach spaces] Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0df0443ecd4fe0fb","tex":"c^q","aliases":"c^q","display":"c^q","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c205c00dfeb99d0b","tex":"C_1","aliases":"C_1","display":"C_1","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0712b486b3e6cec7","tex":"C_2","aliases":"C_2","display":"C_2","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-86bc40a9dfa0a3bf","tex":"d_0","aliases":"d_0","display":"d_0","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-866007381d5c48a8","tex":"d_1","aliases":"d_1","display":"d_1","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-729ca5dae856a3ff","tex":"d_2","aliases":"d_2","display":"d_2","kind":"introduction-passage","excerpt":"Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is m","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a9f51566bd6705f7","tex":"E","aliases":"E","display":"E","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":84},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-115ca4a8b4972e65","tex":"\\operatorname{ev}_n","aliases":"ev_n operatorname","display":"\\operatorname{ev}_n","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6d014ef8b92c530c","tex":"f^{(m)}","aliases":"f^(m)","display":"f^{(m)}","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-095b358cbc4ad8dd","tex":"f_i","aliases":"f_i","display":"f_i","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d4479d142c3b63e5","tex":"f_j","aliases":"f_j","display":"f_j","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e3f4ba757b6d2cf8","tex":"f_n","aliases":"f_n","display":"f_n","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":51},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ae1726ace684b30c","tex":"G_{B_j}","aliases":"G_B_j","display":"G_{B_j}","kind":"introduction-passage","excerpt":"For each j whose output lies in the family, choose B_j⊆ N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a83dd0ccbffe39d0","tex":"I","aliases":"I","display":"I","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":100},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f18b3c49fd8055ae","tex":"i_0","aliases":"i_0","display":"i_0","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-291335565d353600","tex":"i_1","aliases":"i_1","display":"i_1","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d529fadc23d1fc12","tex":"i_k","aliases":"i_k","display":"i_k","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-56e19bddcbcb5ec7","tex":"i_{k+1}","aliases":"i_k+1","display":"i_{k+1}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d822ff9312b40594","tex":"i_{k-1}","aliases":"i_k-1","display":"i_{k-1}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2ee8ac8c8d5446ba","tex":"i_m","aliases":"i_m","display":"i_m","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-45e9122f2e536e8b","tex":"I_{n,m}","aliases":"I_n,m","display":"I_{n,m}","kind":"introduction-passage","excerpt":"8 Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6a50e5637ed9a812","tex":"I_{n,m+1}","aliases":"I_n,m+1","display":"I_{n,m+1}","kind":"introduction-passage","excerpt":"nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4dab5e48ba9132bb","tex":"I_{n,s_n}","aliases":"I_n,s_n","display":"I_{n,s_n}","kind":"introduction-passage","excerpt":"roposition Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently, so only finitely many strict decreases are possible. There is an index s_n≥ n such that I_n,m=I_n,s_n for all m≥ s_n. Consequen","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e87d80e4627c3234","tex":"\\im","aliases":"im","display":null,"kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8254c329a92850f6","tex":"k","aliases":"k","display":"k","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":163},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c4748b1ae1b9971c","tex":"L^m","aliases":"L^m","display":"L^m","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7054bd921fca19d4","tex":"L^\\perp","aliases":"L^⊥ perp","display":"L^\\perp","kind":"introduction-passage","excerpt":"For a closed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ed76d6b3d182baf7","tex":"L_n","aliases":"L_n","display":"L_n","kind":"introduction-passage","excerpt":"nly if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-584e1f4fc366a43f","tex":"\\Longleftrightarrow","aliases":"Longleftrightarrow","display":"\\Longleftrightarrow","kind":"introduction-passage","excerpt":"[Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonze","count":20},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e0987528a0170db9","tex":"\\Longrightarrow","aliases":"Longrightarrow","display":"\\Longrightarrow","kind":"introduction-passage","excerpt":"[Dyadic determination and finite coordinate recovery] In the actual dyadic system, all detail coordinates jointly determine the residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-df701b27f24bbe71","tex":"M_1","aliases":"M_1","display":"M_1","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1 · =S_2M_1+M_2C_1. All terms map the first residual space into the last candidate space, in the written order. Finally, for fixed bounded ","count":13},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e945b5ddcecdc573","tex":"M_2","aliases":"M_2","display":"M_2","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1 · =S_2M_1+M_2C_1. All terms map the first residual space into the last candidate space, in the written order. Finally, for fixed bounded ","count":14},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8200e811d73eda66","tex":"m_m^2","aliases":"m_m^2","display":"m_m^2","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ce78c2a469d42dfa","tex":"m_n","aliases":"m_n","display":"m_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a42932dc6aaac88d","tex":"m_n^2","aliases":"m_n^2","display":"m_n^2","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-43d85ec54b4d02c3","tex":"\\mid_{G_A}","aliases":"mid_G_A mid","display":null,"kind":"introduction-passage","excerpt":"[Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonze","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f6045ff40df65e73","tex":"\\nmid","aliases":"nmid","display":null,"kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7bc2fe684c29431d","tex":"\\norm","aliases":"norm","display":null,"kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":334},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e942b1c655fe6bde","tex":"\\Obs","aliases":"Obs F mathcal","display":"{\\mathcal F}","kind":"introduction-passage","excerpt":"The observation contract. Let F(G) denote the unmarked inverse system of finite quotient groups and their transition maps, considered up to isomorphism as a pro-object. The source group G is not supplied again as part of the input. Nor do we retain evaluations of named source elements. Supplying the source itself would make a reconstruction problem","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-41dacfd07761cf94","tex":"\\overline r","aliases":"overline r overline","display":"\\overline r","kind":"introduction-passage","excerpt":"The endomorphism ring and its units. Write overline r_p for reduction of a rational r at a prime not dividing its denominator. The phrase ``for almost every selected prime'' below means outside a finite set, which may include those denominator primes.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-53f019d1026f6086","tex":"\\overline r_p","aliases":"overline r_p overline","display":"\\overline r_p","kind":"introduction-passage","excerpt":"The endomorphism ring and its units. Write overline r_p for reduction of a rational r at a prime not dividing its denominator. The phrase ``for almost every selected prime'' below means outside a finite set, which may include those denominator primes.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-26e4083a07ea4a31","tex":"\\overline{S(nB_X)}","aliases":"overlineS(nB_X) overline","display":"\\overline{S(nB_X)}","kind":"introduction-passage","excerpt":":inverse-tool Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lem","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f21822f96efaf869","tex":"P_A","aliases":"P_A","display":"P_A","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":33},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-363d67fd271736f2","tex":"P_L","aliases":"P_L","display":"P_L","kind":"introduction-passage","excerpt":"For a closed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7970473ade3c6476","tex":"P_L^2","aliases":"P_L^2","display":"P_L^2","kind":"introduction-passage","excerpt":"sed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2=P_L. For","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1cc1c79952eb4cf2","tex":"\\pmod","aliases":"pmod","display":"\\pmod","kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ddc0eef4732461d5","tex":"Q_U","aliases":"Q_U","display":"Q_U","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e52a322b6d2467f2","tex":"q_v","aliases":"q_v","display":"q_v","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1f4ac7c68b8abd53","tex":"\\mathcal R","aliases":"R mathcal","display":"\\mathcal R","kind":"introduction-passage","excerpt":"The topology is defined without assuming effectivity. For a composable pair q=(vxrightarrow f wxrightarrow g z), let its support be |q|={v,w,z}. A sieve R on W is covering when |q|⊆ W Longrightarrow (|q|↪ W)∈ R for every composable q. This condition depends only on the view category, not on the coefficient groups or on a promised global realization","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a03f214a14787ce8","tex":"R_j","aliases":"R_j","display":"R_j","kind":"introduction-passage","excerpt":"ery] In the actual dyadic system, all detail coordinates jointly determine the residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the res","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b64d075ceafeb44b","tex":"r_k","aliases":"r_k","display":"r_k","kind":"introduction-passage","excerpt":"[Dyadic determination and finite coordinate recovery] In the actual dyadic system, all detail coordinates jointly determine the residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9993dc43cc574b8d","tex":"r_X","aliases":"r_X","display":"r_X","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ca43c3cdcb5f5a6f","tex":"r_Y","aliases":"r_Y","display":"r_Y","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fe39e36813ced2eb","tex":"S^{\\mathsf T}","aliases":"S^ T mathsf","display":"S^{\\mathsf T}","kind":"introduction-passage","excerpt":"rdinate equation is a covector In finite dimensions, use orthonormal coordinates in the ambient candidate and residual spaces. Let the columns of B form an arbitrary basis of H=ker p, not necessarily an orthonormal one. Write a=Bc, S=dB, and G=B^ TB. As before, G is positive definite. For an inexact coefficient vector c, the residual of the coordin","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-88d1f82215e67d59","tex":"s^{(j)}_{2m}","aliases":"s^(j)_2m","display":"s^{(j)}_{2m}","kind":"introduction-passage","excerpt":"residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0,","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7f37d047e0568165","tex":"S_1","aliases":"S_1","display":"S_1","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-db9af401ac382039","tex":"S_2","aliases":"S_2","display":"S_2","kind":"introduction-passage","excerpt":"For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-91ec47101c58ab98","tex":"s_n","aliases":"s_n","display":"s_n","kind":"introduction-passage","excerpt":"ery later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently, so only finitely many strict decreases are possible. There is an index s_n≥ n such that I_n,m=I_n,s_n for all m≥ s_n. Con","count":17},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-38dbe99c73eeaafd","tex":"s_X","aliases":"s_X","display":"s_X","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-89069deb6219b9ac","tex":"s_Y","aliases":"s_Y","display":"s_Y","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0ef3dbe5a13282d7","tex":"\\srank","aliases":"srank operatorname","display":"{\\operatorname{srank}}","kind":"introduction-passage","excerpt":"If all endomorphisms are allowed, this is module generation over the unital ring End(M). We write srank_End(M)(M) for its least generating cardinal. An infinite value can therefore express a strong obstruction even when the operator collection itself is unrestricted.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9567a4e9a9c809fe","tex":"\\srank_{\\End(M)}","aliases":"srank_End(M) srank End operatorname operatorname","display":"{\\operatorname{srank}}_{{\\operatorname{End}}(M)}","kind":"introduction-passage","excerpt":"If all endomorphisms are allowed, this is module generation over the unital ring End(M). We write srank_End(M)(M) for its least generating cardinal. An infinite value can therefore express a strong obstruction even when the operator collection itself is unrestricted.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-729e41c77e53a59d","tex":"\\sup","aliases":"sup","display":"\\sup","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3efb077b7b4a3c92","tex":"\\sup_n","aliases":"sup_n sup","display":"\\sup_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1084f45e6b3e947e","tex":"\\supp","aliases":"supp operatorname","display":"{\\operatorname{supp}}","kind":"introduction-passage","excerpt":"[Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonze","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-51a977c8b2b5f464","tex":"\\mathsf T","aliases":"T mathsf","display":"\\mathsf T","kind":"introduction-passage","excerpt":"The residual of the coordinate equation is a covector. In finite dimensions, use orthonormal coordinates in the ambient candidate and residual spaces. Let the columns of B form an arbitrary basis of H=ker p, not necessarily an orthonormal one. Write a=Bc, S=dB, and G=B^ TB. As before, G is positive definite. For an inexact coefficient vector c, the","count":28},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-10d6fa1215b924bf","tex":"T^*","aliases":"T^*","display":"T^*","kind":"introduction-passage","excerpt":"There are three different residuals. Keep H=ker p, T=d|_H, and a residual datum r. Fix α>0. Write a_α for the unique minimizer of Theorem~[source reference] and let a∈ H be any proposed correction. Three quantities now have different meanings: v=r-Ta, g=T^*(Ta-r)+α a, e=a-a_α. The vector v is the fit residual. The vector g is the normal-equation re","count":50},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-def7ea4ab088787d","tex":"t^2","aliases":"t^2","display":"t^2","kind":"introduction-passage","excerpt":"orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norm","count":21},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-79b896ccd56f4035","tex":"T_q","aliases":"T_q","display":"T_q","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bde3b1705b34e69b","tex":"U_0","aliases":"U_0","display":"U_0","kind":"introduction-passage","excerpt":"[The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-72b2bc9e12d0fa8e","tex":"U_k","aliases":"U_k","display":"U_k","kind":"introduction-passage","excerpt":"tion[The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from ","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f6951bc1a235a45d","tex":"U_{k+1}","aliases":"U_k+1","display":"U_{k+1}","kind":"introduction-passage","excerpt":". The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cd263f633c7dde0b","tex":"\\widetilde s","aliases":"widetilde s widetilde","display":"\\widetilde s","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":16},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5027fc9a4ade4dc0","tex":"X_i","aliases":"X_i","display":"X_i","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2d9731b0ef10fdc5","tex":"x_{i_0}","aliases":"x_i_0","display":"x_{i_0}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e2a78aa8cf2a6130","tex":"x_{i_k}","aliases":"x_i_k","display":"x_{i_k}","kind":"introduction-passage","excerpt":"of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels eac","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a47c3bd800d8c2d7","tex":"x_{i_{k+1}}","aliases":"x_i_k+1","display":"x_{i_{k+1}}","kind":"introduction-passage","excerpt":"of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels eac","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fe618b7cfb9110a0","tex":"x_{i_m}","aliases":"x_i_m","display":"x_{i_m}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2a1409e6c836cba0","tex":"X_j","aliases":"X_j","display":"X_j","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f3be737b42076753","tex":"X_m","aliases":"X_m","display":"X_m","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7a7064d4243ed8b7","tex":"X_{m+1}","aliases":"X_m+1","display":"X_{m+1}","kind":"introduction-passage","excerpt":"nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fcfbe991e8a25e0f","tex":"X_n","aliases":"X_n","display":"X_n","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":16},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-98f50cf2afdfc449","tex":"X_{n+1}","aliases":"X_n+1","display":"X_{n+1}","kind":"introduction-passage","excerpt":"We next prove that restriction induces a surjection Y_n+1→ Y_n, even though X_n+1→ X_n need not be surjective. For membership, let z∈ Y_n+1. Given m≥ n+1, write z=ρ_n+1,m(w) for some w∈ X_m. Then ρ_n z=ρ_n,m(w). Together with the automatic condition at m=n, this shows ρ_nz∈ Y_n.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-caa3e2ac7047f01e","tex":"x_v^{-1}","aliases":"x_v^-1","display":"x_v^{-1}","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f9adc8efb1f46cfd","tex":"X_{\\phi(n)}","aliases":"X_ϕ(n) phi","display":"X_{\\phi(n)}","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-18f5384d58bcb1bb","tex":"Y","aliases":"Y","display":"Y","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":86},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dcdf8b93520408f3","tex":"y_0","aliases":"y_0","display":"y_0","kind":"introduction-passage","excerpt":"with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lemma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f63c0b46cff9ca02","tex":"y_m","aliases":"y_m","display":"y_m","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3859ee776b04fc90","tex":"Y_n","aliases":"Y_n","display":"Y_n","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":16},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-00fe9b383f124db5","tex":"y_n","aliases":"y_n","display":"y_n","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3f18b642e61a8904","tex":"Y_{n+1}","aliases":"Y_n+1","display":"Y_{n+1}","kind":"introduction-passage","excerpt":"We next prove that restriction induces a surjection Y_n+1→ Y_n, even though X_n+1→ X_n need not be surjective. For membership, let z∈ Y_n+1. Given m≥ n+1, write z=ρ_n+1,m(w) for some w∈ X_m. Then ρ_n z=ρ_n,m(w). Together with the automatic condition at m=n, this shows ρ_nz∈ Y_n.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-594e519ae499312b","tex":"z","aliases":"z","display":"z","kind":"introduction-passage","excerpt":"residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0,","count":125},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-42dba447a7ef210c","tex":"z_m","aliases":"z_m","display":"z_m","kind":"introduction-passage","excerpt":"residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0,","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-365f822c88edb476","tex":"|_L","aliases":"|_L","display":"|_L","kind":"introduction-passage","excerpt":"chapQuantitative descent in a common Hilbert model Local exactness and matching are separate operators. Let E be a real Hilbert space of raw local candidates. Let D:E→ R and C:E→ Y be continuous linear maps into Banach spaces, measuring local equation defects and overlap mismatch, respectively. Put L=ker D, B=C|_L, M=ker D∩ker C. The exact matched ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8779068734acaece","tex":"\\cdot","aliases":"· cdot","display":"\\cdot","kind":"introduction-passage","excerpt":"Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do n","count":14},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-67610b0632683369","tex":"\\gamma","aliases":"γ gamma","display":"\\gamma","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-583cf6290785d72c","tex":"\\lambda","aliases":"λ lambda","display":"\\lambda","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":34},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-492247e1bab15272","tex":"\\mu","aliases":"μ mu","display":"\\mu","kind":"introduction-passage","excerpt":"For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, e","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dabec2cdc19c3167","tex":"\\rho","aliases":"ρ rho","display":"\\rho","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":79},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-61e9838ce05b2305","tex":"\\rho^X_{i,j}","aliases":"ρ^X_i,j rho","display":"\\rho^X_{i,j}","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3f6a68400896d8e4","tex":"\\rho^X_{\\phi(i),\\phi(j)}","aliases":"ρ^X_ϕ(i),ϕ(j) rho phi phi","display":"\\rho^X_{\\phi(i),\\phi(j)}","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-12f1c3dae45bea10","tex":"\\rho^Y_{i,j}","aliases":"ρ^Y_i,j rho","display":"\\rho^Y_{i,j}","kind":"introduction-passage","excerpt":"An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bcdc1fcd14dba16f","tex":"\\rho_{i,i}","aliases":"ρ_i,i rho","display":"\\rho_{i,i}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-919c94d29949aa9a","tex":"\\rho_{i,j}","aliases":"ρ_i,j rho","display":"\\rho_{i,j}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f9ea4d96db178cea","tex":"\\rho_{i,k}","aliases":"ρ_i,k rho","display":"\\rho_{i,k}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0019b82ccb43f3b4","tex":"\\rho_{j,k}","aliases":"ρ_j,k rho","display":"\\rho_{j,k}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-18406af98fd43046","tex":"\\rho_m","aliases":"ρ_m rho","display":"\\rho_m","kind":"introduction-passage","excerpt":"nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c03851211fa61002","tex":"\\rho_n","aliases":"ρ_n rho","display":"\\rho_n","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0da056617ca0ff1c","tex":"\\rho_{n+1,m}","aliases":"ρ_n+1,m rho","display":"\\rho_{n+1,m}","kind":"introduction-passage","excerpt":"We next prove that restriction induces a surjection Y_n+1→ Y_n, even though X_n+1→ X_n need not be surjective. For membership, let z∈ Y_n+1. Given m≥ n+1, write z=ρ_n+1,m(w) for some w∈ X_m. Then ρ_n z=ρ_n,m(w). Together with the automatic condition at m=n, this shows ρ_nz∈ Y_n.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-98944c463486b854","tex":"\\rho_{n,m}","aliases":"ρ_n,m rho","display":"\\rho_{n,m}","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-714e0157858a58aa","tex":"\\rho_{\\phi(n),\\phi(m)}","aliases":"ρ_ϕ(n),ϕ(m) rho phi phi","display":"\\rho_{\\phi(n),\\phi(m)}","kind":"introduction-passage","excerpt":"[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-55d1823545b5b4c3","tex":"\\tau","aliases":"τ tau","display":"\\tau","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f0f378bf6e9cc5de","tex":"\\tau_{i_0i_1}","aliases":"τ_i_0i_1 tau","display":"\\tau_{i_0i_1}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-744199ca7fcbd805","tex":"\\tau_{i_{k-1}i_k}","aliases":"τ_i_k-1i_k tau","display":"\\tau_{i_{k-1}i_k}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5ea924893eca4bcf","tex":"\\tau_{i_ki_{k+1}}","aliases":"τ_i_ki_k+1 tau","display":"\\tau_{i_ki_{k+1}}","kind":"introduction-passage","excerpt":". The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f0dc60cd5b6e9148","tex":"\\ldots","aliases":"… ldots","display":"\\ldots","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5b86eb1897c44804","tex":"\\ell","aliases":"ℓ ell","display":"\\ell","kind":"introduction-passage","excerpt":"to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t","count":51},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1ad74f1dd2eab943","tex":"\\ell^2","aliases":"ℓ^2 ell","display":"\\ell^2","kind":"introduction-passage","excerpt":"orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norm","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-df67d8f7c7daff26","tex":"\\mapsto","aliases":"↦ mapsto","display":"\\mapsto","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":22},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4da6085de379a13b","tex":"\\hookrightarrow","aliases":"↪ hookrightarrow","display":"\\hookrightarrow","kind":"introduction-passage","excerpt":"The topology is defined without assuming effectivity. For a composable pair q=(vxrightarrow f wxrightarrow g z), let its support be |q|={v,w,z}. A sieve R on W is covering when |q|⊆ W Longrightarrow (|q|↪ W)∈ R for every composable q. This condition depends only on the view category, not on the coefficient groups or on a promised global realization","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4bab02d584adcb84","tex":"\\varnothing","aliases":"∅ varnothing","display":"\\varnothing","kind":"introduction-passage","excerpt":"For each j whose output lies in the family, choose B_j⊆ N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4471c87ad5a9031e","tex":"\\notin","aliases":"∉ notin","display":"\\notin","kind":"introduction-passage","excerpt":"A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defi","count":13},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4289d0aabfe27afc","tex":"\\prod_{v\\in U}","aliases":"∏_v∈ U prod in","display":"\\prod_{v\\in U}","kind":"introduction-passage","excerpt":"The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If anot","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-56dadbdf217ad92b","tex":"\\sum","aliases":"∑ sum","display":"\\sum","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":47},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b745b62212003b48","tex":"\\sum_{k=0}^{m-1}","aliases":"∑_k=0^m-1 sum","display":"\\sum_{k=0}^{m-1}","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c04523e0e91c10a1","tex":"\\perp","aliases":"⊥ perp","display":"\\perp","kind":"introduction-passage","excerpt":"For a closed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2","count":33},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0ac7e4b05dea168a","tex":"\\bigcap","aliases":"⋂ bigcap","display":"\\bigcap","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e519c34da7accd5d","tex":"\\bigcap_{m\\ge n}","aliases":"⋂_m≥ n bigcap ge","display":"\\bigcap_{m\\ge n}","kind":"introduction-passage","excerpt":"[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) f","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9c20590cf5841569","tex":"\\bigcap_n","aliases":"⋂_n bigcap","display":"\\bigcap_n","kind":"introduction-passage","excerpt":"[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection i","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-036096cd7e9ecc74","tex":"\\bigcup","aliases":"⋃ bigcup","display":"\\bigcup","kind":"introduction-passage","excerpt":":inverse-tool Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lem","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ad4b0793f8d2d37d","tex":"\\bigcup_{n\\ge1}","aliases":"⋃_n≥1 bigcup ge","display":"\\bigcup_{n\\ge1}","kind":"introduction-passage","excerpt":":inverse-tool Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lem","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4f3867cf1346eed7","tex":"\\cdots","aliases":"⋯ cdots","display":"\\cdots","kind":"introduction-passage","excerpt":"Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport conv","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-437de0b3cc0e5a5f","tex":"\\langle","aliases":"⟨ langle","display":"\\langle","kind":"introduction-passage","excerpt":"orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norm","count":51},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7b52d298f1764e49","tex":"\\rangle","aliases":"⟩ rangle","display":"\\rangle","kind":"introduction-passage","excerpt":"orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norm","count":51},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-22449108dc31ebd8","tex":"\\longmapsto","aliases":"⟼ longmapsto","display":"\\longmapsto","kind":"introduction-passage","excerpt":"Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling a","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1e403224a9d979ef","tex":"\\mathbf0","aliases":"0 mathbf","display":"\\mathbf0","kind":"use","excerpt":"ystem presentation of Chapter~[source reference]. Choose any set A of the desired degree and apply Theorem~[source reference]. All the structural and completion properties were proved in Chapters~[source reference]--[source reference]. For A=∅, the copy is computable. For A=K, the group has no computable copy and its least degree is the halting deg","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0b5274a74be9e8df","tex":"\\mathbf0'","aliases":"0' mathbf","display":"\\mathbf0'","kind":"use","excerpt":"ystem presentation of Chapter~[source reference]. Choose any set A of the desired degree and apply Theorem~[source reference]. All the structural and completion properties were proved in Chapters~[source reference]--[source reference]. For A=∅, the copy is computable. For A=K, the group has no computable copy and its least degree is the halting deg","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c0eca90b4aa4b1d1","tex":"\\mathbf1","aliases":"1 mathbf","display":"\\mathbf1","kind":"use","excerpt":"Two boundary examples. On ℓ^∞(N,R), let C remove the first coordinate and let J prepend zero. Then Q_Nx is the tail of x, placed at its original coordinates with zeros before N. Every fixed coordinate is eventually zero in the remainder, and normQ_N≤1. Nevertheless the constant sequence 1 satisfies normQ_N1=1 for every N.","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-232f12a78d23a1e5","tex":"\\mathbf1^{\\mathsf T}","aliases":"1^ T mathbf mathsf","display":"\\mathbf1^{\\mathsf T}","kind":"use","excerpt":"-1 · 0 · -1. The ordinary cycle block satisfies A^ TA=3I-11^ T. It is zero on the constant line and equals 3I on its orthogonal complement. The reflected block satisfies B^ TB=I+vv^ T, v=(1,-1,1)^ T. Since norm v^2=3, this matrix has eigenvalue four in the direction of v and eigenvalue one on its orthogonal complement. In particular it is positive ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-03eeee33e0ea1dca","tex":"\\mathbf1_N","aliases":"1_N mathbf","display":"\\mathbf1_N","kind":"use","excerpt":"A complete calculation of the finite repairs. Let 1_N be one in coordinates 1,…,N and zero elsewhere. Put a_N=( 1_N,0), f_N=f_0-a_N=(- 1_N,1). Then p(f_N)=1, and the first N residual coordinates vanish exactly. The remaining residual and movement satisfy d(f_N)_n=0 · n≤ N,","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-06f7344d1b73ecc4","tex":"A'","aliases":"A'","display":"A'","kind":"use","excerpt":"Constructing the new certificate. Assume the target R is Banach and A:R→ E is a continuous linear right inverse of D, with DA=I, norm A≤ a, and aε<1. Set T=(D-D')A, Z=(I-T)^-1, A'=AZ. The inverse is supplied by the convergent Neumann series because norm T≤ aε<1.","count":14},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-92312bec7dc0bc70","tex":"a'_{ij}","aliases":"a'_ij","display":"a'_{ij}","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-54034542483dca3c","tex":"A^{\\mathsf T}","aliases":"A^ T mathsf","display":"A^{\\mathsf T}","kind":"use","excerpt":"-1 · 0 · -1. The ordinary cycle block satisfies A^ TA=3I-11^ T. It is zero on the constant line and equals 3I on its orthogonal complement. The reflected block satisfies B^ TB=I+vv^ T, v=(1,-1,1)^ T. Since norm v^2=3, this matrix has eigenvalue four in the direction of v and eigenvalue one on its orthogonal complement. In particular it is positive ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-40d113efdc774b45","tex":"a^{(m)}","aliases":"a^(m)","display":"a^{(m)}","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-13c13d5b2dd869b7","tex":"a^{(m)}_n","aliases":"a^(m)_n","display":"a^{(m)}_n","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0f8c9c5387a417b4","tex":"a^{-1}","aliases":"a^-1","display":"a^{-1}","kind":"use","excerpt":"For the groupoid, an arrow a:c→ d means the equation a· c=d. If b:d→ e, then (ba)· c=b·(a· c)=e, giving composition. The identity label is 1 and the inverse label is a^-1. Associativity and the two inverse identities now follow from the corresponding multiplication laws in the product group. The label ba is therefore forced by the order of applicat","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f8ccb1fd0b1219e4","tex":"a^2","aliases":"a^2","display":"a^2","kind":"use","excerpt":"stable repair requires a stable matching problem Suppose ran D and ran B are closed, with residual gains a,b. Then norm u≤ anormDx, norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum no","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-004a43f65c127c71","tex":"a^\\dagger","aliases":"a^† dagger","display":"a^\\dagger","kind":"use","excerpt":"[The regularization limit alternative] As αdownarrow0, Ta_α⟶ b, normr-Ta_α⟶ q, αnorma_α^2⟶0. If b∈ran T, let a^† be its unique minimum norm preimage. Then a_α⟶ a^† in norm. If b∉ran T, then norma_α→∞. Step 1: identify the unavoidable component. Since r-b⊥overlineran T, the adjoint identity gives T^*(r-b)=0. Thus a_α(r)=a_α(b). For every correction ","count":20},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1db92fcd3339ffa0","tex":"a^\\dagger_n","aliases":"a^†_n dagger","display":"a^\\dagger_n","kind":"use","excerpt":"The same operator has an attainable datum. Change only the residual to b_n=1/n^2. Its minimum norm preimage is a^†_n=1/n∈ℓ^2. At α=m^-2, the regularized correction is c^(m)_n=fracm^2n(m^2+n^2), a^†_n-c^(m)_n=fracnm^2+n^2. The bias square is exactly the previously computed B_m. This links the two cases without conflating their existence hypotheses: ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-241d658a47ae5a64","tex":"a_0","aliases":"a_0","display":"a_0","kind":"use","excerpt":"Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The auto","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-47b34af59b56e555","tex":"A_1","aliases":"A_1","display":"A_1","kind":"use","excerpt":"Suppose now that A_1⊇ A_2⊇⋯ are nonempty closed affine subsets of a real Hilbert correction space H. A typical A_n consists of corrections satisfying the first n bounded linear observation equations. The sets are affine because the equations are linear, and closed because the observations are continuous.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1426674c44e8876f","tex":"a_1","aliases":"a_1","display":"a_1","kind":"use","excerpt":"Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The auto","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2941a7c766320c3c","tex":"a_1'","aliases":"a_1'","display":"a_1'","kind":"use","excerpt":"s a_1,a_2 at budgets M_1,M_2 and t∈[0,1]. The candidate a_t=ta_1+(1-t)a_2 satisfies norma_t≤ tM_1+(1-t)M_2, and its residual is t(r-Ta_1)+(1-t)(r-Ta_2). The triangle inequality therefore gives β(tM_1+(1-t)M_2)≤ tβ(M_1)+(1-t)β(M_2). Monotonicity follows from inclusion of balls. If 0≤ M_1≤ M_2 and M_2>0, choose a minimizer a_2 at M_2 and put a_1'=(M_","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bd409829ef36d868","tex":"a_1^{-1}","aliases":"a_1^-1","display":"a_1^{-1}","kind":"use","excerpt":"Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The auto","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e5cda8318149bdb8","tex":"a_2","aliases":"a_2","display":"a_2","kind":"use","excerpt":"For convexity, take minimizers a_1,a_2 at budgets M_1,M_2 and t∈[0,1]. The candidate a_t=ta_1+(1-t)a_2 satisfies norma_t≤ tM_1+(1-t)M_2, and its residual is t(r-Ta_1)+(1-t)(r-Ta_2). The triangle inequality therefore gives β(tM_1+(1-t)M_2)≤ tβ(M_1)+(1-t)β(M_2). Monotonicity follows from inclusion of balls. If 0≤ M_1≤ M_2 and M_2>0, choose a minimize","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-17398d1ee9f4bc45","tex":"A_a","aliases":"A_a","display":"A_a","kind":"use","excerpt":"2. A residual chart can change without changing the defect. Let d:R^2→R be d(u,r)=r. For a∈R, let A_a(r)=(ar,r). Compute the exact repair P_a, compare it with P_b, and decide when they commute.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6cfcb2dc290af84d","tex":"a_a","aliases":"a_a","display":"a_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-66be131002b8b5af","tex":"A_C","aliases":"A_C","display":"A_C","kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩),","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-68421187e198542c","tex":"a_c","aliases":"a_c","display":"a_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-65235a7c0d0ee96d","tex":"A_{C'}","aliases":"A_C'","display":"A_{C'}","kind":"use","excerpt":"If C≠ C', choose j at which they differ. Rule [source reference] makes A_C and A_C' differ at infinitely many odd indices. Theorems~[source reference] and [source reference] then give nonisomorphism and nonembeddability. The remaining properties hold for every paired-prime group.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-41ae46808180b866","tex":"a_e","aliases":"a_e","display":"a_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dc727ddbdff7e587","tex":"a_f","aliases":"a_f","display":"a_f","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-487f58499940e0c4","tex":"a_{ii}","aliases":"a_ii","display":"a_{ii}","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-daf800c8ae3d0681","tex":"a_{ij}","aliases":"a_ij","display":"a_{ij}","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1,","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9f6bca5c1c7a0354","tex":"a_{ij}^{-1}","aliases":"a_ij^-1","display":"a_{ij}^{-1}","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ae3cae372c62870b","tex":"a_{ik}","aliases":"a_ik","display":"a_{ik}","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1, a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-599f350727f7f626","tex":"a_{ji}","aliases":"a_ji","display":"a_{ji}","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0b50cb77343ec39b","tex":"a_{jk}","aliases":"a_jk","display":"a_{jk}","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1, a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e0115225a292e38f","tex":"A_m","aliases":"A_m","display":"A_m","kind":"use","excerpt":"[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B","count":16},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ba209639e02504e4","tex":"a_M","aliases":"a_M","display":"a_M","kind":"use","excerpt":"To prove sharpness, let a_M be an attaining correction from Theorem~[source reference] and put v=r-Ta_M. If v=0, then β(M)=0 and z=0 attains the dual bound. Assume henceforth v≠0 and set z=v/norm v.","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e66d0e178e1abc69","tex":"A_{m+1}","aliases":"A_m+1","display":"A_{m+1}","kind":"use","excerpt":"Step 3: move from regularization parameters to arbitrary budgets. Set M_m=sqrtA_m. Every positive coordinate m^2/(m^2+n^2) increases strictly with m, so A_m, and hence M_m, increases strictly. Since A_m/m→ c=π/4>0, these budgets tend to infinity and (fracM_m+1M_m)^2 =fracA_m+1/(m+1)A_m/mfracm+1m⟶1. Proposition~[source reference] identifies β(M_m)=s","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ffad8741cff39b12","tex":"A_{m,K}","aliases":"A_m,K","display":"A_{m,K}","kind":"use","excerpt":"s] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-758554b79826abfb","tex":"a_{m^{-2}}","aliases":"a_m^-2","display":"a_{m^{-2}}","kind":"use","excerpt":"lta with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By Proposition~[source reference], norma_m^-2(b)-a^†≤sqrt2/m. Theorem~[source reference] at the same ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9d04bb2931d68492","tex":"A_N","aliases":"A_N","display":"A_N","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-07112285aabb492d","tex":"a_N","aliases":"a_N","display":"a_N","kind":"use","excerpt":"A complete calculation of the finite repairs. Let 1_N be one in coordinates 1,…,N and zero elsewhere. Put a_N=( 1_N,0), f_N=f_0-a_N=(- 1_N,1). Then p(f_N)=1, and the first N residual coordinates vanish exactly. The remaining residual and movement satisfy d(f_N)_n=0 · n≤ N,","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-99d30db923da3b41","tex":"A_{N+1}","aliases":"A_N+1","display":"A_{N+1}","kind":"use","excerpt":"Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0c376ef200c86635","tex":"a_n^{(m)}","aliases":"a_n^(m)","display":"a_n^{(m)}","kind":"use","excerpt":"fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and w=(1/n)∈ℓ^2, hence the residual sequence i","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cba9f3864736fa91","tex":"a_n^0","aliases":"a_n^0","display":"a_n^0","kind":"use","excerpt":"displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve the intended-data problem. All three corrections, including the pro","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-803b820dd09c733f","tex":"a_{n_k}","aliases":"a_n_k","display":"a_{n_k}","kind":"use","excerpt":"The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 b","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-db660538430f2d4b","tex":"A_p","aliases":"A_p","display":"A_p","kind":"use","excerpt":"Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a9457f85e2123199","tex":"a_p","aliases":"a_p","display":"a_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b2e272b5b4c3d27e","tex":"a_t","aliases":"a_t","display":"a_t","kind":"use","excerpt":"For convexity, take minimizers a_1,a_2 at budgets M_1,M_2 and t∈[0,1]. The candidate a_t=ta_1+(1-t)a_2 satisfies norma_t≤ tM_1+(1-t)M_2, and its residual is t(r-Ta_1)+(1-t)(r-Ta_2). The triangle inequality therefore gives β(tM_1+(1-t)M_2)≤ tβ(M_1)+(1-t)β(M_2). Monotonicity follows from inclusion of balls. If 0≤ M_1≤ M_2 and M_2>0, choose a minimize","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4f1eee202dc651a9","tex":"A_u","aliases":"A_u","display":"A_u","kind":"use","excerpt":"Three maps that must not be confused. Keep the additive view system of Section~[source reference]. For a prescribed domain U, there are three distinct maps: p_U:E⟶∏_u∈ UA_u, r_U:Γ(A)⟶Γ_U(A), o_U:Γ_U(A)⟶ Q_U. The first restricts a raw family, whether or not it is coherent. The second restricts a global coherent family. The third detects whether an a","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-84e075bd0dff7e4d","tex":"a_u","aliases":"a_u","display":"a_u","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a6b72d71c89e03f5","tex":"a_v^{-1}","aliases":"a_v^-1","display":"a_v^{-1}","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1)","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-509b6f92fab6cf2c","tex":"a_w","aliases":"a_w","display":"a_w","kind":"use","excerpt":"a coordinate preserving equivalence Γ(B_a,b)≃ B(a,b). Under a transition cover, matching local bridge sections glue to a unique global bridge with both prescribed projections unchanged. We must first check that the displayed fibers really form a view system. Let m∈ B_a,b(w) and f:v→ w. Naturality of p and coherence of a give p_v(f_M^*m)=f_F^*p_w(m)","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9f9cdba8dad3a848","tex":"a_z","aliases":"a_z","display":"a_z","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitu","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-edac8f4090c64b8b","tex":"A_\\alpha","aliases":"A_α alpha","display":"A_\\alpha","kind":"use","excerpt":"gularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-753ee05230ab589b","tex":"a_{\\alpha(\\delta)}","aliases":"a_α(δ) alpha delta","display":"a_{\\alpha(\\delta)}","kind":"use","excerpt":"For the limiting statement, insert the exact-data correction at the selected parameter: norma_α(δ)(widetilde r)-a^† ≤fracδ2sqrtα(δ) +norma_α(δ)(r)-a^†. The first term tends to zero by the noise/parameter hypothesis. The second tends to zero by the attained-projection case of Theorem~[source reference], because the selected parameter tends to zero. ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b10de445db1a0c15","tex":"a_{\\alpha_n}","aliases":"a_α_n alpha","display":"a_{\\alpha_n}","kind":"use","excerpt":"Step 5: in the unattained case the norms tend to infinity. If b∉ran T and the norms failed to tend to infinity, there would exist M<∞ and positive α_n→0 with norma_α_n≤ M. Their images still tend to b by Step 2. Theorem~[source reference] would then put b in the actual range, a contradiction. This proves divergence of the norms, not just unboundedn","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-92645d9edf80cd55","tex":"a_{\\alpha_{n_k}}","aliases":"a_α_n_k alpha","display":"a_{\\alpha_{n_k}}","kind":"use","excerpt":"Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity giv","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7d7d4c82e2b393ad","tex":"A_\\gamma","aliases":"A_γ gamma","display":"A_\\gamma","kind":"use","excerpt":"a_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9fdf6a2a563742c0","tex":"a_\\gamma","aliases":"a_γ gamma","display":"a_\\gamma","kind":"use","excerpt":"Parameter monotonicity and the fixed-budget problem. [Regularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-75bb730778c58ee6","tex":"a_{\\kappa(v),i}","aliases":"a_κ(v),i kappa","display":"a_{\\kappa(v),i}","kind":"use","excerpt":"rded. Step 1: normalize the supplied identifications. The triple-overlap equation with all indices equal gives a_ii=a_iia_ii, hence a_ii=1. With indices i,j,i it gives a_jia_ij=1, hence a_ji=a_ij^-1. Composition coverage implies object coverage by inserting identity arrows. Choose a containing patch κ(v) for each view and put b_i(v)=a_κ(v),i(v) (v∈","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4faecfcca4a3cf4c","tex":"\\abs","aliases":"abs","display":null,"kind":"use","excerpt":"How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b645d837ca596c1a","tex":"\\aleph","aliases":"aleph","display":null,"kind":"use","excerpt":"[Countable-code bound] Let G be a finite-description grammar as above. Then its class of realized group isomorphism types has cardinal at most aleph_0. For each length n, the set of strings of length n is countable. The union over n∈ N is countable. The valid strings form a subset, and a single-valued image of a countable set is countable. This is ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8cafa9e3d0a185fe","tex":"\\aleph_0","aliases":"aleph_0 aleph","display":null,"kind":"use","excerpt":"[Countable-code bound] Let G be a finite-description grammar as above. Then its class of realized group isomorphism types has cardinal at most aleph_0. For each length n, the set of strings of length n is countable. The union over n∈ N is countable. The valid strings form a subset, and a single-valued image of a countable set is countable. This is ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ea0b1d9a333c4d54","tex":"\\mathfrak B","aliases":"B mathfrak","display":"\\mathfrak B","kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e77f9eceea598e09","tex":"B'","aliases":"B'","display":"B'","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6bf383288f967d00","tex":"b'","aliases":"b'","display":"b'","kind":"use","excerpt":"Replacing b by an isomorphic boundary b' also gives an equivalence, by composing the identification with b→ b'. For R=id, the marked realization groupoid has center (b,1_b), and between (x,ϕ) and (y,ψ) there is exactly one compatible arrow ψ^-1ϕ. This is a useful example of contractible realization data, not a claim that the original groupoid has n","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7a0bd25f60ee2ba9","tex":"b'_i","aliases":"b'_i","display":"b'_i","kind":"use","excerpt":"The correct uniqueness statement. Suppose (c,b_i) and (c',b'_i) realize the same descent datum. The only possible compatible comparison is g_v=b'_i(v)^-1b_i(v), using any patch through v. The overlap identities make it independent of the patch, and b'_ig=b_i. It is therefore the unique global comparison respecting the fixed local identifications.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-73ec8038e3175b44","tex":"B^{-1}","aliases":"B^-1","display":"B^{-1}","kind":"use","excerpt":"The inverse therefore exists. Its linearity follows from uniqueness of preimages, and its bound is normB^-1y≤α^-1norm y. This construction does not assume closedness of ran T. It uses the positive term α I in the normal operator.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-28bca16a818c103c","tex":"b^2","aliases":"b^2","display":"b^2","kind":"use","excerpt":"stable repair requires a stable matching problem Suppose ran D and ran B are closed, with residual gains a,b. Then norm u≤ anormDx, norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum no","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c73740af47bea4cf","tex":"b_0","aliases":"b_0","display":"b_0","kind":"use","excerpt":"A failed triple. Let t≠1 be an involution and ask for frames b_0,b_1,b_2 satisfying b_1=tb_0, b_2=tb_1, b_2=tb_0. The first two give b_2=b_0, while the third forces t=1. Each displayed transition is individually invertible, but they cannot be the overlap identifications of a global frame. The defect lies in the triple relation.","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6c531ab121c021a5","tex":"B_1","aliases":"B_1","display":"B_1","kind":"use","excerpt":"[Baire's complete-space principle] A nonempty complete metric space cannot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open an","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-74bf6daef0c8b4f8","tex":"b_1","aliases":"b_1","display":"b_1","kind":"use","excerpt":"A failed triple. Let t≠1 be an involution and ask for frames b_0,b_1,b_2 satisfying b_1=tb_0, b_2=tb_1, b_2=tb_0. The first two give b_2=b_0, while the third forces t=1. Each displayed transition is individually invertible, but they cannot be the overlap identifications of a global frame. The defect lies in the triple relation.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bfdaa023faec0e4e","tex":"b_2","aliases":"b_2","display":"b_2","kind":"use","excerpt":"A failed triple. Let t≠1 be an involution and ask for frames b_0,b_1,b_2 satisfying b_1=tb_0, b_2=tb_1, b_2=tb_0. The first two give b_2=b_0, while the third forces t=1. Each displayed transition is individually invertible, but they cannot be the overlap identifications of a global frame. The defect lies in the triple relation.","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fe7473d969f990ef","tex":"B_{a,b}","aliases":"B_a,b","display":"B_{a,b}","kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a34b13535dfa2adc","tex":"b_i","aliases":"b_i","display":"b_i","kind":"use","excerpt":"rded. Step 1: normalize the supplied identifications. The triple-overlap equation with all indices equal gives a_ii=a_iia_ii, hence a_ii=1. With indices i,j,i it gives a_jia_ij=1, hence a_ji=a_ij^-1. Composition coverage implies object coverage by inserting identity arrows. Choose a containing patch κ(v) for each view and put b_i(v)=a_κ(v),i(v) (v∈","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-63b99e36c4729a30","tex":"b_j","aliases":"b_j","display":"b_j","kind":"use","excerpt":"s equal gives a_ii=a_iia_ii, hence a_ii=1. With indices i,j,i it gives a_jia_ij=1, hence a_ji=a_ij^-1. Composition coverage implies object coverage by inserting identity arrows. Choose a containing patch κ(v) for each view and put b_i(v)=a_κ(v),i(v) (v∈ U_i). At a point of U_i∩ U_j, the triple law, now on patches κ(v),i,j, gives b_j(v)=a_ij(v)b_i(v","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-87f4e4b9b8d7159b","tex":"B_K","aliases":"B_K","display":"B_K","kind":"use","excerpt":"Define B_K={2n:n∈ N}∪{2n+1:n∉ K}, H=D_B_K. This is an explicit mathematical definition of a countably infinite, locally cyclic abelian group.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1063fe4f772a9631","tex":"b_k","aliases":"b_k","display":"b_k","kind":"use","excerpt":"[Summable movement gives exactification] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-28ef9ae6743fe0cd","tex":"b_l","aliases":"b_l","display":"b_l","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1a1e8dccad8be342","tex":"B_m","aliases":"B_m","display":"B_m","kind":"use","excerpt":"[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B","count":13},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5df488c3f3b35a2b","tex":"B_{m,K}","aliases":"B_m,K","display":"B_{m,K}","kind":"use","excerpt":"s] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d6ed8162dea38ef2","tex":"B_n","aliases":"B_n","display":"B_n","kind":"use","excerpt":"orem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0","count":13},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4c553faf293d9e77","tex":"B_{n+1}","aliases":"B_n+1","display":"B_{n+1}","kind":"use","excerpt":"annot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open and meets every nonempty open set. Recursively choose a closed ball B_n","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-01ddcddf337569c9","tex":"B_n^2","aliases":"B_n^2","display":"B_n^2","kind":"use","excerpt":"p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really project","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9474dae99bcf0491","tex":"B_t","aliases":"B_t","display":"B_t","kind":"use","excerpt":"Subtract q_0=diag(0,1), obtaining the symmetric matrix B_t=q_t-q_0=frac11+t^2 t^2 · -t -t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5fe27b9bc4a36cd0","tex":"B_t^{\\mathsf T}","aliases":"B_t^ T mathsf","display":"B_t^{\\mathsf T}","kind":"use","excerpt":"-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives e","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1dbc5d4c102370ca","tex":"B_t^2","aliases":"B_t^2","display":"B_t^2","kind":"use","excerpt":"-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives e","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2dd50629910aaca0","tex":"b_v^{-1}","aliases":"b_v^-1","display":"b_v^{-1}","kind":"use","excerpt":"Suppose the orbit is neutral. Equality of orbit classes means there exists an actual based gauge b with b· c^q=1. At an arrow f:v→ w, this equation is b_vq_v^-1c_ff^*(q_w)[f^*(b_w)]^-1=1. Multiply on the left by q_vb_v^-1. Homomorphic preservation of products then gives c_ff^*(q_wb_w^-1)=q_vb_v^-1. Therefore x_v=q_vb_v^-1 is a twisted coherent fami","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ee397f54c2fce937","tex":"B_W","aliases":"B_W","display":"B_W","kind":"use","excerpt":"Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-44eebffdd43cc2d2","tex":"b_w","aliases":"b_w","display":"b_w","kind":"use","excerpt":"· =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitution gives (a·(b· c))_f =a_vb_vc_f[f^*(b_w)]^-1[f^*(a_w)]^-1 =((ab)· c)_f. The reversed order in the two inverse factors is exactly the formula for the inverse of a product. The identity gauge acts trivially.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9c8b5d749b8019c4","tex":"b_w^{-1}","aliases":"b_w^-1","display":"b_w^{-1}","kind":"use","excerpt":"Suppose the orbit is neutral. Equality of orbit classes means there exists an actual based gauge b with b· c^q=1. At an arrow f:v→ w, this equation is b_vq_v^-1c_ff^*(q_w)[f^*(b_w)]^-1=1. Multiply on the left by q_vb_v^-1. Homomorphic preservation of products then gives c_ff^*(q_wb_w^-1)=q_vb_v^-1. Therefore x_v=q_vb_v^-1 is a twisted coherent fami","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-86c749fe35cadfc1","tex":"B_Y","aliases":"B_Y","display":"B_Y","kind":"use","excerpt":"chapBanach repair and constructed split coordinates An actual complete candidate space. Let X_n,Y_n be Banach spaces over a complete nontrivially normed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formal","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-012fc6190ee8df8e","tex":"\\bits","aliases":"bits {0,1}^ N mathbb","display":"{\\{0,1\\}^{{\\mathbb N}}}","kind":"use","excerpt":"Give Sub(D) the topology inherited from {0,1}^D by its membership functions. Membership of a vector with finite support depends on finitely many selector bits. Conversely, membership of e_p_i recovers the ith bit. Consequently the order isomorphism above is also a homeomorphism from Cantor space {0,1}^ N to Sub(D).","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a155fd0fa9678db2","tex":"\\bmod","aliases":"bmod","display":"\\bmod","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-76ac79294e9f6a80","tex":"\\bottomrule","aliases":"bottomrule","display":null,"kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0fab8c9c7c3b2de1","tex":"C'","aliases":"C'","display":"C'","kind":"use","excerpt":"chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.","count":17},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9f9c8470618c69e9","tex":"\\mathcal C'","aliases":"C' mathcal","display":"\\mathcal C'","kind":"use","excerpt":"Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5c37345158ceafa2","tex":"c'","aliases":"c'","display":"c'","kind":"use","excerpt":"The correct uniqueness statement. Suppose (c,b_i) and (c',b'_i) realize the same descent datum. The only possible compatible comparison is g_v=b'_i(v)^-1b_i(v), using any patch through v. The overlap identities make it independent of the patch, and b'_ig=b_i. It is therefore the unique global comparison respecting the fixed local identifications.","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-97c15171699972db","tex":"C^{\\mathsf T}","aliases":"C^ T mathsf","display":"C^{\\mathsf T}","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5eda86663b484271","tex":"c^{\\mathsf T}","aliases":"c^ T mathsf","display":"c^{\\mathsf T}","kind":"use","excerpt":"covector-norm g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0. Thus G is positive definite and invertible. The matrix P=BG^-1B","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3f9e12aa522f08b3","tex":"c^{(m)}_n","aliases":"c^(m)_n","display":"c^{(m)}_n","kind":"use","excerpt":"The same operator has an attainable datum. Change only the residual to b_n=1/n^2. Its minimum norm preimage is a^†_n=1/n∈ℓ^2. At α=m^-2, the regularized correction is c^(m)_n=fracm^2n(m^2+n^2), a^†_n-c^(m)_n=fracnm^2+n^2. The bias square is exactly the previously computed B_m. This links the two cases without conflating their existence hypotheses: ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b0e3969b1f1175ac","tex":"C^{-1}","aliases":"C^-1","display":"C^{-1}","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f076934b22cf32d0","tex":"c^q_f","aliases":"c^q_f","display":"c^q_f","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a20db09b4ab0b0ba","tex":"C_0","aliases":"C_0","display":"C_0","kind":"use","excerpt":"egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-00ae14fd999981ad","tex":"c_0","aliases":"c_0","display":"c_0","kind":"use","excerpt":"[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar s","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2ceb4df5993cba2b","tex":"c_1","aliases":"c_1","display":"c_1","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6e40ae85b89b68e6","tex":"C_{12}","aliases":"C_12","display":"C_{12}","kind":"use","excerpt":"For m=12, only coordinates 2 and 3 matter. The six representatives are 0, e_2, e_3, 2e_3, e_2+e_3, e_2+2e_3. The vector e_2+e_3 generates this quotient, which is C_6, not C_12. Repeated prime powers in the multiplier do not create additional coordinates.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-06a932faf8646c6e","tex":"c_2","aliases":"c_2","display":"c_2","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0413afc6a75dd0cb","tex":"C_{210}","aliases":"C_210","display":"C_{210}","kind":"use","excerpt":"4 · 2,3,5,7 · 210 · C_210","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-155e8a6991cddb21","tex":"C_{2310}","aliases":"C_2310","display":"C_{2310}","kind":"use","excerpt":"5 · 2,3,5,7,11 · 2310 · C_2310","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3134b412f41254b7","tex":"C_{2k}","aliases":"C_2k","display":"C_{2k}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7d25b1661025a3ae","tex":"C_{2k+1}","aliases":"C_2k+1","display":"C_{2k+1}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aa663e271377445d","tex":"C_3","aliases":"C_3","display":"C_3","kind":"use","excerpt":"At the unselected prime 3, the coordinate of u_A is zero. In this case 3(1/3,0)=(1,0). The three roots are (1/3,0), (1/3,e_3), (1/3,2e_3). Their differences form G_A[3]=C_3, as predicted by Proposition~[source reference].","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2341bcb68032a02e","tex":"C_{30}","aliases":"C_30","display":"C_{30}","kind":"use","excerpt":"3 · 2,3,5 · 30 · C_30","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dcb4e0ba5e2088bf","tex":"C_6","aliases":"C_6","display":"C_6","kind":"use","excerpt":"2 · 2,3 · 6 · C_6","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2dc3b3739f1aa480","tex":"C_{a,b}","aliases":"C_a,b","display":"C_{a,b}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-63159a30f6ea5ab4","tex":"c_f^{-1}","aliases":"c_f^-1","display":"c_f^{-1}","kind":"use","excerpt":"Automorphisms and ambiguity are not the same question. The stabilizer of a cocycle is the group of families with a_v=c_f f^*(a_w)c_f^-1. If one gauge a:c→ d is known, all gauges from c to d are uniquely of the form ah, with h∈Stab(c). Indeed a^-1b stabilizes c for every other gauge b. This is the ordinary Hom torsor principle in a groupoid.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7081746b55ffe251","tex":"c_f^q","aliases":"c_f^q","display":"c_f^q","kind":"use","excerpt":"relative extension] The obstruction is independent of the raw lift. It is neutral exactly when s extends to a global section of T_c. The normalized label c^q is the gauge transform of c by q^-1, so Proposition~[source reference] makes it a cocycle. If f:v→ w is internal to U, the partial-section equation says q_v=c_ff^*(q_w), and therefore c_f^q=1 ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d4c67adf201d51ce","tex":"c_f^t","aliases":"c_f^t","display":"c_f^t","kind":"use","excerpt":"If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1 · =t_v^-1c_ff^*(q_w) [f^*(q_w)]^-1f^*(t_w) =t_v^-1c_ff^*(t_w)=c_f^t. Thus changing the raw lift leaves the based orbit unchanged.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-55a47c61b9403260","tex":"c_i","aliases":"c_i","display":"c_i","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1,","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-60b3c954bda772f4","tex":"c_j","aliases":"c_j","display":"c_j","kind":"use","excerpt":"A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1, a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cb563fdc61e28b56","tex":"c_j^2","aliases":"c_j^2","display":"c_j^2","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ab6ab9255d7967c3","tex":"C_k","aliases":"C_k","display":"C_k","kind":"use","excerpt":"[Variable-rate exactification] If Q_n→0 and ∑_n C_nQ_n<∞, then (f_n) has an exact limit and dist(f_n,x)≤ω(f_0)∑_k=n^∞C_kQ_k. For constant q_n=q∈[0,1) and constant C_n=C, the sharper restarted estimate is dist(f_n,x)≤fracC1-q ω(f_n). The estimate ω(f_n)≤ Q_nω(f_0) follows by induction. At n=0 it is equality. If it holds at n, nonnegativity of q_n pe","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6772b00ce756f502","tex":"C_m","aliases":"C_m","display":"C_m","kind":"use","excerpt":"[Finitely generated subgroups] Every finitely generated subgroup of G_A is either finite cyclic or isomorphic to Z⊕ C_m for a squarefree positive integer m, with C_1 trivial. Let L≤ G_A be generated by finitely many elements. Its image in Q is a finitely generated subgroup of Q, hence either zero or infinite cyclic. If it is zero, then L≤ D and its","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-646fe15eb212b5b3","tex":"c_m","aliases":"c_m","display":"c_m","kind":"use","excerpt":"⟨ x-c,z-c⟩≤0 (z∈ C). If C=c_0+L is a closed affine subspace with direction space L, the condition is x-c⊥ L. Produce a Cauchy minimizing sequence. Put d=inf_z∈ Cnormx-z. Nonemptiness makes this a finite nonnegative real number. Choose c_n∈ C so that normx-c_n→ d. Convexity places (c_n+c_m)/2 in C. The parallelogram identity gives normc_n-c_m^2 =2no","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b08948d76a77d953","tex":"C_{M-1}","aliases":"C_M-1","display":"C_{M-1}","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1e15d2e27313753c","tex":"C_N","aliases":"C_N","display":"C_N","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-68a9d5a8339e4038","tex":"c_n","aliases":"c_n","display":"c_n","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bc287c0044452641","tex":"C_{N,M}","aliases":"C_N,M","display":"C_{N,M}","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3573f0e85f5731cd","tex":"C_{N-1}","aliases":"C_N-1","display":"C_{N-1}","kind":"use","excerpt":"egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3d0472df3e9d0eb7","tex":"C_{p_i}","aliases":"C_p_i","display":"C_{p_i}","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5281a9b0294e8bc6","tex":"C_{p_n}","aliases":"C_p_n","display":"C_{p_n}","kind":"use","excerpt":"A purely finite-state boundary. Consider a restricted grammar in which a finite automaton reads the unary word 1^n and selects the factor C_p_n exactly when it accepts.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e1111c7a3d137309","tex":"C_{\\rad(m)}","aliases":"C_rad(m) rad operatorname","display":"C_{{\\operatorname{rad}}(m)}","kind":"use","excerpt":"The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with t","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-53faef096d1a0acc","tex":"c_v","aliases":"c_v","display":"c_v","kind":"use","excerpt":"[Joint determination and the universal property] The evaluations jointly determine a realization. If a set T carries maps c_v:T→ F_v satisfying f^*c_w=c_v, there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-06e1595497d8f5f2","tex":"c_w","aliases":"c_w","display":"c_w","kind":"use","excerpt":"[Joint determination and the universal property] The evaluations jointly determine a realization. If a set T carries maps c_v:T→ F_v satisfying f^*c_w=c_v, there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-31ddf16fe06684fb","tex":"C_\\varepsilon","aliases":"C_ε varepsilon","display":"C_\\varepsilon","kind":"use","excerpt":"It remains to prove exactness of the limit. Fix ε>0. Since ω(f_m)→0, an entire tail of the sequence belongs to the sublevel set C_ε={z:ω(z)≤ε}. Lower semicontinuity makes this set closed, so the limit of that tail belongs to it. Hence ω(x)≤ε for every positive ε. If ω(x)>0, choosing ε=ω(x)/2 contradicts this inequality. Since ω is nonnegative, ω(x)","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-78f0eff5d5a1c58a","tex":"C_\\phi","aliases":"C_ϕ phi","display":"C_\\phi","kind":"use","excerpt":"Actual tower resampling gives a defect map. Return to Banach towers with contractive source and target bonds. Let ϕ be a monotone schedule. Candidate sampling is (S_ϕ f)_n=f_ϕ(n). The sampled bonding map is the actual composite restriction between the selected levels. The residual map C_ϕ is not pointwise sampling of a scalar error score. It accumu","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4a21d8457df5b3c8","tex":"C_{\\phi(n),\\phi(n+1)}","aliases":"C_ϕ(n),ϕ(n+1) phi phi","display":"C_{\\phi(n),\\phi(n+1)}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-269cd99c3ea8db81","tex":"\\code","aliases":"code","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bb28e6eb35641c53","tex":"\\mathrm{comp}","aliases":"comp mathrm","display":"\\mathrm{comp}","kind":"use","excerpt":"[The composition topology] Condition~[source reference] defines a Grothendieck topology J_comp. The sieve generated by a family of inclusions U_i⊆ W is covering exactly when the family covers every composition support in W. In the inclusion poset, identify a sieve on W with a downward-closed collection of subsets of W. Pullback along Z⊆ W consists ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cef2f13a7966bbb0","tex":"\\coprod","aliases":"coprod","display":null,"kind":"use","excerpt":"Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coord","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8760e5ffe5f16a41","tex":"\\coprod_{\\substack{t\\in\\Gamma_W(A)\\\\t|_U=s}}","aliases":"coprod_substackt∈Γ_W(A) t|_U=s coprod substack in Gamma t","display":null,"kind":"use","excerpt":"Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coord","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d6f362ed203647f9","tex":"\\cos","aliases":"cos","display":"\\cos","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8db6134243ea03f4","tex":"\\cos^2","aliases":"cos^2 cos","display":"\\cos^2","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-27513713e8c146a0","tex":"D'","aliases":"D'","display":"D'","kind":"use","excerpt":"chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.","count":38},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c2a814548eab63ec","tex":"\\mathcal D'","aliases":"D' mathcal","display":"\\mathcal D'","kind":"use","excerpt":"Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6c038250227f91ca","tex":"d^{\\mathsf T}","aliases":"d^ T mathsf","display":"d^{\\mathsf T}","kind":"use","excerpt":"The restricted Hilbert normal residual is g=P d^ T(dBc-r)+α Bc. Substitute S=dB, G=B^ TB and the expression for P: g=BG^-1(S^ T(Sc-r)+α Gc)=BG^-1q. Equivalently q=B^ Tg records the pairings of g with the basis columns. Its squared Hilbert norm is g^ Tg =q^ TG^-1B^ TBG^-1q =q^ TG^-1q, using symmetry of G^-1. This proves both coordinate formulas.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d4ef1713d263e2f3","tex":"d^*","aliases":"d^*","display":"d^*","kind":"use","excerpt":"The adjoint below always means T^*:Y→ H, taken with the inherited Hilbert metric on H. In ambient coordinates this is the projection of d^* onto H. A coordinate basis of ker p need not be orthonormal. Its Gram matrix must then be included in any calculation of correction energy.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9329fd1deca2e237","tex":"d^2","aliases":"d^2","display":"d^2","kind":"use","excerpt":"If C=c_0+L is a closed affine subspace with direction space L, the condition is x-c⊥ L. Produce a Cauchy minimizing sequence. Put d=inf_z∈ Cnormx-z. Nonemptiness makes this a finite nonnegative real number. Choose c_n∈ C so that normx-c_n→ d. Convexity places (c_n+c_m)/2 in C. The parallelogram identity gives normc_n-c_m^2 =2normx-c_n^2+2normx-c_m^","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-04edc0a682114fdd","tex":"D_0","aliases":"D_0","display":"D_0","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c4f88bd0811b1b2f","tex":"\\mathcal D_1","aliases":"D_1 mathcal","display":"\\mathcal D_1","kind":"use","excerpt":"The proposed objective is 11/16. Its actual excess over the optimum is 1/48. The residual witness gives D_1(3/4)=frac2132, J(a)- D_1(3/4)=frac132. The duality gap is an upper bound for suboptimality, not necessarily the exact suboptimality. Both repaired candidates preserve p(f)=1.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-943ac188116d74d8","tex":"d_a","aliases":"d_a","display":"d_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2b84d7d41efdc0b1","tex":"d_b","aliases":"d_b","display":"d_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e5cba341fdb2a255","tex":"D_{B_K}","aliases":"D_B_K","display":"D_{B_K}","kind":"use","excerpt":"Define B_K={2n:n∈ N}∪{2n+1:n∉ K}, H=D_B_K. This is an explicit mathematical definition of a countably infinite, locally cyclic abelian group.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-13f2e328ffd83191","tex":"D_{B\\cap C}","aliases":"D_B∩ C cap","display":"D_{B\\cap C}","kind":"use","excerpt":"e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic d","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-30566182622f13f1","tex":"D_{B\\cup C}","aliases":"D_B∪ C cup","display":"D_{B\\cup C}","kind":"use","excerpt":"e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic d","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f7fb93f3c3255dcc","tex":"D_C","aliases":"D_C","display":"D_C","kind":"use","excerpt":"[Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P( N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modul","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b757b9d04b228415","tex":"d_c","aliases":"d_c","display":"d_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d3b4cd694374cac7","tex":"d_e","aliases":"d_e","display":"d_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fac4e28b7bf50eb9","tex":"d_f","aliases":"d_f","display":"d_f","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitu","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-59c0124e24487c12","tex":"d_g","aliases":"d_g","display":"d_g","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitu","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2b510e902537ebcf","tex":"d_{g\\circ f}","aliases":"d_g∘ f circ","display":"d_{g\\circ f}","kind":"use","excerpt":"Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitu","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b1670fb432754538","tex":"D_j","aliases":"D_j","display":"D_j","kind":"use","excerpt":"If all h_j vanish, the finite telescope gives Q_Nx=x for every N. Conversely, if every Q_Nx=x, then L_jh_j=Q_jx-Q_j+1x=0. Apply the left inverse D_j of L_j to obtain h_j=0. The injectivity of the restoration composite is the step that recovers the original detail from its transported value. sourceKS26","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a6fe151e79c7d1a9","tex":"d_l","aliases":"d_l","display":"d_l","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f1be8e49b81c9b1e","tex":"D_M","aliases":"D_M","display":"D_M","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-688ebedc522ae64c","tex":"d_m","aliases":"d_m","display":"d_m","kind":"use","excerpt":"Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3fb8837a44922746","tex":"D_N","aliases":"D_N","display":"D_N","kind":"use","excerpt":"egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a457d6261b7d4571","tex":"d_n","aliases":"d_n","display":"d_n","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-44c6bb8c38959e5b","tex":"D_{\\N\\setminus B}","aliases":"D_N∖ B D_ N∖ B N setminus mathbb setminus","display":"D_{{\\mathbb N}\\setminus B}","kind":"use","excerpt":"e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic d","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0efdddd015383ecb","tex":"d_p","aliases":"d_p","display":"d_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3252f119fdefacda","tex":"d_r","aliases":"d_r","display":"d_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-acd8deecbfbf287e","tex":"D_s","aliases":"D_s","display":"D_s","kind":"use","excerpt":"-t · 1. The common right inverse r↦(r,0) supplies gain at most one for every t, and normD_s-D_t=|s-t|. The general theorem gives normq_s-q_t≤|s-t|.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4a61f76ad851f02d","tex":"D_t","aliases":"D_t","display":"D_t","kind":"use","excerpt":"An all-real worked model. Let D_t(x,y)=x+ty. Its kernel is the line {(-tb,b):b∈R}, and its orthogonal projector is q_t=frac11+t^2 t^2 · -t","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b377ac1e0c05ff0f","tex":"d_t","aliases":"d_t","display":"d_t","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-30d2e95df9b4aa17","tex":"D_\\perp","aliases":"D_⊥ perp","display":"D_\\perp","kind":"use","excerpt":"such that every x has some exact y with normx-y≤ CnormDx. When the range is closed, the least admissible constant is the operator norm of the inverse of D:K^⊥→ran D. In the zero operator case that inverse has zero dimensional domain and its norm is zero. Let K=ker D and K^⊥ carry the induced Hilbert norm. Define D_⊥ u=Du, but regard its target as r","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1d2d51d62f51421c","tex":"D_\\perp^{-1}","aliases":"D_⊥^-1 perp","display":"D_\\perp^{-1}","kind":"use","excerpt":"Suppose first that the range is closed. Because R is Banach, ran D is then Banach. The domain K^⊥ is also complete. The bounded inverse theorem, proved in Appendix~[source reference], gives a bounded inverse A=D_⊥^-1. Applying it to Dx=D(x-qx) yields x-qx=A(Dx), normx-qx≤norm AnormDx. Hence condition 2 holds. If any C≥0 satisfies condition 2, then ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-25ff7c28c66e7d47","tex":"\\deg","aliases":"deg","display":"\\deg","kind":"use","excerpt":"[Uniform construction relative to the selector] If A≤_T X, then the normal-form construction gives an X-computable copy of G_A. In its canonical normal-form presentation, the group operation has Turing degree exactly deg_T(A). To add (r,d) and (t,e), compute the prime divisors of den(r)den(t). At each of these finitely many primes, query the corres","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0d469a888ac8840b","tex":"\\deg_T","aliases":"deg_T deg","display":"\\deg_T","kind":"use","excerpt":"[Uniform construction relative to the selector] If A≤_T X, then the normal-form construction gives an X-computable copy of G_A. In its canonical normal-form presentation, the group operation has Turing degree exactly deg_T(A). To add (r,d) and (t,e), compute the prime divisors of den(r)den(t). At each of these finitely many primes, query the corres","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fbaa37d22c49a1eb","tex":"\\den","aliases":"den operatorname","display":"{\\operatorname{den}}","kind":"use","excerpt":"[A finite support bound] For all r,t∈ Q, supp c_A(r,t)⊆ {p:pmidden(r)den(t)}. At a prime not dividing either denominator, all three rationals have well-defined reductions. On a selected coordinate, reduction respects addition, so the defect is zero. On an unselected coordinate, all section values are zero. The reduced denominator of r+t has no prim","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e48b519eb9071503","tex":"\\operatorname{diag}","aliases":"diag operatorname","display":"\\operatorname{diag}","kind":"use","excerpt":"[Repair of the identity family] The bounded candidate f_n=I is not exact. Its repair is (Pf)_n=diag(2^-n,2^-(n-1),…,2^-1,1). The new last output row is retained at every stage while inherited rows are propagated by the factor 1/2. At level zero the value is one. Exactness forces every inherited row at the next level to be one half of its previous v","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1891403917eb20ac","tex":"\\dim","aliases":"dim","display":"\\dim","kind":"use","excerpt":"is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the positive case is dim H=n-rankA P. Set T=A","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a4b7e80acc67bb28","tex":"\\dim_{\\Q}","aliases":"dim_Q dim_ Q dim Q dim mathbb","display":"\\dim_{{\\mathbb Q}}","kind":"use","excerpt":"A splitting would embed the nonzero divisible group W into P. Theorem~[source reference] excludes every such subgroup. The rank of the torsion-free quotient is exactly dim_ QW. This gives countable examples of every finite rank and of countably infinite rank with the same finite-quotient system. The rank-one family studied next is chosen because it","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-558c1dc971656de6","tex":"\\dist","aliases":"dist","display":null,"kind":"use","excerpt":"A small residual far from exactness. On ℓ^2(N), index the standard basis from one and define De_n=n^-1e_n. The map is bounded and injective, so its exact space is zero. Yet norme_n=1, normDe_n=1/n⟶0. No finite C can bound dist(e_n,ker D) by CnormDe_n. The range is not closed. For example finite truncations of (1/n)_n≥1 belong to the range and conve","count":22},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-868837ca3316be6b","tex":"\\downarrow","aliases":"downarrow","display":"\\downarrow","kind":"use","excerpt":"C and b-Ta∈ C, every correction satisfies normr-Ta^2=q^2+normb-Ta^2. Thus β(M)≥ q. Given ϵ>0, range density in C supplies one correction a with normb-Ta<ϵ. For every M≥norm a it follows that q≤β(M)≤sqrtq^2+ϵ^2. First letting the budget increase and then letting ϵdownarrow0 proves the limit.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b87f37426ee94627","tex":"E^2","aliases":"E^2","display":"E^2","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-89d50d3d0bc30bea","tex":"e^2","aliases":"e^2","display":"e^2","kind":"use","excerpt":"0≤ J_α(a)-J_α(a_α) · ≤fracnorm g^2α. Let e=a-a_α and B=T^*T+α I. The normal equation of the constructed optimizer gives Be=(T^*T+α I)a-T^*r=g. Pairing with e yields the exact identity normTe^2+αnorm e^2=⟨ g,e⟩ ≤norm gnorm e. If e=0, all three conclusions hold with zero error. Otherwise drop the first nonnegative term and divide by αnorm e>0 to obta","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6a07855f650ce9bb","tex":"e_1","aliases":"e_1","display":"e_1","kind":"use","excerpt":"Every expression a(x)+a(y)-a(x+y) is symmetric in x,y, because G is abelian. With e_1=(1,0) and e_2=(0,1), however, c(e_1,e_2)=0, c(e_2,e_1)=1. This contradicts the symmetry required by [source reference]. sourceKS80 The proof rules out all sixteen possible scalar functions a, not just linear functions or selected trial gauges. The source codeProje","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-90781bb8dda5cb7f","tex":"e_2","aliases":"e_2","display":"e_2","kind":"use","excerpt":"Every expression a(x)+a(y)-a(x+y) is symmetric in x,y, because G is abelian. With e_1=(1,0) and e_2=(0,1), however, c(e_1,e_2)=0, c(e_2,e_1)=1. This contradicts the symmetry required by [source reference]. sourceKS80 The proof rules out all sixteen possible scalar functions a, not just linear functions or selected trial gauges. The source codeProje","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2cf646d1eeab02b1","tex":"e_3","aliases":"e_3","display":"e_3","kind":"use","excerpt":"[The same group with a different marked point] Let A=∅ and B={0}. Only the first prime pair changes. Then G_A=G_B inside P, but u_A-u_B=e_2-e_3. The distinguished element is not preserved. In particular, the statement G_A≅ G_Biff A=B would be false for unmarked groups.","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-384a7e923a8d2319","tex":"e_7","aliases":"e_7","display":"e_7","kind":"use","excerpt":"A finite tail exception. Consider an infinite-order element x=s_A(5/6)+2e_7. An exception set for its signature is {2,3,5,7}. Outside these primes, the denominator is invertible, the numerator is nonzero, and the torsion correction vanishes. Thus the root pattern of x agrees with that of u_A in every later prime pair.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8db0cea46fdd60f3","tex":"E_A","aliases":"E_A","display":"E_A","kind":"use","excerpt":"[Endomorphism ring] Under coordinate multiplication, End(G_A)≅ E_A={λ∈ P:∃ r∈ Q, λ_p=overline r_p for almost every p∈ S_A}. The rational r is unique. It is the scalar induced on G_A/D, and the map E_A→ Q sending λ to r is a surjective ring homomorphism. Apply Theorem~[source reference] with B=A. At unselected coordinates, both λ u_A and s_A(r) vani","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b6266893a62573e7","tex":"e_b","aliases":"e_b","display":"e_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6500ef20b6c4932a","tex":"e_c","aliases":"e_c","display":"e_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b49fd93911355d0a","tex":"e_e","aliases":"e_e","display":"e_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-420aa7a6632049fa","tex":"e_f","aliases":"e_f","display":"e_f","kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-53a44fa3ddda48c6","tex":"e_g","aliases":"e_g","display":"e_g","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-54973d23787208ab","tex":"e_i","aliases":"e_i","display":"e_i","kind":"use","excerpt":"Flatness does not remove amplification. Consider a scalar chain with n+1 vertices, where n≥1, and coordinates e_i(z)=2^iz. Set τ_ij=e_ie_j^-1. This connection is flat, but reverse transport along one edge multiplies norms by two. Let x_i=8·2^i-1, y_i=7·2^i. The family y is exact and preserves x_o=7. On a forward edge the residual of x is 1/2, and o","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-16726df327d12b14","tex":"e_j","aliases":"e_j","display":"e_j","kind":"use","excerpt":", every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many vectors a_n. Finite rational linear combinations of those vectors form a countable dense set, so H_0 is separable. Apply Gram--Schmidt to a countable dense sequence, omitting zero remainders. This gives either a finite orthonormal","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-39cc49921fc88ad3","tex":"e_j^{-1}","aliases":"e_j^-1","display":"e_j^{-1}","kind":"use","excerpt":"Flatness does not remove amplification. Consider a scalar chain with n+1 vertices, where n≥1, and coordinates e_i(z)=2^iz. Set τ_ij=e_ie_j^-1. This connection is flat, but reverse transport along one edge multiplies norms by two. Let x_i=8·2^i-1, y_i=7·2^i. The family y is exact and preserves x_o=7. On a forward edge the residual of x is 1/2, and o","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-76853ffdc4938f3f","tex":"\\mathcal E_{Kb}","aliases":"E_Kb mathcal","display":"\\mathcal E_{Kb}","kind":"use","excerpt":"[Invariance of the marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-16230be50f47f21c","tex":"e_l","aliases":"e_l","display":"e_l","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0e681b3e1a1af15b","tex":"e_m","aliases":"e_m","display":"e_m","kind":"use","excerpt":"Existence does not make an arbitrary sequence converge strongly. Take H=Y=ℓ^2 and Te_n=e_n/n. The vectors a_n=e_n have norm one and satisfy Ta_n→0. An exact answer exists, namely zero. Yet norme_n-e_m=sqrt2 for n≠ m, so no subsequence converges in norm.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fe4e3f48e06859e0","tex":"e_n","aliases":"e_n","display":"e_n","kind":"use","excerpt":"A small residual far from exactness. On ℓ^2(N), index the standard basis from one and define De_n=n^-1e_n. The map is bounded and injective, so its exact space is zero. Yet norme_n=1, normDe_n=1/n⟶0. No finite C can bound dist(e_n,ker D) by CnormDe_n. The range is not closed. For example finite truncations of (1/n)_n≥1 belong to the range and conve","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-66c22a9ceb430f6a","tex":"e_o","aliases":"e_o","display":"e_o","kind":"use","excerpt":"For necessity, suppose a constant C satisfies dist(w,ker d)≤ Cnormdw_∞ for every vertex family w. Fix z∈ K^⊥ and take w=R_pz. Every exact family e has e_o∈ K, and root evaluation has norm at most one. Therefore norm z=dist(z,K)≤normz-e_o≤normR_pz-e_∞. Taking the infimum over e∈ker d is legitimate without assuming that this global nearest point is a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-86ea2a47e4e2f4df","tex":"e_{p_i}","aliases":"e_p_i","display":"e_{p_i}","kind":"use","excerpt":"Let B be the set of indices of these factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice oper","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e12eda7995a7cdef","tex":"e_r","aliases":"e_r","display":"e_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-caa961746837d921","tex":"e_s","aliases":"e_s","display":"e_s","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7cf754fef645cd67","tex":"\\endhead","aliases":"endhead","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-63474ae27b72cd77","tex":"\\operatorname{ev}_k","aliases":"ev_k operatorname","display":"\\operatorname{ev}_k","kind":"use","excerpt":"When global behavior determines the finite presentation. Say that X has surjective point projections when ∀ k ev_k:Point(X)→ X_k is onto. This means that every finite input state really occurs in some coherent global input. It is stronger than nonemptiness of the global space.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0cab149d070716dc","tex":"\\operatorname{ev}_w","aliases":"ev_w operatorname","display":"\\operatorname{ev}_w","kind":"use","excerpt":"The evaluations jointly determine a realization. If a set T carries maps c_v:T→ F_v satisfying f^*c_w=c_v, there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for x gives f^*(ev_w x)=f^*(x_w)=x_v=ev_v x. If x,y∈","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-564edd8700e5a5d8","tex":"\\Ext_{U'}^{H^*F}","aliases":"Ext_U'^H^*F Ext","display":null,"kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2c95751a9919cb1b","tex":"\\Ext_U^F","aliases":"Ext_U^F Ext","display":null,"kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-44cbdf7e5a5067f6","tex":"\\Ext_W","aliases":"Ext_W Ext","display":null,"kind":"use","excerpt":"Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coord","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-17dde28ef37430b3","tex":"\\mathbb F","aliases":"F mathbb","display":"\\mathbb F","kind":"use","excerpt":"F_2 · Only zero · Two","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b6bb9a2af76d66fc","tex":"\\mathbf F","aliases":"F mathbf","display":"\\mathbf F","kind":"use","excerpt":"A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-77dd4d71ed008fa6","tex":"\\mathcal F","aliases":"F mathcal","display":"\\mathcal F","kind":"use","excerpt":"Convergence has its own theorem. Call x norm recoverable when Q_Nx→0. This is equivalent to S_Nx→ x, but it does not follow merely from vanishing persistent subspace. Let F=⋃_Nker Q_N be the finite core.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2f4583c8b7c73ba0","tex":"f'","aliases":"f'","display":"f'","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5fefedf3e6c697a3","tex":"f'_j","aliases":"f'_j","display":"f'_j","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e5c12fbb78423d79","tex":"f'_{\\psi'(n)}","aliases":"f'_ψ'(n) psi","display":"f'_{\\psi'(n)}","kind":"use","excerpt":"nd g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7622cd705b527bdd","tex":"f^{(0)}","aliases":"f^(0)","display":"f^{(0)}","kind":"use","excerpt":"[Constructed geometric repair] Starting from f^(0)=f, define f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-51f77419951a1a0f","tex":"f^{(m+1)}","aliases":"f^(m+1)","display":"f^{(m+1)}","kind":"use","excerpt":"[Constructed geometric repair] Starting from f^(0)=f, define f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e1adf13ef9f0afde","tex":"f^{(m+j)}","aliases":"f^(m+j)","display":"f^{(m+j)}","kind":"use","excerpt":"nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-463fe71a6222bdb1","tex":"f^{(m+j+1)}","aliases":"f^(m+j+1)","display":"f^{(m+j+1)}","kind":"use","excerpt":"nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2715e8330c28c57a","tex":"f^{(n)}","aliases":"f^(n)","display":"f^{(n)}","kind":"use","excerpt":"Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a75dbfeaae9c1c65","tex":"f_0","aliases":"f_0","display":"f_0","kind":"use","excerpt":"[Split tower correction] The recurrence defines a continuous linear map A:R→ E with ∂ A=I_R, norm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial co","count":23},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cbeb5f4ee667d5ce","tex":"F_1","aliases":"F_1","display":"F_1","kind":"use","excerpt":"[Baire's complete-space principle] A nonempty complete metric space cannot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open an","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e2c2ad7fac592939","tex":"F_2","aliases":"F_2","display":"F_2","kind":"use","excerpt":"[Baire's complete-space principle] A nonempty complete metric space cannot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open an","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9fb747c54494ef7e","tex":"\\mathbb F_2","aliases":"F_2 mathbb","display":"\\mathbb F_2","kind":"use","excerpt":"F_2 · Only zero · Two","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-710b65b51795073d","tex":"\\mathbf F_2","aliases":"F_2 mathbf","display":"\\mathbf F_2","kind":"use","excerpt":"A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bb1d20c582e7f72b","tex":"\\mathbf F_2^2","aliases":"F_2^2 mathbf","display":"\\mathbf F_2^2","kind":"use","excerpt":"A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-08c122811477fa77","tex":"\\F_3","aliases":"F_3 F mathbb","display":"{\\mathbb F}_3","kind":"use","excerpt":"A torsion correction that cannot be discarded. Take the purely torsion target e_3 and multiplier 2. The root criterion succeeds because the target has zero 2-coordinate. Its rational part is zero, but the answer is not the zero element: 2(0,2e_3)=(0,e_3). In F_3, multiplication by 2 sends 2 to 1. This is the simplest example showing why the finite ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-54c7a07f66f1253f","tex":"\\mathbb F_3","aliases":"F_3 mathbb","display":"\\mathbb F_3","kind":"use","excerpt":"F_3 · Every element · One","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0a27cb068825ed9a","tex":"\\F_5","aliases":"F_5 F mathbb","display":"{\\mathbb F}_5","kind":"use","excerpt":"bottomrule Each residue is exact. At p=2, the lift of 1/2 is set to zero because the denominator is not invertible. At p=5, its value is 3 because 2·3=1 in F_5.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b314e2300364246a","tex":"f_a","aliases":"f_a","display":"f_a","kind":"use","excerpt":"a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[s","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0b8eb6ad7f8b7acf","tex":"f_b","aliases":"f_b","display":"f_b","kind":"use","excerpt":"a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[s","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-088f015dc54be4e4","tex":"f_F^*","aliases":"f_F^*","display":"f_F^*","kind":"use","excerpt":"Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ac9da4be99230127","tex":"f_G^*","aliases":"f_G^*","display":"f_G^*","kind":"use","excerpt":"Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2f6b5c0f84584144","tex":"f_k","aliases":"f_k","display":"f_k","kind":"use","excerpt":"uad normC_ϕ≤ m, ∂_ϕ S_ϕ=C_ϕ∂. Exact candidates remain exact under every monotone common resampling, even without a gap bound. If A and B are bounded right inverses for the fine and sampled differentials, then normP_BS_ϕ f-S_ϕ P_Af ≤(norm A+mnorm B)norm∂ f. For a bounded candidate family, sup_nnormf_ϕ(n)≤sup_knormf_k, proving the sampling bound. For","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0be3a2ded0c0a234","tex":"f_{k+1}","aliases":"f_k+1","display":"f_{k+1}","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-46e315d365be6a6c","tex":"f_m","aliases":"f_m","display":"f_m","kind":"use","excerpt":"on] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k=n^∞b_k. Given ε>0, convergence of th","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dc4a59cf35c9bb32","tex":"f_M^*","aliases":"f_M^*","display":"f_M^*","kind":"use","excerpt":"a coordinate preserving equivalence Γ(B_a,b)≃ B(a,b). Under a transition cover, matching local bridge sections glue to a unique global bridge with both prescribed projections unchanged. We must first check that the displayed fibers really form a view system. Let m∈ B_a,b(w) and f:v→ w. Naturality of p and coherence of a give p_v(f_M^*m)=f_F^*p_w(m)","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c57cafb013c6775f","tex":"F_n","aliases":"F_n","display":"F_n","kind":"use","excerpt":"The centers form a Cauchy sequence: all centers after stage n lie in B_n, whose diameter is at most 2^1-n. Completeness gives a limit. For each fixed n, a tail of centers lies in the closed set B_n, so the limit lies there. It therefore belongs to no F_n, contradicting that these sets cover the space.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-43135e362a6370f0","tex":"f_N","aliases":"f_N","display":"f_N","kind":"use","excerpt":"A complete calculation of the finite repairs. Let 1_N be one in coordinates 1,…,N and zero elsewhere. Put a_N=( 1_N,0), f_N=f_0-a_N=(- 1_N,1). Then p(f_N)=1, and the first N residual coordinates vanish exactly. The remaining residual and movement satisfy d(f_N)_n=0 · n≤ N,","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7817222f46df5e65","tex":"F_{n+1}","aliases":"F_n+1","display":"F_{n+1}","kind":"use","excerpt":"interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open and meets every nonempty open set. Recursively choose a closed ball B_n+1 of positive radius at most 2^-(n+1), contained in the open ball with the same center and radius as B_n, and","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7fc893a0531e77d4","tex":"f_{N+1}","aliases":"f_N+1","display":"f_{N+1}","kind":"use","excerpt":"The residual tends to zero, but the corrections grow like sqrt N. Successive candidates differ by a unit coordinate, so normf_N+1-f_N=1. The movement series diverges and the sequence is not Cauchy. This is exactly the hypothesis missing from the summable exactification theorem.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-af40e62da957c071","tex":"f_{n+1}","aliases":"f_n+1","display":"f_{n+1}","kind":"use","excerpt":"A candidate transport from X to Y consists of maps f_n:X_n→ Y_n. It is exact when ρ^Y_n f_n+1=f_nρ^X_n for all n.","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d2c04c9aaf92eb55","tex":"f_{n+j}","aliases":"f_n+j","display":"f_{n+j}","kind":"use","excerpt":"For constant rates, the global estimate can be sharpened by restarting at the current index. Fix n. A new induction on j gives ω(f_n+j)≤ q^jω(f_n), dist(f_n+j,f_n+j+1)≤ Cq^jω(f_n). Sum the finite chain from n to n+m, and then let m→∞ using the already established limit. The geometric series equals 1/(1-q), including q=0, where its first term is one","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-837249c7d9881c82","tex":"f_{n+j+1}","aliases":"f_n+j+1","display":"f_{n+j+1}","kind":"use","excerpt":"For constant rates, the global estimate can be sharpened by restarting at the current index. Fix n. A new induction on j gives ω(f_n+j)≤ q^jω(f_n), dist(f_n+j,f_n+j+1)≤ Cq^jω(f_n). Sum the finite chain from n to n+m, and then let m→∞ using the already established limit. The geometric series equals 1/(1-q), including q=0, where its first term is one","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b83ac00bc683ab3c","tex":"f_n^{(a)}","aliases":"f_n^(a)","display":"f_n^{(a)}","kind":"use","excerpt":"ct and protected value one, but their middle coordinates differ. They are distinct points of the same feasible fiber. Constant sequences at those points meet exactness and preservation but not a vanishing comparison bound, so these conditions alone cannot imply path independence. sourceKS74 A stronger worked example uses actual convergent paths fro","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-82848474eefdb909","tex":"f_n^{(b)}","aliases":"f_n^(b)","display":"f_n^{(b)}","kind":"use","excerpt":"eKS74 A stronger worked example uses actual convergent paths from the same initial input: f_n^(a)=(s,a(1-2^-n),2^-n) (a∈R). All paths begin at (s,0,1), keep p=s, and have residual 2^-n. Their maximum-norm movements are frac12max{|a|,1}2^-n, so every path satisfies geometric exactification. Their limits are (s,a,0). For different a,b, dist(f_n^(a),f","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2bbdafafd965559e","tex":"\\F_{p_{b_j}}","aliases":"F_p_b_j F mathbb","display":"{\\mathbb F}_{p_{b_j}}","kind":"use","excerpt":"Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finit","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3dd40d0b9ff1c8ed","tex":"F_t","aliases":"F_t","display":"F_t","kind":"use","excerpt":"A global view and a useful warning. [Terminal evaluation] If t is terminal in V, evaluation induces an equivalence Γ(F)≃ F_t. For a∈ F_t, take its restriction along the unique arrow v→ t at each view. Uniqueness of the arrow to t, together with the functor laws, proves coherence. Evaluation at t recovers a. Conversely, any coherent family is the re","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7ae800f554542e07","tex":"f_{\\phi(n)}","aliases":"f_ϕ(n) phi","display":"f_{\\phi(n)}","kind":"use","excerpt":"Actual tower resampling gives a defect map. Return to Banach towers with contractive source and target bonds. Let ϕ be a monotone schedule. Candidate sampling is (S_ϕ f)_n=f_ϕ(n). The sampled bonding map is the actual composite restriction between the selected levels. The residual map C_ϕ is not pointwise sampling of a scalar error score. It accumu","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c1c93a868bfd9d76","tex":"f_{\\psi(n)}","aliases":"f_ψ(n) psi","display":"f_{\\psi(n)}","kind":"use","excerpt":"The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9a7b27c4eeefb975","tex":"f_\\infty","aliases":"f_∞ infty","display":"f_\\infty","kind":"use","excerpt":"ed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formalization derives the required completeness from the operator spaces and their bounded product. The same differential satisfies norm∂ f_∞≤(B_X+B_Y)norm ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8096f24b23ced63d","tex":"\\fixedplate","aliases":"fixedplate","display":null,"kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-15e2ee7354a2849e","tex":"\\mathcal G","aliases":"G mathcal","display":"\\mathcal G","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5e1a1a36d46632a4","tex":"G'","aliases":"G'","display":"G'","kind":"use","excerpt":"Motion of the actual descent repair. Compare pairs (D,C) and (D',C') on the same raw Hilbert space. Suppose each pair has closed local range and closed overlap range on its own locally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤ma","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-02b0b5ff2a25a5d9","tex":"g'","aliases":"g'","display":"g'","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7dd96346a453bb05","tex":"g'_n","aliases":"g'_n","display":"g'_n","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ccf2aef1e39540d4","tex":"g^{\\mathsf T}","aliases":"g^ T mathsf","display":"g^{\\mathsf T}","kind":"use","excerpt":"The restricted Hilbert normal residual is g=P d^ T(dBc-r)+α Bc. Substitute S=dB, G=B^ TB and the expression for P: g=BG^-1(S^ T(Sc-r)+α Gc)=BG^-1q. Equivalently q=B^ Tg records the pairings of g with the basis columns. Its squared Hilbert norm is g^ Tg =q^ TG^-1B^ TBG^-1q =q^ TG^-1q, using symmetry of G^-1. This proves both coordinate formulas.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-565b30cbcdc2ca2a","tex":"G^{-1}","aliases":"G^-1","display":"G^{-1}","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-59c5c6a8e55697c1","tex":"g^2","aliases":"g^2","display":"g^2","kind":"use","excerpt":"A posteriori correction and fit bounds. [Normal residual as a solver certificate] For every proposed a∈ H, norma-a_α · ≤fracnorm gα, normT(a-a_α) · ≤fracnorm g2sqrtα, 0≤ J_α(a)-J_α(a_α) · ≤fracnorm g^2α. Let e=a-a_α and B=T^*T+α I. The normal equation of the constructed optimizer gives Be=(T^*T+α I)a-T^*r=g. Pairing with e yields the exact identity","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0c06bd5499e4979b","tex":"G_1","aliases":"G_1","display":"G_1","kind":"use","excerpt":"uad norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum norm on R× Y, two sufficient gains are therefore G_2=sqrta^2+b^2(1+anorm C)^2 ≤ G_1=a(1+bnorm C)+b. The inequality follows from sq","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3b9dcaf46a46092c","tex":"G_2","aliases":"G_2","display":"G_2","kind":"use","excerpt":"uad norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum norm on R× Y, two sufficient gains are therefore G_2=sqrta^2+b^2(1+anorm C)^2 ≤ G_1=a(1+bnorm C)+b. The inequality follows from sq","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3806cb12e1e3e734","tex":"G_{A_C}","aliases":"G_A_C","display":"G_{A_C}","kind":"use","excerpt":"[Escape within one profinite-completion class] For every C⊆ N, the group G_A_C is not isomorphic to any output of G. Distinct C give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion P. Fix a description index j. If R_j is outside the family, it ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-74c538d30573ec20","tex":"G_B","aliases":"G_B","display":"G_B","kind":"use","excerpt":"The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with t","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7ccf548c96ad41ac","tex":"g_d","aliases":"g_d","display":"g_d","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-82170a73bb5aa973","tex":"G_K","aliases":"G_K","display":"G_K","kind":"use","excerpt":"The group G_K still has a finite mathematical definition. Its noncomputability is not a proof that it cannot be named in a finite semantic language. The theorem concerns effective copies with computable operations and equality.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-15e892a32e1d61ee","tex":"g_n","aliases":"g_n","display":"g_n","kind":"use","excerpt":"[The exact tower category] Identity maps and composites of exact transports are exact. An exact transport induces a map of coherent point spaces by x↦(f_nx_n)_n, preserving identities and composition. For two exact transports, substitute ρ^Z_ng_n+1=g_nρ^Y_n and ρ^Y_nf_n+1=f_nρ^X_n into the composite square. For a coherent point, ρ^Y_n f_n+1x_n+1 =f","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dbbbaca70e55accd","tex":"g_{n+1}","aliases":"g_n+1","display":"g_{n+1}","kind":"use","excerpt":"[The exact tower category] Identity maps and composites of exact transports are exact. An exact transport induces a map of coherent point spaces by x↦(f_nx_n)_n, preserving identities and composition. For two exact transports, substitute ρ^Z_ng_n+1=g_nρ^Y_n and ρ^Y_nf_n+1=f_nρ^X_n into the composite square. For a coherent point, ρ^Y_n f_n+1x_n+1 =f","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f046713e329fd428","tex":"g_p","aliases":"g_p","display":"g_p","kind":"use","excerpt":"[Root or obstruction] Let g=s_A(r)+d∈ G_A and let m≥1. Then g∈ mG_A Longleftrightarrow g_p=0 for every pmid m. If the condition fails, a prime pmid m with g_p≠0 is a nonexistence certificate. If it holds, a root is obtained by the finite procedure in the proof. Necessity follows because multiplication by m is zero in every coordinate C_p with pmid ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-87a3fa4dd69238cf","tex":"g_r","aliases":"g_r","display":"g_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-caf9c919454201a8","tex":"\\mathcal G_U","aliases":"G_U mathcal","display":"\\mathcal G_U","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-99b7a1e1268a1b18","tex":"g_v","aliases":"g_v","display":"g_v","kind":"use","excerpt":"The correct uniqueness statement. Suppose (c,b_i) and (c',b'_i) realize the same descent datum. The only possible compatible comparison is g_v=b'_i(v)^-1b_i(v), using any patch through v. The overlap identities make it independent of the patch, and b'_ig=b_i. It is therefore the unique global comparison respecting the fixed local identifications.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6d584c042dd0c359","tex":"G_w","aliases":"G_w","display":"G_w","kind":"use","excerpt":"descentA set of solutions forgets the witnesses relating equivalent solutions. The next layer retains those witnesses, their composition, and their automorphisms. Gluing then has to respect the overlap identifications themselves. Nonabelian coherence and relative trivialization. Coherent transition labels. Fix groups G_v and restriction homomorphis","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-66d4cedfcc852439","tex":"G_z","aliases":"G_z","display":"G_z","kind":"use","excerpt":"groupoid] Equation~[source reference] defines an action on cocycles. A gauge carrying c to d induces an equivalence of twisted section sets by x_v↦ a_vx_v. The cocycles and actual gauges form a groupoid: an arrow a:c→ d followed by b:d→ e has label ba. First, T_c really is a view system. For f:v→ w, g:w→ z, and x∈ G_z, the homomorphism and cocycle ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1bf12537a68f2d94","tex":"\\mathfrak H","aliases":"H mathfrak","display":"\\mathfrak H","kind":"use","excerpt":"Monotonicity under grammar extension. [Admissible expansion] Suppose every valid description in G translates to a valid description in H with the same semantic group. Then every G-realizable group is H-realizable. If every translated description has length at most f(lvert drvert), then K_ H(G)≤ f(K_ G(G)) whenever K_ G(G) is finite. Translate a sho","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c3d50e4b12d0681c","tex":"H^*","aliases":"H^*","display":"H^*","kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-daaa917c98a2902e","tex":"H^{-1}","aliases":"H^-1","display":"H^{-1}","kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-876dbc1f5c6d7de5","tex":"H^2","aliases":"H^2","display":"H^2","kind":"use","excerpt":"chapMultiscale residuals and the limits of reconstruction A split coarsener retains its missing detail. Let C:R→overline R and J:overline R→ R be linear maps with CJ=I. Put H=I-JC. Then r=JCr+Hr, CH=0, H^2=H. The two coordinates (Cr,Hr) retain all information. The coarse coordinate alone generally does not.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-61ca152e4d27708a","tex":"h^2","aliases":"h^2","display":"h^2","kind":"use","excerpt":". It satisfies T^*(r-Ta_α)=α a_α. For every a∈ H, J_α(a)-J_α(a_α) =normT(a-a_α)^2+αnorma-a_α^2. Step 1: the normal operator has an actual bounded inverse. Let B=T^*T+α I:H→ H. It is bounded and self-adjoint. For every h∈ H, ⟨ Bh,h⟩=normTh^2+αnorm h^2 ≥αnorm h^2. If h≠0, Cauchy--Schwarz and division by norm h give normBh≥αnorm h. The same inequality","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0b14fec69d50fa98","tex":"h_*","aliases":"h_*","display":"h_*","kind":"use","excerpt":"[Nearest feasible protected repair] Every feasible x has a unique closest exact answer y_* preserving p(x). Its correction h_*=x-y_* is the unique vector in the orthogonal complement of ker B inside M satisfying Bh_*=Dx. For every other feasible answer y, normx-y^2=normx-y_*^2+normy_*-y^2. Moreover, the following conditions are equivalent: ran B is","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-eba0c94b0e6f7476","tex":"H_0","aliases":"H_0","display":"H_0","kind":"use","excerpt":"ent [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many ","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2b0800e4e3bac109","tex":"h_0","aliases":"h_0","display":"h_0","kind":"use","excerpt":"ormx-y_*^2+normy_*-y^2. Moreover, the following conditions are equivalent: ran B is closed Longleftrightarrow ∃ C≥0 ∀ feasible x: normx-y_*≤ CnormDx. When these conditions hold, the least nonnegative constant is the transverse inverse norm of B, including the zero-range case. Feasibility gives some h_0=x-y_0∈ M with Bh_0=Dx. Conversely, every h∈ M ","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2dfab4a78393f05c","tex":"h_1","aliases":"h_1","display":"h_1","kind":"use","excerpt":"The choice of x_0 only supplies an origin for the parameters. If x_1=x_0+h_1 is used instead, the parameter of x changes from x-x_0 to x-x_1=(x-x_0)-h_1. The affine set of answers itself has not changed. sourceKS10 An extension fiber is therefore an affine copy, or torsor, of the invisible coherent group. Selecting one solution gives coordinates on","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-da28811766777fc7","tex":"h_J","aliases":"h_J","display":"h_J","kind":"use","excerpt":"To verify weak convergence against h∈ H_0, choose a finite basis combination h_J close to h. Then |⟨ a_n_k-a,h⟩| ≤ |⟨ a_n_k-a,h_J⟩| +2Mnormh-h_J. For a prescribed tolerance, choose h_J first to control the second term and then k to control the first by finitely many coordinate limits. For a general h in the ambient space, write h=P_H_0h+h_⊥. The or","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a57a2041d5b95f96","tex":"h_j","aliases":"h_j","display":"h_j","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-082db4c82cd55c3c","tex":"h_m","aliases":"h_m","display":"h_m","kind":"use","excerpt":"To prove closedness of its range, suppose Bh_n→ y. Applying the lower bound to h_n-h_m gives normh_n-h_m≤α^-1normBh_n-Bh_m. Thus h_n is Cauchy. Completeness of H supplies a limit h, and continuity gives Bh=y. If y is orthogonal to the range of B, then for every h, 0=⟨ Bh,y⟩=⟨ h,By⟩. Taking h=By gives By=0, and injectivity gives y=0. A closed subspa","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-348cda0e12172b60","tex":"H_n","aliases":"H_n","display":"H_n","kind":"use","excerpt":"Three hypotheses that cannot simply be dropped. Summable movement. In maximum-norm R^3, let f_n=(s,H_n,(n+1)^-1), H_n=∑_j=1^nfrac1j. The first coordinate is protected and the residual is the last coordinate. It tends to zero with the same ratios as the polynomial example. The movement equals 1/(n+1), however, and the unprotected coordinate diverges","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-785ae493560363c1","tex":"h_n","aliases":"h_n","display":"h_n","kind":"use","excerpt":"To prove closedness of its range, suppose Bh_n→ y. Applying the lower bound to h_n-h_m gives normh_n-h_m≤α^-1normBh_n-Bh_m. Thus h_n is Cauchy. Completeness of H supplies a limit h, and continuity gives Bh=y. If y is orthogonal to the range of B, then for every h, 0=⟨ Bh,y⟩=⟨ h,By⟩. Taking h=By gives By=0, and injectivity gives y=0. A closed subspa","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d54b1a6be7c911b2","tex":"h_p","aliases":"h_p","display":"h_p","kind":"use","excerpt":"Root conditions do not depend on the subspace. [Ambient root criterion] For every rational subspace W, every h∈ H_W, and every m≥1, h∈ mH_W Longleftrightarrow h_p=0 for all primes pmid m. Multiplication by m on P is zero at primes dividing m and invertible on every other coordinate. Thus the right-hand condition is equivalent to h∈ mP. Apply purity","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3ea834c81d3dfd1d","tex":"H_V","aliases":"H_V","display":"H_V","kind":"use","excerpt":"D agree with h on the primes dividing m and vanish elsewhere. Then h-d∈ mP∩ H. Purity gives h-d=mx for some x∈ H, so q(x)=q(h)/m. Thus q(H) is closed under rational scalars. Since D⊆ H, the equality H=q^-1(q(H)) holds. This proves that the constructions are inverse. Both preserve inclusion. The bottom object is H_{0}=D, and the top object is H_V=P.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f823eeb6cd5f23d3","tex":"h_v","aliases":"h_v","display":"h_v","kind":"use","excerpt":"Let H_U={h:f^*(h_w)=h_v for every f, h|_U=1}.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f020f3626fdf00d6","tex":"h_w","aliases":"h_w","display":"h_w","kind":"use","excerpt":"Let H_U={h:f^*(h_w)=h_v for every f, h|_U=1}.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2bf812a56567a3f8","tex":"H_{\\{0\\}}","aliases":"H_{0}","display":"H_{\\{0\\}}","kind":"use","excerpt":"Given h∈ H and m≥1, let d∈ D agree with h on the primes dividing m and vanish elsewhere. Then h-d∈ mP∩ H. Purity gives h-d=mx for some x∈ H, so q(x)=q(h)/m. Thus q(H) is closed under rational scalars. Since D⊆ H, the equality H=q^-1(q(H)) holds. This proves that the constructions are inverse. Both preserve inclusion. The bottom object is H_{0}=D, a","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-87b87a125d1e88e4","tex":"h_\\perp","aliases":"h_⊥ perp","display":"h_\\perp","kind":"use","excerpt":"To verify weak convergence against h∈ H_0, choose a finite basis combination h_J close to h. Then |⟨ a_n_k-a,h⟩| ≤ |⟨ a_n_k-a,h_J⟩| +2Mnormh-h_J. For a prescribed tolerance, choose h_J first to control the second term and then k to control the first by finitely many coordinate limits. For a general h in the ambient space, write h=P_H_0h+h_⊥. The or","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4c5b1d0f000ac0ff","tex":"\\hline","aliases":"hline","display":null,"kind":"use","excerpt":"category Consider four vertices A,B,C,D, and the free path category generated by Axrightarrow0B, Bxrightarrow0A, Axrightarrow2C, Cxrightarrow5A. There are no arrows incident to D, apart from its identity. A path costs the sum of its edge labels. Concatenation is additive and empty paths cost zero, so all cost axioms hold. The directed distances are","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5bcdae19d97f07ce","tex":"\\Hom","aliases":"Hom Hom operatorname","display":null,"kind":"use","excerpt":"The actual residual of a transport. Assume C is preadditive, so differences of maps make sense. The tower differential is (∂ f)_n=ρ^Y_nf_n+1-f_nρ^X_n. Its target has components Hom(X_n+1,Y_n), not Hom(X_n,Y_n).","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-996aae2a581f4236","tex":"\\Hull","aliases":"Hull operatorname","display":"{\\operatorname{Hull}}","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b0ad653a11ea5542","tex":"\\Hull_{\\mathcal O}","aliases":"Hull_ O Hull mathcal operatorname mathcal","display":"{\\operatorname{Hull}}_{\\mathcal O}","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a3dbd696a54cd3ed","tex":"I_E","aliases":"I_E","display":"I_E","kind":"use","excerpt":"A section is a choice of residual coordinates. Let d:E→ R be a continuous linear map between normed spaces over a nontrivially normed field. Suppose it has a continuous linear right inverse A:R→ E, so dA=I_R. Set K=ker d, P_A=I_E-Ad. Then dP_A=0, P_A fixes K, and P_A^2=P_A. The right inverse chooses where each residual is represented inside the can","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0625c4dc07eb960a","tex":"i_j","aliases":"i_j","display":"i_j","kind":"use","excerpt":"The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-33548fbd7d20a181","tex":"i_{j+1}","aliases":"i_j+1","display":"i_{j+1}","kind":"use","excerpt":"The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-714b73df80781bff","tex":"i_{m-1}","aliases":"i_m-1","display":"i_{m-1}","kind":"use","excerpt":"Distinct walks are retained as arrows of the view category. The product along a walk q=(i_0,…,i_m) is T_q=τ_i_0i_1⋯τ_i_m-1i_m. Concatenation gives the restriction composition law. An exact vertex family defines a coherent walk-view section, and a coherent walk-view section satisfies every edge equation. Thus this graph construction returns to the o","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5dc7ccbeace33e78","tex":"I_R","aliases":"I_R","display":"I_R","kind":"use","excerpt":"[Split tower correction] The recurrence defines a continuous linear map A:R→ E with ∂ A=I_R, norm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial co","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f3d67654dfc71a7c","tex":"I_{R_j}","aliases":"I_R_j","display":"I_{R_j}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3a19cea3d43d3b3d","tex":"I_{R_N}","aliases":"I_R_N","display":"I_{R_N}","kind":"use","excerpt":"j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent ma","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ad3476e646d4f9c7","tex":"I_\\V","aliases":"I_V V","display":null,"kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b1185bc920eea10a","tex":"\\id_A","aliases":"id_A id","display":null,"kind":"use","excerpt":"Extended directed distance. Let C be a category and let W assign each arrow a value in the extended nonnegative reals [0,∞]. Suppose W(id_A)=0, W(gf)≤ W(f)+W(g). The order of ordinary composition is first f, then g. Define d_W(A,B)=inf_f:A→ BW(f). The infimum of an empty family is ∞. No assumption of connectedness, finite cost, symmetry, or attainm","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-011988ed9bdfec11","tex":"\\id_F","aliases":"id_F id","display":null,"kind":"use","excerpt":"[Functorial transport] These induced maps preserve identities and composition: Γ(id_F)=id_Γ(F), Γ(βα)=Γ(β)Γ(α). Evaluate each side at an arbitrary view and apply the corresponding identity for the coordinate maps. Theorem~[source reference] then identifies the maps on the whole realization set. sourceKS02 An arbitrary collection of fiber maps need ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-974a5ed6832aaea3","tex":"\\id_\\W","aliases":"id_W id W","display":null,"kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-14fac349a823622b","tex":"\\id_{\\Gamma(F)}","aliases":"id_Γ(F) id Gamma","display":null,"kind":"use","excerpt":"[Functorial transport] These induced maps preserve identities and composition: Γ(id_F)=id_Γ(F), Γ(βα)=Γ(β)Γ(α). Evaluate each side at an arbitrary view and apply the corresponding identity for the coordinate maps. Theorem~[source reference] then identifies the maps on the whole realization set. sourceKS02 An arbitrary collection of fiber maps need ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4a54158041a35b80","tex":"\\iff","aliases":"iff","display":"\\iff","kind":"use","excerpt":"The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with t","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f3d77613c0887bee","tex":"\\inf_{b\\in B}","aliases":"inf_b∈ B inf in","display":"\\inf_{b\\in B}","kind":"use","excerpt":"Distance to the actual correction image. Let R be a normed additive group and B≤ R an additive subgroup. In the protected linear model, B=d(ker p). For a residual Ω∈ R, define ν_B(Ω)=inf_b∈ BnormΩ-b. The subgroup contains zero, so this is a finite nonnegative real number. It is an infimum, not an assertion that a nearest admissible residual exists.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1f4466ecc1e7344a","tex":"\\inf_{f:A\\to B}","aliases":"inf_f:A→ B inf to","display":"\\inf_{f:A\\to B}","kind":"use","excerpt":"Extended directed distance. Let C be a category and let W assign each arrow a value in the extended nonnegative reals [0,∞]. Suppose W(id_A)=0, W(gf)≤ W(f)+W(g). The order of ordinary composition is first f, then g. Define d_W(A,B)=inf_f:A→ BW(f). The infimum of an empty family is ∞. No assumption of connectedness, finite cost, symmetry, or attainm","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-82ee963b450755be","tex":"\\inf_{\\norm a\\le M}","aliases":"inf_norm a≤ M inf norm le","display":null,"kind":"use","excerpt":"chapThe attainable residual at a fixed correction budget The unrestricted floor dist(r,ran T) forgets how large a correction must be. To retain that information, fix a nonnegative budget M and define β_T,r(M)=inf_norm a≤ Mnormr-Ta. The norm is the actual correction norm in H=ker p. It is not the norm of arbitrary coordinates used to represent that ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-99f24b2b87ddb9f1","tex":"\\inf_{z\\in C}","aliases":"inf_z∈ C inf in","display":"\\inf_{z\\in C}","kind":"use","excerpt":"d convex, and fix x∈ H. There is a unique c∈ C minimizing normx-c. It is characterized by ⟨ x-c,z-c⟩≤0 (z∈ C). If C=c_0+L is a closed affine subspace with direction space L, the condition is x-c⊥ L. Produce a Cauchy minimizing sequence. Put d=inf_z∈ Cnormx-z. Nonemptiness makes this a finite nonnegative real number. Choose c_n∈ C so that normx-c_n→","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-485db41ee683b54c","tex":"J_0","aliases":"J_0","display":"J_0","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-694a109c6881c1d9","tex":"J_{\\mathrm{comp}}","aliases":"J_comp mathrm","display":"J_{\\mathrm{comp}}","kind":"use","excerpt":"[The composition topology] Condition~[source reference] defines a Grothendieck topology J_comp. The sieve generated by a family of inclusions U_i⊆ W is covering exactly when the family covers every composition support in W. In the inclusion poset, identify a sieve on W with a downward-closed collection of subsets of W. Pullback along Z⊆ W consists ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-147f12c21f03b9ea","tex":"J_{M-1}","aliases":"J_M-1","display":"J_{M-1}","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6cca5b503f09502b","tex":"J_N","aliases":"J_N","display":"J_N","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2501a2cd4507c654","tex":"J_{N,M}","aliases":"J_N,M","display":"J_{N,M}","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9ec98c07d696ff5a","tex":"J_{N-1}","aliases":"J_N-1","display":"J_{N-1}","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5cbd3ac214345870","tex":"J_\\alpha","aliases":"J_α alpha","display":"J_\\alpha","kind":"use","excerpt":"The normal equation is derived in the protected space. For α>0, define on H=ker p J_α(a)=normTa-r^2+αnorm a^2. Both norms are the declared Hilbert norms. All minimizers below are corrections in H, so preservation is exact for every parameter, not just in the limit. This is ordinary quadratic Tikhonov regularization applied to the actual protected c","count":14},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d028b14938c1ba39","tex":"J_{\\alpha_n}","aliases":"J_α_n alpha","display":"J_{\\alpha_n}","kind":"use","excerpt":"Why a vanishing normal residual is not enough. Let T=0 on R, let r=0, and take α_n=n^-2 for n≥1. The exact optimizer is zero for every such n. The fixed proposal a_n=1 instead has v_n=0, g_n=α_n⟶0, J_α_n(a_n)=α_n⟶0, absa_n-a_α_n=1. Thus the fit residual, the normal residual, and even the absolute objective value can all tend to zero without converg","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a7e0c315ce3cf160","tex":"\\K","aliases":"K","display":null,"kind":"use","excerpt":"For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0,","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9da7de32ab512a2d","tex":"\\mathcal K","aliases":"K mathcal","display":"\\mathcal K","kind":"use","excerpt":"Internal realization and intended objects. Suppose a separate category S contains the intended semantic objects of an application, and a realization construction gives a comparison functor R: S⟶ K, where K is the relevant category or groupoid of coherent data. Defining K does not establish that R is faithful, full, or essentially surjective. These ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fc2228c8ece1f90d","tex":"\\K^{1\\times m}","aliases":"K^1× m K times","display":null,"kind":"use","excerpt":"[A finite certificate alternative] Exactly one of the following occurs: item There is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the po","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0834cc9214348f2c","tex":"\\K^{1\\times p}","aliases":"K^1× p K times","display":null,"kind":"use","excerpt":"[A finite certificate alternative] Exactly one of the following occurs: item There is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the po","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fa8695fc56eb58b4","tex":"k^2","aliases":"k^2","display":"k^2","kind":"use","excerpt":"Every other correction can be written uniquely as h_*+k with k∈ker B. Orthogonality gives normh_*+k^2=normh_*^2+norm k^2. Writing y_*=x-h_* and y=x-h_*-k, we have x-y=h_*+k and y_*-y=k. Substitution proves the stated Pythagorean identity. Its last term vanishes only when y=y_*, establishing the unique closest answer.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-16b09a3431d14563","tex":"K^3","aliases":"K^3","display":"K^3","kind":"use","excerpt":"is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9ead51bf85f85425","tex":"\\K^{m+p}","aliases":"K^m+p K","display":null,"kind":"use","excerpt":"m has zero left side and nonzero right side, assign arbitrary values to its free variables and solve for its pivot variables. This supplies a solution y_0. For any vector y, Ty=b Longleftrightarrow T(y-y_0)=0. Translation by y_0 is therefore a bijection from ker T to the answer set, with inverse y↦ y-y_0. Rank--nullity applied to the actual linear ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a16ce646a1aaff3e","tex":"\\K^{m\\times n}","aliases":"K^m× n K times","display":null,"kind":"use","excerpt":"For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0,","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fe0fe70909341737","tex":"\\K^n","aliases":"K^n K","display":null,"kind":"use","excerpt":"For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0,","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-473d859b7a40335f","tex":"\\K^{p\\times n}","aliases":"K^p× n K times","display":null,"kind":"use","excerpt":"For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0,","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3ff8bbec26df05be","tex":"K^\\perp","aliases":"K^⊥ perp","display":"K^\\perp","kind":"use","excerpt":"ntitative exactness] The following are equivalent: item ran D is closed. item There is C≥0 with normx-qx≤ CnormDx for all x. item There is C≥0 such that every x has some exact y with normx-y≤ CnormDx. When the range is closed, the least admissible constant is the operator norm of the inverse of D:K^⊥→ran D. In the zero operator case that inverse ha","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-01259f2441a815dd","tex":"K_{\\mathfrak H}","aliases":"K_ H mathfrak","display":"K_{\\mathfrak H}","kind":"use","excerpt":"Monotonicity under grammar extension. [Admissible expansion] Suppose every valid description in G translates to a valid description in H with the same semantic group. Then every G-realizable group is H-realizable. If every translated description has length at most f(lvert drvert), then K_ H(G)≤ f(K_ G(G)) whenever K_ G(G) is finite. Translate a sho","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9817d114fe256b80","tex":"k_0","aliases":"k_0","display":"k_0","kind":"use","excerpt":"Let k_0=P^M_ker Bh_0 and h_*=h_0-k_0. Then Bh_*=Bh_0=Dx and h_*⊥ker B in M. If a different initial correction is chosen, it differs from h_0 by an element of ker B. Its orthogonal-complement component is therefore again h_*. This proves that the construction is independent of the feasibility witness.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b783228933c35b11","tex":"k_i","aliases":"k_i","display":"k_i","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b35f8ec2c58ce5a0","tex":"k_n","aliases":"k_n","display":"k_n","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-264918da85881789","tex":"\\mathcal L","aliases":"L mathcal","display":"\\mathcal L","kind":"use","excerpt":"chapBanach repair and constructed split coordinates An actual complete candidate space. Let X_n,Y_n be Banach spaces over a complete nontrivially normed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formal","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-acac86c0e609ca90","tex":"l","aliases":"l","display":"l","kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-42ada035ee0b4eaf","tex":"L^3","aliases":"L^3","display":"L^3","kind":"use","excerpt":"The tails require a bound independent of sufficiently fine discretization. Since N≥ mL/2, the inequalities f(x)≤ x^-4 and g(x)≤ x^-2 give frac1m∑_n>Nf(n/m) ≤ m^3∑_n>Nn^-4≤fracm^33N^3 ≤frac83L^3, frac1m∑_n>Ng(n/m) ≤ m∑_n>Nn^-2≤frac mN≤frac2L. The integral tails are at most 1/(3L^3) and 1/L. Given an error tolerance, choose L first to control both di","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-64a393c37e7bfedd","tex":"L_0","aliases":"L_0","display":"L_0","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b3f55c1a3fc262cb","tex":"L_{0,N}","aliases":"L_0,N","display":"L_{0,N}","kind":"use","excerpt":"Now Q_Nr=L_0,Nx_N, so the support assertion applies to the restored remainder. Since S_Nr=r-Q_Nr, it gives the reconstruction equation. Finally, if two residuals have identical details at every scale, the finite telescope makes their S_N reconstructions equal for every N. At any fixed coordinate choose N with k+1<2^N. The reconstruction equation th","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4808d25c7c0c8b6f","tex":"l_b","aliases":"l_b","display":"l_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ed4117db0a118834","tex":"l_c","aliases":"l_c","display":"l_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0672c5d4c101de32","tex":"l_d","aliases":"l_d","display":"l_d","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1459f22abbadb5f5","tex":"l_g","aliases":"l_g","display":"l_g","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a00df4a1263286e2","tex":"l_i","aliases":"l_i","display":"l_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-834fb94f0edd7b4c","tex":"L_j","aliases":"L_j","display":"L_j","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a260dece327539f2","tex":"L_{j+1}","aliases":"L_j+1","display":"L_{j+1}","kind":"use","excerpt":"Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1e5f817a12a89680","tex":"L_{j+1,N}","aliases":"L_j+1,N","display":"L_{j+1,N}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c119997d7f38f38c","tex":"L_{j,0}","aliases":"L_j,0","display":"L_{j,0}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ce13126dfae5eb88","tex":"L_{j,N}","aliases":"L_j,N","display":"L_{j,N}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6ad58f1d489a5ca5","tex":"L_{j,N+1}","aliases":"L_j,N+1","display":"L_{j,N+1}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-06367b05ddac23e5","tex":"l_l","aliases":"l_l","display":"l_l","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a47d9642859cb109","tex":"L_M","aliases":"L_M","display":"L_M","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7e66b64743d52741","tex":"l_m","aliases":"l_m","display":"l_m","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c91dabfc8b700b3b","tex":"L_N","aliases":"L_N","display":"L_N","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a113110bb27da5fa","tex":"l_n","aliases":"l_n","display":"l_n","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b08b1edd5a95d60d","tex":"l_t","aliases":"l_t","display":"l_t","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4bd606a0f9ec9dda","tex":"l_w","aliases":"l_w","display":"l_w","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-abd805d5765e70d2","tex":"\\lceil","aliases":"lceil","display":"\\lceil","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-316e91d8c0b468a8","tex":"\\lfloor","aliases":"lfloor","display":null,"kind":"use","excerpt":")=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a mod","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-381a1c27b7e4a3e2","tex":"\\lim","aliases":"lim","display":"\\lim","kind":"use","excerpt":"Continuity of d implies d(P_Af)=lim_m r_m=0. Protection holds before taking a limit: pA=0 gives pf^(m+1)=pf^(m), hence pf^(m)=pf for every m. Continuity of p gives p(P_Af)=pf. This explains why continuity of the observable belongs to this analytic statement.","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-538b1d43a9783a43","tex":"\\lim_k","aliases":"lim_k lim","display":"\\lim_k","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ccb8a376eaddc6e8","tex":"\\lim_m","aliases":"lim_m lim","display":"\\lim_m","kind":"use","excerpt":"Continuity of d implies d(P_Af)=lim_m r_m=0. Protection holds before taking a limit: pA=0 gives pf^(m+1)=pf^(m), hence pf^(m)=pf for every m. Continuity of p gives p(P_Af)=pf. This explains why continuity of the observable belongs to this analytic statement.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-76d061defd0810fd","tex":"\\lim_{M\\to\\infty}","aliases":"lim_M→∞ lim to infty","display":"\\lim_{M\\to\\infty}","kind":"use","excerpt":"sidual minimum exists [Attainment and shape of the budget profile] For a bounded linear map between real Hilbert spaces, the infimum defining β(M) is attained for every M≥0. The profile is nonincreasing and convex. For 0≤ M_1≤ M_2, 0≤β(M_1)-β(M_2)≤norm T(M_2-M_1). Writing C=overlineran T, b=P_Cr, and q=normr-b, one has lim_M→∞β(M)=q. The equality β","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-53f3164e2d42bc85","tex":"\\lim_n","aliases":"lim_n lim","display":"\\lim_n","kind":"use","excerpt":"Now assume condition 2. To prove closedness, take an arbitrary sequence r_n∈ran D with r_n→ r in R. Choose x_n with Dx_n=r_n and put u_n=x_n-qx_n. Then u_n∈ K^⊥ and Du_n=r_n. Applying the estimate to u_n-u_m gives normu_n-u_m≤ Cnormr_n-r_m. Thus (u_n) is Cauchy in the complete space K^⊥. Let its limit be u. Continuity gives Du=lim_n r_n=r, so r∈ran","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e5c19d75887aa542","tex":"\\liminf","aliases":"liminf","display":"\\liminf","kind":"use","excerpt":"The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 b","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bef3746ed99e451b","tex":"\\liminf_k","aliases":"liminf_k liminf","display":"\\liminf_k","kind":"use","excerpt":"The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 b","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-45e90aa85790529d","tex":"\\linewidth","aliases":"linewidth","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bfbb661daa93662b","tex":"\\log","aliases":"log","display":"\\log","kind":"use","excerpt":"Solution. The sampled value is η_ϕ(k)=2^-k. Recovering the original index gives k=log_2(n+1) on the sampled stages, so the value is still 1/(n+1). The schedule preserves vanishing but changes the meaning of the horizontal axis. No faster original process has been proved. Proposition~[source reference] is a convergence equivalence under a monotone c","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-356d34f32f81e480","tex":"\\log_2","aliases":"log_2 log","display":"\\log_2","kind":"use","excerpt":"Solution. The sampled value is η_ϕ(k)=2^-k. Recovering the original index gives k=log_2(n+1) on the sampled stages, so the value is still 1/(n+1). The schedule preserves vanishing but changes the meaning of the horizontal axis. No faster original process has been proved. Proposition~[source reference] is a convergence equivalence under a monotone c","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-768071a8b8aba296","tex":"m^{-1}","aliases":"m^-1","display":"m^{-1}","kind":"use","excerpt":"infinite problem, lim_M→∞Mβ_T,w(M)=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniform","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e615f08a31cd0f9c","tex":"m^{-1/2}","aliases":"m^-1/2","display":"m^{-1/2}","kind":"use","excerpt":"Thus the correction norm grows on the order of sqrt m, the residual norm decays on the order of m^-1/2, and the penalty α A_m=A_m/m^2 tends to zero. The smooth correction avoids the hard discontinuity at a prefix boundary, but it still approaches the forbidden constant sequence coordinate by coordinate.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2d9b06f7801188ec","tex":"m^{-2}","aliases":"m^-2","display":"m^{-2}","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-be3882d8931e393d","tex":"M^2","aliases":"M^2","display":"M^2","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-068897646ca3bf5a","tex":"m^2","aliases":"m^2","display":"m^2","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9155222104384cb9","tex":"m^3","aliases":"m^3","display":"m^3","kind":"use","excerpt":"For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-af37a2ae9e4efb41","tex":"m^4","aliases":"m^4","display":"m^4","kind":"use","excerpt":"[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bb04371f38e2158f","tex":"M_{21}","aliases":"M_21","display":"M_{21}","kind":"use","excerpt":"For two consecutive compatible changes (S_1,C_1), (S_2,C_2), with compatible choices of sections at the three ends, their mismatches satisfy M_21=S_2M_1+M_2C_1. If dx_j→0 in norm for a fixed change and fixed sections, the repair discrepancy tends to zero. First DS=Cd sends ker d into ker D, and DM=DSA-DBC=CdA-C=C-C=0. Thus both terms in the propose","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bc4c24fa42db60db","tex":"m_a","aliases":"m_a","display":"m_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0f5cef98cfcf15fe","tex":"M_c","aliases":"M_c","display":"M_c","kind":"use","excerpt":"sectionA two-coordinate model in which the routes differ Use E=F=R^2, write candidates as (u,r), and take d(u,r)=D(u,r)=r, A(r)=B(r)=(0,r). The exact space is the horizontal axis and both repairs are P(u,r)=(u,0). For a real parameter c, set S_c(u,r)=(u+cr,r), C=I. The defect square commutes because the second coordinate is unchanged. The mismatch ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f1ff17ffbc6328a0","tex":"M_{c_1}","aliases":"M_c_1","display":"M_{c_1}","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4c6336cd396007b7","tex":"M_{c_2}","aliases":"M_c_2","display":"M_{c_2}","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4a4cfce379dd77c2","tex":"M_{c_2+c_1}","aliases":"M_c_2+c_1","display":"M_{c_2+c_1}","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-df92634da3a3e068","tex":"M_i","aliases":"M_i","display":"M_i","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-82c895a6150460c2","tex":"m_i","aliases":"m_i","display":"m_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bf2227566ef624f5","tex":"M_j","aliases":"M_j","display":"M_j","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3cc0ed3324bf0e59","tex":"M_m","aliases":"M_m","display":"M_m","kind":"use","excerpt":"Step 3: move from regularization parameters to arbitrary budgets. Set M_m=sqrtA_m. Every positive coordinate m^2/(m^2+n^2) increases strictly with m, so A_m, and hence M_m, increases strictly. Since A_m/m→ c=π/4>0, these budgets tend to infinity and (fracM_m+1M_m)^2 =fracA_m+1/(m+1)A_m/mfracm+1m⟶1. Proposition~[source reference] identifies β(M_m)=s","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3ba0c3b06de090fa","tex":"M_{m+1}","aliases":"M_m+1","display":"M_{m+1}","kind":"use","excerpt":"Step 3: move from regularization parameters to arbitrary budgets. Set M_m=sqrtA_m. Every positive coordinate m^2/(m^2+n^2) increases strictly with m, so A_m, and hence M_m, increases strictly. Since A_m/m→ c=π/4>0, these budgets tend to infinity and (fracM_m+1M_m)^2 =fracA_m+1/(m+1)A_m/mfracm+1m⟶1. Proposition~[source reference] identifies β(M_m)=s","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ea260e96669ea599","tex":"m_N","aliases":"m_N","display":"m_N","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c1f72b92c8bb119d","tex":"m_N^2","aliases":"m_N^2","display":"m_N^2","kind":"use","excerpt":"Suppose these energies are bounded, and let L=lim_n m_n^2. Given ϵ>0, choose N with L-m_N^2<ϵ^2. For m≥ n≥ N, norma_m-a_n^2=m_m^2-m_n^2≤ L-m_N^2<ϵ^2. Thus (a_n) is Cauchy and has a norm limit a_*. For each fixed n, all terms from that index onward belong to the closed set A_n, so a_*∈ A_n. This places a_* in their entire intersection.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-19afb20993a1d995","tex":"m_r","aliases":"m_r","display":"m_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6f3ae7d6b37bdd61","tex":"m_t","aliases":"m_t","display":"m_t","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4ff979fe78820d20","tex":"M_v","aliases":"M_v","display":"M_v","kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cc2f5f4bfec5fb41","tex":"M_\\alpha","aliases":"M_α alpha","display":"M_\\alpha","kind":"use","excerpt":"Parameter monotonicity and the fixed-budget problem. [Regularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing ","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f7520c75e0aae3a9","tex":"\\Match","aliases":"Match","display":null,"kind":"use","excerpt":"[Strict gluing] Under [source reference], restriction induces an equivalence Γ(F)≃Match(F,(U_i)). Every fixed matching family has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an ex","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e481658e9c6ce30a","tex":"\\max","aliases":"max","display":"\\max","kind":"use","excerpt":"ctions, two reindexed maps out of X are eventually equal exactly when they induce the same realized function. Surjective adjacent bonding maps are a sufficient condition for surjective point projections. Realized equality follows from eventual equality by the preceding theorem, so only its converse remains. Suppose |f|=|g|. Fix the output level n, ","count":15},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bae859577c210728","tex":"\\max_{0\\le i\\le n}","aliases":"max_0≤ i≤ n max le le","display":"\\max_{0\\le i\\le n}","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7be0073546f6c028","tex":"\\max_i","aliases":"max_i max","display":"\\max_i","kind":"use","excerpt":"The maps are linear. On the finite vertex product, normR_pz_∞≤(max_inormT_i^-1)norm z, normx_o≤norm x_∞. These bounds also hold on the subspaces K and ker d, proving continuity of both directions. No completeness is needed for these finite bounds.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e3e0c903e06afb04","tex":"\\max_{\\norm z\\le1}","aliases":"max_norm z≤1 max norm le","display":null,"kind":"use","excerpt":"A dual witness can rule out every correction in the budget. [A sharp budget certificate] For every M≥0, β_T,r(M)= max_norm z≤1(⟨ r,z⟩-MnormT^*z). Thus any single z with norm z≤1 and ⟨ r,z⟩-MnormT^*z>ε certifies that no correction of norm at most M has residual at most ε. For every norm a≤ M and norm z≤1, the adjoint identity and Cauchy--Schwarz giv","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3f9db2ae5742966b","tex":"\\midrule","aliases":"midrule","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b75f3a902c5f7e60","tex":"n^{-1}","aliases":"n^-1","display":"n^{-1}","kind":"use","excerpt":"A small residual far from exactness. On ℓ^2(N), index the standard basis from one and define De_n=n^-1e_n. The map is bounded and injective, so its exact space is zero. Yet norme_n=1, normDe_n=1/n⟶0. No finite C can bound dist(e_n,ker D) by CnormDe_n. The range is not closed. For example finite truncations of (1/n)_n≥1 belong to the range and conve","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c0769f9271add232","tex":"n^{-2}","aliases":"n^-2","display":"n^{-2}","kind":"use","excerpt":"Finally, for K≥ m, both omitted tails consist of nonnegative terms. Bound the first by m^4∑_n>Kn^-4 and the second by ∑_n>Kn^-2. The same upper integral comparisons give the two certified remainders in [source reference]. No finite truncation is substituted for an equality of infinite energies. writtenWR10","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-076d94003a73a0df","tex":"n^{-3}","aliases":"n^-3","display":"n^{-3}","kind":"use","excerpt":"Solution. The noise term is at most 1/(2n), but the solver term is at most one. Thus the displayed tolerance does not provide the required guarantee. The scalar counterexample T=0, r=0, a_n=1 meets it and does not approach the exact optimizer. Replacing it by n^-3 makes the solver term at most 1/n, so the triangle inequality and the assumed vanishi","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-55d7cf56037490a0","tex":"n^{-4}","aliases":"n^-4","display":"n^{-4}","kind":"use","excerpt":"Finally, for K≥ m, both omitted tails consist of nonnegative terms. Bound the first by m^4∑_n>Kn^-4 and the second by ∑_n>Kn^-2. The same upper integral comparisons give the two certified remainders in [source reference]. No finite truncation is substituted for an equality of infinite energies. writtenWR10","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-20cba99e4019fcab","tex":"N^2","aliases":"N^2","display":"N^2","kind":"use","excerpt":"Escape is not an accident of this particular algorithm. Suppose an arbitrary protected candidate (u,1) satisfies normd(u,1)≤ε. On its first N coordinates, ∑_n=1^N|u_n+1|^2 ≤ N^2∑_n=1^Nfrac|u_n+1|^2n^2 ≤ N^2ε^2. The reverse triangle inequality in R^N gives norm u≥sqrt N-Nε (N≥1). If a sequence of protected candidates has residual tending to zero, th","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6ac2b834a040f058","tex":"\\N^2","aliases":"N^2 N mathbb","display":"{\\mathbb N}^2","kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩),","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-12dcb271d25c745a","tex":"n^2","aliases":"n^2","display":"n^2","kind":"use","excerpt":"Unbounded gaps cause a different issue. With ϕ(n)=n^2, the bounded residual family r_k=1 has block sums 2n+1. These are unbounded. Therefore block accumulation cannot be a map from every bounded residual family to a bounded sampled residual family in this example. The constant residual is not claimed to be the differential of a bounded candidate on","count":24},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8cb4842d183cfba8","tex":"N^3","aliases":"N^3","display":"N^3","kind":"use","excerpt":"The tails require a bound independent of sufficiently fine discretization. Since N≥ mL/2, the inequalities f(x)≤ x^-4 and g(x)≤ x^-2 give frac1m∑_n>Nf(n/m) ≤ m^3∑_n>Nn^-4≤fracm^33N^3 ≤frac83L^3, frac1m∑_n>Ng(n/m) ≤ m∑_n>Nn^-2≤frac mN≤frac2L. The integral tails are at most 1/(3L^3) and 1/L. Given an error tolerance, choose L first to control both di","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ac5e944c84a2e799","tex":"n^4","aliases":"n^4","display":"n^4","kind":"use","excerpt":"fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and w=(1/n)∈ℓ^2, hence the residual sequence i","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b18687907767a3b7","tex":"\\N_{>0}","aliases":"N_>0 N","display":null,"kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-99f3c225ef6f9c62","tex":"n_c","aliases":"n_c","display":"n_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b9588331116f694e","tex":"n_e","aliases":"n_e","display":"n_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ae50884201122a48","tex":"n_i","aliases":"n_i","display":"n_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cf9404d3f534ee7e","tex":"n_k","aliases":"n_k","display":"n_k","kind":"use","excerpt":"The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 b","count":15},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-35b602543437bbe8","tex":"N_n","aliases":"N_n","display":"N_n","kind":"use","excerpt":"Protected information beyond the seed. When the target grows, its new kernel directions are independent information. Define N_n(f)=(I-s_nρ^Y_n)f_n+1, (p_Lf)_0=f_0, (p_Lf)_n+1=N_n(f). Each N_n(f) lands in kerρ^Y_n. The observable p_L retains both the seed and every new output component, relative to the chosen sections.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aaec29026bff78cd","tex":"n_o","aliases":"n_o","display":"n_o","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-26511a3b0a02612d","tex":"N_p","aliases":"N_p","display":"N_p","kind":"use","excerpt":"Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5670a09f4b5b441b","tex":"N_p^{-1}","aliases":"N_p^-1","display":"N_p^{-1}","kind":"use","excerpt":"dist(w,ker d)≤ Cnormdw_∞ for every vertex family w. Fix z∈ K^⊥ and take w=R_pz. Every exact family e has e_o∈ K, and root evaluation has norm at most one. Therefore norm z=dist(z,K)≤normz-e_o≤normR_pz-e_∞. Taking the infimum over e∈ker d is legitimate without assuming that this global nearest point is attained. It gives norm z≤ CnormdR_pz_∞ ≤ Cnorm","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bc7de739b4277782","tex":"\\neqfin","aliases":"neqfin ≠^* ne","display":"{\\ne^{*}}","kind":"use","excerpt":"[Torsion-image obstruction] If A≠^* B, every homomorphism G_A→ G_B has image contained in D. Consequently neither group embeds in the other, and neither is a quotient of the other. A homomorphism preserves torsion and induces a map Q→ Q on the quotients, necessarily multiplication by a rational r. If r=0, its image lies in D.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-64824291a60cddb8","tex":"\\newline","aliases":"newline","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c4694f2e93d5c4e7","tex":"O","aliases":"O","display":"O","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-218f4a32e8adb732","tex":"o_a","aliases":"o_a","display":"o_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5e3f9866da12f022","tex":"o_f","aliases":"o_f","display":"o_f","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7fc03b5db3831c26","tex":"o_i","aliases":"o_i","display":"o_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d1a47c863ac7fc79","tex":"o_p","aliases":"o_p","display":"o_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-490c00130e07d0d8","tex":"\\mathfrak o_W","aliases":"o_W mathfrak","display":"\\mathfrak o_W","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0bb121817ecaf82e","tex":"o_z","aliases":"o_z","display":"o_z","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4761badfcfad5d29","tex":"\\one_{A_C}","aliases":"one_A_C 1_A_C one mathbf","display":"{\\mathbf 1}_{A_C}","kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩), 1_A_C(2⟨ j,k⟩+1) · = 1_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd position","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-af8729ce8553c931","tex":"\\one_{B_j}","aliases":"one_B_j 1_B_j one mathbf","display":"{\\mathbf 1}_{B_j}","kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩), 1_A_C(2⟨ j,k⟩+1) · = 1_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd position","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e49bb6af86c95a09","tex":"\\one_C","aliases":"one_C 1_C one mathbf","display":"{\\mathbf 1}_C","kind":"use","excerpt":"Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩), 1_A_C(2⟨ j,k⟩+1) · = 1_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd position","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ca3ea4943ff504c7","tex":"\\one_{S_A}","aliases":"one_S_A 1_S_A one mathbf","display":"{\\mathbf 1}_{S_A}","kind":"use","excerpt":"[One cocycle value reads one coordinate] For every prime p, c_A(1/p,(p-1)/p)=- 1_S_A(p)e_p. The support bound leaves only the coordinate p. At that coordinate both fractional lifts are zero, and the lift of their sum, which is one, is 1_S_A(p). Dropping the cocycle is therefore not a harmless simplification. The direct product operation on Q× D rem","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-54037e9a059bedc5","tex":"\\overline B","aliases":"overline B overline","display":"\\overline B","kind":"use","excerpt":"[The floor detects closure] For every b∈ B, 0≤ν_B(Ω)≤normΩ-b. Moreover, ν_B(Ω)=0 Longleftrightarrow Ω∈overline B. If B is closed, this is equivalent to Ω∈ B. The set of candidate distances is nonempty because 0∈ B, bounded below by zero, and contains the finite value normΩ. Its infimum is therefore a finite nonnegative real, and the defining lower-","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f5ab84da5f379ea9","tex":"\\overline{\\mathcal F}","aliases":"overline F overline mathcal","display":"\\overline{\\mathcal F}","kind":"use","excerpt":"[Density criterion for norm recovery] Suppose sup_NnormQ_N≤ K<∞. Then the norm recoverable vectors are exactly overline F. In particular all vectors are norm recoverable if and only if the finite core is dense. The nested identities first imply that the kernels increase: if Q_My=0 and N≥ M, then Q_Ny=Q_NQ_My=0. Thus the finite core really is an inc","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e4b45cd66b9c8f4e","tex":"\\overline L","aliases":"overline L overline","display":"\\overline L","kind":"use","excerpt":"Equivalently, if r-r'∈ d(ker p), then L(r-r')=0, so L(r)=L(r'). Hence overline L([r])=L(r) is a well-defined homomorphism on R/d(ker p). Its value on [dx] is nonzero. This supplies the quotient interpretation of the same contradiction, without assuming that every nonzero quotient class can be detected by a map into one fixed preferred coefficient g","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9d8a1921f2a37e0a","tex":"\\overline M","aliases":"overline M overline","display":"\\overline M","kind":"use","excerpt":"All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4f9e74405a47101e","tex":"\\overline M_\\lambda","aliases":"overline M_λ overline lambda","display":"\\overline M_\\lambda","kind":"use","excerpt":"All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1a02ef2c814691f5","tex":"\\overline R","aliases":"overline R overline","display":"\\overline R","kind":"use","excerpt":"chapMultiscale residuals and the limits of reconstruction A split coarsener retains its missing detail. Let C:R→overline R and J:overline R→ R be linear maps with CJ=I. Put H=I-JC. Then r=JCr+Hr, CH=0, H^2=H. The two coordinates (Cr,Hr) retain all information. The coarse coordinate alone generally does not.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-553c7221d52abe12","tex":"\\overline{d(\\ker p)}","aliases":"overlined(ker p) overline ker","display":"\\overline{d(\\ker p)}","kind":"use","excerpt":"The bridge back to exactification. Proposition~[source reference] and Proposition~[source reference] fit together closely. If corrections produce a norm-convergent, protected path with vanishing residual, the actual quotient class vanishes. If only the corrected residuals tend to zero, all that follows is membership in overlined(ker p).","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cd341e70441d1406","tex":"\\overline{p_i}","aliases":"overlinep_i overline","display":"\\overline{p_i}","kind":"use","excerpt":"Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-67b35b32b305e3c7","tex":"\\overline{\\ran T}","aliases":"overlineran T overline ran","display":null,"kind":"use","excerpt":"An immediate contrapositive is useful. If r∈overlineran T∖ran T, then every sequence with Ta_n→ r satisfies norma_n→∞. Otherwise infinitely many terms would lie in one bounded ball, contradicting the theorem along that subsequence. This recovers the escape mechanism of Section~[source reference] without using the diagonal formula.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4af27a350f0c4ed1","tex":"\\overline{S(2nB_X)}","aliases":"overlineS(2nB_X) overline","display":"\\overline{S(2nB_X)}","kind":"use","excerpt":"mma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆overlineS(nB_X). Here B(y_0,ρ) is an open ball. If normh<ρ, both y_0+h and y_0 belong to this closure. Choose approximating image sequences for both. Subtracting their preimages shows h∈overlineS(2nB_X).","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6e36290a91c7877b","tex":"\\overline{S(B_X)}","aliases":"overlineS(B_X) overline","display":"\\overline{S(B_X)}","kind":"use","excerpt":"ere are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆overlineS(nB_X). Here B(y_0,ρ) is an open ball. If normh<ρ, both y_0+h and y_0 belong to this closure. Choose approximating image sequences for both. Subtracting their preimages shows h∈overlineS(2nB_X). Scaling by 2n gives a constant c=ρ/(2n)>0 with B(0,c)⊆overlineS(B_X). This only gives approximate ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-efa8224a327a3d68","tex":"\\overline{\\langle u\\rangle}","aliases":"overline⟨ u⟩ overline langle rangle","display":"\\overline{\\langle u\\rangle}","kind":"use","excerpt":"A single topological generator after completion. [Topological cyclicity] Let u=(1,1,1,…)∈ P. Then overline⟨ u⟩=P. In particular, trank(P)=1. A basic open set prescribes a tuple on finitely many distinct prime coordinates. The Chinese remainder theorem supplies an integer k with the specified residues. The element ku belongs to the given open set. T","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c33eb003dc4eec09","tex":"\\mathcal P","aliases":"P mathcal","display":"\\mathcal P","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8c8e7aefa6cd14a4","tex":"P^2","aliases":"P^2","display":"P^2","kind":"use","excerpt":"Set P=I-A∂. Then ∂ P=∂-∂ A∂=0, P^2=I-2A∂+A(∂ A)∂=I-A∂=P. The expression 2A∂ abbreviates addition of the same operator twice and uses no division by two. Every exact f is fixed because ∂ f=0. Since (Ar)_0=0, the seed of Pf is f_0. Finally f-Pf=A∂ f, and the bound on A gives the asserted residual estimate. The denominators above are justified exactly","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-79867b3b5190754f","tex":"p^2","aliases":"p^2","display":"p^2","kind":"use","excerpt":"For H=S_3, these elements are the identity and the three transpositions. There are exactly two orbit classes. The identity cocycle has automorphism group S_3, which is noncommutative. A nonidentity involution gives a coherent cocycle whose regular twisted section equation x=px has no solution. The only nontrivial cocycle relation is p^2=1. At a sin","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-26f9b4dce778eaea","tex":"P^M_{\\ker B}","aliases":"P^M_ker B ker","display":"P^M_{\\ker B}","kind":"use","excerpt":"Let k_0=P^M_ker Bh_0 and h_*=h_0-k_0. Then Bh_*=Bh_0=Dx and h_*⊥ker B in M. If a different initial correction is chosen, it differs from h_0 by an element of ker B. Its orthogonal-complement component is therefore again h_*. This proves that the construction is independent of the feasibility witness.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-db6b2ac0be28aba6","tex":"p_0","aliases":"p_0","display":"p_0","kind":"use","excerpt":"Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ S","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a39d2ca036172648","tex":"p_1","aliases":"p_1","display":"p_1","kind":"use","excerpt":"Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ S","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d5c8ee73677dc4ea","tex":"p_2","aliases":"p_2","display":"p_2","kind":"use","excerpt":"Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ S","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f5af782990149da7","tex":"p_{2n}","aliases":"p_2n","display":"p_{2n}","kind":"use","excerpt":"The information retained by a marked element. For the distinguished element u_A=s_A(1), the coordinate criterion simplifies to pmid_G_Au_A Longleftrightarrow p∉ S_A. Consequently, for each n, p_2nmid_G_Au_A · Longleftrightarrow n∈ A, p_2n+1mid_G_Au_A · Longleftrightarrow n∉ A. Exactly one root equation is solvable. This binary pattern is the select","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-06182cfbeb52a59d","tex":"p_{2n+1}","aliases":"p_2n+1","display":"p_{2n+1}","kind":"use","excerpt":"The information retained by a marked element. For the distinguished element u_A=s_A(1), the coordinate criterion simplifies to pmid_G_Au_A Longleftrightarrow p∉ S_A. Consequently, for each n, p_2nmid_G_Au_A · Longleftrightarrow n∈ A, p_2n+1mid_G_Au_A · Longleftrightarrow n∉ A. Exactly one root equation is solvable. This binary pattern is the select","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-77a74a36475e5eda","tex":"p_{2n+a_n}","aliases":"p_2n+a_n","display":"p_{2n+a_n}","kind":"use","excerpt":"Choosing a rational line. Let A⊆ N, write a_n= 1_A(n), and define S_A={p_2n+a_n:n∈ N}. Exactly one prime from each pair is selected. Both S_A and its complement are infinite. Let u_A∈ P be its indicator: (u_A)_p=1, · p∈ S_A,","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7c2443102c181d52","tex":"p_3","aliases":"p_3","display":"p_3","kind":"use","excerpt":"Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ S","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e82429366fa8634c","tex":"P_a","aliases":"P_a","display":"P_a","kind":"use","excerpt":"2. A residual chart can change without changing the defect. Let d:R^2→R be d(u,r)=r. For a∈R, let A_a(r)=(ar,r). Compute the exact repair P_a, compare it with P_b, and decide when they commute.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e7d1a3be90d27945","tex":"p_A","aliases":"p_A","display":"p_A","kind":"use","excerpt":"Preservation changes the admissible arrows. One response to collapse is to restrict arrows to those satisfying a declared preservation contract. For objects equipped with maps p_A:A→ S, a morphism can be required to satisfy p_Bf=p_A. Identities and composition preserve this equation. Unless p_A=0, the zero arrow is not admissible. In other applicat","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-93f05451e0c22e43","tex":"P_A^2","aliases":"P_A^2","display":"P_A^2","kind":"use","excerpt":"n allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, every iterate equals f, hence P_Af=f. Its image is contained in ker d and it fixes that kernel, proving surjectivity onto its stated image and P_A^2=P_","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a5c35e870d97f9c4","tex":"P_B","aliases":"P_B","display":"P_B","kind":"use","excerpt":"[The split defect chart] The maps χ_A:E≃_L K× R, χ_A(x)=(P_Ax,dx), χ_A^-1(u,r)=u+Ar are inverse continuous linear equivalences. In these coordinates, P_A sends (u,r) to (u,0). If B is another right inverse, then A-B takes values in K, and P_Bx-P_Ax=(A-B)dx. The projections commute if and only if they are equal, and they are equal if and only if A=B","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5e3a3e3d08311a9a","tex":"P_b","aliases":"P_b","display":"P_b","kind":"use","excerpt":"2. A residual chart can change without changing the defect. Let d:R^2→R be d(u,r)=r. For a∈R, let A_a(r)=(ar,r). Compute the exact repair P_a, compare it with P_b, and decide when they commute.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d072d8979775d522","tex":"p_B","aliases":"p_B","display":"p_B","kind":"use","excerpt":"Preservation changes the admissible arrows. One response to collapse is to restrict arrows to those satisfying a declared preservation contract. For objects equipped with maps p_A:A→ S, a morphism can be required to satisfy p_Bf=p_A. Identities and composition preserve this equation. Unless p_A=0, the zero arrow is not admissible. In other applicat","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d04253126b67c473","tex":"p_b","aliases":"p_b","display":"p_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-24b885024a7a3479","tex":"p_{b_j}","aliases":"p_b_j","display":"p_{b_j}","kind":"use","excerpt":"Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finit","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0b67a2140e399c4e","tex":"P_C","aliases":"P_C","display":"P_C","kind":"use","excerpt":"ubsectionAt every finite budget the residual minimum exists [Attainment and shape of the budget profile] For a bounded linear map between real Hilbert spaces, the infimum defining β(M) is attained for every M≥0. The profile is nonincreasing and convex. For 0≤ M_1≤ M_2, 0≤β(M_1)-β(M_2)≤norm T(M_2-M_1). Writing C=overlineran T, b=P_Cr, and q=normr-b,","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4b003ef2553d3052","tex":"p_c","aliases":"p_c","display":"p_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-afaca47f8ebd7708","tex":"P_{D',C'}","aliases":"P_D',C'","display":"P_{D',C'}","kind":"use","excerpt":"ally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤max(ε_D,ε_C). Applying Theorem~[source reference] to the joint operators, and using the actual two-stage identity of Theorem~[source reference], gives normP_D,C-P_D',C' ≤max(G,G')max","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a5e7737709b11f98","tex":"P_{D,C}","aliases":"P_D,C","display":"P_{D,C}","kind":"use","excerpt":"ally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤max(ε_D,ε_C). Applying Theorem~[source reference] to the joint operators, and using the actual two-stage identity of Theorem~[source reference], gives normP_D,C-P_D',C' ≤max(G,G')max","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b9e70c157342a064","tex":"P_H","aliases":"P_H","display":"P_H","kind":"use","excerpt":"The adjoint in g is the restricted adjoint on H, not the unrestricted ambient adjoint. Equivalently, T^*v=P_Hd^*v. Thus the correction certificate must measure the part of the normal equation that belongs to the permitted correction space. A numerical program that forgets this projection can incorrectly reject an exact protected optimizer.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-237e766422ac8978","tex":"P_{H_0}","aliases":"P_H_0","display":"P_{H_0}","kind":"use","excerpt":"To verify weak convergence against h∈ H_0, choose a finite basis combination h_J close to h. Then |⟨ a_n_k-a,h⟩| ≤ |⟨ a_n_k-a,h_J⟩| +2Mnormh-h_J. For a prescribed tolerance, choose h_J first to control the second term and then k to control the first by finitely many coordinate limits. For a general h in the ambient space, write h=P_H_0h+h_⊥. The or","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-429e7f1087bafa6e","tex":"P_i","aliases":"P_i","display":"P_i","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6158f77c3833268f","tex":"p_i","aliases":"p_i","display":"p_i","kind":"use","excerpt":"default. In a declared linear model, the compatibility equations become a bounded operator. Closed range, holonomy, and conditioning then determine what small residuals actually guarantee. Graph views, holonomy, and reconstruction. Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-26af5e2e76634d6f","tex":"P_j","aliases":"P_j","display":"P_j","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c5ca4f817be17bab","tex":"P_K","aliases":"P_K","display":"P_K","kind":"use","excerpt":"chapClosed range is the stability boundary Projection exists before a residual estimate. Let E be a real Hilbert space and R a real Banach space. Let D:E→ R be continuous linear. The kernel K=ker D is closed, so its orthogonal projection q=P_K exists~[source reference].","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f5e4a53d4c21d2f3","tex":"P_{\\ker D}","aliases":"P_ker D ker","display":"P_{\\ker D}","kind":"use","excerpt":"ready exact under permitted correction and the least correction is zero, so the optimal gain is zero. sourceKS56 The relevant stability boundary is therefore D(ker p), not automatically D(E). Restricting permitted changes can turn a well-conditioned unconstrained problem into an obstructed or poorly conditioned protected one. The nearest protected ","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-78436549de06ca5c","tex":"P_{\\ker D'}","aliases":"P_ker D' ker","display":"P_{\\ker D'}","kind":"use","excerpt":"chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8e5399faab2aeb9e","tex":"p_L^2","aliases":"p_L^2","display":"p_L^2","kind":"use","excerpt":"g p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zero of p_L is the identity, and every later coordinate applies B_n, so p_L^2=p_L and normp_L≤1+SB_Y. In particular the","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fe1b7ade23d1e9a1","tex":"P_M","aliases":"P_M","display":"P_M","kind":"use","excerpt":"), write ℓ=P_Lx and let Px be the projection of ℓ onto ker B inside L. Then Px∈ M, u=x-ℓ∈ L^⊥, and v=ℓ-Px∈ L is orthogonal to M. Since M⊆ L, the vector u is also orthogonal to M. Their sum x-Px=u+v is therefore orthogonal to M. A vector in M with that error is the unique orthogonal projection of x, so P=P_M.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-346cf456a77b1ce5","tex":"p_n","aliases":"p_n","display":"p_n","kind":"use","excerpt":"A purely finite-state boundary. Consider a restricted grammar in which a finite automaton reads the unary word 1^n and selects the factor C_p_n exactly when it accepts.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ff4cecd8ff7a194d","tex":"p_o","aliases":"p_o","display":"p_o","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b0692a1c334c82d7","tex":"P_{\\rm anchor}","aliases":"P_rm anchor rm","display":"P_{\\rm anchor}","kind":"use","excerpt":"Preserving only the feasible projection of the anchor gives a different answer. Orthogonally project (2,1) to (2,0), then reconstruct along the tree. The result is P_rm anchorx=((2,0),(2,0),(2,0)). It is the unique exact family preserving the first component of the original anchor. It does not preserve its second component, and makes no claim to do","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1d00be659bf68fc1","tex":"P_{\\rm nearest}","aliases":"P_rm nearest rm","display":"P_{\\rm nearest}","kind":"use","excerpt":"For global nearest-point repair, declare the Euclidean product norm on (R^2)^3. The closest member of [source reference] uses the mean of the three first coordinates, namely three: P_rm nearestx=((3,0),(3,0),(3,0)). To verify this without appealing only to a general theorem, write the squared error at parameter a: (2-a)^2+(4-a)^2+(3-a)^2+1^2+(-1)^2","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b9cbd5c85d37e703","tex":"P_{T(\\{a:\\norm a\\le M\\})}","aliases":"P_T({a:norm a≤ M}) norm le","display":null,"kind":"use","excerpt":"The nearest reachable image y_M=P_T({a:norm a≤ M})r is unique, so the best residual vector r-y_M is unique as well. Corrections mapping to y_M can differ by vectors in ker T. Among those corrections there is a unique least norm correction. These two uniqueness statements should not be confused with injectivity of T.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-26a9ec94734fbef0","tex":"p_v","aliases":"p_v","display":"p_v","kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aacabfcb3613b64e","tex":"p_W","aliases":"p_W","display":"p_W","kind":"use","excerpt":"Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6634b2cbd59c3d1f","tex":"p_w","aliases":"p_w","display":"p_w","kind":"use","excerpt":"a coordinate preserving equivalence Γ(B_a,b)≃ B(a,b). Under a transition cover, matching local bridge sections glue to a unique global bridge with both prescribed projections unchanged. We must first check that the displayed fibers really form a view system. Let m∈ B_a,b(w) and f:v→ w. Naturality of p and coherence of a give p_v(f_M^*m)=f_F^*p_w(m)","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a9cab240c51624bb","tex":"\\mathcal P_\\infty","aliases":"P_∞ mathcal infty","display":"\\mathcal P_\\infty","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7eb28ed8746e003c","tex":"\\path","aliases":"path","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-74be1e0574f29a94","tex":"\\operatorname{Point}","aliases":"Point operatorname","display":"\\operatorname{Point}","kind":"use","excerpt":"The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-28b6406ccead341b","tex":"q'","aliases":"q'","display":"q'","kind":"use","excerpt":"chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.","count":23},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-34d43de423598186","tex":"q^{\\mathsf T}","aliases":"q^ T mathsf","display":"q^{\\mathsf T}","kind":"use","excerpt":"[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-be871d9e7245f709","tex":"q^2","aliases":"q^2","display":"q^2","kind":"use","excerpt":"[Exact orthogonal repair] For every x∈ E, D(qx)=0, q^2=q, normqx≤norm x, and qx is the unique closest point to x in K. Boundedness of D makes K=ker D closed. Lemma~[source reference] constructs q=P_K and gives the orthogonal decomposition x=qx+(x-qx). The first term belongs to K, so Dqx=0. Projection fixes every point of K, giving q(qx)=qx.","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-142e69180c7d1697","tex":"q^j","aliases":"q^j","display":"q^j","kind":"use","excerpt":"nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4a3a203830edc472","tex":"q^m","aliases":"q^m","display":"q^m","kind":"use","excerpt":"the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j ≤fracnorm A1-qnormr_m. Since normr_m≤ q^mnormdf tends to zero, thi","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1928485a8127cf57","tex":"q^N","aliases":"q^N","display":"q^N","kind":"use","excerpt":"[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar s","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-102bb645d8547d75","tex":"q^n","aliases":"q^n","display":"q^n","kind":"use","excerpt":"[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar s","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f9d31c937ed8d6a4","tex":"Q_0","aliases":"Q_0","display":"Q_0","kind":"use","excerpt":"Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b8bbb3a7ebd17a45","tex":"q_0","aliases":"q_0","display":"q_0","kind":"use","excerpt":"[Exact motion of the line] For every real t, norm(q_t-q_0)x^2=fract^21+t^2norm x^2, normq_t-q_0=frac|t|sqrt1+t^2. In the small perturbation interface, kernel transport sends (0,b) to (-tb,b). Orthogonal projection instead sends it to (-tb/(1+t^2),b/(1+t^2)). The vector v_t=(-t,1) spans ker D_t and has squared norm 1+t^2>0. Projection onto this line","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0306f7ccb079ea2e","tex":"Q_{2k}","aliases":"Q_2k","display":"Q_{2k}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8dfb1493f0f1cd4b","tex":"q_{2k}","aliases":"q_2k","display":"q_{2k}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aeeb84c561790b3a","tex":"Q_{2k+1}","aliases":"Q_2k+1","display":"Q_{2k+1}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4c1a7de923feba0c","tex":"q_{2k+1}","aliases":"q_2k+1","display":"q_{2k+1}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-345638a01538c367","tex":"q_f","aliases":"q_f","display":"q_f","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2fdfdde7570c43b1","tex":"q_i","aliases":"q_i","display":"q_i","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a05d3c840afe4a0d","tex":"Q_j","aliases":"Q_j","display":"Q_j","kind":"use","excerpt":"Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-01982e51585e4e66","tex":"Q_{j+1}","aliases":"Q_j+1","display":"Q_{j+1}","kind":"use","excerpt":"Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a9af1147d28b7af5","tex":"Q_k","aliases":"Q_k","display":"Q_k","kind":"use","excerpt":"[Variable-rate exactification] If Q_n→0 and ∑_n C_nQ_n<∞, then (f_n) has an exact limit and dist(f_n,x)≤ω(f_0)∑_k=n^∞C_kQ_k. For constant q_n=q∈[0,1) and constant C_n=C, the sharper restarted estimate is dist(f_n,x)≤fracC1-q ω(f_n). The estimate ω(f_n)≤ Q_nω(f_0) follows by induction. At n=0 it is equality. If it holds at n, nonnegativity of q_n pe","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cba1a0b5de3eccd8","tex":"Q_M","aliases":"Q_M","display":"Q_M","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4a77ffe628ff771a","tex":"q_n","aliases":"q_n","display":"q_n","kind":"use","excerpt":"Variable rates and a movement budget. Let q_n,C_n≥0, and define the prefix product Q_0=1, Q_n+1=q_nQ_n. Assume ω(f_n+1)≤ q_nω(f_n), dist(f_n,f_n+1)≤ C_nω(f_n).","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bf313ea6d2a73b75","tex":"Q_{n+1}","aliases":"Q_n+1","display":"Q_{n+1}","kind":"use","excerpt":"Variable rates and a movement budget. Let q_n,C_n≥0, and define the prefix product Q_0=1, Q_n+1=q_nQ_n. Assume ω(f_n+1)≤ q_nω(f_n), dist(f_n,f_n+1)≤ C_nω(f_n).","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5266bc3b7e8fe0cc","tex":"Q_N^2","aliases":"Q_N^2","display":"Q_N^2","kind":"use","excerpt":"Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e11eab697b8866ee","tex":"q_s","aliases":"q_s","display":"q_s","kind":"use","excerpt":"-t · 1. The common right inverse r↦(r,0) supplies gain at most one for every t, and normD_s-D_t=|s-t|. The general theorem gives normq_s-q_t≤|s-t|.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-919224812081875b","tex":"q_t","aliases":"q_t","display":"q_t","kind":"use","excerpt":"An all-real worked model. Let D_t(x,y)=x+ty. Its kernel is the line {(-tb,b):b∈R}, and its orthogonal projector is q_t=frac11+t^2 t^2 · -t -t · 1. The common right inverse r↦(r,0) supplies gain at most one for every t, and normD_s-D_t=|s-t|. The general theorem gives normq_s-q_t≤|s-t|.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8c29958522a9d167","tex":"q_v^{-1}","aliases":"q_v^-1","display":"q_v^{-1}","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ee3e599e7035de50","tex":"Q_W","aliases":"Q_W","display":"Q_W","kind":"use","excerpt":"Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2ac44abe0fa0cba1","tex":"q_w","aliases":"q_w","display":"q_w","kind":"use","excerpt":"The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed o","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2eee74810f9c985d","tex":"q_z","aliases":"q_z","display":"q_z","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b53f9683364fb5fd","tex":"\\R","aliases":"R","display":null,"kind":"use","excerpt":"A growing example and the gap boundary. Use X_n=Y_n=R^n+1 with maximum norms. Let π_n delete the last coordinate, take ρ^X_n=tfrac12π_n, ρ^Y_n=π_n, and let s_n append zero. Then S=B_Y=1, B_X=1/2, and K=2.","count":46},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aabc6d88cbd006dc","tex":"R'","aliases":"R'","display":"R'","kind":"use","excerpt":"Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-94029690dcc6c9ab","tex":"r'","aliases":"r'","display":"r'","kind":"use","excerpt":"Equivalently, if r-r'∈ d(ker p), then L(r-r')=0, so L(r)=L(r'). Hence overline L([r])=L(r) is a well-defined homomorphism on R/d(ker p). Its value on [dx] is nonzero. This supplies the quotient interpretation of the same contradiction, without assuming that every nonzero quotient class can be detected by a map into one fixed preferred coefficient g","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aff4a4490ea52474","tex":"r^{-1}","aliases":"r^-1","display":"r^{-1}","kind":"use","excerpt":"Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The auto","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cd6a51568c71e4ad","tex":"\\R^2","aliases":"R^2 R","display":null,"kind":"use","excerpt":"A two-coordinate model in which the routes differ. Use E=F=R^2, write candidates as (u,r), and take d(u,r)=D(u,r)=r, A(r)=B(r)=(0,r). The exact space is the horizontal axis and both repairs are P(u,r)=(u,0). For a real parameter c, set S_c(u,r)=(u+cr,r), C=I. The defect square commutes because the second coordinate is unchanged. The mismatch is M_c","count":12},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-78d6b9680d84cb1c","tex":"\\R^3","aliases":"R^3 R","display":null,"kind":"use","excerpt":"Three hypotheses that cannot simply be dropped. Summable movement. In maximum-norm R^3, let f_n=(s,H_n,(n+1)^-1), H_n=∑_j=1^nfrac1j. The first coordinate is protected and the residual is the last coordinate. It tends to zero with the same ratios as the polynomial example. The movement equals 1/(n+1), however, and the unprotected coordinate diverges","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-72b3b4ee79240b62","tex":"\\R^4","aliases":"R^4 R","display":null,"kind":"use","excerpt":"Here the general protected construction is completely explicit. With candidate space E=R^4, defect d=ℓ, and protected observable p(r)=(r_0,r_1,r_2), the allowed direction is the last coordinate axis. The restricted operator d|_ker p has gain one. Without protection, the least gain for distance to kerℓ is 1/2, since normℓ=2. The different constants ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b0a9532805ec4d05","tex":"\\R^N","aliases":"R^N R","display":null,"kind":"use","excerpt":"Escape is not an accident of this particular algorithm. Suppose an arbitrary protected candidate (u,1) satisfies normd(u,1)≤ε. On its first N coordinates, ∑_n=1^N|u_n+1|^2 ≤ N^2∑_n=1^Nfrac|u_n+1|^2n^2 ≤ N^2ε^2. The reverse triangle inequality in R^N gives norm u≥sqrt N-Nε (N≥1). If a sequence of protected candidates has residual tending to zero, th","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b74c15fecc6efa87","tex":"\\R^{n+1}","aliases":"R^n+1 R","display":null,"kind":"use","excerpt":"A growing example and the gap boundary. Use X_n=Y_n=R^n+1 with maximum norms. Let π_n delete the last coordinate, take ρ^X_n=tfrac12π_n, ρ^Y_n=π_n, and let s_n append zero. Then S=B_Y=1, B_X=1/2, and K=2.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8d86c98f6680ae2f","tex":"\\R^{\\N_{>0}}","aliases":"R^N_>0 R N","display":null,"kind":"use","excerpt":"A coherent inverse-limit point still exists in a larger space. At each finite level, the protected exact equations have the unique solution (-1,…,-1;1)∈R^N⊕R. Truncation maps the solution at level N+1 to the one at level N. These finite solution sets are singletons, so their inverse limit is also a singleton. Its coordinate description is (-1,-1,…;","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9eb12de9c970740e","tex":"R_0","aliases":"R_0","display":"R_0","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a6be10a383c0ff11","tex":"r_0","aliases":"r_0","display":"r_0","kind":"use","excerpt":"nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0c95d3415f0ad112","tex":"r_{01}","aliases":"r_01","display":"r_{01}","kind":"use","excerpt":"0 · -1. For one chosen orientation of each edge, the residuals are r_01=x_1-x_0, r_12=x_2-x_1, r_20=Jx_0-x_2. The reverse transports are inverses, so this defines an inverse-consistent isometric connection. Exactness first forces x_1=x_0 and x_2=x_0, and the remaining edge then forces Jx_0=x_0. Hence every exact family is ((a,0),(a,0),(a,0)), a∈R. ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8996cf56d8c9d652","tex":"R_1","aliases":"R_1","display":"R_1","kind":"use","excerpt":"An equivalence between the moving kernels. [Explicit kernel transport] Under the hypotheses of Theorem~[source reference], J:ker D≃_Lker D', Jx=x-A'D'x, with inverse y↦ y-ADy. Moreover normJx-x≤fracaε1-aεnorm x. Let R_0=I-AD and R_1=I-A'D'. From DA=D'A'=I we obtain DR_0=0 and D'R_1=0. Each R_i fixes its corresponding kernel. Thus R_1 restricts to a","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7c77127f7fae6fe1","tex":"r_1","aliases":"r_1","display":"r_1","kind":"use","excerpt":"nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-64ec1404fdffc06e","tex":"r_{12}","aliases":"r_12","display":"r_{12}","kind":"use","excerpt":"0 · -1. For one chosen orientation of each edge, the residuals are r_01=x_1-x_0, r_12=x_2-x_1, r_20=Jx_0-x_2. The reverse transports are inverses, so this defines an inverse-consistent isometric connection. Exactness first forces x_1=x_0 and x_2=x_0, and the remaining edge then forces Jx_0=x_0. Hence every exact family is ((a,0),(a,0),(a,0)), a∈R. ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6d6362ab00b61fc6","tex":"r_2","aliases":"r_2","display":"r_2","kind":"use","excerpt":"nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-628aa4eb9fec0e5b","tex":"r_{20}","aliases":"r_20","display":"r_{20}","kind":"use","excerpt":"0 · -1. For one chosen orientation of each edge, the residuals are r_01=x_1-x_0, r_12=x_2-x_1, r_20=Jx_0-x_2. The reverse transports are inverses, so this defines an inverse-consistent isometric connection. Exactness first forces x_1=x_0 and x_2=x_0, and the remaining edge then forces Jx_0=x_0. Hence every exact family is ((a,0),(a,0),(a,0)), a∈R. ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b95bc35897dc3f06","tex":"r_{2n}","aliases":"r_2n","display":"r_{2n}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-77c3d448c4fd635b","tex":"r_{2n+1}","aliases":"r_2n+1","display":"r_{2n+1}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d5c7051b56820f95","tex":"r_3","aliases":"r_3","display":"r_3","kind":"use","excerpt":"nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4bdf782a5897d64d","tex":"r_b","aliases":"r_b","display":"r_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-567e1106095a8a61","tex":"r_c","aliases":"r_c","display":"r_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-00c00f8247bf728a","tex":"R_i","aliases":"R_i","display":"R_i","kind":"use","excerpt":"tween the moving kernels [Explicit kernel transport] Under the hypotheses of Theorem~[source reference], J:ker D≃_Lker D', Jx=x-A'D'x, with inverse y↦ y-ADy. Moreover normJx-x≤fracaε1-aεnorm x. Let R_0=I-AD and R_1=I-A'D'. From DA=D'A'=I we obtain DR_0=0 and D'R_1=0. Each R_i fixes its corresponding kernel. Thus R_1 restricts to a map from ker D to","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f231812574787070","tex":"r_i","aliases":"r_i","display":"r_i","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8505e2b9e062c506","tex":"r_{ij}","aliases":"r_ij","display":"r_{ij}","kind":"use","excerpt":"Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-35e4b1420672d392","tex":"r_j","aliases":"r_j","display":"r_j","kind":"use","excerpt":"a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-53e3e9d1c2a6131f","tex":"R_{j+1}","aliases":"R_j+1","display":"R_{j+1}","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2a1d688e5f76248c","tex":"r_{j+1}","aliases":"r_j+1","display":"r_{j+1}","kind":"use","excerpt":"r a nonzero residual r∈ Y, set λ=2normr/c. The vector r/λ has norm c/2, so [source reference] supplies z∈ B_X with normSz-r/λ≤ c/4. Then u=λ z satisfies normu≤ 2normr/c, normr-Su≤normr/2. If the residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-41b9e630b25443f1","tex":"R_{j+N}","aliases":"R_j+N","display":"R_{j+N}","kind":"use","excerpt":"Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-272cb3ca57c00592","tex":"r_m","aliases":"r_m","display":"r_m","kind":"use","excerpt":"belKS19 Starting from f^(0)=f, define f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2b9d5b78881f1c09","tex":"r_{m+1}","aliases":"r_m+1","display":"r_{m+1}","kind":"use","excerpt":"fine f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-48ae5eb670db3065","tex":"r_{m+j}","aliases":"r_m+j","display":"r_{m+j}","kind":"use","excerpt":"nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4aa81ec436e2173c","tex":"R_N","aliases":"R_N","display":"R_N","kind":"use","excerpt":"egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8d214535a6b8b511","tex":"r_N","aliases":"r_N","display":"r_N","kind":"use","excerpt":"residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages converges in X by completeness, to a vector x with the displayed norm bound. Continuity of S, together with the finite telescoping equation S(∑_j=0^N-1u_j)=y-r_N, ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a4ff961cd74d1869","tex":"r_n","aliases":"r_n","display":"r_n","kind":"use","excerpt":"A right inverse from target splittings. Suppose each target bonding map has a continuous linear section s_n:Y_n→ Y_n+1, ρ^Y_ns_n=I, norms_n≤ S. Assume the propagation gap SB_X<1, K=fracS1-SB_X. For r∈ R, recursively set (Ar)_0=0, (Ar)_n+1=s_n(r_n+(Ar)_nρ^X_n).","count":15},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fd828431f1a05a6f","tex":"r_o","aliases":"r_o","display":"r_o","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-10d3f7c1355c5340","tex":"r_r","aliases":"r_r","display":"r_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-adea9ba150e9752c","tex":"r_U","aliases":"r_U","display":"r_U","kind":"use","excerpt":"The exact sequence behind the criterion. The additive structures make restriction a homomorphism r_U:Γ(A)→Γ_U(A). Using the zero extension, or lift independence, o_U is also a homomorphism.","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c16fbef0f6898c20","tex":"r_v","aliases":"r_v","display":"r_v","kind":"use","excerpt":"Step 4: retain every morphism. Let c,d be global cocycles. Their canonical restrictions have identity overlap gauges. A morphism between those restrictions is consequently a family r_i satisfying r_j(v)=r_i(v) on each overlap. Assemble r_v from any containing patch. For an arrow choose a patch containing both endpoints. Its gauge equation proves d_","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6c74d6c0c951e9a8","tex":"r_w","aliases":"r_w","display":"r_w","kind":"use","excerpt":"Step 4: retain every morphism. Let c,d be global cocycles. Their canonical restrictions have identity overlap gauges. A morphism between those restrictions is consequently a family r_i satisfying r_j(v)=r_i(v) on each overlap. Assemble r_v from any containing patch. For an arrow choose a patch containing both endpoints. Its gauge equation proves d_","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e81875b79f13e65e","tex":"R_\\alpha","aliases":"R_α alpha","display":"R_\\alpha","kind":"use","excerpt":"et frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ca0ed80fd48b48de","tex":"R_\\gamma","aliases":"R_γ gamma","display":"R_\\gamma","kind":"use","excerpt":"a_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f5cfdb40005f62a6","tex":"r_\\infty","aliases":"r_∞ infty","display":"r_\\infty","kind":"use","excerpt":"orm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial coordinate has norm zero. If norm(Ar)_n≤ Knorm r_∞, submultiplicativity of operator norms gives ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3f0bb881a2e80b27","tex":"\\R_{\\ge0}^{\\infty}","aliases":"R_≥0^∞ R ge infty","display":null,"kind":"use","excerpt":"stances from A to C and from C to A is bounded by d_W^sym(A,B)+d_W^sym(B,C). Taking their maximum proves the symmetric triangle law. Zero diagonal and symmetry give reflexivity and symmetry of the zero-distance relation. The triangle inequality and nonnegativity give transitivity. sourceKS78 The formal source proves these laws directly in R_≥0^∞ an","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-512d411adf8ee1ff","tex":"\\ran","aliases":"ran","display":null,"kind":"use","excerpt":"ion gives norm x^2=normqx^2+normx-qx^2, hence normqx≤norm x. For any y∈ K, the vector qx-y lies in K, whereas x-qx lies in its orthogonal complement. Consequently normx-y^2=normx-qx^2+normqx-y^2≥normx-qx^2. Equality holds only when normqx-y=0, or y=qx. This proves nearest-point uniqueness without any closedness assumption on ran D. Closedness of th","count":39},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-517575feda8f300b","tex":"\\operatorname{rank}","aliases":"rank operatorname","display":"\\operatorname{rank}","kind":"use","excerpt":"is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the positive case is dim H=n-rankA P. Set T=A","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-87db0642ef898b44","tex":"\\rceil","aliases":"rceil","display":"\\rceil","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bf4c43a79117c6ae","tex":"\\operatorname{Reach}","aliases":"Reach operatorname","display":"\\operatorname{Reach}","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ea9d64aac68c5482","tex":"\\operatorname{Reach}_E","aliases":"Reach_E operatorname","display":"\\operatorname{Reach}_E","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-452acf9313788e80","tex":"\\rfloor","aliases":"rfloor","display":null,"kind":"use","excerpt":")=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a mod","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-46238367ef18be35","tex":"\\rightharpoonup","aliases":"rightharpoonup","display":"\\rightharpoonup","kind":"use","excerpt":"Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity giv","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dc5c5f4f378f97aa","tex":"\\mathcal S","aliases":"S mathcal","display":"\\mathcal S","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a43e0d8c06476226","tex":"S'","aliases":"S'","display":"S'","kind":"use","excerpt":"For the basis change B'=BC, the same ambient vector is represented by c'=C^-1c. Also S'=SC and G'=C^ TGC. Substitute these formulas into the new coordinate normal equation: q' · =((S')^ TS'+α G')c'-(S')^ Tr","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c7833b61651f298f","tex":"s^2","aliases":"s^2","display":"s^2","kind":"use","excerpt":"In the previous example, B=(1,1)^ T, G=2, and the proposal c=1/4 gives q=-1/4. Thus q^ TG^-1q=1/32, as required. Replacing B by sB changes q to sq and G to s^2G. The Euclidean norm of the coordinate covector can be made arbitrarily small or large by changing the nonzero scalar s. The actual correction, normal residual, and certified error do not ch","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-913791bf1e0ffd22","tex":"s_{2n}","aliases":"s_2n","display":"s_{2n}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cdc525abf7a300b5","tex":"S_3","aliases":"S_3","display":"S_3","kind":"use","excerpt":"A small noncommutative model. [A C_2 loop with S_3 coefficients] Take one view with a loop σ satisfying σ^2=1, and constant coefficients in a group H. A cocycle is determined by an element p∈ H whose square is the identity. Gauge equivalence is conjugacy, and its automorphisms form the centralizer of p.","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1dcf95b2a026d667","tex":"s_a","aliases":"s_a","display":"s_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-450f6a43de6765bd","tex":"S_B","aliases":"S_B","display":"S_B","kind":"use","excerpt":"ghtarrow A=^* B. When A=^* B, the two groups are equal as subgroups of the fixed product P. Suppose f:G_A→ G_B is an isomorphism. The image of u_A has infinite order. An isomorphism preserves the existence of a pth root for every prime p. Combining the exact signature [source reference] with Lemma~[source reference] for f(u_A) gives S_A=^* S_B. Sin","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fb3ff7699e0538a9","tex":"s_B","aliases":"s_B","display":"s_B","kind":"use","excerpt":"Conversely, suppose A=^* B. For every rational r, the vectors s_A(r) and s_B(r) differ only on the finitely many prime pairs where the selectors differ. Thus their difference belongs to D. The descriptions G_A=s_A( Q)+D and G_B=s_B( Q)+D define the same subset of P. The normal forms need not be identical. Their coordinate change is (r,d)⟼ (r,d+s_A(","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5df9b8f58b7161eb","tex":"s_b","aliases":"s_b","display":"s_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8ec8b0faab9c7fd8","tex":"S_c","aliases":"S_c","display":"S_c","kind":"use","excerpt":"A two-coordinate model in which the routes differ. Use E=F=R^2, write candidates as (u,r), and take d(u,r)=D(u,r)=r, A(r)=B(r)=(0,r). The exact space is the horizontal axis and both repairs are P(u,r)=(u,0). For a real parameter c, set S_c(u,r)=(u+cr,r), C=I. The defect square commutes because the second coordinate is unchanged. The mismatch is M_c","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a64b77f6851063e5","tex":"s_c","aliases":"s_c","display":"s_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ae27dbb473a126cc","tex":"S_{c_2}","aliases":"S_c_2","display":"S_{c_2}","kind":"use","excerpt":"Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7aa9c815c9ae3803","tex":"s_e","aliases":"s_e","display":"s_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1550303e0c5e431e","tex":"s_g","aliases":"s_g","display":"s_g","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e51f187cf7ad66bf","tex":"s_{HLw}","aliases":"s_HLw","display":"s_{HLw}","kind":"use","excerpt":"For the relative claim, pullback clearly sends an extension of s to an extension of H^*s on U'=H^-1U. Conversely suppose y extends H^*s. If w∈ U, the isomorphism w≅ HLw and isomorphism closure imply HLw∈ U. Hence Lw∈ U' and y_Lw=s_HLw. Both endpoints of t_w lie in U, so partial coherence gives (Ry)_w=F(t_w)s_HLw=s_w. Recovery therefore preserves th","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c8ea458b12d6729a","tex":"s_i","aliases":"s_i","display":"s_i","kind":"use","excerpt":"Let (U_i)_i∈ I be full patches. A matching family consists of s_i∈Γ_U_i(F) with equal values on every overlap. The relevant covering condition is for every f:v→ w, some U_i contains both v and w. Identities imply that this condition also covers every object.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-02cad8c82d44ed28","tex":"s_{i(v)}","aliases":"s_i(v)","display":"s_{i(v)}","kind":"use","excerpt":"rem[Strict gluing] Under [source reference], restriction induces an equivalence Γ(F)≃Match(F,(U_i)). Every fixed matching family has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-86ceeed3000b96d6","tex":"s_j","aliases":"s_j","display":"s_j","kind":"use","excerpt":"Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U_j, matching on the overlap U_i(v)∩ U_j gives x_v=(s_j)_v. t","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7b9f4293854c6f56","tex":"\\mathcal S_L","aliases":"S_L mathcal","display":"\\mathcal S_L","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ef445d6d03f47b90","tex":"s_n'","aliases":"s_n'","display":"s_n'","kind":"use","excerpt":"What is canonical. For fixed splitting and protected observable, the repair above is unique. Changing the splitting can change both the observable and the repaired answer. In the growing example, replace s_n(y)=(y,0) by s_n'(y)=(y,y_0). At the first positive level, repair of the identity becomes 1/2 · 0","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cc0051a685f15a7f","tex":"s_{n+1}","aliases":"s_n+1","display":"s_{n+1}","kind":"use","excerpt":"For surjectivity, fix y∈ Y_n and choose m≥max{s_n,s_n+1,n+1}. Since y∈ I_n,m, there is w∈ X_m with ρ_n,m(w)=y. Put z=ρ_n+1,m(w). Stabilization at level n+1 gives z∈ Y_n+1, and composition gives ρ_nz=y.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-efc33e545a71481e","tex":"s_s","aliases":"s_s","display":"s_s","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5dce6ea50bf9e787","tex":"s_u","aliases":"s_u","display":"s_u","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1d641eb69668a366","tex":"s_v","aliases":"s_v","display":"s_v","kind":"use","excerpt":"is neutral. Equality of orbit classes means there exists an actual based gauge b with b· c^q=1. At an arrow f:v→ w, this equation is b_vq_v^-1c_ff^*(q_w)[f^*(b_w)]^-1=1. Multiply on the left by q_vb_v^-1. Homomorphic preservation of products then gives c_ff^*(q_wb_w^-1)=q_vb_v^-1. Therefore x_v=q_vb_v^-1 is a twisted coherent family. On U, b_v=1 an","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7735513ecabe8703","tex":"s_w","aliases":"s_w","display":"s_w","kind":"use","excerpt":"For the relative claim, pullback clearly sends an extension of s to an extension of H^*s on U'=H^-1U. Conversely suppose y extends H^*s. If w∈ U, the isomorphism w≅ HLw and isomorphism closure imply HLw∈ U. Hence Lw∈ U' and y_Lw=s_HLw. Both endpoints of t_w lie in U, so partial coherence gives (Ry)_w=F(t_w)s_HLw=s_w. Recovery therefore preserves th","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f8039ab5de609643","tex":"S_\\phi","aliases":"S_ϕ phi","display":"S_\\phi","kind":"use","excerpt":"Actual tower resampling gives a defect map. Return to Banach towers with contractive source and target bonds. Let ϕ be a monotone schedule. Candidate sampling is (S_ϕ f)_n=f_ϕ(n). The sampled bonding map is the actual composite restriction between the selected levels. The residual map C_ϕ is not pointwise sampling of a scalar error score. It accumu","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-72398ff18d39e1f6","tex":"\\sec","aliases":"sec","display":"\\sec","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c3f001b6b563f398","tex":"\\sec^2","aliases":"sec^2 sec","display":"\\sec^2","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-87e5f1044d3860c9","tex":"\\operatorname{Set}","aliases":"Set operatorname","display":"\\operatorname{Set}","kind":"use","excerpt":"chapA genuine stack on the composition site The preceding theorem can be placed in a standard site, rather than called stack-like. We use the usual effective-descent meaning of a stack~[source reference]. The base of this site is the inclusion poset of full patches P(V)=(Set(ObV),⊆). This patch category is different from the original category of vi","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5fded382aa33f0f0","tex":"\\sin","aliases":"sin","display":"\\sin","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4ff996026bf09795","tex":"\\sin^2","aliases":"sin^2 sin","display":"\\sin^2","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-452a3350693f9f7b","tex":"\\operatorname{span}","aliases":"span operatorname","display":"\\operatorname{span}","kind":"use","excerpt":"In matrix form, the difference operator is D= -1 · 1 · 0 · 0 0 · -1 · 1 · 0 0 · 0 · -1 · 1 1 · 0 · 0 · -1 , ker D=span{(1,1,1,1)}. The explicit construction proves ran D=kerℓ, ℓ(r)=∑_i r_i. Thus ℓ is a complete feasibility test for this prescribed-difference problem. It annihilates every actual difference family, but takes a nonzero value on an inc","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d53379e798f70907","tex":"\\srank_{\\End(G_A)}","aliases":"srank_End(G_A) srank End operatorname operatorname","display":"{\\operatorname{srank}}_{{\\operatorname{End}}(G_A)}","kind":"use","excerpt":"Infinite seed rank before completion. [Seed-rank separation] For every A, srank_End(G_A)(G_A)=aleph_0. No finite family topologically generates G_A, even after arbitrary endomorphisms are admitted. Yet its completion has ordinary topological generating rank one. Each element s_A(r)+d has only finitely many nonzero coordinates outside S_A. A finite ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-310b249a090cc66d","tex":"\\Stab","aliases":"Stab","display":null,"kind":"use","excerpt":"Automorphisms and ambiguity are not the same question. The stabilizer of a cocycle is the group of families with a_v=c_f f^*(a_w)c_f^-1. If one gauge a:c→ d is known, all gauges from c to d are uniquely of the form ah, with h∈Stab(c). Indeed a^-1b stabilizes c for every other gauge b. This is the ordinary Hom torsor principle in a groupoid.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5f9d50c526ca5e23","tex":"\\Sub","aliases":"Sub operatorname","display":"{\\operatorname{Sub}}","kind":"use","excerpt":"[Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P( N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modul","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-848c3c3d231100c2","tex":"\\subsection","aliases":"subsection","display":null,"kind":"use","excerpt":"[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a240bb4eb2bf25a3","tex":"\\sup_k","aliases":"sup_k sup","display":"\\sup_k","kind":"use","excerpt":"uad normC_ϕ≤ m, ∂_ϕ S_ϕ=C_ϕ∂. Exact candidates remain exact under every monotone common resampling, even without a gap bound. If A and B are bounded right inverses for the fine and sampled differentials, then normP_BS_ϕ f-S_ϕ P_Af ≤(norm A+mnorm B)norm∂ f. For a bounded candidate family, sup_nnormf_ϕ(n)≤sup_knormf_k, proving the sampling bound. For","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-61335044b48ba968","tex":"\\sup_N","aliases":"sup_N sup","display":"\\sup_N","kind":"use","excerpt":"[Density criterion for norm recovery] Suppose sup_NnormQ_N≤ K<∞. Then the norm recoverable vectors are exactly overline F. In particular all vectors are norm recoverable if and only if the finite core is dense. The nested identities first imply that the kernels increase: if Q_My=0 and N≥ M, then Q_Ny=Q_NQ_My=0. Thus the finite core really is an inc","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f07ba46881cb0326","tex":"\\sup_{\\norm{z}\\le1}","aliases":"sup_normz≤1 sup norm le","display":null,"kind":"use","excerpt":"linear map between real Hilbert spaces. For fixed z∈ R, the functional h↦⟨ Th,z⟩ is bounded by normTnormznormh. The lemma therefore defines a unique vector T^*z∈ H with ⟨ Th,z⟩=⟨ h,T^*z⟩. Uniqueness of representatives proves that T^* is linear. The same bound proves normT^*z≤normTnormz. Conversely, normTh=sup_normz≤1|⟨ Th,z⟩| ≤normT^*normh, where t","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d0f4101430d2716c","tex":"\\sup_{n\\ge N}","aliases":"sup_n≥ N sup ge","display":"\\sup_{n\\ge N}","kind":"use","excerpt":"[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar s","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c6f5e71887472d54","tex":"\\mathcal T","aliases":"T mathcal","display":"\\mathcal T","kind":"use","excerpt":"For transitivity, suppose R covers W, and that for every U∈ R the pullback of a sieve T to U covers U. Fix a composable pair q in W. Its support S=|q| belongs to R. Since patches are full, q is also a composable pair in S. The covering condition for the pullback of T to S therefore puts the identity inclusion S⊆ S in that pullback. By the definitio","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-104f4e0a72089cc5","tex":"T^0","aliases":"T^0","display":"T^0","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8a06dd992beb5ab4","tex":"t^3","aliases":"t^3","display":"t^3","kind":"use","excerpt":"-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives e","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-450286f521499783","tex":"t^4","aliases":"t^4","display":"t^4","kind":"use","excerpt":"-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives e","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9ea7696d8ec4186b","tex":"T^j","aliases":"T^j","display":"T^j","kind":"use","excerpt":"Neumann inversion with both inverse identities. [A strict perturbation radius] Let W be a real Banach space and T:W→ W bounded linear, with normT≤θ<1 and θ≥0. Then Z=∑_j=0^∞T^j exists in operator norm and satisfies (I-T)Z=Z(I-T)=I, normZ≤frac11-θ. First recall why the space of bounded operators on W is complete. An operator-norm Cauchy sequence A_n","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-910092ae6df8d19a","tex":"T^n","aliases":"T^n","display":"T^n","kind":"use","excerpt":"The perturbed map is surjective, with D'A'=I, normA'≤fraca1-aε. Its exact projection satisfies normx-q'x≤fraca1-aεnormD'x, normq-q'≤fracaε1-aε. The map T=(D-D')A acts on the Banach target R and satisfies norm T≤ aε<1. The operator series Z=∑_n=0^∞ T^n converges in operator norm and obeys norm Z≤(1-aε)^-1. Multiplying its finite partial sums on eith","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-756b0e0aef75f3d4","tex":"T^{N+1}","aliases":"T^N+1","display":"T^{N+1}","kind":"use","excerpt":"1-aεnormD'x, normq-q'≤fracaε1-aε. The map T=(D-D')A acts on the Banach target R and satisfies norm T≤ aε<1. The operator series Z=∑_n=0^∞ T^n converges in operator norm and obeys norm Z≤(1-aε)^-1. Multiplying its finite partial sums on either side by I-T gives I-T^N+1. Passing to the operator-norm limit proves (I-T)Z=Z(I-T)=I. Appendix~[source refe","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4a9e32fb56cb7f6c","tex":"t_a","aliases":"t_a","display":"t_a","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d1367c12dada8011","tex":"t_b","aliases":"t_b","display":"t_b","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bd730f6c2c2e8897","tex":"t_d","aliases":"t_d","display":"t_d","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c78b7188b7b592c0","tex":"t_e","aliases":"t_e","display":"t_e","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c377758d754d2ccd","tex":"t_{Hv}","aliases":"t_Hv","display":"t_{Hv}","kind":"use","excerpt":"If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-84eac95fe9a2a4fb","tex":"T_i^{-1}","aliases":"T_i^-1","display":"T_i^{-1}","kind":"use","excerpt":"Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-95211c71e87133e2","tex":"T_j^{-1}","aliases":"T_j^-1","display":"T_j^{-1}","kind":"use","excerpt":"Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-733902ab0028bba1","tex":"t_m","aliases":"t_m","display":"t_m","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4505604ff161e9a8","tex":"T_N","aliases":"T_N","display":"T_N","kind":"use","excerpt":"Every finite operator is invertible, but the gains diverge. On R^N, let T_N=diag(1,1/2,…,1/N). It is invertible and normT_N^-1=N in the Euclidean norm. The upper bound follows by multiplying each coordinate by an integer at most N, and equality is attained at the last coordinate vector. Full finite rank therefore supplies no uniform gain as N→∞.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3aeff80b8378ac4b","tex":"T_N^{-1}","aliases":"T_N^-1","display":"T_N^{-1}","kind":"use","excerpt":"Every finite operator is invertible, but the gains diverge. On R^N, let T_N=diag(1,1/2,…,1/N). It is invertible and normT_N^-1=N in the Euclidean norm. The upper bound follows by multiplying each coordinate by an integer at most N, and equality is attained at the last coordinate vector. Full finite rank therefore supplies no uniform gain as N→∞.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dcee229e3aaf9526","tex":"t_o","aliases":"t_o","display":"t_o","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-81549140e4fc02bc","tex":"t_p","aliases":"t_p","display":"t_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dd4ff4382ddb04f0","tex":"T_{p_i}","aliases":"T_p_i","display":"T_{p_i}","kind":"use","excerpt":"Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aaffcbbdb2c59acf","tex":"t_r","aliases":"t_r","display":"t_r","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-387970c466cde48f","tex":"t_v^{-1}","aliases":"t_v^-1","display":"t_v^{-1}","kind":"use","excerpt":"If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4ab411d9663afe22","tex":"t_w","aliases":"t_w","display":"t_w","kind":"use","excerpt":"If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1 · =t_v^-1c_ff^*(q_w) [f^*(q_w)]^-1f^*(t_w) =t_v^-1c_ff^*(t_w)=c_f^t. Thus changing the raw lift leaves the based orbit unchanged.","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b04a44cd7b0fa729","tex":"t_{w'}","aliases":"t_w'","display":"t_{w'}","kind":"use","excerpt":"ce and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Contravariance of F a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b00cb8546f0d5d6a","tex":"t_w^{-1}","aliases":"t_w^-1","display":"t_w^{-1}","kind":"use","excerpt":"If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1 · =t_v^-1c_ff^*(q_w) [f^*(q_w)]^-1f^*(t_w) =t_v^-1c_ff^*(t_w)=c_f^t. Thus changing the raw lift leaves the based orbit unchanged.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-75177911206476aa","tex":"\\tan","aliases":"tan","display":"\\tan","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3000c3bd47e8007c","tex":"\\toprule","aliases":"toprule","display":null,"kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d97b932ee16cbf3d","tex":"\\Tor","aliases":"Tor operatorname","display":"{\\operatorname{Tor}}","kind":"use","excerpt":"The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with t","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ec61c8382b2d3710","tex":"U'","aliases":"U'","display":"U'","kind":"use","excerpt":"[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ec7d40a789aa2d83","tex":"u'","aliases":"u'","display":"u'","kind":"use","excerpt":"For any h∈ H, the vector h-ℓ(h)v/ℓ(v) belongs to L. Pairing it with v therefore gives ⟨ h,v⟩=fracℓ(h)ℓ(v)normv^2. Consequently u=ℓ(v)v/normv^2 represents ℓ. If u' also represents it, pairing the difference of the two formulas with h=u-u' gives normu-u'^2=0, hence uniqueness.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6cf86c8bfef7f92f","tex":"U'_x","aliases":"U'_x","display":"U'_x","kind":"use","excerpt":"Changing lifts to U'_x=(-1)^a(x)U_x changes the phase to c'(x,y)=c(x,y)+a(x)+a(y)-a(x+y). No such change makes c' vanish identically. The antisymmetric part is unchanged by every scalar change of lift. In this model it is κ(x,y)=c(x,y)-c(y,x) =x_2y_1-y_2x_1. This is a directly observable obstruction certificate for the model. Its nonzero value at (","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c3517972a5630870","tex":"u^2","aliases":"u^2","display":"u^2","kind":"use","excerpt":"[The two-stage repair is the joint projection] The operator P is the orthogonal projection onto M, equivalently onto the kernel of Δ=(D,C). If u=x-P_Lx and v=P_Lx-Px, then u⊥ v, normx-Px^2=norm u^2+norm v^2. The subspace L=ker D is closed, hence Hilbert with the inherited inner product. The restriction B=C|_L is continuous, so its kernel is a close","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-305733ca1f18133f","tex":"U_{(p,q)}","aliases":"U_(p,q)","display":"U_{(p,q)}","kind":"use","excerpt":"The phase is visible in actual matrices. Over R, define X=0 · 1 1 · 0, Z=1 · 0 0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-924579a9b7ed33d9","tex":"u_B","aliases":"u_B","display":"u_B","kind":"use","excerpt":"[Marked classification] An isomorphism (G_A,u_A)≅(G_B,u_B) exists if and only if A=B. A marked isomorphism preserves the exact two-prime root test [source reference]--[source reference] at every index. Therefore every bit agrees. The converse is the identity map.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-652da1418f4adb94","tex":"U_{e_1}","aliases":"U_e_1","display":"U_{e_1}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a7dee972cf3034ae","tex":"U_{e_2}","aliases":"U_e_2","display":"U_{e_2}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ddf54f09d9f17b33","tex":"U_i","aliases":"U_i","display":"U_i","kind":"use","excerpt":"Let (U_i)_i∈ I be full patches. A matching family consists of s_i∈Γ_U_i(F) with equal values on every overlap. The relevant covering condition is for every f:v→ w, some U_i contains both v and w. Identities imply that this condition also covers every object.","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2231f0f6ca7a3024","tex":"U_{i(v)}","aliases":"U_i(v)","display":"U_{i(v)}","kind":"use","excerpt":"has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U_j, matching on the overlap U_i(v)","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8e0ecb4b8f25eabc","tex":"U_j","aliases":"U_j","display":"U_j","kind":"use","excerpt":"_i)). Every fixed matching family has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8a3e27ecb912ad75","tex":"u_j","aliases":"u_j","display":"u_j","kind":"use","excerpt":"r a nonzero residual r∈ Y, set λ=2normr/c. The vector r/λ has norm c/2, so [source reference] supplies z∈ B_X with normSz-r/λ≤ c/4. Then u=λ z satisfies normu≤ 2normr/c, normr-Su≤normr/2. If the residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-25121c20156ac918","tex":"U_m","aliases":"U_m","display":"U_m","kind":"use","excerpt":"τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels each interior value once with each sign. The remaining terms are U_mx_i_m-U_0x_i_0=T_qx_i_m-x_i_0, pro","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8d86088ae482ad75","tex":"u_m","aliases":"u_m","display":"u_m","kind":"use","excerpt":"Now assume condition 2. To prove closedness, take an arbitrary sequence r_n∈ran D with r_n→ r in R. Choose x_n with Dx_n=r_n and put u_n=x_n-qx_n. Then u_n∈ K^⊥ and Du_n=r_n. Applying the estimate to u_n-u_m gives normu_n-u_m≤ Cnormr_n-r_m. Thus (u_n) is Cauchy in the complete space K^⊥. Let its limit be u. Continuity gives Du=lim_n r_n=r, so r∈ran","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6dc51ddfc0590754","tex":"u_n","aliases":"u_n","display":"u_n","kind":"use","excerpt":"Now assume condition 2. To prove closedness, take an arbitrary sequence r_n∈ran D with r_n→ r in R. Choose x_n with Dx_n=r_n and put u_n=x_n-qx_n. Then u_n∈ K^⊥ and Du_n=r_n. Applying the estimate to u_n-u_m gives normu_n-u_m≤ Cnormr_n-r_m. Thus (u_n) is Cauchy in the complete space K^⊥. Let its limit be u. Continuity gives Du=lim_n r_n=r, so r∈ran","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6ae21ddbb7e958a2","tex":"U_t","aliases":"U_t","display":"U_t","kind":"use","excerpt":"Small holonomy can still be poorly conditioned. In R^2, the rational rotation U_t=frac11+t^2 1-t^2 · -2t 2t · 1-t^2 satisfies norm(U_t-I)z^2=frac4t^21+t^2norm z^2. For t≠0 its fixed space is zero and its residual gain is sqrt1+t^2/(2|t|), which diverges as t→0. At t=0, the entire fiber is fixed. Therefore small loop discrepancy does not supply a un","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-07ca84b023e0812f","tex":"U_x","aliases":"U_x","display":"U_x","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-931251200eb190e2","tex":"U_{x+y}","aliases":"U_x+y","display":"U_{x+y}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cf906e4bc6834869","tex":"U_{x+y+z}","aliases":"U_x+y+z","display":"U_{x+y+z}","kind":"use","excerpt":"Associativity of three lifted matrices requires the same total sign on both parenthesizations. Equation~[source reference] is exactly that requirement: (U_xU_y)U_z =(-1)^c(x,y)+c(x+y,z)U_x+y+z, U_x(U_yU_z) =(-1)^c(y,z)+c(x,y+z)U_x+y+z. Thus a nonzero phase is not automatically a failed associativity law. It can be the data that make the composition","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6df8bf7a9378fe0f","tex":"U_y","aliases":"U_y","display":"U_y","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-810dc694a1dc15a1","tex":"U_z","aliases":"U_z","display":"U_z","kind":"use","excerpt":"Associativity of three lifted matrices requires the same total sign on both parenthesizations. Equation~[source reference] is exactly that requirement: (U_xU_y)U_z =(-1)^c(x,y)+c(x+y,z)U_x+y+z, U_x(U_yU_z) =(-1)^c(y,z)+c(x,y+z)U_x+y+z. Thus a nonzero phase is not automatically a failed associativity law. It can be the data that make the composition","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a8b2a9721419c0b6","tex":"v^{\\mathsf T}","aliases":"v^ T mathsf","display":"v^{\\mathsf T}","kind":"use","excerpt":"-1 · 0 · -1. The ordinary cycle block satisfies A^ TA=3I-11^ T. It is zero on the constant line and equals 3I on its orthogonal complement. The reflected block satisfies B^ TB=I+vv^ T, v=(1,-1,1)^ T. Since norm v^2=3, this matrix has eigenvalue four in the direction of v and eigenvalue one on its orthogonal complement. In particular it is positive ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ff15a0be2cf55ac8","tex":"v^{(m)}","aliases":"v^(m)","display":"v^{(m)}","kind":"use","excerpt":"[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6042bf2f76597e2e","tex":"v^{(m)}_n","aliases":"v^(m)_n","display":"v^{(m)}_n","kind":"use","excerpt":"The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cafb796619c588dd","tex":"v^{-1}","aliases":"v^-1","display":"v^{-1}","kind":"use","excerpt":"Conversely, if x is a twisted extension of the same prescribed data, define h_v=(x_0)_v^-1x_v. On U its value is one. For every arrow, h_v · =[c_f f^*((x_0)_w)]^-1[c_f f^*(x_w)]","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-16120d386ed4594b","tex":"v^2","aliases":"v^2","display":"v^2","kind":"use","excerpt":"[The two-stage repair is the joint projection] The operator P is the orthogonal projection onto M, equivalently onto the kernel of Δ=(D,C). If u=x-P_Lx and v=P_Lx-Px, then u⊥ v, normx-Px^2=norm u^2+norm v^2. The subspace L=ker D is closed, hence Hilbert with the inherited inner product. The restriction B=C|_L is continuous, so its kernel is a close","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9c8da2b101444307","tex":"v^\\perp","aliases":"v^⊥ perp","display":"v^\\perp","kind":"use","excerpt":"These two identities show that the least nonnegative gain for the full defect-to-distance estimate in this Euclidean model is one. The lower bound is attained, for example, on a candidate with zero first components and second-component vector (1,1,0), which lies in v^⊥. The squared input distance and squared residual are both two.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aea73305506ee49e","tex":"v_n","aliases":"v_n","display":"v_n","kind":"use","excerpt":"Why a vanishing normal residual is not enough. Let T=0 on R, let r=0, and take α_n=n^-2 for n≥1. The exact optimizer is zero for every such n. The fixed proposal a_n=1 instead has v_n=0, g_n=α_n⟶0, J_α_n(a_n)=α_n⟶0, absa_n-a_α_n=1. Thus the fit residual, the normal residual, and even the absolute objective value can all tend to zero without converg","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7be15d086b46f651","tex":"v_t","aliases":"v_t","display":"v_t","kind":"use","excerpt":"[Exact motion of the line] For every real t, norm(q_t-q_0)x^2=fract^21+t^2norm x^2, normq_t-q_0=frac|t|sqrt1+t^2. In the small perturbation interface, kernel transport sends (0,b) to (-tb,b). Orthogonal projection instead sends it to (-tb/(1+t^2),b/(1+t^2)). The vector v_t=(-t,1) spans ker D_t and has squared norm 1+t^2>0. Projection onto this line","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c3a76197560b59e1","tex":"v_\\alpha","aliases":"v_α alpha","display":"v_\\alpha","kind":"use","excerpt":"Parameter monotonicity and the fixed-budget problem. [Regularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-27aee75937940a51","tex":"\\varinjlim","aliases":"varinjlim","display":null,"kind":"use","excerpt":"Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2e2fc35414ba9c6b","tex":"\\varinjlim_N","aliases":"varinjlim_N varinjlim","display":null,"kind":"use","excerpt":"Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6c5fe5f64f631d7a","tex":"\\varprojlim","aliases":"varprojlim","display":null,"kind":"use","excerpt":"The distinction is easiest to express in terms of an embedding G⟶ varprojlim G/N. Residual finiteness makes this map injective. The inverse limit includes all compatible tuples. The image of the original G can nevertheless be a proper dense subgroup. Forgetting that image can forget the object even while retaining enough coordinates to distinguish ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d8963e7f8e8a07d5","tex":"\\varprojlim_N","aliases":"varprojlim_N varprojlim","display":null,"kind":"use","excerpt":"Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bb5783d1e38c64af","tex":"\\vec E","aliases":"vec E vec","display":"\\vec E","kind":"use","excerpt":"The candidate and residual spaces are finite products with maximum norms: E=∏_i∈ VH, R=∏_(i,j)∈vec EH, (dx)_ij=τ_ijx_j-x_i. Both orientations of an edge are retained. If M bounds the norms of all measured edge maps, then norm d≤ M+1. The empty edge case has the usual zero residual.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c30dc958e19e4c13","tex":"\\W","aliases":"W","display":null,"kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b99d2e5f37ae7e17","tex":"w'","aliases":"w'","display":"w'","kind":"use","excerpt":"[ Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2bbe13df91f37646","tex":"w^{-1}","aliases":"w^-1","display":"w^{-1}","kind":"use","excerpt":"Conversely, if x is a twisted extension of the same prescribed data, define h_v=(x_0)_v^-1x_v. On U its value is one. For every arrow, h_v · =[c_f f^*((x_0)_w)]^-1[c_f f^*(x_w)] · =f^*((x_0)_w)^-1c_f^-1c_f f^*(x_w) · =f^*((x_0)_w^-1x_w)=f^*(h_w). The cancellation uses inverse multiplication in its actual order. No commuting factors have been moved ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-561fc36c5e029d32","tex":"w^2","aliases":"w^2","display":"w^2","kind":"use","excerpt":"usR, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n has operator norm one. Since d(u,t)=Tu+tw, normd(u,t)≤norm u+|t|norm w ≤sqrt1+norm w^2 norm(u,t)_E. Thus bo","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bd5ef9a5ba57a080","tex":"W_A","aliases":"W_A","display":"W_A","kind":"use","excerpt":"0, · p∉ S_A. Because u_A has infinite support, q(u_A)≠0. We can therefore define a rational line and its preimage by W_A= Q q(u_A), G_A=q^-1(W_A). This definition produces an actual group for every set A. It does not claim an effective method for obtaining the selector bits of an arbitrary A.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8945947ef031a855","tex":"w_n","aliases":"w_n","display":"w_n","kind":"use","excerpt":"The obstruction class is nonzero while its floor is zero. The admissible correction space is ker p=ℓ^2⊕{0}, with correction image B=d(ker p)=ran T ={r∈ℓ^2:(nr_n)_n≥1∈ℓ^2}. It contains every finitely supported sequence, so it is dense in R. But w∉ B, because (nw_n)_n=(1,1,…) is not square summable. Therefore [w]≠0 in R/B, ν_B(w)=0. This is the middl","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d1655f35bb316a56","tex":"\\widehat D","aliases":"widehat D widehat","display":"\\widehat D","kind":"use","excerpt":"The complete finite-quotient system. [Profinite completion of the direct sum] The canonical embedding of D into P identifies P with widehat D. Let f:D→ F be a homomorphism to a finite group. The image of a prime-order factor C_p is trivial unless p divides lvert Frvert. Thus f vanishes on all but finitely many prime components and factors through a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1fc32b24cc091ee1","tex":"\\widehat f","aliases":"widehat f widehat","display":"\\widehat f","kind":"use","excerpt":"[Reflection within the paired-prime family] For a homomorphism f:G_A→ G_B, the induced completion map widehat f is an isomorphism if and only if f is an isomorphism. One implication follows from functoriality. Conversely, suppose widehat f is an isomorphism. By Theorem~[source reference], its coordinate action is multiplication by a diagonal sequen","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-644e773747d7540b","tex":"\\widehat r","aliases":"widehat r widehat","display":"\\widehat r","kind":"use","excerpt":"To repair the residual, now regard the vector of edge data as a candidate in its own right. Its defect is ℓ(r). The exact subspace is the zero-sum hyperplane. With the ordinary Euclidean norm and no protected edge coordinates, the nearest compatible residual is widehat r=r-fracℓ(r)41 =(frac34,-frac14,-frac14,-frac14). The correction is perpendicula","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e8d6f1354dafb866","tex":"\\widehat x","aliases":"widehat x widehat","display":"\\widehat x","kind":"use","excerpt":"Anchor the repaired signal at zero. Its primitive is widehat x=(0,frac34,frac12,frac14). The next partial sum is zero, so it closes. If instead the desired representative has mean zero, subtract the mean 3/8: widehat x_rm mean =(-frac38,frac38,frac18,-frac18). Both primitives realize exactly widehat r. They differ by an element of ker D. The anchor","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e2110cbb1980b3b4","tex":"\\widehat x_{\\rm mean}","aliases":"widehat x_rm mean widehat rm","display":"\\widehat x_{\\rm mean}","kind":"use","excerpt":"Anchor the repaired signal at zero. Its primitive is widehat x=(0,frac34,frac12,frac14). The next partial sum is zero, so it closes. If instead the desired representative has mean zero, subtract the mean 3/8: widehat x_rm mean =(-frac38,frac38,frac18,-frac18). Both primitives realize exactly widehat r. They differ by an element of ker D. The anchor","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-70dd49fe06536922","tex":"\\widehat{G_A}","aliases":"widehatG_A widehat","display":"\\widehat{G_A}","kind":"use","excerpt":"The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with t","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9fb7a711460d4e2d","tex":"\\widehat{H_W}","aliases":"widehatH_W widehat","display":"\\widehat{H_W}","kind":"use","excerpt":"em[One finite-quotient system for every preimage] For every rational subspace W and every m≥1, the coordinate map induces an isomorphism H_W/mH_W ≅ ∏_pmid mC_p. For mmid n, the quotient transition map is the coordinate restriction from primes dividing n to primes dividing m. These quotients are cofinal among all finite quotients, and widehatH_W≅ P.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2b26d57eb5a9595f","tex":"\\widetilde a","aliases":"widetilde a widetilde","display":"\\widetilde a","kind":"use","excerpt":"arepsilon_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve th","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7a98ca23b7e037e5","tex":"\\widetilde a_n","aliases":"widetilde a_n widetilde","display":"\\widetilde a_n","kind":"use","excerpt":"arepsilon_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve th","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0a4e15e563cc5107","tex":"\\widetilde b","aliases":"widetilde b widetilde","display":"\\widetilde b","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2500eabb3d3f0489","tex":"\\widetilde c","aliases":"widetilde c widetilde","display":"\\widetilde c","kind":"use","excerpt":"Step 2: obtain literal matching labels. Define widetilde c_i(f)=b_i(v)^-1c_i(f)f^*(b_i(w)) (f:v→ w in U_i). This is a gauge transform and hence a cocycle on U_i. If the arrow lies also in U_j, substitute the original overlap equation and (*) at its two endpoints: widetilde c_j(f) · =b_i(v)^-1a_ij(v)^-1 a_ij(v)c_i(f)[f^*a_ij(w)]^-1 f^*(a_ij(w)b_i(w)","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c53d21b2c0934aea","tex":"\\widetilde c_i","aliases":"widetilde c_i widetilde","display":"\\widetilde c_i","kind":"use","excerpt":"Step 2: obtain literal matching labels. Define widetilde c_i(f)=b_i(v)^-1c_i(f)f^*(b_i(w)) (f:v→ w in U_i). This is a gauge transform and hence a cocycle on U_i. If the arrow lies also in U_j, substitute the original overlap equation and (*) at its two endpoints: widetilde c_j(f) · =b_i(v)^-1a_ij(v)^-1 a_ij(v)c_i(f)[f^*a_ij(w)]^-1 f^*(a_ij(w)b_i(w)","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-eff9a685778c43a7","tex":"\\widetilde c_j","aliases":"widetilde c_j widetilde","display":"\\widetilde c_j","kind":"use","excerpt":"Step 2: obtain literal matching labels. Define widetilde c_i(f)=b_i(v)^-1c_i(f)f^*(b_i(w)) (f:v→ w in U_i). This is a gauge transform and hence a cocycle on U_i. If the arrow lies also in U_j, substitute the original overlap equation and (*) at its two endpoints: widetilde c_j(f) · =b_i(v)^-1a_ij(v)^-1 a_ij(v)c_i(f)[f^*a_ij(w)]^-1 f^*(a_ij(w)b_i(w)","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d4a13a7052f06326","tex":"\\widetilde D","aliases":"widetilde D widetilde","display":"\\widetilde D","kind":"use","excerpt":"[Measurement and operator uncertainty] With the reference certificate above, normx-qx≤ C(norm y+η) if normDx-y≤η. More generally, let widetilde D:E→ R be the bounded linear operator actually used in the measurement. If normD-widetilde D≤ε, normwidetilde D x-y≤η, then normx-qx≤ C(norm y+η+εnorm x). The projection in this conclusion is still the proj","count":19},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-20e75cd544fc942d","tex":"\\widetilde D_t","aliases":"widetilde D_t widetilde","display":"\\widetilde D_t","kind":"use","excerpt":"plate22_gain_budgetFor the kernel-preserving family widetilde D_t(a,b)=(1-t)a, the exact residual gain is 1/(1-t). It deteriorates as the strict budget is exhausted. At the excluded endpoint the kernel changes, so the finite-radius estimate does not extend by substituting t=1.fig:gain-budget","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-929c170d8af8ade4","tex":"\\widetilde G","aliases":"widetilde G widetilde","display":"\\widetilde G","kind":"use","excerpt":"Retaining the phase restores an ordinary multiplication law. Instead of trying to discard the phase, include it in the state. On the eight-element set widetilde G= F_2× G, put (a,x)·(b,y)=(a+b+c(x,y),x+y). The first coordinate of the left-associated product of three elements is a+b+d+c(x,y)+c(x+y,z). The right-associated coordinate is a+b+d+c(y,z)+","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-818401732c1dd8d1","tex":"\\widetilde r","aliases":"widetilde r widetilde","display":"\\widetilde r","kind":"use","excerpt":"How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde","count":17},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2c8e01763b77b415","tex":"\\widetilde r_n","aliases":"widetilde r_n widetilde","display":"\\widetilde r_n","kind":"use","excerpt":"arepsilon_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve th","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-78ff0ae9d511f4cd","tex":"\\widetilde s'","aliases":"widetilde s' widetilde","display":"\\widetilde s'","kind":"use","excerpt":"The proof is independent of the selected raw lift. Indeed, if widetilde s' is another lift, k=widetilde s'-widetilde s lies in ker p_U, and δwidetilde s'-δwidetilde s=δ k∈ B_U. Thus the two quotient classes agree before either feasibility direction is invoked. A raw lift always exists in this additive model by using zero outside U. Constructing a r","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9fcf9ea0bdd21c78","tex":"\\widetilde T","aliases":"widetilde T widetilde","display":"\\widetilde T","kind":"use","excerpt":"How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e24a9d0c560ae65e","tex":"\\widetilde t","aliases":"widetilde t widetilde","display":"\\widetilde t","kind":"use","excerpt":"Γ(A)xrightarrowr_UΓ_U(A) xrightarrow o_UQ_U is exact at Γ_U(A). Thus ker o_U=im r_U, and restriction is onto precisely when o_U is the zero map. First verify that o_U is a homomorphism, since exactness is being asserted in an additive setting. Given s,t∈Γ_U(A), choose raw lifts widetilde s,widetilde t. Their sum lifts s+t. Lift independence and add","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6c8741428acf5118","tex":"\\widetilde T^*","aliases":"widetilde T^* widetilde","display":"\\widetilde T^*","kind":"use","excerpt":"ct the budget [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde T^*z -δ-η M. Fix a feasible correction a. ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c77170c848a2e8b4","tex":"\\widetilde x","aliases":"widetilde x widetilde","display":"\\widetilde x","kind":"use","excerpt":"Protecting three edges forces a different repair. Suppose the first three values (r_0,r_1,r_2)=(1,0,0) are required to remain fixed. The admissible corrections then affect only the final edge. Compatibility forces widetilde r=(1,0,0,-1), widetilde x=(0,1,1,1) for anchor zero. This is the unique repair preserving those three edges. Its squared corre","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-65e8ec76ae768d6c","tex":"x^{-1}","aliases":"x^-1","display":"x^{-1}","kind":"use","excerpt":"erent cocycle whose regular twisted section equation x=px has no solution. The only nontrivial cocycle relation is p^2=1. At a single view the gauge action is p↦ apa^-1. All three transpositions are conjugate and none is conjugate to the identity. The stabilizer is therefore the centralizer. Finally x=px implies p=1 after multiplication by x^-1. so","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d1d16f551dc7199c","tex":"x^{-2}","aliases":"x^-2","display":"x^{-2}","kind":"use","excerpt":"For the lower bound, n>m implies m^2+n^2≤2n^2, hence fracn^2(m^2+n^2)^2≥frac14n^2. The other direction of the decreasing-function integral comparison gives ∑_n>mfrac1n^2≥∫_m+1^∞ x^-2 dx=frac1m+1. This proves B_m≥1/(4(m+1)).","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-226a286fa7079023","tex":"x^{-4}","aliases":"x^-4","display":"x^{-4}","kind":"use","excerpt":"For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0c94f8f562469cbb","tex":"X^2","aliases":"X^2","display":"X^2","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5b029881cc5d477a","tex":"x^2","aliases":"x^2","display":"x^2","kind":"use","excerpt":"Pythagoras applied to that decomposition gives norm x^2=normqx^2+normx-qx^2, hence normqx≤norm x. For any y∈ K, the vector qx-y lies in K, whereas x-qx lies in its orthogonal complement. Consequently normx-y^2=normx-qx^2+normqx-y^2≥normx-qx^2. Equality holds only when normqx-y=0, or y=qx. This proves nearest-point uniqueness without any closedness ","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-baf58144b77f8bed","tex":"X^p","aliases":"X^p","display":"X^p","kind":"use","excerpt":"The phase is visible in actual matrices. Over R, define X=0 · 1 1 · 0, Z=1 · 0 0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3463f1fd3454cd7b","tex":"X^{y_1}","aliases":"X^y_1","display":"X^{y_1}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aaeae6c7bc84b797","tex":"X_0","aliases":"X_0","display":"X_0","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3078a09641b0267f","tex":"x_0","aliases":"x_0","display":"x_0","kind":"use","excerpt":"[Classification of a nonempty extension fiber] If x_0∈Ext_U(s), translation gives an equivalence H_U≃Ext_U(s), h⟼ x_0+h. Consequently the extension is unique exactly when o_U(s)=0 and H_U=0. Define Θ:H_U→Ext_U(s) by Θ(h)=x_0+h. It lands in the extension set because δ(x_0+h)=δ x_0+δ h=0, p_U(x_0+h)=s+0=s. Define Ψ:Ext_U(s)→ H_U by Ψ(x)=x-x_0. Its di","count":38},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-68998c0c84fa2ca0","tex":"x_0^{-1}","aliases":"x_0^-1","display":"x_0^{-1}","kind":"use","excerpt":"· =f^*((x_0)_w^-1x_w)=f^*(h_w). The cancellation uses inverse multiplication in its actual order. No commuting factors have been moved past one another. Hence h∈ H_U. Pointwise group cancellation shows that h↦ x_0h and x↦ x_0^-1x are inverse maps.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-97e232caf37f1610","tex":"X_1","aliases":"X_1","display":"X_1","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d9b1567a52bb6c42","tex":"x_1","aliases":"x_1","display":"x_1","kind":"use","excerpt":"The choice of x_0 only supplies an origin for the parameters. If x_1=x_0+h_1 is used instead, the parameter of x changes from x-x_0 to x-x_1=(x-x_0)-h_1. The affine set of answers itself has not changed. sourceKS10 An extension fiber is therefore an affine copy, or torsor, of the invisible coherent group. Selecting one solution gives coordinates on","count":15},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bc31453ed51955e3","tex":"X_2","aliases":"X_2","display":"X_2","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ace7c93ba6e4cbba","tex":"x_2","aliases":"x_2","display":"x_2","kind":"use","excerpt":"An initially free parameter can become an obstruction Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k,","count":18},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-af6e3ccbb4a25be4","tex":"x_{2n+1}","aliases":"x_2n+1","display":"x_{2n+1}","kind":"use","excerpt":"For the relator description, introduce a commuting generator x_i for each prime index, impose p_i x_i=0, and impose x_2n+1=0 whenever program n is observed to halt on input n. The resulting quotient retains exactly the prime factors indexed by B_K. All relators are effectively enumerable. Deciding whether an odd generator has become trivial is not ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-67bad659e38592f5","tex":"x_3","aliases":"x_3","display":"x_3","kind":"use","excerpt":"For D(x_1,x_2,x_3)=x_1-x_2, p(x)=(x_1,x_3), and x=(2,0,5), the ordinary orthogonal answer is (1,1,5). The protected answer is (2,2,5). The respective squared errors are two and four. Both have zero residual, but only the latter preserves the declared data. The theorem is formalized in codeProtectedHilbertRepair.lean, using the actual operator on ke","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-73a4e2e74cee4766","tex":"X_b","aliases":"X_b","display":"X_b","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-702e017c46584077","tex":"x_c","aliases":"x_c","display":"x_c","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ebb507729655e93c","tex":"x_{HLw}","aliases":"x_HLw","display":"x_{HLw}","kind":"use","excerpt":"If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cd9406cbc1af71a5","tex":"x_{Hv}","aliases":"x_Hv","display":"x_{Hv}","kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4f0f1e4a3167d257","tex":"x_i","aliases":"x_i","display":"x_i","kind":"use","excerpt":"Bitwise complementation is represented at level n by complementing the first n bits. The left shift F(x)_i=x_i+1 uses the schedule ϕ(n)=n+1. A nearest-neighbor difference over F_2, G(x)_i=x_i+x_i+1, also uses n+1 input bits for n output bits. Applying G twice uses n+2 bits, and the middle terms cancel: G(G(x))_i=x_i+x_i+2. The compatible finite map","count":10},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3eb6a4bc2837a67f","tex":"x_{i+1}","aliases":"x_i+1","display":"x_{i+1}","kind":"use","excerpt":"Bitwise complementation is represented at level n by complementing the first n bits. The left shift F(x)_i=x_i+1 uses the schedule ϕ(n)=n+1. A nearest-neighbor difference over F_2, G(x)_i=x_i+x_i+1, also uses n+1 input bits for n output bits. Applying G twice uses n+2 bits, and the middle terms cancel: G(G(x))_i=x_i+x_i+2. The compatible finite map","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4e65ecaade8e1cc1","tex":"x_{i+2}","aliases":"x_i+2","display":"x_{i+2}","kind":"use","excerpt":"Bitwise complementation is represented at level n by complementing the first n bits. The left shift F(x)_i=x_i+1 uses the schedule ϕ(n)=n+1. A nearest-neighbor difference over F_2, G(x)_i=x_i+x_i+1, also uses n+1 input bits for n output bits. Applying G twice uses n+2 bits, and the middle terms cancel: G(G(x))_i=x_i+x_i+2. The compatible finite map","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aab9db6e286302fc","tex":"x_j","aliases":"x_j","display":"x_j","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0430cf91ce15859e","tex":"x_{j+1}","aliases":"x_j+1","display":"x_{j+1}","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f88b14242652e8b9","tex":"X_k","aliases":"X_k","display":"X_k","kind":"use","excerpt":"Equality is witnessed at finite stages. Two different schedules can represent the same transformation. To compare (ϕ,f) and (ψ,g), require that for every output level n there be a finite level k≥ϕ(n),ψ(n) with f_nρ^X_ϕ(n),k=g_nρ^X_ψ(n),k as functions on all of X_k. Call this eventual equality. The witnessing k may depend on n. Equality is required ","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1581d72466c85316","tex":"x_k","aliases":"x_k","display":"x_k","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c7df6d1ed132831f","tex":"x_{k+1}","aliases":"x_k+1","display":"x_{k+1}","kind":"use","excerpt":"For the adjacent equation at k, choose one n with k+1≤ϕ(n). Then ρ_k,k+1x_k+1 =ρ_k,k+1ρ_k+1,ϕ(n)y_n =ρ_k,ϕ(n)y_n=x_k. The recovered family is therefore a point of the original tower.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f9c58e09df34b610","tex":"x_{k_n}","aliases":"x_k_n","display":"x_{k_n}","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5efdca41184961ce","tex":"x_m","aliases":"x_m","display":"x_m","kind":"use","excerpt":"Finally, fix a level n. Every global point satisfies x_n=ρ_n,m(x_m) for each m≥ n, hence x_n∈ Y_n. Conversely, start with any y∈ Y_n, lift recursively above level n through the surjections just proved, and define the lower coordinates by x_i=ρ_i,n(y). The composite restriction laws verify coherence across and below level n. The resulting point eval","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2ce545975574982d","tex":"x_{m+1}","aliases":"x_m+1","display":"x_{m+1}","kind":"use","excerpt":"Suppose instead that x=(x_n) is a global point. Equality along each inclusion says that the underlying natural numbers of x_n+1 and x_n are equal. Induction therefore makes every underlying coordinate equal to the number m underlying x_0. Membership of x_m+1 in X_m+1 now gives m≥ m+1, a contradiction. Thus no global point exists.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ec84613b712fe298","tex":"x_N","aliases":"x_N","display":"x_N","kind":"use","excerpt":"A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls recons","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-08a31aaffef905c3","tex":"x_n","aliases":"x_n","display":"x_n","kind":"use","excerpt":"information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composi","count":21},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4715e03ab2461874","tex":"x_{n+1}","aliases":"x_n+1","display":"x_{n+1}","kind":"use","excerpt":"sidual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions tur","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-964b3165f4897ae1","tex":"x_o","aliases":"x_o","display":"x_o","kind":"use","excerpt":"Conversely, let x∈ker d. The path telescope along p_i has all zero summands, hence T_ix_i=x_o for every vertex. Applying T_i^-1 gives x_i=T_i^-1x_o, so x=R_p(x_o). The first part then places x_o in K. We have proved both inverse equations and uniqueness of an exact extension of a root value.","count":16},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-53687bf254c09aec","tex":"x_p","aliases":"x_p","display":"x_p","kind":"use","excerpt":"Conversely, suppose mx=0 for a positive integer m. Whenever pnmid m, multiplication by m is invertible in C_p, and hence x_p=0. The support of x is contained in the finite set of prime divisors of m. Thus x∈ D. The all-one vector has infinite support, so it belongs to P∖ D.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a487d9d7f59851e2","tex":"x_z","aliases":"x_z","display":"x_z","kind":"use","excerpt":"[Differential identities] The map δ:E→ R is additive, Γ(A)=kerδ, and (δ x)_1_v=0, (δ x)_g∘ f=f^*((δ x)_g)+(δ x)_f. Additivity follows coordinate by coordinate. Vanishing is exactly the defining coherence equation. For composition, add and subtract f^*x_w: f^*g^*x_z-x_v =f^*(g^*x_z-x_w)+(f^*x_w-x_v). The identity equation uses (1_v)^*=id. sourceKS08","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-145fa231c2e954bf","tex":"X_\\theta","aliases":"X_θ theta","display":"X_\\theta","kind":"use","excerpt":"Cofinal changes also preserve the morphism calculus. A monotone unbounded schedule θ gives the sampled resolution X_θ. There is a transport from X to X_θ that reads the selected level. For the reverse direction choose, for each n, a level j(n) with n≤θ(j(n)), and restrict the available value down to X_n. Taking a least such index gives a monotone s","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-94f101af9ed70d0b","tex":"x_{\\phi(i)}","aliases":"x_ϕ(i) phi","display":"x_{\\phi(i)}","kind":"use","excerpt":"The remaining obligation is naturality on all finite states. Fix i≤ j and u∈ X_ϕ(j). By surjectivity of that point projection, choose x with x_ϕ(j)=u. Equation (*) and output coherence yield ρ^Y_i,jf_j(u) · =ρ^Y_i,jF(x)_j=F(x)_i, f_iρ^X_ϕ(i),ϕ(j)(u) · =f_i(x_ϕ(i))=F(x)_i. The state u was arbitrary. Hence the two finite-level functions are equal, wh","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4bf2d988c0ac047e","tex":"X_{\\phi(j)}","aliases":"X_ϕ(j) phi","display":"X_{\\phi(j)}","kind":"use","excerpt":"The remaining obligation is naturality on all finite states. Fix i≤ j and u∈ X_ϕ(j). By surjectivity of that point projection, choose x with x_ϕ(j)=u. Equation (*) and output coherence yield ρ^Y_i,jf_j(u) · =ρ^Y_i,jF(x)_j=F(x)_i,","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-95e421cf839b657d","tex":"x_{\\phi(j)}","aliases":"x_ϕ(j) phi","display":"x_{\\phi(j)}","kind":"use","excerpt":"The remaining obligation is naturality on all finite states. Fix i≤ j and u∈ X_ϕ(j). By surjectivity of that point projection, choose x with x_ϕ(j)=u. Equation (*) and output coherence yield ρ^Y_i,jf_j(u) · =ρ^Y_i,jF(x)_j=F(x)_i,","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c2bf72b185a8cab0","tex":"x_{\\phi(n)}","aliases":"x_ϕ(n) phi","display":"x_{\\phi(n)}","kind":"use","excerpt":"Sampling a full point and recovering its kth coordinate gives ρ_k,ϕ(n)x_ϕ(n)=x_k by its original coherence. Recovering a sampled point and evaluating at k=ϕ(n) allows the index n itself, giving ρ_ϕ(n),ϕ(n)y_n=y_n. These are both inverse identities. No surjectivity of the bonding maps or of the point evaluations was used. sourceKS18 This theorem is ","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a9479d7be2af7aae","tex":"x_{\\phi(\\psi(n))}","aliases":"x_ϕ(ψ(n)) phi psi","display":"x_{\\phi(\\psi(n))}","kind":"use","excerpt":"Finally, evaluate a witness equation at the kth coordinate of a coherent point x. Its two restricted arguments become x_ϕ(n) and x_ψ(n), proving equality of the realized outputs. Thus realization is well defined on the quotient. For a composite, direct evaluation gives |g∘ f|(x)_n=g_n(f_ψ(n)(x_ϕ(ψ(n)))) =(|g|(|f|(x)))_n. The identity law is equally","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-030c2588c09d3e01","tex":"x_{\\psi(n)}","aliases":"x_ψ(n) psi","display":"x_{\\psi(n)}","kind":"use","excerpt":"Finally, evaluate a witness equation at the kth coordinate of a coherent point x. Its two restricted arguments become x_ϕ(n) and x_ψ(n), proving equality of the realized outputs. Thus realization is well defined on the quotient. For a composite, direct evaluation gives |g∘ f|(x)_n=g_n(f_ψ(n)(x_ϕ(ψ(n)))) =(|g|(|f|(x)))_n. The identity law is equally","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b99185167014e4eb","tex":"x_\\infty","aliases":"x_∞ infty","display":"x_\\infty","kind":"use","excerpt":"The maps are linear. On the finite vertex product, normR_pz_∞≤(max_inormT_i^-1)norm z, normx_o≤norm x_∞. These bounds also hold on the subspaces K and ker d, proving continuity of both directions. No completeness is needed for these finite bounds.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-55abe5dbff13226b","tex":"\\xleftarrow","aliases":"xleftarrow","display":null,"kind":"use","excerpt":"Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2355745dec81ea53","tex":"\\xRightarrow","aliases":"xRightarrow","display":null,"kind":"use","excerpt":"Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e5f6938d2e45504d","tex":"y_*","aliases":"y_*","display":"y_*","kind":"use","excerpt":"[Nearest feasible protected repair] Every feasible x has a unique closest exact answer y_* preserving p(x). Its correction h_*=x-y_* is the unique vector in the orthogonal complement of ker B inside M satisfying Bh_*=Dx. For every other feasible answer y, normx-y^2=normx-y_*^2+normy_*-y^2. Moreover, the following conditions are equivalent: ran B is","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d842f0833fbfd1bc","tex":"Y_0","aliases":"Y_0","display":"Y_0","kind":"use","excerpt":"Choose x_0∈ Y_0 and recursively choose x_n+1∈ Y_n+1 above x_n. The preceding surjectivity justifies each step. Adjacent coherence implies every composite coherence equation, so this gives a point. This is an existence construction using countably many choices, not an asserted effective procedure for locating the stabilization indices.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c887b3828dd4ba10","tex":"y_1","aliases":"y_1","display":"y_1","kind":"use","excerpt":"A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dd412919846eece9","tex":"y_2","aliases":"y_2","display":"y_2","kind":"use","excerpt":"on c(x,y)+c(x+y,z)=c(y,z)+c(x,y+z). There is no function a:G→ F_2 such that c(x,y)=a(x)+a(y)-a(x+y) for every x,y∈ G. Normalization follows from multiplication by zero. Expanding the two sides of [source reference] gives, respectively, x_2y_1+(x_2+y_2)z_1, y_2z_1+x_2(y_1+z_1), which are equal by distributivity.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-536459ef72a0484d","tex":"Y_a","aliases":"Y_a","display":"Y_a","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-274ec82875bc15b2","tex":"y_i","aliases":"y_i","display":"y_i","kind":"use","excerpt":"Flatness does not remove amplification. Consider a scalar chain with n+1 vertices, where n≥1, and coordinates e_i(z)=2^iz. Set τ_ij=e_ie_j^-1. This connection is flat, but reverse transport along one edge multiplies norms by two. Let x_i=8·2^i-1, y_i=7·2^i. The family y is exact and preserves x_o=7. On a forward edge the residual of x is 1/2, and o","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-120d8dd974f4bf08","tex":"y_{LHv}","aliases":"y_LHv","display":"y_{LHv}","kind":"use","excerpt":"If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1f5b22a9aa4cdefc","tex":"y_{Lw}","aliases":"y_Lw","display":"y_{Lw}","kind":"use","excerpt":"sm and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturali","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ecfc6f094e651c7b","tex":"y_{Lw'}","aliases":"y_Lw'","display":"y_{Lw'}","kind":"use","excerpt":"sition Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Contravariance of F and coherence of y therefore give F(f)(Ry)_w' · =F(f)F(t_w')y_Lw' · =","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a916367bec9f924f","tex":"y_M","aliases":"y_M","display":"y_M","kind":"use","excerpt":"The nearest reachable image y_M=P_T({a:norm a≤ M})r is unique, so the best residual vector r-y_M is unique as well. Corrections mapping to y_M can differ by vectors in ker T. Among those corrections there is a unique least norm correction. These two uniqueness statements should not be confused with injectivity of T.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b3b03c916a5da762","tex":"y_N","aliases":"y_N","display":"y_N","kind":"use","excerpt":"Suppose Q_Nx→0. Put y_N=S_Nx=x-Q_Nx. Idempotence gives Q_Ny_N=Q_Nx-Q_N^2x=0, so y_N∈ F. Since x-y_N=Q_Nx→0, these are finite-core approximants converging to x. Hence x∈overline F.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7a4d74bf293c7c99","tex":"y_p","aliases":"y_p","display":"y_p","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1ecb58e66eedd43b","tex":"y_t","aliases":"y_t","display":"y_t","kind":"use","excerpt":"[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4p","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-82ed34227c018c62","tex":"y_v","aliases":"y_v","display":"y_v","kind":"use","excerpt":"), there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for x gives f^*(ev_w x)=f^*(x_w)=x_v=ev_v x. If x,y∈Γ(F) have equal evaluations, then x_v=y_v for every v. Equality of functions into a dependent product gi","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c3e3ca27df8cd514","tex":"y_w","aliases":"y_w","display":"y_w","kind":"use","excerpt":"If d=a· c and x is a section of T_c, put y_v=a_vx_v. Then d_ff^*(y_w) =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(x_w) =a_vx_v=y_v. The inverse map is multiplication by a_v^-1 at each view, proving an equivalence of the actual section sets.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7286e65db353f4e6","tex":"Y_\\eta","aliases":"Y_η eta","display":"Y_\\eta","kind":"use","excerpt":"[Cofinal invariance of eventual transports] Sampling and recovery are inverse in the eventual transport category. Consequently, for monotone unbounded schedules θ and η, there is a bijection EventualHom(X,Y) ≃ EventualHom(X_θ,Y_η). On point spaces it agrees with the previously constructed cofinal realization equivalences. Let s:X→ X_θ be sampling, ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6b722730f1539238","tex":"Y_{\\psi(n)}","aliases":"Y_ψ(n) psi","display":"Y_{\\psi(n)}","kind":"use","excerpt":"The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0eb2c4237a529e5e","tex":"y_\\ell","aliases":"y_ℓ ell","display":"y_\\ell","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bbeebd879e1dff69","tex":"Z","aliases":"Z","display":"Z","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":41},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7f2315ed2fa0ec08","tex":"\\Z","aliases":"Z Z mathbb","display":null,"kind":"use","excerpt":"[Parity and an independent ambiguity] For two views with f^*(m)=2m on Z, a prescribed value k at the coarse view extends exactly when k is even. The extension is (k,k/2) and is unique. In particular 1 is obstructed and 2 extends nontrivially. The parity statement is an instance of Theorem~[source reference]. If a separate, disconnected integer view","count":14},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-544ac8aed4a2c573","tex":"Z^1","aliases":"Z^1","display":"Z^1","kind":"use","excerpt":"to e, then (ba)· c=b·(a· c)=e, giving composition. The identity label is 1 and the inverse label is a^-1. Associativity and the two inverse identities now follow from the corresponding multiplication laws in the product group. The label ba is therefore forced by the order of application, not an arbitrary notational convention. sourceKS30 The action","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1a49e6bec04c7b56","tex":"Z^2","aliases":"Z^2","display":"Z^2","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ddecadd3df3a219e","tex":"\\Z^2","aliases":"Z^2 Z","display":null,"kind":"use","excerpt":"An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c52a560429a7daf2","tex":"z^2","aliases":"z^2","display":"z^2","kind":"use","excerpt":"2t · 1-t^2 satisfies norm(U_t-I)z^2=frac4t^21+t^2norm z^2. For t≠0 its fixed space is zero and its residual gain is sqrt1+t^2/(2|t|), which diverges as t→0. At t=0, the entire fiber is fixed. Therefore small loop discrepancy does not supply a uniform stable realization estimate. The fixed directions and their conditioning must be analyzed together.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c8d007212f447c06","tex":"Z^q","aliases":"Z^q","display":"Z^q","kind":"use","excerpt":"The phase is visible in actual matrices. Over R, define X=0 · 1 1 · 0, Z=1 · 0 0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d322c4a01ae30ded","tex":"Z^{x_2}","aliases":"Z^x_2","display":"Z^{x_2}","kind":"use","excerpt":"0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6f46feb7dedc52ce","tex":"z_1","aliases":"z_1","display":"z_1","kind":"use","excerpt":"on c(x,y)+c(x+y,z)=c(y,z)+c(x,y+z). There is no function a:G→ F_2 such that c(x,y)=a(x)+a(y)-a(x+y) for every x,y∈ G. Normalization follows from multiplication by zero. Expanding the two sides of [source reference] gives, respectively, x_2y_1+(x_2+y_2)z_1, y_2z_1+x_2(y_1+z_1), which are equal by distributivity.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9fdca6236d286603","tex":"Z_n","aliases":"Z_n","display":"Z_n","kind":"use","excerpt":"The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-53947a836b78402e","tex":"z_v","aliases":"z_v","display":"z_v","kind":"use","excerpt":"Given u∈Γ(B_a,b), forget each fiber-membership condition and call the underlying family z_v∈ M_v. Coherence of u is precisely coherence of z, so z∈Γ(M). The equations p_vz_v=a_v hold at every coordinate. Joint determination in Γ(F) yields Γ(p)z=a, and the equations for q similarly yield Γ(q)z=b. Thus z is a global bridge.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1011bf33a5864682","tex":"z_\\alpha","aliases":"z_α alpha","display":"z_\\alpha","kind":"use","excerpt":"[Exact regularized primal and dual gap] For every a∈ H and z∈ Y, J_α(a)- D_α(z) =normTa-r+z^2+ fracnormT^*z-α a^2α. Consequently D_α(z)≤ J_α(a) for every pair. At a=a_α and z=z_α=r-Ta_α, equality holds, so the dual maximum is attained and equals the primal minimum. Moreover, αnorma-a_α^2 ≤ J_α(a)- D_α(z). For the easily computed choice z=r-Ta, J_α(","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-baa7daabefd268a7","tex":"|^2","aliases":"|^2","display":"|^2","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-02ad10b5fe941a06","tex":"|_{\\ker D}","aliases":"|_ker D ker","display":"|_{\\ker D}","kind":"use","excerpt":"[Closed range for the joint defect] Assume ran D is closed. Then ran(D,C) is closed Longleftrightarrow ran(C|_ker D) is closed. Under either condition, a finite joint residual-to-repair bound exists. Suppose the range of the joint map is closed. To test closedness of ran B, take a sequence Bk_n→ y with k_n∈ker D. The joint images are (Dk_n,Ck_n)=(0","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-00398672ab8c08af","tex":"|_{\\ker p}","aliases":"|_ker p ker","display":"|_{\\ker p}","kind":"use","excerpt":"Here the general protected construction is completely explicit. With candidate space E=R^4, defect d=ℓ, and protected observable p(r)=(r_0,r_1,r_2), the allowed direction is the last coordinate axis. The restricted operator d|_ker p has gain one. Without protection, the least gain for distance to kerℓ is 1/2, since normℓ=2. The different constants ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-89bc28b8775c3cb7","tex":"|_M","aliases":"|_M","display":"|_M","kind":"use","excerpt":"Stable repair with protected data. The preceding orthogonal projection does not generally preserve a separately declared observable. To join the analytic theory to Section~[source reference], let p:E→ S be continuous linear into a normed space. Define M=ker p, B=D|_M:M→ R, H=M∩ker D. The closed subspace M inherits its Hilbert structure from E. A ca","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-17be83227c76198b","tex":"|_U","aliases":"|_U","display":"|_U","kind":"use","excerpt":"chapPartial data, gluing, and correspondence fibers Prescribed views. For U⊆ObV, use the full subcategory on U. A partial section s∈Γ_U(F) satisfies all transition equations whose endpoints lie in U. Define Ext_U(s)={x∈Γ(F):x|_U=s}. This fiber can be empty, a singleton, or much larger. A partial section is already coherent on its prescribed domain.","count":9},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-62b2ba1a7f4ec5e9","tex":"|_{U_j}","aliases":"|_U_j","display":"|_{U_j}","kind":"use","excerpt":"Step 3: assembly and restriction are inverse. Formula (*) proves x|_U_j=s_j for every j. Conversely, start with y∈Γ(F) and assemble its restrictions. The resulting coordinate at v is y_v, regardless of which containing patch is chosen. Joint determination gives the original y. These two identities are exactly the inverse identities for the displaye","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2b9f0248ab6db655","tex":"|_W","aliases":"|_W","display":"|_W","kind":"use","excerpt":"The actual patch pseudofunctor. On a patch W, use K_G(W)=Z^1(V|_W;G|_W)//∏_v∈ WG_v. Restriction keeps the existing cocycle values and gauge values on the smaller patch. It first gives a strict category valued functor on the opposite patch poset. Its standard pseudofunctor form has identity comparison isomorphisms, with the required coherence laws.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e64495f92493d8b0","tex":"\\pm","aliases":"± pm","display":"\\pm","kind":"use","excerpt":"Projection (a,x)↦ x is a homomorphism onto G. Its kernel is {(0,0),(1,0)}, which commutes with every element. The map (a,x)⟼(-1)^aU_x is a homomorphism by [source reference]. Its eight matrices are distinct: they are ± I,± X,± Z,± XZ. It is therefore faithful. The phase extension is not an assertion that G itself has acquired an ordinary representa","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-349ba76d907adfab","tex":"\\times","aliases":"× times","display":"\\times","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":17},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3f7f41ad8028cb36","tex":"\\alpha^{-1}","aliases":"α^-1 alpha","display":"\\alpha^{-1}","kind":"use","excerpt":"To prove closedness of its range, suppose Bh_n→ y. Applying the lower bound to h_n-h_m gives normh_n-h_m≤α^-1normBh_n-Bh_m. Thus h_n is Cauchy. Completeness of H supplies a limit h, and continuity gives Bh=y. If y is orthogonal to the range of B, then for every h, 0=⟨ Bh,y⟩=⟨ h,By⟩. Taking h=By gives By=0, and injectivity gives y=0. A closed subspa","count":7},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8bd745916dd5cb08","tex":"\\alpha^{-1/2}","aliases":"α^-1/2 alpha","display":"\\alpha^{-1/2}","kind":"use","excerpt":"fixedplate37_solver_amplificationA fixed normal-equation tolerance has different consequences at different regularization parameters. The curves are universal certified upper bounds for a declared normal-residual norm, not observed errors of a particular experiment. The correction estimate grows as α^-1 while the fitted-data estimate grows as α^-1/","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d830d9bc7e9c803c","tex":"\\alpha^2","aliases":"α^2 alpha","display":"\\alpha^2","kind":"use","excerpt":"To obtain the sharper fitted-data bound, retain the quadratic dependence on norm e instead of substituting the first estimate. Multiplying [source reference] by α and rearranging gives αnormTe^2 · ≤αnorm gnorm e-α^2norm e^2 · =fracnorm g^24 -(αnorm e-fracnorm g2)^2 ≤fracnorm g^24. Division by α and square roots give the factor 1/(2sqrtα).","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a823753ea2193233","tex":"\\alpha_n","aliases":"α_n alpha","display":"\\alpha_n","kind":"use","excerpt":"Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity giv","count":16},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-adddd1027194ff95","tex":"\\alpha_{n_k}","aliases":"α_n_k alpha","display":"\\alpha_{n_k}","kind":"use","excerpt":"Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity giv","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8d6e19f38149c347","tex":"\\alpha_v","aliases":"α_v alpha","display":"\\alpha_v","kind":"use","excerpt":"Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cda0d89220d5cb7b","tex":"\\alpha_w","aliases":"α_w alpha","display":"\\alpha_w","kind":"use","excerpt":"Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-17d9a794b7ae4113","tex":"\\beta_n","aliases":"β_n beta","display":"\\beta_n","kind":"use","excerpt":"When two repair paths select the same point. [Path independence requires a comparison] Let f_n→ x and g_n→ y in a metric space. If dist(f_n,g_n)≤β_n, β_n→0, then x=y. In contrast, exactness and preservation of a nonzero observable alone do not force a unique answer. For each n, the triangle inequality gives dist(x,y)≤dist(x,f_n)+dist(f_n,g_n)+dist(","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9d155779febbf3e2","tex":"\\beta_{T,w}","aliases":"β_T,w beta","display":"\\beta_{T,w}","kind":"use","excerpt":"[A sharp budget asymptotic for the diagonal datum] As m→∞, fracA_mm⟶fracπ4, mB_m⟶fracπ4. For the exact budget profile of this infinite problem, lim_M→∞Mβ_T,w(M)=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-44fa3cd067a59627","tex":"\\beta_{\\widetilde T,\\widetilde r}","aliases":"β_widetilde T,widetilde r beta widetilde widetilde","display":"\\beta_{\\widetilde T,\\widetilde r}","kind":"use","excerpt":"How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aed62ca958ae84bc","tex":"\\beta_x","aliases":"β_x beta","display":"\\beta_x","kind":"use","excerpt":"[Invariance of the marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c47cb5bddd3fc8d3","tex":"\\beta_y","aliases":"β_y beta","display":"\\beta_y","kind":"use","excerpt":"), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(ϕ)σ_x^-1=β_x. So E(a) is a marked arrow. Identity and compo","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8c55dcbd044066e1","tex":"\\Gamma_{U_i}","aliases":"Γ_U_i Gamma","display":"\\Gamma_{U_i}","kind":"use","excerpt":"Let (U_i)_i∈ I be full patches. A matching family consists of s_i∈Γ_U_i(F) with equal values on every overlap. The relevant covering condition is for every f:v→ w, some U_i contains both v and w. Identities imply that this condition also covers every object.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1d3aa5383bcccfb0","tex":"\\Gamma_W","aliases":"Γ_W Gamma","display":"\\Gamma_W","kind":"use","excerpt":"Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coord","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f1941b975ffcc891","tex":"\\Delta","aliases":"Δ Delta","display":"\\Delta","kind":"use","excerpt":"[The two-stage repair is the joint projection] The operator P is the orthogonal projection onto M, equivalently onto the kernel of Δ=(D,C). If u=x-P_Lx and v=P_Lx-Px, then u⊥ v, normx-Px^2=norm u^2+norm v^2. The subspace L=ker D is closed, hence Hilbert with the inherited inner product. The restriction B=C|_L is continuous, so its kernel is a close","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1f70da82e50b4385","tex":"\\delta^{-2/3}","aliases":"δ^-2/3 delta","display":"\\delta^{-2/3}","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0b92c8f7e3920710","tex":"\\delta^{1/3}","aliases":"δ^1/3 delta","display":"\\delta^{1/3}","kind":"use","excerpt":"[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By ","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-172c530a1ef915fa","tex":"\\delta^2","aliases":"δ^2 delta","display":"\\delta^2","kind":"use","excerpt":"e=widetilde r-r. Subtract the two normal equations to obtain (T^*T+α I)h=T^*e. Pair with h and use the adjoint identity: normTh^2+αnorm h^2 =⟨ e,Th⟩≤δnormTh. Put t=normTh≥0. If t>0, dropping the nonnegative penalty and dividing by t gives t≤δ. If t=0, that inequality is already true. Completing the scalar square gives αnorm h^2≤δ t-t^2 =fracδ^24-(t","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b2b4dc127ad4bd0a","tex":"\\delta^{4/3}","aliases":"δ^4/3 delta","display":"\\delta^{4/3}","kind":"use","excerpt":"lceilδ^-2/3rceil and α=m^-2 as before. If an inexact solver satisfies ε≤καδ^1/3 (κ≥0), then the same argument gives norma-a^†≤(sqrt2+1+κ)δ^1/3. Thus the inner solve can retain the proven outer accuracy if its normal residual decreases on the scale αδ^1/3. Since α is on the scale δ^4/3, the corresponding normal-residual scale is δ^5/3, with the stat","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6b477b6ce1da12ae","tex":"\\delta^{5/3}","aliases":"δ^5/3 delta","display":"\\delta^{5/3}","kind":"use","excerpt":"an inexact solver satisfies ε≤καδ^1/3 (κ≥0), then the same argument gives norma-a^†≤(sqrt2+1+κ)δ^1/3. Thus the inner solve can retain the proven outer accuracy if its normal residual decreases on the scale αδ^1/3. Since α is on the scale δ^4/3, the corresponding normal-residual scale is δ^5/3, with the stated ceiling factors retained in the exact r","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ae5d1ee4ca82d29f","tex":"\\delta_n","aliases":"δ_n delta","display":"\\delta_n","kind":"use","excerpt":"[A consistent inexact stopping rule] Assume the projected intended datum belongs to the actual range of T, with minimum-norm preimage a^†. Along a sequence of positive parameters and measured problems, suppose α_n→0, fracδ_nsqrtα_n→0, fracε_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8bc4bfc2bdebfaac","tex":"\\epsilon","aliases":"ϵ epsilon","display":"\\epsilon","kind":"use","excerpt":"Conversely, fix x∈overline F and ϵ>0. If K=0, the common operator bound gives normQ_Nx=0 for every N, so convergence is immediate. Suppose K>0. Choose y∈ F satisfying normx-y<ϵ/(K+1). Some M has Q_My=0. For every N≥ M, nesting and the common operator bound give normQ_Nx=normQ_N(x-y)≤ Knormx-y <ϵfracKK+1≤ϵ. The approximation y is chosen before the d","count":17},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-daec86669d42946c","tex":"\\varepsilon","aliases":"ε varepsilon","display":"\\varepsilon","kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":79},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fb43c73461568514","tex":"\\epsilon^2","aliases":"ϵ^2 epsilon","display":"\\epsilon^2","kind":"use","excerpt":"Suppose these energies are bounded, and let L=lim_n m_n^2. Given ϵ>0, choose N with L-m_N^2<ϵ^2. For m≥ n≥ N, norma_m-a_n^2=m_m^2-m_n^2≤ L-m_N^2<ϵ^2. Thus (a_n) is Cauchy and has a norm limit a_*. For each fixed n, all terms from that index onward belong to the closed set A_n, so a_*∈ A_n. This places a_* in their entire intersection.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-96c7394b81eab434","tex":"\\varepsilon^2","aliases":"ε^2 varepsilon","display":"\\varepsilon^2","kind":"use","excerpt":"Now write (q-q')x=q(I-q')x-(I-q)q'x. The two displayed terms lie in ker D and (ker D)^⊥, respectively, so they are orthogonal. The input components (I-q')x and q'x are also orthogonal. Applying the dual estimate to the first and the original cross estimate to the second gives norm(q-q')x^2 · =normq(I-q')x^2+norm(I-q)q'x^2 · ≤ (C'ε)^2norm(I-q')x^2 +","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5fafc40bd0085548","tex":"\\varepsilon_C","aliases":"ε_C varepsilon","display":"\\varepsilon_C","kind":"use","excerpt":"Motion of the actual descent repair. Compare pairs (D,C) and (D',C') on the same raw Hilbert space. Suppose each pair has closed local range and closed overlap range on its own locally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤ma","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7879135e402b1ed2","tex":"\\varepsilon_D","aliases":"ε_D varepsilon","display":"\\varepsilon_D","kind":"use","excerpt":"Motion of the actual descent repair. Compare pairs (D,C) and (D',C') on the same raw Hilbert space. Suppose each pair has closed local range and closed overlap range on its own locally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤ma","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dc06eebc88a8b163","tex":"\\varepsilon_n","aliases":"ε_n varepsilon","display":"\\varepsilon_n","kind":"use","excerpt":"[A consistent inexact stopping rule] Assume the projected intended datum belongs to the actual range of T, with minimum-norm preimage a^†. Along a sequence of positive parameters and measured problems, suppose α_n→0, fracδ_nsqrtα_n→0, fracε_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2289375dae5eef50","tex":"\\varepsilon_w^{-1}","aliases":"ε_w^-1 varepsilon","display":"\\varepsilon_w^{-1}","kind":"use","excerpt":"simeqΓ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-59a94416a2a319ca","tex":"\\eta","aliases":"η eta","display":"\\eta","kind":"use","excerpt":"[Cofinal invariance of eventual transports] Sampling and recovery are inverse in the eventual transport category. Consequently, for monotone unbounded schedules θ and η, there is a bijection EventualHom(X,Y) ≃ EventualHom(X_θ,Y_η). On point spaces it agrees with the previously constructed cofinal realization equivalences. Let s:X→ X_θ be sampling, ","count":39},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-52f9cbbff7441c6e","tex":"\\eta_{2n}","aliases":"η_2n eta","display":"\\eta_{2n}","kind":"use","excerpt":"For failure without monotonicity, take η_2n=0, η_2n+1=1, and ϕ(n)=2n. All sampled values vanish, but infinitely many original values equal one. sourceKS75 Even in the decreasing case, the numerical rate changes. The profile η_n=1/(n+1), sampled at ϕ(n)=2^n-1, becomes 2^-n. This is exponential decay in the number of observations, not a proof of a fa","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6d059a7e178c6271","tex":"\\eta_{2n+1}","aliases":"η_2n+1 eta","display":"\\eta_{2n+1}","kind":"use","excerpt":"For failure without monotonicity, take η_2n=0, η_2n+1=1, and ϕ(n)=2n. All sampled values vanish, but infinitely many original values equal one. sourceKS75 Even in the decreasing case, the numerical rate changes. The profile η_n=1/(n+1), sampled at ϕ(n)=2^n-1, becomes 2^-n. This is exponential decay in the number of observations, not a proof of a fa","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5ed478059bcd3710","tex":"\\eta_n","aliases":"η_n eta","display":"\\eta_n","kind":"use","excerpt":"Changing the sampling schedule does not preserve a rate. A residual profile η_n indexed by resolution is not an iteration trajectory. Nevertheless a useful convergence property is invariant under a cofinal schedule when the profile is monotone.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fdf52ba77f1cb30f","tex":"\\eta_v","aliases":"η_v eta","display":"\\eta_v","kind":"use","excerpt":"If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4f321af5eb71fc0f","tex":"\\eta_{\\phi(k)}","aliases":"η_ϕ(k) eta phi","display":"\\eta_{\\phi(k)}","kind":"use","excerpt":"Solution. The sampled value is η_ϕ(k)=2^-k. Recovering the original index gives k=log_2(n+1) on the sampled stages, so the value is still 1/(n+1). The schedule preserves vanishing but changes the meaning of the horizontal axis. No faster original process has been proved. Proposition~[source reference] is a convergence equivalence under a monotone c","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-333f69862cebb794","tex":"\\eta_{\\phi(n)}","aliases":"η_ϕ(n) eta phi","display":"\\eta_{\\phi(n)}","kind":"use","excerpt":"[Cofinal sampling of a decreasing profile] Suppose η_n≥0 is nonincreasing and ϕ:N→N is monotone and unbounded. Then η_n→0 Longleftrightarrow η_ϕ(n)→0. Without the monotonicity of η, the reverse implication can fail even for a cofinal schedule. The schedule tends to infinity, so convergence of the full profile implies convergence of the sampled one.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c7b34ab03e848fc4","tex":"\\Theta","aliases":"Θ Theta","display":"\\Theta","kind":"use","excerpt":"[Classification of a nonempty extension fiber] If x_0∈Ext_U(s), translation gives an equivalence H_U≃Ext_U(s), h⟼ x_0+h. Consequently the extension is unique exactly when o_U(s)=0 and H_U=0. Define Θ:H_U→Ext_U(s) by Θ(h)=x_0+h. It lands in the extension set because δ(x_0+h)=δ x_0+δ h=0, p_U(x_0+h)=s+0=s. Define Ψ:Ext_U(s)→ H_U by Ψ(x)=x-x_0. Its di","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2b4af44cb17349bb","tex":"\\theta","aliases":"θ theta","display":"\\theta","kind":"use","excerpt":"[Split tower correction] The recurrence defines a continuous linear map A:R→ E with ∂ A=I_R, norm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial co","count":33},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0c2176b912a83c9c","tex":"\\theta^j","aliases":"θ^j theta","display":"\\theta^j","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1c71e2a304d9a62a","tex":"\\theta^{N+1}","aliases":"θ^N+1 theta","display":"\\theta^{N+1}","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fdc2dc9f687c22a4","tex":"\\lambda_p","aliases":"λ_p lambda","display":"\\lambda_p","kind":"use","excerpt":"[Complete homomorphism description] Every homomorphism f:G_A→ G_B has the form f(g)_p=λ_p g_p (p prime) for a unique λ∈ P. Conversely, coordinate multiplication by λ defines such a homomorphism exactly when q(λ u_A)∈ Q q(u_B). Equivalently, there is a rational r for which λ u_A-s_B(r) has finite support. For a homomorphism f, the composite G_A→ G_B","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-831c58f6031d9f05","tex":"\\lambda_p^{-1}","aliases":"λ_p^-1 lambda","display":"\\lambda_p^{-1}","kind":"use","excerpt":"[Automorphisms] An endomorphism with diagonal λ∈ E_A is an automorphism if and only if every coordinate λ_p is nonzero. In that case its rational quotient scalar is nonzero, and its inverse has diagonal λ_p^-1. An automorphism restricts to an automorphism of each torsion factor, so every scalar is nonzero. Conversely, if every scalar is nonzero, th","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c381b1e9f2cf6895","tex":"\\nu_2","aliases":"ν_2 nu","display":"\\nu_2","kind":"use","excerpt":"[The dyadic support pattern] For the same lifts, 2^Nnmid(k+1) Longrightarrow (Q_Nr)_k=0. Thus coordinate k is certainly recovered once N>ν_2(k+1), where ν_2(m) is the largest power-of-two exponent dividing the positive integer m. The only possible unrecovered coordinates at depth N are 2^N-1, 2·2^N-1,quad3·2^N-1, …. Prove the assertion for L_j,N by","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6037193d07157d39","tex":"\\nu_{B_n}","aliases":"ν_B_n nu","display":"\\nu_{B_n}","kind":"use","excerpt":"[Uniformly positive floors obstruct asymptotic repair] Suppose c>0 and eventually c≤ν_B_n(Ω_n). For any choices b_n∈ B_n, it is impossible that normΩ_n-b_n⟶0. Every allowed correction satisfies normΩ_n-b_n≥ν_B_n(Ω_n). The corrected norms are therefore eventually at least c. They cannot converge to zero. sourceKS77 The quantifier covers all admissib","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-804bed66bc586497","tex":"\\pi_m","aliases":"π_m pi","display":"\\pi_m","kind":"use","excerpt":"The quotient formula. [All integer quotients] For m≥1, the coordinate map π_m:G_A⟶∏_pmid mC_p is surjective with kernel mG_A. Thus G_A/mG_A≅∏_pmid mC_p≅ C_rad(m). For mmid n, the transition map is the corresponding coordinate projection. A vector supported on the prime divisors of m realizes any tuple in the target, proving surjectivity. The root c","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-38449ee7b10fdb4d","tex":"\\pi_n","aliases":"π_n pi","display":"\\pi_n","kind":"use","excerpt":"A growing example and the gap boundary. Use X_n=Y_n=R^n+1 with maximum norms. Let π_n delete the last coordinate, take ρ^X_n=tfrac12π_n, ρ^Y_n=π_n, and let s_n append zero. Then S=B_Y=1, B_X=1/2, and K=2.","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-51f8c0dd58b64765","tex":"\\pi_U","aliases":"π_U pi","display":"\\pi_U","kind":"use","excerpt":"[Extension criterion] For every coherent partial section, o_U(s)=0 Longleftrightarrow Ext_U(s)≠∅. Let π_U:R→ R/B_U be the quotient homomorphism. By the definition of a quotient by a subgroup, π_U(r)=0 Longleftrightarrow r∈ B_U. Here B_U is the image of ker p_U under δ. Consequently membership of δwidetilde s means the existence of a specific h∈ E s","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cf29efe3e08463bd","tex":"\\pi_{U,U}","aliases":"π_U,U pi","display":"\\pi_{U,U}","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7377cb30d6304a53","tex":"\\pi_{Z,U}","aliases":"π_Z,U pi","display":"\\pi_{Z,U}","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-264156b9b1294150","tex":"\\pi_{Z,W}","aliases":"π_Z,W pi","display":"\\pi_{Z,W}","kind":"use","excerpt":"[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescri","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b48f947c0ca70892","tex":"\\rho^X_{2n+1}","aliases":"ρ^X_2n+1 rho","display":"\\rho^X_{2n+1}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-755f7ceba9b9722a","tex":"\\rho^X_{a,b}","aliases":"ρ^X_a,b rho","display":"\\rho^X_{a,b}","kind":"use","excerpt":"a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[s","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-73d72c2cfcbcc383","tex":"\\rho^X_k","aliases":"ρ^X_k rho","display":"\\rho^X_k","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b10f6fc6b14d2593","tex":"\\rho^X_{k+1,b}","aliases":"ρ^X_k+1,b rho","display":"\\rho^X_{k+1,b}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-779098801d242d63","tex":"\\rho^X_{k,b}","aliases":"ρ^X_k,b rho","display":"\\rho^X_{k,b}","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cd764dd5def48bd9","tex":"\\rho^X_{k,\\ell}","aliases":"ρ^X_k,ℓ rho ell","display":"\\rho^X_{k,\\ell}","kind":"use","excerpt":"ality is an equivalence relation and is preserved by composition. Its quotient therefore gives a category of resolutions and eventual transports. Realization descends to a functor into sets, preserving identities and composition. A finite-stage equality remains true after moving to a later stage. Indeed, if it holds on X_k and ℓ≥ k, precompose both","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4279f193653756cd","tex":"\\rho^X_{k_n,\\phi(n)}","aliases":"ρ^X_k_n,ϕ(n) rho phi","display":"\\rho^X_{k_n,\\phi(n)}","kind":"use","excerpt":"For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9dcbb5c17aafa2b5","tex":"\\rho^X_n","aliases":"ρ^X_n rho","display":"\\rho^X_n","kind":"use","excerpt":"A candidate transport from X to Y consists of maps f_n:X_n→ Y_n. It is exact when ρ^Y_n f_n+1=f_nρ^X_n for all n.","count":11},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-19aa1ab1f48b5e8b","tex":"\\rho^X_{n,\\theta(j(n))}","aliases":"ρ^X_n,θ(j(n)) rho theta","display":"\\rho^X_{n,\\theta(j(n))}","kind":"use","excerpt":"with the previously constructed cofinal realization equivalences. Let s:X→ X_θ be sampling, with schedule θ and identity level maps. For recovery choose the least j(n) with n≤θ(j(n)). Unboundedness makes this set nonempty. Increasing n only strengthens the inequality, so j is monotone. The level map of r:X_θ→ X is ρ^X_n,θ(j(n)). Composite restricti","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f193474520572543","tex":"\\rho^X_{\\theta(n),\\theta(j(\\theta(n)))}","aliases":"ρ^X_θ(n),θ(j(θ(n))) rho theta theta theta","display":"\\rho^X_{\\theta(n),\\theta(j(\\theta(n)))}","kind":"use","excerpt":"For sr:X_θ→ X_θ, the schedule at output n is j(θ(n)), and the level map is ρ^X_θ(n),θ(j(θ(n))). Compare it with the identity at a sampled index k≥max{n,j(θ(n))}. Restricting either route from level k of the sampled system gives the same original restriction ρ^X_θ(n),θ(k). This proves eventual equality even when θ has repeated values. Hence the two ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-be807838c5137d76","tex":"\\rho^X_{\\theta(n),\\theta(k)}","aliases":"ρ^X_θ(n),θ(k) rho theta theta","display":"\\rho^X_{\\theta(n),\\theta(k)}","kind":"use","excerpt":"For sr:X_θ→ X_θ, the schedule at output n is j(θ(n)), and the level map is ρ^X_θ(n),θ(j(θ(n))). Compare it with the identity at a sampled index k≥max{n,j(θ(n))}. Restricting either route from level k of the sampled system gives the same original restriction ρ^X_θ(n),θ(k). This proves eventual equality even when θ has repeated values. Hence the two ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0507e0b0818d9ad2","tex":"\\rho^X_{\\phi'(j),k}","aliases":"ρ^X_ϕ'(j),k rho phi","display":"\\rho^X_{\\phi'(j),k}","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-eb28761e19c3c138","tex":"\\rho^X_{\\phi'(\\psi'(n)),k}","aliases":"ρ^X_ϕ'(ψ'(n)),k rho phi psi","display":"\\rho^X_{\\phi'(\\psi'(n)),k}","kind":"use","excerpt":"nd g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-28b1ed6f8781b936","tex":"\\rho^X_{\\phi(j),k}","aliases":"ρ^X_ϕ(j),k rho phi","display":"\\rho^X_{\\phi(j),k}","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8448b479f4cbb66a","tex":"\\rho^X_{\\phi(n),k}","aliases":"ρ^X_ϕ(n),k rho phi","display":"\\rho^X_{\\phi(n),k}","kind":"use","excerpt":"Equality is witnessed at finite stages. Two different schedules can represent the same transformation. To compare (ϕ,f) and (ψ,g), require that for every output level n there be a finite level k≥ϕ(n),ψ(n) with f_nρ^X_ϕ(n),k=g_nρ^X_ψ(n),k as functions on all of X_k. Call this eventual equality. The witnessing k may depend on n. Equality is required ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-000f6c73dd1d8b34","tex":"\\rho^X_{\\phi(\\psi(n)),k}","aliases":"ρ^X_ϕ(ψ(n)),k rho phi psi","display":"\\rho^X_{\\phi(\\psi(n)),k}","kind":"use","excerpt":"nd g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8bf3e7c4ceb8d89f","tex":"\\rho^X_{\\psi(n),k}","aliases":"ρ^X_ψ(n),k rho psi","display":"\\rho^X_{\\psi(n),k}","kind":"use","excerpt":"Equality is witnessed at finite stages. Two different schedules can represent the same transformation. To compare (ϕ,f) and (ψ,g), require that for every output level n there be a finite level k≥ϕ(n),ψ(n) with f_nρ^X_ϕ(n),k=g_nρ^X_ψ(n),k as functions on all of X_k. Call this eventual equality. The witnessing k may depend on n. Equality is required ","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-665c37b658f9761f","tex":"\\rho^Y_{2n}","aliases":"ρ^Y_2n rho","display":"\\rho^Y_{2n}","kind":"use","excerpt":"The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed constru","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-00533f34433c2de0","tex":"\\rho^Y_{a,b}","aliases":"ρ^Y_a,b rho","display":"\\rho^Y_{a,b}","kind":"use","excerpt":"a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[s","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c2351fd63f9dbda2","tex":"\\rho^Y_{a,k}","aliases":"ρ^Y_a,k rho","display":"\\rho^Y_{a,k}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d7c357e7e2c7d8e9","tex":"\\rho^Y_{a,k+1}","aliases":"ρ^Y_a,k+1 rho","display":"\\rho^Y_{a,k+1}","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a2cac7880cf1ce17","tex":"\\rho^Y_k","aliases":"ρ^Y_k rho","display":"\\rho^Y_k","kind":"use","excerpt":"To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-67648e382c5950ed","tex":"\\rho^Y_n","aliases":"ρ^Y_n rho","display":"\\rho^Y_n","kind":"use","excerpt":"A candidate transport from X to Y consists of maps f_n:X_n→ Y_n. It is exact when ρ^Y_n f_n+1=f_nρ^X_n for all n.","count":19},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d2886b332952430a","tex":"\\rho^Y_{\\psi'(n),j}","aliases":"ρ^Y_ψ'(n),j rho psi","display":"\\rho^Y_{\\psi'(n),j}","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-58ef887bb9f1247b","tex":"\\rho^Y_{\\psi(n),j}","aliases":"ρ^Y_ψ(n),j rho psi","display":"\\rho^Y_{\\psi(n),j}","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dc9254e83eae53d1","tex":"\\rho^Z_n","aliases":"ρ^Z_n rho","display":"\\rho^Z_n","kind":"use","excerpt":"[The exact tower category] Identity maps and composites of exact transports are exact. An exact transport induces a map of coherent point spaces by x↦(f_nx_n)_n, preserving identities and composition. For two exact transports, substitute ρ^Z_ng_n+1=g_nρ^Y_n and ρ^Y_nf_n+1=f_nρ^X_n into the composite square. For a coherent point, ρ^Y_n f_n+1x_n+1 =f","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dc9b392a4945bb35","tex":"\\rho_0","aliases":"ρ_0 rho","display":"\\rho_0","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a32d3695336eb34b","tex":"\\rho_{0,1}","aliases":"ρ_0,1 rho","display":"\\rho_{0,1}","kind":"use","excerpt":"An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3417c2f73c9f7d40","tex":"\\rho_{0,2}","aliases":"ρ_0,2 rho","display":"\\rho_{0,2}","kind":"use","excerpt":"An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-737cdd19f67dfe28","tex":"\\rho_1","aliases":"ρ_1 rho","display":"\\rho_1","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3f0529e276ae445a","tex":"\\rho_{1,2}","aliases":"ρ_1,2 rho","display":"\\rho_{1,2}","kind":"use","excerpt":"An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d7ba0ffb2518dd65","tex":"\\rho_2","aliases":"ρ_2 rho","display":"\\rho_2","kind":"use","excerpt":"ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xright","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-362dfb6c4b153cec","tex":"\\rho_A","aliases":"ρ_A rho","display":"\\rho_A","kind":"use","excerpt":"Structure of the extension. By Chapter~[source reference], the group is pure in P, reduced, residually finite, and dense. Its torsion is D, and the rational coordinate gives the exact sequence 0⟶ D⟶ G_Axrightarrowρ_A Q⟶0, ρ_A(s_A(r)+d)=r. It does not split.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-14b329400c5c31fb","tex":"\\rho_{i,n}","aliases":"ρ_i,n rho","display":"\\rho_{i,n}","kind":"use","excerpt":"Finally, fix a level n. Every global point satisfies x_n=ρ_n,m(x_m) for each m≥ n, hence x_n∈ Y_n. Conversely, start with any y∈ Y_n, lift recursively above level n through the surjections just proved, and define the lower coordinates by x_i=ρ_i,n(y). The composite restriction laws verify coherence across and below level n. The resulting point eval","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d51449a1464f8b5e","tex":"\\rho_{k+1,\\phi(n)}","aliases":"ρ_k+1,ϕ(n) rho phi","display":"\\rho_{k+1,\\phi(n)}","kind":"use","excerpt":"For the adjacent equation at k, choose one n with k+1≤ϕ(n). Then ρ_k,k+1x_k+1 =ρ_k,k+1ρ_k+1,ϕ(n)y_n =ρ_k,ϕ(n)y_n=x_k. The recovered family is therefore a point of the original tower.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-bc61c5b129ff9df6","tex":"\\rho_{k,k+1}","aliases":"ρ_k,k+1 rho","display":"\\rho_{k,k+1}","kind":"use","excerpt":"For the adjacent equation at k, choose one n with k+1≤ϕ(n). Then ρ_k,k+1x_k+1 =ρ_k,k+1ρ_k+1,ϕ(n)y_n =ρ_k,ϕ(n)y_n=x_k. The recovered family is therefore a point of the original tower.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-beae91d16a7df6a7","tex":"\\rho_{k,\\phi(m)}","aliases":"ρ_k,ϕ(m) rho phi","display":"\\rho_{k,\\phi(m)}","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a2096330a2542786","tex":"\\rho_{k,\\phi(n)}","aliases":"ρ_k,ϕ(n) rho phi","display":"\\rho_{k,\\phi(n)}","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fb8f4b50aa209d1f","tex":"\\rho_{k,\\phi(\\ell)}","aliases":"ρ_k,ϕ(ℓ) rho phi ell","display":"\\rho_{k,\\phi(\\ell)}","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-997f0b9609eaf253","tex":"\\rho_{\\phi(n),\\phi(n)}","aliases":"ρ_ϕ(n),ϕ(n) rho phi phi","display":"\\rho_{\\phi(n),\\phi(n)}","kind":"use","excerpt":"Sampling a full point and recovering its kth coordinate gives ρ_k,ϕ(n)x_ϕ(n)=x_k by its original coherence. Recovering a sampled point and evaluating at k=ϕ(n) allows the index n itself, giving ρ_ϕ(n),ϕ(n)y_n=y_n. These are both inverse identities. No surjectivity of the bonding maps or of the point evaluations was used. sourceKS18 This theorem is ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aeb949f3ceb46809","tex":"\\rho_{\\phi(n),\\phi(\\ell)}","aliases":"ρ_ϕ(n),ϕ(ℓ) rho phi phi ell","display":"\\rho_{\\phi(n),\\phi(\\ell)}","kind":"use","excerpt":"For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9c3245dfb4ac54c1","tex":"\\sigma","aliases":"σ sigma","display":"\\sigma","kind":"use","excerpt":"A small noncommutative model. [A C_2 loop with S_3 coefficients] Take one view with a loop σ satisfying σ^2=1, and constant coefficients in a group H. A cocycle is determined by an element p∈ H whose square is the identity. Gauge equivalence is conjugacy, and its automorphisms form the centralizer of p.","count":14},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-914fbbb84366f58b","tex":"\\sigma^2","aliases":"σ^2 sigma","display":"\\sigma^2","kind":"use","excerpt":"A small noncommutative model. [A C_2 loop with S_3 coefficients] Take one view with a loop σ satisfying σ^2=1, and constant coefficients in a group H. A cocycle is determined by an element p∈ H whose square is the identity. Gauge equivalence is conjugacy, and its automorphisms form the centralizer of p.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2792c54819372788","tex":"\\sigma_p","aliases":"σ_p sigma","display":"\\sigma_p","kind":"use","excerpt":"The entire subgroup lattice. For B⊆ N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8939774e8d5b06af","tex":"\\sigma_{p_i}","aliases":"σ_p_i sigma","display":"\\sigma_{p_i}","kind":"use","excerpt":"Let B be the set of indices of these factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice oper","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7499d2052ab744b7","tex":"\\sigma_x","aliases":"σ_x sigma","display":"\\sigma_x","kind":"use","excerpt":"he marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-51ded5aa59c649df","tex":"\\sigma_x^{-1}","aliases":"σ_x^-1 sigma","display":"\\sigma_x^{-1}","kind":"use","excerpt":"[Invariance of the marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ae6135878257b347","tex":"\\sigma_y","aliases":"σ_y sigma","display":"\\sigma_y","kind":"use","excerpt":"he marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8bbe064464725f7e","tex":"\\sigma_y^{-1}","aliases":"σ_y^-1 sigma","display":"\\sigma_y^{-1}","kind":"use","excerpt":"he marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cd33bebc9c1e4ef9","tex":"\\tau_{i_ji_{j+1}}","aliases":"τ_i_ji_j+1 tau","display":"\\tau_{i_ji_{j+1}}","kind":"use","excerpt":"The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-92ea019ce8093a2c","tex":"\\tau_{i_{m-1}i_m}","aliases":"τ_i_m-1i_m tau","display":"\\tau_{i_{m-1}i_m}","kind":"use","excerpt":"Distinct walks are retained as arrows of the view category. The product along a walk q=(i_0,…,i_m) is T_q=τ_i_0i_1⋯τ_i_m-1i_m. Concatenation gives the restriction composition law. An exact vertex family defines a coherent walk-view section, and a coherent walk-view section satisfies every edge equation. Thus this graph construction returns to the o","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-14b801f95f5f5a95","tex":"\\tau_{ij}","aliases":"τ_ij tau","display":"\\tau_{ij}","kind":"use","excerpt":"lly guarantee. Graph views, holonomy, and reconstruction. Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear equivalences τ_ij, with τ_","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e3e46cde9b77d238","tex":"\\tau_{ij}^{-1}","aliases":"τ_ij^-1 tau","display":"\\tau_{ij}^{-1}","kind":"use","excerpt":"tionGraph views, holonomy, and reconstruction Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear equivalences τ_ij, with τ_ji=τ_ij^-1. ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-7370b2cce0bbcf1a","tex":"\\tau_{ji}","aliases":"τ_ji tau","display":"\\tau_{ji}","kind":"use","excerpt":"tionGraph views, holonomy, and reconstruction Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear equivalences τ_ij, with τ_ji=τ_ij^-1. ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9a2ae80e5efa1bd7","tex":"\\Phi","aliases":"Φ Phi","display":"\\Phi","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-97267862925829aa","tex":"\\phi'","aliases":"ϕ' phi","display":"\\phi'","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-18f9ac23eff67597","tex":"\\Phi^{-1}","aliases":"Φ^-1 Phi","display":"\\Phi^{-1}","kind":"use","excerpt":"[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n,","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-3dd15563f7adbbf0","tex":"\\chi_A^{-1}","aliases":"χ_A^-1 chi","display":"\\chi_A^{-1}","kind":"use","excerpt":"[The split defect chart] The maps χ_A:E≃_L K× R, χ_A(x)=(P_Ax,dx), χ_A^-1(u,r)=u+Ar are inverse continuous linear equivalences. In these coordinates, P_A sends (u,r) to (u,0). If B is another right inverse, then A-B takes values in K, and P_Bx-P_Ax=(A-B)dx. The projections commute if and only if they are equal, and they are equal if and only if A=B","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f3e9049b2d128d00","tex":"\\chi_B","aliases":"χ_B chi","display":"\\chi_B","kind":"use","excerpt":"[Triangular transport and repair mismatch] In split defect coordinates, the change of representation is χ_BSχ_A^-1(u,r)=(Su+Mr,Cr). The two repair routes differ by P_BSx-SP_Ax=M(dx). Consequently they agree on exact inputs, and normP_BSx-SP_Ax≤norm Mnormdx, norm M≤norm Snorm A+norm Bnorm C. They agree on all inputs if and only if SA=BC.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f1920a0117831d5f","tex":"\\Psi","aliases":"Ψ Psi","display":"\\Psi","kind":"use","excerpt":"10 If x_0∈Ext_U(s), translation gives an equivalence H_U≃Ext_U(s), h⟼ x_0+h. Consequently the extension is unique exactly when o_U(s)=0 and H_U=0. Define Θ:H_U→Ext_U(s) by Θ(h)=x_0+h. It lands in the extension set because δ(x_0+h)=δ x_0+δ h=0, p_U(x_0+h)=s+0=s. Define Ψ:Ext_U(s)→ H_U by Ψ(x)=x-x_0. Its differential is zero, and its restriction is s","count":3},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a1d181fcbd5bda98","tex":"\\psi'","aliases":"ψ' psi","display":"\\psi'","kind":"use","excerpt":"We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applyin","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-1ca12a002e4da1b8","tex":"\\psi^{-1}","aliases":"ψ^-1 psi","display":"\\psi^{-1}","kind":"use","excerpt":"Replacing b by an isomorphic boundary b' also gives an equivalence, by composing the identification with b→ b'. For R=id, the marked realization groupoid has center (b,1_b), and between (x,ϕ) and (y,ψ) there is exactly one compatible arrow ψ^-1ϕ. This is a useful example of contractible realization data, not a claim that the original groupoid has n","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d6aef031749de4a7","tex":"\\Omega","aliases":"Ω Omega","display":"\\Omega","kind":"use","excerpt":"Distance to the actual correction image. Let R be a normed additive group and B≤ R an additive subgroup. In the protected linear model, B=d(ker p). For a residual Ω∈ R, define ν_B(Ω)=inf_b∈ BnormΩ-b. The subgroup contains zero, so this is a finite nonnegative real number. It is an infimum, not an assertion that a nearest admissible residual exists.","count":21},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-097ee86d56b24dd4","tex":"\\omega_f","aliases":"ω_f omega","display":"\\omega_f","kind":"use","excerpt":"stead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e9d6d9121cd11235","tex":"\\Omega_n","aliases":"Ω_n Omega","display":"\\Omega_n","kind":"use","excerpt":"A persistent floor blocks every correction sequence. The correction image can vary with the resolution or the admissibility rule. Let B_n≤ R and let Ω_n∈ R be given.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-84655d740c6d10f2","tex":"\\dagger","aliases":"† dagger","display":"\\dagger","kind":"use","excerpt":"[The regularization limit alternative] As αdownarrow0, Ta_α⟶ b, normr-Ta_α⟶ q, αnorma_α^2⟶0. If b∈ran T, let a^† be its unique minimum norm preimage. Then a_α⟶ a^† in norm. If b∉ran T, then norma_α→∞. Step 1: identify the unavoidable component. Since r-b⊥overlineran T, the adjoint identity gives T^*(r-b)=0. Thus a_α(r)=a_α(b). For every correction ","count":22},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-837406a230d80642","tex":"\\ell^\\infty","aliases":"ℓ^∞ ell infty","display":"\\ell^\\infty","kind":"use","excerpt":"Two boundary examples. On ℓ^∞(N,R), let C remove the first coordinate and let J prepend zero. Then Q_Nx is the tail of x, placed at its original coordinates with zeros before N. Every fixed coordinate is eventually zero in the remainder, and normQ_N≤1. Nevertheless the constant sequence 1 satisfies normQ_N1=1 for every N.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0d9104cdc1fe8aed","tex":"\\ell^\\infty_n","aliases":"ℓ^∞_n ell infty","display":"\\ell^\\infty_n","kind":"use","excerpt":"chapBanach repair and constructed split coordinates An actual complete candidate space. Let X_n,Y_n be Banach spaces over a complete nontrivially normed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formal","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ae3660560a54702c","tex":"\\Rightarrow","aliases":"⇒ Rightarrow","display":"\\Rightarrow","kind":"use","excerpt":"chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4ec4830bef761b7a","tex":"\\forall","aliases":"∀ forall","display":"\\forall","kind":"use","excerpt":"When global behavior determines the finite presentation. Say that X has surjective point projections when ∀ k ev_k:Point(X)→ X_k is onto. This means that every finite input state really occurs in some coherent global input. It is stronger than nonemptiness of the global space.","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-935f9c58c2ce6b38","tex":"\\partial_\\phi","aliases":"∂_ϕ partial phi","display":"\\partial_\\phi","kind":"use","excerpt":"[Bounded-gap resampling and selected repair] Suppose ϕ(n+1)-ϕ(n)≤ m for a fixed natural number m. Then the actual sampling maps are bounded and satisfy normS_ϕ≤1, normC_ϕ≤ m, ∂_ϕ S_ϕ=C_ϕ∂. Exact candidates remain exact under every monotone common resampling, even without a gap bound. If A and B are bounded right inverses for the fine and sampled di","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b161bf60d6b2d19d","tex":"\\exists","aliases":"∃ exists","display":"\\exists","kind":"use","excerpt":"[Semantics preserving correction] For x∈ E, the following conditions are equivalent: · [dx]=0 in R/d(ker p), · ∃ h∈ E: p(h)=0, d(h)=d(x), · ∃ y∈ E: d(y)=0, p(y)=p(x). The class of dx is unchanged when an admissible correction is subtracted from x. The image d(ker p) is a subgroup of R. The defining property of the quotient says [dx]=0 Longleftright","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4e78af967b166a06","tex":"\\prod_{(i,j)\\in\\vec E}","aliases":"∏_(i,j)∈vec E prod in vec","display":"\\prod_{(i,j)\\in\\vec E}","kind":"use","excerpt":"The candidate and residual spaces are finite products with maximum norms: E=∏_i∈ VH, R=∏_(i,j)∈vec EH, (dx)_ij=τ_ijx_j-x_i. Both orientations of an edge are retained. If M bounds the norms of all measured edge maps, then norm d≤ M+1. The empty edge case has the usual zero residual.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-aaf9cd5f4f9933fc","tex":"\\prod_{f:v\\to w}","aliases":"∏_f:v→ w prod to","display":"\\prod_{f:v\\to w}","kind":"use","excerpt":"The compatibility differential. Assume each fiber A_v is an abelian group and each restriction is additive. Set E=∏_v A_v, R=∏_f:v→ wA_v, (δ x)_f=f^*(x_w)-x_v. The group R retains an entry for every arrow. It is not silently identified with a cohomology group.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e3684783b225d098","tex":"\\prod_{i<N}","aliases":"∏_i<N prod","display":"\\prod_{i<N}","kind":"use","excerpt":"Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4dd9169b011221a2","tex":"\\prod_{i\\in V}","aliases":"∏_i∈ V prod in","display":"\\prod_{i\\in V}","kind":"use","excerpt":"The candidate and residual spaces are finite products with maximum norms: E=∏_i∈ VH, R=∏_(i,j)∈vec EH, (dx)_ij=τ_ijx_j-x_i. Both orientations of an edge are retained. If M bounds the norms of all measured edge maps, then norm d≤ M+1. The empty edge case has the usual zero residual.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-badead017d25968f","tex":"\\prod_{j<N}","aliases":"∏_j<N prod","display":"\\prod_{j<N}","kind":"use","excerpt":"nce. This proves equality of the two sets. Applying it to every x∈ R_0 gives the density criterion. sourceKS27 There are other sufficient conditions. If the transported details are absolutely summable in the Banach space R_0, their sum exists and the remaining boundary lies in P_∞. If that persistent subspace is zero, the sum equals x. Alternativel","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-709a1d5e851efb47","tex":"\\prod_{j=0}^{k-1}","aliases":"∏_j=0^k-1 prod","display":"\\prod_{j=0}^{k-1}","kind":"use","excerpt":"The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-92351115e501ee93","tex":"\\prod_{p\\mid m}","aliases":"∏_pmid m prod mid","display":null,"kind":"use","excerpt":"[One finite-quotient system for every preimage] For every rational subspace W and every m≥1, the coordinate map induces an isomorphism H_W/mH_W ≅ ∏_pmid mC_p. For mmid n, the quotient transition map is the coordinate restriction from primes dividing n to primes dividing m. These quotients are cofinal among all finite quotients, and widehatH_W≅ P. T","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-37de85f98def7f5b","tex":"\\prod_{u\\in U}","aliases":"∏_u∈ U prod in","display":"\\prod_{u\\in U}","kind":"use","excerpt":"Three maps that must not be confused. Keep the additive view system of Section~[source reference]. For a prescribed domain U, there are three distinct maps: p_U:E⟶∏_u∈ UA_u, r_U:Γ(A)⟶Γ_U(A), o_U:Γ_U(A)⟶ Q_U. The first restricts a raw family, whether or not it is coherent. The second restricts a global coherent family. The third detects whether an a","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d212f3e24adbf170","tex":"\\prod_v","aliases":"∏_v prod","display":"\\prod_v","kind":"use","excerpt":"The compatibility differential. Assume each fiber A_v is an abelian group and each restriction is additive. Set E=∏_v A_v, R=∏_f:v→ wA_v, (δ x)_f=f^*(x_w)-x_v. The group R retains an entry for every arrow. It is not silently identified with a cohomology group.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-70c07b4410f6643a","tex":"\\prod_{v\\in W}","aliases":"∏_v∈ W prod in","display":"\\prod_{v\\in W}","kind":"use","excerpt":"The actual patch pseudofunctor. On a patch W, use K_G(W)=Z^1(V|_W;G|_W)//∏_v∈ WG_v. Restriction keeps the existing cocycle values and gauge values on the smaller patch. It first gives a strict category valued functor on the opposite patch poset. Its standard pseudofunctor form has identity comparison isomorphisms, with the required coherence laws.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-91e6cb6007134ac7","tex":"\\sum_i","aliases":"∑_i sum","display":"\\sum_i","kind":"use","excerpt":"Every edge map is bijective. Around the cycle their composite is translation by -∑_i r_i. A global realization must be fixed by that translation. Thus the example makes local invertibility and global compatibility visibly different without needing a noncommutative group.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6cb04d84bd2adbc3","tex":"\\sum_{j<N}","aliases":"∑_j<N sum","display":"\\sum_{j<N}","kind":"use","excerpt":"[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I g","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b9a7896e38144f2e","tex":"\\sum_{j=0}^{N}","aliases":"∑_j=0^N sum","display":"\\sum_{j=0}^{N}","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-586bc3571b381e64","tex":"\\sum_{j=0}^{N-1}","aliases":"∑_j=0^N-1 sum","display":"\\sum_{j=0}^{N-1}","kind":"use","excerpt":"residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages converges in X by completeness, to a vector x with the displayed norm bound. Continuity of S, together with the finite telescoping equation S(∑_j=0^N-1u_j)=y-r_N, ","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-b8194ba746cad45a","tex":"\\sum_{j=0}^{n-m-1}","aliases":"∑_j=0^n-m-1 sum","display":"\\sum_{j=0}^{n-m-1}","kind":"use","excerpt":"Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-c579ae579c844347","tex":"\\sum_{j=0}^{\\infty}","aliases":"∑_j=0^∞ sum infty","display":"\\sum_{j=0}^{\\infty}","kind":"use","excerpt":"Neumann inversion with both inverse identities. [A strict perturbation radius] Let W be a real Banach space and T:W→ W bounded linear, with normT≤θ<1 and θ≥0. Then Z=∑_j=0^∞T^j exists in operator norm and satisfies (I-T)Z=Z(I-T)=I, normZ≤frac11-θ. First recall why the space of bounded operators on W is complete. An operator-norm Cauchy sequence A_n","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-16d89af80aea902e","tex":"\\sum_{j=1}^J","aliases":"∑_j=1^J sum","display":"\\sum_{j=1}^J","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-ef75e15ec0684e46","tex":"\\sum_{j=1}^{n}","aliases":"∑_j=1^n sum","display":"\\sum_{j=1}^{n}","kind":"use","excerpt":"Three hypotheses that cannot simply be dropped. Summable movement. In maximum-norm R^3, let f_n=(s,H_n,(n+1)^-1), H_n=∑_j=1^nfrac1j. The first coordinate is protected and the residual is the last coordinate. It tends to zero with the same ratios as the polynomial example. The movement equals 1/(n+1), however, and the unprotected coordinate diverges","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-71e4a7d709b5778b","tex":"\\sum_{j=N+1}^{M}","aliases":"∑_j=N+1^M sum","display":"\\sum_{j=N+1}^{M}","kind":"use","excerpt":"Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fix","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6d14accaece49594","tex":"\\sum_{j\\le J}","aliases":"∑_j≤ J sum le","display":"\\sum_{j\\le J}","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-49afb54019149b8f","tex":"\\sum_{j\\ge0}","aliases":"∑_j≥0 sum ge","display":"\\sum_{j\\ge0}","kind":"use","excerpt":"λ=2normr/c. The vector r/λ has norm c/2, so [source reference] supplies z∈ B_X with normSz-r/λ≤ c/4. Then u=λ z satisfies normu≤ 2normr/c, normr-Su≤normr/2. If the residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f1c2ff812e2b7828","tex":"\\sum_{k=0}^{\\infty}","aliases":"∑_k=0^∞ sum infty","display":"\\sum_{k=0}^{\\infty}","kind":"use","excerpt":"Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many incre","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4a94609cba901bff","tex":"\\sum_{k=a}^{b-1}","aliases":"∑_k=a^b-1 sum","display":"\\sum_{k=a}^{b-1}","kind":"use","excerpt":"For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-baa8f7877711f9a7","tex":"\\sum_{k=n}^{m-1}","aliases":"∑_k=n^m-1 sum","display":"\\sum_{k=n}^{m-1}","kind":"use","excerpt":"∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k=n^∞b_k. Given ε>0, convergence of the nonnegative serie","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f26710096ba8e9ae","tex":"\\sum_{k=n}^{\\infty}","aliases":"∑_k=n^∞ sum infty","display":"\\sum_{k=n}^{\\infty}","kind":"use","excerpt":"[Summable movement gives exactification] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k","count":4},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-802368a4fc45105f","tex":"\\sum_n","aliases":"∑_n sum","display":"\\sum_n","kind":"use","excerpt":"[Variable-rate exactification] If Q_n→0 and ∑_n C_nQ_n<∞, then (f_n) has an exact limit and dist(f_n,x)≤ω(f_0)∑_k=n^∞C_kQ_k. For constant q_n=q∈[0,1) and constant C_n=C, the sharper restarted estimate is dist(f_n,x)≤fracC1-q ω(f_n). The estimate ω(f_n)≤ Q_nω(f_0) follows by induction. At n=0 it is equality. If it holds at n, nonnegativity of q_n pe","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-8773b980988ccc84","tex":"\\sum_{n=0}^\\infty","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^\\infty","kind":"use","excerpt":"The perturbed map is surjective, with D'A'=I, normA'≤fraca1-aε. Its exact projection satisfies normx-q'x≤fraca1-aεnormD'x, normq-q'≤fracaε1-aε. The map T=(D-D')A acts on the Banach target R and satisfies norm T≤ aε<1. The operator series Z=∑_n=0^∞ T^n converges in operator norm and obeys norm Z≤(1-aε)^-1. Multiplying its finite partial sums on eith","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a7626aeba75a509f","tex":"\\sum_{n=0}^{\\infty}","aliases":"∑_n=0^∞ sum infty","display":"\\sum_{n=0}^{\\infty}","kind":"use","excerpt":"[Summable movement gives exactification] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-cdd7cb47a2dff84a","tex":"\\sum_{n=1}^m","aliases":"∑_n=1^m sum","display":"\\sum_{n=1}^m","kind":"use","excerpt":"For n≤ m, the residual square is at most n^2/m^4. The sum-of-squares estimate ∑_n=1^m n^2=fracm(m+1)(2m+1)6≤ m^3 holds for m≥1, so this part contributes at most 1/m. For n>m, each residual square is at most 1/n^2, whose tail is at most 1/m. This proves B_m≤2/m.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-a3ec24382fc8477a","tex":"\\sum_{n=1}^N","aliases":"∑_n=1^N sum","display":"\\sum_{n=1}^N","kind":"use","excerpt":"1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-afbea6d048f6cfef","tex":"\\sum_{n=1}^{N}","aliases":"∑_n=1^N sum","display":"\\sum_{n=1}^{N}","kind":"use","excerpt":"Escape is not an accident of this particular algorithm. Suppose an arbitrary protected candidate (u,1) satisfies normd(u,1)≤ε. On its first N coordinates, ∑_n=1^N|u_n+1|^2 ≤ N^2∑_n=1^Nfrac|u_n+1|^2n^2 ≤ N^2ε^2. The reverse triangle inequality in R^N gives norm u≥sqrt N-Nε (N≥1). If a sequence of protected candidates has residual tending to zero, th","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-67e8088ae0f488e2","tex":"\\sum_{n>K}","aliases":"∑_n>K sum","display":"\\sum_{n>K}","kind":"use","excerpt":"Finally, for K≥ m, both omitted tails consist of nonnegative terms. Bound the first by m^4∑_n>Kn^-4 and the second by ∑_n>Kn^-2. The same upper integral comparisons give the two certified remainders in [source reference]. No finite truncation is substituted for an equality of infinite energies. writtenWR10","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6c1ff4839728583e","tex":"\\sum_{n>m}","aliases":"∑_n>m sum","display":"\\sum_{n>m}","kind":"use","excerpt":"For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9b846ee3188af45a","tex":"\\sum_{n>N}","aliases":"∑_n>N sum","display":"\\sum_{n>N}","kind":"use","excerpt":"1/n · n>N, norma_N^2=N, normf_N^2=N+1. For every N≥1, telescoping comparisons give frac1N+1≤normd(f_N)^2 =∑_n>Nfrac1n^2≤frac1N. Indeed 1/[n(n+1)]≤1/n^2≤1/[n(n-1)] for n≥2. Summing the two telescoping bounds proves [source reference] without numerically evaluating an infinite series.","count":6},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0a31cd44b28cdc9b","tex":"\\sum_{n\\ge1}","aliases":"∑_n≥1 sum ge","display":"\\sum_{n\\ge1}","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-069c58c03dba2dbd","tex":"\\sum_{n\\ge2}","aliases":"∑_n≥2 sum ge","display":"\\sum_{n\\ge2}","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0c8a7313d6794de5","tex":"\\sum_{p\\mid m}","aliases":"∑_pmid m sum mid","display":null,"kind":"use","excerpt":"Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0e301b22b90725c4","tex":"\\setminus","aliases":"∖ setminus","display":"\\setminus","kind":"use","excerpt":"Ω∈overline B∖ B · Nonzero · Arbitrarily small corrected residuals exist, but no exact cancellation exists.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d2ecc6688baa55b4","tex":"\\cup","aliases":"∪ cup","display":"\\cup","kind":"use","excerpt":"e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic d","count":2},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e0395c0dd9f66f1e","tex":"\\int","aliases":"∫ int","display":"\\int","kind":"use","excerpt":"The category of elements ∫_V F and Γ(F) also have different roles. The former organizes individual appearances and their transitions. The latter selects coherent families across all views. The category of elements serves here as explanatory notation, while the Lean formalization directly represents families and their coherence.","count":5},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-f21fa71dce8d6708","tex":"\\int_0^{N/m}","aliases":"∫_0^N/m int","display":"\\int_0^{N/m}","kind":"use","excerpt":"1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dad46397774b52e2","tex":"\\int_0^{\\pi/2}","aliases":"∫_0^π/2 int pi","display":"\\int_0^{\\pi/2}","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-adc46c2b3119d58b","tex":"\\int_0^\\infty","aliases":"∫_0^∞ int infty","display":"\\int_0^\\infty","kind":"use","excerpt":"Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-5f9323382a61523c","tex":"\\int_{m+1}^\\infty","aliases":"∫_m+1^∞ int infty","display":"\\int_{m+1}^\\infty","kind":"use","excerpt":"For the lower bound, n>m implies m^2+n^2≤2n^2, hence fracn^2(m^2+n^2)^2≥frac14n^2. The other direction of the decreasing-function integral comparison gives ∑_n>mfrac1n^2≥∫_m+1^∞ x^-2 dx=frac1m+1. This proves B_m≥1/(4(m+1)).","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fb3ba7923792333d","tex":"\\int_m^\\infty","aliases":"∫_m^∞ int infty","display":"\\int_m^\\infty","kind":"use","excerpt":"For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-dd7b189feddbc76a","tex":"\\int_\\V","aliases":"∫_V int V","display":null,"kind":"use","excerpt":"The category of elements ∫_V F and Γ(F) also have different roles. The former organizes individual appearances and their transitions. The latter selects coherent families across all views. The category of elements serves here as explanatory notation, while the Lean formalization directly represents families and their coherence.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-4d3a285fd7941bf1","tex":"\\ne","aliases":"≠ ne","display":"\\ne","kind":"use","excerpt":"[Extension criterion] For every coherent partial section, o_U(s)=0 Longleftrightarrow Ext_U(s)≠∅. Let π_U:R→ R/B_U be the quotient homomorphism. By the definition of a quotient by a subgroup, π_U(r)=0 Longleftrightarrow r∈ B_U. Here B_U is the image of ker p_U under δ. Consequently membership of δwidetilde s means the existence of a specific h∈ E s","count":32},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d2d5bf05abcf90be","tex":"\\subset","aliases":"⊂ subset","display":"\\subset","kind":"use","excerpt":"[Bounded approximate feasibility] For a bounded linear map T:H→ Y between real Hilbert spaces and r∈ Y, the following are equivalent: · r∈ran T, · there is a bounded sequence (a_n)⊂ H with normTa_n-r⟶0. If all norma_n≤ M, an exact solution exists with norm at most M. If Ta=r, the constant sequence a_n=a is bounded and has zero residual. For the con","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9f55fb1c51ddcd12","tex":"\\supseteq","aliases":"⊇ supseteq","display":"\\supseteq","kind":"use","excerpt":"Suppose now that A_1⊇ A_2⊇⋯ are nonempty closed affine subsets of a real Hilbert correction space H. A typical A_n consists of corrections satisfying the first n bounded linear observation equations. The sets are affine because the equations are linear, and closed because the observations are continuous.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-0b887c3f37e371d2","tex":"\\oplus","aliases":"⊕ oplus","display":"\\oplus","kind":"use","excerpt":"The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n ha","count":8},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e8a1aa147b4b8b89","tex":"\\bigoplus_{i\\in B}","aliases":"⊕_i∈ B bigoplus in","display":"\\bigoplus_{i\\in B}","kind":"use","excerpt":"The entire subgroup lattice. For B⊆ N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-fa0b3e595fc9f5b4","tex":"\\bigoplus_{i\\in I}","aliases":"⊕_i∈ I bigoplus in","display":"\\bigoplus_{i\\in I}","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-2610e51a095ab37e","tex":"\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}","aliases":"⊕_j∈Reach_E(F) bigoplus in operatorname","display":"\\bigoplus_{j\\in\\operatorname{Reach}_E(F)}","kind":"use","excerpt":"[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-e2983c1eebc209b8","tex":"\\bigoplus_{p\\mid m}","aliases":"⊕_pmid m bigoplus mid","display":null,"kind":"use","excerpt":"[All roots and their number] If my=g is solvable, its solution set is the coset y_0+G_A[m], G_A[m]=⊕_pmid mC_p. It has exactly rad(m) elements. Two solutions differ by an element annihilated by m, and adding such an element to a solution gives another. The ambient product has a nonzero m-torsion coordinate only when pmid m. All those finite coordin","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-652adc7f79a6e08d","tex":"\\perp_M","aliases":"⊥_M perp","display":"\\perp_M","kind":"use","excerpt":"If ran B is closed, Theorem~[source reference] applied on M gives a bounded inverse of B on (ker B)^⊥_M. The vector h_* is exactly that inverse applied to Dx, so its norm obeys the required bound.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-9a4555c51b02f0a3","tex":"\\bigcap_{(i,j)\\in\\vec E}","aliases":"⋂_(i,j)∈vec E bigcap in vec","display":"\\bigcap_{(i,j)\\in\\vec E}","kind":"use","excerpt":"Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-d24cfe84a3e98784","tex":"\\bigcup_N","aliases":"⋃_N bigcup","display":"\\bigcup_N","kind":"use","excerpt":"Convergence has its own theorem. Call x norm recoverable when Q_Nx→0. This is equivalent to S_Nx→ x, but it does not follow merely from vanishing persistent subspace. Let F=⋃_Nker Q_N be the finite core.","count":1},{"work":"realization-calculus-twin-edition","url":"notation/realization-calculus-twin-edition.html#symbol-6ef7ee64cfa071c4","tex":"\\rangle^2","aliases":"⟩^2 rangle","display":"\\rangle^2","kind":"use","excerpt":"For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.","count":1}]}